| Título: | Critical Random Graphs: Limiting Constructions and Distributional Properties |
| Autores: |
Addario-Berry, Louigi; McGill University Broutin, Nicolas; INRIA Goldschmidt, Christina; University of Warwick |
| Fecha: | 2010-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
random graph; real tree; scaling limit; Gromov--Hausdorff distance; Brownian excursion; continuum random tree; Poisson process; urn model 05C80;60C05 |
| Descripción: | 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 |
| Idioma: | No aplica |
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