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

Abstract

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. For averages of mixed trace products, a Charlier--Ewens convolution separates the stable Wick term from its full finite-$N$ correction. A Fredholm factorisation gives sharp total-variation asymptotics for the normalised trace in four regimes, from fixed powers to $m\geq N$. For a fixed trace power $p$, coherent summation of the leading Laguerre correction yields the exact factor $e^{2/p}$. At the mean-spacing scale, we obtain a sharp Poisson limit for the unfolded spectrum of $U^m$ on a fixed interval as $N, m \rightarrow \infty$; its first-order correction, obtained via the second Charlier polynomial, quantifies the residual under-dispersion of local eigenvalue counts.