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

5.1 Connective constant

Given a graph $\Gamma $ , a vertex v, and an integer k, let $N_\Gamma (v,k)$ denote the number of self-avoiding walks in $\Gamma $ of length k starting from v. Following [Reference Sinclair, Srivastava, Štefankovič and Yin46], we consider the connective constant $\mu (\mathcal {G})$ of a family of finite graphs $\mathcal {G}$ . Here, $\mu (\mathcal {G})$ is defined as the infimum over all $\mu> 0$ for which there exist $a,c> 0$ such that for any $\Gamma \in \mathcal {G}$ and any $v \in V(\Gamma )$ , it holds that $\sum ^\ell _{k=1} N_\Gamma (v,k) \leq c\mu ^\ell $ for all $\ell \geq a\log \lvert V(\Gamma ) \rvert $ . This definition extends the more usual definition of connective constant for a single infinite almost transitive graph $\Gamma $ , which is given by

$$ \begin{align*} \mu(\Gamma) := \max_{v \in V(\Gamma)} \lim_{\ell \to \infty} N_\Gamma(v,\ell)^{1/\ell}. \end{align*} $$

Indeed, if $\Gamma $ is almost transitive, then $\mu (\Gamma ) = \mu (\mathcal {G}(\Gamma ))$ , where $\mathcal {G}(\Gamma )$ denotes the family of finite subgraphs of $\Gamma $ . Notice that $\mu (\Gamma )$ exists due to Fekete’s lemma and that, if $\Gamma $ is connected, then $\mu (\Gamma ) = \lim _{\ell \to \infty } N_\Gamma (v,\ell )^{1/\ell }$ for arbitrary v. Roughly, the connective constant measures the growth rate of the number of self-avoiding walks according to their length or, equivalently, the branching of $T_{\mathrm {SAW}}(\Gamma ,v)$ . In general, it is not an easy task to compute $\mu (\Gamma )$ (e.g., see [Reference Duminil-Copin and Smirnov15]).

Considering this, we extend the definition of strong spatial mixing to families of graphs as follows: given a family of graphs $\mathcal {G}$ and a family of activity functions $\Lambda = \{\unicode{x3bb} ^\Gamma \}_{\Gamma \in \mathcal {G}}$ with $\unicode{x3bb} ^\Gamma : V(\Gamma ) \to \mathbb {R}_{>0}$ , we say that $(\mathcal {G}, \Lambda )$ satisfies strong spatial mixing 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 $\Gamma \in \mathcal {G}$ , for all $U \Subset V(\Gamma )$ , $v \in U$ , and $y,z \in X(\Gamma )$ ,

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

where $\pi ^y_{\Gamma , U}$ denotes the specification element corresponding to the hardcore model $(\Gamma , \unicode{x3bb} ^\Gamma )$ . We translate into this language the following result from [Reference Sinclair, Srivastava, Štefankovič and Yin46].

Theorem 5.8. [Reference Sinclair, Srivastava, Štefankovič and Yin46]

Let $\mathcal {G}$ be a family of almost transitive locally finite graphs and $\Lambda = \{\unicode{x3bb} ^{\Gamma }\}_{\Gamma \in \mathcal {G}}$ a set of activity functions such that

$$ \begin{align*} \sup_{\Gamma \in \mathcal{G}} \unicode{x3bb}^\Gamma_+ < \unicode{x3bb}_{c}(\mu(\mathcal{G})+1). \end{align*} $$

Then, $(\mathcal {G}, \Lambda )$ exhibits exponential SSM.

Notice that if a graph has maximum degree $\Delta $ , then $\mu (\Gamma ) \leq \Delta -1$ . In addition, observe that $N_\Gamma (v,\ell ) = N_{T_{\mathrm {SAW}}(\Gamma ,v)}(\rho ,\ell )$ . We have the following corollary.

Corollary 5.9. If $(\Gamma ,\unicode{x3bb} )$ is a hardcore model such that

$$ \begin{align*} \unicode{x3bb}_+ < \unicode{x3bb}_c(\mu(\Gamma)+1), \end{align*} $$

then $(T_{\mathrm {SAW}}(\Gamma ,v),\overline {\unicode{x3bb} })$ exhibits (exponential) SSM for every $v \in V(\Gamma )$ . In particular, $(\Gamma ,\unicode{x3bb} )$ exhibits (exponential) SSM and for every $v \in V(\Gamma )$ , $x \in X(\Gamma $ ), and $U \subseteq V(\Gamma )$ ,

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

6.1 Orderable groups

Let $\prec $ be a strict total order on G. We say that $\prec $ is an invariant (right) order if, for all $h_1,h_2,g \in G$ ,

$$ \begin{align*} h_1 \prec h_2 \!\implies\! h_1g \prec h_2g. \end{align*} $$

We call the pair $(G,\prec )$ a (linearly) ordered group. The associated algebraic past of an ordered group $(G,\prec )$ is the set $\Phi _\prec := \{g \in G: g \prec 1_G\}$ and it is a semigroup which satisfies that

$$ \begin{align*} G = \Phi_\prec \sqcup \{1_G\} \sqcup \Phi_\prec^{-1}. \end{align*} $$

Notice that $h \prec g \iff hg^{-1} \in \Phi _\prec $ , so $\Phi _\prec $ fully determines $\prec $ and vice versa.

The class of orderable groups contains all torsion-free abelian groups and it is closed under the operations of taking subgroups, and forming extensions, arbitrary direct products, and directed unions.

A group G is called virtually orderable (respectively ordered) if there exists an orderable (respectively ordered) subgroup $H \leq G$ of finite index $[G:H]$ . Notice that if $G \curvearrowright \Gamma $ is almost transitive and free with fundamental domain $U_0$ , then $H \curvearrowright \Gamma $ is also almost transitive and free with fundamental domain $KU_0$ , where $K \in \mathcal {F}(G)$ is any finite set of representatives. In particular, $\lvert \Gamma /H \rvert = \lvert \Gamma /G \rvert [G:H]$ . For this reason, since we are interested in almost transitive actions, there is no loss of generality if, given a virtually orderable group, we assume that it is just orderable: the free energies $f_G(\Gamma ,\unicode{x3bb} )$ and $f_H(\Gamma ,\unicode{x3bb} )$ will be equal and the only effect of passing to a finite-index subgroup will be that the size of the fundamental domain of the action is multiplied by a constant factor (that is, the index of the subgroup $[G:H]$ ). This last point is relevant, since the size of the fundamental domain will play a role later when measuring the computational complexity of some problems.

Given a finitely generated group G, a generating set S, and its corresponding Cayley graph $\Gamma = \mathrm {Cay}(G,S)$ , we define the volume growth function as $g_\Gamma (n) = \lvert B_\Gamma (1_G,n) \rvert $ . We say that G has polynomial growth if $g_\Gamma (n) \leq p(n)$ for some polynomial p. It is well known that groups with polynomial growth are amenable and a classic result due to Gromov asserts that they are virtually nilpotent [Reference Gromov19]. Without further detail, from Schreier’s lemma, it is also well known that finite index subgroups of finitely generated groups are also finitely generated [Reference Lyndon and Schupp32, Proposition 4.2] and finitely generated nilpotent groups have a torsion-free nilpotent subgroup with finite index [Reference Segal43, Proposition 2]. From this, and since torsion-free nilpotent groups are orderable [Reference Mura and Rhemtulla39, p. 37], it follows that any finitely generated group G with polynomial growth is amenable and virtually orderable. In particular, all our results that apply to amenable and virtually orderable groups will hold for groups of polynomial growth, but they will also hold in groups of super-polynomial—namely, exponential—growth. This includes solvable groups that are not virtually nilpotent [Reference Milnor38] and, more concretely, cases like the Baumslag–Solitar groups $\mathrm {BS}(1,n)$ that can also be ordered. However, not every amenable group is virtually orderable; for example, the direct sum of countably many non-trivial finite groups always results in a countable group that is amenable but not virtually orderable. To address these cases, we introduce a randomized generalization of invariant orders.

6.2 Random orders

Consider now the set of relations $\{0,1\}^{G \times G}$ endowed with the product topology and the closed subset $\mathrm {Ord}(G)$ of strict total orders $\prec $ on G. We will consider the action $G \curvearrowright \mathrm {Ord}(G)$ given by

$$ \begin{align*} h_1(g ~\cdot \prec)h_2 \!\iff (h_1g)\prec(h_2g) \end{align*} $$

for $h_1,h_2,g \in G$ and $\prec \in \mathrm {Ord}(G)$ . An invariant random order on G is a G-invariant Borel probability measure on $\mathrm {Ord}(G)$ . Notice that a fixed point for the action $G \curvearrowright \mathrm {Ord}(G)$ corresponds to a (deterministic) invariant order on G. The space of invariant random orders will be denoted by $\mathcal {M}_G(\mathrm {Ord}(G))$ .

Invariant random orders were introduced in [Reference Alpeev, Meyerovitch and Ryu1] to answer problems about predictability in topological dynamics through what they called the Kieffer–Pinsker formula for the Kolmogorov–Sinai entropy of a group action.

Now, as in the deterministic case, we can also define a notion of past for the group. An invariant random past on G is a random function $\tilde {\Phi }: G \to \{0,1\}^G$ or, equivalently, a Borel probability measure on $(\{0,1\}^G)^G$ that satisfies, for almost every instance of $\tilde {\Phi }$ , the following properties:

1. (1) for all $g \in G$ , the condition $g \notin \tilde {\Phi }(g)$ holds;

2. (2) for all $g,h \in G$ , if $g \in \tilde {\Phi }(h)$ , then $\tilde {\Phi }(g) \subseteq \tilde {\Phi }(h)$ ;

3. (3) if $g \neq h$ , then either $g \in \tilde {\Phi }(h)$ or $h \in \tilde {\Phi }(g)$ ; and

4. (4) for all $g \in G$ , the random subsets $\tilde {\Phi }(g)$ and $\tilde {\Phi }(1_G)g$ have the same distribution.

Notice that if $\prec $ is an invariant random order, then the random function $g \mapsto \{h \in G: h \prec g\}$ defines an invariant random past.

In contrast to deterministic invariant orders, every countable group G admits at least one invariant random total order. Namely, consider the random process $(\chi _g)_{g \in G}$ of independent random variables such that each $\chi _g$ has uniform distribution on $[0,1]$ . This process is invariant and each realization of it induces an order on G almost surely. We call such order the uniform random order.

7.1 A pointwise Shannon–McMillan–Breiman-type theorem

The next theorem establishes a pointwise Shannon–McMillan–Breiman-type theorem for Gibbs measures (related results can be found in [Reference Briceño9, Reference Gurevich and Tempelman20]). To prove it, we use the pointwise ergodic theorem [Reference Lindenstrauss28], which requires tyhe Følner sequence $\{F_n\}_n$ to be tempered, a technical condition that is satisfied by every Følner sequence up to a subsequence and that we will assume without further detail.

Theorem 7.1. Let G be a countable amenable group. For every $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G$ and every $\mathbb {P} \in \mathcal {M}_{\mathrm {Gibbs}}(\Gamma ,\unicode{x3bb} )$ ,

$$ \begin{align*} \lim_n \bigg[ - \frac{1}{\lvert F_nU_0 \rvert} \log \mathbb{P}([x_{F_nU_0}]) \bigg] = -\int{\phi_\unicode{x3bb}} \,d\mathbb{Q} + f_G(\Gamma,\unicode{x3bb}) \quad \mathbb{Q}(x)\text{-almost surely in } x \end{align*} $$

for any tempered Følner sequence $\{F_n\}_n$ and any $\mathbb {Q} \in \mathcal {M}_G^{\mathrm {erg}}(X(\Gamma ))$ .

Proof. Consider the sets $U_n = F_nU_0$ and $M_n = U_n \cup \partial U_n$ . Notice that, by amenability, $\lim _{n \to \infty } {\lvert M_n \rvert }/{\lvert U_n \rvert } = 1$ . Indeed, define $K {\kern-1pt}={\kern-1pt} \{g {\kern-1pt}\in{\kern-1pt} G: \mathrm {dist}_\Gamma (U_0,gU_0) {\kern-1pt}\leq{\kern-1pt} 1\}$ . Then, $1_G \in K$ and $U_0 \cup \partial U_0 \subseteq KU_0$ . Since $\Gamma $ is locally finite and the action is free, K is finite. In addition, $U_n \cup \partial U_n \subseteq F_nKU_0$ . Therefore, by amenability,

$$ \begin{align*} 1 \leq \lim_n \frac{\lvert U_n \cup \partial U_n \rvert}{\lvert U_n \rvert} \leq \lim_n \frac{\lvert F_nKU_0 \rvert}{\lvert F_nU_0 \rvert} = 1, \end{align*} $$

so $\lim _n {\lvert \partial U_n \rvert }/{\lvert U_n \rvert } = 0$ . Fix independent sets $x \in X(\Gamma [U_n])$ , $z_1,z_2 \in X(\Gamma [M_n \setminus U_n])$ , and $y \in X(\Gamma )$ such that $xz_iy \in X(\Gamma )$ for $i = 1,2$ . Then,

$$ \begin{align*} \frac{\pi^y_{M_n}(xz_1)}{\pi^y_{M_n}(xz_2)} & = \frac{\mathrm{w}^y_\unicode{x3bb}(xz_1,M_n) Z^y_\Gamma(M_n,\unicode{x3bb})^{-1}}{\mathrm{w}^y_\unicode{x3bb}(xz_2,M_n) Z^y_\Gamma(M_n,\unicode{x3bb})^{-1}} \\ & = \frac{\mathrm{w}_\unicode{x3bb}(xz_1,M_n)1_{[y_{M_n^c}]}(xz_1)}{\mathrm{w}_\unicode{x3bb}(xz_2,M_n) 1_{[y_{M_n^c}]}(xz_2)} \\ & = \frac{\prod_{v \in M_n \setminus U_n} \unicode{x3bb}(v)^{z_1(v)}}{\prod_{v \in M_n \setminus U_n} \unicode{x3bb}(v)^{z_2(v)}} \\ & \leq \prod_{v \in M_n \setminus U_n} \unicode{x3bb}_+^{z_1(v)}\unicode{x3bb}_-^{-z_2(v)}. \end{align*} $$

Taking $z_2 = 0^\Gamma $ and adding over all possible $z_1$ , we obtain that

$$ \begin{align*} 1 \leq \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^y_{M_n}(x_{U_n}0^{M_n \setminus U_n})} = \sum_{\substack{z \in \{0,1\}^{M_n \setminus U_n}:\\ x_{U_n}z \in X(\Gamma[M_n])}}\frac{\pi^y_{M_n}(x_{U_n}z)}{\pi^y_{M_n}(x_{U_n}0^{M_n \setminus U_n})} \leq (2\max\{1,\unicode{x3bb}_+\})^{\lvert M_n \setminus U_n \rvert}. \end{align*} $$

However, we have that

$$ \begin{align*} \frac{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} = \frac{\mathrm{w}^y_\unicode{x3bb}(x_{U_n}0^{M_n \setminus U_n},M_n) Z^y_\Gamma(M_n,\unicode{x3bb})^{-1}}{\mathrm{w}^{0^\Gamma}_\unicode{x3bb}(x_{U_n}0^{M_n \setminus U_n},M_n) Z^{0^\Gamma}_\Gamma(M_n,\unicode{x3bb})^{-1}} = \frac{Z^y_\Gamma(M_n,\unicode{x3bb})}{Z_\Gamma(M_n,\unicode{x3bb})}, \end{align*} $$

since

$$ \begin{align*} \mathrm{w}^y_\unicode{x3bb}(x_{U_n}0^{M_n \setminus U_n},M_n) = \mathrm{w}^{0^\Gamma}_\unicode{x3bb}(x_{U_n}0^{M_n \setminus U_n},M_n) = \mathrm{w}_\unicode{x3bb}(x_{U_n},U_n) \end{align*} $$

and $Z^{0^\Gamma }_\Gamma (M_n,\unicode{x3bb} ) = Z_\Gamma (M_n,\unicode{x3bb} )$ . In addition,

$$ \begin{align*} 1 \leq \frac{Z^y_\Gamma(M_n,\unicode{x3bb})}{Z_\Gamma(M_n,\unicode{x3bb})} \leq (2\max\{1,\unicode{x3bb}_+\})^{\lvert M_n \setminus U_n \rvert}, \end{align*} $$

since

$$ \begin{align*} Z^y_\Gamma(M_n,\unicode{x3bb}) & \leq Z_\Gamma(M_n,\unicode{x3bb}) \\ & = \sum_{x \in X(\Gamma[U_n])}\sum_{\substack{z \in X(\Gamma[M_n \setminus U_n]):\\ xz \in X(\Gamma[M_n])}} \prod_{v \in U_n} \unicode{x3bb}(v)^{x(v)}\prod_{v \in M_n \setminus U_n} \unicode{x3bb}(v)^{z(v)} \\ & \leq \sum_{x \in X(\Gamma[U_n])}\sum_{\substack{z \in X(\Gamma[M_n \setminus U_n]):\\ xz \in X(\Gamma[M_n])}} \prod_{v \in U_n} \unicode{x3bb}(v)^{x(v)} \max\{1,\unicode{x3bb}_+\}^{\lvert M_n \setminus U_n \rvert} \\ & \leq 2^{\lvert M_n \setminus U_n \rvert} \max\{1,\unicode{x3bb}_+\}^{\lvert M_n \setminus U_n \rvert}\sum_{x \in X(\Gamma[U_n])} \prod_{v \in U_n} \unicode{x3bb}(v)^{x(v)} \\ & = (2\max\{1,\unicode{x3bb}_+\})^{\lvert M_n \setminus U_n \rvert} Z_\Gamma(U_n,\unicode{x3bb}) \\ & \leq (2\max\{1,\unicode{x3bb}_+\})^{\lvert M_n \setminus U_n \rvert} Z^y_\Gamma(M_n,\unicode{x3bb}). \end{align*} $$

Therefore,

$$ \begin{align*} 1 & \leq \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \\ & = \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}\frac{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \\ & = \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}\frac{Z^y_\Gamma(M_n,\unicode{x3bb})}{Z_\Gamma(M_n,\unicode{x3bb})} \\ & \leq (2\max\{1,\unicode{x3bb}_+\})^{2\lvert M_n \setminus U_n \rvert}. \end{align*} $$

In particular, since $\mathbb {P}([x_{F_nU_0}]) = \mathbb {E}_{\mathbb {P}}(\mathbb {P}([x_{U_n}] \vert \mathcal {B}_{M_n^c})(y)) = \mathbb {E}_{\mathbb {P}}(\pi ^y_{M_n}([x_{U_n}]))$ , we have that

$$ \begin{align*} 1 \leq \frac{\mathbb{P}([x_{F_nU_0}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \leq (2\max\{1,\unicode{x3bb}_+\})^{2\lvert M_n \setminus U_n \rvert}, \end{align*} $$

so

$$ \begin{align*} \lvert \log\mathbb{P}([x_{F_nU_0}]) - \log\pi^{0^\Gamma}_{M_n}(x_{U_n}0^{M_n \setminus U_n}) \rvert \leq 2\lvert M_n \setminus U_n \rvert\log(2\max\{1,\unicode{x3bb}_+\}). \end{align*} $$

Now, since $\mathrm {w}^{0^\Gamma }_\unicode{x3bb} (x_{M_n},M_n) = \mathrm {w}_\unicode{x3bb} (x_{M_n},M_n)$ for every x, we have that

$$ \begin{align*} \pi^{0^\Gamma}_{M_n}(x_{U_n}0^{M_n \setminus U_n}) = \mathrm{w}_\unicode{x3bb}(x_{U_n}0^{M_n \setminus U_n},M_n) Z_\Gamma(M_n,\unicode{x3bb})^{-1} = \mathrm{w}_\unicode{x3bb}(x_{U_n},U_n) Z_\Gamma(M_n,\unicode{x3bb})^{-1}. \end{align*} $$

Therefore,

$$ \begin{align*} \lvert \log\mathbb{P}([x_{F_nU_0}]) - (\log\mathrm{w}_\unicode{x3bb}(x_{U_n},U_n) - \log Z_\Gamma(M_n,\unicode{x3bb})) \rvert \leq 2\lvert M_n \setminus U_n \rvert\log(2\max\{1,\unicode{x3bb}_+\}), \end{align*} $$

so

$$ \begin{align*} & \lim_{n \to \infty} \lvert -\frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert U_n \rvert} + \bigg(\frac{\log\mathrm{w}_\unicode{x3bb}(x_{U_n},U_n)}{\lvert U_n \rvert} - \frac{\log Z_\Gamma(M_n,\unicode{x3bb})}{\lvert U_n \rvert}\bigg)\rvert \\ &\quad\leq \lim_n 2\frac{\lvert M_n \setminus U_n \rvert}{\lvert U_n \rvert}\log(2\max\{1,\unicode{x3bb}_+\}) = 0, \end{align*} $$

and we conclude that

$$ \begin{align*} -\lim_n \frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_nU_0 \rvert} & = \lim_n \frac{-\log\mathrm{w}_\unicode{x3bb}(x_{U_n},U_n)}{\lvert U_n \rvert} + \frac{\log Z_\Gamma(M_n,\unicode{x3bb})}{\lvert U_n \rvert} \\ & = -\lim_n \bigg(\frac{1}{\lvert F_n \rvert} \sum_{g \in F_n} \frac{1}{\lvert U_0 \rvert} \sum_{v \in gU_0} x(v)\log\unicode{x3bb}(v)\bigg) + f_G(\Gamma,\unicode{x3bb}), \end{align*} $$

where we have used that $Z_\Gamma (U_n,\unicode{x3bb} ) \leq Z_\Gamma (M_n,\unicode{x3bb} ) \leq (2\max \{1,\unicode{x3bb} _+\})^{\lvert M_n \setminus U_n \rvert }Z_\Gamma (U_n,\unicode{x3bb} )$ . Finally, notice that

$$ \begin{align*} \frac{1}{\lvert F_n \rvert} \sum_{g \in F_n} \frac{1}{\lvert U_0 \rvert} \sum_{v \in gU_0} x(v)\log\unicode{x3bb}(v) = \frac{1}{\lvert F_n \rvert} \sum_{g \in F_n} \phi_\unicode{x3bb}(g \cdot x) \end{align*} $$

and by the pointwise ergodic theorem, we obtain that

$$ \begin{align*} \lim_n \frac{1}{\lvert F_n \rvert} \sum_{g \in F_n} \phi_\unicode{x3bb}(g \cdot x) = \int{\phi_\unicode{x3bb}}\,d\mathbb{Q} \quad \mathbb{Q}\text{-almost surely}, \end{align*} $$

so

$$ \begin{align*} -\lim_n \frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_nU_0 \rvert} = - \int{\phi_\unicode{x3bb}}\,d\mathbb{Q} + f_G(\Gamma,\unicode{x3bb}), \end{align*} $$

and we conclude the proof.

We have the following lemma.

Lemma 7.2. Given $\Delta \in \mathbb {N}$ and $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}^\Delta $ , there exists a constant $C\hspace{-1pt} =\hspace{-1pt} C(\Delta ,\unicode{x3bb} _-, \unicode{x3bb} _+)> 0$ such that for every $U \Subset V(\Gamma )$ , $x \in X(\Gamma )$ , and $v \in U$ ,

$$ \begin{align*} \pi^x_U([x_v]) \geq C. \end{align*} $$

Proof. Fix $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G^\Delta $ , $U \Subset V(\Gamma )$ , $x \in X(\Gamma )$ , and $v \in U$ . Notice that if $U^c \cap \partial \{v\} \neq \emptyset $ and $u \in U^c \cap \partial \{v\}$ is such that $x(u) = 1$ , then necessarily $x_v = 0^v$ and $\pi ^x_U([x_v]) = 1$ . However, if $U \cap \partial \{v\} = \emptyset $ , then $\pi ^x_U([x_v]) = \mathbb {P}_{\Gamma [U'],\unicode{x3bb} }([x_v])$ for $U' = U \setminus \{u \in U: x(u') = 1 \text { for some } u' \in U^c \cap \partial \{u\}\}$ , so, by Lemma 5.3,

$$ \begin{align*} \pi^x_U([x_v]) = \mathbb{P}_{\Gamma[U'],\unicode{x3bb}}([x_v]) \geq \min\bigg\{\frac{1}{1+\unicode{x3bb}_+}, \frac{\unicode{x3bb}_-}{\unicode{x3bb}_- + (1+\unicode{x3bb}_+)^\Delta}\bigg\} \geq \frac{\min\{1,\unicode{x3bb}_-\}}{\unicode{x3bb}_- + (1+\unicode{x3bb}_+)^\Delta}> 0. \end{align*} $$

Therefore, by taking $C = \min \{1, {\min \{1,\unicode{x3bb} _-\}}/({\unicode{x3bb} _- + (1+\unicode{x3bb} _+)^\Delta })\} = {\min \{1,\unicode{x3bb} _-\}}/ (\unicode{x3bb} _- + (1+\unicode{x3bb} _+)^\Delta )$ , we conclude.

7.2 A randomized sequential cavity method

Suppose now that $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G$ is such that the Gibbs $(\Gamma ,\unicode{x3bb} )$ -specification satisfies SSM and let $\mathbb {P}$ be the unique Gibbs measure. Considering this, we define the function $I_{\mathbb {P}}: X(\Gamma ) \times (\{0,1\}^G)^G \to \mathbb {R}$ given by

$$ \begin{align*} I_{\mathbb{P}}(x,\Phi) := \limsup_{n \to \infty} I_{\mathbb{P}, n}(x,\Phi), \end{align*} $$

where

$$ \begin{align*} I_{\mathbb{P},n}(x,\Phi) := -\frac{1}{\lvert \Gamma/G \rvert}\log\mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}]) \end{align*} $$

and $\{F_n\}_n$ is any exhaustion of G (not necessarily Følner). Here, $\Phi \in (\{0,1\}^G)^G$ should be understood as an instance of a random invariant past and $I_{\mathbb {P},n}$ as a rudimentary information function. In this sense, notice that $I_{\mathbb {P},n}(x,\Phi )$ only depends on the values of x restricted to $\Phi (1_G) \in \{0,1\}^G$ , so $\Phi $ carries redundant information. This redundancy will play a role in the next results when taking averages.

Lemma 7.3. If $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G$ is such that the Gibbs $(\Gamma ,\unicode{x3bb} )$ -specification satisfies SSM and $\mathbb {P}$ is the unique Gibbs measure, then the function $I_{\mathbb {P}}$ is measurable, non-negative, defined everywhere, and bounded. Moreover,

$$ \begin{align*} I_{\mathbb{P}}(x,\Phi) = -\frac{1}{\lvert \Gamma/G \rvert}\log\mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G)U_0}]). \end{align*} $$

Proof. Since $\mathbb {P}([x_{U_0}] \vert [x_{(\Phi (1_G) \cap F_n)U_0}])$ depends on finitely many coordinates in both $X(\Gamma )$ and $(\{0,1\}^G)^G$ , $0 < \mathbb {P}([x_{U_0}] \vert [x_{(\Phi (1_G) \cap F_n)U_0}]) < 1$ , and $-\log (\cdot )$ is a continuous function, $I_{\mathbb {P},n}$ is measurable and since $I_{\mathbb {P}}$ is a limit superior, it is measurable as well.

By SSM and Proposition 4.1,

$$ \begin{align*}\lim_{n \to \infty} \mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}]) = \mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G)U_0}])\end{align*} $$

is always a well-defined limit. By Lemma 7.2 and since $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}^\Delta _G$ for some $\Delta $ , there exists a constant $C> 0$ such that for every $v \in V$ , $U \Subset V \setminus \{v\}$ , and $x \in X(\Gamma )$ ,

$$ \begin{align*} \pi^x_U([x_v]) \geq C. \end{align*} $$

Now, combined with the SSM property, this implies that for every $v \in V$ , $U \subseteq V$ , and $x \in X(\Gamma )$ , we have that

$$ \begin{align*} \mathbb{P}([x_v] \vert [x_U]) \geq C. \end{align*} $$

Indeed, if $v \in U$ , this is direct, since $\mathbb {P}([x_v] \vert [x_U]) = 1$ . However, if $v \notin U$ , by SSM,

$$ \begin{align*} \mathbb{P}([x_v] \vert [x_U]) = \lim_{\ell \to \infty} \mathbb{P}([x_v] \vert [x_{U \cup B_\Gamma(v,\ell)^c}]) = \lim_{\ell \to \infty} \pi_{U \cup B_\Gamma(v,\ell)^c}^x([x_v]) \geq \lim_{\ell \to \infty} C = C. \end{align*} $$

Therefore, by conditioning and iterating, we obtain that

$$ \begin{align*} 1 \geq \mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G) U_0}]) = \prod_{i=1}^{\lvert U_0 \rvert} \mathbb{P}([x_{v_i}] \vert [x_{\Phi U_0 \cup \{v_1,\ldots,v_{i-1}\}}]) \geq \prod_{i=1}^{\lvert U_0 \rvert} C = C^{\lvert \Gamma / G \rvert}, \end{align*} $$

so

$$ \begin{align*} 0 \leq I_{\mathbb{P}}(x,\Phi) = -\log\mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G)U_0}]) \leq -\lvert \Gamma / G \rvert\log C < +\infty, \end{align*} $$

that is, $I_{\mathbb {P}}(x,\Phi )$ is bounded.

Following [Reference Alpeev, Meyerovitch and Ryu1], given an invariant random past $\tilde {\Phi }: G \to \{0,1\}^G$ on G with law $\tilde {\nu } \in \mathcal {M}_G((\{0,1\}^G)^G)$ , we denote

$$ \begin{align*} \mathbb{E}_{\tilde{\Phi}}f(\tilde{\Phi}) = \int{f(\tilde{\Phi})}d\tilde{\nu}(\tilde{\Phi}) \end{align*} $$

for $f \in L^1(\tilde {\nu })$ . Now, since $I_{\mathbb {P}}$ is measurable, non-negative, and bounded, we have that for every $\mathbb {Q} \in \mathcal {M}_G(X(\Gamma ))$ , the function $I_{\mathbb {P}}$ is integrable with respect to $\mathbb {Q} \times \tilde {\nu }$ and by Tonelli’s theorem, the function $\mathbb {E}_{\tilde {\Phi }}I_{\mathbb {P}}: X(\Gamma ) \to \mathbb {R}$ is integrable, defined $\mathbb {Q}$ -almost everywhere, and satisfies that

$$ \begin{align*} \int{\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(x,\tilde{\Phi}})\,d\mathbb{Q}(x) = \mathbb{E}_{\tilde{\Phi}}\int{I_{\mathbb{P}}(x,\tilde{\Phi}})\,d\mathbb{Q}(x). \end{align*} $$

We call $\mathbb {E}_{\tilde {\Phi }}I_{\mathbb {P}}$ the random $\mathbb {P}$ -information function (with respect to $\tilde {\Phi }$ ).

Lemma 7.4. For any tempered Følner sequence $\{F_n\}_n$ , any invariant random past $\tilde {\Phi }$ , and any $\mathbb {Q} \in \mathcal {M}_G(X(\Gamma ))$ ,

$$ \begin{align*} \lim_n \bigg[ - \frac{1}{\lvert F_nU_0 \rvert} \log \mathbb{P}([x_{F_nU_0}]) - \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi}) \bigg] = 0 \end{align*} $$

for $\mathbb {Q}$ -almost every x.

Proof. Fix a (tempered) Følner sequence $\{F_n\}_n$ . By the properties of $\tilde {\Phi }$ , for $\tilde {\nu }$ -almost every instance $\Phi $ , every $F_n$ can be ordered as $g_1, \ldots ,g_{\lvert F_n \rvert }$ so that $\Phi (g_i) \cap F_n = \{g_1,\ldots ,g_{i-1}\}$ . Then,

$$ \begin{align*} \mathbb{P}([x_{F_nU_0}]) = \prod_{g \in F_n} \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]). \end{align*} $$

Given $\ell> 0$ , let $K_\ell = \{g \in G: \mathrm {dist}_\Gamma (U_0,gU_0) \leq \ell \}$ . If $g \in \mathrm {Int}_{K_\ell }(F_n) = \{g \in G: K_\ell g \subseteq F_n\}$ , then

$$ \begin{align*} \lvert \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap K_\ell g)U_0}]) \rvert \leq \lvert U_0 \rvert\delta(\ell) \end{align*} $$

and

$$ \begin{align*} \lvert \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap K_\ell g)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) \rvert \leq \lvert U_0 \rvert\delta(\ell), \end{align*} $$

so

$$ \begin{align*} \lvert \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) \rvert \leq 2\lvert U_0 \rvert\delta(\ell). \end{align*} $$

However, by Lemma 7.2 and the discussion after it, for every $g \in G$ and $U \subseteq V$ ,

$$ \begin{align*} \mathbb{P}([x_{gU_0}] \vert [x_U]) \geq C^{\lvert \Gamma / G \rvert}> 0. \end{align*} $$

Therefore, by the mean value theorem,

$$ \begin{align*} \lvert \log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) \rvert \leq \frac{2\lvert U_0\rvert}{C^{\lvert \Gamma / G\rvert}}\delta(\ell). \end{align*} $$

Notice that

$$ \begin{align*} \log\mathbb{P}([x_{F_nU_0}]) & = \sum_{g \in \mathrm{Int}_{K_\ell}(F_n)} \log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) \\ & \quad + \sum_{g \in F_n \setminus \mathrm{Int}_{K_\ell}(F_n)} \log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]), \end{align*} $$

so

$$ \begin{align*} \lvert \log\mathbb{P}([x_{FU_0}]) - \sum_{g \in F_n} \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])\rvert & \leq \lvert \mathrm{Int}_{K_\ell}(F_n)\rvert\frac{2\lvert U_0\rvert}{C^{\lvert \Gamma / G\rvert}}\delta(\ell) \\ & \quad + 2\lvert F_n \setminus \mathrm{Int}_{K_\ell}(F_n)\rvert\log(C^{-\lvert \Gamma / G\rvert}). \end{align*} $$

Now, given $\epsilon> 0$ , there exists $\ell $ and $n_0$ such that for every $n \geq n_0$ ,

$$ \begin{align*} \bigg\lvert \frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_n\rvert} - \frac{1}{\lvert F_n\rvert}\sum_{g \in F_n} \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) \bigg\rvert \leq \epsilon. \end{align*} $$

By G-invariance of $\mathbb {P}$ ,

$$ \begin{align*} \mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) & = \mathbb{P}(g^{-1} \cdot [(g \cdot x)_{U_0}] \vert [x_{\Phi(g)U_0}]) \\ & = \mathbb{P}([(g \cdot x)_{U_0}] \vert g \cdot [x_{\Phi(g)U_0}]) \\ & = \mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\Phi(g)g^{-1}U_0}]), \end{align*} $$

and combining this fact with the previous estimate, we obtain that

$$ \begin{align*} \bigg\lvert \frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_n\rvert} - \frac{1}{\lvert F_n\rvert}\sum_{g \in F_n} \log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\Phi(g)g^{-1}U_0}]) \bigg\rvert \leq \epsilon. \end{align*} $$

Integrating against $\tilde {\nu }$ , we obtain that, for $\mathbb {Q}$ -almost every x,

$$ \begin{align*} \bigg\lvert \frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_n\rvert} - \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(g)g^{-1}U_0}]) \bigg\rvert \leq \epsilon \end{align*} $$

and since $\tilde {\Phi }(g)$ has the same distribution as $\tilde {\Phi }(1_G)g$ , we get that

$$ \begin{align*} \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(g)g^{-1}U_0}]) & = \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(1_G)gg^{-1}U_0}]) \\ & = \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(1_G)U_0}]) \\ & = -\lvert \Gamma/G \rvert\mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(g \cdot x, \tilde{\Phi}), \end{align*} $$

so

$$ \begin{align*} \bigg\lvert -\frac{\log\mathbb{P}([x_{F_nU_0}])}{\lvert F_nU_0 \rvert} - \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(g \cdot x, \tilde{\Phi}) \bigg\rvert \leq \epsilon, \end{align*} $$

and since $\epsilon $ is arbitrary and the limit exists $\mathbb {Q}$ -almost surely, we conclude.

We have the following representation theorem for free energy, which can be regarded as a randomized generalization of the results in [Reference Briceño9, Reference Gamarnik and Katz16, Reference Marcus and Pavlov36] tailored for the specific case of the hardcore model. Notice that, in contrast with [Reference Briceño9, Reference Gamarnik and Katz16, Reference Marcus and Pavlov36], the representation holds in every amenable group and not just virtually orderable groups.

Theorem 7.5. Let $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G$ such that the Gibbs $(\Gamma ,\unicode{x3bb} )$ -specification satisfies SSM and $\mathbb {P}$ is the unique Gibbs measure. Then,

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \int{(\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}} + \phi_\unicode{x3bb})}\,d\mathbb{Q} \end{align*} $$

for any $\mathbb {Q} \in \mathcal {M}_G(X(\Gamma ))$ and for any invariant random past $\tilde {\Phi }$ of G. In particular,

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(0^\Gamma,\tilde{\Phi}). \end{align*} $$

Proof. First, notice that if the statement holds for every $\mathbb {Q} \in \mathcal {M}^{\mathrm {erg}}_G(X(\Gamma ))$ , then it holds for every $\mathbb {Q} \in \mathcal {M}_G(X(\Gamma ))$ by the ergodic decomposition theorem. Then, without loss of generality, we can assume that $\mathbb {Q}$ is G-ergodic. Considering this, by Theorem 7.1, for $\mathbb {Q}$ -almost every x,

$$ \begin{align*} \lim_n \bigg[ - \frac{1}{\lvert F_nU_0 \rvert} \log \mathbb{P}([x_{F_n}]) \bigg] = -\int{\phi_\unicode{x3bb}} \,d\mathbb{Q} + f_G(\Gamma,\unicode{x3bb}). \end{align*} $$

By Lemma 7.4, for $\mathbb {Q}$ -almost every x,

$$ \begin{align*} \lim_n \bigg[ - \frac{1}{\lvert F_nU_0 \rvert} \log \mathbb{P}([x_{F_n}]) \bigg] = \lim_n \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(g \cdot x, \tilde{\Phi}). \end{align*} $$

Therefore, for $\mathbb {Q}$ -almost every x,

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) - \int{\phi_\unicode{x3bb}} \,d\mathbb{Q} = \lim_n \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(g \cdot x, \tilde{\Phi}). \end{align*} $$

Integrating against $\mathbb {Q}$ , we obtain that

$$ \begin{align*} \int{ \lim_n \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}\,d\mathbb{Q} & = \lim_n\int{ \frac{1}{\lvert F_n \rvert}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}\,d\mathbb{Q} \\ & = \lim_n\frac{1}{\lvert F_n \rvert}\sum_{g \in F_n}\mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}\,d\mathbb{Q} \\ & = \lim_n\frac{1}{\lvert F_n \rvert}\sum_{g \in F_n}\mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(x,\tilde{\Phi})}\,d\mathbb{Q} \\ & = \mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(x,\tilde{\Phi})}\,d\mathbb{Q} \\ & = \int{\mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(x,\tilde{\Phi})}\,d\mathbb{Q}, \end{align*} $$

where the first equality is due to the dominated convergence theorem, the second and last equalities are due to Tonelli’s theorem, and the third equality is due to the G-invariance of $\mathbb {Q}$ . We conclude that

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \int{(\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}} + \phi_\unicode{x3bb})}\,d\mathbb{Q}. \end{align*} $$

In particular, if $\mathbb {Q} = \delta _{0^\Gamma } \in \mathcal {M}_G(X(\Gamma ))$ , the Dirac measure supported on $0^\Gamma $ , then

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(0^\Gamma,\tilde{\Phi}) + \phi_\unicode{x3bb}(0^\Gamma) = \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(0^\Gamma,\tilde{\Phi}).\\[-41pt] \end{align*} $$

7.3 An arboreal representation of free energy

The following theorem tell us that, under some special conditions, $f_G(\Gamma ,\unicode{x3bb} )$ can be expressed using $\lvert \Gamma / G \rvert $ terms depending on the probability that the roots of some particular trees are unoccupied. Roughly, the trees involved are the trees of self-avoiding walks that are rooted at the vertices of a given fundamental domain and explore the graph $\Gamma $ without entering to the ‘past graph’ induced by an invariant random past.

Theorem 7.6. Let $(\Gamma ,\unicode{x3bb} ) \in \mathcal {H}_G$ such that the Gibbs $(T_{\mathrm {SAW}}(\Gamma ,v),\overline {\unicode{x3bb} })$ -specification satisfies SSM for every $v \in V(\Gamma )$ and let $v_1,\ldots ,v_{\lvert \Gamma / G \rvert }$ be an arbitrary ordering of a fundamental domain $U_0$ . Given an invariant random past $\tilde {\Phi }$ of G, denote by $\Gamma _i(\tilde {\Phi })$ the random graph given by $\Gamma \setminus (\tilde {\Phi }(1_G)U_0 \cup \{v_1,\ldots ,v_{i-1}\})$ . Then,

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = -\frac{1}{\lvert \Gamma/G\rvert}\sum_{i=1}^{\lvert \Gamma / G \rvert} \mathbb{E}_{\tilde{\Phi}} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\tilde{\Phi}),v_i),\overline{\unicode{x3bb}}}([0^{\rho_i}]), \end{align*} $$

where $\rho _i$ denotes the root of $T_{\mathrm {SAW}(\Gamma _i(\Phi ),v_i)}$ . In particular, if $\prec $ is a deterministic invariant order of G,

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = -\frac{1}{\lvert \Gamma/G \rvert}\sum_{i=1}^{\lvert \Gamma / G \rvert} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i),\overline{\unicode{x3bb}}}([0^{\rho_i}]). \end{align*} $$

Proof. By Theorem 7.5, we know that

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}_{\Gamma,\unicode{x3bb}}}(0^\Gamma,\tilde{\Phi}) = -\frac{1}{\lvert \Gamma/G \rvert}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma,\unicode{x3bb}}([0^{U_0}] \vert [0^{\tilde{\Phi}(1_G)U_0}]). \end{align*} $$

By iterating conditional probabilities, linearity of expectation, and Proposition 4.1 (see Figure 4),

$$ \begin{align*} \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma,\unicode{x3bb}}([0^{U_0}] \vert [0^{\tilde{\Phi}(1_G)U_0}]) & = \sum_{i=1}^{\lvert \Gamma / G \rvert}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma,\unicode{x3bb}}([0^{v_i}] \vert [0^{\tilde{\Phi}(1_G)U_0 \cup \{v_1,\ldots,v_{i-1}\}}]) \\ & = \sum_{i=1}^{\lvert \Gamma / G \rvert}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma \setminus (\tilde{\Phi}(1_G)U_0 \cup \{v_1,\ldots,v_{i-1}\}),\unicode{x3bb}}([0^{v_i}]) \\ & = \sum_{i=1}^{\lvert \Gamma / G \rvert}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma_i(\tilde{\Phi}),\unicode{x3bb}}([0^{v_i}]). \end{align*} $$

Figure 4 The graph $\Gamma \setminus \Phi _\prec U_0$ and the corresponding graphs $\Gamma _i(\Phi _\prec )$ for $G = \mathbb {Z}^2$ and the lexicographic order $\prec $ .

Finally, by Theorem 5.6, we obtain

$$ \begin{align*} \sum_{i=1}^{\lvert \Gamma / G \rvert}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma_i(\tilde{\Phi}),\unicode{x3bb}}([0^{v_i}]) = \sum_{i=1}^{\lvert \Gamma / G \rvert} \mathbb{E}_{\tilde{\Phi}}\log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\tilde{\Phi}),v_i),\overline{\unicode{x3bb}}}([0^{\rho_i}]). \end{align*} $$

Figure 5 A representation of $T_{\mathrm {SAW}(\Gamma _i(\Phi _\prec ),v_i)}$ and a logarithmic depth truncation for $G = \mathbb {Z}^2$ and $\Gamma = \mathrm {Cay}(\mathbb {Z}^2,\{\pm (1,0),\pm (0,1)\})$ .

In particular, if $\prec $ is a deterministic invariant order on G (see Figure 5), we have that

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = -\frac{1}{\lvert \Gamma/G \rvert}\sum_{i=1}^{\lvert \Gamma / G \rvert} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i),\overline{\unicode{x3bb}}}([0^{\rho_i}]).\\[-46pt] \end{align*} $$

8.1 Weitz’s algorithm and a computational phase transition

Observe that if G is the trivial group $\{1\}$ , then $\mathcal {H}_{\{1\}}$ is exactly the family of finite hardcore models. In this case, we have that

$$ \begin{align*} f_G(\Gamma,\unicode{x3bb}) = \frac{\log Z_\Gamma(\unicode{x3bb})}{\lvert V(\Gamma)\rvert}, \end{align*} $$

and we can translate an approximation of $Z_\Gamma (\unicode{x3bb} )$ into an approximation of $f_G(\Gamma ,\unicode{x3bb} )$ and vice versa.

In this finite context, it is common to consider a fully polynomial-time approximation scheme. Given a family $\mathbb {M} \subseteq \mathcal {H}_{\{1\}}$ of finite hardcore models, we will say that $\mathbb {M}$ admits an FPTAS for $Z_\Gamma (\unicode{x3bb} )$ if there is an algorithm such that, given an input $(\Gamma ,\unicode{x3bb} ) \in \mathbb {M}$ and $\epsilon> 0$ , outputs $\hat {Z}$ with

$$ \begin{align*} Z_\Gamma(\unicode{x3bb})e^{-\epsilon} \leq \hat{Z} \leq Z_\Gamma(\unicode{x3bb})e^{\epsilon} \end{align*} $$

in polynomial time in $\lvert V(\Gamma ) \rvert $ and $\epsilon ^{-1}$ . An FPTAS is regarded as an efficient and uniform approximation algorithm for $Z_\Gamma (\unicode{x3bb} )$ .

If we take logarithms and divide by $\lvert V(\Gamma ) \rvert $ in the previous equation, we obtain that

$$ \begin{align*} \lvert f_G(\Gamma,\unicode{x3bb}) - \hat{f} \rvert = \bigg\lvert \frac{\log Z_\Gamma}{\lvert V(\Gamma) \rvert} - \frac{\log \hat{Z}}{\lvert V(\Gamma) \rvert} \bigg\rvert \leq \frac{\epsilon}{\lvert V(\Gamma) \rvert }, \end{align*} $$

where $\hat {f} = {\log \hat {Z}}/{\lvert V(\Gamma ) \rvert }$ , so an FTPAS for $Z_\Gamma (\unicode{x3bb} )$ is equivalent to an additive FPTAS for $f_G(\Gamma ,\unicode{x3bb} )$ , since a polynomial in $\lvert V(\Gamma ) \rvert $ and $\epsilon ^{-1}$ is also a polynomial in $\lvert \langle\Gamma ,\unicode{x3bb}\rangle \rvert $ and $\lvert V(\Gamma ) \rvert \epsilon ^{-1}$ and vice versa. The same correspondence holds between the natural randomized counterparts (FPRAS and additive FPRAS, respectively).

We will fix a positive integer $\Delta $ and $\unicode{x3bb} _0> 0$ . Given these parameters, we aim to develop a fully polynomial-time additive approximation on $\mathbb {M} = \mathcal {H}_G^\Delta (\unicode{x3bb} _0)$ .

The main theorem in [Reference Weitz55] was the development of an FPTAS for $Z_\Gamma (\unicode{x3bb} _0)$ on $\mathcal {H}_{\{1\}}^\Delta (\unicode{x3bb} _0)$ for $\unicode{x3bb} _0 < \unicode{x3bb} _c(\Delta )$ . It is not difficult to see that the theorem extends to non-constant activity functions $\unicode{x3bb} $ . Then, and also translated into the language of free energy, we have the following result.

This theorem was subsequently refined in [Reference Sinclair, Srivastava, Štefankovič and Yin46] by considering connective constants instead of maximum degree $\Delta $ . A very interesting fact is that when classifying graphs according to their maximum degree, then Theorem 8.2 is in some sense optimal due to the following theorem.