Non-Hermitian Random Matrices with a Variance Profile (II): Properties and Examples

For each n, let A(n) = (sigma(ij)) be an n x n deterministic matrix and let X-n = (X-ij) be an n x 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(Y)(n) of the rescaled entry-wise product Y-n = 1/root n A(n) circle dot X-n = (1/root n sigma X-ij(ij)) 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) (i/n, j/n), where (x, y) bar right arrow sigma(2) (x, y) is a given function on [0, 1](2). Associated examples are provided where mu(Y)(n) 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 Poincare 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 root rho(V-n), where V-n = (1/n sigma(2)(ij)) and rho(V-n) is its spectral radius.