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.     

The p-adic Potts model on the Cayley tree of order three

We study a phase transition problem for the q-state p-adic Potts model on the Cayley tree of order three. We find certain conditions for the existence of p-adic Gibbs measures and then establish the existence of a phase transition.     

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...     

Translation-invariant <Emphasis Type="Italic">p</Emphasis>-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

We consider the p-adic Ising–Vannimenus model on the Cayley tree of order k = 2. This model contains nearest-neighbor and next-nearest-neighbor interactions. We investigate the model using a new approach based on measure theory (in the p-adic sense) and describe all translation-invariant p-adic quasi-Gibbs measures associated with the model. As a consequence, we can prove that a phase transition exists in the model. Here, “phase transition” means that there exist at least two nontrivial p-adic quasi-Gibbs measures such that one is bounded and the other is unbounded. The methods used are inapplicable in the real case.     

Construction of a set of <Emphasis Type="Italic">p</Emphasis>-adic distributions

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ?_p of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions).     

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.     

<Emphasis Type="Italic">p</Emphasis>-Adic solid-on-solid model on a Cayley tree

We consider a p-adic solid-on-solid (SOS) model with a nearest-neighbor coupling, m+1 spins, and a coupling constant J ∈ Q _p on a Cayley tree. We find conditions under which a phase transition does not occur in the model. We show that if p | m + 1 for some J, then a phase transition occurs. Moreover, we formulate a criterion for the boundedness of p-adic Gibbs measures for the (m+1)-state SOS model.     

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.     

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.     

Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order $$k=3$$

Theoretical and Mathematical Physics - We study fertile hard-core models with three states and an activity parameter $$\lambda&gt;0$$ on the Cayley tree of order $$k=3$$ . It is known that...     

Von Neumann algebra corresponding to one phase of the inhomogeneous Potts model on a Cayley tree

We prove that the von Neumann algebra generated by the GNS construction associated with the extremal disordered phase of a class of inhomogeneous Potts models on a Cayley tree is a factor of type III₁.     

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...     

Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree

We study translation-invariant Gibbs measures for the ferromagnetic Potts model with q states on the Cayley tree of order k and generalize some earlier results. We consider the question of the extremality of the known translation-invariant Gibbs measures for the Potts model with three states on the Cayley tree of order k = 3.     

A description of harmonic functions via properties of the group representation of the Cayley tree

We introduce a natural generalization of the notion of harmonic functions on a Cayley tree and use some properties of the group representation of the Cayley tree to describe the set of harmonic functions periodic with respect to normal subgroups of finite index.     

Thermodynamics of interacting systems of DNA molecules

Theoretical and Mathematical Physics - We define a DNA molecule as a sequence of the numbers 1 and 2 and embed it on a path of a Cayley tree such that each vertex of the Cayley tree belongs to only...     

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本文的目的是要研究Cayley树图上奇偶马氏链场的渐近均分割性\bd 首先我们给出Cayley树 图上奇偶马氏链场关于状态和状态序偶出现频率的强大数定律, 然后证明其具有a.e.收敛性 的渐近均分割性\bd     

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.     

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.     

Harmonic models and spanning forests of residually finite groups

We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.     

Sparse <Emphasis Type="Italic">p</Emphasis>-adic data coding for computationally efficient and effective big data analytics

We develop the theory and practical implementation of p-adic sparse coding of data. Rather than the standard, sparsifying criterion that uses the L ₀ pseudo-norm, we use the p-adic norm.We require that the hierarchy or tree be node-ranked, as is standard practice in agglomerative and other hierarchical clustering, but not necessarily with decision trees. In order to structure the data, all computational processing operations are direct reading of the data, or are bounded by a constant number of direct readings of the data, implying linear computational time. Through p-adic sparse data coding, efficient storage results, and for bounded p-adic norm stored data, search and retrieval are constant time operations. Examples show the effectiveness of this new approach to content-driven encoding and displaying of data.