Nearest neighbours in a Poisson ensemble

Abstract

Every point of a Poisson ensemble has a unique nearest neighbour. However, every point is not necessarily the nearest neighbour of precisely one other point. The problem considered here is that of determining the probability p n that a point of a Poisson ensemble is the nearest neighbour of precisely n other points. The problem is discussed for Poisson ensembles in 1, 2 and 3 dimensions. Except in the one-dimensional case 1 p n has not been found analytically. Some bounds on p n are provided and estimates of p n are obtained by a Monte Carlotechnique.