Geodesic
A sharp rate of convergence for the empirical spectral measure of a random unitary matrix
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 276-285

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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 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. 
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     author = {E. S. Meckes and M. W. Meckes},
     title = {A sharp rate of convergence for the empirical spectral measure of a random unitary matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {276--285},
     publisher = {mathdoc},
     volume = {457},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a14/}
}
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E. S. Meckes; M. W. Meckes. A sharp rate of convergence for the empirical spectral measure of a random unitary matrix. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 276-285. http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a14/