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

Abstract

Let Sn;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 P(Sn;k connected) > n-γ11 then the probability that Sn;k contains a pair of 'small' components 'close' to each other is o(n-c1) (in a precise sense of 'small' and 'close'), for some absolute constants 1 > 0 and c1 > 0. This answers a question of Walters [13]. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of Sn;k below the connectivity threshold is asymptotically Poisson.