Limits of random tree-like discrete structures

IF 1.3 Q2 STATISTICS & PROBABILITY Probability Surveys Pub Date : 2016-12-08 DOI:10.1214/19-ps338
Benedikt Stufler
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引用次数: 35

Abstract

We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits describing local convergence around fixed and random points in this general context, limit theorems for component sizes when $\mathcal{R}$ is a composite class, and a Gromov--Hausdorff scaling limit of random metric spaces patched together from independently drawn metrics on the $\mathcal{R}$-structures. Our main applications treat a selection of examples encompassed by this model. We consider random outerplanar maps sampled according to arbitrary weights assigned to their inner faces, and classify in complete generality distributional limits for both the asymptotic local behaviour near the root-edge and near a uniformly at random drawn vertex. We consider random connected graphs drawn according to weights assigned to their blocks and establish a Benjamini--Schramm limit. We also apply our framework to recover in a probabilistic way a central limit theorem for the size of the largest $2$-connected component in random graphs from planar-like classes. We prove Benjamini--Schramm convergence of random $k$-dimensional trees and establish both scaling limits and local weak limits for random planar maps drawn according to Boltzmann-weights assigned to their $2$-connected components.
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随机树状离散结构的极限
我们研究了一个随机$\mathcal{R}$-富树模型,该模型基于$\mathcal{R}$-结构的权重,并允许对大量随机离散结构进行统一处理。在这种情况下,我们建立了描述固定点和随机点周围局部收敛的分布极限,当$\mathcal{R}$是复合类时组件大小的极限定理,以及由$\mathcal{R}$-结构上独立绘制的度量拼接在一起的随机度量空间的Gromov—Hausdorff缩放极限。我们的主要应用程序处理这个模型所包含的一些示例。我们考虑随机的外平面映射,根据其内部面的任意权值进行抽样,并对根边附近的渐近局部行为和随机绘制的均匀顶点附近的渐近局部行为进行完全一般分布极限分类。我们考虑根据分配给其块的权重绘制的随机连接图,并建立Benjamini—Schramm极限。我们还应用我们的框架以概率的方式恢复了类平面随机图中最大$2$连通分量大小的中心极限定理。我们证明了随机$k$维树的Benjamini—Schramm收敛性,并建立了根据boltzmann权重分配给其$2$连通分量的随机平面图的缩放极限和局部弱极限。
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
期刊最新文献
Probabilistic representations of fragmentation equations L2-small ball asymptotics for Gaussian random functions: A survey Models of random subtrees of a graph Numerical methods for backward stochastic differential equations: A survey From Markov processes to semimartingales
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