Details

Autor(en) / Beteiligte

• Gupta, Bhupender
• Iyer, Srikanth K.

Titel

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

Ist Teil von

• Advances in applied probability, 2010-09, Vol.42 (3), p.631-658

Erscheinungsjahr

2010

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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 → ∞.