Multivariate Central Limit Theorems for Random Simplicial Complexes

TL;DR

This paper establishes multivariate central limit theorems for volume-based functionals of Vietoris-Rips and Čech complexes built over Poisson point processes, advancing understanding of their asymptotic distributions.

Contribution

It introduces novel multivariate CLTs for volume functionals of random simplicial complexes, extending previous univariate results and covering both Vietoris-Rips and Čech complexes.

Findings

Proves multivariate CLTs for volume functionals as intensity increases.

Provides asymptotic normality results for both Vietoris-Rips and Čech complexes.

Extends univariate CLTs to multivariate settings for random simplicial complexes.

Abstract

Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most $\delta$. Summing powers of the volume of all $k$-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips complex is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. Univariate and multivariate central limit theorems are proven. Analogous results for the \v{C}ech complex are given.