Let $G \sim D(n,p)$ be a directed random graph with $n$ vertices and the probability $p$ that there is a directed edge between any ordered pair of vertices.
Strong connectivity is the property that any pair of vertices are connected with a path of edges in both directions. It is known that a threshold for this property is $$\hat{p}_n=\frac{\log n}{n},$$ which means that if $\frac{p_n}{\hat{p}_n}\to 0$ (resp. $\infty$) then the probability that $G\sim D(n,p_n)$ be strongly connected tends to $0$ (resp. $1$). (See here)
As a consequence, if $0<p<1$ is fixed, then with high probability, $G$ is strongly connected. I would like to know if the convergence speed of this probability to $1$ is known. For example, in the non-directed case, the probability is equivalent to $1-n(1-p)^{n-1}$. (See here)
My goal would be to apply the Borel-Cantelli lemma to show that a.s., for sufficiently large $n$, $G$ is strongly connected.