{
  "abstract": "Let $V_{g,n}$ be the Weil--Petersson volume of the moduli space of hyperbolic surfaces of genus $g$ with $n$ cusps. The fixed-$n$ large-genus expansion of Mirzakhani and Zograf has correction terms polynomial in $n$ of degree $2i$ at order $g^{-i}$, and ceases to be uniform when $n$ is of order $\\sqrt g$. We prove an asymptotic expansion in powers of $1/g$ that holds uniformly in all $n\\ge0$, with coefficients analytic in a density parameter that ranges over a compact interval, including the limits $n/g\\to0$ and $n/g\\to\\infty$. Its leading term is explicit, contains no undetermined constant and has relative error $O(1/g)$; truncation at an order proportional to $\\sqrt g$ gives relative error $O(\\exp(-a\\sqrt g))$ for some $a>0$. For fixed $n$ this confirms Zograf's conjecture that the constant in the Mirzakhani--Zograf asymptotics is $1/\\sqrtπ$. We also determine, at every cusp density, the limits of the volume ratios $V_{g,n}/V_{g-1,n+2}$ and $V_{g,n+1}/(2gV_{g,n})$ and, in all degrees, the asymptotics of the ratios of adjacent coefficients of the one-boundary volume polynomial; this answers two questions of Mirzakhani. The proof deforms mixed $κ_1$--$ψ$ intersection numbers into pure $ψ$ intersection numbers along a family whose recursion has nonnegative coefficients, fixes the overall constant through Painlevé I without assuming resurgence, and recovers individual volumes by a local limit theorem in the number of cusps. As an application, the spectral gap of a Weil--Petersson random surface with $n\\gg\\sqrt g$ cusps tends to zero in probability; this removes the restriction $n/g\\to0$ from a theorem of Shen and Wu.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35492",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiyi Li"
    },
    {
      "name": "Qihang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6z6k47ws2m9vn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35492",
  "fulltext_url": null,
  "id": "6z6k47ws2m9vn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35492",
  "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.AG"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223866226",
  "pdf_url": "https://arxiv.org/pdf/2610.35492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6z6k47ws2m9vn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.35492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Uniform large-genus asymptotics of Weil-Petersson volumes",
  "title_text": "Uniform large-genus asymptotics of Weil-Petersson volumes",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:28Z",
  "url": "https://research.draconitic.com/paper/6z6k47ws2m9vn",
  "venue": "arXiv",
  "versions": []
}