### 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 N^{d} 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 p_{c}(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 p_{c}(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 p_{c}(N, 2) converges, as N → ∞, to the critical density p_{c} 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 p_{c}(N, 2) – pc = (1/N)^{1/v+o(1)} as N → ∞, showing an interesting relation with near-critical percolation .

Original language | English (US) |
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Pages (from-to) | 980-999 |

Number of pages | 20 |

Journal | Electronic Journal of Probability |

Volume | 13 |

DOIs | |

Publication status | Published - Jan 1 2008 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty