Probability Abstracts 48
This document contains abstracts 1153-1193.
They have been mailed on December 28, 1998.
1153. PERCOLATION ON TRANSITIVE GRAPHS AS A COALESCENT PROCESS:
RELENTLESS MERGING FOLLOWED BY SIMULTANEOUS UNIQUENESS
Olle H\"aggstr\"om, Yuval Peres and Roberto H. Schonmann
Consider i.i.d. percolation with retention parameter $p$ on
an infinite graph $G$. There is a well known critical parameter
$p_c \in [0,1]$ for the existence of infinite open clusters.
Recently, it has been shown that when $G$ is quasi-transitive,
there is another critical value $p_u \in [p_c,1]$ such that the
number of infinite clusters is a.s.\ $\infty$ for $p\in(p_c,p_u)$,
and a.s. one for $p>p_u$. We prove a simultaneous version of this
result in the canonical coupling of the percolation processes for all
$p\in[0,1]$. Simultaneously for all $p\in(p_c, p_u)$, we also prove
that each infinite cluster has uncountably many ends. For $p > p_c$ we
prove that all infinite clusters are indistinguishable by robust
properties. Under the additional assumption that $G$ is unimodular,
we prove that a.s. for all $p_1<p_2$ in $(p_c,p_u)$, every infinite
cluster at level $p_2$ contains infinitely many infinite clusters
at level $p_1$. We also show that any Cartesian product $G$ of $d$
infinite connected graphs of bounded degree satisfies
$p_u(G) \leq p_c(\Z^d)$.
olleh@math.chalmers.se peres@stat.berkeley.edu rhs@math.ucla.edu
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here.
- Or
here.
1154. PERCOLATION IN $\infty + 1$ DIMENSIONS
AT THE UNIQUENESS THRESHOLD
Roberto H. Schonmann
For independent density $p$ site percolation on the (transitive
non-amenable) graph $\T_b \times \Z$, where $\T_b$ is a homogeneous tree
of degree $b+1$, and $b$ is supposed to be large, it is shown that for
$p= p_u = \inf \{p \colon \text{a.s. there is a unique infinite cluster}\}$
there are a.s. infinitely many infinite clusters. This contrasts with a
recent result of Benjamini and Schramm, according to whom for transitive
non-amenable planar graphs there is a.s. a unique infinite cluster at $p_u$.
rhs@math.ucla.edu
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1155. AN EMBEDDING FOR THE KESTEN-SPITZER RANDOM WALK IN
RANDOM SCENERY
Endre Cs\'aki, Wolfgang K\"onig and Zhan Shi
Consider the Kesten-Spitzer random walk in random scenery
$K(n)=\sum_{k=0}^n \sigma(S_k)$, where $(S_k)$ is a simple
symmetric random walk and $(\sigma_x)_{x\in Z}$ is an i.i.d.
sequence, such that $(S_k)$ and $(\sigma_x)$ are
independent. Let $G(t)=\int_R L(t,x) dW(x)$, where $W$ is a
Brownian motion, and $L$ is a Brownian local time
independent of $W$.
Under certain conditions, we construct $K(n)$ and $G(n)$
such that with probability one,
$|K(n)-G(n)| = o(n^{5/8+\epsilon})$, as $n$ goes to
infinity. This extends a recent result for Gaussian
sceneries due to Khoshnevisan and Lewis. We also identify
the constant in the law of the iterated logarithm for $K$
and $G$.
csaki@math-inst.hu wkoenig@fields.utoronto.ca shi@ccr.jussieu.fr
1156. A GENERALIZATION OF HASMINISKII'S THEOREM ON EXISTENCE OF INVARIANT
MEASURES FOR LOCALLY INTEGRABLE DRIFTS.
V. Bogachev, M. R\"ockner
We prove existence of invariant measures for diffusions on $\R^n$ whose drifts
are merely locally in $L^p$ with $p>n$. Our only global assumption is that a
Liapunov--type functions exists. Subsequently, we generalize this result to the
case where $\R^n$ is replaced by a
Riemannian manifold.
kraetzsc@Physik.Uni-Bielefeld.DE
1157. CONTROLLING ORTHOGONAL SERIES WITH IRRATIONAL ROTATIONS
Michel Weber
We compute covering numbers associated to the set of partial
sums of orthogonal series by means of irrational rotations.
The device is a new metric inequality linking the increment's
norm of partial sums to the one of ergodic averages of rotations
acting on a suitable
$L^2$-element of the torus. This allows to compute the number
of balls covering the whole set of partial sums, by means of rotations.
weber@math.u-strasbg.fr
1158. SYSTEMES QUASI-ORTHOGONAUX ET THEOREME CENTRAL LIMITE PRESQUE SUR
Michel Weber
In a recent work, we indicated another formulation of the Almost Sure
Central Limit Theorem (A.S.C.L.T.), with series in place of averages,
by showing that the property of the A.S.C.L.T. directly follows from
the theory of orthogonal sums. For, we used the notion of quasi-orthogonal
systems introduced earlier by R. Bellmann, and later developped by
Kac-Salem-Zygmund. The main object of this paper is to prove a similar
result for irrational rotations of the torus. We prove the existence of
a generalized moment version of the A.S.C.L.T., with a speed of convergence.
In our strategy, we use again the notion of quasi-orthogonal system,
and purpose a gaussian randomization technic, new at least in this context.
The proof avoid notably the use of Volny's result on the existence of good
gaussian approximations in aperiodic dynamical systems, and should also
permit to be able to treat problems of comparable nature.
weber@math.u-strasbg.fr
1159. DOUBLE POINTS OF THE BROWNIAN SHEET IN ${\bf R}^d$
AND THE GEOMETRY OF THE PARAMETER SPACE
Peter Imkeller and Ferenc Weisz
For the Brownian sheet $W$ with values in ${\bf R^d}$ the set of
double points
$(s,t)\in A\times B$, i.e. points for which $W_s =
W_t$, is investigated, in
terms of the corresponding self intersection local time.
Its existence is seen
to depend sensitively upon the geometric constellation
of the compact sets
$A, B$ in $[0,1]^2$. We suppose that $A$ and $B$
intersect at exactly one point $p$, and can be
separated locally by an axial parallel line. We
further assume that their boundaries in a vicinity of
$p$ are given by power type functions. We compute the
critical dimension below which self intersection
local time exists and describe it in terms of the
powers of the boundary curves of $A$ and $B$.
imkeller@mathematik.hu-berlin.de
fweisz@mathematik.hu-berlin.de
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1160. INTERSECTION EXPONENTS FOR PLANAR BROWNIAN MOTION
Greg Lawler and Wendelin Werner
We derive properties concerning
all intersection exponents for planar Brownian motion
and we define generalized exponents that
loosely speaking correspond to non-integer numbers of
Brownian paths.
Some of these properties lead to general conjectures
concerning
the exact value of these exponents.
jose@math.duke.edu wendelin.werner@math.u-psud.fr
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here.
1161. STOCHASTIC VERSIONS OF THE LASALLE THEOREM
Xuerong Mao
In 1892, Lyapunov introduced the concept of stability of
a dynamic system and created a very powerful tool known
as Lyapunov's second method in the study of stability.
The Lyapunov method has been developed and applied by
many authors during the past century. One of the important
developments in this direction is the LaSalle theorem for
locating limit sets of nonautonomous systems, from which
follow many of the classical Lyapunov results on stability.
On the other hand, since It\^o introduced his stochastic
calculus about 50 years ago, the theory of stochastic
differential equations has been developed very quickly.
In particular, the Lyapunov method has been developed to
deal with stochastic stability by many authors e.g.
Arnold, Friedman, Has'minskii, Kushner, Kolmanovskii,
Myshkis, Ladde, Lakshmikantham, Mohammed and myself.
The classical results by Has'minskii, Kushner et al.
typically state that solutions of a stochastic differential
equation converge in probability to some invariant set
if the initial condition approaches this invariant set.
However, so far there seems to be no stochastic version
of the LaSalle theorem that locates limit sets of a system,
and the main aim of this paper is to extend it from
ordinary differential equations to stochastic differential
equations. We shall show that many classical results
on stochastic stability follow from our stochastic versions
of the LaSalle theorem. This shows clearly the power of
our new results.
xuerong@stams.strath.ac.uk
1162. STOCHASTIC SYMMETRY-BREAKING IN A GAUSSIAN HOPFIELD-MODEL
Anton Bovier, Aernout C.D. van Enter and Beat Niederhauser
We study a ``two-pattern'' Hopfield model
with Gaussian disorder. We find that there are infinitely many pure states
at low temperatures in this model, and we find that the metastate is
supported on an infinity of symmetric pairs of pure states. The origin of this
phenomenon is the random breaking of a rotation symmetry of the distribution of
the disorder variables.
bovier@wias-berlin.de
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1163. STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS
WITH MARKOVIAN SWITCHING
Xuerong Mao
Stability of stochastic differential equations
with Markovian switching has recently received a lot
of attention. For example, stability of linear or
semi-linear type of such equations has been studied by
Basak et al., Ji \& Chizeck and Mariton. The aim of this
paper is to discuss the exponential stability for general
nonlinear stochastic differential equations with Markovian
switching. By aplying the theory of M-matrices, the
stability criteria established in this paper are
explicitly described in terms of given parameters
of the system and can be verified easily so the results
are very useful in applications.
xuerong@stams.strath.ac.uk
1164. A TEST FOR NONLINEARITY OF TIME SERIES WITH INFINITE VARIANCE
Sidney Resnick and Eric van den Berg
A heavy tailed time series that can be represented as an infinite moving
average has the property that the sample autocorrelation function (ACF)
at lag $h$ converges in probability to a constant $\rho(h)$, although the
mathematical correlation typically does not exist. For many nonlinear heavy
tailed models, however, the sample ACF at lag $h$ converges in distribution
to a nondegenerate random variable. In this paper, a test for (non)linearity
of a given infinite variance time series is constructed, based on subsample
stability of the sample ACF. The test is applied to several real and simulated
datasets.
sid@orie.cornell.edu ericvdb@cam.cornell.edu
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1165. GROWTH RATES OF SAMPLE COVARIANCES
OF STATIONARY SYMMETRIC $\alpha$-STABLE PROCESSES
ASSOCIATED WITH NULL RECURRENT MARKOV CHAINS
Sidney Resnick, Gennady Samorodnitsky and Fang Xue
A null recurrent Markov chain is associated with a
stationary mixing symmetric $\alpha$ stable process.
The resulting process exhibits
such strong dependence that its sample covariance
grows at a surprising rate which is slower than one
would expect based on the fatness of the
marginal distribution tails. An additional feature of
the process is that the sample autocorrelations
converge to non-random limits.
sid@orie.cornell.edu gennady@orie.cornell.edu xue@cam.cornell.edu
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1166. A HEAVY TRAFFIC LIMIT THEOREM FOR WORKLOAD PROCESSES
WITH HEAVY TAILED SERVICE REQUIREMENTS
Sidney Resnick and Gennady Samorodnitsky
A system with heavy tailed service requirements under
heavy load having a single server, has an equilibrium waiting time
distribution which is approximated by the Mittag--Leffler distribution.
This fact is understood by a direct analysis of the weak convergence of
a sequence of negative drift random walks with heavy right tail and the
associated all time maxima of these random walks. This approach
complements the recent transform view of Boxma and Cohen (1997).
sid@orie.cornell.edu gennady@orie.cornell.edu
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1167. LAWS OF THE ITERATED LOGARITHM FOR SOME SYMMETRIC DIFFUSION PROCESSES
Richard F. Bass and Takashi Kumagai
We prove three results concerning LIL for certain symmetric diffusions.
If the symmetric diffusion has transition densities satisfying Aronson-type
estimates, then a Strassen-like functional LIL holds. We give some conditions
under which a continuous functional of any Markov process satisfies the upper
bound for an LIL. We give LILs for the range and local times
of diffusions on affine nested
fractals and Sierpinski carpets.
bass@math.uconn.edu tkuma@kusm.kyoto-u.ac.jp
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1168. ASYMPTOTIC BEHAVIOUR OF THE DENSITY IN A PARABOLIC SPDE
Arturo Kohatsu-Higa, David Marquez-Carreras
and Marta Sanz-Sole
Consider the density of the solution $X(t,x)$ of a stochastic
heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$.
In the paper we study the asymptotics of this density as the noise is
vanishing. A kind of Taylor expansion in powers of the noise
parameter is obtained. The coefficients and the residue of the expansion
are explicitly calculated.
In order to obtain this result some type of exponential estimates of
tail probabilities of the difference between the approximating
process and the limit one is proved. Also a suitable local integration
by parts formula is developped.
kohatsu@upf.es marquez@cerber.mat.ub.es sanz@mat.ub.es
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1169. REFLECTING BROWNIAN SNAKE AND A NEUMANN-DIRICHLET PROBLEM
Romain Abraham
The paper deals with a Markov path-valued process: the
reflecting Brownian snake. It is a particular case of the
path-valued process former introduced by Le Gall. Here the
spatial motion (which is for Le Gall any Markov process)
is a reflecting Brownian motion in a domain $D$ of $\R ^d$.
Using this probabilistic tool, we construct an explicit
function $v$ solution of an integral equation which is,
under some hypothesis on the regularity of $v$, equivalent
to a semi-linear partial differential equation in $D$ with
some mixed Neumann-Dirichlet conditions on the boundary. As
the hypothesis on $v$ are not always verified, we also prove
that $v$ is solution of a weak formulation of the
Neumann-Dirichlet problem.
abraham@math-info.univ-paris5.fr
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information provided by the author
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1170. ON THE EXCURSIONS OF TWO-DIMENSIONAL RANDOM WALK AND
WIENER PROCESS
Endre Csaki, Antonia Foldes, Pal Revesz and Zhan Shi
Consider a simple symmetric random walk on the plane. Its
portion between two consecutive returns to zero are called
excursions. We study the almost sure asymptotic behaviours
of the sum of the excursion lengths when the two largest
ones are eliminated from the sum. Similar investigations
are carried out for two-dimensional Wiener process.
csaki@math-inst.hu afoldes@broadway.gc.cuny.edu
reveszp@math-inst.hu shi@ccr.jussieu.fr
1171. SELF-SIMILAR COMMUNICATION MODELS AND VERY HEAVY TAILS
Sidney Resnick and Holger Rootzen
Several studies of file sizes either being downloaded or stored in the
world wide web have commented that tails can be so heavy that not only
are variances infinite, but so are means. Motivated by this fact,
we study the infinite node
Poisson model under the assumption that transmission times are heavy
tailed with infinite mean. The model is unstable but we are able to
provide growth rates. Self-similar but non-stationary
Gaussian process approximations are provided for the number of active
sources, cumulative input, buffer content and time to buffer overflow.
sid@orie.cornell.edu rootzen@math.chalmers.edu
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1172. A NOTE ON SELF-EXCITING POINT PROCESSES
Mathieu Plante
The self-exciting property for point processes on the half-
line was defined by Kwieci\'{n}ski and Szekli~(1996) for
each one of three partial orders. We unify these definitions
of the self-exciting property through a single treatment of
an arbitrary closed partial order on the space of integer
measures. The self-exciting property is characterized by the
behaviour of the compensator of the point process over a
class (specific to the order) of ``echelon'' sets. Our defi-
nition recovers and generalizes those of Kwieci\'{n}ski and
Szekli.
s596100@matrix.cc.uottawa.ca
1173. METASTABILITY IN STOCHASTIC DYNAMICS OF DISORDERED
MEAN-FIELD MODELS
Anton Bovier, Michael Eckhoff, Veronique Gayrard and Markus Klein
We study a class of Markov chains
that describe reversible stochastic dynamics of a large class of
disordered mean field models at low temperatures. Our main purpose is
to give a precise relation between the metastable time scales
in the problem to the properties of the rate functions of the
corresponding Gibbs measures. We derive the analog of the
Wentzell-Freidlin theory in this case, showing that any transition
can be decomposed, with probability exponentially close to one, into
a deterministic sequence of ``admissible transitions''. For these
admissible transitions we give upper and lower bounds on the
expected transition times that differ only by a factor $\sqrt N$, where
$N$ denotes the volume of the system. The distribution
rescaled transition times are shown
to converge to the exponential distribution.
We exemplify our
results in the context of the random field Curie-Weiss model.
bovier@wias-berlin.de
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1174. BROWNIAN MOTION PENETRATING FRACTALS
- AN APPLICATION OF THE TRACE THEOREM OF BESOV SPACES
Takashi Kumagai
For a closed connected set $F$ in $R^n$, assume that there
is a local regular Dirichlet form (a symmetric diffusion
process) on $F$ whose domain is included in a Lipschitz
space or a Besov space on $F$. Under some condition for the
order of the space, we prove that there exists a symmetric
diffusion process on $R^n$ which moves like the process on
$F$ and like Brownian motion on $R^n$ outside $F$. As an
application, we will show that if $F$ is a nested fractal
or a Sierpinski carpet, we can construct Brownian motion
penetrating the fractal. For the proof, we apply the
technique developed in the theory of Besov spaces.
kumagai@math.ubc.ca
1175. THE CONJUGACY OF STOCHASTIC AND RANDOM DIFFERENTIAL
EQUATIONS AND THE EXISTENCE OF GLOBAL ATTRACTORS
Peter Imkeller, Bjoern Schmalfuss
We consider stochastic differential equations in $d$-dimensional Euclidean space
driven by an $m$-dimensional Wiener process, determined by the drift vector
field $f_0$ and the diffusion vector fields $f_1,\cdots, f_m$, and investigate
the existence of global random attractors for the associated flows $\phi$.
For this purpose $\phi$ is decomposed into a stationary diffeomorphism
$\P$ given by the stochastic differential equation on the space of smooth
flows on $\r^d$ driven by $m$ independent stationary Ornstein Uhlenbeck
processes $z^1,\cdots, z^m$ and the vector fields $f_1,\cdots, f_m$, and a
flow $\chi$ generated by the non-autonomous ordinary differential equation
given by the vector field $(\parder{\P_t}{x})^{-1} [f_0(\P_t) + \suim
f_i(\P_t)\,z_t^i]$. In this setting, attractors of $\chi$ are canonically
related with attractors of $\phi$. For $\chi$, the problem of existence of attractors
is then considered as a perturbation problem. Conditions on the vector fields
are derived under which a Lyapunov function for the deterministic
differential equation determined by the vector field $f_0$ is still a
Lyapunov function for $\chi$, yielding an attractor this way. The criterion
is finally tested in various prominent examples.
imkeller@mathematik.hu-berlin.de
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information provided by the author
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1176. ALMOST SURE FUNCTIONAL LIMIT THEOREMS.
PART I. THE GENERAL CASE
Peter Major
We formulate and prove the almost sure
functional limit theorem in fairly general
cases. This limit theorem is a result
which states that if a stochastic process
$X(t,\omega)$, $t\ge0$, is given on a
probability space with some nice properties,
then an appropriate probability measure
$\bar\lambda_T$ can be defined on the
interval $[1,T]$ for all $T>1$ in such a way
that for almost all $\omega$ the
distributions of the appropriate
normalizations of the trajectories
$X_t(\cdot,\omega)=X(t\cdot,\omega)$, ---
considered as random variables $\xi_T(t)$,
$t\in[1,T]$, on the probability spaces
$([1,T],{\cal A},\lambda_T)$ with values in a
function space --- have a weak limit
independent of $\omega$ as $T\to\infty$.
We shall consider self-similar processes
which appear in different limit
theorems. The almost sure functional limit
theorem will be formulated and proved for
them and their appropriate discretization
under weak conditions. We also formulate and
prove a coupling argument which makes
it possible to prove the almost sure
functional limit theorem for certain
processes which converge to a self-similar
process. In the second part of this work we
shall prove and generalize --- with the
help of the results of the first part ---
some known almost sure functional limit
theorems for independent random variables.
major@math-inst.hu
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1177. ALMOST SURE FUNCTIONAL LIMIT THEOREMS.
PART II. THE CASE OF INDEPENDENT RANDOM VARIABLES
Peter Major
In the first part of this paper we formulated
and proved the almost sure functional limit
theorem under general conditions. In this
paper we prove with its help that the usual
conditions of limit theorems for the
distribution of appropriately normalized sums of independent random
variables are also sufficient for the almost
sure functional limit theorem for these
independent random variables.
major@math-inst.hu
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1178. VARIABLY SKEWED BROWNIAN MOTION
Martin Barlow, Krzysztof Burdzy, Haya Kaspi and Avi Mandelbaum
Given a standard Brownian motion $B$, we show that
the equation $X_t = x_0 + B_t + \beta (L_t^X)$ has
a unique strong solution $X$. Here $L^X$ is the
symmetric local time of $X$ at $0$, and $\beta$
is a given continuously differentiable function
with $\beta (0)=0$, $- 1 < \beta' < 1$.
(For linear $\beta$, the solution is the familiar
skew Brownian motion). For linear $\beta$,
we construct a solution in an ``explicit'' way.
barlow@math.ubc.ca burdzy@math.washington.edu
iehaya@techunix.technion.ac.il avim@techunix.technion.ac.il
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1179. LOCAL TIME FLOW RELATED TO SKEW BROWNIAN MOTION
Krzysztof Burdzy and Zhen-Qing Chen
Suppose $B_t$ is a Brownian motion with $B_0=0$
and $\beta\in(-1,1)$ is a fixed constant.
Then the equation $X_t = x_0 + B_t + \beta L_t^X$
has a unique strong solution $X$. Here $L^X$ is the
symmetric local time of $X$ at $0$. We analyze
the family of the local times $L^X$ corresponding
to a single driving Brownian motion $B$ but
different initial positions $x_0$, or various
values of the skewness parameter $\beta$.
Our results include a Ray-Knight-type theorem.
burdzy@math.washington.edu zchen@math.washington.edu
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1180. CONTINUUM-SITES STEPPING-STONE MODELS, COALESCING EXHCANGEABLE
PARTITIONS, AND RANDOM TREES
Peter Donnelly, Steven N. Evans, Klaus Fleischmann, Thomas G. Kurtz
and Xiaowen Zhou
Analogues of stepping--stone models are considered where the site--space is
continuous, the migration process is a general Markov process, and the
type--space is infinite. Such processes were defined in previous work of the
second author by specifying a Feller transition semigroup in terms of
expectations of suitable functionals for systems of coalescing Markov
processes. An alternative representation is obtained here in terms of a limit
of interacting particle systems. It is shown that, under a mild condition on
the migration process, the continuum--sites stepping--stone process has
continuous sample paths. The case when the migration process is Brownian motion
on the circle is examined in detail using a duality relation between coalescing
and annihilating Brownian motion. This duality relation is also used to show
that a random compact metric space that is naturally associated to an infinite
family of coalescing Brownian motions on the circle has Hausdorff and packing
dimension both almost surely equal to 1/2 and, moreover, this space is capacity
equivalent to the middle--1/2 Cantor set (and hence also to the Brownian zero
set).
evans@stat.berkeley.edu
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information provided by the author
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1181. INVARIANT MEASURES AND CONVERGENCE FOR CELLULAR AUTOMATON 184 AND
RELATED PROCESSES
Vladimir Belitsky and Pablo A. Ferrari
For a class of one-dimensional cellular automata, we review and complete the
characterization of the invariant measures (in particular, all invariant phase
separation measures), the rate of convergence to equilibrium, and the
derivation of the hydrodynamic limit. The most widely known representatives of
this class of automata are: Automaton 184 from the classification of S.
Wolfram, an annihilating particle system and a surface growth model.
pablo@ime.usp.br
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1182. A COMPARISON INEQUALITY FOR SUMS OF INDEPENDENT RANDOM VARIABLES
Stephen Montgomery-Smith and Alexander R. Pruss
We give a comparison inequality that allows one to estimate the tail
probabilities of sums of independent Banach space valued random variables in
terms of those of independent identically distributed random variables. More
precisely, let X_1,...,X_n be independent Banach-valued random variables. Let I
be a random variable independent of X_1,...,X_n and uniformly distributed over
{1,...,n}. Put Z_1 = X_I, and let Z_2,...,Z_n be independent identically
distributed copies of Z_1. Then, P(||X_1+...+X_n|| > t) < c P(||Z_1+...+Z_n|| >
t/c), for all t>0, where c is an absolute constant.
pruss+@pitt.edu
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1183. JEFFREYS PRIORS VERSUS EXPERIENCED PHYSICIST PRIORS - ARGUMENTS AGAINST
OBJECTIVE BAYESIAN THEORY
G. D'Agostini
I review the problem of the choice of the priors from the point of view of a
physicist interested in measuring a physical quantity, and I try to show that
the reference priors often recommended for the purpose (Jeffreys priors) do not
fit to the problem. Although it may seem surprising, it is easier for an
``experienced physicist'' to accept subjective priors, or even purely
subjective elicitation of probabilities, without explicit use of the Bayes'
theorem. The problem of the use of reference priors is set in the more general
context of ``Bayesian dogmatism'', which could really harm Bayesianism.
Giulio.Dagostini@roma1.infn.it
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1184. RANDOM UNITARY MATRICES, PERMUTATIONS AND PAINLEVE
Craig A. Tracy and Harold Widom
This paper is concerned with certain connections between the ensemble of n x
n unitary matrices -- specifically the characteristic function of the random
variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning
the distribution of the length of the longest increasing subsequence in
permutation groups -- and the appearance of Painleve functions in the answers
to apparently unrelated questions. Among the results is a representation in
terms of a Painleve V function for the characteristic function of tr(U) and
(using recent results of Baik, Deift and Johansson) an expression in terms of a
Painleve II function for the limiting distribution of the length of the longest
increasing subsequence in the hyperoctahedral group.
tracy@itd.ucdavis.edu
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1185. BAYESIAN REASONING VERSUS CONVENTIONAL STATISTICS IN HIGH ENERGY PHYSICS
G. D'Agostini
The intuitive reasoning of physicists in conditions of uncertainty is closer
to the Bayesian approach than to the frequentist ideas taught at University and
which are considered the reference framework for handling statistical problems.
The combination of intuition and conventional statistics allows practitioners
to get results which are very close, both in meaning and in numerical value, to
those obtainable by Bayesian methods, at least in simple routine applications.
There are, however, cases in which ``arbitrary'' probability inversions produce
unacceptable or misleading results and in these cases the conscious application
of Bayesian reasoning becomes crucial. Starting from these considerations, I
will finally comment on the often debated question: ``is there any chance that
all physicists will become Bayesian?''
Giulio.Dagostini@roma1.infn.it
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1186. NOISE SENSITIVITY OF BOOLEAN FUNCTIONS AND APPLICATIONS TO PERCOLATION
Itai Benjamini, Gil Kalai, Oded Schramm
It is shown that a large class of events in a product probability space are
highly sensitive to noise, in the sense that with high probability, the
configuration with an arbitrary small percent of random errors gives almost no
prediction whether the event occurs. On the other hand, weighted majority
functions are shown to be noise-stable. Several necessary and sufficient
conditions for noise sensitivity and stability are given.
Consider, for example, bond percolation on an $n+1$ by $n$ grid. A
configuration is a function that assigns to every edge the value 0 or 1. Let
$\omega$ be a random configuration, selected according to the uniform measure.
A crossing is a path that joins the left and right sides of the rectangle, and
consists entirely of edges $e$ with $\omega(e)=1$. By duality, the probability
for having a crossing is 1/2. Fix an $\epsilon\in(0,1)$. For each edge $e$, let
$\omega'(e)=\omega(e)$ with probability $1-\epsilon$, and
$\omega'(e)=1-\omega(e)$ with probability $\epsilon$, independently of the
other edges. Let $p(\tau)$ be the probability for having a crossing in
$\omega$, conditioned on $\omega'=\tau$. Then for all $n$ sufficiently large,
$P\{\tau : |p(\tau)-1/2|>\epsilon\}<\epsilon$.
schramm@wisdom.weizmann.ac.il
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1187. INDISTINGUISHABILITY OF PERCOLATION CLUSTERS
Russell Lyons, Oded Schramm
We show that when percolation produces infinitely many infinite clusters on a
Cayley graph, one cannot distinguish the clusters from each other by any
invariantly defined property. This implies that uniqueness of the infinite
cluster is equivalent to non-decay of connectivity (a.k.a. long-range order).
We then derive applications concerning uniqueness in Kazhdan groups and in
wreath products, and inequalities for $p_u$.
schramm@wisdom.weizmann.ac.il
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1188. RANDOM WALK ON KNOT DIAGRAMS, COLORED JONES POLYNOMIAL AND IHARA-SELBERG
ZETA FUNCTION
Xiao-Song Lin and Zhenghan Wang
A model of random walk on knot diagrams is used to study the Alexander
polynomial and the colored Jones polynomial of knots. In this context, the
inverse of the Alexander polynomial of a knot plays the role of an
Ihara-Selberg zeta function of a directed weighted graph, counting with weights
cycles of random walk on a 1-string link whose closure is the knot in question.
The colored Jones polynomial then counts with weights families of
``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As
a consequence of such interpretations of the Alexander and colored Jones
polynomials, the computation of the limit of the renormalized colored Jones
polynomial when the coloring (or cabling) parameter tends to infinity whereas
the weight parameter tends to 1 leads immediately to a new proof of the
Melvin-Morton conjecture, which was first established by Rozansky and by
Bar-Natan and Garoufalidis.
xl@math.ucr.edu
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1189. STATISTICS OF KNOTS AND ENTANGLED RANDOM WALKS
Sergei Nechaev
The lectures review the state of affairs in modern branch of mathematical
physics called probabilistic topology. In particular we consider the following
problems: (i) We estimate the probability of a trivial knot formation on the
lattice using the Kauffman algebraic invariants and show the connection of this
problem with the thermodynamic properties of 2D disordered Potts model; (ii) We
investigate the limit behavior of random walks in multi-connected spaces and on
non-commutative groups related to the knot theory. We discuss the application
of the above mentioned problems in statistical physics of polymer chains. On
the basis of non-commutative probability theory we derive some new results in
statistical physics of entangled polymer chains which unite rigorous
mathematical facts with more intuitive physical arguments.
nechaev@ipno.in2p3.fr
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1190. GROWTH RATES IN THE QUAQUAVERSAL TILING
Brimstone Draco, Lorenzo Sadun and Douglas Van Wieren
Conway and Radin's ``quaquaversal'' tiling of R^3 is known to exhibit
statistical rotational symmetry in the infinite volume limit. A finite patch,
however, cannot be perfectly isotropic, and we compute the rates at which the
anisotropy scales with size. In a sample of volume N, tiles appear in
O(N^{1/6}) distinct orientations. However, the orientations are not uniformly
populated. A small (O(N^{1/84})) set of these orientations account for the
majority of the tiles. Furthermore, these orientations are not uniformly
distributed on SO(3). Sample averages of functions on SO(3) seem to approach
their ergodic limits as N^{-1/336}. Since even macroscopic patches of a
quaquaversal tiling maintain noticable anisotropy, a hypothetical physical
quasicrystal whose structure was similar to the quaquaversal tiling could be
identified by anisotropic features of its electron diffraction pattern.
sadun@fireant.ma.utexas.edu
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1191. AN APPLICATION OF STOCHASTIC FLOWS TO RIEMANNIAN FOLIATIONS
Alan Mason
A stochastic flow is constructed on a frame bundle adapted to a Riemannian
foliation on a compact manifold. The generator A of the resulting transition
semigroup is shown to preserve the basic functions and forms, and there is an
essentially unique strictly positive smooth function phi satisfying A^* phi =
0. This function is used to perturb the metric, and an application of the
ergodic theorem shows that there exists a bundle-like metric for which the
basic projection of the mean curvature is basic-harmonic.
alanndg@aol.com
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1192. ON RANDOM SECTIONS OF THE CUBE
Yossi Lonke
Let $f(j,k,n)$ denote the expected number of $j$-faces of a random
$k$-section of the $n$-cube. A formula for $f(0,k,n)$ is presented, and for
$j\geq 1$, a lower bound for $f(j,k,n)$ is derived, which implies a precise
asymptotic formula for $f(n-m,n-l,n)$ when $1\leq l<m$ are fixed integers and
$n\to\8$.
jrl16@po.cwru.edu
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1193. STABILITY FOR A CONTINUOUS SOS-INTERFACE MODEL IN A RANDOMLY PERTURBED
PERIODIC POTENTIAL
Christof Kuelske
We consider the Gibbs-measures of continuous-valued height configurations on
the $d$-dimensional integer lattice in the presence a weakly disordered
potential. The potential is composed of Gaussians having random location and
random depth; it becomes periodic under shift of the interface perpendicular to
the base-plane for zero disorder. We prove that there exist localized
interfaces with probability one in dimensions $d\geq 3+1$, in a
`low-temperature' regime. The proof extends the method of
continuous-to-discrete single- site coarse graining that was previously applied
by the author for a double-well potential to the case of a non-compact image
space. This allows to utilize parts of the renormalization group analysis
developed for the treatment of a contour representation of a related
integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of
the disorder, the infinite volume Gibbs measures then have a representation as
superpositions of massive Gaussian fields with centerings that are distributed
according to the infinite volume Gibbs measures of the disordered
integer-valued SOS-model with exponentially decaying interactions.
kuelske@wias-berlin.de
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