{
  "id": "1702.08368",
  "version": "v1",
  "published": "2017-02-27T16:44:46.000Z",
  "updated": "2017-02-27T16:44:46.000Z",
  "title": "The Local Limit of Random Sorting Networks",
  "authors": [
    "Omer Angel",
    "Duncan Dauvergne",
    "Alexander E. Holroyd",
    "Bálint Virág"
  ],
  "comment": "35 pages, 5 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "A sorting network is a shortest path from $12 \\dots n$ to $n \\dots 21$ in the Cayley graph of $S_n$ generated by adjacent transpositions. We establish existence of a local limit of sorting networks in a neighbourhood of $an$ for $a\\in[0,1]$ as $n\\to\\infty$, with time scaled by a factor of $n$. The limit is a swap process $U$ on $\\mathbb{Z}$. We show that $U$ is stationary and mixing with respect to the spatial shift and has time-stationary increments. Moreover, the only dependence on $a$ is through time scaling by a factor of $\\sqrt{a(1-a)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-27T16:44:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05E10",
      "68P10"
    ],
    "keywords": [
      "random sorting networks",
      "local limit",
      "cayley graph",
      "adjacent transpositions",
      "time-stationary increments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}