The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees - Conchon-Kerjan, G; Goldschmidt, C (Institute of Mathematical Statistics, 2023)
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail be- haviour with exponent α + 1, where α ∈ (1,2). The limiting components are constructed from random R-trees encoded by the excursions above its run- ning infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive α-stable Lévy process. These spanning R-trees are measure-changed α-stable trees. In each such 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˝os–Rényi ran- dom graph) and where the role of the α-stable Lévy process is played by a Brownian motion.