# SCIENTIFIC PROGRAMS AND ACTIVITIES

March  7, 2014

## 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

 2008 Speaker and Talk Title June 16 4:10-5 Fields Library Eckhard Schlemm, FU Berlin (visiting U of T) will present a talk about his masters thesis (Diplomarbeit) on First-passage percolation on widh-two stretches 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. Monday, March 24, 2008 4:00 pm, Stewart Libary Fields Lincoln Chayes, UCLA On the absence of ferromagnetizm in typical 2D ferromagnets. Monday, March 17, 2008 10:10 am, 215 Huron St. B. Valko and B. Virag, University of Toronto The Brownian Carousel In the fourth and final part of this epic trilogy we explain some details of the proof of that connects random matrices to hyperbolic Brownian motion. Monday, March 10, 2008 4:00 pm, Stewart Libary Fields 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 Monday, March 3, 2008 4:00 pm, Room TBA B. Valko and B. Virag, University of Toronto The Brownian Carousel, part 2 Monday, Feb. 25, 2008 4:30 pm, Stewart Libary Fields B. Valko and B. Virag, 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, Feb. 11, 2008 4:30 pm, Stewart Libary Fields 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. Monday, Feb. 4, 2008 4:10pm, Stewart Libary Fields Omer Angel (University of Toronto) Monday, Dec. 10, 2007 4:10pm, Stewart Libary Fields 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, Dec. 3, 2007 4:10pm, Stewart Libary Fields 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 pocesses. I will give some bounds on the optimal expected weight. Monday, Nov. 26, 2007 4:10pm, Room 210 Fields Balazs 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 andrandomness. We present recent results in this topic. This is joint work with Laszlo Lovasz. Monday, Nov. 19, 2007 4:10pm, Stewart Library Fields Manjunath Krisnapur (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, Oct. 29, 2007 4:10pm, Stewart Library Fields 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, Oct. 15, 2007 4:10pm, Stewart Library Fields 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, Oct. 1, 2007 4:10pm, Stewart Library Fields 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.