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

For each n, let $$A_n=(\sigma _{ij})$$ be an $$n\times n$$ deterministic matrix and $$X_n=(X_{ij})$$ random 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 empirical spectral distribution $$\mu _n^Y$$ rescaled entry-wise product $$\begin{aligned} Y_n = \frac{1}{\sqrt{n}} A_n\odot X_n \left( \sigma _{ij}X_{ij}\right) \end{aligned}$$ provided a sequence probability measures _n$$ such that difference ^Y_n - \mu converges weakly to zero measure. A key feature Cook (2018) was allow some $$\sigma _{ij}$$ vanish, standard deviation profiles $$A_n$$ satisfy certain quantitative irreducibility property. present article, provide more information on $$(\mu _n)$$ , described by family Master Equations. We consider these equations important special cases as sampled variance ^2_{ij} ^2\left( \frac{i}{n}, \frac{j}{n} \right) $$ where $$(x,y)\mapsto ^2(x,y)$$ is given function $$[0,1]^2$$ . Associated examples are genuine limit. study ’s behavior at zero. As consequence, identify yield circular law. Finally, building upon recent results from Alt (Ann Appl 28(1):148–203, 2018; Ann Inst Henri Poincaré Stat 55(2):661–696, 2019), prove that, except possibly origin, admits positive density disc radius $$\sqrt{\rho (V_n)}$$ $$V_n=(\frac{1}{n} _{ij}^2)$$ $$\rho (V_n)$$ its radius.