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 d e−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−1 log n)1−1/α d n - b n converges weakly to the Gumbel distribution, where b n ∼ ((d − 1)/λα) log log n. We also prove the following strong law for the normalized nearest-neighbor distance d̃ n = (λ−1 log n)1−1/α d 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 → ∞.