Equidistribution of points in the Harmonic ensemble for the Wasserstein distance

We study the asymptotics of the expected Wasserstein distance between the empirical measure of a Determinantal Point Process and the background volume form. The main DPP studied is the harmonic ensemble, where we get the optimal rate of convergence for homogeneous manifolds of dimension d≥ 3, and for two-point homogeneous manifolds. We also discuss some variations of this process on the torus.