论文标题
贝蒂（Betti）数量的大量均质和空间独立的随机简单复合物
Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes
论文作者
论文摘要
毛线壳复合物模型是Erdős-rényi图模型的天然高维类似物。近年来，Hinial和peled建立了对基础参数的适当缩放的BETTI数量的BETTI数量的限制定理。本文旨在将结果扩展到更全一随机的简单复杂模型。我们介绍了一类同质和空间独立的随机简单复合物，包括外线壳复合物模型和随机集团复杂模型作为特殊情况，我们研究了其贝蒂数字的渐近行为。此外，我们获得了其拉普拉斯主义者的经验谱分布的融合。该参数的关键要素是简单复合物的局部弱收敛性。受外线和沮丧的工作的启发，我们建立了均质和空间独立的随机简单复合物的局部弱极限定理。
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erdős-Rényi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate scaling of the underlying parameter. The present paper aims to extend that result to more-general random simplicial complex models. We introduce a class of homogeneous and spatially independent random simplicial complexes, including the Linial-Meshulam complex model and the random clique complex model as special cases, and we study the asymptotic behavior of their Betti numbers. Moreover, we obtain the convergence of the empirical spectral distributions of their Laplacians. A key element in the argument is the local weak convergence of simplicial complexes. Inspired by the work of Linial and Peled, we establish the local weak limit theorem for homogeneous and spatially independent random simplicial complexes.
