Stationary distribution and cover time of random walks on random digraphs

Abstract

We study properties of a simple random walk on the random digraph D_{n,p} when np={d\log n},\; d>1. We prove that whp the stationary probability pi_v of a 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) tends to infinity 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 d\log (d/(d-1))n\log n. If d=d(n) tends to infinity with n, then the cover time is asymptotic to n\log n.