Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold

Let S_{n,k} denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k=k(n) points of the process nearest to it. In this paper we show that if Pr(S_{n,k} connected) > n^{-\gamma_1} then the probability that S_{n,k} contains a pair of `small' components `close' to each other is o(n^{-c_1}) (in a precise sense of `small' and 'close'), for some absolute constants \gamma_1>0 and c_1 >0. This answers a question of Walters. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of S_{n,k} below the connectivity threshold is asymptotically Poisson.