The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory

Details The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory

2002-12-17

arxiv.org/abs/math/0212230v2

Mathematics

Probability

6 pages (REVTex4), minor changes in content, typing errors corrected, references added

Scientific paper

We study different ways of determining the mean distance $ < r_n >$ between a reference point and its $n$-th 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 $ < r_n >$. Next we describe two alternative means of deriving the exact expression of $$: we review the method using absolute probability and develop an alternative method using conditional probability. Finally we obtain an approximation to $ < r_n >$ from the mean volume between the reference point and its $n$-th neighbour and compare it with the heuristic and exact results.

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