Non-uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

TL;DR

This paper constructs models with uncountably many spin values on a Cayley tree, demonstrating the non-uniqueness of Gibbs measures and revealing complex phase behavior in such systems.

Contribution

It introduces new models with uncountable spin sets on Cayley trees and proves the existence of multiple translational-invariant Gibbs measures.

Findings

Multiple Gibbs measures exist for the models.

Models with uncountable spin values exhibit non-uniqueness.

Results contribute to understanding phase transitions in complex systems.

Abstract

In this paper we construct several models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 2$. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.