Premium
Component structure of the configuration model: Barely supercritical case
Author(s) -
van der Hofstad Remco,
Janson Svante,
Luczak Malwina
Publication year - 2019
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20837
Subject(s) - supercritical fluid , component (thermodynamics) , giant component , vertex (graph theory) , connected component , mathematics , branching process , statistical physics , combinatorics , physics , random graph , quantum mechanics , thermodynamics , graph
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.