Electronic Journal of Probability

Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process

Abstract

We study Mandelbrot's percolation process in dimension $d \geq 2$. 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 $d \geq 2$, and in terms of $(d-1)$-dimensional sheets" for all $d \geq 3$.

For any $d \geq 2$, 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 $N \to \infty$. 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 $N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation.

In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N \to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $\nu$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$, showing an interesting relation with near-critical percolation.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 33, 980-999.

Dates
Accepted: 12 June 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819106

Digital Object Identifier
doi:10.1214/EJP.v13-511

Mathematical Reviews number (MathSciNet)
MR2413292

Zentralblatt MATH identifier
1191.60109

Rights

Citation

Broman, Erik; Camia, Federico. Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process. Electron. J. Probab. 13 (2008), paper no. 33, 980--999. doi:10.1214/EJP.v13-511. https://projecteuclid.org/euclid.ejp/1464819106

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