{
  "id": "2101.00533",
  "version": "v1",
  "published": "2021-01-03T00:24:43.000Z",
  "updated": "2021-01-03T00:24:43.000Z",
  "title": "Cutoff phenomenon for the warp-transpose top with random shuffle",
  "authors": [
    "Subhajit Ghosh"
  ],
  "comment": "32 pages, 3 figures, 2 tables",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{G_n\\}_{1}^{\\infty}$ be a sequence of non-trivial finite groups, and $\\widehat{G}_n$ denote the set of all non-isomorphic irreducible representations of $G_n$. In this paper, we study the properties of a random walk on the complete monomial group $G_n\\wr S_n$ generated by the elements of the form $(\\text{e},\\dots,\\text{e},g;\\text{id})$ and $(\\text{e},\\dots,\\text{e},g^{-1},\\text{e},\\dots,\\text{e},g;(i,n))$ for $g\\in G_n,\\;1\\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is $O\\left(n\\log n+\\frac{1}{2}n\\log (|G_n|-1)\\right)$. We show that this shuffle presents $\\ell^2$-pre-cutoff at $n\\log n+\\frac{1}{2}n\\log (|G_n|-1)$. We also show that this shuffle exhibits $\\ell^2$-cutoff phenomenon with cutoff time $n\\log n+\\frac{1}{2}n\\log (|G_n|-1)$ if $|\\widehat{G}_n|=o(|G_n|^{\\delta}n^{2+\\delta})$ for all $\\delta>0$. We prove that this shuffle has total variation cutoff at $n\\log n+\\frac{1}{2}n\\log (|G_n|-1)$ if $|G_n|=o(n^{\\delta})$ for all $\\delta>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-03T00:24:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60B15",
      "60C05"
    ],
    "keywords": [
      "cutoff phenomenon",
      "random shuffle",
      "warp-transpose",
      "total variation cutoff",
      "non-trivial finite groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}