Ergodicity of non-equilibrium Glauber dynamics in continuum

We study asymptotic properties of the continuous Glauber dynamics with unbounded death and constant birth rates. In particular, an information about the location of the spectrum for the symbol of the Markov generator is obtained. The latter fact is used for the proof of the ergodicity of this process. We show that the speed of convergence to the equilibrium is exponential.     

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

On non-equilibrium stochastic dynamics for interacting particle systems in continuum

We propose a general scheme for construction of Markov stochastic dynamics on configuration spaces in continuum. An application to the Glauber-type dynamics with competitions is considered.     

Holomorphic Bogoliubov functionals for interacting particle systems in continuum

Combinatorial harmonic analysis techniques are used to develop new analytical methods for the study of interacting particle systems in continuum based on a Bogoliubov functional approach. Concrete applications of the methods are presented, namely, conditions for the existence of Bogoliubov functionals, a uniqueness result for Gibbs measures in the high temperature regime. We also propose a new approach to the study of non-equilibrium stochastic dynamics in terms of evolution equations for Bogoliubov functionals.     

Mixing of the Glauber dynamics for the ferromagnetic Potts model

We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely‐matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ‐regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 21–52, 2016     

The mixing time of Glauber dynamics for coloring regular trees

We consider Metropolis Glauber dynamics for sampling proper q‐colorings of the n‐vertex complete b‐ary tree when 3 ≤ q ≤ b/(2lnb). We give both upper and lower bounds on the mixing time. Our upper bound is n^{O(b/ log b)} and our lower bound is n^{Ω(b/(q log b))}, where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Existence of equilibrium for infinite system of interacting diffusions

We develop and implement new probabilistic strategy for proving basic results about long-time behavior for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process as a solution to suitable infinite-dimensional martingale problem. However, the techniques allow us to consider cases where the generator of the particle is degenerate elliptic operator. As a model example, we present the situation where the operator arises from Heisenberg group. In the last section, we provide further examples that can be handled using our methods.     

Fluctuation analysis for the loss from default

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy and computational efficiency of the approximation.     

The limit theorems for maxima of stationary Gaussian processes with random Index

Let {X(t), t ≥ 0} be a standard(zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(TT) exists under some additional conditions related to the correlation function r(·).     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Characterization of equilibrium measures for critical reversible Nearest Particle Systems

We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than and obeys some natural regularity conditions. The research of this author is partially supported by the FNS Grant #200021-107475/1     

On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Critical Value of the Contact Process with Added and Removed Edges

We show that the critical value for the contact process on a vertex-transitive graph with finitely many edges added and/or removed is the same as the critical value for the contact process on . This gives a partial answer to a conjecture of Pemantle and Stacey.     

Uniqueness of limit cycles in predator-prey system: the role of weight functions

We consider a Gause type model of interactions between predator and prey populations. Using the ideas of Cheng and Liou we give a sufficient condition for uniqueness of the limit cycle, which is more general than their condition. That is, we include a kind of weight function in the condition. It was motivated by a result due to Hwang, where the prey isocline plays a role of weight function. Moreover, we show that the interval where the condition from Hwang's result is to be fulfilled can be narrowed.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

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A limit theorem for the argument of zeta-functions of certain cusp forms

We obtain a limit theorem for the modulus of the argument of zeta-functions near the critical line of normalized eigenforms. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 501–512, October–December, 2006.     

A two-dimensional limit theorem for Mellin transforms of the Riemann zeta-function

In this paper, we obtain a two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for Mellin transforms of the square and the fourth power of the Riemann zeta-function. Partially supported by the Lithuanian Foundation of Studies and Science.     

Estimating an interaction parameter of an infinite particle system

This article deals with the estimation of a parameter in the stochastic motion affecting an infinite number of particles. An estimator, based on a nonstationary time-series is considered and shown to be consistent. A comparison with more well-known estimates, via asymptotic variances, is also carried out.