临界区欧拉示性数过程的泛函极限定理
Functional limit theorems for the Euler characteristic process in the critical regime
摘要：
本文给出了Vietoris-Rips复形的欧拉示性数的泛函极限定理。这些点是从$\mathbb{R}^d$上的非齐次泊松过程中得到的，控制单形形成的连通半径是时间参数$t$的函数，这使得我们可以把欧拉示性数看作一个随机过程.发生这种情况的背景是临界状态，其中单纯复形是高度连通的，并且具有非平凡拓扑。我们建立了两个“功能级”的极限定理，一个强大数律和一个中心极限定理适当规范的欧拉示性数过程。
英文摘要：
This study presents functional limit theorems for the Euler characteristic of Vietoris-Rips complexes. The points are drawn from a non-homogeneous Poisson process on $\mathbb{R}^d$, and the connectivity radius governing the formation of simplices is taken as a function of time parameter $t$, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have non-trivial topology. We establish two "functional-level" limit theorems, a strong law of large numbers and a central limit theorem for the appropriately normalized Euler characteristic process.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
