Large deviations for the volume of $k$-nearest neighbor balls

TL;DR

This paper establishes large deviation principles for the Euclidean volume of k-nearest neighbor balls in random point processes, revealing different behaviors depending on the growth rate of centering terms.

Contribution

It introduces two types of large deviation results for k-nearest neighbor volumes, expanding understanding of their probabilistic behavior in geometric point processes.

Findings

Donsker-Varadhan large deviation principle derived for slow-growing centering terms.

Large deviations in the al topology for fast-growing centering terms.

Application to large deviations in the degree distribution of geometric graphs.

Abstract

This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.