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arXiv · 2509.08439

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

Louigi Addario-Berry·Sasha Bell·Prabhanka Deka·Serte Donderwinkel·Sourish Maniyar·Minmin Wang·Anita Winter
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Abstract

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 $n \to \infty$.

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Louigi Addario-Berry, Sasha Bell, Prabhanka Deka, Serte Donderwinkel, Sourish Maniyar, Minmin Wang, Anita Winter. 2025-09-10. Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments. https://arxiv.org/abs/2509.08439

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