On Gibbs Measures of the Potts Model with Three Competing Interactions on Cayley Tree of Order 3

Abstract

In this paper, we consider the Potts model with competing interactions on the Cayley tree of order three. We give the Potts model on the Cayley tree and its recursion relation. We construct the Gibbs states corresponding to the model by using Markov random field method. We calculate the critical curve, such that there is a phase transition for the model. We show that there are phase transition of the model for some given parameters. We extend the results obtained by Akin and Temir (Condensed Matt.

Gibbs measures with memory of length 2 on an arbitrary-order Cayley tree

“…In [13], we studied the existence, uniqueness and non-uniqueness of the Gibbs measures associated with the Potts model on a Bethe lattice of order three with three coupling constants by using Markov random field method. In [14], we have obtained the exact solution of a phase transition problem by means of Gibbs state of the same Potts model in [14].…”

“…It is well known that the Potts model is a generalization of the Ising model, but the Potts model on a Cayley tree is not well studied, compared to the Ising model [35]. In the last decade, many researches have investigated Gibbs measures associated with Potts model on Cayley trees [12,13,14,15,16,27]. In [13], we studied the existence, uniqueness and non-uniqueness of the Gibbs measures associated with the Potts model on a Bethe lattice of order three with three coupling constants by using Markov random field method.…”

“…In [13], we studied the existence, uniqueness and non-uniqueness of the Gibbs measures associated with the Potts model on a Bethe lattice of order three with three coupling constants by using Markov random field method. In [14],…”

“…we have obtained the exact solution of a phase transition problem by means of Gibbs state of the same Potts model in [14].…”