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Translation-Invariant Gibbs Measures For Potts Model With Competing Interactions With A Countable Set Of Spin Values On Cayley Tree, Zarinabonu Mustafoyeva
Translation-Invariant Gibbs Measures For Potts Model With Competing Interactions With A Countable Set Of Spin Values On Cayley Tree, Zarinabonu Mustafoyeva
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
In the present paper we consider of an infinite system of functional equations for the Potts model with competing interactions and countable spin values Φ = {0, 1, ..., } on a Cayley tree of order k. We study translation-invariant Gibbs measures that gives the description of the solutions of some infinite system of equations. For any k ≥ 1 and any fixed probability measure ν we show that the set of translation-invariant splitting Gibbs measures contains one and two points for odd k and even k, respectively, independently on parameters of the Potts model with a countable …
Gibbs Measures Of Models With Uncountable Set Of Spin Values On Lattice Systems, Farhod Haydarov
Gibbs Measures Of Models With Uncountable Set Of Spin Values On Lattice Systems, Farhod Haydarov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
In this paper, we shall discuss the construction of Gibbs measures for models with uncountable set of spin values on Cayley trees. It is known that "translation-invariant Gibbs measures" of the model with an uncountable set of spin values can be described by positive fixed points of a nonlinear integral operator of Hammerstein type. The problem of constructing a kernel with non-uniqueness of the integral operator is sufficient in Gibbs measure theory. In this paper, we construct a degenerate kernel in which the number of solutions does not exceed 3, and in turn, it only gives us a chance to …