Narrow Results

In this paper, we focus on studying the non-probability Gibbs measures for a Hard-Core (HC) model on a Cayley tree of order $k\ge 2$, where the set of integers $\mathbb{Z}$ is the set of spin values. It is well known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function (with strictly positive coordinates) defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalizable boundary law corresponds to a Gibbs measure. However, a non-normalizable boundary law can define the gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC model on a Cayley tree of any order $k\ge 2$.

In mathematical statistical mechanics, Gibbs measures have been designed to represent equilibrium states and to model phase transition. In the literature, Gibbs measures associated with the Ising model on Cayley trees have been extensively studied. However, the Ising model has not been studied on $(i,k)$-ary trees before. In this paper, the phase transition problem is investigated for the Ising model on $(i,k)$-ary trees. Specifically, the model on ${\mathrm{\Gamma }}^{(1,k)}$ and ${\mathrm{\Gamma }}^{(2,3)}$, respectively, displays three distinct translation-invariant Gibbs measures in both the ferromagnetic and anti-ferromagnetic states, whereas the traditional Ising model lacks translation-invariant Gibbs measures in the anti-ferromagnetic state. Moreover, non-translation invariant Gibbs measures are constructed as well. Furthermore, for the considered model, the standard free energy does not exist, in contrast to its existence over regular trees.

In this paper, we extend the investigation of Kittel’s molecular zipper model on the infinite Cayley tree. We focus on the weakly periodic Gibbs measures for this model. Under specific parameter conditions, we show that all periodic Gibbs measures coincide with translation-invariant measures. Additionally, we identify distinct parameters that give rise to non-translation-invariant weakly periodic Gibbs measures.

Some results on the relaxation processes (Glauber dynamics) obtained in the last decade are presented. This article is intended to be a short guided tour through these results for readers without prior knowledge of rigorous statistical mechanics or stochastic processes.

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 examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc.

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 parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.

Kittel’s 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained.