On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices
Abstract
For a Haar-distributed matrix U ∈ U(N), Rains's power-map theorem represents the spectrum of Um, m ∈ 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≥ N. For a fixed trace power p, coherent summation of the leading Laguerre correction yields the exact factor e2/p. At the mean-spacing scale, we obtain a sharp Poisson limit for the unfolded spectrum of Um on a fixed interval as N, m → ∞; its first-order correction, obtained via the second Charlier polynomial, quantifies the residual under-dispersion of local eigenvalue counts.
Related
- Averages of traces and ratios of characteristic polynomials of random unitary matrices
- Sums of von Mangoldt convolutions via trace-sum averages over U(N)
- Trace ideals of exterior powers of the module of differentials
- Local Gan-Gross-Prasad conjecture for unitary groups: Trace formula approach for epsilon dichotomy
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BibTeX
@misc{irfan2026trace,
title = {{On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices}},
author = {Ayesha Irfan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32570},
eprint = {2610.32570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices
AU - Ayesha Irfan
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32570
UR - https://arxiv.org/abs/2610.32570
AB - For a Haar-distributed matrix U ∈ U(N), Rains's power-map theorem represents the spectrum of U^m, m ∈ 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≥ 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 → ∞; its first-order correction, obtained via the second Charlier polynomial, quantifies the residual under-dispersion of local eigenvalue counts.
ER -