Probability Abstracts 79

This document contains abstracts 2449-2505. They have been mailed on February 29, 2004.

2449. A SHAPE THEOREM FOR THE SPREAD OF AN INFECTION 

Harry Kesten and Vladas Sidoravicius

We consider the following interacting particle system: There is a ``gas'' of
particles, each of which performs a continuous time simple random walk on the
d-dimensional lattice. These particles are called A-particles and move
independently of each other. We assume that we start the system with a Poisson
number of particles at each lattice site x, with the number of particles at
different x's i.i.d. In addition, there are a finite number of B-particles
which perform the same continuous time simple random walks as the A-particles.
A- and B-particles are interpreted as individuals who are healthy or infected,
respectively. The B-particles move independently of each other. The only
interaction is that when a B-particle and an A-particle coincide, the latter
instantaneously turns into a B-particle. Let B(t) be the set of sites visited
by a B-particle during [0,t]. We show that B(t) grows linearly in time and has
an asymptotic shape; more precisely, there exists a non-random convex, compact
set B_0 such that almost surely, for all 0 < a <1, (1-a)tB_0 is contained in
B(t) and B(t) is contained in (1+a)tB_0 eventually.

kesten@math.cornell.edu

2450. EXTREME EXCHANGEABLE RANDOM ORDER PROCESSES BY POSITIVE DEFINITE FUNCTIONS ON SEMIGROUPS 

Ulrich Hirth

Based on previous work of Paul Ressel and myself, I show that the space of
all "continuous" exchangeable probability measures on a certain set of order
processes is a Bauer simplex, and for the special case of total orders I
moreover present a parametrised family of paint-box distributions, showing that
they belong to the extreme boundary of the Bauer simplex and that the parameter
set can naturally be endowed with a complete metric.

Ulrich.Hirth@UniBw-Muenchen.de

2451. THE GENEALOGY OF SELF-SIMILAR FRAGMENTATIONS WITH NEGATIVE INDEX AS A CONTINUUM RANDOM TREE 

Benedicte Haas, Gregory Miermont

We encode a certain class of stochastic fragmentation processes, namely
self-similar fragmentation processes with a negative index of self-similarity,
into a metric family tree which belongs to the family of Continuum Random Trees
of Aldous. When the splitting times of the fragmentation are dense near 0, the
tree can in turn be encoded into a continuous height function, just as the
Brownian Continuum Random Tree is encoded in a normalized Brownian excursion.
Under mild hypotheses, we then compute the Hausdorff dimensions of these trees,
and the maximal H\"older exponents of the height functions.

miermont@dma.ens.fr

2452. ON VERVAAT AND VERVAAT-ERROR TYPE PROCESSES FOR PARTIAL SUMS AND RENEWALS 

Endre Cs\'aki, Mikl\'os Cs\"{o}rg\H{o}, Zdzis{\l}aw Rychlik
 and Josef Steinebach

We study the asymptotic behaviour of stochastic processes that are generated
by sums of partial sums of i.i.d. random variables and their renewals. We
conclude that these processes cannot converge weakly to any nondegenerate
random element of the space $D[0,1]$. On the other hand we show that their
properly normalized integrals as Vervaat-type stochastic processes converge
weakly to a squared Wiener process. Moreover, we also deal with the asymptotic
behaviour of the deviations of these processes, the so-called Vervaat-error
type processes.

csaki@renyi.hu

2453. MULTIPLE SHOCKS IN A ZERO RANGE TYPE MODEL 

Marton Balazs

In a model similar to zero range we show that a nontrivial class of product
distributions is closed under the time-evolution of the process. This class
also includes measures fitting to shock data of the limiting PDE. In
particular, we show that shocks of this type with discontinuity of size one
perform ordinary nearest neighbor random walks only interacting, in an
attractive way, via their jump rates. Our results are related to those of
Belitsky and Sch\"utz on the simple exclusion process, although we do not use
quantum formalism as they do. The structures we find are described from a fixed
position. Similar ones were found earlier by Bal\'azs, as seen from the random
position of the second class particle.

balazs@math.wisc.edu

2454. MODERATE DEVIATIONS FOR PARTICLE FILTERING 

Arnaud Guillin, Randal Douc and Jamal Najim

Consider the state space model $(X_t,Y_t)$ where $(X_t)$ is a Markov chain
and $(Y_t)$ are the observations. In order to solve the so-called filtering
problem, one has to compute ${\mathcal L}(X_t|Y_1,..., Y_t)$, the law of $X_t$
given the observations $(Y_1,..., Y_t)$. The particle filtering method gives an
approximation of the law ${\mathcal L}(X_t|Y_1,..., Y_t)$ by an empirical
measure $\frac 1n \sum_1^n \delta_{x_{i,t}}$. In this paper, we establish the
Moderate Deviation Principle for the empirical mean $\frac 1n \sum_1^n
\psi(x_{i,t})$ (centered and properly rescaled) when the number of particles
grows to infinity, enhancing the Central Limit theorem. Several extensions and
examples are also studied.

najim@tsi.enst.fr

2455. RANDOM SUBGRAPHS OF FINITE GRAPHS: I. THE SCALING WINDOW UNDER THE TRIANGLE CONDITION 

Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, 
 Gordno Slade, Joel Spencer

We study random subgraphs of an arbitrary finite connected transitive graph
$\mathbb G$ obtained by independently deleting edges with probability $1-p$.
Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their
degree. We define the critical threshold $p_c=p_c(\mathbb G,\lambda)$ to be the
value of $p$ for which the expected cluster size of a fixed vertex attains the
value $\lambda V^{1/3}$, where $\lambda$ is fixed and positive. We show that
for any such model, there is a phase transition at $p_c$ analogous to the phase
transition for the random graph, provided that a quantity called the triangle
diagram is sufficiently small at the threshold $p_c$. In particular, we show
that the largest cluster inside a scaling window of size
$|p-p_c|=\Theta(\cn^{-1}V^{-1/3})$ is of size $\Theta(V^{2/3})$, while below
this scaling window, it is much smaller, of order
$O(\epsilon^{-2}\log(V\epsilon^3))$, with $\epsilon=\cn(p_c-p)$. We also obtain
an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster
above the window. In addition, we define and analyze the percolation
probability above the window and show that it is of order $\Theta(\cn(p-p_c))$.
Among the models for which the triangle diagram is small enough to allow us to
draw these conclusions are the random graph, the $n$-cube and certain Hamming
cubes, as well as the spread-out $n$-dimensional torus for $n>6$.

rhofstad@win.tue.nl

2456. RANDOM SUBGRAPHS OF FINITE GRAPHS: II. THE LACE EXPANSION AND THE TRIANGLE CONDITION 

Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad,
 Gordon Slade, Joel Spencer

In a previous paper, we defined a version of the percolation triangle
condition that is suitable for the analysis of bond percolation on a finite
connected transitive graph, and showed that this triangle condition implies
that the percolation phase transition has many features in common with the
phase transition on the complete graph. In this paper, we use a new and
simplified approach to the lace expansion to prove quite generally that for
finite graphs that are tori the triangle condition for percolation is implied
by a certain triangle condition for simple random walks on the graph.
  The latter is readily verified for several graphs with vertex set $\{0,1,...,
r-1\}^n$, including the Hamming cube on an alphabet of $r$ letters (the
$n$-cube, for $r=2$), the $n$-dimensional torus with nearest-neighbor bonds and
$n$ sufficiently large, and the $n$-dimensional torus with $n>6$ and
sufficiently spread-out (long range) bonds. The conclusions of our previous
paper thus apply to the percolation phase transition for each of the above
examples.

rhofstad@win.tue.nl

2457. RANDOM SUBGRAPHS OF FINITE GRAPHS: III. THE PHASE TRANSITION FOR THE $N$-CUBE 

Christian Borgs, Jennifer T. Chayes, Remco van der Hofstad, 
 Gordon Slade, Joel Spencer

We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor
edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for
which the expected cluster size of a fixed vertex attains the value $\lambda
2^{n/3}$, where $\lambda$ is a small positive constant. Let
$\epsilon=n(p-p_c(n))$. In two previous papers, we showed that the largest
cluster inside a scaling window given by $|\epsilon|=\Theta(2^{-n/3})$ is of
size $\Theta(2^{2n/3})$, below this scaling window it is at most $2(\log2)
n\epsilon^{-2}$, and above this scaling window it is at most $O(\epsilon 2^n)$.
In this paper, we prove that for $p - p_c(n) \geq e^{-cn^{1/3}}$ the size of
the largest cluster is at least $\Theta(\epsilon 2^n)$, which is of the same
order as the upper bound. This provides an understanding of the phase
transition that goes far beyond that obtained by previous authors. The proof is
based on a method that has come to be known as ``sprinkling,'' and relies
heavily on the specific geometry of the $n$-cube.

rhofstad@win.tue.nl

2458. EXPANSION IN $N^{-1}$ FOR PERCOLATION CRITICAL VALUES ON THE $N$-CUBE AND $Z^N$: THE FIRST THREE TERMS 

Remco van der Hofstad, Gordon Slade

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values
for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n =
\{0,1\}^n$ and on $\Z^n$, respectively. Let $\Omega = n$ for $\mathbb{G} =
\mathbb{Q}_n$ and $\Omega = 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the
degree of $\mathbb{G}$. We use the lace expansion to prove that for both
$\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$,
   $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$
 This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} +
O(\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides
a simplified proof of a previous result of Hara and Slade for $\mathbb{Z}^n$.

rhofstad@win.tue.nl

2459. ASYMPTOTIC EXPANSIONS IN $N^{-1}$ FOR PERCOLATION CRITICAL VALUES ON THE $N$-CUBE AND $\MATHBB{Z}^N$ 

Remco van der Hofstad, Gordon Slade

We use the lace expansion to prove that the critical values for
nearest-neighbour bond percolation on the $n$-cube $\{0,1\}^n$ and on
$\mathbb{Z}^n$ have asymptotic expansions, with rational coefficients, to all
orders in powers of $n^{-1}$.

rhofstad@win.tue.nl

2460. GAUSSIAN FLUCTUATIONS OF EIGENVALUES IN THE GUE 

Jonas Gustavsson

Under certain conditions on k we calculate the limit distribution of the k:th
largest eigenvalue, x_k, of the Gaussian Unitary Ensemble (GUE). More
specifically, if n is the dimension of a random matrix from the GUE and k is
such that both k and n-k tends to infinity as n tends to infinity then x_k is
normally distributed in the limit. We also consider the joint limit
distribution of x_k_1 < ... < x_k_m where we require that k_i and n-k_i,
i=1..m, tends to infinity with n. The result is an m-dimensional Normal
Distribution.

jonasgu@math.kth.se

2461. THE SKOROKHOD PROBLEM AND ITS OFFSRING 

Jan Obloj

This is a survey about the Skorokhod embedding problem. It presents all known
solutions together with their properties and some applications. Some of the
solutions are just described, while others are studied in detail and their
proofs are presented. Some new facts which appeared in a natural way when
different solutions were cross-examined, are reported. The Azema and Yor's and
Root's solutions are studied extensively. A possible use of the latter is
suggested together with a conjecture. The study starts with a methodology
discussion and then continues with a description of particular solutions. Among
applications, the link with the optimal stopping theory is underlined.

obloj@mimuw.edu.pl

2462. WEAK CONVERGENCE OF RANDOM P-MAPPINGS AND THE EXPLORATION PROCESS OF INHOMOGENEOUS CONTINUUM RANDOM TREES 

David J. Aldous, Gregory Miermont, Jim Pitman

We study the asymptotics of the $p$-mapping model of random mappings on $[n]$
as $n$ gets large, under a large class of asymptotic regimes for the underlying
distribution $p$. We encode these random mappings in random walks which are
shown to converge to a functional of the exploration process of inhomogeneous
random trees, this exploration process being derived (Aldous-Miermont-Pitman
2003) from a bridge with exchangeable increments. Our setting generalizes
previous results by allowing a finite number of ``attracting points'' to
emerge.

miermont@dma.ens.fr

2463. ESSENTIAL EDGES IN POISSON RANDOM HYPERGRAPHS 

Christina Goldschmidt and James Norris

Consider a random hypergraph on a set of N vertices in which, for k between 1
and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all
subsets of size k. We collapse the hypergraph by running the following
algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse
the hyperedges over that vertex onto their remaining vertices; repeat until
there are no 1-edges left. We call the vertices removed in this process
"identifiable". Also any hyperedge all of whose vertices are removed is called
"identifiable". We say that a hyperedge is "essential" if its removal prior to
collapse would have reduced the number of identifiable vertices. The limiting
proportions, as N tends to infinity, of identifiable vertices and hyperedges
were obtained by Darling and Norris. In this paper, we establish the limiting
proportion of essential hyperedges. We also discuss, in the case of a random
graph, the relation of essential edges to the 2-core of the graph, the maximal
sub-graph with minimal vertex degree 2.

christina@proba.jussieu.fr

2464. STOCHASTIC PROCESSES WITH SHORT MEMORY 

D.N. Zhabin

The mathematical model of a linear system with the short memory about own
stochastic behavior is proposed. It is assumed that the system is under a
continual influence of independent stochastic impulses. In a short memory
approximation the expression of the stochastic process is found. An application
of the model proposed to capital market processes is examined. The approach
allows form a stochastic differential for processes concerned. The analog of
the Black-Scholes equation for assets dealt on a market with the memory is
expressed.

zhabin@phys.tsu.ru

2465. A SURVEY OF RESULTS ON RANDOM RANDOM WALKS ON FINITE GROUPS 

Martin Hildebrand

A number of papers have examined various aspects of "random random" walks on
finite groups; the purpose of this article is to provide a survey of this work
and to show, bring together, and discuss some of the arguments and results in
this work. This article also provides a number of exercises. Some exercises
involve straightforward computations; others involve proving details in proofs
or extending results proved in the article. This article also describes some
problems for further study.

martinhi@math.albany.edu

2466. CRITICAL RANDOM HYPERGRAPHS: THE EMERGENCE OF A GIANT SET OF IDENTIFIABLE VERTICES 

Christina Goldschmidt

We consider a model for random hypergraphs with "identifiability", an
analogue of connectedness. This model has a phase transition in the proportion
of identifiable vertices when the underlying random graph becomes critical. The
phase transition takes various forms, depending on the values of the parameters
controlling the different types of hyperedges. It may be continuous as in a
random graph (in fact, when there are no higher-order edges, it is exactly the
the emergence of the giant component). In this case, there is a sequence of
possible sizes of `components' (including but not restricted to N^(2/3)).
Alternatively, the phase transition may be discontinuous. We are particularly
interested in the nature of the discontinuous phase transition and are able to
exhibit precise asymptotics. Our method extends a result of Aldous on component
sizes in a random graph.

christina@proba.jussieu.fr

2467. CHARACTER EXPANSION METHOD FOR THE FIRST ORDER ASYMPTOTICS OF A MATRIX INTEGRAL 

Alice Guionnet and Mylene Maida

The estimation of various matrix integrals as the size of the matrices goes
to infinity is motivated by theoretical physics, geometry and free probability
questions. On a rigorous ground, only integrals of one matrix or of several
matrices with simple quadratic interaction (called AB interaction) could be
evaluated so far. In this article, we follow an idea widely developped in the
physics litterature, which is based on character expansion, to study more
complex interaction. We more specifically consider a model defined in the
spirit of the 'dually weighted graph model' studied by V. A. Kazakov, M.
Staudacher and T. Wynter, but with a cutoff function such that the matrix
integral and its character expansion converge. We prove that the free energy of
this model converges as the size of the matrices go to infinity and study the
saddle points of the limit.

mmaida@umpa.ens-lyon.fr

2468. MAXIMAL LOCAL TIME OF A D-DIMENSIONAL SIMPLE RANDOM WALK ON SUBSETS 

Endre Cs\'{a}ki, Ant\'{o}nia F\"{o}ldes and P\'al R\'ev\'esz

Strong theorems are given for the maximal local time on balls and subspaces
for the $d$-dimensional simple symmetric random walk.

csaki@renyi.hu

2469. LOGARITHMIC SOBOLEV INEQUALITY FOR ZERO-RANGE DYNAMICS 

Paolo Dai Pra and Gustavo Posta

We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a
box of diameter $L$ grows as $L^2$.

guspos@mate.polimi.it

2470. RELATIVE FATOU'S THEOREM FOR $(-\DELTA)^{\ALPHA/2}$-HARMONIC FUNCTIONS IN BOUNDED $\KAPPA$-FAT OPEN SET 

Panki Kim

We give a probabilistic proof of relative Fatou's theorem for
$(-\Delta)^{\alpha/2}$-harmonic functions (equivalently for symmetric
$\alpha$-stable processes) in bounded $\kappa$-fat open set where $\alpha \in
(0,2)$. That is, if $u$ is positive $(-\Delta)^{\alpha/2}$-harmonic function in
a bounded $\kappa$-fat open set $D$ and $h$ is singular positive
$(-\Delta)^{\alpha/2}$-harmonic function in $D$, then non-tangential limits of
$u/h$ exist almost everywhere with respect to the Martin-representing measure
of $h$.
  It is also shown that, under the gaugeability assumption, relative Fatou's
theorem is true for operators obtained from the generator of the killed
$\alpha$-stable process in bounded $\kappa$-fat open set $D$ through non-local
Feynman-Kac transforms. As an application, relative Fatou's theorem for
relativistic stable processes is also true if $D$ is bounded $C^{1,1}$-open
set.

pkim@math.washington.edu

2471. MULTITYPE CONTACT PROCESS WITH FROZEN STATES 

Nicolas Lanchier

We study a generalization of the two colors multitype contact process
intended to mimic an example of interspecific competition called allelopathy.

nicolas.lanchier@univ-rouen.fr

2472. A GENERAL DIVERGENCE MEASURE FOR MONOTONIC FUNCTIONS AND APPLICATIONS IN INFORMATION THEORY 

Sever Silvestru Dragomir

A general divergence measure for monotonic functions is introduced. Its
connections with the f-divergence for convex functions are explored. The main
properties are pointed out.

2473. THE SQUARE OF WHITE NOISE AS A JACOBI FIELD 

E. Lytvynov

We identify the representation of the square of white noise obtained by L.
Accardi, U. Franz and M. Skeide in [Comm. Math. Phys. 228 (2002), 123--150]
with the Jacobi field of a L\'evy process of Meixner's type.

2474. A NOTE ON RANDOM BULGARIAN SOLITAIRE 

Serguei Popov

We consider a stochastic variant of the game of Bulgarian solitaire [M.
Gardner (1983), Sci. Amer. 249, 12-21]. For the stationary measure of the
random Bulgarian solitaire, we prove that most of its mass is concentrated on
(roughly) triangular configurations of certain type.

2475. A SURVEY OF MAX-TYPE RECURSIVE DISTRIBUTIONAL EQUATIONS 

Antar Bandyopadhyay and David J. Aldous

In certain problems in a variety of applied probability settingss (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form $X =
g((\xi_i, X_i), i \geq 1)$, where equality means in distribution. Here
$(\xi_i)$ and $g(\cdot)$ are given and the $(X_i)$ are inddependent copies of
the unknown distribution $X$. We survey this area, emphasizing examples where
the function $g$ is essentially a "maximum" or "minimum" function. We draw
attention to the theoretical question of endogeny : in the associated recursive
tree process $X_{{\bf i}}$, are the $X_{{\bf i}}$ measurable functions of the
innovation process $(\xi_{{\bf i}})$ ?

2476. BIVARIATE UNIQUENESS IN THE LOGISTIC RECURSIVE DISTRIBUTIONAL EQUATION 

Antar Bandyopadhyay

In this work we prove the bivariate uniqueness property of the logistic
recursive distributional equation, which arise in the study of the random
assignment problem, as described by Aldous (2001). Using this and the genral
framework of Aldous and Bandyopadhyay (2003), we then conclude that the
associated recursive tree process is endogenous. Thus the logistic variables
defined by Aldous (2001) for finding the limiting constant for the optimal cost
in random assignment problem turns out to be measurable with respect the edge
weights. This then ansers the question raised by Aldous (2001). The method
involves construction of an explicit recursion to show that the associated
integral equation has unique solution.

2477. MARKOV PROPERTY OF MONOTONE L\'EVY PROCESSES 

Uwe Franz and Naofumi Muraki

  Monotone L\'evy processes with additive increments are defined and studied.
It is shown that these processes have a natural Markov structure and their
Markov transition semigroups are characterized using the monotone
L\'evy-Khintchine formula. Monotone L\'evy processes turn out to be related to
classical L\'evy processes via Attal's ``remarkable transformation.'' A
monotone analogue of the family of exponential martingales associated to a
classical L\'evy process is also defined.

2478. CONDITIONAL INTENSITY AND GIBBSIANNESS OF DETERMINANTAL POINT PROCESSES 

Hans-Otto Georgii, Hyun Jae Yoo

The Papangelou intensities of determinantal (or fermion) point processes are
investigated. These exhibit a monotonicity property expressing the repulsive
nature of the interaction, and satisfy a bound implying stochastic domination
by a Poisson point process. We also show that determinantal point processes
satisfy the so-called condition $(\Sigma_{\lambda})$ which is a general form of
Gibbsianness. In the absence of percolation, the Gibbsian conditional
probabilities can be identified explicitly.

2479. DEGREES OF TRANSIENCE AND RECURRENCE AND HIERARCHICAL RANDOM WALKS 

D.A. Dawson, L.G. Gorostiza, A. Wakolbinger

The notion of degree and related notions concerning recurrence and transience
for a class of L'evy processes on metric Abelian groups are studied. The case
of random walks on a hierarchical group is examined with emphasis on the role
of the ultrametric structure of the group and on analogies and differences with
Euclidean random walks. Applications to separation of time scales and
occupation times of multilevel branching systems are discussed.

2480. HYDRODYNAMIC LIMIT FOR PERTURBATION OF A HYPERBOLIC EQUILIBRIUM POINT IN TWO-COMPONENT SYSTEMS 

Benedek Valko

We consider one-dimensional, locally finite interacting particle systems with
two conservation laws. The models have a family of stationary measures with
product structure and we assume the existence of a uniform bound on the inverse
of the spectral gap which is quadratic in the size of the system. Under
Eulerian scaling the hydrodynamic limit for the macroscopic density profiles
leads to a two-component system of conservation laws. The resulting pde is
hyperbolic inside the physical domain of the macroscopic densities, with
possible loss of hyperbolicity at the boundary. We investigate the propagation
of small perturbations around a \emph{hyperbolic} equilibrium point. We prove
that the perturbations essentially evolve according to two \emph{decoupled}
Burgers equations. The scaling is not Eulerian: if the lattice constant is
$n^{-1}$, the perturbations are of order $n^{-\beta}$ then time is speeded up
by $n^{1+\b}$. Our derivation holds for $0<\beta< \frac15$. The proof relies on
Yau's relative entropy method, thus it applies only in the regime of smooth
solutions.

2481. SOME PROCESSES ASSOCIATED WITH FRACTIONAL BESSEL PROCESSES 

Yaozhong Hu and David Nualart

Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional
fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=%
\sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process.
It\^{o}'s formula for the fractional Brownian motion leads to the equation $
R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}%
dB_{s}^{i}+H(d-1)\int_{0}^{t}\frac{s^{2H-1}}{R_{s}}ds . $ In the Brownian
motion case ($H=1/2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{%
R_{s}}dB_{s}^{i}$ is a Brownian motion. In this paper it is shown that $X_{t}$
is \underbar{not} a fractional Brownian motion if $H\not=1/2$. We will study
some other properties of this stochastic process as well.

2482. A LATTICE ANIMAL APPROACH TO PERCOLATION 

Alan Hammond

We examine the percolation model on $\mathbb{Z}^d$ by an approach involving
lattice animals and their surface-area-to-volume ratio. For $\beta \in
[0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of
edges of the number of lattice animals containing the origin which have
surface-area-to-volume ratio $\beta$. The function $f$ is bounded above by a
function which may be written in an explicit form. For low values of $\beta$
($\beta \leq 1/p_c - 1$), equality holds, as originally demonstrated by
F.Delyon. For higher values ($\beta > 1/p_c - 1$), the inequality is strict.
  We introduce two critical exponents, one of which describes how quickly $f$
falls away from the explicit form as $\beta$ rises from $1/p_c - 1$, and the
second of which describes how large clusters appear in the marginally
subcritical regime of the percolation model. We demonstrate that the pair of
exponents must satisfy certain inequalities, while other such inequalities
yield sufficient conditions for the absence of an infinite cluster at the
critical value. The first exponent is related to one of a more conventional
nature in the scaling theory of percolation, that of correlation size. In
deriving this relation, we find that there are two possible behaviours,
depending on the value of the first exponent, for the typical
surface-area-to-volume ratio of an unusually large cluster in the marginally
subcritical regime.

2483. PERCOLATION AND LATTICE ANIMALS: EXPONENT RELATIONS, AND CONDITIONS FOR $\THETA(P_C)=0$ 

Alan Hammond

We examine the percolation model in $\mathbb{Z}^d$ by an approach involving
lattice animals, in which their relevant characteristic is
surface-area-to-volume ratio. Two critical exponents are introduced. The first
is related to the growth rate in size of the number of lattice animals up to
translation whose surface-area-to-volume ratio is marginally greater than
$1/p_c -1$. The second describes how unusually large clusters form in the
percolation model at parameter values slightly below $p_c$. Certain
inequalities on the pair of exponents cannot be satisfied, while others imply
the continuity of the percolation probability. The first exponent is related to
one of a more conventional nature, that of correlation size.
  In this paper, the central aspects of the approach are described, and the
proofs of the main results are presented. The report located at math.PR/0402026
gives complete proofs of all of the assertions.

2484. GAUSSIAN SCALING FOR THE CRITICAL SPREAD-OUT CONTACT PROCESS ABOVE THE UPPER CRITICAL DIMENSION 

Remco van der Hofstad, Akira Sakai

We consider the critical spread-out contact process in $\Zd$ with $d\geq 1$,
whose infection range is denoted by $L\geq1$. The two-point function
$\tau_t(x)$ is the probability that $x\in\Zd$ is infected at time $t$ by the
infected individual located at the origin $o\in\Zd$ at time 0. We prove
Gaussian behavior for the two-point function with $L\geq L_0$ for some finite
$L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field
limit to obtain Gaussian behaviour for $\tau_{tT}$ with $t>0$ fixed and $T\to
\infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any
$b>(4-d)/2d$.
  The proof is based on the lace expansion and an adaptation of the inductive
approach applied to the discretized contact process. We prove the existence of
several critical exponents and show that they take on mean-field values. The
results in this paper provide crucial ingredients to prove convergence of the
finite-dimensional distributions for the contact process towards the canonical
measure of super-Brownian motion, which we defer to a sequel of this paper.

2485. CRITICAL POINTS FOR SPREAD-OUT SELF-AVOIDING WALK, PERCOLATION AND THE CONTACT PROCESS ABOVE THE UPPER CRITICAL DIMENSIONS 

Remco van der Hofstad, Akira Sakai

We consider self-avoiding walk and percolation in $\Zd$, oriented percolation
in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being
the coupling function whose range is denoted by $L<\infty$. For percolation,
for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The
above models are known to exhibit a phase transition when the parameter $p$
varies around a model-dependent critical point $\pc$. We investigate the value
of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and
$L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$, where the
universal term 1 is the mean-field critical value, and the model-dependent term
$C(D)=O(L^{-d})$ is written explicitly in terms of the function $D$. Our proof
is based on the lace expansion for each of these models.

2486. FROM N+1-LEVEL ATOM CHAINS TO N-DIMENSIONAL NOISES 

Stephane Attal and Yan Pautrat

In quantum physics, the state space of a countable chain of (n+1)-level atoms
becomes, in the continuous field limit, a Fock space with multiplicity n. In a
more functional analytic language, the continuous tensor product space over R
of copies of the space C^{n+1} is the symmetric Fock space Gamma_s(L^2(R;C^n)).
In this article we focus on the probabilistic interpretations of these facts.
We show that they correspond to the approximation of the n-dimensional normal
martingales by means of obtuse random walks, that is, extremal random walks in
R^n whose jumps take exactly n+1 different values. We show that these
probabilistic approximations are carried by the convergence of the basic matrix
basis a^i_j(p) of $\otimes_N \CC^{n+1}$ to the usual creation, annihilation and
gauge processes on the Fock space.

2487. PERFECT SIMULATION FOR UNILATERAL FIELDS 

Emilio De Santis, Mauro Piccioni

In this paper we consider two-point unilateral Markov fields on a
two-dimensional lattice as considered by Pickard, Galbraith and Walley. We show
that, under various ergodicity conditions, they can be perfectly simulated in
the stationary state on any finite window. The techniques which are used
connect perfect simulation with oriented percolation through suitable coupling
constructions.

2488. CHANGE POINT MODELS AND CONDITIONALLY PURE BIRTH PROCESSES; AN INEQUALITY ON THE STOCHASTIC INTENSITY 

Emilio De Santis, Fabio Spizzichino

We analyze several aspects of a class of simple counting processes, that can
emerge in some fields of applications where the presence of a change-point
occurs. Under simple conditions we, in particular, prove a significant
inequality for the stochastic intensity.

2489. CRITICALITY IN UNBOUNDED-TYPES BRANCHING PROCESSES 

G.T. Tetzlaff

Conditions for almost sure extinction are studied in discrete time branching
processes with an infinite number of types. It is not assumed that the expected
number of children is a bounded function of the parent's type. There might also
be no integer m such that there is a lower positive bound, uniform over the
ancestor's type, for the probability that a population is extinct at the m-th
generation. A weaker condition than the existence of such an m is seen to lead
to extinction almost surely if the sequence of expected generation sizes does
not tend to infinity. Some criteria for a positive probability of nonextinction
are given. Examples are provided by extending to our setting two applications,
namely Leslie population dynamics and processes arising in continuum
percolation in which the offsprings follow Poisson point distributions.

2490. MAXIMAL CLUSTERS IN NON-CRITICAL PERCOLATION AND RELATED MODELS 

Remco van der Hofstad, Frank Redig

We investigate the maximal non-critical cluster in a big box in various
percolation-type models. We investigate its typical size, and the fluctuations
around this typical size. The limit law of these fluctuations are related to
maxima of independent random variable with law described by a single cluster.

2491. ONE-DIMENSIONAL RANDOM FIELD KAC'S MODEL: LOCALIZATION OF THE PHASES 

Marzio Cassandro, Enza Orlandi, Pierre Picco, Maria Eulalia Vares

We study the typical profiles of a one dimensional random field Kac model,
for values of the temperature and magnitude of the field in the region of the
two absolute minima for the free energy of the corresponding random field Curie
Weiss model. We show that, for a set of realizations of the random field of
overwhelming probability, the localization of the two phases corresponding to
the previous minima is completely determined. Namely, we are able to construct
random intervals tagged with a sign, where typically, with respect to the
infinite volume Gibbs measure, the profile is rigid and takes, according to the
sign, one of the two values corresponding to the previous minima. Moreover, we
characterize the transition from one phase to the other.

2492. STOCHASTIC PROCESSES IN RANDOM GRAPHS 

Anatolii A. Puhalskii

We study the asymptotics of large, moderate and normal deviations for the
connected components of the sparse random graph by the method of stochastic
processes. We obtain the logarithmic asymptotics of large deviations of the
joint distribution of the number of connected components, of the sizes of the
giant components, and of the numbers of the excess edges of the giant
components. For the supercritical case, we obtain the asymptotics of normal
deviations and the logarithmic asymptotics of large and moderate deviations of
the joint distribution of the number of components, of the size of the largest
component, and of the number of the excess edges of the largest component. For
the critical case, we obtain the logarithmic asymptotics of moderate deviations
of the joint distribution of the sizes of connected components and of the
numbers of the excess edges. Some related asymptotics are also established. The
proofs of the large and moderate deviation asymptotics employ methods of
idempotent probability theory. As a byproduct of the results, we provide some
additional insight into the nature of phase transitions in sparse random
graphs.

2493. PROBABILITIES OF RANDOMLY CENTERED SMALL BALLS AND QUANTIZATION IN BANACH SPACES 

S. Dereich, M. A. Lifshits

We investigate the Gaussian small ball probabilities with random centers,
find their deterministic a.s.-equivalents and establish a relation to
infinite-dimensional high-resolution quantization.

2494. ASYMPTOTIC LAWS FOR NONCONSERVATIVE SELF-SIMILAR FRAGMENTATIONS 

Jean Bertoin and Alexander Gnedin

We consider a self-similar fragmentation process in which the generic
particle of mass $x$ is replaced by the offspring particles at probability rate
$x^\alpha$, with positive parameter $\alpha$. The total of offspring masses may
be both larger or smaller than $x$ with positive probability. We show that
under certain conditions the typical mass in the ensemble is of the order
$t^{-1/\alpha}$ and that the empirical distribution of masses converges to a
random limit which we characterise in terms of the reproduction law.

2495. RAYLEIGH PROCESSES, REAL TREES, AND ROOT GROWTH WITH RE-GRAFTING 

Steven N. Evans and Jim Pitman and Anita Winter

The real trees form a class of metric spaces that extends the class of trees
with edge lengths by allowing behavior such as infinite total edge length and
vertices with infinite branching degree. Aldous's Brownian continuum random
tree, the random tree-like object naturally associated with a standard Brownian
excursion, may be thought of as a random compact real tree. The continuum
random tree is a scaling limit as N tends to infinity of both a critical
Galton-Watson tree conditioned to have total population size N as well as a
uniform random rooted combinatorial tree with N vertices. The Aldous--Broder
algorithm is a Markov chain on the space of rooted combinatorial trees with N
vertices that has the uniform tree as its stationary distribution. We construct
and study a Markov process on the space of all rooted compact real trees that
has the continuum random tree as its stationary distribution and arises as the
scaling limit as N tends to infinity of the Aldous--Broder chain. A key
technical ingredient in this work is the use of a pointed Gromov--Hausdorff
distance to metrize the space of rooted compact real trees.

2496. A NOTE ON COMPACT MARKOV OPERATORS 

Fabio Zucca

The analytic properties of the Markov operator associated to a random walk
are common tools in the study of the behaviour and some probabilistic features
related to the walk. In this paper we consider a class of Markov operators
which generalizes the class of compact Markov operators and we study some
probabilistic properties of the associated random walk.

2497. LOWER ESTIMATES ON TRANSITION DENSITIES AND BOUNDS ON EXPONENTIAL ERGODICITY FOR STOCHASTIC PDE'S 

B.Goldys and B. Maslowski

A formula for the transition density of a process defined by an
infinite-dimensional stochastic equation is given in terms of the Ornstein
Uhlenbeck Bridge, and lower estimate on the density is provided. Then we prove
uniform exponential ergodicity and V-ergodicity for a large class of equations.
The method allows us to find computable bounds on the respective convergence
rates and on spectral gap for the Markov semigroups defined by the equations.
The bounds turn out to be uniform with respect to a large family of nonlinear
drift coefficients. Examples of finite dimensional stochastic equations and
semilinear parabolic equations are given.

2498. TWO RECURSIVE DECOMPOSITIONS OF BROWNIAN BRIDGE 

David Aldous and Jim Pitman

Aldous and Pitman (1994) studied asymptotic distributions, as n tends to
infinity, of various functionals of a uniform random mapping of a set of n
elements, by constructing a mapping-walk and showing these mapping-walks
converge weakly to a reflecting Brownian bridge. Two different ways to encode a
mapping as a walk lead to two different decompositions of the Brownian bridge,
each defined by cutting the path of the bridge at an increasing sequence of
recursively defined random times in the zero set of the bridge. The random
mapping asymptotics entail some remarkable identities involving the random
occupation measures of the bridge fragments defined by these decompositions. We
derive various extensions of these identities for Brownian and Bessel bridges,
and characterize the distributions of various path fragments involved, using
the theory of Poisson processes of excursions for a self-similar Markov process
whose zero set is the range of a stable subordinator of index between 0 and 1.

2499. NONCLASSICAL BROWNIAN MOTIONS, STOCHASTIC FLOWS, CONTINUOUS PRODUCTS. I 

Boris Tsirelson

Contrary to the classical wisdom, processes with independent values (defined
properly) are much more diverse than white noise combined with Poisson point
processes, as well as product systems are much more diverse than Fock spaces.
  This text is the first part of a survey of recent progress in constructing
and investigating nonclassical Brownian motions, stochastic flows and
continuous products of probability spaces and Hilbert spaces.

2500. FEYNMAN-KAC FORMULAE, GENEALOGICAL AND INTERACTING PARTICLE SYSTEMS WITH APPLICATIONS 

Pierre Del Moral

This book contains a systematic and self-contained treatment of 
Feynman-Kac path measures, their genealogical and interacting 
particle interpretations, and their applications to a variety of problems 
arising in statistical physics, biology,and advanced engineering 
sciences.  Topics include spectral analysis of Feynman-Kac-
Schrodinger operators, Dirichlet problems with boundary conditions, 
finance, molecular analysis, rare events and directed polymers simulation, 
genetic algorithms, Metropolis-Hastings type models, as well as 
filtering problems and hidden Markov chains. 

This text takes readers in a clear and progressive format from simple 
to recent and advanced topics in pure and applied probability such as 
contraction and annealed properties of non linear semi-groups, 
functional entropy inequalities, empirical process convergence, 
increasing propagations of chaos, central limit,and Berry
Esseen type theorems as well as large deviations principles 
for strong topologies on path-distribution spaces. Topics also include 
a body of powerful branching and interacting particle methods and 
worked out illustrations of the key aspect of the theory. 

delmoral@cict.fr

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

2501. THE VOTER MODEL WITH ANTI-VOTER BONDS 

Nina Gantert, Matthias Lowe and Jeffrey Steif

We study the voter model with both positive and negative 
bonds on a general locally finite connected infinite graph. 
We obtain various results concerning ergodicity of the process.

steif@math.chalmers.se

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

2502. COUNTABLE SYSTEMS OF DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO SUPER-MARKOV CHAINS 

Richard F. Bass and Edwin Perkins

We prove well-posedness of the martingale problem       
for an infinite-dimensional degenerate elliptic 
operator under appropriate Holder continuity 
conditions on the coefficients.  These martingale 
problems include large population limits of 
branching particle systems on a countable state 
space in which the particle dynamics and branching 
rates may depend on the entire population in a         
Holder fashion. This extends an approach originally 
used by the authors in finite dimensions.

bass@math.uconn.edu  perkins@math.ubc.ca

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

2503. SOME NEW RESULTS ON BROWNIAN DIRECTED POLYMERS IN RANDOM ENVIRONMENT 

Francis Comets and Nobuo Yoshida

We prove some new results on Brownian directed polymers in random environment 
recently introduced by the authors. The directed polymer in this model is a 
$d$-dimensional Brownian motion (up to finite time $t$) viewed under a Gibbs 
measure which is built up with a Poisson random measure on $\R_+ \times \R^d$ 
(=time $\times$ space). Here, the Poisson random measure plays the role of the 
random environment which is independent both in time and in space. We prove that

(i) For $d \ge 3$ and the inverse temperature $\beta$ smaller than a certain 
positive value $\beta_0$, the central limit theorem for the directed polymer 
holds almost surely with respect to the environment.

(ii) If $d=1$ and $\beta \neq 0$, the fluctuation of the free energy diverges 
with a magnitude not smaller than $t^{1/8}$ as $t$ goes to infinity. The argument 
leading to this result strongly supports the inequalities $\chi(1)\geq 1/5$ for
the fluctuation exponent for the free energy, and $\xi(1)\geq 3/5$ for the 
wandering exponent.

We provide necessary background by reviewing some results in the previous paper:
`Brownian Directed Polymers in Random Environment'

comets@math.jussieu.fr nobuo@math.kyoto-u.ac.jp

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

2504. A BROWNIAN-TIME EXCURSION INTO FOURTH ORDER PDEs, LINEARIZED KURAMOTO-SIVASHINSKY, AND BTP-SPDEs on $\Rp\times\Rd$ 

Hassan Allouba

In recent articles we have introduced the class of Brownian-time processes (BTPs)
and the Linearized Kuramoto-Sivashinsky process (LKSP).  Probabilistically, BTPs 
represent a unifying class for some different exciting processes like the iterated 
Brownian motion (IBM) of Burdzy (a process with fourth order properties) and 
the Brownian-snake of Le Gall (a second order process); they also include many 
additional new and quite interesting processes.  The LKSP is intimately connected 
to the Kuramoto-Sivashinsky PDEs, one of the most celebrated PDEs in modern applied 
mathematics.  We start by surveying the fourth order PDE connections to BTPs and the 
LKSP that we uncovered in two recent articles.

In the second part of the article we introduce BTP-SPDEs, these are SPDEs in which 
the PDE part is that solved by running a BTP.   We consider a BTP-SPDE driven by 
an additive space-time white noise on the time-space set $\Rp\times\Rd$; and we prove 
the existence of a unique {\sl real-valued}, $L^p(\Omega,\P)$ for all $p\ge1$, BTP 
solution to such BTP-SPDEs for $1\le d\le3$.  This contrasts starkly with the standard 
theory of reaction-diffusion type SPDEs driven by space-time white noise, in which 
real-valued solutions are confined to one spatial dimension.  Like the PDEs case, 
BTP-SPDEs also provide a valuable insight into other fourth order SPDEs of applied 
science.  We carry out such a program in upcoming articles.

allouba@math.kent.edu

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

2505. DIFFERENTIAL OPERATORS AND SPECTRAL DISTRIBUTIONS OF INVARIANT ENSEMBLES FROM THE CLASSICAL ORTHOGONAL POLYNOMIALS. THE DISCRETE CASE 

Michel Ledoux

We examine the Charlier, Meixner, Krawtchouk and Hahn
discrete orthogonal polynomial ensembles, deeply investigated by
K. Johansson, using integration by parts, differential equations on Laplace
transforms and moment equations. As for the matrix ensembles, equilibrium
measures of orthogonal polynomials are described as limits of empirical
spectral distributions. In particular, a new description of the equilibrium
measures as adapted mixtures of the universal arcsine law with an
independent uniform distribution is emphasized. Factorial moment identities
on mean spectral measures may be used towards small deviation inequalities
on the associated shape functions at the rate given by the Tracy-Widom
asymptotics.

ledoux@math.ups-tlse.fr
http://www.lsp.ups-tlse.fr/Ledoux/
Home Page | Mirror: France | Latest Abstracts | Search | Abstract Archive | Submissions | Subscriptions
stefano . iacus at unimi . it