Probability Abstracts 74

This document contains abstracts 2177-2237. They have been mailed on April 29, 2003.

2177. STOCHASTIC RESONANCE IN TWO-STATE MARKOV CHAINS 

Peter Imkeller, Ilya Pavlyukevich

In this paper we introduce a model which provides a new approach to the
phenomenon of stochastic resonance. It is based on the study of the properties
of the stationary distribution of the underlying stochastic process. We derive
the formula for the spectral power amplification coefficient, study its
asymptotic properties and dependence on parameters.

pavljuke@mathematik.hu-berlin.de

2178. ON THE CHARACTERISATION OF DUAL STATISTICALLY MONOTONE METRICS 

P. Gibilisco, T. Isola

Hasegawa and Petz introduced the notion of dual statistically monotone
metrics. They also gave a characterisation theorem showing that
Wigner-Yanase-Dyson metrics are the only members of the dual family. In this
paper we show that the characterisation theorem holds true under more general
hypotheses.

isola@mat.uniroma2.it

2179. RANDOM WALKS ON FKG-HORIZONTALLY ORIENTED LATTICES 

Nadine Guillotin-Plantard, Arnaud Le Ny

We study the asymptotic behavior of the simple random walk on oriented
version of $\mathbb{Z}^2$. The considered latticesare not directed on the
vertical axis but unidirectional on the horizontal one, with symmetric random
orientations which are positively correlated. We prove that the simple random
walk is transient and also prove a functionnal limit theorem in the space of
cadlag functions, with an unconventional normalization.

leny@eurandom.tue.nl

2180. BERNOULLI SIEVE 

Alexander Gnedin

Bernoulli sieve is a recursive construction of a random composition (ordered
partition) of integer $n$. This composition can be induced by sampling from a
random discrete distribution which has frequencies equal to the sizes of
component intervals of a stick-breaking interval partition of $[0,1]$. We
exploit Markov property of the composition and its renewal representation to
derive asymptotics of the moments and to prove a central limit theorem for the
number of parts.

gnedin@math.uu.nl

2181. LATE POINTS FOR RANDOM WALKS IN TWO DIMENSIONS 

Amir Dembo, Yuval Peres, Jay Rosen, Ofer Zeitouni

Let $T_n(x)$ denote the time of first visit of a point $x$ on the lattice
torus $Z_n^2=Z^2/nZ^2$ by the simple random walk. The size of the set of
$\alpha,n$-late points $L_n(\alpha)=\{x\in Z_n^2: T_n(x)\geq 4\alpha(n\log
n)^2/\pi\}$ is approximately $n^{2(1-\alpha)}$, for $\alpha\in (0,1)$
($L_n(\alpha)$ is empty if $\alpha>1$ and $n$ is large enough). These sets have
interesting clustering and fractal properties: we show that for $\beta \in
(0,1)$ a disc of radius $n^\beta$ centered at non-random $x$ typically contains
about $n^{2 \beta(1-\alpha/\beta^2)}$ points from $L_n(\alpha)$ (and is empty
if $\beta < \sqrt{\alpha}$), whereas choosing the center $x$ of the disc
uniformly in $L_n(\alpha)$ boosts the typical number $\alpha,n$-late points in
it to $n^{2\beta(1-\alpha)}$. We also estimate the typical number of pairs of
$\alpha,n$-late points within distance $n^\beta$ of each other; this typical
number can be significantly smaller than the expected number of such pairs,
calculated by Brummelhuis and Hilhorst (1991). On the other hand, our results
show that the number of ordered pairs of late points within distance $n^\beta$
of each other, is larger than what one might predict by multiplying the total
number of late points by the number of late points in a disc of radius
$n^\beta$ centered at a typical late point.

zeitouni@ee.technion.ac.il

2182. SURFACE ORDER LARGE DEVIATIONS FOR 2D FK-PERCOLATION AND POTTS MODELS 

O. Couronne, R. J. Messikh

By adapting the renormalization techniques of Pisztora, we establish surface
order large deviations estimates for FK-percolation on $\Z^2$ with parameter
$q\geq 1$ and for the corresponding Potts models. Our results are valid up to
the exponential decay threshold of dual connectivities which is widely believed
to agree with the critical point.

reda-jurg.messikh@math.u-psud.fr

2183. DISCRETE LOEWNER EVOLUTION 

Robert O. Bauer

We study a one parameter family of discrete Loewner evolutions driven by a
random walk on the real line. We show that it converges to the stochastic
Loewner evolution (SLE) under rescaling. We show that the discrete Loewner
evolution satisfies Markovian-type and symmetry properties analogous to SLE,
and establish a phase transition property for the discrete Loewner evolution
when the parameter equals 4.

rbauer@math.uiuc.edu

2184. $SLE(\KAPPA,\RHO)$ MARTINGALES AND DUALITY 

Julien Dubedat

Various features of the two-parameter family of Schramm-Loewner Evolutions
$\SLE(\kappa,\rho)$ are studied. In particular, we derive certain restriction
properties leading to a ``strong duality'' conjecture, which is an identity in
law between the outer boundary of a variant of the $\SLE (\kappa)$ process for
$\kappa \ge 4$ and a variant of the $\SLE(16/\kappa)$ process. In this context,
a natural path-decomposition of the process $\SLE(\kappa,\kappa-4)$ is derived,
which leads to define a family of stable, hull-valued subordinators.

dubedat@clipper.ens.fr

2185. POINCARE INEQUALITIES FOR INHOMOGENEOUS BERNOULLI MEASURES 

Jeremy Quastel, Horng-Tzer Yau 

We consider inhomogeneous Bernoulli measures of the form $\prod_{x\in\Lambda}
p_x$ where $p_x$ are prescribed and uniformly bounded above and below away from
$0$ and $1$. Poincare (spectral gap) inequalities are proved for the Glauber
and Kawasaki dynamics, with constants of the same order as in the homogeneous
case.

quastel@math.toronto.edu

2186. ON $L_1$-DISTANCE BETWEEN FIRST EXIT TIMES FROM TWO REGIONS 

Nikolai Dokuchaev

First exit times from regions and their dependence on variations of
boundaries are discussed for diffusion processes. The paper presents an
estimate of $L_1$-distance between exit times from two regions via expectations
of exit times.

ndokuch@uwimona.edu.jm

2187. RENEWAL THEORY ON THE ORIENTED TREE 

Sara Brofferio

The affine group of a tree is the group of the isometries of a homogeneous
tree that fix an end of its boundary. Consider a probability measure on this
group and the associated random walk. The main goal of this paper is to
determine the accumulation points of the potential kernel at the infinity. In
particular we show that under suitable regularity hypotheses this kernel can be
continuously extended to the tree boundary and we determine the limit measures.

brofferio@finanz.math.tu-graz.ac.at

2188. ANCHORED EXPANSION, PERCOLATION AND SPEED 

Dayue Chen and Yuval Peres, with an appendix by Gabor Pete

Benjamini, Lyons and Schramm (1999) considered properties of an infinite
graph G, and the simple random walk on it, that are preserved by random
perturbations. In this paper we solve several problems raised by those authors.
The anchored expansion constant is a variant of the Cheeger constant; its
positivity implies positive lower speed for the simple random walk, as shown by
Vir\'ag (2000). We prove that if G has a positive anchored expansion constant
then so does every infinite cluster of independent percolation with parameter p
sufficiently close to 1; a better estimate for the parameters p where this
holds is in the appendix. We also show that positivity of the anchored
expansion constant is preserved under a random stretch if, and only if, the
stretching law has an exponential tail. We then study simple random walk in the
infinite percolation cluster in Cayley graphs of certain amenable groups known
as ``lamplighter groups''. We prove that zero speed for random walk on a
lamplighter group implies zero speed for random walk on an infinite cluster,
for any supercritical percolation parameter p. For p large enough, we also
establish the converse.

peres@stat.berkeley.edu

2189. BEHAVIOR DOMINATED BY SLOW PARTICLES IN A DISORDERED ASYMMETRIC EXCLUSION PROCESS 

Ilie Grigorescu, Min Kang, and Timo Seppalainen

We study the large space and time scale behavior of a totally asymmetric,
nearest-neighbor exclusion process in one dimension with random jump rates
attached to the particles. When slow particles are sufficiently rare the system
has a phase transition. At low densities there are no equilibrium
distributions, and on the hydrodynamic scale the initial profile is transported
rigidly. We elaborate this situation further by finding the correct order of
the correction from the hydrodynamic limit, together with distributional bounds
averaged over the disorder. We consider two settings, a macroscopically
constant low density profile and the outflow from a large jam.

seppalai@math.wisc.edu

2190. RANDOM PLANAR CURVES AND SCHRAMM-LOEWNER EVOLUTIONS 

Wendelin Werner

We review some of the results that have been derived in the last years on
conformal invariance, scaling limits and properties of some two-dimensional
random curves. In particular, we describe the intuitive ideas that lead to the
definition of the Schramm-Loewner evolutions SLE, we define these objects,
study its various properties, show how to compute (probabilities, critical
exponents) using SLE, relate SLE to planar Brownian motions (i.e. the
determination of the critical exponents), planar self-avoiding walks, critical
percolation, loop-erased random walks and uniform spanning trees.

wendelin.werner@math.u-psud.fr

2191. EXPLICIT FORMULAS FOR HOOK WALKS ON CONTINUAL YOUNG DIAGRAMS 

Dan Romik

We consider, following the work of S. Kerov, random walks which are
continuous-space generalizations of the Hook Walks defined by
Greene-Nijenhuis-Wilf, performed under the graph of a continual Young diagram.
The limiting point of these walks is a point on the graph of the diagram. We
present several explicit formulas giving the probability densities of these
limiting points in terms of the shape of the diagram. This partially resolves a
conjecture of Kerov concerning an explicit formula for the so-called Markov
transform. We also present two inverse formulas, reconstructing the shape of
the diagram in terms of the densities of the limiting point of the walks. One
of these two formulas can be interepreted as an inverse formula for the Markov
transform. As a corollary, some new integration identities are derived.

romik@wisdom.weizmann.ac.il

2192. RANDOM SURFACES: LARGE DEVIATIONS PRINCIPLES AND GRADIENT GIBBS MEASURE CLASSIFICATIONS 

Scott Sheffield

We study (discretized) ``random surfaces,'' which are random functions from
$\Bbb Z^d$ (or large subsets of $\Bbb Z^d$) to $E$, where $E$ is $\Bbb Z$ or
$\Bbb R$. Their laws are determined by convex, nearest-neighbor, gradient Gibbs
potentials (which may assume the value $\infty$) that are invariant under
translation by a full-rank sublattice $\mathcal L$ of $\Bbb Z^d$; they include
many discrete and continuous height function models (e.g., domino tilings,
square ice, the harmonic crystal, the Ginzburg-Landau $\nabla \phi$ interface
model, the linear solid-on-solid model) as special cases.
  A {\it gradient phase} is an $\mathcal L$-ergodic gradient Gibbs measure with
finite specific free energy. A gradient phase is {\it smooth} if it is the
gradient of an ordinary Gibbs measure; otherwise it is {\it rough}. We prove a
{\it variational principle}--characterizing gradient phases of a given slope as
minimizers of the specific free energy--and an empirical measure large
deviations principle (with a unique rate function minimizer) for random
surfaces on mesh approximations of bounded domains.
  We also prove that the surface tension is strictly convex and that if $u$ is
in the interior of the space of finite-surface tension slopes, then there
exists a minimal energy gradient phase $\mu_u$ of slope $u$. This $\mu_u$ is
unique if at least one of the following holds: $E = \Bbb R$, $d \in \{1, 2 \}$,
there exists a rough gradient phase of slope $u$, or $u$ is irrational. When
$d=2$ and $E = \Bbb Z$, the slopes of {\it all} smooth phases (a.k.a. {\it
facets}) lie in the dual lattice of $\mathcal L$.

scott@math.stanford.edu

2193. STABILITY OF NONLINEAR FILTERS IN NON-MIXING CASE 

P. Chigansky, R. Liptser

The nonlinear filtering equation is said to be stable if it ``forgets'' the
initial condition. It is known that the filter might be unstable even if the
signal is ergodic Markov chain. In general, the filtering stability requires
stronger signal ergodicity provided by, so called, mixing condition. The latter
is formulated in terms of the transition probability density of the signal. The
most restrictive requirement of the mixing condition is a uniform positiveness
of this density. We show that this requirement might be relaxed regardless of
an observation process structure.

pavelm@eng.tau.ac.il

2194. RANDOM WALKS IN RANDOM ENVIRONMENT: WHAT A SINGLE TRAJECTORY TELLS 

Omer Adelman, Nathana\"el Enriquez

We present a procedure that determines the law of a random walk in an iid
random environment as a function of a single "typical" trajectory. We indicate
when the trajectory characterizes the law of the environment, and we say how
this law can be determined. We then show how independent trajectories having
the distribution of the original walk can be generated as functions of the
single observed trajectory.

enriquez@ccr.jussieu.fr

2195. A RIGOROUS DERIVATION OF SMOLUCHOWSKI'S EQUATION IN THE MODERATE LIMIT 

Stefan Grosskinsky, Christian Klingenberg and Karl Oelschlaeger

Smoluchowski's equation is a macroscopic description of a many particle
system with coagulation and shattering interactions. We give a microscopic
model of the system from which we derive this equation rigorously. Provided the
existence of a unique and sufficiently regular solution of Smoluchowski's
equation, we prove the law of large numbers for the empirical processes. In
contrast to previous derivations we assume a moderate scaling of the particle
interaction, enabling us to estimate the critical fluctuation terms by using
martingale inequalities. This approach can be justified in the regime of high
temperatures and particle densities, which is of special interest in
astrophysical studies and where previous derivations do not apply.

stefang@mathematik.tu-muenchen.de

2196. THE BROWNIAN WEB: CHARACTERIZATION AND CONVERGENCE 

L. R. G. Fontes, M. Isopi, C. M. Newman, K. Ravishankar

The Brownian Web (BW) is the random network formally consisting of the paths
of coalescing one-dimensional Brownian motions starting from every space-time
point in ${\mathbb R}\times{\mathbb R}$. We extend the earlier work of Arratia
and of T\'oth and Werner by providing characterization and convergence results
for the BW distribution, including convergence of the system of all coalescing
random walkssktop/brownian web/finale/arXiv submits/bweb.tex to the BW under
diffusive space-time scaling. We also provide characterization and convergence
results for the Double Brownian Web, which combines the BW with its dual
process of coalescing Brownian motions moving backwards in time, with forward
and backward paths ``reflecting'' off each other. For the BW, deterministic
space-time points are almost surely of ``type'' $(0,1)$ -- {\em zero} paths
into the point from the past and exactly {\em one} path out of the point to the
future; we determine the Hausdorff dimension for all types that actually occur:
dimension 2 for type $(0,1)$, 3/2 for $(1,1)$ and $(0,2)$, 1 for $(1,2)$, and 0
for $(2,1)$ and $(0,3)$.

isopi@mat.uniroma1.it

2197. ON THE DISTRIBUTION OF LATTICE POINTS IN THIN ANNULI 

C.P. Hughes and Z. Rudnick

We show that the number of lattice points lying in a thin annulus has a
Gaussian value distribution if the width of the annulus tends to zero
sufficiently slowly as we increase the inner radius.

hughes@aimath.org

2198. ERGODICITY OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION 

Martin Hairer

We study the ergodic properties of finite-dimensional systems of SDEs driven
by non-degenerate additive fractional Brownian motion with arbitrary Hurst
parameter $H\in(0,1)$. A general framework is constructed to make precise the
notions of ``invariant measure'' and ``stationary state'' for such a system. We
then prove under rather weak dissipativity conditions that such an SDE
possesses a unique stationary solution and that the convergence rate of an
arbitrary solution towards the stationary one is (at least) algebraic. A lower
bound on the exponent is also given.

martin.hairer@math.unige.ch

2199. NEW COINS FROM OLD: COMPUTING WITH UNKNOWN BIAS 

Elchanan Mossel and Yuval Peres

Suppose that we are given a function f : (0,1) -> (0,1) and, for some unknown
p in (0,1), a sequence of independent tosses of a p-coin (i.e., a coin with
probability p of ``heads'').
  For which functions f is it possible to simulate an f(p)-coin?; This question
was raised by S. Asmussen and J. Propp. A simple simulation scheme for the
constant function 1/2 was described by von Neumann (1951); this scheme can be
easily implemented using a finite automaton. We prove that in general, an
f(p)-coin can be simulated by a finite automaton for all p in (0,1), if and
only if f is a rational function over Q. We also show that if an f(p)-coin can
be simulated by a pushdown automaton, then f is an algebraic function over Q;
however, pushdown automata can simulate f(p)-coins for certain non-rational
functions such as the square root of p. These results complement the work of
Keane and O'Brien (1994), who determined the functions $f$ for which an
f(p)-coin can be simulated when there are no computational restrictions on the
simulation scheme.

mossel@stat.berkeley.edu

2200. ASYMPTOTIC SHAPE FOR THE CHEMICAL DISTANCE AND FIRST-PASSAGE PERCOLATION IN RANDOM ENVIRONMENT 

Olivier Garet, Regine Marchand

The aim of this paper is to generalize the well-known asymptotic shape result
for first-passage percolation on $\Zd$ to first-passage percolation on a random
environment given by the infinite cluster of a supercritical Bernoulli
percolation model. We prove the convergence of the renormalized set of wet
points to a deterministic shape that does not depend on the random environment.
As a special case of the previous result, we obtain an asymptotic shape theorem
for the chemical distance in supercritical Bernoulli percolation. We also prove
a flat edge result. Some various examples are also given.

Olivier.Garet@labomath.univ-orleans.fr

2201. WIGNER-YANASE INFORMATION ON QUANTUM STATE SPACE:THE GEOMETRIC APPROACH 

P. Gibilisco, T. Isola

In the search of appropriate riemannian metrics on quantum state space the
concept of statistical monotonicity, or contraction under coarse graining, has
been proposed by Chentsov. The metrics with this property have been classified
by Petz. All the elements of this family of geometries can be seen as quantum
analogues of Fisher information. Although there exists a number of general
theorems sheding light on this subject, many natural questions, also stemming
from applications, are still open. In this paper we discuss a particular member
of the family, the Wigner-Yanase information.
 Using a well-known approach that mimics the classical pull-back approach to
Fisher information, we are able to give explicit formulae for the geodesic
distance, the geodesic path, the sectional and scalar curvatures associated to
Wigner-Yanase information. Moreover we show that this is the only monotone
metric for which such an approach is possible.

isola@mat.uniroma2.it

2202. ON THE P-LOGARITHMIC AND ALPHA-POWER DIVERGENCE MEASURES IN INFORMATION THEORY 

Sever Silvestru Dragomir

In this paper we introduce the concepts of p-logarithmic and Alpha-power
divergence measures and point out a number of basic results.

sever.dragomir@vu.edu.au

2203. TWO-DIMENSIONAL POISSON TREES CONVERGE TO THE BROWNIAN WEB 

P. A. Ferrari, L. R. G. Fontes, Xian-Yuan Wu

The two-dimensional Poisson tree is a family of continuous time
one-dimensional random walks with bounded uniform jumps in R. The walks start
at the space-time points of a homogeneous Poisson process in R^2 and are in
fact constructed as a function of the point process. This tree was introduced
by Ferrari, Landim and Thorisson. The Brownian web can be roughly described as
a family of coalescing one-dimensional Brownian motions starting at all times
in R and at all points of R. It was introduced by Arratia; a variant was then
studied by Toth and Werner. Fontes, Isopi, Newman and Ravishankar have recently
derived criteria for weak convergence to the Brownian web under a suitable
topology. By verifying those criteria, we prove that, when properly rescaled,
the Poisson tree converges weakly to the Brownian web.

pablo@ime.usp.br

2204. FOCK FACTORIZATIONS, AND DECOMPOSITIONS OF THE $L^2$ SPACES OVER GENERAL LEVY PROCESSES 

Anatoly Vershik, Natalia Tsilevich

We explicitly construct and study an isometry between the spaces of square
integrable functionals of an arbitrary Levy process and a vector-valued
Gaussian white noise. In particular, we obtain explicit formulas for this
isometry at the level of multiplicative functionals and at the level of
orthogonal decompositions, as well as find its kernel. We consider in detail
the central special case: the isometry between the $L^2$ spaces over a Poisson
process and the corresponding white noise. The key role in our considerations
is played by the notion of measure and Hilbert factorizations and related
notions of multiplicative and additive functionals and logarithm. The obtained
results allow us to introduce a canonical Fock structure (an analogue of the
Wiener--Ito decomposition) in the $L^2$ space over an arbitrary Levy process.
An application to the representation theory of current groups is considered. An
example of a non-Fock factorization is given.

natalia@pdmi.ras.ru

2205. IDENTIFYING SEVERAL BIASED COINS ENCOUNTERED BY A HIDDEN RANDOM WALK 

David A. Levin and Yuval Peres

Suppose that attached to each site z in Z is a coin with bias theta(z), and
only finitely many of these coins have non-zero bias. Allow a simple random
walker to generate observations by tossing, at each move, the coin attached to
its current position. Then we can determine the biases {theta(z) : z in Z},
using only the outcomes of these coin tosses and no information about the path
of the random walker, up to a shift and reflection of Z. This generalizes a
result of Harris and Keane.

levin@math.utah.edu

2206. ON SOME INEQUALITIES FOR GAUSSIAN MEASURES 

Rafa{\l} Lata{\l}a

We review several inequalities concerning Gaussian measures - isoperimetric
inequality, Ehrhard's inequality, Bobkov's inequality, S-inequality and
correlation conjecture.

rlatala@mimuw.edu.pl

2207. RANDOM PATH REPRESENTATION AND SHARP CORRELATIONS ASYMPTOTICS AT HIGH-TEMPERATURES 

M. Campanino, D. Ioffe and Y. Velenik

We recently introduced a robust approach to the derivation of sharp
asymptotic formula for correlation functions of statistical mechanics models in
the high-temperature regime. We describe its application to the nonperturbative
proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding
walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the
proof, in the Ising case, to arbitrary odd-odd correlation functions. We
discuss the fluctuations of connection paths (invariance principle), and relate
the variance of the limiting process to the geometry of the equidecay profiles.
Finally, we explain the relation between these results from Statistical
Mechanics and their counterparts in Quantum Field Theory.

velenik@cmi.univ-mrs.fr

2208. RIGOROUS NON-PERTURBATIVE ORNSTEIN-ZERNIKE THEORY FOR ISING FERROMAGNETS 

M. Campanino, D. Ioffe and Y. Velenik

We rigorously derive the Ornstein-Zernike asymptotics of the pair-correlation
functions for finite-range Ising ferromagnets in any dimensions and at any
temperature above critical.

velenik@cmi.univ-mrs.fr

2209. AGING AND SPIN-GLASS DYNAMICS 

G\'erard Ben-Arous

We survey the recent mathematical results about aging in certain simple
disordered models. We start by the Bouchaud trap model. We then survey the
results obtained for simple models of spin-glass dynamics, like the REM (the
Random Energy Model, which is well approximated by the Bouchaud model on the
complete graph), then the spherical Sherrington-Kirkpatrick model. We will
insist on the differences in phenomenology for different types of aging in
different time scales and different models. This talk is based on joint works
with A.Bovier, J.Cerny, A.Dembo, V.Gayrard, A.Guionnet, as well as works by
C.Newman, R.Fontes, M.Isopi, D.Stein.

gerard.benarous@epfl.ch

2210. SOME ASPECTS OF ADDITIVE COALESCENTS 

Jean Bertoin

We present some aspects of the so-called additive coalescence, with a focus
on its connections with random trees, Brownian excursion, certain bridges with
exchangeable increments, L\'evy processes, and sticky particle systems.

jbe@ccr.jussieu.fr

2211. LOCALIZATION-DELOCALIZATION PHENOMENA FOR RANDOM INTERFACES 

Erwin Bolthausen

We consider d-dimensional random surface models which for d=1 are the
standard (tied-down) random walks (considered as a random ``string''). In
higher dimensions, the one-dimensional (discrete) time parameter of the random
walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it.
The random surface is represented by real-valued random variables \phi_i, where
i is in \Z^d. A class of natural generalizations of the standard random walk
are gradient models whose laws are (formally) expressed as P(d\phi) =1/Z
\exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i,
  V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been
introduced in theoretical physics as (simplified) models for random interfaces
separating different phases. Of particular interest are
localization-delocalization phenomena, for instance for a surface interacting
with a wall by attracting or repulsive interactions, or both together. Another
example are so-called heteropolymers which have a noise-induced interaction.
Recently, there had been developments of new probabilistic tools for such
problems. Among them are: o Random walk representations of Helffer-Sj\"ostrand
type, o Multiscale analysis, o Connections with random trapping problems and
large deviations We give a survey of some of these developments.

eb@amath.unizh.ch

2212. ERGODIC CONVERGENCE RATES OF MARKOV PROCESSES--EIGENVALUES, INEQUALITIES AND ERGODIC THEORY 

Mu-Fa Chen

This paper consists of four parts. In the first part, we explain what
eigenvalues we are interested in and show the difficulties of the study on the
first (non-trivial) eigenvalue through examples. In the second part, we present
some (dual) variational formulas and explicit bounds for the first eigenvalue
of Laplacian on Riemannian manifolds or Jacobi matrices (Markov chains). Here,
a probabilistic approach--the coupling methods is adopted. In the third part,
we introduce recent lower bounds of several basic inequalities; these are based
on a generalization of Cheeger's approach which comes from Riemannian geometry.
In the last part, a diagram of nine different types of ergodicity and a table
of explicit criteria for them are presented. These criteria are motivated by
the weighted Hardy inequality which comes from Harmonic analysis.

mfchen@bnu.edu.cn

2213. TOEPLITZ DETERMINANTS, RANDOM GROWTH AND DETERMINANTAL PROCESSES 

Kurt Johansson

We summarize some of the recent developments which link certain problems in
combinatorial theory related to random growth to random matrix theory.

kurtj@math.kth.se

2214. CONFORMAL INVARIANCE, UNIVERSALITY, AND THE DIMENSION OF THE BROWNIAN FRONTIER 

Gregory Lawler

This paper describes joint work with Oded Schramm and Wendelin Werner
establishing the values of the planar Brownian intersection exponents from
which one derives the Hausdorff dimension of certain exceptional sets of planar
Brownian motion. In particular, we proof a conjecture of Mandelbrot that the
dimension of the frontier is 4/3. The proof uses a universality principle for
conformally invariant measures and a new process, the stochastic Loewner
evolution ($SLE$), introduced by Schramm. These ideas can be used to study
other planar lattice models from statistical physics at criticality. I discuss
applications to critical percolation on the triangular lattice, loop-erased
random walk, and self-avoiding walk.

lawler@math.cornell.edu

2215. BROWNIAN INTERSECTIONS, COVER TIMES AND THICK POINTS VIA TREES 

Yuval Peres

There is a close connection between intersections of Brownian motion paths
and percolation on trees. Recently, ideas from probability on trees were an
important component of the multifractal analysis of Brownian occupation
measure, in joint work with A. Dembo, J. Rosen and O. Zeitouni. As a
consequence, we proved two conjectures about simple random walk in two
dimensions: The first, due to Erd\H{o}s and Taylor (1960), involves the number
of visits to the most visited lattice site in the first $n$ steps of the walk.
The second, due to Aldous (1989), concerns the number of steps it takes a
simple random walk to cover all points of the $n$ by $n$ lattice torus. The
goal of the lecture is to relate how methods from probability on trees can be
applied to random walks and Brownian motion in Euclidean space.

peres@stat.berkeley.edu

2216. RENORMALIZATION, LARGE DEVIATIONS AND PHASE SEPARATION IN ISING AND PERCOLATION MODELS 

Agoston Pisztora

Phase separation is a fairly common physical phenomenon with examples
including the formation of water droplets from humid air (fog, rain), the
separation of a crystalline structure from an isotropic material such as a
liquid or even the formation of the sizzling gas bubbles when a soda can is
opened. It was recognized long ago (at least on a phenomenological level) that
systems exhibiting several phases in equilibrium can be described with an
appropriate variational principle: the phases arrange themselves in such a way
that the energy associated with the phase boundaries is minimal. Typically this
leads to an almost deterministic behavior and the phase boundaries are fairly
regular. However, when looked at from a microscopic point of view, the system
consists of a bunch of erratically moving molecules with relatively strong
short-range interaction and the simplicity of the above macroscopic description
looks more than miraculous. Indeed, when starting from the molecular level,
there are many more questions to be asked and understood: which are the phases
which we will see? why do only those occur? why are the phase boundaries sharp?
how should we find (define) the energy associated with the interfaces? Only
then can we ask the question: why does the system minimize this energy? It is
only in the last decade that a mathematically satisfactory understanding of
this phenomenon has been achieved. The main goal of the talk is to present the
current state of affairs focusing thereby on results obtained in joint works
with Raphael Cerf. The connection to fields of mathematics other than
probability theory or statistical mechanics will be highlighted; namely, to
geometric measure theory and to the calculus of variations.

pisztora@andrew.cmu.edu

2217. BIOLOGICAL SEQUENCE ANALYSIS 

T. P. Speed

This talk will review a little over a decade's research on applying certain
stochastic models to biological sequence analysis. The models themselves have a
longer history, going back over 30 years, although many novel variants have
arisen since that time. The function of the models in biological sequence
analysis is to summarize the information concerning what is known as a motif or
a domain in bioinformatics, and to provide a tool for discovering instances of
that motif or domain in a separate sequence segment. We will introduce the
motif models in stages, beginning from very simple, non-stochastic versions,
progressively becoming more complex, until we reach modern profile HMMs for
motifs. A second example will come from gene finding using sequence data from
one or two species, where generalized HMMs or generalized pair HMMs have proved
to be very effective.

terry@stat.berkeley.edu

2218. ESTIMATES FOR THE STRONG APPROXIMATION IN MULTIDIMENSIONAL CENTRAL LIMIT THEOREM 

A. Yu. Zaitsev

In a recent paper the author obtained optimal bounds for the strong Gaussian
approximation of sums of independent $\R^d$-valued random vectors with finite
exponential moments. The results may be considered as generalizations of
well-known results of Koml\'os--Major--Tusn\'ady and Sakhanenko. The dependence
of constants on the dimension $d$ and on distributions of summands is given
explicitly. Some related problems are discussed.

zaitsev@pdmi.ras.ru

2219. RANDOM WALKS IN RANDOM ENVIRONMENTS 

Ofer Zeitouni

Random walks in random environments (RWRE's) have been a source of surprising
phenomena and challenging problems since they began to be studied in the 70's.
Hitting times and, more recently, certain regeneration structures, have played
a major role in our understanding of RWRE's. We review these and provide some
hints on current research directions and challenges.

zeitouni@ee.technion.ac.il

2220. ADDITIVE FUNCTIONALS ON RANDOM SEARCH TREES 

Nevin Kapur

Search trees are fundamental data structures in computer science. We study
functionals on random search trees that satisfy recurrence relations of a
simple additive form. Many important functionals including the space
requirement, internal path length, and the so-called shape functional fall
under this framework. Our goal is to derive asymptotics of moments and identify
limiting distributions of these functionals under two commonly studied
probability models -- the random permutation model and the uniform model.
  For the random permutation model, our approach is based on establishing
transfer theorems that link the order of growth of the input into a particular
deterministic recurrence to the order of growth of the output. For the uniform
model, our approach is based on the complex-analytic tool of singularity
analysis. To facilitate a systematic analysis of these additive functionals we
extend singularity analysis, a class of methods by which one can translate on a
term-by-term basis an asymptotic expansion of a functional around its dominant
singularity into a corresponding expansion for the Taylor coefficients of the
function. The most important extension is the determination of how
singularities are composed under the operation of Hadamard product of analytic
power series.
  The transfer theorems derived are used in conjunction with the method of
moments to establish limit laws for m-ary search trees under the random
permutation model. For the uniform model on binary search trees, the extended
singularity analysis toolkit is employed to establish the asymptotic behavior
of the moments of a wide class of functionals. These asymptotics are used,
again in conjunction with the method of moments, to derive limit laws.

nevin@jhu.edu

2221. ON THE SCALING OF THE CHEMICAL DISTANCE IN LONG RANGE PERCOLATION MODELS 

Marek Biskup

We consider the (unoriented) long-range percolation on $\Z^d$ in
dimensions $d\ge1$, where distinct sites $x,y\in\Z^d$ get connected with
probability $p_{xy}\in[0,1]$. Assuming $p_{xy}=|x-y|^{-s+o(1)}$
as $|x-y|\to\infty$, where $s>0$ and $|\cdot|$ is a norm distance on $\Z^d$,
and supposing that the resulting random graph contains an infinite connected
component $\scrC_\infty$, we let $D(x,y)$ be the graph distance between $x$
and $y$ measured on $\scrC_\infty$. Our main result is that, for $s\in(d,2d)$,
$D(x,y)=(\log|x-y|)^{\Delta+o(1)}, \qquad x,y\in \scrC_\infty,
|x-y|\to\infty,$ where $\Delta^{-1}$ is the binary logarithm of $2d/s$
and $o(1)$ is a quantity tending to zero in probability as $|x-y|\to\infty$.
Besides its interest for general percolation theory, this result settles an
open question that has recently surfaced in the context of ``small-world''
phenomena. As part of the proof we also establish tight bounds on the
probability that the largest connected component in a finite box contains a
positive fraction of all sites in the box.

biskup@math.ucla.edu

2222. THE BROWNIAN LOOP SOUP 

Gregory F. Lawler, Wendelin Werner

We define a natural conformally invariant measure on unrooted Brownian loops
in the plane and study some of its properties. We relate this measure to a
measure on loops rooted at a boundary point of a domain and show how this
relation gives a way to ``chronologically add Brownian loops'' to simple curves
in the plane.

wendelin.werner@math.u-psud.fr

2223. MDP FOR INTEGRAL FUNCTIONALS OF FAST AND SLOW PROCESSES WITH AVERAGING 

A. Guillin and R. Liptser

We establish large deviation principle (LDP) for the family of vector-valued
random processes $(X^\epsilon,Y^\epsilon),\epsilon\to 0$ defined as  $$
X^\epsilon_t=\frac{1}{\epsilon^\kappa}\int_0^t
H(\xi^\epsilon_s,Y^\epsilon_s)ds,
dY^\epsilon_t=F(\xi^\epsilon_t,Y^\epsilon_t)dt+
D\epsilon^{1/2-\kappa}G(\xi^\epsilon_t,Y^\epsilon_t)dW_t,$$ where $W_t$ is
Wiener process and $\xi^\epsilon_t$ is fast ergodic diffusion. We show that,
under $\kappa<{1/2}$ or less and Veretennikov-Khasminskii type condition for
fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.

liptser@eng.tau.ac.il

2224. MARKOV MORTALITY MODELS: IMPLICATIONS OF QUASISTATIONARITY AND VARYING INITIAL DISTRIBUTIONS 

David Steinsaltz and Steven N. Evans

This paper explains some implications of markov-process theory for
models of mortality.  We show, on the one hand, that an important
qualitative feature which has been found in certain models --- the
convergence to a ``mortality plateau'' --- is a generic
consequence of the convergence to a ``quasistationary
distribution'', which has been explored extensively in the
mathematical literature.  This serves not merely to free these
results from some irrelevant specifics of the models, but also to
offer a new explanation of the convergence to constant mortality.
At the same time that we show that the late behavior --- convergence
to a finite asymptote --- of these models is almost logically
immutable, we also show that the early behavior of the mortality
rates can be more flexible than has been generally acknowledged. 
We show, in particular, that an appropriate choice of initial
conditions enables one popular model to approximate any reasonable
hazard-rate data.  This suggests how precarious it might be to
judge the appropriateness of mortality models by a
perceived consilience with a favored hazard-rate function, such as
the Gompertz exponential.

dstein@demog.berkeley.edu   evans@stat.berkeley.edu

  • To see a preprint or other information provided by the author click here.
  • Or here.

2225. QUASISTATIONARY DISTRIBUTIONS FOR ONE-DIMENSIONAL DIFFUSIONS WITH KILLING 

David Steinsaltz and Steven N. Evans

We extend some results on the convergence of one-dimensional 
diffusions killed at the boundary, conditioned on extended survival, 
to the case of general killing on  the interior.  
In the traditional problem on the half-infinite line, a 
sufficiently strong drift is required to keep the process sufficiently 
close to the origin to allow convergence in distribution.  An 
alternative that arises when general killing is allowed, is that the 
conditioned process is held near the origin by a high rate of killing 
near $\infty$.

dstein@demog.berkeley.edu   evans@stat.berkeley.edu

  • To see a preprint or other information provided by the author click here.
  • Or here.

2226. FLUID MODEL FOR A NETWORK OPERATING UNDER A FAIR BANDWIDTH-SHARING POLICY 

F. P. Kelly and R. J. Williams

We consider a model of Internet congestion control,
that represents the randomly varying number of flows 
present in a network where bandwidth is shared fairly  
between document transfers. We study critical fluid models,
obtained as formal limits under law of large numbers 
scalings when the average load on at least one resource 
is equal to its capacity. We establish convergence to 
equilibria for fluid models, and identify the invariant 
manifold. The form of the invariant manifold gives insight 
into the phenomenon of entrainment, whereby congestion at
some resources may prevent other resources from working 
at their full capacity.

f.p.kelly@statslab.cam.ac.uk williams@math.ucsd.edu

  • To see a preprint or other information provided by the author click here.

2227. BROWNIAN DIRECTED POLYMERS IN RANDOM ENVIRONMENT 

Francis Comets and Nobuo Yoshida

We study the thermodynamics of a continuous model of directed polymers 
in random environment.The environment is given by a space-time Poisson 
point process, whereas the polymer is defined in terms of the Brownian motion. 
We mainly discuss:
(i) The normalized partition function, its positivity in the limit 
which characterizes the phase diagram of the model.
(ii) The existence of quenched Lyapunov exponent, its positivity, and 
its agreement with the annealed Lyapunov exponent;
(iii) The longitudinal fluctuation of the free energy, some of its relations 
with the overlap between replicas and with the transversal fluctuation of the path. 
The model considered here, enables us to use stochastic calculus, with
respect to both Brownian motion and Poisson process, leading to handy formulas 
for fluctuations analysis and qualitative properties of the phase diagram.
 We also relate our model to some formulation of the Kardar-Parisi-Zhang equation,
 more precisely, the stochastic heat equation. Our fluctuation results are 
interpreted as bounds on various exponents and provide a circumstantial evidence 
of super-diffusivity in dimension one. We also obtain an almost sure large deviation 
principle for the polymer measure.
 
comets@math.jussieu.fr  nobuo@kusm.kyoto-u.ac.jp

  • To see a preprint or other information provided by the author click here.
  • Or here.
  • Or here.

2228. FROM UNIFORM DISTRIBUTION TO BENFORD'S LAW. 

Elise Janvresse and Thierry de la Rue

We define the mantissa (base 10) of a positive real number $x$  as the 
unique real number  $M(x) \in [1,10[$, such that $ x = M(x) 10^k$, for 
some integer $k \in \ZZ$.  
Benford's law says that the proportion of numbers $x > 0$ such that 
$M(x) \in [a,b[$ is, for any $1 \leq a< b \leq 10$, 
$ P( M(x) \in [a,b[ ) = \Pb([a,b[) $, 
where $\Pb([a,b[) := \log_{10}b - \log_{10}a $. 
We link Benford's law to what could be the uniform distribution in $\RR^+$ 
as defined by R\'enyi, which leads us to construct a Markov chain admitting 
$\Pb$ as an invariant measure. We prove by a coupling method that 
Benford's law is the unique invariant measure. We also estimate the speed 
of convergence of the chain. Eventually, we connect our results to the 
theorem of B. J. Flehinger about the initial digit of a random integer. 

Elise.Janvresse@univ-rouen.fr Thierry.Delarue@univ-rouen.fr

  • To see a preprint or other information provided by the author click here.

2229. HEAT KERNEL ESTIMATES AND LAW OF THE ITERATED LOGARITHM FOR SYMMETRIC RANDOM WALKS ON FRACTAL GRAPHS 

Ben M. Hambly  and Takashi Kumagai

We study two-sided heat kernel estimates on a class of fractal graphs
which arise from a subclass of finitely ramified fractals.  These
fractal graphs do not have spatial symmetry in general, and we find
that there is a dependence on direction in the estimates. We will give
a new form of expression for the heat kernel estimates using a family
of functions which can be thought of as a ``distance for each
direction''. As an application, we give a law of the iterated
logarithm for the random walk. The directional dependence shows that
there is non-uniform behaviour in the paths.

hambly@maths.ox.ac.uk   kumagai@kurims.kyoto-u.ac.jp

  • To see a preprint or other information provided by the author click here.
  • Or here.

2230. COMPARISON THEOREM AND ESTIMATES FOR TRANSITION PROBABILITY DENSITIES OF DIFFUSION PROCESSES 

Zhongmin Qian,  Francesco Russo and Weian Zheng

We establish several comparison
theorems for the transition probability density $p_{b}(x,t,y)$ of Brownian
motion with drift $b$, and deduce explicit, sharp lower and upper bounds for 
$p_{b}(x,t,y)$ in terms of the norms of the vector field $b$. The main
results are obtained through carefully estimating the mixed moments of
Bessel processes. All constants are explicit in our lower and upper bounds,
which is different from most of the previous estimates, and is important in
many applications for example in statistical inferences for diffusion
processes. 

qian@math.ups-tlse.fr    russo@math.univ-paris13.fr   wzheng@uci.edu

  • To see a preprint or other information provided by the author click here.

2231. VERIFICATION THEOREMS FOR STOCHASTIC OPTIMAL CONTROL PROBLEMS VIA A GENERALIZED FUKUSHIMA - DIRICHLET DECOMPOSITION 

Fausto Gozzi and Francesco Russo

This paper is devoted to present a method of proving
verification theorems for stochastic optimal control of finite dimensional
diffusion processes without control in the diffusion term. The value
function is assumed to be continuous in time and one time continuously 
differentiable in the space variable ($C^{0,1}$) instead of one time continuously 
differentiable in time and twice in space ($C^{1,2}$), like in the classical results. 
The results are obtained via a pathwise Fukushima-Dirichlet decomposition
obtained in the stochastic calculus via regularization. As side effect of
this paper we continue the development of that ''pathwise'' stochastic
calculus. A comparison with other verification techniques is performed, in
particular with the results in the viscosity solution setting. 

fausto.gozzi@uniroma1.it  russo@math.univ-paris13.fr

  • To see a preprint or other information provided by the author click here.

2232. COALESCENT PROCESSES OBTAINED FROM SUPERCRITICAL GALTON-WATSON PROCESSES 

Jason Schweinsberg

Consider a population model in which there are $N$ individuals in
each generation.  One can obtain a coalescent tree by sampling $n$
individuals from the current generation and following their ancestral
lines backwards in time.  It is well-known that under certain conditions
on the joint distribution of the family sizes, one gets a limiting coalescent
process as $N \rightarrow \infty$ after a suitable rescaling.  Here we
consider a model in which the numbers of offspring for the individuals are
independent, but in each generation only $N$ of the offspring are chosen
at random for survival.  We assume further that if $X$ is the number of
offspring of an individual, then $P(X \geq k) \sim Ck^{-a}$ for some $a > 0$
and $C > 0$.  We show that, depending on the value of $a$, the limit may
be Kingman's coalescent, in which each pair of ancestral lines merges at
rate one, a coalescent with multiple collisions, or a coalescent with
simultaneous multiple collisions.

jasonsch@math.cornell.edu

  • To see a preprint or other information provided by the author click here.

2233. SELF-SIMILAR FRAGMENTATIONS AND STABLE SUBORDINATORS 

Gregory Miermont and Jason Schweinsberg

Let $(Y(t), t \geq 0)$ be the fragmentation process introduced by Aldous and
Pitman that can be obtained by time-reversing the standard additive
coalescent.  Let $(\sigma_{1/2}(t), t \geq 0)$ be the stable subordinator of
index $1/2$.  Aldous and Pitman showed that the distribution of the sizes of 
the fragments of $Y(t)$ is the same as the conditional distribution of the jump
sizes of $\sigma_{1/2}$ up to time $t$,  given $\sigma_{1/2}(t) = 1$.  We show
that this is a special property of the stable subordinator of index $1/2$, in the
sense that if $\alpha \neq 1/2$ and $\sigma_{\alpha}$ is the stable
subordinator of index $\alpha$, then there exists no self-similar fragmentation
for which the distribution of the sizes of the fragments at time $t$ equals the
conditional distribution of the jump sizes of $\sigma_{\alpha}$ up to time $t$,
given $\sigma_{\alpha}(t) = 1$. We also show that a property relating the
distribution of a size-biased pick from $Y(t)$ to the distribution of
$\sigma_{1/2}(t)$ is similarly particular to the $\alpha = 1/2$ case.
However, we show that for each $\alpha \in (0,1)$, there is a family of
self-similar fragmentations whose behavior as $t \downarrow 0$ is
related to the stable subordinator of index $\alpha$ in the same way
that the behavior of $Y(t)$ as $t \downarrow 0$ is related to the
stable subordinator of index $1/2$.

Gregory.Miermont@ens.fr    jasonsch@math.cornell.edu

  • To see a preprint or other information provided by the author click here.
  • Or here.

2234. THE PASCAL ADIC TRANSFORMATION IS LOOSELY BERNOULLI. 

Elise Janvresse and Thierry de la Rue

The Pascal adic transformation is one of the simplest examples of adic 
transformations. We recall its construction by cutting and stacking and 
prove that it is loosely Bernoulli.

Elise.Janvresse@univ-rouen.fr Thierry.Delarue@univ-rouen.fr

  • To see a preprint or other information provided by the author click here.

2235. A FATOU THEOREM FOR $\alpha$-HARMONIC FUNCTIONS 

Richard F. Bass and Dahae You

We study  functions which are harmonic in the 
upper half space with respect to(-\Delta)^{\al/2}, 
0<\al<2. We prove a Fatou theorem when the boundary
function is L^p-Holder continuous of order \beta and 
\beta p>1. We give examples to show this condition is 
sharp.

bass@math.uconn.edu  you@math.uconn.edu

  • To see a preprint or other information provided by the author click here.

2236. BACKWARD SDE'S WITH TWO BARRIERS AND CONTINUOUS COEFFICIENT. AN EXISTENCE RESULT. 

Jean-Pierre Lepeltier and Jaime San Martin

In this work we prove the existence of a solution for a reflected backward
SDE with a continuous, linear increasing coefficient, in the case where
the barriers $L$ and $U$ are such that $L<U$ on $\left[0,T\right)$ and 
there exists and It\^o process between $L$ and $U$. Our method is based
on a penalisation argument and could serve as the basis of a numerical
scheme.

jsanmart@dim.uchile.cl

2237. MACROSCOPIC EVOLUTION OF PARTICLE SYSTEMS WITH RANDOM FIELD KAC INTERACTIONS 

Mustapha  Mourragui, Enza Orlandi and Ellen Saada 

We consider a  lattice gas  interacting via a  Kac interaction
$J_\gamma(|x-y|)$ of range $\gamma^{-1}$, $\gamma>0$, $x,y\in\Z^d$ 
and under the influence of an external random  field given by independent 
bounded random variables with a translation invariant distribution. We 
study the evolution of the system through a conservative dynamics, i.e. 
particles jump to nearest neighbor empty sites, with rates satisfying a 
detailed balance condition with respect to the equilibrium measure.  We 
prove that rescaling space as $\gamma^{-1}$ and time as $\gamma^{-2}$, 
in the limit $\gamma\to 0$, for dimension $d\ge 3$, the macroscopic 
density profile $\rho$ satisfies, a.s. with respect to the random field,  
a nonlinear integral differential equation, with a diffusion matrix 
determined  by the statistical properties of the external random field. 
The result holds for all values of the density, also in the  presence of 
phase segregation, and the equation is in the form of the flux gradient 
for the energy functional. 

Ellen.Saada@univ-rouen.fr
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