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Non-Hermitian random matrices with a variance profile (II): properties and examples

For each $n$, let $A_n=(σ_{ij})$ 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., we considered the empirical spectral distribution $μ_n^Y$ of the rescaled entry-wise product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right) \] 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. 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 separable variance profiles $σ^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $σ^2_{ij} = σ^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto σ^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $μ_n^Y$ converges to a genuine limit. We study $μ_n$'s behavior at zero and provide examples where $μ_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $μ_n$ admits a positive density on the centered disc of radius $\sqrt{ρ(V_n)}$, where $V_n=(\frac 1n σ_{ij}^2)$ and $ρ(V_n)$ is its spectral radius.