  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.
