Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

Abstract

Let n points be placed independently in d-dimensional space according to the density f(x) = A_de^-λ||x||α, λ, α > 0, x ∈ R^d, d ≥ 2. Let d_n be the longest edge length of the nearest-neighbor graph on these points. We show that (λ^-1logn)^1-1/αd_n - b_n converges weakly to the Gumbel distribution, where b_n ∼ ((d - 1)/λα)log logn. We also prove the following strong law for the normalized nearest-neighbor distance d̃_n = (λ^-1logn)^1-1/αd_n/log log n: (d - 1)/αλ ≤ lim inf_n→∞d̃_n ≤ lim sup_n→∞d̃_n ≤ d/αλ almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that, for α > 1, d_n → 0, whereas, for α ≤ 1, d_n → ∞ almost surely as n → ∞.