Phase transition for the non-symmetric Continuum Potts model
PPhase transition for the non-symmetric ContinuumPotts model
Pierre Houdebert
Department of Mathematics, University of Potsdam [email protected]
Abstract
We prove a phase transition for the non-symmetric continuum Pottsmodel with background interaction, by generalizing the methods intro-duced in the symmetric case by Georgii and H¨aggstr¨om [13]. The proofrelies on a Fortuin-Kasteleyn representation, percolation and stochasticdomination arguments.
Key words: Gibbs point process, continuum Potts model, DLR equations,continuum percolation, generalized continuum random cluster model, Fortuin-Kasteleyn representation, stochastic domination.AMS MSC 2010:
In Gibbs point processes theory one of the main question of interest is the studyof phase transition. Indeed Gibbs point processes are defined through a family ofequations, the Dobrushin-Lanford-Ruelle equations, and it is a natural questionto ask whether there exists only one or several solutions to these equations.Although phase transition is conjectured for most continuum models, it hasbeen rigorously proved only in a few cases. The first such result was obtainby Ruelle [23] for the symmetric Widom-Rowlinson model, which is a two typeparticles system with an hard-core repulsion between particles of different types,using a continuum version of the Peierls argument. This technique was lattergeneralized to the soft-core Widom-Rowlinson interaction in [20].In the 1990’s Chayes, Chayes & Koteck´y [3] and Georgii & H¨aggstr¨om [13]generalized for continuum models the idea of the Fortuin-Kasteleyn represen-tation [10] introduced for the lattice Ising and Potts models, and proved phasetransition results respectively for the symmetric Widom-Rowlinson model andfor the continuum Potts model with background interaction. This idea wasthen used in a variety of articles, for instance to prove phase transition for thesymmetric Widom-Rowlinson model with unbounded radii [9, 18]. The idea of1 a r X i v : . [ m a t h . P R ] J u l he Fortuin-Kasteleyn representation is generalized to the non-symmetric casein the present article.For the non-symmetric case where each type of particles have different inten-sities, even fewer results are proved. A few results are proved for the Widom-Rowlinson model using the Pirogov-Sinai technique, see for instance [2, 21].Recently a sharp phase transition result for the Widom Rowlinson model wasobtained in [8], giving an almost complete picture of the phase diagram.In this article we are interested in the Continuum Potts model with back-ground interaction, as introduced by Georgii and H¨aggstr¨om [13]. We provethat for any initial proportion of particles ˜ α = ( α , . . . , α q ) and for the activityparameter z large enough, there is at least as many distinct Potts measuresas there are α i , i = 1 . . . q which are maximal in ˜ α . This result and its proofis a generalization of the proof of the symmetric case done by Georgii andH¨aggstr¨om [13].The proof relies on a Fortuin-Kasteleyn representation which expresses thecolouring correlation as the connectivity in the so-called generalized ContinuumRandom Cluster model. This is done using stochastic domination tools. There-fore by proving a percolation-type bound for this process, one can construct dif-ferent Potts measures obtained by having different boundary conditions. Suchan idea was used in [1] for the lattice nearest neighbour Potts model, and weare generalizing it for the continuum setting, to obtain a phase transition resultwith the exact same assumptions as in [13].The article is organized as follow: in Section 2 we introduced the modeland the tools needed later on. In Section 3 we give the assumptions and statethe theorems. In Section 4 is introduced the Fortuin-Kasteleyen representationand we state and prove the percolation bound for the generalized ContinuumRandom Cluster model. In Section 5 we prove the main theorems, and finallyin the appendix Section 6 we give the proofs of classical and technical lemmas. Through the paper the dimension d ≥ q ≥ Λ ) oflocally finite configurations ω in R d (respectively Λ). We will often considerconfigurations marked by a colour. For those we are using the notation σ : (cid:40) ω −→ { , . . . , q } x (cid:55)−→ σ x and we write respectively (cid:101) Ω = { (cid:101) ω = ( ω, σ ) } The configurations spaces Ω and (cid:101)
Ω are embedded with the usual sigma-algebras generated by the counting vari-ables.For Λ ⊆ R d , we write ω Λ as a shorthand for ω ∩ Λ. This notation naturallyextends to (cid:101) ω Λ . We write N Λ ( ω ) (respectively N Λ ( (cid:101) ω )) for the cardinality of therespected configuration inside Λ. We write | Λ | for the Lebesgue measure of2 ⊆ R d , and | j | for the sup norm of j ∈ Z d . We write ω (cid:48) ω (respectively (cid:101) ω (cid:48) (cid:101) ω )has a shorthand for ω (cid:48) ∪ ω (respectively (cid:101) ω (cid:48) ∪ (cid:101) ω ).Let τ x be the translation of vector x ∈ R d . This means that τ x ( ω ) = { y − x, y ∈ ω } . We denote by (cid:101) P θ (respectively (cid:101) P θ δ ) the set of probabilitymeasures on (cid:101) Ω which are invariant under all translations of R d (respectively alltranslation in δ Z d ). For δ >
0, we write ∆ j = j ⊕ ] − δ/ , δ/ d with j ∈ δ Z d . Let π z be the distribution on Ω of the homogeneous Poisson point process withintensity z >
0. Recall that it means • for every bounded Borel set Λ, the distribution of the number of pointsin Λ under π z is a Poisson distribution of mean z | Λ | ; • given the number of points in a bounded Λ, the points are independentand uniformly distributed in Λ.We refer to [4] for details on Poisson point processes. We write (cid:101) π z, ˜ α for thedistribution on (cid:101) Ω of the Poisson point process of intensity z with independentcolour marks distributed according to a probability measure ˜ α = ( α , . . . , α q )on { , . . . , q } . We have (cid:101) π z, ˜ α ∈ (cid:101) P θ . We are assuming, without loss of generality,that ˜ α has non-zero marginals, i.e. α i > i .For Λ ⊆ R d , we denote by π z Λ (respectively (cid:101) π z, ˜ α Λ ) the restriction of π z (re-spectively (cid:101) π z, ˜ α ) on Λ. For Λ ⊆ R d bounded, we define the Λ-Hamiltonian H Λ such that, for (cid:101) ω ∈ (cid:101) Ω, H Λ ( (cid:101) ω ) := (cid:88) { x,y }⊆ ωσ x (cid:54) = σ y { x,y }∩ Λ (cid:54) = ∅ φ ( x − y ) + (cid:88) { x,y }⊆ ω { x,y }∩ Λ (cid:54) = ∅ ψ ( x − y ) := H φ Λ ( (cid:101) ω ) + H ψ Λ ( ω ) , where φ, ψ : R d → ] −∞ , + ∞ ] are even measurable functions. The first potential φ describes a repulsion between points of different colours. The second ψ is atype-independent pair potential. The most classical Potts model is the Widom-Rowlinson model [24], for which ψ = 0 and φ ( x ) = + ∞ | x | small . Definition 2.1.
For a boundary condition (cid:101) ω , we define the Potts specificationon a bounded Λ ⊆ R d as Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) = exp( − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )) Z z, ˜ α Λ ( (cid:101) ω ) (cid:101) π z, ˜ α Λ ( dω (cid:48) Λ ) , where Z z, ˜ α Λ ( (cid:101) ω ) = (cid:82) (cid:101) Ω exp( − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )) (cid:101) π z, ˜ α Λ ( dω (cid:48) Λ ) is the partition function. At this point nothing ensures the well-definedness of the Potts specifica-tion. Conditions ensuring the well-definedness of the Potts specification will beintroduced later. 3 efinition 2.2.
A probability measure P on (cid:101) Ω is a Potts measure of potentials φ, ψ , of activity z and of colour proportion ˜ α , written P ∈ G potts ( z, ˜ α ) , if forevery bounded Λ ⊆ R d and every bounded measurable function f , we have Remark 2.1. In [13], they define the Potts measures on the set of temperedconfigurations . In our proof the measure built will be supported on the temperedconfigurations. However it is not necessary to impose Potts measures to be sup-ported on the set of tempered configurations. The existence of a Potts measurewhich is not supported on the set of tempered configurations remains an openproblem. In the theory of infinite volume Gibbs probability measures, the Gibbs measuresare defined through a family of equations, the DLR equations (2.1). Withsuch definition the questions of existence and uniqueness/non-uniqueness of thedefined objects are natural and interesting questions studied by the statisticalmechanics community for a variety of interactions. In the following we arestating an existence result and a phase transition (meaning the non-uniquenessof the Potts measures) result.We are considering the following assumptions on φ, ψ : there exist u > ≤ r ≤ r < r ≤ r < ∞ such that(A1) (strict repulsion of φ ) φ ≥ φ ( x ) ≥ u when | x | ≤ r ;(A2) (finite range of φ ) φ ( x ) = 0 when | x | ≥ r ;(A3) (strong stability and regularity of ψ ) either ψ ≥ 0, or ψ is superstable andlower regular in the sense of Ruelle, meaning that – (superstability) there exist constants a, b > ω , H ψ ( ω ) := H ψ R d ( ω ) ≥ (cid:88) j ∈ Z d (cid:0) aN ∆ j ( ω ) − bN ∆ j ( ω ) (cid:1) ;4 (lower regularity) there exist positive numbers ψ n , n ∈ N , such that (cid:80) n ∈ N n d − ψ n < ∞ and such that for every configuration ω , (cid:88) x ∈ ω ∆ k (cid:88) y ∈ ω ∆ j ψ ( x − y ) ≥ − ψ δ − | j − k | N ∆ k ( ω ) N ∆ j ( ω );(A4) (short range of repulsion for ψ ) ψ ( x ) ≤ | x | > r , and the positivepart ψ + of ψ satisfies (cid:90) | x |≥ r ψ + ( x ) dx < ∞ ;(A5) (scale relations) r < r / √ d + 3, and r is sufficiently small (but inde-pendent of ˜ α , see (4.5)).A classical model satisfying these assumptions is the Widom-Rowlinson model [24], for which Ψ = 0 and Φ( x ) = + ∞ | x |≤ .Those assumptions are exactly the same as the one considered by Georgiiand H¨aggstr¨om in [13]. In their paper they are considering the symmetric case(i.e. α i = 1 /q for all i ) and are proving a phase transition result. Our resultgeneralized their approach to prove phase transition for the non-symmetric case.Our first theorem states the existence of at least one translation invariantPotts measure. Theorem 1. Assume that assumptions (A1) to (A3) are satisfied. Then forevery z and every ˜ α , there exists at least one Potts measure P ∈ G pottsθ ( z, ˜ α ) ,which is ergodic with respect to the translation group ( τ x ) x ∈ R d . The second theorem states a phase transition for large enough z . To state it,let us first define ˜ α max as the number of colours that have maximal proportionin ˜ α , i.e. ˜ α max = card { i = 1 . . . q | α i ≥ α i (cid:48) for all i (cid:48) (cid:54) = i } . This quantity is between 1 and q . For simplicity we are assuming that the colour1 is one of the colours with maximal proportion, i.e α ≥ α i for all i = 1 . . . q .In the symmetric case when α i = 1 /q for all i , which means that q = ˜ α max ,Georgii and H¨aggstr¨om [13] proved for large enough z the existence of at least q ergodic Potts measures. The following theorem generalizes their result to thenon-symmetric case. Theorem 2. Assume that assumptions (A1) to (A5) are satisfied. Then for z large enough, depending on q , u and r to r , but independent of ˜ α , there existsat least ˜ α max Potts measures for φ, ψ, z, ˜ α which are ergodic with respect to thetranslation group ( τ x ) x ∈ R d . This theorem does not give any indication in the case when ˜ α max = 1. Weare conjecturing that in this case there is no phase transition, at least when q is not too large. This conjecture is motivated by similar result proved for5he (lattice) nearest neighbour Ising model, see [11] for instance. Recently thisconjecture was partially solved in the specific case of Widom-Rowlinson model( q = 2, ψ = 0 and φ ( x ) = ∞ | x | small ) in [8]: they proved that for large activity z , phase transition is only possible in the symmetric case.The idea of the proof of Theorem 2 is the same as in [13]: a Fortuin-Kasteleyn representation and a percolation bound uniform in the volume whichpass through the limit. The novelty is the introduction of the generalized Con-tinuum Random Cluster model . This is a random connection model with aninteraction depending on the number of connected components and their sizes.This model allows the construction of a Fortuin-Kasteleyn representation, evenin the non-symmetric case. Such an idea was already used for the (lattice)nearest-neighbour Potts model in [1], from which the terminology generalizedRandom Cluster Model was taken from.One other question is the uniqueness of the Potts measure. It is in generalconjectured that uniqueness occurs when the activity z is small enough. WhenΨ ≥ 0, the Potts specification is stochastically dominated by a Poisson pointprocess. This stochastic domination leads, using the technique of disagreementpercolation , to the uniqueness of the Potts measure when z is small. Indeed inthe case Ψ = 0, the Potts model falls into the general assumptions of the resultproved in [17]. In the general case where Ψ can be negative, disagreement per-colation does not apply anymore. In [13] the authors claim that an extension ofthe Dobrushin uniqueness criterion could be used in order to prove uniqueness.However, to the best of our knowledge, no such result exists in the literature.One alternative could be to consider using cluster expansion , which is bettersuited for potential with negative part.The rest of the article is divided as follows: in Section 4 we are introducingthe Fortuin-Kasteleyn representation and proving a percolation bound for thegeneralized Continuum Random Cluster model. In Section 5 we are provingTheorem 1 and Theorem 2. Finally in the appendix in Section 6 we are provingsome technical lemmas used during the previous sections. At the core of the proof of Theorem 2 lies a representation of the Potts modelcalled Fortuin-Kasteleyn representation which provides the mean proportionof each colour in the Potts model, expressed as connectivity probabilities in apercolation model. This representation was introduced first by Edwards andSokal and then used to prove phase transition results in many models, includingthe symmetric Widom-Rowlinson model [3] and more generally continuum Pottsmodels [13]. It needs to study connectivity in the so-called Continuum RandomCluster model, which is a Gibbs model with an interaction depending only onthe number of connected components. This model was first introduced in [19]and then used in [13] and [3] to prove phase transition, by providing an uniformbound (with respect to the finite volume box Λ) of the percolative probability6hat the boundary is connected to the origin. The Continuum Random Clustermodel was also studied on its own in [7, 18].In our approach we are generalizing this method to the non-symmetric caseand introducing the generalized Continuum Random Cluster model . This is acontinuum version of the generalized Random Cluster model, used in [1] toprove phase transition for the non-symmetric lattice nearest neighbour Pottsmodel. We consider the set E of locally finite families of edges of the form E = ∪ i ∈ I x i , y i with x i (cid:54) = y i are in R d . This set is endowed with the classical σ -algebra gen-erated by the counting variables. We write E ω ⊆ E for the families of edgesbetween points of ω .From now on we are fixing Λ ⊆ R d bounded. We are defining a point process P z, ˜ α Λ on (cid:101) Ω × E in the following way: • The distribution of points is given by the potential ψ : P z,ψ Λ ( dω ) = exp (cid:16) − H ψ Λ ( ω ) (cid:17) Z ψ Λ ( ∅ ) π z Λ ( dω ) , where Z ψ Λ ( ∅ ) is the corresponding partition function, which is well definedthanks to the stability of the potential ψ . The ” ∅ ” is there to emphasizethat this point process measure is free of boundary condition. Notice herethat P z,ψ Λ is a point process measure on Ω Λ . • when the locations are known, the colours are independent random vari-ables of law ˜ α with deterministic colour for the points too close to theboundary of Λ, i.e λ ˜ α, ω, Λ ( (cid:101) ω ) = 1 Z ˜ α Λ ,ω (1) (cid:32) (cid:89) x ∈ ω Λ α σ x (cid:33) A r ( (cid:101) ω ) , (4.1)where A r = { (cid:101) ω | σ x = 1 , ∀ x s.t. dist ( x, Λ c ) ≤ r } and Z ˜ α Λ ,ω (1) is thecorresponding normalizing constant. The ”1” in the partition function isthere to emphasis on the fact that the points close to the boundary of Λare coloured deterministically. • Finally the edge drawing mechanism between points of ω is the probability µ φω on E ω such that µ φω ( E ) = (cid:89) { x,y }⊆ ω { x,y }∈ E (cid:16) − e − φ ( x − y ) (cid:17) (cid:89) { x,y }⊆ ω { x,y }(cid:54)∈ E e − φ ( x − y ) . (4.2)The probability measure P z, ˜ α Λ is then defined as the product measure P z, ˜ α Λ ( d (cid:101) ω, dE ) = µ φω ( E ) λ ˜ α, ω, Λ ( (cid:101) ω ) P z,ψ Λ ( dω ) . A on (cid:101) ω × E of authorized configurations where everyconnected points have the same colour. This event has positive P z, ˜ α Λ -probability,as the configuration empty of points in Λ is authorized. We can then considerthe probability measure P z, ˜ α Λ , A := P z, ˜ α Λ ( . |A ). Remark 4.1. In [13], they constructed the measure with a periodic boundarycondition of points of type 1. We could have made the same but we do believethat our construction, forcing points close to the boundary to be of colour 1, iseasier to understand and closer to the constructions made for the Ising modelfor instance.The indicator in (4.1) should be understood as if all points close to theboundary of Λ are connected to a imaginary point ”at infinity” which is of colour1. The connected component of points connected to this imaginary point ”atinfinity” will be called the infinite cluster and written C ∞ . A formal definitionwill be given later on. Furthermore, in the definition on the event A r , one doesnot need to take the same radius as in the condition (A2). Every choice of finiteradius would work as well. Proposition 4.1. The projection of the measure P z, ˜ α Λ , A on (cid:101) Ω is Ξ z, ˜ α Λ , ( d (cid:101) ω Λ ) := A r ( (cid:101) ω ) exp( − H Λ ( (cid:101) ω Λ )) Z z, ˜ α Λ (1) (cid:101) π z, ˜ α Λ ( d (cid:101) ω Λ ) . The measure Ξ z, ˜ α Λ , has to be understood the following way: one can imaginethat at the boundary of Λ there are a continuum of boundary points of colour1, forcing points too close to Λ to be of colour 1. But this continuum of pointsdo not give an interaction coming from ψ , only a colour exclusion. Proof. Let f be a measurable bounded function on (cid:101) Ω. (cid:90) (cid:101) Ω ×E f ( (cid:101) ω ) P z, ˜ α Λ , A ( d (cid:101) ω, dE )= P z, ˜ α Λ ( A ) − (cid:90) Ω (cid:90) (cid:101) Ω f ( (cid:101) ω ) (cid:20)(cid:90) E A ( (cid:101) ω, E ) µ φω ( E ) (cid:21) λ ˜ α, ω, Λ ( (cid:101) ω ) P z,ψ Λ ( dω ) , and thanks to simple computation we have (cid:90) E A ( (cid:101) ω, E ) µ φω ( dE )= (cid:88) E ∈E ω ,σ x = σ y ∀{ x,y }∈ E (cid:89) { x,y }⊆ ω { x,y }∈ E (cid:16) − e − φ ( x − y ) (cid:17) (cid:89) { x,y }⊆ ω { x,y }(cid:54)∈ E e − φ ( x − y ) = e − H φ Λ ( (cid:101) ω ) and therefore (cid:90) (cid:101) Ω ×E f ( (cid:101) ω ) P z, ˜ α Λ , A ( d (cid:101) ω, dE ) = κ (cid:90) (cid:101) Ω f ( (cid:101) ω Λ ) e − H Λ ( (cid:101) ω Λ ) A r ( (cid:101) ω ) (cid:101) π z, ˜ α Λ ( d (cid:101) ω Λ )= (cid:90) (cid:101) Ω f ( (cid:101) ω Λ )Ξ z, ˜ α Λ , ( d (cid:101) ω (cid:48) Λ ) . 8e are considering now the projection of P z, ˜ α Λ , A on Ω × E . We say that twopoints x, y ∈ ω are connected in ( ω, E ) if there is a path x , . . . , x n with x = x , x n = y and such that { x i , x i +1 } ∈ E for all i . A point x such that dist ( x, Λ c ) ≤ r is said to be linked (with an imaginary edge) to an imaginary point ”atinfinity”. We are then considering the connected components with respect tothis connectivity rule, with the particularity that all points x connected toinfinity, i.e. such that x is connected to a point y linked to infinity, are said tobe in the infinite connected component C ∞ .The number of connected components is P z, ˜ α Λ − a.s. finite, with at mostone component connected ”at infinity” C ∞ . But let us emphasize that thecardinality | C ∞ | of C ∞ is P z, ˜ α Λ − a.s. finite since C ∞ is a configuration containedin Λ.Let us now consider the measure C z, ˜ α Λ ,wired on Ω × E , called generalized Con-tinuum Random Cluster model on Λ with wired boundary condition, definedas C z, ˜ α Λ ,wired ( dω, dE ) = α | C ∞ | Z g Λ ( ˜ α ) (cid:89) C ∞ (cid:54) = C ⊆ ω cluster of ( ω,E ) (cid:32) q (cid:88) i =1 α | C | i (cid:33) µ φω ( E ) P z,ψ Λ ( dω ) , with Z g Λ ( ˜ α ) being the associated partition function. Proposition 4.2. The projection of P z, ˜ α Λ , A on ω × E is C z, ˜ α Λ ,wired .Proof. Let f be a measurable bounded function on Ω × E . (cid:90) (cid:101) Ω ×E f d P z, ˜ α Λ , A = P z, ˜ α Λ ( A ) − (cid:90) Ω (cid:90) E f ( ω, E ) (cid:20)(cid:90) (cid:101) Ω A ( (cid:101) ω, E ) λ ˜ α, ω, Λ ( (cid:101) ω ) (cid:21) µ φω ( E ) P z,ψ Λ ( dω ) , but thanks to the product structure of the measure λ ˜ α, ω, Λ , and denoting by C i ⊆ ω the finite (i.e. not connected ”at infinity”) connected components of( ω, E ), we have (cid:90) (cid:101) Ω A ( (cid:101) ω, E ) λ ˜ α, ω, Λ ( (cid:101) ω ) = α | C ∞ | Z ˜ α Λ ,ω (1) (cid:88) (cid:101) ω C | ω C · · · (cid:88) (cid:101) ω Cn | ω Cn n (cid:89) i =1 A ( (cid:101) ω C i , E ) (cid:89) x ∈ ω Ci α σ x = α | C ∞ | Z ˜ α Λ ,ω (1) (cid:89) C ∞ (cid:54) = C ⊆ ω cluster of ( ω,E ) (cid:32) q (cid:88) i =1 α | C | i (cid:33) , which implies the wanted result. Here the sum (cid:80) (cid:101) ω | ω is over all coloured configu-rations (cid:101) ω whose projection onto Ω is ω .So from both propositions, the colour of one particle in the Potts model isdirectly related to the connectivity of this point in the generalized ContinuumRandom Cluster model. In particular the points connected ”at infinity” (i.e.those in C ∞ ) have fixed deterministic colour 1.For a configuration (cid:101) ω = ( ω, σ ) ∈ (cid:101) Ω, and for ∆ ⊆ Λ ⊆ R d , we write N ∆ , ( (cid:101) ω )for the number of points of colour 1 inside ∆. We also write N ∆ ↔∞ ( ω, E ) forthe number of points in C ∞ ∩ ∆. 9 roposition 4.3. Assume that ˜ α max > and that i (cid:54) = 1 is one of the othercolours with maximal proportion. Then (cid:90) ( N ∆ , − N ∆ ,i ) d Ξ z, ˜ α Λ , = (cid:90) N ∆ ↔∞ d C z, ˜ α Λ ,wired . Proof. From Proposition 4.1 the left hand side is (cid:90) ( N ∆ , − N ∆ ,i ) d Ξ z, ˜ α Λ , = (cid:90) Ω (cid:90) E (cid:90) (cid:101) Ω A ( (cid:101) ω, E ) P z, ˜ α Λ ( A ) ( N ∆ , ( (cid:101) ω ) − N ∆ ,i ( (cid:101) ω )) λ ˜ α, ω, Λ ( (cid:101) ω ) µ φω ( E ) P z,ψ Λ ( dω )= (cid:90) Ω (cid:90) E (cid:88) x ∈ ω ∆ x ∈ C ∞ (cid:20)(cid:90) (cid:101) Ω A ( (cid:101) ω, E ) P ( A ) λ ˜ α, ω, Λ ( (cid:101) ω ) (cid:21) µ φω ( E ) P z,ψ Λ ( dω ) . The integrated quantity does no longer depend on the colouring of the config-uration, and from Proposition 4.2 we get the result.Intuitively if the quantity (cid:82) ( N ∆ , − N ∆ ,i ) d Ξ z, ˜ α Λ , is bounded from below uni-formly in Λ by some (cid:15) > 0, then when Λ goes to R d the limit of Ξ z, ˜ α Λ , will bea Potts measure with more particles of colour 1 than any other colour. Byrepeating the same with a boundary condition of different colour, we get theexistence of several different Potts measures. From Proposition 4.3, to controlthe quantity (cid:82) ( N ∆ , − N ∆ ,i ) d Ξ z, ˜ α Λ , we need to study the connectivity in the gen-eralized Continuum Random Cluster model C z, ˜ α Λ ,wired . The following propositionis the key tool in proving Theorem 2. Proposition 4.4. Assume that assumptions ( A ) to ( A ) are satisfied, and z is large enough (depending on the parameters, but not on ˜ α ). Then there exists (cid:15) > such that (cid:90) N ∆ ↔∞ ( ω, E ) C z, ˜ α Λ ,wired ( dω, dE ) ≥ (cid:15) for every cell ∆ = ∆ j defined after equation (5.3) and every Λ finite union ofcells ∆ j (cid:48) . The general idea is to use stochastic domination to compare our model to amixed site-bond Bernoulli percolation model. First we are decoupling the edges E and constructing a probability measure ¯ C z, ˜ α Λ the following way: • The distribution of particle positions is given by M z, ˜ α Λ = C z, ˜ α Λ ,wired ( . × E ) . • Given the points ω , we draw between two points x, y ∈ ω such that | x − y | ≤ r an edge with probability¯ p = 1 − e − u q e − u + 1 − e − u (4.3)10here r and u come from assumption (A1). The equation (4.3) defineson E ω the edges distribution ¯ µ ω .We therefore define the measure ¯ C z, ˜ α Λ ( dω, dE ) = ¯ µ ω ( dE ) M z, ˜ α Λ ( dω ). Remark 4.2. First remark that we have C z, ˜ α Λ ,wired ( dω, dE ) = µ ˜ αω, Λ ( dE ) M z, ˜ α Λ ( dω ) with µ ˜ αω, Λ ( dE ) ∼ α | C ∞ | (cid:89) C ∞ (cid:54) = C ⊆ ω Λ cluster of ( ω,E ) (cid:88) i =1 ..q α | C | i µ φω ( E ) being the ”discrete” generalized Random Cluster model. The definition of µ ˜ αω, Λ depends on Λ only through the definition of the infinite connected component C ∞ .Finally the choice q in (4.3) is not optimal, but is uniform with respect to ˜ α . In [13], the value q was enough for the symmetric case. Definition 4.1. For two probability measures µ, µ (cid:48) on E , we say that µ (cid:48) domi-nates µ , written µ (cid:48) (cid:23) µ , if (cid:82) f dµ (cid:48) ≥ (cid:82) f dµ for all measurable increasing function(with respect to the natural order on E ).This notion of domination naturally extends to probability measures in Ω ×E . Lemma 4.1. Assume that assumption ( A ) is satisfied. Then for all ˜ α and ω we have µ ˜ αω, Λ (cid:23) ¯ µ ω and therefore C z, ˜ α Λ ,wired (cid:23) ¯ C z, ˜ α Λ . This lemma is one the principal improvement with respect to the work ofGeorgii and H¨aggstr¨om [13]. Proof. The second assertion is a direct consequence of the first one. For thefirst assertion we will use the well-known Holley inequality , see for instance [16,Th. 2.3]. Let e = { x, y } with x, y ∈ ω , we have¯ µ ω ( e ∈ E | E e c ) = ¯ µ ω ( e ∈ E ) = (cid:40) ˜ p = − e − u q e − u +1 − e − u if | x − y | ≤ r | x − y | > r with ¯ µ ω ( e ∈ E | E e c ) being the probability that e is an edge of E conditioned onknowing all the other edges. From easy computations we also get µ ˜ αω, Λ ( e ∈ E | E e c ) = − e − φ ( x − y ) if x ↔ y in ( ω, E e c ) (cid:18) e − φ ( x − y ) − e φ ( x − y ) (cid:80) i (cid:16) α i α (cid:17) | C y | (cid:19) − if x (cid:54)↔ y and x ↔ ∞ (cid:18) e − φ ( x − y ) − e − φ ( x − y ) (cid:80) i α | Cx | i (cid:80) j α | Cy | j (cid:80) i α | Cx | + | Cy | i (cid:19) − if x (cid:54)↔ y and x, y (cid:54)↔ ∞ where ↔ denotes the connectivity in ( ω, E e c ) and C x , C y are the connectedcomponent of x, y in ( ω, E e c ). Remember that the connectivity of two pointscan be through the imaginary point at infinity.To apply Holley’s inequality, we have to check that ¯ µ ω ( e ∈ E | E e c ) ≤ µ ˜ αω, Λ ( e ∈ E | E e c ). We will only do it for the last expression of µ ˜ αω, Λ ( e ∈ E | E e c ).11his inequality is trivially true when | x − y | > r . Otherwise from assump-tion (A1) we have e − φ ( x − y ) − e − φ ( x − y ) ≤ e − u − e − u . Furthermore (cid:80) i α | C x | i (cid:80) j α | C y | j (cid:80) i α | C x | + | C y | i = (cid:80) i ( α i /α ) | C x | (cid:80) j ( α j /α ) | C y | (cid:80) i ( α i /α ) | C x | + | C y | ≤ (cid:18) α (cid:19) ˜ αmax ≤ q , which implies the wanted inequality. The other cases can be treated the same.From Lemma 4.1 it is enough to prove Proposition 4.4 for the measure ¯ C z, ˜ α Λ .This will be done by a discretization and a comparison to the random connectionmodel. For this remember the definition of the cells ∆ j done just before thebeginning of Section 2.2. Definition 4.2. Starting now we take δ = r √ d +3 , to ensure that any two pointsin two adjacent cells ∆ j , ∆ j (cid:48) are at distance at most r . • We call a cell good if it contains at least n ∗ points forming (with theedges) a connected graph. • Two cells are said linked if there exists an edge connecting two points inthe two cells. This defines a correlated site-bond percolation on Z d . The next lemmastates the usual percolation result for the independent site-bond percolationmodel. Lemma 4.2. Consider on Z d the Bernoulli site-bond percolation model whereeach site and each edge between sites at distance 1 is open with probability p and closed otherwise, independently of everything else. Let us write P rob p theprobability measure associated to this model.There exists p c = p c ( d ) ∈ ]0 , such that for p > p c , θ ( p ) = P rob p ( the origin is connected to infinity ) > . The proof of this lemma is done in the appendix in Section 6. In order tocontrol the probability of a cell being good, we will use the following lemma. Lemma 4.3. For a positive integer n and for < p < , we consider therandom graph G n,p of n vertices where each pair of vertices independently formsan edge with probability p . Then γ ( n, p ) = P rob ( G n,p is connected ) −→ n →∞ , and therefore γ ( p ) = inf { γ ( n, p ) | n ≥ } > . h Λ ( ω ) = (cid:88) E ∈E ω α | C ∞ | Z g Λ ( ˜ α ) (cid:89) C ∞ (cid:54) = C ⊆ ω cluster of ( ω,E ) (cid:88) i =1 ..q α | C | i µ φω ( E ) . The function h Λ is the probability density of M z, ˜ α Λ with respect to P z,ψ Λ . Forfixed ω , h Λ ( ω ) is also the partition function of the discrete generalized RandomCluster model µ ˜ αω, Λ . Lemma 4.4. Under assumption (A2), there exists a constant ˜ κ > such thatfor every ω ∈ Ω Λ and every x ∈ Λ , h Λ ( ω ∪ x ) ≥ ˜ κ × h Λ ( ω ) . This lemma is one of the principal improvement of the initial work of Georgiiand H¨aggstr¨om [13]. Furthermore it is the only part of the article where thefinite range assumption (A2) on φ was used. Proof. In the following, E is an edge configuration between points in ω , i.e. E ∈ E ω , and E (cid:48) is an edge configuration between x and points of ω . The union E ∪ E (cid:48) is in E ω ∪ x . We will denote C j , j = 1 ..n the connected components of( ω, E ), one of which can be infinite (if so it will be the first one C ), which areconnected together in ( ω ∪ x, E ∪ E (cid:48) ). Then we have h Λ ( ω ∪ x ) h Λ ( ω ) = (cid:88) E ∈E ω µ ˜ αω, Λ ( E ) (cid:88) E (cid:48) α (cid:80) j =1 ..n | C j | + C finite (cid:80) i =2 ..q α (cid:80) j =1 ..n | C j | i (cid:32) α | C | + C finite (cid:80) i =2 ..q α | C | i (cid:33) (cid:81) j =2 ..n (cid:80) i =1 ..q α | C j | i µ φω ∪ x ( E (cid:48) ) ≥ (cid:88) E ∈E ω µ ˜ αω, Λ ( E ) (cid:88) E (cid:48) α (cid:80) j =1 ..n | C j | (cid:81) j =1 ..n qα | C j | µ φω ∪ x ( E (cid:48) )= (cid:88) E ∈E ω µ ˜ αω, Λ ( E ) (cid:88) E (cid:48) α µ φω ∪ x ( E (cid:48) ) q number of cc of ( ω,E ) connected to x . For a connected component to be connected to x it must, from assumption (A2),contain a point at distance less than r . We split the closed ball B ( x, r ) intoa minimal number k of disjoint sets B j , j = 1 ..k of diameter less than r . Oneach B j , we consider the event A j that the graph ( ω B j , E ∩ E ω Bj ) is connected.The events A j are increasing and we have h Λ ( ω ∪ x ) h Λ ( ω ) ≥ α q k µ ˜ αω, Λ (cid:92) j =1 ..k A j ≥ α q k ¯ µ ω (cid:92) j =1 ..k A j ≥ α q k γ (˜ p ) k := ˜ κ > , A j with respect to ¯ µ ω , and where γ (˜ p ) is defined in Lemma4.3.We now have all the tools to compare ¯ C z, ˜ α Λ to the independent site-bondBernoulli percolation model. Let us fix p ∗ > p c , where p c is defined in Lemma4.2. Let us define λ ( n, ˜ p ) = 1 − (1 − ˜ p ) n as a lower bound for the ¯ µ ω -probabilitythat there exists at least one edge between points in two neighbouring cellscontaining at least n points each. From Lemma 4.3, we have the existence of n ∗ such that γ ( n, ˜ p ) ≥ (cid:112) p ∗ and λ ( n, ˜ p ) ≥ p ∗ for all n ≥ n ∗ . (4.4)We are now in position to make clear the requirement on r from ( A r < r / √ d + 3 and ( n ∗ − | B (0 , r ) | < ( δ − r ) d . (4.5)Let us define M z, ˜ α Λ , ∆ j ,ω the conditional probability, according to M z, ˜ α Λ , of particlesinside ∆ j , knowing the the configuration in Λ \ ∆ j is ω . Lemma 4.5. Assume that r , r , r satisfy assumption ( A ), and that assump-tions (A2) and (A4) is satisfied. Then for z large enough we have M z, ˜ α Λ , ∆ j ,ω ( N ∆ j ≥ n ∗ ) ≥ (cid:112) p ∗ for all Λ finite union of cells, for all ∆ j cells included in Λ , and for all config-urations ω on Λ \ ∆ j . The proof of this lemma is done in the appendix in Section 6. Using thislemma, and with (4.4) we obtain M z, ˜ α Λ , ∆ j ,ω (∆ j is good) ≥ p ∗ . Furthermore by construction and from (4.4) the probability that two givenneighbouring cells are connected, conditioned on the fact that they are good, isat least p ∗ .Therefore by applying Holley’s inequality, see Theorem 2.3 in [16] we have (cid:90) N ∆ ↔∞ ( ω, E ) C z, ˜ α Λ ,wired ( dω, dE ) ≥ n ∗ × θ ( p ∗ ) := (cid:15) > . Both theorems rely on the standard construction of an infinite volume Pottsmeasure, as a limit of a stationarized finite volume Potts measures consideredon a sequence of increasing boxes. We prove that this sequence admits anaccumulation point, for the topology of local convergence using a now stan-dard tightness tool which is the specific entropy developed by Georgii [12] andadapted to the continuum case by Georgii and Zessin [15]. Then we prove thatthis accumulation point is a continuum Gibbs measure, proving Theorem 1.14inally from this construction and the Proposition 4.4, the phase transition isstraightforward. This type of construction is now standard and have been donein many articles, for the symmetric continuum Potts model [13] or for othertypes of interaction [5, 6, 7, 9].As before we are considering δ = r / √ d + 3 as in Definition 4.2. We considerthe square box Λ n =] − δ ( n + 1 / , δ ( n + 1 / d which is containing exactly (2 n + 1) d disjoints cells ∆ j , for j ∈ L n := δ Z d ∩ Λ n .On (cid:101) Ω Λ n consider the measure P n ( d (cid:101) ω Λ n ) := Ξ z, ˜ α Λ n , ( d (cid:101) ω Λ n ) = A r ( (cid:101) ω Λ n ) exp( − H Λ n ( (cid:101) ω Λ n )) Z z, ˜ α Λ n (1) (cid:101) π z, ˜ α Λ n ( d (cid:101) ω Λ n )defined in Proposition 4.1. Finally we consider the measureˆ P n := 1(2 n + 1) d (cid:88) j ∈ L n ¯ P n ◦ τ − j , where ¯ P n = (cid:79) k ∈ n Z d P n ◦ τ − k . By construction the measures ˆ P n are invariant by the translation in δ Z d : ˆ P n ∈ (cid:101) P θ δ . Definition 5.1. A measurable function f : (cid:101) Ω → R is said local and tame ifthere exists a bounded Λ ⊆ R d and a constant κ ≥ such that f ( (cid:101) ω ) = f ( (cid:101) ω Λ ) and | f ( (cid:101) ω ) | ≤ κ (1 + N Λ ( (cid:101) ω )) for all configurations (cid:101) ω ∈ (cid:101) Ω .A sequence of measures ν n converge to ν in the local convergence topology if (cid:82) f dν n → (cid:82) f dν for all local and tame functions f . Proposition 5.1. The sequence ( ˆ P n ) admits a cluster point ˆ P with respectto the local convergence topology. This cluster point is invariant under thetranslation τ x , x ∈ δ Z d , and it is a Potts measure: ˆ P ∈ G pottsθ δ ( z, ˜ α ) . Remark 5.1. In the following, to lighten the notation, we will avoid to take asubsequence and assume that ( ˆ P n ) converges to ˆ P . We will first admit this proposition and conclude the proofs of Theorem 1and Theorem 2. 15 .1 Proof of Theorem 1 Proposition 5.1 is not enough to conclude directly the proof of Theorem 1, sincethe measure ˆ P is not invariant under all translation of R d . But considering themeasure (cid:101) P := 1 δ d (cid:90) ] − δ/ ,δ/ d ˆ P ◦ τ − x dx, we obtain a measure which is by construction invariant under all translation of R d . This measure satisfies 0 < Z z, ˜ α Λ ( . ) < ∞ (cid:101) P -almost surely for every boundedΛ, and from the translation invariance of the interaction, we obtain (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) (cid:101) P ( d (cid:101) ω )= (cid:90) ] − δ , δ ] d (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ τ x ( (cid:101) ω ) Λ c ) e − H Λ ( (cid:101) ω (cid:48) Λ τ x ( (cid:101) ω ) Λ c ) δ d Z z, ˜ α Λ ( τ x ( (cid:101) ω )) π z, ˜ α Λ ( d (cid:101) ω (cid:48) ) ˆ P ( d (cid:101) ω ) dx = (cid:90) ] − δ , δ ] d (cid:90) (cid:90) f ◦ τ x (cid:16)(cid:101) ω (cid:48) τ − x (Λ) (cid:101) ω τ − x (Λ) c (cid:17) e − H Λ ◦ τ x (cid:18)(cid:101) ω (cid:48) τ − x (Λ) (cid:101) ω τ − x (Λ) c (cid:19) δ d Z z, ˜ α Λ ( τ x ( (cid:101) ω )) dπ z, ˜ α d ˆ P dx = 1 δ d (cid:90) ] − δ , δ ] d (cid:90) f ◦ τ x d ˆ P dx = (cid:90) f d (cid:101) P and therefore (cid:101) P is a Potts measure invariant under all translation of R d .Since the set of translation invariant Potts measures is a convex set withextremal elements being the ergodic Potts measures, see [12], the theorem isproved. Let us consider ∆ = ∆ = Λ =] − δ/ , δ/ d . Let i (cid:54) = 1 an other colour withmaximal proportion (i.e. α = α i ) , we obtain from Proposition 4.3 that (cid:90) ( N ∆ , − N ∆ ,i ) d ˆ P n = 1(2 n + 1) d (cid:88) j ∈ L n (cid:90) ( N ∆ j , − N ∆ j ,i ) dP n = 1(2 n + 1) d (cid:88) j ∈ L n N ∆ j ↔∞ d C z, ˜ α Λ n ,wired . Finally we obtain from Proposition 4.4 that for z large enough (but independentof ˜ α ) that (cid:90) ( N ∆ , − N ∆ ,i ) d ˆ P n ≥ (cid:15) > . Since the integrated function is local and tame, the same bound is valid for theprobability measure ˆ P . 16or the measure (cid:101) P , even if the translated τ x (∆), x ∈ ] − δ/ , δ/ d is not acell as defined before, using the translation invariance of ˆ P by vectors j ∈ ( δ Z ) d ,one can translate back τ x (∆) into ∆, which proves that (cid:90) ( N ∆ , − N ∆ ,i ) d (cid:101) P ≥ (cid:15) > , see Figure 5.2.Figure 1: Left: The cell ∆ and its translation τ x ( δ ). Right: how to translateback piece of τ x (∆) to obtain again ∆.So the probability measure (cid:101) P is a Potts measure with on average moreparticles of colour 1 that any other colours. By repeating the same constructionfor every colour i with maximal proportion, we obtain ˜ α max Potts measuresinvariant by all translations in R d and which are different, since one differentcolour dominates the others for each measure.From the ergodic decomposition of translation invariant (by a vector of R d )Potts measures, see [12], it is clear that one can find at least ˜ α max differentergodic Potts measures. The theorem is proved. This type of construction is classical. It was done for instance for the symmetricPotts model in [13], for the quermass-interaction model in [5], and for manyother cases [6, 7, 9]. The first step is to construct a good candidate. This isdone using the specific entropy as a tightness tool. Then one has to prove thatthis good candidate is indeed a Potts measure, which is done by approximationof the interaction. • Step 1 : Construction of a good candidate Definition 5.2. For a measure P ∈ (cid:101) P θ δ with finite first moment, meaning that (cid:82) N ∆ dP < ∞ , we define the specific entropy I ( P ) := lim n →∞ | Λ n | I Λ n ( P | π z, ˜ α ) , (5.1)17 ith I Λ n ( P | π z, ˜ α ) = (cid:82) log (cid:18) dP Λ n dπ z, ˜ α Λ n ( (cid:101) ω Λ n ) (cid:19) P ( d (cid:101) ω ) , or + ∞ if P Λ n (the restrictionof P in (cid:101) Ω Λ n ) is not absolutely continuous with respect to π z, ˜ α Λ n . The convergence in (5.1) is proved in [15]. The next proposition, also provedin [15], stated the tightness of the level sets of the specific entropy. Proposition 5.2. On the set of probability measure P ∈ (cid:101) P θ δ with finite firstmoment, the specific entropy is affine and upper semi-continuous . Furthermorefor all κ ≥ , the level set I ( P ) ≤ κ is compact and sequentially compact withrespect to the local convergence topology. So from this proposition it is enough to prove that the specific entropy ofthe sequence ( ˆ P n ) is uniformly bounded. It is clear that each ˆ P n has finite firstmoment (but it is not clear yet that one can find an uniform bound). From thefact that the specific entropy is affine, we obtain that I ( ˆ P n ) = 1 | Λ n | I Λ n (cid:16) P n | π z, ˜ α Λ n (cid:17) = − log( Z z, ˜ α Λ n (1)) − (cid:82) (cid:101) Ω H Λ n ( (cid:101) ω Λ n ) P n ( d (cid:101) ω Λ n ) | Λ n | For the assumptions (A1) and (A3) we have H Λ n ( (cid:101) ω Λ n ) ≥ H ΨΛ n ( ω Λ n ) ≥ (cid:88) j ∈ L n aN ∆ j ( ω ) − bN ∆ j ( ω ) ≥ − (2 n + 1) d κ, (5.2)where κ = B / A . Furthermore from standard computation we obtain Z z, ˜ α Λ n (1) ≥ e − z | Λ n | . Therefore we obtain I ( ˆ P n ) ≤ c for a positive finite constant c independent of n . Hence we have the existenceof a cluster point P with respect to the local convergence topology. In thefollowing we omit to take a subsequence to lighten the notation. Remark 5.2. To obtain (5.2) we used the superstability and regularity fromassumption (A3). In the case when the potential ψ is non-negative, the Hamil-tonian is non-negative and we obtain again the existence of a constance c . • Step 2 : the partition function Z z, ˜ α Λ is ˆ P -a.s non degenerate.To prove this, let us first prove that ˆ P has finite second moment. Lemma 5.1. Under assumptions (A1) and (A3), we have for all n (cid:90) N d ˆ P < ∞ and (cid:90) N d ˆ P n < ∞ . roof. Let us now consider the second moment of ˆ P n : (cid:90) (cid:101) Ω N ∆ ( (cid:101) ω ) ˆ P n ( d (cid:101) ω ) = 1(2 n + 1) d (cid:90) (cid:101) Ω (cid:88) j ∈ L n N ∆ j ( ω ) P n ( d (cid:101) ω ) ≤ a (2 n + 1) d (cid:90) (cid:101) Ω H Λ n ( (cid:101) ω Λ n ) P n ( d (cid:101) ω )+ 1(2 n + 1) d (cid:90) (cid:101) Ω (cid:88) j ∈ L n (cid:18) ba N ∆ j ( ω ) − N ∆ j ( ω ) (cid:19) P n ( d (cid:101) ω ) , where the last inequality is a consequence of assumptions (A1) and (A3), usedas in (5.2). From the non-negativity of the local entropy, we have that (cid:90) (cid:101) Ω H Λ n ( (cid:101) ω Λ n ) P n ( d (cid:101) ω ) ≤ − log(( Z z, ˜ α Λ n (1)) ≤ z | Λ n | . Furthermore there exists a constant κ ≥ (cid:90) (cid:101) Ω (cid:88) j ∈ L n (cid:18) ba N ∆ j ( ω ) − N ∆ j ( ω ) (cid:19) P n ( d (cid:101) ω ) ≤ κ (2 n + 1) d . Putting everything together we obtain (cid:90) (cid:101) Ω N ∆ ( ω ) ˆ P n ( d (cid:101) ω ) ≤ ˜ κ, where ˜ κ < ∞ is independent of n .Now the function N is local but not tame. However this is the monotonelimit of local and tame functions, which is enough to conclude that (cid:90) (cid:101) Ω N ∆ ( ω ) ˆ P ( d (cid:101) ω ) ≤ ˜ κ. In the case where ψ ≥ 0, one can prove immediately from the stochasticdomination result of Georgii and K¨uneth [14] that the measure P n is stochasti-cally dominated by π z, ˜ α Λ n and therefore the uniform bound is straightforward.Now let us define on (cid:101) Ω the space of tempered configuration T = (cid:101) ω ∈ (cid:101) Ω , sup n ≥ n d (cid:88) | i |≤ n N j ( (cid:101) ω ) < ∞ (5.3) Lemma 5.2. Under assumptions (A1) and (A3), and for all (cid:101) ω ∈ T , the parti-tion function is non-degenerate: < Z z, ˜ α Λ ( (cid:101) ω ) < ∞ . Furthermore we have ˆ P ( T ) = 1 . • Step 3 : the measure ˆ P satisfies the DLR equations.It is enough to proves the DLR equations only for the Λ n , and starting nowwe fix Λ = Λ n for a fixed n ∈ N . Let us consider f a measurable functionbounded by one, which we can assume without loss of generality that is is local,i.e f ( (cid:101) ω ) = f ( (cid:101) ω Λ n ) for a fixed n . we are interested in proving that the followingquantity κ := (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f d ˆ P − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) ˆ P ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) is small. The first issue is that the probability measures ˆ P n do not satisfy theDLR equations, except in the particular case of Λ ⊆ ∆ . We are introducingthe new sequence of measuresˆ P Λ n := 1(2 n + 1) d (cid:88) j ∈ L N Λ ⊆ τ j (Λ n ) P n ◦ τ − j . Those are not probability measure, but from the following lemma they satisfythe DLR(Λ) equation and are converging to ˆ P . Lemma 5.3. Each ˆ P Λ n satisfies the DLR( Λ ) equation. Furthermore if (A1)and (A3) are satisfied, for all local and tame functions f , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) f d ˆ P Λ n − (cid:90) f d ˆ P n (cid:12)(cid:12)(cid:12)(cid:12) −→ n →∞ . Proof. The first point is a consequence of the compatibility of the Gibbs kernelsand the translation invariance of the interaction. The second point has beentreated in [5] for the quermass interaction model or in [7] for the ContinuumRandom Cluster model, and we are omitting the proof here. Lemma 5.4. Under assumptions (A1), (A2) and (A3) we have for all N largeenough and for all n (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) ˆ P ( d (cid:101) ω ) − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω Λ N ( d (cid:101) ω (cid:48) Λ ) ˆ P ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) and (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) ˆ P Λ n ( d (cid:101) ω ) − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω Λ N ( d (cid:101) ω (cid:48) Λ ) ˆ P Λ n ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) The proof is classical and is done in the appendix in Section 6. Let us nowconclude the proof of Proposition 5.1. Let us fix (cid:15) > 0. From Lemma 5.4 thereis N large enough such that κ ≤ (cid:15) + (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f d ˆ P − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω Λ N ( d (cid:101) ω (cid:48) Λ ) ˆ P ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) . n large enough κ ≤ (cid:15) + (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f d ˆ P Λ n − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω Λ N ( d (cid:101) ω (cid:48) Λ ) ˆ P Λ n ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) , and applying again Lemma 5.4 and the first point of Lemma 5.3, κ ≤ (cid:15) + (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f d ˆ P Λ n − (cid:90) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) ˆ P Λ n ( d (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) = 3 (cid:15), and the proof is concluded. 21 Appendix: proof of the intermediary lemmas The mixed site-bound Bernoulli percolation model is clearly monotone in theparameter p , which gives the existence of p c . It remains to prove that p c isin ]0 , p c > 0, since the absence of percolation in the siteBernoulli percolation model of parameter p implies the same for the mixedsite-bond model of parameter p .The second inequality comes from the observation that for any graph G , thesite percolation threshold is not greater than the bond percolation threshold(both for the Bernoulli model). Hence p c is smaller than the square root of thesite Bernoulli percolation model. Let us denote x , . . . , x n the n points of the graph. By considering the eventthat x is connected to every other points by a path of length exactly 2, weobtain γ ( n, p ) ≥ − ( n − − p ) n − , which proves the result. It is easy to see that M z, ˜ α Λ , ∆ j ,ω ( dω (cid:48) ) = h Λ ( ω (cid:48) ω ) Z Λ , ∆ j ,ω exp (cid:16) − H ψ ∆ j ( ω (cid:48) ω ) (cid:17) π z ∆ j ( dω (cid:48) ) , where Z Λ , ∆ j ,ω is the corresponding partition function. Therefore we obtain for n ≥ M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = n + 1) M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = n ) = zn + 1 (cid:90) g i ( ω (cid:48) ω ) M z, ˜ α Λ , ∆ j ,ω ( dω (cid:48) | N ∆ j = n )with g i ( ω (cid:48) ω ) = (cid:90) ∆ j exp − (cid:88) y ∈ ω (cid:48) ω ψ ( x − y ) h Λ ( ω (cid:48) ω ∪ x ) h Λ ( ω (cid:48) ω ) dx ≥ l (cid:90) ∆ i exp − (cid:88) y ∈ ω (cid:48) ω ψ ( x − y ) dx, with the last inequality coming from Lemma 4.4. Consider now the reduce cell∆ j obtain from ∆ j by removing a boundary layer of width r . By assumption22 A j has positive volume. Then g i ( ω (cid:48) ω ) ≥ l (cid:90) ∆ j exp − (cid:88) y ∈ ω (cid:48) ω ψ ( x − y ) dx ≥ l (cid:90) ∆ j exp − (cid:88) y ∈ ω (cid:48) ψ ( x − y ) dx, where the last inequality comes from assumption ( A ω . The next estimates is directly takenfrom [13], and goes back originally to Dobrushin and Minlos. Let∆ ω (cid:48) = { x ∈ ∆ j , | x − y | ≥ r for all y ∈ ω (cid:48) } . Assume by contradiction that N ∆ j ( ω (cid:48) ) < n ∗ . Then | ∆ ω (cid:48) | ≥ | ∆ j | − ( n ∗ − | B (0 , r ) | := v ∗ , and v ∗ is positive thanks to assumption (A5), see equation (4.5). Furthermore,applying Markov’s inequality to the Lebesgue measure, we obtain for all κ > |{ x ∈ ∆ ω (cid:48) , (cid:88) y ∈ ω (cid:48) ψ ( x − y ) ≥ κ }| ≤ κ (cid:88) y ∈ ω (cid:48) (cid:90) ∆ ω (cid:48) ψ + ( x − y ) dx ≤ n ∗ − κ (cid:90) | x |≥ r ψ + ( x ) dx := b ( n ∗ , r ) κ , with b ( n ∗ , r ) < ∞ thanks to assumption (A4). Adding everything together weobtain when N ∆ j ( ω (cid:48) ) < n ∗ that g i ( ω (cid:48) ω ) ≥ le − κ (cid:18) v ∗ − b ( n ∗ , r ) κ (cid:19) . By choosing κ large enough, there exists a constant ˜ l such that g i ( ω (cid:48) ω ) ≥ ˜ ln ∗ .Hence M z, ˜ α Λ , ∆ j ,ω ( N ∆ j < n ∗ ) = n ∗ − (cid:88) n =0 M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = n )= n ∗ − (cid:88) n =0 M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = n ∗ ) n ∗ − (cid:89) k = n M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = k ) M z, ˜ α Λ , ∆ j ,ω ( N ∆ j = k + 1) ≤ n ∗ − (cid:88) n =0 (cid:18) lz (cid:19) n ∗ − n ≤ lz − , and this quantity goes to 0 when z goes to infinity. Therefore the result isproved. 23 .4 Proof of Lemma 5.2 For the first point, let us consider Λ ⊂ R d bounded, (cid:101) ω ∈ T and (cid:101) ω (cid:48) ∈ (cid:101) ω . Fromstandard computation we obtain Z z, ˜ α Λ ( (cid:101) ω ) ≥ exp ( − z | Λ | ) > . Using assumption(A1) we have H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) ≥ H ψ Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) . Now let us consider assumption (A3). In the case ψ ≥ Z z, ˜ α Λ ( (cid:101) ω ) ≤ < ∞ . In theother case we obtain from assumption (A3) that H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) ≥ (cid:88) j (cid:48) aN ∆ j (cid:48) ( (cid:101) ω (cid:48) ) − b + (cid:88) j ψ δ − | j − j (cid:48) | N ∆ j ( (cid:101) ω ) N ∆ j (cid:48) ( (cid:101) ω (cid:48) ) + error, where the sum is over j (cid:48) such that ∆ j (cid:48) ∩ Λ (cid:54) = 0 and j such that ∆ j ∩ Λ = 0. Theerror term comes from the fact that we did not take into consideration points x ∈ (cid:101) ω (cid:48) Λ and y ∈ (cid:101) ω c Λ which are in the same cell ∆ j . But since the superstabilityand regularity does not depend on the choice of the discretization, we canassume without loss of generality that this error term is null, which is to saythat Λ is exactly the union of a finite number of cells ∆ j . Now from the factthat (cid:101) ω is tempered, we have the existence of a constant κ ≥ (cid:88) k ψ δ − | j − k | N ∆ k ( (cid:101) ω ) ≤ (cid:88) n ∈ N ψ n (cid:88) k,δ − | j − k | = n N ∆ k ( (cid:101) ω ) ≤ κ (cid:88) n ∈ N n d − ψ n := B (cid:48) < ∞ and therefore H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) is bounded from below uniformly in (cid:101) ω (cid:48) , and so Z z, ˜ α Λ ( (cid:101) ω ) < ∞ . From construction and from Lemma 5.1, the measure ˆ P is invariant un-der the translation of ( δ Z ) d and satisfies (cid:82) N d ˆ P < ∞ . Therefore, from theergodic theorem, see [22], the sequence of random variables n (cid:55)→ n d (cid:88) | i |≤ n N ∆ ( (cid:101) ω ) converges ˆ P almost surely towards a finite random variable. The result isproved. 24 .5 Proof of Lemma 5.4 Consider (cid:101) ω ∈ T . Then c ( (cid:101) ω ) := (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) Λ ) − (cid:90) f ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c )Ξ z, ˜ α Λ , (cid:101) ω Λ N ( d (cid:101) ω (cid:48) Λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) Z z, ˜ α Λ ( (cid:101) ω ) − e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ N \ Λ ) Z z, ˜ α Λ ( (cid:101) ω Λ N ) (cid:12)(cid:12)(cid:12)(cid:12) π z, ˜ α Λ ( d (cid:101) ω (cid:48) ) ≤ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) − e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ N \ Λ ) Z z, ˜ α Λ ( (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) π z, ˜ α Λ ( d (cid:101) ω (cid:48) ) + (cid:12)(cid:12)(cid:12)(cid:12) Z z, ˜ α Λ ( (cid:101) ω Λ N ) Z z, ˜ α Λ ( (cid:101) ω ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ c ) − e − H Λ ( (cid:101) ω (cid:48) Λ (cid:101) ω Λ N \ Λ ) Z z, ˜ α Λ ( (cid:101) ω ) (cid:12)(cid:12)(cid:12)(cid:12) π z, ˜ α Λ ( d (cid:101) ω (cid:48) ) . Now using the mean value theorem and by considering N large enough weobtain from assumptions (A2), (A3) and (A4) c ( (cid:101) ω ) ≤ (cid:88) j, ∆ j (cid:54)⊆ Λ N N ∆ j ( (cid:101) ω ) (cid:88) j (cid:48) , ∆ j (cid:48) ⊆ Λ ψ | j − j (cid:48) | (cid:90) N ∆ j (cid:48) ( (cid:101) ω (cid:48) Λ ) P z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) ) . Remark 6.1. In the last bound, we used from (A4) that the potential ψ ( x ) isnon-positive when | x | is large. One can do without this assumption and wouldget an extra factor 2. In the following, when not specified, j, k are indexes such that ∆ j , ∆ k (cid:54)⊆ Λand j (cid:48) , k (cid:48) are such that ∆ j (cid:48) , ∆ k (cid:48) ⊆ Λ.Now let B (cid:101) ω := sup { b + (cid:80) j, ∆ j (cid:54)⊆ Λ ψ | j − k (cid:48) | N ∆ j ( (cid:101) ω ) | ∆ k (cid:48) ⊆ Λ } with a, b comingfrom assumption (A3). Then we have (cid:90) N ∆ j (cid:48) ( (cid:101) ω (cid:48) Λ ) P z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) ) ≤ B (cid:101) ω a + (cid:90) N ∆ j (cid:48) ( (cid:101) ω (cid:48) Λ ) { N ∆ k (cid:48) ( (cid:101) ω (cid:48) ) > B (cid:101) ωa , ∀ k (cid:48) } P z, ˜ α Λ , (cid:101) ω ( d (cid:101) ω (cid:48) ) (cid:124) (cid:123)(cid:122) (cid:125) ∗ . But from assumption (A1) and (A3) ∗ ≤ e z | Λ | (cid:90) N ∆ j (cid:48) ( (cid:101) ω (cid:48) Λ ) { N ∆ k (cid:48) ( (cid:101) ω (cid:48) ) > B (cid:101) ωa , ∀ k (cid:48) } exp (cid:32)(cid:88) k (cid:48) − aN ∆ k (cid:48) ( (cid:101) ω (cid:48) ) + B (cid:101) ω N ∆ k (cid:48) ( (cid:101) ω (cid:48) ) (cid:33) π z, ˜ α Λ ( d (cid:101) ω ) ≤ e z | Λ | (cid:90) N ∆ j (cid:48) ( (cid:101) ω (cid:48) Λ ) exp (cid:32) − a (cid:88) k (cid:48) N ∆ k (cid:48) ( (cid:101) ω (cid:48) ) (cid:33) π z, ˜ α Λ ( d (cid:101) ω ) ≤ κ < ∞ and we finally obtain c ( (cid:101) ω ) ≤ (cid:32) κ (cid:48) + κ (cid:48)(cid:48) sup { (cid:88) k ψ | k − k (cid:48) | N ∆ k ( (cid:101) ω ) | ∆ k (cid:48) ⊆ Λ } (cid:33) (cid:88) j, ∆ j (cid:54)⊆ Λ N (cid:88) j (cid:48) ψ | j − j (cid:48) | N ∆ j ( (cid:101) ω ) , Acknowledgement: This work was supported in part by the ANR projectPPPP (ANR-16-CE40-0016) and by Deutsche Forschungsgemeinschaft (DFG)through grant CRC 1294 ”Data Assimilation”, Project A05. 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