A critical constant for the k-nearest neighbour model

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, ⌊c log ⁡n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c>c_crit, G_n ⌊c log ⁡n⌋ is connected with probablity tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).