{
  "id": "1208.4014",
  "version": "v2",
  "published": "2012-08-20T14:07:39.000Z",
  "updated": "2012-08-22T12:05:59.000Z",
  "title": "On the size of the largest cluster in 2D critical percolation",
  "authors": [
    "Jacob van den Berg",
    "Rene Conijn"
  ],
  "comment": "12 pages, 3 figures, minor changes",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is between an^2 pi(n) and bn^2 pi(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the argument.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-22T12:05:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60C05"
    ],
    "keywords": [
      "2d critical percolation",
      "largest cluster",
      "probability",
      "similar result holds",
      "largest open cluster"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.4014V"
    }
  }
}