{
  "abstract": "A random geometric graph is a random graph model whose vertex set is a binomial point process, where two vertices are connected by an edge if each lies within the ball of radius $r$ centered at the other. In this paper, we investigate the asymptotic fluctuations of the threshold radius required for the maximum degree of the graph to reach a target value. Assuming this target value grows with the number of vertices $n$ at an order of at least $\\log n$ but strictly less than $(\\log n)^d$, we show that these fluctuations follow a Gumbel distribution. We also investigate the spatial configurations of vertices with extremely large degrees, and show that the point process marked by their degrees converges in distribution to a Poisson point process.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15624",
  "author_total": 1,
  "authors": [
    {
      "name": "Junpei Otsuka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eh11verkknehr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15624",
  "fulltext_url": null,
  "id": "eh11verkknehr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15624",
  "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"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7221833808",
  "pdf_url": "https://arxiv.org/pdf/2610.15624",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eh11verkknehr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15624",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15624"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Maximum degree for random geometric graphs in the connectivity regime and beyond",
  "title_text": "Maximum degree for random geometric graphs in the connectivity regime and beyond",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/eh11verkknehr",
  "venue": "arXiv",
  "versions": []
}