On the largest component of the critical random digraph

Abstract

We consider the largest component 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 probabilities that $|\mathcal{C}_1|n^{-1/3}$ is very large or very small that are analogous to those given by Nachmias and Peres for $G(n,p)$.