Mesoscopic Rates of Convergence for Complex Wishart Matrices at the Leftmost Spectrum Edge

Abstract

This paper establishes mesoscopic rates of convergence in the $L^1$-Wasserstein distance for eigenvalue determinantal point processes (DPPs) derived from the Laguerre Unitary Ensemble (LUE) to the corresponding limiting point process (Airy process) as the dimension goes to infinity. Specifically, we prove convergence rates at the leftmost edge of the LUE spectrum, which corresponds to the least eigenvalue. These results are termed mesoscopic because they allow for a direct comparison of point counts between the convergent DPPs and their limits over a range of scales.