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 $k\ge 2$.

The purpose of this review paper is to present systematically all known results on Gibbs measures on Cayley trees (Bethe lattices). There are about 150 papers which contain mathematically rigorous results about Gibbs measures on Cayley trees. This review is mainly based on the recently published mathematical papers. The method used for the description of Gibbs measures on Cayley trees is the method of Markov random field theory and recurrent equations of this theory, but the modern theory of Gibbs measures on trees uses new tools such as group theory, information flows on trees, node-weighted random walks, contour methods on trees and nonlinear analysis. We discuss all the mentioned methods which were developed recently. Thus, the paper informs the reader about what is (mathematically) done in the theory of Gibbs measures on trees and about where the corresponding results were published. We only give proofs which were not published in literature. Moreover, we give several open problems.

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.

In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a $\sigma$-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.

We introduce a variant of the Bethe lattice, the fully connected Bethe lattice (FCBL); with every site in each shell of the Bethe lattice connected to every other site in the shell. A recently introduced decimation renormalization group (RG) method is utilized to obtain the quantum transmission through the Bethe lattice and FCBL. Although classically the extra intra-shell connections would not change the transmission, they have a large effect on the quantum transport and the wave functions.

Different types of the lattice spin systems with the competing interactions have rich and interesting phase diagrams. In this study a system with competing nearest-neighbor interaction J₁, prolonged next-nearest-neighbor interaction J_p and ternary prolonged interaction J_tp is considered on a Cayley tree of arbitrary order k. To perform this study, an iterative scheme is developed for the corresponding Hamiltonian model. At finite temperatures several interesting properties are presented for typical values of α = T/J₁, β = −J_p/J₁ and γ = -J_tp/J₁. This study recovers as particular cases, previous work by Vannimenus¹ with γ = 0 for k = 2 and Ganikhodjaev et al.² in the presence J₁, J_p, J_tp with k = 2. The variation of the wavevector q with temperature in the modulated phase and the Lyapunov exponent associated with the trajectory of our iterative system are studied in detail.

The object of the present investigation is an ensemble of self-avoiding and directed graphs belonging to eight-branching Cayley tree (Bethe lattice) generated by the Fuchsian group of a Riemann surface of genus two and embedded in the Poincaré unit disk. We consider two-parametric lattices and calculate the multifractal scaling exponents for the moments of the graph lengths distribution as functions of these parameters. We show the results of numerical and statistical computations, where the latter are based on a random walk model.

Ising model with competing nearest–neighbors (NN) and prolonged next–nearest–neighbors (NNN) 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 model with competing NN and prolonged NNN interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures (TIGMs) for the model. We show that some of the measures are extreme Gibbs distributions. In this paper, we try to determine when phase transition does occur.

The ferromagnetic properties of the spin-1 BEG model on finite-size Cayley tree are investigated using the exact recursion method. The spontaneous magnetization of the system is studied in detail for different values of the reduced crystal-field interaction D/J, and it is found that there is an unusual behavior (anti-Curie temperature) when D/J > 2.0. We also obtain the Curie temperature of this finite-size system. When the system size is large enough, our results will fit well with that in the thermodynamic limit.

In this work, we study a system of Kuramoto oscillators with identical frequencies in a Cayley tree. Heterogeneity in the frequency distribution is introduced in the root of the tree, allowing for analytical calculations of the phase evolution. In this work, we study a system of Kuramoto oscillators with identical frequencies in a Cayley tree. Heterogeneity in the frequency distribution is introduced in the root of the tree, allowing for analytical calculations of the phase evolution. This simple case can be regarded as a starting point in order to understand how to extract topological features of the connectivity pattern from the dynamic state of the system, and vice versa, for the general situation of a set of phase oscillators located on a tree-like network.

It has been demonstrated that excitable media with a tree structure performed better than other network topologies, therefore it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.

In this paper, we study linear cellular automata (CAs) on Cayley tree of order $k$ over the field ${𝔽}_p$ (the set of prime numbers modulo $p$). After revealing the rule matrix corresponding to cellular automata on Cayley tree with the null boundary condition, we analyze the reversibility problem of these cellular automata for some given values of $a,b,c,d\in {𝔽}_p$ and the levels $n$ of Cayley tree. The necessary and sufficient conditions for determining whether a CA is reversible or not are demonstrated. Furthermore, we compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree, the measure entropy with respect to uniform Bernoulli measure is infinity.

In this paper, we propose a new matching (called a conditional matching), where the condition refers to the matching of the new constructed network which includes all the nodes in the original network. We then enumerate the conditional matchings of the new network and prove that the number of conditional matchings is just the product of degree sequences of the original network. We choose two families of fractal networks to show our obtained results, including the pseudofractal network and Cayley tree. Finally, we calculate the entropy of the conditional matchings on the considered networks and see that the entropy of Cayley tree is smaller than that of the pseudofractal network.

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction J_xy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ J_xy with | J_xy |< p^{-1/(p -1)}. (ii) For q∈p ℕ, p≥3 one can choose J_xy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, such that , there exists a unique symmetric TISGM μ* and there are exactly three symmetric TISGMs: (a "bottom" symmetric TISGM), (a "middle" symmetric TISGM) and (a "top" symmetric TISGM). For we also construct a continuum of distinct, symmertric SGMs which are non-TI.

Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on C*-algebras defined by general graphs. As examples of generalized d-Markov chains, we construct the entangled Markov fields on tree graphs. The concrete examples of generalized d-Markov chains on Cayley trees are also investigated.

In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction, we investigate QMC associated with XY-model on a Cayley tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.