Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree

We study translation-invariant Gibbs measures on a Cayley tree of order k = 3 for the ferromagnetic three-state Potts model. We obtain explicit formulas for translation-invariant Gibbs measures. We also consider periodic Gibbs measures on a Cayley tree of order k for the antiferromagnetic q-state Potts model. Moreover, we improve previously obtained results: we find the exact number of periodic Gibbs measures with the period two on a Cayley tree of order k ≥ 3 that are defined on some invariant sets.     

Periodic Gibbs measures for the Potts model on the Cayley tree

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.     

Nonuniqueness of a Gibbs measure for the Ising ball model

We study a new model, the so-called Ising ball model on a Cayley tree of order k ≥ 2. We show that there exists a critical activity \(\lambda _{cr} = \sqrt[4]{{0.064}}\) such that at least one translation-invariant Gibbs measure exists for λ ≥ λ _cr , at least three translation-invariant Gibbs measures exist for 0 < λ < λ _cr , and for some λ, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor \(\hat G\) of index 2 of the group representation on the Cayley tree, we study \(\hat G\) -periodic Gibbs measures. We prove that there exists an uncountable set of \(\hat G\) -periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.     

Localization of translation-invariant Gibbs measures for the Potts and “solid-on-solid” models on a Cayley tree

We study Potts and “solid-on-solid” models with q ≥ 2 states on the Cayley tree of order k ≥ 1. For any values of the parameter q in the Potts model and q ≤ 6 in the “solid-on-solid” model, we find sets containing all translation-invariant Gibbs measures.     

Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree

We consider the Potts model with competing interactions on the Cayley tree of order k with k ≥ 2. We describe the sets of periodic and weakly periodic ground states corresponding to normal subgroups of the group representation of the Cayley tree of index 4.     

Four competing interactions for models with an uncountable set of spin values on a Cayley tree

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.     

On a generalized p-adic Gibbs measure for Ising model on trees

In this paper we consider a p-adic Ising model on an arbitraty tree. We show the uniqueness and boundedness of the p-adic Gibbs measure for the model. Moreover, we consider translational invariant and periodic generalized p-adic Gibbs measures for the model on the Cayley tree of order two.     

The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model

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.     

Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree

For the Potts model with competing interactions, we describe the set of weakly periodic ground states corresponding to index-two normal divisors of the Cayley tree group representation. We also study some weakly periodic Gibbs measures.     

On periodic Gibbs measures of <Emphasis Type="Italic">p</Emphasis>-adic Potts model on a Cayley tree

In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.     

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. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 109–117, April, 1997.     

Free energies of the Potts model on a Cayley tree

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.     

Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

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 . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 292–302, August, 2008.     

Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree

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.     

A <Emphasis Type="Italic">p</Emphasis>-adic hard-core model with three states on a Cayley tree

We examine the p-adic hard-core model with three states on a Cayley tree. Translationinvariant and periodic p-adic Gibbs measures are studied for the hard-core model for k = 2. We prove that every p-adic Gibbs measure is bounded for p ≠ 2. We show in particular that there is no strong phased transition for a hard-core model on a Cayley tree of order k.     

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

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.     

Weakly Periodic Ground States for the λ-Model

Ukrainian Mathematical Journal - For the λ-model on a Cayley tree of order k ≥ 2, we describe the set of periodic and weakly periodic ground states corresponding to normal divisors of...     

Weakly Periodic Gibbs Measures for HC-Models on Cayley Trees

We study hard-core (HC) models on Cayley trees. Given a 2-state HC-model, we prove that exactly two weakly periodic (aperiodic) Gibbs measures exist under certain conditions on the parameters. Moreover, we consider fertile 4-state HC-models with the activity parameter λ > 0. The three types of these models are known to exist. For one of the models we show that the translationinvariant Gibbs measure is not unique.     

Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree

Theoretical and Mathematical Physics - We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order $${k\geq 2}$$ . For the Potts model in a...