The mixing time of the giant component of a random graph1. Issue 3 (14th May 2014) - Benjamini, Itai; Kozma, Gady; Wormald, Nicholas
Abstract We show that the total variation mixing time of the simple random walk on the giant component of supercritical <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9616h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />G(n, p)</alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9611r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />G(n, m)</alternatives> is <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v96060" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Θ(log2n)</alternatives>. This statement was proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 383–407, 2014