The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees

Abstract

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 L\'evy 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 Erd\H{o}s--R\'enyi random graph) and where the role of the $\alpha$-stable L\'evy process is played by a Brownian motion.