Gibbs measures of models with uncountable set of spin values on lattice systems

Authors

Farhod Haydarov, V.I.Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan; National University of Uzbekistan, Tashkent, UzbekistanFollow

Abstract

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 check the existence of phase transitions.

First Page

166

Last Page

178

References

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Recommended Citation

Haydarov, Farhod (2023) "Gibbs measures of models with uncountable set of spin values on lattice systems," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 6: Iss. 3, Article 5.
DOI: https://doi.org/10.56017/2181-1318.1256