@article{EJP772,
author = {Louigi Addario-Berry and Nicolas Broutin and Christina Goldschmidt},
title = {Critical Random Graphs: Limiting Constructions and Distributional Properties},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {15},
year = {2010},
keywords = {random graph; real tree; scaling limit; Gromov--Hausdorff distance; Brownian excursion; continuum random tree; Poisson process; urn model},
abstract = {We consider the Erdös-Rényi random graph $G(n,p)$ inside the critical window, where $p=1/n+\lambda n^{-4/3}$ for some $\lambda\in\mathbb{R}$. 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 $n\to\infty$ yields a non-trivial sequence of limit metric spaces $C=(C_1,C_2,\ldots)$. 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 $\mathbb{R}_+$. 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},
pages = {no. 25, 741-775},
issn = {1083-6489},
doi = {10.1214/EJP.v15-772},
url = {http://ejp.ejpecp.org/article/view/772}}