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