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.
Highlights
The Erdos–Rényi random graph G(n, p) is the random graph on vertex set {1, 2, . . . , n} in which each of the n possible edges is present independently of the others with probability p
(See the books [14, 26] for a small sample of this corpus.) In a previous paper [1], we considered the rescaled global structure of G(n, p) for p in the critical window – that is, where p = 1/n + λn−4/3 for some λ ∈ – when individual components are viewed as metric spaces with the usual graph distance. (See [1] for a discussion of the significance of the random graph phase transition and the critical window.) The subject of the present paper is the asymptotic behavior of individual components of G(n, p), again viewed as metric spaces, when p is in the critical window
Write n−1/3 n to mean the sequence of components viewed as metric spaces with the graph distance in each multiplied by n−1/3
Summary
The Erdos–Rényi random graph G(n, p) is the random graph on vertex set {1, 2, . . . , n} in which each of the n possible edges is present independently of the others with probability p. The limiting sequence of metric spaces has a surprisingly simple description as a collection of random real trees (given below) in which certain pairs of vertices have been identified (vertex-identification being the natural analog of adding a surplus edge, since edge-lengths are converging to 0 in the limit). The major contribution of this paper is the description and justification of two construction procedures for building the components of directly, conditional on their size and surplus The importance of these new procedures is that instead of relying on a decomposition of a component into a spanning tree and surplus, they rely on a decomposition according to the cycle structure, which from many points of view is much more natural. Before we move on to the precise description of the constructions, we introduce them informally and discuss their relationship with various facts about random graphs and the Brownian continuum random tree
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