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2008 Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process
Erik Broman, Federico Camia
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Electron. J. Probab. 13: 980-999 (2008). DOI: 10.1214/EJP.v13-511

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.

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Erik Broman. Federico Camia. "Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process." Electron. J. Probab. 13 980 - 999, 2008. https://doi.org/10.1214/EJP.v13-511

Information

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

zbMATH: 1191.60109
MathSciNet: MR2413292
Digital Object Identifier: 10.1214/EJP.v13-511

Subjects:
Primary: 60K35
Secondary: 28A80, 60D05, 82B43

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