Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in mandelbrot’s fractal percolation process

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 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 → ∞. 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 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) – pc = (1/N)^1/v+o(1) as N → ∞, showing an interesting relation with near-critical percolation .