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

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