{
  "abstract": "Using results of Leighton and Shor and of Shor and Yukich, we present a connection between $d$-dimensional bandwidth and (sharp) thresholds in $d$-dimensional random geometric graphs: namely, that the sharp threshold for a given graph property in random geometric graphs is determined by its $d$-dimensional bandwidth, and vice versa, whenever the $d$-dimensional bandwidth of this property grows sufficiently fast (at least polylogarithmically) with the order of the graphs. This connection provides a number of corollaries, such as new information about the sharp thresholds for containing different spanning subgraphs in $1$-dimensional random geometric graphs, as well as asymptotically tight bounds for the $d$-dimensional bandwidth of structures such as graphs of bounded treewidth or genus, giving an asymptotic resolution to several problems raised by Chung in 1988.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37408",
  "author_total": 1,
  "authors": [
    {
      "name": "Alberto Espuny Díaz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/emqb64bq2apjn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37408",
  "fulltext_url": null,
  "id": "emqb64bq2apjn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37408",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7224105641",
  "pdf_url": "https://arxiv.org/pdf/2610.37408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/emqb64bq2apjn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "High powers of grids in random geometric graphs and thresholds for wide spanning structures",
  "title_text": "High powers of grids in random geometric graphs and thresholds for wide spanning structures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/emqb64bq2apjn",
  "venue": "arXiv",
  "versions": []
}