{
  "id": "1708.05626",
  "version": "v1",
  "published": "2017-08-18T14:23:57.000Z",
  "updated": "2017-08-18T14:23:57.000Z",
  "title": "Block sizes in two families of random permutations",
  "authors": [
    "Irina Cristali",
    "Vinit Ranjan",
    "Jake Steinberg",
    "Erin Beckman",
    "Rick Durrett",
    "Matthew Junge",
    "James Nolen"
  ],
  "comment": "21 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "An interval of integers $[1,n]$ is said to be a block in an infinite permutation if the permutation maps $[1,n]$ onto itself. We study the size of the smallest block in $\\mathbf p$-biased permutations that were recently introduced by Pitman and Tang, and contrast their behavior with results of Basu and Bhatnagar on block sizes in Mallows($q$) permutations. When $\\mathbf p = \\text{geometric}(p)$ we show for an explicit constant $b\\approx 1.1524$ that $p \\log K \\to b$ in contrast to $p \\log K \\to \\pi^2/6$ for Mallows($1-p$) permutations. For $\\mathbf p$ obtained through i.i.d. stick breaking we give a general upper bound on $\\mathbf E K$. This includes the case of geometric($p$)- and GEM($\\theta$)-biased permutations. The approach uses a recursive distributional equation which, in principle, could be used to get very precise results about $\\mathbf E K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-18T14:23:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "60J10",
      "60K05"
    ],
    "keywords": [
      "block sizes",
      "random permutations",
      "general upper bound",
      "biased permutations",
      "precise results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}