A critical constant for the k nearest neighbour model

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.