Regularity of Gibbs measures for unbounded spin systems on general graphs

Regularit y of Gibbs measures for un b ounded spin systems on general graphs Christoforos P anagiotis 1 and William V eitc h 2 1,2 Departmen t of Mathematical Sciences, Universit y of Bath, UK Abstract W e consider a general class of spin systems with p otentially un b ounded real-v alued spins, deﬁned via a single-site potential with sup er-Gaussian tails on general graphs, allo wing for b oth short- and long-range in teractions. This class includes all P ( φ ) mo dels, in particular the w ell-studied φ 4 mo del. W e construct an inﬁnite-volume extremal measure called the plus measure as the limit of ﬁnite-volume Gibbs measures with weakly gro wing b oundary conditions and sho w that it is regular, in the sense that it admits a bounded Radon-Nik o dym deriv ativ e with resp ect to a product measure of single-site distributions with sup er-Gaussian tails. Moreo ver, we pro vide an alternativ e construction of the plus measure as the limit of ﬁnite-volume Gibbs measures that are regular up to the b oundary . As a key intermediate step, we establish regularity and tigh tness of ﬁnite-volume Gibbs measures for a large class of gro wing b oundary conditions ξ . Our regularit y estimates are enco ded in terms of a function A ( ξ ) , which provides precise control on the change of measure induced by b oundary p erturbations, and can th us be viewed as an analogue of the Cameron–Martin theorem for non-Gaussian ﬁelds. In the nearest- neigh b our case, this class includes boundary conditions that grow at most double- exp onen tially in the distance to the b oundary when the single-site measure has tails of the form e − a | u | n for some n > 2 . In contrast, when the single-site measure has Gaussian tails, the allo wed gro wth is at most exp onen tial. Our results apply to ar- bitrary graphs and improv e up on earlier results of Lebowitz and Presutti [10], and R uelle [12, 13], whic h apply in the context of Z d and allo w only logarithmically grow- ing b oundary conditions, as w ell as subsequen t extensions to vertex-transitiv e graphs of p olynomial gro wth [7]. 1 In tro duction W e consider a general family of spin systems with pair in teractions where each spin tak es v alues in R . On a ﬁnite graph (Λ , E ) with ferromagnetic nearest-neigh b our in- teractions and free boundary conditions, this consists of probabilit y measures on spin conﬁgurations φ ∈ R Λ deﬁned so that the expectation of a b ounded measurable function f : R Λ → R is given by ⟨ f ⟩ = 1 Z Z R Λ f ( φ ) exp   X { x,y }∈ E β φ x φ y   Y x ∈ Λ d ρ ( φ x ) , where β ≥ 0 is the inverse temp er atur e , ρ is th e single-site measure (c hosen so that the ab o v e integral is ﬁnite), and Z is the appropriate normalisation constant, called the p arti- tion function . This framew ork includes sev eral w ell-studied mo dels in statistical ph ysics: 1 ( i ) The Ising and φ 4 mo dels, by c ho osing, resp ectiv ely ρ = δ − 1 + δ 1 , d ρ ( φ x ) = exp( − g φ 4 x − aφ 2 x )d φ x , where for t ∈ R , δ t is the Dirac measure at t , and where g > 0 and a ∈ R . ( ii ) The Gaussian free ﬁeld, b y c ho osing d ρ ( φ x ) = exp( − aφ 2 x )d φ x for a large enough constan t a that dep ends on β . ( iii ) General P ( φ ) models, b y choosing d ρ ( φ x ) = exp( − P ( φ x ))d φ x , (1.1) where P is an even p olynomial of degree at least 4 and of p ositiv e leading co eﬃcien t. F or a comprehensive introduction to these mo dels, the interested reader can consult [4 – 6, 14]. The deﬁnition of the mo del can b e extended to include boundary conditions and long- range interactions; see Section 2.1 for the more general deﬁnition. W e will write ν ξ Λ ,β ,ρ,J for the ﬁnite-volume measure on Λ with in verse temp erature β , single-site measure ρ , in teractions J and b oundary conditions ξ . Inﬁnite-v olume Gibbs measures, deﬁned via the Dobrushin–L anfor d–Ruel le (DLR) e quation (see Deﬁnition 1.4 b elo w), are objects of central interest in statistical ph ysics, arising naturally as limits of ﬁnite-volume measures as (Λ , E ) tends to an inﬁnite graph suc h as the lattice Z d . Since spins are, in general, un b ounded in our setting, this raises the following question: for which b oundary conditions is the sequence of measures tigh t? In the case of the massless Gaussian free ﬁeld, the Cameron–Martin formula implies that the conﬁguration on Λ under b oundary conditions ξ has the same distribution as the con- ﬁguration with free b oundary conditions on Λ shifted b y the harmonic extension of ξ . In particular, if the b oundary spins gro w to inﬁnit y as | Λ | → ∞ , then the sequence is not tigh t. In contrast, for the massive Gaussian free ﬁeld, the faster decay of the density of the single-site measure leads to exp onen tial deca y of correlations, whic h in turn allo ws for tightness of ﬁnite-v olume measures as long as the boundary conditions grow weakly enough. Answ ering this question is more c hallenging in the non-Gaussian case, for example, when the single-site measure ρ is such that ∀ a > 0 0 < Z R e a | u | 2 d ρ ( u ) < ∞ . (1.2) The problem w as studied on the lattice Z d b y Leb o witz and Presutti [10] who pro v ed tigh tness for b oundary conditions that gro w like p log( ∥ x ∥ ∞ ) . Their approac h utilises a regularit y estimate dev elop ed b y Ruelle [12, 13], whic h b ounds the densit y at a spin conﬁguration φ in terms of the density at φ of a non-interacting system (i.e. a system with β = 0 ). See also [1 – 3] and references therein for related results on inﬁnite-v olume Gibbs measures supp orted on conﬁgurations of temp ered growth. The metho ds of [10] were applied in [7] to the φ 4 mo del on vertex-transitiv e graphs of p olynomial gro wth. The main result of [7] in this con text is that any translation in v arian t Gibbs measure is a conv ex combination of tw o extremal measures, and regularity is needed to construct these extremal measures as limits of ﬁnite-volume measures. It w as observ ed 2 in [7] that this result should extend to the setting of vertex-transitiv e amenable graphs, as previously established for the Ising mo del [11], but it is not clear how to generalise the regularit y estimates of [10] to this case. This pro vides the motiv ation for us to dev elop alternativ e arguments for proving regularity . The main result of the present article is a regularity estimate that applies to b oth nearest-neigh b our and long-range interactions on an arbitrary graph. The theorem b ounds the Radon–Nikodym deriv ative of the system in a ﬁnite domain Λ at parameter β > 0 , with b oundary conditions ξ that are allow ed to grow to inﬁnity as | Λ | → ∞ , with resp ect to a pro duct measure associated with a non-interacting system with a modiﬁed single- site measure. The b ound is expressed in terms of a function A ( x, Λ , ξ , C ) , where x ∈ Λ is a vertex and C ≥ 1 is a parameter. This fun ction is related to the mean of each spin φ x and compared to the massless Gaussian free ﬁeld, it can b e in terpreted as an analogue of the harmonic extension of ξ . More generally , it plays the role of a non-Gaussian analogue of the Cameron–Martin form ula, quan tifying the c hange of measure induced by the b oundary conditions. The function A ( x, Λ , ξ , C ) may tak e large v alues when x is close to the b oundary , but it decreases as x mo ves further in to the bulk, and provided the b oundary conditions ξ do not grow too rapidly , A ( x, Λ , ξ , C ) remains b ounded in the bulk of Λ . The formal deﬁnition can be found in Section 2.2. F or now, let us mention for concreteness that in the case of nearest-neighbour in teractions, A ( x, Λ , ξ , C ) ≈ max    1 , max z ∈ ∂ Λ  | ξ z | C  ( n − 1) − d ( x,z )    . W e observe a qualitative c hange in the b eha viour of A ( x, Λ , ξ , C ) dep ending on the tails of the single-site measure ρ . In order to mak e this p oin t clearer, let us state the result in the case where ρ satisﬁes the stronger assumption 0 < Z R e a | u | n d ρ ( u ) < ∞ , (1.3) for some a > 0 and n > 2 , a condition satisﬁed by all the P ( φ ) mo dels. Here we let V b e a coun table set, and we consider interactions ( J x,y ) x,y ∈ V on V . W e call the interactions admissible if they satisfy: (C1) (Symmetry) J x,y = J y ,x for all x, y ∈ V ; (C2) (In tegrabilit y) There exists f : R → [1 , ∞ ) and constants δ f , M f > 0 suc h that f ( t ) ≥ log ( | t | − 1 ) 1 /n for all t ∈ ( − δ f , δ f ) , and P y ∈ V | J x,y | f ( J x,y ) ≤ M f for all x ∈ V . W e now state our regularit y result, which do es not require the in teractions J to be ferro- magnetic. Below φ | Λ ′ denotes the restriction of the ﬁeld to Λ ′ , and d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ = ψ ] the densit y of the restriction. Theorem 1.1. L et V b e a c ountable set and ( J x,y ) x,y ∈ V b e admissible inter actions on V . L et β ≥ 0 , a > 0 , n > 2 and let ρ b e a single-site me asur e satisfying (1.3) . Ther e exist c onstants C ≥ 1 , ˜ C > 0 such that for any Λ ⊂ V ﬁnite, Λ ′ ⊂ Λ , ψ ∈ R Λ ′ , and any b oundary c onditions ξ ∈ R V with P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ , d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ = ψ ] ≤   Y x ∈ Λ ′ e ˜ C A ( x, Λ ,ξ ,C ) n   d ν 0 Λ ′ , 0 ,ρ a 2 , 0 [ ψ ] , wher e ρ a 2 is deﬁne d by d ρ a 2 ( u ) = e a 2 | u | n d ρ ( u ) . Mor e over, one c an take C = C 1 β 1 n − 2 + C 2 and ˜ C = ˜ C 1 β n n − 2 + ˜ C 2 , wher e C 1 , C 2 , ˜ C 1 , ˜ C 2 dep end only on δ f , M f and ρ . 3 Theorem 1.1 can b e generalised to allo w for single-site measures that dep end on the v ertex — see Remark 4.2 — and to more general Hamiltonians or ev en conditional mea- sures — see Theorem 4.3 and Remark 4.4. W e also exp ect our arguments to b e robust enough to apply to models with k -bo dy in teractions, pro vided (1.3) is satisﬁed with n > k , but we do not pursue this here. F urthermore, Theorem 1.1 applies to arbitrary graphs, th us extending the regularity results of [10] and [7]. In particular, it op ens the w ay for c haracterising the translation in v ariant Gibbs measures for the ferromagnetic φ 4 mo del on an y v ertex-transitiv e amenable graph, but verifying this is b ey ond the scop e of the current pap er. T o illustrate the p o wer of Theorem 1.1, let us ﬁrst state a simple but useful application in practice. Be low ˜ ρ denotes a v ertex-dep enden t single-site measure deﬁned at vertex x ∈ Λ ′ b y d ˜ ρ x ( u ) = 1 { u ≥ B x } d ρ a 2 ( u − B x ) , where B x is a constan t which is roughly equal to A ( x, Λ , ξ , C ) . A more precise version of this result can b e found in Corollary 5.10. Corollary 1.2. L et Λ ⊂ V b e ﬁnite and let ξ b e any b oundary c onditions such that P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ . Then ν ξ Λ ,β ,ρ is sto chastic al ly dominate d by ν 0 Λ , 0 , ˜ ρ ; henc e ν ξ Λ ,β ,ρ is sto chastic al ly dominate d by ν 0 Λ ,β , ˜ ρ . Remark 1.3. In c ontr ast to the Camer on–Martin formula for Gaussian ﬁelds, our The o- r em 1.1 and Cor ol lary 1.2 yield only an ine quality r ather than an exact e quality. If ξ ≥ 0 , one c an obtain a r everse ine quality by noting that ν ξ Λ ,β ,ρ sto chastic al ly dominates ν 0 Λ ,β ,ρ , as long as the inter actions ar e ferr omagnetic and ρ is an even me asur e. W e now outline the pro of of Theorem 1.1, which we b elieve oﬀers a clearer probabilistic in tuition than earlier approaches to regularit y . The proof is based on an exploration argumen t in which w e consider the cluster C of Λ ′ consisting of vertices x for which the spin φ x tak es a large v alue. T o accommo date p oten tially large b oundary conditions, the threshold for including a vertex x in the cluster is allow ed to gradually gro w as x mo ves further aw ay from Λ ′ , so that it matc hes the b oundary condition ξ z at a v ertex z on the b oundary of Λ . This is where the function A ( x, Λ , ξ , C ) comes into play; its precise deﬁnition enables ﬁne control of the b ehaviour of C . In particular, A ( x, Λ , ξ , C ) is deﬁned so that certain technical conditions relating parents and c hildren in the exploration pro cess are satisﬁed (see Lemmas 3.1 and 3.2). These conditions allo w us to isolate each v ertex of C from its neigh b ours at a ﬁnite cost, thereby yielding a non-interacting system. Finally , w e control the size of C b y comparing it to the total progeny of a sub critical branc hing pro cess; to carry this out, the parameter C in the deﬁnition of A ( x, Λ , ξ , C ) m ust b e chosen appropriately . In contrast to [10] and [7], our arguments do not require any assumption of p olynomial growth or amenability of the underlying graph. As an immediate consequence of Theorem 1.1, we obtain tigh tness for any boundary conditions such that A ( x, Λ , ξ , C ) remains b ounded for each x ∈ V as Λ ↗ V . In the case of ferromagnetic nearest-neighbour in teractions, this includes b oundary conditions gro wing lik e a double exponential of the form K ( n − 1) d ( o,x ) , where o is a ﬁxed origin. This impro ves on the results of Leb o witz and Presutti [10], whic h allo w only logarithmically gro wing b oundary conditions. In Prop osition 5.2, we show that this result is optimal in the case of non-negative b oundary conditions for ρ deﬁned as in (1.1), in the sense that tigh tness do es not o ccur for non-negative b oundary conditions that grow ev en faster than a constant to the p ow er ( n − 1) d ( o,x ) . In the case of long-range interactions, the rate of deca y of the interactions J comes into play , as any v ertex x can aﬀect the v alue of φ o through the edge ox . This makes characterising the b oundary conditions that lead to tigh tness challenging. Nevertheless, in Prop osition 5.6, w e pro ve that if the in teractions J are well b eha ved, in the sense that | J x,y | ≤ d ( x, y ) − r for some r > 0 , then we hav e 4 tigh tness for b oundary conditions growing lik e f ( d ( o, x ) − r ) , where f is a function that enco des the rate of decay of J , i.e. it satisﬁes (C2) and some additional assumptions. Coming back to the dependence of the behaviour of A ( x, Λ , ξ , C ) on the tails of the single-site measure ρ , let us men tion that a similar statement (see Theorem 4.1) to that of Theorem 1.1 holds if w e relax the assumptions on the single-site measure ρ to allo w for an y ρ that satisﬁes (1.2). In th is case, w e observ e a qualitative change in the b eha viour of A ( x, Λ , ξ , C ) . F or example, in the nearest-neighbour case, we obtain tigh tness for any b oundary conditions growing at most exp onen tially in the distance, so we observ e a jump in the threshold for tigh tness from exp onen tial to double-exp onential at n = 2 . Theorems 1.1 and 4.1 can b e used to obtain regularit y for inﬁnite-volume measures, whic h we deﬁne b elo w. Deﬁnition 1.4. L et a > 0 , β ≥ 0 , and assume J satisﬁes (C1) , (C2) and ρ satisﬁes (1.2) . W e say that a pr ob ability me asur e ν on R V with the σ − algebr a gener ate d by Bor el events dep ending on ﬁnitely many vertic es is • a -r e gular if ther e exists a c onstant B ∈ [0 , ∞ ) such that for every Λ ⊂ V ﬁnite and ψ ∈ R Λ , d ν [ φ | Λ = ψ ] ≤ e B | Λ | d ν 0 Λ , 0 ,ρ a , 0 [ ψ ] , wher e ρ a is deﬁne d by d ρ a ( u ) = e a | u | 2 d ρ ( u ) . • A Gibbs me asur e if for every ﬁnite Λ ⊂ V and any b ounde d me asur able function g : R Λ → R , the DLR e quation ν [ g ] = Z ξ ∈ R V ⟨ g ⟩ ξ Λ ,β ,ρ,J d ν ( ξ ) holds. In p articular, we assume that ν is almost sur ely supp orte d on c onﬁgur ations ξ such that ⟨·⟩ ξ Λ ,β ,ρ,J is wel l-deﬁne d. In Section 5.3, w e giv e conditions on ξ that ensure that the limiting measure (if it exists) as Λ ↗ V is an a -regular Gibbs measure. F or ferromagnetic in teractions, w e construct the plus measure ν + as the limit of ﬁnite-v olume measures and sho w that it is a -regular for some a > 0 . W e also sh o w that ν + is maximal, hence extremal, in the sense that if ν is an a ′ -regular Gibbs measure for some a ′ > 0 , then ν is sto c hastically dominated by ν + . F urthermore, in the nearest-neighbour case, we introduce a family of ﬁnite-v olume measures with random b oundary conditions that conv erge to ν + and are regular up to the b oundary , in contrast to the constructions in [10] and [7], whic h rely on logarithmically growing b oundary conditions. These ﬁnite-volume measures are also sto c hastically decreasing in the volume, similarly to the case of the Ising mo del. W e exp ect that this may help a v oid c hallenges arising from the absence of maximal b oundary conditions at ﬁnite volume, leading to simpliﬁcations of the arguments in [7] and [8], as w ell as to applications in future w orks. 1.1 P ap er organisation In Section 2, we deﬁne the notation that will b e used throughout the rest of the paper. Section 3 is dedicated to the pro of of Theorem 1.1. The metho ds dev elop ed here can b e applied to a range of other similar mo dels, some examples of which are given in Section 4. In Section 5, we examine for which b oundary conditions tightness can b e obtained and construct the inﬁnite-volume plus measure as a limit of ﬁnite-volume measures. 5 A c knowledgemen ts W e thank T rishen Gunaratnam, Dmitrii Krac hun, Romain P anis and F ranco Severo for useful discussions. CP w as supp orted by an EPS R C New In vesti- gator A ward (UKRI1019). 2 Deﬁnitions and preliminaries In this section, w e deﬁne the mo del in full generality , as well as introduce some addi- tional notation and results that will b e used in the pro ofs. 2.1 Deﬁnition of the mo del Let V b e a countably inﬁnite set of vertices. Let β ≥ 0 b e the inv erse temp erature and let ρ b e a single-site measure satisfying (1.2). A t some p oin ts, including in Theorem 1.1, w e assume ρ satisﬁes the stronger condition (1.3) with resp ect to some constants a > 0 and n > 2 . F or b ∈ R , we will write ρ b for the measure with densit y e b | u | n with resp ect to ρ , where we implicitly assume n = 2 when we do not require ρ to satisfy (1.3). Consider in teractions ( J x,y ) x,y ∈ V on V that satisfy conditions (C1) and (C2). W e will sometimes assume also that the interactions are ferromagnetic, meaning that J x,y ≥ 0 for all x, y ∈ V , but this is not required in our regularit y results. Given a ﬁnite subset Λ ⊂ V , w e denote by E (Λ , J ) the set of unordered pairs of v ertices x, y ∈ V with at least one v ertex in Λ such that J x,y  = 0 , and write elemen ts of E (Λ , J ) in the form xy . W e deﬁne the mo del on Λ with boundary conditions ξ ∈ R V , which w e assume satisfy P y ∈ V | J x,y ξ y | < ∞ for all x ∈ Λ . Deﬁnition 2.1. The ﬁnite-volume spin mo del on Λ is the me asur e ν ξ Λ ,β ,ρ,J on R Λ given by d ν ξ Λ ,β ,ρ,J [ φ ] = 1 Z ξ Λ ,β ,ρ,J exp( − β H ξ Λ ,J ( φ )) Y x ∈ Λ d ρ ( φ x ) , (2.1) wher e the p artition function Z ξ Λ ,β ,ρ,J is the normalising c onstant that makes ν ξ Λ ,β ,ρ,J a pr ob ability me asur e and H ξ Λ ,J ( φ ) is the Hamiltonian, given by H ξ Λ ,J ( φ ) = − X xy ∈ E (Λ ,J ) x,y ∈ Λ J x,y φ x φ y − X xy ∈ E (Λ ,J ) x ∈ Λ , y ∈ V \ Λ J x,y φ x ξ y . Let us men tion that assumption (1.2) is necessary for the mo del to b e well-deﬁned for all β ≥ 0 due to the quadratic nature of the in teractions: φ x φ y = − ( φ x − φ y ) 2 2 + φ 2 x + φ 2 y 2 . W e write ⟨·⟩ ξ Λ ,β ,ρ,J for the exp ectation with respect to the measure ν ξ Λ ,β ,ρ,J , and for Λ ′ ⊂ Λ write ν ξ (Λ | Λ ′ ) ,β ,ρ,J for the restriction of ν ξ Λ ,β ,ρ,J to ev ents that only dep end on spins in Λ ′ . F or a sequence (Λ i ) i ≥ 1 of ﬁnite subsets of V , w e sa y Λ i ↗ V if Λ i ⊂ Λ i +1 for all i and S ∞ i =1 Λ i = V . W e write Λ ⋐ V to denote that Λ is a ﬁnite subset of V and say that the family of measures ( ν ξ Λ ,β ,ρ,J ) Λ ⋐ V is tigh t if for any Λ ′ ⋐ V , the measures ν ξ (Λ | Λ ′ ) ,β ,ρ,J for Λ ′ ⊂ Λ ⋐ V are tight in the usual sense. The measure ν ξ Λ ,β ,ρ,J satisﬁes the domain Marko v prop ert y , which states that for an y Λ ′ ⊂ Λ , ψ ∈ R Λ ′ , η ∈ R Λ \ Λ ′ , d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ = ψ | φ | Λ \ Λ ′ = η ] = d ν η ∪ ξ Λ ′ ,β ,ρ,J [ ψ ] , 6 where η ∪ ξ ∈ R V is the conﬁguration which is equal to η on Λ \ Λ ′ and is equal to ξ elsewhere. Another useful prop erty of the mo del is monotonicity in b oundary conditions. Before stating this, we m ust ﬁrst in tro duce the notion of an increasing function. Deﬁnition 2.2. W e say that g : R Λ → R is an incr e asing function if for any φ, φ ′ ∈ R Λ with φ x ≤ φ ′ x for al l x ∈ Λ , then g ( φ ) ≤ g ( φ ′ ) . W e say an event E is an incr e asing event if 1 E is an incr e asing function. F or me asur es ν, ν ′ on R Λ , we say that ν is sto chastic al ly dominate d by ν ′ and write ν ⪯ ν ′ if ν [ g ] ≤ ν ′ [ g ] for any incr e asing function g : R Λ → R . Prop osition 2.3. Supp ose J is ferr omagnetic and ξ , ξ ′ ∈ R V ar e such that ξ x ≤ ξ ′ x for al l x ∈ V . Then ν ξ Λ ,β ,ρ,J ⪯ ν ξ ′ Λ ,β ,ρ,J . Pr o of. Let g : R Λ → R b e an increasing function, and note that F ( φ ) : = exp( β H ξ Λ ,J ( φ ) − β H ξ ′ Λ ,J ( φ )) = exp X xy ∈ E (Λ ,J ) x ∈ Λ , y ∈ V \ Λ β J x,y φ x ( ξ ′ y − ξ y ) ! is an increasing function. Using the FKG inequality [6, Theorem 4.4.1], we obtain ⟨ g ⟩ ξ ′ Λ ,β ,ρ,J = ⟨ F g ⟩ ξ Λ ,β ,ρ,J ⟨ F ⟩ ξ Λ ,β ,ρ,J ≥ ⟨ g ⟩ ξ Λ ,β ,ρ,J . 2.2 Deﬁnition of A ( x, Λ) In this section, we in tro duce the functions A ( x, Λ) and ˜ A ( x, Λ) . Before stating the formal deﬁnitions, w e giv e the following rough description. Consider a walk from x to some v ertex z ∈ V and assign a v alue to each vertex along the walk, with the v alue at z prop ortional to | ξ z | . The v alue at each vertex mo ving a wa y from z drops b y an amoun t dep ending on the in teraction strength b et ween the v ertices, and we set ˜ A ( x, Λ) to b e the maxim um ov er all walks of the v alue at x . The function A ( x, Λ) is similar, but here we view the b oundary conditions ξ as an external ﬁeld b y combining the contributions from all the v ertices in V \ Λ that interact with a given v ertex x ∈ Λ in to a single p oin t h x, Λ , deﬁned as h x, Λ = X y ∈ V \ Λ J x,y ξ y . Doing this will ensure that A ( x, Λ) is ﬁnite for an y ﬁnite subset Λ ⊂ V , since our assump- tions on the boundary conditions ξ imply that | h x, Λ | < ∞ for all x ∈ Λ . W e now giv e the deﬁnitions of A ( x, Λ) and ˜ A ( x, Λ) , ﬁ rst stating what w e mean by a w alk. Let R , S, T ⊂ V . • W e say that a sequence of (not necessarily distinct) v ertices x 0 , x 1 , . . . , x m ∈ V is a w alk from S to T in ( R, J ) if x 0 ∈ S, x m ∈ T , x 1 , . . . , x m − 1 ∈ R , and J x i − 1 ,x i  = 0 for all i ∈ { 1 , . . . , m } . • W e say that R is J -connected if for any x, y ∈ R , there exists a w alk from x to y in ( R, J ) . 7 When discussing walks to or from a singleton { x } , we ma y write it as x instead. W e ﬁrst treat the n > 2 case. W e give concrete examples in Sections 5.1 and 5.2. F or n > 2 , R ⊂ V , x ∈ R and C ≥ 1 , deﬁne ˜ A ( x, R ) = ˜ A ( x, R, ξ , C , J, f , n ) , to b e the smallest A ≥ 1 such that for an y z ∈ V and an y walk x 0 , . . . , x m from x to z in ( R, J ) , | ξ z | ≤ C A ( n − 1) m m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m − i . if such A exists, and otherwise we say that ˜ A ( x, R ) = ∞ . Note that ˜ A ( x, R ) is increasing in R . W e also deﬁne A ( x, R ) = A ( x, R, ξ , C , J, f , n ) to b e the smallest A ≥ 1 suc h that for an y y ∈ R and any walk x 0 , x 1 , . . . , x m from x to y in ( R , J ) , | h y ,R | ≤   X z ∈ V \ R | J y ,z | f ( J y ,z )   C A ( n − 1) m +1 m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m +1 − i , and say that A ( x, R ) = ∞ if no suc h A exists. W e now giv e the deﬁnition in the n = 2 case. F or R ⊂ V , x ∈ R , C ≥ 1 and λ ≥ 1 , w e deﬁne ˜ A ( x, R ) = ˜ A ( x, R, λ, ξ , C , J, f ) to b e the smallest A ≥ 1 such that for any z ∈ V and any walk x 0 , x 1 , . . . , x m from x to z in ( R, J ) , | ξ z | ≤ C Aλ m m Y i =1 f ( J x i − 1 ,x i ) . W e also deﬁne A ( x, R ) = A ( x, R, λ, ξ , C , J, f ) to b e the smallest A ≥ 1 suc h that for an y w alk x 0 , x 1 , . . . , x m from x to R in ( R , J ) , | h x m ,R | ≤   X z ∈ V \ R | J x m ,z | f ( J x m ,z )   C Aλ m +1 m Y i =1 f ( J x i − 1 ,x i ) . W e will frequen tly drop the parameters ξ , C, J, f , n, λ from the notation when they are clear from the context. In both cases n > 2 and n = 2 , it follows (from Theorem 1.1 and Theorem 4.1 resp ectiv ely) that we ha ve tightness if A ( x, Λ) is b ounded ab o ve by a function of x that do es not depend on Λ . When considering whether this is the case for a particular c hoice of b oundary conditions, it may b e more con venien t to work with the function ˜ A and use the fact that for any Λ ⋐ V and an y x ∈ Λ we ha ve A ( x, Λ) ≤ ˜ A ( x, Λ) ≤ ˜ A ( x, V ) . T o see why this is the case, let y ∈ Λ and let x 0 , . . . , x m b e a w alk from x to y in (Λ , J ) . Supp ose n > 2 (the n = 2 case is similar). F or any z ∈ V with J y ,z  = 0 , the deﬁnition of ˜ A ( x, Λ) applied to the w alk x 0 , . . . , x m , z implies that | ξ z | ≤ C ˜ A ( x, Λ) ( n − 1) m +1 f ( J y ,z ) m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m +1 − i , hence | h y , Λ | ≤ X z ∈ V \ Λ | J y ,z || ξ z | ≤   X z ∈ V \ Λ | J y ,z | f ( J y ,z )   C ˜ A ( x, Λ) ( n − 1) m +1 m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m +1 − i , 8 whic h implies A ( x, Λ) ≤ ˜ A ( x, Λ) . Since ˜ A ( x, R ) is increasing in R , w e obtain tigh tness if ˜ A ( x, V ) is ﬁnite for all x ∈ V . F or n > 2 , deﬁne Ξ = Ξ( V , J, f , n ) to be the set of b oundary conditions for which this is the case. Similarly , for n = 2 , deﬁne Ξ( λ ) to b e the set of b oundary conditions for which ˜ A ( x, V , λ ) is ﬁnite for all x ∈ V . See Sections 5.1 and 5.2 for examples of boundary conditions that are in Ξ for diﬀerent c hoices of in teractions J . W e now state a lemma that allows us to compare the v alues of A ( x, R ) and A ( y , R ) or ˜ A ( x, R ) and ˜ A ( y , R ) . In the case when V is J -connected, this means that to determine whether given b oundary conditions are in Ξ , it suﬃces to c heck whether ˜ A ( x, V ) is ﬁnite for one vertex x . Lemma 2.4. L et R ⊂ V , C ≥ 1 , and ξ ∈ R V . F or any walk x 0 , x 1 , . . . , x k in ( R, J ) with x 0 , x k ∈ R , we have (i) If n > 2 , ˜ A ( x k , R ) ≤ ˜ A ( x 0 , R ) ( n − 1) k k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k − i , (ii) If n > 2 , A ( x k , R ) ≤ A ( x 0 , R ) ( n − 1) k k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k − i , (iii) If n = 2 , ˜ A ( x k , R ) ≤ ˜ A ( x 0 , R ) λ k k Y i =1 f ( J x i − 1 ,x i ) , (iv) If n = 2 , A ( x k , R ) ≤ A ( x 0 , R ) λ k k Y i =1 f ( J x i − 1 ,x i ) . Pr o of. The proofs of the ﬁrst t wo and last t wo statements are very similar, so w e only pro ve (i) and (iv) here. F or (i), let z ∈ V and let y 0 , y 1 , . . . , y j b e a walk from x k to z in ( R, J ) . Considering the walk x 0 , . . . , x k , y 1 , . . . , y j , then b y deﬁnition of ˜ A ( x 0 , R ) , we ha ve | ξ z | ≤ C ˜ A ( x 0 , R ) ( n − 1) k + j k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k + j − i !   j Y i =1 f ( J y i − 1 ,y i ) ( n − 1) j − i   . Hence ˜ A ( x 0 , R ) ( n − 1) k + j Q k i =1 f ( J x i − 1 ,x i ) ( n − 1) k + j − i satisﬁes the requirements for ˜ A ( x k , R ) ( n − 1) j , so ˜ A ( x k , R ) ( n − 1) j ≤ ˜ A ( x 0 , R ) ( n − 1) k + j k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k + j − i ! = ˜ A ( x 0 , R ) ( n − 1) k k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k − i ! ( n − 1) j . T aking b oth sides to the p o wer 1 / ( n − 1) j yields (i). T o pro ve (iv), let y ∈ R and let y 0 , y 1 , . . . , y j b e a w alk from x k to y in ( R, J ) . Considering the walk x 0 , . . . , x k , y 1 , . . . , y j , then by deﬁnition of A ( x 0 , R ) , we ha ve | h y ,R | ≤   X z ∈ V \ R | J y ,z | f ( J y ,z )   C A ( x 0 , R ) λ k + j +1 k Y i =1 f ( J x i − 1 ,x i ) j Y i =1 f ( J y i − 1 ,y i ) . This means that A ( x 0 , R ) λ k Q k i =1 f ( J x i − 1 ,x i ) satisﬁes the requiremen ts for A ( x k , R ) , whic h yields (iv). 9 2.3 Branc hing pro cesses In the proof of our main regularit y theorem, we will use a standard result on branching pro cesses from [9] to bound the size of the cluster where the spins tak e large v alues. Below w e give the deﬁnition of a branching pro cess and then state this result. F or a random v ariable X taking v alues in the non-negativ e integers, a branc hing pro cess with oﬀspring distribution X and initial p opulation k ∈ N is a sequence of random v ariables ( Z n ) n ≥ 0 suc h that Z 0 = k and for all n ≥ 1 , Z n = P Z n − 1 i =1 X n,i , where X n,i are independent random v ariables with the same distribution as X . W e call T := P ∞ n =0 Z n the total progen y of the branching pro cess. Theorem 2.5 ( [9, Theorem 3.13]) . F or a br anching pr o c ess with oﬀspring distribution X and initial p opulation k , the distribution of the total pr o geny T is given by P [ T = n ] = k n P [ X 1 + . . . + X n = n − k ] , wher e X 1 , . . . , X n ar e indep endent r andom variables with the same distribution as X . 3 Pro of of Theorem 1.1 In this section, w e prov e Theorem 1.1 by emplo ying an exploration argument. W e start b y gathering some useful inequalities. Lemma 3.1. L et β > 0 , n ≥ 2 , α 0 ≥ 1 and C ≥ α 0 . F or every x, y ∈ V , | φ x | ≥ C , and t x , t y ∈ [ − α 0 , α 0 ] , β ( | φ x || φ y | + | t x || t y | ) ≤ 2 β C n − 2 ( | φ x | n + | φ y | n ) . Pr o of. By Y oung’s inequality and the fact that | t x | , | t y | ≤ | φ x | , β ( | φ x || φ y | + | t x || t y | ) ≤ β 2 ( | φ x | 2 + | φ y | 2 + | t x | 2 + | t y | 2 ) ≤ β 2 (3 | φ x | 2 + | φ y | 2 ) . (3.1) If | φ y | < C , then (3.1) implies that β ( | φ x || φ y | + | t x || t y | ) ≤ 2 β | φ x | 2 ≤ 2 β C n − 2 | φ x | n . If | φ y | ≥ C , then (3.1) giv es β ( | φ x || φ y | + | t x || t y | ) ≤ β 2 C n − 2 (3 | φ x | n + | φ y | n ) ≤ 3 β 2 C n − 2 ( | φ x | n + | φ y | n ) . This completes the proof. Lemma 3.2. L et β , a > 0 , n > 2 , α 0 ≥ 1 and C ≥ α 0 +  4 M f β a  1 n − 2 . If | φ x | ≥ C and | φ y | ≤ f ( J x,y ) C n − 2 | φ x | n − 1 , then for any t ∈ [ − α 0 , α 0 ] , β ( | φ x || φ y | + | t || φ y | ) ≤ af ( J x,y ) 2 M f | φ x | n . 10 Pr o of. Note that since | φ y | ≤ f ( J x,y ) C n − 2 | φ x | n − 1 , we hav e β | φ y | ( | φ x | + | t | ) − af ( J x,y ) 2 M f | φ x | n ≤ β f ( J x,y ) | φ x | n − 1 C n − 2 ( | φ x | + | t | ) − af ( J x,y ) 2 M f | φ x | n ≤ f ( J x,y ) | φ x | n β C n − 2  1 + α 0 | φ x |  − a 2 M f ! . F or every C ≥ α 0 + (4 a − 1 M f β ) 1 n − 2 , the ab ov e expression is negative when | φ x | ≥ C . W e now pro ceed with the proof of Theorem 1.1. Recall that we consider Λ ′ ⊂ Λ . Our approac h is based on an exploration process that builds the cluster C of vertices x such that there exists a walk from Λ ′ to x where the spins tak e large v alues at each v ertex along the walk. T o accommo date the p oten tially large b oundary conditions, we allow the minim um spin v alue needed to b e in C to grow progressively as w e mov e aw ay from Λ ′ , ensuring that no vertices in V \ Λ are included in C . This gradual growth ensures that the conditions of Lemmas 3.1 and 3.2 are satisﬁed, which in turn enables us to isolate the v ertices of C from their neighbours, at the cost of mo difying the single-site measure. F or those vertices in Λ ′ where the spins take small v alues, we estimate their con tribution to the Radon–Nikodym deriv ativ e directly . Pr o of of The or em 1.1. Fix a ﬁnite subset Λ ⊂ V and let Λ ′ ⊂ Λ . W e write E for { xy ∈ E (Λ , J ) : x, y ∈ Λ } and A x for A ( x, Λ , ξ , C , J, f , n ) , where C ≥ 1 is a constant to b e determined. Deﬁne C to b e the set of v ertices x ∈ Λ suc h that for some m ∈ { 0 , 1 , . . . } there exists a w alk x 0 , x 1 , . . . , x m from Λ ′ to x in (Λ , J ) that satisﬁes ∀ k ∈ S m | φ x k | ≥ C A ( n − 1) k x 0 k Y i =1 f ( J x i − 1 ,x i ) ( n − 1) k − i , (3.2) where S 0 = { 0 } and S m = { 1 , . . . , m } for m ≥ 1 . F or each i ∈ { 0 , 1 , . . . } , let C i ⊂ C denote the set of vertices for which i is the smallest v alue of m such that there exists a w alk satisfying (3.2). Note that if x ∈ C , then | φ x | ≥ C , and if y ∈ Λ ′ , then Lemma 2.4 implies that y ∈ C if and only if y ∈ C 0 , that is, if and only if | φ y | ≥ C A y . Our aim is to prov e that there exists K ≥ 0 such that for an y V 1 , V 2 , . . . pairwise disjoin t subsets of Λ \ Λ ′ w e hav e d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ , ( C i ) i ≥ 1 = ( V i ) i ≥ 1 ] ≤ (3.3) exp  K | Λ ′ ∪ V ′ |    Y x ∈ Λ ′ e α 1 M f A n x d ρ a 2 ( φ x )   ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y , where α 1 = 2 β C 2 , V 0 = Λ ′ , V ′ = S ∞ i =1 V i , and p x,y = ( R | u |≥ C f ( J x,y ) d ρ a 2 ( u ) if J x,y  = 0 , 0 if J x,y = 0 . Here ( p x,y ) x,y ∈ V can b e interpreted as the oﬀspring distribution of a branc hing pro cess in the sense that y is a child of x with probability p x,y . The distribution of the n umber of c hildren of vertex x in this pro cess dep ends on x , but we can get a uniform con trol of the distribution by tuning the v alue of C . 11 Assume that (3.3) holds for no w. Summing (3.3) o ver all p ossibilities ( V j ) for ( C j ) and noting that Λ ′ ∩ V ′ = ∅ on the ev ent { ( C i ) i ≥ 1 = ( V i ) i ≥ 1 } , we obtain d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ ] ≤   Y x ∈ Λ ′ e α 1 M f A n x d ρ a 2 ( φ x )   X ( V j ) e K ( | Λ ′ | + | V ′ | ) ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y . W e can th us conclude by applying Lemma 3.4, whic h b ounds the righ t-hand side. Let us no w pro ve (3.3). Let α 0 ≥ 1 b e suc h that ρ a ([ − α 0 , α 0 ]) > 0 . Given ( C i ) i ≥ 1 = ( V i ) i ≥ 1 with S ∞ i =1 V i = V ′ , let t ∈ [ − α 0 , α 0 ] Λ ′ ∪ V ′ and deﬁne ˜ φ ∈ R Λ to b e the conﬁguration with ˜ φ x = t x for x ∈ Λ ′ ∪ V ′ and ˜ φ x = φ x for x ∈ Λ \ (Λ ′ ∪ V ′ ) . Comparing the v alue of the integrand at φ with its v alue at ˜ φ and setting α 1 = 2 β C 2 , w e will see that w e can c ho ose C large enough that β J x,y φ x φ y ≤ β J x,y ˜ φ x ˜ φ y (3.4) + | J x,y | f ( J x,y ) a | φ x | n 2 M f 1 x ∈C + α 1 A n x 1 x ∈ Λ ′ \C + a | φ y | n 2 M f 1 y ∈C + α 1 A n y 1 y ∈ Λ ′ \C ! . W e no w verify (3.4) by considering an edge xy ∈ E and splitting into cases based on whether x and y are in C , Λ ′ \ C , or Λ \ (Λ ′ ∪ C ) . In eac h case we will use the b ound β J x,y ( φ x φ y − ˜ φ x ˜ φ y ) ≤ β | J x,y | ( | φ x || φ y | + | ˜ φ x || ˜ φ y | ) . • If x, y ∈ C , then | φ x | , | φ y | ≥ C , so by Lemma 3.1, if C ≥ α 0 +  4 M f β a  1 n − 2 , then β ( | φ x || φ y | + | t x || t y | ) ≤ af ( J x,y ) 2 M f ( | φ x | n + | φ y | n ) . • If x ∈ C and y ∈ Λ \ (Λ ′ ∪ C ) , then there exists a walk x 0 , x 1 , . . . , x m from Λ ′ to x in (Λ , J ) satisfying (3.2), but there is no suc h walk from Λ ′ to y . Hence the walk x 0 , . . . , x m , y do es not satisfy (3.2), so | φ y | ≤ C A ( n − 1) m +1 x 0 f ( J x,y ) m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m +1 − i = f ( J x,y ) C n − 2 C A ( n − 1) m x 0 m Y i =1 f ( J x i − 1 ,x i ) ( n − 1) m − i ! n − 1 ≤ f ( J x,y ) C n − 2 | φ x | n − 1 . Com bining the latter with Lemma 3.2 w e get β ( | φ x || φ y | + | t x || φ y | ) ≤ af ( J x,y ) 2 M f | φ x | n . • If x ∈ C and y ∈ Λ ′ \ C , then applying Lemma 3.1 with C ≥ α 0 +  4 M f β a  1 n − 2 and using that | φ y | ≤ C A y , we get β ( | φ x || φ y | + | t x || t y | ) ≤ 2 β C n − 2 ( | φ x | n + ( C A y ) n ) ≤ af ( J x,y ) 2 M f | φ x | n + 2 β C 2 A n y . • If x, y ∈ Λ ′ \ C , then | φ x | ≤ C A x and | φ y | ≤ C A y ≤ C A n − 1 x f ( J x,y ) by Lemma 2.4, so β ( | φ x || φ y | + | t x || t y | ) ≤ 2 β C 2 A n x f ( J x,y ) . 12 • If x ∈ Λ ′ \ C and y ∈ Λ \ (Λ ′ ∪ C ) , then | φ x | ≤ C A x and | φ y | ≤ C A n − 1 x f ( J x,y ) , so β ( | φ x || φ y | + | t x || φ y | ) ≤ 2 β C 2 A n x f ( J x,y ) . The terms e β h x, Λ φ x coming from interaction with the spins outside Λ can b e b ounded similarly , using that if x ∈ C , then by deﬁnition of C and A x w e hav e | h x, Λ | ≤   X y ∈ V \ Λ | J x,y | f ( J x,y )   | φ x | n − 1 C n − 2 , so Lemma 3.2 applies. If x ∈ Λ ′ \ C then we b ound | h x, Λ | using the deﬁnition of A x . Ov erall, we obtain β h x, Λ φ x ≤ β h x, Λ ˜ φ x + a | φ x | n 2 M f 1 x ∈C + α 1 A n x 1 x ∈ Λ ′ \C ! X y ∈ V \ Λ | J x,y | f ( J x,y ) . (3.5) No w (3.3) follows from Lemma 3.3 b elo w, whic h we also state in the case n = 2 . It only remains to show that we can c ho ose C and ˜ C of the required form. Note that C depends on β only through the condition C ≥ α 0 +  4 M f β a  1 n − 2 when applying Lemmas 3.1 and 3.2. W e can choose ˜ C = log ( α 2 R R d ρ a 2 ( u )) + α 1 M f , where α 2 is given b y Lemma 3.4 and do es not dep end on β . Hence the only dep endence of ˜ C on β comes from α 1 = 2 β C 2 . Lemma 3.3. A ssume that (C1) , (C2) , (3.4) and (3.5) hold for some n ≥ 2 , a > 0 , α 1 > 0 and A x ≥ 1 , wher e ˜ φ in (3.4) and (3.5) is deﬁne d as ab ove for α 0 ≥ 1 such that ρ a ([ − α 0 , α 0 ]) > 0 . Then ther e exists a c onstant K = K ( a, α 0 ) ≥ 0 such that d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ , ( C i ) i ≥ 1 = ( V i ) i ≥ 1 ] ≤ exp  K | Λ ′ ∪ V ′ |    Y x ∈ Λ ′ e α 1 M f A n x d ρ a 2 ( φ x )   ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y . Pr o of. F or ease of notation, write P Λ ( φ ) = Q x ∈ Λ e − a | φ x | n and π E ( φ ) =   Y xy ∈ E e β J x,y φ x φ y     Y x ∈ Λ e β h x, Λ φ x   . Note that (2.1) giv es that d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ , ( C i ) i ≥ 1 = ( V i ) i ≥ 1 ] = 1 Z ξ Λ ,β ,ρ,J Z R Λ \ Λ ′ 1 { ( C i ) i ≥ 1 =( V i ) i ≥ 1 } π E ( φ ) P Λ ( φ ) Y x ∈ Λ d ρ a ( φ x ) , where ρ a satisﬁes 0 < ρ a ( R ) < ∞ . W e estimate π E ( φ ) b y applying (3.4) to each element of the ﬁrst product and (3.5) to eac h element of the second pro duct. This yields π E ( φ ) ≤ Y x ∈ Λ ′ \C exp   α 1 A n x X y ∈ V | J x,y | f ( J x,y )   Y x ∈C exp   a | φ x | n 2 M f X y ∈ V | J x,y | f ( J x,y )   π E ( ˜ φ ) . Using (C2) the ab o ve inequality simpliﬁes to π E ( φ ) ≤   Y x ∈ Λ ′ e α 1 M f A n x   Y x ∈C e a 2 | φ x | n ! π E ( ˜ φ ) . 13 Com bining this with the pro duct ov er v ertices of the terms coming from the single-site measure, and using that | t x | ≤ α 0 for any x ∈ Λ ′ ∪ V ′ , we get π E ( φ ) P Λ ( φ ) ≤   Y x ∈ Λ ′ e α 1 M f A n x     Y x ∈ Λ ′ ∪ V ′ e − a 2 | φ x | n   π E ( ˜ φ ) P Λ \ (Λ ′ ∪ V ′ ) ( ˜ φ ) (3.6) ≤ exp  aα n 0 | Λ ′ ∪ V ′ |    Y x ∈ Λ ′ e α 1 M f A n x     Y x ∈ Λ ′ ∪ V ′ e − a 2 | φ x | n   π E ( ˜ φ ) P Λ ( ˜ φ ) . W e no w in tegrate with resp ect to φ x for eac h x ∈ Λ \ Λ ′ o ver the ev en t { ( C i ) i ≥ 1 = ( V i ) i ≥ 1 } . Observ e that, on this even t, if y ∈ V i +1 for some i ∈ { 0 , 1 , . . . } , then | φ y | ≥ C f ( J x,y ) for some x ∈ V i with J x,y  = 0 . Hence, ignoring the requiremen t for spins outside C to b e small, we hav e Z R V ′ 1 { ( C i ) i ≥ 1 =( V i ) i ≥ 1 } Y x ∈ V ′ e − a 2 | φ x | n d ρ a ( φ x ) ≤ ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y . (3.7) Note that for an y function F : R Λ ′ ∪ V ′ → R ≥ 0 , ρ a ([ − α 0 , α 0 ]) | Λ ′ ∪ V ′ | min t ∈ [ − α 0 ,α 0 ] Λ ′ ∪ V ′ F ( t ) ≤ Z [ − α 0 ,α 0 ] Λ ′ ∪ V ′ F ( t ) Y x ∈ Λ ′ ∪ V ′ d ρ a ( t x ) ≤ Z R Λ ′ ∪ V ′ F ( t ) Y x ∈ Λ ′ ∪ V ′ d ρ a ( t x ) . (3.8) Th us, since t ∈ [ − α 0 , α 0 ] Λ ′ ∪ V ′ is arbitrary in the deﬁnition of ˜ φ , integrating (3.6) and dividing by Z ξ Λ ,β ,ρ,J , applying (3.8) for F b eing π E ( ˜ φ ) P Λ ( ˜ φ ) , and using (3.7) yields (3.3), with K = max { 0 , aα n 0 − log ( ρ a ([ − α 0 , α 0 ])) } . W e now state and pro v e the second lemma used in the pro of of Theorem 1.1 ab ov e, whic h we also state in the case n = 2 . W e use the same notation as in the proof of Theorem 1.1, and w e recall that the deﬁnition of p x,y in volv es a constant C . Lemma 3.4. L et n ≥ 2 and K ≥ 0 . Then ther e exist C 0 , α 2 ≥ 1 that do not dep end on β such that for every C ≥ C 0 we have X ( V j ) e K ( | Λ ′ | + | V ′ | ) ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y ≤ α | Λ ′ | 2 . Pr o of. W e aim to compare Q ∞ i =0 Q y ∈ V i +1 max x ∈ V i p x,y with the probability that W i = V i for all i ≥ 1 , where ( W i ) i ≥ 0 is the exploration pro cess deﬁned as follows. First set W 0 = Λ ′ and ensure that C is chosen large enough that R | u |≥ C d ρ a 2 ( u ) ≤ 1 . Assuming that W i has already b een constructed, for eac h vertex x ∈ W i and y ∈ Λ \ ( W 0 ∪ . . . ∪ W i ) , say that the edge xy is op en with probability p x,y , indep enden tly of all other edges. Otherwise, sa y xy is closed. Then set W i +1 to b e the set of v ertices y ∈ Λ \ ( W 0 ∪ . . . ∪ W i ) such that xy is op en for some x ∈ W i . Once W i has b een constructed for all i ∈ { 0 , 1 , . . . } , set W = S ∞ i =1 W i . With this deﬁnition, we ha ve P [ W i +1 = V i +1 | W 1 = V 1 , . . . , W i = V i ] = Y y ∈ Λ \ ( V 0 ∪ ... ∪ V i +1 ) P   \ x ∈ V i { xy closed }   Y y ∈ V i +1 P   [ x ∈ V i { xy op en }   ≥ b | V i | Y y ∈ V i +1 max x ∈ V i p x,y , 14 where b = inf x ∈ V Y y ∈ V \{ x } (1 − p x,y ) . Com bining ov er all generations of the exploration pro cess, we obtain P [( W i ) i ≥ 1 = ( V i ) i ≥ 1 ] = ∞ Y i =0 P [ W i +1 = V i +1 | W 1 = V 1 , . . . , W i = V i ] ≥ ∞ Y i =0 b | V i | Y y ∈ V i +1 max x ∈ V i p x,y = b | Λ ′ | + | V ′ | ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y . (3.9) W e now claim that sup x ∈ V P y ∈ V \{ x } p x,y tends to 0 as C tends to inﬁnit y , which in turn implies that b tends to 1 . F or an y x, y ∈ V with J x,y  = 0 , we hav e p x,y = Z | u |≥ C f ( J x,y ) e − a 2 | u | n d ρ a ( u ) ≤ exp  − a 2 ( C f ( J x,y )) n  ρ a ( R ) . Giv en x, y ∈ V with | J x,y | < δ f , we hav e by (C2) that p x,y ≤ ρ a ( R ) | J x,y | a 2 C n , and we can c ho ose C > 0 to b e large enough so that ρ a ( R ) | J x,y | a 2 C n ≤ | J x,y | C M f . Since P y ∈ V | J x,y | ≤ M f , there are at most M f /δ f v ertices y ∈ V such that | J x,y | ≥ δ f and for these v ertices, w e can use the bound p x,y ≤ e − aC n / 2 ρ a ( R ) . The claim follows. Let α 3 = b − 1 e K ∈ [ 1 , ∞ ) . It follo ws from summing (3.9) ov er V 1 , V 2 , . . . that X ( V j ) e K ( | Λ ′ | + | V ′ | ) ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y ≤ α | Λ ′ | 3 ∞ X k =0 α k 3 P [ | W | = k ] . (3.10) W e w ant to sto c hastically dominate ( W i ) i ≥ 0 b y a branching pro cess ( Z i ) i ≥ 0 . By choosing the v alue of C to b e large enough, we can ensure that b ≥ 1 / 2 . W e can then deﬁne a random v ariable X b y P [ X = k ] =        b if k = 0 , 1 − b − (1 − b ) 2 b if k = 1 , (1 − b ) k if k ∈ { 2 , 3 , . . . } , and let ( Z i ) i ≥ 0 b e a branching pro cess with initial p opulation | Λ ′ | and oﬀspring distribution X . If x ∈ W i for some i ∈ { 0 , 1 , . . . } , let X x b e the num b er of vertices y ∈ W i +1 suc h that the edge xy is op en. F or all k ≥ 0 , we hav e P [ X x ≥ k ] ≤ P [ X x ≥ 1] k ≤ (1 − b ) k ≤ P [ X ≥ k ] , whic h prov es the desired sto c hastic domination. In particular, since α 3 ≥ 1 w e hav e that ∞ X k =0 α k 3 P [ | W | = k ] ≤ ∞ X k =0 α k 3 P [ T = k + | Λ ′ | ] , where T is the total progen y of ( Z i ) i ≥ 0 . Applying Theorem 2.5 gives that ∞ X k =0 α k 3 P [ | W | = k ] ≤ α | Λ ′ | 3 + ∞ X k = | Λ ′ | α k 3 | Λ ′ | k + | Λ ′ | P [ X 1 + . . . + X k + | Λ ′ | = k ] ≤ α | Λ ′ | 3 + 1 2 ∞ X k = | Λ ′ | α k 3 P [ X 1 + . . . + X 2 k ≥ k ] , (3.11) where X 1 , X 2 , . . . are indep enden t with the same distribution as X . W e now b ound the latter probability as follows. Setting θ = 2 log (2 α 3 ) , w e ha ve that E [ e θX ] → 1 as b → 1 . 15 By increasing the v alue of C , we can make b as close to 1 as desired, so b y choosing C large enough, we can ensure that E [ e θX ] < e θ/ 4 . W e then hav e b y the exp onential Marko v inequalit y and indep endence P [ X 1 + . . . + X 2 k ≥ k ] ≤ e − θk E [ e θX ] 2 k ≤ e − θk / 2 = (2 α 3 ) − k . (3.12) Com bining (3.11) and (3.12) yields ∞ X k =0 α k 3 P [ | W | = k ] ≤ α | Λ ′ | 3 + 2 −| Λ ′ | , and substituting this in (3.10) completes the pro of. Remark 3.5. In the c ase of ne ar est-neighb our inter actions on a gr aph of b ounde d de gr e e, the pr o of of L emma 3.4 c an b e simpliﬁe d by using the fact that the numb er of ways to cho ose C \ Λ ′ so that |C \ Λ ′ | = k is at most exp onential in k + | Λ ′ | . 4 Regularit y for related mo dels In this section, we aim to generalise Theorem 1.1. W e ﬁrst show that our arguments can also b e applied when ρ satisﬁes (1.3) with n = 2 for some a ≥ 4 β M f , with the other assumptions from Section 2.1 unc hanged. Recall from Section 2.2 the deﬁnition of A ( x, Λ) in the case n = 2 . The assumption that a ≥ 4 β M f is stronger than is necessary for our argumen ts or the arguments of [10] to apply , but in later applications we only consider the case when ρ satisﬁes (1.2), so w e include this assumption for simplicit y . The theorem b elo w is the analogue of Theorem 1.1 in the case n = 2 . Theorem 4.1. L et a ≥ 4 β M f and assume ρ satisﬁes R R e a | u | 2 d ρ ( u ) < ∞ . Ther e exist C ≥ 1 , ˜ C > 0 dep ending only on β , δ f , M f , ρ and a such that for any Λ ⋐ V , Λ ′ ⊂ Λ , ψ ∈ R Λ ′ , λ ≤ a 4 β M f , and any b oundary c onditions ξ ∈ R V with P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ , d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ = ψ ] ≤ Y x ∈ Λ ′ e ˜ C A ( x, Λ ,ξ ,C ) 2 d ν 0 Λ ′ , 0 ,ρ a 2 , 0 [ ψ ] . When ρ satisﬁes (1.2), it follows from Theorem 4.1 that we hav e tigh tness for an y b oundary conditions that are in Ξ( λ ) for some λ ≥ 1 . F or nearest-neigh b our interactions on a graph G , this includes exp onen tially growing b oundary conditions of the form | ξ x | ≤ C λ d G ( o,x ) (see (5.3)), so w e observ e that the threshold for tigh tness jumps from exp onential when n = 2 to double exp onen tial when n > 2 . The pro of of Theorem 4.1 is essen tially the same as that of Theorem 1.1 but with diﬀeren t deﬁnitions of C and A . Pr o of of The or em 4.1. Fix a ﬁnite subset Λ ⊂ V and let Λ ′ ⊂ Λ . W e write E for { xy ∈ E (Λ , J ) : x, y ∈ Λ } and A x for A ( x, Λ , λ, ξ , C , J, f ) , where C ≥ 1 is a constan t to b e determined. Deﬁne C to b e the set of v ertices x ∈ Λ suc h that for some m ∈ { 0 , 1 , . . . } there exists a w alk x 0 , x 1 , . . . , x m from Λ ′ to x in (Λ , J ) that satisﬁes ∀ k ∈ S m | φ x k | ≥ C A x 0 λ k k Y i =1 f ( J x i − 1 ,x i ) , (4.1) 16 where S 0 = { 0 } and S m = { 1 , . . . , m } for m ≥ 1 . F or each i ∈ { 0 , 1 , . . . } , let C i ⊂ C denote the set of vertices for which i is the smallest v alue of m such that there exists a w alk satisfying (4.1). Let α 0 ≥ 1 b e suc h that ρ a ([ − α 0 , α 0 ]) > 0 . Giv en ( C i ) i ≥ 1 = ( V i ) i ≥ 1 with S ∞ i =1 V i = V ′ , let t ∈ [ − α 0 , α 0 ] Λ ′ ∪ V ′ and deﬁne the conﬁguration ˜ φ ∈ R Λ as in the pro of of Theorem 1.1. Setting α 1 = aC 2 2 M f ≥ 2 β C 2 λ , one can sho w that when C ≥ α 0 β J x,y φ x φ y ≤ (4.2) β J x,y ˜ φ x ˜ φ y + | J x,y | f ( J x,y ) a | φ x | 2 2 M f 1 x ∈C + α 1 A 2 x 1 x ∈ Λ ′ \C + a | φ y | 2 2 M f 1 y ∈C + α 1 A 2 x 1 y ∈ Λ ′ \C ! , and β h x, Λ φ x ≤ β h x, Λ ˜ φ x + a | φ x | 2 2 M f 1 x ∈C + α 1 A 2 x 1 x ∈ Λ ′ \C ! X y ∈ V \ Λ | J x,y | f ( J x,y ) . (4.3) The ﬁrst inequality (4.2) can b e veriﬁed in a similar wa y to (3.4). Indeed, if | φ x | , | φ y | ≥ α 0 , then applying Lemma 3.1 and using that a ≥ 4 β M f giv es that β ( | φ x || φ y | + | t x || t y | ) ≤ af ( J x,y ) 2 M f ( | φ x | 2 + | φ y | 2 ) . W e also use that if x ∈ C and y ∈ Λ \ C , then | φ x | ≥ α 0 and | φ y | ≤ λf ( J x,y ) | φ x | , whic h implies β ( | φ x || φ y | + | t x || φ y | ) ≤ 2 β λf ( J x,y ) | φ x | 2 ≤ af ( J x,y ) 2 M f | φ x | 2 . (4.4) T o prov e (4.3), if x ∈ C then | h x, Λ | ≤   X y ∈ V \ Λ | J x,y | f ( J x,y )   λ | φ x | , so (4.4) applies. If x ∈ Λ ′ \ C then we b ound | h x, Λ | using the deﬁnition of A x to obtain | h x, Λ | ≤ C λA x   X y ∈ V \ Λ | J x,y | f ( J x,y )   . Ha ving obtained (4.2) and (4.3), we use Lemmas 3.3 and 3.4 as in the pro of of Theorem 1.1 to conclude. Remark 4.2. The or em 4.1 c an b e gener alise d by al lowing the single-site me asur e to dep end on the vertex. W e wil l use such a gener alisation to c onstruct the inﬁnite-volume plus me asur e as the limit of systems with a shifte d single-site me asur e at the b oundary. Supp ose the single-site me asur e at vertex x is given by dρ x, Λ ( u ) = e − a x, Λ | u | 2 dµ x, Λ ( u ) and ther e exist a b ounde d subset T ⊂ R and c onstants a min , a max , M 1 , M 2 > 0 such that for al l Λ ⋐ V and x ∈ Λ , (A1) a min ≤ a x, Λ ≤ a max , (A2) µ x, Λ ( T ) ≥ M 1 , (A3) µ x, Λ ( R ) ≤ M 2 . 17 Then ther e exist C ≥ 1 , ˜ C > 0 such that for any Λ ⋐ V , Λ ′ ⊂ Λ , ψ ∈ R Λ ′ , λ ≤ a min 4 β M f , and any b oundary c onditions ξ ∈ R V with P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ , d ν ξ Λ ,β ,ρ,J [ φ | Λ ′ = ψ ] ≤   Y x ∈ Λ ′ e ˜ C A ( x, Λ ,ξ ,C ) 2   d ν 0 Λ ′ , 0 , ˜ ρ, 0 [ ψ ] , wher e ˜ ρ is given by d ˜ ρ x ( u ) = e − 1 2 a x, Λ | u | 2 dµ x, Λ ( u ) . A ssumption (A2) is use d in (3.8) while assumptions (A1) and (A3) ar e use d to b ound p x,y = Z | u |≥ C f ( J x,y ) e − a y, Λ 2 d µ y , Λ ( u ) (4.5) for J x,y  = 0 . The or em 1.1 also holds for any single site me asur es satisfying (A1) , (A2) , (A3) . Changing the Hamiltonian can also b e considered. W e will still restrict our atten tion to pairwise in teractions and will assume further that in teractions occur only b et ween neigh b ours on a graph with b ounded degree, so there is an upp er b ound on the num b er of vertices that an y given vertex can interact with. Let G = ( V , E ) b e a graph with b ounded degree. F or Λ ⋐ V and b oundary conditions ξ ∈ R V , deﬁne the measure ν ξ Λ ,U,ρ b y d ν ξ Λ ,U,ρ [ φ ] = 1 Z ξ Λ ,U,ρ Y xy ∈ E x,y ∈ Λ U xy ( φ x , φ y ) Y xy ∈ E x ∈ Λ ,y ∈ V \ Λ U xy ( φ x , ξ y ) Y x ∈ Λ d ρ x, Λ ( φ x ) , for φ ∈ R Λ , where Z ξ Λ ,U,ρ is the partition function and for eac h xy ∈ E , U xy : R 2 → R + is a function. Theorem 4.3. L et D ≥ 1 and let G = ( V , E ) b e a gr aph such that deg( x ) ≤ D for al l x ∈ V . L et ρ x, Λ b e single-site me asur es satisfying (1.2) and assumptions (A1) , (A2) , (A3) . F or e ach xy ∈ E , let U xy : R 2 → R + b e a function satisfying the fol lowing assumptions for some c onstants C ≥ 1 , λ ≥ 1 , and function F : [ 1 , ∞ ) → [1 , ∞ ) : (i) If | φ x | ≥ C , t x ∈ T and | φ y | ≤ λ | φ x | , then U xy ( φ x , φ y ) ≤ U xy ( t x , φ y ) exp  a x, Λ 2 D | φ x | 2  . (ii) If t x , t y ∈ T and | φ x | ≤ C A x , | φ y | ≤ C λA x for some A x ≥ 1 , then max ( U xy ( t x , φ y ) U xy ( t x , t y ) , U xy ( φ x , φ y ) U xy ( t x , t y ) , U xy ( φ x , φ y ) U xy ( t x , φ y ) ) ≤ e F ( A x ) . Then ther e exist c onstants C 1 , C 2 such that for any Λ ⊂ V ﬁnite, Λ ′ ⊂ Λ , ψ ∈ R Λ ′ , and any b oundary c onditions ξ ∈ R V satisfying P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ , d ν ξ Λ ,U,ρ [ φ | Λ ′ = ψ ] ≤ Y x ∈ Λ ′ exp  C 1 F ( A ( x, Λ , λ, ξ , C 2 )) − 1 2 a x, Λ | ψ x | 2  d µ x, Λ ( ψ x ) . 18 Pr o of of The or em 4.3. Let f (1) = 1 and write A x for A ( x, Λ , λ, ξ , C 2 , J, f ) , where C 2 ≥ 1 is a constan t to be determined. Deﬁne C to b e the set of vertices x ∈ Λ such that for some m ∈ { 0 , 1 , . . . } there exists a walk x 0 , x 1 , . . . , x m from Λ ′ to x in (Λ , J ) that satisﬁes ∀ k ∈ S m | φ x k | ≥ C 2 A x 0 λ k , (4.6) where S 0 = { 0 } and S m = { 1 , . . . , m } for m ≥ 1 . F or each i ∈ { 0 , 1 , . . . } , let C i ⊂ C denote the set of vertices for which i is the smallest v alue of m such that there exists a w alk satisfying (4.6). Giv en ( C i ) i ≥ 1 = ( V i ) i ≥ 1 with S ∞ i =1 V i = V ′ , let t ∈ T Λ ′ ∪ V ′ and deﬁne the conﬁgurations φ ′ , ˜ φ ∈ R V b y φ ′ x = ( φ x if x ∈ Λ , ξ x otherwise , ˜ φ x = ( t x if x ∈ Λ ′ ∪ V ′ , φ ′ x otherwise . Assumption (i) in the statement of the theorem is analogous to (4.4), and (ii) allows us to b ound U xy ( φ ′ x , φ ′ y ) in terms of U xy ( ˜ φ x , ˜ φ y ) when x or y is in Λ ′ \ C . T ogether with the deﬁnitions of C and A ( o, Λ) , they imply that for any xy ∈ E U xy ( φ ′ x , φ ′ y ) ≤ U xy ( ˜ φ x , ˜ φ y ) exp  a x, Λ 2 D | φ x | 2 1 x ∈C + F ( A x ) 1 x ∈ Λ ′ \C + a y , Λ 2 D | φ y | 2 1 y ∈C + F ( A y ) 1 y ∈ Λ ′ \C  . Com bining ov er all edges and using that eac h vertex has degree at most D , w e get Y xy ∈ E U xy ( φ ′ x , φ ′ y ) ≤   Y xy ∈ E U xy ( ˜ φ x , ˜ φ y )   Y x ∈C e a x, Λ 2 | φ x | 2 Y x ∈ Λ ′ e DF ( A x ) . By com bining the ab o v e with the terms coming from the single-site measure and in tegrat- ing, we can sho w as in the pro of of Lemma 3.3 that for some K ≥ 0 , d ν ξ Λ ,U,ρ [ φ | Λ ′ , ( C i ) i ≥ 1 = ( V i ) i ≥ 1 ] ≤ exp  K | Λ ′ ∪ V ′ |    Y x ∈ Λ ′ e DF ( A x ) e − a x, Λ 2 | φ x | 2 d µ x, Λ ( φ x )   ∞ Y i =0 Y y ∈ V i +1 max x ∈ V i p x,y , with p x,y deﬁned as in (4.5). Applying Lemma 3.4 concludes the pro of. Remark 4.4. One example wher e The or em 4.3 is useful is the r andom cluster r epr esen- tation of the φ 4 mo del, intr o duc e d in [8], which is a me asur e on p airs ( a , ω ) , wher e a is the absolute value ﬁeld and ω is a p er c olation c onﬁgur ation. W e may wish to c onsider the distribution of a in this mo del c onditional on observing a given p er c olation c onﬁgur ation ω , similarly to [8, L emma 6.8]. In this c ase, the functions U xy ar e given by U xy ( a x , a y ) = ( e − β a x a y if ω xy = 0 , e β a x a y − e − β a x a y if ω xy = 1 . W e now che ck that assumptions (i) and (ii) in The or em 4.3 ar e satisﬁe d and that the choic e of C, λ and F do es not dep end on ω . A ssume ρ x, Λ ar e single-site me asur es supp orte d on R + that satisfy the assumptions of The or em 4.3 and let λ = a min 4 Dβ . L et C ≥ 1 b e a c onstant to b e determine d and let T = [ t min , t max ] , wher e 0 < t min < t max ≤ C and T satisﬁes (A2) . L et xy ∈ E . If ω xy = 0 , then the inter action term U xy ( a x , a y ) is of the same form as in The or em 4.1, so (i) fol lows fr om (4.4) . Now supp ose ω xy = 1 and observe that sinc e the 19 spins a x only take p ositive values in this mo del, U xy is incr e asing in b oth ar guments. Note that if a x ≥ C ≥ t x ≥ t min and a y ≤ λ a x , then U xy ( a x , a y ) U xy ( t x , a y ) is an incr e asing function of a y . Henc e, U xy ( a x , a y ) U xy ( t x , a y ) ≤ U xy ( a x , λ a x ) U xy ( t x , λ a x ) ≤ exp( β λ a 2 x ) U xy ( t min , C λ ) . The choic e of λ implies the right hand side ab ove is at most exp  a min 2 D a 2 x  for al l C lar ge enough, so (i) holds. F or (ii), we use that if ω xy = 1 , the maximum in (ii) is at most U xy ( C A x , C λA x ) U xy ( t min , t min ) , and if ω xy = 0 then it is at most exp(2 β C 2 λA 2 x ) by (4.2) . 5 Corollaries and applications In this section, w e giv e some examples of in teractions J and boundary conditions ξ for whic h w e can apply our regularit y results, and we then apply them to construct inﬁnite v olume measures. Recall from Section 2.2 that Ξ is the set of boundary conditions ξ for which ˜ A ( x, V , ξ ) < ∞ for all x ∈ V , and that we ha ve tightness for any ξ ∈ Ξ . F or nearest-neighbour interactions, we will giv e a full characterisation of Ξ and sho w for certain choices of ρ that ξ ∈ Ξ is necessary to obtain tightness in the case of non-negative b oundary conditions. W e also giv e examples of b oundary conditions that are in Ξ for diﬀeren t forms of long-range interactions. Later w e giv e conditions on ξ that ensure the measures ν ξ Λ ,β ,ρ,J con verge to an a -regular Gibbs measure as Λ ↗ V and construct the extremal regular Gibbs measures ν + and ν − . W e begin b y deﬁning some notation that will be used throughout this section. Assume G = ( V , E ) is an inﬁnite connected graph such that every v ertex has ﬁnite degree, and ﬁx an origin o ∈ V . Let d G : V × V → N 0 b e the graph distance in G , and for S ⊂ V let d S denote the graph distance in the subgraph of G induced by S . F or x ∈ V , let deg( x ) b e the degree of x in the graph G . 5.1 Results for nearest-neighbour interactions W e ﬁrst consider the case of (ferromagnetic) nearest-neigh b our in teractions, which are deﬁned as follows when G has bounded degree. Deﬁnition 5.1. If ther e exists a c onstant D such that deg( x ) ≤ D for al l x ∈ V , then we deﬁne ne ar est-neighb our inter actions J G on G by ( J G ) x,y = ( 1 if xy ∈ E , 0 otherwise . In the nearest-neighbour case, we ma y choose f to b e so that f (1) = 1 . Then, in the n > 2 case, A ( x, R ) = max    1 , max y ∈ ∂ R | h y ,R | C | N y ,V \ R | ! ( n − 1) − d R ( x,y ) − 1    , 20 where N y ,V \ R = { z ∈ V \ R : d G ( y , z ) = 1 } and ∂ R = { y ∈ R : N y ,V \ R  = ∅} . W e also hav e ˜ A ( x, R ) = max    1 , max z ∈ V  | ξ z | C  ( n − 1) − d R ∪{ z } ( x,z )    . As a consequence of this and connectedness, it follows from Lemma 2.4 that Ξ = { ξ ∈ R V : ∃ A ξ ∈ (0 , ∞ ) suc h that | ξ z | ≤ A ( n − 1) d G ( o,z ) ξ ∀ z ∈ V } . (5.1) In the n = 2 case, we hav e A ( x, R ) = max ( 1 , max y ∈ ∂ R | h y ,R | C | N y ,V \ R | λ d R ( x,y )+1 !) , (5.2) and ˜ A ( x, R ) = max  1 , max z ∈ V  | ξ z | C λ d R ∪{ z } ( x,z )  , so that Ξ( λ ) = { ξ ∈ R V : ∃ C ξ ∈ (0 , ∞ ) suc h that | ξ z | ≤ C ξ λ d G ( o,z ) ∀ z ∈ V } . (5.3) The remainder of the subsection is dev oted to justifying that our regularit y results are optimal. More precisely , w e aim to sho w that for the P ( φ ) mo dels, any non-negativ e b oundary conditions for which we ha ve tightness are in Ξ . T o simplify the calculations, w e only consider the case when P ( u ) = ˜ a | u | n here. Note that in this case ρ satisﬁes (1.3) for any a < ˜ a . Prop osition 5.2. A ssume n > 2 , ˜ a > 0 , d ρ ( u ) = e − ˜ a | u | n d u and G has b ounde d de gr e e. If ξ ∈ ( R + ) V \ Ξ , then the family of me asur es ( ν ξ Λ ,β ,ρ,J G ) Λ ⋐ V is not tight. When V = Z w e can obtain the same result as Proposition 5.2 for mixed p ositive and negativ e b oundary conditions. Prop osition 5.3. A ssume n > 2 , ˜ a > 0 , d ρ ( u ) = e − ˜ a | u | n d u and G = ( Z , { xy : | x − y | = 1 } ) . If ξ ∈ R Z \ Ξ , then the family of me asur es ( ν ξ Λ ,β ,ρ,J G ) Λ ⋐ V is not tight. In the pro ofs of the ab ov e prop ositions, we will use the following monotonicity prop erty . Lemma 5.4. A ssume ρ is an even me asur e satisfying (1.3) . Supp ose J, J ′ ar e inter actions on V satisfying (C1) , (C2) and 0 ≤ J x,y ≤ J ′ x,y for al l x, y ∈ V . Supp ose also that ξ , ξ ′ ar e b oundary c onditions on Λ such that 0 ≤ ξ x ≤ ξ ′ x for al l x ∈ V and P y ∈ V J ′ x,y | ξ ′ y | < ∞ for al l x ∈ Λ . L et u ≥ 0 and x ∈ Λ . Then ν ξ Λ ,β ,ρ,J [ φ x ≥ u ] ≤ ν ξ ′ Λ ,β ,ρ,J ′ [ φ x ≥ u ] . Pr o of. Monotonicit y in ξ has already b een established in Prop osition 2.3, so we just need to prov e that ν ξ Λ ,β ,ρ,J [ φ x ≥ u ] ≤ ν ξ Λ ,β ,ρ,J ′ [ φ x ≥ u ] . W riting σ x for the sign of φ x and using that 1 { σ x =1 } = 1 2 (1 + σ x ) , we hav e ν ξ Λ ,β ,ρ,J [ φ x ≥ u ] = ν ξ Λ ,β ,ρ,J [ | φ x | ≥ u ] ν ξ Λ ,β ,ρ,J [ σ x = 1 | | φ x | ≥ u ] = 1 2 ν ξ Λ ,β ,ρ,J [ | φ x | ≥ u ](1 + ⟨ σ x | | φ x | ≥ u ⟩ ξ Λ ,β ,ρ,J ) . Conditional on the absolute v alue ﬁeld, σ is distributed according to an Ising mo del with coupling constan ts determined by the absolute v alue ﬁeld. As the b oundary conditions are p ositiv e, monotonicit y of the Ising mo del in J follows by diﬀerentiating and using Griﬃths’ inequality [5, Theorem 3.20], and monotonicity of the absolute v alue ﬁeld was pro ved in [8, Prop osition 4.10]. 21 W e are no w ready to pro ceed with the pro ofs of Prop ositions 5.2 and 5.3. Pr o of of Pr op osition 5.2. Using the c haracterisation (5.1) of Ξ , ξ ∈ ( R + ) V \ Ξ implies that there exists a sequence of v ertices ( z i ) i ≥ 1 suc h that ξ ( n − 1) − m i z i → ∞ as i → ∞ , where m i = d G ( o, z i ) . Since there are only ﬁnitely many vertices at any ﬁxed distance from o , b y passing to a subsequence, we may assume that 1 ≤ m i < m j for an y i < j . Given i ≥ 1 , let Λ i = { x ∈ V : d G ( o, x ) < m i } and let y i, 0 , y i, 1 , . . . , y i,m i b e a w alk from o to z i in (Λ i , J ) . Also let α = β ˜ an 2 n − 1 , and for j ∈ { 0 , . . . , m i } , deﬁne D i,j = α 1 − ( n − 1) j − m i n − 2 ξ ( n − 1) j − m i z i . Then D i,m i = ξ z i and D i,j +1 = α − 1 D n − 1 i,j . W e will show that there exists ε > 0 suc h that for all i suﬃcien tly large, ν ξ Λ i ,β ,ρ,J G [ φ o ≥ D i, 0 ] ≥ ε . Since D i, 0 ≥ min { 1 , α 1 n − 2 } ξ ( n − 1) − m i z i → ∞ as i → ∞ , this implies that the sequence is not tigh t. The strategy for the p roof is to condition in turn on the ev ents { φ y i,j ≥ D i,j } . W e can then use Lemma 5.4 to set J x,y = 0 ev erywhere except for the edge b etw een y i,j and y i,j +1 , meaning that we only ha ve to calculate a one-dimensional in tegral in eac h step. W e ha v e ν ξ Λ i ,β ,ρ,J G [ φ o ≥ D i, 0 ] ≥ ν ξ Λ i ,β ,ρ,J G [ φ y i,j ≥ D i,j ∀ j ∈ { 0 , . . . , m i − 1 } ] = m i − 1 Y j =0 ν ξ Λ i ,β ,ρ,J G [ φ y i,j ≥ D i,j | φ y i,k ≥ D i,k ∀ k ∈ { j + 1 , . . . , m i − 1 } ] . Deﬁne J ( i,j ) b y ( J ( i,j ) ) x,y = 1 if { x, y } = { y i,j , y i,j +1 } and ( J ( i,j ) ) x,y = 0 otherwise. Also let ξ ( i,j ) b e deﬁned by ξ ( i,j ) y i,j +1 = D i,j +1 and ξ ( i,j ) x = 0 for all x ∈ V \ { y i,j +1 } . Using the domain Marko v property and Lemma 5.4, w e hav e ν ξ Λ i ,β ,ρ,J G [ φ y i,j ≥ D i,j | φ y i,k ≥ D i,k ∀ k ∈ { j + 1 , . . . , m i − 1 } ] ≥ ν ξ ( i,j ) { y i,j } ,β ,ρ,J ( i,j ) [ φ y i,j ≥ D i,j ] . W e now estimate the probabilit y on the righ t hand side. Let r = β D i,j +1 φ y i,j − ˜ a | φ y i,j | n . Then d r d φ y i,j     φ y i,j = D i,j = β D i,j +1 − ˜ anD n − 1 i,j = β D i,j +1  1 − 1 2 n − 1  , where the second equality is from our choice of α . This is greater than 1 for all j if i is large enough. Then b ecause r is a concav e function of φ y i,j , d r d φ y i,j ≥ 1 whenever φ y i,j ≤ D i,j . Hence, Z D i,j −∞ e β D i,j +1 φ y i,j − ˜ a | φ y i,j | n d φ y i,j ≤ Z β D i,j D i,j +1 − ˜ a ( D i,j ) n −∞ e r d r = exp( β D i,j D i,j +1 − ˜ a ( D i,j ) n ) . (5.4) Note that the maximum v alue of r o ccurs when φ y i,j = ( β ˜ an D i,j +1 ) 1 / ( n − 1) = 2 D i,j , and r is increasing when φ y i,j < 2 D i,j . Consequen tly , when φ y i,j ∈ [ D i,j , 2 D i,j ] , the v alue of r is at least β D i,j D i,j +1 − ˜ a ( D i,j ) n , which implies Z ∞ D i,j e β D i,j +1 φ y i,j − ˜ a | φ y i,j | n d φ y i,j ≥ D i,j exp( β D i,j D i,j +1 − ˜ a ( D i,j ) n ) . (5.5) Com bining (5.4) and (5.5) w e obtain that ν ξ ( i,j ) { y i,j } ,β ,ρ,J ( i,j ) [ φ y i,j ≥ D i,j ] ≥ D i,j D i,j + 1 . 22 T o conclude, w e need to take the product o ver j and verify that this is b ounded b elo w for all i suﬃciently large by a positive constan t that do es not dep end on i . W e hav e ν ξ Λ i ,β ,ρ,J G [ φ o ≥ D i, 0 ] ≥ m i − 1 Y j =0 D i,j D i,j + 1 = exp   m i − 1 X j =0 log( D i,j ) − log( D i,j + 1)   . T aylor expanding log ( D i,j + 1) around D i,j , we get log( D i,j + 1) ≤ log( D i,j ) + 1 D i,j , so ν ξ Λ i ,β ,ρ,J G [ φ o ≥ D i, 0 ] ≥ exp   − m i − 1 X j =0 1 D i,j   ≥ exp   − ∞ X j =0 1 min { 1 , α 1 n − 2 } ( ξ ( n − 1) − m i z i ) ( n − 1) j   . Since ξ ( n − 1) − m i z i → ∞ as i → ∞ , The last sum ab ov e con verges for all i suﬃciently large and decreases to 0 as i → ∞ , from whic h the desired result follows. Pr o of of Pr op osition 5.3. If ξ / ∈ Ξ , then using (5.1) there exists a sequence of v ertices ( z i ) i ≥ 1 suc h that | ξ z i | ( n − 1) −| z i | → ∞ as i → ∞ . W e will pro ceed with the pro of in the case where for inﬁnitely man y i , z i and ξ z i are p ositiv e (the other cases are similar). By taking an appropriate subsequence, we can assume that ξ z i ≥ 0 and 1 ≤ z i < z j for all 1 ≤ i < j . Deﬁne Λ i = { x ∈ Z : | x | < z i } . W e ﬁrst consider the case when there exists a subse- quence ( z i k ) k ≥ 1 suc h that ξ − z i k < − ξ z i k for all k ≥ 1 , which implies that ν ξ Λ i k ,β ,ρ,J G [ φ 0 ≤ 0] ≥ 1 2 . T o see wh y this is true, note that if ξ − z i k = − ξ z i k , then φ 0 and − φ 0 ha ve the same distribution, so the probability that φ 0 is negativ e is 1 / 2 . Now reducing ξ − z i k increases the probability of the ev ent { φ 0 ≤ 0 } b y Prop osition 2.3. Conditionally on { φ 0 ≤ 0 } , the subgraph {− 1 , − 2 , . . . , − ( z i k − 1) } has non-positive b oundary conditions. Hence − φ is distributed according to a measure with non-negative b oundary conditions, and w e can apply Prop osition 5.2 to deduce that there exists ε > 0 suc h that ν ξ Λ i k ,β ,ρ,J G [ φ − 1 ≤ − D k | φ 0 ≤ 0] ≥ ε for all k suﬃciently large, where D k → ∞ as k → ∞ . It follo ws that ν ξ Λ i k ,β ,ρ,J G [ | φ − 1 | ≥ D k ] ≥ ε 2 , so the sequence of measures ( ν ξ Λ i k ,β ,ρ,J G ) k ≥ 1 is not tight. If no suc h subsequence exists, then for all i large enough w e hav e that ξ − z i ≥ − ξ z i , so ν ξ Λ i ,β ,ρ,J G [ φ 0 ≥ 0] ≥ 1 2 . W e can no w conclude the pro of s imilarly using that if φ 0 ≥ 0 then w e hav e non-negative b oundary conditions on the subgraph { 1 , 2 , . . . , z i k − 1 } . 5.2 Results for more general in teractions In this subsection, we determine which b oundary conditions are in Ξ in the case of long-range interactions that satisfy some additional assumptions. Deﬁnition 5.5. W e say that inter actions ( J x,y ) x,y ∈ V ar e r e asonable if they satisfy the fol lowing assumptions in addition to (C1) and (C2) : • V is J -c onne cte d. • Ther e exists r > 0 such that for al l x  = y ∈ V , | J x,y | ≤ d G ( x, y ) − r . • f is an even function and is de cr e asing on (0 , ∞ ) . • Ther e exists c ∈ (0 , 1) such that f (2 r t ) ≥ cf ( t ) for al l t ∈ [0 , ∞ ) . 23 Note that this includes the nearest-neigh b our interactions J G as we are assuming G is connected. The next prop osition gives a suﬃcient condition to hav e ξ ∈ Ξ when J is reasonable. Belo w c and r are the constants of Deﬁnition 5.5. Prop osition 5.6. Supp ose that J is r e asonable and ξ ∈ R V is such that ther e exists M ξ ∈ R with | ξ x | ≤ M ξ f ( d G ( o, x ) − r ) for al l x ∈ V \ { o } . Then ξ ∈ Ξ( λ ) for any λ ≥ 1 c . Before pro ving the prop osition, we giv e tw o examples where it can b e applied. Firstly , the mildest function satisfying the assumption (C2) is the function f 1 giv en by f 1 ( t ) = ( log( | t | − 1 ) 1 / 2 if | t | < e − 1 , 1 otherwise . Prop osition 5.6 implies that if | ξ x | gro ws at most like p log( d G ( o, x )) , then ξ ∈ Ξ( λ ) for an y reasonable J and λ large enough. F or the second example, we consider Z d , or an y v ertex-transitive graph of dimension d . Let J b e translation in v ariant in teractions satisfying | J x,y | ≤ C J d G ( x, y ) − d − ε for all x  = y , where C J , ε > 0 are constants. This is the setting in which the regularity results of [10] and [7] were pro ved. In this case (after making C J = 1 by c hanging the v alue of β ), for any α < ε d + ε w e can apply Prop osition 5.6 with r = d + ε to the function f α giv en by f α ( t ) = ( | t | − α if | t | < 1 , 1 otherwise . This implies that an y ξ with | ξ x | ≤ M ξ d G ( x, y ) α ( d + ε ) is in Ξ( λ ) for all λ ≥ 2 ε , so w e hav e tigh tness for b oundary conditions gro wing at most lik e d G ( o, x ) δ for δ < ε . W e now give the pro of of Prop osition 5.6. Pr o of of Pr op osition 5.6. Since V is J -connected, w e just need to show that ˜ A ( o, V , λ ) is ﬁnite, as Lemma 2.4 then implies ˜ A ( x, V , λ ) < ∞ for all x ∈ V . First observe that if λ ≥ 1 c , then the assumptions on f imply that for an y walk in ( V , J ) consisting of distinct v ertices x 0 , x 1 , . . . , x m with m ≥ 2 , λf ( d G ( x m − 2 , x m − 1 ) − r ) f ( d G ( x m − 1 , x m ) − r ) ≥ λf  1 2 d G ( x m − 2 , x m )  − r ! (5.6) ≥ f ( d G ( x m − 2 , x m ) − r ) . Rep eatedly applying (5.6) and using that f ( J x i − 1 ,x i ) ≥ f ( d G ( x i − 1 , x i ) − r ) by Deﬁnition 5.5, we deduce that λ m m Y i =1 f ( J x i − 1 ,x i ) ≥ λf ( d G ( x 0 , x m ) − r ) . (5.7) The ab o ve inequalit y is in fact v alid for an y w alk x 0 , x 1 , . . . , x m with m ≥ 1 b ecause rep eating a v ertex in the walk mak es the left hand side of (5.7) larger. No w consider z ∈ V \ { o } with a walk x 0 , x 1 , . . . , x m from o to z in ( V , J ) . Applying (5.7) to reduce x 0 , x 1 , . . . , x m to a one-step w alk o, z yields | ξ z | ≤ M ξ f ( d G ( o, z ) − r ) ≤ M ξ λ m m Y i =1 f ( J x i − 1 ,x i ) , so if A ≥ M ξ C , then A satisﬁes the requirements for ˜ A ( o, V , λ ) for any z ∈ V \ { o } . The case z = o can also b e included by increasing the v alue of A further if necessary . 24 5.3 Results for inﬁnite-volume measures In this section, we show ho w our results can b e applied to measures deﬁned on R V . Recall that G = ( V , E ) is an inﬁnite connected graph where every v ertex has ﬁnite degree and with a ﬁxed origin o ∈ V . F or y ∈ V , w e let B k ( y ) = { x ∈ V : d G ( x, y ) ≤ k } . Throughout this section, w e assume that J , f , β and ρ are ﬁxed with ρ satisfying (1.2) and that V is J -connected. W e will also drop J from the subscripts. If ρ satisﬁes the stronger assumption (1.3) for some a > 0 , n > 2 , then sligh tly stronger versions of some statemen ts in this section can b e obtained b y using the mac hinery of Theorem 1.1 instead of Theorem 4.1. Recall from Deﬁnition 1.4 the deﬁnitions of a -regular measures and Gibbs measures. As a corollary of Theorem 4.1, we obtain regularity when ν is a limit of ﬁnite-volume measures with b oundary conditions gro wing slo wly enough that A ( x, Λ) is b ounded b y a constan t for all x suﬃciently far from the b oundary of Λ . Corollary 5.7. L et (Λ i ) i ≥ 1 b e a se quenc e of ﬁnite subsets of V such that Λ i ↗ V as i → ∞ and assume that the b oundary c onditions ξ satisfy for some λ ≥ 1 ∃ A max ∈ [ 1 , ∞ ) such that lim sup Λ ↗ V A ( x, Λ , λ, ξ ) ≤ A max ∀ x ∈ V . (5.8) If ν ξ Λ i ,β ,ρ c onver ges we akly to a pr ob ability me asur e ν as i → ∞ , then ν is an a − r e gular Gibbs me asur e for any a ≥ 2 β M f λ . Pr o of. Fix Λ ′ ⋐ V and a ≥ 4 β M f λ . By Theorem 4.1, we ha ve for an y i large enough that Λ ′ ⊂ Λ i , d ν ξ Λ i ,β ,ρ [ φ | Λ ′ = ψ ] ≤   Y x ∈ Λ ′ exp( ˜ C A ( x, Λ i , λ, ξ ) 2 )   d ν 0 Λ ′ , 0 ,ρ a 2 [ ψ ] . T aking i → ∞ and using (5.8), w e hav e that ν is a 2 − regular with B = ˜ C A 2 max . The fact that ν is a Gibbs measure follows from the domain Marko v prop erty for the measures ν ξ Λ i ,β ,ρ . Let us consider some examples where (5.8) is satisﬁed. Let x ∈ V and assume that Λ is large enough that d G ( x, z ) ≥ 1 2 d G ( o, z ) for any z ∈ V \ Λ . F or nearest-neighbour in teractions, supp ose ξ ∈ Ξ( λ ) . Then b y (5.3), there exists C ξ ∈ (0 , ∞ ) such that for any z ∈ V \ Λ | ξ z | ≤ C ξ λ d G ( o,z ) ≤ C ξ λ 2 d G ( x,z ) , whic h implies that for any y ∈ ∂ Λ , | h y , Λ | C | N y ,V \ Λ | λ 2( d Λ ( x,y )+1) ≤ max z ∈ N y,V \ Λ  | ξ z | C λ 2 d G ( x,z )  ≤ C ξ C . It follo ws from (5.2) that A ( x, Λ , λ 2 , ξ ) ≤ max { 1 , C ξ /C } , and since this is true for an y Λ large enough, (5.8) holds with λ 2 in place of λ . F or reasonable in teractions, (5.8) holds with λ ≥ 1 c for an y b oundary conditions satis- fying the assumptions of Prop osition 5.6. Indeed, the choice of Λ and the assumptions on f in Deﬁnition 5.5 imply that for any z ∈ V \ Λ , f ( d G ( x, z ) − r ) ≥ f (2 r d G ( o, z ) − r ) ≥ cf ( d G ( o, z ) − r ) . 25 Hence by the assumptions on ξ , w e hav e for any w alk x 0 , . . . , x m from x to z | ξ z | ≤ M ξ f ( d G ( o, z ) − r ) ≤ M ξ c f ( d G ( x, z ) − r ) ≤ M ξ λ m m Y i =1 f ( J x i − 1 ,x i ) , (5.9) where w e ha ve used (5.7) and the fact that f ( J x i − 1 ,x i ) ≥ f ( d G ( x i − 1 , x i ) − r ) in the last inequalit y . F rom (5.9) we see that A ( x, Λ , λ, ξ ) ≤ M ξ C . W e now show ho w we can use regularity to make sense of “maximal” b oundary condi- tions, whic h will allow us to construct the inﬁnite-v olume plus measure. Deﬁne ξ + ∈ R V b y ξ + x = q log( | B d G ( o,x ) ( o ) | ) and let ξ − = − ξ + . Belo w r and c are as in Deﬁnition 5.5. Prop osition 5.8. A ssume the inter actions J ar e r e asonable and ferr omagnetic. Supp ose also that ther e exists a c onstant c 0 > 0 such that for any j, k ≥ 1 , f ( k − r ) ≥ c 0 s log( | B k + j ( o ) | ) log( | B j ( o ) | ) . (5.10) Then ther e exist 2 β M f c − r e gular Gibbs me asur es ν + β ,ρ , ν − β ,ρ such that lim Λ ↗ V ν ξ + Λ ,β ,ρ = ν + β ,ρ , lim Λ ↗ V ν ξ − Λ ,β ,ρ = ν − β ,ρ . Mor e over, for any a > 0 , any a − r e gular Gibbs me asur e ν satisﬁes ν − β ,ρ ⪯ ν ⪯ ν + β ,ρ . Prop osition 5.8 includes the case of nearest-neighbour interactions, as f can b e c ho- sen arbitrarily in this case. Also note that if G is vertex-transitiv e then | B k + j ( o ) | ≤ | B k ( o ) || B j ( o ) | and the condition on f simpliﬁes to f ( k − r ) ≥ c 0 p log( | B k ( o ) | ) . F or the φ 4 mo del, as a corollary of the Lee–Y ang theorem, ν + β ,ρ coincides with the measure de- ﬁned with an external ﬁeld h b y ﬁrst taking Λ ↗ V and then taking the limit as h ↘ 0 (see [7, Prop. 2.6]). See also [2, 10] for constructions of the plus measure when V = Z d . The main ingredien t in the pro of of Prop osition 5.8 is the following lemma, whic h allo ws us to obtain monotonicity in Λ for the measures ν ξ + Λ ,β ,ρ up to an error term which tends to 0 as Λ ↗ V . Lemma 5.9. Consider the event F Λ , Λ ′ = {| φ x | ≤ ξ + x ∀ x ∈ Λ \ Λ ′ } . If J and f satisfy the assumptions of Pr op osition 5.8, then ν ξ + Λ ,β ,ρ [ F c Λ , Λ ′ ] → 0 uniformly in Λ ⊃ Λ ′ as Λ ′ ↗ V . Pr o of. W e will drop β , ρ from the notation and just write ν ξ Λ for the ﬁnite-v olume measure on Λ with b oundary conditions ξ . Let λ = 1 c , where c is as in Deﬁnition 5.5. Applying Theorem 4.1 with a = 4 β M f λ , together with a union b ound, yields ν ξ + Λ [ F c Λ , Λ ′ ] ≤ X x ∈ Λ \ Λ ′ ν ξ + Λ [ | φ x | > ξ + x ] ≤ 1 ρ a 2 ( R ) X x ∈ Λ \ Λ ′ exp( ˜ C A ( x, Λ , λ ) 2 ) ρ a 2 [ | φ x | > ξ + x ] . Let a ′ > 0 b e a constant to b e determined. Applying Mark o v’s inequality to the random v ariable e a ′ φ 2 x , we hav e ν ξ + Λ [ F c Λ , Λ ′ ] ≤ ρ a 2 [ e a ′ φ 2 x ] ρ a 2 ( R ) X x ∈ Λ \ Λ ′ exp( ˜ C A ( x, Λ , λ ) 2 ) | B d G ( o,x ) ( o ) | − a ′ . (5.11) W e will show that there exists a constan t C ′ ≥ 1 suc h that for any ﬁnite Λ ⊂ V containing o and any x ∈ Λ \ { o } , A ( x, Λ , λ ) ≤ q C ′ log( | B d G ( o,x ) ( o ) | ) . (5.12) 26 Com bining (5.11) and (5.12) giv es for an y Λ ′ con taining o ν ξ + Λ [ F c Λ , Λ ′ ] ≤ ρ a 2 [ e a ′ φ 2 x ] ρ a 2 ( R ) X x ∈ Λ \ Λ ′ | B d G ( o,x ) ( o ) | ˜ C C ′ − a ′ ≤ ρ a 2 [ e a ′ φ 2 x ] ρ a 2 ( R ) ∞ X i = g (Λ ′ ) | B i ( o ) | 1+ ˜ C C ′ − a ′ , where g (Λ ′ ) = min x ∈ V \ Λ ′ d G ( o, x ) . By c ho osing a ′ appropriately , the last sum conv erges and decreases to 0 as Λ ′ ↗ V . It remains to pro ve (5.12). Let C ′ ≥ 1 b e a constan t to be determined and set A x = q C ′ log( | B d G ( o,x ) ( o ) | ) . W e aim to show that C ′ can b e c hosen so that for an y Λ ⋐ V , x ∈ Λ \ { o } and an y w alk x 0 , . . . , x m from x to a v ertex z ∈ V \ Λ , C A x λ m m Y i =1 f ( J x i − 1 ,x i ) ≥ | ξ + z | , (5.13) whic h implies that A ( x, Λ , λ ) ≤ A x . As in the pro of of Prop osition 5.6, w e can reduce the w alk x 0 , . . . , x m to a one-step w alk x 0 , x m b y using that f ( J x i − 1 ,x i ) ≥ f ( d G ( x i − 1 , x i ) − r ) and applying the inequalit y (5.7). This giv es C A x λ m m Y i =1 f ( J x i − 1 ,x i ) ≥ C √ C ′ c q log( | B d G ( o,x ) ( o ) | ) f ( d G ( x, z ) − r ) , (5.14) and (5.10) implies that 1 c 0 q log( | B d G ( o,x ) ( o ) | ) f ( d G ( x, z ) − r ) ≥ q log( | B d G ( o,x )+ d G ( x,z ) ( o ) | ) ≥ | ξ + z | . (5.15) Com bining (5.14) and (5.15), we see that it is p ossible to choose C ′ so that (5.13) is satisﬁed, completing the proof of (5.12) and of the lemma. W e now pro ceed with the pro of of Prop osition 5.8. Pr o of of Pr op osition 5.8. W e only prov e the statements for ν + β ,ρ . As w e hav e ﬁxed β , ρ we will drop them from the notation and just write ν ξ Λ for the ﬁnite-volume measure on Λ with b oundary conditions ξ . Let H b e an increasing even t that dep ends only on spins inside a ﬁnite subset Λ ′ ⊂ V and let (Λ i ) i ≥ 1 b e a sequence of ﬁnite subsets of V with Λ i ↗ V as i → ∞ . F or k ≥ i , write F k,i for the ev ent F Λ k , Λ i deﬁned in Lemma 5.9. Using the domain Mark ov prop erty and the fact that if φ ∈ F k,i , then Prop osition 2.3 implies that ν φ Λ i [ H ] ≤ ν ξ + Λ i [ H ] , w e hav e for any k ≥ i such that Λ ′ ⊂ Λ i , ν ξ + Λ k [ H ] ≤ ν ξ + Λ i [ H ] + ν ξ + Λ k [ F c k,i ] . Sending ﬁrst k to inﬁnit y , and then i to inﬁnit y , and using Lemma 5.9, we get that lim sup k →∞ ν ξ + Λ k [ H ] ≤ lim inf i →∞ ν ξ + Λ i [ H ] . Hence lim Λ ↗ V ν ξ + Λ [ H ] exists for any increasing ev ent H dep ending only on ﬁnitely many spins. As these even ts generate the σ − algebra, w e obtain con vergence of ν ξ + Λ to a measure ν + as Λ ↗ V . The assumption (5.10) implies that the b oundary conditions ξ + satisfy the assumptions of Prop osition 5.6. Hence, (5.8) is satisﬁed with λ = 1 c , and we can apply Corollary 5.7 to deduce that ν + is an a -regular Gibbs measure for any a ≥ 2 β M f λ . No w consider any a > 0 and supp ose ν is an a − regular Gibbs measure. Then for an y ﬁnite Λ ⊂ V , the DLR equation gives that ν [ H ] = ν [ F Λ ] Z φ ∈ R V ν φ Λ [ H ]d ν ( φ | F Λ ) + ν [ F c Λ ] Z φ ∈ R V ν φ Λ [ H ]d ν ( φ | F c Λ ) , (5.16) 27 where F Λ = { φ : φ x ≤ ξ + x ∀ x ∈ V \ Λ } . W e can show that ν [ F c Λ ] → 0 as Λ ↗ V in the same w ay as in the pro of of Lemma 5.9: ﬁrst using a union b ound, regularit y , and Marko v’s inequalit y , w e hav e for any a ′ > 0 ν [ F c Λ ] ≤ X x ∈ V \ Λ ν [ | φ x | ≥ ξ + x ] ≤ B ρ a [ e a ′ φ 2 x ] ρ a ( R ) X x ∈ V \ Λ | B d G ( o,x ) ( o ) | − a ′ . It then follows that ν [ F c Λ ] → 0 as Λ ↗ V , provided that a ′ is chosen large enough. Hence taking Λ ↗ V in (5.16) and using Proposition 2.3, w e ha ve ν [ H ] ≤ ν + [ H ] . 5.4 An alternativ e construction of the plus measure One p otential disadv antage of the construction in Prop osition 5.8 is that it relies on gro wing b oundary conditions, so the ﬁnite-volume measures are not regular up to the b oundary . In this section, we provide a new wa y of constructing the inﬁnite-volume plus measure without growing b oundary conditions in the case of nearest-neighbour in terac- tions. W e start b y stating the following corollary of Theorem 4.1, which applies for any in teractions J satisfying (C1) and (C2) and is of a similar nature to [8, Prop. 2.6]. Given a > 0 and B x > 0 , we deﬁne ζ x = ζ x,a,B x to b e a single-site measure that dep ends on the v ertex x (see Remark 4.2), deﬁned b y d ζ x ( u ) = 1 { u ≥ B x } d ρ a ( u − B x ) . Belo w, we let B x =  2 a  ˜ C A ( x, Λ , C ) 2 + log ( ρ a ( R )) − log( ρ a 2 ( R ))   1 2 , with C , ˜ C the constants from Theorem 4.1. Corollary 5.10. L et a ≥ 4 β M f , λ ≤ a 4 β M f , and assume that ρ ([0 , ∞ )) > 0 . F or any ﬁnite Λ ⊂ V and Λ ′ ⊂ Λ , and any b oundary c onditions ξ such that P y ∈ V | J x,y ξ y | < ∞ for al l x ∈ Λ , ν ξ (Λ | Λ ′ ) ,β ,ρ is sto chastic al ly dominate d by ν 0 Λ ′ , 0 ,ζ . Pr o of. W e aim to sho w that for an y u ∈ R Λ ′ , ν ξ Λ ,β ,ρ [ φ x ≥ u x , ∀ x ∈ Λ ′ ] ≤ ν 0 Λ ′ , 0 ,ρ a [ φ x + B x ≥ u x , ∀ x ∈ Λ ′ | φ x ≥ 0 , ∀ x ∈ Λ ′ ] . The probabilit y on the right hand side abov e is equal to ν 0 Λ , 0 ,ζ [ φ x ≥ u x , ∀ x ∈ Λ ′ ] , so this implies the desired stochastic domination. Applying Theorem 4.1 with Λ u := { x ∈ Λ ′ : u x ≥ B x } in place of Λ ′ , we obtain ν ξ Λ ,β ,ρ [ φ x ≥ u x , ∀ x ∈ Λ ′ ] ≤ ν ξ Λ ,β ,ρ [ φ x ≥ u x , ∀ x ∈ Λ u ] ≤   Y x ∈ Λ u exp( ˜ C A ( x, Λ) 2 )   ν 0 Λ u , 0 ,ρ a 2 [ φ x ≥ u x , ∀ x ∈ Λ u ] ≤ Y x ∈ Λ u 1 ρ a 2 ( R ) Z ∞ u x exp  ˜ C A ( x, Λ) 2 − a 2 | φ x | 2  d ρ a ( φ x ) . The choice of B x giv es that exp  ˜ C A ( x, Λ) 2 − a 2 | φ x | 2  ≤ ρ a 2 ( R ) ρ a ( R ) whenev er φ x ≥ B x , so since u x ≥ B x for x ∈ Λ u , ν ξ Λ ,β ,ρ [ φ x ≥ u x , ∀ x ∈ Λ ′ ] ≤ Y x ∈ Λ u 1 ρ a ( R ) Z ∞ u x d ρ a ( φ x ) ≤ ν 0 Λ ′ , 0 ,ρ a [ φ x + B x ≥ u x , ∀ x ∈ Λ ′ | φ x ≥ 0 , ∀ x ∈ Λ ′ ] . 28 W e no w sho w ho w the abov e corollary allows us to construct the plus-measure without the need for growing b oundary conditions. W e give t wo constructions, one with random b oundary conditions, the other by making the single-site measure dep end on the vertex. W e will work on a graph G of degree b ounded by some constan t D > 0 and consider nearest-neigh b our interactions on G . Denote by ∂ Λ the set of vertices in Λ that are adjacen t to V \ Λ . Let a = 8 β D and write B =  2 a  2 ˜ C + log( ρ a ( R )) − log( ρ a 2 ( R ))  1 2 , where ˜ C is the constan t given by Theorem 4.1. Recall the deﬁnition of the measure ζ = ζ a,B . Let us introduce the measure ˜ ν 0 Λ ,β ,ρ deﬁned on any b ounded measurable function g : R Λ \ ∂ Λ → R as ˜ ν 0 Λ ,β ,ρ [ g ] = Z ξ ∈ R ∂ Λ ⟨ g ⟩ ξ Λ \ ∂ Λ ,β ,ρ d ν 0 ∂ Λ , 0 ,ζ ( ξ ) . In w ords, ˜ ν 0 Λ ,β ,ρ is a measure with random b oundary conditions sampled from the pro duct measure ν 0 ∂ Λ , 0 ,ζ . In the next proposition, w e show that ˜ ν 0 Λ ,β ,ρ satisﬁes a form of mono- tonicit y in the volume, which we b eliev e to b e of independent in terest. Prop osition 5.11. L et G = ( V , E ) b e a gr aph of b ounde d de gr e e. F or every β > 0 ther e exists r > 0 such that the fol lowing holds. Consider Λ ′ ⊂ Λ ⋐ V such that d G ( ∂ Λ , Λ ′ ) > r . Then ˜ ν 0 Λ ′ ,β ,ρ sto chastic al ly dominates ˜ ν 0 (Λ | Λ ′ ) ,β ,ρ . Pr o of. Let ξ ∼ ν 0 ∂ Λ , 0 ,ζ , write R = Λ \ ∂ Λ and consider a set S ⊂ ∂ Λ ′ . By Theorem 4.1, d ν ξ R,β ,ρ [ φ | S = ψ ] ≤ Y x ∈ S e ˜ C A ( x,R,ξ ,C ) 2 d ν 0 S, 0 ,ρ a 2 [ ψ ] . It suﬃces to pro ve that ν 0 ∂ Λ , 0 ,ζ " Y x ∈ S e ˜ C A ( x,R,ξ ,C ) 2 # ≤ e 2 ˜ C | S | . (5.17) Indeed, (5.17) implies that ˜ ν 0 (Λ | S ) ,β ,ρ is a (ﬁnite-volume) a/ 2 -regular measure with con- stan t B = 2 ˜ C , and since S is an arbitrary subset of ∂ Λ ′ , we can apply the argumen t of Corollary 5.10 to deduce that ν 0 ∂ Λ ′ , 0 ,ζ = ˜ ν 0 (Λ ′ | ∂ Λ ′ ) ,β ,ρ sto c hastically dominates ˜ ν 0 (Λ | ∂ Λ ′ ) ,β ,ρ . Then the desired sto c hastic domination in the whole of Λ ′ follo ws from monotonicity in b oundary conditions. Let us now pro ve (5.17). Note that A ( x, R, ξ , C ) 2 ≤ max ( 1 , max z ∈ ∂ Λ | ξ z | 2 C 2 λ 2 d Λ ( x,z ) !) ≤ 1 + X z ∈ ∂ Λ | ξ z | 2 C 2 λ 2 d Λ ( x,z ) . Hence X x ∈ S A ( x, R, ξ , C ) 2 ≤ | S | + X z ∈ ∂ Λ | ξ z | 2 C 2 X k ≥ d Λ ( z ,S ) | B k ( z ) | λ 2 k ≤ | S | + X z ∈ ∂ Λ | ξ z | 2 λ d Λ ( z ,S ) , pro vided λ ≥ 2 D , where D is the maxim um degree, and d G ( ∂ Λ , Λ ′ ) is large enough. Th us, ν 0 ∂ Λ , 0 ,ζ   Y x ∈ ∂ Λ ′ e ˜ C A ( x, Λ ,ξ ,C ) 2   ≤ e ˜ C | S | exp   C ′ X z ∈ ∂ Λ λ − d Λ ( z ,S )   29 for some constan t C ′ > 0 , where here we used indep endence and the inequalit y E ( e tX ) ≤ 1 + t E ( e X ) ≤ e t E ( e X ) for any t ∈ (0 , 1) and any random v ariable X ≥ 0 . By decomp osing according to the points in S that attain d Λ ( z , S ) w e get that X z ∈ ∂ Λ λ − d Λ ( z ,S ) ≤ X x ∈ S X k ≥ d G ( ∂ Λ , Λ ′ ) | B k ( x ) | λ − k ≤ ˜ C | S | /C ′ , pro vided that d G ( ∂ Λ , Λ ′ ) is large enough. This implies (5.17) and concludes the pro of. Prop osition 5.12. L et G = ( V , E ) b e a c onne cte d gr aph of de gr e e b ounde d by D > 0 . Consider ne ar est-neighb our inter actions on G and let ρ b e an even me asur e. F or every β ≥ 0 , ˜ ν 0 Λ ,β ,ρ → ν + β ,ρ as Λ ↗ V . F urthermor e, for every β ≥ 0 ther e exist a ′ , B ′ > 0 that dep end c ontinuously on β such that for every Λ ⋐ V , ˜ ν 0 Λ ,β ,ρ is sto chastic al ly dominate d by ν 0 Λ , 0 ,ζ a ′ ,B ′ . F or our second construction, we deﬁne a single-site measure ˜ ρ x, Λ b y d ˜ ρ x, Λ ( u ) = ( d ζ a,B ( u ) if x ∈ ∂ Λ , d ρ ( u ) otherwise. Prop osition 5.13. L et G = ( V , E ) b e a c onne cte d gr aph of de gr e e b ounde d by D > 0 . Consider ne ar est-neighb our inter actions on G and let ρ b e an even me asur e. F or every β ≥ 0 , ν 0 Λ ,β , ˜ ρ → ν + β ,ρ as Λ ↗ V . F urthermor e, for every β ≥ 0 ther e exist a ′ , B ′ > 0 that dep end c ontinuously on β such that for every Λ ⋐ V , ν 0 Λ ,β , ˜ ρ is sto chastic al ly dominate d by ν 0 Λ , 0 ,ζ a ′ ,B ′ . T o prov e Prop osition 5.13, we will need to use monotonicit y in β of the measures ν 0 (Λ | ∂ Λ) ,β , ˜ ρ , which is pro vided by the follo wing lemma. Lemma 5.14. If ρ is an even me asur e, then for any β ′ ≥ β ≥ 0 and any Λ ⋐ V , ν 0 (Λ | ∂ Λ) ,β , ˜ ρ ⪯ ν 0 (Λ | ∂ Λ) ,β ′ , ˜ ρ . Pr o of. Let A b e an increasing even t dep ending only on v ertices in ∂ Λ . Without loss of generalit y , assume that ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ A ] > 0 . Diﬀerentiating with respect to β we obtain d ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ A ] d β = X xy ∈ E ⟨ 1 A φ x φ y ⟩ 0 Λ ,β , ˜ ρ − ⟨ 1 A ⟩ 0 Λ ,β , ˜ ρ ⟨ φ x φ y ⟩ 0 Λ ,β , ˜ ρ , so it suﬃces to sho w that ⟨ φ x φ y | A ⟩ 0 Λ ,β , ˜ ρ − ⟨ φ x φ y ⟩ 0 Λ ,β , ˜ ρ ≥ 0 for all xy ∈ E . If x, y ∈ ∂ Λ then φ x , φ y ≥ 0 , so φ x φ y is an increasing function and the desired inequalit y follo ws from the FKG inequality . No w supp ose that x, y ∈ Λ \ ∂ Λ . W e use the domain Marko v property and the fact that ˜ ρ = ρ on Λ \ ∂ Λ to get ⟨ φ x φ y | A ⟩ 0 Λ ,β , ˜ ρ = Z η ∈ ( R + ) ∂ Λ ⟨ φ x φ y ⟩ η Λ \ ∂ Λ ,β ,ρ d ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ η | A ] . Since ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ · | A ] ⪰ ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ · ] by the FKG inequality , and ⟨ φ x φ y ⟩ η Λ \ ∂ Λ ,β ,ρ is an increasing function of η (whic h follows by diﬀerentiating and using Griﬃths’ inequality), the right hand side ab o v e is at least ⟨ φ x φ y ⟩ 0 Λ ,β , ˜ ρ . 30 It remains to consider the case when the edge xy has one endp oint in ∂ Λ and the other endp oin t in Λ \ ∂ Λ . If x ∈ Λ \ ∂ Λ and y ∈ ∂ Λ , then ⟨ φ x φ y | A ⟩ 0 Λ ,β , ˜ ρ = Z η ∈ ( R + ) ∂ Λ η y ⟨ φ x ⟩ η Λ \ ∂ Λ ,β ,ρ d ν 0 (Λ | ∂ Λ) ,β , ˜ ρ [ η | A ] . Prop osition 2.3 together with the fact that the b oundary conditions η are p ositiv e and ρ is an ev en measure implies that η y ⟨ φ x ⟩ η Λ \ ∂ Λ ,β ,ρ is an increasing function of η , so the right hand side is again at least ⟨ φ x φ y ⟩ 0 Λ ,β , ˜ ρ . W e now prov e conv ergence of the measures ν 0 Λ ,β , ˜ ρ and ˜ ν 0 Λ ,β ,ρ to the plus measure. Pr o of of Pr op osition 5.13. First note that the single-site measures ˜ ρ x, Λ satisfy the assump- tions of Remark 4.2 with a min = a 2 , and that the result of Theorem 4.1 still applies in this case. Therefore, there exists ˜ C 2 > 0 suc h that for all Λ ⋐ V , Λ ′ ⊂ Λ \ ∂ Λ and ψ ∈ R Λ ′ , d ν 0 Λ ,β , ˜ ρ [ φ | Λ ′ = ψ ] ≤ e ˜ C 2 | Λ ′ | d ν 0 Λ ′ , 0 ,ρ a 2 [ ψ ] . (5.18) It follows that the family of measures ( ν 0 Λ ,β , ˜ ρ ) Λ ⋐ V is tight, so there exists a sequence (Λ i ) i ≥ 1 with Λ i ↗ V such that ν 0 Λ i ,β , ˜ ρ con verges to some measure ˜ ν as i → ∞ . The measure ˜ ν is a 2 -regular b y (5.18) and is a Gibbs measure by the domain Mark ov prop erty . Hence ˜ ν ⪯ ν + β ,ρ b y Prop osition 5.8. It remains to prov e that ν + β ,ρ ⪯ ˜ ν , whic h implies that ˜ ν = ν + β ,ρ and shows that w e hav e full conv ergence to ν + β ,ρ . Setting λ = 2 and using (5.2), we see that for eac h i ≥ 1 we can ﬁnd k ( i ) ≥ i suc h that A ( x, Λ k ( i ) , ξ + , C ) = 1 for any x ∈ Λ i . No w using Corollary 5.10 and Lemma 5.14, ν ξ + (Λ k ( i ) | ∂ Λ i ) ,β ,ρ ⪯ ν 0 ∂ Λ i , 0 , ˜ ρ ⪯ ν 0 (Λ i | ∂ Λ i ) ,β , ˜ ρ . Let Λ ′ ⋐ V and consider an increasing even t H that dep ends only on spins inside Λ ′ . The domain Marko v prop ert y and Prop osition 2.3 together with the ab o ve sto c hastic domination imply that whenev er i is large enough that Λ ′ ⊂ Λ i \ ∂ Λ i , ν ξ + Λ k ( i ) ,β ,ρ [ H ] = Z R ∂ Λ i ν η Λ i \ ∂ Λ i ,β ,ρ [ H ]d ν ξ + (Λ k ( i ) | ∂ Λ i ) ,β ,ρ [ η ] ≤ Z R ∂ Λ i ν η Λ i \ ∂ Λ i ,β ,ρ [ H ]d ν 0 (Λ i | ∂ Λ i ) ,β , ˜ ρ [ η ] = ν 0 Λ i ,β , ˜ ρ [ H ] . T aking i → ∞ , w e hav e ν + β ,ρ [ H ] ≤ ˜ ν [ H ] . The desired sto chastic domination follows from Corollary 5.10. Pr o of of Pr op osition 5.12. First observe that the monotonicity in volume of Prop osition 5.11 implies conv ergence of the measures ˜ ν 0 Λ ,β ,ρ to an inﬁnite-v olume limit ˜ ν as Λ ↗ V . F rom Lemma 5.14 and the domain Marko v property , w e ha ve that ˜ ν 0 Λ ,β ,ρ ⪯ ν 0 Λ ,β , ˜ ρ for any Λ ⋐ V , so ˜ ν ⪯ ν + β ,ρ b ecause ν 0 Λ ,β , ˜ ρ → ν + β ,ρ b y Proposition 5.13. The pro of that ν + β ,ρ ⪯ ˜ ν is sim- ilar to that of Prop osition 5.13, and in fact, ev en simpler, as one do es not need to use Lemma 5.14. References [1] S. Alb ev erio, Y. G. K ondratiev, M. Rö c kner, and T. V. T sikalenko. A priori esti- mates for symmetrizing measures and their applications to Gibbs states. Journal of F unctional A nalysis , 171(2):366–400, 2000. 31 [2] J. Bellissard and R. Høegh-Krohn. Compactness and the maximal Gibbs state for ran- dom Gibbs ﬁelds on a lattice. Communic ations in Mathematic al Physics , 84(3):297– 327, 1982. [3] M. Cassandro, E. Olivieri, A. Pellegrinotti, and E. Presutti. Existence and uniqueness of DLR measures for unbounded spin systems. Zeitschrift für W ahrscheinlichkeits- the orie und verwandte Gebiete , 41(4):313–334, 1978. [4] H. Duminil-Copin. Lectures on the Ising and Potts mo dels on the h yp ercubic lattice. In PIMS-CRM Summer Scho ol in Pr ob ability , pages 35–161. Springer, 2017. A v ailable at [5] S. F riedli and Y. V elenik. Statistic al Me chanics of L attic e Systems: A Concr ete Mathematic al Intr o duction . Cambridge Universit y Press, 2017. [6] J. Glimm and A. Jaﬀe. Quantum Physics : A F unctional Inte gr al Point of V iew . Springer New Y ork, 2nd edition, 1987. [7] T. S. Gunaratnam, C. P anagiotis, R. P anis, and F. Severo. Random tangled curren ts for φ 4 : T ranslation in v ariant Gibbs measures and con tin uity of the phase transi- tion. Journal of the Eur op e an Mathematic al So ciety (to app e ar) , 2025. A v ailable at h [8] T. S. Gunaratnam, C. P anagiotis, R. P anis, and F. Severo. The sup ercrit- ical phase of the φ 4 mo del is w ell b eha v ed. Pr eprint , 2025. A v ailable at h [9] R. v an der Hofstad. R andom gr aphs and c omplex networks. V olume 1 , v olume 43 of Cambridge series in statistic al and pr ob abilistic mathematics . Cambridge Universit y Press, 2017. [10] J. L. Leb owitz. and E. Presutti. Statistical mec hanics of systems of unbounded spins. Communic ations in Mathematic al Physics , 50(3):195–218, 1976. [11] A. Raouﬁ. T ranslation-inv ariant Gibbs states of the Ising mo del: General setting. The A nnals of Pr ob ability , 48(2):760–777, 2020. [12] D. Ruelle. Sup erstable interactions in classical statistical mechanics. Communic ations in Mathematic al Physics , 18(2):127–159, 1970. [13] D. Ruelle. Probabilit y estimates for contin uous sp in systems. Communic ations in Mathematic al Physics , 50(3):189–194, 1976. [14] W. W erner and E. P ow ell. L e ctur e Notes on the Gaussian F r e e Field , v olume 28 of Cours Spécialisés . Société Mathématique de F rance, P aris, 2021. 32

Regularity of Gibbs measures for unbounded spin systems on general graphs

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