# Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree

## TL;DR

This paper investigates the existence of weakly periodic Gibbs measures for two and three state HC models on a Cayley tree, revealing new measures and conditions for non-uniqueness of extreme measures.

## Contribution

It introduces new weakly periodic Gibbs measures for HC models on Cayley trees under specific parameter conditions, expanding understanding of measure diversity.

## Key findings

- Existence of weakly periodic (non periodic) Gibbs measures under certain conditions.
- Identification of parameter regions where extreme measure non-uniqueness occurs.
- Extension of known Gibbs measure classifications for HC models on Cayley trees.

## Abstract

In this paper we study two and three state HC-model on a Cayley tree. Under some conditions on parameters of the HC-model, in the case of normal divisor of the index four, we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures. Moreover, we find some regions for the $\lambda$ parameter ensuring that a extreme measure is not unique.