Untitled

hep-lat/9210032 22 Oct 92RegularityPropertiesandPathologiesofPosition-SpaceRenormalization-GroupTransformationsAernoutC.D.vanEnterInstituteforTheoreticalPhysicsRijksuniversiteitGroningenP.O.Box00AGGroningenTHENETHERLANDSAENTER@RUGTH.TH.RUG.NLRobertoFernandezInstitutdePhysiqueTheoriqueEcolePolytechniqueFederaledeLausannePHB{EcublensCH{0LausanneSWITZERLANDFERNANDE@ELDP.EPFL.CHAlanD.SokalDepartmentofPhysicsNewYorkUniversityWashingtonPlaceNewYork,NY000USASOKAL@ACF.NYU.EDUOctober,Shorttitle:Renormalization-GroupPathologiesKEYWORDS:Renormalizationgroup;position-spacerenormalization;real-spacerenormalization;decimationtransformation;majority-ruletransformation;Kadanotransformation;block-spintransformation;relativeentropy;largedeviations;Griths-Pearcepathologies;Gibbsmeasure;non-Gibbsianmeasure;quasilocality;Pirogov-Sinaitheory;Fermat'slasttheorem.

AbstractWereconsidertheconceptualfoundationsoftherenormalization-group(RG)formalism,andprovesomerigoroustheoremsontheregularitypropertiesandpossiblepathologiesoftheRGmap.Regardingregularity,weshowthattheRGmap,denedonasuitablespaceofinteractions(=formalHamiltonians),isalwayssingle-valuedandLipschitzcontinuousonitsdomainofdenition.ThisrulesoutarecentlyproposedscenariofortheRGdescriptionofrst-orderphasetransitions.Onthepathologicalside,wemakerigoroussomeargumentsofGrif-ths,PearceandIsrael,andproveinseveralcasesthattherenormalizedmeasureisnotaGibbsmeasureforanyreasonableinteraction.ThismeansthattheRGmapisill-dened,andthattheconventionalRGdescriptionofrst-orderphasetransitionsisnotuniversallyvalid.FordecimationorKadanotransformationsappliedtotheIsingmodelindimensiond,thesepathologiesoccurinafullneighborhoodf>0;jhj<()gofthelow-temperaturepartoftherst-orderphase-transitionsurface.Forblock-averagingtransformationsappliedtotheIsingmodelindimensiond,thepathologiesoccuratlowtemperaturesforarbitrarymagnetic-eldstrength.PathologiesmayalsooccurinthecriticalregionforIsingmodelsindimensiond.Wediscussindetailthedistinc-tionbetweenGibbsianandnon-Gibbsianmeasuresandthepossibleoccurrenceofthelatterinothersituations,andgivearathercompletecatalogueoftheknownexamples.Finally,wediscusstheheuristicandnumericalevidenceonRGpathologiesinthelightofourrigoroustheorems.ContentsIntroductionandSummaryofResults.GeneralIntroduction::::::::::::::::::::::::::::.PlanofThisPaper(Or,WhattoReadandWhattoSkip):::::::.SummaryofFirstandSecondFundamentalTheorems::::::::::0.SummaryofGriths-Pearce-IsraelPathologies::::::::::::::Innite-VolumeLatticeSystems:GeneralFormalism.Congurations,Events,Functions,Measures:::::::::::::::..CongurationsandEvents:::::::::::::::::::::..Functions(=Observables):::::::::::::::::::::..Measures::::::::::::::::::::::::::::::0.InteractionsandHamiltonians:::::::::::::::::::::::

.SpecicationsandGibbsMeasures::::::::::::::::::::..Specications::::::::::::::::::::::::::::..GibbsianSpecicationsandGibbsMeasures:::::::::::..Quasilocality::::::::::::::::::::::::::::0..FellerProperty:::::::::::::::::::::::::::..PhysicalEquivalenceintheDLRSense::::::::::::::..StructureoftheSpaceG()::::::::::::::::::::..ConditioningonanArbitrarySubsetofSpins::::::::::.TranslationInvariance:::::::::::::::::::::::::::..VanHoveConvergence:::::::::::::::::::::::..Translation-InvariantMeasures::::::::::::::::::..DividingOutTranslationInvariance:::::::::::::::..SpacesofTranslation-InvariantInteractions:::::::::::..TheObservablefCorrespondingtoanInteraction::::::..PhysicalEquivalenceintheRuelleSense:::::::::::::..EstimatesonHamiltonians:BulkversusSurfaceEects:::::0..HowtoObtainanInteractionfromaGibbsMeasure::::::..Translation-InvariantSpecicationsandGibbsMeasures::::.Entropy,LargeDeviationsandtheVariationalPrinciple:Finite-VolumeCase:::::::::::::::::::::::::::::::::::::..FreeEnergy:::::::::::::::::::::::::::::..RelativeEntropy::::::::::::::::::::::::::..LargeDeviations::::::::::::::::::::::::::..VariationalPrinciple::::::::::::::::::::::::0.Entropy,LargeDeviationsandtheVariationalPrinciple:Innite-VolumeCase:::::::::::::::::::::::::::::::::::::..FreeEnergyDensity(\Pressure"):::::::::::::::::..RelativeEntropyDensity::::::::::::::::::::::..LargeDeviations::::::::::::::::::::::::::..VariationalPrinciple::::::::::::::::::::::::..WhatisaPhaseTransition?::::::::::::::::::::0..WhenistheRelativeEntropyDensityZero? :::::::::::..PathologiesinVariousInteractionSpacesBh:::::::::::Position-SpaceRenormalizationTransformations:RegularityProp-erties.BasicSet-Up::::::::::::::::::::::::::::::::..RenormalizationTransformationActingonMeasures::::::..Examples::::::::::::::::::::::::::::::..RenormalizationTransformationActingonInteractions:::::..ARemarkonSystemsofUnboundedSpins::::::::::::.FirstFundamentalTheorem:Single-ValuednessoftheRTMap:::::.SecondFundamentalTheorem:ContinuityPropertiesoftheRTMap:

ProvablyPathologicalRenormalizationTransformations.Griths-Pearce-IsraelPathologiesI:Israel'sExample::::::::::..Introduction:::::::::::::::::::::::::::::..Israel'sExample:Decimationind=::::::::::::::.Griths-Pearce-IsraelPathologiesII:GeneralMethod::::::::::0.Griths-Pearce-IsraelPathologiesIII:SomeFurtherExamples:::::0..Israel'sExampleinDimensiond:::::::::::::::0..DecimationwithSpacingb::::::::::::::::::..KadanoTransformationwithpFinite::::::::::::::..Majority-RuleTransformation:::::::::::::::::::..Block-AveragingTransformations:::::::::::::::::..GeneralizationtoNonzeroMagneticField::::::::::::.Large-CellRenormalizationMapsinDimensiond()::::::::::..Non-GibbsiannessoftheSignFieldofan(An)harmonicCrystal..Non-GibbsiannessofLocalNonlinearFunctionsofan(An)harmonicCrystal::::::::::::::::::::::::::::::::..PhysicalInterpretation:::::::::::::::::::::::..ApplicationtotheRenormalizationGroup::::::::::::.OtherResultsonNon-GibbsiannessandNon-Quasilocality:::::::..TrivialExample:ConvexCombinationofGibbsMeasuresforDierentInteractions::::::::::::::::::::::::..RestrictionoftheTwo-DimensionalIsingModeltoanAxis:::..Fortuin-KasteleynRandom-ClusterModel::::::::::::..StationaryMeasuresinNonequilibriumStatisticalMechanics::..ComparisonofMethodsforProvingNon-Gibbsianness:::::..Are\Most"MeasuresNon-Gibbsian?:::::::::::::::Discussion.NumericallyObservedDiscontinuitiesoftheRGMap::::::::::..StatementoftheProblem:::::::::::::::::::::..AnIdealizedModelofParameterEstimation:::::::::::..ApplicationtotheRenormalizationGroup::::::::::::.ARemarkonDangerousIrrelevantVariables:::::::::::::::ConclusionsandOpenQuestions.Conclusions:::::::::::::::::::::::::::::::::..HowMuchoftheStandardPictureoftheRGMapisTrue?::..ResponsestoSomeObjections:::::::::::::::::::..WhereDoesAllThisLeaveRGTheory? :::::::::::::..TowardsaNon-GibbsianPointofView::::::::::::::.SomeOpenQuestions:::::::::::::::::::::::::::

AProofsofSomeTheoremsfromSectionA.ProofsandReferencesforSection.:::::::::::::::::::A.ProofsandReferencesforSection.:::::::::::::::::::A.ProofsandReferencesforSection.:::::::::::::::::::A..VanHoveConvergence:::::::::::::::::::::::A..Translation-InvariantMeasures::::::::::::::::::A..ADigressiononSubadditivity:::::::::::::::::::A..ALemmaonSumsofTranslates:::::::::::::::::0A..TheQuotientSeminorm::::::::::::::::::::::A..ClosedandCompactSetsinB0::::::::::::::::::A..PhysicalEquivalence::::::::::::::::::::::::A..EstimatesonHamiltoniansandGibbsMeasures:::::::::A.ProofsandReferencesforSection.:::::::::::::::::::A.ProofsandReferencesforSection. :::::::::::::::::::A..TheInnite-VolumeLimit:Proofs:::::::::::::::::A..TheInnite-VolumeLimit:Counterexamples:::::::::::BLow-TemperaturePhaseDiagramsandPirogov-SinaiTheoryB.GeneralitiesonPhaseDiagrams::::::::::::::::::::::B.Zero-TemperatureLatticeSystems.GeneralFormalism:::::::::B..Zero-TemperatureGibbsMeasures::::::::::::::::00B..Ground-StateCongurations.SupportPropertiesofZero-TemperatureGibbsMeasures:::::::::::::::::::::::::::00B..RigidGround-StateCongurations::::::::::::::::0B..ConvivialCongurations.Zero-TemperatureEntropy::::::0B..SuperuousGround-StateCongurations:::::::::::::0B..NonuniquenessofSpecicationsandInteractions:::::::::0B..Stabilityandw-Stability::::::::::::::::::::::0B..Variational-PrincipleApproach::::::::::::::::::0B..InniteRangeandLackofQuasilocality:::::::::::::B.PhaseDiagrams:::::::::::::::::::::::::::::::B..RegularPhaseDiagrams::::::::::::::::::::::B..Zero-TemperatureRegularPhaseDiagrams:::::::::::B..Low-TemperaturePhaseDiagrams.ScopeofPirogov-SinaiTheoryB.Pirogov-SinaiTheory::::::::::::::::::::::::::::B..BoundaryofaConguration.ThePeierlsCondition:::::::B..Contours.TheGeneralizedPeierlsCondition:::::::::::B..ResultsoftheTheory::::::::::::::::::::::::B..ExtensionsoftheTheory.TheRandomCase::::::::::B.ApplicationtotheExamplesofSection:::::::::::::::::B..GeneralStrategies:::::::::::::::::::::::::B..Internal-SpinSystemswithUniqueGround-StateCongurationsB..InternalSpinsunderDecimation::::::::::::::::::

B..InternalSpinsundertheKadanoTransformation.Uniformity:B..InternalSpinsunderMajorityRule::::::::::::::::B..InternalSpinsunderBlock-AveragingTransformations:::::B..InternalSpinswhenh=0.RandomField::::::::::::CSolutionoftheDiophantineEquation(. )References

IntroductionandSummaryofResults.GeneralIntroductionAprincipaltenetoftherenormalization-group(RG)theoryofphasetransitions[,,,]isthattheRGmap,denedonasuitablespaceofHamiltonians,issmooth(i.e.analyticoratleastseveral-timesdierentiable),evenonphase-transitionsurfaces.Thesingularitiesassociatedwithcriticalpoints[,,,]andrst-orderphasetransitions[0,,]arethenexplainedintermsofthebehavioroftheRGmapunderinniteiteration.ThispictureofasmoothRGmaphas,however,beenquestioned,particularlyasregardsthebehavioratorneararst-orderphasetransition.Ontheonehand,theexistenceofseveralphasesraisesthepossibilitythattheRGmapmaybediscontinuousormulti-valued[,,0]ontherst-ordertransitionsurface,asthenumericalevidencereportedbyseveralgroups[,,,]seemstosuggest.Ontheotherhand,GrithsandPearce[,,]havepointedoutsome\peculiarities"ofthecommonlyuseddiscrete-spinposition-spaceRGtransformations(decimation,majorityrule,etc. );inparticulartheysuggestedthattheRGmapforthetwo-dimensionalIsingmodelmusthavesingularities(orotherstrangebehavior)inaratherlargepartof(;h)-plane(seealso[,]).Inanimportantbutapparentlylittle-knownpaper,Israel[0]clariedthenatureoftheGriths-Pearcepeculiarities:heshowedthatinatleastonecasetherenormalizedsystemcannotbedescribedbyaBoltzmann-GibbsprescriptionforanyreasonableHamiltonian,i.e.therenormalizedmeasureisnon-Gibbsian.InthispaperwereconsidertheconceptualfoundationsoftheRGformalism,andprovesomerigoroustheoremsonthenatureoftheRGmap.Ontheonehand,weprovetwoFundamentalTheoremsonthesingle-valuednessandcontinuityoftheRGmap;thesetheoremsruleoutthediscontinuous-owscenarioproposedinreferences[,,,,,0].Ontheotherhand,weprove,completingandextendingIsrael'sargument,thatinseveralcasestheRGmapisill-denedforamuchmorebasicreason:therenormalizedHamiltonianmayfailtoexistaltogether.ThisimpliesthattheconventionalRGdescriptionofrst-orderphasetransitions[0,,]isnotvalideither(atleastinthesemodelsandfortheseRGtransformations).Moreover,thispathologycanoccurinthevicinityof|notonlyat|arst-orderphasetransition:fortheIsingmodelindimensionditoccursinafullneighborhoodf>0;jhj<()gofthelow-temperaturepartoftherst-orderphase-transitionsurface.Indeed,forcertainblock-averagingtransformationsweareabletoshowthatthepathologyoccursatlowtemperatureandallmagneticeldsh.Ourpointofviewisthefollowing:AnRGmapisdenedinitiallyasarule(whichSimilarpeculiarities,andalsodierentones,havebeenfoundbyHasenfratzandHasenfratz[].ThephenomenafoundinSectionoftheirpaperareverycloselyrelatedtothoseofGrithsandPearce,whilethoseinSectionsandseemtobequitedierent.Briefsummariesofourresultshaveappearedpreviously[,,,].

maybeeitherdeterministicorstochastic)forgeneratingaconguration!0of\blockspins"givenaconguration!of\originalspins".MathematicallythisisgivenbyaprobabilitykernelT(!!!0).Usingsuchamap,onecanimmediatelydeneaprobabilitydistribution0(!0)ofblockspinsfromanygivenprobabilitydistribution(!)oforiginalspins,namely0(!0)=(T)(!0)X!(!)T(!!!0):(:)Inotherwords,theRGmapiseasilydenedasamapfrommeasurestomeasures.Ontheotherhand,mostapplicationsoftherenormalizationgroupassume(andinfactneed)thattheRGmapisdenedasamapfromHamiltonianstoHamiltonians.Thatis,ifistheGibbsmeasureforastatistical-mechanicalsystemwithHamiltonianH,thenoneusuallyassumesthat0istheGibbsmeasureforasystemwithsomeHamiltonianH0;thisistakentodeneanRGmapRonsomesuitablespaceofHamiltonians,bythediagramT!0T"#HR!H0(:)FormallytherelationbetweenaHamiltoniananditscorrespondingGibbsmeasureisgivenby=consteH,andhencetheRGmaponthespaceofHamiltoniansisdenedformallybyH0(!0)=(RH)(!0)=log"X!eH(!)T(!!!0)#+const:(:)However,thisformulaisvalidonlyinnitevolume;ininnitevolume,theHamiltonianH(!)isill-dened(itsvalueisalmostsurely),andtheconnectionbetweenaformalHamiltonian(moreprecisely,aninteraction)anditscorrespondingGibbsmeasure(s)ismuchmorecomplicated.Weemphasizethatthisisnotameremathematicalnicety,butcontainsthefundamentalphysicsofphasetransitions.Innitevolume,wheretheformula=consteHmakessense,allthermodynamicfunctionsaremanifestlyanalyticfunctionsoftheparametersintheHamiltonian,soaphasetransitionisim-possible.Phasetransitionsoccuronlyforinnite-volumesystems.Now,onefeatureoftheinnite-volumelimitisthepossibilitythattheGibbsmeasuremaybenon-unique:correspondingtoagivenformalHamiltonian(=interaction)theremayexistseveraldistinctGibbsmeasures,eachonecorrespondingtoadistinctthermodynamicallysta-ble\purephase"ofthesystem.Indeed,suchamultiple-phasecoexistencecanserveasonedenitionofarst-orderphasetransition.Therefore,forHamiltoniansHwithanon-uniqueGibbsmeasure(=Hamiltonianslyingonarst-orderphase-transitionsurface),theupwardverticalarrowin(.)maywellbeamulti-valuedmap;andonemightfearthatthiscouldcausetheputativeRGmapRtobecomemulti-valuedaswell.(Weshallseelater,however,thatthispathologycannotoccur.)Evenmoresub-tleproblemsarisefromthedownwardverticalarrowin(. ):thoughatmostoneH0

cancorrespondtoagiven0[],itcanhappenthatnoH0correspondstothegiven0|thatis,itcanoccurthattheimagemeasure0isnotaGibbsmeasureforanyreasonableHamiltonian.InSectionweshallshowthatsuchnon-GibbsiannessistheonlywaythattheRGmapcanbecomegrossly\pathological".InSectionweshallshowthatthispathologydoesinfactoccurinaratherwidevarietyofexamples.(Ofcourse,wemustmakeprecisewhatwemeanbya\reasonable"Hamiltonian,andconvincethereaderthatourclassissucientlywidetocapturefullytheintuitivenotionof\physicalreasonableness".ThiswillbediscussedindetailinSections(especially..,..and..)and...Suceittosaynowthatweallowinteractionsofarbitrarilylongrangeandinvolvingarbitrarilymanyspins,subjectonlytotheconditionofabsolutesummability.)TheseresultsleaveRGtheoryinroughlythefollowingsituation:TheRGmaphasbeenproventobewell-denedandanalyticathightemperature[0,]and,insomecases,atlargemagneticeld[]|regionsinwhichphasetransitionsareabsent,andRGtheoryisunnecessary.TheRGmaphasbeenproveninsomecasestobeill-denedatlowtemperature(Section).Nearthecriticalpoint|whereRGtheoryisofthemostinterest|verylittleisknownaboutthebehavioroftheRGmap,buttherearesomeindicationsofpossiblepathologiesindimensionsd()(Sections.and.).Nevertheless,RGideashavebeenofgreatvalueeveninsituationsinwhichthestrictWilsonprescription(.)hasnotbeen|andmaybeevencannotbe|implemented[,,0,,,,,,,,,,,,,,,,,,,,].WediscusstheseissuesfurtherinSection...PlanofThisPaper(Or,WhattoReadandWhattoSkip)Wehopethatthispaperwillberead(andreadable)bothbytheoreticalphysicists|particularlythosedoingreal-spaceRGandMonteCarloRGcalculations|andbymathematicalphysicistsinterestedinthestatisticalmechanicsoflatticesystems.ForthisreasonwehavegiveninSectionaratherdetailed(and,wehope,comprehensible)summaryofthegeneraltheoryofinnite-volumelatticesystems,inwhichwemakepre-cisetheconceptsof\interaction",\Hamiltonian",\Gibbsmeasure"and\equilibriummeasure"andtheconnectionsbetweenthem.Aswehaveargued,acarefultreatmentoftheinnite-volumeproblemisessentialforacorrectphysicalunderstandingofphasetransitionsingeneral,andoftherenormalizationgroupinparticular.WehopethatSectionwillbeusefultophysicistswhomaynotbefamiliarwiththeseideas.Someabstractmathematicsisofnecessityinvolved;wehavetriedhardtominimize\mathe-maticsforthesakeofmathematics",andtointroduceonlythosemathematicalobjectswhichcorrespondtoclearphysicalconcepts.Thereadercanjudgewhetherwehavesucceeded.The\experts"willnoticeafewinnovationsandnewresultsinSectionandtheassociatedAppendixA:theextensivediscussionofphysicalequivalence(Sections..,..,..,..,A..,A..andA..);somepreciseestimatesonbulkvs.surfaceeects(Sections..,..andA.. );

InSectionwedeneourgeneralframeworkforstudyingrenormalizationtransfor-mations,andprovethetwoFundamentalTheoremsonsingle-valuednessandcontinuityoftheRGmap.ThesetheoremsshowthattheRGmapRcanneverbecomemulti-valuedordiscontinuous;butitcanbecomenon-valued,whichoccursiftheimagemeasure0isnon-Gibbsian.Thisfocusonnon-Gibbsianness|whichistherealmes-sageofourpaper|isaprofoundinsightduetoIsrael[0].InSectionwecompleteandextendIsrael'sargument,andshowthatinalargeclassofexamples(alwaysatlowtemperature,butnotonlyonphase-transitionsurfaces)theimagemeasure0isindeednon-Gibbsian.Wealsodiscusssomeotheroperationsthatcanleadtonon-Gibbsianmeasures,includingonewhichisrelevantto\large-cell"majority-rulemaps;andwegivearathercompletecatalogueoftheknownexamplesofnon-Gibbsianness.InSec-tionwediscusstheheuristicandnumericalevidenceonRGpathologiesinthelightofourrigoroustheorems.WealsodiscusssomeheuristicargumentsforthepossibleexistenceofRGpathologiesinthecriticalregionforIsing-to-IsingRGmapsindimen-siond().InSectionwesummarizeourresultsanddiscusstheirimplications.Weconcludewithalistofopenquestions.InAppendixAwesupplytheproofsofsometheoremsthatarestatedwithoutproofinSection.InAppendixBweprovideabriefsummaryofPirogov-Sinaitheory,whichisneededasatechnicaltoolinSection.InAppendixCwesolveaDiophantineequationarisinginourstudyofthemajority-rulemap.LetusagainexpressourhopethatthereaderwillatleastperuseSection.(Hey,wespentalongtimeonit,andwethinkitisrathergoodpedagogy.)However,forthereaderwhoistrulyallergictoabstractmathematics,weoerthefollowingadvice:readtheremainderofthisIntroduction,followedbySections(skippingtheproofs),..,.(skippingtheproofs),.and.Finally,forthereaderwhoisallergicbothtoabstractmathematicsandto0-pagepapers,weoer\RGlite":readtheremainderofthisIntroduction,andthenskiptotheConclusion(Section.)..SummaryofFirstandSecondFundamentalTheoremsWewouldlikenexttosummarizethetwoFundamentalTheoremsandgivethephysicalintuitionbehindtheirproofs.Consider,forconcreteness,theIsingmodelindimensiondatlowtemperature(c)andzeromagneticeld.Atsuchapointtherearepreciselytwo[]purephases(extremaltranslation-invariantGibbsmeasures):thepositivelymagnetized(or\+")phase+,andthenegativelymagnetized(or\")phase.Thesepurephasescanbeobtainedbytakingtheinnite-volumelimitaconsistentuseofvanHoveconvergenceandcompletesubadditivity(Sections..,A..,A..andA..);andsomeinterestingcounterexamplesconcerningthepressureandentropy(AppendixA..).ThersttwooftheseinnovationsplayacrucialroleinourproofoftheSecondFundamentalTheorem(Section. ).The\experts"willnoticesomesmallinnovationsinourpresentationofPirogov-Sinaitheory,notablyouremphasisonquestionsofuniformity.ThisplaysanimportantroleinourapplicationtotheKadanotransformation:seeSection..andAppendixB...0

using\+"or\"boundaryconditions,respectively.BothofthesephaseshavealargemagnetizationM0andasmallcorrelationlength.Nowletusapplysomeblock-spintransformationT,suchasthemajority-ruletransformationonblocksofsized.Thentheimagemeasures0=TwillhaveayetlargermagnetizationM00(sinceminoritiestendtoget\outvoted")andayetsmallercorrelationlength0(weexpectroughly0=,sincedistancesarebeingscaledbyafactorof).Wenowask:Theseimagemeasures0aretypicalofwhatkindofHamiltonian(ifany)?Onepossibility|andtheoneconventionallyassumed[0,,]|isthattheRGowistowardlowertemperaturesalongtheh=0line.ThispictureiscertainlyconsistentwiththeintuitiveideathatmagnetizationincreasesandcorrelationlengthdecreasesundertheRGmap.Inthisscenario[Figure(a)],thetwoimagemeasures0wouldbeGibbsianforthesameHamiltonianH0,andthisHamiltonianwouldbeinvariantunderthe!symmetry.AdierentpossibilitywasadvocatedbyDecker,HasenfratzandHasenfratz[].Inthisscenario[Figure(b)],theRGowisdiscontinuousatthephase-transitionlineh=0:HamiltoniansHwithaninnitesimalpositive(resp.negative)magneticeldhgetmappedbyasingleRGsteptorenormalizedHamiltoniansH0havingastrictlypositive(resp.strictlynegative)magneticeldh0.Furthermore,ath=0therenormalizedHamiltonianH0dependsonwhichpurephase,+or,oneusesinthetopleftcornerof(.):theimagemeasure0+wouldbeGibbsianforsomeHamiltonianH0+having(amongothercouplings)astrictlypositivemagneticeld,whiletheimagemeasure0wouldbeGibbsianforsomeHamiltonianH0havingastrictlynegativemagneticeld.(ObviouslyH0+andH0wouldberelatedbythe!symmetry,i.e.byreversingthesignsofalloddcouplings. )Inthisscenario,therefore,theRGmapRisdiscontinuousasoneapproachesthephase-transitionline,andmulti-valuedonthatline.ThispictureisalsoconsistentwiththeintuitiveideathatmagnetizationincreasesandcorrelationlengthdecreasesundertheRGmap.Howcanwedistinguishbetweenthesetwoscenarios?Otherwiseput:Supposewearegivenameasure0withalargepositivemagnetizationandasmall(butnonzero)correlationlength.DoesthismeasurecomefromaHamiltonianH0withlargeandh=0,ordoesitcomefromaHamiltonianwithnotsolarge(possiblyevensmall)andhlargeandpositive?Bothoftheseregionsinthe(;h)-planecorrespondtoalargepositivemagnetizationandasmallcorrelationlength.Howcanwedistinguishbetweenthetwo?Theanswerhastodowiththelarge-deviationpropertiesofthemeasure0.LetbealargecubicalboxofsideL,andletMPxxbethetotalspininMoreprecisely,theowwouldtakeplaceinaninnite-dimensionalspaceofcouplings,butwouldrespectthe!symmetry.Thatis,second-nearest-neighborandlonger-distancepaircouplings,four-spincouplings,six-spincouplingsandsoforthwouldcertainlybeinduced;butnomagneticelds,three-spincouplingsorotheroddinteractionswouldarise.Thispossibilitywassuggestedearlier,inthecontextofthe-statePottsmodelinthreedimensions,byBloteandSwendsen[]andwithespecialclaritybyRebbiandSwendsen[0,p.0].Itwasalsosuggested,inthecontextofamean-eldcomputation,byHudak[].

h = +h = 0 β = 0β = (a)h=βc∞∞∞h = +h = 0 β = 0β = (b)βc∞∞h= ∞Figure:TwopossiblescenariosfortheRGowintheIsingmodelatlowtemperature.(a)RGmapiscontinuousandsingle-valuedonthephase-transitionline. (b)RGmapisdiscontinuousandmulti-valuedonthephase-transitionline.

(arandomvariable).ClearlythereisanoverwhelmingprobabilitythatMwillbepositive(andinfactveryclosetoitsmeanvalueLdM00=Ldhi0);buthowrareisittohaveMnegative?If0isaGibbsmeasureforsomeHamiltonianwithh>0,thentheeventM<0issuppressedbythebulkmagneticeld:Prob0(M<0)eO(Ld):(:)Ontheotherhand,if0isaGibbsmeasureforsomeHamiltonianwithh=0and>c,thentheeventM<0issuppressedonlybyasurfaceenergy:Prob0(M<0)eO(Ld):(:)ItisnoweasytodecidebetweenthetwoscenariosfortheRGow.Inthestartingmeasure+,theoccurrenceofalargeregionwithnegativetotalspinissuppressedonlylikeeO(Ld);roughlyspeaking,themeasure+\knows"thatitisdegeneratewiththemeasure.Butthenintheblock-spinmeasure0+=+T,theremustalsobeaprobability>eO(Ld)ofobservinganegativetotalspin(sinceanetnegativeoriginalspinimplies,withhighprobability,anetnegativeblockspin).Sincethiscontradicts(. ),weconcludethat0+cannotbetheGibbsmeasureofaHamiltonianwithstrictlypositivemagneticeld.Picturesquely,theimagemeasure0+\remembers"thatitarosefromanoriginalHamiltonianHwithcoexistingphases.Therefore,thediscontinuous-owscenarioisimpossible;theRGmapcannotbemulti-valuedordiscontinuous.Itisarelativelyshortstepfromtheseintuitiveideastoarigorousproof.InSectionweprove,ingreatgenerality,thefollowingtwotheorems:Firstfundamentaltheorem.IfandareGibbsmeasuresforthesameinteraction,theneitherTandTarebothnon-Gibbsian,orelsethereexistsaninteractionforwhichbothTandTareGibbsmeasures.Inthelattercase,thisistheonlyinteractionforwhicheitherTorTisaGibbsmeasure.Therefore,therenormalization-groupmapRcannotbemulti-valued.Secondfundamentaltheorem.Therenormalization-groupmapRiscontinuous(infact,Lipschitzcontinuous)onthedomainwhereitisdened.Ofcourse,thesesummariesofthetheoremsarenotquiteprecise:weneedtomakeclear,forexample,inwhatspaceofinteractionsweareworking,andinwhatnormwearedeningcontinuity.ThedetailedstatementoftheFundamentalTheoremscanbefoundinSections.and.,respectively.TheproofsoftheFundamentalTheoremsarebasedonthegeneraltheoryofinnite-volumelatticesystemsdevelopedinSection.ThesetwotheoremsmakeclearthattheonlywayinwhichtheRGmapcanbecomegrosslypathologicalisforittobeundened,i.e.fortheimagemeasure0tobenon-Gibbsian.

.SummaryofGriths-Pearce-IsraelPathologiesThisisnot,however,theendofthestory:althoughthediscontinuous-owscenariofortheRGmapinthelow-temperatureIsingmodelisnotcorrect,thetraditionalscenarioisinmanycasesnotcorrecteither!TheFirstFundamentalTheoremleavesopenthepossibilitythattheimagemeasure0=Tmaybenon-Gibbsian,inwhichcasetheRGmapRwouldbeundened.Itturnsoutthatthispathologydoesinfactoccurinaratherwidevarietyofexamples.Theoccurrenceofnon-Gibbsiannessfortheimagemeasure0wasrstpointedoutbyIsrael[0]inoneofthecasessuggestedbyGrithsandPearce[,].InSectionwecompleteandextendIsrael'sargument,andshowthatinalargeclassofexamples(alwaysatlowtemperature,butnotonlyonphase-transitionsurfaces)theimagemeasure0isnon-Gibbsian.Thenon-Gibbsiannessarisesfromthefact|alreadynotedbyGrithsandPearce|thatthe\internalspins"(thevariablesbeingintegratedoverintheRGtransfor-mation)mayundergoarst-orderphasetransitionforsomexedblock-spincongu-ration!0special.Moreover,insomecasesthedierentphases(=Gibbsmeasures)oftheinternal-spinsystemcanbeselectedbyanappropriatechoiceofblock-spinboundaryconditions.Inthisway,informationcanbetransmittedfromdistantblockspinstotheblockspinattheoriginviatheinternalspinsintheintermediateregion,evenwhentheblockspinsintheintermediateregionarexed.Asaconsequence,therenormalizedmeasure0violatesaveryweaklocalitycondition|quasilocality,seeSection..|whichisobeyedbyeveryGibbsmeasurecomingfromareasonableinteraction.Itfollowsimmediatelythattherenormalizedmeasure0mustbenon-Gibbsian.Itisatrstsurprisingthattheexistenceofpathologiesforasingleblock-spinconguration!0special|whichhas,ofcourse,probabilityzero|canneverthelesscausethenon-Gibbsiannessoftherenormalizedmeasure;andindeed,thisfactaloneisnotsucientforconcludingnon-Gibbsianness.Rather,whathappensintheseexamplesisthatforblock-spincongurationswhicharenear(intheproducttopology)to!0special|namely,thosewhichagreewith!0specialinalargecubeanddierfromitoutside|theinternal-spinphasedependssensitivelyontheblockspinsoutsidethecube.Thesecongurationshaveasmallbutnonzeroprobability,andthisturnsouttobesucientforprovingnon-Gibbsianness.ThedetailsoftheproofaregiveninSections. {..Weprovenon-Gibbsiannessatlowtemperatureandzeromagneticeldinthefol-lowingexamples:Decimationwithanyspacingb,fortheIsingmodelinanydimensiond.TheKadanotransformationwithnitepandarbitraryblocksizeb,fortheIsingmodelindimensiond.Themajority-ruletransformationwith(or,,:::)blocksforthetwo-dimensionalIsingmodel.Inearlierversionsofthiswork[,],weclaimedthisresultonlyforsmallp.Subsequentlywefoundaproofvalidforall0

Averagingtransformationwithanyevenblocksizeb,fortheIsingmodelinanydimensiond.Moreover,inseveralcaseswecanprovethatthesepathologiesarepresentalsoatnonzeromagneticeld.Forthersttwoexamples,weprovenon-Gibbsiannessindi-mensiondinafullneighborhoodf>0;jhj<()gofthelow-temperaturepartoftherst-orderphase-transitionsurface.Inthelastexample,thepathologiescanbeproveninanydimensiondandforarbitraryvaluesofthemagneticeld,againatlowtemperature.TheselatterresultsmakeclearthattheGriths-Pearce-Israelpathologiesarenotassociatedwiththefactthattheoriginalmodelissittingonaphase-transitionsurface.Rather,itsucesthatarst-orderphasetransitioncanbeinducedintheinternal-spinsystembychoosinganappropriateblock-spincongura-tion.Forthisweneedtoworkatlowtemperaturebutnotnecessarilyclosetothephase-transitionsurface.Thoughwehavenotyetbeenabletodemonstratenon-Gibbsiannessforthemajority-ruletransformationonorblocks,orforanyblocksizeindimensiond,wefeelthattheobstaclesaretechnicalratherthanfundamental.Indeed,theresultsinSectionsuggestthatnon-GibbsiannessmaybethenormalsituationforRGmapsatlowtemperatureand/orneararst-orderphasetransition.Thereaderwillprobablynotbesurprisedthatthedecimationtransformationis\pathological":thistransformation,unlikeotherRGtransformations,doesnotinanysenseintegrateoutthe\high-momentummodes"andleavethe\low-momentummodes";itmerelyintegratesoutonesublatticeandleavesanother.Inparticular,ifthesublatticeofinternal(integrated-out)spinsisconnected,itishardlysurprisingthattheinternal-spinsystemcanexhibitaphasetransition,andthatthiscangiverisetoRGpathologies.Wethereforewanttostressthatthesamepathology|non-Gibbsiannessafteronerenormalizationstep|isalsopresentatlowtemperatureforatleastsomeKadano,majority-ruleandblock-averagingtransformations.Theselattertransformationsdo(atleastseemingly)integrateoutthe\high-momentummodes"andleavethe\low-momentummodes",andtheyhavebeengenerallyconsideredtobewell-behaved.Indeed,nearlyallreal-spaceRGstudiesofIsingmodelshaveusedsomevariantofthesetransformations.Itisthusahighlynon-trivialfactthattheseRGmapscanbeill-denedatlowtemperature.Innite-VolumeLatticeSystems:GeneralFor-malismConsideraclassicalstatistical-mechanicalsystemwithcongurationspace,Hamil-tonianHandapriorimeasure0.TheBoltzmann-GibbsdistributionBGforthissysteminthecanonicalensembleatinversetemperaturecanbecharacterizedineitheroftwoways:(a)Explicitformula.dBG(!)=ZeH(!)d0(! );(:)

whereofcourseZ=ZeH(!)d0(! ):(:)(b)Variationalprinciple.BGisthatprobabilitymeasurewhichmaximizesentropyminustimesmeanenergy:BGmaximizesS(j0)E(H;);(:)whereS(j0)=Z logdd0!d=Z dd0logdd0!d0(.)andE(H;)=hHiZH(!)d(!):(:)Theequivalenceofthesetwocharacterizationsisasimplecomputationinthecalculusofvariations.Unfortunately,thiselementarytheorydoesnotapplytoinnite-volumesystems,becausetheHamiltonianH(!)isill-dened:foralmostanyconguration!wehaveH(!)=.Nevertheless,non-trivialanaloguesofthesetwocharacterizationscanbedevelopedforinnite-volumesystems.TheanalogueoftheexplicitformulaisthetheoryofspecicationsandGibbsmeasures:aninnite-volumeGibbsmeasureisonewhoseconditionalprobabilitiesfornitesubsystemsaregivenbytheBoltzmann-Gibbsformula.Theanalogueofthevariationalapproachisthetheoryofequilibriummeasures:anequilibriummeasureisatranslation-invariantmeasurethatmaximizesentropydensityminustimesmeanenergydensity.TheseapproachesarereviewedinSections.and.

{.,respectively.Thefundamentalfeatureofinnite-volumesystems,whichdistinguishesthemfromnite-volumesystems,isthatthemapbetween\Hamiltonians"(moreprecisely,interactions)andGibbsmeasures(orequilibriummea-sures)isneithersingle-valuednoronto:someinteractionshavemultipleGibbsmea-sures,whilesomemeasuresarenotGibbsianforanyinteraction.Thesefactsareattheheartofthetheoryofphasetransitions,andoftherenormalizationgroup.ThestandardreferencesforthematerialinthissectionarethebooksofGeorgii[],Preston[]andIsrael[0].GeorgiiandPrestondealprincipallywiththetheoryofGibbsmeasures,whileIsraeldealsprincipallywiththetheoryofequilibriummeasures.WeassumeinthissectionthatthereaderhassomeknowledgeofmetricspacesandBanachspaces,ideallyatthelevelofRoyden[0]orReedandSimon[0],andofmeasuretheoryandprobabilitytheory,ideallyatthelevelofBauer[]orKrickeberg[].However,werealizethatformanyreadersthesetheoriesbelongtoonlyfaintlyrememberedmathematicscoursesandareratherdistantfromtheirday-to-daywork

intheoreticalphysics.Nevertheless,weurgesuchreadersnottobediscouragedbytheabstractjargon,andtousetheexamplesweprovideasameanstograsptheessentialphysicalideasunderlyingthemathematics.Inthissectiontheemphasisisonconceptsandideas(bothphysicalandmathe-matical),notontechniquesofproof.Therefore,alldenitionsandtheoremsarestatedprecisely,butproofsareomitted.InAppendixAweprovide,foreachtheorem,eitherapublishedreference(iftheresultisknown)oraproof(ifitisnew).HenceforthweabsorbintotheHamiltonianH;thissimpliesthenotation.Letusalsoremarkthatalthoughourexpositioniscouchedinthelanguageofthecanonicalensemble,theformalismisequallyapplicabletothegrandcanonicalensemble:itsucestointerpretourHtomean\HN".Infact,thisformalismappliestoanarbitrary\generalized(grand)canonicalensemble"withparameters;:::;nconjugatetoobservablesH;:::;Hn..Congurations,Events,Functions,MeasuresClassicalstatisticalmechanicsisabranchofprobabilitytheory.Thebasicstruc-turesofprobabilitytheoryare:Acongurationspace|thisisthesetofallpossible(microscopic)congura-tionsofthesystemunderstudy.A-eldFofsubsetsof|thisisthesetofallevents(=yes-noquestions)thataremeasurablebysomeconceivable(possiblyextremelyidealized)experiment.Varioussub--eldsAFmaycorrespondtorestrictedclassesofexperiments(e.g.experimentsperformedwithinaspeciedregionofspace).Observables(=randomvariables=real-valuedF-measurablefunctionson)|thesecorrespondtoexperimentswhichgivearealnumberasananswer.Varioussubclassesofobservables(e.g.thosemeasurablewithrespecttoaspeciedsub--eldA)maycorrespondtorestrictedclassesofexperiments(e.g.experimentsperformedwithinaspeciedregionofspace).Aprobabilitymeasure(=probabilitydistribution)on(;F)|thisdescribeseitherourstateofpartialknowledgeofthesystem(ifwetakea\subjective"inter-pretationofprobabilitytheory)oranensembleof\identicallyprepared"randomsystems(ifwetakean\objective"interpretationofprobabilitytheory).Themathematicsofstatisticalmechanicsdoesnotdependonanyparticularinterpre-tationofitsmathematicalobjects,sothereaderisurgedtoemploywhicheverinterpretationhe/sheprefers.Inthissectionwedescribetheparticularcaseofthisstructurethatisappropriatefortheequilibriumstatisticalmechanicsofaninnite-volumeclassicallatticesystem.AreferenceforthissectionisGeorgii[,IntroductionandSections.and. ].

..CongurationsandEventsThecongurationspaceofaninnite-volumelatticesystemisspeciedbythefollowingingredients:Thesingle-spinspace0.Thisisthespaceofpossiblecongurationsofthephysicalvariable(s)atasinglelatticesite.(Forbrevitywecallthesevariablesa\spin".)Examples:Isingmodel,0=f;g;N-vectormodel,0=SN=unitsphereinRN;N-componentGaussianor'model,0=RN;solid-on-solid(SOS)ordiscreteGaussianmodel,0=Z.Sincestatisticalmechanicsisbasedonprobabilitytheory,weshallalwaysas-sume0tobeequippedwitha-eldF0of\measurablesets".Usually0willalsocomeequippedwithaphysicallynaturaltopology;infact,0willalmostalwaysbeacompleteseparablemetricspace,andF0willbethe-eldofBorelsets.If0isacompactmetricspace,wesaythatthesystemhasboundedspins;otherwisewesaythatthesystemhasunboundedspins.Examples:TheIsingandN-vectormodelshaveboundedspins;theGaussian,',SOSanddiscreteGaussianmodelshaveunboundedspins.ThelatticeL|acountablyinnitesetof\sites".ForthemomentweneednotgiveLanygeometricstructure,butforconcretenessthereadercanimagineLtobesomed-dimensionallattice.Theinnite-volumecongurationspaceisthendenedtobetheCartesianproduct(0)L;thatis,itisthesetofallcongurations!=(!x)xLwith!x0foreachsitex.Thespaceisequippedwiththeproduct-eldF=(F0)Landwiththeproducttopology.Theproducttopologymeansthatasequence(ornet)ofcongurations(!n)convergestoaconguration!ifandonlyif!nx!!xforallxL.If0ismetrizable(resp.separable,completemetric,compact),thensois.Itisimportanttounderstandphysicallywhattheproducttopologymeans.Supposeforsimplicitythat0isametricspace.Thenatypicalneighborhoodof!isthesetN!;;=f!0:dist(!x;!0x)0andisanitesubsetofL.0Thatis,atypicalneighborhoodof!intheproducttopologyisthesetofcongurationsthatarecloseto!onsomenitesetofsites,butarearbitraryoutside.Inparticular,if0isdiscrete(ase.g.intheIsingmodel),thenaneighborhoodof!isthesetofcongurationsthatagreewith!onsomenitesetofsites,butarearbitraryoutside.Thesefactswillplayanimportantroleinourdiscussionofnon-GibbsiannessforRGimagemeasures(Sections.{.).If0isaseparablemetricspace,thentheproduct-eldoftheindividualBorel-eldscoincideswiththeBorel-eldfortheproducttopology.0Moreprecisely,thesetsN!;;formaneighborhoodbasisof!,i.e.everyneighborhoodof!containsoneofthesetsN! ;;.

ForeachsubsetL,weletFFbethesub--eldcorrespondingtoeventsdependingonlyonthespins!=(!x)x;thatis,Fisthe-eldofeventsmeasurablewithinthesubset.WedenotebyStheclassofallnonemptynitesubsetsofL.WedenotebycthecomplementofinL.Remark.TheCartesianproduct(0)Lisnotthemostgeneralcongurationspace.Oftenonewishestostudyalatticemodelwithlocalconstraints(e.g.hard-coreexclu-sions).Oneway(nottheonlyone)totreattheseconstraintsistocuttheexcludedcongurationsoutofthecongurationspace:thatis,weletthecongurationspacebeanappropriatesubsetoftheproductspace(0)L.Wedonotallowthismuchgeneralityhere,butmuchofthepresenttheorygoesthrough(withsomemodication)inthissituation[,,]...Functions(=Observables)Anobservableissimplyareal-valuedmeasurablefunctionon.Weconsidervariousspacesofsuchfunctions:ThespaceB()=B(;F)ofboundedmeasurablefunctions.Thisisthelargestspaceoffunctionsweshallconsider.ThespaceBloc()=SSB(;F)ofboundedlocalfunctions.Afunctionislocalifitdependsononlynitelymanyspins.ThespaceBql()=Bloc()ofboundedquasilocalfunctions.Afunctionisquasilocalifitistheuniformlyconvergentlimitofsomesequenceoflocalfunc-tions.Equivalently,afunctionisquasilocalifit\dependsweaklyondistantspins"inthesensethatlim"Lsup!;!0!=!0jf(!)f(!0)j=0:(:)ThespaceC()ofboundedcontinuousfunctions.ThespaceCloc()Bloc()\C()ofboundedcontinuouslocalfunctions.ThespaceCql()Bql()\C()ofboundedcontinuousquasilocalfunctions.Examples..For0=R,thefunctionf(')=sgn('0)isboundedandlocal(hencequasilocal)butnotcontinuous.Analogousfunctionscanobviouslybecon-structedfor0=SNorRN,andindeedonanysingle-spinspacewhichisnotdiscrete.Thestatementlim"LF()=(whereRorC)meansthatforeach>0,thereexistsanitesubsetKLsuchthatjF()j

.If0=L=Z,any(bounded)functionof0is(boundedand)continuousbutnotquasilocal.(Thisexample,whichwassuggestedtousbyHans-OttoGeorgii,isfurtherdiscussedinAppendixA..)Weequipeachoftheabovespaceswiththe\supremumnorm"(or\uniformnorm")kfk=kfksup!jf(!)j:(:)Soequipped,thespacesB(),Bql(),C()andCql()areBanachspaces.Letusnoticethat:(a)Ifthesingle-spinspace0isacompactmetricspace,theneverycontinuousfunctionisboundedandquasilocal.HenceC()=Cql()Bql().(b)Ifthesingle-spinspace0isdiscrete,theneveryquasilocalfunctioniscontinuous.Inparticular,Bql()=Cql()C().(c)Ifthesingle-spinspace0isnite,thenquasilocalityandcontinuityareequiva-lent(andimplyboundedness).HenceC()=Cql()=Bql()...MeasuresNextwestudymeasureson.LetM()=M(;F)bethespaceofnitesignedmea-sureson,andM+()=M+(;F)M()bethespaceofprobabilitymeasures.Thereisanaturaldualitybetweenspacesoffunctionsandspacesofmeasures,namelyh;fi(f)Zfd:(:)Ifiscompact,theneveryboundedlinearfunctionalonC()arisesinthiswayfromanitesignedmeasure(Riesz-Markovtheorem);otherwiseput,theBanach-spacedualofC()isexactlyM().Letbeaprobabilitymeasureon;thenthesupportof(denotedsupp)isaclosedsubsetofthatcanbedenedinanyofthreeequivalentways:(a)Thesetofall!suchthateveryneighborhoodof!hasnonzeromeasure.(b)Theintersectionofallclosedsetsofmeasure. (c)Thecomplementoftheunionofallopensetsofmeasurezero.Thekeytheoremis:ifisaseparablemetricspace,then(supp)=,sothatsuppisthesmallestclosedsethavingmeasure.Weneedtodiscusswhatitmeansforasequence(ornet)ofmeasuresntoconvergetoalimitingmeasure;inotherwords,weneedtoequipthespacesM()andM+()withatopology.Infact,thereareseveralmathematicallynaturaltopologies,eachwith0

adistinctphysicalmeaning.Thesimplesttopologyisthenormtopologydenedbythetotalvariationnormkk=supfB(;F)kfkj(f)(f)j=supfC()kfkj(f)(f)j=supAFj(A)(A)j:(.0a)Asequence(ornet)nconvergesinvariationnormtoifknk!0.Physically,normconvergenceofntomeansthatexpectationvaluesinnconvergetothosein,uniformlyforallboundedobservablesf.Thisisanextremelystrongnotionofconvergence,whichoccursonlyrarelyinphysicalapplications.Therefore,weintroducealsotheweaktopologiesinducedbythevariousclassesoffunctionsdenedinSection..:Theboundedmeasurabletopology:n!ifn(f)!(f)forallfB(;F).[Ifthenareprobabilitymeasures,itsucestocheckthatn(A)!(A)forallAF.]Theboundedquasilocaltopology:n!ifn(f)!(f)forallfBql(;F).[Ifthenareprobabilitymeasures,itsucestocheckconvergenceforfBloc(;F),oralternativelyforallASSF.]The(ordinary)weaktopology:n!ifn(f)!(f)forallfC().Theweakquasilocaltopology:n!ifn(f)!(f)forallfCql(;F).[Ifthenareprobabilitymeasures,itsucestocheckconvergenceforfCloc(;F).]Weemphasizethattheconvergenceisrequiredtooccurforeachobservablefinthedesignatedclass,buttheconvergenceisnotrequiredtobeuniforminf.Thisisimportant,sincefcouldequallywellbethelocalenergydensityinNewYorkorthelocalenergydensityonAndromeda;andoneshouldnotexpect,inmostsituations,theconvergencetobeuniformonallsuchobservables.Thisreasoningalsosuggeststhatthetwoquasilocaltopologiesarelikelytobetheonesofgreatestphysicalrelevance.Examples..Let0=R,andletn(resp. )betheDiracdeltameasureconcen-tratedonthecongurationinwhichallthespinstakethevalue=n(resp.0).Thenn!intheweakandweakquasilocaltopologies,butnotintheboundedmeasurableorboundedquasilocaltopologies..Let0=f;g,andletnbetheDiracdeltameasureconcentratedonthecongurationwhichis+forallspinsatadistancenfromtheoriginandforallotherspins.LetbetheDiracdeltameasureconcentratedontheconguration

whichisall+.Thenn!intheboundedquasilocal,weakandweakquasilocaltopologies,butnotintheboundedmeasurabletopology.Georgiibaseshistheoryontheboundedquasilocaltopology(whichhecallsthe\topologyoflocalconvergence"orthe\L-topology")[,Chapter];Israelrestrictsattentiontocompactmetricsingle-spinspaces,andusesmainlytheweak(=weakquasilocal)topology[0,ChaptersIIandIV].Finally,letusremarkthatwithrespecttothe(ordinary)weaktopology,M+()isseparableandmetrizable(resp.completemetrizable,compactmetrizable)ifandonlyifis.Letusalsoremarkthatif0isseparableandmetrizable(resp.countableanddiscrete),thentheboundedquasilocaltopologyisstrongerthan(resp.equalto)the(ordinary)weaktopology;thisistrueeventhoughthesehypothesesdonotimplythatC()Bql()..InteractionsandHamiltoniansAsdiscussedintheIntroductiontothissection,theHamiltonianH(!)foraninnite-volumesystemisanill-denedobject.Thereforewemustproceedmorecau-tiously.Wedenersttheconceptofaninteraction,whichcorrespondsroughlytotheideaofa\formalHamiltonian"ora\setofcouplingconstants".Thenwedenethenite-volumeHamiltonianscorrespondingtoagiveninteractionandgivenboundaryconditions.The(meaningless)Hamiltonianofaninnite-volumesystemiswrittenformallyasasumoftermscorrespondingtovariousnitesubsetsofthelattice:one-bodyterms,two-bodyterms,three-bodytermsandsoforth.Mathematicallythisideaismadepreciseasfollows:Denition.Aninteraction(orinteractionpotentialorpotential)isafamily=(A)ASoffunctionsA:!R,suchthatforeachAS,thefunctionAisFA-measurable(i.e.dependsonlyonthespinsinthenitesubsetA).Remark.NotethatwedonotallowtheinteractionAtotakethevalue+.Therefore,a\hard-coreinteraction"isnotincludedinourformulation.Example.ConsidertheIsingmodelwhoseformal(i.e.meaningless)HamiltonianisH(!)\="XhxyiJxy!x!yXxhx!x:(:)Thismodelisdened(meaningfully!)bytheinteractionA(!)=<:hx!xifA=fxgJxy!x!yifA=fx;yg0otherwise(:)ReferencesforthissectionareGeorgii[,Section.]andIsrael[0,SectionsI.andI. ].

ThenextstepistodenetheHamiltonianHcorrespondingtoaninteractionactinginanitevolume.Thisdepends,however,onwhatboundaryconditionsonechooses.Thesimplestcaseisfreeboundaryconditions:Denition.Letbeaninteraction.Then,foreachS,theHamiltonianH;freeforvolumewithfreeboundaryconditionsisthefunctionH;free=XASAA:(:)Notethatthisisalwaysanitesum,sothefree-b.c.Hamiltonianisalwayswell-dened.NotealsothatH;freedependsonlyonthespinsinside.Freeboundaryconditionsarenot,however,sucient:formanypurposesweneedHamiltoniansinwhichtheinteriorofthevolumeisallowedtointeractwiththeexterior.Todothis,wemustconsiderthebondsthatcoupleagivenvolumewithitsexterior;thesegiveacontributionoftheformW;c=XASA\=?A\c=?A:(:)Notethatnowwearedealingwithaninnitesum;thereforewemustbecarefulaboutitsconvergence.Inanycase,theHamiltonianforvolumewithgeneralexternalboundaryconditionscorrespondstoaddingthecontributions(.)and(.):Denition.Letbeaninteraction.Then,foreachS,theHamiltonianHforvolumewithgeneralexternalboundaryconditionsisthefunctionH(!)=XASA\=?A(!)(.a)H;free(!)+W;c(!);(.b)providedthatthissumconvergestoanitelimitforall!,inwhichcasewecalltheinteractionconvergent.Heretheconvergenceisnotrequiredtobeabsolute,norisitrequiredtobeuniformin!;weinsistonlythatthenite-volumeHamiltonianH(!)H(!;!c)bewell-denedforallcongurations!(i.e.allpairsconsistingofaninternalcongurationMoreprecisely,whatthismeansisthatthenet0BBBBB@PASA\=?AA(!)CCCCCASconvergestoanitelimit(foreach! )as"L.

!andanexternalconguration!c).Thisisaverymodestrequirement.Itrulesout,however,theuseofthisformalismforaCoulombsystem,inwhichtheinteractiondecaystooslowlytobesummableinanyreasonablesense.Formanypurposesitisconvenienttothinkofthecongurationoutsideasxed(the\boundarycondition")andthecongurationinsideasvariable.Therefore,foranyxed,wedenetheHamiltonianH;whichusesboundaryconditionoutsidethevolumetobeH;(!)=H(!c):(:)Here!cisthecongurationwhichagreeswith!onandwithonc.NotethatH;(!)dependsonlyonthebehaviorof!inside.ItisalsopossibletodenetheHamiltonianwithotherboundaryconditions(e.g.periodic),butweshallhavenoneedforthese.ThesummabilitypropertiesoftheHamiltonian(.a)haveimportantimplicationsforthecharacteristicsofthemeasuresconstructedwiththem.InadditiontothenotionofconvergenceintroducedinDenition.above,wewishtodistinguishtwostrongernotionsofsummability:Denition.Wecalltheinteractionuniformlyconvergentif,foreveryS,thesum(.a)convergesuniformlyin!;absolutelysummableif,foreveryS,thesum(.a)convergesinB()norm.ThisisequivalenttotheconditionthatPASAikAk

)Anexampleofauniformlyconvergentinteractionwhichisnotabsolutelysummableisthefollowingone-dimensionalIsingmodel[]:A(!)=>><>>:()ncnifAisanon-emptysetofnadjacentpointsand!x=+forallxA0otherwise(:)forasuitablesequenceofnon-negativenumbers(cn)n.Ifncn#0,thisinteractionisuniformlyconvergent;butitisnotabsolutelysummableunlessPnncn<.Thus,cn=nwithR.Foranite-rangeinteraction,thesum(.a)isanitesum,sois(trivially)auniformlyconvergentinteraction.If,inaddition,eachAisaboundedfunction,thenisabsolutelysummable.)Letusnowintroducetwonaturalpiecesofterminology:First,weshallcallaninteractionboundedifeachAisaboundedfunction.Notethatifisbounded(resp.absolutelysummable),theneachHamiltonianH;free(resp.H)isaboundedlocal(resp.boundedquasilocal)function.Aboundedinteraction,however,mayfailtobeabsolutelysummableiftheboundskAkdonotdecayfastenough.)Second:if,asisusual,thespace0(andhence)comesequippedwithatopology,thenwecallaninteractioncontinuousifeachAisacontinuousfunction.Notethatifiscontinuous(resp.continuousanduniformlyconvergent),theneachHamiltonianH;free(resp.H)isacontinuousfunction.Alltheinteractionsconsid-eredinthiswork(andanoverwhelmingmajorityofthoseconsideredelsewhere)arecontinuous.)If0(andhence)iscompact,theneverycontinuousinteractionisautomaticallybounded.Thisisonereasonwhysystemsofboundedspinsareeasiertoworkwiththansystemsofunboundedspins.)Nevertheless,aswediscusslater(Section.. ),allthepropertiesofaninter-actionmustbeinterpretedmodulophysicalequivalence.Inthisregard,theapparentsummabilitypropertiesmayturnouttobemisleading,astheymaychangewidelyfromonephysicallyequivalentinteractiontoanother[]..SpecicationsandGibbsMeasuresReferencesforthissectionareGeorgii[,Chapters{]andPreston[,Chapters,and].

Wenowcometotheheartofthetheoryofinnite-volumelatticesystems,whichistomakeprecisewhatwemeanbyaninnite-volumeGibbsmeasureforagiveninter-action.Wecannotsimplyusetheexplicitformula(.),becausetheinnite-volumeHamiltonianHisill-dened.Thetraditionalsolutionistodeneaninnite-volumeGibbsmeasuretobeanymeasurewhichisalimit(inasuitabletopology)ofnite-volumeGibbsmeasureswithsomechosenboundaryconditions.Thedisadvantageofthisdenitionisthatitiscumbersometocheck:givenameasureontheinnite-volumecongurationspace,howdowedeterminewhetherthereexistssomesequenceofnite-volumeGibbsmeasuresconvergingto?Wewouldprefer,therefore,tohaveamoredirectconditionontheinnite-volumemeasure.SuchaconditionwasrstproposedbyDobrushin[]andLanfordandRuelle[]:theirideaistodeneaninnite-volumeGibbsmeasuretobeonewhoseconditionalprobabilitiesfornitesub-systems,conditionedonthecongurationoutside,aregivenbytheBoltzmann-GibbsformulabasedontheHamiltonianH.Thisistheapproachweshalltake;thetraditionalapproachcanthenbejustiedaposteriori(Propositions.and.).Letusnotethat,ingeneral,wemustconditiononthecongurationintheentireexteriorof|thatis,wemustspecifyacomplete\externalcondition".However,inthespecialcaseofanearest-neighborinteraction(resp.aninteractionofniterangeR),itsucestospecifythespinsimmediatelyadjacentto(resp.thespinsatadistanceRfrom)|hencetheterm\boundarycondition".Weshallusuallybowtotraditionandcallourexternalcongurations\boundaryconditions",butweemphasizethatinthegeneralcaseofaninnite-rangeinteractionitisessentialtospecifythecongurationintheentireexteriorregion.Letusalsoremindthereaderofthephysicalroleplayedbyboundaryconditions:ininnitevolumetheGibbsmeasure(tobedenedshortly)maynotbeunique,andtheboundaryconditionsservetoselectaparticularGibbsmeasure(i.e.aparticular\phase").Allthiswillbedescribedingreaterdetailinwhatfollows...SpecicationsWebeginbyformalizingtheideaof\conditioningontheexteriorofavolume",irrespectiveofanyparticularformulafortheseconditionalprobabilities.Thepointisthatforagivenexternalconguration!c,wewishtospecifythe(conditional)probabilitydistributionofthespinsinsidethevolume:thatis,wewanttospecifyProb!c(d!).Suchanobjectiscalledaprobabilitykernel.Ingeneral,aprobabilitykernelfromaspace(;F)toanotherspace(0;F0)isanobject(!;A)withtwo\slots":an\input"slotthatacceptsaninputconguration!,andan\output"slotthatacceptsasetAF0andreturnsitsprobability.Moreformally,aprobabilitykernelfrom(;F)to(0;F0)isamap:F0![0;]satisfying:(a)Foreachxed!,(! ;)isaprobabilitymeasureon(0;F0).Foramoreextensiveintroductiontoprobabilitykernelsandtheirproperties,see[,Section]or[,SectionIII{].

(b)ForeachxedAF0,(;A)isaF-measurablefunctionon.Weshallwritesuchaprobabilitykernelequivalentlyas(!;A)(Aj!)!(A).Therstnotationemphasizesthatisakindof\transitionprobability"(asinthetheoryofMarkovprocesses);thesecondnotationemphasizesthatwilllaterbein-terpretedasaconditionalprobability;andthethirdnotationemphasizesthat!isaparameter(\boundarycondition")indexingtheprobabilitymeasureon0.Thus,inourcaseweneedtospecifyaprobabilitykernelfrom(c;Fc)to(;F).Fortechnicalreasons,however,itisconvenienttodeneinsteadasaprobabilitykernelfromthefullspace(;F)toitself:wethenimposeexplicitlytheconditionthat(!;)dependon!onlythroughitscomponents!c(i.e.itisFc-measurable),andthatitreproducethe\boundarycondition"!cwhenthequestionfedintoitssecondslotconcernsonlyspinsoutside(i.e.whenAFc).Wearethusledtothefollowingdenition:Denition.Aspecicationisafamily=()Sofprobabilitykernelsfrom(;F)toitself,satisfyingthefollowingconditions:(a)ForeachAF,thefunction(;A)isFc-measurable.(b)isFc-proper,i.e.foreachBFc,(!;B)=B(!).(c)If0,then0=0.Physically,theideaisthat(!;)istheequilibriumprobabilitydistributionforvolumesubjecttotheboundarycondition!outside.Condition(a)statesthatthismeasuredepends,infact,onlyonthebehaviorof!outside.Condition(b)statesthatforobservationsoutside,thismeasureequalsthedeltameasure!,i.e.See[,Section]foramoreleisurelydiscussionofthesepoints.Insomemathematical-physicsliterature(e.g.[])theterm\localspecication"isused.Weemphasizethatthisadjective\local"issuperuous;theconceptsdiscussedhereandin[]areidentical.Inparticular,thereadershouldnotconfusethis(redundant)useoftheword\local"withourconceptof\quasilocalspecication"tobeintroducedinSection...Theproductoftwoprobabilitykernelsisaprobabilitykernel:()(!;A)Z(!;d!0)(!0;A):Forfuturereferencewealsodenetwowaysofmultiplyingameasurebyaprobabilitykernel:()(A)Z(d!)(!;A)()(B)Z(d!)(!;d!0)B(! !0)whereAFandBFF0.Thus,isaprobabilitymeasureontheproductspace(0;FF0),whileisitsmarginalonthesecondspace(0;F0).

itreproducestheboundarycondition.Condition(c)isacompatibilityconditionforpairsofvolumes0:itstatesthatifavolume0isinequilibriumwithitsexterior,thenallsubsetsof0areinequilibriumwiththeirexteriors.Denition.Aprobabilitymeasureonissaidtobeconsistentwiththespec-ication=()Sifitsconditionalprobabilitiesfornitesubsystemsaregivenbythe()S:thatis,ForeachSandAF;E(AjFc)=(;A)-a.e.(:)WedenotebyG()thesetofallmeasuresconsistentwith.Thefollowingpropositiongivestwoapparentlyweaker,butinfactequivalent,for-mulationsofthecondition(.):Proposition.Let=()Sbeaspecication,letbeaprobabilitymeasureon,andletS.Thenthefollowingareequivalent:(a)ForeachAF,E(AjFc)=(;A)-a.e.(b)Thereexistsameasuresuchthat=.(c)=.Physically,(b)statesthatistheequilibriumprobabilitydistributionforvolumewithsome(possiblystochastic)boundarycondition,while(c)statesthatcanitselfplaytheroleof.LetusnotethatG(),thesetofallmeasuresconsistentwith,isaconvexset:if;:::;nbelongtoG(),thensodoesanyconvexcombinationofthem.Thephysicalinterpretationofsuchconvexcombinations,andoftheextremalpointsofG(),willbediscussedinSection...Wealsomakethe(trivial)remarkthatifthelatticeLwerenite,thentherewouldbeauniquemeasureconsistentwith,namelythemeasureL(!;)whichmustbeindependentof!.Thisisoneaspectofthefactthatphasetransitionscannotoccurinnitesystems...GibbsianSpecicationsandGibbsMeasuresAnimportantexampleofaspecicationistheGibbsianspecication=()Scorrespondingtoagiveninteraction.Moreprecisely,letbeaconvergentinter-action,sothatwecandenetheHamiltoniansHwithgeneralexternalboundaryconditions.Let0=QxL0xbeaprobabilitymeasure,calledtheapriorimeasure.WethendenetheconditionalpartitionfunctionZ(!c)=Zexp[H(! )]Yxd0x(!x):(:)

[NotethattheHamiltonianH(!)dependsonboththespins!insideandonthe\boundaryconditions"!c.Afterintegratingoutthespins!,weobtainafunctionof!c.]SinceHiseverywherenite,itfollowsthatZ(!c)>0forall!.IfmoreoverZ(!c)<+forallSandall!,wesaythattheinteractionis0-admissible.NoteinparticularthatifeachHisboundedbelow|whichcertainlyoccursifisabsolutelysummable,sincethismakeseachHbounded|thenisautomatically0-admissible.Also,ifthesingle-spinspace0isnite,theneveryconvergentinteractionisautomatically0-admissible[becausetheintegral(.)isthenanitesumofniteterms].Denition.Let0=QxL0xbeaprobabilitymeasure,andletbeaconvergent,0-admissibleinteraction.Thentheprobabilitymeasure(!;)onFdenedby(!;A)=Z(!c)ZA(!)exp[H(!)]Yxd0x(!x)(:0)iscalledtheGibbsdistributioninvolumewithboundarycondition!ccorrespondingtotheinteractionandtheapriorimeasure0.Itisstraightforwardtoverifythatthefamily=()Sisindeedaspecication;itiscalledtheGibbsianspecicationfor(and0).AmeasureconsistentwithiscalledaGibbsmeasurefor(and0).ByProposition.,ameasureisaGibbsmeasureforifandonlyif=forall,i.e.Zd()Z(c)ZA(!c)exp[H(!c)]Yxd0x(!x)=(A)(:)forallAFandallS.Theequation(.)iscalledtheDobrushin-Lanford-Ruelle(DLR)equation.AslightlysimplerequationisobtainedbyrestrictingAtoF:dd0(!)=Zd()Z(c)exp[H(!c)]0-a.e.(:)Ingeneral(.)isweakerthan(.);theformerisanecessarybutnotsucientconditionfortobeaGibbsmeasurefor.However,innearlyallpracticalsituationsthetwoconditionsareequivalent:seeRemarkattheendofSection..below.Atthispointthereadermaybewondering:Whyhavewebotheredtointroducetheverygeneral(andabstract)conceptofaspecication,whenvirtuallyalloftheconcretemodelsstudiedinstatisticalmechanicscorrespondtoGibbsianspecications?Wehavetworeasons:Firstly,non-Gibbsianspecicationsmustbeemployedinsomeinterestingstatistical-mechanicalproblems,notablythoseinvolvinghard-coreexclusions(whichwedonotconsiderinthispaper)orzerotemperature(AppendixB.. ).Butperhapsmoreimportantly,wewanttobeconsistentwiththeunderlyingmessageofthiswork,

whichisthatnoteverythingintheworldisGibbsian.Therefore,wemustintroduceconceptswhicharegeneralenoughsothattheproblemswewishtostudywillnothavebeenexcludedsimplybydenition.Havingdoneso,wewillthenbeabletoinvestigate,withoutaprioripreconceptions,whichproblemsgiverisetoGibbsianspecicationsandwhichonesdonot...QuasilocalityInalltheoreticalphysics,afundamentalroleisplayedbytheconceptofan\isolatedsystem".Acompletelyisolatedsystemisofcourseanidealization,butonecaningeneralrenderasystemasclosetoisolatedasdesiredbymovingitalargedistanceawayfromallotherobjects.(Hereweneglectcosmologicaleects,aswellascouplingstoeldsthatcouldcarryoradiation.)Thisasymptoticisolationispossible,ofcourse,becausetheinteractionpotentialsdecaytozeroasthespatialseparationtendstoinnity.Onecanevenarguethatthisdecayofinteractionsisanessentialpreconditionforthepossibilityofdoingscience:withoutit,theresultsofexperimentsonEarthwoulddependsensitivelyonconditionsonAndromeda,andtherepeatabilitythatisfundamentaltothescienticmethodwouldbeabsent.Theseremarksjustifytheintroductionofaclassofspecicationsthatwillplayacentralroleintheremainderofthispaper:Denition.Aspecication=()Sissaidtobequasilocalif,foreachS,fBql()impliesfBql().[Equivalently:fBloc()impliesfBql().]Notethat(f)(!)R(!;d!0)f(!0)isthemeanvalueoffintheequilibriumprobabilitydistributionforvolumewithboundarycondition!c.Therefore,aspeci-cationisquasilocalifthemeanvaluesof(quasi)localobservablesdependveryweaklyontheexternalspinsfarfrom(e.g.outsideaverylargevolume0)whentheexternalspinsintheintermediateregion0narexed,i.e.lim0"Lsup!;!(!)0=(!)0j(f)(!)(f)(!)j=0(:)forallfBql()[orBloc()].0Weemphasizethat(.)constrainsonlythedirectinuenceofthespinsoutside0(sincethespinsinthe\annulus"0narexed).Inparticular,(.)isperfectlycompatiblewiththeoccurrenceoflong-rangeorder:itsaysmerelythatanyinuenceonfromthespinsoutside0hastobetransmittedbytheintermediateregion.Weemphasizealsothatthisconditionof\weakdependence"isformulatedinthesupremumnorm,i.e.itisa\worst-case"condition.0Ifthestatespace0isnite,itsucestocheck(.)forfB(;F),becauseanyfBloc()[say,fB(;Fe)forsomee]correspondstonitelymanydierentfunctionsinB(;F)whenonexestheconguration!en.Ifthestatespace0isinnite,wedonotknowwhetherornotthisweakerconditionisequivalentto(. ).0

Examples..IfalltheHamiltoniansHarelocalfunctions,thenisaquasilocalspecication.Thisoccurs,inparticular,ifisanite-range(and0-admissible)interaction..IfalltheHamiltoniansHarequasilocalfunctions,thenisaquasilocalspec-ication.Thisoccurs,inparticular,ifisauniformlyconvergent(and0-admissible)interaction..Althoughwehavenotshownexplicitlyherehowtotreatmodelswithconstraints(e.g.hard-coreexclusions),itiseasytoseethatlocalconstraintsdonotdisruptquasilo-cality.Examplesandcoverallreasonablesystems(ofeitherboundedorunboundedspins)withnite-rangeinteractions.Examplesandcoverallreasonablesystemsofboundedspins.Therefore,wearguethatallsystemsofphysicalinterestarequasilocalwiththeexceptionofmodelsofunboundedspinswithinnite-rangeinteractions.Theselattersystemsare,unfortunately,usuallynotquasilocal:.Consideramodelofreal-valuedspinsf'ig|forexample,aGaussianor'model|withformalHamiltonianH=Xi;jJij'i'j(:)whereJhasinniterange.Thentheresultingspecicationisnotquasilocal,becauseanexternalspinarbitrarilyfarawayfromthevolumecan,bytakingextremelylargevalues,havelargeeectsinside.Thetroublehereisthatquasilocalityisdenedinthesupremumnorm,whichistoostrongaconditionforsystemswithunboundedHamiltonians.(Thereisinfactamoreseriousdicultyinthisexample:forsomeexternalconditionstheHamiltonianHisdivergent.Therefore,totreatthesesystemsitisnecessarytoenlargeslightlytheconceptofspecicationinordertoallowsomeexternalconditionstobe\forbidden"[,pp.{and][,0],orelsetoplaysomeminortrickery[,pp.,{and{].)Wesummarizethemainconclusionfromthisdiscussion:Theorem.0Letbeauniformlyconvergentand0-admissibleinteraction.[Inparticularthishappensifisabsolutelysummable,orifisnite-rangeand0-admissible.]Thenthespecicationisquasilocal.TheGibbsianspecicationarisingfromamodelwithnite(resp.bounded)Hamil-tonianshasanadditionalcharacteristicproperty:Denition.Aspecication=()Sissaidtobenonnull(withrespectto0)if,foreachSandeachAF,0(A)>0=)(!;A)>0forall! :(:)

uniformlynonnull(withrespectto0)if,foreachS,thereexistconstants0<

tobewell-dened,butonlytherelativeHamiltoniansHrel;(!;!0)=XASA\=?[A(!)A(!0)](:)forcongurations!;!0thatagreeoutside.Itturnsout[,Proposition]thatforinteractionswhoserelativeHamiltoniansareuniformlyconvergent(Sullivancallstheseinteractions\L-convergent"),thecorrespondingspecicationisagainquasilocalandnonnull(atleastfornitesingle-spinspace).Sothisgeneralizationdoesnotprovideexamplesofphysicallyinterestingnon-quasilocalspecications.Indeed,wecancombinethisresultwith(c)=)(a)ofTheorem.,andconcludethatforany\relativelyuniformlyconvergent"interaction(atleastonanitesingle-spinspace)thereisanabsolutelysummableinteraction0suchthat=0.Roughlyspeakingthismeansthatand0are\physicallyequivalent"(seeSection..)..Ifisaquasilocalspecication,thenthecriterionfortobeconsistentwithcanbeweakenedslightly:insteadofrequiring=[Proposition.(c)],itsucestohave=onthe-eldF[,Remark.].Thus,ifisauniformlyconvergent(and0-admissible)interaction,thenthealternateDLRequation(.)isequivalenttothestandardDLRequation(.)..Inrathergreatgeneralityitcanbeproven[,00,0]thateverymeasureisconsistentwithsomespecication.However,thisspecicationwillingeneralnotbequasilocal.Indeed,inSectionweshallgivenumerousexamplesofmeasuresthatarenotconsistentwithanyquasilocalspecication...FellerPropertyItisusefultosingleoutaclassofspecicationsinwhichthenite-volumeGibbsmeasure(!;)dependsina\sucientlycontinuous"wayontheboundarycondition! :Denition.Aspecication=()SissaidtobeFellerif,foreachS,fC()impliesfC().Example.Iftheinteractioniscontinuousanduniformlyconvergent(and0-admissible),thenthespecicationisFeller.Thus,nearlyallspecicationsofphys-icalinterestareFeller.ItisworthremarkingthatthedenitionoftheFellerpropertyformallyresemblesthatofquasilocality:indeed,Denition.isidenticaltoDenition.,withBql()replacedeverywherebyC().Inparticular,ifthesingle-spinspace0isnite,thenBql()=C(),sotheconceptsof\quasilocalspecication"and\Fellerspecication"coincide.Wecannowstateaveryimportantuniquenesstheorem:

Theorem.LetbeaprobabilitymeasurethatgivesnonzeromeasuretoeveryopensetU.ThenthereisatmostoneFellerspecicationwithwhichisconsistent.Inparticular,ifthesingle-spinspace0isnite,thenthereisatmostonequasilocalspecicationwithwhichisconsistent.Thistheoremhasanimportantconsequencefortherenormalizationgroup:itshowsthatthedownwardverticalarrowin(.)cannotbeamulti-valuedmap,providedthatweinterpretH0asstandingforaspecication.Remark.Suchuniquenessdoesnotholdingeneralfornon-Fellerspecications.In-deed,if;;:::isanyniteorcountablyinnitesetofprobabilitymeasuresthataredistinguishableatinnity,thereexistsaspecication(ingeneralnon-Fellerandnon-quasilocal)withwhichallthesemeasuresareconsistent.Forexample,let;;:::beGibbsmeasuresofthetwo-dimensionalIsingmodelatanarbitrarysequenceoftemperatures;;:::[;+];thenthereexistsaspecicationwithwhichallthesemeasuresareconsistent!(ByTheorem.,suchaspecicationisofnecessitynon-Fellerandnon-quasilocal.)Thisremarkshowsthatnon-quasilocalspecicationscanbeextremelypathologicaland\unphysical";itisanadditionalargumentfortheimportanceofquasilocality...PhysicalEquivalenceintheDLRSenseThesamephysicalsituationcanbedescribedbymanydierentinteractions.Forexample,theinteractionsA(!)=<:h!iifA=figJ!i!i+ifA=fi;i+g0otherwise(:)and0A(!)=h!iJ!i!i+ifA=fi;i+g0otherwise(:0)bothdescribetheone-dimensionalIsingmodelwithnearest-neighborinteractionJandmagneticeldh;theyareobviously\physicallyequivalent".Thereasontheyare\physicallyequivalent"isthattheydenethesamespecication|anditisthespecicationthatdeterminesthephysics.Reectingalittlebitonthisandsimilarexamples,onecomestothefollowingdenition[,Section.]:Thismeans,roughlyspeaking,thateverycongurationinis\possible",i.e.thereareno\hard-coreexclusions".ThismeansthatthereexistdisjointsetsF;F;:::bFTSFcsuchthatk(Fk)=foreachk.Proof:Formthemeasure=Pkckk,wherec;c;:::>0isanysequencewithsum.By[,00,0]thereexistsaspecicationwithwhichisconsistent.Butthenk=ckFkisalsoconsistentwith[,Lemma. ].

Denition.Letand0beconvergentinteractions.Wesaythatand0arephysicallyequivalentintheDLRsenseif,forallS,thefunctionHH0isFc-measurable(i.e.dependsonlyonthespinsoutside).Onecanthenprovethefollowingtheorem:Theorem.Letand0beconvergent0-admissibleinteractions.Considerthefollowingstatements:(a)and0arephysicallyequivalentintheDLRsense.(b)=0,i.e.thespecicationsforand0coincide.Then(a)=)(b).Moreover,if0(U)>0foreveryopensetU,andtheinteractionsand0arecontinuous,then(b)=)(a).Corollary.(Griths{Ruelle)Letand0beuniformlyconvergent,continu-ous,0-admissibleinteractions;andassumethat0(U)>0foreveryopensetU.IfthereexistsameasurethatisGibbsianforbothand0,thenand0arephysicallyequivalentintheDLRsense,and=0[henceand0haveexactlythesameGibbsmeasures].Thereareseveralwaystodealwiththeambiguitycausedbyphysicalequivalence.Onewayistoselectasingle\preferred"representativefromeachclassofphysicallyequivalentinteractions:intheIsingmodelthisisexempliedbythepossibilityofusing\spin"interactionsA=JAAor\lattice-gas"interactionsA=JAAJA+A[0,];andmoregenerallyitisexempliedbytheconceptsof\-normalized"interactionsand\gas"interactions[,Sections.and.].However,forinteractionswhicharenotnite-range,thisapproachcangiverisetoconvergenceproblems[].Theotherapproachistoaccepttheambiguityasinevitable,andtoworkwithequivalenceclassesofinteractionsmodulophysicalequivalence.Weshalltakethislatterapproach.ThekeyresulthereisCorollary.,dueoriginally(albeitinaveryslightlyweakerform)toGrithsandRuelle[].Thisresulthasanimportantconsequencefortherenormalizationgroup:itshowsthatthedownwardverticalarrowin(. )cannotbeamulti-valuedmap,providedthatweinterpretH0asstandingforanequivalenceclassofinteractionsmodulophysicalequivalence.Toavoidtrivialities,weassumehenceforththat0(U)>0foreveryopensetU.Thismeans,roughlyspeaking,thateverycongurationinis\possible".Ifitwerenotso,thenthetruecongurationspacewouldbeaproperclosedsubsetF=QxLsupp0x.WecouldthenmaketheconditionholdsimplybyredeningthecongurationspacetobeFratherthan.Sotheconditionmeanssimplythatthecongurationspacedoesnotcontainany\uselesspoints".Somesuchconditionisneededfor(b)=)(a)tohold,becausetheinteractioniscompletelyarbitraryatthe\uselesspoints"!nF.

..StructureoftheSpaceG()Physicalsystemsexhibitingeneraloneormorepossible\macrostates",dependingonthevaluesofsomecontrolparameters.Forinstance,watercanbeinaliquid,solidorgaseous\macrostate"dependingontemperatureandpressure;andtherearepointsonthetemperature-pressurephasediagramwheretwoorevenallthreeofthese\macrostates"arepossible.Thephysicalrelevanceofthetheorydevelopedintheprecedingsubsectionsreliesontheassumptionthatforeachphysicalsystemthereexistsaspecicationfromwhichallthestatistical-mechanicalinformationaboutthesystemcanbeobtained:thatis,suchthatthespaceG()ofmeasuresconsistentwithdescribesallthe\macrostates"ofthephysicalsystemthatarepossibleforthegivenchoiceofcontrolparameters.Therefore,wemustbeabletotranscribealltheexpectedpropertiesofthesetofthese\macrostates"intermsofpropertiesofthespaceG().Webrieydiscussherethistranscription.InconsistencywithourmainmessagethatnoteverythingintheworldisGibbsian,everythinginthissubsectionholdsforgeneralspecications,whichneednotbeGibbsian.(Thisgeneralitywillalsobeusefulwhendiscussingstatisticalmechanicsatzerotemperature:seeAppendixB... )However,forthesakeofbrevityandfamiliarity,wewillsometimesrefertothemeasuresconsistentwithasthe\Gibbsmeasures"for|whichisaslightabuseoflanguagewhenisnotGibbsian.Therearetwoimportantpropertiesthatcharacterizethemacroscopicsystemsob-servedinnature.Firstly,thesesystemsinvolveahugenumberofdegreesoffreedom,solargethatonlyastatisticaldescriptionispossible.However,thesestatisticalaspectsdonotmanifestthemselvesatamacroscopiclevel:thatis,macroscopicobservablesdonotuctuate;thesystembehavesdeterministicallywithrespecttothem.Thesecondpropertyreferstothemicroscopicobservables:theydouctuate,buttheiructua-tionsareonlylocal,notaectinglargeregions.Equivalently,localobservationsmadefarawayonefromtheotherarealmostindependent.Totranslatethesepropertiesintoprecisemathematicalstatements,weneedrsttospecifywhatamacroscopicobservableis.Asisusualwithlong-usedconcepts,thereismorethanonepossiblemeaning.Somepeopleconsideramacroscopicobservabletobeanytranslation-invariantmeasurablefunction.Atthispoint,however,wewouldliketoremainatagenerallevel,leavingtheaspectsrelatedtotranslation-invarianceuntilthenextsection.Soweadoptanalternativedenition,whichcorrespondstowhatcouldbecalled\global"observables,namelyobservablesthatdonotdependonwhathappenstonitelymanyspins.Recallthatifisanitesubsetofthelattice,thenFcisthe-eldconsistingofalleventsthataremeasurablebyobservationsmadeThese\macrostates"arealsoreferredtoas\phases"inthechemicalandphysicalliterature.Here,followinganestablishedmathematical-physicsnomenclature,wereservetheword\phase"forthenotionof\purephase",tobedenedinSection..below.Forthisinformaldiscussionweprefertousetheword\macrostate",butkeepingthequotationmarkstoemphasizetheinformalityoftheconcept.Wedonotwanttogetentangledwiththemanydierentsensesadoptedintheliteraturefortheword\state".

solelyoutside;thatis,theyaretheeventsthatdonotdependonthebehaviorofthespinsinside.NowconsidertheeventsthatbelongtoFcforeverynitesubset:theseeventsconstitutea-eldbF\SFc;(:)whichconsistsofallthoseeventswhosedenitionisnotaectedbychangesonanynitenumberofspins.Thiseldisusuallycalledinmathematicsthetaileld,andcouldbethoughtastheeldofglobalevents.Thefunctionsmeasurablewithrespecttothiseldarecalledobservablesatinnityandcanbeinterpretedasglobalobservables.Examplesofglobalobservables..All\macroscopicaverages",forinstanceobservablesoftheformf<:limn!jnjPxnf(!x)ifthelimitexists0(orwhatever)otherwise(:)where(n)isasuitableincreasingsequenceofnitesubsetsofLwhichtogetherexhaustL(wewilldiscussthisfurtherinSection..),andf:0!Risameasurablefunction.Amacroscopicaverageasin(.)isobviouslyunaectedbyalteringnitelymanyspins,sofisindeedanobservableatinnity..InanIsingmodel,considerg(!)=><>:ifthereexistsaninniteconnectedclusterof+spins0otherwise(:)Theexistenceofaninniteclusterisobviouslyunaectedbyalteringnitelymanyspins,sogisindeedanobservableatinnity.(Thisobservableisofparticularimpor-tanceinpercolationtheory. ).InanIsingmodel,considerthedierenceinmagnetizationbetweentheevenandoddsublattices:Mstagg<:limn!jnjPxn()jxj!xifthelimitexists0otherwise(:)where(n)isasbefore.Thisalsoisobviouslyanobservableatinnity.Thus,theusualmacroscopicmeasurementsperformedonrealsystemscorrespondtoglobalobservables,buttheconverseisnottrue:asExampleillustrates,ourcon-ceptof\globalobservables"includessomequantitiesthatareexperimentallynotveryaccessible.Forexample,intheantiferromagneticIsingmodel,thesignofthestaggeredmagnetizationisanobservableatinnity,whichdetectswhichofthetwosublatticesispositivelymagnetizedandwhichisnegativelymagnetized.Butitisveryunlikely

thatanexperimentercouldsucceedinreliablylabellingthetwosublattices,muchlessinmeasuringseparatelytheirmagnetizations.Afterthepreviousdiscussion,wecannowstatemorepreciselywhichpropertiesameasurerepresentinga\macrostate"ofaphysicalsystemmusthave:(i)Itmustbedeterministiconglobalevents,thatis(A)canonlytakethevalues0orforaneventAbF;and(ii)itsexpectationforspatiallydistanteventsmust,insomesense,asymptoticallyfactorize(=short-rangecorrelations=(sometypeof)clusterproperty).Itturnsoutthatthesetwopropertiesareequivalent:Proposition.LetM+().Thenthefollowingpropertiesareequivalent:(a)hastrivialtaileld,thatis,ifAbFthen(A)equalseither0or.(b)hasshort-rangecorrelations,thatis,foreachAFwehavelim"LSsupBFcj(A\B)(A)(B)j=0:(:)Property(a)states,roughlyspeaking,thatalltheobservablesatinnity(=globalobservables)takeaconstantvaluefromthepointofviewofthemeasure.Forinstance,thefactthatallthesetsoftheformf!:f(!)Bghavemeasureeither0ormeansthatthereisaprecisevaluefsuchthatf=fwith-probability.Property(b)isastrong\clusterproperty":itstatesthatdistantregionsofthelatticeareasymptoticallyindependent(evenifoneoftheregionsinvolvesinnitelymanyspins),uniformlyintheobservablemeasuredinthesecondregion.Nowxaspecication,andletusconsiderthestructureofthesetG().WeknowthatG()isaconvexset,soitisnaturaltoaskwhatareitsextremepoints.Theansweris:Proposition.0LetG().Thenthefollowingpropertiesareequivalent:(a)isanextremepointofG().(b)hastrivialtaileld. (c)hasshort-rangecorrelations.Theupshotoftheprecedingdiscussionisthatthe\macrostates"ofaphysicalsystemdescribedbyacertainspecicationcorrespondtotheextremalGibbsmeasuresforthisspecication.Whatistheinterpretationofthenon-extremalmeasuresofG()?For\nice"convexsets,everypointinthesetcanberepresentedasthebarycenterofaprobabilitymeasureconcentratedontheextremepoints(thisisakindof\integral"Werecallthattheextremepointsofaconvexsetarethosethatcannotbewrittenasanon-trivialconvexcombinationofotherpointsintheset.

convexcombination).ItturnsoutthatG()isniceinthissense.Thus,everynon-extremalmeasureinG()isan(integral)convexcombinationofextremalones.Infact,adeepresult([,Theorem.],[,Theorem.])statesthatthisdecompositionisunique,thatis,thatG()isasimplex.Theseresultsmean,inexperimentalterms,thatanon-extremalGibbsmeasurecorrespondssimplytothepreparationofarandomlychosenextremalGibbsmeasure.Theprobabilitiesforthischoicearegivenbythe\coecients"oftheconvexcombination.Thisextrarandomnesscanbeinterpretedasrepresentingignoranceonthepartoftheexperimenteraboutthesystem's\macrostate"(i.e.overandabovehis/herunavoidableignoranceaboutitsmicrostate).Fromthispointofview,thephysicalsystemitselfcanalwaysbeconsideredtobeinawell-dened\macrostate"describedbyanextremalGibbsmeasure.Thus,theextremalGibbsmeasuresarethe\pure"physicalobjects.Asaconsequenceoftheprecedingdiscussion,weconcludethatthecardinalityofthesetofextremalmeasuresofG()representsthenumberofphysical\macrostates"availabletothesystem.Achangeinthisnumberasthecontrolparametersarevariedcorrespondstoaphasetransition(moreprecisely,tooneofthenotionsofphasetransi-tion,seeSection..);andthevariationofthisnumberasafunctionofthesecontrolparameters(temperature,magneticeld,chemicalpotential,etc.)canberecordedintheformofaphasediagram.Therefore,thestudyofthesetofextremalmeasuresofG()isacentralprobleminstatisticalmechanics.Asarststep,itisessentialtodetermineconditionsunderwhichthesetG()isnonempty,i.e.underwhichthereexistsatleastoneinnite-volumeGibbsmeasure.Contrarytowhatonemightinitiallythink,thisisanon-trivialproblem,sincethereexistphysicallyquitereasonablemodelsforwhichtherearenoinnite-volumeGibbsmeasures.Thetypicalexamplesaretheshort-rangemasslessGaussianmodels(harmoniccrystals)indimensiond,andthesolid-on-solidorthediscreteGaussianmodelsind=.TheessentialpointhereisthattheexistenceofGibbsmeasuresinthesemodelsisequivalenttothebreakingofanon-compactsymmetryofthesingle-spinspace(theshiftofallthespinvaluesbyaconstant);and,asiswellknown,itisimpossibletobreakdiscretesymmetriesind=orcontinuoussymmetriesind.Wereferto[,Chapter]forprecisestate-ments,referencesandfurtherexamples.Inanycase,thefollowingtheoremsucesforvirtuallyallapplicationstomodelsofboundedspins:Proposition.Letbeacompactmetricspace,andlet=()SbeaFellerspecication.ThenG()isnonempty.Thisresultis,infact,animmediateconsequenceofProposition.below:takeanysequencewhatsoeverofboundaryconditions(n);bycompactness,thesequence(nn)musthaveatleastonelimitpoint,andProposition.thenguaranteesthatG().ForsystemsofboundedspinsthiscanbeprovenbyappealingtotheChoquettheorem[,0].Forgeneralsystemsitcanbeproventhroughdirectprobabilisticarguments[,pp. {][,Section.andtheassociatednotes][0].

Ifthereareseveral\macrostates"availabletothesystem,andanexperimenterwantstoselectaparticularonewithabsolutecertainty,howmusthe/sheproceed?Therearebasicallytwoways:OneapproachistoaddtotheHamiltoniansomeaddi-tionalelds,suchthataninnitesimalvalueoftheseelds|moreprecisely,alimitprocessconsistinginturningthemonandthenslowlyoinsomeappropriatesequence|selectsoneortheotherofthe\macrostates".Forexample,inanIsingmodelatlowtemperature,onemayaddtotheHamiltonianamagneticeldh;thelimitsh#0andh"0thenselecttheextremalGibbsmeasures+andofthezero-eldIsingmodel.Analternativeapproachistoimmersethe(nite)sampleinacongurationtypicaloftheintended\macrostate"(selectionviaboundaryconditions).Forexam-ple,intheIsingcasewecoulduseboundaryconditionsinwhichthespinsoutsidethevolumearexedtobeall+orall;takingthelimit"Lwiththeseboundaryconditionsagainselects+or,respectively.Inrelationwiththissecondpointofviewwepresenttwopropositions,therstofwhichjustiesaposteriorithetraditionalapproachtoinnite-volumelatticesystemsbasedoninnite-volumelimits:Proposition.Let=()SbeaFellerspecication.Let(n)nbeanin-creasingsequenceofnitevolumeswhoseunionisL,andlet(n)nbeanarbitrarysequenceofprobabilitymeasureson(i.e.arbitrarydeterministicorrandomboundaryconditions).Letbeanylimitpoint(intheweaktopology)ofthesequence(nn)n.Thenisconsistentwith.Inparticular,G()isaclosedsubsetofM+().Proposition.Letbeacompactmetricspace,let=()SbeaFellerspecication,andletbeanextremepointofG().Then,for-a.e. !,lim"LS!=(:)intheweaktopology.Proposition.statesthatanyweaklimitofnite-volumeGibbsmeasures,witharbitrarydeterministicorrandomboundaryconditions,isaninnite-volumeGibbsmeasure.ThisisthelinkbetweentheDLRapproachandthetraditionalapproachvialimitsofcorrelations.Proposition.isaverystrongconversestatement,forthespecialcaseofextremalGibbsmeasures:itstatesthatifonetakesany\typical"congurationfromthemeasureandusesitasaboundarycondition,intheinnite-volumelimitonerecovers.Thisisthemathematicaltranscriptionoftheprocessofselectinga\macrostate"bypreparingthesamplewithanappropriateboundarycondition.Infact,thereisarevealinggeneralizationofthis,thatstatesthatifisanyGibbsmeasure,thenifonetakesa\typical"congurationfromthemeasureandusesitasaboundarycondition,intheinnite-volumelimitonerecoversoneoftheextremalGibbsmeasuresinthedecompositionof[].Thistheoremcanbeinterpretedassayingthattheresultofameasurementonalarge(strictlyspeakinginnite)systemwillalwaysyieldavaluecharacteristicofoneoftheextremalGibbs0

measures:forexample,ameasurementofthemagnetizationinalow-temperatureIsingmodelatzeromagneticeldwillalwaysyieldM0,notanintermediatevalue.Finally,theconsistencybetweenthephysicalpictureandthemathematicalformal-ismrequiressomediscussionoftheissueofdistinguishabilityof\macrostates".Phys-ically,two\macrostates"shouldbeconsidereddierentonlyifthereissomemacro-scopicmeasurementthatcantellthedierence.Intermsoftheformalismdiscussedsofar,thiscorrespondstotherequirementthatglobalobservablesbeabletodistinguishamongthedierentextremalmeasuresforagivenspecication.Thefollowingtheo-remshowsthatevenmoreistrue:theglobalobservablesuniquelycharacterizeeachmeasure|extremalornot|consistentwithagivenspecication.Theorem.Letbeanspecication.Then:(a)TheextremalmeasuresofG()aremutuallysingularwhenrestrictedtothetaileld.Thatis,ifandaredistinctextremalmeasuresofG(),thereexistsasetAbFsuchthat(A)=and(A)=0.(b)EachmeasureG()isuniquelydetermined[amongthemeasuresofG()]bytheeventsinthetaileld.Thatis,ifandaremeasuresinG()suchthat(A)=(A)foreachAbF,then=.Fortheproof,see[,Theorem.]...ConditioningonanArbitrarySubsetofSpinsTheDLRequationstellushowtoconditiononthespinsinthecomplementofaniteset.However,inSectionweshallneedtoconditiononsetsofspinswhicharenotcomplementsofnitesets.Therefore,weneedthefollowingtechnicalconstruction,whichcanbeskippedonarstreading.Let=()Sbeaspecication.LetbeasubsetofL(notnecessarilyco-nite!).Let!beaconguration(butonlyitscomponents!willplayanyrole).WethendenethesystemrestrictedtothevolumeLn,withcongurationspace(0)Ln:thespecicationforvolumeLnwithexternalspinssetto!isthefamily!=(!)S;Lndenedby!(!0;A)=(!!0;A)(:)where!0(0)LnandAFLn.Clearlythefunctions!(;A)areF(Ln)n-measurable.Itiseasytoseethatthefamily!denesaspecicationonthesystemwithlatticeLn.Letnowbeameasureconsistentwith.Let!bearegularconditionalproba-bilityforgivenF.(Suchregularconditionalprobabilitiesalwaysexistif(;F)is,forexample,astandardBorelspace.Thisincludesallexamplesofphysicalinterest.)Wethenhavethefollowingintuitivelyobviousresult:Proposition.For-a.e. !,themeasure!FLnisconsistentwith!.

.TranslationInvarianceUntilnowthelatticeLhasbeensimplyacountablyinnitesetofsites,devoidofanygeometricstructure.Inmostapplications,however,Lisaregulard-dimensionallattice;thisadditionalstructureallowsustodenethenotionoftranslationinvarianceformeasures,interactions,specicationsandsoforth.ForsimplicityweshalltakeLtobethesimple(hyper)cubiclatticeZd.Thisisnoreallossofgenerality,becauseotherregularlatticescanbemappedtoZdbyanappropriatelabellingofsites...VanHoveConvergenceAnimportantroleinthestatisticalmechanicsoftranslation-invariantsystemsisplayedbysequencesofvolumes(n)whichgrowinsuchawaythatthesurface-to-volumeratiotendstozero.Wethereforemakethefollowingdenitions:Denition.Letr>0,andletZd.Wethendenetheinnerr-boundary@r=fx:dist(x;c)rgtheouterr-boundary@+r=fxc:dist(x;)rgther-boundary@r=@r[@+rWecanthenstatethedesiredconditioninanumberofequivalentways:Proposition.Let(n)nbeasequenceofnonemptynitesubsetsofZd.Thenthefollowingareequivalent:(a)limn!j@nj=jnj=0.(b)limn!j@+nj=jnj=0.(c)Foreachr>0,limn!j@rnj=jnj=0.(d)ForeachaZd,limn!jnn(n+a)j=jnj=0.(e)ForeachaZd,limn!j(n+a)nnj=jnj=0. (f)ForeachnitesubsetAZd,limn!jn(n+A)j=jnj=0.ReferencesforthissectionareGeorgii[,Chapter],Israel[0,ChapterIV],Preston[,Chapter]andRuelle[,Chapter].WhatisreallyrelevanthereisnotthatLequalsZd,butmerelythattheadditivegroupZdactsonL:thatis,thereshouldexistbijectionsta:L!L(aZd)suchthattatb=ta+bandt0=identity.Theformulaebelowcaneasilybegeneralizedtothiscase,byreplacingeachoccurrenceofxabyta(x).

Moreover,alloftheseconditionsimplythat:()limn!jnj=.()ThereexistvectorsanZdsuchthatthetranslatesnanlloutZdinthefollowingsense:foreachnitesubsetAZd,thereexistsn0(A)

Proposition.0LetM+;inv().Thenthefollowingpropertiesareequivalent:(a)isanextremepointofM+;inv().(b)Everytranslation-invariantfunctionfB()is-a.e.constant.(c)limn!ndPaCn(fTag)=(f)(g)forallf;gB()[orBql()orBloc()orC()],whereCnisacubeofsiden.(d)lim%jjPa(fTag)=(f)(g)forallf;gB()[orBql()orBloc()orC()].Wenoticethatthe\clusterproperty"embodiedbyproperties(c)and(d)ismuchweakerthantheonepresentedinSection..[part(b)ofProposition.]:(c)and(d)statethatdistantregionsofthelatticeareasymptoticallyindependent,butonlyinanaveragedsense.AmeasureM+;inv()havingthepropertieslistedinProposition.0issaidtobeergodic.Therefore,byconsiderationsanalogoustothoseofSection..,ifweconsiderthetranslation-invariantfunctionstobetheonly\macroscopic"observables,thentheer-godicmeasuresareassociatedtophysical\macrostates"andtheirconvexcombinationsto\mixtures"representingignoranceonthepartoftheexperimenter.Notethat,asintherstpartofSection..(throughProposition.),wehavenotmadeanyreferencetointeractions,specicationsorGibbsianness;thepresentcommentshavegeneralvalidity.Wehavenowintroducedtwodistinctclassesofobservablesthatcouldplausiblybecalled\macroscopic":theglobalobservables(Section..)andthetranslation-invariantobservables(presentsection).Whichclasstrulycorrespondstothe\experi-mentallyaccessible"observables?Thisquestiondoesnothaveacanonicalanswer:italldependsonthesystemandtheexperiments.Itisknown[,Proposition.]thatforatranslation-invariantmeasure,everytranslation-invariantfunctionismeasur-ableatinnity,moduloasetof-measurezero.Theconverseisnottrue.Bylimitingourselvestotranslation-invariantobservables,weeliminatesomenot-very-accessibleglobalobservables,likethestaggeredmagnetizationmentionedinSection...AnalogousquestionscouldbeposedinrelationtowhethertheextremalmeasuresofG()ortheextremalmeasuresofM+;inv()shouldrepresentphysical\macrostates".Weshallcommentbrieyonthispointoncewedenethenotionoftranslation-invariantspecications(Section..).Fornow,letuscommentthattheergodicmeasureshavetheadditionalappealofbeingpreciselythoseforwhich\spaceaveragesequalensembleaverages":Proposition. (Ergodictheorem)Letbeanergodictranslation-invariantprob-abilitymeasureon,andletfL().Then:(a)lim%jjPaTaf=RfdinL()norm.

(b)limn!ndPaCnTaf=Rfdpointwise-a.e.Part(a)iscalledtheL(ormean)ergodictheorem;itiseasilygeneralizedtoLpforallp<.Part(b),whichismuchdeeper,iscalledtheBirkho(orindividual)ergodictheorem.ThesimplexM+;inv()oftranslation-invariantmeasureshasthepropertythatitsextremalelements|namely,theergodicmeasures|aredenseinthewholeset,intheboundedquasilocalorweakquasilocaltopology.Inotherwords,anytranslation-invariantmeasurecanbeapproximatedarbitrarilyclosely,withregardtoanynitesetof(quasi)localobservables,byergodicmeasures.Physicallythismeansthatthroughobservationsinanynitevolume,nomatterhowlarge,onecannotlearnthelong-rangecorrelationpropertiesofthemeasure(ergodicityorthelackthereof).Theproofofthisfactisreallyquitesimple:PaveZdbycubesofsiden;letnbeequaltooneachcube,butindependentbetweencubes(i.e.cutthecorrelationsbetweendistinctcubes);andnally,letenbenaveragedoverthendpossibletranslates(soastomakeittranslation-invariant).Thenitiseasytoseethatenisergodic,andthatlimn!en=intheboundedquasilocaltopology.Wehavejustsketchedtheproofof:Proposition.TheergodicmeasuresareadensesubsetofM+;inv(),intheboundedquasilocaltopology[andhencealsointheweakquasilocaltopology].Thedensityoftheergodicmeasuresisthusanintrinsicandnaturalfeatureofinnite-volumephysics.Geometrically,however,asimplexwithdenseextremepoints(aso-calledPoulsensimplex)ishighlyunintuitive.Indeed,ourusualintuition,derivedfromnite-dimensionalgeometry,isthattheextremepointsshouldformaclosedsubset(ase.g.theverticesofatriangle,ofatetrahedron,etc.).TheunusualbehaviorofM+;inv()ispossibleonlyininnitedimensions.Itwillbeattheoriginofmanyofthe\pathologies"tobediscussedinSection...Remark.Itisanamazingmathematicalfactthata(compactmetrizable)sim-plexwithdenseextremepointsisessentiallyunique:allPoulsensimplicesareanelyhomeomorphictoeachother[,].Ifwethinkoftheergodicmeasuresasrepresentingallthe\macrostates"availabletothesystem,itisnaturaltoinquirewhetherthetranslation-invariantobservablesdistinguishbetweendierentsuchmeasures,asisdesirableonphysicalgrounds(seetheanalogousdiscussionattheendofSection..).Theanswerisyes:Theorem. (a)TheextremalmeasuresofM+;inv()(i.e.theergodicmeasures)aremutuallysingularwhenrestrictedtothe-eldFinvoftranslation-invariantevents.Thatis,ifandaredistinctergodicmeasures,thereexistsasetAFinvsuchthat(A)=and(A)=0.

(b)EachmeasureM+;inv()isuniquelydetermined[amongthemeasuresofM+;inv()]bythetranslation-invariantevents.Thatis,andaremeasuresinM+;inv()suchthat(A)=(A)foreachAFinv,then=.Fortheproof,see[,Theorem.].Infact,thistheoremisalsotruewiththeinvarianteldFinvreplacedeverywherebythetaileldbF;thisfollowsfrom[,Proposition.]...DividingOutTranslationInvarianceTranslationinvariancebringsalongsomenaturalnotionsof\equivalence".Forin-stance,dierentobservablescannotalwaysbedistinguishedwhenlookedatinatrans-lation-invariantmeasure.(Example:0versus.)Inthissectionwediscussthecentralobjectgeneratingallthesenotionsof\equivalence",namelythesetoffunctionsthathavezeroaveragewithrespecttoalltranslation-invariantmeasures.FromnowonuntiltheendofSection,weshallgenerallyassumethatthesingle-spinspace0isacompactmetricspace,i.e.werestrictattentiontomodelsofboundedspins.Thecongurationspaceisthenalsocompact.Thisrestrictionismadeprimarilytosimplifytheexposition;inAppendixAwepartiallyremovethisrestriction.Thefunctionsofinterestherearecharacterizedbythefollowingproposition:Proposition.Let0beacompactmetricspace,andletfC().Thenthefollowingpropertiesareequivalent:(a)fhaszeromeanwithrespecttoeverytranslation-invariantprobabilitymeasure,i.e.Rfd=0forallM+;inv().(b)fhaszeromeanwithrespecttoeverytranslation-invariantnitesignedmeasure,i.e.Rfd=0forallMinv().(c)fliesintheclosedlinearspanofthefamilyoffunctionsfgTag:gC();aZdg.(d)limn!ndPaCnTaf=0.(e)lim%jjPaTaf=0.WedenotebyItheclassoffunctionshavingthepropertiesspeciedintheforegoingproposition;itisaclosedlinearsubspaceofC(),andisexactlytheannihilatorofMinv().ThespaceIwillplayaveryimportantroleinthetheoryoftranslation-invariantequilibriummeasures,andinparticularinthediscussionof\physicalequiv-alence".Wedenethequotient(semi)norms:kfkC()=constinfcRkfck=(supfinff)(.)kfkC()=IinfgIkfgk(.)kfkC()=(I+const)infgI+constkfgk(.)

Thequotient(semi)normsinC()=IandC()=(I+const)aregivenbysimpleexplicitformulae:Proposition.LetfC().Then:(a)lim%jjPaTafexistsandequalskfkC()=I.(b)lim%jjPaTafC()=constexistsandequalskfkC()=(I+const)...SpacesofTranslation-InvariantInteractionsWithL=Zd,italsomakessensetodiscusstranslation-invarianceofinteractions:Denition.Aninteraction=(A)issaidtobetranslation-invariantifA+x=TxAforallAS;xZd:(:)Forexample,theIsinginteraction(.)istranslation-invariantiJxy=J(xy)andhx=h=constant.WenowintroducesomeimportantBanachspacesofinteractions:Denition.Foreach0,wedenotebyBthespaceoftranslation-invariantcontinuousinteractionswithnormkkBXX0jXjkXk<:(:)Moregenerally,foranytranslation-invariantfunctionh:S![;),weletBhbethespaceoftranslation-invariantcontinuousinteractionswithnormkkBhXX0h(X)jXjkXk<:(:)ThemostimportantofthesespacesareB0(\Israel'sbigBanachspace")andB(\Is-rael'ssmallBanachspace").Indeed,B0isnaturallyrelatedtoC()[seeProposition.0below],andsowillbethenaturalspaceonwhichtodevelopthetheoryofequilib-riummeasures(Section.);whileBisthespaceoftranslation-invariantabsolutelysummablecontinuousinteractions(seeDenition. ),andsoisanaturalspaceforthetheoryofGibbsmeasures.NotethatourassumptionhimpliesthatkkBhkkB0andhenceBhB0;soB0isthelargestspaceofinteractionsthatweshallconsider.LetusalsointroducethespaceBniteconsistingofallnite-rangetranslation-invariantcontinuousinteractions.BniteisadenselinearsubspaceofeachoftheBanachspacesBh.Itwillsometimesbeconvenienttocarryoutproofsrstforsomeclassof\nice"interactions|e.g.nite-rangeones|andthenextendtomoregeneralinter-actionsbyadensityargument.

Remark.Thehypothesisofcontinuityoftheinteractionplaysaroleinsomebutnotallofthetheoremsbelow(themathematicallyinclinedreaderisinvitedtogureoutwhichones).Toavoidcomplicatingthenotation,wehaveincludedcontinuityaspartofthedenitionofthespacesB,BhandBnite.WeemphasizethatallthespacesBpermittwo-body(ormoregenerallyn-body)interactionsofarbitrarilylongrange,providedonlythattheyareabsolutelysummable.Indeed,forapuren-bodyinteraction,thenormskkBareallequivalent:wehavekkB=nkkB0.ThedierencebetweenthespacesBisthatlowervaluesofpermitinteractionswhichcontainheaviercontributionsfromlargen,i.e.whichare\morestronglymany-body".Ifwewanttoforcetobe\short-range",wemusttakeh(X)togrowto+asthediameterofX(andnotjustitscardinality)tendstoinnity[,]:Denition.Wewritehif,foreachK<,thereexistsR=R(K)

ItisobviousfromthedenitionofB0thatthissumisconvergentinkknorm,andthatkfkkkB0.Notealsothatfisaquasilocalfunction,i.e.fCql().Thisdenitionoffisnotunique:onecouldequallywelluseinsteadf0XXmin0X(:)whereXmin0denotesthat0isthesmallestelementofXinlexicographicorder,ormanyotherdenitions[,Section.].Theimportantpointisthatallsuchdenitionsgivethesamevalueforthemeanoffwithrespecttoanytranslation-invariantmeasure(thatis,theygivethesame\meanenergypersite");inotherwords,anytwosuchdenitionsoffdierbyanelementofthespaceIdenedinProposition..Therefore,whatisdenednaturallyisnotthemap!fofB0intoC(),butratherthemap![f]ofB0intothequotientspaceC()=I.WecanthendenethefollowingsubspacesofB0Const=f:f=constantg(.)J=f:fIg(.0)J+Const=f:fI+constg(.)andthecorrespondingquotient(semi)normskkB0=Const=infConstkkB0(.)kkB0=J=infJkkB0(.)kkB0=(J+Const)=infJ+ConstkkB0(.)Itisthennotdiculttoverifythat:Proposition.0Let0beacompactmetricspace.Thenthemap[]![f]isanisometryofB0=JontoC()=I,andofB0=(J+Const)ontoC()=(I+const)...PhysicalEquivalenceintheRuelleSenseThediscussionintheprecedingsectionmotivatesthefollowingdenition:Denition.Let;0B0.Wesaythatand0arephysicallyequivalentintheRuellesenseif0J+Const,i.e.ifff0I+const.Ruelle[]wastherst,toourknowledge,tohighlightthecentralroleplayedbythesubspaceIinthevariationaltheory(seealso[,0]).Wehavenowdenedtwodistinctnotionsof\physicalequivalence"forinteractions:TheDLRsense(Section..),whichisdenedforarbitraryconvergent(butnotnecessarilytranslation-invariant)interactions,andwhichguaranteestheequalityofthespecications(Theorem. ).

TheRuellesense,whichisdenedforarbitrarytranslation-invariant(butnotnec-essarilyabsolutelysummableorevenconvergent)interactionsinB0,andwhichguaranteestheequalityofthefamilyofequilibriummeasures(Proposition.below).Itisnaturaltoask,therefore,whetherthesetwonotionsareequivalentontheircommondomainofdenition.Theanswer,fortunately,isyes:Theorem.Letthesingle-spinspace0beacompleteseparablemetricspace,andlet;0beinteractionsinB.Thenand0arephysicallyequivalentintheDLRsenseifandonlyiftheyarephysicallyequivalentintheRuellesense.InSections.and..wewillneedaversionofProposition.\modulophysicalequivalence".Unfortunately,wehavenotbeenabletoprovesucharesultforB(oranyspaceB),andwedonotknowwhetheritittrue.AllwehaveisaresultforspacesBhofshort-rangeinteractions:Proposition.Ifhandthesingle-spinspace0isnite,thenforeachM

(c)kH;freekC()=const=jjkkB0=(J+Const)+o(jj)=jjkfkC()=(I+const)+o(jj)(.)as%(vanHove).(d)H;freeXxTxfo(jj)(:)as%(vanHove).Note,inparticular,part(d)ofthisproposition:sincefis(roughly)\thecontributiontotheenergyfromtheneighborhoodoftheorigin",itfollowsthatPxTxfoughttobe(roughly)\thecontributiontotheenergyfromthevolume".Andindeeditis:whilethissumdoesnotexactlyequalH;free,itdiersfromitonlybya\surface"term.Inthissense,PxTxfcanbethoughtofasyetanotherHamiltonianforvolume,correspondingtosomenewtypeof\boundarycondition".InordertocontroltheHamiltonianswithgeneralexternalboundaryconditions,itisnecessarytotaketolieinthe\small"BanachspaceB:Proposition.LetB.Then:(a)isabsolutelysummable,andkHkjjkkB:(:)(b)kHk=jjkkB0=J+o(jj)=jjkfkC()=I+o(jj)(.0)as%(vanHove).(c)kHkC()=const=jjkkB0=(J+Const)+o(jj)=jjkfkC()=(I+const)+o(jj)(.)as%(vanHove). (d)kW;ckkHH;freek=supkH;H;freeko(jj)(.a)kHXxTxfksupkH;XxTxfko(jj)(.b)as%(vanHove).Insummary,B0sucestocontroltheHamiltonianwithfreeboundarycondi-tions,butBisneededinordertocontroltheHamiltonianwithexternalboundaryconditionsandhencetoapplythetheoryofspecicationsandGibbsmeasures.

..HowtoObtainanInteractionfromaGibbsMeasureIfisaGibbsmeasureforaninteractionB,thentheDLRequationspermitustoreadotheinteraction,modulophysicalequivalence,fromthemeasure:Proposition.(a)LetbeaGibbsmeasure(notnecessarilytranslation-invariant)foraninteractionB.Thenlogdd0XxTxfC()=consto(jj)(:)as%(vanHove).Infact,thisboundisuniformforG().(b)Let;beGibbsmeasures(notnecessarilytranslation-invariant)forinterac-tions;B,respectively.ThenlogddjjkkB0=(J+Const)+o(jj)(.)logddC()=const=jjkkB0=(J+Const)+o(jj)(.)as%(vanHove).Infact,thisboundisuniformforG()andG().Part(a)ofthispropositiontellsusthattheinteractioncanbereconstructedbytakingthelogarithmofthenite-volumedensities.ThiscorrespondstothefactthatBoltzmannfactorsareexponentialsofHamiltonians.Animmediateconsequenceofthisispart(b).Oneimplicationofpart(b)isthatthereconstructedinteractionisuniquemodulophysicalequivalence(Griths-Ruelletheorem):justtake=in(.)toconcludethatkkB0=(J+Const)=0.Inotherwords,ifisaGibbsmeasureforinteractions;B,thenandmustbephysicallyequivalentintheRuellesense.Ofcourse,wealreadyknewthis(Corollary.plusTheorem.).Itiscuriousthatalthough;arerequiredtobelongtothe\small"BanachspaceB,thenalestimateisintermsoftheB0=Jnorm,hencemuchstronger.Thereasonisthat;Bisneededinordertoensurethattheboundaryenergycontributionsareindeedo(jj);butoncethisisdone,thenthebulkenergycontributionisdeterminedbytheB0=Jnorm,asinProposition. (b)...Translation-InvariantSpecicationsandGibbsMeasuresWecannowexaminethetheoryofspecicationsandGibbsmeasuresunderthehy-pothesisoftranslationinvariance.

Denition.Aspecication=()Sissaidtobetranslation-invariantif(!;A)=+a(Ta!;TaA)(:)forallS,!,AFandaZd.Inparticular,ifisatranslation-invariant(andconvergent,0-admissible)interaction,thenisobviouslyatranslation-invariantspecication.Fixatranslation-invariantspecication.WedenotebyGinv()G()\M+;inv()thesetofalltranslation-invariantmeasuresconsistentwith.Ginv()isaconvexset,anditsextremepointsarecharacterizedbythefollowingtheorem:Proposition.Letbeatranslation-invariantspecication.Then:(a)AmeasureGinv()isextremalinGinv()ifandonlyifitisextremalinM+;inv(),i.e.ifandonlyifitisergodic.(b)Ginv()isafaceofM+;inv():thatis,if;M+;inv()and0<

(iii)TheextremalpointsofGinv().Thisisamuchlargersetthantheonedis-cussedin(ii).Inparticularitisneveremptyforcompact-spinmodels(seebe-low).Forinstance,intheexampleoftheIsingantiferromagnet,thereisonlyonetranslation-invariantGibbsmeasure|(+)|whichisobviouslyextremalinGinv()butnotinG().ThesemeasuressatisfythecomparativelyweakerpropertiesofProposition.0:theyaredeterministicforthesmallersetoftranslation-invariantobservables,andtheyexhibittheclusterpropertyonlyintheweakest(Cesaro-averaged)sense,namelyergodicity.Inthemathematicalstatistical-mechanicsliterature,thesemeasures|theextremalpointsofGinv(),orequivalentlytheergodicelementsofGinv()|arecalledpurephasesforthespecication.(Unfortunately,theterm\purephase"issometimesusedwithdierentbutcloselyrelatedmeanings:seee.g.AppendixB...)Whichsetisinterpretedasrepresentingthephysical\macrostates"isaproblem-dependentissue.Inproblemswherenon-translation-invariantmeasuresarerelevant(interfaces,surfacetension,crystalshape,wetting,systemswithdisorder,quasicrys-tals),itismandatorytoconsiderthesetG()ofallGibbsmeasures.Thenthe\macrostates"shouldcorrespondtothemeasuresin(i),andthetranslation-invariant\macrostates"shouldcorrespondtothemeasuresin(ii).Ontheotherhand,ifonelim-itsoneselftomeasuringbulkobservables(i.e.macroscopicaverages),thenitisnaturaltoconsideronlythetranslation-invariantGibbsmeasuresGinv()andtheirextremepoints:thatis,(iii)isthenaturalchoice[(ii)beingoftentoosmall,e.g.empty].Inthisregard,theuseofthecatchylabel\purephases"forthemeasuresin(iii)isontheonehandnatural,giventhetraditionalinterestin\macrostates"withsymmetryundertranslations,butontheotherhandunfortunateforthecurrentinterestinmoregeneralphenomena.Anomenclaturemoreconsistentwithourpurposescouldbetocallex-tremalGibbsmeasuresthosein(i),translation-invariantextremalGibbsmeasuresthosein(ii),andjustergodicGibbsmeasuresthoseof(iii)[orextremaltranslation-invariantGibbsmeasures,providedthatwepayattentiontothesubtletiesofword-ordering].Inanycase,intheremainderofthispaperweshallusetheterm\phase"or\purephase"todenotethemeasuresin(iii),withoneexception:inAppendixB(andonlythere!)weshallsuccumbtothecustomaryterminologyofPirogov-Sinaitheory(aswellasbrevity)andusetheterm\purephase"todenotethemeasuresin(ii)[infactaslightgeneralizationofthem].RegardingtheconditionsunderwhichthesetGinv()isnon-empty,itsucestomentionaresultanalogoustoProposition.(Section.. ):Proposition.Letbeacompactmetricspace,andlet=()Sbeatranslation-invariantFellerspecication.ThenGinv()isnonempty.BecausethetranslationsformanAbeliangroup,thisisanimmediateconsequenceofProposition.andtheMarkov-Kakutanitheorem[0,0].Theideaisthat,givenameasureG(),wecanconstructameasureinGinv()byaveragingovertranslations(andextracting,ifnecessary,aconvergentsubsequence).

Remark.Onewouldliketohaveatranslation-invariantversionoftheGibbsRep-resentationTheorem(Theorem.).Thatis,ifisaquasilocal,uniformlynonnullandtranslation-invariantspecication,onewouldliketoprovethatthereexistsanabsolutelysummabletranslation-invariantinteractionsuchthat=.However,itseemstobeanopenquestionwhetherthisistrueornot.Sullivan[,CorollarytoTheorem]constructedatranslation-invariantwhichis\relativelyabsolutelysummable"(seeRemarkattheendofSection..),whileKozlov[,Theorem]constructedatranslation-invariantabsolutelysummableunderaconditiononstrongerthanquasilocality.0.Entropy,LargeDeviationsandtheVariationalPrinciple:Finite-VolumeCaseWenowbeginthestudyofthesecondapproachtoclassicalstatisticalmechanics,namelytheonebasedonthevariationalprinciple,whichstatesthattheBoltzmann-Gibbsmeasureistheonethatmaximizesentropyminusmeanenergy.Thetheorydevelopedinthissectionisapplicabletoanarbitraryclassical-statistical-mechanicalsystemforwhichtheHamiltonianHmakessense.Inpracticethisusuallymeansanite-volumesystem.Firstweintroducethefreeenergy;nextweintroducetheconceptofrelativeentropyanditsinterpretationintermsoflargedeviations;nallyweprovethevariationalprinciplethatconnectsthesetwoquantities.InSection.wewilldeveloptheanalogoustheoryfortranslation-invariantinnite-volumelatticesystems.Inthissectionweareworkinginacompletelygeneralclassical-statistical-mechanical(=probabilistic)context:(;)isanarbitrarymeasurablespace...FreeEnergyDenition.0Letbeaprobabilitymeasureon(;),andletfbeaboundedmeasurablefunctionon.WethendeneP(fj)=logZefd:(:)Physically,P(fj)isminusthefreeenergyforasystemwithHamiltonianH=fandapriorimeasure.Ourchoiceofsignconventionmakestheformulaeslightlymoreelegant.Itiseasytoprovethefollowingpropertiesofthefreeenergy:Proposition.Letbeaprobabilitymeasureon(;).ThenP(j)hasthefollowingproperties:0Kozlov'sTheoremuses(atleastintheEnglishtranslation)thewords\necessaryandsucient",butinfactheprovesonlythesuciency.ReferencesforthissectionareGeorgii[,Section.],Israel[0,SectionI.andII. ],Preston[,Chapter]andEllis[0,ChaptersI,II,VIIandVIII].

(a)P(0j)=0.(b)fg=)P(fj)P(gj).(c)P(f+cj)=P(fj)+cforanyrealnumberc.(d)P(fj)P(gj)kfgk.Thatis,P(j)isLipschitzcontinuouswithLipschitzconstant.(e)P(j)isconvex.(f)P(j)isstrictlyconvexindirectionscorrespondingtofunctionswhicharenot-a.e.constant...RelativeEntropyDenition.Letandbeanytwoprobabilitymeasureson(;).Thentherelativeentropy(orinformationgainorKullback-Leiblerinformation)ofrelativetoisdenedasI(j)=<:Zlogddd=Zddlogdddif+otherwise(:)Moregenerally,ifAisanysub--eldof,thenwedeneIA(j)=IAA:(:)Actually,ourI(j)isthenegativeoftheusualrelativeentropyS(j);butitismoreconvenienttoworkwithIthanwithS,anditistoocumbersometokeepsayingthewords\negativeof".SoweshalljustcallIthe\relativeentropy"toutcourt.Butthissigndierenceshouldbeborneinmindwheninterpretingthevariationalprinciple!(SeealsotheRemarksattheendofthissubsectionforacomparisonwiththeusualphysicists'entropy.)Itisnothardtoprovethefollowingpropertiesoftherelativeentropy:Proposition.Let;beprobabilitymeasureson(;).Then:(a)0I(j)Imaxlogmin,wheremin=inf?=A(A).[Forexample,ifisnormalizedcountingmeasureonanitespace,thenImax=logjj.Ifisaninnitespace,thenImax=+.](b)I(j)=0ifandonlyif=. (c)I(j)isaconvexfunctionofthepair(;).

(d)Forxed,I(j)is\almost"aconcavefunctionof,inthesensethatInXi=iijnXi=iI(ij)+nXi=ilogi(.0a)nXi=iI(ij)logn(.0b)foranyprobabilitymeasures;:::;nandnumbers;:::;n0withPni=i=.(e)Forxed,I(j)isalowersemicontinuousfunctionofintheboundedmea-surabletopology,andintheweaktopologyifisacompleteseparablemetricspace.(f)Forxedandxedc<,thesetf:I(j)cgiscompactandsequentiallycompactintheboundedmeasurabletopology(andhencealsointheweaktopology).(g)IA(j)isanincreasingfunctionofA.(h)IfAA,and!A(resp.!A)isaregularconditionalprobabilityfor(resp.)givenA,thenIA(j)=IA(j)+Zd(!)IA(!Aj!A):(:)[Thisobviouslyrenes(g).](i)(Strongsuperadditivity)LetA;A;Abesub--eldsofwhichareindependentwithrespectto.ThenIA[A[A(j)+IA(j)IA[A(j)+IA[A(j):(:)Remarks..Thestandardstatistical-mechanicstextbooks(e.g.[,Chapters,and],[0,Section.B],[0,Chapter])introduceaquantitywhichisapparentlytheentropyofasinglemeasure,withoutreferencetoabasemeasure:Sbooks()\="(P! !log!ifisdiscreteR(x)log(x)dxifiscontinuous(:)However,closerexaminationrevealsthatabasemeasurehasbeenintroducedsurrep-titiouslyintheseformulae,namelycountingmeasureinthediscretecaseorLebesguemeasureinthecontinuouscase.Thisbasemeasuredoesplayaphysicalroleinthetheory:thephysicswouldbedierentifcountingorLebesguemeasurewerereplacedRecallthatanetfgconvergestointheboundedmeasurabletopologyifRfd!RfdforallfB(;).

bysomeothermeasure.Thus,theformulae(.),inwhichthebasemeasureishidden,arequitemisleading.(Theyarealsoinelegant,ascanbeseenfromthein-compatibletreatmentgiventothediscreteandcontinuouscases.)Whatisinvolvedhereisthecommonsinoffailingtodistinguishbetweenameasureandadensity(=Radon-Nikodymderivative):thelatterisdenedonlyrelativetoaspeciedbasemeasure.Inmanysituations,thissinisharmless,becausethereisa\natural"anduniversally-agreedchoiceofbasemeasure.Butnothere.Wethereforefeelstronglythatinstatisticalmechanicsthebasemeasureshouldbeintroducedexplicitly.Notealsothatthedenition(.)usesunnormalizedcountingorLebesguemeasureasthebasemeasure,whilewealwaystakethebasemeasuretobeaprobabilitymeasure.Thiscausesan(irrelevant)additiveshiftintheentropy:e.g.fornite,I(j)=Sbooks()+logjj(:)whenisnormalizedcountingmeasure[(f!g)==jjforeach!].Thus,bothI(j)andSbooks()takevaluesintheinterval[0;logjj],butlargevaluesofI(j)correspondtosmallvaluesofSbooks(),andviceversa.Thereaderisurgedtorememberthetwonotationaldierences|thesignandtheadditiveconstant|wheninterpretingourresults..TherelativeentropyI(j)playsanimportantroleininformationtheoryandinmathematicalstatistics(large-sampleasymptotictheoryofhypothesistestingandmaximum-likelihoodestimation);thisfollowsfromthelarge-deviationstheorytobediscussedinthenextsubsection.Seee.g.[0,pp.{]and[,,].Therelationshipwithmaximum-likelihoodestimationisdiscussedalsoinSection..be-low...LargeDeviationsThephysicalinterpretationofrelativeentropyisassociatedwiththeproblemoflargedeviations,whichconcerns,roughlyspeaking,theestimationofthe(verysmall)prob-abilitiesoflargesimultaneousuctuationsinasystemconsistingofalargenumberofrandomvariables.Inthissectionwewillconsiderthecaseofindependent,identicallydistributed(i.i.d. )randomvariables.SoletX;X;:::beasequenceofindependentsamplesfromtheprobabilitydistribution;andletfbeanyboundedreal-valuedmea-surablefunctionon.Thenf(X);f(X);:::isasequenceofindependent,identicallydistributedreal-valuedrandomvariables.InsuchasituationtheweaklawoflargenumbersstatesthatthesamplemeanSfnnPni=f(Xi)is,withhighprobability,veryclosetothetheoreticalmeanvaluemRfd:moreprecisely,ifAisanyclosedInsomecases,countingorLebesguemeasuremayplayaprivilegedrolebyvirtueofsomesym-metry:e.g.spin-ipsymmetryintheIsingmodel,orsymplecticsymmetryinaclassicalHamiltoniansystem.Inothercases,however,theprivilegedmeasurecouldbesomeothermeasure:e.g.HaarmeasureonaLiegroupisnotLebesguemeasureexceptinsomeveryspecialparametrizations.Thisisyetanotherreasonformakingthebasemeasureexplicit:itclarieswhetherornotthereisasymmetryargumentthatprivilegesonechoiceofoveranother.

subsetofthereallinenotcontainingm,thenProb(SfnA)!0asn!.Large-deviationtheorems[0,0,]areastrengtheningoftheweaklawoflargenumbers,inthattheygivethepreciserateofconvergenceofthisprobabilitytozeroasn!.Itturnsoutthatthisprobabilityisexponentiallysmallinn,thatis,Prob(SfnA)enconst(f;;A)(:)whereconst(f;;A)>0wheneverAisaclosedsetnotcontainingm.Moreprecisely,itcanbeshownthatlimn!nlogProb(SfnA)><>:inf:RfdAI(j)ifAisaclosedsetinf:RfdAI(j)ifAisanopenset(:)whereI(j)istherelativeentropy.Intheprecedingthought-experiment,welookedatonlyonereal-valuedobservablef.Moregenerally,wecouldlookatavector-valuedobservablef=(f;:::;fk),andaskfortheprobabilitythatSfnliesinsomesubsetARk.Notsurprisinglywehavelimn!nlogProb(SfnA)><>:inf:RfdAI(j)ifAisaclosedsetinf:RfdAI(j)ifAisanopenset(:)TheseresultscanbewritteninamoresuccinctwaybynotingthetrivialidentitynnXi=f(Xi)= nnXi=Xi! (f)(:)(herexisthedeltameasureatx),whichcanbewrittenasSfn=Ln(f)ZfdLn(:)whereLnnnXi=Xi(:0)iscalledtheempiricalmeasure.WeemphasizethatLnisarandommeasure:itdependsontherandomsampleX;:::;Xn.Inthislanguage,theweaklawoflargenumberscanbereformulatedassayingthattheempiricalmeasureLnis,withhighprobability,veryclosetothetheoreticalmeasure,when\closeness"isunderstoodintheboundedmeasurabletopology(thatis,theweaktopologygeneratedbytheboundedmeasurablefunctions).Moreprecisely,ifAisanyclosedsubsetofM+()notcontaining,thenProb(LnA)!0asn!.Thelarge-deviationtheorem[,,]thenstatesthatthisprobabilityisinfactexponentiallysmallinn,namelyProb(LnA)enconst(;A)(:)Inthisparticulartopology,abasisfortheneighborhoodsofisgivenbythesetsB;f;:ZfidZfid

whereconst(;A)>0wheneverAisaclosedsetofmeasuresnotcontaining.Moreprecisely,limn!nlogProb(LnA)><>:inf:AI(j)ifAisaclosedsetinf:AI(j)ifAisanopenset(:)Infact,thisresultismerelyasophisticatedrestatementof(.),sinceeveryclosed(resp.open)setofmeasuresAiscontainedin(resp.contains)oneoftheformf:RfdAgforsomef=(f;:::;fk)andsomeAclosed(resp.open)Rk.Formulas(.)/(.)provideaphysicalinterpretationoftherelativeentropy.Indeed,wecansay(roughlyspeaking)thattheprobabilitythatasampleX;:::;Xn,takenfromtheprobabilitydistribution,\lookslikeatypicalsamplefrom"decaysexponentiallywithrateI(j):Prob(X;:::;Xnistypicalfor)enI(j):(:)Intheprobabilisticliterature,(.)/(.)arecalledlevel-large-deviationfor-mulae,and(.)iscalledalevel-large-deviationformula...VariationalPrincipleThefreeenergyandtherelativeentropyarerelatedbythefollowingvariationalprin-ciple:Theorem. (Variationalprinciple)Fixaprobabilitymeasureon(;).ThenP(j)andI(j)areconjugateconvexfunctions,inthesensethatP(fj)=supM+(;)ZfdI(j)(.a)I(j)=supfB(;)ZfdP(fj)(.b)Moreover,thesupremumisachievedifandonlyifequalstheBoltzmann-Gibbsmea-sureforHamiltonianH=f(andapriorimeasure),namelyBG;f;efdRefd:(:)wheref=(f;:::;fk)runsoverallnitefamiliesofboundedmeasurablefunctions,andrunsoverallstrictlypositivenumbers.BytheusualweaklawoflargenumberswehaveProb(Ln=B;f;)kXi=Prob(jSfinmij)!0asn!,sincekisnite.SinceanyclosedsetAiscontainedinthecomplementofsomesetB;f;,theclaimProb(LnA)!0isproven.0

Thiscomplementarypairofvariationalprinciplesestablishestheequivalenceof(.)and(.)fornite-volumestatistical-mechanicalsystems.Indeed,RfdisminusthemeanenergyforasystemwithHamiltonianH=f,andI(j)isminustheentropy;therefore,(.a)statesthattheBoltzmann-Gibbsmeasureistheonethatminimizesenergyminusentropy,andthattheminimumvalueofenergyminusentropyequalsthefreeenergy.(Inthermodynamicnotation,F=ETS;recallthatwearetaking=.).Entropy,LargeDeviationsandtheVariationalPrinciple:Innite-VolumeCaseThevariationalapproachdevelopedintheprecedingsectionisadequatefornite-volumestatistical-mechanicalsystems,inwhichtheHamiltonianHiswell-denedandnite.Butitis(notsurprisingly)insucientfortheinnite-volumecase,inwhichalltherelevantquantities|Hamiltonian,freeenergy,meanenergyandrelativeentropy|arealmostcertainlyinnite.Nevertheless,onemighthopethatfortranslation-invariantinnite-volumesystemstherewouldexistananalogoustheoryinwhichtheconceptsoffreeenergy,meanenergyandrelativeentropyarereplacedbythesesamequantitiesperunitvolume;onecouldthendeneanequilibriummeasuretobeatranslation-invariantmeasurethatmaximizestheentropydensityminusmeanenergydensity.Inthissectionweshalldevelopsuchatheory.Butthisinnite-volumetheoryisconsiderablymoresubtlethanitsnite-volumecounterpart:thissubtletyarisesfromthephysicalpossibilityofphasetransitions,aswellasfromadditionalmathematicalpathologiestobeexplainedinSection..below.Thevariationalapproachtoinnite-volumelatticesystemsislessgeneralthantheonebasedontheDLRequations,becauseofitsrestrictiontotranslation-invariantmeasures,butwithinitsrestricteddomainitisequivalenttotheDLRtheory:thekeytheorem(Corollary. )statesthat,foranyinteractionB,theequilibriummeasurescoincidewiththetranslation-invariantGibbsmeasures...FreeEnergyDensity(\Pressure")Welookrstatthefreeenergydensity,orwhatisequivalent,the\pressure":Denition.Letbeatranslation-invariantprobabilitymeasureon=(0)Zd,andletfbeaboundedmeasurablefunction.ThenthepressureoffrelativetoisReferencesforthissectionareGeorgii[,Chaptersand],Israel[0,ChaptersI,IIandV],Preston[,Chaptersand],Ruelle[,Chaptersand]andEllis[0,ChaptersIVandVandAppendixC].Eveniftheinteractionistranslation-invariant,theremayexistnon-translation-invariantGibbsmeasures(e.g.fortheIsingmodelindimensiond[,]),andtheseareofinterestindescribinginterfaces.

denedasp(fj)=limn!ndlogZexpXxCnTxfd(:)ifthislimitexists.Similarly,ifisaninteractioninB0,thenthepressureofrelativetoisdenedasp(j)=limn!ndlogZexphHCn;freeid(:)ifthislimitexists.Thisquantityshouldreallybecalled\minusthefreeenergydensity".Theterm\pres-sure"arisesfromtheinterpretationofthecanonical-ensembleIsingmodelasequivalenttoagrand-canonical-ensemblelatticegas;inthegeneralcasetheterm\pressure"isnotreallyappropriate,butithasbecomestandardamongmathematicalphysicists.Ithas,atleast,thevirtueofbrevity.Weemphasizethattheexistenceofthelimit(.)[or(.)]isanontrivialprob-lem;infact,thereexistexamplesoftranslation-invariantmeasuresforwhichthelimitdoesnotexist,evenforsimplelocalfunctionsf(seeAppendixA..).Therefore,weshallrestrictattentiontotwocases:whenisaproductmeasure,andmoregenerally,whenisaGibbsmeasureforatranslation-invariantinteraction.Proposition.Letbeaproductmeasure.Thenthepressurep(fj)existsforallboundedquasilocalfunctionsf;infact,thelimitexistsalsoinvanHovesense,namelyp(fj)=lim%jjlogZexp"XxTxf#d:(:)Moreover,p(j)hasthefollowingproperties:(a)p(0j)=0.(b)fg=)p(fj)p(gj).(c)p(f+cj)=p(fj)+cforanyrealnumberc.(d)p(fj)p(gj)kfgk.Thatis,p(j)isLipschitzcontinuouswithLipschitzconstant.(e)p(f+hj)=p(fj)foranyhI. (f)p(j)isconvex.Weemphasize,inparticular,part(e):thepressureisconstantwithin\subspacesofphysicalequivalence".

Proposition.Letbeatranslation-invariantGibbsmeasureforaninteractionB(andapriorimeasure0).Thenthepressurep(fj)existsforallboundedquasilocalfunctionsf;infact,thelimitexistsalsoinvanHovesense,namelyp(fj)=lim%jjlogZexp"XxTxf#d:(:)Moreover,thelimitisgivenbyp(fj)=p(ffj0)p(fj0):(:0)Inparticular,p(j)hasalltheproperties(a){(f)ofProposition..Thepressureofafunctionfisthesimplestobjectfromamathematicalpointofview,butthepressureofaninteractionisperhapsmorefamiliartophysicists.Infactthesetwoobjectsareessentiallyidentical:Proposition.LetB0,andletbeatranslation-invariantmeasuresatisfyingtheconditionsofProposition.or..Then:(a)p(j)existsandequalsp(fj).Infact,thelimitexistsalsoinvanHovesense,i.e.p(j)=lim%jjlogZexphH;freeid:(:)(b)IfinadditionB,thenforany,lim%jjlogZexphH;idalsoexistsandequalsp(fj).Part(b)statesthat,forinteractionsB,thepressureisindependentofboundaryconditions.Thereaderwillnotethatwehavenotassertedthestrictconvexityofp(j);thisisbecause,insharpcontrasttothenite-volumecase,theinnite-volumepressureisnotstrictlyconvex(notevenmodulophysicalequivalence).Indeed,thisfailureofstrictconvexityisattheoriginofsomerathersurprisingpathologiesoftheinnite-volumevariationaltheoryinthe\large"spaceofinteractionsB0(seeSection..below).However,inthesmallerspaceBthesepathologiesdonotarise:Proposition. (Griths{Ruelle[])Letbeatranslation-invariantmea-suresatisfyingtheconditionsofProposition.or..Thenthepressurep(j),restrictedtothespaceofinteractionsB,isstrictlyconvexindirections=J+Const.

Note,inparticular,thecontrapositiveofthisproposition:ifp(j)isnotstrictlyconvexonBindirections=J+Const,thenisnottheGibbsmeasureforanyinteractioninB.Thisgivesamethodforprovingnon-Gibbsianness,whichwillbeexploitedinSection..Remark.ThefailureofstrictconvexityinB0wasrstpointedoutbyFisher[],whoprovidedafamilyofexactlysolubleone-dimensionalIsingmodelsinwhichthepressurecanbeexplicitlyseentohavestraightsegments.ThesemodelsarelatticeversionsoftheFisher-Felderhof[0,,,]clustermodels.Thefailureofstrictconvexitycanherebegivenaphysicalinterpretationintermsoftheformationofaperfectlyrigidcrystal.ThisindicatesthatB0nBdoescontainsomeinteractionsofphysicalinterest,ifonlyfortheirratherstrangethermodynamicproperties...RelativeEntropyDensityForbrevitywehenceforthwritetherelativeentropyinvolumeasI(j),insteadofthemorepedanticIF(j).Wenowdenetherelativeentropydensity:Denition.0Let;betranslation-invariantprobabilitymeasureson=(0)Zd.Therelativeentropydensity(orrelativeentropyperunitvolume)ofrelativetoisdenedasi(j)=limn!ndICn(j)(:)ifthislimitexists.Weemphasizethattheexistenceofthelimit(.)isanontrivialproblem;infact,thereexistexamplesoftranslation-invariantmeasures;forwhichthelimitdoesnotexist(seeAppendixA..).Therefore,justasforthepressure,weshallrestrictattentiontotwocases:whenisaproductmeasure,andmoregenerally,whenisaGibbsmeasureforatranslation-invariantinteraction.Proposition.Letbeaproductmeasure.Thentherelativeentropydensityi(j)existsforalltranslation-invariantprobabilitymeasures;infact,thelimitexistsinvanHovesenseandalsoasasupremum:i(j)=lim%jjI(j)(.a)=supSjjI(j):(.b)Moreover,i(j)hasthefollowingproperties:(a)0i(j)imaxlogmin;0,wheremin;0=inf?=AFf0g(A).[Forexample,ifistheproductofnormalizedcountingmeasureonanitesingle-spinspace0,thenimax=logj0j.Ifthesingle-spinspace0isinnite,thenimax=+.]

(b)i(j)isananefunctionof,i.e.inXi=iij=nXi=ii(ij)(:)foranymeasures;:::;nM+;inv()andnumbers;:::;n0withPni=i=.(c)Forxed,i(j)isalowersemicontinuousfunctionofintheboundedquasilo-caltopology,andintheweakquasilocaltopologyif0isacompleteseparablemetricspace.(d)Forany,thereexistsasequence(n)nsuchthatn!intheboundedquasilocaltopology,andi(nj)=imaxforalln.Itfollowsthati(j)isadiscontinuousfunctionofintheboundedquasilocaltopology(andhencealsointheweakquasilocaltopology)ateachsatisfyingi(j)

Moreover,thislimitisgivenbyi(j)=i(j0)+p(fj0)+Zfd:(:)Moreover,i(j)hasproperties(b)and(c)ofProposition..Notethat,by(.),therelativeentropydensityi(j)dependsononlyviatheinteraction:thatis,ifandaretranslation-invariantGibbsmeasuresforthesameinteractionB,wehavei(j)=i(j)forall.Thereaderwillnotethatwehavenotassertedthati(j)=0ifandonlyif=.Indeed,thisnaiveconjectureisfalse:aswehavejustseen,i(j)=0alsoholdswheneverandaretranslation-invariantGibbsmeasuresforthesameinteraction.InSection..weshallshowthat,roughlyspeaking,i(j)=0onlywhenandareGibbsmeasuresforthesameinteraction.ThisfactwillplayacrucialroleintheproofoftheFirstFundamentalTheorem(seeSection.).Remark.Wehaveproventheexistenceofi(j)whenisaGibbsmeasure,butthisdoesnotexhaustthecasesforwhichi(j)exists.Indeed,bycombiningTheorem.withourconstructioninSection.,weprovideanexplicitexampleofnon-Gibbsiantranslation-invariantmeasuresandforwhichi(j)exists(andisinfactzero):namely,(resp.)istheimageofthe+(resp.)phaseofthetwo-dimensionalIsingmodel(atlowenoughtemperature)undertheb=decimationtransformation.Itisaninteresting(andprobablydicult)mathematicalproblemtocharacterizethepairs(;)forwhichi(j)exists...LargeDeviationsInSection..wedevelopedthetheoryoflargedeviationsforindependentrepetitionsofanarbitraryprobabilisticexperiment.Thistheoryprovidedaphysical(andstatis-tical)interpretationfortheconceptofrelativeentropy.Itisnaturaltoaskwhetherthereisananalogue,fortranslation-invariantmeasuresonaclassicallatticesystem,inwhich\timeaverages"arereplacedby\spaceaverages".Thatis,insteadofconsideringlargedeviationsforthesamplemeaninalargenumberofindependentrepetitionsofthesameexperiment,onemightinsteadconsiderlargedeviationsfromspatialmeans(physically,largeuctuationsofextensivequantities)inasingleinnite-volumere-alization.Suchalarge-deviationtheorywouldthen,itishoped,provideaphysicalinterpretationoftherelativeentropydensity.Inthissectionwedescribe(withoutproof! )thebasicfeaturesofsuchalarge-deviationtheory.Weemphasizethatthistheoryismuchmoresubtlethanthetheoryfortheindependent-repetitionscase,becausethespinsindisjointregionsofspaceneednotbeprobabilisticallyindependent.Indeed,forgeneraltranslation-invariantmeasureson,nosatisfactorylarge-deviationtheoryisknown.Therefore,weshallrestrictattentiontothecaseinwhichisanergodictranslation-invariantGibbsmeasureforaninteractionB.OurexpositionisbasedontherecentworkofFollmerandOrey

[],Olla[,],Comets[]andGeorgii[](seealso[]),whichinturnisinspiredbythepioneeringworkofDonskerandVaradhan[,00,0,0,0].Inthephysicsliterature,therelationbetweenthermodynamicsandlargedeviationswaspointedoutlongagobyLanford[0].Iffisaboundedmeasurablefunctionon,thenthemeanergodictheoremstatesthatthespatialaveragesSfjjPaTafconvergeinL()normtotheexpectedvaluemRfd,as%.Inparticular,ifAisanyclosedsubsetofthereallinenotcontainingm,thenProb(SfA)!0as%.Themeanergodictheoremis,therefore,anaturalgeneralizationoftheweaklawoflargenumbers.Thelarge-deviationtheoremsstrengthentheergodictheorembygivingapreciserateofconvergenceofProb(SfA)tozeroas%.Ifwerstrestrictattentiontosingle-siteobservablesf(i.e.functionsofasinglespin),thenthelarge-deviationtheoremsforspatialaverages(level)andforthesingle-siteempiricalmeasure(level)aredirectanaloguesof(.)and(.):lim%jjlogProb(SfA)><>:infM+;inv():RfdAi(j)ifAisaclosedsetinfM+;inv():RfdAi(j)ifAisanopenset(:)andlim%jjlogProb(LA)>><>>:infM+;inv():Ff0gAi(j)ifAisaclosedsetinfM+;inv():Ff0gAi(j)ifAisanopenset(:)wherei(j)istherelativeentropydensity,andforeachconguration!thesingle-siteempiricalmeasureinvolumeisdenedtobeLjjPi!i.TheempiricalmeasureLisatoolforstudyingeventsoccurringatasinglesiteonly.Theseeventswouldcompletelycharacterizethemeasureifitwereaproductmeasure(asinthei.i.d.casestudiedinSection..),butinthegeneralcaseoneclearlyneedsmulti-siteobservables(i.e.functionsofseveralspins)inordertodescribecorrelations.Thestudyofsuchobservablesgivesrisetothe\level-"large-deviationtheory.ItisbasedonthetrivialidentityjjXa(Taf)(! )= jjXaTa!!(f);(:)whichcanbewrittenasSf=R(f)ZfdR(:00)whereRjjXaTa!

(:0)Inthemathematicalliteraturethelarge-deviationtheoremsareusuallyprovenforsequencesofcubes,butthesameargumentsoughttoworkforgeneralvanHovesequences.

iscalledtheempiricaleld.WeemphasizethatRisarandommeasure(ontheinnite-volumecongurationspace),sinceitdependsontherandomconguration!.Inthislanguage,theergodictheoremcanbereformulatedasimplyingthattheem-piricaleldRis,withhighprobability,veryclosetothetheoreticalmeasure,when\closeness"isunderstoodintheboundedquasilocaltopology(i.e.theweaktopologygeneratedbytheboundedquasilocalfunctions).Moreprecisely,ifAisanyclosedsubsetofM+()notcontaining,thenProb(RA)!0as%.Thelarge-deviationtheorem[]thenstatesthatthisprobabilityisinfactexponentiallysmallinjj,namelyProb(RA)ejjconst(;A)(:0)whereconst(;A)>0wheneverAisaclosedsubsetofM+()notcontaining.Indetail,lim%jjlogProb(RA)><>:inf:A\M+;inv()i(j)ifAisaclosedsetinf:A\M+;inv()i(j)ifAisanopenset(:0)Theseformulaeprovideaphysicalinterpretationfortherelativeentropydensity.Roughlyspeaking,theprobabilitythataconguration!,takenfromtheprobabilitydistribution,\looksinlikeatypicalcongurationfrom"decaysexponentiallyinthevolumeofwithratei(j):Prob(!istypicalfor)ejji(j):(:0)ThisinterpretationoftherelativeentropydensitywillplayakeyroleinmotivatingtheFirstFundamentalTheorem(Section.).Remarks..Someofthelarge-deviationtheoremsuseaperiodizedempiricaleldR(per),whichisatranslation-invariantmeasureon.OneexpectsRandR(per)tobehaveinthesameway..OurresultsinSectiongiveexamplesofsomenon-Gibbsianmeasuresforwhichalarge-deviationstheorycanbedeveloped,e.g.=Twhereisatwo-dimensionalIsing-modelGibbsmeasureatlowtemperature,andTisasuitablerenormalizationmap.Ofcourse,oneisabletocontrolthelargedeviationsforonlybyreducingittothesameproblemforthebetter-behavedmeasure...VariationalPrincipleThepressureandtherelativeentropydensityarerelatedbythefollowingvariationalprinciple:Theorem. (Variationalprinciple)Fixatranslation-invariantmeasuresat-isfyingtheconditionsofProposition.or..Thenp(j)andi(j)areconjugate

convexfunctions,inthesensethatp(fj)=supM+;inv(;F)Zfdi(j)(.0a)i(j)=supfBql(;F)Zfdp(fj)(.0b)Writtenintermsofinteractions,thisreadsp(j)=supM+;inv(;F)Zfdi(j)(.0a)i(j)=supB0Zfdp(j)(.0b)Thisvariationalprinciplegivesusanotherwaytoassociate(innite-volume)prob-abilitymeasurestoagiveninteraction:Denition.LetB0andM+;inv().Wesaythatisanequilibriummeasurefor(andapriorimeasure0)ifthepair(;)saturatesthevariationalprinciple(.0)with=0,i.e.ifp(j0)+i(j0)=Zfd:(:0)Wehavenowlaidouttwodistinctapproachestoinnite-volumephysics:)TheDLRapproach,whichsayswhatitmeansfora(notnecessarilytranslation-invariant)measuretobeaGibbsmeasureforaconvergentand0-admissible(butnotnecessarilytranslation-invariant)interaction.Thisistheinnite-volumeanalogueoftheexplicitformula(.).Thisapproachisconstructedpurelyonthebasisofprobabilitytheory,andhenceitcanbecalledthestatistical-mechanicalapproach.)Thevariationalapproach,whichsayswhatitmeansforatranslation-invariantmeasuretobeanequilibriummeasureforatranslation-invariant(butnotnec-essarilyconvergent)interaction.Thisistheinnite-volumeanalogueofthevariationalprinciple(.).Thisapproachisbasedonoptimizationofthermo-dynamicpotentials,andhenceitcanbecalledthethermodynamicapproach.However,asremarkedbyWightman[],conventionalthermodynamicsreferstotheoptimizationofpotentialswithrespecttoaratherreducednumberofpa-rameters(temperature,chemicalpotential,etc. ).Incontrast,theoptimizationofthepreviouspropositioniswithrespecttoaninnite-dimensionalspaceofpossibleinteractions.Fortranslation-invariantinteractionsandtranslation-invariantmeasures,thismeansinpracticethefollowing:TheDLRapproachappliestoamorerestrictedclassof

interactions,butinreturnprovidesmuchmoreinformationonthemeasures.Thatis,itrequiresB,butgivesstrongcontrolonviatheDLRequations(.)/(.).Ontheotherhand,thevariationalapproachneedsonlyB0,butprovidesmuchweakercontrolover.Inanycase,thetwoapproachesareequivalentintheircommondomainofapplicability:ifisatranslation-invariantinteractioninBandisatranslation-invariantmeasure,thenisaGibbsmeasureforifandonlyifitisanequilibriummeasurefor.WewillprovethisinCorollary.below.Atthispoint,thereadermaybewondering:IftheDLRandvariationalapproachesareequivalent(forinteractionsinB),thenwhybotherintroducingbothofthem?Whynotstickwithoneortheother,andshortenthisarticlebyatleast0pages?TheansweristhatmanydeepresultsarebasedontheinterplaybetweenDLRandvariationalideas.ThisisthecaseforTheorem.below,anditisalsothecaseformanyofourRGresults(notablythoseinSections.,.and.).Beforeleavingthesubjectofthevariationalprinciple,letusnoteasimplecorollary.LetF(;)betheamountbywhichthepair(;)failstosatisfythevariationalprinciple,i.e.F(;)p(j0)+i(j0)+Zfd0:(:0)ThenitiseasytoseethatjF(;)F(;0)jk0kB0=(J+Const);(:0)indeed,thisisanimmediateconsequenceofPropositions.(c){(e)and.(a).Inparticular,ifisanequilibriummeasurefor,thentheamountbywhichfailstosatisfythevariationalprinciplefor0isatmostk0kB0=(J+Const).NotealsothatifB,thenF(;)canbeinterpretedasarelativeentropy:F(;)=i(j)foranyGinv():(:0)Thisisthecontentofequation(. ).Aspecialcaseof(.0)[whichisalsoeasytoseedirectly]isthefollowing:Proposition.Let;B0bephysicallyequivalentintheRuellesense(i.e.0J+Const).Thenand0haveexactlythesameequilibriummeasures...WhatisaPhaseTransition?Informally,theoccurrenceofaphasetransitionisassociatedtooneorbothofthefol-lowingphenomena:asingularityofsomethermodynamicpotentialand/orthechangeinthenumberof\macrostates"availabletothesystem.Historically,therstpointofviewwasprimarilyassociatedwithEhrenfest,whilethesecondpointofviewwasprimarilyassociatedwithGibbs.However,thefullformalizationofthesecondpointofview|inparticular,givingaprecisemeaningto\macrostate"|andtheclaricationoftherelationbetweenthesethermodynamicconceptsandtheunderlying(microscopic)0

statistical-mechanicalconceptshadtoawaitthedevelopmentoftheDLRandrigorousvariationalapproaches.Thegeneralinterpretationofphasetransitionsassingularitiesofthe(whatturnedouttobeinnite-volume)freeenergy(=pressure)gaverisetotheEhrenfestclassi-cation:asystemissaidtoexhibitannth-orderphasetransitionifsomenthderivativeofthefreeenergyisdiscontinuous(andallthederivativesoflowerorderarecontin-uous).Forexample,thetwo-dimensionalIsingmodelatlowtemperaturesundergoesarst-orderphasetransitionasthemagneticeldpassesthroughzero,becausethemagnetization(=rstderivativeofthefreeenergywithrespecttotheeld)hasadiscontinuity.Ontheotherhand,iftheeldiskeptequaltozeroandthetemperatureislowered(startingfromahighvalue),thesystemundergoesasecond-orderphasetransitionatthecriticaltemperature,becausethemagnetizationandenergy(=rstderivativesofthefreeenergy)remaincontinuousbutthesusceptibilityandspecicheat(=secondderivativesofthefreeenergy)blowup.Fromthepointofviewofmathematicalphysics,however,theEhrenfestclassicationisbothtoodetailedandtoocrudeforourcurrentlevelofunderstanding.Itistoodetailedbecause,aswediscussbelow,onlythedistinctionbetweenrst-orderandtheresthasbeenputontoarmbasis.Consequently,authorsusuallygroupallthetransitionsofordertwoorhigherintoasingleclassandcallallofthemcontinuousphasetransitions|becausetheorderparameter,e.g.themagnetization(seebelow),remainscontinuous.Ontheotherhand,theEhrenfestclassicationistoocrude,becausethepossiblesingularitiesofthefreeenergyaremuchtoovariedtobecapturedinasingleintegern.Someexamplesare:Theone-dimensionalIsingmodelwith=rinteraction,inwhichitisbelieved[,,]thatthefreeenergyf(;h=0)isCbutnonanalyticatthecriticalpointc,atthesametimeasthespontaneousmagnetizationM(;h=0)=@f=@hjh=0isdiscontinuousatc(Thoulesseect)[].Thetwo-dimensionalXYmodel(Kosterlitz-Thoulesstransition),inwhichitisbelieved[0]thatthefreeenergyf(;h=0)isCbutnonanalyticatc;herethespontaneousmagnetizationM(;h=0)=@f=@hjh=0vanishesidentically,whilethezero-eldsusceptibility(;h=0)=@f=@hjh=0isbelievedtoblowupatcandremaininniteforallc.Systemswithdisorder,inwhichitisexpectedingeneral(andsometimesproven)thatathightemperaturethefreeenergyiseverywhereCbutnowhereanalytic,asafunctionoftemperatureand/ormagneticeld.ThisphenomenonisknownasaGrithssingularity[].Thedescriptionoftransitionswherethenumberof\macrostates"changesisbasedontheuseoforderparameters.Theseareobservablesacquiringdierentexpectationvaluesforthedierent\macrostates".Each\macrostate"canbeselectedeitherbyintroducingsomeextraeldthatisturnedointhelimit,orbyusingtherightboundaryconditions.Theconnectionbetweenthispointofviewandtheexistenceof

singularitiesinthepressure(freeenergy)wasinformallyknownsincethebeginningoftheeld:Thepressurehastobeconvex|forthesytemtobestable|henceitsonlypossiblediscontinuitiesaretheexistenceof\sharpcorners"wherethevariousone-sidedderivativesofthepressuretakedierentvalues.Eachofthesevaluesdenesadierent\macrostate".Forexample,inthecaseoftheIsingmodel,therightandleftderivativeswithrespecttothemagneticeldgivethetwopossiblemagnetizations.Onecanselectoneofthemagnetizationsbyturningoapositivemagneticeld(i.e.comingfromtheright)oranegativeone(leftlimit),or,alternatively,bysurroundingthesamplebyspinspolarizedinthedesiredform.Itturnsoutthatthisintuitioncanbeformalizedintheframeworkofthevariational-principleapproach.UsingtheabstractnotionoftangenttoaconvexfunctionalinaBanachspace,GallavottiandMiracle-Sole[]andLanfordandRobinson[]showedinthemid-0'showtheexistenceofmorethanonepurephase(ergodicequilibriummeasure)isequivalenttolackof(G^ateaux)dierentiabilityofthepressure(seee.g. [0]or[,Chapter]).Moreover,incompleteagreementwiththeaboveexampleoftheIsingmodel,thedirectioninwhichthedierentiabilityfailsispreciselythedirectionoftheeldconjugatetotherelevantorderparameter,andthedierentdirectionalderivativesgivetheexpectationsofthisobservableinthedierentpurephases.Therefore,ifwerestrictourselvestotranslation-invariantspecicationsandmea-sures,wehavetheimportantdistinctionthatrst-orderphasetransitionscorrespondtoachangeinthenumberofergodicequilibriummeasures(purephases),whilecontin-uoustransitionsdonotnecessarilychangethisnumberandcorrespondtomuchmoresubtlephenomena(e.g.slowdecayofcorrelations=uctuationspropagatingovermacroscopicscales=criticalopalescence).Thepointsinparameterspacewherethereisasecond-(orhigher-)orderphasetransitionarecustomarilycalledcriticalpoints,inanalogytothecriticalpointofliquid-gassystems,whichwastheearliest-knownexampleofthisphenomenon.Forphenomenainwhichonehastogobeyondtranslationinvariance,theconnectionbetweenfree-energysingularitiesandpropertiesofthesetofextremalGibbsmeasuresislessclear.Nevertheless,transitionsinvolvingachangeinthenumberofextremalGibbsmeasuresareusuallycalled(byanalogyratherthanlogic)\rst-order"alsointhisgeneralcase.Correspondingtothetwodierentnotionsof\phasetransition"mentionedatthebeginningofthissubsection,therearetwodierenttypesofresulton\absenceofphasetransitions":Ontheonehand,thereareresultsprovingtheuniquenessoftheGibbsmeasure(jG()j=)[,0,,]orofthetranslation-invariantGibbsmeasure(jGinv()j=)[,].Ontheotherhand,thereareresultsonanalyticityofthefreeenergyandcorrelations[0,0,,,,].Inthelasttworeferences,DobrushinandShlosmanintroducedanextremelystrongnotionofabsenceofphasetransitions,whichtheycallthecompleteanalyticitycondition.Itcorrespondsroughlytotheanalyticityofallthenite-volumefreeenergiesuniformlyinthevolumeandintheboundaryconditions.Itisknownthatingeneralthedierentnotionsofpresenceandabsenceofphase

transitionsarenotequivalent.Thisnon-equivalenceisprobablyduetophysicalreasonsinmostofthecases,butsometimesitseemsanartifactofthemathematicalformalism[,]...WhenistheRelativeEntropyDensityZero?Wenowcometoakeyquestion(whichwillplayacrucialroleinourRGtheory):Underwhatconditionsdoesi(j)=0?Thatis,underwhatconditionsistherelativeentropyinvolumeaquantityo(jj),i.e.a\surfaceeect"?Theanswerissimple:ifisaGibbsmeasureforsomeinteraction,theni(j)=0whenandonlywhenisaGibbsmeasureforthesameinteraction.Thefollowingtwotheoremsmakethisprecise,inaratherstrongform:Theorem.Let;beGibbsmeasures(notnecessarilytranslation-invariant)forinteractions;B,respectively.Thenlimsup%jjI(j)kkB0=(J+Const):(:)Ifandaretranslation-invariant,thismeansthati(j)kkB0=(J+Const):(:)Inparticular,ifandaretranslation-invariantGibbsmeasuresforthesamein-teractionB,theni(j)=0.Theorem.Letbeaquasilocalspecication,letGinv(),andletM+;inv().SupposethatthereexistsavanHovesequence(n)nsuchthatlimn!jnjIn(j)=0:(:)ThenGinv().Theorem.isanimmediateconsequenceofestimate(.)inProposition.(b).Note,again,thatalthough;arerequiredtobelongtothe\small"BanachspaceB,thenalestimateisintermsoftheB0=(J+Const)norm,hencemuchstronger.Theorem.is,ontheotherhand,adeepandsurprising(atleasttous)result:fromahypothesisonthebehaviorperunitvolumeintheinnite-volumelimitoneobtainsaconclusionvalidforeveryvolume(namely=).ThecombinationofTheorems.and.willplayakeyroleintheproofoftheFirstFundamentalTheorem(seeSection. ).CombiningTheorems.and.,wededucethekeyresultrelatingtheDLRandvariationalapproachestoclassicallatticesystems:Corollary.LetBandletM+;inv().ThenisaGibbsmeasureforifandonlyifitisanequilibriummeasurefor.

..PathologiesinVariousInteractionSpacesBhInSection..weintroducedalargeclassofinteractionspacesBh,ofwhichthemostimportantareB0andB.Nowwewouldliketodiscussthephysicaldierencesbetweenthesespaces.Thisisanimportantissue,becauseweneedtojustifyourviewthat(roughlyspeaking)Bisthelargest\physicallyreasonable"spaceofinteractions.Ourpointofviewisthatthefundamentalphysicalprinciplesofinnite-volumeequilibriumstatisticalmechanicsaregivenbythetheoryofspecicationsandGibbsmeasures.(Weconsiderthevariationaltheoryoftranslation-invariantequilibriummea-surestobeonlyausefultechnicaltool.)Furthermore,wearguedinSection..that,atleastforsystemsofboundedspins(including,inparticular,allmodelswithnitesingle-spinspace),aphysicallyreasonablespecicationmustbequasilocal.Ifthenweputasidehard-coreinteractions,itfollowsfromTheorem.thataphysicallyrea-sonablespecicationmustbetheGibbsianspecicationforsomeabsolutelysummableinteraction.SinceBisthespaceoftranslation-invariantabsolutelysummablecontinu-ousinteractions,thisjustiesourcontentionthatBisthelargestphysicallyreasonablespaceofinteractions.Fromamathematicalpointofview,B0isthenaturalspaceofinteractionsonwhichtodevelopthevariationaltheoryofequilibriummeasures.WeneverthelessclaimthatB0is,fromaphysicalpointofview,muchtoolarge;eventhevariationaltheoryonB0is\pathological".(ThisisconnectedwiththefactthatinteractionsinB0nBdonotingeneraldenespecications,sotherearenoDLRequations.Forthisreason,Corollary.andPropositions.and.donotholdingeneralinB0,andthelarge-deviationtheorydoesnotapplytoequilibriummeasureswhicharenotGibbsmeasures.)ToemphasizethatB0isanunphysicallylargespaceofinteractions,welistheresomeofthestrangephenomenathatcanbeprovenforinteractionsinthisspace:)ThereisadensesetofinteractionsinB0withuncountablymanyextremalequilibriummeasures[0,TheoremV..(c)].(Itisperhapsnotsurprisingthathighlyfrustratedinteractionscouldproduceuncountablymanypurephases;butinB0thishappensarbitrarilyclosetozerointeraction,i.e.atwhatoughttocorrespondto\hightemperature".))Foranynitefamily;:::;nofergodictranslation-invariantmeasuresofniteentropydensity(relativeto0),thereexistsaninteractioninB0forwhichallofthesemeasuresaresimultaneouslyequilibriummeasures[0,TheoremV..(a)].0(Wendthisresultabsolutelyabbergasting:itimplies,forexample,thatthereexistsaninteractioninB0forwhichtheGibbsmeasuresoftheinnite-temperatureandzero-temperatureIsingmodelsarecoexistingpurephases! )ItfollowsthatinB0theinteractioncannotbereconstructeduniquelyfromtheequilibriummeasure:foranygivenmeasure,therearemanydierentinteractionsinB0havingasanequilibriummeasure.ThisisinsharpcontrasttoProposition.,whichassertstheuniqueness0Thisresultisreminiscentofthecorrespondingresultinthetheoryof(non-quasilocal)specica-tions:seetheremarkattheendofSection...

(modulophysicalequivalence)oftheinteraction(ifoneexistsatall)withinB.)ThepressureisnowhereFrechet-dierentiableinB0[0].Bycontrast,thepres-sureisFrechetdierentiableoforderninaneighborhoodoftheorigin(\hightemper-ature")inBn(n)[,,].EventhespaceBisincrediblylarge,inthatitallowsinteractionswhicharestronglymany-body(thoughnotquitesostronglyasinB0)andofarbitrarilylongrange(pro-videdonlythattheyareabsolutelysummable).ThismeansthateveninBsomeratherstrangephenomenaoccur:)Atlowtemperature,theGibbsphaseruleisgenericallyviolatedinallofthespacesBn.Thisisbecausearst-orderphasetransitioncanbedestroyedbyanarbi-trarilyweak(in`norm)butverylong-rangetwo-bodyinteraction[0,,,0].TheGibbsphaserulecanholdonlyinspacesBhwheretheweighth(X)growssu-cientlyfastwiththediameterofX(andnotmerelyitscardinality).)ThepressureisnotanalyticinanyopensetinanyofthespacesBn[];inparticular,itisnotanalyticevenat\hightemperature"(i.e.aneighborhoodoftheorigin).Infact,forspacesBhinwhichh(X)dependsonlyonthecardinalityofX,thepressureisanalyticinaneighborhoodoftheoriginifandonlyifh(X)constejXjforsome>0[0,].Remark.IntheIsingmodel,analyticitydoesholdinBnormforthesubspacesofBcorrespondingtointeractionswritteninlattice-gasorspinform(X=JXXorX=JXX,respectively)[0].Thisisaverysurprisingresult,whichwedonotcompletelyunderstandfromaphysicalpointofview.ItisrelatedtothefactthatItseemstobeanopenquestionwhetherthepressureisonceFrechetdierentiableinaneigh-borhoodoftheorigininB.Theproofsofhigher-orderdierentiabilityin[,,]usetheDobrushinuniquenesstheorem,whichappliesonlyinBorhigher.Seealso[,Chapterandthecorrespondingnotes].Thisstatementisaslightlie.WhatDobrushinandMartirosyan[]actuallyproveisthefollowing:Letthesingle-spinspace0benite;leth(X)dependonlyonthecardinalityofX,andnotsatisfyh(X)constejXjforany>0;andletBChbethecomplexicationofBh.Then,ineveryopensetUBChcontainingarealpoint,thereexistsacomplexinteractionUandasequenceofcubesn%suchthatthenite-volumepartitionfunctionsZn()RexphHn;freeiareallzero.Thus,thenite-volumefreeenergieshave(complex)singularitiesarbitrarilyclosetoevery(real)pointinBh.Thisresultmakesitveryunlikelythattheinnite-volumepressurecouldbeanalyticinanopensetofBh;butstrictlyspeakingitdoesnotruleitout,becauseconceivablythesingularitiespresentinnitevolumecouldmiraculouslydisappearinthepassagetotheinnite-volumelimit.[Hereisasimpleexampleinonecomplexvariable:LetZn(z)=zz0foralln,wherez0CnR.Thenlimn!nlogZn(z)=0forallzR(providedthatthebranchcutischosentoavoidtherealaxis).Andthefunction0certainlydoeshaveananalyticcontinuationfromRtoC! ]Weproposeasanopenproblemtomathematicalstatisticalmechanicians:provethattheinnite-volumepressure,whichiswell-denedonthespaceBhofrealinteractions,hasnoanalyticcontinuationtoanyopensetUBChcontainingarealpoint.Inanycase,theresultofDobrushinandMartirosyandoesshowthattheDobrushin-Shlosman[,]completeanalyticityconditiondoesnotholdforanyopenneighborhoodinBh,forthespeciedclassofh.

physicallyequivalentinteractionscanhavewidelydieringnormsinanygivenspaceBh;inparticular,forlattice-gasorspininteractions,onecanhavekkBkkB=J[].InSectionweshallprovethatcertainrenormalizedmeasuresarenotGibbsianforanyinteractioninB.ThefactthatnoteveninB|aspacelargeenoughtosupportmuchpeculiarbehavior|doesaninteractionexistisanindicationofhowstrongthisresultis.Position-SpaceRenormalizationTransformations:RegularityPropertiesInthissectionwedeneourgeneralframeworkforstudyingrenormalizationtransfor-mations(RTs),andprovethetwoFundamentalTheoremsonsingle-valuednessandcontinuityoftheRTmap.WeconsideronlyasingleapplicationoftheRTmap.Therefore,thesemigrouppropertyofthe\renormalization(semi)group"playsnoroleforus.Inparticular,weneednotassumethattheimagesystemisofthesametypeastheoriginalsystem.Nevertheless,weshalloccasionally(byabuseoflanguage)usetheterm\RGmap",forreasonsoffamiliarityandbrevity..BasicSet-Up..RenormalizationTransformationActingonMeasuresWeconsidera\renormalizationmap"Tfromanoriginal(orobject)system(=Zd0;F;0)toanimage(orrenormalized)system(0=0Zd00;F0;00).Thesingle-spinspaces0and00neednotbethesame;indeed,wewillpresentanimportantexampleinwhichtheyarenotthesame(seeExamplebelow,andSection..).Althoughourtheoryinthissectionworksonlywhenthespatialdimensionsdandd0arethesame|seethediscussionofExamplebelow,aswellasSection..|wenditnotationallyconvenienttokeeptheprimeonallimage-systemquantities,asthismakesiteasytoseewhichquantityreferstowhichsystem.WeassumethefollowingpropertiesforT:T)Tisaprobabilitykernelfrom(;F)to(0;F0).T)Tcarriestranslation-invariantmeasuresonintotranslation-invariantmeasureson0.[Thatis,ifMinv(),thenTMinv(0). ]T)Tisstrictlylocalinpositionspace,withasymptoticvolumecompressionfactorK<.Moreprecisely,thereexistvanHovesequences(n)Zdand(0n)Zd0suchthat:

(a)Thebehavioroftheimagespinsin0ndependsonlyontheoriginalspinsinn,i.e.ForeachAF00n;thefunctionT(;A)isFn-measurable.(:)(b)limsupn!jnjj0njK.(T)allowstherenormalizationmaptobeeitherdeterministicorstochastic.Inthedeterministiccase,theconguration!0oftheimagesystemisafunction!0=t(!)oftheoriginalconguration.Themostconspicuousexamplesofthesetypeoftransformationsaredecimation,linearblock-spintransformations,andmajorityruleforblockswithanoddnumberofspins(seeExamples,andbelow).Forthegeneralcaseofastochastictransformation,givenanoriginal-systemconguration!,wechooseanimage-systemconguration!0withacertainprobabilityT(!;d!0).Thespecialcaseofadeterministicmapt:!0correspondstosettingT(!;)tobethedelta-measuret(!)[i.e.theconguration!0=t(!)ischosenwithprobability].Examplesofstochastictransformationsarethemajority-ruletransformationforblockswithevennumberofspins,andmoregenerallytheKadanotransformation(seeExamples,andbelow).Themainpointof(T)istoexcludetransformationswithnegativeweights,whichhavenosensibleprobabilisticinterpretation.(T)isself-explanatory.Typicallytranslationsoftheimagesystemcorrespondtosomesubgroupoftranslationsoftheoriginalsystem.Thatis,theretypicallyexistsahomomorphismR:Zd0!ZdsuchthatT(TR(x)!;)=TxT(!;)(:)forallxZd0andall!.Forexample,aRTemployingbbblockswillhaveR(x)=bx.ThusthetranslationgroupZd0oftheimagesystemcorrespondstothesubgroupR[Zd0]Zdoftranslationsoftheoriginalsystem.Property(T)triviallyfollowsfromthis.Someexamplesaregivenbelow.Properties(T)and(T)makerigoroustheequation(.):themap!Tisawell-denedmapfromM+;inv()intoM+;inv(0).ThisjustiestheclaimmadeintheIntroduction,thatitiseasytodenetheRTmapfrommeasurestomeasures.ThemoredicultandsubtleproblemofdeningtheRTmapfrominteractionstointeractionswillbediscussedinSection...Property(T)|thestrictlocalityoftherenormalizationmap|iscrucialforourproofsoftheFirstandSecondFundamentalTheorems.Mostoften(althoughweshallnotrequirethis)theprobabilitymeasureT(!;)hasaproductstructureT(!;d!0)=YxZd0T(!Bx;d!0x);(:)Transformationswithnegativeweightshaveoccasionallybeenusedinthephysicsliterature,notnecessarilyintentionally:seee.g.[].Seealsothecommentsin[,footnoteonp.andtextonp.].Inmoredetail,T(TR(x)!;A)=T(! ;Tx[A])forallxZd0,!andAF0.

whereBxisthenitesetoforiginalspinswhichtogetherdeterminetheimagespin!0x.NowletussupposethatBx=B0+R(x)[i.e.B0translatedbyR(x)],whereR:Zd0!ZdisahomomorphismsatisfyingdetR=0(obviouslythisneedsd0=d).Wethenclaimthat(T)holdswithK=jdetRj.Proof:Let(0n)beanyvanHovesequenceinZd0.Whatsets(n)Zdshouldwetaketosatisfy(T)?ClearlytheimagespinsinndependonlyontheoriginalspinsinthesetnR[0n]+B0Zd.Soatrstonemightthinktotaken=n.Thetroubleisthatthe(n)neednotformavanHovesequence,becausetheymayhaveanonzerodensityof\holes".[Consider,forexample,decimationwithspacingb>:hereB0=f0gandR(x)=bx. ]Sowetakeinsteadn=Zd\convexhullofn:(:)Then,usingthefactthatdetR=0,itisnothardtoconvinceoneselfthat(n)isavanHovesequence,andthatlimn!jnjj0nj=jdetRj:(:)Twopointsarerelevanthere:Firstly,weneeddetR=0(andinparticulard0=d)inordertoguaranteethatthesets(n)aresuciently\fat"toformavanHovesequence(seethediscussionofExamplebelowforwhatcanhappenifthisdoesnothold).Secondly,thequantityKlimsupn!jnj=j0njisbydenitiontheasymptoticvolumecompressionfactor:assuch,itisdeterminedsolelybyR;itdoesnotdependonthesizeofB0aslongasB0isnite.WeconjecturethatthetwoFundamentalTheoremsholdalsoforquasilocalrenor-malizationmaps|i.e.mapsinwhich!0xdependssucientlyweaklyondistantspins!y|butwearenotabletoprovethiswithourpresentmethods.Quasilocalrenor-malizationmapsareofgreatpracticalimportance:forexample,in\momentum-space"renormalizationoneoftenusesadeterministictransformation!0x=XyF(bxy)!y(:)withsomelengthrescalingfactorb>andsomekernelF.Inparticular,ifoneusesa\soft"cutoinmomentumspace[,],thenthekernelFisrapidlydecreasingatinnityinx-space(e.g.decreasingfasterthananyinversepowerofitsargument).Itisanimportantopenproblemtoextendourresultstosuchmaps...Examples)Decimationtransformation[0,].Let0=andd0=d,andletbbeaninteger.DenethedeterministicRTmap!0x=!bx:(:)Actually,allwereallyneedisthatR,consideredasadd0matrix,haverankd.Thus,wecouldallowsomecaseswithd0>d.Buttheseareoflittleinterest.Theinterestingcaseswithd0=dhaved0

Thismapisstrictlylocal[infact,oftheproductform(.)]withasymptoticvolumecompressionfactorK=bd.Itisoftheform(.)withR(x)=bx.Moregenerally,let0=andd0=dandletRbeanyhomomorphismfromZd0toZdsatisfyingdetR=0.DenethedeterministicRTmap!0x=!R(x):(:)Thismapisstrictlylocal[infact,oftheproductform(.)]withasymptoticvolumecompressionfactorK=jdetRj.SomeexamplesareshowninFigure(a){(b).)Majority-ruletransformationfortheIsingmodel[,,].Letbbeaninteger,letB0beaxednitesubsetofZd(theblock),andletBx=B0+bx(i.e.B0translatedbybx).Denethemap0x=><>:+ifPyBxy>0ifPyBxy<0ifPyBxy=0(:)where\"denotesarandomchoicewithprobabilitiesof=each.Thistransforma-tionisdeterministicifbisodd,stochasticifbiseven. )KadanotransformationfortheIsingmodel[0].AlargeclassofnonlinearRTmapsfortheIsingmodel=0=f;gZdcanberepresentedinthefollowingform:ConsiderthesameblocksBxasinthepreviousexample,andletp>0.DenethestochasticRTmapT(;0)=YxZd0exp p0xPyBxy!cosh pPyBxy!:(:0)Thismapisstrictlylocal[andclearlyoftheproductform(.)]withasymptoticvolumecompressionfactorK=bd,andisoftheform(.)withR(x)=bx.Manywell-knownRTmapsarespecialcasesof(.0):(a)WithB0=f0gandb=,(.0)ismodelIofGrithsandPearce[,],akindof\copyingwithnoise".(Thismapalsoarisesinapplicationstoimageprocessing[,].)Asp!ittendstotheidentitytransformation.(b)WithB0=f0gandb,(.0)ismodelIIofGrithsandPearce[,],akindof\decimationwithnoise".Asp!ittendstotheordinarydecimationtransformation(.).(c)WithB0=f0;;:::;bgd(ahypercubeofsideb)andb,(.0)istheKadanotransformation[0].Inthelimitp!ittendstothemajority-ruletransformation(.

).

(a)(b)→(c)→(d)→(e)Figure:SomeexamplesofRTmapsindimensiond=.(a)Decimationwithb=andK=.(b)DecimationwithR(x;x)=(x+x;xx)andK=(\checkerboarddeci-mation").(c)Blocktransformationwithb=,B0=f(0;0);(;0);(0;);(;)gandK=.(d)BlocktransformationwithR(x;x)=(xx;x+x),B0=f(0;0);(;)gandK=[]. (e)BlocktransformationwithR(x;x)=(x+x;x+x),B0=f(0;0);(;0);(;)gandK=[,].0

Asinthedecimationtransformation,wecanreplacebxbyamoregeneralnonsin-gularhomomorphismR(x).ThenK=jdetRj.SomeexamplesareshowninFigure(c){(e).)KadanotransformationfortheN-vectormodel.FortheN-vectormodel,inwhichthespinsareunitvectorsinRN,thenaturalgeneralizationofthemajority-ruletransformationisthe\rescaledblock-spintransformation"[]0x=PyBxyPyBxy;(:)whichisdeterministic.(InprincipleoneshouldspecifywhathappenswhenPyBxy=0:forexample,onecouldchoosesomeparticularvalueof0x,oronecouldlet0xbeuniformlydistributedontheunitsphere.Butthissituationoccurswithprobabilityzero,soitisirrelevantwhatchoiceonemakes. )Similarly,theKadanotransformationhasanaturalgeneralization[0]:T(;d0)=YxZd0exp p0xPyBxy!ZN pPyBxy!d(0x);(:)whereZN(h)ZSNehd()=N jhj!NIN(jhj)(:)andddenotesuniformmeasureontheunitsphereinRN.Asp!thistendstothedeterministicmap(.).AnalogousformulaecanbeusedtodeneaKadanotransformationfortheq-statePottsmodel,usingtherepresentationofPottsspinsasunitvectorsinRqpointingfromthecenterofa\hypertetrahedron"toitsvertices.Asp!thistransformationtendstothe\plurality-rule"transformationwithrandomtie-breakers.ThePottsmodelwithvacancies[,0]canalsobetreatedinthisframework,byrepresentingthe\vacancy"stateastheorigininRq.

)Linearblock-spintransformations.Anaturalchoiceofadeterministiclineartransformationistheaveragingtransformation0x=cXyBxy;(:)forasuitablychosenrescalingfactorc.TypicallywechoosejB0jc.Weobservethatifc>jB0j,thistransformationdoesnotmapanymodelofboundedspinstoitself:if0=[M;M],wemusttake00=[M0;M0]withM0jB0jcM>M.As

aconsequence,thexedpoint(s)(ifany)forsuchatransformationmustcorrespondtomodel(s)ofunboundedspins(i.e.0=R).Forthisreason,itismostnaturaltoconsider(.)asacting,rightfromthestart,onsuchasystemofreal-valuedspins.However,inthispaperwearenotconcernedwithxedpoints;ourinterestisinwhethertherstapplicationoftheRTmapiswell-dened.Forthispurposewemayworkentirelywithmodelsofboundedspins,providedthatwearewillingtoaccept00=0.Forexample,inthetwo-dimensionalIsingmodelwithblocks(andc=),wehave0=f;gbut00=f;;0;;g.ForunboundedspinswithvaluesinR(orRN),onecanuseeitheradeterministiclinearblock-spintransformation[]'0x=cXyBx'y(:)orthestochasticlinearblock-spintransformation[,]T(';d'0)=YxZd0constexp0@'0xcXyBx'yAd'0x;(:)whichcorrespondstoaddingGaussianwhitenoiseofvariancetothedeterministicblockspins(.).Inbothcases,therescalingfactorcmustbechosenappropriatelyifthetransformationistohaveaxedpoint:e.g.forhypercubicblocksofsidebonetakesc=><>:bd=tohaveahigh-temperaturexedpointbdtohavealow-temperaturexedpointb(d+)=tohaveacriticalxedpoint(:)Thisneedtoxaparameterischaracteristicoflinearrenormalizationtransformations.Linearblock-spintransformationshaveattractedtheattentionofmathematicalphysicistsbecauseoftheirconnectionswithcentral-limittheorems:seeforexample[00,,,,].)Linearblock-spintransformationwithlarge-eldcuto.Evenwhenthelinearblock-spintransformation(.)doesnotmapmodelsofboundedspinstothemselves,oneexpectsthecorrespondingxed-pointmeasure(s)tohaverapidlydecaying(e.g.Gaussianorfaster)densitiesatlarge'.Therefore,itmaybeareasonableapproxima-tiontomodify(.)bycuttingotheeldsexplicitlyatjj=M,whereMissomexedlargenumber.Thatis,onthespace=0=[M;M]ZdonecanconsiderthedeterministicRTmap'0x=><>:cPyBx'yifcPyBx'yMMsgnPyBx'yifcPyBx'y>M(:)[ThisworksalsoforN-componentspins,ifoneinterpretssgn(')='=j'j.]Toourknowledge,thistransformationhasnotbeenconsideredpreviously.(ButseeCam-marota[]forarelatedidea.)

)Restrictiontoahyperplane[].Let00=0buttaked00[seeSection..],contrarytowhatwouldhappenif(T)weretohold[cf.(.0)]...RenormalizationTransformationActingonInteractionsWecannowdenepreciselytherenormalizationmapRactingonthespaceofinter-actions,makingrigorousthediagram(.).AsarguedinSection..,thelargest\physicallyreasonable"spaceofinteractionsisB,thespaceoftranslation-invariantcontinuousabsolutelysummableinteractions.Therefore,indeningR,weshallrestrictattentiontointeractionsBsuchthatthereexistsanimageinteraction0B.Sinceaprioriwewishtoadoptacompletelyopen-mindeddenition|allowingforthepossibilityofmulti-valuedness|wemustdeneRasarelationratherthanafunction.Denition.LetTbeanRTmapsatisfyingproperties(T)and(T).WethendenethecorrespondingmapR=RTtobetherelationR=f(;0)BB:thereexistsGinv()suchthatTGinv(0)g:(:0)WecanalsothinkofRasamulti-valuedfunction:wewrite0R()asasynonymfor(;0)R.WedenethedomainofRtobethesetdomR=f:thereexists0with(;0)Rg=f:R()=?g:(.)ApriorithemapRcouldbemulti-valued.Indeed,thewaywehavedenedit,itsurelyismulti-valued,becauseofphysicalequivalence:if0R()and0B\(J+Const),thenalso0+0R().ThemoreinterestingquestioniswhetherRcanbemulti-valuedapartfromthe\trivial"multi-valuednesscausedbyphysicalequivalence.Thescenarioproposedin[]ispreciselytheclaimthatthiscanhappen;weshallproveinourFirstFundamentalTheorem(Theorem. )thatinfactitcannot

happen.Thatis,weshallprovethatthemapRissingle-valuedmodulophysicalequivalence.Weshallmoreoverprovethatthephrase\thereexists"in(.0)canbereplacedequivalentlyby\forall".ForRTmapssatisfyingaverymildcontinuitycondition,wecansaysomethingabouttheclosurepropertiesofthemulti-valuedmapR.Toavoidbothersometopo-logicalcomplexities,werestrictattentiontocompactmetricsingle-spinspaces0.Theorem.Let0beacompactmetricspace,andassumethatTsatises(T)and(T)andisFeller(i.e.Tfiscontinuousiffis).ThenRisaclosedsubsetofBBwithrespecttotheB0=(J+Const)B0=(J+Const)seminorm.Proof.Assumethat(n;0n)Rand(;0)BB,withlimn!knkB0=(J+Const)=limn!k0n0kB0=(J+Const)=0:(:)Weneedtoprovethat(;0)R.Choose,foreachn,atranslation-invariantGibbsmeasurenforn.Bypassingtoasubsequence,wecanassumewithoutlossofgeneralitythatnconvergesweaklytosomemeasure;andsincen!inB0=(J+Const)seminorm,itiseasytoseethatisatranslation-invariantGibbsmeasurefor.NowtheFellerhypothesisonTguaranteesthatnT!Tweakly.SincenTisatranslation-invariantGibbsmeasurefor0n,and0n!0inB0=(J+Const)seminorm,itfollowsthatTisatranslation-invariantGibbsmeasurefor0.Butthisimpliesthat(;0)R...ARemarkonSystemsofUnboundedSpinsTheresultstobeproveninSections.and.areinprincipleapplicabletosystemsofeitherboundedorunboundedspins.Butforunboundedspinsourresultsarenotofmuchinterest,becausewerestrictattentiontoboundedHamiltonians(i.e.absolutelysummableinteractions).Thetrouble,asdiscussedattheendSection..,isthatwelackatpresentanadequategeneraltheoryofunboundedspinsystems:weareunabletospecify,forexample,aspaceofinteractionsthatincludesall\reasonable"interactions.Thedevelopmentofsuchageneraltheoryisanimportantopenproblem;itwouldbearststeptowardsputtingthestandardWilson-styleRGtheory[]onarigorousfooting.Inparticular,insuchaframeworkonecouldtrytoproveanaloguesofourFirstandSecondFundamentalTheorems.InthisregarditshouldberemarkedthattheimportantworkofGawedzkiandKupiainen[,,0,]onrigorousRGtheorydoesnotimplementexactlythestandardWilsonprescription,atleastforbosonictheories:whilethesmall-eldpartoftheGibbsmeasureisrepresentedbyaHamiltonianoftheusualkind,thelarge-eldpartisrepresentedinsteadbyapolymerexpansion[]. (Inrecentwork,Brydges

andYau[]systematizethisidea,andformulatetheRGpurelyintermsofapolymerexpansion.)Forfermionictheories,wherethereisno\large-eldregion",GawedzkiandKupiainen[]doimplementthefullWilsonprescription;however,fermionictheorieshavenodirectprobabilisticinterpretation.Also,forbosons,KochandWittwer[,]implementtheWilsonprescription,butsofaronlyinthehierarchicalmodel..FirstFundamentalTheorem:Single-ValuednessoftheRTMapAmongthepossiblepathologiesoftheRTappliedatthelevelofHamiltonians,thefollowingscenariohasbeenproposed[]:ConsideraHamiltonianHlyingonarst-orderphase-transitionsurface,thatis,oneforwhichthereexistatleasttwodistinctpurephases(extremaltranslation-invariantGibbsmeasures),callthemand.NowperformarenormalizationtransformationTasindicatedin(.).Theresultingrenormalizedmeasures0Tand0Tmaythen,itisclaimed,beGibbsianfortwodierentrenormalizedHamiltoniansH0=H0.Inotherwords,therenormalizationmapRfromHamiltonianstoHamiltonians,denedby(. ),maybemulti-valued.Herewedisprovesuchascenario.WeshowthatiftwoinitialGibbsmeasurescorrespondtothesameinteraction,thentherenormalizedmeasuresareeitherbothGibbsianforthesamerenormalizedinteraction0,orelsetheyarebothnon-Gibbsian(inwhichcasethereisnorenormalizedinteractionatall).Thistheoremfollowsfromcomparingthelarge-deviationpropertiesofdierentGibbsmeasuresaccordingtowhethertheybelongtothesameordierentinteractions.Heuristically,ifandaretwoGibbsmeasurescorrespondingtodierentinteractions,thentheprobabilityofndinginalargedropletlookinglikeatypicalcongurationforthemeasureisexponentiallysmallinthevolumeofthedroplet:Prob(!istypicalfor)eO(jj):(:)Ontheotherhand,ifandcorrespondtothesameinteraction,thisprobabilityissub-exponential:Prob(!istypicalfor)eo(jj):(:)Mathematically,asseeninSection.,thisisexpressedinthefactthattherelativeentropydensitysatisesi(j)(>0ifandareGibbsmeasuresfordierentinteractions=0ifandareGibbsmeasuresforthesameinteraction(:)ThisscenarioisstatedveryclearlyintheMonteCarlopaperofDecker,HasenfratzandHasenfratz[,p.,lines{].Ontheotherhand,theanalyticargumentsinthecompanionpaperofHasenfratzandHasenfratz[]concern\singularities"whoseprecisenatureisunspecied.Weareunabletomakeaconnectionbetweenthetwolinesofreasoning.

Now,underrenormalizationonelooksonlyattheblockspinsandforgetsabouttheinternalspins,henceProb(blockspinsin!aretypicalfor)Prob(allspinsin!aretypicalfor):(.)Therefore,ifinitiallytheprobabilitywassubexponential(sameinteraction),thenunderrenormalizationitremainssoandwecanneverobtaintheexponentialdecay(.)characteristicofdierentinteractions.Mathematically,thisisexpressedbythefactthattherelativeentropydecreasesundertheapplicationofarbitrarydeterministicorstochastictransformations,inparticularundertheRT:Lemma.Let(;)and(0;0)bemeasurablespaces,andletTbeaprobabilitykernelfrom(;)to(0;0).Then,ifandareprobabilitymeasureson,I(TjT)I(j):Proof.Thisisawell-knownresult,althoughitisratherdiculttondacompleteproofintheliterature.(Mostofthepublishedproofsconcernoneoranotherspecialcase:Tdeterministic,T=,discretestatespace,etc.)TherstcompleteproofofwhichweareawareisduetoCsiszar()[];however,wewouldnotbesurprisedtolearnthatthisresultwasknownmuchearlier.Seealso,forinstance,[]and[,Theorem.];andsee[]forsomestrongerresults.Fortheconvenienceofthereader,letusgiveaone-lineproof:I(j)=I(TjT)I(TjT):(:)HeretherstequalityisProposition.(h):themeasuresTandThavethesameregularconditionalprobabilitygiven,namelyT.[Theintuitiveideaisthatthepair(T;T)containsatleastasmuchinformationasthepair(;),sincethelatteristherestrictionoftheformertothesub--eld0;butitcontainsnomoreinformation,becausethesameprobabilitykernelhasbeenusedtogeneratebothTandTfromand.]AndtheinequalityisProposition.(g),sinceT(resp.T)istherestrictionofT(resp.T)tothesub--eld00.Theorem. (Firstfundamentaltheorem)Letandbetranslation-invariantGibbsmeasureswithrespecttothesameinteractionB,andletTbeanRTmapsatisfyingproperties(T){(T).Then:(a)EitherTandTarebothnon-quasilocal(i.e.notconsistentwithanyquasilocalspecication),orelsethereexistsaquasilocalspecication0withwhichbothTandTareconsistent.Inthelattercase,ifthesingle-spinspaceisnite,then0istheuniquequasilocalspecicationwithwhicheitherTorTisconsistent,anditistranslation-invariant.

(b)EitherTandTarebothnon-Gibbsian(forabsolutelysummableinteractions),orelsethereexistsanabsolutelysummableinteraction0forwhichbothTandTareGibbsmeasures.Inthelattercase,if0iscontinuous[asitalwaysise.g.foradiscretesingle-spinspace],0istheuniquecontinuousabsolutelysummableinteraction(modulophysicalequivalenceintheDLRsense)forwhicheitherTorTisaGibbsmeasure.Proof.Let(n)Zdand(0n)Zd0bevanHovesequenceshavingtheproperties(T)assumedinSection..Now,byTheorem.,thefactthatandareGibbsmeasuresforthesameinteractionimpliesthatlimn!jnjIn(j)=0:(:)Ontheotherhand,theimagespinsin0ndependonlyontheoriginalspinsinn:thatis,(T)F00nistheimageunderTofFn,andlikewisefor.Hence,byLemma.wehaveI0n(TjT)In(j):(:)Itfollowsthat0limsupn!j0njI0n(TjT)limn!KjnjIn(j)=0:(.0)Therefore,byTheorem.,ifTisconsistentwithaquasilocalspecication0,thenTmustalsobeconsistentwiththissamespecication0.Thesameargumentcanbemadeinterchangingtherolesofand.Thus,eitherTandTarebothnon-quasilocal,orelsethereexistsaquasilocalspecication0withwhichbothTandTareconsistent.Inthelattercase,ifthesingle-spinspaceisnite,Theorem.guaranteestheuniquenessof0.Inparticular,sinceTandT(beingtranslation-invariant)areobviouslyconsistentwithanytranslateof0,weconcludethat0istranslation-invariant.Aspecialcaseoftheforegoingis:ifT(resp.T)isGibbsianwithrespecttoanabsolutelysummableinteraction0,thenT(resp.T)mustalsobeGibbsianwithrespecttothissameinteraction0.Theuniquenessmodulophysicalequivalenceof0isthenguaranteedbyCorollary..TheFirstFundamentalTheoremshowsthattheRTmapRissingle-valuedmodulophysicalequivalence.Italsoshowsthatthephrase\thereexists"inthedenition(.0)canbereplacedequivalentlyby\forall".Remarks..Therststepofthisproof(usingTheorem.)doesnotrequireandtobetranslation-invariant.Butthesecondstep(usingTheorem. )doesseemtorequireatleastTandTtobetranslation-invariant.Sowedonotknowwhether

thehypothesisoftranslation-invarianceofandcanbeomittedinthistheorem.(Note:Wealwaysassumethattheinteractionistranslation-invariant.).Inpart(b),theinteraction0,ifitexists,oughttobephysicallyequivalentintheDLRsensetoatranslation-invariantinteraction.Unfortunately,wearenotabletoprovethis.Fromtheuniquenessweknowthat0isphysicallyequivalenttoallofitstranslates;butitseemstobeanopenquestionwhetherthisguaranteesthat0isphysicallyequivalentintheDLRsensetoatranslation-invariantinteraction.AnarmativeanswerwouldalsoallowKozlov's[]GibbsRepresentationTheoremtobegivenasatisfactorytranslation-invariantversion(seetheRemarkattheendofSection..)..SecondFundamentalTheorem:ContinuityPropertiesoftheRTMapAsecondaspectofthescenarioproposedbyDecker,HasenfratzandHasenfratz[]isthattheRTmapmaybediscontinuousatanoriginalHamiltonianH0lyingonarst-orderphase-transitionsurface:namely,fororiginalHamiltoniansHarbitrarilyclosetoH0onoppositesidesofthephase-transitionsurface,itisclaimedthatthecorrespondingrenormalizedHamiltoniansH0maybeanitedistanceapart.Herewedisprovethisscenariotoo.WeshowthattheRTmapisalwayscontinuous(inasuitablenorm)onthesetofHamiltonianswhereitiswell-dened,thatis,onthesetofHamiltoniansforwhichtheimagemeasuresareGibbsian.Thekeyideaunderlyingourproofisthefactthat,ifisaGibbsmeasureforaninteractionB,thentheDLRequationsallowthereconstructionoftheinteraction(modulophysicalequivalence)fromthemeasure:logdd0=XxTxf+const()+o(jj)(:)(seeSection.. ).Therefore,ifandareGibbsmeasuresforinteractions;B,wehavelogdd=XxTxf+const()const()+o(jj)(:)andinparticularlogddB()=const=jjkkB0=(J+Const)+o(jj):(:)Nowtheprobabilitydensitiesofrenormalizedmeasuresare(particular)weightedav-eragesoftheoriginaldensities,sothesupremumoftherenormalizeddensitycannotexceedthatoftheoriginaldensity.Thatis,klog(d=d)kB()=constcanonlydecreaseundertheRT:

Lemma.Let(;)and(0;0)bemeasurablespaces,andletTbeaprobabilitykernelfrom(;)to(0;0).Letandbeprobabilitymeasureson,with.ThenTTandinfactlogd(T)d(T)L(T)logddL()(.)logd(T)d(T)L(T)=constlogddL()=const(.)Proof.SupposethattheRadon-Nikodymderivative(=density)d=dsatises0addb+-a.e.(:)Thenab(inthesenseoftheusualorderingonpositivemeasures),soobviouslya(T)Tb(T).Itfollowsthatad(T)d(T)b(T)-a.e.(:)SincelogddL()=max(logb;loga)(.)logddL()=const=(logbloga)(.)[whereaandbarethesharpestvaluesmaking(.)true],withananalogousformulaford(T)=d(T),thelemmaisproven.Theorem.(Secondfundamentaltheorem)LetTbeanRTmapsatisfyingprop-erties(T){(T),andlet;domR.Then,forall0R()and0R(),k00kB0=(J+Const)KkkB0=(J+Const):(:0)Thatis,onitsdomainthemapRisLipschitzcontinuous(withLipschitzconstantK)intheB0=(J+Const)norm.Proof.Let(n)Zdand(0n)Zd0bevanHovesequenceshavingtheproperties(T)assumedinSection..LetGinv()andGinv().BytheFirstFundamentalTheorem(Theorem. )wehaveTGinv(0)andTGinv(0).Nowtheimagespinsin0ndependonlyontheoriginalspinsinn:thatis,(T)F00nistheimageunderTofFn,andlikewisefor.Therefore,byLemma.wehavelogd(T)0nd(T)0nB(0)=constlogd()nd()nB()=const:(:)

(SincethemeasuresandTarebothGibbsian,theygivenonzeromeasuretoeveryopenset;andmoreovertheinteractions,,0and0areallcontinuous.Thereforewecanreplacetheessentialsupnormsbythetruesupnorms.)Thenk00kB0=(J+Const)=limn!j0njlogd(T)0nd(T)0nB(0)=constlimn!Kjnjlogd()nd()nB()=const=KkkB0=(J+Const);(.)wherewehavetwiceusedProposition.(b).Itiscuriousthatalthoughalltheinteractions,,0and0arehererequiredtobelongtoB,theLipschitzestimate(.0)isstatedinB0norm.ThisisbecauseB0(ormorepreciselyitsquotientbyJorJ+Const)isthenaturalnormformeasuringbulkenergycontributions,asdiscussedinSection...TherestrictiontoBisneededsolelytoensurethattheboundaryenergycontributionsareo(jj),soastoavoidthepathologiesdiscussedinSection...Inanycase,wewouldliketoemphasizethatalltheBnormsareequivalent(uptoaboundedfactor)forinteractionsinvolvingboundedlymanyspinsatatime(e.g.two-spininteractions),evenwhentheyareofarbitrarilylongrange.ThedierencebetweentheBnormsconcernshowtheytreatinteractionsthatareverystronglymulti-body.Theorem.constrainsverystronglythewaysinwhichtheRTmapcanblowupasapproachestheboundaryofitsdomain.Indeed,supposethat(n)nisasequenceindomRBthatconvergesinB0norm[ormoregenerally,inB0=(J+Const)seminorm]toB0.(WeneednotrequireconvergenceinBnorm,norneedwerequirethatbelongtoB.)Nextlet(0n)nbeanychoiceofrenormalizedinterac-tions[i.e.0nR(n)B];herethechoiceconcernstheselectionofrepresentativesmodulophysicalequivalence.Then(.0)guaranteesthat(0n)isaCauchysequenceintheB0=(J+Const)seminorm,henceconvergesinB0=(J+Const)seminormtosome0B0;moreover,thislimitisuniquemodulophysicalequivalence(i.e.moduloJ+Const).Now,ifand0(oranyinteractionsintheirphysical-equivalenceclasses)arebothinB,thenitfollowsfromTheorem.that(;0)R,hencedomR.Therefore,ifBndomR,itmustbethat0isnotphysicallyequivalenttoanyinteractioninB.OnewouldliketoconcludefromthisthattheB(semi)normsk0nkB=(J+Const)mustdivergeasn!.Unfortunately,wearenotquiteabletoprovethis,becausewehavenotbeenabletoproveaversionofProposition.(a)modulophysicalequivalence(cf.Proposition. ).Thebestwehavebeenabletoproveisthefollowing:Corollary.Let0beacompactmetricspace,andassumethatTsatises(T){(T)andisFeller.Let(n)nbeasequenceindomRBthatconvergesinB0=(J+Const)seminormtoBndomR.Foreachn,let0nbeanyinteractioninR(n)B.Then:0

(a)Either(0n)nfailstoconvergeinB0,orelsek0nkB!.Ifthesingle-spinspace00isnite,thenwealsohave:(b)Foranyh,k0nkBh=(J+Const)!.Proof.(a)Supposethat0n!0inB0,butk0nkB!.Sothereisasubse-quenceof(0n)onwhichtheBnormisbounded,saybyM;andProposition.(a)thenimpliesthat0B(withk0kBM).ButbyTheorem.thismeansthat(;0)R,contrarytothehypothesisthat=domR.(b)Asarguedabove,theequivalenceclasses[0n]0n+J+ConstconvergeinB0=(J+Const)tosomeequivalenceclass[0].Itfollowsthatonecanchoosenewrepresentatives^0n[0n]and^0[0]suchthat^0n!^0inB0.Nowsupposethatk^0nkBh=(J+Const)k0nkBh=(J+Const)!.Thenthereisasubsequenceof(^0n)onwhichtheBh=(J+Const)seminormisbounded,saybyM;andProposition.thenimpliesthat^0Bh+J+Const(withk^0kBh=(J+Const)M).Butthismeansthatthereexists^^0BhB(withk^^0kBhM)suchthat^^0[^0]=[0].AndbyTheorem.thismeansthat(;^^0)R,contrarytothehypothesisthat=domR.OurinabilitytoprovethedivergenceoftheBseminormisnotasseriousasitmayseem:aswillbediscussedinSection..,oneprobablywantsanywaytoformulateRGtheoryinaspaceBhofshort-rangeinteractions,andforsuchaspaceourresult(b)issucient(when00isnite).ProvablyPathologicalRenormalizationTrans-formations.Griths-Pearce-IsraelPathologiesI:Israel'sExample..IntroductionGrithsandPearce[,,]werethersttopointoutthepossibleexistenceofwhattheycalled\peculiarities"oftheRT.Thesepeculiaritieswereexhibitedinmodelsinwhichtheinternalspinsundergoaphasetransitionforsomexedblock-spinconguration.Theyobservedthatinsuchasituationthecorrelationfunctionsoftheinternal-spinsystemcouldbecomediscontinuousfunctionsoftheblockspins,whichimpliesthateachofthetermsofthe(formal)expansionyieldingtherenormalizedHamiltonian(. )couldbediscontinuous.Thiscastsdoubtsontheconvergenceofsuchanexpansion,andhenceoneithertheexistenceorthecontinuitypropertiesoftherenormalizedHamiltonian.ThissituationwasfurtherclariedbyIsrael[0]intheparticularcaseoftheb=decimationtransformation.Hearguedthatwhensuchpeculiaritiesexist,a

veryweaklocalityconditionisviolatedbytherenormalizedmeasure:theconditionalexpectationforasinglesiteisadiscontinuousfunction(intheproducttopology)oftheboundaryconditions.Thatis,itispossibletoxtheblock-spincongurationinanarbitrarilylargevolumearoundtheorigininsuchawaythatwhathappensattheorigindependsstronglyontheblockspinswhichareoutsideofthevolume.Thesetofcongurationsforwhichthispathologyoccursisimprobable,butnotofzeromeasure.Inourterminology,therenormalizedmeasureisnon-quasilocal:thatis,itisnotconsistentwithanyquasilocalspecication.Inparticular,therenormalizedmeasureisnottheGibbsmeasureforanyuniformlyconvergentinteraction.InthissectionwellinthetechnicaldetailsofIsrael'sargument,therebyconvertingitintoarigorousproof.InthefollowingsectionsweshallgeneralizeIsrael'sargumenttoothermodelsandotherrenormalizationtransformations.Inallcases,theunderlyingphysicalmechanismcausingthenon-Gibbsiannessoftherenormalizedmeasureisthesame:theinuencefromtheblockspinsoutsidethespeciedvolumeistransmittedtotheoriginviatheinternalspinsintheintermediateregion,by-passingtheblockspinsintheniteenvironmentoftheorigin.Thisoccursbecausetheinternalspinshaveaphasetransition,andtheblock-spinboundaryconditionscanpickdierentphasesoftheseinternalspins...Israel'sExample:Decimationind=LetuspresentnowIsrael'sexample|thetwo-dimensionalIsingmodelatlowtemper-atureandzeromagneticeld,usingtheb=decimationtransformation|togetherwiththeproofthatafteronerenormalizationsteptherenormalizedmeasureisnolongerGibbsian.Thestrategyoftheproofistoshowthattherenormalizedmeasureexhibitsgrosslynon-localcorrelations,inthesensethattheconditionalprobabilitydis-tributionofthespinattheorigin,asafunctionofalltheotherspins,dependsstronglyonthespinsarbitrarilyfarawayfromtheorigin.Moreprecisely,weshallshowthatifwetakeanarbitrarilylargecubeandxalltheblockspinsinside,excepttheorigin,inafullyalternatingconguration,thentherenormalizedmagnetizationattheorigindependsstronglyontheblock-spincongurationoutsideofthecube.Theferromagneticnearest-neighborIsingmodelinZisdenedbytheformalHamiltonianH=JXhijiij;(:)wherehijidenotesnearest-neighborpairsandJplaystheroleofaninversetempera-ture.Weshallusethedecimationtransformationwithscalefactorb=.Theimage(orblock)spinsarethosespinswithbothcoordinateseven,whiletheremainingspinsItiswellknownthatthedecimationtransformationisbadlybehavedinthelimitofinnitelymanydecimations[0,]:forexample,anyxedpointmusthaveatwo-pointcorrelationfunctionh0;xiwhichisindependentofx(soinparticulardoesn'tdecayasjxj! ).Butthepresentexampleismuchmoredrastic,astheproblemsappearafterasinglestep.

aretheinternalspins.Wedenoteby(Z)image(resp.(Z)int)thesetofimage(resp.in-ternal)spins.Moregenerally,ifisasubsetofZ,wedenotebyimage\(Z)image(resp.int\(Z)int)thesetofimage(resp.internal)spinsin.Theproofofnon-quasilocalityoftherenormalizedmeasuregoesinfoursteps:Step0.Computationoftheconditionalprobabilitiesfortheimagesystem.Theseconditionalprobabilitiesturnouttoberelatedtoexpectationvaluesinasystemofinternalspins,withxedimagespins!0.Step.Selectionofanimage-spinconguration!0special.Wendanimage-spinconguration!0specialsuchthatthecorrespondingsystemofinternalspinshasanon-uniqueGibbsmeasure(i.e.arst-orderphasetransition).Step.Studyofaneighborhoodof!0special.Westudytheinternal-spinsystemforimage-spincongurations!0inaneighborhoodof!0special,andshowthattheinternal-spinorderparameterisadiscontinuousfunctionof!0.Inphysicalterms,thismeansthattheinternal-spinorderparameterdependssensitivelyontheimage-spincongura-tionarbitrarilyfarfromtheorigin,iftheimage-spincongurationintheintermediateregionissetto!0special.Step.\Unxing"ofthespinattheorigin.Thisisatechnicalsteprelatingtheimagespinattheorigintotheinternalspinsnearby.(Afterall,wewanttheconditionalprobabilitiesforimagespins,notinternalspins.)Letusnowdiscussthesestepsindetail:Step0.Computationoftheconditionalprobabilitiesfortheimagesystem.LetbeanyGibbsmeasurefortheferromagneticIsingmodelinZwithnearest-neighborcouplingJ.Ourgoalistoshowthat,forJsucientlylarge,theimage(decimated)measureThasnon-quasilocalconditionalprobabilities.Therefore,ourrstorderofbusinessmustbetocomputetheseconditionalprobabilities.Todothis,weusetheonlyfactweknowaboutthemeasure,namelythatitsatisestheDLRequationsfortheferromagneticnearest-neighborIsingmodel.Thepresentcaseisrelativelysimple,becausetheimagespinsaresimplyasubsetoftheoriginalspins.Let,therefore,0beanitesubsetofZ;wewishtocomputetheconditionalprobabilitiesET(fjf0jgj0c)forfunctionsfofthespinsf0igi0.ButthesearejusttheconditionalprobabilitiesE(fjflgl(0c))forfunctionsfofthespinsfkgk0.Thereisaslightcomplicationnow,becausetheset(0c)=(Z)imagen0isnotthecomplementofaniteset;itscomplementconsistsoftheimagespinsin0plusalltheinternalspins.Therefore,theDLRequationsfortheoriginalmodeldonotimmediatelytellushowtoconditiononflgl(0c).However,wehavestudiedthisprobleminSection..;theconclusion(Proposition. )isthattheconditionalprobabilitymeasure(jflgl(0c))is,for-almost-everyflgl(0c),aGibbsmeasurefortheIsingmodelrestrictedtovolume(0)[(Z)intwithexternalspinssettoflgl(0c).ThislattersystemisspeciedbythesameformalHamiltonian(andhencethesameinteraction)astheoriginalIsingmodel,exceptthatnowonlythespinsin(0)[(Z)intareconsideredtoberandomvariables,andthespinsflgl(0c)areconsideredtobexed.Notethatweknowonlythat(jflgl(0c))issomeGibbsmeasureforthere-

strictedinteraction:iftherestrictedinteractionhappenstohavemorethanoneGibbsmeasure,thenwehavenowayofknowingwhichoneis(jflgl(0c)).Therefore,weshallhavetoproveboundswhicharevaliduniformlyforallGibbsmeasuresoftherestrictedinteraction.ThisiswhatweshalldoinStepsandbelow.Notealsothatthiscomputationoftheconditionalprobabilitiesisassertedtobevalidonlyfor-almost-everyflgl(0c);indeed,conditionalprobabilitiesareonlydeneduptomodicationsonasetofmeasurezero.Therefore,inordertoprovenon-quasilocalitywemustprovenotonlythatthisparticularversionoftheconditionalprobabilitiesisadiscontinuousfunction,butthatnofunctionobtainedfromthisonebymodicationonasetof-measurezerocanbecontinuous.Thatis,wemustprovethattheconditionalprobabilitiesareessentiallydiscontinuous.WeshalldothisinStepsandbelow.Itisconvenienttostudyrstthesystemofinternalspinsalone,i.e.thesystemin(Z)intwithallimagespinsf0jgjZsettoxedvalues.Wecallthissystemthemodiedobjectsystemforimage-spincongurationf0jgjZ.InStepbelowwewill\unx"theimagespinsinthevolume0.Infact,itsucestoconsiderjustoneparticularvolume0,whichweshalltaketobef0g.Step.Selectionofanimage-spinconguration!0special.Ourgoalistoshowthattheconditionalprobabilities(jflgl(0c))areessentiallydiscontinuousfunctionsofflgl(0c).Therefore,wemustndapoint!0=f0jgjZofessentialdiscontinuity.Agoodcandidatewouldbeanimage-spinconguration!0specialsuchthatthecorrespond-ingsystemofinternalspinshasanon-uniqueGibbsmeasure.Indeed,non-uniquenessoftheGibbsmeasuremeansthattheinternalspinsinvolume0dependsensitivelyontheinternalspinsarbitrarilyfarfromthevolume0(albeitwiththeintermediateinternalspinsfreetouctuate);soitisareasonableguessthattheGibbsmeasuremightdependsensitivelyalsoontheimagespinsarbitrarilyfaraway(butwiththeintermediateimagespinsheldxedat!0special),andthisispreciselythestatementofessentialdiscontinuity(seeStepbelow).Fortheb=decimationtransformation,suchaconguration!0specialwasfoundbyGrithsandPearce[,]:itisthefullyalternatingconguration!0altdenedby0i;ii;i=()i+i(:)[seeFigure(a)].Noticethateachinternalspinisadjacenteithertotwoimagespinsofoppositesign|inwhichcasetheeectivemagneticeldscancel|orelsetonoimagespin.Therefore,themodiedobjectsystemissimplyaferromagneticIsingmodelinzeroeldonadecoratedlattice[],asshowninFigure(b).Nowwecanexplicitlyintegrateoutthespinsinthedecoratedlatticethathaveexactlytwoneighbors,yieldinganeectivecouplingJ0=logcoshJbetweenthosetwoneighbors.TheresultisanordinaryferromagneticIsingmodelonZ,withnearest-neighborcouplingJ0andzeromagneticeld[Figure(c)].IfJ0>Jc=log(+p)=0:0:::,thatis,J>cosh(+p)=0:::::Jc,thenthemodiedobjectsystemforimage-spinconguration!0althastwodistinctGibbsmeasures,a\+"phaseandaphase(obtainablebyusing\+"orboundaryconditions,respectively).

(a)JJJJ(b)J'J'(c)Figure:(a)Thefullyalternatingimage-spinconguration!0alt.(b)Thedecoratedsystemofinternalspins. (c)Theequivalentnearest-neighborinteractionon(Z).

Step.Studyofaneighborhoodof!0special=!0alt.Thenextstepistostudyimage-spincongurationsinaneighborhood(intheproducttopology)of!0alt.Toshowthattheorderparameterhii!0isanessentiallydiscontinuousfunctionoftheimage-spinconguration!0,itsucestoshowthatthereexistsaconstant>0suchthatineachneighborhoodof!0alttheessentialoscillationofhii!0isatleast.Moreprecisely,itsucestoshowthatthereexists>0suchthatineachneighborhoodN!0altthereexistnonemptyopensetsN+;NNandconstantsc+>cwithc+csuchthathii!0c+whenever!0N+(.a)hii!0cwhenever!0N(.b)NowabasisfortheneighborhoodsN!0altisgivenbysetsoftheformNR=f!0:!0=!0altonR;!0=arbitraryoutsideRg;(:)whereRisasquareofsideR+centeredattheorigin(hereRisanunprimeddistance).WeshalltakeN+;NtobesetsoftheformNR;R0;+=f!0:!0=!0altonR;!0=+onR0nR;!0=arbitraryoutsideR0g(.a)NR;R0;=f!0:!0=!0altonR;!0=onR0nR;!0=arbitraryoutsideR0g(.b)withR0chosenappropriatelyasafunctionofR(R

(ofbothimageandinternalspins)outsideR+.Wewanttoconvincethereaderthatallthemeasuressoobtainedhavelocalmagnetizationshi;iifor(i;i)intR+whichareboundedbelowbyastrictlypositiveconstant,uniformlyinR(sucientlylarge)anduniformlyintheboundaryconditionoutsideR+.ThesequenceofboundsusedinourproofissummarizedinFigure.EachinternalspininintR+feelsan\eectivemagneticeld"Jfromeachimagespinadjacenttoit;butbecausetheimage-spincongurationinRisalternating,these\eectivemagneticelds"areallzeroexceptatsomesitesinlayersR+andR+:(i)AninternalspininlayerR+feelsaneectiveeld+Jifitisadjacenttotwo\+"imagespins.(ii)AninternalspininlayerR+feelsaneectiveeld+Jor+JdependingonwhethertheadjacentspininlayerR+(whichisalwaysaninternalspin)happenstobe\+"or\".WethereforeconsiderthesystemofinternalspinsinintR+withthemagneticeldsdescribedin(i)and(ii)above[Figure(b)].NextwenoticethatbytheFKGinequality[](oralternativelytheGrithsIIinequality[]),thelocalmagnetizationshi;iifor(i;i)intR+areboundedbelowbythevaluesthattheywouldtakeifthemagneticelds+Jin(i)werechangedtozero,andtheelds+Jin(ii)changedto+J.WenowhaveasystemconsistingofthespinsinintR+,withamagneticeld+JoneachspininlayerintR+[Figure(c)].Thislattersystemlivesonanitesubsetofthedecoratedlattice.WecanexplicitlyintegrateoutthespinsinintR+thathaveexactlytwoneighbors(namely,thespinsthathaveonecoordinateevenandonecoordinateodd),yieldinganeectivecouplingJ0=logcoshJbetweenthosetwoneighbors.Similarly,wecanintegrateoutthespinsinintR+,yieldinganeectivemagneticeldh0=logcoshJ>0oneachremainingspininintR+,exceptthattheeldish0atthecorners[Figure(d)].Butthislastsystemisequivalenttoasquarelatticeofsize(R+)(R+)withnearest-neighborcouplingJ0and+boundaryconditions[Figure(e)].AsR!thissystemtendstothe\+"phaseforanIsingmodelwithcouplingJ0.Inparticular,themagnetizationhi;iiofanyspinremaininginthissystem(i.e.anyspinwithiandibothodd)tendstothespontaneousmagnetizationM0(J0),whichis>0ifJ0>Jc.Wecannowreturntothedecoratedlattice,tocomputethemagnetizationhi;iiontheinternalspinsthatgotintegratedout(i.e.theoneswithievenandioddorviceversa):hi;iiintR+=htanh(J(0+00))iR+=h(tanhJ)(0+00)iR+!(tanhJ)M0(J0)>0(. )where0and00arethetwointernalspinsadjacenttoi;i.Wehavethereforeprovenourclaimthatthemagnetizationshi;iifor(i;i)intR+areboundedbelowbyastrictlypositiveconstant[namely()(tanhJ)M0(J0)forany>0],uniformlyin

mm≥(a)=2R+12R+1(b)h=2Jh=J or 3Jm2R+1h=J=m(c)R+2m→(d)(e)h'=2J'h'=J'J'J'Figure:Sequenceofboundsprovingalowerboundonthemagnetizationfortheinternalspins.

h=−Jh=−Jh=−Jh=−JJJJJFigure:Decoratedlatticewhenthespinattheoriginisfreetouctuate.RR0(i;i)anduniformlyintheboundaryconditionoutsideR+.RepeatingtheargumentbutwiththeimagespinsinR+chosenas\",weobtainthe\"phasefortheinternalspinsandthusastrictlynegativeupperboundonhi;ii.Thisprovesthatthelocalmagnetization,sayforthefourinternalspinsneighboringtheorigin,isdeterminedbytheimagespinsatfarawaydistances.Step.Unxingofthespinattheorigin.Wenowhavetomakeaslightmodi-cationintheprecedingargument,asthesystemwereallywanttostudyisthesystemconsistingoftheinternalspinsinR+andthespinattheorigin,withtheimagespinsinimageRotherthantheoneattheoriginxedinthealternatingconguration!0alt,theimagespinsinlayerimageR+settobe\+",andthespinsoutsideR+(bothimageandinternal)xedinsomearbitraryconguration.Bythesamereasoningasbefore,weobtainasystemonthedecoratedlatticeplustheorigin,withacouplingJbetweentheoriginanditsfourneighborsandanadditionalmagneticeldJontheneighborsoftheorigin[Figure].Denotingbyhi+(resp.hi+)theexpectationintheold(resp.Becausewehavexedtheimagespinattheorigintobe+,thetwosituationsarenotquitesymmetric.ButtheonlychangeisashiftinthelocationoftheinternalspinsinR+whichfeela

new)decoratedsystem,itiseasytoseethath0;0i+=P0;0=h0;0exp[J(0;0)(0;+0;+;0+;0)]i+P0;0=hexp[J(0;0)(0;+0;+;0+;0)]i+=hexp[J(0;+0;+;0+;0)]i++hexp[J(0;+0;+;0+;0)]i+:(.)Similarly,fortheanalogoussystemwiththeimagespinsinimageR+setto\",wehaveh0;0i=hexp[J(0;+0;+;0+;0)]i+hexp[J(0;+0;+;0+;0)]i=hexp[+J(0;+0;+;0+;0)]i++hexp[+J(0;+0;+;0+;0)]i+:(.)Therefore,h0;0i+h0;0i=(yx)(+x)(+y);(:)wherex=hexp[J(0;+0;+;0+;0)]i+(.0)y=hexp[+J(0;+0;+;0+;0)]i+(.)Nowyx=hsinhJ(0;+0;+;0+;0)i+=Xk=kodd(J)kk!h(0;+0;+;0+;0)ki+Jh(0;+0;+;0+;0)i+=Jh0;i+;(.)sincethecontributionsfromk=;;:::areallnonnegativebyGriths'rstinequality.Ontheotherhand,thedenominatorin(. )isboundedbetweenand(+eJ).SinceweprovedpreviouslythatforJ0>Jc,thelocalmagnetizationh0;i+isboundedbelowbyastrictlypositiveconstant,uniformlyinR(sucientlylarge)andinthecongurationoutsideR+,wecanconcludethath0;0i+h0;0i>0(:)uniformlyinR(sucientlylarge)andinthecongurationoutsideR+.nonzeroeectiveeld;andthisisirrelevant,sincewereplacetheseeldsbyzeroanyway.00

Conclusionoftheargument.Insummary,wehaveshownthatatzeromagneticeldandanysucientlylowtemperature,givenanyGibbsmeasure,therenormalizedmeasureThasthefollowingproperty:Let!0altbethefullyalternatingconguration0i;i=()i+i;letAR;+NR=;(R=)+;+bethesetofallcongurationsf0gthatarealternatinginR=,\+"in(R=)+andarbitraryoutside;andletAR;beanalo-gouslydenedbutwith\"in(R=)+.IntheproducttopologyAR;+andAR;areopensets|inparticular,theyhavestrictlypositive(T)-measure|andgivenanyneighborhoodofN!0altwealwayschooseRlargeenoughsothatAR;+[AR;N.Moreover,wehaveproventhatforall!0AR;+and!0AR;,wehaveET00;0jf0i;ig(i;i)=(0;0)(!0)ET00;0jf0i;ig(i;i)=(0;0)(!0)>0:(:)Thismeans|asrstpointedoutbyIsrael[0]|thattheconditionalexpectationsof00;0arediscontinuousasafunctionoftheboundaryconditions.Moreprecisely,theyareessentiallydiscontinuous:nomodicationonasetof(T)-measurezerocanmakethemcontinuousat!0alt.Now,forsystemswithanitesingle-spinspace(suchastheIsingmodel),continuityisequivalenttoquasilocality.Therefore,whatwehavereallyprovenisthattherenormalizedmeasureTisnotconsistentwithanyquasilocalspecication.(Inourlanguage,Tisnon-quasilocal;intheterminologyofSullivan[],itisnon-almostMarkovian.)Inparticular,TisnotGibbsianforanyuniformlyconvergentinteraction.Theorem.LetbeanyGibbsmeasureforthetwo-dimensionalIsingmodelwithnearest-neighborcouplingJ>cosh(+p)=0:::::Jcandzeromagneticeld.LetTbethedecimationtransformationwithspacingb=.ThenthemeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.Inphysicalterms,wehaveshownthatthevalueoftherenormalizedspinattheorigin,00;0,dependsstronglyonthevaluesoftherenormalizedspinsarbitrarilyfarfromtheorigin,iftherenormalizedspinsintheintermediateregionarexedtobealternating.Suchalong-rangedependenceisincompatiblewiththemeasureTbeingGibbsianforanyreasonableinteraction..Griths-Pearce-IsraelPathologiesII:GeneralMethodInthissectionweabstracttheessentialfeaturesoftheGriths-Pearce-Israelargument,inordertopreparethewayforgeneralizationstomorecomplicatedexamples.Step0.Computationoftheconditionalprobabilities.Thisstepistechnicalandmessy,butthenalresultistheobviousone[cf.(.)/(. )below].Thereaderisthereforeinvitedtoskipthissteponarstreading.Fordecimation,thecomputationoftheconditionalprobabilitiesofTwasanimmediateapplicationofProposition..FormoregeneralRTmaps,itwillbea0

morecomplicatedapplicationofthissameproposition:theideaistoconsiderrstajointsystemofinteractingspins!and!0,andthendecimatethissystemtothespace0.Ifisanymeasureonthesystemoforiginalspins(i.e.on),andTisanyprob-abilitykernelfromto0,thenthejointmeasureTon0iswell-denedby(T)(A)=Zd(!)ZT(!;d!0)A(!;!0)(:)formeasurablesetsA0.Now,if=fgisaspecicationforthesystemoforiginalspins,wewishtodeneaspecicationT=f(T);0gforthejointsystem,withthepropertythatconsistentwith=)TconsistentwithT:(:)(Infact,theconverseshouldalsohold,i.e.ameasureon0shouldbeconsistentwithTifandonlyifFisconsistentwithand=T.)ForsimplicityletusassumethattheprobabilitykernelThasthefollowingform:T(!;d!0)=YxL0eTx(!Bx;!0x)dx(!0x)(:)wherethexareprobabilitymeasures,andtheBxarenitesetsoforiginalspinswhichtogetherdeterminetheimagespin!0x.WealsoassumethatthefamilyofsetsfBxgxL0islocallynite,i.e.onlynitelymanyimagespinsxdependonanygivenoriginalspiny.Now,tomotivatetheconstruction,supposethatisaGibbsmeasureforaninteraction(andapriorimeasure0).Then,formallythemeasureTisgivenby(T)(d!;d!0)\="constYXLeX(!)YxL0eTx(!Bx;!0x)YxLd0x(!x)YxL0dx(!0x):(.)Ofcourse,thersttwoinniteproducts(theonesoverfunctionseX(!)andeTx)aremeaningless,butweknowwhattodo:todescribetheconditionalprobabilitydistributionT,with!xedoutsideanitesetand!0xedoutsideaniteset0,weretainintheproductsonlythosetermsthatintersectand/or0,i.e.(T)(d!;d!00j!c;!00c)=const(!c;!00c)YX\=?eX(!)Yx:x0orBx\=?orbotheTx(!Bx;!0x)Yxd0x(!x)Yx0dx(!0x):(. )Thatis,thesetfx:BxygisniteforeachyL.0

Now,therstproductisjusteH(!;!c),andtherstandthirdproductstogetheryield(whenproperlynormalized)thekernel(!c;d!).Therefore,thespecicationTshouldbedenedas(T)(de!;de!00j!c;!00c)=Z;0(!c;!00c)(!c;de!)Yx:x0orBx\=?orbotheTx(!ce!)Bx;(!00ce!00)xYx0dx(!0x);(.0)whereZ;0(!c;!00c)=Z00(!c;de!)Yx:x0orBx\=?orbotheTx(!ce!)Bx;(!00ce!00)xYx0dx(!0x)(.)andwehaveassumed,ofcourse,thatZ;0(!c;!00c)>0.[IfZ;0(!c;!00c)=0,then(!c;!00c)isa\forbiddenboundarycondition",whichhastobedealtwithasinthetheoryoflatticesystemswithhard-coreconstraints[,].]Wemustnowcheckthat:(a)T,thusdened,isindeedaspecication.(b)Ifisanymeasureconsistentwith,thenTisconsistentwithT.Thesetwovericationsaremessycalculations,whichtheauthorsareconvincedwillworkout(althoughmentalexhaustionpreventedthemfromwritingoutthefulldetails).ThingsbecomemuchsimplerwhenistheGibbsianspecicationforaninteractionandapriorimeasure0,andtheeTxareallnonvanishing.ThenitiseasytoseethatTistheGibbsianspecicationfortheinteractione(onthelatticeL[L0)denedbyeX;X0(!;!0)=<:X(!)ifX0=?logeTx(!Bx;!0x)ifX=BxandX0=fxg0otherwise(:)andapriorimeasure0.(Inparticular,itfollowsimmediatelyfromthegeneraltheoryinSection..thatTisindeedaspecication.)ThisistheinteractioncorrespondingtotheformalHamiltonianHjoint(!;!0)=XXLX(!)XxL0logeTx(!Bx;!0x)=Horiginal(!)XxL0logeTx(!Bx;!0x):(. )0

IftheeTxcanvanish,thenemaytakethevalue+,whichisnot(strictlyspeaking)permittedinourformulation;butthesamealgebrashowsthatTisindeedaspecication,atleastwhenZ;0(!c;!00c)>0.0HavingconstructedthespecicationTonthelatticeL[L0,wecannowapplythesameargumentasinthedecimationcase,basedonProposition.(seeSection..,Step0).Indeed,therenormalizedmeasureTisobtainedbydecimatingthejointmeasureT,i.e.restrictingittothelatticeL0.Wehopethatsomeonewillcomealongandsimplifyour\abstractnonsense"con-cerningStep0.Butwehavenodoubtthatourconcreteargumentsinthispaperarecorrect.Step.Selectionofanimage-spinconguration!0specialforwhichthecorrespondinginternal-spinsystemhasanon-uniqueGibbsmeasure.Weneedtondanimage-spinconguration!0specialsuchthattheresultingsystemofinternalspins(the\modiedobjectsystem")hasatleasttwodistinctGibbsmea-sures,callthem+and.Howwedothisdependsonthedetailsofthemodelandtherenormalizationtransformation.Fortheb=decimationtransformationonthenearest-neighborIsingmodel,thefullyalternatingconguration!0altdoesthetrick.Forthemajority-ruletransformationweshallneedamorecomplicatedconguration(Section..).Nowletfbealocalobservablesuchthat+(f)>(f);weshallcallfthe\internal-spinorderparameter".(Forthedecimationexample,fisthespinataneighboroftheorigin. )Step.Discontinuityoftheinternal-spinorderparameterasafunctionoftheimage-spincongurationinaneighborhoodof!0special.Thenextstepistostudythebehavioroftheinternal-spinsystemforimage-spincongurationsinaneighborhood(intheproducttopology)of!0special.Ourgoalistoshowthattheorderparameterfortheinternal-spinsystemisessentiallydiscontinuousasafunctionoftheimage-spinconguration!0.Todothis,werstchooseimage-spincongurations!0+and!0whichwehopewill\selectthephases+and".Wethenstudyimage-spincongurations!0whichareequalto!0specialonsomelargeboxR,whichareequalto!0+[or!0]onsomeannulusR0nR(R

Inmathematicalterms,ourgoalistoshowthatthereexistsanumber>0suchthatineachneighborhoodN!0special(intheproducttopology)thereexistnonemptyopensetsN+;NNandnumbersc+;cwithc+csuchthatforevery!0N+[resp.!0N]andeveryGibbsmeasurefortheinternal-spinsystemwithimagespinssetto!0,wehave(f)c+[resp. (f)c].NowabasisfortheneighborhoodsN!0altisgivenbysetsoftheformNR=f!0:!0=!0altonR;!0=arbitraryoutsideRg;(:)WeshalltakeN+;NtobesetsoftheformNR;R0;+=f!0:!0=!0altonR;!0=!0+onR0nR;!0=arbitraryoutsideR0g(.a)NR;R0;=f!0:!0=!0altonR;!0=!0onR0nR;!0=arbitraryoutsideR0g(.b)WethenhavetoprovethatR0canbechosenasafunctionofR(R

Step.Unxingofthespinattheorigin.Thenalstepistoshowthatifthesystemofinternalspinsisslightlymodiedbychangingtheinteractionwithafew(intheourexamplesjustone)imagespinsclosetotheorigin,theorderparameterattheseextraspinsdierslittlefromthevalueatinternalspinsclosetotheorigin.Thisisthestepof\unxing"someimagespinsdiscussedabove.Conclusionoftheargument.CombiningtheconclusionsofStepsand,wehavethatforallpossibleimage-spincongurationsoutsideR0,theorderparameteratimagespinsclosetotheoriginisdeterminedbytheimagespinsinthearbitrarilyfarawayannulusR0nR.Inmathe-maticalterms,theconditionalprobabilitydistributionoftheimagespinattheoriginisanessentiallydiscontinuousfunctionoftheotherimagespins,inaneighborhoodof!0specific.Thus,therenormalizedmeasurehasnon-quasilocalconditionalprobabilities:itisnotconsistentwithanyquasilocalspecication,andinparticularisnottheGibbsmeasureofanyuniformlyconvergentinteraction..Griths-Pearce-IsraelPathologiesIII:SomeFurtherEx-amplesInthissectionweapplytheGriths-Pearce-Israelmethodtoprovenon-Gibbsiannessoftherenormalizedmeasureinthefollowingadditionalexamples:b=decimationfortheIsingmodelindimensiond.Decimationwithspacingb,fortheIsingmodelinanydimensiond.TheKadanotransformationwithnitepandarbitraryblocksizeb,fortheIsingmodelinanydimensiond.Somecasesofthemajority-ruletransformationfortheIsingmodelindimensiond=.Block-averaging,withevenblocksizeb,fortheIsingmodelinanydimensiond.Finally,andmoststrikingly,wecanshowthatinalloftheseexamplesexcept(andthisprobablyonlyfortechnicalreasons)themajority-rulecase,thereisinfactanopenregioninthe(J;h)-planeforwhichtherenormalizedmeasuresarenon-Gibbsian.Therefore,theGriths-Pearce-Israelpathologiesarenotassociatedwiththefactthattheoriginalmodelissittingonaphase-transitionsurface.Rather,itsucesthatarst-orderphasetransitioncanbeinducedintheinternal-spinsystembychoosinganappropriateblock-spinconguration.Forthisweneedtoworkatlowtemperaturebutnotnecessarilyatzeromagneticeld.0

..Israel'sExampleinDimensiondInthissectionwestudytheb=decimationtransformationfortheIsingmodelindimensiond.Step0.Computationoftheconditionalprobabilities.Thishasalreadybeendone.Step.Choiceof!0special.Asinthetwo-dimensionalcase,wechoose!0specialtobethefullyalternatingconguration!0alt.Thesystemofinternalspinsforafullyalternatingimage-spincongurationagaincorrespondstoaperiodicallydilutedferromagnet:aninternalspinwithallbutoneofitscoordinateseven|thatis,onewhichisadjacenttotwoimagespins|hastwolessneighborscoupledtoitself,whileallotherinternalspinsareunaected.Theonlydierencefromthetwo-dimensionalcaseisthattheresultinglatticeisnotmerelyadecoratedversionofanexactlysolubleIsingmodel,sowecannotwriteanexplicitformulaforitscriticaltemperature.Nevertheless,itiseasytoshowthattheinternal-spinsystemdoeshaveaphasetransition,andthatatlowenoughtemperaturethereexistdistinctGibbsmeasures+andwithstrictlypositiveandstrictlynegativemagnetization,respectively;thesephasescanbeselectedbyusing,forexample,\+"or\"boundaryconditions.TheseclaimsfolloweasilyfromaPeierlsargument(foradescriptionofsucharguments,seee.g.[]).Theycanalternativelybeprovenbyobservingthatthedilutedsystemisacollectionof(d)-dimensionaldilutedandundilutedIsingmodels,ferromagneticallycoupled.Inparticular,thed-dimensionaldilutedsystemismoreferromagneticthanthe(d)-dimensionalundilutedIsingmodel,andhence[]exhibitsspontaneousmagnetizationforalltemperaturesbelowthecriticaltemperatureJc;dofthe(d)-dimensionalundilutedIsingmodel.Step.Studyofaneighborhoodof!0special=!0alt.Nextwemustndimage-spincongurations!0+and!0thatwill\select"thephases+andoftheinternal-spinsystem.Thechoiceisobvious:asinthetwo-dimensionalcase,wetake!0+(resp. !0)tobethecongurationwithallspins+(resp.allspins).WeneedthentoshowthatiftheimagespinsinimageRarexedinafullyalternatingconguration,andthoseinanannulusimageR0nimageRaresettoall+(orall),thenforR0largeenough(dependingonR)theimagespinsintheannulusarecapableofdeterminingtheinternal-spinphase.IntwodimensionswewereabletotakeR0=R+.Thatis,wewereabletoshieldoavolumebyxingarounditasinglelayerofimagespins:namely,bysettingtheimagespinsonlyinlayerR+tobe+,wewereabletoguaranteethattheeectivemagneticeldsfeltbytheinternalspinsinintR+areallnonnegative,evenifallthespins(bothimageandinternal)outsideR+aresettobe[seeFigures(a){(b)].Thissituationdoesnot,however,persistinhigherdimensions:alayerimageR+of+imagespinsdoesnotprotectalloftheinternalspinsinintR+fromthepossiblespinsinlayerR+[seeFigure].Therefore,wehavetoresorttoamoregeneralargumenttoshowthatthereexistsashieldinglayer,thoughthicker.Consider,therefore,thesystemofinternalspinsinvolumeintR0,withtheimagespinsinimageRxedinthe0

ΓR+3(worst case)ΓR+2h=−J<0Figure:Whyasinglelayerdoesnotworkind.Forthe\worst"congurationofthenextexternallayer(imageandinternalspinsall\"),someoftheinternalspinsinlayerintR+pickupanegativemagneticeld.0

alternating()conguration,theimagespinsintheannulusimageR0nimageRxedtobeall+,andthespinsoutsideR0(bothimageandinternal)xedinanarbitraryconguration.WedenoteexpectationsinthissystembyhiR;R0;+;.Wewanttoshowthat,forJsucientlylarge,thereexistsc>0suchthatforallR>0thereexistsR0>R(dependingonR)suchthathiiR;R0;+;hiiR;R0;+;c>0(:)andbysymmetryhiiR;R0;;hiiR;R0;;+c<0;(:)foreverycongurationoutsideR0andeveryiintR.ThiswillbeprovenusingcorrelationinequalitiestogetherwiththeuniquenessoftheGibbsmeasurefortheinternal-spinsystemwithimagespinssettoall+orall.Moreprecisely,theproofof(. )willinvolveasequenceofinequalitiescomparingthefollowingsystemsofinternalspins:ThesystemofinternalspinsinvolumeintR0describedabove,whichwedenoteby R;R0;+;!.Theinnite-volumesystemofinternalspinsint(Zd)int,withtheimagespinsinimageRxedinthealternating()congurationandtheimagespinsoutsideRxedtobeall+.Wedenotethissystemby R;;+!.Theinnite-volumesystemofinternalspinsint(Zd)int,withtheimagespinseverywherexedinthealternating()conguration.Wedenotethissystemby != R;;!.Weshallprovethefollowing:Step.

)R;R0;+;convergesasR0!toaGibbsmeasureforthesystem R;;+!.Step. )Thesystem R;;+!hasauniqueGibbsmeasure,callitR;;+.Step.

)ThemeasureR;;+islarger(inFKGsense)thanallGibbsmeasuresforthesystem !.Step. )Let(+)bethe+phase(i.e.themaximalGibbsmeasureinFKGsense)forthesystem !.Then(+)(i)c>0.0

Fromtheseresultswewillthendeduce(. ).Step..ThelimitR0!.WewishtoconsiderthelimitasR0!ofthemeasuresR;R0;+;(forxedR).Bycompactness,thissequenceofmeasureshasatleastonelimitpointintheweaktopology(infact,anysubsequencehasalimitpoint).WeclaimthatanylimitpointofthemeasuresR;R0;+;(witharbitrary)isnecessarilyaGibbsmeasureforthesystem R;;+!.Theproofistrivial:foranyvolumeR0,theDLRequationsforthesystems R;R0;+;!and R;;+!areidentical(i.e.the'sarethesame);soforlargeenoughR0,themeasureR;R0;+;satisestheDLRequationinvolumealsoforthesystem R;;+!.Sincethelattersystem'sspecicationisFeller,theDLRequationsarepreservedunderaweaklimit.NotethatwehavenotyetproventhatthelimitasR0!exists;dierentcon-vergentsubsequencesmightapriorihavedierentlimits.ButinthenextstepwewillprovethattheGibbsmeasureforthesystem R;;+!isunique,soinfactthelimitdoesexist.Step..UniqueGibbsmeasureforthesystem R;;+!.Considertheinnite-volumesystemofinternalspins(Zd)int,withtheimagespinsinimageRxedinthealternating()conguration,andtheimagespinsoutsideRxedtobeall+.WeclaimthatthissystemhasauniqueGibbsmeasure.(Weonlyneeduniquenessatlowenoughtemperature,butinfacttheGibbsmeasureisuniqueatalltemperatures.)Thisuniquenessisintuitivelyobvious:theeectivemagneticeldsinducedbythe+imagespinsoutsideRaresucienttopushthesystemintothe+phase.Unfortunately,theproofwehavetooerisabittoocomplicatedforourtaste.Itgoesasfollows.First,wenoticethatitisenoughtoproveuniquenessoftheGibbsmeasurewhenalltheimagespins(includingthoseinsideR)aresetinthe\+"position.Indeed,changingtheimagespinsinsideRamountstoanite-volumeperturbationofthesystemandhenceitdoesnotalterthenumberofGibbsmeasures[,section.].[Infact,everyGibbsmeasure0fortheperturbedinteractioncomesfromauniquelydenedGibbsmeasureoftheunperturbedinteraction:ifWistheperturbation,then0()=(eW)=(eW).

]ToprovetheuniquenessoftheGibbsmeasureforthesystemwithallimagespins\+",weprovidetwoarguments.Firstargument,provinguniquenessonlyatlowtemperature:Pirogov-Sinaitheory[,0,]impliesthatthephasediagramatlowenoughtemperatureisasmalldeformationofthatatzerotemperature,butinthiscasethereisonlyonegroundstate(namely,allspins\+").Secondargument,provinguniquenessatalltemperatures:Theinternal-spinsystemisanIsingmodelonaperiodiclattice,withnearest-neighborcouplingJ>0andaperiodicmagneticeldhx=hn.i.x(heren.i.x=ifxneighborsanimagespin,and0otherwise),specialized0

toh=+J.BytheLee-Yangtheorem([0,Section.]or[0]andreferencescitedtherein)andaresultofLebowitzandPenrose[](seealso[,Theorem.]),itcanbeshownthatthepressureofsuchanIsingmodelisajointlyanalyticfunctionofJandhonthedomainJ;h>0.ItfollowsthatallperiodicGibbsmeasuresgivethesamemeanvaluetotheobservablesconjugatetoJandh:theseobservablesare,respectively,Pijwherethesumrunsoverallnearest-neighborpairshijiinaunitcelloftheperiodiclattice,andPkwherethesumrunsoverallsiteskinthisunitcellthatarenearestneighbortoanimagespin.ByGriths'comparisoninequality,itfollowsthat(ij)=+(ij)(.a)(k)=+(k)(.b)foreverypairhijiofnearestneighborsandforeverysitekneighboringanimagespin;here+andarethemeasurescorrespondingto\+"and\"boundaryconditions,respectively.Wethenresorttotheinequality[]+(A)(A)+(B)(AB)(B)+(AB)(:)validforanysetsA;BZd(wedenoteA=QiAi).From(.)and(.)weconcludethat(A)=+(A)(:0)wheneverAisaproductoffunctionsoftheformijwithi;jnearestneighborsandkwithkbeinganeighbortoaimage-spinsite.(Inotherwords,Amustbethesymmetricdierenceofafamilyofsuchsetsfi;jgand/orfkg.)ButitisnothardtoseethatallsetsA(Zd)intareofthisform,hence=+:(:)NowbytheFKGinequality+inFKGsenseforeveryGibbsmeasure,hencethereisauniqueGibbsmeasureatalltemperatures.(ThisargumentisessentiallyduetoLebowitz[,],withminoralterationstoaccommodateperiodicsystems. )Step..Comparisontothe !system.WeclaimthatthemeasureR;;+islarger(inFKGsense)thanallGibbsmeasuresforthesystem !.ThisisanimmediateconsequenceoftheFKGinequalitycombinedwiththeuniquenessprovedinStep..Indeed,bytheFKGinequality,thenite-volumeGibbsmeasureforthe R;;+!Wewrite0incasei0iforallsitesi.Anobservablefissaidtobeincreasingiff()f(0)whenever0.WesaythatinFKGsenseincase(f)(f)forallincreasinglocalobservablesf.

systemwithany(internal-spin)boundaryconditionislargerinFKGsensethanthenite-volumeGibbsmeasureforthe !systemwiththesameboundaryconditions.Thisinequalitypassesdirectlytotheinnite-volumelimit.Step..Spontaneousmagnetizationforthe+phaseofthe !system.The !systemispreciselytheIsingmodelonaperiodicallydilutedlattice.AsdiscussedinStep,thismodelhasspontaneousmagnetizationforJsucientlylarge.Step.Unxingofthespinattheorigin.Finally,wecan\unx"thespinattheorigininthesamewayasinthe-dimensionalexample.Conclusionoftheargument.Weconcludethatineveryneighborhoodof!0altthereareopensetsN+;NsuchthatET(00jf0igi=0)(!)ET(00jf0igi=0)(! )>0(:)for!N+and!N.Asinthe-dimensionalcase,thisimpliesthenon-quasilocalityoftherenormalizedmeasureT,foranyoriginalGibbsmeasure.Thisworksforanytemperaturebelowthecriticaltemperatureoftheundiluted(d)-dimensionalIsingmodel.Wehavethereforeproven:Theorem.Letd.ThenforallJ>Jc;d,thefollowingholds:LetbeanyGibbsmeasureforthed-dimensionalIsingmodelwithnearest-neighborcouplingJandzeromagneticeld.LetTbethedecimationtransformationwithspacingb=.ThenthemeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction...DecimationwithSpacingbTheconclusionsofTheorem.fordecimationwithspacingb=holdalsoforlargerspacings.Themaindierencefromtheb=caseisthatforbthesystemofinternalspinsobtainedwith!0=!0altisnolongersimplyaperiodicallydilutedIsingmodelinzeromagneticeld;rather,itcontainsaperiodicalternatingmagneticeldwhichisnonzeroatthesitesneighboringanimagespin.Asaconsequence,weneedamoresophisticatedtechniquetoconcludethatthereisindeedaphasetransition(Step).TheappropriatetoolforthispurposeisPirogov-Sinaitheory[,],whichissummarizedinAppendixB.TheupshotofP-Stheoryisthatthephasediagramofalatticesystematlowtemperaturecaninsomecasesbededucedfromthephasediagramatzerotemperature.Moreprecisely,ifthereareanitenumberofperiodicgroundstates,andthesegroundstatessatisfyasuitable\Peierlscondition",thenthephasediagramofperiodicGibbsmeasuresatlowtemperatureisasmallperturbationofthephasediagramofgroundstates.Inthecaseathand,onecanshowthatforthefullyalternatingblock-spinconguration,thesystemofinternalspinshasonly

twoperiodicgroundstates|namely,theonewithallinternalspins+,andtheonewithallinternalspins|andthatthesegroundstatessatisfythePeierlscondition.ItfollowsfromP-StheorythatatlowtemperaturetherearepreciselytwoperiodicGibbsmeasures,+and,characterizedrespectivelybyastrictlypositiveorstrictlynegativemagnetization.ThedetailsofthispartoftheargumentarepresentedintheAppendixB(SectionB..).Stepsandarethenproveninamannerexactlyidenticaltotheb=case.TheanalysisofSectionB..yieldsa(veryweak)estimateoftherangeoftemperaturesforwhichthepathologiesarepresent[formula(B.)].Wehavethusproventhefollowing:Theorem.Letdandb.ThenforallJsucientlylarge(dependingondandb),thefollowingholds:LetbeanyGibbsmeasureforthed-dimensionalIsingmodelwithnearest-neighborcouplingJandzeromagneticeld.LetTbethedecimationtransformationwithspacingb.ThenthemeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.Remark.Checkerboarddecimation,asshowninFigure(b),isaverydierentsituation:theinternalspinsarenotconnected,andhencetheycannotcooperatetohaveaphasetransition.Infact,inthiscasetherstiterationofthetransformationiswell-dened[][,p.].However,theseconditerationofthistransformationcorrespondstoasingleiterationoftheb=decimationtransformation,andsoisill-denedatlowenoughtemperature...KadanoTransformationwithpFiniteInsomesensetheresultsthusfarshouldnotbesurprising:thedecimationtransforma-tion,unlikeotherRGtransformations,doesnotinanysenseintegrateoutthe\high-momentummodes"andleavethe\low-momentummodes";itmerelyintegratesoutonesublatticeandleavesanother.Inparticular,ifthesublatticeofinternal(integrated-out)spinsisconnected,itishardlysurprisingthattheinternal-spinsystemcanexhibitaphasetransition,andthatthiscangiverisetoRGpathologies.Inthissectionweshowsomethingconsiderablymoresurprising:thatthesamepathology|non-Gibbsiannessafteronerenormalizationstep|ispresentatlowtemperaturefortheKadanotransformationwithanynite(butnonzero)p.ThisresultisinclearconictwiththeRGideology,whichstatesthatintegrationoverhigh-momentummodescannotproducesingularities.(Indeed,ourproofmakesnodistinctionbetweenblocksizesbandb=|andforb=oneisnotintegratingoverany\modes",high-momentumorotherwise! )Inthenextsubsectionweshallproveasimilarresultforsomemajority-ruletransformations(i.e.Kadanowithp=).Inearlierversionsofthiswork[,],weclaimedthisresultonlyforsmallp.Subsequentlywefoundaproofvalidforall0

ConsidertheKadanotransformation(.0)withblocksizebandparameterp.From(.0)onereadilyconcludes[]thatforeachchoiceofblockspins0thecondi-tionalprobabilitiesoftheinternalspinscorrespondtoaHamiltonianHe()=JXhijiijpXx0xXiBxi+XxlogcoshpXiBxi:(:)ThisistheoriginalIsing-modelHamiltonianperturbedbyablock-dependentmagneticeldandanantiferromagneticmulti-spincoupling.Toobtainnon-trivialresultsweconsiderblocksatleastofsizeineachcoordinatedirection.Itisnaturaltoexpectthat,foranyxedp<,forsucientlylargeJ(i.e.lowenoughtemperature)theperturbationbecomeeectivelysmall,andthephasediagramasmalldeformationtothatoftheoriginalIsingmodel.Wenotice,however,thatthereisasmalldierencewiththeoriginalperturbativesettinginthatthelasttwotermsin(.)donotincludeatemperaturefactor.Inthestudyofdeformationsofphasediagrams,oneconsidersaxedvalueofmultiplyingallthetermsoftheHamiltonian,andanalyzestheconsequencesofchanging(perturbing)someoftheremainingparameters.Theproofthatthedeformationsaresmoothusuallyrequiresthatthesizeofthisperturbationnotexceedacertain-dependentbound.Inourcase,afterpullingoutacommonfactor,theparametersoftheperturbationacquirea-dependenceandoneisconfrontedwiththeproblemofverifyingthatthis-dependentsizeissmallerthanthe-dependentbound.Thisproblemisespeciallyseriousinthecaseofthelasttermin(.),whichdoesnothaveanysmallparameterprecedingit,sothatitssizedecreasesonlyas=.WeconcludethattosuccessfullycompleteStepweneedaslightstrengtheningoftheusualPStheory,involvingfamiliesofinteractions,andshowingthatthedeformationsofthephasediagramaresmalluniformlyinmembersofthisfamily.SuchastrengtheningisdiscussedinAppendixB(CorollariesB.andB.).ForStep,then,wechooseaconguration!0specialfortheblockspinssothatthemiddletermintheRHSof(.)doesnotfavoranyoverallinternalspinorientation|forexample,afullyalternatingconguration.The\uniform"versionofPStheoryimplies(AppendixB..)thatatlowenoughtemperaturetherearetwocoexistingphases+and.ThisistheendofStepoftheGriths-Pearce-Israelargument.Stepsandarethencompletedalmostidenticallytothepreviousexamples.Inthiswayweconclude:Theorem.Letd,band0J0thefollowingholds:LetbeanyGibbsmeasureforthed-dimensionalIsingmodelwithnearest-neighborcouplingJandzeromagneticeld.LetTbetheKadanotransformationwithparameterpandblocksizeb.ThentherenormalizedmeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.A(poor)estimateofthesmallnessofthetemperatureisgiveninformula(B. ).WeemphasizethatourestimateJ0isnonuniforminp.Asaresult,wearenotableto

takep!atanyxedJ,andtherebytreatthemajority-rulemap.(Ourpartialresultsonthemajority-rulemap,obtainedbyadierentmethod,willbedescribedinthenextsubsection.)Remarks..Theoccurrenceof\peculiarities"intheKadanotransformationatsmallpandlowtemperaturewassuggestedalreadybyGrithsandPearce[,]..InterestingapplicationsoftheKadanotransformation(withblocksizeb=! )ariseinimageprocessing[,,,,],speechrecognition[0]andothereldsofappliedprobabilitytheory.ThebasictheoreticalconstructintheseeldsisaclassofmodelstermedhiddenMarkovmodels[0,,];inourlanguagethesearesimplytheimagesofMarkovian(i.e.nearest-neighbor)spinmodelsunderlocalrenormalizationtransformations.IthasbeenlongrecognizedthatsuchmeasurescanbeveryfarfromMarkovian;herewehaveshownthattheycanevenbenon-Gibbsian.Consider,forexample,anIsing-modelGibbsmeasurecorruptedbywhitenoise:withprobabilityaspinisobservedincorrectly,independentlyateachsite.ThisismodelIofGrithsandPearce[,],andisequivalenttotheKadanotrans-formationwithp=tanh()onblocksofsizeb=.Forany>0,wehaveproventhattheimagemeasureisnon-Gibbsianfor(J;h)inan(-dependent)openneighborhoodofthelow-temperaturezero-eldregion.Thissystemisofinterestinapplicationstoimageprocessing[,]...Majority-RuleTransformationNextwewishtoshowthatGriths-Pearce-Israelpathologiesoccuralsoforthemajority-ruletransformation(i.e.theKadanotransformationwithp=).Forsimplicityletusconsiderthecaseofanoddblocksizeb,soastoavoidthecomplicationscausedbyties.Inviewoftheforegoingexamples,itisnaturaltotryafullyalternatingblock-spinconguration!0alt.UsingPirogov-Sinaitheory,onemighthopetoprovethatatlowtemperaturetheinternal-spinsystemhaspreciselytwoextremalperiodicGibbsmeasures:a\+"phaseinwhichtheinternalspinsshowanoverwhelmingmajorityof+spinsinblockswheretheblockspinis+butonlyaweakmajorityofspinswheretheblockspinis,anda\"phasewiththereversebehavior.Thisresultwouldinfactfollowifonecouldshowthattherearepreciselytwoperiodicgroundstates:a\+"-likestateinwhichtheinternalspinsareunanimously+inblockswheretheblockspinis+andshowabaremajoritywheretheblockspinis,anda\"-likestatewiththereversebehavior[Figure(a)].Unfortunately,neithertheshapenortheposi-tionwithinablockofthese\minimalislands"ofminorityspinsisingeneraluniquelydetermined[Figure(b)];therefore,thisfamilyofstatesisinnitelydegenerate,andwecannotapplyP-Stheory(atleastinitsusualform).Moreover,itturnsoutthatthesecongurationsarenotevengroundstates:thereare\strip-like"statesoflowerenergydensity[Figure(c)].Webelievethatthesestrip-likestatesaretrulygroundstates(thoughwehavenotprovenit);andsincetheytooareinnitelydegenerate,P-Stheorycannotbeapplied.

(a)(and rotated)(b)(c)Figure:(a)Thehoped-forstructureofthe\+"-likegroundstate.(b)Indeterminacyoftheshapeandpositionoftheminimalislandsofspins,forthecaseofablock.Theenergyperislandis0J,irrespectiveofitsshape.Theenergydensityis0Jperblock. (c)Strip-likestateswithanenergydensityofJperblock.Thesestatesalsohaveanindeterminacyineachblock.

Wesuspectthatforeachoddbtheredoexistblock-spincongurations(morecomplicatedthan!0alt)forwhichtheGriths-Pearce-Israelargumentcanbecarriedthrough,butforb=;wehavebeenunabletondany.Thesimplestcaseinwhichwemanagedtoavoidtheseproblemsisb=.Hereabaremajorityinablockconsistsofspins,andtheuniqueminimal-energycongurationforanislandoformorespinsisasquare.Bytakingadoubly-alternatingblock-spinconguration[Figure(a)],wecanforcethesesquarestobepositionedinauniqueminimal-energyway[Figure(b)].Theenergyofthisarrangementis0Jpergroupofeightblocks,or0Jperblock.Ontheotherhand,strip-likestateswouldcostatleastJperblock.Therefore,withthisblock-spincongurationtheinternal-spinsystemhaspreciselytwogroundstates:the\+"-likestatedepictedinFigure(b),andthereverse\"-likestate.ItthenfollowsfromP-Stheorythatatlowenoughtemperaturetheinternal-spinsystemhastwoextremalperiodicGibbsmeasures,+and,characterizedbyanonzeroposition-dependentmagnetizationofoppositesigns.Theingredientsoftherigorousargumentshowingthatindeedthe\+"-and\"-likecongurationsofthetypeofFigure(b)aretheonlygroundstates,andthatPStheoryisapplicable,aresummarizedinSectionB...ThiscompletesStep,whichisthehardpartoftheproof.TheproofofStepreliesagainonP-StheoryandtheFKGinequality.Step.isprovenintheusualway.Thefactthatthesystemwith+blockmagnetizationoutsideasquareRhasauniqueGibbsmeasureisaconsequenceofP-Stheory:atzerotemperaturethissystemhasauniquegroundstate,namelythestatewithallspins+,andP-Simplies(seeSectionB..)thatthistrivialphasediagrampersistsatlowenoughtemperature.ThisprovesStep..Finally,weclaimthatthesystemwith+blockspinsoutsideasquareR(anddoublyalternatingblockspinsinside)hasalargermagnetizationthanthesystemwithdoublyalternatingblockspinseverywhere.Indeed,theconstraintthatthemajorityofinternalspinsinablockBbe(resp.+)canbeimposedbyincludingintheHamiltonianatermhBsgn(PiBi)withhB! (resp.hB!+).Sincesgn(PiBi)isanincreasingfunctionofthespins,theFKGinequalityimpliesthatthemagnetizationatanysiteisanincreasingfunctionofhB.ThisprovesStep..Stepisprovenintheusualway.Wethereforeconclude:Theorem.ForallJsucientlylarge,thefollowingholds:LetbeanyGibbsmeasureforthetwo-dimensionalIsingmodelwithnearest-neighborcouplingJandzeromagneticeld.LetTbethemajority-ruletransformationonsquareblocks.ThenthemeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.Itisofcourseunnaturalandunpleasantforthisresulttoberestrictedtothespecialcaseofblocks.ThisrestrictionwasnecessaryonlyinStep(theproofofaphasetransitionforsomexedblock-spinconguration);itarosefromthenecessitytoobtainanitenumberofperiodicgroundstatesinordertoapplyP-Stheory.AlltheotherstepsintheproofremainvalidforblocksofarbitrarysizeandforIsingmodelsin

(a)557(b)Figure:Majorityruleforblocks.(a)Thedoubly-alternatingblock-spincong-uration. (b)Theuniqueminimal-energyarrangementofislandsofspinsina+sea.Theenergyis0Jperblocks,or0Jperblock.

arbitrarydimension.Diggingalittledeeperweseethatthe\rigidity"intheshapeandpositionoftheislandsofminorityspins,andhencetheboundednessofthenumberofperiodicgroundstates,isaconsequenceofthefollowingnumerological\miracle":theblocksizeb=andtheislandsizec=satisfytheDiophantineequation+b=c:(:)Theproofextendsautomaticallytoanyblocksizebforwhichcdenedby(.)isaninteger.InAppendixCwendthegeneralsolutiontothisDiophantineequation:theadmissibleblocksizesturnouttobeebk=h(+p)k++(p)k+i(:)fork=;;;:::.Therstfewebkare,,,,,:::.Forotherblocksizes,aproofofnon-Gibbsiannessusingourmethodswouldrequireeitheramorecleverchoiceofblock-spinconguration!0special,orelseamoresophisticatedversionofP-Stheorycapableofdealingwithinnitelymanyperiodicgroundstates[,,,,].Irrespectiveofthesetechnicaldetails,itseemsplausibletoexpectthattheconclusionofTheorem.remainsvalidforallblocksizesb.Remark.GrithsandPearce[,]andlaterHasenfratzandHasenfratz[,Section]havepresentedaratherdierentclassofcasesinwhichthemajority-ruletransformationisexpectedtohave\peculiarities":intheseexamplestheblock-spinconguration!0specialistakentobepurely+,andthemagneticeldistakentobenegative(withanorder-strengthchosentoexactlycompensatetheeectoftheblockspins).Ourschemeofproofdoesnotapplyintheseexamples,fortworeasons:Firstly,thereareinnitelymanyperiodicgroundstates,soP-Stheoryinitsusualformdoesnotapply.Secondly(andperhapsmoreseriously),intheconguration!0specialalloftheblockspinsarealready+,andbyconstructionthecorrespondinginternal-spinsystemdoesnothaveauniqueGibbsmeasure;soitisclearlyimpossibleto\select"the+phase(i.e.makeitunique)bysettingtheblockspinsinanannulustobe+.ThislatterfactwasalreadynotedbyIsrael[0,p. ]...Block-AveragingTransformationsIncontrasttoourpreviousexample,inthiscaseourproofworksforevenblocksizes(andonlythese)preciselybecauseofthepossibilityofties.Wediscussherethesimplestcase,namelytheblock-averagingtransformationforthetwo-dimensionalnearest-neighborIsingmodelatlowtemperatures.WedivideZintoblocksBj,anddene0j=XiBji:(:)Wenoticethatalthoughtheoriginalvariablestaketwovalues(),therenormalizedspins0takevevalues(0;;).Usuallytheaveragespinsarerescaled,butsucha

(1)(2)(3)(4)Figure:Thefourperiodicgroundstatesoftheinternal-spinsystemobtainedbyconstrainingtheblockspinstobezero.rescalingisirrelevantforourdiscussionbecauseweonlyconsiderasingleapplicationofthetransformationanddonotiterate.Step.Wechoosetheconguration!0specialdenedby0j=0foralljZ.Theresultingsystemofinternalspinshas,atlowtemperatures,fourperiodicGibbsmeasurescorrespondingtofourgroundstatesformedbyinnitealternatingstripsofthickness(seeFig.).ThisfollowsimmediatelyfromPirogov-Sinaitheory(seeAppendixB..).Step.LetbeaNNsquare.Takeblock-spinboundaryconditionsasfollows:+fortherowsofblockspinsimmediatelyaboveandbelow,+forthecolumnsimmediatelytotherightandleftof,and+forthecolumnsjusttotherightandleftofthese[seeFig.0(a)].AslightmodicationoftheusualPeierlsargumentprovesthattheseboundaryconditionsinduceatlowtemperaturetheGibbsmeasureassociatedtoground-state#inFig.[seeFig.0(b)].Step.Weunxtwonearest-neighborblockspins:theoneattheoriginandtheoneimmediatelyaboveit.Then,atsucientlylowtemperature,onehaswithhighprobabilitytheboundaryconditionofFig.forthetwo-blocksystem(00;00){(00;0)(forasuitablepositioningofthevolume).Noticethatthisboundaryconditionhaseight+spinsandonlyfourspins;therefore,itisclearthatthespinsinsidethetwoblocksarebiasedtowards+,sothat0;!0special;@;;(0(00;00))=0;!0special;@;;(0(00;0))c+(J)>0(:)atzeromagneticeld.[Indeed,atlowtemperaturethereisaprobabilityofhavingastripcongurationwith0(00;00)=0(00;0)=0andaprobabilityofhavinganall-+congurationwith0(00;00)=0(00;0)=+,sothatlimJ!c+(J)=+. ]Similarly,byreversingthesignoftheblockspinsontheboundary,weobtain0;!0special;@;;(0(00;00))=0;!0special;@;;(0(00;0))c(J)<0(:)whereofcoursec(J)=c+(J)bysymmetry.Thiscompletestheargument.Remark.NoticethatitdoesnotsucetounxasingleblockspintodistinguishamongthefourGibbsmeasures,becauseinallfourmeasurestheboundarycondition0

+4+4+4+4+2+2(a)(b)Figure0:Block-spinboundaryconditionschosenforStep.Figure:Boundaryconditionsfortheblock-spinobservableofStep.

ontheunxedblockwouldbesymmetricbetween+and(i.e.four+spinsandfourspins),andtheexpectationoftheblockspinwouldbezero.Theargumentgivenhereclearlyworksforanyevenblocksizeb,inanylatticedimensiond.Inthiswayweconclude:Theorem.Letd,andletbbeeven.ThenforallJsucientlylarge(dependingondandb),thefollowingholds:LetbeanyGibbsmeasureforthed-dimensionalIsingmodelwithnearest-neighborcouplingJandzeromagneticeld.LetTbetheblock-averagingtransformationwithblocksizeb.ThenthemeasureTisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction...GeneralizationtoNonzeroMagneticFieldItmightappearfromourresultsthusfarthattheRGpathologyissomehowassociatedwiththefactthattheoriginalHamiltonianHliesonthephase-coexistencecurve(whichintheIsingmodelmeanszeromagneticeld).Thisisinfactnotthecase.Inthissectionweincludeamagneticeldh,andshowthatindimensiondthereisanopenregioninthe(J;h)-plane|namely,lowenoughtemperatureandsmallenougheld|wherethedecimationtransformationproducesanon-Gibbsianmeasureafteroneiteration.(Wesuspectthattheresultistruealsoford=,butitwillrequireadierentproof.)Ourargumentworksfordecimationwitharbitraryscalefactorb,andfortheKadanotransformationwithanyp<.Moreover,forblock-averagingwehaveanevenstrongerresult:therenormalizedmeasuresarenon-Gibbsianforarbitrarystrengthoftheeldindimensionsd,atlowtemperatures.WeconcludethattheGriths-Pearce-Israelpathologiesarenotassociatedwiththefactthattheoriginalmodelissittingonaphase-transitionsurface.Rather,itisthesystemofinternalspinsconstrainedbytheconguration!0specialwhichmusthaveaphasetransition.Ifthistransitionisofasimilartypeasthatoftheoriginalsystem,thenitisnaturaltoexpectthattheoriginalsystemmustatleastbeclosetoaphasetransition,insomesense.Buteventhisneednotbethecase,astheexampleofblock-averagingtransformationswillshow,ifthesetwotransitionsareofdierentnature.Letusrsttreatthecaseofdecimationtransformations.Consider,therefore,aninteractionH=JXhijiijhXii:(:)(Notethatinournormalization,themagneticeldisnotexplicitlymultipliedbyanyfactorofJor. )Theideaisthatforsuitable(small)valuesofehh=J,wecanndanimage-spinconguration!0specialforwhichthecorrespondinginternal-spinsystemhasanon-uniqueGibbsmeasure.Roughlyspeaking,!0specialmustbesuchthatit\compensates"theeectofthemagneticeld,sothatthesystemofinternalspinsForthisreason,thetitleofourearlierreport[]isinretrospectsomewhatmisleading.

subjectedbothtothehomogeneouseldhandtotheinhomogeneouseldduetotheimagespinshastwoormoreextremalGibbsmeasures.Asinallthepreviouscases,weformalizethisideaintwosteps:werstshowthatitworksatzerotemperature,namelythattherearechoicesofehand!0specialforwhichthegroundstateisnotunique;andsecondweshowthatPirogov-SinaitheoryisapplicablesothatitimpliesnonuniquenessofGibbsmeasuresatlowbutnonzerotemperatures.Thesimplestcaseistolet!0specialbeperiodic.Inthissituationonecanndthe\compensating"eldeh0neededtoobtainmorethanonegroundstatebystudyingcongurationsinsideaperiod.Wedonotwanttoenterintothedetails,aswelateroerabetterandmoregeneralprocedure,butwemaketheratherobviousremarkthattheeldmustbetakeninadirectionoppositetothatofthemajorityofinternalspinswithinaperiod.Thevalueoftheeldmustbesuchthatiftheinternalspinsfollowit,theenergygainisexactlycompensatedbythepenaltypaidbytheinternalspinsneighboringtheimagespinsofoppositesign.(Inotherwords,thesumofalltheelds|externalorduetoimagespins|feltbytheinternalspinsinaperiodmustbezero. )Thiseldstrengtheh0istooweaktofavortheippingofsmallregionsofinternalspins,andonlyacollectiveipisenergeticallyacceptable.Ofcourse,thisdelicatebalanceisbrokeniftheeldischanged,nomatterhowlittle.Therefore,weconcludethatthisvalueeh0eh0(!0special)issuchthatforeh>eh0(resp.ehJ0thereisacontinuouscurveeh=eh(J),withlimJ!+eh(J)=eh0,onwhichtheinternal-spinsystemhaspreciselytwoperiodicextremalGibbsmeasures,namelya\+"phaseanda\"phase.Aslongas!0specialisnotall+orall,StepcanbeprovenusingtheFKGinequality,asinSection...Wethereforeconcludethatforeh=eh(J)therenormalizedmeasureTisnon-Gibbsian.Thechieflimitationofthisprocedureisthatitproducesonlyrationalvaluesofeh0andthatthereisnouniformityineh0fortherangeoftemperaturesforwhichthenonuniquenesspersists.Hence,byletting!0specialrangeoverallperiodiccongurations,weprovenon-Gibbsiannessonlyforaregionofthephasediagramformedbycountablymanycurveseh=eh(J).Wecanprovethatthesetofeh0valuesisdenseinsomeintervaljehj

Ifwewanttoextendthisargumenttomoregeneralchoicesof!0special,weareconfrontedwiththelimitationimposedbythepresentversionsofPirogov-Sinaithe-ory.Onepossiblegeneralizationofthisconstructionistolet!0specialbequasiperiodic.ThenonecanuseanextensionofPirogov-SinaitheoryduetoKoukiou,PetritisandZahradnk[].(Actually,theseauthorsrequirethequasiperiodicpartoftheinterac-tiontobesmall;sowecannothandledecimation,butcanhandletheKadanotrans-formationwithpsmall.)Inthiswayweobtainuncountablymanycurveseh=eh(J)onwhichtherenormalizedmeasureisnon-Gibbsian.(IftheresultsofKoukiouetal.canbeextendedtofrequencieswhichareDiophantineofarbitrarytypel<|atpresenttheytreatonlyl=|thenthecorrespondingsetofeh0valueswouldcontainsomeintervaljehjJ0;jhj<0Jginthe(J;h)-plane,asoriginallyconjecturedbyGrithsandPearce[,]andIsrael[0].Thekeyingredientisamechanismtogenerateacontinuumofimage-spincongurations!0specialsuchthatP-Stheoryisapplicabletotheresultinginternal-spinsystem.Atpresentthisisonlypossibleifweresorttorandomness:Zahradnk[,0,],andwithlessgeneralityBricmontandKupiainen[,],extendedP-Stheorytosystemswithsuperimposed(small)randominteractionsfordimensionsd.Ourconstructionwill,therefore,bebasedona(slightly)randomchoiceoftheconguration!0specialandwillbelimitedtod.Consider,forstarters,decimationwithsomespacingb,appliedtoanIsingmodelwithferromagneticnearest-neighborinteractionJandmagneticeldh=Jeh>0.Weconsiderablock-spincongurationwhichisequaltothefullyalternatingcon-gurationexceptthatthespinsthatwouldcorrespondtoa\+"haveaprobability=Jofbecominga\",independentlyforeachsuchspin.Wewishtoshowthatforeachsucientlysmallpositiveeh,thereexistsansuchthattherandommagneticeldinducedbytheblockspins(whoseneteectisnegative)exactlycompensatesthepositiveuniformeld,inthesensethatforalmostallsuchimage-spincongurationstherearetwodistinctGibbsmeasures+and.Todothis,weapplyanas-yet-unpublishedtheoremofZahradnk[0,],whichgeneralizesPirogov-Sinaitheorytosmallrandominteractions,ifthelatticedimensionis.(Inthepreprint[0],therandominteractionsareassumedtobesmalluniformlyinallrealizationsoftherandomness.Thisconditionisnotsatisedinourcase,asonehaslargeterms(ofstrengthJ),albeitoccurringwithsmallprobability(=J).Inaprivatecommuni-cation[],Zahradnkhasinformedusthatminormodicationsofhisproofssucetocoveralsothiscase.)WeapplyZahradnk'stheorywiththeoriginalHamiltonianH0takentobethesystemofinternalspinswithfullyalternatingimagespins(i.e.aferromagneticnearest-neighborIsingmodelinaperiodicallydilutedlatticeandwithaperiodicmagneticeldofmeanzero,seeSection.. );thesymmetry-breaking\elds"aretakentobethe

uniformmagneticeld,andtherandomnegativemagneticeldscomingfromthoseblockspinsthatwereippedfrom\+"to\"accordingtotheprocedureexplainedabove.Theanalysisoftheground-statestructureofH0,andtheproofofthePeierlsconditionforit,werealreadycarriedoutinSections..andB...Zahradnk'stheory(TheoremB.)thenassuresus(SectionB..)thatforeachJsucientlylargeandeachsucientlysmall,thephasediagramis,withprobability,asmalldeformationofthatoftheHamiltonianH0;thatis,foreachsuchpair(J;)thereexistsauniqueeh(J;)>0suchthatthesystemhastwodistinctGibbsmeasures+and(whichcanbeobtained,forexample,bytakingeh#ehoreh"eh,respectively).Moreover,thevalueeh0()atwhichthe\+"and\"congurationsaresimultaneouslygroundstatesisastrictlyincreasinglinearfunctionof.(Thisfollowsfromanargumentsimilarto,albeitmoreelaboratethan,theonepresentedatthebeginningofthesectionforperiodicchoicesof!0special.Theslopedependsontheblock-sizebandthedimensionalityd.)AsZahradnk'stheorytellsusthatthelow-temperaturephasediagramisasmoothdeformationofthezero-temperatureone,weconcludethatthatehisacontinuousandstrictlyincreasingfunctionof.Obviouslythecaseh<0canbehandledbythesameargumentwith\+"and\"reversed.Thebottomlineis,therefore,thatthereexists|foreachJsucientlylarge|acontinuousandmonotoniccurve(eh)throughtheorigin,denedforjehjsmall,suchthatforalmostallchoicesoftherandomblock-spincongurationthesystempresentsmultipleGibbsmeasuresonthecurveandauniqueGibbsmeasuretoeachsideofthecurve(Figure).Thus,fortheIsingmodelwithJsucientlylargeandjehjsucientlysmall,wecanproveStepbychosingas!0specialanyoneofthecongurationsfromtheprobability-setcorrespondingto=(eh).TheproofofthevalidityofStepsandisessentiallyidenticaltothatofthecaseh=0(Sections..and..).Wenoticethatduetothesmoothnessofthephasediagramdeformations,theboundjehj<(J)forwhichthesesteps,andhencetheexistenceofRGpathologies,canbeprovenisgivenbyacontinuousfunction(J).Moreover,wehaveliminfJ! (J)0>0.Thenalresultisthefollowing:Theorem.Foreachdandb,thereexistsaJ00(depend-ingondandb)suchthatforallJ>J0andjhj<0Jthefollowingistrue:LetbeanyGibbsmeasureforthed-dimensionalIsingmodelwithnearest-neighborcouplingJandmagneticeldh.ThentherenormalizedmeasureTarisingfromthedecimationtransformationwithspacingbisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.Similarresultsarevalid,byasimilarargument,fortheKadanotransformationwithanyxed0

hεunique Gibbs measureunique Gibbs measureJ largemultiple Gibbs measuresFigure:Phasediagramforarandom-eldIsingmodelatlowtemperatures(d).

areexactlythesameforallvaluesoftheeld,becausetheconstraintofzeroblockspinsremovesalleld-dependenceinsidesuchblocks.(Thepointisthattheground-statecongurations,aswellastheblock-spinboundaryconditionsneededtoselectthem,arethesameforallvaluesofthemagneticeld.)InStep,thepresenceofaeldcausesanasymmetrybetween\+"and\"boundaryconditions|i.e.wenolongerhavec(J)=c+(J)|aswellasasmallervalueforthedierencec+(J)c(J)betweenthetwomagnetizations.Butthisdierenceisstillboundedawayfromzerouniformlyin(theextrafactorinvolveddependsonlyontheeldsattheeightsitesofthetwo-blockobservable),sotheresultisstillvalid.Alternatively,onecould\unx"astripofNblocks.Therefore,wehave:Theorem.Foreachdandeachevenb,thereexistsaJ0(dependingondandb)suchthatthefollowingistrue:ForanyGibbsmeasureofthed-dimensionalIsingmodelwithnearest-neighborcouplingJ>J0andarbitrarymagneticeldh,therenormalizedmeasureTarisingfromtheblock-averagingtransformationisnotconsistentwithanyquasilocalspecication.Inparticular,itisnottheGibbsmeasureforanyuniformlyconvergentinteraction.ThisresultisincontrastwiththeresultsobtainedabovefordecimationandKadanotransformations,wherewewereabletoprovenon-Gibbsiannessforh=0onlyfordandonlyforjhj=Jsmall.Therestrictiontoweakeldsis,fortheseexamples,essen-tial,becauseitisknownthatinastrongeldtherenormalizedmeasureisGibbsian[,,0].Moreover,MartinelliandOlivieri[]haveproventhatforany(J;h)withh=0,thedecimationtransformationresultsinaGibbsianmeasurewhenthespacingbislargeenough(howlargedepends,ofcourse,onJandh).Finally,wenoteaninterestingconsequenceofourTheorem.:fortheIsingmodelindimensiond,intheregionfJ>J0;jhj<0Jg,theDobrushin-Shlosman[,]completeanalyticityconditionisviolated..Large-CellRenormalizationMapsinDimensiond()FourFiveyearsago,LebowitzandMaes[]constructedaverydierentexampleofanon-Gibbsianmeasure,arisinginthestudyofentropicrepulsionofasurfacebyawall.Subsequently,DorlasandvanEnter[0]generalizedthisexample,andpointedoutitsrelevancefortherenormalization-grouptheoryofIsing-likemodelsindimensiond().InthissectionwepresentaslightlygeneralizedversionoftheLebowitz-Maes-Dorlas-vanEntertheoremonnon-Gibbsianness,andthendiscussitsrelevanceforRGtheory.Thereaderinterestedprimarilyintheresults(resp.intheapplicationtoRGtheory)shouldreadupthroughthestatementofTheorem.,andthenskipdirectlytoSection..(resp.toSection.. ).

..Non-GibbsiannessoftheSignFieldofan(An)harmonicCrystalConsiderasystemofreal-valuedspinsf'xgxZd,anddenex=sgn('x).ClearlyfxgxZdisaeldofIsingspins.WeshallshowthatforcertainmasslessGibbsmeasuresonthesystemoff'gspins,theprojectionofsuchameasureonthefgspinsisnon-Gibbsian.Themeasureswehaveinmindarethosepossessingaspontaneouslybrokenglobalshiftsymmetry'!'+c.Moreprecisely,considerasystemdenedformallybytheHamiltonianH(')=Xx=yVxy('x'y);(:0)wherethefunctionsVxyareeven,andVxy=Vx+a;y+aforallx;y;aZd.Suchasystemistermedananharmoniccrystal(orifthefunctionsVxyareallquadratic,aharmoniccrystal).[Morerigorously,suchasystemisdenedbytheinteractionA(')=Vxy('x'y)ifA=fx;yg0otherwise(:)wheretheapriorimeasured0x('x)istakentobeLebesguemeasure.Lebesguemea-sureisnotnormalizable,butifthepotentialsVxyarechosensuitably,thenonehasZ('c)<,andthespecicationisthenwell-dened.Intheinnite-rangecasetherearesomesubtletiesassociatedwithrapidlygrowingboundaryconditions,asdiscussedinExampleofSection... ]Foran(an)harmoniccrystal,aninnite-volumeGibbsmeasureneednotexist;andindeed,itwillnotexistinlowenoughdimension,e.g.dforshort-rangeinteractions[,,].However,ifaGibbsmeasuredoesexist,thenitpossessesaspontaneouslybrokenglobalshiftsymmetryinthesensethatcisalsoaGibbsmeasureforthesameinteraction(herecisthemapthatshiftsallspinsbyaconstantc),butc=forc=0.ThatcisaGibbsmeasureisanimmediateconsequenceoftheDLRequations,whilec=followsfromtheimpossibilityoftheprobabilitydistributionof'0beinginvariantunderanon-trivialshift.Furtherinformationonthepropertiesof(an)harmoniccrystalscanbefoundinreferences[,].EveryharmoniccrystalisamasslessGaussianmodel,andtheconverseisverynearlytrue.Toseethis,consideratranslation-invariantGaussianmeasureonRZdwithmeanmandcovarianceh'x;'yi=Cxy=()dZ[;]dbc(p)eip(xy)dp;(:)Strictlyspeakingwemustdenexalsointheambiguouscase'x=0.Thesimplestchoiceistosetx=+byat;themostelegantchoiceistosetx=withprobabilities.However,thischoicewillinfactplaynorole,aseverymeasurethatwewillconsiderhasthepropertyProb('x=0foratleastonex)=0.

wherebc(p)isanonnegative,even,integrablefunctionofp[;]d.NowletusdeneBxy=()dZ[;]dbc(p)eip(xy)dp;(:)assumingthatbc(p)isanintegrablefunctionofp.ThenBistheinversematrixofthecovariancematrixC.Now,supposethatXyjBxyj<;(:)aswilloccurifbc(p)isatleastmodestlysmooth.Inthatcase,thereisawell-denedspecicationcorrespondingtotheformalHamiltonianH(')=Xx;yBxy'x'yhXx'x(:)withh=m=bc(0)andapriorimeasuretakentobeLebesguemeasure;andisaGibbsmeasureforthisspecication.Inparticular,ifbc(0)=|thisisthe\massless"case|thenthecouplingsBxysatisfyXyBxy=0:(:)ThismeansthattheHamiltoniancanberewrittenasH(')=Px=yBxy('x'y)(:)(notethathereh=0).Thus,everymasslessGaussianmeasure(satisfyingmildreg-ularityconditions)istheGibbsmeasureforsomeharmoniccrystal,andconversely.FurtherdetailsontheGibbsrepresentationofGaussianmeasurescanbefoundinreferences[,]and[,Chapter].Nowletbeanytranslation-invariantGibbsmeasureofthe(an)harmoniccrystal,andletebeitsIsingprojection.UndermildtechnicalconditionsonthepotentialsVxy,wewillprovethateisanon-Gibbsianmeasure.ThebasicideaoftheproofistousethespontaneouslybrokenshiftsymmetrytoshowthatProb('x>0forallxA)eo(jAj)(:)asA%(vanHove).Thatis,theprobabilitythatallthespinsinaregionAaresimultaneouslypositiveisexponentiallysuppressedatarateslowerthanthevolumeofA(roughlyspeaking,itissuppressedbya\surfaceterm").Thismeansthati(+je)=0;(:)AnecessaryconditionforPyjBxyj

where+isthedeltameasureconcentratedonthecongurationwithallspins+.IfewereGibbsian,thenbyProposition.,+wouldhavetobeGibbsianforthesameinteraction.But+isobviouslynon-Gibbsian(noabsolutelysummableinteractioncanforceaspintobe+),soemustalsobenon-Gibbsian.Theproofof(.)proceedsinthreesteps:Step.UsingtheDLRequations,oneprovestheidentityZF(')d(')=ZF('+kA)eHrel('+kA;')d(')(:0)foranyboundedfunctionFandanyGibbsmeasure.HereAisanarbitrarynitesetofsites,Aisitsindicatorfunction,kisanarbitraryrealnumber,andHreldenotestheenergydierencebetweenthetwocongurations.(Sincethetwocongurationsdieronanitesetofsites,thisenergydierenceisnite-a.e.)Inessence,thisidentitysaysthataconguration'+kAhasaprobabilityeHrel('+kA;')timesaslargeasthatoftheconguration'.Step.OneestimatestheenergydierenceHrel,andattemptstoremovethefactoreHrelfromtheright-handsideof(.0)atthepriceofaprefactoreo(jAj).Step.SpecializingtothecaseF(')=('>0onA),oneattemptstoprovealowerboundonRF('+kA)d(')thatisoftheformejAjf(k),wheref(k)!0ask!+.Sincetheleft-handsideof(.0)isindependentofk,wecantakek!+andthuscompletetheproof.Unfortunately,Stepsandareslightlytricky(thoughnotterriblycomplicated),andthedetailsoftheproofdependontheexactformofthepotentialsVxy.Infact,wehavethreedistinctproofs,eachonevalidforadistinctclassofVxy:(a)EachVxyisconvex,andPz=0kV00zk<.(b)EachVxyisquadratic(ofeithersign),andthemeasureisamasslessGaussiansatisfying(.)and(.).(c)EachVxyisconvex,themodelisnite-range(i.e.onlynitelymanyoftheV0zarenonzero),andthemodelisdominatedbyastableGaussianinthesensethatthevectorsfz:inf'V000z(')>0gspanasubspaceofRdofdimension>.(Weconjecturethatthesetechnicalconditionscanberemovedoratleastweakened.)Case(a)istheeasiestcase,astheenergyshiftisuniformlybounded;unfortunately,thesupnormconditiononV0doesnotallowpotentialsgrowingfasterthanlinearlyatinnity(suchasGaussians!).Allthetechnicaldetailsincases(b)and(c)areattemptsAlternativeargument:IfewereGibbsian,thenbyProposition.,+wouldhavetobeGibbsianforthesameinteraction,andmoreoveri(ej+)wouldhavetobezero.Butinfacti(ej+)=.Secondalternativeargument:IfewereGibbsian,thenbyProposition.,thepressurep(gje)wouldhavetobestrictlyconvexindirectionsg=f=I+constarisingfrominteractionsB.Butforg()=0(i.e.amagneticeld),itiseasytoseefrom(.)thatp(gje)=forall0,contradictingthestrictconvexity.(Foramoregeneralversionofthislatterargument,seeSection... )0

tocontrolanenergyshiftthatisboundedonlyinsomeaveragesense.Weurgethereadertostudyrsttheproofforcase(a),beforeproceedingtocases(b)and(c).Case(b)istheonetreatedbyDorlasandvanEnter[0];wefollowtheirproofalmostverbatim.Case(c)isaminorgeneralizationoftheonetreatedbyLebowitzandMaes[];theproofwegiveisslightlydierentfromtheirs,buttheunderlyingideasandtricksarethesame.Theorem.Letbeatranslation-invariantGibbsmeasureforan(an)harmoniccrystalsatisfyingoneoftheconditions(a){(c)listedabove.Incase(c),assumeinadditionthatissymmetricarounditsmean.ThenforeachM<,wehaveProb('x>MforallxA)eo(jAj)(:)asA%(vanHove).Itfollowsthate,theprojectionofontheIsingspinsxsgn('x),isnottheGibbsmeasureforanyinteractioninB.Remarks..Oneconsequenceofthistheoremisthatanarbitrarilyweakpertur-bationoftheformH!HPxf('x),wherefisnondecreasingandnonconstant,willdrivethespins'xto+.Asaresult,thustheperturbedmodelwillhavenoinnite-volumetranslation-invariantGibbsmeasures.Thisisthephenomenonofen-tropicrepulsionofasurfacebyasoftwall,studiedbyLebowitzandMaes[]..Itisnaturaltoaskwhethereisnon-quasilocal(andnotmerelynon-Gibbsian).Wediscussthisquestion,insomewhatgreatergenerality,inSection...ProofofTheorem..Sincethehypothesesofthetheoremareinvariantunderauniformshift'!'+c,itsucestoconsiderthecaseM=0;thislightensthenotation.(Forourapplicationtothesignfunction,weneedonlyM=0anyway.ButwewillexploittheformulationwithgeneralMinSection... )Step.LetbeanyGibbsmeasureforanymodelofreal-valuedspins(notneces-sarilyananharmoniccrystal).ThentheDLRequationsforvolumesaythatd('j'c)=Z('c)eH(';'c)d':(:)NowletFbeanybounded(forsimplicity)measurablefunction,andlet beanyeldwhichvanishesoutside.ThenZF(')d('j'c)=Z('c)ZF(')eH(';'c)d'=Z('c)ZF('+ )eH('+ ;'c)d'=ZF('+ )e[H('+ ;'c)H(';'c)]d('j'c);(.

)whereinthemiddlelineweusedtheshiftinvarianceofLebesguemeasure.Nowintegrateoverdc('c):weobtainZF(')d(')=ZF('+ )eHrel('+ ;')d(');(:)

whereHrel('+ ;')H('+ ;'c)H(';'c):(:)NotethatHrelisindependentofassoonassupp .Theidentity(. )isthusvalidforany ofboundedsupport.Inparticular,ifwetake =kA,weobtain(.0).Inthecaseoftheanharmoniccrystal(.0)wehavethefollowingexpressionforHrel:Hrel('+kA;')=XxAyAc[Vxy('x+k'y)Vxy('x'y)](.a)=XxAyAcZk0V0xy('x'y+ )d :(.b)Step.ThegoalofthisstepistoprovethatZF(')d(')eok(jAj)ZF('+kA)d(')(:)(orsomesimilarformula)forsomesuitableclassofnonnegativefunctionsF.Hereok(jAj)denotesatermthatmaydependinanarbitrarywayonk,butforeachrealkitshouldbeo(jAj)asA%.Case(a):Thisistheeasycase,astheenergyshift(.)canbeboundedinsupnorm:kHrelkXxAyAcjkjkV0xykjkjo(jAj)asA%(.)bytheusualargumentbasedonPz=0kV00zk<(seee.g.theproofofProposition.inAppendixA..).Substituting(.

)intotheidentity(.0),weconcludethatZF(')d(')ejkjo(jAj)ZF('+kA)d(')(:)uniformlyforallnonnegativeboundedfunctionsF.Case(b):HereweapplytheSchwarzinequalitytotheright-handsideoftheidentity(.0):ZF('+kA)eHrel('+kA;')d(')hRF('+kA)=d(')iRe+Hrel('+kA;')d(')(:0)foranyF0.Inparticular,ifFistheindicatorfunctionofsomeset,thenF==F.Nowincase(b)wehaveHrel('+kA;')=k(';BA)+k(A;BA);(:)

andisaGaussianmeasurewithmeanmandcovariancematrixC=B.WecanthereforecalculateexactlyZe+Hrel('+kA;')d(')=exp[k(A;BA)+km(;BA)]=exp[k(A;BA)](.)sinceB=0by(.).Nowk(A;BA)=kXxAyAcBxyko(jAj)asA%(.)bytheusualargumentbasedonPz=0jB0zj<.HenceZF(')d(')eko(jAj)ZF('+kA)d(')(:)uniformlyforallindicatorfunctionsF.Thisisaslightvariantof(.).Case(c):Thiscaseisalittlebittrickier.LetFbeanynonnegativefunctionsupportedonthesetf':a'bonAanda0'b0on@+rAg,whereristherangeoftheinteraction.Thenontheright-handsideof(.0)theintegrandisnonvanishingonlywhenak'xbkforxA,anda0'yb0fory@+rA.Now,sinceVxyisconvex,V0xyisincreasing,soVxy('x'y+k)Vxy('x'y)isanincreasing(resp.decreasing)functionof'x'yfork0(resp.k0),asseenfrom(.b).Therefore,fork0(whichisthecasethatwillinterestus)wehaveHrel('+kA;')Xx@rAy@+rA[Vxy(ba0)Vxy(ba0k)]C(a0;b;k)j@rAj;(.)whereC(a0;b;k)Pz[V0z(ba0)V0z(ba0k)]isniteforalla0;b;k(sinceonlynitelymanytermsinthissumarenonzero).HenceZF(')d(')eC(a0;b;k)j@rAjZF('+kA)d(')(:)uniformlyforallnonnegativeFsatisfyingthesupportcondition.This,too,isavariantof(.).Step,Case(a):Weapply(. )toF(')=('>0onA),sothatF('+kA)=('>konA).SincetheVxyareconvex,itfollowsimmediatelyfromtheDLRequationthattheFKGinequality[]holdsforeveryGibbsmeasure.Thereforewehaveh('>konA)iYxAh('x>k)i=Prob('0>k)jAj(:)

Combiningthiswith(.),wegetliminfA%jAjlogProb('>0onA)logProb('0>k):(:)Buttakingk!+,theright-handsidegoestozero.Step,Case(b):Weapply(. )toF(')=('>0onA).WecontrolProb('>konA)usingtheBrascamp-Liebinequality[,],whichisvalidforarbitraryGaussianmeasures,combinedwiththeChebyshevinequality:Prob('>konA)Prob(j'mj

[byBrascamp-Lieb]= C00(m+k)!jAj:(.)Combiningthiswith(. ),wegetliminfA%jAjlogProb('>0onA)log C00(m+k)!:(:0)Nowtakek!+.Step,Case(c):Weapply(.)toF(')=(a'bonAanda0'b0on@+rA),withthechoicesa=0,b=m+k,a0=k,b0=m+k,k0.WethereforeneedtocontrolProb(ak'bkonAanda0'b0on@+rA)=Prob(j'mjm+konA[@+rA):(.)

Todothis,weemploytheBrascamp-LiebandChebyshevinequalitiesasincase(b)[theBrascamp-LiebinequalityisvalidbecausealltheVxyareconvex].Hereitisimportantthatbeevenaboutitsmean,becauseBrascamp-Liebreferstovariancesratherthantoexpectationsofsquares;weneedtoknowthatconditioningonasetsymmetricaroundthemeandoesnotdisplacethemean.Theonlyotherchangefromcase(b)isthatvar('xij)isboundedabovenotbyvar('xi),butratherbythevarianceof'xiinthedominatingGaussian,whichbyhypothesisisnite(callitC00).Thus,wehaveProb(j'mjm+konA[@+rA) C00(m+k)!jA[@+rAj:(:)Combiningthiswith(. ),wegetliminfA%jAjlogProb('>0onA)liminfA%jAjlogProb(0<'

'x+kintheregionA,weshouldapplyanonlinearmap'x!f('x)thatwouldproducealargeupwardshiftwhen'xisnegative,butasmallershiftwhen'xislargeandpositive.Inthiswaywemayhopetohaveanenergyshiftthatisuniformlyboundedabove.Of

course,inthecaseofanonlinearmapfwemustalsodealwithaJacobian,butthisturnsouttobemanageable.Theideamaybecrazy,butitseemstowork,atleastforsomeratherlargeclassofpotentialsVxy.However,thispaperisalreadymuchtoolong,andwehavenothadtimetoworkoutallthedetails,soweleavefurtherdevelopmentofthiscircleofideastotheinterestedreader...Non-GibbsiannessofLocalNonlinearFunctionsofan(An)harmonicCrystalThemethodoftheprecedingsectionapplies,infact,tolocalnonlinearfunctionsmuchmoregeneralthanthesign.Indeed,let00beacompactmetricspace,andletf:R!00beanyfunction(notnecessarilycontinuous)suchthatlim'!+f(')=!exists.WeshallshowthatfortheclassofmasslessGibbsmeasuresonthesystemoff'gspinsconsideredintheprecedingsection,theprojectionofsuchameasureonthef!gspinsisnon-Gibbsian.Theorem.0Letbeanytranslation-invariantmeasureonRZdsatisfyingthees-timate(.)forallM<.Let00beacompactmetricspace,andletf:R!00beafunction(notnecessarilycontinuous)suchthatlim'!+f(')=!exists.Letebetheimagemeasureofunderthemapfappliedtoeachspin.TheneisnottheGibbsforanyinteractioninB,withrespecttoanyapriorimeasuresupportedonmorethanonepoint.Proof.LetU;Vbeopensetsin00satisfying!UUV.Thenletg0:00![0;]beacontinuousfunctionsatisfyingg0Uandg0Vc0;theexistenceofsuchafunctionisguaranteedbyUrysohn'slemma.Nowdeneg:00Zd![0;]byg(f!xgxZd)=g0(!0).Thatis,gisthefunctiong0appliedtothespinattheorigin.Nowletuscomputethepressurep(gje)for0:p(gje)limn!ndlogZexpXxCng0(!x)de(!)=limn!ndlogZexpXxCng0(f('x))d(')(.)ifthislimitexists.Sinceg0,clearlythelimsupis.Ontheotherhand,theliminfisliminfn!ndloghendProb(f('x)UforallxCn)iliminfn!ndloghendProb('x>MforallxCn)i=;(.)

whereMischosensothat'>Mimpliesf(')U;herethenalequalityusesthefundamentalestimate(.).Sowehavep(gje)=forall0:(:)ButthisviolatesthestrictconvexityofthepressurewhichmustholdifeisaGibbsmeasureforaninteractioninB(Griths-Ruelletheorem,Proposition.).Henceeisnon-Gibbsian...PhysicalInterpretationWehaveproventhateisnottheGibbsmeasureforanyinteractioninB,butisthisenough?Weknowthatnon-Gibbsiannesscansometimesoccurfor\trivial"rea-sons,e.g.iftherearehard-coreexclusions,orfor\semi-trivial"reasons,e.g.iftheHamiltonianHisquasilocalbutunbounded.(Thislattercanhappenonlywhenthesingle-spinspaceisinnite.)Ifwecontendthateis\pathological",thenwereallyoughttoprovenotmerelythateisnon-Gibbsian,butalsothatitisnon-quasilocal.Wearenotableatpresenttoprovenon-quasilocality,butwecanargueheuristicallythatinatleastsomecasesthenon-Gibbsiannessdoesinvolvesomestronglynon-localeect.Considerthesignofthe(an)harmoniccrystal.RecallingTheorem.togetherwithRemarkfollowingit,itisnaturaltoconjecturethatlimR0!Esgn('0)'x>0forallxhavingRjxjR0=(:)forallR,nomatterhowlarge.(Atleastincase(c)ofSection..,weareabletoprovethisusingtheFKGinequality,viaaslightextensionoftheargumentsofLebowitzandMaes[].)Thatis,ifweconditiononthespinsinanannulusRjxjR0beingall>0,asR0!thisdrivesallthespinsto+,andinparticularforcesthesignofthespinattheorigintobe+(withprobability!).FortheIsingmeasuree,thismeansheuristicallythatthespinattheoriginisfeelinganinniteenergy.However,sincetheeectoccursforallR,nomatterhowlarge,thisinniteenergymustarisefromtheinteractionbetweenthespinattheoriginandarbitrarilydistantspins.(Crudelyspeaking,theinteraction,ifitexists,isnon-summable.)Thus,wedonothaveheremerelythe\semi-trivial"situationofaHamiltonianwhichisquasilocalbutunbounded(whichanywayisimpossibleforamodelwithnitesingle-spinspace);somestronglynon-localeectistakingplace.Itmayevenbethat(.)impliesnon-quasilocality;oritmaybethatnon-quasilocalitycanbeprovenbyadierentargument.Theseareopenquestions.AsimilarsituationprobablyholdsinthesetupofSection..,whenevertheimagesingle-spinspace00isnite.AverydierentsituationarisesiffisabijectivemapofRonto00(ofcourse00mustthenbeuncountablyinnite! ).Inthiscasefismerelyaone-to-onerelabellingofspinvalues;thephysicsoftheimagemeasureeisobviouslyidenticaltothatofthe

originalmeasure.Inparticular,iftheoriginal(an)harmoniccrystalhasnite-rangeinteractions,theneisconsistentwithaGibbsianspecicationforaparticularnite-rangebutunboundedinteraction,namelytheonegottenbymappingthe(an)harmonic-crystalspecicationviathefunctionf.Suchaspecicationisalwaysquasilocal;theinteractionisuniformlyconvergentbutnotabsolutelysummable.Finally,letusremarkthatthelocalnonlinearmapsconsideredhereareaspe-cialcaseoftherenormalizationtransformationsconsideredinSectionsand.{.:namely,oneinwhichtheblocksaresinglesites,thetransformationisdeterministic,andtheimagespaceisingeneraldierentfromtheoriginalspace.Suchtransforma-tionstriviallyobeyproperties(T){(T)ofSection..Ofcourse,iffisone-to-one,thenthetransformationistrivial(justarelabellingofspincongurations).However,iffismany-to-one,thenthetransformationisnotsodierentinnaturefromtheusual(block-spin)renormalizationtransformations:both\discarddetails"fromtheoriginalspinconguration.Thesedetailsmaybeinthenestructureofasinglespin,orinthelocalnestructureofasmallblockofspins,butqualitativelytheredoesnotseemtobeanygreatintrinsicdierence.OurtheoremsbothinSections.{.andinthecurrentsubsectionareofthegeneraltype:anRTmapwhichdiscards(important)informationmakestheimagemeasure(sometimes)non-Gibbsian(andpossiblyevennon-quasilocal)...ApplicationtotheRenormalizationGroupInthissectionweapplyTheorem.totheRG,followingcloselyDorlasandvanEnter[0].LetusconsideranIsingmodelindimensiond>atthecriticalpoint,andapplyblock-averagingtransformationsonvariousblocksizesb.ThenDeConinckandNewman[]andShlosman[,andprivatecommunication]haveshownthatthereexistsab-dependentchoiceofnormalizationsuchthattheblock-spinmeasuresconvergeasb!toamasslessGaussianmeasure;thisisaslightvariantoftheAizenman-Frohlichtrivialitytheorem.Nowthekeyobservationisthatablock-averagingtransformationfollowedbyaprojectiononIsingcongurationsisidenticaltoamajority-ruletransformation.Soconsiderapplyingthemajority-ruletransformationusinglargerandlargerblocksizesb.Sincetheblock-averagedspins(withasuitableb-dependentnormalization)convergeasb!toamasslessGaussian,itisnotdiculttoshowthatthemajority-ruleimagespinsconvergeasb!tothesignofthissamemasslessGaussian.ButbyTheorem.,thislattermeasureisnon-Gibbsian!(Fordetails,see[0]. )Thisnon-Gibbsianscalinglimitisnotaxedpointinthestrictsense,asthesequenceofmajority-ruletransformationslacksthesemigroupproperty:themajorityConventionalwisdomholdsthatthenormalizationcanbechosentobebpforasuitablepowerp[infactonepredictsp=(d+)==(d+)=].Ifthisisthecase,thenthelimitingmeasurecanalsobeobtainedbyrepeatedapplicationoftheblock-averagingtransformationwithaxedblocksizeb,andhenceisaself-similarGaussianmeasure[,,].However,thisconventionalwisdomhasnotyet(asfarasweknow)beenprovenrigorously.

ruleonblocksizebisnotequaltotheseconditerationofmajorityruleonblocksizeb(aspoliticianswellknow!).Therefore,theexistenceofpathologiesforthexedpointarisingfromtheb!limitdoesnotguaranteethatthecorrespondingpathologieswilloccurforthexedpointarisingfromiterationofamajority-rulemapwithaxedblocksizeb.Butitdoesmakeitplausible:theredoesnotseemtobesomuchdierencebetweenmajorityruleonablockofsizebnandniterationsofmajorityruleonablockofsizeb.And,inanycase,the\large-cellmajority-rule"approachisclearlypartoftheRGenterprise[,],soitisinterestingtoseethatitcanfail.Finally,aswediscussinSection.,thereareotherreasonstoexpectthatthisbehaviorisinsomesensetypical.Indeed,weconjecturethatthexed-pointmeasuresofnonlinearRGtransformationsford()du(uppercriticaldimensionofthemodel)willbenon-Gibbsianinconsiderablegenerality.Finally,weremarkthattheresultsdiscussedhereford>areexpectedtoholdalsoford=,providedthatthe\trivialityconjecture"[,]istrue..OtherResultsonNon-GibbsiannessandNon-QuasilocalityInSections.{.,wehavegivenanumberofexamplesofnon-Gibbsian(orwhatisslightlystronger,non-quasilocal)measures,withparticularattentiontothosearisinginRGtheory.Itisnaturaltoaskwhetherthephenomenonofnon-Gibbsianness(ornon-quasilocality)ismorewidespread.Unfortunately,verylittleisknownatpresentaboutthepropertiesofnon-quasilocalmeasures,andveryfewexamplesofnon-quasilocalmeasuresareknown.Inthissectionwetrytomakeacompletesurveyofallknownphysicallyinterestingexamplesofnon-quasilocality.(Thelistisshortenoughthatsuchacomprehensivesurveyisfeasible. )..TrivialExample:ConvexCombinationofGibbsMeasuresforDif-ferentInteractionsTheseareperhapsrathersillyexamples:ifonemakesaconvexcombinationofGibbsmeasuresfortheIsingmodelattwodierenttemperatures,thenitishardlysurprisingthattheresultingmeasurewillnotbeGibbsianatall.TheproofsaysroughlythatiftheresultingmeasurewereGibbsianforsomeinteraction,thenthetwooriginalmeasureswouldalsohavetobeGibbsianfor.Butthisisimpossible,becausetheGriths-RuelletheoremtellsusthatameasurecanbeGibbsianforatmostoneinteraction(modulophysicalequivalence).Weneedapreliminaryresult,concerningtheconditionsunderwhicha\reweight-ing"ofaGibbsmeasureremainsaGibbsmeasure:Lemma.LetbeaspecicationandameasureinG().Ameasureoftheform=fbelongsalsotoG()ifandonlyiffisbF-measurable(modulo-nullsets).

(WerecallthatbFTSFcisthe-eldofobservablesatinnity:seeSection...)Theproofofthislemmaisgiven,forinstance,in[,Lemma.]andin[,Theorem.].Wecannowprovethemainresult:Proposition.Let;;:::beaniteorcountablyinnitefamilyofmeasures(notnecessarilytranslation-invariant)whicharearedistinguishableatinnity,i.e.thereexistdisjointsetsF;F;:::bFsuchthatk(Fk)=foreachk.Assumefurtherthateachofthemeasures;;:::givesnonzeromeasuretoeveryopensetin.Nowformaconvexcombination=Pkckkwithallck>0.Ifisconsistentwithaspecication,thensoare;;:::;andifisFeller,thenthisistheonlyFellerspecicationwithwhichanyofthesemeasuresisconsistent.Thus,ifsometwoofthefkg|say,iandj|happentobeconsistentwithdierentFellerspecications(i=j),thenitfollowsthatisnotconsistentwithanyFellerspecication.Inparticular,isnotaGibbsmeasureforanycontinuous,uniformlyconvergentinteraction.Ifthesingle-spinspace0isnite,thismeansthatisnotconsistentwithanyquasilocalspecication,andinparticularthatisnotaGibbsmeasureforanyuniformlyconvergentinteraction.Remark.Itisnotdiculttoshowthatifthemeasures;;:::arepairwisedistinguishableatinnity,thentheyarejointlydistinguishableatinnityinthesenseofProposition..Hereitiscrucialthatwearedealingwithacountablefamily.ProofofProposition..Supposethatisconsistentwithaspecication.Then,byLemma.,themeasuresk=ckFkarealsoconsistentwith.TheuniquenessfollowsfromTheorem..InordertoapplyProposition.,weneedtoverifythatthemeasures;;:::aredistinguishableatinnity(thesupporthypothesisisusuallytrivialtocheck).Oneeasywaytoobtainsuchmeasuresistorecallthatdistinctergodictranslation-invariantmeasuresaredistinguishableatinnity(Theorem.andtheremarkfollowingit).Wethereforehave:Corollary.Let;;:::beaniteorcountablyinnitefamilyofergodictranslation-invariantGibbsmeasuresforinteractions;;:::B,respectively.Nowformaconvexcombination=Pkckkwithallck>0.IfisconsistentwithaFellerspeci-cation,thenalltheinteractionskmustbephysicallyequivalentintheDLRsense(andhencealsointheRuellesense).Proof.Ifi=j,theniandjmustbephysicallyequivalentintheDLRsense(Corollary. ).Sowecanassumewithoutlossofgeneralitythatthemeasures;;:::arealldistinct.Sincedistinctergodicmeasuresaredistinguishableatin-nity,andGibbsmeasuresforanabsolutelysummableinteractionalwaysgivenonzero0

measuretoeveryopenset,wecanapplytheprecedingpropositiontoconcludethat===:::.TherestfollowsfromTheorems.and..Therefore,(non-trivial)niteorcountablyinniteconvexcombinationsofergodictranslation-invariantGibbsmeasuresfornon-physically-equivalentinteractionscannotbeGibbsian;andfornitesingle-spinspacetheycannotevenbequasilocal...RestrictionoftheTwo-DimensionalIsingModeltoanAxisSchonmann[]gaveanotherexampleofanon-Gibbsianmeasurethatcanbeobtainedbyapplyingasimpletransformationtoawell-knownGibbsianmeasure.Heprovedthatif+isthe\+"phaseofthetwo-dimensionalIsingmodelatzeroeldandatanytemperaturebelowcritical,thenitsrestriction+Ptotheaxisf(i;0):iZgisanon-Gibbsianone-dimensionalIsingmodel.Hisargumentisbasedontworesults:R)Foralltemperaturesbelowthecriticaltemperatureforthed=Isingmodel,i(Pj+P)=0.R)Let0n;Ndenotethespincongurationonthe\annulus"f(i;0):njijNg.ThenforeachnthereexistsanN(n)suchthat(j0n;N(n)=)!asn!(:)forallGibbsmeasuresoftheoriginalmodel.Asaconsequence(+P)(j0n;N(n)=)!Pasn!:(:)Result(.)impliesthatif+Pisconsistentwithsomequasilocal(=Feller)specication,thenPmustbeconsistentwiththatsamespecication.Heuristicallythisisduetothefactthatameasureobtainedjustbyachangeintheboundaryconditionsmustbeadierentphaseforthesameinteraction.Toseeitmathematically,let=()beaquasilocalspecicationwithwhich+Pisconsistent.Thenforeachsetcontainedintheinterval(n;n)wehavethat(+P)(j0n;N(n)=)=(+P)(j0n;N(n)=)(:0)byproperty(b)ofDenition.;andpassingtothelimitn! (sinceisFeller)weobtain(P)=P:(:)Therefore,if+PwereaGibbsmeasureforsome(uniformlyconvergent)interaction,thensowouldbeP.Butthiscontradictstheresult(R),becauseGibbsmeasuresforthesame(absolutelysummabletranslation-invariant)interactionhavezerorelativeentropydensity.ThisstatementeasilyfollowsfromSchonmann'sLemma.

Schonmann'srestrictionPdoesnottintotheframeworkconsideredinSection,becausethevolumecompressionfactorKisnotnite(seeExampleinSection.).Ontheotherhand,Schonmann'sproofofnon-GibbsiannessseemstoberatherdierentfromourproofsinSections.{..Weshowherethat,nevertheless,hisresultcanbeobtainedbyfollowingbasicallythestepsdiscussedinSections.{.(althoughatpresentweareabletodoitonlyfortemperatureslowenough).Thiswillprovethat+Pisnotmerelynon-Gibbsian,butinfactnon-quasilocal.Theproofwilluse(R)butnot(R).Inourlanguage,theimagespinsforthistransformationarethespinsonthehori-zontalline,andtheinternalspinsareallthespinsoftheplaneexceptthoseoftheline.WerstnoticethatSchonmann'sresult(R)correspondsexactlytoourStep:thatis,(. )showsthattheannulus[N;n][[n;N]ofimagespinsselectsthephaseoftheinternalspins.Physically,thisisakindofwettingphenomenon:imposingspinsonalargesegmentoftheaxis(ofsizeN)giverisetoadropletofthephaseinaneighborhoodoftheaxis,evenwhenthebulkboundaryconditionsare+;asN!thewidthofthedropletgrowstoinnity,andmoreovertheleftandrightdropletsjoin,therebyenforcingthephasethroughouttheinnitesystem.WesketchnowhowourStepcanbeprovenviaacontourargument,sothatweobtainthenon-quasilocalityoftheimagesystemwithoutmakinguseofthelarge-deviationestimate(R).Weconsidertheoriginunxedfromthestart(soStepissuperuous),andconsider!0specialtobeanalternatingcongurationsuchthattheneighborsoftheoriginareofoppositesign:(!0special)(i;0)=(()iifi>0()i+ifi<0:(:)Weshallprovethefollowing:thereexists>0suchthatforallkthereexistn(k)andN(k)suchthat+(0j0;k=!0special;0n(k);N(k)=+)>0(.a)+(0j0;k=!0special;0n(k);N(k)=)<0(.b)Itisclearthat(.a)and(.b)togetherimplythenon-quasilocalityof+P,fortheyshowthatinanarbitrarilysmallneighborhoodof!0specialf;gZ(namely,Nkf0:0;k=!0specialg),thereexistopensubsetsNk;+=f0:0;k=!0specialand0n(k);N(k)=+g(.a)Nk;=f0:0;k=!0specialand0n(k);N(k)=g(.b)suchthatthe(+P)-averagevalueofE+P(00jf0xgx=0)overNk;+(resp.Nk;)is(resp.=).ThisisincompatiblewithE+P(00jf0xgx=0)havinganycontinuous(quasilocal)version.

Inordertoprove(.a)and(.b),weshallprovethefollowingintermediateresult:thereexists>0suchthat+(0j0;k=!0special)(:)forallk.Thistriviallyimplies(.a),bytheFKGinequality,foranychoiceofnandN.Toseethatitalsoimplies(.b),weuse(.)with=+andappliedtothefunctionsfk=(0;k=!0special)(.a)gk=0(0;k=!0special)(.b)Weobtainlimn!+(fkj0n;N(n)=)=(fk)(.a)limn!+(gkj0n;N(n)=)=(gk)(.b)Dividing(.b)by(.a)wegetlimn!+(0j0;k=!0special;0n;N(n)=)=(0j0;k=!0special):(:)Now,by(.)andspin-ipsymmetry,theRHSis.0Therefore,foreachkthereexistsann(k)suchthat+(0j0;k=!0special;0n(k);N(n(k))=);(:)whichis(.b).Sonowletusprove(.)|atlowenoughtemperatures|byamore-or-lessstandardPeierlsargument[].Herethecontoursaredenedastheboundaries(intheduallattice)ofregionswherethespinsdierfromtheground-stateconguration(thatis,all\+"exceptfortherequiredalternating\").Incountingtheenergyofsuchcontoursonemustsubtracttheenergyofthecontoursalreadyexistinginthegroundstate(squaressurroundingthealternating\").Aftersomethought,oneconcludesthattheenergyofthecontoursisatleastproportionaltoNv+Nh,whereNv(resp.Nh)isthethenumberofvertical(resp.horizontal)bondsinthecontour.Ontheotherhand,thenumberofpossiblecontoursisevenlessthanthatfortheunconditionedIsingmodel.AsinthestandardPeierlsargument,thesefactsimplythattheprobabilityofndingacontoursurroundingtheorigin|thatis,ofhavinga\"attheorigin|goestozeroasgoestoinnity.0Ifweapplyspin-ipsymmetryto(. ),wenotonlychange+toandto,butmustalsochange!0specialto!0special.Butthislatterisjust!0specialreectedinthex-axis(i.e.x!x),andthemeasures+andareinvariantunderthisreection.

Remarks..IncontrasttotheRGexamplesgiveninSections.{.,herethenon-Gibbsiannessoccursonlyforinteractionsontherst-orderphase-transitioncurve,i.e.zeromagneticeld.Indeed,MaesandvandeVelde[]haveproventhatifeitherh=0orissucientlysmall,therestrictionofthetwo-dimensionalIsingmodeltoanaxisisGibbsian..Itisnaturaltogeneralizethisexample:considerad-dimensionalIsingmodelandad0-dimensionalcoordinateplane(d00areparameters;hereN0(n)[resp.N(n)]isthenumberofbondsbwithnb=0[resp.nb=],andC(n)isthenumberof\clusters"(i.e.connectedcomponentsofvertices)inthegraphGnwhosevertexsetisVandwhoseedgesaretheoccupied(nb=)bonds.Forq=thismodelreducestoordinary(independent)bondpercolation,whileforintegerqthereareidentitiesrelatingtherandom-clustermodeltotheq-statePottsmodel[,,0].Letusnowtrytoformulatetherandom-clustermodelonacountablyinnitegraphG=(V;B)[forexample,V=ZdandB=nearest-neighborbondsinZd],followingtheDLRapproach.The\lattice"ishereB,andthecongurationspaceisf0;gB.LetbeanitesubsetofB,andletVbethesetofallverticestouchingatleastonebondb.Weneedtospecifytheconditionalprobabilitiesoffnbgbgivenfnb0gb0Bn.Butthisiseasy,bythesamemethodasforspinsystems:wewritedowntheformal(meaningless)Boltzmannfactorfortheinnitelattice,andthendropalltermsthatdon'tinvolvefnbgb.Theresultissimple:itisProb(fnbgbjfnb0gb0Bn)=const(fnb0gb0Bn)pN(n)(p)N0(n)qC(n);(:)whereN0(n)[resp.N(n)]isthenumberofbondsbwithnb=0[resp.nb=],whileC(n)isthenumberofclusterscontainingatleastoneelementof,inthegraphwhoseedgesaretheoccupied(nb=)bonds(boththoseinsideandoutside).Itiseasytoseethat(. )denesaspecication(i.e.itisconsistentfordierent).Itisalsoeasytoseethatthedependenceonfnb0gb0Bnisonlyviathesetofanswerstothefollowingquestions:foreachpairx;y,onewantstoknowwhetherxandycanbeconnectedbyapathofoccupiedbondslyinginBn.Note,however,thattheanswertothisquestioncoulddependonbondsnb0arbitrarilyfarawayfrom

(providedthatthegraphGcontainsarbitrarilylargeclosedloops).Therefore,forq=,thespecicationdenedby(.)isnotquasilocal(aswaspreviouslynotedin[]).Aizenman,Chayes,ChayesandNewman[]haveproventheexistenceoftheinnite-volumelimitforthe\Gibbs"measuresoftherandom-clustermodeltakenwitheitherfree(nb00)orwired(nb0)boundaryconditions.However,sincethespeci-cation(.)isnotquasilocal(hencenotFeller),itisnotimmediatethattheselimitingmeasuresfandwareindeedconsistentwiththespecication(.)[sinceProposi-tion.doesnotapply],althoughitseemsveryplausible.Indeed,itisnotclearthatthereexistanymeasuresconsistentwiththespecication(.).Wethereforeposethefollowingopenquestion:Provethattheinnite-volumelimitmeasurestakenwithfreeorwiredboundaryconditionsareconsistentwiththespecication(.).Assumingthattheredoexistmeasuresconsistentwiththespecication(.),wecannowprovethatallthesemeasuresarenon-quasilocal(hencenon-Gibbsian).Denition.Letbeametricspace.Wecallafunctionf:!Rstronglydiscontinuousifeverycontinuousfunctiondiersfromfonasethavingnonemptyinterior.[Indetail:foreverygC(),thesetf!:f(!)=g(!)ghasnonemptyinterior.]Wecallaspecicationstronglynon-FellerifthereexistsSandfC()suchthatfisstronglydiscontinuous.Asucientconditionforstrongdiscontinuityofafunctionfisthefollowing:thereexistsan!andan>0suchthatforeveryneighborhoodN!thereexistopensetsN+;NNsuchthatinf!N+f(!)sup!Nf(!).Itisnoweasytoprovethatthespecication(.)isstronglynon-Feller.Toavoiduninterestinggraph-theoreticcomplexities,weprovethetheoremforthespecialcaseV=ZdandB=nearest-neighborbondsinZd.ThereadercaneasilygeneralizethistoasuitableclassofcountablyinnitegraphsG.Proposition.Letq=.Thenthespecication(.)fortherandom-clustermodelisstronglynon-Feller,whenV=ZdandB=nearest-neighborbondsinZd.Proof.Letbeasetcontainingasinglebondb0=fx0;xg,andletf(n)=nb0.Nowlet!bethecongurationwhichsetsnb=onparallelraysrunningfromx0andxtoinnity,perpendiculartothebondb0,andwhichsetsnb=0onallotherbonds.NowanyneighborhoodN! (intheproducttopology)containstheparticularneighborhoodNR=fn:n=!onRg;(:)whereRisthesetofallbondsinasquareofsideR+centeredattheorigin.WethenchooseNR;+tobethesubsetofNRinwhichanoccupiedbondinR+nRconnectsthetwoparallelrays;andwechooseNR;tobethesubsetofNRinwhichall

thebondsinR+nRarevacant(sothatthetwoparallelrayscannotbeconnected,nomatterwhathappensoutsideR+).Itiseasytoseethat(fb0gf)(!)(pforall!NR;+pp+(p)qforall!NR;(:)forallR.Since00),weconclude:Corollary.Letq=,andletbeanymeasureconsistentwiththeFKspeci-cation(.)[forV=ZdandB=nearest-neighborbondsinZd].ThenisnotconsistentwithanyFeller(quasilocal)specication.Wenotethatthemethodusedheretoprovenon-quasilocalityisessentiallythesameasthatusedinSections.{.ontheRGexamples.Theonlydierenceisthathereweareworkingwithanexplicitspecication,sothatwecanproveboundsoverthewholesetsN+andN;whereasinSections. {.wewereworkingwiththeconditionalprobabilitiesofagivenmeasure0,whicharedenedonlyuptomodicationon0-nullsets,andthereforewecouldonlyprovetheboundsoverN+andNinthe0-a.e.sense.Finally,weremarkthatforintegerq,thereexistsajointmodelofinteractingPottsspinsandbondoccupationvariables|thatis,amodelwhosestatespaceisf;:::;qgVf0;gB|whosemarginalsonthespinandbondvariablesarethePotts

andrandom-clustermodels,respectively[0].Thisjointmodelhaslocalinteractions,soitsspecicationobviouslyquasilocal.(Theonlyreasonitisn'tGibbsianisthattherearesomeexclusions.)Theidentitiesrelatingthejoint,Pottsandrandom-clustermodelsareeasilyproveninnitevolume,buttheycanpresumablybemaderigorousininnitevolumebymethodslikethosesketchedinSection.,Step0.Ifso,thenanyGibbsmeasureofthejointmodelwouldproduce,upon\decimation"tothebondvariables,anon-quasilocalmeasure(namely,ameasureconsistentwiththerandom-cluster-modelspecication).Thiswouldthenbeanotherexampleinwhich\decimation"ofaquasilocalmeasureyieldsanon-quasilocalmeasure...StationaryMeasuresinNonequilibriumStatisticalMechanicsConsideraninnite-volumelatticesystemevolvingstochastically,ineithercontinuoustimeordiscretetime,accordingto(quasi)localruleswhichdonotsatisfydetailedbalance.Thus,incontinuoustimewehaveinmindaninteractingparticlesystem[]:forexample,asystemofthespin-ip(resp.spin-exchange)type,inwhicheachspinips(resp.eachnearest-neighborpairofspinsexchangesvalues)independently,atPoissonrandomtimes,withratesdependingina(quasi)localwayontheotherspins.Examplesofsuchdynamicsinclude:(a)Thevotermodel[]:independentlyateachsitex,atPoissonrandomtimesthespin(\voter")atxchangesitsvaluetothatofarandomlychosenneighbor.(b)AnIsingmodelwithcompetingdynamics:forexample,amixtureofGlauberdy-namicsfortwodierenttemperatures[],oramixtureofGlauberdynamicsforonetemperatureandKawasakidynamicsforadierenttemperature[].(ThelattermodelhasbeenconsideredbyLebowitzandhiscollaboratorsinconnectionwiththehydrodynamiclimit[]. )Indiscretetimewehaveinmindaprobabilisticcellularautomaton(PCA)[,0]:simultaneouslyateachclocktick,eachspinattemptsindependentlytoip,againwithratesdependingina(quasi)localwayontheotherspins.Anexampleis:(c)TheToommodel[,0]:eachspinchangesitsvalue,withprobabilityp,tothemajorityofitsnorthernneighbor,itseasternneighbor,anditself,andwithprobabilities(p)=to.Thus,thePCAsarethediscrete-timeanalogueofthespin-ipinteractingparticlesystems.LebowitzandSchonmann[,p.0]havearguedthatinboththecontinuous-timeanddiscrete-timecases,thestationarymeasure(s)shouldgenerallybeexpectedtobenon-Gibbsianandindeednon-quasilocal:for\systemsmaintainedinanonequilibriumThiswouldprobablyalsogiveamethodforprovingthatfandwareconsistentwiththerandom-cluster-modelspecication,atleastforintegerq.

statebycontactswithoutsidesources:::[themeasuresdescribing]stationarynon-equilibriumstatescannotbeexpectedtobehaveinaquasi-Markovian[inourlanguage,quasilocal]way|isolatingapart[ofthesystemfromtherest]willgenerallychangeitsbehaviordrastically."ThisconjecturehasbeenprovenbyLebowitzandSchonmann[]inthecaseofthevotermodel.Moreprecisely,theyhaveproven[,equation(.)]thati(+j)=0where(0<<)isanextremaltranslation-invariantstationarymeasureofthevotermodelinZd(d).Thisshowsthatisnon-Gibbsian(asremarkedalsoin[0]).Itisinterestingtonotethatthisisthesamelarge-deviationsargumentemployedintheLebowitz-Maes-Dorlas-vanEnterexamples(Section.).MartinelliandScoppola[]havegivenanotherexampleofadynamicsinwhichthestationarymeasureisnon-Gibbsian:againtheprobabilityofaregioninwhichallthespinsare+decaysmoreslowlythanexponentiallyinthevolumeoftheregion,sothemeasurecannotbeGibbsian.However,theMartinelli-Scoppoladynamicsishighlynon-local|itinvolvescollectiveipsofarbitrarilylargeclusters|soperhapsthenon-Gibbsiannessisnotsosurprising.(TheMartinelli-ScoppoladynamicssuperciallyresemblestheSwendsen-Wang[0]dynamics;butintruththeresemblanceisnotsoclose,sincethestationarymeasureoftheformerisnon-Gibbsian,whilethestationarymeasureofthelatteristhenearest-neighborIsingmodel!)Finally,MaesandRedig[]havedescribedan(anisotropic)localspin-exchangedynamicsinwhichthestationarymeasureisexpectedtohavenon-summablelong-rangecorrelationsinthe\high-noise"regime(i.e.atwhatoughttocorrespondto\hightemperature").Suchunusualbehaviorwouldsuggest,thoughitwouldnotprove,thatthestationarymeasureisnon-Gibbsian.Thelong-rangespatialcorrelationsareindicatedinthismodelbyaperturbationcalculation,butamoregeneralphysicalintuitionseemstobethefollowing:Transportpropertiesforspin-exchangeprocessesarediusive,andthecorrelationfunctionsareexpectedtoexhibitslow(power-law)decayintime(\long-timetails").Now,oneexpectsspatialandtemporalcorrelationstohaveroughlysimilardecay|i.e.bothexponentialorbothpower-law|exceptinveryspecialcasessuchasmodelssatisfyingdetailedbalance.Thissuggeststhatspin-exchangeprocessesnotsatisfyingdetailedbalanceshouldhave,quitegenerally,stationarymeasureswithlong-rangecorrelations,andverylikely,stationarymeasuresthatarenon-Gibbsian.InthePCAmodels,theprobabilitymeasureonthespace-timehistoriesistheGibbsmeasurefora(d+)-dimensionallatticemodelwithinteractionswhichcanbeexpressedintermsofthetransitionrulesofthePCAmodel[,0].Thestation-arymeasureofthePCAmodelthuscorrespondstotherestrictionofthisspace-timemeasuretoad-dimensional(equal-time)hyperplane.WhenthePCAisinthe\high-noise"regime|sothattheassociated(d+)-dimensionalequilibriummodelisintheDobrushin-Shlosmanhigh-temperatureregime|thestationarymeasureisknowntobeuniqueandGibbsian[0].(Asimilartheoremhasrecentlybeenprovenalsoforcontinuous-timespin-ipsystems[].)However,byanalogywiththeSchonmannexample(Section.. ),onemaysuspectthatinthe\non-ergodic"(phase-transition)

regimeofthePCAmodel|wherethestationarymeasureisnotunique|eachsta-tionarymeasurewouldtypicallybenon-Gibbsian.Inparticular,onemayconjecturethatthisissofortheToommodel.WethussuspectthatLiggett'sconjecture[,p.],totheeectthateverytranslation-invariantnite-rangedynamicswithstrictlypositiverateshasaGibbsianstationarymeasure,ismostlikelyfalse.Remark.Theforegoingconsiderationsareforratesthatdonotsatisfydetailedbalance.Iftheratessatisfydetailedbalance,thenoneexpectsallthestationarymea-surestobeGibbsian(foranexplicitGibbsianspecicationthatiseasytowritedowngiventherates);however,thishasnotyetbeenprovenrigorouslyevenintheGlauberdynamicsforthenearest-neighborIsingmodelindimensiond[,ProblemIV..].Finally,letusquotearesultofKunsch[]forcontinuous-timelocalspin-ipprocesseswithstrictlypositiverates:ifthereexistsatranslation-invariantstationarymeasurewhichisGibbsianforsome(absolutelysummable)interaction,theneveryothertranslation-invariantstationarymeasuremustbeGibbsianforthesameinterac-tion...ComparisonofMethodsforProvingNon-GibbsiannessAnytheoremoftheform\everyGibbsmeasurehasthepropertyP"providesamethodforprovingnon-Gibbsiannessviathecontrapositive:ameasurenothavingthepropertyPmustbenon-Gibbsian.Wehaveseenfourpropertiesofthissort:(i)AGibbsmeasure(foranabsolutelysummableinteraction)mustbeuniformlynon-null.Thisisaconsequenceofthe\easyhalf"oftheGibbsrepresentationtheorem[Theorem.(a)=)(b)].(ii)AGibbsmeasure(forauniformlyconvergentinteraction)mustbequasilocal[Theorem.0].(iii)AmeasurecanbeGibbsianforatmostone(uniformlyconvergent,continuous)interaction,upto\physicalequivalence"[Corollary.].(iv)Translation-invariantGibbsmeasures(fortranslation-invariantabsolutelysummableinteractions)have\good"large-deviationproperties:theprobabilitythatspinsinacertainregionuctuateintoacongurationcharacteristicofanothertranslation-invariantmeasuredecreasesexponentiallyinthevolumeoftheregion,exceptifthisothermeasureisalsoGibbsianforthesame(absolutelysummable)interac-tion.Inprecisemathematicalterms:atranslation-invariantmeasurehaszerorelativeentropydensityrespecttoanothertranslation-invariantmeasurewhichisGibbsianforaninteraction,ifandonlyifisalsoGibbsianforthesameinteraction.ThisisoneoftheconsequencesofthediscussionofSection...Itisalsocloselyrelatedtothestrictconvexityofthepressure[Proposition. ].Foreachoftheseconditions,wehaveseenexamplesinwhichthenon-Gibbsiannessisprovenbyitsviolation:

(i)Lackofuniformnonnullness.Thishastwomanifestations:Ameasurecanbenonnullbutnotuniformlyso(seeDenition.).ThistypicallymeansthattheHamiltoniansareunboundedfunctionsandonecannotusetheformalismdevelopedforabsolutelysummableinteractions.Thisisthegenericsituationforunbounded-spinmodels,anditgetsdelicateforinnite-rangeinteractions.Inthesecases,oftenthenotionofGibbsiannesscanbepreservedifoneexcludes\byhand"problematiccongurations[,0].Ontheotherhand,themeasuremayfailtobenonnull,whichmeansthatsomecylindersetshavezeromeasure.Thisisarathersimplecaseofnon-GibbsiannessinwhichtheGibbsiannesscanberestoredbyallowinghard-coreinteractionsorworkingonamorerestrictedcongurationspace(seeforexample[]).Wementionthat,inthesettingofcomplexinteractions,thereareexamplesofGibbsianmeasuresthatafteronerenormalizationstepremainquasilocalbutlosenonnullness[].(ii)Violationofquasilocality.Mostofthecasesofpathologicalrenormalizationtransformationsanalyzedabove(Sections.{.and..)fallintothiscategory.Thisphenomenonappearswhentherearesome\hiddenspins"thattransmitinformationfromarbitrarilyfarawayevenifthe\non-hidden"spinsarexed.Intherenormal-izationtransformationsthe\hiddenvariables"aretheuctuationsoftheoriginalorinternalspinsthatremainoncetheblockspinsarexed.InSchonmann'sexample(Section..),the\hiddenvariables"areallthespinsoutsidethex-axis,whichare\hidden"bytheprocessofrestriction.(iii)Threatenedviolationofuniqueness.Weusedthismethodtostudythe\trivial"examplesofnon-GibbsiannessdiscussedinSection...Consideraniteorcountablefamilyofdierent(non-physically-equivalent)interactionsandpickforeachoneaner-godictranslation-invariantGibbsmeasure.ThenanontrivialconvexcombinationofthesemeasurescannotbeGibbsianforany(uniformlyconvergent,continuous)interac-tion,becauseifitwere,theneachoftheoriginalmeasureswouldbeaGibbsmeasurealsoforthisnewinteraction,violatinguniqueness.Inthecaseofanitesingle-spinspace,thismethodalsoprovesnon-quasilocality.(iv)Wronglargedeviationproperties.Thereseemtobetworatherdierenttypesof\bad"large-deviationproperties:()Sub-exponentialdecayforeventswhoseprobability\should"decayexponentiallyinthevolume.Thisappliestothesigneldofthe(an)harmoniccrystal(Section.),andthestationarymeasuresforthevoterandMartinelli-ScoppolamodelsmentionedinSection...Hereoneshowsthattheprobabilityofallthespinsinalargeregionbecoming+decayssub-exponentiallyinthevolumeoftheregion;thisisincompatiblewithbeingGibbsianforanyabsolutelysummableinteraction.Inotherwords,oneshowsthatthemeasuresatisesi(+j)=0,where+isthedelta-measureconcentratedontheall-+conguration.Asthismeasureisobviouslynon-Gibbsian(itisnotnonnull!),neitheris. ()Exponentialdecayforeventswhoseprobability\should"decaysub-exponentially.TheoriginalproofofSchonmann'sexample[]isbasedonanargumentofthis0

kind.Hereoneshowsthat,inthe+phase,theprobabilityofhavinganetnegativemagnetizationinalargeregiondecaysexponentiallyinthevolumeoftheregion.Inotherwords,themeasuresobtainedvia\+"and\"boundaryconditionshaveastrictlypositiverelativeentropy.IfeitherofthesemeasureswereGibbsian(foranabsolutelysummableinteraction),theotherwouldhavetobeGibbsianforthesameinteraction(becausetheydieronlybyboundaryconditions);butthentherelativeentropywouldhavetobezero(Theorem.).Therefore,theycannotbeGibbsian.Oftenwewouldliketoprovenotonlythatameasureisnon-Gibbsian,butalsothatitisnon-quasilocal(whichisstronger).Innearlyallcaseswehavedonethis\byhand":thatis,byprovingboundsontheconditionalprobabilitieswhichareincompatiblewiththeirhavinganyquasilocalversion(seeSections.{.and..).Inonlyonecasewereweabletoprovenon-quasilocalitybyanabstract\trick":thiswasthe\trivial"convex-combinationexample(Section..),whereweusedmethod(iii)above.Itwouldbeinterestingtohaveavailableothermethodsforprovingnon-quasilocality...Are\Most"MeasuresNon-Gibbsian?Thetraditionalbeliefamongphysicists(includingourselvesuntilrecently)isthatall(ornearlyall)physicallyinterestingmeasuresareGibbsian.Indeed,thisbeliefissomuchtakenforgrantedthatitisrarelystatedexplicitly.TheprofoundmessageofIsrael'spioneeringwork[0],andoftheexamplesgivenhere,isthatthistraditionalbeliefisfalse:manyphysicallyinterestingmeasuresarenon-Gibbsian.Infact,wenowsuspectthatGibbsiannessshouldbeconsideredtobetheexceptionratherthantherule|that,insomesense,mostmeasuresarenon-Gibbsian.ItisthereforeofatleastmathematicalinteresttostudythesetGSBGinv()ofalltranslation-invariantmeasureswhichareGibbsianforsometranslation-invariantabsolutelysummablecontinuousinteraction.IsGa\big"ora\small"subsetofthespaceM+;inv()ofalltranslation-invariantmeasures?Therearemanyexamplesofthisinthephysicsliterature:see,forexample,[,].Oneexceptionistherecentstatementbyanotedmathematicalphysicistthat\everygoodran-domeldisGibbsian"[].Inasimilarvein,amathematiciansays:\theGibbsianformoflocalconditionaldistributionsisaratherweakcondition,butitisdiculttocheckit."[,p.0]Arelatedthoughsomewhatweakerintuitioncanbefoundinawell-knownmonographoninteractingparticlesystems:\IsittruethateverytranslationinvariantstrictlypositivespinsystemonZdwithniterangehasaninvariantmeasurewhichisaGibbsstate?Thisisplausible:::[because]thestrictpositivityoftheratesshouldimplythataninvariantmeasureissomewhatsmooth."[,p.]Onthissameconjecture,anothermathematiciansays:\Wecouldn'tproveingeneraltheexistenceofastationaryGibbsmeasure,althoughthisisverylikelytohold."[,p.0]AsdiscussedinSection..,thisconjectureisstillanopenproblem,butthereisgoodreasontosuspectthatitisfalse.(Theseexamples,togetherwiththoseoftheprecedingfootnote,illustratethedierencebetweenphysicistsandmathematicians:bothoftenhaveerroneousintuitions,butthemathematiciansstatethemexplicitly.)

Itisa\big"setinaveryweaksense,namelythatofbeingdenseintheweaktopol-ogy.Infact,theGibbsmeasuresfornite-rangecontinuousinteractionsarealreadydense:Proposition.Assumethatthesingle-spinspace0isacompactmetricspace,andthattheapriorisingle-spinmeasure0xgivesnonzeromeasuretoeveryopensetof0.ThenGfinite[BniteGinv()isdenseinM+;inv()intheweaktopology.Proof.Theproofgoesinthreesteps:First,theergodicmeasuresofniteentropydensity(relativeto0)aredenseinM+;inv()[Proposition.(e)].Secondly,Israel[0]hasshown,usingtheBishop-Phelpstheorem,thateachergodicmeasureofniteentropydensityisan(extremal)equilibriummeasureforsomeinteractionB0(seeiteminSection..).Finally,thenite-rangeinteractionsformadensesubsetBniteB0;anditfollowsfromatheoremofLanfordandRobinson[](seealsoSokal[])thateveryextremalequilibriummeasureforB0canbeapproximatedintheweaktopologybyequilibriummeasuresforinteractionsnBnitewithknkB0!0.Weemphasizethatdensityintheweaktopologyisanextremelyweakproperty:itmeansonlythatanarbitrarymeasureM+;inv()canbeapproximatedarbitraryclosely,withregardtoanynitefamilyoflocalobservables,byameasureinGfinite.Inparticular,thelong-range-orderpropertiesoftheapproximatingmeasurescanbetotallydierentfromthoseofthelimitingmeasure.Thus,Proposition.isveryfarfromsayingthat\most"measuresareGibbsian.InamoreprofoundsenseweexpectthatGisinfactarather\small"subsetofM+;inv().Forexample,weconjecture:Conjecture.(a)GisasetofrstBairecategoryinM+;inv().[Thatis,GisacountableunionofsetswhicharenowheredenseinM+;inv().](b)G\exM+;inv()isasetofrstBairecategoryinexM+;inv().[Here\ex"denotestheextremepoints,i.e.theergodicmeasures.]FirstBairecategoryisaclassicnotionof\smallness"intopology[].WecanmakesomesmallstepstowardprovingConjecture. (a):Proposition.0Ghasemptyinterior.Proposition.Assumethatthesingle-spinspace0isacompactmetricspace,andthattheapriorisingle-spinmeasure0xgivesnonzeromeasuretoeveryopensetof0.LetSbeacompactsubsetofB0,andletESbethesetofequilibriummeasuresforinteractionsinS.ThenESisacompactsubsetofM+;inv().

Corollary.Assumethatthesingle-spinspace0isacompactmetricspace,andthattheapriorisingle-spinmeasure0xgivesnonzeromeasuretoeveryopensetof0.IfSisa-compactsubsetofB0,withSB,thenES=GSSSGinv()is-compactandofrstBairecategoryinM+;inv().Inparticular,thisoccursforS=Bhwithh.ProofofProposition.0.LetGandM+;invnG.Then,byProposition.(b),()+=Gfor0<.But()+!weaklyas#0.HenceGcannotcontainanyopenneighborhoodof.[InthisproofwecouldequallywellhavetakentobeaGibbsmeasureforaninteractionnotphysicallyequivalenttotheoneforwhichisGibbsian,andthenapplyProposition.(b)andtheGriths-Ruelletheorem.]ProofofProposition..M+;inv()iscompact,soweneedonlyshowthatESisclosed.LetnbeanequilibriummeasurefornS,withn!weakly.Then,sinceSiscompact,thereexistsasubsequencenithatconverges(inB0norm)tosomeS.Butthenisanequilibriummeasurefor.ProofofProposition..TherststatementisanimmediateconsequenceofPropositions.0and..ThesecondstatementfollowsfromProposition. (b).WethankS.R.S.Varadhanforsuggestingtheselatterresultsandsketchingtheproofs.Discussion.NumericallyObservedDiscontinuitiesoftheRGMap..StatementoftheProblemInseveralMonteCarlorenormalizationgroup(MCRG)studies[,,,],ithasbeenfoundthatthenumericallycomputedrenormalizationtransformationR:H!H0isdiscontinuousatarst-orderphase-transitionsurface.However,thisbehaviorisThemodelsinwhichthisbehaviorhasbeen(atleasttentatively)observedincludethetwo-dimensionalIsingmodelatlowtemperature[],the0-statePottsmodelintwodimensions[],the-statePottsmodelinthreedimensions[],theZlatticegaugetheoryinfourdimensions[]andtheU()latticegaugetheoryinfourdimensions[,].However,inamorerecentstudyofthetwo-dimensionalIsingmodelatlowtemperature[],theobserveddiscontinuitywasalwayslessthantheestimatedtruncationerror,anditdecreasedasmoretermswereincludedintherenormalizedHamiltonian;thiswasinterpretedasevidenceagainstadiscontinuityintheexactrenormalizationmap.

rigorouslyexcludedbyourSecondFundamentalTheorem(Theorem.).Inthissec-tionwewouldliketooerourinterpretationofthenumericallyobserveddiscontinuities.AMCRGstudy[,]proceedsasfollows:WechooseanoriginalHamiltonianH,andgeneratealongsequenceofrandomsamples!;!;:::fromtheGibbsmeasure=consteHusingsomeMonteCarloprocedure.Oneachofthese\original-spin"congurations!iweapplytherenormalizationmapTtogeneratethecorrespondingblock-spinconguration!0i.Inthiswaywehavegeneratedarandomsample!0;!0;:::fromtherenormalizedmeasure0=T.Itisnowassumedthat0istheGibbsmeasureforsomerenormalizedHamiltonianH0belongingtoaxednite-parameterfamilyH(;:::;N),andsomestatisticalmethod[,,]isemployedtoestimatetheunknownparameters;:::;N.Suchaprocedurehasthreesourcesoferror:)StatisticalerrorarisingfromtheniteMonteCarlosample.)Systematicerrorarisingfromthenitelatticesize.(Wetakethepointofviewthatourgoalistolearnaboutthebehavioroftheinnite-volumesystem.))SystematicerrorarisingfromtruncationoftherenormalizedHamiltonian:0maynotbe(infact,inalmostallcasesisnot)aGibbsmeasureforanyHamiltonianintheassumedN-parameterfamily.Weincludeherethepossibility|studiedindetailinSection|that0isnottheGibbsmeasureforanyreasonableHamiltonian.Itisusefultostudythesethreesourcesoferrorseparately.Inparticular,wewouldliketostudytheproblemoftruncationoftherenormalizedHamiltonian,independentlyoftheproblemsofstatisticalandnite-sizeerrors.Therefore,webeginbyformulatinganidealizedmodeloftheparameter-estimationprobleminwhichweassumethattheexperimenterknowsexactlytheexpectationvaluesofanappropriatesetofobservables(tobespeciedlater)intheinnite-volumerenormalizedmeasure0.Thisidealizedsituationcanbeapproximatedtoarbitraryaccuracywithsucientcomputertime,bymakinglongMonteCarlorunsonlargesystems.(Inprincipleweshouldthendiscussthestabilityofourtheoryrelativetosmallstatisticalornite-sizeerrors.Butwefeelthatourconsiderationsarestilltoopreliminarytojustifyenteringintosuchtechnicalities. )..AnIdealizedModelofParameterEstimationLetusrstconsidertheparameter-estimationprobleminageneralprobabilistic(=statistical-mechanical)context,withoutregard(forthemoment)totherenormalization-groupapplication.Let,therefore,(;F)beanarbitrarymeasurablespace,letFbesomefamilyofprobabilitymeasureson(;F),andletbeanotherprobabilitymea-sureon(;F).WewishtondthemeasureinFwhichisinsomesense\closestto"(or\bestapproximates").Howshouldwedene\closeness"?Anydenitionis,ofcourse,somewhatarbitrary,butweclaimthatthefollowingdenitionisverynatural:

ThemeasureinFclosestto,denotedF,istheonewhichminimizestherelativeentropyI(j),assumingthataminimizerwithniterelativeentropyexists(itmayormaynotbeunique).Notethattheunknownmeasureistakenhereasthereferencemeasure(secondar-gument)intherelativeentropy.Insupportofthisdenition,wecitethefollowingproperties:)IfF,thenF=,uniquely.Thisisarathertrivialproperty,butitisatleastanecessaryconditionforanyreasonabledenitionof\closeness".)Supposethatonegeneratesalargerandomsamplefrom,andconstructsmaximum-likelihoodestimates[]basedonthe(false)assumptionthatthesam-plearosefromsomemeasureinF.Inthelarge-samplelimit,thismaximum-likelihoodestimatewillconvergetoF.Thiscanbeprovenundersuitabletechnicalhypotheses[],butitiseasytoseeintuitivelywhyitistrue:therelativeentropyI(j)is,uptoanadditiveconstant,preciselyminusthemean(underthetruemeasure)oftheloglikelihoodfunction:I(j)=Zdlogdd=constZdlog\d";(.)somaximizingthelikelihoodisequivalenttominimizingtherelativeentropy.Thus,Fistheestimatethatwouldbegeneratedbyanexperimenterpossessinganinniterandomsamplefromandusingtheoptimalestimationmethod(namely,maximumlikelihood).(Notealsothatthemaximum-likelihoodestimateforanitesample!;:::;!nisthemeasureinFclosesttotheempiricalmeasureLnnPni=!iforthegivensample.Thiscloserelationbetweenmaximum-likelihoodestimationandrelativeentropyhasbeennoticedbypreviousauthors.)SupposenowthatthesetFisoftheBoltzmann-Gibbsform(=exponentialfamily)F=(Z()exp"NXi=iHi#0:RN)(:)forsomespeciedfamilyH;:::;HNandapriorimeasure0.Wecanassumewithoutlossofgeneralitythatthefunctions;H;:::;HNarelinearlyindependent(0-a.e.).ThisfollowsCencov[].(Note,however,thatCencov'snotationfortheargumentsofI(j)isthereverseofours. )Bycontrast,Csiszar[,]considersthequitedierentprobleminwhichtheunknownmeasureistherstargumentintherelativeentropy.Thisassertionisperhapssomewhatmisleading:Themaximum-likelihoodmethodisoptimalasregardsstatisticalerrors(inthelarge-samplelimit)[],whilehereweareconcernedwiththesystematicerrorsduetotruncation.Indeed,theproblemhereistodenewhatwemeanbythe\optimal"truncation.Inanycase,weclaimthatmaximum-likelihoodestimationisasensibleidealizedmodelofwhatagoodexperimenterwouldactuallydoifhe/shecould.

ThentherelativeentropyI(j)=logZ()+NXi=iZHid+const(:)isastrictlyconvexfunctionof;inparticular,themeasureFisuniqueifitexistsatall,anditisdenedbytheconditionshHii=hHiiforalli=;:::;N:(:)Theforegoingtheoryisadequateforparameterestimationinnite-volumesystems;butforinnite-volumesystemsitisinapplicable,becausetherelativeentropyoftwotranslation-invariantmeasuresisinnearlyallcases+.Theproblemhereisthat,asdiscussedinSection.,therelativeentropyinvolumetypicallygrowsproportionallytothevolume(unlessthetwomeasureshappentobeGibbsmeasuresforthesameinteraction).Thisvolumefactorisuninterestinginthepresentcontext,becauseitdoesnotdependontheparametersoverwhichwewanttooptimize.Therefore,itmakessensetojustdivideoutthisvolumefactorandminimizetherelativeentropydensity.Thatis,ifFM+;inv()andM+;inv(),wedene:ThemeasureinFclosestto,denotedF,istheonewhichminimizestherelativeentropydensityi(j),assumingthattheserelativeentropydensitiesarewell-denedandthataminimizerwithniterelativeentropydensityexists(itmayormaynotbeunique).SupposenowthatFisthesetoftranslation-invariantGibbsmeasuresforinterac-tionsV(andapriorimeasure0),whereV=span(;:::;N)issomespeciednite-dimensionallinearsubspaceofB.Then,from(.)wehavei(j)=p(fj0)+Zfd+i(j0)F()(.)wheneverisaGibbsmeasurefor.Thus,i(j)dependsononlyviatheinteraction;infact,F()ispreciselytheamountbywhichthepair(;)failstosatisfythevariationalprinciple.Therefore,ifoneGibbsmeasureforhappenstominimizei(j),thenallGibbsmeasuresfordoso.SothemeasureinFclosesttomaynotbeunique.Nevertheless,thecorrespondinginteractionisnecessarilyunique(modulophysicalequivalence)ifitexistsatall,becauseFisstrictlyconvexonVBTheminimizerFcouldfailtoexist:Consider,forexample,=f;g,0=(++),=,N=andH(! )=!.Thentheminimumis\at=+";thereisnominimizeratnite.Again,aminimizercouldfailtoexist,iftheminimumis\atinnity".Thisoccurs,forexample,ifisaground-statemeasureforsomeinteractionV:thenF()!0as!+.

(Proposition.).Thisisgood,becauseitisafteralltheinteractionthatwewouldliketoestimate.Now,aninteractionminimizesFVifandonlyiff[V]isatangentfunctionalatftothepressurerestrictedtof[V].[Heref[V]denotestheimageofVunderthemap!f;itisalinearsubspaceofC().]Now,bytheHahn-Banachtheorem,everytangentfunctionaltopf[V]canbeextendedtoatangentfunctionaltop,i.e.toanequilibrium(=Gibbs)measure.ItfollowsthataninteractionminimizesFVifandonlyifthereexistsatranslation-invariantGibbsmeasureforsuchthathfii=hfiiforalli=;:::;N:(:)Whathappensifweconsiderlargerandlargersubspacesofinteractions?LetVV:::beanincreasingsequenceofnite-dimensionallinearsubspacesofB,whoseunionisdenseinB.LetnbetheinteractioninVnthatminimizesFVn.Thenitisnaturaltoconjecturethefollowing:Conjecture.(i)IfisaGibbsmeasureforsomeinteractionB,thenn!inBnormasn!.(ii)IfisnotaGibbsmeasureforanyinteractioninB,thenknkB!asn!.Wearenotabletoprovethismuch(andwesuspectthatitmaynotbetruewithoutadditionalhypotheses).Regardingconjecture(i),whatwecanproveisthefollowing:Proposition.Letbeanergodictranslation-invariantmeasureofniteentropydensity(relativeto0);andletVV:::beanincreasingsequenceofsubsetsofB0,whoseunionisdenseinB0.Then:(a)ThereexistsaninteractionbB0(notnecessarilyinB!)forwhichisanequilibriummeasure.(b)LetbbeanyinteractioninB0forwhichisanequilibriummeasure.ThenthereexistsasequencebnVnwhichconvergestobinB0norm.If,inaddition,bbelongstosomespaceBhB0and[n=VnisdenseinBh(inBhnorm),thenthereexistsasequencebnVnwhichconvergestobinBhnorm.(c)LetbbeanyinteractioninB0forwhichisanequilibriummeasure,andlet(bn)beanysequenceconvergingtobinB0norm.ThenF(bn)!0.[Inparticular,wehavelimn!infVnF()=0.](d)Conversely,let(bn)beanysequenceforwhichF(bn)!0andwhichconvergesinB0norm,saytob.Thenisanequilibriummeasureforb.Inparticular,if(kbnkB)isbounded,thenbBandisaGibbsmeasureforb.TherequiredversionoftheHahn-Banachtheorem[,p.,A..]canbededucedeasilyfromtheseparating-hyperplaneversion([,p.,TheoremII..]or[0,p.,Theorem. (a)])byconsideringepigraphs.

ThisshowsthatifisaGibbsmeasureforB,thenthereexistsasequencebnVnofapproximateminimizerswhichconverges(inBnorm)to.Unfortunately,thereisnoguaranteethattheexactminimizersn(ifsuchexist)convergeto;theymightfailtoconverge,ortheymightconvergeinsteadtosomeinteractionbB0nBforwhichisanequilibrium(butnotGibbs!)measure.Itisanimportantopenproblemtondconditionsunderwhichconjecture(i),orsomethinglikeit,canbeproven.Remark.ItiscertainlypossibleforasequencebnofapproximateminimizersofFtofailtoconvergeevenwhenisaGibbsmeasureforsomeinteractioninB.Consider,forexample,anIsingmodel:take=0=productmeasure,andletbnbeaferromagnetictwo-bodyinteractionndf(n(xy))xy,wherefissomexednonnegativesmoothfunctionwith00.0Thissituationisreminiscentof\mean-eld-like"interactions.However,insuchsituationsoneusuallyexpects(andinsomecasescanprove,asinthepreviousremark)thatthelimitingmeasureisGibbsianforsomeinteractionBwhichhaspickedupamagneticeld.WewishtothankBobGrithsandBobSwendsenforadiscussionofthispoint.

ProofofProposition..(a)isaspecialcaseofatheoremofIsrael[0,TheoremV..(a)].(b)followsfromthedensityof[n=VninB0(orBh).(c)followsfromthe(Lipschitz)continuityofthefunctionFinB0norm.(d)isalsoaconsequenceofthecontinuityofF,sincethehypothesesimplythatF(b)=0.ThelaststatementisaconsequenceofProposition.(a)appliedtoBh=B.ProofofProposition..Supposethat(bn)convergesinB0normtob.ThenF(b)=0,soisanequilibriummeasureforb.Byhypothesisthismeansthatb=Bh,i.e.kbkBh=.NowassumethatkbnkBh!;thenthereisasubsequenceof(bn)onwhichtheBhnormisbounded,saybyM;butbyProposition.(a)thisimpliesthatkbkBhM,acontradiction.Thisprovesthateither(a)or(b)[orboth]mustbetrue.Finally,supposethatthesingle-spinspaceisnite,thath,andthatthereisasubsequenceof(bn)onwhichtheBhnormisbounded,saybyM.ThenbyProposition.(b)thereexistsasub-subsequencewhichconvergesinB0normtosomebwithkbkBhM;andisanequilibriummeasureforb;butthiscontradictsthehypothesisoftheproposition.Remarks..SimilarideasappearintheworkofHugenholtz[]..ApartiallyalternateproofofthesecondhalfofProposition.,whenisergodic(oraniteconvexcombinationofergodicmeasures),goesasfollows:BytheBishop-Phelpstheorem[0,CorollaryV.. ]thereexistsenB0withkenbnkB0CF(bn)suchthatisanequilibriummeasureforen[if=mPi=iiistheergodicdecompositionof,thenC=(minimi)].NowassumethatkbnkBh!,sothatthereisasubsequenceof(bn)onwhichtheBhnormisbounded,saybyM.WehavethenshownthatthereexistinteractionseninB0,arbitrarilyclosetof:kkBhMg,forwhichisanequilibriummeasure.WhenthisBh-balliscompactinB0(i.e.0niteandh),thenitiseasytoshowthatthereexistsaninteractioninthisballforwhichisanequilibriummeasure[we'vedoneitabove,byextractingasub-subsequenceof(bn)convergenttob;forwhatit'sworth,thecorrespondingsub-subsequenceof(en)alsoconvergestob].Unfortunately,iftheBh-ballisnotcompact,wedonotseeanywaytoconcludethatthereexistsaninteractionintheball(oreveninBh)forwhichisanequilibriummeasure.Sothismethodstilldoesnotsucetoproveconjecture(ii)...ApplicationtotheRenormalizationGroupNowletusapplytheseideastotherenormalizationgroup,bytakingtobetherenormalizedmeasure0.Weassumethattheexperimenterusestheschemedescribedintheprevioussectiontoconstructanestimatedrenormalizedinteraction0nVn.Wecontinuetoignorestatisticalandnite-sizeerrors.

Theexpectedbehavioroftheestimates0ndependscriticallyonwhether0isGibbsianornon-Gibbsian.AssumingthevalidityofConjecture.(orsomethinglikeit),wehavethefollowingscenario:Case(i):0isGibbsianfor0B.Thenweexpecttheestimatedrenormalizedinteractions0ntoconvergeinBnormto0.Now,bytheFirstFundamentalTheorem(Theorem.),therenormalizedmeasuresarisingfromdistinctphasesoftheoriginalmodelmustbeGibbsianforthesameinteraction0.Therefore,anyobservedmulti-valuednessoftheRGmapmustdisappearasymptoticallyastheassumedinteractionspaceVngrows.Case(ii):0isnon-Gibbsian.Inthiscaseweexpecttheestimatedrenormalizedinteractions0ntodivergeinBnorm,i.e.k0nkB!.Thisbehaviorisalmostrigorouslyproven(Proposition.).Thisdichotomyprovides,atleastinprinciple,aclearmethodfordistinguishingexperimentallytheGibbsiannessornon-Gibbsiannessoftherenormalizedmeasure0.Whetheritwillworkinpracticeislessclear:theproofsofnon-GibbsiannessinSection(andoftheFundamentalTheoremsinSection)involveextremelyrareeventsinlargevolumes;sothedistinctionbetweenGibbsiannessandnon-Gibbsiannessmightturnouttobevisibleonlywithextremelyhighstatisticsandwhenusinganextremelylargespaceofrenormalizedinteractions(thatis,includinginteractionsinvolvingmanyspinssimultaneously).Ontheotherhand,itisatleastconceivablethatthisdependenceonrareeventsisanartifactoftheproofandnotoftheresult.Itwouldbeinteresting,therefore,toperformahigh-precisionMCRGtest,usingalargespaceofrenormalizedinteractions,tocompareacaseinwhich0isexpectedtobeGibbsian(e.g.thed=Isingmodelatatemperaturenottoofarbelowcritical)withacaseinwhich0isexpectedorproventobenon-Gibbsian(e.g.thed=Isingmodelatlowtemperature).Wearesomewhatpessimisticaboutwhethertheasymptotic(n!)behaviorcanbeseenwithanycurrentlyfeasibleexpenditureofresources,butitcannothurttotry.IntheexistingMCRGstudies,theinteractionspaceVisusuallytakentobequitesmall:typicallydimV<0.CanweexplaintheobserveddiscontinuityoftheRGmapasanartifactofthistruncationtoasmallspaceofinteractions?Inouropiniontheanswerisyes.Noterstthattheestimatedrenormalizedinteraction0is,accordingto(.)/(. ),justaproxyfortherenormalizedexpectationvalueshfi0,V.Theselatterexpectationvaluesare,ofcourse,discontinuousatarst-orderphase-transitionsurface(andmulti-valuedonthatsurface).Thatinitselfdoesnotimplythediscontinuityandmulti-valuednessof0,becausethemapfrominteractionstoexpectationvaluesisitselfdiscontinuousandmulti-valued.However,formostrenor-malizationtransformationsweexpecttherenormalizedexpectationvaluestobemorediscontinuousthantheoriginalexpectationvalues;anditisfarfromclearthatthislargerdiscontinuitycanberealized,simultaneouslyforallobservablesf(V),atanyinteractioninthegivenspaceV.Ifitcannot,thentheRGmaponthespaceofinteractionswillappeartobediscontinuous.Consider,forexample,anIsingmodelath=0and>c,andusethemajority-ruletransformation.ThentherenormalizedmagnetizationM0willundoubtedlybe0

largerthantheoriginalmagnetizationM=M(;0+),sinceminoritiestendtogetoutvoted.Nowsupposethatoneisusing,asintheworkofDecker,HasenfratzandHasenfratz[,Section],onlyasinglerenormalizedcouplingh0,with0xedtoequal.(Thatis,Visaone-dimensionalanesubspace.)Thenonewillinevitablyndarenormalizedcouplingh0>0fortheimagemeasure0+(resp.h0<0for0),sinceonlyinthiswaycanoneaccountforarenormalizedmagnetizationM0=M(;h0)>M.Deckeretal.dorecognizethisobjection,andtrytoarguethatallowing0tovarywouldnotproduceaneectlargeenoughtoaccountfortheobserveddiscontinuityinh0,butwedonotndtheirargumentconvincing.Thesituationismoresubtleifoneconsidersthetwo-dimensionalspaceofcouplings0andh0.Thenonehastochoosethepair(0;h0)soastomatchtheobservedrenormalizedmagnetizationM0andtheobservedrenormalizedenergyE0.Todothis,letusdeterminersttheuniquevalue0suchthattherenormalizedenergycanbematchedatzeromagneticeld,i.e.E0=E(0;0).ThenweaskhowtherenormalizedmagnetizationM0comparestothespontaneousmagnetizationat0:(a)IfM0>M(0;0+),thenitisimpossibletomatchbothM0andE0atzeromagneticeld.Therefore,therenormalizedcouplingh0willbefoundtobe>0(resp.<0)fortheimagemeasure0+(resp.0).(b)IfM0M(0;0+),thenM0andE0canbematchedbytaking=0,h0=0.[IfM0

PluggingthisintoM,wehaveM(u0;0+)=uupu=+O(u);(:0)whichisequaltoM0(u;0+)atleadingorderandgreaterthanM0(u;0+)atorderu.WeconcludethatatlowtemperatureM0andE0canbematchedath=0inthetwo-dimensionalIsingmodelwiththemajority-ruletransformation.However,wedonotknowwhatwillhappenwithothertransformations.Similarremarksapplyinthecaseofhigher-dimensionalinteractionspacesV.Whileweareunabletoprovethatadiscontinuitywillinevitablybeobserved,neitherdoweseeanyreasontobelievethattherenormalizedexpectationvalueshfi0canalwaysbematched,simultaneouslyforallVandallphases0,bysomeinteraction0V.Therefore,wemustexpectthattypicallytheobservedRGmapwillbemulti-valuedanddiscontinuousatarst-orderphase-transitionsurface,purelyasanartifactofthetruncationoftherenormalizedinteraction.Ofcourse,iftheimagemeasure0isGibbsian,thenthisdiscontinuityshouldgotozeroasymptoticallyastheassumedinteractionspaceVngrows.If0isnon-Gibbsian,thenweexpecttheestimatedrenormalizedinteractions0ntodivergeinBnorm,anditisperfectlylikelythatthe0ncorrespondingtodierentphaseswilldivergeindierentways.WethinkthatthisexplainsthenumericallyobserveddiscontinuitiesoftheRGmap,irrespectiveofwhethertherenormalizedmeasure0isGibbsianornot..ARemarkonDangerousIrrelevantVariablesTherenormalization-groupdescriptionofcriticalbehaviorinitssimplestformseemstoimplyhyperscalingrelationssuchasd=0+(.)d=(.)d=(.)=d+d+(.)where;;;;0;;;arecriticalexponentsanddisthespatialdimension.Itisawell-knownfact,however,thathyperscalingdoesnotholdforsystemsabovetheiruppercriticaldimensiondu:ford>duthecriticalexponentsareexpectedtobethoseofmean-eldtheory,andtheseexponentssatisfythehyperscalingrelationsonlyatd=du.Indeed,thehyperscalingrelations(.){(. )havebeenprovenrigorouslytofailforIsing-likemodelsindimensiond>[,0,,,,].Thetraditionalexplanationofhyperscaling|andofitsfailure|isthefollowing[,,]:UnderanRGtransformationH0=R(H)withlinearscalefactorl,thecorrelationlengthandfreeenergydensityftransformas(H)=l(H0)(.a)f(H)=g(H)+ldf(H0)(.b)

wheregisnonsingular.(Infact,formostRGmapstheseidentitiesareonlyapproxi-mate. )NearaxedpointHweparametrizetheHamiltonianbyscalingeldsg;g;:::witheigenvaluesly;ly;:::;thevariablegiissaidtoberelevant(resp.irrelevant)ifyi>0(resp.yi<0).Thecriticalsurfacecorrespondstosettingalltherelevantscalingeldstozero.Wecanassumewithoutlossofgeneralitythatgisarelevantvariable(y>0).Theasymptoticscalinglawsthenread(g;g;:::)l(lyg;lyg;:::)(.a)fsing(g;g;:::)ldfsing(lyg;lyg;:::)(.b)Makingthechoicel=g=y,weobtain(g;g;g;:::)jgj=y ;gjgjy=y;gjgjy=y;:::!

(.a)fsing(g;g;g;:::)jgjd=yfsing ;gjgjy=y;gjgjy=y;:::!(.b)Ifnowgiisanirrelevantvariable(yi<0),thengi=jgjyi=y!0asg!0.Itappearsatrstglance,therefore,thatforthepurposeofdeterminingtheleadingscalingbehavior,thequantitygi=jgjyi=yontheright-handsidesof(. )canbereplacedbyzero.Forexample,ifonlythersttwoeldsarerelevant(thecaseofanordinarycriticalpoint),wewouldobtain(g;g;g;g;:::)jgj=y ;gjgjy=y;0;0;:::!

(.a)fsing(g;g;g;g;:::)jgjd=yfsing ;gjgjy=y;0;0;:::!(.b)Inparticular,supposethatwesetg=t(thetemperaturedeviationfromcriticality)andg=h(themagneticeld).Then(.a)yieldsthescalingbehaviorofthecorrelationlength:(t;h=0;g;g;:::)(tast!0+(t)0ast!0)with=0==yt:(:)Likewise,(.b)anditsderivativesyieldthescalingbehaviorforthethermodynamicquantities:(a)Dierentiating(.b)twicewithrespecttotandsettingh=0,weobtainthecriticalexponentsforthespecicheat:=0=d=yt.Combiningthiswith(. )yieldsthehyperscalinglawd=.Foraposition-spaceRGmap,thiscanofcoursebedoneonlyapproximately,sincelmustbeapowerofthebasicblocksizeb.However,thisisgoodenoughforthepurposeofobtainingcriticalexponents:bychoosinglwithinafactorbofthedesiredvalue,oneobtainsthedesiredequalitywithinaboundedmultiplicativeconstant.

(b)Dierentiating(.b)onceortwicewithrespecttoh,thensettingh=0,weobtainthecriticalexponentsforthespontaneousmagnetizationandthesuscep-tibility:=(dyh)=ytand=0=(yhd)=yt.Combiningthiswith(.)yieldsthehyperscalinglawd=0+.(c)Dierentiating(.b)fourtimeswithrespecttoh,thensettingh=0(witht>0),weobtainthecriticalexponentforthefour-pointcumulant:+=(yhd)=yt.Combiningthiswith(.)andtheformulaforyieldsthehyperscalinglawd=.(Relationsforexponentsonthecriticalisothermcanbeobtainedinasimilarmannerbysettingg=handg=t=0.)However,Fisher[]pointedoutthatthisreasoningiscorrectonlyiff(g;g;g;g;:::)anditslow-orderderivativeshavenitelimitsasg;g;:::!0wheng=.Ifforoneofitslow-orderderivativesdivergesasg;g;:::!0,thenthehyperscalingrelationscanfail.AvariablegiwhichisirrelevantintheRGsensebutwhichprovokesadivergenceofthefreeenergydensity(oroneofitslow-orderderivatives)istermedadangerousirrelevantvariable.Weemphasizethatthefreeenergyisherebeingevaluatedwellawayfromthecriticalpoint,namelyatg=.Thestandardexampleofsuchabehavioristhe'modelindimensiond>.HerethexedpointisGaussian,withrelevanteldsg=tandg=h;the'couplingconstantg=uisirrelevantintheRGsense.However,theGaussianmodelisunstableatnonzeromagneticeldonthecriticalisotherm,andalsoatzeromagneticeldbelowthecriticaltemperature,andtheirrelevant'termisneededtostabilizeit.Amean-eldcalculation(whichisexpectedtogivethecorrectscalingford>)predictsthatthefreeenergydivergesasu#0,asf(t=;h;u)uW(hu=)(.0)f(t=0;h;u)u=h=(.)whereWisawell-behavedfunction.Insertingthisbehaviorinto(.b),onendsmodiedhyperscalinglawswhichdierfrom(.){(.)andwhichareconsistentwiththemean-eldexponents.ThisbehavioroccursbecausethexedpointHisontheboundaryofthestabilityregion,andthefreeenergydivergesasthisboundaryisapproached.Herewewouldliketomakethetrivialobservationthatsuchablow-upofthefreeenergyispossibleonlyinmodelswithunboundedHamiltonians(suchasthe'model).Indeed,weknowthatforabsolutelysummableinteractions(B),thefreeenergydensityisaLipschitzcontinuousfunctionoftheinteraction(Propositions.SeealsoFisher[,AppendixD]andMa[,SectionVII. ].Aqualitativelysimilarbehaviorisexpectedtooccuralsoindimensiond=.Herethedangerousirrelevantvariableg=uisonlymarginallyirrelevant(i.e.y=0,butsecond-ordereectsmakegirrelevant),sothattheviolationsofhyperscalingareonlylogarithmic.

and.).Thismeansthatthefreeenergydensityanditsrstderivativesarealwaysbounded.Thissituationprevailsinallphysicallysensiblemodelsofboundedspins.Theseconsiderationsdonotquiteruleoutthepossibilityofdangerousirrelevantvariables:inprincipleitcouldhappenthatf(;g;g;:::)anditsrstderivativesarebounded,butthathigherderivativesblowup.Thiswouldcausesomeorallofthehyperscalingrelationstofail.ThisisindeedwhathappensintheXYmodelindimensiond=(andprobablyalsod=)ifweletgbethecoecientofacosnsingle-siteterm,wherenisevenand[,0].SuchatermisirrelevantintheRGsense(atleastifissmallenough),butforT[].Othercasesinwhichanapparentlyirrelevantterm(intheRGsense)changesthephasediagramhavebeenstudiedin[0,0,0].However,wehavenotbeenabletoconstructanyplausibleAnsatzforsuchabe-haviorinanIsing-to-IsingRGmapfortheIsingmodelindimensiond>.Nordoweknowofanyplausiblecandidateforthedangerousirrelevantvariable.(IntheIsinglanguagethereisnotermintheHamiltoniancorrespondingtothe\'coupling";suchatermisbuiltintotheapriorisingle-spinmeasure.)Weconcludethatthedangerous-irrelevant-variablesscenarioisprobablynotthecorrectdescriptionofwhatishappeningintheIsingmodelindimensiond>,atleastinthecontextofanIsing-to-IsingRGmap.Ontheotherhand,weknowthatthehyperscalingrelations(.){(.)dofailforIsingmodelsindimensiond>.Therefore,oneoftheotherassumptionsmadeintheconventionalRGtheorymustfailwhenappliedtoIsing-to-IsingRGmapsindimensiond>.Forlarge-cellRGmaps(b!),theresultssummarizedinSection.showthatwhatfailsistheGibbsiannessofthexed-pointmeasure.Nowthereisaveryclosesimilaritybetweenthedangerous-irrelevant-variablesscenarioandthenon-Gibbsiannessproofforlarge-cellRGmaps:bothhingeonthefactthatamasslessGaussianeldisunstabletomagnetic-eldperturbations.Thisreasoningsuggeststhatnon-Gibbsiannessofthexed-pointmeasuremayoccuralsoforiteratedIsing-to-Isingtransformationswithxedblocksizeb(e.g.majorityruleortheKadanotransformation).Ifthiswerethecase,thentheRGmapRfromHamiltonianstoHamiltonianswouldbeill-denedatthecriticalIsingmodel,andtheputativexed-Fisher[,p. ]statesthatthederivationofthehyperscalingrelationsreliesimplicitlyontheassumptionthatthefreeenergyf(;g;g;g;:::)hasawell-denednitelimitasg;g;:::!0.However,thisstatementisslightlymisleading,becauseitistooweak:infact,asisclearfrom(a){(c)above,toderiveahyperscalingrelationoneneedstoknowthatatleastthesecondderivativeoffwithrespecttotorhhasagoodlimit.

pointHamiltonianHwouldsimplynotexist.ConclusionsandOpenQuestions.Conclusions..HowMuchoftheStandardPictureoftheRGMapisTrue?WecanclassifytheevidenceregardingthevalidityorfailureofthestandardpictureofRGtransformationsinthreecategories:)Positiveresults.SomeRGmapsarewell-denedinpartsoftheone-phaseregion.Thepublishedproofsrefertothefollowingcases:(i)High-eldresults.DecimationandKadanotransformationsforabsolutelysum-mablelattice-gas[]andIsing-spin[0]interactions.(ii)High-temperatureresults.Decimation[0,],Kadano[0]andaveraging[,]transformationsforabsolutelysummableIsing-spininteractions.(iii)Small-eldresults.TheseresultsrefertodecimationtransformationsoftheIsingmodelinanydimension[]:Foranyxedtemperatureandnonzerovalueofthemagneticeld,thereexistsaminimumblocksizebminbeyondwhichtherenormalizationtransformationiswell-dened(theminimumblocksizedivergesasapowerof=hwhenh!0).(iv)Resultsinonedimension.Thedecimationtransformationiswell-denedindi-mensiond=forlattice-gasinteractionswithmany-bodyandlong-rangecou-plingssatisfyingthesummabilityconditionPA0(diamA+)jAjkAk<[].Forinstance,thisincludesallthetwo-bodyIsinginteractionsdecreasingstrictlyfasterthan=r.Theseresultsare,however,oflimitedinterest,astheycorrespondtowell-understoodregionsofthephasediagram,deepwithintheregimeinwhichuniquenessoftheGibbsmeasure,andevenanalyticityofthefreeenergyandcorrelationfunctions,canbeproven.Iftheseweretheonlypositiveresults,thenonewouldconclude,inagreementwithGrithsandPearce'spessimist[],thatthemethodonlyworkswhereonedoesnotreallyneedit(and,wemayadd,sometimesnoteventhere,givenTheorem.).Wecanalsomention,aspositiveresults(ofasort),ourFundamentalTheoremsofSectionwhichsaythattheRGmapissingle-valuedandcontinuous|inaccordancewiththestandardpicture|ifitexistsatall. )Non-negativeresults.ThereisatpresentnoevidenceofRGpathologiesaboveoratthecriticaltemperatureformodelsstrictlybelowtheuppercriticaldimensiondu(=forIsing-likemodels).Infact,thereexistmodelsforwhichthecriticalpointhasbeenrigorouslystudiedusingthestandardRGprescription:thehierarchicalmodels

[,,]andtheGross-Neveumodel[].ThehierarchicalmodelspresentthemostfaithfultranscriptionofWilson'sprescription,butfromourpointofviewtheyaresomewhatarticialasthepossiblepathologiesareremoved\byhand".TheGross-Neveumodelisfermionic,andthushasnodirectprobabilisticinterpretation.Wealsoshouldmentionheresomeveryinterestingpreliminaryresults[]indicatingthatforthetwo-dimensionalIsingmodelatzeroeld,themajority-ruletransformationmightbewell-denedat(aswellasslightlybelow)thecriticaltemperature.Theseresultsarepartiallyrigorousandpartiallynumerical,andsofartheyconcernonlysomeselected(albeitjudiciouslyselected)block-spincongurations.Wefeelthatthisworkprovidessomesupportforthestandardpicture,butitsresultsarestillinconclusive.)Negativeresults.Therearepathologiesatlowtemperature(notonlyatzeromagneticeld)inalldimensions,andquitepossiblyatthecriticalpointindimen-siond()du.Intheformercasethesepathologiesconsistinthenon-Gibbsiannessoftherenormalizedmeasure,thatis,intheimpossibilityofconstructingarenormal-izedHamiltonianafterevenasingleRGtransformation.InSections.{.wehaveshownexamplesofsuchpathologiesforallthestandardreal-spacetransformations(decimation,Kadano,majority-rule,averaging).Therangeoftemperatureswherethesepathologiesareproventoexistdoesnotincludethecriticaltemperature,butontheotherhandthepathologicalregionextendsothephase-coexistencecurve,i.e.tononzero(andinsomecaseslarge)magneticeld(Section..).Finally,inSections.and.wehavegivenargumentsindicatingthatford()theremaybepathologiesatthecriticalpoint.Intheselattercasesourargumentssuggestthatthexed-pointHamiltonianmaybeill-dened.Takentogether,theseresultssuggestthatnon-GibbsiannessmaybethenormalsituationforRGmapsatlowtemperatureand/orneararst-orderphase-transitionsurface,oratthecriticalpointinhighdimensions.ThisisindirectconictwiththeconventionalRGideology(comparetherstparagraphoftheIntroduction)...ResponsestoSomeObjectionsManyofourcolleagues,uponhearingourresults,haveinitiallyreactedbysaying:\If0isnotGibbsianforsomeinteractioninB,thenthatjustmeansitisGibbsianforsomeinteractionnotinB.Youhavetousealargerspaceofinteractions. "Thisviewseemsapriorireasonable|anditisevenconceivablethatitiscorrect|butunfortunatelythingsarenotquitesosimple.Beforeassertingthat0isGibbsianforsomeinteraction0=B,onersthastodenewhatitmeansforameasuretobe\Gibbsian"foranon-absolutely-summableinteraction.OurnotionofGibbsmeasurereliesontheDLRequations,andiftheinteractionfailstobeabsolutelysummable(oratleastconvergent),thentheseequationssimplydonotmakesense.Itisthusincumbentontheadvocateof\largerinteractionspaces"tomakeprecisewhatisthecorrespondencebetweenmeasuresandinteractionsthatistosubstitutefortheDLRequations(andbeequivalenttothemwhentheinteractionisabsolutelysummable).Now,asisusualwhenoneislookingforthesolutionofanequation,therearetwo

complementaryaspects|existenceanduniqueness|andonewantspreferablyforbothpropertiestohold.Theexistenceisfavoredbyenlargingthespaceofpossiblesolu-tions,whiletheuniquenessisfavoredbynarrowingit;anditisfarfromclear,apriori,whetherthereexistsaspaceinwhichthesolutionsbothexistandareunique.Thesegeneralremarkscanbeexempliedinourstatistical-mechanicalproblem.WithintheclassofFellerspecications(andhenceafortioriwithinB),theGriths-Ruellethe-orem(Theorem.,Corollary.andProposition.)guaranteestheuniquenessofthespecicationforagivenmeasure(andhencetheuniquenessmodulophysicalequivalenceoftheinteraction).Buttheexistencemayfail,asweshowedthroughnu-merousexamplesinSection.Ontheotherhand,ifweenlargethespaceofallowedspecicationsbydroppingtheFeller(quasilocality)property,thentheexistenceholdsbuttheuniquenessfailsspectacularly(seetheRemarkattheendofSection..).Similarly,ifweenlargethespaceofallowedinteractionsfromBtoB0,andgeneralize\Gibbsmeasure"to\equilibriummeasure",theneveryergodicmeasureofniteentropydensityistheequilibriummeasureforsomeinteraction,buttheunique-nessagainfailsspectacularly(seeiteminSection..).OnecertainlycannotdevelopasatisfactoryRGtheoryinsuchpathologicalspaces.Furthermore,wehavegivenstrongargumentsthatanyphysicallyreasonablespeci-cationmustbequasilocal,atleastinsystemsofboundedspins(seeSection..).Ontheotherhand,inSections.{.and..wehaveprovendirectlythattherenor-malizedmeasuresarenotconsistentwithanyquasilocalspecication.SoevenifthereweretoexistquasilocalspecicationscorrespondingtointeractionsnotinB,suchspecicationscouldnotbeofanyrelevanceforourrenormalization-groupproblem.OurnalobjectiontoconsideringspacesofinteractionslargerthanBisthatBisalreadytoobig!Indeed,thestandardRGideology[]isthattheRGowshouldtakeplaceinsomespaceof\short-range"interactions,e.g.interactionswhichdecayexponentiallyoratleastlikeasucientlylargepower(e.g.jxjpwithpd+).Thisideologyisnotamerewhim,butresultsfromtheneedtoexplainuniversalityofcriticalbehavior:oneneedstohaveaninteractionspaceinwhichtheunstablemanifoldofagivenxedpointisnite-dimensional(i.e.therearenitelymanyrelevantscalingelds).Now,suchabehaviorisimpossibleinaspaceoflong-rangeinteractions(suchasB),sinceingeneralthecriticalexponentswillbealteredbyanyperturbationthatdecayslikejxj(d+)with>thecriticalexponentoftheoriginalmodel[,Section0. ].Moreover,eventhequalitativephasediagramisunstabletolong-rangebutsummablepairinteractions[,,0]:thatis,theGibbsphaserulecannotholdinBoreveninanyBn.InordertohaveanyhopeofconstructingasatisfactoryRGtheory,itisnecessarytoworkinaspaceof\short-range"interactions,suchasthespaceBhforsomeh.Asecondcommentwhichisoftenmadeisthefollowing:\TheRGmapisalwaysNotealsothatSullivan[]andKozlov[]havealmostproventhateveryquasilocalspeci-cationarisesfromaninteractioninB:seeTheorem.andtheRemarksfollowingit,plustheRemarkattheendofSection...

well-denedasamapfrommeasurestomeasures;thepathologiescomefromtryingtoliftittoamapfromHamiltonianstoHamiltonians.SowhynotjuststickwiththeRGmap(.)onthespaceofmeasures?"Thisisasensiblequestion,whichwasalreadyraisedbyGrithsandPearce[,p. {],andouranswerisessentiallythesameastheirs:Manyinterestingthingscan,indeed,belearnedbystudyingtheactionofRGtransformationsonmeasures.ForlinearRGtransformations,thisisanancientbranchofprobabilitytheorythatgoesbacktoGauss'andDeMoivre'sinvestigationsofthecentrallimittheoremforin-dependentrandomvariables,andwhichcontinuestothisdayinstudiesofcentralandnon-centrallimittheoremsfordependentrandomelds[0,0,,,];itiscloselyrelatedtostudiesoftrivialityandnon-trivialityforscalinglimitsinstatisticalmechanicsandquantumeldtheory[,,].FornonlinearRGtransformations,thisstudyisonlybeginning[,],butweexpectittobefruitfulaswell.Unfor-tunately,notalloftheRGtheorycanbecarriedoutonthespaceofmeasuresalone.Forexample,thecriticalexponentmeasurestherateofdivergenceofthesusceptibil-ityasthetemperatureapproachesthecriticaltemperature.Now,thesusceptibilityistheintegralofthe-pointcorrelationfunction,andthuscanbereadothemeasure;whilethe(inverse)temperatureisthecoecientofsometermintheHamiltonian(e.g.thenearest-neighborterminthecaseoftheIsingmodel).Therefore,theexponentcanbededucedonlyfromatheorythatrelatesthemeasuretotheHamiltonian(orinteraction);itcannotbededucedsolelyfromanRGmapactingonthespaceofmea-sures.Thesamegoesfortheexponent,whichmeasurestherateofdivergenceofthecorrelationlengthasthetemperatureapproachescriticality.Ontheotherhand,theexponentratio=measurestherelativerateofdivergenceoftwodierentaspectsofthe-pointcorrelationfunction,andsocanpotentiallybededucedfromameasures-to-measuresRGmap.Likewise,thecriticalexponentmeasurestherateofdecayofthe-pointfunctionatthecriticalpoint,makingnoreferencewhatsoevertothetemperature;therefore,ittoocanpotentiallybededucedfromameasures-to-measuresRGmap.Itfollowsthatthescalinglaw==alsoliespotentiallywithinthepurviewofameasures-to-measuresRGtheory...WhereDoesAllThisLeaveRGTheory?Afterthemore-or-lesscoldexpositionoffactsofSection..,andtheadditionalclar-ication(pre-emptivedefense)oftheprevioussubsection,letuspresentsomegeneralremarksabouttheconsequencesofthepresentworkfortheRGenterprise.WethinkthatthereisalreadyasubstantialbodyofevidenceindicatingthattheconventionalRGtheory,initsnarrowsenseofHamiltonian-to-Hamiltonianmaps,needstobereexamined.However,thisdoesnot,initself,detractinanywayfromthevalueandsignicanceoftheRGideasthathavepervadedmuchoftoday'sstatis-ticalmechanicsandquantumeldtheory.TheRGphilosophy|interpretedbroadlytoincludevariouskindsof\multi-length-scale"and\coarse-graining"arguments|hasbeen,andwillcontinuetobe,ourmaintooltoanalyzetheotherwiseinaccessible\intermediatetemperature"regions,whichfallbeyondthereachofseriesexpansions

orperturbativeargumentsandyetaretheregionsinwhichthemostinterestingphe-nomenatakeplace.ThemainissueintheproperapplicationofRGtheorytoaparticularproblemisthechoiceofvariablesinwhichtoexpressthemodel,alongwiththechoiceoftheRGmap.Thiswasalreadyunderstoodbythefoundingfathersoftheeld.ItcorrespondstowhatMichaelFishercalls\aptnessorfocusability"ofthetransformation,andhisownwordsareespeciallyclear:ForanygivenHamiltonianorclassofHamiltoniansthereisnotjustonerenormalizationgroup|\therenormalizationgroup"assomepeoplesay|butrathertherearemanythatmightbeintroduced,andonemustquestion,forexample,whethertheprocessisbestcarriedoutinrealspaceormo-mentumspaceandsoon.A\good"renormalizationgroupmustbe\apt"orappropriatefortheproblemathand,anditmust,inparticular,\focus"properlyonthecriticalphenomenaofinterest. [,page]Letusmentionanillustrativeexample:Theusualtransformationsinvolvingaveraging(orotherkindsof\voting")oversquareblocksaredesignedmostlyhavingferromag-neticsystemsinmind.Theyareecientforselectingthezero-momentummodes,whichareindeedthemodesthatbecomecriticalinanordinaryferromagnetictransi-tion.Ontheotherhand,thesetransformationsareunsuitableforstudyingantiferro-magnetsbecausetheydonotdistinguishtheoppositelymagnetizedsublattices.ThiswasremarkedbyvanLeeuwen[],whoshowedhowamorecarefuldesignoftheblockshapescouldovercomethisdeciency(hisproposalisdepictedinFigure(d)).Thus,whilemostpeopleimagineRGmapsasactinginahugespaceofHamiltonians|includingregionsexhibitingvariousdierenttypesofphasetransitions(ferromag-netic,antiferromagneticandmanyothers)|itisunlikelythatanysingleRGmapcanexhibitwell-behavedxedpointscorrespondingtoallofthesetransitions.Rather,onemust\custom-make"theRGmapforeachnewphysicalsituation.Inthisregard,ourwork|buildingonthatofGriths,PearceandIsrael[,,,0]|canbeconsideredanextensionoftheprecedingobservations:\aptness"and\focusability"areneedednotjusttoensuretheusefulnessofthemap,butevenitsveryexistence.Ontheotherhand,thesuccessstoriesofrigorousRGstudiesteachusthatthesearchforthis\aptness"mayrequireaveryopen-mindedattitude,inthesensethat,inmanycases,theappropriatevariablesarenotnecessarilyspinvariablesand,infact,notevenlocalobjects.Indeed,withtheexceptionofhierarchical[,,]andfermionic[]models,rigorousRGstudieshavenotimplementedthestrictWilsonprescriptioninvolvinganRGtransformationofHamiltonianswrittenintermsofspinvariables.Rather,theyhaveemployedacombinationofspinvariablesandpolymerensembles[,,0,,,]orapurepolymerensemble[]whenstudyingcriticalphenomena,oranensembleofPeierls-likecontours[,,]whenstudyingrst-orderphasetransitions.Moregenerally,\multi-scale"and\coarse-graining"ideashavebeenusedinawidevarietyofproblems,including:ultravioletstabilityofthe'[,]andYang-Mills[]quantumeldtheories;0

theKosterlitz-Thoulesstransitioninthetwo-dimensionalXYandrelatedmodels[,];theferromagnetictransitionintheone-dimensional=rIsingmodel[];connementinthethree-dimensionalU()latticegaugetheory[];localizationforrandomSchrodingeroperators[];thephasetransitioninplaquettepercolation[];theintersectionpropertiesofordinaryrandomwalksandofBrownianmotion[,,];andthecriticalbehaviorofself-avoidingwalks[,,,],percolation[,]andbranchedpolymers[]inhighdimensions.Inmanyoftheseexamples,the\coarse-graining"isappliedatthelevelofobjectswithsomegeometriccontent,suchasrandomwalks,clusters,surfaces,contours,etc.Thus,ourworkisinnowayanattackontheessentialphysicalideasbehindtheRGapproach.Itsimplypointsouttheneedforamoregeneraldenitionoftheirscope...TowardsaNon-GibbsianPointofViewLetusclosewithsomegeneralremarksonthesignicanceof(non-)Gibbsiannessand(non-)quasilocalityinstatisticalphysics.OurrstobservationisthatGibbsiannesshasheretoforebeenubiquitousinequilibriumstatisticalmechanicsbecauseithasbeenputinbyhand:nearlyallthemeasuresthatphysicistsencounterareGibbsianbecausephysicistshavedecidedtostudyGibbsmeasures!However,wenowknowthatnaturaloperationsonGibbsmeasurescansometimesleadoutofthisclass:amongsuchop-erationsaresomerenormalizationtransformations(Sections.{.and..),somenonlinearlocalfunctions(Section.),convexcombinations(Section..),andweaklimits(Section..).Itisthusofgreatinteresttostudywhichtypesofoperationspreserve,orfailtopreserve,theGibbsianness(orquasilocality)ofameasure.Thisstudyiscurrentlyinitsinfancy.Moregenerally,inareasofphysicswhereGibbsiannessisnotputinbyhand,oneshouldexpectnon-Gibbsiannesstobeubiquitous.Thisisprobablythecaseinnonequilibriumstatisticalmechanics(Section.. ).SinceonecannotexpectallmeasuresofinteresttobeGibbsian,thequestionthenariseswhetherthereareweakerconditionsthatcapturesomeormostofthe\good"physicalpropertiescharacteristicofGibbsmeasures.Forexample,thestationarymea-sureofthevotermodelappearstohavethecriticalexponentspredicted(underthehypothesisofGibbsianness)bytheMonteCarlorenormalizationgroup[],eventhoughthismeasureisprovablynon-Gibbsian[].Onemayalsoinquirewhetherthereisaclassicationofnon-Gibbsianmeasuresaccordingtotheir\degreeofnon-Gibbsianness".JoelLebowitzhassuggestedtous

theanalogywiththerationalandrealnumbers:althoughthesetofrationalsisvery\small"inmanysenses(e.g.rstBairecategory,zeroLebesguemeasure),itis\large"intheweaksensethatanyrealnumbercanbeapproximatedbyasequenceofrationalnumbers;andtheirrationalnumberscanbeclassiedaccordingtotherateatwhichtheycanbeapproximatedbyrationals(Diophantineapproximation).InSection..weconjecturedasimilarscenariofortheGibbsianmeasureswithinthespaceofallmeasures.Itwouldthenbenaturaltoclassifythenon-Gibbsianmeasuresaccordingtohowwell(orhowrapidly)theycanbeapproximatedbyGibbsianones.Finally,thereisaphilosophicalquestion,raisedbyoneofourcolleaguesinRome(towhomweapologizebecausewecannotrememberhisname):Allmathematicalmodelling,inanybranchofscience,involvesselectingthe\important"variablesinthedescriptionofasystemandneglectingthevariablesjudged\unimportant".Inastatisticalsystemthismeansthatthe\unimportant"variablesareintegratedout,i.e.oneperformsakindof\decimation"transformation.Now,ifthedecimatedvariablesareonlyweaklycoupledtotheothers,thenonemayhopethatthedecimationwillleadtoaGibbsmeasure(althoughrigoroustheoremsguaranteeingthisseemtobelacking).However,onecouldalsofearthattheresultofthedecimationmightbeanon-Gibbsianmeasure,especiallyifthedecimatedvariablesarestronglycoupledtotheothers.(Suchvariablesmightstillbedeemed\unimportant"iftheywerebelievedtoaectonlyuninterestingquantitativedetailsoftheproblem,withoutchangingthefeaturesofinterest.)Inthiscase,notonlywouldonebemakinganapproximationindescribingthesystembyaparticular\modelHamiltonian",buteventhedescriptionofthedecimatedsystembyanyHamiltonianwoulditselfbeanapproximation.Andonewouldhavetoinvestigatehowgoodthisapproximationis..SomeOpenQuestionsWeendwithalistofopenquestionsforfutureresearch:)CleanupthecircleofresultsconnectedwiththeGibbsRepresentationTheorem(Theorem.),particularlyinthetranslation-invariantcase(Sections..,..andA.).)Determinewhetherf:kkBhMg+JisaclosedsubsetofB0,ifh,andinparticularforBh=B(Sections..and..).ThisaectsthewaysinwhichtheRTmapcanblowupattheboundaryofitsdomain(Section.),andarisesalsoinourtheoryofparameterestimation(Section..).)Deviseacleangeneraltheoryforsystemsofunboundedspins,analogoustothespacesB0andBforsystemsofboundedspins(Sections..and..).)InvestigaterigorouslytheGibbsiannessornon-Gibbsiannessoftherenormalizedmeasureinthefollowingmodels:(a)FerromagneticIsingmodel,usingthedecimationtransformationwithspacingb:doesthecutotemperaturefornon-GibbsiannesstendtoJcasb!?(SeeSection...)

(b)FerromagneticIsingmodel,usingthemajority-ruletransformationwithblocksizesbnotcoveredbytheconstructioninSection...(Fordimensiond,itappearsthatnoblocksizesbarecoveredbythisconstruction:seeAppendixC.)(c)FerromagneticIsingmodelatlowtemperatureandnonzeromagneticeld,indimensiond=,usingthedecimation,Kadanoormajority-ruletransformation.(d)Antiferromagneticnearest-neighborIsingmodelinauniformmagneticeld,ontheparamagnetic-antiferromagneticcriticalsurface:comparethemajority-rule(orKadano)transformationonsquare(bb)blockstothesametransformationonvanLeeuwen's-spinblocks[,].(e)q-statePottsmodelwithqlarge,at(ornear)therst-orderphasetransition,usingeithertheordinary\plurality-rule"(orKadano)transformation[0]orthemodiedtransformationincludingvacan-cies[,0].(f)Othermodelsatorneararst-orderphasetransition.(g)FerromagneticIsingmodelatthecriticalpointindimensiond>,usingamajority-rule(orKadano)transformationwithxedblocksizeb(Section.).)Improve/generalizethetheoremsonnon-Gibbsiannessoflocalnonlinearfunc-tionsofananharmoniccrystal,andinparticulartrytoprovenon-quasilocality(Section.).)TrytogeneralizeSchonmann'sexample(Section..)todimensionsd;d0otherthand=,d0=.)Prove(ordisprove)theexistenceofmeasuresconsistentwiththeFortuin-Kasteleynrandom-cluster-modelspecication(.);inparticular,prove(ordisprove)thattheinnite-volumelimitmeasurestakenwithfreeorwiredboundaryconditionsarecon-sistentwiththisspecication(Section..).)InvestigatetheGibbsiannessornon-Gibbsiannessofthestationarymeasure(s)invariousstochasticevolutionsnotsatisfyingdetailedbalance(Section..).)InvestigatetheabstractpropertiesofthesetGofGibbsianmeasures(Sections..and..).0)InvestigaterigorouslythemodelofparameterestimationintroducedinSection..;inparticular,trytoproveConjecture.orsomeweakenedversionofit.)Makeahigh-precisionMCRGtest,usingalargespaceofrenormalizedinterac-tions,tocompareacaseinwhichtherenormalizedmeasureisexpectedtobeGibbsian(e.g.thed=Isingmodelatatemperaturenottoofarbelowcritical)withacaseinwhichtherenormalizedmeasureisexpectedorproventobenon-Gibbsian(e.g.thed=Isingmodelatlowtemperature)[Section.. ].

)ClarifytherelationshipbetweenRGtransformationsactingoncontours[,]orpolymers[,,0,,,,],andthetraditionalRGtransforma-tionsactingonspins.)DiscusstheGibbsiannessornon-Gibbsiannessofvariousstatesofquantumlatticesystems[,].Hereoneproblemistounderstandbettertherelationshipsbetweenthevariousalternativenotionsof\Gibbsianness"inthequantumcase.)ProveConjectureC..AProofsofSomeTheoremsfromSectionA.ProofsandReferencesforSection.TheremarksmadeinSection..areallwell-knownresults.Herearesomereferences:(a)0compact=)compact=)everycontinuousfunctiononisbounded[0,Proposition.].ThedensityofCloc()inC()isaneasyconsequenceoftheStone-Weierstrasstheorem[0,Theorem.].(b)If0isdiscrete,theneverylocalfunctioniscontinuous;andcontinuityispreservedunderuniformconvergence.(c)Thisisanimmediateconsequenceof(a)and(b).FurtherRemark.Ifthesingle-spinspace0isnoncompact,theremayexistboundedcontinuousfunctionswhicharenotquasilocal.Hans-OttoGeorgiiprovideduswiththefollowingexample:takeL=0=Zandletf()=0(!);thenletgbeaboundedfunctionoff,sayg=jfj=(+jfj).Infact,thisconstructioncanbeimitatedwhenever0isanoncompactmetricspaceandLisinnite:Letf;f;:::C(0)havedisjointsupportsS;S;:::withSi\Sj=iSj=?andkfik=(suchfunctionsareeasilyconstructedusingUrysohn'slemma);letx0;x;x;:::bedistinctsitesinL;letgC(0)benon-constant;anddeneh()=Pn=g(xn)fn(x0).StandardreferencesforthetheoryofprobabilitymeasuresonmetricspacesarethebooksofParthasarathy[]andBillingsley[].Probabilitymeasuresongeneral(notnecessarilymetrizable)topologicalspacesaretreatedin[,,0].TheRiesz-Markovtheoremis[0,Theorem.]or[,TheoremsII..andII..].Thetheoremonsupportofameasureis[,TheoremII..].TheboundedmeasurabletopologyonM()andM+()isdiscussedin[].TheweaktopologyonM+()isdiscussedindetailin[,,];inparticular,thetopologicalpropertiesofM+()fordierentclassesofspacesarediscussedin[,PartII],[,SectionII.]and[,TheoremIII{0].If0isaseparablemetricspace,theneveryuniformlycontinuousfunctiononisquasilocal[,Remark. ()].Sincetheboundeduniformlycontinuousfunctionsaresucienttogeneratethe(ordinary)weaktopology(thisisthefamous\portmanteau

theorem"[,Theorem.]),itfollowsthattheboundedquasilocaltopologyisstrongerthantheweaktopology.Ontheotherhand,if0isalsodiscrete(hencecountable),theneveryquasilocalfunctioniscontinuous,sothetwotopologiesinfactcoincide.See[,Remark0.].A.ProofsandReferencesforSection.Proposition.isessentially[,Remark.].ExamplesandinSection..are[,Proposition.andExample.].Theorem.0andrelatedresultsarediscussedin[,Section.].Theorem.isprovenbyKozlov[];seealsoSullivan[].Remarks..ThefollowingconjecturedextensionsofTheorem.appeartobeopenquestions:(a)Ifisquasilocalandnonnull(butnotuniformlynonnull),and0isnotnite,doesthereexistauniformlyconvergentinteractionsuchthat=?[Thistheoremmightberelevanttomodelsofunboundedspinswithnite-rangeinteractions.](b)Ifisquasilocal,uniformlynonnullandstronglyFellerinthesensethatfB(;F)impliesfCql(),and0isnotnite,doesthereexistacontinuousabsolutelysummableinteractionsuchthat=?.RegardingtherelationbetweenquasilocalityandtheFellerproperty,thefol-lowingappearstobeanopenquestion:If0iscompactbutnotnite,canaFellerspecicationfailtobequasilocal?ProofofTheorem..LetbeconsistentwithFellerspecicationsand.ThenE(fjFc)(!)=(f)(!)=(f)(!)-a.e.(A:)foreachfC().Now,sincegivesnonzeromeasuretoeveryopenset,twocon-tinuousfunctionswhichagree-a.e.mustinfactagreeeverywhere.Sowemusthave(f)(!)=(f)(!)forall!.Butifthetwomeasures(!;)and(!;)giveequalexpectationstoeachcontinuousfunctionf,thentheymustbeequal.Furtherexamplesofpathologicalnon-quasilocalspecications,alongthelinesoftheRemarkattheendofSection..,aregivenbyGeorgii[,pp.{].Theorem.andCorollary.areprovenin[,Theorem.].Propositions.and.0are[,Proposition.andTheorem.].Proposition.isalmostimmediatefromthedenitionofFellerspecicationandweakconvergence;forrelatedresults,see[,Sections.and.].Proposition.isprovenin[,Theorem.].ProofofProposition..Recallthat!()isaregularconditionalprobabil-ityforgivenF,i.e.itdependson!onlythrough! ;andweareinterestedonlyin

itsrestrictiontoFc,i.e.wewanttostudythemeasure!(d!0c).Theclaimisnowthatfor-a.e.!,wehaveZ!(d!0c)!(!0c;A)=!(A)(A:)forallAFcandallc.BothsidesofthisequationareF-measurable.SoitsucestoprovethatforallfB(;F)wehaveZd(!)f()Z!(d!0c)!(!0c;A)=Zd(!)f()!(A):(A:)Nowtheright-handsideof(A.)isRfAd,bydenitionofregularconditionalprob-ability.Asfortheleft-handside,letusrewriteitasZ[d(!)!(d!0c)]f()!(!0c;A);(A:)thispassagefromaniteratedintegraltoasingleintegralontheproductspaceisjustiedby[,PropositionIII{{].ButthemeasureinbracketsinA.ispreciselyd(!;!0c);sotheleft-handsideof(A.)equalsZd(!;!0c)f()(!!0c;A);(A:)wherewehavenowinsertedthedenition(.)of!.Wenowusethefactthatisconsistentwith,andthatc(soc);itfollowsthat(A. )equalsRfAd.A.ProofsandReferencesforSection.A..VanHoveConvergenceProofofProposition..Itiseasytoseethatx=)dist(x;c)=dist(x;@+)(A.a)xc=)dist(x;)=dist(x;@)(A.b)Therefore,j@rj(r+)dj@+j(A.a)j@+rj(r+)dj@j(A.b)Itfollowsthat(a){(c)areequivalent.Nextnoticethatxn(+a)=)xanddist(x;c)jaj(A.a)x(+a)n=)xcanddist(x;)jaj(A.b)

Therefore(c)implies(d)and(e).Conversely,@=[jaj=[n(+a)](A.a)@+=[jaj=[(+a)n](A.b)so(d)=)(a)and(e)=)(b).Finally,(+A)[aA[(+fag)];(A:0)so(d)and(e)togetherimply(f).Ontheotherhand,takingA=fagshowstriviallythat(f)implies(d)and(e).Thiscompletestheproofofequivalenceof(a){(f).Nextweprovethatlimn!jnj=:thisfollowsimmediatelyfrom(a)andthefactthatj@jwheneverandcarebothnonempty.Finally,letusprovestatement():Foreachn,chooseanZdandrnZ+sothatBrn(an)fxZd:jxanjrngisamaximum-sizedballcontainedinn.Weclaimthatlimn!rn=.Proof:Fixanyr>0.Sincelimn!jnj=andlimn!j@rnj=jnj=0,weclearlyhavelimn!jnn@rnj=andhenceinparticularnn@rn=?forallsucientlylargen.Butnn@rn=?isjustanotherwayofsayingthatrnr.Remarks..Manybooks[,0,0]useamorecomplicateddenitionofvanHoveconvergence,basedonapavingofZdbycubesofsidea.Itiseasytoseethatthisdenitionisequivalenttoconditions(a){(f)..WhatphysicistscallvanHoveconvergenceistermedFlnerconvergencebymathematicians.Muchofthetheoryextends,infact,tolocallycompactamenable(semi)groups[].See[,Section.]forergodictheoremsinthiscontext.A..Translation-InvariantMeasuresProposition.0is[,Theorem.andProposition.].Proposition.is[,Corollary.AandTheorem.A].Formoreinformationonergodictheorems,alongwithsomerelevantcounterexamples,see[,pp.{].Proposition.isprovenin[,Theorem.]or[0,LemmaIV..];astrongerformwillbeprovenasProposition. (e)below.InformationonthePoulsensimplexcanbefoundin[,].A..ADigressiononSubadditivityAnimportantroleinthetheoryoftranslation-invariantlatticesystemsisplayedbytheconceptofasubadditivesetfunction.Subadditivityargumentswillbeusedtoprovetheexistenceoftheinnite-volumelimitforthepressure,theentropydensity,

andquantitiesconnectedwiththequotientnorm.Wethereforecollectheretheneededresults.DenitionA.LetSbetheclassofallnonemptynitesubsetsofZd,andletS=S[f?g.AfunctionF:S![;)iscalledsubadditiveifF(A[A)F(A)+F(A)wheneverA;ASwithA\A=?completelysubadditiveifF(A)nPi=iF(Ai)wheneverA;A;:::;AnSwithA=nPi=iAiandalli0stronglysubadditiveifF(A[A)+F(A\A)F(A)+F(A)wheneverA;ASClearly,completesubadditivityimpliessubadditivity.Thekeynontrivialfactis:LemmaA.([,Theoreme])IfFisstronglysubadditiveandF(?)0,thenFiscompletelysubadditive.Remark.IfFissubadditive,theneitherF(?)0orelseF.InourapplicationswewillalwayshaveF(?)=0.Wecannowstatethetwoprincipaltheoremsontheexistenceoftheinnite-volumelimit:PropositionA.LetF:S![;)betranslation-invariantandcompletelysub-additive.Thenlim%jjF()existsandequalsinfSjjF().PropositionA.LetF:S! [;)betranslation-invariantandsubadditive.Thenlimn!jnjF(n)existsforanyvanHovesequence(n)satisfyingtheaddi-tionalconditionjnj=diam(n)d>0forsome>0.Moreover,thislimitequalsinfnjCnjF(Cn).WenotethatordinarysubadditivityisnotsucientfortheexistenceofthevanHovelimit;acounterexamplehasbeengivenin[].ProofofPropositionA..Thisresultisstatedin[,Theoreme0]andprovenin[,Corollaire0],buttheproofisratherdiculttofollow.Forcompletenessletusgiveanelementaryproof[]:LetA;BS;withoutlossofgeneralityletussupposethat0B.NowconsiderthedecompositionA=Xa:B+aAjBjB+a+XxAxfxg(A:)

wherex=jfa:B+axandB+aAgjjBj:(A:)Clearly0x;andbysumming(A.)overyZdwendXxAx=jAj\bB(Ab)=An\bB(Ab)[since0B]mB(A):(A.)Bycompletesubadditivityandtranslation-invarianceitfollowsfrom(A.)and(A. )thatF(A)Xa:B+aAF(B+a)jBj+XxAxF(fxg)=F(B)jBj\bB(Ab)+ XxAx!F(f0g)=F(B)jBjhjAjmB(A)i+mB(A)F(f0g):(A.)NowdividebyjAjandtakeA%(vanHove):byProposition.

(d)wehavelimA%mB(A)=jAj=0.ThereforelimsupA%F(A)jAjF(B)jBj:(A:)SincethisholdsforallBS,wehavelimsupA%F(A)jAjinfBSF(B)jBjliminfB%F(B)jBj:(A:)ProofofPropositionA..Thisisessentially[,Proposition.0].Seealso[,].Remarks..TheimportantconceptofcompletesubadditivitywasapparentlyrstintroducedbyMoulin-OllagnierandPinchon[,]..Theproofsgivenhereactuallywork(afterslightnotationalchanges)inanarbitrarydiscreteamenablegroup[].AslightlydierentproofofPropositionA.,alsovalidfordiscreteamenablegroups,isimplicitin[,proofofTheoreme].Foranextensiontolocallycompactamenablegroups,see[].

A..ALemmaonSumsofTranslatesNextweusesubadditivityargumentstoproveanimportantlemmaconcerningtheinnite-volumelimitofsumsoftranslatesofafunctionf.Thislemmawillplayanimportantroleinourstudyofthequotientseminorms.Firstletusintroduceaconvenientnotation:foranygB(),letusdenethemaximum,minimumandmidpointvaluesofgbysupgsup!g(!)(A.a)infginf!g(!)(A.b)midg"sup!g(!)+inf!g(!)#=(supg+infg)(A.c)Clearlywehavekgk=max(supg;infg)(A.a)kgkB()=const=(supginfg)=kgmidgk:(A.b)LemmaA.LetfB().Then:(a)lim%jjsupPaTafexistsandequalsinfSjjsupPaTaf.(b)lim%jjinfPaTafexistsandequalssupSjjinfPaTaf.(c)lim%jjPaTafexistsandequalsinfSjjPaTaf.(d)lim%jjPaTafB()=constexistsandequalsinfSjjPaTafB()=const. (e)lim%jjmid PaTaf!existsandliesintheinterval[inff;supf].Proof.(a)ConsiderthesetfunctionF+()supPaTaf,denedfornitesub-setsZd.ClearlyFisnite-valuedandtranslation-invariant.Moreover,itiscompletelysubadditive(DenitionA.):ifA;A;:::;AnSwithA=nPi=iAiandalli0,thenF(A)sup!XaA(Taf)(!)=sup!nXi=iXaAi(Taf)(!

)0

nXi=isup!XaAi(Taf)(!)nXi=iF+(Ai):(A.)PropositionA.thenimpliesthatlim%jjF+()existsandequalsinfSjjF+().(b)issimply(a)appliedtothefunctionf.(c)ConsiderthesetfunctionF()kPaTafk;theproofisthenasin(a).(d)Thisisanimmediateconsequenceof(a)and(b)togetherwith(A.b).[OritcanbeprovendirectlybyapplyingcompletesubadditivitytoFc()kPaTafkB()=const.](e)Thisisanimmediateconsequenceof(a)and(b).Remark.ForfBql()wecanprovethislemmabyaslightlydierentargumentbasedonthefactthatthesetfunctionsF+,FandFcare\almostadditive"(andnotmerelysubadditive).SincetheargumentisvirtuallyidenticaltothatusedbyIsraelinprovingtheexistenceofthepressure[0,TheoremsI..andI..],wegiveonlyabriefsketch.Forsimplicityletusconsiderpart(c);theotherpartsaresimilar.Supposerstthatfisaboundedlocalfunction,i.e.thatfB(;FX)wherediam(X)

(c0)fliesinIfclosedlinearspanofffTaf:aZdg.(c00)fliesinIB()closedlinearspanoffgTag:gB();aZdg.(d)limn!ndPaCnTaf=0.(e)lim%jjPaTaf=0.Then(a)()(b)(=(c0)()(c00)()(d)()(e).Moreover,if0isacom-pactmetricspaceandfC(),thenallthesepropertiesareequivalent.[Inthiscaseproperty(c)ofProposition.isintermediatebetween(c0)and(c00),hencealsoequivalent.]Proof.(a)=)(b):IfMinv(),then+;Minv()[otherwisetheJordandecompositionofintopositiveandnegativepartswouldn'tbeunique].SoeveryMinv()isalinearcombinationoftwomeasuresinM+;inv().(b)=)(a):Trivial.(c0)=)(c00):Trivial.(c00)=)(e):Assumethatf=gTagwithgB().ThenkXxTxfk=kXxTxgXx+aTxgkj(+a)jkgk(A.0)wheredenotessymmetricdierence.ByProposition.,j(+a)j=jj!0as%(vanHove).Thisprovestheclaimforfunctionsfofthegivenform.Thesameobviouslyholdsfornitelinearcombinations.Itisthenroutinetopasstonormlimits.(e)=)(d):Trivial.(d)=)(c0):hnfndPaCnTafliesinthelinearspanofffTaf:aZdg,andlimn!khnfk=0.(d)=)(b):Sinceistranslation-invariant,(f)=ndPaCn(Taf)foralln.Hencej(f)jkkkndPaCnTafk.Nowletn!.(b)=)(c)=)(c00),if0iscompactandfC():Supposethatf=IC()closedlinearspanoffgTag:gC();aZdg.Then,bytheHahn-Banachtheorem,thereexistslC()suchthatlIC()0andl(f)=.BytheRiesz-Markovtheorem,larisesfromsomeM()andlIC()0meanspreciselythatMinv().Butthen(b)impliesthatl(f)=0,acontradiction.Remarks..VariantsofthisPropositionseemstobewellknown[,p.](seealso[,pp.{0]forasimilarargument),butwehavenotbeenabletondapublishedproof.Seealso[,Exercise. ]forarelatedresult..Wedonotknowwhether(a){(b)areequivalentto(c0){(e)ingeneral;orifnot,underwhatminimalextraconditionsthisequivalencecanbeproven.Foraestheticreasons,ifnoother,itwouldbedesirabletoresolvethisquestion.

NextweproveananalogueofPropositionA.inwhichwequotientoutconstantfunctions:PropositionA.LetfB().Considerthefollowingproperties:(a)fhasthesamemeanwithrespecttoeverytranslation-invariantprobabilitymea-sure,i.e.Rfd=Rfdforall;M+;inv().(b)fhaszeromeanwithrespecttoeverytranslation-invariantnitesignedmeasureofzerototalmass,i.e.Rfd=0forallMinv()satisfying()=0.(c0)fliesinIf+constclosedlinearspanofffTaf:aZdgandconstantfunctions.(c00)fliesinIB()+constclosedlinearspanoffgTag:gB();aZdgandconstantfunctions.(d)limn!ndPaCnTafB()=const=0.(e)lim%jjPaTafB()=const=0.Then(a)()(b)(=(c0)()(c00)()(d)()(e).Moreover,if0isacompactmetricspaceandfC(),thenallthesepropertiesareequivalent.Proof.(a)=)(b):IfMinv()with()=0,then+;Minv()with+()=()=0.If=0wearedone;if>0,apply(a)tothemeasures+;M+;inv().(b)=)(a):Justapply(b)to.(c0)=)(c00):Trivial.(c00)=)(e):Assumethatf=gTag+cwithgB()andcR.ThenXxTxfB()=const=XxTxgXx+aTxg+cjjB()=constXxTxgXx+aTxgj(+a)jkgk:(A.)TherestisasinPropositionA..(e)=)(d):Trivial.(d)=)(c0):Letclimn!ndmid(PaCnTaf)asguaranteedbyLemmaA.(e).ThenhnfndPaCnTafliesinthelinearspanofffTaf:aZdg,andlimsupn!k(hn+c)fk=limsupn!ndXaCnTafclimsupn!ndXaCnTafB()=const=0:(A.)

(d)=)(b):AtrivialmodicationofthecorrespondingproofinPropositionA..(b)=)(c)=)(c00),if0iscompactandfC():SameasinPropositionA.,butusethesubspaceIC()+constinplaceofIC();thesignedmeasurewillthenhavezerototalmass.NextweproveastrengthenedversionofProposition..Againwecanallowanarbitrary(notnecessarilycompact)single-spinspace0,andanarbitrary(notnecessarilycontinuousorquasilocal)functionf.TheonlysubtletyisthatinthiscasewemustchoosethecorrectdenitionofI,since(a){(b)and(c0){(e)arenotnecessarilyequivalent.Therightdenitionturnsouttobe(c0){(e).PropositionA.(=Proposition.0)LetfB().Thenlim%jjXaTaf=infSjjXaTaf(A.a)=kfkB()=eI(A.b)andlim%jjXaTafB()=const=infSjjXaTafB()=const(A.a)=kfkB()=(eI+const)(A.b)forallclosedlinearsubspaceseIsatisfyingIfeIIB().Proof.InLemmaA.(c,d)wehaveproventheexistenceofthelimitsandtheirequalitytothecorrespondinginma.Nowwewanttoidentifythelimitswiththequotientseminorms.LetusdenotebyLfthelimit(A.a).ClearlyLfkfk.Moreover,byProposi-tionA.(c0)=)(e),Lf=Lf0wheneverff0IB();henceLfkfkB()=IB()kfkB()=eI.Toprovethereverseinequality,notethatbyaneasycorollaryoftheHahn-Banachtheorem[0,CorollaryofSectionIII. ]thereexistslB()suchthatklk,l(f)=kfkB()=eIandleI0.Ontheotherhand,foreverylB()thatannihilateseIIfwehavel(f)=l ndXaCnTaf!n!!Lfklk:(A:)HencekfkB()=eILf.Thisproves(A.b).Acompletelyanalogousargumenthandles(A.):itsucestoreplaceeIandIB()everywherebyeI+constandIB()+const,respectively.Remark.Theproofgivenhereof(A.)isaslightelaborationofonesketchedbyHugenholtz[,p.

];byusingcompletesubadditivityweareabletodeducethefullvanHoveconvergence.

A..ClosedandCompactSetsinB0ProofofProposition..(a)Weshallactuallyprovesomethingslightlystronger,namelythatf:kkBhMgisclosedintheproducttopologyQXSC(X)(whichisweakerthantheB0normtopology).Solet(n)beasequenceinf:kkBhMg,andletbeanotherinteraction;andsupposethatk(n)XXk!0foreachX.(Thiswouldoccur,inparticular,ifn!inB0norm.)ThenkkBh=XX0h(X)jXjkXk=XX0h(X)jXjlimn!k(n)Xkliminfn!XX0h(X)jXjk(n)Xk=liminfn!knkBhM;(A.)whereinthekeyinequalitywehaveusedFatou'slemma.(b)Let(n)beasequenceinf:kkBhMg.Sincethesingle-spinspaceisnite,eachspaceC(X),Xnite,isnite-dimensional.Therefore,bycompactnessoftheballinC(X)togetherwiththeusualdiagonalargument,wecanextractasubsequence(n0)suchthat(n0)Xconverges(inkknorm)foreachX,saytoX.Let=fXg.Inpart(a)wehaveshownthatkkBhM.Nowwewishtoshowthatn0!inB0norm.SoxK<;wethenhavekn0kB0=XX0h(X)

Now,foreachsequence`,let=Pn=nn(thissumisabsolutelyconvergentinBh).ThenkkBh=kk`andk0kBh=k0k`.(Thatis,!isanisometricisomorphismof`ontoaclosedlinearsubspaceofBh.)NowletS=fng,andletT=f:0nn;Xn=n=g:(A:0)TistheclosedconvexhullofSinBh.Now,kkBh=forallT,so0=T.Ontheotherhand,0doesbelongtotheclosureofTinB0,sincelimn!knkB0=0.ThenaturalsettingfordiscussingthespacesB0andBhisthatofweighted`directsumsofBanachspaces.LetY;Y;:::beBanachspaces,andleth:N!(0;).ThenwedeneYhtobethespaceofsequencesy=(y;y;:::),witheachyiYi,forwhichthenormkykYhXi=h(i)kyikYi(A:)isnite.ForhwewriteYh=Y.ItiseasytoprovethatallthespacesYhareBanachspaces.Thecanonicalprojectionpi:Yh!Yidenedbypi(y)=yihasnorm=h(i).Wethenhavethefollowingresults:PropositionA.Theclosedballfy:kykYhMgisclosedintheproducttopologyQiYi.PropositionA.0LetSY.Thenthefollowingareequivalent:(a)ShascompactclosureinY.(b)Sisbounded,pi[S]hascompactclosureinYiforeachi,andlimN!supySXi=NkyikYi=0:(A:)(c)Sisbounded,pi[S]hascompactclosureinYiforeachi,andthereexistsafunctionh:N![;)suchthatlimi!h(i)=+andsupySkykYh<:(A:)NotethatifYiisnite-dimensional,thenSbounded=)pi[S]bounded=)pi[S]hascompactclosureinYi.TheproofofPropositionA.iscompletelyanalogoustothatofProposition. (a).LetussketchtheproofofPropositionA.0:

(a)=)(b):LetSbecompactinY.Thenclearlypi[S]pi[S]iscompactinYi.Moreover,foreach>0thereexistsanitesety();:::;y(n)YsuchthatSnSk=B(y(k);).ItfollowsthatsupySXi=NkyikYi+maxknXi=Nky(k)ikYi:(A:)TakingN!,wegetlimsupN!supySXi=NkyikYi:(A:)Sincewasarbitrary,theproofiscomplete.(b)=)(c):ChooseN

A..PhysicalEquivalenceHereweproveTheorem.ontheequivalenceofthetwonotionsofphysicalequiv-alence(DLRandRuelle).Sincebothsensesofphysicalequivalencearestatementsaboutthedierence0,itsucestoconsiderthecase0=0.ProofofTheorem.,DLR=)Ruelle.WewishtomeasurehowstronglyH(!;!c)dependson!c.LetusthereforedenetheoscillationofHwithrespectto!cbyoscc(H)sup!;!0!=!0jH(!)H(!0)j=sup!;!c;!0cjH(!;!c)H(!;!0c)j:(A.)ConsideringnowthedenitionH(!)=PA:A\=?A(!),itiseasytoseethatoscc(H)getscontributionsonlyfromsetsAthatintersectbothandc,sothatoscc(H)kW;ck:(A:0)Inparticular,forBwehaveoscc(H)o(jj)as%(vanHove);(A:)by(.a).Supposenowthatisphysicallyequivalentto0intheDLRsense,i.e.thatHisFc-measurableforall.(Actually,itsucestoassumethisforsomevanHovesequenceofsets.)ThenH(!;!c)isindependentof!,sooscc(H)isequaltotheunrestrictedoscillationosc(H)supHinfH(A.)kHkB()=const:(A.)Combining(A.)and(A.),weconcludethatkHkB()=consto(jj)as%(vanHove):(A:)ByProposition. (c),weconcludethatkkB0=(J+Const)=0,i.e.J+Const|thatis,isphysicallyequivalenttozerointheRuellesense.ProofofTheorem.,Ruelle=)DLR.Supposethatthesingle-spinspace0isastandardBorelspace(e.g.acompleteseparablemetricspace),andthat;0BarephysicallyequivalentintheRuellesense.Thenby[,Theorems.and.andthecommentsafterthem],thereexistsatranslation-invariantGibbsmeasurefor

,callit.ByCorollary.,isanequilibriummeasurefor.ByProposition.,isanequilibriummeasurealsofor0.ByCorollary.again,isaGibbsmeasurefor0.ButthenCorollary.impliesthatand0arephysicallyequivalentintheDLRsense.Remark.TheproofgivenhereofRuelle=)DLRisaestheticallyunsatisfying:thetwonotionsofphysicalequivalencearestatementspurelyaboutinteractionsandHamiltonians,sothereoughttobeapurely\algebraic"proofoftheirequivalenceinvolvingonlytheseconcepts,withoutdragginginthewholetheoryofequilibriummeasures,Gibbsmeasuresandtheirequivalence.Inparticular,itisgallingtohavetoassumethat0isastandardBorelspace,foraresultthatobviouslyhasnothingtodowithtopology.However,wehavebeenunabletondsuchanalgebraicproof;wehopethatsomereaderwilldoso.A..EstimatesonHamiltoniansandGibbsMeasuresInSection..westatedProposition.0forthecaseofacompactmetricsingle-spinspace.Hereweproveamoregeneraltheoreminwhichthisrestrictionisremoved.(Westillconsideronlycontinuousinteractionsandfunctions,butthatrestrictiontoocouldberemovedifwereallycared.)PropositionA.(=Proposition.00)Themap[]![f]isanisometryofB0=JontoCql()=Iql,andofB0=(J+Const)ontoCql()=(Iql+const).HereIqlI\Cql().Proof.Itisconvenient(followingRuelle[,Section.])tointroducethemodiedobservablef00XXmid0X;(A:)whereXmid0denotesthat0istheb(jXj+)=cthelement(\middleelement")ofXinlexicographicorder.Clearlyff00Iql.Theadvantagesoff00areduetothefollowingeasilyveriedfacts[,p. ]:a)ff00:Bniteg=Cloc().b)ff00:B0g=Cql().c)ForallfCql(),kfk=infB0:f00=fkkB0:(A:)Moreover,forfCloc()thereexistsaBnitethatattainsthisminimum.

Inparticular,themap![f]=[f00]isontoCql()=Iql.Now,fromkfkkkB0weeasilydeducethatk[f]kC()=Ik[]kB0=J:(A:)Toprovethereverseinequality,notethatbyPropositionA.wehave,foranyfCql(),k[f]kC()=I=lim%jjPaTaf.Now,byproperty(c)above,foreachandeach>0wecanchooseB0suchthatf00=jjPaTafandkkB0jjPaTaf+.(Inparticular,wehave[f]=[f00]=[f].)Then,bytaking%and#0weconcludethatforallfCql(),k[f]kC()=IinfB0:[f00]=[f]kkB0:(A:)Inparticular,takingf=f,wegetk[f]kC()=Ik[]kB0=J:(A:)Thisprovesthatthemap[]![f]isanisometryofB0=JintoC()=I.Repeatingthesameargumentwithfreplacedbyf+c,andthenoptimizingoverc,weconcludethat[]![f]isalsoanisometryofB0=(J+Const)intoC()=(I+const).ProofofProposition..(a)iseasyandwellknown:see[0,p.].(d)BydenitionwehaveH;free=XXX(A:0)andXxTxf=XxXX0jXjTxX=XxXX0jXjX+x=XxXYxjYjY=XYjY\jjYjY(A. )(thedoublesumisabsolutelyconvergentandhencecanberearrangedfreely).ThusH;freeXxTxf=XX\=?X\c=?jX\jjXjX:(A:)0

Takingnorms,wehavekH;freeXxTxfkXX\=?X\c=?jX\jjXjkXk=XxXXxX\c=?jXjkXk=XxXY0(Y+x)\c=?jYjkYk=XY0j(cY)\jjYjkYk(A.)Nowdividebyjj:jjkH;freeXxTxfkXY0j(cY)\jjjkYkjYj:(A:)Thissumisdominateduniformlyin,sincej(cY)\j=jjandB0.Ontheotherhand,foreachxednitesetY,wehavej(cY)\jjjXyYj\(cy)jjj=XyYjn(y)jjj;(A.)whichtendstozeroas%(vanHove).Hence,bythedominatedconvergencetheorem,(A.)tendstozeroas%(vanHove).(b)and(c)areimmediateconsequencesof(d)togetherwithPropositionsA.andA..Remark.See[,p.]foranalternateproofof(c),carriedoutrstforBniteandthenextendedtoB0bydensity.ProofofProposition..(a)iseasyandwellknown:see[0,p.]or[,p.]. (d)Bydenition,HH;free=W;c=XX\=?X\c=?X:(A:)

Takingnorms,wehavekHH;freekXX\=?X\c=?kXkXxXXxX\c=?kXk=XxXY0(Y+x)\c=?kYk=XY0j(cY)\jkYk:(A.)TheremainderoftheargumentiscompletelyparalleltotheproofofProposition.(d),butusingBratherthanB0.Thisproves(.a).Asfor(.b),theleftmosttermiso(jj)asanimmediateconsequenceof(.a)and(.).Themiddletermisproventobeo(jj)in[,pp.0{].(Thatproofisstatedonlyforcubes,butitisvalidforarbitraryvanHovesequences.)(b)isthenanimmediateconsequenceof(.a)and(.).(c)islikewiseanimmediateconsequenceof(.a)and(.).ProofofProposition..LetbeaGibbsmeasureforaninteractionBandapriorimeasure0.ThentheDLRequation(.)statesthatdd0(!)=Zd()Z(c)exp[H(!c)];(A:)whereZ(c)=Zexp[H(!c)]Yxd0x(!x):(A:)Now,byProposition.(d),wecanreplaceH(!c)everywherebyH;free(!),incurringanerrorwhichiso(jj)uniformlyin!and.Therefore,logdd0+H;free+logZ;freeo(jj):(A:0)ButkH;freePxTxfko(jj)byProposition.(c),andjlogZ;freejjp(j0)jo(jj)byProposition. (a).Hencelogdd0+XxTxf+jjp(j0)o(jj):(A:)

Inparticular,logdd0+XxTxfC()=consto(jj):(A:)Thisproves(.).ThisboundisuniformforallG().Nowlet(resp.)beGibbsianforinteractions(resp.)inB,withthesameapriorimeasure0.Combining(A.)forthetwocases,wegetlogdd=XxTxf+jj[p(j0)p(j0)]+o(jj):(A:)ButXxTxf=jjkkB0=J+o(jj)(A:)byPropositionsA.andA.,whilep(j0)p(j0)kkB0=J(A:)byPropositions.(d,e)and.(a).HencelogddjjkkB0=J+o(jj):(A:)ButbyPropositions.(c)and.(a),theright-handsideof(A.)isunchangedifwereplaceby+withConst(i.e.iffconst).Thus,in(A.)wecanreplacetheB0=JnormbyB0=(J+Const).Thisproves(.).Inasimilarwaywededuce(.)fromthetwocasesof(A.)togetherwith(A.).Remarks..Wewishtoemphasizethat(.)isanequality.ThisfactplaysacrucialroleinourproofoftheSecondFundamentalTheorem(Section.)..TheproofsofPropositions.and.donotusetheseestimates,sothereasoningisnotcircular.A.ProofsandReferencesforSection.Proposition.iseasytoprove:seee.g.[0,LemmaI..].ProofofProposition..(a),(b),(c)and(g)areprovenin[,Proposition.].(d)isatrivialgeneralizationofwhatisprovenin[,Proposition.()].(e)isprovenfortheboundedmeasurabletopologyin[,Corollary.andproofofProposition.()].Fortheweaktopology,see[0,pp.{];thoughstatedthereforcompactmetricspaces,theproofisinfactvalidforarbitrarycompleteseparablemetricspaces.Seealso[,p. ].

(f)Weknowfrompart(e)thatf:I(j)cgisclosedintheboundedmea-surabletopology.In[,proofofProposition.]itisshownthatthedensitiesfd=d:I(j)cgareuniformly-integrable(seealso[,TheoremII{]);thisimplies,bytheDunford-Pettistheorem,thatf:I(j)cgisrelativelycompactandrelativelysequentiallycompactintheboundedmeasurabletopology([,Theo-remII{]or[,PropositionIV{{]).Sincetheweaktopologyisweakerthantheboundedmeasurabletopology,thelaststatementisanimmediateconsequence.(h)isprovenin[0,Theorem.andLemma.].(i)isanabstractionoftheusualstatementofstrongsuperadditivity[,Proposi-tion.0].Remarks..Statement(d)isnottruejointlyinand.Counterexample:Let=fa;bg,==a,==b,==.ThenI(j)=I(j)=+,whileI(+j+)=0..Forsomeimprovementsof(d)iftheihave\almostdisjoint"supports,see[,Proposition.andCorollary.]and[,Theorem.]..Ifthe-eldiscountablygenerated,thenthesetf:I(j)cgisinfactcompactandmetrizableintheboundedmeasurabletopology:thisfollowsfrom[,TheoremII{]..Additionalusefulpropertiesoftherelativeentropyaregivenin[,Proposition.andCorollary.].Thenite-volumevariationalprinciple(Theorem.)iswellknown:seee.g.[0,p.]or[,Lemma.].A.ProofsandReferencesforSection.A..TheInnite-VolumeLimit:ProofsProofofProposition..Forallbutpart(e),see[0,TheoremsI..andI..].Part(e)isanimmediateconsequenceofProposition.(e).Proposition.is[,Proposition.].Proposition.isanimmediateconse-quenceofPropositions.and.togetherwiththeestimates(.)and(.b).ProofofProposition..Whenisaproductmeasure,thisis[,Corollary.(b)].WhenisaGibbsmeasure,thisfollowsfromtheproduct-measurecasetogetherwith(.0).ProofofProposition..TheexistenceofthevanHovelimit,anditsequalitytothesupremum,bothfollowfromthestrongsuperadditivityofI(j)asafunctionof,whenisaproductmeasure[Proposition. (i)].Onewaytoseethisistonotethatstrongsuperadditivity

impliescompletesuperadditivity(LemmaA.);theclaimthenfollowsfromPropositionA..Alternatively,onecanmakeadirectargumentusingthestrongsuperadditivity[0,TheoremII..].TheanenessisanimmediateconsequenceofProposition.(c,d),andthelowersemicontinuityisanimmediateconsequenceofProposition.(e)andequation(.b);see[0,TheoremII..]or[,Proposition.].Theproofof(d)employsthefollowingconstruction:PaveZdbyacubeCnanditsdisjointtranslates.Now,givenatranslation-invariantmeasure,letnbeameasurewhichequalswhenrestrictedtoeachofthesecubes,andinwhichthecopiesofthespinsinthevariouscubesarerigidlyforcedtobeequal.Thenletn=ndPaCnTan.Byconstructionnistranslation-invariant;andwithalittleworkonecanprovethati(nj)=imax.Ontheotherhand,itiseasytoseethatlimn!n=limn!n=intheboundedquasilocaltopology.(e)isprovenin[0,LemmaIV..].When0isastandardBorelspace(e.g.acompleteseparablemetricspace,oraBorelsubsetthereof),thecompactnessintheboundedquasilocaltopologyisprovenin[,Proposition.()].(Wedonotknowwhethertheresultistrueformoregeneralspaces0.)Sincetheweakquasilocaltopologyisweakerthantheboundedquasilocaltopology,thelaststatementisanimmediatecorollary.Proposition.isprovenin[,Theorem.0(b)].Remark.Follmer[]hasgivenabeautifulformulafori(j)intermsoftherelativeentropy(notrelativeentropydensity!)oftheconditionaldistributionsofandgiventhelexicographicpast.Seealso[,Proposition.andTheorem.0].Theorem.isessentially[,Theorems.0(b)and.].Remark.AratherweakconversetoTheorem.isthefollowing:Let;M+;inv()withGibbsianforB,i(j)Kandergodic.ThenthereexistsaninteractionB0(notB!)withkkB0K=suchthatisanequilibriummeasurefor.ThiscanbeprovenusingtheBishop-Phelpstheorem[0,CorollaryV..].Thesameistrueifisaniteconvexcombinationofergodicmeasures,butthentheconstantK=isreplacedbyaworseone.Theorem.isprovenin[,Theorem. ].Theproofisgiventhereforasequenceofcubes,butthesameproofworksforanarbitraryvanHovesequence.ProofofCorollary..Ginv()=?,soletGinv()anduse(.0).ThenGibbs=)equilibriumisTheorem.,andequilibrium=)GibbsisTheorem..

A..TheInnite-VolumeLimit:CounterexamplesAsmentionedinSections..and..,theexistenceofthelimitsdeningtheinnite-volumepressurep(fj)andtheinnite-volumerelativeentropydensityi(j)isahighlynontrivialproblem:contrarytowhatmightbesupposedatrstglance,theselimitsdonotalwaysexist.TherstcounterexamplesbearingonthisproblemareduetoKieer[].HerewegiveasimpliedversionofKieer'scounterexample,duetoSokal[]:Let=f;gZ.Letnbethemeasurewhichgivesweight=ntoeachoftheperiodicsequencesofperiodnconsistingofn'sfollowedbyn's.LetbetheconvexcombinationPn=ann.Weshallshowthatforasuitablechoiceofthecoecientsfang:(a)Forthefunctionf(! )=!0,thepressurelimk!klogRexp kPi=!i!d(!)doesnotexist.

(b)Forthemeasure=+deltameasureconcentratedonthesequenceofall+'s,therelativeentropydensitylimk!kIf;:::;kg(j)doesnotexist.Proofof(a).Letgn(k)Rexp kPi=!i!dn(! ).Itiseasytoseethatgnisaperiodicfunctionofperiodn,andsatisesthe(crude)boundsneFn(k)gn(k)eFn(k);(A:)whereFn(k)njk(modn)nj(A:)isthesawtoothfunctiontakingthevalue0atk=0;n;n;:::andthevaluenatk=n;n;n;:::.Henceg(k)Zexp kXi=!i!d(!)=Xn=angn(k)>><>>:Pn=anneFn(k)Pn=aneFn(k)(A.

)Nowchoosethesequencefangtohavehugegaps:an=constenifn=lforsomeintegerl0otherwise(A:0)where>0willbechosenlater.Thenfork=lwehavethelowerboundg(k)akgk(k)akkeFk(k)=ke()k(A:)

andhenceliminfl!llogg(l):(A:)Ontheotherhand,forl

)Letusagaintakean=constenifn=lforsomeintegerl0otherwise(A:)Thenfork=lwehaveh(k)logakk=k+log(k):(A:)

Ontheotherhand,forl0weconcludethatlimk!kh(k)doesnotexist.WenotealsothatVaradhan[]andNewman[]havegivenanexampleofamixingGaussianprocessforwhichthepressuredoesnotexist.BLow-TemperaturePhaseDiagramsandPirogov-SinaiTheoryB.GeneralitiesonPhaseDiagramsThecentralprobleminequilibriumstatisticalmechanicsisthedescriptionofthesetofGibbsmeasuresforagiveninteraction.Moregenerally,familiesofinteractions(orofspecications)areconsidered,withmemberslabelledbycertainparameters:inversetemperature,magneticeld,chemicalpotential,etc.TheultimategoalisthentodescribethesetofGibbsmeasures,inparticularthenumberofextremalGibbsmeasures,asafunctionoftheseparameters.ThepartitionoftheparameterspaceintoregionswithdierentnumbersofextremalGibbsmeasuresiscalledaphasediagramofthefamilyofinteractions,andthemanifoldsdelimitingsuchregionsarecalledphase-transitionmanifolds.Anaturalapproachtothedicultproblemofdeterminingthefullphasediagramistoxrstsomeoftheparameterssothattheresulting\restricted"phasediagramisamenabletoacomparativelysimpleanalysis.Then,onestudieswhetherthisphasediagramis\stable",thatis,whetherasmallchangeinthexedparametersproducesAswewanttoexplicitlydiscusstheroleofthisparameter,throughoutthisappendixweun-absorbfrominteractionsandHamiltonians.

onlyasmalldeformationofthediagramkeepingunalteredthemainpropertiesoftheextremalGibbsmeasures.Themostwidelyused\restricted"phasediagramsarethehigh-temperature(=0)andlow-temperature(=)limits.Intheformer,thesituationisparticularlysimple:Thenite-volumeGibbsdistribution(.0)becomesfor=0justtheproductmeasureQxd0xindependentlyoftheboundarycondition.Hence,thereisauniqueGibbsmeasure,namely0=QxLd0x,whichcorrespondstoindependentspins,theoneatsitexdistributedaccordingtotheapriorimeasure0x.[Notethatfortranslation-invariantGibbsmeasuresthesameconclusionfollowsfromthevariationalprinciple(.0a),whichforf=0requiresi(j0)=infi(j0)=0,hence=0. ]Itiswellknownthatthisinnite-temperaturephasediagramisstableinasuitablespaceofinteractions:forsmalltheGibbsmeasureremainsuniqueanditcorrespondstoweaklydependentspins.Thishasbeenprovenforlattice-gas[0]or,moregenerally,spin-/[0]interactionsinB,andforgeneralinteractionsinB[,].ItisnotknownwhetheritistrueforgeneralinteractionsinB.Thephasediagramforthezero-temperaturelimitis,ingeneral,morecomplicatedtodescribe;itsstabilityisthesubjectofPirogov-Sinaitheory.InthisappendixwegiveabriefoverviewoftheconclusionsofthistheorywithaneyeontheapplicationsneededinSection.Itsunderstandingrequires,ofcourse,apropergraspofthebasicnotionsinvolvedintheconstructionofzero-temperaturephasediagrams.AsremarkedalreadyintheseminalworkofRuelle[],theformalismforzero-temperaturestatisticalmechanicshassomeimportantdierenceswiththeonefornitetemperaturesreviewedinSection.Moreover,thenomenclatureadoptedthroughouttheexistingliteratureisoftenasourceofconfusion,withdierentauthorsassigningdierentmeaningstothesamewords.Therefore,fortheconvenienceofthereaderandtoxtheterminology,westartwithareviewofthezero-temperatureformalism.Forthispartoftheappendix,thereferenceclosesttoourneeds|andfromwhichwehavetakenmanyoftheideas|isthereviewbyDobrushinandShlosman[].However,forthesakeofconsistencywiththerestofourwork,weadoptanomenclatureslightlydierentfromtheirs.Weshallparentheticallycontrastthesedierencesbothforthebenetofthereaderfamiliarwith[]andasatokenoftheconfusingstateofthenomenclature.Letusstateonceandforallthatinthisappendix,weconsideronlythecaseofperiodicinteractionsandnitesingle-spinspace,i.e.j0jnite.Moreover,exceptinSectionsB..andB..,theinteractionsareassumedtobeofniterange.B.Zero-TemperatureLatticeSystems.GeneralFormalismHeuristically,as!onlycongurationswithminimalenergy\survive",theothersbeingexponentiallydampedbytheBoltzmannfactor.However,inthegeneraltheoryofzero-temperaturestatisticalmechanics|asinstatisticalmechanicsquitegenerally|thecentralobjectsarenotindividualcongurationsbutratherprobabilitymea-suresdescribingarandomdistributionofcongurations[0].Justasfornon-zerotemperature,suchmeasurescanbedenedeitherviaspecicationsorviaavariational

principle.B..Zero-TemperatureGibbsMeasuresLetusstartwiththeapproachbasedonspecications.Weseethatthe!limitofthenite-volumeGibbsdistribution(.0)withaxedboundaryconditionproducesameasureconcentratedonthecongurationsof\minimalenergy"forthegivenboundaryconditionandgivingequalprobabilitytoeachsuchconguration.Thatis,foranyinteraction,anynitevolumeandanyboundaryconditionc,wehavelim!;(A)=0(A\;)0(;);T=0;(A)(B:)forallsetsAF,where;isthesetofcongurations!inminimizingtheenergyH(!c):;=!:H(!c)=infe!H(e!c):(B:)Wecall;T=0=(;T=0;)Sthezero-temperaturespecication(orground-statespec-ication[])fortheinteraction.DenitionB.Azero-temperatureGibbsmeasureforisameasureconsistentwiththespecication(B.).Weremarkthatthespecications(B.)arequasilocal(sinceweonlyconsidernite-rangeinteractions),butnotuniformlynonnull,hencetheyarenotGibbsian.Therefore,zero-temperatureGibbsmeasureshappennottobehonestGibbsmeasures.Infact,thepossibilityofincluding(B.)inthegeneralframeworkisoneoftheadvantagesofintroducingthegeneralnotionofspecication(Section..),ratherthanjustthemorerestricted(andpopular)classofGibbsianspecications(Section..).Thezero-temperatureGibbsmeasuresforagiveninteractionforma(weakly)closed|hence(weakly)compact|convexsubsetofthecompactmetricspaceM+()M().Therefore,byChoquet'stheorem[]anysuchmeasurecanbewrittenasthebarycenterofaprobabilitymeasureconcentratedontheextremepoints.Infact,thegeneraltheoryofspecicationsguaranteesusthatthisdecompositionintoextremalmeasuresisunique[,Theorem.],i.e.thatthesetofzero-temperatureGibbsmeasuresisasimplex.B..Ground-StateCongurations.SupportPropertiesofZero-TemperatureGibbsMeasuresThespecications(B.)satisfy(!j!c)=0unless! ;!c:(B:)00

Thispropertyimpliesthatthezero-temperatureGibbsmeasures|whichsatisfy=foreveryniteset|aresupportedbythesetofcongurations!suchthat!;!cforeverynite,i.e.isofcongurationsthatminimizethelocalenergywhentheythemselvesaretheboundarycondition.Congurationswiththispropertyarecalledground-statecongurations.By(B.)theycanbecharacterizedasthosecongurationswhoseenergycannotbeloweredbyanychangeinvolvingonlyanitenumberofspins.Thatis,!isaground-statecongurationforaninteractionifandonlyifforeveryandeveryconguration!0suchthat!c=!0c,wehaveH(!0)H(!)XASA\=?[A(!0)A(!)]0:(B:)Thesetofground-statecongurationsisclosed(hencecompact)becausethecondi-tions(B.)involvenite-volumeHamiltonianswhicharecontinuousfunctionsofthecongurations.Thisfactofbeingaclosedsetjustiestheuseaboveoftheexpression\issupportedby"(=\itssupportisasubsetof").Werecallthatthesupportofameasureisthesmallestclosedsetoffullmeasure(Section..).Thefactofbeingsupportedoncongurationssatisfying(B.)isnotequivalenttobeingconsistentwiththespecications(B.)|itisweaker.Themoregeneralmea-surescharacterizedonlybythissupportpropertyturnouttoplayanimportantroleinthestudyofthestabilityofzero-temperaturephasediagrams(TheoremB.be-low).Inspiredby[],wecallthesemeasuresw-(forweak)zero-temperaturemeasures.Formally:DenitionB.Aw-zero-temperaturemeasureforaninteractionisameasuresatisfying(fground-statecongurationsforg)=:(B:)InSectionB..wediscussanaturallimitprocessthatproducesw-zero-temperaturemeasures,andwepresentanexample(fortheIsingantiferromagnetwithamagneticeld)inwhichthislimitprocessproducesatranslation-invariantw-zero-temperaturemeasurewhichisnotazero-temperatureGibbsmeasure.Obviouslythesetofw-zero-temperaturemeasuresforagiveninteractionis(weakly)closed|hence(weakly)compactandconvex.Theextremepointsaresimplythedeltameasures!concentratedonasingleground-stateconguration!.Thissetisthereforetriviallyasimplex.Thepreviousdiscussioncanbesummarizedinthefollowingway:TheoremB.Everyzero-temperatureGibbsmeasureisaw-zero-temperaturemeasureforthecorrespondinginteraction,i.e.itsatises(B. ).Thistheoremconstitutesthepreciseversionoftheideathatonlycongurationswithminimalenergy\survive"atzerotemperature.0

B..RigidGround-StateCongurationsThesetofground-statecongurationsisingeneralratherlarge.Alreadytheferro-magneticIsingmodelprovidesarichillustration.Thismodelhasexactlytwoperiodic(infacttranslation-invariant)ground-statecongurations:theall-\+"andtheall-\"congurations.Butinadditionitpresentsinnitelymanynon-periodiccongurationsexhibitinginterfacesbetween\+"and\"spins.Inalldimensionswehavetheat-interfacecongurations:!x=(+forx0forx<0(B:)(andtranslated,0-rotatedand0-rotatedversionsofthis).Inhigherdimensionswehaveagrowingzoo:Fordimensionsdwehavecongurationswithinterfacesintheformofstaircases;fordthereappearcongurationsresembling\booksonatable"or\booksonastaircase"[].Seethislastreferenceforapartialcatalogue.Notallthesecongurationsareequallyrelevantforzero-andlow-temperaturephasediagrams.Wecandistinguishthreemutuallyexclusivecategoriesroughlyrepresentingdierent(forusdecreasing)levelsofrelevance.Weshallcallthemrigid,convivialandsuperuous.Therigidcongurationsareusuallythemostimportantones(albeitnotthemostnumerous);theyareassociatedtodeterministiczero-temperatureGibbsmeasures:DenitionB.Foragiveninteraction,aground-stateconguration!iscalledrigid[]ifthemeasure!concentratedon!isazero-temperatureGibbsmeasurefor,i.e.isconsistentwiththespecication(B.).Asimplecalculationprovesthefollowing:PropositionB.Aground-stateconguration!isrigidifandonlyifj;!cj=(B:)forallnite.Inwords,thistheoremsaysthat!placedasaboundaryconditiondeterminesuniquelytheminimal-energycongurationinsideanygivenvolume(therebyjustifyingthequal-ier\rigid").Equivalently,anylocalchangeof!producesastrictlypositivechangeofenergy.Usualphase-diagramstudies|inparticularPirogov-Sinaitheory|dealonlywiththesedeterministiczero-temperatureGibbsmeasuresandtheirlow-temperatureperturbations.(Warning:Reference[]reservesthename\ground-statecongura-tions"onlyfortherigidones.)FortheIsingmodel(ferromagnetic,zeromagneticeld),itisclearthattheall-\+"andall-\"congurationssatisfy(B.)andhencetheyarerigidinanydimension.Thecaseofthenon-periodicground-statecongurations(at-interface,staircase-interface,etc. )ismoredelicate.Thereis,however,asimpleargument[]showingthatif!isaground-statecongurationforthed-dimensionalIsingmodel,thenitscylindrical0

extensiontoanextradimension|denedbye!(x;:::;xd;xd+)!(x;:::;xd)|isarigidground-statecongurationforthe(d+)-dimensionalIsingmodel.Indeed,ifwethinkoftheextradimensionas\vertical",anylocalchangeofe!consistsofanitestackoflocalchangesof!.Thebottomandtopd-dimensionalsectionsofthisstackfacesectionswherethecongurationisequalto!withoutchanges.Thus,someofthecorresponding\vertical"bondsjoinantiparallelspins,whichproducesastrictlypositivecontributiontothechangeinenergy.Thisproves(B. )andhencetherigidityofe!.Asaconsequenceofthisargument,weconcludethattheat-interfacecongurationsarerigidford,thestaircase-interfaceonesarerigidford,andsoon.Theproofthattherigiditydoesnotextendbelowsuchdimensionsrequiresfurtherarguments.Weshallcommentonthisbelow.Remark.Rigidityofaground-stateconguration!doesnotexcludeitsbelongingalsotothesupportofsomezero-temperatureGibbsmeasurethatisnotdeterministic.Forinstance,ifthereismorethanonerigidconguration,onecanofcoursetakeconvexcombinationsofthecorrespondingdelta-measures.Thepossibilityofalesstrivialexamplewillbediscussedbelow,after(B.0).B..ConvivialCongurations.Zero-TemperatureEntropyHowever,notallisdeterministicinzero-temperaturelife.Ournexttypeofcongura-tionsarethosethatbelongonlytothesupportofanon-deterministiczero-temperatureGibbsmeasure.Werecallthatthesupportofameasureisthecomplementoftheunionofallthezero-measureopensets(thatis,thesmallestclosedsetwithfullmeasure).DenitionB.Foragiveninteraction,aground-stateconguration!iscalledcon-vivialif!isnotazero-temperatureGibbsmeasurebutthereexistsazero-temperatureGibbsmeasurehaving!initssupport.Theseground-statecongurations,whichindividuallyhavelittleornoweightbutarerelevantasanensemble,andtheassociatednon-deterministicGibbsmeasuresupportedonsuchanensemble,areprobablynotwhatthephysicist-in-the-streethasinmindwhenthinkingaboutzerotemperature.Oneexpectsthemincaseswherethereisalargedegeneracyinthegroundstate.Thepreciseconceptmeasuringsuchdegeneracyisthezero-temperatureentropy(alsocalledresidualentropy).Forthesakeofcompleteness,webrieyreviewthedenitionandprincipalpropertiesofthisquantity.OurmainreferenceistheclassicarticlebyAizenmanandLieb[].Therearesomesubtletiesinvolvedintherightnotionofzero-temperatureentropy.Heuristically,itscomputationrequiresalimitprocess:onemustcompute(ormeasure)asequenceoflow-temperatureentropiesandtakethelimitasthetemperaturegoestozero.Theso-called\thirdlawofthermodynamics"claimsthatsuchalimitmustbezero;suchbehaviorisindeedseeninsimplemodels,butnotalways.Itsviola-tionmustbeinterpretedassignalingalarge\degeneracyofthegroundstate".Theformalizationoftheseideasrequiresaconsiderationoftheroleoftheinnite-volumelimit.Aspointedoutbysomeauthors(seereferencesin[]),thevolumemustbe0

senttoinnitybeforetakingthelimitT!0.Butinthiscase,onemustconsiderwithsomecaretheboundaryconditions.If,motivatedbythe\ground-state-energy"(=variational)approach[seeeq.(B.)below],oneworkswithpre-xed|forin-stancefree|boundaryconditions,thenthereareexampleswherethecontributionofsomeexcitedcongurationssurvivesthezero-temperaturelimit,sothattheresidualentropyseemstobemeasuringmorethanjustthe\degeneracyofthegroundstate".Thecorrectwaytoconsidertheboundaryconditions,andhencetherightdenitionof\degeneracy",waspointedoutbyAizenmanandLieb[].Atthesametime,theyprovidedaremarkableformulaforthezero-temperatureentropypurelyintermsofzero-temperatureconcepts,withnoreferencetolimitsfromnitetemperatures.Weshalltakethisformulaasthedenition.Foranitesetandaninteraction,letusdenoteGthesetofrestrictionstooftheground-statecongurationsfor.DenitionB.Thezero-temperatureentropyforaninteractionisthelimits=lim%jjlogjGj:(B:)Inwords,thisformulasaysthatasystemhasnon-zeroresidualentropyithenum-berofdistinctground-statecongurations,asviewedwithinanitevolume,growsexponentiallywiththisvolume.Following[],itissuggestivetocallsuchmodelssuper-degenerate.Intuitively,thisfeaturerequiresthepresenceofcompetinginterac-tionstoproduceasucientlargenumberofground-statecongurations.Indeed,itcanbeproven[]thatallferromagneticmodelshavezeroresidualentropy.Thekeyresultestablishingtheconnectionbetweennon-zeroresidualentropyandexistenceofconvivialground-statecongurationsisthefollowing.Toabbreviate,foratranslation-invariant(orperiodic)measurewedenotes()i(j0)+logj0j,wherei(j)istherelativeentropydensitydenedinSection..,and0istheproductoverallsitesofnormalizedcountingmeasure.(Thisthethephysicists'usualentropy,whichisdenedrelativetounnormalizedcountingmeasureonthesingle-spinspace0|thisaccountsfortheadditiveconstantlogj0j.)PropositionB.Fixaninteraction.Then:(a)Ifisatranslation-invariantw-zero-temperaturemeasurefor,thens()s. (b)Thereexistsforatranslation-invariantzero-temperatureGibbsmeasuresuchthats()=s.Wesummarizebelowtheresultsonwhichthispropositionisbased(PropositionB.andTheoremsB.,B.andB.part(b)).Wenotethatifthesupportofisaniteset,thens()=0[isoftheformPii!i,hences()(=jj)Piilogi!0].Therefore,weconcludethefollowing:PropositionB.Asuper-degeneratesystemwithnitelymanyrigidground-statecongurationsexhibitsinnitelymanyconvivialground-statecongurations.0

Thispropositioncoversallthecasesweknowofinwhichtheexistenceofconvivialground-statecongurationshasbeenproven.Consider,forexample,themodelwithspins!i=;0;andHamiltonianH=Xjijj=(!i!j):(B:)Theground-statecongurationsforthismodelareallthecongurationswithnospinequaltozero,andtheall-\0"conguration.Thezero-temperatureentropyforthismodelisexactlylog.Astheall-\0"congurationistheonlyrigidone,weconclude,bythepreviousproposition,thattheremustbeinnitelymanyconvivialground-statecongurations.AnotherimportantexampleistheIsingmodelwithnearest-neighborantiferromagneticcouplingofstrengthJandmagneticeldh=djJj.Itsground-statecongurationsarethoseinwhichnotwonearest-neighborspinsaresimultaneously\",afactthatproducesanon-zeroresidualentropy.Therearenorigidcongurations,hencethepropositionimpliestheexistenceofmanyconvivialones.Forthismodel,suchafactcanbeprovenalsobyadierentargumentwhichyieldssomeadditionalinsight.Indeed,byidentifyinga\"spinwiththepresenceofaparticle,theensembleofground-statecongurations|withtheassociatedconditionalprobabilitiesgivingequalweighttoallofthem|isseentocorrespondtothegrand-canonicalensemblefortheideallatticegaswithnearest-neighborexclusionandchemicalpotentialequaltozero.Usingabeautifulcomputer-assistedproof,Dobrushin,KolafaandShlosman[]provedthatsuchasystemhasanuniqueGibbsmeasure.Asnoneoftheground-statecongurationsarerigid,thisGibbsmeasureisnon-deterministicandthereforesupportedonconvivialcongurations.Thisexampleshowsaway(infact,theonlyoneweknowof)tointerpretandunderstandthecharacteristicsofnon-deterministiczero-temperatureGibbsmeasuressupportedon(verymany)convivialground-statecongurations:Onemapsitintoastatistical-mechanicalproblemforanother,betterunderstood,equivalentsystem.Astheoriginalensembleinvolvescongurationssat-isfyingsomeconditionderivedfromtheminimal-energyrequirement,thisequivalentsystemwill,ingeneral,beamodelwithexclusions.Thatis,itwillnottintothegeneralformalismdevelopedinChapter.PropositionB.doesnotyieldanyinformationonmodelswithzeroresidualen-tropy,forinstanceonferromagneticsystems.Inparticular,thequestionremainsofwhetherconvivialityrequiressuper-degeneracy.Apossiblecounterexampleispresentedinreference[]:Consider,forthethree-dimensionalferromagneticIsingmodel,theensembleofground-statecongurationsthatdieronlylocally(i.e.innitevolumes)fromthe\zig-zaginterface"one:!zigzag(t;t;t)=(+ift+t+t>0ift+t+t0:(B:0)Suchanensemblecanbemappedontoanappropriatesolid-on-solidmodel.IfthismodelcanbeproventohaveatleastoneGibbsmeasure(aproblemstillopen),thenitwouldimplythattheabovecongurationsareconvivial.Wemustacknowledgethat0

thestandingconjecture[,]isthatsuchGibbsmeasuresdonotexist.NotethatifthisGibbsmeasureexists,thentheall-\+"andall-\"congurationswouldbeatthesametimerigidandinthe(boundaryofthe)supportofahighlynon-deterministiczero-temperatureGibbsmeasure.Nevertheless,thereisaninterestingresult(Propositionof[])involvingmodelswithzeroresidualentropy:PropositionB.0Ifs=0,everytranslation-invariantw-zero-temperaturemeasureforissupportedonthesetofrigidground-statecongurations.Thatis,ifamodelwithzeroresidualentropydoesinfactpossessconvivialground-statecongurations,thensuchcongurationscanlieinthesupportonlyofnon-translation-invariantzero-temperatureGibbsmeasures.Remark.Ontheotherhand,Radin[0]hasshownexamplesofsuper-degeneratesystemswithauniquetranslation-invariantw-zero-temperaturemeasureentirelysup-portedonthesetofrigidground-statecongurations.Intheseexamples,thesetofground-statecongurationsdoesnothaveanyclosedtranslation-invariantpropersub-set,henceitisformedbyallthetranslatesofasingle(non-periodic)conguration,andlimitsofsuch.Thenon-zeroresidualentropyimpliesthatanytwosuchtranslatesmustdierininnitelymanysites,andhencetheyallmustberigidgroundstates.However,thisphenomenoncanhappenonlyinthepresenceofinnite-rangeinteractions(albeitdecreasingarbitrarilyfastwiththerange)[0].B..SuperuousGround-StateCongurationsThelasttypeofground-statecongurationsarethosethatarenotinthesupportofanyzero-temperatureGibbsmeasure.Theseareobviouslytheleastinterestingones,andweshallcallthemsuperuousground-statecongurations.Themostimmediateexampleisprovidedbytheone-dimensionalferromagneticIsingmodel.Itsground-statecongurationsaretheall-\+",all-\"andtheat-interfacecongurations.However,allthezero-temperatureGibbsmeasuresareoftheform++();theat-interfacecongurations(B.)aresuperuous.Heuristicallythisisbecausetheinterfaceisfreetowanderatnoenergycost;intheinnite-volumelimititwandersto.Theproofgoesasfollows:Denoteby+x0theindicatorfunctionofthecongurationwhichis+forx>><>>>:=(N+)if(N+)=+;N+=andNx0N+0otherwise(B:)[TherstlineisduetotheN+possiblepositionsx0forthe\kink"(lackofrigidity). ]Therefore,ifisazero-temperatureGibbsmeasure,thenforeveryNjx0jwehave(+x0)=;T=0(+x0j)0

=N+(!(N+)=+and!N+=)N+:(B.)LettingN!weconcludethat(+x0)=0.Thesameholds,ofcourse,fortheat-interfacecongurationswhichgofromto+.Astheground-statecongurationshereformacountableset(labelledbyx0[;]andthepolarityofthekink),itsmeasureisthesumofthemeasureofeachofitspoints.Therefore,(B.)impliesthatgivesfullmeasuretothesetformedonlybytheall-\+"andall-\"congurations;theat-interfacecongurationsaresuperuous.Combiningthiswiththeresultsstatedabove,weconcludethattheat-interfacecongurations(B. )aresuperuousind=,andrigidind.Theprecedingargumentrequiresnotonlythattherebeagrowingdegeneracyinthepositionoftheinterface,butalsothatthenumberofground-statecongurationsbenottoolarge,i.e.atmostcountable.Thissecondfactisnottrueforhigherdimen-sions.Indimensiontwo,forinstance,the\staircase-interface"congurationsformanuncountableset.Tobesure,thesetofstaircaseswithnitelymanystairsiscountable,andtheaboveargumentcanbeusedtoprovethatthissethasmeasurezeroforallzero-temperatureGibbsmeasures.Thiswouldprovethatthesupportofsuchmea-suresisalwayscontainedinthesetformedbytheall-\+",all-\",at-interfaceandinnite-staircasecongurations(thisbeingaclosedsetwhosecomplementhasmeasurezero).Butitdoesnotruleouttheoccurrenceofnon-deterministiczero-temperatureGibbsmeasuressupportedoninnite-staircasecongurations,similarlytowhatmayhappeninthethree-dimensionalIsingmodelforcongurationscloseto!zigzag.Nev-ertheless,wemustkeepinmindthatweareprimarilyinterestedinthosefeaturesofzero-temperaturephasediagramsthatsurviveat(cantellussomethingabout)lowbutnonzerotemperature.Therefore,forthetwo-dimensionalIsingmodelthepossibleexistenceofsuchazero-temperatureGibbsmeasurewithsupportoninnite-staircasecongurationsisaratherirrelevantissue,sinceithasbeenproven[,]thatonlytheGibbsmeasuresoftheform++()\survive"atnon-zerotemperatures.Thequestionofnon-deterministicGibbsmeasuresbecomesreallyimportantonlyfordimensiond.B..NonuniquenessofSpecicationsandInteractionsWeshallnowcommentononeimportantdierencebetweenthezero-temperatureandnonzero-temperatureformalisms:Atzerotemperaturethe\inverseproblem"|givenameasure,determinethespecicationand/ortheinteraction|isnolongerwell-posed:themapfrominteractions(orspecications)tozero-temperatureGibbsmeasuresis,ingeneral,many-to-one.Thislackofuniquenessappearsatthreedierentlevels:A)Therearemeasuresconsistentwithseveraldierentzero-temperaturespeci-cationssimultaneously.Theorem.doesnotapplyatzerotemperaturebecause0

thereare(large)opensetshavingzeromeasureforallzero-temperatureGibbsmea-sures.Therefore,byredeningthespecicationmoreorlessarbitrarilyonsuchopensetswecanobtainseveraldierentspecicationsforthesamezero-temperatureGibbsmeasure.Letuspresentanexplicitexample.Considerthenearest-neighborIsingmodelwithformalHamiltonianJPhxyi!x!yhPx!x.Thenthemeasure+isazero-temperatureGibbsmeasureforthefollowingzero-temperaturespecications:.Thespecication;T=0whereisdenedbyJ=0andsomeh>0..Thespecication;T=0whereisdenedbysomeJ>0andh=0.Nevertheless,;T=0=;T=0becausetheformerhas+asitsonlyzero-temperatureGibbsmeasure,whilethelatterhasboth+andaszero-temperatureGibbsmea-sures.B)Therearezero-temperaturespecicationswhicharisefromseveralnon-physically-equivalentinteractions(inotherwords,thenotionofphysicalequivalencebecomesmeaninglessatT=0).Wegivetwoexamples:.Trivialexample:Consideranyinteractionandanynumber>0.Thenandarenot(usually)physicallyequivalent,buttheyhavethesamezero-temperaturespecications..Lesstrivialexample:AllIsing-typepairinteractions(notnecessarilyferromag-netic)suchthathx>PyjJxyjforallxgiverisetothesamezero-temperaturespecication,namelytheonethatforeachnitesetandeveryboundarycon-ditiongives,inside,themeasureconcentratedintheall-\+"conguration.C)Thevariationalprinciple(SectionB..below)reducestotheminimizationofthespecicenergy,whichisnotastrictlyconvexfunctionalonBoranyofitssubspacesBh.B..Stabilityandw-StabilityZerotemperatureisinitselfunattainable.Soonereallyisinterestedinthosezero-temperaturefeaturesthat\survive"atlowbutnonzerotemperatures.Forinstance,oneisinterestedindeterminingwhicharethemeasuresthatcanbeobtainedasa!limitofpositive-temperatureGibbsmeasuresforaxedinteraction.Weshallrefertothesemeasuresasstablemeasuresfortheinteraction.Itissimpletocheckthatallthesestablemeasuresmustbezero-temperatureGibbsmeasuresfor:TheoremB.LetnbeGibbsmeasuresforaxedinteractionandasequenceofinversetemperaturesnwithn!+.Ifn!,thenthemeasureisazero-temperatureGibbsmeasurefor.However,noteveryzero-temperatureGibbsmeasureforagivenpotentialisneces-sarilystable.WecanillustratethisconceptwiththecaseoftheIsingmodel.Ford=,0

noneofthedeterministiczero-temperatureGibbsmeasures(+and)arestable.Infact,theonlystablemeasureis(++)=.FortheIsingmodelindimension,onlythemeasuresoftheform++()arestable.ThedeterministicGibbsmeasuresassociatedwiththerigidat-interfacecongurationsareunstable:atanynonzerotem-perature,theinterface\wanders"toandweareleftwithaconvexcombinationofthe\+"and\"phases[,].Fordimension,theat-interfacemeasureswereproventobestablebyDobrushin[](seealso[]).Remark:Thelow-temperatureGibbsmeasurefortheat-interfacephaseseems,atleastinnumericalexperiments,todisappearatatemperaturestrictlybelowthecriticaltemperature,givingrisetoarougheningtransition.Iftheabove-mentionedzero-temperaturemeasuresupportedneartheconguration!zigzaghappenstoexist,wecouldasktwoquestions:(i)DoesitsurviveforT>0(stability)?;and,ifso,(ii)DoesitfailtosurvivetoT=Tc?.Iftheanswertobothquestionswereyes,thentheIsingmodelwouldexhibitasecondrougheningtransition.However,asoureventualgoalisthestudyofhowthefullphasediagramdeformsasthetemperatureisraised,wemustconsideramoregeneralsituationinwhichtheinteractionisalsovariedasthetemperaturegoestozero.Thatis,wemustcon-siderthemoregeneralclassofmeasuresthatcanbeobtainedasa!limitofpositive-temperatureGibbsmeasuresforinteractionsn!.Weshallrefertosuchmeasuresasw-stablemeasuresfortheinteraction[].(Warning:Reference[]callsthesemeasuresstable. )Ingeneral,suchmeasuresneednotbezero-temperatureGibbsmeasuresfor.Forexample,iftotheantiferromagneticIsingmodelwitheldh=djJjconsideredaboveweaddanadditionaleldhn==n,weobtain,inthelimitn!,aGibbsmeasurecorrespondingtoalatticegaswithchemicalpotential.Allthesemeasuresaredierentamongthemselves,anddierentfromtheuniquezero-temperatureGibbsmeasurefortheIsingantiferromagnetinaeldh=djJj,whichcorrespondsto=0.Amoredramaticexamplewouldbetoadd,tothesamemodel,aeldh==pn.Themeasureobtainedinthelimitn!wouldthenbethemeasure+(alltheconditionalprobabilitiesT=0areequalto+),whichisnotazero-temperatureGibbsmeasureforbecausethereisnorigidground-statecongurationforthismodel.Thisexampleshowsthatthenotionofw-stabilityisper-hapsalittletoogeneral;forinterestingapplicationsoneusuallyconstrainsoneselftow-stabilitywithrespecttoapre-xedsetofperturbedinteractions.InSectionB..weshallmakeprecisethedesirablepropertiesofsuchperturbations.Atanyrate,itisimmediatethatallw-stablemeasureshavetheweakerpropertyofbeingsupportedontheground-statecongurationsforthegiveninteraction,i.e.theyareweakzero-temperaturemeasures:TheoremB.LetnbeGibbsmeasuresforasequenceofinteractionsnandasequenceofinversetemperaturesnsuchthatn!andn!+.Ifn!,then(fground-statecongurationsforg)=;(B:)i.e.isaweakzero-temperaturemeasurefor.0

Thenotionofzero-temperatureentropyinvolvesazero-temperaturelimit,henceitmusthavesomethingtosayaboutstability.Indeed,AizenmanandLieb[]haveproventhefollowing:PropositionB.Ifisastabletranslation-invariantzero-temperatureGibbsmea-surefor,thens()=s:Thisresult,togetherwithTheoremB.andthefactthatthesetoftranslation-invariantGibbsmeasuresisnon-emptyatalltemperatures,provesPropositionB.(b)above.Inthecaseofsuper-degeneratesystems(i.e.systemsforwhichs>0),PropositionB.canbeusedtoruleoutthestabilityofsomemeasures:CorollaryB.Forasuper-degeneratesystem,everytranslation-invariantzero-temperatureGibbsmeasuresupportedonanitesetisunstable.Forinstance,forthesystem(B.),theall-\0"Gibbsstateisunstable.B..Variational-PrincipleApproachThevariational-principleapproachforzero-temperaturemeasureswashistoricallytherstonetobeconsidered[].Atzerotemperatureitprovidesanevensimplercrite-rionthanatnon-zerotemperatures,becauseitreducestoaminimal-energycondition(F=ETSreducestoF=EifT=0).Foratranslation-invariantinteraction,itisnothardtoshowthatthelimitelim%jjinf!XAA(!)(B:)exists;wecallittheminimalspecicenergy(orground-stateenergy).Atranslation-invariantmeasuresatisfying(f)=e(B:)iscalledazero-temperatureequilibriummeasurefortheinteraction.Thedenitioncanbeextendedtoperiodicmeasuresiffincludesanaverageoverallthesitesofabasicperiod:IfisinvariantunderasubgroupSofZd,withZd=SisomorphictoanitesetPZd,thenonemustdenefjPjPxPPXxjXjX.Weshallassumethisextensioninthesequel.Schraderhasproven[]:TheoremB.(a)Everytranslation-invariantw-zero-temperaturemeasureforisazero-temperatureequilibriummeasurefor,i.e.satises(B.). (b)Conversely,everyzero-temperatureequilibriummeasureforisaw-zero-temperaturemeasurefor.0

Thatis,fortranslation-invariantmeasurestobesupportedon(local)ground-statecongurationsisequivalenttohavingminimalaverageenergydensity.Wenoticethat,unlikethenite-temperaturecase,wedonothaveanequivalencebetweenthevaria-tionalandtheGibbsian-specicationsapproaches;onlythemoregeneralw-measuresappearintheprevioustheorem.Therelationshipbetweenzero-temperatureGibbsmeasuresandequilibriummeasuresismuchmoreproblematic.Thevariationalapproachyieldsalsoacharacterizationofperiodicground-statecongurations:TheoremB..Foranyperiodicconguration!,thespecicenergy(energypersite)e(!)=lim%jjXAA(!)(B:)exists..Theinmumofe(!)overallperiodiccongurations!isniteandequalsthevalueedenedin(B.)..!isaperiodicground-statecongurationifandonlyife(!)=e[,].Forcompleteness,wementionalsotwovariationalprinciplesinvolvingtheminimalenergydensityandtheresidualentropy:TheoremB.(a)e=infM+;per(;F)(f)(B:)(b)[]s=supfs()jM+;per(;F)and(f)=eg(B.a)=supfs()jisaw-zerotemperaturemeasureforg:(B.b)Inparticular,(B.b)provesPropositionB. (a).The\inf"inpart(a)andthe\sup"inpart(b)areinfact\min"and\max",respectively.Theyarerealizedbythezero-temperatureequilibriummeasures.

B..InniteRangeandLackofQuasilocalityThevariational-principleapproachtozero-temperatureclassicallatticesystemscanbeextendedwithoutdicultytointeractionsinB0[,].TheextensionoftheDLRapproachtoinnite-rangeinteractions(e.g.inB)is,however,moreproblematic.Inparticular,thevalidityoftheimportantTheoremB.isanopenquestion:Asequenceofpositive-temperatureGibbsmeasuresforcouldconceivablyconvergetoalimitingmeasurethatisnotconsistentwiththezero-temperaturespecication(B.).Ifthislatterspecicationwerequasilocal,suchaphenomenoncouldnotoccur[,Theorem.];however,forlong-rangeinteractionsthespecication(B. )isingeneralnotquasilocal.Letusconcludethissectionwithanexampleshowingthislackofquasilocality.Consideranylong-rangeone-dimensionalIsingmodelwithpairinteractionsJxy=JjxyjsatisfyingPnjJnj<.Themodelhastobetrulylong-rangeinthesensethattheremustbeinnitelymanynonzerocouplingsJn;forsimplicityofnotationweassumethatJn=0foralln.Weclaimthatthezero-temperaturespecicationofsuchamodelisnon-quasilocal.Indeed,thezero-temperatureconditionalprobabilityforthespinattheoriginsatises:;T=0f0g(!0=+j)=><>:ifPx=0Jxx>0=ifPx=0Jxx=00ifPx=0Jxx<0:(B:)Toprovethatthisisnotaquasilocalfunctionoftheboundarycondition,weneedtoshowthatthereexistssome">0forwhichthefollowingistrue:Foraninnitesequenceofnestednitesetsthereexisttwoopensetsofcongurations,NandN0,formedbycongurationswhichareallidenticalinside,butsuchthat;T=0f0g(!0=+j);T=0f0g(!0=+j0)"(B:0)ifNand0N0.SuchsetsN,N0areconstructedasfollows:TakeN=[N;N]andxN0>NsuchthatXx>N0jJxj>><>>>:+ifxNifNxsgnJxifN+jxjN0anythingifjxj>N0:(B:)

ThesetsetN0isdenedanalogouslybutreplacingsgnJxbysgnJx.Wethenhave:Xx=0Jxx=Xjxj>NJxx=>>>>>>>><>>>>>>>>:N0Xjxj=N+jJxj+Xjxj>N0Jxx>0forNN0Xjxj=N+jJxj+Xjxj>N0Jxx<0forN0;(B:)wherethelastinequalitiesfollowfrom(B.).Therefore,by(B.),;T=0f0g(!0=+j);T=0f0g(!0=+j0)=(B:)ifNand0N0,andthespecicationisnotquasilocal.B.PhaseDiagramsB..RegularPhaseDiagramsThewords\phasediagram"areusuallyassociatedwithnicepicturesinwhichtwoconditionsaresatised:)OnlyperiodicextremalGibbsmeasuresareconsidered.Weemphasizethattheorderofthequaliershasbeencarefullychosen:themeasuresrelevantherearethoseextremalGibbsmeasuresthathappentobeperiodic;wearenotreferringtothemeasuresthatareextremalamongtheperiodicones(thislatterisalargerandless-well-behavedclass).Forshort,weshallcallthesemeasurespurephases,butweemphasizethatthisembodiesadoublechangewithrespecttotheterminologyadoptedintherestofthispaper:First,weconsiderallperiodicGibbsmeasuresonthesamefooting,whethertheyareinvariantunderthewholetranslationgroupZdormerelyanontriviald-dimensionalsubgroupofit.Second,weinverttheorderofthequaliers,thatis,wecallpurephaseanextremalmeasureinthesenseof(ii)inSection..,ratherthaninthemorecustomarysense(iii).Weshallfulllthisconditionthroughouttherestofthisappendix:by\phasediagram"wewillmeanthepartitionofacertainparameterspaceintoregionswithagivennumberandtypeofpurephases. )TheGibbsphaserule[]isobeyed.Letusexplaininalittlemoredetailwhatthismeans.AnexampleofaphasediagramsatisfyingtheGibbsphaseruleispresentedinFigurebelow:Thereisapointwherethreepurephasescoexist(pointofmaximalcoexistence),fromwhichthereemanatethreelineswheretwopurephasescoexist,whichinturnboundthreeopenregionsinwhichthereisonlyoneperiodicextremalGibbsmeasure.Suchaphasediagramwillbecalledregular.Moregenerally,anr-regularphasediagramconsistsof[0,AppendixA]:()apointofmaximalcoexistencewhererpurephasescoexist;

()rone-dimensionalopenmanifolds,eachboundedbythismaximal-coexistencepoint,whereexactlyrphasescoexist;()r(r)=two-dimensionalopenmanifolds,eachboundedbypairsofthepreviousone-dimensionalmanifolds,whereexactlyrpurephasescoexist;...(r)ropen(r)-dimensionalmanifolds,eachboundedbythe(r)-dimensional-phase-coexistencemanifolds,andsuchthattheclosureoftheirunionisthewholeparameterspace,wherethereisonlyonepurephase.Usually,thepurephasesaredenedbyxingtheboundaryconditionsaccordingtosomeparameter-independentsetKofreferencecongurations(or,moregenerally,mea-sures).Typically,Kisthesetofground-statecongurationsatthepointofmax-imalcoexistenceatT=0.Onecanthenlabeleachpurephaseaccordingtotheboundaryconditionemployedinitsdenition.OnecallstheK-stratum(KK)themanifoldinparameterspacewherethecoexistingphasesarepreciselythosela-belledbyelementsofK.Forinstance,inFigure,thedierentstrataarelabelledbytheboundaryconditions\+",\0"and\".Thereare,therefore,sevenstrata:f+g;f0g;fg;f+;0g;f+;g;f0;g;f+;0;g.Amoreabstract(topological)wayofvisualizingsuchaphasediagramisprovidedbythefollowingequivalentcharacterization:ar-regularphasediagramisadiagramthatcanbehomeomorphicallymappedontotheboundaryofthepositiveoctantinrdimensions,@Qr=n(t;:::;tr)Rd0:minirti=0o;(B:)insuchawaythatthepointofmaximalcoexistencecorrespondstotheorigin,thecurvesof(r)-phasecoexistencecorrespondtothepositivecoordinateaxesexcludingtheorigin,:::,theopensetswithonlyonepurephasecorrespondtothe(r)-dimensionalcoordinatehyperplanesexcludingtheir(r)-dimensionalboundaries.Inbrief,thedierentstrataaremappedintothedierentsubmanifoldsoftheboundaryofther-octant.GeneralphasediagramsneednotobeytheGibbsphaserule.Atypicalsituationisforsomeofthepurephasestoalwaysappeartogetherthroughoutthediagram.Suchasituationiscalledadegeneracy,anditisusuallyassociatedtosomesymmetryofthesystem(ifnosymmetrycanexplainit,thedegeneracyiscalledfortuitous).Theadditionoffurtherinteractions(notrespectingthesymmetry)canproduceaphasediagramwithoutdegeneracy.Theseextrainteractionsaresaidtobreakthedegeneracyofthepurephasesinquestion.Aninteractionissaidtocompletelybreakthedegeneracyofthepurephasesifitsadditionyieldsaregularphasediagram.B..Zero-TemperatureRegularPhaseDiagramsForzero-temperaturephasediagrams,itisrelativelysimpletogiveconditionsontheextrainteractionsneededtoensurearegularphasediagram.Indeed,atzerotem-

peraturedegeneracymeansequalspecicenergyforallvaluesoftheparameters,anditsbreakinginvolvesaddinginteractionsproducingadierentsetofspecicenergiesforeachoftheinitiallydegeneratepurephases.Thisisusuallydoneperturbatively,thatis,eachadditionalinteractionismultipliedbyanoverall\turn-on"parameter.Supposewestartwithaninteraction0havingrdegeneratezero-temperaturepurephases;:::r.Then,tocompletelybreakthedegeneracyoneusuallyconsidersradditionalinteractions;:::;randconstructsthe\perturbed"interactions=0+rXi=ii:(B:)[Examples:(i)FortheIsingmodelatzeroeld,=h;(ii)fortheBlume-Capelinteractiondenedby(B.)below,=gand=hinthe\perturbation"(B.).]Theparameters=(;:::;r)usuallytakevaluesinacertainneighborhoodoftheorigin.Thedegreeofdegeneracyfortheperturbedinteractiondependsonther-tupleofspecicenergiese()=(e();:::;e(r)):(B:)Infact,ifwedenoteQ()=fi:iminimizese()g:(B:)thenthestrataofthezero-temperaturephasediagramarethesetsSK=f:Q()=Kg(B:)foreachsubsetoflabelsKf;:::;rg.Theperturbedinteractioncompletelybreaksthedegeneracyifthephasediagramformedbythestrata(B. )isr-regular.Itisofinteresttotranslatetherequirementofregularityintoconditionsontheperturbationsi.Onewaytodoitistonoticethat,asthespecicenergydependslinearlyontheparametersi,itcanbewrittenintheforme()=rXi=ie(i);(B:0)withe(i)=(ei();:::;ei(r)):(B:)Oneoftheconditionsforthephasediagramtober-regularisthattheorigin=0betheonlypointofmaximalcoexistence.Thisimpliesthatnononzerovectoroftheform(B.0)canhaveallitscoordinatesequal.Alittlebitoflinearalgebrashowsthatalltheotherconditionsforregularityaresatisedifthevectorsfe(i)girare,inaddition,linearlyindependent.Therefore,theperturbations;:::rcompletelybreakthedegeneracyof0ifandonlyifthevectorse(i)arelinearlyindependentandtheydonotspanthevector(;:::;)Rr.Alternatively,ifweresorttothepreviousgeometricaldescriptionofregularity,weconcludethatitisequivalenttorequirethatthevectore()|shiftedsoitalwayshas

atleastonecoordinateequaltozero|sweepsovertheboundary@Qrofthepositiveoctant.Preciselystated,ifwedenoteee(i)=e(i)minjre(j);(B:)theperturbationcompletelybreaksthedegeneracyof0ifandonlyifthemap!(ee();:::ee(r))(B:)isone-to-one.Inotherwords,ifsuchamapisabijectionfromaneighborhoodof0Rdtoaneighborhoodof0@Qr.Foreachparticularvalueof,thecoexistingpurephasesarethoseiwithee(i)=0.B..Low-TemperaturePhaseDiagrams.ScopeofPirogov-SinaiTheoryIfnatureisfair,oneexpectsthatlow-temperaturephasediagramslookverysimilartothecorrespondingzero-temperatureones.Thisisnotalwaysso,however,andthequestionofstabilityorw-stabilityofGibbsmeasuresisanimportantissue.Pirogov-Sinaitheoryhasbeenpreciselydesignedtosingleoutsomeimportantcasesinwhichindeedthelow-temperaturediagramsareonlyasmalldeformationoftheonesatzerotemperature.Whenthetheoryapplies,oneisguaranteedthattheregularityofthediagramispreservedatleastforsmalltemperatures;and,furthermore,thatthelow-temperaturepurephaseslookvery\similar"tothezero-temperatureones.AsaninputtothePirogov-Sinaitheoryonemustdeterminethezero-temperaturephasediagramandshowthattwokeyhypothesesaresatised.Thersthypothesisreferstothenumberofzero-temperaturedeterministicpurephases:Initsoriginalversion[,],Pirogov-Sinai(PS)theoryappliestoasystemwithanite-rangeperiodicinteraction,exhibitinganitenumberofperiodicrigidground-statecongu-rations.(Thishassubsequentlybeengeneralizedtosomeextent:seeSectionB... )Thesecondhypothesisisthattheinteractionsatisfytheso-called\Peierlscondition",tobestatedmorepreciselybelow,whichroughlyrequiresthatforeachrigidperiodicground-statecongurationtheenergycostofintroducingadropletofspinsalignedasinadierentgroundstatemustgrowtypicallyastheareaoftheboundaryofthedroplet.Thisconditionallowstheenergycostofcreatingexcitationstobeattheen-tropygain,preservingthelong-rangeorderobservedatzerotemperature.However,thePeierlsconditionhasthisdesiredeectonlyford.Thetroubleisthatford=thesizeoftheboundaryofasetdoesnotgrowwithitsvolume.Therefore,Pirogov-Sinaitheoryisnotapplicabletoone-dimensionalmodels.Ontheotherhand,fordthePeierlsconditioniscertainlystrongerthannecessary:thereexistmodelswithanitenumberofrigidperiodicground-statecongurationswhichhaveanon-trivialphasediagramandwhichdonotsatisfythePeierlscondition[0,].Nevertheless,Insomesenseitisthestrongestpossiblecondition:seethecommentsafterDenitionB.below.

thePeierlsconditionappliesinalargenumberofinterestingmodels,andallowsaveryprecisedescriptionofthelow-temperaturebehavior.Theoutputofthetheoryisafamilyofresultsinvolvingextensionstonon-zerotemperatures.ThemainresultofthetheoryisthatforasystemsatisfyingthePeierlsconditionthephasediagraminvolvingtheseperiodicdeterministicmeasuresisstable:Asthetemperatureincreases,thecoexistencemanifoldsdeformcontinuously(infactanalytically).Moreover,thetheorymakesrigoroustheintuitivepictureofwhateachlow-temperaturepurephaselookslike:itstypicalcongurationsconsistofa\sea"ofspinsalignedasintheground-statecongurationwithsmallandsparse\islands"ofoverturnedspins.Weremarkthatthetheorydoesnothaveanythingtosayaboutthestabilityofthe(possiblyinnitelymany)non-periodicground-statecongurationsandthezero-temperatureGibbsmeasurestheysupport(butseeSectionB..).Othertechniquesareneededtoshow,forexample,thattheat-interfaceground-statecongurations(B. )|whicharerigidford|areunstableford=[,,]andstableford[,].Moreover,theoriginalversionofPStheorygivesonlyverylimitedinformationastothespecicsofthedeformationofthephasediagram;inparticularitdoesnotproduceausefulcriteriontodeterminewhichpurephasesarestableforthedierentregionsofthezero-temperaturephasediagram.Thatis,itdoesnottellusinwhichdirectionthephaseboundariesmovewhenthetemperatureisraisedfromzero.Therefore,forinteractionslyingonaphase-transitionmanifoldofthezero-temperaturephasediagram,theoriginalPStheorydoesnottellusinwhichphase(s)endsupatT>0;thatis,itdoesnottelluswhichone(s)ofthecoexistingzero-temperaturepurephasesis/arestable,andwhichareonlyw-stableforthefamilyofinteractionsadopted.However,Slawny'sextensionofPStheory[]providesthisadditionalinformation.Toclarifythispoint,letusborrowaveryinstructiveexamplefromthereviewbySlawny[].Considerthespin-Blume-CapelmodeldenedbytheformalHamilto-nianH0=Xhxyi(!x!y);(B:)where!x=;0;andthesumisoverpairsofnearest-neighborsitesinZd,d>.Suchamodelhasthreeperiodic(infacttranslation-invariant)ground-statecongu-rations:all-\+",all-\0"andall-\".Theyareallrigid.Toobtaina-regularphasediagramonecanconsider,forinstance,thefamilyofinteractionsdenedbytheformalHamiltoniansH(g;h)=H0gXx!xhXx!x:(B:)Thecorrespondingzero-temperaturephasediagramispresentedinFigure(a).Pirogov-SinaitheorytellsusthatforT>0lowenoughthephasediagramisjustacontinuousdeformationoftheonedepicted,buttoconcludethatsuchdeformationslookasinFigure(b)weneedsomeextrainformationwhichisnotdirectlyobtainablefromPStheory,althoughitisprobablycontainedinit.Thisextrainformationispresentedex-

plicitly,forinstance,inSlawny'stheoryofasymptoticsofphasediagrams[].Fromthelatterdiagramwesee,forinstance,thatofthethreedeterministicpurephasesofH(g=0;h=0)onlytheall-\0"isstable,whiletheothertwopurephasesarew-stable.Thewell-studiedferromagneticIsingmodelprovidesanexampleofanexceptionalna-ture:itsphasediagramremainsundeformedatlowtemperatures;forallvaluesofthemagneticeldtheperiodiczero-temperatureGibbsmeasuresarestable.B.Pirogov-SinaiTheoryWesummarizenowthemainaspectsofPStheory.Inthersttwosubsectionswecarefullydiscussthebasichypothesesrequiredbythetheory;inthethirdsubsectionwepresentasomewhatdetailedaccountoftheresults(fornite-rangeinteractions).Ofcourse,weomitallproofs;thesecanbefoundinthereferencescited.Asalreadypointedout,thetheorydoesnotapplyford=,thereforeintherestofthisappendixwerestrictourselvestod.B..BoundaryofaConguration.ThePeierlsConditionTypicalcongurationsofalow-temperaturepurephaseareexpectedtobesmalluc-tuationsaroundthoseofacorrespondingzero-temperaturepurephase.Theseuctua-tionsresultintheappearanceof\droplets"(\bubbles",\islands")ofspinsalignedac-cordingtoadierentzero-temperaturepurephase|or,moregenerally,a\metastablephase"[]aswediscussbelow.Thesedropletsaresurroundedbyatransitionalre-gionor\boundary"ofsetsofspinsnotalignedaccordingtoanyzero-temperaturepurephase,whichthereforeraisestheenergyoftheconguration.Theprobabilityofsuchuctuationsisdeterminedbythecompetitionbetweentwofactors:theenergycostofintroducingaboundaryandtheentropygainduetothedierentpossibleshapesandlocationsofthedroplets.Iftheenergycostislargeenoughtoovercome,atlowtemperatures,theentropygain,theneachzero-temperaturepurephasegivesrisetoalow-temperatureonewhichdiersonlyinthepresenceoffewandsmalldropletsofoverturnedspins.Inparticular,thiswouldprovethattherearepreciselyasmanycoexistingpurephasesatlowtemperaturesasthereareatzerotemperature,andhencethatthereisaphasetransition.ThistypeofargumentwasrstintroducedbyPeierls[,,]toprovetheexistenceofaphasetransitioninthed-dimensionalIsingmodelford,andhenceitisoftenreferredtoasthe\Peierlsargument".Pirogov-Sinaitheoryisageneralization(and,thus,amoreabstractversion)ofsuchanargument.ToformulatethePeierlsargumentinarigorousformweneedacriteriontodeter-minewhentheenergycostofaboundaryis\largeenough"todefeattheentropygain.ThePeierlsconditionispreciselyonesuch(sucient)criterion.Itrelies,however,onasuitabledenitionofthe\boundary"ofaconguration,whichisnotauniquelydenedconcept.Infact,twocomplementarynotionsareintroducedatthisstage:thebound-ary,whichroughlycorrespondstothecollectionofsiteswherethespinsaremisaligned,andthecontours,whicharethedierentcomponentsofthisboundarytogetherwith

hg(a)"+"-pure phase"−"-pure phase"0"-pure phasetwo-phase coexistencethree-phase coexistencehg(b)"0"-pure phase"+"-pure phase"−"-pure phaseFigure:Phasediagramsofthemodelwithinteraction(B.).(a)Zerotemperature. (b)Lowtemperature.

thecorrespondingspincongurations.Thelatterallowacompletedeterminationoftheenergyofagivenconguration.Letusmotivatethegeneraldenitionsviaexamples.Intheoriginalcaseofthefer-romagneticIsingmodel,theboundaryofacongurationcan,forinstance,bedenedasallthepairsofnearest-neighborsiteswithoppositespins.Aseachsuchpaircontributesequallytotheenergyoftheconguration,regardlessofwhichspinofthepairisupandwhichisdown,onedoesnotneedtospecifytheactualcongurationontheboundarytocomputetheenergy.Thereforecontoursaredenedwithnoreferencetocongura-tions,byconsideringthepolyhedralsurfaceformedbyplaquettesperpendiculartothebondsjoiningmisalignedspins,andtakingitsconnectedcomponents[,].ThesamedenitionofboundaryworksfortheBlume-Capelmodel(B.),buttocomputetheenergywenowmustspecifythecongurationofeachpairofmisalignedspins,asdierentcombinationshavedierentenergies.Thedenitionofcontoursrequireshencetoconsiderpolyhedralabeledbythecongurationontheimmediatelyadjacent(internalandexternal)shellsofspins.Thenextcomplicationappearsformodelswithinteractionsextendingbeyondnearestneighborsand/orinvolvingmorethantwospinsatatime.Anexampleofpracticalinterestistheantiferromagnetonaface-centeredcubiclattice[,p.andreferencestherein].Suchmodelsrequire\thicker"bound-ariesandcontoursdenedspecifyingthecongurationsoflargergroupsofspins.Therefore,todeneboundaryandcontoursinageneralfashion,wemustcheckwhethersetsofspinsarealignedormisaligned,butthischeckinghastobedoneonsucientlylargecollectionsofspinsatatime.Followingclosely[,ChapterII],weconsiderasetK=f!();:::;!(r)gofperiodiccongurations(r).ForthestandardstatementofthePeierlsconditionKwillbethesetofperiodic(deterministic)ground-statecongurationsofsomeinteraction,butthedenitioncanbedone(andmustbedone,asweshalldiscussinnextsection),forgeneralsetsofperiodiccongurations.LetuscallKthesetofreferencecongurations[].Wealsoconsiderforsomexeda0thecubesWa(x)=fyZd:;jyixijaforidg|thesamplingcubes.DenitionB.Theboundaryofaconguration!|withrespecttothesetofref-erencecongurationsKandsamplingcubesWa(x)|isthesetofsites@!=[xZdnWa(x):!jWa(x)=!jWa(x)!Ko:(B:)Typicallywewillconsidercongurations!equaltosome!Kexceptforanitesetofspins.Inthissituationtheboundaryisaniteset.LetusnowstatethesimplestandmostpopularversionofPeierlscondition;inthefollowingsectionwediscussamoregeneraldenition.Weconsideraninteraction0and,foreachxed!KconstructtherelativeHamiltonianH0(!j!)=XA:AZdnite[0A(!)0A(! )];(B:)denedonlyforcongurations!coincidingwith!exceptonaniteset.LetusdenoteGperT=0(0)thesetofperiodicground-statecongurationsof0.0

DenitionB.Theinteraction0satisesthe(original)Peierlsconditionifthereexistsaconstant0>0suchthatforeach!GperT=0(0)H0(!j!)0j@!j(B:)foreveryconguration!coincidingwith!exceptpossiblyonanitesetofsites.Here@!istheboundaryof!withrespecttoK=GperT=0(0)andsamplingcubesdenedbysomexedchoiceofa0.Weshallcallaconstant0satisfying(B.)aPeierlsconstantfortheinteraction0(andthechosenKanda).ThePeierlsconditionimmediatelyimpliesthateachperiodicground-statecongurationisrigid,andhencedenesadeterministiczero-temperaturepurephase(SectionB..).Theconverseisnottrue[0,].WealsonoticethatanupperboundoftheformH0(!j!)e0j@!jisalwaystrue,hencethePeierlsconditionisbasicallyarequirementfortheenergycosttogrowasfastaspossiblewiththesizeoftheboundaryoftheconguration.Thereareimportantmodelswherethisisnottrue,i.e.inwhichtheenergycostgrowsmoreslowlythantheareaoftheboundary:forexample,thebalancedmodel[,]andtheANNNImodel[,andreferencestherein]).WenoticethatthevalidityofthePeierlsconditiondoesnotdependontheparticularchoiceoftheparametera0adoptedforthedenitionoftheboundary,buttheactualvalueofthePeierlsconstantdoes.Indeed,following[]wenoticethatif@0!indicatestheboundarydenedviasamplingcubesWa0witha0a(othercaseslefttothereader),then@!@0![x@!Wa0(x)(B:)thus,j@!jj@0!j(a0+)dj@!j:(B:0)Therefore,dierentchoicesofachangetheactualvalueof0:0(a0+)d00=0sup!j@!jj@0!j0;(B:)butnotitsnonzerocharacter.Onehasthefreedomofadjustingaaccordingtofutureconvenience.However,theactualvalueof0isrelatedtotherangeoftemperatureswherePStheoryisvalid(thisrangeisproportionalto0).Hence,forthesakeofquan-titativepredictionsoneshouldemployavalueof0aslargeaspossible,whichmeansaassmallaspossible.AnextremelyfavorablecaseisexempliedbytheferromagneticIsingmodel,anditsgeneralizationstohigherspins,forwhichtheboundaryofcong-urationscanbedenedviapolyhedraof\zerowidth"[andwemayevenhaveequalityin(B. )].Strictlyspeaking,thecorresponding\zero-width"(orthin)contoursarenotincludedintheformalismtobeintroducedbelow,butweshallkeepthemwithinourdiscussionthroughappropriatecomments.TheactualvericationofthePeierlsconditionisamodel-dependent,generallynontrivial,procedure.Thestartingpointis,inprinciple,thedeterminationofall

periodicground-statecongurations|anoftentediousprocess.Aslightsimplicationfollowsfromtheobservationthatifwendthat(B.)issatisedforsomenitesetKofperiodiccongurations,automaticallythesemustbealltheperiodicgroundstates.Indeed,(B.)impliesthatsuchcongurations!aregroundstates,andiftherewereothers(B.)wouldnotbesatisedbecausearbitrarilylargeboundariescouldbeconstructedwithoutextraenergycost,simplybyinterposingregionsoccupiedbythegroundstatesnotaccountedfor.Inpractice,thisobservationisoflittlehelp,asthedeterminationofgroundstatesismadeusingsomesortofcontourideas,socheckingthePeierlsconditionandndingtheground-statecongurationsarealmostsimultaneousprocesses(however,see[]).TheonlyrealshortcutavailableisasucientconditionduetoHolsztynskiandSlawny[]whichwewilluseforalmostalltheapplicationsinthispaper.DenitionB.0Apotentialisanm-potentialifthereexistsaconguration!simultaneouslyminimizingeachjAj-bodyfunction:A(!)=mine!A(e!)AS:(B:)ForsuchaninteractionletusdenotebyGT=0()the(nonempty)setofcongurationsminimizingallA.Thesucientconditionis:TheoremB. (Holsztynski-Slawny)Anite-rangem-potentialwithGT=0()nitesatisesthePeierlscondition.Resortingtoanalternative|andsuggestive|terminology,wecansaythatanm-potentialisoneforwhichtherearegroundstates\satisfying"allbonds.Anim-mediateexampleisanyIsingmodelwithferromagneticinteractions(A=JAAwithJA0forallA):clearlytheall-\+"congurationsimultaneouslyminimizesallA.Theoppositecaseisthatofthepotentialswith\frustration",i.e.forwhicheverycongurationhasbondsthatgiveanenergycontributionlargerthantheminimumpossible(\frustratedbonds").However,thesenotionsofm-potentialsand\frustra-tion"mustbetakenmodulophysicalequivalence,becauseequivalentpotentialshavethesamestatistical-mechanicalproperties.Thisaddsanextratwisttothematter.ApopularexampleistheantiferromagneticIsingmodelinatriangularlattice.Itiseasytoseethatwhenthemodelisgivenitsusualformulationintermsoftwo-spininter-actions,nocongurationcan\satisfy"simultaneouslythethreebondsofatriangularplaquette.Butthisseeminglyfrustratedpotentialcanequivalentlybewrittenbycon-sideringthetriangularplaquettesthemselvesasthebonds,withanenergycontributionobtainedbyasuitablecombinationofthecontributionsoftheoriginaltwo-spinbondsaroundtheplaquette.Inthisformulationthemodelisnowanm-potential(althoughonecannotuseTheoremB.becausethereareinnitelymanyperiodicground-statecongurations).Inthisregard,probablythemostdicultaspectoftheapplicationofthisveryconvenienttheoremisthevericationofwhetherthepotentialofinterestcanberewrittenas(i.e.isphysicallyequivalentto)anm-potential.ItwouldbeverynicetocomplementTheoremB.withsomesimplesucientcriterionforaninteractionto

bephysicallyequivalenttoanm-potential,butthismaynotbeaneasytask.Forin-stance,thenaturalconjecturethateverynite-rangetranslation-invariantinteractionisequivalenttoa(translation-invariantnite-range)m-potentialisfalse[].Atanyrate,oncethem-potentialcharacterhasbeenveried,TheoremB.isanextremelyconvenienttool.Ithas,however,animportantdrawback:itsproofisnotconstructive,soitdoesnotprovideanyexplicitexpressionforthePeierlsconstant.Therefore,argumentsbasedonthistheoremdonotallowanydeterminationoftherangeoftemperatureswherethePStheoryremainsvalid.B..Contours.TheGeneralizedPeierlsConditionInthepresenceofthePeierlsconditionforaninteraction0,theusualPeierlsargumentcanberepeatedforthosezero-temperaturepurephasesof0forwhichtheentropyfactorcanbeshowntogrowatmostexponentiallywiththesizeoftheboundary.Indeed,thePeierlsconditionensuresthattheenergycostgrowsasleastasfastbutwithanexponentincludingafactor,hencetheenergycostbeatstheentropygainforlargeenough,andonlysmallboundariesarepresent.However,thisenergy-beats-entropyphenomenonisingeneralnottrueforallthepurephases,onlyforthestableones.Itturnsoutthattoobtainasituationinwhichtheentropyisbeatenbytheenergyforalltherigidperiodicground-statecongurationsof0|andhenceallofthemcoexist|onemustconsideraperturbedinteraction=0+eforasuitablyadjustede(shiftinthepointofmaximalcoexistence).Ingeneral,notalltheground-statecongurationsfor0areground-statecongurationsfor,hencethisprocessof\tuning"requiresustoconsiderasetKnotreducedjusttocongurationswithminimal-energy.AnotherreasontogeneralizethePeierlsconditionappearswhenstudyingwholeregionsofthephasediagram.Insuchasituationoneisinterestedinestimatesvaliduniformlythroughouttheregion;butauniformPeierlscondition,asstatedinDe-nitionB.,isnotingeneralpossible.Forexample,considertheIsingmodelinthepresenceofastrictlypositivemagneticeld.Theonlyground-statecongurationistheall-\+"|tobedenoted!(+)|andhencej@!jisproportionaltothenumberof\"present.Forinstance,forthecongurations!Wequalto+everywhereexceptinsideacubeW,therelativeenergyisH(!Wj! (+))=Jj@Wj+hvol(W),whilej@!Wjvol(W).AsimplecalculationshowsthatforthePeierlsconditiontobevalidforallthese!Wweneed

uniformityrequirementisimportantforourexampleoftheKadanotransformation(Section..).Infact,thisapplicationdemandsonlyaparticularcaseofuniformity(CorollaryB.below),and,moreover,theresultweneedisexactlygivenbyatheoremduetoZahradnk(TheoremB.0below).However,weshalltakeherethetimetodiscusstheuniformityissueinsomegenerality,becausewefeelthatithasnotbeensucientlyemphasizedintheliterature.Letusrstintroducethenotionofcontour.WexasetKofreferencecongura-tionsandachoiceofsamplingcubes(valueofa).Theideaistodecomposethebound-aryincomponents:twosetsAandBofsitesarecalledconnectedifdist(A;B)inlatticeunits.Acontourofaconguration!isapair=(M;!M)whereMisamaximallyconnectedcomponentoftheboundaryof!.ThesetMisoftencalledthesupportofthecontour.Atthispointwestartintroducingconstraintsonthesizeofthesamplingcubes.Werequire:(C)Thevaluea+mustbestrictlylargerthanalltheperiodsofthereferencecongurations!K.Sucharequirementimpliesthefollowingextensionproperty(nomenclaturetakenfrom[]):ifaconguration!coincideswithareferenceconguration!KonthesamplingcubeWa(x)andwith!0KonthecubeWa(y)withdist(x;y),then!=!0.Thishasthekeyconsequencethatwecanreconstructuniquelyaconguration!startingfromitsfamilyofcontours.EachcontourwithanitesupportdividesZdnMintoseveraldisconnectedcom-ponents:Oneofthemisunbounded,andiscalledtheexteriorofthecontour;theothersareboundedandarecollectivelycalledtheinteriorofthecontour.Eachofthesecomponentshasareferencecongurationassociatedtoit,namelythatofthesamplingcubescenteredonsitesadjacenttothesupportofthecontour.Thecontourisa!-contourifitsexteriorcorrespondstothereferenceconguration!.Ontheotherhand,the!(i)-interior|denotedInt!(i)()|istheunionofthecomponentsoftheinteriorofassociatedtoareferenceconguration!(i).Ingeneral,!willhaveothercontoursbesides,someofwhichmaybeintheinteriorof.Hence!maynotcoincidewith!(i)onthewholeInt!(i)ThegeneralizedPeierlsconditionisarequirementontheminimumenergycostofintroducingacontour.Thiscanbeestimatedbyconsideringtheconguration!thathasasitsonlycontour.If=(M;!M)isa!-contour,!coincideswith!ontheexteriorof,with!(i)onthewholeInt!(i),andwith!MonthesupportMLetusintroducenowaperiodicinteraction.Theenergycostofthe!-contourisgivenbytherelativeenergyH(!j!),whichcanbedecomposedintheform:H(!j!)=()+rXi=[e(!(i))e(!)]jInt!(i)j:(B:)ThesecondtermintheRHSis,uptotermsproportionaltoj@Mj,theenergycontri-butionduetothecongurationsintheinteriorof.Thistermisabsentifallthe! (i)areground-statecongurationsof.Thecontourfunctional()isdenedbythe

identity(B.);itisroughlyequaltoPAM[A(!)A(!)],butitalsoincludesthejustmentionedtermsproportionaltoj@Mj.DenitionB.AninteractionsatisesthegeneralizedPeierlscondition|withrespecttoasetKofreferencecongurations|ifthereexistsaconstant>0suchthatforeachcontour=(M;!M).()jMj(B:)The(original)Peierlscondition(B.)correspondstotheparticularcaseinwhichK=GperT=0().Weremarkthatthisgeneralizedconditionissometimescalledjust\Peierlscondition",or\Gerzik-Pirogov-Sinai"condition.WeshallalsocallaPeierlsconstant|fortheinteraction|aconstantsatisfying(B.).TounderstandwhyDenitionB.hasthedesireduniformity,letusreturntotheexampleoftheIsingmodelwithmagneticeldh>0.WemustnowconsiderK=f!(+);!()g,where!(+)and!()aretheall-\+"andall-\"congurationsrespectively.Wenotice,however,that!()isnotagroundstate.WiththischoiceofK,thecontourscanbetakentobe\thin"asinthezero-eldcase,andwehavethatforany!(+)-contourH(!j!(+))=Jj@Wj+hjInt!()()j;(B:)whileforan!()-contourH(!j!())=Jj@WjhjInt!(+)()j:(B:)So,comparingwith(B.)weseethatthegeneralizedPeierlsconditionissatisedwith=J,uniformlyinh.Weseethatthisuniformityisgainedbyincludingtheextraconguration!()whichisnotagroundstate,butrathercouldbeinterpretedasa\metastablestate".Ingeneral,theuniformitypropertyofthegeneralizedPeierlsconditionisaconse-quenceofanestimatevalidforsamplingcubeslargerthantheperiodandrangeoftheinteraction;thatis,weimposethefollowingextraconditiononthesamplingcubes:(C)Thevaluea+mustbestrictlylargerthantheperiodandtherangeoftheinteraction.Weemphasizethatduetorequirements(C)and(C),thevaluechosenfora|thatis,thedenitionofthecontours|dependsonthesetKandontheinteraction(s)present.Thereadershouldkeepthisinmindespeciallybecause,tokeepformulassimpletoread,thenotationwillnotmakethisdependenceexplicit.Inparticular,achangeintheinteraction|forinstancetheadditionofanarbitrarilysmallperturbation|willrequiretheredenitionofthecontours.Undercondition(C),XA:A\hInt!(i)[@extInt!(i)i=?A(!)jInt!(i)je(!(i))(a+)dkkB0j@extInt!(i)j(B.)

whichimpliesthefollowingkeyestimate.If=(M;!M)isaa!-contour,thenforanyperiodicinteractioneHe(!j!)rXi=jInt!(i)j[ee(!(i))ee(!)](a+)dkekB0jMj:(B:)Therefore:TheoremB.(Uniformityproperty)Ifaperiodicinteraction0satisesthegeneralizedPeierlsconditionwithconstant,thenforanyinteractionewithkekB0c=(a+)d,thesum0+esatisesthegeneralizedPeierlsconditionwithconstant(c).Anotherusefulresult,whichbasicallyfollowsfrom(B.),isthefollowing[,Lemma.]:PropositionB.Consideraperiodicinteraction0satisfyingtheoriginalPeierlscondition(B.)withconstant0.Then,foranyotherperiodicinteractionekekB0<=(a+)d=)GperT=0(0+e)GperT=0(0):(B:)WepresenttwocorollariesofTheoremB..ForourstudyoftheKadanotransfor-mationweneedthefollowingtrivialconsequence:CorollaryB.Consideraperiodicinteraction0satisfyingtheoriginalPeierlscondition(B.)withconstant0,andanotherinteractionesuchthatGperT=0(0)=GperT=0(0+e).ThenifkekB0c0=(a+)d,thesum0+esatisestheoriginalPeierlsconditionwithconstant0(c).However,thecorollarymoreoftenusedis:CorollaryB.If0satisestheoriginalPeierlsconditionwithconstant0[andK=GperT=0(0)],thena\perturbed"interaction=0+esatisesthegeneralizedPeierlsconditionwithconstant0(c)[andthesameK]ifkekB0c0=(a+)d.ThiscorollarygeneralizeswhatwasobservedregardingtheIsingmodelinnon-zeroeld.Astheinclusionin(B.)isingeneralstrict,thelastcorollaryimpliesthat,fromthepointofviewof=0+e,theuniformityisgainedatthecostofincludingsomeextrareferencecongurationsthatarenotgroundstates(e.g.! ()intheaboveexample).Theseextracongurationscanbeinterpretedas\metastablestates"or\localgroundstates"for[].Ontheotherhand,anysystemwithanitenumberofperiodicground-statecongurationsoughttosatisfythegeneralizedPeierlsconditionifoneaddsallthelocalgroundstatesofthemodel[](orallowmorecomplicatedtypesofreferencestates).Attheriskofbeingconsideredalmostpatronizing,weemphasizeagainthatthesizeainthepreviousresultsischosensoastosatisfy(C)and(C)forthetotal

interaction0+e.Often,0isasimplerormorestandardinteractionthatonestudiesindependentlyorforwhichonecanborrowresultsfromtheliterature.ThePeierlsconstantdeterminedinthismannercorresponds,hence,toavaluesofachosenwithoutreferencetoanythingbut0.Whenconsideringinadditionperturbationse,nomatterhowsmall,thissizemayneedtoberedenedtoanewvaluea0suitableforthetotalinteraction.Ifso,thePeierlsconstant0appearinginthepreviousresultsissmallerthantheoneinitiallydetermined.Thesimplestprocedureatthispoint,ifonedoesnotwanttocompletelyredotheanalysiswiththenewdenitionofcontours,istoadoptfor0theinitialvaluedividedby(a0+)d[leftmostinequalityin(B.)].NotethatthePeierlsconstantchoseninthiswaygoestozerowithincreasingrangeoftheperturbations.Infact,ingeneralonecannotdomuchbetterthanthis.Inparticular,itisknown(cf.RemarkinSection..)thatarbitrarilyweakperturbationswithlong-rangeinteractionscandestroythephasediagram.Toconcludethissection,weobservethatthenotionofcontourcanbepresentedinaslightlymoregeneral(andabstract)fashion.Indeed,thekeypropertiessupportingtherestofthetheoryaretheuniquereconstructionofacongurationfromasetofcontours[hereaconsequenceoftheextensionproperty,requirement(C)],estimates(B.)and(B.),andthattheentropygainbebeatenbytheenergycostatlowtem-peratures.Aslongasthesepropertiesaresatised,contoursneednotbedenedviasamplingcubes.Anillustrationofthisobservationistheuseof\thin"contoursinferro-magneticnearest-neighborIsingmodelsor,moregenerally,modelswhoseground-statecongurationsareconstant.Theboundaryinsuchamodelcanbedenedasasetofpolyhedra,andthecontoursarenon-self-intersectingclosed(hyper)surfaces(uniquelydenedviasuitablexedprescriptionstohandleintersections),labelledbythecong-urationsoftheadjacentspins.Thelabellingallowsforauniquereconstructionoftheconguration,andthethincontourssatisfyestimate(B.)withjMjreplacedbyjj=areaofthepolyhedra=numberofplaquettesformingitsfaces,andestimate(B. )withadeterminedonthebasisofe.Moreover,theyhavesmallerentropythanthe\thick"contours.Note,however,thattheremarkdiscussedinthepreviousparagraphisespeciallyrelevantinconnectionwiththincontours:ingeneral,iftheinteractionisperturbed,onecannotusethevalueof0determinedviathincontours;onemust,forinstance,divideitbyafactor(a+)d,whereadependsontheperturbationeconsidered.B..ResultsoftheTheoryWepresentherethemainresultsofPStheory.Weincludesomegeneralcommentsontheunderlyingideas,butwedonotdiscussthedetailsoftheproofs.Thesecanbeconsultedinthebibliography.WementionthattherearetwoapproachestoPStheory:theoriginalone,basedon\contourmodelswithparameters",andthemorerecentone,duetoZahradnk,basedinsteadonaclassicationofcontoursinto\stable"and\unstable"ones.Referencesfortherstapproacharetheseminalpapers[,],Sinai'sbook[],andSlawny'sreviewarticle[].Thesecondapproachwas

introducedin[];aconcisepresentationisgivenin[]andapedagogicalonein[].Thissecondapproachisintuitivelymoreappealing,providessomemoreinformation|asforinstancethecompleteness[]andanalyticity[]ofthephasediagram|andhasservedasabasisforfurtherextensionsandapplicationsofthetheory[,,,,,].Inthecommentsbelowwemostlyhaveinmindsuchanapproach.TheessenceofPirogov-Sinaitheory|inheritedfromthePeierlsargument|isthedenitionofmapsfromtheoriginalspinensembleintoensemblesofcontoursthatinteractonlybyvolume-exclusion,thatis,intogasesofcontours.Thefamiliesofcon-toursinthelatterdonotnecessarilycorrespondtoanactualcollectionofcontoursofaspinconguration,becausetheyarenotrequiredto\match"exteriorswithinteriors.Forinstance,asetoftwo\"-contours,oneinsidetheother,isanallowedelementofoneofthecontourensembles,evenwhenthereisnospincongurationhavingitasitsfamilyofcontours(inaspincongurationtherewouldbeanintermediate\+"-contour).Thislackof\matching"requirementmakesthecontourensemblesmuchsimplersystemstoworkwith.Themapsaredenedsothateachstablepurephaseisequivalenttoacontourensembleinthesensethatbothhavethesamedistributionofexternalcontours.Thelow-temperaturepictureofonlysmall\islands"ofover-turnedspinscanthenbepreciselyprovenbyestimatingtheprobabilitiesof(external)boundariesusingthecontourensembles.Oneconsidersrdierentcontourensembles,oneforeachreferenceconguration!(i)K.Thei-thensembleisformedbyallthe!(i)-contoursinteractingonlyviatherestrictionofbeingseparatedbyormorelatticeunits.Thestatisticalweightofeachcontourisgivenbyanactivityexp[F(i)()]withafunctionalF(i)()determinedviaarelation(formulas(.)or(.)in[])thatroughlycomparesthe\workneededtoinstallacontour"[]inthespinandcontourensembles.[IntheoriginalPStheory,someextraweightsexp[b(i)jInt()j]areassignedtotheexternalcontours[],andboththe\parameters"b(i)andthefunctionalF(i)arealsodeterminedbycomparing\works"(formula(.)in[]).WeprefertofollowhereZahradnk'sapproachinwhichthe\parameterdegreeoffreedom"isabsorbedintothefunctionalF(i).]Eachcontourensembleisastatistical-mechanicalsystemofitsown,whichcanbestudiedwithoutanyreferencetotheoriginalspinsystem.Propertiesofthesecontourmodelscanthenbetranscribedintoresultsforthespinsystemviatheidenticationbetweentheensembles.Thisistheusualpolicyinthestandardexpositionsofthetheory,allofwhichincludean\interlude"inwhichabstractcontourensemblesareanalyzedperse(SectionstoinChapterof[],Sectionin[],etc).Basically,contourmodelsarestudiedviacluster-expansiontechniques:thisisthemethodofchoiceforsystemsat\hightemperature"or\lowdensity".AllthecontourensemblessatisfyoneofthekeyingredientsofthePeierlsargument:theentropyfactorgrowsatmostexponentiallywiththesizeofthecontours[,Lemma.](thisfactisfalseford=! ).Therefore,thereisamarkeddierenceaccordingtowhetherthefunctionalF(i)deningthecontouractivitysatisesaboundoftheformF(i)()(i)jMj(B:0)

with(i)>0.Ifthisisthecase,itiscustomarytosaythatF(i)isa(i)-functional.[FortheoriginalPSapproach,thebigdierenceiswhetherthecorrespondingparameterb(i)iszero;allthefunctionalsF(i)inthePSapproachare-functionals.]Thecontourmodelsdenedby-functionalsenjoyseveralremarkablepropertiesifislargeenoughtoovercometheentropygrowth.Thisgrowthischaracterizedbyanexponentialfactorboundedby[,Lemma.]=maxndlog(a+);logj0j+do:(B:)Ifthecontourmodelhasaconvergentclusterexpansion,whichoccursatleastif[,Lemma.andPropositions.and.](i);(B:)thenithasawell-denedthermodynamiclimit,withawell-denedpressureandinnite-volumeprobabilitymeasure.Forthismeasure,innitecontourshavezeroprobabilityofoccurrence,moregenerally,theprobabilityforagivencontourtobepresentdecreasesexponentiallywiththesizeofitssupport.Moreover,themeasuresatisesexponentialmixingconditionsfordisjointfamiliesofexternalcontours.(See,forinstance,Sections-of[].)Furthermore,eachofsuchcontourmeasuresisequivalenttoaGibbsmeasureinthespinsystem:ifF(i)isa(i)-functional,with(i),thenthe(innite-volume)probabilitydensityofexternalcontoursofthecontourensembleisequaltothatoftheGibbsmeasure|atinversetemperature|ofthespinmodeldenedbythe!(i)boundarycondition.Thus,thisGibbsmeasureinheritsthesparsityof(external)contourscharacterizingthecontourensembleanditsmixingproperties.Itis,therefore,anextremalperiodicGibbsmeasure(purephase)whichisonlyasmallperturbationofthereference(infactground-state)conguration!(i).Thepreciseresultofthisargumentis:TheoremB.(Pirogov-Sinai-Zahradnk)Assumed.Ifanite-rangepe-riodicinteractionsatisesthegeneralizedPeierlscondition(B.)withrespectotanitesetofperiodicreferencecongurationsK=f!();:::;!(r)g,thenthereexist0

Zahradnk[](\completeness").Oneoftheconsequencesofthiscompletenessisthatiftheinteractionhasauniqueperiodicground-stateconguration,anditsatisesthePeierlscondition,thenthereisalsoauniquepurephaseatlowtemperature.Infact,Martirosyan[0]hasproventhat,inthissituation,indtherearenootherextremalGibbsmeasures,periodicornot.Theparameters(i)characterizingthepurephasesareoftheform(seetheproofofProposition.in[])(i)(i)with(i)e(i)d;(B:)whereisthePS-constantoftheinteraction.Fromthisexpressiononecanobtainsome(farfromoptimal)estimatesoftheparametersinvolved.Indeed,forexamplewecanchoose(i)=(B:)withsatisfying=ed:(B:)Then,bytherequirement(B.)weobtainthebounde(B:)andhence,0=+=(e):(B:)Notethatas!onealsoobtains,from(B.)=+O(e):(B:)Inprinciple,TheoremB.(a)providesacriterionforthestabilityofaground-stateconguration,butitisquiteuselessforpracticalapplications.TheworkofZahradnk[,]providesadierentcriterionwhichcouldbeemployedforacomputer-basedprocedure.Itisbasedonthecomputationofthepressureep(F(i))forthe!(i)-contourensemblebutincludingonlysmall(orstable)contours.Thesearecontourswhoseinteriorvolumeisatmostproportionaltothesizeofthesupport.Atlowtemperature,thecoexistingpurephasesarethoseminimizingh(i)e(!(i))ep(F(i)):(B:0)Itcanbeproven[]thatep(F(i))!0as(i)!(i.e.! ),thustheminimizingcongurationsaregroundstates.Hence,(B.0)meansthatthestableground-statecongurationsarethosemaximizingthecontour-ensemblepressure;thatis,thosead-mittingthelargernumberoflow-energysmallcontours.ArelatedstabilitycriterionwasdevelopedbySlawny[],employingthepressureofagasofelementaryexcita-tionsinsteadofthecontour-ensemblepressure.Thestablephasesarethereforedeter-minedasthosewiththelargernumberoflow-energyexcitedcongurations(dominant0

ground-statecongurations).Thiscriterionissimplertoapplyforpaper-and-pencilcalculations.Theprecedingtheoremisthemaintoolusedtoprovethestabilityofthewholephasediagram.LetusdenoteU"(0)=fRd:Pri=ji0ij<"g.TheoremB.(Pirogov-Sinai-Zahradnk)Consideranite-rangeperiodicinter-action0indimensiondsuchthat:(i)ithasr

[IntheoriginalPSformulation,theparametersb(i)playedtheroleoftheh(i)here.]Suchamapisingeneralnotanalyticbecauseifitwereitwouldimplythatthefreeenergyofapurephasecouldbeanalyticallycontinuedinintothemetastabilityregion;andalreadyfortheIsingmodelithasbeenshown[0]thatsuchananalyticmetastableextensiondoesnotexist.Ontheotherhand,themap(B.)isoflimitedphysicalsignicance,becausefor!(i)notdeningapurephase,thecorrespondingquantityh(i)()isonlyanauxiliaryconcept,notevenuniquelydened[].ThephysicallyinterestingobjectsarethestrataSK()=f:Q()=Kg(B:)foreachKK,whereQ()=ni:h(i)()=minjrh(j)()o:(B:)Thesestratadeform(locally)analyticallywiththetemperature,byPart(c)ofTheoremB..Non-optimalestimationsofthelimitvalues0and"0ofTheoremB.canbeobtainedcombiningCorollaryB.with(B.):If0satisesthePeierlsconditionwithconstant0,thenatmost0=+=(e)0(c)(B:)ifatleast"0=c0(a+)d:(B:)TheseboundsareanexplicitexampleofageneralfactabouttheproofofTheoremB..Asalltheresultsfollowfromstudyingtheequivalentcontourensembles,therelevantmagnitudesarethoseactuallyusedtoconstructtheseensembles:thePeierlsconstant0andtheexponentialentropyfactor.Inaddition,thedimensiondandthesizeaofthesamplingcubesappearviatheuniformityproperty.Asaconsequence,wehave:CorollaryB.Considerafamilyoforiginalinteractionsf0(p)gpPRmsatisfyingthePeierlsconditionuniformlythatis,withthesameconstant0andthesamefamilyofperiodicground-statecongurationsKforallpP(e.g.intheconditionsofCorollaryB. ).ThenTheoremB.holdsalsouniformlyforalltheinteractions0(p)[i.e.,onecanchosethesame0and"0inParts(a)and(b),andthesame-and-intervalsinPart(c)].Moregenerally,TheoremB.canbeextendedtosituationsinwhichthereisafurthersmoothdependenceoftheinteractionsontheextraparametersp.TheoremB.0Assumedandconsiderinteractions0(p);(p);:::;r(p)dependinganalyticallyonparametersptakingvaluesonanopensetPRm,andwith

boundedperiodandrange.Assumethatthereexistsap0Psuchthat(i)0(p0)hasr

a\pure-entropy"restrictedensemble[],asopposedtothe\pure-energy"(justonegroundstateconguration)or\almost-pure-energy"(groundstateplusexcitations)usedinmostoftheapplications.Ingeneral,therestrictedensemblesarechosensotohaveminimal(restricted)freeenergyatthetemperatureofinterest.Often,interactingcontoursandreferencemeasuresarealternativeprocedures;itisamatteroftastetochooseoneortheother.AthirddirectioninwhichPStheoryhasbeenextended|andonewhichiscrucialforourapplicationtotheproofofRGpathologiesinnonzeromagneticeld(Section..)|istowardstheincorporationofrandominteractions.Inworkunpublishedsofar,Zahradnk[0,]|generalizingtheworkofBricmontandKupiainen[,]|hasproventhatfordtheadditionofasmallenoughrandominteractiononlyproducessmalldeformationsinthephasediagram.Animportantissueisthemeaningof\smallenough".Intheoriginalwork[0],therandominteractionwasrequiredtobeuniformlysmallrespecttothenonrandompart.Laterwewereinformed[]thattheproofalsoappliestorandomcontributionsthataresmallinprobability.Westatethelaterversion,whichistheonesuitedtoourapplications.TheoremB.ConsiderthelatticeZd,d,andaninteraction=0+Pri=iisatisfyingthehypothesisofTheoremB..Addanite-rangerandomin-teractionrdom=nrdomA(;)oAS,whereisarandomvariablewithprobabilitydis-tributionP,suchthattherandomvariablesrdomA(;)andrdomA0(;)havethesamedistributionifA0isatranslateofAandareindependentifA\A0=?.Assume,inaddition,thefollowingsmallnesscondition:Foreach>0thereexists()smallenoughsuchthatP(jrdomA(! ;)j>)(B:)forallAS,!.Then,forlargeenoughthephasediagramsforand+rdomarehomeomorphic.Moreprecisely,forlargeanduniformlysmall,thereexistsanhomeomorphismL;:V0;!V00;(B:0)betweentwoopensetsV0;;V00;Rrmappinganr-regularphasediagramontoanr-regular+rdomphasediagram.ThehomeomorphismL;tendstotheidentityas!0uniformly.B.ApplicationtotheExamplesofSectionB..GeneralStrategiesInprinciple,thevericationofthePeierlsconditionforacertaininteraction0isatwo-stageprocess:(A)Findalltheperiodicground-statecongurationsof0.Thisstageusuallyinvolvescountinghowmany\frustratedbonds"eachcandidatecongurationhas.Thisshouldbefollowedbyaproofshowingthatindeednootherperiodiccongurationhasthesameorlessenergydensity,butthisproofisusuallyomittedbecauseitiseither

consideredtobeobviousortoomessytowritedown.Furthermore,asremarkedbeforeDenitionB.0,thisproofisnotreallynecessaryifthePeierlscondition(nextstage)canbesuccessfullyveried.Inthisregard,thetediousprocessofndingthesereferencecongurationsisanaturalcandidateforacomputer-assistedprocedure.However,thismaynotbepossible,ingeneral:theproblemofcheckingwhetheragivenperiodiccongurationisagroundstatemaynotbealgorithmicallydecidable(see[,Section.]and[]andreferencesthereinforsomerelatedundecidabilityresults,andseealso[]).Inanycase,thisproblemmaybealleviatedinpracticeifoneworksintheframeworkofPStheory.Indeed,thefurtherstepsofthetheoryworkasacorrectingmechanism:iftoofewcongurationshavebeenfound,thePeierlsconditionwillfailandthethecongurationsoflargecontourswithverylowenergydensitywillgiveahintofhowadditionalground-statecongurationslooklike.Ontheotherhand,iftoomanycongurationshavebeenselected,thespuriousoneswillbeeventuallyruledoutinthesensethattheywillnotgiverisetopurephases;theyleadtocontourensembleswithhighfree-energycost,whicharenotassociatedto-functionals.[WeowethisinsighttoconversationswithMilosZahradnk.](B)Deviseasuitablenotionofcontourandshowthattheenergygrowspropor-tionallytoitssupport.Thisisanextremelymodel-dependentprocess.Often,thedeterminationofground-statecongurationsandofcontourenergiesaredonesimultaneously:Thecomparisonbetweentheenergiesofdierentcandidatecongurationsisdonealreadywiththehelpofcontours.Thisisamanifestationofwhatwehaverepeatedlycommentedupon:thecontourenergyistheessentialquantity;thecrucialstageis(B).Ifwemanagetoshowthatcontourenergiesarelargeenoughthenwedonotneedtocareaboutthenatureofthereferencecongurations,theyaregroundstatesbyforce.Yetagain,thesameargumentusedtoprovetheappropriategrowthofcontourenergies,usuallyshowstheground-statecharacterofthecongurations.Proving(B)directlyisnotasubstantialsavingofmisery.Below,weshallusethiscanonicalapproachfordecimationandKadano-ppre-scriptions,wherewecanresorttotheIsingtypeofcontours:polyhedradrawnontheduallatticewithplaquettesperpendiculartoeachfrustratedbond.ForthestudiesonotherRGtransformationsweshalluseinsteadtheHolsztynski-Slawnycriterion(The-oremB. )basedonthenotionofm-potentials(DenitionB.0).Thecorrespondingprocedureusuallystartsbyrewritingtheinteractionintoanequivalentformwhichisindeedanm-potential.Thisisthehardpartofthegame;itinvolvessomeknowledgeofwhattheground-statecongurationslooklike.Oncetherewritingisdone,theac-tualvericationofwhichcongurationshaveminimalenergyisingeneralsimple;onestudiesonebondatatime.Asalreadyremarked,thedisadvantageofthisapproachisthatitdoesnotsupplyavalueforthePeierlsconstant,afactthat,inourcase,preventsusfromproducinganyestimateoftherangeoftemperaturesforwhichthepathologiesoftheRGtransformationsoccur.WeobservethatfortheSteps{intheproofsofexistenceofpathologies,wedonotneedthefullinformationonphasediagramsprovidedbyPStheory.Rather,weareonlyinterestedinthepointofmaximumcoexistence(forStep),andregionsofuniqueness

ofGibbsmeasure(forStep).Moreover,bothtypesofquestionsrefertodierentsystems:Themaximumcoexistencepointmustbedeterminedforinternal-spinsystemsobtainedwhentheimagespinsdonotfavoranyphaseoftheoriginalsystem(thisamounts,ingeneral,toblockspinschoseninanalternatingorrandomway).Inthesecases,thesymmetry-breakingperturbationischosensimplyasauniformmagneticeldandsymmetryconsiderationsimplythatmaximumcoexistenceisachievedonlywhenthiseldiszero.Therefore,thissymmetry-breakinginteractionplaysanalmostinvisiblerole,anditisonlybrieymentioned.Theonlycaseinwhichthesymmetry-breakingdeservescarefulconsiderationisthatofsystemswhichalreadyinitiallyhaveamagneticeld(Sections..andB..).Ontheotherhand,theuniquenessofGibbsmeasuresisofinterestforinternal-spinsystemscorrespondingtoblockspinschosensoastodenitelyfavoroneofthephases.Itisnotwisetothinkofthesetwointernal-spinsystems(determinedbyblockspinseitherfavoringornotfavoringonepurephase)aslivinginthesamephasediagramwithblock-spinippingassymmetry-breakinginteraction.Theproblemisthatsuchaninteractioncannotbeconsideredaperturbation:itgoesbynitestepsandhencemaythrowusoutofthesmall-parameterregionV(seeTheoremB.)ofthephasediagramwherePStheoryholds.Letusnowdiscussthedierentapplicationsstartingfromthesimplestones.B..Internal-SpinSystemswithUniqueGround-StateCongurationsInalltheapplicationsofSectionthereisastep(Step.)whichinvolvesshowingthatatlowtemperaturetheensembleofinternal(ororiginal)spinshasauniqueGibbsmeasureforsomeparticularchoiceofimagespins.Theseareallcasesinwhichthereisonlyone(periodic)ground-stateconguration,duetothepresenceofaperiodicsingle-signmagneticeld.TheuniquenessoftheGibbsstateis,basically,aconsequenceofPStheoryplusZahradnk'scompletenessresult(seethecommentimmediatelyfollowingTheoremB.).However,thiswouldonlyproveuniquenessamongtheperiodicGibbsmeasures;weneedMartirosyan'sextensionofthisresult[0]toproveuniquenessamongthesetofallGibbsmeasures.B..InternalSpinsunderDecimationInthiscasewecanapplythecanonicaltwo-stageprocessdescribedabovetoverifythePeierlscondition.AsdiscussedindetailinSections..and. (Step0),theensembleofinternalspinsforagivenimage-spincongurationisjusttheensembleofcongurationsoforiginalspinswithsomeofthespinsconstrainedtobexed.TothelatterwecanapplytheusualPeierlsconstructionofpolyhedralcontours.Weshallthereforeusethe\thin"notionofcontoursdiscussedattheendofthetwopreviousSectionsB..andB...Theinternal-spincontoursareobtainedfromtheoriginal-spincontoursbyremovingtheplaquettesadjacenttoanimagespin[thicklinesinFigure(a)].Theargumentthatfollowsisthusbasedoncomparinginternal-spinwithoriginal-spin(=internal+image)quantities.

Wextheimagespininthefullyalternating\+="conguration,sothetwoobvi-ouscandidatestoground-statecongurationsintheresultingsystemofinternalspinsarethesameasfortheoriginalsystem:!(+)equalto+everywhere,and!()equaltoeverywhere.Thatis,intermsoforiginalspins,!(+)(resp.!())correspondstoallspins\+"(\")exceptforasublatticeofperiodb|withbbeingthedecimationspacing|wherethespinsareipped.Thesymmetry-breakingperturbationcanbetakentobeamagneticeldateach(internal)site.However,thesymmetryoftheproblem(i.e.ofthechoiceofblockspins),impliesthatthecoexistenceofthe\+"and\"measuresoccursatzerovaluesofthiseld.Wethereforeforgetaboutthisextraeld,andconcentrateonprovingthatthezero-temperaturephasediagramdeformslittleforlowtemperatures.Ifwewishonlytoprovethat!(+)and!()arealltheinternal-spinground-statecongurations,wecanforinstanceconsiderthecorrespondingsetoforiginal-spincon-tours,whichisjustanarrayofb-spacedunitcubessurroundingeachippedspin,andshowthatallotheroriginal-spincongurationsleadtoasystemof(original-spin)contourswithalargerarea.Thisisnothardtodo,forinstancewecanargueasfollows:Observethateveryintervalparalleltotheaxisbetweentwonearest-neighborimagespinsnecessarilycontainsatleastonebrokenbond,astwonearest-neighborimagespinsalwayshaveoppositesigns.Thechoiceofallinternalspinseitherall+orallhaspreciselythisminimalchoiceofexactlyonebrokenbondineachinterval,andnootherbrokenbonds.However,suchanargumentisnotreallyneeded.Ascommentedabove,amorecon-venientapproachforourpurposesistoshowdirectlythattheinsertionofacontourin!(+)hasanenergycostproportionaltoitsarea.Considerthenthe(internal-spin)conguration!obtainedbyinsertinga\+"(internal-spin)contourinsidethecon-guration!(+),thatis,aregionof\"boundedby[Figure(a)].Itsrelativeenergycanbewrittenintheform:H(!j! (+))=JjjE+E+;(B:)wherejjistheareaofthecontour[numberofplaquettes,orlengthofthethicklinesforthetwo-dimensionalexampleofFigure(a)],Eistheenergygainduetothefactthatthe\"imagespinsinsideof,orvisitedby,acquire\"neighbors[e.g.thesitesaiinFigure(a)],andE+istheadditionalenergydueto\+"imagespinsin.ToprovethePeierlsconditionwehavetocheckthattheextracontributionEE+isproportionaltotheareaofthecontourwithanottoolargeproportionalityconstant.Theintuitionisclear:Thecontributionscorrespondingto\+"and\"imagespinsinsidethevolumecanceleachother,exceptpossiblyforalayerofimagespinsplacedclosetothecontour.Thiscorrectionishenceoftheorderoftheareaofthecontour,withproportionalityconstantgivenroughlybytheinverseoftheseparationbetweentheseimagespins.However,thislastboundisnotapplicableifthecontourinvolvesfew(e.g.one)imagespins.Inthissense,itisnaturaltodistinguishbetweencontourssurroundingandcontoursavoidingtheimagespins.Whiletheformer\feel"

a1a2a4a3(a)(b)Figure:Systemofinternalspins(smallcircles)underdecimation,whentheimagespins(squares)arexedinthefullyalternating\+="conguration.(a)A\"contour(thicklines). (b)Thefundamentalbonds(boundedbythicklines)ofanequivalentm-potential.

thesublatticeofimagespins,theenergycontributionofthelatterisalmostthesameasfortheusualIsingmodel.ThisproducesanestimationofthePeierlsconstant(andhenceofthecriticaltemperature),withtwocompetingterms:onedependingonthedecimationperiodb(andtendingto0asbtendstoinnity),andanotherindependentofbandclosetothePeierlsconstantfortheIsingmodel(thecloserthehigherthedimension).Toformalizetheseideaswechooseonecoordinateaxis,saytheonelabelled,andperformthecancellationsbysweepinginorderalongit.Toabbreviate,letuscall\left"thedirectioninZdtowardssmaller-components,and\right"theoppositedirection.Also,weshallsaythatanimagespinis\in"thecontourifitisvisitedbyitoritiscontainedinitsvolume.Morespecically,an\internalspinwithlplaquettesinthecontour"isaninternalspinsuchthatthecontoursurroundsloftheplaquettesoftheunitcubecenteredonit(e.g.inFig.(a),theinternalspinatahasplaquetteinthecontour,theoneatahas,etc.).Notethat,insuchacase,theenergycontributionoftheimagespinisE=jJjl(B.a)E+=0(B.b)ifitisa\"spin,andtheconversefora\+"imagespin.Thecancellationcanbedone,forinstance,asfollows:Foreachlineintheleft-rightdirectionintersectingthecontour,wechoosetheimagespininthecontourfurthertotheleftandlookforthenextimagespininthecontour,ofoppositesignandlocatedtotherightandalongthesameline.Ifthe\+"imagespinhasthesamenumberofplaquettesormoreinthecontourthanthe\",wecancelbothcontributions(obtainingalowerboundfortheenergyifthenumberofplaquettesisnotthesame).Otherwisewedonothing.Wethenproceedtothenextuncancelledimagespinalongthesameline,alwaystravellingtowardstheright.Oncealltheleft-rightlineshavebeenscanned,weobtainalowerboundfortheenergyoftheform(B. )butwhereEandE+referonlytoalayerofimagespinsinthecontouratadistancenotexceedingb+fromit(remainingspins).Theenergygainduetotheseremainingspinscannotexceedthatofthecaseinwhichthereareno\+"imagespinsleftandallthe\"spinshavetheirdplaquettesinthecontour.WethereforeboundEE+dJN;(B:)whereNisthenumberof\"imagespinsinsidetheabove-mentionedlayer.TocompletethePeierlsbound,wehavetorelateNtothecontourareajj.AtthispointwemustdistinguishbetweencontourswithN(\widecontours")andcontourswithN(\narrowcontours").Forthewidecontours,thekeyobservationisthateachtwo\"imagespinsmustbeatleastadistancebapartineachcoordinatedirectionandthecontourmustpassatadistanceb+orlessofeachofthem.Therefore,thenumberofremaining\"imagespinsforagivenvalueofcannotexceedthatofthecaseinwhichallofthespinsarelocatedsoastoforma\tube",separatedb

unitsfromeachother,andbeingthewallofsucha\tube":Njj(d)(b):(B:)(Notethattheboundcontainsb,ratherthanb,becausetheplaquettescorrespondingto\+"imagespinsarenotpartofthecontour.)From(B.){(B.),weconcludethatH(!j!(+))Jd(d)(b)jj;N:(B:)Thisboundisnotusefulforthelimitcased=,b=;butforitwehavealreadygoodboundsforthecriticaltemperature(Section..).Letusnowconsiderthenarrowcontours(N).Itisnothardtoconvinceoneselfthattheworstcaseiswhenthecontourvisitsonlyone\"imagespin,whichhaslplaquettesinthecontour.Thecontourmustthenincludetheoppositelplaquettesplustheplaquettesneededtojointheseamongthemselvesor/andtothecubesotoformaclosedsurface.Onecancheckthattheworstsituation(smallestratiol=jj)iswhenthelplaquettesofthecubearenotconsecutive,inwhichcasethecontourincludestheloppositeplaquettesandthe(d)lplaquettesneededtojointhemtothecube.Hence,jjl[(d)+]:Therefore,using(B.),EE+JdjjandH(!j!(+))Jdjj;N:(B:)(AnotherwaytointerpretthisPeierlsboundisbynotingthatthenarrowcontourwiththeleastenergycostistheoneproducedbyasingleippedinternalspinadjacenttoa\"imagespin.)Formulas(B.)and(B.)showboththatthecongurations!(+)and!()areindeedtheonlyperiodicground-statecongurationsandthatthePeierlsconditionissatisedfortheinternal-spinsystemwithPeierlsconstant0J(Md;b);(B:)whereMd;b=max(d;d(d)(b)):(B:)Thisvalueof0,togetherwiththeestimates(B.)for0and(B. )fortheentropyfactorofthincontours,impliesthereisaphasetransitionatleastforlog(d)+=(e)J(Md;b):(B:)0

Thisestimateisprobablyveryweak,butatleastitincreaseswithdandwithbasitshould.Infact,ifbislarge,thealternatingeldsareveryfarapart,andthesystembecomesalmostindistinguishablefromazero-eldIsingmodel.Thereforeweexpect,butcannotprove,thatthislimittemperatureapproachestheIsing-modelcriticaltemperaturewhenbtendstoinnity.Inequality(B.)determinestherangeoftemperaturesforwhichwecanprovethatthedecimationtransformationhaspathologies(Sections.and..).Asawarm-upforthefollowingsections,letussketchhowthePeierlsconditioncanbeveriedinthepresentexampleusingtheHolsztynski-Slawnycriterion(TheoremB.).Theargumentdependsslightlyonthedecimationspacingb.Forb=theinternal-spininteractionisalreadyanm-potentialbecauseitisjustaferromagnetictwo-bodyinteractionina\diluted"lattice(Sections.and..).Forbaperiodicm-potentialisobtainedbygroupingallthebondsinsidecubescontainingatleastoneperiodoftheimage-spinconguration[Figure(b)].Explicitly,ifinternalistheinteractionfortheinternal-spinsystem,theequivalentb-periodicm-potentialmpothasmpotA=0unlessAisaperiodictranslate,withperiodb,ofthecubeV="$b%;$b%#d(B:0)(herebcdenotesintegerpart),oroftheinter-cubebondsformedbynearest-neighborpairshx;yiwith,say,xinthe(internal)boundaryofV.Forthesepairsmpotfxyg=fxyg=J,andforthecubeVmpotV=XAVinternalA:(B:)Itisnothardtoverifythatthecongurations!(+)and!()aretheonlyminimizersofthempotV,andtheyobviouslyalsominimizetheenergyoftheinter-cubebonds.Therefore,mpotisanm-potentialwithanitenumberofground-statecongurations.ByTheoremB.,thePeierlsconditionissatised.Noestimationof0followsfromthisapproach.B..InternalSpinsundertheKadanoTransformation.UniformityForthiscasewehavetoapplytheuniformityresultsthatwesocarefullystatedabove.TheHamiltonian(. )canbedecomposedintheformHe=H0+fHp(B:)whereH0istheusualIsingHamiltonianandfHpcorrespondstotheinteractione(p;)denedby(eA(p;))()=>><>>:(p=)0xiifA=figandiBx(=)logcoshpPiBxiifA=Bx0otherwise:(B:)

(Werecallthatinthisappendixweare\unabsorbing"whichin(.)isabsorbedonlyinJ.)Wechoose!0specialassomealternatingconguration|withasmanyplusesasminuses|sothat,bysymmetry,thecoexistenceofthe\+"and\"internal-spinGibbsmeasuresdoesnotrequireanyadditionaleld.Thatis,asbeforeweforgetaboutsymmetry-breakinginteractions.Theinteraction0satisestheoriginalPeierlsconditionwith0=J(thincon-tours,K=f!(+);!()g).WecanthenresorttoCorollaryB.toconcludethatthewholeinteraction0+e(p;)satisestheoriginalPeierlscondition.However,toes-timatethePeierlsconstantwemustcorrect0soastosatisfy(C)and(C)forthetotalinteraction.Forinstance(seeremarksattheendofSectionB..),wecanreplaceitby0=J(a+)d;withaequaltohalfthelengthofthelargestsideoftheblock.Wethenconcludethatforeachpthereexistsavalue(p)denedbyp+log(coshpjBj)=cJ(a+)d;(B:)suchthatfor(p)theeectiveinternal-spininteractionforthep-Kadanotrans-formationsatisesthePeierlsconditionwithconstant=J(c)(a+)d:(B:)Inthelasttwoformulas,theconstantcisarbitraryaslongas0

Formula(B.)givesalowerboundforthetemperatureuptowhichaKadanop-transformationexhibitspathologies(Section..).Thisboundgoestozeroifptendstoinnity,henceitisuselessformajority-ruletransformations.B..InternalSpinsunderMajorityRuleForthiscase,weusetheHolsztynski-Slawnycriterion(TheoremB.).Thesystemofinternalspinssubjectedtotheconstraintofadoubly-alternatingblock-spincongurationcanbewrittenasaperiodicm-potentialwithperiod.ThefundamentalbondsofthispotentialarethesquaresofsizedepictedinFigure(a),andallthebondsconnectingneighboringsquares.ItisstraightforwardtocheckthatthecongurationsofFigure(b)aretheonlyonessatisfyingallthebondsofthism-potential.Hence,thesystemhasanitenumber(two)ofgroundstateswhich,byTheoremB.andPStheory,giverisetodierentandcoexistingGibbsmeasuresatlow-enoughtemperature.Bysymmetryconsiderationsthiscoexistencetakesplaceatzerovaluesofthesymmetry-breakingeld.Ananalogousargumentcanbeusedforalltheotherblock-sizesbkgivenin(.).B..InternalSpinsunderBlock-AveragingTransformationsAgain,weresorttotheHolsztynski-Slawnycriterion(TheoremB. ).Thesystemofinternalspinssubjectedtotheconstraintofzeroaveragespinineveryblockcanagainbewrittenasanm-potentialwhichisperiodic(withperiod).Thefundamentalbondsarethesquaresandthebondsconnectingthem,andtheonlyfourperiodicgroundstatessatisfyingeverybondareeasilyseentobetheonesdepictedinFigure.ThusatsucientlylowtemperaturePS-theoryprovidesthephasetransitionneededinStep.Againsymmetryallowsustodisposeofanysymmetry-breakingeldtofollowthecoexistencepoint.B..InternalSpinswhenh=0.RandomFieldTheresultneededtoprovethepresenceofpathologiesfornon-zeroeldinthedeci-mationandKadanoexamplesofSection.,isadirectconsequenceofZahradnk'sTheoremB..Weapplythistheoremwith0equaltotheinteractionforthesystemofinternalspinswithfullyalternatingimagespins,andthesymmetry-breakingpertur-bationtakentobeauniformmagneticeld.Therandominteractionistherandommagneticeldinducedbythoseblockspinsthatwereippedfrom\+"to\"withprobability=(J),asdiscussedinSection...ByTheoremB.,theresultinglow-temperatureandlow-phasediagramisonlyasmallperturbationofthephasediagramforthenon-randompart,whichisitselfasmallperturbationofitszero-temperaturephasediagram.Inparticular,theground-stateenergyforalmostallrealizationsoftheHamiltonianisanalmostlinearfunctionoftheparametershand.Forsucientlysmall,thelinearityisonlyweaklyviolated,andthecompensatinguniformeld(that

is,thevalueofhasafunctionofneededtokeepthesystemonacoexistencesurface)isanalmostlinear|hencestrictlyincreasing|function.CSolutionoftheDiophantineEquation(.)ConsiderthepairofDiophantineequationsb=c(b;cintegers).ThefollowingintuitionwassuggestedtousbyVincentRivasseau:If(b;c)satiseseitheroftheseequations,thenb=cmustbeanexcellentrationalapproximationtop,inthesensethatb=cp=jb=cjb=c+ppc:(C:)(Notethat,bycontrast,for\typical"integerdenominatorsc,onehasinfbZjb=cpj=c=c.)Now,thebestrationalapproximationstopcanbeobtainedfromthecontinuedfraction[,]p=+++:(C:)Thissuggeststoconsidertherecursionxn+=+xn;(C:)whichconvergestopasn! (foranyx0>);equivalently,deningyn=xn+,wendtherecursionyn+=+yn+yn;(C:)whichconvergestop(foranyy0>).Inparticular,settingyn=bn=cnwithbn;cnpositiveintegers,wendthelinearrecursionbn+=bn+cn(C.a)cn+=bn+cn(C.b)Nowthisrecursionhastheremarkablepropertythatbn+cn+=(bncn);(C:)inparticular,ifbn=cn,thenbn+=cn+.Therefore,ifwestartfrom(b0;c0)=(;),wegeneratepairs(bn;cn)whichsatisfybn=cn(resp.bn=cn+)foreven(resp.odd)valuesofn.Theexplicitformulaisbn=h(+p)n++(p)n+i(C.a)cn=ph(+p)n+(p)n+i(C.b)

Toprovethatthissequenceconstitutesthecompletesetofintegersolutionstob=c,weruntheiteration(C.)backwards:given(b;c),wedeneb0=b+c(C.a)c0=bc(C.b)andshowthatrepeatedapplicationofthismapmusteventuallyleadtothepair(;).LemmaC.Letb;cbeintegerssatisfyingb=c.Thenb0;c0areintegerssatisfying00.Alsocc0=cb>0,soc0

Unfortunately,wesuspectthatfordtherearenopositive-integersolutionsto(C.)otherthanb=c=.ThebestresultswehavebeenabletogleanfromthemathematicalliteraturearesummarizedinTheoremsC.andC.:TheoremC.Letlbeapositiveintegersatisfyinganyoneofthefollowingthreeconditions:(a)l=;or(b)l=;or(c)lisaregularprimesuchthattheexponentofmodliseither(l)=oreven,andsuchthatl(modl).Letdbeanymultipleofl(includinglitself).Thentheonlypositive-integersolutionstoxd+yd=zdarex=y=z.Inparticular,theonlypositive-integersolutionto+bd=cdisb=c=.TheoremC.hasalonghistory.Obviously,ifthetheoremholdsforanygivenpowerl,ittriviallyholdsalsoformultiplesofthatpower.Thecasel=wasprovenbyEulersometimebefore0[,p.];muchmoregeneralresultsarenowknown[,pp.,0,0].Thecasel=isaspecializationofatheoremprovenbySchopisin[,p.][,p.];again,moregeneralresultsarenowknown[,pp.,,].Thecasel=wasproveninthemid-nineteenthcentury,buttheproperattributionisunclear.Denes[]creditsDirichlet[],butourreadingofDirichlet'spaperindicatesthathetreatednumerouscasesofx+y=AzbutnotA=(seealso[,p.]).ThecorrectattributionseemstobeV.A.Lebesguein[][,p.].Seealso[,pp.{]and[,p.]forgeneralizations.Thecase(c)wasprovenbyDenes[]in.Tointerpretit,notethattherstirregularprimesare;;;0;:::.Therstregularprimesforwhichtheexponentconditionfailsare;;;;:::.Finally,theonlyprimesl<0(andindeedtheonlyonescurrentlyknown)forwhichl(modl)are0and[0,pp.{].Thus,therstprimesforwhichcondition(c)failsare;;;;;;0;:::.Inparticular,TheoremC.holdsforallexponentsd00exceptpossibly;;;;;;;.0TheoremC.Forarbitraryd,thereisatmostonepositive-integersolutionto+bd=cdotherthanb=c=.Aprimel>iscalledregularifitdoesnotdivideanyofthenumeratorsoftheBernoullinumbersB;B;:::;Blexpressedinlowestterms.Theexponentofmodlisthesmallestintegernsuchthatn(modl).0Denes'paper[]containsalistofallprimesl

TheoremC.isaspecialcaseofaresultofDomar[],whoprovesthatforarbitraryintegersA;Bandd,theequationjAxdBydj=hasatmosttwosolutionsinpositiveintegersx;y.Seealso[].Theconjecturedunsolvabilityofxd+yd=zdfordisaspecialcaseofan\extendedFermat'slasttheorem"whichmightconceivablybetrue[,]:ConjectureC.Letdandabeintegers,withdandad.Thentherearenosolutionsofxd+yd=azdinnonzerointegers,exceptforx=y=zinthecasea=.WenditamusingthatarealprobleminphysicsshouldbeconnectedwithFermat'slasttheorem,butwethinkthatthisisanartifactofourmethodofproofandnotanintrinsicfactaboutmajority-ruletransformations.Indeed,wesuspectthatTheorem.holdsforallblocksizesbandalldimensionsd,withoutregardforsubtlenumber-theoreticproperties.Butwecouldbewrong.AcknowledgmentsWeareverygratefultoJeanBricmont,AnnaHasenfratz,MichaelKiessling,JesusSalas,DanSteinandDougToussaintformanyhelpfulcommentsonarstdraftofthispaper.Inaddition,wewishtothankPavelBleher,CarlodiCastro,JoelCohen,RolandDobrushin,MichaelFisher,JurgFrohlich,StuartGeman,Hans-OttoGeorgii,AntonioGonzalez-Arroyo,BobGriths,RobertIsrael,TomKennedy,RomanKotecky,AnttiKupiainen,JoelLebowitz,ChristianMaes,FabioMartinelli,JacekMiekisz,ChuckNewman,EnzoOlivieri,CharlesRadin,VincentRivasseau,RobertoSchonmann,SenyaShlosman,BobSwendsen,SrinivasaVaradhan,MarinusWinninkandMilosZahradnkforhelpfulconversationsandcorrespondence.Twooftheauthors(R.F.andA.D.S.)thanktheUniversitadiRoma\TorVergata"andtheRijksuniversiteitGroningenforhospitalitywhilethispaperwasbeingwritten.Theotherauthor(A.C.D.v.E.)thankstheETH{HonggerbergandtheEPFL{Lausanneforhospitalityduringthissameperiod.Lastbutnotleast,wewishtothankJoelLebowitzforbeingwillingeventoconsiderapaperofthislengthforpublicationintheJournalofStatisticalPhysics.Theresearchoftherstauthor(A.C.D.v.E.)hasbeenmadepossiblebyafellowshipoftheRoyalNetherlandsAcademyofArtsandSciences(KNAW).Theresearchofthesecondauthor(R.F.)wassupportedinpartbytheSchweizerNationalfondsandbytheFondsNationalSuisse.Theresearchofthethirdauthor(A.D.S. )wassupportedinpartbyU.S.NationalScienceFoundationgrantsDMS{0,DMS{andDMS{00andbyU.S.DepartmentofEnergycontractDE-FG0-0ER0.

References[]M.Aizenman.TranslationinvarianceandinstabilityofphasecoexistenceinthetwodimensionalIsingsystem.Commun.Math.Phys.,:{,0.[]M.Aizenman.Geometricanalysisof'eldsandIsingmodels.PartsIandII.Commun.Math.Phys.,:{,.[]M.Aizenman.TheintersectionofBrownianpathsasacasestudyofarenormal-izationgroupmethodforquantumeldtheory.Commun.Math.Phys.,:{0,.[]M.Aizenman,J.T.Chayes,L.Chayes,J.Frohlich,andL.Russo.Onasharptransitionfromarealawtoperimeterlawinasystemofrandomsurfaces.Com-mun.Math.Phys.,:{,.[]M.Aizenman,J.T.Chayes,L.Chayes,andC.M.Newman.Discontinuityofthemagnetizationinone-dimensional=jxyjIsingandPottsmodels.J.Stat.Phys.,0:{0,.[]M.AizenmanandR.Fernandez.Onthecriticalbehaviorofthemagnetizationinhigh-dimensionalIsingmodels.J.Stat.Phys.,:{,.[]M.AizenmanandR.Graham.Ontherenormalizedcouplingconstantandthesusceptibilityin'eldtheoryandtheIsingmodelinfourdimensions.NuclearPhysics,B[FS]:{,.[]M.AizenmanandE.Lieb.Thethirdlawofthermodynamicsandthedegeneracyofthegroundstateforlatticesystems.J.Stat.Phys.,:{,.[]M.P.AlmeidaandB.Gidas.AvariationalmethodforestimatingtheparametersofMRFfromcompleteorincompletedata.BrownUniversitypreprint,0.[0]D.J.AmitandL.Peliti.Ondangerousirrelevantoperators.Ann.Phys.,0:0{,.[]P.W.AndersonandG.Yuval.SomenumericalresultsontheKondoproblemandtheinverse-squareone-dimensionalIsingmodel.J.Phys.,C:0{0,.[]P.W.Anderson,G.Yuval,andD.R.Hamann.ExactresultsintheKondoproblem.II.Scalingtheory,qualitativelycorrectsolution,andsomenewresultsonone-dimensionalclassicalstatisticalmodels.Phys.Rev.,B:{,0.[]C.Arag~aodeCarvalho,S.Caracciolo,andJ.Frohlich.Polymersandg'theoryinfourdimensions.Nucl.Phys.,B[FS]:0{,. []M.Asorey,J.G.Esteve,R.Fernandez,andJ.Salas.Rigorousanalysisofrenor-malizationgrouppathologiesinthe-stateclockmodel.Preprint,.

[]M.B.Averintsev.Gibbsdescriptionofrandomeldswhoseconditionalproba-bilitiesmayvanish.Probl.Inform.Transmission,:{,.[]R.R.Bahadur.SomeLimitTheoremsinStatistics.SIAM,Philadelphia,.[]G.A.BakerandS.Krinsky.Renormalizationgroupstructureoftranslationinvariantferromagnets.J.Math.Phys.,:0{0,.[]T.Balaban.Ultravioletstabilityineldtheory.Themodel.InJ.Frohlich,ed-itor,ScalingandSelf-SimilarityinPhysics.Birkhauser,Boston{Basel{Stuttgart,.[]T.Balaban.Largeeldrenormalization.II.Localization,exponentiationandboundsfortheRoperation.Commun.Math.Phys.,:{,.[0]R.Balian.FromMicrophysicstoMacrophysics:MethodsandApplicationsofStatisticalPhysics.Springer-Verlag,Berlin{NewYork,.[]A.G.Basuev.Completephasediagramswithrespecttoexternaleldsatlowtemperaturesformodelswithnearest-neighborinteractioninthecaseofaniteorcountablenumberofgroundstates.Theor.Math.Phys.,:{,.[]A.G.Basuev.Hamiltonianofthephaseseparationborderandphasetransitionsoftherstkind.I.Theor.Math.Phys.,:{,.[]A.G.Basuev.Hamiltonianofthephaseseparationborderandphasetransitionsoftherstkind.II.Thesimplestdisorderedphases.Theor.Math.Phys.,:{,.[]G.A.BattleandL.Rosen.TheFKGinequalityfortheYukawaquantumeldtheory.J.Stat.Phys.,:{,0.[]H.Bauer.ProbabilityTheoryandElementsofMeasureTheory.Holt,RinehartandWinston,NewYork,.[]R.T.Baumel.OnspontaneouslybrokensymmetryintheP()modelquantumeldtheory.PhDthesis,PrincetonUniversity,June.[]T.L.BellandK.G.Wilson.Finite-latticeapproximationstorenormalizationgroups.Phys.Rev.,B:{,.[]G.Benfatto,M.Cassandro,G.Gallavotti,F.Nicolo,E.Olivieri,E.Presutti,andE.Scacciatelli.UltravioletstabilityinEuclideanscalareldtheories.Commun.Math.Phys.,:{0,0.[]P.Billingsley.ConvergenceofProbabilityMeasures.Wiley,NewYork,. [0]P.Billingsley.ProbabilityandMeasure.Wiley,NewYork,.

[]M.J.Bissett.Aperturbation-theoreticderivationofWilson's`incompleteinte-gration'theory.J.Phys.C,:0{00,.[]P.M.BleherandP.Major.Criticalphenomenaanduniversalexponentsinstatisticalphysics.OnDyson'sHierarchicalmodel.Ann.Prob.,:{,.[]H.W.J.BloteandR.H.Swendsen.First-orderphasetransitionsandthethree-statePottsmodel.Phys.Rev.Lett.,:{0,.[]E.Bolthausen.Markovprocesslargedeviationsinthe-topology.Stoch.Proc.Appl.,:{0,.[]C.BorgsandJ.Z.Imbrie.Auniedapproachtophasediagramsineldtheoryandstatisticalmechanics.Commun.Math.Phys.,:0{,.[]C.BorgsandR.Kotecky.Arigoroustheoryofnite-sizescalingatrst-orderphasetransitions.J.Stat.Phys.,:{,0.[]H.J.BrascampandE.H.Lieb.SomeinequalitiesforGaussianmeasures.InA.M.Arthurs,editor,FunctionalIntegrationanditsApplications,Oxford,.ClarendonPress.[]H.J.BrascampandE.H.Lieb.OnextensionsoftheBrunn-MinkowskiandPrekopa-Leindlertheorems,includinginequalitiesforlogconcavefunctions,andwithanapplicationtothediusionequation.J.Funct.Anal.,:{,.[]H.J.Brascamp,E.H.Lieb,andJ.L.Lebowitz.Thestatisticalmechanicsofanharmoniclattices.Bull.Internat.Statist.Inst.,(Book):{0,.[0]O.BratteliandD.W.Robinson.OperatorAlgebrasandQuantumStatisticalMechanicsII.Springer-Verlag,NewYork{Heidelberg{Berlin,.[]J.Bricmont,J.-R.Fontaine,andL.J.Landau.Ontheuniquenessoftheequi-libriumstateforplanerotators.Commun.Math.Phys.,:{0,.[]J.Bricmont,J.R.Fontaine,J.L.Lebowitz,andT.Spencer.LatticesystemswithacontinuoussymmetryI.Perturbationtheoryforunboundedspins.Commun.Math.Phys.,:{0,0.[]J.BricmontandA.Kupiainen.LowercriticaldimensionfortherandomeldIsingmodel.Phys.Rev.Lett.,:{,.[]J.BricmontandA.Kupiainen.PhasetransitioninthedrandomeldIsingmodel.Commun.Math.Phys.,:{,. []J.Bricmont,K.Kuroda,andJ.L.Lebowitz.ThestructureofGibbsstatesandphasecoexistencefornon-symmetriccontinuumWidom-Rowlinsonmodel.Z.WahrscheinlichkeitstheorieVerw.Geb.,:{,.0

[]J.Bricmont,K.Kuroda,andJ.L.Lebowitz.Firstorderphasetransitionsinlatticeandcontinuoussystems:ExtensionofPirogov-Sinaitheory.Commun.Math.Phys.,0:0{,.[]J.BricmontandJ.Slawny.Firstorderphasetransitionsandperturbationthe-ory.InT.C.Dorlas,N.M.Hugenholtz,andM.Winnink,editors,StatisticalMe-chanicsandFieldTheory:MathematicalAspects(Proceedings,Groningen),Berlin{Heidelberg{NewYork,.Springer-Verlag(LectureNotesinPhysics#).[]J.BricmontandJ.Slawny.Phasetransitionsinsystemswithanitenumberofdominantgroundstates.J.Stat.Phys.,:{,.[]D.A.BrowneandP.Kleban.Equilibriumstatisticalmechanicsforkineticphasetransitions.Phys.Rev..,A0:{,.[0]A.D.BruceandA.Aharony.Coupledorderparameters,symmetry-breakingirrelevantscalingelds,andtetracriticalpoints.Phys.Rev.,B:{,.[]W.Bryc.Onthelargedeviationprincipleforstationaryweaklydependentran-domelds.Ann.Prob.,0:00{00,.[]D.BrydgesandT.Spencer.Self-avoidingwalksinormoredimensions.Com-mun.Math.Phys.,:{,.[]D.BrydgesandH.-T.Yau.Grad'perturbationsofmasslessGaussianelds.Commun.Math.Phys.,:{,0.[]T.W.Burkhardt.Random-eldsingularitiesinposition-spacerenormalization-grouptransformations.Phys.Rev.Lett.,:{,.[]T.W.BurkhardtandJ.M.J.vanLeeuwen.Progressandproblemsinreal-spacerenormalization.InT.W.BurkhardtandJ.M.J.vanLeeuwen,editors,Real-SpaceRenormalization.Springer-Verlag,Berlin{Heidelberg{NewYork,.[]C.Cammarota.Thelargeblockspininteraction.NuovoCim.,B:{,.[]J.L.Cardy.One-dimensionalmodelswith=rinteractions.J.Phys.,A:0{,.[]E.A.CarlenandA.Soer.Entropyproductionbyblockspinsummationandcentrallimittheorems.Commun.Math.Phys.,0:{,. []M.CassandroandE.Olivieri.Renormalizationgroupandanalyticityinonedimension:aproofofDobrushin'stheorem.Commun.Math.Phys.,0:{,.

[0]M.Cassandro,E.Olivieri,A.Pellegrinotti,andE.Presutti.ExistenceanduniquenessofDLRmeasuresforunboundedspinsystems.Z.Wahrscheinlichkeit-stheorieverw.Geb.,:{,.[]N.N.Cencov.StatisticalDecisionRulesandOptimalInference.AmericanMath-ematicalSociety(TranslationsofMathematicalMonographs,Vol.),Provi-dence,R.I.,.[]S.ChowlaandM.Cowles.RemarksonequationsrelatedtoFermat'slastthe-orem.InN.Koblitz,editor,NumberTheoryRelatedtoFermat'sLastTheorem,Boston{Basel{Stuttgart,.Birkhauser.[]F.S.CohenandD.B.Cooper.Simpleparallelhierarchicalandrelaxationalgo-rithmsforsegmentingnoncausalMarkovianrandomelds.IEEETrans.PatternAnal.MachineIntell.,:{,.[]J.E.Cohen,Y.Iwasa,G.Rautu,M.B.Ruskai,E.Seneta,andG.Zbaganu.Relativeentropyundermappingsbystochasticmatrices.ToappearinLin.Alg.Appl.[]F.Comets.GrandesdeviationspourdeschampsdeGibbssurZd.ComptesRendusAcad.Sci.ParisSerieI,0:{,.[]I.P.Cornfeld,S.V.Fomin,andYa.G.Sinai.ErgodicTheory.Springer-Verlag,NewYork{Heidelberg{Berlin,.[]I.Csiszar.EineinformationstheoretischeUngleichungundihreAnwendungaufdenBeweisderErgodizitatvonMarkoschenKetten.MagyarTud.Akad.Mat.KutatoInt.Kozl.,:{0,.[SeealsoMath.Reviews,#().].[]I.Csiszar.I-divergencegeometryofprobabilitydistributionsandminimizationproblems.Ann.Prob.,:{,.[]I.Csiszar.Sanovproperty,generalizedI-projectionandaconditionallimitthe-orem.Ann.Prob.,:{,.[0]H.A.M.DanielsandA.C.D.vanEnter.Dierentiabilitypropertiesofthepressureinlatticesystems.Commun.Math.Phys.,:{,0.[]J.deConinckandC.M.Newman.Themagnetization-energyscalinglimitinhighdimension.J.Stat.Phys.,:{,0.[]K.Decker,A.Hasenfratz,andP.Hasenfratz.Singularrenormalizationgrouptransformationsandrstorderphasetransitions(II).MonteCarlorenormaliza-tiongroupresults.Nucl.Phys.,B[FS]:{,. []C.DellacherieandP.-A.Meyer.ProbabilitiesandPotential.North-Holland,Amsterdam{NewYork{Oxford,.

[]P.Denes.UberdieDiophantischeGleichungxl+yl=czl.ActaMath.,:{,.[]J.-D.DeuschelandD.W.Stroock.LargeDeviations.AcademicPress,SanDiego{London,.[]L.E.Dickson.HistoryoftheTheoryofNumbers,volumeII.Chelsea,NewYork,.[]E.L.DinaburgandA.E.Mazel.Low-temperaturephasetransitionsinANNNImodel.InM.MebkhoutandR.Seneor,editors,ProceedingsthInternationalCongressonMathematicalPhysics,Singapore,.WorldScientic.[]E.L.DinaburgandA.E.Mazel.Analysisoflow-temperaturephasediagramofthemicroemulsionmodel.Commun.Math.Phys.,:{,.[]E.L.Dinaburg,A.E.Mazel,andYa.G.Sinai.ANNNImodelandcontourmodelswithinteractions.InS.P.Novikov,editor,MathematicalPhysicsReviews,Sov.Sci.ReviewsSect.C,Vol..GordonandBreach,NewYork,.[0]E.L.DinaburgandYa.G.Sinai.AnanalysisofANNNImodelbyPeierl's[sic]contourmethod.Commun.Math.Phys.,:{,.[]P.G.L.Dirichlet.Memoiresurl'impossibilitedequelquesequationsindetermineesducinquiemedegre.J.ReineAngew.Math.(Crelle'sJournal),:{,.[]R.L.Dobrushin.Existenceofaphasetransitioninthetwo-dimensionalandthree-dimensionalIsingmodels.SovietPhys.Doklady,0:{,.[]R.L.Dobrushin.Thedescriptionofarandomeldbymeansofconditionalprobabilitiesandconditionsofitsregularity.Theor.Prob.Appl.,:{,.[]R.L.Dobrushin.TheproblemofuniquenessofaGibbsrandomeldandtheproblemofphasetransitions.Funct.Anal.Appl.,:0{,.[]R.L.Dobrushin.Gibbsstatesdescribingcoexistenceofphasesforathree-dimensionalIsingmodel.Theor.Prob.Appl.,:{00,.[]R.L.Dobrushin.Gaussianrandomelds{Gibbsianpointofview.InR.L.DobrushinandYa.G.Sinai,editors,MulticomponentRandomSystems(AdvancesinProbabilityandRelatedTopics,Vol.),pages{,NewYork{Basel,0.MarcelDekker. []R.L.Dobrushin,J.Kolafa,andS.B.Shlosman.Phasediagramofthetwo-dimensionalIsingantiferromagnet(computer-assistedproof).Commun.Math.Phys.,0:{0,.

[]R.L.DobrushinandM.R.Martirosyan.NonniteperturbationsofGibbselds.Theor.Math.Phys.,:0{0,.[]R.L.DobrushinandM.R.Martirosyan.Possibilityofhigh-temperaturephasetransitionsduetothemany-particlenatureofthepotential.Theor.Math.Phys.,:{,.[0]R.L.DobrushinandE.A.Pecherski.Uniquenessconditionfornitelydependentrandomelds.InRandomFields.Esztergom(Hungary),Vol.I,Amsterdam,.North-Holland.[]R.L.DobrushinandE.A.Pecherski.AcriterionfortheuniquenessofGibbsianeldsinthenon-compactcase.InProb.Th.andMath.Statistics,volumeLNM0,pages{0,.[]R.L.DobrushinandS.B.Shlosman.Nonexistenceofone-andtwo-dimensionalGibbseldswithnoncompactgroupofcontinuoussymmetries.InR.L.Do-brushinandYa.G.Sinai,editors,MulticomponentRandomSystems(AdvancesinProbabilityandRelatedTopics,Vol.),pages{0,NewYork{Basel,0.MarcelDekker.[]R.L.DobrushinandS.B.Shlosman.Completelyanalyticrandomelds.InStatisticalMechanicsandDynamicalSystems,Bostonetc.,.Birkhauser.[]R.L.DobrushinandS.B.Shlosman.Constructivecriterionfortheuniquenessofagibbseld.InStatisticalMechanicsandDynamicalSystems,Bostonetc.,.Birkhauser.[]R.L.DobrushinandS.B.Shlosman.TheproblemoftranslationinvarianceofGibbsstatesatlowtemperatures.Sov.Sci.Rev.CMath.Phys.,:{,.[]R.L.DobrushinandS.B.Shlosman.Completelyanalyticalinteractions:con-structivedescription.J.Stat.Phys.,:{0,.[]R.L.DobrushinandM.Zahradnk.Phasediagramsforcontinuousspinsystems.InR.L.Dobrushin,editor,MathematicalProblemsofStatisticalPhysicsandDynamics.Reidel,.[]Y.Domar.OntheDiophantineequationjAxnBynj=.Math.Scand.,:{,.[]M.D.DonskerandS.R.S.Varadhan.AsymptoticevaluationofcertainMarkovprocessexpectationsforlargetime,I.Comm.PureAppl.Math.,:{,. [00]M.D.DonskerandS.R.S.Varadhan.AsymptoticevaluationofcertainMarkovprocessexpectationsforlargetime,II.Comm.PureAppl.Math.,:{0,.

[0]M.D.DonskerandS.R.S.Varadhan.AsymptoticevaluationofcertainMarkovprocessexpectationsforlargetime,III.Comm.PureAppl.Math.,:{,.[0]M.D.DonskerandS.R.S.Varadhan.AsymptoticevaluationofcertainMarkovprocessexpectationsforlargetime,IV.Comm.PureAppl.Math.,:{,.[0]T.C.DorlasandA.C.D.vanEnter.Non-Gibbsianlimitforlarge-blockmajority-spintransformations.J.Stat.Phys.,:{,.[0]N.DunfordandJ.T.Schwartz.LinearOperators.Interscience,NewYork,.[0]F.Dunlop.Correlationinequalitiesformulticomponentrotators.Commun.Math.Phys.,:{,.[0]F.DunlopandC.M.Newman.Multicomponenteldtheoriesandclassicalrotators.Commun.Math.Phys.,:{,.[0]E.B.Dynkin.Sucientstatisticsandextremepoints.Ann.Prob.,:0{0,.[0]R.E.Edwards.FourierSeries:AModernIntroduction,volumeI.Holt,RinehartandWinston,NewYork,.[0]R.G.EdwardsandA.D.Sokal.GeneralizationoftheFortuin-Kasteleyn-Swendsen-WangrepresentationandMonteCarloalgorithm.Phys.Rev.,D:00{0,.[0]R.S.Ellis.Entropy,LargeDeviationsandStatisticalMechanics.Springer-Verlag,NewYork{Heidelberg{Berlin{Tokyo,.[]J.H.Evertse.UpperBoundsfortheNumbersofSolutionsofDiophantineEqua-tions.MathematischCentrum(MathematicalCentreTracts#),Amsterdam,.[]G.FelderandJ.Frohlich.Intersectionpropertiesofsimplerandomwalks:Arenormalizationgroupapproach.Commun.Math.Phys.,:{,.[]B.U.Felderhof.Phasetransitionsinone-dimensionalcluster-interactionuids.III.Correlationfunctions.Ann.Phys.,:{00,0.[]B.U.FelderhofandM.E.Fisher.Phasetransitionsinone-dimensionalcluster-interactionuids.II.Simplelogarithmicmodel.Ann.Phys.,:{0,0. []R.Fernandez,J.Frohlich,andA.D.Sokal.RandomWalks,CriticalPhenomena,andTrivialityinQuantumFieldTheory.Springer-Verlag,Berlin{Heidelberg{NewYork,.

[]M.E.Fisher.Ondiscontinuityofthepressure.Commun.Math.Phys.,:{,.[]M.E.Fisher.Crossovereectsandoperatorexpansions.InJ.D.GuntonandM.S.Green,editors,RenormalizationGroupinCriticalPhenomenaandQuan-tumFieldTheory:ProceedingsofaConference,pages{.TempleUniversity,Philadelphia,.[]M.E.Fisher.Scaling,universalityandrenormalizationgrouptheory.InF.J.W.Hahne,editor,CriticalPhenomena(Stellenbosch),pages{.Springer-Verlag(LectureNotesinPhysics#),Berlin{Heidelberg{NewYork,.[]M.E.FisherandA.N.Berker.Scalingforrst-orderphasetransitionsinther-modynamicandnitesystems.Phys.Rev.,B:0{,.[0]M.E.FisherandB.U.Felderhof.Phasetransitionsinone-dimensionalcluster-interactionuids.IA.Thermodynamics.Ann.Phys.,:{,0.[]M.E.FisherandB.U.Felderhof.Phasetransitionsinone-dimensionalcluster-interactionuids.IB.Criticalbehavior.Ann.Phys.,:{,0.[]H.Follmer.Onentropyandinformationgaininrandomelds.Z.Wahrschein-lichkeitstheorieverw.Geb.,:0{,.[]H.Follmer.Randomeldsanddiusionprocesses.InP.Hennequin,editor,Ecoled'etedeProbabilitesdeSaint-FlourXV{XVII,Berlin{Heidelberg{NewYork,.Springer-Verlag(LectureNotesinMathematics#).[]H.FollmerandS.Orey.LargedeviationsfortheempiricaleldofaGibbsmeasure.Ann.Prob.,:{,.[]C.M.Fortuin.Ontherandomclustermodel.III.Thesimplerandomclustermodel.Physica,:{0,.[]C.M.Fortuin,J.Ginibre,andP.W.Kasteleyn.Correlationinequalitiesonsomepartiallyorderedsets.Commun.Math.Phys.,:{0,.[]C.M.FortuinandP.W.Kasteleyn.Ontherandomclustermodel.I.Introductionandrelationtoothermodels.Physica,:{,.[]Z.FriedmanandJ.Felsteiner.KadanoblocktransformationbytheMonte-Carlotechnique.Phys.Rev.B,:{,.[]A.FrigessiandM.Piccioni.Parameterestimationfortwo-dimensionalIsingeldscorruptedbynoise.Stoch.Proc.Appl.,:{,0. [0]J.Frohlich.Onthetrivialityof'theoriesandtheapproachtothecriticalpointind>(=)dimensions.NuclearPhysics,B00[FS]:{,.

[]J.Frohlich.Mathematicalaspectsofthephysicsofdisorderedsystems.InK.Os-terwalderandR.Stora,editors,CriticalPhenomena,RandomSystems,GaugeTheories[LesHouches],PartII,pages{,Amsterdam,.North-Holland.[]J.Frohlich,R.B.Israel,E.H.Lieb,andB.Simon.Phasetransitionsandreec-tionpositivity.II.Latticesystemswithshort-rangeandCoulombinteractions.J.Stat.Phys.,:{,0.[]J.FrohlichandC.E.Pster.Ontheabsenceofspontaneoussymmetrybreakingandofcrystallineorderingintwo-dimensionalsystems.Commun.Math.Phys.,:{,.[]J.FrohlichandT.Spencer.TheKosterlitz-Thoulessphasetransitioninthetwo-dimensionalplanerotatorandCoulombgas.Phys.Rev.Lett.,:00{00,.[]J.FrohlichandT.Spencer.TheKosterlitz-Thoulesstransitionintwo-dimensionalAbelianspinsystemsandtheCoulombgas.Commun.Math.Phys.,:{0,.[]J.FrohlichandT.Spencer.Phasediagramsandcriticalpropertiesof(classical)Coulombsystems.InRigorousAtomicandMolecularPhysics,pages{0,NewYorkandLondon,.PlenumPress.[]J.FrohlichandT.Spencer.Thephasetransitionintheone-dimensionalIsingmodelwith=rinteractionenergy.Commun.Math.Phys.,:{0,.[]J.FrohlichandT.Spencer.AbsenceofdiusionintheAndersontightbindingmodelforlargedisorderorlowenergy.Commun.Math.Phys.,:{,.[]G.GallavottiandS.Miracle-Sole.Statisticalmechanicsoflatticesystems.Com-mun.Math.Phys.,:{,.[0]G.GallavottiandS.Miracle-Sole.Correlationfunctionsoflatticesystems.Com-mun.Math.Phys.,:{,.[]G.GallavottiandS.Miracle-Sole.EquilibriumstatesoftheIsingmodelinthetwo-phaseregion.Phys.Rev.,B:{,.[]J.M.Gandhi.OnFermat'slasttheorem.Amer.Math.Monthly,:{00,. []P.Ganssler.Compactnessandsequentialcompactnessinspacesofmeasures.Z.Wahrscheinlichkeitstheorieverw.Geb.,:{,.

[]P.L.Garrido,A.Labarta,andJ.Marro.StationarynonequilibriumstatesintheIsingmodelwithlocallycompetingtemperatures.J.Stat.Phys.,:{,.[]K.Gawedzki.Rigorousrenormalizationgroupatwork.Physica,0A:{,.[]K.Gawedzki,R.Kotecky,andA.Kupiainen.Coarsegrainingapproachtorstorderphasetransitions.J.Stat.Phys.,:0{,.[]K.GawedzkiandA.Kupiainen.Arigorousblockspinapproachtomasslesslatticetheories.Commun.Math.Phys.,:{,0.[]K.GawedzkiandA.Kupiainen.Blockspinrenormalizationgroupfordipolegasand(r).Ann.Phys.,:{,.[]K.GawedzkiandA.Kupiainen.Gross-Neveumodelthroughconvergentpertur-bationexpansions.Commun.Math.Phys.,0:{0,.[0]K.GawedzkiandA.Kupiainen.Masslesslatticetheory:Rigorouscontrolofarenormalizableasymptoticallyfreemodel.Commun.Math.Phys.,:{,.[]K.GawedzkiandA.Kupiainen.Asymptoticfreedombeyondperturbationtheory.InK.OsterwalderandR.Stora,editors,CriticalPhenomena,RandomSystems,GaugeTheories[LesHouches],PartI,pages{,Amsterdam,.North-Holland.[]D.Geman.Randomeldsandinverseproblemsinimaging.InEcoled'EtedeProbabilitesdeSaint-FlourXVIII{,pages{.Springer-Verlag(LectureNotesinMathematics#),Berlin{Heidelberg{NewYork,0.[]S.Geman.HiddenMarkovmodelsforimageanalysis.LectureatIstitutoperleApplicazionidelCalcolo,Roma(July0).[]S.GemanandD.Geman.Stochasticrelaxation,GibbsdistributionsandtheBayesianrestorationofimages.IEEETrans.PatternAnal.MachineIntell.,:{,.[]H.-O.Georgii.LargedeviationsandmaximumentropyprincipleforinteractingrandomeldsonZd.ToappearinAnn.Prob.[]H.-O.Georgii.Tworemarksonextremalequilibriumstates.Commun.Math.Phys.,:0{,.[]H.-O.Georgii.GibbsMeasuresandPhaseTransitions.WalterdeGruyter(deGruyterStudiesinMathematics,Vol. ),Berlin{NewYork,.

[]B.Gidas.Arenormalizationgroupapproachtoimageprocessingproblems.IEEETrans.PatternAnal.Mach.Intell.,:{0,.[]J.GlimmandA.Jae.Positivityofthe'Hamiltonian.FortschrittederPhysik,:{,.[0]J.GlimmandA.Jae.QuantumPhysics:AFunctionalIntegralPointofView.Springer-Verlag,Berlin{Heidelberg{NewYork,.[]S.Goldstein.Anoteonspecications.Z.Wahrscheinlichkeitstheorieverw.Geb.,:{,.[]S.Goldstein,R.Kuik,J.L.Lebowitz,andC.Maes.FromPCA'stoequilibriumsystemsandback.Commun.Math.Phys.,:{,.[]A.Gonzalez-Arroyo,M.Okawa,andY.Shimizu.MonteCarlorenormalization-groupstudyofthefour-dimensionalZgaugetheory.Phys.Rev.Lett.,0:{0,.[]A.Gonzalez-ArroyoandJ.Salas.ComputingthecouplingsofIsingsystemsfromSchwinger-Dysonequations.Phys.Lett.,B:{,.[]A.Gonzalez-ArroyoandJ.Salas.Renormalizationgroupowofthetwo-dimensionalIsingmodelatT

[]R.B.GrithsandP.A.Pearce.Mathematicalpropertiesofposition-spacerenormalization-grouptransformations.J.Stat.Phys.,0:{,.[]R.B.GrithsandD.Ruelle.Strictconvexity(\continuity")ofthepressureinlatticesystems.Commun.Math.Phys.,:{,.[]C.GrillenbergerandU.Krengel.OnthespatialconstantofsuperadditivesetfunctionsinRd.InH.Michel,editor,ErgodicTheoryandRelatedTopics,Berlin,.Akademie-Verlag.[]P.Groeneboom,J.Oosterho,andF.H.Ruymgaart.Largedeviationtheoremsforempiricalprobabilitymeasures.Ann.Prob.,:{,.[]L.Gross.Absenceofsecond-orderphasetransitionsintheDobrushinuniquenessregion.J.Stat.Phys.,:{,.[]L.Gross.Thermodynamics,statisticalmechanics,andrandomelds.InEcoled'EtedeProbabilitesdeSaint-FlourX{0.Springer-Verlag(LectureNotesinMathematics#),Berlin{Heidelberg{NewYork,.[]C.GruberandA.Sut}o.Phasediagramsoflatticesystemsofresidualentropy.J.Stat.Phys.,:{,.[0]P.HallandC.C.Heyde.MartingaleLimitTheoryanditsApplication.AcademicPress,NewYork,0.[]T.Hara.Arigorouscontroloflogarithmiccorrectionsinfour-dimensional'spinsystems.I.TrajectoryofeectiveHamiltonians.J.Stat.Phys.,:{,.[]T.Hara.Mean-eldcriticalbehaviourforcorrelationlengthforpercolationinhighdimensions.Prob.Th.Rel.Fields,:{,0.[]T.HaraandG.Slade.Mean-eldcriticalbehaviourforpercolationinhighdimensions.Commun.Math.Phys.,:{,0.[]T.HaraandG.Slade.Ontheuppercriticaldimensionoflatticetreesandlatticeanimals.J.Stat.Phys.,:{0,0.[]T.HaraandG.Slade.Criticalbehaviourofself-avoidingwalkinveormoredimensions.Bull.Amer.Math.Soc.,:{,.[]T.HaraandG.Slade.Thelaceexpansionforself-avoidingwalkinveormoredimensions.Rev.Math.Phys.,:{,. []T.HaraandG.Slade.Self-avoidingwalkinveormoredimensions.I.Thecriticalbehaviour.Commun.Math.Phys.,:0{,.0

[]T.HaraandH.Tasaki.Arigorouscontroloflogarithmiccorrectionsinfour-dimensional'spinsystems.II.Criticalbehaviorofsusceptibilityandcorrelationlength.J.Stat.Phys.,:{,.[]A.HasenfratzandP.Hasenfratz.Singularrenormalizationgrouptransformationsandrstorderphasetransitions(I).Nucl.Phys.,B[FS]:{0,.[0]A.Hasenfratz,P.Hasenfratz,U.Heller,andF.Karsch.ImprovedMonteCarlorenormalizationgroupmethods.Phys.Lett.,B0:{,.[]T.Heath.AHistoryofGreekMathematics,volumeI,pages{and0.ClarendonPress,Oxford,.[]Y.Higuchi.Ontheabsenceofnon-translationallyinvariantGibbsstatesforthetwo-dimensionalIsingsystem.InJ.Fritz,J.L.Lebowitz,andD.Szasz,editors,RandomFields:RigorousResultsinStatisticalMechanicsandQuantumFieldTheory(Esztergom).North-Holland,Amsterdam,.[]Y.HiguchiandR.Lang.OntheconvergenceoftheKadanotransformationtowardstrivialxedpoints.Z.Wahrscheinlichkeitstheorieverw.Geb.,:0{,.[]P.Holicky,R.Kotecky,andM.Zahradnk.Rigidinterfacesforlatticemodelsatlowtemperatures.J.Stat.Phys.,0:{,.[]W.HolsztynskiandJ.Slawny.Peierlsconditionandthenumberofgroundstates.Commun.Math.Phys.,:{0,.[]P.J.Huber.Thebehaviorofmaximumlikelihoodestimatesundernonstandardconditions.InL.M.LeCamandJ.Neyman,editors,ProceedingsoftheFifthBerkeleySymposiumonMathematicalStatisticsandProbability,vol.I,pages{.UniversityofCaliforniaPress,Berkeley{LosAngeles,.[]O.Hudak.Onthecharacterofpeculiaritiesintheposition-spacerenormalization-grouptransformations.Phys.Lett.,A:{,.[]N.M.Hugenholtz.C-algebrasandstatisticalmechanics.InProceedingsofSymposiainPureMathematics,Volume,Part,pages0{,Providence,R.I.,.AmericanMathematicalSociety.[]N.M.Hugenholtz.Ontheinverseprobleminstatisticalmechanics.Commun.Math.Phys.,:{,. [00]D.IagolnitzerandB.Souillard.Randomeldsandlimittheorems.InJ.Fritz,J.L.Lebowitz,andD.Szasz,editors,RandomFields(Esztergom,),Vol.II,pages{.North-Holland,Amsterdam,.

[0]I.A.IbragimovandYu.V.Linnik.IndependentandStationarySequencesofRandomVariables.Wolters{Noordho,Groningen,.[0]J.Z.Imbrie.PhasediagramsandclusterexpansionsforlowtemperatureP()models.I.Thephasediagram.Commun.Math.Phys.,:{0,.[0]J.Z.Imbrie.PhasediagramsandclusterexpansionsforlowtemperatureP()models.II.TheSchwingerfunction.Commun.Math.Phys.,:0{,.[0]S.N.Isakov.NonanalyticfeaturesoftherstorderphasetransitionintheIsingmodel.Commun.Math.Phys.,:{,.[0]R.B.Israel.High-temperatureanalyticityinclassicallatticesystems.Commun.Math.Phys.,0:{,.[0]R.B.Israel.ConvexityintheTheoryofLatticeGases.PrincetonUniv.Press,Princeton,N.J.,.[0]R.B.Israel.BanachalgebrasandKadanotransformations.InJ.Fritz,J.L.Lebowitz,andD.Szasz,editors,RandomFields(Esztergom,),Vol.II,pages{0.North-Holland,Amsterdam,.[0]R.B.Israel.Generictrivialityofphasediagramsinspacesoflong-rangeinter-actions.Commun.Math.Phys.,0:{,.[0]R.B.IsraelandR.R.Phelps.Someconvexityquestionsarisinginstatisticalmechanics.Math.Scand.,:{,.[0]L.P.KadanoandA.Houghton.Numericalevaluationsofthecriticalpropertiesofthetwo-dimensionalIsingmodel.Phys.Rev.,B:{,.[]J.-P.Kahane.SeriesdeFourierAbsolumentConvergentes.Springer-Verlag,Berlin{Heidelberg{NewYork,0.[]I.A.Kashapov.Justicationoftherenormalization-groupmethod.Theor.Math.Phys.,:{,0.[]A.Katz.PrinciplesofStatisticalMechanics.Freeman,SanFrancisco{London,.[]T.Kennedy.Somerigorousresultsonmajorityrulerenormalizationgrouptrans-formationsnearthecriticalpoint.UniversityofArizonapreprint,.[]T.KennedyandE.H.Lieb.Anitinerantelectronmodelwithcrystallineormagneticlong-rangeorder.Physica,A:0{,. []J.C.Kieer.AcounterexampletoPerez'sgeneralizationoftheShannon-McMillantheorem.Ann.Prob.,:{,.Correctionin:{().

[]W.Klein,D.J.Wallace,andR.K.P.Zia.Essentialsingularitiesatrst-orderphasetransitions.Phys.Rev.Lett.,:{,.[]H.KochandP.Wittwer.Anon-Gaussianrenormalizationgroupxedpointforhierarchicalscalarlatticeeldtheories.Commun.Math.Phys.,0:{,.[]H.KochandP.Wittwer.Ontherenormalizationgrouptransformationforscalarhierarchicalmodels.Commun.Math.Phys.,:{,.[0]J.M.Kosterlitz.Thecriticalpropertiesofthetwo-dimensionalxymodel.J.Phys.C,:0{00,.[]F.Koukiou,D.Petritis,andM.Zahradnk.ExtensionofthePirogov-Sinaitheorytoaclassofquasiperiodicinteractions.Commun.Math.Phys.,:{,.[]O.K.Kozlov.Gibbsdescriptionofasystemofrandomvariables.Probl.Inform.Transmission,0:{,.[]U.Krengel.ErgodicTheorems.WalterdeGruyter(deGruyterStudiesinMath-ematics,Vol.),Berlin{NewYork,.[]K.Krickeberg.ProbabilityTheory.Addison-Wesley,Reading,Mass.,.[]S.Kullback.InformationTheoryandStatistics.Wiley,NewYork,.[]S.KullbackandR.A.Leibler.Oninformationandsuciency.Ann.Math.Stat.,:{,.[]H.Kunsch.ThermodynamicsandstatisticalanalysisofGaussianrandomelds.Z.Wahrscheinlichkeitstheorieverw.Geb.,:0{,.[]H.Kunsch.DecayofcorrelationsunderDobrushin'suniquenessconditionanditsapplications.Commun.Math.Phys.,:0{,.[]H.Kunsch.Non-reversiblestationarymeasuresforinniteinteractingparticlesystems.Z.Wahrscheinlichkeitstheorieverw.Geb.,:0{,.[0]O.E.LanfordIII.Entropyandequilibriumstatesinclassicalstatisticalme-chanics.InA.Lenard,editor,StatisticalMechanicsandMathematicalProblems(BattelleSeattleRencontres),pages{.Springer-Verlag(LectureNotesinPhysics#0),Berlin{Heidelberg{NewYork,. []O.E.LanfordIIIandD.W.Robinson.Statisticalmechanicsofquantumspinsystems.III.Commun.Math.Phys.,:{,.

[]O.E.LanfordIIIandD.Ruelle.Observablesatinnityandstateswithshortrangecorrelationsinstatisticalmechanics.Commun.Math.Phys.,:{,.[]C.B.Lang.RenormalizationstudyofcompactU()latticegaugetheory.Nucl.Phys.,B0[FS]:{,.[]S.Lang.IntroductiontoDiophantineApproximations.Addison-Wesley,Reading,Mass.,.[]V.A.Lebesgue.Surl'equationindetermineex+y=az.J.deMaths.PuresAppl.,:{0,.[]J.L.Lebowitz.CoexistenceofphasesinIsingferromagnets.J.Stat.Phys.,:{,.[]J.L.Lebowitz.Numberofphasesinonecomponentferromagnets.InG.Dell'Antonio,S.Doplicher,andG.Jona-Lasinio,editors,MathematicalProb-lemsinTheoreticalPhysics.Springer-Verlag(LectureNotesinPhysics#0),Berlin{Heidelberg{NewYork,.[]J.L.Lebowitz.Microscopicoriginofhydrodynamicequations:Derivationandconsequences.Physica,0A:{,.[]J.L.LebowitzandC.Maes.Theeectofanexternaleldonaninterface,entropicrepulsion.J.Stat.Phys.,:{,.[0]J.L.Lebowitz,C.Maes,andE.R.Speer.Statisticalmechanicsofprobabilisticcellularautomata.J.Stat.Phys.,:{0,0.[]J.L.LebowitzandO.Penrose.RigoroustreatmentoftheVanderWaals-Maxwelltheoryoftheliquid-vaportransition.J.Math.Phys.,:{,.[]J.L.LebowitzandO.Penrose.Analyticandclusteringpropertiesofthermo-dynamicfunctionsanddistributionfunctionsforclassicallatticeandcontinuoussystems.Commun.Math.Phys.,:{,.[]J.L.LebowitzandO.Penrose.Divergentsusceptibilityofisotropicferromagnets.Phys.Rev.Lett.,:{,.[]J.L.Lebowitz,M.K.Phani,andD.F.Styer.PhasediagramofCu-Au-typealloys.J.Stat.Phys.,pages{,.[]J.L.LebowitzandE.Presutti.Statisticalmechanicsofunboundedspinsystems.Commun.Math.Phys.,0:{,. []J.L.LebowitzandR.H.Schonmann.Pseudo-freeenergiesandlargedeviationsfornon-GibbsianFKGmeasures.Prob.Th.Rel.Fields,:{,.

[]D.H.Lehmer.Reviewof[].Math.Reviews,:0,.[]B.G.Leroux.Maximum-likelihoodestimationforhiddenMarkovmodels.Stoch.Proc.Appl.,0:{,.[]A.L.Lewis.Latticerenormalizationgroupandthermodynamiclimit.Phys.Rev.B,:{,.[0]E.H.LiebandA.D.Sokal.AgeneralLee-Yangtheoremforone-componentandmulti-componentferromagnets.Commun.Math.Phys.,0:{,.[]T.M.Liggett.InteractingParticleSystems.Springer-Verlag,NewYork{Berlin{Heidelberg{Tokyo,.[]J.Lindenstrauss,G.Olsen,andY.Sternfeld.ThePoulsensimplex.Ann.Inst.Fourier(Grenoble),:{,.[]J.LorincziandM.Winnink.SomeremarksonalmostGibbsstates.TobepublishedintheproceedingsoftheNATOAdvancedStudiesInstituteworkshoponCellularAutomataandCooperativeSystems[LesHouches].N.Boccara,E.Goles,S.MartinezandP.Picco,editors.Kluwer,Dordrecht,.[]S.K.Ma.ModernTheoryofCriticalPhenomena.Benjamin,Reading,Mass.,.[]C.MaesandF.Redig.Anisotropicperturbationsofthesimplesymmetricexclu-sionprocess:longrangecorrelations.J.PhysiqueI,:{,.[]C.MaesandK.vandeVelde.DeningrelativeenergiesfortheprojectedIsingmeasure.ToappearinHelv.Phys.Acta.[]C.MaesandK.vandeVelde.Theinteractionpotentialofthestationarymeasureofahigh-noisespinipprocess.Leuvenpreprint,.[]F.MartinelliandE.Olivieri.Privatecommunicationandpaperinpreparation,.[]F.MartinelliandE.Scoppola.Asimplestochasticclusterdynamics:rigorousresults.J.Phys.A,:{,.[0]D.G.Martirosyan.UniquenessofGibbsstatesinlatticemodelswithonegroundstate.Theor.Mat.Phys.,:{,.[]A.E.MazelandYu.M.Suhov.Randomsurfaceswithtwo-sidedconstraints:anapplicationofthetheoryofdominantgroundstates.Preprint,. []A.Messager,S.Miracle-Sole,andC.-E.Pster.Correlationinequalitiesanduniquenessoftheequilibriumstatefortheplanerotatorferromagneticmodel.Commun.Math.Phys.,:{0,.

[]J.Miekisz.Theglobalminimumofenergyisnotalwaysasumoflocalminima|anoteonfrustration.Preprint,0.[]J.Miekisz.Classicallatticegasmodelwithauniquenondegeneratebutunstableperiodicgroundstateconguration.Commun.Math.Phys.,:{,.[]L.J.Mordell.DiophantineEquations.AcademicPress,London{NewYork,.[]J.Moulin-Ollagnier.Theoremeergodiquepresquesous-additifetconvergenceenmoyennedel'information.Ann.Inst.HenriPoincare,B:{,.[]J.Moulin-OllagnierandD.Pinchon.MesuresdeGibbsinvariantesetmesuresd'equilibre.Z.Wahrscheinlichkeitstheorieverw.Geb.,:{,.[]J.Moulin-OllagnierandD.Pinchon.Filtremoyennantetvaleursmoyennesdescapacitesinvariantes.Bull.Soc.Math.France,0:{,.[]J.Moussouris.GibbsandMarkovrandomsystemswithconstraints.J.Stat.Phys.,0:{,.[0]C.Mugler.PlatonetlaRechercheMathematiquedesonEpoque,pages{.P.H.Heitz,Strasbourg{Zurich,.[]D.R.Nelson.Coexistence-curvesingularitiesinisotropicferromagnets.Phys.Rev.,B:{0,.[]J.Neveu.BasesMathematiquesduCalculdesProbabilites,emeed.Masson,Paris,0.[Englishtranslationofrstedition:MathematicalFoundationsoftheCalculusofProbability(Holden-Day,SanFrancisco,).].[]C.M.Newman.NormaluctuationsandtheFKGinequalities.Commun.Math.Phys.,:{,0.[]C.M.Newman.AgeneralcentrallimittheoremforFKGsystems.Commun.Math.Phys.,:{0,.[]C.M.Newman.Privatecommunication,.[]T.NiemeijerandJ.M.J.vanLeeuwen.Wilsontheoryforspinsystemsonatriangularlattice.Phys.Rev.Lett.,:{,.[]T.NiemeijerandJ.M.J.vanLeeuwen.RenormalizationtheoryforIsing-likespinsystems.InC.DombandM.S.Green,editors,PhaseTransitionsandCriticalPhenomena,Vol..AcademicPress,London{NewYork{SanFrancisco,. []T.NiemeyerandJ.M.J.vanLeeuwen.Wilsontheoryfor-dimensionalIsingspinsystems.Physica,:{0,.

[]B.Nienhuis,A.N.Berker,E.K.Riedel,andM.Schick.First-andsecond-orderphasetransitionsinPottsmodels:Renormalization-groupsolution.Phys.Rev.Lett.,:{0,.[0]B.NienhuisandM.Nauenberg.First-orderphasetransitionsinrenormalization-grouptheory.Phys.Rev.Lett.,:{,.[]G.L.O'Brien.Scalingtransformationsforf0;g-valuedsequences.Z.Wahrscheinlichkeitstheorieverw.Geb.,:{,0.[]S.Olla.LargedeviationsforalmostMarkovianprocesses.Probab.Th.Rel.Fields,:{0,.[]S.Olla.LargedeviationsforGibbsrandomelds.Probab.Th.Rel.Fields,:{,.[]G.H.Olsen.OnsimplicesandthePoulsensimplex.InFunctionalAnalysis:Sur-veysandRecentResultsII[Proceedingsoftheconferenceonfunctionalanalysis,Paderborn,Germany,],pages{,Amsterdam{NewYork{Oxford,0.North-Holland(North-HollandMathematicsStudies#).[]J.C.Oxtoby.MeasureandCategory.Springer-Verlag,NewYork{Heidelberg{Berlin,.[]Y.M.Park.Clusterexpansionforclassicalandquantumlatticesystems.J.Stat.Phys.,:{,.[]Y.M.Park.ExtensionofPirogov-SinaitheoryofphasetransitionstoinniterangeinteractionsI.Clusterexpansion.Commun.Math.Phys.,:{,.[]Y.M.Park.ExtensionofPirogov-Sinaitheoryofphasetransitionstoinniterangeinteractions.II.Phasediagram.Commun.Math.Phys.,:{,.[]K.R.Parthasarathy.ProbabilityMeasuresonMetricSpaces.AcademicPress,NewYork{SanFrancisco{London,.[0]E.A.Pecherski.ThePeierlscondition(GPScondition)isnotalwayssatised.Sel.Math.Sov.,:{,/.[]R.Peierls.Ising'smodelofferromagnetism.Proc.CambridgePhilos.Soc.,:{,.[]P.PfeutyandG.Toulouse.IntroductiontotheRenormalizationGroupandtoCriticalPhenomena.Wiley,Chichester{NewYork,. []R.R.Phelps.LecturesonChoquet'sTheorem.VanNostrand,Princeton,.

[]R.R.Phelps.GenericFrechetdierentiabilityofthepressureincertainlatticesystems.Commun.Math.Phys.,:{,.[]S.A.Pirogov.Coexistenceofphasesinamulticomponentlatticeliquidwithcomplexthermodynamicparameters.Theor.Math.Phys.,:{,.[]S.A.PirogovandYa.G.Sinai.Phasediagramsofclassicallatticesystems.Theor.Math.Phys.,:{,.[]S.A.PirogovandYa.G.Sinai.Phasediagramsofclassicallatticesystems.Continuation.Theor.Math.Phys.,:{,.[]C.Prakash.High-temperaturedierentiabilityoflatticeGibbsstatesbyDo-brushinuniquenesstechniques.J.Stat.Phys.,:{,.[]C.Preston.RandomFields.Springer-Verlag(LectureNotesinMathematics#),Berlin{Heidelberg{NewYork,.[00]C.Preston.Constructionofspecications.InL.Streit,editor,QuantumFields,Algebras,Processes,Wien{NewYork,0.Springer-Verlag.[0]Proclus.CommentairesurlaRepublique,volumeII,pages{.LibrairiePhilosophiqueJ.Vrin,Paris,0.TranslationandnotesbyA.J.Festugiere.[0]L.R.Rabiner.AtutorialonhiddenMarkovmodelsandselectedapplicationsinspeechrecognition.Proc.IEEE,:{,.[0]C.Radin.Lowtemperatureandtheoriginofcrystallinesymmetry.Int.J.Mod.Phys.B,:{,.[0]C.Radin.Disorderedgroundstatesofclassicallatticemodels.Rev.Math.Phys.,:{,.[0]C.RebbiandR.H.Swendsen.MonteCarlorenormalization-groupstudiesofq-statePottsmodelsintwodimensions.Phys.Rev.,B:0{0,0.[0]M.ReedandB.Simon.MethodsofModernMathematicalPhysicsI:FunctionalAnalysis.AcademicPress,NewYork{London,.[0]L.E.Reichl.AModernCourseinStatisticalPhysics.Univ.ofTexasPress,Austin,0.[0]P.Ribenboim.TheBookofPrimeNumberRecords.Springer-Verlag,NewYork{Berlin{Heidelberg,secondedition,.[0]H.L.Royden.RealAnalysis.Macmillan,NewYork,secondedition,. [0]W.Rudin.FunctionalAnalysis.McGraw-Hill,NewYork,.

[]D.Ruelle.Someremarksonthegroundstateofinnitesystemsinstatisticalmechanics.Commun.Math.Phys.,:{,.[]D.Ruelle.StatisticalMechanics:RigorousResults.Benjamin,Reading,Mas-sachusetts,.[]D.Ruelle.ThermodynamicFormalism.Addison-Wesley,Reading,Mas-sachusetts,.[]J.Salas.Privatecommunication..[]H.H.Schaefer.TopologicalVectorSpaces.Springer-Verlag,NewYork{Heidelberg{Berlin,0.[]A.G.Schlijper.Tilingproblemsandundecidabilityintheclustervariationmethod.J.Stat.Phys.,0:{,.[]W.Schmidt.DiophantineApproximation.Springer-Verlag(LectureNotesinMathematics#),Berlin{Heidelberg{NewYork,0.[]R.H.Schonmann.ProjectionsofGibbsmeasuresmaybenon-Gibbsian.Com-mun.Math.Phys.,:{,.[]R.Schrader.Groundstatesinclassicallatticesystemswithhardcore.Commun.Math.Phys.,:{,0.[0]L.Schwartz.RadonMeasuresonArbitraryTopologicalSpacesandCylindricalmeasures.TataInstituteofFundamentalResearchandOxfordUniversityPress,Oxford,.[]S.H.ShenkerandJ.Tobochnik.MonteCarlorenormalization-groupanalysisoftheclassicalHeisenbergmodelintwodimensions.Phys.Rev.,B:{,0.[]S.B.Shlosman.Uniquenessandhalf-spacenonuniquenessofGibbsstatesinCzechmodels.Theor.Math.Phys.,:{,.[]S.B.Shlosman.GaussianbehaviorofthecriticalIsingmodelindimensiond>.SovietPhys.Dokl.,:0{0,.[]S.B.Shlosman.Relationsamongthecumulantsofrandomeldswithattraction.TheoryProb.Appl.,:{,.[]S.D.Silvey.StatisticalInference.ChapmanandHall,London,. []B.SimonandA.D.Sokal.Rigorousentropy-energyarguments.J.Stat.Phys.,:{,.Addendumin:().

[]Ya.G.Sinai.Self-similarprobabilitydistributions.Theor.Prob.anditsAppl.,:{0,.[]Ya.G.Sinai.TheoryofPhaseTransitions:RigorousResults.PergamonPress,Oxford{NewYork,.[]J.Slawny.Low-temperaturepropertiesofclassicallatticesystems:phasetransi-tionsandphasediagrams.InC.DombandJ.L.Lebowitz,editors,PhaseTran-sitionsandCriticalPhenomena,Vol..AcademicPress,London{NewYork,.[0]A.D.Sokal.Existenceofcompatiblefamiliesofproperregularconditionalprob-abilities.Z.Wahrscheinlichkeitstheorieverw.Geb.,:{,.[]A.D.Sokal.Unpublished,.[]A.D.Sokal.Moresurprisesinthegeneraltheoryoflatticesystems.Commun.Math.Phys.,:{,.[]A.D.Sokal.Subadditivesetfunctionsonadiscreteamenablegroup.Unpublishedmanuscript,.[]A.Stella.Singularitiesinrenormalizationgrouptransformations.Physica,0A:{0,.[]K.Subbarao.RenormalizationgroupforIsingspinsonanitelattice.Phys.Rev.,B:{,.[]W.G.Sullivan.PotentialsforalmostMarkovianrandomelds.Commun.Math.Phys.,:{,.[]R.H.Swendsen.MonteCarlorenormalization.InT.W.BurkhardtandJ.M.J.vanLeeuwen,editors,Real-SpaceRenormalization,pages{.Springer-Verlag,Berlin{Heidelberg{NewYork,.[]R.H.Swendsen.Monte-Carlorenormalizationgroup.InM.Levy,J.-C.LeGuil-lou,andJ.Zinn-Justin,editors,PhaseTransitions(Cargese0),pages{.Plenum,NewYork{London,.[]R.H.Swendsen.MonteCarlocalculationofrenormalizedcouplingparameters.I.d=Isingmodel.Phys.Rev.,B0:{,.[0]R.H.SwendsenandJ.-S.Wang.Non-universalcriticaldynamicsinMonteCarlosimulations.Phys.Rev.Lett.,:{,. []G.S.Sylvester.InequalitiesforcontinuousspinIsingferromagnets.J.Stat.Phys.,:{,.0

[]I.Syozi.TransformationofIsingmodels.InC.DombandM.S.Green,editors,PhaseTransitionsandCriticalPhenomena,Vol..AcademicPress,London{NewYork,.[]P.Tannery.ReviewofH.Konen,GeschichtederGleichungtDu=.Bull.desSci.Math.,:{,0.Seepage.[]TheondeSmyrne.ExpositiondesConnaissancesMathematiquesUtilespourlaLecturedePlaton,pages{.Hachette,Paris,.TranslatedbyJ.Dupuis.[]C.J.Thompson.MathematicalStatisticalMechanics.PrincetonUniversityPress,Princeton,NJ,.[]C.J.Thompson.ClassicalEquilibriumStatisticalMechanics.ClarendonPress,Oxford,.[]A.L.Toom.Stableandattractivetrajectoriesinmulticomponentsystems.InR.L.DobrushinandYa.G.Sinai,editors,MulticomponentRandomSystems(AdvancesinProbabilityandRelatedTopics,Vol.),pages{,NewYork{Basel,0.MarcelDekker.[]H.vanBeijeren.InterfacesharpnessintheIsingsystem.Commun.Math.Phys.,0:{,.[]A.C.D.vanEnter.Anoteonthestabilityofphasediagramsinlatticesystems.Commun.Math.Phys.,:{,.[0]A.C.D.vanEnter.Instabilityofphasediagramsforaclassof\irrelevant"perturbations.Phys.Rev.,B:{,.[]A.C.D.vanEnterandR.Fernandez.Aremarkondierentnormsandanalyticityformany-particleinteractions.J.Stat.Phys.,:{,.[]A.C.D.vanEnter,R.Fernandez,andA.D.Sokal.Regularitypropertiesandpathologiesofposition-spacerenormalization-grouptransformations.Nucl.Phys.B(Proc.Suppl.),0:{,.[]A.C.D.vanEnter,R.Fernandez,andA.D.Sokal.Renormalizationtransfor-mationsinthevicinityofrst-orderphasetransitions:Whatcanandcannotgowrong.Phys.Rev.Lett.,:{,. []A.C.D.vanEnter,R.Fernandez,andA.D.Sokal.Non-Gibbsianstates.TobepublishedintheproceedingsofthePragueWorkshoponPhaseTransitions.R.Kotecky,editor.WorldScientic,Singapore,.

[]A.C.D.vanEnter,R.Fernandez,andA.D.Sokal.Non-Gibbsianstatesforrenormalization-grouptransformationsandbeyond.Tobepublishedinthepro-ceedingsoftheNATOAdvancedStudiesInstituteworkshoponCellularAu-tomataandCooperativeSystems[LesHouches].N.Boccara,E.Goles,S.MartinezandP.Picco,editors.Kluwer,Dordrecht,.[]A.C.D.vanEnterandJ.Miekisz.Breakingofperiodicityatpositivetempera-tures.Commun.Math.Phys.,:{,0.[]J.M.J.vanLeeuwen.SingularitiesinthecriticalsurfaceanduniversalityforIsing-likespinsystems.Phys.Rev.Lett.,:0{0,.[]V.S.Varadarajan.Measuresontopologicalspaces[inRussian].Mat.Sbornik(N.S.),():{00,.[EnglishtranslationinAmer.Math.Soc.Transla-tions(Ser.),{().].[]S.R.S.Varadhan.Privatecommunication,.[0]S.R.S.Varadhan.LargeDeviationsandApplications.SIAM,Philadelphia,.[]J.Voigt.Stochasticoperators,informationandentropy.Commun.Math.Phys.,:{,.[]J.-S.WangandJ.L.Lebowitz.Phasetransitionsanduniversalityinnonequilib-riumsteadystatesofstochasticIsingmodels.J.Stat.Phys.,:{0,.[]A.S.Wightman.Convexityandthenotionofequilibriumstateinthermody-namicsandstatisticalmechanics.IntroductiontoR.B.Israel,ConvexityintheTheoryofLatticeGases,PrincetonUniversityPress,.[]K.G.Wilson.Therenormalizationgroup:CriticalphenomenaandtheKondoproblem.Rev.Mod.Phys.,:{0,.[]K.G.WilsonandJ.Kogut.Therenormalizationgroupandthe-expansion.Phys.Reports,C:{00,.[]M.Zahradnk.AnalternateversionofPirogov-Sinaitheory.Commun.Math.Phys.,:{,.[]M.Zahradnk.LowtemperaturecontinuousspinGibbsstatesonalatticeandtheinterfacesbetweenthem|aPirogov-Sinaitypeapproach.InT.C.Dorlas,N.M.Hugenholtz,andM.Winnink,editors,StatisticalMechanicsandFieldThe-ory:MathematicalAspects(Proceedings,Groningen),Berlin{Heidelberg{NewYork,.Springer-Verlag(LectureNotesinPhysics#). []M.Zahradnk.Analyticityoflow-temperaturephasediagramsoflatticespinmodels.J.Stat.Phys.,:{,.

[]M.Zahradnk.Phasediagramsoflatticespinmodels.CoursdeTroisiemeCycledelaPhysiqueenSuisseRomande(Lausanne),May.[0]M.Zahradnk.Lowtemperaturephasediagramsoflatticemodelswithrandomimpurities.Preprint,. []M.Zahradnk.Privatecommunication.0.

출처: arXiv:9210.032 • 원문 보기