Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices

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. Combined with a result of Lee-Schnelli [26], some quantitative estimates also hold for more general separable sample covariance matrices $X^* \Sigma X$ with general diagonal population $\Sigma$.