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,⌊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 probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).