Tracy-Widom law for sample covariance matrices under quadratic form concentration

Abstract

We study sample covariance matrices formed from vectors $q_i = \Sigma^{1/2}u_i \in \mathbb{R}^M$, $1 \leq i \leq N$, where $u_1, \ldots, u_N$ are independent, centered, isotropic random vectors whose coordinates may not be independent. We assume that the random vectors $q_i$ satisfy a concentration bound for their quadratic forms $q_i^T A q_i$. Under this assumption and a regular right edge condition on $\Sigma$, we establish edge universality and obtain the Tracy-Widom law for the largest eigenvalue of the sample covariance matrix after appropriate centering and scaling. In particular, for log-concave ensembles, our result removes the unconditionality assumption imposed in existing edge universality results~\cite{BX}, resolving the question that originally motivated our work. The concentration assumption also allows us to extend the result beyond the log-concave class.