The component sizes of a critical random graph with given degree sequence

Consider a critical random multigraph Gn with n vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution . (criticality means that the second moment of . is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of Gn as n tends to infinity in different cases. When . has finite third moment, the components sizes rescaled by n.2/3 converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when . is a power law distribution with exponent ..(3,4), the components sizes rescaled by n.(..2)/(..1) converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when . has finite third moment.