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

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 ${\theta }_{\mathrm{c}}$ such that for $\theta \le {\theta }_{\mathrm{c}}$ there is a unique translation-invariant splitting Gibbs measure. For ${\theta }_{\mathrm{c}}<\theta$ 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.

9 pages