on the set of complex $N \times N$ matrices. Here $d {\mathbf}{M}$ denotes the Lebesgue measure on the $2N^2$ real entries $$\{ {\operatorname{Re}}(m_{ij}) : 1 \leq i, j \leq N\} \cup \{ {\operatorname{Im}}(m_{ij}) : 1 \leq i, j \leq N\}$$ of ${\mathbf}{M}=(m_{ij})_{i,j=1}^N$. The measure ${\mathbb{P}}(d {\mathbf}{M})$ is known as the [*complex Ginibre ensemble*]{}. The [*real Ginibre ensemble*]{} is defined analogously. Following Ginibre [@Gi], one may compute the joint density of the eigenvalues of a random matrix ${\mathbf}{Y}_N$ drawn from the complex Ginibre ensemble. Indeed, $(\lambda_1({\mathbf}{Y}_N), \ldots, \lambda_N({\mathbf}{Y}_N))$ has density $$\label{eq:ginibre}
    p_N(z_1, \ldots, z_N) := \frac{1}{\pi^N \prod_{i=1}^N k! } \exp \left( - \sum_{k=1}^N |z_k|^2 \right) \prod_{1 \leq i < j \leq N} |z_i - z_j |^2.$$

Mehta [@M; @M:B] used the joint density function to compute the limiting spectral measure of the complex Ginibre ensemble. In particular, he showed that if ${\mathbf}{Y}_N$ is drawn from the complex Ginibre ensemble, then the ESD of $\frac{1}{\sqrt{N}} {\mathbf}{Y}_N$ converges to the [*circular law*]{} $F_{\mathrm{circ}}$ as $N \to \infty$, where $$F_{\mathrm{circ}}(x,y) := \mu_{\mathrm{circ}} \left( \left\{ z \in \mathbb{C} : {\operatorname{Re}}(z) \leq x, {\operatorname{Im}}(z) \leq y \right\} \right)$$ and $\mu_{\mathrm{circ}}$ is the uniform probability measure on the unit disk in the complex plane. Edelman [@Ed-cir] verified 