Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

TL;DR

This paper constructs and analyzes gradient Gibbs measures for a height-periodic SOS model on Cayley trees, revealing symmetric and asymmetric boundary laws and their properties.

Contribution

It introduces explicit boundary laws for the SOS model on Cayley trees, including symmetric and asymmetric solutions, advancing understanding of phase structures.

Findings

Existence of vertex-independent boundary laws

Construction of mirror symmetric boundary laws

Identification of non-symmetric boundary laws

Abstract

For the solid-on-solid (SOS) model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror symmetric coordinates.