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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.08439 |
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| _version_ | 1866914031320694784 |
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| author | Addario-Berry, Louigi Bell, Sasha Deka, Prabhanka Donderwinkel, Serte Maniyar, Sourish Wang, Minmin Winter, Anita |
| author_facet | Addario-Berry, Louigi Bell, Sasha Deka, Prabhanka Donderwinkel, Serte Maniyar, Sourish Wang, Minmin Winter, Anita |
| contents | 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 $n \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments Addario-Berry, Louigi Bell, Sasha Deka, Prabhanka Donderwinkel, Serte Maniyar, Sourish Wang, Minmin Winter, Anita Probability 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 $n \to \infty$. |
| title | Revisiting scaling limits for critical inhomogeneous random graphs with finite third moments |
| topic | Probability |
| url | https://arxiv.org/abs/2509.08439 |