{
  "id": "0808.3516",
  "version": "v1",
  "published": "2008-08-26T13:50:48.000Z",
  "updated": "2008-08-26T13:50:48.000Z",
  "title": "Edge percolation on a random regular graph of low degree",
  "authors": [
    "Boris Pittel"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AOP361 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2008, Vol. 36, No. 4, 1359-1389",
  "doi": "10.1214/07-AOP361",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a uniformly random regular graph of a fixed degree $d\\ge3$, with $n$ vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$, respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$ is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around $p^*$ has width roughly of order $n^{-1/3}$. More precisely, suppose that $p=p(n)$ is such that $\\omega:=n^{1/3}|p-p^*|\\to\\infty$. If $p<p^*$, then with high probability (whp) the largest component has $O((p-p^*)^{-2}\\log n)$ vertices. If $p>p^*$, and $\\log\\omega\\gg\\log\\log n$, then whp the largest component has about $n(1-(p\\pi+q)^d)\\asymp n(p-p^*)$ vertices, and the second largest component is of size $(p-p^*)^{-2}(\\log n)^{1+o(1)}$, at most, where $\\pi=(p\\pi+q)^{d-1},\\pi\\in(0,1)$. If $\\omega$ is merely polylogarithmic in $n$, then whp the largest component contains $n^{2/3+o(1)}$ vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-26T13:50:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B27",
      "60G42",
      "82C20"
    ],
    "keywords": [
      "edge percolation",
      "low degree",
      "largest component contains",
      "second largest component",
      "uniformly random regular graph"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.3516P"
    }
  }
}