Percolation in the k-nearest neighbor graph

Let P be a Poisson process of intensity one in R2. For a fixed integer k, join every point of P to its k nearest neighbors, creating a directed random geometric graph ~ Gk(R). We prove bounds on the values of k that, almost surely, result in an infinite connected component in ~ Gk(R) for various definitions of “component”. We also give high confidence results for the exact values of k needed. In particular, for percolation on the underlying (undirected) graph of ~ Gk(R), we prove that k = 11 is sufficient, and show with high confidence that k = 3 is the actual threshold for percolation.