Component structure of the configuration model: barely supercritical case

Abstract

We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\nu_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $\nu_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=\nu_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \rho_n {\mathbb E} D_n (1+o(1))$, where $\rho_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \emph{critical} regime, where $\varepsilon_n = O(n^{-1/3} ({\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\cite{HatamiMolloy}.