{
  "abstract": "We prove quantitative limit theorems for the edge density $Q$ in the dense edge--triangle exponential random graph model at criticality and under parameter perturbations of order $n^{-1}$. At the fixed critical parameters $(α_c,h_c),$ the centered fluctuations satisfy $n^{2/3}(Q-\\mathbb{E} Q)\\Rightarrow \\mathcal{N}(0,σ_0^2)$, with explicit variance $σ_0^2>0$. This disproves the quartic fluctuation conjecture of Bianchi, Collet and Magnanini (2024). Gaussian limits also hold $(α_c+ λ/n,h_c + γ/n)$, except when the coefficients satisfy $-\\frac{11}{16}+\\frac{2}{9}(γ+\\frac49λ)=0$. Under this linearity condition, the limiting laws of $\\sqrt n(Q-2/3)$ have densities proportional to $\\exp\\{-81z^4/64+9θz^2/8\\}$, where $θ=-57/16+8λ/27$. We obtain a sharp $n^{-1/3}$ rate in $1$-Wasserstein distance for the Gaussian limits and an $O(n^{-1/2})$ bound for the quartic limits. The proofs combine Stein's method for exchangeable pairs with conditional row estimates and a local Dobrushin contraction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41781",
  "author_total": 4,
  "authors": [
    {
      "name": "Song-Hao Liu"
    },
    {
      "name": "Qi-Man Shao"
    },
    {
      "name": "Hao Shi"
    },
    {
      "name": "Jing-Yu Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ygwjhmr16622.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41781",
  "fulltext_url": null,
  "id": "4ygwjhmr16622",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41781",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ygwjhmr16622.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41781",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41781"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical fluctuations in the edge--triangle exponential random graph models",
  "title_text": "Critical fluctuations in the edge--triangle exponential random graph models",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/4ygwjhmr16622",
  "venue": "arXiv",
  "versions": []
}