论文标题
欧几里得最小跨越氧气的中心极限定理
Central Limit Theorem for Euclidean Minimal Spanning Acycles
论文作者
论文摘要
我们在$ \ mathbb {r}^d，d \ geq 2 $上，在固定的泊松过程中研究了（alpha）-delaunay综合体（alpha）-delaunay综合体的最小跨越氧气的渐近学。跨越最小的氧气是最小的跨越树木的拓扑（或更高维）的概括。我们建立了一个中心限制定理，用于在泊松 - 二奈尼综合体上最小的跨度弧形的总重量。我们的方法还使我们能够在Delaunay综合体的持续图中建立中央限制定理，以实现出生时间和寿命的总和。我们证明的关键是显示了最小跨越氧气的所谓弱稳定化，这些抗氧气是通过建立合适的链图并使用最小跨越抗体的矩形特性来进行的。与通过渗透理论估计值对欧几里得最小跨越树木的弱稳定的证明相反，我们的弱稳定性证明本质上是代数，即使在最小的跨越树木的情况下，也提供了另一种证据。
We investigate asymptotics for the minimal spanning acycles of the (Alpha)-Delaunay complex on a stationary Poisson process on $\mathbb{R}^d, d \geq 2$. Minimal spanning acycles are topological (or higher-dimensional) generalization of minimal spanning trees. We establish a central limit theorem for total weight of the minimal spanning acycle on a Poisson-Delaunay complex. Our approach also allows us to establish central limit theorems for sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of minimal spanning acycles which proceeds by establishing suitable chain maps and uses matroidal properties of minimal spanning acycles. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.
