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

Consider a critical random multigraph $\mathcal{G}_n$ 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 $\mathcal{G}_n$ 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 $γ\in(3,4)$, 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.

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