A critical constant for the k nearest-neighbour model

Abstract

Let 𝓟 be a Poisson process of intensity 1 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_crit such that, for c‹c_crit, G_{n,⌊clogn⌋} is disconnected with probability tending to 1 as n →∞ and, for c‹c_crit, G_{n,⌊clogn⌋} is connected with probability tending to 1 as n →∞. This answers a question posed in Balister et al. (2005).