{
  "id": "1201.6529",
  "version": "v4",
  "published": "2012-01-31T12:55:53.000Z",
  "updated": "2012-09-23T09:14:08.000Z",
  "title": "The phase transition in random graphs - a simple proof",
  "authors": [
    "Michael Krivelevich",
    "Benny Sudakov"
  ],
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \\epsilon>0 and p=(1-\\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while for p=(1+\\epsilon)/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p=(1+\\epsilon)/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-09-23T09:14:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05"
    ],
    "keywords": [
      "simple proof",
      "experiences sharp phase transition",
      "random graph models",
      "connected component",
      "high probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.6529K"
    }
  }
}