Gradient Gibbs measures of an SOS model with alternating magnetism on Cayley trees

TL;DR

This paper investigates gradient Gibbs measures for an SOS model with alternating magnetism on Cayley trees, providing explicit constructions of periodic invariant measures using boundary law equations.

Contribution

It introduces new periodic gradient Gibbs measures for the SOS model with alternating magnetism on Cayley trees, expanding understanding of such measures.

Findings

Constructed q-height-periodic invariant GGMs for q=2,3,4

Applied boundary law equations to derive these measures

Extended the class of known gradient Gibbs measures for this model

Abstract

The work is devoted to gradient Gibbs measures (GGMs) of a SOS model with countable set $\mathbb Z$ of spin values and having alternating magnetism on Cayley trees. This model is defined by a nearest-neighbor gradient interaction potential. Using K\"ulske-Schriever argument, based on boundary law equations, we give several $q$-height-periodic translations invariant GGMs for $q=2,3,4$.