Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold value in Mandelbrot's fractal percolation process

Journal article, 2008

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 of all d >= 2, and in terms of (d-1)-dimensional "sheets" of 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 ->infinity. As an immediate consequence, we otain 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 >= 2, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p(c)(N, 2) converges, as N ->infinity, 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)-p(c)=(1/N)(1/v+o(1)) as N ->infinity, show an interseting relation with near-critical percolation.