Sizes of the largest clusters for supercritical percolation on random recursive trees. Issue 1 (17th July 2012) - Bertoin, Jean
Abstract We consider Bernoulli bond‐percolation on a random recursive tree of size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4n9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />n≫1</alternatives>, with supercritical parameter <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4cw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />p(n)=1−t/ln⁡n+o(1/ln⁡n)</alternatives> for some <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4jn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />t>0</alternatives> fixed. We show that with high probability, the largest cluster has size close to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j464" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />e−tn</alternatives> whereas the next largest clusters have size of order <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j487" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />n/ln⁡n</alternatives>only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014