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

Published online by Cambridge University Press:  01 July 2016


Bhupender Gupta
Affiliation:
Indian Institute of Information Technology
Srikanth K. Iyer
Affiliation:
Indian Institute of Science
Corresponding
E-mail address:

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 n = (λ−1 log n)1−1/α d n / log log n: (d − 1)/αλ ≤ lim inf n→∞ n ≤ lim sup n→∞ 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 → ∞.


Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported in part by UGC SAP IV and a grant from the DRDO-IISc program on Mathematical Engineering.


References

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