{
  "abstract": "Let $D=2^n$ and equip $\\operatorname{PU}(D)$ with the one-step-cliff Nielsen metric, with quadratic metric coefficients one in Pauli directions of weights one and two and $D^2$ in all higher weights. We prove that the distance from the identity of a Haar-random element, normalized by $D$, converges to $π/\\sqrt3$ in probability and in $L^p$ for every $1\\le p<\\infty$. Quantitatively, it lies within $O(D^{-1/8}(\\log D)^{1/2})$ of this limit outside a set of Haar measure at most $\\exp\\{-Ω(D^{7/4}\\log D)\\}$. For each fixed $0<x<π/\\sqrt3$, the ball of radius $xD$ has Haar measure $\\exp\\{-Θ_x(D^2)\\}$; for $x>π/\\sqrt3$, its complement has measure at most $e^{-c_xD^2}$ for some $c_x>0$. As $x\\uparrowπ/\\sqrt3$, the lower and upper logarithmic rates are both asymptotic to $(π^2/3-x^2)^2/(16ζ(3))$. The small-ball upper bound follows from a comparison of Jacobi determinants, obtained by rescaling the linearized geodesic equations and applying Kato transport. Weyl integration reduces the remaining integral to an Abel-regularized logarithmic-energy estimate on the circle. A centered principal logarithm and concentration of the circular unitary ensemble eigenangle second moment give the distance upper bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09642",
  "author_total": 1,
  "authors": [
    {
      "name": "Honghuai Fang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13bn0k7ayvz96.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09642",
  "fulltext_url": null,
  "id": "13bn0k7ayvz96",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09642",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212110906",
  "pdf_url": "https://arxiv.org/pdf/2609.09642",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13bn0k7ayvz96.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09642",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09642"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry",
  "title_text": "Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/13bn0k7ayvz96",
  "venue": "arXiv",
  "versions": []
}