Extreme Gaps in the Spectrum of Random Matrices

MEMBERS SEMINAR
Topic:Extreme Gaps in the Spectrum of Random Matrices
Speaker:Gerard Arous
Affiliation:Courant Institute, New York University
Date:Monday, May 3
Time/Room:2:00pm - 3:00pm/S-101

I will present a recent joint work with Paul Bourgade (Paris) about the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, we show that the smallest gaps when rescaled by N^-4/3, are Poissonian and we give the limiting distribution of the k-th smallest gap. We also show that the largest gap, when normalized by log N/N, converges in Lp to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.