{
  "id": "1504.01999",
  "version": "v1",
  "published": "2015-04-08T15:02:57.000Z",
  "updated": "2015-04-08T15:02:57.000Z",
  "title": "Random walks on the random graph",
  "authors": [
    "Nathanael Berestycki",
    "Eyal Lubetzky",
    "Yuval Peres",
    "Allan Sly"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We study random walks on the giant component of the Erd\\H{o}s-R\\'enyi random graph ${\\cal G}(n,p)$ where $p=\\lambda/n$ for $\\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $\\log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(\\log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $(\\nu {\\bf d})^{-1}\\log n \\pm (\\log n)^{1/2+o(1)}$, where $\\nu$ and ${\\bf d}$ are the speed of random walk and dimension of harmonic measure on a ${\\rm Poisson}(\\lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-08T15:02:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B10",
      "60J10",
      "60G50",
      "05C80"
    ],
    "keywords": [
      "random graph",
      "study random walks",
      "cutoff phenomenon occurs",
      "giant component",
      "prescribed degree sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150401999B"
    }
  }
}