Central limit theorem for euclidean minimal spanning acycles

Abstract

In this paper, we investigate asymptotics for the minimal spanning acycles (MSAs) of the (Alpha)-Delaunay complex on a stationary Poisson process on ${ℝ}^d,d\ge 2$. MSAs are topological (or higher-dimensional) generalizations of minimal spanning trees. We establish a central limit theorem (CLT) for total weight of the MSA on a Poisson Alpha-Delaunay complex. Our approach also allows us to establish CLTs for the 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 MSAs which proceeds by establishing suitable chain maps and uses matroidal properties of MSAs. 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.