{
  "id": "0706.0497",
  "version": "v3",
  "published": "2007-06-04T18:49:03.000Z",
  "updated": "2014-06-26T18:10:14.000Z",
  "title": "Local Limit Theorems and Number of Connected Hypergraphs",
  "authors": [
    "Michael Behrisch",
    "Amin Coja-Oghlan",
    "Mihyun Kang"
  ],
  "comment": "24 pages",
  "journal": "Combinatorics, Probability and Computing 23 (2014), 331-366 and 367-385",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$ vertices and $m$ edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of $H_d(n,p)$ and $H_d(n,m)$ for the regime ${{n-1}\\choose{d-1}} p,dm/n >(d-1)^{-1}+\\epsilon$. As an application, we obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in $H_d(n,p)$ given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-06-26T18:10:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "connected hypergraphs",
      "derive local limit theorems",
      "largest component",
      "probability",
      "asymptotic formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.0497B"
    }
  }
}