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Unique Bernoulli Gibbs states and g-measures

Paul Hulse

TL;DR

The work provides a sufficient condition under which Gibbs states of a one-dimensional, shift-invariant specification are $g$-chains for a continuous $g$, and uses this to derive criteria guaranteeing a unique, shift-invariant, Bernoulli Gibbs measure. The methodology centers on a ratio-bound framework that translates Gibbs-state structure into a $g$-measure problem via a limit $g_{\mathcal{V}}$, combined with lacunary-variation controls that ensure Bernoullicity through results of Johansson–Öberg–Pollicott. For interaction potentials $\Phi$ with summable variation, the paper yields explicit conditions implying uniqueness and Bernoulli mixing, strengthening classical results (Dobrushin, Ruelle, Coelho–Quas). It also discusses weaker, non-Bernoulli uniqueness results, clarifying the trade-offs between uniqueness and strong mixing in low-dimensional lattice systems. Overall, the results provide practically checkable criteria for uniqueness and Bernoulli behavior in 1D Gibbsian systems, with implications for models like the Ising chain.

Abstract

A sufficient condition for the Gibbs states of a shift-invariant specification on a one-dimensional lattice to be the $g$-chains for some continuous function $g$ is obtained. This is then used to derive criteria under which there is a unique Gibbs state, which is also shift-invariant and Bernoulli.

Unique Bernoulli Gibbs states and g-measures

TL;DR

The work provides a sufficient condition under which Gibbs states of a one-dimensional, shift-invariant specification are -chains for a continuous , and uses this to derive criteria guaranteeing a unique, shift-invariant, Bernoulli Gibbs measure. The methodology centers on a ratio-bound framework that translates Gibbs-state structure into a -measure problem via a limit , combined with lacunary-variation controls that ensure Bernoullicity through results of Johansson–Öberg–Pollicott. For interaction potentials with summable variation, the paper yields explicit conditions implying uniqueness and Bernoulli mixing, strengthening classical results (Dobrushin, Ruelle, Coelho–Quas). It also discusses weaker, non-Bernoulli uniqueness results, clarifying the trade-offs between uniqueness and strong mixing in low-dimensional lattice systems. Overall, the results provide practically checkable criteria for uniqueness and Bernoulli behavior in 1D Gibbsian systems, with implications for models like the Ising chain.

Abstract

A sufficient condition for the Gibbs states of a shift-invariant specification on a one-dimensional lattice to be the -chains for some continuous function is obtained. This is then used to derive criteria under which there is a unique Gibbs state, which is also shift-invariant and Bernoulli.

Paper Structure

This paper contains 6 sections, 11 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Specifications and Gibbs states
  3. A ratio bound
  4. Gibbs states and g-chains
  5. Proof of main results
  6. Uniqueness without the Bernoulli property

Key Result

Theorem 1.1

Let ${\mathcal{V}}=\{\phi_\Lambda\}_{\Lambda\in{\mathcal{C}}}$ be a positive, continuous, $T$-invariant specification on $S^{\mathbb Z}$, and let for some $s\in S$. If, for some $0<\alpha\leq1$, and then ${\mathcal{V}}$ has a unique Gibbs state $\mu$, which is $T$-invariant, and $(T,\mu)$ is Bernoulli.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['prop:rb']}
  • Proposition 4.1
  • proof
  • ...and 11 more