The continuum limit of critical random graphs

We consider the ErdAs-R,nyi random graph G(n, p) inside the critical window, that is when p = 1/n + lambda n (-4/3), for some fixed lambda is an element of 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.