To see the convergence in action, increase n from small to large
If we have a large $n \times n$ real symmetric (or complex
Hermitian) matrix, $H$, where the entries are random variables
with mean $0$ and variance $1/n$.
We know that the eigenvalues of $H$ will be real (random variables) from the Spectral Theorem .
The Semicircle Law in Random Matrix Theory says that given any realization of such random object,
the histogram of the eigenvalues will roughly
have the shape of a semicircle.
The limiting semicircle distribution can be view as a counterpart of Gaussian Distribution for
non-commutative random variables, which are the
main objects of interest in Free
Probability, a field
that is closely related to Random Matrix Theory.
The Semicircle Law
Empirical Spectral Measure
The histogram of eigenvalues of a $n\times n$ matrix $H_n$ are really just a
pictorial representation of
the empirical spectral distribution of $H_n$
Let $\lambda_{i}$ denote the (real) eigenvalues of $H_n$, with $\lambda_{1} \leq \lambda_{2} \leq \cdots \leq
\lambda_{n}$.
The empirical Spectral distribution of $H_n$ is defined as the probability measure on
$\mathbb{R}$
$$
\mu_{H_n}:=\frac{1}{n} \sum_{i=1}^{n} \delta_{\lambda_{i}}
$$
If $H_n$ is a random matrix, then $\mu_{H_n}$ is a random probability measure.
If for a matrix $H$, the eigenvalues are
$$[-0.72, -0.14, 0.46, 0.87, 1.42]$$
Then the empirical distribution is
$$\mu_{H}:=\frac{1}{5}(\delta_{-0.72}+\delta_{-0.14}+\delta_{0.46}+\delta_{0.87}+\delta_{1.42})$$
If $H$ is a $2\times 2$ matrix, with random entries
$$
H = \begin{pmatrix}
x_1 & x_2\\
0 & x_3
\end{pmatrix},\qquad x_1,x_2 \sim \mathcal{N}(0,1), \qquad x_3 \sim \mathcal{Bin}(1,0.5)
$$
For an open interval $I$, the empirical spectral distribution is given by
$$\mu_{H}(I) = \frac{1}{2} \left[\mathbb{P}_{ \mathcal{N}(0,1)}(x_1 \in I) + \mathbb{P}_{
\mathcal{Bin}(1,0.5)}(x_3 \in I)\right]$$
For example, if $I=(-2,2)$, then
$$ \P{\mu_{H}(I)= m} \simeq \begin{cases}
\frac{1}{2} (0.95+1),\qquad & m=1\\
\frac{1}{2} (0.05+1),\qquad & m=0.5
\end{cases}$$
as we can see, the measure $\mu_{H}$ is random.
Wigner Matrices
For some random matrix models, a matrix version of the Central Limit Theorem exist.
A Wigner Hermitian (Symmetric) matrix is a random matrix
$$H_{n}=\left(\xi_{i j}\right)_{1 \leq i, j \leq n}$$ with
the upper-triangular entries $\xi_{i j}, i>j$ are iid complex (real) random variables with mean $0$ and
variance $\frac{1}{n}$.
and the diagonal entries $\xi_{i i}$ are iid real variables, independent of the upper-triangular entries, with
bounded mean and variance.
The following is an example of a $3\times 3$ Wigner Hermitian Matrix, with entries complex Gaussian.
$$\begin{equation}
\left[
\begin{array}{ccc}
0.59+0.0\textit{i} & 0.17-0.38\textit{i} & -0.15+0.08\textit{i} \\
0.17+0.38\textit{i} & -0.27+0.0\textit{i} & 0.17+0.24\textit{i} \\
-0.15-0.08\textit{i} & 0.17-0.24\textit{i} & 0.78+0.0\textit{i} \\
\end{array}
\right]
\end{equation}$$
Semicircle Distribution
When asked what is the counterpart of the ubiquitous Normal Distribution in Random Matrix
Theory, there are several candidates.
For the empirical spectral measure of Wigner matrices, it is the semi-circle distribution:
The semicircle distribution is defined by the density
$$\mu_{sc}(x)=\frac{1}{2 \pi} \sqrt{4-x^{2}} \mathbf{1}_{|x| \leq 2}$$
We are now ready to state the first Central Limit Theorem in Random Matrix Theory:
Let $\left\{H_{n}\right\}_{n=1}^{\infty}$ be a
sequence of
Wigner
matrices.
Then
$$
\mu_{H_n} \rightharpoonup \mu_{sc} \qquad a.s.
$$
Method of Moments
The method of moments is the technique used by Wigner in his
1958 paper
On the Distribution of the
Roots of
Certain Symmetric
Matrices. Wigner is a nobel prize winner in Physics, and is often regarded
as the founder of Random Matrix Theory.
In his 1958 paper, Wigner assumed weaker conditions, he required the entries to have
symmetrical distributions
identical second moment
and, all moments bounded
but the technique can be extended to the more general version. The version above, which only requires a finite
second moment assumption, is due to Bai.
Moments of the Semi-circle Law
It is a standard calculus exercise to show that the moments, $\mu_{sc,m}$ of $\mu_{sc}$
$$\mu_{sc,m}:=\int x^m \ d\mu_{sc}=\begin{cases}0 \qquad &m \text{ odd} \\
\left(\begin{array}{c}
m \\
\frac{1}{2}m \\
\end{array}\right) \frac{1}{m+1} &m \text{ even} \end{cases} $$
The sequence of even moments are known at the Catalan
Numbers . The famous combinatorist Richard Stanley has an entire book
dedicated to
the Catalan numbers, where he demonstrate the ubiquitousness of the Catalan numbers by giving over a hundred
examples of where Catalan Number can appear.
The moments of the Semicircle Law Satisfies the Carleman's Condition
$$\sum_{m=1}^{\infty} \frac{1}{\mu_{sc,m}^{\frac{1}{2m}}}=\infty$$
Thus there is only one measure, namely $\mu_{sc}$, with the moments $\mu_{sc,m}$.
Moments of the Empirical Distribution
To study the random measures $\mu_{H_n}$ via its moments, it is important how the moments are related to the
entries of $H_n$.
We will use $X_n$ to denote the random variable characterized $\mu_{H_n}$.
By definition, the $m$-th moment of $\mu_{H_n}$, denote it by $X_{n,m}$, is given by the following (random
object)
$$ \begin{align} X_{n,m} : &= \int x^m d\mu_{H_n} \\
&=\frac{1}{n}\sum_{i=1}^{n} \lambda_i^m \\
&= \frac{1}{n} \operatorname{tr} H_n^m
\\&=\frac{1}{n} \sum H_{i_{1}, i_{2}} H_{i_{2}, i_{3}} \cdots H_{i_{m}, i_{1}}
\end{align}$$
Note that the moments are random, thus it make sense to talk about its expectation and variance
$$\mathbb{E}(X_{n,m})=\frac{1}{n} \sum \mathbb{E}( H_{i_{1}, i_{2}} H_{i_{2}, i_{3}} \cdots H_{i_{m}, i_{1}})$$
$$\mathbb{Var}(X_{n,m})=\frac{1}{n^2} \sum \operatorname{Cov} [(H_{i_{1}, i_{2}} H_{i_{2}, i_{3}} \cdots H_{i_{m},
i_{1}}),( H_{j_{1}, j_{2}} H_{j_{2}, j_{3}} \cdots H_{j_{m}, j_{1}})]$$
One can show that $$\mathbb{E}(X_{n,m}) \to \mu_{sc,m}$$
$$\mathbb{Var}(X_{n,m}) = O_m(\frac{1}{n})$$
Which implies, by Borel-Catelli Lemma $$X_{n,m}\to \mu_{sc,m}\quad a.s.$$
and that the sequence $\{d\mu_{H_n}\}$ is tight. Thus every subsequence $\{d\mu_{H_{n_k}}\}$ has a further
convergence subsequence.
Since the moments of the full sequence (and thus any subsequence) converges to the moments of the Semicircle
Distribution, by the uniqueness guaranteed by Carleman's condition,
all the converging subsequence, and thus the full sequence, must be converging weakly to the Semicircle law almost
surely.
Stieljes Transform
Definition
Another useful, and often more flexible, technique in proving results in Random Matrix Theory is the Stieljes
Transform.
The Stieltjes Transform of a real measure $\mu$ is defined by
$$
s_{\mu}(z):=\int_{\mathbb{R}} \frac{1}{x-z} d \mu(x)
$$
for all $z \in \mathbb{C}$ outside the support of $\mu$.
The Stieljes Transform of the Semicircle Law is given by
$$s_{s c}(z)=\frac{-z+\sqrt{z^{2}-4}}{2}$$
The Stieljes Transform of the empirical distribution of a Hermitian Matrix $H_n$, using Spectral
Theorem, is given by
$$s_{n}(z)=\int_{\mathbb{R}} \frac{1}{x-z} d \mu_{n}(x)=\frac{1}{n} \operatorname{tr}\left(\frac{1}{\sqrt{n}}
H_{n}-z I_{n}\right)^{-1}$$
The term, $ \left(\frac{1}{\sqrt{n}}
H_{n}-z I_{n}\right)^{-1}$ is known as the Resolvent of $H_n$, which plays an important role in the
developments of later techniques..
Inversion Formula
Note that the imaginary part of $s_{\mu}(z)$ can be written as a convolution,
$$\operatorname{Im} s_{\mu}(\lambda+i \epsilon)=\int \frac{\epsilon}{(x-\lambda)^{2}+\epsilon^{2}} d
\mu(x)=\pi\left(\mu * f_{\epsilon}\right)(\lambda)$$
Where $f_{\epsilon}(y)=\frac{1}{\pi}\frac{\epsilon}{y^2+\epsilon^2}$ is the density of the Cauchy distribution,
$Y_\epsilon$ , with location parameter $0$ and scale parameter $\epsilon$.
In other words, denoting $X$ a random variable with density $\mu$, we have $$X+Y_{\epsilon}\sim \operatorname{Im}
s_{\mu}(\lambda+i \epsilon)$$
Let $\epsilon \to 0$, we get
inversion formula
$$\operatorname{Im} s_{\mu}(\lambda+i \epsilon) \lambda \rightharpoonup \pi \mu(\lambda) \quad \text { as } \epsilon
\searrow 0$$
Continuity Theorem
Let $\mu_n$ be a sequence of probability measures with corresponding Stieljes Transform $s_{n}$.
Suppose
there exist $m:\C_+ \to \C_+$ such that $$ \lim_{n\to \infty} m_n(z) \to m(z)\qquad \forall z\in \C_+$$
and $m$ is the stieljes transform of a probability measure $\mu$.
then $$
\mu \weakly \mu_n
$$
Matrix Dyson Equation (MDE)
The Matrix Dyson Equation (MDE) can be expressed as follows:
$$G(z) = (-zI + A + S[G(z)])^{-1}$$
G(z) is the resolvent or Green's function of the random matrix at spectral parameter $z \in
\mathbb{C}^+$.
A is typically a deterministic Hermitian matrix representing a structured component or expectation.
S is a linear operator acting on matrices, encoding the variance structure and correlations of the
random matrix entries.
I is the identity matrix.
The solution G(z) captures essential spectral information about the eigenvalue distribution of
large-dimensional random matrices.
For classical Wigner matrices, the MDE reduces to
$$
m(z) = \frac{1}{-z - m(z)}, \quad z \in \mathbb{C}^+
$$
whose solution yields Wigner's famous semicircle distribution for eigenvalues.