Uniqueness and nonuniqueness conditions for weakly periodic Gibbs measures for the hard-core model
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UNIQUENESS AND NONUNIQUENESS CONDITIONS FOR WEAKLY PERIODIC GIBBS MEASURES FOR THE HARD-CORE MODEL R. M. Khakimov∗ and M. T. Makhammadaliev†
We study a “hard-core” model on a Cayley tree. In the case of a normal divisor of index 4, we show the uniqueness of weakly periodic Gibbs measures under certain conditions on the parameters. Moreover, we prove that there exist weakly periodic (nonperiodic) Gibbs measures different from those previously known.
Keywords: Cayley tree, configuration, hard-core model, Gibbs measure, translation-invariant measure, periodic measure, weakly periodic measure DOI: 10.1134/S0040577920080073
1. Introduction Problems that arise in studying the thermodynamic properties of physical and biological systems are mainly solved in the framework of Gibbs measure theory. The Gibbs measure is a fundamental concept that determines the probability of a microscopic state of a given physical system. It is known that each Gibbs measure is associated with one phase of the physical system, and if the Gibbs measure is not unique, then we say that there is a phase transition. For a fairly wide class of Hamiltonians, it is known that the set of all limit Gibbs measures (corresponding to a given Hamiltonian) forms a nonempty convex compact subset in the set of all probability measures (see, e.g., [1]–[3]), and each point of this convex set is uniquely decomposed at its extreme points. In this connection, it is particularly interesting to describe all the extreme points of this convex set, i.e., the extreme Gibbs measures. The definition of the Gibbs measure and other concepts related to Gibbs measure theory can be found, for example, in [1]–[4]. Although there are many works devoted to studying Gibbs measures, a complete description of all limit Gibbs measures has not yet been obtained for any of the models. This problem is well understood for the Ising model on a Cayley tree. For example, an uncountable set of extreme Gibbs measures was constructed in [5], and necessary and sufficient conditions for the extremum of the disordered phase of the Ising model on the Cayley tree were found in [6]. Mazel and Sukhov introduced and studied the “hard-core” (HC) model on the d-dimensional lattice d Z [7]. Studying Gibbs measures for the HC model with two states on the Cayley tree was the topic in [8]– [15]. In [8], the uniqueness of the translation-invariant measure and the nonuniqueness of periodic Gibbs measures for the HC model were proved. For the parameters of the HC model, a sufficient condition was also found in [8] under which the translation-invariant Gibbs measure is nonextreme. In the case where ∗
Namangan Department, Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, Namangan, Uzbekistan, e-mail: [email protected] (corresponding author). †
Namangan State University, Namangan, Uzbekistan, e-mail: [email protected].
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 258–279, August, 2020. Received September 12, 2019. Revised February 23, 2020. Accepted
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