THE STABLE GRAPH: THE METRIC SPACE SCALING LIMIT OF A CRITICAL RANDOM GRAPH WITH IID POWER-LAW DEGREES

• 成果类型: Article
• 署名作者: Conchon-Kerjan, Guillaume; Goldschmidt, Christina
• 署名单位: University of Bath; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1587
• 发表日期: 2023
• 页码: 1-69
• 关键词: continuum random trees; bounded-size rules; multiplicative coalescent; giant component; percolation

摘要

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 alpha + 1, where alpha & ISIN; (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 alpha-stable Levy process. These spanning R-trees are measure-changed alpha-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 Erdo?s-Renyi ran-dom graph) and where the role of the alpha-stable Levy process is played by a Brownian motion.