  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 $\nu$ (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 $\mathcal{G}_n$ as $n$ tends to infinity 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
above past minima, 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 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 $\nu$ has finite third moment.
