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