Gibbs measures for a Hard-Core model with a countable set of states

TL;DR

This paper investigates the conditions under which unique or multiple non-probability Gibbs measures exist for a Hard-Core model with countably infinite states on Cayley trees, focusing on boundary laws and their properties.

Contribution

It provides new criteria for the existence and uniqueness of non-probability Gibbs measures in a Hard-Core model with infinite states on Cayley trees.

Findings

Conditions for uniqueness of Gibbs measures identified

Non-normalisable boundary laws can define Gibbs measures

Results applicable to Cayley trees of any order k ≥ 2

Abstract

In this paper, we focus on studying non-probability Gibbs measures for a Hard Core (HC) model on a Cayley tree of order $k\geq 2$, where the set of integers $\mathbb Z$ is the set of spin values. It is well-known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalisable boundary law corresponds to a Gibbs measure. However, a non-normalisable boundary law can define gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC-model on a Cayley tree of any order $k\geq 2$.