Phase Transitions for a model with uncountable set of spin values on a Cayley tree

TL;DR

This paper investigates phase transitions in a spin model with a continuous set of values on a Cayley tree, establishing conditions for multiple Gibbs measures indicating different phases.

Contribution

It introduces a nonlinear functional equation approach to analyze Gibbs measures for a continuous-spin model on Cayley trees, proving non-uniqueness under certain conditions.

Findings

Existence of phase transitions for k=2 and 3

Non-uniqueness of translation-invariant Gibbs measures

Conditions on parameters leading to multiple phases

Abstract

In this paper we consider a model with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k \geq 2$. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For $k=2$ and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e. there are phase transitions).