We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W \sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.

In this paper, we prove a CLT for the sample canonical correlation coefficients between two high-dimensional random vectors with finite rank correlations. More precisely, consider two random vectors $\wt{\bx}=\mathbf x + A \mathbf z $ and $\wt{\by}=\mathbf y + B \mathbf z $, where $\mathbf x \in \R^p$, $\mathbf y \in \R^q$ and $\mathbf z\in \R^r$ are independent random vectors with i.i.d.\;entries of mean zero and variance one, and $A \in \R^{p\times r}$ and $B\in \R^{q\times r}$ are two arbitrary deterministic matrices. Given $n$ samples of $\wt{\bx}$ and $\wt{\by}$, we stack them into two matrices $\cal X= X+AZ$ and $\cal Y= Y+BZ$, where $X\in \R^{p\times n}$, $Y\in \R^{q\times n}$ and $Z\in \R^{r\times n}$ are random matrices with i.i.d.\;entries of mean zero and variance one. Let $\wt\lambda_1 \ge \wt\lambda_2\ge \cdots \ge \wt\lambda_{r}$ be the largest $r$ eigenvalues of the sample canonical correlation (SCC) matrix $\cal C_{\cal X\cal Y}=(\cal X\cal X^\top)^{-1/2}\cal X\cal Y^\top (\cal Y\cal Y^\top)^{-1}\cal Y \cal X^\top (\cal X\cal X^\top)^{-1/2}$, and let $t_1\ge t_2 \ge \cdots\ge t_r$ be the squares of the population canonical correlation coefficients between $\wt{\bx}$ and $\wt{\by}$. Under certain moment assumptions, we show that there exists a threshold $t_c \in(0, 1)$ such that if $t_i>t_c$, then \smash{$\sqrt{n} (\wt\lambda_i-\theta_i)$} converges weakly to a centered normal distribution, where $\theta_i $ is a fixed outlier location determined by $t_i$. Our proof uses a self-adjoint linearization of the SCC matrix and a sharp local law on the inverse of the linearized matrix.

We introduce a class of M×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IM may depend on M in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of Q. We derive precise large deviation estimates on the generalized components ⟨w,ξi⟩ of the outlier and non-outlier eigenvectors ξi. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \emph{subcritical} spikes of Σ. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q, satisfying EQ=IM, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.