the roots in $\mathbb{C}$ of the characteristic polynomial $\det( {\mathbf}{M} - z {\mathbf}{I})$, where ${\mathbf}{I}$ is the identity matrix. We let $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ denote the eigenvalues of ${\mathbf}{M}$. In this case, the *empirical spectral measure* $\mu_{{\mathbf}{M}}$ is given by $$\mu_{\mathbf}{M} := \frac{1}{N} \sum_{i=1}^N \delta_{\lambda_i({\mathbf}{M})}.$$ The corresponding *empirical spectral distribution* (ESD) is given by $$F^{{\mathbf}{M}}(x,y) := \frac{1}{N} \# \left\{1 \leq i \leq N: {\operatorname{Re}}(\lambda_i({\mathbf}{M})) \leq x, {\operatorname{Im}}(\lambda_i({\mathbf}{M})) \leq y \right\}.$$ Here $\# E$ denotes the cardinality of the set $E$.

If the matrix ${\mathbf}{M}$ is Hermitian, then the eigenvalues $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ are real. In this case the ESD is given by $$F^{{\mathbf}{M}}(x) := \frac{1}{N} \# \left\{ 1 \leq i \leq N : \lambda_i({\mathbf}{M}) \leq x \right\}.$$

One of the simplest random matrix ensembles is the class of random matrices with independent and identically distributed (iid) entries.

Let $\xi$ be a complex random variable. We say ${\mathbf}{Y}_N$ is an $N \times N$ *iid random matrix* with atom variable $\xi$ if the entries of ${\mathbf}{Y}_N$ are iid copies of $\xi$.

When $\xi$ is a standard complex Gaussian random variable, ${\mathbf}{Y}_N$ can be viewed as a random matrix drawn from the probability distribution $${\mathbb{P}}(d {\mathbf}{M}) = \frac{1}{\pi^{N^2}} e^{- \operatorname{tr}({\mathbf}{M} {\mathbf}{M}^\ast)} d {\mathbf}{M}$$ 