Anatomy of the giant component: The strictly supercritical regime

Abstract

In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdős-Rényi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+ε)/n$ where $ε^3 n \to\infty$ and $ε=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = λ/n$ where $λ>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; Attach i.i.d. Poisson Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim's Poisson-cloning method and the Pittel-Wormald local limit theorems.

19 pages. arXiv admin note: text overlap with arXiv:0906.1839