The scaling limit of a critical random directed graph

Abstract

We consider the random directed graph $\overrightarrow{\mathit{G}}(\mathit{n},\mathit{p})$ with vertex set $\{1,2,\dots ,\mathit{n}\}$ in which each of the $\mathit{n}(\mathit{n}-1)$ 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 $\mathit{p}=1/\mathit{n}$, with critical window $\mathit{p}=1/\mathit{n}+\mathit{\lambda }{\mathit{n}}^{-4/3}$ for $\mathit{\lambda }\in \mathbb{R}$. We show that, within this critical window, the strongly connected components of $\overrightarrow{\mathit{G}}(\mathit{n},\mathit{p})$, ranked in decreasing order of size and rescaled by ${\mathit{n}}^{-1/3}$, converge in distribution to a sequence $({\mathcal{C}}_1,{\mathcal{C}}_2,\dots )$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in 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 $\mathit{G}(\mathit{n},\mathit{p})$, whose scaling limit is well understood. We show that the limiting sequence $({\mathcal{C}}_1,{\mathcal{C}}_2,\dots )$ 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.