Gradient Gibbs measures for the SOS model with integer spin values on a Cayley tree

Abstract

In the present paper we continue the investigation from Bovier and Külske (1996 J. Stat. Phys. 83 751-59) and consider the SOS (solid-on-solid) model on the Cayley tree of order k 2. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.

Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws

“…Remark 1. The equation (2.6) was also considered in [3]. Lemma 1 improves their result (see Theorem 5.2 of [3]), because we found explicit formula for the critical value τ c .…”

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

“…the Ising model, the Potts model and the solid-on-solid (SOS) model), E is a finite set (i.e. with a finite underlying measure λ) and ξ x has a physical interpretation as the spin of a particle at location x in a crystal lattice (details in [1,4,[7][8][9][10][11][12][13][14]).…”

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We consider gradient Gibbs measures corresponding to a periodic boundary law for a generalized solid-on-solid (SOS) model with spin values from a countable set on a Cayley tree. On the Cayley tree, detailed information on gradient Gibbs measures for models of SOS type is given in Botirov and Haydarov (2020

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