Singularities of the density of states of random Gram matrices

TL;DR

This paper investigates the eigenvalue density of large random Gram matrices with independent, non-identically distributed entries, revealing the nature of their singularities and extending existing local laws near these points.

Contribution

It characterizes the types of singularities in the eigenvalue density of such matrices and extends the bulk local law to include neighborhoods of these singularities.

Findings

Density has only square and cubic-root singularities away from zero.

Extended local law applies near these singularities.

Eigenvalue density approximated by a deterministic measure.

Abstract

For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R}$. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [arXiv:1606.07353] to the vicinity of these singularities.