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

For each n, let $$A_n=(\sigma _{ij})$$ A n = ( σ ij ) be an $$n\times n$$ n × n deterministic matrix and let $$X_n=(X_{ij})$$ X n = ( X ij ) be an $$n\times n$$ 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 $$\mu _n^Y$$ μ n Y of the rescaled entry-wise product $$\begin{aligned} Y_n = \frac{1}{\sqrt{n}} A_n\odot X_n = \left( \frac{1}{\sqrt{n}} \sigma _{ij}X_{ij}\right) \end{aligned}$$ Y n = 1 n A n ⊙ X n = 1 n σ ij X ij and provided a deterministic sequence of probability measures $$\mu _n$$ μ n such that the difference $$\mu ^Y_n - \mu _n$$ μ n Y - 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 $$\sigma _{ij}$$ σ ij to vanish, provided that the standard deviation profiles $$A_n$$ A n satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $$(\mu _n)$$ ( μ n ) , described by a family of Master Equations. We consider these equations in important special cases such as sampled variance profiles $$\sigma ^2_{ij} = \sigma ^2\left( \frac{i}{n}, \frac{j}{n} \right) $$ σ ij 2 = σ 2 i n , j n where $$(x,y)\mapsto \sigma ^2(x,y)$$ ( x , y ) ↦ σ 2 ( x , y ) is a given function on $$[0,1]^2$$ [ 0 , 1 ] 2 . Associated examples are provided where $$\mu _n^Y$$ μ n Y converges to a genuine limit. We study $$\mu _n$$ 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, $$\mu _n$$ μ n admits a positive density on the centered disc of radius $$\sqrt{\rho (V_n)}$$ ρ ( V n ) , where $$V_n=(\frac{1}{n} \sigma _{ij}^2)$$ V n = ( 1 n σ ij 2 ) and $$\rho (V_n)$$ ρ ( V n ) is its spectral radius.