This article is cited in 6 papers

Gibbs measures for fertile hard-core models on the Cayley tree

R. M. Khakimov

Institute for Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: We study fertile hard-core models with the activity parameter $\lambda>0$ and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter $\lambda$ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of $\lambda$ at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten–Stigum condition) for a measure not to be extreme.

Keywords: Cayley tree, configuration, fertile graph, hard-core model, Gibbs measure, translation-invariant measure.

Received: 16.02.2015

DOI: 10.4213/tmf8886

 English version: Theoretical and Mathematical Physics, 2016, 186:2, 294–305

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