A strict undirected model for the $k$-nearest neighbour graph

Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each pair of points which are both $k$ nearest neighbours of each other. Then $G_{n,k}$ can be used as a model for wireless networks, and has some advantages in terms of applications over the two previous $k$-nearest neighbour models studied by Balister, Bollob\'{a}s, Sarkar and Walters, who proved good bounds on the connectivity models thresholds for both. However their proofs do not extend straightforwardly to this new model, since it is now possible for edges in different components of $G$ to cross. We get around these problems by proving that near the connectivity threshold, edges will not cross with high probability, and then prove that $G$ will be connected with high probability if $k>0.9684\log n$, which improves a bound for one of the models studied by Balister, Bollob\'{a}s, Sarkar and Walters too.