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

Gupta, Bhupender and Iyer, Srikanth K (2010) Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs. In: Advances in Applied Probability, 42 (3). pp. 631-658.

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## Abstract

Let n points be placed independently in d-dimensional space according to the density f(x) = A(d)e(-lambda parallel to x parallel to alpha), lambda, alpha > 0, x is an element of R-d, d >= 2. Let d(n) be the longest edge length of the nearest-neighbor graph on these points. We show that (lambda(-1) log n)(1-1/alpha) d(n) - b(n) converges weakly to the Gumbel distribution, where b(n) similar to ((d - 1)/lambda alpha) log log n. We also prove the following strong law for the normalized nearest-neighbor distance (d) over tilde (n) = (lambda(-1) log n)(1-1/alpha) d(n)/log log n: (d - 1)/alpha lambda <= lim inf(n ->infinity) (d) over tilde (n) <= lim sup(n ->infinity) (d) over tilde (n) <= d/alpha lambda almost surely. Thus, the exponential rate of decay alpha = 1 is critical, in the sense that, for alpha > 1, d(n) -> 0, whereas, for alpha <= 1, d(n) -> infinity almost surely as n -> infinity.

Item Type: Journal Article Copyright of this article belongs to Applied Probability Trust. Random geometric graph; nearest-neighbor graph; Poisson point process; largest nearest-neighbor link; vertex degree Division of Physical & Mathematical Sciences > Mathematics Id for Latest eprints 04 Nov 2010 11:41 04 Nov 2010 11:41 http://eprints.iisc.ac.in/id/eprint/33582

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