Component structure of the configuration model: Barely supercritical case: VAN DER HOFSTAD et al..

We study near-critical behavior in the configuration model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ->infinity, but nu n-1 n-1/3(E[Dn3])2/3. Let Dn* denote the size-biased version of D-n. We prove that there is a unique giant component of size n rho nEDn(1+o(1)), where rho(n) denotes the survival probability of a branching process with offspring distribution Dn*-1. This extends earlier results of Janson and Luczak, as well as those of Janson, Luczak, Windridge, and House, to the case where the third moment of D-n is unbounded. We further study the size of the largest component in the critical regime, where nu n-1=O(n-1/3(EDn3)2/3), extending and complementing results of Hatami and Molloy.