The stable graph: The metric space scaling limit of a critical random graph with i.i.d. power-law degrees
The stable graph: The metric space scaling limit of a critical random graph with i.i.d. power-law degrees
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DOI：
10.1214/22-aop1587
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发表时间：
2020-02
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通讯作者：
Guillaume Conchon--Kerjan-Guillaume-Conchon--Kerjan-1455031601;C. Goldschmidt
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作者：
Guillaume Conchon--Kerjan-Guillaume-Conchon--Kerjan-1455031601;C. Goldschmidt
我们证明了一个度量空间的标度极限的临界随机图具有独立同分布度具有幂律尾行为的指数为$\alpha+1$，其中$\alpha \in（1，2）$。的限制组件是从随机$\mathbb{R}$-树编码的运行下确界以上的过程，其法律是局部绝对连续的一个频谱正$\alpha$-稳定的Levy过程的行程。这些生成$\mathbb{R}$-树是测度改变的$\alpha$-稳定树。在每一个这样的$\mathbb{R}$-树中，我们做一个随机数的顶点标识，其位置由一个辅助泊松过程确定。这概括的结果已经知道的情况下，度分布有一个有限的三阶矩（一个模型，位于相同的普遍性类的Erdensys-Renyi随机图）和其中的作用的$\alpha$-稳定的Levy过程是由布朗运动。
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive $\alpha$-stable Levy process. These spanning $\mathbb{R}$-trees are measure-changed $\alpha$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős--Renyi random graph) and where the role of the $\alpha$-stable Levy process is played by a Brownian motion.