Criticality of the Exponential Rate of Decay for the Largest Nearest Neighbor Link in Random Geometric Graph

Abstract

Let n points be placed independently in d-dimensional space according to the densities $f(x) = A_d e^{-λ\|x\|^α}, λ> 0, x \in \Re^d, d \geq 2.$ Let $d_n$ be the longest edge length for the nearest neighbor graph on these points. We show that $(\log(n))^{1-1/α}d_n -b_n$ converges weakly to the Gumbel distribution where $b_n \sim \log \log n.$ We also show that the strong law result, % \lim_{n \to \infty} \frac{(λ^{-1}\log(n))^{1-1/α}d_n}{\sqrt{\log \log n}} \to \frac{d}{αλ}, a.s. % Thus, the exponential rate of decay i.e. $α= 1$ is critical, in the sense that for $α> 1, d_n \to 0,$ where as $α< 1, d_n \to \infty$ a.s. as $n \to \infty.$