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semicircle law, Wigner random matrices, random Schroedinger operator, density of states, localization, extended states
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\title{Semicircle law on short scales
and delocalization of eigenvectors for Wigner random matrices}

\author{L\'aszl\'o Erd\H os${}^1$, Benjamin
Schlein${}^1$\thanks{Supported by Sofja-Kovalevskaya Award of
the Humboldt Foundation. On leave from Cambridge University, UK}\;
and Horng-Tzer Yau${}^2$\thanks{Partially supported
by NSF grant DMS-0602038} \\
\\
Institute of Mathematics, University of Munich, \\
Theresienstr. 39, D-80333 Munich, Germany${}^1$ \\ \\
Department of Mathematics, Harvard University\\
Cambridge MA 02138, USA${}^2$ \\ \\
\\}
\begin{document}

\maketitle

\begin{abstract}

We consider  $N\times N$ Hermitian random matrices with i.i.d.
entries.  The matrix is normalized so that
the average spacing between consecutive eigenvalues is of order $1/N$.
We study the connection between eigenvalue statistics
on microscopic energy scales $\eta\ll 1$ and (de)localization
properties of the eigenvectors.
Under suitable assumptions on the distribution
of the single matrix elements, we first give an upper bound on
the density of states on short energy scales of order $\eta
\sim \log N/N$. We then prove that the density of states
concentrates around the Wigner semicircle law
on energy scales $\eta \gg N^{-2/3}$.
We  show that most eigenvectors are fully delocalized
in the sense that their $\ell^p$-norms are comparable with $N^{\frac{1}{p}
-\frac{1}{2}}$ for $p\ge 2$,
 and we obtain the weaker bound $N^{\frac{2}{3}\big(\frac{1}{p}
-\frac{1}{2}\big)}$ for all eigenvectors whose eigenvalues
are separated away from the spectral edges.
We also prove that, with  a
probability very close to one,
 no eigenvector can be  localized. Finally, we give
an optimal bound on the second moment of the
Green function.
\end{abstract}

{\bf AMS Subject Classification:} 15A52, 82B44

\medskip

{\it Running title:} Semicircle law on short scales

\medskip

{\it Key words:} Semicircle law, Wigner random matrix,
random Schr\"odinger operator, density of states, localization,
extended states.

%\received{}

%\maketitle


\section{Introduction}

Denote the $(ij)$-th entry of an $N\times N$  matrix $H$  by $h_{ij}$.
We shall assume that the matrix is  Hermitian,
i.e., $h_{ij} = \overline {h_{ji}}$.
These matrices form a {\it Hermitian Wigner ensemble} if
\be
h_{i j} =   N^{-1/2} [ x_{ij} + \sqrt{-1}\;  y_{ij}],
\quad (i < j), \quad \text{and} \quad   h_{i i} =  N^{-1/2}  x_{ii},
\label{wig}
\ee
where $x_{ij},   y_{ij}$ ($i<j$) and $x_{ii}$
are independent real  random variables with  mean zero.
We assume that $x_{ij},   y_{ij}$ ($i< j$)
all have a common distribution $\nu$ with variance 1/2
and with a strictly positive density function:
$\rd \nu(x)=(\const.) e^{-g(x)}\rd x$.
The diagonal elements, $x_{ii}$,  also have  a
common distribution, $\rd\wt \nu(x)=(\const.) e^{-\wt g(x)}\rd x$,
that may be different from $\rd \nu$.
We remark that the special case $g(x)=x^2$ and $\wt g(x) = x^2/2$
 is called the {\it Gaussian Unitary Ensemble (GUE)}.
Let  $\P$ and $\E$
denote the probability and the expectation value, respectively,
 w.r.t the joint distribution of all matrix elements.

\bigskip

E. Wigner has
introduced random matrices to
model Hamiltonians, $H$, of atomic nuclei. Lacking
precise knowledge about the interaction among
different quantum states, he assumed that
the matrix elements $(\varphi, H \psi)$
for any two orthogonal states $\varphi, \psi$ are
identically distributed and as maximally independent
as the unitary symmetry group acting on the
Hilbert space of states allows. These assumptions
already imply that the distribution of $H$
is GUE (modulo changing the  expectation value
and the variance). Astonishingly, this simple model
very accurately reproduced the energy level statistics
of various large nuclei.

Random Hamiltonians are also used in solid state
physics to study electrons in disordered metallic lattices.
The simplest example is the {\it Anderson model} on a discrete lattice,
where  the disorder is modelled
by i.i.d. on-site potentials.
The Anderson model can be generalized to continuous space
and to include magnetic fields. These models are commonly referred to as
 {\it random Schr\"odinger operators}. Their key feature is that
they have an underlying  spatial structure
and only matrix elements connecting nearby sites
are non-zero, in contrast to the mean-field character
of the Wigner ensembles.

The conductance properties of metallic lattices
are strongly influenced by the spatial localization
of the eigenfunctions  of the corresponding Hamiltonian.
Depending on the energy range, on the disorder strength
and on the spatial dimension, random Schr\"odinger operators
are believed to exhibit a transition between localized ($L^2$-normalizable)
and delocalized eigenstates. These two regimes
can also be characterized by pure point or absolutely
continuous spectrum, respectively.  While the localization regime
is fairly well understood, it remains an outstanding
open problem to prove the existence of the delocalization regime.
An even more ambitious conjecture states that
the level spacing statistics of consecutive eigenvalues
(of the finite dimensional approximation) of the
random Schr\"odinger operator
also characterizes these two regimes.
In the localization regime, consecutive eigenvalues
should be independent and should follow the statistics
of a Poisson point process. In the delocalization regime,
the level spacing statistics is believed to be identical
to that of the GUE.

\bigskip

Random matrices are mostly studied from the point of view
of eigenvalue statistics such as density of states
(e.g. Wigner semicircle law) or level statistics
of consecutive eigenvalues.
The density of states is well understood for general
Wigner matrices on
macroscopic energy windows where the number of
eigenstates is proportional to $N$.
In our normalization, this corresponds to
energy windows of order one.
On the finest energy scale of order $1/N$, where individual
eigenvalues are observed, a universal level spacing distribution
is believed to emerge, that is called the Wigner-Dyson statistics.
This has been proven only for  Gaussian and related models
(see \cite{D} and references therein) and
for Wigner matrices where the distribution
of the matrix elements were Gaussian convolutions
\cite{J}. The proofs use explicit formulae for
the eigenvalue correlation functions which are
available only for Gaussian related models.


The eigenvalue distribution of general  Wigner
matrices is poorly understood on microscopic
energy scales $\eta\ll 1$ due to the lack of explicit
formulae for the eigenvalue distribution. The fluctuations
of the density of states are known to be negligible
down to energy windows of order $N^{-1/2}$ \cite{GZ}.
and the expected value  is also known to
follow the semicircle law on scales $N^{-1/2}$ and larger
\cite{BMT}.
It is an open problem to show
that  both the fluctuation and the expected value
of the density of states can be controlled down to energy
scales of order $1/N$. This would be the first step
toward the proof of Wigner-Dyson universality for
Wigner matrices.
 Moreover, given the presumed connection between eigenvalue
statistics and eigenfunction localization
in the case of random Schr\"odinger operators,
it is natural to investigate the (de)localization properties
of the eigenvectors of random  matrices.
Due to the mean field character of the Wigner matrix,
the eigenvectors are believed to be extended; a conjecture
that is consistent with the expected
repulsion of neighboring eigenvalues.


We remark that in finite dimensional Hilbert spaces,
extended states are characterized by
$\ell^p$-norms with $p\neq 2$ instead of the absolute continuity
of the spectrum. If all components
of an $\ell^2$-normalized vector $\bv\in \bC^N$ are equal,
then $\| \bv\|_p = N^{\frac{1}{p}-\frac{1}{2}}$.
Thus deviations of the $\ell^p$-norm of an eigenvector
from $N^{\frac{1}{p}-\frac{1}{2}}$ can be used to quantify
the delocalization properties of the state.
In particular, T. Spencer
has posed the question to prove that the $\ell^4$-norm of
all eigenvectors are of order $N^{-1/4}$.

\bigskip

In this paper we prove several results  in these
directions for general Hermitian Wigner matrices.
In Theorem \ref{du} we give
an  upper bound on the eigenvalue density
down to energy scales of order $\eta \ge \frac{\log N}{N}$.


Theorem \ref{thm:fluc}  states that the
density of states concentrates around its mean
in probability sense
down to energy windows of order $\eta \gg N^{-2/3}$ (modulo
logarithmic corrections), improving  the fluctuation
result of \cite{GZ} from scales $\eta\gg N^{-1/2}$.
In Theorem \ref{thm:sc}
we prove that the expectation value of the
density of states on scales $\eta \gg N^{-2/3}$
 converges to the Wigner semicircle law.
The previous best result \cite{BMT} was valid for scales
$\eta\gg N^{-1/2}$.
These two theorems establish the validity
of the Wigner semicircle law for all energy windows
of  order $\eta \gg N^{-2/3}$.



In Theorem \ref{thm:loc} we show that most
eigenvectors are fully extended in the sense
that their $\ell^\infty$-norm is of
order $N^{-1/2}$ (modulo logarithmic corrections).
We remark that
this result can be easily obtained for all eigenvectors in the
GUE case, by using the underlying unitary symmetry group.
The reason why our proof of Theorem \ref{thm:loc}
  does not apply to  all eigenvectors
is the lack of the  lower bound on
the density of states on the very short scales of $\eta\gg 1/N$.
However, the results of Section \ref{sec:fluc} imply
a bound of order $N^{\frac{2}{3}\big(\frac{1}{p}-\frac{1}{2}\big)}$
 for the $\ell^p$-norm ($p\ge 2$) of
{\it all} eigenvectors away from the spectral edge
(Corollary \ref{cor:linfty}).



In Theorem \ref{thm:green}, by
using the bounds on the eigenvectors,
we give an estimate on the second moment
of the Green function. Finally, in
Theorem \ref{thm:Meta} we prove that
no eigenvector is strongly localized
in the sense that no eigenvector can
be essentially supported on a small
percentage of the sites. As a corollary,
we show that the $\ell^p$-norm of
the eigenvectors is $N^{\frac{1}{p}-\frac{1}{2}}$
for $1\leq p <2$.


We remark that all our results hold for Wigner ensemble
of real symmetric matrices as well. We will present
the Hermitian case only, the proofs for the
real case require only obvious modifications.



\bigskip
In some of our results, we need to assume further conditions
on the distributions
of the matrix elements in addition to \eqref{wig}.
For convenience, we list the conditions we use in some of our
theorems:
\begin{itemize}
\item[{\bf C1})] The functions $g$ and $\wt g$ are twice differentiable
and they satisfy
\be
   g''(x) \leq M, \qquad \wt g''(x) \leq M
\label{gM}
\ee
with some finite $M$.
\item[{\bf C2)}] There exists a $\delta>0$ such that
\be
  \int e^{\delta x^2}\rd \nu(x)  <\infty,
\label{x2}
\ee
and the same holds for $\wt\nu$.
\item[{\bf C3)}]
 The measures $\nu, \wt\nu$ satisfy the spectral gap inequality, i.e.
there exists a constant $C$ such that for  any function $u$
\be
   \int \Big|u-\int u\; \rd\nu\Big|^2   \rd \nu
  \leq C \int|\nabla u|^2 \rd \nu, \qquad
 \label{gap}
\ee
and the same holds for $\wt \nu$.
\item[{\bf C4)}] The measures $\nu, \wt \nu$ satisfy the logarithmic
Sobolev inequality, i.e. there exists a constant $C$ such that for
any density function $u>0$ with $\int u\, \rd\nu =1$,
\be
  \int u\log u\;  \rd \nu \leq C\int |\nabla \sqrt{u}|^2 \rd \nu\, ,
\label{logsob}
\ee
and the same holds for $\wt \nu$.
\end{itemize}
We remark that C4) implies C3) and
that all conditions are satisfied if $c_1 \leq g'',\wt g''\leq c_2$
for some positive constants $c_1, c_2$.






\bigskip

{\it Notation.} We will use the notation $|A|$ both for
the Lebesgue measure of a set $A\subset \bR$ and  for
the cardinality of a discrete set $A\subset \bbZ$.
The usual Hermitian scalar product for vectors $\bx,\by\in \bC^N $ will be
denoted by $\bx\cdot \by$ or by $( \bx, \by)$.
We will use the convention that $C$ denotes generic large
constants and $c$ denotes generic small positive constants
whose values may change from line to line. Since we
are interested in large matrices, we always assume that $N$
is sufficiently large.











\section{Upper bound on the density of states}

The typical number of eigenvalues in an interval $I$
within the spectrum is expected to be of order $N|I|$.
The following theorem proves the corresponding  upper bound.



\begin{theorem}\label{du} Let $H$ be an $N\times N$
 Wigner matrix as described in \eqref{wig} and
we assume condition \eqref{gM}.
 Let $I \subset \bR$ be an interval
with $|I|\ge (\log N)/N$
and denote by $\cN_I$ the number of
eigenvalues of $H$ in the interval $I$. Then there exists a constant
$c>0$ such that  for any $K$ large enough
\be
\P \big\{ \cN_I \geq KN|I| \big\}\leq  e^{-c K N|I|}.
 \label{cn}
\ee
\end{theorem}


For a fixed spectral parameter,
$z=E+i\eta$ with $E\in\bR$, $\eta>0$,
we denote $G_z = (H-z)^{-1}$
the Green function.
 Let $\mu_1\leq \mu_2\leq \ldots \leq \mu_N$
be the eigenvalues of $H$ and let $F(x)$ be the empirical counting
function of the eigenvalues
\be
  F(x)= \frac{1}{N}\big| \, \big\{ \al \; : \; \mu_\al \leq x\big\}\Big|\; .
\label{Fdef}
\ee
We define the Stieltjes transform of $F$ as
\be
   m= m(z) =\frac{1}{N}\text{Tr} \; G_z = \int_\bR \frac{\rd F(x)}{x-z}\, ,
\label{Sti}
\ee
and we let
\be
\rho=\rho_{\eta}(E) = \frac{ \text{Im} \;  m(z)}{\pi}=
 \frac{1}{N\pi} \text{Im} \; \text{Tr} \; G_z
 =\frac{1}{N\pi}\sum_{\al=1}^N \frac{\eta}{(\mu_\al-E)^2+\eta^2}
\label{rhodef}
\ee
 be the normalized density of states of $H$
around energy $E$ and regularized on scale $\eta$.
The random variables $m$ and $\varrho$  also depend
on $N$, when necessary, we will indicate this fact  by
writing $m_N$ and $\varrho_N$.


The counting function $\cN_I$ for intervals of
length $|I|=\eta$ and the regularized density of
states are closely related.
On one hand, for the interval $I= [E-\frac{\eta}{2},
E+\frac{\eta}{2}]$, we obviously have
\be
\cN_I\leq CN|I|\varrho_\eta(E)\,.
\label{Nrho}
\ee
On the other hand,
Theorem \ref{du} provides the following upper bound
for $m(z)$
under an additional assumption.

\begin{corollary}\label{ducor}
Let $z=E+i\eta$ with $E\in \bR$ and $\eta \ge \log N/N$.
We assume conditions \eqref{gM} and \eqref{x2}.
Then there exists $c>0$ such that for any sufficiently
large $K$
\be
  \P \Big\{ \sup_E |m(E+i\eta)| \leq K\Big\}\ge 1- e^{-cKN\eta}\,.
\label{rholde}
\ee
In particular, there exists a universal constant $C$
such that
\be
  \sup_E \E \, |m(E+i\eta)| \leq C\, .
\label{Erho}
\ee
The same bounds hold for the density  since
 $\varrho_\eta(E)\leq \pi^{-1} |m(E+i\eta)|$.
\end{corollary}

\begin{proof}
It is well known that if the tail of the distribution of the matrix elements
decays sufficiently fast, then
the eigenvalues of $H$ lie within a compact set with
the exception of an exponentially  small probability. For
completeness we will prove in Lemma \ref{lm2} that
there is a universal constant $c_0$ depending only on $\delta$
in \eqref{x2}
such that for any sufficiently large $K_0$ we have
\be
     \P \big\{\max_\al |\mu_\al| \ge K_0\big\}\leq e^{-c_0K_0^2N}\, .
\label{tail}
\ee
Cover  the interval $[-K_0,K_0]$ by the union
of subintervals $I_n=[ (n-\frac{1}{2})\eta, (n+\frac{1}{2})\eta]$ of
length $\eta$ where the integer index $n$ runs from
$-[K_0\eta^{-1}]-1$ to $[K_0\eta^{-1}]+1$ (here $[\, \cdot\, ]$
denotes the integer part). Clearly
$$
  |m(E+i\eta)|\leq \frac{\pi}{N\eta}\; \max_n \cN_{I_n}
$$
assuming that $\max_\al |\mu_\al| \le K_0$. Adding up the
probabilities of the exceptional sets where $\cN_{I_n}\ge K_0N\eta$
and recalling $\eta\ge\log N/N$, we proved \eqref{rholde}.
The proof of \eqref{Erho} obviously follows from
\eqref{rholde} and from the deterministic bound
 $|m(E+i\eta)|\leq  \eta^{-1}$.
This completes
the proof of Corollary \ref{ducor}.
\end{proof}

\bigskip



In order to prove Theorem \ref{du}, we start with the following
lemma:

\begin{lemma}\label{lm:BL} Suppose that $x_j$ and $y_j$,
$j=1,2,\ldots, N$, are
i.i.d. real random variables with mean zero and
with a density function $(\const.)e^{-g(x)}$.
The expectation w.r.t. their joint probability measure
$\rd\mu=(\const.)\prod_{j=1}^N e^{-g(x_j)- g(y_j)}\rd x_j\rd y_j$
is denoted by $\E$.
We assume that
$g$ satisfies
\be
 g^{\prime \prime}(x) < M
\label{gprime}
\ee
with some finite constant $M$.
We set $z_j =x_j + \sqrt{-1}\; y_j$ and let
 $\bz = (z_1, \ldots, z_N)\in \bC^N$.
Let  $P$ be an orthogonal projection of rank $m$
  in $\bC^N$.
Then for any constant $c>0$ there exists a
positive constant $\tilde c$, depending only on $c$ and $M$, such that
\[
\E \exp \left [ - c X      \right ]  \le e^{- \tilde c m},
\qquad X=  (  P \bz \, ,  \,  P \bz   )  \; .
 \]
\end{lemma}
\begin{proof}
Let   $\mu_t$ be  the probability measure on $\bR^{2N}\cong\bC^N$ given by
\[
\rd \mu_t : = Z_t^{-1}  \exp \left [ - t
X    \right ] \rd \mu, \qquad Z_t = \int \exp \left [ - t
 X    \right ] \rd \mu
\]
and denote the expectation  w.r.t  $\mu_t$  by $\E_t$. In case
$t=0$, we shall drop the subscript. The covariance of two random
vectors $\bY,\bZ\in \bC^N$
 w.r.t. the measure $\mu_t$
is denoted by
$$
 \langle \bY; \bZ\rangle_{\mu_t} : = \E_t (\bY, \bZ)
 - ( \E_t \bY, \E_t \bZ).
$$


Simple differentiation gives
\[
\partial_t  \log \E \exp \left [ - t
 X     \right ]  = - \E_t \, X
 = -  \langle \, P \bz\, ; \,  P \bz \, \rangle_{\mu_t} -
 ( \E_t \,  P \bz, \;  \E_t\, P \bz ) \le
 -  \, \langle \, P \bz\, ; \,  P \bz \, \rangle_{\mu_t}.
 \]
Let $\nu_t$ denote the product measure on $\bR^{2N}\cong\bC^N$
with density for $z_j=x_j+\sqrt{-1}\; y_j$
to be proportional to $e^{ -  (M+2t) |z_j|^2/2}$, $j=1,2, \ldots, N$.
We can rewrite
\[
\rd \mu_t = Z_t^{-1}  \exp \left [ - t
X    \right ] \rd \mu =  \frac {\rd \mu_t } {\rd \nu_t}      \rd \nu_t\; .
\]
{F}rom the assumption \eqref{gprime}
on $g$ and from $0\leq P \leq I$,
we obtain that  $\frac {d \mu_t } {d \nu_t}$ is log convex
on $\bR^{2N}$.
{F}rom the Brascamp-Lieb inequality
 (Theorem 5.4 in \cite{BL})
 we have
\[
\langle \, P \bz\, ; \,  P \bz \,
 \rangle_{\mu_t} \ge \langle \, P \bz\, ; \,  P \bz \, \rangle_{\nu_t}.
\]
By computing the Gaussian covariance explicitly, there exists
a constant $c'>0$, depending only on $M$ and $c$, such that
$$
\langle \, P \bz\, ; \,  P \bz \, \rangle_{\nu_t}
\ge c' m \qquad \forall \; t\in [0, c].
$$
We have thus obtained that
\[
\partial_t  \log \E \exp \left [ - t
 X     \right ]   \le -  c' m \qquad \forall \; t\in [0, c].
 \]
Integrating this inequality from $t=0$ to $c$, we obtain the Lemma.
\end{proof}

\bigskip

We will use this result in the following setup. Let
$\bv_1, \bv_2, \ldots \bv_{N-1}$ form an orthonormal basis
in $\bC^{N-1}$. Let
$$
   \xi_\al : = | \bz\cdot \bv_\al|^2,
$$
where the components of $\bz= \bx + \sqrt{-1}\; \by\in \bC^{N-1}$
 are distributed
according to $(\const.)\prod_j e^{-g(x_j)-g(y_j)}\rd x_j \rd y_j$.
With this notation, a
standard  large deviation argument yields the following corollary
to Lemma \ref{lm:BL}:
\begin{corollary} \label{cor:BL} Under the condition \eqref{gprime},
there exists a positive $c$ such that for any $\delta$ small enough
\begin{equation}\label{ld}
     \P \left( \sum_{\alpha \in A} \xi_\alpha \leq \delta
 m \right) \le e^{- c m}\;
\end{equation}
for all $A \subset \{1, \cdots, N-1 \}$ with cardinality $|A|=m$.
\end{corollary}

\bigskip

{\bf Proof of Theorem \ref{du}.}
To prove \eqref{cn}, we decompose the Hermitian $N\times
N$
matrix $H$ as follows \be\label{Hd} H = \begin{pmatrix} h & \ba^* \\
\ba & B
\end{pmatrix}
\ee
where  $\ba= (h_{12}, \dots h_{1N})^*$  and  $B$ is the $(N-1)
\times (N-1)$ matrix obtained by removing the first row and first column
from $H$.  Recall that $\mu_1\leq \mu_2\leq \ldots \leq \mu_N$
denote the eigenvalues of $H$ and let $\lambda_1\leq \lambda_2 \leq
\ldots \leq \lambda_{N-1}$ denote the eigenvalues of $B$.
Note that $B$ is an $(N-1)\times (N-1)$ Hermitian Wigner
matrix with a normalization off by a factor $(1-\frac{1}{N})^{1/2}$.
The following Lemma is well-known, we include a short
proof for completeness.

\begin{lemma}\label{lm:interlace}  (i) With probability one,
the eigenvalues of any Hermitian Wigner
matrix \eqref{wig} are simple.

(ii) The eigenvalues of $H$ and $B$ are interlaced:
\be\label{interlace}
  \mu_1 < \lambda_1 < \mu_2 < \lambda_2 < \mu_3 < \ldots
\ldots \mu_{N-1} < \lambda_{N-1} <\mu_N.
\ee
\end{lemma}

\begin{proof} The proof of (i) follows directly from
the continuity of the distribution of the matrix elements
and is left to the reader. For the proof of (ii), suppose
 that $\mu$ is one of the eigenvalues of $H$. Let $\bv= (v_1, \dots,
v_N)^t$ be a normalized eigenvector associated with $\mu$.
{F}rom the continuity of the distribution it
also follows that $v_1\neq 0$ almost surely.
 {F}rom the eigenvalue equation $H \bv = \mu \bv$
and from \eqref{Hd} we find that
\be\label{ee}
h v_1 + \ba \cdot \bw = \mu v_1, \quad \text{and } \quad \ba v_1 + B
\bw = \mu \bw
\ee
with $\bw= (v_2, \dots ,v_N)^t$. From these equations we obtain
\be\label{ee1}
 \bw = (\mu-B)^{-1} \ba v_1\qquad \text{ and thus } \quad
(\mu-h)v_1 = \ba \cdot   (\mu-B)^{-1} \ba v_1
  =  \frac{v_1}{N}\sum_{\al} \frac{\xi_\al}{\mu-\lambda_\al}
\ee
using the spectral representation of $B$, where we set
$$
\xi_{\alpha} = |\sqrt{N} \ba \cdot \bu_{\alpha}|^2,
$$
 with $\bu_{\alpha}$ being the normalized
eigenvector of $B$ associated with the eigenvalue
$\lambda_{\alpha}$. Since $v_1\neq0$, we have
\be
\mu-h
  =  \frac{1}{N}\sum_{\al} \frac{\xi_\al}{\mu-\lambda_\al},
\label{inter}
\ee
where  $\xi_\al$'s are strictly positive almost surely
(notice that $\ba$ and $\bu_\al$ are independent).
In particular, this shows that $\mu\neq \lambda_\al$ for any $\al$.
In the open interval $\mu\in (\lambda_{\al-1}, \lambda_\al)$
the function
$$
     \Phi(\mu): =\frac{1}{N}\sum_{\al} \frac{\xi_\al}{\mu-\lambda_\al}
$$
is strictly decreasing from $\infty$ to $-\infty$, therefore
there is exactly one solution to the equation $\mu-h = \Phi(\mu)$.
 Similar argument shows that there
is also exactly one solution below $\lambda_1$ and above $\lambda_{N-1}$.
This completes the proof.
\end{proof}

\bigskip

We continue the proof of Theorem \ref{du}.
Using the decomposition \eqref{Hd},
we obtain the following formula for the Green
function $G_z = (H-z)^{-1}$, $z=E+i\eta$ with $E\in\bR$, $\eta>0$:
\be
   G_z(1,1)= \frac{1}{h -z -  \ba \cdot (B-z)^{-1}\ba}
 =\Big[ h-z - \frac{1}{N}
\sum_{\al=1}^{N-1}\frac{\xi_\al}{\lambda_\al- z}\Big]^{-1}.
\label{G11}
\ee
This formula in this context has already appeared in \cite{B}.
In particular, by considering only the imaginary part, we obtain
$$
  |G_z(1,1)|\leq \eta^{-1}\Big| 1+
\frac{1}{N}\sum_{\al=1}^{N-1}
\frac{\xi_\al}{(\lambda_\al- E)^2+\eta^2}\Big|^{-1}.
$$


Similarly, for any $k=1,2,\ldots, N$, we
define $B^{(k)}$ to be the $(N-1)\times (N-1)$ minor of $H$
obtained after removing the $k$-th row and $k$-th column.
Let $\ba^{(k)} = (h_{k1}, h_{k2}, \ldots h_{k,k-1}, h_{k,k+1}, \ldots
h_{kN})^*$ be the $k$-th column of $H$ without the $h_{kk}$ element.
Let $\lambda_1^{(k)} < \lambda_2^{(k)}< \ldots$ be
the eigenvalues and $\bu_1^{(k)}, \bu_2^{(k)}, \ldots$
the corresponding eigenvectors of $B^{(k)}$ and
set $\xi_\al^{(k)}:= N|\ba^{(k)}\cdot \bu_\al^{(k)}|^2$. Then we have
the estimate
\be
  |G_z(k,k)|  \leq \eta^{-1} \Big| 1+ \frac{1}{N}
\sum_{\al=1}^{N-1}\frac{\xi_\al^{(k)}}{(\lambda_\al^{(k)}-E)^2
+\eta^2 }\Big|^{-1}.
\label{gz}
\ee



For the interval $I\in \bR$ given in Theorem \ref{du},
set $E$ to be its midpoint and $\eta=|I|$, i.e. $I= [E-\frac{\eta}{2},
E+\frac{\eta}{2}]$.
{F}rom \eqref{rhodef}, \eqref{Nrho} and \eqref{gz}  we obtain
\be
  \cN_I\leq C\eta\sum_{k=1}^N |G_z(k,k)| \leq
 CN\eta^2\sum_{k=1}^N \Big|
\sum_{\al: \lambda_\al^{(k)}\in I}\xi_\al^{(k)}\Big|^{-1},
\label{nxi}
\ee
where we restricted the $\al$ summation in \eqref{gz}
only to eigenvalues
lying in $I$.

For each $k=1,2,\ldots N$, we define the event
$$
  \Omega_k :=\Big\{ \sum_{\al: \lambda_\al^{(k)}\in I}\xi_\al^{(k)}
  \leq \delta (\cN_I-1)\Big\}
$$
for some small $\delta>0$.
By the interlacing property of the $\mu_\al$ and $\lambda_\al^{(k)}$
eigenvalues, we know that there
is at least $\cN_I-1$ eigenvalues of $B^{(k)}$ in $I$.
By Corollary \ref{cor:BL}, there exists a positive universal constant $c$
such that $\P(\Omega_k)\leq e^{-c(\cN_I-1)}$.
Setting $\wt\Omega
= \bigcup_{k=1}^N\Omega_k$, we see that
\be
\P(\wt\Omega \; \text{and} \; \cN_I\ge KN|I|)
\leq N e^{-c(\cN_I-1)}\leq e^{-c'KN|I|}
\label{wt}
\ee
if $K$ is sufficiently large, recalling that $\eta=|I|\ge \log N/N$.
On the complement event, $\wt\Omega^c$, we have from \eqref{nxi} that
$$
  \cN_I\leq \frac{CN^2\eta^2}{ \delta(\cN_I-1)}
$$
i.e. $\cN_I\leq (C/\delta)^{1/2} N\eta$. Choosing $K$
sufficiently large, we obtain \eqref{cn} from \eqref{wt}.
 This proves Theorem \ref{du}. \qed

\section{Fluctuations of the density of states}\label{sec:fluc}


\begin{theorem}\label{thm:fluc}
Let $H$ be an $N\times N$ Wigner matrix as described in \eqref{wig}
and we assume the condition \eqref{gM} and \eqref{x2}.
Fix $E,\eta\in \bR$ with $(\log N) / N\le \eta \le 1$
and set $z=E+i\eta$.
\begin{itemize}
\item[i)] Suppose that the measures $\nu, \wt\nu$
satisfy the spectral gap condition \eqref{gap},
then there exists a constant $C$ such that
the covariance of the
Stieltjes transform of the empirical eigenvalue distribution
\eqref{Sti}
 satisfies
\[
\big\langle \; m(z) ; m(z)\; \big\rangle =
\E \left| m(z) - \E m(z)\right|^2 \leq
\frac{C }{N^2 \eta^3} \, . \]
\item[ii)] Suppose that  the measures $\nu$ and $\wt\nu$ satisfy
the logarithmic Sobolev inequality \eqref{logsob},
 then there exists $c>0$ such that
\be \P \left\{ |m(z) - \E m(z)| \geq\e
\right\} \leq e^{-cN\eta\e \, \min \{1, N\eta^2\e\}}
\label{ldem}
\ee
holds for any $\e>0$.
\end{itemize}
The same bounds hold if $m(z)$ is replaced with the
 density of states $\varrho_\eta(E)=\frac{1}{\pi}\, \text{Im}\, m(z)$.
\end{theorem}

We remark that estimates on the covariance were obtained in
\cite{B}, \cite{BMT} down to scale $\eta \gg N^{-1/2}$.
Concentration estimates down to the same scale were proven
in \cite{GZ}.


\bigskip

\begin{proof}
We start proving i). Denote by $\mu_{\alpha}, \alpha=1, \dots ,N$,
the eigenvalues of $H$. Since, by first order perturbation theory,
\begin{equation}
\begin{split}
\frac{\partial \mu_{\alpha}}{\partial \text{Re} \, h_{ij}} &=
\overline\bv_{\alpha} (i) \bv_{\alpha} (j) +
\overline\bv_{\alpha} (j) \bv_{\alpha} (i) = 2 \text{Re} \,
\overline\bv_{\alpha} (i) \bv_{\alpha} (j) \\
\frac{\partial \mu_{\alpha}}{\partial \text{Im} \, h_{ij}} &= \sqrt{-1}
\big[ \overline\bv_{\alpha} (i) \bv_{\alpha} (j) -
\overline\bv_{\alpha} (j) \bv_{\alpha} (i) \big] = 2
\text{Im} \, \overline\bv_{\alpha} (j) \bv_{\alpha} (i)
\end{split}
\end{equation}
for all $1\leq i<j \leq N$ and
\[\frac{\partial \mu_{\alpha}}{\partial h_{ii}} =
\overline\bv_{\alpha} (i) \bv_{\alpha} (i) \] for all
$i=1,\dots,N$, we obtain
\begin{equation}\label{eq:covrho}
\begin{split} \big\langle \; m(z) ; m(z)\;\big\rangle
\leq \; &C \sum_{i < j}^N \E
\left( \left|\frac{\partial m(z)}{\partial \sqrt{N}
\text{Re} \, h_{ij}} \right|^2 + \left|\frac{\partial
m(z)}{\partial \sqrt{N} \text{Im} \, h_{ij}} \right|^2
\right) + C \sum_{i=1}^N \E \left|\frac{\partial
m(z)}{\partial \sqrt{N}  \, h_{ii}} \right|^2 \\
= \; &\frac{C}{N^3} \; \sum_{i < j}^N \E \left( \left|\sum_{\alpha}
\frac{1}{(\mu_{\alpha}-z)^2}\;
\frac{\partial \mu_{\alpha}}{\partial \text{Re} \, h_{ij}} \right|^2  +
\left|\sum_{\alpha}
\frac{1}{(\mu_{\alpha}-z)^2}\;
\frac{\partial \mu_{\alpha}}{\partial \text{Im} \, h_{ij}} \right|^2
 \right) \\ &+\frac{C}{N^3} \sum_{i=1}^N \E
\left|\sum_{\alpha} \frac{1}{(\mu_{\alpha}-z)^2}\;
\frac{\partial \mu_{\alpha}}{\partial \, h_{ii}} \right|^2
\\ = \; &\frac{C}{N^3} \E \sum_{i
< j}^N \sum_{\alpha,\beta} \frac{1}{(\mu_{\alpha}-z)^2}
\frac{1}{(\mu_{\beta}-\bar z)^2}\\
&\hspace{4cm} \times \left(\text{Re} \, \overline\bv_{\alpha}
(i) \bv_{\alpha} (j) \text{Re} \, \overline\bv_{\beta} (i)
\bv_{\beta} (j) +\text{Im} \, \overline\bv_{\alpha} (j) {\bf
u}_{\alpha} (i) \text{Im} \,
\overline\bv_{\beta} (j) \bv_{\beta} (i) \right) \\
&+ \frac{C}{N^3} \sum_{i=1}^N \sum_{\alpha, \beta}
 \frac{1}{(\mu_{\alpha}-z)^2}
\frac{1}{(\mu_{\beta}-\bar z)^2}
|\bv_{\alpha} (i)|^2 |\bv_{\beta} (i)|^2\\
= \; &\frac{C}{N^3} \E \sum_{\alpha,\beta}  \frac{1}{(\mu_{\alpha}-z)^2}
\frac{1}{(\mu_{\beta}-\bar z)^2}
\sum_{i,j} \bv_{\alpha} (i)
\overline{\bv_{\beta} (i)}
\bv_{\beta} (j) \overline{\bv_{\alpha} (j)}  \\
= \; &\frac{C}{N^3} \E \sum_{\alpha} \frac{1}{|\mu_{\alpha}-z|^4}.
\end{split}
\end{equation}
Now, for arbitrary $n \in \bbZ$, we define the interval
\begin{equation}\label{eq:intervals}
I_n = \left[E + (n-\frac{1}{2}) \, \eta \, ; \, E + (n +
\frac{1}{2}) \, \eta \right] \, .
\end{equation}
Let  $\cN_{I_n}$
denote the number of eigenvalues of $H$ in the interval $I_n$.
For any $\eta \geq
(\log N)/N$ it follows from Theorem \ref{du}
that
\[ \P \Big\{ \cN_{I_n} \geq K N\eta \Big\} \leq e^{-cKN\eta}\, .
\]

Therefore, for any fixed
$K_0$ large enough, we find a constant $D$ such that
$D\eta^{-1}$ is an integer and
\begin{equation}\label{K0}
\begin{split}
\sum_{\alpha} \frac{1}{|\mu_{\alpha}-z|^4}
 & \leq \sum_{n = - D \eta^{-1}}^{D\eta^{-1}}
\sum_{\alpha: \mu_{\alpha} \in I_n}  \frac{1}{|\mu_{\alpha}-z|^4}
 + \sum_{\alpha:
|\mu_{\alpha}| \geq K_0} \frac{1}{\eta^4} \\ & \leq \frac{C}{\eta^4}
\sup_{|n| \leq D \eta^{-1}} \; |\{ \alpha : \mu_{\alpha} \in I_n\}|
+ \frac{1}{\eta^4} |\{ \alpha : |\mu_{\alpha}| \geq K_0\}| \,.
\end{split}
\end{equation}
{F}rom (\ref{eq:covrho}),
 we obtain
\begin{equation}
\begin{split}
\big\langle \; m(z) ; \; m(z)\; \big\rangle \leq \; &
\frac{C}{N^3\eta^4} \E \sup_{|n| \leq D \eta^{-1}} \; |\{ \alpha :
\mu_{\alpha} \in I_n\}| +
\frac{1}{N^3 \eta^4} \E |\{ \alpha : |\mu_{\alpha}| \geq K_0 \}| \\
\leq \; &\frac{CK }{N^2 \eta^3} + \frac{C}{N^2 \eta^4} \P \left\{
\sup_{|n| \leq D\eta^{-1}} \; |\{\alpha : \mu_{\alpha} \in I_n\}|
\geq K N \eta  \right\} \\
&+ \frac{1}{N^2 \eta^4} \P \left\{ \exists \text{ eigenvalue $\mu$
with } | \mu | \geq K_0\right\}
\\ \leq \; &\frac{CK}{N^2 \eta^3} + \frac{C D}{N^2\eta^5}
e^{-cKN\eta}
+ \frac{1}{N^2 \eta^{4}} e^{-c_0K_0^2  N}\, ,
\end{split}
\end{equation}
where we applied \eqref{cn} for the second term with a sufficiently
large $K$
and we used \eqref{tail}
in the third term to estimate the probability of finding
an eigenvalue $|\mu| \geq K_0$.
 This proves part i) of Theorem \ref{thm:fluc}.

\bigskip

Next, we prove ii). Let $\rd\P$ denote the probability measure
of the Hermitian Wigner matrix described in \eqref{wig}.
Remark that
\begin{equation}\label{eq:dbeta} \frac{\rd}{\rd \beta}
\Bigg[ e^{-\beta}
\log \int e^{e^{\beta}
|m(z) - \E \,m(z)|} \rd \P\Bigg] = e^{-\beta} \int u \log
u \, \rd \P,
 \end{equation}
where we defined the probability density
\[ u= \frac{e^{e^{\beta}|m(z) - \E \, m(z)|}}{ \int
e^{e^{\beta}|m(z) - \E \, m(z)|} \rd \P}.
\]
{F}rom
(\ref{eq:dbeta}), we find, using the logarithmic Sobolev inequality
and the bounds \eqref{eq:covrho}, \eqref{K0},
\begin{equation}\label{eq:dbeta2}
\begin{split}
\frac{\rd}{\rd \beta} & \Bigg [ e^{-\beta} \log \int e^{e^{\beta}
| m(z) - \E\, m(z)|} \rd \P\Bigg] \leq Ce^{-\beta} \int
|\nabla \sqrt{u}|^2 \; \rd \P \\ \leq \; &Ce^{\beta} \int
\left\{ \sum_{i<j}^N \left[ \left| \frac{\partial
m(z)}{\partial \sqrt{N} \text{Re} \, h_{ij}} \right|^2 +
\left| \frac{\partial m(z)}{\partial \sqrt{N} \text{Im} \,
h_{ij}} \right|^2\right] + \sum_{i=1}^N \left|\frac{\partial
m(z)}{\partial \sqrt{N} h_{ii}} \right|^2 \right\} u \; \rd\P \\
\leq \; & \frac{C e^{\beta}}{N^3 \eta^4}  \int \sup_{|n| \leq D
\eta^{-1}} |\{ \alpha : \mu_{\alpha} \in I_n \}| u \; \rd \P + \frac{C
e^{\beta}}{N^3 \eta^4} \int |\{ \alpha: |\mu_{\alpha}| \geq K_0 \}| u \;
\rd \P\\
\leq \; & \frac{C e^{\beta}}{N^2 \eta^3}
\\ &+ \sum_{\ell \geq 1} \frac{ Ce^{\beta} (k+1)}{N^2 \eta^3}
\\ & \hspace{1cm} \times \int {\bf 1} \left( K \ell \eta N
\leq \sup_{|n| \leq D \eta^{-1}} |\{ \alpha : \mu_{\alpha} \in I_n
\}| \leq K (\ell+1) \eta N \right)
\; {\bf 1} \left(\max_\al|\mu_{\alpha}| \leq K_0  \right) u \; \rd \P \\
&+ \frac{C e^{\beta}}{N^3 \eta^4}
\|u\|_\infty \P\big\{ \exists \alpha :
|\mu_{\alpha}| \geq K_0 \big\} \; ,
\end{split}
\end{equation}
where we used the same intervals $I_n$ introduced in
(\ref{eq:intervals}), and where the constants $K$, $D$ and $K_0$ have to be
chosen sufficiently large.


To bound the second term on
the r.h.s. of  \eqref{eq:dbeta2} we observe that, if $\sup_{|n| \leq
D \eta^{-1}} |\{ \alpha:\mu_{\alpha} \in I_n\}| \leq K N \eta (\ell +1)$
 and if there is no $\alpha$ with $|\mu_{\alpha}| \geq K_0$,
then, similarly to the proof of Corollary \ref{ducor},
\[
u \leq e^{e^{\beta}|m(z) - \E \, m(z)|} \leq
e^{4K \ell e^{\beta} } \]
where we also used \eqref{Erho} and that $K$ is sufficiently large.
Therefore we obtain, for a large $K$,
\begin{equation}
\begin{split}
\sum_{\ell \geq 1}  &\frac{C e^{\beta} (\ell+1)}{N^2 \eta^3}
\\ &\hspace{1cm}\times \int {\bf 1}
\left( K \ell \eta N  \leq \sup_{|n| \leq D
\eta^{-1}} |\{ \alpha : \mu_{\alpha} \in I_n \}| \leq K (\ell+1) \eta N
 \right) \; {\bf 1} \left(\max_\al|\mu_{\alpha}| \leq K_0 \right)
u\; \rd \P \\ \leq & \sum_{\ell \geq 1}  \frac{C
e^{\beta} (\ell+1) }{N^2 \eta^3} e^{4 K\ell e^{\beta} } \P
\left(\sup_{|n| \leq D \eta^{-1}} |\{ \alpha : \mu_{\alpha} \in I_n
\}| \geq K\ell \eta N  \right)
\\ \leq \; & \sum_{\ell \geq 1}  \frac{C
e^{\beta} (\ell+1) }{N^2 \eta^3} e^{- K\ell (cN\eta-4e^{\beta}) } \\
\leq \; &  \frac{C e^\beta }{N^2 \eta^3}
\end{split}
\end{equation}
as long as $N\eta \ge 8c^{-1} e^\beta$, where $c>0$
is the constant from Theorem \ref{du}.




To bound the last term on the r.h.s. of (\ref{eq:dbeta2})
we use that $|m(z)|\leq \eta^{-1}$ and \eqref{tail}:
$$
  \frac{C e^{\beta}}{N^3 \eta^4}
\|u\|_\infty \P\big\{ \exists \alpha :
|\mu_{\alpha}| \geq K_0 \big\} \leq \frac{C e^{\beta}}{N^3 \eta^4}
 e^{C\eta^{-1}e^\beta} e^{-c_0K_0^2N} \leq
\frac{Ce^{\beta}}{N^3 \eta^4}
$$
as long as $N\eta \ge C_0e^{\beta}$ with a sufficiently big $C_0$.


Putting everything together, we obtain, from (\ref{eq:dbeta2}),
\begin{equation}
\begin{split}
\frac{\rd}{\rd \beta} &e^{-\beta} \log \int e^{e^{\beta}
|m(z) - \E \, m(z)|} \rd \P \leq \frac{C
e^{\beta}}{N^2 \eta^3}
\end{split}
\end{equation}
for all $\beta$ such that $N\eta \ge C_1e^{\beta}$
with a sufficiently big $C_1$.
Integrating this inequality from $\beta=-\infty$ to $\beta=\log L$  and
exponentiating, we find that
\[ \E \; e^{L |m(z) - \E \, m(z)|} \leq
\exp{ \big( CL^2N^{-2} \eta^{-3} \big)}
\]
holds for any $L$ satisfying $0<L \leq C_1^{-1} N\eta$.
Therefore
\begin{equation}
\P \big\{ |m(z) - \E \, m(z)| \geq \e \big\}
\le \exp{ \big( C L^2N^{-2} \eta^{-3}-\e L\big) }\leq
 e^{-cN\eta \e \min\{ 1, N\eta^2\e\}}
\end{equation}
with a sufficiently small $c>0$
after optimizing for $L$ under the condition $L \leq C_1^{-1} N\eta$.
\end{proof}

\section{Semicircle law on short scales}

For any $z=E+i\eta$ we let
$$
 m_{sc}= m_{sc}(z) = \int_\bR \frac{\varrho_{sc}(x)\rd x}{x - z}
$$
be the Stieltjes transform of the Wigner semicircle
distribution function whose density is given by
$$
  \varrho_{sc}(x) = \frac{1}{2\pi} \sqrt{4-x^2}
   {\bf 1}(|x|\leq 2)\rd x .
$$
For
 $\kappa, \wt\eta>0$ we define the set
$$
S_{N,\kappa,\wt\eta}:= \Big\{ z=E+i\eta\in \bC\; : \; |E|\leq 2-\kappa, \;
\wt\eta\leq \eta \leq 1\Big\}
$$
and for $\wt\eta = N^{-2/3}\log N$ we write
$$
S_{N,\kappa}:= \Big\{ z=E+i\eta\in \bC\; : \; |E|\leq 2-\kappa, \;
\frac{\log N}{N^{2/3}}\leq \eta \leq 1\Big\}.
$$

\begin{theorem}\label{thm:sc}
Let $H$ be an $N\times N$ Wigner matrix as described in \eqref{wig}
and assume the conditions \eqref{gM}, \eqref{x2} and \eqref{logsob}.
Then for any $\kappa>0$,  the Stieltjes transform $m_N(z)$ (see \eqref{Sti})
of the empirical
eigenvalue distribution of the $N\times N$ Wigner matrix  satisfies
\be
  \lim_{N\to \infty} \sup_{z\in S_{N,\kappa}}
|\E m_N(z) - m_{sc}(z)| =0.
\label{eq:sc}
\ee
\end{theorem}

Combining this result with Theorem \ref{thm:fluc}, we obtain

\begin{corollary}\label{cor:sc}
Let $\kappa>0$ and $\eta \in [N^{-2/3}\log N, 1]$ and
assume the conditions of Theorem \ref{thm:sc}.
Then we have
\be
 \P \Big\{ \sup_{z\in S_{N,\kappa,\eta}} |m_N(z)- m_{sc}(z)| \ge \e\Big\}
 \leq e^{-cN\eta \e \min\{ 1, N\eta^2\e\}}
\label{mcont}
\ee
for any $\e>0$. In particular, the density of states
$\varrho_\eta(E)$  converges to the
Wigner semicircle law in probability uniformly for all energies away from
the spectral edges and for all energy windows at least $N^{-2/3}\log N$.


Let $\eta^*=\eta^*(N)$ such that
$\eta\ll\eta^*\ll 1$ as $N\to \infty$, then we
have the convergence of the counting function as well:
\be
  \P \Big\{ \sup_{|E|\leq 2-\kappa} \Big| \frac{\cN_{\eta^*}(E)}{2N\eta^*}
  - \varrho_{sc}(E)\Big|\ge \e\Big\}\leq
 e^{-cN\eta \e \min\{ 1, N\eta^2\e\}}
\label{ncont}
\ee
for  any $\e>0$, where $\cN_{\eta^*}(E)= |\{ \al\; : \; |\mu_\al - E|
\leq \eta^*\}|$ denotes the number of eigenvalues in
the interval $[E-\eta^*, E+\eta^*]$.
\end{corollary}


We remark, that Bai et. al. \cite{BMT} have investigated the speed of
convergence of the empirical eigenvalue distribution
to the semicircle law. Their results directly imply
\eqref{eq:sc} for $\eta =\text{Im} \, z \gg N^{-1/2}$
and \eqref{ncont} for $\eta \ge N^{-2/5}$.

\bigskip

{\bf Proof of Corollary \ref{cor:sc}.} For any two
points $z, z'\in S_{N, \kappa,\eta}$ we have
$$
  |m_N(z)-m_N(z')|\leq CN^{4/3}|z-z_j|
$$
since the gradient of $m_N(z)$ is bounded by $C |\text{Im}\; z|^{-2}\leq
CN^{4/3}$ on $S_{N, \kappa,\eta}$.
We can choose a set of at most $M= C\e^{-2}N^{4}$ points, $z_1, z_2, \ldots,
z_M$, in $S_{N, \kappa,\eta}$  such that for any $z\in S_{N,\kappa,\eta}$
there exists a point $z_j$ with $|z-z_j|\leq \e N^{-2} $.
In particular, $|m_N(z)-m_N(z_j)|\le \e/4$ if $N$ is large enough.
Then using \eqref{ldem} we obtain
$$
     \P \Big\{ \sup_{z\in S_{N,\kappa,\eta}} |m_N(z)- \E \, m_N(z)|
 \ge \e\Big\}
   \leq \sum_{j=1}^M
  \P \Big\{ |m_N(z_j)- \E \, m_N(z_j)| \ge \frac{\e}{2}\Big\}
  \leq e^{-cN\eta \e \min\{ 1, N\eta^2\e\}}
$$
under the condition that $\eta\ge N^{-2/3}\log N$ since $\text{Im} \, z_j\ge
\eta$.
Combining this estimate with  \eqref{eq:sc}, we have proved \eqref{mcont}.

To prove \eqref{ncont}, we  set
\[
  R(\lambda)=
 \frac{1}{\pi}\int_{E-M\eta}^{E+M\eta} \frac{\eta}{(\lambda-x)^2+\eta^2}
\; \rd x = \frac{1}{\pi}\Bigg[\arctan \Big( \frac{E-\lambda}{\eta}+M\Big)
  -  \arctan \Big( \frac{E-\lambda}{\eta}-M\Big) \Bigg]\\
\]
and let $ {\bf 1}_{I^*}(\lambda)$ denote the characteristic
function of the interval $I^*= [E-\eta^*, E+\eta^*]$ with
$\eta^*=M\eta$.
From elementary
calculus it follows that $ {\bf 1}_{I^*}-R $ can be decomposed
into a sum of three functions, $ {\bf 1}_{I^*}-R= T_1+ T_2+T_3$ with the
following properties:
$$
   |T_1|\leq CM^{-1/2}, \qquad
\text{supp}(T_1)\in I_1= [E-2\eta^*, E+2\eta^*];
$$
$$
  |T_2|\leq 1, \qquad \text{supp}(T_2) = J_1\cup J_2
$$
where $J_1$ and $J_2$ are two intervals
of length $M^{1/2}\eta$ with midpoint at $E-\eta^*$ and
at $E+\eta^*$, respectively; and
$$
   |T_3(\lambda)|\leq \frac{C\eta\eta^*}{(\lambda-E)^2+[\eta^*]^2}, \qquad
\text{supp}(T_3)\in I_1^c \, .
$$
We thus have
\be
     \frac{\cN_{\eta^*}(E)}{2N\eta^*}
    =  \frac{1}{2\eta^*}\int   {\bf 1}_{I^*} (\lambda) \rd F(\lambda)
   =  \frac{1}{2\eta^*}\int   R(\lambda) \rd F(\lambda)
  + \frac{1}{2\eta^*}\int   \big[ T_1(\lambda)+T_2(\lambda)+T_3(\lambda)\big]
\rd F(\lambda)\, .
\label{R}\ee
The last three terms are estimated trivially by
\[
\begin{split}
   \frac{1}{2\eta^*}\int   \big| T_1+T_2+T_3\big|
\rd F& \leq \| T_1\|_\infty\frac{\cN_{I_1}}{2N\eta^*} +
\frac{\cN_{J_1}+\cN_{J_2}}{2N\eta^*} + \frac{C\eta}{\eta^*}
\varrho_{\eta^*}(E) \\
&\leq \frac{C}{M^{1/2}}
\big[\varrho_{2\eta^*}(E) + \varrho_{M^{1/2}\eta}(E-\eta^*)
+ \varrho_{M^{1/2}\eta}(E+\eta^*) +\varrho_{\eta^*}(E)\big]
\end{split}
\]
Using the bound \eqref{rholde}, this error term is bounded
by $CM^{-1/2}$ uniformly in $E$ apart from an event
of exponentially small probability. In particular,
this term is smaller than $\e/3$ if $M=\eta^*/\eta$ is
sufficiently large.

The main term in \eqref{R} is computed as
$$
   \frac{1}{2\eta^*}\int   R(\lambda) \rd F(\lambda)
   = \frac{1}{2\eta^*}\int_{E-\eta^*}^{E+\eta^*} \varrho_\eta(x) \, \rd x
  =  \frac{1}{2\eta^*}\int_{E-\eta^*}^{E+\eta^*} \varrho_{sc}(x) \, \rd x
  + \frac{1}{2\eta^*}\int_{E-\eta^*}^{E+\eta^*} \big[ \varrho_\eta(x)-
\varrho_{sc}(x)\big] \, \rd x
$$
and the first term converges to $\varrho_{sc}(E)$ as long as $\eta^*\to 0$.
Using \eqref{mcont}, the second term is smaller than $\e/3$ apart from
a set of probability $\exp{(-cN\eta \e \min\{ 1, N\eta^2\e\})}$.
Putting these estimates together, we arrive at \eqref{ncont}.
\qed

\bigskip


{\bf Proof of Theorem \ref{thm:sc}.}
Recall from the proof of Theorem \ref{du}
that $B^{(k)}$ denotes the $(N-1)\times (N-1)$ minor
of $H$ after removing the $k$-th row and $k$-th column.
Similarly to the definition of $m(z)$ in \eqref{Sti},
we also define the Stieltjes transform
of the density of states of $B^{(k)}$
$$
 m^{(k)}= m^{(k)}(z) = \frac{1}{N-1}\, \text{Tr}\, \frac{1}{B^{(k)}-z}
   =\int_\bR \frac{\rd F^{(k)}(x)}{x - z}
$$
with the empirical counting function
$$
    F^{(k)}(x) = \frac{1}{N-1} \big| \, \big\{ \al \; : \;
   \lambda_{\al}^{(k)}\leq x \big\}\big|,
$$
where $\lambda_{\al}^{(k)}$ are the eigenvalues of $B^{(k)}$.
The spectral parameter $z$ is fixed throughout the proof
and we will omit from the argument of the Stieltjes transforms.

{F}rom a formula analogous to \eqref{G11} but applied to
the $k$-th minor we get
\be
m = \frac{1}{N}\sum_{k=1}^N G_z(k,k)
=\frac{1}{N}\sum_{k=1}^N \frac{1}{ h_{kk} - z -
\ba^{(k)} \cdot\frac{1}{B^{(k)}-z} \ba^{(k)}},
\label{mm}
\ee
where recall that $\ba^{(k)}$ is the $k$-th column without the diagonal.
Let $\E_k$ denote the expectation value w.r.t
the random vector $\ba^{(k)}$.
Define the random variable
\be
  X_k: =\ba^{(k)}\cdot \frac{1}{B^{(k)}-z} \ba^{(k)}
 - \E_k\; \ba^{(k)} \cdot \frac{1}{B^{(k)}-z} \ba^{(k)}
\label{def:X}
\ee
and note that
$$
   \E_k\; \ba^{(k)} \cdot \frac{1}{B^{(k)}-z} \ba^{(k)} = \frac{1}{N}
   \sum_\al\frac{1}{\lambda_{\al}^{(k)}-z} = \Big( 1-\frac{1}{N}\Big)m^{(k)}.
$$


With this notation, it follows from \eqref{mm} that
\be
  \E \, m =-\E  \; \Bigg[ \frac{1}{X_1
 +(1-\frac{1}{N})\big[ m^{(1)} - \E m^{(1)}\big] + \big[ (1-\frac{1}{N})
\E m^{(1)} -\E m\big]
+\big[ \E m + z\big] -h_{11} -\frac{1}{N} \E m^{(1)}  } \Bigg] \; ,
\label{m-m}
\ee
where we used that the
distribution of $X_k$ and $m^{(k)}$ is independent of $k$.


Fix $\e >0$.
The first term in the denominator of \eqref{m-m}
is estimated in the following
lemma whose proof is given at the end of the section.

\begin{lemma}\label{lm:x4}
For the random variable $X_1$ from \eqref{def:X} we have
\be
      \E |X_1|^4 \leq \frac{C(\log N)^2}{N^2\eta^2},
\label{x4}\ee
in particular
$$
    \P\big\{ |X_1|\ge \e\big\}\leq \frac{C(\log N)^2}{N^2\eta^2\e^4}.
$$
\end{lemma}


For the second term in the denominator in \eqref{m-m} we
 apply the large deviation estimate from Theorem
\ref{thm:fluc} to the Stieltjes transform
of $B^{(1)}$
$$
\P \Big\{ \big| m^{(1)} - \E m^{(1)}\big|\ge \e\Big\} \leq
e^{-cN\eta\e\, \min\{ 1, N\eta^2\e\}}\; .
$$
For the third term, we use that
$$
     \Big|  m -  \Big(1-\frac{1}{N}\Big)m^{(1)}\Big|
  =\Big| \int \frac{\rd F(x)}{x-z}
  -  \Big(1-\frac{1}{N}\Big)\int \frac{\rd F_1(x)}{x-z}\Big|
  = \frac{1}{N}\Big| \int \frac{NF(x)-(N-1)F_1(x)}{(x-z)^2} \rd x\Big|.
$$
By the interlacing property between the eigenvalues of $H$ and $B^{(1)}$,
we have $\max_x|NF(x)-(N-1)F_1(x)|\leq 1$, thus
$$
   \Big|  m -  \Big(1-\frac{1}{N}\Big)m^{(1)}\Big|
\leq \frac{1}{N} \int \frac{\rd x}{|x-z|^2}
  \leq  \frac{C}{N\eta}\, ,
$$
and therefore $|\E m - (1-N^{-1})\E m^{(1)}|\leq C(N\eta)^{-1}$.
Finally, from $\E \, x_{11}^2 <\infty$ we have
$$
  \P \big\{ |h_{11}|\ge \e \big\} \le \frac{C}{N\e^2}.
$$



We define the set of events
$$
  \Omega: = \big\{ \big| X_1\big|\ge \e
  \big\} \cup
\big\{ \big| m^{(1)} - \E m^{(1)}\big|\ge \e
  \big\} \cup \big\{  \big| h_{11}\big|\ge \e
  \big\}
$$
then
$$
  \P (\Omega)\leq e^{-cN\eta\e\, \min\{ 1, N\eta^2\e\}} +\frac{C}{N\e^2}+
 \frac{C(\log N)^2}{N^2\eta^2\e^4}.
$$
Let
$$
   Y = X_1  + (1-N^{-1})\big[ m^{(1)} - \E m^{(1)}\big] + \big[ (1-N^{-1})
\E m^{(1)} -\E m\big]
 -h_{11},
$$
then, similarly to \eqref{gz}, we have
$$
   \Big| Y + \E \, m  + z\Big| \ge \Big| \text{Im} z + \ba^{(k)}
\cdot \frac{1}{B^{(k)}-z} \ba^{(k)}\Big| \ge \eta \; .
$$
We also have $|\E \; m + z|\ge \eta$ since $\text{Im} \, m \ge 0$.
Set $\wt Y: = Y\cdot {\bf 1}_{\Omega^c}$, then obviously $|\wt Y|\leq 4\e$.
Moreover, from
\eqref{m-m} we have
\be
  \E \, m +  \frac{1}{ \E \, m + z }
= \E \; {\bf 1}_{\Omega^c} \Big[ \frac{1}{ \E \, m + z }
 -\frac{1}{ \E \, m + z + \wt Y}\Big]
+ \E \; {\bf 1}_\Omega\Big[  \frac{1}{ \E \, m + z } -\frac{1}{ \E \, m + z +  Y}
\Big]\, .
\label{EE}
\ee
The second term is bounded by
$$
   \Bigg| \E \; {\bf 1}_\Omega\Big[  \frac{1}{ \E \, m + z }
-\frac{1}{ \E \, m + z +  Y} \Big]\Bigg| \leq 2\eta^{-1} P(\Omega) \leq
  \frac{C}{\e^4 \log N} \leq C\e
$$
uniformly for $z\in S_{N,\kappa}$ if $N\ge N(\e)$.
In the first term we use the stronger bounds
$$
    |\E \, m + z|\ge \text{Im} \, m(z) +\eta \qquad
   |\E \, m + z + \wt Y|\ge \text{Im} \, m(z) +\eta - 4\e
$$
on the denominators. Thus, from \eqref{EE} we obtain
\be
    \Big|  \E \, m +  \frac{1}{ \E \, m + z } \Big|
  \leq \frac{C\e}{ \big[ \text{Im} \, m(z)+\eta \big]
 \big[ \text{Im} \, m(z)+\eta -4\e\big]} + C\e\;
\label{cont1}
\ee
uniformly for $z\in S_{N,\kappa}$.

We note that the equation
\be
   M+ \frac{1}{M+z} =0
\label{stab1}
\ee
has a unique solution for any $z\in S_{N,\kappa}$ with $\text{Im} \, M>0$,
 namely
$M= m_{sc}(z)$, the Stieltjes transform of the semicircle law.
Note that there exists $c(\kappa)>0$ such that
$\text{Im} \, m_{sc}(E+i\eta) \ge c(\kappa)$ for any $|E|\leq 2-\kappa$,
uniformly in $\eta$.

The equation \eqref{stab1} is stable in the following sense.
For any small $\delta$, let $M=M(z,\delta)$ be a solution to
\be
   M + \frac{1}{M+z} = \delta
\label{stab2}
\ee
with $\text{Im}\, M>0$.
Subtracting \eqref{stab1} with
$M=m_{sc}$ from \eqref{stab2}, we have
$$
   (M-m_{sc})\Big[m_{sc}+z - \frac{1}{M+z}\Big] = \delta(m_{sc}+z)
$$
and
$$
  \text{Im}\Big[m_{sc}+z - \frac{1}{M+z}\Big] \ge \text{Im}\, m_{sc} \ge c(\kappa)
$$
Since the function $m_{sc}+z$ on the compact set $z\in S_{N,\kappa}$
is bounded, we get that
\be
   | M-m_{sc}| \leq C_\kappa \delta \,
\label{cont3}
\ee
for some constant $C_\kappa$ depending only on $\kappa$.




Now we perform a continuity argument in $\eta$ to prove that
\be
        |\E \, m(E+i\eta) - m_{sc}(E+i\eta)|\leq C^*\e
\label{cont}\ee
uniformly in $z\in S_{N,\kappa}$  with a sufficiently large constant $C^*$.
Fix $E$ with $|E|\leq 2-\kappa$.
For $\eta=[\frac{1}{2}, 1]$,
 \eqref{cont} follows from \eqref{cont1} with some small $\e$, since
the right hand side of \eqref{cont1} is bounded by $C\e$.
Suppose now that \eqref{cont} has been proven for some $\eta\in [2N^{-2/3}\log N,\, 1]$
and we want to prove it for $\eta/2$.
By integrating the inequality
$$
    \frac{\eta/2}{(x-E)^2 + (\eta/2)^2}
\ge \frac{1}{2} \frac{\eta}{(x-E)^2+\eta^2}
$$
with respect to $\rd F(x)$ we obtain that
$$
 \text{Im}\, m\big(E+i\frac{\eta}{2}\big) \ge \frac{1}{2} \text{Im}\,
  m(E+i\eta)
\ge \frac{1}{2}c(\kappa)- C^*\e > \frac{c(\kappa)}{4}\,
$$
for sufficiently small $\e$. Thus the r.h.s of \eqref{cont1}
for $z=E+i\frac{\eta}{2}$
is bounded by $C\e$, the constant depending only on $\kappa$.
Applying the stability bound \eqref{cont3}, we get \eqref{cont}
for $\eta$ replaced with $\eta/2$.
This completes the proof of Theorem \ref{thm:sc}.
\qed


\bigskip

{\bf Proof of Lemma \ref{lm:x4}.}
Recall that $\lambda_\al^{(1)}$ denote the eigenvalues
and $\bu_\al^{(1)}$ denote the eigenvectors of $B^{(1)}$
for $\al=1,2, \ldots , N-1$. We also
defined $\xi_\al^{(1)}
 = |\bb^{(1)}\cdot \bu_\al{(1)}|^2$
with  the vector $\bb^{(1)} =(b_1, \ldots b_{N-1})= \sqrt{N}\ba^{(1)}
= \sqrt{N}(h_{12}, h_{13}, \ldots h_{1N})^*$ whose components
are i.i.d. random variables with real and imaginary parts
 distributed according to $\nu$.
Dropping the sub- and superscripts, we have
$$
   X =   \frac{1}{N}
\sum_{\al =1}^{N-1}  \frac{\xi_\al-1}{\lambda_\al-z}
= \frac{1}{N}
\sum_{\al}  \frac{ \sum_{i,j} b_i\bar b_j \bu_\al(i)\bar \bu_\al (j)
-1}{\lambda_\al-z} \, ,
$$
where all summations run from 1 to $N-1$.

Since the distribution $\nu$ satisfies the
spectral gap inequality \eqref{gap}, we have
\be
  \E |X|^2  \leq C \E \sum_k  \Bigg[
 \Big| \frac{\pt X}{\pt b_k} \Big|^2 +
\Big| \frac{\pt X}{\pt \bar b_k} \Big|^2 \Bigg]\;,
\label{spg}
\ee
where $\pt/\pt b = \frac{1}{2}\big[\pt/\pt(\text{Re} \, b)
 -i \pt/\pt ( \text{Im}\, b)\big]$ and
$\pt/\pt \bar b = \frac{1}{2}\big[\pt/\pt(\text{Re} \, b)
 +i \pt/\pt ( \text{Im}\, b)\big]$.
We compute
\be
\begin{split}
\sum_k  \Bigg[
 \Big| \frac{\pt X}{\pt b_k} \Big|^2 +
\Big| \frac{\pt X}{\pt \bar b_k} \Big|^2 \Bigg]&=
\sum_k \Bigg[ \Bigg| \frac{1}{N}
\sum_{\al,j}
\frac{ \bar b_j \bu_\al(k)\bar \bu_\al (j)}{\lambda_\al-z}\Bigg|^2
+  \Bigg| \frac{1}{N}\sum_{\al,i}
\frac{ b_i \bu_\al(i)\bar \bu_\al (k)}{\lambda_\al-z}\Bigg|^2
\Bigg]\\
&=\frac{1}{N^2} \sum_k
\sum_{\al,\beta,i,j} \Bigg[
\frac{ \bar b_j b_i
 \bu_\al (k)\bar \bu_\beta(k) \bar \bu_\al (j)\bu_\beta(i)}
{(\lambda_\al-z)(\lambda_\beta-\bar z)}
+ \frac{ b_i\bar b_j \bu_\al(i)\bar \bu_\beta(j)
 \bar\bu_\al (k) \bu_\beta(k) }
{(\lambda_\al-z)(\lambda_\beta-\bar z)}\Bigg]\\
&=
\frac{2}{N^2}\sum_{\al,i,j}
\frac{ \bar b_j b_i
 \bar \bu_\al (j)\bu_\al(i)}
{|\lambda_\al-z|^2}\;.
\end{split}
\label{pt2}
\ee
Here we  used the orthonormality of the eigenfunctions,
$\sum_k \bu_\al (k)\bar\bu_\beta(k) = \delta_{\al,\beta}$.
We insert this into \eqref{spg} and take the expectation
with respect to the $\bb$ variables, $\E\, \bar b_j b_i=\delta_{ij}$,
by using the fact that
 the components of $\bb$ are independent
of the $\lambda_\al$'s and $\bu_\al$'s.
\[
\begin{split}
  \E |X|^2 & \leq \frac{C}{N^2} \E \sum_{\al,i,j}
\frac{ \bar b_j b_i
 \bar \bu_\al (j)\bu_\al(i)}
{|\lambda_\al-z|^2} \\
&= \frac{C}{N^2} \E \sum_\al
\frac{ 1}{|\lambda_\al-z|^2} \leq \frac{C}{N\eta} \E\; \frac{1}{N} \sum_\al
\frac{1}{|\lambda_\al -z|}\;.
\end{split}
\label{lon}
\]
To estimate
the last term,
we have
\be
 \E \; \frac{1}{N} \sum_\al
\frac{1}{|\lambda_\al -z|}  \leq \int_{|\lambda|\le K_0}
\frac{\E \, \varrho_\eta(\lambda)}{|\lambda-z|} \; \rd \lambda
 + \frac{1}{\eta} \P \{\max |\lambda_\al |\ge K_0\} \leq C\log N \, .
\label{1mom}
\ee
In the last step, by choosing $K_0$ sufficiently large, we used
the uniform estimate \eqref{Erho} on $\E \, \varrho_\eta(\lambda)$
and the bound  \eqref{tail} for the eigenvalues
of the $(N-1)\times (N-1)$ Wigner matrix $B^{(1)}$.
Thus we have showed that
\be
  \E \, |X|^2 \leq \frac{C\log N}{N\eta} \; .
\label{xsec}
\ee

To estimate  the fourth moment, we have
$$
  \E \, |X|^4 = \big[ \E\, |X|^2\big]^2 + \E \big[ |X|^2- \E\, |X|^2\big]^2
  \leq \frac{(C\log N)^2}{(N\eta)^2}
  +  C \E \sum_k  \Bigg[
 \Bigg| \frac{\pt |X|^2}{\pt b_k} \Bigg|^2 +
\Bigg| \frac{\pt |X|^2}{\pt \bar b_k} \Bigg|^2 \Bigg] \, .
$$
We will compute only the first term in the summation,
 the second one is identical.
We have
$$
  C\E \sum_k
 \Bigg| \frac{\pt |X|^2}{\pt b_k} \Bigg|^2
\leq 2C\E \Bigg[ \; |X|^2
 \sum_k  \Bigg(\Big| \frac{\pt X}{\pt b_k}\Big|^2 +
\Big| \frac{\pt X}{\pt \bar b_k}\Big|^2\Bigg)\Bigg]
 \leq \frac{1}{4} \E \, |X|^4 + C \E \Bigg[
\sum_k  \Bigg(\Big| \frac{\pt X}{\pt b_k}\Big|^2+
\Big| \frac{\pt X}{\pt \bar b_k}\Big|^2\Bigg) \Bigg]^2\, ,
$$
therefore
\be
  \frac{1}{2} \E |X|^4 \leq  \frac{(C\log N)^2}{(N\eta)^2}
  +  C \E \sum_k  \Bigg[
 \Big| \frac{\pt |X|^2}{\pt b_k} \Big|^2 +
\Big| \frac{\pt |X|^2}{\pt \bar b_k} \Big|^2 \Bigg] \, .
\label{tre}
\ee
For the last term, we use \eqref{pt2}:
\be
\begin{split}
 \E \Bigg[
\sum_k  \Bigg(\Big| \frac{\pt X}{\pt b_k}\Big|^2+
\Big| \frac{\pt X}{\pt \bar b_k}\Big|^2\Bigg) \Bigg]^2
 &= \frac{1}{N^4}\E \Bigg[\sum_{\al,i,j}
\frac{ \bar b_j b_i
 \bar \bu_\al (j)\bu_\al(i)}
{|\lambda_\al-z|^2}\Bigg]^2\\
&=  \frac{1}{N^4}\E \sum_{\al,\beta}\sum_{i,j,\ell,m}
  \frac{ \E \big[\bar b_j b_i \bar b_\ell b_m\big]
 \bar \bu_\al (j)\bu_\al(i) \bar \bu_\beta(\ell)\bu_\beta(m)}
{|\lambda_\al-z|^2|\lambda_\beta-z|^2}\\
&=  \frac{1}{N^4}\E \sum_{\al,\beta}\sum_{i\neq\ell}
  \frac{ |\bu_\al(i)|^2|\bu_\beta(\ell)|^2}
{|\lambda_\al-z|^2|\lambda_\beta-z|^2}
+ \frac{1}{N^4}\E \sum_{\al,\beta}\sum_{i\neq j}
  \frac{ \bar \bu_\al(j)\bu_\al(i)\bar \bu_\beta(i)\bu_\beta(j)}
{|\lambda_\al-z|^2|\lambda_\beta-z|^2}\\
& \hskip.5cm +\frac{c_4}{N^4} \, \E \sum_{\al,\beta}\sum_{i}
  \frac{ |\bu_\al(i)|^2|\bu_\beta(i)|^2}
{|\lambda_\al-z|^2|\lambda_\beta-z|^2}\\
&\leq \frac{C}{N^4}\E \sum_{\al,\beta}\sum_{i,\ell}
  \frac{ |\bu_\al(i)|^2|\bu_\beta(\ell)|^2}
{|\lambda_\al-z|^2|\lambda_\beta-z|^2}\\
&\leq \frac{C}{(N\eta)^2} \E \Bigg[ \frac{1}{N}\sum_\al
 \frac{1}{|\lambda_\al-z|}\Bigg]^2 .
\end{split}
\label{lo}
\ee
In the second line we used that
$$
  \E \big[\bar b_j b_i \bar b_\ell b_m\big] =\delta_{ij}\delta_{\ell m}
(1-\delta_{i\ell}) + \delta_{i\ell}\delta_{jm}(1-\delta_{im})
   + c_4\delta_{ij}\delta_{j\ell}\delta_{\ell m},
$$
where $c_4 = \E |b|^4 =\int (x^2+y^2)^2 \rd\nu(x)\rd\nu(y)$.
Finally,
the last expectation value is  estimated as
$$
\E \Bigg( \frac{1}{N} \sum_\al \frac{1}{|\lambda_\al-z|}
 \Bigg)^2 \leq \E \Bigg( \int_{|\lambda|\leq K_0} \frac{\varrho_\eta(\lambda)}
{|\lambda-z|}\rd\lambda \Bigg)^2 + \eta^{-2} \;
\P\big\{ \max |\lambda_\al|\ge K_0\big\} \; .
$$
The second term is exponentially small by \eqref{tail}.
In the first term we use \eqref{rholde} to conclude that
$\varrho_\eta(\lambda) \leq K$ uniformly in $\lambda$
apart from an event of exponentially small probability.
Inserting this bound into \eqref{lo} and \eqref{tre},
we obtain the desired bound $\E |X|^4\leq C(\log N)^2/(N\eta)^2$
in Lemma \ref{lm:x4}.
\qed



\section{Extended states}


Recall that the eigenvalues  of $H$ are denoted by $\mu_1 < \mu_2 <
\cdots < \mu_N$ and the corresponding normalized eigenvectors by
$\bv_1, \bv_2,\ldots \bv_N$.

\begin{theorem}\label{thm:loc}
Let $H$ be an $N\times N$ Wigner matrix as described in \eqref{wig}
and satisfying the conditions (\ref{gM}) and (\ref{x2}). Then there
exist  positive constants, $C_1, C_2$ and $c$, depending only on
the constants $M$ in (\ref{gM}) and $\delta$ in (\ref{x2}), such
that for any $q>0$ \be\label{eq:thmloc} \P \Big\{ \frac{1}{N}\Big|
\big\{ \beta\; :\; \max_j |\bv_\beta(j)|^2
 \ge \frac{C_1 q^2(\log N)^2}{N}\big\}\Big| \ge \frac{C_2}{q}\Big\} \leq
e^{-c(\log N)^2}. \ee
\end{theorem}


{\it Remark.} Suppose that  $\|\bv \|_\infty^2 \leq 1/L$ holds
 for an $\ell^2$-normalized vector $\bv=(v_1, v_2, \ldots)$.
Then the support of $\bv$ contains at least $L$ elements. Thus the
quantity $\| \bv\|_\infty^{-2}$ can be intrepreted as the
localization length of $\bv$. With this interpretation, Theorem
\ref{thm:loc} states that the density of eigenstates with a
localization length $L\le Nq^{-2}$ (with logarithmic corrections) is
bounded from above by $C/q$.

It also follows from Theorem \ref{thm:loc}, that, for every $p \geq
2$, \begin{equation}\label{eq:lp>2} \P \Big\{ \frac{1}{N}\Big|
\big\{ \beta\; :\; \| \bv_\beta \|_{\ell^p} \geq C_1
N^{\frac{1}{p}-\frac{1}{2}} (\log N)^{2-\frac{4}{p}} \big\} \Big|
\ge \frac{C_2}{\log N}\Big\} \leq e^{-c (\log N)^2}.
\end{equation}
In other words, with high probability, all the $N$ eigenvectors
apart from a fraction converging to zero as $N \to \infty$, have the
expected delocalization properties up to logarithmic corrections.

Note that, by duality, (\ref{eq:lp>2}) immediately implies that
\begin{equation}\label{eq:lp<2}
\P \Big\{ \frac{1}{N}\Big| \big\{ \beta\; :\; \| \bv_\beta
\|_{\ell^p} \leq C_1^{-1} N^{\frac{1}{p}-\frac{1}{2}} (\log
N)^{2-\frac{4}{p}} \big\} \Big| \ge \frac{C_2}{\log N}\Big\} \leq
e^{-c (\log N)^2}.
\end{equation}
for all $1 \leq p \leq 2$. In Section \ref{sec:Meta}, we will
improve (\ref{eq:lp<2}), by showing, in Corollary \ref{cor:lp},
that, up to an event with exponentially small probability, every
eigenvector $\bv$ of $H$ satisfies $\| \bv \|_p \leq c
N^{\frac{1}{p} - \frac{1}{2}}$ for all $1 \leq p \leq 2$.

\bigskip

\begin{proof} For brevity, we introduce the notation
\[
 \theta = [\log N]^2,
\]
where $[\, \cdot \,]$ denotes the integer part. For $q>0$, let $O_q$
denote the set of eigenvalue indices $\al$ such that the distance
between the eigenvalues $\mu_{\alpha+\theta}$ and
$\mu_{\alpha-\theta}$ is less than   $q\theta/N$: \be\label{Od} O_q
= \Big\{\alpha: |\mu_{\alpha-\theta} - \mu_{\alpha+\theta}|
 \le \frac{q\theta}{N}
 \Big\}.
\ee Here we used the notation $\mu_{\alpha} = \mu_1$ if $\alpha <1$
and $\mu_{\alpha} = \mu_N$ if $\alpha >N$. Given $K_0>0$, we define
$\Omega$ to be the event characterized by all eigenvalues of $H$
being in the interval $[-K_0, K_0]$, that is
\[ \Omega = \{ \omega : \sigma (H) \subset [-K_0, K_0] \}\, . \]
By (\ref{tail}), we have
\[ \P ( \Omega) \geq 1 - e^{-cN} \] if
$K_0$ is sufficiently large. We have
\begin{equation}\begin{split} \P
\Big\{ \frac{1}{N}\Big| &\big\{ \beta\; :\; \max_j |\bv_\beta(j)|^2
\ge \frac{C_1 q^2 \theta}{N}\big\} \Big| \ge \frac{C_2}{q}\Big\} \\
\leq \; &\P \Big\{ \frac{1}{N}\Big| \big\{ \beta\; :\; \max_j
|\bv_\beta(j)|^2 \ge \frac{C_1 q^2 \theta}{N}\big\} \Big| \ge
\frac{C_2}{q} \text{ and } \Omega \Big\} + e^{-cN} \\ \leq  \; &\P
\Big\{ \frac{1}{N}\Big| \big\{ \beta\; :\; \max_j |\bv_\beta(j)|^2
\ge \frac{C_1 q^2\theta}{N}\big\} \cap O_q \Big| \ge \frac{C_2}{2q}
\text{ and } \Omega \Big\} + \P \Big\{ \Big|O^c_q \Big| \ge
\frac{C_2 N}{2q} \text{ and } \Omega \Big\} + e^{-cN}
\end{split}
\end{equation}
A simple counting shows that the cardinality of the complement of
$O_q$ is bounded by
$$
|O_q^c|\leq \frac{CN}{q} .
$$
on $\Omega$. Therefore, by choosing $C_2$ sufficiently large, we have
\begin{equation*}
\begin{split}
\P \Big\{ \frac{1}{N}\Big| \big\{ \beta\; :\; \max_j
|\bv_\beta(j)|^2 \ge \frac{C_1 q^2\theta}{N}\big\} \Big| \ge
\frac{C_2}{q}\Big\} \leq  \; &\P \Big\{ \frac{1}{N}\Big| \big\{
\beta \in O_q\; :\; \max_j |\bv_\beta(j)|^2 \ge \frac{C_1
q^2\theta}{N}\big\}\Big| \ge \frac{C_2}{2q}\Big\} + e^{-cN}
\\ \leq  \; &\P \Big\{ \exists \beta \in O_q \; :\; \max_j
|\bv_\beta(j)|^2 \ge \frac{C_1 q^2\theta}{N} \Big\} + e^{-cN}
\\ \leq  \; & N \sup_{\beta} \, \P \Big\{ \beta \in O_q \text{ and } \max_j
|\bv_\beta(j)|^2 \ge \frac{C_1 q^2\theta}{N} \Big\} + e^{-cN},
\end{split}
\end{equation*}
where we used that $q \ll N$ (if $q \geq N^{1/2}$, (\ref{eq:thmloc})
is trivial). The theorem now follows from Lemma \ref{lm:ext} below.
\end{proof}

\bigskip

\begin{lemma}\label{lm:ext}
Under the assumptions of Theorem \ref{thm:loc}, there exists a
constant $c>0$ such that for any sufficiently large $C$ and for any
$q>0$ we have
\begin{equation}\label{ll}
\sup_{\beta} \P \Big\{ \beta \in O_q \text{ and } \max_j
|\bv_\beta(j)|^{2 } \ge \frac {C\theta q^2} N \Big\} \le    e^{- c
\theta}.
\end{equation}
\end{lemma}
\begin{proof} It is enough to prove
that, for arbitrary $\beta \in \{ 1, \dots, N
\}$,
\[
\P \Big\{ \beta \in O_q \text{ and } \max_j |\bv_\beta(j)|^{2
} \ge \frac {C\theta q^2} N \Big\} \le e^{- c \theta} .
\]
Therefore
we fix $\beta \in O_q$ and we consider first the $j=1$
component $v_1=\bv_{\beta} (1)$ of $\bv_{\beta}$; for brevity we
drop the index $\beta$ from the notation $\mu_\beta$ and
$\bv_\beta$. Set $\kappa:=q\theta/N$. Recall that $\lambda_\alpha$
denotes the eigenvalues of $B$ in the decomposition \eqref{Hd}.
 Denote by $A$ the  set
$$
  A:= \{\al\; :\; |\mu-\lambda_\al|\leq \kappa\}\; .
$$
{F}rom the interlacing property of
the eigenvalues, $|A|\ge \theta$
(if $\theta \leq \beta \leq N-\theta$, then actually $|A|=2\theta$).

Recall the equations \eqref{ee} and \eqref{ee1} obtained from the
eigenvalue equation $H\bv = \mu\bv$ and from the decomposition
\eqref{Hd}. In particular, from \eqref{ee1} we find \be\label{w}
\|\bw\|^2= \bw\cdot \bw = |v_1|^2 \ba^* (\mu -B)^{-2} \ba \, . \ee
Since $\|\bw\|^2 = 1 - |v_1|^2$, we obtain \be\label{v2} |v_1|^2 =
\frac{1}{1+ \ba^* (\mu -B)^{-2} \ba} = \frac{1}{1 + \frac{1}{N}
\sum_{\alpha} \frac{\xi_{\alpha}}{(\mu - \lambda_{\alpha})^2}} \, ,
\ee recalling the notation $\xi_\al=N|\ba\cdot \bu_\al|^2$ where
$\bu_\al$ is the normalized eigenvector of $B$ associated with the
eigenvalue $\lambda_\al$. Thus we have \be\label{v3} |v_1|^2 \le
\frac 1 { 1 + N^{-1} \kappa^{-2} \sum_{\alpha \in A} \xi_\alpha } =
\frac {N \kappa^{2}} { N \kappa^{2} +  \sum_{\alpha \in A}
\xi_\alpha }. \ee

Fix a small $\delta>0$. Let $Q$ be the following event
\[
Q= \Big\{ \sum_{\alpha \in A} \xi_\alpha >  \theta
\delta \Big\}.
\]
On this set $Q$ we have the  bound for $|v_1|^2$
\[
{\bf 1}_Q|v_1|^2\leq {\bf 1}_Q \frac {N \kappa^{2}} { N \kappa^{2} +
\sum_{\alpha\in A} \xi_\alpha } \le \delta^{-1} N \kappa^{2}
\theta^{-1} = \frac {\theta q^2} {N\delta} \; ,
\]
and for $\delta$ small enough, we have
\[
\P(Q^c) \le e^{- c \theta}
\]
by Corollary \ref{cor:BL}.

So far we have considered the $j=1$ component of $\bv$. We can
repeat the argument for each $j=1,2,\ldots, N$. Thus $Q$ should
carry a subscript $1$ and we can define $Q_j$ accordingly. Clearly,
$\P\{(\cap_j Q_j)^c\} \le  N e^{- c \theta}\le e^{-c'\theta}$.
 On the other hand,
on the set $\cap_j Q_j$ we have \[
 \max_j |\bv_\beta(j)|^2  \le
\frac {\theta q^2} {N \delta}\qquad \mbox{for any}\quad\beta\in
O_q\, . \]
\end{proof}

\bigskip

Theorem \ref{thm:loc} implies that all eigenvectors of $H$, apart
from a fraction vanishing in the limit $N \to \infty$, are
completely extended, in the sense that, up to logarithmic
corrections, $\| \bv \|_{\infty} \leq \const / N^{1/2}$. The reason
we cannot prove this bound for all eigenvectors of $H$ is the lack
of information about the microscopic distribution of the eigenvalues
of $H$ (and of its minors) on scales of order $O(1/N)$. {F}rom
Corollary \ref{cor:sc}, which gives precise information on the
eigenvalue distribution up to scales of order $O(N^{-2/3}\log N)$,
we can nevertheless get a non-optimal bound on $\| \bv \|_{\infty}$
for all eigenvectors of $H$.


\begin{proposition}\label{cor:linfty} Let $H$ be an $N\times N$
Wigner matrix as described in \eqref{wig} and satisfying the
conditions (\ref{gM}), (\ref{x2})  and (\ref{logsob}).
Fix $\kappa>0$, and assume that $C$ is large enough. Then there
exists $c>0$ such that
\[
\P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$,
$\| \bv \|=1$, $\mu \in [-2+\kappa, 2-\kappa]$ and } \| \bv
\|_\infty \ge \frac{C(\log N)}{N^{1/3}} \Bigg\} \leq e^{-c(\log
N)^2}\;. \]
\end{proposition}

\bigskip

{\it Remark.} The bound $\|\bv\|_\infty\leq CN^{-1/3}\log N$,
obtained in this proposition
trivially implies the upper bound $\|\bv \|_p\leq C(\log N)^{1-2/p}
N^{\frac{2}{3}\big(\frac{1}{p}-\frac{1}{2}\big)}$ for $2\leq p < \infty$
as well.

\bigskip

\begin{proof}
Let $\eta = N^{-2/3} (\log N)$ and define \[ I_n = [ -2 + \kappa +
(n-1) \eta ; -2 + \kappa + n \eta ] \qquad \text{for } n = 1,\dots,
n_{\text{max}} = [ (4-2\kappa) \eta^{-1} ] + 1, \] where $[x]$
denotes the integer part of $x\in\bR$. Then
\[ \bigcup_{n=1}^{n_{\text{max}}} I_n \supset [-2 + \kappa, 2 - \kappa]
\, \text{ and } |I_n| = \eta = N^{-2/3} (\log N)  \qquad \text{for
all } n=1,\dots, n_{\text{max}}.
\]
As before, let $\cN_{I}=|\{ \beta\; : \; \mu_\beta\in I\}|$ for any
$I\subset \bR$.
 By choosing $\eta^*= \eta\log N$
in \eqref{ncont} in Corollary \ref{cor:sc}, and using $\cN_{I_n}\leq
\cN_{I_n^*}$ with $I_n^*$ being the interval of length $2\eta^*$
with the same center as $I_n$,
we have
\[
\P \left\{\max_n \cN_{I_n} \leq \frac{N\eta}{\log N} \right\}
 \leq e^{-c (\log N)^2}.
\]
Suppose that $\mu
\in I_n$, and that $H\bv = \mu \bv$. {F}rom (\ref{v2}), we obtain
\[ |v_1|^2 = \frac{1}{1+\frac{1}{N} \sum_{\alpha}
\frac{\xi_{\alpha}}{(\lambda_{\alpha}-\mu)^2}} \leq
\frac{1}{1+\frac{1}{4N\eta^2} \sum_{\lambda_\alpha \in I_n} \xi_{\alpha}}
\leq \frac{4 N \eta^2}{\sum_{\lambda_\alpha \in I_n} \xi_{\alpha}}\,.
 \]
and from the interlacing
property, there exist at least $\cN_{I_n}-1$ eigenvalues $\lambda_\al$
in $I_n$.
Therefore
\begin{equation}
\begin{split}
\P \Big( \exists &\text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$,
$\mu \in [-2+\kappa, 2-\kappa]$ and } \| \bv \|_\infty \ge
\frac{C(\log N)}{N^{1/3}} \Big)  \\
&\leq \sum_{n=1}^{n_{\text{max}}} \sum_{j=1}^N \, \P \Big( \exists
\text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I_n$ and }
|v_j|^2 \ge \frac{C(\log N)^2}{N^{2/3}} \Big) \\
&\leq N n_{\text{max}} \sup_n \P \Big( \exists \text{ $\bv$ with
$H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I_n$ and } |v_1|^2 \ge
\frac{C(\log N)^2}{N^{2/3}} \Big)\\
&\leq \const \,  N \eta^{-1} \sup_n \P \left( \sum_{\lambda_\alpha \in I_n}
\xi_{\alpha} \leq \frac{4 N^{1/3}}{C} \right) \\
&\leq  \const \, N \eta^{-1} \sup_n \P \left( \sum_{\lambda_\alpha \in I_n}
\xi_{\alpha} \leq \frac{4 N^{1/3}}{C} \text{ and } \cN_{I_n} \geq
N^{1/3} \right) + \const \,  N \eta^{-1}  \sup_n \, \P
\left(\cN_{I_n} \leq \frac{N\eta}{\log N} \right)
\\&\leq \const \,  N^{5/3} e^{-\delta N^{1/3}} + \const \, N^{5/3} e^{-c (\log
N)^2} \leq e^{-c' (\log N)^2}\,.
\end{split}
\end{equation}
using Corollary \ref{cor:BL} and choosing $C$ sufficiently large.
\end{proof}


\section{Second moment of the Green function}

In this section, we use the result of Theorem \ref{thm:loc} to
obtain bounds on the second moment of the diagonal elements of the
Green function of $H$.
 Recall the notation $\theta
=[\log N]^2$.

\begin{theorem}\label{thm:green}
Let $H$ be an $N\times N$ Wigner matrix as described in \eqref{wig}
and satisfying the conditions (\ref{gM}) and (\ref{x2}). Let
$z=E+i\eta$ be the spectral parameter of the Green function
$G_{E,\eta}=G_z=(H-z)^{-1}$. Then there exist $c,C>0$ such that for
any $\eta$, \be \P\Bigg( \Big| \Big\{ E \, :\,
\frac{1}{N}\sum_{j=1}^N
  |G_{E,\eta}(j,j)|^2 \ge C(\log N)^{12}\Big\}\Big|\ge \frac{C}{\log N} \Bigg)
\leq e^{-c(\log N)^2}. \ee
\end{theorem}

{\it Remark.} This theorem states that, with the exception of a very
small probability, the second moment of the Green function, averaged
over all sites, remains bounded (modulo logarithmic corrections) for
all but a negligible set of energies in the sense of Lebesgue
measure.

\bigskip

\begin{proof}
For any $k\in \bbZ$, we define the random sets
$$
   M_k: = \Big\{ \al\, :\, \frac{2^k}{N} < |\mu_{\al-\theta}
-\mu_{\al + \theta}|\leq \frac{2^{k+1}}{N}\Big\}\, .
$$
where we used again the convention that $\mu_{\alpha} = \mu_1$ for
all $\alpha \leq 1$ and $\mu_{\alpha} = \mu_N$ for all $\alpha \geq
N$. For given $\kappa, K_0 >0$, let
$$
  \Omega_1: = \Big\{\bigcup_{k=0}^{\kappa \log N}
   M_k=\{1,2, \ldots N\} \Big\}
\cap \Big\{  \sigma (H) \subset [ - K_0 , K_0] \Big\} .
$$
From \eqref{tail} we know that
$$
 \P\Big\{ \sigma(H)\in [-K_0 , K_0]\Big\}\ge 1- e^{-cN},
$$
for a sufficiently large $K_0$, so we
obtain that $M_k=\emptyset$ for all $k\ge \kappa \log N$, if
$\kappa$ is large enough, apart from an exponentially small event.
{F}rom Theorem \ref{du} we obtain that
$$
\P \Big\{ M_k =\emptyset \; \mbox{for all}\; k <0 \Big\}\ge 1 -
e^{-c\theta}
$$
and therefore, if $K_0$ and $\kappa$ are large enough,
$$
   \P (\Omega_1)\ge 1- e^{-c\theta}.
$$
In the sequel we will work on the set $\Omega_1$, i.e. we can assume
that the index $k$ runs from $0$ to $(\const)\log N$ and that all
eigenvalues lie in $[-K_0, K_0]$.

By a simple counting,  the cardinality of $M_k$ is bounded by \be
  |M_k|\leq (\const)2^{-k}N\theta\,
\label{Mk} \ee on the event $\Omega_1$.


For any $\alpha \in M_k$, denote
\be
  \Omega_k(\alpha): = \Big\{\max_j |\bv_\al(j)|^2
\le C \frac{2^{2k}}{4N\theta}\Big\},
\label{omal}
\ee
where $\bv_\al$ is the normalized eigenvector to
the eigenvalue $\mu_\al$. {F}rom Lemma \ref{lm:ext}, we obtain, for
every $k=0,\dots,\kappa \log N$,
$$
   \P \Big\{ \bigcup_{\alpha \in M_k} \Omega^c_k(\alpha) \Big\} \le
  e^{-c\theta}
$$
for some $c>0$. Let
$$
   \Omega: = \Omega_1\cap \bigcap_k\bigcap_{\alpha\in M_k} \Omega_k(\alpha),
$$
then
$$
 \P (\Omega)\ge 1- e^{-c\theta},
$$
for some $c>0$. In the sequel
 we will work on the event $\Omega$.



Define the following random set of energies:
$$
   \cE :=\bR\setminus\bigcup_k \bigcup_{\al\in M_k}
   \Big\{ E\; :\; |\mu_\al -E|\leq \frac{2^k}{N\theta^2}\Big\}.
$$
The Lebesgue measure of the complement of $\cE$ is bounded by
$$
  |\cE^c|\leq \sum_k \sum_{\al\in M_k} \frac{2^{k+1}}{N\theta^2}
\leq \frac{C}{\log N}.
$$

Let $E\in \cE$ and $\omega\in \Omega$. We compute \be
\begin{split}
  \frac{1}{N}\sum_{j=1}^N |G_{E,\eta}(j,j)|^2
\leq&  \frac{1}{N}\sum_{j=1}^N\sum_{k,\ell=0}^{\kappa \log N}
   \sum_{\al\in M_k}\sum_{\beta\in M_\ell}
    \frac{|\bv_\al(j)|^2}{|\mu_\al-E|}\frac{|\bv_\beta(j)|^2}{|\mu_\beta-E|}
\\
\leq & \frac{2}{N}\sum_{j=1}^N\sum_{k\le\ell}^{\kappa \log N}
   \sum_{\al\in M_k}\sum_{\beta\in M_\ell}
    \frac{|\bv_\al(j)|^2}{|\mu_\al-E|}\frac{|\bv_\beta(j)|^2}{|\mu_\beta-E|}
\\
\leq & \frac{2}{N}\sum_{k\le\ell}^{\kappa \log N}
   \frac{2^{2k}C}{4N\theta}
   \sum_{\al\in M_k}\sum_{\beta\in M_\ell}
    \frac{1}{|\mu_\al-E|}\frac{1}{|\mu_\beta-E|}
\end{split}
\label{albeta} \ee In the second line we used the symmetry between
$\al$ and $\beta$, in the third line we used the estimate on
$|\bv_\al(j)|^2$ from \eqref{omal} and that $\sum_j
|\bv_\beta(j)|^2=1$.

We now perform the $\al\in M_k$ summation; the $\beta\in M_\ell$
summation will be identical. Let $I$ be an arbitrary interval of
length $|I|= 2^{k}/N$. We claim that the number of eigenvalues
$\mu_\al \in I$ with $\al\in M_k$ is at most $2\theta$. We label the
elements of $M_k$ in increasing order; $\al_1<\al_2< \ldots
<\al_{|M_k|}$. Let $\mu_{\alpha_i}$ be the smallest eigenvalue in
the set $I$ with index in $M_k$. If $i > |M_k| - 2\theta$, then
there cannot be more than $2\theta$ eigenvalues with indices in
$M_k$ in $I$. Otherwise, if $i \leq |M_k| - 2\theta$, we have
$$
    \mu_{\al_{i+2\theta}} -\mu_{\al_i}\ge \mu_{\al_{i+\theta}+\theta}
   - \mu_{\al_{i+\theta}-\theta} >\frac{2^k}{N}
$$
and therefore, since $|I| = 2^k/N$, $\mu_{\al_{i+2\theta}}$ cannot
be in $I$.





We now define the intervals
$$
      I_m: = \Big [E+\frac{2^k(m-\frac{1}{2})}{N},
E+\frac{2^k(m+\frac{1}{2})}{N}\Big]
$$
for each $m\in \bbZ$, $|m|\leq CN\cdot 2^{-k}$. Clearly, each $I_m$
contains at most $2\theta$ eigenvalues $\mu_\al$ with index $\al\in
M_k$.

Notice that for any $\mu\in I_m$, $m\neq 0$, we have $|\mu-E|\ge
2^{k-1}m/N$. For $\mu_\al\in I_0$, with $\al\in M_k$,
 by the choice of $E\in \cE$,
we have $|\mu_\al-E|\ge  2^k/(N\theta^3)$. Therefore \be
\begin{split}
  \sum_{\al\in M_k}  \frac{1}{|\mu_\al-E|} & \leq
  2\theta \sum_{|m|\leq CN\cdot 2^{-k}}
   \max\Big\{ \frac{1}{|\mu_\al-E|} \; : \; \al\in M_k, \;
  \mu_\al\in I_m\Big\}\\
 &\leq 2\theta\Big[ \max\Big\{ \frac{1}{|\mu_\al-E|} \; : \; \al\in M_k\Big\}
  + 2\sum_{m=1}^{ CN\cdot 2^{-k}} \frac{N}{2^{k-1}m}\Big]
  \\
  & \leq \frac{ C\theta^3 N}{2^k}.
\label{alsum}
\end{split}
\ee


Using \eqref{alsum} both for the $\al$ and $\beta$ summations in
\eqref{albeta}, we obtain
$$
  \frac{1}{N}\sum_{j=1}^N |G_{E,\eta}(j,j)|^2
\leq  \frac{2}{N}\sum_{k=0}^{\kappa \log N} \sum_{\ell=k}^{\kappa
\log N}
   \frac{2^{2k}}{4N\theta} \frac{ C\theta^3 N}{2^k}\frac{ C\theta^3 N}{2^\ell}
  \leq C\theta^6
$$
for any $E\in \cE$ and $\om\in \Omega$. This completes the proof of
Theorem \ref{thm:green}.
\end{proof}


\section{Absence of localized eigenvectors}\label{sec:Meta}

In this section we show that eigenvectors of Wigner random matrices,
up to events with exponentially small probability, cannot be
localized in a strong sense given by the following definition.

\begin{definition} Let $L\ge 1$ be an integer
and $\eta>0$.  We say that an $\ell^2$-normalized vector $\bv =(v_1,
\ldots, v_N)\in \bC^N$ exhibits $(L,\eta)$-localization  if there
exists a set $A \subset \{ 1, 2, \dots ,N\}$ such that $|A|=L$ and
$\sum_{j \in A^c} |v_j|^2 \leq \eta$.
\end{definition}

\begin{theorem}\label{thm:Meta} Let $ H$ be
an $N\times N$ Hermitian random matrix from the Wigner ensemble
defined in \eqref{wig}, satisfying also the condition (\ref{gM}) and
(\ref{x2}). Suppose that $\eta$ and $\nu=L/N$ are sufficiently
small. Then, with a constant $c>0$ that depends only on $M$ and
$\delta$ from \eqref{gM}, \eqref{x2}), we have
\[ \P \big\{ \exists \text{ a normalized eigenvector
$\bv$ of $H$ exhibiting $(L,\eta)$-localization} \big\} \leq e^{-c
N} .
\]
\end{theorem}

\begin{proof}
Since, by (\ref{tail}), \[ \P \Big\{ \exists \text{ eigenvalue $\mu$
of $H$ with } |\mu| \geq K_0 \Big\} \leq e^{-cN} \] if $K_0$ is
large enough, it is sufficient to prove that
\begin{equation}\label{eq:Met1} \sup_{\beta \in \{ 1, \dots ,N \}} \,
\P \Big\{ \text{$\bv_{\beta}$ exhibits $(L,\eta)$ localization and
}|\mu_{\beta}|\leq K_0 \Big\} \leq e^{-cN} \end{equation} where
$\mu_1 \leq \mu_2 \leq \dots \leq \mu_N$ denote the eigenvalues of
$H$, and $\bv_1, \bv_2,\dots, \bv_N$ the corresponding normalized
eigenvectors. To prove (\ref{eq:Met1}), we fix $\beta$, and consider
the eigenvector $\bv_{\beta}$ associated with the eigenvalue
$\mu_{\beta}$; for brevity, we drop the index $\beta$ from
$\mu_{\beta}$ and $\bv_{\beta}$.

By the definition of $(L,\eta)$-localization and by the permutation
symmetry
\begin{equation}\label{eq:Meta1}
\begin{split}
\P \big\{\text{$\bv$ is $(L,\eta)$-localized and $|\mu| \leq
K_0$}\big\} &=\P\Big\{\exists \; A \subset \{1, \dots ,N\} : |A|= L
\text{ and } \sum_{j\in A^c}
|v_j|^2 \leq \eta  \text{ and $|\mu| \leq K_0$} \Big\} \\
&\leq {N \choose L} \, \P \Big\{ \sum_{j=L+1}^N  |v_j|^2 \leq \eta
\text{ and $|\mu| \leq K_0$} \Big\}\,.
\end{split}
\end{equation}
We introduce the notation $\bu= (v_1,\dots v_L)^t$, $\bw=(v_{L+1},
\dots , v_N)^t$, and, for $j=L+1,\dots,N$,
\[
\bc_j = \frac{1}{\sqrt{N}} \, (h_{j1}, h_{j2}, \dots, h_{jL})^* \in
\bC^L \quad \text{and} \quad \bd_j = \frac{1}{\sqrt{N}} \,
(h_{j,L+1}, \dots , h_{jN})^* \in \bC^{N-L}.
\]
{F}rom the eigenvalue equation $H\bv = \mu \bv$, we obtain, for all
$j \geq L+1$,
\[
\mu v_j = \bc_j \cdot \bu + \bd_j \cdot \bw
% \qquad \Rightarrow \qquad \bc_j \cdot \bu = \mu v_j - \bd_j \cdot \bw
\]
and thus
\[
\sum_{j=L+1}^N |\bc_j \cdot \bu|^2 = \sum_{j=L+1}^N |\mu v_j - \bd_j
\cdot \bw|^2 \leq 2 \mu^2 \| \bw \|^2 + 2 \sum_{j=L+1}^N |\bd_j
\cdot \bw|^2.
\]
Denoting by $D_1$ the $(N-L) \times L$ matrix with rows given by
$\bc_{L+1}^*, \dots, \bc_N^*$ and by $D_2$ the $(N-L) \times (N-L)$
matrix with rows given by $\bd_{L+1}^*, \dots , \bd_N^*$, last
equation implies
\[
( \bu, D_1^* D_1 \bu ) \leq 2 \mu^2 \| \bw \|^2 + 2 ( \bw , D_2^*
D_2 \bw ) \leq 2 \| \bw \|^2 \left( \mu^2 + \lambda_{\text{max}}
(D_2^* D_2) \right) .
\]
Thus, from (\ref{eq:Meta1}), we conclude that
\begin{equation}\begin{split}\label{eq:lbd}
 \P \big\{\text{$\bv$ is $(L,\eta)$-localized}
 &\text{ and $|\mu| \leq K_0$} \big\}
\\ &\leq {N \choose L} \, \P \{ \| \bw \|^2 \leq \eta \text{ and $|\mu|
\leq K_0$} \}  \\ &\leq {N \choose L} \, \P \left\{( \bu, D_1^* D_1
\bu) \leq 2 \eta \left( \mu^2 + \lambda_{\text{max}} (D_2^* D_2)
\right) \text{ and $|\mu| \leq K_0$} \right\}
\\ &\leq {N \choose L}
\, \P \left\{ (1-\eta) \lambda_{\text{min}} (D_1^* D_1) \leq 2 \eta
\left( K_0^2 + \lambda_{\text{max}} (D_2^* D_2) \right) \right\}
\\ &\leq {N \choose L} \, \P \left\{ (1-\eta) \frac{N-L}{N}
\lambda_{\text{min}} (X_1^* X_1) \leq 2 \eta \left( K_0^2 +
\frac{N-L}{N} \lambda_{\text{max}} (X_2^* X_2) \right) \right\}
\\ &\leq {N \choose L} \, \Big[ \P \big\{\lambda_{\text{min}} (X_1^* X_1)
\leq c \big\} + \P \big\{\lambda_{\text{max}} (X_2^* X_2) \geq
C\big\}
 \Big]
\end{split}
\end{equation}
for any positive constants $c$ and $C$ if $\eta$ and $\nu=L/N$ are
sufficiently small (because $(1-\eta)(1-\nu) c \geq 2 \eta (K_0^2 +
(1-\nu) C)$ if $\eta, \nu$ are sufficiently small). Here
$\lambda_{\text{min}} (F)$ and $\lambda_{\text{max}} (F)$ denotes
the minimal and, respectively, the maximal eigenvalue of the
Hermitian matrix $F$, and $X_1 = \sqrt{N/(N-L)} \, D_1$, $X_2 =
\sqrt{N/(N-L)} \, D_2$. {F}rom Lemma \ref{lm1} and Lemma~\ref{lm2}
below,
we know that, for any sufficiently small $\nu=L/N$, for sufficiently
large $C$, and for $c<1/2$, there exists $\alpha
>0$ such that
\begin{equation}
\begin{split}
\P\big\{\lambda_{\text{min}} (X_1^* X_1) \leq c\big\} &\leq
e^{-\alpha (N-L)}
\qquad \text{and} \\
\P\big\{\lambda_{\text{max}} (X_2^* X_2) \geq C\big\} &\leq
e^{-\alpha (N-L)}.
\end{split}
\end{equation}
Thus, from (\ref{eq:lbd}), we obtain that, for $\eta>0$ and $\nu =
L/N$ small enough,
\begin{equation}
\begin{split} P \big\{\text{$\bv$ is $(L,\eta)$-localized and $|\mu| \leq K_0$}\big\}
\leq 2 {N \choose L} \, e^{-\alpha (N-L)} \leq \left(
\frac{e}{\nu}\right)^{\nu N} e^{-\alpha N (1- \nu)} \leq e^{-\alpha
N/4}.
\end{split}
\end{equation}
Since the constant $\alpha$ is independent of the eigenvalue $\mu$,
(\ref{eq:Met1}) follows.
\end{proof}

\bigskip

\begin{corollary}\label{cor:lp}
Suppose that the random matrix $H$ satisfies the same assumptions as
in Theorem \ref{thm:Meta}.  Then, for every $\kappa >0$
sufficiently small, there exists a constant $c>0$ such that
\[ \P \left\{
 \exists \text{ normalized } \bv \in \bC^N \text{ such that } H \bv = \mu
\bv \text{ and } \| \bv \|_p \leq \kappa N^{\frac{1}{p} -
\frac{1}{2}} \right\} \leq e^{-cN} \] for any $1\leq p \leq 2$.
\end{corollary}

{\it Remark.} If the eigenvector $\bv$ is uniformly extended, i.e.
$|v_j|^2=\frac{1}{N}$, then $\| \bv \|_p = N^{1/p-1/2}$. This
Corollary indicates that the behavior of all eigenvectors is
consistent with the  extended states hypothesis as far as the low
$\ell^p$-norms ($1\leq p\leq 2$) are concerned.

\bigskip

\begin{proof}
{F}rom \eqref{tail} with a sufficiently large $K_0$ we have
\begin{equation}\label{eq:lp-Meta}
\begin{split}
 \P \Big( \exists &\text{ normalized }
\bv \in \bC^N \text{ such that } H \bv = \mu
\bv \text{ and } \| \bv \|_p \leq \kappa N^{\frac{1}{p} -
\frac{1}{2}} \Big) \\
%\leq \; &\P \left( \exists \text{ eigenvalue }
%\mu \text{ of $H$ with } |\mu| \geq K_0 \right) \\ &+ \P \Big(
%\exists \text{ normalized } \bv \in \bC^N \text{ such that } H \bv =
%\mu \bv, |\mu| \leq K \text{ and } \| \bv \|_p \leq \kappa
%N^{\frac{1}{p} - \frac{1}{2}} \Big) \\
\leq \; &e^{-cK_0N} + \P \Big(
\exists \text{ normalized } \bv \in \bC^N \text{ such that } H \bv =
\mu \bv, |\mu| \leq K_0 \text{ and } \| \bv \|_p \leq \kappa
N^{\frac{1}{p} - \frac{1}{2}} \Big)\,.
\end{split}
\end{equation}
Now, if $\bv$ is a normalized eigenvector of $H$, associated with an
eigenvalue $|\mu| \leq K_0$, we can apply Theorem~\ref{thm:Meta}. To
this end, we fix $\nu$ and $\eta$ small enough, and let $L = \nu N$.
After relabeling, we can assume that $|v_1| \geq |v_2| \geq \dots
\geq |v_L| \geq |v_{L+1}| \geq \dots \ge |v_N|$. Then, by Theorem
\ref{thm:Meta},
\[
\P \Big\{ \sum_{j\leq L} |v_j|^2 \geq \eta\Big\} \leq e^{- c N} .
\]
Thus, with the exception of an event with exponentially small
probability,
\[
L |v_L|^2 \leq \sum_{j=1}^L |v_j|^2 \leq \eta
\]
This implies that $|v_L| \leq \sqrt{\eta/L}$. Therefore
\[
1-\eta \leq \sum_{j \geq L+1} |v_j|^2 \leq |v_L|^{2-p} \sum_{j\geq
L+1} |v_j|^p \leq (\eta/L)^{1-p/2} \sum_{j=1}^N |v_j|^p
\]
and hence
\[
\P \left( \sum_{j=1}^N |v_j|^p \leq L^{1-p/2}
\frac{1-\eta}{\eta^{1-p/2}} = \kappa^p  N^{1-p/2} \right) \leq e^{-c
N}
\]
which, together with (\ref{eq:lp-Meta}), completes the proof.
\end{proof}

\bigskip

In the next two lemmas, we prove effective large deviation estimate
on the largest and the smallest eigenvalue of some covariance
matrices.

\begin{lemma}\label{lm1} Let $X=(X_{ij})$ be a complex $N\times L$ matrix,
with $N >L$, such that, for all $i=1,\dots,N$ and $j=1,\dots,L$,
$\text{Re}\, X_{ij}, \text{Im} \, X_{ij}$ are i.i.d. random
variables with
\[
\E \, X_{ij} = 0,  \qquad \E \, |X_{ij}|^2 = \frac{1}{2N} \qquad
\text{and } \quad \E \, e^{\delta N|X_{ij}|^2} \leq K_\delta <\infty
\]
for some $\delta>0$ and with $K_\delta$ independent of $N$.
\begin{itemize}
\item[i)] For $C>0$ large enough
\[ \P (\lambda_{\text{max}} (X^* X) \geq C) \leq e^{-c_0 C N} \] for
a constant $c_0$ depending only on $\delta$.
\item[ii)] For $\nu= L/N$ sufficiently small and for all $0 <c<1/2$
there exists $\alpha_0 = \alpha_0 (\delta, c , \nu) >0$ such that
\[ \P (\lambda_{\text{min}} (X^* X) \leq c) \leq e^{-\alpha N} \]
for all $\alpha < \alpha_0$.
\end{itemize}
\end{lemma}

{\it Remark.} The precise large deviation rate
function for $\lambda_{\min}$ and $\lambda_{\max}$ was determined
recently in \cite{FHK} in the limit $N\to\infty$ under the
additional condition that $L=o(N/\log\log N)$. Our proof is somewhat
different and it also applies to the case $L\leq \nu N$, with $\nu$
small enough, but the decay rate we obtain is not precise.
The history and earlier results in this direction was reviewed
in \cite{FHK} and we shall not repeat it here.

\bigskip

\begin{proof}
We begin by proving i). First, fix $\bz \in \bC^L$, with $\| \bz \|
=1$. We claim that
\begin{equation}\label{Meta1b}
\P\big\{ ( \bz,X^* X \bz) \geq C\big\} \leq e^{-c_1 CN}
\end{equation} for a constant $c_1$ depending only on $\delta$.
In fact, for arbitrary $\kappa >0$,
\begin{equation}
\begin{split}
\P \big\{ (\bz, X^* X \bz) \geq C\big\}
 &\leq e^{-\kappa C N} \E \, e^{\kappa N
(\bz, X^* X \bz)} \\
&=e^{-\kappa C N} \E \, e^{\kappa N \sum_{j=1}^N |\bX_j \cdot
\bz|^2}
\end{split}
\end{equation}
where, for $j=1, \dots ,N$, $\bX_j = (X_{j1}, \dots ,X_{jL})^*$
denotes the adjoint of the $j$-th row of $X$. Since different rows
of $X$ are independent and identically distributed, we find
\begin{equation}\label{Meta2}
\begin{split}
\P \big\{ (\bz, X^* X \bz) \geq C\big\}
 &\leq e^{-\kappa C N} \prod_{j=1}^N \E\,
e^{\kappa N |\bX_j \cdot \bz|^2} = e^{-\kappa C N} \left( \E \,
e^{\kappa N |\bX_1 \cdot \bz|^2} \right)^N.
\end{split}
\end{equation}
Consider now the random vector $\bY= \sqrt{N} \bX_1 = (y_1, \dots ,
y_L)^*$ with i.i.d. components. We have
\begin{equation}\label{Meta3}
\begin{split}
\E \, e^{\kappa |\bY \cdot \bz|^2} &= \const \int_{\bR\times\bR} \rd
q \rd p \, e^{-\frac{q^2 + p^2}{4}} \, \E\, e^{\sqrt{\kappa} \left(q
\text{Re} \, (\bY\cdot \bz) + p \, \text{Im} \, (\bY \cdot \bz)
\right) } \\ &= \const \int_{\bR\times\bR}
 \rd q \rd p \, e^{-\frac{q^2 + p^2}{4}} \, \prod_{i=1}^L \E \,
e^{\sqrt{\kappa} \left( q \, \text{Re} (z_i y_i) + p \, \text{Im}
(z_i y_i) \right)}\\ &= \const \int_{\bR\times\bR}
 \rd q \rd p \, e^{-\frac{q^2 +
p^2}{4}} \, \prod_{i=1}^L \E \, e^{\sqrt{\kappa} \left( q \,
\text{Re} z_i  + p\, \text{Im} z_i \right) \text{Re} y_i} \, \E \,
e^{\sqrt{\kappa} \left(- q \, \text{Im} z_i  + p \, \text{Re} z_i
\right) \text{Im} y_i}
\end{split}
\end{equation}
with an appropriate normalization constant. Since $\E \, \text{Re}
y_i = 0$ we find, for arbitrary $r \in \bR$,
\begin{equation}
\E \, e^{ r \text{Re} \, y_i} = \sum_{n \geq 0} \frac{r^n}{n!} \E
(\text{Re} \,y_i)^n = 1 + \sum_{n \geq 1}\frac{r^{2n}}{(2n)!} \E
(\text{Re} \, y_i)^{2n} + \sum_{n \geq 1}  \frac{r^{2n+1}}{(2n+1)!}
\E (\text{Re} \, y_i)^{2n+1}.
\end{equation}
Using that, for all $n \geq 1$,
\[
\frac{r^{2n+1}}{(2n+1)!} \E (\text{Re} \, y_i)^{2n+1} \leq
\frac{r^{2n}}{(2n)!} \E (\text{Re}\, y_i)^{2n} +
\frac{r^{2n+2}}{(2n+2)!} \E (\text{Re}\, y_i)^{2n+2},
\]
 we obtain
that
\begin{equation}
\begin{split}
\E \, e^{ r \text{Re} \, y_i} &= \sum_{n \geq 0} \frac{r^n}{n!} \E
(\text{Re} \,y_i)^n = 1 + 3 \sum_{n \geq 1}\frac{r^{2n}}{(2n)!} \E
(\text{Re} \, y_i)^{2n} \leq 1 + \sum_{n \geq 1}\frac{n! \,
(3r)^{2n}}{\delta^{2n} (2n)!} \E e^{\delta (\text{Re} \, y_i)^2} \\
& \leq 1 + \sum_{n \geq 1} \frac{(3 r)^{2n} K_{\delta}^n}{n!
\delta^{2n}} \leq e^{9 K_{\delta} r^2/ \delta^2},
\end{split}
\end{equation}
where we chose $\delta >0$ small enough, and we used that
$K_{\delta} = \E e^{\delta y^2}= \int e^{\delta y^2}e^{ - g(y)}\rd y
 < \infty$.
Since $\|\bz\|=1$, from (\ref{Meta3}) we obtain
\begin{equation}\label{Meta4}
\E \, e^{\kappa |\bY \cdot \bz|^2} \leq \const \int_{\bR\times\bR}
 \rd q \rd p \, e^{-(q^2+p^2) \left(\frac{1}{4} - 36
\kappa (K_{\delta}/\delta^2)\right)} \leq \const
\end{equation}
by choosing $\kappa >0$ small enough.
 Inserting in
(\ref{Meta2}), and choosing $C$ large enough, we find
(\ref{Meta1b}).

\medskip

Now, for fixed $0< \e <1/4$, we choose a family $\{ \bz_j \}_{j\in
I}$ with $\bz_j \in \bC^L$, $\| \bz_j\| \leq 1$ for all $j \in I$,
such that $|I| \leq (2/\e)^{2L}$, and such that, for all $\bz \in
\bC^{L}$ with $\| \bz \|=1$ there exists $j \in I$ with $\| \bz -
\bz_j \| \leq \e$. For a suitable $j \in I$, we have
\begin{equation}\begin{split}
\| X^* X \| &= \sup_{\bz\in \bC^N} (\bz, X^* X \bz) =
(\bz_{\text{max}}, X^* X \bz_{\text{max}}) \\ &\leq 2
\|\bz_{\text{max}} - \bz_j \| \| X^* X \| + (\bz_j, X^* X \bz_j)
\leq 2 \e \| X^* X \| + (\bz_j, X^* X \bz_j)
\end{split}\end{equation}
 and thus, if
$\lambda_{\text{max}} (X^* X) \geq C$, there must be at least one
$j\in I$ such that
\[
(\bz_j, X^* X \bz_j) \geq (1-2\e) C.
 \]
Therefore, since $|I| \leq (2/\e)^{2L}$, we can apply (\ref{Meta1b})
to obtain
\begin{equation}\begin{split}
\P \big\{ \lambda_{\text{max}} (X^* X) \geq C\big\} &\leq \P
\big\{\exists j \in I : (\bz_j, X^* X \bz_j) \geq (1-2\e) C\big\} \\
&
\leq (2/\e)^{2L}  \sup_{j} \, \P\big\{(\bz_j, X^* X \bz_j) \geq (1-2\e) C\big\} \\
&\leq (2/\e)^{2L} e^{-c_1 C N} ,
\end{split} \end{equation}
and thus, for $C$ large enough (and since $L \leq N$),
\[
\P \big\{\lambda_{\text{max}} (X^* X) \geq C\big\} \leq e^{-(c_1/2)
C N} .
\]

Next, we prove ii). Again, we first fix $\bz \in \bC^L$, and prove
that, for $0<c<1/2$, and for all $\alpha$ sufficiently small,
\begin{equation}
\label{eq:ii1} \P\big\{ (\bz, X^*X \bz) \leq c\big\} \leq e^{-\alpha
N}.
\end{equation}
To this end, we observe that, for $\beta >0$
\begin{equation}
\begin{split}
\P\big\{ (\bz, X^*X \bz) \leq c\big\}  &\leq e^{\beta N c} \, \E
\,e^{-\beta N (\bz, X^*X \bz)} =
 \left( e^{\beta c} \, \E \, e^{-\beta | \bY \cdot \bz|^2} \right)^N,
\end{split}\label{eq:end}
\end{equation}
where we defined, as before, $\bY=\sqrt{N} \bX_1 = (y_1, \dots ,
y_L)^*$. Since $e^{-\beta r} \leq 1 - \beta r + \beta^2 r^2/2$ for
all $r \geq 0$, we obtain
\[
\E \, e^{-\beta | \bY \cdot \bz|^2} \leq 1 - \beta \E \, |\bY
\cdot\bz|^2 + \frac{\beta^2}{2} \E |\bY \cdot \bz|^4 = 1 -
\frac{\beta}{2} + O(\beta^2) \leq e^{-\beta/2 + O(\beta^2)}
\]
if $\beta>0$ is sufficiently small
depending only on $\E \, y_1^4$. Therefore, we find
\[
e^{\beta c}\, \E \, e^{-\beta | \bY \cdot \bz|^2} \leq e^{-\beta
\big(\frac{1}{2}-c\big)+ O(\beta^2)}
\]
which proves (\ref{eq:ii1}) from \eqref{eq:end}
with a sufficiently small $\al$, depending on $c$.


 To conclude the proof of ii), we fix $\e >0$
and a family $\{ \bz_j \}_{j\in I}$ with $\bz_j \in \bC^L, \|
\bz_j\| \leq 1$ for all $j \in I$, such that, for all $\bz \in
\bC^{L}$ with $\| \bz \|=1$ there exists $j \in I$ with $\| \bz -
\bz_j \| \leq \e$ and $|I| \leq (2/\e)^{2L}$. Then, for a suitable
$j \in I$,
\begin{equation}\begin{split}
\lambda_{\text{min}} (X^*X) &=  \inf_{\| \bz \| = 1} (\bz, X^* X
\bz) = (\bz_{\text{min}}, X^* X \bz_{\text{min}}) \\ &\geq (\bz_j,
X^* X \bz_j) - 2 \| \bz_{\text{min}} - \bz_j \| \lambda_{\text{max}}
(X^*X) \\ &\geq (\bz_j, X^* X \bz_j) - 2 \e \lambda_{\text{max}}
(X^*X).
\end{split}\end{equation} Therefore, we find
\begin{equation}
\begin{split}
\P\big\{ \lambda_{\text{min}} (X^* X) \leq c\big\} \leq \; &\P
\big\{\lambda_{\text{min}} (X^*X) \leq c \; \text{and} \;
\lambda_{\text{max}} (X^*X) \leq C\big\} + \P
\big\{\lambda_{\text{max}} (X^*X) \geq C\big\} \\ \leq \; &\P
\big\{\exists j : (\bz_j, X^* X \bz_j) \leq c + 2 \e C\big\} + \P
\big\{\lambda_{\text{max}} (X^*X) \geq C\big\} \\ \leq \; &
\left(\frac{2}{\e}\right)^{2L} \, \P \big\{(\bz_1, X^* X \bz_1) \leq
c + 2 \e C\big\} + \P \big\{\lambda_{\text{max}} (X^*X) \geq
C\big\}.
\end{split}
\end{equation}
Part ii) now follows using the result of part i) with a sufficiently
large $C$, choosing $\e >0$ sufficiently small and using that $L/N
=\nu$ is small enough.
\end{proof}


\begin{lemma}\label{lm2} Let $X$ be a $N\times N$
Hermitian random matrix as described in \eqref{wig}
and we assume condition \eqref{x2}.
%{\bf such that $\text{Re} X_{ij}, \text{Im}
%X_{ij}$ and $X_{ii}$ are independent real random variables for all
%$1\leq i < j \leq N$, such that $\text{Re} X_{ij}, \text{Im} X_{ij}$
%are identically distributed for all $1\leq i <j \leq N$, and such
%that}
%\[ \E X_{ij} = 0, \quad \E e^{\delta N|X_{ij}|^2}
%\leq K_\delta  \quad \text{for all} \quad 1 \leq i \leq j \leq N \]
%for a sufficiently small $\delta >0$ and with a finite constant
%$K_\delta$ independent of $N$.
Then, for $K_0>0$ large enough
\[
\P \big\{ \lambda_{\text{max}} (X) \geq K_0\big\} \leq e^{-c_0 K_0^2N}
\]
with a constant $c_0$ depending only on $\delta$ in \eqref{x2}.
\end{lemma}
\begin{proof}
Fix $\bz \in \bC^N$ with $\| \bz\| =1$. Then, with the notation
$\bX_j = (X_{j1}, \dots , X_{jN})^*$ for $j=1, \dots, N$,
\begin{equation}\label{1}
\begin{split}
\P \big\{ (\bz,X^* X \bz) \geq C\big\} &\leq e^{-\kappa C N} \E \,
e^{\kappa N \sum_j |\bX_j \cdot \bz|^2} \\ & \leq e^{-\kappa C N} \E
\, e^{2 \kappa N \sum_j |\sum_{l\leq j} X_{jl} \cdot z_l|^2} e^{2
\kappa N\sum_j |\sum_{l > j} X_{jl} \cdot z_l|^2}
\\ & \leq e^{-\kappa C N} \left(\E \, e^{4 \kappa N \sum_j
|\sum_{l\leq j} X_{jl} \cdot z_l|^2}\right)^{1/2} \left( \E \, e^{4
\kappa N \sum_j |\sum_{l > j} X_{jl} \cdot z_l|^2} \right)^{1/2}
\end{split}
\end{equation}
Next, choosing $\kappa>0$ sufficiently small, we can show that,
similarly to (\ref{Meta4}),
\begin{equation}
\E \, e^{4 \kappa N \sum_j |\sum_{l\leq j}  X_{jl} \cdot z_l|^2} =
\prod_{j=1}^N \E \, e^{4\kappa N |\sum_{l \leq j} X_{jl} \cdot
z_l|^2} \leq \const^N
\end{equation}
and
\begin{equation}
\E \, e^{4 \kappa N \sum_j |\sum_{l\geq j} X_{jl} \cdot z_l|^2} =
\prod_{j=1}^N  \E \, e^{4\kappa N |\sum_{l \leq j} X_{jl} \cdot
z_l|^2} \leq \const^N
\end{equation}
because $\sum_{l\leq j} |z_l|^2 \leq 1$ and $\sum_{l >j} |z_l|^2
\leq 1$. Thus, from (\ref{1}), we have, for $C$ large enough,
\[
\P \big\{ (\bz,X^* X \bz) \geq C\big\} \leq e^{-c_1 C N}
\]
 for
a constant $c_1$ only depending on $\delta$. {F}rom the last
equation, the lemma follows with $C=K_0^2$ by
the same argument that was used at the end
of the proof of part i) of Lemma \ref{lm1}.
\end{proof}




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