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

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.

arXiv comment24 pages, 3 figures

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Cited by in corpus (8)

• Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees
• Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
• Existence of gradient Gibbs measures on regular trees which are not translation invariant
• Convergence of deterministic growth models
• A HC model with countable set of spin values: uncountable set of Gibbs measures
• Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws
• Infinite-volume states with irreducible localization sets for gradient models on trees
• Fixed points of an infinite dimensional operator related to Gibbs measures