# Gibbs measures and free energies of Ising-Vannimenus Model on the Cayley tree

## TL;DR

This paper rigorously analyzes the Ising-Vannimenus model on a Cayley tree, establishing conditions for Gibbs measures, phase transitions, and calculating free energies and entropies, advancing the mathematical understanding of this complex system.

## Contribution

It introduces a measure-theoretical framework for the model and provides rigorous proofs for the existence of Gibbs measures and phase transitions.

## Key findings

- Conditions for existence of Gibbs measures established
- Phase transition confirmed through rigorous proofs
- Free energies and entropies computed for invariant measures

## Abstract

In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition. Moreover, the free energies and entropies, associated with translation invariant Gibbs measures, are calculated.