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.
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