The mixing time of the giant component of a random graph

We show that the total variation mixing time of the simple random walk on the giant component of supercritical G(n, p) and G(n,m) is Θ(log n). This statement was only recently 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.