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 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 $d\geq 3$, and for two-point homogeneous manifolds. We also discuss some variations of this process on the torus. Regarding other point processes, we find the optimal rate for the spherical ensemble and the zeros of Gaussian Analytic Functions.