On the value of the critical point in fractal percolation

We derive a new lower bound p_c>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 C_n 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 C_n 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.