Covariance matrices of volume power functionals of random simplicial complexes -- an asymptotic analysis

Abstract

This work analyzes and compares the asymptotic properties of the covariance matrices of vectors of volume power functionals of random Vietoris-Rips complexes, as the intensity of the underlying homogeneous Poisson point process grows. Several key results are established which, in particular, generalize well-known facts on random graphs. Findings regarding rank, definiteness, determinant, eigenspaces, and related decompositions are presented within three distinct regimes. Moreover, we derive stochastic applications of these algebraic properties, leading to interesting results for vectors of volume power functionals.