RANDOM WALKS ON THE RANDOM GRAPH

Abstract

We study random walks on the giant component of the Erdos-Renyi random graph G(n, p) where p = lambda/n for lambda > 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 log(2) n. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to O(log n) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (nu d)(-1) log n +/-(log n)(1/2+o(1)), where nu and d are the speed of random walk and dimension of harmonic measure on a Poisson(lambda)-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.