[Pas] Probability Abstracts 94
pas at www2.economia.unimi.it
pas at www2.economia.unimi.it
Sun Oct 1 06:46:01 CEST 2006
Oct 1st, 2006
Letter 94
Probability Abstract Service
Abstracts from Aug-1-2006 to Set-30-2006
html version here: http://www2.economia.unimi.it/PAS/Letters/
letter_94.shtml
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4514. AN ISOPERIMETRIC INEQUALITY ON THE ELL_P BALLS
Sasha Sodin
The normalised volume measure on the $ell_p^n$ unit ball (for p
between 1 and
2) satisfies the following isoperimetric inequality: the boundary
measure of a
set of measure $a$ is at least $c n^1/p a' log^{1-1/p} (1/a')$, where
$a' =
min(a, 1 - a)$.
http://front.math.ucdavis.edu/math.PR/0607398
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4515. CONVERGENCE RATES OF RANDOM WALK ON IRREDUCIBLE REPRESENTATIONS
OF FINITE GROUPS
Jason Fulman
Random walk on the set of irreducible representations of a finite
group is
investigated. For the symmetric and general linear groups, a sharp
convergence
rate bound is obtained and a cutoff phenomenon is proved. As a
related result,
an asymptotic description of Plancherel measure of the finite general
linear
groups is given.
http://front.math.ucdavis.edu/math.PR/0607399
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4516. MIRROR COUPLINGS AND NEUMANN EIGENFUNCTIONS
Rami Atar and Krzysztof Burdzy
We analyze a pair of reflected Brownian motions in a planar domain $D
$, for
which the increments of both processes form mirror images of each
other when
the processes are not on the boundary. We show that for $D$ in a
class of
smooth convex planar domains, the two processes remain ordered forever,
according to a certain partial order.
This is used to prove that the second eigenvalue is simple for the
Laplacian
with Neumann boundary conditions for the same class of domains.
http://front.math.ucdavis.edu/math.PR/0607400
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4517. LAW OF LARGE NUMBERS FOR PRODUCTS OF RANDOM MATRICES WITH
COEFFICIENTS IN THE MAX-PLUS SEMI-RING
Glenn Merlet (IRMAR)
We analyze the asymptotic behavior of random variables $x(n,x\_0)$
defined by
$x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A(n)x(n,x\_0)$, where $\sAn$ is a
stationary
and ergodic sequence of random matrices with entries in the semi-ring
\mbox{$\R\cup\{-\infty\}$} whose addition is the $\max$ and whose
multiplication is $+$. Such sequences modelize a large class of
discrete event
systems, among which timed event graphs, 1-bounded Petri nets, some
queuing
networks, train or computer networks. We give necessary conditions for
$(\frac{1}{n}x(n,x\_0))\_{n\in\N}$ to converge almost surely. Then,
we prove a
general scheme to give partial converse theorems. When $\max\_{A\_{ij}
(0)\neq
-\infty}|A\_{ij}(0)|$ is integrable, it allows us: - to give a
necessary and
sufficient condition for the convergence of $(\frac{1}{n}x(n,0))\_{n
\in\N}$
when the sequence $(A(n))\_{n\in\N}$ is i.i.d., - to prove that, if $
(A(n)
)\_{n\in\N}$ satisfy a condition of reinforced ergodicity and a
condition of
fixed structure (i.e. $\P(A\_{ij}(0)=-\infty)\in\{0,1\}$), then
$(\frac{1}{n}x(n,0))\_{n\in\N}$ converges almost-surely, - and to
reprove the
convergence of $(\frac{1}{n}x(n,0))\_{n\in\N}$ if the diagonal
entries are
never $-\infty$.
http://front.math.ucdavis.edu/math.PR/0607406
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4518. FEEDBACK STABILIZATION FOR OSEEN FLUID EQUATIONS:A STOCHASTIC
APPROACH
Jinqiao Duan and Andrei V. Fursikov
The authors consider stochastic aspects of the stabilization problem
for two
and three-dimensional Oseen equations with help of feedback control
defined on
a part of the fluid boundary. Stochastic issues arise when inevitable
unpredictable fluctuations in numerical realization of stabilization
procedures
are taken into account and they are supposed to be independent
identically
distributed random variables. Under this assumption the solution to the
stabilization problem obtained via boundary feedback control can be
described
by a Markov chain or a discrete random dynamical system. It is shown
that this
random dynamical system possesses a unique, exponentially attracting,
invariant
measure, namely, this random dynamical system is ergodic. This gives
adequate
statistical description of the stabilization process on the stage when
stabilized solution has to be retained near zero (i.e. near unstable
state of
equilibrium).
http://front.math.ucdavis.edu/math.AP/0607429
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4519. ATTRACTORS AND TIME AVERAGES FOR RANDOM MAPS
Vitor Araujo
Considering random noise in finite dimensional parameterized families of
diffeomorphisms of a compact finite dimensional boundaryless manifold
M, we
show the existence of time averages for almost every orbit of each
point of M,
imposing mild conditions on the families. Moreover these averages are
given by
a finite number of physical absolutely continuous stationary probability
measures.
We use this result to deduce that situations with infinitely many
sinks and
Henon-like attractors are not stable under random perturbations, e.g.,
Newhouse's and Colli's phenomena in the generic unfolding of a quadratic
homoclinic tangency by a one-parameter family of diffeomorphisms.
http://front.math.ucdavis.edu/math.DS/0607433
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4520. INFINITELY MANY STOCHASTICALLY STABLE ATTRACTORS
Vitor Araujo
Let f be a diffeomorphism of a compact finite dimensional boundaryless
manifold M exhibiting infinitely many coexisting attractors. Assume
that each
attractor supports a stochastically stable probability measure and
that the
union of the basins of attraction of each attractor covers Lebesgue
almost all
points of M. We prove that the time averages of almost all orbits
under random
perturbations are given by a finite number of probability measures.
Moreover
these probability measures are close to the probability measures
supported by
the attractors when the perturbations are close to the original map f.
http://front.math.ucdavis.edu/math.DS/0607434
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4521. ESTIMATES AND STRUCTURE OF $\ALPHA$-HARMONIC FUNCTIONS
Krzysztof Bogdan and Tadeusz Kulczycki and Mateusz Kwa\'snicki
We prove a uniform boundary Harnack inequality for nonnegative harmonic
functions of the fractional Laplacian on arbitrary open set $D$. This
yields a
unique representation of such functions as integrals against measures on
$D^c\cup \{\infty\}$ satisfying an integrability condition. The
corresponding
Martin boundary of $D$ is a subset of the Euclidean boundary
determined by an
integral test.
http://front.math.ucdavis.edu/math.PR/0607561
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4522. THE TOPOLOGICAL STRUCTURE OF SCALING LIMITS OF LARGE PLANAR MAPS
Jean-Francois Le Gall
We discuss scaling limits of large bipartite planar maps. If p is a
fixed
integer strictly greater than 1, we consider a random planar map M(n)
which is
uniformly distributed over the set of all 2p-angulations with n
faces. Then, at
least along a suitable subsequence, the metric space M(n) equipped
with the
graph distance rescaled by the factor n to the power -1/4 converges in
distribution as n tends to infinity towards a limiting random compact
metric
space, in the sense of the Gromov-Hausdorff distance. We prove that the
topology of the limiting space is uniquely determined independently
of p, and
that this space can be obtained as the quotient of the Continuum
Random Tree
for an equivalence relation which is defined from Brownian labels
attached to
the vertices. We also verify that the Hausdorff dimension of the
limit is
almost surely equal to 4.
http://front.math.ucdavis.edu/math.PR/0607567
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4523. A CHARACTERIZATION OF THE SET-INDEXED FRACTIONAL BROWNIAN
MOTION BY INCREASING PATHS
Erick Herbin and Ely Merzbach
We prove that a set-indexed process is a set-indexed fractional Brownian
motion if and only if its projections on all the increasing paths are
one-parameter time changed fractional Brownian motions. As an
application, we
present an integral representation for such processes.
http://front.math.ucdavis.edu/math.PR/0607575
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4524. STATIONARY SYMMETRIC ALPHA-STABLE DISCRETE PARAMETER RANDOM FIELDS
Parthanil Roy and Gennady Samorodnitsky
We establish a connection between the structure of a stationary
symmetric
alpha-stable random field (0 < alpha < 2) and ergodic theory of non-
singular
group actions, elaborating on a previous work by Rosinski (2000).
With the help
of this connection, we study the extreme values of the field over
increasing
boxes. Depending on the ergodic theoretical and group theoretical
structures of
the underlying action, we observe different kinds of asymptotic
behavior of
this sequence of extreme values.
http://front.math.ucdavis.edu/math.PR/0607587
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4525. STOCHASTIC GEOMETRY OF CRITICAL CURVES, SCHRAMM-LOEWNER
EVOLUTIONS, AND CONFORMAL FIELD THEORY
Ilya A. Gruzberg
Conformally-invariant curves that appear at critical points in
two-dimensional statistical mechanics systems, and their fractal
geometry have
received a lot of attention in recent years. On the one hand, Schramm
has
invented a new rigorous as well as practical calculational approach
to critical
curves, based on a beautiful unification of conformal maps and
stochastic
processes, and by now known as Schramm-Loewner evolution (SLE). On
the other
hand, Duplantier has applied boundary quantum gravity methods to
calculate
exact multifractal exponents associated with critical curves.
In the first part of this paper I provide a pedagogical
introduction to SLE.
I present mathematical facts from the theory of conformal maps and
stochastic
processes related to SLE. Then I review basic properties of SLE and
provide
practical derivation of various interesting quantities related to
critical
curves, including fractal dimensions and crossing probabilities.
The second part of the paper is devoted to a way of describing
critical
curves using boundary conformal field theory (CFT) in the so-called
Coulomb gas
formalism. This description provides an alternative (to quantum
gravity) way of
obtaining the multifractal spectrum of critical curves using only
traditional
methods of CFT based on free bosonic fields.
http://front.math.ucdavis.edu/math-ph/0607046
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4526. CHARACTERIZATION OF THE OPTIMAL PLANS FOR THE MONGE-
KANTOROVICH TRANSPORT PROBLEM
Christian L\'{e}onard (MODAL'X and CMAP)
We present a general method, based on conjugate duality, for solving
a convex
minimization problem without assuming unnecessary topological
restrictions on
the constraint set. It leads to dual equalities and characterizations
of the
minimizers without constraint qualification. As an example of
application, the
Monge-Kantorovich optimal transport problem is solved in great
detail. In
particular, the optimal transport plans are characterized without
restriction.
This characterization improves the already existing literature on the
subject.
http://front.math.ucdavis.edu/math.OC/0607604
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4527. PATHWISE ASYMPTOTIC BEHAVIOR OF RANDOM DETERMINANTS IN THE
JACOBI ENSEMBLE
Alain Rouault (LM-Versailles)
This is a companion paper of arxiv math.PR/050921. It concentrates on
asymptotic properties of determinants of random matrices in the Jacobi
ensemble. Let $M \in {\cal M}\_{n\_1 + n\_2,r}(`R)$ (with $r \leq n
\_1 + n\_2$)
be a matrix whose entries are standard i.i.d. Gaussian. If $M^T = (M
\_1^T,
M\_2^T)$ with $M\_1 \in {\cal M}\_{n\_1,r}$ and $M\_2 \in {\cal M}\_{n
\_2,r}$,
then, $W\_1 := M\_1^T M\_1$ and $W\_2 := M\_2^T M\_2$ are independent
$r\times
r$ Wishart matrices with parameters $n\_1$ and $n\_2$ and $M^T M = W
\_1 + W\_2$
is Wishart with parameter $n\_1+ n\_2$. Then ${\cal Z} := (W\_1 + W
\_2)^{-1/2}
W\_1 (W\_1 + W\_2)^{-1/2}$ has a Beta matrix variate distribution with
parameters $n\_1/2, n\_2/2$ (sometimes called the Jacobi
distribution). We set
$n\_1 = \lfloor n\tau\_1 \rfloor$, $n\_2 = \lfloor n\tau\_2 \rfloor$,
$r=
\lfloor nt\rfloor$ $t\in [0, \tau\_1)$ and let $n \to \infty$; we
define ${\cal
Z}\_n (t)$ as the corresponding matrix and $\Theta\_n (t) := |{\cal Z}
\_n(t)|$
as its determinant. In the Jacobi ensemble, the Kshirsagar's theorem
decomposes
$\Theta\_n (t)$ into a product of independent beta distributed
variables. This
allows us to study the process $\frac{1}{n} (n^{-1} \log \Theta\_n
(t), t \in
[0,\tau\_1))$ and the asymptotic behavior of the sequence $\{\frac{1}{n}
n^{-1}\log \Theta\_n \}\_n$ as $n\to \infty$ with $\tau\_1$ and $\tau
\_2$ fixed
: a.s. convergence, fluctuations, large deviations. We connect the
results for
marginals (fixed $t$) with those obtained by the study of the empirical
spectral distribution. In the whole paper, we consider the problem of
general
$\beta$, where the particular cases $\beta = 1,2,4$ correspond to real,
complex, and quaternionic matrices.
http://front.math.ucdavis.edu/math.PR/0607767
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4528. MEAN-VARIANCE HEDGING IN THE DISCONTINUOUS CASE
Jianming Xia
The results on the mean-variance hedging problem in Gouri\'eroux,
Laurent and
Pham (1998), Rheinl\"ander and Schweizer (1997) and Arai (2005) are
extended to
discontinuous semimartingale models. When the num\'eraire method is
used, we
only assume the Radon-Nikodym derivative of the variance-optimal signed
martingale measure (VSMM) is non-zero almost surely (but may be strictly
negative). When discussing the relation between the solutions and the
Galtchouk-Kunita-Watanabe decompositions under the VSMM, we only
assume the
VSMM is equivalent to the reference probability.
http://front.math.ucdavis.edu/math.PR/0607775
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4529. TRANSLATED POISSON APPROXIMATION USING EXCHANGEABLE PAIR COUPLINGS
Adrian R\"ollin
It is shown that the method of exchangeable pairs introduced by Stein
(1986)
for normal approximation can effectively be used for translated Poisson
approximation. Introducing an additional smoothness condition, one
can obtain
approximation results in total variation and also in a local limit
metric. The
result is applied in particular to the anti-voter model on finite
graphs as
analysed by Rinott and Rotar (1997), obtaining the same rate of
convergence,
but now for a stronger metric.
http://front.math.ucdavis.edu/math.PR/0607781
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4530. STOCHASTIC STOKES' DRIFT OF A FLEXIBLE DUMBBELL
Kalvis M. Jansons
We consider the stochastic Stokes' drift of a flexible dumbbell. The
dumbbell
consists of two isotropic Brownian particles connected by a linear
string with
zero natural length, and is advected by a sinusoidal wave. We find an
asymptotic approximation for the Stokes' drift in the limit of a weak
wave, and
find good agreement with the results of a Monte Carlo simulation.
Interestingly, the dependence of the Stokes' drift on the strength of
the
spring is not monotonic.
http://front.math.ucdavis.edu/math.PR/0607797
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4531. ODE METHODS FOR SKIP-FREE MARKOV CHAIN STABILITY WITH
APPLICATIONS TO MCMC
Gersende Fort (TSI) and Sean Meyn and Eric Moulines (TSI) and
Pierre Priouret (PMA)
Fluid limit techniques have become a central tool to analyze queueing
networks over the last decade, with applications to performance
analysis,
simulation, and optimization. In this paper some of these techniques are
extended to a general class of skip-free Markov chains. As in the
case of
queueing models, a fluid approximation is obtained by scaling time,
space, and
the initial condition by a large constant. The resulting fluid limit
is the
solution of an ordinary differential equation (ODE) in ``most'' of
the state
space. Stability and finer ergodic properties for the stochastic
model then
follow from stability of the set of fluid limits. Moreover, similar
to the
queueing context where fluid models are routinely used to design control
policies, the structure of the limiting ODE in this general setting
provides an
understanding of the dynamics of the Markov chain. These results are
illustrated through application to Markov Chain Monte Carlo.
http://front.math.ucdavis.edu/math.PR/0607800
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4532. ISOPERIMETRIC INEQUALITIES AND MIXING TIME FOR A RANDOM WALK ON
A RANDOM POINT PROCESS
P. Caputo and A. Faggionato
We consider the random walk on a simple point process on R^d, d>1,
whose jump
rates decay exponentially in the A-power of jump length. The case A=1
corresponds to the phonon-induced variable-range hopping in
disordered solids
in the regime of strong Anderson localization. Under mild assumptions
on the
point process, we show for A in (0,d) that the random walk confined
to a cubic
box of side L has a.s. Cheeger constant of order at least L^{-1} and
mixing
time of order L^2. For the Poisson point process we prove that at A=d
there is
a transition from diffusive to subdiffusive behavior of the random walk.
http://front.math.ucdavis.edu/math.PR/0607805
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4533. TUG OF WAR WITH NOISE: A GAME THEORETIC VIEW OF THE P-LAPLACIAN
Yuval Peres and Scott Sheffield
Fix a bounded domain Omega in R^d, a continuous function F on the
boundary of
Omega, and constants epsilon>0, p>1, and q>1 with p^{-1} + q^{-1} =
1. For each
x in Omega, let u^epsilon(x) be the value for player I of the following
two-player, zero-sum game. The initial game position is x. At each
stage, a
fair coin is tossed and the player who wins the toss chooses a vector
v of
length at most epsilon to add to the game position, after which a random
``noise vector'' with mean zero and variance (q/p)|v|^2 in each
orthogonal
direction is also added. The game ends when the game position reaches
some y on
the boundary of Omega, and player I's payoff is F(y).
We show that (for sufficiently regular Omega) as epsilon tends to
zero the
functions u^epsilon converge uniformly to the unique p-harmonic
extension of F.
Using a modified game (in which epsilon gets smaller as the game
position
approaches the boundary), we prove similar statements for general
bounded
domains Omega and resolutive functions F.
These games and their variants interpolate between the tug of war
games
studied by Peres, Schramm, Sheffield, and Wilson (p=infinity) and the
motion-by-curvature games introduced by Spencer and studied by Kohn
and Serfaty
(p=1). They generalize the relationship between Brownian motion and the
ordinary Laplacian and yield new results about p-capacity and p-harmonic
measure.
http://front.math.ucdavis.edu/math.AP/0607761
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4534. LARGE DEVIATIONS FOR SEMIFLOWS OVER A NON-UNIFORMLY EXPANDING BASE
Vitor Araujo
We obtain a large deviation bound for continuous observables on
suspension
semiflows over a non-uniformly expanding base transformation with non-
flat
singularities or criticalities, where the roof function defining the
suspension
behaves like the logarithm of the distance to the singular/critical
set of the
base map. That is, given a continuous function we consider its space
average
with respect to a physical measure and compare this with the time
averages
along orbits of the semiflow, showing that the Lebesgue measure of
the set of
points whose time averages stay away from the space average tends to
zero
exponentially fast as time goes to infinity. Suspension semiflows
model the
dynamics of flows admitting cross-sections, where the dynamics of the
base is
given by the Poincar\'e return map and the roof function is the
return time to
the cross-section. The results are applicable in particular to semiflows
modeling the geometric Lorenz attractors and the Lorenz flow, as well
as other
semiflows with multidimensional non-uniformly expanding base with non-
flat
singularities and/or criticalities under slow recurrence rate
conditions to
this singular/critical set. We are also able to obtain exponentially
fast
escape rates from subsets without full measure.
http://front.math.ucdavis.edu/math.DS/0607771
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4535. A NOTE ON THE MENCHOV-RADEMACHER INEQUALITY
Witold Bednorz
We improve constants in the Rademacher-Menchov inequality.
http://front.math.ucdavis.edu/math.PR/0608023
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4536. SLOW MOVEMENT OF RANDOM WALK IN RANDOM ENVIRONMENT ON A REGULAR
TREE
Yueyun Hu (LAGA) and Zhan Shi (PMA)
We consider a recurrent random walk in random environment on a
regular tree.
Under suitable general assumptions upon the distribution of the
environment, we
show that the walk exhibits an unusual slow movement: the order of
magnitude of
the walk in the first $n$ steps is $(\log n)^3$.
http://front.math.ucdavis.edu/math.PR/0608036
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4537. EQUILIBRIUM GLAUBER DYNAMICS OF CONTINUOUS PARTICLE SYSTEMS AS
A SCALING LIMIT OF KAWASAKI DYNAMICS
Dmitri L. Finkelshtein and Yuri G. Kondratiev and Eugene W. Lytvynov
A Kawasaki dynamics in continuum is a dynamics of an infinite system of
interacting particles in $\mathbb{R}^d$ which randomly hop over the
space. In
this paper, we deal with an equilibrium Kawasaki dynamics which has a
Gibbs
measure $mu$ as invariant measure. We study a scaling limit of such a
dynamics,
derived through a scaling of the jump rate. Informally, we expect
that, in the
limit, only jumps of ``infinite length'' will survive, i.e., we
expect to
arrive at a Glauber dynamics in continuum (a birth-and-death process in
$\mathbb{R}^d$). We prove that, in the low activity-high temperature
regime,
the generators of the Kawasaki dynamics converge to the generator of
a Glauber
dynamics. The convergence is on the set of exponential functions, in the
$L^2(\mu)$-norm. Furthermore, additionally assuming that the
potential of pair
interaction is positive, we prove the weak convergence of the
finite-dimensional distributions of the processes.
http://front.math.ucdavis.edu/math.PR/0608051
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4538. MEASURE CONCENTRATION OF HIDDEN MARKOV PROCESSES
Leonid Kontorovich
We prove what appears to be the first concentration of measure result
for
hidden Markov processes. Our bound is stated in terms of the contraction
coefficients of the underlying Markov process, and strictly
generalizes the
Markov process concentration results of Marton (1996) and Samson (2000).
Somewhat surprisingly, the bound turns out to be the same as for
ordinary
Markov processes; this property, however, fails for general hidden/
observed
process pairs.
http://front.math.ucdavis.edu/math.PR/0608064
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4539. ON TWO BIASED GRAPH PROCESSES
Gideon Amir and Eyal Lubetzky
In [Amir et al.], the authors consider the generalization $\Gor$ of the
Erd\H{o}s-R\'enyi random graph process $G$, where instead of adding
new edges
uniformly, $\Gor$ gives a weight of size 1 to missing edges between
pairs of
isolated vertices, and a weight of size $K\in[0,\infty)$ otherwise.
This can
correspond to the linking of settlements or the spreading of an
epidemic. The
authors investigate $\tgor(K)$, the critical time for the appearance
of a giant
component as a function of $K$, and prove that
$\tgor=(1+o(1))\frac{4}{\sqrt{3K}}$, using a proper timescale.
In this work, we show that a natural variation of the model $\Gor$
has
interesting properties. Define the process $\Gand$, where a weight of
size $K$
is assigned to edges between pairs of non-isolated vertices, and a
weight of
size 1 otherwise. We prove that the asymptotical behavior of the giant
component threshold is essentially the same for $\Gand$, and namely $
\tgand /
\tgor$ tends to $\frac{64\sqrt{6}}{\pi(24+\pi^2)}\approx 1.47$ as $K
\to\infty$.
However, the corresponding thresholds for connectivity satisfy $\tcand /
\tcor=\max\{{1/2},K\}$ for every $K>0$. Following the methods of
[Amir et al.],
$\tgand$ is characterized as the singularity point to a system of
differential
equations, and computer simulations of both models agree with the
analytical
results as well as with the asymptotic analysis. In the process, we
answer the
following question: when does a giant component emerge in a graph
process where
edges are chosen uniformly out of all edges incident to isolated
vertices,
while such exist, and otherwise uniformly? This corresponds to the
value of
$\tgand(0)$, which we show to be ${3/2}+\frac{4}{3\mathrm{e}^2-1}$.
http://front.math.ucdavis.edu/math.CO/0608097
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4540. ON THE ABSENCE OF FERROMAGNETISM IN TYPICAL 2D FERROMAGNETS
Marek Biskup and Lincoln Chayes and Steven A. Kivelson
We consider the Ising systems in $d$ dimensions with nearest-neighbor
ferromagnetic interactions and long-range repulsive (antiferromagnetic)
interactions which decay with a power, $s$, of the distance. The
physical
context of such models is discussed; primarily this is $d=2$ and $s=3
$ where,
at long distances, genuine magnetic interactions between genuine
magnetic
dipoles are of this form. We prove that when the power of decay lies
above $d$
and does not exceed $d+1$, then for all temperatures, the spontaneous
magnetization is zero. In contrast, we also show that for powers
exceeding
$d+1$ (with $d\ge2$) magnetic order can occur.
http://front.math.ucdavis.edu/math-ph/0608009
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4541. INVASION PERCOLATION ON REGULAR TREES
Omer Angel and Jesse Goodman and Frank den Hollander and Gordon Slade
We consider invasion percolation on a rooted regular tree. For the
infinite
cluster invaded from the root, we identify the scaling behaviour of its
$r$-point function for any $r \ge 2$ and of its volume both at a
given height
and below a given height. In addition, we derive scaling estimates
for simple
random walk on the cluster starting from the root. We find that while
the power
laws of the scaling are the same as for the incipient infinite
cluster for
ordinary percolation, the scaling functions differ. Thus, somewhat
surprisingly, the two clusters behave differently. We show that the
invasion
percolation cluster is stochastically dominated by the incipient
infinite
cluster. Far above the root, the two clusters have the same law
locally, but
not globally. A key ingredient in the proofs is an analysis of the
forward
maximal weights along the backbone of the invasion percolation
cluster. These
weights decay towards the critical value for ordinary percolation,
but only
slowly, and this slow decay causes an anomalous scaling behaviour.
http://front.math.ucdavis.edu/math.PR/0608132
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4542. SYMMETRIC AND CENTERED BINOMIAL APPROXIMATION OF SUMS OF
LOCALLY DEPENDENT RANDOM VARIABLES
Adrian R\"ollin
Stein's method is used to approximate sums of discrete and locally
dependent
random variables by a centered and symmetric Binomial distribution.
Under
appropriate smoothness properties of the summands, the same order of
accuracy
as in the Berry-Essen Theorem is achieved. The approximation of the
total
number of points of a point processes is also considered. The results
are
applied to the exceedances of the $r$-scans process and to the Mat\'ern
hardcore point process type I.
http://front.math.ucdavis.edu/math.PR/0608138
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4543. EVOLUTION OF THE INTERFACES IN A TWO DIMENSIONAL POTTS MODEL
Glauco Valle
We investigate the evolution of the random interfaces in a two
dimensional
Potts model at zero temperature under Glauber dynamics for some
particular
initial conditions. We prove that under space-time diffusive scaling
the shape
of the interfaces converges in probability to the solution of a non-
linear
parabolic equation. This Law of Large Numbers is obtained from the
Hydrodynamic
limit of a coupling between an exclusion process and an inhomogeneous
one
dimensional zero range process with asymmetry at the origin.
http://front.math.ucdavis.edu/math.PR/0608142
---------------------------------------------------------------
4544. RANDOM WALK ON THE INCIPIENT INFINITE CLUSTER FOR ORIENTED
PERCOLATION IN HIGH DIMENSIONS
Martin T. Barlow and Antal A. Jarai and Takashi Kumagai and Gordon
Slade
We consider simple random walk on the incipient infinite cluster for the
spread-out model of oriented percolation on $Z^d \times Z_+$. In
dimensions
$d>6$, we obtain bounds on exit times, transition probabilities, and
the range
of the random walk, which establish that the spectral dimension of the
incipient infinite cluster is 4/3, and thereby prove a version of the
Alexander--Orbach conjecture in this setting. The proof divides into
two parts.
One part establishes general estimates for simple random walk on an
arbitrary
infinite random graph, given suitable bounds on volume and effective
resistance
for the random graph. A second part then provides these bounds on
volume and
effective resistance for the incipient infinite cluster in dimensions
$d>6$, by
extending results about critical oriented percolation obtained
previously via
the lace expansion.
http://front.math.ucdavis.edu/math.PR/0608164
---------------------------------------------------------------
4545. STATIONARY AND NONEQUILIBRIUM FLUCTUATIONS IN BOUNDARY DRIVEN
EXCLUSION PROCESSES
Claudio Landim (LMRS) and Aniura Milan\'{e}s (ICEX) and Stefano
Olla (CEREMADE)
We prove nonequilibrium fluctuations for the boundary driven
symmetric simple
exclusion process. We deduce from this result the stationary
fluctuations.
http://front.math.ucdavis.edu/math.PR/0608165
---------------------------------------------------------------
4546. AN IMPACT OF STOCHASTIC DYNAMIC BOUNDARY CONDITIONS ON THE
EVOLUTION OF THE CAHN-HILLIARD SYSTEM
Desheng Yang and Jinqiao Duan
Nonlinear systems are often subject to random influences. Sometimes
the noise
enters the system through physical boundaries and this leads to
stochastic
dynamic boundary conditions. A dynamic, as opposed to static, boundary
condition involves the time derivative as well as spatial derivatives
for the
system state variables on the boundary. Although stochastic \emph
{static}
(Neumann or Dirichet type) boundary conditions have been applied for
stochastic
partial differential equations, not much is known about the dynamical
impact of
stochastic \emph{dynamic} boundary conditions. The purpose of this
paper is to
study possible impacts of stochastic dynamic boundary conditions on
the long
term dynamics of the Cahn-Hilliard equation arising in the materials
science.
We show that the dimension estimation of the random attractor
increases as the
coefficient for the dynamic term in the stochastic dynamic boundary
condition
decreases. However, the dimension of the random attractor is not
affected by
the corresponding stochastic static boundary condition.
http://front.math.ucdavis.edu/math.DS/0608133
---------------------------------------------------------------
4547. RANDOM DYNAMICAL SYSTEMS
Vitor Araujo
The concept of random dynamical system is a comparatively recent
development
combining ideas and methods from the well developed areas of
probability theory
and dynamical systems.
Due to our inaccurate knowledge of the particular physical system
or due to
computational or theoretical limitations (lack of sufficient
computational
power, inefficient algorithms or insufficiently developed
mathematical or
physical theory, for example), the mathematical models never
correspond exactly
to the phenomenon they are meant to model. Moreover when considering
practical
systems we cannot avoid either external noise or measurement or
inaccuracy
errors, so every realistic mathematical model should allow for small
errors
along orbits not to disturb too much the long term behavior. To be
able to cope
with unavoidable uncertainty about the ``correct'' parameter values,
observed
initial states and even the specific mathematical formulation
involved, we let
randomness be embedded within the model to begin with.
We present the most basic classes of models in what follows, then
define the
general concept and present some developments and examples of
applications.
http://front.math.ucdavis.edu/math.DS/0608162
---------------------------------------------------------------
4548. HIGH ORDER EXPANSION OF MATRIX MODELS AND ENUMERATION OF MAPS
Edouard Maurel-Segala
Perturbation of the GUE are known in physics to be related to
enumeration of
graphs on surfaces. We investigate this idea and show that for a
small convex
perturbation, we can perform a genus expansion: the moments of the
empirical
measure can be developed into a series whose g-th term is a
generating function
of graphs on a surface of genus g.
http://front.math.ucdavis.edu/math.PR/0608192
---------------------------------------------------------------
4549. UNITARY MATRIX INTEGRALS
Benoit Collins and Alice Guionnet and Edouard Maurel-Segala
We prove that the limit of various unitary matrix integrals,
including the
Itzykson-Zuber integral, exists in a small parameters region and is
analytic in
these parameters.
http://front.math.ucdavis.edu/math.PR/0608193
---------------------------------------------------------------
4550. DAMAGE SEGREGATION AT FISSIONING MAY INCREASE GROWTH RATES: A
SUPERPROCESS MODEL
Steven N. Evans and David Steinsaltz
A fissioning organism may purge unrepairable damage by bequeathing it
preferentially to one of its daughters. We propose a superprocess
model, and
show that when damage accumulates deterministically, optimal growth
is achieved
by unequal division of damage between the daughters.
http://front.math.ucdavis.edu/q-bio.PE/0608008
---------------------------------------------------------------
4551. ON THE ASYMPTOTIC BEHAVIOUR OF RANDOM RECURSIVE TREES IN
RANDOM ENVIRONMENT
Konstantin Borovkov and Vladimir Vatutin
We consider growing random recursive trees in random environment, in
which at
each step a new vertex is attached (by an edge of a random length) to an
existing tree vertex according to a probability distribution that
assigns the
tree vertices masses proportional to their random weights. The main
aim of the
paper is to study the asymptotic behaviour of the distance from the
newly
inserted vertex to the tree's root and that of the mean numbers of
outgoing
vertices as the number of steps tends to infinity. Most of the
results are
obtained under the assumption that the random weights have a product
form with
independent identically distributed factors.
http://front.math.ucdavis.edu/math.PR/0608211
---------------------------------------------------------------
4552. PROOF OF A CONJECTURE OF N. KONNO FOR THE 1D CONTACT PROCESS
J. van den Berg and O. H\"{a}ggstr\"{o}m and J. Kahn
Consider the one-dimensional contact process. About ten years ago, N.
Konno
stated the conjecture that, for all positive integers $n,m$, the upper
invariant measure has the following property: Conditioned on the
event that $O$
is infected, the events $\{$All sites $-n,...,-1$ are healthy$\}$ and
$\{$All
sites $1,...,m$ are healthy$\}$ are negatively correlated. We prove
(a stronger
version of) this conjecture, and explain that in some sense it is a dual
version of the planar case of one of our results in \citeBHK.
http://front.math.ucdavis.edu/math.PR/0608216
---------------------------------------------------------------
4553. A NOTE ON PERCOLATION IN COCYCLE MEASURES
Ronald Meester
We describe infinite clusters which arise in nearest-neighbour
percolation
for so-called cocycle measures on the square lattice. These measures
arise
naturally in the study of random transformations. We show that infinite
clusters have a very specific form and direction. In concrete
situations, this
leads to a quick decision whether or not a certain cocycle measure
percolates.
We illustrate this with two examples which are interesting in their
own right.
http://front.math.ucdavis.edu/math.PR/0608217
---------------------------------------------------------------
4554. RANDOM WALK IN RANDOM SCENERY: A SURVEY OF SOME RECENT RESULTS
Frank den Hollander and Jeffrey E. Steif
. In this paper we give a survey of some recent results for random
walk in
random scenery (RWRS). On $\mathbb {Z}^d$, $d\geq 1$, we are given a
random
walk with i.i.d. increments and a random scenery with i.i.d.
components. The
walk and the scenery are assumed to be independent. RWRS is the
random process
where time is indexed by $\mathbb {Z}$, and at each unit of time both
the step
taken by the walk and the scenery value at the site that is visited are
registered. We collect various results that classify the ergodic
behavior of
RWRS in terms of the characteristics of the underlying random walk
(and discuss
extensions to stationary walk increments and stationary scenery
components as
well). We describe a number of results for scenery reconstruction and
close by
listing some open questions.
http://front.math.ucdavis.edu/math.PR/0608219
---------------------------------------------------------------
4555. LINEARLY EDGE-REINFORCED RANDOM WALKS
Franz Merkl and Silke W. W. Rolles
We review results on linearly edge-reinforced random walks. On finite
graphs,
the process has the same distribution as a mixture of reversible
Markov chains.
This has applications in Bayesian statistics and it has been used in
studying
the random walk on infinite graphs. On trees, one has a
representation as a
random walk in an independent random environment. We review recent
results for
the random walk on ladders: recurrence, a representation as a random
walk in a
random environment, and estimates for the position of the random walker.
http://front.math.ucdavis.edu/math.PR/0608220
---------------------------------------------------------------
4556. INVARIANCE PRINCIPLES FOR FRACTIONALLY INTEGRATED NONLINEAR
PROCESSES
Xiaofeng Shao and Wei Biao Wu
We obtain invariance principles for a wide class of fractionally
integrated
nonlinear processes. The limiting distributions are shown to be
fractional
Brownian motions. Under very mild conditions, we extend earlier ones
on long
memory linear processes to a more general setting. The invariance
principles
are applied to the popular R/S and KPSS tests.
http://front.math.ucdavis.edu/math.PR/0608223
---------------------------------------------------------------
4557. WEAK STABILITY AND GENERALIZED WEAK CONVOLUTION FOR RANDOM
VECTORS AND STOCHASTIC PROCESSES
Jolanta K. Misiewicz
A random vector ${\bf X}$ is weakly stable iff for all $a,b\in \mathbb
{R}$
there exists a random variable $\Theta$ such that $a{\bf X}+b{\bf
X}'\stackrel{d}{=}{\bf X}\Theta$. This is equivalent (see \cite{MOU})
with the
condition that for all random variables $Q_1,Q_2$ there exists a random
variable $\Theta$ such that $$ X Q_1 + X' Q_2 \stackrel{d}{=} X
\Theta, $$
where ${\bf X},{\bf X}',Q_1,Q_2,\Theta$ are independent. In this
paper we
define generalized convolution of measures defined by the formula $$ L
(Q_1)
\oplus_{\mu} L(Q_2) = L(\Theta), $$ if the equation $(*)$ holds for $
{\bf
X},Q_1,Q_2,\Theta$ and $\mu ={\cal L}(\Theta)$. We study here basic
properties
of this convolution, basic properties of $\oplus_{\mu}$-infinitely
divisible
distributions, $\oplus_{\mu}$-stable distributions and give a series of
examples.
http://front.math.ucdavis.edu/math.PR/0608225
---------------------------------------------------------------
4558. ON RANDOM WALKS IN RANDOM SCENERY
F. M. Dekking and P. Liardet
This paper considers 1-dimensional generalized random walks in random
scenery. That is, the steps of the walk are generated by an arbitrary
stationary process, and also the scenery is a priori arbitrary
stationary.
Under an ergodicity condition--which is satisfied in the classical
case--a
simple proof of the distinguishability of periodic sceneries is given.
http://front.math.ucdavis.edu/math.DS/0608218
---------------------------------------------------------------
4559. RECURRENCE OF COCYCLES AND STATIONARY RANDOM WALKS
Klaus Schmidt
We survey distributional properties of $\mathbb{R}^d$-valued cocycles of
finite measure preserving ergodic transformations (or, equivalently, of
stationary random walks in $\mathbb{R}^d$) which determine recurrence or
transience.
http://front.math.ucdavis.edu/math.DS/0608221
---------------------------------------------------------------
4560. SOLVING NON-UNIQUENESS IN AGGLOMERATIVE HIERARCHICAL CLUSTERING
USING MULTIDENDROGRAMS
Alberto Fernandez and Sergio Gomez
In agglomerative hierarchical clustering, pair-group methods suffer
from a
problem of non-uniqueness when two or more distances between
different clusters
coincide during the amalgamation process. The traditional approach
for solving
this drawback has been to take any arbitrary criterion in order to
break ties
between distances, which results in different hierarchical
classifications
depending on the criterion followed. In this article we propose a
variable-group algorithm that consists in grouping more than two
clusters at
the same time when ties occur. We give a tree representation for the
results of
the algorithm, which we call a "multidendrogram", as well as a
generalisation
of the Lance and Williams' formula which enables the implementation
of the
algorithm in a recursive way.
http://front.math.ucdavis.edu/cs.IR/0608049
---------------------------------------------------------------
4561. NON COMMUTATIVE LAPLACE TRANSFORMS, H\"ORMANDER'S TYPE
OPERATORS AND LOCAL INDEX THEOREMS
Fabrice Baudoin
The purpose of this work is to provide a general formalism for the
study in
small times of heat evolution semigroups associated to operators that
can be
written as sum of squares. We give a representation of such heat
kernels as the
averaging over the set of Brownian paths of the exponential of an
infinite Lie
series. The method we develop is an alternative to It\^o's theory of
stochastic
differential equations for small times problems and can be applied in
a more
general setting. In order to illustrate the method, we apply this
formalism to
give a new short proof of Atiyah-Singer local index theorem.
http://front.math.ucdavis.edu/math.PR/0608231
---------------------------------------------------------------
4562. STRONG INVARIANCE PRINCIPLE FOR DEPENDENT RANDOM FIELDS
Alexander Bulinski and Alexey Shashkin
A strong invariance principle is established for random fields which
satisfy
dependence conditions more general than positive or negative
association. We
use the approach of Cs\"{o}rg\H{o} and R\'{e}v\'{e}sz applied
recently by Balan
to associated random fields. The key step in our proof combines new
moment and
maximal inequalities, established by the authors for partial sums of
multiindexed random variables, with the estimate of the convergence
rate in the
CLT for random fields under consideration.
http://front.math.ucdavis.edu/math.PR/0608237
---------------------------------------------------------------
4563. HEAVY TAIL PROPERTIES OF STATIONARY SOLUTIONS OF
MULTIDIMENSIONAL STOCHASTIC RECURSIONS
Yves Guivarc'h
We consider the following recurrence relation with random i.i.d.
coefficients
$(a_n,b_n)$: $$ x_{n+1}=a_{n+1} x_n+b_{n+1} $$ where $a_n\in
GL(d,\mathbb{R}),b_n\in \mathbb{R}^d$. Under natural conditions on $
(a_n,b_n)$
this equation has a unique stationary solution, and its support is
non-compact.
We show that, in general, its law has a heavy tail behavior and we
study the
corresponding directions. This provides a natural construction of
laws with
heavy tails in great generality. Our main result extends to the
general case
the results previously obtained by H. Kesten in [16] under positivity or
density assumptions, and the results recently developed in [17] in a
special
framework.
http://front.math.ucdavis.edu/math.PR/0608239
---------------------------------------------------------------
4564. CHARACTERIZATION OF TALAGRAND'S LIKE TRANSPORTATION-COST
INEQUALITIES ON THE REAL LINE
Nathael Gozlan (MODAL'X)
In this paper, we give necessary and sufficient conditions for
Talagrand's
like transportation cost inequalities on the real line. This brings a
new wide
class of examples of probability measures enjoying a dimension-free
concentration of measure property. Another byproduct is the
characterization of
modified Log-Sobolev inequalities for Log-concave probability
measures on R.
http://front.math.ucdavis.edu/math.PR/0608241
---------------------------------------------------------------
4565. MARKOVIANITY IN SPACE AND TIME
M. N. M. van Lieshout
. Markov chains in time, such as simple random walks, are at the
heart of
probability. In space, due to the absence of an obvious definition of
past and
future, a range of definitions of Markovianity have been proposed. In
this
paper, after a brief review, we introduce a new concept of
Markovianity that
aims to combine spatial and temporal conditional independence.
http://front.math.ucdavis.edu/math.PR/0608242
---------------------------------------------------------------
4566. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR THE M/GI/INFINITY QUEUE
Laurent Decreusefond and Pascal Moyal
In this paper, we present a functional fluid limit theorem and a
functional
central limit theorem for a queue with an infinity of servers M/GI/$
\infty$.
The system is represented by a point-measure valued process keeping
track of
the remaining processing times of the customers in service. The
convergence in
law of a sequence of such processes is proved by compactness-uniqueness
methods, and the deterministic fluid limit is the solution of an
integrated
equation in the space $\S^{\prime}$ of tempered distributions. We then
establish the corresponding central limit theorem, i.e. the
approximation of
the normalized error process by a $\S^{\prime}$-valued diffusion.
http://front.math.ucdavis.edu/math.PR/0608258
---------------------------------------------------------------
4567. COVERAGE OF SPACE IN BOOLEAN MODELS
Rahul Roy
For a marked point process $\{(x_i,S_i)_{i\geq 1}\}$ with $\{x_i\in
\Lambda:i\geq 1\}$ being a point process on $\Lambda \subseteq \mathbb
{R}^d$
and $\{S_i\subseteq R^d:i\geq 1\}$ being random sets consider the region
$C=\cup_{i\geq 1}(x_i+S_i)$. This is the covered region obtained from
the
Boolean model $\{(x_i+S_i):i\geq 1\}$. The Boolean model is said to be
completely covered if $\Lambda \subseteq C$ almost surely. If $\Lambda
$ is an
infinite set such that ${\bf s}+\Lambda \subseteq \Lambda$ for all $
{\bf s}\in
\Lambda$ (e.g. the orthant), then the Boolean model is said to be
eventually
covered if ${\bf t}+\Lambda \subseteq C$ for some ${\bf t}$ almost
surely. We
discuss the issues of coverage when $\Lambda$ is $\mathbb{R}^d$ and when
$\Lambda$ is $[0,\infty)^d$.
http://front.math.ucdavis.edu/math.CO/0608238
---------------------------------------------------------------
4568. DECOMPOSITIONS OF THE FREE ADDITIVE CONVOLUTION
Romuald Lenczewski
We introduce and study a new type of convolution of probability measures
called the orthogonal convolution, which is related to the monotone
convolution. Using this convolution, we derive alternating
decompositions of
the free additive convolution of compactly supported probability
measures in
free probability. These decompositions are directly related to
alternating
decompositions of the associated subordination functions. In
particular, they
allow us to compute free additive convolutions of compactly supported
measures
without using free cumulants or $R$-transforms. In simple cases,
representations of the corresponding Cauchy transforms as continued
fractions
are obtained in a natural way. Moreover, this approach establishes a
clear
connection between convolutions and products associated with the main
notions
of independence (free, monotone and boolean) in noncommutative
probability.
Finally, our result leads to natural decompositions of the free
product of
rooted graphs.
http://front.math.ucdavis.edu/math.OA/0608236
---------------------------------------------------------------
4569. BRANCHING RANDOM WALK WITH EXPONENTIALLY DECREASING STEPS, AND
STOCHASTICALLY SELF-SIMILAR MEASURES
Itai Benjamini and Ori Gurel-Gurevich and and Boris Solomyak
We consider a Branching Random Walk on $\R$ whose step size decreases
by a
fixed factor, $0<b<1$, with each turn. This process generates a random
probability measure on $\R$, that is, the limit of uniform
distribution among
the $2^n$ particles of the $n$-th step. We present an initial
investigation of
the limit measure and its support. We show, in particular, that (1)
for almost
every $b>1/2$ the limit measure is almost surely (a.s.) absolutely
continuous
with respect to the Lebesgue measure, but for Pisot $1/b$ it is a.s.
singular;
(2) for all $b > (\sqrt{5}-1)/2$ the support of the measure is a.s.
the closure
of its interior; (3) for Pisot $1/b$ the support of the measure is
``fractured'': it is a.s. disconnected and the components of the
complement are
not isolated on both sides.
http://front.math.ucdavis.edu/math.PR/0608271
---------------------------------------------------------------
4570. INVERTING RANDOM FUNCTIONS III: DISCRETE MLE REVISITED
Mike A. Steel and Laszlo A. Szekely
This paper continues our earlier investigations into the inversion of
random
functions in a general (abstract) setting. In Section 2 we investigate a
concept of invertibility and the invertibility of the composition of
random
functions. In Section 3 we resolve some questions concerning the
number of
samples required to ensure the accuracy of parametric maximum likelihood
estimation (MLE). A direct application to phylogeny reconstruction is
given.
http://front.math.ucdavis.edu/math.PR/0608273
---------------------------------------------------------------
4571. DYNAMICS & STOCHASTICS: FESTSCHRIFT IN HONOR OF M. S. KEANE
Dee Denteneer and Frank den Hollander and Evgeny Verbitskiy
The present volume is a Festschrift for Mike Keane, on the occasion
of his
65th birthday on January 2, 2005. It contains 29 contributions by Mike's
closest colleagues and friends, covering a broad range of topics in
Dynamics
and Stochastics. To celebrate Mike's scientific achievements, a
conference
entitled ``Dynamical Systems, Probability Theory and Statistical
Mechanics''
was organized in Eindhoven, The Netherlands, during the week of
January 3--7,
2005. This conference was hosted jointly by EURANDOM and by Philips
Research.
It drew over 80 participants from 5 continents, which is a sign of
the warm
affection and high esteem for Mike felt by the international mathematics
community.
http://front.math.ucdavis.edu/math.PR/0608289
---------------------------------------------------------------
4572. POISSON REPRESENTATION OF A EWENS FRAGMENTATION PROCESS
Alexander Gnedin and Jim Pitman
A simple explicit construction is provided of a partition-valued
fragmentation process whose distribution on partitions of $[n]=
\{1,...,n\}$ at
time $\theta \ge 0$ is governed by the Ewens sampling formula with
parameter
$\theta$. These partition-valued processes are exchangeable and
consistent, as
$n$ varies. They can be derived by uniform sampling from a
corresponding mass
fragmentation process defined by cutting a unit interval at the
points of a
Poisson process with intensity $\theta x^{-1} \diff x$ on ${\mathbb R}
_+$,
arranged to be intensifying as $\theta$ increases.
http://front.math.ucdavis.edu/math.PR/0608307
---------------------------------------------------------------
4573. ON THE CORRELATION MEASURE OF A FAMILY OF COMMUTING HERMITIAN
OPERATORS WITH APPLICATIONS TO PARTICLE DENSITIES OF THE QUASI-FREE
REPRESENTATIONS OF
THE CAR AND CCR
Eugene Lytvynov and Lin Mei
Let $X$ be a locally compact, second countable Hausdorff topological
space.
We consider a family of commuting Hermitian operators $a(\Delta)$
indexed by
all measurable, relatively compact sets $\Delta$ in $X$ (a quantum
stochastic
process over $X$). For such a family, we introduce the notion of a
correlation
measure. We prove that, if the family of operators possesses a
correlation
measure which satisfies some condition of growth, then there exists a
point
process over $X$ having the same correlation measure. Furthermore, the
operators $a(\Delta)$ can be realized as multiplication operators in the
$L^2$-space with respect to this point process. In the proof, we
utilize the
notion of $\star$-positive definiteness, proposed in [Y. G.
Kondratiev and T.\
Kuna, {\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\bf 5}
(2002),
201--233]. In particular, our result extends the criterion of
existence of a
point process from that paper to the case of the topological space $X
$, which
is a standard underlying space in the theory of point processes. As
applications, we discuss particle densities of the quasi-free
representations
of the CAR and CCR, which lead to fermion, boson, fermion-like, and
boson-like
(e.g. para-fermions and para-bosons of order 2) point processes.
In particular, we prove that any fermion point process
corresponding to a
Hermitian kernel may be derived in this way.
http://front.math.ucdavis.edu/math.PR/0608334
---------------------------------------------------------------
4574. IMAGE OF THE SPECTRAL MEASURE OF A JACOBI FIELD AND THE
CORRESPONDING OPERATORS
Yurij M. Berezansky and Eugene W. Lytvynov and Artem D. Pulemyotov
By definition, a Jacobi field $J=(J(\phi))_{\phi\in H_+}$ is a family of
commuting selfadjoint three-diagonal operators in the Fock space $
\mathcal
F(H)$. The operators $J(\phi)$ are indexed by the vectors of a real
Hilbert
space $H_+$. The spectral measure $\rho$ of the field $J$ is defined
on the
space $H_-$ of functionals over $H_+$. The image of the measure $\rho
$ under a
mapping $K^+:T_-\to H_-$ is a probability measure $\rho_K$ on $T_-$.
We obtain
a family $J_K$ of operators whose spectral measure is equal to $\rho_K
$. We
also obtain the chaotic decomposition for the space $L^2(T_-,d\rho_K)$.
http://front.math.ucdavis.edu/math.PR/0608335
---------------------------------------------------------------
4575. LAPLACE OPERATORS AND DIFFUSIONS IN TANGENT BUNDLES OVER
POISSON SPACES
S. Albeverio and A. Daletskii and E. Lytvynov
Spaces of differential forms over configuration spaces with Poisson
measures
are constructed. The corresponding Laplacians (of Bochner and de Rham
type) on
1-forms and associated semigroups are considered. Their probabilistic
interpretation is given.
http://front.math.ucdavis.edu/math.PR/0608337
---------------------------------------------------------------
4576. DE RHAM COHOMOLOGY OF CONFIGURATION SPACES WITH POISSON MEASURE
S. Albeverio and A. Daletskii and E. Lytvynov
The space $\Gamma_X$ of all locally finite configurations in a
Riemannian manifold $X$ of infinite volume is considered. The
deRham complex
of square-integrable differential forms over $\Gamma_X$, equipped
with the
Poisson measure, and the corresponding deRham cohomology are studied.
The
latter is shown to be unitarily isomorphic to a certain Hilbert
tensor algebra
generated by the $L^2$-cohomology of the underlying manifold $X$.
http://front.math.ucdavis.edu/math.PR/0608338
---------------------------------------------------------------
4577. OPERATORS OF GAMMA WHITE NOISE ANALYSIS
Yu. Kondratiev and E. Lytvynov
The paper is devoted to the study of Gamma white noise analysis. We
define an
extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, d\sigma)$, and
show how to
include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We
introduce
in $\Gama(\Ha)$ operators $a(\xi)=\int_{\R^d} dx \xi(x)a(x)$, $\xi\in
S$, with
$a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and
$\di_x$ are the creation and annihilation operators at $x$. We show that
$(a(\xi))_{\xi\in S}$ is a family of commuting selfadjoint operators in
$\Gama(\Ha)$ and construct the Fourier transform in generalized joint
eigenvectors of this family. This transform is a unitary $I$ between
$\Gama(\Ha)$ and the $L^2$-space $L^2(S',d\mu_{\mathrm G})$, where
$\mu_{\mathrm G}$ is the measure of Gamma white noise with intensity $
\sigma$.
The image of $a(\xi)$ under $I$ is the operator of multiplication by
$\la\cdot,\xi\ra$, so that $a(\xi)$'s are Gamma field operators. The
Fock
structure of the Gamma space determined by $I$ coincides with that
discovered
in {\bf [}{\it Infinite Dimensional Analysis,
Quantum Probability and Related Topics} {\bf 1} (1998), 91--117
{\bf ]}. We
note that $I$ extends in a natural way the multiple stochastic
integral (chaos)
decomposition of the ``chaotic'' subspace of the Gamma space. Next, we
introduce and study spaces of test and generalized functions of Gamma
white
noise and derive explicit formulas for the action of the creation,
neutral, and
Gamma annihilation operators on these spaces.
http://front.math.ucdavis.edu/math.PR/0608340
---------------------------------------------------------------
4578. ON A SPECTRAL REPRESENTATION FOR CORRELATION MEASURES IN
CONFIGURATION SPACE ANALYSIS
Yu. M. Berezansky and Yu. G. Kondratiev and T. Kuna and E. Lytvynov
The paper is devoted to the study of configuration space analysis by
using
the projective spectral theorem. For a manifold $X$, let $\Gamma_X$,
resp.\
$\Gamma_{X,0}$ denote the space of all, resp. finite configurations
in $X$. The
so-called $K$-transform, introduced by A. Lenard, maps functions on
$\Gamma_{X,0}$ into functions on $\Gamma_{X}$ and its adjoint $K^*$ maps
probability measures on $\Gamma_X$ into $\sigma$-finite measures on
$\Gamma_{X,0}$. For a probability measure $\mu$ on $\Gamma_X$,
$\rho_\mu:=K^*\mu$ is called the correlation measure of $\mu$. We
consider the
inverse problem of existence of a probability measure $\mu$ whose
correlation
measure $\rho_\mu$ is equal to a given measure $\rho$. We introduce an
operation of $\star$-convolution of two functions on $\Gamma_{X,0}$
and suppose
that the measure $\rho$ is $\star$-positive definite, which enables
us to
introduce the Hilbert space ${\cal H}_\rho$ of functions on $\Gamma_
{X,0}$ with
the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_{\rho}}=
\int_{\Gamma_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(\eta) \rho(d\eta)$.
Under a
condition on the growth of the measure $\rho$ on the $n$-point
configuration
spaces, we construct the Fourier transform in generalized joint
eigenvectors of
some special family $A=(A_\phi)_{\phi\in\D}$, $\D:=C_0^\infty(X)$, of
commuting
selfadjoint operators in ${\cal H}_\rho$. We show that this Fourier
transform
is a unitary between ${\cal H}_{\rho}$ and the $L^2$-space
$L^2(\Gamma_X,d\mu)$, where $\mu$ is the spectral measure of $A$.
Moreover,
this unitary coincides with the $K$-transform, while the measure $\rho
$ is the
correlation measure of $\mu$.
http://front.math.ucdavis.edu/math.PR/0608343
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4579. ANALYSIS AND GEOMETRY ON MARKED CONFIGURATION SPACES
S. Albeverio and Yu. G. Kondratiev and E. W. Lytvynov and g. F. Us
We carry out analysis and geometry on a marked configuration space
$\Omega^M_X$ over a Riemannian manifold $X$ with marks from a space $M
$. We
suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a
transformation group $\frak A$ on $\Omega_X^M$ we take the
``lifting'' to
$\Omega_X^M$ of the action on $X\times M$ of the semidirect product
of the
group $\operatorname{Diff}_0(X)$ of diffeomorphisms on $X$ with
compact support
and the group $G^X$ of smooth currents, i.e., all $C^\infty$ mappings
of $X$
into $G$ which are equal to the identity element outside of a compact
set. The
marked Poisson measure $\pi_\sigma$ on $\Omega_X^M$ with L\'evy measure
$\sigma$ on $X\times M$ is proven to be quasiinvariant under the
action of
$\frak A$. Then, we derive a geometry on $\Omega_X^M$ by a natural
``lifting''
of the corresponding geometry on $X\times M$. In particular, we
construct a
gradient $\nabla^\Omega$ and a divergence $\operatorname{div}^\Omega
$. The
associated volume elements, i.e., all probability measures $\mu$ on
$\Omega_X^M$ with respect to which $\nabla^\Omega$ and
$\operatorname{div}^\Omega$ become dual operators on $L^2(\Omega_X^M;
\mu)$, are
identified as the mixed marked Poisson measures with mean measure
equal to a
multiple of $\sigma$. As a direct consequence of our results, we
obtain marked
Poisson space representations of the group $\frak A$ and its Lie
algebra $\frak
a$. We investigate also Dirichlet forms and Dirichlet operators
connected with
(mixed) marked Poisson measures.
http://front.math.ucdavis.edu/math.PR/0608344
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4580. ANALYSIS AND GEOMETRY ON $R_+$-MARKED CONFIGURATION SPACES
Yu. G. Kondratiev and E. W. Lytvynov and G. F. Us
We carry out analysis and geometry on a marked configuration space
$\Omega_X^{R_+}$ over a Riemannian manifold $X$ with marks from the
space $R_+$
as a natural generalization of the work {\bf [}{\it J. Func. Anal}.
{\bf 154}
(1998),
444--500{\bf ]}. As a transformation group $\mathfrak G$ on this
space, we
take the ``lifting'' to $\Omega_X^{R_+}$ of the action on $X\times R_+
$ of the
semidirect product of the group Diff of diffeomorphisms on $X$ with
compact
support and the group $R_+^X$ of smooth currents, i.e., all $C^\infty
$ mappings
of $X$ into $R_+$ which are equal to one outside a compact set. The
marked
Poisson measure $\pi$ on $\Omega_X^{R_+}$ with L\'evy measure $\sigma
$ is
proven to be quasiinvariant under the action of $\mathfrak G$. Then,
we derive
a geometry on $\Omega_X^{R_+}$ by a natural ``lifting'' of the
corresponding
geometry on $X\times R_+$. In particular, we construct a gradient
$\nabla^\Omega$ and divergence $div^\Omega$. The associated volume
elements,
i.e., all probability measures $\mu$ on $\Omega_X^{R_+}$ with respect
to which
$\nabla^\Omega$ and $div^\Omega$ become dual operators on $L^2
(\Omega_X^{R_+}
,\mu)$ are identified as the mixed Poisson measures with mean measure
equal to
a multiple of $\sigma$. As a direct consequence of our results, we
obtain
marked Poisson space representations of the group $\mathfrak G$ and
its Lie
algebra $\mathfrak g$. We investigate also Dirichlet forms and Dirichlet
operators connected with (mixed) marked Poisson measures. In
particular, we
obtain conditions of ergodicity of the semigroups generated by the
Dirichlet
operators. A possible generalization of the results of the paper to
the case
where the marks belong to a homogeneous space of a Lie group is noted.
http://front.math.ucdavis.edu/math.PR/0608347
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4581. LAPLACE OPERATORS ON DIFFERENTIAL FORMS OVER CONFIGURATION SPACES
S. Albeverio and A. Daletskii and E. Lytvynov
Spaces of differential forms over configuration spaces with Poisson
measures
are constructed. The corresponding Laplacians (of Bochner and de Rham
type) on
forms and associated semigroups are considered. Their probabilistic
interpretation is given.
http://front.math.ucdavis.edu/math.PR/0608349
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4582. MULTIAGENT MODELS IN TIME-VARYING AND RANDOM ENVIRONMENT
Biao Wu
In this paper we study multiagent models with time-varying type change.
Assume that there exist a closed system of $N$ agents classified into
$r$ types
according to their states of an internal system; each agent changes
its type by
an internal dynamics of the internal states or by the relative
frequency of
different internal states among the others, e.g., multinomial
sampling. We
investigate the asymptotic behavior of the empirical distributions of
the
agents' types as $N$ goes to infinity, by the weak convergence
criteria for
time-inhomogeneous Markov processes and the theory of Volterra integral
equations of the second kind. We also prove convergence theorems of
these
models evolving in random environment.
http://front.math.ucdavis.edu/math.PR/0608352
---------------------------------------------------------------
4583. COINCIDENCE OF LYAPUNOV EXPONENTS FOR RANDOM WALKS IN WEAK
RANDOM POTENTIALS
Markus Flury
We investigate the free energy of nearest-neighbor random walks on $
\mathbb
Z^d$, endowed with a drift along the first axis, and evolving in a
nonnegative
random potential given by i.i.d. random variables. Our main result
concerns the
ballistic regime in dimensions $d\geq 4$, at what we show that
quenched and
annealed Lyapunov exponents are equal, as soon as the strength of the
potential
is small enough.
http://front.math.ucdavis.edu/math.PR/0608357
---------------------------------------------------------------
4584. MOTT LAW FOR MOTT VARIABLE--RANGE RANDOM WALK
A. Faggionato and P. Mathieu
We consider a random walk on the support of an ergodic simple point
process
on R^d, d>1, furnished with independent energy marks. The jump rates
of the
random walk decay exponentially in the jump length and depend on the
energy
marks via a Boltzmann-type factor. This is an effective model for the
phonon-induced hopping of electrons in disordered solids in the
regime of
strong Anderson localization. Under mild assumptions on the point
process we
prove an upper bound of the asymptotic diffusion matrix of the random
walk in
agreement with Mott law. A lower bound in agreement with Mott law was
proved in
\cite{FSS}.
http://front.math.ucdavis.edu/math-ph/0608033
---------------------------------------------------------------
4585. ON PROCESSES WHICH CANNOT BE DISTINGUISHED BY FINITARY OBSERVATION
Yonatan Gutman and Michael Hochman
A function $J$ defined on a family $C$ of stationary processes is
finitely
observable if there is a sequence of functions $s_n$ such that $s_n
(x_1 ...
x_n)\to J(X)$ in probability for every process $X=(x_n)\in C$. Recently,
Ornstein and Weiss roved the striking result that if $C$ is the class of
aperiodic ergodic finite valued processes, then the only finitely
observable
isomorphism invariant on $C$ is entropy. We sharpen this in several
ways. Our
main theorem is that if $X \to Y$ is a zero-entropy extension of
finite entropy
ergodic systems and $C$ is the family of processes arising from $X$
and $Y$,
then every finitely observable function on $C$ is constant. This implies
Ornstein and Weiss' result, and extends it to many other families of
processes,
e.g. it shows that there are no nontrivial finitely observable
isomorphism
invariants for processes arising from Kronecker systems, mild and
strong mixing
zero entropy systems. It also implies that any finitely observable
isomorphism
invariant defined on the family of processes arising from irrational
rotations
must be constant for rotations belonging to a set of full Lebesgue
measure.
http://front.math.ucdavis.edu/math.DS/0608310
---------------------------------------------------------------
4586. UPCROSSING INEQUALITIES FOR STATIONARY SEQUENCES AND
APPLICATIONS TO ENTROPY AND COMPLEXITY
Michael Hochman
An empirical statistic for a class $C$ of stationary processes is a
function
$g$ which assigns to each process $(X_n)\in C$ with distribution $P$
and to
each sample $X_1,...,X_n$ of the process a real number $g_P
(X_1,...,X_n)$. We
describe a condition on $g$ which implies that the sequence
$(g_P(X_1,...,X_n))_{n=1}^{\infty}$ obeys a (universal) upcrossing
inequality,
that is, that the probability that this sequence fluctuates across some
interval $k$ times decays to zero with $k$. As applications we get
upcrossing
inequalities for the ergodic theorem (recovering known results), and get
upcrossing inequalities for the Shannon-McMillan-Breiman theorem and
for the
Kolmogorov complexity statistic.
http://front.math.ucdavis.edu/math.DS/0608311
---------------------------------------------------------------
4587. FUNCTIONAL SPACES AND OPERATORS CONNECTED WITH SOME L\'EVY NOISES
E. Lytvynov
We review some recent developments in white noise analysis and quantum
probability. We pay a special attention to spaces of test and
generalized
functionals of some L\'evy white noises, as well as as to the
structure of
quantum white noise on these spaces.
http://front.math.ucdavis.edu/math.PR/0608380
---------------------------------------------------------------
4588. A NOTE OF SPACES OF TEST AND GENERALIZED FUNCTIONS OF POISSON
WHITE NOISE
E. Lytvynov
The paper is devoted to construction and investigation of some
riggings of
the $L^2$-space of Poisson white noise. A particular attention is
paid to the
existence of a continuous version of a function from a test space,
and to the
property of an algebraic structure under pointwise multiplication of
functions
from a test space.
http://front.math.ucdavis.edu/math.PR/0608383
---------------------------------------------------------------
4589. ORDER OF CURRENT VARIANCE AND DIFFUSIVITY IN THE ASYMMETRIC
SIMPLE EXCLUSION PROCESS
Marton Balazs and Timo Seppalainen
We prove that the variance of the current across a characteristic is
of order
t^{2/3} in a stationary asymmetric simple exclusion process, and that
the
diffusivity has order t^{1/3}. The proof proceeds via couplings to
show the
corresponding results for the expected deviations and variance of a
second
class particle.
http://front.math.ucdavis.edu/math.PR/0608400
---------------------------------------------------------------
4590. LEVY PROCESSES, GENERATORS
Sakhnovich Lev
For a broad class of the Levy processes the new form (convolution
type) of
the infinitesimal generators is introduced. It leads to the new
notions: a
truncated generator, a quasi-potential. The probability of the Levy
process
remaining within the given domain is estimated.
http://front.math.ucdavis.edu/math.PR/0608402
---------------------------------------------------------------
4591. EQUIVALENCE OF ENSEMBLES FOR TWO-SPECIES ZERO-RANGE INVARIANT
MEASURES
Stefan Grosskinsky
We study the equivalence of ensembles for stationary measures of
interacting
particle systems with two conserved quantities and unbounded local
state space.
The main motivation is a condensation transition in the zero-range
process
which has recently attracted attention. Establishing the equivalence of
ensembles via convergence in specific relative entropy, we derive the
phase
diagram for the condensation transition, which can be understood in
terms of
the domain of grand-canonical measures. Of particular interest, also
from a
mathematical point of view, are the convergence properties of the
Gibbs free
energy on the boundary of that domain, involving large deviations and
multivariate local limit theorems of subexponential distributions.
http://front.math.ucdavis.edu/math-ph/0608029
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4592. VARIATIONAL INEQUALITIES IN HILBERT SPACES WITH MEASURES AND
OPTIMAL STOPPING
Viorel Barbu and Carlo Marinelli
We study the existence theory for parabolic variational inequalities in
weighted $L^2$ spaces with respect to excessive measures associated
with a
transition semigroup. We characterize the value function of optimal
stopping
problems for finite and infinite dimensional diffusions as a generalized
solution of such a variational inequality. The weighted $L^2$ setting
allows us
to cover some singular cases, such as optimal stopping for stochastic
equations
with degenerate diffusion coefficient. As an application of the
theory, we
consider the pricing of American-style contingent claims. Among
others, we
treat the cases of assets with stochastic volatility, of path-dependent
payoffs, and of interest-rate derivatives.
http://front.math.ucdavis.edu/math.AP/0608379
---------------------------------------------------------------
4593. MIXED POWERS OF GENERATING FUNCTIONS
Manuel Lladser
Given an integer m>=1, let || || be a norm in R^{m+1} and let S
denote the
set of points with nonnegative coordinates in the unit sphere with
respect to
this norm. Consider for each 1<= j<= m a function f_j(z) that is
analytic in an
open neighborhood of the point z=0 in the complex plane and with
possibly
negative Taylor coefficients. Given a vector n=(n_0,...,n_m) with
nonnegative
integer coefficients, we develop a method to systematically associate a
parameter-varying integral to study the asymptotic behavior of the
coefficient
of z^{n_0} of the Taylor series of (f_1(z))^{n_1}...(f_m(z))^{n_m},
as ||n||
tends to infinity. The associated parameter-varying integral has a
phase term
with well specified properties that make the asymptotic analysis of the
integral amenable to saddle-point methods: for many directions d in
S, these
methods ensure uniform asymptotic expansions for the Taylor
coefficient of
z^{n_0} of (f_1(z))^{n_1}...(f_m(z))^{n_m}, provided that n/||n|| stays
sufficiently close to d as ||n|| blows up to infinity. Our method finds
applications in studying the asymptotic behavior of the coefficients
of a
certain multivariable generating functions as well as in problems
related to
the Lagrange inversion formula for instance in the context random
planar maps.
http://front.math.ucdavis.edu/math.CO/0608398
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4594. LOCALIZED LARGE SUMS OF RANDOM VARIABLES
Kevin Ford and Gerald Tenenbaum
We study large partial sums, localized with respect to the sums of
variances,
of a sequence of centered random variables. An application is given
to the
distribution of prime factors of typical integers.
http://front.math.ucdavis.edu/math.PR/0608411
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4595. BOUNDARY PARTITIONS IN TREES AND DIMERS
Richard W. Kenyon and David B. Wilson
We study groves on planar graphs, which are forests in which every tree
contains one or more of a special set of vertices on the outer face,
referred
to as nodes. Each grove partitions the set of nodes. When a random
grove is
selected, we show how to compute the various partition probabilities as
functions of the electrical properties of the graph when viewed as a
resistor
network. We prove that for any partition sigma, Pr[grove has type
sigma] /
Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise
resistances between the nodes, and Pr[grove has type sigma] / Pr
[grove has
maximal number of trees] is an integer-coefficient polynomial in the
entries of
the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient
polynomial formulas for the pairings of chains in the double-dimer
model. We
show that the distribution of pairings of contour lines in the
Gaussian free
field with certain natural boundary conditions is identical to the
distribution
of pairings in the scaling limit of the double-dimer model. These
partition
probabilities are relevant to multichordal SLE_2, SLE_4, and SLE_8.
http://front.math.ucdavis.edu/math.PR/0608422
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4596. EXACT CONNECTIONS BETWEEN CURRENT FLUCTUATIONS AND THE SECOND
CLASS PARTICLE IN A CLASS OF DEPOSITION MODELS
Marton Balazs and Timo Seppalainen
We consider a large class of nearest neighbor attractive stochastic
interacting systems that includes the asymmetric simple exclusion,
zero range,
bricklayers' and the symmetric K-exclusion processes. We provide
exact formulas
that connect particle flux (or surface growth) fluctuations to the
two-point
function of the process and to the motion of the second class
particle. Such
connections have only been available for simple exclusion where they
were of
great use in particle current fluctuation investigations.
http://front.math.ucdavis.edu/math.PR/0608437
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4597. ON PRODUCTS OF RANDOM MATRICES AND CERTAIN HECKE ALGEBRAS
ASSOCIATED WITH GROUPS OF $2\TIMES 2$ MATRICES
Jafar Shaffaf
The determination of the density functions for products of random
elements
from specified classes of matrices is a basic problem in random
matrix theory
and is also of interest in theoretical physics. For connected simple
Lie groups
of $2\times 2$ matrices and conjugacy and spherical classes a
complete solution
is given here. The problem/solution can be re-stated in terms of the
structure
of certain Hecke algebras attached to groups of $2\times 2$ matrices.
http://front.math.ucdavis.edu/math.RT/0608440
---------------------------------------------------------------
4598. ASYMPTOTIC BEHAVIOR OF A GENERALIZED TCP CONGESTION AVOIDANCE
ALGORITHM
Teunis J. Ott and Jason Swanson
The Transmission Control Protocol (TCP) is a Transport Protocol used
in the
Internet. Ott has introduced a more general class of candidate Transport
Protocols called "protocols in the TCP Paradigm". The long run
objective of
studying this larger class is to find protocols with promising
performance
characteristics. This paper studies Markov chain models derived from
protocols
in the TCP Paradigm. Protocols in the TCP Paradigm, as TCP, protect
the network
from congestion by reducing the "Congestion Window" (the amount of
data allowed
to be sent but not yet acknowledged) when there is packet loss or packet
marking, and increasing it when there is no loss. When loss of different
packets are assumed to be independent events and the probability p of
loss is
assumed to be constant, the protocol gives rise to a Markov chain
{W_n}, where
W_n is the size of the congestion window after the transmission of
the n-th
packet. For a wide class of such Markov chains, we prove weak
convergence
results, after appropriate rescaling of time and space, as p tends to
0. The
limiting processes are defined by stochastic differential equations.
Depending
on certain parameter values, the stochastic differential equation can
define an
Ornstein-Uhlenbeck process or can be driven by a Poisson process.
http://front.math.ucdavis.edu/math.PR/0608476
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4599. MEASURE CONCENTRATION OF MARKOV TREE PROCESSES
Leonid Kontorovich
We prove an apparently novel concentration of measure result for
Markov tree
processes. The bound we derive reduces to the known bounds for Markov
processes
when the tree is a chain, thus strictly generalizing the known Markov
process
concentration results. We employ several techniques of potential
independent
interest, especially for obtaining similar results for more general
directed
acyclic graphical models.
http://front.math.ucdavis.edu/math.PR/0608511
---------------------------------------------------------------
4600. AN INTRINSIC METRIC FOR POWER SPECTRAL DENSITY FUNCTIONS
Tryphon T. Georgiou
We present an intrinsic metric that quantifies distances between power
spectral density functions. The metric was derived by the author in a
recent
arXiv-report (math.OC/0607026) as the geodesic distance between spectral
density functions with respect to a particular pseudo-Riemannian metric
motivated by a quadratic prediction problem. We provide an independent
verification of the metric inequality and discuss certain key
properties of the
induced topology.
http://front.math.ucdavis.edu/math.OC/0608486
---------------------------------------------------------------
4601. ON VARYING INCUBATION PERIODS IN A DYNAMICAL MODEL
Arni S. R. Srinivasa Rao
We consider previously well-known models in epidemiology where the
parameter
for incubation period is used as one of the important components to
explain the
dynamics of the variables. Such models are extended here to explain the
dynamics with respect to a given therapy that prolongs the incubation
period. A
deconvolution method is demonstrated for estimation of parameters in the
situations when no-therapy and multiple therapies are given to the
infected
population. The models and deconvolution method are extended in order
to study
the impact of therapy in age-structured populations. A generalisation
for a
situation when n- types of therapies are available is given.
http://front.math.ucdavis.edu/q-bio.QM/0608028
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4602. TIME CHANGE APPROACH TO GENERALIZED EXCURSION MEASURES, AND
ITS APPLICATION TO LIMIT THEOREMS
P. J. Fitzsimmons and K. Yano
It is proved that generalized excursion measures can be constructed
via time
change of Ito's Brownian excursion measure. A tightness-like
condition on
strings is introduced to prove a convergence theorem of generalized
excursion
measures. The convergence theorem is applied to obtain a conditional
limit
theorem, a kind of invariance principle where the limit is the Bessel
meander.
http://front.math.ucdavis.edu/math.PR/0608530
---------------------------------------------------------------
4603. GROWTH AND ROUGHNESS OF THE INTERFACE FOR BALLISTIC DEPOSITION
Mathew D. Penrose
In ballistic deposition (BD), $(d+1)$-dimensional particles fall
sequentially
at random towards an initially flat, large but bounded $d$-
dimensional surface,
and each particle sticks to the first point of contact. For both
lattice and
continuum BD, a law of large numbers in the thermodynamic limit
establishes
convergence of the mean height and surface width of the interface to
constants
$h(t)$ and $w(t)$, respectively, depending on time $t$. We show that
$h(t)$ is
asymptotically linear in $t$, while $w(t)$ grows at least
logarithmically in
$t$ when $d=1$. We also give duality results saying that the height
above the
origin for deposition onto an initially flat surface is
equidistributed with
the maximum height for deposition onto a surface growing from a
single site.
http://front.math.ucdavis.edu/math.PR/0608540
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4604. COMPLETE LOCALISATION IN THE PARABOLIC ANDERSON MODEL WITH
PARETO-DISTRIBUTED POTENTIAL
Wolfgang Konig and Peter Morters and Nadia Sidorova
The parabolic Anderson problem is the Cauchy problem for the heat
equation
$\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times
{\mathbb
Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We
consider
independent and identically distributed potential variables, such that
Prob$(\xi(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is
initially
localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at
any large
time $t$, the solution is completely localised in a single point with
high
probability. More precisely, we find a random process $(Z_t \colon t
\ge 0)$
with values in $\Z^d$ such that $\lim_{t \uparrow\infty}
u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify
the
asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.
http://front.math.ucdavis.edu/math.PR/0608544
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4605. PROPAGATION TIME IN STOCHASTIC COMMUNICATION NETWORKS
Jonathan Rowe and Boris Mitavskiy
Dynamical processes taking place on networks have received much
attention in
recent years, especially on various models of random graphs
(including small
world and scale free networks). They model a variety of phenomena,
including
the spread of information on the Internet; the outbreak of epidemics
in a
spatially structured population; and communication between randomly
dispersed
processors in an ad hoc wireless network. Typically, research has
concentrated
on the existence and size of a large connected component
(representing, say,
the size of the epidemic) in a percolation model, or uses differential
equations to study the dynamics using a mean-field approximation in
an infinite
graph. Here we investigate the time taken for information to
propagate from a
single source through a finite network, as a function of the number
of nodes
and the network topology. We assume that time is discrete, and that
nodes
attempt to transmit to their neighbors in parallel, with a given
probability of
success. We solve this problem exactly for several specific
topologies, and use
a large-deviation theorem to derive general asymptotic bounds, which
apply to
any family of networks where the diameter grows at least
logarithmically in the
number of nodes. We use these bounds, for example, to show that a
scale-free
network has propagation time logarithmic in the number of nodes, and
inversely
proportional to the transmission probability.
http://front.math.ucdavis.edu/math.PR/0608561
---------------------------------------------------------------
4606. RANDOM WALK ON GRAPHS WITH REGULAR RESISTANCE AND VOLUME GROWTH
Andras Telcs
In this paper characterizations of graphs satisfying heat kernel
estimates
for a wide class of space-time scaling functions are given. The
equivalence of
the two-sided heat kernel estimate and the parabolic Harnack
inequality is also
shown via the equivalence of the upper (lower) heat kernel estimate
to the
parabolic mean value (and super mean value) inequality.
http://front.math.ucdavis.edu/math.PR/0608594
---------------------------------------------------------------
4607. LOOPS STATISTICS IN THE TOROIDAL HONEYCOMB DIMER MODEL
C\'edric Boutillier and B\'eatrice de Tili\`ere
The dimer model on a graph embedded in the torus can be interpreted as a
collection of random self-avoiding loops. We prove that when the mesh
of the
graph tends to zero, and the aspect of the torus is fixed, the
winding number
of this collection of loops converges in law to a two-dimensional
discrete
Gaussian distribution. This is the first mathematical proof of a
result known
to physicists in the context of toroidal 2-D critical models, and
their mapping
to the massless free field on the torus.
http://front.math.ucdavis.edu/math.PR/0608600
---------------------------------------------------------------
4608. SUB-GAUSSIAN SHORT TIME ASYMPTOTICS FOR MEASURE METRIC
DIRICHLET SPACES
Andras Telcs
This paper presents estimates for the distribution of the exit time from
balls and short time asymptotics for measure metric Dirichlet spaces.
The
estimates cover the classical Gaussian case, the sub-diffusive case
which can
be observed on particular fractals and further less regular cases as
well. The
proof is based on a new chaining argument and it is free of volume
growth
assumptions.
http://front.math.ucdavis.edu/math.PR/0608615
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4609. SMILE ASYMPTOTICS II: MODELS WITH KNOWN MOMENT GENERATING FUNCTION
Shalom Benaim and Peter Friz
In a recent article the authors obtained a formula which relates
explicitly
the tail of risk neutral returns with the wing behavior of the Black
Scholes
implied volatility smile. In situations where precise tail
asymptotics are
unknown but a moment generating function is available we first
establish, under
easy-to-check conditions, tail asymptoics on logarithmic scale as soft
applications of standard Tauberian theorems. Such asymptotics are
enough to
make the tail-wing formula work and we so obtain a version of Lee's
moment
formula with the novel guarantee that there is indeed a limiting
slope when
plotting implied variance against log-strike. We apply these results to
time-changed Levy models and the Heston model. In particular, the
term-structure of the wings can be analytically understood.
http://front.math.ucdavis.edu/math.PR/0608619
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4610. CONSTRAINED EXCHANGEABLE PARTITIONS
Alexander Gnedin
For a class of random partitions of an infinite set a de Finetti-type
representation is derived, and in one special case a central limit
theorem for
the number of blocks is shown.
http://front.math.ucdavis.edu/math.PR/0608621
---------------------------------------------------------------
4611. THE PROBLEM OF SMALL UNILATERAL DEVIATIONS: THE EXISTENCE OF
DECAY EXPONENTS
G.Molchan
Let x(s), s in R^d be a Gaussian self-similar random process of index
H. We
consider the problem of log-asymptotics for the probability p(T) that
x(s),
x(0)=0 does not exceed a fixed level in a star-shaped increasing
domain T*U as
T >> 1. General conditions are given to guarantee the existence of
the limit of
(-log p(T))/L(T) as T >> 1 for a slowly increasing function L(T).
http://front.math.ucdavis.edu/math.PR/0608630
---------------------------------------------------------------
4612. SMALL-TIME AND TAIL ASYMPTOTICS FOR A TIME-CHANGED DIFFUSION,
WITH APPLICATIONS TO LOCAL VOLATILITY AND CEV-HESTON MODELS
Martin Forde
Building on an insight in Carr&Lee\cite{CarrLee03}, we establish a
simple
relationship between the prices of Eigenfunction contracts and the
mgf of the
time-change, under a model where the Stock price is a diffusion process
evaluated at an independent stochastic clock. In particular, we
characterize
the tail behaviour (Theorems \ref{thm:CEVtail}, \ref
{thm:CEVstocvoltail}) and
the small-time behaviour (Theorem \ref{thm:CEVLargeDev}) of a CEV
diffusion,
and a time-changed CEV diffusion. We describe the small-time
behaviour of the
Heston subordinator (Theorem \ref{thm:HestonLDP}) using large deviations
theory, which shows that the previous three results are applicable to
the
CEV-Heston stochastic volatility model discussed in Atlan&Leblanc\cite
{Atlan}.
We also use a general result by Norris&Stroock\cite{NorrisStroock} to
characterize the tail behaviour of the transition densities for a
general
Dupire local volatility model\cite{Dupire94}, in terms of an Energy
functional
(Corollary \ref{cor:SN}). Finally, in section 3, we discuss
calibration issues
for a time-changed diffusion model. Specifically, for the time-
changed CEV
model, we show that if we wish to apply an extended version of the
Carr-Lee\cite{CarrLee03} methodology to infer the characteristic
function of
the time-change from an observed single-maturity smile, then the
tails of the
distribution of the time-change have to have sub-exponential
behaviour, or else
we have to use \textit{analytic continuation}
http://front.math.ucdavis.edu/math.PR/0608634
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4613. CENTRAL LIMIT THEOREMS FOR NON-INVERTIBLE MEASURE PRESERVING MAPS
Michael C. Mackey and Marta Tyran-Kaminska
We establish a new functional central limit theorem result for non-
invertible
measure preserving maps that are not necessarily ergodic, using the
Perron-Frobenius operator. We apply the result to asymptotically
periodic
transformations and give an extensive specific example of asymptotically
periodic transformations by using the tent map.
http://front.math.ucdavis.edu/math.PR/0608637
---------------------------------------------------------------
4614. FIRST-PASSAGE COMPETITION WITH DIFFERENT SPEEDS: POSITIVE
DENSITY FOR BOTH SPECIES IS IMPOSSIBLE
Olivier Garet (MAPMO) and R\'{e}gine Marchand (IECN)
Consider two epidemics whose expansions on $\mathbb{Z}^d$ are
governed by two
families of passage times that are distinct and stochastically
comparable. We
prove that when the weak infection survives, the space occupied by
the strong
one is almost impossible to detect: for instance, it could not be
observed by a
medium resolution satellite. We also recover the same fluctuations
with respect
to the asymptotic shape as in the case where the weak infection
evolves alone.
In dimension two, we prove that one species finally occupies a set
with full
density, while the other one only occupies a set of null density. We
also prove
that the H\"{a}ggstr\"{o}m-Pemantle non-coexistence result "except
perhaps for
a denumerable set" can be extended to families of stochastically
comparable
passage times indexed by a continuous parameter.
http://front.math.ucdavis.edu/math.PR/0608667
---------------------------------------------------------------
4615. CAPACITIVE FLOWS ON A 2D RANDOM NET
Olivier Garet (MAPMO)
This paper concerns maximal flows on $\mathbb{Z}^2$ traveling from a
convex
set to infinity, the flows being restricted by a random capacity. For
every
compact convex set $A$, we prove that the maximal flow $\Phi(nA)$
between $nA$
and infinity is such that $\Phi(nA)/n$ almost surely converges to the
integral
of a deterministic function over the boundary of $A$. The limit can
also be
interpreted as the optimum of a deterministic continuous max-flow
problem. We
derive some properties of the infinite cluster in supercritical
Bernoulli
percolation.
http://front.math.ucdavis.edu/math.PR/0608676
---------------------------------------------------------------
4616. MODIFIED LOG-SOBOLEV INEQUALITIES AND ISOPERIMETRY
Alexander V. Kolesnikov
We find sufficient conditions for a probability measure $\mu$ to
satisfy an
inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_
{\R^d} f^2 d
\mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|
f|}
\Bigr) d \mu + A \int_{\R^d} f^2 d \mu, $$ where $F$ is concave and $c
$ (a cost
function) is convex. In particular, for every convex $\mu$ satisfying
$\int_{\R^d} e^{\epsilon |x|^{\alpha}} d\mu < \infty$ for some $
\epsilon>0$, $1
< \alpha \le 2$, we establish a family of tight inequalities
interpolating
between the $F$-Sobolev and modified log-Sobolev inequalities.
http://front.math.ucdavis.edu/math.PR/0608681
---------------------------------------------------------------
4617. CHARACTERIZATION OF LIL BEHAVIOR IN BANACH SPACE
Uwe Einmahl and Deli Li
In a recent paper by the authors a general result characterizing two-
sided
LIL behavior for real valued random variables has been established.
In this
paper, we show that there are analogous results in the Banach space
setting.
One of our main new tools is an improved Fuk-Nagaev type inequality
in Banach
space which should be of independent interest.
http://front.math.ucdavis.edu/math.PR/0608687
---------------------------------------------------------------
4618. TIGHTNESS FOR THE INTERFACES OF ONE-DIMENSIONAL VOTER MODELS
Samir Belhaouari and Thomas Mountford and Glauco Valle
We show that for the voter model on $\{0,1\}^{\mathbb{Z}}$
corresponding to a
random walk with kernel $p(\cdot)$ and starting from unanimity to the
right and
opposing unanimity to the left, a tight interface between 0's and 1's
exists if
$p(\cdot)$ has finite second moment but does not if $p(\cdot)$ fails
to have
finite moment of order $\alpha$ for some $\alpha <2$.
http://front.math.ucdavis.edu/math.PR/0608690
---------------------------------------------------------------
4619. RANDOM WALK IN RANDOM ENVIRONMENT WITH ASYMPTOTICALLY ZERO
PERTURBATION
M. V. Menshikov and Andrew R. Wade
We give criteria for ergodicity, transience and null recurrence for the
random walk in random environment on {0,1,2,...}, with reflection at the
origin, where the random environment is subject to a vanishing
perturbation.
Our results complement existing criteria for random walks in random
environments and for Markov chains with asymptotically zero drift,
and are
significantly different to these previously studied cases. Our method
is based
on a martingale technique - the method of Lyapunov functions.
http://front.math.ucdavis.edu/math.PR/0608696
---------------------------------------------------------------
4620. LOGARITHMIC SPEEDS FOR ONE-DIMENSIONAL PERTURBED RANDOM WALK IN
RANDOM ENVIRONMENT
M. V. Menshikov and Andrew R. Wade
We study the random walk in random environment on {0,1,2,...}, where the
environment is subject to a vanishing (random) perturbation. The two
particular
cases we consider are: (i) random walk in random environment
perturbed from
Sinai's regime; (ii) simple random walk with random perturbation. We
give
almost sure results on how far the random walker will be from the
origin after
a long time t, for almost every environment. We give both upper and
lower
almost sure bounds. These bounds are of order $(\log t)^\beta$, for $
\beta \in
(1,\infty)$, depending on the perturbation. In addition, in the
ergodic cases,
we give results on the rate of decay of the stationary distribution.
http://front.math.ucdavis.edu/math.PR/0608697
---------------------------------------------------------------
4621. ENTROPY METHOD FOR THE LEFT TAIL
Hyungsu Kim and Chul Ki Ko and Sungchul Lee
When we use the entropy method to get the tail bounds, typically the
left
tail bounds are not good comparing with the right ones. Up to now this
asymmetry has been observed many times. Surprisingly we find an
entropy method
for the left tail that works in the exactly same way that it works
for the
right tail.
http://front.math.ucdavis.edu/math.PR/0608706
---------------------------------------------------------------
4622. LIMITING LAWS OF LINEAR EIGENVALUE STATISTICS FOR UNITARY
INVARIANT MATRIX MODELS
L. Pastur
We study the variance and the Laplace transform of the probability
law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming
that the test
function of statistics is smooth enough and using the asymptotic
formulas by
Deift et al for orthogonal polynomials with varying weights, we show
first that
if the support of the Density of States of the model consists of two
or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining
the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the
Central
Limit Theorem would be valid, and we find the asymptotic form of the
Laplace
transform of the probability law in certain cases.
http://front.math.ucdavis.edu/math.PR/0608719
---------------------------------------------------------------
4623. FLUCTUATION PROPERTIES OF THE TASEP WITH PERIODIC INITIAL
CONFIGURATION
Alexei Borodin (1) and Patrik L. Ferrari (2) and Michael Pr\"ahofer
(2) and Tomohiro Sasamoto (3) ((1) Caltech, (2) TU-Muenchen, (3)
Chiba University)
We consider the joint distributions of particle positions for the
continuous
time totally asymmetric simple exclusion process (TASEP). They are
expressed as
Fredholm determinants with a kernel defining a signed determinantal
point
measure. We then consider certain periodic initial conditions and
determine the
kernel in the scaling limit. This result has been announced first in
a letter
by one of us and here we provide a self-contained derivation.
Connections to
last passage directed percolation and random matrices are also briefly
discussed.
http://front.math.ucdavis.edu/math-ph/0608056
---------------------------------------------------------------
4624. TAIL ESTIMATES FOR SUMS OF VARIABLES SAMPLED FROM A RANDOM WALK
Roy Wagner
We prove a tail estimate for the variable $\sum f(X_i)$, where $(X_i)
_i$ is
the trajectory of a random walk on a graph (or a reversible Markov
chain). The
estimate is in terms of the maximum of the function, its variance,
and the
spectral gap of the graph. Our proof is more elementary than other
proofs in
the literature, and for some parameter regimes our results are
sharper. We
obtain Bernstein and Bennett-type inequalitis, as well as an
inequality for
subgaussian variables.
http://front.math.ucdavis.edu/math.PR/0608740
---------------------------------------------------------------
4625. TWO PARAMETERS CIRCULAR ENSEMBLES AND JACOBI-TRUDI TYPE
FORMULAS FOR JACK FUNCTIONS OF RECTANGULAR SHAPES
Sho Matsumoto
Jack function theory is useful for the calculation of the moment of the
characteristic polynomials in Dyson's circular $\beta$-ensembles (C$
\beta$E).
We define a $q$-analogue of the C$\beta$E and calculate moments of
characteristic polynomials via Macdonald function theory. By this
$q$-deformation, the asymptotics calculation of these moments becomes
simple
and the ordinary C$\beta$E case is recovered as $q \to 1$. Further,
by using a
hyperdeterminant which is a simple generalization of a determinant,
we give a
Jacobi-Trudi type formula for Jack symmetric functions of rectangular
shapes.
http://front.math.ucdavis.edu/math.PR/0608751
---------------------------------------------------------------
4626. NUMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS
Bernard Shiffman and Steve Zelditch
Our main results are asymptotic formulas for the variance of the number
$\mathcal{N}^U_N$ of zeros of $m$ Gaussian random polynomials of
degree $N$ in
an open set $U\subset C^m$ with smooth boundary as the degree $N\to
\infty$, and
more generally for the zeros of $m$ random holomorphic sections of
high powers
of any positive line bundle over any $m$-dimensional compact K\"ahler
manifold.
Our result for number statistics states that the variance of the number
$\mathcal{N}^U_N$ of zeros in $U$ is asymptotic to $N^{m-1/2} \nu_{mm}
Vol(\partial U)$, where $\nu_{mm}$ is a universal constant depending
only on
the dimension $m$. We also give variance results for $Vol(Z^k_N\cap U)
$, where
$Z^k_N$ denotes the set of simultaneous zeros of $k<m$ random degree-$N$
polynomials, as well as for the linear statistics $(Z^k_N,\phi)$,
where $\phi$
is a smooth test form.
http://front.math.ucdavis.edu/math.CV/0608743
---------------------------------------------------------------
4627. QUANTUM STOCHASTIC CONVOLUTION COCYCLES
Adam Skalski
A concept of quantum stochastic convolution cocycle is introduced and
studied
in two different contexts -- purely algebraic and operator space
theoretic. A
quantum stochastic convolution cocycle is a quantum stochastic
process on a
coalgebra satisfying the convolution cocycle relation and the initial
condition
given by the counit. The notion generalises that of quantum Levy
process, which
in turn is a noncommutative probability counterpart of classical Levy
process
on a group.
Convolution cocycles arise as solutions of quantum stochastic
differential
equations. In turn every sufficiently regular cocycle satisfies an
equation of
that type. This is proved along with the corresponding existence and
uniqueness
of solutions for coalgebraic quantum stochastic differential
equations. The
stochastic generators of unital *-homomorphic cocycles are
characterised in
terms of structure maps on a *-bialgebra. This yields a simple proof
of the
Schurmann Reconstruction Theorem for a quantum Levy process; it also
yields a
topological version for a quantum Levy process on a C*-bialgebra.
Precise
characterisation of the stochastic generators of completely positive and
contractive quantum stochastic convolution cocycles in the C*-
algebraic context
is given, leading to some dilation results. A few examples are
presented and
some interpretations offered for quantum stochastic convolution
cocycles and
their stochastic generators on different types of *-bialgebra.
http://front.math.ucdavis.edu/math.OA/0608756
---------------------------------------------------------------
4628. THE BURKHOLDER-DAVIS-GUNDY INEQUALITY FOR ENHANCED MARTINGALES
Peter Friz and Nicolas Victoir
Multi-dimensional continuous local martingales, enhanced with their
stochastic area process, give rise to geometric rough paths with a.s.
finite
homogenous p-variation, p>2. Here we go one step further and establish
quantitative bounds of the p-variation norm in the form of a BDG
inequality.
Our proofs are based on old ideas by Lepingle. We also discuss
geodesic and
piecewise linear approximations.
http://front.math.ucdavis.edu/math.PR/0608783
---------------------------------------------------------------
4629. COMPLEX DETERMINANTAL PROCESSES AND H1 NOISE
Brian Rider and Balint Virag
For the plane, sphere, and hyperbolic plane we consider the canonical
invariant determinantal point processes with intensity rho dnu, where
nu is the
corresponding invariant measure. We show that as rho converges to
infinity,
after centering, these processes converge to invariant H1 noise. More
precisely, for all functions f in the interesection of H1(nu) and L1
(nu) the
distribution of sum f(z) - rho/pi integral f dnu converges to
Gaussian with
mean 0 and variance given by ||f||_H1^2 / (4 pi).
http://front.math.ucdavis.edu/math.PR/0608785
---------------------------------------------------------------
4630. STOCHASTIC LAGRANGIAN TRANSPORT AND GENERALIZED RELATIVE ENTROPIES
Peter Constantin and Gautam Iyer
We discuss stochastic representations of advection diffusion
equations with
variable diffusivity, stochastic integrals of motion and generalized
relative
entropies.
http://front.math.ucdavis.edu/math.AP/0608797
---------------------------------------------------------------
4631. ON UNIFORMLY SUBELLIPTIC OPERATORS AND STOCHASTIC AREA
Peter Friz and Nicolas Victoir
We consider uniformly subelliptic operators on certain unimodular Lie
groups
of polynomial growth. It was shown by Saloff-Coste and Stroock that
classical
results of De Giorgi, Nash, Moser, Aronson extend to this setting. It
was then
observed by Sturm that many proofs extend naturally to the setting of
locally
compact Dirichlet spaces. We relate these results to what is known as
rough
path theory by showing that they provide a natural and powerful analytic
machinery for construction and study of (random) geometric Hoelder
rough paths.
(In particular, we obtain a simple construction of the Lyons-Stoica
stochastic
area for a diffusion process with uniformly elliptic generator in
divergence
form.) Our approach then enables us to establish a number of far-
reaching
generalizations of classical theorems in diffusion theory including
Wong-Zakai
approximations, Freidlin-Wentzell sample path large deviations and the
Stroock-Varadhan support theorem. The latter was conjectured by T.
Lyons in his
recent St. Flour lecture.
http://front.math.ucdavis.edu/math.PR/0609007
---------------------------------------------------------------
4632. COUNTING KNIGHT'S TOURS THROUGH THE RANDOMIZED WARNSDORFF RULE
H\'ector Cancela (INCO and UdelaR) and Ernesto Mordecki (CMAT and
UdelR)
We give an estimate of the number of geometrically distinct open
tours $\G$
for a knight on a chessboard. We use a randomization of Warnsdorff
rule to
implement importance sampling in a backtracking scheme, correcting
the observed
bias of the original rule, according to the proposed principle that
``most
solutions follow Warnsdorff rule most of the time''. After some
experiments in
order to test this principle, and to calibrate a parameter,
interpreted as a
distance of a general solution from a Warnsdorff solution, we
conjecture that
$\G=1.22\times 10^{15}$.
http://front.math.ucdavis.edu/math.PR/0609009
---------------------------------------------------------------
4633. CONLEY INDEX FOR RANDOM DYNAMICAL SYSTEMS
Zhenxin Liu
Conley index theory is a very powerful tool in the study of dynamical
systems, differential equations and bifurcation theory. In this
paper, we make
an attempt to generalize the Conley index to discrete random
dynamical systems.
And we mainly follow the Conley index for maps given by Franks and
Richeson in
\cite{Fra}. Furthermore, we simply discuss the relations of isolated
invariant
sets between time-continuous random dynamical systems and the
corresponding
time-$h$ maps. For applications we give several examples to
illustrate our
results.
http://front.math.ucdavis.edu/math.DS/0609011
---------------------------------------------------------------
4634. THE CHOQUET-DENY EQUATION IN A BANACH SPACE
W. Jaworski and M. Neufang
Let $G$ be a locally compact group and $\pi$ a representation of $G$ by
weakly^* continuous isometries acting in a dual Banach space $E$.
Given a
probability measure $\mu$ on $G$ we study the Choquet-Deny equation
$\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation
form the
range of a projection of norm 1 and can be represented by means of a
``Poisson
formula'' on the same boundary space that is used to represent the
bounded
harmonic functions of the random walk of law $\mu$. The relation
between the
space of solutions of the Choquet-Deny equation in $E$ and the space
of bounded
harmonic functions can be understood in terms of a construction
resembling the
$W^*$-crossed product and coinciding precisely with the crossed
product in the
special case of the Choquet-Deny equation in the space $E=B(L^2(G))$
of bounded
linear operators on $L^2(G)$. Other general properties of the Choquet-
Deny
equation in a Banach space are also discussed.
http://front.math.ucdavis.edu/math.FA/0609035
---------------------------------------------------------------
4635. THE EMERGENCE OF THE DETERMINISTIC HODGKIN-HUXLEY EQUATIONS AS
A LIMIT FROM THE UNDERLYING STOCHASTIC ION-CHANNEL MECHANISM
Tim D. Austin (UC and Los Angeles)
The mechanism of transmission of an action potential along the axon of a
neuron has been heavily studied by biophysicists for over fifty
years, and
several detailed models now exist to describe axonal behaviour. Older
models
have been purely deterministic, predicting behaviour by various
representative
quantities evolving according to differential equations. More recently,
however, stochastic elements have been included to represent more
faithfully a
large number of unpredictable sub-processes at work at the scale of
individual
protein molecules within the axon.
In this paper we consider the classical differential equations of
Hodgkin and
Huxley and a natural refinement of them to include a layer of stochastic
behaviour, modelled by a large number of finite-state-space Markov
processes
coupled to a simple modification of the original Hodgkin-Huxley PDE.
We first
prove existence, uniqueness and some regularity for the stochastic
process, and
then show that in a suitable limit as the number of stochastic
components of
the stochastic model increases and their individual contributions
decrease the
process that they determine converges to the trajectory predicted by the
deterministic PDE, uniformly up to finite time horizons in
probability. In a
sense, this verifies the consistency of the deterministic and stochastic
processes.
http://front.math.ucdavis.edu/math.PR/0609068
---------------------------------------------------------------
4636. A TANAKA FORMULA FOR THE DERIVATIVE OF INTERSECTION LOCAL TIME
IN $\REALS^1$
Greg Markowsky
Let $B_t$ be a one dimensional Brownian motion, and let $\alpha'$
denote the
derivative of the intersection local time of $B_t$ as defined in Jay
Rosen's
work (see references). The object of this paper is to prove the
following
formula $(1/2)\alpha'_t(x) + (1/2)sgn(x)t = \int_0^t L_s^{B_s - x}dB_s -
\int_0^t sgn(B_t - B_u - x) du$ which was given as a formal identity
by Rosen
without proof.
http://front.math.ucdavis.edu/math.PR/0609084
---------------------------------------------------------------
4637. THE SMALL-TIME BEHAVIOUR OF DIFFUSION AND TIME-CHANGED
DIFFUSION PROCESSES ON THE LINE
Martin Forde
Using a result by Doss\cite{Doss77} and the G\"{a}rtner-Ellis
theorem, we
prove, by bounding the It\^{o} map, that under certain bounds on the
diffusion
coefficients, the transition densities of a one-dimensional diffusion
process
satisfy the \textit{large deviation principle} (Theorem \ref
{thm:Tails}). We
prove a similar result for a diffusion proces on the line evaluated
at an
independent stochastic clock, when the arithmetic average of the time-
change
also satisfies the LDP (Theorem (\ref{thm:stocvoltail}), as it does
for the
well know Cox-Ingersoll-Ross subordinator (Theorem \ref{thm:HestonLDP}).
http://front.math.ucdavis.edu/math.PR/0609117
---------------------------------------------------------------
4638. APPROXIMATION BY THE DICKMAN DISTRIBUTION AND QUASI-
LOGARITHMIC COMBINATORIAL STRUCTURES
Bruno Nietlispach
Quasi-logarithmic combinatorial structures are a class of decomposable
combinatorial structures which extend the logarithmic class. In order
to obtain
asymptotic approximations to their component spectrum, it is
necessary first to
establish an approximation to the sum of an associated sequence of
independent
random variables in terms of the Dickman distribution. This in turn
requires an
argument that refines the Mineka coupling by incorporating a blocking
construction, leading to exponentially sharper coupling rates for the
sums in
question. Applications include distributional limit theorems for the
size of
the largest component and for the vector of counts of the small
components in a
quasi-logarithmic combinatorial structure.
http://front.math.ucdavis.edu/math.CO/0609129
---------------------------------------------------------------
4639. ASYMPTOTIC DENSITY IN QUASI-LOGARITHMIC ADDITIVE NUMBER SYSTEMS
Bruno Nietlispach
We show that in quasi-logarithmic additive number systems all
partition sets
have asymptotic density, and we obtain a corresponding monadic second-
order
limit law for adequate classes of relational structures.
http://front.math.ucdavis.edu/math.CO/0609143
---------------------------------------------------------------
4640. EXPLORATION TREES AND CONFORMAL LOOP ENSEMBLES
Scott Sheffield
We construct and study the conformal loop ensembles CLE(kappa),
defined for
all kappa between 8/3 and 8, using branching variants of SLE(kappa)
called
exploration trees. The conformal loop ensembles are random
collections of
countably many loops in a planar domain that are characterized by
certain
conformal invariance and Markov properties. We conjecture that they
are the
scaling limits of various random loop models from statistical physics,
including the O(n) loop models.
http://front.math.ucdavis.edu/math.PR/0609167
---------------------------------------------------------------
4641. ANALYSIS OF TOP-SWAP SHUFFLING FOR GENOME REARRANGEMENTS
Nayantara Bhatnagar and Pietro Caputo and Prasad Tetali and Eric
Vigoda
We study Markov chains which model genome rearrangements. These
models are
useful for studying the equilibrium distribution of chromosomal
lengths, and
are used in methods for estimating genomic distances. The primary
Markov chain
studied in this paper is the top-swap Markov chain. The top-swap
chain is a
card-shuffling process with n cards divided over k decks, where the
cards are
ordered within each deck. A transition consists of choosing a random
pair of
cards, and if the cards lie in different decks, we cut each deck at
the chosen
card and exchange the tops of the two decks. We prove precise bounds
on the
relaxation time (inverse spectral gap) of the top-swap chain. In
particular, we
prove the relaxation time is of order n+k. This resolves an open
question of
Durrett.
http://front.math.ucdavis.edu/math.PR/0609171
---------------------------------------------------------------
4642. ESTIMATING HEAVY-TAIL EXPONENTS THROUGH MAX SELF-SIMILARITY
Stilian A. Stoev and George Michailidis and Murad S. Taqqu
In this paper, a novel approach to the problem of estimating the
heavy-tail
exponent alpha>0 of a distribution is proposed. It is based on the
fact that
block-maxima of size m of the independent and identically distributed
data
scale at a rate of m^{1/alpha}. This scaling rate can be captured
well by the
max-spectrum plot of the data that leads to regression based estimators.
Consistency and asymptotic normality of these estimators is
established under
mild conditions on the behavior of the tail of the distribution. The
results
are obtained by establishing bounds on the rate of convergence of
moment-type
functionals of heavy-tailed maxima. Such bounds often yield exact
rates of
convergence and are of independent interest. Practical issues on the
automatic
selection of tuning parameters for the estimators and corresponding
confidence
intervals are also addressed. Extensive numerical simulations show
that the
proposed method proves competitive for both small and large sample
sizes and
for a large range of tail exponents. The method is shown to be more
robust than
the classical Hill plot and is illustrated on two data sets of
insurance claims
and natural gas field sizes.
http://front.math.ucdavis.edu/math.ST/0609163
---------------------------------------------------------------
4643. A SIMPLE STABILITY CONDITION FOR RED USING TCP MEAN-FIELD MODELING
Julien Reynier (INRIA Rocquencourt)
Congestion on the Internet is an old problem but still a subject of
intensive
research. The TCP protocol with its AIMD (Additive Increase and
Multiplicative
Decrease) behavior hides very challenging problems; one of them is to
understand the interaction between a large number of users with delayed
feedback. This article will focus on two modeling issues of TCP which
appeared
to be important to tackle concrete scenarios when implementing the model
proposed in [Baccelli McDonald Reynier 02] firstly the modeling of
the maximum
TCP window size: this maximum can be reached quickly in many
practical cases;
secondly the delay structure: the usual Little-like formula behaves
really
poorly when queuing delays are variable, and may change dramatically the
evolution of the predicted queue size, which makes it useless to study
drop-tail or RED (Random Early Detection) mechanisms. Within proposed
TCP
modeling improvements, we are enabled to look at a concrete example
where RED
should be used in FIFO routers instead of letting the default drop-
tail happen.
We study mathematically fixed points of the window size distribution
and local
stability of RED. An interesting case is when RED operates at the
limit when
the congestion starts, it avoids unwanted loss of bandwidth and delay
variations.
http://front.math.ucdavis.edu/cs.NI/0609014
---------------------------------------------------------------
4644. ADAPTIVE WEAK APPROXIMATION OF DIFFUSIONS WITH JUMPS
E. Mordecki and A. Szepessy and R. Tempone and G. E. Zouraris
This work develops Monte Carlo Euler adaptive time stepping methods
for the
weak approximation problem of jump diffusion driven stochastic
differential
equations. The main result is the derivation of a new expansion for the
omputational error, with computable leading order term in a
posteriori form,
based on stochastic flows and discrete dual backward problems which
extends the
results in [STZ]. These expansions lead to efficient and accurate
computation
of error estimates. Adaptive algorithms for either stochastic time
steps or
quasi-deterministic time steps are described. Numerical examples show
the
performance of the proposed error approximation and of the described
adaptive
time-stepping methods.
http://front.math.ucdavis.edu/math.NA/0609186
---------------------------------------------------------------
4645. VERTEX DEGREE OF RANDOM GEOMETRIC GRAPH ON EXPONENTIALLY
DISTRIBUTED POINTS
Bhupendra Gupta
Let $X_1,X_2,...$ be an infinite sequence of i.i.d. random vectors
distributed exponentially with parameter $\lam .$ For each $y$ and $n
\geq 1,$
form a graph $G_n(y)$ with vertex set $V_n = \{X_1,...,X_n\},$ two
vertices are
connected if and only if edge distance between them is greater then $y
$, i.e,
$\|X_i-X_j\| \leq y.$ Almost-sure asymptotic rates of convergence/
divergence
are obtained for the minimum and maximum vertex degree of the random
geometric
graph, as the number of vertices becomes large $n,$ and the edge
distance
varies with the number of vertices.
http://front.math.ucdavis.edu/math.PR/0609193
---------------------------------------------------------------
4646. A FORMULA OF TOTAL PROBABILITY WITH INTERFERENCE TERM AND THE
HILBERT SPACE REPRESENTATION OF THE CONTEXTUAL KOLMOGOROVIAN MODEL
Andrei Khrennikov
We compare the classical Kolmogorov and quantum probability models.
We show
that the gap between these model is not so huge as it was commonly
believed.
The main structures of quantum theory (interference of probabilities,
Born's
rule, complex probabilistic amplitudes, Hilbert state space,
representation of
observables by operators) are present in a latent form in the
Kolmogorov model.
In particular, we obtain ``interference of probabilities'' without to
appeal to
the Hilbert space formalism. We interpret ``interference of
probabilities'' as
a perturbation (by a $\cos$-term) of the conventional formula of total
probability. Our classical derivation of quantum probabilistic
formalism can
stimulate applications of quantum methods outside of microworld : in
psychology, biology, economy,...
http://front.math.ucdavis.edu/math.PR/0609197
---------------------------------------------------------------
4647. LOWER BOUNDS FOR TAILS OF SUMS OF INDEPENDENT SYMMETRIC RANDOM
VARIABLES
Lutz Mattner
The approach of Kleitman (1970) and Kanter (1976) to multivariate
concentration function inequalities is generalized in order to obtain
for
deviation probabilities of sums of independent symmetric random
variables a
lower bound depending only on deviation probabilities of the terms of
the sum.
This bound is optimal up to discretization effects, improves on a
result of
Nagaev (2001), and complements the comparison theorems of Birnbaum
(1948) and
Pruss (1997). Birnbaum's theorem for unimodal random variables is
extended to
the lattice case.
http://front.math.ucdavis.edu/math.PR/0609200
---------------------------------------------------------------
4648. ON RANDOM CAMEO GRAPHS WITH INDEPENDENT EDGES PART I: PATH
CONNECTIVITY AND ESSENTIAL DIAMETER
Philippe Blanchard and Tyll Krueger and Madeleine Sirugue-Collin
We study growth properties of the number of paths of lenght k for a
variant
of Cameo graphs introduced in an earlier paper. Sharp results are
obtained for
threshold for the k-path connectivity and the essential diameter.
http://front.math.ucdavis.edu/math.PR/0609202
---------------------------------------------------------------
4649. THE EXACT VALUE FOR EUROPEAN OPTIONS ON A STOCK PAYING A
DISCRETE DIVIDEND
Jo\~{a}o Amaro de Matos and Rui Dil\~{a}o and Bruno Ferreira
In the context of a Black-Scholes economy and with a no-arbitrage
argument,
we derive arbitrarily accurate lower and upper bounds for the value
of European
options on a stock paying a discrete dividend. Setting the option
price error
below the smallest monetary unity, both bounds coincide, and we
obtain the
exact value of the option.
http://front.math.ucdavis.edu/math.PR/0609212
---------------------------------------------------------------
4650. A MARKOV CHAIN ON PERMUTATIONS WHICH PROJECTS TO THE PASEP
Sylvie Corteel and Lauren K. Williams
The partially asymmetric exclusion process (PASEP) is an important
model from
statistical mechanics which describes a system of interacting
particles hopping
left and right on a one-dimensional lattice of N sites. It is partially
asymmetric in the sense that the probability of hopping left is q
times the
probability of hopping right. Additionally, particles may enter from
the left
with probability alpha and exit from the right with probability beta.
It has been observed that the (unique) stationary distribution of
the PASEP
has remarkable connections to combinatorics -- see for example the
papers of
Derrida, Duchi and Schaeffer, and Corteel. Most recently we proved
that in fact
the (normalized) probability of being in a particular state of the
PASEP can be
viewed as a certain weight generating function for permutation
tableaux of a
fixed shape. (This result implies the previous combinatorial
results.) However,
our proof relied on the matrix ansatz of Derrida et al, and hence did
not give
an intuitive explanation of why one should expect the steady state
distribution
of the PASEP to involve such nice combinatorics.
In this paper we define a Markov chain -- which we call the PT
chain -- on
the set of permutation tableaux which projects to the PASEP in a very
strong
sense. This gives a new proof of our previous result which bypasses
the matrix
ansatz altogether. Furthermore, via the bijection from permutation
tableaux to
permutations, the PT chain can also be viewed as a Markov chain on the
symmetric group. Another nice feature of the PT chain is that it
possesses a
certain symmetry which extends the "particle-hole symmetry" of the
PASEP. More
specifically, this is a graph-automorphism on the state diagram of
the PT chain
which is an involution; this has a simple description in terms of
permutations.
http://front.math.ucdavis.edu/math.CO/0609188
---------------------------------------------------------------
4651. SHARP PROBABILITY ESTIMATES FOR GENERALIZED SMIRNOV STATISTICS
Kevin Ford
We give sharp, uniform estimates for the probability that the empirical
distribution function for n uniform-[0,1] random variables stays to
one side of
a given line.
http://front.math.ucdavis.edu/math.PR/0609224
---------------------------------------------------------------
4652. RENORMALIZATION AND CONVERGENCE IN LAW FOR THE DERIVATIVE OF
INTERSECTION LOCAL TIME IN R^2
Greg Markowsky
In this paper we will examine the derivative of intersection local
time of
Brownian motion and symmetric stable processes in $R^2$. These
processes do not
exist when defined in the canonical way. The purpose of this paper is to
exhibit the correct rate for renormaliztion of these processes.
http://front.math.ucdavis.edu/math.PR/0609265
---------------------------------------------------------------
4653. A SPECIAL SET OF EXCEPTIONAL TIMES FOR DYNAMICAL RANDOM WALK ON
$\Z^2$
Gideon Amir and Christopher Hoffman
Benjamini, \olle, Peres and Steif introduced the model of dynamical
random
walk on $\Z^d$ \cite{ds}. This is a continuum of random walks indexed
by a
parameter $t$. They proved that for $d=3,4$ there almost surely exist
$t$ such
that the random walk at time $t$ visits the origin infinitely often,
but for $d
\geq 5$ there almost surely do not exist such $t$.
Hoffman showed that for $d=2$ there almost surely exists $t$ such
that the
random walk at time $t$ visits the origin only finitely many times
\cite{H1}.
We refine the results of \cite{H1} for dynamical random walk on $\z^2
$, showing
that with probability one the are times when the origin is visited
only a
finite number of times while other points are visited infinitely often.
http://front.math.ucdavis.edu/math.PR/0609267
---------------------------------------------------------------
4654. OCCUPATION TIME LIMITS OF INHOMOGENEOUS POISSON SYSTEMS OF
INDEPENDENT PARTICLES
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
We prove functional limits theorems for the occupation time process of a
system of particles moving independently in $R^d$ according to a
symmetric
$\alpha$-stable L\'evy process, and starting off from an
inhomogeneous Poisson
point measure with intensity measure
$\mu(dx)=(1+|x|^{\gamma})^{-1}dx,\gamma>0$, and other related
measures. In
contrast to the homogeneous case $(\gamma=0)$, the system is not in
equilibrium
and ultimately it vanishes, and there are more different types of
occupation
time limit processes depending on arrangements of the parameters $
\gamma, d$
and $\alpha$. The case $\gamma<d<\alpha$ leads to an extension of
fractional
Brownian motion.
http://front.math.ucdavis.edu/math.PR/0609290
---------------------------------------------------------------
4655. ON STOCHASTIC CONTINUITY OF GENERALIZED DIFFUSION PROCESSES
CONSTRUCTED AS THE STRONG SOLUTION TO AN SDE
Ludmila L. Zaitseva
The comparison theorem for skew Brownian motions is proved. As the
corollary
we get the estimate on ${\Cal L}_1-$distance between two skew
Brownian motions
started from different points. Using this result we prove the continuous
dependence on starting point of one class of generalized diffusion
processes
constructed as the strong solution to an SDE.
http://front.math.ucdavis.edu/math.PR/0609305
---------------------------------------------------------------
4656. ON THE MARKOV PROPERTY OF STRONG SOLUTIONS TO SDE WITH
GENERALIZED COEFFICIENTS
Ludmila L. Zaitseva
We show the complete proof of the Markov property of the strong
solution to a
multidimensional SDE whose coefficients involve local time on a
hyperplane of
the unknown process.
http://front.math.ucdavis.edu/math.PR/0609307
---------------------------------------------------------------
4657. STATISTICAL ASPECTS OF THE FRACTIONAL STOCHASTIC CALCULUS
Ciprian A. Tudor and Frederi G. Viens
We apply the techniques of stochastic integration with respect to the
fractional Brownian motion and the theory of regularity and supremum
estimation
for stochastic processes to study the maximum likelihood estimator
(MLE) for
the drift parameter of stochastic processes satisfying stochastic
equations
driven by fractional Brownian motion with any level of Holder-
regularity (any
Hurst parameter). We prove existence and strong consistency of the
MLE for
linear and nonlinear equations. We also prove that a version of the
MLE using
only discrete observations is still a strongly consistent estimator.
http://front.math.ucdavis.edu/math.ST/0609295
---------------------------------------------------------------
4658. THE SIMPLE RANDOM WALK AND MAX-DEGREE WALK ON A DIRECTED GRAPH
Ravi Montenegro
We show bounds on total variation and $L^{\infty}$ mixing times,
spectral gap
and magnitudes of the complex valued eigenvalues of a general (non-
reversible
non-lazy) Markov chain with a minor expansion property. This leads to
the first
known bounds for the non-lazy simple and max-degree walks on a
(directed)
graph, and even in the lazy case they are the first bounds of the
optimal
order. In particular, it is found that within a factor of two or
four, the
worst case of each of these mixing time and eigenvalue quantities is
a walk on
a cycle with clockwise drift.
http://front.math.ucdavis.edu/math.CO/0609303
---------------------------------------------------------------
4659. REGULARITY OF TRANSITION SEMIGROUPS ASSOCIATED TO A 3D
STOCHASTIC NAVIER-STOKES EQUATION
F. Flandoli and M. Romito
A 3D stochastic Navier-Stokes equation with a suitable non degenerate
additive noise is considered. The regularity in the initial
conditions of every
Markov transition kernel associated to the equation is studied by a
simple
direct approach. A by-product of the technique is the equivalence of all
transition probabilities associated to every Markov transition kernel.
http://front.math.ucdavis.edu/math.PR/0609317
---------------------------------------------------------------
4660. EXISTENCE OF MARTINGALE AND STATIONARY SUITABLE WEAK SOLUTIONS
FOR A STOCHASTIC NAVIER-STOKES SYSTEM
M. Romito
The existence of suitable weak solutions of 3D Navier-Stokes equations,
driven by a random body force, is proved. These solutions satisfy a
local
balance of energy. Moreover it is proved also the existence of a
statistically
stationary solution.
http://front.math.ucdavis.edu/math.PR/0609318
---------------------------------------------------------------
4661. CLASSES OF SKOROKHOD EMBEDDINGS FOR THE SIMPLE SYMMETRIC RANDOM
WALK
Alexander M.G. Cox and Jan Obloj (PMA)
The Skorokhod Embedding problem is well understood when the underlying
process is a Brownian motion. We examine the problem when the
underlying is the
simple symmetric random walk and when no external randomisation is
allowed. We
prove that any measure on Z can be embedded by means of a minimal
stopping
time. However, in sharp contrast to the Brownian setting, we show
that the set
of measures which can be embedded in a uniformly integrable way is
strictly
smaller then the set of centered probability measures: specifically
it is a
fractal set which we characterise as an iterated function system.
Finally, we
define the natural extension of several known constructions from the
Brownian
setting and show that these constructions require us to further
restrict the
sets of target laws.
http://front.math.ucdavis.edu/math.PR/0609330
---------------------------------------------------------------
4662. ASYMPTOTICS FOR ROOTED PLANAR MAPS AND SCALING LIMITS OF TWO-
TYPE SPATIAL TREES
Mathilde Weill (DMA)
We prove some asymptotic results for the radius and the profile of large
random bipartite planar maps. Using a bijection due to Bouttier, Di
Francesco
and Guitter between rooted bipartite planar maps and certain two-type
trees
with positive labels, we derive our results from a conditional limit
theorem
for two-type spatial trees. Finally we apply our estimates to separating
vertices of bipartite planar maps: with probability close to one when
$n$ goes
to infinity, a random $2\ka$-angulation with $n$ faces has a
separating vertex
whose removal disconnects the map into two components each with size
greater
that $n^{1/2-\vep}$.
http://front.math.ucdavis.edu/math.PR/0609334
---------------------------------------------------------------
4663. ON PERFORMANCE OF EVENT-TO-SINK TRANSPORT IN TRANSMIT-ONLY
SENSOR NETWORKS
Bartlomiej Bartek Blaszczyszyn (INRIA Rocquencourt) and Bozidar
Radunovic (INRIA Rocquencourt)
We consider a hybrid wireless sensor network with regular and
transmit-only
sensors. The transmit-only sensors do not have receiver circuit,
hence are
cheaper and less energy consuming, but their transmissions cannot be
coordinated. Regular sensors, also called cluster-heads, are
responsible for
receiving information from transmit-only sensors and forwarding it to
sinks.
The main goal of such a hybrid network is to reduce the cost of
deployment
while achieving some performance constraints (minimum coverage,
sensing rate,
etc). In this paper we are interested in the communication between
transmit-only sensors and cluster-heads. We develop a detailed
analytical model
of the physical and MAC layer using tools from queuing theory and
stochastic
geometry. (The MAC model, that we call Erlang's loss model with
interference,
might be of independent interest as adequate for any non-slotted; i.e.,
unsynchronized, wireless communication channel.) We give an explicit
formula
for the frequency of successful packet reception by a cluster-head,
given
sensors' locations. We further define packet admission policies at a
cluster-head, and we calculate the optimal policies for different
performance
criteria. Finally we show that the proposed hybrid network, using the
optimal
policies, can achieve substantial cost savings as compared to
conventional
architectures.
http://front.math.ucdavis.edu/cs.NI/0609038
---------------------------------------------------------------
4664. CHAOTIC TEMPERATURE DEPENDENCE AT ZERO TEMPERATURE
A. C. D. van Enter and W. M. Ruszel
We present a class of examples of nearest-neighbour, boubded-spin
models, in
which the low-temperature Gibbs measures do not converge as the
temperature is
lowered to zero, in any dimension.
http://front.math.ucdavis.edu/math-ph/0609024
---------------------------------------------------------------
4665. INTERPOLATION OF RANDOM HYPERPLANES
Ery Arias-Castro
Let {(Z_i,W_i):i=1,...,n} be uniformly distributed in [0,1]^d * G
(k,d), where
G(k,d) denotes the space of k-dimensional linear subspaces of R^d. For a
differentiable function f from [0,1]^k to [0,1]^d we say that f
interpolates
(z,w) in [0,1]^d * G(k,d) if there exists x in [0,1]^k such that f(x)
= z and
vec{f}(x) = w, where vec{f}(x) denotes the tangent space at x defined
by f. For
a smoothness class F of H\"older type, we obtain probability bounds
on the
maximum number of points a function f in F interpolates.
http://front.math.ucdavis.edu/math.PR/0609340
---------------------------------------------------------------
4666. PATHWISE STATIONARY SOLUTIONS OF STOCHASTIC BURGERS EQUATIONS
WITH $L^2[0,1]$-NOISE AND STOCHASTIC BURGERS INTEGRAL EQUATIONS ON
INFINITE
HORIZON
Yong Liu and Huaizhong Zhao
In this paper, we show the existence and uniqueness of the stationary
solution $u(t,\omega)$ and stationary point $Y(\omega)$ of the
differentiable
random dynamical system $U:R\times L^2[0,1]\times \Omega\to L^2[0,1]$
generated
by the stochastic Burgers equation with $L^2[0,1]$-noise and large
viscosity,
especially, $u(t,\omega)=U(t,Y(\omega),\omega)=Y(\theta(t,\omega))$, and
$Y(\omega) \in H^1[0,1]$ is the unique solution of the following
equation in
$L^2[0,1]$
$$
Y(\omega)={1/2}\int_{-\infty}^0T_\nu(-s)\frac{\partial
(Y(\theta(s,\omega))^2}{\partial x}ds +\int_{-\infty}^0T_\nu(-s)dW_s
(\omega),
$$ where $\theta$ is the group of $P$-preserving ergodic
transformation on
the canonical probability pace $(\Omega, {\cal F}, P)$ such that
$\theta(t,\omega)(s)=W(t+s)-W(t)$.
http://front.math.ucdavis.edu/math.PR/0609344
---------------------------------------------------------------
4667. THE SIZE OF RANDOM FRAGMENTATION TREES
S. Janson and R. Neininger
We study a random fragmentation process and its associated random
tree. The
process has earlier been studied by Dean and Majumdar (J. Phys. A:
Math. Gen.,
vol. 35, L501--L507), who found a phase transition: the number of
fragmentations is asymptotically normal in some cases but not in others,
depending on the position of roots of a certain characteristic
equation. This
parallels the behaviour of discrete analogues with various random
trees that
have been studied in computer science. We give rigorous proofs of
this phase
transition, and add further details.
The proof uses the contraction method. We extend some previous
results for
recursive sequences of random variables to families of random
variables with a
continuous parameter; we believe that this extension has independent
interest.
http://front.math.ucdavis.edu/math.PR/0609350
---------------------------------------------------------------
4668. FAST SIMULATED ANNEALING IN $\R^D$ AND AN APPLICATION TO
MAXIMUM LIKELIHOOD ESTIMATION
Sylvain Rubenthaler (JAD) and Tobias Ryd\'{e}n (CENTRE for
Mathematical Sciences), Magnus Wiktorsson (CENTRE for Mathematical
Sciences)
Using classical simulated annealing to maximise a function $\psi$
defined on
a subset of $\R^d$, the probability $\p(\psi(\theta\_n)\leq
\psi\_{\max}-\epsilon)$ tends to zero at a logarithmic rate as $n$
increases;
here $\theta\_n$ is the state in the $n$-th stage of the simulated
annealing
algorithm and $\psi\_{\max}$ is the maximal value of $\psi$. We
propose a
modified scheme for which this probability is of order $n^{-1/3}\log n
$, and
hence vanishes at an algebraic rate. To obtain this faster rate, the
exponentially decaying acceptance probability of classical simulated
annealing
is replaced by a more heavy-tailed function, and the system is cooled
faster.
We also show how the algorithm may be applied to functions that
cannot be
computed exactly but only approximated, and give an example of
maximising the
log-likelihood function for a state-space model.
http://front.math.ucdavis.edu/math.PR/0609353
---------------------------------------------------------------
4669. A LAW OF LARGE NUMBERS FOR FINITE-RANGE DEPENDENT RANDOM MATRICES
Greg Anderson and Ofer Zeitouni
We consider random hermitian matrices in which distant above-diagonal
entries
are independent but nearby entries may be correlated. We find the
limit of the
empirical distribution of eigenvalues by combinatorial methods. We
also prove
that the limit has algebraic Stieltjes transform by an argument based on
dimension theory of noetherian local rings.
http://front.math.ucdavis.edu/math.PR/0609364
---------------------------------------------------------------
4670. RATES OF CONVERGENCE OF MEANS OF EUCLIDEAN FUNCTIONALS
Yooyoung Koo (Sungkyunkwan Univ) and Sungchul Lee (Yonsei Univ)
Let $L$ be the Euclidean functional with $p$-th power-weighted edges.
Examples include the sum of the $p$-th power-weighted lengths of the
edges in
minimal spanning trees, traveling salesman tours, and minimal matchings.
Motivated by the works of Steele, Redmond and Yukich (1994, 1996)
have shown
that for $n$ i.i.d. sample points $\{X_1,...,X_n\}$ from $[0,1]^d$,
$L(\{X_1,...,X_n\})/n^{(d-p)/d}$ converges a.s. to a finite constant.
Here we
bound the rate of convergence of $EL(\{X_1,...,X_n\})/n^{(d-p)/d}$.
http://front.math.ucdavis.edu/math.PR/0609382
---------------------------------------------------------------
4671. A FUNCTIONAL LIMIT THEOREM FOR THE PROFILE OF SEARCH TREES
Michael Drmota and Svante Janson and Ralph Neininger
We study the profile $X_{n,k}$ of random search trees including
binary search
trees and $m$-ary search trees. Our main result is a functional limit
theorem
of the normalized profile $X_{n,k}/\E X_{n,k}$ for $k = \lfloor
\alpha \log
n\rfloor$ in a certain range of $\alpha$.
A central feature of the proof is the use of the contraction
method to prove
convergence in distribution of certain random analytic functions in a
complex
domain. This is based on a general theorem on the contraction method
for random
variables in an infinite dimensional Hilbert space. As part of the
proof, we
show that the Zolotarev metric is complete for a Hilbert space.
http://front.math.ucdavis.edu/math.PR/0609385
---------------------------------------------------------------
4672. ON RANDOM MEASURES, UNORDERED SUMS AND DISCONTINUITIES OF THE
FIRST KIND
Frank Oertel
By investigating in detail discontinuities of the first kind of real-
valued
functions and the analysis of unordered sums, where the summands are
given by
values of a positive real-valued function, we develop a measure-
theoretical
framework which in particular allows us to describe \textit
{rigorously} the
representation and meaning of sums of jumps of type $\sum_{0 < s \leq
t} \Phi
\circ | \Delta X_s |$, where $X : \Omega \times \R_+ \longrightarrow
\R$ is a
stochastic process with regulated trajectories, $t \in \R_+$ and $
\Phi : \R_+
\longrightarrow \R_+$ is a strictly increasing function which maps 0
to 0 (cf.
Proposition \ref{prop:sum of jumps on R+ with invertible function}).
Moreover,
our approach enables a natural extension of the jump measure of c\`{a}
dl\`{a}g
and adapted processes to an integer-valued random measure of optional
processes
with regulated trajectories which need not necessarily to be right- or
left-continuous (cf. Theorem \ref{thm:optional random measures}). In
doing so,
we provide a detailed and constructive proof of the fact that the set
of all
discontinuities of the first kind of a given real-valued function on $
\R$ is at
most countable (cf. Lemma \ref{lemma:right limits and left limits},
Theorem
\ref{thm:at most countably many jumps on compact intervals} and Theorem
\ref{thm:at most countably many jumps on R+}).
By using the powerful analysis of unordered sums, we hope that our
contributions fill an existing gap in the literature, since neither a
detailed
proof of (the frequently used) Theorem \ref{thm:at most countably
many jumps on
compact intervals} nor a precise definition of sums of jumps seems to be
available yet.
http://front.math.ucdavis.edu/math.PR/0609395
---------------------------------------------------------------
4673. UTILITY-BASED SUPER-REPLICATION PRICES OF UNBOUNDED CONTINGENT
CLAIMS AND DUALITY OF CONES
Frank Oertel and Mark Owen
Consider a financial market in which an agent trades with utility-
induced
restrictions on wealth. We prove that the utility-based super-
replication price
of an unbounded (but sufficiently integrable) contingent claim is
equal to the
supremum of its discounted expectations under pricing measures with
finite
entropy. Central to our proof is the representation of a cone $C_\V$ of
utility-based super-replicable contingent claims as the polar cone of
the set
of finite entropy separating measures. $C_\V$ is shown to be the
closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal
wealth. As
our approach shows, those terminal wealths need {\it not} necessarily
stem from
{\it admissible} trading strategies only.
We investigate also the natural dual of this result, and show that
the polar
cone of $C_\V$ is the cone generated by separating measures with {\it
finite
loss-entropy}. For an agent whose utility function is unbounded from
above, the
set of pricing measures with finite loss-entropy can be slightly
larger than
the set of pricing measures with finite entropy. Indeed, we prove
that the
former set is the closure of the latter under a suitable weak topology.
Finally, we show how our framework can be applied to another field of
mathematical economics and how it sheds a different light on Farkas'
Lemma and
its infinite dimensional version there.
http://front.math.ucdavis.edu/math.PR/0609402
---------------------------------------------------------------
4674. ON UTILITY-BASED SUPER-REPLICATION PRICES OF CONTINGENT CLAIMS
WITH UNBOUNDED PAYOFFS
Frank Oertel and Mark Owen
Consider a financial market in which an agent trades with utility-
induced
restrictions on wealth. For a utility function which satisfies the
condition of
reasonable asymptotic elasticity at $-\infty$ we prove that the
utility-based
super-replication price of an unbounded (but sufficiently integrable)
contingent claim is equal to the supremum of its discounted
expectations under
pricing measures with finite {\it loss-entropy}. For an agent whose
utility
function is unbounded from above, the set of pricing measures with
finite
loss-entropy can be slightly larger than the set of pricing measures
with
finite entropy. Indeed, the former set is the closure of the latter
under a
suitable weak topology.
Central to our proof is the representation of a cone $C_U$ of
utility-based
super-replicable contingent claims as the polar cone to the set of
finite
loss-entropy pricing measures. The cone $C_U$ is defined as the
closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal
wealth.
We investigate also the natural dual of this result and show that
the polar
cone to $C_U$ is generated by those separating measures with finite
loss-entropy. The full two-sided polarity we achieve between measures
and
contingent claims yields an economic justification for the use of the
cone
$C_U$, and an open question.
http://front.math.ucdavis.edu/math.PR/0609403
---------------------------------------------------------------
4675. THE GEOMETRY OF INFORMATION OF A SINGLE MATRIX RANDOM MATRIX MODEL
Dan Shiber
In this paper we develop the geometry of information for a single matrix
random matrix model, with two goals: proving a Cramer-Rao theorem for
estimators on random matrices, and calculating the Legendre transform of
pressure and entropy with respect to a metric duality. In our
development we
recover several quantities from free probability: Voiculescu's conjugate
variable is the tangent vector to the GUE pertrubation model, giving
rise to a
metric which turns out to be Voiculescu's Free Fisher Information
measure;
Hiai's Legendre transform of free pressure agrees with our Legendre
transform
of pressure; and Speicher's covariance of fluctuations naturally
arises as the
metric on perturbations of the random matrix model. Incidentally, we
obtain a
new kind of convexity for the free entropy of the limit of a random
matrix
model.
http://front.math.ucdavis.edu/math.OA/0609372
---------------------------------------------------------------
4676. ON THE DYNAMIC PROGRAMMING APPROACH FOR THE 3D NAVIER-STOKES
EQUATIONS
Luigi Manca
The dynamic programming approach for the control of a 3D flow
governed by the
stochastic Navier-Stokes equations for incompressible fluid in a
bounded domain
is studied. By a compactness argument, existence of solutions for the
associated Hamilton-Jacobi-Bellman equation is proved. Finally,
existence of an
optimal control through the feedback formula and of an optimal state is
discussed.
http://front.math.ucdavis.edu/math.OC/0609389
---------------------------------------------------------------
4677. STOCHASTIC NONLINEAR SCHRODINGER EQUATIONS DRIVEN BY A
FRACTIONAL NOISE - WELL POSEDNESS, LARGE DEVIATIONS AND SUPPORT
Eric Gautier
We consider stochastic nonlinear Schrodinger equations driven by an
additive
noise. The noise is fractional in time with Hurst parameter H in
(0,1). It is
also colored in space and the space correlation operator is assumed
to be
nuclear. We study the local well-posedness of the equation. Under
adequate
assumptions on the initial data, the space correlations of the noise
and for
some saturated nonlinearities, we prove a sample path large deviations
principle and a support result. These results are stated in a space of
exploding paths which are Holder continuous in time until blow-up. We
treat the
case of Kerr nonlinearities when H > 1/2.
http://front.math.ucdavis.edu/math.PR/0609423
---------------------------------------------------------------
4678. SMALL NOISE ASYMPTOTIC OF THE TIMING JITTER IN SOLITON
TRANSMISSION
Arnaud Debussche and Eric Gautier
We consider random perturbations of the focusing cubic one dimensional
nonlinear Schrodinger equation. The noises, either additive or
multiplicative,
are white in time and colored in space. In the additive case, a white
noise
limit is considered. We study the small noise asymptotic of the tails
of the
center and mass of a pulse at a fixed coordinate when the initial
datum is null
or a soliton profile. Our main tools are large deviation results at
the level
of paths. Upper and lower bounds are obtained from bounds for the
optimal
control problems derived from the rate function of the large deviation
principles. Our results are in perfect agreement with several results
from
physics. These results had been obtained with arguments which seem
difficult to
fully justify mathematically. Some results are new.
http://front.math.ucdavis.edu/math.PR/0609424
---------------------------------------------------------------
4679. SMALL NOISE ASYMPTOTIC OF THE TIMING JITTER IN SOLITON
TRANSMISSION
Arnaud Debussche (IRMAR) and Eric Gautier (IRMAR)
We consider random perturbations of the focusing cubic one dimensional
nonlinear Schr\"{o}dinger equation. The noises, either additive or
multiplicative, are white in time and colored in space. In the
additive case, a
"white noise limit" is considered. We study the small noise
asymptotic of the
tails of the center and mass of a pulse at a fixed coordinate when
the initial
datum is null or a soliton profile. Our main tools are large
deviation results
at the level of paths. Upper and lower bounds are obtained from
bounds for the
optimal control problems derived from the rate function of the large
deviation
principles. Our results are in perfect agreement with several results
from
physics.These results had been obtained with arguments which seem
difficult to
fully justify mathematically. Some results are new.
http://front.math.ucdavis.edu/math.PR/0609434
---------------------------------------------------------------
4680. CONVEXITY OF THE MEDIAN IN THE GAMMA DISTRIBUTION
Christian Berg and Henrik L. Pedersen
We show that the median $m(x)$ in the gamma distribution with
parameter $x$
is a strictly convex function on the positive half-line
http://front.math.ucdavis.edu/math.PR/0609442
---------------------------------------------------------------
4681. LARGE DEVIATIONS FOR A SCALAR DIFFUSION IN RANDOM ENVIRONMENT
P. Chigansky and R. Liptser
Let $\xi(u)$, $u\in \Real$ be an ergodic stationary Markov chain,
taking a
finite number of values, and consider the diffusion process generated
by the
SDE $$ dX^\eps_t = b(X^\eps_t)dt +\eps^\kappa\xi\big(X^\eps_t/\eps
\big)dB_t $$
with a small positive scaling parameter $\eps$, where $B=(B_t)_{t\in
\Real_+}$
is a Brownian motion, independent of $\xi$, and $\kappa\ge 0$ is a fixed
constant. Such model describes evolution of a particle, perturbed by
a small
white noise disturbance, whose intensity is switched by the random
environment
$\xi$. We show that for $\kappa\in (0,1/6)$, the process $X^\eps$
satisfies the
same Large Deviations Principle (LDP) of the Freidlin-Wentzell type
as the
process $\hat{X}^\epsilon$: $$ dX^\eps_t = b(\hat{X}^\eps_t)dt +
\eps^\kappa\sqrt{\mathbf{a}}dB_t, $$ with $\mathbf{a}=\dfrac{1}{\E
\xi^{-2}(0)}$. For $\kappa=0$, $X^\epsilon$ converges weakly to the the
solution of the SDE $dX_t=b(X_t)dt+\sqrt{\mathbf{a}}dB_t.$
http://front.math.ucdavis.edu/math.PR/0609443
---------------------------------------------------------------
4682. STATIONARY ALGORITHMIC PROBABILITY
Markus Mueller
Kolmogorov complexity and algorithmic probability quantify the
randomness and
universal a priori probability of finite binary strings.
Nevertheless, they
share the disadvantage of depending on the choice of the universal
computer
which is used as a reference computer to count the program lengths.
In this
paper, we propose an approach to algorithmic probability that tries to
eliminate this machine-dependence.
Elaborating on the idea that computers with ``atypical'' algorithmic
probability should be hard to emulate, we define the notion of emulation
complexity. This naturally leads to a Markov process of universal
computers
that randomly emulate each other, yielding stationary probability
distributions
on the computers and finite binary strings.
By proving symmetry relations with respect to input and output
transformations, we show that properties of individual computers are
successfully eliminated. Our approach is not limited to prefix-free
computers,
but can be applied to more general sets of computers. The question
for what
computer sets such stationary distributions exist remains open in
general, but
is answered in some special cases and is shown to be closely related
to the
aforementioned symmetry relations.
http://front.math.ucdavis.edu/cs.IT/0608095
---------------------------------------------------------------
4683. L\'EVY PROCESSES AND FOURIER MULTIPLIERS
Rodrigo Ba\~nuelos and Krzysztof Bogdan
We study Fourier multipliers which result from modulating jumps of L
\'evy
processes. Using the theory of martingale transforms we prove that these
operators are bounded in $L^p(\Rd)$ for $1<p<\infty$ and we obtain
the same
explicit bound for their norm as the one known for the second order
Riesz
transforms.
http://front.math.ucdavis.edu/math.FA/0609432
---------------------------------------------------------------
4684. ESCAPE OF MASS IN ZERO-RANGE PROCESSES WITH RANDOM RATES
Pablo A. Ferrari and Valentin V. Sisko
We consider zero-range processes in Z^d with site dependent jump
rates. The
rate for a particle jump from site x to y in Z^d is given by
\lambda_x g(k)
p(y-x), where p(\cdot) is a probability in Z^d, g(k) is a bounded
nondecreasing
function of the number k of particles in x and \lambda = \{\lambda_x
\} is a
collection of i.i.d. random variables with values in (c,1], for some
c>0. For
almost every realization of the environment \lambda the zero-range
process has
product invariant measures \{\nu_{\lambda,v}: 0\le v \le c\}
parametrized by v,
the average total jump rate from any given site. The density of a
measure,
defined by the asymptotic average number of particles per site, is an
increasing function of v. There exists a product invariant measure
\nu_{\lambda,c}, with maximal density. Let \mu be a probability measure
concentrating mass on configurations whose number of particles at
site x grows
less than exponentially with \|x\|. Denoting by S_{\lambda}(t) the
semigroup of
the process, we prove that all weak limits of \{\mu S_{\lambda}(t), t
\ge 0 \}
as t \to \infty are dominated, in the natural partial order, by
\nu_{\lambda,c}. In particular, if \mu dominates \nu_{\lambda,c},
then \mu
S_{\lambda}(t) converges to \nu_{\lambda,c}. The result is particularly
striking when the maximal density is finite and the initial measure
has a
density above the maximal.
http://front.math.ucdavis.edu/math.PR/0609469
---------------------------------------------------------------
4685. DISTRIBUTIONS OF FUNCTIONALS OF THE TWO PARAMETER POISSON-
DIRICHLET PROCESS
L. F. James and A. Lijoi and I. Pruenster
The present paper provides exact expression for the probability
distribution
of linear functionals of the two--parameter Poisson-Dirichlet process
PD$(\alpha,\theta)$. Distributional results that follow from the
application of
an inversion formula for a (generalized) Cauchy--Stieltjes transform are
achieved. Moreover, several interesting integral identities are
obtained by
exploiting a correspondence between the mean functional of a Poisson--
Dirichlet
process and the mean functional of a suitable Dirichlet process.
Finally, some
distributional characterizations in terms of mixture representations are
illustrated. Our formulae are relevant to occupation time phenomena
connected
with Brownian motion and more general Bessel processes, as well as to
models
arising in Bayesian nonparametric statistics.
http://front.math.ucdavis.edu/math.PR/0609488
---------------------------------------------------------------
4686. ON THE VARIANCE OF THE NUMBER OF OCCUPIED BOXES
L. V. Bogachev and A. V. Gnedin and Yu.V. Yakubovich
We consider the occupancy problem where balls are thrown
independently at
infinitely many boxes with fixed positive frequencies. It is well
known that
the random number of boxes occupied by the first n balls is
asymptotically
normal if its variance V_n tends to infinity. In this work, we mainly
focus on
the opposite case where V_n is bounded, and derive a simple necessary
and
sufficient condition for convergence of V_n to a finite limit, thus
settling a
long-standing question raised by Karlin in the seminal paper of 1967.
One
striking consequence of our result is that the possible limit may
only be a
positive integer number. Some new conditions for other types of
behavior of the
variance, like boundedness or convergence to infinity, are also
obtained. The
proofs are based on the poissonization techniques.
http://front.math.ucdavis.edu/math.PR/0609498
---------------------------------------------------------------
4687. ASYMPTOTIC OPTIMALITY IN BAYESIAN CHANGE-POINT DETECTION
PROBLEMS UNDER GLOBAL FALSE ALARM PROBABILITY CONSTRAINT
Alexander G. Tartakovsky
In 1960s Shiryaev developed Bayesian theory of change detection in
independent and identically distributed (i.i.d.) sequences. In
Shiryaev's
classical setting the goal is to minimize an average detection delay
under the
constraint imposed on the average probability of false alarm. Recently,
Tartakovsky and Veeravalli (2005) developed a general Bayesian
asymptotic
change-point detection theory (in the classical setting) that is not
limited to
a restrictive i.i.d. assumption. It was proved that Shiryaev's detection
procedure is asymptotically optimal under traditional average false
alarm
probability constraint, assuming that this probability is small. In
the present
paper, we consider a less conventional approach where the constraint
is imposed
on the global, supremum false alarm probability. An asymptotically
optimal
Bayesian change detection procedure is proposed and thoroughly
evaluated for
both i.i.d. and non-i.i.d. models when the global false alarm
probability
approaches zero.
http://front.math.ucdavis.edu/math.ST/0609467
---------------------------------------------------------------
4688. EUCLIDEAN GIBBS STATES OF INTERACTING QUANTUM ANHARMONIC
OSCILLATORS
Y. Kozitsky and T. Pasurek
A rigorous description of the equilibrium thermodynamic properties of an
infinite system of interacting $\nu$-dimensional quantum anharmonic
oscillators
is given. The oscillators are indexed by the elements of a countable set
$\mathbb{L}\subset \mathbb{R}^d$, possibly irregular; the anharmonic
potentials
vary from site to site. The description is based on the
representation of the
Gibbs states in terms of path measures -- the so called Euclidean Gibbs
measures. It is proven that: (a) the set of such measures $\mathcal{G}
^{\rm t}$
is non-void and compact; (b) every $\mu \in \mathcal{G}^{\rm t}$
obeys an
exponential integrability estimate, the same for the whole set
$\mathcal{G}^{\rm t}$; (c) every $\mu \in \mathcal{G}^{\rm t}$ has a
Lebowitz-Presutti type support; (d) $\mathcal{G}^{\rm t}$ is a
singleton at
high temperatures. In the case of attractive interaction and $\nu=1$
we prove
that $|\mathcal{G}^{\rm t}|>1$ at low temperatures. The uniqueness of
Gibbs
measures due to quantum effects and at a nonzero external field are
also proven
in this case. Thereby, a qualitative theory of phase transitions and
quantum
effects, which interprets most important experimental data known for the
corresponding physical objects, is developed. The mathematical result
of the
paper is a complete description of the set $\mathcal{G}^{\rm t}$,
which refines
and extends the results known for models of this type.
http://front.math.ucdavis.edu/math-ph/0609045
---------------------------------------------------------------
4689. CHANGING THE BRANCHING MECHANISM OF A CONTINUOUS STATE
BRANCHING PROCESS USING IMMIGRATION
Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS)
We construct a continuous state branching process with immigration (CBI)
whose immigration depends on the CBI itself and we recover a
continuous state
branching process (CB). This provides a dual construction of the
pruning at
nodes of CB introduced by the authors in a previous paper. This
construction is
a natural way to model neutral mutation. Using exponential formula,
we compute
the probability of extinction of the original type population in a
critical or
sub-critical quadratic branching, conditionally on the non extinction
of the
total population.
http://front.math.ucdavis.edu/math.PR/0609518
---------------------------------------------------------------
4690. MAFIA : A THEORETICAL STUDY OF PLAYERS AND COALITIONS IN A
PARTIAL INFORMATION ENVIRONMENT
Mark Braverman and Omid Etesami and Elchanan Mossel
In this paper we study a game called {\em mafia} in which different
players
have different types of information, communication, and
functionality. The
players communicate and function in a way that resembles some real life
situations. We consider two types of operations. First, there are
operations
that follow an open democratic discussion. Second, some subgroups of the
players who may have different interests, make decisions based on
their own
group interest. A key ingredient here is that the identity of each
subgroup is
known only to the members of that group.
http://front.math.ucdavis.edu/math.PR/0609534
---------------------------------------------------------------
4691. A POSITIVSTELLENSATZ WHICH PRESERVES THE COUPLING PATTERN OF
VARIABLES
Jean B. Lasserre
We specialize Schm\"udgen's Positivstellensatz and its Putinar and
Jacobi and
Prestel refinement, to the case of a polynomial $f\in R[X,Y]+R[Y,Z]$,
positive
on a compact basic semi algebraic set $K$ described by polynomials in
$R[X,Y]$
and $R[Y,Z]$ only, or in $R[X]$ and $R[Y,Z]$ only (i.e. $K$ is a
cartesian
product). In particular, we show that the preordering $P(g,h)$ (resp.
quadratic
module $Q(g,h)$) generated by the polynomials $\{g_j\}\subset R[X,Y]$
and
$\{h_k\}\subset R[Y,Z]$ that describe $K$, is replaced with $P(g)+P(h)
$ (resp.
$Q(g)+Q(h)$), so that the absence of coupling between $X$ and $Z$ is
also
preserved in the representation. A similar result applies with Krivine's
Positivstellensatz involving the cone generated by $\{g_j,h_k\}$.
http://front.math.ucdavis.edu/math.AC/0609529
---------------------------------------------------------------
4692. RANDOM SORTING NETWORKS
Omer Angel and Alexander E. Holroyd and Dan Romik and Balint Virag
A sorting network is a shortest path from 12...n to n...21 in the Cayley
graph of S_n generated by nearest-neighbour swaps. We prove that for
a uniform
random sorting network, as n->infinity the space-time process of swaps
converges to the product of semicircle law and Lebesgue measure. We
conjecture
that the trajectories of individual particles converge to random sine
curves,
while the permutation matrix at half-time converges to the projected
surface
measure of the 2-sphere. We prove that, in the limit, the
trajectories are
Holder-1/2 continuous, while the support of the permutation matrix
lies within
a certain octagon. A key tool is a connection with random Young
tableaux.
http://front.math.ucdavis.edu/math.PR/0609538
---------------------------------------------------------------
4693. LIMITING DYNAMICS FOR SPHERICAL MODELS OF SPIN GLASSES AT HIGH
TEMPERATURE
Amir Dembo and Alice Guionnet and Christian Mazza
We analyze the coupled non-linear integro-differential equations whose
solutions is the thermodynamical limit of the empirical correlation and
response functions in the Langevin dynamics for spherical p-spin
disordered
mean-field models. We provide a mathematically rigorous derivation of
their FDT
solution (for the high temperature regime) and of certain key
properties of
this solution, which are in agreement with earlier derivations based on
physical grounds.
http://front.math.ucdavis.edu/math.PR/0609546
---------------------------------------------------------------
4694. NEAR-MINIMAL SPANNING TREES: A SCALING EXPONENT IN PROBABILITY
MODELS
David Aldous and Charles Bordenave and Marc Lelarge
We study the relation between the minimal spanning tree (MST) on many
random
points and the "near-minimal" tree which is optimal subject to the
constraint
that a proportion $\delta$ of its edges must be different from those
of the
MST. Heuristics suggest that, regardless of details of the
probability model,
the ratio of lengths should scale as $1 + \Theta(\delta^2)$. We prove
this
scaling result in the model of the lattice with random edge-lengths.
In the
2-dimensional Euclidean model, by exploiting the well-known
connection between
MSTs and continuum percolation we can prove the scaling result up to
an Ansatz
that a known technical result for lattice percolation extends to
continuum
percolation.
http://front.math.ucdavis.edu/math.PR/0609547
---------------------------------------------------------------
4695. MIXING TIMES VIA SUPER-FAST COUPLING
Robert Burton and Yevgeniy Kovchegov
We provide a coupling proof that the transposition shuffle on a deck
of n
cards is mixing of rate $n\log(n)$ with a moderate constant. This has
already
been shown by Diaconis and Shahshahani but no natural coupling proof
has been
demonstrated to date. We also enlarge the methodology of coupling to
include
intuitive but nonadapted coupling rules, for example, to take in
account future
events and to prepare for their occurrence.
http://front.math.ucdavis.edu/math.PR/0609568
---------------------------------------------------------------
4696. AN INVARIANCE PRINCIPLE FOR THE LAW OF THE ITERATED LOGARITHM
FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS
Guangyu Yang and Yu Miao
In this paper, we prove Strassen's strong invariance principle for a
vector-valued additive functionals of a Markov chain via the martingale
argument and the theory of fractional coboundaries. The hypothesis is
a moment
bound on the resolvent.
http://front.math.ucdavis.edu/math.PR/0609593
---------------------------------------------------------------
4697. MERGING PERCOLATION AND CLASSICAL RANDOM GRAPHS: PHASE
TRANSITION IN DIMENSION 1
Tatyana S. Turova and Thomas Vallier
We study a random graph model which combines properties of the edge
percolation model on Z^d and a classical random graph G(n,c/n). We
show that
this model, being a homogeneous random graph, has a natural relation
to the
so-called "rank 1 case" of inhomogeneous random graphs. This allows
us to use
the newly developed theory of inhomogeneous random graphs to describe
completely the phase diagram in the case d=1. The phase transition is
similar
to the classical random graph, it is of the second order. We also
find the
scaled size of the largest connected component above the phase
transition.
http://front.math.ucdavis.edu/math.PR/0609594
---------------------------------------------------------------
4698. EULER HYDRODYNAMICS OF ONE-DIMENSIONAL ATTRACTIVE PARTICLE SYSTEMS
C. Bahadoran and H. Guiol and K. Ravishankar and E. Saada
We consider attractive irreducible conservative particle systems on
$\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit
invariant
measures. We prove that for such systems, the hydrodynamic limit
under Euler
time scaling exists and is given by the entropy solution to some scalar
conservation law with Lipschitz-continuous flux. Our approach is a
generalization of Bahadoran et al. [Stochastic Process. Appl. 99
(2002) 1--30],
from which we relax the assumption that the process has explicit
invariant
measures.
http://front.math.ucdavis.edu/math.PR/0609605
---------------------------------------------------------------
4699. DIAMETER OF RANDOM CAYLEY GRAPH OF Z_Q
Gideon Amir and Ori Gurel-Gurevich
Consider the Cayley graph of the cyclic group of prime order q with k
uniformly chosen generators. For k fixed, we prove that the diameter
of said
graph is asymptotically (in q) of order q^(1/k).
http://front.math.ucdavis.edu/math.PR/0609620
---------------------------------------------------------------
4700. LOOP MODELS AND THEIR CRITICAL POINTS
Paul Fendley
Loop models have been widely studied in physics and mathematics, in
problems
ranging from polymers to topological quantum computation to Schramm-
Loewner
evolution. I present new loop models which have critical points
described by
conformal field theories. Examples include both fully-packed and
dilute loop
models with critical points described by the superconformal minimal
models and
the SU(2)_2 WZW models. The dilute loop models are generalized to
include
SU(2)_k models as well.
http://front.math.ucdavis.edu/cond-mat/0609435
---------------------------------------------------------------
4701. LEVY PROCESSES ON A FIRST ORDER MODEL
Siu-Ah Ng
The classical notion of a Levy process is generalized to one that takes
values in an arbitrary model of a first order language. This is
achieved by
defining a convolution product and the infinite divisibility with
respect to
it.
http://front.math.ucdavis.edu/math.LO/0609608
---------------------------------------------------------------
4702. SUMS OF EXTREME VALUES OF SUBORDINATED LONG-RANGE DEPENDENT
SEQUENCES: MOVING AVERAGES WITH FINITE VARIANCE
Rafal Kulik
In this paper we characterize the limiting behavior of sums of
extreme values
of long range dependent sequences defined as functionals of linear
processes
with finite variance. The extremal sums behave completely different
by compared
to the i.i.d case. In particular, though we still have asymptotic
normality,
the scaling factor is relatively bigger than in the i.i.d case,
meaning that
the maximal terms have relatively smaller contribution to the whole
sum. Also,
the scaling need not depend on the tail index of the underlying marginal
distribution, as it is well-known to be so in the i.i.d. situation.
Furthermore, subordination may completely change the asymptotic
properties of
sums of extremes.
http://front.math.ucdavis.edu/math.PR/0609625
---------------------------------------------------------------
4703. A CHAOTIC REPRESENTATION PROPERTY OF THE MULTIDIMENSIONAL
DUNKL PROCESSES
L\'{e}onard Gallardo and Marc Yor
Dunkl processes are martingales as well as c\`{a}dl\`{a}g homogeneous
Markov
processes taking values in $\mathbb{R}^d$ and they are naturally
associated
with a root system. In this paper we study the jumps of these
processes, we
describe precisely their martingale decompositions into continuous
and purely
discontinuous parts and we obtain a Wiener chaos decomposition of the
corresponding $L^2$ spaces of these processes in terms of adequate mixed
multiple stochastic integrals.
http://front.math.ucdavis.edu/math.PR/0609679
---------------------------------------------------------------
4704. ON THE SECOND MOMENT OF THE NUMBER OF CROSSINGS BY A STATIONARY
GAUSSIAN PROCESS
Marie F. Kratz and Jos\'{e} R. Le\'{o}n
Cram\'{e}r and Leadbetter introduced in 1967 the sufficient condition
\[\frac{r''(s)-r''(0)}{s}\in L^1([0,\delta],dx),\qquad \delta>0,\] to
have a
finite variance of the number of zeros of a centered stationary Gaussian
process with twice differentiable covariance function $r$. This
condition is
known as the Geman condition, since Geman proved in 1972 that it was
also a
necessary condition. Up to now no such criterion was known for counts of
crossings of a level other than the mean. This paper shows that the
Geman
condition is still sufficient and necessary to have a finite variance
of the
number of any fixed level crossings. For the generalization to the
number of a
curve crossings, a condition on the curve has to be added to the Geman
condition.
http://front.math.ucdavis.edu/math.PR/0609682
---------------------------------------------------------------
4705. ON GIBBSIANNESS OF RANDOM FIELDS
Serguei Dachian and Boris Nahapetian (IMNASA)
The problem of characterization of Gibbs random fields is considered.
Various
Gibbsianness criteria are obtained using the earlier developed one-point
framework which in particular allows to describe random fields by
means of
either one-point conditional or one-point finite-conditional
distributions. The
main outcome are the criteria in terms of one-point finite-conditional
distribution, on the basis of which a simple and comprehensible
definition of
Gibbs random field is given.
http://front.math.ucdavis.edu/math.PR/0609688
---------------------------------------------------------------
4706. A NOTE ABOUT KHOSHNEVISAN--XIAO CONJECTURE
Martynas Manstavi\v{c}ius
Khoshnevisan and Xiao showed in [Ann. Probab. 33 (2005) 841--878]
that the
statement about almost surely vanishing Bessel--Riesz capacity of the
image of
a Borel set $G\subset\mathbb{R}_+$ under a symmetric L\'{e}vy process
$X$ in
$\mathbb{R}^d$ is equivalent to the vanishing of a deterministic $f$-
capacity
for a particular function $f$ defined in terms of the characteristic
exponent
of $X$. The authors conjectured that a similar statement is true for all
L\'{e}vy processes in $\mathbb{R}^d$. We show that the conjecture is
true
provided we extend the definition of $f$ and require certain
integrability
conditions which cannot be avoided in general.
http://front.math.ucdavis.edu/math.PR/0609696
---------------------------------------------------------------
4707. QUELQUES APPROXIMATIONS DU TEMPS LOCAL BROWNIEN
Blandine Berard Bergery (IECN) and Pierre Vallois (IECN)
We give some approximations of the local time process $(L\_t^x)\_{t
\geqslant
0}$ at level $x$ of the real Brownian motion $(X\_t)$. We prove that $
\frac{1}{\epsilon}\int\_0^t (\indi\_{\{x<X\_{s+\epsilon}\}} -
\indi\_{\{x<X\_{s}\}}) (X\_{s+\epsilon}-X\_{s})ds$ goes to $L\_t^x$
in the ucp
sense as $\epsilon \to 0$, and that the rate of convergence in $L^2
(\Omega)$ is
of order $\epsilon^\alpha$, for any $\alpha < {1/4}$. Moreover,
approximations
of some Brownian stochastic integrals are given.
http://front.math.ucdavis.edu/math.PR/0609701
---------------------------------------------------------------
4708. SIMPLE SYSTEMS WITH ANOMALOUS DISSIPATION AND ENERGY CASCADE
Jonathan C. Mattingly and Toufic Suidan and Eric Vanden-Eijnden
We analyze a class of linear shell models subject to stochastic
forcing in
finitely many degrees of freedom. The unforced systems considered
formally
conserve energy. Despite being formally conservative, we show that these
dynamical systems support dissipative solutions (suitably defined)
and, as a
result, may admit unique (statistical) steady states when the forcing
term is
nonzero. This claim is demonstrated via the complete characterization
of the
solutions of the system above for specific choices of the coupling
coefficients. The mechanism of anomalous dissipations is shown to
arise via a
cascade of the energy towards the modes ($a_n$) with higher $n$; this is
responsible for solutions with interesting energy spectra, namely $
\EE |a_n|^2$
scales as $n^{-\alpha}$ as $n\to\infty$. Here the exponents $\alpha$
depend on
the coupling coefficients $c_n$ and $\EE$ denotes expectation with
respect to
the equilibrium measure. This is reminiscent of the conjectured
properties of
the solutions of the Navier-Stokes equations in the inviscid limit
and their
accepted relationship with fully developed turbulence. Hence, these
simple
models illustrate some of the heuristic ideas that have been advanced to
characterize turbulence, similar in that respect to the random
passive scalar
or random Burgers equation, but even simpler and fully solvable.
http://front.math.ucdavis.edu/math-ph/0607047
---------------------------------------------------------------
4709. STOCHASTIC PRECONDITIONING FOR ITERATIVE LINEAR EQUATION SOLVERS
Haifeng Qian and Sachin S. Sapatnekar
This paper presents a new stochastic preconditioning approach. For
symmetric
diagonally-dominant M-matrices, we prove that an incomplete LDL
factorization
can be obtained from random walks, and used as a preconditioner for an
iterative solver, e.g., conjugate gradient. It is argued that our factor
matrices have better quality, i.e., better accuracy-size tradeoffs, than
preconditioners produced by existing incomplete factorization methods.
Therefore the resulting preconditioned conjugate gradient (PCG)
method requires
less computation than traditional PCG methods to solve a set of linear
equations with the same error tolerance, and the advantage increases
for larger
and denser sets of linear equations. These claims are verified by
numerical
tests, and we provide techniques that can potentially extend the
theory to more
general types of matrices.
http://front.math.ucdavis.edu/math.NA/0609672
---------------------------------------------------------------
4710. CORRECTION NOTE. TYPICAL CONFIGURATION FOR ONE-DIMENSIONAL
RANDOM FIELD KAC MODEL
Marzio Cassandro and Enza Orlandi and Pierre Picco
Estimate (3.39) which appears in the proof of Proposition 3.4 in [Ann.
Probab. 27 (1999) 1414--1467, doi:10.1214/aop/1022677454] is wrong.
We present
below a corrected proof which introduces an extra factor 2 in
equations (3.34)
and (3.35). This has no consequence in the rest of the paper since
Proposition
3.4 is used to estimate only ratios; see (3.23) and (3.25).
http://front.math.ucdavis.edu/math.PR/0609719
---------------------------------------------------------------
4711. VITESSE DE CONVERGENCE DANS LE TH\'EOR\`EME LIMITE CENTRAL
POUR CHA\^INES DE MARKOV DE PROBABILIT\'E DE TRANSITION QUASI-COMPACTE
Lo\"ic Herv\'e (IRMAR)
Let $Q$ be a transition probability on a measurable space $E$, let
$(X\_n)\_n$ be a Markov chain associated to $Q$, and let $\xi$ be a
real-valued
measurable function on $E$, and $S\_n = \sum\_{k=1}^{n} \xi(X\_k)$.
Under
functional hypotheses on the action of $Q$ and its Fourier kernels $Q
(t)$, we
investigate the rate of convergence in the central limit theorem for the
sequence $(\frac{S\_n}{\sqrt n})\_n$. According to the hypotheses, we
prove
that the rate is, either $O(n^{-\frac{\tau}{2}})$ for all $\tau<1$, or
$O(n^{-{1/2}})$. We apply the spectral method of Nagaev which is
improved by
using a perturbation theorem of Keller and Liverani and a method of
martingale
difference reduction. When $E$ is not compact or $\xi$ is not
bounded, the
conditions required here are weaker than the ones usually imposed
when the
standard perturbation theorem is used. For example, in the case of
$V$-geometric ergodic chains or Lipschitz iterative models, the rate of
convergence in the c.l.t is $O(n^{-{1/2}})$ under a third moment
condition on
$\xi$.
http://front.math.ucdavis.edu/math.PR/0609720
---------------------------------------------------------------
4712. EXPONENTIAL CONCENTRATION FOR FIRST PASSAGE PERCOLATION THROUGH
MODIFIED POINCARE INEQUALITIES
Michel Benaim and Raphael Rossignol
We provide a new exponential concentration inequality for
First Passage Percolation valid for a wide class of edge times
distributions.
This improves and extends a result by Benjamini, Kalai and Schramm
which gave a
variance bound for Bernoulli edge times. Our approach is based on some
functional inequalities extending the work of Rossignol and Falik and
Samorodnitsky.
http://front.math.ucdavis.edu/math.PR/0609730
---------------------------------------------------------------
4713. INTERMITTENT RANDOM WALKS FOR AN OPTIMAL SEARCH STRATEGY: ONE-
DIMENSIONAL CASE
G.Oshanin (1) and H.S.Wio (2) and K.Lindenberg (3) and
S.F.Burlatsky (4)((1) LPTMC, Universite Paris 6, France; (2)
Instituto de Fisica de Cantabria,
Santander, Spain; (3) Department of Chemistry and Biochemistry,
University of
California at San Diego, USA; (4) United Technologies Research
Center, UT
Corp, USA)
We study the search kinetics of an immobile target by a concentration of
randomly moving searchers. The object of the study is to optimize the
probability of detection within the constraints of our model. The
target is
hidden on a one-dimensional lattice in the sense that searchers have
no a
priori information about where it is, and may detect it only upon
encounter.
The searchers perform random walks in discrete time n=0,1,2, ..., N,
where N is
the maximal time the search process is allowed to run. With
probability \alpha
the searchers step on a nearest-neighbour, and with probability (1-
\alpha) they
leave the lattice and stay off until they land back on the lattice at
a fixed
distance L away from the departure point. The random walk is thus
intermittent.
We calculate the probability P_N that the target remains undetected
up to the
maximal search time N, and seek to minimize this probability. We find
that P_N
is a non-monotonic function of \alpha, and show that there is an
optimal choice
\alpha_{opt}(N) of \alpha well within the intermittent regime, 0 <
\alpha_{opt}(N) < 1, whereby P_N can be orders of magnitude smaller
compared to
the "pure" random walk cases \alpha =0 and \alpha = 1.
http://front.math.ucdavis.edu/cond-mat/0609641
---------------------------------------------------------------
4714. LARGE DEVIATIONS FOR THE LARGEST EIGENVALUE OF RANK ONE
DEFORMATIONS OF GAUSSIAN ENSEMBLES
Myl\`ene Ma\"{\i}da
We establish a large deviation principle for the largest eigenvalue
of a rank
one deformation of a matrix from the GUE or GOE. As a corollary, we
get another
proof of the phenomenon, well-known in learning theory and finance,
that the
largest eigenvalue separates from the bulk if the perturbation is
large enough.
A large part of the paper is devoted to an auxiliary result on the
continuity
of spherical integrals, in the case when one of the matrix is of rank
one, as
studied in a previous work.
http://front.math.ucdavis.edu/math.PR/0609738
---------------------------------------------------------------
4715. A CENTRAL LIMIT THEOREM FOR A LOCALIZED VERSION OF THE SK MODEL
Sergio De Carvalho Bezerra (IECN) and Samy Tindel (IECN)
In this note, we consider a SK (Sherrington--Kirkpatrick)-type model
on Z^d
for d greater or equal to 1, weighted by a function allowing to any
single spin
to interact with a small proportion of the other ones. In the
thermodynamical
limit, we investigate the equivalence of this model with the usual SK
spin
system, through the study of the fluctuations of the free energy.
http://front.math.ucdavis.edu/math.PR/0609754
---------------------------------------------------------------
4716. ON THE STRONG LAW OF LARGE NUMBERS FOR L-STATISTICS WITH
DEPENDENT DATA
Evgeny Baklanov (Novosibirsk State University)
The strong law of large numbers for linear combinations of functions
of order
statistics ($L$-statistics) based on weakly dependent random
variables is
proven. We also establish the Glivenko--Cantelli theorem for $\phi$-
mixing
sequences of identically distributed random variables.
http://front.math.ucdavis.edu/math.PR/0609758
---------------------------------------------------------------
4717. LARGE DEVIATIONS AND PHASE TRANSITION FOR RANDOM WALKS IN
RANDOM NONNEGATIVE POTENTIALS
Markus Flury
We establish large deviation principles and phase transition results
for both
quenched and annealed settings of nearest-neighbor random walks with
constant
drift in random nonnegative potentials on $\mathbb Z^d$. We
complement the
analysis of \cite{Zer}, where a shape theorem on the Lyapunov
functions and a
large deviation principle in absence of the drift are achieved for
the quenched
setting.
http://front.math.ucdavis.edu/math.PR/0609766
---------------------------------------------------------------
4718. ON ASYMPTOTIC EXPONENTIALITY OF THE DISTRIBUTION OF FIRST EXIT
TIMES FOR A CLASS OF MARKOV PROCESSES
Moshe Pollak and Alexander G. Tartakovsky
We consider the first exit time of a nonnegative Harris-recurrent Markov
process from the interval $[0,A]$ as $A\to\infty$. We provide a
method of proof
of asymptotic exponentiality of the first exit time (suitably
standardized)
that does not rely on embedding a regeneration process. We provide
examples for
which regeneration embedding fails to yield a proof, whereas our method
succeeds. We show that under certain conditions the moment generating
function
of a suitably standardized version of the first exit time converges
to that of
$\Exp(1)$. The results are applied to the evaluation of a
distribution of run
length to false alarm in change-point detection problems.
http://front.math.ucdavis.edu/math.PR/0609780
---------------------------------------------------------------
4719. CONVERGENCE IN DISTRIBUTION OF RANDOM METRIC MEASURE SPACES: ($
\LAMBDA$-COALESCENT MEASURE TREES)
Andreas Greven and Peter Pfaffelhuber and Anita Winter
We consider the space of complete and separable metric spaces which are
equipped with a probability measure. A notion of convergence is given
based on
the philosophy that a sequence of metric measure spaces converges if
and only
if all finite subspaces sampled from these spaces converge. This
topology is
metrized following Gromov's idea of embedding two metric spaces
isometrically
into a common metric space combined with the Prohorov metric between
probability measures on a fixed metric space. We show that for this
topology
convergence in distribution follows - provided the sequence is tight
- from
convergence of all randomly sampled finite subspaces. We give a
characterization of tightness based on quantities which are
reasonably easy to
calculate. Subspaces of particular interest are the space of real
trees and of
ultra-metric spaces equipped with a probability measure. As an
example we
characterize convergence in distribution for the (ultra-)metric
measure spaces
given by the random genealogies of the $\Lambda$-coalescents. We show
that the
$\Lambda$-coalescent defines an infinite (random) metric measure
space if and
only if the so-called ``dust-free''-property holds.
http://front.math.ucdavis.edu/math.PR/0609801
---------------------------------------------------------------
4720. MEIXNER POLYNOMIALS AND RANDOM PARTITIONS
Alexei Borodin and Grigori Olshanski
The paper deals with a 3-parameter family of probability measures on
the set
of partitions, called the z-measures. The z-measures first emerged in
connection with the problem of harmonic analysis on the infinite
symmetric
group. They are a special and distinguished case of Okounkov's Schur
measures.
It is known that any Schur measure determines a determinantal point
process on
the 1-dimensional lattice. In the particular case of z-measures, the
correlation kernel of this process, called the discrete
hypergeometric kernel,
has especially nice properties. The aim of the paper is to derive the
discrete
hypergeometric kernel by a new method, based on a relationship
between the
z-measures and the Meixner orthogonal polynomial ensemble. The
present paper
can be viewed as an introduction to another our paper where the same
approach
is applied to studying a dynamical model related to the z-measures
(Markov
processes on partitions, Prob. Theory Rel. Fields 135 (2006), 84-152;
arXiv:
math-ph/0409075).
http://front.math.ucdavis.edu/math.PR/0609806
---------------------------------------------------------------
4721. WEAK COUPLING LIMIT OF A POLYMER PINNED AT INTERFACES
Nicolas Petrelis
We consider a simple random walk of length N denoted by $(S_{i})_{i\in
\{1,...,N\}}$, and we define independently a double sequence
$(\gamma^{j}_{i})_{i\geq 1,j\geq 1}$ of i.i.d. random variables and
$(w_i)_{i\geq 1}$ a sequence of centered i.i.d. random variables. We set
$\beta\geq 0$, $\lambda\geq 0$, $h\geq 0$ and $K \in \mathbb{N}$ and
transform
the measure of each random trajectory with the Hamiltonian $\lambda
\sum_{i=1}^{N} (w_i+h) \sign(S_i)+\beta \sum_{j=-K}^{K}\sum_{i=1}^{N}
\gamma_{i}^{j} \boldsymbol{1}_{\{S_{i}=j\}}$. This new path measure
describes
an hydrophobic homopolymer interacting with a layer of width $2K$
around an
interface between oil and water.
In this article we prove the convergence at weak coupling (namely
when $h$
and $\beta$ go to 0) of this discrete model towards its continuous
counterpart.
To that aim we develop a technique of coarse graining introduced by
Bolthausen
and den Hollander in \cite{BDH}. This result shows in particular that
the
randomness of the pinning around the interface vanishes as the
coupling becomes
weaker.
We also introduce a new model of polymer interacting with
infinitely many
horizontal interfaces located at heights $(P_k)_{k\in\mathbb{Z}}$
through the
Hamiltoninan $\beta\sum_{i=1}^{N}\sum_{j\in\mathbb{Z}}\gamma_i^j\
\ind_{\{S_i=P_k\}}$ and we extend the former convergence result to a
particular
case of this model, namely when the widths between successive
interfaces are
equal.
http://front.math.ucdavis.edu/math.PR/0609814
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