Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

Abstract

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. Recently, solving this integral equation some periodic (in particular translation-invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of non-translation-invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)