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

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 $\log n$.