Poisson fluctuations for edge counts in high-dimensional random geometric graphs

Abstract: 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 (2016). As an application of this Poisson limit theorem, we consider a stationary Poisson point process in $\mathbb{R}^d$ 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, showing that the phase transition phenomenon holds also in the high-dimensional set-up.