Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

MODIFIED LOG-SOBOLEV INEQUALITIES, CONCENTRA TION BOUNDS AND UNIQUENESS OF GIBBS MEASURES Y ANNIC STEENBECK Abstract. W e pro v e that there is only one translation-in v ariant Gibbsian point pro cess w.r.t. to a c hosen in teraction if an y of them satisfies a certain b ound re- lated to concen tration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In par- ticular, for natural examples with non-uniqueness regimes, a mo dified logarithmic Sob olev inequality cannot b e satisfied. Therefore, in these situations, the free- energy dissipation in related contin uous-time birth-and-death dynamics in R d is not exp onen tially fast. 1. Modified logarithmic Sobolev inequalities, rela ted concentra tion bounds and uniqueness of Gibbs measures via specific rela tive entropy dist ance There is a w ell-kno wn meta-corresp ondence b etw een certain functional inequali- ties, the so-called concentration-of-measure phenomenon and trend to equilibrium for related dynamics. Eac h of these sub jects is of ma jor interest on its own, but their connections can be particularly exciting. Perhaps the most prominen t example of functional inequalities in this con text are the lo garithmic Sob olev ine qualities dis- co vered b y Leonard Gross [ Gro75 ] in their simplest form reading as follows. Letting Φ : R + → R b e given via Φ( x ) = x log x , x ∈ R + , we define En t ν [ F ] = ν  Φ( F )  − Φ  ν [ F ]  , where ν is the standard Gaussian measure on R d and F : R d → R a smo oth and com- pactly supported function. F urthermore, the (formal) Diric hlet form corresponding to the Ornstein–Uhlenb e ck semigr oup , admitting the standard Gaussian measure ν as reversible measure, is giv en b y E ν ( F , G ) = ν  ∇ F · ∇ G  = ν  F ( − L G )  , with the (formal) generator ( L F )( x ) = ∆ F ( x ) − x · ∇ F ( x ) . Date : March 27, 2026. 2020 Mathematics Subje ct Classific ation. 82C21, 82B21, 60F10, 46E35; Secondary 60G55, 60E15, 60H07, 39B62, 60K35, 60J25. K ey wor ds and phr ases. Gibbs measures, modified logarithmic Sobolev inequalities, concentra- tion inequalities, p oint pro cesses, relativ e en tropy , entrop y dissipation, spatial birth and death pro cesses. 1 2 Y. STEENBECK Then, the following logarithmic Sob olev inequality En t ν [ F 2 ] ≤ C E ν ( F , F ) (1.1) holds with C = 1 . It turns out that this inequalit y generalizes to probabilit y measures on R d other than the standard Gaussian one. Ira Herbst observ ed that a necessary condition for (1.1) to hold for a probability measure ν on R d is that ν  e λ ( F − µ [ F ])  ≤ exp  C 4 λ 2 ∥ F ∥ 2 Lip  holds for every λ ∈ R and Lipschitz-con tinuous F . The connection with concen- tration inequalities via Chernoff b ounds, i.e. the exp onential Marko v inequality , is w ell-understo o d. The deep links betw een this family of functional inequalities and concen tration-of-measure b ounds for reversible measures as well as conv ergence of asso ciated dynamics to equilibrium are for example discussed in [ Led99 ]. Since then, m uch researc h has o ccurred in this direction and the general correspondence can be transferred to many differen t situations. W e will now recall a tiny bit of the corresp onding explorations in the con text of Poisson p oint pr o c esses . But first let us recall what a P oisson point pro cess π of in tensity one on R d is and embed it into a more general setup of notation. W e will use the shorthand Λ ⋐ R d to denote that Λ ⊆ R d is b ounded and Borel- measurable. Let Ω b e the set of all simple, lo cally finite coun ting measures (p oint configurations) on R d , equipp ed with the σ -algebra F which is induced by all the coun ting v ariables { N Λ : Λ ⋐ R d } , where N Λ : Ω → [0 , ∞ ) , ω 7→ #( ω ∩ Λ) . W e ma y iden tify η ∈ Ω with its supp ort and write η Λ = η ∩ Λ for the restriction of η to Λ ⊆ R d . Similarly , for η , ζ ∈ Ω the point configuration η Λ ζ ∆ ∈ Ω is then giv en by the union of all the p oin ts of η in Λ and all the p oin ts of ζ in ∆ . It is common to sa y that a measurable F : Ω → R is lo c al iff there is a Λ ⋐ R d suc h that F = F ( · Λ ) . W e will o ccasionally use the notations ν  F ( η )  = ν  F  for the exp ectation R F ( η ) ν (d η ) of a measurable F : Ω → R w.r.t. a probability measure ν on (Ω , F ) ; the integration v ariable is then alwa ys called η . No w let P θ = P θ (Ω , F ) b e the space of translation-in v ariant probability measures on (Ω , F ) , i.e. of those probabilit y measures on (Ω , F ) whic h are inv ariant under all translation maps ( θ x ) x ∈ R d , acting as θ x : Ω → Ω , ω = P i δ x i 7→ θ x ω := P i δ x i − x . No w, the Poisson p oint pr o c ess of intensity one on R d , here denoted by π , is the unique elemen t ν of P θ suc h that, under ν , N Λ is P oisson-distributed with mean | Λ | (where |·| means Leb esgue-measure on R d ) and N Λ 1 , . . . , N Λ k are independent for all k ≥ 2 and disjoint Λ , Λ 1 , . . . , Λ k ⋐ R d . No w, in [ W u00 ], the author pro ves sev eral functional inequalities for P oisson p oin t processes and deriv es concentration bounds for P oisson functionals from them. Among these functional inequalities, we wan t to recapitulate the following mo difie d lo garithmic Sob olev ine quality . Replacing the differen tial op erator ∇ in the Gaussian w orld, the discrete gradients D x , acting as ( D x F )( η ) = F ( η + δ x ) − F ( η ) , MODIFIED LOG-SOBOLEV, CONCENTRA TION, UNIQUE GIBBS MEASURES 3 no w come into pla y and the corresp onding Dirichlet form is given b y E π ( f , g ) = ν  Z R d ( D x f )( D x g ) d x  . Note that the definition of Ent µ [ F ] as En t µ [ F ] := µ  Φ( F )  − Φ  µ [ F ]  , still mak es sense for an y probabilit y measure µ ∈ P θ and measurable functions F : Ω → (0 , ∞ ) of sufficient integrabilit y . As already discussed in the introduction of [ W u00 ] and first noted in [ Sur84 ], the naïve translation En t π [ F 2 ] ≲ E π ( F , F ) of (1.1) to the P oisson point pro cess of intensit y one π fails due to the strictly-fatter- than-Gaussian tails of Poisson random v ariables. Ho wev er, the follo wing inequalit y do es hold, c.f. [ W u00 , Corollary 2.2], En t π [ F ] ≤ E π ( F , log F ) (1.2) for every F ∈ L 1 ( ν ) with F > 0 π -a.s.. In fact, the work [ W u00 ] even contains im- pro vemen ts ov er this inequality . Nevertheless, the inequalit y (1.2) o ccurs v ery nat- urally in the context of the Ornstein–Uhlenb e ck semigr oup ( T t ) t ≥ 0 (see e.g. [ Las14 ] for some results related to this semigroup) and that is wh y w e will fo cus on it. There, it gov erns the trend to equilibrium and exp onen tial deca y of relative en tropy as follows. Given the (formal) generator ( L F )( η ) = Z R d { F ( η + δ x ) − F ( η ) } d x + X x ∈ η { F ( η − δ x ) − F ( η ) } , of ( T t ) t ≥ 0 , describing dynamics where p oin ts are indep endently b orn into space and die with unit exponential rate, we ha ve indeed E π ( F , G ) = ν [ F ( − L G )] b y the Meck e equation. One quan tity which in a wa y measures distance to the P oisson point pro cess π , but also is of indep endent in terest in statistical mechanics, is the sp e cific r elative entr opy I ( µ | π ) of a measure µ ∈ P θ w.r.t. π . It is giv en b y I ( µ | π ) = inf Λ ⋐ R d 1 | Λ | I ( µ Λ | π Λ ) = lim n →∞ 1 | Λ n | I Λ n ( µ | π ) . where Λ n = [ − n, n ] d for n ∈ N . Herein and alw ays in this work, we denoted b y µ Λ , Λ ⋐ R d , the restriction of µ to the sub- σ -algebra F Λ generated by the lo cal coun ting v ariables { N ∆ : ∆ ⋐ R d , ∆ ⊆ Λ } , and I Λ ( µ | ν ) = I ( µ Λ | ν Λ ) with I ( P | Q ) =    Q  Φ( f )  , if f = d P d Q exists , ∞ , otherwise . A formal calculation no w shows that d d t I ( µT t | π ) = E π ( T t f , log T t f ) , (1.3) 4 Y. STEENBECK where µ ∈ P θ is another probabilit y measure and f = d µ d π . Hence, the mo dified loga- rithmic Sobolev inequality (1.2) is (morally) equiv alent to exponential decay of the (sp ecific) relative en tropy along tra jectories of the Ornstein–Uhlen b eck semigroup. A generalization of the ab o v e situation can b e made to the world of Gibbs me a- sur es , a v ast landscap e of mathematical statistical mechanics and probabilit y theory . T extb o ok references are e.g. [ Geo11 , Der19 ]. Recall that a (translation-inv arian t) Gibbs p oint pr o c ess , or sometimes just Gibbs me asur e , is an elemen t ν of P θ whic h satisfies the DLR e quations Z f ( ω ) ν (d ω ) = Z  Z f ( η ) G Λ ,ω (d η )  ν (d ω ) for ev ery measurable f ≥ 0 and ev ery Λ ⋐ R d . Herein, G Λ ,ω is the probabilit y measure defined by Z f ( η ) G Λ ,ω (d η ) = Z − 1 Λ ,ω Z f ( η Λ ω Λ c ) e − H Λ ( η Λ ω Λ c ) π (d η ) where π is the Poisson p oint pro cess of intensit y one, Z − 1 Λ ,ω is a normalization con- stan t and H Λ ( η ) := lim n →∞  H ( η Λ n ) − H ( η Λ n \ Λ )  is the conditional energy asso ciated with a (translation-in v ariant) ener gy function H . W e will not consider issues of well-definedness and existence of Gibbs measures in the present work. Let us just mention tw o examples of energy functions for which these prop erties are w ell-known. Example 1.1 (V ery nice pair p otentials) . Let ϕ : R d → R + b e a non-negative, compactly supp orted and ev en function. W e can then consider the energy function H ( ω ) = X { x,y }⊆ ω , x  = y ϕ ( x − y ) for totally finite point configurations ω . Example 1.2 (Area interac tion) . Let R > 0 and γ ∈ R \ { 0 } be some fixed num b er. W e can then consider the energy function H ( ω ) = γ | B R ( ω ) | for totally finite p oin t configurations ω and B R ( ω ) = S x ∈ ω B R ( x ) , B R ( x ) = { y ∈ R d : | x − y | ≤ R } . It is well-kno wn, compare with e.g. [ R ue71 , GLM95 , CCK95 ], that there are parameters γ > 0 , R > 0 such that more than one Gibbs measures exists for the corresponding area interaction H giv en ab o ve. Example 1.3 (Sup erstable pair in teractions) . W e use the same definitions as in [ Geo94 ]. Here, the energy function is again giv en as H ( ω ) = X { x,y }⊆ ω , x  = y ϕ ( x − y ) MODIFIED LOG-SOBOLEV, CONCENTRA TION, UNIQUE GIBBS MEASURES 5 for totally finite point configurations ω , but with a m uch more general class of ϕ : R d → R than in Example 1.1 . W e assume ϕ to b e sup erstable and r e gu- lar . This is e.g. fulfilled if ϕ is non-inte gr ably diver gent at the origin and r e gular , meaning resp ectively that there is some decreasing function χ : (0 , ∞ ) → R + with R 1 0 χ ( r ) r d − 1 d r = ∞ suc h that ϕ ( x ) ≥ χ ( | x | ) whenever | x | is small enough and that there is some decreasing function ψ : R + → R + with R ∞ 0 ψ ( r ) r d − 1 d r < ∞ such that (1) ϕ ( x ) ≥ − ψ ( | x | ) for all x ∈ R d , (2) ϕ ( x ) ≤ ψ ( | x | ) for some r ( ϕ ) < ∞ and all x ∈ R d with | x | ≥ r ( ϕ ) . F or well-definedness of the conditional energies and hence finite-v olume Gibbs mea- sures, we consider the class Ω ∗ ⊂ Ω of temp er e d c onfigur ations Ω ∗ := [ t> 0  X i ∈ Λ n N 2 i +[ − 1 , 1] d ≤ t | Λ n | for all n ∈ N  . W e will call a probability measure ν ∈ P θ a temp er e d Gibbs me asur e if it is a temp er e d pr ob ability me asur e , i.e. if ν (Ω ∗ ) = 1 , and a Gibbs measure using the definition via DLR equations given ab o ve. No w, if we denote b y h ( x, η ) the c onditional ener gy h ( x, η ) := lim n ↑∞  H ( η Λ n + δ x ) − H ( η Λ n )  of a p oint x ∈ R d in a configuration η ∈ Ω , the birth r ate b is given b y b ( x, η ) := e − h ( x,η ) . In p oin t pro cess language, what we call a birth rate here is a Pap angelou intensity . It gives rise to the following formal generator ( L F )( η ) = Z R d b ( x, η ) { F ( η + δ x ) − F ( η ) } d x + X x ∈ η { F ( η − δ x ) − F ( η ) } , whic h, at least formally , generates a semigroup admitting the corresp onding Gibbs measures as reversible measures. This rev ersibility is no w a consequence of the GNZ e quations in place of the Meck e equation. Rigorous results regarding such contin uum birth-and-de ath dynamics in that direction are contained e.g. in the articles [ JKSZ25 , JKSZ26 ]. No w, we see as in (1.3) that a relev ant quan tity regarding the decay of relativ e en tropy w.r.t. ν , along the trajectories of the dynamics asso ciated with L , is giv en b y the formal Dirichlet form expression E ν ( F , log F ) = ν  F ( − L log F )  = ν  Z R d b ( x, · ) ( D x F )( D x log F ) d x  . (1.4) Note that in Example 1.1 and Example 1.2 , the birth rate b is b ounded. Hence, at least in these settings, the following inequality will b e the relev ant one. W e will 6 Y. STEENBECK sa y that ν satisfies the mo dified logarithmic Sob olev inequalit y ( MLSI–1 ) with constan t c ν > 0 , if En t ν [ F ] ≤ c ν ν  Z R d ( D x F )( D x log F ) d x  (1.5) for all b ounded lo cal F > 0 . Finite-v olume v ersions of this inequalit y are, among other things, discussed in the article [ DPP13 ]. As an example, it is sho wn there that Gibbs measures for Example 1.1 ob ey suc h a b ound for sufficiently low in teraction strength (e.g. in high-temp erature regimes). In [ JKSZ26 ], these tec hniques are applied to sa y the same for Example 1.2 and sufficiently high temp eratures. In the case that the birth rate b is unbounded, ( MLSI–1 ) do es not follow from En t ν [ F ] ≤ c ν E ν ( F , log F ) = c ν ν  Z R d b ( x, · ) ( D x F ) ( D x log F ) d x  (1.6) for all bounded lo cal F > 0 . In case that (1.6) holds for all b ounded lo cal F > 0 , we will sa y that ν satisfies the the mo dified logarithmic Sob olev inequalit y ( MLSI–b ) with constant c ν > 0 . A t high temp eratures, there will be, as is w ell-kno wn for these examples, just one Gibbs measure. On the other hand, for Example 1.2 also a phase transition, i.e. non-uniqueness of Gibbs measures at some parameters, is well-kno wn. No w, a natural question is: Do es the fact that some Gibbs me asur e ν satisfies a ( MLSI–1 ) or a ( MLSI–b ) with c onstant c ν > 0 alr e ady imply that no other Gibbs me asur e w.r.t. the same chosen inter action (and p ar ameters) exists? In [ CMR U20 , CR22 ] the authors presen ted the follo wing idea in the con text of statistical mec hanics of lattice systems, whic h w e shall adapt to the presen t sub ject of inv estigation. A ccording to the Gibbs variational principle , the sp ecific relativ e entrop y of a Gibbs measure µ w.r.t. another Gibbs measure ν for the same sp ecification is zero: I ( µ | ν ) = lim n →∞ 1 | Λ n | I ( µ Λ n | ν Λ n ) = 0 and I ( e µ | ν ) = 0 for another probability measure e µ already implies that e µ is in fact a Gibbs measure to o. Hence, if ν is any Gibbs measure, an equiv alent condition to its uniqueness is I ( µ | ν ) > 0 for every µ ∈ P θ \ { ν } . Emplo ying the Donsker–V aradhan form ula for the relative en tropy , w e see that 1 | Λ n | I ( µ Λ n | ν Λ n ) ≥ 1 | Λ n | n µ  F n  − ν  F n   − log ν  e F n − ν [ F n ]  o for every measurable F n of our choice. Insp ecting the righ t-hand side of this in- equalit y , w e can already see that it will be uniformly b ounded from below if w e are able to construct ( F n ) n ∈ N with roughly the follo wing prop erties (1) 1 | Λ n | µ [ F n ] − 1 | Λ n | ν [ F n ] ≫ 0 , MODIFIED LOG-SOBOLEV, CONCENTRA TION, UNIQUE GIBBS MEASURES 7 (2) F n concen trates sufficiently well, on an exp onential scale, around its mean ν [ F n ] under ν . The first prop erty is easily achiev ed b y c ho osing a bounded lo cal observ able f whic h separates the distributions µ, ν as in µ [ f ] − ν [ f ] > 0 and then, exploiting stationarit y , lo oking at its space-av erages F n = R Λ n f ◦ θ x d x . F or the second prop erty , we will ha ve to use the corresp onding concentration assumptions on ν and take more care with the initial c hoice of f to apply them. The contribution of this small w ork is the insight that ideas from [ CMR U20 , CR22 ] carry o ver from the lattice situation to the con tin uum p oint process setting, also replacing the Gaussian concentration assumption with a P oisson-type one, and the technical w ork needed to enable this. Indeed, we presen t the following Theorem 1.4 (MLSI–1 implies strictly p ositive sp ecific relativ e en tropy distance) . L et ν ∈ P θ satisfy ( MLSI–1 ) with some c onstant c ν > 0 . Then, I ( µ | ν ) > 0 for al l µ ∈ P θ \ { ν } . This theorem now yields the following easy corollaries. Corollary 1.5. L et the Gibbs variational principle hold in the fol lowing sense: if ν ∈ P θ is a Gibbs me asur e w.r.t. the fixe d ener gy function H and µ ∈ P θ is any pr ob ability me asur e with I ( µ | ν ) = 0 , then µ is Gibbs me asur e w.r.t. H to o. Then, if any Gibbs me asur e ν w.r.t. H satisfies ( MLSI–1 ) with some c onstant c ν > 0 , ther e is no other Gibbs me asur e w.r.t. H . Corollary 1.6. Ther e ar e p ar ameters γ ∈ R \ { 0 } and R > 0 such that none of the Gibbs me asur es ν (which ar e mor e than one) w.r.t. the inter action in Example 1.2 c an satisfy ( MLSI–1 ) with some c onstant c ν > 0 . In the case of un b ounded birth rates, the following theorem b ecomes relev ant. Theorem 1.7 (MLSI–b implies strictly p ositiv e sp ecific relativ e entrop y distance) . L et ν b e a (temp er e d, tr anslation-invariant) Gibbs me asur e in the setting of Exam- ple 1.3 and assume that ν ∈ P θ satisfies ( MLSI–b ) with some c onstant c ν > 0 . Then, I ( µ | ν ) > 0 for al l µ ∈ P θ \ { ν } . In p articular, ther e is no other (temp er e d, tr anslation-invariant) Gibbs me asur e. Remark 1.8. The the or em is formulate d for the c ase of sup erstable p air inter actions as define d in Example 1.3 , but the pr o of do es in fact not dir e ctly r ely on this sp e cific form of the ener gy function. A n additional input ne e de d for this pr o of is just an upp er b ound of the form ν  ( N Λ n / | Λ n | ) 1 N Λ n ≥ t | Λ n |  ≲ e − r ( t ) | Λ n | wher e sup t> 0 r ( t ) > 0 . It is her e, in the setting of sup erstable p air inter actions, pr ovide d by the upp er b ounds of an LDP for stationary empiric al fields c onne cte d to the Gibbs variational principle, which is c ontaine d in [ Geo94 ] . 8 Y. STEENBECK No w turning to pro ofs, our Theorem 1.4 is a direct consequence of the following t wo prop ositions. Let us say that ν satisfies ( MGF–CI ) with constant c ν > 0 if ν  e λ ( F − ν [ F ])  ≤ exp n c ν α 2 λ e β λ − 1 β o . for every F with D x F ≤ β ( ν ⊗ d x ) -a.s. and R R d | D x F | 2 d x ≤ α 2 ν -a.s., for α 2 , β > 0 . The first prop osition is simply a restatemen t of Theorem 1.4 with replaced as- sumptions: Prop osition 1.9 (MGF b ounds imply distance in sp ecific relative en tropy) . L et ν ∈ P θ satisfy ( MGF–CI ) with some c onstant c ν > 0 . Then, I ( µ | ν ) > 0 for al l µ ∈ P θ \ { ν } . The second prop osition shows that an MLSI for ν implies MGF b ounds under ν for certain observ ables. Hence, Prop osition 1.9 is applicable with the assumptions of Theorem 1.4 . The deriv ed bounds are of course equiv alent to corresp onding concen tration-of-measure inequalities. Prop osition 1.10 (MGF b ounds from MLSI–1) . A ssume that ν ∈ P θ satisfies ( MLSI–1 ) with some c onstant c ν > 0 . Then, it also satisfies ( MGF–CI ) with c onstant c ν > 0 . The pro ofs of Prop osition 1.9 and Prop osition 1.10 are given in the next Section 2 , sp ecifically in Section 2.1 . Subsequen tly and building on the preceding arguments, Theorem 1.7 is pro ven in Section 2.2 . As a concluding remark, w e w ant to say that this result further solidifies the need for different and new techniques to understand quantitatively trend to equilibrium of sto c hastic dynamics suc h as the considered birth-and-death dynamics and esp ecially deca y of sp ecific relative en tropy in regimes of non-uniqueness of (the rev ersible) Gibbs measures. Some of the qualitative basic building blo cks in regimes of non- uniqueness were already laid in [ JKSZ25 , JKSZ26 ]. 2. Proofs The pro ofs of Prop osition 1.9 , Prop osition 1.10 and hence Theorem 1.4 are con- tained in Section 2.1 . Enhancing the approac h there with some new ideas to deal with the unbounded birth rate b , Section 2.2 contains the proof of Theorem 1.7 . 2.1. Diric hlet form with birth rate 1 . In our first proof, we detail ho w w e adapt ideas of [ CMRU20 , CR22 ] to our setting. Heuristics regarding this were already giv en in Section 1 . Pr o of of Pr op osition 1.9 . Consider µ ∈ P θ \ { ν } . Then, w e can separate µ and ν as follo ws. There is an r > 0 and g ≥ 0 , g ∈ C c ( R d ) with supp g ⊆ Λ r suc h that ϱ := µ  f  − ν  f  > 0 MODIFIED LOG-SOBOLEV, CONCENTRA TION, UNIQUE GIBBS MEASURES 9 for f ( η ) = ± e − P x ∈ η g ( x ) , where the sign is c hosen to make ϱ p ositiv e. No w set F n := Z Λ n ( f ◦ θ x ) d x. (2.1) Then, ( D z f )( η ) = ( e − g ( z ) − 1) f ( η ) and hence | ( D z F n )( η ) | =     Z Λ n ( D z − x f )( θ x η ) d x     ≤ Z Λ n    1 − e − g ( z − x )    | f ( θ x η ) | d x ≤ Z R d    1 − e − g ( x )    d x =: β . With ψ ( x ) := sup ω | ( D x f )( ω ) | , w e also get the following b ound which we prov e in Lemma 2.1 b elow: Z R d | D z F n | 2 d z ≤ ∥ ψ ∥ 2 L 1 ( R d ) | Λ n | ≤ β 2 | Λ n | =: α 2 n Then, b y the Donsk er–V aradhan formula for the relativ e en tropy in combination with F Λ n + r -measurabilit y of F , and by translation-in v ariance 1 | Λ n | I Λ n + r ( µ | ν ) ≥ 1 | Λ n | n µ  F n  − ν  F n   − log ν  e F n − ν [ F n ]  o =  µ  f  − ν  f   − 1 | Λ n | log ν  e F n − ν [ F n ]  . No w replacing f by λf for λ > 0 yields 1 | Λ n | I Λ n + r ( µ | ν ) ≥ sup λ> 0 n ϱλ − 1 | Λ n | log ν  e λ ( F n − ν [ F n ])  o . F rom the assumption ( MGF–CI ) on ν , w e get ν  e λ ( F n − ν [ F n ])  ≤ exp n c ν α 2 n λ e β λ − 1 β o = exp n c ν β 2 | Λ n | λ e β λ − 1 β o . Finally , 1 | Λ n | I Λ n + r ( µ | ν ) ≥ sup λ> 0 n ϱλ − ( c ν β 2 e β λ − 1 β λ ) λ 2 o > 0 , indep enden tly of n ∈ N , giving I ( µ | ν ) = inf n ∈ N 1 | Λ n + r | I Λ n + r ( µ | ν ) = inf n ∈ N 1 | Λ n | I Λ n + r ( µ | ν ) > 0 . □ The following technical lemma w as needed in the ab ov e pro of. The statement and pro of are v ery close to [ CMR U20 , Lemma A.1] in the lattice setting, but we still incorp orate it for conv enience of the reader. 10 Y. STEENBECK Lemma 2.1. L et f b e a b ounde d lo c al function and F := Z Λ ( f ◦ θ x ) d x. Then Z R d | D z F | 2 d z ≤ ∥ ψ ∥ 2 L 1 ( R d ) | Λ | with ψ ( x ) := sup ω | ( D x f )( ω ) | . Pr o of. W e hav e | ( D z F )( η ) | =     Z Λ f ( θ x η + δ z − x ) d x − Z Λ f ( θ x η ) d x     ≤ Z ψ ( z − x ) 1 Λ ( x ) d x = ( ψ ∗ 1 Λ )( z ) Y oung’s conv olution inequality no w implies Z R d | D z F | 2 d z ≤ ∥ ψ ∗ 1 Λ ∥ 2 L 2 ( R d ) ≤ ∥ ψ ∥ 2 L 1 ( R d ) ∥ 1 Λ ∥ 2 L 2 ( R d ) = ∥ ψ ∥ 2 L 1 ( R d ) | Λ | . □ Finally , we sho w ho w to con vert MLSI into corresponding MGF / concentration b ounds. Pr o of of Pr op osition 1.10 . W e follo w the so-called Herbst’s argument, as seen in the pro of of e.g. [ W u00 , Prop osition 3.1]. Set H ( λ ) := 1 λ log ν  e λF  and notice that H (0 + ) = ν [ F ] as well as H ′ ( λ ) = En t ν [ e λF ] λ 2 ν [e λF ] . No w, En t ν [ e λF ] ≤ c ν ν  Z R d ( D x e λF ) ( D x λF ) d x  ≤ c ν ν  Z R d ( D x λF ) 2 e λF e λ | D x F | d x  ≤ c ν α 2 λ 2 e β λ ν  e λF  , where we used the mean-v alue theorem for the second inequalit y . It follows that H ′ ( λ ) ≤ c ν α 2 e β λ and hence ν  e λF  = exp  λH ( λ )  = exp n λH (0) + λ Z λ 0 H ′ ( e λ )d e λ o ≤ exp n λ ν [ F ] + c ν α 2 λ e β λ − 1 β o . (2.2) □ MODIFIED LOG-SOBOLEV, CONCENTRA TION, UNIQUE GIBBS MEASURES 11 2.2. Diric hlet form with birth rate b . W e will finally analyze the case of ( MLSI– b ), where the generally unbounded birth rate / Papangelou intensit y b app ears in the corresp onding Diric hlet form. It is recommended to first read the pro ofs in Section 2.1 to facilitate understanding. Pr o of of The or em 1.4 . W e follo w the reasoning of the pro of of Prop osition 1.9 and adapt the technique of the pro of of Prop osition 1.10 to the sp ecific F n as in (2.1) . Indeed, we see as in (2.2) that ν  e λ ( F n − ν [ F n ])  ≤ exp  c ν λ e β λ Z λ 0 K n ( e λ ) d e λ  with K n ( λ ) := 1 ν  e λF n  ν  e λF n  Z R d b ( z , · ) | D z F n | 2 d z  . Our task is now to giv e b ounds for K n ( λ ) / | Λ n | which are somewhat unif orm in n and v alid for small λ . Define ϕ ( x ) = 1 − e − g ( x ) and observe that ν  e λF n  Z R d b ( z , · ) | D z F n | 2 d z  ≤ ν  e λF n  Z R d b ( z , · ) | ϕ ∗ 1 Λ n ( z ) | 2 d z  = ν  X z ∈ η e λF n ( η − δ z ) | ϕ ∗ 1 Λ n ( z ) | 2  = ν  e λF n ( η ) X z ∈ η e λ ( D z F n )( η − δ z ) | ϕ ∗ 1 Λ n ( z ) | 2  ≤ e β λ ν  e λF n ( η ) X z ∈ η | ϕ ∗ 1 Λ n ( z ) | 2  ≤ β e β λ ν  e λF n N Λ n +supp ϕ  , where the first iden tity is b y the GNZ equations. No w define the stationary empiric al fields R n,ω ∈ P θ b y R n,ω := 1 | Λ n | Z Λ n δ θ x ω ( n ) d x, where ω ( n ) := X x ∈ ω Λ n , i ∈ Z d δ x +2 ni is the Λ n -p erio dization of ω Λ n . No w let k b e the smallest j ∈ N suc h that Λ n + supp g ⊆ Λ n + j and let us use the follo wing crude b ound v alid for an y T > 0 and λ ∈ (0 , 1) : 1 ν  e λF n  ν  e λF n N Λ n +supp ϕ  ≤ 1 ν  e λF n  ν  e λF n N Λ n + k  = | Λ n + k | 1 ν  e λF n  ν  e λF n R n + k, · [ N C ]  ≤ | Λ n + k |  T + e 2 λ ∥ f ∥ ∞ | Λ n + k | ν h | Λ n + k | R n + k, · [ N C ] 1 R n + k, · [ N C ] ≥ T i  12 Y. STEENBECK ≤ | Λ n + k |  T + e − ( T − 2 ∥ f ∥ ∞ ) | Λ n + k | ν h e 2 | Λ n + k | R n + k, · [ N C ] i  with C = [0 , 1] d . W e will no w apply the upp er b ound of the LDP [ Geo94 , Theorem 3 (b)] for R n, · under ν with rate function I 1 , 1 , for all P ∈ P θ giv en b y I 1 , 1 ( P ) = I ( P | π ) + H ( P ) − min Q ∈P θ [ I ( Q | π ) + H ( Q )] ≥ 0 where H ( P ) := lim n →∞ 1 | Λ n | P [ H Λ n ] . W e get that lim sup n →∞ 1 | Λ n + k | log ν h e 2 | Λ n + k | R n + k, · [ N C ] i ≤ − inf P ∈P θ J ( P ) (2.3) with J ( P ) := I 1 , 1 ( P ) − 2 P [ N C ] = I ( P | π ) + H ( P ) − 2 P [ N C ] − c, where c ∈ R is a finite constant. W e now only hav e to argue that the infimum on the right-hand side of (2.3) is not −∞ . T o see that: stability of H yields H ( P ) ≥ − bP [ N C ] for some fixed b > 0 . But, for any a > 0 , aP [ N C ] = 1 | Λ n | P [ aN Λ n ] ≤ 1 | Λ n | I ( P Λ n | π Λ n ) + 1 | Λ n | log π [e aN Λ n ] and hence I ( P | π ) ≥ aP [ N C ] − (e a − 1) . It follows, with e.g. a = b + 2 , that J ( P ) ≥ ( a − b − 2) P [ N C ] − (e a − 1) − c ≥ − (e b +2 − 1) − c. for all P ∈ P θ . 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