Critical Random Graphs: Limiting Constructions and Distributional Properties

We consider the Erdös-Rényi random graph G (n, p) inside the critical window, where p=1/n+ n^-4/3 for some. We proved in 1 that considering the connected components of G (n, p) as a sequence of metric spaces with the graph distance rescaled by n^-1/3 and letting n yields a non-trivial sequence of limit metric spaces C= (C₁, C₂, ). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of 29 by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component