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

TL;DR

This paper proves that local eigenvalue statistics in the bulk of the spectrum for complex sample covariance matrices are universal under certain conditions, regardless of the specific distribution of matrix entries, as the matrix size grows large.

Contribution

It establishes universality of local eigenvalue statistics in the bulk for complex sample covariance matrices with general entry distributions.

Findings

Universality holds for a wide class of distributions F.

Results apply as N and p grow large with p/N approaching a positive constant.

Local eigenvalue statistics are shown to be independent of the distribution F under regularity conditions.

Abstract

We consider complex sample covariance matrices $M_N=\frac{1}{N}YY^*$ where $Y$ is a $N \times p$ random matrix with i.i.d. entries $Y_{ij}, 1\leq i\leq N, 1\leq j \leq p$ 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 $\gamma \in (0, \infty)$.