Uniqueness of the translation-invariant Gibbs measure for Hard-Core models with four-states on the Cayley tree

TL;DR

This paper proves the uniqueness of the translation-invariant Gibbs measure for four-state hard-core models on a Cayley tree of order two, extending understanding of phase behavior in such lattice models.

Contribution

It establishes the uniqueness of the Gibbs measure for all three types of four-state hard-core models on a Cayley tree of order two, which was previously unknown.

Findings

Uniqueness of Gibbs measure proven for all three model types

Applicable to Cayley trees of order two

Advances understanding of phase transitions in hard-core models

Abstract

We consider fertile HC-models with four-states and the parametre activity on a Cayley tree. It is known that three types of such models exist. For each of these models we prove uniqueness of the translation-invarinat Gibbs measure on a Cayley tree of order two.