{
  "id": "1210.4150",
  "version": "v2",
  "published": "2012-10-15T19:57:40.000Z",
  "updated": "2013-10-07T13:42:00.000Z",
  "title": "New methods to bound the critical probability in fractal percolation",
  "authors": [
    "Henk Don"
  ],
  "comment": "22 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\\geq p_c(M). For all M\\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those already known. To find lower bounds, we compare fractal percolation with site percolation. A fundamentally new result is that for all M we construct an increasing sequence that converges to p_c(M). The terms in the sequence can in principle be calculated algorithmically. These ideas lead to (computer aided) proofs that p_c(2)> 0.881 and p_c(3)>0.784. For the upper bounds, we introduce the idea of classifications. The fractal percolation iteration process now induces an iterative random process on a finite alphabet, which is easier to analyze than the original process. This theoretical framework is the basis of computer aided proofs for the following upper bounds: p_c(2)<0.993, p_c(3)<0.940 and p_c(4)<0.972.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-07T13:42:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22"
    ],
    "keywords": [
      "critical probability",
      "fractal percolation iteration process",
      "upper bounds",
      "integer subdivision index",
      "computer aided proofs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4150D"
    }
  }
}