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

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 \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right) \] and provided a deterministic sequence of probability measures ${\mu }_n$ such that the difference ${\mu }_n^Y-{\mu }_n$ 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 ${\sigma }_{ij}^2=d_i{\tilde{d}}_j$ and sampled variance profiles ${\sigma }_{ij}^2={\sigma }^2\left(\frac{i}{n},\frac{j}{n}\right)$ 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 $V_n=(\frac{1}{n}{\sigma }_{ij}^2)$ and $\rho (V_n)$ is its spectral radius.

35 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1612.04428