Equidistribution and $β$ ensembles

Abstract

We find the precise rate at which the empirical measure associated to a $β$-ensemble converges to its limiting measure. In our setting the $β$-ensemble is a random point process on a compact complex manifolds distributed according to the $β$ 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.