SCIENTIFIC PROGRAMS AND ACTIVITIES

Toronto Probability Seminar 2007-08
held at the Fields Institute

Organizers
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

Speaker and Talk Title

Tuesday,
April 8, 2008
4:30 p.m.
215 Huron,
Room 1018
*Note Unusual Time and Place*

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.

B. Valko and B. Virag, University of Toronto
The Brownian Carousel, part 2b.
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

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-orderdifferential 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. Virag, and myself on the RMT "soft" edge.

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 pocesses. I will give some bounds on the optimal expected weight.

Gabor Pete (Microsoft Research)
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.