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Semicircle law, Wigner random matrix, random Schroedinger operator, density of states, localization, extended states.
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\title{Local semicircle law and complete
delocalization  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 $1/N$.
Under suitable assumptions on the distribution
of the single matrix element, we prove that,  away from the spectral
edges,
 the density of eigenvalues concentrates around the Wigner semicircle law
on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic
factor, this is the smallest energy scale
for which the semicircle law may be valid.
We  also prove  that for all eigenvalues away
from the spectral edges, the $\ell^\infty$-norm  of the
corresponding  eigenvectors
is of order $O(N^{-1/2})$, modulo logarithmic corrections.
The upper bound  $O(N^{-1/2})$
implies that  every eigenvector is completely de-localized, i.e.,
 the maximum size of the components of the eigenvector is of the
same order as their average size.

In the Appendix, we include a lemma by J. Bourgain which removes  one of our  assumptions on the distribution  of the matrix elements.
\end{abstract}

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

\medskip

{\it Running title:} Local semicircle law

\medskip

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

%\received{}

%\maketitle


\section{Introduction}


The  Wigner semicircle law states that the empirical
density of the eigenvalues of a random matrix
is given by the universal semicircle distribution.
 This statement has been proved for many different ensembles,
in particular for the case when the distributions of
the entries of the matrix
are independent, identically distributed (i.i.d.).
To fix the scaling, we normalize the matrix so that
the bulk of the spectrum lies in the energy interval $[-2,2]$, i.e.
the average spacing between consecutive eigenvalues is of order $1/N$.
We now consider a window of size $\eta$ in the bulk so that the typical
number
of eigenvalues is $N \eta$.
In the  usual statement of the semicircle law,
$\eta$ is a  fixed number independent of $N$
and it is taken to zero only
after the limit $N\to \infty$.
 This can be viewed as the largest scale
on which  the semicircle law is valid. On the other extreme,
for the smallest scale, one may take $\eta = k /N $ and take the limit
 $N\to \infty$ followed by $k \to \infty$.
 If the semicircle law is valid in this sense,
 we shall say that the {\it local semicircle law} holds.
 Below this smallest scale, the eigenvalue  distribution is
 expected to be governed by the Dyson's statistics related to sine
kernels.
 The Dyson statistics was proved for many ensembles (see  \cite{AGZ, D}
for a review), including Wigner matrices with Gaussian convoluted
 distributions \cite{J}.


In this paper, we establish
the local semicircle law up to logarithmic factors in the scales, i.e.,
for
$\eta \sim  N^{-1} (\log N)^8$. The result holds for any energy
window in the bulk spectrum away from the spectral edges.
In \cite{ESY} we have proved the same statement for
$\eta \gg N^{-2/3}$ (modulo
logarithmic corrections). Prior to our work the best result was obtained
in \cite{BMT}
 for  $\eta\gg N^{-1/2}$. See also \cite{GZ} and \cite{Kh} for
related and earlier results.
As a corollary, our result also proves that no gap between consecutive
bulk
eigenvalues can be bigger than $(\log N)^8/N$, to
be compared with the expected average $1/N$ behavior
given by Dyson's law.

It is widely believed that the eigenvalue distribution of the
Wigner random matrix
and the random Schr\"odinger operator in the
 extended (or delocalized) state regime are the same
up to normalizations. Although this conjecture
is far from the reach of the current method,
a  natural question arises as to
 whether the eigenvectors of random matrices are extended.
More precisely, if $\bv= (v_1, \cdots, v_N)$
is a normalized  eigenvector, we say that
$\bv$ is {\it completely delocalized}
 if $\| \bv\|_\infty = \max_j |v_j|$ is bounded from above by
$CN^{-1/2}$,  the average size of  $|v_j|$.
In this paper, we shall prove that
all eigenvectors with eigenvalues away from the spectral
edges  are completely delocalized
 (modulo logarithmic corrections) in probability.   Similar results, but
with
$C N^{-1/2}$ replaced by  $C N^{-1/3}$ were proved in  \cite{ESY}.
 Notice that our new result,
in particular, answers (up to logarithmic factors) the question
posed by T. Spencer that $\| \bv\|_4$ should be of order  $N^{-1/4}$.



\bigskip




  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 $\rd \nu$.
Let  $\P$ and $\E$
denote the probability and the expectation value, respectively,
 w.r.t the joint distribution of all matrix elements.

\bigskip


We need to assume further conditions
on the distributions
of the matrix elements in addition to \eqref{wig}.
\begin{itemize}
\item[{\bf C1})] The function $g$ is twice differentiable
and it satisfies
\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 C4)}] The measure $\nu$ satisfies 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}
Here we have followed the convention in \cite{ESY} to use the label C4)
and reserved  C3) for an spectral gap condition in \cite{ESY}.


\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 $A\subset \bZ$.
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.





\bigskip

Let $H$ be the $N\times N$ Wigner matrix with eigenvalues
$\mu_1\leq \mu_2 \leq\ldots \leq \mu_N$.  For any spectral
parameter $z= E+i\eta\in \bC$, $\eta>0$, we denote the Green function
by $G_z= (H-z)^{-1}$. Let $F(x)=F_N(x)$ be
the empirical distribution 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 $\varrho_N$.


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
$$
  \varrho_{sc}(x) = \frac{1}{2\pi} \sqrt{4-x^2}
   {\bf 1}(|x|\leq 2)\; .
$$
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 $\wt\eta = N^{-1}(\log N)^8$ we write
$$
S_{N,\kappa}:= \Big\{ z=E+i\eta\in \bC\; : \; |E|\leq 2-\kappa, \;
\frac{(\log N)^8}{N}\leq \eta \leq 1\Big\}.
$$
The following two theorems are the main results of this paper.


\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 $\kappa>0$ and $\e>0$,
  the Stieltjes transform $m_N(z)$ (see \eqref{Sti})
of the empirical
eigenvalue distribution of the $N\times N$ Wigner matrix  satisfies
\be
 \P \Big\{ \sup_{z\in S_{N,\kappa}} |m_N(z)- m_{sc}(z)| \ge \e\Big\}
 \leq e^{-c(\log N)^2}
\label{mcont}
\ee
where $c>0$ depends on $\kappa, \e$. 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 $N^{-1}(\log N)^8$.

Furthermore, let $\eta^*=\eta^*(N)$ such that
$(\log N)^8/N\ll\eta^*\ll 1$ 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^{-c(\log N)^2}
\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{theorem}



\begin{theorem}\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
\[
\P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$,
$\| \bv \|=1$, $\mu \in [-2+\kappa, 2-\kappa]$ and } \| \bv
\|_\infty \ge \frac{C(\log N)^{9/2}}{N^{1/2}} \Bigg\} \leq e^{-c(\log
N)^2}\;. \]
\end{theorem}

\noindent
Condition C1) is needed only because we
 use Lemma 2.3 of \cite{ESY} in the following proof.
J. Bourgain \cite{B} has informed us that it
can also be proved without this condition.
We include his proof
%thank for his permission to reproduce his argument
in the Appendix.


\section {Proof of Theorem \ref{thm:sc}}


The proof of \eqref{ncont} follows from \eqref{mcont} exactly as
in Corollary 4.2 of \cite{ESY}, so we focus on proving
\eqref{mcont}. We first  remove the  supremum in   \eqref{mcont}.

For any two
points $z, z'\in S_{N, \kappa,\eta}$ we have
$$
  |m_N(z)-m_N(z')|\leq N^{2}|z-z'|
$$
since the gradient of $m_N(z)$ is bounded by $|\text{Im}\; z|^{-2}\leq
N^{2}$ on $S_{N, \kappa}$.
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 \frac{1}{4}\e N^{-2} $.
In particular, $|m_N(z)-m_N(z_j)|\le \e/4$ if $N$ is large enough
and $|m_{sc}(z)-m_{sc}(z_j)|\leq \e/4$.
Then
$$
     \P \Big\{ \sup_{z\in S_{N,\kappa}} |m_N(z)-  m_{sc}(z)|
 \ge \e\Big\}
   \leq \sum_{j=1}^M
  \P \Big\{ |m_N(z_j)-   m_{sc}(z_j)| \ge \frac{\e}{2}\Big\}
  \leq e^{-c(\log N)^2}
$$
under the condition that $\eta\ge N^{-1}(\log N)^8$
 since $\text{Im} \, z_j\ge
\eta$. So it is enough to prove
\be
 \P \Big\{ |m_N(z)- m_{sc}(z)| \ge \e\Big\}
 \leq e^{-c(\log N)^2}
\label{mcont1}
\ee
 for each fixed $z\in S_{N,\kappa}$.








Let $B^{(k)}$ denote the $(N-1)\times(N-1)$ minor of $H$ after
removing the $k$-th row and $k$-th column. Note that
$B^{(k)}$ is an $(N-1)\times(N-1)$ Hermitian Wigner matrix
with a normalization factor off by $(1-\frac{1}{N})^{1/2}$.
 Let
$\lambda_1^{(k)}\leq \lambda_2^{(k)}\leq \ldots \leq \lambda_{N-1}^{(k)}$
denote its eigenvalues and $\bu_1^{(k)},\ldots , \bu_{N-1}^{(k)}$
the corresponding normalized eigenvectors.


Let $\ba^{(k)}=(h_{k1}, h_{k2}, \ldots h_{k(k-1)}, h_{k(k+1)}, \ldots
h_{kN})^*
\in \bC^{N-1}$, i.e. the $k$-th column after removing the diagonal
element $h_{kk}$. Computing the $(k,k)$ diagonal element of
the resolvent $G_z$, we have
\be
   G_z(k,k)= \frac{1}{h_{kk}-z-\ba^{(k)}\cdot (B^{(k)}-z)^{-1}\ba^{(k)}}
   = \Big[ h_{kk}-z-\frac{1}{N}\sum_{\alpha=1}^{N-1}\frac{\xi_\al^{(k)}}
  {\lambda_\al^{(k)}-z}\Big]^{-1}
\label{mm}
\ee
where we defined
$$
    \xi_\al^{(k)} : = \big| \sqrt{N}\ba^{(k)}\cdot \bu_\al^{(k)}\big|^2.
$$




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
$$
    F^{(k)}(x) = \frac{1}{N-1} \big| \, \big\{ \al \; : \;
   \lambda_{\al}^{(k)}\leq x \big\}\big|.
$$
The spectral parameter $z$ is fixed throughout the proof
and we will omit from the argument of the Stieltjes transforms.


It follows from \eqref{mm} that
\be
m=m(z) = \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(B^{(k)}-z)^{-1} \ba^{(k)}}\, .
\label{mm1}
\ee
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)}
  = \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\xi_\al^{(k)} -1}
   {\lambda_\al^{(k)}-z}
\label{def:X}
\ee
where we used that $\E_k \xi_\al^{(k)}=\| \bu_\al^{(k)}\|^2=1$.
We 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\label{recur}
   m =  \frac{1}{N}\sum_{k=1}^N \frac{1}{ h_{kk} -z -
   \big(1-\frac{1}{N}\big)m^{(k)} - X_k} \; .
\ee


We use that
$$
     \Big|  m -  \Big(1-\frac{1}{N}\Big)m^{(k)}\Big|
  =\Big| \int \frac{\rd F(x)}{x-z}
  -  \Big(1-\frac{1}{N}\Big)\int \frac{\rd F^{(k)}(x)}{x-z}\Big|
  = \frac{1}{N}\Big| \int \frac{NF(x)-(N-1)F^{(k)}(x)}{(x-z)^2} \rd
x\Big|.
$$
We recall that the eigenvalues of $H$ and $B^{(k)}$ are interlaced,
\be
 \mu_1\leq \lambda_1^{(k)}\leq \mu_2 \leq
\lambda_2^{(k)} \leq \ldots \leq \lambda_{N-1}^{(k)}
\leq \mu_N,
\label{interlace}
\ee
(see e.g. Lemma 2.5 of \cite{ESY}), therefore
we have $\max_x|NF(x)-(N-1)F^{(k)}(x)|\leq 1$.  Thus
$$
   \Big|  m -  \Big(1-\frac{1}{N}\Big)m^{(k)}\Big|
\leq \frac{1}{N} \int \frac{\rd x}{|x-z|^2}
  \leq  \frac{C}{N\eta}\, .
$$


We postpone the proof of the following lemma:
\begin{lemma}\label{lm:x}
Suppose that $\bv_\alpha$ and $\lambda_\alpha$ are eigenvectors
and eigenvalues
of an $N\times N$ random matrix with a
law satisfying the assumption of Theorem \ref{thm:sc}.
Let
$$
    X = \frac{1}{N} \sum_\al \frac{\xi_\al-1}{\lambda_\al-z}
$$
with $z=E+i\eta$, $\xi_\al = |\bb\cdot \bv_\al|^2$, where
the components of $\bb$ are i.i.d.
 random variables satisfying
\eqref{x2} and \eqref{logsob}.
Then in the joint product probability space of $\bb$ and the law
 of the random matrices we have
$$
   \P[ |X|\ge \e] \leq e^{- c\e (\log N)^2}
$$
with a sufficiently small $c$ and $\e\leq \e_0$
for any $\eta \ge (\log N)^8/N$.
\end{lemma}


For given $\e>0$ we define the event
$$
   \Omega = \bigcup_{k=1}^N \{|X^{(k)}|\ge \e/3\}\cup \{ |h_{kk}|\ge
\e/3\}\,.
$$
Since $h_{kk} = N^{-1/2} b_{kk}$ with $b_{kk}$  satisfying
\eqref{x2}, we have
$$
  \P \{ |h_{kk}|\ge \e/3\} \leq Ce^{-\delta\e^2 N/9}.
$$
We now apply Lemma \ref{lm:x} for each $X^{(k)}$
and conclude that
$$
  \P(\Omega) \leq e^{- c\e (\log N)^2}
$$
with a sufficiently small $c>0$.
On the complement $\Omega^c$ we have from \eqref{recur}
$$
    m= \frac{1}{N}\sum_{k=1}^N \frac{1}{-m -z +\delta_k}
$$
where $\delta_k$ are random variables satisfying
 $|\delta_k|\leq \e$. After expansion
\be
   \Big| m + \frac{1}{ m +z }\Big|\leq \frac{\e}{|m+z|^2}\,.
\label{cont1}
\ee



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 $\text{Im}\, M>0$. Explicitly, we have
$$
M = \frac {-z + \sqrt {z^2 - 4 + 2 z \delta + \delta^2 } } 2 +
\frac{\delta}{2},
$$
where we have chosen the square root so that $\im M > 0$  when $\delta=0$
and $\im z > 0$. On the compact set $z\in S_{N,\kappa}$,
  $|z^2 - 4|$ is bounded away from zero
and thus
\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
        |\, 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
 $\eta\in [2N^{-1}(\log N)^8,\, 1]$
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 $\e$, where \eqref{cont}
and $\im\, m_{sc}(E+i\eta)\ge c(\kappa)$ were
used. Thus the right hand side 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 $\eta/2$. \qed










\section{ Proof of Lemma \ref{lm:x}}



Let $I_n = [n\eta, (n+1)\eta]$ and $K_0$
be a sufficiently  large number.
We have $[-K_0, K_0] \subset \cup_{ n = -m}^m  I_n$ with $m \le
CK_0/\eta$.
Denote by $\Omega$ the event
$$
  \Omega : = \Big\{ \max_n \cN_{I_n} \ge N\eta (\log N)^2\Big\}
\cup \{ \max_\al |\lambda_\al|\ge K_0\}
$$
where $\cN_{I_n}=|\{\al\; : \; \lambda_\al\in I_n\}|$
 is the number of eigenvalues in the interval $I_n$.
{F}rom Theorem 2.1 and Lemma 7.4  of \cite{ESY},
  the probability of $\Omega$ is bounded by
$$
   \P(\Omega) \leq e^{-c(\log N)^2}.
$$
Therefore,
\be
\begin{split}
  \P[ |X|\ge \e] & \leq e^{-c(\log N)^2} + \E \Big[ {\bf 1}_{\Omega^c}
   \P_\bb  [ |X|\ge \e] \Big]\\
  & \leq  e^{-c(\log N)^2} +
   e^{-\e T} \E \Big[ {\bf 1}_{\Omega^c}
   \cdot \E_\bb e^{ T|X|} \Big]
\end{split}\label{T}
\ee
for any $T>0$,
where $\P_\bb$ and $\E_\bb$ denote
the probability and  the expectation w.r.t. the variable $\bb$.

On $\Omega^c$ we have
$$
   \frac{\rd}{\rd\beta} \Big[ e^{-\beta} \log
\E_\bb \exp{\big( e^\beta  \re X\big)}\Big]
   = e^{-\beta} \E_\bb u\log u \leq Ce^{-\beta} \E_\bb |\nabla \sqrt u|^2
$$
with
$$
      u = \frac{ \exp{\big( e^\beta  \re X\big)}}{\E_\bb
 \exp{\big( e^\beta  \re X\big)}}
$$
and $C$ being the  constant in the logarithmic Sobolev inequality
\eqref{logsob}.
Simple computation yields that
\be\nonumber
\begin{split}
  e^{-\beta} \E_\bb |\nabla \sqrt u|^2 & \leq e^\beta \E_\bb   \Bigg[ u\;
  \sum_k  \Big( \Big| \frac{\partial X}{\partial\; \re  b_k}\Big|^2
  + \Big|\frac{\partial X}{\partial\; \im  b_k}\Big|^2 \Big)\Bigg]  \\
 &  =  \frac{e^\beta}{N^2} \E_\bb\Bigg[  u\;
  \sum_k \sum_{\al,\beta} \sum_{i,j} \Big(
\frac{b_i\bar b_j \bar \bv_\al(k)\bv_\beta(k)
\bv_\al(j)\bar \bv_\beta(i)}{(\lambda_\al -z)(\lambda_\beta-\bar z)}
+\frac{b_i\bar b_j \bar \bv_\al(i)\bv_\beta(j) \bv_\al(k)
 \bar\bv_\beta(k)}{(\lambda_\al -z)(\lambda_\beta-\bar z)}
\Big)\Bigg]\\
 &  =  \frac{2e^\beta}{N^2} \E_\bb \Bigg[ u
  \sum_{\al} \sum_{i,j} \frac{b_i\bar b_j \bar \bv_\al(i)
   \bv_\al(j)}{|\lambda_\al -z|^2} \Bigg]\\
& = \frac{e^\beta}{N^2} \E_\bb \Bigg[ u
  \sum_{\al}  \frac{\xi_\al}{|\lambda_\al -z|^2}\Bigg] \leq
    \frac{e^\beta}{N\eta} \E_\bb \big[ u Y\big],
\end{split}
\ee
where $Y$ denotes
$$
   Y = \frac{1}{N}  \sum_{\al}  \frac{\xi_\al}{|\lambda_\al -z|}\; .
$$
Recall the entropy inequality that for any $\gamma > 0$,
$$
\E_\bb \left [  uY \right ] \le
 \gamma^{-1} \E_\bb \, u \log u + \gamma^{-1}  \E_\bb \, e^{\gamma Y}
$$
Let  $\gamma = 2Ce^{2\beta}/N\eta$ where $C\ge 1$ is the log-Sobolev
constant from \eqref{logsob}. We thus have
$$
  e^{-\beta} \E_\bb \, |\nabla \sqrt u|^2
 \leq \frac{e^\beta}{N\eta\gamma}
\Big[ \E_\bb \, u \log u + \E_\bb e^{\gamma Y}
\Big] \leq \frac{e^{-\beta}}{2} \E_\bb\,  |\nabla \sqrt u|^2
  + \frac{e^{-\beta}}{2} \E_\bb \, e^{\gamma Y}.
$$
Hence
$  \E_\bb \,|\nabla \sqrt u|^2
 \leq   \E_\bb \, e^{\gamma Y}$ and we have
$$
\frac{\rd}{\rd\beta} \Big[ e^{-\beta} \log \E_\bb \,
\exp{\big[ e^\beta X\big]}\Big]
\leq  \frac{e^{-\beta}}{2} \E_\bb \, e^{\gamma Y}.
$$
We now choose $\beta_0$ such that
$$
   e^{\beta_0} = \frac{(N\eta)^{1/2}}{(\log N)^2}.
$$
Integrating $\beta$ from $M$ to $\beta_0$ and using that
$$
  \gamma \leq \frac{2C}{(\log N)^4}
$$
for all $\beta \le \beta_0$,  we have
\be
  e^{-\beta_0} \log \E_\bb \exp{\big[ e^{\beta_0} \re X\big]}
\leq e^{-M} \log \E_\bb \exp{\big[ e^M  \re X\big]}
  + e^{-\beta_0}\E_\bb \exp{\left[ \frac{2C}{(\log N)^4}Y\right]}.
\label{betaM}
\ee
The first term on the right hand side is expressed as
$$
    e^{-M} \log \E_\bb \, \exp{\big[ e^M \re X\big]}
 = e^{-M}\log \Big[ 1+ \E_\bb \big[ e^M \re X\big]
   + \E_\bb \Big(\exp{\big[ e^M \re X\big]}-1-e^M \re X\Big)\Big]\; .
$$
Since $\xi_\al =\big| \sum_j \bar b_j v_\al(j)\big|^2 \leq \sum_j
|b_j|^2$,
we have
$$
  |X| \leq \eta^{-1} + \frac{1}{N\eta} \sum_\al \xi_\al
  \leq \eta^{-1} \sum_j (|b_j|^2+1)\; .
$$
{F}rom the assumption \eqref{x2}, we have
$ \E_\bb\;  e^{ \tau |X|}  < \infty$
for $\tau$ small enough (depending on $N$ and $\eta$). Therefore,
$$
\lim_{M \to  - \infty} \E_\bb \Big(\exp{\big[ e^M \re X\big]}-1-e^M \re
X\Big)
= 0.
$$

Using $\E_\bb  X=0$  and $|\log (1+c)|\leq 2|c|$
for any sufficiently small $c$,
we  have
\be
   e^{-M} \log \E_\bb\,  \exp{\big[ e^M \re X\big]}
 \leq  2  e^{-M} \Big | \E_\bb \Big(\exp{\big[ e^M \re X\big]}
-1-e^M \re X\Big) \Big | \; .
\label{2nd}
\ee
Since $|e^c-1-c|\leq |c|^2 e^{|c|}$  for any real $c$,
we can bound the right hand side of \eqref{2nd} by
$$
 2   e^{-M} \E_\bb |e^MX|^2 e^{e^M|X|} =
 e^M \E_\bb |X|^2\exp{\big[ e^M|X|\big]}.
$$
Since $|X|$ is exponentially integrable, the last term vanishes in
the limit $M \to  - \infty$.  {F}rom \eqref{betaM}
we have thus proved that
\be
  e^{-\beta_0} \log \E_\bb \, \exp{\big[ e^{\beta_0} \re X\big] }\leq
 e^{-\beta_0}\E_\bb \exp{\left[\frac{2C}{(\log N)^4}Y\right]}.
\label{lo}
\ee

Denote by $\nu$ the constant $\nu = 2C/(\log N)^4$.
By H\"older inequality, we can estimate
\be
  \E_\bb e^{\nu Y} = \E_\bb \prod_\al
\exp{\Big[\frac{\nu }{N|\lambda_\al -z|}\xi_\al\Big]}
\leq      \prod_\al  \Bigg( \E_\bb
\exp{\Big[\frac{\nu c_\al}{N|\lambda_\al -z|}\xi_\al\Big]}
\Bigg)^{1/c_\al},
\label{hold}
\ee
where $\sum_\al \frac{1}{c_\al}=1$. We shall choose
$$
   c_\al = \varrho \frac{N|\lambda_\al -z|}{\nu}
$$
where $\varrho$ is given by
$$
      \varrho = \frac{\nu}{N}\sum_\al \frac{1}{|\lambda_\al -z|} \leq
      \frac{\nu \log N}{N\eta}\max_n \cN_{I_n} \leq \nu (\log N)^3
    = \frac{2C}{\log N} \, .
$$
Here we have used  $\max_n \cN_{I_n} \le N \eta (\log N)^2$
due to that we are in the set  $\Omega^c$.
Notice that with this choice,
$$\frac{ \nu c_\al}{N |\lambda_\al -z|}\leq\frac{2C}{\log N}
$$
is a small number. In the proof of Lemma 7.4 of \cite{ESY}
(see equation (7.13) of \cite{ESY})  we
showed that
$$
     \E_\bb \; e^{\tau \xi_\al} < K
$$
with a universal constant $K$
if $\tau$ is sufficiently small depending on $\delta$ in \eqref{x2}.
Thus from \eqref{lo} and \eqref{hold} we obtain
$$
  \log \E_\bb \, \exp{\big[ e^{\beta_0} \re X\big]}
\leq \E_\bb \, e^{\nu Y}\leq  \prod_\al K^{1/c_\al} =K\, .
$$
The same inequality holds for $ - \re X$
 as well as for the imaginary part of $X$.
Thus we have proved that
$$
\E_\bb e^{e^{\beta_0}|X|} \leq e^K.
$$
Choosing an optimal $T$ in
\eqref{T}, this proves that
$$
  \P[ |X|\ge \e] \leq  e^{-c(\log N)^2} +
   e^{-\e e^{\beta_0}} \leq  e^{-c\e(\log N)^2}
$$
as long as $N\eta \ge (\log N)^8$ and we thus conclude the Lemma.  \qed



\section{Delocalization of eigenvectors}

\bigskip



Here we prove Theorem \ref{cor:linfty},
the argument follows the same line as in \cite{ESY}
(Proposition 5.3).
Let $\eta^* = N^{-1} (\log N)^9$ and partition the
interval $[-2+\kappa, 2-\kappa]$ into $n_0= O(1/\eta^*)\leq O(N)$
intervals $I_1, I_2, \ldots I_{n_0}$
of length $\eta^*$.
As before, let $\cN_{I}=|\{ \beta\; : \; \mu_\beta\in I\}|$
denote the eigenvalues in $I$.
 By choosing \eqref{ncont} in Theorem \ref{thm:sc},
we have
\[
\P \left\{\max_n \cN_{I_n} \leq \e N\eta^* \right\}
 \leq e^{-c (\log N)^2}.
\]
if $\e$ is sufficiently small (depending on $\kappa$).
Suppose that $\mu
\in I_n$, and that $H\bv = \mu \bv$. Consider the decomposition
\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 $H$. Let  $\lambda_\al$ and $\bu_\al$ (for $\al=1,2,\ldots , N-1$)
 denote the eigenvalues and the normalized eigenvectors  of $B$.
 {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 $\bw= (v_2, \dots ,v_N)^t$. {F}rom these equations
we obtain
$ \bw = (\mu-B)^{-1} \ba v_1 $ and thus
$$
\|\bw\|^2= \bw\cdot \bw = |v_1|^2 \ba\cdot (\mu -B)^{-2} \ba
$$
Since $\|\bw\|^2 = 1 - |v_1|^2$, we obtain
\be\label{v2} |v_1|^2 =
\frac{1}{1+ \ba\cdot(\mu -B)^{-2} \ba} = \frac{1}{1 + \frac{1}{N}
\sum_{\alpha} \frac{\xi_{\alpha}}{(\mu - \lambda_{\alpha})^2}}
\leq \frac{4 N [\eta^*]^2}{\sum_{\lambda_\alpha \in I_n} \xi_{\alpha}} \,
,
\ee
where in the second equality we set
$\xi_{\alpha} = |\sqrt{N} \ba \cdot \bu_{\alpha}|^2$
 and used
the spectral representation of $B$. By the interlacing property
of  the eigenvalues of $H$ and $B$,
there exist at least $\cN_{I_n}-1$ eigenvalues $\lambda_\al$
in $I_n$.
Therefore, using that the components of any eigenvector are identically
distributed,
\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)^{9/2}}{N^{1/2}} \Big)  \\
&\leq N n_0 \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)^9}{N} \Big)\\
&\leq \const \,  N^2 \sup_n \P \left( \sum_{\lambda_\alpha \in I_n}
\xi_{\alpha} \leq \frac{4N\eta^*}{C} \right) \\
&\leq  \const \, N^2 \sup_n \P \left( \sum_{\lambda_\alpha \in I_n}
\xi_{\alpha} \leq \frac{4 N\eta^*}{C} \text{ and } \cN_{I_n} \geq
\e N\eta^*\right) + \const \,  N^2  \sup_n \, \P
\left(\cN_{I_n} \leq \e N\eta^* \right)
\\&\leq \const \,  N^2 e^{-c(\log N)^9} + \const \, N^2 e^{-c (\log
N)^2} \leq e^{-c' (\log N)^2}\,.
\end{split}
\end{equation}
using Corollary 2.4 of \cite{ESY} and choosing $C$ sufficiently large,
depending on $\kappa$ via $\e$. Here we used
 Corollary 2.4. of \cite{ESY} that states that under condition \eqref{gM}
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 $|A|=m$. \qed




%\end{proof}

%\appendix

\bigskip

\centerline{\Large\bf{Appendix: Removal of the assumption C1)}}

%\centerline{
%\section{Removal of the assumption (C1)}\label{app}
%}

\bigskip

\centerline{\Large Jean Bourgain}
\centerline{\large School of Mathematics}
\centerline{\large Institute for Advanced Study}
\centerline{\large Princeton, NJ 08540, USA}


\bigskip

The following Lemma shows that  the assumption C1)
in Lemma 2.3 and its corollary
in \cite{ESY}  can be removed.

\begin{lemma}\label{lm:BL} Suppose that $z_1, \dots ,z_N$ are bounded,
complex valued i.i.d. random variables with $\E \, z_i = 0$ and
$\E \, |z_i|^2 = a>0$.
 Let $P: \bC^N \to \bC^N$ be a rank-$m$ projection,
and $\bz = (z_1, \dots ,z_N)$. Then, if $\delta$
is small enough, there exists $c>0$ such that
\[ \P \, \left( |P \bz|^2 \leq \delta m \right) \leq e^{-cm} \, .\]
\end{lemma}


\medskip

\begin{proof}
It is enough to prove that
\begin{equation}\label{eq:clb} \P \left( \left| |P \bz|^2 - a m \right| >
 \tau m \right) \leq e^{-c \tau^2 m} \, \end{equation}
for all $\tau$ sufficiently small. Introduce the notation
$\| X \|_q=\big[\E |X|^q\big]^{1/q}$. Since
\begin{equation}
\begin{split}
\P \left( \left| |P\bz|^2 - a m \right| > \tau m \right)
\leq \frac{\left\| \, |P \bz|^2 - a m \right\|_q^q}{(\tau m)^q},
\end{split}
\end{equation}
the bound (\ref{eq:clb}) follows by showing that
\begin{equation}\label{eq:clb2} \| | P \bz |^2 - a m \|_q \leq
 C \sqrt{q} \, \sqrt{m} \qquad \text{for all $q < m$} \end{equation}
(and then choosing $q = \alpha \tau^2 m$ with a small
enough $\alpha$). To prove (\ref{eq:clb2}), observe that
 (with the notation $\bbe_i = ( 0, \dots , 0 ,1 , 0 \dots, 0)$
 for the standard basis of $\bC^N$)
\begin{equation}\label{eq:clb3}
\begin{split}
| P \bz |^2 &= \sum_{i=1}^N |z_i|^2 \, |P \bbe_i|^2 + \sum_{i \neq j}^N
\overline{z}_i \, z_j \, P \bbe_i \cdot P \bbe_j = a m + \sum_{i=1}^N
\left( |z_i|^2 - \E |z_i|^2 \right) \, |P \bbe_i|^2 + \sum_{i\neq j}
 \overline{z}_i z_j \, P \bbe_i \cdot P \bbe_j
\end{split}
\end{equation}
and thus
\begin{equation}\label{eq:clb4}
\begin{split}
\left\| |P \bz|^2 - a m \right\|_q \leq \left\| \, \sum_{i=1}^N
 \left( |z_i|^2 - \E |z_i|^2 \right) |P \bbe_i|^2 \right\|_q + \left\| \,
\sum_{i \neq j}^N \overline{z}_i z_j \, P \bbe_i \cdot P \bbe_j \right\|_q
\,.
\end{split}
\end{equation}
To bound the first term, we use that for arbitrary i.i.d. random variables
 $x_1, \dots ,x_N$ with $\E \, x_j =0$ and
$\E \, e^{\delta |x_j|^2} < \infty$ for some $\delta >0$, we have the
bound
\begin{equation}\label{eq:bern}
\| X \|_q \leq C \sqrt{q} \, \| X \|_2 \end{equation} for $X =
\sum_{j=1}^N a_j x_j$, for arbitrary $a_j \in \bC$.
The bound (\ref{eq:bern}) is an extension of Khintchine's inequality
 and it can be proven as follows using the representation
\begin{equation}\label{eq:bern1}
  \| X \|_q^q = q\int_0^\infty   \rd y  \; y^{q-1} P( |X|\ge y) \, .
\end{equation}
Writing $a_j = |a_j|e^{i\theta_j}$, $\theta_j\in \bR$, and decomposing
$e^{i\theta_j}x_j$ into real and imaginary parts,
it is clearly sufficient to prove \eqref{eq:bern} for the case when
$a_j, x_j\in \bR$ are real and $x_j$'s are independent with $\E \, x_j =0$
and
$\E \, e^{\delta |x_j|^2} < \infty$.
To bound the probability $\P (|X| \geq y)$ we observe that
\[    \P(X\ge y) \leq e^{-ty} \, \E \, e^{tX} = e^{-t y} \, \prod_{j=1}^N
  \E \, e^{t a_j x_j} \leq e^{-t y} \, e^{C t^2\sum_{j=1}^N a_j^2} \]
because $\E  \, e^{\tau x} \leq e^{C \tau^2}$  from the moment assumptions
on $x_j$
with a sufficiently large $C$ depending on
$\delta$. Repeating this argument for $- X$, we find
\[ \P(|X|\ge y) \leq 2 \, e^{-ty} \, e^{Ct^2\sum_{j=1}^N a_j^2}
 \leq e^{-y^2/(2 C \sum_{j=1}^N a_j^2)} \] after optimizing in $t$.
The estimate (\ref{eq:bern}) follows then by plugging the last bound
into (\ref{eq:bern1}) and computing the integral.

\medskip

Applying (\ref{eq:bern}) with $x_i = |z_i|^2 - \E \, |z_i|^2$
($\E \, e^{\delta x_i^2} < \infty$ follows from the assumption
$\| z_i \|_{\infty} < \infty$), the first term on the r.h.s. of
(\ref{eq:clb4}) can be controlled by
\begin{equation}\label{eq:clb5}
\left\| \, \sum_{i=1}^N \left( |z_i|^2 - \E |z_i|^2 \right)
 |P \bbe_i|^2 \right\|_q \leq C \sqrt{q} \left( \sum_{i=1}^N
 |P \bbe_i |^4 \right)^{1/2} \leq C \sqrt{q} \left(
 \sum_{i=1}^N |P \bbe_i |^2 \right)^{1/2} = C \sqrt{q} \sqrt{m}\,.
\end{equation}
As for the second term on the r.h.s. of (\ref{eq:clb4}), we define
the functions $\xi_j (s), s \in [0,1], j=1,\dots, N$ by
\[ \xi_j (s) =  \left\{  \begin{array}{ll} 1 \quad &\text{if }
 s \in \bigcup_{j=0}^{2^{k-1} - 1} \, \left[ \frac{2j}{2^k},
\frac{2j+1}{2^k} \right) \\ 0 \quad &\text{otherwise} \end{array} \right.
\, .\]
 Since
\[ \int_0^1 \rd s \; \xi_i (s) (1- \xi_j (s)) = \frac{1}{4} \] for all
 $i \neq j$, the second term on the r.h.s. of (\ref{eq:clb4}) can be
 estimated by
\begin{equation}\label{eq:clb6}
\left\| \, \sum_{i \neq j}^N \overline{z}_i z_j \, P \bbe_i \cdot
P \bbe_j \right\|_q \leq  4 \int_0^1 \rd s \; \left\| \, \sum_{i \neq j}^N
 \xi_i (s) \, (1- \xi_j (s)) \, \overline{z}_i z_j \,
 P \bbe_i \cdot P \bbe_j \right\|_q \,.
\end{equation}
For fixed $s \in [0,1]$, set \[ I (s) = \{ 1 \leq i
 \leq N : \xi_i (s) = 1 \} \qquad \text{and } \quad J (s) = \{ 1, \dots ,N
\}
\backslash I (s) \, . \] Then
\begin{equation*}
\left\|\, \sum_{i \neq j}^N  \xi_i (s) \, (1- \xi_j (s))
 \, \overline{z}_i z_j \, P
\bbe_i \cdot P \bbe_j \right\|_q = \left\| \,
\sum_{i \in I (s), j \in J (s)} \overline{z}_i z_j \, P \bbe_i
\cdot P \bbe_j  \right\|_q =
 \left\| \, \sum_{j \in J(s)} z_j \left( \sum_{i \in I(s)}
 z_i P \bbe_i \right)
\cdot \bbe_j \right\|_q \, .
\end{equation*}
Since by definition $I \cap J = \emptyset$,
the variable $\{ z_i \}_{i \in I}$ and the variable
 $\{ z_j \}_{j \in J}$ are independent. Therefore,
 we can apply Khintchine's inequality (\ref{eq:bern})
 in the variables $\{ z_j \}_{j \in J}$
 (separating the real and imaginary parts) to conclude that
\begin{equation}
\begin{split}
\left\|\, \sum_{i \neq j}^N  \xi_i (s) \, (1- \xi_j (s)) \,
 \overline{z}_i z_j \, P
\bbe_i \cdot P \bbe_j \right\|_q   &\leq C \sqrt{q}
\left\| \, \left( \, \sum_{j \in J(s)} \left|
\left(\sum_{i \in I (s)} z_i P \bbe_i \right) \cdot
 \bbe_j \right|^2 \right)^{1/2} \right\|_q \\ & \leq C \sqrt{q}
\, \left\| \, \sum_{i \in I (s)} z_i P \bbe_i \right\|_q
\leq C \sqrt{q} \, \| P \bz \|_q  \,
\end{split}
\end{equation}
for every $s \in [0,1]$. It follows from (\ref{eq:clb6}) that
\[  \left\| \, \sum_{i \neq j}^N \overline{z}_i z_j \, P \bbe_i
\cdot P \bbe_j \right\|_q \leq C \, \sqrt{q} \, \| P\bz \|_q \, . \]
 Inserting the last equation and (\ref{eq:clb5}) into the r.h.s. of
 (\ref{eq:clb4}), it follows that
\[ \left\| |P \bz|^2 - a m \right\|_q \leq C \, \sqrt{q} \,
\left( \sqrt{m} + \| P \bz \|_q \right) \, . \]
Since clearly \[  \| P \bz \|_q \leq \left\|
|P \bz|^2 - a m \right\|^{1/2}_q + \sqrt{am} \]
the bound (\ref{eq:clb2}) follows immediately.
\end{proof}


\thebibliography{hhh}


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