  A sequence of Markov chains is said to exhibit (total variation) cutoff if
the convergence to stationarity in total variation distance is abrupt. We
consider reversible lazy chains. We prove a necessary and sufficient condition
for the occurrence of the cutoff phenomena in terms of concentration of hitting
time of "worst" (in some sense) sets of stationary measure at least $\alpha$,
for some $\alpha \in (0,1)$.
  We also give general bounds on the total variation distance of a reversible
chain at time $t$ in terms of the probability that some "worst" set of
stationary measure at least $\alpha$ was not hit by time $t$. As an application
of our techniques we show that a sequence of lazy Markov chains on finite trees
exhibits a cutoff iff the ratio of their relaxation-times and their (lazy)
mixing-times tends to 0.
