Voronoi cells in random split trees

TL;DR

This paper investigates the size distribution of Voronoi cells in random split trees, revealing that one cell dominates while others are exponentially smaller, contrasting with uniform distribution results in other tree models.

Contribution

It demonstrates that in split trees, the largest Voronoi cell contains most vertices, with others being exponentially smaller, even under various modifications.

Findings

Largest Voronoi cell contains most vertices

Remaining cells are of order n*exp(-const*sqrt(log n))

Discrepancy persists with modifications

Abstract

We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $n\exp(-\mathrm{const}\sqrt{\log n})$. This discrepancy persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different "influence" parameters (called "speeds" in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.