Thursday, May 10,
2:10pm
Liliana Borcea (Rice University)
Array Imaging in Random Media
In array imaging, we wish to find strong reflectors in a medium,
given measurements of the time traces of the scattered echoes at a
remote array of receivers. I will discuss array imaging in cluttered
media, modeled with random processes, in regimes with significant
multipathing of the waves by the inhomogeneities in the clutter. In
such regimes, the echoes measured at the array are noisy and exhibit
a lot of delay spread. This makes imaging difficult and the usual
techniques give unreliable, statistically unstable results. I will
present a coherent interferometric imaging approach for random media,
which exploits systematically the spatial and temporal coherence in
the data to obtain statistically stable images. I will discuss theresolution
of this method and its statistical stability and I will illustrate
its performance with numerical simulations.
Monday, April 23
Rowan Killip (UCLA)
From the cicular moment problem to random matrices
I will begin by reviewing some classical topics in analysis
then segue into my recent work on random matrices.
Monday, April 16
Dan Romik (Bell Laboratories)
Gravitational allocation to Poisson points
An allocation rule for the standard Poisson point process in R^d
is a translation-invariant way of allocating to the Poisson points
mutually disjoint cells of volume 1 that cover almost all R^d. I
will describe a new construction in dimensions 3 and higher of an
allocation rule based on Newtonian gravitation: each Poisson point
is thought of as a star of unit mass, and the cell allocated to
a star is its basin of attraction with respect to the flow induced
by the total gravitational force exerted by all the stars. This
allocation rule is efficient, in the sense that the distance a typical
point has to move is a random variable with exponentially decreasing
tails.
The talk is based on joint work with Sourav Chatterjee, Ron Peled
and Yuval Peres.
Monday, March 26, 16:10, 2007, 4:10 pm
Thomas Bloom (University of Toronto):
Random Polynomials and (Pluri)-Potential Theory
I will report on results on the expected distribution of zeros
of random polynomials in one and several (complex) variables.The
results will involve concepts from potential and pluripotential
theory. In particular,a recent result(joint with B.Shiffman)showing
that the expected distribution of the common zeros of m random Kac
polynomials (i.e.polynomials with standard Gaussians as coefficients)
in m variables tends,as the degree increases,to the product of the
angular measures on each of the m unit circles.This generalizes
a classical result of Hammarsley.
Monday, March 12
Márton Balázs (Technical University Budapest)
Order of current variance in the simple exclusion process
The simple exclusion process is one of the simplest stochastic
interacting particle systems: particles try to perform nearest neighbor
jumps on the integer line Z, but only succeed when the destination
site is not occupied by another particle. It is somewhat surprising
that such a system shows very exotic, time^{1/3}-scaling properties
when turning to these particles' current fluctuations. Limiting
distribution results have existed in this direction for the totally
asymmetric case (particles only try to jump to their right neighboring
site), and heavy combinatoric and analytic tools were used to prove
them.
By a joint work with T. Seppäläinen, we managed to prove
this scaling (but not the limiting distribution) for the general
nearest neighbor asymmetric case, with the use of purely probabilistic
ideas. I will introduce the process, define the objects we worked
with in probabilistic coupling arguments, and summarize the method
that led to the proof of the scaling.
(This work is related to recent results of Jeremy Quastel and Benedek
Valkó.)
Thursday, March 8, 2007, 4:10 pm,
Alan Hammond (Courant Institute)
Resonances in the cycle rooted spanning forest on a two-dimensional
torus
Consider an n by m discrete torus with a directed graph structure,
in which one edge, pointing north or east with probability one-half,
independently, emanates from each vertex. The behaviour of the cycle
structure of this graph depends sensitively on the aspect ratio
m/n of the torus. The expected total number of edges contained in
cycles, for example, is peaked when m/n is close to a small rational.
This work, joint with Rick Kenyon, complements an earlier paper
of Kenyon and Wilson, that analyses resonance among paths in a model
that is equivalent to a honeycomb dimer model on a discrete torus.
Monday, February 26, 2007, 4:10 pm
Elena Kosygina (Baruch College and the CUNY Graduate Center)
Stochastic Homogenization of Hamilton-Jacobi-Bellman Equations
We consider a homogenization problem for Hamilton-Jacobi-Bellman
equations in a stationary ergodic random media. After a brief review
of the standard approach for periodic Hamiltonians, we shall discuss
the difficulties and current methods of stochastic homogenization
for such equations and explain the connection with large deviations
for diffusions in a random medium. This is a joint work with F.
Rezakhanlou and S.R.S. Varadhan.
Monday, February 12, 2007, 4:10 pm
Jeremy Quastel (University of Toronto)
White Noise and the Korteweg-de Vries Equation
In joint work with Benedek Valko (Toronto) we found that Gaussian
white noise is an invariant measure for KdV on the circle. In this
talk we will describe the relevant concepts, what the result means
both mathematically and physically, and give some ideas of the proof.
(The preprint may be downloaded from here
Monday, February 5, 2007, 4:10 pm
Manjunath Krsihnapur (University of Toronto)
Zeros of random analytic functions and Determinantal point processes
On each of the plane, the sphere and the unit disk, there is
exactly a one-parameter family of Gaussian analytic functions whose
zeros have isometry-invariant distributions (Sodin). Of these there
is only one whose zero set is a determinantal point process (Peres-Virag).
By using Gaussian analytic functions as building blocks, we construct
many non-Gaussian random analytic functions with invariant zero
sets. We pick out certain candidates among these, whose zero sets
may be expected to be determinantal. We prove that this is indeed
the case for a family of random polynomials on the sphere, and partially
prove the same for a family of random analytic functions on the
unit disk. No prior knowledge of determinantal point processes or
random analytic functions is necessary. These results are from my
thesis.
Monday, January 29, 2007, 14:10
Bálint Virág (University of Toronto)
Scaling Limits of Random Matrices
Recently, it has become clear that the sine and Airy point processes
arising from random matrix eigenvalues play a fundamental role in
probability theory, partly due to their connection to Riemann zeta
zeros and random permutations. I will describe recent work on the
Stochastic Airy and Stochastic sine differential equations, which
are shown to describe these point processes and can be thought of
as scaling limits of random matrices. This new approach resolves
some open problems, e.g. it generalizes these point processes for
all values of the parameter beta.
Wednesday, December 6, 2006, 15:10
Dimitris Cheliotis (University of Toronto)
Patterns for the 1-dimensional random walk in the random environment
- a functional LIL
We start with a one dimensional random walk (or diffusion) in
a Wiener-like environment. We look at its graph at different, increasing
scales natural for it. What are the patterns that appear repeatedly?
We characterize them through a functional law of the iterated logarithm
analogous to Strassen's result for Brownian motion and simple random
walk.
The talk is based on joint work with Balint Virag.
Monday, November 27, 2006, 4:10 pm
Antal Járai (Carleton University)
Random walk on the incipient infinite cluster for oriented percolation
in high dimensions
We consider simple random walk on the incipient infinite cluster
for the spread-out model of oriented percolation in d+1 dimensions.
For d > 6, we obtain bounds on exit times, transition probabilities,
and the range of the random walk, which establish that the spectral
dimension of the incipient infinite cluster is 4/3, and thereby
prove a version of the Alexander-Orbach conjecture in this setting.
The proof divides into two parts. One part establishes general estimates
for simple random walk on an arbitrary infinite random graph, given
suitable bounds on volume and effective resistance for the random
graph. A second part then provides these bounds on volume and effective
resistance for the incipient infinite cluster in dimensions d >
6, by extending results about critical oriented percolation obtained
previously via the lace expansion.
Monday, November 20, 2006, 4:30 pm
Alexander Holroyd (University of British Columbia)
Bootstrap Percolation - a case study in theory versus experiment
Cellular automata arise naturally in the study of physical systems,
and exhibit a seemingly limitless range of intriguing behaviour.
Such models lend themselves naturally to computer simulation, but
rigorous analysis can be notoriously difficult, and can yield highly
unexpected results. Bootstrap percolation is a very simple model
for nucleation and growth which turns out to hold many surprises.
Sites in a square grid are initially declared "infected"
independently with some fixed probability. Subsequently, healthy
sites become infected if they have at least two infected neighbours,
while infected sites remain infected forever. The model undergoes
a phase transition at a certain threshold whose asymptotic value
differs from numerical predictions by more than a factor of two!
This discrepancy points to a previously unsuspected phenomenon called
"crossover", and leads to further intriguing questions.
Monday, November 13, 2006, 4:10 pm
Balázs Szegedy (University of Toronto)
Limits of discrete structures and group invariant measures
An important branch of statistics studies networks (structures)
that grow randomly according to some law. A natural question is whether
there is a natural limit object for the process. We present a group
theoretic approach to this problem.
Monday, October 30, 2006, 4:10 pm
Bálint Tóth (Technical University Budapest)
Tagged particle diffusion in 1d Rayleigh-gas - old and new results
I will consider the M -> 0 limit for tagged particle diffusion
in a 1-dimensional Rayleigh-gas, studied originaly by Sinai and
Soloveichik (1986), respectively, by Szász and Tóth
(1986). In this limit we derive a new type of model for tagged paricle
diffusion, with Calogero-Moser-Sutherland (i.e. inverse quadratic)
interaction potential between the two central particles. Computer
simulations on this new model reproduce exactly the numerical value
of the limiting variance obtained by Boldrighini, Frigio and Tognetti
(2002). I will also present new bounds on the limiting variance
of tagged particle diffusion in (variants of) 1D Rayleigh gas which
improve some bounds of Szász, Tóth (1986). The talk
will be based on joint work of the following three authors: Péter
Bálint, Bálint Tóth, Péter Tóth.
Friday, October 27, 2006, 2:10pm
Bernard Shiffman (John Hopkins University)
Complex zeros of random multivariable polynomial systems
I will discuss the distribution of zeros of systems of independent
Gaussian random polynomials in n complex variables. Results on the
distribution of the number N(U) of zeros in a complex domain U of
a random polynomial of one complex variable were given in recent
papers of Sodin-Tsirelson and Forrester-Honner. They showed that
the variance of N(U) grows like the square root of the degree d,
and thus the number of zeros in U is "self-averaging"
in the sense that its fluctuations are of smaller order than its
typical values. A natural question is whether self-averaging occurs
for zeros of systems of n independent Gaussian random polynomials
of n complex variables. To answer this question, I will give asymptotic
formulas for the variance of the number of simultaneous zeros in
a domain U in C^n as the degree d of the polynomials goes to infinity.
I will explain how "correlation currents" for zeros and
complex potential theory are used to compute variances for complex
zeros. This talk involves joint work with Steve Zelditch.
Monday, October 16, 2006, 4:10 pm
Vladimir Vinogradov (Ohio University)
On Local Approximations For Two Classes of Distributions
We derive local approximations along with estimates of the remainders
for two classes of integer-valued variables. One of them is comprised
of Pólya-Aeppli distributions, while members of the other
class are the convolutions of a zero-modified geometric law. We
also derive the closed-form representation for the probability function
of the latter convolutions and investigate its properties. This
provides the distribution theory foundation for the studies on branching
diffusions. Our techniques involve a Poisson mixture representation,
Laplace's method and upper estimates in the local Poisson theorem.
The parallels with Gnedenko's method of accompanying infinitely
divisible laws are established.
Monday, October 2, 2006, 4:10 pm,
Omer Angel (University of Toronto)
Invasion Percolation on Trees
We consider the invasion percolation cluster (IPC) in a regular
tree. We calculate the scaling limit of $r$-point functions, the
volume at a given level and up to a level. While the power laws
governing the IPC are the same as for the incipient infinite cluster
(IIC), the scaling functions differ. We also show that the IPC stochastically
dominates the IIC. Given time I will discuss the continuum scaling
limit of the IPC.
Monday, September 25, 2006, 4:10 pm,
Paul Federbush (Ann Arbor)
A random walk on the permutation group, some formal long-time
asymptotic expansions
We consider the group of permutations of the vertices of a lattice.
A random walk is generated by unit steps that each interchange two
nearest neighbor vertices of the lattice. We study the heat equation
on the permutation group, using the Laplacian associated to the
random walk. At t = 0 we take as initial conditions a probability
distribution concentrated at the identity. A natural conjecture
for the probability distribution at long times is that it is "approximately"
a product of Gaussian distributions for each vertex. That is, each
vertex diffuses independently of the others. We obtain some formal
asymptotic results in this direction. The problem arises in certain
ways of treating the Heisenberg model in statistical mechanics.
Monday, September 18, 2006, 4:10 pm,
Siva Athreya (Indian Statistical Institute, Bangalore)
Age-Dependent Superprocesses
In this talk I will discuss an age dependent branching particle
system and its rescaled limit the super-process. The above systems
are non-local in nature (i.e. the position of the offspring is not
the same as that of the parent) and some specific difficulties arise
in this setting. We shall begin with a review of the literature,
discuss the above difficulties and present some new observations.
Tuesday, September 5, 2006, 4:10pm
Wilfrid Kendall (Warwick)
Coupling all the Levy stochastic areas of multidimensional Brownian
motion
I will talk about how to construct a successful co-adapted coupling
of two copies of an n-dimensional Brownian motion (B1, ... , Bn)
while simultaneously coupling all corresponding copies of Levy stochastic
areas.
