Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


We consider a model where the spin takes values in the set [0,1]d, and is assigned to the vertexes of the Cayley tree. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some non-linear integral equation. For a concrete form of the Kernel of the integral equation we show the uniqueness of solution.

First Page


Last Page



1. Baxter R.J. Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1982.

2. Jahnel B., Kuelske Ch., Botirov G.I. Phase transition and critical values of a nearest-neighbor system with uncountable local state space on Cayley trees. Math. Phys. Anal. Geom., 2014, V. 17, No. 3-4, pp.323–331.

3. Ganikhodjaev N.N., Rozikov U.A. The Potts Model with Countable Set of Spin Values on a Cayley Tree. Letters Math. Phys., 2006, V. 75, pp.99–105.

4. Eshkabilov Yu.Kh., Rozikov U.A., Botirov G.I. Phase transition for a model with uncountable set of spin values on Cayley tree. Lobachevskii Journal of Mathematics, 2013, V. 34, No. 3, pp.256–263.

5. Rozikov U.A., Eshkabilov Yu.Kh. On models with uncountable set of spin values on a Cayley tree: Integral equations. Math. Phys. Anal. Geom., 2010, V. 13, pp.275–286.

6. Rozikov U.A., Eshkobilov Yu.Kh., Haydarov F.H. Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree. Jour. Stat. Phys., 2012, V. 147, No. 4, pp.779–794.

7. Rozikov U.A. Gibbs measures on Cayley trees. World Scientific, 2013. – 404 p.

Included in

Analysis Commons



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.