{
  "abstract": "We study the homology of percolation clusters on disk-shaped patches of the regular $\\{3,q\\}$ triangle tilings for $q\\ge7$. Varying $q$ gives a one-parameter family of hyperbolic tilings, from mildly curved ($q=7$) to deeply hyperbolic ($q=20$), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter $λ(q)$, the asymptotic ratio of successive ring sizes, governs every result. Total persistent $H_1$ of the bond-percolation complex, per vertex, is measured across $q=7,\\dots,20$ and found to be a nearly linear function of $1/λ(q)$. An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in $1/λ(q)$ by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as $5/4-2π/(3\\sqrt3)$, and agrees with the measurements to within the size of the next-order term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30347",
  "author_total": 1,
  "authors": [
    {
      "name": "Zachary Treisman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29nmj9s8rxmsj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30347",
  "fulltext_url": null,
  "id": "29nmj9s8rxmsj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30347",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7214598950",
  "pdf_url": "https://arxiv.org/pdf/2609.30347",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29nmj9s8rxmsj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2609.30347",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30347"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Loop content of bond percolation on hyperbolic triangulations: an exact identity and a $1/λ$ expansion",
  "title_text": "Loop content of bond percolation on hyperbolic triangulations: an exact identity and a 1/λ expansion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/29nmj9s8rxmsj",
  "venue": "arXiv",
  "versions": []
}