The evolution of the mixing rate of a simple random walk on the giant component of a random graph

In this article we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most O ( √ln n ), proving that the mixing time in this case is Θ(( n / d ) 2 ) asymptotically almost surely. As the average degree grows these become negligible and it is the diameter of the largest component that takes over, yielding mixing time Θ( n / d ) a.a.s.. We proved these results during the 2003–04 academic year. Similar results but for constant d were later proved independently by Benjamini et al. in 3 .