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We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set of spin values on a Cayley tree. This operator is defined by a countable set of parameters , , . 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. [ABSTRACT FROM AUTHOR]

We study the Blume–Capel model with a countable set of spin values and a force of interaction between the nearest neighbors on a Cayley tree of order . The following results are obtained. Let , , be the temperature. For , there exist no translation invariant Gibbs measures or -periodic Gibbs measures. For , we prove the uniqueness of a translation-invariant Gibbs measure. Let and . If , then there exists exactly one -periodic Gibbs measure that is translation invariant. For , there exist exactly three -periodic Gibbs measures, one of which is a translation-invariant Gibbs measure. [ABSTRACT FROM AUTHOR]

We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order . For the Potts model in a translation-invariant external field for , the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order . Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three. [ABSTRACT FROM AUTHOR]

We define a DNA molecule as a sequence of the numbers 1 and 2 and embed it on a path of a Cayley tree such that each vertex of the Cayley tree belongs to only one DNA and each DNA has its own countable set of neighboring DNA. The Hamiltonian of this set of DNA is a model with two spin values regarded as DNA base pairs. We describe translation-invariant Gibbs measures (TIGMs) of the model on the Cayley tree of order two and use them to study the thermodynamic properties of the model of DNA. We show that there is a critical temperature such that if the temperature , then there is a unique TIGM, and if , then there are three TIGMs. Each TIGM gives a phase of the set of DNA. In the cases of very high and very low temperatures, we find stationary distributions and typical configurations of the model. [ABSTRACT FROM AUTHOR]

We study translation-invariant Gibbs measures for the ferromagnetic Potts model with q states on the Cayley tree of order k and generalize some earlier results. We consider the question of the extremality of the known translation-invariant Gibbs measures for the Potts model with three states on the Cayley tree of order k = 3. [ABSTRACT FROM AUTHOR]

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ℤ of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions). [ABSTRACT FROM AUTHOR]

We consider models with four competing interactions ( external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique. [ABSTRACT FROM AUTHOR]

For the Potts model on the Cayley tree, we obtain some explicit formulas for the free energies and entropies in the case of vector-valued boundary conditions. These formulas include translation-invariant, periodic, and Dobrushin-like boundary conditions and also those corresponding to weakly periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures [ABSTRACT FROM AUTHOR]

We study the hard-core model on the Cayley tree. We show that this model admits only periodic Gibbs measures with the period two. We find sufficient conditions for all weakly periodic Gibbs measures to be translation invariant. [ABSTRACT FROM AUTHOR]

We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x + ax on the space of complex p-adic numbers. [ABSTRACT FROM AUTHOR]

We propose a model on the Cayley tree and prove that a uncountable set of Ĝ-periodic Gibbs measures exists for this model, in contrast to models studied previously. [ABSTRACT FROM AUTHOR]

We describe some p-harmonic functions on the Cayley tree constructively and also study linear relations between such functions. [ABSTRACT FROM AUTHOR]

We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices 2 and 4 of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model. [ABSTRACT FROM AUTHOR]

We consider fertile hard-core (HC) models with three states on a homogeneous Cayley tree. It is known that four types of such models exist. For these models, we describe the translation-invariant and periodic HC Gibbs measures. We also construct a uncountable set of nonperiodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T c < T cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T c or T > T cr , three weakly periodic Gibbs measures for T = T c , and one weakly periodic Gibbs measure for T c < T ≤ T cr . [ABSTRACT FROM AUTHOR]

We consider the Potts model with three spin values and with competing interactions of radius r = 2 on the Cayley tree of order k = 2. We completely describe the ground states of this model and use the contour method on the tree to prove that this model has three Gibbs measures at sufficiently low temperatures. [ABSTRACT FROM AUTHOR]

We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric. [ABSTRACT FROM AUTHOR]

We describe a wide class of normal divisors of infinite index of the group representation of the Cayley tree and study the structure of partitions of the Cayley tree w.r.t. any normal divisor of infinite index. We prove that for a specific normal divisor of infinite index, there are three periodic and uncountably many nonperiodic Gibbs measures for an inhomogeneous Ising model. [ABSTRACT FROM AUTHOR]

We find the sufficient condition on the parameters of the inhomogeneous Potts model on the Cayley tree under which the disordered phase becomes extremal. [ABSTRACT FROM AUTHOR]

We prove the existence of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model on a Cayley tree and describe them constructively. [ABSTRACT FROM AUTHOR]

In the group representation of the Cayley tree, the distribution of elements of the partition into conjugate classes of finite-index, normal subgroups is described. For the inhomogeneous Ising model, it is proved that there exist only three H0-periodic Gibbs distributions, where H0 is a normal subgroup of finite index. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]