The component sizes of a critical random graph with pre-described degree sequence
Résumé
Consider a critical random multigraph $\mathcal{G}_n$ constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_n$ in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift, whereas when $\nu$ is a power law distribution with exponent $\gamma \in (3,4)$, the components sizes rescaled by $n^{-(\gamma-2)/(\gamma-1)}$ converge to the excursion lengths of a drifted process with independent increments that will be characterized.