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

Abstract

We study Mandelbrot's percolation process in dimension $d\ge 2$. 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 $d\ge 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d\ge 3$. For any $d\ge 2$, 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 $N\to \infty$. 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 $N\ge 2$). For $d\ge 3$, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N\to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $\nu$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N}{)}^{1/\nu +o(1)}$ as $N\to \infty$, showing an interesting relation with near-critical percolation.

24 pages, 2 figures

References in corpus (1)

• Near-critical percolation in two dimensions