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

We study expected Riesz ss-energies and linear statistics of some determinantal processes on the sphere SdSd. 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 SdSd. 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 22-energies of points defined through determinantal point processes associated with isotropic kernels. As a corollary we get that the Riesz 22-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 (in S2S2).