Content-Type: multipart/mixed; boundary="-------------0811161127871" This is a multi-part message in MIME format. ---------------0811161127871 Content-Type: text/plain; name="08-217.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-217.comments" 35 pages ---------------0811161127871 Content-Type: text/plain; name="08-217.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-217.keywords" Semicircle law, Wigner random matrices, level repulsion, Wegner estimate, density of states, localization, extended states ---------------0811161127871 Content-Type: application/x-tex; name="lev1111.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="lev1111.tex" \documentclass[draft]{article} %\usepackage{amsmath,amsfonts,latexsym, amsrefs,amssymb} \usepackage{amsmath,amsfonts,latexsym, amssymb} \usepackage{color} %\newcommand{\sidenote}[1]{} \newcommand{\sidenote}[1]{\marginpar{\color{red}\footnotesize #1}} \oddsidemargin=0in \evensidemargin=0in \textwidth=6.5in %\usepackage[notref,notcite]{showkeys} %\usepackage{showkeys} \newcommand{\ov}{\overline} \newcommand{\const}{\mbox{const}} \newcommand{\La}{\Lambda} \newcommand{\e}{\varepsilon} \newcommand{\eps}{\varepsilon} \newcommand{\pt}{\partial} \newcommand{\rd}{{\rm d}} \newcommand{\bR}{{\mathbb R}} \newcommand{\bbZ}{{\mathbb Z}} \newcommand{\bke}[1]{\left( #1 \right)} \newcommand{\bkt}[1]{\left[ #1 \right]} \newcommand{\bket}[1]{\left\{ #1 \right\}} \newcommand{\norm}[1]{\| #1 \|} \newcommand{\Norm}[1]{\left\Vert #1 \right\Vert} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\bka}[1]{\left\langle #1 \right\rangle} \newcommand{\vect}[1]{\begin{bmatrix} #1 \end{bmatrix}} \newcommand{\bs}{{\bf{s}}} \newcommand{\ba}{{\bf{a}}} \newcommand{\bb}{{\bf{b}}} \newcommand{\bx}{{\bf{x}}} \newcommand{\by}{{\bf{y}}} \newcommand{\bt}{{\bf{t}}} \newcommand{\bu}{{\bf{u}}} \newcommand{\bv}{{\bf{v}}} \newcommand{\Be}{{\bf{e}}} \newcommand{\bw}{{\bf{w}}} \newcommand{\bz}{{\bf{z}}} \newcommand{\bc}{{\bf{c}}} \newcommand{\bd}{{\bf{d}}} \newcommand{\bh}{{\bf{h}}} \newcommand{\bbe}{{\bf{e}}} \newcommand{\wh}{\widehat} \newcommand{\ui}{{\underline i}} \newcommand{\uj}{{\underline j}} \newcommand{\ual}{{\underline \al}} \newcommand{\uz}{{\underline z}} \newcommand{\bX}{{\bf{X}}} \newcommand{\bY}{{\bf{Y}}} \newcommand{\bZ}{{\bf{Z}}} \newcommand{\wG}{{\widehat G}} \newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ga}{{\gamma}} \newcommand{\Ga}{{\Gamma}} \newcommand{\la}{\lambda} \newcommand{\Om}{{\Omega}} \newcommand{\om}{{\omega}} \newcommand{\si}{\sigma} \renewcommand{\th}{\theta} \newcommand{\td}{\tilde} \newcommand{\ze}{\zeta} \newcommand{\cA}{{\cal A}} \newcommand{\cL}{{\cal L}} \newcommand{\cE}{{\cal E}} \newcommand{\cN}{{\cal N}} \newcommand{\cP}{{\cal P}} \newcommand{\cQ}{{\cal Q}} \newcommand{\bP}{{\bf P}} \newcommand{\bQ}{{\bf Q}} \newcommand{\im}{{\text{Im} }} \newcommand{\re}{{\text{Re} }} \newcommand{\E}{{\mathbb E }} \newcommand{\R}{{\mathbb R }} \newcommand{\N}{{\mathbb N}} \renewcommand{\P}{{\mathbb P}} \newcommand{\bC}{{\mathbb C}} \newcommand{\pd}{{\partial}} \newcommand{\nb}{{\nabla}} \newcommand{\lec}{\lesssim} \newcommand{\ind}{{\,\mathrm{d}}} %\newcommand{\qed}{\hfill\fbox{}\par\vspace{0.3mm}} \newcommand{\ph}{{\varphi}} \renewcommand{\div}{\mathop{\mathrm{div}}} \newcommand{\curl}{\mathop{\mathrm{curl}}} \newcommand{\spt}{\mathop{\mathrm{spt}}} \newcommand{\wkto}{\rightharpoonup} \newenvironment{pf}{{\bf Proof.}} {\hfill\qed} \newcommand{\wt}{\widetilde} \newcommand{\lv}{{\bar v}} \newcommand{\lp}{{\bar p}} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} %\theoremstyle{definition} \newtheorem{remark}{Remark} \newtheorem{definition}{Definition} \newcommand{\qed}{\hfill\fbox{}\par\vspace{0.3mm}} \newenvironment{proof}{{\bf Proof.}} {\hfill\qed} % NUMBERING SCHEME \numberwithin{equation}{section} \numberwithin{theorem}{section} \numberwithin{definition}{section} %\numberwithin{corollary}{section} %\numberwithin{lemma}{section} % set the depth for the table of contents (0-2) \setcounter{tocdepth}{1} \title{Wegner estimate and level repulsion for Wigner random matrices} \author{L\'aszl\'o Erd\H os${}^1$\thanks{Partially supported by SFB-TR 12 Grant of the German Research Council}, Benjamin Schlein${}^2$\thanks{Partially supported by Sofja-Kovalevskaya Award of the Humboldt Foundation.}\; and Horng-Tzer Yau${}^3$\thanks{Partially supported by NSF grants DMS-0602038, 0757425, 0804279} \\ \\ Institute of Mathematics, University of Munich, \\ Theresienstr. 39, D-80333 Munich, Germany${}^1$ \\ \\ Department of Pure Mathematics and Mathematical Statistics \\ University of Cambridge \\ Wilberforce Rd, Cambridge CB3 0WB, UK${}^2$ \\ \\ Department of Mathematics, Harvard University\\ Cambridge MA 02138, USA${}^3$ \\ \\ \\} \begin{document} \date{Nov 11, 2008} \maketitle \begin{abstract} We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are 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 first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{ESY2}. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture. \end{abstract} {\bf AMS Subject Classification:} 15A52, 82B44 \medskip {\it Running title:} Wegner estimate and level repulsion \medskip {\it Key words:} Semicircle law, Wigner random matrix, level repulsion, Wegner estimate, density of states, localization, extended states. %\received{} \bigskip \section{Introduction} Let $H=(h_{ij})$ be an $N\times N$ hermitian matrix with eigenvalues $\mu_1\leq \mu_2 \leq\ldots \leq \mu_N$. These matrices form a {\it hermitian Wigner ensemble} if the matrix elements, \be h_{i j} = \bar{h}_{ji}= N^{-1/2}z_{ij}\in \bC, \quad (1\leq i < j\leq N), \quad \text{and} \quad h_{i i} = N^{-1/2} x_{ii}\in \bR, \quad (1\leq i\leq N) \label{wig} \ee are independent random variables with mean zero. We assume that $z_{ij}$ ($i< j$) all have a common distribution $\nu$ with variance $\int_\bC |z|^2 \rd\nu(z)=1$ and with a strictly positive density function $h:\bR^2\to \bR_+$, i.e. $$ \rd \nu(z)= \mbox{(const.)}h(x,y)\rd x\rd y\quad \mbox{where}\quad x=\re \, z,\;\;\ y=\im \, z. $$ We will often denote $g:= -\log h$. Throughout the paper we also assume that \be \mbox{either} \quad h(x,y)= h^*(x)h^*(y), \qquad \mbox{or} \quad h(x,y)= h^* (x^2+y^2) \label{hass} \ee with some positive function $h^*:\bR\to\bR_+$, i.e. either the real and imaginary parts of the random variables $z_{ij}$, $i0$ such that \be D:=\int_\bC \exp \big[ \delta_0 |z|^2\big] \rd \nu(z) <\infty , \qquad \wt D:=\int_\bR \exp{\big[\delta_0 x^2\big]}\rd \wt\nu(x) <\infty \; . \label{x2} \ee \end{itemize} To establish the Wegner estimate and the level repulsion, we need some smoothness property of the density function $h$. We assume that \begin{itemize} \item[{\bf C2)}] The Fourier transform of the functions $h$ and $h(\Delta g)$, with $g=-\log h$, satisfies the decay estimate \be |\wh h(t,s)|\leq \frac{1}{\left[1+\om_a(t^2+s^2)\right]^a}, \qquad |\wh{h\Delta g}(t,s)| \leq \frac{1}{\left[1+\wt\om_a(t^2+s^2)\right]^a} \label{charfn} \ee with some exponent $a\ge 1$ and constants $\om_a, \wt\om_a> 0$. (Note that $a\om_a\leq \frac{1}{4}$ by the condition that the variance is 1.) \end{itemize} \medskip \noindent In our previous papers \cite{ESY, ESY2} we assumed that $\rd\nu$ satisfies the logarithmic Sobolev inequality for the proof of the analogue of Lemma \ref{lm:x-old} (Lemma 2.1 of \cite{ESY2}). M. Ledoux has kindly pointed out to us that by applying a theorem of Hanson and Wright \cite{HW}, this lemma also holds under the moment condition {\bf C1)} only. We remark that the original paper \cite{HW} assumed that $\rd\nu$ was symmetric; this conditon was later removed by Wright \cite{Wr}. Another assumption we made in \cite{ESY, ESY2} states that either the Hessian of $g=-\log\, h$ is bounded from above or the distribution is compactly supported. This was needed because we used Lemma 2.3 of \cite{ESY}, whose original proof required the condition on $\mbox{Hess}\; g$. An alternative proof of this lemma was given by Bourgain (the proof reproduced in the Appendix of \cite{ESY2}) under the additional condition that the support of $\rd \nu$ is compact. In this paper, we extend the results of \cite{HW, Wr} and apply them to prove a weaker but for our purposes still sufficient version of Lemma 2.3 in \cite{ESY}. This approach requires no additional condition apart from {\bf C1)}. Condition ${\bf C2)}$ will play a role only in Theorem \ref{thm:wegner} and Theorem \ref{thm:repul}. In our previous papers \cite{ESY, ESY2} we assumed that the real and imaginary parts of $z_{ij}$ are independent. It is straightforward to check that all results of \cite{ESY, ESY2} hold for the case of radially symmetric distributions (second condition in \eqref{hass}) as well. \medskip {\it Convention.} We assume condition {\bf C1)} throughout the paper and every constant may depend on the constants $\delta_0, D, \wt D$ from \eqref{x2} without further notice. \bigskip {\it Acknowledgement.} The authors are grateful to M. Ledoux for his remark that Lemma 2.1 of \cite{ESY2} follows from a result of Hanson and Wright \cite{HW}. \section{Notation and the basic formula}\label{sec:not} For any spectral parameter $z= E+i\eta\in \bC$, $\eta>0$, we denote the Green function by $G_z= (H-z)^{-1}$. Let $F(E)=F_N(E)$ be the empirical distribution function of the eigenvalues \be F(E):= F_N(E)= \frac{1}{N}\big| \, \big\{ \al \; : \; \mu_\al \leq E\big\}\Big|\; \label{Fdef} \ee (in physics it is called the integrated density of states). Its derivative is the empirical density of states measure $$ \varrho(E): = F'(E) = \frac{1}{N}\sum_{\al=1}^N \delta(E-\mu_\al) . $$ Its statistical average, $\E \, \varrho(E)$, is called the {\it averaged density of states.} We define the Stieltjes transform of $F$ as \be m= m(z) =\frac{1}{N}\text{Tr} \; G_z = \int_\bR \frac{\rd F(E)}{E-z}\,, \label{Sti} \ee and we let \be \varrho_{\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 $\eta$. We note that $\varrho(E) = \lim_{\eta\to 0+0}\varrho_\eta(E)$. The random variable $m$ and the random measures $\varrho$ and $\varrho_\eta$ also depend on $N$, when necessary, we will indicate this fact by writing $m_N$, $\varrho_N$ and $\varrho_{\eta,N}$. 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 \be \varrho_{sc}(E) = \frac{1}{2\pi} \sqrt{4-E^2} {\bf 1}(|E|\leq 2)\; . \label{scform} \ee \bigskip 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 $\ba^{(k)}=(h_{k,1}, h_{k,2}, \ldots h_{k,k-1}, h_{k,k+1}, \ldots h_{k,N})^* \in \bC^{N-1}$, i.e. the $k$-th column after removing the diagonal element $h_{k,k}=h_{kk}$. 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 $$ \xi_\al^{(k)} : = \big| \sqrt{N}\ba^{(k)}\cdot \bu_\al^{(k)}\big|^2 $$ and note that $\E \, \xi_\al^{(k)}=1$. Thus \be m(z) =\frac{1}{N}\sum_{k=1}^N \Bigg[ h_{kk} - z - \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\xi^{(k)}_\al}{\lambda_\al^{(k)}-z}\Bigg]^{-1}\;. \label{mm1} \ee 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 in most of the proofs and we will often omit it from the argument of the Stieltjes transforms. Let $\E_k$ denote the expectation value w.r.t the random vector $\ba^{(k)}$. The distribution of $B^{(k)}$, $\ba^{(k)}$ and $\xi_\al^{(k)}$ does not depend on $k$, so we will often omit this superscript when it is unnecessary. For any spectral interval $I\subset \bR$, we denote $$ \cN_I: =\#\{ \alpha\; :\; \mu_\al\in I\} $$ $$ \cN_I^{(k)};= \#\{ \alpha\; : \; \lambda_\al^{(k)}\in I\} $$ the number of eigenvalues in $I$ of $H$ and $B^{(k)}$, respectively. When we are interested only in the distribution of $\cN_I^{(k)}$, we drop the superscript $k$, but to avoid confusion with $\cN_I$, we denote by $\cN^\lambda_I$ a random variable with the common distribution of $\cN_I^{(k)}$. \bigskip With these notations, the following basic upper bound on $\cN_I$ follows immediately: \begin{proposition}\label{prop:basic} Let $I=[E-\eta/2, E+\eta/2]$ be an interval of length $\eta$ about the spectral point $E$ and let $z=E+i\eta$. Then we have the following estimate on the number of eigenvalues in $I$: \be \begin{split} \cN_I \leq &\; C\eta \, \im \sum_{k=1}^N \Bigg[ h_{kk} - z - \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\xi^{(k)}_\al}{\lambda_\al^{(k)}-z}\Bigg]^{-1}\; . \label{basic} \end{split} \ee \end{proposition} {\it Proof.} We have $$ \cN_I = N\int_I \rd F(x) \leq \frac{5}{4}N\eta\int_{E-\eta/2}^{E+\eta/2} \frac{\eta\rd F(x)}{(x-E)^2+\eta^2} \leq \frac{5}{4}N\eta \, \im \, m(z) $$ that completes the proof of the first inequality using \eqref{mm1}. \qed \section{Main results} The first main result establishes the semicircle law on the optimal scale $\eta\ge O(1/N)$; the proof will be given in Section \ref{sec:sc}. \begin{theorem}\label{thm:sc1} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}. Let $\kappa>0$ and fix an energy $E\in [-2+\kappa, 2-\kappa]$. Set $c_0=\pi \varrho_{sc}(E)>0$ and let $K = 300/c_0$. Let $z=E+i\eta$ denote the spectral parameter with imaginary part satisfying $\eta\ge K/N$. Then there are constants $C$ and $c$, depending on $\kappa$, such that \be \P ( |m(z)-m_{sc}(z)|\ge \delta) \leq C\, e^{-c\delta\sqrt{N\eta}} \label{sc:new} \ee for any sufficiently small $\delta$ (depending on $\kappa$) and any sufficiently large $N\ge N_0(\delta)$. Furthermore, if $\cN_{\eta^*}(E)= |\{ \al\; : \; |\mu_\al - E| \leq \eta^*/2\}|$ denotes the number of eigenvalues in the interval $[E-\eta^*/2, E+\eta^*/2]$, then for any $\delta >0$ there is a constant $K_\delta$ such that \be \P \Big\{ \Big| \frac{\cN_{\eta^*}(E)}{N\eta^*} - \varrho_{sc}(E)\Big|\ge \delta\Big\}\leq C\, e^{-c\delta\sqrt{N\eta^*}} \label{ncont} \ee holds for all $\eta^*$ satisfying $K_\delta\leq N\eta^*$ and for all sufficiently large $N\ge N_0$ (depending on $\delta$ and $\kappa$). \end{theorem} \medskip As a corollary to this theorem, we can formulate a result on the eigenvectors: \begin{corollary}\label{cor:linfty} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}, then the following hold: \begin{itemize} \item[(i)] For any $\kappa>0$ and $K>0$ there exist constants $M_0=M_0(K,\kappa)$, $N_0=N_0(K,\kappa)$ and $c=c(K, \kappa)$ such that for any interval $I\subset [-2+\kappa, 2-\kappa]$ of length $|I|\leq K/N$ we have \be \P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I$ and } |v_1| \ge \frac{M}{N^{1/2}} \Bigg\} \leq e^{-c \sqrt{M}}\; \label{efn} \ee for all $M \geq M_0$ and $N\ge N_0$. \item[(ii)] For any $\kappa>0$ and $2\leq p<\infty$ there exist $M_0=M_0(\kappa,p)$, $N_0=N_0(\kappa,p)$ and $c=c(\kappa,p)>0$ such that for any interval $I\subset [-2+\kappa, 2-\kappa]$ of length $|I|=1/N$ we have \be \P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I$ and } \|\bv\|_p \ge M N^{\frac{1}{p}-\frac{1}{2}} \Bigg\} \leq e^{-c\sqrt{M}}\; \label{efn1} \ee for all $M \geq M_0$, all $N\ge N_0$. \item[(iii)] For any $\kappa>0$ there exist $M_0=M_0(\kappa)$, $N_0=N_0(\kappa)$ and $c=c(\kappa)$ such that \be \P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I$ and } \|\bv\|_\infty \ge \frac{M}{N^{1/2}} \Bigg\} \leq e^{-c \sqrt{M}}\; \label{efn2} \ee for all $M \geq M_0(\log N)^2$, all $N\ge N_0$. \end{itemize} \end{corollary} The second main result is an upper bound on the tail distribution of the eigenvalue gap; the proof is given in Section \ref{sec:tail}. \begin{theorem}\label{thm:gap} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}. Fix an energy $E\in [-2+\kappa, 2-\kappa]$. Denote by $\lambda_\al$ the largest eigenvalue below $E$ and assume that $\al\leq N-1$. Then there are positive constants $C$ and $c$ depending on $\kappa$ such that \be \P \Big(\lambda_{\al+1} -E\ge \frac{K}{N}, \; \al\leq N-1\Big) \leq C\; e^{-c\sqrt{K}} \label{gapdec} \ee for any $N\ge 1$ and any $K\ge 0$. \end{theorem} The third main result is the Wegner estimate for the averaged density of states: \begin{theorem}\label{thm:wegner} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)} and condition {\bf C2)} with an exponent $a=5$ in \eqref{charfn}. Let $\kappa>0$, choose an energy $|E|<2-\kappa$ and consider an energy interval $I=[E-\eta/2, E+\eta/2]$ and set $\e = N\eta$, with $\e>0$. Let $\cN_I$ be the number of eigenvalues in $I$ and assume $N\ge 10$. Then \be \P(\cN_I\ge 1)\leq \E \; \cN_I^2\leq C \, \e \label{n1} \ee uniformly in $N$ and $E$. In particular, \be \sup_{I\subset [-2+\kappa, 2-\kappa]} \sup_{N\ge 10} \; \E \Big[ \frac{\cN_I}{N|I|}\Big]\leq C \; , \label{eq:ds} \ee and therefore the averaged density of states, $\E \, \varrho_N(E)$, is an absolutely continuous measure with a uniformly bounded density, i.e. \be \sup_{|E|\leq 2-\kappa} \sup_{N\ge 10} \E \, \varrho_N(E) \leq C\, \label{ids} \ee (with a slight abuse of notations, $\E \, \varrho_N(E)$ denotes the measure and its density as well). %the averaged integrated density %of states (see \eqref{def:ids}) $I_N(E)$, is Lipschitz %continuous on the scale $1/N$; %\be % \sup_{|E|\leq 2-\kappa} % \sup_{N} N\big[ I_N(E+\e/N)- I_N(E)\big]\leq C \, \e\; . %\label{holder} %\ee The constant $C$ in \eqref{n1}, \eqref{eq:ds} and \eqref{ids} depends only on $\kappa$ and on the constants characterizing the distribution $\rd \nu$ via the conditions {\bf C1)}--{\bf C2)}. The estimate \eqref{n1} holds for $N\le 10$ as well if, instead of {\bf C1)} and {\bf C2)}, we assume that the density function, $(const.)\exp(-\wt g)$, of the diagonal matrix elements satisfies $\int_\bR |\wt g'(x)|\exp(-\wt g(x))\rd x <\infty$. \end{theorem} Finally, the following theorem establishes an upper bound on the level repulsion. \begin{theorem}\label{thm:repul} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}. Let $\kappa>0$, fix an energy $E$ with $|E|<2-\kappa$, set $\eta = \e/N$ and let $\cN_\eta$ be the number of eigenvalues in $I_{\eta} = [E-\eta/2, E+\eta/2]$. Fix $k \in \N$, and assume that condition ${\bf C2)}$ holds with $a=k^2+5$. Then, there exists a constant $C >0$, depending on $k$ and $\kappa$, such that \be \P ( \cN_\eta\ge k ) \leq C \; \e^{k^2} \label{want2} \ee for all $\e >0$ and uniformly for all $N\ge N_0(k)$. \end{theorem} The common starting point of all proofs is Proposition \ref{prop:basic}. Using the estimate $\text{Im} (a+bi)^{-1} \leq (a^2 + b^2)^{-1/2}$ on the right hand side of \eqref{basic}, we have \be \cN_I \leq C \eta\sum_{k=1}^N \frac{1}{(a_k^2 + b_k^2)^{1/2}} \label{out} \ee with $$ a_k := \eta + \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\eta \xi_\al^{(k)}}{(\lambda_\al^{(k)}-E)^2 + \eta^2}, \qquad b_k:= h_{kk} - E - \frac{1}{N}\sum_{\al=1}^{N-1} \frac{ (\lambda_\al^{(k)}-E) \xi_\al^{(k)}}{(\lambda_\al^{(k)}-E)^2 + \eta^2}\; , $$ where $a_k$ and $b_k$ are the imaginary and real part, respectively, of the reciprocal of the summands in \eqref{basic}. Theorems \ref{thm:sc1} and \ref{thm:gap} rely only on the imaginary part, i.e. $b_k$ in \eqref{out} will be neglected. In the proofs of Theorems \ref{thm:wegner} and \ref{thm:repul}, however, we make an essential use of $b_k$ as well. Since typically $1/N \lesssim |\lambda_\al^{(k)}-E|$, we note that $a_k^2$ is much smaller than $b_k^2$ if $\eta\ll 1/N$ and this is the relevant regime for the Wegner estimate and for the level repulsion. Assuming a certain smoothness condition on the distribution $\rd \nu$ (condition {\bf C2)}), the distribution of the variables $\xi_\al^{(k)}$ will also be smooth. Although $\xi_\al^{(k)}$ are not independent for different $\al$'s, they are sufficiently decorrelated so that the distribution of $b_k$ inherits some smoothness which will make the expectation value $(a_k^2 + b_k^2)^{-p/2}$ finite for certain $p>0$. This will give a bound on the $p$-th moment on $\cN_I$ which will imply \eqref{n1} and \eqref{want2}. \bigskip \section{Semicircle law and delocalization on intermediate scales}\label{sec:larger} In this section we review the proof of the convergence to the semicircle law on intermediate energy scales of the order $\eta \geq (\log N)^4/ N$. This convergence has already been established in our previous work \cite{ESY2} but with a speed of convergence uniform in $\eta$, for $\eta \geq (\log N)^8/ N$. Our new estimate shows that the speed of convergence becomes faster as $\eta$ increases (and we also reduce the power of the logarithm from 8 to 4). Moreover, we show that the results hold under the condition {\bf C1)} only. Thus we obtain a stronger version of our earlier results under weaker assumptions. The following result is an analogue of Theorem \ref{thm:sc1} for intermediate scales. It states that the density of states regularized on any scale $\eta\ge N^{-1}(\log N)^4$ converges to the Wigner semicircle law in probability uniformly for all energies away from the spectral edges. Note, however, that the estimate for larger scales is sufficiently strong so that uniformity in the spectral parameter $z$ can be obtained which is not expected for short scales. \begin{theorem}\label{thm:sc-old} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}. Let the energy scale $\eta$ be chosen such that $(\log N)^4/N\leq \eta\leq1$. Then for any $\kappa>0$ there exists a constant $c=c(\kappa)$ such that 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_{\text{Re}\, z \in [-2 + \kappa, 2 - \kappa]} |m_N(z)- m_{sc}(z)| \ge \delta \Big\} \leq e^{-c \delta \sqrt{N\eta}} \label{mcont-old} \ee for any $\delta>0$ if $N\ge N_0(\kappa, \delta)$ is large enough. Furthermore, on the scale $\eta^*$ with $(\log N)^4 /N\ll \eta^*\ll 1$ we have the convergence of the counting function as well. More precisely, let $\cN_{\eta^*}(E)= |\{ \al\; : \; |\mu_\al - E| \leq \eta^*/2\}|$ denote the number of eigenvalues in the interval $[E-\eta^*/2, E+\eta^*/2]$. Then, for any $\delta >0$ there is a constant $K_\delta$ such that \be \P \Big\{ \sup_{|E|\leq 2-\kappa} \Big| \frac{\cN_{\eta^*}(E)}{N\eta^*} - \varrho_{sc}(E)\Big|\ge \delta\Big\}\leq e^{-c\delta\sqrt{N\eta^*}} \label{ncont-old} \ee holds for all $\eta^*$ satisfying $K_\delta(\log N)^4/N\leq \eta^* \leq 1/K_\delta$ and for all sufficiently large $N\ge N_0$ (depending on $\delta$ and $\kappa$). \end{theorem} {\it Proof.} This theorem is proven exactly as Theorem 1.1 in \cite{ESY2} after replacing the key Lemma 2.1 of \cite{ESY2} by the following Lemma \ref{lm:x-old}. M. Ledoux has informed us that Lemma 2.1 of \cite{ESY2} follows from a result of Hanson and Wright \cite{HW}. We will reproduce his argument in the proof of Proposition \ref{prop:x}. This requires only Proposition \ref{prop:HW} below, which is a mild extension of the Hanson-Wright theorem to the complex case. \begin{lemma}\label{lm:x-old} Let $E\in [-2+\kappa, 2-\kappa]$. Suppose that $\bv_\alpha$ and $\lambda_\alpha$ are eigenvectors and eigenvalues of an $N\times N$ random hermitian matrix $B$ with a law satisfying the assumption of Theorem \ref{thm:sc-old}. 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, independent of $B$ and satisfying the condition {\bf C1)}. Then there exists a positive constant $c$ (depending on $\kappa$) so that for every $\delta >0$, we have \begin{equation}\label{eq:claim} \P[ |X|\ge \delta] \leq 5\, e^{- c \min\{\delta \sqrt{N\eta}, \, \delta^2N\eta\}} \end{equation} if $N\eta \geq (\log N)^2$ and $N$ is sufficiently large. \end{lemma} \noindent For simplicity, we formulated the lemma for $N\times N$ matrices, but it will be applied for the $(N-1)\times (N-1)$ minors of $H$. \medskip {\it Proof of Lemma \ref{lm:x-old}.} Define the intervals $I_n= [E- 2^{n-1}\eta, E+2^{n-1}\eta]$ and let $M$ and $K_0$ be sufficiently large fixed numbers. We have $[-K_0, K_0] \subset I_{n_0}$ with $n_0= C\log (K_0/\eta)\le C\log (NK_0)$. Denote by $\Omega$ the event \be \Omega : = \Omega(M, K_0)= \Big\{ \max_n \frac{\cN_{I_n}}{N|I_n|} \ge M \Big\} \cup \{ \max_\al |\lambda_\al|\ge K_0\} \; , \label{omegadef} \ee where $\cN_{I_n}=|\{\al\; : \; \lambda_\al\in I_n\}|$ is the number of eigenvalues in the interval $I_n$. Therefore, if $\P_\bb$ denotes the probability w.r.t. the variable $\bb$, we find $$ \P[ |X|\ge \delta] \leq \E \Big[ {\bf 1}_{\Omega^c}\cdot \P_\bb \big[ |X|\ge \delta] \Big] + \P(\Omega) \; . $$ We will prove below the following two propositions which complete the proof of Lemma \ref{lm:x-old}. \qed \begin{proposition}\label{prop:x} Assume condition {\bf C1)}. Let $\Omega = \Omega(M, K_0)$ be given by \eqref{omegadef} and let $\eta\ge 1/N$. Then for sufficiently large and fixed $M, K_0$ there is a positive $c=c(M, K_0)$ such that for any $\delta>0$ $$ \E \Big[ {\bf 1}_{\Omega^c}\cdot \P_\bb \big[ |X|\ge \delta] \Big] \leq 4\, e^{-c\min\{ \delta\sqrt{N\eta}, \, \delta^2N\eta\}} \; . $$ \end{proposition} \begin{proposition}\label{prop:omega} Assume condition {\bf C1)}. Let $\eta$ be chosen such that $(\log N)^2/N\leq \eta\leq1$. Then for sufficiently large and fixed $M$ and $K_0$ there is a positive constant $c$ such that \be \P\big[\Omega(M, K_0)\big] \leq e^{-c\sqrt{MN\eta}} \; . \label{Pom} \ee for all $N$ sufficiently large. \end{proposition} Both results are based on a theorem of Hanson and Wright \cite{HW}, extended to non-symmetric variables by Wright \cite{Wr}. The result was formulated for real valued random variables. We do not know if their theorems hold for general complex random variables, but they hold true in two special cases, namely when either the real and imaginary parts of $b_j$ are i.i.d. or if the distribution of $b_j$ is rotationally symmetric (see \eqref{hass}). We formulate this easy extension of their result and we give the proof in the Appendix. \begin{proposition}\label{prop:HW} Let $b_j$, $j=1,2,\ldots N$ be a sequence of complex i.i.d. random variables with distribution $\rd\nu$ satisfying the Gaussian decay \eqref{x2} for some $\delta_0>0$. Suppose that condition \eqref{hass} holds, i.e. either both the real and imaginary parts are i.i.d. or the distribution $\rd\nu$ is rotationally symmetric. Let $a_{jk}$, $j,k=1,2,\ldots N$ be arbitrary complex numbers and let $\cA$ be the $N\times N$ matrix with entries $\cA_{jk}:= |a_{jk}|$. Define $$ X=\sum_{j,k=1}^N a_{jk} \big[ b_j\ov{b}_k -\E b_j\ov{b}_k\big]\; . $$ Then there exists a constant $c>0$, depending only on $\delta_0, D$ from \eqref{x2}, such that for any $\delta>0$ $$ \P (|X|\ge \delta)\leq 4\exp\big( -c\min\{\delta/A,\; \delta^2/A^2\}\big)\;, $$ where $A:= (\text{Tr}\, \cA\cA^t)^{1/2}=\big[\sum_{j,k} |a_{jk}|^2\big]^{1/2}$. \end{proposition} {\it Proof of Proposition \ref{prop:x}.} Write $X$ in the form $$ X= \sum_{j,k=1}^N a_{jk} \big[ b_j\overline{b_k} - \E b_j\overline{b_k}\big]\; , $$ where $$ a_{jk} = \frac{1}{N}\sum_\al \frac{ \overline{ u_\al(j)} u_\al(k)}{\lambda_\al-z}\; . $$ We have $$ A^2: = \sum_{j,k=1}^N |a_{jk}|^2 =\frac{1}{N^2} \sum_\al\frac{1}{|\lambda_\al-z|^2} \; . $$ On the set $\Omega^c$ we have \be \begin{split} A^2 =\frac{1}{N^2} \sum_{n= 0}^{n_0} \sum_{ \lambda_\al\in I_n\setminus I_{n-1}} \frac{1}{|\lambda_\al-z|^2} \leq \frac{1}{N^2}\sum_{n= 0}^{n_0} \frac{\cN_{I_n}}{(2^n\eta)^2} \leq \frac{2M}{N\eta} \; \label{omc} \end{split} \ee where we estimated the number of eigenvalues in $I_n\setminus I_{n-1}$ by $\cN_{I_n}$ and we set $I_{-1}:=\emptyset$. Using Proposition \ref{prop:HW} we obtain that $$ \E \Big[ {\bf 1}_{\Omega^c}\cdot \P_\bb [ |X|\ge \delta] \Big] \leq 4\exp\big( -c\min \{ \delta\sqrt{N\eta}, \, \delta^2 N\eta\}\big)\; $$ where the constant $c$ depends on $M$ and on $\delta_0, D$ from \eqref{x2}. This completes the proof of Proposition \ref{prop:x}. \qed \bigskip {\it Remark.} The same result can be proven by assuming that the distribution $\rd\nu$ satisfies the logarithmic Sobolev inequality, see Lemma 2.1 of \cite{ESY2}; the bound $\exp{(-c\delta(\log N)^2)}$ obtained there can be easily improved to $C\exp{(-c\delta\sqrt{N\eta})}$ since the exceptional set $\Omega$ is defined differently. \medskip {\it Proof of Proposition \ref{prop:omega}.} Under condition {\bf C1)}, we showed in Lemma 7.4 of \cite{ESY} that \be \P\{ \max_\al |\lambda_\al|\ge K_0\} \leq e^{-cK_0^2N} \label{tail} \ee for sufficiently large $K_0$. To estimate the large deviation of $\cN_{I_n}$, we use the following weaker version of Theorem 2.1 of \cite{ESY}: \begin{theorem}\label{thm:upp1} Assume condition {\bf C1)}. Let $I\subset \bR$ be an interval with length $|I| \ge (\log N)/N$. Then there is a positive constant $c>0$ such that for any $K$ large enough \be \P \big\{ \cN_I\ge KN|I|\} \leq e^{-c\sqrt{KN|I|}}\; . \label{uppp} \ee \end{theorem} Combining \eqref{tail} and \eqref{uppp} and recalling $N\eta\ge (\log N)^2$, we have $$ \P (\Omega) \leq C\log (NK_0) e^{-c\sqrt{MN\eta}} + e^{-cK_0^2N} \leq e^{-\wt c\sqrt{MN\eta}} $$ completing the proof of Proposition \ref{prop:omega}. \qed \bigskip {\it Proof of Theorem \ref{thm:upp1}.} The proof is the same as the proof of Theorem 2.1 in \cite{ESY} but in the estimate (2.20) at the end of the proof we use the following lemma instead of Corollary 2.4 to Lemma 2.3 \cite{ESY}: \begin{lemma}\label{lm:bour} Assume condition {\bf C1)}. Let the components of the vector $\bb\in \bC^{N-1}$ be complex i.i.d. variables with a common distribution $\rd \nu$ and let $\xi_\al = |\bb \cdot \bv_\al|^2$, where $\{ \bv_\al\}_{\al\in {\cal I}}$ is an orthonormal set in $\bC^{N-1}$. Then for $\delta\leq 1/2$ there is a constant $c>0$ such that \be \P\big\{ \sum_{\al\in {\cal I}}\xi_\al \leq \delta m\big\} \leq e^{-c\sqrt{m}} \; \label{lm:xii} \ee holds for any ${\cal I}$, where $m=|{\cal I}|$ is the cardinality of the index set ${\cal I}$. \end{lemma} We remark that a stronger bound of the form $e^{-cm}$ was proven in Lemma 2.3 \cite{ESY} under the condition that $\mbox{Hess} \; g $ is bounded and in the special case when $g(x,y)$ was in the form $g(x)+g(y)$. An alternative proof under the condition that the support of $\rd\nu$ is compact is due to J. Bourgain and it is reproduced in the Appendix of \cite{ESY2}. Using the stronger $e^{-cm}$ bound in \eqref{lm:xii}, the bound in \eqref{uppp} can be improved to $e^{-cKN|I|}$. Here we present a proof that gives the weaker bound but it uses no additional assumption apart from {\bf C1)} and \eqref{hass}. \bigskip {\it Proof of Lemma \ref{lm:bour}.} Let $$ X:= \sum_{i, j=1}^N a_{ij} \big[ b_i\ov{b}_j - \E \, b_i\ov b_j\big], \qquad \mbox{with}\qquad a_{ij}: = \sum_{\al\in {\cal I}} \ov{v}_\al(i)v_\al(j)\, . $$ Notice that $\sum_{\al\in {\cal I} }\xi_\al = X + | {\cal I}|=X+m$ since $\E \,\xi_\al =1$. By $\delta\le 1/2$ we therefore obtain $$ \P\big\{ \sum_{\al\in {\cal I}}\xi_\al \leq \delta m\big\} \leq \P\big\{ |X| \ge \frac{m}{2}\big\}\; . $$ Since $$ A^2:= \sum_{i,j=1}^N |a_{ij}|^2 = \sum_{\al,\beta\in {\cal I}} \sum_{i,j=1}^N\ov{v}_\al(i)v_\al(j)v_\beta(i)\ov{v}_\beta(j) = m \; , $$ by Proposition \ref{prop:HW}, we obtain $$ \P\big\{ \sum_{\al\in {\cal I}}\xi_\al \leq \delta m\big\} \leq \P\big\{ |X| \ge \frac{m}{2}\big\}\; \leq 4\exp\Big( -c\min\big\{ \frac{m}{2A}, \frac{m^2}{4A^2} \big\} \Big)\leq e^{-c\sqrt{m}}. $$ for some $c>0$. \qed \bigskip Using Theorem \ref{thm:sc-old}, we can prove delocalization of the eigenvectors of $H$. In Theorem 1.2 of \cite{ESY2} we proved that $\| \bv \|_{\infty} \leq (\log N)^{9/2} / N^{1/2}$ holds for all eigenvectors with probability bigger than $1- e^{-c(\log N)^2}$. The following theorem is a generalization of this result using the stronger estimates from Theorem \ref{thm:sc-old}. \begin{theorem}\label{cor:linfty-old} Let $H$ be an $N\times N$ hermitian Wigner matrix satisfying the condition {\bf C1)}. For any $\kappa>0$ there exists $c>0$, depending on $\kappa$ such that \[ \P \Bigg\{\exists \text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in [-2+\kappa, 2-\kappa]$ and } \| \bv \|_\infty \ge \frac{M}{N^{1/2}} \Bigg\} \leq e^{-c \sqrt{M}}\; \] for all $M \geq (\log N)^4$ and all $N\ge N_0(\kappa)$ large enough. \end{theorem} {\it Proof.} Let $\eta^* = M/N$ and partition the interval $[-2+\kappa, 2-\kappa]$ into $n_1= O(1/\eta^*)\leq O(N)$ intervals $I_1, I_2, \ldots, I_{n_1}$ of length $\eta^*$. As before, let $\cN_{I}=|\{ \beta\; : \; \mu_\beta\in I\}|$ denote the number of eigenvalues in $I$. Let $$ c_1 :=\varrho_{sc}(2-\kappa)= \min \big\{ \varrho_{sc}(E) \; : \; E\in [-2+\kappa, 2-\kappa]\big\} >0 \; . $$ By using \eqref{ncont-old} in Theorem \ref{thm:sc-old} and the fact that $N\eta^*\ge (\log N)^4$, we have \be \P \left\{\max_n \cN_{I_n} \leq \frac{c_1}{2}\, N\eta^*\right\} \leq CN e^{- c \sqrt{N\eta^*}} \leq e^{- \wt{c} \sqrt{N\eta^*}} \; . \label{pmax} \ee Suppose that $\mu \in I_n$, and that $H\bv = \mu \bv$. Consider the decomposition \be \label{Hd-old} H = \begin{pmatrix} h & \ba^* \\ \ba & B \end{pmatrix} \ee where $\ba= (h_{1,2}, \dots h_{1,N})^*$ 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 $B$. Similarly to \cite{ESY2}, from the eigenvalue equation $H \bv = \mu \bv$ and from \eqref{Hd-old} we find for the first component of $\bv =(v_1, v_2, \ldots , v_N)$ that %that %\be %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{v1} |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 $B$. We recall that the eigenvalues of $H$, $\mu_1\leq \mu_2 \leq \ldots\leq \mu_N$, and the eigenvalues of $B$ are interlaced: $\mu_1 \leq \lambda_1\leq \mu_2\leq \lambda_2 \leq \ldots \leq \lambda_{N-1} \leq \mu_N$ and the inequalities are strict with probability one (see Lemma 2.5 of \cite{ESY}). This means that there exist at least $\cN_{I_n}-1$ eigenvalues of $B$ in $I_n$. Therefore, using that the components of any eigenvector are identically distributed, we have \begin{equation} \begin{split}\label{lon} \P \Big( \exists &\text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in [-2+\kappa, 2-\kappa]$ and } \| \bv \|_\infty \ge \frac{M}{N^{1/2}} \Big) \\ &\leq N n_1 \, \sup_n \P \Big( \exists \text{ $\bv$ with $H\bv=\mu\bv$, $\| \bv \|=1$, $\mu \in I_n$ and } |v_1|^2 \ge \frac{M^2}{N} \Big)\\ &\leq C\, N^2 \sup_n \P \left( \sum_{\lambda_\alpha \in I_n} \xi_{\alpha} \leq 4 \right) \\ &\leq C \, N^2 \sup_n \P \left( \sum_{\lambda_\alpha \in I_n} \xi_{\alpha} \leq 4 \text{ and } \cN_{I_n} \geq \frac{c_1}{2}\, N \eta^* \right) + C\, N^2 \sup_n \, \P \left(\cN_{I_n} \leq \frac{c_1}{2}\, N \eta^* \right) \\&\leq C \, N^2 e^{- \wt{c}\sqrt{ N\eta^*}} + C\, N^2 e^{- \wt{c} \, \sqrt{N\eta^*}} \\ &\leq e^{-c \sqrt{M}}, \end{split} \end{equation} for a sufficiently small $\wt{c} >0$ (we also used that $N\eta^* = M \ge(\log N)^4$ in the last step). Here we used Lemma \ref{lm:bour} to estimate the first probability in the fourth line of \eqref{lon} and \eqref{pmax} to estimate the second one. \qed \section{Upper bound for the density on short scales}\label{sec:upp} Now we start our analysis on short scales $\eta\ge 1/N$. As before, we always assume condition {\bf C1)} in addition to \eqref{hass}. We first show a large deviation upper bound on the number of eigenvalues on short scales about a fixed energy $E$ away from the spectral edges. This complements the estimate in Theorem \ref{thm:upp1} that was valid for larger scales. \begin{theorem}\label{thm:upp} Let $\kappa>0$ and fix an energy $E\in [-2+\kappa, 2-\kappa]$. Let $\eta>0$ with $1\leq N\eta\leq CN^{1/10}$. Let $$ \cN := \#\{ \al\; :\; \mu_\al \in I_\eta:=[E-\eta/2, E+\eta/2]\}\; . $$ Then for any $2\leq M\leq \frac{CN^{1/10}}{N\eta}$, we have \be \P \Big( \frac{\cN}{N\eta} \ge M\Big) \leq \Big(\frac{C}{M}\Big)^{\frac{1}{8}MN\eta} \label{upperprob} \ee and for any $1\leq p\leq CN^{1/20}$ \be \E \Big[\frac{\cN}{N\eta}\Big]^p \leq C^p\Big(1+ \frac{p}{N\eta}\Big)^p \; . \label{upperexp} \ee All constants depend on $\kappa$. \end{theorem} \bigskip {\it Proof of Theorem \ref{thm:upp}.} It is sufficent to prove \eqref{upperprob}, since \eqref{upperexp} easily follows from it and from \eqref{uppp}: \be \begin{split} \E \Big[\frac{\cN}{N\eta}\Big]^p \leq \; & 2^p +p\int_{2}^\infty M^{p-1} \P \Big( \frac{\cN}{N\eta} \ge M\Big) \rd M\\ \leq \; & 2^p + p\int_2^{\Lambda} M^{p-1-\frac{1}{16} MN\eta}\; \rd M + \int_{\Lambda}^\infty M^{p-1-c\sqrt{MN\eta}}\;\rd M\\ \leq \; & C^p\Big(1+ \frac{p}{N\eta}\Big)^p\; \end{split} \ee with $\Lambda=CN^{1/10}/N\eta$ and with a sufficiently large constant $C$. To prove \eqref{upperprob}, we use \eqref{basic} to obtain \be \frac{\cN_I}{N\eta} \leq C\sum_{k=1}^N \frac{1}{\eta + \frac{1}{N}\sum_\al \frac{\eta\xi^{(k)}_\al} {(\lambda_\al^{(k)}-E)^2+\eta^2}} \leq\frac{C}{N}\sum_{k=1}^N\frac{1}{\eta+ \frac{Z^{(k)}}{N\eta }} \; , \label{keyestimate} \ee where we defined \be Z^{(k)}:=Z^{(k)}(\eta) = \sum_{\al: \lambda_\al^{(k)} \in I_\eta} \xi_\al^{(k)}\; . \label{def:Z} \ee To estimate the large deviation of $Z^{(k)}$, we will later prove the following lemma: \begin{lemma}\label{lemma:lde} Let $\nu, \beta$ be nonnegative numbers such that $\nu+7\beta<1$. Then for any $\delta \ge N^{-\nu}$ and $m\leq N^\beta$, we have \be \P\Big\{ \frac{1}{m}\sum_{\al=1}^m \xi_\al \le \delta\Big\} \leq (C\delta)^{m} \label{new} \ee with a constant $C$ depending on $\nu$ and $\beta$. \end{lemma} Note that the estimate in this lemma is more precise than \eqref{lm:xii}, but the stronger estimate is valid only if $m$ is not too large. In the proof we will use information about the eigenfunctions obtained in Theorem \ref{cor:linfty-old}. \medskip Let $$ \cN^{(k)}:=\cN^{(k)}_{I_\eta}=\#\{ \al: \lambda_\al^{(k)} \in I_\eta\} $$ denote the number of eigenvalues of the minor $B^{(k)}$ in the interval $I_\eta$ (see Section \ref{sec:not} for the definitions). By the interlacing property of the eigenvalues, $\cN\ge MN\eta$ implies $\cN^{(k)}\ge MN\eta-1 \ge \frac{1}{2}MN\eta\ge N\eta$ for any $k$ (since $M\ge 2$ thus $MN\eta\ge 2$). Therefore, from \eqref{keyestimate} we have for any $q\ge1$ that \be \begin{split}\label{Mest} \P\Big( \frac{\cN}{N\eta}\ge M\Big) \leq \; & \P\Big(\frac{C}{N}\sum_k \frac{{\bf 1}(\cN^{(k)}\ge \frac{1}{2}MN\eta)}{\frac{Z^{(k)}}{N\eta} + \eta} \ge M\Big) \\ \leq \; & \Big(\frac{C}{M}\Big)^q \E \Bigg[ \frac{{\bf 1}(\cN^{(1)}\ge\frac{1}{2}M N\eta)}{\frac{Z^{(1)}}{N\eta} + \eta}\Bigg]^q \\ \leq \; &\Big(\frac{C}{M}\Big)^q \int_0^\infty \P \Bigg[ \cN^{(1)}\ge \frac{1}{2}MN\eta, \; \frac{Z^{(1)}}{N\eta} + \eta\leq t^{-1/q}\Bigg]\rd t \\ \leq \; &\Big(\frac{C}{M}\Big)^q +\Big(\frac{C}{M}\Big)^q \int_1^{(1/\eta)^q} \P\Bigg( \sum_{\al=1}^{MN\eta/2} \xi_\al^{(1)} \leq N\eta t^{-1/q}\Bigg)\rd t\\ \leq \; &\Big(\frac{C}{M}\Big)^q +\Big(\frac{C}{M}\Big)^q \int_1^{(1/\eta)^q} \big[C\max \{t^{-1/q}, N^{-\nu}\}\big]^{\frac{1}{2}MN\eta} \rd t\\ \leq \; & \Big(\frac{C}{M}\Big)^{\frac{1}{8}MN\eta} \end{split} \ee if we use Lemma \ref{lemma:lde} with e.g. $\nu =1/4$ (noticing that $\frac{1}{2}MN\eta\leq N^\beta$ with $\beta=1/10$) and we choose $q=\frac{1}{8}MN\eta$ in the last line (we use that $[\eta^{-1} N^{-4\nu}]^q\leq 1$). $\Box$ \bigskip {\it Proof of Lemma \ref{lemma:lde}.} We will present the proof under the first condition in \eqref{hass}; the proof under the second condition is analogous. With the notation $\bb=\sqrt{N}\ba$, the components of $\bb$ can thus be written as $b_j = x_j+iy_j$ where $x_j$, $y_j$ are i.i.d. random variables with expectation zero and variance 1/2. Similarly we decompose the eigenvectors into real and imaginary parts, i.e. we write $\bu_\al= \bv_\al+ i\bw_\al$ and we have $$ \xi_\al = |\bb \cdot \bu_\al|^2 = \Big( \sum_{j=1}^N (x_jv_\al(j)+ y_jw_\al(j)) \Big)^2 +\Big( \sum_{j=1}^N(x_jw_\al(j)- y_jv_\al(j)) \Big)^2 \; . $$ The probability and expectation w.r.t. $\bb$ are denoted by $\P_\bb$ and $\E_\bb$. We define the event $$ \Omega:=\Big\{ \| \bu_\al\|_\infty \leq CN^{2\beta-1/2}(\log N)^4\; :\; \al=1,2,\ldots, m\Big\} \; , $$ where $\bu_\al$ are the eigenvectors of $B=B^{(1)}$. Note that $\Omega$ is independent of the vector $\bb = \sqrt{N} \ba^{(1)}$, thus $$ \P\{\sum_{\al=1}^m \xi_\al \leq m\delta\} \leq \P (\Omega^c) + \E\Big[ {\bf 1}(\Omega)\P_\bb \big(\sum_{\al=1}^m \xi_\al \leq m\delta\big) \Big]\;. $$ By Theorem \ref{cor:linfty-old}, $$ \P(\Omega^c)\leq e^{-cN^\beta (\log N)^2}\le (C\delta)^m\; . $$ On the event $\Omega$, the probability $\P_\bb \big(\sum_{\al=1}^m \xi_\al \leq m\delta\big)$ will be estimated as follows, where we introduced $t:=\delta^{-1}\leq N^{\nu}$: \be \begin{split} \P_\bb\Big\{ \sum_{\al=1}^m \xi_\al \le m\delta\Big\}\leq \; & e^{m}\E_\bb e^{-t\sum_{\al=1}^m \xi_\al} \\ =\; & e^{m}\E_\bb \prod_{\al =1}^me^{-t|\bb\cdot \bu_\al|^2}\\ =\; & e^{m}\E_\bb \prod_{\al=1}^m \int_{\bR^2} \frac{\rd\tau_\al\rd s_\al}{\pi} e^{-i\sqrt{t}\big[ \tau_\al \sum_j(x_jv_\al(j)+ y_jw_\al(j)) + s_\al \sum_j(x_jw_\al(j)- y_jv_\al(j)) \big] - \tau_\al^2/4 - s_\al^2/4}\\ =\; & e^{m} \Big(\prod_{\al=1}^m \int_{\bR^2} e^{- \frac{1}{4} (\tau_\al^2+ s_\al^2)} \frac{\rd\tau_\al\rd s_\al}{\pi} \Big) \\ &\quad \times \prod_{j=1}^N \E_{x_j}\E_{y_j} e^{-i\sqrt{t} \sum_\al \Big[ x_j (\tau_\al v_\al(j) + s_\al w_\al(j)) - y_j(s_\al v_\al(j) - \tau_\al w_\al(j))\Big]}\\ \leq\; & e^{m}\Big(\prod_{\al=1}^m \int_{\bR^2}e^{- \frac{1}{4} (\tau_\al^2+s_\al^2)} {\bf 1}\big( |\tau_\al|+|s_\al|\le N^{\beta/2} \log N\big) \; \frac{\rd\tau_\al\rd s_\al}{\pi} \Big)\\ & \quad\times \prod_{j=1}^N \Bigg(1 - \frac{t}{8}\Big[ \big( \sum_\al (\tau_\al v_\al(j) + s_\al w_\al(j))\big)^2 + \big( \sum_\al (s_\al v_\al(j) - \tau_\al w_\al(j))\big)^2\Big] \Bigg)\\ & \qquad\quad + % o(tN^{-1}) \sum_\al (\tau_\al^2+s_\al^2)\Bigg) + m(Ce)^me^{-cN^\beta(\log N)^2}\; .\label{pb} \end{split} \ee The last term comes from the Gaussian tail of the restriction $|\tau_\al|, |s_\al| \leq N^{\beta/2}\log N$ for all $\al$. In estimate \eqref{pb} we have used that $\Big|\E\big[ e^{iY} - 1- iY + \frac{1}{2}Y^2\big]\Big| \leq \E |Y^3|$, thus for any real random variable $Y$ with $\E\, Y=0$ and $|Y|\leq \frac{1}{4}$ we have $$ |\E \, e^{iY}| \leq 1- \frac{1}{2} \E\; Y^2 + \E |Y^3| \leq 1-\frac{1}{4} \E\; Y^2 \; . $$ We applied this to $$ Y =Y_j= -\sqrt{t}\sum_{\al=1}^m \Big[ x_j (\tau_\al v_\al(j) + s_\al w_\al(j)) + y_j(s_\al v_\al(j) - \tau_\al w_\al(j))\Big] $$ with $$ \E \, Y^2_j = \frac{t}{2}\Big[ \big( \sum_\al (\tau_\al v_\al(j) + s_\al w_\al(j))\big)^2 + \big( \sum_\al (s_\al v_\al(j) - \tau_\al w_\al(j))\big)^2\Big] $$ and we also used \be \begin{split} %\Big|\sqrt{t}\sum_{\al=1}^m \Big[ x_j (\tau_\al v_\al(j) + s_\al w_\al(j)) % + y_j(s_\al v_\al(j) - \tau_\al w_\al(j))\Big]\Big| |Y_j| \leq \; C m\sqrt{t}\,N^{5\beta/2-1/2}(\log N)^{5} \leq \; C N^{(\nu+7\beta-1)/2} (\log N)^{5} \leq \frac{1}{4} % o( tN^{-1})\sum_\al (\tau_\al^2+s_\al^2)\; \label{third} \end{split} \ee on the event $\Omega$ and in the regime where $|\tau_\al|, |s_\al| \leq N^{\beta/2}\log N$ for all $\al$. Reexponentiating $1-\frac{1}{4} \E \, Y^2_j \leq \exp(- \frac{1}{8} \E Y^2_j)$ and using that by the orthogonality of $\bu_\al$ we have, $$ \sum_{j=1}^N \Big[\big( \sum_{\al=1}^m (\tau_\al v_\al(j) + s_\al w_\al(j))\big)^2 + \big( \sum_{\al=1}^m (s_\al v_\al(j) - \tau_\al w_\al(j))\big)^2\Big] =\sum_{\al=1}^m (\tau_\al^2 +s_\al^2) \; $$ and we obtain (with $t=\delta^{-1}$) \be \begin{split} \P_\bb\Big\{ \sum_{\al=1}^m \xi_\al \le m\delta\Big\} \leq \; & e^{m}\Big(\prod_{\al=1}^m \int_{\bR^2}e^{- \frac{1}{4}(\tau_\al^2 + s_\al^2) } \frac{\rd\tau_\al\rd s_\al}{2\pi} \Big) e^{ - \frac{t}{16}\sum_\al (\tau_\al^2 +s_\al^2) }+ Ce^{-cN^\beta(\log N)^2} \\ \leq \; & \Big(\frac{C}{1+t}\Big)^m + Ce^{-cN^\beta(\log N)^2} \\ \leq \; & (C\delta)^m \; .\qquad \Box \end{split} \ee \bigskip \section{Proof of the semicircle law on short scales}\label{sec:sc} In this section we prove the semicircle law on the shortest possible scale $\eta\ge O(1/N)$. \bigskip {\it Proof of Theorem \ref{thm:sc1}.} We will prove only \eqref{sc:new}, the proof of \eqref{ncont} can be obtained from \eqref{sc:new} exactly as in Corollary 4.2 of \cite{ESY}. We can assume that $\eta\leq (\log N)^4/N$, since the regime $\eta\ge (\log N)^4/N$ has been covered in Theorem \ref{thm:sc-old}. The constants in this proof depend on $\kappa$ (in addition to $\delta_0, D$ from \eqref{x2}) and we will not follow their precise dependence. For $k=1,2, \ldots, N$ define the random variables \be X_k(z)=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 $\E_k \xi_\al^{(k)}=\| \bu_\al^{(k)}\|^2=1$ and we recall that $\E_k$ denotes the expectation w.r.t. the random vector $\ba^{(k)}$ (see Section \ref{sec:not} for notation). We note that $$ \E_k\; \ba^{(k)} \cdot \frac{1}{B^{(k)}-z} \ba^{(k)} = \frac{1}{N}\sum_\al \frac{1}{\lambda_\al^{(k)}-z} = \Big(1-\frac{1}{N}\Big) m^{(k)} \;. $$ It follows from \eqref{Sti} and \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 $$ \Big| m - \Big(1-\frac{1}{N}\Big)m^{(k)}\Big| =\Big| \int \frac{\rd F(x)}{x-z} - \Big(1-\frac{1}{N}\Big)\int \frac{\rd F^{(k)}(x)}{x-z}\Big| = \frac{1}{N}\Big| \int \frac{NF(x)-(N-1)F^{(k)}(x)}{(x-z)^2} \rd x\Big|. $$ and 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 \be \Big| m - \Big(1-\frac{1}{N}\Big)m^{(k)}\Big| \leq \frac{1}{N} \int \frac{\rd x}{|x-z|^2} = \frac{\pi}{N\eta}\, . \label{mmm} \ee Let $M\ge 2$ be sufficiently large and fixed. Fix $E\in[-2+\kappa, 2-\kappa]$ away from the spectral edge. Assume for the moment only that $1/N\leq \eta\le 1$. Define $I_n = [E- 2^{n-1} \eta, E+ 2^{n-1}\eta]$, and let $K_0$ be a sufficiently large fixed number. For some constant $C=C(K_0)$ we have $[-K_0, K_0] \subset \bigcup_{ n = 0}^{C\log N} I_n$. Denote by $\Omega$ the event \be\label{def:Omega} \Omega : = \Big\{ \max_{n\leq C\log N}\; \frac{\cN_{I_n}}{N|I_n|} \ge M\Big\} \cup \{ \max_\al |\lambda_\al|\ge K_0\} \; . \ee Let $n_0$ be the largest non-negative integer such that $2^{n_0}N\eta\leq (\log N)^4$, recall that we assumed $N\eta\leq (\log N)^4$. Similarly to the proof of Proposition \ref{prop:omega}, by using \eqref{upperprob} for short scales and \eqref{uppp} for larger scales, we get \be \begin{split} \P(\Omega)\leq & \; e^{-cN}+\sum_{n=0}^{n_0} \Big(\frac{C}{M} \Big)^{2^{n-3}MN\eta} +\sum_{n=n_0+1}^{C\log N} e^{-c \sqrt{2^nMN\eta}} \\ \leq &\; e^{-cN}+ \Big(\frac{C}{M}\Big)^{cMN\eta} + e^{-c\sqrt{MN\eta}} \leq 3\, e^{-c\sqrt{N\eta}} \label{omegaest} \end{split} \ee with some $c>0$ (first term coming from the probability of $\max_\al |\lambda_\al|\ge K_0$). {F}rom now on, we additionally assume that $K/N\leq \eta \leq (\log N)^4/N$. For $n\leq n_0$ define $z_n= E+ i\eta_n $ with $\eta_n= 2^n\eta$, i.e. $z=z_0$ and $2^n\eta \leq (\log N)^4/N$ for all $n\leq n_0$. We have from \eqref{recur} \be \begin{split}\label{mplusm} m(z_n) = \; &\frac{1}{N}\sum_{k=1}^N \frac{1}{-m(z_n) -z_n +\delta_k}\\ = \; & \frac{1}{-m(z_n) - z_n} - \frac{1}{N} \sum_{k=1}^N \frac{1}{-m(z_n) - z_n}\; \frac{\delta_k}{h_{kk} - z_n - \frac{1}{N} \sum_{\al=1}^{N-1} \frac{\xi_\al^{(k)}} {\lambda_\al^{(k)}-z_n} }\; , \end{split} \ee where $$ \delta_k =\delta_k(z_n): = h_{kk} + m(z_n) - \Big(1-\frac{1}{N}\Big) m^{(k)}(z_n) - X_k(z_n). $$ Recall $c_0=\pi \varrho_{sc}(E)$, thus $\mbox{Im}\; m_{sc}(z)=c_0+ O(\eta)$. Define the event $$ \Xi_n: =\{ \mbox{Im}\; m(z_n) \ge c_0/10\}\; . $$ On the event $\Xi_n$, by using \eqref{mmm} and that $N\eta\ge K= 300/c_0$, we have $\mbox{Im}\; m^{(k)}(z_n)\ge c_0/20$ for any $k$. Thus, on the event $\Xi_n\cap \Omega^c$ and for any positive integer $r$, we have \be \begin{split} \frac{c_0}{20}\leq \; & \frac{1}{N}\sum_\al \frac{\eta_n}{(\lambda^{(k)}_\al - E)^2+\eta_n^2}\\ \leq\; & \frac{\cN^{(k)}_{I_{n+r}}}{N\eta_n}+ \frac{1}{N}\sum_{\ell = n+r+1}^{C\log N} \sum_{\al\; : \; \lambda_\al^{(k)} \in I_\ell \setminus I_{\ell-1}} \frac{\eta_n}{(\lambda^{(k)}_\al - E)^2+\eta_n^2}\\ \leq\; & \frac{\cN^{(k)}_{I_{n+r}}}{N\eta_n}+ \frac{1}{N}\sum_{\ell =n+r+1}^{C\log N} \frac{2^n\eta \cN_{I_\ell}^{(k)}}{ (2^{\ell-2}\eta)^2}\\ \leq\; & \frac{\cN^{(k)}_{I_{n+r}}}{N\eta_n}+ 16\sum_{\ell= n+r+1}^{C\log N} \frac{\cN_{I_\ell}+1}{N|I_\ell|}\frac{1}{2^{\ell-n}}\\ \leq\; & \frac{\cN^{(k)}_{I_{n+r}}}{N\eta_n}+ 2^{5-r}M\; , \end{split} \ee where we used that from the interlacing property we have $\cN^{(k)}_{I}\leq \cN_I+1$, for any interval $I$. Thus, on $\Xi_n\cap \Omega^c$, with the choice $r= [\log_2(1280M/c_0)]+1$, we have the lower bound $$ \cN^{(k)}_{I_{n+r}}\ge \gamma_n\quad \mbox{with} \quad \gamma_n:= \frac{c_0}{40}N\eta_n $$ for any $n\leq n_0$ and for any $k=1,2, \ldots N$. Hence from \eqref{mplusm} and recalling the definition \eqref{def:Z} we get, for any $p\ge 1$, that \be \begin{split}\label{mm1new} \E \Big| m(z_n) + \frac{1}{m(z_n) + z_n}\Big|^p{\bf 1}\big(\Xi_n\cap \Omega^c\big) \leq \; & \E \Bigg[ \frac{10}{c_0}\; \frac{1}{N} \sum_{k=1}^N \frac{|\delta_k|\cdot {\bf 1}(\cN_{I_{n+r}}^{(k)}\ge \gamma_n) }{ \eta_n + \frac{1}{N} \sum_\al \frac{\eta_n\xi_\al^{(k)} }{ (\lambda_\al^{(k)}-E)^2+\eta^2_n}} \Bigg]^p \\ \ \leq \; & \E \Bigg[ \frac{10}{c_0}\; \frac{|\delta_1|\cdot {\bf 1}(\cN_{I_{n+r}}^{(1)}\ge\gamma_n) }{ \eta_n + \frac{1}{2^{2r}N\eta_n}Z^{(1)}(\eta_{n+r}) } \Bigg]^p\\ \leq \; & 2^{2pr}C_1^p \big[ \E \; |\delta_1|^{2p} \big]^{1/2} \Bigg[ \E \Bigg|\frac{1}{ \eta_n + \gamma_n^{-1} \sum_{\al =1}^{\gamma_n} \xi_\al^{(1)}} \Bigg|^{2p} \Bigg]^{1/2} \end{split} \ee (with $C_1 = (const)c_0^{-2}$). The second term can be estimated similarly to \eqref{Mest}. For any $1\leq p\leq c_0N\eta/300$ we have that \be \begin{split} \E \Bigg|\frac{1}{ \eta_n + \gamma_n^{-1} \sum_{\al =1}^{\gamma_n} \xi_\al^{(1)}} \Bigg|^{2p} \leq \; & \int_0^{(1/\eta_n)^{2p}}\P \Big(\sum_{\al=1}^{\gamma_n/2}\xi^{(1)}_\al \leq \gamma_n t^{-1/2p}\Big)\rd t\\ \leq \; & 1 +\int_1^{(1/\eta)^{2p}} \big[ C\max\{t^{-1/2p}, N^{-\nu}\}\big]^{\gamma_n/2}\rd t\\ \leq \; & C_\nu \end{split} \ee where we chose $\nu>3/4$ and used that $4p/\nu \leq \gamma_n \leq C(\log N)^4$. For the first term on the r.h.s. of \eqref{mm1new}, we use $\E |h_{kk}|^{2p} \leq C^pN^{-p}$ and \eqref{mmm} to get \be \E \; |\delta_1|^{2p} \leq C^pN^{-p} + \Big(\frac{C}{N\eta_n}\Big)^{2p} + C^p\E|X_1(z_n)|^{2p}. \label{delta1} \ee To estimate $\E|X_1(z_n)|^{2p}$, we will need the following extension of Lemma \ref{lm:x-old} to $\eta\ge O(1/N)$. \begin{lemma}\label{lm:x} Let $E\in [-2+\kappa, 2-\kappa]$. 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-old}. 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 the condition {\bf C1)}. Then there exist two positive constants $K$, $C$ and $c$ (depending on $\kappa$) so that for every $0<\delta \leq 1$, we have \begin{equation}\label{eq:Xbound} \P[ |X|\ge \delta] \leq C\; e^{- c \, \min\{ \delta \sqrt{N\eta},\; \delta^2N\eta\} } \end{equation} if $K\leq N\eta \leq(\log N)^4$. \end{lemma} {\it Proof of Lemma \ref{lm:x}.} We follow the proof of Lemma \ref{lm:x-old} but with the redefined set $\Omega$ (see \eqref{def:Omega} instead of \eqref{omegadef}). Using the improved bounds from Theorem \ref{thm:upp} we have already proved in \eqref{omegaest} that $P(\Omega)\leq 3\, e^{-c\sqrt{N\eta}}$. To estimate $\E\big[ 1_{\Omega^c}\cdot \P_\bb (|X|\ge \delta)\big]$, we follow the proof of Proposition \ref{prop:x}. The only difference is that in \eqref{omc} the summation runs from $n=0$ to $n=C\log N$, but the estimate on the right hand side of \eqref{omc} is still valid. This completes the proof of Lemma \ref{lm:x}. \qed \bigskip Given the bound \eqref{eq:Xbound}, we have $$ \E|X_1(z_n)|^{2p} \leq \frac{(Cp^2)^p}{(N\eta_n)^p} $$ and, from \eqref{delta1}, we get $$ \E \; |\delta_1|^{2p} \leq \frac{(Cp^2)^p}{(N\eta_n)^p}\; . $$ Thus \be \E \Big| m(z_n) + \frac{1}{m(z_n) + z_n}\Big|^p{\bf 1}\big( \Xi_n\cap\Omega^c\big) \leq \frac{(Cp)^p}{(N\eta_n)^{p/2}} \; . \label{concn} \ee For any $\delta$, set the event $$ \Lambda_n(\delta)=\Lambda_n:= \Big\{ \Big| m(z_n) + \frac{1}{m(z_n) + z_n}\Big|\ge \delta\Big\} $$ then from \eqref{concn} $$ \P(\Lambda_n\cap \Omega^c)\leq \P(\Xi_n^c\cap\Omega^c) + \frac{(Cp)^p}{(N\eta_n\delta^2)^{p/2}} \; . $$ We recall the stability of the equation $m+(m+z)^{-1}=0$, i.e. that there exists a constant $C_\kappa\ge 1$, depending only on $\kappa$, so that for any $\delta$ \be \Big| m + \frac{1}{m+z}\Big|\leq \delta \quad \Longrightarrow \quad |m-m_{sc}(z)|\leq C_\kappa\delta\; . \label{stabb} \ee Choosing $\delta \leq c_0/10C_\kappa$ and recalling that $\mbox{Im}\; m_{sc}(z)=c_0 + o(1)$ as $N\to\infty$, we also see that \be \Big| m + \frac{1}{m+z}\Big|\leq \delta \quad \Longrightarrow \quad \mbox{Im} \; m \ge c_0/2 \label{lowb} \ee for sufficiently large $N$. We also know (e.g. from \cite{ESY}) $$ \mbox{Im} \; m(z_{n}) \ge \frac{1}{2} \mbox{Im} \; m(z_{n+1}) \; . $$ Thus, on the event $\Xi_n^c$ we have $\mbox{Im}\; m(z_{n+1})\le c_0/5$, which by \eqref{lowb} implies that $$ \Xi_n^c \subset \Lambda_{n+1} $$ assuming that $\delta \leq 4c_0/5C_\kappa$. Thus, we get $$ \P (\Lambda_n\cap \Omega^c)\leq \P(\Lambda_{n+1}\cap\Omega^c) + \frac{(Cp)^p}{(2^n\eta N\delta^2)^{p/2}} \; . $$ After summing up this inequality for $0\leq n \leq n_0$, and using the result from \cite{ESY2} on the scale $\eta_{n_0}\sim (\log N)^4/N$, we get $$ \P (\Lambda_n\cap \Omega^c)\leq\frac{(Cp)^p}{(\eta N\delta^2)^{p/2}} +e^{-c(\log N)^2} $$ for sufficiently large $N\ge N_0(\delta)$. Thus, combining this with \eqref{omegaest}, for sufficiently small $\delta$, we have $$ \P \Big( \Big| m(z_0) + \frac{1}{m(z_0)+ z_{0}}\Big|\ge \delta \Big) \leq \frac{(Cp)^p}{(\eta N\delta^2)^{p/2} }+ C\, e^{-c\sqrt{N\eta}} +e^{-c(\log N)^2} $$ Choosing $p=\min\{ 1, \; c\delta \sqrt{N\eta}\}$ with some small constant $c$ and using the stability bound \eqref{stabb}, we obtain Theorem \ref{thm:sc1} for the remaining case of $\eta\leq (\log N)^4/N$. $\Box$. \bigskip {\it Proof of Corollary \ref{cor:linfty}.} Part (i) follows from \eqref{v1} and \eqref{lon} by noticing that no $N^2$ entropy factor in \eqref{lon} is needed. In estimating $\P (\cN_{I_n} \leq \frac{1}{2} c_1 N\eta^*)$ in \eqref{lon} we infer to the semicircle law \eqref{ncont} which now holds on the $O(1/N)$ scale. Part (ii) follows from part (i) and from $$ \P (\| \bv \|_p^p \ge M^p N^{1-\frac{p}{2}} ) = \P \Big( \frac{1}{N}\sum_{j=1}^N |v_j|^p \ge \frac{M^p}{N^{p/2}} \Big) \leq \Big(\frac{N^{p/2}}{M^p}\Big)^q \E |v_1|^{pq} \leq e^{-c\sqrt{M}} \; . $$ with the choice of $q=c\sqrt{M}$ where $c=c(\kappa, p)>0$ is sufficiently small. Here we used that from part (i) we have that for any $m\ge 1$ $$ \E \, (N^{1/2}|v_1|)^m \leq M_0^m + m\int_{M_0}^\infty t^{m-1} e^{-c\sqrt{t}}\rd t \leq (Cm)^{2m} $$ where $C=C(\kappa)$. Part (iii) also follows from part (i) after summing up the estimate \eqref{efn} for all spectral intervals and for all coordinates $v_j$ of $\bv$ by using that the distribution of $v_j$ is independent of $j$. \qed \section{Proof of the tail of the gap distribution}\label{sec:tail} {\it Proof of Theorem \ref{thm:gap}.} First notice that for any $K_0(\kappa)$ it is sufficient to prove the theorem for all $K\ge K_0(\kappa)$, by adjusting the prefactor $C=C(\kappa)$ in \eqref{gapdec}. Second, it is sufficient to consider the case of sufficient large $N\ge N_0(\kappa)$. By increasing $K_0(\kappa)$ to ensure $K_0(\kappa)\ge N_0(\kappa)^2$ if necessary, we can estimate \be \P(\lambda_{\alpha+1}-E\ge K/N, \; \al\leq N-1) \leq \P(\max_\beta \lambda_\beta \ge \sqrt{K}-2) \label{gg} \ee for any $K\ge K_0$ and $N\leq N_0$. We recall part i) of Lemma 7.3 of \cite{ESY}, i.e. that there is a constant $c>0$ such that \be \P \{ \max_\beta \lambda_\beta\ge L\} \leq e^{-cL^2N} \label{extre} \ee for all $L\ge L_0$ sufficiently large (both $c$ and $L_0$ depend on the constants in \eqref{x2}). Thus the probability in \eqref{gg} can be estimated by $C\exp(-c\sqrt{K})$. Next we treat the case $K\ge CN$ with some large constant $C$. Since $\lambda_{\al+1}\ge E+ K/N$ implies $\max_\beta \lambda_\beta \ge K/N-2\ge L_0$ for a sufficiently large $C$, and using \eqref{extre}, we obtain much stronger bound of the form $\exp(-cK^2N)$ for the tail probability of $\lambda_{\al+1}$. For the rest of the proof we can thus assume that $K\leq CN$ and both $K$ and $N$ are sufficiently large, depending on $\kappa$. The event $\lambda_{\al+1}\ge E+K/N$ implies that there is a gap of size $K/N$ about $E' = E+K/2N$. Fix a sufficiently large $M$ (depending on $\kappa$) and let $z'= E'+i\eta$, with $\eta=K/(NM^2)$ and denote $$ \cN_j = \#\{ \beta\; : \; 2^{j-1}K/N\leq |\lambda_\beta-E'|\leq 2^{j}K/N\} \; , \quad j=0, 1,2, \ldots $$ On the set where $\max_\al |\lambda_\al|\leq K_0$, with some large constant $K_0$, we can estimate \be \begin{split} \mbox{Im}\; m(z') = & \frac{1}{N}\sum_{\beta=1}^{N-1} \frac{\eta}{(\lambda_\beta -E')^2 +\eta^2} \\ \leq & \frac{\eta}{N}\sum_{j=0}^{C\log N} \frac{\cN_j}{(2^{j-1} K/N)^2}\; . %\\ \leq & \; %\frac{8}{M^2}\sum_{j=0}^{C\log N} 2^{-j} \frac{\cN_j }{2^{j+1} K}\; . \end{split}\label{im} \ee Define $$ \Omega : = \max_\al\{ |\lambda_\al|\leq K_0\} \cup \bigcup_{j=0}^{C\log N} \{ \cN_j \leq 2^{j+1}KM\}, $$ with a sufficiently large $K_0$, then, similarly to the estimate \eqref{omegaest}, and together with $K\leq C N$, we get $$ \P(\Omega^c)\le Ce^{-c\sqrt{K}} \; . $$ Then, on the set $\Omega$, we have from \eqref{im} $$ \mbox{Im}\; m(z')\leq \frac{16}{M}\; . $$ For large $M$ this implies that $ |\mbox{Im}\; m(z')- \mbox{Im}\; m_{sc}(z')|\ge \frac{1}{2}\mbox{Im}\; m_{sc}(z')=:c_0>0$ and from Theorem \ref{thm:sc1} we know that $$ \P (|m(z')-m_{sc}(z')|\ge c_0) \leq e^{-c\sqrt{N\eta}} =e^{-c'\sqrt{K}}\; , $$ where the constants depend on $\kappa$. Thus, recalling that $\al$ was defined to be the index of the largest eigenvalue below $E$, we have $$ \P(\lambda_{\alpha+1}-E\ge K/N, \; \al\leq N-1)\leq \P(\Omega^c) + \P (|m(z')-m_{sc}(z')|\ge c_0) \le Ce^{-c\sqrt{K}}\; . $$ This proves Theorem \ref{thm:gap}. $\Box$ \section{Proof of the Wegner estimate}\label{sec:wegner} {\it Proof of Theorem \ref{thm:wegner}.} We can assume that $\e<1/2$, otherwise the statement is trivial. {F}rom the basic formulae \eqref{basic}, \eqref{out} and using the Schwarz inequality, we obtain that \be \begin{split} \E \; \cN_I^2 \leq &\; C(N\eta)^2\; \E \Bigg[ \im\; \frac{1}{h - z- \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\xi_\al}{\lambda_\al-z}} \Bigg]^2\\ \leq &\;C\e^2\; \E\; \Bigg[ \Big( \eta+ \frac{1}{N}\sum_{\al=1}^{N-1} \frac{\eta\xi_\al}{(\lambda_\al-E)^2+\eta^2} \Big)^2 + \Big(h - E -\frac{1}{N}\sum_{\al=1}^{N-1} \frac{(\lambda_\al-E)\xi_\al}{(\lambda_\al-E)^2+\eta^2} \Big)^2\Bigg]^{-1}\;, \end{split}\label{basic1} \ee where $h = h_{11}$ and $\lambda_\al =\lambda_\al^{(1)}$, i.e. the eigenvalues of the minor $B=B^{(1)}$ obtained from $H$ by removing the first row and column, and $\xi_\al = \xi_\al^{(1)}= |\bb \cdot \bu_\al^{(1)}|^2$ where $\bb \equiv (b_1, \dots ,b_{N-1}) := \sqrt{N}(h_{12},h_{13}, \ldots , h_{1N})$. Introducing the notation \[ d_{\alpha}: = \frac{N(\lambda_{\al} - E)}{N^2 (\lambda_{\al} - E)^2 + \e^2}, \qquad c_\al: = \frac{\e}{N^2 (\lambda_{\al} - E)^2 + \e^2}, \] we have \be \E \; \cN_I^2 \leq C\e^2\; \E\; \Bigg[ \Big( \sum_{\al=1}^{N-1} c_\al \xi_\al \Big)^2 + \Big(h - E -\sum_{\al=1}^{N-1} d_\al \xi_\al\Big)^2\Bigg]^{-1}. \label{n12} \ee Let $\gamma$ be defined so that $$ \lambda_{\gamma} - E = \min \Big\{ \lambda_\al - E \; : \; \lambda_\al - E \ge \frac{\e}{N}\Big\}, $$ i.e. $\lambda_{\gamma}$ is the first eigenvalue above $E+\e/N$. Thus $\lambda_{\gamma} \leq \lambda_{\gamma+1} \leq \lambda_{\gamma+2}\leq \lambda_{\gamma+3}$ are the first four eigenvalues above $E+\e/N$. If there are no four eigenvalues above $E+\e/N$, then we use the four consecutive eigenvalues below $E-\e/N$, as it will be clear from the proof, what matters is only that the signs of $d_{\gamma+j}$, $j=0,1,2,3$, are identical. At the end of the proof we will consider the exceptional case when there are less than four $\lambda$-eigenvalues both above $E+\e/N$ and below $E-\e/N$, i.e. in this case all but at most six eigenvalues are in $[E-\e/N, E+\e/N]$. We define then \be \Delta := N(\lambda_{\gamma+3} - E). \label{def:DELTA} \ee Note that, by definition, $$ \e \leq N(\lambda_{\gamma}-E)\leq \ldots \leq N(\lambda_{\gamma+3} - E) =\Delta\;, $$ in particular $d_{\gamma}\ge d_{\gamma+1}\ge d_{\gamma+2}\ge d_{\gamma+3}$ (since the function $x\to x/(x^2+\e^2)$ is decreasing for $x\ge \e$) and $c_{\gamma}\ge c_{\gamma+1}\ge c_{\gamma+2}\ge c_{\gamma+3}$ thus \be \min_{j=0,1,2,3} d_{\gamma+j} = \frac{\Delta}{\Delta^2+\e^2}\ge \frac{1}{2\Delta}, \qquad \min_{j=0,1,2,3} c_{\gamma+j} \ge \frac{\e}{\Delta^2}\;. \label{mind} \ee \medskip Next, we discard, in the first term in the denominator of (\ref{n12}), all contributions but the ones from $\al = \gamma,\gamma+1$. We find \begin{equation*} \begin{split} \E \; \cN_I^2\leq & \; C\e^2 \; \E \; \Bigg[ \Big( c_{\gamma}\xi_{\gamma} + c_{\gamma+1}\xi_{\gamma+1} \Big)^2 + \, \Big( h- E - \sum_{\al=1}^{N-1} d_{\alpha} \xi_\al \Big)^2\Bigg]^{-1}. \end{split} \end{equation*} Note that $c_\al$ and $d_\al$ depend on the minor $B$ and are independent of the vector $\bb$, so we can first take the expectation value with respect to $\bb$. In Lemma \ref{lm:Ebb} below we give a general estimate for such expectation values. Applying \eqref{eq:r>1} from Lemma \ref{lm:Ebb} with $r=p=2$, $\beta_1=\gamma+2$, $\beta_2=\gamma+3$, and using the estimates \eqref{mind}, we have \begin{equation*} \begin{split} \E\; \cN_I^2\leq & \; C \e \E\; \Delta^3\; . \end{split} \end{equation*} To estimate the tail probability of $\Delta$, we note that for any $K\ge \e$, the event $\Delta \ge K$ means that there must be an interval of size $(K-\e)/4N$ between $E+\e/N$ and $E+K/N$ with no $\lambda$-eigenvalue. {F}rom Theorem \ref{thm:gap} we have $$ \P(\cN^\lambda_J =0)\leq C\, e^{-c\sqrt{N|J|}} $$ for any interval $J$ with length $|J|\geq 1/N$. Thus $$ \P(\Delta\ge t)\leq C\, e^{-c\sqrt{t}}, \qquad t\ge 1. $$ Therefore $\E \; \Delta^3$ is finite and thus $\E \; \cN_I^2 \leq C\e$ is proven. The other statements in Theorem \ref{thm:wegner} are easy consequences of this estimate. Finally, we have to consider the case, when all but at most six $\lambda$-eigenvalues are within $[E-\e/N, E+\e/N]$. For all these eigenvalues $\lambda_\al$ we have $\frac{1}{2}\e^{-1}\leq c_\al\leq \e^{-1}$. If $N-1\ge 9$, then there are at least three eigenvalues in $[E-\e/N, E+\e/N]$, we denote them by $\lambda_{\gamma_1}, \lambda_{\gamma_2}$, and $\lambda_{\gamma_3}$. Then we have from \eqref{n12} and from \eqref{eq:sigma0} of Lemma \ref{lm:Ebb} below that $$ \E \; \cN_I^2 \leq \e^2 \E \Big( c_{\gamma_1}\xi_{\gamma_1} + c_{\gamma_2}\xi_{\gamma_2} + c_{\gamma_3}\xi_{\gamma_3} \Big)^{-2} \leq C\e^4\; . $$ This completes the proof for $N\ge 10$. The case $N<10$ requires a different argument. Let $f$ be a smooth cutoff function supported on $[-1,1]$, $0\leq f\leq 1$ and $f(x)\equiv 1$ for $|x|\leq 1/2$, and let $F(s)=\int_{-\infty}^s f(x) \rd x$ its antiderivative, clearly $0\leq F(s)\leq 2$. Write $$ \E \, \cN_I^2 \leq N \sum_{\al=1}^N \E^*\Big[ \E^{**} f\Big( \frac{\mu_\al-E}{\e/N}\Big)\Big]\; , $$ where $\E^*$ is the expectation with respect to the off-diagonal matrix elements and $\E^{**}$ is the expectation with respect to the diagonal elements $x_{ii}$, $i=1,2,\ldots N$. Since $N$ is bounded, it is sufficient to show that the expectation inside the square bracket is bounded by $C\e$. Let $\bx = (x_{11}, x_{22}, \ldots, x_{NN})$ and viewing $\mu_\al$ as a function of $\bx$, we have \be \nabla_\bx\Big[ F\Big( \frac{\mu_\al-E}{\e/N}\Big)\Big] = N\e^{-1} f\Big( \frac{\mu_\al-E}{\e/N}\Big)\nabla_\bx \mu_\al \; . \label{dere} \ee Simple first order perturbation shows that $$ \frac{\partial \mu_\al}{\partial x_{ii}} = \frac{2}{\sqrt N} |\bv_\al(i)|^2 \; $$ where $\bv_\al$ is the eigenvector of $H$ belonging to $\mu_\al$. Notice that the components of the gradient in \eqref{dere} are nonnegative and their sum is $2/\sqrt{N}$. Thus, summing up each component of \eqref{dere}, we get \be \begin{split} \E^{**} f\Big( \frac{\mu_\al-E}{\e}\Big) & = \frac{\e}{2\sqrt{N}} \sum_{i=1}^N \int_{\bR^N} \Big[\prod_{j=1}^N \rd \wt\nu(x_{jj})\Big] \frac{\partial}{\partial x_{ii}} \Big[ F\Big( \frac{\mu_\al-E}{\e/N}\Big)\Big] \\ & = (\mbox{const.})\frac{\e \sqrt{N}}{2} \int_\bR \rd x_{11}\; e^{-\wt g(x_{11})} \frac{\partial}{\partial x_{11}} \Bigg[ \int_{\bR^{N-1}} F\Big( \frac{\mu_\al-E}{\e/N}\Big) \prod_{j=2}^N \rd \wt\nu(x_{jj}) \Bigg] \\ &\leq C\e\sqrt{N}. \end{split} \ee In the last step we used integration by parts, the boundedness of $F$, the fact that $\rd \wt\nu$ is a probability measure and that $\int_\bR |\wt g'(x)|\exp (-\wt g(x))\rd x$ is finite. Thus we obtained the Wegner estimate for the small values of $N$ as well. \qed \bigskip The proof actually shows the following stronger result that will be needed in Section \ref{sec:level}. As before, let $\mu$'s be the eigenvalues of an $N\times N$ Wigner matrix, and let $\gamma = \gamma(N)$ defined as $$ \mu_{\gamma} - E = \min \Big\{ \mu_\al - E \; : \; \mu_\al - E \ge \frac{\e}{N}\Big\}\; . $$ For any positive integer $d$, let \be \Delta_{d}^{(\mu)} = N(\mu_{\gamma(N)+d-1}-E) \label{def:deltamu} \ee i.e. the rescaled distance from $E$ to the $d$-th $\mu$-eigenvalue above $E+\e/N$. If there are no $d$ $\mu$-eigenvalues above $E+\e/N$, then we use the eigenvalues below $E-\e/N$ to define $\gamma =\gamma(N)$ as $$ \mu_{\gamma} - E = \max \Big\{ \mu_\al - E \; : \; \mu_\al - E \le -\frac{\e}{N}\Big\} $$ and \be \Delta_{d}^{(\mu)} = N(E- \mu_{\gamma(N)-d+1}) \; . \label{def:deltamu1} \ee To unify the notation, let us introduce the symbol \be \Delta_d^{(\mu)} =\infty \label{def:deltamu2} \ee for the extreme case, when there are at most $d-1$ eigenvalues above $E+\e/N$ and at most $d-1$ eigenvalues below $E-\e/N$; in particular in this case all but at most $2d-2$ eigenvalues are between $E-\e/N$ and $E+\e/N$. \begin{corollary}\label{cor:weg} With the notation above, for any $d\ge 5$, $N\ge 10$ and $M\in \N$ there is a constant $C=C_{M,d}$ such that \be \E \Big[ {\bf 1}(\cN_I\ge 1) \cdot \big[\Delta_d^{(\mu)}\big]^M\cdot {\bf 1}( \Delta_{d}^{(\mu)} <\infty) \Big] \leq C\,\e \; . \label{genweg} \ee \end{corollary} {\it Proof.} We proceed as in the proof of Theorem \ref{thm:wegner} above. By ${\bf 1}(\cN_I\ge 1)\leq\cN_I^2$ and following the estimates \eqref{basic1}--\eqref{n12}, we have $$ \E \Big[ {\bf 1}(\cN_I\ge 1) \cdot \big[\Delta_d^{(\mu)}\big]^M\cdot {\bf 1}( \Delta_{d}^{(\mu)} <\infty) \Big]\leq C\e^2\; \E\; \frac{\big[\Delta_d^{(\mu)}\big]^M\cdot {\bf 1}(\Delta_{d}^{(\mu)} <\infty)}{ \Big( \sum_{\al=1}^{N-1} c_\al \xi_\al \Big)^2 + \Big(h - E -\sum_{\al=1}^{N-1} d_\al \xi_\al\Big)^2}\;. $$ With the notation \eqref{def:deltamu}, the $\Delta$ in \eqref{def:DELTA} is actually $\Delta = \Delta_4^{(\lambda)}$, where the superscript indicates that it is defined in the $\lambda$-spectrum. By the interlacing property and by $d\ge 5$, we have $$ \Delta = \Delta_4^{(\lambda)}\leq \Delta_d^{(\mu)} \leq \Delta_{d+1}^{(\lambda)} , $$ thus $$ \E \Big[ {\bf 1}(\cN_I\ge 1) \cdot \big[\Delta_d^{(\mu)}\big]^M\cdot {\bf 1}( \Delta_{d}^{(\mu)} <\infty) \Big] \leq \e^2\; \E\; \frac{\big[\Delta_{d+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{d+1}^{(\lambda)} <\infty)}{ \Big( \sum_{\al=1}^{N-1} c_\al \xi_\al \Big)^2 + \Big(h - E -\sum_{\al=1}^{N-1} d_\al \xi_\al\Big)^2} \; . $$ Now we perform the expectation with respect to the $\bb$ variables as before; the numerator is independent of $\bb$. We get $$ \E \Big[ {\bf 1}(\cN_I\ge 1) \cdot \big[\Delta_d^{(\mu)}\big]^M\cdot {\bf 1}( \Delta_{d}^{(\mu)} <\infty) \Big] \leq C\e \; \E \big[\Delta^{(\lambda)}_4\big]^3 \big[\Delta_{d+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{d+1}^{(\lambda)} <\infty) \leq C\e $$ since the tail distribution of any $\Delta_{d}^{(\lambda)}$ decays faster than any polynomial. \qed \begin{lemma}\label{lm:Ebb} Fix $p \in \N / \{ 0 \}$ and let $N\ge p+3$. Let $\bu_1, \bu_2, \ldots , \bu_{N-1}$ be an arbitrary orthonormal basis in $\bC^{N-1}$ and set $\xi_\al = |\bb\cdot \bu_\al|^2$, where the components of $\bb$ are i.i.d complex variables with distribution $\nu$ with density $h$ satisfying the condition {\bf C2)} with an exponent $a=p+3$ in \eqref{charfn}. Fix different indices $\al_1, \dots ,\al_p, \beta_1, \beta_2 \in \{ 1, 2, \dots ,N-1 \}$. Assume that $c_j > 0$, for $j =1 ,\dots ,p$. Let $d_{\alpha} \in \bR$ for all $1 \leq \al \leq N-1$ be arbitrary numbers such that $d_{\beta_1}, d_{\beta_2} >0$. Then, for every $1 < r < p+1$, there exists a constant $C_{r,p} < \infty$ such that \begin{equation}\label{eq:r>1} \E_{\bb} \, \left[ \Big( \sum_{j=1}^p c_j \xi_{\al_j} \Big)^2 + \Big(E - \sum_{\al=1}^{N-1} d_{\al} \xi_\al\Big)^2 \right]^{-\frac{r}{2}} \leq \frac{C_{p,r}}{ (\prod_{j=1}^p c_j)^{\frac{r-1}{p}} \, \min (d_{\beta_1}, d_{\beta_2})} \,. \end{equation} Moreover, for every $p \geq 3$, we also have the improved bound \begin{equation}\label{eq:impr} \E_{\bb} \, \left[ \Big( \sum_{j=1}^p c_j \xi_{\al_j} \Big)^2 + \Big(E - \sum_{\al=1}^{N-1} d_{\al} \xi_\al\Big)^2 \right]^{-\frac{p}{2}} \leq \frac{C_{p}}{ (\prod_{j=1}^{p-2} c_j) \, \min (c_{p-1}, c_p) \, \min (d_{\beta_1}, d_{\beta_2})} \,. \end{equation} for a constant $C_p$ depending only on $p$. %For $p\ge2$, the constant in this inequality can be chosen %$$ % C_{p,r} =\frac{C^p}{(p!)^2(p-r+1)\om_{p+3}^{p+2}}. %$$ %For the case $r=1$ and $p\ge 2$ we find %\begin{equation}\label{eq:r1} %\sup_E \E_{\bb} \, \left[ \Big( \sum_{j=1}^p c_j \xi_{\al_j} % \Big)^2 + (E - \sum_{\al=1}^{N-1} d_{\al} \xi_\al)^2 \right]^{-\frac{1}{2}} %\leq 1 + C\; \frac{1+|\log \min c_j|} %{ \min (d_{\beta_1}, d_{\beta_2})} %\end{equation} \medskip Without the second term in the denominator, we have the following estimates: For all $1\leq r < p$, there exists a constant $C_{p,r} < \infty$ such that \begin{equation}\label{eq:sigma0} \E_{\bb} \left[\; \sum_{j=1}^p c_j \xi_{\al_j}\right]^{-r} \leq \frac{C_{p,r}}{(\min c_j)^{r}}\; . \end{equation} %The constant in this inequality can be chosen %$$ % C_{p,r} =\frac{C^p}{(p!)^2(p-r)\om_{p+1}^p}. %$$ %For $r = p-1$ (if $p \geq 2$) we also have, with %$c_1 = \min c_j$, %\begin{equation}\label{eq:sigma0log} %\E_{\bb} \left[\;\sum_{j=1}^p c_j \xi_{\al_j}\right]^{-(p-1)} \leq % \frac{C_p \, |\log \, c_1|}{(\min_{j=2,\dots ,p} c_j)^{p-1}} %\end{equation} %for an appropriate constant $C_p$. \end{lemma} %[WE can remove \eqref{eq:r1} and \eqref{eq:sigma0log}, %the other two are the clean statements] {\it Remark.} For (\ref{eq:sigma0}), it is enough to assume that \begin{equation} |\wh{h} (t,s)| \leq \frac{1}{(1 + \om_{p+1}( t^2+s^2))^{p+1}} \end{equation} instead of both conditions in \eqref{charfn} with exponent $a=p+3$. \bigskip {\it Proof.} To prove (\ref{eq:r>1}), we perform a change variables from $\bb =(b_1, \ldots , b_{N-1})$ to $\bz =(z_1, \ldots z_{N-1})$ by introducing $$ \bz = U^* \bb $$ where $U$ is the unitary matrix with columns $(\bu_1, \ldots, \bu_{N-1})$. Notice that the Jacobian is one, thus \be \begin{split} \text{I} := \; & \E_{\bb} \, \left[ \left( \sum_{j=1}^p c_j \xi_{\al_j} \right)^2 + \Big(E - \sum_{\al=1}^{N-1} d_{\al} \xi_\al\Big)^2 \right]^{-r/2} = \int \frac{\rd\mu(\bz)}{[ P(\bz)]^{r/2}} \end{split}\label{eq:bb} \ee with $$ \rd\mu(\bz) : = e^{-\Phi(\bz)}\prod_{\al=1}^{N-1} \rd z_\al \rd \overline{z}_\al, \qquad \Phi(\bz):= \sum_{\ell=1}^{N-1} g\left( \re \, (U\bz)_{\ell}, \; \im \, (U\bz)_\ell \right) $$ and $$ P(\bz) : = \Big( \sum_{j=1}^p c_{j} |z_{\al_j}|^2 \Big)^2 + \Big(E - \sum_{\al=1}^{N-1} d_{\al} |z_\al|^2 \Big)^2. $$ We define, for $t \in \bR$, \be \label{def:F} F (t) := \int_{-\infty}^t \rd s \; \left( \Big( \sum_{j=1}^p c_j |z_{\al_j}|^2 \Big)^2 + s^2 \right)^{-r/2} \;. \ee Note that, for every $r >1$, there exists a constant $C_r <\infty$, such that \be\label{eq:Fbd} 0 \leq F (t) \leq \, \frac{C_r}{\left( \sum_{j=1}^p c_{j} |z_{\al_j}|^2 \right)^{r-1}} \ee for every $t \in \bR$. For $j=1,2$, we have \begin{equation*} \begin{split} z_{\beta_j} \frac{\rd}{\rd z_{\beta_j}} \; F \left( E- \sum_{\al=1}^{N-1} d_\al |z_{\al}|^2 \right) = - \frac{ d_{\beta_j} |z_{\beta_j}|^2 }{ [P(\bz)]^{r/2}}\; . \end{split} \end{equation*} Introducing the first order differential operator $$ D : = z_{\beta_1} \frac{\rd}{\rd z_{\beta_1}} + z_{\beta_2} \frac{\rd}{\rd z_{\beta_2}} \;, $$ we find \begin{equation}\label{eq:der1} D \left[ F \Big(E- \sum_{\al=1}^{N-1} d_\al |z_{\al}|^2\Big) \right] = - \frac{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2}{ [P(\bz)]^{r/2}}\; . \end{equation} {F}rom (\ref{eq:bb}), we get \begin{equation} \text{I} = \, - \int \, \rd\mu(\bz) \frac{1}{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2} \; D \left[ F \Big(E- \sum_{\al=1}^{N-1} d_\al |z_{\al}|^2\Big) \right] \,. \end{equation} Integrating by parts and using the fact that $$ \frac{\rd}{\rd z_{\beta_1}} \; \frac{z_{\beta_1}}{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2} + \frac{\rd}{\rd z_{\beta_2}} \;\frac{z_{\beta_2}}{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2} = \frac{1}{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2} , $$ we find \begin{equation}\label{eq:bb2} \begin{split} \text{I} = \; & \int \, \rd\mu(\bz) \frac{F \left( E- \sum_{\al=1}^{N-1} d_\al |z_{\al}|^2 \right)}{d_{\beta_1} |z_{\beta_1}|^2 + d_{\beta_2} |z_{\beta_2}|^2} \Big(1- D \Phi(\bz)\Big). \end{split} \end{equation} Clearly $$ |D\Phi(\bz)|^2\leq (|z_{\beta_1}|^2+ |z_{\beta_2}|^2) \Bigg( \Big|\frac{\partial \Phi(\bz)}{\partial z_{\beta_1}}\Big|^2+ \Big|\frac{\partial \Phi(\bz)}{\partial z_{\beta_2}}\Big|^2\Bigg). $$ By a Schwarz inequality in (\ref{eq:bb2}) and using (\ref{eq:Fbd}), we have \be\label{eq:A+B} \text{I} \leq C_r \;\frac{ A + B_1 +B_2}{ \min(d_{\beta_1}, d_{\beta_2})}, \ee where \be\label{eq:A,B} \begin{split} A:= & \int \rd\mu(\bz) \frac{1}{\left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{r-1}} \frac{1}{ |z_{\beta_1}|^2 + |z_{\beta_2}|^2} \\ B_k:= & \int \rd\mu(\bz) \frac{1}{\left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{r-1}} \Big|\frac{\partial \Phi(\bz)}{\partial z_{\beta_k}}\Big|^2. \end{split} \ee The integral $\text{A}$ can be bounded as follows \begin{equation}\label{eq:sumA} \text{A} \leq A_1 + A_2 + A_3 \ee with \be \begin{split} A_1:= \; & \int \rd\mu(\bz) \frac{{\bf 1}\big(\sum_{j=1}^p c_j |z_{\al_j}|^2\leq \kappa \big) } {\left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{r-1}} \\ A_2:= \; & \frac{1}{\kappa^{r-1}}\int \rd\mu(\bz) \frac{1}{ |z_{\beta_1}|^2 + |z_{\beta_2}|^2}\\ A_3:= \; & \int \rd\mu(\bz) \frac{{\bf 1}(|z_{\beta_1}|^2 + |z_{\beta_2}|^2\leq 1) \cdot {\bf 1} \left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \leq \kappa \right)} {\left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{r-1} (|z_{\beta_1}|^2 + |z_{\beta_2}|^2)} \end{split} \end{equation} for any $\kappa>0$. We start with the estimate of $A_3$. Decompose $z_{\al_j} = x_j + iy_j$ and $z_{\beta_j}= x_{p+j}+ iy_{p+j}$ into real and imaginary parts. We define the function \begin{equation}\label{eq:f} f(x_1, \ldots x_{p+2}, y_1, \ldots , y_{p+2}) : = \frac{ {\bf 1}\left(\sum_{j=1}^p c_j (x_j^2 + y_j^2) \leq \kappa\right)\cdot {\bf 1}\left(\sum_{j=p+1}^{p+2} (x_j^2 + y_j^2)\leq 1\right)} {\Big[\sum_{j=1}^p c_j (x_j^2 + y_j^2)\Big]^{r-1} \sum_{j=p+1}^{p+2} (x_j^2 + y_j^2)} \end{equation} on $\bR^{2p+4}$. Changing variables $c^{1/2}_j x_j \to x_j$, $c_j^{1/2} y_j \to y_j$ and using that $r < p+1$, it is simple to check that $$ \| f \|_1 \leq \frac{C_{r,p} \, \kappa^{p+1-r}}{\prod_{j=1}^p c_j}\, . $$ Thus, recalling that $z_{\al}= (U^*\bb)_\al$ and since the indices $\al_1, \ldots, \al_p$, $\beta_1, \beta_2$ are all distinct, we find, by taking the Fourier transformation in the $x_1, \ldots x_{p+2}$, $ y_1, \ldots , y_{p+2}$ variables, that \be \begin{split}\label{a3est} A_3 \leq & \; \|\wh{f} \|_{\infty} \int_{\bR^{2p+4}} \prod_{j=1}^{p+2} \rd t_j \rd s_j \; \Big| \E_\bb e^{-i\sum_{j=1}^p[ t_j \re (U^*\bb)_{\al_j} +s_j \im (U^*\bb)_{\al_j}]-i\sum_{j=p+1}^{p+2}[ t_j \re (U^*\bb)_{\beta_j} +s_j \im (U^*\bb)_{\beta_j}] } \Big| \\ \leq & \; \|f\|_1 \int_{\bR^{2p+4}} \prod_{j=1}^{p+2} \rd t_j \rd s_j \Big| \E_\bb e^{- i [\re (U\bt')+\im (U\bs')]\cdot \re \bb - i[\re (U\bt')-\im (U\bs')]\cdot \im \bb }\Big| \\ \leq &\;\|f\|_1 \int_{\bR^{2p+4}} \prod_{j=1}^{p+2} \rd t_j \rd s_j \frac{1}{ \left( 1+ \om_{p+3} \|U\bt'\|^2 + \om_{p+3}\|U\bs'\|^2\right)^{p+3}} \\ \leq &\;\|f\|_1 \int_{\bR^{2p+4}} \prod_{j=1}^{p+2} \rd t_j \rd s_j \frac{1}{ \left( 1+ \om_{p+3} \|\bt\|^2 + \om_{p+3}\|\bs\|^2\right)^{p+3}} \\ \leq \; &\frac{C_{r,p} \, \kappa^{p+1-r}}{\prod_{j=1}^p c_j} \end{split} \ee for an appropriate constant $C_p$. Here the components $t'_j$ of the vector $\bt'\in \bR^{N-1}$ are defined to be all zero except $t'_{\al_j}: = t_j$, $t'_{\beta_{p+1}}: = t_{p+1}$, $t'_{\beta_{p+2}}: = t_{p+2}$; the vector $\bs'$ is defined similarly. In the last but one step we used the bound \eqref{charfn} with exponent $p+3$ for the Fourier transform of the distribution of $\bb$. In the last step, we used that for the Euclidean norm $\|U\bt'\|=\| \bt'\| = \| \bt \|$ with $\bt = (t_1, \ldots t_{p+2})$ and similarly $\|U\bs'\|= \| \bs \|$ with $\bs = (s_1, \ldots, s_{p+2})$. Using similar arguments to bound the terms $A_1$ and $A_2$, we conclude that $$ A \leq C_{r,p} \left( \frac{1}{\kappa^{r-1}} + \frac{\kappa^{p+1-r}}{\prod_{j=1}^p c_j} \right) $$ for arbitrary $\kappa >0$. Optimizing over $\kappa$, we find \be \label{Abound} A \leq \frac{C_{r,p}}{\left( \prod_{j=1}^p c_j \right)^{\frac{r-1}{p}}} \, . \ee \medskip To control the integrals $\text{B}_k$, $k=1,2$ in \eqref{eq:A+B}, we integrate by parts and we use that $\beta_k\neq \al_j$: $$ B_k = -\int \prod_{\al=1}^{N-1} \rd z_\al \rd \bar z_\al \; \left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{-(r-1)} \frac{\partial \Phi(\bz)}{\partial z_{\beta_k}} \; \frac{\partial e^{-\Phi(\bz)}}{\partial \bar z_{\beta_k}} = \int \rd\mu(\bz) \left( \sum_{j=1}^p c_j |z_{\al_j}|^2 \right)^{-(r-1)} \frac{\partial^2\Phi(\bz)}{\partial z_{\beta_k}\partial\bar z_{\beta_k}}. $$ Simple calculation shows that $$ \frac{\partial^2\Phi(\bz)}{\partial z_{\beta}\partial\bar z_{\beta}} = \frac{1}{4} \sum_{\ell=1}^{N-1} |\bu_{\beta}(\ell)|^2 \Delta g\left( \re \, (U\bz)_{\ell}, \; \im \, (U\bz)_\ell \right) $$ thus \be B_k = \frac{1}{4} \sum_\ell | \bu_{\beta_k}(\ell) |^2 \; \E_\bb \Bigg[ \left( \sum_{j=1}^p c_j |(U^*\bb)_{\al_j}|^2 \right)^{-(r-1)} \Delta g(\re \, b_\ell, \im\, b_\ell) \Bigg]\; . \label{bk} \ee For each fixed $\ell$, the estimate of the expectation value is identical to that of $A_1$ if the density function $e^{-g}$ for $b_\ell$ is replaced with $e^{-g}\Delta g$ (and all other $b_m$, $m \neq \ell$, are still distributed according to $e^{-g}$). Although $e^{-g}\Delta g$ is not a probability density, it is only the decay of its Fourier transform that is relevant to proceed similarly to the estimate \eqref{a3est}. Having obtained uniform bound on the expectation in \eqref{bk}, we can perform the summation over $\ell$ and we obtain $$ B_k \leq \frac{C_{p,r}}{\left( \prod_{j=1}^p c_j \right)^{\frac{r-1}{p}}} \, . $$ Combining this with \eqref{Abound} and \eqref{eq:A+B}, we have proved \eqref{eq:r>1}. \medskip To prove (\ref{eq:impr}), we proceed as before up to (\ref{eq:A+B}). This time, however, we bound the term $\text{A}$ in (\ref{eq:A,B}), with $r = p$, by $$ A \leq A_4 + A_5 + A_6 + A_7, $$ where \begin{equation} \begin{split} A_4 &= \int \rd \mu(\bz) \frac{{\bf 1} \left( |z_{\al_{p-1}}|^2 + |z_{\al_p}|^2 \leq 1 \right)}{\left( \sum_{j=1}^{p-2} c_j |z_{\al_j}|^2 + \wt c \left( |z_{\al_{p-1}}|^2 + |z_{\al_p}|^2 \right) \right)^{p-1}} \\ A_5 &= \int \rd \mu(\bz) \frac{1}{\left( \sum_{j=1}^{p-2} c_j |z_{\al_j}|^2 + \wt c \right)^{p-1}} \\ A_6 &= \int \rd \mu(\bz) \frac{ {\bf 1} ( |z_{\beta_1}|^2 + |z_{\beta_2}|^2 \leq 1)}{\left( \sum_{j=1}^{p-2} c_j |z_{\al_j}|^2 + \wt c \right)^{p-1}} \frac{1}{ |z_{\beta_1}|^2 + |z_{\beta_2}|^2} \\ A_7 &= \int \rd\mu(\bz) \frac{{\bf 1} \left( |z_{\al_{p-1}}|^2 + |z_{\al_p}|^2 \leq 1 \right) {\bf 1} ( |z_{\beta_1}|^2 + |z_{\beta_2}|^2 \leq 1)}{\left( \sum_{j=1}^{p-2} c_j |z_{\al_j}|^2 + \wt c \left( |z_{\al_{p-1}}|^2 + |z_{\al_p}|^2 \right) \right)^{p-1}} \frac{1}{ |z_{\beta_1}|^2 + |z_{\beta_2}|^2} \end{split} \end{equation} with $\wt c = \min (c_{p-1}, c_p)$. We consider first the term $A_7$. We decompose $z_{\al_j} = x_j + iy_j$ and $z_{\beta_j}= x_{p+j}+ iy_{p+j}$ into real and imaginary parts. We define the function \begin{equation}\label{eq:f1} f(x_1, \ldots x_{p+2}, y_1, \ldots , y_{p+2}) : = \frac{{\bf 1} \left( \sum_{j=p-1}^p x_j^2 + y_j^2 \leq 1 \right) {\bf 1} \left( \sum_{j=p+1}^{p+2} (x_j^2 + y_j^2) \leq 1\right)} {\Big[ \sum_{j=1}^{p-2} c_j (x_j^2 + y_j^2) + \wt c \sum_{j=p-1}^p (x_j^2 + y_j^2) \Big]^{p-1} \sum_{j=p+1}^{p+2} (x_j^2 + y_j^2)} \end{equation} on $\bR^{2p+4}$. Changing variables $c^{1/2}_j x_j \to x_j$, $c^{1/2} y_j \to y_j$ for $j=1, \dots ,p-2$, and then letting $r = \sum_{j=1}^{p-2} (x_j^2 + y_j^2)$ and $w = \sum_{j=p-1}^p (x_j^2 + y_j^2)$, we find that \begin{equation} \begin{split} \| f \|_1 \leq \frac{C_p}{\prod_{j=1}^{p-2} c_j} \int_0^1 \rd w \, w \int_0^{\infty} \rd r \frac{r^{p-3}}{(r + \wt c \, w)^{p-1}} %\leq \frac{C_p}{\prod_{j=1}^{p-2} c_j} % \int_0^1 \rd w \, w \int_0^{\infty} \frac{\rd r}{(r+\wt c \, w)^2} \leq \leq \frac{C_p}{\wt c \, \prod_{j=1}^{p-2} c_j } \end{split} \end{equation} for an appropriate constant $C_p$. Proceeding as in (\ref{a3est}), we conclude that $$ A_7 \leq \frac{C_p}{\wt c \, \prod_{j=1}^{p-2} c_j}. $$ The terms $A_4, A_5,A_6$ can be controlled similarly. Hence $$ A \leq \frac{C_p}{\wt c\, \prod_{j=1}^{p-2} c_j} \;. $$ The bound for $B$ in (\ref{eq:A,B}) can be obtained analogously as in the proof of (\ref{eq:r>1}) (with the same modifications used for the term $A$). The proof of (\ref{eq:sigma0}) is similar (but much simpler). \qed \section{Proof of the level repulsion}\label{sec:level} {\it Proof of Theorem \ref{thm:repul}.} We can assume that $\e<1/2$ and that $k\ge 2$, the $k=1$ case was proven in Theorem \ref{thm:wegner}. We recall the notation $\Delta^{(\mu)}_d$ from \eqref{def:deltamu}--\eqref{def:deltamu2} and we split $$ \P (\cN_\eta\ge k) \leq (I) + (II) $$ with \be \begin{split} (I) := & \P (\cN_\eta\ge k, \Delta^{(\mu)}_d=\infty) \\ (II) := &\P (\cN_\eta\ge k, \Delta^{(\mu)}_d<\infty) \; \end{split} \ee for some positive integer $d$. {F}rom the basic formula \eqref{basic} we have \begin{equation} \begin{split} \cN_\eta %\eta \, \sum_{j=1}^N \im \, %\frac{1}{h_{jj} - E - i\eta - \frac{1}{N} \sum_{\al} % \frac{\xi^{(j)}_\al}{\lambda_\al^{(j)} - e - i \eta}} \\ \leq \; &\frac{C\e}{N} \sum_{j=1}^N \left[ \left( \eta + \frac{\eta}{N} \sum_{\al=1}^{N-1} \frac{\xi^{(j)}_\al}{(\lambda_\al^{(j)} - E)^2 + \eta^2} \right)^2 + \left( E -h_{jj} + \frac{1}{N} \sum_{\al=1}^{N-1} \frac{(\lambda_\al^{(j)} - E) \, \xi^{(j)}_\al}{(\lambda_\al^{(j)} -E)^2 + \eta^2}\right)^2 \right]^{-1/2} \; . \end{split} \end{equation} We introduce the notations $\xi_\al = \xi^{(1)}_\al$, $\lambda_\al = \lambda_{\al}^{(1)}$, $h = h_{11}$, and $$ c_\al = \frac{\e}{N^2 (\lambda_\al - E)^2 + \e^2}, \qquad d_\al = \frac{N(\lambda_{\al} - E)}{N^2 (\lambda_\al - E)^2 + \e^2} $$ as before. Using a moment inequality, we get $$ (I) \leq C_k \e^{k^2} \, \E \; \frac{{\bf 1}(\Delta_{d}^{(\mu)} =\infty)}{ \left( \sum_{\al=1}^{N-1} c_\al \, \xi_\al \right)^{k^2}} \; . $$ This term represents the extreme case, when all but at most $2d-2$ eigenvalues are in $[E-\e/N, E+\e/N]$. Choosing $d=2k$ and assuming that $N\ge k^2+4k$, we see that for at least $k^2+1$ different $\al$-indices we have $\lambda_\al\in[E-\e/N, E+\e/N]$, i.e. $\frac{1}{2}\e^{-1}\leq c_\al\leq \e^{-1}$. Using \eqref{eq:sigma0} with $r=k^2$, $p=k^2+1$, we get \be (I) \leq C_k \e^{2k^2} \; . \label{eq:I} \ee Now we turn to the estimate of (II) and we will consider the following somewhat more general quantity: $$ I_N(M, k,\ell): = \E \Big[{\bf 1} (\cN^{\mu}_\eta \geq k) \cdot \big[ \Delta_{\ell}^{(\mu)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+4}^{(\mu)} <\infty)\Big] $$ for any $M\in\N$ and $4\leq\ell \leq k+2$. The index $N$ refers to the fact that the $\mu$'s are the eigenvalues of an $N\times N$ Wigner matrix. The superscript $\mu$ in $\cN^{\mu}_\eta$ indicates that it counts the number of $\mu$-eigenvalues. Since by definition $ \Delta_{\ell}^{(\mu)}\ge1$, we know that $I_N(M, k, \ell)$ is monotone increasing in $M$. Moreover, with the choice $M=0$, $\ell=4$ we have \be (II)\leq I_N(0, k, 4) \; . \label{III} \ee Since the existence of $k$ $\mu$-eigenvalues in the interval $I_{\eta}$ implies that $\cN_\eta^{(j)} \geq k-1$ for all $j=1, \dots ,N$ (where $\cN_{\eta}^{(j)}$ denotes the number of eigenvalues $\lambda^{(j)}_{\al} \in I_\eta$), we obtain that \begin{equation} \begin{split} I_N(M,k,\ell) \leq \; & C_k \, \eps^{k+1} \, \E\; \frac{{\bf 1} (\cN^{(1)}_\eta \geq k-1) \cdot \big[ \Delta_{\ell}^{(\mu)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+4}^{(\mu)} <\infty)}{ \left[ \left( \sum_{\al=1}^{N-1} c_\al \, \xi_\al \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{(k+1)/2}} \;. \end{split} \end{equation} By the interlacing property we have $\Delta_{2k-\ell+3}^{(\lambda)} \leq \Delta_{2k-\ell+4}^{(\mu)}$ and $ \Delta_{\ell}^{(\mu)} \leq \Delta_{\ell+1}^{(\lambda)}$, thus we have \begin{equation} \begin{split} I_N(M,k,\ell) \leq \; & C_k \, \eps^{k+1} \, \E\; \frac{{\bf 1} (\cN^{(1)}_\eta \geq k-1) \cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)}{ \left[ \left( \sum_{\al=1}^{N-1} c_\al \, \xi_\al \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{(k+1)/2}}\; . \end{split} \end{equation} We split this quantity into two terms: $$ I_N(M,k,\ell) \leq\left(\text{A} + \text{B} \right) $$ with \begin{equation}\label{eq:A,Brep} \begin{split} \text{A} &: = C_k \, \eps^{k+1} \, \E \frac{{\bf 1} (\cN^{(1)}_\eta \geq k+2) \cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)}{ \left[ \left( \sum_{\al=1}^{N-1} c_\al \xi_\al \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{(k+1)/2}} \\ \text{B} &:= C_k \eps^{k+1} \, \E \frac{{\bf 1} (k-1 \leq \cN^{(1)}_\eta < k+2) \cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)}{ \left[ \left( \sum_{\al=1}^{N-1} c_\al \xi_\al \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{(k+1)/2}} \, . \end{split} \end{equation} To control the first term, we denote by $\lambda_{\al_1}, \dots ,\lambda_{\al_{k+2}}$ the first $k+2$ $\lambda$-eigenvalues in the set $I_{\eta}$. Then $c_{\al_j} \geq \frac{1}{2}\e^{-1}$, for all $j=1, \dots ,k+2$ and therefore, by (\ref{eq:sigma0}), \begin{equation} \label{eq:Ares} \text{A} \leq C_k \, \eps^{k+1} \E \frac{{\bf 1} (\cN^{(1)}_\eta \geq k+2) \cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)}{ \left( \e^{-1} \sum_{j=1}^{k+2} \xi_{\al_j} \right)^{k+1}} \leq C_k \, \e^{2k+2} I_{N-1}(M, k-1, \ell+1)\; \end{equation} using that $\Delta^{(\lambda)}_m$ is monotone increasing in $m$. To control the term $\text{B}$ in (\ref{eq:A,Brep}), we choose the indices $\al_1, \dots , \al_{k-1}$ so that $\lambda_{\al_j} \in I_{\eta}$ for all $j=1, \dots ,k-1$. Since we know that there are at most $k+1$ eigenvalues in $I_{\eta}$, there must be, either on the right or on the left of $E$, $\lambda$-eigenvalues at distances larger than $\e/N$ from $E$ if $N\ge k+8$. Let us suppose, for example, that there are four such eigenvalues on the right of $E$. Then, we define the index $\al_k$ so that $$ \lambda_{\al_k} - E = \min \Big \{ \lambda_\al - E \; : \; \lambda_\al - E > \frac{\e}{N} \Big\} $$ i.e. $\lambda_{\al_k}$ is the first eigenvalue above $E+\e/N$. Moreover, let $\al_{k+1} = \al_k + 1$, $\beta_1 = \al_k + 2$ and $\beta_2 = \al_{k+1}+3$. Recalling the notation \eqref{def:deltamu}, we set $\Delta:=\Delta^{(\lambda)}_4 = N(\lambda_{\beta_2} - E)$. By definition $$ \e\le N(\lambda_{\al_{k}} -E) \leq N (\lambda_{\al_{k+1}} - E) \leq N (\lambda_{\beta_1} - E) \leq N (\lambda_{\beta_2} - E) = \Delta. $$ and $\min (d_{\beta_1}, d_{\beta_2})\ge \frac{1}{2\Delta}$. Therefore \begin{equation}\label{eq:node} \begin{split} \text{B} \leq C_k \, \eps^{k+1} \, \E \;\frac{{\bf 1} \left(\cN^{(1)}_\eta \geq k-1\right)\cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)}{ \left[ \left(\sum_{j=1}^{k-1} \e^{-1} \xi_{\al_j} + \frac{\e}{\Delta^2} (\xi_{\al_k} + \xi_{\al_{k+1}}) \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{(k+1)/2}} \; . \end{split} \end{equation} {F}rom (\ref{eq:node}), we find \begin{equation} \begin{split} \text{B} \leq \; & C_k \e^{k+1}\;\E_{\lambda,h} \Bigg\{ \Big[ {\bf 1} \left(\cN^{(1)}_\eta \geq k-1\right)\cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)\Big] \\ & \times \E_{\bb} \left[ \left(\sum_{j=1}^{k-1} \e^{-1} \xi_{\al_j} + \e \Delta^{-2} (\xi_{\al_k} + \xi_{\al_{k+1}}) \right)^2 + \left( E -h + \sum_{\al=1}^{N-1} d_\al \, \xi_\al \right)^2 \right]^{-\frac{k+1}{2}} \Bigg\}\; . \end{split} \end{equation} Using (\ref{eq:impr}) from Lemma \ref{lm:Ebb} (with $p=k+1$, $c_j =\e^{-1}$ for all $j=1, \dots, k-1$, $c_k = c_{k+1} = \e \Delta^{-2}$), it follows that \begin{equation} \begin{split} \text{B} \leq & \; C_k \, \e^{2k-1}\, \E_{\lambda} \Big[ {\bf 1} \left(\cN^{(1)}_\eta \geq k-1\right) \big[\Delta_4^{(\lambda)}\big]^3\cdot \big[ \Delta_{\ell+1}^{(\lambda)}\big]^M\cdot {\bf 1}(\Delta_{2k-\ell+3}^{(\lambda)} <\infty)\Big]\\ \leq & \; C_k \, \e^{2k-1}\,I_{N-1}(M+3, k-1, \ell+1)\; , \end{split} \end{equation} where we used that $\min (d_{\beta_1}, d_{\beta_2}) \geq 1/2\Delta$ and that $\Delta_4^{(\lambda)} \leq \Delta_{\ell+1}^{(\lambda)}$. Together with (\ref{eq:Ares}) and the monotonicity of $I_{N-1}$ in $M$, we obtain that $$ I_N(M, k, \ell) \leq C_k \e^{2k-1} I_{N-1}(M+3, k-1, \ell+1)\; . $$ Iterating this inequality, we arrive at $$ I_N(M, k, \ell) \leq C_k \e^{k^2-1} I_{N-k+1}(M+3(k-1), 1, \ell +k-1)\;. $$ Recalling \eqref{III}, we have $$ (II)\leq I_N(0, k, 4) \leq C_k \e^{k^2-1} I_{N-k+1}(3(k-1), 1, k+3) \; . $$ Finally, $I_{N-k+1}(M, 1, d)$ was exactly the quantity that has been estimated by $C\e$ for any $M$ and $d\ge 5$ in Corollary \ref{cor:weg} (replacing $N$ by $N-k+1$), thus we have $$ (II) \leq C_k\e^{k^2} \; . $$ Together with \eqref{eq:I}, this completes the proof of Theorem \ref{thm:repul}.\qed \appendix \section{Proof of Proposition \ref{prop:HW}} We first consider the case, when the real and imaginary parts of $b_j$ are i.i.d. (first condition in \eqref{hass}). We split $a_{jk}$ and $b_j$ into real and imaginary parts, $a_{jk} = p_{jk} + i q_{jk}$, $b_j = x_j + iy_j$, and form the vector $\bw = (x_1, \ldots x_N, y_1, \ldots y_N)\in \bR^{2N}$ with i.i.d. components. We write $X= X_1 + iX_2$ where $X_1 = \bw\cdot \bP \bw - \E \bw\cdot \bP \bw$, $X_2= \bw \cdot \bQ \bw -\E \bw \cdot \bQ \bw$ with symmetric real $(2N)\times (2N)$ matrices $\bP$ and $\bQ$, written in a block-matrix form as $$ \bP = \frac{1}{2}\begin{pmatrix} P+P^t & Q-Q^t\cr Q^t-Q & P+P^t \end{pmatrix}, \qquad \bQ = \frac{1}{2}\begin{pmatrix} Q+Q^t & P^t-P\cr P-P^t & Q+Q^t \end{pmatrix} \; , $$ where $P=(p_{jk})$ and $Q=(q_{jk})$. We define $\cP$ to be the symmetric matrix whose entries are the absolute values of the matrix entries of $\bP$: $$ \cP = \frac{1}{2}\begin{pmatrix} P^\dagger & P^\# \cr P^\# & P^\dagger \end{pmatrix}, \qquad (P^\dagger)_{jk} = |p_{jk}+p_{kj}|, \qquad ( P^\#)_{jk} = |q_{kj}-q_{jk}| \; . $$ Then $$ \mbox{Tr}\, \cP^2 = \frac{1}{2} \sum_{j, k} \Big( |p_{jk}+p_{kj}|^2 + |q_{kj}-q_{jk}|^2\Big) \leq 2\sum_{j,k} \big[ p_{jk}^2 +q_{jk}^2\big] \; . $$ We apply the non-symmetric version of of the Hanson-Wright theorem \cite{Wr} for $X_1$ and $X_2$ separately; note that the components of $\bw$ are i.i.d. Together with the bound $\|\cP\|\leq \sqrt{\text{Tr}\, \cP^2}$ we have $$ \P ( |X_1|\ge \delta) \leq 2\exp{(-c \min\{ \delta/\sqrt{ \text{Tr}\, \cP^2}, \delta^2/\text{Tr}\, \cP^2\})} $$ for some constant $c$ depending on $\delta_0$ and $D$ from \eqref{x2}. Similar estimate holds for $X_2$, so we have $$ \P ( |X|\ge \delta) \leq 4\exp{(-c \min\{ \delta/A, \delta^2/A^2\})} $$ where $A^2= \sum_{j,k} |a_{jk}|^2 =\sum_{j,k}\big[ |p_{jk}|^2 + |q_{jk}|^2\big]$. In the second case in \eqref{hass}, when the distribution of the complex random variable $b_j$ is rotationally symmetric, we can directly extend the proof \cite{HW} (note that \cite{HW} uses the notation $X_j$ for $b_j$). We first symmetrize the quadratic form $X$ by replacing $a_{jk}$ with $\frac{1}{2}[a_{jk} + \ov{a}_{kj}]$. We then follow the proof in \cite{HW} and note that the only change is that $Z$ used starting from Lemma 2 in \cite{HW} will be a standard complex Gaussian random variable and instead of $Z^2$ or $Z^{2n}$ we consider $|Z|^2=Z\ov Z$ and $|Z|^{2n}$, and similarly $X^{2n}$ is replaced by $|X|^{2n}$, $n=1,2,\ldots$. With these changes, Lemma 1--6 in \cite{HW} hold true for the complex case as well. In the proof of the theorem, starting on page 1082 of \cite{HW}, instead of $\prod_i \E X_i^{\al_i} (X_i^2 - \E X_i^2)^{\beta_i}$ the expansion will contain terms of the form $\prod_i \E X_i^{\al_i} \ov X_i^{\al_i'} (|X_i|^2 - \E |X_i|^2)^{\beta_i}$. Due to the rotational symmetry of the distribution, these terms are all zero (case (i) on page 1082 of \cite{HW}) unless $\al_i=\al_i'$ for all $i$. In the latter case, the bound $|\E |X_i|^{2\al_i}(|X_i|^2 - \E |X_i|^2)^{\beta_i}| \leq \lambda^{2\al_i+2\beta_i} \E |Z_i|^{2\al_i}(|Z_i|^2 - 1)^{\beta_i}$ holds with a sufficiently large $\lambda$ (depending on $\delta_0$ from \eqref{x2}) exactly as in case (ii) on page 1082 of \cite{HW}. From now on the proof is unchanged and we obtain $$ \P \Big( \Big| \sum_{jk} a_{jk}(b_j\ov{b}_k -\E b_j\ov{b}_k) \Big|\ge\delta\Big)\leq 2\exp\big(-c\min(\delta/A, \delta^2/A^2)\big)\; , $$ where $\sum_{jk} \big|\frac{1}{2}[a_{jk}+\ov{a}_{kj}]\big|^2$ was estimated by $A^2=\sum_{jk}|a_{jk}|^2$ from above. \qed \thebibliography{hhh} \bibitem{AGZ} Anderson, G. W., Guionnet, A., Zeitouni, O.: Lecture notes on random matrices. Book in preparation. %\bibitem{B} Bai, Z.: Convergence rate of expected spectral %distributions of large random matrices. Part I. Wigner matrices. %{\it Ann. Probab.} {\bf 21} (1993), No.2. 625--648. \bibitem{BMT} Bai, Z. D., Miao, B., Tsay, J.: Convergence rates of the spectral distributions of large Wigner matrices. {\it Int. Math. J.} {\bf 1} (2002), no. 1, 65--90. \bibitem{BG} Bobkov, S. G., G\"otze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. {\it J. Funct. 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Phys.} {\bf 207} (1999), no.3. 697-733. \bibitem{W} Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. {\it Ann. of Math.} {\bf 62} (1955), 548-564. \bibitem{Wr} Wright, F.T.: A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. {\it Ann. Probab.} {\bf 1} No. 6. (1973), 1068-1070. \end{document} \newpage \section{Improved bounds for smooth distribution} [THIS section may not be needed] The convergence to the semicircle law in Theorem \ref{thm:sc1} was shown to be polynomial in the large parameter $N\eta$ and as a consequence, the estimate of the tail of the gap distribution in Theorem \ref{thm:gap} was also only power law. Since Theorem \ref{thm:sc1} is a large deviation estimate, one expects exponential convergence. Moreover, the universality conjecture for the tail of the Wigner surmise predicts Gaussian decay. In this section we improve the bounds in Theorem \ref{thm:sc1} and Theorem \ref{thm:gap} assuming that the distribution of the matrix elements are smooth. More precisely, we will use the bound (\ref{eq:sigma0}) in Lemma \ref{lm:Ebb} for all $p$, which requires a lower bound on $\inf_{a\in\N}a\om_a$, where, recall, that $\om_a$ is the constant in \eqref{charfn}. This is a control on the smoothness of the density function $h$. Note, for example, that if $\wh h(t,s)$ has a Gaussian decay, $|\wh h (t,s)|\leq \exp(-c(t^2+s^2))$ for some $c>0$, then \eqref{charfn} holds with $\inf_{a\in\N}a\om_a=c>0$. The key for this improvement is the observation that using the explicit constant in (\ref{eq:sigma0}), we can remove the conditions $\delta \ge N^{-\nu}$ and $m\leq N^{\beta}$ in Lemma \ref{lemma:lde} at the expense of lowering the exponent of $\delta$ by one. Since $$ \P \Big\{ \frac{1}{m}\sum_{\al=1}^m \xi_\al \leq \delta\Big\} \leq (m\delta)^{m-1} \E \Big( \sum_{\al=1}^m \xi_\al\Big)^{-(m-1)} \leq \frac{ (C\delta)^{m-1}}{[ (m+1)\om_{m+1}]^m} $$ from (\ref{eq:sigma0}), we have proved \begin{lemma}\label{lm:lde1} Assume that the constants $\om_a$ in \eqref{charfn} satisfy $C_\om:=\sup_{a\in \N}[a\om_a]^{-1}<\infty$. Then for any $m\in \N$ and $\delta>0$ we have \be \P \Big\{ \frac{1}{m}\sum_{\al=1}^n \xi_\al \leq \delta\Big\} \leq (C_\om\delta)^{m-1} \label{ldegen} \ee \end{lemma} The estimate $P(\xi\leq \delta)\lesssim \delta$ for some $\delta\ll 1$ is a statement about the continuity of the distribution of $\xi$. Since $\xi = [\bu \cdot \bb]^2$, such a statement can be obtained either by assuming smoothness on the distribution of $\bb$ (this is the mechanism in Lemma \ref{lm:lde1}) or by assuming that the nonzero components of $\bu$ are sufficiently spread so that $\bu\cdot \bb$ is effectively becomes a smooth distribution by central limit theorem (this is Lemma \ref{lemma:lde}). In the latter case the estimate cannot hold for very small $\delta$ since the possible discontinuity of the distribution of $\bb$ eventually has an effect on very small scale. This explains the condition $\delta\ge N^{-\nu}$ in Lemma \ref{lemma:lde}. \end{document} \section{ Proof of Lemma \ref{lm:x}} We fix a sufficiently large $M$. Recall the set $\Omega$ from \eqref{def:Omega} and the estimate \eqref{omegaest}. Therefore, if $\P_\bb$ denotes the probability w.r.t. the variable $\bb$, we find \be \begin{split} \P[ |X|\ge \e] & \leq \Big(\frac{C}{M}\Big)^{cMN\eta} + \E \Big[ {\bf 1}_{\Omega^c} \P_\bb [ |X|\ge \e] \Big]\\ & \leq \Big(\frac{C}{M}\Big)^{cMN\eta} + \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Re} X \le - \e /2 ] \Big] + \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Re} X \ge \e /2 ] \Big] \\ & \hspace{.5cm} + \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Im} X \le - \e /2 ] \Big] + \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Im} X \ge \e /2 ] \Big]. \end{split}\label{T} \ee The last four terms on the r.h.s. of the last equation can all be handled with similar arguments; we show, for example, how to bound the last term. For any $T>0$, we have \begin{equation}\label{eq:T2new} \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Im} X \ge \e /2 ] \Big] \leq e^{- T \e / 2} \, \E \Big[ {\bf 1}_{\Omega^c} \, \E_\bb \, e^{T \, \text{Im} X} \Big] \end{equation} where $\E_\bb$ denotes the expectation w.r.t. the variable $\bb$. Using the fact that the distribution of the components of $\bb$ satisfies the log-Sobolev inequality (\ref{logsob}) with Sobolev constant $C_S$, it follows from the concentration inequality (Theorem 2.1 from \cite{BG}) that \begin{equation} \E_{\bb} \, e^{T \, \text{Im} X} \leq \E_{\bb} \, \exp \left( \frac{C_{\text{S}} T^2}{2} \, |\nabla ( \text{Im} X)|^2 \right) \, \end{equation} where \begin{equation} \begin{split} |\nabla ( \text{Im} X)|^2 = \; & \sum_k \Big( \Big| \frac{\partial \, (\text{Im} X)}{\partial\, (\re \, b_k)}\Big|^2 + \Big| \frac{\partial \, (\text{Im} X)}{\partial \, ( \text{Im} \, b_k)} \Big|^2 \Big) \\ =\; & \sum_k \Big( \Big| \frac{\eta}{N} \sum_{\al} \frac{1}{|\lambda_{\al} - z|^2} \left( (\bb \cdot \bv_{\al} )\, \overline{\bv}_{\al} (k) + \overline{(\bb \cdot \bv_{\al})} \, \bv_{\al} (k) \right) \Big|^2 \\ & \hspace{2cm}+ \Big| \frac{\eta}{N} \sum_{\al} \frac{1}{|\lambda_{\al} - z|^2} \left((\bb \cdot \bv_{\al}) \overline{\bv}_{\al} (k) - \overline{(\bb \cdot \bv_{\al})} \, \bv_{\al} (k) \right) \Big|^2 \Big) \\ = \; &4 \frac{\eta^2}{N^2} \sum_{\al} \frac{\xi_{\alpha}}{|\lambda_{\alpha} - z|^4} \\ \leq \; & \frac{4}{N^2} \sum_{\al} \frac{\xi_{\alpha}}{|\lambda_{\alpha} - z|^2} \,. \end{split} \end{equation} {F}rom (\ref{eq:T2new}) and using that $N\eta \geq K$, we obtain \begin{equation}\label{eq:Y2new} \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Im} X \ge \e /2 ] \Big] \leq e^{- \e T/2} \, \E \Big[ {\bf 1}_{\Omega^c} \, \E_{\bb} \, \exp \left( \frac{8C_ST^2}{ N^2} \sum_{\al} \frac{\xi_{\alpha}}{|\lambda_{\alpha} - z|^2} \right) \Big]\,. \end{equation} By H\"older inequality, we can estimate \be \E_{\bb} \, \exp \left( \frac{8C_{\text{S}}T^2}{ N^2} \sum_{\al} \frac{\xi_{\alpha}}{|\lambda_{\alpha} - z|^2} \right) \leq \prod_\al \Bigg( \E_\bb \exp{\Big[\frac{8C_ST^2 c_\al}{N^2|\lambda_\al -z|^2}\xi_\al\Big]} \Bigg)^{1/c_\al}, \label{holdnew} \ee where $\sum_\al \frac{1}{c_\al}=1$. We shall choose $$ c_\al = \varrho \frac{N^2|\lambda_\al -z|^2}{8C_ST^2} $$ where $\varrho$ is given by $$ \varrho = \frac{8C_ST^2}{N^2}\sum_\al \frac{1}{|\lambda_\al -z|^2} $$ We need that $$ \E e^{\varrho \xi}\leq C $$ which holds only if $\varrho$ is a sufficiently small (Proof of Lemma 7.4 of \cite{ESY}). On the set $\Omega^c$ we can estimate \be \varrho\leq \frac{8C_ST^2}{N^2}\sum_{n\ge 0} \frac{\cN_{I_n}}{(2^n\eta)^2} \leq \frac{16C_S T^2M}{N\eta} \label{omc} \ee Choosing $$ T = c\Big( \frac{N\eta}{16C_S M} \Big)^{1/2} $$ with a sufficiently small $c$, we ensure the finiteness of $\E e^{\varrho \xi}$. {F}rom (\ref{eq:Y2new}), it follows that \begin{equation}\label{sqrt} \E \Big[ {\bf 1}_{\Omega^c} \, \cdot \P_\bb [ \text{Im} X \ge \e /2 ] \Big] \leq C e^{- c\e\sqrt{N\eta} } \end{equation} Since similar bounds hold for the other terms on the r.h.s. of (\ref{T}) as well, this concludes the proof of the lemma (the estimate \eqref{omegaest} can be easily absorbed into \eqref{sqrt} since $N\eta\ge 1$ and $M\ge 2$. \qed \bigskip {\it Proof of Lemma \ref{lm:x} under the condition \eqref{x2}.} M. Ledoux has kindly pointed out to us the following alternative proof of Lemma \ref{lm:x} that uses only the weaker sub-Gaussian assumption \eqref{x2} and the symmetry of the distribution $\rd\nu$ instead of the logarithmic Sobolev inequality \eqref{logsob}. Write $X$ in the form $$ X= \sum_{i,j=1}^N a_{ij} \big[ b_i\overline{b_j} - \E b_i\overline{b_j}\big] $$ where $$ a_{ij} = \frac{1}{N}\sum_\al \frac{ \overline{ u_\al(i)} u_\al(j)}{\lambda_\al-z}\; . $$ We have $$ A^2: = \sum_{i,j=1}^N |a_{ij}|^2 =\frac{1}{N^2} \sum_\al\frac{1}{|\lambda_\al-z|^2} $$ On the set $\Omega^c$ we have (see \eqref{omc}) $$ A^2 \leq \frac{2M}{N\eta} $$ Using the result of Hanson and Wright \cite{HW}, we obtain that $$ \P (|X|\ge \e)\leq C\exp\big( -c\e/A\big)\; . $$ where the constant $c$ depends on $\delta_0$ in \eqref{x2} (in the original paper \cite{HW} the result was formulated for real valued random variables; the complex case can be reduced to this by separating real and imaginary parts). Combining this with the estimate \eqref{omegaest} on $\P(\Omega)$, we conclude $$ \P(|X|\ge \e) \leq Ce^{-c\e\sqrt{N\eta}} \; . \qquad \mbox{\qed} $$ \section{Case of the slowly decaying characteristic function} [This is probably to be omitted, just mention the result] We show how to extend Lemma \ref{lm:N2} and Lemma \ref{lm:exp} for the case when the characteristic function $\varphi(t)$ of the distribution $\rd \nu$ satisfies only the decay bound \eqref{charfn} with $a=1$, i.e. \be |\varphi(t)|\leq \frac{1}{1+ c|t|^2} \label{phidec} \ee \begin{lemma}\label{lm:N2strong} We fix $E$, set $\eta = \e/N$ with $\e<1/2$ and $\cN_\eta$ is the number of eigenvalues in $[E-\eta/2, E+\eta/2]$. Assuming \eqref{phidec} we have \be \P ( \cN_\eta\ge 2)\leq C\e^2|\log \e|^2 \label{want} \ee \end{lemma} {\it Proof.} Define $$ \Omega:=\Big\{ \forall j, \al\; : \; \|\bu_\al^{(j)}\|_\infty\leq \frac{(\log N)^4}{N^{1/2}} \Big\} $$ Then we know that \be P(\Omega^c)\leq e^{-c(\log N)^2} \label{omcom} \ee and we will also use \be \P(\cN\ge 2) \leq\; \P (\Omega^c) + C\e^2\E_\lambda\Bigg[ \Big( [N(\lambda_{\al+1}-E)]^2+1\Big) \;\E_\bb \Big( \frac{{\bf 1}(\Omega)}{\e^2/N + \xi_\al + \e^2\xi_{\al +1}}\Big)\Bigg] \\ \label{PE1} \ee instead of \eqref{PE}. The key ingredient in the proof is the following strengthening of Lemma \ref{lm:exp} \begin{lemma}\label{lm:expstrong} Under the condition \eqref{phidec}, we have for any $t\ge 2$, that \be \P(\xi_\al\leq 1/t)\leq Ct^{-1}|\log t| \label{prob1} \ee and \be \E_\bb \Big( \frac{ 1 }{\e^2/N + \xi_\al + \e^2\xi_{\al +1}}\Big)\leq C|\log \e|^2 \qquad \mbox{if} \quad N\leq \e^{-100} \label{bb1} \ee \be \E_\bb \Big( \frac{ {\bf 1}(\Omega) }{\e^2/N + \xi_\al + \e^2\xi_{\al +1}}\Big)\leq C|\log \e|^2 \qquad \mbox{if} \quad N\ge \e^{-100} \label{bb} \ee \end{lemma} Using this lemma, \eqref{want} will follow from \eqref{PE1} (and using \eqref{omcom} in case of $a=1$, $N\ge \e^{-100}$) and the fast decay of the gap distribution given in Theorem \ref{thm:gap}. This completes the proof of Lemma \ref{lm:N2strong}. \bigskip {\it Proof of Lemma \ref{lm:expstrong}.} Following the same calculation from \eqref{compu} with $a=1$ above, we have \be \begin{split} \P(\xi_\al\le 1/t)\leq \; & \int_{\bR^2}\frac{\rd \tau \rd s}{4\pi} \frac{e^{-\tau^2-s^2}}{1+ t(\tau^2+s^2)}\\ = \; & \int_0^\infty \rd q \frac{e^{- q}}{1+ 4tq} \\ \leq \; & \frac{C\log t}{t} \end{split} \ee after splitting the integration regime according to whether $q \le 1$ or $\ge 1$. This proves \eqref{prob1}. For the proof of \eqref{bb1} it is sufficient to use only one term by using the estimate \eqref{prob1} \be \begin{split} \E_\bb \Big( \frac{1}{\e^2/N + \xi_\al + \e^2\xi_{\al +1}}\Big)\leq\; & \E \; \Big( \frac{1}{\e^{102} + \xi_\al }\Big)\\ \leq \; & 2 + \int_2^{\e^{-102}} \P(\xi \leq t^{-1})\rd t\\ \leq \;& 2 + C\int_1^{\e^{-102}} t^{-1}\log t\rd t\\ \leq \;& C|\log \e|^2 \end{split} \ee The proof of \eqref{bb} is more involved, from now on we assume that $N\ge \e^{-100}$. We split \be \begin{split}\label{3split} \E_\bb \Big( \frac{1}{\e^2/N + \xi_\al + \e^2\xi_{\al +1}}\Big) \leq 1 + \int_1^{N^\nu}\P(\xi_\al+\e^2\xi_{\al+1}\leq 1/t) \rd t + \int_{N^\nu}^{N/\e^2}\P(\xi_\al+\e^2\xi_{\al+1}\leq 1/t) \rd t \end{split} \ee where $\nu<1$. In Lemma 1.2 we proved that $$ \P (\frac{1}{m}\sum_\al \xi_\al \leq \delta)\leq (C\delta)^m $$ if $\delta\ge N^{-\nu}$. Following the same proof until the end, but for the generalized sums $\sum_\al c_\al \xi_\al$ with $c_\al\le 1$, we have (see last line of that proof) \be \begin{split} \P(\sum_\al c_\al \xi_\al \leq 1/t)\leq \; & C^m \Big(\prod_\al\int_{\bR^2} e^{-\frac{1}{4}(\tau_\al^2+s_\al^2)} \frac{\rd \tau_\al \rd s_\al}{4\pi}\Big) e^{-\frac{t}{4}\sum_\al c_\al(\tau_\al^2+ s_\al^2) + o(t)\sum_\al c_\al(\tau_\al^2+ s_\al^2)} + Ce^{-cN^\beta}\\ \leq \; & C^m \Big(\prod_\al \int_0^\infty \rd q_\al\Big) e^{-\sum_\al (1+tc_\al/2) q_\al}+ Ce^{-cN^\beta}\\ \leq \; & C^m \prod_\al \frac{1}{1+ tc_\al} + Ce^{-cN^\beta} \end{split} \ee where $\beta <\frac{1}{17}(1-\nu)$. Thus \be \begin{split}\label{middle} \int_1^{N^\nu}\P(\xi_\al+\e^2\xi_{\al+1}\leq 1/t) \rd t \leq \; & C\int_1^{N^\nu} \frac{\rd t}{(1+t)(1+\e^2 t)} + N^\nu Ce^{-cN^\beta}\\ \leq \; & C|\log \e| \end{split} \ee Finally, we estimate the last term in \eqref{3split}. We follow the calculation \eqref{compu} with $a=1$ but use \eqref{trivi} in \eqref{1} only once: \be \begin{split} \P\{ \sum_{\al=1}^m c_\al^2\xi_\al \le 1/t\} \leq \; & e^m \Big(\prod_\al \int_{\bR^2}e^{- \tau_\al^2/4 - s_\al^2/4} \frac{\rd\tau_\al\rd s_\al}{4\pi} \Big) \prod_j\frac{1}{1 + t C_j} \end{split}\label{improve} \ee where $$ C_j = \Big[ \sum_\al c_\al (\tau_\al v_\al(j) + s_\al w_\al(j))\Big]^2 + \Big[ \sum_\al c_\al (s_\al v_\al(j) - \tau_\al w_\al(j))\Big]^2 $$ We will use the estimates on the eigenfunctions, i.e. that $$ \| \bu_\al\|_\infty\leq \frac{(\log N)^4}{N^{1/2}} $$ that holds apart from probability $e^{-c(\log N)^{3/2}}\ll \e^2|\log \e|^2$. Using this bound and cutting off the regime where $\max_\al (\tau_\al^2 + s_\al^2)\ge |\log t|^2$ (on that regime we have the trivial estimate $\exp(-|\log t|^2)\ll t^{-4}$ yielding an integrable bound), i.e. assuming that $\max_\al (\tau_\al^2 + s_\al^2)\le |\log t|^2\leq |\log N|^2$ in the given integration regime, we have for all $j$ $$ C_j \leq \frac{C (\log N)^9}{N}\sum_\al c_\al (\tau_\al^2 + s_\al^2) $$ On the other hand, we know that $\sum C_j = \sum_\al c_\al(\tau_\al^2 + s_\al^2)$. This means that there exists at least $cN/(\log N)^9$ indices of $j$ such that $C_j$ is bigger than half of the average, i.e. if we set $$ \Omega = \Omega(\tau, s) =\Big\{ j\; : \; C_j \ge \frac{1}{2N} \sum_\al c_\al(\tau_\al^2 + s_\al^2)\Big\} $$ then $$ |\Omega|\ge T=:\frac{cN}{(\log N)^9} $$ Now we keep only these indices in \eqref{improve}: \be \begin{split} \P\{ \sum_{\al=1}^m c_\al \xi_\al \le 1/t\} \leq \; & C^m \Big(\prod_\al \int_{\bR^2}e^{- \tau_\al^2/4 - s_\al^2/4} \frac{\rd\tau_\al\rd s_\al}{4\pi} \Big) \prod_{j\in \Omega(\tau,s)} \frac{1}{1 + \frac{t}{2N}\sum_\al c_\al(\tau_\al^2+s_\al^2)}\\ \leq \; & C^m \Big(\prod_\al \int_{\bR^2} \frac{\rd\tau_\al\rd s_\al}{4\pi} \Big)e^{-\sum_\al(\tau_\al^2+s_\al^2)/4}\Bigg( \frac{1}{1 + \frac{t}{2N}\sum_\al c_\al (\tau_\al^2+s_\al^2)} \Bigg)^T\\ \leq \; & C^m \Big( \prod_\al \int_0^\infty \rd q_\al\Big)e^{-\sum_\al q_\al} \Bigg( \frac{1}{1 + \frac{2t}{N}\sum_\al c_\al q_\al} \Bigg)^T \end{split} \ee Thus for the estimate the last term in \eqref{3split}, using $N\ge \e^{-100}$, we have the trivial estimate \be \begin{split} \int_{N^\nu}^{N/\e^2}\P(\xi_\al+\e^2\xi_{\al+1}\leq 1/t) \rd t \leq \; & C \int_0^\infty \rd p \rd q e^{-p-q}\int_{N^\nu}^{N/\e^2} \Bigg( \frac{1}{1 + \frac{2t}{N}(p+\e^2q)} \Bigg)^T\rd t \\ \leq \; & C \int_0^\infty \rd p \rd q e^{-p-q}\int_{N^\nu}^{N^{1.02}} \Bigg( \frac{1}{1 + \frac{2t}{N^{1.02}}(p+q)} \Bigg)^T\rd t\\ \leq \; & C \int_0^\infty \rd x \; x e^{-x}\int_{N^\nu}^{N^{1.02}} \Bigg( \frac{1}{1 + \frac{2t}{N^{1.02}}x} \Bigg)^T\rd t\\ \leq \; & C \int_0^1 \rd x \; x \int_{N^\nu}^{N^{1.02}} \Bigg( \frac{1}{1 + \frac{2t}{N^{1.02}}x} \Bigg)^T\rd t\\ & +C \int_1^\infty \rd x \; x e^{-x}\int_{N^\nu}^{N^{1.02}} \Bigg( \frac{1}{1 + \frac{2t}{N^{1.02}}} \Bigg)^T\rd t\\ \leq \; & C \int_0^1 \rd x \; x \int_{N^\nu}^{N^{1.02}} e^{-\frac{2tx}{N^{0.02}(\log N)^9}}\rd t\\ & +C \int_1^\infty \rd x \; x e^{-x} \int_{N^\nu}^{N^{1.02}}e^{-\frac{2t}{N^{0.02}(\log N)^9}} \rd t\\ \leq \; & C \int_0^1 \rd x \int_{N^\nu}^{\infty} e^{-\frac{2t}{N^{0.02}(\log N)^9}}\rd t\\ & +C \int_1^\infty \rd x \; x e^{-x} \int_{N^\nu}^{N^{1.02}}e^{-N^{\nu-0.03}} \rd t\\ \leq \; & C \end{split}\label{last} \ee Combining \eqref{middle} and \eqref{last} we obtain \eqref{bb}. $\Box$ \thebibliography{hhh} \bibitem{AGZ} Anderson, G. 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