Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}))$ by $\mu_n^{A_n} = \frac{1}{n} \sum\limits_{j=1}^n \delta_{\lambda_j(A_n)}$. For many interesting random matrix ensembles $(A_n)_{n \in \mathbb{N}}$ there are results of the form $$ n\big( \mu^{A_n}_n(f) - \mu(f) \big) \xrightarrow{n \rightarrow \infty}_{\mathcal{D}} X_f $$ for certain (sufficiently smooth) functions $f : \mathbb{C} \rightarrow \mathbb{R}$, a determinsitic measure $\mu$ and a Gaussian process $X_f$.

My question is if anyone knows if such a result is known for iid matrices, that is when the entries of $A_n$ are all independent identically distributed with mean zero and variance one. By the Circular Caw we know, that in this case $\mu$ would need to be the uniform distribution on the unit sphere $B_1^{\mathbb{C}}(0)$.

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\mathbb{C}))$ by $\mu_n^{A_n} = \frac{1}{n} \sum\limits_{j=1}^n \delta_{\lambda_j(A_n)}$. For many interesting random matrix ensembles $(A_n)_{n \in \mathbb{N}}$ there are results of the form $$ n\big( \mu^{A_n}_n(f) - \mu(f) \big) \xrightarrow{n \rightarrow \infty}_{\mathcal{D}} X_f $$ for certain (sufficiently smooth) functions $f : \mathbb{C} \rightarrow \mathbb{R}$, a determinsitic measure $\mu$ and a Gaussian process $X_f$.

My question is if anyone knows if such a result is known for iid matrices, that is when the entries of $A_n$ are all independent identically distributed with mean zero and variance $\frac{1}{n}$. By the Circular Caw we know, that in this case $\mu$ would need to be the uniform distribution on the unit sphere $B_1^{\mathbb{C}}(0)$.

An example for a random matrix ensemble, where such a result is known, is the Wigner ensemble. See Theorem 9.2 of Bai and Silversteins 'Spectral Analysis of Large Dimensional Random Matrices'.