On the value of the critical point in fractal percolation

Abstract

AbstractWe 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∞) and adding to it a countable number of straight line segments, chosen in a certain (nonrandom) way as to simplify greatly the connectivity structure. We denote the modified model thus obtained by C∞, 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 finite n. For p=0.8107 and n=360 we obtain a value less than 10−5; using some topological arguments it follows that 0.8107 is subcritical for C∞ and hence (since ∞ dominates A∞) for A∞. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 332–345, 2001.