Mathematical Physics, Analysis and Geometry

, Volume 13, Issue 3, pp 275–286 | Cite as

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

  • Utkir A. Rozikov
  • Yusup Kh. Eshkobilov


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.


Cayley tree Configuration Gibbs measures Potts model 

Mathematics Subject Classifications (2010)

Primary 82B05 82B20; Secondary 60K35 


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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

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

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