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 A1 ) 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 modified model thus obtained by C1 , 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 \Gamma5 ; using some topological arguments it follows that 0.8107 is subcritical for C1 and hence (since C1 dominates A1 ) for A1 . 1 A new lower bound via a new model The dyadic fractal percolation model [5] can be described informally as follows. Fix 0 p 1. Divide the unit square I = [0; 1] 2 into 4 equal smaller squares, and in..