No Directed Fractal Percolation in Zero Area

We consider the fractal percolation process on the unit square, with fixed decimation parameter N and level dependent retention parameters {pk}; that is, for all k ≥ 1, at the kth stage every retained square of side-length N1−k is partitioned into N2 congruent subsquares, and each of these is retained with probability pk, independently of all others. We show that if ∏ k pk = 0 (i.e., if the area of the limiting set vanishes a.s.) then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down and to the right).