{
  "id": "0909.4490",
  "version": "v1",
  "published": "2009-09-24T16:23:03.000Z",
  "updated": "2009-09-24T16:23:03.000Z",
  "title": "Critical percolation: the expected number of clusters in a rectangle",
  "authors": [
    "Clément Hongler",
    "Stanislav Smirnov"
  ],
  "comment": "27 pages, 14 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and $SLE$ techniques, and in principle should provide a new approach to establishing conformal invariance of percolation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-24T16:23:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "30C35",
      "81T40",
      "82B43"
    ],
    "keywords": [
      "expected number",
      "critical percolation",
      "explicit conformal invariant",
      "triangular lattice",
      "conformally invariant scaling limits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.4490H"
    }
  }
}