On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices

For a Haar-distributed matrix $U \in U(N)$, Rains's power-map theorem represents the spectrum of $U^m$, $m \in \mathbb{N}$, as a superposition of independent lower-dimensional CUE spectra. We determine how this exact decomposition appears in trace moments and in sharp Gaussian and Poisson limit laws