# Non-translation-invariant Gibbs Measures for Models With Uncountable Set of Spin Values on a Cayley Tree

## TL;DR

This paper introduces three new methods to construct non-translation-invariant Gibbs measures for models with uncountable spin values on a Cayley tree, expanding understanding beyond previously known translation-invariant solutions.

## Contribution

The paper presents novel constructions of non-translation-invariant Gibbs measures for models with uncountable spins on Cayley trees, based on existing solutions of a nonlinear integral equation.

## Key findings

- Three new sets of non-translation-invariant Gibbs measures constructed.
- Extensions of known solutions to generate diverse Gibbs measures.
- Enhanced understanding of phase structures in models with uncountable spins.

## 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. 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.