# Student Probability Seminar

#### Eigenvectors of symmetric mean-field random matrices

Speaker: Lucas Benigni

Location: Warren Weaver Hall 1314

Date: Wednesday, November 15, 2017, 10 a.m.

Synopsis:

Consider a $N \times N$ symmetric matrix whose entries are, up to the symmetry, iid random variables. It is of interest to look at spectral elements of these matrices. In this talk, we will focus on eigenvectors and gives an overview of some local statistics such as delocalization, asymptotic distribution or quantum unique ergodicity. We will see how a local control over the resolvent and properties of the Dyson Brownian motion, a matrix whose entries are Brownian motions, allow us to control eigenvectors of these matrices. In particular, we will see that we can study eigenvectors dynamically by looking at their moments, a multi-particle random walk in a random environment then appears.