U. A. Rozikov

Gibbs measures on Cayley trees

Properties of a Group Representation of the Cayley Tree Ising Model on Cayley Tree Ising Type Models with Competing Interactions Information Flow on Trees The Potts Model The Solid-on-Solid Model Models with Hard Constraints Potts Model with Countable Set of Spin Values Models with Uncountable Set of Spin Values Contour Arguments on Cayley Trees Other Models.

QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

Description of periodic extreme gibbs measures of some lattice models on the Cayley tree

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

AbstractThe paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ∀λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λc=1/(k−1)×(k/(k−1))k. Then: (i) for λ≤λc, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ>λc, in addition to μ*, there exist two distinct translation-periodic measures, μ+and μ−, taken to each other by the unit space shift. Measures μ+and μ−are extreme ∀λ>λc. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For

F-quadratic stochastic operators

\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k

A Contour Method on Cayley Trees

, measure μ*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λeand λo, for even and odd sites. We discuss open problems and state several related conjectures.

On disordered phase in the ferromagnetic Potts model on the Bethe lattice

In the present paper a model with competing ternary (J2) and binary (J1) interactions with spin values ±1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J1,J2 (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation-invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the unordered phase is a factor of type IIIλ, where λ=exp{−2βJ2}, β>0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type IIIδ, otherwise they have type III1.

On the Uniqueness of Gibbs Measures for p-Adic Nonhomogeneous λ-Model on the Cayley Tree

In this paper, we introduce the notion of an F-quadratic stochastic operator. It is shown that each F-quadratic operator has a unique fixed point. Besides, it is proved that any trajectory of an F-quadratic stochastic operator exponentially rapidly converges to this fixed point.