Quantitative equidistribution of eigenvalues of Random Normal Matrices in the Wasserstein distance

The object of study in this paper is the expected $2$-Wasserstein distance between the empirical measures of several point processes and their respective limit.

For this, the main tool developed is a smoothing procedure in Euclidean spaces using the heat equation with Neumann boundary conditions.

It is applied to the spectrum of Random Normal Matrices with \textit{reasonable} assumptions on the potential, as well as to Hyperuniform Point Processes such as the infinite Ginibre ensemble and the zero set of the planar Gaussian Analytic Function.

In both of these cases, the technique obtains the optimal rate of convergence.