Content-Type: multipart/mixed; boundary="-------------0503251335392" This is a multi-part message in MIME format. ---------------0503251335392 Content-Type: text/plain; name="05-117.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-117.keywords" Random Schroedinger operators. ---------------0503251335392 Content-Type: application/x-tex; name="rsz2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rsz2.tex" \documentclass[12pt]{amsart} \usepackage{amssymb,amsfonts,latexsym,amscd,epsfig,psfrag} %\usepackage{graphicx} %\pagestyle{myheadings} \setlength\textwidth{6.5 in} \setlength\textheight{8.5 in} \voffset=-0.6in \hoffset = -0.6in \parindent = 0.4in \pagestyle{plain} \begin{document} \def\A{{\mathcal A}} \def\alg{{\mathfrak A}} \def\amp{{\rm Amp}} \def\ampi{{A_{\dex,\tau,J,\lambda,\e}}} \def\Bound{{\mathcal B}} \def\bra{\big\langle} \def\cj{K_\dex(J)} \def\C{{\Bbb C}} \def\Ch{C^{(h)}} \def\tCh{\tilde C^{(h)}} \def\Cv{C^{(v)}} \def\Detau{{\alg}_L(I_\tau)} \def\cDetau{{\alg}_L(I_\tau^c)} \def\Dell{{\alg}_{L}^{(\omega)}(\e,\delta,\ell;I_\tau)} \def\cDell{\Detau\setminus\Dell} \def\Dom{\mathfrak{Dom}} \def\dist{{\rm dist}} \def\dex{{\sigma}} \def\Exp{{\Bbb E}} \def\Expnd {{\Bbb E}_{n-d}} \def\Exptc{{\Bbb E}_{2-conn}} \def\Expd{{\Bbb E}_{disc}} \def\e{\varepsilon} \def\en{{e_\Delta}} \def\Fou{{\mathcal F}} \def\H{{\mathcal H}} \def\HLL{{H_\omega^{(\LL)}}} \def\Hpl{{\Bbb H}} \def\ids{{\overline{{\rm IDS}}}} \def\lb{\left[} \def\Ie{I} \def\Im{{ Im}} \def\Je{J_{\lambda^2}} \def\Jeta{J_\eta} \def\ket{\big\rangle} \def\LL{\Lambda_L} \def\tLL{\tilde \LL} \def\LLinv{\frac{1}{|\LL|}} \def\mat{{\mathcal M}} \def\mes{{\rm mes}} \def\mset{K} \def\N{{\Bbb N}} \def\p{r} \def\pip{\tau} \def\qm{q^{(m+1)}} \def\rb{\right]} \def\R{{\Bbb R}} \def\Rem{ R} \def\Sc{{\mathcal S}} \def\supp{{\rm supp}} \def\tk{{\tilde k}} \def\Tor{\Bbb T} \def\tr{{\rm Tr}} \def\uj{\underline{j}} \def\up{\underline{\vp}} \def\uk{\underline{\vk}} \def\utk{\underline{\tilde\vk}} \def\uvw{\underline{\vw}} \def\ux{\underline{x}} \def\Z{{\Bbb Z}} \def\vx{{ x}} \def\vy{{ y}} \def\ve{{ e}} \def\vk{{ k}} \def\vl{{ l}} \def\vm{{ m}} \def\vn{{ n}} \def\vp{{ p}} \def\vQ{{ Q}} \def\vq{{ q}} \def\vr{{ r}} \def\vv{{ v}} \def\vw{{ w}} \def\1{{\bf 1}} \def\eqnn{\begin{eqnarray*}} \def\eeqnn{\end{eqnarray*}} \def\eqn{\begin{eqnarray}} \def\eeqn{\end{eqnarray}} \def\bal{\begin{align}} \def\eal{\end{align}} \def\prf{\begin{proof}} \def\endprf{\end{proof}} %\def\prf{\noindent{\em Proof.}$\;$} %\def\endprf{\hspace*{\fill}\mbox{$\Box$}} %%\numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{definition}{Definition}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}{Lemma}[section] \title{Localization lengths for Schr\"odinger operators on $\Z^2$ with decaying random potentials} \author{Thomas Chen} \address{Department of Mathematics, Princeton University, 807 Fine Hall, Washington Road, Princeton, NJ 08544, U.S.A.} \email{tc@math.princeton.edu} \date{} \maketitle \begin{abstract} We study a class of Schr\"odinger operators on $\Z^2$ with a random potential decaying as $|x|^{-\dex}$, $0<\dex\leq\frac12$, in the limit of small disorder strength $\lambda$. For the critical exponent $\dex=\frac12$, we prove that the localization length of eigenfunctions is bounded below by $2^{\lambda^{-\frac14+\eta}}$, while for $0<\dex<\frac12$, the lower bound is $\lambda^{-\frac{2-\eta}{1-2\dex}}$, for any $\eta>0$. These estimates "interpolate" between recent results of Bourgain on the one hand, and of Schlag-Shubin-Wolff on the other hand. \end{abstract} %\newpage \section{Introduction} We study the discrete random Schr\"odinger operator \eqn H_\omega=\Delta+\lambda V_\omega \eeqn on $\ell^2(\Z^2)$, where $\Delta$ is the (centered) nearest neighbor Laplacian, with spectrum supported in $[-4,4]$, and $\lambda$ is a small parameter (the disorder strength). The random potential is given by $V_\omega(x)=v_\dex(x)\omega_x$, where $v_\sigma(x)\sim|x|^{-\dex}$ and $\{\omega_x\}_{x\in\Z^2}$ are Gaussian i.i.d. random variables. The restriction to Gaussian randomness has expository advantages, but is not essential for our techniques to apply. Extension of our methods to non-Gaussian random potentials can be accessed along the lines demonstrated in \cite{ch}. The purpose of this paper is to derive lower bounds on the localization lengths of eigenfunctions of $H_\omega$. In the supercritical case $\dex>\frac12$, it was proven by Bourgain in \cite{bo1} that with large probability, $H_\omega$ (with Bernoulli or Gaussian randomness) has pure a.c. spectrum in $(-4+\tau,-\tau)\cup(\tau,4-\tau)$ ($\tau>0$ arbitrary, but fixed), and the wave operators were constructed. The (generalized) eigenfunctions are therefore delocalized. A number of other types of lattice Schr\"odinger operators with decaying random potentials have been proven to exhibit a.c. spectrum and scattering by Bourgain in \cite{bo2}, and by Rodnianski and Schlag in \cite{rosc}. We also note the contextually related work of Denissov in \cite{de}. In the case $\dex=0$, Schlag, Shubin and Wolff have proven lower bounds on the localization length of eigenfunctions of the form $\lambda^{-2+\eta}$, for any $\eta>0$, \cite{shscwo}. For $\dex=0$ and $d=3$, lower bounds of the form $\lambda^{-2}|\log\lambda|^{-1}$ were derived in \cite{ch}. We shall here address the case $0<\dex\leq\frac12$ in dimension two, which is at present open. Our main results are as follows. For the critical decay exponent $\dex=\frac12$, the problem is {\em marginal} in the language of renormalization group theory. Accordingly, we obtain a comparison of the {\em logarithm} of the localization length to powers of $\lambda$, yielding lower bounds on the localization length that are {\em exponential} in $\frac1\lambda$, of the form $2^{\lambda^{-\frac14+\eta}}$ ($\eta>0$ arbitrary). In the subcritical case $0<\dex<\frac12$, it is suspected that the model exhibits a significant component of point spectrum. In the language of renormalization group theory, the potential scales like a {\em relevant} perturbation, whereby we obtain a comparison of the localization length to powers of $\lambda$. Consequently, our lower bounds on the localization lengths are {\em polynomial} in $\frac1\lambda$ for $0<\dex<\frac12$, of the form $\lambda^{-\frac{2-\eta}{1-2\dex}}$ ($\eta>0$ arbitrary). On the one hand, our strategy employs graph expansion methods due to Erd\"os and Yau \cite{erd, erdyau}, and further elaborated on by the author \cite{ch, ch1}. On the other hand, we use smoothing of resolvent multipliers by dyadic restriction, which is similar to Bourgain's approach in \cite{bo1}. Our methods can be extended to higher dimensions, but we will here only focus on the case $d=2$. The following works, which determine macroscopic hydrodynamic limits of the quantum dynamics in the Anderson model at small disorders (without spatial decay, i.e. $\dex=0$), are closely related to the topics discussed here. In an important early work, Spohn proved in \cite{sp} that the kinetic macroscopic scaling and low coupling limit is determined by a linear Boltzmann equation, locally in macroscopic time. Erd\"os and Yau proved the corresponding global in macroscopic time result for the continuum model in $\R^d$, $d=2,3$, and Gaussian randomness, \cite{erdyau}, which was extended by Erd\"os to the case of a Schr\"odinger electron interacting with a phonon heat bath, \cite{erd}. The author derived the corresponding result for the lattice $\Z^3$ and non-Gaussian randomness, \cite{ch}, and proved that the mode of convergence can be extended to $r$-th mean, for any $r\in\R_+$ (the previous works proved convergence in expectation), \cite{ch1}. Eng and Erd\"os proved the corresponding result for the kinetic macroscopic and low density limit, \cite{engerd}. Very recently, Erd\"os, Salmhofer and Yau established the breakthrough result that beyond kinetic scaling, the macroscopic dynamics is governed by a diffusion equation, \cite{erdsalmyau}. \section{Definition of the model and statement of the main results} \label{intro-sect-1} We consider the discrete random Schr\"odinger operator \eqn H_\omega = \Delta + \lambda V_\omega \; \label{Homega-def} \eeqn on $\ell^2(\Z^2)$, with a radially decaying potential function \eqn V_{\omega}(x) =v_\dex(x) \omega_x \;, \eeqn where $\{\omega_x\}_{x\in\Z^2}$ are independent, identically distributed Gaussian random variables normalized by $\Exp[\omega_x]=0$, $\Exp[\omega_x^2]=1$, for all $x\in\Z^2$. Expectations of higher powers of $\omega_x$ satisfy Wick's theorem, see \cite{erdyau}, and our discussion below. We shall use the convention \eqn \Fou(f)(k)\;\equiv\;\hat f(k)&=&\sum_{x\in\Z^2} e^{-2\pi i kx} f(\vx) \nonumber\\ \Fou^{-1}(g)(x)\;\equiv\;\check g(x) &=&\int_{\Tor^2} dk\, e^{2 \pi i k x} g(k) \eeqn for the Fourier transform and its inverse, where $\Tor:=[-\frac12,\frac12]$. We introduce a partition of unity $\sum_{j=0}^\infty P_j=1$ on $\Z^2$, where $P_j\sim \chi(2^j<|x|\leq2^{j+1})$, $j\in\N_0$, is an approximate characteristic functions for a dyadic shell of scale $2^j$. We require that $|\Fou(P_j P_{j'})|$, for $|j-j'|\leq1$, are bump functions on $\Tor^2$ at the dual scale $2^{-j}$ satisfying $\|\Fou(P_j P_{j'})\|_{L^1(\Tor^2)}\sim 1$. We shall assume that $v_\dex$ is such that for any $j,j'\in\N_0$ with $|j-j'|\leq 1$, the Fourier transform of $P_j P_{j'} v_\dex^2$ satisfies \eqn |\Fou(P_j P_{j'} v_\dex^2)| \leq C 2^{-2\dex j}|\Fou(P_j^2)| \;, \label{Fouv-dyad-est-1} \eeqn for a constant $C$ independent of $j,j'$. Since \eqn \|P_jv_\dex\|_{\ell^\infty(\Z^2)}= \|P_j^2 v_\dex^2\|_{\ell^\infty(\Z^2)}^{1/2} \leq\|\Fou(P_j^2 v_\sigma^2)\|_{L^1(\Tor^2)}^{1/2} \sim 2^{-\dex j}\;, \eeqn this in particular implies that \eqn |x|^{\dex}|v_\dex(x)|\leq C \;, \eeqn for $0<\dex\leq\frac12$. The centered nearest neighbor lattice Laplacian $\Delta$ defines the Fourier multiplier \eqn \Fou(\Delta f) (\vk) = \en(\vk) \hat f(\vk) \;, \eeqn where \eqn \en(\vk) = 2\cos(2\pi k_1)+2\cos(2\pi k_2) \label{kinendef} \eeqn is the quantum mechanical kinetic energy of the electron. For almost every realization of $V_\omega$, $H_\omega$ is a selfadjoint operator on $\ell^2(\Z^2)$. We shall use the same argument for the determination of the localization length of eigenfunctions of $H_\omega$ as in \cite{ch1}. Let $L> e^{\lambda^{-2}}$, and \eqn \Lambda_L:=[-L,L]^2\cap\Z^2 \;. \eeqn For $\ell\ll L$ and $x\in\LL$, let \eqn R_{x,\delta, \ell}\sim \chi\big(\,\big\{y\in\Z^2\big|\,\delta\ell<|x_i-y_i|<\ell\,,\, i=1,2\big\}\,\big) \eeqn denote an approximate characteristic function supported on a cubical shell centered at $x$, of side length $2\ell$ and thickness $(1-\delta)\ell$. We shall here make the same choice for $R_{x,\delta, \ell}$ (a product of differences of Fej\'er kernels) as in \cite{ch}. Its explicit form only plays a role in connection with a result that can be straightforwardly adapted from \cite{ch} (Eq. (~\ref{free-evol-est-1})), hence we shall not write it out here. Given a fixed realization of the random potential for which $H_\omega$ is selfadjoint on $\ell^2(\Z^2)$, let $\HLL$ denote the restriction of $H_\omega$ to $\LL$. Moreover, let $\{\psi_\alpha^{(L)}\}_{\alpha\in\alg_L}$ denote an orthonormal $\HLL$-eigenbasis in $\ell^2(\LL)$ \eqn (\HLL\psi_\alpha^{(L)})(x)&=&e_\alpha^{(L)}\psi_\alpha^{(L)}(x) \;\;\; (x\in\LL) \;, \eeqn satisfying Dirichlet boundary conditions \eqn \psi_\alpha^{(L)}(x)=0 \;\;\; (x\in\partial\LL :=\Lambda_{L+1}\setminus\LL)\;. \label{eigenL} \eeqn The number of eigenfuntions is given by \eqn |\alg_L|=|\LL| \;. \eeqn Let, for $\tau>0$ arbitrary but fixed, and independent of $\lambda$ and $\dex$, \eqn I_\tau:= (-4+\tau,-\tau)\cup(\tau,4-\tau) \;. \eeqn Let \eqn \Detau:=\{\alpha\in\alg_L\big|\,e_\alpha^{(L)}\in I_\tau\}\;, \eeqn and similarly as in \cite{ch1}, let for $\e$ small \eqn \Dell&:=&\big\{\,\alpha\in\Detau\big|\, \nonumber\\ &&\hspace{1cm} \sum_{x\in\LL} |\psi_\alpha^{(L)}(x)| \, \big\| R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)} < \e\,\big\} \;. \eeqn As pointed out in \cite{ch}, the key observation is that $\{\psi_\alpha^{(L)}\}_{\alpha\in\Dell}$ contains the class of localized eigenstates with energies in $I_\tau$ that are concentrated in balls of radius $O(\frac{ \delta \ell }{ \log \ell })$, with $\delta$ independent of $\ell$. Our main result is the following theorem. \begin{theorem} \label{thm-main-1} For $\delta>0$ sufficiently small, $0<\lambda\ll\delta$, any fixed $\tau$ with $\lambda\ll\tau<\delta$, and any arbitrary $\eta>0$, \eqn \liminf_{L\rightarrow\infty}\Exp \left[ \frac {|\alg_L\setminus\alg_L(\delta^{\frac45},\delta,\ell_\dex(\lambda);I_\tau)|} {|\alg_L|} \right]\ge 1 - \delta^{\frac{1}{5}} \;. \eeqn The lower bound on the localization length $\ell_\dex(\lambda)$ satisfies the following bounds: \begin{itemize} \item In the subcritical case $0<\dex<\frac12$, there exist positive constants $\lambda_0(\dex,\eta)\ll1$ and $C_\dex$ for every fixed $0<\dex<\frac12$ such that \eqn \ell_\dex(\lambda)\geq C_\dex \lambda^{-\frac{2-2\eta}{1-2\dex}} \eeqn for all $\lambda<\lambda_0(\dex,\eta)$. \item In the critical case $\dex=\frac12$, there exists a positive constant $\lambda_0(\eta)\ll1$ such that \eqn \ell_{\dex=\frac12}(\lambda)\geq 2^{\lambda^{-\frac14+\eta}} \eeqn for all $\lambda<\lambda_0(\eta)$. \end{itemize} \end{theorem} Trivially, the bound \eqn {\Bbb P}\Big[\liminf_{L\rightarrow\infty} \frac {|\alg_L\setminus\alg_L(\e_\delta,\delta,\ell_\dex(\lambda);I_\tau)|} {|\alg_L|}>1-\delta^{\frac{1}{10}}\Big]>1-\delta^{\frac{1}{10}} \eeqn follows. Spectral restriction to the interval $I_\tau$ is necessary in our analysis for the following two reasons: On the one hand, it suppresses infrared singularities. On the other hand, it ensures certain smoothing properties of $\frac{1}{\en-z}$ as observed in \cite{bo1}. Only a slight modification of the bounds used in our analysis of the subcritical case is necessary to yield the lower bound $\lambda^{-2+\eta}$ for $\dex=0$. Including a classification of graphs argument as in \cite{erdyau, ch} would improve the lower bound to $\lambda^{-2}|\log\e|^{-1}$. We shall not further discuss these matters here, since the argument is the same as the one presented in \cite{ch} for the 3-D problem. \section{Proof of Theorem {~\ref{thm-main-1}}} Our starting point is the following key lemma. It is an extension of a joint result with L. Erd\"os and H.-T. Yau in \cite{ch1}. \begin{lemma}\label{ceylemma} Let $\e,\delta>0$ be small and $\lambda\lll1$. Assume that there exists $t^*(\delta,\ell)>0$, such that \eqn \label{mainest} &&\Exp \Big[\frac{1}{|\alg_L|} \sum_{x\in\LL}\big\| R_{x, \delta, \ell}\chi_{I_\tau}(\HLL) e^{-i t^*(\delta,\ell) \HLL } \delta_x\big\|_{\ell^2(\LL)} ^2\Big] \nonumber\\ &&\hspace{3cm}\ge 1- \e - \Exp\Big[\frac{|\cDetau|}{|\alg_L|}\Big] -C\frac{\ell}{L} \;. \eeqn Then, \eqn \liminf_{L\rightarrow\infty}\Exp\left[ \frac {|\alg_L\setminus \Dell| } {|\alg_L|}\right]\ge 1 - 4 \e^{\frac12}\;. \eeqn \end{lemma} \prf The proof follows closely a line of arguments presented in \cite{ch1}, but comprises key modifications due to the restriction of the energy range to $I_\tau$. We expand $\delta_x$ in the eigenbasis $\{\psi_\alpha^{(L)}\}$, \eqnn \delta_x &=& \sum_\alpha {a_x^\alpha} \psi_\alpha^{(L)} \; \;\\ a_x^\alpha &=& \overline{\big\langle \delta_\vx \, , \, \psi_\alpha^{(L)} \big\rangle } = \overline{\psi_\alpha^{(L)}(x)} \;, \eeqnn so that in particular, \eqn \|\delta_x\|_{\ell^2(\LL)}^2=\sum_{\alpha\in\alg_L}|a_x^\alpha|^2=1\;. \label{axalphl2norm} \eeqn Applying the Schwarz inequality, \eqn \Big\| R_{x, \delta, \ell}\chi_{I_\tau}(\HLL) e^{-i t \HLL } \delta_x\Big\|_{\ell^2(\LL)}^2 \leq (1+ \e^{-\frac12} )(A)+ (1+\e^{\frac12}) (B) \; , \label{CSest1} \eeqn where \eqn (A)&:=&\Big\|R_{x, \delta, \ell}e^{-i t \HLL } \sum_{\alpha \in \Dell}{a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \nonumber\\ &\leq& \Big\|R_{x, \delta, \ell} \sum_{\alpha \in \Dell} e^{-i t e_\alpha^{(L)} }{a_x^\alpha} \psi_\alpha^{(L)}\Big\|_{\ell^2(\LL)} \nonumber\\ &\leq& \sum_{\alpha\in \Dell} |\psi_\alpha^{(L)}(x)| \big\| R_{x, \delta, \ell} \psi_\alpha^{(L)} \big\|_{\ell^2(\LL)} \;, \eeqn using the a priori bound \eqn (A)&\leq& \Big\| \sum_{\alpha \in \Dell}e^{-i t e_\alpha^{(L)} }{a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \nonumber\\ &=&\sum_{\alpha\in\Dell}|a_x^\alpha|^2 \;\leq \; 1 \;, \eeqn which follows from $\|R_{x, \delta, \ell}\|_\infty=1$, orthonormality of $\{\psi_\alpha^{(L)}\}_{\alpha\in\alg_L}$, and (~\ref{axalphl2norm}). Moreover, \eqn (B)&:=&\Big\| R_{x, \delta, \ell} e^{-i t \HLL } \sum_{\alpha \in \cDell} {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \nonumber\\ &\leq& \Big\|\sum_{\alpha \in \cDell}e^{-i t e_\alpha^{(L)} } {a_x^\alpha} \psi_\alpha^{(L)} \Big\|_{\ell^2(\LL)}^2 \nonumber\\ &=& \sum_{\alpha \in \cDell}|a_x^\alpha|^2 \nonumber\\ &=& \sum_{\alpha \in \cDell} |\psi_\alpha^{(L)}(x)|^2 \; . \eeqn Summing over $x\in\LL$, \eqn \sum_{x\in\LL} \big\| R_{x, \delta, \ell} e^{-i t \HLL } \delta_x\big\|_{\ell^2(\LL)}^2 &\leq& (1+\e^{\frac12})\, \big|\Detau\setminus\Dell\big| \label{DbarDsplitest}\\ &+&\e (1+ \e^{-\frac12} )\, |\Dell| \; , \nonumber \eeqn using the definition of $\Dell$. Let $I_\tau^c:=\R\setminus I_\tau$. We thus get \eqn \frac {|\alg_L\setminus\Dell | } {|\alg_L|} &=&\frac{|\cDetau|}{|\alg_L|} + \frac {|\Detau\setminus\Dell | } {|\alg_L|} \nonumber\\ &\geq& \frac{|\cDetau|}{|\alg_L|} \nonumber\\ &+& \frac{1-\e^{\frac12}}{|\alg_L|}\sum_{x\in\LL} \big\| R_{x, \delta, \ell} \chi_{I_\tau}(\HLL) e^{-i t \HLL } \delta_x\big\|_{\ell^2(\LL)}^2 \nonumber\\ &-& (1+ \e^{-\frac12})\, \e -C\frac{\ell}{L} \;. \label{fracAcAlowbd} \eeqn Taking expectations and using (~\ref{mainest}), \eqn \Exp\Big[\frac {|\alg_L\setminus\Dell | } {|\alg_L|}\Big] &\geq&1-\e^{\frac{1}{2}}\Exp\Big[\frac{|\cDetau|}{\alg_L}\Big]-3\e^{\frac12} -C\frac{\ell}{L}\;. \eeqn Since $\frac{|\cDetau|}{|\alg_L|}\leq1$, this implies the claim. \endprf Our strategy therefore is to find large values for $\ell$ and $t^*(\delta,\ell)$ such that (~\ref{mainest}) is satisfied. The following lemma controls the free Schr\"odinger evolution. We adopt here the choice for the cutoff function $R_{x,\delta,\ell}$ from \cite{ch1}, which is a product of differences of Fej\'er kernels with $\|R_{x,\delta,\ell}\|_{\ell^\infty(\LL)}=1$. \begin{lemma}\label{fundestlemma} Let for $\lambda$ small and $0<\delta<1$ \eqn t^*(\delta ,\lambda):=\delta^{\frac45}\ell \;. \eeqn Then, the free evolution satisfies \eqn\label{fundest0} && \frac{1}{|\alg_L|}\sum_{x\in\LL} \big\| R_{x,\delta, \ell_\dex(\lambda)}\chi_{I_\tau}(\HLL) e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\LL^2)}^2 \nonumber\\ &&\hspace{3.5cm}\geq 1 - \delta^{\frac{3}{10}} - \frac{|\cDetau|}{|\alg_L|} -C\frac{\ell }{L}\;. \eeqn \end{lemma} \prf We note that \eqn &&\sum_{x\in\LL} \big\| R_{x,\delta, \ell_\dex(\lambda)}\chi_{I_\tau}(\HLL) e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\LL^2)}^2 \nonumber\\ &&\hspace{2cm}\geq \;(I)-(II) \eeqn where \eqn (I)&:=&\sum_{x\in\LL}\|R_{x,\delta, \ell_\dex(\lambda)} e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\LL)}^2 \nonumber\\ (II)&:=&\sum_{x\in\LL} \big\|\chi_{I_\tau^c}(\HLL) e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\LL)}^2 \;. \eeqn This follows from $\chi R^2\chi=\chi^2-\chi\overline{R^2}\chi= 1-\overline{\chi^2}-\chi\overline{R^2}\chi \geq 1-\overline{R^2}-\overline{\chi^2}=R^2-\overline{\chi^2}$, where $R\equiv R_{x,\delta, \ell_\dex(\lambda)}$, $\chi\equiv \chi_{I_\tau}(\HLL)$, and $\bar{A}:=1-A$ (so that $\overline{\chi^2}=\chi_{I_\tau^c}^2(\HLL)$). Replacing $\|\,\cdot\,\|_{\ell^2(\LL)}$ by $\|\,\cdot\,\|_{\ell^2(\Z^2)}$ in $(I)$ costs a boundary term of size $O(\ell L)$ or smaller. Since $|\alg_L|\sim L^2$, \eqn &&\frac{1}{|\alg_L|}\sum_{x\in\LL}\|R_{x,\delta, \ell } e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2(\LL)} \nonumber\\ &=&\frac{1}{|\alg_L|}\sum_{x\in\LL}\|R_{x,\delta, \ell } e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2(\Z^2)} +O(\frac{\ell}{L}) \;. \eeqn A corresponding argument in \cite{ch} for dimension 3 applied to the present case yields \eqn \|R_{x,\delta, \ell } e^{-i t^*(\delta,\lambda)\Delta } \delta_x \big\|_{\ell^2(\Z^2)} \geq 1-\delta^{\frac{3}{10}} \;. \label{free-evol-est-1} \eeqn In \cite{ch}, it was shown for a specific choice of $R_{x,\delta,\ell}$ which we also use here. On the other hand, \eqn (II)&\leq&\sum_{x\in\LL} \big\| \chi_{I_\tau^c}(\HLL) e^{-i t^*(\delta, \lambda) \Delta } \delta_x \big\|_{\ell^2(\LL)}^2 \nonumber\\ &=&\tr\Big[e^{i t^*(\delta, \lambda) \Delta }\chi_{I_\tau^c}(\HLL)e^{-i t^*(\delta, \lambda) \Delta }\Big] \nonumber\\ &=&\tr\Big[\chi_{I_\tau^c}(\HLL) \Big] \nonumber\\ &=&|\cDetau| \;. \eeqn Recalling that $|\LL|=|\alg_L|$, this completes the proof. \endprf Our result is implied by the following key lemma. It controls the interaction of the electron with the impurity potential over a time $t^*$ comparable to the localization length $\ell_\dex(\lambda)$. \begin{lemma} \label{Lemma-main-0} Let for $0<\delta<1$ \eqn t^*_{\delta,\dex,\lambda}=\delta^{\frac45}\ell_{\dex}(\lambda) \;. \label{tstar-def-1} \eeqn Then, for any arbitrary, but fixed $\tau>0$, \eqn\label{fundest10} &&\limsup_{L\rightarrow\infty} \Exp \Big[\frac{1}{|\Detau|}\sum_{x\in\LL}\big\|\chi_{I_\tau}(\HLL)\big( e^{-i t^*_{\delta,\dex,\lambda} \HLL }\delta_x- e^{-i t^*_{\delta,\dex,\lambda} \Delta } \delta_x \big) \big\|_{\ell^2(\LL)}^2\Big] \nonumber\\ &&\hspace{3cm}\leq C\tau^{\frac12}+\lambda^{\eta} \, \;. \eeqn The definition of $\ell_\dex(\lambda)$ is given in Theorem {~\ref{thm-main-1}}. \end{lemma} To establish Lemma {~\ref{Lemma-main-0}}, it suffices to prove the following estimate. \begin{lemma} \label{Lemma-main-1} Under the assumptions of Lemma {~\ref{Lemma-main-0}}, \eqn &&\sup_{\phi\in\ell^2(\Z^2)\atop\|\phi\|_{\ell^2(\LL)}=1} \Exp\big[\|\chi_{I_\tau}(H_\omega)\big( e^{-i t^*_{\delta,\dex,\lambda} H_\omega }- e^{-i t^*_{\delta,\dex,\lambda} \Delta } \big)\phi\|_{\ell^2(\Z^3)}^2\big]0$, in our bounds), our methods require an expression of the above form for $R_{N,t}$ (because we will apply the time partitioning trick used in \cite{erdyau} and \cite{ch}). To this end, we claim that \eqn R_{N,t}&=&R_{N,t}^{(0)}+R_{N,t}^{(1)} \label{RNt-def-2} \eeqn with \eqn R_{N,t}^{(0)}&:=&e^{-itH_\omega}\frac{-\lambda}{2\pi i}\int_{\Ch} d\alpha \frac{1}{H_\omega-\alpha-i\e}V\tilde\phi_{N,\e}(\alpha) \\ R_{N,t}^{(1)}&:=&-i\lambda\int_0^tds e^{-i(t-s)H_\omega}V \phi_{N,s}\;. \eeqn To see this, we note that (~\ref{RNt-def-1}) implies \eqn \partial_t R_{N,t} = -iH_\omega R_{N,t} -i\lambda V \phi_{N,t} \;, \eeqn which is solved by the variation of constants formula (~\ref{RNt-def-2}). We note that if in place of $\Ch$, we were given a connected $\alpha$-integration contour that encloses the spectrum of $H_\omega$, $R_{N,t}^{(0)}$ would vanish. This is because the integration contour could then be arbitrarily deformed away from the spectrum of $H_\omega-i\e$ into the upper half plane, as there is no obstructing phase factor $e^{-it\alpha}$. Furthermore, due to the truncation of the integration contour to $\Ch$, it is also necessary to control \eqn &&\|\chi_{I_\tau}(H_\omega)\big(\phi_{0,t}- e^{-it\Delta}\phi_0\big)\|_{\ell^2(\Z^2)}^2 \nonumber\\ &&\hspace{2cm}\leq \int_{\Tor^2}dp \Big|\int_{C\setminus \Ch}d\alpha e^{-it\alpha} \frac{1}{\en(p)-\alpha-i\e}\Big|^2 \;, \label{free-evol-error-1} \eeqn where \eqn \tilde C&:=&[-4-\e,4+\e]\cup(4+\e-2i\e[0,1])\cup \nonumber\\ &&\hspace{2cm}([-4-\e,4+\e]-2i\e)\cup(-4-\e-2i\e[0,1])\;. \eeqn We write $\tilde C\setminus \Ch =\tilde C_-\cup \tilde C_0\cup \tilde C_+$, where $\tilde C_{\pm}:=\{z\in \tilde C\setminus\Ch\big| \pm\Re(z)>2\}$. $\tilde C_-$ and $\tilde C_+$ are connected arcs, while $\tilde C_0$ consists of two disjoint, parallel lines, all of length $O(\tau)$. We claim that \eqn &&\Big|\int_{\tilde C_-\cup \tilde C_0\cup \tilde C_+}d \alpha e^{-it\alpha} \frac{1}{\en(p)-\alpha-i\e}\Big| \nonumber\\ &&\hspace{2cm}< C\Big[\chi(|\en(p)+4|<2\tau)+\chi(|\en(p)+4|<2\tau) \nonumber\\ &&\hspace{3cm} +\chi(|\en(p)|<4\tau) + \frac{\e}{\tau}\Big]\;. \eeqn For fixed $p$, the size of \eqn \int_{\tilde C_- }d\alpha e^{-it\alpha} \frac{1}{\en(p)-\alpha-i\e} \eeqn can be estimated as follows. If $|\en(p)-4|<2\tau$, $\tilde C_-$ can be deformed into a loop that encloses $\en(p)-i\e$, and a disjoint arc of length $O(\e)$ connecting the endpoints of $\tilde C_-$. The loop picks up the resolvent at $\en(p)-i\e$, yielding a factor $e^{-it(\en(p)-i\e)}$. The integral over the arc can be bounded by its length $O(\e)$, multiplied with the bound $\frac1\e$ on the resolvent. Both contributions are $O(1)$. If $|\en(p)+4|>2\tau$, $\tilde C_-$ can be deformed into a straight line of length $2\e$ connecting its endpoints, which has a distance $\geq\tau$ from $\en(p)$. Thus, the modulus of the resolvent is bounded by $O(\frac1\tau)$, and we get an error of order $O(\frac\e\tau)$ from integration. The cases $\tilde C_0$ and $\tilde C_+$ are similar. Thus, \eqn (~\ref{free-evol-error-1})&<&C\Big[\mes\{|\en(p)+4|<2\tau\}+ \mes\{|\en(p)|<4\tau\} \nonumber\\ &&\hspace{2cm}+\mes\{|\en(p)-4|<2\tau\}+ \frac\e\tau\Big] \nonumber\\ &<& C\tau^{\frac12}\;, \eeqn as $\e$ will be chosen $\ll\tau$ in the end. The Schwarz inequality thus yields \eqn &&\Exp\Big[\|\chi_{I_\tau}(H_\omega)\big(\phi_t^{(h)}- e^{-it\Delta}\phi_0\big)\|_{\ell^2(\Z^2)}^2\Big] \nonumber\\ &&\hspace{3cm}\leq\;C\tau^{\frac12} + 2\, \Exp\Big[ \big\| \sum_{n=1}^N \phi_{n,t} \big\|_2^2 \Big] +2 \,\Exp\Big[ \big\| \chi_{I_\tau}(H_\omega)\Rem_{N,t} \big\|_2^2 \Big] \nonumber\\ &&\hspace{3cm}=\;C\tau^{\frac12} + 2\sum_{n,n'=1}^N \Exp\Big[ \langle\phi_{n',t},\phi_{n,t}\rangle \Big] +2\, \Exp\Big[ \big\| \chi_{I_\tau}(H_\omega)\Rem_{N,t} \big\|_2^2 \Big] \; . \label{exp-phi-ell2-Schwarz-1} \eeqn Clearly, if $n+n'\not\in2\N$, $\Exp[\langle\phi_{n',t},\phi_{n,t}\rangle]=0$. We partition $V$ into dyadic shells, \eqn V=\sum_{j=0}^{J+1} V_j \;, \eeqn where \eqn V_j(x)&=&P_j(x) v_\dex(x)\omega_x \eeqn for $0\leq j\leq J$. The cutoff functions $P_j$ are defined at the beginning of section {~\ref{intro-sect-1}}. For $j>J$, we rename $P_j\rightarrow \tilde P_j$, and define \eqn P_{J+1}&:=&\sum_{j=J+1}^\infty \tilde P_j \label{tildPi-def-1} \eeqn Hence, the functions $V_j$ are supported on dyadic annuli of radii and thicknesses $\sim 2^j$ centered at the origin, $j=1,\dots,J$, while $V_{J+1}$ is the part of $V$ supported in regions with a distance larger than $2^{J+1}$ from the origin. Let \eqn R_z:=\frac{1}{\Delta-z}\;. \eeqn Then, we have \eqn \Exp\lb \langle\phi_{n',t},\phi_{n,t}\rangle \rb &=& \sum_{j_1,\dots,j_{2\bar n}=1}^{J+1} \frac{e^{2\e t} \lambda^{2\bar n}}{(2\pi)^2} \int_{\Ch\times \overline \Ch} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{0.5cm}\Exp\Big[\langle\phi_0\,,\, R_{\alpha+i\e} V_{j_1} R_{\beta-i\e}V_{j_2} R_{\beta-i\e}\cdots\cdots \nonumber\\ &&\hspace{1.5cm}\cdots\cdots V_{j_n} R_{\beta-i\e} R_{\alpha+i\e} V_{j_{n+2}} \cdots \cdots V_{j_{2\bar n}} R_{\alpha+i\e} \phi_0\rangle\Big] \label{exp-phinn-res-1} \eeqn for $1\leq n,n' \leq N$, and $\bar n:=\frac{n+n'}{2}\in\N$. $\overline \Ch$ denotes the complex conjugate of $\Ch$, and is taken in the counterclockwise direction by the variable $\beta$. For $1\leq n,n' \leq N$, and $\bar n:=\frac{n+n'}{2}\in\N$, let \eqn \up&=&(p_0,\dots,p_n,p_{n+1},\dots,p_{2\bar n+1}) \eeqn and \eqn (\alpha_j,\sigma_j) &=& \left\{\begin{array}{ll}(\alpha,1)& 0\leq j\leq n\\ (\beta,-1)&n0$ and $\pi\in\Pi_{n,n'}$, there exists a finite constant $C_\tau$ depending only on $\tau$ such that defining \eqn \ampi:=C_\tau(\cj \lambda^2\log\frac1\e +\e^{-1}\dex^{-1} 2^{-2\dex J}\lambda^2\log\frac1\e) \label{ampi-def-1} \eeqn and \eqn \cj:=\left\{ \begin{array}{cl} J&{\rm if}\;\dex=\frac12\\ \frac{ 2^{(1-2\dex){J+1}} -1 }{ 2^{(1-2\dex)}-1 }&{\rm if}\;0<\dex<\frac12\;, \end{array}\right. \eeqn one gets \eqn |\amp(\pi)|<(\log\frac1\e)^2(\ampi)^{\bar n} \;. \eeqn \end{lemma} \prf We choose a spanning tree $T$ on $\pi$ that contains all contraction lines between the pairs of random potentials, and $\bar n$ out of all particle lines. In addition, $T$ shall include those particle lines labeled by the momenta $p_n,p_{2\bar n+1}$, but not those labeled by $p_0,p_{n+1}$. We then call $T$ {\em admissible}. Momenta (resolvents) supported on $T$ are referred to as tree momenta (resolvents), and momenta (resolvents) supported on its complement $T^c$ are called loop momenta (resolvents). We shall then group together every tree resolvent with one adjacent contraction line carrying a factor $\Fou(P_{j_i}P_{j_{i'}}v_\dex^2)$, $|j_i-j_{i'}|\leq1$, and estimate the corresponding convolution integral of the form (~\ref{convol-est-1}) below. All loop resolvents supported on $T^c$ are estimated in $L^1(\Tor^2)$. We recall that \eqn \amp(\pi)&=&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1} \frac{e^{2\e t} \lambda^{2\bar n}}{(2\pi)^2} \int_{\Ch\times \overline \Ch} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{0.5cm} \int_{(\Tor^3)^{2\bar n+2}} d\up \,\delta (p_n-p_{n+1}) \delta_{\pi}(\uj;\up;v_\dex) \nonumber\\ &&\hspace{3.5cm} \overline{\Fou(\phi_0)(p_0)}\Fou(\phi_0)(p_{2\bar n+1}) \nonumber\\ &&\hspace{2cm} \prod_{l=0}^{2\bar n+1} \frac{1}{\en(p_l)-\alpha_l-i\sigma_l\e} \; . \label{exp-sum-graphs-2} \eeqn for $\uj=(j_1,\dots,j_{2\bar n})$. We integrate out the variable $p_{n+1}$, and apply the coordinate transformation $p_j\mapsto p_j+p_n$, for all $j=0,\dots,n-1,n+2,\dots,2\bar n+1$. It is easy to see that thereby, $\delta_{\pi}(\uj;\up;v_\dex)$ becomes independent of $p_n$ and $p_{n+1}$. We obtain \eqn \amp(\pi)&=&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1} \frac{e^{2\e t} \lambda^{2\bar n}}{(2\pi)^2} \int_{\Ch\times \overline \Ch} d\alpha d\beta e^{-it(\alpha-\beta)} \nonumber\\ &&\hspace{0.5cm} \int_{(\Tor^3)^{2\bar n}} d\up' \, \delta_{\pi}'(\uj;\up';v_\dex) \nonumber\\ &&\hspace{0.5cm} \int_{\Tor^2}dp_n\frac{1}{\en(p_n)-\alpha-i\e}\frac{1}{\en(p_n)-\beta+i\e} \nonumber\\ &&\hspace{3.5cm} \overline{\Fou(\phi_0)(p_0+p_n)}\Fou(\phi_0)(p_{2\bar n+1}+p_n) \nonumber\\ &&\hspace{2cm} \prod_{l=0\atop l\neq n,n+1}^{2\bar n+1} \frac{1}{\en(p_l+p_n)-\alpha_l-\sigma_l\e} \; , \label{exp-sum-graphs-3} \eeqn where \eqn \up':=(p_0,\dots,p_{n-1},p_{n+2},\dots,p_{2\bar n+1}) \eeqn and \eqn \delta_{\pi}'(\uj;\up';v_\dex):= \delta_{\pi}(\uj;\up;v_\dex)\Big|_{p_{n+1}, p_n\rightarrow0 }\;. \eeqn Clearly, \eqn |\amp(\pi)|&\leq&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1} \frac{e^{2\e t} \lambda^{2\bar n}}{(2\pi)^2} \Big[\sup_{q,q'\in\Tor^2}\int_{\Ch\times \overline \Ch} |d\alpha|\,|d\beta| \nonumber\\ &&\hspace{0.5cm} \int_{\Tor^2}dp_n\frac{1}{\en(p_n)-\alpha-i\e}\frac{1}{\en(p_n)-\beta+i\e} \nonumber\\ &&\hspace{3.5cm} \overline{\Fou(\phi_0)(p_0+q)}\Fou(\phi_0)(p_{2\bar n+1}+q')\Big] \nonumber\\ &&\hspace{0.5cm} \sup_{\alpha\in\Ch}\sup_{\beta\in\overline{\Ch}}\sup_{p_n\in\Tor^2} \Big[\int_{(\Tor^3)^{2\bar n}} d\up' \, \delta_{\pi}'(\uj;\up';v_\dex) \nonumber\\ &&\hspace{3.5cm} \prod_{l=0\atop l\neq n,n+1}^{2\bar n+1} \frac{1}{|\en(p_l+p_n)-\alpha_l-\sigma_l\e|} \Big] \; . \label{exp-sum-graphs-4} \eeqn Thus, dividing the resolvents into tree and loop terms and defining \eqn \delta_\pi(\uj):= \prod_{m=1}^{\bar n} \delta_{|j_{i}-j_{i'}|\leq1} \Big|_{i\sim_m i'}\;, \label{deltapi-def-2} \eeqn (see also (~\ref{deltapi-def-1})), one gets \eqn |\amp(\pi)|&\leq&\sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1} \frac{e^{2\e t} \lambda^{2\bar n}}{(2\pi)^2} \delta_{\pi}(\uj) \nonumber\\ &&\hspace{0.5cm} \Big[\sup_{q,q'\in\Tor^2}\int_{\Tor^2}dp_{n}|\phi_0(p_n+q)|\,|\phi_0(p_n+q')|\Big] \nonumber\\ &&\hspace{1cm}\Big[\sup_{p_n\in\Tor^2}\int_{\Ch}|d\alpha|\,\frac{1}{|\en(p_n)-\alpha-i\e|} \nonumber\\ &&\hspace{2.5cm}\int_{\overline{\Ch}}|d\beta|\, \frac{1}{|\en(p_{n})-\beta+i\e|}\Big] \nonumber\\ && \sup_{\alpha\in\Ch}\sup_{\beta\in\overline{\Ch}}\sup_{p_n\in\Tor^2} \Big\{\; \Big[\prod_{ T^c}\Big\|\frac{1}{\en(p_r)-\alpha_i\pm i\e}\Big\|_{L^1(\Tor^2)}\Big] \nonumber\\ &&\hspace{2cm} \Big[\prod_{ T} \Big\|\,\Big|\frac{1}{\en-\alpha_i\pm i\e}\Big|* \big|\Fou(P_{j_i}^2 v_\dex^2)\big|\,\Big\|_{L^\infty(\Tor^2)}\Big] \; \Big\} \;. \eeqn Assuming (~\ref{Pj-ass-Rnorm-1}) and using Lemma {~\ref{R-Lnorm-bounds-lemma-1}}, one finds \eqn \sup_{q\in\Tor^2} \int_{\Tor^2} dp\Big|\frac{1}{\en(p)-\alpha-i\e}\Big|\,\big|\Fou(P_j^2 v_\dex^2)(p-q)\big| \leq C_\tau 2^{(1-2\dex)j} \label{convol-est-1} \eeqn if $0\leq j\leq J$, and \eqn \sup_{q\in\Tor^2} \int_{\Tor^2} dp\Big|\frac{1}{\en(p)-\alpha-i\e}\Big|\,\big|\Fou(P_{J+1}^2 v_\dex^2)(p-q)\big| \leq \e^{-1}\dex^{-1}2^{-2\dex J} \label{convol-est-2} \eeqn if $j=J+1$. Hence, \eqn |\amp(\pi)|&\leq&(C\log\frac1\e)^2\|\phi_0\|^2_{L^2(\Tor^2)} \sum_{j_0,\dots,j_{2\bar n+1}=0}^{J+1} \delta_{\pi}(\uj) (C\log\frac1\e)^{|T^c|} \nonumber\\ &&\hspace{1cm}\prod_{i=1}^{2\bar n}\Big(2^{(1-2\dex) j_i }\chi(j\leq J) + \dex^{-1}\e^{-1}2^{-2\dex J} \delta_{j_i,J+1}\Big)^{1/2} \;, \label{exp-sum-graphs-5} \eeqn where we have used \eqn \sup_{p_n\in\Tor^2}\int_{\Ch}|d\alpha|\,\frac{1}{|\en(p_n)-\alpha-i\e|}0$ be arbitrary but fixed. Setting \eqn \e&=&2^{-J} \label{eps-def-lambda-1}\\ JK_{\dex}(J)&=&\lambda^{-2+2\eta} \label{eps-def-lambda-2} \eeqn we find \eqn K_{\dex}(J)\lambda^2\log\frac1\e&=&J K_\dex(J)\lambda^2\;\leq\;\lambda^{2\eta} \nonumber\\ \e^{-1}\dex^{-1}2^{-2\dex J}\lambda^2\log\frac1\e&=& \dex^{-1}2^{(1-2\dex)J}\lambda^2\log\frac1\e \nonumber\\ &=&\dex^{-1}J K_\dex(J) \;, \eeqn so that \eqn \ampi&<&\lambda^{ 1.9\eta } \;, \eeqn for $\lambda$ sufficiently small (depending on $\dex$). Choosing \eqn N&=&\frac{\eta\log\frac1\lambda}{10\log\log\frac1\lambda} \;, \eeqn one gets (noting that $\e>\lambda^2$) \eqn \sum_{ n=1}^N n! (\log\frac1\e)^2 (\ampi)^{ n} &<&C(\log\frac1\lambda)^2\sum_{ n=1}^N (N\ampi)^{ n} \nonumber\\ &<&C(\log\frac1\lambda)^2\sum_{ n=1}^N \lambda^{1.5 \eta n} \;<\;\lambda^{1.1\eta} \eeqn and \eqn N!\frac{\lambda^2}{\tau^2}(\log\frac1\e)^2 (\ampi)^{N}&<&C(\log\frac1\lambda)^2 (N\ampi)^N \;<\;\lambda \eeqn for $\tau\gg\lambda$. Choosing \eqn \kappa&=&(\log\frac1\lambda)^{\frac{30}{\eta(1-2\dex)}} \;, \eeqn one gets \eqn \lambda^2 (3\kappa N)^2(\log\frac1\e)^2\sum_{n=N+1}^{4N-1} n! (\ampi)^{n}&<&C\lambda^2 (\log\frac1\lambda)^{\frac{100}{(1-2\dex)\eta}} (4N\ampi)^N \nonumber\\ &<&\lambda^{2\eta} \;. \eeqn Furthermore, since \eqn \kappa^{(1-2\dex)N}\;>\;\lambda^{-3} \;, \eeqn one finds \eqn (4N)!\frac{\lambda^2}{\e^2\kappa^{(1-2\dex)N}} (\log\frac1\e)^2 (\ampi)^{4N}&<&(4N\ampi)^{4N} \;<\;\lambda^{2\eta}\;. \eeqn Thus, for $\lambda$ sufficiently small (depending on $\dex$ and $\eta$), \eqn l.h.s.\;of\;(~\ref{Lemma-main-est-1}) 0$ be arbitrary (small) but fixed. Setting \eqn J&=&N\;=\;\lambda^{-\frac14+\eta} \nonumber\\ \e&=&2^{-\lambda^{-\frac14+\eta}} \;=\;2^{-N}\;=\;2^{-J} \;, \label{eps-def-lambda-3} \eeqn we get, for sufficiently small $\lambda>0$, \eqn \ampi &=&C_\tau\Big(J\lambda^2\log\frac1\e+2\e^{-1}2^{-J}\log\frac1\e\Big) \nonumber\\ &<&2C_\tau N^2\lambda^2 \eeqn and \eqn N^2\ampi&<&\lambda^{3\eta}\;. \eeqn Then, \eqn \sum_{n=1}^N n!(\log\frac1\e)^2(\ampi)^n&<&N^2\ampi+\sum_{n=2}^N N^2(N\ampi)^n \;<\;\lambda^{2\eta} \eeqn and \eqn N!\frac{\lambda^2}{\tau^2}(\log\frac1\e)^2(\ampi)^N &<&\frac{\lambda^{2}}{\tau^2}N^2(N\ampi)^N \;<\;\lambda \;. \eeqn Furthermore, \eqn N!\e^{-2}\lambda^2(\log\frac1\e)^2(\ampi)^N&<& \lambda^2 N^2 (4N\ampi)^N \nonumber\\ &<&\lambda^{3\eta}(4\lambda^{2\eta})^{\lambda^{-\frac23+\eta}} \;<\;\lambda \;. \eeqn In conclusion, \eqn l.h.s.\;of\;(~\ref{Lemma-main-est-1}) > >> }if }if exch {imagemask}{image}ifelse end }bd /cguidfix{statusdict begin mark version end {cvr}stopped{cleartomark 0}{exch pop}ifelse 2012 lt{dup findfont dup length dict begin {1 index/FID ne 2 index/UniqueID ne and {def} {pop pop} ifelse}forall currentdict end definefont pop }{pop}ifelse }bd /t_array 0 def /t_i 0 def /t_c 1 string def /x_proc{ exch t_array t_i get add exch moveto /t_i t_i 1 add store }bd /y_proc{ t_array t_i get add moveto /t_i t_i 1 add store }bd /xy_proc{ t_array t_i 2 copy 1 add get 3 1 roll get 4 -1 roll add 3 1 roll add moveto /t_i t_i 2 add store }bd /sop 0 def /cp_proc/x_proc ld /base_charpath { /t_array xs /t_i 0 def { t_c 0 3 -1 roll put currentpoint t_c cply sop cp_proc }forall /t_array 0 def }bd /sop/stroke ld /nop{}def /xsp/base_charpath ld /ysp{/cp_proc/y_proc ld base_charpath/cp_proc/x_proc ld}bd /xysp{/cp_proc/xy_proc ld base_charpath/cp_proc/x_proc ld}bd /xmp{/sop/nop ld /cp_proc/x_proc ld base_charpath/sop/stroke ld}bd /ymp{/sop/nop ld /cp_proc/y_proc ld base_charpath/sop/stroke ld}bd /xymp{/sop/nop ld /cp_proc/xy_proc ld base_charpath/sop/stroke ld}bd /refnt{ findfont dup length dict copy dup /Encoding 4 -1 roll put definefont pop }bd /renmfont{ findfont dup length dict copy definefont pop }bd L3? dup dup{save exch}if /Range 0 def /Domain 0 def /Encode 0 def /Decode 0 def /Size 0 def /DataSource 0 def /mIndex 0 def /nDomain 0 def /ival 0 def /val 0 def /nDomM1 0 def /sizem1 0 def /srcEncode 0 def /srcDecode 0 def /nRange 0 def /d0 0 def /r0 0 def /di 0 def /ri 0 def /a0 0 def /a1 0 def /r1 0 def /r2 0 def /dx 0 def /Nsteps 0 def /sh3tp 0 def /ymax 0 def /ymin 0 def /xmax 0 def /xmin 0 def /min { 2 copy gt {exch pop}{pop}ifelse }bd /max { 2 copy lt {exch pop}{pop}ifelse }bd /inter { 1 index sub 5 2 roll 1 index sub 3 1 roll sub 3 1 roll div mul add }bd /setupFunEvalN { begin /nDomM1 Domain length 2 idiv 1 sub store /sizem1[ 0 1 nDomM1 { Size exch get 1 sub }for ]store /srcEncode currentdict/Encode known { Encode }{ [ 0 1 nDomM1 { 0 sizem1 3 -1 roll get }for ] }ifelse store /srcDecode currentdict/Decode known {Decode}{Range}ifelse store /nRange Range length 2 idiv store end }bd /FunEvalN { begin nDomM1 -1 0 { 2 mul/mIndex xs Domain mIndex get max Domain mIndex 1 add get min Domain mIndex get Domain mIndex 1 add get srcEncode mIndex get srcEncode mIndex 1 add get inter round cvi 0 max sizem1 mIndex 2 idiv get min nDomM1 1 add 1 roll }for nDomM1 1 add array astore/val xs nDomM1 0 gt { 0 nDomM1 -1 0 { dup 0 gt { /mIndex xs val mIndex get 1 index add Size mIndex 1 sub get mul add }{ val exch get add }ifelse }for }{ val 0 get }ifelse nRange mul /ival xs 0 1 nRange 1 sub { dup 2 mul/mIndex xs ival add DataSource exch get 0 255 srcDecode mIndex 2 copy get 3 1 roll 1 add get inter Range mIndex get max Range mIndex 1 add get min }for end }bd /sh2 { /Coords load aload pop 3 index 3 index translate 3 -1 roll sub 3 1 roll exch sub 2 copy dup mul exch dup mul add sqrt dup scale atan rotate /Function load setupFunEvalN clippath {pathbbox}stopped {0 0 0 0}if newpath /ymax xs /xmax xs /ymin xs /xmin xs currentdict/Extend known { /Extend load 0 get { /Domain load 0 get /Function load FunEvalN sc xmin ymin xmin abs ymax ymin sub rectfill }if }if /dx/Function load/Size get 0 get 1 sub 1 exch div store gsave /di ymax ymin sub store /Function load dup /Domain get dup 0 get exch 1 get 2 copy exch sub dx mul exch { 1 index FunEvalN sc 0 ymin dx di rectfill dx 0 translate }for pop grestore currentdict/Extend known { /Extend load 1 get { /Domain load 1 get /Function load FunEvalN sc 1 ymin xmax 1 sub abs ymax ymin sub rectfill }if }if }bd /shp { 4 copy dup 0 gt{ 0 exch a1 a0 arc }{ pop 0 moveto }ifelse dup 0 gt{ 0 exch a0 a1 arcn }{ pop 0 lineto }ifelse fill dup 0 gt{ 0 exch a0 a1 arc }{ pop 0 moveto }ifelse dup 0 gt{ 0 exch a1 a0 arcn }{ pop 0 lineto }ifelse fill }bd /calcmaxs { xmin dup mul ymin dup mul add sqrt xmax dup mul ymin dup mul add sqrt xmin dup mul ymax dup mul add sqrt xmax dup mul ymax dup mul add sqrt max max max }bd /sh3 { /Coords load aload pop 5 index 5 index translate 3 -1 roll 6 -1 roll sub 3 -1 roll 5 -1 roll sub 2 copy dup mul exch dup mul add sqrt /dx xs 2 copy 0 ne exch 0 ne or { exch atan rotate }{ pop pop }ifelse /r2 xs /r1 xs /Function load dup/Size get 0 get 1 sub /Nsteps xs setupFunEvalN dx r2 add r1 lt{ 0 }{ dx r1 add r2 le { 1 }{ r1 r2 eq { 2 }{ 3 }ifelse }ifelse }ifelse /sh3tp xs clippath {pathbbox}stopped {0 0 0 0}if newpath /ymax xs /xmax xs /ymin xs /xmin xs dx dup mul r2 r1 sub dup mul sub dup 0 gt { sqrt r2 r1 sub atan /a0 exch 180 exch sub store /a1 a0 neg store }{ pop /a0 0 store /a1 360 store }ifelse currentdict/Extend known { /Extend load 0 get r1 0 gt and { /Domain load 0 get/Function load FunEvalN sc { { dx 0 r1 360 0 arcn xmin ymin moveto xmax ymin lineto xmax ymax lineto xmin ymax lineto xmin ymin lineto eofill } { r1 0 gt{0 0 r1 0 360 arc fill}if } { 0 r1 xmin abs r1 add neg r1 shp } { r2 r1 gt{ 0 r1 r1 neg r2 r1 sub div dx mul 0 shp }{ 0 r1 calcmaxs dup r2 add dx mul dx r1 r2 sub sub div neg exch 1 index abs exch sub shp }ifelse } }sh3tp get exec }if }if /d0 0 store /r0 r1 store /di dx Nsteps div store /ri r2 r1 sub Nsteps div store /Function load /Domain load dup 0 get exch 1 get 2 copy exch sub Nsteps div exch { 1 index FunEvalN sc d0 di add r0 ri add d0 r0 shp { d0 0 r0 a1 a0 arc d0 di add 0 r0 ri add a0 a1 arcn fill d0 0 r0 a0 a1 arc d0 di add 0 r0 ri add a1 a0 arcn fill }pop /d0 d0 di add store /r0 r0 ri add store }for pop currentdict/Extend known { /Extend load 1 get r2 0 gt and { /Domain load 1 get/Function load FunEvalN sc { { dx 0 r2 0 360 arc fill } { dx 0 r2 360 0 arcn xmin ymin moveto xmax ymin lineto xmax ymax lineto xmin ymax lineto xmin ymin lineto eofill } { xmax abs r1 add r1 dx r1 shp } { r2 r1 gt{ calcmaxs dup r1 add dx mul dx r2 r1 sub sub div exch 1 index exch sub dx r2 shp }{ r1 neg r2 r1 sub div dx mul 0 dx r2 shp }ifelse } } sh3tp get exec }if }if }bd /sh { begin /ShadingType load dup dup 2 eq exch 3 eq or { gsave newpath /ColorSpace load scs currentdict/BBox known { /BBox load aload pop 2 index sub 3 index 3 -1 roll exch sub exch rectclip }if 2 eq {sh2}{sh3}ifelse grestore }{ pop (DEBUG: shading type unimplemented\n)print flush }ifelse end }bd {restore}if not dup{save exch}if L3?{ /sh/shfill ld /csq/clipsave ld /csQ/cliprestore ld }if {restore}if end setpacking %%EndFile %%EndProlog %%BeginSetup %%EndSetup %%Page: 1 1 %%PageBoundingBox: 0 0 446 59 %%BeginPageSetup cg_md begin bp sdmtx %RBIBeginFontSubset: DUXVBD+Helvetica %!PS-TrueTypeFont-1.0000-0.0000-2 14 dict begin/FontName /DUXVBD+Helvetica def /PaintType 0 def /Encoding 256 array 0 1 255{1 index exch/.notdef put}for dup 33 /one put dup 34 /n put dup 35 /plus put dup 36 /two put readonly def 42/FontType resourcestatus{pop pop false}{true}ifelse %APLsfntBegin {currentfile 0(%APLsfntEnd\n)/SubFileDecode filter flushfile}if /FontType 42 def /FontMatrix matrix def /FontBBox[2048 -342 1 index div -914 2 index div 2036 3 index div 2100 5 -1 roll div]cvx def /sfnts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def /CharStrings 5 dict dup begin /.notdef 0 def /plus 1 def /one 2 def /two 3 def /n 4 def end readonly def currentdict dup/FontName get exch definefont pop end %APLsfntEnd 42/FontType resourcestatus{pop pop true}{false}ifelse {currentfile 0(%APLT1End\n)/SubFileDecode filter flushfile}if /FontType 1 def /FontMatrix [ 0.00048828125 0 0 0.00048828125 0 0 ] def /FontBBox{-342 -914 2036 2100}def /UniqueID 4045371 def currentdict currentfile eexec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cleartomark end %APLT1End %RBIEndFontSubset /DUXVBD+Helvetica cguidfix /F1.1/DUXVBD+Helvetica renmfont %RBIBeginFontSubset: LWIDED+Helvetica-Oblique %!PS-TrueTypeFont-1.0000-0.0000-2 14 dict begin/FontName /LWIDED+Helvetica-Oblique def /PaintType 0 def /Encoding 256 array 0 1 255{1 index exch/.notdef put}for dup 33 /p put dup 34 /zero put dup 35 /two put dup 36 /n put dup 37 /plus put dup 38 /one put readonly def 42/FontType resourcestatus{pop pop false}{true}ifelse %APLsfntBegin {currentfile 0(%APLsfntEnd\n)/SubFileDecode filter flushfile}if /FontType 42 def /FontMatrix matrix def /FontBBox[2048 -342 1 index div -914 2 index div 2036 3 index div 2100 5 -1 roll div]cvx def /sfnts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def /CharStrings 7 dict dup begin /.notdef 0 def /plus 1 def /zero 2 def /one 3 def /two 4 def /n 5 def /p 6 def end readonly def currentdict dup/FontName get exch definefont pop end %APLsfntEnd 42/FontType resourcestatus{pop pop true}{false}ifelse {currentfile 0(%APLT1End\n)/SubFileDecode filter flushfile}if /FontType 1 def /FontMatrix [ 0.00048828125 0 0 0.00048828125 0 0 ] def /FontBBox{-342 -914 2036 2100}def /UniqueID 4045371 def currentdict currentfile eexec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cleartomark end %APLT1End /LWIDED+Helvetica-Oblique /LWIDED+Helvetica-Oblique findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall FontMatrix [1 0 0.212557 1 0 0] matrix concatmatrix /FontMatrix exch def currentdict end definefont pop %RBIEndFontSubset /LWIDED+Helvetica-Oblique cguidfix /F2.1/LWIDED+Helvetica-Oblique renmfont [ /CIEBasedA 5 dict dup begin /WhitePoint [ 0.9505 1.0000 1.0891 ] def /DecodeA { { 1.8008 exp } bind exec} bind def /MatrixA [ 0.9642 1.0000 0.8249 ] def /RangeLMN [ 0.0 2.0000 0.0 2.0000 0.0 2.0000 ] def /DecodeLMN [ { 0.9857 mul} bind { 1.0000 mul} bind { 1.3202 mul} bind ] def end ] /Cs1 exch/ColorSpace dr pop [ /CIEBasedABC 4 dict dup begin /WhitePoint [ 0.9505 1.0000 1.0891 ] def /DecodeABC [ { 1.8008 exp } bind { 1.8008 exp } bind { 1.8008 exp } bind ] def /MatrixABC [ 0.4294 0.2332 0.0202 0.3278 0.6737 0.1105 0.1933 0.0938 0.9580 ] def /RangeLMN [ 0.0 0.9505 0.0 1.0000 0.0 1.0891 ] def end ] /Cs2 exch/ColorSpace dr pop %%EndPageSetup /Cs1 SC 1 sc q 0 0 446 59 rc 0 59 m 446 59 l 446 0 l 0 0 l h f 0.60000002 i 0 sc 1 0 0 -1 -88.5 120.5 cm 96 87 m 199 87 l 516 87 l S /Cs2 SC 0 0 0 sc CM 181.5 30.5 m 187.5 30.5 l 187.5 36.5 l 181.5 36.5 l h 181.5 30.5 m f /Cs1 SC 0 sc 1 0 0 -1 -88.5 120.5 cm 270 90 m 276 90 l 276 84 l 270 84 l h 270 90 m S 1 sc CM 331.5 30.5 m 337.5 30.5 l 337.5 36.5 l 331.5 36.5 l h 331.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 420 90 m 426 90 l 426 84 l 420 84 l h 420 90 m S 1 sc CM 61.5 30.5 m 67.5 30.5 l 67.5 36.5 l 61.5 36.5 l h 61.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 150 90 m 156 90 l 156 84 l 150 84 l h 150 90 m S 1 sc CM 121.5 30.5 m 127.5 30.5 l 127.5 36.5 l 121.5 36.5 l h 121.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 210 90 m 216 90 l 216 84 l 210 84 l h 210 90 m S 1 sc CM 151.5 30.5 m 157.5 30.5 l 157.5 36.5 l 151.5 36.5 l h 151.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 240 90 m 246 90 l 246 84 l 240 84 l h 240 90 m S 1 sc CM 271.5 30.5 m 277.5 30.5 l 277.5 36.5 l 271.5 36.5 l h 271.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 360 90 m 366 90 l 366 84 l 360 84 l h 360 90 m S 1 sc CM 241.5 30.5 m 247.5 30.5 l 247.5 36.5 l 241.5 36.5 l h 241.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 330 90 m 336 90 l 336 84 l 330 84 l h 330 90 m S 1 sc CM 211.5 30.5 m 217.5 30.5 l 217.5 36.5 l 211.5 36.5 l h 211.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 300 90 m 306 90 l 306 84 l 300 84 l h 300 90 m S [ 4 4 ] 0 d 183 90 m 183 99 l 243 99 l 243 90 l S 0 i 1 0 0 -1 37.5 45.5 cm /F1.1[ 12 0 0 -12 0 0]sf -4 3 m (!)s 1 0 0 -1 154.5 45.5 cm -4 3 m (")s 1 0 0 -1 215.5 24.5 cm -11 3 m ("#!)[ 6.673828 7.007812 6.673828 ] xS 1 0 0 -1 391 45.5 cm -7.5 3 m ($")[ 6.673828 6.673828 ] xS [] 0 d 0.60000002 i 1 0 0 -1 -88.5 120.5 cm 480 69 m 486 69 l S 1 sc CM 391.5 30.5 m 397.5 30.5 l 397.5 36.5 l 391.5 36.5 l h 391.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 480 90 m 486 90 l 486 84 l 480 84 l h 480 90 m S 1 sc CM 361.5 30.5 m 367.5 30.5 l 367.5 36.5 l 361.5 36.5 l h 361.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 450 90 m 456 90 l 456 84 l 450 84 l h 450 90 m S 1 sc CM 301.5 30.5 m 307.5 30.5 l 307.5 36.5 l 301.5 36.5 l h 301.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 390 90 m 396 90 l 396 84 l 390 84 l h 390 90 m S 1 sc CM 34.5 30.5 m 40.5 30.5 l 40.5 36.5 l 34.5 36.5 l h 34.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 123 90 m 129 90 l 129 84 l 123 84 l h 123 90 m S 1 sc CM 91.5 30.5 m 97.5 30.5 l 97.5 36.5 l 91.5 36.5 l h 91.5 30.5 m f 0 sc 1 0 0 -1 -88.5 120.5 cm 180 90 m 186 90 l 186 84 l 180 84 l h 180 90 m S [ 4 4 ] 0 d 153 84 m 153 75 l 213 75 l 213 84 l S 303 84 m 303 66 l 363 66 l 363 84 l S 126 90 m 126 105 l 423 105 l 423 90 l S 333 84 m 333 75 l 393 75 l 393 84 l S 453 93 m 453 99 l 483 99 l 483 93 l S 0 i 1 0 0 -1 13.5 24.5 cm /F2.1[ 12 0 0 -12 0 0]sf -4 3 m (!)s 1 0 0 -1 19.5 15.5 cm -4 3 m (")s 1 0 0 -1 409.5 24.5 cm -4 3 m (!)s 1 0 0 -1 423 12.5 cm -14.5 3 m (#$%&)[ 6.673828 6.673828 7.007812 6.673828 ] xS [] 0 d 0.60000002 i 1 0 0 -1 -88.5 120.5 cm 507 101 m 513 101 l S ep end %%Trailer %%EOF ---------------0503251335392--