Using New Approaches to obtain Gibbs Measures of Vannimenus model on a Cayley tree

TL;DR

This paper introduces a novel approach to derive Gibbs measures for the Vannimenus model on Cayley trees, revealing new measures and analyzing their properties, including extremality and invariance.

Contribution

The paper develops a new method to obtain Gibbs measures for the Ising-Vannimenus model on Cayley trees, identifying measures not previously reported.

Findings

New sets of Gibbs measures identified

Some measures are extreme Gibbs distributions

Analysis of translation-invariant and periodic measures

Abstract

In this paper, we consider Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree. For this model we define Markov random fields with memory of length 2. By using a new approach, we obtain new sets of Gibbs measures of Ising-Vannimenus model on Cayley tree of order 2. We construct the recurrence equations corresponding Ising-Vannimenus model. We prove the Kolmogorov consistency condition. We investigate the translation-invariant and periodic non transition-invariant Gibbs measures for the model. We find new sets of Gibbs measures different from the Gibbs measures given in the references \cite{NHSS,FreeMA}. We show that some of the measures are extreme Gibbs distributions.