Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

TL;DR

This paper establishes a sufficient condition for the uniqueness of Gibbs measures in models with uncountable spin values on Cayley trees, using solutions of a nonlinear integral equation.

Contribution

It provides a new criterion ensuring the uniqueness of splitting Gibbs measures for models with continuous spins on Cayley trees.

Findings

Derived a sufficient condition for the uniqueness of solutions to the integral equation.

Proved that under this condition, the model has a unique splitting Gibbs measure.

Extended understanding of phase transition phenomena in models with uncountable spin sets.

Abstract

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ 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.