  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.
