The continuum limit of critical random graphs

Abstract

We consider the Erdős–Renyi random graph G(n, p) inside the critical window, that is when p = 1/n + λn−4/3, for some fixed \({\lambda \in \mathbb{R}}\) . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n−1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n−1/3 converges in distribution to an absolutely continuous random variable with finite mean.