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 ν (criticality means that the second moment of ν 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 ν 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 ν is a power law distribution with exponent γ∈(3,4), the components sizes rescaled by n-(γ-2)/(γ-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 ν has finite third moment.