Gaussian fluctuations for edge counts in high-dimensional random geometric graphs

TL;DR

This paper establishes a quantitative central limit theorem for the number of edges with midpoints in the unit ball in high-dimensional random geometric graphs generated by Poisson point processes, as both dimension and intensity grow large.

Contribution

It provides the first quantitative CLT for edge counts in high-dimensional random geometric graphs with simultaneous growth in dimension and intensity.

Findings

Edge count fluctuations converge to a Gaussian distribution in high dimensions.

Explicit bounds on the rate of convergence are derived.

Results extend understanding of geometric graph behavior in high-dimensional regimes.

Abstract

Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric graph. The number of edges of this graph is counted that have midpoint in the $d$-dimensional unit ball. A quantitative central limit theorem for this counting statistic is derived, as the space dimension $d$ and the intensity of the Poisson point process tend to infinity simultaneously.