Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space

TL;DR

This paper establishes Gaussian limit theorems for geometric functionals of the nearest neighbour embracing graph constructed over Poisson points in Euclidean and hyperbolic spaces, revealing the probabilistic behavior of these structures.

Contribution

It provides the first quantitative Gaussian fluctuation results for the nearest neighbour embracing graph in both Euclidean and hyperbolic spaces.

Findings

Gaussian fluctuations for total edge length

Limit theorems for length-power functionals

Distributional results for vertex outdegree counts

Abstract

Consider a stationary Poisson process $\eta$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $\eta$ as follows. First, each point $x\in\eta$ is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until $x$ is contained in the convex hull of the points already connected to $x$. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered.