Probability Abstracts 78
This document contains abstracts 2399-2448.
They have been mailed on December 31, 2003.
2399. GIBBS PROPERTIES OF THE FUZZY POTTS MODEL ON TREES AND IN MEAN FIELD
Olle Haeggstroem, Christof Kuelske
We study Gibbs properties of the fuzzy Potts model in the mean field case
(i.e on a complete graph) and on trees. For the mean field case, a complete
characterization of the set of temperatures for which non-Gibbsianness happens
is given. The results for trees are somewhat less explicit, but we do show for
general trees that non-Gibbsianness of the fuzzy Potts model happens exactly
for those temperatures where the underlying Potts model has multiple Gibbs
measures.
kuelske@wias-berlin.de
2400. THE NOISY VOTER-EXCLUSION PROCESS
Paul Jung
The symmetric exclusion process and the voter model are two interacting
particle systems for which a dual finite particle system allows one to
characterize its invariant measures. Adding spontaneous births and deaths to
the two processes still allows one to use the dual to obtain information about
the original process. We study the noisy voter-exclusion process which
generalizes these processes by allowing for all of these interactions to take
place. The dual process is used to characterize its invariant measures under
various circumstances. Finally, an ergodic theorem for a related process is
proved using couplings.
pjung@math.cornell.edu
2401. STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS: I.
PATHWISE UNIQUENESS AND LARGE DEVIATION
Shizan Fang, Tusheng Zhang
We study a class of stochastic differential equations with non-Lipschitzian
coefficients.A unique strong solution is obtained and a large deviation
principle of Freidln-Wentzell type has been established.
fang@u-bourgogne.fr
2402. STOCHASTIC DIFFERENTIAL EQUTIONS WITH NON-LIPSCHITZ COEFFICIENTS:II.
DEPENDENCE WITH RESPECT TO INITIAL VALUES
Shizan Fang, Tusheng Zhang
The existence of the unique strong solution for a class of stochastic
differential equations with non-Lipschitz coefficients was established
recently. In this paper, we shall investigate the dependence with respect to
the initial values. We shall prove that the non confluence of solutions holds
under our general conditions. To obtain a continuous version, the modulus of
continuity of coefficients is assumed to be less than $\dis
|x-y|\log{1\over|x-y|}$. In this case, it will give rise to a flow of
homeomorphisms if the coefficients are compactly supported.
fang@u-bourgogne.fr
2403. A PHASE TRANSITION AND STOCHASTIC DOMINATION IN PIPPENGER'S
PROBABILISTIC FAILURE MODEL FOR BOOLEAN NETWORKS WITH UNRELIABLE GATES
Maxim Raginsky
We study Pippenger's model of Boolean networks with unreliable gates. In this
model, the conditional probability that a particular gate fails, given the
failure status of any subset of gates preceding it in the network, is bounded
from above by some $\epsilon$. We show that if we pick a Boolean network with
$n$ gates at random according to the Barak-Erd\H{o}s model of a random acyclic
digraph, such that the expected edge density is $c n^{-1}\log n$, and if
$\epsilon$ is equal to a certain function of the size of the largest reflexive,
transitive closure of a vertex (with respect to a particular realization of the
random digraph), then Pippenger's model exhibits a phase transition at $c=1$.
Namely, with probability $1-o(1)$ as $n\to\infty$, we have the following: for
$0 \le c \le 1$, the minimum of the probability that no gate has failed, taken
over all probability distributions of gate failures consistent with Pippenger's
model, is equal to $o(1)$, whereas for $c >1$ it is equal to
$\exp(-\frac{c}{e(c-1)}) + o(1)$. We also indicate how a more refined analysis
of Pippenger's model, e.g., for the purpose of estimating probabilities of
monotone events, can be carried out using the machinery of stochastic
domination.
maxim@ece.northwestern.edu
2404. ON THE LOCATION OF THE 1-PARTICLE BRANCH OF THE SPECTRUM OF THE
DISORDERED STOCHASTIC ISING MODEL
M. Gianfelice, M. Isopi
We analyse the lower non trivial part of the spectrum of the generator of the
Glauber dynamics, which we consider a positive operator, for a d-dimensional
nearest neighbour Ising model with a bounded random potential. We prove
conjecture 1 in a paper by Albeverio et al.(referred as [AMSZ]) that is, for
sufficently large values of the temperature, the first band of the spectrum of
the generator of the process coincides with a closed non random segment of the
real line.
gianfeli@dm.unibo.it
2405. BOOTSTRAP PERCOLATION ON INFINITE TREES AND NON-AMENABLE GROUPS
Jozsef Balogh, Yuval Peres, Gabor Pete
Bootstrap percolation on an arbitrary graph has a random Bernoulli$(p)$
initial configuration of occupied sites and a deterministic spreading rule with
a fixed parameter $k$: if a vacant site has at least $k$ occupied neighbors at
a certain time step, then it becomes occupied in the next step. This process is
well-studied on $\Z^d$; here we investigate it on infinite trees and on
non-amenable Cayley graphs. The critical probability is the infimum of those
values of $p$ for which the process achieves complete occupation with positive
probability. On general trees, we find the following discontinuity: if the
branching number of a tree is strictly smaller than $k$, then the critical
probability is 1, while it is $1-1/k$ on the $k$-ary tree. A related result is
that in any rooted tree $T$ there is a way of erasing $k$ children of the root,
together with all their descendents, and repeating this for all remaining
children, and so on, such that the remaining tree $T'$ has branching number
$\br(T')\leq \max\{\br(T)-k, 0\}$. We also prove that on any $2k$-regular
non-amenable graph, the critical probability for the $k$-rule is strictly
positive.
gabor@stat.berkeley.edu
2406. UNIFORM INFINITE PLANAR TRIANGULATION AND RELATED TIME-REVERSED CRITICAL
BRANCHING PROCESS
Maxim Krikun
We establish a connection between the uniform infinite planar triangulation
and some critical time-reversed branching process. This allows to find a
scaling limit for the principal boundary component of a ball of radius R for
large R (i.e. for a boundary component separating the ball from infinity). We
show also that outside of R-ball a contour exists that has length linear in R.
krikun@lbss.math.msu.su
2407. RENEWAL THEORY AND GEOMETRIC INFINITE DIVISIBILITY
E. Sandhya and R. N. Pillai
The role of geometrically infinitely divisible laws in renewal equations and
superposition of renewal processes are explored here. Some examples are also
discussed.
esandhya@hotmail.com
2408. NONSTATIONARY QUEUES: ESTIMATION OF THE RATES OF CONVERGENCE
Boris L. Granovsky and Aleksandr I. Zeifman
The paper is devoted to the estimation of the rate of of exponential
convergence of nonhomogeneous queues exhibiting different types of ergodicity.
The main tool of our study is the method, which was proposed by the second
author in the late 1980-s and was subsequently extended and developed in
different directions in a series of joint papers by the authors of the present
paper. The method originated from the idea of Gnedenko and Makarov to employ
the logarithmic norm of a matrix to the study of the problem of stability of
nonhomogeneous Markov chains.
In the present paper, we apply the method to a class of Markov queues with a
special form of nonhomogenuity that is common in applications.
mar18aa@techunix.technion.ac.il
2409. A MODEL OF THE TERM STRUCTURE OF INTEREST RATES BASED ON L\'EVY FIELDS
Sergio Albeverio, Eugene Lytvynov, Andrea Mahnig
An extension of the Heath--Jarrow--Morton model for the development of
instantaneous forward interest rates with deterministic coefficients and
Gaussian as well as L\'evy field noise terms is given. In the special case
where the L\'evy field is absent, one recovers a model discussed by
D.P.~Kennedy.
lytvynov@wiener.iam.uni-bonn.de
2410. INSTABILITY IN STOCHASTIC AND FLUID QUEUEING NETWORKS
David Gamarnik and John Hasenbein
The fluid model has proven to be one of the most effective tools for the
analysis of stochastic queueing networks, specifically for the analysis of
stability. It is known that stability of a fluid model implies positive
(Harris) recurrence (stability) of a corresponding stochastic queueing network,
and weak stability implies rate stability of a corresponding stochastic
network. These results have been established both for cases of specific
scheduling policies and for the class of all work conserving policies.
However, only partial converse results have been established and in certain
cases converse statements do not hold. In this paper we close one of the
existing gaps. For the case of networks with two stations we prove that if the
fluid model is not weakly stable under the class of all work conserving
policies, then a corresponding queueing network is not rate stable under the
class of all work conserving policies. We establish the result by building a
particular work conserving scheduling policy which makes the associated
stochastic process transient. An important corollary of our result is that the
condition $\rho^*\leq 1$, which was proven in \cite{daivan97} to be the exact
condition for global weak stability of the fluid model, is also the exact
global rate stability condition for an associated queueing network. Here
$\rho^*$ is a certain computable parameter of the network involving virtual
station and push start conditions.
gamarnik@watson.ibm.com
2411. 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 earlierwork of Arratia
and of T\'oth and Werner by providing a new characterization which is then used
to obtain convergence results for the BW distribution, including convergence of
the system ofall coalescing random walks to the BW under diffusive space-time
scaling.
isopi@mat.uniroma1.it
2412. BESSEL PROCESSES, THE INTEGRAL OF GEOMETRIC BROWNIAN MOTION, AND ASIAN
OPTIONS
M. Schr\"oder, P. Carr
This paper is motivated by questions about averages of stochastic processes
which originate in mathematical finance, originally in connection with valuing
the so-called Asian options. Starting with research of Yor's in 1992, these
questions about exponential functionals of Brownian motion have been studied in
terms of Bessel processes using Yor's 1980 Hartman-Watson theory. Consequences
of this approach for valuing Asian options proper have been spelled out by
Geman and Yor in 1993 whose Laplace transform results were in fact regarded as
a noted advance. Unfortunately, a number of difficulties with the key results
of this last contribution have surfaced which are now addressed in this paper.
One of them in particular is of a principal nature and originates with the
Hartman-Watson approach itself: this approach is in general applicable without
modifications only if it does not involve Bessel processes of negative indices.
The main mathematical contribution of this paper is the developement of three
principal ways to overcome these restrictions, in particular by merging
stochastics and complex analysis in what seems a novel way, and the discussion
of their consequences for the valuation of Asian options proper.
schroeder@math.uni-mannheim.de
2413. STABILITY OF RANDOM SUMS
S. Satheesh, N. Unnikrishnan Nair and E. Sandhya
When the distribution of a random (N) sum of independent copies of a r.v X is
of the same type as that of X we say that X is N-sum stable. In this paper we
consider a generalization of stability of geometric sums by studying
distributions that are stable under summation w.r.t Harris law. We show that
the notion of stability of random sums can be extended to include the case when
X is discrete. Finally we propose a method to identify the probability law of N
for which X is N-sum stable. See also Satheesh and Nair (2002), (Some classes
of distributions on ther non-negative lattice, J. Ind. Statist. Assoc., 2002,
40, 41-58) for a study of discrete laws of the same type and stability of
geometric sums of discrete laws.
esandhya@hotmail.com
2414. SOME CLASSES OF DISTRIBUTIONS ON THE NON-NEGATIVE LATTICE
S. Satheesh and N. Unnikrishnan Nair
A method for constructing distributions on the non negative integers as
discrete analogue of continuous distributions on the non negative real is
presented. A justification of the definition of discrete self decomposable laws
is provided. Discrete analogue of distributions of the same type and the role
of Bernoulli law in this context is discussed. Generalizations of some discrete
laws and their properties are given. The geometric compounding problem for
discrete distributions is studied by introducing discrete semi Mittag Leffler
laws.
ssatheesh@sancharnet.in
2415. INFINITE INTERACTING DIFFUSION PARTICLES I: EQUILIBRIUM PROCESS AND ITS
SCALING LIMIT
Yuri Kondratiev, Eugene Lytvynov, Michael R\"ockner
A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system
is a stochastic process which takes values in the space $\Gamma$ of all locally
finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $\mu$
as an invariant measure. We assume that $\mu$ corresponds to a symmetric pair
potential $\phi(x-y)$. An important class of stochastic dynamics of a classical
continuous system is formed by diffusions. Till now, only one type of such
dynamics--the so-called gradient stochastic dynamics, or interacting Brownian
particles--has been investigated. By using the theory of Dirichlet forms, we
construct and investigate a new type of stochastic dynamics, which we call
infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal
E}_\mu^\Gamma$ on $L^2(\Gamma;\mu)$, and under general conditions on the
potential $\phi$, prove its closability. For a potential $\phi$ having a
``weak'' singularity at zero, we also write down an explicit form of the
generator of ${\cal E}_\mu^\Gamma$ on the set of smooth cylinder functions. We
then show that, for any Dirichlet form ${\cal E}_\mu^\Gamma$, there exists a
diffusion process that is properly associated with it. Finally, we study a
scaling limit of interacting diffusions in terms of convergence of the
corresponding Dirichlet forms, and we also show that these scaled processes are
tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of
${\cal D}{:=}C_0^\infty({\Bbb R})$.
lytvynov@wiener.iam.uni-bonn.de
2416. STATISTICAL HAUSDORFF DIMENSION OF LABELLED TREES AND QUADRANGULATIONS
Philippe Chassaing, Bergfinnur Durhuus
Exploiting a bijective correspondence between planar quadrangulations and
so-called well labelled trees, we define a random ensemble of infinite
surfaces, as a limit of uniformly distributed ensembles of quadrangulations of
fixed finite volume. The limit random surface can be described in terms of a
birth and death process and a sequence of multitype Galton Watson trees. As a
consequence, we find that the volume of the ball of radius $r$ around a marked
point in the limit random surface is $\Theta(r^{4})$, leading to statistical
Hausdorff dimension 4 for this random ensemble of infinite surfaces.
chassain@iecn.u-nancy.fr
2417. THE DEAD LEAVES MODEL : GENERAL RESULTS AND LIMITS AT SMALL SCALES
Yann Gousseau and Francois Roueff
In this work, we introduce a random field in view of natural image modeling,
obtained as a limit of sequences of dead leaves models, when considering
arbitrarily small or big objects. The dead leaves model, introduced by the
Mathematical Morphology school, consists in the superposition of random closed
sets, and enables to model the occlusion phenomena. When combined with specific
sizes distributions for objects, they are known to provide adequate models for
natural images. However this framework yields a small scales cutoff and a limit
random field is introduced by letting this cutoff tend to zero. We first give a
rigorous definition of the dead leaves model, and compute the probability that
n compacts are included in distinct visible parts, which characterizes the
model. Then, we derive our limit model and some of its property, and study its
regularity.
gousseau@tsi.enst.fr
2418. SHARP INTEGRABILITY FOR BROWNIAN MOTION IN PARABOLA-SHAPED REGIONS
Rodrigo Banuelos and Tom Carroll
We study the sharp order of integrability of the exit position of Brownian
motion from the planar domains ${\cal P}_\alpha = \{(x,y)\in \bR\times
\bR\colon x> 0, |y| < Ax^{\alpha}\}$, $0<\alpha<1$. Together with some simple
good-$\lambda$ type arguments, this implies the order of integrability for the
exit time of these domains; a result first proved for $\alpha =1/2$ by
Ba\~nuelos, DeBlassie and Smits \cite{ba} and for general $\alpha$ by Li
\cite{li}. A sharp version of this result is also proved in higher dimensions.
banuelos@math.purdue.edu
2419. DEVIATIONS FROM THE CIRCULAR LAW
Brian Rider
Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in
which all entries are independent complex Gaussians of mean zero and variance
$\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the
eigenvalues converges to the uniform measure on the unit disk in the complex
plane. In this note we describe fluctuations about this {\em Circular Law}.
First we obtain finite $N$ formulas for the covariance of certain linear
statistics of the eigenvalues. Asymptotics of these objects coupled with a
theorem of Costin and Lebowitz then result in central limit theorems for a
variety of these statistics.
rider@math.duke.edu
2420. NONPARAMETRIC ESTIMATION IN THE MODEL OF MOVING AVERAGE
Alexander Alekseev
The subject of robust estimation in time series is widely discussed in
literature. One of the approaches is to use GM-estimation. This method
incorporates a broad class of nonparametric estimators which under suitable
conditions includes estimators robust to outliers in data. For the linear
models the sensitivity of GM-estimators to outliers have been studied in the
work by Martin and Yohai [5], and influence functionals for this estimator were
derived. In this paper we follow this direction and examine the asymptotical
properties of the class of M-estimators, which is narrower than the class of
GM-estimators, but gives more insight into asymptotical properties of such
estimators. This paper gives an asymptotic expansion of the residual weighted
empirical process, which allows to prove asymptotic normality of these
estimators in case of non-smooth objective functions. For simplicity MA(1)
model is considered, but it will be shown that even in this case mathematical
techniques used to derive these asymptotic properties appear to be rather
complicated.However, the approach used in this paper could be applied to
GM-estimators and to more realistic models.
AAlekseev@cefir.ru
2421. A CRITERION FOR TALAGRAND'S QUADRATIC TRANSPORTATION COST INEQUALITY
Patrick Cattiaux and Arnaud Guillin
We show that the quadratic transportation cost inequality $T_2$ is equivalent
to both a Poincar\'e inequality and a strong form of the Gaussian concentration
property. In particular if a logarithmic Sobolev inequality implies $T_2$, we
are able to give examples for which $T_2$ holds but the logarithmic Sobolev
inequality does not hold. This answers to a question left open by Otto and
Villani \cite{OV00} and Bobkov, Gentil and Ledoux \cite{BGL}, and furnishes (in
a Riemannian setting) the analogue of the well known criterion by Bobkov and
G\"{o}tze for the linear transportation cost inequality $T_1$ \cite{BG99} (also
see \cite{DGW}). The main ingredient in the proof is a new family of
inequalities, called modified quadratic transportation cost inequalities in the
spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux
\cite{BL97}, that are shown to hold as soon as a Poincar\'e inequality is
satisfied.
guillin@ceremade.dauphine.fr
2422. GAUSSIAN FREE FIELDS FOR MATHEMATICIANS
Scott Sheffield
The d-dimensional Gaussian free field (GFF), also called the (continuous)
massless free field, is a natural d-dimensional analog of Brownian motion. We
present a short overview of the GFF and some of its mathematical properties,
including the conformal Markov property (d=2), Fock spaces and Wiener
decompositions for the GFF, and the emergence of the GFF as a scaling limit of
discrete versions of the GFF (also called "harmonic crystals") on lattice
graphs.
sheff@microsoft.com
2423. APPROXIMATION OF INTEGRALS OVER ASYMPTOTIC SETS WITH APPLICATIONS TO
PROBABILITY AND STATISTICS
Philippe Barbe
In this monograph, we prove an asymptotic approximation for integrals of
probability densities over sets in finite dimensional euclidean space, which
are far away from the origin (asymptotic sets). We use this approximation to
investigate tails of quadratic forms of random vectors, supremum of random
linear forms among others. Applications to the study of finite size random
matrices, finite sample statistics of autoregressive processes, and supremum of
some stochastic processes.
barbe@math.u-cergy.fr
2424. A FRACTAL VALUED RANDOM ITERATION ALGORITHM AND FRACTAL HIERARCHY
Michael Barnsley, John E. Hutchinson, \"Orjan Stenflo
We describe new families of random fractals, referred to as "V-variable",
which are intermediate between the notions of deterministic and of standard
random fractals. The parameter V describes the degree of "variability" : at
each magnification level any V-variable fractals has at most V key "forms" or
"shapes". V-variable random fractals have the surprising property that they can
be computed using a forward process. More precisely, a version of the usual
Random Iteration Algorithm, operating on sets (or measures) rather than points,
can be used to sample each family. To present this theory, we review relevant
results on fractals (and fractal measures), both deterministic and random. Then
our new results are obtained by constructing an iterated function system (a
super IFS) from a collection of standard IFSs together with a corresponding set
of probabilities. The attractor of the super IFS is called a superfractal; it
is a collection of V-variable random fractals (sets or measures) together with
an associated probability distribution on this collection. When the underlying
space is for example $\mathbb{R}^{2}$, and the transformations are
computationally straightforward (such as affine transformations), the
superfractal can be sampled by means of the algorithm, which is highly
efficient in terms of memory usage. The algorithm is illustrated by some
computed examples. Some variants, special cases, generalizations of the
framework, and potential applications are mentioned.
stenflo@math.su.se
2425. EIGENVALUE SPACING DISTRIBUTION FOR THE ENSEMBLE OF REAL SYMMETRIC
TOEPLITZ MATRICES
Christopher Hammond, Steven J. Miller
Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv
from a fixed probability distribution p of mean 0, variance 1, and finite
higher moments. The limiting spectral measure (the density of normalized
eigenvalues) converges weakly to a new universal distribution with unbounded
support, independent of p. This distribution's moments are almost those of the
Gaussian's; the deficit may be interpreted in terms of Diophantine
obstructions. With a little more work, we obtain almost sure convergence. An
investigation of spacings between adjacent normalized eigenvalues looks
Poissonian, and not GOE.
sjmiller@math.ohio-state.edu
2426. PERTURBATION OF SINGULAR EQUILIBRIA OF HYPERBOLIC TWO-COMPONENT SYSTEMS:
A UNIVERSAL HYDRODYNAMIC LIMIT
Balint Toth, Benedek Valko
We consider one-dimensional, locally finite interacting particle systems with
two conservation laws which under Eulerian hydrodynamic limit lead to
two-by-two systems of conservation laws:
\pt \rho +\px \Psi(\rho, u)=0
\pt u+\px \Phi(\rho,u)=0,
with $(\rho,u)\in{\cal D}\subset\R^2$, where ${\cal D}$ is a convex compact
polygon in $\R^2$. The system is typically strictly hyperbolic in the interior
of ${\cal D}$ with possible non-hyperbolic degeneracies on the boundary
$\partial {\cal D}$. We consider the case of isolated singular (i.e. non
hyperbolic) point on the interior of one of the edges of ${\cal D}$, call it
$(\rho_0,u_0)=(0,0)$ and assume ${\cal D}\subset\{\rho\ge0\}$. This can be
achieved by a linear transformation of the conserved quantities. We investigate
the propagation of small nonequilibrium perturbations of the steady state of
the microscopic interacting particle system, corresponding to the densities
$(\rho_0,u_0)$ of the conserved quantities. We prove that for a very rich class
of systems, under proper hydrodynamic limit the propagation of these small
perturbations are \emph{universally} driven by the two-by-two system
\pt\rho + \px\big(\rho u\big)=0
\pt u + \px\big(\rho + \gamma u^2\big) =0
where the parameter $\gamma:=\frac12 \Phi_{uu}(\rho_0,u_0)$ (with a proper
choice of space and time scale) is the only trace of the microscopic
structure. The proof is valid for the cases with $\gamma>1$.
[truncated]
valko@math.bme.hu
2427. GLOBAL EXISTENCE OF BELL'S TIME-INHOMOGENEOUS JUMP PROCESS FOR LATTICE
QUANTUM FIELD THEORY
Hans-Otto Georgii, Roderich Tumulka
We consider the time-inhomogeneous Markovian jump process introduced by John
S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on
the associated configuration space. Its jump rates, tailored to give the
process the quantum distribution $|\Psi_t|^2$ at all times $t$, typically
exhibit singularities. We establish the existence of a unique such process for
all times, under suitable assumptions on the Hamiltonian or the initial state
vector $\Psi_0$. The proof of non-explosion takes advantage of the special role
of the $|\Psi_t|^2$ distribution.
tumulka@rz.mathematik.uni-muenchen.de
2428. BINDWEEDS OR RANDOM WALKS IN RANDOM ENVIRONMENTS ON MULTIPLEXED TREES
AND THEIR ASYMPOTICS
Mikhail Menshikov, Dimitri Petritis, Serguei Popov
We report on the asymptotic behaviour of a new model of random walk, we term
the bindweed model, evolving in a random environment on an infinite multiplexed
tree. The term \textit{multiplexed} means that the model can be viewed as a
nearest neighbours random walk on a tree whose vertices carry an internal
degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The
consequence of the internal degree of freedom is an enhancement of the tree
graph structure induced by the replacement of ordinary edges by multi-edges,
indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the
information on the internal degree of freedom of the vertices contiguous to
each edge. The term \textit{random environment} means that the jumping rates
for the random walk are a family of edge-indexed random variables, independent
of the natural filtration generated by the random variables entering in the
definition of the random walk; their joint distribution depends on the index of
each component of the multi-edges. We study the large time asymptotic behaviour
of this random walk and classify it with respect to positive recurrence or
transience in terms of a specific parameter of the probability distribution of
the jump rates. This classifying parameter is shown to coincide with the
critical value of a matrix-valued multiplicative cascade on the ordinary tree
(\textit{i.e.} the one without internal degrees of freedom attached to the
vertices) having the same vertex set as the state space of the random walk.
Only results are presented here since the detailed proofs will appear
elsewhere.
dimitri.petritis@univ-rennes1.fr
2429. V-VARIABLE FRACTALS AND SUPERFRACTALS
Michael Barnsley, John E. Hutchinson, \"{O}rjan Stenflo
Deterministic and random fractals, within the framework of Iterated Function
Systems, have been used to model and study a wide range of phenomena across
many areas of science and technology. However, for many applications
deterministic fractals are locally too similar near distinct points while
standard random fractals have too little local correlation. Random fractals are
also slow and difficult to compute. These two major problems restricting
further applications are solved here by the introduction of V-variable fractals
and superfractals.
stenflo@math.su.se
2430. SOME JUMP PROCESSES IN QUANTUM FIELD THEORY
Roderich Tumulka
A jump process on configuration space whose transition rates are governed by
the state vector of a quantum field theory was first considered by J.S. Bell.
This process and its variants involving ``minimal'' jump rates are reviewed,
and a recent proof of global existence of Bell's process is sketched. As an
outlook, it is suggested how methods of this proof could be applied to similar
global existence questions, and the particular usefulness of minimal jump rates
on manifolds with boundaries is underlined.
tumulka@rz.mathematik.uni-muenchen.de
2431. A FUNCTIONAL CENTRAL LIMIT THEOREM IN EQUILIBRIUM FOR A LARGE NETWORK IN
WHICH CUSTOMERS JOIN THE SHORTEST OF SEVERAL QUEUES
Carl Graham
We consider N single server infinite buffer queues with service rate beta.
Customers arrive at rate N times alpha, choose L queues uniformly, and join the
shortest one. The stability condition is alpha strictly less than beta. We
study in equilibrium the sequence of the fraction of queues of length at least
k, in the large N limit. We prove a functional central limit theorem on an
infinite-dimensional Hilbert space with its weak topology, with limit a
stationary Ornstein-Uhlenbeck process. We use ergodicity and justify the
inversion of limits of long times and large sizes N by a compactness-uniqueness
method. The main tool for proving tightness of the ill-known invariant laws and
ergodicity of the limit is a global exponential stability result for the
nonlinear dynamical system obtained in the functional law of large numbers
limit.
carl@cmapx.polytechnique.fr
2432. OUT OF EQUILIBRIUM FUNCTIONAL CENTRAL LIMIT THEOREMS FOR A LARGE NETWORK
WHERE CUSTOMERS JOIN THE SHORTEST OF SEVERAL QUEUES
Carl Graham
Customers arrive at rate N times alpha on a network of N single server
infinite buffer queues, choose L queues uniformly, join the shortest one, and
are served there in turn at rate beta. We let N go to infinity. We prove a
functional central limit theorem (CLT) for the tails of the empirical measures
of the queue occupations, in a Hilbert space with the weak topology, with limit
given by an Ornstein-Uhlenbeck process. The a priori assumption is that the
initial data converge. This completes a recent functional CLT in equilibrium
result for which convergence for the initial data was not known in advance, but
was deduced a posteriori from the functional CLT.
carl@cmapx.polytechnique.fr
2433. ON THE APPROXIMATION OF ONE MARKOV CHAIN BY ANOTHER
Mark Jerrum
Motivated by applications in Markov chain Monte Carlo, we discuss what it
means for one Markov chain to be an approximation to another. Specifically
included in that discussion are situations in which a Markov chain with
continuous state space is approximated by one with finite state space. A simple
sufficient condition for close approximation is derived, which indicates the
existence of three distinct approximation regimes. Counterexamples are
presented to show that these regimes are real and not artifacts of the proof
technique. An application to the ``ball walk'' of Lov\'asz and Simonovits is
provided as an illustrative example.
mrj@inf.ed.ac.uk
2434. NONUNIQUENESS FOR \SPECS IN $\ELL^{2+\EPSILON}$
Noam Berger, Christopher Hoffman and Vladas Sidoravicius
Keane, Berbee and others have studied the question of which specifications
(aka $g$-functions) admit a unique Gibbs measure. Bramson and Kalikow
constructed the first example of a regular and continuous specification which
admits multiple measures. For every $p>2$, we construct a regular and
continuous specification, whose variation is in $\ell^p$, that admits multiple
Gibbs measures. This shows that a recent condition of \"Oberg and Johansson is
tight.
noam@stat.berkeley.edu
2435. COEXISTENCE IN TWO-TYPE FIRST-PASSAGE PERCOLATION MODELS
Olivier Garet, Regine Marchand
We study the problem of coexistence in a two-type competition model governed
by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli
percolation. Actually, we prove for a large class of ergodic stationary passage
times that for distinct points $x,y\in\Zd$, there is a strictly positive
probability that $\{z\in\Zd;d(y,z)<d(x,z)\}$ and $\{z\in\Zd;d(y,z)>d(x,z)\}$
are both infinite sets. We also show that there is a strictly positive
probability that the graph of time-minimizing path from the origin in
first-passage percolation has at least two topological ends. This generalizes
results obtained by H{\"a}ggstr{\"o}m and Pemantle for independent exponential
times on the square lattice.
Olivier.Garet@labomath.univ-orleans.fr
2436. HARNESS PROCESSES AND HARMONIC CRYSTALS
Pablo A. Ferrari, Beat M. Niederhauser
We consider the long-term behaviour of infinite-volume Hammersley's harness
processes in continuous time. In this process, the height at each site is
updated at rate 1 to an average of the neighboring heights plus a centered
random variable (the noise). We show that the process started from the flat
configuration and viewed from the height at the origin converges to an
invariant measure. In dimension three and higher, the process itself converges
to an invariant measure. When the noise is Gaussian the limiting measures are
Gaussian fields (harmonic crystals) and are also reversible for the process.
The construction is used to show almost sure and $L^2$ versions of the infinite
volume limit of those fields.
pablo@ime.usp.br
2437. QUICKSORT WITH UNRELIABLE COMPARISONS: A PROBABILISTIC ANALYSIS
L. Alonso, P. Chassaing, F. Gillet, S. Janson, E. M. Reingold and R.
Schott
We provide a probabilistic analysis of the output of Quicksort when
comparisons can err.
chassain@iecn.u-nancy.fr
2438. THE DECAY RATE OF THE SOJOURN TIME IN THE M/G/1 FB QUEUE WITH
LIGHT-TAILED SERVICE TIMES
Misja Nuyens
The asymptotic decay rate of the sojourn time of a customer in the stationary
M/G/1 queue under the Foreground Background (FB) service discipline is studied.
The FB discipline gives service to those customers that have received the least
service so far. It is shown that this decay rate is the lowest possible for a
work-conserving discipline.
mnuyens@science.uva.nl
2439. CONTINUOUS AND DISCONTINUOUS PHASE TRANSITIONS IN HYPERGRAPH PROCESSES
R. W. R. Darling, D. A. Levin, J. R. Norris
Let V denote a set of N vertices. To construct a "hypergraph process", create
a new hyperedge at each event time of a Poisson process; the cardinality K of
this hyperedge is random, with arbitrary probability generating function r[x],
except that we assume P[K=1] +P[K=2]>0. Given K=k, the k vertices appearing in
the new hyperedge are selected uniformly at random from V. Hyperedges of
cardinality 1 are called patches, and serve as a way of selecting root
vertices. Identifiable vertices are those which are reachable from these root
vertices, in a strong sense which generalizes the notion of graph component.
Hyperedges are called reducible if all of their vertices are identifiable. We
use "fluid limit" scaling: hyperedges arrive at rate N, and we study structures
of size O(1) and O(N). After division by N, numbers of identifiable vertices
and reducible hyperedges exhibit phase transitions, which may be continuous or
discontinuous depending on the shape of the structure function -log[1 -
x])/r'[x], for x in (0,1). Both the case P[K=1]>0 and the case P[K=1] = 0 <
P[K=2] are considered; for the latter, a single extraneous patch is added to
mark the root vertex.
rwrd@afterlife.ncsc.mil
2440. THE DIVERGENCE OF BANACH SPACE VALUED RANDOM VARIABLES ON WIENER SPACE
E. Mayer-Wolf M. Zakai
The domain of definition of the divergence operator \delta on an abstract
Wiener space (W, H, \mu) is extended to include W-valued and W\otimesW-valued
"integrands". The main properties and characterizations of this extension are
derived and it is shown that in some sense the added elements in \delta's
extended domain have divergence zero. These results are then applied to the
analysis of quasiinvariant flows induced by W-valued vector fields and, among
other results, it turns out that these divergence-free vector fields "are
responsible" for generating measure preserving flows.
zakai@ee.technion.ac.il
2441. THE STRUCTURE OF FINITE CLUSTERS IN HIGH INTENSITY POISSON BOOLEAN STICK
PROCESS
Rahul Roy and Hideki Tanemura
Sticks at one of different orientation are placed in an i.i.d. fashion at
points of a Poisson point process of intensity $\lambda$. Sticks of the same
direction have the same length, while sticks in different directions may have
different lengths. We study the geometry of finite cluster as $\lambda \to
\infty$. The asymptotic shape of the custer being determined by the
probabilities of the sticks in various direction and their lengths and
orientations. We also obtain the limiting geometric structure of this
component.
tanemura@math.s.chiba-u.ac.jp
2442. THE SPREAD OF A RUMOR OR INFECTION IN A MOVING POPULATION
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 also perform continuous time simple random walks. A- and B-particles are
interpreted as individuals who are healthy or infected, respectively (or as
individuals who are ignorant or have heard a certain rumor, 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. It is assumed for most of the paper that the A- and B-particles
have the same jump rate. We show that if B(t) denotes the set of sites visited
by a B-particle during [0,t], then there exist strictly positive constants C
and D, such that almost surely B(t) contains a cube of linear size Ct and is
contained in a cube of linear size Dt.
kesten@math.cornell.edu
2443. ON THE NOISE-INDUCED PASSAGE THOUGH AN UNSTABLE PERIODIC
ORBIT I: TWO-LEVEL MODEL
Nils Berglund, Barbara Gentz
We consider the problem of stochastic exit from a planar domain, whose
boundary is an unstable periodic orbit, and which contains a stable
periodic orbit. This problem arises when investigating the distribution
of noise-induced phase slips between synchronized oscillators, or when
studying stochastic resonance far from the adiabatic limit. We introduce
a simple, piecewise linear model equation, for which the distribution of
first-passage times can be precisely computed. In particular, we obtain
a quantitative description of the phenomenon of cycling: The distribution
of first-passage times rotates around the unstable orbit, periodically
in the logarithm of the noise intensity, and thus does not converge
in the zero-noise limit. We compute explicitly the cycling profile,
which is universal in the sense that in depends only on the product
of the period of the unstable orbit with its Lyapunov exponent.
berglund@cpt.univ-mrs.fr
- To see a preprint or other
information provided by the author
click here.
2444. LARGE DEVIATIONS FOR RENORMALIZED SELF-INTERSECTCION
LOCAL TIMES OF STABLE PROCESSES
Richard F. Bass, Xia Chen, and Jay Rosen
We study large deviations for the renormalized
self-intersection local time of $d$-dimensional
stable processes of index $\beta\in (2d/3,d]$.
We find a difference between the upper and
lower tail. In addition, we find that the behaviour
of the lower tail depends critically on whether
$\beta<d$ or $\beta=d$.
bass@math.uconn.edu xchen@math.utk.edu jrosen3@earthlink.net
- To see a preprint or other
information provided by the author
click here.
2445. SOME REMARKS ON THE ELLIPTIC HARNACK INEQUALITY
Martin T. Barlow
This note gives three short results concerning the
elliptic Harnack inequality (EHI), in the context of random
walks on graphs. The first is that the EHI implies polynomial
growth of the number of points in balls, and the second that
the EHI is equivalent to an annulus type Harnack inequality
for Green's functions. The third result uses the lamplighter
group to give a counterexample concerning the relation of
coupling with the EHI. The paper concludes with an open
problem concerning the characterization of the EHI in terms of
electrical resistances.
barlow@math.ubc.ca
- To see a preprint or other
information provided by the author
click here.
2446. FIELDS WITH EXCEPTIONAL TANGENT FIELDS
Celine Lacaux
The asymptotic self-similarity property describes the local structure of a
random field. In this paper, we introduce a locally asymptotically
self-similar field $X_{H,\be}$ whose local structures at $x=0$ and at
$x\ne 0$ are very far from each other. More precisely, at $x\ne 0$ the
tangent field is a Fractional Brownian Motion. Furthermore, even if
$X_{H,\be}$ has finite second order moments, its tangent field at $x=0$ is
a Fractional Stable Motion. In addition, this field is asymptotically
self-similar at infinity with a Gaussian field, which is not a Fractional
Brownian Motion, as tangent field. Then, the trajectories regularity of
$X_{H,\be}$ is studied. Finally, the Hausdorff dimension of the graphs of
$X_{H,\be}$ is given.
celine.lacaux@math.ups-tlse.fr
- To see a preprint or other
information provided by the author
click here.
2447. LAW OF LARGE NUMBERS FOR A CLASS OF SUPERDIFFUSIONS
Janos Englander, Anita Winter
Pinsky (1996) proved that the finite mass superdiffusion X corresponding
to the semilinear operator Lu+\beta u-\alpha u^2 exhibits local
extinction if and only if \lambda_c \le 0, where \lambda_c:=\lambda_c(L+\beta)
is the generalized principal eigenvalue of L+\beta on R^d.
For the case when \lambda_c > 0, it has been shown in Englander
and Turaev (2000) that in law the superdiffusion locally behaves
like \exp[t\lambda_c] times a nonnegative non-degenerate random variable,
provided that the operator L+\beta-\lambda_c satisfies a certain spectral
condition (`product-criticality'), and that \alpha and \mu=X_0 are `not too large'.
In this article we will prove that the convergence in law can actually
be replaced by convergence in probability. Furthermore we replace
R^d by an arbitrary Euclidean domain.
As far as the proof of our main theorem is concerned, the heavy analytic
method of Englander and Turaev (2000) is replaced by a different,
simpler and more probabilistic one. We introduce a space-time weighted
superprocess (H- transformed superprocess) and use it in the proof along
with some elementary probabilistic arguments.
englander@pstat.ucsb.edu winter@mi.uni-erlangen.de
- To see a preprint or other
information provided by the author
click here.
2448. HEAT KERNEL ESTIMATES AND PARABOLIC HARNACK
INEQUALITIES ON GRAPHS AND RESISTANCE FORMS
Takashi Kumagai
We summarize recent work on heat kernel estimates and parabolic Harnack inequalities
for graphs, where the time scale is the $\beta$-th power of the space scale for some $\beta\ge 2$.
We then discuss self-adjoint operators induced by resistance forms.
Using a resistance metric, we give a simple condition for detailed heat kernel estimates and
parabolic Harnack inequalities. As an application, we show that on trees a detailed two-sided
heat kernel estimate is equivalent to some volume growth condition.
kumagai@kurims.kyoto-u.ac.jp
- To see a preprint or other
information provided by the author
click here.
