{
  "id": "math/0512201",
  "version": "v4",
  "published": "2005-12-09T20:51:37.000Z",
  "updated": "2007-11-02T19:34:06.000Z",
  "title": "The critical random graph, with martingales",
  "authors": [
    "Asaf Nachmias",
    "Yuval Peres"
  ],
  "comment": "13 pages, 1 figure. Revised version. Contains stronger probability deviation bounds and handles the entire scaling window. To appear in Israel Journal of Mathematics",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We give a short proof that the largest component of the random graph $G(n, 1/n)$ is of size approximately $n^{2/3}$. The proof gives explicit bounds for the probability that the ratio is very large or very small.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-11-02T19:34:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical random graph",
      "martingales",
      "short proof",
      "largest component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12201N"
    }
  }
}