On the phase transition to sheet percolation in random Cantor sets

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 intoM^d 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 probabilityp_s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M^d) asM, where p_c(M^d) is the critical probability of site percolation on the lattice (M^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. 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.