Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with markovian structure

TL;DR

This paper proves the uniqueness of Gibbs states for certain one-dimensional lattice systems with Markovian structure, showing they are Gibbs-Bowen measures and satisfy a central limit theorem, thus revealing their statistical properties.

Contribution

It establishes the uniqueness of Gibbs states for a broad class of one-dimensional Markovian lattice systems with continuous potentials, and demonstrates their statistical properties.

Findings

Uniqueness of the Gibbs state $mbda_$ for the specified systems.

Gibbs state is a Gibbs-Bowen measure.

The Gibbs state satisfies a central limit theorem.

Abstract

Let $M$ be a compact metric space and $X = M^{\mathbb{N}}$, we consider a set of admissible sequences $X_{A, I} \subset X$ determined by a continuous admissibility function $A : M \times M \to \mathbb{R}$ and a compact set $I \subset \mathbb{R}$. Given a Lipschitz continuous potential $\varphi : X_{A, I} \to \mathbb{R}$, we prove uniqueness of the Gibbs state $\mu_\varphi$ and we show that it is a Gibbs-Bowen measure and satisfies a central limit theorem.