The scaling limit of a critical random directed graph

We consider the random directed graph G→(n,p) with vertex set {1,2,…,n} in which each of n(n−1) possible edges is present independently probability p. are interested strongly connected components this graph. A phase transition for emergence a giant component known to occur at p=1/n, critical window p=1/n+λn−4/3 λ∈R. show that, within window, G→(n,p), ranked decreasing order size and rescaled by n−1/3, converge distribution sequence (C1,C2,…) finite multigraphs edge lengths either 3-regular or loops. The convergence occurs sense an ℓ1 metric two close if there compatible isomorphisms between their sets roughly preserve lengths. Our proofs rely on depth-first exploration enables us relate particular spanning forest undirected Erdős–Rényi G(n,p), whose scaling limit well understood. that limiting contains only finitely many not If we ignore lengths, any fixed positive probability.