A new set of Gibbs measures for the SOS model on a Cayley tree

The phase transition phenomenon is one of the central problems of statistical mechanics. It occurs when the model possesses multiple Gibbs measures. In this paper, we consider a three-state SOS (solid-on-solid) model on a Cayley tree. We reduce description of Gibbs measures to solving of a non-linear functional equation, which each solution of the equation corresponds to a Gibbs measure. We give some sufficiency conditions on the existence of multiple Gibbs measures for the model. We give a review of some known (translation-invariant, periodic, non-periodic) Gibbs measures of the model and compare them with our new measures. We show that the Gibbs measures found in the paper differ from the known Gibbs measures, i.e, we show that these measures are new

On Ground States and Phase Transition for λ\lambda-Model with the Competing Potts Interactions on Cayley Trees

In this paper, we consider the $\lambda$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $\lambda$-function is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model in more general setting. Furthermore, we find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1704.0193

The Potts model on a Cayley tree: the new class of Gibbs measures

For the Potts model on Cayley trees, a very wide class of new Gibbs measures is given. We give a review of all known Gibbs measures of the Potts model on trees and compare them with our new measures.Comment: 18 pages, in Russian language, 4 figure

GROUND STATES FOR A MODEL WITH NON-ZERO EXTERNAL FIELD ON THE CAYLEY TREE OF ORDER THREE

We consider the SOS model with non-zero external field on the Cayley tree of order 3 k . Described translation-invariant ground states for the model of SOS with translation-invariant external field

Основные состояния для модели SOS c конкурирующими взаимодействиями

We study periodic and weakly periodic ground states for the SOS model with competing interactions on the Cayley tree of order two and three. Further, we study non periodic ground states for the SOS model with competing interactions on the Cayley tree of order twoВ работе для нормального делителя индекса 2 изучены слабо-периодические основные состояния для модели SOS с конкурирующими взаимодействиями на дереве Кэли порядка 2 и порядка 3. Далее изучены непериодические основные состояния для модели SOS с конкурирующими взаимодействиями на дереве Кэли второго порядк