On the phase transition to sheet percolation in random Cantor sets

Abstract

Thed-dimensional random Cantor set is a generalization of the classical “middle-thirds” Cantor set. Starting with the unit cube [0, 1] d , at every stage of the construction we divide each cube remaining intoMd equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d−1)-dimensional “sheets”. We examine the critical probabilityps(M, d) marking the existence of these sheet crossings, and show that ps(M,d)→1−pc(Md) asM→∞, where pc(Md) is the critical probability of site percolation on the lattice (Md) obtained by adding the diagonal edges to the hypercubic lattice ℤd. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.