Poisson and Gaussian fluctuations for the $\mathbf{f}$-vector of high-dimensional random simplicial complexes

TL;DR

This paper studies the asymptotic distribution of the face counts in high-dimensional random Vietoris-Rips complexes generated over Poisson point processes, revealing Poisson and Gaussian fluctuation behaviors as dimensions grow.

Contribution

It provides a detailed analysis of the limiting distributional behavior of the -vector in high-dimensional random complexes, a novel contribution to topological data analysis.

Findings

Identification of Poisson fluctuations in face counts.

Discovery of Gaussian fluctuations under certain conditions.

Asymptotic behavior as dimension and intensity tend to infinity.

Abstract

We investigate the high-dimensional asymptotic distributional behavior of the $\mathbf{f}$-vector of a random Vietoris-Rips complex, that is generated over a stationary Poisson point process in $[-\frac{1}{2}, \frac{1}{2}]^d$ as the space dimension and the intensity tend to infinity while the radius parameter tends to zero simultaneously.