Diameter and long paths in critical digraph

We study the random directed graph $\vec G(n,p)$ in which each of the $n(n-1)$ possible directed edges are present with probability $p$. We show that in the critical window the longest self avoiding oriented paths in $\vec G(n,p)$ have length $O_{\mathbb{P}}(n^{1/3})$ so $\vec G(n,p)$ has diameter $O_{\mathbb{P}}(n^{1/3})$.