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Extremality of Translation-Invariant Phases for a Three-State SOS-Model on the Binary Tree

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Abstract

We consider the solid-on-solid model, with spin values \(0,1,2\), on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model. These measures are labeled by solutions to a nonlinear vector-valued functional equation. We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always non-extremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz.

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Notes

  1. These measures are called in [16] Markov chains and in [22] entrance laws. The notation splitting Gibbs measures was used in [18], to emphasize the property that, in addition to the Markov property, these measures satisfy the following condition: given a configuration \(\sigma _n\) in \(V_n\), the values \(\sigma (y)\) at sites \(y\in W_{n+1}\) are conditionally independent.

  2. See (3.8) for an exact value.

  3. Ergodic means irreducible and aperiodic Markov chain. Therefore has a unique stationary distribution \(\pi =(\pi _1,\dots ,\pi _q)\) with \(\pi _i>0\) for all \(i\).

  4. Permissive means that for arbitrary finite \(A\) and boundary condition outside \(A\) being \(\eta \) the conditioned Gibbs measure on \(A\), corresponding to the channel is positive for at least one configuration.

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Acknowledgments

U.A. Rozikov thanks the DFG Sonderforschungsbereich SFB \(|\) TR12-Symmetries and Universality in Mesoscopic Systems and the Ruhr-University Bochum (Germany) for financial support and hospitality. We thank both referees for their useful suggestions.

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  1. Fakultät für Mathematik, Ruhr-University of Bochum, Postfach 102148, 44721, Bochum, Germany

    C. Kuelske

  2. Institute of mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan

    U. A. Rozikov

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  1. C. Kuelske
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Kuelske, C., Rozikov, U.A. Extremality of Translation-Invariant Phases for a Three-State SOS-Model on the Binary Tree. J Stat Phys 160, 659–680 (2015). https://doi.org/10.1007/s10955-015-1279-9

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