Toronto Probability

The study of the empirical law of eigenvalues of a random matrix is an important step in understanding its asymptotic behaviour. But while in the self-adjoint or normal case the proof is often straightforward, streamlined by a collection of existing results, in the general case it is usually an especially tricky problem due to the necessary use of the so-called Brown measure. The aim of this talk is to investigate one of those models. In the late nineties, Philippe Biane studied the asymptotic behaviour of Brownian motions on Lie Group. While in the unitary case he fully described their limit and proved the convergence, in the case of GL(N,C), this problem remained open until recently, where we managed to prove it in its entirety. This talk will be divided into three part, first an introduction to explain the difficulties arising from handling the Brown measure, then an explanation of the model and the result that we prove, and finally some elements of proof. In particular, one of the key element to this proof is a new approach to computing matrix integrals with the help of free probability that has yielded pretty general results in the last few years. This talk is based on a joint work ( https://arxiv.org/pdf/2511.10535 ) with Tatiana Brailovskaya, Nicholas Cook, and Todd Kemp.