  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.
