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On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations

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Abstract

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ⩾ 1. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some nonlinear integral equation. For k = 1 we show that the integral equation has a unique solution. In case k ⩾ 2 some models (with the set [0, 1] of spin values) which have a unique splitting Gibbs measure are constructed. Also for the Potts model with uncountable set of spin values it is proven that there is unique splitting Gibbs measure.

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Authors and Affiliations

  1. Institute of Mathematics and Information Technologies, Tashkent, Uzbekistan

    Utkir A. Rozikov

  2. National University of Uzbekistan, Tashkent, Uzbekistan

    Yusup Kh. Eshkobilov

Authors
  1. Utkir A. Rozikov
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  2. Yusup Kh. Eshkobilov
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Correspondence to Utkir A. Rozikov.

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Rozikov, U.A., Eshkobilov, Y.K. On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations. Math Phys Anal Geom 13, 275–286 (2010). https://doi.org/10.1007/s11040-010-9079-6

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  • DOI: https://doi.org/10.1007/s11040-010-9079-6

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