The Wasserstein distance to the circular law

We investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the circular law. For Ginibre matrices, we obtain an optimal rate of convergence n-1/2 in 1-Wasserstein distance. This shows that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster (due to the repulsive behaviour) compared to that of i.i.d. points, which is known to include a logarithmic factor. For non-Gaussian entry distributions with finite moments, we also show that the rate of convergence nearly attains this optimal rate.