Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

We propose the construction of a quantum Markov chain that corresponds to a “forward” quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an XY-model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain, i.e., we show that the state is independent of the boundary conditions.     

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.     

Phase transitions for a model with uncountable spin space on the Cayley tree: the general case

Positivity - In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J...     

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.     

On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three

In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators {K_áx,y?}{\{K_{\langle x,y\rangle}\}}.     

On the scheme of rare events for a homogeneous Markov chain

We analyze the asymptotic behavior of a homogeneous, Markov chain with discrete time under a certain scheme of rare events. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

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.     

A model with uncountable set of spin values on a Cayley tree: phase transitions

In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order two. This model depends on two parameters \(n\in \mathbb N\) and \(\theta \in [0,1)\). We prove that if \( 0 \le \theta \le \frac{2n+3}{2(2n+1)}\), then for the model there exists a unique translational-invariant Gibbs measure; If \(\frac{2n+3}{2(2n+1)}< \theta <1\), then there are three translational-invariant Gibbs measures (i.e. phase transition occurs).     

Convergence rates for Markov chain order estimates using EDC criterion

The Efficient Determination Criterion (EDC) generalizes the AIC and BIC criteria and provides a class of consistent estimators for the order of a Markov chain with finite state space. In this note, we derive rates of convergence for the EDC estimates. *Partially supported by CNPq, CAPES/PROCAD, FAPDF/PRONEX, FINATEC and FUNPE/UnB. **Partially supported by CAPES.     

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.     

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...     

THE POINT PROCESS OF STATETRANSITIONS IN AREGULAR MARKOV CHAIN

1.IntroductionInreliabilitytheory,inordertocalculatethefailurefrequencyofarepairablesystem,Shily]firstintroducedandstudiedthetransitionfrequencybetweentwodisjointstatesetsforafiniteMarkovchainandavectorMarkovprocesswithfinitediscretestatespaceandobtainedageneralformulaoftransitionfrequency.Then,ontheconditionthatthegeneratormatrixofMarkovchainisuniformlybounded,Shi[8'9]againprovedthetransitionfrequencyformulaandobtainedthreeotherusefulformulas.Obviously,thepoint(orcalledcounting)processofsta…     

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.     

非齐次树上马氏链的若干强偏差定理

通过构造适当的非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,给出了一类非齐次树上马氏链场加权和滑动平均的若干强偏差定理.     

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.     

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

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.     

Phase transitions in Mandelbrot's percolation process in three dimensions

Summary We study the phase structure and transitions in three-dimensional Mandelbrot percolation—a process which generates random fractal sets. We establish the existence of three distinct phase transitions, and we show that two of these transitions, corresponding to percolation across the initial set by paths and sheets, are discontinuous.Work partially supported by NSF Grant: DMS 8806552Work partially supported by NSF Grant: DMS 8902152