New methods to bound the critical probability in fractal percolation

Abstract

Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\geq p_c(M). For all M\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those already known. To find lower bounds, we compare fractal percolation with site percolation. A fundamentally new result is that for all M we construct an increasing sequence that converges to p_c(M). The terms in the sequence can in principle be calculated algorithmically. These ideas lead to (computer aided) proofs that p_c(2)> 0.881 and p_c(3)>0.784. For the upper bounds, we introduce the idea of classifications. The fractal percolation iteration process now induces an iterative random process on a finite alphabet, which is easier to analyze than the original process. This theoretical framework is the basis of computer aided proofs for the following upper bounds: p_c(2)<0.993, p_c(3)<0.940 and p_c(4)<0.972.