{
  "id": "2412.09535",
  "version": "v1",
  "published": "2024-12-12T18:27:28.000Z",
  "updated": "2024-12-12T18:27:28.000Z",
  "title": "Local limit theorem for joint subgraph counts",
  "authors": [
    "Ashwin Sah",
    "Mehtaab Sawhney",
    "Daniel G. Zhu"
  ],
  "comment": "33 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Extending a previous result of the first two authors, we prove a local limit theorem for the joint distribution of subgraph counts in the Erd\\H{o}s-R\\'{e}nyi random graph $G(n,p)$. This limit can be described as a nonlinear transformation of a multivariate normal distribution, where the components of the multivariate normal correspond to the graph factors of Janson. As an application, we show a number of results concerning the existence and enumeration of proportional graphs and related concepts, answering various questions of Janson and collaborators in the affirmative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-12T18:27:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "05C80",
      "60C05"
    ],
    "keywords": [
      "local limit theorem",
      "joint subgraph counts",
      "multivariate normal distribution",
      "multivariate normal correspond",
      "nonlinear transformation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}