  The cutoff phenomenon is an abrupt transition from out of equilibrium to
equilibrium undergone by certain Markov processes in the limit where the size
of the state space tends to infinity: instead of decaying gradually over time,
their distance to equilibrium remains close to the maximal value for a while
and suddenly drops to zero as the time parameter reaches a critical threshold.
Despite the accumulation of many examples, this phenomenon is still far from
being understood, and identifying the general conditions that trigger it has
become one of the biggest challenges in the quantitative analysis of finite
Markov chains. Very recently, the author proposed a general sufficient
condition for the occurrence of a cutoff, based on a certain
information-theoretical statistics called varentropy. In the present paper, we
demonstrate the sharpness of this approach by showing that the cutoff
phenomenon is actually equivalent to the varentropy criterion for all sparse,
fast-mixing chains. Reversibility is not required.
