Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in mandelbrot’s fractal percolation process

Erik I. Broman, Federico Camia

Research output: Contribution to journalArticle

Abstract

We study Mandelbrot’s percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0,1]d in Nd subcubes, and independently retaining or discarding each subcube with probability p or 1 – p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d – 1)-dimensional “sheets” for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value pc(N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0,1]d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at pc(N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that pc(N, 2) converges, as N → ∞, to the critical density pc of site percolation on the square lattice. Assuming the existence of the correlation length exponent v for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that pc(N, 2) – pc = (1/N)1/v+o(1) as N → ∞, showing an interesting relation with near-critical percolation .

Original languageEnglish (US)
Pages (from-to)980-999
Number of pages20
JournalElectronic Journal of Probability
Volume13
DOIs
StatePublished - Jan 1 2008

Keywords

  • Critical probability
  • Crossing probability
  • Enhancement/diminishment percolation
  • Fractal percolation
  • Near-critical percolation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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