y_{ji}) : 1 \leq i < j \leq N\}$ is a collection of iid copies of $(\xi_1,\xi_2)$.

-   (diagonal entries) $\{y_{ii} : 1 \leq i \leq N\}$ is a collection of iid copies of $\zeta$.

Real elliptic random matrices generalize iid random matrices. Indeed, if $\xi_1, \xi_2, \zeta$ are iid, then ${\mathbf}{Y}_N$ is just an iid random matrix. On the other hand, if $\xi_1 = \xi_2$ almost surely, then ${\mathbf}{Y}_N$ is a real symmetric matrix. In this case, the eigenvalues of ${\mathbf}{Y}_N$ are real and ${\mathbf}{Y}_N$ is known as a *real symmetric Wigner matrix* [@W].

Suppose $\xi_1, \xi_2$ have mean zero and unit variance. Set $\rho := {\mathbb{E}}[\xi_1 \xi_2]$. When $|\rho| < 1$ and $\zeta$ has mean zero and finite variance, it was shown in [@NgO] that the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges almost surely to the elliptic law $F_\rho$ as $N \to \infty$, where $$F_\rho(x,y) = \mu_{\rho}( \{z \in \mathbb{C} : {\operatorname{Re}}(z) \leq x, {\operatorname{Im}}(z) \leq y\})$$ and $\mu_\rho$ is the uniform probability measure on the ellipsoid $$\mathcal{E}_{\rho} = \left\{ z \in {\mathbb{C}}: \frac{{\operatorname{Re}}(z)^2}{(1+\rho)^2} +\frac{{\operatorname{Im}}(z)^2}{(1-\rho)^2} < 1 \right\} .$$ This is a natural generalization of the circular law (Theorem \[thm:tvcirc\]). Figure \[fig:elliptic\] displays a numerical simulation of the eigenvalues 