The continuum limit of critical random graphs

By Louigi Addario-Berry, Nicolas Broutin, Christina Goldschmidt. 2009

153 citations recorded by the source corpus.

We consider the Erdős–Rényi 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/146bc848fb5913e7067a92227e9a1cebb56b3f3e

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The local weak limit of the minimum spanning tree of the complete graph (2013, background): “Results from [1, 2] (described by Nicolas Broutin elsewhere in this report) tell us that for fixed λ, the Erdős–Rényi graph possesses a scaling limit when the graph distance is rescaled by n−1/3 within each component.”