Gaussian ensembles and Dyson Brownian motion

Christian Serio
Fri, Feb 2 2024, 4:00pm

Abstract: I will introduce the GOE (orthogonal) and GUE (unitary) Gaussian ensembles, which are special Wigner matrices with Gaussian entries leading to nice symmetries. The main result will be the Ginibre formula for the density of the eigenvalues of these ensembles. There are several ways to arrive at this formula. I will discuss an argument from Tao using Dyson Brownian motion, which is the eigenvalue process of a time dependent GOE/GUE matrix whose entries are independent Brownian motions. It is possible to derive an SDE for this process, which leads to a heat equation for the transition density of the eigenvalues that can be explicitly solved. Interestingly, this eigenvalue process is equal in law to independent Brownian motions conditioned never to intersect; time permitting I will try to explain why this is true.