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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*
Indian Institute of Information Technology
Srikanth K. Iyer*
Indian Institute of Science
Postal address: Department of Computer Science and Engineering, Indian Institute of Information Technology, Jabalpur 482011, India.
∗∗ Postal address: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Email address:
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Let n points be placed independently in d-dimensional space according to the density f(x) = Ade−λ||x||α, λ, α > 0, x ∈ ℝd, d ≥ 2. Let dn be the longest edge length of the nearest-neighbor graph on these points. We show that (λ−1 log n)1−1/α dn - bn converges weakly to the Gumbel distribution, where bn ∼ ((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 dn/ log log n: (d − 1)/αλ ≤ lim infn→∞n ≤ lim supn→∞nd/αλ almost surely. Thus, the exponential rate of decay α = 1 is critical, in the sense that, for α > 1, dn → 0, whereas, for α ≤ 1, dn → ∞ almost surely as n → ∞.

Stochastic Geometry and Statistical Applications
Copyright © Applied Probability Trust 2010 


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


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