The phase transition in the evolution of random digraphs

Let $ \mathop {\rm D}\limits^ \to $(n, M) denote a digraph chosen at random from the family of all digraphs on n vertices with M arcs. We shall prove that if M/n ≤ c < 1 and ω(n) → ∞, then with probability tending to 1 as n → ∞ all components of $ \mathop {\rm D}\limits^ \to $(n, M) are smaller than ω(n), whereas when M/n ≥ c > 1 the largest component of $ \mathop {\rm D}\limits^ \to $(n, M) is of the order n with probability 1 - o(1).