Probability Abstracts 56

1479. BETWEEN SOBOLEV AND POINCAR\'E

Rafa{\l} Lata{\l}a, Krzysztof Oleszkiewicz 

We establish a family of functional inequalities interpolating between the
classical logarithmic Sobolev and Poincar\'e inequalities. We prove that they
imply the concentration of measure phenomenon intermediate between Gaussian and
exponential. Our bounds are close to optimal.

rlatala@mimuw.edu.pl

• This article is available from the xxx mathematics archive as math.PR/0003043.

1480. PRECISE PROPAGATION OF UPPER AND LOWER PROBABILITY BOUNDS IN SYSTEM P

Angelo Gilio 

In this paper we consider the inference rules of System P in the framework of
coherent imprecise probabilistic assessments. Exploiting our algorithms, we
propagate the lower and upper probability bounds associated with the
conditional assertions of a given knowledge base, automatically obtaining the
precise probability bounds for the derived conclusions of the inference rules.
This allows a more flexible and realistic use of System P in default reasoning
and provides an exact illustration of the degradation of the inference rules
when interpreted in probabilistic terms. We also examine the disjunctive Weak
Rational Monotony of System P+ proposed by Adams in his extended probability
logic.

gilio@dmi.unict.it

• This article is available from the xxx mathematics archive as math.PR/0003046.

1481. HYDRODYNAMIC PROFILES FOR THE TOTALLY ASYMMETRIC EXCLUSION PROCESS WITH A SLOW BOND

Timo Seppalainen

We study a totally asymmetric simple exclusion process where jumps happen at
rate one, except at the origin where the rate is lower. We prove a hydrodynamic
scaling limit to a macroscopic profile described by a variational formula. The
limit is valid for all values of the slow rate. The only assumption required is
that a law of large numbers holds for the initial particle distribution. This
includes also deterministic initial configurations. The hydrodynamic
description contains as an unknown parameter the macroscopic rate at the
origin, which is strictly larger than the microscopic slow rate. The limit is
proved by the variational coupling method.

seppalai@iastate.edu

• This article is available from the xxx mathematics archive as math.PR/0003049.

1482. SUPER-BROWNIAN MOTION WITH REFLECTING HISTORICAL PATHS

Krzysztof Burdzy and Jean-Francois Le Gall

We consider super-Brownian motion whose historical paths reflect from each
other, unlike those of the usual historical super-Brownian motion. We prove
tightness for the family of distributions corresponding to a sequence of
discrete approximations but we leave the problem of uniqueness of the limit
open. We prove a few results about path behavior for processes under any limit
distribution. In particular, we show that for any $\gamma>0$, a "typical"
increment of a reflecting historical path over a small time interval $\Delta t$
is not greater than $(\Delta t)^{3/4 - \gamma}$.

burdzy@math.washington.edu

• This article is available from the xxx mathematics archive as math.PR/0003056.

1483. RECONSTRUCTION OF GRAY-SCALE IMAGES

Pablo A. Ferrari, Marco D. Gubitoso, E. Jordao Neves

We present an algorithm to reconstruct gray scale images corrupted by noise.
We use a Bayesian approach. The unknown original image is assumed to be a
realization of a Markov random field on a finite two dimensional region. This
image is degraded by some noise, which is assumed to act independently in each
site and to have the same distribution on all sites. For the estimator we use
the mode of the posterior distribution: the so called maximum a posteriori
(MAP) estimator. The algorithm, that can be used for both gray-scale and
multicolor images, uses the binary decomposition of the intensity of each color
and recovers each level of this decomposition using the identification of the
problem of finding the two color MAP estimator with the min-cut max-flow
problem in a binary graph, discovered by Greig, Porteous and Seheult (1989).

pablo@ime.usp.br

• This article is available from the xxx mathematics archive as math.PR/0003098.

1484. RELIABLE CELLULAR AUTOMATA WITH SELF-ORGANIZATION

Peter Gacs

In a probabilistic cellular automaton in which all local transitions have
positive probability, the problem of keeping a bit of information indefinitely
is nontrivial, even in an infinite automaton. Still, there is a solution in 2
dimensions, and this solution can be used to construct a simple 3-dimensional
discrete-time universal fault-tolerant cellular automaton. This technique does
not help much to solve the following problems: remembering a bit of information
in 1 dimension; computing in dimensions lower than 3; computing in any
dimension with non-synchronized transitions.
  Our more complex technique organizes the cells in blocks that perform a
reliable simulation of a second (generalized) cellular automaton. The cells of
the latter automaton are also organized in blocks, simulating even more
reliably a third automaton, etc. Since all this (a possibly infinite hierarchy)
is organized in ``software'', it must be under repair all the time from damage
caused by errors. A large part of the problem is essentially self-stabilization
recovering from a mess of arbitrary size and content. The present paper
constructs an asynchronous one-dimensional fault-tolerant cellular automaton,
with the further feature of ``self-organization''. The latter means that unless
a large amount of input information must be given, the initial configuration
can be chosen homogeneous.

peter.gacs@cwi.nl

• This article is available from the xxx mathematics archive as math.PR/0003117.

1485. LIMITING DISTRIBUTIONS FOR A POLYNUCLEAR GROWTH MODEL WITH EXTERNAL SOURCES

Jinho Baik and Eric Rains

The purpose of this paper is to investigate the limiting distribution
functions for a polynuclear growth model with two external sources, which was
considered by Pr\"ahofer and Spohn. Depending on the strength of the sources,
the limiting distribution functions are either the Tracy-Widom functions of
random matrix theory, or a new explicit function which has the special property
that its mean is zero. Moreover, we obtain transition functions between pairs
of the above distribution functions in suitably scaled limits. There are also
similar results for a discrete totally asymmetric exclusion process.

jbaik@math.princeton.edu

• This article is available from the xxx mathematics archive as math.PR/0003130.

1486. VALUES OF BROWNIAN INTERSECTION EXPONENTS II: PLANE EXPONENTS

Gregory F. Lawler, Oded Schramm, Wendelin Werner

We derive the exact value of intersection exponents between planar Brownian
motions or random walks, confirming predictions from theoretical physics by
Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple
random walks) in the plane, started from distinct points. We prove that the
probability that the paths B[0,t] and B'[0,t] do not intersect decays like
t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)|
=1, for all t \ge 1, $$ c^{-1} t^{-5/8} \le \P[ B[0,t] \cap B'[0,t] = \emptyset
] \le c t^{-5/8}. $$ One consequence is that the set of cut-points of B[0,1]
has Hausdorff dimension 3/4 almost surely. The values of other exponents are
also derived. Using an analyticity result, which is to be established in a
forthcoming paper, it follows that the Hausdorff dimension of the outer
boundary of B[0,1] is 4/3, as conjectured by Mandelbrot.
  The proofs are based on a study of SLE_6 (stochastic Loewner evolution with
parameter 6), a recently discovered process which conjecturally is the scaling
limit of critical percolation cluster boundaries. The exponents of SLE_6 are
calculated, and they agree with the physicists' predictions for the exponents
for critical percolation and self-avoiding walks. From the SLE_6 exponents the
Brownian intersection exponents are then derived.

schramm@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0003156.

1487. EXPONENTIAL AND MOMENT INEQUALITIES FOR U-STATISTICS

Evarist Gin\'e, Rafa{\l} Lata{\l}a, and Joel Zinn 

A Bernstein-type exponential inequality for (generalized) canonical
U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-J{\o}rgensen
inequalities for sums of independent random variables are extended to
(generalized) U-statistics of any order whose kernels are either nonnegative or
canonical

rlatala@mimuw.edu.pl

• This article is available from the xxx mathematics archive as math.PR/0003228.

1488. MALLIAVIN CALCULUS AND SKOROHOD INTEGRATION FOR QUANTUM STOCHASTIC PROCESSES

Uwe Franz, Remi Leandre and Rene Schott

A derivation operator and a divergence operator are defined on the algebra of
bounded operators on the symmetric Fock space over the complexification of a
real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar
properties as the derivation and divergence operator on the Wiener space over
$\eufrak{h}$. The derivation operator is then used to give sufficient
conditions for the existence of smooth Wigner densities for pairs of operators
satisfying the canonical commutation relations. For
$\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide
with the Hudson-Parthasarathy quantum stochastic integral for adapted
integrable processes and with the non-causal quantum stochastic integrals
defined by Lindsay and Belavkin for integrable processes.

franz@mail.uni-greifswald.de

• This article is available from the xxx mathematics archive as math.PR/0004088.

1489. HYPOTHESIS TESTING WITH THE GENERAL SOURCE

Te Sun Han

The asymptotically optimal hypothesis testing problem with the general
sources as the null and alternative hypotheses is studied under
exponential-type error constraints on the first kind of error probability. Our
fundamental philosophy in doing so is first to convert all of the hypothesis
testing problems completely to the pertinent computation problems in the large
deviation-probability theory. It turns out that this kind of methodologically
new approach enables us to establish quite compact general formulas of the
optimal exponents of the second kind of error and correct testing
probabbilities for the general sources including all nonstationary and/or
nonergodic sources with arbitrary abstract alphabet (countable or uncountable).
Such general formulas are presented from the information-spectrum point of
view.

han@toraja.hn.is.uec.ac.jp

• This article is available from the xxx mathematics archive as math.PR/0004121.

1490. STATISTICAL PROPERTIES OF CONVEX MINORANTS OF RANDOM WALKS AND BROWNIAN MOTIONS

Toufic Suidan

This paper calculates several useful statistical properties of the convex
minorant process generated by symmetric random walk processes. In particular,
we calculate the statistics of the longest segment in the convex minorant of a
random walk of a given length. In addition, we calculate the probability that
the convex minorant of a random walk of length N is composed of exactly m
segments. We obsevere that some of this analysis can be meaningful for the case
of Brownian motion on finite intervals; we can calculate exact formulas for the
density of the length of the longest segment in the convex minorant of Brownian
motion on finite intervals.

tmsuidan@math.princeton.edu

• This article is available from the xxx mathematics archive as math.PR/0004131.

1491. NO MORE THAN THREE FAVOURITE SITES FOR SIMPLE RANDOM WALK

Balint Toth

We prove that, with probability one, eventually there are no more than three
favourite (i.e. most visited) sites of simple random walk. This partially
answers a relatively long standing question of Pal Erdos and Pal Revesz.

balint@renyi.hu

• This article is available from the xxx mathematics archive as math.PR/0004164.

1492. CRITICAL LARGE DEVIATIONS IN HARMONIC CRYSTALS WITH LONG RANGE INTERACTIONS

Pietro Caputo and Jean-Dominique Deuschel

We continue our study of large deviations of the 
empirical measures of a massless Gaussian field on 
$\bbZ^d$, whose covariance is given by the Green function
of a long range random walk, \cite{CD}. 
In this paper
we extend techniques and results of \cite{BD} to the 
{\em non-local} case
of a random walk in the domain of attraction of the
symmetric $\al$-stable law, with $\al\in(0,2\wedge d)$. 
In particular, we show that critical 
large deviations occur at the capacity scale $N^{d - \al}$,
with a rate function given by the Dirichlet form of the
embedded $\al$-stable process. 
We also prove that if we impose zero boundary 
conditions, the rate function is  
given by the Dirichlet form of the
killed $\al$-stable process.

caputo@tx.technion.ac.il   deuschel@math.tu-berlin.de

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

1493. THE NONNEGATIVITY/POSITIVITY AND ASYMMETRIC SYSTEM CONTRIBUTION CONJECTURES

Osher Doctorow

The weak equivalence prinicple or Uniqueness of Free Fall (see Misner, 
Thorne, and Wheeler Gravitation 1973, Freeman: San Francisco)which underlie 
most gravitational theories is actually the tip of an iceberg represented 
presently by the Nonnegativity and Asymmetric System Contribution CConjectures 
(N and ASC respectively) which maintain that the variation of the action, 
the Lagrangian density itself, the (generalized) covariant derivative, and 
the logic-based probability (LBP) derivative are all nonnegative or even positive 
for events which contribute to rather than detract from the system (universe 
and its (sub-)contributing subsystems).  The LBP derivative roughly expresses 
the variation of random variables with each other, and its positivty roughly 
indicates that they are positively correlated.  LBP itself was developed by 
the author in a series of papers most of which (30 at last count) are 
abstracted on the internet via the Institute for Logic at the university of Vienna!

It is closely related to Lukasiewicz and Godel logics and fuzzy logics, 
fractals, and Bayesian conditional probability.  The asymmetry of the universe 
with respect to events that contribute to it versus detract from it is a typical 
conjecture which LBP makes possible, largely because LBP is a probability on a 
set as opposed to conditional probability which is a ratio of probabilities.  
Another advantage of LBP, its definition on sets/events of probability zero, 
enables it to study surfaces and boundaries of objects, very rare events, 
lower dimensional subsets of three dimensional physical objects provided that a 
continuous probability distribution exists on a region of spacetime containing the 
events in question.  It generalizes Hilbert Space of quantum theory to 
Banach spaces and even beyond, which A. Bohm has recently done in a more 
limited way via Rigged Hilbert Spaces and Lattices of Hilbert and Banach Spaces.  
One advantage of this is the replacement of Heisenberg's U!

Uncertainty Principle (UP) by (e.g., 2+1 dimensional) modular
transformations.  Jammer (1974) has already indicated considerable logical 
problems with the UP. This inturn enables intersections and hence ordinary 
probabilities in neo quantum logic, a major defect of classical quantum 
logic that led to its near demise in the 1960s and 1970s.   

osher@ix.netcom.com

1494. BRANCHING RANDOM WALK IN RANDOM ENVIRONMENT: FULLY QUENCHED CASE

Stanislav Volkov

The purpose of this report is to introduce a branching 
random walk in random environment on $\Z^d$ where particles 
perform independent simple random walks and branch,
according to a given law, which is obtained by fixing 
branching numbers at each point in the space. These numbers 
represent a realization of an integer-valued random field
on $\Z^d$ with the value at each point being independent 
of all other points.

With just one particle starting at the origin, we identify 
the conditions which separate transience and recurrence, 
i.e.,  the progeny hits the origin with probability $<1$ and 
respectively 1, in the same manner as it is done by 
F. den Hollander, M.V. Menshikov and S.Yu. Popov (1999),
and previously by M.V. Menshikov and S.E. Volkov (1997).

svolkov@euridice.tue.nl

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

1495. BRANCHING MARKOV CHAINS: QUALITATIVE CHARACTERISTICS

Mikhail Menshikov and Stanislav Volkov

In this paper we study random walks with branching. We 
introduce the notion of recurrence and transience for these 
processes and provide criteria for them. For the Lamperti 
problem and many-dimensional random walks with branching we 
find the critical (for transience vs. recurrence) speed of 
decay of the average number of off-springs at a point with 
respect to its distance  to the origin.

svolkov@euridice.tue.nl menshikv@facet.inria.msu.ru 

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

1496. RANDOM WALKS IN RANDOM LABYRINTHS

G.R. Grimmet, M.V. Menshikov and S.E. Volkov

A random labyrinth is a disordered environment of scatterers on a
lattice. A beam of light travels through the medium, and is reflected
off the scatterers. The set of illuminated vertices is studied, under
the assumption that there is a positive density of points, called
`normal points', at which the light behaves in the manner of a simple
symmetric random walk. The ensuing `random walk in a labyrinth' is
found to be recurrent in two dimensions, and also non-localised under certain
extra assumptions on the underlying probability distribution. The walk
is shown to be transient (with strictly positive probability)
in three and higher dimensions, subject to the assumption that 
the density of `non-trivial' scatterers is
sufficiently small. The principal arguments used in deriving such
results originate in percolation theory. In addition, we utilise the
relationship between random walks and electrical networks, namely that
a random walk is recurrent if and only if a certain electrical network has
infinite resistance.

g.r.grimmett@statslab.cam.ac.uk  menshikv@facet.inria.msu.ru 
svolkov@euridice.tue.nl

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

1497. VERTEX-REINFORCED RANDOM WALK ON ARBITRARY GRAPHS

Stanislav Volkov

Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988) 
is a random process in a continuously changing environment 
which is more likely to visit states  it has visited before.
We consider VRRW on arbitrary graphs and show that on almost 
all of them, VRRW visits only finitely many vertices with 
a positive probability. 

We conjecture that on all graphs of bounded degree, this 
actually happens a.s., and provide the proof only for trees 
of this type. We distinguish between several different 
patterns of localization and explicitly  describe the 
long-run structure of VRRW, which depends on whether the graph  
contains triangles or not. 

svolkov@euridice.tue.nl

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

1498. RIGIDITY PERCOLATION AND BOUNDARY CONDITIONS

Alexander E. Holroyd

We study the effects of boundary conditions in 
two-dimensional rigidity percolation.  Specifically, we 
consider generic rigidity in the bond percolation model on 
the triangular lattice.  We introduce a theory of boundary 
conditions, and define two different notions of `rigid 
clusters', called $r^0$-clusters and $r^1$-clusters, which 
correspond to free boundary conditions and wired boundary 
conditions respectively.  The definition of an $r^0$-cluster
turns out to be equivalent to the definition of a rigid 
component used in earlier papers by Holroyd and Haggstrom.  
We define two critical probabilities, associated with the
appearance of infinite $r^0$-clusters and infinite 
$r^1$-clusters respectively, and we prove that these two 
critical probabilities are in fact equal.  Furthermore, we 
prove that for all parameter values $p$ except possibly this
unique critical probability, the set of $r^0$-clusters
equals the set of $r^1$-clusters almost surely.  It is an
open problem to determine what happens {\it at} the critical
probability.

holroyd@math.ucla.edu

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

1499. A LARGE WIENER SAUSAGE FROM CRUMBS

Omer Angel, Itai Benjamini and Yuval Peres

Let $B(t)$ denote  Brownian motion in $\R^d$.
It is a classical fact that for any Borel set 
$A$ in $\R^d$, the volume $V_1(A)$ of the Wiener sausage
$B[0,1]+A$ has nonzero expectation iff $A$ is nonpolar. 
We show that for any nonpolar $A$, the random variable
$V_1(A)$ is unbounded.

omer@wisdom.weizmann.ac.il itai@wisdom.weizmann.ac.il peres@math.huji.ac.il 

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

1500. SPATIALLY PERIODIC EQUILIBRIA FOR A NON LOCAL EVOLUTION EQUATION

S. R. M. Barros, A. L. Pereira, C. Possani and A. Simonis

In this work we prove the existence of a global attractor 
for  the non local evolution equation 
$ { { \partial m ( r , t ) } \over { \partial t } } = 
- m ( r , t ) 
+ \tanh \left( \beta J * m ( r , t )  \right) $
in  the space of $\tau$-periodic functions, for $\tau$
sufficiently large. We also show the existence of non 
constant (unstable) equilibria in these spaces. 
 
saulo@ime.usp.br alpereir@ime.usp.br 
cpossani@ime.usp.br asimonis@ime.usp.br

1501. DIMENSION GAP FOR CONTINUED FRACTIONS WITH INDEPENDENT DIGITS

Yuri Kifer, Yuval Peres and Benjamin Weiss

Kinney and Pitcher (1966) determined the dimension of measures 
on [0,1] which make the digits in the continued fraction 
expansion i.i.d. Surprisingly, from their formula it is not clear 
that these dimensions are less than 1. We prove that these 
dimensions are, in fact, bounded away from 1. More generally, 
we consider f-expansions  with a corresponding absolutely 
continuous measure \mu under which the digits form a stationary 
process. Denote by E_\delta the set of reals where the
asymptotic frequency of some digit in the f-expansion
differs by at least \delta from the frequency  prescribed by \mu.
Then E_\delta has Hausdorff dimension less than 1  for any \delta>0.

kifer@math.huji.ac.il peres@math.huji.ac.il weiss@math.huji.ac.il

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

1502. IMMANANTS AND FINITE POINT PROCESSES

Persi Diaconis and Steven N. Evans

Given a Hermitian, non-negative definite kernel $K$ 
and a character $\chi$ of the symmetric group on $n$ 
letters, define the corresponding immanant function
$K^\chi[x_1, \ldots, x_n] := \sum_{\sigma}
\chi(\sigma) \prod_{i=1}^n K(x_i, x_{\sigma(i)})$,
where the sum is over all permutations $\sigma$ of 
$\{1, \ldots, n\}$.
When $\chi$ is the sign character (resp.the trivial 
character), then $K^\chi$ is a determinant (resp. permanent).
The function $K^\chi$ is symmetric and
non-negative, and, under suitable conditions, is also
non-trivial and integrable with respect to the
product measure $\mu^{\otimes n}$ for a given measure $\mu$.
In this case, $K^\chi$ can be normalised to be a symmetric 
probability density. The determinantal and permanental cases 
of this construction correspond to the fermion and boson 
point processes which have been studied extensively in 
the literature.

The case where $K$ gives rise to an orthogonal projection of
$L^2(\mu)$ onto a finite--dimensional subspace
is studied here in detail.  The determinantal instance
of this special case has a substantial literature because 
of its role in several problems in mathematical physics, 
particularly as the distribution of eigenvalues for various
models of random matrices. The representation theory of the
symmetric group is used to compute the normalisation 
constant and identify the $k^{th}$--order marginal
densities for $1 \le k \le n$ as linear combinations of 
analogously defined immanantal densities.
Connections with inequalities for immanants, particularly
the permanental dominance conjecture of Lieb,
are considered, and asymptotics when the dimension of the 
subspace goes to infinity are presented.

diaconis@math.stanford.edu  evans@stat.berkeley.edu

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

1503. MULTIPLICITY OF PHASE TRANSITIONS AND MEAN FIELD CRITICALITY ON HIGHLY NON-AMENABLE GRAPHS

Roberto H. Schonmann

We consider independent percolation, Potts (especially Ising)
models and the contact process on infinite, locally finite,
connected graphs, with emphasis on transitive graphs.

The presence of more than one critical point, separating qualitatively
distinct regimes, is investigated for each one of these models.
For each model, separation of critical points is proved under the
assumption that the edge-isoperimetric constant (Cheeger constant)
of the graph is large in some sense which only depends on the degrees
of the vertices of the graph.

In the case of unimodular transitive graphs, such conditions are proved
to imply also mean field critical behavior for independent percolation,
the Ising model and the contact process.

For Potts models on unimodular transitive graphs, we prove the monotonicity
in the temperature of the property that the free Gibbs measure is extremal
in the set of automorphism invariant Gibbs measures. We also prove that the
corresponding critical temperature is positive if and only if the threshold
for uniqueness of the infinite cluster in independent bond percolation on
the same graph is less than 1.

The finite island property for independent percolation on infinite,
locally finite, connected graphs is proved to hold at large density
under the assumption that the anchored vertex-isoperimetric constant
is positive, or under the assumption that the graph is transitive,
has one end, and has a property that we call the quasi-connected
minimal cut sets property (and which is known to hold for all the
Cayley graphs of finitely generated, finitely presented groups).
This allows us to show that for a large class of graphs the $q$-Potts
model has a low temperature regime in which the free Gibbs measure
decomposes as the uniform mixture of the $q$ ordered phases.

In the case of non-amenable transitive planar graphs with one end, we show
that the $q$-Potts model has a well defined non-trivial critical point
separating a regime of high temperatures in which the free Gibbs measure
is extremal in the set of automorphism invariant Gibbs measures from a
regime of low temperatures in which the free Gibbs measure decomposes as
the uniform mixture of the $q$ ordered phases.

rhs@math.ucla.edu

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

1504. HEAT EQUATION AND REFLECTED BROWNIAN MOTION IN TIME DEPENDENT DOMAINS

K. Burdzy, Z. Chen and J. Sylvester

The paper is divided into three major parts. Their main
results include
(a) strong and weak formulations of an initial boundary
value problem representing heat flow in a domain with a moving
insulated boundary, and existence and uniqueness of solutions;
(b) various aspects of the relationship between
reflected Brownian motion and the heat equation
in varying domains; a Feynman-Kac type formula derived
in the second part is a basis for several effective quantitative
and qualitative arguments presented in the last part;
(c) a study of two types of possible singularities of the
solution to the heat equation at the boundary of a moving
domain; many explicit results on ``heat atoms'' and ``heat
singularities'' are given.

burdzy@math.washington.edu zchen@math.washington.edu  
sylvest@math.washington.edu

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

1505. EULER CHARACTERISTICS FOR GAUSSIAN FIELDS ON MANIFOLDS

Jonathan E. Taylor and Robert J. Adler 

We are interested in the geometric properties of 
real-valued Gaussian random fields defined on manifolds. 
Our manifolds, $M$, are of class $C^3$ and the random 
fields $f$ are smooth.

Our interest in these fields focuses on their excursion 
sets, $f^{-1}[u, +\infty)$, and their geometric properties.
Specifically, we derive the expected Euler characteristic 
$E[\chi(f^{-1}[u, +\infty))]$ of an excursion set of a 
smooth Gaussian random field.

Part of the motivation for this comes from the fact that 
$E[\chi(f^{-1}[u,+\infty))]$ relates global properties 
of $M$ to the intrinsic geometry induced by $f$. Of further
interest is the relation between the expected Euler 
characteristic of an excursion set above a level $u$ and 
$P[ \sup_{p \in M} f(p) > u ]$.
 
As a corollary to our main results, we give a simple proof 
of the Chern-Gauss-Bonnet Theorem for manifolds with or 
without boundary. Our proofsrely on results from random 
fields on $\Re^n$ as well as standard tools from 
differential and Riemannian geometry.

jtaylor@math.mcgill.ca robert@ieadler.technion.ac.il

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

1506. OSCILLATION OF THE TRANSITION DENSITY FOR BROWNIAN MOTION ON RANDOM RECURSIVE SIERPINSKI GASKETS

Ben M. Hambly and Takashi Kumagai

We consider a class of random recursive Sierpinski gaskets and 
examine the oscillation in the short time asymptotics of the on-diagonal
transition density for a natural Brownian motion. Using the resolvent
density we are able to explicitly show that there are fluctuations 
in time at typical points in the fractal and, by considering the supremum
and infimum of the on-diagonal transition density over all points 
in the fractal, we show that there are also fluctuations in space.

b.hambly@bris.ac.uk  kumagai@i.kyoto-u.ac.jp

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

1507. LIMIT THEOREMS FOR MANDELBROT'S MULTIPLICATIVE CASCADES

Quansheng Liu and Alain Rouault

Let $W\geq 0$ be a random variable with $EW=1$, and let 
$Z^{(r)}\,(r\geq 2) $ be the limit of a Mandelbrot's 
martingale, defined as sums of product of independent random
weights having the same distribution as $W$, indexed by 
nodes of a homogeneous $r$-ary tree. We study asymptotic 
properties of $Z^{(r)}$ as $r\rightarrow \infty$: we obtain 
a law  of large numbers, a central limit theorem, a result 
for convergence of moment generating functions and a theorem
of large deviations. Some results are extended to the case 
where the number of branches is a random variable whose 
distribution depends on a parameter $r$.

Quansheng.Liu@univ-rennes1.fr rouault@math.uvsq.fr

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

1508. LIMIT THEOREMS FOR MULTIPLICATIVE PROCESSES

Quansheng Liu and Alain Rouault

Let $W$ be a non-negative random variable with $EW=1$, and let $\{W_{\bf i}\}$ be a family 
of independent copies of $W$, indexed by all the finite sequences ${\bf i}=i_1...i_n$ of 
positive integers. We are interested to a sequence of random measures,
$\{\mu_r^n\}_{(r\geq 2, n\geq 1)}$, on $[0,1]$, and their weak limits $\{\mu_r^\infty\}$ 
as $n\rightarrow \infty$; by definition, for fixed $r$ and $n$, the measure $\mu_r^n$ has, 
on each $r$-adic interval $A^r_{i_1...i_n}$ at $n$-th level, the density  
$W_{i_1} ... W_{i_1...i_n}$ with respect to the Lebesgue measure, so that $\{\mu_r^1\}_r$ 
is the well-known smoothed partial-sum process associated with the random walk 
$W_1+...+W_r$, $\{\mu_r^n\}$ is its natural extension to a tree-indexed process at $n$-th 
level, and $\{\mu_r^\infty\}_r$ are the famous Mandelbrot measures for multiplicative 
cascades. In a preceding paper, we proved limit theorems for the total masses 
$Z_r^\infty$ of $\mu_r^\infty$ as $r\rightarrow\infty$. Here we study asymptotic 
properties for the measures $\{\mu_r^\infty\}_r$ instead of their masses, and we find 
laws of large numbers, convergence in $L^p$, central limit theorems, laws of iterated 
logarithm and large deviation principles. The results are also established for each 
multiplicative process $\{\mu_r^n\}_{r}$ at $n$-th level, for fixed $1\leq n <\infty$.  
  
Quansheng.Liu@univ-rennes1.fr rouault@math.uvsq.fr

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

1509. SAMPLE PATH LARGE DEVIATIONS AND CONVERGENCE PARAMETERS

Irina Ignatiouk-Robert

We prove the local sample path large deviation estimates for a general 
class of Markov chains with discontinuous statistics. The local rate 
function is represented in terms of the convergence parameter of associated 
local transform matrices. Our methods rely on two different descriptions of 
the convergence parameter: (1) in terms of Perron-Frobenius eigenvalues, 
and (2) by using $\rho$-superharmonic functions. The results are applied 
to identify the sample path large deviation rate function for perturbated 
random walks on $Z^k$. 

Irina.Ignatiouk@math.u-cergy.fr

1510. A SURVEY OF ONE-DIMENSIONAL RANDOM POLYMERS

Remco van der Hofstad, Wolfgang Koenig

In the last decade there has been an
enormous progress in the mathematical 
understanding of one-dimensional polymer 
measures, which are path measures that
suppress self-intersections. We currently 
seem to be in the situation that the 
interesting questions have either been
answered, or the answers are not to be 
expected soon. In this survey paper, 
we discuss the most relevant results,
open questions, and heuristics.

R.W.vanderHofstad@its.tudelft.nl  koenig@math.TU-Berlin.DE

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

1511. ON THE INFINITE CLUSTER OF BERNOULLI BOND PERCOLATION IN THE SCHERK'S GRAPH

Dayue Chen

The Scherk's graph is a subgraph of  $Z^3$ with the same set of vertices.
Two vertices $(x, y, z)$ and $(x', y' z')$ are adjacent if $x=x'$ and 
$|y-y'| + |z- z'| = 1$ or $z = z' = 0$ and $|x-x'| + |y-y'| = 1$.   It is 
shown by Markvorsen, McGuinness & Thomassen that the Scherk's graph is 
transient.    Consider now the Bernoulli bond percolation in the Scherk's 
graph. Itai Benjamini points out that the critical point $p_c<1/2$ for 
the Scherk's graph  (Private communication via Yuval Peres).  It is not 
difficult to verify that there is a unique infinite cluster when $p> 1/2$.
We prove that the infinite cluster of Bernoulli bond percolation in the 
Scherk's graph is transient for $p > 1/2$ and is recurrent for $p < 1/2$. 
Since the Scherk's graph is a subgraph of $Z^3$, the theorem implies the well 
known result of Grimmett, Kesten and Zhang that the infinite cluster of
the Bernoulli bond percolation in $Z^3$ is transient[3] for $p > 1/2$.
Their result holds for all $p >p_c$.  On the other hand, the Scherk's graph 
exhibits a new dichotomy in the supercritical region. 

dayue@pku.edu.cn

1512. ON A RANDOM GRAPH WITH IMMIGRATING VERTICES: EMERGENCE OF THE GIANT COMPONENT

David Aldous and Boris Pittel

A randomly evolving graph, with vertices immigrating at rate $n$ and 
each possible edge appearing at rate $1/n$, is studied. The detailed 
picture of emergence of giant components with $O(n^{2/3})$ vertices 
is shown to be the same as in the Erdos - Renyi graph process 
with the number of vertices fixed at $n$ at the start. A major difference 
is that now the transition occurs about a time $t=\pi/2$, rather 
than $t=1$. The proof has three ingredients. The size of the largest 
component in the subcritical phase is bounded by comparison with a 
certain multitype branching process. With this bound at hand, the growth 
of the sum-of-squares and sum-of-cubes of component sizes is shown, via
martingale methods, to follow closely a solution of the Smoluchovsky-type 
equations. The approximation allows us to apply results of Aldous (1997) on 
emergence of giant components in the multiplicative coalescent, i.e. a 
non-uniform random graph process.

aldous@stat.berkeley.edu

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

1513. LEE-YANG MODELS, SELFDECOMPOSABILITY AND NEGATIVE-DEFINITE FUNCTIONS

Joel De Coninck and Zbigniew J. Jurek

A class $\mathcal{L}$ of Ising models is introduced via L\'evy class L 
characteristic functions. The critical temperature for these new models
is associated with the weak law
of large numbers and it is proved that the critical exponent $\delta$ is 
greater or equal one. New inequalities  for the Ursell 
functions are proposed via the Schoenberg Theorem. Moreover, with the 
functions $u_0$ and $u_1$ one associates some Fourier transforms as 
functions of the external field.

zjjurek@math.uni.wroc.pl (Preprints or Latex files can be obtained)

1514. 1-D ISING MODELS, COMPOUND GEOMETRIC DISTRIBUTIONS AND SELFDECOMPOSABILITY

Zbigniew J. Jurek

It is shown that the inverse of the partition 
function in 1-D Ising model, as a function of the external field, is 
a product of Fourier transforms of compound geometric distributions. 
These are random sums (randomly stopped random walks) with the probability 
of a success depending \emph{only} on the interaction constant K between 
sites. Moreover, it is proved that those distribution belong to the 
L\'evy class $L$ of selfdecomposable probability measures, therefore they 
have the BDLP's, i.e., the background driving L\'evy processes. It is 
important that the general structure of class L characteristic functions is 
well-known and that it is much more specific than the L\'evy-Khintchine 
formula for infinite divisible variables.

zjjurek@math.uni.wroc.pl (Preprints or Latex files can be obtained)

1515. A NOTE ON GAMMA RANDOM VARIABLES AND DIRICHLET SERIES.

Zbigniew J. Jurek

We give necessary and sufficient conditions for 
convergence of series of centered gamma random variables. Those series 
provide distributions of L\'evy class $L$ of selfdecomposable probability 
distributions. Relations to Dirichlet series and the background driving
L\'evy processes (BDLP's) are investigated.

zjjurek@math.uni.wroc.pl (Preprints or Latex files can be obtained)

1516. SOME STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISTRIBUTIONAL DRIFT

Franco Flandoli, Francesco Russo and Jochen Wolf

In dimension 1 we study a martingale
 problem related to a parabolic
PDE operator $L$ with continuous (non-degenerate) diffusion term
and with drift being the derivative of a continuous function.
In several situations, this problem turns out to correspond to a true
SDE.
We study  existence and uniqueness and other properties of
the solution.
We state a necessary and sufficient characterization for
the solution $X$ (or $f(X)$ ) to be a semimartingale.
When $X$ is a semimartingale, we also establish an It\^o formula
under very weak conditions for $f(X)$.

russo@math.univ-paris13.fr

1517. GENERALIZED INTEGRATION AND STOCHASTIC ODE's. PART I

Franco Flandoli and Francesco Russo 

Stochastic forward integrals for processes more general than semimartingales
are shown to exist, generalized forms of It\^{o}-Wentzell formula and
covariation formula are proved, and one-dimensional stochastic equations
driven by finite quadratic variation processes and semimartingales are
solved. This generalized stochastic calculus is motivated by applications to
uniqueness and dependence on parameters for stochastic equations with non
regular drift.

russo@math.univ-paris13.fr

1518. GENERALIZED INTEGRATION AND STOCHASTIC ODE's. PART II

Franco Flandoli and Francesco Russo 

The generalized stochastic calculus developed in part I (stochastic forward
integrals for processes more general than semimartingales, It\^{o}-Wentzell
formula and covariation formulae, stochastic equations driven by finite
quadratic variation processes and semimartingales) is applied here to treat
in a new way uniqueness and regular dependence on parameters for stochastic
equations with non regular drift.

russo@math.univ-paris13.fr