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• W2152880689 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 $lambdainmathbb{R}$. 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 $ntoinfty$ 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" @default.
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• W2152880689 date "2010-01-01" @default.
• W2152880689 modified "2023-09-25" @default.
• W2152880689 title "Critical Random Graphs: Limiting Constructions and Distributional Properties" @default.
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• W2152880689 doi "https://doi.org/10.1214/ejp.v15-772" @default.
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