# Translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree

## TL;DR

This paper investigates translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree, identifying a critical temperature where the number of such measures changes and analyzing their extremality properties.

## Contribution

It determines the approximate critical temperature for phase transition and characterizes the number and extremality of Gibbs measures in the Blum-Kapel model on Cayley trees.

## Key findings

- Unique Gibbs measure for T ≥ T_cr
- Three Gibbs measures for 0 < T < T_cr
- Analysis of extremality of the Gibbs measure

## Abstract

In this paper we consider translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree of order k. An approximate critical temperature T_{cr} is found such that for T\geq T_{cr} there exists a unique translation-invariant Gibbs measure and for 0<T<T_{cr} there are exactly three translation-invariant Gibbs measures. In addition, we studied the problem of (not) extremality for the unique Gibbs measure.