论文标题
乘法不均匀的随机图和Lévy树的限制：限制定理
Limits of multiplicative inhomogeneous random graphs and Lévy trees: Limit theorems
论文作者
论文摘要
我们考虑了一种不均匀随机图的自然模型，该模型扩展了经典的ERD \ H OS-Rényi图，并与Aldous [AOP 1997]指出，与乘法合并有密切的联系。在此模型中，将顶点分配给了其形成边缘趋势的权重。通过查看Aldous和Limic [EJP 1998]鉴定出乘法结合的入口边界，这是通过这些图的连接组件的质量（权重总和）的渐近分布，该分布与某些Lévy-type过程的外观长度密切相关。取而代之的是，我们查看这些组件的度量结构，并证明其Gromov-Hausdorff-Prokhorov收敛到一类随机紧凑的测量公制空间，这些公寓已在同伴论文中引入。我们的渐近方案直接与Aldous和Limic的工作中出现的一般收敛条件有关。我们的技术为这种一般的“关键”制度提供了一种统一的方法，并依赖于两种关键要素：通过某些Lévy过程对图的编码以及将其连接组件的嵌入到Galton-Watson Forests中。这种嵌入渐近嵌入极限物体中的嵌入到莱维树的森林中，这使我们能够从Lévy-type工艺的偏移中明确构造极限对象。电源结果与另一篇论文中的结果相结合，使我们能够扩展和补充几个通过政权特定证明获得的结果，例如：ERD \ H OS-Rényi随机图是通过Addario-berry，Goldschmidt和B. [Ptrf 2012]，Addario-berry-berry-berry-berry-grors-fore and the Assed as sen ass sen or n hamer case as sen or sen和whang the and the sen和wang，由Bhamidi，Sen和van der Hofstad [PTRF 2018]。
We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 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 [EJP 1998] have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-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. 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 Lévy 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 Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-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 regime-specific proofs, for instance: the case of Erd\H os-Rényi random graphs obtained by Addario-Berry, Goldschmidt and B. [PTRF 2012], the asymptotic homogeneous case as studied by Bhamidi, Sen and Wang [PTRF 2017], or the power-law case as considered by Bhamidi, Sen and van der Hofstad [PTRF 2018].
