Riesz energy, L2$L^2$ 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 Sd$\mathbb {S}^d$ and the flat torus Td$\mathbb {T}^d$ , and the so‐called spherical ensemble on S2$\mathbb {S}^2$ , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz s$s$ ‐energy of the harmonic ensemble to the nonsingular regime s<0$s<0$ , and as a corollary find the expected value of the spherical cap L2$L^2$ discrepancy via the Stolarsky invariance principle. We find the expected value of the L2$L^2$ discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on Td$\mathbb {T}^d$ . We also show that the spherical ensemble and the harmonic ensemble on S2$\mathbb {S}^2$ and T2$\mathbb {T}^2$ with N$N$ points attain the optimal rate N−1/2$N^{-1/2}$ in expectation in the Wasserstein metric W2$W_2$ , in contrast to independent and identically distributed random points, which are known to lose a factor of (logN)1/2$(\log N)^{1/2}$ .