Spectral measures of powers of random matrices

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This paper considers the empirical spectral measure of a power of a random matrix drawn uniformly from one of the compact classical matrix groups.

• We give sharp bounds on the $L_p$-Wasserstein distances between this empirical measure and the uniform measure on the circle, which show a smooth transition in behavior when the power increases and yield rates on almost sure convergence when the dimension grows.
• Along the way, we prove the sharp logarithmic Sobolev inequality on the unitary group.