## Electronic Journal of Probability

### Critical Random Graphs: Limiting Constructions and Distributional Properties

#### 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

#### Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 25, 741-775.

Dates
Accepted: 24 May 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819810

Digital Object Identifier
doi:10.1214/EJP.v15-772

Mathematical Reviews number (MathSciNet)
MR2650781

Zentralblatt MATH identifier
1227.05224

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability

Rights

#### Citation

Addario-Berry, Louigi; Broutin, Nicolas; Goldschmidt, Christina. Critical Random Graphs: Limiting Constructions and Distributional Properties. Electron. J. Probab. 15 (2010), paper no. 25, 741--775. doi:10.1214/EJP.v15-772. https://projecteuclid.org/euclid.ejp/1464819810

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