{
  "id": "1104.0613",
  "version": "v1",
  "published": "2011-04-04T16:28:06.000Z",
  "updated": "2011-04-04T16:28:06.000Z",
  "title": "The phase transition in the configuration model",
  "authors": [
    "Oliver Riordan"
  ],
  "comment": "37 pages",
  "journal": "Combinatorics, Probability and Computing 21 (2012), 265--299",
  "doi": "10.1017/S0963548311000666",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a random $r$-regular graph where $r\\ge 3$ is fixed and $n=|G|\\to\\infty$. We study the percolation phase transition on such graphs $G$, i.e., the emergence as $p$ increases of a unique giant component in the random subgraph $G[p]$ obtained by keeping edges independently with probability $p$. More generally, we study the emergence of a giant component in $G(d)$ itself as $d$ varies. We show that a single method can be used to prove very precise results below, inside and above the `scaling window' of the phase transition, matching many of the known results for the much simpler model $G(n,p)$. This method is a natural extension of that used by Bollobas and the author to study $G(n,p)$, itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-04T16:28:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "configuration model",
      "percolation phase transition",
      "unique giant component",
      "regular graph",
      "random graph"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.0613R"
    }
  }
}