  We link two phenomena concerning the asymptotical behavior of stochastic
processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape
behavior usually associated to exit from metastability. The former is
characterized by convergence at asymptotically deterministic times, while the
convergence times for the latter are exponentially distributed. We compare and
study both phenomena for discrete-time birth-and-death chains on Z with drift
towards zero. In particular, this includes energy-driven evolutions with energy
functions in the form of a single well. Under suitable drift hypotheses, we
show that there is both an abrupt convergence towards zero and escape behavior
in the other direction. Furthermore, as the evolutions are reversible, the law
of the final escape trajectory coincides with the time reverse of the law of
cut-off paths. Thus, for evolutions defined by one-dimensional energy wells
with sufficiently steep walls, cut-off and escape behavior are related by time
inversion.
