---
abstract: |
    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 Erdös-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from [@bourgade2018extreme] 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 [@lee2016tracy], some quantitative estimates also hold for more general separable sample covariance matrices $ X^* \Sigma X $ with general diagonal population $ \Sigma $.
address: 'Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012'
author:
- Haoyu Wang
bibliography:
- 'CovarianceTW.bib'
title: Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices
---

Introduction
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Overview and main results
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Edge universality of sample covariance matrices has been a classical problem in random matrix theory. It is well known that the distribution of the largest eigenvalue (after appropriate rescaling) converges to the Tracy-Widom 