Edge Universality of Correlation Matrices

Let $\widetilde X_{M\times N}$ be a rectangular data matrix with entries with independent real valued entries $[\widetilde x_{ij}]$ satisfying $\mathbb E \,\widetilde{x}_{ij} = 0$ and $\mathbb E \,\widetilde{x}^2_{ij} = {1 \over M}$, $N, M\to \infty$ and these entries have a sub-exponential decay at the tails. We will be working in the regime $N/M = d_N \in (0,\infty), \lim_{N\to \infty}d_N \neq 1$. In this paper we prove the edge universality of correlation matrices $X^\dagger X$, where the rectangular matrix $X$ is obtained by normalizing each column of the data matrix $\widetilde X$ by its Euclidean norm. Our main result states that, asymptotically the $k$-point ($k \geq 1$) correlation functions of the extreme eigenvalues (at both edges of the spectrum) of the correlation matrix $X^\dagger X$ converge to those of the Gaussian correlation matrix, i.e., Tracy-Widom law, and thus in particular, the largest and the smallest eigenvalue of $X^\dagger X$ after appropriate centering and rescaling converges to the Tracy-Widom distribution. The proof is based on the comparison of Green functions, but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix. We achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix. Our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest.