The continuum limit of critical random graphs

TL;DR

This paper studies the structure of critical Erdős-Rényi random graphs, showing that their scaled components converge to continuous metric spaces and analyzing their diameter distribution.

Contribution

It establishes the continuum limit of critical random graph components using bijections with random walks and continuum trees, providing new insights into their metric properties.

Findings

Convergence of scaled graph components to continuum metric spaces

Distributional limit of the graph diameter

Finite mean of the limiting diameter distribution

Abstract

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, that is when p=1/n+ lambda*n^{-4/3}, for some fixed lambda in R. Then, as a metric space with the graph distance rescaled by n^{-1/3}, the sequence of connected components G(n,p) 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.