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

TL;DR

This paper proves mesoscopic convergence rates in Wasserstein distance for eigenvalue processes at the spectrum edge of complex Wishart matrices, specifically the Laguerre Unitary Ensemble, as the matrix dimension grows.

Contribution

It establishes explicit convergence rates at the spectrum's edge for eigenvalue point processes, advancing understanding of their asymptotic behavior.

Findings

Convergence rates are established in Wasserstein distance at the spectrum edge.

Results apply to eigenvalue point processes from the Laguerre Unitary Ensemble.

The work enables comparison of eigenvalue counts over various scales.

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.