Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres

TL;DR

This paper analyzes the energy and statistical properties of determinantal point processes on high-dimensional spheres, showing optimality of the harmonic ensemble's Riesz 2-energy and comparing variances of linear statistics.

Contribution

It computes expected energies for spherical harmonic-based determinantal processes and proves their optimality among isotropic kernels, advancing understanding of point configurations on spheres.

Findings

Harmonic ensemble's Riesz 2-energy is optimal among isotropic kernels.

Improved estimates for minimal energy configurations on the sphere.

Variance analysis of linear statistics compared to spherical ensemble.

Abstract

We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on the sphere. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated to isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble.