Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012). It was proved that when the degrees obey a power law with exponent $$\tau \in (3,4)$$τ∈(3,4), the sequence of clusters ordered in decreasing size and multiplied through by $$n^{-(\tau -2)/(\tau -1)}$$n-(τ-2)/(τ-1) converges as $$n\rightarrow \infty $$n→∞ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237–269, 2001) for the Erdős–Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

中文翻译：

---

临界幂律非齐次随机图的簇尾

最近，Bhamidi 等人获得了具有有限方差但无限三阶矩度的 rank-1 类型的临界非齐次随机图的簇大小的缩放限制。(Ann Probab 40:2299–2361, 2012)。证明了当度数服从指数为 $$\tau \in (3,4)$$τ∈(3,4) 的幂律时，簇序列按大小递减排序并乘以 $$n^ {-(\tau -2)/(\tau -1)}$$n-(τ-2)/(τ-1) 收敛为 $$n\rightarrow \infty $$n→∞ 到一个递减的序列非退化随机变量。在这里，我们研究重新缩放的最大簇的极限的尾部，即最大簇的缩放极限取大值 u 的概率，作为 u 的函数。这扩展了 Pittel (J Combin Theory Ser B 82(2):237–269, 2001) 将 Erdős-Rényi 随机图设置为具有无限三阶矩度的 1 级非齐次随机图。我们利用微妙的大偏差和弱收敛论点。