论文标题
限制动态多参数简单复合物的拓扑不变的定理
Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex
论文作者
论文摘要
现有随机简单复合物的拓扑研究是非平凡的，已导致了几项开创性的作品。但是，此类研究的适用性受到限制，因为其随机性通常受单个参数约束。考虑到这一点，我们将重点放在最近提出的多参数随机简单复合物的拓扑上，更重要的是，我们在此处介绍了它的动态类似物。在这种动态设置中，简单的时间演变是由具有更新结构的固定和可能的非马克维亚过程确定的。集团复合物和外线雕刻复合物的动态版本是我们设置的特殊情况。我们的关键结果涉及特定维度占主导地位的面对面的制度。我们表明，与此维度相对应的BETTI数量和Euler特征满足了大数字和功能性中心极限定理的功能强度。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的非平凡维度中的动力学。
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.
