No directed fractal percolation in zero area |
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| Authors: | L Chayes Robin Pemantle Yuval Peres |
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| Institution: | (1) Department of Mathematics, University of California, 90024 Los Angles, California;(2) Department of Mathematics, University of Wisconsin, 53706 Madison, Wisconsin;(3) Mathematics Institute, The Hebrew University, Jerusalem, Israel;(4) Department of Statistics, University of California, Berkeley, California |
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| Abstract: | We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p
k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN
1−
k is partitioned intoN
2 congruent subsquares, and each of these is retained with probabilityp
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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| Keywords: | Fractal percolation oriented percolation branching process in varying environment |
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