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

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

2006-04-27

arxiv.org/abs/math/0604599v3

Mathematics

Probability

Communicated to 'Stochastic Processes and Their Applications'. Sep. 11, 2006: replaced paper uploaded on Apr. 27, 2006 by a co

Scientific paper

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 $n \to \infty.$

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