One of the most basic examples in random matrix theory is the Ginibre ensemble, in which all the entries of an NxN matrix are chosen independently with a normal distribution of mean zero and variance 1/N. When N is large, the eigenvalues are, with high probability, almost uniformly distributed over the unit disk in the complex plane.
One can also think of the Ginibre ensemble as the distribution at time 1 of a Brownian motion in the space of all NxN matrices. I will then discuss a “multiplicative” analog of this model, consisting of Brownian motion in the group of invertible matrices. In this case, the eigenvalues cluster for large N into a certain region Sigma_t introduced by Biane. I will describe these regions and the distribution of eigenvalues in them. The talk will be self-contained and have lots of pictures. This is joint work with Bruce Driver and Todd Kemp.