The large-N limit of the threshold values in Mandelbrot's fractal percolation process

Abstract

The authors consider Mandelbrot's fractal percolation process, characterised by a density parameter p in (0,1) and an integer subdivision index N>1. For each N, the process is known to have a percolation transition at density p_c(N) in (0,1). They prove that lim P_c(N)=P_c where p_c is the threshold value of the ordinary square lattice site percolation model.