Random walks on the random graph

We study random walks on the giant component of the Erd.s.Rényi random graph G(n,p) where p=./n for .>1 fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order log2n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(logn) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (.d).1logn±(logn)1/2+o(1), where . and d are the speed of random walk and dimension of harmonic measure on a Poisson(.)-Galton.Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the nonbacktracking random walk.