A HC model with countable set of spin values: uncountable set of Gibbs measures

TL;DR

This paper studies a hard core model with countably infinite spin states on a Cayley tree, analyzing Gibbs measures, their periodicity, and constructing an uncountable set of such measures.

Contribution

It provides the limit points of the dynamical system for the model and constructs an uncountable set of Gibbs measures, revealing complex measure structures.

Findings

Periodic Gibbs measures are either translation-invariant or have period two.

The model admits an uncountable set of Gibbs measures.

Limit points of the associated dynamical system are characterized.

Abstract

We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters $\lambda_{i}>0, i \in \mathbb{Z}\setminus\{0\}$. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.