  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 $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$,
and the so-called spherical ensemble on $\mathbb{S}^2$, which originates in
random matrix theory. We extend results of Beltr\'an, Marzo and Ortega-Cerd\`a
on the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime
$s<0$, and as a corollary find the expected value of the spherical cap $L^2$
discrepancy via the Stolarsky invariance principle. We find the expected value
of the $L^2$ discrepancy with respect to axis-parallel boxes and Euclidean
balls of the harmonic ensemble on $\mathbb{T}^d$. We also show that the
spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and
$\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ in
expectation in the Wasserstein metric $W_2$, in contrast to i.i.d. random
points, which are known to lose a factor of $(\log N)^{1/2}$.
