A critical constant for the k nearest neighbour model

Details A critical constant for the k nearest neighbour model A critical constant for the k nearest neighbour model

2007-08-29

arxiv.org/abs/0708.4007v1

Mathematics

Probability

14 pages; 4 postscript figures

Scientific paper

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 connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.

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