No directed fractal percolation in zero area

Details No directed fractal percolation in zero area No directed fractal percolation in zero area

Sep 1996

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1996jsp....88.1353c&link_type=abstract

Journal of Statistical Physics, Volume 88, Issue 5-6, pp. 1353-1362

Physics

Fractal Percolation, Oriented Percolation, Branching Process In Varying Environment

Scientific paper

We consider the fractal percolation process on the unit square with fixed decimation parameter N and level-dependent retention parameters { p k}; that is, for all k ⩾ 1, at the k th stage every retained square of side length N 1- k is partitioned into N 2 congruent subsquares, and each of these is retained with probability p k. independent of all others. We show that if Πk p k =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).

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