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 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 modiied model thus obtained by C 1 , 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 nite 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 1 and hence (since C 1 dominates A 1) for A 1. 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 the natural way retain each of these squares with probability p, or else remove it with probability 1?p. Iterate this procedure (suitably scaled) of subdivision and random removal on each of the retained squares; in this way we obtain a nested sequence A 0 (I), A 1 , A 2 ; : : : of random (compact) subsets of I. The intersection of this sequence, which we shall denote A 1 , is a random fractal set. For 0 n 1 let n = n (p) denote the probability that there is a left-right crossing of A n in I, that is, that there is a connected component of A n that intersects both the left side f0g 0; 1] and the right side f1g 0; 1] of the unit square. It is well known (see 1], 3]) that there is a critical value p c , with 0 < p c < 1, such that 1 (p) is zero if and only if p < p c. In particular, 1 (p) is discontinuous at p c .