  We find the precise rate at which the empirical measure associated to a
$\beta$-ensemble converges to its limiting measure. In our setting the
$\beta$-ensemble is a random point process on a compact complex manifolds
distributed according to the $\beta$ power of a determinant of sections in a
positive line bundle. A particular case is the spherical ensemble of
generalized random eigenvalues of pairs of matrices with independent
identically distributed Gaussian entries.
