Limits of multiplicative inhomogeneous random graphs and Levy trees: limit theorems

• 成果类型: Article
• 署名作者: Broutin, Nicolas; Duquesne, Thomas; Wang, Minmin
• 署名单位: Sorbonne Universite; University of Sussex
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01075-z
• 发表日期: 2021
• 页码: 865-973
• 关键词: scaling limits; UNIVERSALITY

摘要

We consider a natural model of inhomogeneous random graphs that extends the classical Erdos-Renyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812-854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1-59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Levy-type processes. We, instead, look at the metric structure of these components and prove their Gromov-Hausdorff-Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Levy trees: the continuum graphs. , 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general critical regime, and relies upon two key ingredients: an encoding of the graph by some Levy process as well as an embedding of its connected components into Galton-Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Levy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Levy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdos-Renyi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367-406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565-641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387-474, 2018).