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We prove new results on lazy random walks on finite graphs. To start, weobtain new estimates on return probabilities $P^t(x,x)$ and the maximumexpected hitting time $t_{\rm hit}$, both in terms of the relaxation time. Wealso prove a discrete-time version of the first-named author's ``Meeting timelemma"~ that bounds the probability of random walk hitting a deterministictrajectory in terms of hitting times of static vertices. The meeting timeresult is then used to bound the expected full coalescence time of multiplerandom walks over a graph. This last theorem is a discrete-time version of aresult by the first-named author, which had been previously conjectured byAldous and Fill. Our bounds improve on recent results by Lyons andOveis-Gharan; Kanade et al; and (in certain regimes) Cooper et al.

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