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Two models of a random digraph on $n$ vertices,$D(n,\text{Prob}(\text{arc})=p)$ and $D(n,\text{number of arcs}=m)$ arestudied. In 1990, Karp for $D(n,p)$ and independently T. \L uczak for$D(n,m=cn)$ proved that for $c>1$, with probability tending to 1, there is anunique strong component of size of order $n$. Karp showed, in fact, that thegiant component has likely size asymptotic to $n\theta^2$, where$\theta=\theta(c)$ is the unique positive root of $1-\theta=e^{-c \theta}$. Inthis paper we prove that, for both random digraphs, the joint distribution ofthe number of vertices and number of arcs in the giant strong component isasymptotically Gaussian with the same mean vector $n\boldsymbol{\mu}(c)$,$\boldsymbol{\mu}(c):=(\theta^2, c\theta^2)$ and two distinct $2\times 2$covariance matrices, $n\mathbf{B}(c)$ and $n[\mathbf{B}(c)+c(\boldsymbol{\mu}'(c))^T (\boldsymbol{\mu}'(c)))]$. To this end, we introduceand analyze a randomized deletion process which determines the directed$(1,1)$-core, the maximal digraph with minimum in-degree and out-degree atleast 1. This $(1,1)$-core contains all non-trivial strong components. However,we show that the likely numbers of peripheral vertices and arcs in the$(1,1)$-core, those outside the largest strong component, are of log-polynomialorder, thus dwarfed by anticipated fluctuations, on the scale of $n^{1/2}$, ofthe giant component parameters. By approximating the likely realization of thedeletion algorithm with a deterministic trajectory, we obtain our main resultvia exponential supermartingales and Fourier-based techniques.

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs