This article is cited in 1 paper

Fixed points of an infinite-dimensional operator related to Gibbs measures

U. R. Olimov^a, U. A. Rozikov<sup>abc

a</sup> Romanovskiy Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
^b AKFA University, Tashkent, Uzbekistan
^c National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan

Abstract: We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set $\mathbb N$ of spin values on a Cayley tree. This operator is defined by a countable set of parameters $\lambda_i>0$, $a_{ij}\in\{0,1\}$, $i,j\in\mathbb N$. We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model.

Keywords: fixed point, Cayley tree, Gibbs measure, HC model.

Received: 28.06.2022
Revised: 28.06.2022

DOI: 10.4213/tmf10331

 English version: Theoretical and Mathematical Physics, 2023, 214:2, 282–295

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