Stationary distribution and cover time of random walks on random digraphs

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 value π_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 d log(d/(d − 1))n log n . If d=d(n)→∞ with n , then the cover time is asymptotic to n log n