Component structure of the configuration model: Barely supercritical case. Issue 1 (23rd January 2019) - van der Hofstad, Remco; Janson, Svante; Luczak, Malwina
Abstract : We study near‐critical behavior in the configuration model. Let D n be the degree of a random vertex and ν n = E [ D n ( D n − 1 ) ] / E [ D n ] ; we consider the barely supercritical regime, where ν n →1 as n → ∞, but ν n − 1 ≫ n − 1 / 3 ( E [ D n 3 ] ) 2 / 3 . Let D 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 distribution D 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, where ν n − 1 = O ( n − 1 / 3 ( E D n 3 ) 2 / 3 ), extending and complementing results of Hatami and Molloy.