{
  "abstract": "Let $T_n$ be the number of triangles in the Erdős-Rényi random graph $G(n,p)$ with $n^{-1/2}\\ll p\\ll1$ and fix $η\\in(0,1)$. We study the large deviation rate of the lower tail event $T_n\\leη\\mathbb{E}[T_n]$, and the typical structure of $G(n,p)$ conditioned on this event. By earlier work [16,19,21,49], both the rate and the structure are governed by a variational problem over graphons. In this work we solve this variational problem for every $η\\in(0,1)$. We show that every minimizer is either a constant graphon or a step graphon with two blocks of equal size. Correspondingly, the conditioned graph is close to an Erdős-Rényi graph or to a symmetric stochastic block model with two blocks. These two regimes are separated by a first order phase transition at the threshold $η_*=0.009217\\ldots$. This resolves a conjecture of Zhao [78] and an open problem of Janson and Warnke [39] on triangles. It thereby gives a complete description of the lower tail of triangle counts in this range of $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32879",
  "author_total": 2,
  "authors": [
    {
      "name": "Dong Huang"
    },
    {
      "name": "Shuangping Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/079g1qgwvpjft.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32879",
  "fulltext_url": null,
  "id": "079g1qgwvpjft",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32879",
  "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": "W7223692285",
  "pdf_url": "https://arxiv.org/pdf/2610.32879",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/079g1qgwvpjft.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32879",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32879"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The lower tail of triangle counts in sparse random graphs",
  "title_text": "The lower tail of triangle counts in sparse random graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/079g1qgwvpjft",
  "venue": "arXiv",
  "versions": []
}