No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {pk}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN1−k is partitioned intoN2 congruent subsquares, and each of these is retained with probabilitypk. independent of all others. We show that if Πkpk =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).