Component structure of the configuration model: Barely supercritical case

We study near-critical behavior in the configuration model. Let Dn be the degree of a random vertex and (Formula presented.); we consider the barely supercritical regime, where νn→1 as n→∞, but (Formula presented.). Let (Formula presented.) denote the size-biased version of Dn. We prove that there is a unique giant component of size (Formula presented.), where ρn 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 Dn 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. © 2019 The Authors. Random Structures and Algorithms published by Wiley Periodicals, Inc.