A critical constant for the k nearest-neighbour model

Abstract

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).