Gravitational allocation for uniform points on the sphere

Abstract

Given a collection $\mathcal{L}$ of $n$ points on a sphere $\mathbf{S}^{2}_{n}$ of surface area $n$, a fair allocation is a partition of the sphere into $n$ cells each of area $1$, and each associated with a distinct point of $\mathcal{L}$. We show that if the $n$ points are chosen uniformly at random and the partition is defined by considering a “gravitational” potential defined by the $n$ points, then the expected distance between a point on the sphere and the associated point of $\mathcal{L}$ is $O(\sqrt{\log n})$. We use our result to define a matching between two collections of $n$ independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is $O(\sqrt{\log n})$, which is optimal by a result of Ajtai, Komlós and Tusnády. Furthermore, we prove that the expected number of maxima for the gravitational potential is $\Theta (n/\log n)$. We also study gravitational allocation on the sphere to the zero set $\mathcal{L}$ of a particular Gaussian polynomial, and we quantify the repulsion between the points of $\mathcal{L}$ by proving that the expected distance between a point on the sphere and the associated point of $\mathcal{L}$ is $O(1)$.