动态多参数单纯复形的拓扑不变量的极限定理
Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex
摘要：
现有的随机单纯复形的拓扑研究是不平凡的，并导致了一些开创性的工作。然而，这种研究的适用性是有限的，因为那里的随机性通常是由一个单一的参数。考虑到这一点，我们集中在这里的拓扑结构最近提出的多参数随机单纯复形，更重要的是，它的动态模拟，我们在这里介绍。在这个动态设置中，单形的时间演化是由固定的，可能是非马尔可夫过程的更新结构。集团复合体和线性网状复合体的动态版本是我们设置的特殊情况。我们的主要结果涉及的制度，在一个特定的维度的面孔计数占主导地位。证明了对应于该维数的贝蒂数和欧拉示性数线满足泛函强大数律和泛函中心极限定理.令人惊讶的是，在后一个结果中，限制高斯过程仅取决于最小非平凡维度中的动力学。
英文摘要：
Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex and, more importantly, of its dynamic analogue that we introduce here. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. The dynamic versions of the clique complex and the Linial-Meshulum complex are special cases of our setup. Our key result concerns the regime where face-counts of a particular dimension dominate. We show that the Betti numbers corresponding to this dimension and the Euler characteristic satisfy functional strong law of large numbers and functional central limit theorems. Surprisingly, in the latter result, the limiting Gaussian process depends only upon the dynamics in the smallest non-trivial dimension.
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