Gaussian and non-Gaussian limits for the total edge-length of random directional nearest-neighbour graphs

We study planar random spatial graphs defined by inserting an edge between every point of a homogeneous Poisson process on the unit square and a nearest-neighbour within a fixed cone of directions (if there is such a point). We prove large-sample distributional limit theorems for the total Euclidean edge length of the graph, where the limit distribution is either Gaussian (due to contributions from the bulk) or, when one or both of the boundaries of the cone of directions align with the edges of the square, a convolution of Gaussian with a non-Gaussian law arising from boundary effects due to long edges near a boundary. This work extends previously known cases, including the ordinary nearest-neighbour graph (full cone of directions), where the Gaussian limit theorem was obtained by Bickel & Breiman and Avram & Bertsimas, and Bhatt & Roy's minimal directed spanning forest (with 'south-west' directions), where the non-Gaussian limit theorem was obtained by Penrose & Wade.