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. in Electron J Probab 23:Paper No. 110, 61, 2018), we considered the empirical spectral distribution \(\mu _n^Y\) of the rescaled entry-wise product $$\begin{aligned} Y_n = \frac{1}{\sqrt{n}} A_n\odot X_n = \left( \frac{1}{\sqrt{n}} \sigma _{ij}X_{ij}\right) \end{aligned}$$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. (2018) 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 sampled variance profiles \(\sigma ^2_{ij} = \sigma ^2\left( \frac{i}{n}, \frac{j}{n} \right) \) where \((x,y)\mapsto \sigma ^2(x,y)\) is a given function on \([0,1]^2\). Associated examples are provided where \(\mu _n^Y\) converges to a genuine limit. We study \(\mu _n\)’s behavior at zero. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al. (Ann Appl Probab 28(1):148–203, 2018; Ann Inst Henri Poincaré Probab Stat 55(2):661–696, 2019), we prove that, except possibly at the origin, \(\mu _n\) 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.