Riesz energy, L2 discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere (Formula presented.) and the flat torus (Formula presented.), and the so-called spherical ensemble on (Formula presented.), which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega-Cerdà on the Riesz (Formula presented.) -energy of the harmonic ensemble to the nonsingular regime (Formula presented.), and as a corollary find the expected value of the spherical cap (Formula presented.) discrepancy via the Stolarsky invariance principle. We find the expected value of the (Formula presented.) discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on (Formula presented.). We also show that the spherical ensemble and the harmonic ensemble on (Formula presented.) and (Formula presented.) with (Formula presented.) points attain the optimal rate (Formula presented.) in expectation in the Wasserstein metric (Formula presented.), in contrast to independent and identically distributed random points, which are known to lose a factor of (Formula presented.).