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Theoretical and Mathematical Physics

, Volume 191, Issue 3, pp 910–923 | Cite as

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

  • U. A. Rozikov
  • F. H. Haydarov
Article
  • 21 Downloads

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 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and Information TechnologiesTashkentUzbekistan
  2. 2.National University of UzbekistanTashkentUzbekistan

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