On the value of the critical point in fractal percolation

Abstract

We derive a new lower bound pc > 0:8107 for the critical value of Mandelbrot's dyadic fractal percolation model. This is achieved by taking the random fractal set (to be denoted A 1) and adding to it a countable number of straight line segments, chosen in a certain (non-random) way as to simplify greatly the connectivity structure. We denote the modied model thus obtained by C 1, and write Cn for the set formed after n steps in its construction. Now it is possible, using an iterative technique, to compute the probability of percolating through Cn for any parameter value p and any nite n. For p = 0:8107 and n = 360 we obtain a value less than 10