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arXiv · 2209.14797

A non-linear second-order difference equation related to Gibbs measures of a SOS model

U. A. Rozikov
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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.

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U. A. Rozikov. 2022-09-29. A non-linear second-order difference equation related to Gibbs measures of a SOS model. https://arxiv.org/abs/2209.14797

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