Related papers: Gibbs measure for mixed spins and mixed types mode…

In this paper, we consider a Hard-Core $(HC)$ model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we…

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.

We consider fertile HC-models with four-states and the parametre activity on a Cayley tree. It is known that three types of such models exist. For each of these models we prove uniqueness of the translation-invarinat Gibbs measure on a…

In this paper we show that under some conditions on the parameter of the Potts model with three states with zero external field on the Cayley tree of order $k>2$, there are exactly two periodic (non translation-invariant) Gibbs measures.

We consider fertile HC-models with three states on a Cayley tree. It is known that four types of such models exist. For these models we describe the translation-invarinat Gibbs measures on a Cayley tree of order three.

In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We…

In this paper, we investigate tree-indexed Markov chains (Gibbs measures) defined by a Hamiltonian that couples two Ising layers: hidden spins \(s(x) \in \{\pm 1\}\) and observed spins \(\sigma(x) \in \{\pm 1\}\) on a Cayley tree. The…

We consider the Ising model on a Cayley tree . For the Ising model found new weakly periodic measures corresponding normal subgroups of index 2 of the group representation of the tree Cayley.

Work is a continuation of work TMPh, 175(2), 2013. We study the Potts model with zero external field on the Cayley tree. It is shown that for any values of the parameter all periodic Gibbs measures are translation-invariant for the q-state…

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work \cite{KRK} we gave a…

In the present paper the Ising model with competing binary $J$ and $J_1$ interactions with spin values $\pm 1$, on a Cayley tree is considered. We study translation-invatiant Gibbs measures and corresponding free energies ones.

In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of…

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two. We reduce the…

Ising model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree has long been studied but there are still many problems untouched. This paper tackles new Gibbs measures of Ising-Vannimenus…

We investigate the finite-state $p$-solid-on-solid model, for $p=\infty$, on Cayley trees of order $k\geq 2$ and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our…

In this paper we study the extremality of translation-invariant Gibbs measure for the HC-model on a Cayley tree. It is known that for this model the translation-invariant measure is unique. We give a new proof of this statement and found…

In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some…

In the paper, we consider the $\lambda$-model with spin values $\{1, 2, 3\}$ on the Cayley tree of order two. We first describe ground states of the model. Moreover, we also proved the existence of translation-invariant Gibb measures for…

We consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value $0,1$ or $2$ and the pairwise interaction of neighboring vertices is given by their spin difference to the power $p>0$.…

In this paper we consider translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree of order k. An approximate critical temperature T_{cr} is found such that for T\geq T_{cr} there exists a unique translation-invariant…