1 Introduction

We consider Gibbs states on sequence spaces, that is, probability measures for which there is a specified set of functions that determine the probabilities of configurations on any finite subset conditioned on the configuration outside that subset. Let $X=S^I$ be a sequence space, where S is a finite set with the discrete topology, and I is a countable (infinite) set. The topology on X is the product topology, with respect to which X is compact, and ${\mathcal B}$ denotes the corresponding Borel $\sigma $ -algebra. If $x\in S^{\Lambda ^\prime }\, (\Lambda ^\prime \subseteq I)$ and $\Lambda \subseteq \Lambda ^\prime $ , then $x_\Lambda $ denotes the natural projection of x onto $S^\Lambda $ and $[x]_\Lambda $ is the cylinder set ${\{y\in X:y_\Lambda =x_\Lambda \}}$ ; similarly, if $s\in S$ and $i\in I$ , $[s]_i$ denotes $\{y\in X:y_i=s\}$ . The sub- $\sigma $ -algebra of ${\mathcal B}$ generated by $\{[x]_\Lambda : x\in X\}$ is denoted by ${\mathcal B}_\Lambda $ , and ${\mathcal C}$ denotes the set of finite, non-empty subsets of I.

A specification on X is a collection ${\mathcal V}=\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ of non-negative Borel functions on X such that, for all $\Lambda \in {\mathcal C}$ and $x\in X$ ,

(1.1) $$ \begin{align} \sum_{y\in[x]_{\Lambda^c}}\phi_\Lambda(y)=1, \end{align} $$

and for $K\subseteq \Lambda $ ,

(1.2) $$ \begin{align} \phi_\Lambda(x)=\phi_K(x)\sum_{y\in[x]_{K^c}}\phi_\Lambda(y). \end{align} $$

The specification ${\mathcal V}$ is said to be continuous if $\phi _\Lambda $ is continuous for all $\Lambda \in {\mathcal C}$ , and positive if $\phi _\Lambda>0$ for all $\Lambda \in {\mathcal C}$ . If $I={\mathbb Z}^d\, (d\geq 1)$ , then $T=\{T_i\}_{i\in I}$ denotes the d-dimensional shift on X, that is, $(T_ix)_j=x_{i+j}\, (x\in X,\ i,j\in {\mathbb Z}^d)$ , and ${\mathcal V}$ is said to be T-invariant if $\phi _{i+\Lambda }(x)=\phi _\Lambda (T_ix)$ for all $i\in {\mathbb Z}^d$ and $\Lambda \in {\mathcal C}$ . (As usual, when $d=1$ , we equate T with $T_1$ .)

A probability measure $\mu $ on $(X,{\mathcal B})$ is a Gibbs state for a specification ${\mathcal V}=\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ if

$$ \begin{align*} \mu([x]_\Lambda\,|\,{\mathcal B}_{\Lambda^c})(x) = \phi_\Lambda(x)\quad\text{a.e. } (\mu) \end{align*} $$

for all $\Lambda \in {\mathcal C}$ (by a.e., we mean almost everywhere). The set of Gibbs states is denoted by ${\mathcal G}({\mathcal V})$ .

If ${\mathcal V}$ is continuous, there exists at least one Gibbs state, and natural questions concern the structure of ${\mathcal G}({\mathcal V})$ , in particular, uniqueness and any ergodic or mixing properties of Gibbs states. In general, ${\mathcal G}({\mathcal V})$ is a compact, convex subset of $M(X)$ (the set of probability measures on $(X,{\mathcal B})$ ), and its extreme points are those states for which the tail $\sigma $ -field $\bigcap _{\Lambda \in {\mathcal C}}{\mathcal B}_{\Lambda ^c}$ is trivial. If, in addition, $I={\mathbb Z}^d$ and ${\mathcal V}$ is T-invariant, then at least one of the Gibbs states is T-invariant; the T-invariant Gibbs states form a convex subset of ${\mathcal G}(V)$ , the extreme points of which are the ergodic ones. Thus, a unique T-invariant Gibbs state $\mu $ is necessarily ergodic, but if it is also the only Gibbs state, then it follows from the triviality of the tail field that $(T,\mu )$ has some strong mixing properties. However, from the dynamical systems perspective, the system with the strongest mixing properties is one that is isomorphic to, and hence indistinguishable from, a Bernoulli shift (by which we mean the invertible dynamical system generated by a sequence of independent identically distributed random variables taking values in a finite set). While it is known that for those specifications that are attractive (or ferromagnetic), uniqueness of the Gibbs state implies the Bernoulli property (see [Reference Adams1, Reference Hulse12, Reference Hulse13]), in general, this is not the case.

Dobrushin [Reference Dobrushin7] showed that if ${\mathcal V}=\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ is a continuous specification such that

$$ \begin{align*} \sup_{i\in I}\sum_{s\in S}\sum_{j\neq i} \sup_{\substack{x_{\{j\}^c}=y_{\{j\}^c}\\x_i=y_i=s}} |\phi_{\{i\}}(x)-\phi_{\{i\}}(y)|<2, \end{align*} $$

then ${\mathcal V}$ has a unique Gibbs state $\mu $ . This result holds for all countable I and does not depend on I having a group structure. If $I={\mathbb Z}^d$ (in fact, if I is any amenable group) and ${\mathcal V}$ is T-invariant, then $\mu $ is T-invariant and the corresponding dynamical system $(T,\mu )$ is Bernoulli (see [Reference Hulse13]).

Of particular interest in statistical mechanics are those specifications determined by an interaction potential $\Phi =\{\Phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ , that is, a set of functions on X such that $\Phi _\Lambda $ is ${{\mathcal B}}_\Lambda $ -measurable. The corresponding specification ${\mathcal V}_\Phi $ is given by

(1.3) $$ \begin{align} \phi_\Lambda(x)=\exp(-H^\Phi_\Lambda(x))\bigg\{\sum_{y\in[x]_{\Lambda^c}} \exp(-H^\Phi_\Lambda(y))\bigg\}^{-1}, \end{align} $$

where

$$ \begin{align*} H^\Phi_\Lambda(x)=\sum_{\{K\in{\mathcal C}:K\cap\Lambda\neq\emptyset\}}\Phi_K(x). \end{align*} $$

Usually, $\Phi $ is required to satisfy a summability condition that ensures $H^\Phi _\Lambda $ is continuous, and hence that ${\mathcal V}_\Phi $ is continuous and positive. When $I={\mathbb Z}^d$ , the interaction potential $\Phi $ is said to be T-invariant if $\Phi _{\Lambda +i}(x)=\Phi _\Lambda (T_ix)$ for all $\Lambda \in {\mathcal C}$ and $i\in {\mathbb Z}^d$ , in which case, ${\mathcal V}_\Phi $ is also T-invariant. An example that has been studied extensively is the well-known Ising model on $\{-1,1\}^{{\mathbb Z}^d}$ , where $\Phi _\Lambda \equiv 0$ for $|\Lambda |\neq 2$ . In particular, we note that in dimensions $d\geq 2$ , the nearest-neighbour model (where $\Phi _{\{i,j\}}\equiv 0$ unless $|i-j|=1$ ) gives examples of non-unique Gibbs states. When $d=1$ , the situation is quite different. Here, the nearest-neighbour model has a unique Gibbs state; more generally, if $\Phi _{\{i,j\}}(x)=-J(|i-j|)x_ix_j$ , where $\{J(n)\}$ is a non-negative sequence, there is a unique Gibbs state provided $\sum _{i=1}^niJ(i)=o(\sqrt {\log n})$ [Reference Rogers and Thompson17]. In contrast, if $J(n)=\beta n^{-\alpha }$ , there is a phase transition (that is, non-uniqueness) for large enough $\beta $ when $1<\alpha \leq 2$ [Reference Fröhlich and Spencer9]. (A more thorough survey of such results can be found in [Reference Georgii11].)

For potentials in general, Ruelle [Reference Ruelle19] showed that when $I={\mathbb Z}$ and $\Phi =\{\Phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ is T-invariant, there is a unique T-invariant Gibbs state provided

(1.4) $$ \begin{align} \sum_{\Lambda\ni0}\text{diam}(\Lambda)\|\Phi_\Lambda\|<\infty. \end{align} $$

Gallavotti [Reference Gallavotti10] and Ledrappier [Reference Ledrappier16] independently proved it has the Bernoulli property. Coelho and Quas [Reference Coelho and Quas6] pointed out that by combining Ruelle’s characterization [Reference Ruelle18] of the T-invariant Gibbs states of $\Phi $ as equilibrium states of the function

$$ \begin{align*} A_\Phi(x)=\sum_{\min\Lambda=0}\Phi_\Lambda(x) \end{align*} $$

with the results of Walters [Reference Walters21], (1.4) could be replaced by the weaker hypothesis

(1.5) $$ \begin{align} \sum_{\min\Lambda=0}\text{diam}(\Lambda)\|\Phi_\Lambda\|<\infty. \end{align} $$

Berbee [Reference Berbee3] considered continuous specifications $\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ determined as follows. There is a sequence $\{\Lambda _n\}_{n=0}^\infty \subset {\mathcal C}$ such that $\Lambda _n\nearrow I$ and positive ${\mathcal B}_{\Lambda _{n-1}^c}$ -measurable functions $f_n\in C(X)\, (n\geq 0)$ (with $\Lambda _{-1}=\emptyset $ ) such that

$$ \begin{align*} \phi_{\Lambda_n}(x) =\bigg(\prod_{j=0}^nf_j(x)\bigg) \bigg(\sum_{y\in[x]_{\Lambda_n^c}}\prod_{j=0}^nf_j(y)\bigg)^{-1} \quad(n\geq 0,\,x\in X). \end{align*} $$

Any positive continuous specification can be described in this way (see §2). For a positive function f on X and $\Lambda \subseteq I$ , let

$$ \begin{align*} r_\Lambda(f)=\sup\{f(x)/f(y):x_\Lambda=y_\Lambda\}, \quad\bar r_\Lambda(f)=\inf\{f(x)/f(y):x_\Lambda=y_\Lambda\}. \end{align*} $$

(Note that $\bar r_\Lambda (f)=r_\Lambda (f)^{-1}$ .) Berbee showed that if

(1.6) $$ \begin{align} \sum_{n=0}^\infty\prod_{k=0}^n\inf_{i\geq0}\bar r_{\Lambda_{i+k}}(f_i) =\infty, \end{align} $$

then there is a unique Gibbs state. Like Dobrushin’s result, this holds for any countable I and for non-invariant, as well as invariant, specifications. Taking $I={\mathbb Z}$ and

(1.7) $$ \begin{align} \Lambda_{2k}=[-k,k],\quad\Lambda_{2k+1}=[-k,k+1] \quad(k\geq0) \end{align} $$

(here and later, we use the usual notation for an interval ${\mathcal I}\subseteq {\mathbb R}$ to denote ${\mathcal I}\cap {\mathbb Z}$ ), and comparing hypotheses for the relevant $f_n$ for an interaction potential (2.7), it can be seen that Berbee’s hypothesis is weaker than Ruelle’s (1.4); however, while in some cases it is weaker than that of Coelho and Quas (1.5), in others, it is not.

The hypotheses (1.4)–(1.5) can be viewed as measures of the continuity of $A(\Phi )$ in terms of its variation on sites outside finite subsets, or equivalently, in terms of the rate at which $\|\phi _\Lambda \|$ decreases as $\text {diam}(\Lambda )$ increases. Similarly, (1.6) is a measure of the continuity of the functions $f_n$ , which, in the case of an interaction potential $\Phi $ with the $f_n$ defined by (2.7), is also related to the continuity of $A_\Phi $ . In this paper, we consider whether this type of continuity hypothesis can be relaxed. We restrict our attention to the case $I={\mathbb Z}$ since (1.4) and (1.5) are only valid in this case, and while Berbee’s result holds for any I, in the case of T-invariant specifications, it would seem that (1.6) rules out any genuinely multi-dimensional examples. Indeed, as has already been noted, the nearest-neighbour Ising model provides examples of non-uniqueness in dimensions greater than one. From hereon, X denotes $S^{\mathbb Z}$ .

As Dobrushin [Reference Dobrushin7] pointed out, any positive specification ${\mathcal V}=\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ is completely determined by $\{\phi _{\{n\}}:n\in {\mathbb Z}\}$ ; consequently, if ${\mathcal V}$ is also T-invariant, it is determined by $\phi _{\{0\}}$ alone. Thus, it seems natural to seek conditions for uniqueness and Bernoullicity in terms of $\phi _{\{0\}}$ , or equivalently, the ratios $\{\phi _{\{0\}}(s_0x)/\phi _{\{0\}}(t_0x)\colon s,t\in S,\,x\in X\}$ , where $s_nx\in X\, (s\in S,\,x\in X,\,n\in {\mathbb Z})$ is defined by

$$ \begin{align*} (s_nx)_i= \begin{cases} s,&i=n,\\ x_i,&i\neq n. \end{cases} \end{align*} $$

We show the following theorem.

Theorem 1.1. Let ${\mathcal V}=\{\phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ be a positive, continuous, T-invariant specification on X, and let

$$ \begin{align*} f(x)=\phi_{\{0\}}(x)/\phi_{\{0\}}(s_0x)\quad(x\in X) \end{align*} $$

for some $s\in S$ . If, for some $0<\alpha \leq 1$ ,

$$ \begin{align*} \prod_{i=0}^nr_{(-\infty,i]}(f) =O(n^{1-\alpha}) \end{align*} $$

and

$$ \begin{align*} \sum_{i=2^n}^{2^{n+1}} (\log r_{[-i,\infty)}(f))^{2\alpha}=o(1), \end{align*} $$

then ${\mathcal V}$ has a unique Gibbs state $\mu $ , which is T-invariant, and $(T,\mu )$ is Bernoulli.

In particular, there is a unique, Bernoulli Gibbs state if

$$ \begin{align*} \max\{\log r_{(-\infty,n]}(f),\,\log r_{[-n,\infty)}(f)\} \leq\beta n^{-1}+a_n \quad(n\geq1) \end{align*} $$

for some $\beta <1/2$ and non-negative series $\sum a_n<\infty $ . In fact, we prove Theorem 1.1 under slightly weaker hypotheses (Theorem 5.1), which allows the following version to be deduced when the specification is given in terms of an interaction potential.

Theorem 1.2. Let $\Phi =\{\Phi _\Lambda \}_{\Lambda \in {\mathcal C}}$ be a T-invariant interaction potential on X. If

(1.8) $$ \begin{align} \sum_{\substack{\Lambda\ni0\\\Lambda\cap[n,\infty)\neq\emptyset}} \mathrm{var}(\Phi_\Lambda) \leq\beta n^{-1}+a_n \quad(n\geq1) \end{align} $$

for some $\beta <1/2$ and $a_n\geq 0$ such that $\sum a_n<\infty $ , then $\Phi $ has a unique Gibbs state $\mu $ , which is T-invariant, and $(T,\mu )$ is Bernoulli.

This includes all interaction potentials satisfying (1.4). In some cases, such as when $\sup \{|\Lambda | :\Phi _\Lambda \not \equiv 0\}<\infty $ , (1.8) is also weaker than (1.5), but in others, it is not. Comparison with (1.6) is less straightforward, since (1.6) applies to non-invariant specifications and arbitrary increasing sequences $\{\Lambda _n\}_{n=0}^\infty \subset {\mathcal C}$ . Consideration of the Ising model suggests Theorems 1.1 and 1.2 can give better results for uniqueness in some cases, but the main difference is that the Bernoulli property is obtained.

The idea of the proof of Theorem 5.1 is to characterize a Gibbs state $\mu $ in terms of its conditional probabilities

$$ \begin{align*} g(x)=\mu([x]_{\{0\}}\,|\,{\mathcal B}_{(-\infty,-1]})(x)\quad(x\in X), \end{align*} $$

in other words, as a g-chain; uniqueness, T-invariance and the Bernoulli property can then be determined from similar results for g-chains. Here, we use a result (Theorem 4.5) that is a formulation based on the techniques of Johansson, Öberg and Pollicott [Reference Johansson, Öberg and Pollicott15]. The estimate of the variation of $\log g$ needed for Theorem 4.5 is provided by Theorem 4.2. This result may be of some interest in its own right as it gives a sufficient condition for the set of Gibbs states of a T-invariant specification to be the set of g-chains for some continuous function g. (A characterization of g-chains which are also Gibbs states, as well as a discussion of the problem of determining whether a Gibbs state is a g-chain, can be found in [Reference Berghaut, Fernandez and Verbitskiy5].) Theorem 4.2 follows from Proposition 4.1, which gives a general criterion for the set of Gibbs states to be the same as the set of g-chains for some continuous function g, and Proposition 3.1, which is an adaptation of Berbee’s method of ratio bounds in [Reference Berbee3].

Lastly, we consider some alternative hypotheses which guarantee a unique T-invariant Gibbs state, but not the Bernoulli property.