{
  "abstract": "For a $Δ$-regular connected graph $H$ we obtain the upper tail large deviations speed and rate function for the number of copies of $H$ in $\\mathbb{G}_n^d$, a random $d$-regular graph on $n$ vertices, in the regime $np^{Δ/2} \\gg (\\log n)^{1/(v_H-2)}$ and $p \\ll 1$, where $p:=d/n$ and $v_H$ is the number of vertices, and also identify the typical structure of $\\mathbb{G}_n^d$ conditioned on this upper tail event. For $Δ\\geq 3$, the speed and the rate function are same with those in the Erdős-Rényi setting. In contrast, for $Δ=2$ (i.e.~in case of cycles), due to the degree constraint in $\\mathbb{G}_n^d$, although speed remains the same the rate function turns out to be different. Moreover, conditioned on the upper tail event of cycle counts in $\\mathbb{G}_n^d$, we show that the random graph contains a disjoint collection of clique and biclique like structures with probability approaching to one, a stark contrast compared to the Erdős-Rényi setting. In the complementary regime, i.e.~for $1 \\ll n^{v_H}p^{e_H} \\ll (\\log n)^{α^*_H/(α^*_H-1)}$, where $e_H$ is the number of edges in $H$, $α^*_H$ is the fractional independence number of $H$, and $H$ is a strictly balanced graph (and thus including the regular case), we show that the upper tail large deviations speed and rate function for the number of copies of $H$ match with those in the Erdős-Rényi setting. In this regime, we further show that almost all of the unlabeled copies of $H$ conditioned on the upper tail event are edge disjoint.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31590",
  "author_total": 2,
  "authors": [
    {
      "name": "Anirban Basak"
    },
    {
      "name": "Shaibal Karmakar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40qzb9zgdarbj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31590",
  "fulltext_url": null,
  "id": "40qzb9zgdarbj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31590",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223348086",
  "pdf_url": "https://arxiv.org/pdf/2610.31590",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40qzb9zgdarbj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31590",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31590"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Upper tail large deviations of regular subgraph counts in sparse random regular graphs",
  "title_text": "Upper tail large deviations of regular subgraph counts in sparse random regular graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/40qzb9zgdarbj",
  "venue": "arXiv",
  "versions": []
}