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

TL;DR

This paper establishes new bounds on hitting, meeting, coalescing, and returning times for lazy random walks on finite graphs, improving upon recent theoretical results and confirming a conjecture by Aldous and Fill.

Contribution

It introduces novel bounds on return probabilities and hitting times, and provides a discrete-time version of the meeting time lemma, advancing understanding of random walk behaviors.

Findings

New estimates on return probabilities in terms of relaxation time.

Bounds on expected hitting and coalescence times.

A discrete-time version of the meeting time lemma.

Abstract

We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, 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 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.