# 2005

### 2005 Volume 11 Issue 1

#### H.-O. Georgii and R. TumulkaGlobal Existence of Bell's Time-Inhomogeneous Jump Process for Lattice Quantum Field Theory pp. 1-18

We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [J.S. Bell, Beables for quantum field theory, Phys. Rep., 1986, v.137, 49-54] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $|\Psi_t|^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector $\Psi_0$. The proof of non-explosion takes advantage of the special role of the $|\Psi_t|^2$ distribution.
Keywords: Markov jump processes, non-explosion, time-dependent jump rates, equi-variant distributions, Bell's process, lattice quantum field theory

#### D. PanchenkoA Note on the Free Energy of the Coupled System in the Sherrington - Kirkpatrick Model pp. 19-36

In this paper we consider a system of spins that consists of two configurations $\vec{\sigma}^1,\vec{\sigma}^2\in\Sigma_N=\{-1,+1\}^N$ with Gaussian Hamiltonians $H_N^1(\vec{\sigma}^1)$ and $H_N^2(\vec{\sigma}^2)$ correspondingly, and these configurations are coupled on the set where their overlap is fixed $\{R_{1,2}=N^{-1}\sum_{i=1}^N \sigma_i^1\sigma_i^2 = u_N\}$. We prove the existence of the thermodynamic limit of the free energy of this system given that $\lim_{N\to\infty}u_N = u\in[-1,1]$ and give the analogue of the Aizenman - Sims - Starr variational principle that describes this limit via random overlap structures.
Keywords: spin glasses, Sherrington - Kirkpatrick model

#### M. Menshikov, D. Petritis and S. PopovA Note on Matrix Multiplicative Cascades and Bindweeds pp. 37-54

We study the problem of Mandelbrot's multiplicative cascades, but with random matrices instead of random variables. Then, we introduce a new model (which we call the bindweed model), which can be viewed as a random string in a random environment on a tree, and show that the classification of this model from the point of view of positive recurrence can be obtained from the corresponding classification of the matrix-valued multiplicative cascades.
Keywords: Markov chain, trees, random environment, recurrence criteria, matrix multiplicative cascades

#### I. KurkovaFluctuations of the Free Energy and Overlaps in the High-Temperature p-Spin SK and Hopfield Models pp. 55-80

We study the fluctuations of the free energy and overlaps of $n$ replicas for the p-spin Sherrington - Kirkpatrick and Hopfield models of spin glasses in the high temperature phase. For the first model we show that at all inverse temperatures $\beta$ smaller than Talagrand's bound $\beta_p$ the free energy on the scale $N^{1-(p-2)/2}$ converges to a Gaussian law with zero mean and variance $\b^4 p!/2$; and that the law of the overlaps $\s\cdot \s'=\sum_{i=1}^{N}\s_i\s'_i$ of $n$ replicas on the scale $\sqrt{N}$ under the product of Gibbs measures is asymptotically the one of $n(n-1)/2$ independent standard Gaussian random variables. For the second model we prove that for all $\beta$ and the load of the memory $t$ with $\beta(1+\sqrt{t})<1$ the law of the overlaps of $n$ replicas on the scale $\sqrt{N}$ under the product of Gibbs measures is asymptotically the one of $n(n-1)/2$ independent Gaussian random variables with zero mean and variance $(1-t\b^2(1-\b)^{-2})^{-1}$.
Keywords: spin glasses, Sherrington - Kirkpatrick model, p-spin model, Hopfield model, overlap, free energy, martingales

#### A.I. Komech, E.A. Kopylova and N.J. MauserOn Convergence to Equilibrium Distribution for Schrodinger Equation pp. 81-110

Consider the Schrodinger equation with constant or variable coefficients in $R^3$. We study a distribution $\mu_t$ of a random solution at a time $t \in R$. An initial measure $\mu_0$ has translation-invariant correlation matrices, zero mean and finite mean density of charge. It also satisfies a Rosenblatt- or Ibragimov - Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t \to\pm\infty$ which gives the Central Limit Theorem for the Schrodinger equation. The proof for the case of constant coefficients is based on a spectral cutoff and an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's room-corridor' argument. The case of variable coefficients is reduced to that of constant ones by a version of a scattering theory for the solutions with infinite charge.
Keywords: Schrodinger equation, Cauchy problem, random initial solution, mixing condition, Gaussian measure, scattering theory, correlation functions

#### Yu.G. Kondratiev and O.V. KutoviyExistence and Some Properties of Gibbs Measures in the Continuum pp. 111-132

We develop a modified approach to the study of the existence problem for Gibbs measures of continuous systems with infinite-range pair interactions. We prove the existence of Gibbs measures possessing more a priori properties compared with [R.L. Dobrushin, Gibbsian random fields for particles without hard core, Theor. Math. Fyz., 1970, v.4, 101-118], [E. Pechersky and Yu. Zhukov, Uniqueness of Gibbs state for nonideal gas in $R^d$: The case of pair potentials, J. Stat. Phys., v. 97, 145-172]. A technical advantage of the new proposed approach is that we use appropriate compact functions which come out from the metrical structure of the finite volume configuration space.
Keywords: configuration space, Gibbs measure, existence, specification

#### M. Shcherbina and B. TirozziCentral Limit Theorems for the Free Energy of the Modified Gardner Model pp. 133-144

Fluctuations of the free energy of the modified Gardner model for any $\alpha<\alpha_c$ are studied. It is proved that they converge in distribution to a Gaussian random variable.
Keywords: central limit theorem, Gardner problem

#### A.E. KyprianouAsymptotic Radial Speed of the Support of Supercritical Branching Brownian Motion and Super-Brownian Motion in $R^d$ pp. 145-156

It has long been known that the left-most or right-most particle in a one dimensional dyadic branching Brownian motion with constant branching rate $\beta >0$ has almost sure asymptotic speed $\sqrt{2\beta }$, (cf. [H.P. McKean, Application of Brownian motion to the equation of Kolmogorov - Petrovskii - Piscounov, Comm. Pure and Appl. Math., 1975, v.128, 323-331]). Recently similar results for higher dimensional branching Brownian motion and super-Brownian motion have also been established but in the weaker sense of convergence in probability; see [R.G. Pinsky, On large time growth rate of the support of supercritical super-Brownian motion, Ann. Prob., 1995, v.23, 1748-1754] and [J. Englander and F. den Hollander, On branching Brownian motion in a Poissonian trap field, Markov Processes Relat. Fields, 2003, v.9, 363-389. In this short note we confirm the folklore' for higher dimensions and establish an asymptotic radial speed of the support of the latter two processes in the almost sure sense. The proofs rely on Pinsky's local extinction criterion, martingale convergence, projections of branching processes from higher to one dimensional spaces together with simple geometrical considerations.
Keywords: spatial branching processes, super-Brownian motion, branching Brownian motion, local extinction

#### S. Albeverio and S. LiangA Limit Theorem for the Wick Exponential of the Free Lattice Fields pp. 157-164

Let $G_a$ be the free lattice field measure of mass $m_0$ on $a Z^2$, and let $:\exp\{\alpha \phi_x\}:$ be the corresponding Wick exponential of the lattice field $\phi_x$. Let $A \subset R^2$ be a bounded region and $a'(a) \ge a$ satisfy: $\lim_{a \to 0} a'(a) = 0$. In this paper, a limit theorem for the distribution of $a'^2 \sum_{x \in a' Z^2 \cap A} :\exp\{\alpha \phi_x\}:$ under $G_a$ is given, under the condition $\lim_{a \to 0} a'^4 |\log a| = \infty$. The corresponding problem for the $:\phi_x^4:$-field has been studied by Albeverio and Zhou [S. Albeverio and X.Y. Zhou, A central limit theorem for the fourth Wick power of the free lattice field, Commun. Math. Phys., 1996, v.181, no.1, 1-10].
Keywords: free lattice fields, continuum limit, asymptotics of random fields, extreme events, i.i.d. random variables, quantum fields

#### D. DenisovA Note on the Asymptotics for the Maximum on a Random Time Interval of a Random Walk pp. 165-169

Let $\xi,\xi_1$, $\xi_2$, \ldots\ be independent random variables with a common distribution $F$ and negative mean $\E\xi=-m$. Consider the random walk $S_0=0$, $S_n=\xi_1+\cdots+\xi_n$, the stopping time $\tau = \min\{n\ge 1: S_n\le 0\}$ and let $M_\tau=\max_{0\le i\le \tau} S_i.$ S. Asmussen found the asymptotics of $\P(M_\tau>x)$ as $x \to\infty$, when $F$ is subexponential. We give a short proof of this result. We also provide comments on the light-tailed case.
Keywords: random walk, cycle maximum, heavy-tailed distribution, stopping time

### 2005 Volume 11 Issue 2

#### Inhomogeneous Random Systems: Quantum Information Theory. Low Energy States in Quantum Many Body Systems. Interacting Particle Systems: New Trends, with Applications in Biology and Economy pp. 171-176

The present issue of Markov Processes and Related Fields contains papers presented at the two meetings Inhomogeneous Random Systems'' held at the University of Cergy-Pontoise, on January 28-29, 2003, and at the Institut Henri Poincare, Paris, on January 28, 2004. These meetings, which concentrate each year on different topics, bring together an interdisciplinary audience of mathematicians and physicists.

#### P. Briet, H.D. Cornean and D. LouisGeneralized Susceptibilities for a Perfect Quantum Gas pp. 177-188

The system we consider here is a charged fermions gas in the effective mass approximation, and in grand-canonical conditions. We assume that the particles are confined in a three dimensional cubic box $\Lambda$ with side $L\geq 1$, and subjected to a constant magnetic field of intensity $B \geq 0$. Define the grand canonical generalized susceptibilities $\chi_L^N$, $N\geq 1$, as successive partial derivatives with respect to $B$ of the grand canonical pressure $P_L$. Denote by $P_{\infty}$ the thermodynamic limit of $P_L$. Our main result is that $\chi_L^N$ admit as thermodynamic limit the corresponding partial derivatives with respect to $B$ of $P_{\infty}$. In this paper we only give the main steps of the proofs, technical details will be given elsewhere.
Keywords: quantum gas, magnetic field, thermodynamic limit

#### P. CaputoEnergy Gap Estimates in XXZ Ferromagnets and Stochastic Particle Systems pp. 189-210

This expository article is a survey of recent results [P. Caputo and F. Martinelli, Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model. Commun. Math. Phys., 2002, v.226, 323-375. P. Caputo and F. Martinelli, Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models. Ann. Appl. Probab., 2003, v.13, no.2, 691-721] on the energy gap above the interface ground states of XXZ ferromagnets. Main ideas and techniques are reviewed with special emphasis on the equivalence between the quantum spin models and classical stochastic particle systems.
Keywords: simple exclusion, Heisenberg model, quantum interfaces, spectral gap

#### O. Bolina, P. Contucci and B. NachtergaelePath Integral Representations for the Spin-Pinned Quantum XXZ Chain pp. 211-221

Two discrete path integral formulations for the ground state of a spin- pinned quantum anisotropic XXZ Heisenberg chain are introduced. Their properties are discussed and two recursion relations are proved.
Keywords: Heisenberg model, path integral representation, ground state, pinned spin

#### T. KennedyExpansions for Droplet States in the Ferromagnetic XXZ Heisenberg Chain pp. 223-236

We consider the highly anisotropic ferromagnetic spin 1/2 Heisenberg chain with periodic boundary conditions. In each sector of constant total z component of the spin, we develop convergent expansions for the lowest band of eigenvalues and eigenfunctions. These eigenstates describe droplet states in which the spins essentially form a single linear droplet which can move. Our results also give a convergent expansion for the dispersion relation, i.e., the energy of the droplet as a function of its momentum. The methods used are from [N. Datta and T. Kennedy, Expansions for one quasiparticle states in spin 1/2 systems, J. Stat. Phys., 2002, v.108, 373-399, arXiv:cond-mat/0104199] and [N. Datta and T. Kennedy, Instability of interfaces in the antiferromagnetic XXZ chain at zero temperature, Commun. Math. Phys., 2003, v.236, 477-511, arXiv:math-ph/0208026], and this short paper should serve as a pedagogic introduction to those papers.
Keywords: droplets, XXZ ferromagnet, quantum spin chain

#### T. Michoel and B. NachtergaeleThe Large-Spin Asymptotics of the Ferromagnetic XXZ Chain pp. 237-266

We present new results and give a concise review of recent previous results on the asymptotics for large spin of the low-lying spectrum of the ferromagnetic XXZ Heisenberg chain with kink boundary conditions. Our main interest is to gain detailed information on the interface ground states of this model and the low-lying excitations above them. The new and most detailed results are obtained using a rigorous version of bosonization, which can be interpreted as a quantum central limit theorem.
Keywords: XXZ chain, Heisenberg ferromagnet, large-spin limit, bosonization

#### V.A. ZagrebnovOne-Mode Bose - Einstein Condensation and Bogoliubov Theory pp. 267-282

The first ansatz of the Bogoliubov theory is based on hypothesis of existence of the one-mode Bose - Einstein (BE) condensation and on its stability with respect to two-body particle interaction. We present here some rigorous results, which are relevant to this hypothesis.
Keywords: Bose - Einstein condensation, Bogoliubov weakly imperfect gas, one-particle excitations, generalized condensation

#### C. MaesNew Trends in Interacting Particle Systems pp. 283-288

Since about a decade some new trends in the study of interacting particle systems can be identified. New is meant here with respect to some standard references and plays both in new mathematical challenges and in new models or phenomena. Examples are collected in the present volume. I give them here a short introduction.
Keywords: interacting particle systems, stochastic dynamics

#### B. DerridaStatistical Properties of Genealogical Trees pp. 289

One can associate to the genealogical tree of each individual the distribution of repetitions of his ancestors, at a number $g$ of generations in the past. By solving a simple model of a population of constant size $N$ with random mating, one can show that this distribution of repetitions reaches a stationary shape, with a non trivial power law which can be calculated analytically. By comparing the genealogical trees of two distinct individuals, one can also show that the two trees become identical within a number $\log N$ of generations.
Keywords: genealogical tree, distribution of repetitions of ancestors

#### P. HogewegMultilevel Particle Systems and the Study of Biological Evolution pp. 291-312

Biotic systems are preeminently multilevel systems in which processes at different space- and time-scales interact. In order to study such processes we use particle based systems in two ways. First we review results in which larger scale spatial patterns formed by local interaction between the model particles feed back, via Darwinian evolution, on the local interactions. The conclusion from this work is that multiple levels of selection occur, and that the basic properties of earlier recognized major transition in evolution'' are generically exhibited in such model systems. Secondly we examine particle' models which are defined as multilevel systems. The particles' here extend over many grid cells, and are flexible in shape. Shape changes take place at the grid-level, however these changes are co-determined by properties of the extended particles. We use this formalism to study two examples of major transitions in evolution in more detail. We demonstrate the power of this formulation for modeling multilevel evolution and development.
Keywords: multilevel particle systems, biomathematical modelling, stochastic processes, spatial self-organization, cellular automata

#### A.A. JaraiThermodynamic Limit of the Abelian Sandpile Model on $Z^d$ pp. 313-336

We review basic properties of the Abelian sandpile model and describe recent progress made regarding its infinite volume limit on $Z^d$. In particular, we discuss the existence of the infinite volume limit of the stationary measure for $d \ge 2$, existence of infinite volume addition operators for $d \ge 3$, and construction of an infinite volume process for $d \ge 5$. We give an overview of the techniques relevant for these constructions.
Keywords: Abelian sandpile model, uniform spanning tree, thermodynamic limit, waves, two-component spanning forest

#### V.A. MalyshevFixed Points for Stochastic Open Chemical Systems pp. 337-354

In the first part of this paper we give a short review of the hierarchy of stochastic models, related to physical chemistry. In the basement of this hierarchy there are two models - stochastic chemical kinetics and the Kac model for Boltzman equation. Classical chemical kinetics and chemical thermodynamics are obtained as some scaling limits in the models, introduced below. In the second part of this paper we specify some simple class of open chemical reaction systems, where one can still prove the existence of attracting fixed points. For example, Michaelis - Menten kinetics belongs to this class. At the end we present a simplest possible model of the biological network. It is a network of networks (of closed chemical reaction systems, called compartments), so that the only source of nonreversibility is the matter exchange (transport) with the environment and between the compartments.
Keywords: chemical kinetics, chemical thermodynamics, Kac model, mathematical biology

#### R. Meester and C. QuantConnections Between Self-Organised' and Classical' Criticality pp. 355-370

We investigate the nature of the self-organised critical behaviour in the Abelian sandpile model and in the Bak - Sneppen evolution model. We claim that in either case, the self-organised critical behaviour can be explained by the careful choice of the details of the model: they are designed in such a way that the models are necessarily attracted to the critical point of a conventional parametrised equilibrium system. In the case of the Abelian sandpile we prove this connection to conventional criticality rigorously in one dimension, and provide evidence for a similar result in higher dimensions. In the case of the Bak - Sneppen evolution model, we give an overview of the current results, and explain why these results support our claim. We conclude that the term self-organised criticality is somewhat confusing, since the tuning of parameters in a model has been replaced by the careful choice of a suitable model. Viewed as such, we can hardly call this critical behaviour spontaneous.
Keywords: self-organised criticality, Bak - Sneppen evolution model, Abelian sandpile, classical criticality, particle system

#### J. MiekiszLong-Run Behavior of Games with Many Players pp. 371-388

We discuss similarities and differences between systems of many interacting players maximizing their individual payoffs and particles minimizing their interaction energy. We analyze long-run behavior of stochastic dynamics of many interacting agents in spatial and adaptive population games. We review results concerning the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. In particular, we present examples of games in which when the number of players or the noise level increases, a population undergoes a transition between its equilibria.
Keywords: evolutionary game theory, Nash equilibria, stochastic adaptive dynamics, spatial games, equilibrium selection, long-run behavior, stochastic stability, ensemble stability

#### L. TrioloSpace Structures and Different Scales for Many-Component Biosystems pp. 389-404

The right scale of description, and more generally, the matching between different scales, in modelling physical or biological systems, presents a substantial interest and related difficulties. This applies for instance to competition/diffusion models of different species, or to the invasion process by an aggressive population against a native one, like tumor cells in a normal tissue. Under a suitable scaling the system may be described by PDE's where space structures are visible, as is shown and discussed here in a tumor growth model. On the other side, the separation of scales may be not so sharp: events occurring in microscopic" regions may affect the macroevolution considerably. This is shown here in a model of metastatic spreading which can be studied like a collection of proliferating" random walks on the positive one-dimensional lattice.
Keywords: scaling limits, stochastic modelling, tumor growth model

### 2005 Volume 11 Issue 3

#### A. Dermoune and P. HeinrichEquivalence of Ensembles for Coloured Particles in a Disordered Lattice Gas pp. 405-424

We consider a system of coloured particles in $Z^d$ driven by a disordered Markov generator similar to the one of [A. Faggionato and F. Martinelli, Hydrodynamic limit of a disordered lattice gas. Probab. Theory and Relat. Fields, 2003, 127, 535-608]. We establish the closeness between grand-canonical and canonical measures.
Keywords: lattice gas, equivalence of ensembles, disordered system, Markov process

#### A. Fey-den Boer and F. RedigOrganized Versus Self-Organized Criticality in the Abelian Sandpile Model pp. 425-442

We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the Abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable.
Keywords: self-organized criticality, Abelian sandpile model, activated random walkers, stabilizability

#### P. Collet, D. Duarte and A. GalvesBootstrap Central Limit Theorem for Chains of Infinite Order via Markov Approximations pp. 443-464

We present a new approach to the bootstrap for chains of infinite order taking values on a finite alphabet. It is based on a sequential Bootstrap Central Limit Theorem for the sequence of canonical Markov approximations of the chain of infinite order. Combined with previous results on the rate of approximation this leads to a Central Limit Theorem for the bootstrapped estimator of the sample mean which is the main result of this paper.
Keywords: bootstrap, chains of infinite order, canonical Markov approximations, central limit theorem

#### D. KhmelevOn Convergence to Equilibrium of Infinite Closed Jackson Networks pp. 467-488

This paper studies an infinite asymmetric close Jackson network, which is a generalization of a zero range interaction process at Bose - Einstein speeds. The network is expected to approach equilibrium if deterministic initial configuration has density. It is shown that in some networks on $\mathbf{N}$ this is indeed the case, moreover, there exist irregular initial configurations such that marginal distributions of the network oscillate with time. However one can construct examples of two networks on $\mathbf{Z}$ such that their finite restrictions share the same invariant distributions, while the infinite networks starting from the same initial configuration converge to different invariant distributions.
Keywords: interacting particle systems, zero range process, convergence to equilibrium

#### A. Manita and V. ShcherbakovAsymptotic Analysis of a Particle System with Mean-Field Interaction pp. 489-518

We study a system of N interacting particles on $\mathbf{Z}$. The stochastic dynamics consists of two components: a free motion of each particle (independent random walks) and a pairwise interaction between particles. The~interaction belongs to the class of mean-field interactions and models a rollback synchronization in asynchronous networks of processors for a distributed simulation. First of all we study an empirical measure generated by the particle configuration on $\mathbf{R}$. We prove that if space, time and a parameter of the interaction are appropriately scaled (hydrodynamical scale), then the empirical measure converges weakly to a deterministic limit as $N$ goes to infinity. The limit process is defined as a weak solution of some partial differential equation. We also study the long time evolution of the particle system with fixed number of particles. The Markov chain formed by individual positions of the particles is not ergodic. Nevertheless it is possible to introduce relative coordinates and prove that the new Markov chain is ergodic while the system as a whole moves with an asymptotically constant mean speed which differs from the mean drift of the free particle motion.
Keywords: particle systems, hydrodynamic limit, martingale problem, ergodicity

#### K. Fleischmann and J. XiongLarge Deviation Principle for the Single Point Catalytic Super-Brownian Motion pp. 519-533

In the single point catalytic super-Brownian motion particles'' branch only if they meet the position of the single point catalyst. If the branching rate tends to zero, the model degenerates to the heat flow. We are concerned with large deviation probabilities related to this law of large numbers. To this aim the well-known explicit representation of the model by excursion densities is heavily used. The rate function is described by the Fenchel - Legendre transform of log-exponential moments described by a log-Laplace equation.
Keywords: point catalyst, superprocess, large deviations, exponential moments, singular catalytic medium, log-Laplace equation, representation by excursion densities

#### E. JarpeAn Ising-Type Model for Spatio-Temporal Interactions pp.535-552

A model which possesses both spatial and time dependence is the Markov chain Markov field [X. Guyon, Random Fields on a Network, Springer-Verlag, New York, 1995]. Here inference about the parameter for spatio-temporal interaction of a special case of a Markov chain Markov field model is considered. A statistic which is minimal sufficient for the spatio-temporal interaction parameter and its asymptotic distribution are derived. A condition for stationarity of the sufficient statistic process and the stationary distribution are given. Likelihood based inference methods such as estimation, hypothesis testing and monitoring are briefly examined.
Keywords: asymptotic distribution, interaction, Markov chain Markov field, perfect simulation, stationarity, sufficient statistic

### 2005 Volume 11 Issue 4

#### A. Gaudilliere, E. Olivieri and E. Scoppola Nucleation Pattern at Low Temperature for Local Kawasaki Dynamics in Two Dimensions pp.553-628

We study the first transition between metastability and stability for a two-dimensional Ising lattice gas evolving at low temperature under a local version of the Kawasaki conservative dynamics. We describe geometrically the configurations along paths typically followed during the transition, and show that the whole evolution goes with high probability from quasi-squares' to larger quasi-squares'. Moreover, along these paths, between two successive quasi-squares', the fluctuations in the dimensions of the clusters are bounded: if an $l\times L$ rectangle, with $l \leq L$, circumscribes one of these clusters then we have $L-l \leq 1+2\sqrt{L}$. Finally we show that fluctuations of this order cannot be neglected: such fluctuations occur with a probability non-exponentially small' in the inverse temperature $\beta$. This nucleation process thus substantially differs from that which takes place under the Glauber dynamics, especially in its supercritical part.
Keywords: metastability, conservative dynamics, Kawasaki dynamics, nucleation

#### W. Hachem, P. Loubaton and J. Najim The Empirical Eigenvalue Distribution of a Gram Matrix: From Independence to Stationarity pp.629-648

Consider a $N\times n$ matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary Gaussian random field: $Z^n_{j_1 j_2} = \frac{1}{\sqrt{n}} \sum_{(k_1,k_2) \in \mathrm{Z}^2} h(k_1,k_2)\, U(j_1-k_1,j_2-k_2),$ where $h\in\ell^1(\mathbf{Z}^2)$ is a deterministic complex summable sequence and $(U(j_1,j_2)$; $(j_1,j_2)\in \mathbf{Z}^2)$ is a sequence of independent complex Gaussian random variables with mean zero and unit variance. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n )^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $n\rightarrow \infty$ and $N/n \rightarrow c \in (0,\infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature [A. Boutet de Monvel, A. Khorunzhy and V. Vasilchuk, Limiting eigenvalue distribution of random matrices with correlated entries. Markov Processes Relat. Fields, 1996, v. 2, N4, 607-636. V.L. Girko, Theory of Stochastic Canonical Equations. Vol. I. Volume 535 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2001], etc.
Keywords: random matrix, empirical eigenvalue distribution, Stieltjes transform

#### L.R.G. Fontes, M. Vachkovskaia and A. Yambartsev A Dynamical Surface Interacting with Rarefied Walls pp.649-660

We consider the motion of a discrete random surface interacting by exclusion with a rarefied wall. The dynamics is given by the serial harness process. We prove that the process delocalizes iff the mean number of visits to the set of sites where the wall is present by a random walk is infinite. When the surface delocalizes, bounds on its average speed are obtained.
Keywords: harness process, surface dynamics, entropic repulsion, random environment

#### F. Zhang Ergodicity and Reversibility of the Minimal Diffusion Process pp.661-676

In this paper, Prigogine's non-equilibrium steady state is modelled by a minimal diffusion process with only smoothness condition on the coefficients. We show that for such a process, the time reversal process is still a minimal diffusion process. We define an entropy production rate and provide three equivalent conditions of the reversibility of the process. Thus the existence of a reversible probability measure may be checked directly from the coefficients. For this paper to be self-contained, we first recall the construction of a minimal diffusion process whose Markov semigroup acts on a separable Banach space, which could not be the space of functions vanishing at infinity $C_0 (R^d)$. Since the coefficients may increase rapidly, $C_0 (R^d)$ is not invariant under the Markov semigroup. For this process, we show the Foguel alternative, i.e. the ergodicity of the Markov semigroup. Moreover, the ergodic theorem in the path space of a stationary process holds.
Keywords: diffusion process, ergodicity, irreversibility, entropy production rate

#### C. Albanese and S. Lawi Laplace Transforms for Integrals of Markov Processes pp.677-724

Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein - Uhlenbeck, the Cox - Ingerssol - Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitional probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree.
Keywords: Laplace transforms, diffusion processes, continuous-time Markov chains, closed form solutions, hypergeometric functions, orthogonal polynomials, spectral expansions, Askey scheme