IMS Medallion Lecture: Ramon van Handel (Princeton University)

Thursday 9:30-10:30

Title: Nonasymptotic random matrix theory

Abstract: Suppose we are given a random matrix with an essentially arbitrary pattern of entry means and variances, dependencies, and distributions. What can we say about its spectrum? It may appear hopeless that anything useful can be proved at this level of generality, which lies far outside the scope of classical random matrix theory. The aim of my lecture is to describe the basic ingredients of a new theory that provides sharp nonasymptotic information on the spectrum in an extremely general setting. This is made possible by an unexpected phenomenon: the spectra of essentially arbitrarily structured random matrices turn out to be accurately captured by certain models of free probability theory under surprisingly minimal assumptions. (Based on joint works with Afonso Bandeira, March Boedihardjo, and Tatiana Brailovskaya.)