{
  "id": "1212.5414",
  "version": "v3",
  "published": "2012-12-21T12:26:46.000Z",
  "updated": "2015-05-28T07:14:55.000Z",
  "title": "Asymptotic domino statistics in the Aztec diamond",
  "authors": [
    "Sunil Chhita",
    "Kurt Johansson",
    "Benjamin Young"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1021 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 3, 1232-1278",
  "doi": "10.1214/14-AAP1021",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-21T09:23:22.000Z",
      "abstract": "We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominos. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominos close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.",
      "comment": "Expanded and revised version. Removed and retracted the Gaussian free field statement and outlined proof from the previous preprint version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-05-28T07:14:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "aztec diamond",
      "asymptotic domino statistics",
      "southern domino process converges",
      "inverse kasteleyn matrix",
      "study random domino tilings"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5414C"
    }
  }
}