{
  "abstract": "Consider a random geometric graph $G_M(n;r)$ on a compact Riemannian manifold $M$, whose vertices are a cloud of $n$ independently sampled points, and whose edges connect vertices at distance $\\le r$. We show that, in the thermodynamic (i.e. bounded expected average degree) regime, if $G_M(n;r)$ is supercritical in the sense of continuum percolation, then the first homology group $H_1(M)$ of $M$ can be correctly inferred from $G_M(n;r)$ with high probability as $n \\to \\infty$. Specifically, one can obtain $H_1(M)$ by taking the cycle space of $G_M(n;r)$ and quotienting out all the cycles of metric diameter $O(r|\\log r|)$ (or of graph diameter $O(|\\log r|)$). Our method of estimating $H_1(M)$ exploits a coarse-topological fact about supercritical percolation, as opposed to usual methods, which examine the topology of neighborhoods of the point cloud. Whereas previous methods use combinatorial models which require $O(n \\log n)$ edges, our method only requires $O(n)$ edges. We also show that, in all phases of the thermodynamic regime, if one instead takes the quotient by cycles of metric diameter $o(r|\\log r|)$, with high probability, one will not recover $H_1(M)$. Thus $Θ(r|\\log r|)$ is the ``right scale.'' On the way, we show that an arbitrary compact $d$-dimensional Riemannian manifold has a \\emph{first homological percolation threshold} in the sense of Bobrowski and Skraba \\cite{BS2020} which coincides with the continuum percolation threshold on $\\R^d$, a result previously only known for the flat torus. This strongly suggests that our results are optimal, in the sense that $H_1(M)$ cannot be inferred from $G_M(n;r)$ in the subcritical thermodynamic regime. All results hold for homology with arbitrary coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25065",
  "author_total": 1,
  "authors": [
    {
      "name": "Christian Gorski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/efm73f07n48wn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25065",
  "fulltext_url": null,
  "id": "efm73f07n48wn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25065",
  "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.AT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7204491021",
  "pdf_url": "https://arxiv.org/pdf/2608.25065",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/efm73f07n48wn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.25065",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25065"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Detection of first homology via random geometric graphs in the thermodynamic regime",
  "title_text": "Detection of first homology via random geometric graphs in the thermodynamic regime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/efm73f07n48wn",
  "venue": "arXiv",
  "versions": []
}