{
  "id": "1512.09107",
  "version": "v1",
  "published": "2015-12-30T20:35:21.000Z",
  "updated": "2015-12-30T20:35:21.000Z",
  "title": "Critical Percolation and the Minimal Spanning Tree in Slabs",
  "authors": [
    "Charles M. Newman",
    "Vincent Tassion",
    "Wei Wu"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The minimal spanning forest on $\\mathbb{Z}^{d}$ is known to consist of a single tree for $d \\leq 2$ and is conjectured to consist of infinitely many trees for large $d$. In this paper, we prove that there is a single tree for quasi-planar graphs such as $\\mathbb{Z}^{2}\\times {\\{0,\\ldots,k\\}}^{d-2}$. Our method relies on generalizations of the \"Gluing Lemma\" of arXiv:1401.7130. A related result is that critical Bernoulli percolation on a slab satisfies the box-crossing property. Its proof is based on a new Russo-Seymour-Welsh type theorem for quasi-planar graphs. Thus, at criticality, the probability of an open path from $0$ of diameter $n$ decays polynomially in $n$. This strengthens the result of arXiv:1401.7130, where the absence of an infinite cluster at criticality was first established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-30T20:35:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B27",
      "82B43",
      "82B44"
    ],
    "keywords": [
      "minimal spanning tree",
      "critical percolation",
      "single tree",
      "quasi-planar graphs",
      "russo-seymour-welsh type theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151209107N"
    }
  }
}