Uniqueness of Gibbs Measure 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. For arbitrary k ≥ 2 we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)