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 andν n = E [ D n ( D n − 1 ) ] / E [ D n ] ; we consider the barely supercritical regime, where ν n →1 as n → ∞ , butν n − 1 ≫ n − 1 / 3( E [ D n 3 ] ) 2 / 3 . LetD n ∗denote the size‐biased version of D n . We prove that there is a unique giant component of size n ρ n E D n ( 1 + o ( 1 ) ) , where ρ n denotes the survival probability of a branching process with offspring distributionD n ∗ − 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, whereν n − 1 = O ( n − 1 / 3( E D n 3 ) 2 / 3 ) , extending and complementing results of Hatami and Molloy.