{
  "abstract": "Let $M$ be a Weil-Petersson random hyperbolic surface of genus $g$, and let $Γ\\subset \\mathbb{Z}^{2g}$ be a lattice of prime index $q$. We study the distribution of primitive closed geodesics in homology classes mod $Γ$ in the large genus limit. Averaging over all lattices of index $q$, with $q \\to \\infty$, we compute all the centered moments of the corresponding weighted counting functions, and exhibit a transition between Poisson and Gaussian regimes (depending on whether $\\frac{X}{q\\log X}$, the expected number of primitive geodesics in a given homology class mod $Γ$, tends to $λ>0$ or $\\infty$). We also study the unnormalized variance $G_M(X,Γ)$ of the counts among homology classes, and show that as $X \\to \\infty$, averaged over all lattices of prime index $q$, it is asymptotic to $X\\log X$ in the large genus limit. These results are analogous to phenomena arising in the distribution of primes in arithmetic progressions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26311",
  "author_total": 1,
  "authors": [
    {
      "name": "Noam Pirani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1vkhxv7ec2xgy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26311",
  "fulltext_url": null,
  "id": "1vkhxv7ec2xgy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26311",
  "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.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204580162",
  "pdf_url": "https://arxiv.org/pdf/2608.26311",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1vkhxv7ec2xgy.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2608.26311",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26311"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Closed geodesics in homology classes modulo sublattices",
  "title_text": "Closed geodesics in homology classes modulo sublattices",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/1vkhxv7ec2xgy",
  "venue": "arXiv",
  "versions": []
}