Explicit laws of large numbers for random nearest-neighbour type graphs

Abstract

Under the unifying umbrella of a general result of Penrose & Yukich [Ann. Appl. Probab., (2003) 13, 277--303] we give laws of large numbers (in the $L^p$ sense) for the total power-weighted length of several nearest-neighbour type graphs on random point sets in $\R^d$, $d\in\N$. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.