2004

2004
Volume 10
Issue 1


O. Hryniv and D. Ioffe
Self-Avoiding Polygons: Sharp Asymptotics of
Canonical Partition Functions Under the Fixed Area Constraint
 pp. 1-64

We study the self-avoiding polygons (SAP) connecting the
vertical and the horizontal semi-axes  of the
positive quadrant of $\Z^2$. For a fixed $\beta>0$, assign to
each such polygon $\omega$ the weight $\exp\{-\beta |\omega|\}$, 
$|\omega|$ denoting the length of $\omega$, and consider
the sum ${\cal Z}_{Q,+}$ of these weights for all SAP enclosing area
$Q>0$. We study the statistical properties of such SAP and, in
particular, derive the exact asymptotics for the partition function
${\cal Z}_{Q,+}$ as $Q\to\infty$.
The results are valid for any $\beta>\beta_c$, $\beta_c$ being the
critical value for the 2D self-avoiding walks.
Keywords: self-avoiding random walks, phase boundaries,
sharp local limit theorems, equilibrium crystal shapes

A. Le Ny and F. Redig
Large Deviation Principle at Fixed Time in Glauber Evolutions
 pp. 65-74

We consider the evolution of an asymptotically
decoupled probability measure $\nu$ on Ising spin configurations
under a Glauber dynamics. We prove that for any $t>0$, $\nu_t$ is
asymptotically decoupled and hence satisfies a large deviation
principle with the relative entropy density as rate function.
Keywords: Glauber dynamics, large deviations, abstract cluster expansion

C. Kulske
Regularity Properties of Potentials for Joint Measures
of Random Spin Systems
 pp. 75-88

We consider general quenched disordered lattice spin models on compact
local spin spaces with possibly dependent disorder. We discuss their
corresponding joint measures on the product space of disorder variables
and spin variables in the infinite volume. These measures often possess 
pathologies in a low temperature region reminiscent of renormalization 
group pathologies in the sense that they are not Gibbs measures on the 
product space. Often the joint measures are not even almost Gibbs, but 
it is known that there is always a potential for their conditional 
expectations that may however only be summable on a full measure set, and
not everywhere. In this note we complement the picture from the non-pathological 
side. We show regularity properties for the potential in the region of 
interactions where the joint potential is absolutely summable everywhere.
We prove unicity and Lipschitz-continuity, much in analogy to the two 
fundamental regularity theorems proved by van Enter, Fernandez, Sokal for
renormalization group transformations.
Keywords: disordered systems, Gibbs measures, non-Gibbsian measures, 
joint measures, random field model

O. Benois, R. Esposito, R. Marra and M.Mourragui
Hydrodynamics of a Driven Lattice Gas with Open Boundaries: the Asymmetric Simple Exclusion
 pp. 89-112

We consider the asymmetric simple exclusion process in $d\ge 3$ with
open boundaries. The particle reservoirs of constant densities are
modeled by birth and death processes at the boundary. We prove that,
if the initial density and the densities of the boundary reservoirs
differ for order of $\epsilon$ from $1/2$, the empirical density field,
rescaled as $\epsilon^{-1}$, converges to the solution of the
initial-boundary value problem for the viscous Burgers equation in a
finite domain with given density on the boundary.
Keywords: interacting particle systems, open systems, hydrodynamic limit

P. Dai Pra and S. Roelly
An Existence Result for Infinite-Dimensional 
Brownian Diffusions with Non-Regular and Non-Markovian Drift
 pp. 113-136

We prove  in this paper an existence result for infinite-dimensional
stationary  interactive Brownian diffusions. The interaction is
supposed to be small in the norm $\|\cdot\|_{\infty}$ but otherwise
is very general, being possibly non-regular and non-Markovian.
Our method consists in using the characterization of such diffusions
as space-time Gibbs fields so that we construct them by
space-time cluster expansions in the small coupling parameter.
Keywords: infinite-dimensional Brownian diffusion, space-time Gibbs field, cluster expansion

M.V. Menshikov, S.Yu. Popov, V. Sisko and M. Vachkovskaia
On a Many-Dimensional Random Walk in a Rarefied Random Environment
 pp. 137-160

We consider a modification of the Simple Random Walk (SRW), which can be
described as follows. Initially, any $x\in\Z^d$ becomes ``special''
with probability $p(x)$; then, in all special sites
we modify the transition probabilities in order to create
a drift which is directed outwards the origin (in the case of one-
or two-dimensional  SRW) or towards the origin (for higher dimensions),
thus constructing a random environment.
Then, based on the asymptotic behaviour of the function $p(x)$,
we give some sufficient conditions for transience and recurrence.
Keywords: recurrence, transience, Lyapunov functions, random
environment

L.M. Morato
Some New Conditions for the Existence of Singular 
Non-Symmetric Diffusions
pp. 161-174

Some new conditions which ensure the existence of diffusion
processes with values in $\R ^{d}$ properly associated to
non-symmetric Dirichlet forms are given. The results are extended
to the case of diffusions taking values in Wiener spaces. Some of
them can be expressed in terms of dynamical potentials appearing
in Schroedinger operators so that they are suitable for
application to Stochastic Mechanics both in finite and infinite
dimension.
Keywords: non-symmetric diffusions, Dirichlet forms, Hamiltonians

K. Duffy and W.G. Sullivan
Logarithmic Asymptotics for Unserved Messages at a FIFO
 pp. 175-189

We consider an infinite-buffered single server First In, First Out
(FIFO) queue. Messages arrive at stochastic intervals and take
random amounts of time to process. Logarithmic asymptotics are
proved for the tail of the distribution of the number of messages
awaiting service, under general large deviation and stability
assumptions, and formulae presented for the asymptotic decay rate.
Keywords: FIFO-system, logarithmic asymptotics, large deviations




2004
Volume 10
Issue 2


C. Tremoulet
Equilibrium Fluctuations for an Interacting Brownian Particles Process
 pp. 191-215

In this paper, we study the equilibrium fluctuations
for an interacting Brownian particles process. The
interaction between particles is given by a two-body
superstable potential. The result is old and it is
due to Spohn, [H. Spohn, Equilibrium fluctuations for interacting
 Brownian particles, Commun. Math. Phys., 1986, v. 103, 1-33]. 
He proved that the time-dependent density fluctuations field converges
in law to the solution of a stochastic partial differential
equation driven by white noise. In this paper, we give a new
and simpler proof introduced by Chang in [C.-C. Chang, Equilibrium fluctuations of gradient reversible
particles systems, Probab. Theory and Relat. Fields, 1994, v. 100, 269-283]. Moreover,
in [R. Lang, Unendlich-dimensionale Wienerprozesse mit
Wechselwirkung Teil I, Z. Wahrsch. verw. Geb., 1977, v. 38, 819-834], the existence of the dynamics of the process is proven.
But, in this paper, we also add a new and less difficult proof of the
existence of the dynamics at equilibrium.
Keywords: interacting particles process, equilibrium fluctuations, bulk diffusion

M. Deijfen, O. Haggstrom and J. Bagley
A Stochastic Model for Competing Growth on R^d. 
 pp. 217-248
A stochastic model, describing the growth of two
competing infections on $R^d$, is introduced. The growth
is driven by outbursts in the infected region, an outburst in the
type 1 (2) infected region transmitting the type 1 (2) infection
to the previously uninfected parts of a ball with stochastic
radius around the outburst point. The main result is that with the
growth rate for one of the infection types fixed, mutual unbounded
growth has probability zero for all but at most countably many
values of the other infection rate. This is a continuum analog of
a result of Haggstrom and Pemantle. We also extend a shape
theorem of Deijfen for the corresponding model with just one type
of infection.
Keywords: spatial spread, Richardson's model, shape theorem, 
competing growth

T. Kuneth
Large Deviations for Random Fields on $Z^d$ with Unbounded Interaction
 pp. 249-288

We prove a Large Deviation Principle (LDP) for the empirical
fields of Gibbs random fields with arbitrary state space. The
stationary manybody interaction $\varphi$ may be unbounded, but
must satisfy strong regularity and stability conditions. The
underlying topology on the set of all stationary probability
measures is defined by the local functions with a suitable growth
condition. Along the way we prove existence and various properties
of specific energy and entropy.
Keywords: large deviations, Gibbs fields, unbounded interaction

E.E. Dyakonova, J. Geiger and V.A. Vatutin
On the Survival Probability and a Functional Limit Theorem 
for Branching Processes in Random Environment 
 pp. 289-306

Let $(Z_n)_{n\ge 0}$ be a  branching process in
i.i.d. random environment. We consider a generalization of the so-called
critical case assuming  that the distribution of the logarithmic
conditional mean offspring number is attracted without centering
 to a stable law.  We show that subject to moment assumptions
the exact asymptotics of $P\{Z_n>0\}$
 is proportional to $n^{-(1-\rho)} L_1(n)$,  where
$\rho\in (0,1)$ can be expressed in terms of the index and the skewness
 parameter of the attracting stable
law and $L_1$ is some slowly varying function. Moreover, we prove
 a conditional functional limit
law for the suitably rescaled generation size process.
Keywords: branching process, random environment,
survival probability, functional limit theorem,
conditioned random walk

E. Zhizhina
Convergence Properties of Quasi-Particles of
Various Species in the Stochastic Blume-Capel Model
 pp. 307-326

We construct two leading invariant subspaces of the generator of
the stochastic Blume - Capel model under high temperatures,
describing states of two various quasi-particles.
We show that the corresponding spectrum branches of the generator
occupy small vicinities of the point $-1$ and prove that the restrictions of the
generator to these invariant subspaces have different spectral gaps.
Keywords: Glauber dynamics, invariant subspaces of the
generator, spectral gap

Y. Kovchegov
The Brownian Bridge Asymptotics in the Subcritical 
Phase of Bernoulli Bond Percolation Model
 pp. 327-344

For a given point $\vec{\mathbf{a}}$ in $Z^d$, we prove that a cluster
in the $d$-dimensional subcritical Bernoulli bond percolation model conditioned
on connecting points $(0,\dots,0)$ and $n \vec{\mathbf{a}}$ if scaled by
$1 / (n \| \vec{\mathbf{a}} \|)$ along $\vec{\mathbf{a}}$ and by
$1 / \sqrt{n}$ in the orthogonal directions converges
asymptotically to Time $\times$ ($d-1$)-dimensional Brownian bridge.
Keywords: percolation, Brownian bridge, cluster

P. Eichelsbacher and M. Lowe
Moderate Deviations for a Class of Mean-Field Models
 pp. 345-366

We derive moderate deviation principles for partial sums $S_n$ for triangular
arrays of dependent random variables, known as Curie - Weiss models. Outside the
critical inverse temperature $\beta_c=1$ we obtain a quadratic rate function, at the critical temperature
the rate function is non-Gaussian: under appropriate assumptions on the underlying measure
there exists a positive real number $\lambda$, and a positive integer $k$ such that
$S_n/n^{\alpha}$ satisfies a moderate deviations principle with speed $n^{1-2k+2 \alpha k}$ and rate function
$- \lambda x^{2k}/(2k!)$ for every $ 1- 1/(2k) < \alpha <1$. Moreover, we analyze the moderate
deviations behaviour as the temperature $1/\beta_n$ converges to one and obtain a threshold for the
speed of this convergence to one: if $\beta_n$ converges to $\beta_c$ fast enough,
faster than ${\mathcal O}(n^{2 \alpha (k-1)+ 2 -2k})$, then the non-Gaussian rate function persists,
whereas for $\beta_n$ converging to one slowly, slower than ${\mathcal O}(n^{2 \alpha (k-1)+ 2 -2k})$,
the moderate deviations principle
is given by the Gaussian rate. At the borderline the moderate deviation rate function
is the one at criticality plus an additional term.
Keywords: moderate deviations, mean-field, Curie - Weiss model, critical temperature,
large deviations

N. Konno, R.B. Schinazi and H. Tanemura
Coexistence Results for a Spatial Stochastic Epidemic Model
 pp. 367-376

We introduce a spatial stochastic model for infectious 
diseases, such as influenza, that do not confer immunity and from which one 
usually recovers.
We prove the existence of an endemic state in any dimension for any
strictly positive value of the recovery rate. This is in sharp contrast with
what happens for the model with no recovery in
$d=1$ for which it is known that
the disease dies out for all parameters values.
Keywords: epidemic, infectious diseases, influenza, spatial
stochastic model



2004
Volume 10
Issue 3

Special issue containing proceedings of the workshop
``Gibbs vs. Non-Gibbs in Statistical Mechanics and Related Fields''.
Editors: Aernout C.D. van Enter, Frank Redig and Arnaud Le Ny.
Introduction
 pp. 377-379

L. Bertini, E.N.M. Cirillo and E. Olivieri
Gibbsian Properties and Convergence of the Iterates for the Block Averaging Transformation
 pp. 381-394

We analyze the Block Averaging Transformation applied to the 
two-dimensional Ising model in the uniqueness region. We discuss the
Gibbs property of the renormalized measure and the convergence of
renormalized potential under iteration of the map. It turns out that for
any temperature $T$ higher than the critical one $T_c$ the renormalized
measure is strongly Gibbsian, whereas for $TT_c$ and in a weak sense for $TKeywords: lattice systems, cluster expansion, disordered systems,
renormalization group

D. Dereudre and S. Roelly
On Gibbsianness of Infinite-Dimensional Diffusions
 pp. 395-410

We analyse different Gibbsian properties of interactive
Brownian diffusions $X$ indexed by the lattice
$Z^d : X=(X_i(t)$, $i \in Z^d$, $t \in [0,T]$, $0Keywords: infinite-dimensional Brownian diffusion, Gibbs field, cluster
expansion

A.C.D. van Enter and E.A. Verbitskiy
On the Variational Principle for Generalized Gibbs Measures 
 pp. 411-434

We present  a novel  approach to establishing  the variational  principle for
Gibbs  and generalized  (weak  and  almost) Gibbs  states.  Limitations of  a
thermodynamic formalism  for generalized Gibbs  states will be  discussed.  A
new  class  of  intuitively  Gibbs  measures is  introduced,  and  a 
typical  example  is studied.   Finally,  we present  a  new  example of  a
non-Gibbsian measure arising from an industrial application.
Keywords: Gibbs versus non-Gibbs, generalized Gibbs measures, variational 
principle, relative entropy density

R. Fernandez and G. Maillard
Chains and Specifications
 pp. 435-456

We review four types of results combining or
relating the theories of discrete-time stochastic processes and of
one-dimensional specifications.  First we list some general
properties of stochastic processes which are extremal among those
consistent with a given set of transition probabilities. They
include: triviality on the tail field, short-range correlations,
realization via infinite-volume limits and ergodicity.  Second we
detail two new uniqueness criteria for stochastic processes and
discuss corresponding mixing bounds.  These criteria are analogous
to those obtained by Dobrushin and Georgii for Gibbs measures.
Third, we discuss conditions for a stochastic process to define a
Gibbs measure and vice versa, that generalize well known
equivalence results between ergodic Markov chains and fields.
Finally we state a (re)construction theorem for specifications
starting from single-site conditioning, which applies in a rather
general setting.
Keywords: discrete-time stochastic processes, Gibbs measure,
chains with complete connections, Markov chains, ergodicity and rates of mixing

H. Guiol
About the Long Range Exclusion Process
 pp. 457-476

Introduced by Spitzer [F. Spitzer, Interaction of Markov processes,
Adv. Math., 1970, v. 5, 246-290] and studied by
Liggett [T.M. Liggett, Long range exclusion processes,
Ann. Prob., 1980, v. 8, N 5, 861-889], the Long Range Exclusion Process
(LREP) is an interacting particle system with truly long range interaction.
Informally speaking: each particle on a lattice hops at
independent random times following instantaneously a random
dynamic on the lattice until finding a vacant site (if any). These
instantaneous, potentially long jumps prevent the process to have
the Feller property. In this paper we review the main
results about the LREP including recent developments
obtained in [X. Zheng, Ergodic theorem for generalized long-range
exclusion processes with positive recurrent transition probabilities.
Acta Mathematica Sinica (N.S.), 1988, v. 4, N 3, 193-209;
H. Guiol, Un resultat pour le processus d'exclusion a longue portee,
Ann. Inst. H. Poincare, Probabilites et Statistiques, 1997,
v. 33, N 4, 387-405] and [E. Andjel and H. Guiol,
Long range exclusion processes, generator and invariant measures,
To appear in Ann. Prob., 2004].
New results on Feller approximations and about the
regularity set of the LREP are also provided.
Finally we briefly discuss some connections of the LREP
with the discrete Hammersley process introduced
in [P.A. Ferrari, Limit theorems for tagged particles,
Markov Processes Relat. Fields, 1996, v. 2, N 1, 17-40]
and the sandpile process in infinite volume developed in
[C. Maes, F. Redig, E. Saada and A. Van Moffaert, On the thermodynamic limit for a one-dimensional
sandpile process, Markov Processes Relat. Fields, 2000, v. 6, N 1,
1-21] and [C. Maes, F. Redig and E. Saada, The abelian sandpile model on an infinite tree,
Ann. Prob., 2002, v. 30, N 4, 2081-2107].
Keywords: infinite particle systems, non-Feller process, long
range exclusion, invariant measures, formal generator

O. Haggstrom and C. Kulske
Gibbs Properties of the Fuzzy Potts Model on Trees and in Mean Field
 pp. 477-506

We study Gibbs properties of the fuzzy Potts model in the mean field
case (i.e. on a complete graph) and on trees. For the mean field case,
a complete characterization of the set of temperatures for which
non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do show for general
trees that non-Gibbsianness of the fuzzy Potts model happens exactly
for those temperatures where the underlying Potts model has multiple
Gibbs measures.
Keywords: Potts model, fuzzy Potts model, Gibbs measures,
non-Gibbsian measures, trees, mean-field models

F. den Hollander
Gibbs under Stochastic Dynamics?
 pp. 507-516

This paper is a mini-overview of some recent results on the
evolution of Ising-spin systems under Glauber spin-flip dynamics,
in particular, the question whether Gibbsianness is preserved,
lost or recovered during the dynamics. Examples of all three
scenarios are given, with an explanation of what drives the
behavior. Some open problems are formulated as well.
Keywords: Ising-spin systems, Glauber spin-flip
dynamics, Gibbs measure, bad configuration, preservation, loss or
recovery of Gibbsianness

R.B. Israel
Some Generic Results in Mathematical Physics
 pp. 517-521

We discuss three consequences of the Baire Category Theorem. In spaces
of long-range interactions, typical phase diagrams are trivial. Most 
ergodic states of a lattice system are not Gibbs states. 
Unitarily-equivalent self-adjoint operators on Hilbert space with
purely discrete spectrum usually can't be added.
Keywords: generic sets, very sparse sets, generic nonGibbsianness, 
generic lack of self-adjointness properties

R. Kuhn
Gibbs vs. Non-Gibbs in the Equilibrium Ensemble Approach to
Disordered Systems
 pp. 523-545

We describe the salient ideas of the equilibrium ensemble approach to
disordered systems, paying due attention to the appearance of non-Gibbsian
measures. A canonical scheme of approximations --- constrained annealing ---
is described and characterised in terms of a Gibbs' variational principle
for the free energy functional. It provides a family of increasing exact
lower bounds of the quenched free energy of disordered systems, and is
shown to avoid the use of non-Gibbsian measures. The connection between
the equilibrium ensemble approach and conventional low-concentration
expansions or perturbation expansions about ordered reference systems is
also explained. Finally applications of the scheme to a number of
disordered Ising models and to protein folding are briefly reviewed.
Keywords: disordered systems, non-Gibbsianness, variational bounds, Morita method

C. Kulske
How Non-Gibbsianness Helps a Metastable Morita Minimizer 
to Provide a Stable Free Energy
 pp. 547-564

We analyze a simple approximation scheme based
on the Morita-approach for  the example of the
mean field random field Ising model
where it is claimed to be exact in some of the physics literature.
We show that the approximation scheme is flawed, but
it provides a set of equations whose metastable solutions surprisingly
yield the correct solution of the model. We explain how the same equations
appear in a different way  as rigorous consistency equations.  We
clarify the relation between the validity of their
solutions and the almost surely discontinuous behavior of the single-site
conditional probabilities.
Keywords: disordered systems, Morita approach,
non-Gibbsian measures, mean field models, random field model



2004
Volume 10
Issue 4


P. Jung
Perturbations of the Symmetric Exclusion Process
 pp. 565-584

This paper gives results concerning the asymptotics of the
invariant measures, $\mathcal{I},$ for exclusion processes where
$p(x,y)=p(y,x)$ except for finitely many $x,y\in\mathcal{S}$ and
$p(x,y)$ corresponds to a transient Markov chain on $\mathcal{S}$.
As a consequence, a complete characterization of $\mathcal{I}$ is
given for the case where $p(x,y)=p(y,x)$ for all but a single
ordered pair $(u,v)$. Also, this paper addresses the question:
When do local changes to a symmetric kernel $p(x,y)=p(y,x)$ affect
the evolution of the exclusion process globally?
Keywords: interacting particle system, exclusion process,
infinitesimal coupling, invariant measures

T.C. Dorlas and W.M.B. Dukes
Fluctuations of the Local Magnetic
Field in Frustrated Mean-Field Ising Models
 pp. 585-606

We consider fluctuations of the
local magnetic field in frustrated mean-field Ising models.
Frustration can come about due to randomness of the interaction
as in the Sherrington - Kirkpatrick model, or through fixed
interaction parameters but with varying signs. We consider
central limit theorems for the fluctuation of the local magnetic
field values w.r.t. the a priori spin distribution for both
types of models. We show that, in the case of the
Sherrington - Kirkpatrick model there is a central limit theorem
for the local magnetic field, a.s. with respect to the
randomness. On the other hand, we show that, in the case of
non-random frustrated models, there is no central limit theorem
for the distribution of the values of the local field, but that
the probability distribution of this distribution does converge.
We compute the moments of this probability distribution on the
space of measures and show in particular that it is not
Gaussian.
Keywords: spin glasses, frustrated spin systems, 
probability measures on infinite-dimensional spaces, limit theorems, 
occupation measures

A. Procacci, B. Scoppola, G.A. Braga and R. Sanchis
Percolation Connectivity in the Highly Supercritical Regime
 pp. 607-628

We  prove  that  the two point finite connectivity function
$\t^{\rm f}(x,y)$  in the $d$-dimensional Bernoulli bond percolation
is analytic in $p$ around $1$. We also provide upper and lower bounds
for this function in the case $d\ge 3$ and near $p=1$. The gap between
lower bound and upper bound is sufficiently narrow to conclude
that  the rate of exponential decay, i.e. the inverse correlation length
$m(p)$, is, for $p$ sufficiently near to $1$ and for $x-y$ in the coordinate
axis directions, of the form $m(p)=2(d-1)|\ln (1-p)|+O(1-p)$, as expected
by intuition based on low temperature expansion arguments.
Keywords: supercritical percolation, finite connectivity

A.M. Alhakim, J. Kawczak and S. Molchanov
On the Class of Nilpotent Markov Chains, I.
The Spectrum of Covariance Operator
 pp.629-652

We study the central limit theorem and the structure of the
corresponding covariance operator for the Markov chains generated
by successive (overlapping) $k$-tuples $(X_{n+1},\ldots,X_{n+k})$,
$n=0,1,\ldots$ formed from the i.i.d.r.v. $\{X_n\}$. The potential
application of the theory includes the design of statistical
tests. In particular, we present the explicit spectral analysis of
the covariance matrices related to Marsaglia's $k$ permutation
test for $k=2, 3, 4, 5$.
Keywords: CLT for the nilpotent Markov chain,
spectral decomposition, testing random number generators

S. Roelly and M. Sortais
Space-Time Asymptotics of an
 Infinite-Dimensional Diffusion Having a Long-Range Memory
 pp.653-686

We develop a cluster expansion in space-time for an
 infinite-dimensional system of interacting diffusions
where the drift term of each diffusion depends on the whole past of
the trajectory; these interacting diffusions arise when considering 
the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.
Keywords: random field Ising model, Langevin dynamics, interacting 
diffusion processes, space-time cluster expansions

M. Hildebrand
Rates of Convergence of the Diaconis - Holmes - Neal
Markov Chain Sampler with a V-Shaped Stationary Probability
 pp.687-704

The Diaconis - Holmes - Neal Markov chain sampler is a modification
of the Metropolis algorithm based on the nearest neighbor random walk.
For certain V-shaped stationary probabilities $\pi$ and a choice of a
parameter $\theta$, this paper proves that the Diaconis - Holmes - Neal
Markov chain approaches its stationary distribution faster than the
Markov chain of the corresponding Metropolis algorithm.
Keywords: Metropolis algorithm, Diaconis - Holmes - Neal sampler

A. Hinojosa
Exit Time for a Reaction Diffusion Model
 pp.705-744

We  consider an interacting particle system, the Glauber+Kawasaki model.
This model is the result of the combination of a fast stirring, the Kawasaki
part, and a spin flip process, the Glauber part. This process has a
Reaction Diffusion equation as hydrodynamic limit. The ergodicity of this
dynamics in the presence of a metastable state (double well potential) was
recently proven, for any dimension. In this article we obtain the asymptotic
exponential distribution of certain exit time from a subset of the basin of
attraction of one of the wells.
Keywords: exit times, interacting particle systems, Glauber - Kawasaki
dynamics, Reaction Diffusion equations, hydrodynamic limits