April 22, 2018

Toronto Probability Seminar 2006-07
held at the Fields Institute

Bálint Virág , Benedek Valkó
University of Toronto, Mathematics and Statistics

For questions, scheduling, or to be added to the mailing list, contact the organizers at probsem-at-math-dot-toronto-dot-edu

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.