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2,500 |
Coupling and Harnack inequalities for Sierpinski carpets
|
math.PR
|
Uniform Harnack inequalities for harmonic functions on the pre- and graphical
Sierpinski carpets are proved using a probabilistic coupling argument. Various
results follow from this, including the construction of Brownian motion on
Sierpinski carpets embedded in $\R^d$, $d\geq 3$, estimates on the fundamental
solution of the heat equation, and Sobolev and Poincar\'e inequalities.
|
math
|
2,501 |
A general decomposition theory for random cascades
|
math.PR
|
This announcement describes a probabilistic approach to cascades which, in
addition to providing an entirely probabilistic proof of the Kahane-Peyri\`ere
theorem for independent cascades, readily applies to general dependent
cascades. Moreover, this unifies various seemingly disparate cascade
decompositions, including Kahane's T-martingale decomposition and dimension
disintegration.
|
math
|
2,502 |
The dimension of the Brownian frontier is greater than 1
|
math.PR
|
Consider a planar Brownian motion run for finite time. The frontier or
``outer boundary'' of the path is the boundary of the unbounded component of
the complement. Burdzy (1989) showed that the frontier has infinite length. We
improve this by showing that the Hausdorff dimension of the frontier is
strictly greater than 1. (It has been conjectured that the Brownian frontier
has dimension $4/3$, but this is still open.) The proof uses Jones's Traveling
Salesman Theorem and a self-similar tiling of the plane by fractal tiles known
as Gosper Islands.
|
math
|
2,503 |
No directed fractal percolation in zero area
|
math.PR
|
We show that fractal (or "Mandelbrot") percolation in two dimensions produces
a set containing no directed paths, when the set produced has zero area. This
improves a similar result by the first author in the case of constant retention
probabilities to the case of retention probabilities approaching 1.
|
math
|
2,504 |
Markov chains in a field of traps
|
math.PR
|
A general criterion is given for when a Markov chain trapped with probability
p(x) in state x will be almost surely trapped. The quenched (state x is a trap
forever with probability p(x)) and annealed (state x traps with probability
p(x) on each visit) problems are shown to be equivalent.
|
math
|
2,505 |
Vertex-reinfoced random walk on Z visits finitely many states
|
math.PR
|
Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a
random walk that is biased to visit sites it has already visited a lot. We show
that this reinforcement scheme, in contrast to the scheme of
edge-reinforcement, causes random walk on a line to get trapped in a finite
set.
|
math
|
2,506 |
Sets avoided by Brownian motion
|
math.PR
|
Any fixed cylinder is hit almost surely by a 3-dimensional Brownian motion,
but is there a random cylinder that is in the complement? We answer this for
cylinders, and then replacing a cylinder with a more general set.
|
math
|
2,507 |
First passage percolation and a model for competing spatial growth
|
math.PR
|
We generalize Richardson's model by starting with two sites of different
colors and giving each new site the color of the site that spawned it. We show
that co-existence is possible.
|
math
|
2,508 |
Paths with exponential intersection tails and oriented percolation
|
math.PR
|
We show that oriented percolation occurs whenever a condition is satisfied
called "exponential intersection tails". This condition says that a measure on
paths exists for which the probability of two independent paths intersecting in
more than k sites is exponentially small in k.
|
math
|
2,509 |
The probability that Brownian motion almost covers a line
|
math.PR
|
Lower and upper estimates are given for the probability that the
epsilon-enlargement of planar Brownian motion to time 1 (the epsilon sausage)
contains a unit line segment. The estimates imply that Brownian motion to time
1 itself contains no line segment.
|
math
|
2,510 |
On minimal parabolic functions and time-homogeneous parabolic h-transforms
|
math.PR
|
Does a minimal harmonic function $h$ remain minimal when it is viewed as a
parabolic function? The question is answered for a class of long thin
semi-infinite tubes $D\subset \R^d$ of variable width and minimal harmonic
functions $h$ corresponding to the boundary point of $D$ ``at infinity.''
Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and
$f$ is a Lipschitz function. The answer to the question is affirmative if and
only if $\int^\infty f^3(u)du = \infty$. If the test fails, there exist
parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite
lifetime which are not time-homogenous.
|
math
|
2,511 |
Completely regular multivariate stationary process and the Muckenhoupt condition
|
math.PR
|
We give necessary and sufficient conditions for a multivariate stationary
stochastic process to be completely regular. We also give the answer to a
question of V.V. Peller concerning the spectral measure characterization of
such processes.
|
math
|
2,512 |
A support property for infinite dimensional interacting diffusion processes
|
math.PR
|
The Dirichlet form associated with the intrinsic gradient on Poisson space is
known to be quasi-regular on the complete metric space $\ddot\Gamma=$
$\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions,
the set $\ddot\Gamma\setminus\Gamma$ is $\e$-exceptional, where $\Gamma$ is the
space of locally finite configurations in $\IR^d$, that is, measures
$\gamma\in\ddot\Gamma$ satisfying $\sup_{x\in\IR^d}\gamma(\{x\})\leq 1$. Thus,
the associated diffusion lives on the smaller space $\Gamma$. This result also
holds for Gibbs measures with superstable interactions.
|
math
|
2,513 |
Strong uniqueness for certain infinite dimensional Dirichlet operators and applications to stochastic quantization
|
math.PR
|
Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on
rigged Hilbert spaces are studied. An analytic approach based on a--priori
estimates is used. The extension of the problem to the $L^p$-setting is
discussed. As a direct application essential self--adjointness and strong
uniqueness in $L^p$ is proved for the generator (with initial domain the
bounded smooth cylinder functions) of the stochastic quantization process for
Euclidean quantum field theory in finite volume $\Lambda \subset \R^2$.
|
math
|
2,514 |
Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent
|
math.PR
|
Sufficient conditions are given for existence and uniqueness in
Smoluchowski's coagulation equation, for a wide class of coagulation kernels
and initial mass distributions. An example of non-uniqueness is constructed.
The stochastic coalescent is shown to converge weakly to the solution of
Smoluchowski's equation.
|
math
|
2,515 |
Stochastic bifurcation models
|
math.PR
|
We study an ordinary differential equation controlled by a stochastic
process. We present results on existence and uniqueness of solutions, on
associated local times (Trotter and Ray-Knight theorems), and on time and
direction of bifurcation. A relationship with Lipschitz approximations to
Brownian paths is also discussed.
|
math
|
2,516 |
Limit Theorems for Sums of p-Adic Random Variables
|
math.PR
|
We study p-adic counterparts of stable distributions, that is limit
distributions for sequences of normalized sums of independent identically
distributed p-adic-valued random variables. In contrast to the classical case,
non-degenerate limit distributions can be obtained only under certain
assumptions on the asymptotic behaviour of the number of summands in the
approximating sums. This asymptotics determines the ``exponent of stability''.
|
math
|
2,517 |
Existence and regularity for a class of infinite-measure $(ξ,ψ,K)$-superprocesses
|
math.PR
|
We extend the class of $(\xi,\psi,K)$-superprocesses known so far by applying
a simple transformation induced by a \lq\lq weight function\rq\rq\ for the
one-particle motion. These transformed superprocesses may exist under weak
conditions on the branching parameters, and their state space automatically
extends to a certain space of possibly infinite Radon measures. It turns out
that a number of superprocesses which were so far not included in the general
theory fall into this class. For instance, we are able to extend the hyperbolic
branching catalyst of Fleischmann and Mueller to the case of $\beta$-branching.
In the second part of this paper, we discuss regularity properties of our
processes. Under the assumption that the one-particle motion is a Hunt process,
we show that our superprocesses possess right versions having \cadlag\ paths
with respect to a natural topology on the state space. The proof uses an
approximation with branching particle systems on Skorohod space.
|
math
|
2,518 |
Rademacher's theorem on configuration spaces and applications
|
math.PR
|
We consider an $L^2$-Wasserstein type distance $\rho$ on the configuration
space $\Gamma_X$ over a Riemannian manifold $X$, and we prove that
$\rho$-Lipschitz functions are contained in a Dirichlet space associated with a
measure on $\Gamma_X$ satisfying some general assumptions. These assumptions
are in particular fulfilled by a large class of tempered grandcanonical Gibbs
measures with respect to a superstable lower regular pair potential. As an
application we prove a criterion in terms of $\rho$ for a set to be
exceptional. This result immediately implies, for instance, a quasi-sure
version of the spatial ergodic theorem. We also show that $\rho$ is optimal in
the sense that it is the intrinsic metric of our Dirichlet form.
|
math
|
2,519 |
A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions
|
math.PR
|
The growth exponent $\alpha$ for loop-erased or Laplacian random walk on the
integer lattice is defined by saying that the expected time to reach the sphere
of radius $n$ is of order $n^\alpha$. We prove that in two dimensions, the
growth exponent is strictly greater than one. The proof uses a known estimate
on the third moment of the escape probability and an improvement on the
discrete Beurling projection theorem.
|
math
|
2,520 |
On increasing subsequences of iid samples
|
math.PR
|
We study the fluctuations, in the large deviations regime, of the longest
increasing subsequence of a random i.i.d. sample on the unit square. In
particular, our results yield the precise upper and lower exponential tails for
the length of the longest increasing subsequence of a random permutation.
|
math
|
2,521 |
Markov Processes with Identical Bridges
|
math.PR
|
Let X and Y be time-homogeneous Markov processes with common state space E,
and assume that the transition kernels of X and Y admit densities with respect
to suitable reference measures. We show that if there is a time t>0 such that,
for each x\in E, the conditional distribution of (X_s)_{0 < s < t}, given X_0 =
x = X_t, coincides with the conditional distribution of (Y_s)_{0 < s < t},
given Y_0 = x = Y_t, then the infinitesimal generators of X and Y are related
by [L^Y]f = \psi^{-1}[L^X](\psi f)-\lambda f, where \psi is an eigenfunction of
L^X with eigenvalue \lambda. Under an additional continuity hypothesis, the
same conclusion obtains assuming merely that X and Y share a ``bridge'' law for
one triple (x,t,y). Our work entends and clarifies a recent result of I.
Benjamini and S. Lee.
|
math
|
2,522 |
A Simple Path to Biggins' Martingale Convergence for Branching Random Walk
|
math.PR
|
We give a simple non-analytic proof of Biggins' theorem on martingale
convergence for branching random walks.
|
math
|
2,523 |
The Stable Manifold Theorem for Stochastic Differential Equations
|
math.PR
|
We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic
differential equations (sde's) that are driven by spatial Kunita-type
semimartingales with stationary ergodic increments. Both Stratonovich and
It\^o-type equations are treated. Starting with the existence of a stochastic
flow for a sde, we introduce the notion of a hyperbolic stationary trajectory.
We prove the existence of invariant random stable and unstable manifolds in the
neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the
stable and unstable manifolds are dynamically characterized using forward and
backward solutions of the anticipating sde. The proof of the stable manifold
theorem is based on Ruelle-Oseledec multiplicative ergodic theory.
|
math
|
2,524 |
Stochastic analysis on configuration spaces: basic ideas and recent results
|
math.PR
|
The purpose of this paper is to provide a both comprehensive and summarizing
account on recent results about analysis and geometry on configuration spaces
$\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to a
complete description of the so--called ``lifting--procedure'', Markov resp.
strong resp. $L^1$--uniqueness results, the non--conservative case, the
interpretation of the constructed diffusions as solutions of the respective
classical ``heuristic'' stochastic differential equations, and a
self--contained presentation of a general closability result for the
corresponding pre--Dirichlet forms. The latter is presented in the general case
of arbitrary (not necessarily pair) potentials describing the singular
interactions. A support property for the diffusions, the intrinsic metric, and
a Rademacher theorem on $\Gamma_X$, recently proved, are also discussed.
|
math
|
2,525 |
Measures on contour, polymer or animal models. A probabilistic approach
|
math.PR
|
We present a new approach to study measures on ensembles of contours,
polymers or other objects interacting by some sort of exclusion condition. For
concreteness we develop it here for the case of Peierls contours. Unlike
existing methods, which are based on cluster-expansion formalisms and/or
complex analysis, our method is strictly probabilistic and hence can be applied
even in the absence of analyticity properties. It involves a Harris graphical
construction of a loss network for which the measure of interest is invariant.
The existence of the process and its mixing properties depend on the absence of
infinite clusters for a dual (backwards) oriented percolation process which we
dominate by a multitype branching process. Within the region of subcriticality
of this branching process the approach yields: (i) exponential convergence to
the equilibrium (=contour) measures, (ii) standard clustering and finite-effect
properties of the contour measure, (iii) a particularly strong form of the
central limit theorem, and (iv) a Poisson approximation for the distribution of
contours at low temperature.
|
math
|
2,526 |
Cheeger's inequalities for general symmetric forms and existence criteria for spectral gap
|
math.PR
|
In this paper, some new forms of the Cheeger's inequalities are established
for general (maybe unbounded) symmetric forms, the resulting estimates improve
and extend the ones obtained by Lawler and Sokal (1988) for bounded jump
processes. Furthermore, some existence criteria for spectral gap of general
symmetric forms or general reversible Markov processes are presented, based on
the Cheeger's inequalities and a relationship between the spectral gap and the
first Dirichlet and Neumann eigenvalues on local region.
|
math
|
2,527 |
On the conditioned exit measures of super-Brownian motion
|
math.PR
|
In this paper we present a martingale related to the exit measures of
super-Brownian motion. By changing measure with this martingale in the
canonical way we have a new process associated with the conditioned exit
measure. This measure is shown to be identical to a measure generated by a
non-homogeneous branching particle system with immigration of mass. An
application is given to the problem of conditioning the exit measure to hit a
number of specified points on the boundary of a domain. The results are similar
in flavor to the "immortal particle" picture of conditioned super-Brownian
motion but more general, as the change of measure is given by a martingale
which need not arise from a single harmonic function.
|
math
|
2,528 |
Concrete representation of martingales
|
math.PR
|
Let (f_n) be a mean zero vector valued martingale sequence. Then there exist
vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n)
dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the
same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . Similar results for tangent
sequences and sequences satisfying condition (C.I.) are presented. We also
present a weaker version of a result of McConnell that provides a Skorohod like
representation for vector valued martingales. This paper may be found at
http://math.missouri.edu/~stephen/preprints
|
math
|
2,529 |
Unitary Brownian motions are linearizable
|
math.PR
|
Brownian motions in the infinite-dimensional group of all unitary operators
are studied under strong continuity assumption rather than norm continuity.
Every such motion can be described in terms of a countable collection of
independent one-dimensional Brownian motions. The proof involves continuous
tensor products and continuous quantum measurements. A by-product: a Brownian
motion in a separable F-space (not locally convex) is a Gaussian process.
|
math
|
2,530 |
Some properties of the range of super-Brownian motion
|
math.PR
|
We consider a super-Brownian motion $X$. Its canonical measures can be
studied through the path-valued process called the Brownian snake. We obtain
the limiting behavior of the volume of the $\epsilon$-neighborhood for the
range of the Brownian snake, and as a consequence we derive the analogous
result for the range of super-Brownian motion and for the support of the
integrated super-Brownian excursion. Then we prove the support of $X_t$ is
capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as
well as the support of the integrated super-Brownian excursion are
capacity-equivalent to $[0,1]^4$ in $\R^d$, $d\geq 5$.
|
math
|
2,531 |
Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations
|
math.PR
|
\def\R{\mathbb R}
We give a characterization of G-regularity for super-Brownian motion and the
Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times
\R^d$, which is not invariant by translation. We then prove that the hitting
probability of a Borel set $A\subset E$ for the graph of the Brownian snake
starting at $(0,0)$ is comparable, up to multiplicative constants, to its
capacity. This implies that super-Brownian motion started at time 0 at the
Dirac mass $\delta_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for
$A^c$) if and only if its capacity is infinite. As a direct consequence, if
$Q\subset E$ is a domain such that $(0,0)\in \partial Q$, we give a necessary
and sufficient condition for the existence on $Q$ of a positive solution of
$\partial_t u+{1/2}\Delta u =2u^2$ which blows up at $(0,0)$. We also give an
estimation of the hitting probabilities for the support of super-Brownian
motion at fixed time. We prove that if $d\geq 2$, the support of super-Brownian
motion is intersection-equivalent to the range of Brownian motion.
|
math
|
2,532 |
Non-degenerate conditionings of the exit measures of super-Brownian motion
|
math.PR
|
We introduce several martingale changes of measure of the law of the exit
measure of super Brownian motion. These changes of measure include and
generalize one arising by conditioning the exit measures to charge a point on
the boun dary of a 2-dimensional domain. In the case we discuss this is a
non-degenerate conditioning. We give characterizations of the new processes in
terms of "immortal particle" branching processes with immigration of mass, and
give application s to the study of solutions to Lu = cu^2 in D. The
representations are related to those in an earlier paper, which treated the
case of degenerate conditionings.
|
math
|
2,533 |
The Repeated Solicitation Model
|
math.PR
|
This paper presents a probabilistic analysis of what we call the "repeated
solicitation model". To give a specific context, suppose B is a direct
marketing company with a list of S sales prospects. At epoch 1, B sends a
solicitation to every prospect on the list, and elicits X(1) replies. The
company deletes the respondents from the list, and at epoch 2 sends a
solicitation to the other prospects, of whom X(2) respond, and so on. This
continues until an epoch n such that X(n) = 0, which we call epoch T, and then
B makes no further solicitations. We seek (a) the probability distribution of
T; (b) the distribution of the total number of respondents; (c) the expected
total number of solicitations. All three quantities are explicitly computed,
assuming that (i) prospects' response times are independent, and (ii) S is
Poisson distributed.
|
math
|
2,534 |
Finite time extinction of super-Brownian motions with catalysts
|
math.PR
|
Consider a catalytic super-Brownian motion $X=X^\Gamma$ with finite variance
branching. Here `catalytic' means that branching of the reactant $X$ is only
possible in the presence of some catalyst. Our intrinsic example of a catalyst
is a stable random measure $\Gamma $ on $R$ of index $0< gamma <1$.
Consequently, here the catalyst is located in a countable dense subset of $R$.
Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is
shown to die in finite time. Our probabilistic argument uses the idea of good
and bad historical paths of reactant `particles' during time periods
$[T_{n},T_{n+1})$. Good paths have a significant collision local time with the
catalyst, and extinction can be shown by individual time change according to
the collision local time and a comparison with Feller's branching diffusion. On
the other hand, the remaining bad paths are shown to have a small expected mass
at time $T_{n+1}$ which can be controlled by the hitting probability of point
catalysts and the collision local time spent on them.
|
math
|
2,535 |
Fractional Brownian motion and the Markov Property
|
math.PR
|
Fractional Brownian motion belongs to a class of long memory Gaussian
processes that can be represented as linear functionals of an infinite
dimensional Markov process. This representation leads naturally to: - An
efficient algorithm to approximate the process. - An infinite dimensional
ergodic theorem which applies to functionals of the type $integral_0^t
phi(V_h(s)) ds $ where $V_h(s)=integral_0^t h(t-u) dB_u$ and $B$ is a standard
Brownian motion.
|
math
|
2,536 |
Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes
|
math.PR
|
Martin boundaries and integral representations of positive functions which
are harmonic in a bounded domain $D$ with respect to Brownian motion are well
understood. Unlike the Brownian case, there are two different kinds of
harmonicity with respect to a discontinuous symmetric stable process. One kind
are functions harmonic in $D$ with respect to the whole process $X$, and the
other are functions harmonic in $D$ with respect to the process $X^D$ killed
upon leaving $D$. In this paper we show that for bounded Lipschitz domains, the
Martin boundary with respect to the killed stable process $X^D$ can be
identified with the Euclidean boundary. We further give integral
representations for both kinds of positive harmonic functions. Also given is
the conditional gauge theorem conditioned according to Martin kernels and the
limiting behaviors of the $h$-conditional stable process, where $h$ is a
positive harmonic function of $X^D$. In the case when $D$ is a bounded $C^{1,
1}$ domain, sharp estimate on the Martin kernel of $D$ is obtained.
|
math
|
2,537 |
Intrinsic Ultracontractivity, Conditional Lifetimes and Conditional Gauge for Symmetric Stable Processes on Rough Domains
|
math.PR
|
For a symmetric $\alpha$-stable process $X$ on $\RR^n$ with $0<\alpha <2$,
$n\geq 2$ and a domain $D \subset \RR^n$, let $L^D$ be the infinitesimal
generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class
function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a
H\"older domain $D$ of order 0. This is then used to establish the conditional
gauge theorem for $X$ on bounded Lipschitz domains in $\RR^n$. It is also shown
that the conditional lifetimes for symmetric stable process in a H\"older
domain of order 0 are uniformly bounded.
|
math
|
2,538 |
Free probability for probabilists
|
math.PR
|
This is an introduction to some of the most probabilistic aspects of free
probability theory.
|
math
|
2,539 |
Some function spaces related to the Brownian motion on simple nested fractals
|
math.PR
|
In this paper we identify the domain of the Dirichlet form associated with
the Brownian motion on simple nested fractals with an integral Lipschitz space.
This result generalizes such an identification on the Sierpi\'nski gasket,
carried on by Jonsson.
|
math
|
2,540 |
On the thermodynamic limit for a one-dimensional sandpile process
|
math.PR
|
Considering the standard abelian sandpile model in one dimension, we
construct an infinite volume Markov process corresponding to its thermodynamic
(infinite volume) limit. The main difficulty we overcome is the strong
non-locality of the dynamics. However, using similar ideas as in recent
extensions of the standard Gibbs formalism for lattice spin systems, we can
identify a set of `good' configurations on which the dynamics is effectively
local. We prove that every configuration converges in a finite time to the
unique invariant measure.
|
math
|
2,541 |
The restriction of the Ising model to a layer
|
math.PR
|
We discuss the status of recent Gibbsian descriptions of the restriction
(projection) of the Ising phases to a layer. We concentrate on the projection
of the two-dimensional low temperature Ising phases for which we prove a
variational principle.
|
math
|
2,542 |
A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums of independent identically distributed random variables
|
math.PR
|
Let X_1,X_2,... be a sequence of independent and identically distributed
random variables, and put S_n=X_1+...+X_n. Under some conditions on the
positive sequence tau_n and the positive increasing sequence a_n, we give
necessary and sufficient conditions for the convergence of sum_{n=1}^infty
tau_n P(|S_n|>t a_n) for all t>0, generalizing Baum and Katz's (1965)
generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also
allowing us to characterize the convergence of the above series in the case
where tau_n=1/n and a_n=(n log n)^{1/2} for n>1, thereby answering a question
of Spataru. Moreover, some results for non-identically distributed independent
random variables are obtained by a recent comparison inequality. Our basic
method is to use a central limit theorem estimate of Nagaev (1965) combined
with the Hoffman-Jorgensen inequality(1974).
|
math
|
2,543 |
Continuum-sites stepping-stone models, coalescing exhcangeable partitions, and random trees
|
math.PR
|
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).
|
math
|
2,544 |
A comparison inequality for sums of independent random variables
|
math.PR
|
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.
|
math
|
2,545 |
On random sections of the cube
|
math.PR
|
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
$n\to\8$.
|
math
|
2,546 |
An embedding for the Kesten-Spitzer random walk in random scenery
|
math.PR
|
For one-dimensional simple random walk in a general i.i.d. scenery and its
limiting process we construct a coupling with explicit rate of approximation
extending a recent result for Gaussian sceneries due to Khoshnevisan and Lewis.
Furthermore we explicity identify the constant in the law of iterated
logarithm.
|
math
|
2,547 |
Necessary and Sufficient Conditions for the Strong Law of Large Numbers for U-statistics
|
math.PR
|
Under some mild regularity on the normalizing sequence, we obtain necessary
and sufficient conditions for the Strong Law of Large Numbers for (symmetrized)
U-statistics. We also obtain nasc's for the a.s. convergence of series of an
analogous form.
|
math
|
2,548 |
Trees, not cubes: hypercontractivity, cosiness, and noise stability
|
math.PR
|
Noise sensitivity of functions on the leaves of a binary tree is studied, and
a hypercontractive inequality is obtained. We deduce that the spider walk is
not noise stable.
|
math
|
2,549 |
A pattern theorem for lattice clusters
|
math.PR
|
We consider general classes of lattice clusters, including various kinds of
animals and trees on different lattices. We prove that if a given local
configuration ("pattern") of sites and bonds can occur in large clusters, then
it occurs at least cN times in most clusters of size n, for some constant c>0.
An analogous theorem for self-avoiding walks was proven in 1963 by Kesten. The
results also apply to weighted sums, and in particular we can take a$sub n$ to
be the probability that the percolation cluster containing the origin consists
of exactly n sites. Another consequence is strict inequality of connective
constants for sublattices and for certain subclasses of clusters.
|
math
|
2,550 |
Fourier-Walsh coefficients for a coalescing flow (discrete time)
|
math.PR
|
A two-dimensional array of independent random signs produces coalescing
random walks. The position of the walk, starting at the origin, after N steps
is a highly nonlinear, noise sensitive function of the signs. A typical term of
its Fourier-Walsh expansion involves the product of about square roof of N
signs.
|
math
|
2,551 |
Extinction for two parabolic stochastic PDE's on the lattice
|
math.PR
|
It is well known that, starting with finite mass, the super-Brownian motion
dies out in finite time. The goal of this article is to show that with some
additional work, one can prove finite time die-out for two types of systems of
stochastic differential equations on the lattice Z^d. Our first system involves
the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma
dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When
gamma = 1/2, the measure which puts mass u(t,x) at x is a super-random walk and
it is well-known that the process becomes extinct in finite time a.s.
Finite-time extinction is known to be a.s. false if gamma = 1. For 1/2 < gamma
< 1, we show finite-time die-out by breaking up the solution into pieces, and
showing that each piece dies in finite time. Our second example involves the
mutually catalytic branching system of stochastic differential equations on
Z^d, which was first studied by Dawson and Perkins. Roughly speaking, this
process consists of 2 superprocesses with the continuous time simple random
walk as the underlying spatial motion. Furthermore, each process stimulates
branching and dying in the other process. By using a somewhat different
argument, we show that, depending on the initial conditions, finite time
extinction of one type may occur with probability 0, or with probability
arbitrarily close to 1.
|
math
|
2,552 |
Scaling limit of Fourier-Walsh coefficients (a framework)
|
math.PR
|
Independent random signs can govern various discrete models that converge to
non-isomorphic continuous limits. Convergence of Fourier-Walsh spectra is
established under appropriate conditions.
|
math
|
2,553 |
The Expected Number of Real Roots of a Multihomogeneous System of Polynomial Equations
|
math.PR
|
Theorem 1 is a formula expressing the mean number of real roots of a random
multihomogeneous system of polynomial equations as a multiple of the mean
absolute value of the determinant of a random matrix. Theorem 2 derives closed
form expressions for the mean in special cases that include earlier results of
Shub and Smale (for the general homogeneous system) and Rojas (for ``unmixed''
multihomogeneous systems). Theorem 3 gives upper and lower bounds for the mean
number of roots, where the lower bound is the square root of the generic number
of complex roots, as determined by Bernstein's theorem. These bounds are
derived by induction from recursive inequalities given in Theorem 4.
|
math
|
2,554 |
Sample path large deviations for a class of Markov chains related to disordered mean field models
|
math.PR
|
We prove a large deviation principle on path space for a class of discrete
time Markov processes whose state space is the intersection of a regular domain
$\L\subset \R^d$ with some lattice of spacing $\e$. Transitions from $x$ to $y$
are allowed if $\e^{-1}(x-y)\in \D$ for some fixed set of vectors $\D$. The
transition probabilities $p_\e(t,x,y)$, which themselves depend on $\e$, are
allowed to depend on the starting point $x$ and the time $t$ in a sufficiently
regular way, except near the boundaries, where some singular behaviour is
allowed. The rate function is identified as an action functional which is given
as the integral of a Lagrange function. %of time dependent relativistic
classical mechanics. Markov processes of this type arise in the study of mean
field dynamics of disordered mean field models.
|
math
|
2,555 |
The LIL for canonical U-statistics of order 2
|
math.PR
|
Let X,X_1,X_2,... be independent identically distributed random variables and
let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that
the bounded law of the iterated logarithm, $\limsup_n (n\log\log
n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the
following three conditions are satisfied: h is canonical for the law of X (that
is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both,
$E\min(h^2(X_1,X_2),u)<C\log\log u$ for all large u and
$sup\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\|g(X)\|_2<1, \|f\|_\infty<\infty,
\|g\|_\infty<\infty\}< C$.
|
math
|
2,556 |
Stationary Measures for Random Walks in a Random Environment with Random Scenery
|
math.PR
|
Let $\Gamma$ act on a countable set V with only finitely many orbits. Given a
$\Gamma$-invariant random environment for a Markov chain on V and a random
scenery, we exhibit, under certain conditions, an equivalent stationary measure
for the environment and scenery from the viewpoint of the random walker. Such
theorems have been very useful in investigations of percolation on
quasi-transitive graphs.
|
math
|
2,557 |
Lattice trees, percolation and super-Brownian motion
|
math.PR
|
This paper surveys the results of recent collaborations with Eric Derbez and
with Takashi Hara, which show that intergrated super-Brownian excursion (ISE)
arises as the scaling limit of both lattice trees and the incipient infinite
percolation cluster, in high dimensions. A potential extension to oriented
percolation is also mentioned.
|
math
|
2,558 |
Monotonicity property for a class of semilinear partial differential equations
|
math.PR
|
We establish a monotonicity property in the space variable for the solutions
of an initial boundary value problem concerned with the parabolic partial
differential equation connected with super-Brownian motion.
|
math
|
2,559 |
A variational coupling for a totally asymmetric exclusion process with long jumps but no passing
|
math.PR
|
We prove a weak law of large numbers for a tagged particle in a totally
asymmetric exclusion process on the one-dimensional lattice. The particles are
allowed to take long jumps but not pass each other. The object of the paper is
to illustrate a special technique for proving such theorems. The method uses a
coupling that mimics the Hopf-Lax formula from the theory of viscosity
solutions of Hamilton-Jacobi equations.
|
math
|
2,560 |
Noise sensitivity on continuous products: an answer to an old question of J. Feldman
|
math.PR
|
A relation between sigma-additivity and linearizability, conjectured by Jacob
Feldman in 1971 for continuous products of probability spaces, is established
by relating both notions to a recent idea of noise stability/sensitivity.
|
math
|
2,561 |
Lévy Processes on $U_q(g)$ as Infinitely Divisible Representations
|
math.PR
|
L\'evy processes on bialgebras are families of infinitely divisible
representations. We classify the generators of L\'evy processes on the compact
forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then
we show how the processes themselves can be reconstructed from their generators
and study several classical stochastic processes that can be associated to
these processes.
|
math
|
2,562 |
Vertex-reinforced random walk on arbitrary graphs
|
math.PR
|
Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988a), is a
random process in a continuously changing environment which is more likely to
visit states it has visited before. We consider VRRW on arbitrary graphs and
show that on almost all of them, VRRW visits only finitely many vertices with a
positive probability. We conjecture that on all graphs of bounded degree, this
happens a.s., and provide a proof only for trees of this type. We distinguish
between several different patterns of localization and explicitly describe the
long-run structure of VRRW, which depends on whether a graph contains triangles
or not. While the results of this paper generalize those obtained by Pemantle
and Volkov (1998) for Z,ideas of proofs are different and typically based on a
large deviation principle rather than a martingale approach.
|
math
|
2,563 |
The Propagation of Molecular Chaos by Markov Transitions
|
math.PR
|
We establish a necessary and sufficient condition for the propagation of
chaos by a family of many-particle Markov processes, if the particles live in a
Polish space: a sequence of n-particle Markov transition functions propagates
chaos if and only if it propagates chaos for pure initial states.
|
math
|
2,564 |
Singularity of Some Random Continued Fractions
|
math.PR
|
We prove singularity of some distributions of random continued fractions that
correspond to iterated function systems with overlap and a parabolic point.
These arose while studying the conductance of Galton-Watson trees.
|
math
|
2,565 |
Loss of tension in an infinite membrane with holes distributed by Poisson law
|
math.PR
|
If one randomly punches holes in an infinite tensed membrane, when does the
tension cease to exist? This problem was introduced by R. Connelly in
connection with applications of rigidity theory to natural sciences. We outline
a mathematical theory of tension based on graph rigidity theory and introduce
several probabilistic models for this problem. We show that if the ``centers''
of the holes are distributed in R^2 according to Poisson law with parameter
\lambda>0, and the distribution of sizes of the holes is independent of the
distribution of their centers, the tension vanishes on all of R^2 for any value
of \lambda. In fact, it follows from a more general result on the behavior of
iterative convex hulls of connected subsets of R^d, when the initial
configuration of subsets is distributed according to Poisson law and the sizes
of the elements of the original configuration are independent of this Poisson
distribution. For the latter problem we establish the existence of a critical
threshold in terms of the number of iterative convex hull operations required
for covering all of R^d. The processes described in the paper are somewhat
related to bootstrap and rigidity percolation models.
|
math
|
2,566 |
Quasi-invariance and reversibility in the Fleming-Viot process
|
math.PR
|
Reversible measures of the Fleming-Viot process are shown to be characterized
as quasi-invariant measures with a cocycle given in terms of the mutation
operator. As applications, we give certain integral characterization of
Poisson-Dirichlet distributions and a proof that the stationary measure of the
step-wise mutation model of Ohta-Kimura with periodic boundary condition is
nonreversible.
|
math
|
2,567 |
Phase transition and percolation in Gibbsian particle models
|
math.PR
|
We discuss the interrelation between phase transitions in interacting lattice
or continuum models, and the existence of infinite clusters in suitable
random-graph models. In particular, we describe a random-geometric approach to
the phase transition in the continuum Ising model of two species of particles
with soft or hard interspecies repulsion. We comment also on the related
area-interaction process and on perfect simulation.
|
math
|
2,568 |
How to Couple from the Past Using a Read-Once Source of Randomness
|
math.PR
|
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.
|
math
|
2,569 |
Splitting: Tanaka's SDE revisited
|
math.PR
|
The weak solution of Tanaka's SDE is not a function of the driving Brownian
motion, and therefore it has no Wiener chaos expansion. However in some sense
explained here it has a generalised chaos expansion involving infinite products
of stochastic differentials accumulating at the minimum of the Brownian path.
This is related to the existence of a non-classical noise richer than the usual
white noise.
|
math
|
2,570 |
Coalescence of skew Brownian motions
|
math.PR
|
We prove that two skew Brownian motions with the same skewness parameter
(different from 0) and driven by the same Brownian motion coalesce a.s.
|
math
|
2,571 |
Layered Multishift Coupling for use in Perfect Sampling Algorithms (with a primer on CFTP)
|
math.PR
|
In this article we describe a new coupling technique which is useful in a
variety of perfect sampling algorithms. A multishift coupler generates a random
function f() so that for each real x, f(x)-x is governed by the same fixed
probability distribution, such as a normal distribution. We develop the class
of layered multishift couplers, which are simple and have several useful
properties. For the standard normal distribution, for instance, the layered
multishift coupler generates an f() which (surprisingly) maps an interval of
length L to fewer than 2+L/2.35 points --- useful in applications which perform
computations on each such image point. The layered multishift coupler improves
and simplifies algorithms for generating perfectly random samples from several
distributions, including the autogamma distribution, posterior distributions
for Bayesian inference, and the steady state distribution for certain storage
systems. We also use the layered multishift coupler to develop a Markov-chain
based perfect sampling algorithm for the autonormal distribution.
At the request of the organizers, we begin by giving a primer on CFTP
(coupling from the past); CFTP and Fill's algorithm are the two predominant
techniques for generating perfectly random samples using coupled Markov chains.
|
math
|
2,572 |
Diffeomorphic flows driven by Levy processes
|
math.PR
|
We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x)
+ \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u},
Y_{s,s}(x)=x\in\R^d. $$ driven by a L\'evy process whose paths have finite
p-variation almost surely for some $p\in[1,2)$ defines a flow of locally
C^1-diffeomorphisms provided the vector field f is $\alpha$-Lipschitz for some
$\alpha>p$. Using a path- wise approach we relax the smoothness condition
normally required for a class of discontinuous semi-martingales.
|
math
|
2,573 |
Path-wise solutions of SDE's driven by Levy processes
|
math.PR
|
In this paper we show that a path-wise solution to the following integral
equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under
the assumption that X_t is a L\'evy process of finite p-variation for some $p
\geq1$ and that f is an $\alpha$-Lipschitz function for some alpha>p. There are
two types of solution, determined by the solution's behaviour at jump times of
the process X, one we call geometric the other forward. The geometric solution
is obtained by adding fictitious time and solving an associated integral
equation. The forward solution is derived from the geometric solution by
correcting the solution's jump behaviour. L\'evy processes, generally, have
unbounded variation. So we must use a pathwise integral different from the
Lebesgue-Stieltjes integral. When X has finite p-variation almost surely for
p<2 we use Young's integral. This is defined whenever f and g have finite p and
q-variation for 1/p+1/q>1 (and they have no common discontinuities). When p>2
we use the integral of Lyons. In order to use this integral we construct the
L\'evy area of the L\'evy process and show that it has finite (p/2)-variation
almost surely.
|
math
|
2,574 |
Random Walks and Electric Networks
|
math.PR
|
A popular account of the connection between random walks and electric
networks.
|
math
|
2,575 |
Markov Transitions and the Propagation of Chaos
|
math.PR
|
The propagation of chaos is a central concept of kinetic theory that serves
to relate the equations of Boltzmann and Vlasov to the dynamics of
many-particle systems. Propagation of chaos means that molecular chaos, i.e.,
the stochastic independence of two random particles in a many-particle system,
persists in time, as the number of particles tends to infinity.
We establish a necessary and sufficient condition for a family of general
n-particle Markov processes to propagate chaos. This condition is expressed in
terms of the Markov transition functions associated to the n-particle
processes, and it amounts to saying that chaos of random initial states
propagates if it propagates for pure initial states.
Our proof of this result relies on the weak convergence approach to the study
of chaos due to Sznitman and Tanaka. We assume that the space in which the
particles live is homeomorphic to a complete and separable metric space so that
we may invoke Prohorov's theorem in our proof.
We also show that, if the particles can be in only finitely many states, then
molecular chaos implies that the specific entropies in the n-particle
distributions converge to the entropy of the limiting single-particle
distribution.
|
math
|
2,576 |
q-probability: I. Basic discrete distributions
|
math.PR
|
For basic discrete probability distributions, $-$ Bernoulli, Pascal, Poisson,
hypergeometric, contagious, and uniform, $-$ $q$-analogs are proposed.
|
math
|
2,577 |
The supremum of Brownian local times on Holder curves
|
math.PR
|
For $f: [0,1]\to \R$, we consider $L^f_t$, the local time of space-time
Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions
whose H\"older norm of order $\al$ is less than or equal to 1. We show that the
supremum of $L^f_1$ over $f$ in $\sS_\al$ is finite is $\al>\frac12$ and
infinite if $\al<\frac12$.
|
math
|
2,578 |
On the cover time of planar graphs
|
math.PR
|
The cover time of a finite connected graph is the expected number of steps
needed for a simple random walk on the graph to visit all the vertices. It is
known that the cover time on any n-vertex, connected graph is at least (1+o(1))
n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for
bounded-degree planar graphs the cover time is at least c n(log n)^2, and at
most 6n^2, where c is a positive constant depending only on the maximal degree
of the graph. The lower bound is established via use of circle packings.
|
math
|
2,579 |
Some measure-preserving point transformations on the Wiener space and their ergodicity
|
math.PR
|
Suppose that T is a map of the Wiener space into itself, of the following
type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also
that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work
we prove that if T preserves the Wiener measure, then necessarily u is in the
first Wiener chaos and the transformation corresponding to it is a rotation in
the sense of [9]. Afterwards the ergodicity and mixing of such transformations,
which are second quantizations of the unitary operators on the Cameron-Martin
space, are characterized. Finally, the ergocity of the transformation
dY_t=gamma(t)dW_t, 0 \le t \le 1 where W is n-dimensional Wiener and gamma is
non random is characterized
|
math
|
2,580 |
Precise Propagation of Upper and Lower Probability Bounds in System P
|
math.PR
|
In this paper we consider the inference rules of System P in the framework of
coherent imprecise probabilistic assessments. Exploiting our algorithms, we
propagate the lower and upper probability bounds associated with the
conditional assertions of a given knowledge base, automatically obtaining the
precise probability bounds for the derived conclusions of the inference rules.
This allows a more flexible and realistic use of System P in default reasoning
and provides an exact illustration of the degradation of the inference rules
when interpreted in probabilistic terms. We also examine the disjunctive Weak
Rational Monotony of System P+ proposed by Adams in his extended probability
logic.
|
math
|
2,581 |
Super-Brownian motion with reflecting historical paths
|
math.PR
|
We consider super-Brownian motion whose historical paths reflect from each
other, unlike those of the usual historical super-Brownian motion. We prove
tightness for the family of distributions corresponding to a sequence of
discrete approximations but we leave the problem of uniqueness of the limit
open. We prove a few results about path behavior for processes under any limit
distribution. In particular, we show that for any $\gamma>0$, a "typical"
increment of a reflecting historical path over a small time interval $\Delta t$
is not greater than $(\Delta t)^{3/4 - \gamma}$.
|
math
|
2,582 |
Exponential and moment inequalities for U-statistics
|
math.PR
|
A Bernstein-type exponential inequality for (generalized) canonical
U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-J{\o}rgensen
inequalities for sums of independent random variables are extended to
(generalized) U-statistics of any order whose kernels are either nonnegative or
canonical
|
math
|
2,583 |
Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes
|
math.PR
|
A derivation operator and a divergence operator are defined on the algebra of
bounded operators on the symmetric Fock space over the complexification of a
real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar
properties as the derivation and divergence operator on the Wiener space over
$\eufrak{h}$. The derivation operator is then used to give sufficient
conditions for the existence of smooth Wigner densities for pairs of operators
satisfying the canonical commutation relations. For
$\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide
with the Hudson-Parthasarathy quantum stochastic integral for adapted
integrable processes and with the non-causal quantum stochastic integrals
defined by Lindsay and Belavkin for integrable processes.
|
math
|
2,584 |
No more than three favourite sites for simple random walk
|
math.PR
|
We prove that, with probability one, eventually there are no more than three
favourite (i.e. most visited) sites of simple random walk. This partially
answers a relatively long standing question of Pal Erdos and Pal Revesz.
|
math
|
2,585 |
The identification capacity and resolvability of channels with input cost constraint
|
math.PR
|
Given a general channel, we first formulate the idetification capacity
problem as well as the resolvability problem with input cost constraint in as
the general form as possible, along with relevant fundamental theorems. Next,
we establish some mild sufficient condition for the key lemma linking the
identification capacity with the resolvability to hold for the continuous input
alphabet case with input cost constraint. Under this mild condition, it is
shown that we can reach the {\em continuous}-input fundamental theorem of the
same form as that for the fundamental theorem with {\em finite} input alphabet.
Finally, as important examples of this continuous-input fundamental theorem, we
show that the identification capacity as well as the resolvability coincides
with the channel capacity for stationary additive white (and also non-white)
Gaussian noise channels.
|
math
|
2,586 |
On the critical exponents of random k-SAT
|
math.PR
|
There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from satisfiability to unsatisfiability; in the case of 2-SAT this happens when
m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The
sharpness of this transition is characterized by a critical exponent, sometimes
called \nu=\nu_k (the smaller the value of \nu the sharper the transition).
Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05,
\nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k
--> 1 as k --> infinity. We give here a simple proof that each of these
exponents is at least 2 (provided the exponent is well-defined). This result
holds for each of the three standard ensembles of random k-SAT formulas: m
clauses selected uniformly at random without replacement, m clauses selected
uniformly at random with replacement, and each clause selected with probability
p independent of the other clauses. We also obtain similar results for
q-colorability and the appearance of a q-core in a random graph.
|
math
|
2,587 |
Occupation Time Fluctuations in Branching Systems
|
math.PR
|
We consider particle systems in locally compact Abelian groups with particles
moving according to a process with symmetric stationary independent increments
and undergoing one and two levels of critical branching. We obtain long time
fluctuation limits for the occupation time process of the one-and two-level
systems. We give complete results for the case of finite variance branching,
where the fluctuation limits are Gaussian random fields, and partial results
for an example of infinite variance branching, where the fluctuation limits are
stable random fields. The asymptotics of the occupation time fluctuations are
determined by the Green potential operator G of the individual particle motion
and its powers $G^2, G^3$, and by the growth as $t\to\infty$ of the operator
$G_t=\int^t_0T_sds$ and its powers, where $T_t$ is the semigroup of the motion.
The results are illustrated with two examples of motions: the symmetric
$\alpha$-stable L\'evy process in $\erre^d$ $(0<\alpha\leq2)$,and the so called
c-hierarchical random walk in the hierarchical group of order N (0<c<N). We
show that the two motions have analogous asymptotics of $G_t$ and its powers
that depend on an order parameter $\gamma$ for their transience /recurrence
behavior. This parameter is $\gamma=d/\alpha-1$ for the $\alpha$-stable motion,
and $\gamma=\log c/\log (N/c)$ for the c-hierarchical random walk. As a
consequence of these analogies, the asymptotics of the occupation time
fluctuations of the corresponding branching particle systems are also
analogous. In the case of the c-hierarchical random walk, however, the growth
of $G_t$ and its powers is modulated by oscillations on a logarithmic time
scale.
|
math
|
2,588 |
Mixing times for Markov chains on wreath products and related homogeneous spaces
|
math.PR
|
We develop a method for analyzing the mixing times for a quite general class
of Markov chains on the complete monomial group G \wr S_n (the wreath product
of a group G with the permutation group S_n) and a quite general class of
Markov chains on the homogeneous space (G \wr S_n) / (S_r \times S_{n - r}).
We derive an exact formula for the L^2 distance in terms of the L^2 distances
to uniformity for closely related random walks on the symmetric groups S_j for
1 \leq j \leq n or for closely related Markov chains on the homogeneous spaces
S_{i + j} / (S_i \times S_j) for various values of i and j, respectively. Our
results are consistent with those previously known, but our method is
considerably simpler and more general.
|
math
|
2,589 |
Random polynomials having few or no real zeros
|
math.PR
|
Consider a polynomial of large degree n whose coefficients are independent,
identically distributed, nondegenerate random variables having zero mean and
finite moments of all orders. We show that such a polynomial has exactly k real
zeros with probability n^{-b+o(1)}$ as n --> infinity through integers of the
same parity as the fixed integer k >= 0. In particular, the probability that a
random polynomial of large even degree n has no real zeros is n^{-b+o(1)}. The
finite, positive constant b is characterized via the centered, stationary
Gaussian process of correlation function sech(t/2). The value of b depends
neither on k nor upon the specific law of the coefficients. Under an extra
smoothness assumption about the law of the coefficients, with probability
n^{-b+o(1)} one may specify also the approximate locations of the k zeros on
the real line. The constant b is replaced by b/2 in case the i.i.d.
coefficients have a nonzero mean.
|
math
|
2,590 |
Random walks on wreath products of groups
|
math.PR
|
We bound the rate of convergence to uniformity for certain random walks on
the complete monomial groups G \wr S_n for any group G. These results provide
rates of convergence for random walks on a number of groups of interest: the
hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n,
and S_m \wr S_n. These results provide benchmarks to which many other random
walks, modeling a wide range of phenomena, may be compared using the comparison
technique, thereby yielding bounds on the rates of convergence to uniformity
for previously intractable random walks.
|
math
|
2,591 |
A signed generalization of the Bernoulli-Laplace diffusion model
|
math.PR
|
We bound the rate of convergence to stationarity for a signed generalization
of the Bernoulli-Laplace diffusion model; this signed generalization is a
Markov chain on the homogeneous space (Z_2 \wr S_n) / (S_r \times S_{n-r}).
Specifically, for r not too far from n/2, we determine that, to first order in
n, 1/4 n \log n steps are both necessary and sufficient for total variation
distance to become small. Moreover, for r not too far from n/2, we show that
our signed generalization also exhibits the ``cutoff phenomenon.''
|
math
|
2,592 |
Favourite sites of simple random walk
|
math.PR
|
We survey the current status of the list of questions related to the
favourite (or: most visited) sites of simple random walk on Z, raised by Pal
Erdos and Pal Revesz in the early eighties.
|
math
|
2,593 |
Stationary Markov chains with linear regressions
|
math.PR
|
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.
|
math
|
2,594 |
Critical exponents, conformal invariance and planar Brownian motion
|
math.PR
|
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.
|
math
|
2,595 |
Gaussian Random Matrix Models for q-deformed Gaussian Random Variables
|
math.PR
|
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.
|
math
|
2,596 |
Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K-exclusion processes
|
math.PR
|
We study aspects of the hydrodynamics of one-dimensional totally asymmetric
K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove
that the weak solution chosen by the particle system is the unique one with
maximal current past any fixed location. A uniqueness result is needed because
we can prove neither differentiability nor strict concavity of the flux
function, so we cannot use the Lax-Oleinik formula or jump conditions to define
entropy solutions. Next we prove laws of large numbers for a second class
particle in K-exclusion. The macroscopic trajectories of second class particles
are characteristics and shocks of the conservation law for the particle
density. In particular, we extend to K-exclusion Ferrari's result that the
second class particle follows a macroscopic shock in the Riemann solution. The
technical novelty of the proofs is a variational representation for the
position of a second class particle, in the context of the variational coupling
method.
|
math
|
2,597 |
The Randomness Recycler: A new technique for perfect sampling
|
math.PR
|
For many probability distributions of interest, it is quite difficult to
obtain samples efficiently. Often, Markov chains are employed to obtain
approximately random samples from these distributions. The primary drawback to
traditional Markov chain methods is that the mixing time of the chain is
usually unknown, which makes it impossible to determine how close the output
samples are to having the target distribution. Here we present a new protocol,
the randomness recycler (RR), that overcomes this difficulty. Unlike classical
Markov chain approaches, an RR-based algorithm creates samples drawn exactly
from the desired distribution. Other perfect sampling methods such as coupling
from the past use existing Markov chains, but RR does not use the traditional
Markov chain at all. While by no means universally useful, RR does apply to a
wide variety of problems. In restricted instances of certain problems, it gives
the first expected linear time algorithms for generating samples. Here we apply
RR to self-organizing lists, the Ising model, random independent sets, random
colorings, and the random cluster model.
|
math
|
2,598 |
A signal-recovery system: asymptotic properties and construction of an infinite volume limit
|
math.PR
|
We consider a linear sequence of `nodes', each of which can be in state 0
(`off') or 1 (`on'). Signals from outside are sent to the rightmost node and
travel instantaneously as far as possible to the left along nodes which are
`on'. These nodes are immediately switched off, and become on again after a
recovery time. The recovery times are independent exponentially distributed
random variables.
We present properties for finite systems and use some of these properties to
construct an infinite-volume extension, with signals `coming from infinity'.
This construction is related to a question by D. Aldous and we expect that it
sheds some light on, and stimulates further investigation of, that question.
|
math
|
2,599 |
Microscopic shape of shocks in a domain growth model
|
math.PR
|
Considering the hydrodynamical limit of some interacting particle systems
leads to hyperbolic differential equation for the conserved quantities, e.g.
the inviscid Burgers equation for the simple exclusion process. The physical
solutions of these partial differential equations develop discontinuities,
called shocks. The microscopic structure of these shocks is of much interest
and far from being well understood. We introduce a domain growth model in which
we find a stationary (in time) product measure for the model, as seen from a
defect tracer or second class particle, travelling with the shock. We also show
that under some natural assumptions valid for a wider class of domain growth
models, no other model has stationary product measure as seen from the moving
defect tracer.
|
math
|
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