Periodic Gibbs Measures for Models with Uncountable Set of Spin Values on a Cayley Tree

TL;DR

This paper investigates the existence of periodic Gibbs measures for models with uncountable spin values on Cayley trees, establishing conditions under which such measures do or do not exist, especially focusing on period-two measures.

Contribution

It provides new conditions for the absence or presence of period-two Gibbs measures in models with uncountable spins on Cayley trees, including explicit constructions.

Findings

No period-two Gibbs measures for k=1

Sufficient conditions for non-existence of period-two measures when k≥2

Existence of models with at least two periodic Gibbs measures

Abstract

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. We study periodic Gibbs measures of the model with period two. For $k=1$ we show that there is no any periodic Gibbs measure. In case $k\geq 2$ we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures.