{
  "abstract": "We revisit a classic topic in probabilistic combinatorics, the lower-tail large-deviation problem for triangles in the random graph $G(n,p)$. Here we aim for first-order asymptotics for the quantity $\\mathbb{P}[X=k]$ with $X$ the number of triangles in $G(n,p)$ and $0 \\le k \\le (1-ε) \\mathbb{E}X$, in the sparse regime in which the logarithmic asymptotics are Poissonian. When $k=0$ (the case of triangle-freeness) and $p = p(n)$ is sufficiently small, first-order asymptotics are known via Janson's inequality and results of Stark and Wormald; when $k$ is sufficiently close to $\\mathbb{E}X$, first-order asymptotics are known via local central limit theorems. Our main result gives first-order asymptotics for all $k$ in the above range when $p = o(n^{-2/3})$, improving upon the result of Frieze that required $p = o(n^{-4/5})$. We also characterize, up to vanishing total variation distance, the distribution of the $k$ triangles in the corresponding conditional distribution and give an efficient algorithm to approximately sample from this conditional distribution. Notably, when $k = Θ(μ)$ there is a transition at $p = Θ(n^{-4/5})$ from an asymptotically uniform triangle distribution to a distribution asymptotically singular to uniform.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21890",
  "author_total": 3,
  "authors": [
    {
      "name": "Jade Lintott"
    },
    {
      "name": "Will Perkins"
    },
    {
      "name": "Corrine Yap"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5djv3ys0jfaag.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21890",
  "fulltext_url": null,
  "id": "5djv3ys0jfaag",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21890",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213876357",
  "pdf_url": "https://arxiv.org/pdf/2609.21890",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5djv3ys0jfaag.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21890",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21890"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Local large deviations for triangles in sparse random graphs",
  "title_text": "Local large deviations for triangles in sparse random graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/5djv3ys0jfaag",
  "venue": "arXiv",
  "versions": []
}