Tracy Widom law for sample covariance matrices under quadratic form concentration

arXiv:2610.41416v1 Announce Type: cross Abstract: We study sample covariance matrices formed from i.i.d., centered, isotropic random vectors $q_1, \dots, q_N \in \mathbb{R}^M$, whose coordinates are allowed to be dependent. We assume the random vectors satisfy a concentration bound for their quadratic forms $q_i^T A q_i$. Under this assumption, 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, our result removes the unconditionality assumption in existing edge universality results for log-concave ensembles~\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.