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

For each n , let A_n=(σ _ij) be an n× n deterministic matrix and let X_n=(X_ij) be an n× 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 μ _n^Y of the rescaled entry-wise product Y_n = 1/√(n) A_n⊙ X_n = ( 1/√(n)σ _ijX_ij) and provided a deterministic sequence of probability measures μ _n such that the difference μ ^Y_n - μ _n converges weakly in probability to the zero measure. A key feature in Cook et al. (2018) was to allow some of the entries σ _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 (μ _n) , described by a family of Master Equations . We consider these equations in important special cases such as sampled variance profiles σ ^2_ij = σ ^2( i/n, j/n) where (x,y)↦σ ^2(x,y) is a given function on [0,1]^2 . Associated examples are provided where μ _n^Y converges to a genuine limit. We study μ _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, μ _n admits a positive density on the centered disc of radius √(ρ (V_n)) , where V_n=(1/nσ _ij^2) and ρ (V_n) is its spectral radius.