Local law for random Gram matrices

TL;DR

This paper establishes a precise local spectral law for large random Gram matrices with independent entries, extending classical results to more general variance profiles and matrix shapes.

Contribution

It proves a local law for the spectrum of general random Gram matrices with arbitrary variances, including edge cases, and determines the limiting eigenvalue density via nonlinear equations.

Findings

Optimal local laws with sharp error bounds

Extension of Marchenko-Pastur law to general variance profiles

Existence of a spectral gap in the rectangular case

Abstract

We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.