Related papers: Gibbs measures for HC-model with a countable set o…

In this paper, we study the Hard Core (HC) model with a countable set $\mathbb Z$ of spin values on a Cayley tree of order $k=2$. This model is defined by a countable set of parameters (that is, the activity function $\lambda_i>0$, $i\in…

We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters $\lambda_{i}>0, i \in \mathbb{Z}\setminus\{0\}$. For all possible values of…

In this paper we investigate generalized Gibbs measure (GGM) for $p$-adic Hard-Core(HC) model with a countable set of spin values on a Cayley tree of order $k\geq 2$. This model is defined by $p$-adic parameters $\lambda_i$, $i\in \mathbb…

In this paper, we focus on studying non-probability Gibbs measures for a Hard Core (HC) model on a Cayley tree of order $k\geq 2$, where the set of integers $\mathbb Z$ is the set of spin values. It is well-known that each Gibbs measure,…

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear…

In this paper, we investigate translation-invariant splitting Gibbs measures (TISGMs) for the HC-Blume-Capel model on a "wand" graph embedded in the Cayley tree of arbitrary order $k \geq 2$. It is known that there is the exact critical…

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 consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. We study periodic Gibbs measures of the model with period two. For $k=1$ we show that there is no any…

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 an Ising model on the Cayley tree $\Gamma_k$ of arbitrary order $k\ge1$ with three spin species of values $(\tfrac12,1,\tfrac32)$ distributed deterministically with period three along the generations. Within the framework of…

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear…

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 is studied HC-models on a Cayley tree. two states HC-model on a Cayley tree and Under some conditions on parameters of the two state HC-model we prove existence exactly two of the weakly periodic (non periodic) Gibbs measures.…

The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We…

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…

We consider fertile three-state hard core models with the activity parameter on an order-three Cayley tree. It is known that there exist four types of such models: in two of them the regions of extremality of the unique…

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"\,. In cases "wand"\, and…

We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq 2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For $m=2$, in the…

We consider a nearest-neighbor $p$-adic $\l$-model with spin values $\pm 1$ on a Cayley tree of order $k\geq 1$. We prove for the model there is no phase transition and as well as the unique $p$-adic Gibbs measure is bounded if and only if…

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…