Siblings in d-dimensional nearest neighbour trees

Abstract

Pick a sequence of uniform points on the $d$-dimensional sphere. Then, link the $n$th point to its closest one that arrives in the past. This constructs a labelled tree called the nearest neighbour tree on the $d$-dimensional sphere. These trees share some properties with the random recursive tree: the height of the last arrival node, the mean degree of the root, etc. On the contrary, the number of leaves seems to depend on dimension $d$, but no such properties have been proved yet. In this article, we prove that the mean number of siblings depends on $d$. In particular, we give explicit calculations of this number. In dimension $1$, it is $1 + \ln 2$ and, in any dimension $d$, it has an explicit integral form, but unfortunately, it does not give an explicit number. Nevertheless, we show that it converges to $2$ when $d \to \infty$ exponentially quick at a rate of $\sqrt{3}/2$. To prove these results, we look at the local limit of those trees and we do some fine computations about the intersection of two balls in dimension $d$. In particular, we obtain a non-trivial upper bound for those intersections in some precise cases.