**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 joint work with Guillaume Cébron, following related work of Thierry Lévy and Mylène Maïda, we showed that the bulk fluctuations (a.k.a. linear statistics) of the eigenvalues are Gaussian, with an explicit covariance that generalizes the Haar unitary case studied by Evans and Diaconis.
- In joint work with Benoît Collins and Antoine Dahlqvist, we showed that the largest (angle) eigenvalue of the Brownian motion has an explicit almost sure limit.