LIMITS OF MULTIPLICATIVE INHOMOGENEOUS RANDOM GRAPHS AND LEVY TREES: THE CONTINUUM GRAPHS

• 成果类型: Article
• 署名作者: Broutin, Nicolas; Duquesne, Thomas; Wang, Minmin
• 署名单位: Universite Paris Cite; Sorbonne Universite; University of Sussex
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1737
• 发表日期: 2022
• 页码: 2448-2503
• 关键词: scaling limits; UNIVERSALITY

摘要

Motivated by limits of critical inhomogeneous random graphs, we construct a family of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the Erdos-Renyi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton-Watson forest. The new representation allows us to demonstrate that a process that was already present in the pioneering work of Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59] about the multiplicative coalescent actually also essentially encodes the limiting metric. The discrete embedding of random graphs into a Galton-Watson forest is paralleled by an embedding of the encoding process into a Levy process which is crucial in proving the very existence of the local time functionals on which the metric is based; it also yields a transparent approach to compactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous multiplicative graphs are indeed the scaling limit of inhomogeneous random graphs.