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Counting independent sets in amenable groups

Published online by Cambridge University Press:  24 May 2023

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
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Given a locally finite graph $\Gamma $ , an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $ , consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $ . Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$ , there exists a randomized $\epsilon $ -additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$ , where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $ -regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$ , there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$ . This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

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1 Introduction

Suppose that we are given a finite simple graph $\Gamma = (V,E)$ and we are asked to count its number of independent sets. An independent set is a subset $I \subseteq V$ such that $(v,v') \notin E$ (that is, $(v,v')$ is not an edge) for all $v,v' \in I$ . For example, if $\Gamma $ is the $4$ -cycle $C_4$ with $V = \{v_1,v_2,v_3,v_4\}$ and $E = \{(v_1,v_2),(v_2,v_3),(v_3,v_4),(v_4,v_1)\}$ , it can be checked that there are exactly seven different independent sets, namely $\emptyset $ , $\{v_1\}$ , $\{v_2\}$ , $\{v_3\}$ , $\{v_4\}$ , $\{v_1,v_3\}$ , and $\{v_2,v_4\}$ . A common generalization of this question is to ask for the ‘number’ of weighted independent sets in $\Gamma $ : given a parameter $\unicode{x3bb}> 0$ —usually called activity or fugacity—we ask for the value of the summation

$$ \begin{align*} Z_\Gamma(\unicode{x3bb}) := \sum_{I \in X(\Gamma)} \unicode{x3bb}^{\lvert I \rvert}, \end{align*} $$

where $X(\Gamma )$ denotes the collection of all independent sets in $\Gamma $ and $\lvert I \lvert $ , the cardinality of a given independent set I. Notice that we recover the original problem—that is, to compute $\lvert X(\Gamma ) \rvert $ —if we set $\unicode{x3bb} = 1$ . The sum $Z_\Gamma (\unicode{x3bb} )$ corresponds to the normalization factor of the probability distribution $\mathbb {P}_{\Gamma ,\unicode{x3bb} }$ on $X(\Gamma )$ that assigns to each $I \in X(\Gamma )$ a probability proportional to $\unicode{x3bb} ^{\lvert I \rvert }$ , that is, the so-called partition function (also known as the independence polynomial) of the well-known hardcore model from statistical physics.

In general, it is not possible to compute exactly $Z_\Gamma (\unicode{x3bb} )$ efficiently [Reference Luby and Vigoda31], even for the case $\unicode{x3bb} = 1$ [Reference Xia, Zhao, Cai, Cooper and Li56]; technically, to compute $Z_\Gamma (\unicode{x3bb} )$ is an $\mathrm {NP}$ -hard problem and to compute $\lvert X(\Gamma ) \rvert $ is a $\#\mathrm {P}$ -complete problem. Therefore, one may attempt to at least find ways to approximate $Z_\Gamma (\unicode{x3bb} )$ efficiently.

In recent years, there has been a great deal of attention given to the complexity of approximating partition functions of spin systems (e.g., see [Reference Barvinok4]). Among these systems, the hardcore model, possibly together with the Ising model [Reference Sinclair, Srivastava and Thurley47], occupies the most important place. One of the most notable results, due to Weitz [Reference Weitz55], and then Sly [Reference Sly48] and Sly and Sun [Reference Sly and Sun49], is the existence of a computational phase transition for having a fully polynomial-time approximation scheme (FPTAS) for the approximation of $Z_\Gamma (\unicode{x3bb} )$ . In simple terms, Weitz developed an FPTAS, a particular kind of efficient deterministic approximation algorithm, on the family of finite graphs with bounded degree $\Delta $ provided $\unicode{x3bb} < \unicode{x3bb} _c(\Delta )$ , where $\unicode{x3bb} _c(\Delta ) := {(\Delta -1)^{(\Delta -1)}}/{(\Delta -2)^{\Delta }}$ denotes the critical activity for the hardcore model on the $\Delta $ -regular tree $\mathbb {T}_\Delta $ . Conversely, a couple of years later, Sly and Sun managed to prove that the existence of even a fully polynomial-time randomized approximation scheme (FPRAS)—which is a probabilistic and therefore weaker version of an FPTAS—for $\unicode{x3bb}> \unicode{x3bb} _c(\Delta )$ would imply that $\mathrm {NP} = \mathrm {RP}$ , the equivalence of two well-known computational complexity classes which are widely believed to be different [Reference Arora and Barak2].

The work of Weitz exploited a technique based on trees of self-avoiding walks and a special notion of correlation decay known as strong spatial mixing that, in particular, holds when the graph is $\mathbb {T}_\Delta $ and $\unicode{x3bb} < \unicode{x3bb} _c(\Delta )$ . Later, Sinclair et al [Reference Sinclair, Srivastava, Štefankovič and Yin46] studied refinements of Weitz’s result by considering families of finite graphs parameterized by their connective constant instead of their maximum degree, and established that there exists an FPTAS for $Z_\Gamma (\unicode{x3bb} )$ for families of graphs with connective constant bounded by $\mu $ , whenever $\unicode{x3bb} < {\mu ^\mu }/{(\mu -1)^{(\mu +1)}}$ .

Now, if $\Gamma $ is an infinite graph, most of these concepts stop making sense. One way to deal with this issue is by choosing an appropriate normalization and by using the DLR formalism. The idea is roughly the following: suppose that we have a sequence $\{\Gamma _n\}_n$ of finite subgraphs that ‘exhausts’ $\Gamma $ in some way. This sequence induces two other sequences: a sequence $\{Z_{\Gamma _n}(\unicode{x3bb} )\}_n$ of partition functions and a sequence $\{\mathbb {P}_{\Gamma _n,\unicode{x3bb} }\}_n$ of probability distributions. A way to extend the idea of ‘number of weighted independent sets (per site)’ in $\Gamma $ is by considering the sequence $\{Z_{\Gamma _n}(\unicode{x3bb} )^{1/\lvert \Gamma _n \rvert }\}_n$ and hoping that it converges. Under the right assumptions on $\Gamma $ and $\{\Gamma _n\}_n$ , this is exactly the case and something similar happens to $\{\mathbb {P}_{\Gamma _n,\unicode{x3bb} }\}_n$ . Moreover, there is an intimate connection between the properties of the limit measures and our ability to estimate the value of $\lim _n \lvert \Gamma _n \rvert ^{-1} \log Z_{\Gamma _n}(\unicode{x3bb} )$ , that is, to ‘approximately count’ it. We denote this limit—which, a priori, may depend on the sequence $\{\Gamma _n\}_n$ —by $f_{\{\Gamma _n\}_n}(\Gamma ,\unicode{x3bb} )$ and call it the free energy of the hardcore model $(\Gamma ,\unicode{x3bb} )$ , one of the most crucial quantities in statistical physics [Reference Baxter6, Reference Georgii17, Reference Simon44].

It can be checked that if $\Gamma $ is finite, to approximate the partition function $Z_\Gamma (\unicode{x3bb} )$ with a multiplicative error (in polynomial time) is equivalent to approximate the free energy $f_{\{\Gamma \}_n}(\Gamma ,\unicode{x3bb} )$ —where $\{\Gamma \}_n$ is the constant sequence which immediately exhausts the graph—with an additive error [Reference Liu, Sinclair and Srivastava30] (in polynomial time). Therefore, the problem of approximating $f_{\{\Gamma \}_n}(\Gamma ,\unicode{x3bb} )$ recovers the problem of approximating the partition function in the finite case and, at the same time, extends the problem to the infinite setting.

The main goal of this paper is to establish a computational phase transition for the free energy on ensembles of—possibly infinite—hardcore models. In other words, we aim to prove the existence of an efficient additive approximation algorithm for the free energy when the activity is low and to establish that there is no efficient approximation algorithm for the free energy when the activity is high, unless $\mathrm {NP} = \mathrm {RP}$ .

There have been many recent works related to the study of correlation decay properties and their relation to approximation algorithms for the free energy (and related quantities such as pressure, capacity, and entropy) in the infinite setting [Reference Briceño8, Reference Gamarnik and Katz16, Reference Marcus and Pavlov34Reference Marcus and Pavlov36, Reference Pavlov40, Reference Wang, Yin and Zhong53]. In this work, we put all these results in a single framework, which also encompasses the results from Weitz, Sly and Sun, and Sinclair et al, and at the same time generalizes them.

In 2009, Gamarnik and Katz [Reference Gamarnik and Katz16] introduced what they called the sequential cavity method, which can be regarded as a sort of infinitary self-reducibility property [Reference Jerrum, Valiant and Vazirani24]. Combining this method with Weitz’s results, they managed to prove that the free energy of the hardcore model in the Cayley graph of $\mathbb {Z}^d$ with canonical generators admits a (deterministic) $\epsilon $ -additive approximation algorithm that runs in time polynomial in $\epsilon ^{-1}$ whenever $\unicode{x3bb} < \unicode{x3bb} _c(2d)$ , where $2d$ is the maximum degree of the graph. Related results were also proven by Pavlov in [Reference Pavlov40], who developed an approximation algorithm for the hard square entropy, that is, the free energy of the hardcore model in the Cayley graph of $\mathbb {Z}^2$ with canonical generators and activity $\unicode{x3bb} = 1$ . Later, there were also some explorations due to Wang et al [Reference Wang, Yin and Zhong53] in Cayley graphs of $\mathbb {Z}^2$ with respect to other generators (e.g., the non-attacking kings system) in the context of information theory and algorithms for approximating capacities.

In this paper, we prove that all these results fit and can be generalized to hardcore models $(\Gamma ,\unicode{x3bb} )$ such that: (1) $\Gamma $ is a locally finite graph; (2) $G \curvearrowright \Gamma $ is free and almost transitive for some countable amenable subgroup $G \leq \mathrm {Aut}(\Gamma )$ ; and (3) $\unicode{x3bb} : V \to \mathbb {Q}_{>0}$ is a—not necessarily constant—G-invariant activity function. In addition, for the algorithmic implications, we assume that G satisfies some of the recursion-theoretic assumptions described below. Given this setting, we consider a Følner sequence $\{F_n\}_n$ , a fundamental domain $U_0 \subseteq V$ of $G \curvearrowright \Gamma $ , and the sequence of finite subgraphs $\{\Gamma _n\}_n$ induced by $\{F_nU_0\}_n$ . First, we show that $f_{\{\Gamma _n\}_n}(\Gamma ,\unicode{x3bb} )$ is independent of $\{F_n\}_n$ and $U_0$ , and that the limit $f_{\{\Gamma _n\}_n}(\Gamma ,\unicode{x3bb} )$ —which we denote by $f_G(\Gamma ,\unicode{x3bb} )$ to emphasize the independence of $\{F_n\}_n$ and $U_0$ —can be expressed as an infimum over some suitable family of finite subgraphs of $\Gamma $ . Next, based on results from [Reference Briceño9, Reference Gurevich and Tempelman20], we prove in Theorem 7.1 that $f_G(\Gamma ,\unicode{x3bb} )$ can be obtained as the pointwise limit of a Shannon–McMillan–Breiman-type ratio with regards to any Gibbs measure on $X(\Gamma )$ . In Theorem 7.5, we prove that if $\unicode{x3bb} $ is such that $(\Gamma ,\unicode{x3bb} )$ satisfies strong spatial mixing, then $f_G(\Gamma ,\unicode{x3bb} )$ corresponds to the evaluation of a random information function, based on ideas about random invariant orders and the Kieffer–Pinsker formula for measure-theoretical entropy introduced in [Reference Alpeev, Meyerovitch and Ryu1]. Then, in Theorem 7.6, using the previous representation theorem and the techniques from [Reference Weitz55], we provide a formula for $f_G(\Gamma ,\unicode{x3bb} )$ in terms of trees of self-avoiding walks in $\Gamma $ . These first three theorems can be regarded as a preprocessing treatment of $f_G(\Gamma ,\unicode{x3bb} )$ to obtain an arboreal representation of free energy to develop approximation techniques, but we believe that they are of independent interest.

Later, we consider a finitely generated amenable group G with a prescribed set of generators S such that its word problem can be solved in exponential time. This last requirement seems to be natural and many groups satisfy it (for example, any linear group, including all abelian, all nilpotent groups, and, more generally, all virtually polycyclic groups). Given a positive integer $\Delta $ and $\unicode{x3bb} _0> 0$ , we denote by $\mathcal {H}_G^\Delta (\unicode{x3bb} _0)$ the ensemble of hardcore models $(\Gamma ,\unicode{x3bb} )$ such that $G \curvearrowright \Gamma $ is free and almost transitive, the maximum degree of $\Gamma $ is bounded by $\Delta $ , and the values of $\unicode{x3bb} $ are bounded from above by $\unicode{x3bb} _0$ . Then, in Theorem 8.5, we establish the following algorithmic implication: if $\unicode{x3bb} _0 < \unicode{x3bb} _c(\Delta )$ , there exists an additive FPRAS on $\mathcal {H}_G^{\Delta }(\unicode{x3bb} _0)$ for $f_G(\Gamma ,\unicode{x3bb} )$ , where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $ -regular tree $\mathbb {T}_\Delta $ . This can be considered as a confirmation in the amenable setting of the ‘algorithmic version’—as called in [Reference Weitz55]—of [Reference Sokal50, Conjecture 2.1]. In addition, under the extra assumption that G has a finite index linearly ordered subgroup $(H, \prec )$ such that its algebraic past $\Phi _\prec = \{g \in H: g \prec 1_G\}$ can be decided in exponential time, we prove that the algorithm can be chosen to be deterministic, that is, there exists an additive FPTAS. Groups that satisfy this extra condition include all finitely generated abelian groups, nilpotent groups like the Heisenberg group $H_3(\mathbb {Z})$ , and solvable groups like the Baumslag–Solitar groups $BS(1,n)$ . However, in Corollary 8.8, we observe that if $\unicode{x3bb} _0> \unicode{x3bb} _c(\Delta )$ , there is no additive FPRAS unless $\mathrm {NP} = \mathrm {RP}$ . In particular, we obtain that the results from Weitz, Sly, and Sun correspond to the special case when G is the trivial (and orderable) group.

By an additive FPRAS, we mean a probabilistic algorithm that given $(\Gamma ,\unicode{x3bb} )$ and $\epsilon> 0$ , outputs a number $\hat {f}$ such that $\lvert f_G(\Gamma ,\unicode{x3bb} ) - \hat {f} \rvert < \epsilon $ with probability greater than $3/4$ in time polynomial in $\lvert \Gamma / G \rvert $ and $\epsilon ^{-1}$ . Here, $\lvert \Gamma / G \rvert $ denotes the size of some (or any) fundamental domain of the action $G \curvearrowright \Gamma $ , and therefore, all the information we need to construct $\Gamma $ . However, by an additive FPTAS, we mean an additive FPRAS with success probability equal to $1$ instead of just $3/4$ . We assume throughout the paper that the standard functions and arithmetic operations of the numerical values involved can be computed exactly in one unit of time.

Finally, as an application in symbolic dynamics, we show how to use these results to establish representation formulas and efficient approximation algorithms for the topological entropy of nearest-neighbor subshifts of finite type with enough safe symbols. Also, we consider the pressure of single-site potentials with a vacuum state, which includes systems such as the Widom–Rowlinson model and some other weighted graph homomorphisms from $\Gamma $ to any finite graph, among others. These results can also be regarded as an extension of the works by Marcus and Pavlov in $\mathbb {Z}^d$ (see [Reference Marcus and Pavlov34Reference Marcus and Pavlov36]), who developed additive approximation algorithms for the entropy and free energy (or pressure) of general $\mathbb {Z}^d$ -subshifts of finite type, with special emphasis in the case $d=2$ . We believe that these implications are relevant, especially in the light of results like that from Hochman and Meyerovitch. In [Reference Hochman and Meyerovitch23], Hochman and Meyerovitch proved that the set of topological entropies that a nearest-neighbor $\mathbb {Z}^2$ -subshift of finite type can achieve coincides with the set of non-negative right-recursively enumerable real numbers. This class of numbers includes numbers that are poorly computable or even not computable. In addition, we discuss the case of the monomer–dimer model and counting independent sets of line graphs, which is a special case that does not exhibit a phase transition. As a byproduct of our results, we also give sufficient conditions for the existence of a unique measure of maximal entropy for subshifts on arbitrary amenable groups.

We remark that our results—considering related ones, like those obtained by Gamarnik and Katz in [Reference Gamarnik and Katz16]—are novel in at least three aspects.

  1. (1) Almost transitive framework. The generalization to the almost transitive case provides enough flexibility so that (i) other systems (such as subshifts of finite type, matchings, etc.) can be represented through reductions in terms of independent sets in suitable graphs and (ii) the measurement of (the size of) fundamental domains as a way to measure computational complexity provides a way to obtain a computational phase transition. These aspects—to our knowledge—are new, even in the relevant case $G = \mathbb {Z}^d$ , that is, the family of graphs such that $\mathbb {Z}^d$ acts almost transitively on them.

  2. (2) Algorithms for graphs with exponential growth. Our approach, which provides polynomial time approximation algorithms, works for amenable groups not only of polynomial growth but also exponential growth. A relevant case that is fully explored in §8 is the family of Baumslag–Solitar groups $BS(1,n)$ for $n \geq 2$ , which have exponential growth but admit even a deterministic approximation algorithm for free energy.

  3. (3) Lack of orderability. If a group does not have an orderable subgroup of finite index, it is less clear how to obtain a sequential cavity method as in [Reference Gamarnik and Katz16], which exploits the existence of an invariant deterministic order of the group at hand (like, for example, the lexicographic order in $\mathbb {Z}^d$ ). Our free energy representation formulas, in terms of invariant random orders, provide a way to develop randomized approximation algorithms for groups that are not necessarily orderable.

The paper is organized as follows: in §2, we introduce the basic concepts regarding graphs, homomorphisms, independent sets, group actions, Cayley graphs, and partition functions; in §3, we rigorously define free energy based on the notion of amenability and show some robustness properties of its definition; in §4, we define Gibbs measures and relevant spatial mixing properties; in §5, we develop the formalism based on trees of self-avoiding walks and discuss some of their properties; in §6, we present the formalism of invariant (deterministic and random) orders of a group; in §7, we prove Theorems 7.1, 7.5, and 7.6, which provide a randomized sequential cavity method that allows us to obtain an arboreal representation of free energy; in §8, we prove Theorem 8.5 and establish the algorithmic implications of our results; in §9, we provide reductions that allow us to translate the problem of approximating pressure of a single-site potential and the topological entropy of a subshift into the problem of counting independent sets, and discuss other consequences that are implicit in our results.

2 Preliminaries

2.1 Graphs

A graph will be a pair $\Gamma = (V,E)$ such that V is a countable set—the vertices—and $E \subseteq V \times V$ is a symmetric relation—the edges. Let $\leftrightarrow $ be the equivalence relation generated by E, that is, $v \leftrightarrow v'$ if and only if there exist $n \in \mathbb {N}_0$ and $\{v_i\}_{0 \leq i \leq \ell }$ such that $v = v_0$ , $v' = v_n$ , and $(v_i,v_{i+1}) \in E$ for every $0 \leq i < n$ . Denote by $n(v,v')$ the smallest n with this property. This induces a notion of distance in $\Gamma $ given by

$$ \begin{align*} \mathrm{dist}_\Gamma(v,v') = \begin{cases} n(v,v') & \text{if } v \leftrightarrow v', \\ +\infty & \text{otherwise.} \end{cases} \end{align*} $$

Given a set $U \subseteq V$ , we define its boundary $\partial U$ as the set $\{v \in V: \mathrm {dist}_\Gamma (v,U) = 1\}$ , where $\mathrm {dist}_\Gamma (U,U') = \inf _{v \in U, v' \in U'} \mathrm {dist}_\Gamma (v,v')$ . In addition, given $\ell \geq 0$ and $v \in V$ , we define the ball centered at v with radius $\ell $ as $B_\Gamma (v,\ell ) := \{v' \in V: \mathrm {dist}_\Gamma (v,v') \leq \ell \}$ .

A graph $\Gamma $ is:

  • loopless, if E is anti-reflexive (that is, there is no vertex related to itself);

  • connected, if $v \leftrightarrow v'$ for every $v,v' \in V$ ; and

  • locally finite, if every vertex is related to only finitely many vertices.

Sometimes we will write $V(\Gamma )$ and $E(\Gamma )$ —instead of just V and E—to emphasize $\Gamma $ .

2.2 Homomorphisms

Consider graphs $\Gamma _1$ and $\Gamma _2$ . A graph homomorphism is a map $g: V(\Gamma _1) \to V(\Gamma _2)$ such that

$$ \begin{align*} (v,v') \in E(\Gamma_1) \implies (g(v), g(v')) \in E(\Gamma_2). \end{align*} $$

We denote by $\mathrm {Hom}(\Gamma _1,\Gamma _2)$ the set of graph homomorphisms from $\Gamma _1$ to $\Gamma _2$ .

A graph isomorphism is a bijective map $g: V(\Gamma _1) \to V(\Gamma _2)$ such that

$$ \begin{align*} (v,v') \in E(\Gamma_1) \iff (g(v), g(v')) \in E(\Gamma_2). \end{align*} $$

If a map like this exists, we say that $\Gamma _1$ and $\Gamma _2$ are isomorphic, denoted by $\Gamma _1 \cong \Gamma _2$ .

A graph automorphism is a graph isomorphism from a graph $\Gamma $ to itself. We denote by $\mathrm {Aut}(\Gamma )$ the set of graph automorphisms of $\Gamma $ . This set is a group when considering composition $\circ $ as the group operation and the identity map $\mathrm {id}_\Gamma : V \to V$ as the identity group element $1_{\mathrm {Aut}(\Gamma )}$ . In this case, instead of writing $g_1 \circ g_2$ , we will simply write $g_1g_2$ to emphasize the group structure.

2.3 Independent sets

Given a subset $U \subseteq V$ , the induced subgraph by U, denoted $\Gamma [U]$ , is the graph with a set of vertices U and set of edges $E \cap (U \times U)$ . A subset $I \subseteq V$ is called an independent set if $\Gamma [I]$ has no edges. We can also represent an independent set by its indicator function, that is, by the map $x: V \to \{0,1\}$ so that

$$ \begin{align*} [x(v) = 1 \text{ and } (v,v') \in E ] \implies x(v') = 0. \end{align*} $$

In addition, if we consider the finite graph $H_0 := (\{0,1\},\{(0,0),(0,1),(1,0)\})$ , then x can be also understood as a graph homomorphism from $\Gamma $ to $H_0$ (see Figure 1).

Figure 1 The graph $H_0$ .

We denote by $X(\Gamma )$ the set of independent sets of $\Gamma $ . Notice that $X(\Gamma ) \subseteq \{0,1\}^{V}$ can be identified with the set $\mathrm {Hom}(\Gamma ,H_0)$ and that the empty independent set $0^{V}$ always belongs to $X(\Gamma )$ . Sometimes we will denote this independent set by $0^\Gamma $ .

2.4 Group actions

Let G be a subgroup of $\mathrm {Aut}(\Gamma )$ . Given $g \in G$ and $v \in V$ , the map $(g,v) \mapsto g \cdot v := g(v)$ is a (left) group action, this is to say, $1_G \cdot v = v$ and $(gg') \cdot v = g \cdot (g' \cdot v)$ for all $g' \in G$ , where $1_G = 1_{\mathrm {Aut}(\Gamma )}$ . In this case, we say that G acts on $\Gamma $ and denote this fact by $G \curvearrowright \Gamma $ .

The group G also acts on $\{0,1\}^{V}$ by precomposition. Given $g \in G$ and $x \in \{0,1\}^{V}$ , consider the map $(g,x) \mapsto g \cdot x := x \circ g^{-1}$ . A subset $X \subseteq \{0,1\}^{V}$ is called G-invariant if $g \cdot X = X$ for all $g \in G$ , where $g \cdot X := \{g \cdot x: x \in X\}$ . Clearly, if $x \in X(\Gamma )$ , then $g \cdot x$ and $g^{-1} \cdot x$ also belong to $X(\Gamma )$ , since $g \in \mathrm {Aut}(\Gamma )$ and $x \in \mathrm {Hom}(\Gamma ,H_0)$ . Therefore, $X(\Gamma )$ is G-invariant and the action $G \curvearrowright X(\Gamma )$ is well defined.

We will usually use the letter v to denote vertices in V, the letter g to denote graph automorphisms in G, and the letter x to denote independent sets in $X(\Gamma )$ . To distinguish the action of G on V from the action of G on $X(\Gamma )$ , we will write $g v$ instead of $g \cdot v$ , without risk of ambiguity.

The action $G \curvearrowright \Gamma $ is always faithful, that is, for all $g \in G \setminus \{1_G\}$ , there exists $v \in V$ such that $g v \neq v$ . The G-orbit of a vertex $v \in V$ is the set $Gv := \{g v: g \in G\}$ . The set of all G-orbits of $\Gamma $ , denoted by $\Gamma /G$ , is a partition of V and it is called the quotient of the action.

We say that a subset $\emptyset \neq U \subseteq V$ is dynamically generating if $GU = V$ , where $GF := \{g v: g \in F, v \in U\}$ for any $F \subseteq G$ , and a fundamental domain if it is also minimal, that is, if $U' \subsetneq U$ , then $GU' \subsetneq V$ . The action $G \curvearrowright \Gamma $ is almost transitive if $\lvert \Gamma /G \rvert < +\infty $ and transitive if $\lvert \Gamma /G \rvert = 1$ . A graph $\Gamma $ is called almost transitive (respectively transitive) if $\mathrm {Aut}(\Gamma ) \curvearrowright \Gamma $ is almost transitive (respectively transitive).

The index of a subgroup $H \leq G$ , denoted by $[G:H]$ , is the cardinality of the set of cosets $\{Hg: g \in G\}$ . We will usually consider subgroups of finite index. In this case, we have that $\lvert \Gamma /H \rvert = \lvert \Gamma /G \rvert [G:H]$ .

The G-stabilizer of a vertex $v \in V$ is the subgroup $\mathrm {Stab}_G(v) := \{g \in G: g v = v\}$ . Notice that, since $\mathrm {Stab}_G(gv) = g\mathrm {Stab}_G(v)g^{-1}$ for every $g \in G$ , we have that $\lvert \mathrm {Stab}_G(v) \rvert = \lvert \mathrm {Stab}_G(v') \rvert $ for all $v' \in Gv$ . If $\lvert \mathrm {Stab}_G(v) \rvert < \infty $ for all v, we say that the action is almost free, and if $\lvert \mathrm {Stab}_G(v) \rvert = 1$ (that is, if $\mathrm {Stab}_G(v) = \{1_G\}$ ) for all $v,$ we say that the action is free.

A relevant observation is that if we assume that $\Gamma $ is countable and $G \curvearrowright \Gamma $ is almost transitive and almost free, then G must be a countable group. In this work, we will only consider almost free and almost transitive actions. In this case, there exists a finite fundamental domain $U_0 \subseteq V$ such that $\lvert U_0 \rvert = \lvert \Gamma /G \rvert $ and, if $\Gamma $ is locally finite, then $\Gamma $ must have bounded degree, that is, there is a uniform bound on the number of vertices to which each vertex is related. In this case, we denote by $\Delta (\Gamma )$ the maximum degree among all vertices of the graph $\Gamma $ .

2.5 Transitive case: Cayley graphs

Consider a subset $S \subseteq G$ that we assume to be symmetric, that is, $S = S^{-1}$ , where $S^{-1} = \{s^{-1} \in S: s \in S\}$ . We define the (right) Cayley graph as $\mathrm {Cay}(G,S) = (V,E)$ , where

$$ \begin{align*} V = G \quad \text{and} \quad E = \{(g,sg): g \in G, s \in S\}. \end{align*} $$

Cayley graphs are a natural construction used to represent groups in a geometric fashion. In this context, it is common to ask that $1_G \notin S$ , S to be finite, and S to be generating, that is, $G = \langle S \rangle $ , where

$$ \begin{align*} \langle S \rangle := \{s_1 \cdots s_k: s_i \in S \text{ for all } 1 \leq i \leq k \text{ and } k \in \mathbb{N}\}. \end{align*} $$

Groups that have a set S satisfying these conditions are called finitely generated. Notice that if $1_G \notin S$ , then $\mathrm {Cay}(G,S)$ is loopless; if S is finite, then $\mathrm {Cay}(G,S)$ has bounded degree; and if S is generating, then $\mathrm {Cay}(G,S)$ is connected. Now, suppose that $G \curvearrowright \Gamma $ is transitive (and free). Then, there exists a symmetric set $S \subseteq G$ such that

$$ \begin{align*} \Gamma \cong \mathrm{Cay}(G,S). \end{align*} $$

Indeed, it suffices to take $S = \{g \in G: (v,g v) \in E\}$ , where $v \in V$ is arbitrary (see [Reference Sabidussi42]).

We will be interested in Cayley graphs $\Gamma = \mathrm {Cay}(G,S)$ and their subgroup of automorphisms induced by group multiplication as a special and relevant case: given $g \in G$ , we can define $f_g: \Gamma \to \Gamma $ as $f_g(g') = g'g$ and it is easy to check that $f_g \in \mathrm {Aut}(\Gamma )$ . Then G acts (as a group, from the left) on $\Gamma $ so that $g \cdot g' = f_{g^{-1}}(g') = g'g^{-1}$ for all $g' \in G$ and $G \hookrightarrow \mathrm {Aut}(\Gamma )$ by identifying g with $f_{g^{-1}}$ . In addition, via this identification, G acts transitively on $\Gamma $ as a subgroup of graph automorphisms. In particular, every Cayley graph is transitive.

2.6 Partition functions

Given a graph $\Gamma = (V,E)$ , let us consider $\unicode{x3bb} : V \to \mathbb {R}_{>0}$ , an activity function. We call the pair $(\Gamma ,\unicode{x3bb} )$ a hardcore model. We will say that a hardcore model $(\Gamma ,\unicode{x3bb} )$ is finite if $\Gamma $ is finite. If $U \subseteq V$ is a finite subset, a fact that we denote by $U \Subset V$ , and $x \in X(\Gamma )$ is an independent set, we define the $\unicode{x3bb} $ -weight of x on U as

$$ \begin{align*} \mathrm{w}_\unicode{x3bb}(x,U) := \prod_{v \in U} \unicode{x3bb}(v)^{x(v)} \end{align*} $$

and the $(\Gamma ,U,\unicode{x3bb} )$ -partition function as

$$ \begin{align*} Z_\Gamma(U,\unicode{x3bb}) := \sum_{x \in X(\Gamma,U)} \mathrm{w}_\unicode{x3bb}(x,U) = \sum_{x \in X(\Gamma,U)} \prod_{v \in U} \unicode{x3bb}(v)^{x(v)}, \end{align*} $$

where $X(\Gamma ,U) := \{x \in X(\Gamma ): x(v) = 0 \text { for all } v \notin U\}$ is the finite set corresponding to the subset of independent sets of $\Gamma $ supported on U. It is easy to check that there is an identification between $X(\Gamma ,U)$ and $X(\Gamma [U])$ . Then, the quantity $Z_\Gamma (U,\unicode{x3bb} )$ corresponds to the summation of independent sets of $\Gamma [U]$ weighted by $\unicode{x3bb} $ . In the special case $\unicode{x3bb} \equiv 1$ , we have that $Z_\Gamma (U,1) = \lvert X(\Gamma ,U) \rvert = \lvert X(\Gamma [U]) \rvert $ , that is, the partition function is exactly the number of independent sets supported on U. If $(\Gamma ,\unicode{x3bb} )$ is finite, we will simply write $Z_\Gamma (\unicode{x3bb} )$ instead of $Z_\Gamma (V,\unicode{x3bb} )$ .

Remark 2.1. Notice that if $(v,v) \in E$ or $\unicode{x3bb} (v) = 0$ , then $Z_\Gamma (U,\unicode{x3bb} ) = Z_{\Gamma }(U \setminus \{v\},\unicode{x3bb} )$ ; due to this fact, we usually ask $\unicode{x3bb} $ to be strictly positive and that $\Gamma $ is loopless.

3 Free energy

Now, suppose that we have an increasing sequence $\{U_n\}_n$ of finite subsets of vertices exhausting $\Gamma $ , that is, $U_n \subseteq U_{n+1}$ and $\bigcup _n U_n = V$ . Tentatively, we would like to define the exponential growth rate of $Z_\Gamma (V_n,\unicode{x3bb} )$ as

$$ \begin{align*} \lim_n \frac{\log Z_\Gamma(U_n, \unicode{x3bb})}{\lvert U_n \rvert}. \end{align*} $$

To guarantee the existence of this limit, we will provide a self-contained argument based on the particular properties of the hardcore model and amenability. The reader that is familiar with this kind of argument may skim over the next part and go then to §4.

3.1 Amenability


$$ \begin{align*} \mathcal{F}(G) := \{F \subseteq G: 0 < \lvert F \rvert < \infty\} \end{align*} $$

be the set of finite non-empty subsets of G. Given $g \in G$ and $K,F \subseteq G$ , we denote $Fg = \{hg: h {\kern-1pt}\in{\kern-1pt} F\}$ , $gF {\kern-1pt}={\kern-1pt} \{g h: h {\kern-1pt}\in{\kern-1pt} F\}$ , $F^{-1} {\kern-1pt}:={\kern-1pt} \{g^{-1}: g {\kern-1pt}\in{\kern-1pt} F\}$ , and $KF {\kern-1pt}={\kern-1pt} \{hg: h {\kern-1pt}\in{\kern-1pt} K, g {\kern-1pt}\in{\kern-1pt} F\}$ .

We say that $\{F_n\}_n \subseteq \mathcal {F}(G)$ is a right Følner sequence if

$$ \begin{align*} \lim_n \frac{\lvert F_ng \triangle F_n \rvert}{\lvert F_n \rvert} = 0 \quad \text{for all } g \in G, \end{align*} $$

where $\triangle $ denotes the symmetric difference. Similarly, $\{F_n\}_n$ is a left Følner sequence if

$$ \begin{align*} \lim_n \frac{\lvert g F_n \triangle F_n \rvert}{\lvert F_n \rvert} = 0 \quad \text{for all } g \in G, \end{align*} $$

and $\{F_n\}_n$ is a two-sided Følner sequence if it is both a left and a right Følner sequence. The group G is said to be amenable if it has a (right or left) Følner sequence. Notice that $\{F_n\}_n$ is left Følner if and only if $\{F_n^{-1}\}_n$ is right Følner. A Følner sequence $\{F_n\}_n$ is a Følner exhaustion if in addition $F_n \subseteq F_{n+1}$ and $\bigcup _n F_n = G$ . Every countable amenable group has a two-sided Følner exhaustion (see [Reference Kerr and Li26, Theorem 4.10]).

Every virtually amenable group is amenable. Moreover, the class of amenable groups contains all finite and all abelian groups, and it is closed under the operations of taking subgroups and forming quotients, extensions, and directed unions (see [Reference Ceccherini-Silberstein and Coornaert12]).

3.2 Growth rate of independent sets

Given $\emptyset \neq U \Subset V$ , define $\varphi _U: \mathcal {F}(G) \to \mathbb {R}$ as

$$ \begin{align*} \varphi_{U}(F) := \log Z_\Gamma(F U, \unicode{x3bb}). \end{align*} $$

From now on, we will assume that $\unicode{x3bb} : V \to \mathbb {R}_{>0}$ is G-invariant, this is to say,

$$ \begin{align*} \unicode{x3bb}(g v) = \unicode{x3bb}(v) \quad \text{for all } g \in G. \end{align*} $$

In other words, $\unicode{x3bb} $ is constant along the G-orbits, so it achieves at most $\lvert \Gamma /G \rvert $ different values. We denote by $\unicode{x3bb} _+$ and $\unicode{x3bb} _-$ the maximum and minimum among these values, respectively.

Now, let W be an abstract set, M a finite subset of W, and $k \in \mathbb {N}$ . We will say that a finite collection $\mathcal {K}$ of non-empty finite subsets of W, with possible repetitions, is a k-cover of M if , where denotes the indicator function of a set $A \subseteq W$ . The following lemma is due to Downarowicz, Frej, and Romagnoli.

Lemma 3.1. [Reference Downarowicz, Frej, Romagnoli, Kolyada, Möller, Moree and Ward14]

Let Y be a subset of $A^{n}$ , where A is a finite set and $n \in \mathbb {N}$ . Let $\mathcal {K}$ be a k-cover of the set of coordinates $M = \{1,\ldots , n\}$ . For $K \in \mathcal {K}$ , let $Y_{K} = \{y_K: y \in Y\}$ , where $y_K$ is the restriction of y to K. Then,

$$ \begin{align*} \lvert Y \rvert \leq \prod_{K \in \mathcal{K}}\lvert Y_{K}\rvert^{{1}/{k}}. \end{align*} $$

Given $\varphi : \mathcal {F}(G) \to \mathbb {R}$ , we will say that $\varphi $ satisfies Shearer’s inequality if

$$ \begin{align*} \varphi(F) \leq \frac{1}{k} \sum_{K \in \mathcal{K}} \varphi(K) \end{align*} $$

for all $F \in \mathcal {F}(G)$ and for all k-cover $\mathcal {K}$ of F with $K \subseteq F$ for all $K \in \mathcal {K}$ . We have the following theorem.

Theorem 3.2. [Reference Kerr and Li26, Theorem 4.48]

Given a countable amenable group G, suppose that $\varphi : \mathcal {F}(G) \to \mathbb {R}$ satisfies Shearer’s inequality and $\varphi (Fg) = \varphi (F)$ for all $F \in \mathcal {F}(G)$ and $g \in G$ . Then,

$$ \begin{align*} \lim_n \frac{\varphi(F_n)}{\lvert F_n \rvert} = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{\lvert F \rvert} \end{align*} $$

for any Følner sequence $\{F_n\}_n$ .

Considering the two previous results, we obtain the next lemma.

Lemma 3.3. Given a fundamental domain $U_0$ of $G \curvearrowright \Gamma $ and $\unicode{x3bb} : V \to \mathbb {Q}_{> 0}$ such that $\unicode{x3bb} (v) = {p_v}/{q_v}$ with $p_v, q_v \in \mathbb {N}$ for all $v \in V$ , we have that, for any Følner sequence $\{F_n\}_n$ ,

$$ \begin{align*} \lim_n \frac{\varphi(F_n)}{\lvert F_n \rvert} = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{\lvert F \rvert}, \end{align*} $$

where $\varphi : \mathcal {F}(G) \to \mathbb {R}$ is given by $\varphi (F) = \log Z_\Gamma (FU_0, \unicode{x3bb} ) + \lvert F \rvert \sum _{v \in U_0} \log q_v$ .

Proof. Given $F \in \mathcal {F}(G)$ and $k \in \mathbb {N}$ , let $\mathcal {K}$ be a k-cover of F with $K \subseteq F$ for all $K \in \mathcal {K}$ . Notice that

$$ \begin{align*} Z_\Gamma(FU_0,\unicode{x3bb}) & = \sum_{x \in X(\Gamma, FU_0)} \prod_{v \in FU_0} \bigg(\frac{p_v}{q_v}\bigg)^{x(v)} \\ & = \frac{1}{\prod_{v \in FU_0} q_v} \sum_{x \in X(\Gamma, FU_0)} \prod_{v \in FU_0} p_v^{x(v)} q_v^{1-x(v)}. \end{align*} $$

Consider $q := \max _v q_v$ , $p := \max _v p_v$ , $A := \{-q,\ldots ,-1\} \cup \{1,\ldots ,p\}$ , and

$$ \begin{align*} Y := \{y \in A^{FU_0}: -q_v \leq y(v) \leq p_v \text{ and } [y(v) \geq 1 \land (v,v') \in E(\Gamma)] \implies y(v') \leq -1\}. \end{align*} $$

Notice that

$$ \begin{align*} \lvert Y \rvert = \sum_{x \in X(\Gamma,FU_0)} \prod_{v \in FU_0} p_v^{x(v)} q_v^{1-x(v)}. \end{align*} $$

Therefore, by Lemma 3.1, and noticing that $\lvert Y_{KU_0} \rvert = \prod _{v \in KU_0} q_v \cdot Z_\Gamma (KU_0,\unicode{x3bb} )$ , we have that

$$ \begin{align*} \prod_{v \in FU_0} q_v \cdot Z_\Gamma(FU_0,\unicode{x3bb}) = \lvert Y \rvert \leq \prod_{K \in \mathcal{K}}\lvert Y_{KU_0} \rvert^{{1}/{k}} \leq \prod_{K \in \mathcal{K}} \bigg(\prod_{v \in KU_0} q_v \cdot Z_\Gamma(KU_0,\unicode{x3bb})\bigg)^{{1}/{k}}, \end{align*} $$

where we use that $\{KU_0: K \in \mathcal {K}\kern1.2pt\}$ is a k-cover of $FU_0$ . Therefore, by G-invariance of $\unicode{x3bb} $ ,

$$ \begin{align*} \varphi(F) & = \log Z_\Gamma(FU_0, \unicode{x3bb}) + \lvert F \rvert \sum_{v \in U_0} \log q_v \\ & \leq \frac{1}{k} \sum_{K \in \mathcal{K}}\bigg( \log Z_\Gamma(KU_0,\unicode{x3bb}) + \lvert K \rvert\sum_{v \in U_0} \log q_v \bigg) \\ & = \frac{1}{k} \sum_{K \in \mathcal{K}} \varphi(K), \end{align*} $$

so $\varphi $ satisfies Shearer’s inequality. However, by G-invariance of $X(\Gamma )$ and $\unicode{x3bb} $ , it follows that $\varphi (Fg) = \varphi (F)$ for all $F \in \mathcal {F}(G)$ and $g \in G$ . Therefore, by Theorem 3.2, we conclude.

Proposition 3.4. Given a fundamental domain $U_0$ of $G \curvearrowright \Gamma $ , we have that

$$ \begin{align*} \lim_n \frac{\log Z_\Gamma(F_nU_0, \unicode{x3bb})}{\lvert F_n \rvert} = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \unicode{x3bb})}{\lvert F \rvert} \end{align*} $$

for any Følner sequence $\{F_n\}_n$ .

Proof. First, suppose that $\unicode{x3bb} $ only takes rational values, that is, $\unicode{x3bb} {\kern-1.2pt}:{\kern-1.2pt} V {\kern-1.2pt}\to{\kern-1.2pt} \mathbb {Q}_{>0}$ so that $\unicode{x3bb} (v) = {p_v}/{q_v}$ for all $v \in V$ . By Lemma 3.3, for $\varphi (F) = \log Z_\Gamma (FU_0, \unicode{x3bb} ) + \lvert F \rvert \sum _{v \in U_0} q_v$ , we have that

$$ \begin{align*} \lim_{n} \frac{\log Z_\Gamma(F_nU_0, \unicode{x3bb})}{\lvert F_n \rvert} + \sum_{v \in U_0} q_v & = \lim_{n} \frac{\varphi(F_n)}{\lvert F_n \rvert} \\ & = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{\lvert F \rvert} \\ & = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \unicode{x3bb})}{\lvert F \rvert} + \sum_{v \in U_0} q_v, \end{align*} $$

and, after canceling out $\sum _{v \in U_0} q_v$ , we obtain that

$$ \begin{align*} \lim_{n} \frac{\log Z_\Gamma(F_nU_0, \unicode{x3bb})}{\lvert F_n \rvert} = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \unicode{x3bb})}{\lvert F \rvert}. \end{align*} $$

Now, given a general $\unicode{x3bb} $ , we can always approximate it by some G-invariant $\tilde {\unicode{x3bb} }{\kern-1.2pt}:{\kern-1.2pt} V {\kern-1.2pt}\to{\kern-1.2pt} \mathbb {Q}_{>0}$ arbitrarily close in the supremum norm. Given $\epsilon> 0$ , pick $\tilde {\unicode{x3bb} }$ so that $\tilde {\unicode{x3bb} }(v) \leq \unicode{x3bb} (v) \leq (1+\epsilon )\tilde {\unicode{x3bb} }(v)$ for every v. Then,

$$ \begin{align*} \log Z_\Gamma(FU_0,\tilde{\unicode{x3bb}}) & \leq \log Z_\Gamma(FU_0,\unicode{x3bb}) \\ & \leq \log Z_\Gamma(FU_0,(1+\epsilon)\tilde{\unicode{x3bb}}) \\ & \leq \lvert FU_0 \rvert\log(1+\epsilon) + \log Z_\Gamma(FU_0,\tilde{\unicode{x3bb}}), \end{align*} $$


$$ \begin{align*} \frac{\log Z_\Gamma(FU_0,\tilde{\unicode{x3bb}})}{\lvert F \rvert} \leq \frac{\log Z_\Gamma(FU_0,\unicode{x3bb})}{\lvert F \rvert} \leq \lvert U_0 \rvert\log(1+\epsilon) + \frac{\log Z_\Gamma(FU_0,\tilde{\unicode{x3bb}})}{\lvert F \rvert}. \end{align*} $$


$$ \begin{align*} \liminf_n \frac{\log Z_\Gamma(F_nU_0,\unicode{x3bb})}{\lvert F_n \rvert} & \geq \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0,\unicode{x3bb})}{\lvert F \rvert} \\ & \geq \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0,\tilde{\unicode{x3bb}})}{\lvert F \rvert} \\ & = \lim_n \frac{\log Z_\Gamma(F_nU_0,\tilde{\unicode{x3bb}})}{\lvert F_n \rvert} \\ & \geq \limsup_n \frac{\log Z_\Gamma(F_nU_0,\unicode{x3bb})}{\lvert F_n \rvert} - \lvert U_0 \rvert\log(1+\epsilon), \end{align*} $$

and since $\epsilon $ was arbitrary, we conclude.

To fully characterize $\lim _n ({\log Z_\Gamma (U_n, \unicode{x3bb} )}/{\lvert U_n \rvert })$ , we have the following lemma.

Lemma 3.5. Let $\{F_n\}_n$ be a Følner sequence and $U_0$ a fundamental domain. Then, for any Følner sequence $\{F_n\}_n$ ,

$$ \begin{align*} \lim_n \frac{\lvert F_n U_0 \rvert}{\lvert F_n \rvert} = \sum_{v \in U_0} \lvert \mathrm{Stab}_G(v) \rvert^{-1}. \end{align*} $$

Proof. First, pick $v \in U_0$ . Since $\mathrm {Stab}_G(v)$ is finite and $\{F_n\}_n$ is a Følner sequence, we have that $\lim _n {\lvert F_n \mathrm {Stab}_G(v) \rvert }/{\lvert F_n \rvert } = 1$ . However, $F_n \mathrm {Stab}_G(v) v = F_n v$ and for each $v' \in F_n v$ , there exist exactly $\lvert \mathrm {Stab}_G(v) \rvert $ different elements $g \in F_n \mathrm {Stab}_G(v)$ such that $g v = v'$ . In other words,

$$ \begin{align*} \lvert F_n \mathrm{Stab}_G(v) \rvert = \lvert F_n v \rvert \lvert \mathrm{Stab}_G(v) \rvert, \end{align*} $$


$$ \begin{align*} \lim_n \frac{\lvert F_n v \rvert}{\lvert F_n \rvert} = \lim_n \frac{\lvert F_n \mathrm{Stab}_G(v) \rvert}{\lvert F_n \rvert\lvert \mathrm{Stab}_G(v) \rvert} = \lvert \mathrm{Stab}_G(v) \rvert^{-1}. \end{align*} $$


$$ \begin{align*} \lim_n \frac{\lvert F_n U_0 \rvert}{\lvert F_n \rvert} = \sum_{v \in U_0} \lim_n \frac{\lvert F_n v \rvert}{\lvert F_n \rvert} = \sum_{v \in U_0} \lvert \mathrm{Stab}_G(v) \rvert^{-1}.\\[-48pt] \end{align*} $$

Now, given a fundamental domain $U_0$ , define

$$ \begin{align*} f_{G}(\Gamma,U_0,\unicode{x3bb}) := \inf_{F \in \mathcal{F}(G)}\frac{\log Z_\Gamma(FU_0, \unicode{x3bb})}{\lvert FU_0 \rvert}, \end{align*} $$

which, by Proposition 3.4 and Lemma 3.5, is equal to

$$ \begin{align*} \bigg(\sum_{v \in U_0} \lvert \mathrm{Stab}_G(v) \rvert^{-1}\bigg)^{-1} \lim_n \frac{\log Z_\Gamma(F_nU_0, \unicode{x3bb})}{\lvert F_n \rvert} \end{align*} $$

for any Følner sequence $\{F_n\}_n$ and, in particular, for any Følner exhaustion. Notice that, since $GU_0 = V$ , the sequence $\{U_n\}_n$ defined as $U_n = F_nU_0$ is an exhaustion of V in the sense for which we were looking. Now we will see that $f_{G}(\Gamma ,U_0,\unicode{x3bb} )$ is independent of $U_0$ .

Proposition 3.6. Given two fundamental domains $U_0$ and $U_0'$ of $G \curvearrowright \Gamma $ , we have that

$$ \begin{align*} f_{G}(\Gamma,U_0,\unicode{x3bb}) = f_{G}(\Gamma,U_0',\unicode{x3bb}). \end{align*} $$

Proof. Since $V = GU_0 = GU_0'$ , there must exist $K,K' \in \mathcal {F}(G)$ such that $U_0' \subseteq KU_0$ and $U_0 \subseteq K'U_0'$ . Then, for every $F \in \mathcal {F}(G)$ ,

$$ \begin{align*} FU_0 \triangle FU_0' & = (FU_0 \setminus FU_0') \cup (FU_0' \setminus FU_0) \\ & \subseteq (FK'U_0' \setminus FU_0') \cup (FKU_0 \setminus FU_0) \\ & = (FK' \setminus F)U_0' \cup (FK \setminus F)U_0. \end{align*} $$

Therefore, $\lvert FU_0 \triangle FU_0 \rvert \leq \lvert FK' \setminus F \rvert \lvert U_0' \rvert + \lvert FK \setminus F \rvert \lvert U_0 \rvert $ . Now, notice that for $U, U' \Subset V$ , we always have that:

  1. (1) $Z_\Gamma (U \cup U',\unicode{x3bb} ) \leq Z_\Gamma (U,\unicode{x3bb} ) \cdot Z_\Gamma (U',\unicode{x3bb} )$ , provided $U \cap U' = \emptyset $ ;

  2. (2) $Z_\Gamma (U,\unicode{x3bb} ) \leq Z_\Gamma (U',\unicode{x3bb} )$ , provided $U \subseteq U'$ ; and

  3. (3) $Z_\Gamma (U,\unicode{x3bb} ) \leq (2\max \{1,\unicode{x3bb} _+\})^{\lvert U \rvert }$ ,


$$ \begin{align*} \log Z_\Gamma(FU_0,\unicode{x3bb}) & \leq \log Z_\Gamma(FU_0 \cap FU_0',\unicode{x3bb}) + \log Z_\Gamma(FU_0 \setminus FU_0',\unicode{x3bb}) \\ & \leq \log Z_\Gamma(FU_0',\unicode{x3bb}) + \log Z_\Gamma(FU_0 \triangle FU_0',\unicode{x3bb}) \\ & \leq \log Z_\Gamma(FU_0',\unicode{x3bb}) + \lvert FU_0 \triangle FU_0' \rvert\log(2\max\{1,\unicode{x3bb}_+\}) \\ & \leq \log Z_\Gamma(FU_0',\unicode{x3bb}) + (\lvert FK' \setminus F \rvert\lvert U_0' \rvert + \lvert FK \setminus F \rvert\lvert U_0 \rvert)\log(2\max\{1,\unicode{x3bb}_+\}). \end{align*} $$

Finally, since $\lvert U_0 \rvert = \lvert U_0' \rvert $ and $\lvert FU_0 \rvert = \lvert F \rvert \lvert U_0 \rvert $ , it follows by amenability that

$$ \begin{align*} f_G(\Gamma,U_0,\unicode{x3bb}) & = \lim_n \frac{\log Z_\Gamma(F_nU_0,\unicode{x3bb})}{\lvert F_nU_0 \rvert} \\ & \leq \lim_n \frac{\log Z_\Gamma(F_nU_0',\unicode{x3bb})}{\lvert F_nU_0' \rvert} \\ & \quad + \lim_n\bigg( \frac{\lvert F_nK' \setminus F_n \rvert}{\lvert F_n \rvert} + \frac{\lvert F_nK \setminus F_n \rvert}{\lvert F_n \rvert}\bigg)\log(2\max\{1,\unicode{x3bb}_+\}) \\ & = f_{G}(\Gamma,U_0',\unicode{x3bb}), \end{align*} $$

and by symmetry of the argument, we conclude.

Then, we can consistently define the Gibbs $(\Gamma ,\unicode{x3bb} )$ -free energy according to G as

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) := f_G(\Gamma,U_0,\unicode{x3bb}), \end{align*} $$

where $U_0$ is an arbitrary fundamental domain of $G \curvearrowright \Gamma $ . In addition, it is easy to see that if $G_1$ and $G_2$ act almost transitively on $\Gamma $ , and the $G_1$ -orbits and $G_2$ -orbits coincide, that is, $G_1 v = G_2 v$ for all $v \in V$ , then

$$ \begin{align*} f_{G_1}(\Gamma,\unicode{x3bb}) = f_{G_2}(\Gamma,\unicode{x3bb}). \end{align*} $$

In particular, we have that $f_G(\Gamma ,\unicode{x3bb} )$ is equal for all G acting transitively on $\Gamma $ . Then, we can define the Gibbs $(\Gamma ,\unicode{x3bb} )$ -free energy as

$$ \begin{align*} f(\Gamma,\unicode{x3bb}) := \inf_{\emptyset \neq U \Subset V} \frac{\log Z_\Gamma(U, \unicode{x3bb})}{\lvert U \rvert}, \end{align*} $$

which is a quantity that only depends on the graph $\Gamma $ and the activity function $\unicode{x3bb} $ , and satisfies that $f(\Gamma ,\unicode{x3bb} ) = f_G(\Gamma ,\unicode{x3bb} )$ for any $G \leq \mathrm {Aut}(\Gamma )$ acting transitively on $\Gamma $ .

Remark 3.7. In the almost transitive case, $f_G(\Gamma ,\unicode{x3bb} )$ does not necessarily coincide with $f(\Gamma ,\unicode{x3bb} )$ for G acting almost transitively: consider the graph $\Gamma $ obtained by taking the disjoint union of $\Gamma _1 = \mathrm {Cay}(\mathbb {Z},\emptyset )$ and $\Gamma _2 = \mathrm {Cay}(\mathbb {Z},\{1,-1\})$ , and the constant activity function $\unicode{x3bb} \equiv 1$ . Then, $f_{\mathbb {Z}}(\Gamma _1,1) = \log 2$ and $f_{\mathbb {Z}}(\Gamma _2,1) = \log (({1+\sqrt {5}})/{2})$ , so

$$ \begin{align*} f_{\mathbb{Z}}(\Gamma,1) = \frac{1}{2}(f_{\mathbb{Z}}(\Gamma_1,1) + f_{\mathbb{Z}}(\Gamma_2,1))> f_{\mathbb{Z}}(\Gamma_2,1) \geq \inf_{\emptyset \neq U \Subset V} \frac{\log Z_\Gamma(U, 1)}{\lvert U \rvert}. \end{align*} $$

The value of $f_{\mathbb {Z}}(\Gamma _2,1)$ corresponds to the topological entropy of the golden mean shift (see [Reference Lind and Marcus27, Example 4.1.4] and §9).

The main theme of this paper will be to explore our ability to approximate $f_G(\Gamma ,\unicode{x3bb} )$ . From now, and without much loss of generality, we will assume that $G \curvearrowright \Gamma $ is free (see §9.7 for a reduction of the almost free case to the free case).

4 Gibbs measures

Given a graph $\Gamma = (V,E)$ , consider the set $\{0,1\}^{V}$ endowed with the product topology and the set $X(\Gamma )$ , with the subspace topology. The set of independent sets $X(\Gamma )$ is a compact and metrizable space. A base for the topology is given by the cylinder sets

$$ \begin{align*} [x_U] := \{x' \in X(\Gamma): x'_U = x_U\} \end{align*} $$

for $U \Subset V$ and $x \in X(\Gamma )$ , where $x_U$ denotes the restriction of x from V to U. If U is a singleton $\{v\}$ , we will omit the brackets and simply write $x_v$ and the same convention will hold in analogous instances. Given $W \subseteq V$ , we denote by $\mathcal {B}_W$ the smallest $\sigma $ -algebra generated by

$$ \begin{align*} \{[x_U]: U \Subset W, x \in X(\Gamma)\}, \end{align*} $$

and by $\mathcal {B}_\Gamma $ the Borel $\sigma $ -algebra, which corresponds to $\mathcal {B}_{V}$ .

Let $\mathcal {M}(X(\Gamma ))$ be the set of Borel probability measures $\mathbb {P}$ on $X(\Gamma )$ . We say that $\mathbb {P}$ is G-invariant if $\mathbb {P}(A) = \mathbb {P}(g \cdot A)$ for all $A \in \mathcal {B}_\Gamma $ and $g \in G$ , and G-ergodic if $g \cdot A = A$ for all $g \in G$ implies that $\mathbb {P}(A) \in \{0,1\}$ . We will denote by $\mathcal {M}_G(X(\Gamma ))$ and $\mathcal {M}_G^{\mathrm {erg}}(X(\Gamma ))$ the set of G-invariant and the subset of G-invariant measures that are G-ergodic, respectively.

For $\mathbb {P} \in \mathcal {M}(X(\Gamma ))$ , define the support of $\mathbb {P}$ as

$$ \begin{align*} \mathrm{supp}(\mathbb{P}) := \{x \in X(\Gamma): \mathbb{P}([x_U])> 0 \text{ for all } U \Subset V\}. \end{align*} $$

Given $\emptyset \neq U \Subset V$ and $y \in X(\Gamma )$ , we define $\pi ^y_U$ to be the probability distribution on $X(\Gamma ,U)$ given by

$$ \begin{align*} \pi_U^y(x) := \mathrm{w}^y_\unicode{x3bb}(x,U) Z^y_\Gamma(U,\unicode{x3bb})^{-1}, \end{align*} $$

where and $Z^y_\Gamma (U,\unicode{x3bb} ) = \sum _x \mathrm {w}^y_\unicode{x3bb} (x,U)$ . In other words, to each independent set x supported on U, we associate a probability proportional to its $\unicode{x3bb} $ -weight over U, $\prod _{v \in U} \unicode{x3bb} (v)^{x(v)}$ , provided $x_U$ is compatible with $y_{U^{\mathrm { c}}}$ , in the sense that the element $z \in \{0,1\}^{V}$ such that $z_U = x_U$ and $z_{U^{\mathrm { c}}} = y_{U^{\mathrm { c}}}$ is an independent set.

Now, given an activity function $\unicode{x3bb} : V \to \mathbb {R}_{>0}$ , consider the hardcore model $(\Gamma , \unicode{x3bb} )$ and the collection $\pi _{\Gamma ,\unicode{x3bb} } = \{\pi ^y_U: U \Subset V, y \in X(\Gamma )\}$ . We call $\pi _{\Gamma ,\unicode{x3bb} }$ the Gibbs $(\Gamma ,\unicode{x3bb} )$ -specification. A measure $\mathbb {P} \in \mathcal {M}(X(\Gamma ))$ is called a Gibbs measure (for $(\Gamma ,\unicode{x3bb} )$ ) if for all $U \Subset V$ , $U' \subseteq U$ , and $x \in X(\Gamma )$ ,

$$ \begin{align*} \mathbb{P}([x_{U'}] \vert \mathcal{B}_{U^{\mathrm{ c}}})(y) = \pi^y_U([x_{U'}]) \quad \mathbb{P}\text{-almost surely in } y, \end{align*} $$

where $\pi ^y_U([x_U'])$ denotes the marginalization

$$ \begin{align*} \pi^y_U([x_U']) = \sum_{x' \in X(\Gamma[U]): x'_{U'} = x_{U'}} \pi^y_U(x') \end{align*} $$

and for $A \in \mathcal {B}_\Gamma $ . We denote by $\mathcal {M}_{\mathrm {Gibbs}}(\Gamma ,\unicode{x3bb} )$ the set of Gibbs measures for $(\Gamma ,\unicode{x3bb} )$ .

An important question in statistical physics is whether the set of Gibbs measures is empty or not, and if not, whether there is a unique or multiple Gibbs measures [Reference Georgii17].

4.1 The locally finite case

The model described in [Reference Georgii17, Example 4.16] can be understood as an attempt to formalize the idea of a system where there is a single particle $1$ (uniformly distributed) or none, that is, $0$ everywhere. There, it is proven that this model cannot be represented as a Gibbs measure. This example can be also viewed as a hardcore model in a countable graph that is complete (that is, there is an edge between any pair of different vertices) and, in particular, in a non-locally finite graph. In other words, there exist examples of non-locally finite graphs $\Gamma $ such that the $(\Gamma ,\unicode{x3bb} )$ -specification $\pi _{\Gamma ,\unicode{x3bb} }$ has no Gibbs measure.

From now on, we will always assume that $\Gamma $ is locally finite. In this case, the existence of Gibbs measures is guaranteed (see [Reference Brightwell and Winkler10, Reference Dobrushin13]) and, moreover, every Gibbs measure must be a Markov random field that is fully supported.

Indeed, it can be checked that $\pi _{\Gamma ,\unicode{x3bb} }$ is an example of a Markovian specification (see [Reference Georgii17, Example 8.24]). In this case, any Gibbs measure $\mathbb {P} \in \mathcal {M}_{\mathrm {Gibbs}}(\Gamma ,\unicode{x3bb} )$ satisfies the following local Markov property:

$$ \begin{align*} \mathbb{P}([x_U] \vert \mathcal{B}_{U^{\mathrm{ c}}})(y) = \mathbb{P}([x_U] \vert \mathcal{B}_{\partial U})(y) \quad \mathbb{P}\text{-almost surely in } y \end{align*} $$

for any $U \Subset V$ and $x \in X(\Gamma )$ . In other words, $\mathbb {P}$ is a Markov random field, so any event supported on a finite set conditioned to a specific value on its boundary is independent of events supported on the complement.

In addition, it can be checked that any Gibbs measure $\mathbb {P}$ must be fully supported, that is, $\mathrm {supp}(\mathbb {P}) = X(\Gamma )$ . Indeed, it suffices to check that $X(\Gamma ) \subseteq \mathrm {supp}(\mathbb {P})$ ; the other direction follows directly from the definition of $\pi _{\Gamma ,\unicode{x3bb} }$ and Gibbs measures. Now, given $x \in X(\Gamma )$ and $U \Subset V$ , we would like to check that $\mathbb {P}([x_U])> 0$ . To prove this, observe that given $x \in X(\Gamma )$ , we have that $z \in \{0,1\}^{V}$ defined as $z_U = x_U$ , $z_{\partial U} \equiv 0$ , and $z_{W^c} = y_{W^c}$ always belongs to $X(\Gamma )$ for any $y \in X(\Gamma )$ , where $W = U \cup \partial U$ . In particular, $\pi ^y_{W}(z)> 0$ for any $y \in X(\Gamma )$ . Then, considering that $\partial (W^c)$ is finite,

$$ \begin{align*} \mathbb{P}([x_U]) & \geq \mathbb{P}([z_{W}]) \\ & = \sum_{y \in X(\Gamma,\partial W): \mathbb{P}([y_{\partial W}])> 0} \mathbb{P}([z_{W}] \vert [y_{\partial W}])\mathbb{P}([y_{\partial W}]) \\ & = \sum_{y \in X(\Gamma,\partial W): \mathbb{P}([y_{\partial W}]) > 0} \pi^y_{W}(z)\mathbb{P}([y_{\partial W}]) \\ & \geq \pi^{y^*}_{W}(z)\mathbb{P}([y^*_{\partial W}]) > 0, \end{align*} $$

since $\mathbb {P}$ is a probability measure and there must exist $y^* \in X(\Gamma )$ such that $\mathbb {P}([y^*_{\partial W}])> 0$ . In other words, $X(\Gamma )$ satisfies the property (D*) introduced in [Reference Ruelle41, 1.14 Remark], which guarantees full support.

4.2 Spatial mixing and uniqueness

Given a Gibbs $(\Gamma ,\unicode{x3bb} )$ -specification $\pi _{\Gamma ,\unicode{x3bb} }$ , we define two spatial mixing properties fundamental to this work.

Definition 4.1. We say that a hardcore model $(\Gamma ,\unicode{x3bb} )$ exhibits strong spatial mixing (SSM) if there exists a decay rate function $\delta : \mathbb {N} \to \mathbb {R}_{\geq 0}$ such that $\lim _{\ell \to \infty } \delta (\ell ) = 0$ and for all $U \Subset V$ , $v \in U$ , and $y,z \in X(\Gamma )$ ,

$$ \begin{align*} \vert \pi^y_U([0^v]) - \pi^z_U([0^v]) \rvert \leq \delta(\mathrm{dist}_\Gamma(v,D_U(y,z))), \end{align*} $$

where $[0^v]$ denotes the event that the vertex v takes the value $0$ and

$$ \begin{align*} D_U(y,z) := \{v' \in U^c: y(v') \neq z(v')\}. \end{align*} $$

This definition is equivalent (see [Reference Marcus and Pavlov35, Lemma 2.3]) to the—a priori—stronger following property: for all $U' \subseteq U \Subset V$ and $x,y,z \in X(\Gamma )$ ,

$$ \begin{align*} \lvert \pi^y_U([x_{U'}]) - \pi^z_U([x_{U'}]) \rvert \leq \lvert U' \rvert\delta(\mathrm{dist}_\Gamma(U',D_U(y,z))). \end{align*} $$

Similarly, we say that $(\Gamma ,\unicode{x3bb} )$ exhibits weak spatial mixing (WSM) if for all $U' \subseteq U \Subset V$ and $x,y,z \in X(\Gamma )$ ,

$$ \begin{align*} \lvert \pi^y_U([x_{U'}]) - \pi^z_U([x_{U'}]) \rvert \leq \lvert U' \rvert \cdot \delta(\mathrm{dist}_\Gamma(U',U^c)). \end{align*} $$

Clearly, SSM implies WSM. Moreover, it is well known that, in this context, WSM (and therefore, SSM) implies uniqueness of Gibbs measures [Reference Weitz54]. In other words, $\mathcal {M}_{\mathrm {Gibbs}}(\Gamma ,\unicode{x3bb} ) = \{\mathbb {P}_{\Gamma ,\unicode{x3bb} }\}$ , where $\mathbb {P}_{\Gamma ,\unicode{x3bb} }$ denotes the unique Gibbs measure for $(\Gamma ,\unicode{x3bb} )$ . In this case, $\mathbb {P}_{\Gamma ,\unicode{x3bb} }$ is always $\mathrm {Aut}(\Gamma )$ -invariant.

We say that $(\Gamma ,\unicode{x3bb} )$ exhibits exponential SSM (respectively exponential WSM) if there exist constants $C,\alpha> 0$ such that $\pi _{\Gamma ,\unicode{x3bb} }$ exhibits SSM (respectively WSM) with decay rate function $\delta (n) = C \cdot \exp (-\alpha \cdot n)$ .

Given $U \subseteq V$ , we denote by $\Gamma \setminus U$ the subgraph induced by $V \setminus U$ , that is, $\Gamma [V \setminus U]$ . We have the following result due to Gamarnik and Katz.

Proposition 4.1. [Reference Gamarnik and Katz16, Proposition 1]

If a hardcore model $(\Gamma ,\unicode{x3bb} )$ satisfies SSM, then so does the hardcore model $(\Gamma ',\unicode{x3bb} )$ for any subgraph $\Gamma '$ of $\Gamma $ . The same assertion applies to exponential SSM. Moreover, for every $U \subseteq V$ and $v \in V \setminus U$ , the following identity holds:

$$ \begin{align*} \mathbb{P}_{\Gamma,\unicode{x3bb}}([0^v] \vert [0^U]) = \mathbb{P}_{\Gamma \setminus U,\unicode{x3bb}}([0^v]), \end{align*} $$

where $\mathbb {P}_{\Gamma ,\unicode{x3bb} }$ and $\mathbb {P}_{\Gamma \setminus U,\unicode{x3bb} }$ are the unique Gibbs measures for $(\Gamma ,\unicode{x3bb} )$ and $(\Gamma \setminus U,\unicode{x3bb} )$ , respectively, and $[0^U]$ denotes the event that all the vertices in U take the value $0$ . In particular, $\mathbb {P}_{\Gamma ,\unicode{x3bb} }([0^v] \vert [0^U])$ is always well defined, even if U is infinite.

Remark 4.2. Notice that any event of the form $[x_U]$ can be translated into an event of the form $[0^{U' }]$ for a suitable set $U'$ : it suffices to define $U' = U \cup \partial \{v \in U: x_U(v) = 1\}$ since, deterministically, every neighbor of a vertex colored $1$ must be $0$ , so Proposition 4.1 still holds for more general events. We also remark that in [Reference Gamarnik and Katz16], it is assumed that $\unicode{x3bb} $ is a constant function. Here we drop this assumption, but it is direct to check that the same proof of [Reference Gamarnik and Katz16, Proposition 1] also applies to the more general non-constant case.

4.3 Families of hardcore models

We will denote by $\mathcal {H}$ the family of hardcore models $(\Gamma , \unicode{x3bb} )$ such that $\Gamma $ is a countable locally finite graph and $\unicode{x3bb} $ is any activity function $\unicode{x3bb} : V(\Gamma ) \to \mathbb {R}_{>0}$ .

Given a countable group G, we will denote by $\mathcal {H}_G$ the set of hardcore models $(\Gamma ,\unicode{x3bb} )$ in $\mathcal {H}$ for which G is isomorphic to some subgroup of $\mathrm {Aut}(\Gamma )$ such that $G \curvearrowright \Gamma $ is free and almost transitive and $\unicode{x3bb} : V(\Gamma ) \to \mathbb {R}_{>0}$ is a G-invariant activity function.

Given a positive integer $\Delta $ , we will denote by $\mathcal {H}^\Delta $ the set of hardcore models $(\Gamma ,\unicode{x3bb} )$ in $\mathcal {H}$ such that $\Delta (\Gamma ) \leq \Delta $ . Notice that any hardcore model defined on the $\Delta $ -regular (infinite) tree $\mathbb {T}_\Delta $ belongs to $\mathcal {H}^\Delta $ .

Given $\unicode{x3bb} _0> 0$ , we will denote by $\mathcal {H}(\unicode{x3bb} _0)$ the family of hardcore models $(\Gamma ,\unicode{x3bb} )$ in $\mathcal {H}$ such that $\unicode{x3bb} _+ \leq \unicode{x3bb} _0$ .

We will also combine the notation for these families in the natural way; for example, $\mathcal {H}_G^\Delta (\unicode{x3bb} _0)$ will denote the set of hardcore models $(\Gamma ,\unicode{x3bb} )$ in $\mathcal {H}$ such that $G \curvearrowright \Gamma $ is free and almost transitive, $\unicode{x3bb} $ is G-invariant, $\Delta (\Gamma ) \leq \Delta $ , and $\unicode{x3bb} _+ \leq \unicode{x3bb} _0$ .

5 Trees

Given a graph $\Gamma $ , a trail w in $\Gamma $ is a finite sequence $w = (v_1,\ldots ,v_n)$ of vertices such that consecutive vertices are adjacent in $\Gamma $ and the edges $(v_i,v_{i+1})$ involved are not repeated. For a fixed vertex $v \in V(\Gamma )$ , the tree of self-avoiding walks starting from v, denoted by $T_{\mathrm {SAW}}(\Gamma ,v)$ , is defined as follows.

  1. (1) Consider the set $W_0$ of trails starting from v that repeat no vertex and the set $W_1$ of trails that repeat a single vertex exactly once and then stop (that is, the set of non-backtracking walks that end immediately after performing a cycle). We define $T_{\mathrm {SAW}}(\Gamma ,v)$ to be a rooted tree with root $\rho = (v)$ such that the set of vertices $V(T_{\mathrm {SAW}}(\Gamma ,v))$ is $W_0 \cup W_1$ and the set of (undirected) edges $E(T_{\mathrm {SAW}}(\Gamma ,v))$ corresponds to all the pairs $(w,w')$ such that $w'$ is a one-vertex extension of w or vice versa. In simple words, $T_{\mathrm {SAW}}(\Gamma ,v)$ is a rooted tree that represents all self-avoiding walks in $\Gamma $ that start from v. It is easy to check that the set of leaves of $T_{\mathrm {SAW}}(\Gamma ,v)$ contains $W_1$ , but they are not necessarily equal (e.g., see vertex b in Figure 2).

    Figure 2 A representation of a graph $\Gamma $ and its corresponding tree of self-avoiding walks $T_{\mathrm {SAW}}(\Gamma ,v)$ including the conditioning of terminal trails ( $\bot $ ). Here, the order of each neighborhood is alphabetical and every trail/vertex is represented by the final vertex of the trail in $\Gamma $ starting from $v = a$ . See also [Reference Weitz55] for an explanation of the same picture.

  2. (2) For $u \in V(\Gamma )$ , consider an arbitrary ordering $\partial \{u\} = \{u_1, \ldots , u_d\}$ of its neighbors. Given $w \in W_1$ , we can represent this walk as a sequence

    $$ \begin{align*}w = (v,\ldots, u,u_i,\ldots,u_j,u),\end{align*} $$

    with $u_i,u_j \in \partial \{u\}$ . Notice that $i \neq j$ , since we are not repeating edges. Considering this, we condition the ‘terminal’ trail w to be $1$ (occupied) if $i < j$ and to be $0$ (unoccupied) if $i> j$ , inducing the corresponding effect of this conditioning in the graph (that is, removing the vertex and its neighbors or just removing the vertex, respectively).

Given a hardcore model $(\Gamma ,\unicode{x3bb} )$ , a vertex $v \in V(\Gamma )$ , a subset $U \subseteq V(\Gamma )$ , and an independent set $x \in X(\Gamma )$ , we are interested in computing the marginal probability that v is unoccupied in $\Gamma $ given the partial configuration $x_U$ , that is, $\mathbb {P}_{\Gamma ,\unicode{x3bb} }([0^v] \vert [x_U])$ . Notice that if $(\Gamma ,\unicode{x3bb} )$ satisfies SSM (which includes the particular but relevant case of $\Gamma $ being finite), then this probability is always well defined due to Proposition 4.1, even if U is infinite.

To understand better $\mathbb {P}_{\Gamma ,\unicode{x3bb} }([0^v] \vert [x_U])$ , we consider $(T_{\mathrm {SAW}}(\Gamma ,v),\overline {\unicode{x3bb} })$ to be the hardcore model where $\overline {\unicode{x3bb} }(w) = \unicode{x3bb} (u)$ for every trail w ending in u. In this context, a condition $x_U$ in $(\Gamma ,\unicode{x3bb} )$ is translated into the condition $\overline {x_U}$ in $T_{\mathrm {SAW}}(\Gamma ,v)$ , whose support is the set $W(U)$ of trails w that end in u for some $u \in U$ , and $\overline {x}(w) = x(u)$ for all these w. We have the following result from [Reference Weitz55], that we adapt to the more general non-constant $\unicode{x3bb} $ case and we include its proof for completeness.

Theorem 5.1. [Reference Weitz55, Theorem 3.1]

For every finite hardcore model $(\Gamma ,\unicode{x3bb} )$ , every $v \in V(\Gamma )$ , and $U \subseteq V(\Gamma )$ ,

$$ \begin{align*} \mathbb{P}_{\Gamma,\unicode{x3bb}}([0^v] \vert [x_U]) = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\unicode{x3bb}}}([0^\rho] \vert [\overline{x_U}]). \end{align*} $$

Proof. Instead of probabilities, we work with the ratios

$$ \begin{align*} R_{\Gamma,\unicode{x3bb}}(v,x_U) := \frac{\mathbb{P}_{\Gamma,\unicode{x3bb}}([1^v] \vert [x_U])}{\mathbb{P}_{\Gamma,\unicode{x3bb}}([0^v] \vert [x_U])}, \end{align*} $$

where if $v \in U$ and $x_U(v)$ is equal to $1$ or $0$ , we let $R_{\Gamma ,\unicode{x3bb} }(v,x_U)$ be $\infty $ or $0$ , respectively. Notice that

$$ \begin{align*} \mathbb{P}_{\Gamma,\unicode{x3bb}}([0^v] \vert [x_U]) = \frac{1}{1+R_{\Gamma,\unicode{x3bb}}(v,x_U)} \quad \mbox{ and } \quad \mathbb{P}_{\Gamma,\unicode{x3bb}}([1^v] \vert [x_U]) = \frac{R_{\Gamma,\unicode{x3bb}}(v,x_U)}{1+R_{\Gamma,\unicode{x3bb}}(v,x_U)}. \end{align*} $$

Given a finite tree T rooted at $\rho $ , let us denote by $\{\rho _1,\ldots ,\rho _d\}$ the set of neighbors $\partial \{\rho \}$ of $\rho $ and by $T_i$ , for $i=1,\ldots ,d$ , the corresponding subtrees starting from $\rho _i$ , that is, $V(T) = \{\rho \} \cup V(T_1) \cup \cdots \cup V(T_d)$ . If we have a condition $x_U$ on U, we define $U_i = U \cap V(T_i)$ and $x_{U_i} = (x_{U})\vert _{U_i}$ . Considering this, we have that

$$ \begin{align*} R_{T,\unicode{x3bb}}(\rho, x_U) & = \frac{\mathbb{P}_{T,\unicode{x3bb}}([1^\rho] \vert [x_U])}{\mathbb{P}_{T,\unicode{x3bb}}([0^\rho] \vert [x_U])} \\ & = \frac{\unicode{x3bb}(\rho) \cdot Z^{x_U}_{T \backslash \{\rho \cup \partial \{\rho\}\}}(\unicode{x3bb})}{Z^{x_U}_T(\unicode{x3bb})} \cdot \frac{Z^{x_U}_T(\unicode{x3bb})}{Z^{x_U}_{T \backslash \{\rho\}}(\unicode{x3bb})} \\ & = \unicode{x3bb}(\rho) \cdot \prod_{i=1}^{d}\frac{Z^{x_{U_i}}_{T_i \backslash \{\rho_i\}}(\unicode{x3bb}_i)}{Z^{x_{U_i}}_{T_i}(\unicode{x3bb})} \\ & = \unicode{x3bb}(\rho) \cdot \prod_{i=1}^{d} \mathbb{P}_{T_i,\unicode{x3bb}}([0^{\rho_i}] \vert [x_{U_i}]) \\ & = \unicode{x3bb}(\rho) \cdot \prod_{i=1}^{d} \frac{1}{1+R_{T_i,\unicode{x3bb}}(\rho_i, x_{U_i})}, \end{align*} $$


$$ \begin{align*} Z^{x_U}_\Gamma(\unicode{x3bb}) := \sum_{y \in X(\Gamma): y_U = x_U} \prod_{v \in V(\Gamma)} \unicode{x3bb}(v)^{y(v)}. \end{align*} $$

Notice that this gives us a linear recursive procedure for computing $R_{T,\unicode{x3bb} }(\rho , x_U)$ , and therefore $\mathbb {P}_{T,\unicode{x3bb} }([0^\rho ] \vert [x_U])$ , with base cases: $R_{T,\unicode{x3bb} }(\rho , x_U) = 0 \text { or } +\infty $ if $\rho $ is fixed, and $R_{T,\unicode{x3bb} }(\rho , x_U) = \unicode{x3bb} (\rho )$ if $\rho $ is free and isolated.

Now, consider an arbitrary hardcore model $(\Gamma ,\unicode{x3bb} )$ and $v \in V(\Gamma )$ with neighbors $\partial \{v\} = \{u_1,\ldots ,u_d\}$ . We consider the auxiliary hardcore model $(\Gamma ',\unicode{x3bb} ')$ , where:

  • $V(\Gamma ') = V(\Gamma ) \backslash \{v\} \cup \{v_1,\ldots ,v_d\}$ ;

  • $E(\Gamma ') = E(\Gamma ) \backslash \{(v,u_i)\}_{i=1,\ldots ,d} \cup \{(v_i,u_i)\}_{i=1,\ldots ,d}$ ;

  • $\unicode{x3bb} '(v_i) = \unicode{x3bb} (v)^{1/d}$ for $i=1,\ldots ,d$ , and $\unicode{x3bb} '(u) = \unicode{x3bb} (u)$ otherwise.

Notice that

$$ \begin{align*} R_{\Gamma,\lambda}(v, x_U) & = \frac{\mathbb{P}_{\Gamma,\lambda}([1^v] \vert [x_U])}{\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U])} \\ & = \frac{\mathbb{P}_{\Gamma',\lambda'}([1^{\{v_1, \ldots, v_d\}}] \vert [x_U])}{\mathbb{P}_{\Gamma',\lambda'}([0^{\{v_1, \ldots, v_d\}}] \vert [x_U])} \\ & = \prod_{i=1}^{d}\frac{\mathbb{P}_{\Gamma',\lambda'}([0^{\{v_1,\ldots,v_{i-1}\}}1^{\{v_i, \ldots, v_d\}}] \vert [x_U])}{\mathbb{P}_{\Gamma',\lambda'}([0^{\{v_1,\ldots,v_i\}}1^{\{v_{i+1}, \ldots, v_d\}}] \vert [x_U])} \\ & = \prod_{i=1}^{d}\frac{\mathbb{P}_{\Gamma',\lambda'}([1^{v_i}] \vert [x_Uz_i])}{\mathbb{P}_{\Gamma',\lambda'}([0^{v_i}] \vert [x_Uz_i])} \\ & = \prod_{i=1}^{d}R_{\Gamma',\lambda'}(v_i,x_Uz_i), \end{align*} $$

where $z_i = 0^{\{v_1,\ldots ,v_{i-1}\}}1^{\{v_{i+1}, \ldots , v_d\}}$ and $x_Uz_i$ is the concatenation of $x_U$ and $z_i$ . Now, since $v_i$ is connected only to $u_i$ , notice that

$$ \begin{align*} R_{\Gamma',\unicode{x3bb}'}(v_i,x_Uz_i) = \frac{\unicode{x3bb}'(v_i) \cdot Z^{x_Uz_i}_{\Gamma' \backslash \{v_i,u_i\}}(\unicode{x3bb}')}{Z^{x_U z_i}_{\Gamma' \backslash \{v_i\}}(\unicode{x3bb}')} = \frac{\unicode{x3bb}^{1/d}(v)}{1+R_{\Gamma' \backslash \{v_i\},\unicode{x3bb}'}(u_i,x_Uz_i)}. \end{align*} $$


$$ \begin{align*} R_{\Gamma,\unicode{x3bb}}(v, x_U) = \prod_{i=1}^{d}\frac{\unicode{x3bb}^{1/d}(v)}{1+R_{\Gamma' \backslash \{v_i\},\unicode{x3bb}'}(u_i,x_Uz_i)} = \unicode{x3bb}(v) \cdot \prod_{i=1}^{d}\frac{1}{1+ R_{\Gamma' \backslash \{v_i\},\unicode{x3bb}'}(u_i,x_Uz_i)}. \end{align*} $$

Notice that the previous recursion can increase the original number of vertices, but the number of free vertices always decreases, so the recursion ends. Then, we have that:

  1. (1) $R_{T,\unicode{x3bb} }(v, x_U) = \unicode{x3bb} (\rho ) \cdot f(R_{T_1,\unicode{x3bb} }(\rho _1, x_{U_1}),\ldots ,R_{T_d,\unicode{x3bb} }(\rho _d, x_{U_d}))$ ; and

  2. (2) $R_{\Gamma ,\unicode{x3bb} }(v, x_U) = \unicode{x3bb} (v) \cdot f(R_{\Gamma ' \backslash \{v_1\},\unicode{x3bb} '}(u_1,x_U\tau _1),\ldots ,R_{\Gamma ' \backslash \{v_d\},\unicode{x3bb} '}(u_d,x_U\tau _d))$ ,

where $f(r_1,\ldots ,r_d) = \prod _{i=1}^{d}({1}/{(1+r_i)})$ . Now we proceed by induction in the number of free vertices. We can consider the base case where there are no free vertices (besides v) and the theorem is trivial. Then, if we know that the theorem is true when we have n free vertices, we prove it for $n+1$ . Notice that if $R_{\Gamma ',\unicode{x3bb} '}(v,x_U)$ involves $n+1$ free vertices, then $R_{\Gamma ' \setminus \{v\},\unicode{x3bb} '}(v_i,x_Uz_i)$ involves n free vertices, so by the induction hypothesis,

$$ \begin{align*} R_{\Gamma' \backslash \{v_i\},\unicode{x3bb}}(u_i,x_Uz_i) = R_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\unicode{x3bb}}}(\rho_i,\overline{x_{U}z_i}). \end{align*} $$

Then, noticing that the rooted subtree $(T_i,\rho _i)$ and the condition $\overline {x_{U}z_i}$ give exactly the tree of self-avoiding walks of $\Gamma ' \backslash \{v_i\}$ starting from $u_i$ under the condition $x_Uz_i$ , we are done.

Remark 5.2. The recursions presented in the proof of Theorem 5.1 give us a recursive procedure to compute the marginal probability of the root $\rho $ of a tree T being occupied which requires linear time with respect to the size of the tree. However, if $\Gamma $ is such that $\Delta (\Gamma ) \leq \Delta $ , then $T_{\mathrm {SAW}}(\Gamma ,v)$ is a subtree of $\mathbb {T}_\Delta $ and its size of $T_{\mathrm {SAW}}(\Gamma ,v)$ can be (at most) exponential in the size of