Gradient Gibbs measures for the SOS model with integer spin values on a Cayley tree

TL;DR

This paper studies the SOS model on a Cayley tree, identifying multiple periodic gradient Gibbs measures through solutions to boundary law equations, advancing understanding of phase structures in such models.

Contribution

It extends previous work by explicitly finding multiple periodic gradient Gibbs measures for the SOS model on Cayley trees.

Findings

Identified three solutions to period-4 boundary law equations.

Defined up to three periodic gradient Gibbs measures.

Enhanced understanding of phase behavior in SOS models on Cayley trees.

Abstract

In the present paper we continue the investigation from [1] and consider the SOS (solid-on-solid) model on the Cayley tree of order $k \geq 2$. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.