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 , where . The limiting components are constructed from random -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 α-stable Lévy process. These spanning -trees are measure-changed α-stable trees. In each such -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–Rényi random graph) and where the role of the α-stable Lévy process is played by a Brownian motion.
C.G.’s research was supported by EPSRC Fellowship EP/N004833/1.
G.C. would like to thank Thomas Duquesne and Igor Kortchemski for useful discussions. C.G. is very grateful to James Martin and Jon Warren for very helpful discussions, to Serte Donderwinkel for her careful reading of the paper, which resulted in many improvements, especially in the proof of Proposition 5.3, and to Zheneng Xie for an improved proof of Lemma 4.8. She would also like to thank Robin Stephenson, Jean Bertoin, Juan Carlos Pardo Millán, Andreas Kyprianou and Víctor Rivero Mercado for advice and discussions. Work on this paper was considerably facilitated by professeur invité positions at LIX, École polytechnique in November 2016 and November 2017 and at LAGA, Université Paris 13 in September 2018. C.G. would like to thank Marie Albenque and LIX, and Bénédicte Haas and LAGA for their hospitality and the ANR GRAAL for funding. Both authors would like to express their gratitude to the referee for their careful and insightful reading of the paper, and for comments, which led to many improvements.
"The stable graph: The metric space scaling limit of a critical random graph with i.i.d. power-law degrees." Ann. Probab. 51 (1) 1 - 69, January 2023. https://doi.org/10.1214/22-AOP1587