{
  "abstract": "We study the random geometric graph formed by connecting $n$ independent uniform points on $\\mathbb S^{d-1}$ whenever their inner product exceeds a threshold chosen to give edge density $p$. We establish the conjectured dimensionality threshold for indistinguishability from the Erdös-Rényi graph $G(n,p)$ throughout the sparse regime with polynomially growing average degree. Specifically, for every fixed $δ>0$, if $p\\to0$, $p\\ge n^{-1+δ}$, and $d\\gg (np\\log(1/p))^3$, then the Kullback-Leibler (KL) divergence between the two distributions vanishes. Our proof builds on the posterior analysis framework introduced in previous work of the authors, combining a truncated second moment analysis with Fourier and spherical harmonic expansions. As in prior works, we reduce the KL bound to controlling a truncated $χ^2$-divergence $Δ$ for the neighborhood of the last vertex conditional on the graph induced by the preceding vertices. We then bound $Δ$ using a Fourier expansion of the conditional likelihood ratio for this neighborhood, where the high-frequency contribution is made negligible by truncation to a typical-degree event, while a spherical harmonic expansion approximates the low-frequency contribution in terms of posterior moments of bounded-degree harmonic polynomials. The key observation is that these moments and $Δ$ satisfy a coupled system of cavity recurrences, yielding a self-bounding inequality that implies the desired KL bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08517",
  "author_total": 5,
  "authors": [
    {
      "name": "Hang Du"
    },
    {
      "name": "Cheng Mao"
    },
    {
      "name": "Nike Sun"
    },
    {
      "name": "Yihong Wu"
    },
    {
      "name": "Jiaming Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fevvew7ma0nf2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08517",
  "fulltext_url": null,
  "id": "fevvew7ma0nf2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08517",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220987380",
  "pdf_url": "https://arxiv.org/pdf/2610.08517",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fevvew7ma0nf2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08517",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08517"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp detection thresholds for random geometric graphs with polynomial average degrees",
  "title_text": "Sharp detection thresholds for random geometric graphs with polynomial average degrees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/fevvew7ma0nf2",
  "venue": "arXiv",
  "versions": []
}