Equidistribution and $\beta$ ensembles

TL;DR

This paper establishes the exact convergence rate of empirical measures in $eta$-ensembles, a class of random point processes on complex manifolds, with applications to matrix eigenvalue distributions.

Contribution

It provides the first precise quantification of convergence rates for $eta$-ensembles on complex manifolds, including special cases like the spherical ensemble.

Findings

Determined the convergence rate of empirical measures in $eta$-ensembles.

Applied results to eigenvalues of random matrices with Gaussian entries.

Extended understanding of random point processes on complex manifolds.

Abstract

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.