Bulk Fluctuations and the Hard Edge of Unitary Brownian Motion

The Brownian motion on the Unitary group \(\mathrm{U}(N)\) has a large-\(N\) (spectral) limit: for each fixed time, the histogram of eigenvalues converges almost surely to a deterministic law with a (mostly) smooth density on the circle. This was proved by Philippe Biane in the late 1990s. One can think of this as a companion to Wigner's semicircle law for Hermitian Gaussian random matrices: the latter is really about the Brownian motion on the Unitary Lie algebra, and so it is compelling that some of the same behavior carries over to the Lie group. In this lecture, I will talk about two finer properties of the large-\(N\) limit of Unitary Brownian motion. In both cases, we proved multi-dimensional versions of the theorems, which I will also describe if time permits.