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 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 pc(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 pc(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. Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden. E-mail: [email protected] The work of this author was carried out while at the Department of Mathematics of the Vrije Universiteit Amsterdam. Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. E-mail: fede@ few.vu.nl Partially supported by a VENI grant of the NWO.