Cayley Tree Of Arbitrary Order Research Articles

Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

ABSTRACT We consider fertile three-state Hard-Core (HC) models on a Cayley tree. It is known that there exist four types of such models: ‘wrench’, ‘wand’, ‘hinge’ and ‘pipe’. In cases ‘wand’ and ‘hinge’ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The concept of alternative Gibbs measure is introduced and in the case ‘wand’ translational invariance conditions for alternative Gibbs measures are found. Also, we show the existence of alternative Gibbs measures which are not translation-invariant.

The extremality of disordered phases for the mixed spin-(1,1/2) Ising model on a Cayley tree of arbitrary order

This paper continues the exploration of translation-invariant splitting Gibbs measures (TISGMs) within the framework of the Ising model with mixed spin-(1,1/2) (abbreviated as (1,1/2)-MSIM) on a Cayley tree of arbitrary order. Building upon our prior work (Akin and Mukhamedov 2022 J. Stat. Mech. 053204), where we extensively elucidated TISGMs and investigated the extremality of disordered phases employing a Markov chain indexed by a tree on a semi-infinite Cayley tree of order two, our current research extends this analysis. In the present study, we begin by constructing TISGMs and tree-indexed Markov chains tailored to the (1,1/2)-MSIM. Notably, we extend our scope to encompass Cayley trees of varying orders, thereby affording a comprehensive examination of the extremality characteristics inherent to disordered phases. Utilizing the Kesten–Stigum condition, we delve into the non-extremality aspects of disordered phases by scrutinizing the eigenvalues of the stochastic matrix associated with the (1,1/2)-MSIM on Cayley trees with order k ( k⩾3 ). This approach allows for a nuanced exploration of the model’s behavior and stability under different conditions.

Extremality of Disordered Phase of λ-Model on Cayley Trees

In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.

Gradient Gibbs measures for the SOS model with countable values on a Cayley tree

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.

Gibbs measures with memory of length 2 on an arbitrary-order Cayley tree

In this paper, we consider the Ising-Vanniminus model on an arbitrary-order Cayley tree. We generalize the results conjectured by Akın [Chinese J. Phys. 54(4), 635–649 (2016) and Int. J. Mod. Phys. B 31(13), 1750093 (2017)] for an arbitrary-order Cayley tree. We establish the existence and a full classification of translation-invariant Gibbs measures (TIGMs) with a memory of length 2 associated with the model on arbitrary-order Cayley tree. We construct the recurrence equations corresponding to the generalized ANNNI model. We satisfy the Kolmogorov consistency condition. We propose a rigorous measure-theoretical approach to investigate the Gibbs measures with a memory of length 2 for the model. We explain if the number of branches of the tree does not change the number of Gibbs measures. Also, we try to determine when the phase transition does occur.

On Construction of Quantum Markov Chains on Cayley trees

The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.

On p-adic Ising–Vannimenus model on an arbitrary order Cayley tree

In this paper, we continue an investigation of the p-adic Ising–Vannimenus model on the Cayley tree of an arbitrary order k(k ⩾ 2). We prove the existence of p-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional p-adic system of equations. We are also able to show the uniqueness of translation-invariant p-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising–Vannimenus model depending on the order k of the Cayley tree and the prime p. Note that the methods used in the paper are not valid in the real setting, since all of them are based on p-adic analysis and p-adic probability measures.

Two Potts Models on Cayley Tree of Arbitrary Order

On a Cayley tree of arbitrary order k we consider two different Potts models with competing nearest-neighbour interactions J1 and next-nearest-neighbour interactions Jp and Jo, where coupling Jp corresponds to interaction of spins belonging to the same branch of the tree (prolonged) and coupling Jo corresponds to interaction of spins belonging to the same shell of the tree (one-level) and find for each model in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of coupling values where the local magnetization has chaotic oscillatory glass-like behaviour. We also show that the ranges corresponding two models are different.

Strange Attractors in the Vannimenus Model on an Arbitrary Order Cayley Tree

We consider the Vannimenus model on a Cayley tree of arbitrary order k with competing nearest-neighbour interactions J1 and next-nearest-neighbour interactions J2 and J3 in the presence of an external magnetic field h. In this paper we study the phase diagram of the model using an iterative scheme for a renormalized effective nearest-neighbour coupling Kr and effective field per site Xr for spins on the rth level; it recovers, as particular cases, previous works by Vannimenus, Inawashiro et al, Mariz et al and Ganikhodjaev and Uǧuz. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous or abrupt, respectively characterizing second- or first- order phase transitions. We present a few typical attractors and at finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into few disconnected pieces, etc) are exhibited for typical values of parameters.

Potts model with next-nearest-neighbor ternary interactions on an arbitrary-order Cayley tree

We study the phase diagrams for the Potts model with restricted competing nearest-neighbor interactions J 1 and ternary interactions J pt on a Cayley tree of arbitrary order k. At vanishing temperature, the phase diagram is fully determined for all values and signs of J pt /J 1 and T/J 1. The phase diagrams are obtained from stability conditions, and characteristic points in the iteration scheme are numerically analyzed. The wavevectors versus temperature are plotted for some critical points in the modulated phases. Then, we using the Lyapunov exponent to verify the stability of the periods.

LYAPUNOV EXPONENTS AND MODULATED PHASES OF AN ISING MODEL ON CAYLEY TREE OF ARBITRARY ORDER

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 Jp and ternary prolonged interaction Jtp 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 = −Jp/J1 and γ = -Jtp/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, Jp, Jtp 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.

Behaviors of Phase Diagrams of Ising Model with Competing Ternary and Binary Interactions on a Cayley Tree of Arbitrary Order

An Ising model with competing interactions has recently been studied extensively because of the appearance of nontrivial magnetic orderings. In this paper, we study the phase diagrams for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions Jtp on a Cayley tree of arbitrary order k and compare with the phase diagrams obtained in Uguz et al. and Vannimenus results for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions Jp. For some values of k, we obtain phase diagrams of the model. We clarify the role of order k of the Cayley tree. We also plot the variation of the wave vector with temperature.

Competing binary and [formula omitted]-tuple interactions on a Cayley tree of arbitrary order

We study the phase diagram for the Ising model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J1, prolonged next-nearest-neighbor interactions Jp, and one-level k-tuple neighbor interaction Jo. The phase diagram is studied for several ranges of the competing parameters; it shows the appearance of several features and modulated phases arising from the frustration effects introduced by the one-level k-tuple neighbor interaction Jo. The variation of the wavevector with temperature in the modulate phase is studied in detail; it shows narrow commensurate steps between incommensurate regions. Finally, the Lyapunov exponent associated with the trajectory of the system is investigated.