This article is cited in 40 papers

Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree

U. A. Rozikov^a, M. M. Rakhmatullaev<sup>b

a</sup> Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan
^b Namangam State University

Abstract: We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values $T_{\mathrm{c}}<T_{\mathrm{cr}}$ of the temperature $T>0$ such that there exist five weakly periodic Gibbs measures for $0<T<T_{\mathrm{c}}$ or $T>T_{\mathrm{cr}}$, three weakly periodic Gibbs measures for $T=T_{\mathrm{c}}$, and one weakly periodic Gibbs measure for $T_{\mathrm{c}}<T\le T_{\mathrm{cr}}$.

Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic measure.

Received: 26.07.2007
Revised: 23.10.2007

DOI: 10.4213/tmf6248

 English version: Theoretical and Mathematical Physics, 2008, 156:2, 1218–1227

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