Critical random graphs: limiting constructions and distributional properties
Louigi Addario-Berry, McGill University
Nicolas Broutin, INRIA
Christina Goldschmidt, University of Warwick
Abstract
We consider the Erdös--Rényi random graph $G(n,p)$ inside the critical window, where $p=1/n+λ n-4/3$ for some $λ∈ R$. 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 $n → ∞$ yields a non-trivial sequence of limit metric spaces $C=(C_1, C_2, ...)$. 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 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
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