•  
  •  
 

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Abstract

We consider a model where the spin takes values in the set $[0,1]^{\rm d}$, 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

36

Last Page

42

References

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

Share

COinS