Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices

Understand

We prove the first explicit rate of convergence to the Tracy-Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable.

• Our primary focus is matrices of type $ X^*X $ and the proof follows the Erd\"{o}s-Schlein-Yau dynamical method.
• We use a recent approach to the analysis of the Dyson Brownian motion from [5] to obtain a quantitative error estimate for the local relaxation flow at the edge.
• Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence.