{
  "id": "math/0701316",
  "version": "v4",
  "published": "2007-01-11T01:47:59.000Z",
  "updated": "2008-08-27T05:34:18.000Z",
  "title": "Critical random graphs: Diameter and mixing time",
  "authors": [
    "Asaf Nachmias",
    "Yuval Peres"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AOP358 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, 1267-1286",
  "doi": "10.1214/07-AOP358",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{C}_1$ denote the largest connected component of the critical Erd\\H{o}s--R\\'{e}nyi random graph $G(n,{\\frac{1}{n}})$. We show that, typically, the diameter of $\\mathcal{C}_1$ is of order $n^{1/3}$ and the mixing time of the lazy simple random walk on $\\mathcal{C}_1$ is of order $n$. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size $n^{2/3}$ of $p$-bond percolation on any $d$-regular $n$-vertex graph where such clusters exist, provided that $p(d-1)\\le1+O(n^{-1/3})$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2008-08-27T05:34:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "82B43",
      "60C05"
    ],
    "keywords": [
      "critical random graphs",
      "mixing time",
      "lazy simple random walk",
      "vertex graph",
      "bond percolation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1316N"
    }
  }
}