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\centerline{\tt MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH}
\vskip 15mm\relax
\centerline{\tt T\ a\ g\ u\ n\ g\ s\ b\ e\ r\ i\ c\ h\ t\ \ 51/1992}
\vskip 10mm plus 5mm minus 5mm
\centerline{\large Large Deviations and Applications}
\vskip 5mm plus 2mm minus 2mm
\centerline{\large 29.\,11. bis 5.\,12.\,1992}
\vskip 10mm plus 5mm minus 5mm
\noindent
Zu dieser Tagung unter der gemeinsamen Leitung von E.~Bolthausen (Z\"urich),
J.~G\"artner (Berlin) und S.\,R.\,S.~Varadhan (New York) trafen sich
Mathematiker und mathematische Physiker aus den verschiedensten L\"andern mit
einem breiten Spektrum von Interessen.
\smallskip\noindent
Die Theorie vom asymptotischen Verhalten der Wahr\0scheinlichkeiten gro\3er
Abweichungen ist einer der Schwerpunkte der j\"ungeren
wahr\-schein\-lich\-keits\-theo\-re\-ti\-schen Forschung. Es handelt sich um
eine Pr\"a\-zi\-sie\-rung von Gesetzen gro\3er Zahlen. Gegenstand der
Untersuchungen sind sowohl die Skala als auch die Rate des exponentiellen
Abfalls der kleinen Wahrscheinlichkeiten, mit denen ein untypisches Verhalten
eines stochastischen Prozesses auftritt. Die Untersuchung dieser klei\-nen
Wahrscheinlichkeiten ist f\"ur viele Fragestellungen interessant.
\smallskip\noindent
Wichtige Themen der Tagung waren:
\smallskip
\item{--}Anwendungen gro\3er Abweichungen in der Statistik
\item{--}Verschiedene Zug\"ange zur Theorie gro\3er Abweichungen
\item{--}Stochastische Prozesse in zuf\"alligen Medien
\item{--}Wechselwirkende Teilchensysteme und ihre Dynamik
\item{--}Statistische Mechanik, Thermodynamik
\item{--}Hydrodynamischer Grenz\"ubergang
\item{--}Verhalten von Grenzfl\"achen, monomolekulare Schichten
\item{--}Dynamische Systeme und zuf\"allige St\"orungen
\item{--}Langreichweitige Wechselwirkung, Polymere
\item{--}Stochastische Netzwerke
\smallskip\noindent
Die Tagung hatte 47 Teilnehmer, es wurden 42 Vortr\"age gehalten.
\goodbreak
\bigskip
\leftline{\large Abstracts}
\vskip-\medskipamount
\abstract{G.~Ben Arous (joint work with A.~Guionnet)}
{Langevin dynamics for spin glasses}
{We study the dynamics for the Sherrington-Kirkpatrick model of
spin glasses. More precisely, we take a soft spin model
suggested by Sompolinski and Zippelius. Thus we consider
diffusions interacting with a Gaussian random potential
of S-K-type.
\smallskip
\item{(1)}We give an \lq\lq annealed\rq\rq\ large
deviation principle for the empirical measure at the process
level. We deduce from this an annealed law of large numbers
and a central limit theorem. The limit process is a new
object, a nonlinear and non-Markovian process.
\item{(2)}We then show how for certain initial measures this can give
the \lq\lq quenched\rq\rq\ law of large numbers and we
see again that the disorder in the interaction produces
non-Markovianity of the limit.
\smallskip\noindent
All this is valid only above a critical temperature
on a given interval of time or before a critical time
at a given temperature.}
\abstract{Anton Bovier (joint work with V.~Gayrard and P.~Picco)}
{Thermodynamics of the Hopfield model}
{We study the Hopfield model of a neural network in the spirit of disordered
mean field models of spin systems. The disorder here resides in the coupling
matrix $J_{ij}=(1/N)\sum_{\mu=1}^m x_i^\mu x_j^\mu$, where $\{x_i^\mu\}$ is a
family of i.\,i.\,d.\ random variables taking the values $+1$ and $-1$ with
equal probability. Properties of this model depend crucially on the parameter
$m$. We present the following results:
\smallskip
\item{(1)} In the cases where $m/N\searrow 0$, as $N\nearrow\infty$, we prove
that the free energy of this model converges to that of the standard Curie-Weiss
model, almost surely. Moreover, we show that to each of the vectors $x^\mu$
there corresponds, for $T<1$, Gibbs measures in the infinite volume limit that
are concentrated on configurations having overlap $a(T)$ with the vector
$x^\mu$, and overlap zero with all other vectors $x^\nu$, $\nu\not=\mu$.
\item{(2)} In the case where $m/N=\alpha$, with $\alpha$ sufficiently small, we
show that the structure of the Gibbs measures remains the same as before, for
$T\le T(\alpha)<1$, where $T(\alpha)\to1$ as $\alpha\searrow0$.}
\abstract{Francis Comets}
{Erd\"os-R\'enyi laws and Gibbs measures}
{Erd\"os-R\'enyi type of laws state that in a given sample of size $n$, one will
observe in subsamples of size $(1/t)\log n$ all deviations with rate of decay
less or equal to $t$ ($t>0$), with probability 1 as $n\to\infty$.
\smallskip
\item{(1)} We give general formulations of this result, for the
empirical field or process under the condition of uniform large deviation
estimates (or hypermixing processes).\hfill \item{(2)} We give applications to
Gibbs measures, and we study in this case the limit $t\searrow 0$. The result
then yields positive answers to questions like: Can we detect phase transition
from a single (but large) sample? Can we learn some information on the other
Gibbs measures?}
\abstract{Ted Cox (joint work with Andreas Greven and Tokuzo Shiga)}
{Finite and infinite systems of interacting diffusions}
{The subject of this talk is a theorem relating the asymptotic behavior of large finite
systems of interacting diffusions and the corresponding infinite system. The infinite system
$x(t)=\{\,x_i(t),\,i\in\zee^d\,\}$ is the Markov process determined by
$$
dx_i(t)=\biggl[\sum_{j\in\zee^d}a(i,j)x_j(t)-x_i(t)\biggr]dt+\sqrt{g(x_i(t))}\,dW_i(t)\eqno(*)
$$
where $a(i,j)$ is an irreducible random walk kernel on $\zee^d$, $g\from[0,1]\to\re^+$ is
Lipschitz, $g(0)=g(1)=0$, $g>0$ on $(0,1)$, and $\{W_i(t)\}$ is a family of independent
Brownian motions. There is a family $\{\,\nu_\theta,\,\theta\in[0,1]\,\}$ of invariant
measures for $x(t)$ with $E^{\nu_\theta}x_i=\theta$. The finite systems
$x^N(t)=\{\,x_i^N(t),\, i\in(-N,N]^d\,\}$ are defined by an equation like $(*)$ treating
$(-N,N]^d$ as a torus. The main result is that under some conditions, for
$t_N\uparrow\infty$ with $N$, $t_N/(2N)^d\to s\in[0,1]$,
$$
{\cal L}(x^N(t_N))\Rightarrow\int_{[0,1]}Q(\varrho,d\theta)\,\nu_\theta
$$
where $Q(\varrho,\,\cdot\,)$ is the transition of a certain diffusion on $[0,1]$. In
particular, we see that if $t_N=o(N^d)$ as $N\to\infty$ then ${\cal
L}(x^N(t_N))\Rightarrow\nu_\varrho$, so that the invariant measures of the infinite system
describe the behavior of the finite systems for times up to a certain order.}
\abstract{P.~Dai Pra}
{Large deviations for interacting particle systems}
{We study the large deviation of the space-time empirical averages of a $d$-dimensional
stochastic spin system whose Markov semigroup is generated by the operator
$$
Lf(\sigma)=\sum_{i\in\zee^d}c(\vartheta_i\sigma)[f(\sigma^i)-f(\sigma)]
$$
where $\vartheta_i$ is the shift on $\{-1,1\}^{\zee^d}$ and
$\sigma^i(j)=(-1)^{\delta_{ij}}\sigma(j)$. We prove a
$n^{d+1}$-large deviation principle for the empirical process
$$
R_{n,\omega}={1\over
n^{d+1}}\sum_{i\in\{0,1,\dots,n-1\}^d}\int_0^n\delta_{\vartheta_{t,i}\omega}\,dt, $$
where $\omega\in\Omega={\Bbb D}(\re,\{-1,1\}^{\zee^d})$ and $\vartheta_{t,i}$ are the
space-time shift maps on $\Omega$, and we identify the rate function. Moreover,
we prove that the zeros of the rate function correspond to the invariant
measures for the system. We also give results on some related problems, as the
\lq\lq contraction\rq\rq\ to deviations of lower level and critical large
deviations for non-ergodic systems.}
\abstract{Donald~A.~Dawson}
{Some comments on the hierarchical mean-field limit}
{We begin with a system of a large number of components where interactions are organized in a
hierarchical manner. The $k$th level of the hierarchy is comprised of $N$ objects of the
$(k-1)$st level and the strength of the interaction decreases as a function of the
hierarchical distance (and also as a function of $N$). The single level hierarchy in the
limit $N\to\infty$ is known as the mean-field limit. The case in which $N$ is fixed and
$k\to\infty$ corresponds to the thermodynamic limit. The hierarchical mean-field limit
corresponds to the finite or infinite hierarchy in the $N\to\infty$ limit. The effect of
taking the limit $N\to\infty$ is to separate the natural time scales or spatial scales
relevant to the different levels of the hierarchy. To illustrate this we consider two
examples. The first is the continuous spin ferromagnetic model. In joint work with J\"urgen
G\"artner this hierarchical mean-field limit of this ferromagnetic model is analysed using
multilevel large deviation theory as $N\to\infty$. This analysis leads to a notion of discrete
symmetry breaking in the mean-field limit. The second model considered is the stepping stone
model arising in population genetics. This model has been analysed in joint work with Andreas
Greven using multiple time scale analysis. This work shows that the criteria for continuous
symmetry breaking in this model in the hierarchical mean-field and in the thermodynamic limit
sense are in fact equivalent for a large family of interaction strengths.}
\abstract{Frank den Hollander (joint work with A.~Greven)}
{Large deviations for a random walk in random environment}
{Let $\omega=(p_x)_{x\in\zee}$ be an i.\,i.\,d.\ collection of $(0,1)$-valued random
variables. Given $\omega$, let $(X_n)_{n\ge0}$ be the Markov chain on $\zee$ defined by
$X_0=0$ and $X_{n+1}=X_n\pm 1$ with probability $p_{X_n}$ resp.\ $1-p_{X_n}$. It is shown
that $X_n/n$ satisfies a large deviation principle, i.\,e.,
$$
\lim_{n\to\infty}{1\over n}\log P_\omega(X_n=\lfloor\theta_n
n\rfloor)=-I(\theta)\quad\omega\hbox{-a.\,s.\ for any $\theta_n\to\theta\in[-1,1]$.}
$$
First we derive a representation of the rate function $I$ in terms of a variational problem.
Second we solve the latter explicitly in terms of random continued fractions. This leads to
a classification and qualitative description of the shape of $I$. In the recurrent case $I$
is non-analytic at $\theta=0$. In the transient case $I$ is non-analytic at
$\theta=-\theta_c,0,\theta_c$ for some $\theta_c\ge 0$, with linear pieces in
between.}
\abstract{J.-D.~Deuschel (joint work with A.~Pisztora and C.~Newman)}
{Critical large deviations}
{Let $P_0$ be a product measure on $\Omega=E^{\zee^d}$ and denote by
$R_N(\omega)=(1/|V_N|)\sum_{k\in V_N}\delta_{\theta_k\omega}$ the empirical
field of the box $V_N=[1,N]^d$. For a given interaction potential $\varrho$,
define the approximate microcanonical distribution
$\mu_{N,\delta}(\,\cdot\,)=P_0(\,\cdot\mid |U_N-\nu|\le \delta)$, where $U_N$ is
the average energy of $V_N$. Large deviations show that the law of the empirical
field $R_N$ converges at a volume exponential rate on the set of Gibbs
distributions at an appropriate inverse temperature $\beta=\beta(u)$. In case of
phase transition, we expect that $R_N$ concentrates on the extremal Gibbs
states. We show that a surface exponential rate occurs for the Ising model. The
central estimate is a surface-order large deviations for the empirical
magnetization of the free boundary Gibbs distribution. The method uses
F-K-percolation at sufficiently small temperature and the isoperimetric
estimate.}
\abstract{Hermann Dinges}
{Second order large deviations}
{We start with a family of distributions $\{\,{\cal L}(X_\varepsilon)\from\varepsilon\to
0\,\}=\{\,p_\varepsilon(x)\,dx\from\varepsilon\to0\,\}$ on $U\subset\re^d$ of the form
(uniformly on compacts)
$$
\openup-1\jot\eqalign{
p_\varepsilon(x)\,dx&=(2\pi\varepsilon)^{-d/2}\cr
&\qquad\times\exp\Bigl(-{1\over\varepsilon}K(x)-K_0(x)-\varepsilon
K_1(x)+o(\varepsilon)\Bigr)dx_1\dots dx_d.\cr}
$$
$K(\,\cdot\,)$ is not necessarily the Legendre-transform of a cumulant generating function.
Just $K(\,\cdot\,)$ smooth and $K(x^*)=0$ for some $x^*$, $K(x)>0$ for $x\not=x^*$,
$K^{\prime\prime}(\,\cdot\,)$ positive definite ($K_0(\,\cdot\,)$ and $K_1(\,\cdot\,)$ are
required to satisfy certain smoothness conditions as well.)
For nice sets $A=\{\,x\from F(x)\le\hbox{const}\,\}$ we find an asymptotic expansion
$$
\Lambda(\Pr(X_\varepsilon\in A))={1\over\varepsilon}K(\hat x)+(H_0(\hat
x))+\delta(\hat x))+O(\varepsilon),
$$
where $\Lambda(p)=[\Phi^{-1}(p)]^2/2$,
$K(\hat x)=\inf\{\,K(x)\from x\in\partial A\,\}$,
$\delta(\hat x)$ vanishes when $A$ is a half\-space, and
$$
H_0(\hat x)={1\over 2}\ln\biggl[{K'(K^{\prime\prime})^{-1}K'\over 2K}(\hat
x)\biggr]+K_0(\hat x)+{1\over 2}\ln|\hbox{det}K^{\prime\prime}(\hat x)|.
$$
In the second part of the lecture such an asymptotic expansion was given
explicitly in a particular case; we studied an approximation of the so called
non-central $t$-distribution
$$
T^{(n)}:={\Ybar\over\sqrt{{1\over n-1}\sum_{i=1}^n(Y_i\Ybar)^2}}
$$
in the
general case and in the special Gaussian case
$$
\widetilde T^{(n)}={\vartheta+(1/\sqrt n)Z_0\over\sqrt{(1/n)\sum_{i=1}^nZ_i^2}}
$$
with
$Z_0,Z_1,\dots,Z_n$ independent standard normal. Then
$$
\eqalign{
\Pr\nolimits_\vartheta&\biggl(\widetilde
T^{(n)}\le{\vartheta+a\over\sqrt{1-a(a+\vartheta)}}\biggr)\cr
&\approx \Phi\biggl(\pm\sqrt 2\sqrt{n
K(\vartheta,a)-{1\over 2}\ln\left[{2K(\vartheta,a)\over
a^2(1+(1/2)\vartheta(a+\vartheta))}\right]+\hbox{rest}}\,\biggr)\cr}
$$
where $2K(\vartheta,a)=a^2-a(a+\vartheta)-\ln[1-a(a+\vartheta)]$ for
$a\in(-\infty,+\infty)$.}
\goodbreak
\abstract{Richard~S.~Ellis (joint work with Paul Dupuis)}
{A stochastic optimal control approach to the theory of large deviations}
{We present a new and widely applicable approach to the theory of large deviations which is
based on stochastic optimal control theory. In our opinion, this approach reduces many
aspects of the theory of large deviations to the theory of weak convergence of probability
measures. We demonstrate the versatility of the approach by applying it to three diverse
large deviation problems:\hfill\smallskip
\item{(1)} small random perturbations of dynamical systems with continuous statistics,
\item{(2)} small random perturbations of dynamical systems with discontinuous statistics,
\item{(3)} the empirical measures of Markov chains with continuous statistics and with
discontinuous statistics.\hfill\smallskip
\noindent While our main goal is to exhibit a general methodology, the technique allows, in
the examples considered, a weakening of the assumptions that have previously been used in
proving the large deviation principle. We also obtain a number of new results.}
\abstract{Klaus Fleischmann (joint work with Ingemar Haj)}
{Large deviation probabilities for some rescaled superprocesses}
{Large deviations are discussed for the continuous super-Brownian motion in $\re^d$ in the
case of an asymptotically small branching rate. Based on a complete blow-up property for the
related cumulant equation some $L^2$-formula for the rate functional is derived. This
formula might have some applications, as well as might give some hints concerning on eventual
general theory for large deviations for measure-valued diffusions behind this particular
example of a super-Brownian motion.}
\abstract{Mark Freidlin}
{Random perturbations of dynamical systems with conservation laws}
{The evolution of first integrals along the trajectories of the perturbed system is
considered. After proper rescaling of time the first integral converges to a diffusion process
on a graph corresponding to the conservation Law. Under certain assumptions concerning the
non-perturbed system on the level set of the first integral the limiting process turns out
to be Markovian. The limiting process is defined by a family of second order differential
operators and by a collection of gluing conditions in the vertices. The operators are the
result of averaging over the connected components of the level sets. The gluing conditions are
calculated on the vertices of the graph corresponding to the saddle points of the first
integral (if it is defined by a smooth function). The extremal points of the first integral
correspond to those vertices which are inaccessible for the limiting process, and no boundary
conditions should be given at these points.}
\abstract{Tadahisa Funaki}
{Hydrodynamic limit for one-dimensional exclusion processes}
{We consider a particle system on the one-dimensional periodic lattice with hard core
exclusion. The jump rate is spatially homogeneous, non-degenerate and satisfies the detailed
balance condition with respect to a trivial Hamiltonian $H\equiv 0$. The Bernoulli measures
are therefore reversible for the dynamics. For this model, the non-equilibrium fluctuation
problem (in the gradient case by using the method of Chang-Yau) and the hydrodynamic limit
(in the general non-gradient case by applying the method of Varadhan; this part is due to
Uchiyama) are discussed. The basic tools are the logarithmic Sobolev inequality and
the spectral gap for the exclusion process.}
\abstract{Hans-Otto Georgii (joint work (in part) with H.~Zessin)}
{Large deviations for Gibbsian point random fields}
{We present a large deviation principle for the stationary empirical fields for systems of
marked point particles in boxes $\Lambda_n\uparrow\re^d$. The particle distributions are
Gibbsian relative to one of the following types of interaction:\hfill\smallskip
\item{(1)} interactions of possibly infinite range with hard-core repulsion,
\item{(2)} superstable pair interactions of finite range,
\item{(3)} interactions of mean-field type depending on the particle marks,
\item{(4)} nearest-particle interactions for $d=1$.\hfill\smallskip
\noindent In the cases (2) and (4) we impose periodic boundary conditions. Since the
underlying topology is chosen fine enough, the contraction principle then gives us a large
deviation principle for the \lq\lq individual empirical fields\rq\rq\ defined by averaging
over the particle positions. We also present a maximum entropy principle implying a general
version of the equivalence of ensembles.}
\abstract{Andreas Greven (joint work with Frank den Hollander)}
{A variational characterization of the speed of
a one-dimensional self-repellent
random walk} {Let $Q_n^\alpha$ denote the probability measure of an $n$-step random walk
$(0,S_1,\dots,S_n)$ on $\zee$ obtained by weighting the simple random walk with the factor
$(1-\alpha)$ for every self-intersection. This is a model for a one-dimensional polymer. We
prove that for every $\alpha\in(0,1)$ there exists $\theta^*(\alpha)\in(0,1)$ such that
$$
\lim_{n\to\infty}Q_n^\alpha\Bigl({|S_n|\over
n}\in[\theta^*(\alpha)-\varepsilon,\,\theta^*(\alpha)+\varepsilon]\Bigr)=1\quad\hbox{for
every }\varepsilon>0.
$$
We give a characterization of $\theta^*(\alpha)$ in terms of the largest eigenvalue of a
one-parameter family of $\na\times\na$ matrices, which allows us to prove that $\theta^*$ is
an analytic function, $\theta^*(0)=0$, $\theta^*(1)=1$, and $\theta^*(x)\in(0,1)$ for
$x\in(0,1)$. Besides for the speed we prove a limit law for the local times of the walk. The
techniques used enable us to treat more general forms of self-repellence involving multiple
intersections.}
\abstract{C.~Kipnis and S.~Olla (joint work with C.~Landim)}
{Hydrodynamics for the generalized exclusion process}
{The generalized exclusion process with at most two particles per site is one of
the simplest infinite particle systems which is non-gradient with product-form
invariant measures and for which one can prove hydrodynamical limits. The
limiting equation is, as expected, of the form
$\partial_t \varrho=\partial_x(\hat a(\varrho)\partial_x\varrho)$
where $\hat a$ is given by a variational formula.}
\abstract{A.~I.~Kometch (joint work with E.~Kopylova and N.~Ratanov)}
{The stabilization of statistics in wave equations with mixing}
{There exist many statistical equilibrium phenomena in physics related to Hamiltonian
infinite-dimensional systems of mathematical physics, for example Gibbs measures in
statistical mechanics and the black-body emission law in electrodynamics. The phenomena lead
us to a problem of \lq\lq statistical stabilization\rq\rq. This means that these statistics
appear as $t\to\infty$ for the solutions of equations considered when the initial statistics
at $t=0$ is \lq\lq almost arbitrary\rq\rq.
We prove such stabilization for the linear wave equation and also for the Klein-Gordon
equation with constant or variable coefficients in $\re^n$, where $n\ge 2$.\hfill\break
We assume that the initial statistics fit the Rosenblatt-Ibragimov mixing condition and that
they are homogeneous in $x\in\re^n$. In the case of constant coefficients we use the explicit
formula for the solution and apply the extention of the \lq\lq rooms-corridors\rq\rq\ method
of S.\,N.~Bernstein, M.~Rosenblatt, and Ibragimov-Linnik. In the case of variable
coefficients there is no of explicit formula. We reduce the case to the constant coefficients
case by the scattering theory. But the total energy of solutions considered is infinity
almost surely because of the homogeneity of the initial data (and the solutions). Then we must
construct the scattering theory for solutions of infinite energy.\hfill\break
The result is: the statistics of solutions at time $t$ converge to some {\it Gaussian\/}
measure as $t\to\infty$. This is the analogue of the central limit theorem for the
Hamiltonian systems considered. Note that the Gibbs measures for our linear equations
\lq\lq must\rq\rq\ be Gaussian, because their Hamilton functions are quadratic
forms. This means for the large deviations of the solutions, considered in each
bound\-ed region of space $\re^n$: We can almost surely tame the initial data to be very
small bounded functions in $\re^n$. But, as $t\to\infty$, the solution at the considered point
(or the energy in the considered region) may be arbitrary large.}
\abstract{A.~P.~Korostel\"ev}
{Action functional for dynamical systems with discontinuities}
{A well-known \lq\lq continuous mapping\rq\rq\ method is applied to a piecewise smooth
dynamical system having a surface of \lq\lq stable discontinuity\rq\rq. For such a system
disturbed by a standard white Gaussian noise of a small intensity $\varepsilon$, i.\,e.\ for
the solution of the stochastic equation
$$
\dot X^\varepsilon(t)=b(X^\varepsilon(t))+\varepsilon\dot W(t),\quad
0\le t\le T,\ \varepsilon\to0,\ X^\varepsilon(0)=0,
$$
the action functional (i.\,e.\ the rate function governing the large deviations) is
obtained. The basic idea is that there exists a continuous mapping $F\from C_{0,T}\to
C_{0,T}$, which is Lipschitz in the space of continuous functions $C_{0,T}$ and satisfies
$X^\varepsilon=F(\varepsilon W)$. Moreover, there exists another mapping $G\from
C_{0,T}\to C_{0,T}$ such that $G(\varepsilon W)=\pi^\varepsilon$ where
$\pi^\varepsilon(t)=\int_0^t I(X_1^\varepsilon(s)>0)\,ds$, i.\,e.\ $\pi^\varepsilon(t)$ is the
staying-time of $X^\varepsilon(t)$ in the positive half-space (we assume without loss of
generality that the surface of discontinuity is described by $x_1=0$). If $\varphi\in
C_{0,T}$, and $\psi=F(\varphi)$, $\mu=G(\varphi)$, then the inverse mapping has an explicit
expression: $\varphi=\psi-\int b_+(\psi)\,d\mu-\int b_{-}(\psi)\,d(t-\mu)$ where $b_{\pm}$ are
one-sided limits of $b$ on the surface of discontinuity. Thus, appling the large deviation
principle for the Wiener process one gets without any cumbersome calculation the action
functional for the joint process $(X^\varepsilon,\pi^\varepsilon)$:
$$
I(\psi,\mu)={1\over 2}\int_0^T\|\dot\psi-b_+\dot\mu-b_{-}(1-\dot\mu)
\|^2.
$$
The extensions to jumping processes are discussed. It is known that if one of the three
staying-times (in the positive or the negative half-space, or that on the surface of
discontinuity) is vanishing, then the approach applies. In particular, the large deviations
for the solution of $\dot
X^\varepsilon=-c\,\hbox{sgn}(X^\varepsilon)+\dot\xi^\varepsilon$, where
$\xi^\varepsilon$ is the rescaled Poisson process, are governed by the action functional
$$
I(\psi,\mu)=\int_0^T L_0(\dot\psi+c\dot\mu)\quad\hbox{where
}L_0(u)=1+u\log(u/e).
$$
But the same equation noised by the two-sided Poisson process (jumps $\pm 1$ with probability
$1/2$) leads to a problem that has no simple solution.}
\abstract{C.~Landim}
{An application of large deviation principles for the empirical measures of interacting
particle systems}
{We consider the symmetric simple exclusion process for which a large deviation principle
for the empirical measure was proved by Kipnis, Olla and Varadhan in finite volume and
extended to infinite volume by Landim.
We obtain a large deviation principle for the occupation time of a site in this model as a
consequence of the previous result in one dimension.
}
\abstract{Tzong-Yow Lee}
{Large deviations for branching diffusions}
{For a branching Brownian motion starting from the origin with multiplication rate
$\varepsilon^{-1}C$ and diffusivity $\varepsilon D$, write $P^\varepsilon$ the for probability
measure and $E^\varepsilon$ for the expectation. We ask:
$$\openup1\jot\leqalignno{
P^\varepsilon\biggl\{
\setbox0=\hbox{sample tree has at least one $1$-branch in}
\vcenter{\vbox{\normalbaselines\hsize=\wd0\noindent\parfillskip=0pt
sample tree has at least one $1$-branch in a tiny
\lq\lq neighborhood\rq\rq\ of $\varphi(s)$, $0\le s\le1$}}
\biggr\}&\asymp{}?\,,&(1)\cr
%
P^\varepsilon\biggl\{
\setbox0=\hbox{sample tree has at least one $2$-branch}
\vcenter{\vbox{\normalbaselines\hsize=\wd0\noindent
sample tree has at least one $2$-branch in a
tiny \lq\lq neighborhood\rq\rq\ of $(\varphi_1,\varphi_2)$}}
\biggr\}&\asymp{}?\,,&(2)\cr
%
P^\varepsilon\bigl\{R_1\sim b_1,R_2\sim b_2\bigr\}&\asymp{}?\,,&(3)\cr}
$$
where $\asymp$
means logarithmic equivalence as $\varepsilon\searrow 0$ and $R_t$ denotes the position of the
rightmost particle at time $t$. Problems~(1) and (2) are answered, for problem~(3) a partial
solution is given.}
\abstract{J.\,T.~Lewis}
{Thermodynamical aspects of large deviations}
{The use in risk theory of intensive parameters analogous to the thermodynamic temperature
(Martin-L\"of 1986) prompts the question: Under what conditions does the machinery of
equilibrium thermodynamics apply in the theory of large deviations?
In joint work with Ch.~Pfi\-ster (Lausanne), we examine the thermodynamic
formalism of Ruelle (1965) and Lanford (1973) in the setting of probability
measures on Banach spaces. We define a Lanford entropy function and a grand
canonical pressure and give conditions for the equivalence of ensembles.
Motivated by Gibbs' axiomatization of thermodynamics (Gross 1982), we define a
Gibbs entropy function. We give conditions for the Lanford entropy function to
exist and be a Gibbs entropy function; we examine the connection with the large
deviation principle (cf.\ O'Brien and Vervaat 1990). \smallskip
\item{[1]} Martin-L\"of, A.: Entropy: a useful
tool in risk theory. Scand. Actuarial J. 1986, 223--235.
\item{[2]} Ruelle, D.: J. Math.\ Phys.\ 6,
201--209 (1965).
\item{[3]} Lanford, O.\,E.: 1971 Battelle Lectures, LNP 20 (1973).
\item{[4]} Gross, L.:
Saint~Flour X--1980, LNM 929 (1982).
\item{[5]} O'Brien, G., Vervaat, W.: Capacities, Large Deviations
and Log-Log Laws, York Univ.\ Report 90/19 (1990).}
\abstract{Matthias L\"owe}
{Large deviations for $U$-statistics}
{Let $(X_i)_{i\in\na}$ be a sequence of i.\,i.\,d.\ random variables taking values in some
Polish probability space $X$ with common law $\pi$. It is well-known that
$$\eqalignno{
U_n&:={1\over \left({n\atop m}\right)}\sum_{1\le i_1<\cdots< i_m\le
n}h(X_{i_1},\dots,X_{i_m})\cr
\noalignb{and}
V_n&:={1\over n^m}\sum_{1\le i_1,\dots,i_m\le n}h(X_{i_1},\dots,X_{i_m})\cr}
$$
are \lq\lq good\rq\rq\ estimators for $E_{\pi^m}(h)$, where $h\from X^m\to\re^d$ is some
integrable function. Under the condition that the moment generating function of $h$ and every
\lq\lq diagonal\rq\rq\ of $h$ exist, we derive a large deviation principle for the
distributions of $U_n$ and $V_n$. In both of the cases the rate function is given by
$$
I(y)=\inf\Bigl\{H(\varrho\,|\,\pi)\Bigm|\varrho\in{\cal M}_1(X),\,\int
h\,d\varrho^m=y\Bigr\} $$
where $H(\,\cdot\,|\,\pi)$ denotes the usual entropy with respect to $\pi$. Our key tools are
the contraction principle, Sanov's theorem and a graph-theoretic result about the
factorization of complete hypergraphs due to Baranyai.}
\abstract{Peter Major}
{Phase transition in random external magnetic field -- a conjecture}
{We discussed a one-dimensional long-range interaction model with a random external magnetic
field. Our conjecture is that there is a phase transition in this model at low temperatures.
This conjecture follows from a large deviation result about the distribution of the average
spin in this model. We claim that the rate function appearing in this result is not convex
in a certain region. Thus convexity is the cause of the phase transition, and its appearance
is closely related to the long-range interaction of the model.}
\abstract{M.\,B.~Maljutow}
{Large deviations in search for significant variables of a function\nobreak}
{A function $f(x_1,\dots,x_t)$ of a vast number of variables may be expressed in the
form $g(x_{\lambda_1},\dots,x_{\lambda_s})$ where $\lambda_1,\dots,\lambda_s$ is a sequence
of unknown indices and $s$ is small compared to $t$. Choosing the sequence
$\Xbar (i)=(x_1(i),\dots,x_t(i))$, $i=1,\dots,N$, arbitrarily, we observe the values of a
random variable $Z_i$ which are related to the sequence of $Y_i=f(\Xbar(i))$ via
transition probabilities $T(Z_i|Y_i)$. Measurements are independent given the sequence
$\Xbar(1),\dots,\Xbar(N)$. The main quantity of interest is the minimal sample
size $N_l$ which guarantees the correct decision on
$\labar=\lambda_1,\dots,\lambda_s$ with probability of error not exceeding
$\varepsilon$. The cases of static and sequential designs are investigated. In both
cases the upper estimate is $N_l \le\hbox{const}\times\ln t$ when $t\to\infty$ and $s$ is
constant. Of special interest is the additive smooth model
$g(x_1,\dots,x_s)=\sum_{\alpha=1}^s g_\alpha(x_\alpha)$, disturbed by an additive noise.
Under the condition of subgaussian tales of errors A.~Korostelev proved the large deviation
estimate for the rather unexpected statistic-inconsistent estimate of $S_\alpha^2=\int_{-1}^1
g_\alpha^2(x)\,dx$, assuming that all functions $g_\alpha$ and their derivatives are bounded
and $S_\alpha^2\ge\Delta>0$. This estimate is the base for obtaining the estimate for $N_l$
mentioned above. For sequential design the simple lemma on large deviations for
supermartingales allows us to obtain the same asymptotics in a more simple way. Some lower
bounds for $N_l$ are reviewed and cases, where estimates for $N_l$ are precise, are
mentioned.}
\abstract{A.\,A.~Mogulskii (joint work with A.\,A.~Borovkov)}
{Large deviation theorems for likelihood estimators}
{Let $a_1(\theta),a_2(\theta),\dots$ be i.\,i.\,d.\ random fields in $(C(\Theta),B)$, where
$C(\Theta)$ is a linear space of continous functions $f(\theta)$, $\theta\in\Theta$, and
$\Theta$ is a closed bounded subset of $\re^k$.
We call a vector $\theta_n^+\in\Theta$ at which $A_n(\theta)=a_1(\theta)+\cdots+a_n(\theta)$
attains its maximum a {\it maximum point of\/} $A_n(\theta)$:
$$
A_n(\theta_n^+)=\max_{\theta\in\Theta}A_n(\theta).
$$
The vector $\theta_n^+$ is not uniquely defined. Therefore, we define \lq\lq
upper\rq\rq\ and \lq\lq lower\rq\rq\ distributions of $\theta_n^+$ by the
formulae
$$\eqalignno{
P_+(\theta_n^+\in B)&\equiv P\bigl(\max_{\theta\in
B}A_n(\theta)\ge\max_{\theta\in\Theta\setminus B}A_n(\theta)\bigr)\cr
\noalignb{and}
P_{-}(\theta_n^+\in B)&\equiv P\bigl(\max_{\theta\in
B}A_n(\theta)>\max_{\theta\in\Theta\setminus B}A_n(\theta)\bigr).\cr}
$$
In this talk we study the \lq\lq fine\rq\rq\ asymptotics of the sequence
$$
P_\pm(\theta_n^+\in B).
$$}
\abstract{Peter~E.~Ney}
{Large deviations in $\re^d$}
{Let $X_1,X_2,\dots$ be i.\,i.\,d.\ random variables taking values in $\re^d$, $S_n=\sum_1^n
X_i$, $\Lambda(\alpha)=E\,e^{\langle \alpha,X_1\rangle}$ for $\alpha\in\re^d$, and ${\cal
D}(\Lambda)=\{\,\alpha\from\Lambda(\alpha)<\infty\,\}$. If ${\cal D}(\Lambda)$ does not
contain a neighborhood of the origin, then the level sets of
$\Lambda^{\!*}(x)=\sup_\alpha[\langle\alpha,x\rangle-\Lambda(\alpha)]$, $x\in\re^d$, will not
be compact, and the large deviation principle upper bound may fail. However, if the level sets
of $\Lambda^{\!*}$ can be suitably approximated by half-spaces, then an upper bound can be
proved. Necessary and sufficient conditions are given for such an approximation to be
possible. They boil down to the property that the gererating functions of certain marginal
random variables should not be degenerate (i.\,e. $\not\equiv\infty$ away from 0).
The above results are extended to approximation and separation theorems for the conjugate
$f^*$ of an essentially arbitrary convex function $f$. The hypotheses are expressed in terms
of the domain ${\cal D}(f)$. This leads to a classification of the sections of $f^*$ into
\lq\lq elliptic\rq\rq, \lq\lq parabolic\rq\rq\ and \lq\lq hyperbolic\rq\rq\ classes, which
are natural extensions of the conic sections.}
\abstract{Esa Nummelin}
{A matrix representation for the one-dimensional transfer operator}
{We consider the transfer operator ${\cal L}$ defined by
$$
{\cal L} f(i_{-\infty}^{0})=\sum_{i_1} l(i_{-\infty}^{1})f(i_{-\infty}^1)
$$
where $i_{-\infty}^{0}\in A^{\times\na_{-}}$, $A$ is a finite alphabet, $l$ and
$f$ are lower semicontinuous non-negative functions on $A^{\times\na_{-}}$. We
construct a non-negative matrix $Q$ with index set $S$ equal to the collection
of finite sequences of $A$-symbols, and such that $$
{\cal L} I_{j^*}=\sum_{i^*}Q(i^*,j^*)I_{i^*},
$$
where $I_{i^*}$ is the indicator of a
cylinder $i^*\in S$.
Under the usual variation conditions we establish positive and geometric recurrence properties
of~$Q$. These are related to the eigenvalue problem for the transfer operator ${\cal L}$
(Ruelle's Perron-Frobenius theorem).}
\abstract{E.\,A.~Pechersky}
{The large deviations for a simple information network}
{\def\rahmen#1#2{
\vbox{\hrule
\hbox
{\vrule
\hskip#1
\vbox{\vskip#1\relax
#2%
\vskip#1}%
\hskip#1
\vrule}
\hrule}}%
\def\wort#1{\rahmen{1.5pt}{\hbox{#1}}}%
We consider a tandem system as on this picture
$$
\longrightarrow\vcenter{\hbox{\wort{\kern10pt 1\kern10pt}}}
\longrightarrow\vcenter{\hbox{\wort{\kern10pt 2\kern10pt}}}\longrightarrow
$$
defined by i.\,i.\,d.\ vectors $(\tau_i,\xi^1_i,\xi_i^2)$. These $\tau_i$ are intervals
between messages and the $\xi_i^j$ are the times for transmitting messages through the $j$-th
node. We assume that $P(\tau_i>x)=e^{-\lambda x}$, $E\xi_i^j=\mu_j$, and
$\varphi_j(\theta)=\smash{Ee^{\theta\xi_1^j}}$. Then
$(1/x)\log P(\omega>x)\to-\min\{\beta_1,\beta_2\}$, where $\omega$ is the total
waiting time of a message in the tandem, and the $\beta_j$ are defined by the equation
$\beta_j=\lambda[\varphi_j(\beta_j){-1}]$.}
\abstract{Ch.~Pfister}
{Large deviations and the isoperimetric problem in Ising model}
{The rate function of the empirical magnetization is computed ex\-plicit\-ly in the case of
coexistence of phases. The rate function is given by the minimum of a variant of the
classical isoperimetric problem. The computation is done in two dimensions. If $\tau(n)$ is
the surface tension in the direction $n\in\re^2$, $\|n\|=1$, then for $-m^*0$, one has
$$
\Pr\{S_N=b\}={2\over \sqrt{2\pi D_{N,b}}}\exp\{-I_N(b)\}(1+o_N(1)),
$$
where $D_{N,b}$ is the \lq\lq tilted\rq\rq\ variance, and $I_N(\,\cdot\,)$ is
the rate function. In the complementary region $b-E(S_N)\le-c\,N^\gamma$,
$\gamma>\nu/(\nu+1)$, $c>0$, one has
$$
{\ln\Pr\{S_N=b\}\over(E(S_N)-b)^{(\nu-1)/\nu}}=O_N(1).
$$
So the region of deviations around $E(S_N)-N^{\nu/(\nu+1)}$ contains a threshold
where the condensation of microscopic $(\sim \ln N)$ droplets to macroscopic
droplet $(\sim N^\kappa$ with $\kappa\ge 1/(\nu+1))$ takes place.}
\abstract{Herbert Spohn}
{Large scale dynamics in stochastic models for interfaces}
{The statistical mechanics of surfaces is modelled conveniently in terms of effective
interface models. They are given by a real valued field, $\phi$, over the lattice
$\zee^d$. The surface is the graph of this function. The field has the energy
$$
H=\sum_{\langle x,y\rangle}V(\phi(x)-\phi(y)),
$$
where $\langle x,y\rangle$ is a pair of nearest neighbors and $V$ is convex and bounded as
$V(\phi)\ge c|\phi|^{1+\delta}$, $\delta>0$. Clearly $H$ is invariant under the global shift
$\phi(x)\to\phi(x)+a$, which is needed to have the interpretation of a surface
energy. To $H$ there corresponds a $d$-parameter family of Gibbs measures. They should be
thought of being defined on the difference variables $\phi(y)-\phi(x)$, $|x-y|=1$. They are
defined by taking the infinite volume limit at fixed tilt, $\phi(x)=u\cdot x$ for
$x\in\partial\Lambda$.
We consider pure relaxational dynamics
$$
d\phi_t(x)={\partial H\over \partial\phi(x)}(\phi_t)\,dt+dW_t(x)
$$
with independent Brownian motions of each site. The goal is to prove a law of large numbers
in the form
$$
\lim_{\varepsilon\to0}\varepsilon^d\sum_x f(\varepsilon
x)\varepsilon\phi_{\varepsilon^{-2}t}(x)=\int d^d r\,f(r)h(r,t).\eqno(*)
$$
The macroscopic height profile should satisfy
$$
{\partial\over\partial t}h_t=\mu\sum_{\alpha=1}^d{\partial\over\partial
r_\alpha}\sigma_\alpha(\nabla h_t)
$$
with $\mu$ the mobility and $\sigma$ the surface tension, $\sigma_\alpha(u)={\partial \over
\partial u_\alpha}\sigma(u)$. Elements of the proof of ($*$) are discussed.}
\abstract{Josef Steinebach}
{Exponential large deviations of the mean under special spherical
distributions}
{Consider an $n$-dimensional sample $X=(X_1,\dots,X_n)$ under a spherical
distribution, i.\,e.\ $X=\mu+e$, $\mu\in\re^n$, where the distribution of the
error vector $e$ has a $\lambda^n$-density
$$
f(x;g)=c(n;g)g(\|x\|^2),\quad x\in\re^n,\eqno (1)
$$
generated by a nonnegative measurable function $g$ with positive normalization
$c(n;g)$.
We are interested in convergence rates of the least squares estimate of a
possible common mean of $X_1,\dots,X_n$, that is, we want to investigate the large
deviations of
$$
P(A_n)=P(|\Xbar_n-\mubar_{\cdot}|>\varepsilon)=2P\biggl(\sum_{i=1}^n(X_i-\mu_i)>\varepsilon
\biggr),
$$
where $\varepsilon>0$, $\Xbar_n=(1/n)\sum_{i=1}^n X_i$ and $\mubar_{\cdot}=
(1/n)\sum_{i=1}^n \mu_i$.
For a class of spherical distributions generated by a function $g$ of type
$$
g(r^2)=a\,r^b e^{-cr^d},\quad r>0,\eqno(2)
$$
($a$, $b$, $c$, $d$ positive constants), the following large deviation results can be
established:\medskip
\noindent{\bf Theorem. }Under spherical distribution of $X$ according to (1) with $g$ as in
(2), we have as $n\to\infty$,\smallskip
\item{(i)} if $d\ge 2$, then $\log P(A_n)\sim -c(n\varepsilon^2)^{d/2},$
\item{(ii)} if $10$\break (\lq\lq no exponential rate\rq\rq).}
\abstract{Alain-Sol Sznitman}
{Brownian motion in a Poissonian potential}
{I describe in this talk certain large deviation principles which govern the behavior of
Brownian motion moving in a typical Poissonian potential. These large deviation principles
involve the construction, via a shape theorem, very much in the spirit of first passage
percolation, of certain constants which generalize the Lyapounov exponents in one dimension.
These large deviation results then enable us to study Brownian motion with a constant drift
$h$ moving in the same potential and to describe the transition of regime which occurs between
small $h$ and large $h$.}
\abstract{Srinivasa~R.\,S.~Varadhan (joint work with S.~Olla and H.-T.~Yau)}
{Hydrodynamic limit for Hamiltonian systems with noise}
{We consider a Hamiltonian system of $N$ particles in the phase space $({\Bbb T}^3\times
{\Bbb R}^3)^N$ evolving under a short range pair potential of the form $V((x-y)/\varepsilon)$
where $\varepsilon$ is a scale parameter related to $N$ by $N\varepsilon^3=1$. We aim to
establish a relationship between the Hamiltonian dynamics and the corresponding Euler
equation derived by the thermodynamic formalism. In order to achieve this some small noise is
added to the velocity components in such a way as not to destroy the conservation of momenta
and energy. The classical Hamiltonian is replaced by one with bounded velocities. Then in a
regime where the Euler equation has a smooth solution, we show that a suitably prepared local
Gibbs family of densities on the phase space constructed from the solutions of Euler equation
is close to the corresponding time solution of the Hamiltonian system with noise.}
\abstract{Kongming Wang (joint work with J.-D.~Deuschel)}
{Large deviations for the occupation time functional of a Poisson system of independent
Brownian particles}
{Let $\{N_s\}_{s\ge0}$ be the evolution system starting from $N_0$, a Poisson point
process with intensity $dx$, where each particles independently follows the law of a
$d$-dimensional Brownian motion. Take $\varphi\in L^1(\re^d)$ with compact support, and let
$N_s(\varphi)=\smash{\sum_{x\in{\rm supp}(N_0)}\varphi(B_s^x)}$ and $L_T(\varphi)(t)=\int_0^t
N_{Ts}(\varphi)\,ds$.
We study the large deviations and central limit theorems for $L_T(\varphi)(t)$, $t\in[0,1]$.
In the lower (recurrent) dimensions $d=1,2$ we have critical orders $T^{1/2}$ and $T/\log
T$, whereas in higher (transient) dimensions we have the usual order $T$. We
give explicit expressions for the corresponding rate functions and covariance
functionals and derive some asymptotic microcanonical distributions.}
\abstract{W.\,A.~Woyczy\'nski (joint work with J.~Szulga and J.\,A.~Mann)}
{Large deviation techniques in analysis of monomolecular layers}
{The partition function of a statistical mechanical system of hard oval shaped molecules
moving on real line and with rotational degree of freedom, is replaced by its Poissonized
version, which, in turn, can be analyzed via large deviation techniques when considered in
the thermo\-dynamic limit.}
\abstract{H.-T.~Yau (joint work with Shenglin Lu)}
{Spectral gap and logarithmic Sobolev inequality for the Glauber and Ka\-wa\-sa\-ki dynamics}
{We prove that there is a spectral gap uniformly with respect to the volume and boundary
condition for the Glauber dynamics. If the Glauber dynamics is replaced by the Kawasaki
dynamics then the spectral gap is proved to shrink by $1/L^2$. We assume some mixing
conditions for the Gibbs state to hold. Furthermore, we prove a similar result for the
logarithmic Sobolev inequality except for the Kawasaki dynamics for dimension $d>1$.}
\abstract{Sandy~L.~Zabell (joint work with I.\,H.~Dinwoodie)}
{Large deviations for sequences of mixtures}
{Say that a family $\{\,P_\theta^n\from\theta\in\Theta,\,n\ge1\,\}$ is {\it exponentially
continuous\/} if when $\theta_n\to\theta$, one has that $\{P_{\theta_n}^n\}$ satisfies a
large deviation principle with rate function $\lambda(\theta,v)$ for each $\theta\in\Theta$.
In this case, if $\mu$ is a measure on $\Theta$, then $P^n:=\int_\Theta P_\theta^n\,d\mu$
satisfies a large deviation principle with rate function
$\inf\{\,\lambda(\theta,v)\from\theta\in\hbox{supp}(\mu)\,\}$ provided $\Theta$ is compact
and given weak regularity conditions; see Dinwoodie and Zabell, {\it Annals of
Probability,\/} 1992. In this talk I discuss to what extent the conditions of this theorem
can be weakened; a necessary and sufficient condition for exponential continuity is given;
and a relationship with epiconvergence is discussed.}
\bigskip\bigskip\noindent
{\titlebold Berichterstatter:} Gerda Schacher und Uwe Schmock
\vfill\break
\leftline{\large E-Mail Addresses}
\bigskip
\bigskip
\tabskip=0pt
\halign to\hsize{#\quad\hfil&#\hfil\tabskip=\centering\cr
A. Ben Arous&BenArous@dmi.ens.fr\cr
A. Bovier&Bovier@IAAS-Berlin.dbp.de\cr
E. Bolthausen&K563720@czhrzu1a.bitnet\cr
J.\,T. Cox&JTCox@mailbox.syr.edu\cr
D.\,A. Dawson&DADawson@carleton.ca\cr
F. den Hollander&denHolla@math.ruu.nl\cr
J.-D. Deuschel&Deuschel@math.ethz.ch\cr
R.\,S. Ellis&RSEllis@math.umass.edu\cr
K. Fleischmann&Fleischmann@IAAS-Berlin.dbp.de\cr
M. Freidlin&MIF@athena.umd.edu\cr
J. G\"artner&JG@euklid.math.TU-Berlin.dbp.de\cr
F. G\"otze&Goetze@math4.Mathematik.Uni-Bielefeld.de\cr
A. Greven&E91@bhdurz1.bitnet\cr
R. Kotecky&Kotecky@cspuni12\cr
T.-Y. Lee&TYL@hilda.umd.edu\cr
M. Major&h1131Maj@ella.hu\cr
S. Olla&Olla@vaxtvm.infn.it.bitnet\cr
U. Schmock&Schmock@amath.unizh.ch\cr
H. Spohn&Herbert.Spohn@stat.Physik.Uni-Muenchen.dbp.de\cr
A.-S. Sznitman&Sznitman@math.ethz.ch\cr
S.\,R.\,S. Varadhan&Varadhan@csd22.cs.nyu.edu\cr
K. Wang&K563750@czhrzu1a.bitnet\cr
W\kern-.5mm.\,A. Woyczy\'nski&WAW@po.cwru.edu\cr
H.-T. Yau&Yau@acf9.nyu.edu\cr
S.\,L. Zabell&Zabell@math.nwu.edu\cr
}
\vfil\end