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.     

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...     

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.     

The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree

We study the q-state Potts model on a Cayley tree of order k ≥ 2. In the group representation of the Cayley tree for the ferromagnetic Potts model, we single out a set of index-2 subgroups under which each weakly periodic Gibbs measure is translation invariant. For the anti-ferromagnetic Potts model with k ≥ 2 and q ≥ 2, we show that a weakly periodic Gibbs measure that is not translation invariant is not unique.     

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

For the Potts model with competing interactions, we describe the set of weakly periodic ground states corresponding to index-two normal divisors of the Cayley tree group representation. We also study some weakly periodic Gibbs measures.     

Extremality of translation-invariant Gibbs measures for the $$\lambda$$ -model on the binary Cayley tree

Theoretical and Mathematical Physics - We consider the $$\lambda$$ -model on the Cayley tree of order $$k=2$$ . Under some conditions, we study translation-invariant Gibbs measures. Moreover, we...     

Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree

We consider the Potts model with competing interactions on the Cayley tree of order k with k ≥ 2. We describe the sets of periodic and weakly periodic ground states corresponding to normal subgroups of the group representation of the Cayley tree of index 4.     

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.     

On uniqueness of Gibbs measure for -adic countable state Potts model on the Cayley tree

In the present paper we consider countable state of p-adic Potts model on the tree. Under some condition on weights we establish uniqueness of Gibbs measures for the model. Note that this condition does not depend on values of the prime p. An analogous fact is not true when the number of spins is finite.     

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.     

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.     

Gibbs measures for fertile hard-core models on the Cayley tree

We study fertile hard-core models with the activity parameter λ > 0 and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter λ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of λ at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten–Stigum condition) for a measure not to be extreme.     

On periodic Gibbs measures of <Emphasis Type="Italic">p</Emphasis>-adic Potts model on a Cayley tree

In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.     

Translation Invariance of the Periodic Gibbs Measures for the Potts Model on the Cayley Tree

Theoretical and Mathematical Physics - We study the Potts model with a zero external field on the Cayley tree. For the antiferromagnetic Potts model with q states on a second-order Cayley tree and...     

On Gibbs measures of P-adic Potts model on the cayley tree

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ? _p) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ? p and p ≥ 3 (resp. q ? 2², p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ? p if p ≥ 3 and q ? 2² if p = 2.     

p-adic Gibbs measures for the hard core model with three states on the Cayley tree

We study p-adic hard core models with three states on the Cayley tree. It is known that there are four types of such models. We find conditions that must be imposed on the order k of the Cayley tree and on the prime p for a translation-invariant p-adic Gibbs measure to exist.     

Free energies of the Potts model on a Cayley tree

For the Potts model on the Cayley tree, we obtain some explicit formulas for the free energies and entropies in the case of vector-valued boundary conditions. These formulas include translation-invariant, periodic, and Dobrushin-like boundary conditions and also those corresponding to weakly periodic Gibbs measures.     

p-Adic Gibbs quasimeasures for the Vannimenus model on a Cayley tree

We study p-adic Gibbs quasimeasures for the Vannimenus model on the order-two Cayley tree. We especially address the problem of the boundedness of translation-invariant p-adic Gibbs quasimeasures. We also study periodic p-adic Gibbs quasimeasures.