{
  "id": "math/0102193",
  "version": "v2",
  "published": "2001-02-26T05:14:05.000Z",
  "updated": "2002-12-31T20:22:45.000Z",
  "title": "Mixing times of lozenge tiling and card shuffling Markov chains",
  "authors": [
    "David Bruce Wilson"
  ],
  "comment": "39 pages, 8 figures",
  "journal": "Annals of Applied Probability, 14(1):274--325, 2004",
  "doi": "10.1214/aoap/1075828054",
  "categories": [
    "math.PR"
  ],
  "abstract": "We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-12-31T20:22:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60C05"
    ],
    "keywords": [
      "mixing time",
      "card shuffling markov chains",
      "lozenge tiling",
      "upper bound",
      "generate random tilings"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......2193W"
    }
  }
}