Large nearest neighbour balls in hyperbolic stochastic geometry

TL;DR

This paper studies the behavior of the largest hyperbolic volumes of k-th nearest neighbor balls in a hyperbolic Poisson process, establishing a limit theorem with a Gumbel distribution.

Contribution

It introduces a new limit theorem for hyperbolic nearest neighbor volumes, linking them to an inhomogeneous Poisson process and deriving Gumbel distribution convergence.

Findings

Convergence of the point process to an inhomogeneous Poisson process

Quantitative limit theorem for maximum hyperbolic nearest neighbor volume

Gumbel distribution as the limit law for maxima

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 $k$th 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 $k$th nearest neighbour ball with a limiting Gumbel distribution is derived.