Microscopic Rates of Convergence for Hermitian Unitary Ensembles

Abstract: This paper provides microscopic rates of convergence (ROC) with respect to the $L^1$-Wasserstein distance for the eigenvalue determinantal point processes (DPPs) from the three major Hermitian unitary ensembles, the Gaussian Unitary Ensemble (GUE), the Laguerre Unitary Ensemble (LUE), and the Jacobi Unitary Ensemble (JUE) to their limiting point processes. We prove ROCs for the bulk of the GUE spectrum, the hard edge of the LUE spectrum, and the soft edges of the GUE, LUE, and JUE spectrums. These results are called microscopic because we are able to directly compare the point counts between the converging and limit DPPs, the most refined type of information possible in this situation. We are able to achieve these results by controlling the trace class norm of the integral operators determined by the DPP kernels.