New methods to bound the critical probability in fractal percolation. Issue 4 (31st July 2014) - Don, Henk
ABSTRACT We study the critical probability pc(M) in two‐dimensional M‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to pc(M). We prove that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />pc(2)>0.881</alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgphr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />pc(3)>0.784</alternatives>. For the upper bounds, we introduce an iterative random process on a finite alphabet <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpg7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />A</alternatives>, which is easier to analyze than the original process. We show that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpd6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />pc(2)≤0.993, pc(3)≤0.940</alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpb5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />pc(4)≤0.972</alternatives>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015