On the geometry of random Cantor sets and fractal percolation

Abstract

Random Cantor sets are constructions which generalize the classical Cantor set, “middle third deletion” being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]^d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1]^e. This answers a question of Dekking and Meester,⁽⁹⁾ (b) The special case of “fractal percolation” arises when the substitution is as follows: The cube [0,1]^d is divided intoM ^d subcubes of side-lengthM ⁻, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep _c(M, d) ofp, marking the existence of crossings of [0,1]^d contained in the limit set, satisfiesp _c(M, d)→p _c(d) asM→∞, wherep _c(d) is the critical probability of site percolation on a latticeL ^d obtained by adding certain edges to the hypercubic lattice ℤ^d. This result generalizes in an unexpected way a finding of Chayes and Chayes,⁽⁴⁾ who studied the special case whend=2.