Probability Abstracts 63

1726. THICK POINTS FOR INTERSECTIONS OF PLANAR SAMPLE PATHS

Amir Dembo, Yuval peres, Jay Rosen, Ofer Zeitouni

Let $L_n^{X}(x)$ denote the number of visits to $x \in {\bf Z}^2$ of the
simple planar random walk $X$, up till step $n$. Let $X'$ be another simple
planar random walk independent of $X$. We show that for any $0<b<1/(2 \pi)$,
there are $n^{1-2\pi b+o(1)}$ points $x \in {\bf Z}^2$ for which
$L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$. This is the discrete counterpart of
our main result, that for any $a<1$, the Hausdorff dimension of the set of {\it
thick intersection points} $x$ for which $\limsup_{r \to 0} {\mathcal
I}(x,r)/(r^2|\log r|^4)=a^2$, is almost surely $2-2a$. Here ${\mathcal I}(x,r)$
is the projected intersection local time measure of the disc of radius $r$
centered at $x$ for two independent planar Brownian motions run till time 1.
The proofs rely on a `multi-scale refinement' of the second moment method. In
addition, we also consider analogous problems where we replace one of the
Brownian motions by a transient stable process, or replace the disc of radius
$r$ centered at $x$ by $x+rK$ for general sets $K$.

zeitouni@ee.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0105107.

1727. COMPATIBLE SEQUENCES AND A SLOW WINKLER PERCOLATION

Peter Gacs

Two infinite 0-1 sequences are called compatible when it is possible to cast 
out 0's from both in such a way that they become complementary to each other. 
Answering a question of Peter Winkler, we show that if the two 0-1-sequences 
are random i.i.d. and independent from each other, with probability p of 1's, 
then if p is sufficiently small they are compatible with positive probability. 
The question is equivalent to a certain dependent percolation with a power-law 
behavior: the probability that the origin is blocked at distance n but not 
closer decreases only polynomially fast and not, as usual, exponentially. 

peter.gacs@cwi.nl

• This article is available from the xxx mathematics archive as math.PR/0011008.

1728. RANDOM REGULARIZATION OF BROWN SPECTRAL MEASURE

Piotr Sniady

We geralize a recent result of Haagerup; namely we show that a convolution
with a standard Gaussian random matrix regularizes behaviour of
Kadison--Fuglede determinant and Brown spectral distribution measure. In this
way it is possible to establish a connection between limit eigenvalues
distributions of a wide class of random matrices and the Brown measure of the
corresponding limits.

psnia@math.uni.wroc.pl

• This article is available from the xxx mathematics archive as math.PR/0105109.

1729. ASYMPTOTICS OF CERTAIN COAGULATION-FRAGMENTATION PROCESSES AND INVARIANT POISSON-DIRICHLET MEASURES

Eddy Mayer-Wolf, Ofer Zeitouni, Martin P. W. Zerner

We consider Markov chains on the space of (countable) partitions of the
interval $[0,1]$, obtained first by size biased sampling twice (allowing
repetitions) and then merging the parts with probability $\beta_m$ (if the
sampled parts are distinct) or splitting the part with probability $\beta_s$
according to a law $\sigma$ (if the same part was sampled twice). We
characterize invariant probability measures for such chains. In particular, if
$\sigma$ is the uniform measure then the Poisson-Dirichlet law is an invariant
probability measure, and it is unique within a suitably defined class of
``analytic'' invariant measures. We also derive transience and recurrence
criteria for these chains.

zeitouni@ee.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0105111.

1730. ANOMALOUS SLOW DIFFUSION FROM PERPETUAL HOMOGENIZATION

Houman Owhadi

This paper is concerned with the asymptotic behavior solutions of stochastic
differential equations $dy_t=d\omega_t -\nabla V(y_t) dt$, $y_0=0$. When $d=1$
and $V$ reflects media characterized by an infinite number of spatial scales
$V(x) = \sum_{k=0}^\infty U_k(x/R_k)$, where $U_k$ are smooth functions of
period 1, $U_k(0)=0$, and $R_k$ grows exponentially fast with $k$, we can show
that $y_t$ has an anomalous slow behavior and obtain quantitative estimates on
the anomaly using and developing the tools of homogenisation. Pointwize
estimates are based on a new analytical inequality for sub-harmonic functions.
When $d\geq 1$ and $V$ is periodic, quantitative estimates are obtained on the
heat kernel of $y_t$, showing the rate at which homogenization takes place, the
latter result is directly linked to Davies's conjecture and based on a
quantitative estimate for the Laplace transform of martingales that can be used
to obtain similar results for periodic elliptic generator.

owhadi@techunix.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0105165.

1731. ON CONFORMALLY INVARIANT SUBSETS OF THE PLANAR BROWNIAN CURVE

Vincent Beffara 

We define and study a family of generalized non-intersection exponents for
planar Brownian motions that is indexed by subsets of the complex plane: For
each $A\subset\CC$, we define an exponent $\xi(A)$ that describes the decay of
certain non-intersection probabilities. To each of these exponents, we
associate a conformally invariant subset of the planar Brownian path, of
Hausdorff dimension $2-\xi(A)$. A consequence of this and continuity of
$\xi(A)$ as a function of $A$ is the almost sure existence of pivoting points
of any sufficiently small angle on a planar Brownian path.

vincent.beffara@math.u-psud.fr

• This article is available from the xxx mathematics archive as math.PR/0105192.

1732. SUPER-DIFFUSIVITY IN A SHEAR FLOW MODEL FROM PERPETUAL HOMOGENIZATION

G\'{e}rard Ben Arous and Houman Owhadi

This paper is concerned with the asymptotic behavior solutions of stochastic
differential equations $dy_t=d\omega_t -\nabla \Gamma(y_t) dt$, $y_0=0$ and
$d=2$. $\Gamma$ is a $2\times 2$ skew-symmetric matrix associated to a shear
flow characterized by an infinite number of spatial scales
$\Gamma_{12}=-\Gamma_{21}=h(x_1)$, with $h(x_1)=\sum_{n=0}^\infty \gamma_n
h^n(x_1/R_n)$ where $h^n$ are smooth functions of period 1, $h^n(0)=0$,
$\gamma_n$ and $R_n$ grow exponentially fast with $n$. We can show that $y_t$
has an anomalous fast behavior ($\E[|y_t|^2]\sim t^{1+\nu}$ with $\nu>0$) and
obtain quantitative estimates on the anomaly using and developing the tools of
homogenization.

owhadi@techunix.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0105199.

1733. DEVIATION BOUNDS FOR WAVELET SHRINKAGE

Dawei Hong, Jean-Camille Birget 

We analyse the wavelet shrinkage algorithm of Donoho and Johnstone in order
to assess the quality of the reconstruction of a signal obtained from noisy
samples. We prove deviation bounds for the maximum of the squares of the error,
and for the average of the squares of the error, under the assumption that the
signal comes from a H"older class, and the noise samples are independent, of 0
mean, and bounded. Our main technique is Talgrand's isoperimetric theorem. Our
bounds refine the known expectations for the average of the squares of the
error.

birget@camden.rutgers.edu

• This article is available from the xxx mathematics archive as math.PR/0105200.

1734. AGING PROPERTIES OF SINAI'S MODEL OF RANDOM WALK IN RANDOM ENVIRONMENT

Amir Dembo, Alice Guionnet, Ofer Zeitouni

We study in this short note aging properties of Sinai's (nearest neighbour)
random walk in random environment. With $\PP^o$ denoting the annealed law of
the RWRE $X_n$, our main result is a full proof of the following statement due
to P. Le Doussal, C. Monthus and D. S. Fisher: $$\lim_{\eta\to0}
\lim_{n\to\infty} \PP^o (\frac{|X_{n^h} - X_n|}{(\log n)^2} < \eta) =
\frac{1}{h^2} [ {5/3} - {2/3} e^{-(h-1)} ]. $$

amir@stat.stanford.edu

• This article is available from the xxx mathematics archive as math.PR/0105215.

1735. MATHEMATICS OF LEARNING

Natalia Komarova and Igor Rivin

We study the convergence properties of a pair of learning algorithms
(learning with and without memory). This leads us to study the dominant
eigenvalue of a class of random matrices. This turns out to be related to the
roots of the derivative of random polynomials (generated by picking their roots
uniformly at random in the interval [0, 1], although our results extend to
other distributions). This, in turn, requires the study of the statistical
behavior of the harmonic mean of random variables as above, which leads us to
delicate question of the rate of convergence to stable laws and tail estimates
for stable laws. The reader can find the proofs of most of the results
announced here in the paper entitled "Harmonic mean, random polynomials, and
random matrices", by the same authors.

irivin@math.princeton.edu

• This article is available from the xxx mathematics archive as math.PR/0105235.

1736. HARMONIC MEAN, RANDOM POLYNOMIALS AND STOCHASTIC MATRICES

Natalia Komarova and Igor Rivin

Motivated by a problem in learning theory, we are led to study the dominant
eigenvalue of a class of random matrices. This turns out to be related to the
roots of the derivative of random polynomials (generated by picking their roots
uniformly at random in the interval [0, 1], although our results extend to
other distributions). This, in turn, requires the study of the statistical
behavior of the harmonic mean of random variables as above, and that, in turn,
leads us to delicate question of the rate of convergence to stable laws and
tail estimates for stable laws.

irivin@math.princeton.edu

• This article is available from the xxx mathematics archive as math.PR/0105236.

1737. SCALE INVARIANCE OF THE PNG DROPLET AND THE AIRY PROCESS

Michael Praehofer, Herbert Spohn

We establish that the static height fluctuations of a particular growth
model, the PNG droplet, converges upon proper rescaling to a limit process,
which we call the Airy process, A(y). The Airy process is stationary, it has
continuous sample paths, its single "time" (fixed y) distribution is the
Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and
the Airy process has a slow decay of correlations as y^(-2). Roughly the Airy
process describes the last line of Dyson's Brownian motion model for random
matrices. Our construction uses a multi--layer version of the PNG model, which
can be analyzed through fermionic techniques. Specializing our result to a
fixed value of y, one reobtains the celebrated result of Baik, Deift, and
Johansson on the length of the longest increasing subsequence of a random
permutation.

praehofe@mathematik.tu-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/0105240.

1738. APPROXIMATING THE LIMITING QUICKSORT DISTRIBUTION

James Allen Fill and Svante Janson 

The limiting distribution of the normalized number of comparisons used by
Quicksort to sort an array of n numbers is known to be the unique fixed point
with zero mean of a certain distributional transformation S. We study the
convergence to the limiting distribution of the sequence of distributions
obtained by iterating the transformation S, beginning with a (nearly) arbitrary
starting distribution. We demonstrate geometrically fast convergence for
various metrics and discuss some implications for numerical calculations of the
limiting Quicksort distribution. Finally, we give companion lower bounds which
show that the convergence is not faster than geometric.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0105246.

1739. QUICKSORT ASYMPTOTICS

James Allen Fill and Svante Janson 

The number of comparisons X_n used by Quicksort to sort an array of n
distinct numbers has mean mu_n of order n log n and standard deviation of order
n. Using different methods, Regnier and Roesler each showed that the normalized
variate Y_n := (X_n - mu_n) / n converges in distribution, say to Y; the
distribution of Y can be characterized as the unique fixed point with zero mean
of a certain distributional transformation.
  We provide the first rates of convergence for the distribution of Y_n to that
of Y, using various metrics. In particular, we establish the bound 2 n^{- 1 /
2} in the d_2-metric, and the rate O(n^{epsilon - (1 / 2)}) for
Kolmogorov-Smirnov distance, for any positive epsilon.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0105248.

1740. EXTENSION OF FILL'S PERFECT REJECTION SAMPLING ALGORITHM TO GENERAL CHAINS

James Allen Fill, Motoya Machida, Duncan J. Murdoch, Jeffrey S. Rosenthal

By developing and applying a broad framework for rejection sampling using
auxiliary randomness, we provide an extension of the perfect sampling algorithm
of Fill (1998) to general chains on quite general state spaces, and describe
how use of bounding processes can ease computational burden. Along the way, we
unearth a simple connection between the Coupling From The Past (CFTP) algorithm
originated by Propp and Wilson (1996) and our extension of Fill's algorithm.

machida@math.usu.edu

• This article is available from the xxx mathematics archive as math.PR/0105252.

1741. MULTI-SCALE HOMOGENIZATION WITH BOUNDED RATIOS AND ANOMALOUS SLOW DIFFUSION

G\'{e}rard Ben-Arous and Houman Owhadi

This paper is concerned with the asymptotic behavior of effective diffusivity
matrices $D(V_0^n)$, associated to media characterized by an arbitrarily large
number of scales but with ratios bounded independently from their numbers,
$V_0^n=\sum_{k=0}^n U_k(x/R_k)$, where $U_k$ are Holder-continuous functions of
period $T^d_1$ (torus of dimension $d\geq 1$ and side 1), $U_k(0)=0$ and $R_k$
grows exponentially fast with $k$ but has bounded ratios
$\sup_{k}R_{k+1}/R_k<\infty$. 
  $$ ^tlD(V_0^n)l=\inf_{f\in C^\infty(T^d_{R_n})}\int_{T^d_{R_n}}|l-\nabla
f(x)|^2 e^{-2 V_0^n(x)}dx\Big/\int_{T^d_{R_n}} e^{-2 V_0^n(x)}dx $$
  We obtain quantitative estimates on $D(V_0^n)$, putting into evidence its
geometric rate of convergence towards 0. From this we deduce the anomalous slow
behavior of solutions of $dy_t=d\omega_t -\nabla V_0^\infty(y_t) dt$ using the
tools of homogenization.

owhadi@techunix.technion.ac.il

• This article is available from the xxx mathematics archive as math.PR/0105258.

1742. EXTENSION OF FILL'S PERFECT REJECTION SAMPLING ALGORITHM TO GENERAL CHAINS (EXTENDED ABSTRACT)

James Allen Fill, Motoya Machida, Duncan J. Murdoch, Jeffrey S. Rosenthal

We provide an extension of the perfect sampling algorithm of Fill (1998) to
general chains, and describe how use of bounding processes can ease
computational burden. Along the way, we unearth a simple connection between the
Coupling From The Past (CFTP) algorithm originated by Propp and Wilson (1996)
and our extension of Fill's algorithm.

machida@math.usu.edu

• This article is available from the xxx mathematics archive as math.PR/0106002.

1743. ON THE SIMPLEST SPLIT-MERGE OPERATOR ON THE INFINITE-DIMENSIONAL SIMPLEX

Natalia Tsilevich

We consider the simplest split-merge Markov operator $T$ on the
infinite-dimensional simplex $\Sigma_1$ of monotone non-negative sequences with
unit sum. For a sequence $x\in\Sigma_1$, it picks a size-biased sample (with
replacement) of two elements of $x$; if these elements are distinct, it merges
them and reorders the sequence, and if the same element is picked twice, it
splits this element uniformly into two parts and reorders the sequence. We
prove that the means along the $T$-trajectory of the $\de$-measure at the
vector $(1,0,0,{...})$ converge to the Poisson--Dirichlet distribution PD(1).

natalia@pdmi.ras.ru

• This article is available from the xxx mathematics archive as math.PR/0106005.

1744. PERCOLATION ON FINITE GRAPHS

Itai Benjamini

The asymptotic study of percolation on finite transitive graphs is
considered. Several questions and very few answers regarding percolation on
finite graphs are presented.

itai@wisdom.weizmann.ac.il

• This article is available from the xxx mathematics archive as math.PR/0106022.

1745. BASIC PROPERTIES OF SLE

Steffen Rohde and Oded Schramm

SLE is a random growth process based on Loewner's equation with driving
parameter a one-dimensional Brownian motion running with speed $\kappa$. This
process is intimately connected with scaling limits of percolation clusters and
with the outer boundary of Brownian motion, and is conjectured to correspond to
scaling limits of several other discrete processes in two dimensions.
  The present paper attempts a first systematic study of SLE. It is proved that
for all $\kappa\ne 8$ the SLE trace is a path; for $\kappa\in[0,4]$ it is a
simple path; for $\kappa\in(4,8)$ it is a self-intersecting path; and for
$\kappa>8$ it is space-filling.
  It is also shown that the Hausdorff dimension of the SLE trace is a.s. at
most $1+\kappa/8$ and that the expected number of disks of size $\eps$ needed
to cover it inside a bounded set is at least $\eps^{-(1+\kappa/8)+o(1)}$ for
$\kappa\in[0,8)$ along some sequence $\eps\to 0$. Similarly, for $\kappa\ge 4$,
the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most
$1+2/\kappa$, and the expected number of disks of radius $\eps$ needed to cover
it is at least $\eps^{-(1+2/\kappa)+o(1)}$ for a sequence $\eps\to 0$.

schramm@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0106036.

1746. TAILS OF PROBABILITY DENSITY FOR SUMS OF RANDOM INDEPENDENT VARIABLES

Michael I.Tribelsky

The exact expression for the probability density $p_{_N}(x)$ for sums of a
finite number $N$ of random independent terms is obtained. It is shown that the
very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random
terms are distributed according to the Gauss Law. In all other cases the tail
for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms
diverge the non-Gaussian tail is related to a Levy distribution for
$p_{_N}(x)$. However, the tail is not Gaussian even if the variances are
finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At small
and moderate values of $x$ the distribution is Gaussian. At large $x$ the
non-Gaussian tail arises. The crossover between the two asymptotics occurs at
$x$ proportional to $N$. For this reason the non-Gaussian tail exists at finite
$N$ only. In the limit $N$ tends to infinity the origin of the tail is shifted
to infinity, i. e., the tail vanishes. Depending on the particular type of the
distribution of the random terms the non-Gaussian tail may decay either slower
than the Gaussian, or faster than it. A number of particular examples is
discussed in detail.

tribel@scroll.apphy.fukui-u.ac.jp

• This article is available from the xxx mathematics archive as math.PR/0106037.

1747. A DUALITY METHOD IN PREDICTION THEORY OF MULTIVARIATE STATIONARY SEQUENCES

Michael Frank, Lutz P. Klotz

Let W be an integrable positive Hermitian q x q -matrix valued function on
the dual group of a discrete abelian group G such that W^{-1} is integrable.
Generalizing results of T. Nakazi and of A. G. Miamee and M. Pourahmadi for q=1
we establish a correspondence between trigonometric approximation problems in
L^2(W) and certain approximation problems in L^2(W^{-1}). The result is applied
to prediction problems for q-variate stationary processes over G, in
particular, to the case where G is the group of integers Z.

frank@mathematik.uni-leipzig.de

• This article is available from the xxx mathematics archive as math.PR/0106056.

1748. REARRANGEMENT INVARIANT NORMS OF SYMMETRIC SEQUENCE NORMS OF INDEPENDENT SEQUENCES OF RANDOM VARIABLES

Stephen Montgomery-Smith

Let X_1, X_2,..., X_n be a sequence of independent random variables, let M be
a rearrangement invariant space on the underlying probability space, and let N
be a symmetric sequence space. This paper gives an approximate formula for the
quantity || ||(X_i)||_N ||_M whenever L_q embeds into M for some 1 le q <
infty. This extends work of Johnson and Schechtman who tackled the cases when N
= l_1 or N = l_2, and recent work of Gordon, Litvak, Schuett and Werner who
obtained similar results for Orlicz spaces.

stephen@math.missouri.edu

• This article is available from the xxx mathematics archive as math.PR/0106114.

1749. FINITE DIMENSIONAL REALIZATIONS OF STOCHASTIC EQUATIONS

Damir Filipovic, Josef Teichmann

This paper discusses finite-dimensional (Markovian) realizations (FDRs) for
Heath-Jarrow-Morton interest rate models. We consider a d-dimensional
driving Brownian motion and stochastic volatility structures that are
non-degenerate smooth functionals of the current forward rate. In a recent
paper, Bj\"ork and Svensson give sufficient and necessary conditions for the
existence of FDRs within a particular Hilbert space setup. We extend their
framework, provide new results on the geometry of the implied FDRs and classify
all of them. In particular, we prove their conjecture that every short rate
realization is 2-dimensional. More generally, we show that all generic FDRs are
at least (d+1)-dimensional and that all generic FDRs are affine. As an
illustration we sketch an interest rate model, which goes well with the
Svensson curve-fitting method. These results cannot be obtained in the
Bj\"ork-Svensson setting.
  A substantial part of this paper is devoted to analysis on Fr\'echet spaces,
where we derive a Frobenius theorem. Though we only consider stochastic
equations in the HJM-framework, many of the results carry over to a more
general setup.

josef.teichmann@fam.tuwien.ac.at

• This article is available from the xxx mathematics archive as math.PR/0106155.

1750. HOW TO COMBINE FAST HEURISTIC MARKOV CHAIN MONTE CARLO WITH SLOW EXACT SAMPLING

David J. Aldous, Antar Bandyopadhyay 

Use each of n exact samples as the initial state for a MCMC sampler run for m
steps. We give confidence intervals for accuracy of estimators which are always
valid and which, in certain settings, are almost as good as the intervals one
would obtain if the (unknown) mixing time of the chain were known.

aldous@stat.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0106159.

1751. THE POISSON DENSITY

John L. Haller Jr

Describing the Poisson process in both a mathematical and physical system,
the sequence is shown to be a broadcast of information where each observer who
whitnesses an event can obtain 2log(e) of classical information. With
evididence generated by the martingale approach, multiple pathways are used to
derive this result. Further proof of the correctness of the equality between
the counting of the number of events in the Poisson Process and the measure of
information shifted inside the bounds of the process, is suggested to be found
in the actual data that is collected from the Vacuum tube.

jlhaller@stanford.edu

• This article is available from the xxx mathematics archive as math.PR/0106215.

1752. CHANGE INTOLERANCE IN SPANNING FORESTS

Deborah Heicklen and Russell Lyons

Call a percolation process on edges of a graph change 
intolerant if the status of each edge is almost surely 
determined by the status of the other edges. We give 
necessary and sufficient conditions for change
intolerance of the wired spanning forest when the 
underlying graph is a spherically symmetric tree. 

rdlyons@indiana.edu

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

1753. PARTICLE APPROXIMATIONS OF LYAPUNOV EXPONENTS CONNECTED TO SCHRODINGER OPERATORS AND FEYNMAN-KAC SEMIGROUPS.

Pierre Del Moral and Laurent Miclo

We present an interacting particle system methodology for 
the numerical solving of the Lyapunov exponent of 
Feynman-Kac semigroups and for estimating the principal  
eigenvalue of Schrodinger generators. 
The continuous or discrete time models studied in this 
work consists of N interacting particles evolving in an 
environment with soft obstacles related to a potential 
function V. 
These models are related to genetic algorithms and Moran 
type particle schemes. Their choice is not unique. We will 
examine a class of models extending the hard obstacle model  
of  K. Burdzy, R. Holyst and P. March and including the 
Moran type scheme presented by the authors in a previous 
work. We provide precise uniform estimates with respect 
to the time parameter and we analyse the fluctuations 
of continuous time particle models.        

delmoral@cict.fr

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

1754. CHARACTERIZATION OF STATIONARY MEASURES FOR ONE DIMENSIONAL EXCLUSION PROCESSES

Maury Bramson, Thomas M. Liggett and Thomas Mountford

The product Bernoulli measures $\nu_\alpha$ with densities
$\alpha$, $\alpha\in [0,1]$, are the extremal translation invariant
stationary measures for an exclusion process on $\Bbb Z$ with irreducible
random walk kernel $p(\cdot)$.  Stationary measures that are not
translation invariant are known to exist for finite range $p(\cdot)$ with
positive mean.  These measures have particle densities that tend to 1 as
$x\to\infty$ and tend to $0$ as $x\to -\infty$; the corresponding extremal
measures form a one parameter family and are translates of one another.
Here, we show that for an exclusion process where $p(\cdot)$ is irreducible
and has positive mean, there are no other extremal stationary measures.
When $\Sigma_{x<0} x^2 p(x) =\infty$, we show that any nontranslation
invariant stationary measure is not a blocking measure, i.e., there are
always either an infinite number of particles to the left of any site or an
infinite number of empty sites to the right of the site.  This contrasts
with the case where $p(\cdot)$ has finite range and the above stationary
measures are all blocking measures. We also present two results on the
existence of blocking measures when $p(\cdot)$ has positive mean, and
$p(y)\leq p(x)$ and $p(-y)\leq p(-x)$ for $1\leq x\leq y$.  When the left
tail of $p(\cdot)$ has slightly more than a third moment, stationary
blocking measures exist.  When $p(-x)\leq p(x)$ for $x>0$ and
$\Sigma_{x<0}x^2p(x)<\infty$, stationary blocking measures also exist. 

tml@math.ucla.edu

1755. GENERALIZED COVARIATIONS, LOCAL TIME AND STRATONOVICH ITÔ's FORMULA FOR FRACTIONAL BROWNIAN MOTION WITH HURST INDEX GREATER OR EQUAL THAN 1/4

Mihai Gradinaru, Francesco Russo and Pierre Vallois

Given a locally bounded real function g, we examine the 
existence of a 4-covariation [g(B),B,B,B], where B is a
fractional Brownian motion with Hurst index H greater or 
equal than 1/4. We provide two essential applications.
First, in the case H = 1/4, we relate the mentioned 
covariation to one expression involving the derivative of 
local time, generalizing an identity of Bouleau-Yor type,
 well-known for the case of classical Brownian motion.
A second application is an Itô's formula of Stratonovich 
type for f(B). The main difficulty comes from the fact B 
has only a finite 4-variation.

gradinar@iecn.u-nancy.fr  russo@math.univ-paris13.fr   vallois@iecn.u-nancy.fr

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

1756. TRANSITION PROBABILITIES FOR SYMMETRIC JUMP PROCESSES

Richard Bass and David Levin

We consider symmetric Markov chains on the $d$-dimensional
integer lattice, where $\alpha\in (0,2)$ and the
conductance between $x$ and $y$ is comparable to
$|x-y|^{-d-\alpha}$. We establish upper and lower bounds 
on the transition probabilities that are sharp up to
constants.

bass@math.uconn.edu  levin@math.uconn.edu

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

1757. BROWNIAN ANALOGUES OF BURKE'S THEOREM

Neil O'Connell and Marc Yor

We discuss Brownian analogues of a celebrated theorem, due 
to Burke, which states that the output of a (stable, 
stationary) M/M/1 queue is Poisson, and the related notion 
of quasireversibility.  A direct analogue of Burke's theorem 
for the Brownian queue was stated and proved by Harrison and 
Williams (1985).  We present several different proofs of 
this and related results.  We also present an analogous 
result for geometric functionals of Brownian motion.  
By considering series of queues in tandem, these theorems 
can be applied to a certain class of directed percolation 
and directed polymer models. It was recently discovered that 
there is a connection between this directed percolation 
model and the GUE random matrix ensemble.  We extend and 
give a direct proof of this connection in the two 
dimensional case. In all of the above, reversibility plays 
a key role.  

noc@hplb.hpl.hp.com

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1758. A REPRESENTATION FOR NON-COLLIDING RANDOM WALKS

Neil O'Connell and Marc Yor

We define a sequence of mappings $\Gamma_k:D_0(\R_+)^k\to 
D_0(\R_+)^k$ and prove the following result:
Let $N_1,\ldots,N_n$ be the counting functions of
independent Poisson processes on $\R_+$ with
respective intensities $\mu_1<\mu_2<\cdots <\mu_n$.
The conditional law of $N_1,\ldots,N_n$, given that
$$N_1(t)\le\cdots\le N_n(t),\ \mbox{ for all }t\ge 0,$$
is the same as the unconditional law of $\Gamma_n(N)$.
>From this, we deduce the corresponding results for 
independent Poisson processes of equal rates and for 
independent Brownian motions (in both of these cases the 
conditioning is in the sense of Doob).  This extends a 
recent observation, independently due to Baryshnikov (2001) 
and Gravner, Tracy and Widom (2001), which relates the
law of a certain functional of Brownian motion to that  
of the largest eigenvalue of a GUE random matrix.  Our
main result can also be regarded as a generalisation of 
Pitman's representation for the 3-dimensional Bessel process.

noc@hplb.hpl.hp.com

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1759. NON-COLLIDING RANDOM WALKS, TANDEM QUEUES, AND DISCRETE ORTHOGONAL POLYNOMIAL ENSEMBLES

Wolfgang Koenig, Neil O'Connell and Sebastien Roch

We show that the function $h(x)=\prod_{i< j}(x_j-x_i)$ is 
harmonic for any random walk in $\R^k$ with exchangeable
increments, provided the required moments exist. For the 
subclass of random walks which can only exit the Weyl 
chamber $W=\{x\colon x_1< x_2< \cdots< x_k\}$ onto a point 
where $h$ vanishes, we define the corresponding Doob 
$h$-transform. For certain special cases, we show that the 
marginal distribution of the conditioned process at a fixed 
time is given by a familiar discrete orthogonal polynomial 
ensemble. These include the Krawtchouk and Charlier ensembles, 
where the underlying walks are binomial and Poisson, 
respectively. We refer to the corresponding conditioned 
processes in these cases as the Krawtchouk and Charlier 
processes. In [O'Connell and Yor (2001)], a representation 
is obtained for the Charlier process by considering a 
sequence of $M/M/1$ queues in tandem. We present the 
analogue of this representation theorem for the Krawtchouk 
process, by considering a sequence of discrete-time $M/M/1$ 
queues in tandem. We also present related results for random 
walks on the circle, and relate a system of non-colliding 
walks in this case to the discrete analogue of the circular 
unitary ensemble (CUE). 

koenig@math.tu-berlin.de noc@hplb.hpl.hp.com sebastien.roch@polytechnique.org

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• Or here.

1760. A NON-MARKOVIAN VERSION OF PITMAN'S 2M-X THEOREM

B.M. Hambly, James Martin and Neil O'Connell

Let $(\xi_k,\ k\ge 0)$ be a Markov chain on $\{-1,+1\}$ with 
$\xi_0=1$ and transition probabilities $P(\xi_{k+1}=1|\
\xi_k=1)=a$ and $P(\xi_{k+1}=-1|\ \xi_k=-1)=b< a$. Set $X_0=0$,
$X_n=\xi_1+\cdots +\xi_n$ and $M_n=\max_{0\le k\le n}X_k$. 
We prove that the process $2M-X$ has the same law as that of 
$X$ conditioned to stay non-negative. 

noc@hplb.hpl.hp.com

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