{
  "id": "2205.04285",
  "version": "v1",
  "published": "2022-05-09T13:55:43.000Z",
  "updated": "2022-05-09T13:55:43.000Z",
  "title": "A fourth moment phenomenon for asymptotic normality of monochromatic subgraphs",
  "authors": [
    "Sayan Das",
    "Zoe Himwich",
    "Nitya Mani"
  ],
  "comment": "24 pages, 1 figure; comments welcome!",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Given a graph sequence $\\{G_n\\}_{n\\ge1}$ and a simple connected subgraph $H$, we denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \\ge 2$ colors. In this article, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $H$ that we call good joins. Counts of good joins are closely related to the fourth moment of a normalized version of $T(H,G_{n})$, and that connection allows us to show a fourth moment phenomenon for the central limit theorem. Precisely, for $c\\ge 30$, we show that $T(H,G_n)$ (appropriately centered and rescaled) converges in distribution to $\\mathcal{N}(0,1)$ whenever its fourth moment converges to 3 (the fourth moment of the standard normal distribution). We show the convergence of the fourth moment is necessary to obtain a normal limit when $c\\ge 2$. The combination of these results implies that the fourth moment condition characterizes the limiting normal distribution of $T(H,G_n)$ for all subgraphs $H$, whenever $c\\ge 30$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-09T13:55:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60F05",
      "05C15"
    ],
    "keywords": [
      "fourth moment phenomenon",
      "asymptotic normality",
      "monochromatic subgraphs",
      "central limit theorem",
      "fourth moment condition characterizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}