al remarks apply about the sufficiency of identical distribution or a Lyapunov-type condition: holds in the settings of Theorems \[T:circular-law-correlated\] and \[T:circular-law-uncorrelated\] if all the $Y_a^{(n)}$ are identically distributed, or have uniformly bounded $(2 + \delta)$ moments; it holds in the settings of Theorems \[T:semicircle-law-general\] and \[T:semicircle-law-special\] if all the random variables with a given variance assumption satisfy such assumptions.

We now state our main results, deferring the proofs until the end of the section.

\[T:circular-law-correlated\] Let $\alpha \in [0,1]$. Suppose that for each $n$, $\{Y_a^{(n)} \mid
  a \in G^{(n)}\}$ are independent; that $${\mathbb{E}}Y_a^{(n)} = 0, \quad {\mathbb{E}}\bigl\vert Y_a^{(n)}\bigr\vert^2 = 1, \quad
  \text{and} \quad {\mathbb{E}}\bigl(Y_a^{(n)}\bigr)^2 = \alpha$$ for every $a \in G^{(n)}$; and that holds. Suppose further that $\lim_{n\to \infty} p_2^{(n)} = p$ exists. Then $\mu^{(n)}$ converges, in mean and in probability, to $(1 - p) \gamma_{\mathbb{C}}+ p \gamma_\alpha$.

One of the main special cases of interest in Theorem \[T:circular-law-correlated\] is when $\alpha = 1$, that is, when the matrix entries are all real. In that case, the limiting spectral distribution of $M^{(n)}$ is complex Gaussian if the number of $a$ with $a^2 = 1$ is negligible for large $n$. On the other hand, if the fraction of such $a$ is asymptotically constant then, due to the presence of many real-valued characters $\chi$, the limiting spectral distribution will be a mixture of $\gamma_{\mathbb{C}}$ and $\gamma_{\mathbb{R}}$.

The other main special case of interest is when $\alpha = 0$, so that the matrix entries have uncorrelated real and imaginary parts. In that case, which generalizes the setting of Corollary \[T:C-Ginibre-limit\], one can remove the assumption that $p_2^{(n)}$ approaches a limit.

\[T:circular-law-uncorrelated\] Suppose that for each $n$, $\{Y_a^{(n)} \mid a \in G^{(n)}\}$ are independent; that $${\mathbb{E}}Y_a^{(n)} = 0,