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

Abstract

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 $\mathbb{P}(S_{n,k} \textrm{ connected})>n^{-\gamma_1}$ 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.