A critical constant for the k nearest neighbour model

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Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c'<c, G_{n,c'log n} is disconnected with probability tending to 1 as n tends to infinity, and for c'>c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.
14 pages; 4 postscript figures

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Probability, Combinatorics, 60K35; 82B43

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http://salesiana.dossiersoluciones.com/handle/123456789/136939

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0708.4007
 http://arxiv.org/abs/0708.4007

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arXiv