Probability Abstracts 58
This document contains abstracts 1556-1592.
They have been mailed on August 31, 2000.
1556. ALGORITHMIC STATISTICS
Peter Gacs, John Tromp, Paul Vitanyi
While Kolmogorov complexity is the accepted absolute measure of information
content of an individual finite object, a similarly absolute notion is needed
for the relation between an individual data sample and an individual model
summarizing the information in the data, for example,a finite set (or
probability distribution) where the data sample typically came from. The
statistical theory based on such relations between individual objects can be
called algorithmic statistics,in contrast to ordinary statistical theory that
deals with relations between probabilisticensembles.v We develop the
algorithmic theory of typical statistic, sufficient statistic, and minimal
sufficient statistic. This theory is based on two-part codes consisting of the
code for the statistic (the model embodying the regularities, the meaningful
information, in the data) and the model-to-data code. We distinguish implicit
and explicit descriptions of the models, and give characterizations of
algorithmic (a.k.a. Kolmogorov) minimal sufficient statistics for all data
samples for both description modes-in the explicit mode under some constraints.
We also strengthen and elaborate some earlier results by Shen on the
``Kolmogorov structure function'' and ``absolutely non-stochastic
objects''-objects that have no simpler algorithmic (explicit) sufficient
statistics and are literally their own algorithmic (explicit) minimal
sufficient statistics. vWe discuss the implication of the results for potential
applications.
paul.vitanyi@cwi.nl
1557. STATIONARY MARKOV CHAINS WITH LINEAR REGRESSIONS
Wlodzimierz Bryc
In a previous paper we determined one dimensional distributions of a
stationary field with linear regressions and quadratic conditional variances
under a linear constraint on the coefficients of the quadratic expression. In
this paper we show that for stationary Markov chains with linear regressions
and quadratic conditional variances the coefficients of the quadratic
expression are indeed tied by a linear constraint which can take only one of
the two alternative forms.
brycwz@email.uc.edu
1558. CRITICAL EXPONENTS, CONFORMAL INVARIANCE AND PLANAR BROWNIAN MOTION
Wendelin Werner
In this review paper, we first discuss some open problems related to
two-dimensional self-avoiding paths and critical percolation. We then review
some closely related results (joint work with Greg Lawler and Oded Schramm) on
critical exponents for two-dimensional simple random walks, Brownian motions
and other conformally invariant random objects.
wendelin.werner@math.u-psud.fr
1559. GAUSSIAN RANDOM MATRIX MODELS FOR Q-DEFORMED GAUSSIAN RANDOM VARIABLES
Piotr Sniady
We construct a family of random matrix models for the q-deformed Gaussian
random variables G_\mu=a_\mu+a^\star_\mu where the annihilation operators a_\mu
and creation operators a^\star_\nu fulfil the q-deformed commutation relation
a_\mu a^\star_\nu-q a^\star_\nu a_\mu=\Gamma_{\mu\nu}, \Gamma_{\mu\nu} is the
covariance and 0<q<1 is a given number. Important feature of considered random
matrices is that the joint distribution of their entries is Gaussian.
psnia@math.uni.wroc.pl
1560. METASTABILITY AND LOW LYING SPECTRA IN REVERSIBLE MARKOV CHAINS
A. Bovier, M. Eckhoff, V. Gayrard, M. Klein
We study a large class of reversible Markov chains with discrete state space
and transition matrix $P_N$. We define the notion of a set of {\it metastable
points} as a subset of the state space $\G_N$ such that (i) this set is reached
from any point $x\in \G_N$ without return to x with probability at least $b_N$,
while (ii) for any two point x,y in the metastable set, the probability
$T^{-1}_{x,y}$ to reach y from x without return to x is smaller than
$a_N^{-1}\ll b_N$. Under some additional non-degeneracy assumption, we show
that in such a situation: \item{(i)} To each metastable point corresponds a
metastable state, whose mean exit time can be computed precisely. \item{(ii)}
To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is
essentially equal to the inverse mean exit time from this state. The
corresponding eigenfunctions are close to the indicator function of the support
of the metastable state. Moreover, these results imply very sharp uniform
control of the deviation of the probability distribution of metastable exit
times from the exponential distribution.
bovier@wias-berlin.de
1561. ON THE PROBABILISTIC RATIONALE OF I-DIVERGENCE AND J-DIVERGENCE
MINIMIZATION
Marian Grendar Jr and Marian Grendar
A probabilistic rationale for I-divergence minimization (relative entropy
maximization), non-parametric likelihood maximization and J-divergence
minimization (Jeffres' entropy maximization) criteria is provided.
umergren@savba.sk
1562. UNIFORM ASYMPTOTIC ESTIMATES OF TRANSITION PROBABILITIES ON COMBS
Daniela Bertacchi, Fabio Zucca
We investigate the asymptotical behaviour of the transition probabilities of
the simple random walk on the 2-comb. In particular we obtain space-time
uniform asymptotical estimates which show the lack of symmetry of this walk
better than local limit estimates. Our results also point out the impossibility
of getting Jones-type non-Gaussian estimates.
zucca@mat.unimi.it
1563. STATIONARY RANDOM FIELDS WITH LINEAR REGRESSIONS
Wlodzimierz Bryc
We analyze certain stationary fields with linear regressions and quadratic
conditional variances. This classic probabilistic problem leads somewhat
unexpectedly to stationary Markov processes closely tied to non-commutative
probability through the q-Hermite polynomials.
brycwz@email.uc.edu
1564. IMAGES OF EIGENVALUE DISTRIBUTIONS UNDER POWER MAPS
Eric M. Rains
In [earlier work by the author], it was shown that if U is a random n x n
unitary matrix, then for any p>=n, the eigenvalues of U^p are i.i.d. uniform;
similar results were also shown for general compact Lie groups. We study what
happens when p<n instead. For the classical groups, we find that we can
describe the eigenvalue distribution of U^p in terms of the eigenvalue
distributions of smaller classical groups; the earlier result is then a special
case. The proofs rely on the fact that a certain subgroup of the Weyl group is
itself a Weyl group. We generalize this fact, and use it to study the power-map
problem for general compact Lie groups.
rains@research.att.com
1565. EXPLICIT ISOPERIMETRIC CONSTANTS, PHASE TRANSITIONS IN THE
RANDOM-CLUSTER AND POTTS MODELS, AND BERNOULLICITY
Olle Haggstrom, Johan Jonasson, Russell Lyons
The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model with cluster parameter $q \ge
1$. These include an explicit pointwise dynamical coupling of random-cluster
measures for arbitrary graphs, and for unimodular transitive graphs, lack of
percolation for the free random-cluster measure at the lower critical value on
nonamenable graphs, and a number of inequalities for the critical values. Some
of these inequalities lead to considerations of isoperimetric constants in
certain hyperbolic graphs, and the first nontrivial explicit calculations of
such constants are obtained. Applications to the Potts model include
Bernoullicity in the $\Z^d$ case at all temperatures, and non-robust phase
transition in the case of nonamenable regular graphs.
rdlyons@indiana.edu
1566. THE SPECTRAL GAP OF THE 2-D STOCHASTIC ISING MODEL WITH MIXED BOUNDARY
CONDITIONS
Kenneth S. Alexander, Nobuo Yoshida
We establish upper bounds for the spectral gap of the stochastic Ising model
at low temperatures in an n-by-n box with boundary conditions which are not
purely plus or minus; specifically, we assume the magnitude of the sum of the
boundary spins over each interval of length n in the boundary is bounded by
\delta n, where \delta < 1. We show that for any such boundary condition, when
the temperature is sufficiently low (depending on \delta), the spectral gap
decreases exponentially in n.
alexandr@math.usc.edu
1567. PATHWISE DESCRIPTION OF DYNAMIC PITCHFORK BIFURCATIONS WITH ADDITIVE
NOISE
Nils Berglund and Barbara Gentz
The slow drift (with speed $\eps$) of a parameter through a pitchfork
bifurcation point, known as the dynamic pitchfork bifurcation, is characterized
by a significant delay of the transition from the unstable to the stable state.
We describe the effect of an additive noise, of intensity $\sigma$, by giving
precise estimates on the behaviour of the individual paths. We show that until
time $\sqrt\eps$ after the bifurcation, the paths are concentrated in a region
of size $\sigma/\eps^{1/4}$ around the bifurcating equilibrium. With high
probability, they leave a neighbourhood of this equilibrium during a time
interval $[\sqrt\eps, c\sqrt{\eps\abs{\log\sigma}}]$, after which they are
likely to stay close to the corresponding deterministic solution. We derive
exponentially small upper bounds for the probability of the sets of exceptional
paths, with explicit values for the exponents.
berglund@wias-berlin.de
1568. THE SPECTRAL GAP OF THE 2-D STOCHASTIC ISING MODEL WITH NEARLY
SINGLE-SPIN BOUNDARY CONDITIONS
Kenneth S. Alexander
We establish upper bounds for the spectral gap of the stochastic Ising model
at low temperature in an N-by-N box, with boundary conditions which are
``plus'' except for small regions at the corners which are either free or
``minus.'' The spectral gap decreases exponentially in the size of the corner
regions, when these regions are of size at least of order \log N. This means
that removing as few as O(\log N) plus spins from the corners produces a
spectral gap far smaller than the order N^{-2} gap believed to hold under the
all-plus boundary condition. Our results are valid at all subcritical
temperatures.
alexandr@math.usc.edu
1569. CUBE-ROOT BOUNDARY FLUCTUATIONS FOR DROPLETS IN RANDOM CLUSTER MODELS
Kenneth S. Alexander
For a family of bond percolation models on Z^{2} that includes the
Fortuin-Kasteleyn random cluster model, we consider properties of the
``droplet'' that results, in the percolating regime, from conditioning on the
existence of an open dual circuit surrounding the origin and enclosing at least
(or exactly) a given large area A. This droplet is a close surrogate for the
one obtained by Dobrushin, Koteck\'y and Shlosman by conditioning the Ising
model; it approximates an area-A Wulff shape. The local part of the deviation
from the Wulff shape (the ``local roughness'') is the inward deviation of the
droplet boundary from the boundary of its own convex hull; the remaining part
of the deviation, that of the convex hull of the droplet from the Wulff shape,
is inherently long-range. We show that the local roughness is described by at
most the exponent 1/3 predicted by nonrigorous theory; this same prediction has
been made for a wide class of interfaces in two dimensions. Specifically, the
average of the local roughness over the droplet surface is shown to be
O(l^{1/3}(\log l)^{2/3}) in probability, where l = \sqrt{A} is the linear scale
of the droplet. We also bound the maximum of the local roughness over the
droplet surface and bound the long-range part of the deviation from a Wulff
shape, and we establish the absense of ``bottlenecks,'' which are a form of
self-approach by the droplet boundary, down to scale \log l. Finally, if we
condition instead on the event that the total area of all large droplets inside
a finite box exceeds A, we show that with probability near 1 for large A, only
a single large droplet is present.
alexandr@math.usc.edu
1570. PRODUCT INTEGRALS IN QUANTUM STOCHASTIC CALCULUS
Robin Hudson and Sylvia Pulmannov'a
Motivated by the search for solutions of the quantum
Yang-Baxter equation, an algebraic theory of quantum
stochastic product integrals is developed. The product
integrators are formal power series in an indeterminate
$h$ whose coefficients are elements of the Lie algebra
$/mathcal{L}$ labelling the usual integrators of a many-
dimensional quantum stochastic calculus. The product
integrtals are also formal power series in $h$, whose
coefficients are finite iterated additive stochastic
integrals which act on the exponential domain of the
Fock space of the calculus and which represent elements of
the unoversal enveloping algebra of $/mathcal{L}$. They
obey a multiplication rule suggested by the quantum It/^{o}
product formula, and are characterized among all such
formal power series by a group-like property.
rlh@maths.nott.ac.uk pulmann@mau.sav.sk
1571. SYMMETRISED DOUBLE QUANTUM STOCHASTIC PRODUCT INTEGRALS
Robin Hudson and Sylvia Pulmannov'a
A theory is developed of double product integrals
over pairs of disjoint finite subintervals
of the real half-line in which the integrator $g[h]$ is a
formal power series in the
indeterminate $h$ whose constant term is zero and whose
coefficients are elements of $/mathcal{L}/otimes /mathcal
{L}$, where $/mathcal{L}$ is the space of basic
differentials of a multidimensional quantum stochastic
calculus. The product integrals themselves are formal
power series in $h$ whose coefficients are finite sums
of products of iterated stochastic integrals against
the elements of$/mathcal{L}$ over the two subintervals.
They are symmetrised so as to be the images,
under the tensor product representation
of the representations of the universal enveloping algebra
&/mathcal{U}$ of the Lie algebra $/mathcal{L}$
canonically associated with the intervals, of a formal
power series $/prod /prod (1+dg[h])$ whose coefficients
are elements of $/mathcal{U}/otimes /mathcal{U}$.
It is shown that the naturally conjectured multiplication
rule, analogous to the multiplication rule for simple
product integrals, holds in the commutative case.
rlh@maths.nott.ac.uk pulmann@mau.savba.sk
1572. METHOD OF FORMAL POWER SERIES IN QUANTUM STOCHASTIC
CALCULUS
R L Hudson, K R Parthasarathy and S Pulmannova
We consider quantum stochastic differential equations of
various types driven by formal power series in an
indeterminate $h$ with zero constant term, whose
coefficients are quantum stochastic differentials of the
usual form. The solutions by iteration are then also formal
power series in $h$ whose coefficients are finite itereated
integrals. They may thus be freely multiplied without
addressing convergence or domain questions. Necessary
conditions for unitarity of evolutions and for
multiplicativity of flows similar to those of the usual
theory can then be shown to be sufficient by purely
algebraic arguments. The Hochschild cohomological
structure of the generators of multiplicative flows
becomes transparent, and its perturbation-theoretic
interpretation is similarly simplified.
rlh@maths.nott.ac.uk krp@isid.ac.in pulmann@mau.savba.sk
1573. CLASSICAL STOCHASTIC PROCESSES FROM QUANTUM
STOCHASTIC CALCULUS
R L Hudson
After briefly reviewing quantum probability and quantum
stochastic calculus and in particular how the one-
dimensional form of the latter includes the stochastic
calculus of both Brownian motion and the Poisson process,
we consider ''Casimir processes'' which are classical
stochastic processes determined by central elements of
the enveloping algebra of the Lie algebra of a multi
dimensional quantum stochastic calculus.
In the $N$-dimensional case a generating
family of the algebra of such processes consists of
processes whose values are the total edge-length (which
is a Poisson process), the total surface area,..., the
hypervolume of a random hypercube in $/Bbb{Z}^N$ which
labels irreducible representations of the $N$-dimensional
general linear group.
rlh@maths.nott.ac.uk
1574. HARNACK INEQUALITIES FOR JUMP PROCESSES
Richard F. Bass and David A. Levin
We consider a class of pure jump Markov processes
in $R^d$ whose jump kernels are comparable to
those of symmetric stable processes. We establish
a Harnack inequality for nonnegative functions that
are harmonic with respect to these processes. We
also establish regularity for the solutions to
certain integral equations.
bass@math.uconn.edu levin@math.uconn.edu
- To see a preprint or other
information provided by the author
click here.
1575. QUASI-FACTORIZATION OF THE ENTROPY AND LOGARITHMIC
SOBOLEV INEQUALITIES FOR GIBBS RANDOM FIELDS
Filippo Cesi
We show that the entropy functional exhibits a
quasi--factorization property with respect to a pair of
weakly dependent $\sigma$--algebras. As an application we
give a simple proof that the Dobrushin and
Shlosman's complete analyticity condition, for a Gibbs
specification with finite range summable interaction,
implies uniform logarithmic Sobolev inequalities. This
result has been previously proven using several different
techniques. The advantage of our approach is that it relies
almost entirely on a general property of the entropy, while
very little is assumed on the Dirichlet form. No topology
is introduced on the single spin space, thus discrete and
continuous spins can be treated in the same way.
cesi@zephyrus.roma1.infn.it
- To see a preprint or other
information provided by the author
click here.
1576. ON THE CHARACTERISTIC POLYNOMIAL OF A RANDOM UNITARY MATRIX
C.P. Hughes, J.P. Keating and Neil O'Connell
We present a range of fluctuation and large deviations results for the
logarithm of the characteristic polynomial $Z$ of a random $N\times N$
unitary matrix, as $N\to\infty$. First we show that
$\ln Z/\sqrt{\frac{1}{2}\ln N}$,
evaluated at a finite set of distinct points, is asymptotically a collection
of iid complex normal random variables. This leads to a refinement of a recent
central limit theorem due to Keating and Snaith, and also explains the
covariance structure of the eigenvalue counting function. We also obtain
a central limit theorem for $\ln Z$ in a Sobolev space of generalised functions
on the unit circle. In this limiting regime, lower-order terms which reflect
the global covariance structure are no longer negligable and feature in the
covariance structure of the limiting Gaussian measure. Large deviations
results for $\ln Z/A$, evaluated at a finite set of distinct points,
can be obtained for $\sqrt{\ln N} \ll A \ll \ln N$. For higher-order
scalings we obtain large deviations results for $\ln Z/A$ evaluated at
a single point. There is a phase transition at $A=\ln N$ (which only
applies to negative deviations of the real part) reflecting a switch
from global to local conspiracy.
noc@hplb.hpl.hp.com
- To see a preprint or other
information provided by the author
click here.
1577. LINEAR FUNCTIONALS OF EIGENVALUES OF RANDOM MATRICES
Persi Diaconis and Steven N. Evans
Let $M_n$ be a random $n \times n$ unitary matrix
with distribution given by Haar measure on the
unitary group. Using explicit moment calculations,
a general criterion is given for linear combinations of
traces of powers of $M_n$ to converge to a Gaussian limit
as $n \rightarrow \infty$. By Fourier analysis, this result
leads to central limit theorems for the measure on the
circle that places a unit mass at each of the eigenvalues
of $M_n$. For example, the integral of this measure
against a function with suitably decaying Fourier
coefficients converges to a Gaussian limit without any
normalisation. Known central limit theorems for the
number of eigenvalues in a circular arc and the logarithm
of the characteristic polynomial of $M_n$ are also derived
from the criterion. The methods in the paper avoid
the use of Szego-type theorems for determinants of
Toeplitz matrices and apply to the orthogonal and
symplectic groups.
diaconis@math.stanford.edu evans@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1578. FLUCTUATIONS OF THE FREE ENERGY IN THE RE MAND THE P-SPIN SK MODELS
Anton Bovier, Irina Kurkova and Matthias Loewe
We consider the random fluctuations of the free energy in the $p$-spin
version of the Sherrington-Kirkpatrick model in the high temperature regime.
Using the martingale approach of Comets and Neveu as used in the standard SK
model combined with truncation techniques inspired by a recent paper by
Talagrand on the $p$-spin version, we prove that (for $p$ even) the random
corrections to the free energy are on a scale $N^{-(p-2)/4}$ only, and after
proper rescaling converge to a standard Gaussian random variable. This is shown
to hold for all values of the inverse temperature, $\b$, smaller than a
critical $\b_p$. We also show that $\b_p\rightarrow \sqrt{2\ln 2}$ as
$p\uparrow +\infty$. Additionally we study the formal $p\uparrow +\infty$ limit
of these models, the random energy model. Here we compute the precise limit
theorem for the partition function at {\it all} temperatures. For
$\b<\sqrt{2\ln2}$, fluctuations are found at an {\it exponentially small}
scale, with two distinct limit laws above and below a second critical value
$\sqrt{\ln 2/2}$: For $\b$ up to that
value the rescaled fluctuations are Gaussian, while below that there are
non-Gaussian fluctuations driven by the Poisson process of the extreme values
of the random energies. For $\b$ larger than the critical $\sqrt{2\ln 2}$, the
fluctuations of the logarithm of the partition function are on scale one and
are expressed in terms of the Poisson process of extremes. At the critical
temperature, the partition function divided by its expectation converges to
$1/2$.
bovier@wias-berlin.de kurkova@eurandom.tue.nl Lowe@eurandom.tue.nl
- To see a preprint or other
information provided by the author
click here.
1579. CONTINUITY OF STOCHASTIC INTEGRALS WITH RESPECT TO
INFINITELY DIVISIBLE RANDOM MEASURES
Michael B. Marcus and Jan Rosinski
Sufficient conditions for continuity are obtained for
stochastic integrals of the form
X(t) = \int_{S} f(t, s) \, M(ds), t \in T
where M is a zero-mean, independently scattered,
infinitely divisible random measure without Gaussian
component, on a Borel space S, T is a compact metric space
and f: T\times S \to R (or C ) is a deterministic
function.
Let \tau: T^2 \to R^+ be a continuous pseudo--metric on
a compact metric space T. Define the \tau-Lipschitz norm of sections of
f by
\|f\|_{\tau,s}=\sup_{u,v\in T}\frac{|f(u,s)-f(v,s)|}
{\tau(u,v)}.
The continuity conditions are given in terms of defining
properties of M, \|f\|_{\tau,s} and metric entropy
conditions determined by the pseudo--metric \tau.
mbmarcus@earthlink.net rosinski@math.utk.edu
- To see a preprint or other
information provided by the author
click here.
1580. THE MOST VISITED SITES OF CERTAIN LEVY PROCESSES
Michael B. Marcus
Let X be a symmetric Levy process with
Ee^{i\la X_t}=e^{-t\psi(\la)}.
Let
\phi(x )=2/\pi \int_0^\ff \frac{1-\cos\la x }{\psi(\la)}d\la.
Assume that \psi(\la) is regularly varying at
zero with index 1<\al\le 2 and that \phi(x)is increasing on
[0,\ff).
Let L_t^x denote the local time of X at x up to time t.
Let V(t) be such that
L_t^{V(t)}=\sup_{x\in R} L_t^x .
V(t)is called the most visited site of X up to time t.
We show that under the above conditions on X, V(t) is
transient and obtain a lower bound for the rate of growth of
V(t). This paper generalizes ``The most visited sites of
symmetric stable processes'', by R. Bass, N. Eisenbaum and
Z. Shi and uses many of their ideas.
mbmarcus@earthlink.net
- To see a preprint or other
information provided by the author
click here.
1581. n-COVARIATION AND SYMMETRIC SDE's DRIVEN BY FINITE
CUBIC VARIATION PROCESSES
Mohammed Errami and Francesco Russo
The n-variation of a continuous process and the
n-covariation of a vector of continuous processes, are
defined through a regularization procedure. We calculate
explicitly the n-variation process, when it exists, of a
martingale convolution. For processes having finite 3-
variation (also indicated by cubic variation), a basic
stochastic calculus is developed. We prove an It\^o
formula and we study existence and uniqueness of the
solution of a stochastic differential equation, in a
symmetric-Stratonovich sense, with respect to those
processes.
russo@math.univ-paris13.fr
1582. SELF-INTERSECTION LOCAL TIME OF ORDER $k$ FOR GAUSSIAN
PROCESSES IN $S'(R^d)$
Anna Talarczyk
We study existence and continuity of self-intersection local
time (SILT) of any order of Gaussian $S'(R^d)$ processes.
We give a general scheme for proving existence and
path-continuity of SILT. In more detail we study Wiener
and Ornstein-Uhlenbeck processes. We consider processes
associated with several classes of covariances,
corresponding to space-homogeneous as well as space-
-inhomogeneous fields. In the first case we give necessary
and sufficient conditions for existence of SILT of order k,
and in the second case - sufficient conditions only.
We show that for some processes existence of SILT implies
its path-continuity, in other cases we need an additional
assumption. We present several examples, in particular
concerning fluctuation limits of $\alpha$-stable particle
systems.
annatal@mimuw.edu.pl
- To see a preprint or other
information provided by the author
click here.
1583. UNREALITY AND IMPROBABILITY BY EXTRAPOLATION OF $i$.
Zakri Kneebone
The improbability of $i$ as a real number reflects the
difficult task of defining seven quantum dimensions in the
Euclidean-sourced three dimensional frameset by the
insufficience of linear algebraic processes.
By 'true mathematics' [it] can only be described as not
restrained to the recognized set of real numbers.1 It is
far too abstract for even 7D mathematics whereas $i^7=-i$.
Obviously, there is a hint of the unreal in the equation
$0=1$.2
There are examples of unreality in quantum gravity. The
tachyon, for instance, is propelled at an unreal speed by
unknown, improbable force. The positron as well is an
example, making itself known as real-world left-overs and
making possible 12 or 14 or more dimensions. Half of these
are unreality, improbability, and other concepts so
abstract they defy even these descriptions.
I fear that being left with mere extractions of the set of
Unreality will be hindering in its definintion. Indeed, we
may need to wait out our lack of abstract concept. It
necessitates adaption; we will eventually conquer the
improbable and unreality.
c=subset of e=element of e\=not an element of.
Postulate of unreality: $-1eR<=>ie\R$
1 $U={CcR}.:UcR or RcU?$ $U=Ri$
2 $(1/1)-(1/1)=0/1=0^-1$ $(0/1)0=0^0=1$
zkneebone@yahoo.com
1584. TAGGED PARTICLE DISTRIBUTIONS OR HOW TO CHOOSE A HEAD AT RANDOM
Thomas M. Liggett
Thorisson and others have proved results that imply the
following: given an
i.i.d. family of Bernoulli random variables indexed by $Z^d$, there
exists an occupied site $X\in Z^d$ with the property that
relative to it, the other variables are still i.i.d. We
raise the question of how large such an $X$ must be. For
$d=1$, we prove that any $X$ with this property satisfies
$E|X|^{\frac 12}=\infty$. Moreover, there does exist such an $X$
with tails $P(|X|\geq n)$ of order $Cn^{-\frac 12}$, so these
results are essentially best possible. Analogous results for the
Poisson process in one dimension are given as well.
tml@math.ucla.edu
- To see a preprint or other
information provided by the author
click here.
1585. TRUNCATION AND AUGMENTATION OF LEVEL-INDEPENDENT QBD PROCESSES
Guy Latouche and Peter Taylor
In the study of quasi-birth-and-death (QBD) processes, the first
passage probabilities from states in level one to the boundary level
zero are of fundamental importance. These probabilities are organised
into a matrix, usually denoted by $G$.
The matrix $G$ is the minimal nonnegative solution of a matrix
quadratic equation. If the QBD process is recurrent, then $G$ is
stochastic. Otherwise, $G$ is sub-stochastic and the matrix equation
has a second solution $G^*$, which is stochastic. In this paper we
give a physical interpretation of $G^*$ in terms of sequences of
truncated and augmented QBD processes.
As part of the proof of our main result, we derive expressions for the
first passage probabilities that a QBD process will hit level $k$
before level zero and vice versa, which are of interest in their own
right.
The paper concludes with a discussion of the stability of a recursion
naturally associated with the matrix equation which defines $G$ and
$G^*$. In particular, we show that $G$ is a stable equilibrium point
of the recursion while $G^*$ is an unstable equilibrium point.
latouche@ulb.ac.be ptaylor@ossa.maths.adelaide.edu.au
1586. DRIFT CONDITIONS FOR MATRIX-ANALYTIC MODELS
Guy Latouche and Peter Taylor
In his seminal work, Neuts gave drift criteria by which one can
determine whether processes of GI/M/1 or M/G/1 type are positive
recurrent. Recently, a different drift condition to determine the
ergodic character of a quasi-birth-and-death process (QBD) appeared in
the literature, although its justification does not seem to have been
formally established.
In this note, we provide a proof for this new drift condition in a
general context. We also give a simple proof for Neuts' original
condition and establish a number of new drift conditions for the
ergodic character of matrix-analytic models.
latouche@ulb.ac.be ptaylor@ossa.maths.adelaide.edu.au
1587. DISPERSION RATES UNDER FINITE MODE KOLMOGOROV FLOWS
Michael Cranston, Michael Scheutzow
We consider the growth rate of a body of passive tracers
moving in the plane under the influence of a random,
fluctuating velocity field. The velocity fields we consider
are finite mode approximations to Kolmogorov velocity
fields. The latter are commomly used as models for turbulent
diffusion. We show that the diameter of a body of passive
tracers grows linearly in time under the influence of
these velocity fields, provided the initial body has
positive area. This is in contrast to the growth rate
of a single passive tracer 'fixed in advance' which,
due to homogenization results, will move approximately
like a Brownian particle.
cran@math.rochester.edu ms@math.tu-berlin.de
1588. A LARGE DEVIATION RESULT FOR THE RANGE OF A RANDOM
WALK AND FOR THE WIENER SAUSAGE
Yuji Hamana and Harry Kesten
Let $\{S_n\}$ be a random walk on $\mathbb Z^d$
and let $R_n$ be the number of different points among
$\bold 0, S_1, \dots, S_{n-1}$. We prove here that if
$d \ge 2$, then $\psi(x) := \lim_{n \to \infty}
(-1/n) \log P\{R_n \ge nx\}$ exists for $x \ge 0$ and
establish some convexity and monotonicity
properties of $\psi(x)$. The one-dimensional case will be
treated in a separate paper.
We also prove a similar result for the Wiener sausage (with
drift). Let ${B(t)}$ be a $d$-dimensional Brownian motion
with constant drift, and for a bounded set
$A \subset \mathbb R^d$ let $\La_t= \La_t(A)$
be the $d$-dimensional Lebesgue measure of the
`sausage' $\bigcup_{0 \le s \le t} (B(s)+A)$.
Then $\phi(x) := \lim_{t \to \infty}
(-1/t) \log P\{\La_t \ge tx\}$
exists for $x \ge 0$ and has similar properties as $\psi$.
kesten@math.cornell.edu
1589. LARGE DEVIATIONS FOR THE RANGE OF AN INTEGER VALUED RANDOM WALK
Yuji Hamana and Harry Kesten
Let $R_n$ be the number of different points among
$\bold 0, S_1, \dots, S_{n-1}$. We prove that
$\psi(x) := \lim_{n \to \infty} (-1/n) \log P\{R_n \ge nx\}$
exists for $x \ge 0$ and establish some convexity and
monotonicity properties of $\psi(x)$. This is a sequel to a
recent paper which treats random walks on $\mathbb Z^d$
with $d \ge 2$.
kesten@math.cornell.edu
1590. RANDOM PARKING, SEQUENTIAL ADSORPTION, AND THE
JAMMING LIMIT
Mathew D. Penrose
Identical cars are dropped sequentially from above into
a large parking lot. Each car is positioned uniformly
at random, subject to non-overlap with its predecessors,
until jamming occurs. There have been many studies
of the limiting mean coverage as the parking lot becomes
large, but no complete proof that such a limit exists,
until now.
We prove spatial laws of large numbers demonstrating
that for various multidimensional random and cooperative
sequential adsorption schemes such as the one above,
the jamming limit coverage is well-defined. These
are analogous to a well-known one-dimensional result
of Renyi.
mathew.penrose@durham.ac.uk
- To see a preprint or other
information provided by the author
click here.
- Or
here.
1591. SPDEs LAW EQUIVALENCE AND THE COMPACT SUPPORT PROPERTY:
APPLICATIONS TO THE ALLEN-CAHN SPDE
Hassan Allouba
Using our uniqueness in law transfer result for SPDEs,
described in a recent note, we prove the equivalence of
laws of SPDEs differing by a drift, under vastly applicable
conditions. This gives us the equivalence in the compact
support property among a large class of SPDEs. As an
important application, we prove the equivalence in law of
the Allen-Cahn and the associated heat SPDEs; and
we give a criterion for the compact support property to hold
for the Allen-Cahn SPDE with diffusion function
$a(t,x,u)=Cu^\gamma$, with $C\ne0$ and $1/2\le\gamma<1$.
allouba@indiana.edu
- To see a preprint or other
information provided by the author
click here.
1592. PHASE COEXISTENCE IN ISING, POTTS AND PERCOLATION MODELS
Raphael Cerf and Agoston Pisztora
We study phase separation and phase coexistence phenomena in Ising,
Potts and random cluster models in dimensions $d\geq 3$. The
simultaneous occurrence of several phases is typical for systems
with appropriately arranged (mixed) boundary conditions and for
systems conditioned on certain large deviation events. These
phases define a partition of the space. Our main results are
large deviations principles for (the shape of) the empirical
phase partition emerging in these models. More specifically,
we establish a general large deviation principle for the
partition induced by large (macroscopic) clusters in the
Fortuin--Kasteleyn model and transfer it to the Ising-
-Potts model where we obtain a large deviation principle
for the natural partition corresponding to the various phases.
The rate function turns out to be the total surface energy
(associated with the surface tension of the model and with
boundary conditions) which can be naturally assigned to
each reasonable partition. These LDP-s imply a weak law of
large numbers: the law of the phase partition is asymptotically
determined by an appropriate variational problem. More precisely,
the phase partition will be close to some partition which is
compatible with the constraints imposed on the system and which
minimizes the total surface energy. A general compactness argument
guarantees the existence of at least one such minimizing partition.
Our results are valid for temperatures $T$ below a limit of
slab-thresholds $\hat{T}_c$ conjectured to agree with the
critical point $T_c$. Moreover, $T$ should be such that there exists
only one translation invariant infinite volume state in the
corresponding Fortuin--Kasteleyn model; a property which can fail
for at most countably many values and which is conjectured to be
true for every~$T$.
Raphael.Cerf@math.u-psud.fr pisztora@asd4.math.cmu.edu
