Percolation in Random Cantor Sets

Abstract

The d-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 into Nd equal subcubes, and select each of these at random with probability p. The resulting limit set is a random fractal C. We present some of the main probabilistic properties of C, with an emphasis on the existence of large connected components ("percolation") and (d-1)-dimensional surfaces ("sheet-percolation"). We also look at some closely related models. Rigorous proofs are not attempted, but we give some heuristic explanations and further references. These notes formed the basis for a talk at the NATO Advanced Study Institute on Fractal Image Encoding and Analysis in Trondheim, Norway on July 8–17, 1995.