# Gradient Gibbs measures for the SOS model with countable values on a Cayley tree

## TL;DR

This paper investigates translation-invariant gradient Gibbs measures for an SOS model with integer spins on Cayley trees, identifying multiple periodic boundary laws and corresponding Gibbs measures, revealing complex phase structures.

## Contribution

It introduces new solutions for periodic boundary laws in the SOS model on Cayley trees, expanding understanding of Gibbs measures beyond previously known cases.

## Key findings

- Up to five solutions for 4-periodic boundary law equations in the ferromagnetic case.
- Identification of up to four distinct gradient Gibbs measures from these boundary laws.
- Construction of 3-periodic boundary laws leading to new Gibbs measures on arbitrary Cayley tree orders.

## Abstract

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.