{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32570",
  "author_total": 1,
  "authors": [
    {
      "name": "Ayesha Irfan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/832s7c3erwkb8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32570",
  "fulltext_url": null,
  "id": "832s7c3erwkb8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32570",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7223697658",
  "pdf_url": "https://arxiv.org/pdf/2610.32570",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/832s7c3erwkb8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.32570",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices",
  "title_text": "On Trace Averages and Sharp Asymptotics for Powers of Haar-Unitary Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/832s7c3erwkb8",
  "venue": "arXiv",
  "versions": []
}