We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform meas
ure on the unit disc, both in expectation and almost surely. For $1 le p le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.
The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of r
andom Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.
Let $T$ be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of $T$ to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinat
e-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on some aspects of the result of Chatterjee and Ledoux.
