The mean distance to the nth neighbour in a uniform distribution of random points: an application of probability theory

We study different ways of determining the mean distance <rn> between a reference point and its nth neighbour among random points distributed with uniform density in a D-dimensional Euclidean space. First, we present a heuristic method; though this method provides only a crude mathematical result, it shows a simple way of estimating <rn>. Next, we describe two alternative means of deriving the exact expression of <rn> we review the method using absolute probability and develop an alternative method using conditional probability. Finally, we obtain an approximation to <rn> from the mean volume between the reference point and its nth neighbour and compare it with the heuristic and exact results