Large deviations for hyperbolic $k$-nearest neighbor balls

TL;DR

This paper establishes a large deviation principle for the distribution of large $k$-nearest neighbor balls in hyperbolic space, providing a probabilistic framework for rare events in geometric point processes.

Contribution

It introduces a large deviation principle for Poisson $k$-nearest neighbor balls in hyperbolic space, with a novel coarse-graining technique for the proof.

Findings

Large deviation principle for exceedances of $k$-nearest neighbor volumes.

Rate function expressed via relative entropy.

Method applicable to hyperbolic geometric probability models.

Abstract

We prove a large deviation principle for the point process of large Poisson $k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a stationary Poisson point process of unit intensity in a growing sampling window in hyperbolic space. We further take a growing sequence of thresholds such that there is a diverging expected number of Poisson points whose $k$-nearest neighbor ball has a volume exceeding this threshold. Then, the point process of exceedances satisfies a large deviation principle whose rate function is described in terms of a relative entropy. The proof relies on a fine coarse-graining technique such that inside the resulting blocks the exceedances are approximated by independent Poisson point processes.