Counting independent sets in amenable groups

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 $Z_\Gamma (V,\unicode{x3bb} )$ .

3.1 Amenability

Let

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

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,

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

so,

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

Therefore,

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

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,

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

so,

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

Therefore,

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

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

so

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

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).

4.1 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.

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 $\unicode{x3bb} _+ \leq \unicode{x3bb} _0$ .