Gradient Gibbs measures with periodic boundary laws of a generalized SOS model on a Cayley tree

TL;DR

This paper investigates Gradient Gibbs measures for a generalized SOS model with periodic boundary laws on Cayley trees, reducing the problem to a functional equation and explicitly finding measures with 4-periodic boundary laws.

Contribution

It extends previous work by solving a functional equation to explicitly identify all Gradient Gibbs measures with 4-periodic boundary laws for the generalized SOS model.

Findings

Explicit solutions for Gradient Gibbs measures with 4-periodic boundary laws.

Reduction of the measure-finding problem to a solvable functional equation.

Extension of prior results to a generalized SOS model on Cayley trees.

Abstract

We consider Gradient Gibbs measures corresponding to a periodic boundary law for a generalized SOS model with spin values from a countable set, on Cayley trees. On the Cayley tree, detailed information on Gradient Gibbs measures for models of SOS type are given in \cite{3, 16,8,11} and we continue the works for the generalized SOS model. Namely, in this paper, the problem of finding Gradient Gibbs measures that correspond to periodic boundary laws is reduced to a functional equation and by solving the equation all Gradient Gibbs measures with 4 periodic boundary laws are found.