The scaling limit of a critical random directed graph

We consider the random directed graph \(\vec{G}(n,p)\) with vertex set \(\{1,2,\ldots,n\}\) in which each of the \(n(n-1)\) possible directed edges is present independently with probability \(p\). We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at \(p = 1/n\), with critical window \(p= 1/n + \lambda n^{-4/3}\) for \(\lambda \in \mathcal{R}\). We show that, within this critical window, the strongly connected components of \(\vec{G}(n,p)\), ranked in decreasing order of size and rescaled by \(n^{-1/3}\), converge in distribution to a sequence \((\mathcal{C}_1,\mathcal{C}_2,\ldots)\) of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs the sense of an \(\ell^1\) sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge-lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph \(G(n,p)\), whose scaling limit is well understood. We show that the limiting sequence \((\mathcal{C}_1,\mathcal{C}_2,\ldots)\) contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.