2006/04/27 by Bhupendra Gupta, Gupta, Bhupendra, Srikanth K. Iyer +1
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.math/0604599
Communicated to 'Stochastic Processes and Their Applications'. Sep. 11, 2006: replaced paper uploaded on Apr. 27, 2006 by a corrected version; errors/corrections found by the authors themselves
openalex publication_date 2006/04/27 · arxiv created 2009/05/30 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Let n points be placed independently in d-dimensional space according to the densities f(x) = Ad e-λ‖x‖α, λ> 0, x ∈ \Red, d ≥ 2. Let dn be the longest edge length for the nearest neighbor graph on these points. We show that (log(n))1-1/αdn -bn converges weakly to the Gumbel distribution where bn ∼ log log n. We also show that the strong law result, % limn → ∞ \frac(λ-1log(n))1-1/αdn√(log log n) → (d)/(αλ), a.s. % Thus, the exponential rate of decay i.e. α= 1 is critical, in the sense that for α> 1, dn → 0, where as α< 1, dn → ∞ a.s. as n → ∞.