Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population

We consider sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2} X)^*$, where the sample X is an $M\times N$ random matrix whose entries are real independent random variables with variance 1/N and where $\Sigma$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both M and N tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $\Sigma$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that $\Sigma$ is diagonal and that the entries of X have a subexponential decay. We follow a new approach to the edge universality of deformed Wigner matrices introduced in [28].