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Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

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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 c < T cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T c or T > T cr , three weakly periodic Gibbs measures for T = T c , and one weakly periodic Gibbs measure for T c < T ≤ T cr .

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Authors and Affiliations

  1. Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences, Tashkent, Uzbekistan

    U. A. Rozikov

  2. Namangan State University, Namangan, Uzbekistan

    M. M. Rakhmatullaev

Authors
  1. U. A. Rozikov
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  2. M. M. Rakhmatullaev
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Correspondence to U. A. Rozikov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 292–302, August, 2008.

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Rozikov, U.A., Rakhmatullaev, M.M. Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree. Theor Math Phys 156, 1218–1227 (2008). https://doi.org/10.1007/s11232-008-0091-y

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  • DOI: https://doi.org/10.1007/s11232-008-0091-y

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