  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.
