Large nearest neighbour balls in hyperbolic stochastic geometry

Abstract

Consider a stationary Poisson process in a d-dimensional hyperbolic space. For R>0 define the point process xi _R^{(k)} of exceedance heights over a suitable threshold of the hyperbolic volumes of kth nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius R centred at a fixed point. The point process xi _R^{(k)} is compared to an inhomogeneous Poisson process on the real line with intensity function e^{-u} and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum kth nearest neighbour ball with a limiting Gumbel distribution is derived.