Central Limit Theorem for Euclidean Minimal Spanning Acycles

TL;DR

This paper proves a central limit theorem for the total weight of minimal spanning acycles in Euclidean Delaunay complexes, extending classical results to higher dimensions and topological structures.

Contribution

It introduces an algebraic approach to establish weak stabilization for minimal spanning acycles, enabling CLT proofs in higher-dimensional topological complexes.

Findings

Established CLT for total weight of minimal spanning acycles

Proved CLT for sum of birth times and lifetimes in persistent diagrams

Provided an algebraic alternative to percolation-based stabilization proofs

Abstract

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.