Random walks on the random graph

Abstract

We study random walks on the giant component of the Erdős–Renyi random graph G(n,p)G(n,p) where p=λ/np=λ/n for λ>1λ>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 log2nlog2⁡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(logn)O(log⁡n) and concentrates it (the cutoff phenomenon occurs): the typical mixing is at (νd)−1logn±(logn)1/2+o(1)(νd)−1log⁡n±(log⁡n)1/2+o(1), where νν and dd are the speed of random walk and dimension of harmonic measure on a Poisson(λ)Poisson⁡(λ)-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.