数理解析研究所講究録
第 1904 巻 2014 年 105-122

105

Universality in Random Matrix TheoryJI OON LEE
Department of Mathematical Sciences, KAIST

1IntroductionLet

be independent, identically distributed random variables. Assume that $\mathbb{E}X=0$ and. Then, their sum $S_{N}:=X_{1}+X_{2}+\cdots+X_{N}$ satisfies$X_{1},$ $X_{2},$$\mathbb{E}X^{2}=1$$\cdots,$ $X_{N}$(in distribution),$\frac{S_{N}}{\sqrt{N}}arrow N(0,1)$(1.1)

which is the Central Limit Theorem. In principle, all the random variables
can be of order
1, hence $S_{N}\sim 1$ as well, but the probability of having such a rare event is incredibly small. We can evenestimate the bound on the probability for the rare event from the large deviation principle.
A similar phenomenon happens when we form a large matrix from i.i. . random variables and consider
the distribution of its eigenvalues. The simplest example for the random matrix model is the Gaussian
Orthogonal Ensemble (GOE).
$X_{1},$ $X_{2},$$X_{N}$$d$Definition 1,1 (Gaussian Orthogonal Ensemble (GOE)). An $N\cross N$ random matrix $H=(h_{ij})$ is said tobe the Gaussian Orthogonal Ensemble if its entries
are real Gaussian random variables, independent up
to the symmetry constraint $h_{ij}=h_{ji}$ , satisfying$h_{ij}$1.

$\mathbb{E}h_{ij}=0,$2.

$\mathbb{E}|h_{ij}|^{2}=N^{-1}$3.

$\mathbb{E}|h_{ii}|^{2}=2N^{-1}$

for $i\neq j$ , andSimilarly, we can define a model for complex Hermitian matrices, which is known as Gaussian Unitary
Ensemble (GUE).

Definition 1.2 (Gaussian Unitary Ensemble (GUE)). An $N\cross N$ random matrix $H=(h_{ij})$ is said to be
the Gaussian Unitary Ensemble if its entries
are complex Gaussian random variables, independent up to
the symmetry constraint, satisfying$h_{ij}$$h_{ij}=\overline{h_{j\iota’}}$1.

$\mathbb{E}h_{ij}=0,$2.

$\mathbb{E}|h_{ij}|^{2}=N^{-1}$3.

$\mathbb{E}(h_{ij})^{2}=0$, and

if $i\neq j.$

In order to introduce results on GOE or GUE analogous to CLT, we consider the empirical spectral
distribution of a random matrix.

Definition 1.3. Let
of $H$ is defined by$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{N}$be the eigenvalues of $H$ . Then, the empirical (spectral) measure$\mu(H) :=\frac{1}{N}\sum_{j=1}^{N}\delta_{\lambda_{j}}$

.

(1.2)

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Random matrices such as GOE or GUE were first introduced by Wigner [17] for the spectral analysisof heavy nuclei. Later, Gaudin, Mehta, and Dyson [10, 11, 3, 4] proved important results on Gaussian
ensembles such as the bulk universality results. While the early works focused on the properties of theeigenvalue distribution of the Gaussian ensembles, the recent developments on the random matrix theory
showed that in fact most of the properties hold for a larger class of Wigner matrices as well [7, 14]. The
most basic yet the most well-known such a phenomena is the semicircle law proved by Wigner [17]. We firstconsider the GOE case.
be the empirical measure of $GOE$ . Then,
Theorem 1.4. Let
whose density is defined to be
Wigner semicircle distribution$\mu_{GOE}$$\mu_{GOE}$

converges almost surely to the$\mu_{sc}$$\mu_{sc}(x)dx :=\frac{1}{2\pi}\sqrt{(4-x^{2})_{+}}dx$

(1.3)

.

Wigner semicircle law holds for a much larger class of random matrices. The most notable examples are
Wigner matrix ensembles.
Definition 1.5. An $N\cross N$ random matrix $H=(h_{ij})$ is said to be
if its entries
are (real/complex) random variables satisfying(symmetric/Hermitian) Wigner matrix$a$$h_{ij}$

1. The upper right entries $h_{ij}(i<j)$ are i.i. . with$d$

2. The diagonal entries$h_{ii}$are i.i. . with$d$

3. In addition, for the Hermitian case,$\mathbb{E}h_{ii}=0$$\mathbb{E}h_{ij}=0$

and

, and

$\mathbb{E}|h_{ij}|^{2}=N^{-1}$$\mathbb{E}|h_{ii}|^{2}=CN^{-1}$for some constant $C.$$\mathbb{E}(h_{ij})^{2}=0.$

The bulk universality concerns the behavior of eigenvalues in the bulk of the spectrum. Consider thefollowing -point correlation function of the eigenvalues.
$k$Definition 1.6 ( -point correlation function). Let $p_{N}(x_{1}, x_{2}, \cdots, x_{N})$ be the symmetric density function of$N$ , we define the-point correlation function of the. For any $k=1$ , 2,
the eigenvalues
eigenvalues by$k$$\lambda_{1},$$\lambda_{2},$$\cdots,$$k$$\lambda_{N}$$\cdots,$$p_{N}^{(k)}(x_{1}, x_{2}, \cdots, x_{k}) :=\int p_{N}(x_{1}, x_{2}, \cdots, x_{N})dx_{k+1}\cdots dx_{N}$For $GUE$ , it is proved by Gaudin and Mehta [10, 11] thatexists a function
such that$p_{N}^{(k)}$

.

(1.4)

has a determinantal structure, i.e., there$K_{N}$$p_{N}^{(k)}(x_{1}, x_{2}, \cdots, x_{k})=\det[I\iota_{N}’(x_{i}, x_{j})]_{1\leqi,j\leq k}$

Later, Dyson found the formula for$K_{N}$(1.5).

which becomes$K_{N}(x+ \frac{y_{1}}{N\mu_{sc}(x)}, x+\frac{y_{2}}{N\mu_{sc}(x)})=\mu_{sc}(x)K(y_{1}, y_{2})$

,

(1.6)

where $K$ is the function$K(y_{1}, y_{2}):= \frac{\sin(\pi(y_{1}-y_{2}))}{\pi(y_{1}-y_{2})}$

,

(1.7)

which is called the Dyson sine kernel.
As in the Wigner semicircle law, the limiting distribution for the correlation function is universal, undersuitable conditions, in the sense that it does not depend on the specific matrix model. There is, however, one
crucial diﬀerence between the universality of the correlation function and that of the semi-circle law, whichis that the correlation function may depend on whether the given random matrix model is an orthogonalensemble or a unitary ensemble. The Dyson sine kernel can describe the limiting distribution for unitaryensembles but not for orthogonal ensembles. Including this feature, we use the term bulk universalitythat asserts the fact that the limiting distribution for the correlation function depends only on the symmetry
class.

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The bulk universality also implies that the limiting distribution of the gap’, the distance between adjacenteigenvalues, is also universal. In particular, since the adjacent eigenvalues in GUE or GOE tend to keep
distance from each other, we can also find that the adjacent eigenvalues in many random matrix models.
This is called (level repulsion’. Note that this also tells the eigenvalues of random matrix models are strongly
correlated, unlike i.i. . random variables.
The dyson sine kernel, and in general the determinantal structure of the correlation functions, appear
in many other fields of mathematics. For example, if we consider the non-trivial zeros of the Riemann zetafunction on the critical line, then after a proper normalization, the imaginary parts of the zeros behave likethe eigenvalues of GUE.
Near the edge of the spectrum, i.e., for the eigenvalues very near 2 or $-2)$ , the limiting distribution forthe correlation function has diﬀerent asymptotics. In this case,collverges to a diﬀerent kernel, which isusually called the Airy kernel. However, there is another formula, proved by Tracy and Widom [15], for thelimiting distribution for the largest eigenvalue.
$d$

$($

$K_{N}$

Definition 1.7 (Tracy-Widom distribution). Let $q(t)$ be a solution to the Painlev\’e II equation$q”=tq+2q^{3}$

satisfying$q(t)\sim Ai(t)$The function$F_{2}$

as$tarrow\infty.$

is a distribution function defined by(1.8)$F_{2}(t)=\exp(-l^{\infty}(x-t)q(x)^{2}dx)$

The function$F_{2}$

is called the Tracy-Widom distribution.

It is proved by Tracy and Widom that the cumulative distribution function of the rescaled largest eigenvalue of GUE$\mathbb{P}[N^{2/3}(\lambda_{N}-2)\leq t]arrow F_{2}(t)$in distribution (vague convergence). In particular, the size of the fluctuation of the largest eigenvalue of GUE
around 2 is of order $N^{-2/3}$
The subindex 2 inagain denotes the dependence on the symmetry class [16]. For orthogonal ensembles,
the limiting distribution function is diﬀerent from , and it is usually called . As in the bulk universality,
under suitable conditions, the limiting distribution for the largest eigenvalue only depends on the symmetry
class. This fact is what we call the edge universality.
The Tracy-Widom distribution appears in many other applications. For example, if we consider
the length of the longest increasing subsequence of a random permutation on 1, 2,$N\}$ , then $L_{N}/\sqrt{N}$behaves as the largest eigenvalue of GUE, hence the limiting distribution is given by Tracy-Widom.
The universality results, bulk and edge, provide a very important strategy in the random matrix theory.
When a certain property of the eigenvalues of a given random matrix model is hard to achieve, we canfirst consider the property we want to attain in GOE or GUE. Then, by proying a universality result, bycomparing the limiting distribution of some eigenvalues of the given model with that of GOE or GUE, we
can find a desired property in the given random matrix. This is very powerful because there are additional
tools working only on GUE or GOE due to some symmetry or some explicit formulas.
Although omitted in this article, there are many other random matrix models that exhibit similar behaviours as Wigner matrices. Among important random matrix models are the Wishart random matrix (sample
covariance matrix), which is of the form $XX^{*}$ for an $M\cross N$ random matrix $X$ with i.i. . entries, and the
Circular Ensembles COE $/CUE)$ . In some cases, a random matrix ensemble is defined by a probability
distribution on the set of matrices, such as$F_{2}$$F_{2}$$F_{1}$$L_{N},$$\{$$\cdots,$

$d$

$($

$P(M) dM=\frac{1}{Z_{N}}e^{-trQ(M)}dM$

,

(1.9)

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is a real-valued function satisfying
where $dM$ is the Lebesgue measure on the independent entries of $M,$
$Q(x)arrow\infty$ as $xarrow\pm\infty$ , andis the normalization constant (partition function) given by$Q$$Z_{N}$(1.10)$Z_{N}= \int e^{-trQ(M)}dM$ .

The set $\{M\}$ can be a set of all real symmetric matrices or a set of all Hermitian matrices. The former isknown as ‘orthogonal ensembles’ and the latter (unitary ensembles’. As the names suggest, GOE and GUE
belong to orthogonal ensembles and unitary ensembles, respectively, where the ‘potential’ becomes$Q$(1.11)$Q(x)= \frac{\beta N}{4}x^{2}$

with $\beta=1$ for orthogonal ensembles and $\beta=2$ for unitary ensembles. In the invariant ensemble, althougha typical choice for is $\beta=1$ or $\beta=2$ , it is also possible to use diﬀerent other than 1 or 2. (The choices$\beta=1$ , 2, and 4 are called ‘classical’.) In this case, the bulk universality means that the limiting distribution
depends only on under suitable conditions on $Q.$
, and other similar notations will always refer to the limit
Throughout the article, the limit, $O,$
$Narrow\infty$
unless noticed otherwise. In some cases the notation $A<B\sim$ will be used, where it means that$\beta$$\beta$$\beta$$0,$$\gg,$$\ll$$A\leq(\log N)^{C\log\log N}B$

for some constant$C$

independent of $N$ . The notation $A\sim B$ means that
$C^{-1}|B|\leq|A|\leq C|B|$

for some constant2$C$

independent of $N.$

Wigner semicircle law

In this chapter, we prove Wigner semicircle law for Wigner matrices in two diﬀerent ways. The first one by
the moment counting is almost the same as the original proof by Wigner, and it requires some combinatorialarguments. The second one uses the Stieltjes transform, which provides a nice tool in the random matrix
theory in general.

2.1

Proof by moment counting

In the original work of Wigner, a combinatorial proof by moment counting was used. In order to prove thesemicircle law, it suﬃces to show that$\mathbb{P}(|\langle\mu, f\rangle-\langle\mu_{sc}, f\rangle|>\epsilon)arrow 0$. By Weierstrass approximation theorem,
for any
and for any
polynomial. Thus, it suﬃces to prove that$f\in C_{b}(\mathbb{R})$$\epsilon>0$$f$

can be approximated by a$\langle\mu, x^{k}\ranglearrow\langle\mu_{sc}, x^{k}\rangle.$

By an explicit calculation, we can find that the $2k$ -th and $(2k-1)-st$ moments satisfy(2.1)$m_{2k}=C_{k}, m_{2k-1}=0$ ,where$C_{k}$

is the k-th Catalan number defined by$C_{k}= \frac{1}{k+1}(\begin{array}{l}2kk\end{array})=\frac{(2k)!}{(k+1)!k!}$

.

(2.2)

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Thus, it only remains to prove that$\langle\mu,$$x^{k} \rangle=\frac{1}{N}$

tr

(2.3)

$H^{k}arrow\{\begin{array}{ll}C_{k/2} if k is even,0 if k is odd.\end{array}$We first prove the following lemma.
Lemma 2.1. After taking expectation, we have$\mathbb{E}\langle\mu,$(2.4)$x^{k}\ranglearrow\{\begin{array}{ll}C_{k/2} if k is even,0 if k is odd.\end{array}$

Proof. Consider(2.5)$\mathbb{E}\langle\mu,$$x^{k} \rangle=\frac{1}{N}\sum_{i_{1},i_{2},\cdots,i_{k}}\mathbb{E}[H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}]$Since each entry in $H$ is independent (up to the constraint on the symmetry), iffor
For givenconsider a graphwith vertices. From the argument
and edges
above, it suﬃces to consider the graphs where vertices andare connected at least twice. In particular,
if the graphcontains more than $(k/2)+1$ distinct vertices, the contribution from the graph to thesum vanishes.
We now estimate. Each entry of $H$ contains the factor $N^{-1/2}$ , hence$N^{-k/2}$
the expectation contains the factor
If we have distinct vertices in the graph, then, we have
$N$ . Together with the factor$N(N-1)\cdots(N-r+1)$ diﬀerent choices of labeling the vertices using 1, 2,
in front of the summation, we conclude that the conribution from the graphto the sum isunless $r\geq(k/2)+1.$
So far, we have seen that the only case we need to consider is $r=(k/2)+1$ , in particular, the sum
converges to when is odd. When is even, the graphs we want to consider are doubly connected trees,
which can be identified with ordered trees, with $(k/2)+1$ vertices. The number of diﬀerent ordered trees
with $(k/2)+1$ vertices is, thus$\{i_{j}i_{j+1}\}\neq\{i_{j’}i_{j’+1}\}$some $1\leq j\leq k$ and all $j’\neq j$ , then the summand vanishes. (Here, we let $i_{k+1}=i_{1}$$(i_{1}, i_{2}, \cdots, i_{k})$$G_{i_{1}\cdots i_{k}}$$i_{1},$$i_{2},$$\cdots,$$i_{k},$$(i_{1}i_{2}, i_{2}i_{3}, \cdots, i_{k}i_{1})$$i_{j+1}$$i_{j}$$G_{i_{1}\cdots i_{k}}$$\mathbb{E}[H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}]$$r$$\cdots,$$N^{-1}$$G_{i_{1}\cdots i_{k}}$

$0$

$k$

$0$

$k$

$C_{k/2}$(2.6)$\frac{1}{N}\sum_{i_{1},i_{2},\cdot\cdot,i_{k}}.\mathbb{E}[H_{i_{1}i_{2}}H_{i_{2}i_{3}}\cdots H_{i_{k-1}i_{k}}H_{i_{k}i_{1}}]arrow C_{k/2}$

when

$k$

is even.

$\square$

In order to complete the proof of the semicircle law, it is enough to prove that
(2.7)$\mathbb{E}[\langle\mu_{)}x^{k}\rangle^{2}]-[\mathbb{E}\langle\mu, x^{k}\rangle]^{2}arrow 0$

and use Chebyshev inequality. It requires more graph counting, which we omit here.

2.2

Stieltjes transform

We briefly study the basic properties of Stieltjes transform.
Definition 2.2. For a finite measure$v$

on the real line, the Stieltjes transform of$m_{1}$forノ$(z):= \int_{\mathbb{R}}\frac{\nu(dx)}{x-z}$$z\in \mathbb{C}\backslash \mathbb{R}.$If we let $z=E+i\eta$ , then we have the following trivial properties.

$v,$$m_{\nu}$

is defined by
(2.8)

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Lemma 2.3. Let$m_{\nu}$

be the Stieltjes transform of a finite measure$\nu$

on the real line.

1. If $\eta>0$ , then ${\rm Im} m_{\nu}(z)>0.$

2. If$\nu(\mathbb{R})=C_{\nu}$, then $|m_{\nu}(z)|\leq C\eta^{-1}$

The Stieltjes transform
transform.

Lemma 2.4. Assume that$m_{\nu}$$\nu$

is (complex) analytic on$\mathbb{C}\backslash \mathbb{R}$. It is also possible to (invert’ the Stieltjes

is a probability measure without any point mass (atom). Then,$\frac{1}{\pi}{\rm Im} m_{\nu}(\cdot+i\eta)arrow\nu$

vaguely as$\eta\searrow 0$. In particular, for any interval$I,$$\nu(I)=\frac{1}{\pi}\lim_{\eta\searrow 0}\int_{I}{\rm Im} m_{\nu}(E+i\eta)dE$

.

(2.9)

As we can see from the above lemma, we can recover information on the measure from its Stieltjes
transform. Thus, given a sequence of measures, it is natural to consider the Stieltjes transform of themeasures in order to find its limit. The following lemma supports the technique.
be a sequence of random probability measures. Then, $m_{\nu}.(z)arrow m_{\nu}(z)$ almost surely (inLemma 2.5. Let
almost surely (in probability)
probability) for some deterministic probability measure if and only ifplane.
upper
every
in
the
half
for
$(\nu_{i})$$\nu_{i}arrow\nu$

$\nu$

$z$

The Stieltjes transform of a measureshows that$\nu$

contains information on the moments of . A formal calculation$\nu$$S_{\mu}(z)= \int\frac{\mu(dx)}{x-z}=-\frac{1}{z}\sum_{k=0}^{\infty}\frac{1}{z^{k}}\langle x^{k}, \mu\rangle.$

For the semicircle distribution, if we consider$g(z):= \sum_{k=0}^{\infty}m_{k}z^{k},$then, for $|z|<1/2,$ $g(z)$ is analytic. $Moreover_{\}}$ we find that$g(z)=1+z^{2}[g(z)]^{2}= \frac{1-\sqrt{1-4z^{2}}}{2z^{2}}.$We also have for $|z|<1/2$ that$g(z)= \sum_{k=0}^{\infty}z^{k}\int x^{2k}\mu_{sc}(x)dx.$

We thus obtain the Stieltjes transform of the semicircle measure$m_{sc}=- \frac{1}{z}g(\frac{1}{z})=\frac{-z+\sqrt{z^{2}-4}}{2}$2.3.

(2.10)

Local semicircle law

In order to understand the spectrum of a given operator, we use the resolvent, or the Green’s function defined
as follows:

Definition 2.6 (Resolvent(Green’s function
resolvent of $H$ is defined by

For a real symmetric (complex Hermitian) matrix $H$ , the$G$$G(z) :=(H-zI)^{-1}$(2.11)

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The idea of using Stieltjes transform to prove the semicircle law based on the averaged Green’s function$m(z)$

Recall that the empirical measure$\mu$$:= \frac{1}{N}$tr$G(z)= \frac{1}{N}\sum_{j=1}^{N}G_{jj}(z)$(2.12).

was defined by$\mu=\frac{1}{N}\sum_{j=1}^{N}\delta_{\lambda_{j}}.$If we calculate the Stieltjes transform of , then we get$\mu$tr $(H-zI)^{-1}=m(z)$ .

$m_{\mu}(z)= \int\frac{\mu(dx)}{x-z}=\frac{1}{N}\sum_{j=1}^{N}\frac{1}{\lambda_{j}-z}=\frac{1}{N}$

Thus, the semicircle law can be attained once we prove that $marrow m_{sc}$ as $Narrow\infty.$
To estimate the diﬀerence $|m-m_{sc}|$ , we use the following formula:

Lemma 2.7 (Schur complement formula). Let
be the submatrix of $H$ attained by removing the i-th row
and the column of H. (The indices are not changed.) If we letbe the resolvent of, then we have$H^{(i)}$$G^{(i)}$$H^{(i)}$$G_{ii}= \frac{1}{h_{ii}-z-\sum_{s,t}^{(i)}h_{is}G_{st}^{(i)}h_{ti}}.$

From Schur complement formula, together with the fact that $h_{ii}\sim N^{-1/2}$ , we find that
$G_{ii} \simeq\frac{1}{-z-N^{-1}\sum_{j}^{(i)}G_{jj}^{(i)}}\simeq\frac{1}{-z-m(z)}.$(See [6] for more detail.) In particular,$m(z) \simeq\frac{1}{-z-m(z)}.$Since $m_{sc}(z)$ is the solution to the equation$m_{sc}(z)= \frac{1}{-z-m_{sc}(z)},$we can estimate the diﬀerence $|m(z)-m_{SC}(z)|$ in terms of $N.$
For any fixed $z=E+i\eta$ , the best result known [8] is that$|m(z)-m_{sc}(z)|< \sim\frac{1}{N\eta}$,(2.13)

which is enough to prove the semicircle law in the limit $Narrow\infty$ . (It is believed to be optimal.) If we choose$\eta\ll 1$ , we can obtain more information on the distribution of eigenvalues. Because Equation (2.13) describes
the local behaviour of the eigenvalues, it is often called the local semicircle law. The local law has many
important consequences including the complete delocalization of eigenvectors and the rigidity of eigenvalues,
which will be discussed in the following chapters.

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3Green IUnction Comparison

3. 1

Lindeberg replacement strategy

Suppose that we want to prove the central limit theorem. For simplicity, assume thatfor any finite . Let, andrandom variables with$\mathbb{E}Y_{i}=0,$$\mathbb{E}|Y_{i}|^{2}=1$$\mathbb{E}|Y_{i}|^{p}<C_{p}$$(Y_{i})$are i.i. . real
$d$

$p$

$W_{N}= \frac{Y_{1}+Y_{2}+\cdots+Y_{N}}{\sqrt{N}}.$

Suppose that$(X_{i})$are i.i. . real Gaussian random variables with$d$$\mathbb{E}X_{i}=0$and$\mathbb{E}|X_{i}|^{2}=1$. We also let$Z_{N}= \frac{X_{1}+X_{2}+\cdots+X_{N}}{\sqrt{N}}.$is a centered Gaussian with variance 1, which can be easily checked by a Fourier transform
Notice thatmethod. Define$Z_{N}$$W_{N}^{(i)}= \frac{X_{1}+\cdots+X_{i}+Y_{i+1}+\cdots+Y_{N}}{\sqrt{N}})$in particular, $W_{N}^{(0)}=W_{N}$ and $W_{N}^{(N)}=Z_{N}$ . We now consider the m-th moment of, we find thatwith

$W_{N}$. Since (Y) are i.i.

$d.$$\mathbb{E}Y_{i}=0$(3.1)$\mathbb{E}(W_{N})^{m}=\sum_{i_{1},i_{2},\cdot\cdot,i_{k}}.\mathbb{E}(\frac{Y_{i_{1}}}{\sqrt{N}})^{r_{1}}(\frac{Y_{i_{2}}}{\sqrt{N}})^{r_{2}}\ldots(\frac{Y_{i_{k}}}{\sqrt{N}})^{r_{k}}$, $i_{k}>0$ and $i_{1}+i_{2}+\cdots+i_{k}=m.$with. from the argument above, we can easily find that
We now consider$i_{1},$$i_{2},$$\cdots$$\mathbb{E}[(W_{N}^{(j)})^{m}-(W_{N}^{(j-1)})^{m}]$(3.2)$\mathbb{E}[(W_{N}^{(j)})^{m}-(W_{N}^{(j+1)})^{m}]=\sum_{s=2}^{m}C_{s}\mathbb{E}[(\frac{X_{j}}{\sqrt{N}})^{S}-(\frac{Y_{j}}{\sqrt{N}})^{s}]$andThey $may$ depend $on m, but we$ omit $the$ detail. Since
for some finite constants
have matching second moment, we find that $\mathbb{E}[(W_{N}^{(j)})^{m}-(W_{N}^{(j+1)})^{m}]=O(N^{-3/2})$ . Applying the argument$N$ -times, we find from the telescoping series$C_{2},$ $C_{3},$$\cdots$$)$$($$Y_{j}$$X_{j}$$\mathbb{E}[(W_{N})^{m}-(Z_{N})^{m}]=\sum_{j=0}^{N-1}\mathbb{E}[(W_{N}^{(j)})^{m}-(W_{N}^{(j+1)})^{m}]$that $\mathbb{E}[(W_{N})^{m}-(Z_{N})^{m}]=O(N^{-1/2})$ . This proves the central limit theorem.
The advantage of using Lindeberg replacement strategy in proving CLT is that it does not require the. Instead, the moments are simply compared with well-known objects
exact calculation on the moments of-the moments of the Gaussian-and the proof can be done by controlling the error terms in the comparison.
Lindeberg replacement strategy has many important applications in random matrix theory, most notably,
the Green function comparison for the universality.
$W_{N}$

3.2

Eigenvalue counting and Green function

For $E_{1}<E_{2}$ , let$\mathcal{N}(E_{1}, E_{2})=|\{\lambda_{j}:E_{1}\leq\lambda_{j}<E_{2}\}|,$

i.e., the number of eigenvalues in the interval
we have$[E_{1}, E_{2}$). If we consider the characteristic function$\mathcal{N}(E_{1}, E_{2})=tr\chi(H)$.

$\chi=\chi_{[E_{1},E_{2})},$

113
Since is not a continuous function, we want to approximate it by a smooth function. We use the Poisson
kernel in the upper half plane, i.e.,$\chi$$P_{\eta}(x)= \frac{\eta}{\pi(x^{2}+\eta^{2})}.$

Notice thatUsing

$P_{\eta}*\chi$is a smooth function for any $\eta>0.$

instead of , we find$P_{\eta}*\chi$$\chi$$\mathcal{N}(E_{1}, E_{2})=tr\chi(H)\simeq tr(P_{\eta}*\chi)(H)=tr\int_{E_{1}}^{E_{2}}P_{\eta}(H-E)dE$

(3.3)

.

Applying the identity$P_{\eta}(x)= \frac{1}{\pi}{\rm Im}\frac{1}{x-i\eta},$

we get$\mathcal{N}(E_{1}, E_{2})\simeq\frac{1}{\pi}\int_{E_{1}}^{E_{2}}$tr ${\rm Im} \frac{1}{H-E-i\eta}dE=\frac{N}{\pi}\int_{E_{1}}^{E_{2}}{\rm Im} m(E+i\eta)dE$ .

(3.4)

Roughly speaking, this approximation shows that the estimate on the diﬀerence of the Green functions of
two diﬀerent Wigner matrices may imply the universality of eigenvalue distribution.

3.3

Four moment theorem and Green function comparison theorem

Definition 3.1 (Moment matching). For two independent real random variables $X$ and

$Y$

, if

$\mathbb{E}X^{k}=\mathbb{E}Y^{k}$, we say that the firstfor $k=1$ , 2,
use the condition
$p$

$p$

moments of $X$ and$Y$match. For complex random variables, we$\mathbb{E}\overline{X}^{j}X^{k}=\mathbb{E}\overline{Y}^{j}Y^{k}$

for

$j,$

$k=0$ , 1, 2, $\cdots,$ $p$ with $j+k\leq p$ , then we say that the first $p$ moments of $X$ and $Y$ match.

If we have some moment matching condition on the entries of two Wigner matrices $H$ and $H’$ , then the
eigenvalue distributions of them can be easily bounded. The following theorem by Tao and Vu [14] showsthat matching four moments is always suﬃcient.

Theorem 3.2 (Four moment theorem). Let $H$ and $H’$ be Wigner matrices. Assume that the first four
moments ofmatch, the first two moments ofandmatch, and the subexponential decay
andcondition holds for the single entry distributions. Then, there exists a small positive constantsuch that,
positive , for any function
satisfying
for any suficiently
for $j\leq 5$ , and forany selection of -tuple of indices,, we have$h_{ij}$$h_{ii}$$h_{ij}’$$h_{ij}$$c_{0}$$F:\mathbb{R}^{k}arrow \mathbb{R}$$\mathcal{S}mall$$\epsilon$$k$$i_{1},$$i_{2},$$\cdots$$|\nabla^{j}F|\leq N^{\epsilon}$$i_{k}\in[\epsilon N, (1-\epsilon)N]$$|\mathbb{E}F(\lambda_{i_{1}}, \lambda_{i_{2}, )}\lambda_{i_{k}})-\mathbb{E}F(\lambda_{i_{1}}’, \lambda_{i_{2}}’, \cdots, \lambda_{i_{k}}’)|\leq N^{-c_{0}}$(3.5)

for any suficiently large $N.$
In particular, if a real symmetric Wigner matrix $H$ satisfies that$\mathbb{E}x_{ij}=0, \mathbb{E}x_{ij}^{2}=1, \mathbb{E}x_{ij}^{3}=0, \mathbb{E}x_{ij}^{4}=3, Ex_{ii}=0, \mathbb{E}x_{ii}^{2}=2,$then we always have universality. In order to find the statistics of eigenvalues around a fixed point, insteadof considering the eigenvalues with fixed indices, the following theorem can be used [5].

114
and
Theorem 3.3 (Green function comparison theorem). Suppose thatandfying the subexponential decay condition and the first four moments ofa positive constantsuch that, for any $\eta\in[N^{-1-\epsilon}, N]$ and for any
have$H^{v}$$\epsilon>0$

are Wigner matrices satis-$H^{w}$$H^{v}$$z_{1},$

match. Then, there exist
, we
with

$H^{w}$$z_{2}$$\lim_{Narrow\infty}[\mathbb{E}tr G^{v}(z_{1}) tr G^{v}(z_{2})-\mathbb{E}trG^{w}(z_{1}) tr G^{2}(z_{2})]=0$

where$G^{v}$

and$G^{w}$

denote the Green functions of$H^{v}$

and$H^{w}$${\rm Im} z_{1},$${\rm Im} z_{2}=\pm\eta$

(3.6)

,

, respectively.

Both of the theorems can be applied to prove the universality, provided that a nice matrix $H’$ exists.

Idea of proof of Green function comparison theorem. Suppose that we want to estimate $|\mathbb{E}(m-m’)|$ , where
and $m’$ denote the averaged Green functions of $H$ and $H’$ , respectively. Let $\gamma=N(N+1)/2$ . Adopting, wherethe Lindeberg replacement strategy, we consider a sequence of Wigner matrices$H_{0}=H,$ $H_{\gamma}=H’$ , andand $H_{k+1}$ diﬀer at most in two matrix entries. We now consider a telescoping
series$m$$\{H_{0}, H_{1}, \cdot\cdot, H_{\gamma}\}$$H_{k}$(3.7)$\mathbb{E}m-\mathbb{E}m’=\frac{1}{N}\sum_{k=1}^{\gamma}[\mathbb{E}tr(H_{k-1}-z)^{-1}-\mathbb{E}tr(H_{k}-z)^{-1}]$

Suppose thatand $H_{k-1}$ diﬀer only at the $(i,j)$ -position. (It can be a diagonal element, i.e., it is allowed
to have $i=j.$ ) Let be a matrix having zero entries at the $(i, j)$ -position and all the other entries are equal
and $H_{k+1}$ . We letto those of$H_{k}$$Q$$H_{k}$$H_{k}=Q+y_{)} H_{k-1}=Q+W.$

We now expand the resolvents ofand $H_{k-1}$ . Let$H_{k}$$R=(Q-z)^{-1}, S=(H_{k}-z)^{-1}, T=(H_{k-1}-z)^{-1}$Then, we have an identity$S=R-RVS$ .

(3.8)$S=R-RVR+RVRVR-(RV)^{3}R+(RV)^{4}R-(RV)^{5}S$(3.9)

Applying (3.8) successively, we also find thatWe then calculate the expectation of$S_{aa}$. From (3.9), we find that$\mathbb{E}S_{aa}=\mathbb{E}R_{aa}-\mathbb{E}(RVR)_{aa}+\cdots-[(RV)^{5}S]_{aa}.$

Consider the second term in the above expansion. Since the entries of
we have$V$

are all zero except$V_{ij}$

and$V_{ji},$$(RVR)_{aa}=R_{ai}V_{ij}R_{ja}+R_{aj}V_{ji}R_{\dot{n}a}.$

However,
respect to$Q$

is independent of, we find that$V$

, thus $R$ is also independent of $V$ . Thus, by taking partial expectation with

$V_{ij}$$\mathbb{E}(RVR)_{aa}=\mathbb{E}V_{ij}\cdot \mathbb{E}[R_{ai}R_{ja}]+\mathbb{E}V_{ji}\cdot \mathbb{E}[R_{aj}R_{ia}]=0.$

The third term in the expansion (3.9) does not vanish. Nevertheless, we can decouple
the same idea as above, so we find that$\mathbb{E}(RVRVR)_{aa}=\mathbb{E}[V_{ij}V_{ji}]\cdot \mathbb{E}[R_{ai}R_{jj}R_{ia}]+\cdots$

We can also consider the expansion of$T$, alld in the expansion, we find a term$\mathbb{E}(RWRWR)_{aa}=\mathbb{E}[W_{ij}W_{ji}]\cdot \mathbb{E}[R_{ai}R_{jj}R_{\triangleleft a}]+\cdots,$

$V$

and

$R$

by using

115
which will exactly cancel the term above in the diﬀerence
matching condition, we find that$\mathbb{E}S_{aa}-\mathbb{E}T_{aa}$. In this way, from the four moment$\mathbb{E}S_{aa}-\mathbb{E}T_{aa}=-\mathbb{E}[(RV)^{5}S]_{aa}+\mathbb{E}[(RW)^{5}T]_{aa}.$

From the local law, we can attain a priori estimates $|S_{ii}|<C,$ $|S_{ij}|=o(1)$ , and similar results forT. Rom those estimates, we find that$\mathbb{E}S_{\alpha a}-\mathbb{E}T_{aa}=O(N^{-5/2})$

$R$

and

.

Since we have $O(N^{2})$ terms in the telescoping series (3.7), we conclude that$\mathbb{E}m-\mathbb{E}m’=O(N^{-1/2})$.
$\square$Remark 3.4. The four moment theorem as well as the Green function comparison theorem can be extend to
the edge of the spectrum.

Remark 3.5. The fourth moment matching condition (and the two moment matching condition for the
diagonal entries) can be weakened tofor some
In the proof of the bulk universality and the edge universality, the Green function comparison theorem
is applied to compare the eigenvalue distribution of a given Wigner matrix and a reference matrix such as
Gaussian ensembles. In some situations, we even consider a chain of Wigner matrices and compare the Greenfunctions of them one pair at a time.
$\delta>0.$$\mathbb{E}(h_{\iota’j})^{4}-\mathbb{E}(h_{ij}’)^{4}=O(N^{-\delta})$4

Dyson sine kernel and the bulk universality

4.1

Joint density of Gaussian unitary ensemble

Recall that we defined GUE as a Wigner matrix with Gaussian random variables. With a slight abuse ofnotation, we may write it using the Gaussian measures such as$dh_{ii}=C_{1}^{(N)_{e^{-Nh_{t}^{2}}:}/2}dM_{ii}, dh_{ij}^{R}=C_{2}^{(N)}e^{-N(h_{ij}^{R})^{2}}dM_{ij}^{R}, dh_{ij}^{I}=C_{2}^{(N)}e^{-N(h_{ij}^{I})^{2}}dM_{ij}^{I},$, andare Lebesgue measures on . (Here, and stand for ‘real’ and (imaginary’,
respectively.) If we consider the probability measure corresponding to GUE defined on the-dimensional
space of Hermitian matrices, then we obtain that

where$dM_{ii},$$dM_{ij}^{R}$$\mathbb{R}$$dM_{ij}^{I}$

$R$

$I$

$N^{2}$$dH=\prod_{i=1}^{N}dh_{ii}\prod_{i<j}dh_{ij}^{R}dh_{ij}^{I}=C_{N}\exp(-\frac{N}{2}\sum_{i,j=1}^{N}|h_{ij}|^{2})\prod_{i=1}^{N}dM_{ii}\prod_{i<j}dM_{ij}^{R}dM_{ij}^{I}$(4.1)$=C_{N}\exp$ $(- \frac{N}{2}$tr$H^{2}) \prod_{i=1}^{N}dM_{ii}\prod_{i<j}dM_{ij}^{R}dM_{ij}^{I}$

For the GUE matrix, we can diagonalize it and havefor a unitary matrix and a real diagonal
matrix. Notice that there is arbitrariness in the choice of , which corresponds
to the phase factor, i.e., for any. Thus, we consider the, we havequotient group $\tilde{U}=U/T$ , where is the unitary group of order $N$ , and let
be the Haar measure on U.
$H=U\Lambda U\dagger$

$U$

$U$

$\Lambda=dia9(\lambda_{1}, \lambda_{2}, \cdots, \lambda_{N})$$T=diag(e^{i\theta_{1}}, e^{i\theta_{2}}, \cdots, e^{i\theta_{N}})$$U$$T\Lambda\tau\dagger=\Lambda$$d\tilde{U}$We now have
$dH=C_{N}\exp$ $(- \frac{N}{2}$ tr $H^{2}) \mathcal{J}(H)d\Lambda dU=C_{N}\exp(-\frac{N}{2}\sum_{i=1}^{N}\lambda_{i}^{2})|\mathcal{J}(H)|d\Lambda dU$

(4.2)

116
where
that

$\mathcal{J}(H)$is the Jacobian and$d\Lambda=d\lambda_{1}d\lambda_{2}\cdots d\lambda_{N}$is the Lebesgue measure on$\mathcal{J}(H)=\prod_{i<j}(\lambda_{i}-\lambda_{j})^{2}$$\mathbb{R}^{N}$

A calculation shows

,

(4.3)

hence we find the formula$dH=\frac{1}{Z_{N}}\exp(-\frac{N}{2}\sum_{i=1}^{N}\lambda_{i}^{2})\prod_{i<j}(\lambda_{i}-\lambda_{j})^{2}d\Lambda d$

In particular, for any function$f(H)=f(\lambda_{1}, \lambda_{2}, \cdots, \lambda_{N})$invariant underU.

$Harrow UHU^{\uparrow}$(4.4), we have$\int f(H)dH=\frac{1}{Z_{N}}\int_{\mathbb{R}^{N}}f(\lambda_{1)}\lambda_{2}, \cdots, \lambda_{N})\exp(-\frac{N}{2}\sum_{i=1}^{N}\lambda_{i}^{2})\prod_{i<j}(\lambda_{i}-\lambda_{j})^{2}d\Lambda$

.

(4.5)

The expression $\prod_{i<j}(x_{i}-x_{j})$ is known as the van der Monde determinant; using the determinant, we

can rewrite it as$\prod_{i<j}(x_{i}-x_{j})=(-1)^{n(n-1)/2}\det(\begin{array}{llll}1 1 \cdots 1x_{l} x_{2} \cdots x_{N} x_{l}^{N-l} x_{2}^{N-l} \cdots x_{N}^{N-l}\end{array})$

4.2

(4.6)

Orthogonal polynomial

Consider a family of polynomials$\{f_{k}(x)\}$, where$f_{k}$

is a degree$k$monic polynomial with the following

property:$\int_{\mathbb{R}}f_{j}(x)f_{k}(x)e^{-Q(x)}dx=c_{i}c_{j}\delta_{ij}$There are many (classical’ orthogonal polynomials,
for some function and for some constantse.g., Jacobi polynomials, Laguerre polynomials, Hermite polynomials, etc., depending on the choice of $Q.$
When $Q(x)=x^{2}/2$ , we can use Hermite polynomials defined as follows:$Q$$c_{1},$ $c_{2},$$\cdots$

Definition 4.1 (Hermite polynomials). The n-th Hermite polynomial$H_{n}$

is define by(4.7)$H_{n}(x)=(-1)^{n}e^{x^{2}/2} \frac{d^{n}}{dx^{n}}e^{-x^{2}/2}$

We also define the n-th normalized oscillator wavefunction$\psi_{n}$by$\psi_{n}(x)=\frac{e^{-x^{2}/4}}{\sqrt{n!\sqrt{2\pi}}}H_{n}(x)$

.

(4.8)

Remark 4.2. Often in literature, the choice of $Q(x)=x^{2}$ is used when defining Hermite polynomials, which
gives “physicists‘ Hermite polynomials”’
Notice we chose the constant in the definition of
to ensure that$\psi_{n}$$\int_{R}\psi_{m}(x)\psi_{n}(x)dx=\delta_{mn}$.

Recall the van der Monde determinant from the last section. Since(4.9)$H_{n}$

is monic, we also have$\det(\begin{array}{llll}1 1 \cdots 1x_{1} x_{2} \cdots x_{N}\vdots \vdots \vdots x_{l}^{N-l} x_{2}^{N-1} \cdots x_{N}^{N-l}\end{array})=\det(\begin{array}{llll}H_{0}(x_{l}) H_{0}(x_{2}) \cdots H_{0}(x_{N})H_{1}(x_{l}) H_{1}(x_{2}) \cdots H_{l}(x_{N})\vdots H_{N-l}(x_{l}) H_{N-l}(x_{2}) \cdots H_{N-1}(x_{N})\end{array})$117
Thus, if we let(4.10)$K_{N}(x, y):= \sum_{k=1}^{N}\psi_{k-1}(x)\psi_{k-1}(y))$

then we obtain that$\exp(-\sum_{\prime,l=1}^{N}\frac{x_{i}^{2}}{2})\prod_{i<j}(x_{i}-x_{j})^{2}=C[\det\psi_{i-1}(x_{j})]^{2}=C\det K_{N}(x_{i}, x_{j})$.

We remark that$\int K_{N}(x, x)dx=N$ .

(4.11)We also have the following properties:
Lemma 4.3. For$\psi_{j}$anddefined above, we have

$K_{N}$

1. (Reproducing kernel property)$\int K_{N}(x, y)K_{N}(y, z)dy=K_{N}(x, z)$.

2. (Christoﬀel-Darboux formula)$K_{N}(x, y)= \sqrt{N}[\frac{\psi_{N}(x)\psi_{N-1}(y)-\psi_{N}(y)\psi_{N-1}(x)}{x-y}]$Moreover, if $x=y$ , we have
$K_{N}(x, x)=\sqrt{N}(\psi_{N}’(x)\psi_{N-1}(x)-\psi_{N-1}’(x)\psi_{N}(x))$

3. (Asymptotic behaviour)$\lim_{Narrow\infty}N^{1/4}\psi_{n}(N^{-1/2}t)=\frac{1}{\sqrt{\pi}}\cos(t-\frac{n\pi}{2})$locally uniformly, where $n=N-\ell$ for a fixed

4.3$\ell.$

Dyson sine kernel

We first consider the determinantal structure of the correlation functions. If we let$p_{N}^{(k)} (x_{1}, x_{2}, \cdots , x_{k})=\frac{1}{Z}\int\exp(-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2})\prod_{i<j}(x_{i}-x_{j})^{2}dx_{k+1}\cdots dx_{N}$$=C \int\det I\zeta_{N(X_{i},X_{j})dx_{k+1}\cdots dx_{N}},$then the reproducing kernel property shows that the determinantal structure holds forwhen $N=3$ , we have$p_{N}^{(k)}$For example,$\int\det K_{N}(x_{i}, x_{j})dx_{3}$$= \int dx_{3}[K_{N}(x_{3)}x_{1})|\begin{array}{ll}K_{N}(x_{l},x_{2}) K_{N}(x_{l},x_{3})K_{N}(x_{2},x_{2}) K_{N}(x_{2},x_{3})\end{array}|-K_{N}(x_{3)}x_{2})|_{K_{N}(x_{2},x_{1})}^{K_{N}(x_{1},x_{1})} K_{N}(x_{1},x_{3})K_{N}(x_{2},x_{3})]$$+ \int dx_{3}K_{N}(x_{3}, x_{3})|_{K_{N}(x_{2},x_{1})}^{K_{N}(x_{1},x_{1})}$$K_{N}(x_{1},x_{2})K_{N}(x_{2},x_{2})|$$=|\begin{array}{ll}K_{N}(x_{l},x_{2}) K_{N}(x_{1)}x_{1})K_{N}(x_{2},x_{2}) K_{N}(x_{2},x_{l})\end{array}|-|\begin{array}{ll}K_{N}(x_{l},x_{1}) K_{N}(x_{l},x_{2})K_{N}(x_{2},x_{1}) K_{N}(x_{2},x_{2})\end{array}|+N|_{K_{N}(x_{2},x_{1})}^{K_{N}(x_{1},x_{1})} K_{N}(x_{2},x_{2})K_{N}(x_{1},x_{2})$$=|\begin{array}{ll}K_{N}(x_{l},x_{l}) K_{N}(x_{1},x_{2})K_{N}(x_{2},x_{1}) K_{N}(x_{2},x_{2})\end{array}|$118
Collecting the tools we developed in the last section, we can prove the formula established by Dyson. We
first rescale the formulas we have and find that$\exp(-\frac{N}{2}\sum_{i=1}^{N}x_{i}^{2})\prod_{i<j}N^{-\frac{N(N-1)}{2}}(-\frac{N}{2}\sum_{i=1}^{N}x_{i}^{2})\prod_{i<j}(\sqrt{N}x_{i}-\sqrt{N}x_{j})^{2}$$=CN^{-\frac{N(N-1)}{2}}\det K_{N}(\sqrt{N}x_{i}, \sqrt{N}x_{j})$.

We then use the Christoﬀel-Darboux formula to get$K_{N}( \sqrt{N}\sqrt{N})=\frac{\psi_{N}(\sqrt{N}x_{i})\psi_{N-1}(\sqrt{N}x_{j})-\psi_{N}(\sqrt{N}x_{j})\psi_{N-1}(\sqrt{N}x_{i})}{x_{i}-x_{j}}.$When$x_{i}$

and$x_{j}$

are near zero, we let$x_{i}= \frac{\pi y_{i}}{N}, x_{j}=\frac{\pi y_{j}}{N}.$Using the property on the asymptotic behaviour of$\psi_{N}$, we obtain$K_{N}(\sqrt{N}x_{i}, \sqrt{N}x_{j})$$arrow\frac{\sqrt{N}}{\pi}\frac{\cos(\pi y_{i}-N\pi/2)\cos(\pi y_{j}-(N-1)\pi/2)-\cos(\pi y_{j}-N\pi/2)\cos(\pi y_{i}-(N-1)\pi/2)}{\pi\sqrt{N}(x_{i}-x_{j})}$$= \sqrt{N}\frac{\sin(\pi(y_{i}-y_{j}))}{\pi(y_{i}-y_{j})}.$

In particular, if we let$K(y_{i}, y_{j})= \frac{\sin(\pi(y_{i}-y_{j}))}{\pi(y_{i}-y_{j})}$,(4.12)

then we find that$p_{N}^{(k)} (y_{1}, y_{2}, \cdots, y_{k})arrow C\det[K(x_{i}, x_{j})]_{i,j=1}^{k},$fornear zero. This proves the Dyson sine kernel formula near zero.
For general eigenvalues, we use an asymptotic formula$y_{i}$$\lim_{Narrow\infty}N^{1/4}\psi_{n}(\sqrt{N}x+\frac{t}{\sqrt{N}\mu_{sc}(x)})=\mu_{sc}(x)\cos(t-\frac{n\pi}{2})$and proceed as the proof for the eigenvalues near zero.
In order to fix the constant, we may use the following lemma.

Lemma 4.4 (Integrating out lemma). For any $m\leq N$ , we have$\int\det[K_{N}(x_{i}, x_{j})]_{i,j=1}^{m}dx_{m}=(N-m+1)\det[K_{N}(x_{i}, x_{j})]_{i,j=1}^{m-1}$.

(4.13)

Instead of tracking the constants, we can easily find that$\frac{1}{N!}\det K_{N}(x_{i}, x_{j})$defines a probability measure. After applying the integrating out lemma $(N-k)$ -times, we find that$p_{N}(x_{1}, x_{2}, \cdots, x_{N})arrow\frac{(N-k)!}{N!}\det[K(x_{i}, x_{j})]_{i,j=1}^{k}$since the right hand side defines a probability measure.

,(4.14)

119
5

Edge universality and Tracy-Widom distribution

5.1

Tracy-Widom distribution

Recall that the Tracy-Widom distribution is defined by(5.1)$F_{2}(t)= \exp(-\int_{t}^{\infty}(x-t)q(x)^{2}dx)$

where $q(t)$ is a solution to the Painlev\’e II equation$q”=tq+2q^{3}$

satisfying$q(t)\sim Ai(t)$as$tarrow\infty.$

The first result on the distribution of the largest eigenvalue was based on the Fredholm determinant as
in the gap probability [9]. Considering an interval, we can see that$(t, \infty)$$\lim_{Narrow\infty}\mathbb{P}(N^{-2/3}(\lambda_{N}-2)\leq t)=1+\sum_{s=1}^{\infty}\frac{(-z)^{s}}{s!}\int_{(t,\infty)^{s}}\det[K(x_{i}, x_{j})]_{i,j=1}^{s}dx_{1}\cdots dx_{s}$

.

(5.2)

Later, it was shown by Tracy and Widom that the above limit becomes. For a diﬀerent ensemble, thelimiting distribution is diﬀerent, though it is also called the Tracy-Widom distribution. For example,$F_{2}$$F_{1}(s)= \exp(-\frac{1}{2}\int_{S}^{\infty}q(x)dx)(F_{2}(s))^{1/2},$

which is the Tracy-Widom distribution for GOE.
Using the tools as Green function comparison, we can prove that the edge universality holds for a very
general class of Wigner matrices, $e.g.$ , a Wigner matrix with subexponential decay$\mathbb{P}(|\sqrt{N}h_{ij}|\geq x)\leq C_{0}e^{-x^{1/\theta}}$for some positive constantand $\theta>1$ . Notice that this conditions guarantees that the p-th moment of
the random variable $x_{ij}=\sqrt{N}h_{ij}$ is finite for any fixed . On the other hand, if the p-th moment ofisinfinite for some $p>0$ , then the Wigner matrix may exhibit a totally diﬀerent behaviour. For example, it
is known that the Wigner semicircle law fails if the variance ofis infinite. When the variance is finite
but the 4-th moment is infinite, the largest eigenvalue can become larger than 2 although the semicircle law
holds as we can see from the following theorem [2]:$C_{0}$$p$$x_{ij}$$x_{ij}$Theorem 5.1. Let
be the largest eigenvalue of a Wigner matrix $H$ and let $x_{ij}=\sqrt{N}h_{ij}$ . Then,$Narrow\infty$asalmost surely (with probability 1) if and only iffor $i\neq j.$$\lambda_{N}$$\lambda_{N}arrow 2$$\mathbb{E}|x_{ij}|^{4}<\infty$

5.2

Poisson statistics for the largest eigenvalue

If the entries of the Wigner matrix in question have heavy tails, the edge universality may fail, i.e., thelimiting distribution for the largest eigenvalue diﬀers from the Tracy-Widom distribution [13]. More precisely,
let $x_{ij}=\sqrt{N}h_{ij}$ . If the decay ofis given by$x_{ij}$$\mathbb{P}(|x_{ij}|>s)\sim s^{-\alpha}$

for $\alpha<4$ , the largest eigenvalue has Poisson statistics [1]. Notice that the fourth moment ofin this case. For simplicity, we consider real symmetric cases only in the following.

$|x_{ij}|$

is infinite

120
When $\alpha<4$ , setting $\delta=4-\alpha$ , we have$\mathbb{P}(|h_{ij}|>N^{\frac{\delta}{4\alpha})=\mathbb{P}(|x_{ij}|}>N^{\frac{1}{2}+\frac{|f}{4\alpha})\sim N^{-\frac{a}{2}-\frac{\overline{A}}{4}}=N^{-2+\frac{\prime\dagger}{2}}}\gg N^{-2+\frac{\delta}{4}}.$Thus, the probability of having at least one entry bigger than$N^{\frac{\delta}{4a}}$is$1-(\mathbb{P}(|h_{ij}|\leq N^{\frac{\delta}{4\alpha}))^{N(N+1)/2}}\geq 1-(1-N^{-2+\frac{\delta}{4}})^{N(N+1)/2}\geq 1-Ce^{-N^{c}}$and we find that there exists with high probability an entry bigger than
, we consider
Whenfor some$\epsilon>0$$h_{ij}>N^{\epsilon}$$v_{i}=v_{j}= \frac{1}{\sqrt{2}},$$v_{k}=0$$(If i=j, we let v_{i}=1 and all the$ other entries $of v$ vanish.

$)$

if $k\neq i,$

Since

in $H.$
such that$N^{\frac{\delta}{4\alpha}}$$v=(v_{1}, v_{2}, \cdots, v_{N})^{T}\in \mathbb{R}^{N}$$j.$$\Vert v\Vert_{2}=1$

and$\langle v, Hv\rangle=\frac{1}{2}(H_{ij}+H_{ji})=H_{ij}>N^{\epsilon}$

Thus, by using the minimax principle, we find that
In general, when $\alpha<4$ , the limiting distribution of the largest eigenvalue is solely determined by thei.e., the distribution of the largest one among $(N(N+1)/2)$ i.i.d. random
order statistics ofvariables. (This becomes the Frech\’et distribution whose p.d. . is given by) For the-largest eigenvalue, by using the interlacing property of the eigenvalues, we can prove a similar statement,i.e., the limiting distribution of them is given by the order statistics. It is also known that the such an order
statistics corresponds to an inhomogeneous Poisson random point process.
$\lambda_{N}>N^{\epsilon}$$\{|h_{ij}|\}_{1\leq i<j\leq N_{\rangle}}$$\{|h_{ij}|\}_{1\leq i<j\leq N}$$f$$e^{-x^{-c}}.$

$k$

5.3

A necessary and suﬃcient condition for the edge universality

For a Wigner matrix with heavy tails, the following theorem gives a necessary and suﬃcient condition for
the edge universality [12].

Theorem 5.2. Let
be the largest eigenvalue of a Wigner matrix $H$ and let $x_{ij}=\sqrt{N}h_{ij}$ . Then, thedistribution of the rescaled eigenvalue $N^{2/3}(\lambda_{N}-2)$ converges to the Tracy-Widom distribution in probability
as $Narrow\infty$ if and only if$\lambda_{N}$$\lim_{sarrow\infty}s^{4}\mathbb{P}(|x_{ij}|\geq s)=0$

for$i\neq j.$

Notice that the condition is weaker than the existence of 4-th moment. For example, if the density of
decays as $|x|^{-5}(|\log x|^{-1})$ , then the 4-th moment does not exists since$|x_{ij}|$$l^{\infty} \frac{x^{4}}{x^{5}\log x}dx=\int_{\log s}^{\infty}\frac{dt}{t}=\infty.$

However, it satisfies the condition in the above theorem, since$\lim_{sarrow\infty}s^{4}\mathbb{P}(|x_{ij}|\geq s)=\lim_{sarrow\infty}s^{4}\int_{s}^{\infty}\frac{l}{x^{5}\log x}dx=\lim_{sarrow\infty}\frac{s^{-5}(\log_{\mathcal{S}})^{-1}}{-4s^{-5}}=0.$

We remark that the condition in Theorem 5.2 does not contradicts the conclusion in Theorem 5.1 because of
the diﬀerent notion of the convergence.
In order to prove the necessity, one may argue as in the previous section, i.e., the probability of having
an element bigger than 4 is quite large if the condition fails. The proof of the suﬃciency requires the Green121
function comparison theorem. The Green function comparison at the edge features a diﬀerent scaling ofRecall that$\mathcal{N}(E_{1}, E_{2})\simeq\frac{1}{\pi}\int_{E_{1}}^{E_{2}}$$\eta.$

tr ${\rm Im} \frac{1}{H-E-i\eta}dE=\frac{N}{\pi}\int_{E_{1}}^{E_{2}}{\rm Im} m(E+i\eta)dE.$

To see the fluctuation of the largest eigenvalue, we letsatisfy $|E_{1}-2|,$ $|E_{2}-2|\sim N^{-2/3}$ Then,andthe fluctuation of $m$ at this scale can be estimated by using $\eta\sim N^{-2/3}$ . In this scaling, the oﬀ-diagonalterm in the resolvent expansion becomes $G_{ij}\sim N^{-1/3}$ , which gives an extra converging factor in the Greenfunction comparison.
$E_{1}$$E_{2}$

References[1] Auﬃnger, A., Ben Arous, G., P\’ech\’e, S.: Poisson Convergence for the Largest Eigenvalues of Heavy Tailed
Random Matrices, Ann. Inst. Henri Poirlcar\’e-Probab. Stat. 45, 589-610 (2009).

[2] Bai, Z. D., Yin, Y. Q.: Necessary and Suficient Conditions for Almost Sure Convergence of the Largest
Eigenvalue of a Wigner Matrix, Ann. of Probab. 16, 1729-1741 (1988).
[3] Dyson, F. J.: Statistical Theory of Energy Levels of Complex Systems, I, II, and III., J. Math. Phys. 3,
140-156, 157-165, 166-175 (1962).
[4] Dyson, F. J.: Correlations between Eigenvalues of a Random Matrix, Commun. Math. Phys. 19, 235-250,(1970).

[5] Erd\’o’s, L.: Universality of Wigner random matrices: a Survey of Recent Results, arXiv:1004.0861v2(2010).
[6] Erd\’o’s, L., Schlein, B., Yau, H.-T.: Semicircle Law on Short Scales and Delocalization of Eigenvectors for
Wigner Random Matrices, Ann. Probab. 37, 815-852 (2009).
[7] Erd\’o’s, L., Schlein, B., Yau, H.-T.: Universality of Random Matrices and Local Relaxation flow, Invent.
Math. 185, 75-119 (2011).

[8] Erd\’os, L., Yau, H.-T., Yin, J.: Rigidity of Eigenvalues of Generalized Wigner Matrices, Adv. Math. 229,
1435-1515 (2012).
[9] Forrester, P.: The Spectral Edge of Random Matrix Ensembles, Nucl. Phys. B402, 709-728 (1993).
[10] Mehta, M. L.: On the Statistical Properties of the Level-Spacings in Nuclear Spectra, Nuclear Phys. 18,
395-419 (1960).

[11] Mehta, M. L., Gaudin, M.: On the Density of Eigenvalues of a Random Matrix, Nuclear Phys. 18,
420-427 (1960).
[12] Lee, J. O., Yin, J.: A Necessary and Suficient Condition for Edge Universality of Wigner Matrices, To
appear in Duke Math. J., arXiv:1206.2251 (2012).
[13] Soshnikov, A.: Poisson Statistics for the Largest Eigenvalue of Wigner Random Matrices with Heavy
Tails, Elect. Comm. in Probab. 9, 82-91 (2004).

[14] Tao, T., Vu, V.: Random Matrices: Universality of the Local Eigenvalue Statistics, Acta Math, 206,127-204 (2011).
[15] Racy, C., Widom, H.: Level-Spacing Distributions and the Airy Kernel, Commun. Math. Phys. 159,
151-174 (1994).

122[16] Tracy, C., Widom, H.: On Orthogonal and Symplectic Matrix Ensembles, Commun. Math. Phys. 177,

727-754 (1996).
[17] Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62, 548-564(1955).