The spherical ensemble and uniform distribution of points on the sphere

TL;DR

This paper analyzes the spherical ensemble, a rotation-invariant point process on the sphere, demonstrating its effective uniform distribution properties and local repelling behavior, comparable or superior to existing deterministic point arrangements.

Contribution

The paper provides bounds and asymptotic analysis of the spherical ensemble's uniformity metrics, showing its competitive or superior distribution quality compared to deterministic methods.

Findings

The spherical ensemble exhibits strong local repelling properties.

It achieves near-optimal discrepancy bounds for random point distributions.

It outperforms some deterministic configurations in Riesz energy measures.

Abstract

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of an arrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).