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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 33 open journal systems 


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

Erik I Broman, Chalmers University of Technology
Federico Camia, Vrije Universiteit Amsterdam


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 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 ∞. 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 ∞, to the critical density p_c of site percolation on the square lattice. Assuming the existence of the correlation length exponent ν 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=(1/N)1/ν+o(1) as N to ∞, showing an interesting relation with near-critical percolation.


Full text: PDF

Pages: 980-999

Published on: June 12, 2008
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Electronic Journal of Probability. ISSN: 1083-6489