Toronto Probability Seminar

Past talks

(Click here for upcoming talks)

Tuesday, April 8 2008
Mate Matolcsi (Renyi Institute of Mathematics, Hungary)
The real polarization problem

We study a conjecture of Benitez, Sarantopoulos and Tonge concerning a lower bound on the norm of products of real linear functioanls. The conjecture is that the lower-bound is attained if and only if the vectors corresponding to the functionals are orthogonal. There are several approaches to the problem, analytic (Revesz, Pappas, 2004), geometric (Matolcsi, 2005), and probabilistic (Frenkel, 2007), yielding partial results. The probabilistic approach of Fernkel, 2007, deduces a lower bound from the following theorem: If X1, ... , Xn are jointly Gaussian random variables with zero expectation, then E(X1^2 ... Xn^2) >= EX1^2 ... EXn^2. Equality holds if and only if they are independent or at least one of them is almost surely zero. A similar result for higher moments would imply the conjecture.

Monday, February 25, 2008, 4:30pm
Monday, March 3, 2008, 4:00pm
Monday, March 11, 2008, 4:00pm
Wednesday, March 19, 2008, 10:00am
Bálint Virág and Benedek Valkó (University of Toronto)
The Brownian Carousel
The eigenvalues of a random Hermitian matrix form a random set of points on the real line. As the matrix size converges to infinity, the eigenvalues, after appropriate scaling, converge to a point process.
The possible limit processes, called Sine-beta processes, are fundamental objects of probability theory. They are famous for their conjectured relationship to the Riemann zeta zeros, Dirichlet eigenvalues of Euclidean domains, random Young tableaux, and non-colliding walks.
This series of informal talks is about a new description of these processes in terms of Brownian motion in the hyperbolic plane, called the Brownian carousel. We plan to have three lectures:

1. Introduction to random matrix eigenvalues, definition and basic properties of the Brownian Carousel
2. Computing with the Brownian carousel; continuity, phase transitions, Dyson's predictions
3. Convergence of finite random matrix eigenvalues to the Brownian carousel

Monday, February 11, 2008, 4:30pm
Brian Rider(University of Colorado at Boulder)
Diffusion at RMT's hard edge
The RMT hard edge refers to the behavior of the minimal eigenvalues of a (natural) one-parameter generalization of Gaussian sample covariance matrices. We show that, in the large dimensional limit, the law of these points are shared by that of the spectrum of a certain random second-order differential operator. The latter may be viewed as the generator of a Brownian motion with white noise drift. By a Riccati transform, we get a second diffusion description of the hard edge in terms of hitting times.
This is joint work with J. Ramirez and should be compared with slightly less recent results of J. Ramirez, B. Virág, and myself on the RMT "soft" edge.

Monday, February 5, 2008, 4:10pm
Omer Angel (University of Toronto)
TBA
We consider a directed random walk on the permutation group: a random generator (adjacent transposition) is sequentially applied, constrained to increase the inversion number of the permutation. (The model is equivalent to a multi-type TASEP on an interval). This gives a random sorting network - a path from the identity to the reverse identity (n,...,3,2,1). Many properties of this path can be computed. I will describe the particle trajectories and their finishing times.

Monday, December 10, 2007, 4:10pm
James Mingo (Queen's University)
Free Cumulants: First and Second Order
Twenty years ago Voiculescu showed that the limiting distribution of sums and products of some ensembles of random matrices could be computed using some algebraic methods of "free" probability. At the core of free probability are the "free" cumulants.
In recent years I have developed with Roland Speicher a theory of second order cumulants to do for global fluctuations what Voiculescu's theory did for limiting distributions.

Monday, December 3, 2007, 4:10pm
Omer Angel (University of Toronto)
Minimal Spanning Trees revisited
Given a graph with weighted edges it is easy to find the spanning tree with minimal total weight. If the graph is the complete graph K_n and the weights are independent uniform on [0,1] the MST weight converges in distribution to \zeta(3). I will discuss two variation on this result.
If the diameter of the tree is constrained to be at most K, what is the minimal weight? Turns out that there is a transition at K=\log_2\log n.
If the edges are presented sequentially, and an algorythm must make a decision on each edge with only partial information, what can be achieved? Some heuristics lead to algorithms related to coalescent processes. I will give some bounds on the optimal expected weight.

Monday, November 26, 2007, 4:10pm
Balázs Szegedy (University of Toronto)
Forcing Randomness
A surprising theorem by Chung, Graham and Wilson says that if a graph has edge density close to 1/2 and four cycle density close to 1/16 than the structure of the graph is close to "random looking". The natural question arises: What structures can be forced upon a graph by a finite family of subgraph densities? These structures are interesting combinations of algebraic structure and randomness. We present recent results in this topic. This is joint work with Laszlo Lovasz.

Monday, November 19, 2007, 4:10pm
Manjunath Krishnapur (University of Toronto)
From random matrices to random analytic functions
Peres and Virag proved that the zeros of the power series a_0+za_1+z^2a_2+..., with i.i.d. standard complex Gaussian coefficients is a determinantal point process on the unit disk. Extending this result, I proved recently that the singular points of the power series A_0+zA_1+z^2A_2+..., where A_i are k x k matrices with i.i.d. standard complex Gaussian coefficients, is also determinantal. As this was presented as conjecture in earlier talks, the emphasis will be on the proof and its connection to truncations of unitary random matrices sampled according to Haar measure.

Monday, October 29, 2007, 4:10pm
Mathieu Merle (University of British Columbia)
Voter, Lotka-Volterra models and super-Brownian motion
Voter model was initially interpreted as representing the spread of an opinion, but as the Lotka-Volterra model, it can be also be interpreted as a stochastic model for competition species. Super-Brownian motion is a model for population undergoing both spatial displacement and a continuous branching phenomenon. Recently, it was shown by Bramson, Cox, Durrett, Le Gall and Perkins that these objects are closely related, as super-Brownian motion appears at the scaling limit of both voter and Lotka-Volterra models, in dimension greater than two. Then, know properties of super-Brownian motion can be exploited in order to gain information on these discrete models. We will see how this leads to asymptotic results for the hitting probabilities of the voter model started with a single one, in dimensions 2 and 3. We will also briefly survey recent work of Cox and Perkins, who obtain results on survival and coexistence for the Lotka-Volterra model in dimension greater than 3.

Monday, October 15, 2007, 4:10pm
Gidi Amir (University of Toronto)
Excited random walk against a wall
We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, reflects on the plane $z=0$, and behaves like a simple random walk otherwise. a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of $\sqrt{\log n}$ (Joint work with Itai Benjamini and Gady Kozma)

Monday, October 1, 2007, 4:10pm
Gabor Pete (Microsoft)
The exact noise and dynamical sensitivity of critical percolation, via the Fourier spectrum
Let each site of the triangular lattice (or edge of the \Z^2 lattice) have an independent Poisson clock switching between open and closed. So, at any given moment, the configuration is just critical percolation. In particular, the probability of a left-right open crossing in an n*n box is roughly 1/2, and, on the infinite lattice, almost surely there are only finite open clusters.
In the box, how long do we have to wait before we lose essentially all correlation between having a left-right open crossing now and then? In the infinite lattice, are there random exceptional times when there are infinite clusters? In joint work with Christophe Garban and Oded Schramm, we give quite complete answers: e.g., exceptional times do exist on both lattices, and the Hausdorff dimension of their set is computed to be 31/36 for the triangular lattice.
The indicator function of a percolation crossing event is a function on the hypercube {-1,+1}^{sites or edges}, and thus it has a Fourier-Walsh expansion. Our proofs are based on giving sharp estimates on the ``weight'' of the Fourier coefficients at different frequencies.

Thursday, May 10
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.
Click for a picture

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.

Monday, April 3, 2006
Yuri Bakhtin (U of Toronto)
Malliavin Calculus in Infinite Dimension

I will consider an infinite-dimensional differential equation perturbed by a finite-dimensional additive noise, and concentrate on finite-dimensional projections of the transition probabilities for this stochastic system. Namely, I will describe sufficient conditions that guarantee that these projections are absolutely continuous with respect to the Lebesgue measure and the density is infinitely differentiable. I will also explain why the regularity of the transition kernels is important in the problem of uniqueness of stationary solutions for stochastic PDEs like stochastic Navier--Stokes in 2D.
Though the results are based on the Malliavin calculus, I will try to avoid too much technicalities. This is a joint work with Jonathan Mattingly (Duke University).

Monday, March 27, 2006, 12pm
Ehud Friedgut (Hebrew University, Jerusalem)
On the robustness of dictatorships, spectral methods

The Erdos-Ko-Rado theorem is perhaps the most fundamental theorem in extremal set theory. It characterizes the structure and size of a maximal intersecting family of sets. In this talk we show that this characterization, and similar ones, are robust: any intersecting family that is close to maximal size is also close to having the structure guaranteed by the EKR theorem.
We use spectral methods and rely on some recent results concerning Boolean functions on the discrete cube.
The talk will (hopefully) be essentially self contained.

Monday, March 20, 2006
Amir Dembo (Stanford)
Limiting dynamics for spherical models of spin glasses.

We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations, first introduced in the physics literature by Cugliandolo and Kurchan.
In this talk, based on joint works with Gerard Ben Arous, Christian Mazza and Alice Guionnet, I shall also explain the predicted long time behavior of the limiting equations, why it is of interest, and what can be rigorously proved about it.

Monday, March 13, 4pm
Jeremy Quastel(UofT)
Effect of noise on KPP traveling fronts

We study the effect of small noise on the speed of traveling fronts in one of the simplest reaction-diffusion equations, the Kolmogorov-Petrovsky-Piscunov equation. In the mid 90's it was observed numerically that the noise has an unusually large effect on the front speed. Brunet and Derrida have made some very precise conjectures, which we will explain.
This is joint work with Carl Mueller (Rochester) and Leonid Mytnik (Technion).

Monday, March 6, 2006
Konstantin Khanin (University of Toronto)
Directed polymers and KPZ-type scalings

I am going to discuss few problems related to directed polymers in quasi-stationary random potentials. Such potentials correspond to disordered systems interacting with a chaotic external field. We show that transversal fluctuations for such directed polymers are of the same order as the KPZ scaling n^{2/3}, although the system belongs to a different universality class.

Friday, December 2, 2005
Gordon Slade (UBC)
The survival probability for critical oriented percolation above 4 + 1 dimensions

We consider spread-out critical oriented percolation in d + 1 dimensions. We develop a new point-to-plane version of the lace expansion and use it to prove that the probability that the cluster of the origin survives to time n is asymptotic to a multiple of n^{-1}, when d is greater than 4. This is joint work with Remco van der Hofstad and Frank den Hollander.

Monday, November 7, 4:10
Michael Rubinstein (Waterloo)
Statistics of the Riemann zeta function

I will discuss various theorems and conjectures regarding statistical properties of the Riemann zeta function and related number theoretic problems.

Monday, October 31, 2005
Yuri Bakhtin (Fields Institute)
Random trees and stationary solutions of randomly forced 3D Navier--Stokes system

Under a certain smallness condition on the initial data and forcing, a unique solution of 3D Navier--Stokes system can be obtained via a beautiful random tree construction due to Le Jan and Sznitman. I will use this random tree approach to prove an existence and uniqueness theorem for solutions of randomly forced 3D Navier--Stokes. I will also show that under the same smallness condition the solution is uniquely determined by the history of the forcing.

Monday, October 24, 2005
Benedek Valko (U of Toronto)
Limits of random trees from real-world networks

The following random tree model is often used to describe real-world networks. We start with a single vertex and in every step we connect a new vertex randomly to one of the old ones with probability proportional to a function of its degree. We find asymptotic degree distribution, and prove a limit theorem for the tree itself as viewed from a random point. (Joint work with A. Rudas and B. Toth)

Monday, October 17, 4pm
Deniz Sezer (York University)
A Theory of Filtration Shrinkage

Filtration shrinkage is concerned with the following problem: given a stochastic process X , how can we make inferences on X based on the information contained in a subfiltration of its natural filtration? This problem arises in applications, e.g. in medicine and finance, when the main process of interest is hidden and can only be observed indirectly. A filtration shrinkage model is as follows: Given a finite collection of points x_1, ...,x_N, suppose at any given time t we only know if x_i < X < x_{i+1} for each i. We let F be the sub-filtration representing this partial information. We study the martingales and the stopping times of F when the underlying process X is a diffusion.

Tuesday, October 11, 4pm
Martin Kassabov (Cornell)
Rapidly mixing random walks on symmetric groups

The mixing time of a random walk on a Cayley graph of a finite group G is closely related to the representation theory of G. Using this connection I will construct Cayley graphs for the symmetric groups S_n with bounded degree and mixing time n log n.
Organizer's note: This answers a long-standing open question about expander graphs.

Tuesday May 24, 4:10 pm
Frank den Hollander (Scientific Director, EURANDOM)
Metastability for the lattice gas, subject to Kawasaki dynamics.

Wednesday, May 18, 11 am
Carl Mueller (University of Rochester)
Regularity of a one-dimensional stochastic heat equation with extra noise from a stochastic flow.

This is a report on work in progress with Kijung Lee and Jie Xiong. We perturb the one dimensional stochastic PDE for the Dawson-Watanabe superprocess by a stochastic flow. Using results of Krylov, we get regularity of the solution. We also raise some unsolved questions about the fundamental solution of the linear part of the equation.

Monday, April 11, 4:10pm
Stuart Whittington (UofT)
Randomly coloured self-avoiding walks and copolymer localization

Self-avoiding walks are a standard model of the conformational properties of linear polymers in dilute solution and, if the vertices of the walk are coloured, the model can be extended to model copolymers, ie polymers with more than one type of monomer. Both periodic and random colourings are interesting but this seminar will focus on the random case. The particular physical situation to be considered is a random copolymer at an interface between two immiscible solvents. One monomer type prefers one solvent while the other monomer type prefers the other solvent. At low temperatures the polymer localizes close to the interface to optimise the energy of the system while at high temperatures the polymer delocalizes into one of the two solvents to optimise the entropy. We shall show that the system has a phase transition and we shall explore the nature of the phase diagram. There are several important open questions which will be discussed, especially about the order of the phase transition.

Monday, March 21, 4:10pm
Vladimir Vinogradov (Ohio University)
On ''Contagious'' Exponential Dispersion Models Related to Continuous-State Branching

Consider a branching-diffusing particle system that belongs to the domain of attraction of a continuous Dawson-Watanabe process. For a fixed time instant, and in the case when the mechanism of local branching is binary, we demonstrate that the number of alive descendents of this system generates a discrete infinitely divisible exponential dispersion model comprised of ''contagious'' distributions. We name this exponential dispersion model the Polya-Aeppli model and identify its unit variance function. Also, we demonstrate that this model belongs to the domain of attraction of a particular member of the power-variance family of probability distributions. This stipulates an interesting connection between high-density limits of branching-diffusing populations and the convergence theory for exponential dispersion models. Moreover, our theorem is acting on ''the whole positive semi-axis'' in the sense of Yu.V. Linnik. Namely, it incorporates both the results on weak convergence and on large deviations.
We discuss potential applications and generalize our results by deriving a theorem on weak convergence for Poisson-Pascal exponential dispersion models, which are also comprised of ''contagious'' distributions. In conclusion, we consider a relationship between the latter models and Poisson-Tweedie mixtures and generalize this relationship to the case of discontinuous Dawson-Watanabe processes.

Monday, February 28, 4:00pm
Place: N627 Ross Building, York University (see building 28 in the map),
Manuel Morales (York University)
Generalized Risk Models, Levy Processes and the Discounted Penalty Function

We will review, from a historical point of view, the use of Levy processes in ruin theory. We focus on the decomposition for the ruin probability and we argue how its convolution structure is inherited from the Levy family of processes. We will introduce the notion of discounted penalty function and discuss its importance in ruin theory. The problem of finding expressions for this function in a risk model driven by a Levy process will be addressed. Examples for which these expressions are available will also be discussed. Finally, a conjecture on the general form for this function will be presented.

Monday, February 21, 4:10pm
Julien Dubedat (NYU)
Commutation of SLEs

Schramm-Loewner Evolutions (SLEs) have proved a powerful tool to describe the scaling limit of a conformally invariant simple curve. In several instances (percolation, unifrom spanning tree ...), one can define in a discrete setting several simple curves. We will discuss questions pertaining to the joint law of these curves in the scaling limit.

Monday, January 24, 4:00pm
Place: N627 Ross Building, York University (see building 28 in the map),
Pablo Olivares (University of Havana, visiting York)
Martingale Methods for some diffusion with jump stochastic processes

We cover three types of stochastic models following a diffusion movement with random jumps. Martingale methods and the Markovian structure of the processes are applied in the estimation of their parameters and in the calculation of some interesting properties. Firstly we considered a stochastic differential equation with jumps driven by a Compound Poisson Process, we study the parameter estimation problem under discrete observation using maximum likelihood and martingale related estimators. The second model is a branching spatial model, where individual move according a diffusion process then split at random times giving birth to a random number of identical individuals and so on. Again the parametric estimation problem is considered, its asymptotic properties, consistency and asymptotic normality, are obtained.

December 6, 4:10pm
Sharad Goel (Cornell)
Estimating Convergence Rates for Finite Markov Chains

How many times do you need to shuffle a deck of n cards before it is close to random? log n? n? n^3? I plan to discuss similar convergence rate questions for finite Markov chains, and to present two recent approaches to this problem: modified log Sobolev inequalities and Faber-Krahn inequalities. Using comparison techniques for random walks on groups, I will also present an analysis of the top to bottom k_n shuffles, a family of shuffles that includes both the top to random walk and the Rudvalis shuffle.

November 22, 4:10pm
Jacques Verstraete (Waterloo)
Martingale Inequalities and Enumeration

In this talk, I will discuss a few extremal problems in combinatorics, focussing on the probabilistic methods used to solve them. These methods include some Martingale inequalities, a little Fourier analysis, and some purely combinatorial theorems on graphs and hypergraphs.

An exemplary problem of this sort is to determine the number of sets X in {1,2,...,n} such that there is an n-vertex graph whose set of cycle lengths is X, and to determine the number of subsets of Z/nZ closed under taking pairwise differences or sums. The latter problem was solved completely by Ben Green. For the particular problems mentioned above, a combination of all the above-mentioned techniques will be used to show that there are extremely few of these sets.

October 18
Tom Salisbury (York University and the Fields Institute)
Conditioned superprocesses with Levy branching

Consider super-Brownian (SBM) motion with Levy branching in a domain D. In joint work with Siva Athreya, we describe the genealogy of the particles that reach n given points on the boundary of D. More precisely, we condition the SBM to have these given points in the support of its exit measure, and derive an explicit representation for it in terms of this genealogical tree plus additional mass created along it. This generalizes earlier work with John Verzani in the case of binary branching. This particular conditioning blows up in the case of stable branching, and the representation can be used to understand the source of the blowup.

September 27
Leonid Mytnik (Technion, visiting UofT)
On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients

We consider the question of uniqueness of solution to stochastic partial differential equations (SPDEs). We focus on the case of a particular parabolic SPDE --- the heat equation perturbed by a multiplicative noise, or the stochastic heat equation. In this work we establish new pathwise uniqueness results for some stochastic heat equations with non-Lipschitz coefficients.

The first part of the talk will be introductory. A motivating example, where non-Lipschitz stochastic heat equation arises as a limit of certain branching particle systems, will be described.

I will discuss two important types of uniqueness: pathwise uniqueness and iuniqueness in probability law (weaker type) of the solution. Under Lipschitz assumptions on noise coefficients, the pathwise uniqueness for a large class of SPDEs has been known for a long time. For non-Lipschitz SPDEs, uniqueness in law has been known in some very specific cases.

This is a joint work with Edwin Perkins and Anja Sturm.

September 20
Dimitrios Cheliotis (UofT)
Diffusion in a one-dimensional random environment

For a diffusion X in a one-dimensional Wiener medium, it was proved by S. Schumacher and T. Brox that (X_t-b_{logt})/(logt)^2 goes to 0 in probability, as t goes to infinity, where b is a stochastic process having an explicit description and depending only on the environment. I will give a result concerning the distribution of the number of the sign changes for b on a compact interval of (0,+\infty). I will also explain what information one can get about the path of the diffusion from results about the path of the process b.

The first half of the talk will be introductory. I will define the diffusion and its discrete time analog, the so-called Sinai walk, and give some of their basic, well known properties.

September 13
Bal�zs Szegedy (Microsoft Research)
Reflection positivity and limits of dense graph sequences

We say that a sequence of dense graphs G_n is convergent if for every fixed graph F the density of copies of F in G_n tends to a limit f(F). Many theorems and conjectures in extremal graph theory can be formulated as inequalities for the possible values of the function f. We prove that every such inequality follows from the positive definiteness of the so-called connection matrices. Moreover we construct a natural limit object for the sequence G_n namely a symmetric measurable function on the unit square. Along the line we introduce a rather general model of random graphs which seems to be interesting on its own right. Joint work with L. Lov�sz (Microsoft Research).

July 9
Nikolai Dokuchaev (University of Limerick)
Pricing rules for random volatility with uncertainty and modeling of the volatility smile

We investigate impact of popular pricing rules on implied volatility. We show that the most popular existing models allow a possibility that the option price calculated for random volatility with an error in volatility forecasts is lower than the price for the market with zero error of volatility forecast. We suggest and study a pricing rule that eliminate this possibility and is consistent with the volatility smile. The rule is based on maximization of the price via a class of possible equivalent risk-neutral measures. In Markovian setting, it requires to solve a parabolic Bellman equation. For this equation, some existence results and a prior estimates are obtained. In addition, we suggest to calculate two implied parameters: the implied volatility and the implied average cumulative risk free interest rate. They can be found unconditionally from a system of two equations. We found that very simple models with random volatilities allow to generate various shapes of volatility smiles and skews.

May 17
Rinaldo Schinazi (University of Colorado, Colorado Springs)
Branching random walks on finite subsets of Z^d

We show that a branching random walk that is supercritical (that is, starting with a single particle there is a positive probability that there will be particles at all times) on Z^d is also supercritical, on a rather strong sense, on a large enough finite ball of Z^d. This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on Z^d as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

March 22
Krzysztof Burdzy (University of Washington)
Neumann eigenfunctions and Brownian couplings

I will review some recent progress on the "hot spots" conjecture of J. Rauch and related problems concerned with Neumann eigenfunctions. I will also present some results on and problems about Brownian couplings, that is, a probabilistic technique used to study eigenfunctions. The talk will be non-technical (a lot of color pictures), aimed at a general mathematical audience, and accessible to graduate students.

March 29
Jason Schweinsberg (Cornell)
Using random partitions to approximate the effect of beneficial mutations on the genealogy of a population

When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep.
Suppose we sample n individuals at the end of a selective sweep. If we focus on a site on the chromosome that is close to the location of the beneficial mutation, then many of the individuals will likely be descended from the individual that had the beneficial mutation at the beginning of the selective sweep, while others will be descended from a different individual because of recombination between the two sites on the chromosome.
We will describe a random partition of {1,...,n} which gives a very accurate approximation to the effect of the selective sweep on the genealogy of the n sampled individuals.

March 8
Carl Mueller (Rochester)
Properties of the random string and related processes

The random string was first introduced by physicists as a model for the evolution of polymers. Later, Funaki gave a precise mathematical formulation in terms of stochastic partial differential equations. We claim that the string is a basic object in probability, just as Brownian motion is a basic model for random motion of a particle. We will discuss some properties of the string, obtained jointly with R. Tribe. Secondly, there is a mysterious connection between the string and certain stochastic partial differential equations with reflection. We will describe this connection an how to exploit it. The second part is based on work with R. Dalang and L. Zambotti.

March 1
Ana Savu (UofT)
Hydrodynamic Scaling Limit of the fourth order Ginzburg-Landau model

The fourth order Ginzburg-Landau model has been developed to understand the relaxation to equilibrium of surfaces. I will discuss how the evolution of a surface on the macroscopic scale, given by a fourth order nonlinear evolution equation, emerges as a scaling limit of particle dynamics. Since the model is of non-gradient type, a major step in the computation of the limit is finding the right decomposition of the Hilbert space of "closed functions".

February 23
Jeremy Quastel (UofT)
Bulk diffusion for interacting random walks in random environment

We discuss the diffusive limit of the site disorder model, which are reversible dynamics with respect to a family of random Bernoulli measures. The proof is by a type of renormalization which is a variant of the non-gradient method for hydrodynamic limits.

February 2
Serban Nacu (UC Berkeley)
Fast Simulation Of New Coins From Old

You are given a coin with probability of heads p, where p is unknown. Can you use it to simulate a coin with probability of heads 2p? This question was raised by Asmussen in 1991, motivated by an application in the simulation of renewal processes. More generally, if f is a known function, can you use a coin with probability of heads p (p unknown) to simulate a coin with probability of heads f(p)? In 1994, Keane and O'Brien obtained necessary and sufficient conditions for a function f to have such a simulation.
We are looking at the problem of efficient simulation. Let N be the number of p-coin tosses required to simulate a f(p)-coin toss. Typically N will be random; we say the simulation is fast if N has exponential tails. We prove that a function f has a fast simulation if and only if it is real analytic. The proof is constructive, and leads to algorithms that can be implemented. We use tools from the theory of large deviations, approximation theory, and complex analysis.
(joint work with Yuval Peres)

January 26
Assaf Naor (Microsoft Research)
Shannon's problem on the monotonicity of entropy

Let X be a real valued random variable with density f. The entropy of X is defined as Ent(X)=-\int f\log f. A classical inequality of Shannon and Stam states that if X_1 and X_2 are i.i.d. copies of X then Ent(X_1+X_2)/\sqrt{2}>= Ent(X). The problem whether the sequence Ent_n=Ent((X_1+...+X_n)/\sqrt{n}) is increasing for X_1,...,X_n i.i.d. remained open (in particular is wasn't known whether it is always the case that Ent_3>= Ent_2).
In this talk we will show that Ent_n is indeed increasing with n. The proof is based on a new formula for the entropy of a marginal which is motivated by (a proof of) the Brunn-Minkowski inequality.
Joint work with S. Artstein, K. Ball and F. Barthe.

December 8
Dror Bar-Nathan (UofT)
Probability: Fact, Fiction and Quantum

In the Theory of Evolution one separates "the fact of evolution" (that species have evolved) from "the theory of evolution" (natural selection, mutations). Softcore critics accept the fact but attack the theory, often replacing it by things divine (hardcore critics attack even the fact).

In my talk I will formalize in precise terms what I believe is the undisputed "fact" of probability - that stochastic things happen. I will then discuss three theories "explaining" that fact: a tautological theory which explains nothing at all, the classical "Kolmogorov" theory (aka "fiction") and the Quantum Probability theory which seems to be the one really running our universe. I will give a beautiful example that underlines the difference between the classical and the quantum theories and discuss the (proper) inclusion of the former by the latter.

This is a service talk. Everything I will talk about is well known and nothing is original, and I will make every effort to make the talk accessible to anyone not afraid of diagonalizing a matrix.

December 1
Yuval Peres (U.C. Berkeley and Microsoft Research)
A stable marriage of Poisson and Lebesgue

Given a point process M of intensity one in the plane, the well-known Voronoi tesselation assigns a polygon (of different area) to each point of M. The geometry of "fair" allocations (assigning unit area to each point of M) is richer and more mysterious: see here.

There is a unique "fair" allocation that is "stable" in the sense of the Gale-Shapley stable marriage problem, every point of M is assigned a a bounded region with finitely many components, but obtaining any(!) tail estimate for the diameter of these regions is open. These allocations arose from the continuum version of the "extra head" problem. The original problem is to find in a sequence of i.i.d. coins with heads probability p, one coin that landed heads so that all other coins are still i.i.d. with heads probability p [This is possible only when 1/p is an integer]. (Talk based on joint works with C. Hoffman and A. Holroyd).

November 24
Bruce Reed (McGill)
The evolution of the mixing rate

We will discuss the mixing rate of the standard random walk on the giant component of the random graph G(n,p). We tie down the mixing rate precisely for all values of p greater than (1+c)/n for any positive constant c. We need to develop a new bound on the mixing time of general Markov Chains, inspired by and extending work of Kannan and Lovasz. This is joint work with Nick Fountoulakis.

November 10
Benedek Valk� (Technical University, Budapest)
Hydrodynamic limit for perturbation of equilibria

We derive a special class of two-component systems of PDEs (hyperbolic conservation laws) as hydrodynamic limits for interacting particle systems (in the domain where the solution stays smooth). The scaling regime interpolates between the Eulerian scaling and the scaling of equilibrium fluctuations. The PDEs are derived as "universal laws" driving propagation of small perturbations of equilibria. (This is a joint work with Balint Toth.)

November 3
Nick Wormald (Waterloo)
The size of the 2-core in a random graph

Erdos and Renyi first considered the evolution of a random graph, in which n vertices begin life as isolated points and then edges are thrown in randomly one by one. This evolving random graph undergoes a phase transition when the number of edges is around n/2: a "giant" component suddenly appears.

We give a result on the joint distribution of three parameters of the giant component in the phase after it appears: the number of vertices in the 2-core (the largest subgraph of minimum degree 2 or more); the excess (#edges - #vertices) of the 2-core; and the number of vertices not in the 2-core. This uses a combination of combinatorial and probabilistic tools. It is joint work with B. Pittel.

October 27
David Revelle (UC Berkeley)
Mixing times for random walks on finite lamplighter groups

We study random walks on lamplighter groups. In a wide class of examples, we examine how different notions of of mixing time are related to maximal hitting time, expected cover time and the relaxation time of the underlying graph.

For the case of the a lamplighter group over the torus, the relaxation time is of the order n² log n, the total variation mixing time is on the order of n² log² n, and the uniform mixing time is on the order of n⁴.
This is joint work with Y. Peres.

September 29
Shlomo Hoory (UofT computer science)
An Alon-Boppana type bound for irregular graphs

Consider a finite connected graph with average degree d>=2. Using random walks, we will prove that the spectral radius of its universal cover must be at least 2 sqrt(d-1).
Using the above we generalize the Alon-Boppana theorem to irregular graphs. Assume that a graph has an r-robust average degree, i.e. the average degree after deleting any radius r ball is at least d. We give a lower bound on the second largest eigenvalue of the adjacency matrix in terms of r and d.

September 22
Ben Morris (Bloomington)
The mixing time for simple exclusion

We obtain a tight bound of O(L² log r) for the mixing time of the exclusion process in Z^d/LZ^d with r <= L^d/2 particles.