The critical window in random digraphs

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