# Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

## 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