k-Nearest-Neighbor Clustering and Percolation Theory

Abstract Let P be a realization of a homogeneous Poisson point process in ℝd with density 1. We prove that there exists a constant kd, 1<kd<∞, such that the k-nearest neighborhood graph of P has an infinite connected component with probability 1 when k≥kd. In particular, we prove that k2≤213. Our analysis establishes and exploits a close connection between the k-nearest neighborhood graphs of a Poisson point set and classical percolation theory. We give simulation results which suggest k2=3. We also obtain similar results for finite random point sets.

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