On Ground States and Phase Transitions of $\lambda$-Model on the Cayley Tree

TL;DR

This paper investigates the $$-model on a Cayley tree, identifying ground states, proving phase transitions via Gibbs measures, and establishing the existence of 2-periodic Gibbs measures.

Contribution

It introduces a detailed analysis of ground states and phase transitions for the $$-model on Cayley trees, including new proofs of Gibbs measures existence.

Findings

Ground states of the $$-model are characterized.

Existence of translation-invariant Gibbs measures is proven.

2-periodic Gibbs measures are shown to exist.

Abstract

In the paper, we consider the $\lambda$-model with spin values $\{1, 2, 3\}$ on the Cayley tree of order two. We first describe ground states of the model. Moreover, we also proved the existence of translation-invariant Gibb measures for the $\lambda$-model which yield the existence of the phase transition. Lastly, we established the exitance of 2-periodic Gibbs measures for the model.