{
  "id": "1407.7578",
  "version": "v2",
  "published": "2014-07-28T21:49:09.000Z",
  "updated": "2014-12-27T16:44:21.000Z",
  "title": "Lozenge tilings and Hurwitz numbers",
  "authors": [
    "Jonathan Novak"
  ],
  "comment": "9 pages, 3 figures. Version 2 fixes errors, adds clarifications and new material; title changed",
  "categories": [
    "math-ph",
    "math.CO",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-28T21:49:09.000Z",
      "title": "On the Central Limit Theorem for Lozenge Tilings of Sawtooth Domains",
      "abstract": "In this note, we consider the distribution of vertical tiles on a horizontal slice through a uniformly random lozenge tiling of a large sawtooth domain. We conjecture that the joint distribution of vertical tiles asymptotically coincides with the joint distribution of eigenvalues of a GUE random matrix if and only if the height of the slice is negligible in comparison to the square root of the height of the domain. We prove that this conjecture holds with respect to a natural family of observables playing the role of moments.",
      "comment": "11 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-27T16:44:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central limit theorem",
      "lozenge tiling",
      "joint distribution",
      "large sawtooth domain",
      "vertical tiles"
    ],
    "publication": {
      "doi": "10.1007/s10955-015-1330-x",
      "journal": "Journal of Statistical Physics",
      "year": 2015,
      "month": "Oct",
      "volume": 161,
      "number": 2,
      "pages": 509
    },
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JSP...161..509N"
    }
  }
}