Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous ising model

We prove the existence of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model on a Cayley tree and describe them constructively. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 95–104, January, 1999.     

Four competing interactions for models with an uncountable set of spin values on a Cayley tree

We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.     

Weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $$k=2$$

Theoretical and Mathematical Physics - We prove the existence of weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $$k=2$$ with respect to a normal divisor of index 4.     

Nonuniqueness of a Gibbs measure for the Ising ball model

We study a new model, the so-called Ising ball model on a Cayley tree of order k ≥ 2. We show that there exists a critical activity \(\lambda _{cr} = \sqrt[4]{{0.064}}\) such that at least one translation-invariant Gibbs measure exists for λ ≥ λ _cr , at least three translation-invariant Gibbs measures exist for 0 < λ < λ _cr , and for some λ, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor \(\hat G\) of index 2 of the group representation on the Cayley tree, we study \(\hat G\) -periodic Gibbs measures. We prove that there exists an uncountable set of \(\hat G\) -periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.     

Nonuniqueness of a Gibbs measure for a model on the Cayley tree

We propose a model on the Cayley tree and prove that a uncountable set of Ĝ-periodic Gibbs measures exists for this model, in contrast to models studied previously.     

Periodic Gibbs measures for the Potts model on the Cayley tree

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures.     

Fixed points of Lyapunov integral operators and Gibbs measures

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for models with four competing interactions and uncountable (i.e. [0, 1]) set of spin values on a Cayley tree. We prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of a fixed point.     

On a generalized p-adic Gibbs measure for Ising model on trees

In this paper we consider a p-adic Ising model on an arbitraty tree. We show the uniqueness and boundedness of the p-adic Gibbs measure for the model. Moreover, we consider translational invariant and periodic generalized p-adic Gibbs measures for the model on the Cayley tree of order two.     

Stability of the phase transition of critical-field Ising model on Cayley trees under inhomogeneous external fields

We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.     

Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree

We study translation-invariant Gibbs measures on a Cayley tree of order k = 3 for the ferromagnetic three-state Potts model. We obtain explicit formulas for translation-invariant Gibbs measures. We also consider periodic Gibbs measures on a Cayley tree of order k for the antiferromagnetic q-state Potts model. Moreover, we improve previously obtained results: we find the exact number of periodic Gibbs measures with the period two on a Cayley tree of order k ≥ 3 that are defined on some invariant sets.     

Periodic Gibbs measures for the three-state SOS model on a Cayley tree with a translation-invariant external field

Theoretical and Mathematical Physics - We consider a three-state solid-on-solid (SOS) model on a Cayley tree in the presence of an external field. We show that periodic Gibbs measures are either...     

Partition structures of the cayley tree and applications for describing periodic gibbs distributions

In the group representation of the Cayley tree, the distribution of elements of the partition into conjugate classes of finite-index, normal subgroups is described. For the inhomogeneous Ising model, it is proved that there exist only three H₀-periodic Gibbs distributions, where H₀ is a normal subgroup of finite index. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 170–175.     

A p-Adic Generalized Gibbs Measure for the Ising Model on a Cayley Tree

Theoretical and Mathematical Physics - We consider a p-adic Ising model on the Cayley tree of order k ≥ 2. We completely describe all p-adic-translation-invariant generalized Gibbs measures...     

On chaotic behaviour of the p-adic generalized Ising mapping and its application

In the present paper, by conducting research on the dynamics of the p-adic generalized Ising mapping corresponding to renormalization group associated with the p-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.     

On the Constructive Description of Gibbs Measures for the Potts Model on a Cayley Tree

Ukrainian Mathematical Journal - We consider the Potts model on a Cayley tree and prove the existence of Gibbs measures constructed by the method proposed in [H. Akin, U. A. Rozikov, and S. Temir,...     

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices 2 and 4 of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model.     

Fertile HC models with three states on a cayley tree

We consider fertile hard-core (HC) models with three states on a homogeneous Cayley tree. It is known that four types of such models exist. For these models, we describe the translation-invariant and periodic HC Gibbs measures. We also construct a uncountable set of nonperiodic Gibbs measures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 412–424, September, 2008.     

Description of weakly periodic Gibbs measures for the isingmodel on a cayley tree

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T _c < T _cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T _c or T > T _cr , three weakly periodic Gibbs measures for T = T _c , and one weakly periodic Gibbs measure for T _c < T ≤ T _cr . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 292–302, August, 2008.     

Periodic Gibbs Measures for the Ising Model with Competing Interactions

For the Ising model with competing interactions on the second-order Cayley tree, we find the operator corresponding to the periodic Gibbs distributions with period two and determine the invariant subsets of this operator, which are used to describe the periodic Gibbs distributions.     

Gibbs measures for the SOS model with four states on a Cayley tree

We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 18–31, October, 2006.