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Critical Random Graphs: Limiting Constructions and Distributional Properties

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1.	Title	Title of document	Critical Random Graphs: Limiting Constructions and Distributional Properties
2.	Creator	Author's name, affiliation, country	Louigi Addario-Berry; McGill University; France
2.	Creator	Author's name, affiliation, country	Nicolas Broutin; INRIA; France
2.	Creator	Author's name, affiliation, country	Christina Goldschmidt; University of Warwick; United Kingdom
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	random graph; real tree; scaling limit; Gromov--Hausdorff distance; Brownian excursion; continuum random tree; Poisson process; urn model
3.	Subject	Subject classification	05C80;60C05
4.	Description	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
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSERC, EPSRC
7.	Date	(YYYY-MM-DD)	2010-05-24
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/772
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-772
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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