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2019 Gradient Gibbs measures for the SOS model with countable values on a Cayley tree
Florian Henning, Christof Külske, Arnaud Le Ny, Utkir A. Rozikov
Electron. J. Probab. 24: 1-23 (2019). DOI: 10.1214/19-EJP364


We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order $k\geq 2$ and are interested in tree-automorphism 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 period-$4$ height-periodic boundary law equations (in particular, some period-2 height-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some period-$3$ height-periodic boundary laws on the Cayley tree of arbitrary order $k\geq 2$, which define GGMs different from the $4$-periodic ones.


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Florian Henning. Christof Külske. Arnaud Le Ny. Utkir A. Rozikov. "Gradient Gibbs measures for the SOS model with countable values on a Cayley tree." Electron. J. Probab. 24 1 - 23, 2019.


Received: 14 February 2019; Accepted: 18 September 2019; Published: 2019
First available in Project Euclid: 1 October 2019

zbMATH: 07142898
MathSciNet: MR4017122
Digital Object Identifier: 10.1214/19-EJP364

Primary: 82B26
Secondary: 60K35


Vol.24 • 2019
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