发表机构
Mathematical Science Center for Co-Creative Society, Tohoku University（东北大学共创社会数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于同调代数的方法，为泊松和吉布斯点过程构造的链复形推导持久Betti数的中心极限定理，并应用于多种同调情形。
AI 中文摘要
近年来，人们发展了各种持久不变量。本文中，我们发展了一种基于同调代数的方法，用于推导与在$\mathbb{R}^d$上单位强度的齐次泊松点过程构造的$\mathbb{R}$指标链复形相关的持久Betti数的中心极限定理。我们的方法也适用于吉布斯点过程。这种代数方法将所需条件的验证简化为检查链复形之间加一映射的核与余核的性质。作为应用，我们恢复了由单纯复形过滤产生的持久Betti数的已知中心极限定理。此外，我们证明了$\ell_p$-Vietoris-Rips单纯同调的持久Betti数的中心极限定理，包括随机几何图的模糊幅度同调。我们还建立了相对$\ell_p$-Vietoris-Rips同调的中心极限定理，特别是随机几何图的幅度Betti数。这些结果表明，我们的代数方法可以更广泛地应用于推导持久Betti数的中心极限定理。
英文摘要
Various persistent invariants have been developed in recent years. In this paper, we develop a method based on homological algebra for deriving central limit theorems for persistent Betti numbers associated with $\mathbb{R}$-indexed chain complexes constructed from a homogeneous Poisson point process of unit intensity on $\mathbb{R}^d$. Our method also applies to Gibbs point processes. This algebraic method reduces the verification of the required conditions to checking properties of the kernels and cokernels of add one maps between chain complexes. As an application, we recover the known central limit theorem for persistent Betti numbers arising from simplicial complex filtrations. In addition, we prove a central limit theorem for persistent Betti numbers of $\ell_p$-Vietoris-Rips simplicial homology, including blurred magnitude homology of random geometric graphs. We also establish central limit theorems for relative $\ell_p$-Vietoris-Rips homology and, in particular, for magnitude Betti numbers of random geometric graphs. These results suggest that our algebraic method can be applied more broadly to derive central limit theorems for persistent Betti numbers.
Comments22 pages
