Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments

We consider the rank-1 inhomogeneous random graph in the Brownian regime in the critical window. Aldous studied the weights of the components, and showed that this ordered sequence converges in the $\ell^2$-topology to the ordered excursions of a Brownian motion with parabolic drift when appropriately rescaled (http://doi.org/10.1214/aop/1024404421), as the number of vertices $n$ tends to infinity. We show that, under the finite third moment condition, the same conclusion holds for the ordered component sizes. This in particular proves a result claimed by Bhamidi, Van der Hofstad and Van Leeuwaarden (https://doi.org/10.1214/EJP.v15-817). We also show that, for the large components, the ranking by component weights coincides with the ranking by component sizes with high probability as $n \to \infty$.