Limit laws for longest edges in empty region graphs

TL;DR

This paper studies the asymptotic distribution of long edges in empty region graphs generated by a Poisson process, showing convergence to Poisson and Gumbel distributions with explicit error bounds.

Contribution

It establishes the limiting distribution of long edges in various empty region graphs and provides explicit error bounds for the convergence.

Findings

Edge midpoint process converges to a Poisson process in distribution.

Longest edge length converges to a Gumbel distribution.

Results apply to multiple types of empty region graphs, including Gabriel and relative neighbourhood graphs.

Abstract

Empty region graphs are graphs whose vertices are points in $\mathbb{R}^d$ and where two vertices are connected by an edge whenever some associated region does not contain any other vertices. We investigate the asymptotic behaviour of long edges in empty region graphs generated by a stationary Poisson process in $\mathbb{R}^d$. {Letting} the intensity of the underlying Poisson process tend to infinity, we consider the associated point process of edge midpoints, suitably transformed edge lengths, and directions of the edges. We prove that it converges in distribution to a Poisson process on $\mathbb{R}^d \times \mathbb{R}\times\mathbb{L}^d$, where $\mathbb{L}^d$ is the space of lines in $\mathbb{R}^d$ through the origin, and that the suitably transformed length of the longest edge with midpoint in an observation window converges in distribution to a Gumbel distributed random variable. Our approach yields explicit error bounds in Kantorovich--Rubinstein distance for the point process convergence {when restricting to an observation window} and in Kolmogorov distance for the maximal edge length. The results apply uniformly to a broad class of empty region graphs, including the Gabriel graph, the relative neighbourhood graph, the beta-skeleton graph, the Mastercard graph, and the Pacman graph.