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Poisson approximation for large-contours in low-temperature Ising models

We consider the contour representation of the infinite volume Ising model at low temperature. Fix a subset V of Z^d, and a (large) N such that calling G_{N,V} the set of contours of length at least N intersecting V, there are in average one contour in G_{N,V} under the infinite volume"plus"measure. We find bounds on the total variation distance between the law of the contours of lenght at least N intersecting V under the"plus"measure and a Poisson process. The proof builds on the Chen-Stein method as presented by Arratia, Goldstein and Gordon. The control of the correlations is obtained through the loss-network space-time representation of contours due to Fernandez, Ferrari and Garcia.


Introduction
The infinite volume Ising model is one of the most studied in Statistical Physics. One of the fundamental historical fact was the Peierls argument for the existence of more than one phase at low enough temperature -which we assume throughout this paper-in any dimension d ≥ 2. This argument was set by Dobrushin as the existence of at least two extremal Gibbs states, ν + and ν − that are obtained as infinite volume limits of Gibbs measures with + resp. − boundary conditions. In the Physics literature, a configuration of spins which is typical with respect to ν + , is frequently described as a set of islands of − within a sea of + ; the contours being the boundaries of the islands. Various questions can be asked about typical configuration of contours at low temperature. A very simple one that does not seem to have attracted attention is: what is the number of occurrences of contours larger than a fixed (big) N, that intersect a fixed volume V = V (N) say centered at the origin. To describe the physical idea of rarity of the island of − , a Poisson type behavior naturally comes in mind. We prove indeed that the distribution of these occurrences, under the condition that the mean number of occurrences in the volume V (N) is a fixed value λ, approaches sharply a Poisson process of mean λ. The paper is organized as follows: in section 2, we give the definitions, state the main result and the Arratia-Goldstein-Gordon formulation of the Chen-Stein Method used in the proof. In section 3 we recall the graphical construction of the loss-network space-time representation of contours of Fernández-Ferrari-Garcia which we apply to obtain the relevant bounds. In section 4 we prove the main result.

Definitions and the main result
Peierls introduced a map between typical configurations of ν + or ν − into an ensemble of objects -the contours-interacting only by perimeter-exclusion. See, for instance, Section 5B of Dobrushin [6], for a concise and rigorous account of this mapping. Contours are hyper surfaces formed by a finite number of (d − 1)-dimensional unit cubes -links for d = 2, plaquettes for higher dimensions-centered at points of Z d and perpendicular to the edges of the dual lattice Z d +( 1 2 , · · · , 1 2 ). To formalize their definition, let us call two plaquettes adjacent if they share a (d − 2)-dimensional face. A set of plaquettes, γ, is connected if for any two plaquettes in γ there exists a sequence of adjacent plaquettes in γ joining them. The set γ is closed if every (d − 2)-dimensional face is covered by an even number of plaquettes in γ. Contours are connected and closed sets of plaquettes. For example, in two dimensions contours are closed polygonals. Two contours γ and θ are said compatible if no plaquette of γ is adjacent to a plaquette of θ. In this case we write γ ∼ θ. In two dimensions, therefore, contours are compatible if and only if they do not share the endpoint of a link. In three dimensions two compatible contours can share vertices, but not sides of plaquettes. Ising spin configurations in a bounded region with "+" (or "−") boundary condition are in one-to-one correspondence with families of pairwise compatible contours.
The set of possible contours contained in Λ ⊂ Z d will be denoted G[Λ]. The set of configurations of compatible contours is X We denote |γ| the number of plaquettes of the contour γ and define the finite-volume Gibbs measure µ Λ on X [Λ]: for η ∈ X [Λ], We fix β > β * , the solution of α 0 (β) = 1, where α 0 (β) := θ:θ∋0 e −β|θ| . Then there exists a unique weak limit µ = lim Λ→Z d µ Λ [7,8]. Notice that β * is strictly bigger than β P , the Peierls inverse-temperature, which is the infimum of the β making the previous sum finite. Let p γ = µ(dη) η(γ) be the probability of the presence of contour γ under µ. Fix N > 0, let V = V (N) ⊂ Z d and λ > 0 be such that if we define G = G(N, V, λ) := {κ ∈ G[Z d ] : |κ| ≥ N and κ∩V = ∅} as the set of contours with length at least N intersecting V , then γ∈G p γ = λ. Hence V is a set such that in average there are λ contours of length at least N intersecting V . Define the processes X := (η(γ) : γ ∈ G), Y := (Y (γ) : γ ∈ G) where η has distribution µ and Y (γ) are iid with Poisson distribution of mean p γ . We denote L(X) the law of a process X and L(X) − L(Y) TV the total variation distance between the laws of the processes X and Y [2].
The proof of this theorem is an application of the Chen-Stein method [10,5,2,3] as proposed in [2]. In order to describe it, let the distance between contours be d(γ, θ) [2] shows that the total variation distance between the Poisson process Y and the process X is dominated by 2(2b 1 + 2b 2 + b 3 ). To prove Theorem 1 we choose D equal to N times an appropriate constant and then find bounds on b i . The bound on b 1 does not offer problems, but the bound on b 2 requires the inequality p γθ ≤ p γ e −β|θ| +p θ e −β|γ| proved in Lemma 4 below. The Chen-Stein method has been applied mostly to examples where b 3 = 0 [2], while in our case b 3 = 0. To dominate b 3 we control the correlations between "far" contours in Lemma 7 below. The proof of those lemmas is based on the graphical representation of a loss network, a Markov process having µ as invariant measure [7,8]. There are many papers dealing with Poisson approximations; we limit ourselves to quote the books [1] and [3] -with a nice introduction explaining the Chen-Stein method-, the original paper of Chen [5] and a monograph of Stein [10]. There are a few results for random fields. The number of "−" spins in the "+" measure of the Ising model at low temperature and/or high external "+" magnetic field converges to Poisson [4,9]. In this case the variables are positively associated and the Chen-Stein method works via cluster expansions giving explicit convergence rates [9]. In [8] the approximation is established for the contours of the Ising model at low temperature and without magnetic field. The available bounds for the total variation distance are of the order of the inverse of the volume of the observed set V [9,8].

Graphical construction
In this section we quickly review the construction and some results of [7] relevant to our proof. To each contour γ ∈ G(Z d ) we associate an independent (of everything) marked Poisson process N γ on R with rate e −β|γ| . We call . For each occurrence time T i (γ) of the process N γ we choose an independent mark S i (γ) exponentially distributed with mean 1. At the Poisson time-event T i (γ) a contour γ appears and it lasts S i (γ) time units. The resultant object is the random family For t ∈ R we define ξ t (γ, C) = C ∈ C 1{Basis (C) = γ, Life (C) ∋ t}. The above process, called the free network, is a product of independent stationary birth-and-death processes on N G[Z d ] with t ∈ R whose generator is given by The invariant (and reversible) measure for this process is the product measure µ 0 on N G[Z d ] with Poisson marginals: µ 0 {ξ(γ) = k} = exp e −β|γ| (e −β|γ| ) k /k! In particular, for any t ∈ R, Ef (ξ t ) = µ 0 f .
We say that cylinders C and C ′ are incompatible and write C ′ ∼ C if and only if Basis (C) ∼ Basis (C ′ ) and Life (C) ∩ Life (C ′ ) = ∅. We say that two sets of cylinders A and A ′ are incompatible and write A ∼ A ′ if there is a cylinder in A incompatible with a cylinder in A ′ . Otherwise we use the sign ∼ for compatibility. For any cylinder C define the set of ancestors of C as the set of cylinders in C born before C that are incompatible with C: A C 1 = {C ′ ∈ C ; C ′ ∼ C, Birth (C ′ ) < Birth (C)}. Recursively for n ≥ 2, the nth generation of ancestors of C is A C n = {C ′′ : C ′′ ∈ A C ′ 1 for some C ′ ∈ A C n−1 }. Let the clan of C be the union of its ancestors: A C = n≥1 A C n . Under the condition β > β * all cylinders in C have a finite clan with probability one. This property called no backwards oriented percolation is essential to show that the loss network η t can be constructed in a stationary way for t ∈ R. Assume that there is no backwards oriented percolation. The construction is as follows.
All cylinders in C are classified as kept and erased. Since all clans are finite, we can write C = ∪ n≥0 C n , where C n := {C ∈ C : . Let the set of kept cylinders be K = ∪ n K n and the set of erased cylinders be D = ∪ n D n . Clearly K∪D = C. The event {C ∈ K} is measurable with respect to the sigma field generated by A C . In words, it is sufficient to know the (finite) clan of C to know if C is kept or erased. The stationary loss network is defined by η t (γ, C) = C∈K 1 Basis (C) = γ, Life (C) ∋ t . The process η t is Markovian with generator The unique invariant (and reversible) measure for this process is the Gibbs measure µ. In particular, for any t ∈ R, Ef (η t ) = µf . We study properties of µ by studying the law of η 0 , the stationary loss network at time zero.
The presence/absence of contours intersecting a region Λ at time t depends only on the set A Λ,t , the union of the clans of the cylinders of C with basis intersecting Λ and life containing time t, that is A Λ,t := {C ′ ∈ A C : The reason is that in order to determine if C ∈ K it suffices to look at A C . When t = 0 we will use the notation A Λ instead of A Λ,0 .
Let C γ,t,r = (γ, −t, t+r) be a cylinder with Birth (C γ,t,r ) = −t, Basis (C γ,t,r ) = γ, Life (C γ,t,r ) = t + r. Conditioning to the birth time of the (kept) cylinder alive at time zero with basis γ, Coupling of clans. For disjoint sets Λ and Υ of Z d it is possible to construct (A Λ , A Υ , A Λ , A Υ ), a coupling between four sets of cylinders satisfying (a) then the marginals coincide: A Λ = A Λ and A Υ = A Υ . We write P and E for the probability and the expectation of the coupling. The following bound for the probability of incompatibility between clans follows as in the proof of (2.13) of [8]: for any β ′ ∈ (β * , β), 4 Proof of the theorem The proof of the theorem is based on a sequence of lemmata.
Lemma 2 Denote ρ = α 0 (β). Then for γ such that |γ| ≥ N, PROOF. Bounding the conditional probability inside the integral in (4) by one we get the upper bound. For the lower bound notice that the event {C γ,t,r ∈ K} is measurable with respect to the sigma field generated by the cylinders born before −t, the birth time of C γ,t,r . Hence P(C γ,t,r ∈ K | C γ,t,r ∈ C) = P θ:θ ∼γ The first inequality follows from η t (γ) ≤ ξ t (γ); the second identity from the distribution of ξ t described in Section 3. Inserting this inequality in (4) we get the left inequality in (6).

Lemma 3 The following inequalities hold
PROOF. The first inequality is immediate. Bounding below the sum in (9) by one term (choose any γ 0 such that |γ 0 | = N) we dominate the middle term in (9) by |γ 0 |/p γ 0 which using the left inequality in (6) is bounded by the rhs of (9).

Lemma 6
The following bound holds for b 2 PROOF. By Lemma 4, b 2 is bounded above by The first term was controlled in the previous lemma. The second one can be treated in a similar way.

Lemma 7
The following bound holds for b 3 . For any β ′ ∈ (β * , β), letting where the identity comes from property (c) of the coupling defined in Section 2 and the second inequality from property (d). We divide the sum in (20) in two parts. The first one is over those γ ∈ G of length bigger than KN, for some K to be chosen later. Dominating the indicator function by one, this sum is bounded by γ∈G, |γ|≥KN 2p γ ≤ 2λ e (β+ρ)N e −(β−β * )KN The second sum is over those γ ∈ G with length less than KN. Dominating η 0 (·) by one, the summand is bounded by four times using (5) in the first inequality. Hence the second sum is bounded by using (9) to bound |V |. The sum (20) is then bounded above by the sum of (21) and (23). Fixing K = β+ρ β−β ′ + 1, we get (18).
Proof of Theorem 1. It remains to choose D in such a way that 2(2b 1 + 2b 2 + b 3 ) is bounded above by the right hand side of (2). We already know that b 1 and b 2 decay as e −(β−β * −ε)N . If we choose D = δN in (18), where With this bound and the choice ε = β ′ − β * in (13) and (16) we get (2). The constant M(β, β ′ ) is given by