A sharp rate of convergence for the empirical spectral measure of a random unitary matrix

TL;DR

This paper establishes precise bounds on how quickly the empirical spectral measure of random unitary matrices converges to the uniform distribution, revealing a logarithmic rate that is slower than other metrics.

Contribution

It provides the first sharp bounds on the Kolmogorov distance convergence rate for spectral measures of random unitary matrices, using determinantal process techniques.

Findings

Kolmogorov distance converges at rate log N / N

Convergence is slower in Kolmogorov distance than in L1-Kantorovich distance

Bounds hold both in expectation and almost surely

Abstract

We consider the convergence of the empirical spectral measures of random $N \times N$ unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and the uniform measure on the unit circle is of the order $\log N / N$, both in expectation and almost surely. This implies in particular that the convergence happens more slowly for Kolmogorov distance than for the $L_1$-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process.