  A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium
remains close to its initial value over a certain number of iterations and then
abruptly drops to near 0 on a much shorter time scale. Originally discovered in
the context of card shuffling (Aldous-Diaconis, 1986), this remarkable
phenomenon is now rigorously established for many reversible chains. Here we
consider the non-reversible case of random walks on sparse directed graphs, for
which even the equilibrium measure is far from being understood. We work under
the configuration model, allowing both the in-degrees and the out-degrees to be
freely specified. We establish the cutoff phenomenon, determine its precise
window and prove that the cutoff profile approaches a universal shape. We also
provide a detailed description of the equilibrium measure.
