The critical window in random digraphs.

We consider the component structure of the random digraph $D(n,p)$ inside the critical window $p = n^{-1} + lambda n^{-4/3}$.We show that the largest component $mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the tail probabilities of $|mathcal{C}_1|n^{-1/3}$.