## Seminars and Colloquia by Series

Tuesday, October 6, 2015 - 15:05 , Location: Skiles 269 , Katia Meziani , Universite Paris Dauphine , Organizer: Christian Houdre
Thursday, October 1, 2015 - 15:05 , Location: Skiles 006 , Philippe Sosoe , Harvard University , Organizer: Christian Houdre
In the 1970s, Girko made the striking observation that, after centering, traces of functions of large random matrices have approximately Gaussian distribution. This convergence is true without any further normalization provided f is smooth enough, even though the trace involves a number of terms equal to the dimension of the matrix. This is particularly interesting, because for some rougher, but still natural observables, like the number of eigenvalues in an interval, the fluctuations diverge. I will explain how such results can be obtained, focusing in particular on controlling the fluctuations when the function is not very regular.
Thursday, September 24, 2015 - 15:05 , Location: Skiles 006 , Jack Hanson , School of Mathematics, Georgia Tech and CUNY , Organizer: Christian Houdre
The Abelian sandpile was invented as a "self-organized critical" model whose stationary behavior is similar to that of a classical statistical mechanical system at a critical point. On the d-dimensional lattice, many variables measuring correlations in the sandpile are expected to exhibit power-law decay. Among these are various measures of the size of an avalanche when a grain is added at stationarity: the probability that a particular site topples in an avalanche, the diameter of an avalanche, and the number of sites toppled in an avalanche. Various predictions about these exist in the physics literature, but relatively little is known rigorously. We provide some power-law upper and lower bounds for these avalanche size variables and a new approach to the question of stabilizability in two dimensions.
Thursday, September 17, 2015 - 15:05 , Location: Skiles 006 , Karim Lounici , School of Mathematics, Georgia Tech , Organizer: Christian Houdre
Thursday, September 10, 2015 - 15:05 , Location: Skiles 006 , Xuan Wang , School of Mathematics, Georgia Tech , Organizer: Christian Houdre
We consider the first-passage percolation model defined on the square lattice Z^2 with nearest-neighbor edges. The model begins with i.i.d. nonnegative random variables indexed by the edges. Those random variables can be viewed as edge lengths or passage times. Denote by T_n the length (i.e. passage time) of the shortest path from the origin to the boundary of the box [-n,n] \times [-n,n]. We focus on the case when the distribution function of the edge weights satisfies F(0) = 1/2. This is sometimes known as the "critical case" because large clusters of zero-weight edges force T_n to grow at most logarithmically. We characterize the limit behavior of T_n under conditions on the distribution function F. The main tool involves a new relation between first-passage percolation and invasion percolation. This is joint work with Michael Damron and Wai-Kit Lam.
Thursday, September 3, 2015 - 15:05 , Location: Skiles 006 , Michael Damron , School of Mathematics, Georgia Tech , Organizer: Christian Houdre
In two-dimensional critical percolation, the work of Aizenman-Burchard implies that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+\epsilon, for some positive \epsilon. No more precise lower bound has been given so far. Conditioned on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Morrow and Zhang to have volume n^4/3 on the triangular lattice. In 1992, Kesten and Zhang asked how, given the existence of an open crossing, the length of the shortest open crossing compares to that of the lowest; in particular, whether the ratio of these lengths tends to zero in probability. We answer this question positively.
Monday, June 15, 2015 - 14:00 , Location: Skiles 005 , Galyna Livshyts , Kent State University , Organizer: Karim Lounici
It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This talk is based on the joint work with G. Paouris and P. Pivovarov.
Tuesday, May 19, 2015 - 15:05 , Location: Skiles 005 , Umit Islak , University of Minnesota , Organizer: Christian Houdre
For a nonnegative random variable Y with finite nonzero mean \mu, we say that Y^s has the Y-size bias distribution if E[Yf(Y)] = \mu E[f(Y^s)] for all bounded, measurable f. If Y can be coupled to Y^s having the Y-size bias distribution such that for some constant C we have Y^s \leq Y + C, then Y satisfies a 'Poisson tail' concentration of measure inequality. This yields concentration results for examples including urn occupancy statistics for multinomial allocation models and Germ-Grain models in stochastic geometry, which are members of a class of models with log concave marginals for which size bias couplings may be constructed more generally. Similarly, concentration bounds can be shown when one can construct a bounded zero bias coupling or a Stein pair for a mean zero random variable Y. These latter couplings can be used to demonstrate concentration in Hoeffding's permutation and doubly indexed permutations statistics. The bounds produced, which have their origin in Stein's method, offer improvements over those obtained by using other methods available in the literature. This work is joint with J. Bartroff, S. Ghosh and L. Goldstein.
Wednesday, April 29, 2015 - 16:05 , Location: Skiles 005 , J.-C. Breton , University of Rennes , Organizer: Christian Houdre
In this talk, we propose moment identities for point processes. After revisiting the case of Poisson point processes, we propose a direct approach to derive (joint factorial) moment identities for point processes admitting Papangelou intensities. Applications of such identities are given to random transformations of point processes and to their distribution invariance properties.
Thursday, April 23, 2015 - 15:05 , Location: Skiles 006 , Solesne Bourguin , Carnegie Mellon University , Organizer: Christian Houdre
Motivated by understanding the intricate combinatorial structure of the Poisson chaos in order to see whether or not a fourth moment type theorem may hold on that space, we define, construct and study the free Poisson chaos, a non-commutative counterpart of the classical Poisson space, on which we prove the free counter part of the fourth moment theorem. This is joint work with Giovanni Peccati.