Random walks on graphs: New bounds on hitting, meeting, coalescing and returning

Abstract

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities Pt(x,x) and the maximum expected hitting time thit, both in terms of the relaxation time. We also prove a discrete-time version of the first-named author’s “Meeting time lemma” that bounds the probability of a random walk hitting a deterministic trajectory in terms of hitting times of static vertices. The meeting time result is then used to bound the expected full coalescence time of multiple random walks over a graph. This last theorem is a discrete-time version of a result by the first-named author, which had been previously conjectured by Aldous and Fill. Our bounds improve on recent results by Lyons and Oveis-Gharan; Kanade et al; and (in certain regimes) Cooper et al.