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TMF, 2017, Volume 191, Number 3, Pages 503–517 (Mi tmf9215)  

This article is cited in 2 scientific papers (total in 2 papers)

Four competing interactions for models with an uncountable set of spin values on a Cayley Tree

U. A. Rozikova, F. Kh. Khaidarovb

a Institute of Mathematics and Information Technologies, Tashkent, Uzbekistan
b National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set $[0,1]$ of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.

Keywords: Cayley tree, competing interaction, configuration, Gibbs measure, Ising model, Potts model, periodic Gibbs measure, phase transition.

DOI: https://doi.org/10.4213/tmf9215

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English version:
Theoretical and Mathematical Physics, 2017, 191:3, 910–923

Bibliographic databases:

Document Type: Article
Received: 23.04.2016

Citation: U. A. Rozikov, F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley Tree”, TMF, 191:3 (2017), 503–517; Theoret. and Math. Phys., 191:3 (2017), 910–923

Citation in format AMSBIB
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\paper Four competing interactions for models with an uncountable set of
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\pages 503--517
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\jour Theoret. and Math. Phys.
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\pages 910--923
\crossref{https://doi.org/10.1134/S0040577917060095}
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  • http://mi.mathnet.ru/eng/tmf/v191/i3/p503

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Eshkabilov Yu.Kh., Haydarov F.H., “Lyapunov Operator l With Degenerate Kernel and Gibbs Measures”, Nanosyst.-Phys. Chem. Math., 8:5 (2017), 553–558  crossref  isi
    2. Haydarov F.H., “Fixed Points of Lyapunov Integral Operators and Gibbs Measures”, Positivity, 22:4 (2018), 1165–1172  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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