Given $p \in [0,1]$, an Erdos-Renyi graph ${ER}(n,p)$ on $n$ vertices is constructed by defining, for each unordered couple of distinct vertices ${i,j}$ an edge between $i$ and $j$ with probability $p$.

Similarly we define an Erdos-Renyi **digraph**, which we shall denote as ${ER}_d(n,p)$, on $n$ vertices by defining, for each ordered pair of distinct vertices $(i,j)$ an edge $i \to j$ with probability $p$.

Erdos and Renyi proved that $p=\frac{\ln(n)}{n}$ is a sharp threshold for the connectedness of an (undirected) E-R graph, that is :

If ${\displaystyle p<{\tfrac {(1-\varepsilon )\ln n}{n}}}$, then ${ER}(n,p)$ will almost surely contain isolated vertices, and thus be disconnected.

If ${\displaystyle p>{\tfrac {(1+\varepsilon )\ln n}{n}}}$, then a graph ${ER}(n,p)$ will almost surely be connected.

Is there a similar sharp bound for the **strong connectedness** of ${ER}_d(n,p)$?

I collected statistics on ${ER}_d(n,p)$ for $n \le 300$ and a resolution of $0.02$ for $p$, with $200$ generations for each $(n,p)$. I measure a critical probability for (almost sure) strong connectedness asymptotically close to $p=\frac{2\ln(n)}{n}$.