Non-Hermitian random matrices with a variance profile (II): properties and examples

TL;DR

This paper analyzes the spectral properties of non-Hermitian random matrices with variance profiles, providing explicit descriptions of their limiting spectral measures, behaviors at zero, and conditions for the circular law, extending previous work with new examples and detailed analysis.

Contribution

It advances the understanding of spectral distributions of non-Hermitian matrices with variance profiles by deriving Master Equations and exploring their limits, including special cases and conditions for the circular law.

Findings

Convergence of empirical spectral distributions to deterministic measures.

Behavior of spectral density at zero varies with profiles, including bounded, blow-up, or vanishing densities.

Profiles that satisfy certain conditions yield the circular law.

Abstract

For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. In the companion article Cook et al., we considered the empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right) \] and provided a deterministic sequence of probability measures $\mu_n$ such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature in Cook et al. was to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(\mu_n)$, described by a family of Master Equations. We consider these equations in important special cases such as separable variance profiles $\sigma^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $\sigma^2_{ij} = \sigma^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto \sigma^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $\mu_n^Y$ converges to a genuine limit. We study $\mu_n$'s behavior at zero and provide examples where $\mu_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $\mu_n$ admits a positive density on the centered disc of radius $\sqrt{\rho(V_n)}$, where $V_n=(\frac 1n \sigma_{ij}^2)$ and $\rho(V_n)$ is its spectral radius.