Siblings in d-dimensional nearest neighbour trees

TL;DR

This paper investigates the properties of nearest neighbour trees on the d-dimensional sphere, focusing on the mean number of siblings, providing explicit formulas, asymptotic behavior, and bounds related to the dimension.

Contribution

It introduces explicit calculations for the mean number of siblings in d-dimensional nearest neighbour trees and analyzes their asymptotic behavior as the dimension grows.

Findings

Mean number of siblings in 1D is 1 + ln 2

Explicit integral form for any dimension d

Convergence of the mean to 2 as d approaches infinity

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.