Related papers: On three-state Potts model on a Cayley tree: perio…

In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We…

In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $\rho\in\bq_p$.…

In this paper we consider a model with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k \geq 2$. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional…

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 consider the Potts-SOS model where the spin takes values in the set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions.…

We consider nearest-neighbor fertile hard-core models, with three states, on a homogeneous Cayley tree. It is known that there are four type of such models. We investigate all of them and describe translation-invariant and periodic…

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 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…

We consider a nearest-neighbor $p$-adic Potts (with $q\geq 2$ spin values and coupling constant $J\in \Q_p$) model on the Cayley tree of order $k\geq 1$. It is proved that a phase transition occurs at $k=2$, $q\in p\mathbb{N}$ and $p\geq 3$…

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…

In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In…

In the present paper, we study the $(2,q)$-Ising-Potts model on the Cayley tree. We have derived a recurrence equation that shows the existence of a splitting Gibbs measure for this model. Furthermore, we have proven that for the…

For the $q$-state Potts model on a Cayley tree of order $k\geq 2$ it is well-known that at sufficiently low temperatures there are at least $q+1$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures…

We consider fertile three-state Hard-Core (HC) models with the activity parameter $\lambda>0$ on a Cayley tree. It is known that there exist four types of such models: wrench, wand, hinge, and pipe. These models arise as simple examples of…

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the…

We consider Potts model, with competing interactions and countable spin values $\Phi=\{0,1,\dots \}$ on a Cayley tree of order three. We study periodic ground states for this model.

In this paper we construct several models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 2$. We prove that each of the constructed model has at least two translational-invariant…

We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\geq 2$ spin values) model on the Cayley tree of order $k\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $…

In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically,…

We consider the Potts model with two-step interactions and spin values 1,2,3,4 on a Cayley tree. We describe periodic ground states and verify the Peierls condition for the model.