Gibbs measures for SOS models with external field on a Cayley tree

TL;DR

This paper investigates Gibbs measures for a three-state SOS model with external field on Cayley trees, revealing phase transition phenomena and classifying all translation-invariant measures based on interaction type and external influences.

Contribution

It extends prior work on SOS models by analyzing the three-state case with external field, providing a comprehensive classification of Gibbs measures and phase transition behavior.

Findings

Unique TISGM in antiferromagnetic case for all temperatures

Multiple TISGMs (up to seven) in ferromagnetic case depending on parameters

Identification of phase transition phenomena related to external field and interaction strength

Abstract

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,\ldots,m,$ $m\geq2,$ and nonzero external field, on a Cayley tree of degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of \cite{rs} where the SOS model is studied with (mainly) three spin values and zero external field. The SOS model can be treated as a natural generalization of the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three spin values) and study translation-invariant (TI) and splitting (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) For $m=2$, in the antiferromagnet (AFM) case, a TISGM is unique for all temperatures with an external field. In the ferromagnetic (FM) case, for $m=2,$ the number of TISGMs varies with the temperature and the external field: this gives an interesting example of phase transition. Our second result gives a classification of all TISGMs of the Three-State SOS-Model on the Cayley tree of degree two with the presence of an external field. We show uniqueness in the case of antiferromagnetic interactions and the existence of up to seven TISGMs in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength and external field.