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.
We study properties of a simple random walk on the random digraph D"n","p when np=dlogn, d>1. We prove that whp the value @p"v of the stationary distribution at vertex v is asymptotic to deg^-(v) m where deg^-(v) is the in-degree of v and m=n(n-1)p is the expected number of edges of D"n","p. If d=d(n)-> with n, the stationary distribution is asymptotically uniform whp. Using this result we prove that, for d>1, whp the cover time of D"n","p is asymptotic to dlog(d (d-1))nlogn. If d=d(n)-> with n, then the cover time is asymptotic to nlogn.  Let D(n,r) be a random r-out regular directed multigraph on the set of vertices 1,...,n . In this work, we establish that for every r � 2, there existsr > 0 such that diam(D(n,r)) = (1 + �r + o(1))log r n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n,r). In particular, we determine the asymptotic behaviour ofmax andmin, the maximum and the minimum values of the stationary distribution. We show that with