Component structure of the configuration model: barely supercritical case

We study near-critical behavior in the configuration model. Let D _n be the degree of a random vertex and (Formula presented.); we consider the barely supercritical regime, where ν _n→1 as n→∞, but (Formula presented.). Let (Formula presented.) denote the size-biased version of D _n. We prove that there is a unique giant component of size (Formula presented.), where ρ _n denotes the survival probability of a branching process with offspring distribution (Formula presented.). 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 (Formula presented.), extending and complementing results of Hatami and Molloy.