Universality of Covariance Matrices

We prove the universality of covariance matrices of the form $H_{N \times N} = {1 \over N} \tp{X}X$ where $[X]_{M \times N}$ is a rectangular matrix with independent real valued entries $[x_{ij}]$ satisfying $\E \,x_{ij} = 0$ and $\E \,x^2_{ij} = {1 \over M}$, $N, M\to \infty$. Furthermore it is assumed that these entries have sub-exponential tails. We will study the asymptotics in the regime $N/M = d_N \in (0,\infty), \lim_{N\to \infty}d_N \neq 1$. Our main result states that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Marcenko-Pastur (MP) law uniformly with an error of order $(N \eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter. From this strong local MP law, we derive the following results. 1. The \emph{rigidity of eigenvalues}: If $\gamma_j$ denotes the {\it classical location} of the $j$-th e.v. under the MP law ordered in increasing order, then the $j$-th eigenvalue $\lambda_j$ of $H$ is close to $\gamma_j$ such that, $$ P(\exists j: |\lambda_j-\gamma_j| \ge (\log N)^{C\log\log N} [\min \big (\,\min(N,M) - j,j \big) ]^{-1/3} N^{-2/3}) \le C\exp{\big[-(\log N)^{c\log\log N} \big]} $$ 2. The delocalization of the eigenvectors of the matrix $X\tp{X}$ uniformly both at the edge and the bulk. 3. Bulk universality, i.e., $n$-point correlation functions of the e.v. of $\tp{X}X$ coincide with those of the Wishart ensemble, when $N$ goes to infinity. 4. Universality of the eigenvalues of the sample covariance matrix $\tp{X}X$ at \emph{both} edges of the spectrum. Furthermore the first two results are applicable even in the case in which the entries of the column vectors of $X$ are not independent but satisfy a certain large deviation principle. All our results hold for both real and complex valued entries.