Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

Heavy-tailed configuration models at criticality

We study the critical behavior of the component sizes for the configuration model when the tail of the degree distribution of a randomly chosen vertex is a regularly-varying function with exponent $τ-1$, where $τ\in (3,4)$. The component sizes are shown to be of the order $n^{(τ-2)/(τ-1)}L(n)^{-1}$ for some slowly-varying function $L(\cdot)$. We show that the re-scaled ordered component sizes converge in distribution to the ordered excursions of a thinned Lévy process. This proves that the scaling limits for the component sizes for these heavy-tailed configuration models are in a different uni…

重新审视具有有限三阶矩的临界非齐次随机图的标度极限

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

本文在有限三阶矩条件下，证明了临界非齐次随机图的分量大小按标度收敛于带抛物漂移的布朗运动游程，并指出大分量按权重与按大小排序高概率一致。