We prove a Poisson limit theorem in the total variation distance of functionals of a general Poisson point process using the Malliavin-Stein method. Our estimates only involve first and second order difference operators and are closely related to the corresponding bounds for the normal approximation in the Wasserstein distance by Last, Peccati and Schulte, see [LPS16]. As an application of this Poisson limit theorem, we consider a stationary Poisson point process in R d and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have a midpoint in the d-dimensional unit ball. A quantitative Poisson limit theorem for this counting statistic is derived, as the space dimension d and the intensity of the Poisson point process tend to infinity simultaneously, extending our previous work, [GT16] where we derived a central limit theorem, showing that the phase transition phenomenon holds also in the high-dimensional set-up.

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

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that -up to homotopy equivalence -it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.