Odd cutsets and the hard-core model on <span class="mathjax-formula">$\mathbb {Z}^{d}$</span>

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www.imstat.org/aihp DOI:10.1214/12-AIHP535

Odd cutsets and the hard-core model on Z ^d

Ron Peled

^a,1

and Wojciech Samotij

^b,2

aSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel. E-mail:peledron@post.tau.ac.il;

url:http://www.math.tau.ac.il/~peledron

bSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel; and Trinity College, Cambridge CB2 1TQ, UK.

E-mail:samotij@post.tau.ac.il; url:http://www.math.tau.ac.il/~samotij Received 18 June 2011; revised 25 November 2012; accepted 01 December 2012

Abstract. We consider the hard-core lattice gas model onZ^d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceedsCd⁻^1/3(logd)², the model exhibits multiple hard-core measures, thus improving the previous bound ofCd⁻^1/4(logd)^3/4given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets inZ^d, the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model.

We provide a refined combinatorial analysis of the structure of these cutsets yielding a quantitative form of concentration for their possible shapes as the dimensiondtends to infinity. This analysis relies upon and improves previous results obtained by the first author.

Résumé. Nous étudions la structure de phase, en grande dimension, d’un modèle de sphères dures sur le réseauZ^d. Nous prouvons que le modèle présente plusieurs mesures lorsque le paramètre de densité dépasseCd⁻^1/3(logd)², améliorant ainsi la borne de Cd⁻^1/4(logd)^3/4obtenue par Galvin et Kahn. Notre approche repose sur l’étude de certaines classes d’ensembles séparateurs dansZ^d, constituées d’ensembles impaires, qui délimitent la frontière entre différentes phases du modèle de sphères dures. Nous faisons une analyse combinatoire précise de la structure de ces ensembles séparateurs et obtenons une forme quantitative de la concentration des différentes formes possibles prises par ces ensembles lorsque la dimensiondtend vers l’infini. Cette analyse repose sur des méthodes obtenues auparavant par le premier auteur, tout en les améliorant.

MSC:82B20; 60C05; 05C30; 05A16

Keywords:Edge cutsets; Gibbs measures; Hard-core model; Integer lattice

1. Introduction

Thehard-core model(short forhard-core lattice gas model) was originally introduced in statistical mechanics as a simple mathematical model of a gas whose particles have non-negligible size and cannot overlap, see [3,6]. Later, the model was rediscovered in operations research in the context of certain communication networks, see [8,9]. The model has also attracted interest in ergodic theory where it is known by the name “the golden mean subshift” [13], since its topological entropy on the one-dimensional lattice is the logarithm of the golden ratio(1+√

5)/2.

LetGbe a finite graph with vertex setV and letλbe a positive real. Aconfigurationω∈ {0,1}^V is calledfeasible if it is the characteristic function of some independent set inG, i.e., if for no adjacent pair{v1, v2}of vertices ofG,

1Supported by a Marie Curie Reintegration Grant SPTRF from the Commission of the European Communities.

2Supported by ERC Advanced Grant DMMCA.

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it satisfiesω(v₁)=ω(v₂)=1. Thehard-core model onGwith activityλ is the probability measureμon{0,1}^V defined by

μ(ω)=

Z⁻¹

v∈Vλ^ω(v) ifωis feasible,

0 otherwise,

whereZis the appropriate normalizing constant (called thepartition function) which makesμa probability measure, i.e.,Z=

ω

vλ^ω(v), whereωranges over the set of all feasible configurations. In other words,μ(ω)is proportional toλ^|{^v:^ω(v)⁼¹^}|ifωis a feasible configuration andμ(ω)=0 otherwise.

IfG is an infinite graph, then we call a probability measure on {0,1}^V ahard-core measure if for each finite W⊆V, everyω1∈ {0,1}^W, andμ-a.e.ω2∈ {0,1}^V^\^W, we have

μ

ω|W=ω₁ω|V\W=ω₂

=

Z_W⁻¹

v∈Wλ^ω(v) ifω₁ω₂is feasible,

0 otherwise,

whereω1ω2is the configuration onV that agrees withω1onW and withω2onV \W. A standard compactness argument shows that for any infinite but locally finite graphGand anyλ, there exists at least one hard-core measure onGwith activityλ. One of the main questions in the study of the hard-core model is to decide, for givenGand λ, whether there is a unique or multiple hard-core measures onGwith activityλ. An important contribution to this problem was made by van den Berg and Steif [16], continuing previous work of van den Berg [15], who showed that for any connected infinite graph³G, there is a unique hard-core measure with activityλwheneverλ <₁₋^p^c_p^(G)

c(G). Here, p_c(G)stands for the site percolation threshold for the graphG.

The most-studied case of a hard-core model is that whenGis the nearest-neighbor graph of the integer latticeZ^d, i.e., the graph on the vertex setZ^d in which two vertices are adjacent if and only if theirL¹-distance is equal to 1.

See Fig.1below for a simulation of the model in two dimensions. The above-mentioned result of van den Berg and Steif, combined with the simple lower boundpc(Z^d)≥_2d¹₋₁, proves that ifλ < _2d¹₋₂, then there is a unique hard- core measure with activityλonZ^d. The seminal result of Dobrushin [3] says that whend≥2 andλis sufficiently large (depending on d), then Z^d admits multiple hard-core measures with activity λ; Dobrushin’s result was later rediscovered by Louth [10]. The lower bound onλproved by Dobrushin is rather weak (see the discussion in [5]) as it grows withd, quite in contrast with the popular belief that for a givenλ, the existence of multiple hard-core measures in dimensiondshould imply the existence of multiple hard-core measures in all higher dimensions. Almost 40 years had passed since Dobrushin published his result before Galvin and Kahn [5] proved that the threshold activity that implies the existence of multiple hard-core measures onZ^dtends to 0 asd→ ∞.

Theorem 1.1 ([5]). There exist constantsd₀andCsuch that ifd≥d₀andλ≥Cd⁻^1/4(logd)^3/4,then the graphZ^d has multiple hard-core measures with activityλ.

The bound in Theorem1.1 is undoubtedly not best possible. It is commonly believed that multiple hard-core measures should appear when the activityλis either atΘ(1/d)or atΘ(logd/d). A natural question in this line of research is that of the existence of acritical activityλc(d)such that there are multiple hard-core measures as soon as λ > λ_c(d)and a unique measure ifλ < λ_c(d). Even though it is believed that in the case ofZ^d, such critical activity exists, so far it has not been proved or disproved. Interestingly, it was shown in [2] that there are graphs for which there is no such critical activity. Following [5], even if the hard-core model onZ^ddoes not have a critical activity, one can still define a similar quantityλ(d)to be the supremum of those activitiesλfor which there is a unique hard-core measure. With this notation at hand, one can rephrase Theorem1.1asλ(d)=O(d⁻^1/4(logd)^3/4)asd→ ∞.

The aim of this paper is twofold. First, using the counting and structure theorems forodd cutsetsproved by the first author in [11], we give what we think is a shorter and possibly more transparent proof of (a slightly weaker version of) the result of Galvin and Kahn.

Theorem 1.2. There are constantsd₀andCsuch that ifd≥d₀,then λ(d)≤Cd⁻^1/4(logd)².

3More precisely, a countably infinite, locally finite, connected graph.

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Second, we prove a refined version of one of the main results of [11], the so-calledinterior approximationtheorem for odd cutsets inZ^dand use it to improve the bound in Theorem1.1.

Theorem 1.3. There are constantsd₀andCsuch that ifd≥d₀,then λ(d)≤Cd⁻^1/3(logd)².

We would like to point out that, mimicking the general strategy in [11], we try to separate the probabilistic and geometric parts of the argument as much as possible. First of all, we hope that this makes our argument better structured and more easily comprehensible. Secondly, we aim to stress our belief that a further refinement of the geometric part will improve the bound onλ(d). Last but not least, this separation allows us to reuse most of the proof of Theorem1.2 in the proof of Theorem1.3. Finally, we remark that the ideas of using cutsets with the “odd” property and approximating them have been used in several previous works, e.g., in [1,4,5,12]. Moreover, the general strategy which we use in the proof of Theorems1.2and1.3closely follows that of Galvin and Kahn [5], see the more detailed discussion in Section3.

2. Preliminaries

2.1. Notation General

For a setX, we will denote the power set (the set of all subsets) ofXbyP(X)and for a positive integerk, we will abbreviate{1, . . . , k}by[k]. We always write log for the natural logarithm.

Graph theory

Given a graphH, we will denote its vertex and edge sets byV (H )andE(H ), respectively. For the sake of clarity, for two vertices v, w∈V (H ), we will sometimes writevw for the unordered pair{v, w}. IfW⊆V (H ), then we will writeH[W]for the graph induced byH onW andH\W forH[V (H )\W]. IfF ⊆E(H ), then we will write H\F for the graph obtained from H by removing from it all edges inF. For everyv∈V (H ), theneighborhood ofv, denotedN (v), is the set of all vertices that are adjacent tov; for a setS⊆V (H ), we letN_H(S)=

v∈SN (v).

The distance between two verticesv, w∈V (H ), denoted distH(v, w), is the length of the shortest path connectingv andwinH or∞if there is no such path. A setW⊆V (H )isconnectedif each pair of vertices inW is connected by a path, i.e., if distH(v, w) <∞for everyv, w∈W. Sometimes we will drop the subscriptH to keep things less cluttered, provided that the graphH is clear from the context. Theconnected componentof a vertexv(a connected setW) is the largest connected subset ofV (H )that contains the vertexv(the setW). For a positive integerk, thekth power ofH, denotedH^k, is the graph on the vertex setV (H )whose two verticesvandware adjacent if and only if distH(v, w)≤k. Finally, we will write(H )for the maximum degree ofH.

The graphZ^d

Since this graph is clearly connected and bipartite, it is natural to refer to its two color classes as the even and the odd vertices, denotedVêvenandVôddrespectively. The setVêven(Vôdd) consists of those lattice points whose coordinates sum up to an even (odd) number. Recall that two verticesv, w∈Z^dare adjacent if and only ifw=v+f for some f ∈Z^dwhoseL¹-norm is equal to 1. It is therefore convenient to denote all such vectorsf, i.e., all{−1,0,1}-vectors with exactly one non-zero coordinate, byf₁, . . . , f_2d. For example, observe thatN (v)= {v+f_j: j∈ [2d]}for every v∈Z^d. Finally, when a feasible configurationω∈ {0,1}^Z^d is clear from the context,V^vac will denote the set of all vacantvertices, i.e.,V^vac= {v∈Z^d: ω(v)=0}. If a vertex is not vacant, then we say that it isoccupied.

2.2. Finitized version of the problem

As observed by Galvin and Kahn [5], the problem of showing the existence of multiple hard-core measures can be finitized as follows. Let Gn be the subgraph of Z^d induced on the set of lattice points in[−n, n]^d and define the

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boundary B_n of this subgraph by B_n= [−n, n]^d\ [−(n−1), n−1]^d. Letμ_n be the (unique) hard-core measure onG_n with activityλ, letμêven_n beμ_n conditioned on the eventω(v)=1 for allv∈B_n∩Vêven, and defineμôdd_n accordingly. As shown in [5], there are positive constantscandC such that the following holds. If for every fixed x∈Z^dand sufficiently largen,

μ^even_n

ω(x)=1

< c/d ifx∈V^odd and μ^odd_n

ω(x)=1

< c/d ifx∈V^even,

then there are multiple hard-core measures onZ^d with activity λ, provided that λ > C/d. In view of the above, Theorem1.3will be a direct consequence of the next theorem, whose proof is the main part of this paper.

Theorem 2.1. There exist constantsd0andC such that ifd≥d0,λ≥Cd⁻^1/3(logd)²,n≥2,andxis an arbitrary even vertex ofG_n,then

μ^odd_n

ω(x)=1

< c/d.

Moreover,the same result holds when the roles of even and odd vertices are reversed.

Remark. We will first prove Theorem 2.1under the stronger assumption that λ≥Cd⁻^1/4(logd)²,which implies Theorem1.2.Later,we will refine our approach and prove it for the stated regime.

2.3. Tools

In this section, we collect two auxiliary lemmas that will be used in the proof of our main result. Both are fairly standard and can be surely found in the literature. The proofs are given only for the sake of completeness.

Lemma 2.2. LetM be an integer,letH be an arbitrary graph,and let v∈V (H ).The number of connected sets E⊆V (H )such thatv∈Eand|E| =Mdoes not exceed((H ))^2M⁻².

Proof. For every suchE, we fix an arbitrary spanning treeT_EofE. Starting fromv, we perform a depth-first search onT_E, starting and ending atv and passing through every edge exactly twice. Since every spanning tree ofEhas exactlyM−1 edges, the number of possibilities for such a walk (and hence forE) is not larger than the number of

walks of length 2M−2 inH that start atv.

Our second lemma formalizes the following intuition. If an eventA in some discrete probability space(X, μ) admits an expanding transformationT:A→P(X), i.e., a map for whichμ(T (a))is much larger thanμ(a)for every a∈A, thenμ(A)is small, provided that nox∈Xappears in the image ofT too many times.

Lemma 2.3. LetXbe a finite set,letA⊆X,and letμbe a probability measure onX.Suppose that there are positive numberspandqand a mappingT: A→P(X)such that for eacha∈Aand eachx∈X,

μ T (a)

≥q·μ(a) and b∈A: x∈T (b)≤p.

Thenμ(A)≤p/q.

Proof. Our assumptions easily imply that μ(A)=

a∈A

μ(a)≤ 1 q ·

a∈A

μ T (a)

=1 q ·

a∈A

x∈T (a)

μ(x)

=1 q ·

x∈X

a∈A: x∈T (a)·μ(x)≤p q ·

x∈X

μ(x)=p q.

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Fig. 1. Typical configurations with odd boundary conditions forn=20,d=2, and activitiesλ=1 andλ=5, respectively. Odd and even occupied nodes are represented by black and gray circles, respectively. Even occupied nodes are surrounded by their corresponding Breaks. Simulated using coupling from the past [7].

3. Outline of the argument

Fix integersnandd and recall thatG_nis the subgraph ofZ^dinduced on the set of lattice points in[−n, n]^d. We start by observing that if the odd boundary vertices ofG_n are occupied, andx is an even vertex which is occupied, there must be an “outermost” surface inG_n in which the pattern “flips” from odd/occupied to even/occupied (see Fig.1).

Formally, this surface is defined in Section4.1as a set of edgesΓ having the following properties:

(i) Γ forms a minimal cutset separatingBnfromx. In other words,Γ partitionsGninto two connected components, the component of the boundary, denoted byA0(Γ ), and the component ofx, denoted byA1(Γ ).

(ii) The vertices on the outer boundary ofΓ are even and vacant and the vertices on the inner boundary ofΓ are odd and vacant.

Edge cutsets with the above properties play a prominent role in our analysis and we term them OMCut (for odd minimal edge cutsets). We remark that the idea of considering such cutsets also lies at the heart of the approach taken by Galvin and Kahn [5].

In what follows, let x be a fixed even vertex ofGn and let Ω be our “bad” event, i.e., the set of all feasible configurations with all odd boundary vertices (i.e., the vertices in the setV^odd∩B_n) andx occupied. For a given configurationω∈Ω we term the aboveΓ as Break(ω). A simple consequence of the above properties and the fact thatxis even is that|Break(ω)| ≥2d(2d−1)(Proposition4.5).

Our next observation is that applying the following “shift transformation” toω∈Ωyields a feasible configuration (see Fig.2):

Shiftj(ω)(v)=

ω(v+f_j) ifv∈A₁

Break(ω) , ω(v) otherwise.

A similar transformation was used by Galvin and Kahn [5]. The key properties of this transformation are:

(i) Shiftj is measure preserving, i.e.,μ^odd(Shiftj(ω))=μ^odd(ω).

(ii) Denotingω=Shiftj(ω), every vertex in the set{v∈A₁(Break(Γ )): (v, v+f_j)∈Break(ω)}is surrounded by vacant vertices inω. Thus an arbitrary modification ofωon these vertices yields a feasible configuration.

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Fig. 2. The shift transformation. Odd and even occupied nodes are represented by black and gray circles, respectively. The transformationT₁ shifts the occupied nodes inside the cutset Break(ω)to the left. In the new configuration all vertices having an edge of Break(ω)on their right (represented by white circles) necessarily have no occupied neighbors and can thus be set occupied or vacant arbitrarily.

Fig. 3. Odd cutsets approximating the boundary of a cube. On the left, every second boundary vertex can be “pushed out” independently of other vertices, yielding 2⁽^2d¹^−o^d^(1))Lodd cutsets withLedges, asL→ ∞. On the right, for every odd cutset obtained in such way, we have the additional option of independently “pushing out” vertices all of whose 2d−2 neighbors were “pushed out” in the first stage. Combining these two stages shows that asL→ ∞(along a subsequence) there are at least 2⁽^2d¹^+c^d^−o^d^(1))Lodd cutsets withLedges for somec_d>0.

These two properties and an application of Lemma2.3imply that for every fixedΓ ∈OMCut, μ^odd

Break(ω)=Γ

≤(1+λ)^−|^Γ^|^/2d. (1)

At this point one is tempted to conclude the proof of Theorem2.1by the union bound over all possibleΓ ∈OMCut.

This approach unfortunately fails since the number ofΓ ∈OMCut with a given size turns out to be too large. Indeed, if we let OMCutL= {Γ ∈OMCut:|Γ| =L}then|OMCutL| ≥2⁽^2d¹⁺^c^d^)Lfor somec_d>0, at least on a subsequence ofLs. This can be seen by counting those cutsets which approximate closely the boundary of a large cube with sides parallel to the axes ofZ^d(see Fig.3), but we neither prove nor use this fact in our work.

Instead, we introduce a (very coarse) measure of regularity on OMCut (the quantityR_Γ(E₁(Γ ))defined in Sec- tion 4.3) and partition all cutsets into “regular” and “irregular” ones. Theorem 4.7shows that the set of irregular

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cutsets is fairly small and hence the union bound (with the above estimate (1)) is sufficient to bound the probability that Break(ω)is irregular (see Theorem4.15).

In order to bound the probability that Break(ω) is regular, we partition the set of all regular cutsets into a (rela- tively) small number of classes and give an estimate on the probability of Break(ω)belonging to each such class that is strong enough to facilitate a union bound over all classes. We obtain this partition by exploiting a certain concentration of shape phenomenon. It is shown in Theorems4.8and4.9that, informally, for sufficiently larged there is a set of

“shapes” which contain a good “approximation” to every cutset in OMCutL, with the number of such shapes significantly smaller than|OMCutL|. Adapting the shift transformation to this notion of approximation (see Section4.4.2), we are able to bound the probability that Break(ω)belongs to the set of cutsets with a given shape. The bound we get is only strong enough for cutsets which are regular (see Theorems4.16and4.17), necessitating the separate treatment of irregular cutsets given above. This last part of the argument, i.e., separate treatment of regular and irregular cutsets is the main novelty in our approach.

4. The argument

4.1. Locating the cutset

Let d, n≥2 and let x be an arbitrary even vertex ofGn. Let Fnôdd be the set of all feasible configurations with ω(v)=1 for allv∈B_n∩Vôddand letΩ_ndenote the “bad” event, i.e., the set of allω∈Fnôddwithω(x)=1. For the sake of clarity of the presentation, most of the time we will drop the subscriptnfromG_n,B_n,μôdd_n ,F_nôdd, andΩ_n.

Fix an arbitrary “bad” configuration ω∈Ω. Observe that all odd boundary vertices are occupied and all even boundary vertices are vacant, whereasxis an even vertex that is occupied and all its neighbors are odd and vacant. We are going to associate withωan edge cutset inGthat will mark the outermost break in the above boundary pattern, i.e., the outermost contour separating vertices that are odd and occupied from vertices that are even and occupied. More precisely, we letXbe the set of all vacant odd vertices, i.e.,X=V^odd∩V^vac, letA₀be the connected component of Bin the graphG\X(note thatBis connected inGand disjoint fromX), and note thatx /∈A₀asN (x)⊆X. Finally, letA1be the connected component ofx in the graphG\A₀ and letΓ be the set of all edges with one endpoint in A₀and one endpoint inA1. By definition, every path fromxtoBmust use an edge ofΓ and no strict subsetΓ⊆Γ has this property. In other words, we may say thatΓ is aminimal edge cutsetseparatingxfromB. To summarize, we have defined a mapping Break: Ω→P(E(G))that assigns to each configuration inΩ an edge cutset separatingx fromB. Below, we establish some crucial properties of this cutset.

LetΓ =Break(ω). For an arbitrary vertexv∈V (G), letPΓ(v)be the number of edges ofΓ that are incident tov and let

E0= v∈A₀: PΓ(v) >0

= v /∈A1: PΓ(v) >0

and E1= v∈A1: PΓ(v) >0 .

Proposition 4.1. E₀⊆V^even∩V^vacandE₁⊆V^odd∩V^vac.

Proof. Fix some v∈E₁ and let w be an arbitrary vertex with {v, w} ∈Γ; at least one such vertex exists since P_Γ(v) >0. Clearly,w∈A₀and hencev∈Xor otherwisev would also belong toA₀(recall thatA₀ is a maximal connected subset of vertices inG\X). It follows thatE₁⊆X=Vôdd∩V^vac. Since each vertex inE₀is adjacent to a vertex inE₁, it immediately follows thatE₀⊆Vêven. Finally, fix an arbitraryw∈E₀. It remains to be shown thatw is vacant. Sincew∈A₀, then there is a path inA₀connectingwtoB∩Vôdd. The immediate neighbor ofwon this path is an odd vertex that does not belong toXand hence it is occupied. Therefore,wmust be vacant.

Before we proceed, we have to take a little detour and introduce some terminology that will help us deal with cutsets arising in the procedure described above. Following [11], we let MCut be the set of allminimal edge cutsets separatingxandB, i.e., the set of allΓ ⊆E(G)such that any path fromxtoBmust cross an edge ofΓ and no strict subsetΓ⊆Γ shares this property. For everyΓ ∈MCut, letA₀(Γ )andA₁(Γ )denote the connected components of BandxinG\Γ, respectively. By minimality ofΓ, every edge ofΓ must have an endpoint inA0(Γ )and an endpoint

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inA₁(Γ ). It follows thatA₀(Γ )andA₁(Γ )form a partition of the vertex set ofG. For every vertexv∈V (G), let P_Γ(v)be the number of edges inΓ that are incident tov, and let

E₀(Γ )= v∈A₀(Γ ): P_Γ(v) >0

and E₁(Γ )= v∈A₁(Γ ): P_Γ(v) >0 .

Finally, we define OMCut, the set ofodd minimal edge cutsets(or simplyodd cutsets) to be the set of allΓ ∈MCut that satisfyE₁(Γ )⊆V^odd.

Since the new definitions ofA₁,E₀, andE₁coincide with the old ones (more precisely, ifΓ =Break(ω)for some ω∈Ω and the procedure described at the beginning of this section defines setsA₁,E₀, andE₁, thenA₁=A₁(Γ ), E₀=E₀(Γ ), andE₁=E₁(Γ )), then it is easy to see that the mapping Break assigns to each configuration inΩ an odd cutsetΓ =Break(ω)∈OMCut satisfying Proposition4.1.

We need to establish one more crucial property of the mapping Break. To this end, withΓ ∈OMCut fixed, we say that a configurationωis aninterior modificationof another configurationωif they agree everywhere but at most on the setA1(Γ )\E1(Γ ), i.e., ifω(v)=ω(v)for allv /∈A1(Γ )\E1(Γ ). A moment of thought assures us that the following is true about the map Break.

Proposition 4.2. Let Γ ∈OMCut and assume that Γ =Break(ω) for some configuration ω∈ Ω. Then Γ = Break(ω)for everyωthat is an interior modification ofω.

Proof. LetX= {v∈Vôdd:ω(v)=0}and letX= {v∈Vôdd:ω(v)=0}. Sinceωis an interior modification ofω, it follows thatX\A1(Γ )=X\A1(Γ )and Proposition4.1implies thatX, X⊇E1(Γ ). By definition, every path from BtoA₁(Γ )has a vertex inE₁(Γ ). It follows that the connected components ofBinG\XandG\Xare identical and hence Break(ω)=Break(ω). To see this, recall that Break(ω)depends solely on the connected component ofB in the graphG\ {v∈Vôdd: ω(v)=0}, which we denoted byA₀at the beginning of this section.

4.2. Minimal edge cutsets

In this section, we give basic definitions and properties of minimal edge cutsets that will be used throughout the paper.

The following property of the setsE₀(Γ )andE₁(Γ ), which is a direct consequence of the results proved by Timár [14], will be crucial in our considerations.

Proposition 4.3. For everyΓ ∈MCut,the setsE₀(Γ )andE₁(Γ )are connected in the graphG².

Proof. Following [14], for a graphH, anyC⊆V (H ), and av∈V (H ), define the outer boundary ofCvisible from v, denoted∂_H^vis(v, C), to be the set of ally∈NH(C)that are connected tovin the graphH\C. It follows from [14], Lemma 2, that for any connected subsetC⊆V (G)such thatB⊆C orB∩C=∅and anyv∈V (G)\C, the set

∂_G^vis(v, C)is connected inG². The reason for the assumption thatC either containsB or is disjoint fromB is that such setC, when viewed as a subset in the graphZ^d, satisfies∂_G^vis(v, C)=∂_Z^vis_d(v, C)for everyv∈V (G)\C⊆Z^d. To conclude, we simply note that the setsA0(Γ )andA1(Γ )are connected,B⊆A0(Γ ),B∩A1(Γ )=∅,E1(Γ )=

∂_G^vis(x, A0(Γ )), andE0(Γ )=∂_G^vis(b, A1(Γ ))for everyb∈B. For everyj ∈ [2d], we let

E1,j(Γ )= v∈E1(Γ ): {v, v+fj} ∈Γ . We also define

E1,e(Γ )= v∈E1(Γ ): PΓ(v)≥2d−√ d and

E1,j,x(Γ )= v∈E1,e(Γ ): v+fj∈A1(Γ )

= v∈E1,e(Γ ): {v, v+fj}∈/Γ .

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The letterestands forexposedas vertices inE_1,e(Γ )are exposed toΓ from many directions. The setsE_1,j,E_1,e, andE_1,j,xwill play an important role in our considerations. We also let

Γr= vw∈Γ: v∈E1(Γ )\E1,e(Γ ) , for everyj∈ [2d], let

Γ^j= vw∈Γ: v∈E₁(Γ )andw=v+f_j

and Γ_r^j=Γ_r∩Γ^j.

Additionally, note that for eachj∈ [2d], we have that|E1,j| = |Γ^j|, and|E1,j\E1,e| = |Γ_r^j|. We will repeatedly use the following simple facts.

Proposition 4.4. Let Γ ∈OMCut,let δ∈ {0,1},and let v∈E_δ(Γ ). For every j ∈ [2d], if v+f_j ∈A_δ(Γ ) (or, equivalently,{v, v+f_j}∈/Γ),thenN (v+f_j)⊆A_δ(Γ ).

Proof. Assume first thatδ=1. Letv∈E₁(Γ )and letj∈ [2d]be such thatv+f_j∈A₁(Γ ). SinceE₁(Γ )⊆Vôdd, thenv+f_j∈Vêven. SinceN (w)⊆A₁(Γ )∪E₀(Γ )for everyw∈A₁(Γ ),N (v+f_j)⊆Vôdd, andE₀(Γ )⊆Vêven, it

follows thatN (v+f_j)⊆A₁(Γ ). The caseδ=0 follows similarly.

Proposition 4.5. For everyΓ ∈OMCut,|Γ| ≥2d(2d−1).

Proof. For everyj ∈ [2d]andi∈ [2d] \ {j}, consider the pathx, x−f_j, x−f_j+f_i, . . . , x−f_j+f_i, whereis the smallest integer such thatx−f_j+f_i∈B. SinceΓ separatesxfromB, at least one edge on each such path must belong toΓ and this edge is certainly not{x, x−f_j}asx∈V^even∩A₁(Γ )⊆A₁(Γ )\E₁(Γ )and hence bothxand x−f_j belong toA₁(Γ ). Moreover, the edges of the form{x, x−f_j}are the only edges belonging to more than one

of these 2d(2d−1)paths. It follows that|Γ| ≥2d(2d−1).

Proposition 4.6. For everyΓ ∈OMCutandv∈V (G),P_Γ(v)≤2d−1.

Proof. Suppose thatP_Γ(v)=2d for somev∈V (G). Assume first that v∈A₁(Γ ). SinceA₁(Γ ) is connected in G\Γ,x∈A₁(Γ ), and all 2dedges incident tovbelong toΓ, it must be thatA₁(Γ )=E₁(Γ )= {v} = {x}. But recall thatx∈V^even, which contradicts the fact thatE₁(Γ )⊆V^odd(Γ is an odd cutset). Finally, asB⊆A₀(Γ )andA₀(Γ ) is connected, one can quickly rule out the other possibility thatv∈A0(Γ )and all 2dedges incident tovare inΓ. 4.3. Counting odd cutsets

In this section, which borrows heavily from [11], we shall state three theorems that estimate the number of odd cutsets in various settings. Before we do that, we need to introduce some more notation. Following [11], for every Γ ∈OMCut,v∈V (G), andE⊆V (G), we define

RΓ(v)=min PΓ(v),2d−PΓ(v) and R_Γ(E)=

v∈E

R_Γ(v).

For integersMandR, we let

OMCut(M, R)= Γ ∈OMCut:|E₁(Γ )| =MandR_Γ E₁(Γ )

=R . A key observation that may elucidate the above definitions is that, since

2d k

= 2d

2d−k

≤(2d)^min^{^k,2d⁻^k^} for everykwith 0≤k≤2d,

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then for everyv∈V (G), there are at most(2d)^R^Γ^(v) ways to chooseP_Γ(v)out of the 2d edges incident tov. One might think of the parameterRΓ(E1(Γ ))as a measure of the regularity ofΓ. Note thatRΓ(E1(Γ ))≤ |Γ|and that a value ofR_Γ(E₁(Γ ))significantly smaller than|Γ|indicates that most verticesvinE₁(Γ )haveP_Γ(v)close to 2d; this can be interpreted as some roughness ofΓ. The following result is a straightforward corollary of [11, Theorem 4.5].

Theorem 4.7. There exist constantsC andd₀such that for all integersM,R,anddwithd≥d₀, OMCut(M, R)≤exp

C(logd)²

d R

.

Recalling the definition ofE_1,e(Γ ), we say that a setE⊆V (G)is aninterior approximationtoΓ if

E₁(Γ )\E_1,e(Γ )⊆E⊆A₁(Γ ). (2)

The following result is a straightforward corollary of [11, Theorem 4.13].

Theorem 4.8. There exist constantsCandd₀such that for all integersLanddwithd≥d₀,there exists a familyE of subsets ofV (G)satisfying

|E| ≤exp

C(logd)² d^3/2 L

and such that for everyΓ ∈OMCutwith|Γ| =L,there is anE∈Ethat is an interior approximation ofΓ.

Remark. Theorems4.7and4.8were proved in[11]for an alternative definition ofOMCut.There, OMCutwas defined as the set of odd minimal cutsets separating two points(or a set and a point)in a discretetorus.However,the theorems apply to our setting,since we can think thatGn is embedded naturally in the discrete torusZ^d_4nand note that then OMCut (in our definition)is a subset of the set of odd minimal cutsets separatingx from a “far away” point in this torus.In addition,in[11],the upper bound on|E|in Theorem4.8had an additional factor of2.In our application, this factor can be absorbed in the constantCby Proposition4.5.

One of the main ingredients in our proof of Theorem1.3that will allow us to improve the bound onλ(d)given by Theorem 1.2is a refined version of Theorem4.8. The main idea behind this improved interior approximation theorem is specializing the familyEfrom the statement of Theorem4.8to work only for odd cutsets with a particular distribution of edges adjacent to exposed and non-exposed vertices (i.e., the vertices inE_1,eand the vertices inE₁\ E_1,e). To be more precise, for anyεwithd⁻^1/3≤ε≤1/2, recalling the definition ofΓ_r, we let

OMCut(ε)= Γ ∈OMCut:ε|Γ|<|Γ_r| ≤2ε|Γ| .

In other words, OMCut(ε) consists of those odd cutsets whose only aboutε-fraction of edges are adjacent to non- exposed vertices. Since interior approximations “detect” only non-exposed vertices, it should not come at a surprise that asεgets smaller, approximating cutsets in OMCut(ε)becomes easier. We show that the following statement is true.

Theorem 4.9. There exist constants C and d₀ such that for all integers L and d with d ≥d₀, and everyε with d⁻^1/2≤ε≤1/2,there exists a familyEof subsets ofV (G)satisfying

|E| ≤exp C√

ε(logd)² d^3/2 L

and such that for everyΓ ∈OMCut(ε)with|Γ| =L,there is anE∈E that is an interior approximation toΓ. Remark. In fact,our proof of Theorem4.9yields a somewhat stronger property of the family E.We show that for everyΓ ∈OMCut(ε)with|Γ| =L,there is anE∈Esuch that{v∈E₁(Γ ): P_Γ(v) <2d−√

εd} ⊆E⊆A₁(Γ ),see Proposition5.6.

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As we remarked in theIntroduction, the proof of our main result, Theorem1.3, has been split up into the geometric part, which was presented in this section, and the probabilistic part, which we will present in the next few sections.

Since these parts are almost completely independent, we will postpone the proof of Theorem4.9to Section5and use it as a “black box” when we derive the main result.

4.4. Definition of the transformations

In this section, we define the mapping that we alluded to in Section 3 and establish its key properties. Recall the definitions ofΩ andF^odd from Section4.1. Throughout this section, we fix someω∈Ω, letΓ =Break(ω),A₁= A₁(Γ ),E₀=E₀(Γ ),E₁=E₁(Γ ),E_1,e=E_1,e(Γ ), andE_1,j=E_1,j(Γ )andE_1,j,x=E_1,j,x(Γ )for everyj∈ [2d].

Our transformation will take one of two possible forms, depending on the shape ofΓ.

As a preparatory step, for everyj∈ [2d], we define thejthshift transformationShiftj:Ω→F^odd(see Fig.2) by Shiftj(ω)(v)=

ω(v+f_j) ifv∈A1, ω(v) otherwise.

We remark that such a transformation already appeared in [5].

Proposition 4.10. The configurationShiftj(ω)is indeed feasible with the odd boundary vertices occupied.In other words, Shiftj(ω)∈F^odd.

Proof. SinceΓ ∈OMCut, thenB∩A1=∅and therefore Shiftj(ω)(v)=ω(v)=1 for everyv∈B∩V^odd. It remains to check that Shiftj(ω)is feasible, i.e., for nov∈V (G)andi∈ [2d], bothvandv+f_i are occupied. If bothvand v+f_i belong toA₁or bothvandv+f_i are not inA₁, then this follows from the fact thatωis feasible. Otherwise, assume WLOG thatv /∈A₁andv+f_i∈A₁. It follows thatv∈E₀, so Shiftj(ω)(v)=ω(v)=0 by Proposition4.1.

Proposition 4.11. For allv∈E_1,j andi∈ [2d],we haveShiftj(ω)(v+f_i)=0.

Proof. Fix somev∈E_1,j andi∈ [2d]. By definition,v+f_j ∈E₀. Ifv+f_i ∈E₀, then Shiftj(ω)(v+f_i)=ω(v+ f_i)=0 by Proposition 4.1. If v+f_i ∈/E₀, then v+f_i +f_j ∈E₁ (since v+f_i +f_j ∈A₁ and it is adjacent to v+f_j ∈/A₁). It follows that Shiftj(ω)(v+f_i)=ω(v+f_i +f_j)=0, where the last equality again follows from

Proposition4.1.

Proposition 4.12. The shift transformation is preservesμôdd,i.e.,μôdd(Shiftj(ω))=μôdd(ω).

Proof. Since Shiftj(ω)(v)=ω(v) for all v /∈A₁, it suffices to show that |{v∈A₁: Shiftj(ω)(v)=1}| = |{v∈ A₁: ω(v)=1}|. To see this, note that ifv∈A₁ and Shiftj(ω)(v)=1, thenω(v+f_j)=1 and hencev+f_j∈A₁ since otherwisev+f_j∈E₀andω(v+f_j)=0 by Proposition4.1. Conversely, ifv+f_j ∈A₁andω(v+f_j)=1, thenv+f_j∈/E₁by Proposition4.1and hencev∈A₁and Shiftj(ω)(v)=ω(v+f_j)=1.

4.4.1. The shift transformation

We are now ready to define the first expanding transformationT₁: Ω→P(F^odd). First, for everyj∈ [2d], we define the transformationT1,j: Ω→P(F^odd)by lettingT1,j(ω)be the set of all configurationsωof the form

ω(v)=

Shiftj(ω)(v) ifv /∈E_1,j,

ε_v otherwise,

where(εv)vis an arbitrary{0,1}-sequence indexed byE1,j. Propositions4.10and4.11imply that each suchωindeed belongs toF^oddwhereas Proposition4.12implies that

μ^odd

T1,j(ω)

=(1+λ)^|^E^1,j^|μ^odd(ω).

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Next, observe that|Γ| =_2d

j=1|Γ^j|and hence|Γ^j| ≥ |Γ|/(2d)for somej; in fact, sinceΓ ∈OMCut, we have that

|Γ^j| = |Γ|/(2d)for allj, see [11], but we will not need this. We define the transformationT₁byT₁(ω)=T_1,j(ω), wherej=j (Γ )=j (Break(ω))is the smallest indexj satisfying|Γ^j| ≥ |Γ|/(2d). It follows that

μ^odd T₁(ω)

=(1+λ)^|Ê^1,j^|μôdd(ω)=(1+λ)^|^Γ^j^|μôdd(ω)≥(1+λ)^|^Γ^|^/(2d)μôdd(ω). (3) Finally, we show that we can “invert”T₁if we know Break(ω).

Proposition 4.13. For everyΓ ∈OMCut,andω∈F^odd,there is at most oneω∈Ω satisfyingΓ =Break(ω)and ω∈T₁(ω).

Proof. WithΓ ∈OMCut fixed, letA1=A1(Γ ),E1=E1(Γ ), andj=j (Γ ). Letω∈Ω satisfyΓ =Break(ω)and ω∈T₁(ω)=T_1,j(ω). We show that we can recoverωfromω. By the definition ofT_1,j, we haveω(v)=ω(v)for allv /∈A₁andω(v)=ω(v−f_j)for allv∈A₁such thatv−f_j∈A₁. Finally, ifv∈A₁butv−f_j∈/A₁, thenv∈E₁

andω(v)=0 by Proposition4.1.

4.4.2. The shift+erase transformations

Next, we define the second expanding transformation T₂: Ω→P(F^odd). First, for every j ∈ [2d] we define the transformationT_2,j: Ω→P(F^odd)by lettingT_2,j(ω)be the set of all configurationsωof the form

ω(v)=

⎧⎨

⎩

Shiftj(ω)(v) ifv /∈E_1,j∪E_1,e, ε_v ifv∈E_1,j\E_1,e,

0 ifv∈E_1,e,

where(εv)v is an arbitrary {0,1}-sequence indexed byE1,j\E1,e. Again, Propositions4.10 and4.11imply that each suchω indeed belongs to F^odd. With the aim of computingμ^odd(T2,j(ω)), first let Xj(ω) denote the set of exposed vertices that are occupied in Shiftj(ω), butT_2,j(ω)forces them to be vacant, i.e.,X_j(ω)= {v∈E_1,e:ω(v+ f_j)=1}, and observe thatX_j(ω)⊆E_1,j,xsince ifv∈E_1,e\E_1,j,x, thenv+f_j∈E₀and henceω(v+f_j)=0 by Proposition4.1. Now it is not hard to see that

μ^odd

T_2,j(ω)

=(1+λ)^|Ê^1,j^\Ê^1,e^|λ^−|^X^j^(ω)^|μôdd(ω).

The transformationT₂is defined byT₂(ω)=T_2,j(ω), wherej =j (Γ )=j (Break(ω))is the smallest indexj that maximizes the quantity|Γ_r^j| −8|E_1,j,x|. It follows that

μ^odd T2(ω)

=(1+λ)^|Ê^1,j^\Ê^1,e^|λ^−|^X^j^(ω)^|μôdd(ω)=(1+λ)^|^Γ^r^j^|λ^−|^X^j^(ω)^|μôdd(ω). (4) We close this section by showing how we can “invert”T2if we know an interior approximationEto Break(ω)(recall the definition of an interior approximation given in (2)) and the setX(ω)=Xj (Break(ω))(ω). More precisely, we first prove that givenEand anω∈T₂(ω), we can reconstruct Break(ω)and then, if we additionally specify the setX(ω), thenωis uniquely determined.

Proposition 4.14. For every E⊆V (G)and ω∈F^odd, there is at most oneΓ ∈OMCutsuch that the following holds:

(i) Ifω∈T2(ω)for someω∈Ω such thatEis an interior approximation toBreak(ω),thenBreak(ω)=Γ. (ii) For everyX⊆V (G), there is at most one ω∈Ω such that ω∈T₂(ω), E is an interior approximation to

Break(ω),andX_{j (Γ )}(ω)=X.

Proof. Letω∈Ω be any configuration such thatω∈T₂(ω). We first show (i), i.e., that we can recover Break(ω)if we know thatEis an interior approximation to it. To see this, define a configurationωby

ω(v)=

ω(v) ifv /∈E, 0 ifv∈E,

Odd cutsets and the hard-core model on <span class="mathjax-formula">$\mathbb {Z}^{d}$</span>

Odd cutsets and the hard-core model on Z d

Ron Peled

and Wojciech Samotij

Odd cutsets and the hard-core model on Z ^d