A non-linear second-order difference equation related to Gibbs measures of a SOS model
Abstract
For the SOS (solid-on-solid) model with an external field and with spin values from the set of all integers on a Cayley tree each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a non-linear functional equation. Recently some translation-invariant and height-periodic (non-normalisable) solutions to the equation are found. Here our aim is to find non-height-periodic and non-normalisable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalisable boundary laws. Moreover, we reduce the problem to solving of a non-linear, second-order difference equation. We give analytic and numerical analysis of the difference equation.
Keywords
Cite
@article{arxiv.2209.14797,
title = {A non-linear second-order difference equation related to Gibbs measures of a SOS model},
author = {U. A. Rozikov},
journal= {arXiv preprint arXiv:2209.14797},
year = {2022}
}
Comments
17 pages, 13 figures