Universality in the bulk of the spectrum for complex sample covariance matrices

Abstract

We consider complex sample covariance matrices $M_N=\frac{1}{N}Y{Y}^{\ast }$ where $Y$ is a $N\times p$ random matrix with i.i.d. entries $Y_{ij},1\le i\le N,1\le j\le p$ with distribution $F$. Under some regularity and decay assumption on $F$, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where $N\to \infty$ and ${\lim }_{N\to \infty }p/N=\gamma$ for any real number $\gamma \in (0,\infty )$.

arXiv commentTypos corrected, figures and exposition improved