Local law and rigidity for unitary Brownian motion

TL;DR

This paper provides precise probabilistic estimates on the eigenvalue locations of unitary Brownian motion, including at spectral edges and cusps, using dynamical PDE methods related to free probability.

Contribution

It introduces optimal high probability eigenvalue estimates for unitary Brownian motion, especially near spectral edges and cusps, using a novel dynamical PDE approach.

Findings

Eigenvalue locations are tightly estimated with high probability.

Results are optimal up to small polynomial factors in N.

Estimates hold at spectral edges, bulk, and cusp points.

Abstract

We establish high probability estimates on the eigenvalue locations of Brownian motion on the $N$-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are optimal up to arbitrarily small polynomial factors in $N$. Our results hold at the spectral edges (showing that the extremal eigenvalues are within $\mathcal{O} (N^{-2/3+})$ of the edges of the limiting spectral measure), in the spectral bulk, as well as for times near $4$ at which point the limiting spectral measure forms a cusp. Our methods are dynamical and are based on analyzing the evolution of the Borel transform of the empirical spectral measure along the characteristics of the PDE satisfied by the limiting spectral measure, that of the free unitary Brownian motion.