Large nearest neighbour balls in hyperbolic stochastic geometry

Abstract

Consider a stationary Poisson process in a d-dimensional hyperbolic space. For $$R>0$$ R > 0 define the point process $$\xi _R^{(k)}$$ ξ 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 $$\xi _R^{(k)}$$ ξ R ( k ) is compared to an inhomogeneous Poisson process on the real line with intensity function $$e^{-u}$$ 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.