The continuum limit of critical random graphs

Louigi Addario-Berry (1) , Nicolas Broutin (2) , Christina Goldschmidt (3)

1 Department of Mathematics and Statistics [Montréal]
2 RAP - Networks, Algorithms and Probabilities
3 Department of Statistics [Oxford]

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.

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Louigi Addario-Berry, Nicolas Broutin, Christina Goldschmidt. The continuum limit of critical random graphs. Probability Theory and Related Fields, 2012, 152, pp.367-406. ⟨10.1007/s00440-010-0325-4⟩. ⟨hal-00773370⟩