Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 530–536 | Cite as

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

  • E. S. MeckesEmail author
  • M. W. Meckes

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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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Case Western Reserve UniversityClevelandUSA

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