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Probability Abstracts 84
This document contains abstracts 2957-3073.
They have been mailed on December 30, 2004.
Author(s): Jason Fulman
Abstract: Motivated by Bourque and Pevzner's simulation study of the parsimony method
for studying genome rearrangement, Berestycki and Durrett used techniques from
random graph theory to prove that the minimum parsimony distance after
iterating the random transposition shuffle undergoes a transition from Poisson
to normal behavior. This paper establishes an analogous result for minimum
parsimony distance after iterates of riffle shuffles or iterates of riffle
shuffles and cuts. The analysis is elegant and uses different tools: Stein's
method and generating functions. A useful technique which emerges is that of
making a problem more tractable by adding extra symmetry, then using Stein's
method to exploit the symmetry in the modified problem, and from this deducing
information about the original problem.
http://arXiv.org/abs/math/0410622
http://front.math.ucdavis.edu/math.PR/0410622
(alternate)
Author(s): Amanda G. Turner
Abstract: We consider sequences of Markov processes in two dimensions whose fluid limit
is a stable solution of an ordinary differential equation of the form dx/dt =
b(x), where the linear part of b(x) has eigenvalues -mu and lambda for some
lambda, mu > 0. Here the processes are indexed so that the variance of the
fluctuations is inversely proportional to N. The simplest example arises from
the OK Corral gunfight model which was formulated by Williams and McIlroy
(1998) and studied by Kingman (1999). These processes exhibit their most
interesting behaviour at times of order log N so it is necessary to establish a
fluid limit that is valid for large times. We find that this limit is
inherently random and obtain its distribution. Using this, it is possible to
derive scaling limits for the points where these processes hit straight lines
through the origin, and the minimal distance from the origin that they can
attain. The power of N that gives the appropriate scaling somewhat surprisingly
turns out to be mu / 2(lamba + mu).
http://arXiv.org/abs/math/0412051
http://front.math.ucdavis.edu/math.PR/0412051
(alternate)
Author(s): E. Di Nardo and D. Senato
Abstract: We provide an algebraic setting for cumulants and factorial moments through
the classical umbral calculus. Main tools are the compositional inverse of the
unity umbra, connected with the logarithmic power series, and a new umbra here
introduced, the singleton umbra. Various formulae are given expressing
cumulants, factorial moments and central moments by umbral functions.
http://arXiv.org/abs/math/0412052
http://front.math.ucdavis.edu/math.PR/0412052
(alternate)
Author(s): E. Di Nardo and D. Senato
Abstract: Extending the rigorous presentation of the classical umbral calculus given by
Rota and Taylor in 1994, the so-called partition polynomials are interpreted
with the aim to point out the umbral nature of the Poisson random variables.
Among the new umbrae introduced, the main tool is the partition umbra that
leads also to a simple expression of the functional composition of the
exponential power series. Moreover a new short proof of the Lagrange inversion
formula is given.
http://arXiv.org/abs/math/0412054
http://front.math.ucdavis.edu/math.PR/0412054
(alternate)
Author(s): Robert O. Bauer and Roland M. Friedrich
Abstract: We discuss the extension of radial SLE to multiply connected planar domains.
First, we extend Loewner's theory of slit mappings to multiply connected
domains by establishing the radial Komatu-Loewner equation, and show that a
simple curve from the boundary to the bulk is encoded by a motion on moduli
space and a motion on the boundary of the domain. Then, we show that the
vector-field describing the motion of the moduli is Lipschitz. We explain why
this implies that "consistent," conformally invariant random simple curves are
described by multidimensional diffusions, where one component is a motion on
the boundary, and the other component is a motion on moduli space. We argue
what the exact form of this diffusion is (up to a single real parameter
$\kappa$) in order to model boundaries of percolation clusters. Finally, we
show that this moduli diffusion leads to random non-self-crossing curves
satisfying the locality property if and only if $\kappa=6$.
http://arXiv.org/abs/math/0412060
http://front.math.ucdavis.edu/math.PR/0412060
(alternate)
Author(s): Piotr Sniady
Abstract: In this series of articles we study connections between combinatorics of
multidimensional generalizations of Cauchy identity and continuous objects such
as multidimensional Brownian motions and Brownian bridges.
In Part I of the series we present a bijective proof of multidimensional
generalizations of the Cauchy identity. Our bijection uses oriented planar
trees equipped with some linear orders.
http://arXiv.org/abs/math/0412043
http://front.math.ucdavis.edu/math.CO/0412043
(alternate)
Author(s): Dimitris Achlioptas and Cristopher Moore
Abstract: Given any integer d >= 3, let k be the smallest integer such that d < 2k log
k. We prove that with high probability the chromatic number of a random
d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k
then the chromatic number is either k+1 or k+2.
http://arXiv.org/abs/cond-mat/0407278
http://front.math.ucdavis.edu/cond-mat/0407278
(alternate)
Author(s): Lorenzo Finesso and Peter Spreij
Abstract: In this paper we consider the Nonnegative Matrix Factorization (NMF) problem:
given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for
assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and
$H\in\R_+^{k\times n}$ such that $V=WH$. Exact, non trivial, nonnegative
factorizations do not always exist, hence it is interesting to pose the
approximate NMF problem. The criterion which is commonly employed is
I-divergence between nonnegative matrices. The problem becomes that of finding,
for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An
iterative algorithm, EM like, for the construction of the best pair $(W, H)$
has been proposed in the literature. In this paper we interpret the algorithm
as an alternating minimization procedure \`a la Csisz\'ar-Tusn\'ady and
investigate some of its stability properties. NMF is widespreading as a data
analysis method in applications for which the positivity constraint is
relevant. There are other data analysis methods which impose some form of
nonnegativity: we discuss here the connections between NMF and Archetypal
Analysis.
http://arXiv.org/abs/math/0412070
http://front.math.ucdavis.edu/math.OC/0412070
(alternate)
Author(s): Jason Schweinsberg and Rick Durrett
Abstract: When a beneficial mutation occurs in a population, the new, favored allele
may spread to the entire population. This process is known as a selective
sweep. Suppose we sample $n$ individuals at the end of a selective sweep. If we
focus on a site on the chromosome that is close to the location of the
beneficial mutation, then many of the lineages will likely be descended from
the individual that had the beneficial mutation, while others will be descended
from a different individual because of recombination between the two sites. We
introduce two approximations for the effect of a selective sweep. The first one
is simple but not very accurate: flip $n$ independent coins with probability
$p$ of heads and say that the lineages whose coins come up heads are those that
are descended from the individual with the beneficial mutation. A second
approximation, which is related to Kingman's paintbox construction, replaces
the coin flips by integer-valued random variables and leads to very accurate
results.
http://arXiv.org/abs/math/0411069
http://front.math.ucdavis.edu/math.PR/0411069
(alternate)
Author(s): Oliver Delzeith
Abstract: The paper presents a factorization theorem for a certain class of stochastic
processes. Skorohod spaces carry the rich structure of standard Borel spaces
and appear to be suitable universal sample path spaces. We show that, if $\xi$
is a RCLL stochastic process with values in a complete separable metric space
$E$, any other RCLL stochastic process $X$ adapted to the filtration induced by
$\xi$ factors through the Skorohod space $D_E[0,\infty)$. This can be
understood as an extension of a stochastic process to a standard Borel space
enjoying nice properties. Moreover, the trajectories of the factorized
stochastic process defined on $D_E[0,\infty)$ inherit the properties of being
continuous, non-decreasing, and of bounded variation, resp., from those of $X$.
Considering situations which are invariant under the factorization procedure,
the main theorem is a reduction tool to assume the underlying measurable space
be a standard Borel space. In an example, we pick the existence theorem of
regular conditional probabilities on standard Borel spaces to simplify a
conditional expectation appearing in stochastic control problems.
http://arXiv.org/abs/math/0412092
http://front.math.ucdavis.edu/math.PR/0412092
(alternate)
Author(s): Victor Beresnevich and Sanju Velani
Abstract: A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric
number theory is introduced and discussed. The general conjecture is
established modulo the original conjecture. The key result is a Mass
Transference Principle which allows us to transfer Lebesgue measure theoretic
statements for $\limsup$ subsets of $\R^k$ to Hausdorff measure theoretic
statements. In view of this, the Lebesgue theory of $\limsup $ sets is shown to
underpin the general Hausdorff theory. This is rather surprising since the
latter theory is viewed to be a subtle refinement of the former.
http://arXiv.org/abs/math/0412141
http://front.math.ucdavis.edu/math.NT/0412141
(alternate)
Author(s): James Allen Fill and Nevin Kapur and Alois Panholzer
Abstract: We consider the total cost of cutting down a random rooted tree chosen from a
family of so-called very simple trees (which include ordered trees, $d$-ary
trees, and Cayley trees); these form a subfamily of simply generated trees. At
each stage of the process an edge is chose at random from the tree and cut,
separating the tree into two components. In the one-sided variant of the
process the component not containing the root is discarded, whereas in the
two-sided variant both components are kept. The process ends when no edges
remain for cutting. The cost of cutting an edge from a tree of size $n$ is
assumed to be $n^\alpha$. Using singularity analysis and the method of moments,
we derive the limiting distribution of the total cost accrued in both variants
of this process. A salient feature of the limiting distributions obtained
(after normalizing in a family-specific manner) is that they only depend on
$\alpha$.
http://arXiv.org/abs/math/0412155
http://front.math.ucdavis.edu/math.PR/0412155
(alternate)
Author(s): Thierry Huillet (LPTM)
Abstract: Suppose some random resource (energy, mass or space) $\chi \geq 0$ is to be
shared at random between (possibly infinitely many) species (atoms or
fragments). Assume ${\Bbb E}\chi =\theta <\infty $ and suppose the amount of
the individual share is necessarily bounded from above by 1. This random
partitioning model can naturally be identified with the study of infinitely
divisible random variables with L\'{e}vy measure concentrated on the interval%
$.$ Special emphasis is put on these special partitioning models in the
Poisson-Kingman class. The masses attached to the atoms of such partitions are
sorted in decreasing order. Considering nearest- neighbors spacings yields a
partition of unity which also deserves special interest. For such partition
models, various statistical questions are addressed among which: correlation
structure, cumulative energy of the first $K$ largest items, partition
function, threshold and covering statistics, weighted partition, R\'{e}nyi's,
typical and size-biased fragments size. Several physical images are supplied.
When the unbounded L\'{e}vy measure of $\chi $ is $\theta x^{-1}\cdot {\bf I}%
(x\in (0,1)) dx$, the spacings partition has Griffiths-Engen-McCloskey or
GEM$(\theta) $ distribution and $% \chi $ follows Dickman distribution. The
induced partition models have many remarkable peculiarities which are outlined.
The case with finitely many (Poisson) fragments in the partition law is also
briefly addressed. Here, the L\'{e}vy measure is bounded.
http://arXiv.org/abs/cond-mat/0412166
http://front.math.ucdavis.edu/cond-mat/0412166
(alternate)
Author(s): J.-R. Chazottes and P. Collet and B. Schmitt
Abstract: In this paper, we prove an inequality, which we call "Devroye inequality",
for a large class of non-uniformly hyperbolic dynamical systems (M,f). This
class, introduced by L.-S. Young, includes families of piece-wise hyperbolic
maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas),
unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound
for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K
is any separately Holder continuous function of n variables. In particular, we
can deal with observables which are not Birkhoff averages. We will show in
\cite{CCS} some applications of Devroye inequality to statistical properties of
this class of dynamical systems.
http://arXiv.org/abs/math/0412166
http://front.math.ucdavis.edu/math.DS/0412166
(alternate)
Author(s): J.-R. Chazottes and P. Collet and B. Schmitt
Abstract: In this paper, we apply Devroye inequality to study various statistical
estimators and fluctuations of observables for processes. Most of these
observables are suggested by dynamical systems. These applications concern the
co-variance function, the integrated periodogram, the correlation dimension,
the kernel density estimator, the speed of convergence of empirical measure,
the shadowing property and the almost-sure central limit theorem. We proved in
\cite{CCS} that Devroye inequality holds for a class of non-uniformly
hyperbolic dynamical systems introduced in \cite{young}. In the second appendix
we prove that, if the decay of correlations holds with a common rate for all
pairs of functions, then it holds uniformly in the function spaces. In the last
appendix we prove that for the subclass of one-dimensional systems studied in
\cite{young} the density of the absolutely continuous invariant measure belongs
to a Besov space.
http://arXiv.org/abs/math/0412167
http://front.math.ucdavis.edu/math.DS/0412167
(alternate)
Author(s): Misja Nuyens
Abstract: We give an overview of the results in the literature on single-server queues
with the FB discipline. The FB discipline gives service to the customer that
has received the least amount of service. This not so well-known discipline has
some appealing features, and performs well for heavy-tailed service times. We
describe results on the queue length, sojourn time, and the influence of
variability in the service times.
http://arXiv.org/abs/math/0412182
http://front.math.ucdavis.edu/math.PR/0412182
(alternate)
Author(s): Hanene Mohamed (RAP UR-R) and Philippe Robert (RAP UR-R)
Abstract: In this paper a general class of tree algorithms is analyzed. It is shown
that, by using an appropriate probabilistic representation of the quantities of
interest, the asymptotic behavior of these algorithms can be obtained quite
easily without resorting to complex analysis techniques as it is usually the
case. This approach gives a unified probabilistic treatment of these questions.
It simplifies and extends some of the results known in this domain.
http://arXiv.org/abs/math/0412188
http://front.math.ucdavis.edu/math.PR/0412188
(alternate)
Author(s): Djalil Chafai (LSProba and Upte Umr Inra/Envt 181) and Didier Concordet (LSProba, Upte Umr Inra/Envt 181)
Abstract: We present a continuous time model of maturation and survival, obtained as
the limit of a compartmental evolution model when the number of compartments
tends to infinity. We establish in particular an explicit formula for the law
of the system output under inhomogeneous killing and when the input follows a
time-inhomogeneous Poisson process. Identifiability issues are discussed, and
an application to the modelling of the toxicity of anti-cancer drugs is given.
Such models can be seen in particular as generalisations of previous works of
Jacquez & Simon and Schuhmacher & Thieme.
http://arXiv.org/abs/math/0412193
http://front.math.ucdavis.edu/math.PR/0412193
(alternate)
Author(s): Jan Obloj (PMA and Mimuw) and Marc Yor (PMA)
Abstract: We discuss certain facts involving a continuous local martingale $N$ and its
supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic
functions is proposed. This yields an important family of martingales, the
usefulness of which is demonstrated, by means of examples involving the
Skorokhod embedding problem, bounds on the law of the supremum, or the local
time at 0, of a martingale with a fixed terminal distribution, or yet in some
Brownian penalization problems. In particular we obtain new bounds on the law
of the local time at 0, which involve the excess wealth order.
http://arXiv.org/abs/math/0412196
http://front.math.ucdavis.edu/math.PR/0412196
(alternate)
Author(s): Rick Durrett and Jason Schweinsberg
Abstract: When an advantageous mutation occurs in a population, the favorable allele
may spread to the entire population in a short time, an event known as a
selective sweep. As a result, when we sample $n$ individuals from a population
and trace their ancestral lines backwards in time, many lineages may coalesce
almost instantaneously at the time of a selective sweep. We show that as the
population size goes to infinity, this process converges to a coalescent
process called a coalescent with multiple collisions. A better approximation
for finite populations can be obtained using a coalescent with simultaneous
multiple collisions. We also show how these coalescent approximations can be
used to get insight into how beneficial mutations affect the behavior of
statistics that have been used to detect departures from the usual Kingman's
coalescent.
http://arXiv.org/abs/math/0411071
http://front.math.ucdavis.edu/math.PR/0411071
(alternate)
Author(s): Pablo A. Ferrari and James B. Martin and Leandro P. R. Pimentel
Abstract: The competition interface between two growing ``Young clusters'' (diagrams),
in a two-dimensional random cone, is mapped to the path of a second-class
particle in the one-dimensional totally asymmetric simple exclusion process.
Using the asymptotics of the second class particle and hydrodynamic limits for
the exclusion process (Burgers equation), we show that the behavior of the
competition interface depends on the angle of the cone: for angles in [180^o,
270^o) the competition interface has a deterministic inclination, while for
angles in [90^o,180^o) the inclination is random. We relate the competition
model to a model of random directed polymers, and obtain some partial results
for the fluctuations of the competition interface.
http://arXiv.org/abs/math/0412198
http://front.math.ucdavis.edu/math.PR/0412198
(alternate)
Author(s): Annie Millet and Marta Sanz-Sol\'e
Abstract: Starting from the construction of a geometric rough path associated with a
fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by
Coutin and Qian (2002), we prove a large deviation principle in the space of
geometric rough paths, extending classical results on Gaussian processes. As a
by-product, geometric rough paths associated to elements of the reproducing
kernel Hilbert space of the fractional Brownian motion are obtained and an
explicit integral representation is given.
http://arXiv.org/abs/math/0412200
http://front.math.ucdavis.edu/math.PR/0412200
(alternate)
Author(s): Massimo Ostilli and Carlo Presilla
Abstract: We present a large deviation analysis of a recently proposed probabilistic
approach to the study of the ground-state properties of lattice quantum
systems. The ground-state energy, as well as the correlation functions in the
ground state, are exactly determined as a series expansion in the cumulants of
the multiplicities of the potential and hopping energies assumed by the system
during its long-time evolution. Once these cumulants are known, even at a
finite order, our approach provides the ground state analytically as a function
of the Hamiltonian parameters. A scenario of possible applications of this
analyticity property is discussed.
http://arXiv.org/abs/cond-mat/0412157
http://front.math.ucdavis.edu/cond-mat/0412157
(alternate)
Author(s): Ross G. Pinsky
Abstract: The purpose of this article is to give a rather thorough understanding of the
compact support property for measure-valued diffusion processes corresponding
to semi-linear equations of the form \[& u_t=Lu+\beta u-\alpha u^p \text{in}
R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0
\text{in} R^d\times[0,\infty). \] In particular, we shall investigate how the
interplay between the underlying motion (the diffusion process corresponding to
$L$) and the branching affects the compact support property. In \cite{EP99},
the compact support property was shown to be equivalent to a certain analytic
criterion concerning uniqueness of the Cauchy problem for the semilinear
parabolic equation related to the measured valued diffusion. In a subsequent
paper \cite{EP03}, this analytic property was investigated purely from the
point of view of partial differential equations. Some of the results obtained
in this latter paper yield interesting results concerning the compact support
property. In this paper, the results from \cite{EP03} that are relevant to the
compact support property are presented, sometimes with extensions. These
results are interwoven with new results and some informal heuristics. Taken
together, they yield a fairly comprehensive picture of the compact support
property. \it Inter alia\rm, we show that the concept of a measure-valued
diffusion \it hitting\rm a point can be investigated via the compact support
property, and suggest an alternate proof of a result concerning the hitting of
points by super-Brownian motion.
http://arXiv.org/abs/math/0412246
http://front.math.ucdavis.edu/math.PR/0412246
(alternate)
Author(s): Imen Bentahar (CEREMADE) and Bruno Bouchard (CREST and Lfa and Pma)
Abstract: We consider a multivariate financial market with transaction costs and study
the problem of finding the minimal initial capital needed to hedge, without
risk, European-type contingent claims. The model is similar to the one
considered in Bouchard and Touzi (2000), except that some of the assets can be
exchanged freely, i.e. without paying transaction costs. In this context, we
generalize the result of the above paper and prove that the value of this
stochastic control problem is given by the cost of the cheapest hedging
strategy in which the number of non-freely exchangeable assets is kept constant
over time.
http://arXiv.org/abs/math/0412247
http://front.math.ucdavis.edu/math.PR/0412247
(alternate)
Author(s): Boris Tsirelson
Abstract: A counterexample to the conjecture that the automorphisms of an arbitrary
Arveson system act transitively on its normalized units.
http://arXiv.org/abs/math/0411062
http://front.math.ucdavis.edu/math.OA/0411062
(alternate)
Author(s): Russell Lyons and Yuval Peres and and Oded Schramm
Abstract: We study minimal spanning forests in infinite graphs, which are weak limits
of minimal spanning trees from finite subgraphs corresponding to i.i.d. random
labels on the edges. These limits can be taken with free or wired boundary
conditions, and are denoted $\fmsf$ (free minimal spanning forest) and $\wmsf$
(wired minimal spanning forest), respectively. The $\wmsf$ is the union of the
trees that arise from invasion percolation started at all vertices. We show
that on any Cayley graph where critical percolation has no infinite clusters,
all the component trees in the $\wmsf$ have one end a.s. In $\Z^d$ this was
proved by \ref b.Alexander:MSF/, but a different method is needed for the
nonamenable case. We show that on any connected graph, the union of the $\fmsf$
and independent Bernoulli percolation (with arbitrarily small parameter) is
a.s. connected. In conjunction with a recent result of Gaboriau, this implies
that in any Cayley graph, the expected degree of the $\fmsf$ is at least the
expected degree of the $\fsf$ (the weak limit of uniform spanning trees). We
show that on any graph, each component tree in the $\wmsf$ has $\pc = 1$ a.s.,
where $\pc$ denotes the critical probability for having an infinite cluster in
Bernoulli percolation. We show that the number of infinite clusters for
Bernoulli($\pu$) percolation is at most the number of components of the
$\fmsf$, where $\pu$ denotes the critical probability for having a unique
infinite cluster.
http://arXiv.org/abs/math/0412263
http://front.math.ucdavis.edu/math.PR/0412263
(alternate)
Author(s): Wei Biao Wu
Abstract: Empirical processes for stationary, causal sequences are considered. We
establish empirical central limit theorems for classes of indicators of left
half lines, absolutely continuous functions and piecewise differentiable
functions. Sample path properties of empirical distribution functions are also
discussed. The results are applied to linear processes and Markov chains.
http://arXiv.org/abs/math/0412267
http://front.math.ucdavis.edu/math.ST/0412267
(alternate)
Author(s): Wei Biao Wu
Abstract: We study the asymptotic behavior of M-estimates of regression parameters in
multiple linear models where errors are dependent random variables. A Bahadur
representation of the M-estimates is derived and a central limit theorem is
established. The results are applied to linear models with errors being
short-range dependent linear processes, heavy-tailed linear processes and some
widely used nonlinear time series.
http://arXiv.org/abs/math/0412268
http://front.math.ucdavis.edu/math.ST/0412268
(alternate)
Author(s): Eugene Wong
Abstract: Consider an n-fold integrated Brownian motion. We show that a simple change
in time and scale transforms it into a stationary Gaussian process. The
collection of stationary processes so constructed not only constitutes an
interesting family of processes, but their spectral representation is also
useful in dealing with integrated Brownian motion. We illustrate this by
deriving an explicit representation for the joint density function for a family
of integrated Brownian motions and showing some of its properties.
http://arXiv.org/abs/math/0412291
http://front.math.ucdavis.edu/math.PR/0412291
(alternate)
Author(s): A. Singer and Z. Schuss and D. Holcman and R.S. Eisenberg
Abstract: A Brownian particle with diffusion coefficient $D$ is confined to a bounded
domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small
absorbing window. The mean time to absorption diverges as the window shrinks,
thus rendering the calculation of the mean escape time a singular perturbation
problem. We construct an asymptotic approximation for the case of an elliptical
window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is
$E\tau\sim\ds{\frac{V}{2\pi Da}} K(e)$, where $e$ is the eccentricity of the
ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In
the special case of a circular hole the result reduces to Lord Rayleigh's
formula $E\tau\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic
considerations. For the special case of a spherical domain, we obtain the
asymptotic expansion $E\tau=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a}
+ O(\frac{a}{R}) ]$. This problem is important in understanding the flow of
ions in and out of narrow valves that control a wide range of biological and
technological function.
http://arXiv.org/abs/math-ph/0412048
http://front.math.ucdavis.edu/math-ph/0412048
(alternate)
Author(s): A. Singer and Z. Schuss and D. Holcman
Abstract: We consider Brownian motion in a circular disk $\Omega$, whose boundary
$\p\Omega$ is reflecting, except for a small arc, $\p\Omega_a$, which is
absorbing. As $\epsilon=|\partial \Omega_a|/|\partial \Omega|$ decreases to
zero the mean time to absorption in $\p\Omega_a$, denoted $E\tau$, becomes
infinite. The narrow escape problem is to find an asymptotic expansion of
$E\tau$ for $\epsilon\ll1$. We find the first two terms in the expansion and an
estimate of the error. The results are extended in a straightforward manner to
planar domains and two-dimensional Riemannian manifolds that can be mapped
conformally onto the disk. Our results improve the previously derived expansion
for a general smooth domain, $E\tau =
\ds{\frac{|\Omega|}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ ($D$ is the
diffusion coefficient) in the case of a circular disk. We find that the mean
first passage time from the center of the disk is $E[\tau |
\x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}{\epsilon}} + \log 2
+\ds{{1/4}} + O(\epsilon)]$. The second term in the expansion is needed in real
life applications, such as trafficking of receptors on neuronal spines, because
$\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is
small. We also find the singular behavior of the probability flux profile into
$\p\Omega_a$ at the endpoints of $\p\Omega_a$, and find the value of the flux
near the center of the window.
http://arXiv.org/abs/math-ph/0412050
http://front.math.ucdavis.edu/math-ph/0412050
(alternate)
Author(s): A. Singer and Z. Schuss and D. Holcman
Abstract: We consider Brownian motion in a bounded domain $\Omega$ on a two-dimensional
Riemannian manifold $(\Sigma,g)$. We assume that the boundary $\p\Omega$ is
smooth and reflects the trajectories, except for a small absorbing arc
$\p\Omega_a\subset\p\Omega$. As $\p\Omega_a$ is shrunk to zero the expected
time to absorption in $\p\Omega_a$ becomes infinite. The narrow escape problem
consists in constructing an asymptotic expansion of the expected lifetime,
denoted $E\tau$, as $\epsilon=|\partial \Omega_a|_g/|\partial \Omega|_g\to0$.
We derive a leading order asymptotic approximation $E\tau =
\ds{\frac{|\Omega|_g}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)]$. The order 1
term can be evaluated for simply connected domains on a sphere by projecting
stereographically on the complex plane and mapping conformally on a circular
disk. It can also be evaluated for domains that can be mapped conformally onto
an annulus. This term is needed in real life applications, such as trafficking
of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not
necessarily large, even when $\epsilon$ is small. If the absorbing window is
located at a corner of angle $\alpha$, then $E\tau =
\ds{\frac{|\Omega|_g}{D\alpha}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ if near a
cusp, then $E\tau$ grows algebraically, rather than logarithmically. Thus, in
the domain bounded between two tangent circles, the expected lifetime is $E\tau
= \ds{\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\epsilon} + O(1))$.
http://arXiv.org/abs/math-ph/0412051
http://front.math.ucdavis.edu/math-ph/0412051
(alternate)
Author(s): Patrick Bernard (IF) and Boris Buffoni (EPFL)
Abstract: We study optimal transportation of measures on compact manifolds for costs
defined from convex Lagrangians. We prove that optimal transportation can be
interpolated by measured Lipschitz laminations, or geometric currents. The
methods are inspired from Mather theory on Lagrangian systems. We make use of
viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of
Fathi's approach to Mather theory.
http://arXiv.org/abs/math/0412299
http://front.math.ucdavis.edu/math.DS/0412299
(alternate)
Author(s): Jean Bertoin (PMA) and Bernard Roynette (IEC) and Marc Yor (PMA)
Abstract: We describe some connections, via composition, between two functional spaces:
the space of (sub)critical branching mechanisms and the space of Bernstein
functions. The functions ${\bf e}_\alpha: x\to x^{\alpha}$ where $x\geq0$ and
$0<\alpha\leq 1/2$, and in particular the critical parameter $\alpha=1/2$, play
a distinguished role.
http://arXiv.org/abs/math/0412322
http://front.math.ucdavis.edu/math.PR/0412322
(alternate)
Author(s): David Siegmund and Benjamin Yakir
Abstract: An urn model of Diaconis and some generalizations are discussed. A
convergence theorem is proved that implies for Diaconis' model that the
empirical distribution of balls in the urn converges with probability one to
the uniform distribution.
http://arXiv.org/abs/math/0412333
http://front.math.ucdavis.edu/math.PR/0412333
(alternate)
Author(s): Eric Gautier (IRMAR and Crest-Insee Laboratoire De Statistique)
Abstract: Uniform large deviations for the laws of the paths of the solutions of the
stochastic nonlinear Schrodinger equation when the noise converges to zero are
presented. The noise is a real multiplicative Gaussian noise. It is white in
time and colored in space. The path space considered allows blow-up and is
endowed with a topology analogue to a projective limit topology. Thus a large
variety of large deviation principle may be deduced by contraction. As a
consequence, asymptotics of the tails of the law of the blow-up time when the
noise converges to zero are obtained.
http://arXiv.org/abs/math/0412319
http://front.math.ucdavis.edu/math.AP/0412319
(alternate)
Author(s): J.C. Breton and C. Houdr\'e
Abstract: The purpose of these notes is to further complete our understanding of the
stable concentration phenomenon, by obtaining the finite range behavior of
$P(F-E[F]\geq x)$, with $F=f(X)$ where $f$ is a Lipschitz function and $X$ is a
stable random vector or with $F$ a stochastic functional on the Poisson space
equipped with a stable L\'evy measure.
http://arXiv.org/abs/math/0412334
http://front.math.ucdavis.edu/math.PR/0412334
(alternate)
Author(s): Thomas Logan Ritchie
Abstract: We provide an explicit construction for the thermodynamic limit measure for
finite range exclusion schemes on $\mathbb{Z}^{d}$. By means thereof a strong
law of large numbers for occupation densities is accomplished, and, amongst
other results, the so called ``super-exponential'' (i.e. gamma) decay of
pair-correlation functions is established.
http://arXiv.org/abs/math/0412343
http://front.math.ucdavis.edu/math.PR/0412343
(alternate)
Author(s): J.C. Breton and C. Houdr\'e and N. Privault
Abstract: Concentration inequalities are obtained on Poisson space, for random
functionals with finite or infinite variance. In particular, dimension free
tail estimates and exponential integrability results are given for the
Euclidean norm of vectors of independent functionals.
In the finite variance case these results are applied to infinitely divisible
random variables such as quadratic Wiener functionals, including L\'evy's
stochastic area and the square norm of Brownian paths.
In the infinite variance case, various tail estimates such as stable ones are
also presented.
http://arXiv.org/abs/math/0412346
http://front.math.ucdavis.edu/math.PR/0412346
(alternate)
Author(s): Thomas Logan Ritchie and Vladimir Belitsky
Abstract: We present a coupled decreasing sequence of random walks on $ \mathbb Z $
that dominates the edge process of oriented-bond percolation in two dimensions.
Using the concept of "random walk in a strip ", we construct an algorithm that
generates an increasing sequence of lower bounds that converges to the critical
probability of oriented-bond percolation. Numerical calculations of the first
ten lower bounds thereby generated lead to an improved,i.e. higher, rigorous
lower bound to this critical probability, viz. $p_{c} \geq 0.63328 $. Finally a
computer simulation technique is presented; the use thereof establishes 0.64450
as a non-rigorous five-digit-precision (lower) estimate for $p_{c}$.
http://arXiv.org/abs/math/0412348
http://front.math.ucdavis.edu/math.PR/0412348
(alternate)
Author(s): Raluca Balan
Abstract: In this paper, we use the Markov property introduced in Balan and Ivanoff (J.
Theor. Probab. 15, 2002, 553-588) for set-indexed processes and we prove that a
Markov prior distribution leads to a Markov posterior distribution. In
particular, by proving that a neutral to the right prior distribution leads to
a neutral to the right posterior distribution, we extend a fundamental result
of Doksum (Ann. Probab. 2,1974, 183-201) to arbitrary sample spaces.
http://arXiv.org/abs/math/0412349
http://front.math.ucdavis.edu/math.PR/0412349
(alternate)
Author(s): Raluca Balan and Gail Ivanoff
Abstract: We consider a type of Markov property for set-indexed processes which is
satisfied by all processes with independent increments and which allows us to
introduce a transition system theory leading to the construction of the
process. A set-indexed generator is defined such that it completely
characterizes the distribution of the process.
http://arXiv.org/abs/math/0412350
http://front.math.ucdavis.edu/math.PR/0412350
(alternate)
Author(s): R. Averkamp and C. Houdr\'e
Abstract: Unbiased risk estimation, \`a la Stein, is studied for infinitely divisible
laws with finite second moment.
http://arXiv.org/abs/math/0412345
http://front.math.ucdavis.edu/math.ST/0412345
(alternate)
Author(s): Enrique Figueroa-Lopez and Christian Houdre
Abstract: Nonparametric methods for the estimation of the Levy density of a Levy
process are developed. Estimators that can be written in terms of the ``jumps''
of the process are introduced, and so are discrete-data based approximations. A
model selection approach made up of two steps is investigated. The first step
consists in the selection of a good estimator from a linear model of proposed
Levy densities, while the second is a data-driven selection of a linear model
among a given collection of linear models. By providing lower bounds for the
minimax risk of estimation over Besov Levy densities, our estimators are shown
to achieve the ``best'' rate of convergence. A numerical study for the case of
histogram estimators and for variance Gamma processes, models of key importance
in risky asset price modeling driven by Levy processes, is presented.
http://arXiv.org/abs/math/0412351
http://front.math.ucdavis.edu/math.ST/0412351
(alternate)
Author(s): Jason Schweinsberg
Abstract: Suppose a gambler starts with a fortune in (0,1) and wishes to attain a
fortune of 1 by making a sequence of bets. Assume thay whenever the gambler
stakes the amount s, the gambler's fortune increases by s with probability w
and decreases by s with probability 1 - w, where w < 1/2. Dubins and Savage
showed that the optimal strategy, which they called "bold play", is always to
stake min{f, 1-f}, where f is the gambler's current fortune. Here we consider
the problem in which the gambler may stake no more than l at one time. We show
that the bold strategy of always betting min{l, f, 1-f} is not optimal if l is
irrational, extending a result of Heath, Pruitt, and Sudderth.
http://arXiv.org/abs/math/0412362
http://front.math.ucdavis.edu/math.PR/0412362
(alternate)
Author(s): Jinho Baik and Toufic M. Suidan
Abstract: We prove a GUE central limit theorem for random variables with finite fourth
moment. We apply this theorem to prove that the directed first and last passage
percolation problems in thin rectangles exhibit universal fluctuations given by
the Tracy-Widom law. In addition, we conjecture a precise value for the time
constant in the general first and last passage problems.
http://arXiv.org/abs/math/0412369
http://front.math.ucdavis.edu/math.PR/0412369
(alternate)
Author(s): Jonah Blasiak
Abstract: The expected length of longest common subsequences is a problem that has been
in the literature for at least twenty five years. Determining the limiting
constants \gamma_k appears to be quite difficult, and the current best bounds
leave much room for improvement. Boutet de Monvel explores an independent
version of the problem he calls the Bernoulli Matching model. He explores this
problem and its relation to the longest common subsequence problem. This paper
continues this pursuit by focusing on a simplification we term r-reach. For the
string model, L_r(u,v) is the longest common subsequence of u and v given that
each matched pair of letters is no more than r letters apart.
http://arXiv.org/abs/math/0412375
http://front.math.ucdavis.edu/math.PR/0412375
(alternate)
Author(s): Yuval Peres
Abstract: Benjamini, Kalai and Schramm (2001) showed that weighted majority functions
of $n$ independent unbiased bits are uniformly stable under noise: when each
bit is flipped with probability $\epsilon$, the probability $p_\epsilon$ that
the weighted majority changes is at most $C\epsilon^{1/4}$. They asked what is
the best possible exponent that could replace 1/4. We prove that the answer is
1/2. The upper bound obtained for $p_\epsilon$ is within a factor of
$\sqrt{\pi/2}+o(1)$ from the known lower bound when $\epsilon \to 0$ and
$n\epsilon\to \infty$.
http://arXiv.org/abs/math/0412377
http://front.math.ucdavis.edu/math.PR/0412377
(alternate)
Author(s): G. Kordzakhia
Abstract: There are two types of particles interacting on a homogeneous tree of degree
d + 1. The particles of the first type colonize the empty space with
exponential rate 1, but cannot take over the vertices that are occupied by the
second type. The particles of the second type spread with exponential rate
\lambda. They colonize the neighboring vertices that are either vacant or
occupied by the representatives of the opposite type, and annihilate the
particles of the type 1 as they reach them. There exists a critical value
\lambda_c =(2d - 1) + \sqrt{(2d -1)^2 -1} such that the first type survives
with positive probability for \lambda < \lambda_c, and dies out with
probability one for \lambda > \lambda_c.
We also find the growth profile which characterizes the rate of growth of the
type 1 in the space-time on the event of survival.
http://arXiv.org/abs/math/0412392
http://front.math.ucdavis.edu/math.PR/0412392
(alternate)
Author(s): T. Antal (1 and 2) and S. Redner (2 and 1) ((1) Boston University and (2) CNLS and Los Alamos National Laboratory)
Abstract: We study the excited random walk, in which a walk that is at a site that
contains cookies eats one cookie and then hops to the right with probability p
and to the left with probability q=1-p. If the walk hops onto an empty site,
there is no bias. For the 1-excited walk on the half-line (one cookie initially
at each site), the probability of first returning to the starting point at time
t scales as t^{-(2-p)}. Although the average return time to the origin is
infinite for all p, the walk eats, on average, only a finite number of cookies
until this first return when p<1/2. For the infinite line, the probability
distribution for the 1-excited walk has an unusual anomaly at the origin. The
positions of the leftmost and rightmost uneaten cookies can be accurately
estimated by probabilistic arguments and their corresponding distributions have
power-law singularities near the origin. The 2-excited walk on the infinite
line exhibits peculiar features in the regime p>3/4, where the walk is
transient, including a mean displacement that grows as t^{nu}, with nu>1/2
dependent on p, and a breakdown of scaling for the probability distribution of
the walk.
http://arXiv.org/abs/math/0412407
http://front.math.ucdavis.edu/math.PR/0412407
(alternate)
Author(s): Michele L. Baldini
Abstract: Given an one-dimensional positive recurrent diffusion governed by the
Stratonovich SDE \[ X_t=x+\int_0^t\sigma(X_s)\strat db(s)+\int_0^t m(X_s) ds,
\] we show that the associated stochastic flow of diffeomorphisms focuses as
fast as $ \mathrm{exp}(-2t\int_{R}\frac{m^2}{\sigma^2} d\Pi)$, where $d\Pi$ is
the finite stationary measure. Moreover, if the drift is reversed and the
diffeomorphism is inverted, then the path function so produced tends,
independently of its starting point, to a single (random) point whose
distribution is $d\Pi$. Applications to stationary solutions of $X_t$,
asymptotic behavior of solutions of SPDEs and random attractors are offered.
http://arXiv.org/abs/math/0412410
http://front.math.ucdavis.edu/math.PR/0412410
(alternate)
Author(s): Gennady Samorodnitsky
Abstract: This paper elucidates the connection between stationary symmetric
alpha-stable processes with 0
http://arXiv.org/abs/math/0412419
http://front.math.ucdavis.edu/math.PR/0412419
(alternate)
Author(s): Hock Peng Chan and Tze Leung Lai
Abstract: Several classical results on boundary-crossing probabilities of Brownian
motion and random walks are extended to asymptotically Gaussian random fields,
which include sums of i.i.d. random variables with multidimensional indices,
multivariate empirical processes, and scan statistics in change-point and
signal detection as special cases. Some key ingredients in these extensions are
moderate deviation approximations to marginal tail probabilities and weak
convergence of the conditional distributions of certain ``clumps'' around
high-level crossings. We also discuss how these results are related to the
Poisson clumping heuristic and tube formulas of Gaussian random fields, and
describe their applications to laws of the iterated logarithm in the form of
the Kolmogorov--Erdos--Feller integral tests.
http://arXiv.org/abs/math/0412428
http://front.math.ucdavis.edu/math.PR/0412428
(alternate)
Author(s): I. Bjelakovic and J.-D. Deuschel and T. Krueger and R. Seiler and Ra. Siegmund-Schultze, A. Szkola
Abstract: We present a quantum extension of a version of Sanov's theorem focussing on a
hypothesis testing aspect of the theorem: There exists a sequence of typical
subspaces for a given set $\Psi$ of stationary quantum product states
asymptotically separating them from another fixed stationary product state.
Analogously to the classical case, the exponential separating rate is equal to
the infimum of the quantum relative entropy with respect to the quantum
reference state over the set $\Psi$. However, while in the classical case the
separating subsets can be chosen universal, in the sense that they depend only
on the chosen set of i.i.d. processes, in the quantum case the choice of the
separating subspaces depends additionally on the reference state.
http://arXiv.org/abs/quant-ph/0412157
http://front.math.ucdavis.edu/quant-ph/0412157
(alternate)
Author(s): Nicoletta Cancrini and Pietro Caputo and Fabio Martinelli
Abstract: We prove tight bounds on the relaxation time of the so called $L$--reversal
chain, introduced by R. Durrett as a stochastic model for the evolution of
chromosome chains. The process is described as follows: we have $n$ distinct
letters on the vertices of the $n$--cycle ($\bbZ$ mod $n$); at each step a
connected subset of the graph is chosen uniformly at random among all those of
length at most $L$ and the current permutation is shuffled by reversing the
order of the letters over that subset. We show that the relaxation time
$\t(n,L)$, defined as the inverse of the spectral gap of the associated Markov
generator, satisfies $\t(n,L)=O(n \vee \frac{n^3}{L^3})$. Our results can be
interpreted as a strong evidence for a conjecture of R. Durrett of a similar
behavior for the mixing time of the chain.
http://arXiv.org/abs/math/0412449
http://front.math.ucdavis.edu/math.PR/0412449
(alternate)
Author(s): Terence Tao and Van Vu
Abstract: We consider several questions concerning the determinant of a random $\pm 1$
matrix, in particular computing the probability on when this determinant is
zero or close to zero. We present simpler proofs of existing results and
introduce some new ones.
http://arXiv.org/abs/math/0411095
http://front.math.ucdavis.edu/math.CO/0411095
(alternate)
Author(s): Pietro Caputo and Fabio Martinelli
Abstract: Consider a low temperature stochastic Ising model in the phase coexistence
regime with Markov semigroup $P_t$. A fundamental and still largely open
problem is the understanding of the long time behavior of $\d_\h P_t$ when the
initial configuration $\h$ is sampled from a highly disordered state $\nu$
(e.g. a product Bernoulli measure or a high temperature Gibbs measure).
Exploiting recent progresses in the analysis of the mixing time of Monte Carlo
Markov chains for discrete spin models on a regular $b$-ary tree $\Tree^b$, we
tackle the above problem for the Ising and hard core gas (independent sets)
models on $\Tree^b$. If $\nu$ is a biased product Bernoulli law then, under
various assumptions on the bias and on the thermodynamic parameters, we prove
$\nu$-almost sure weak convergence of $\d_\h P_t$ to an extremal Gibbs measure
(pure phase) and show that the limit is approached at least as fast as a
stretched exponential of the time $t$. In the context of randomized algorithms
and if one considers the Glauber dynamics on a large, finite tree, our results
prove fast local relaxation to equilibrium on time scales much smaller than the
true mixing time, provided that the starting point of the chain is not taken as
the worst one but it is rather sampled from a suitable distribution.
http://arXiv.org/abs/math/0412450
http://front.math.ucdavis.edu/math.PR/0412450
(alternate)
Author(s): Dmitry Panchenko
Abstract: We conjecture that the Parisi functional in the SK model is convex in the
functional order parameter and prove a partial result that shows the convexity
along one-sided directions. A consequence of this result is log-convexity of
L_1 norm for a class or random variables.
http://arXiv.org/abs/math/0412463
http://front.math.ucdavis.edu/math.PR/0412463
(alternate)
Author(s): J. van den Berg and R. Brouwer
Abstract: We consider a forest-fire model which, somewhat informally, is described as
follows: Each site (vertex) of the square lattice is either vacant or occupied
by a tree.Vacant sites become occupied at rate 1. Further, each site is hit by
lightningat rate lambda. This lightning instantaneously destroys (makes vacant)
the occupied cluster of the site. This model is closely related to the
Drossel-Schwabl forest-fire model, which has received much attention in the
physics literature. The most interesting behaviour seems to occur when the
lightning rate goes to zero. In the physics literature it is believed that then
the system has so-called self-organized critical behaviour.
We let the system start with all sites vacant and study, for positive but
small lambda,the behaviour near the `critical time' tc; that is, the time after
which in the modified system without lightning an infinite occupied cluster
would emerge.
Intuitively one might expect that if, for fixed t > tc, we let simultaneously
lambda tend to 0 and m to infinity, the probability that some tree at distance
smaller than m from O is burnt before time t goes to 1. However, we show that
under a percolation-like assumption (which we can not prove but believe to be
true) this intuition is false. We compare with the case where the square
lattice is replaced by the directed binary tree, and pose some natural open
problems.
http://arXiv.org/abs/math/0412488
http://front.math.ucdavis.edu/math.PR/0412488
(alternate)
Author(s): Alexander Gnedin and Jim Pitman
Abstract: For two collections of nonnegative and suitably normalised weights
$\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of
partitions of the set $\{1,...,n\}$ is defined by assigning to a generic
partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}...
\W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose
constraints on the weights by assuming that the resulting random partitions
$\Pi_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an
exchangeable partition of the set of all natural numbers. This implies that the
weights $\W$ must be of a very special form depending on a single parameter
$\alpha\in [-\infty,1]$. The case $\alpha=1$ is trivial, and for each value of
$\alpha\neq 1$ the set of possible $\V$-weights is an infinite-dimensional
simplex. We identify the extreme points of the simplex by solving the boundary
problem for a generalised Stirling triangle. In particular, we show that the
boundary is discrete for $-\infty\leq\alpha<0$ and continuous for
$0\leq\alpha<1$. For $\alpha\leq 0$ the extremes correspond to the members of
the Ewens-Pitman family of random partitions indexed by $(\alpha,\theta)$,
while for $0<\alpha<1$ the extremes are obtained by conditioning an
$(\alpha,\theta)$-partition on the asymptotics of the number of blocks of
$\Pi_n$ as $n$ tends to infinity.
http://arXiv.org/abs/math/0412494
http://front.math.ucdavis.edu/math.PR/0412494
(alternate)
Author(s): Anna Karczewska
Abstract: We study stochastic convolutions providing by fundamental solutions of a
class of integrodifferential equations which interpolate the heat and the wave
equations. We give sufficient condition for the existence of function--valued
convolutions in terms of the covariance kernel of a noise given by spatially
homogeneous Wiener process.
http://arXiv.org/abs/math/0412495
http://front.math.ucdavis.edu/math.PR/0412495
(alternate)
Author(s): Anna Karczewska
Abstract: The paper is devoted to estimates for convolutions appearing in some class of
stochastic Volterra equations. Two maximal inequalities and exponential tail
estimate are provided. In the paper the fractional method of infinite
dimensional stochastic calculus has been used.
http://arXiv.org/abs/math/0412496
http://front.math.ucdavis.edu/math.PR/0412496
(alternate)
Author(s): Bela Bollobas and Oliver Riordan
Abstract: Recently, the authors showed that the critical probability for random Voronoi
percolation in the plane is 1/2. A by-product of the method was a short proof
of the Harris-Kesten Theorem concerning bond percolation in the planar square
lattice. The aim of this paper is to show that the same techniques can be
applied to many other planar percolation models, both to obtain short proofs of
known results, and to prove new ones.
http://arXiv.org/abs/math/0412510
http://front.math.ucdavis.edu/math.PR/0412510
(alternate)
Author(s): Anna Karczewska
Abstract: In the paper we study stochastic convolution appearing in Volterra equation
driven by so called L\'evy process. By L\'evy process we mean a process with
homogeneous independent increments, continuous in probability and cadlag.
http://arXiv.org/abs/math/0411148
http://front.math.ucdavis.edu/math.PR/0411148
(alternate)
Author(s): Cyril Banderier (LIPN) and Sylviane Schwer (LIPN)
Abstract: This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.
http://arXiv.org/abs/math/0411128
http://front.math.ucdavis.edu/math.CO/0411128
(alternate)
Author(s): Cyril Banderier (LIPN) and Jean-Marie Le Bars (LIPN and GREYC) and Vlady Ravelomanana (LIPN)
Abstract: In this article, we study directed graphs (digraphs) with a coloring
constraint due to Von Neumann and related to Nim-type games. This is equivalent
to the notion of kernels of digraphs, which appears in numerous fields of
research such as game theory, complexity theory, artificial intelligence
(default logic, argumentation in multi-agent systems), 0-1 laws in monadic
second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead
to numerous difficult questions (in the sense of NP-completeness,
#P-completeness). However, we show here that it is possible to use a generating
function approach to get new informations: we use technique of symbolic and
analytic combinatorics (generating functions and their singularities) in order
to get exact and asymptotic results, e.g. for the existence of a kernel in a
circuit or in a unicircuit digraph. This is a first step toward a
generatingfunctionology treatment of kernels, while using, e.g., an approach "a
la Wright". Our method could be applied to more general "local coloring
constraints" in decomposable combinatorial structures.
http://arXiv.org/abs/math/0411138
http://front.math.ucdavis.edu/math.CO/0411138
(alternate)
Author(s): Francesco Fidaleo
Abstract: We continue the analysis of nontrivial examples of quantum Markov processes.
This is done by applying the construction of entangled Markov chains obtained
from classical Markov chains with infinite state--space. The formula giving the
joint correlations arises from the corresponding classical formula by replacing
the usual matrix multiplication by the Schur multiplication. In this way, we
provide nontrivial examples of entangled Markov chains on $\bar{\cup_{J\subset
Z} \bar{\otimes}_{J}F}^{C^{*}}$, $F$ being any infinite dimensional type $I$
factor, $J$ a finite interval of $Z$, and the bar the von Neumann tensor
product between von Neumann algebras. We then have new nontrivial examples of
quantum random walks which could play a r\^ole in quantum information theory.
In view of applications to quantum statistical mechanics too, we see that the
ergodic type of an entangled Markov chain is completely determined by the
corresponding ergodic type of the underlying classical chain, provided that the
latter admits an invariant probability distribution. This result parallels the
corresponding one relative to the finite dimensional case.
Finally, starting from random walks on discrete ICC groups, we exhibit
examples of quantum Markov processes based on type $II_1$ von Neumann factors.
http://arXiv.org/abs/math/0411202
http://front.math.ucdavis.edu/math.OA/0411202
(alternate)
Author(s): Irina Ignatiouk-Robert
Abstract: The essential spectral radius of a sub-Markovian process is defined as the
infimum of the spectral radiuses of all local perturbations of the process.
When the family of rescaled processes satisfies sample path large deviation
principle, the spectral radius and the essential spectral radius are expressed
in terms of the rate function. The paper is motivated by applications to
reflected diffusions and jump Markov processes describing stochastic networks
for which the sample path large deviation principle has been established and
the rate function has been identified while essential spectral radius has not
been calculated.
http://arXiv.org/abs/math/0411221
http://front.math.ucdavis.edu/math.PR/0411221
(alternate)
Author(s): Francis Comets (PMA) and Nobuo Yoshida (DIVISION of Mathematics and Kyoto University)
Abstract: In this paper, we consider directed polymers in random environment with
discrete space and time. For transverse dimension at least equal to 3, we prove
that diffusivity holds for the path in the full weak disorder region, i.e.,
where the partition function differs from its annealed value only by a
non-vanishing factor. Deep inside this region, we also show that the quenched
averaged energy has fluctuations of order 1. In complete generality (arbitrary
dimension and temperature), we prove monotonicity of the phase diagram in the
temperature.
http://arXiv.org/abs/math/0411223
http://front.math.ucdavis.edu/math.PR/0411223
(alternate)
Author(s): Lancelot F. James
Abstract: In this paper we consider classes of nonparametric priors on spaces of
distribution functions and cumulative hazards that are based on extensions of
the neutral to the right concept. In particular we extend the definition of NTR
processes from the real line to classes of distributions on general spaces.
Representations of the posterior distributions are given using a different type
of calculus than traditionally used in the Bayesian literature. The techniques
are applied to progressively more complex models. Refinements are then given
which describes the underlying properties of spatial NTR models analogous to
those developed for the Dirihclet process. The analysis yields accessible
moment formulae and characterizations of the the posterior distribution and
relavant marginal distributions. In the homogeneous case this work turns out to
be connected to and overlap with recent work on regenerative compositions
defined by a suitable discretisation of subordinators. The results also have
connections to other related work on exponential functionals of subordinators.
In addition, we develop results for spatial NTR mixture models and identify a
class of species sampling models derived from spatial NTR processes.
http://arXiv.org/abs/math/0305053
http://front.math.ucdavis.edu/math.ST/0305053
(alternate)
Author(s): P. Major
Abstract: We prove a multivariate version of Bernstein's inequality about the
probability that degenerate $U$-statistics take a value larger than some number
$u$. This is an improvement of former estimates for the same problem which
yields an asymptotically sharp estimate for not too large numbers $u$. This
paper also contains an analogous bound about the distribution of multiple
Wiener-Ito integrals. Their comparison shows that our results are sharp. The
proofs are based on good estimates about high moments of multiple random
integrals. They are obtained by means of a diagram formula which enables us to
express the product of multiple random integrals as the sum of such
expressions.
http://arXiv.org/abs/math/0411287
http://front.math.ucdavis.edu/math.PR/0411287
(alternate)
Author(s): P. Major
Abstract: We prove a multivariate version of Hoeffding's inequality about the
distribution of homogeneous polynomials of Rademacher functions. The proof is
based on such an estimate about the moments of homogeneous polynomials of
Rademacher functions which can be considered as an improvement of Borell's
inequality in a most important special case.
http://arXiv.org/abs/math/0411288
http://front.math.ucdavis.edu/math.PR/0411288
(alternate)
Author(s): Sho Matsumoto
Abstract: For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$,
we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we
call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the
$\alpha$-determinant. It gives the pfaffian at $\alpha=-1$. We give some
formulas for $\alpha$-pfaffians and study the positivity. Further we define
point processes determined by the $\alpha$-pfaffian. Also we provide a linear
algebraic proof of the explicit pfaffian expression for the correlation
function of the shifted Schur measure.
http://arXiv.org/abs/math/0411277
http://front.math.ucdavis.edu/math.CO/0411277
(alternate)
Author(s): Javier Parcet and Narcisse Randrianantoanina
Abstract: We provide an analogue of Gundy's decomposition for L1-bounded
non-commutative martingales. An important difference from the classical case is
that for any L1-bounded non-commutative martingale, the decomposition consists
of four martingales. This is strongly related with the row/column nature of
non-commutative Hardy spaces of martingales. As applications, we obtain simpler
proofs of the weak type (1,1) boundedness for non-commutative martingale
transforms and the non-commutative analogue of Burkholder's weak type
inequality for square functions. A sequence (x_n) in a normed space X is called
2-co-lacunary if there exists a bounded linear map from the closed linear span
of (x_n) to l2 taking each x_n to the n-th vector basis of l2. We prove (using
our decomposition) that any relatively weakly compact martingale difference
sequence in L1(M,\tau) whose sequence of norms is bounded away from zero is
2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative
L1-spaces.
http://arXiv.org/abs/math/0411296
http://front.math.ucdavis.edu/math.OA/0411296
(alternate)
Author(s): Sergio Albeverio and Carlo Marinelli
Abstract: The problem of reconstructing the drift of a diffusion in $\erre^d$, $d\geq
2$, from the transition probability density observed outside a domain is
considered. The solution of this problem also solves a new inverse problem for
a class of parabolic partial differential equations. This work considerably
extends \cite{jsp} in terms of generality, both concerning assumptions on the
drift coefficient, and allowing for non-constant diffusion coefficient.
Sufficient conditions for solvability of this type of inverse problem for $d=1$
are also given.
http://arXiv.org/abs/math/0411008
http://front.math.ucdavis.edu/math.PR/0411008
(alternate)
Author(s): Julien Dubedat
Abstract: Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth
processes in the plane that have particular conformal invariance properties.
For instance, SLE can define simple random curves in a simply conneccted
domain. In this paper we are interested in questions pertaining to the
definition of several SLEs in a domain (i.e. several random curves). In
particular, one derives infinitesimal commutation conditions, discuss some
solutions, and show how to lift these infinitesimal relations to global
relations in simple cases.
http://arXiv.org/abs/math/0411299
http://front.math.ucdavis.edu/math.PR/0411299
(alternate)
Author(s): Ramon van Handel
Abstract: We develop a method to prove almost global stability of stochastic
differential equations in the sense that almost every initial point (with
respect to the Lebesgue measure) is asymptotically attracted to the origin with
unit probability. The method can be viewed as a dual to Lyapunov's second
method for stochastic differential equations and extends the deterministic
result in [A. Rantzer, Syst. Contr. Lett., 42 (2001), pp. 161--168]. The result
can also be used in certain cases to find stabilizing controllers for
stochastic nonlinear systems using convex optimization. The main technical tool
is the theory of stochastic flows of diffeomorphisms.
http://arXiv.org/abs/math/0411311
http://front.math.ucdavis.edu/math.PR/0411311
(alternate)
Author(s): W. Hachem and P. Loubaton and J. Najim
Abstract: Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are
given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the
$X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a
continuous function called a variance profile. Consider now a deterministic
$N\times n$ matrix $\Lambda_n=(\Lambda_{ij}^{n})$ whose non diagonal elements
are zero. Denote by $\Sigma_n$ the non-centered matrix $Y_n + \Lambda_n$. Then
under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$ \frac{1}{N}
\sum_{i=1}^{N} \delta_{(\frac{i}{N}, (\Lambda_{ii}^n)^2)} \xrightarrow[n\to
\infty]{} H(dx,d\lambda), $$ where $H$ is a probability measure, it is proven
that the empirical distribution of the eigenvalues of $ \Sigma_n \Sigma_n^T$
converges almost surely in distribution to a non random probability measure.
This measure is characterized in terms of its Stieltjes transform, which is
obtained with the help of an auxiliary system of equations. This kind of
results is of interest in the field of wireless communication.
http://arXiv.org/abs/math/0411333
http://front.math.ucdavis.edu/math.PR/0411333
(alternate)
Author(s): Remco van der Hofstad and Nina Gantert and Wolfgang K{\"o}nig
Abstract: Let $(Z_n)_{n\in\N_0}$ be a d-dimensional random walk in random scenery,
i.e., $Z_n=\sum_{k=0}^{n-1}Y_{S_k}$ with $(S_k)_{k\in\N_0}$ a random walk in
$\Z^d$ and $(Y_{z})_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk.
We assume that the random variables $Y_{z}$ have a stretched exponential
tail. In particular, they do not possess exponential moments. We identify the
speed and the rate of the logarithmic decay of $\P(\frac 1n Z_n>t_n)$ for all
sequences $(t_n)_{n\in\N}$ satisfying a certain lower bound. This complements
results of \cite{GKS04}, where it was assumed that $Y_{z}$ has exponential
moments of all orders. Informally, in contrast to the situation \cite{GKS04},
the event $\{\frac 1n Z_n>t_n\}$ is not realized by a homogeneous behavior of
the walk's local times and the scenery, but by many visits of the walker to a
particular site and a large value of the scenery at that site. This reflects a
well-known extreme behavior typical for random variables having no exponential
moments.
http://arXiv.org/abs/math/0411361
http://front.math.ucdavis.edu/math.PR/0411361
(alternate)
Author(s): Davar Khoshnevisan and David A. Levin and Zhan Shi
Abstract: We present an extreme-value analysis of the classical law of the iterated
logarithm (LIL) for Brownian motion. Our result can be viewed as a new
improvement to the LIL.
http://arXiv.org/abs/math/0411376
http://front.math.ucdavis.edu/math.PR/0411376
(alternate)
Author(s): Nathanael Berestycki
Abstract: Make the set of permutations of $n$ objects into a graph $G_n$ by connecting
two permutations that differ by one transposition, and let $\sigma_t$ be the
continuous time simple random walk on this graph. In a previous paper,
Berestycki and Durrett (2004) showed that the limiting behavior of the distance
from the identity at time $cn/2$ has a phase transition at $c=1$. When $c<1$,
it is asymptotically $cn/2$, while for $c>1$ it is $u(c)n$ with $u(c) < c/2$.
Here we investigate some consequences of this result for the geometry of $G_n$.
Our first result is that when we consider the sphere of radius $an$ centered at
the origin, and pick two points independently according to the hitting
distribution, then Gromov hyperbolicity breaks down at critical radius $a=1/4$.
When $a<1/4$ the space is hyperbolic but also displays behavior that is much
different from manifolds of negative curvature - it is shown that there are
many geodesics that may travel much different paths to get to a point. We also
show that the hitting distribution of the sphere of radius $an$ is
asymptotically singular with respect to the uniform distribution. Finally, we
prove that the qualitative behavior of the Gromov hyperbolicity persists if we
pick points independently according to the uniform measure on the sphere of
radius $an$. However, in this case, the critical radius is $a=1-\log 2$.
http://arXiv.org/abs/math/0411011
http://front.math.ucdavis.edu/math.PR/0411011
(alternate)
Author(s): Samuel Herrmann and Peter Imkeller and Dierk Peithmann
Abstract: We consider potential type dynamical systems in finite dimensions with two
meta-stable states. They are subject to two sources of perturbation: a slow
external periodic perturbation of period $T$ and a small Gaussian random
perturbation of intensity $\eps$, and therefore mathematically described as
weakly time inhomogeneous diffusion processes. A system is in stochastic
resonance provided the small noisy perturbation is tuned in such a way that its
random trajectories follow the exterior periodic motion in an optimal fashion,
i.e. for some optimal intensity $\eps(T)$. The physicists' favorite measures of
quality of periodic tuning -- and thus stochastic resonance -- such as spectral
power amplification or signal-to-noise ratio have proven to be defective. They
are not robust w.r.t. effective model reduction, i.e. for the passage to a
simplified finite state Markov chain model reducing the dynamics to a pure
jumping between the meta-stable states of the original system. An entirely
probabilistic notion of stochastic resonance based on the transition dynamics
between the domains of attraction of the meta-stable states -- and thus failing
to suffer from this robustness defect -- was proposed before in the context of
one-dimensional diffusions. It is investigated for higher dimensional systems
here, by using extensions and refinements of the Freidlin-Wentzell theory of
large deviations for time homogeneous diffusions. Large deviation principles
developed for weakly time inhomogeneous diffusions prove to be key tools for a
treatment of the problem of diffusion exit from a domain and thus for the
approach of stochastic resonance via transition probabilities between
meta-stable sets.
http://arXiv.org/abs/math/0411386
http://front.math.ucdavis.edu/math.PR/0411386
(alternate)
Author(s): Kent E. Morrison
Abstract: We investigate several infinite product of cosines and find the closed form
using the Fourier transform. The answers provide limiting distributions for
some elementary probability experiments.
http://arXiv.org/abs/math/0411380
http://front.math.ucdavis.edu/math.CA/0411380
(alternate)
Author(s): T. Imamura and T. Sasamoto
Abstract: We present a random matrix interpretation of the distribution functions which
have appeared in the study of the one-dimensional polynuclear growth (PNG)
model with external sources. It is shown that the distribution, GOE$^2$, which
is defined as the square of the GOE Tracy-Widom distribution, can be obtained
as the scaled largest eigenvalue distribution of a special case of a random
matrix model with a deterministic source, which have been studied in a
different context previously. Compared to the original interpretation of the
GOE$^2$ as ``the square of GOE'', ours has an advantage that it can also
describe the transition from the GUE Tracy-Widom distribution to the GOE$^2$.
We further demonstrate that our random matrix interpretation can be obtained
naturally by noting the similarity of the topology between a certain
non-colliding Brownian motion model and the multi-layer PNG model with an
external source. This provides us with a multi-matrix model interpretation of
the multi-point height distributions of the PNG model with an external source.
http://arXiv.org/abs/math-ph/0411057
http://front.math.ucdavis.edu/math-ph/0411057
(alternate)
Author(s): Tomasz Schreiber
Abstract: We establish a version of the Dobrushin-Kotecky-Schlosman phase separation
theorem for the length-interacting Arak-Surgailis polygonal Markov fields with
V-shaped nodes.
http://arXiv.org/abs/math-ph/0411064
http://front.math.ucdavis.edu/math-ph/0411064
(alternate)
Author(s): Momar Dieng
Abstract: We derive Painlev\'e--type expressions for the distribution of the $m^{th}$
largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the
edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies
the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of
the appropriate Wishart distribution.
http://arXiv.org/abs/math/0411421
http://front.math.ucdavis.edu/math.PR/0411421
(alternate)
Author(s): Haakan Hedenmalm and Nikolai Makarov
Abstract: In this note, we discuss the quantum Hele-Shaw flow, a random measure process
in the complex plane introduced by the physicists P.Wiegmann, A. Zabrodin, et
al. This process arises in the theory of electronic droplets confined to a
plane under a strong magnetic field, as well as in the theory of random normal
matrices. We extend a result of Elbau and Felder to general external field
potentials, and also show that if the potential is $C^2$-smooth, then the
quantum Hele-Shaw flow converges, under appropriate scaling, to the classical
(weighted) Hele-Shaw flow, which can be modeled in terms of an obstacle
problem.
http://arXiv.org/abs/math/0411437
http://front.math.ucdavis.edu/math.PR/0411437
(alternate)
Author(s): Yacine Ait-Sahalia and Jean Jacod (PMA)
Abstract: This paper studies the asymptotic behavior of the Fisher information for a
Levy process discretely sampled at an increasing frequency. We show that it is
possible to distinguish not only the continuous part of the process from its
jumps part, but also different types of jumps, and derive the rates of
convergence of efficient estimators.
http://arXiv.org/abs/math/0411438
http://front.math.ucdavis.edu/math.PR/0411438
(alternate)
Author(s): Alan Hammond
Abstract: Assign to each site of the integer lattice $\Zd$ a real score, sampled
according to the same distribution $F$, independently of the choices made at
all other sites. A lattice animal is a finite connected set of sites, with its
weight being the sum of the scores at its sites. Let $N_n$ be the maximal
weight of those lattice animals of size $n$ that contain the origin. Denote by
$N$ the almost sure finite constant limit of $n^{-1} N_n$, which exists under a
mild condition on the positive tail of $F$. We study certain geometrical
aspects of the lattice animal with maximal weight among those contained in an
$n$-box where $n$ is large, both in the supercritical phase where $N > 0$, and
in the critical case where $N = 0$.
http://arXiv.org/abs/math/0411459
http://front.math.ucdavis.edu/math.PR/0411459
(alternate)
Author(s): Sandrine P\'ech\'e
Abstract: We compute the limiting eigenvalue statistics at the edge of the spectrum of
large Hermitian random matrices perturbed by the addition of small rank
deterministic matrices. To be more precise, we consider random Hermitian
matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various
expectations. We prove that the largest eigenvalue of such random matrices
exhibits, in the large $N$ limit, various limiting distributions depending on
both the eigenvalues of the matrix $(\mathbb{E}M_{ij})_{i,j=1}^N$ and its rank.
http://arXiv.org/abs/math/0411487
http://front.math.ucdavis.edu/math.PR/0411487
(alternate)
Author(s): S.I. Bityukov and V.V. Smirnova and V.A. Taperechkina
Abstract: The reconstruction of the parameter of the model by the measurement of the
random variable depending on this parameter is one of the main tasks of
statistics. In the paper the notion of the statistically dual distributions is
introduced. The approach, based on the properties of the statistically dual
distributions, to resolving of the given task is proposed.
http://arXiv.org/abs/math/0411462
http://front.math.ucdavis.edu/math.ST/0411462
(alternate)
Author(s): H. Soula and G. Beslon and O. Mazet
Abstract: We study in this paper the effect of an unique initial stimulation on random
recurrent networks of leaky integrate and fire neurons. Indeed given a
stochastic connectivity this so-called spontaneous mode exhibits various non
trivial dynamics. This study brings forward a mathematical formalism that
allows us to examine the variability of the afterward dynamics according to the
parameters of the weight distribution. Provided independence hypothesis (e.g.
in the case of very large networks) we are able to compute the average number
of neurons that fire at a given time -- the spiking activity. In accordance
with numerical simulations, we prove that this spiking activity reaches a
steady-state, we characterize this steady-state and explore the transients.
http://arXiv.org/abs/cs/0411052
http://front.math.ucdavis.edu/cs.NE/0411052
(alternate)
Author(s): Marcus Hutter
Abstract: Given i.i.d. data from an unknown distribution, we consider the problem of
predicting future items. An adaptive way to estimate the probability density is
to recursively subdivide the domain to an appropriate data-dependent
granularity. A Bayesian would assign a data-independent prior probability to
"subdivide", which leads to a prior over infinite(ly many) trees. We derive an
exact, fast, and simple inference algorithm for such a prior, for the data
evidence, the predictive distribution, the effective model dimension, and other
quantities.
http://arXiv.org/abs/math/0411515
http://front.math.ucdavis.edu/math.ST/0411515
(alternate)
Author(s): Sourav Chatterjee and Persi Diaconis and Elizabeth Meckes
Abstract: This is a survery paper on Poisson approximation using Stein's method of
exchangeable pairs. We illustrate using Poisson-binomial trials and many
variations on three classical problems of combinatorial probability: the
matching problem, the coupon collector's problem, and the birthday problem.
While many details are new, the results are closely related to a body of work
developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry
Goldstein, and their collaborators. Some comparison with these other approaches
is offered.
http://arXiv.org/abs/math/0411525
http://front.math.ucdavis.edu/math.PR/0411525
(alternate)
Author(s): Alan Hammond and Yuval Peres
Abstract: We consider a planar Brownian loop $B$ that is run for a time $T$ and
conditioned on the event that its range encloses the unusually high area of
$\pi T^2$, with $T$ being large. We study the deviation of the range of the
conditioned process $X$ from a circle of radius $T$, as a model for the
fluctuation of a phase boundary. This deviation is measured by means of the
inradius and outradius of the region enclosed by the range of $X$. We prove
that in a typical realization of the conditioned measure, each of these
quantities differs from $T$ by at most $T^{2/3 + \epsilon}$.
http://arXiv.org/abs/math/0411540
http://front.math.ucdavis.edu/math.PR/0411540
(alternate)
Author(s): Raphael Cerf and Sana Louhichi
Abstract: We consider the 2D stochastic Ising model evolving according to the Glauber
dynamics at zero temperature. We compute the initial drift for droplets which
are discretizations of smooth domains. A specific spatial average of the
derivative at time~0 of the volume variation of a droplet close to a boundary
point is equal to its curvature multiplied by a direction dependent
coefficient. For a boundary point having a tangent with angle $\theta$, this
coefficient is equal to $-\frac{\textstyle 1}{\textstyle 2}|\cos 2\theta|$.
http://arXiv.org/abs/math/0411545
http://front.math.ucdavis.edu/math.PR/0411545
(alternate)
Author(s): Wei-Shih Yang and Aklilu Zeleke
Abstract: Let \zeta be the intersection exponent of random walks in Z^3 and \alpha be a
positive real number. We construct a stochastic process from a simple random
walk by erasing loops of length at most N^\alpha. We will prove that for \alpha
< \frac{1}{1+2\zeta}, the limiting distribution is Gaussian. For \alpha > 2 the
limiting distribution will be shown to be equal to the limiting distribution of
the loop erased walk.
http://arXiv.org/abs/math/0411551
http://front.math.ucdavis.edu/math.PR/0411551
(alternate)
Author(s): Jan Pospisil and Roger Tribe
Abstract: This paper calculates the exact quadratic variation in space and quartic
variation in time for the solutions to a one dimensional stochastic heat
equation driven by a multiplicative space-time white noise.
http://arXiv.org/abs/math/0411552
http://front.math.ucdavis.edu/math.PR/0411552
(alternate)
Author(s): Leandro P.R. Pimentel
Abstract: In this paper we consider first-passage percolation models on Voronoi tilings
of the plane and present a sufficient condition for the compactness of the
limit set. This result is based on a static renormalization technique and also
provide an inequality involving critical probabilities for bond percolation
models.
http://arXiv.org/abs/math/0411560
http://front.math.ucdavis.edu/math.PR/0411560
(alternate)
Author(s): Leandro P.R. Pimentel
Abstract: We consider a competing spatial growth dynamics permitting that more than one
cluster develop in the same environment given by a first-passage percolation
model on a Voronoi tiling of the plane. We focus on the long time behavior of
these competing clusters and derive some limit theorems related to the
morphology of the ``competition interface''. To study the structure of this
interface we use the notion of geodesic in first-passage percolation and
explore the coalescence behavior of semi-infinite geodesics with the same
orientation.
http://arXiv.org/abs/math/0411583
http://front.math.ucdavis.edu/math.PR/0411583
(alternate)
Author(s): Ravi Montenegro
Abstract: We consider the problem of lower bounding a generalized Minkowski measure of
subsets of a convex body with a log-concave probability measure, conditioned on
the set size. A bound is given in terms of diameter and set size, which is
sharp for all set sizes, dimensions, and norms. In the case of uniform density
a stronger theorem is shown which is also sharp.
http://arXiv.org/abs/math/0411018
http://front.math.ucdavis.edu/math.FA/0411018
(alternate)
Author(s): P. Chigansky
Abstract: Exponential stability of the nonlinear filtering recursion is revisited, when
the signal is a finite state Markov chain. An asymptotic upper bound for the
filtering error due to incorrect initial condition is derived for the case of
slowly switching signal.
http://arXiv.org/abs/math/0411596
http://front.math.ucdavis.edu/math.PR/0411596
(alternate)
Author(s): Steffen Dereich
Abstract: We investigate the high resolution coding problem for solutions of stochastic
differential equations in the L^p[0,1]- and the C[0,1]-space. Tight asymptotic
estimates are found under weak regularity assumptions. The main technical tool
is a decoupling method which allows us to relate the complexity of the
diffusion process to that of the Wiener process under certain random
distortions.
http://arXiv.org/abs/math/0411597
http://front.math.ucdavis.edu/math.PR/0411597
(alternate)
Author(s): F. Rassoul-Agha and T. Seppalainen
Abstract: We consider a discrete time random walk in a space-time i.i.d. random
environment. We use a martingale approach to show that the walk is diffusive in
almost every fixed environment. We improve on existing results by proving an
invariance principle and considering environments with an annealed $L^2$ drift.
We also state an a.s. invariance principle for random walks in general random
environments whose hypothesis requires a subdiffusive bound on the variance of
the quenched mean, under an ergodic invariant measure for the environment
chain.
http://arXiv.org/abs/math/0411602
http://front.math.ucdavis.edu/math.PR/0411602
(alternate)
Author(s): F. Rassoul-Agha and T. Seppalainen
Abstract: We prove an invariance principle for a vector-valued additive functional of a
Markov chain for almost every starting point with respect to an ergodic
equilibrium distribution. The hypothesis is a moment bound on the resolvent.
http://arXiv.org/abs/math/0411603
http://front.math.ucdavis.edu/math.PR/0411603
(alternate)
Author(s): J\"urgen Hellmich and Claus K\"ostler and Burkhard K\"ummerer
Abstract: We introduce a non-commutative extension of Tsirelson-Vershik's noises,
called (non-commutative) continuous Bernoulli shifts. These shifts encode
stochastic independence in terms of commuting squares, as they are familiar in
subfactor theory. Such shifts are, in particular, capable of producing
Arveson's product system of type I and type II. We investigate the structure of
these shifts and prove that the von Neumann algebra of a (scalar-expected)
continuous Bernoulli shift is either finite or of type III.
The role of (`classical') stationary flows for Tsirelson-Vershik's noises is
now played by cocycles of continuous Bernoulli shifts. We show that these
cocycles provide an operator algebraic notion for Levy processes. They lead, in
particular, to units and `logarithms' of units in Arveson's product systems.
Furthermore, we introduce (non-commutative) white noises, which are operator
algebraic versions of Tsirelson's `classical' noises. We give examples coming
from probability, quantum probability and from Voiculescu's theory of free
probability.
Our main result is a bijective correspondence between additive and unital
shift cocycles. For the proof of the correspondence we develop tools which are
of interest on their own: non-commutative extensions of stochastic Ito
integration, stochastic logarithms and exponentials.
http://arXiv.org/abs/math/0411565
http://front.math.ucdavis.edu/math.OA/0411565
(alternate)
Author(s): B. Naimark and E. Ostrovsky
Abstract: We study the exact constants in the moment inequalities for sums of centered
independent random variables: improve their asymptotics, low and upper bounds,
calculate more exact asymptotics, elaborate the numerical algorithm for their
calculation, study the class of smoothing etc.
http://arXiv.org/abs/math/0411614
http://front.math.ucdavis.edu/math.PR/0411614
(alternate)
Author(s): B.M. Migdashiev and E.I. Ostrovsky
Abstract: We construct a non - improved exponential bounds for distribution of normed
sums of i.,i.d. random variables with random numbers of summand.
http://arXiv.org/abs/math/0411616
http://front.math.ucdavis.edu/math.PR/0411616
(alternate)
Author(s): E. D. Andjel (Univ. Provence) and H. Guiol (INP Grenoble)
Abstract: We show that if $\mu$ is an invariant measure for the long range exclusion
process putting no mass on the full configuration, $L$ is the formal generator
of that process, and $f$ is a cylinder function, then $Lf \in\mathbf{L}^{1}
(d\mu)$ and $\int Lf d\mu =0$. This result is then applied to determine, i) the
set of invariant and translation invariant measures of the long range exclusion
process on $\Z^d$ when the underlying random walk is irreducible; ii) the set
of invariant measures of the long range exclusion process on $\Z$ when the
underlying random walk is irreducible and has either zero mean or allows jumps
only to the nearest neighbors.
http://arXiv.org/abs/math/0411655
http://front.math.ucdavis.edu/math.PR/0411655
(alternate)
Author(s): Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig
Abstract: We introduce two probabilistic models for $N$ interacting Brownian motions
moving in a trap in $ \R^d $ under mutually repellent forces. The two models
are defined in terms of transformed path measures on finite time intervals
under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The
first pair interaction exhibits a {\it particle} repellency, while the second
one imposes a {\it path} repellency.
We analyse both models in the limit of diverging time with fixed number $ N $
of Brownian motions. In particular, we prove large deviations principles for
the normalised occupation measures. The minimisers of the rate functions are
related to the Hamilton operator for $ N $ interacting trapped particles. More
precisely, in the particle-repellency model, the minimiser is its ground state,
and in the path-repellency model, the minimisers are its ground product-states.
This study is a contribution to the search for a mathematical formulation of
the quantum system of $ N $ trapped interacting bosons as a model for {\it
Bose-Einstein condensation}, motivated by the success of the famous 1995
experiments. Recently, Lieb, et al. described the large-N behaviour of the
ground state in terms of the well-known {\it Gross-Pitaevskii} formula,
involving the scattering length of the pair potential. We prove that the
large-N behaviour of the ground product-states is also described by the
Gross-Pitaevskii formula, however with the scattering length of the pair
potential replaced by its integral.
http://arXiv.org/abs/math/0411660
http://front.math.ucdavis.edu/math.PR/0411660
(alternate)
Author(s): Frederic Dambreville (DGA/CTA/DT/GIP)
Abstract: The Bayesian logic is generally associated to the definition of a prior
probabilistic law. Conditional algebra have been investigated by some authors
though, but somehow the background framework is still probabilistic and the
entire logic is not specified. In this paper, the definition of a Deterministic
Bayesian Logic is proposed. This logic is completely independent of any notion
of probability. The coherence of this logic is proven and various logical
theorems are derived. It is shown that this logic is probabilizable and avoids
the negative result of Lewis. At last the probabilistic Bayesian rule is
recovered by posteriorly probabilizing our logic.
http://arXiv.org/abs/cs/0411097
http://front.math.ucdavis.edu/cs.LO/0411097
(alternate)
Author(s): Craig A. Tracy and Harold Widom
Abstract: The extended Airy kernel describes the space-time correlation functions for
the Airy process, which is the limiting process for a polynuclear growth model.
The Airy functions themselves are given by integrals in which the exponents
have a cubic singularity, arising from the coalescence of two saddle points in
an asymptotic analysis. Pearcey | |
