  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.
