Limit distributions of the threshold radius for the maximum degree and the associated point configurations in random geometric graphs

TL;DR

This paper studies the limiting distribution of the threshold radius in random geometric graphs where the maximum degree exceeds a certain value, revealing different behaviors depending on the degree growth rate.

Contribution

It provides a detailed analysis of the limit distributions of the threshold radius and the point configurations associated with maximum degree in random geometric graphs.

Findings

Limit distribution exhibits compound Poisson behavior when maximum degree is bounded.

Poisson behavior emerges when maximum degree diverges slowly than log n.

The study characterizes the asymptotic behavior of the vertices with maximum degree.

Abstract

A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate the limit distribution of the threshold radius for which the maximum degree of the graph is at least a given value that depends on $n$. In addition, given the radii $(r_n)_{n \in \mathbf{N}}$, we examine the limiting behavior of the point process formed by the vertices that achieve the maximum degree. Roughly speaking, the limiting process exhibits a compound Poisson behavior in the regime where the maximum degree remains bounded, due to local geometric dependencies, whereas it exhibits a Poisson behavior in the regime where the maximum degree diverges more slowly than $\log n$.