0 M ay 2 00 9 Criticality of the Exponential Rate of Decay for the Largest Nearest Neighbor

Let n points be placed independently in d−dimensional space according to the density f(x) = Ade −λ‖x‖α , λ > 0, x ∈ Rd, d ≥ 2. Let dn be the longest edge length of the nearest neighbor graph on these points. We show that (λ−1 log n)dn−bn converges weakly to the Gumbel distribution where bn ∼ (d−1) λα log log n. We also prove the following strong law result for the normalized nearest neighbor distance d̃n := (λ−1 log n) 1−1/α dn log log n . d− 1 αλ ≤ lim inf n→∞ d̃n ≤ lim sup n→∞ d̃n ≤ d αλ , almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that for α > 1, dn → 0, whereas for α ≤ 1, dn → ∞ a.s. as n → ∞. May 30, 2009 AMS 1991 subject classifications: Primary: 60D05, 60G70 Secondary: 05C05, 90C27