The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n, p) for np = 1+ε, where ε = ε(n) = o(n). We show that if εn → −∞, then a.a.s. D(n, p) consists of components which are either isolated vertices or directed cycles, each of size Op(|ε|). On the other hand, if εn →∞, then a.a.s. the structure of D(n, p) is dominated by the unique complex component of size (4 + o(1))εn, while all other components are of size Op(ε −1).