A Critical Constant for the k Nearest-Neighbour Model

Let P be a Poisson process of intensity one 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 logn⌋ is disconnected with probability tending to 1 as n → ∞, and for c > ccrit, Gn,⌊c logn⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = Θ(log n), and we proved in [1] that if k(n) ≤ 0.3043 logn then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139 logn then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a critical constant ccrit such that for c < ccrit, P(Gn,⌊c logn⌋ is connected) → 0 and for c > ccrit, P(Gn,⌊c logn⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture. University of Memphis, Department of Mathematics, 3725 Norriswood, Memphis, TN 38152, USA Trinity College, Cambridge CB2 1TQ, UK Peterhouse, Cambridge CB2 1RD