A Sharp Rate of Convergence for the Empirical Spectral Measure of a Random Unitary Matrix


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We consider the convergence of the empirical spectral measures of random N × N unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and uniform measure on the unit circle is of 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 L1-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process.

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E. Meckes

Case Western Reserve University

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Email: elizabeth.meckes@case.edu
美国, Cleveland, Ohio

M. Meckes

Case Western Reserve University

Email: elizabeth.meckes@case.edu
美国, Cleveland, Ohio

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