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For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrixand let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centeredentries of unit variance. In the companion article Cook et al., we consideredthe empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wiseproduct \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}}\sigma_{ij}X_{ij}\right) \] and provided a deterministic sequence ofprobability 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 etal. was to allow some of the entries $\sigma_{ij}$ to vanish, provided that thestandard deviation profiles $A_n$ satisfy a certain quantitative irreducibilityproperty.  In the present article, we provide more information on the sequence$(\mu_n)$, described by a family of Master Equations. We consider theseequations in important special cases such as separable variance profiles$\sigma^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $\sigma^2_{ij}= \sigma^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto \sigma^2(x,y)$is a given function on $[0,1]^2$. Associate examples are provided where$\mu_n^Y$ converges to a genuine limit.  We study $\mu_n$'s behavior at zero and provide examples where $\mu_n$'sdensity is bounded, blows up, or vanishes while an atom appears. As aconsequence, we identify the profiles that yield the circular law.  Finally, building upon recent results from Alt et al., we prove that exceptmaybe in zero, $\mu_n$ admits a positive density on the centered disc of radius$\sqrt{\rho(V_n)}$, where $V_n=(\frac 1n \sigma_{ij}^2)$ and $\rho(V_n)$ is itsspectral radius.

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