Inhomogeneous Circular Law for Correlated Matrices

TL;DR

This paper extends the circular law to non-Hermitian matrices with correlated entries, providing a deterministic density approximation and analyzing spectral properties at fine scales.

Contribution

It introduces a new inhomogeneous circular law for correlated matrices, characterizing the spectral density via coupled matrix equations and explicit edge behavior.

Findings

Spectral density is radially symmetric and positive inside a disk.

Explicit formula for the disk radius based on entry covariances.

Convergence to the spectral density occurs at near-optimal local scales.

Abstract

We consider non-Hermitian random matrices $X \in \mathbb{C}^{n \times n}$ with general decaying correlations between their entries. For large $n$, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear $n \times n$ matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of $X$. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.