June 2023 The scaling limit of a critical random directed graph
Christina Goldschmidt, Robin Stephenson
Author Affiliations +
Ann. Appl. Probab. 33(3): 2024-2065 (June 2023). DOI: 10.1214/22-AAP1859


We consider the random directed graph G(n,p) with vertex set {1,2,,n} in which each of the n(n1) 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+λn4/3 for λR. We show that, within this critical window, the strongly connected components of G(n,p), ranked in decreasing order of size and rescaled by n1/3, converge in distribution to a sequence (C1,C2,) of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an 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 (C1,C2,) 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.

Funding Statement

This research was supported by EPSRC Fellowship EP/N004833/1.


We would like to thank Nicolas Broutin and Julien Berestycki for helpful discussions. We are very grateful to Éric Brunet for the proof of the second statement in Proposition 4.1. We would like to thank the referees for their careful reading of the paper and insightful comments, which led to many improvements.


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Christina Goldschmidt. Robin Stephenson. "The scaling limit of a critical random directed graph." Ann. Appl. Probab. 33 (3) 2024 - 2065, June 2023. https://doi.org/10.1214/22-AAP1859


Received: 1 October 2019; Revised: 1 August 2021; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583664
zbMATH: 1511.60019
Digital Object Identifier: 10.1214/22-AAP1859

Primary: 05C80 , 60C05
Secondary: 05C20‎ , 60F05

Keywords: phase transition , Random directed graph , Scaling limit

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 3 • June 2023
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