The continuum limit of critical random graphs

Details The continuum limit of critical random graphs The continuum limit of critical random graphs

2009-03-27

arxiv.org/abs/0903.4730v2

Mathematics

Probability

34 pages, 5 figures

Scientific paper

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.

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