Translation-invariant and weakly periodic Gibbs measures for HC-models on a Cayley tree

TL;DR

This paper investigates Gibbs measures for HC-models on Cayley trees, proving the existence of exactly two weakly periodic measures in certain cases and demonstrating non-uniqueness of translation-invariant measures for some four-state models.

Contribution

It establishes the existence of exactly two weakly periodic Gibbs measures for two-state HC-models and shows non-uniqueness of translation-invariant measures for specific four-state models.

Findings

Exactly two weakly periodic Gibbs measures exist under certain conditions.

Non-uniqueness of translation-invariant Gibbs measure for a four-state HC-model.

Results contribute to understanding phase structures in HC-models on Cayley trees.

Abstract

In this paper is studied HC-models on a Cayley tree. two states HC-model on a Cayley tree and Under some conditions on parameters of the two state HC-model we prove existence exactly two of the weakly periodic (non periodic) Gibbs measures. Furthermore, it considered translation-invariant Gibbs measures for fertile four state HC-models and for one of them it proved the non-uniqueness of the translation-invariant Gibbs measure.