# 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