  For each $n$, let $A_n=(\sigma_{ij})$ 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 $\mu_n^Y$ of the rescaled entry-wise
product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}}
\sigma_{ij}X_{ij}\right) \] and provided a deterministic sequence of
probability 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 et
al. was to allow some of the entries $\sigma_{ij}$ 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
$(\mu_n)$, described by a family of Master Equations. We consider these
equations 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 $\mu_n$'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, $\mu_n$ 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 its
spectral radius.
