Characterization of cutoff for reversible Markov chains

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 ., for some ..(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 . 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 product of their spectral gaps and their (lazy) mixing-times tends to ..