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 ε3n→∞ 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.