Phase transitions for a model with uncountable spin space on the Cayley tree: the general case

TL;DR

This paper completes the analysis of phase transitions in a statistical mechanics model with uncountable spin space on Cayley trees, identifying a critical parameter where the number of Gibbs measures changes.

Contribution

It extends previous work by analyzing the full parameter range, establishing the existence of a critical value and multiple Gibbs measures for the model.

Findings

Existence of a critical parameter $ heta_c$ for phase transition.

Unique Gibbs measure for $ heta \,\leq\, \theta_c$.

Three Gibbs measures for $ heta \,>\, \theta_c$.

Abstract

In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12,EsRo10,BoEsRo13,JaKuBo14,Bo17]. The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value $\theta_{\rm c}$ such that for $\theta\le \theta_{\rm c}$ there is a unique translation-invariant splitting Gibbs measure. For $\theta_{\rm c}<\theta$ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.