Speaker: Yi Han (MIT)

Title: Some progress and mysteries in the study of inhomogeneous random matrices

Abstract:

The spectrum of a random matrix is well-understood for an invariant ensemble, and sufficiently strong information has been obtained for mean field type random matrices. There are however far more general situations where much less information is obtained: when the entries have a heavy-tailed distribution or when the variance profile has a specific structure. In this talk I discuss some recent discoveries in the latter regime. The topics include: a sharp description for the edge of a symmetric random matrix when its tail decays precisely like x^{-4} (the transition regime); a very weak condition for determining spectral outliers for finite rank perturbations of non-Hermitian random matrices with a banded variance profile; the smallest singular value for rectangular random matrix with entries in the domain of attraction of alpha-stable law; and on convergence to the circular law for ESDs of some nonhomogeneous matrices. The work employs recent concentration inequalities invented by Bandeira, Boedihardjo, van Handel and Brailovskaya. While satisfying results are obtained for some problems, a sharp understanding is still not obtained for other problems despite significant quantitative improvement.