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

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.

33 pages