Critical random graphs: limiting constructions and distributional properties

TL;DR

This paper studies the limiting metric space structures of critical Erdős-Rényi random graphs, providing new probabilistic constructions and detailed distributional properties of their components in the critical window.

Contribution

It introduces two equivalent probabilistic constructions for the limit metric spaces derived from critical random graphs, enhancing understanding of their distributional properties.

Findings

Characterization of the distribution of component masses and lengths.

Precise distributional convergence for path lengths and cycle lengths.

New probabilistic constructions using Dirichlet variables, Brownian CRT, and Poisson processes.

Abstract

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) 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 go to infinity 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 Luczak, Pittel and Wierman 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.