Content-Type: multipart/mixed; boundary="-------------0505310851255" This is a multi-part message in MIME format. ---------------0505310851255 Content-Type: text/plain; name="05-196.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-196.keywords" Boltzmann equation, mass disorder, Wigner transform, homogenization ---------------0505310851255 Content-Type: application/x-tex; name="rmasses.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rmasses.tex" %\documentclass[11pt,a4paper,draft]{article} \documentclass[11pt,a4paper]{article} %\documentclass[12pt,a4paper,draft]{article} %\usepackage[centertags]{amstex} \usepackage[english,german]{babel} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage{ae,aecompl} \usepackage{times} \usepackage{amsmath} \usepackage{amsfonts,bbm,theorem} \ifx\pdftexversion\undefined \usepackage[dvips]{graphicx} \else \usepackage[pdftex]{graphicx} \fi \usepackage{epsfig,url} \newcommand{\myfigure}[2]{ \includegraphics*[#1]{#2} } % To skip all the figures uncomment the lines below: % \renewcommand{\myfigure}[2]{ % % \framebox[15em]{\raisebox{0pt}[5em][5em]{\url{#2.eps}}} } % For a draft: %\usepackage[inner]{showlabels} %\renewcommand{\baselinestretch}{1.5} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} %Proof-environment without margins \newenvironment{proof}{\begin{trivlist}\item[]{\em Proof:}\/}{% \hfill\mbox{$\Box$}\end{trivlist}} \newenvironment{proofof}[1]{\begin{trivlist}\item[]{\em Proof of #1:}\/}{% \hfill\mbox{$\Box$}\end{trivlist}} % Enumerate-list without extra space \newcounter{jlisti} \newenvironment{jlist}[1][(\thejlisti)]{\begin{list}{{\rm #1}\ \ }{ % \usecounter{jlisti} % \setlength{\itemsep}{0pt} \setlength{\parsep}{0pt} % \setlength{\leftmargin}{0pt} % \setlength{\labelwidth}{0pt} % \setlength{\labelsep}{0pt} % % \settowidth{\labelwidth}{(DR2)} % \setlength{\topsep}{0pt} % }}{\end{list}} \newcounter{printapu} \newcommand{\alphit}[1]{\setcounter{printapu}{#1}\alph{printapu}} %\selectlanguage{german} \selectlanguage{english} \newcommand{\sabs}[1]{\langle #1\rangle} \newcommand{\Bigsabs}[1]{\Bigl\langle #1\Bigr\rangle} %\newcommand{\sabs}[1]{\left\lfloor #1 \right\rfloor_{\! 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\left| #2 % \makebox[0cm]{$\displaystyle\phantom{#1}$}\right.\!\right\rangle} \newcommand{\wvep}{W^{\vep}} \newcommand{\cwvep}{\mathcal{W}^{\vep}} \newcommand{\owvep}{\overline{W}^{\vep}} \newcommand{\owvept}{\overline{W}^{\vep}\!(t)} \newcommand{\owl}{\overline{W}} \newcommand{\ulr}{\underline{r}} %\newcommand{\tmacro}{t^{\text{macro}}} %% \newcommand{\tmacro}{\tilde{t}} %% \newcommand{\nmacro}{\tilde{n}} %% \newcommand{\pmacro}{\tilde{p}} \newcommand{\tmacro}{\bar{t}} \newcommand{\nmacro}{\bar{n}} \newcommand{\pmacro}{\bar{p}} %\newcommand{\tmacro}{\tau} \newcommand{\Kpart}{\mathcal{K}} \newcommand{\Kpamp}{\mathcal{K}^{(\text{amp})}} \newcommand{\Kpmain}{\mathcal{K}^{(\text{main})}} %% \newcommand{\ommax}{\overline{\omega}} %% \newcommand{\ommin}{\omega_0} \newcommand{\ommax}{\omega_{\text{max}}} \newcommand{\ommin}{\omega_{\text{min}}} \newcommand{\totalcs}{\sigma} \newcommand{\csmax}{\sigma_{\text{max}}} \newcommand{\ho}{H} %\newcommand{\Emax}{E_{\text{max}}} \newcommand{\Emax}{E_{0}} \newcommand{\oswvep}{\overline{w}^{\vep}} \newcommand{\VR}{V^{R}} \DeclareMathOperator*{\esssup}{ess\,sup} \newcommand{\talpha}{\tilde{\alpha}} \newcommand{\coupl}{\sqrt{\vep}} \newcommand{\Edens}{\mathcal{E}} \newcommand{\hilb}{\mathcal{H}} \newcommand{\banach}{\mathcal{B}} \newcommand{\cals}{\mathcal{S}} \newcommand{\Qhat}[1]{\widehat{Q}^{(\vep,#1)}} \newcommand{\Bonenorm}[2]{\norm{#1}_{\banach_{#2}}} \newcommand{\tilpsi}{\smash{\tilde{\psi}}} \newcommand{\Uz}{U^{(0)}} \newcommand{\Yj}{Y^{(j)}} \newcommand{\Yy}{Y^{(y)}} \newcommand{\Vj}{V^{(j)}} \newcommand{\Vy}{V^{(y)}} \newcommand{\Wy}{W^{(y)}} \newcommand{\gpath}{{\Gamma_\beta}} \newcommand{\psifen}{\psi^{(0;N)}} \newcommand{\psiocn}{\psi^{(1;N)}} \newcommand{\setm}{\!\setminus } \newcommand{\FT}[1]{\smash{\widehat{#1}}} \newcommand{\IFT}[1]{\smash{\widetilde{#1}}} %\newcommand{\vci}{_{\vc{i}}} \newcommand{\psierr}{\psi_{\text{err}}} \newcommand{\psimain}{\psi_{\text{main}}} \newcommand{\Fmain}{F^\vep_{\text{main}}} \newcommand{\comega}{c_{\omega}} \newenvironment{hlist}{\begin{trivlist}}{\end{trivlist}} \begin{document} \selectlanguage{english} \newcommand{\email}[1]{E-mail: \tt #1} \newcommand{\emailjani}{\email{jlukkari@ma.tum.de}} \newcommand{\addressjani}{\em Zentrum Mathematik, Technische Universit\"at M\"unchen, \\ \em Boltzmannstr. 3, D-85747 Garching, Germany} \newcommand{\emailherbert}{\email{spohn@ma.tum.de}} \title{Kinetic Limit for Wave Propagation in a Random Medium} \author{Jani Lukkarinen\thanks{\emailjani}, Herbert Spohn\thanks{\emailherbert}\\[1em] \addressjani } \maketitle \begin{abstract} We study crystal dynamics in the harmonic approximation. The atom\-ic masses are weakly disordered, in the sense that their deviation from uniformity is of order $\sqrt{\vep}$. The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit $\vep\to 0$ the disorder averaged Wigner function on the kinetic scale, time and space of order $\vep^{-1}$, is governed by a linear Boltzmann equation. \end{abstract} \tableofcontents \section{Introduction} When investigating the propagation of waves, one has to deal with the fact that the supporting medium often is not perfectly homogeneous, but suffers from irregularities. A standard method is then to assume that the material coefficients characterizing the medium are random, being homogeneous only in average. Examples abound: Shallow water waves travelling in a canal with uneven bottom, radar waves propagating through turbulent air, elastic waves dispersing in a random compound of two materials. The arguably simplest prototype is the scalar wave equation \begin{align}\label{eq:randomwaveeq} n^2 \partial_t^2 u = c^2 \Delta u \end{align} with a random index of refraction $n$. We will be interested in the case where the randomness is frozen in, or at most varies slowly on the time scale of the wave propagation. To say, we assume $x\mapsto n(x)$ to be a stationary stochastic process with short range correlations. An important special case is a random medium with a small variance of $n(x)$, which one can write as \begin{align}\label{eq:smalldisord} n(x) = (1 + \sqrt{\vep} \xi(x))^{-1} \end{align} with $\xi(x)$ order $1$ and $\vep\ll 1$. As argued many times, ranging from isotope disordered harmonic crystals to seismic waves propagating in the crust of the Earth, for such weak disorder a kinetic description becomes possible and offers a valuable approximation to the complete equation (\ref{eq:randomwaveeq}) -- we refer to the highly instructive survey by Ryzhik, Keller and Papanicolaou \cite{ryzhik96} for details. In the kinetic limit one considers times of order $\vep^{-1}$ and spatial distances of order $\vep^{-1}$. On that scale, the Wigner function $W$ associated to the solution $u$ of (\ref{eq:randomwaveeq}) is, in a good approximation, governed by the Boltzmann type transport equation \begin{align}\label{eq:genBtransporteq} & \partial_t W(x,k,t) + \nabla \omega(k)\cdot \nabla_x W(x,k,t) \nonumber \\ &\quad = \int \! \rmd k' \left(r(k',k) W(x,k',t) - r(k,k') W(x,k,t) \right) . \end{align} Here $x\in \R^3$, the physical space, and $k$ denotes the wave number. $\omega$ is the dispersion relation, $\omega(k)=c|k|$ with $k\in \R^3$ for (\ref{eq:randomwaveeq}). Note that the left hand side of (\ref{eq:genBtransporteq}) is the semiclassical approximation to (\ref{eq:randomwaveeq}) with $n(x)=1$. The collision operator on the right hand side of (\ref{eq:genBtransporteq}) describes the scattering from the inhomogeneities with a rate kernel $r(k,k')\rmd k'$ which depends on the particular model under consideration. Despite the wide use of the kinetic approximation (\ref{eq:genBtransporteq}), there is no complete mathematical justification for the step from microscopic equations like (\ref{eq:randomwaveeq}), together with (\ref{eq:smalldisord}), to (\ref{eq:genBtransporteq}) apart from one exception: Erd\H{o}s and Yau \cite{erdyau99} (see also \cite{chen03,chen04,eng04,erdos02,erdyau04}) investigate the random Schr\"{o}dinger equation \begin{align}\label{eq:randomSch} \ci \partial_t \psi(x,t) =(-\Delta + \sqrt{\vep} V)\psi(x,t), \end{align} where $\psi$ is the $\C$-valued wave function. This equation can be thought of as a two component wave equation for our purposes. In \cite{erdyau99} it is established that (\ref{eq:genBtransporteq}) becomes valid on the kinetic scale. Of course, the proof exploits special properties of the Schr\"{o}dinger equation. For us one motivation leading to the present investigation was to understand whether the techniques developed in \cite{erdyau99} carry over to standard wave equations such as (\ref{eq:randomwaveeq}). In fact, with the proper adjustments they do, and we are quite confident that also other wave equations with small random coefficients, as e.g.\ discussed in \cite{ryzhik96}, can be treated in the same way. Due to the intricate nature of the estimates, we do not claim this to be an easy exercise, but there is a blue-print which now can be followed. Even restricting to the scalar wave equation (\ref{eq:randomwaveeq}) there are choices to be made. One could add dispersion as $c^2(\Delta u-u)$ or the randomness could sit in the Laplacian as $\nabla\cdot(c(x)^2\nabla u)$ with $c(x)$ random and $n(x)=1$. To have a model of physical relevance, in our contribution we will consider a dielectric crystal in the harmonic approximation. If, for simplicity, the crystal structure is simple cubic, then $u_y$, $y\in\Z^3$, are the displacements of the atoms from their equilibrium position. Their movement is governed by Newton's equations of motion \begin{align}\label{eq:cubicNewt} m_y \frac{\rmd^2}{\rmd t^2} u_y = (\Delta u)_y, \qquad y\in\Z^3. \end{align} Here $\Delta$ is the lattice Laplacian, which corresponds to an elastic coupling between nearest neighbour atoms, and $m_y$ is the mass of the atom at $y$. (\ref{eq:cubicNewt}) can be regarded as the space discretized version of (\ref{eq:randomwaveeq}). Real crystals come as isotope mixtures. For instance, natural silicon consists in 92.23\% of ${}^{28}$Si, 4.68\% of ${}^{29}$Si, and 3.09\% of ${}^{30}$Si. Thus $\text{Var}(m_x)/\text{Av}(m_x)^2 \approx 10^{-4}$ and, in the appropriate units, we set \begin{equation} \label{eq:defmi} m_y = (1+\sqrt{\vep}\, \xi_y)^{-2}, \qquad \vep \ll 1, \end{equation} where $\xi_y$, $y\in\Z^3$, are i.i.d.\ bounded, mean zero, random variables, in slight generalization of our example. For the discretized wave equation the wave vector space is the unit torus $\T^3$. If $\omega$ denotes the dispersion relation for (\ref{eq:cubicNewt}), the Boltzmann transport equation becomes \begin{align}\label{eq:Btransporteq} & \partial_t W(x,k,t) + \frac{\nabla \omega(k)}{2\pi}\cdot \nabla_x W(x,k,t) \nonumber \\ &\quad = 2\pi \E[\xi_0^2] \int \! \rmd k' \omega(k')^2 \delta(\omega(k)-\omega(k')) \left( W(x,k',t) - W(x,k,t) \right) . \end{align} We will establish that the disorder averaged Wigner function on the kinetic scale, space and time of order $\vep^{-1}$, is governed by (\ref{eq:Btransporteq}). In fact, we will allow for more general elastic couplings between the crystal atoms than given in (\ref{eq:cubicNewt}). Our precise assumptions on $\omega$ will be discussed in Section \ref{sec:assumptions}. In passing, let us remark that, to compute the thermal conductivity of real crystals, scattering from isotope disorder contributes only as one part. At least equally important are weak non-linearities in the elastic couplings, see \cite{spohn05} for an exhaustive discussion. In addition, at low temperatures, roughly below $100^\circ \text{K}$ for silicon, lattice vibrations have to be quantized. However, for isotope disorder as in (\ref{eq:defmi}) quantization would not make any difference, since the corresponding Heisenberg equations of motion are also linear. In a loosely related work, Bal, Komorowski, and Ryzhik \cite{bal03} study the high frequency limit of (\ref{eq:randomwaveeq}) and (\ref{eq:smalldisord}), under the assumption that the initial data vary on a space scale $\gamma^{-1}$ with $\gamma\ll \vep\ll 1$. They prove that the Wigner function is well approximated by a transport equation of the form (\ref{eq:genBtransporteq}). Only the Boltzmann collision operator is to be replaced by its small angle approximation. Thus according to the limit equation the wave vector $k$ diffuses on the sphere $|k|=\text{\it const.}$, whereas in (\ref{eq:genBtransporteq}) it would be a random jump process. Their method is disjoint from ours and would not be able to cover the limiting case $\gamma=\vep$. Bal {\it et al.\/}\ also prove self-averaging of the limit Wigner function, while our result will concern only the disorder averaged Wigner function. We expect however to have self-averaging of the Wigner function also in our case, see \cite{chen04} for the corresponding result for the lattice random Schr\"{o}dinger equation (\ref{eq:randomSch}). Wave propagation in a random medium has been studied also away from the weak disorder regime. As the main novelty, at strong disorder, and at any disorder in space dimension $1$, propagation is suppressed. The wave equation has localized eigenmodes. We refer to the review article \cite{klein04}. The regime of extended eigenmodes is still unaccessible mathematically. The kinetic limit can be viewed as yielding some, even though rather modest, information on the delocalized eigenmodes, compare with \cite{chen03}. In the following section we provide a more precise definition of the model, describe in detail our assumptions on the dispersion relation $\omega$ and on the initial conditions, and state the main result. %\section*{\normalsize Acknowledgments} \subsection*{Acknowledgements} Our interest in wave propagation in random media where triggered by discussions of H.S.\ with H.-T.\ Yau during a common stay at the Institute for Advanced Study, Princeton in the spring 2003. We are grateful to L\'{a}szl\'{o} Erd\H{o}s and Thomas Chen for their constant support and encouragement. We also thank A.\ Kupiainen, A.\ Mielke, G.\ Panati, and S.\ Teufel for instructive discussions. J.L. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) project SP~181/19-1 and from the Academy of Finland in 2003. This work has also been supported by the European Commission through its 6th Framework Programme ``Structuring the European Research Area'' and the contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action. \section{Main result} \label{sec:main} \subsection{Discrete wave equation} \label{sec:model} We will study the kinetic limit of the discrete wave equation \begin{align}\label{eq:defDyn} \frac{\rmd}{\rmd t} q_y(t) & = v_y(t), \nonumber \\ (1+\sqrt{\vep}\, \xi_y)^{-2} \frac{\rmd}{\rmd t} v_y(t) & = -\sum_{y'\in\Z^3} \alpha(y-y') q_{y'}(t) \end{align} with $y\in \Z^3$ and $q_y(t),v_y(t)\in \R$. As a shorthand we set %$q(t)=\set{q_y(t),y\in\Z^3}$, $v(t)=\set{v_y(t),y\in\Z^3}$. $q(t)=(q_y(t),y\in\Z^3)$, $v(t)=(v_y(t),y\in\Z^3)$. The mass of the atom at site $y$ is $(1+\sqrt{\vep}\, \xi_y)^{-2}$, where $\xi=(\xi_y,y\in\Z^3)$ is a family of independent, identically distributed random variables. Their common distribution is independent of $\vep$, has zero mean and is supported on the interval $[-\bar{\xi},\bar{\xi}]$. Expectation with respect to $\xi$ is denoted by $\E$. We assume $\vep<\vep_0=\bar{\xi}^{-2}$ throughout. Hence $1+\sqrt{\vep}\,\xi_y>0$ with probability one. The coefficients $\alpha(y)$ are the elastic couplings between atoms, and we require them to have the following properties. \begin{jlist}[(E\thejlisti)] \item\label{it:EC0} $\alpha(y)\ne 0$ for some $y\ne 0$. \item\label{it:EC1} $\alpha(-y)=\alpha(y)$ for all $y$. \item\label{it:EC2} There are constants $C_1,C_2>0$ such that for all $y$ \begin{align}\label{eq:expdec} |\alpha(y)|\le C_1 \rme^{-C_2 |y|}. \end{align} \item\label{it:EC3} Let $\FT{\alpha}$ be the Fourier transform of $\alpha$, which we define by \begin{align}\label{eq:defFT} \FT{\alpha}(k) = \sum_{y\in \Z^3} \rme^{-\ci 2 \pi k \cdot y} \alpha(y). \end{align} Then $\FT{\alpha}:\T^3\to \R$, where $\T^3$ denotes the $3$-torus with unit side length. Mechanical stability demands $\FT{\alpha}\ge 0$. We require here the somewhat stronger condition \begin{align}\label{eq:pinning} \FT{\alpha}(k)> 0, \quad \text{for all }k\in \T^3. \end{align} \end{jlist} If $\vep=0$, Eqs.~(\ref{eq:defDyn}) admit plane wave solutions with wave vector $k\in\T^3$ and frequency \begin{align}\label{eq:defom} \omega(k)=\sqrt{\FT{\alpha}(k)}. \end{align} The function $\omega:\T^3\to \R$ is the {\em dispersion relation\/}. Under our assumptions for $\alpha$, $\omega$ is real-analytic, $\omega(-k)=\omega(k)$, $0<\ommin=\min_k\omega(k)$, and $\ommax=\max_k\omega(k)<\infty$. We solve the differential equations (\ref{eq:defDyn}) as a Cauchy problem with initial data $q(0),v(0)$. The time-evolution (\ref{eq:defDyn}) conserves the energy \begin{align}\label{eq:defHam} E(q,v) = \frac{1}{2} \Bigl( \sum_{y\in \Z^3} (1+\sqrt{\vep}\, \xi_y)^{-2} v^2_y + \sum_{y,y'\in \Z^3} \alpha(y-y') q_y q_{y'} \Bigr). \end{align} The initial data are assumed to have finite energy, $E(q(0),v(0))<\infty$. Since $\ommin >0$, this implies that $q(0),v(0)\in\ell_2(\Z^3,\R)$. For any realization of $\xi$, the generator of the time-evolution (\ref{eq:defDyn}) is a bounded operator on $\ell_2(\Z^3,\R^2)$. Therefore, the Cauchy problem has a unique, norm-continuous solution which remains in $\ell_2(\Z^3,\R^2)$ for all $t\in \R$. The energy depends on the realization of $\xi$, and it will be more convenient to switch to new variables such that the flat $\ell_2$-norm is conserved. For this purpose, let tilde denote the inverse Fourier transform, for which we adopt the convention \begin{equation} \label{eq:definvFT} \IFT{f}_y = \int_{\T^3}\!\rmd k \, \rme^{\ci 2\pi y\cdot k} f(k), \end{equation} %for any $f\in L^1(\T^3)$, and let $\Omega$ denote the bounded operator on $\ell_2(\Z^3,\C)$ defined via \begin{align}\label{eq:defOm} (\Omega \phi)_y = \sum_{y'\in\Z^3} \IFT{\omega}_{y-y'} \phi_{y'}. \end{align} Since $q(t),v(t)\in\ell_2(\Z^3,\R)$, we can introduce the vector $\psi(t)\in \ell_2(\Z^3,\C^2)$ through \begin{align}\label{eq:defPsi} \psi(t)_{\sigma,y} = \frac{1}{2} \left( (\Omega q(t))_y + \ci \sigma (1+\sqrt{\vep}\, \xi_y)^{-1} v(t)_y \right), \end{align} where $\sigma = \pm 1$ and $y\in\Z^3$. From now on, let us denote $\ell_2=\ell_2(\Z^3,\C)$, and $\hilb = \ell_2(\Z^3,\C^2)=\ell_2\oplus \ell_2$. If we regard $\xi$ as a multiplication operator on $\ell_2$, i.e., if we define $(\xi\psi)_y = \xi_y \psi_y$, then $\psi(t)$ satisfies the differential equation \begin{align}\label{eq:hilbevol} \frac{\rmd}{\rmd t} \psi(t) = -\ci H_\vep \psi(t), \quad\text{with}\quad H_\vep = H_0 + \sqrt{\vep} V, \end{align} where \begin{align}\label{eq:defH0} H_0 = \begin{pmatrix} \Omega & 0\\ 0 & -\Omega \end{pmatrix},\quad V = \frac{1}{2} \begin{pmatrix} \Omega\xi + \xi \Omega & -\Omega\xi + \xi \Omega \\ \Omega\xi - \xi \Omega & -\Omega\xi - \xi \Omega \end{pmatrix} . \end{align} Because $H_\vep$ is a self-adjoint operator on $\hilb$, the solution to (\ref{eq:hilbevol}) generates a unitary group on $\hilb$. Unitarity is equivalent to energy conservation, since for all $t$ \begin{align}\label{eq:normisE} \norm{\psi(t)}^2 = E(q(t),p(t)). \end{align} If $\psi(t)$ is one of the ``physical'' states obtained by (\ref{eq:defPsi}), then it satisfies $\psi(t)_{-,y}^*=\psi(t)_{+,y}$ for all $y$ and $t$ due to $q(t),v(t)\in \R$. We will discuss in Section \ref{sec:classical} how information about $\psi(t)$ is transferred to $q(t)$, $v(t)$. \subsection{Lattice Wigner function, initial conditions, and dispersion relation} \label{sec:assumptions} The disorder has strength $\sqrt{\vep}$. Since $\E[\xi_0]=0$, effects of order $\sqrt{\vep}$ vanish in the mean, and a wave packet has a mean free path of the order of $\vep^{-1}$ lattice spacings. In the kinetic limit the speed of propagation of the waves is independent of $\vep$, indicating that the first time-scale, at which the randomness becomes relevant, is also of the order of $\vep^{-1}$. If $\xi=0$, this scaling corresponds to the semiclassical limit in which the Wigner function satisfies the transport equation \begin{align} & \partial_t W(x,k,t) + \frac{\nabla \omega(k)}{2\pi}\cdot \nabla_x W(x,k,t) = 0, \quad x\in\R^3,\ k\in \T^3 . \end{align} We refer to \cite{mielke05} for an exhaustive discussion. From this perspective, the Wigner function is the natural object for studying the kinetic limit. Given a scale $\vep>0$, we define the Wigner function $W^\vep$ of any state $\psi\in \hilb$ as the distributional Fourier transform \begin{align}\label{eq:Wdef} W_{\sigma'\sigma}^\vep(x,k) = \int_{\R^3}\! \!\rmd p\, \rme^{\ci 2\pi x\cdot p} \FT{\psi}_{\sigma'}\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}_{\sigma}\Bigl(k+\frac{1}{2}\vep p\Bigr), \end{align} where $\sigma',\sigma \in \set{\pm 1}$, $x\in\R^3$, $k\in \T^3$. This is also called the Wigner transform of $\psi$ and we denote it by $W^\vep[\psi]$. The Fourier transform of $\psi$, denoted by $\FT{\psi}$, is defined as in (\ref{eq:defFT}) and periodically extended to a function on the whole of $\R^3$. $W$ is an Hermitian $\M_2$-valued (complex $2{\times}2$-matrix) distribution. When integrated against a matrix-valued test-function $J\in \cals(\R^3\times\T^3,\M_2)$, (\ref{eq:Wdef}) becomes \begin{align}\label{eq:defWpsi} & \mean{J,W^\vep} = \int_{\R^3}\! \!\rmd p \int_{\T^3}\! \!\rmd k\, \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr) \cdot \FT{J}(p,k)^* \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr), \end{align} where $\FT{J}$ denotes the Fourier transform of $J$ in the first variable, \begin{align}\label{eq:defFT1J} \FT{J}(p,k) = \int_{\R^3} \!\rmd x\, \rme^{-\ci 2\pi p\cdot x} J(x,k). \end{align} The notation $A^*$ denotes Hermitian conjugation, and the dot is used for a finite-dimensional scalar product: $a\cdot b = \sum_i a_i^* b_i$. We have included a complex conjugation of the test-function in the definition in order to have the same sign convention for the Fourier transform of both test functions and distributions. Let us choose now some initial conditions for (\ref{eq:hilbevol}). In general, it will be $\vep$-dependent and we denote it by $\psi^\vep$. The solution to (\ref{eq:hilbevol}) is then \begin{align}\label{eq:defpsisol} \psi(t) = \rme^{-\ci t H_\vep} \psi^\vep . \end{align} In the following we will be studying a limit where $\vep\to 0^+$ via some arbitrary sequence of values. Our assumptions on the initial conditions are \begin{assumption}[Initial conditions]\label{th:initass} For every $\vep$, there is $\psi^\vep \in\hilb$, independent of $\xi$, such that \begin{jlist}[{\rm (IC\thejlisti)}] \item\label{it:I1} $\displaystyle\sup_\vep \norm{\psi^\vep} < \infty$. \item\label{it:I2} $\displaystyle\lim_{R\to\infty} \limsup_{\vep\to 0} \sum_{|y|> R/\vep} |\psi^\vep_{y}|^2 = 0$. \item\label{it:I3} There exists a positive bounded Borel measure $\mu_0$ on $\R^3\times \T^3$ such that \begin{align}\label{eq:IClim} \lim_{\vep\to 0}\mean{J,W^\vep_{++}[\psi^\vep]} = \int_{\R^3\times \T^3} \!\!\mu_0(\rmd x \,\rmd k)\, J(x,k)^*. \end{align} for all $J\in \cals(\R^3\times \T^3)$. \end{jlist} \end{assumption} These assumptions are rather weak. In fact, as discussed in the Appendix \ref{sec:appWigner}, if we assume (IC\ref{it:I1}), then the existence of the limit in (\ref{eq:IClim}) for all $J$ implies already the existence of the measure $\mu_0$. The condition (IC\ref{it:I2}) means that the sequence $|\psi^\vep_{y}|^2/\norm{\psi^\vep}^2$ of probability measures on $\Z^3$ is tight on the kinetic scale $\vep^{-1}$. Our second set of assumptions deals with the dispersion relation $\omega$. For this we need to introduce the notations \begin{align}\label{eq:defsabs} \sabs{x} = \sqrt{1+x^2} \qand \norm{f}_{N,\infty} = \sup_{|\alpha|\le N} \norm{D^\alpha\! f}_\infty , \end{align} where $N=0,1,\ldots$ and $\alpha$ denotes a multi-index. \begin{assumption}[Dispersion relation]\label{th:disprelass} Let $\omega:\T^3\to\R$ satisfy all of the following: \begin{jlist}[(DR\thejlisti)] \item\label{it:DC1} $\omega$ is smooth and $\omega(-k)=\omega(k)$. \item\label{it:DC2} $\ommin>0$ and $\ommax < \infty$ with $\ommin = \min_{k} \omega(k)$ and $\ommax = \max_{k} \omega(k)$. \item ({\em dispersivity})\label{it:suffdisp}\ \ There are constants $d_1\in\N$ and $\comega>0$ such that for all $t\in \R$ and $f\in C^\infty(\T^3)$, \begin{align}\label{eq:suffdisp} \left| \int_{\T^3} \rmd k\, f(k) \rme^{-\ci t \omega(k)}\right| \le \frac{\comega}{\sabs{t}^{3/2}}\norm{f}_{d_1,\infty} . \end{align} \item\label{it:crossing} ({\em crossings are suppressed})\ \ There are constants $c_2>0$, $0<\gamma\le 1$ and $d_2\in\N$ such that for all $u\in \T^3$, $0<\beta\le 1$, $\alpha\in \R^3$, and $\sigma\in\set{\pm 1}^3$, \begin{align}\label{eq:crossingest} & \int_{(\T^3)^2} \rmd k_1\rmd k_2\, \frac{1}{|\alpha_1-\sigma_1 \omega(k_1)+\ci\beta| |\alpha_2-\sigma_2 \omega(k_2)+\ci\beta|} \nonumber \\ & \qquad\times \frac{1}{|\alpha_3-\sigma_3 \omega(k_1-k_2+u)+\ci\beta|} \le c_2 \beta^{\gamma-1} \sabs{\ln \beta}^{d_2} . \end{align} \end{jlist} \end{assumption} If $\omega$ has only isolated, non-degenerate critical points, i.e., if $\omega$ is a Morse function, then the bound (\ref{eq:suffdisp}) with $d_1=4$ follows by standard stationary phase methods. The crossing condition is more difficult to verify. We will discuss these issues in detail in Sec.~\ref{sec:dispersion}, where examples satisfying (DR\ref{it:suffdisp}) and (DR\ref{it:crossing}) are also provided. \subsection{Main Theorem} \label{sec:mainth} The Boltzmann equation (\ref{eq:Btransporteq}) is the forward equation of a Markov jump process $(x(t),k(t))$, $t\ge 0$. $k(t)$, $t\ge 0$ is governed by the collision rate \begin{align}\label{eq:defnuk} \nu_k(\rmd k') = \rmd k' \delta(\omega(k)-\omega(k')) 2\pi \E[\xi_0^2]\omega(k')^2, \quad k\in\T^3 , \end{align} with a total collision rate \begin{align}\label{eq:deftotcs} \totalcs(k) = \nu_k(\T^3). \end{align} As proved in Appendix \ref{sec:appBoltzmann}, since $\omega$ is continuous and (DR\ref{it:suffdisp}) is satisfied, the map \begin{align}\label{eq:cscont} k\mapsto \int\! \nu_k(\rmd k') g(k') \end{align} is continuous for every $g\in C(\T^3)$. In particular, $\csmax = \sup_k \sigma(k) < \infty$. Now given $k=k(0)$, one has $k(t)=k$ for $0\le t \le \tau$ with $\tau$ an exponentially distributed random variable of mean $\totalcs(k)^{-1}$. At time $\tau$, $k$ jumps to $\rmd k'$ with probability $\nu_k(\rmd k')/\sigma(k)$, etc. To define the joint process $(x(t),k(t))$, $t\ge 0$, one sets \begin{align}\label{eq:freebevol} \frac{\rmd}{\rmd t} x(t) = \frac{1}{2\pi} \nabla\omega(k(t)) . \end{align} We assume the process to start in the measure $\mu_0$ from (IC\ref{it:I3}). Because of continuity in (\ref{eq:cscont}), the process $(x(t),k(t))$, $t\ge 0$, is Feller. Hence there is a well-defined joint distribution at time $t$, which we denote by $\mu_t(\rmd x \,\rmd k)$. We are now ready to state our main result. \begin{theorem}\label{th:main} Let the Assumptions \ref{th:initass} and \ref{th:disprelass} hold and let $\psi(t)$ denote the random vector determined by (\ref{eq:defpsisol}). Then for all $t\ge 0$, $J\in \cals(\R^3\times \T^3)$, one has the limit \begin{equation} \label{eq:Wlimit} \lim_{\vep\to 0} \E[\mean{J,W_{++}^\vep[\psi(t/\vep)]}] = \int_{\R^3\times \T^3} \!\!\mu_t(\rmd x \,\rmd k)\, J(x,k)^*. \end{equation} \end{theorem} As a complete theorem, one would have expected a limit for the Wigner matrix, not just for the ($++$)-component, as stated above. From the evolution equation (\ref{eq:hilbevol}) it follows immediately that Theorem \ref{th:main} also holds for $W_{--}$. One only has to assume (IC\ref{it:I3}) for $W_{--}^\vep$, and replace everywhere $\omega$ by $-\omega$. As the rate kernel remains unchanged, this amounts to changing the sign in (\ref{eq:freebevol}). For the deterministic initial data of Section \ref{sec:model} the off-diagonal components $W^\vep_{+-}$ and $W^\vep_{-+}$ are fastly oscillating. In general, they do not have a pointwise limit, but vanish upon time-averaging, i.e., for any $t\ge 0$, $T>0$ and $\sigma=\pm 1$, one has \begin{align} \lim_{\vep\to 0} \frac{1}{T}\int_{0}^T\!\! \rmd \tau\, \E\!\left[W^\vep_{\sigma,-\sigma}[\psi( (t+\tau)/\vep)]\right] = 0. \end{align} Physically, one would like to avoid the assumption $\ommin>0$, since elastic forces depend only on the relative distances between atoms, and thus $\FT{\alpha}(0)=0$. If $\FT{\alpha}(0)=0$, generically $\omega(k)\approx |k|$ for small $k$. In addition, by (\ref{eq:defH0}), the two bands of $H_0$ touch at $k=0$. On a technical level, the non-smooth crossing of the bands adds another layer of difficulty which we wanted to avoid here. To give a brief outline: In the following section we exploit general properties about weak limits of lattice Wigner transforms to reduce the proof of the main theorem into a Proposition stating that their Fourier transforms converge to the characteristic functions of $(\mu_t)$. These properties concerning the Wigner transform are valid under more general assumptions than those of the main theorem, and we have separated their derivation to Appendix \ref{sec:appWigner}. The core of the paper is the graphical expansion of Section \ref{sec:graphs} where the proof of the above Proposition is done by dividing it into several layers with ever more detailed Lemmas acting as links between the different layers. In particular, we have separated the analysis of the non-vanishing parts of the graph expansion, so called simple graphs, to Section \ref{sec:simple}. The graph expansion follows the outline laid down in the works cited earlier. The new ingredients are the matrix structure and the momentum dependence of the interaction. We also develop here an alternative version for the so-called partial time-integration needed in the estimation of the error terms. The present version, described in Sec.~\ref{sec:duhamel}, facilitates the analysis of the error terms, allowing the use of same estimates for both partially time-integrated and fully expanded graphs. We also consider here more general dispersion relations and initial conditions than before, although it needs to be stressed that in the case of the dispersion relation, the improvement is mainly a matter of more careful bookkeeping. The estimates, which allow the division of the graphs into leading and subleading ones, rely on the decay estimates (DR\ref{it:suffdisp}) and (DR\ref{it:crossing}). In section \ref{sec:dispersion} we discuss proving (DR\ref{it:crossing}) for a given dispersion relation in more detail. In particular, we show there that the taking of the square root, which is necessary for obtaining the dispersion relation from the elastic couplings, in general retains the validity of the crossing estimate. Finally, in the last section we return to the original lattice dynamics (\ref{eq:defDyn}), explain how (IC\ref{it:I1}) -- (IC\ref{it:I3}) relate to the initial positions and velocities, and, in particular, discuss the propagation of the energy density. \section{Proof of the Main Theorem} \label{sec:proofmain} In all of the results in this and the following two sections, unless stated otherwise, we make the assumptions of Theorem \ref{th:main}. In addition, we assume that $\E[\xi_0^2]=1$. This is not a restriction, as it can always be achieved by rescaling $\xi$ by $\E[\xi_0^2]^{-\frac{1}{2}}$ and $\vep$ by $\E[\xi_0^2]$. We study a given sequence $(\vep_k)$, $k=1,2,\ldots$, such that $0<\vep_k<\vep_0$ and $\lim_k \vep_k=0$. For notational simplicity, we will always denote the limits of the type $\lim_{k\to\infty} f(\vep_k)$ by $\lim_{\vep\to 0} f(\vep)$. We will study the limits of the mappings $\owvept$ defined by $\mean{J,\owvept }= \E[\mean{J,W_{++}^\vep[\psi(t/\vep)]}]$ where $\psi(t) = \rme^{-\ci H_\vep t} \psi^\vep$. For any $\vep$ and $t\ge 0$, the mapping $\xi \mapsto \psi(t/\vep)$ lifts the probability measure for $\xi$ to a probability measure $\nu^\vep_t$ on the Hilbert space $\hilb$. For instance, $\nu^\vep_0$ is a Dirac measure concentrated at $\psi^\vep$. Each of the measures $\nu^\vep_t$ is a weak Borel measure. In particular, let us prove next that every $\psi(t/\vep)_{\sigma,y}$ is measurable. For any $R>0$, define $\VR$ as the potential obtained by neglecting far lying perturbations $\xi$, i.e., let \begin{equation} \label{eq:defVL} \VR = \sum_{\norm{y}_\infty \le R} \xi_{y} \Vy \end{equation} where $\Vy$, $y\in\Z^3$, has a Fourier transform given by the integral kernel \begin{equation} \label{eq:VFT} \FT{V}^{(y)}_{\sigma'\sigma}(k',k) = \rme^{-\ci 2\pi y\cdot(k'-k)} v_{\sigma'\sigma}(k',k) , \end{equation} and $v\in L^2(\T^3\times\T^3,\M_2)$ is defined for $\sigma',\sigma\in \set{\pm1}$ and $k',k\in \T^3$ by \begin{equation} \label{eq:vdef} v_{\sigma'\sigma}(k',k) = \frac{\sigma'\sigma}{2} \left( \sigma' \omega(k') + \sigma \omega(k) \right). \end{equation} Then $\VR \to V$ strongly (i.e., for all $\psi\in\hilb$, $\norm{\VR\psi - V\psi}\to 0$) when $R\to\infty$, and, as $\sup_R \norm{\VR}<\infty$, the same is true for any product of $\VR$:s and bounded $R$-independent operators. Therefore, \begin{align}\label{eq:measurability} \psi(t/\vep)_{\sigma,y} = \lim_{R\to\infty} \sum_{N=0}^R \frac{(-\ci t/\vep)^N}{N!} ( (H_0+\sqrt{\vep} \VR)^N\psi^\vep )_{\sigma,y} . \end{align} As the summand is a complex function depending only on finitely many of $(\xi_y)$, it is measurable. For all $|\xi|\le \bar{\xi}$, $\psi(t/\vep)_{\sigma,y}$ is a convergent limit of a sequence of such functions, which implies that also $\psi(t/\vep)_{\sigma,y}$ is measurable. In addition, by the unitarity of $\rme^{-\ci H_\vep t}$, \begin{align} \norm{\psi(t/\vep)}^2 = \norm{\psi^\vep}^2 \end{align} which is uniformly bounded by (IC\ref{it:I1}). By Theorem \ref{th:cwvep}, $\mean{J,W_{\nu_t^\vep}^\vep} = \int\!\nu_t^\vep(\rmd \psi) \mean{J,W^\vep[\psi]}$ defines a distribution in $\cals'(\R^3\times \T^3,\M^2)$ which we call the Wigner transform of the measure $\nu_t^\vep$. These distributions behave very similarly to probability measures on $\R^3\times \T^3$, and we have collected their main properties in Appendix \ref{sec:appWigner}. In particular, we can conclude that $\owvept\in \cals'(\R^3\times\C^3)$, as for all $J\in \cals(\R^3\times\C^3)$, \begin{align} \mean{J,\owvept}=\mean{J,(W_{\nu_t^\vep}^\vep)_{++}}= \int\!\nu_t^\vep(\rmd \psi) \mean{J,W^\vep[\psi_+]}. \end{align} By Proposition \ref{th:Fnuprop}, the Fourier transform of $\owvept$ is determined by the functions \begin{align}\label{eq:FvepFourier2} & F^\vep_t(p,n) = \E_{\nu_t^\vep}\!\!\left[ \int_{\T^3}\! \!\rmd k\, \rme^{\ci 2 \pi n\cdot k} \FT{\psi}_+\!\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}_+\!\Bigl(k+\frac{1}{2}\vep p\Bigr)\right] \end{align} where $p\in \R^3$ and $n\in \Z^3$. The assumptions (IC\ref{it:I1}) -- (IC\ref{it:I3}) allow then applying Theorem \ref{th:weakimpliesborel} to conclude that $F^\vep_0$ converges pointwise to the Fourier transform of $\mu_0$ which, by Theorem \ref{th:FimpliesWweak}, implies \begin{lemma}\label{th:winitlim} For all $p\in \R^3$ and $f\in C(\T^3)$, \begin{align}%\label{eq:Fveptomucont2} \lim_{\vep\to 0} \int_{\T^3} \!\!\rmd k f(k)\, \FT{\psi}^\vep_+\!\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}^\vep_+\!\Bigl(k+\frac{1}{2}\vep p\Bigr) = \int_{\R^3\times \T^3}\!\! \!\!\mu_0(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi p\cdot x} f(k) . \end{align} \end{lemma} Using time-dependent perturbation expansion, we will prove in Section \ref{sec:graphs} that \begin{proposition}\label{th:mainlim} For all $\tmacro>0$, $\pmacro\in \R^3$ and $\nmacro\in \Z^3$ \begin{align}\label{eq:Fveptomut} & \lim_{\vep\to 0} F^\vep_{\tmacro}(\pmacro,\nmacro) = \int_{\R^d\times \T^d}\!\! \!\!\mu_{\tmacro}(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi (\pmacro\cdot x- \nmacro\cdot k)}. \end{align} \end{proposition} Then we can apply Theorem \ref{th:FimpliesWweak}, and conclude that for any $t>0$, the sequence $(\owvept)_\vep$ converges in the weak-$*$ topology to a bounded positive Borel measure whose characteristic function coincides with the limit of $F^\vep_t$. However, then by (\ref{eq:Fveptomut}) this measure is in fact equal to $\mu_t$. This is sufficient to prove Theorem \ref{th:main}, since (\ref{eq:Wlimit}) is valid at $t=0$ by assumption (IC\ref{it:I3}). \section{Graph expansion (proof of Proposition \ref{th:mainlim})} \label{sec:graphs} In this section we assume that all assumptions of Proposition \ref{th:mainlim} are valid. In particular, $\pmacro\in \R^3$, $\nmacro\in \Z^3$ and $\tmacro>0$ will denote the fixed macroscopic parameters. We first derive, using time-dependent perturbation theory, a way of splitting the time-evolved states into two parts, \begin{align} \rme^{-\ci t H_\vep }\psi^\vep = \psimain(t) + \psierr(t). \end{align} The splitting is done in such the way that each part is component-wise measurable, as before, and \begin{align}%\label{eq:errtermbound} & \lim_{\vep\to 0}\E\!\left[\norm{\psierr(\tmacro/\vep)}^2\right] = 0. \end{align} Then we will only need to inspect the limit of the main part. \subsection{Duhamel expansion with soft partial time-integration} \label{sec:duhamel} We begin by deriving the above splitting. Since both $H_0$ and $V$ are bounded operators for any realization of the randomness, the Duhamel formula states that, for any $t\in \R$ we have as vector valued integrals in $\banach(\hilb)$, \begin{gather}\label{eq:baseduh} \rme^{-\ci t H_\vep } = \rme^{-\ci t H_0} + \int_0^{t}\!\! \rmd s\, \rme^{-\ci (t-s) H_\vep } (- \ci \sqrt{\vep} V) \rme^{-\ci s H_0} . \end{gather} This could be iterated to yield the full Dyson series which, however, would become ill-behaved in the kinetic limit. Instead, we will expand the series only partially, up to $N_0$ ``collisions''. For the remainder we use a different method, essentially a version of the ``partial time integration'' introduced in \cite{erdyau99} with a ``soft cut-off'' which allows easier analysis of the error terms. The results will be expressed in terms of the following (random) functions: \begin{definition}\label{th:defFGA} For any $\vep$, and any $\kappa\ge 0$ and $s\in \R$ let \begin{align}\label{eq:defW} W_{s} & = (- \ci \sqrt{\vep} V) \rme^{-\ci s H_0} \qand W_{s,\kappa} = (-\ci \sqrt{\vep} V) \rme^{-\ci s (H_0-\ci \kappa)}, \end{align} and define for any $\kappa\ge 0$, $t>0$, and $N,N',N_0\in\N$ with $N_0\ge 1$, as vector valued integrals in $\banach(\hilb)$, \begin{align} F_{N}(t;\vep) & = \int_{\R_+^{N+1}}\!\rmd s \, \delta\Bigl(t-\sum_{\ell=1}^{N+1} s_\ell\Bigr) \rme^{-\ci s_{N+1} H_0} W_{s_{N}}\cdots W_{s_1}, \label{eq:defFN} \\ %\nonumber \\ & \quad \times G_{N',N}(t;\vep,\kappa) & = \int_{\R_+^{N+N'+1}}\!\!\!\!\!\!\!\!\!\!\! \rmd s \, \delta\Bigl(t-\!\!\!\!\sum_{\ell=1}^{N+N'+1}\!\!\!\! s_\ell\Bigr) %\nonumber \\ & \quad \times \rme^{-\ci s_{N+N'+1} (H_0-\ci\kappa)}\! \prod_{j=N+1}^{N+N'}\!\!\!\! W_{s_{j},\kappa} \prod_{j=1}^{N} W_{s_{j}}, \label{eq:defGN} \\ A_{N',N_0}(t;\vep,\kappa) & = \int_{\R_+^{N_0+N'}}\! \rmd s \, \delta\Bigl(t-\sum_{\ell=1}^{N_0+N'} s_\ell\Bigr) \prod_{j=N_0+1}^{N_0+N'} W_{s_{j},\kappa} \prod_{j=1}^{N_0} W_{s_{j}} \label{eq:defAN}. \end{align} Let us also define \begin{align}\label{eq:defzeroval} F_{N}(0;\vep)=\delta_{N0}\1,\quad G_{N',N}(0;\vep,\kappa)=\delta_{N+N',0}\1,\quad\text{and}\quad A_{N',N_0}(0;\vep,\kappa) = 0. \end{align} \end{definition} In these definitions, the notation $\rmd s\, \delta(t-\sum_{\ell=1}^N s_\ell)$, with $t>0$ and $N\in \N_+$, refers to a bounded positive Borel measure on $\R_+^{N}$ defined naturally by the $\delta$-function by integrating out one of the coordinates $s_\ell$. Explicitly, for any $f\in C(\R^{N})$ we have, for $N=1$, $\int_{0}^\infty\! \rmd s\, \delta(t-s) f(s) = f(t)$, and for $N\ge 1$, \begin{align} & \int_{\R_+^{N}}\!\!\rmd s\, \delta\Bigl(t-\sum_{\ell=1}^N s_\ell\Bigr) f(s) %\nonumber \\ & \quad = \int_{\R_+^{N-1}}\!\!\!\!\!\!\!\rmd s \, \1\!\Bigl(\sum_{\ell=1}^{N-1} s_\ell\le t\Bigr) f\Bigl(s_1,\ldots,s_{N-1},t-\sum_{\ell=1}^{N-1} s_\ell\Bigr). \end{align} The function $\1$ in the integrand restricts the integration region to the standard simplex in $\R^{N-1}$ scaled by the factor $t$. This is a compact set and therefore, as long as the integrand is a continuous mapping from the simplex to a \frechet\ space, it can be used to define vector valued integrals in the sense of \cite{Rudin:FA}, Theorem 3.27. The measure is invariant under permutations of $(s_\ell)$ -- which proves that we could have integrated out any of the coordinates, not only the last one -- and it is bounded by \begin{align}\label{eq:tNbound} \int_{\R_+^{N}}\!\rmd s \,\delta\Bigl(t-\sum_{\ell=1}^N s_\ell\Bigr) = \frac{t^{N-1}}{(N-1)!}. \end{align} The proof that the integrands in the Definition \ref{th:defFGA} are continuous, as well as a number of useful relations between the functions, are given in the following: \begin{lemma}\label{th:FGAcontin} $W_s$ and $W_{s,\kappa}$ are continuous in $\banach(\hilb)$ in the variable $s$, as well as are all of the functions defined in (\ref{eq:defFN}) -- (\ref{eq:defzeroval}) in $t$. They are also related by the following equalities for all $N_0\ge 1$ and $N',N\ge 0$: \begin{align} F_N(t;\vep) & = G_{0,N}(t;\vep,0), \label{eq:FGrel}\\ F_{N_0}(t;\vep) & = \int_0^{t}\!\! \rmd r\, \rme^{-\ci (t-r) H_0} A_{0,N_0}(r;\vep,\kappa), \label{eq:FArel}\\ G_{N',N_0}(t ;\vep,\kappa) & = \int_0^{t }\!\! \rmd r\, \rme^{-\kappa (t -r)} \rme^{-\ci (t -r) H_0} A_{N',N_0}(r;\vep,\kappa),\label{eq:GArel} \\ A_{0,N_0+1}(t;\vep,\kappa) & = \int_0^{t}\!\! \rmd r\, W_{t-r} A_{0,N_0}(r;\vep,\kappa), \label{eq:AA0rel} \\ A_{N'+1,N_0}(t;\vep,\kappa) & = \int_0^{t}\!\! \rmd r\, W_{t-r,\kappa} A_{N',N_0}(r;\vep,\kappa). \label{eq:AArel} \end{align} \end{lemma} \begin{proof} As $H_0$ is bounded, $s\mapsto \rme^{-\ci s H_0}$ is norm-continuous for all $s\in \R$, and so are then $W_s$ and $W_{s,\kappa}$. This proves that the integrands in the definitions (\ref{eq:defFN}) -- (\ref{eq:defAN}) are continuous functions for all real $s$, and thus all of the vector valued integrals are well-defined in $\banach(\hilb)$. We next need to prove the continuity of the functions $F(t)$, $G(t)$ and $A(t)$. Since the proof is essentially identical in all three cases, we shall do it only for $F$. First, for $N=0$, we have $F_0(t) = \rme^{-\ci t H_0}$ which is norm-continuous for $t>0$, and $\lim_{t\to 0^+}F_0(t) =\1$, which proves that $F_0$ is continuous also at $t=0$. When $N>0$, we have explicitly for all $t>0$ \begin{align}\label{eq:defFNexpl} F_N(t) & = \int_{s\in\R_+^{N}}\! \rmd s \, \1\Bigl(\sum_{\ell=1}^N s_\ell \le t\Bigr) %\nonumber \\ & \quad \times \rme^{-\ci \left(t-\sum_{\ell=1}^{N} s_\ell\right)H_0} W_{s_{N}}\cdots W_{s_1}. \end{align} Since $\norm{W_s}, \norm{W_{s,\kappa}}\le \sqrt{\vep}\norm{V}$, % for $s\ge 0$, we have by (\ref{eq:tNbound}), $\norm{F_N(t)}\le (\sqrt{\vep}\norm{V} t)^{N}/N!$. Therefore, $\lim_{t\to 0^+}F_N(t) =0$ which proves that $F_N$ is continuous at $0$. On the other hand, for $t,h>0$ \begin{align} & \norm{F_N(t+h)-F_N(t)} \le \left(\sqrt{\vep}\norm{V}\right)^{N} \Bigl[ \int_{s\in\R_+^{N}}\! \rmd s \, \1\Bigl(\sum_{\ell=1}^N s_\ell \le t\Bigr) \norm{\rme^{-\ci h H_0}-\1} \nonumber \\ & \qquad + \int_{s\in\R_+^{N}}\! \rmd s \, \1\Bigl(t<\sum_{\ell=1}^N s_\ell \le t+h\Bigr) \Bigr]. \end{align} The bound goes to $0$ when $h\to 0$ by dominated convergence, and we have proven that $F_N$ is norm-continuous for all $N\in\N$. The integrands on the right hand side of equations (\ref{eq:FArel}) -- (\ref{eq:AArel}) are, therefore, continuous, and each of the integrals is a vector valued integral in $\banach(\hilb)$. Equation (\ref{eq:FGrel}) is obvious from the definitions, and if we can prove (\ref{eq:GArel}), then (\ref{eq:FArel}) follows from it (note that $A_{0,N_0}(r;\vep,\kappa)$ actually does not depend on $\kappa$). To prove (\ref{eq:GArel}), apply an arbitrary functional $\Lambda\in\banach(\hilb)^*$ to the integral on the right hand side, and use the definition of $A$ to evaluate $\Lambda[ \rme^{-\ci (t -r) H_0} A_{N',N_0}(r;\vep,\kappa)]$. Then Fubini's theorem allows rearranging the integrals so that a change of variables $s_{N'+N_0+1}=t-r$ yields $\Lambda[G_{N',N_0}(t ;\vep,\kappa)]$. The proofs of equations (\ref{eq:AA0rel}) and (\ref{eq:AArel}) are very similar and we skip them here. \end{proof} \begin{theorem}\label{th:stoppedduh} Let $N_0\ge 1$, $N'_0\ge 0$, and $\kappa>0$ be given. Then for any $t>0$ and for any realization of $\xi$, we have as vector valued integrals in $\banach(\hilb)$ \begin{align}\label{eq:stoppedduh} \rme^{-\ci t H_\vep } & = \sum_{N=0}^{N_0-1} F_{N}(t;\vep) + \sum_{N'=0}^{N'_0-1} \kappa \int_0^{\infty}\!\!\rmd r\, \rme^{-\kappa (r-\ulr)} \rme^{-\ci (t-\ulr) H_\vep } G_{N',N_0}(\ulr;\vep,\kappa) \nonumber \\ & \quad + \int_0^{t}\!\! \rmd r\, \rme^{-\ci (t-r) H_\vep } A_{N'_0,N_0}(r;\vep,\kappa) \end{align} where $\ulr = \min(t,r)$. \end{theorem} \begin{proof} Let us suppress the dependence on $\vep$ from the notation in this proof. The first of the above integrals is defined as $T\to\infty$ limit of \begin{align} & \int_0^{T}\!\rmd r\, \rme^{-\kappa (r-\ulr)} \rme^{-\ci (t-\ulr) H} G_{N',N_0}(\ulr;\kappa) \nonumber \\ & \quad = \int_0^{t}\!\rmd r\, \rme^{-\ci (t-r) H} G_{N',N_0}(r;\kappa) + \int_t^{T}\!\rmd r\, \rme^{-\kappa (r-t)} G_{N',N_0}(t;\kappa), \end{align} which is well-defined as, by Lemma \ref{th:FGAcontin}, $G_{N',N_0}(r;\kappa)$ is continuous in $r$. By the same Lemma, also $A$ in the second integrand is continuous showing that the vector valued integral is well-defined. If $N_0'=0$, Eq.~(\ref{eq:stoppedduh}) follows from (\ref{eq:baseduh}) by a straightforward induction in $N_0$ using (\ref{eq:FArel}) and (\ref{eq:AA0rel}). Let us thus fix $N_0\ge 1$, and perform a second induction in $N'_0\ge 0$. Now for any $r'\ge 0$, \begin{align} 1 = \kappa \int_{r'}^{\infty}\!\rmd r\, \rme^{-\kappa (r-r')}, \end{align} which shows that \begin{align} & \int_0^{t}\!\! \rmd r'\, \rme^{-\ci (t-r') H} A_{N'_0,N_0}(r';\kappa) \nonumber \\ & \quad = \int_0^{t}\!\! \rmd r'\, \kappa \int_{r'}^{\infty}\!\rmd r\, \rme^{-\kappa (r-\ulr)} \rme^{-\kappa (\ulr-r')} \rme^{-\ci (t-\ulr) H} \rme^{-\ci (\ulr-r') H} A_{N'_0,N_0}(r';\kappa) \nonumber \\ & \quad = \kappa \int_{0}^{\infty}\!\rmd r\, \rme^{-\kappa (r-\ulr)} \rme^{-\ci (t-\ulr) H} \int_0^{\ulr}\!\! \rmd r'\, \rme^{-\kappa (\ulr-r')} \rme^{-\ci (\ulr-r') H_0} A_{N'_0,N_0}(r';\kappa) \nonumber \\ & \qquad + \int_0^{t}\!\! \rmd r'\, \kappa \int_{r'}^{\infty}\!\rmd r\, \int_0^{\ulr-r'}\!\! \rmd s\, \rme^{-\kappa (r-r')} \rme^{-\ci (t-r'-s) H} W_s A_{N'_0,N_0}(r';\kappa) \end{align} where we applied the Duhamel formula to the term $\rme^{-\ci (\ulr-r') H}$, and all the manipulations can the justified as before, by applying an arbitrary functional and then using Fubini's theorem. By (\ref{eq:GArel}), the first term yields the new term to the sum over $N'$ in (\ref{eq:stoppedduh}). In the second term we first change integration variables from $s$ to $s'=s+r'$, and then use the identity \begin{align} \1(r'\le s'\le \ulr) = \1(r'\le s') \1(s'\le t) \1(r\ge s') \end{align} and Fubini's theorem yielding the following form for the second term: \begin{align} & \int_0^{t}\!\! \rmd s'\, \rme^{-\ci (t-s') H} \int_0^{s'}\!\! \rmd r'\, \kappa \int_{s'}^{\infty}\!\rmd r\, \rme^{-\kappa (r-r')} W_{s'-r'} A_{N'_0,N_0}(r';\kappa) \nonumber \\ & \quad = \int_0^{t}\!\! \rmd s'\, \rme^{-\ci (t-s') H} \int_0^{s'}\!\! \rmd r'\, \rme^{-\kappa (s'-r')} W_{s'-r'} A_{N'_0,N_0}(r';\kappa) \nonumber \\ & \quad = \int_0^{t}\!\! \rmd s'\, \rme^{-\ci (t-s') H} A_{N'_0+1,N_0}(s';\kappa) \end{align} where we have used (\ref{eq:AArel}). This completes the induction step in $N_0'$. \end{proof} Now we are ready to define how we the splitting is done. \begin{definition}\label{th:defkappaetc} Let $\gamma$ be a constant for which the dispersion relation $\omega$ satisfies the crossing assumption (IC\ref{it:crossing}), and let \begin{align}\label{eq:defgammap} \gamma' = \min\Bigl(\frac{1}{2},\gamma\Bigr),\quad a_0 = \frac{\gamma'}{40}\quad\text{and}\quad b_0 = 40 \Bigl( 1 + \frac{2}{\gamma'}\Bigr). \end{align} For any $\vep$ let us then define \begin{align}\label{eq:chooseN0} N_0(\vep) = \max\Bigl(1,\Bigl\lfloor\,\frac{a_0 \, |\ln \vep|}{\ln \sabs{\ln \vep}}\, \Bigr\rfloor\Bigr), \quad N'_0(\vep) = 8 N_0(\vep) \quad\text{and}\quad \kappa(\vep) = \vep \sabs{\ln \vep}^{b_0} , \end{align} where $\lfloor x\rfloor$ denotes the integer part of $x\ge 0$, and let \begin{align} \psimain(t;\vep) = \sum_{N=0}^{N_0(\vep)-1} F_{N}(t;\vep)\psi^\vep \quad\text{and}\quad \psierr(t;\vep) = \rme^{-\ci t H_\vep }\psi^\vep - \psimain(t;\vep). \end{align} \end{definition} For this choice of parameters, in the limit $\vep\to 0$ we have $N_0\to \infty$, $\kappa\to 0$, and \begin{align}\label{eq:N0limits} c^N N! \sabs{\ln \vep}^{N+d} \vep^{\gamma'} \to 0, \quad\text{and}\quad \vep^{-2} \left(\frac{\vep}{\kappa}\right)^{N_0} c^N N! \sabs{\ln \vep}^{N+d} \to 0, \quad \end{align} where $N=r N_0(\vep)$, with $0\le r < 20$ and $c,d\ge 0$ being arbitrary constants. \begin{definition} For $p\in\R^3$ and $f\in C(\T^3,\M_2)$ let $B_{p,f}$ denote the operator defined for all $\phi,\psi\in\hilb$ by \begin{align} \braket{\phi}{B_{p,f}\psi} = \int_{\T^3}\! \!\rmd k\, \FT{\phi}(k-p/2)\cdot f(k) \FT{\psi}(k+p/2). \end{align} \end{definition} Clearly, then \begin{align}\label{eq:Bpfnorm} \left| \braket{\phi}{B_{q,f}\psi} \right| \le \norm{\phi}\, \norm{\psi}\, \norm{f}_{\infty} \end{align} thus $B_{q,f}\in\banach(\hilb)$ and $\norm{B_{q,f}}\le \norm{f}_{\infty}$. Let us also point out that $B_{0,\1}=\1$, and \begin{align}\label{eq:FvepFourier3} & F^\vep_{\tmacro}(\pmacro,\nmacro) = \E_{\nu_{\tmacro}^\vep}\!\!\left[ \braket{\psi}{ B_{\vep\pmacro,e_{\nmacro}\! P_{++}}\psi}\right] = \E\!\left[ \braket{\rme^{-\ci \tmacro H_\vep/\vep }\psi^\vep}{ B_{\vep\pmacro,e_{\nmacro}\! P_{++}} \rme^{-\ci \tmacro H_\vep/\vep }\psi^\vep}\right] \end{align} where $e_n(k)=\rme^{\ci 2 \pi n\cdot k}$ and $P_{++}$ denotes the projection onto the $+$-subspace. Suppose that \begin{align}\label{eq:errtermbound} & \lim_{\vep\to 0}\E\!\left[\norm{\psierr(\tmacro/\vep;\vep)}^2\right] = 0. \end{align} Then we only need to consider the terms coming from $\psimain$, i.e., to inspect the limit of \begin{align}\label{eq:FvepFourier4} & \Fmain(\pmacro,\nmacro,\tmacro) = \E\!\left[ \braket{\psimain(\tmacro/\vep;\vep)}{ B_{\vep\pmacro,e_{\nmacro}\! P_{++}}\psimain(\tmacro/\vep;\vep)}\right]. \end{align} To see this, first note that by (\ref{eq:FvepFourier3}) and $\norm{B_{\vep\pmacro,e_{\nmacro}\! P_{++}}}\le 1$, \begin{align} \label{eq:deltaFbound} & | F^\vep_{\tmacro}(\pmacro,\nmacro)-\Fmain(\pmacro,\nmacro,\tmacro)| \nonumber \\ & \quad \le 2 \E[\norm{\psierr(\tmacro/\vep;\vep)}^2]^{\frac{1}{2}} \E[\norm{\psimain(\tmacro/\vep;\vep)}^2]^{\frac{1}{2}} + \E[\norm{\psierr(\tmacro/\vep;\vep)}^2] . \end{align} On the other hand, by unitarity and the assumption (IC\ref{it:I1}), then \begin{align} \sup_{\vep}\E[ \norm{\psimain(\tmacro/\vep;\vep)}^2] \le 2 (\sup_\vep \norm{\psi^\vep}^2 + \sup_\vep \E[ \norm{\psierr(\tmacro/\vep;\vep)}^2]) <\infty, \end{align} and the bound in (\ref{eq:deltaFbound}) goes to zero as $\vep \to 0$. To prove (\ref{eq:errtermbound}), we apply Theorem \ref{th:stoppedduh} with $\kappa=\kappa(\vep)$ and $N'_0=N'_0(\vep)$. By the Schwarz inequality, then \begin{align} & \E\!\left[\norm{\psierr(t;\vep)}^2\right] \le 2\Bigl( N'_0 \sum_{N'=0}^{N'_0-1} \kappa^2 \, \E\Bigl[\Bigl(\int_0^{\infty}\!\!\rmd r\, \rme^{-\kappa (r-\ulr)} \norm{G_{N',N_0}(\ulr;\vep,\kappa)\psi^\vep}\Bigr)^2\Bigr] \nonumber \\ & \quad + \E\Bigl[ \Bigl( \int_0^{t}\!\! \rmd r\, \norm{A_{N'_0,N_0}(r;\vep,\kappa)\psi^\vep}\Bigr)^2 \Bigr] \Bigr) . \end{align} Now we can use Schwarz again in the form \begin{align}%\label{eq:errbnd2} & \Bigl(\int_0^{\infty}\!\!\rmd r\, \rme^{-\kappa (r-\ulr)} \norm{G_{N',N_0}(\ulr;\vep,\kappa)\psi^\vep}\Bigr)^2 \nonumber \\ &\quad \le \int_0^{\infty}\!\!\rmd r'\, \rme^{-\kappa (r'-\min(t,r'))} \int_0^{\infty}\!\!\rmd r\, \rme^{-\kappa (r-\ulr)} \norm{G_{N',N_0}(\ulr;\vep,\kappa)\psi^\vep}^2, \end{align} and similarly for the term containing $A$, and we then obtain the bound \begin{align}\label{eq:errbnd2} & \E\!\left[\norm{\psierr(\tmacro/\vep;\vep)}^2\right] \le 2 \tmacro^2 \vep^{-2} \sup_{0\le r\le \tmacro/\vep} \E\!\left[ \norm{A_{N'_0,N_0}(r;\vep,\kappa)\psi^\vep}^2\right] \nonumber \\ & \qquad + 2 (\tmacro \kappa/\vep + 1)^2 (N'_0)^2 \sup_{\substack{0\le N'\le N_0'-1\\ 0\le r\le \tmacro/\vep}} \E\!\left[\norm{G_{N'\!,N_0}(r;\vep,\kappa)\psi^\vep}^2\right]. \end{align} Let $\Emax = \sup_\vep \norm{\psi^\vep}^2$ which is finite by (IC\ref{it:I1}). In the following sections we shall prove that \begin{proposition}\label{th:GNbound} There are constants $c$ and $c'$ and $\vep_1$, which depend only on $\omega$ and $\bar{\xi}$, such that, if $0<\vep\le \vep_1$ and $\tmacro>0$, then for all $0\le t\le \tmacro/\vep$ and $0\le N'0$, then for all $0\le t\le \tmacro/\vep$ \begin{align}\label{eq:ANbound} \E\bigl[ \norm{A_{N'_0,N_0}(t;\vep,\kappa)\psi^\vep}^2\bigr] & \le c' \Emax \, (c T)^{\frac{\bar{N}}{2}} \bar{N}!\, \Bigsabs{\ln \frac{T}{\vep}}^{\! \bar{N}} \Bigl[ \vep^3 + \Bigl(\frac{\vep}{\kappa}\Bigr)^{N_0} \Bigr] \end{align} where $N_0$, $N'_0$ and $\kappa$ are as in Definition \ref{th:defkappaetc}, $T=\sabs{\tmacro}$ and $\bar{N}=18 N_0$. \end{proposition} Using these bounds in (\ref{eq:errbnd2}) and then applying (\ref{eq:N0limits}) shows that indeed then (\ref{eq:errtermbound}) holds: for the term containing $\lfloor N_0/2\rfloor!$ this can be seen using, for instance, the property that $|\ln \vep|\le (N_0+1)^2$ for all sufficiently small $\vep$. Therefore, to complete the proof of Theorem \ref{th:mainlim}, we only need to prove the Propositions \ref{th:GNbound} and \ref{th:ANbound} and that \begin{align}\label{eq:Fveptomut2} & \lim_{\vep\to 0} \Fmain(\pmacro,\nmacro,\tmacro) = \int_{\R^d\times \T^d}\!\! \!\!\mu_{\tmacro}(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi (\pmacro\cdot x- \nmacro\cdot k)}. \end{align} \subsection{Graph representation}\label{sec:graphrep} To prove the remaining Propositions, we use a representation of the expectation values as a sum over a finite number of graphs each contributing a term whose magnitude can be estimated. We first present two Lemmas, the first of which is used compute the expectation values, and the second is a standard tool in time-dependent perturbation theory for manipulation of oscillatory integrals. \begin{lemma}[Representation of expectation values]\label{th:mainrep} Let $N',N\ge 0$ and $\vep>0$ be given, and let $s\in\R^{N+1}$ and $s'\in\R^{N'+1}$. Let also $\psi\in\hilb$ be some non-random vector. Then for all $p\in\R^3$ and $f\in C^{\infty}(\T^3,\M_2)$, \begin{align}\label{eq:psiampL5} & \E\left[\braket{\rme^{-\ci s'_{N'+1} H_0} W_{s'_{N'}} \cdots W_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W_{s_{N}} \cdots W_{s_{1}} \psi}\right] \nonumber \\ & %\quad = (-\ci)^{N-N'} \vep^{\frac{N'+N}{2}} \!\!\!\!\!\!\sum_{S\in\pi(I_{N,N'})} \prod_{A\in S} C_{|A|} \!\!\!\!\!\!\sum_{\sigma \in \set{\pm 1}^{N+N'+2}} \int_{\T^3} \rmd \eta_0 \,\FT{\psi}_{\sigma_1}\!(\eta_0)\, \FT{\psi}_{\sigma'_{1}}\!(\eta_0-p)^* \nonumber \\ & \quad \times \int_{\T^{3(N+N'+1)}}\! \rmd\eta\, \delta(\eta_{N+1}+p) \prod_{A\in S} \delta\Bigl(\sum_{\ell\in A} \eta_\ell\Bigr) f_{\sigma_{N+2},\sigma_{N+1}}\!\Bigl(k_{N+1}-\frac{1}{2} p\Bigr) \nonumber \\ & \quad \times \prod_{\ell=1}^{N} v_{\sigma_{\ell+1}\sigma_\ell}(k_{\ell+1},k_\ell) \prod_{\ell=1}^{N'} v_{\sigma'_{\ell+1}\sigma'_\ell}(k'_{\ell+1},k'_\ell) % \nonumber \\ & \qquad \times \prod_{\ell=1}^{N+1} \rme^{-\ci s_\ell \sigma_\ell \omega(k_\ell)} \prod_{\ell=1}^{N'+1} \rme^{\ci s'_\ell \sigma'_\ell \omega(k'_\ell)} \end{align} where $I_{N,N'}=\set{1,\ldots,N}\cup\set{N+2,\cdots,N+1+N'}$, and $\pi(I)$ denotes the set of all partitions of the finite set $I$. In addition, $k_\ell$ and and $k'_\ell$ are functions of $\eta$: for all $\ell=1,\ldots,N+N'+2$, we define \begin{align}\label{eq:defkell} k_\ell(\eta) = \sum_{n=0}^{\ell-1} \eta_n \end{align} and for all $\ell=1,\ldots,N'+1$, we let $k'_\ell(\eta)=k_{N+N'+3-\ell}(\eta)$ and $\sigma'_\ell = \sigma_{N+N'+3-\ell}$. \end{lemma} $\pi(I)$ is defined explicitly in Appendix \ref{sec:appComb}, in Definition \ref{th:defPiI}. The delta-functions here are a convenient notation for denoting restrictions of the integration into subspaces. Like the earlier time-integration delta-functions, they can be resolved by integrating formally out one of the variables: for each $A\in S$ we choose $n\in A$, remove the integral over $\eta_n$ and set $\eta_n = -\sum_{n'\in A:n'\ne n} \eta_{n'}$. In particular, always $k_1=\eta_0$ and $k'_1=\eta_0-p$. \begin{proof} Both sides of the equality (\ref{eq:psiampL5}) are continuous in $\psi$. Therefore, it is enough to prove the Lemma for $\psi$ which have a compact support. Assume such a vector $\psi$. Using (\ref{eq:defVL}) -- (\ref{eq:vdef}), we define for any $R>0$ \begin{align} W^R_{s} & = (- \ci \sqrt{\vep} \VR ) \rme^{-\ci s H_0} \qand \Wy_{s} = (- \ci \sqrt{\vep} \Vy ) \rme^{-\ci s H_0} . \end{align} As already mentioned in Sec.~\ref{sec:proofmain}, any finite product of $\VR$:s and arbitrary $R$-inde\-pendent bounded operators converge strongly when $R\to\infty$ to the expression with $\VR$ replaced by $V$. In addition, since $\norm{\VR}\le \norm{V}\le 2 \ommax \bar{\xi}$, we can now apply dominated convergence to prove that \begin{align}\label{eq:Rcutoff} & \E\!\left[\braket{\rme^{-\ci s'_{N'+1} H_0} W_{s'_{N'}} \cdots W_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W_{s_{N}} \cdots W_{s_{1}} \psi}\right] \nonumber \\ & \quad = \lim_{R\to \infty} \E\!\left[\braket{\rme^{-\ci s'_{N'+1} H_0} W^R_{s'_{N'}} \cdots W^R_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W^R_{s_{N}} \cdots W^R_{s_{1}} \psi}\right] . \end{align} For a fixed $R>0$, let us define $\Lambda_R=\defset{y\in\Z^3}{\norm{y}_\infty\le R}$, and use (\ref{eq:defVL}) to the term on the right yielding \begin{align} & \sum_{y'\in \Lambda_R^{N'}\!\!, y\in\Lambda_R^{N} } \E\Bigl[\prod_{\ell'=1}^{N'} \xi_{y'_{\ell'}} \prod_{\ell=1}^N \xi_{y_\ell}\Bigr] \\ & \qquad \times \braket{\rme^{-\ci s'_{N'+1} H_0} W^{(y'_{N'})}_{s_{N'}} \cdots W^{(y'_{1})}_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W^{(y_{N})}_{s_{N}} \cdots W^{(y_{1})}_{s_{1}} \psi} . \nonumber \end{align} Then we can denote $y_{N+2+N'-\ell'}=y'_{\ell'}$, define the new index set $I=I_{N,N'}$ and apply the moments-to-cumulants formula, Lemma \ref{th:momtocum}, to find that this is equal to \begin{align}\label{eq:remscalp} & \sum_{S\in\pi(I)} \sum_{x\in \Lambda_R^{S}} \prod_{A\in S} C_{|A|} \sum_{y\in (\Lambda_R)^{I_{N,N'}} } \prod_{A\in S} \prod_{\ell\in A} \delta_{y_\ell,x_A} \\ & \quad \times \braket{\rme^{-\ci s'_{N'+1} H_0} W^{(y_{N+2})}_{s_{N'}} \cdots W^{(y_{N+1+N'})}_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W^{(y_{N})}_{s_{N}} \cdots W^{(y_{1})}_{s_{1}} \psi} . \nonumber \end{align} Evaluation of the remaining scalar product in Fourier space yields the following integral representation for it: \begin{align} & (-\ci)^{N-N'} \vep^{\frac{N'+N}{2}} \sum_{\sigma \in \set{\pm 1}^{N+N'+2}} \int_{\T^3} \!\rmd h \, f_{\sigma_{N+2},\sigma_{N+1}}\!(h) \int_{(\T^3)^{N}} \!\rmd k \int_{(\T^3)^{N'}} \!\rmd k' \nonumber \\ & \quad \times \FT{\psi}_{\sigma_1}\!(k_1)\, \FT{\psi}_{\sigma'_{1}}\!(k'_1)^* \prod_{\ell=1}^N \left[ \rme^{-\ci 2 \pi y_\ell\cdot (k_{\ell+1}-k_\ell)} \right] \prod_{\ell=1}^{N'} \left[ \rme^{\ci 2 \pi y_{N+2+N'-\ell}\cdot (k'_{\ell+1}-k'_\ell)} \right] \nonumber \\ & \quad \times \prod_{\ell=1}^{N} v_{\sigma_{\ell+1}\sigma_\ell}(k_{\ell+1},k_\ell) \prod_{\ell=1}^{N'} v_{\sigma'_{\ell+1}\sigma'_\ell}(k'_{\ell+1},k'_\ell) % \nonumber \\ & \qquad \times \prod_{\ell=1}^{N+1} \rme^{-\ci s_\ell \sigma_\ell \omega(k_\ell)} \prod_{\ell=1}^{N'+1} \rme^{\ci s'_\ell \sigma'_\ell \omega(k'_\ell)} \end{align} where we have defined $k_{N+1}= h + p/2$, and $k'_{N'+1}= h - p/2$, and $\sigma'_\ell = \sigma_{N+3+N'-\ell}$ for all $\ell=1,\ldots,N'+1$. We then change integration variables, first $h= k_{N+1}-p/2$, and then from $k$ to \begin{align} \eta_{\ell} = \begin{cases} k_1, & \text{ for }\ell=0,\\ k_{\ell+1} - k_{\ell}, &\text{ for }\ell=1,\ldots,N,\\ k'_{N+2+N'-\ell} - k'_{N+3+N'-\ell}, &\text{ for } \ell=N+2,\ldots,N+1+N',\\ \end{cases} \end{align} and we also define $\eta_{N+1} = k'_{N'+1}-k_{N+1}= -p$. The inverse of this transformation is given by (\ref{eq:defkell}), and thus the change of variables has a Jacobian equal to one. In the new variables we have \begin{align} \prod_{\ell=1}^N \left[ \rme^{-\ci 2 \pi y_\ell\cdot (k_{\ell+1}-k_\ell)} \right] \prod_{\ell=1}^{N'} \left[ \rme^{\ci 2 \pi y_{N+2+N'-\ell}\cdot (k'_{\ell+1}-k'_\ell)} \right] = \prod_{n\in I} \rme^{-\ci 2 \pi y_n\cdot \eta_n}. \end{align} As for each $\ell\in I$ there is a unique $A\in S$ such that $\ell \in A$, we can perform the sum over $y$ in (\ref{eq:remscalp}). Thus (\ref{eq:remscalp}) is equal to \begin{align} & (-\ci)^{N-N'} \vep^{\frac{N'+N}{2}} \sum_{S\in\pi(I)} \prod_{A\in S} C_{|A|} \sum_{\sigma \in \set{\pm 1}^{N+N'+2}} \sum_{x\in \Lambda_R^{S}} \int_{\T^3} \rmd \eta_0 \int_{(\T^3)^I} \!\rmd \eta\, \FT{\psi}_{\sigma_1}\!(\eta_0) \nonumber \\ & \quad \times \FT{\psi}_{\sigma'_{1}}\!\Bigl( \eta_0-p+\sum_{n\in I}\eta_n\Bigr)^* %\nonumber \\ & \quad \times f_{\sigma_{N+2},\sigma_{N+1}}\!(k_{N+1}-p/2) \prod_{A\in S} \rme^{-\ci 2 \pi x_A\cdot \sum_{n\in A}\eta_n} \nonumber \\ & \quad \times \prod_{\ell=1}^{N} v_{\sigma_{\ell+1}\sigma_\ell}(k_{\ell+1},k_\ell) \prod_{\ell=1}^{N'} v_{\sigma'_{\ell+1}\sigma'_\ell}(k'_{\ell+1},k'_\ell) % \nonumber \\ & \qquad \times \prod_{\ell=1}^{N+1} \rme^{-\ci s_\ell \sigma_\ell \omega(k_\ell)} \prod_{\ell=1}^{N'+1} \rme^{\ci s'_\ell \sigma'_\ell \omega(k'_\ell)} . \end{align} Then we can do one more change of variables by choosing for each $A$ a representative $n_A\in A$, and changing the integration variable $\eta_{n_A}$ to $q_A = \sum_{n\in A}\eta_n$ (with unit Jacobian). Then we are left with sums of the form \begin{align} \sum_{x\in \Lambda_R^{S}} \int_{(\T^3)^S}\! \rmd q\, \rme^{-\ci 2 \pi \sum_{A\in S} x_A\cdot q_A} F(q), \end{align} where $F$ denotes the result from first integrating out all the remaining $\eta$-integrals. Since $\psi$ has compact support, $\FT{\psi}$ is smooth, and so is the rest of the $\eta$-integrand, by assumption (DR\ref{it:DC1}). Therefore, by compactness of the integration region, $F$ is a smooth function of $q$, and thus its Fourier transform is pointwise invertible, implying \begin{align} \lim_{R\to\infty} \sum_{x\in \Lambda_R^{S}} \int_{(\T^3)^S}\! \rmd q\, \rme^{-\ci 2 \pi \sum_{A\in S} x_A\cdot q_A} F(q) = F(0). \end{align} Then it is a matter of inspection to check that indeed \begin{align} & \lim_{R\to\infty} \E\left[\braket{\rme^{-\ci s'_{N'+1} H_0} W^R_{s'_{N'}} \cdots W^R_{s'_{1}} \psi}{ B_{p,f} \rme^{-\ci s_{N+1} H_0} W^R_{s_{N}} \cdots W^R_{s_{1}} \psi}\right] \end{align} is equal to the right hand side of (\ref{eq:psiampL5}). \end{proof} \begin{lemma}\label{th:Knprop} For any $N\ge 1$, define $K_N:(0,\infty)\times \C^{N} \to \C$ by \begin{gather} \label{eq:defKn} K_N(t,w) = \int_{\R_+^{N}}\! \rmd s \, \delta\Bigl(t-\sum_{\ell=1}^{N} s_\ell\Bigr) \prod_{\ell=1}^N \rme^{-\ci s_\ell w_\ell} . \end{gather} Then all of the following hold: \newcounter{tempnumi} \begin{enumerate} \item\label{it:Knrecur} $\displaystyle K_{N+1}(t,w) = \int_0^t \!\rmd r\, \rme^{-\ci (t-r)w_{N+1} } K_{N}(r,w_1,\ldots,w_{N})$. \item\label{it:Knbound} $\displaystyle\left|K_N(t,w)\right| \le \frac{t^{N-1}}{(N-1)!} \rme^{R N t}$, where $R=\max(0,\im w_1,\ldots,\im w_N)$. %\item\label{it:Knsmooth} $K_N(t,w)$ is entire in $w$ and smooth in $t$. \item\label{it:KncountourI} Let $D\subset \C$ be compact and let $\Gamma$ be a closed path which goes once anticlockwise around $D$ without intersecting it. Then for all $w\in D^{N}$ and $t>0$, \begin{gather} \label{eq:Knrep1} K_N(t,w) = - \oint_\Gamma \frac{\rmd z}{2\pi} \rme^{-\ci t z} \prod_{\ell=1}^{N} \frac{\ci}{z-w_\ell} . \end{gather} \end{enumerate} \end{lemma} \begin{proof} Now $K_1(t,w) = \rme^{-\ci t w}$, for which the properties \ref{it:Knbound} and \ref{it:KncountourI} hold trivially. When $N\ge 2$, the definition of $K_N$ is explicitly \begin{align} K_N(t,w) = \int_{\R_+^{N-1}}\! \rmd s \, \1\!\Bigl(\sum_{\ell=1}^{N-1} s_\ell\le t\Bigr) \rme^{-\ci \sum_{\ell=1}^{N-1} s_\ell w_\ell} \rme^{-\ci (t-\sum_{\ell=1}^{N-1} s_\ell) w_{N}}, \end{align} from which \ref{it:Knrecur} can be proven by induction. The property in \ref{it:Knbound} % and \ref{it:Knderiv}, follows then by induction from \ref{it:Knrecur}. So does also \ref{it:KncountourI}, after one notices that if $N\ge 2$ and $t=0$, the right hand side of (\ref{eq:Knrep1}) is equal to zero since Cauchy's theorem allows taking the path $\Gamma$ to infinity. \end{proof} \begin{figure} \begin{center} \myfigure{width=0.9\textwidth}{Gbpath} \caption{The integration path $\gpath(c)$. If $c=\ommax$, the shaded area contains all values of the type $\pm\omega(k)-\ci \kappa$ for all $0\le \kappa\le 1$.} \label{fig:intpath} \end{center} \end{figure} To apply the above Lemma we will choose the integration path $\Gamma$ as follows: For any $c>0$ and $0<\beta\le 1$, let $\gpath(c)$ denote the integration contour which follows the path shown in Fig.~\ref{fig:intpath}. Let $\gpath=\gpath(\ommax)$, and we will choose $\Gamma=\gpath$ for some $\beta$. By construction, (\ref{eq:Knrep1}) then holds for all $w_\ell$ of the form $\pm\omega(k_\ell)-\ci \kappa_\ell$ with $k_\ell\in \T^3$ and $0\le\kappa_\ell\le 1$. \begin{lemma}\label{th:EGG} Let $t>0$, $\kappa\ge 0$, and $N',N_1,N_2\in\N$ be given. Then for all $p\in\R^3$, and $f\in C^{\infty}(\T^3,\M_2)$, \begin{align}\label{eq:EGG} & \E\!\left[\braket{G_{N',N_2}(t;\vep,\kappa)\psi^\vep}{ B_{p,f} G_{N',N_1}(t;\vep,\kappa)\psi^\vep}\right] \nonumber \\ & \quad = \sum_{S\in\pi(I)} \prod_{A\in S} C_{|A|} \, \Kpart(t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f) \end{align} where $I=I_{N'+N_1,N'+N_2}$, and for any partition $S\in \pi(I)$ we have defined \begin{align}\label{eq:defKpart} & \Kpart (t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f) \nonumber \\ & \quad = (-\ci)^{\bar{N}_1-\bar{N}_2} \vep^{\bar{N}/2} \int_{\T^3} \rmd \eta_0 \oint_\gpath \frac{\rmd z}{2\pi} \oint_\gpath \frac{\rmd z'}{2\pi} \, \rme^{-\ci t (z+z')} \nonumber \\ & \qquad \times \int_{(\T^{3})^{\bar{N}+1}}\! \rmd\eta\, \delta(\eta_{\bar{N}_1+1}+p) \prod_{A\in S} \delta\Bigl(\sum_{\ell\in A} \eta_\ell\Bigr) \, \FT{\psi}^\vep(\eta_0-p) \cdot \nonumber \\ & \quad \Bigl[ \prod_{\ell=\bar{N}_1+3+N'}^{\bar{N}+2} \Bigl( \frac{\ci}{z'+\ho(k_\ell)} v(k_{\ell},k_{\ell-1}) \Bigr) \prod_{\ell=\bar{N}_1+3}^{\bar{N}_1+2+N'} \Bigl( \frac{\ci}{z'+\ci \kappa+\ho(k_\ell)} v(k_{\ell},k_{\ell-1}) \Bigr) \nonumber \\ & \quad \times \frac{\ci}{z'+\ci \kappa+\ho(k_{\bar{N}_1+2})} f\Bigl(k_{\bar{N}_1+1}-\frac{1}{2} p\Bigr) \frac{\ci}{z+\ci \kappa-\ho(k_{\bar{N}_1+1})} \nonumber \\ & \quad \times \prod_{\ell=N_1+1}^{N_1+N'} \Bigl( v(k_{\ell+1},k_{\ell}) \frac{\ci}{z+\ci \kappa-\ho(k_\ell)} \Bigr) \prod_{\ell=1}^{N_1} \Bigl( v(k_{\ell+1},k_{\ell}) \frac{\ci}{z-\ho(k_\ell)} \Bigr) \FT{\psi}^\vep(\eta_0) \Bigr] \end{align} with $\bar{N}_i=N_i+N'$ for $i=1,2$, and $\bar{N}=\bar{N}_1+\bar{N}_2$. $\ho$ and $v$ are matrix-valued functions, $v$ is defined by (\ref{eq:vdef}) and $\ho(k)_{\sigma'\sigma} = \delta_{\sigma'\sigma} \sigma \omega(k)$. $0<\beta\le 1$ is arbitrary, and $\gpath$ denotes the corresponding integration path. \end{lemma} \begin{proof} First we use the definition of the two $G$-operators, (\ref{eq:defGN}), to express them as integrals over time-variables which we denote by $s$ and $s'$. Since these are vector valued integrals, the scalar product can be taken inside the time-integrations. Then Fubini's theorem allows swapping the order of the time-integrations and the expectation value. For this we need to have measurability with respect to the product measure, which can be proven by showing, as in (\ref{eq:Rcutoff}), that the integrand is a limit of a sequence of measurable functions. We use Lemma \ref{th:mainrep} to express the remaining expectation value as an integral over the $\eta$-variables, and apply Fubini's theorem to exchange the order of the $s$- and $s'$-integrations and the $\eta$-integration. By Lemma \ref{th:Knprop}:\ref{it:KncountourI} we can express the $s$ and $s'$ -integrals as integrals over $z$ and $z'$, and then summing over the $\sigma$-variables we arrive at the integrand in (\ref{eq:defKpart}). The only remaining step is to reorder the $\eta$- and $z$-, $z'$-integrals as given in (\ref{eq:defKpart}) which is allowed by Fubini's theorem. \end{proof} \begin{corollary}\label{th:EFF} %Let $t>0$ and $N_1,N_2\in\N$ be given. Then \begin{align}\label{eq:EFF} & \Fmain(\pmacro,\nmacro,\tmacro) \\ \nonumber & \quad = \sum_{N_1,N_2=0}^{N_0(\vep)-1} \sum_{S\in\pi(I_{N_1,N_2})} \prod_{A\in S} C_{|A|} \, \Kpart(\tmacro/\vep,S;(0,N_2),(0,N_1),\vep,0,\vep \pmacro,e_{\nmacro}P_{++}). \end{align} \end{corollary} \begin{proof} By definition (\ref{eq:FvepFourier4}), \begin{align} \Fmain(\pmacro,\nmacro,\tmacro) = \sum_{N_1,N_2=0}^{N_0(\vep)-1} & \E\!\left[\braket{F_{N_2}(\tmacro/\vep;\vep)\psi^\vep}{ B_{\vep p,e_{\nmacro}P_{++}} F_{N_1}(\tmacro/\vep;\vep)\psi^\vep}\right] \end{align} which yields (\ref{eq:EFF}) by using first (\ref{eq:FGrel}) and then Lemma \ref{th:EGG}. \end{proof} \begin{lemma}\label{th:EAA} Let $t,\kappa>0$, and $N'_0,N_0\in\N$, with $N_0\ge 1$ be given. Then \begin{align}\label{eq:EAA} & \E\!\left[\norm{A_{N'_0,N_0}(t;\vep,\kappa)\psi^\vep}^2\right] % \nonumber \\ & \quad = \sum_{S\in\pi(I)} \prod_{A\in S} C_{|A|} \, \Kpamp (t,S;N'_0,N_0,\vep,\kappa) \end{align} where $I=I_{N,N}$ with $N=N'_0+N_0$, and for any partition $S\in \pi(I)$, \begin{align}\label{eq:defKpamp} & \Kpamp (t,S;N'_0,N_0,\vep,\kappa) %\nonumber \\ & \quad = \vep^{N} \int_{\T^3} \rmd \eta_0 \oint_\gpath \frac{\rmd z}{2\pi} \oint_\gpath \frac{\rmd z'}{2\pi} \rme^{-\ci t (z+z')} \nonumber \\ & \quad \times \int_{(\T^{3})^{2 N+1}}\! \rmd\eta\, \delta(\eta_{N+1}) \prod_{A\in S} \delta\Bigl(\sum_{\ell\in A} \eta_\ell\Bigr) %\nonumber \\ & \quad \times \, \FT{\psi}^\vep(\eta_0) \cdot \nonumber \\ & \quad \Bigl[ \prod_{\ell=N+3+N'_0}^{2 N+2} \Bigl( \frac{\ci}{z'+\ho(k_\ell)} v(k_{\ell},k_{\ell-1}) \Bigr) %\nonumber \\ & \quad \times \prod_{\ell=N+3}^{N+2+N'_0} \Bigl( \frac{\ci}{z'+\ci \kappa+\ho(k_\ell)} v(k_{\ell},k_{\ell-1}) \Bigr) \nonumber \\ & \quad \times \prod_{\ell=N_0+1}^{N_0+N'_0} \Bigl( v(k_{\ell+1},k_{\ell}) \frac{\ci}{z+\ci \kappa-\ho(k_\ell)} \Bigr) % \nonumber \\ & \quad \times \prod_{\ell=1}^{N_0} \Bigl( v(k_{\ell+1},k_{\ell}) \frac{\ci}{z-\ho(k_\ell)} \Bigr) \FT{\psi}^\vep(\eta_0) \Bigr] \end{align} where the matrix-valued functions $\ho$ and $v$, and the path $\gpath$ are defined as in Lemma \ref{th:EGG} and $0<\beta\le 1$ is arbitrary. \end{lemma} \begin{proof} By following the same steps as in the proof of Lemma \ref{th:EGG}. \end{proof} \begin{figure} \begin{center} { \myfigure{width=0.9\textwidth}{graphex} } \caption{An example of a graph for $\Kpart$ corresponding to $N_1=1$, $N_2=3$, $N'=2$, and to a partition $S$ with $|S|=4$. We have also indicated how we chose to label the interaction vertices, and a few momenta belonging to the propagator lines. See the text for a description of the precise meaning of the different components of the graph.} \label{fig:graphex} \end{center} \end{figure} $\Kpart$ is called the {\em amplitude\/} of the partition, or graph, $S$. It will be helpful to think of the amplitudes in terms of planar graphs, where the structure of the graph encodes the inter-dependence of the momenta $k_\ell$, as imposed by the product of delta-functions $\prod_{A\in S} \delta(\sum_{\ell\in A} \eta_\ell)$. The graph is constructed by starting from the left with a circle, denoting the rightmost $\FT{\psi}^\vep$, and then representing the different factors in the matrix product (\ref{eq:defKpart}), in the order they are acting, so that a solid line represents a term $\ci/(z-H)$, a cross a term $v$, a dashed line a term $\ci/(z+\ci\kappa-H)$, until we reach the {\em observable\/} $f$, which will be denoted by a square. After this the same procedure is repeated, except now a solid line denotes $\ci/(z'+H)$ and a dashed line, $\ci/(z'+\ci\kappa+H)$. The line terminates at a circled asterisk which corresponds to $(\FT{\psi}^\vep)^*$. Each of the fractionals is called a {\em propagator,\/} and a cross is called an {\em interaction vertex.\/} Finally, all interaction vertices belonging to the same cluster in the partition $S$ are joined by a dotted line. Fig.\ \ref{fig:graphex} gives an illustration of such a graph. The graph for $\Kpamp$ is constructed similarly. As $\Kpamp$ is missing the propagators attached to the observable, it is called the amplitude of an {\em amputated\/} graph. We divide the graphs into the following categories: \begin{definition} Let $N',N\ge 0$ be given, and let $S\in\pi(I_{N,N'})$. We call the partition $S$ {\em irrelevant\/} if it contains a singlet, i.e., if there is $A\in S$ such that $|A|=1$. Otherwise, the partition $S$ is called {\em relevant,\/} and then it is \begin{description} \setlength{\itemsep}{0pt} \item[higher order,] if there is $A\in S$ such that $|A|>2$. \item[crossing,] if it is a pairing which contains two pairs crossing each other, i.e., there are $\set{i_1,i_2}$, $\set{j_1,j_2}\in S$ such that $i_10$, $\kappa, \vep\in (0,\frac{1}{2}]$, $N_0,N_0'\in\N$ with $N_0\ge 1$, and a relevant $S\in\pi(I_{N,N})$, \begin{align}\label{eq:Abound} & \left| \Kpamp (t,S;N'_0,N_0,\vep,\kappa) \right| \le c' \Emax \, \vep^{N-|S|} \Bigl(\frac{\vep}{\kappa}\Bigr)^{n_S} \left(c \sabs{\vep t}\right)^{N} \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{2N} \end{align} where $N=N'_0+N_0$ and \begin{align}\label{eq:defNS} n_S = \left| \defset{\max A}{A\in S} \cap \set{N_0,\ldots,N_0+1+2 N'_0}\right|. \end{align} \end{lemma} This estimate suffices to prove the bound for the amputated expectation value. \begin{proofof}{Proposition \ref{th:ANbound}} Let $\kappa=\kappa(\vep)$ and $N'_0 = 8 N_0$ as in Definition \ref{th:defkappaetc}, and denote $N=N_0+N'_0 = 9 N_0$. Let also $a=2\bar{\xi}(3\bar{\xi}^2+1)$ as in Lemma \ref{th:highosum}, and assume that $\vep_1 \le \min(1/a^2,1/2)$ is chosen so that $\kappa(\vep)\le 1/2$ for all $\vep\le \vep_1$. We then consider an arbitrary $\vep\le \vep_1$. We can then apply Lemma \ref{th:basicAest}, together with $\sabs{\vep t}\le \sabs{\tmacro}=T$, arriving at \begin{align}%\label{eq:Abound} & \left| \Kpamp (t,S;N'_0,N_0,\vep,\kappa) \right| \le c' \Emax \, \vep^{N-|S|} \Bigl(\frac{\vep}{\kappa}\Bigr)^{n_S} (c T)^{N} \Bigsabs{\ln \frac{T}{\vep}}^{2N} . \end{align} We still need to estimate the sum over the partitions $S$. First we sum over all partitions containing a cluster of size at least $7$. In this case, we estimate $\vep/\kappa\le 1$ and, using $\sqrt{\vep}\le 1/a$, apply Lemma \ref{th:highosum} which proves \begin{align}%\label{eq:Abound} & \sum_{\substack{S\in\pi(I_{N,N}),\\ \exists A\in S : |A|>6}} \prod_{A\in S} \left|C_{|A|} \right| \, \left| \Kpamp (t,S;N'_0,N_0,\vep,\kappa) \right| \nonumber \\ & \qquad \le c' \Emax (c T)^{N} \Bigsabs{\ln \frac{T}{\vep}}^{2N} (2 N)! \, a^6 \vep^3 . \end{align} Let then $S$ be such that for all $A\in S$, $|A|\le 6$. Then $|S|\ge (2 N)/6= 3 N_0$, and thus $n_S\ge |S| - 2 N_0\ge N_0$. Therefore, $(\vep/\kappa)^{n_S}\le (\vep/\kappa)^{N_0}$. To estimate the remaining sum over the partitions, we can neglect the restriction on the size of the clusters. We use Lemma \ref{th:highosum} to bound the sum over higher-order partitions, and compute the estimate for pairings explicitly. Since the number of possible pairings is $(2N)!/(2^N N!) \le (2N)!$ we get \begin{align}%\label{eq:Abound} & \sum_{S\in\pi(I_{N,N})} \prod_{A\in S} \left|C_{|A|} \right| \, \vep^{N-|S|} %\nonumber \\ & \qquad \le (2 N)! ( 1 + \vep a^2 ) \le 2 (2 N)!. \end{align} We have thus proven that \begin{align}%\label{eq:Abound} & \sum_{S\in\pi(I_{N,N})} \prod_{A\in S} \left|C_{|A|} \right| \, \left| \Kpamp (t,S;N'_0,N_0,\vep,\kappa) \right| \nonumber \\ & \qquad \le c' \Emax (c T)^{N} \Bigsabs{\ln \frac{T}{\vep}}^{2N} (2 N)! \left( a^6 \vep^3 + 2 \Bigl(\frac{\vep}{\kappa}\Bigr)^{N_0} \right) \end{align} from which (\ref{eq:ANbound}) follows by Lemma \ref{th:EAA} after redefinition of the constant $c'$. The bound is trivially valid for $t=0$ since $\norm{A_{N'_0,N_0}(t)}=0$. \end{proofof} For the other estimates, we no longer need the additional decay provided by the terms containing $\kappa$. We do however need to make sure that its presence does not spoil any of the estimates for $G$. In the following Lemmas, whose proofs will be postponed until Section \ref{sec:pflemmas}, $N',N_1,N_2\ge 0$ and $N_0\ge 1$, and we use the notations $\bar{N}_1=N'+N_1$, $\bar{N}_2=N'+N_2$, $\bar{N}=\bar{N}_1+\bar{N}_2$. Let also $I=I_{\bar{N}_1,\bar{N}_2}$. \begin{lemma}[Basic estimate]\label{th:basicest} There are constants $c$ and $c'$, which depend only on $\omega$, such that for any $t>0$, $\kappa, \vep\in (0,\frac{1}{2}]$, and every relevant $S\in\pi(I)$, \begin{align}\label{eq:basicbound} & \left| \Kpart(t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f) \right| \nonumber \\ & \quad \le c' \norm{f}_\infty \Emax \left(c\vep \sabs{t}\right)^{\!\frac{\bar{N}}{2}} \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{\bar{N}+2} \vep^{\1(\bar{N}>0)(\bar{N}-2 |S|)/2} . \end{align} \end{lemma} \begin{lemma}[Crossing partition]\label{th:crossingest} Let the assumptions of Lemma \ref{th:basicest} be satisfied. There is a constant $c''$, depending only on $\omega$, such that if $S\in\pi(I)$ is crossing, then the bound on the right hand side of (\ref{eq:basicbound}) is valid also if it is multiplied by \begin{align}\label{eq:crossingextra} c'' \sabs{t}^{-\gamma} \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{\max(0,d_2-2)}, \end{align} where $\gamma$ and $d_2$ are as in the assumption (DR\ref{it:crossing}). \end{lemma} \begin{lemma}[Nested partition]\label{th:nestedest} Let the assumptions of Lemma \ref{th:basicest} be satisfied. There are constants $c''_1$, $c''_2$, depending only on $\omega$, such that if $S\in\pi(I)$ is nested, then the bound on the right hand side of (\ref{eq:basicbound}) is valid also if it is multiplied by \begin{align}\label{eq:nestedextra} c''_1 (c''_2)^{\frac{\bar{N}}{2}} \bar{N}^{d_1} \!\sabs{t}^{-\frac{1}{2}}. \end{align} \end{lemma} These immediately yield the following estimate for the contribution from non-simple partitions: \begin{corollary}\label{th:nonsimple} There are constants $c$ and $c'$ and $\vep'$, which depend only on $\omega$ and $\bar{\xi}$, such that, if $0<\vep\le \vep'$, $0\le \kappa\le 1/2$, $t>0$, and $\bar{N}>0$, then \begin{align}\label{eq:nonsimplebound} & \sum_{\substack{S\in\pi(I),\\ S \text{ not simple}}} \prod_{A\in S} |C_{|A|}| \, |\Kpart(t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f)| \nonumber \\ & \quad \le c' \norm{f}_\infty \Emax \left(c\vep \sabs{t}\right)^{\!\frac{\bar{N}}{2}} \bar{N}! \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{\bar{N}+\max(2,d_2)} \bar{N}^{d_1} \Bigl(\frac{\sabs{\vep t}}{\sabs{t}}\Bigr)^{\gamma'} \end{align} where $\gamma'$ is defined in (\ref{eq:defgammap}), and $d_1,d_2$ are the constants in (DR\ref{it:suffdisp}) and (DR\ref{it:crossing}). \end{corollary} \begin{proof} Let $a=2\bar{\xi}(3\bar{\xi}^2+1)$ as in Lemma \ref{th:highosum}, and let $\vep'=\min(1/a^2,1/2)$. Then for any $0<\vep\le \vep'$, we get from Lemmas \ref{th:basicest} and \ref{th:highosum}, \begin{align}%\label{eq:Abound} & \sum_{\substack{S\in\pi(I),\\ \exists A\in S : |A|>2}} \prod_{A\in S} \left|C_{|A|} \right| \, \left| \Kpart(t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f) \right| \nonumber \\ & \qquad \le c' \norm{f}_\infty \Emax \left(c\vep\sabs{t}\right)^{\!\frac{\bar{N}}{2}} \Bigsabs{\ln \frac{\sabs{t}}{\vep}}^{\bar{N}+2} \bar{N}! \sabs{a}^2 \sqrt{\vep} . \end{align} There are at most $\bar{N}!$ pairings, which can be combined with Lemmas \ref{th:crossingest} and \ref{th:nestedest} to prove (\ref{eq:nonsimplebound}) after redefinition of constants. \end{proof} \begin{figure} \begin{center} { \myfigure{width=0.9\textwidth}{GrSimple} } \caption{An example of a simple graph $S_m(n,n')$ with $\kappa=0$, $m=2$, and $n=(1,2,0)$, $n'=(0,0,1)$. We have also depicted the dependence of the momenta outside the gates, see Sec.\ \ref{sec:gensimple} for details.} \label{fig:gensimple} \end{center} \end{figure} We need one more estimate before we can complete the proof of Proposition \ref{th:GNbound}. \begin{lemma}[Simple partition]\label{th:simpleest} For any $m\in \N$ and $n,n'\in \N^{m+1}$, let $S_m(n',n)$ denote the partition which consists a ladder of $m$ ``rungs'' and where the components of $n$ and $n'$ define the number of ``gates'' between the rungs, see Fig.~\ref{fig:gensimple}. $S\in\pi(I_{\bar{N}_1,\bar{N}_2})$ is simple if and only if there are $m$ and $n,n'$, with $m+2\sum_{j=1}^{m+1} n_j = \bar{N}_1$ and $m+2\sum_{j=1}^{m+1} n'_j = \bar{N}_2$, such that $S=S_m(n,n')$. In addition, there are constants $c$, $c'$, and $c_i$, $i=0,1,2,3$, depending only on $\omega$, such that for any $t>0$, $\kappa, \vep\in (0,\frac{1}{2}]$, \begin{align}\label{eq:simplebound} & \left| \Kpart(t, S_m(n,n');(N',N_2),(N',N_1),\vep,\kappa,p,f) \right| \nonumber \\ & \quad \le \frac{ \Emax \norm{f}_\infty (c \vep t)^{\bar{N}/2}}{ [((\bar{N}_1+m)/2)!((\bar{N}_2+m)/2)!]^{1/2}} % \nonumber \\ & \qquad + c' \norm{f}_\infty \Emax (c_0\vep \sabs{t})^{\bar{N}/2} \nonumber \\ & \quad \times \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{3} (\bar{N}+1) \Bigl(1 + c_1 \frac{|p|}{\sqrt{\vep}} + c_2 \frac{\kappa}{\vep} + c_3 \sqrt{1+\kappa/\vep}\Bigr) \frac{ \sabs{\vep t}}{\sabs{t}^{1/2}}. \end{align} \end{lemma} \begin{proofof}{Proposition \ref{th:GNbound}} Let $\kappa=\kappa(\vep)$ and $N'_0 = 8 N_0$ as in Definition \ref{th:defkappaetc}. Let also $a=2\bar{\xi}(3\bar{\xi}^2+1)$ as in Lemma \ref{th:highosum}, and assume that $\vep_1 \le \min(1/a^2,1/2)$ is chosen so that $\kappa(\vep)\le 1/2$ for all $\vep\le \vep_1$. We then consider an arbitrary $0<\vep\le \vep_1$, $0\le N'1\ge 2\gamma'$ to justify the estimate $\sabs{t}^{\bar{N}/2-\gamma'} \le \sabs{\bar{t}/\vep}^{\bar{N}/2-\gamma'}$. Here applying first $\vep \sabs{\bar{t}/\vep} \le \sabs{\bar{t}}= T$ and then $\bar{N}\le 18 N_0$ yields the first term in (\ref{eq:GNbound}). We estimate the last term in (\ref{eq:GNbound2}) using \begin{align}%\label{eq:} & \sum_{S_m(n,n')\in\pi(I)} \frac{2^{-\bar{N}/2}}{((N_1+m)/2)!} \le \sum_{m=0}^{N_1} \sum_{n\in \N^{m+1}} \sum_{n'\in \N^{m+1}} \frac{1}{((N_1+m)/2)!} \nonumber \\ & \qquad \1\Bigl(N_1=m+2\sum_{j=1}^{m+1} n_j\Bigr) \1\Bigl(N_1=m+2\sum_{j=1}^{m+1} n'_j\Bigr) \frac{1}{2^m} \prod_{j=0}^m \Bigl(\frac{1}{2^{n_j}}\frac{1}{2^{n'_j}}\Bigr) \nonumber \\ & \quad \le \frac{1}{\lfloor N_1/2\rfloor!} \sum_{m=0}^{\infty} 2^{m+2} \frac{1}{\lfloor m/2\rfloor!} \end{align} where the sum over $m$ is finite. Applying $N_1\ge N_0$ and $2 c_0 \vep t \le \sabs{2 c_0} T$, and readjusting the constants finishes then the proof of the Proposition. \end{proofof} \subsection{Consequences of dispersivity} \label{sec:mainDRestimates} In the derivation of the above Lemmas, we will heavily rely on the following estimates, which follow from the assumed sufficiently strong dispersivity of $\omega$. \begin{lemma} \label{th:morseprop} Let $\omega \in L^{\infty}(\T^3)$ be such that it satisfies the assumption (DR\ref{it:suffdisp}) with a constant $\comega$, and assume that $\ommax = \sup_k |\omega(k)|<\infty$. Then, for any $\kappa\ge 0$ and $\beta>0$ such that $\kappa+\beta\le 1$, all of the following propositions hold for $\gpath=\gpath(\ommax)$ and $n\in\N$: \begin{enumerate} \item\label{it:m0} $\displaystyle \sup_{\sigma=\pm 1,k\in\T^3,z\in\gpath} \frac{1}{|z+\ci\kappa-\sigma \omega(k)|} \le \frac{1}{\beta+\kappa}$, \item\label{it:mz1} $\displaystyle \sup_{\sigma=\pm 1,k\in\T^3} \oint_{\gpath} \frac{|\rmd z|}{2\pi} \frac{1}{|z+\ci\kappa-\sigma \omega(k)|} \le |\gpath| \sabs{\ln (\beta+\kappa)}$. \item\label{it:mk1} $\displaystyle\sup_{\sigma=\pm 1,z\in\gpath} \int_{\T^3} \rmd k\, \frac{1}{|z+\ci\kappa-\sigma \omega(k)|} \le 12 \comega \sabs{\ln (\beta+\kappa)}$. \item\label{it:mk2} For all $n\ge 2$, $\displaystyle \sup_{\sigma=\pm 1,z\in\gpath} \int_{\T^3} \rmd k\, \frac{1}{|z+\ci\kappa-\sigma \omega(k)|^n} \le \frac{3 \comega}{(\beta+\kappa)^{n-1}}$. \item\label{it:mpure} For any smooth function $f$, \begin{align} \sup_{\sigma=\pm 1,z\in\gpath} \left| \int_{\T^3} \rmd k\, \frac{f(k)}{z+\ci\kappa-\sigma \omega(k)} \right| \le 3 \comega \norm{f}_{d_1,\infty}, \end{align} and for all $n\ge 2$, and $n_1,n_2\ge 0$ such that $n_1+n_2=n$, \begin{align} & \sup_{\sigma=\pm 1,z\in\gpath} \left| \int_{\T^3} \rmd k\, \frac{f(k)}{(z-\sigma \omega(k))^{n_1} (z+\ci\kappa-\sigma \omega(k))^{n_2}} \right| \le \frac{3 \comega \norm{f}_{d_1,\infty} }{\beta^{n-3/2}}. \end{align} \end{enumerate} \end{lemma} These results are similar to those used in \cite{chen03,erdos02,erdyau04}, as are the ideas behind the proofs. However, we present here a more straightforward way of doing the analysis. The main additional ingredient we need is the following Lemma: \begin{lemma}\label{th:absest} For any $r\in\R$ and $0<\beta\le 1$, \begin{align}\label{eq:absvalrep} \frac{1}{\sqrt{r^2+\beta^2}} = \int_{-\infty}^\infty\!\! \rmd s \, \rme^{\ci s r} \int_0^\infty\! \frac{\rmd x}{\pi} \frac{1}{\sqrt{1+x^2}} \rme^{-\beta |s| \sqrt{1+x^2}} \end{align} where for any $s\in\R$, $s\ne 0$, \begin{align}\label{eq:absvalbound} & \int_0^\infty\! \frac{\rmd x}{\pi} \frac{1}{\sqrt{1+x^2}} \rme^{-\beta |s| \sqrt{1+x^2}} \le \sabs{\ln\beta} \rme^{-\beta |s|} + \1(|s|\le 1)\ln |s|^{-1} \end{align} and the function on the right hand side belongs to $L^1(\rmd s)$. \end{lemma} \begin{proof} Suppose we have proven the bound in (\ref{eq:absvalbound}). Since $\int_{-1}^1 \rmd s \ln|s|^{-1}=2$, the bound on the right hand side belongs to $L^1(\rmd s)$, and we can apply Fubini's theorem in (\ref{eq:absvalrep}) to swap the $s$ and $x$ integrals. Then \begin{align} \int_{-\infty}^\infty\!\! \rmd s \, \rme^{\ci s r-\beta \sqrt{1+x^2} |s|} = \frac{2 \beta \sqrt{1+x^2}}{\beta^2(1+x^2)+r^2} \end{align} from which (\ref{eq:absvalrep}) follows immediately. We thus only need to prove the bound (\ref{eq:absvalbound}). Let first $0<|s|\le 1$. Then $\beta |s|\le 1$, and \begin{align} & \int_0^\infty\!\frac{\rmd x}{\sqrt{1+x^2}} \rme^{-\beta |s| \sqrt{1+x^2}} %\nonumber \\ & \quad \le \int_0^{(\beta|s|)^{-1}}\!\!\!\frac{\rmd x}{\sqrt{1+x^2}} \rme^{-\beta |s|} + \int_{(\beta|s|)^{-1}}^\infty\!\!\rmd x\, \beta|s| \rme^{-\beta |s| x} \nonumber \\ & \quad = {\rm arsinh}((\beta|s|)^{-1} ) \rme^{-\beta |s|} + \rme^{-1} \le (2+\ln\beta^{-1}) \rme^{-\beta |s|} + \ln |s|^{-1} \nonumber \\ & \quad \le 2\sqrt{2} \sabs{\ln\beta} \rme^{-\beta |s|} + \ln |s|^{-1} \le \pi (\sabs{\ln\beta} \rme^{-\beta |s|} + \ln |s|^{-1} ) \end{align} where we used the fact that for all $x\ge 1$ %as $\ln(1+\sqrt{2})\le 1$, \begin{align} {\rm arsinh}\, x = \ln(x+\sqrt{x^2+1}) \le 1 + \ln x. \end{align} If $|s|\ge 1$, similarly \begin{align} & \int_0^\infty\!\!\frac{\rmd x}{\sqrt{1+x^2}} \rme^{-\beta |s| \sqrt{1+x^2}} \le \int_0^{1/\beta}\!\!\frac{\rmd x}{\sqrt{1+x^2}} \rme^{-\beta |s|} + \int_{1/\beta}^\infty\!\!\rmd x\, \beta \rme^{-\beta |s| x} \nonumber \\ & \quad = {\rm arsinh}(1/\beta )\rme^{-\beta |s|} + \rme^{-|s|} \le (2+\ln\beta^{-1})\rme^{-\beta |s|} \le \pi \sabs{\ln\beta}\rme^{-\beta |s|} \end{align} which completes the proof of inequality (\ref{eq:absvalbound}). \end{proof} \begin{proofof}{Lemma \ref{th:morseprop}} The integration path $\gpath$ consists of two pieces: the uppermost part parameterized by $[-1-\ommax ,\ommax +1]\ni \alpha \mapsto z=-\alpha+\ci \beta$, and the remainder whose distance from the set $\defset{|\re z|\le \ommax}{-1\le \im z\le 0}$ is at least $1$, see Fig.\ \ref{fig:intpath}. Therefore, on the first part $|z+\ci\kappa-\sigma \omega(k)|$ is bounded from below by $\beta+\kappa\le 1$, and on the second part by $1$. This proves item \ref{it:m0}. For item \ref{it:mz1}, we first separate a segment of length two in the uppermost part of $\gpath$, corresponding to $|\alpha-\sigma\omega(k)|\le 1$. The integral over the remaining part is then bounded by $(|\gpath|-2)/(2\pi)$. The value of the integral over the segment is equal to $1/\pi$ times \begin{align} & \int_{0}^1\! \rmd r \, \frac{1}{\sqrt{r^2+(\beta+\kappa)^2}} = \Bigl/_{\!\!\!0}^1 \, \ln(r +\sqrt{(\beta+\kappa)^2+r^2}) \le 1+ |\ln (\beta+\kappa)|. \end{align} Thus the total integral is bounded by \begin{align} \frac{|\gpath|+ 2 |\ln (\beta+\kappa)|}{2\pi} \le |\gpath| \frac{1+ |\ln (\beta+\kappa)|}{2\pi} \le |\gpath| \sabs{\ln (\beta+\kappa)}. \end{align} In all of the estimates in items \ref{it:mk1}--\ref{it:mpure} it is sufficient to assume that $z$ belongs to the uppermost part of the integration path, i.e., $z=\alpha+\ci \beta$ for some $\alpha\in\R$, since otherwise we trivially have bounds by $1$ for items \ref{it:mk1} and \ref{it:mk2}, and by $\norm{f}_\infty$ for item \ref{it:mpure}. Let us also denote $\beta'=\beta+\kappa$. Then in item \ref{it:mk1}, the bound follows from applying Lemma \ref{th:absest}, the assumption (DR\ref{it:suffdisp}), and the relations \begin{align}\label{eq:int0bounds} \int_0^1 \!\rmd s\, \ln |s|^{-1} =1 \qand \int_0^\infty \!\rmd s \, \frac{1}{\sabs{s}^{3/2}} \le 1+\int_1^{\infty}\!\rmd s\, s^{-3/2}=3. \end{align} For item \ref{it:mk2}, we first use the trivial bound in item \ref{it:mz1} to see that the integrand is bounded by $(\beta')^{2-n}/|\alpha-\sigma \omega(k)+\ci\beta'|^2$. Then the estimate follows from applying the equality \begin{align}\label{eq:deltaappr} \frac{1}{r^2+(\beta')^2} = \frac{1}{2\beta'} \int_{-\infty}^\infty\!\! \rmd s \, \rme^{\ci s r-\beta' |s|} \end{align} valid for all $r\in\R$, and then estimating the result using the assumption (DR\ref{it:suffdisp}). Finally, for item \ref{it:mpure}, we use the fact that in all of the terms the imaginary part is strictly positive, proving \begin{align} & \int_{\T^3} \rmd k\, \frac{f(k)}{(\alpha+\ci\beta -\sigma \omega(k))^{n_1} (\alpha+\ci(\beta+\kappa)-\sigma \omega(k))^{n_2}} \nonumber \\ & \quad = (-\ci)^n \int_{\R_+^n}\!\rmd s\, \rme^{-\sum_{\ell=1}^{n_2} s_{\ell} \kappa} \rme^{-(\beta-\ci \alpha) \sum_{\ell=1}^{n} s_{\ell} } \int_{\T^3} \rmd k\, f(k) \rme^{-\ci \sigma \omega(k) \sum_{\ell=1}^{n} s_{\ell} } . \end{align} By (DR\ref{it:suffdisp}), this is bounded by \begin{align} & \int_{\R_+^n}\!\rmd s\, \rme^{-\beta \sum_{\ell} s_{\ell}} \frac{\comega}{\Bigsabs{ \sum_{\ell} s_{\ell}}^{3/2}} \norm{f}_{d_1,\infty} = \frac{\comega \norm{f}_{d_1,\infty} }{(n-1)!} \int_0^\infty\!\rmd s \frac{s^{n-1}}{\sabs{s}^{3/2}} \rme^{-\beta s}. \end{align} If $n=1$, the integral over $s$ is bounded by $3$. Otherwise, it is bounded by \begin{align} & 1 + \int_1^\infty\!\rmd s \, s^{n-1-3/2} \rme^{-\beta s} \le 1 + \beta^{3/2-n} \left(1+\int_0^\infty\!\rmd r \, r^{n-2} \rme^{-r}\right) \nonumber \\ & \le 3 \beta^{3/2-n} (n-1)! \end{align} where we have used $\beta\le 1$. This finishes the proof of item \ref{it:mpure}. \end{proofof} \begin{corollary} \label{th:Mmorseprop} Under the same assumptions as in Lemma \ref{th:morseprop}, for both $\sigma=\pm 1$, \begin{enumerate} \item\label{it:Nm0} $\displaystyle \sup_{k\in\T^3,z\in\gpath} \Bigl\Vert\frac{1}{z+\ci\kappa-\sigma \ho(k)} \Bigr\Vert \le \frac{1}{\beta+\kappa}$, \item\label{it:Nmz1} $\displaystyle \sup_{k\in\T^3} \oint_{\gpath} \frac{|\rmd z|}{2\pi} \Bigl\Vert\frac{1}{z+\ci\kappa-\sigma \ho(k)} \Bigr\Vert \le 2 |\gpath| \sabs{\ln \beta}$. \item\label{it:Nmk1} $\displaystyle\sup_{z\in\gpath} \int_{\T^3} \rmd k\, \Bigl\Vert\frac{1}{z+\ci\kappa-\sigma \ho(k)} \Bigr\Vert \le 24 \comega \sabs{\ln \beta}$. \end{enumerate} \end{corollary} \begin{proof} Item \ref{it:Nm0} is clear. The other two follow by first estimating the integrands by \begin{align}\label{eq:normup} \Bigl\Vert\frac{1}{z+\ci\kappa -\sigma \ho(k)} \Bigr\Vert \le \sum_{\sigma'=\pm 1} \Bigl|\frac{1}{z+\ci\kappa-\sigma' \omega(k)} \Bigr| \end{align} and then applying the Lemma. \end{proof} \subsection{Derivation of the basic bounds} \label{sec:pflemmas} Let us now fix the way of resolving the momentum delta-functions. Given a partition $S\in \pi(I_{N,N'})$, we define $M(S)=\defset{\max A}{A\in S}$, and for any index $\ell\in I_{N,N'}$, let $A(\ell)$ denote the unique cluster in $S$ which contains $\ell$. For each $A\in S$ we integrate out $\eta_\ell$ with $\ell=\max A$. Then every $\eta_\ell$ with $\ell\in I_{N,N'}\setm M(S)$ is {\em free\/}, i.e., it is integrated over the whole of $\T^3$ independently of the values of the other integration variables, and for $\ell\in M(S)$ we have \begin{align} \eta_\ell = - \sum_{n\in A(\ell):n<\ell} \eta_{n}. \end{align} Given a propagator index $\ell\in \set{1,\ldots,N+N'+2}$, we call a cluster $A\in S$ {\em broken at $\ell$\/} if $\min A<\ell \le \max A$, and we call an index $n \in I_{N,N'}$ {\em free at $\ell$\/} if $n<\ell$ and $\max A(n)\ge \ell$. The first terminology is explained by Figure \ref{fig:graphex}, and the second comes from the fact that the function $k_\ell(\eta)$ depends only on those free integration variables $\eta_n$ which are free at $\ell$. Explicitly, by (\ref{eq:defkell}) we have for all $\ell$ \begin{align}\label{eq:standardkell} k_\ell(\eta) = \eta_0 + \1(\ell>N+1) \eta_{N+1} + \sum_{\substack{n\in I_{N,N'}: n<\ell\\ \max A(n)\ge \ell} } \eta_n. \end{align} The following Lemma will allow estimating most of the $\eta$-integrals: \begin{lemma}\label{th:iterstep} Let $N',N\in \N$ be given, and assume $S\in\pi(I_{N,N'})$. Define $M'=M(S)\cup\set{N+1}$, and for each $\ell=2,\ldots,N+N'+1$ let $f_\ell\in L^1(\T^3)\cap L^\infty(\T^3)$ with $f_\ell\ge 0$. Then, if $n\in\set{1,\ldots,N+N'}$ is such that $|I_{n}\setm M'|>1$, we have for $n'=\max (I_n \setm M')-1$ \begin{align}\label{eq:fiterest} &\int_{(\T^3)^{I_n\setm M'}}\!\!\!\!\!\rmd \eta\, \prod_{\ell=2}^{n+1} f_\ell(k_\ell(\eta)) %\nonumber \\ & \quad \le \norm{f_{n'+2}}_1 \prod_{\ell =n'+3}^{n+1} \norm{f_{\ell}}_\infty \int_{(\T^3)^{I_{n'}\setm M'}}\!\!\!\! \rmd \eta\, \prod_{\ell=2}^{n'+1} f_\ell(k_\ell(\eta)) . \end{align} \end{lemma} \begin{proof} Since $|I_{n}\setm M'|>1$, $\max (I_n \setm M')\ge 2$ and thus $1\le n'< n$. Clearly, then also $I_{n'}\setm M'$ is a non-empty, proper subset of $I_{n}\setm M'$. If $n'< n-1$, we first use $f_\ell\le \norm{f_\ell}_\infty$ and positivity of $f_\ell$, to estimate the product of the terms with $\ell=n'+3,\ldots,n+1$. Since $I_{n}\setm M' = \set{n'+1} \cup (I_{n'}\setm M')$, we find that the left hand side of (\ref{eq:fiterest}) is less than or equal to \begin{align} & \prod_{\ell =n'+3}^{n+1} \norm{f_{\ell}}_\infty \int_{(\T^3)^{I_{n'}\setm M'}}\! \rmd \eta\, \Bigl[ \int_{\T^3}\! \rmd \eta_{n'+1}\, \prod_{\ell=2}^{n'+2} f_\ell(k_\ell(\eta)) \Bigr]. \end{align} However, if $\ell\le n'+1$, then $k_\ell$ does not depend on $\eta_{n'+1}$, and therefore \begin{align} & \int_{\T^3}\! \rmd \eta_{n'+1}\, \prod_{\ell=2}^{n'+2} f_\ell(k_\ell(\eta)) = \prod_{\ell=2}^{n'+1} f_\ell(k_\ell(\eta)) \int_{\T^3}\! \rmd \eta_{n'+1}\, f_{n'+2}(k_{n'+2}(\eta)). \end{align} As $k_{n'+2}(\eta) = \eta_{n'+1} +( \text{term independent of }\eta_{n'+1})$, the remaining integral is equal to $\norm{f_{n'+2}}_1$. \end{proof} The point of including $N+1$ to $M'$ is that then $I_{N,N'}\setm M(S)=I_{N+N'}\setm M'$ for all $N,N'\ge 0$. Estimating the missing case with $|I_{n}\setm M'|=1$ similarly and using induction proves \begin{corollary}\label{th:genbasicb} If $I_{N,N'}\setm M(S) \ne \emptyset$, \begin{align}%\label{eq:fiterest} &\int_{(\T^3)^{I_{N,N'}\setm M(S)}}\! \rmd \eta\, \prod_{\ell=2}^{N+N'+1} f_\ell(k_\ell(\eta)) %\nonumber \\ & \quad \le \prod_{n\in I_{N+N'}\setm M'} \norm{f_{n+1}}_1 \!\!\prod_{n\in M'\cap I_{N+N'}}\!\! \!\!\norm{f_{n+1}}_\infty . \end{align} \end{corollary} This is sufficient to prove the basic estimates. \begin{proofof}{Lemma \ref{th:basicAest}} Applying the above resolution of delta-functions to (\ref{eq:defKpamp}), and then taking absolute values inside the remaining integrals shows that \begin{align}\label{eq:defKpamp2} & | \Kpamp (t,S;N'_0,N_0,\vep,\kappa)| %\nonumber \\ & \quad \le (2\ommax)^{2 N}\vep^{N} \int_{\T^3} \rmd \eta_0 \norm{\FT{\psi}^\vep(\eta_0)}^2 \nonumber \\ & \times \oint_\gpath \frac{|\rmd z|}{2\pi} \oint_\gpath \frac{|\rmd z'|}{2\pi} \rme^{t \im (z+z')} \Bigl\Vert \frac{1}{z-\ho(\eta_0)} \Bigr\Vert \, \Bigl\Vert \frac{1}{z'+\ho(\eta_0)} \Bigr\Vert \nonumber \\ & \times \int_{(\T^3)^{I_{N,N}\setm M(S)}}\! \rmd \eta\, \prod_{\ell=2}^{N} \Bigl\Vert\frac{1}{z+\ci \kappa_\ell -\ho(k_\ell)} \Bigr\Vert \prod_{\ell=N+3}^{2 N+1} \Bigl\Vert\frac{1}{z'+\ci \kappa_\ell +\ho(k_\ell)} \Bigr\Vert %\nonumber \\ & \quad \times \end{align} where we used the bound $\norm{v(k',k)}\le 2\ommax $, and defined $\kappa_\ell = \kappa$ for $N_0+1\le \ell\le N_0+2+2 N'_0$ and zero elsewhere. We shall now choose \begin{align}\label{eq:defbeta} \beta=\frac{1}{2\sabs{t/2}}, \end{align} when $\beta+\kappa \le 1$. Since $N>0$ and there are no singlets in $S$, we have $1\in I_{N,N}\setm M(S)$. Therefore, by Corollaries \ref{th:Mmorseprop} and \ref{th:genbasicb}, the last line of (\ref{eq:defKpamp2}) is bounded by \begin{align} (24 \comega \sabs{\ln \beta})^{2 N-|S|-1} \frac{1}{(\beta+\kappa)^{n_S}} \frac{1}{\beta^{|S|-n_S}}. \end{align} where $n_S$ is defined by (\ref{eq:defNS}). To arrive at this bound, first note that $f_{N+1},f_{N+2}=1$ and, as by $N>0$ we have $N+1\in M'\cap I_{2N}$, one of the $L^\infty$-estimates is $\norm{f_{N+2}}_{\infty}=1$. For $f_{N+1}$ we have used the property that all bounds coming from Corollary \ref{th:Mmorseprop} are greater than one. The remaining integrals over $z$ and $z'$ are then estimated using $t\im(z+z')\le 2 t \beta\le 2$ and Corollary \ref{th:Mmorseprop}. Since for all $c\ge 1$ and $x\in \R$, $\sabs{c x} \le c \sabs{x}$, now $\vep/\beta \le 2\vep \sabs{t\vep/(2\vep)} \le \sabs{t\vep}$, and thus also $0<-\ln \beta \le \ln (\sabs{t\vep}\!/\vep)$. Using these bounds and $\comega\ge 1$ proves (\ref{eq:Abound}) for $c=\sabs{48 \ommax \comega}^2$ and $c'= \rme^2 2^4 (2 \ommax+5)^2$. \end{proofof} For the rest of this section, we make the assumptions in Lemma \ref{th:basicest}. In particular, we assume that $N_1$, $N_2$ and $N'$ are given as in the Lemma, and we let $\bar{N}_1=N_1+N'$, $\bar{N}_2=N_2+N'$, and $N=(\bar{N}_1 + \bar{N}_2)/2$. Let us also define $\kappa_\ell = \kappa$ for $N_1+1\le \ell\le N_1+2+2 N'$ and zero elsewhere. \begin{proofof}{Lemma \ref{th:basicest}} The proof is almost identical to the one above, except we can ignore the sharper bounds coming from $\kappa>0$. We start from \begin{align}\label{eq:Kpartb1} & \left| \Kpart(t,S;(N',N_2),(N',N_1),\vep,\kappa,p,f) \right| \nonumber \\ & \quad \le \norm{f}_\infty (2\ommax)^{2 N} \vep^{N} \int_{\T^3} \rmd \eta_0 \norm{\FT{\psi}^\vep(\eta_0)}\norm{\FT{\psi}^\vep(\eta_0-p)} \nonumber \\ & \quad \times \oint_\gpath \frac{|\rmd z|}{2\pi} \oint_\gpath \frac{|\rmd z'|}{2\pi} \rme^{2} \Bigl\Vert \frac{1}{z+\ci \kappa_1-\ho(\eta_0)} \Bigr\Vert \, \Bigl\Vert \frac{1}{z'+\ci \kappa_{2 N+2}+\ho(\eta_0-p)} \Bigr\Vert \nonumber \\ & \quad \int_{(\T^3)^{I_{\bar{N}_1,\bar{N}_2}\!\setm M(S)}}\! \rmd \eta\, \prod_{\ell=2}^{\bar{N}_1+1} \Bigl\Vert\frac{1}{z+\ci \kappa_\ell -\ho(k_\ell)} \Bigr\Vert \prod_{\ell=\bar{N}_1+2}^{2 N+1} \Bigl\Vert\frac{1}{z'+\ci \kappa_\ell +\ho(k_\ell)} \Bigr\Vert . %\nonumber \\ & \quad \times \end{align} If $N=0$, then $N'=0=N_1=N_2$ and the last line in the above formula is equal to one, and estimating the first two lines by Corollary \ref{th:Mmorseprop} yields the estimate in (\ref{eq:basicbound}) with $c'= \rme^2 2^4 (2 \ommax+5)^2$. Let then $N>0$. As $S$ is relevant and thus contains no singlets, we have $I_{\bar{N}_1,\bar{N}_2}\!\setm M(S)\ne \emptyset$ and $|S|\le N$. We can thus apply Corollaries \ref{th:Mmorseprop} and \ref{th:genbasicb} and show that the last line in (\ref{eq:Kpartb1}) is bounded by \begin{align}\label{eq:b1b2} (24 \comega \sabs{\ln \beta})^{|I_{2 N}\setm M'|} \beta^{-|M'\cap I_{2 N}|}. \end{align} Now $|M'\cap I_{2 N}|\le |S|$ as $|M'|=|S|+1$ and $2 N+1 \in M'\setm I_{2 N}$ (if $\bar{N}_2=0$, then $2 N +1 = \bar{N}_1+1$, and otherwise $2 N +1 \in I_{\bar{N}_1,\bar{N}_2}$). Using also $|I_{2 N}\setm M'|\le 2 N$, we thus find that (\ref{eq:b1b2}) is bounded by $(24 \comega)^{2 N} \sabs{\ln \beta}^{2 N} \beta^{-|S|}$. The remainder of the integral can be estimated as when $N=0$, and the terms containing $\beta$ majorized as in the previous proof. This proves (\ref{eq:Abound}) for the same $c'$ as above and $c=\sabs{48 \ommax \comega}^2$. \end{proofof} \begin{proofof}{Lemma \ref{th:crossingest}} Let us denote here $I=I_{\bar{N}_1,\bar{N}_2}$, and recall the earlier definitions of $M(S)$ and $M'$. We begin the estimation of the amplitude of the crossing partition $S$ from (\ref{eq:Kpartb1}). A sequence of two pairings $(P,P')$, $P,P'\in S$, is called {\em crossing\/} if $\min P<\min P'<\max P<\max P'$. For convenience we have included the ordering of the pairings in the definition. Furthermore, a crossing sequence is called {\em loose,\/} if for every $n\in I$ with $\min P'N+1$. Here $g$ is the following matrix-valued function: \begin{definition}\label{th:defgate} We define for all $k\in\T^3$, and $w\in\C\setm[-\ommax,\ommax]$, \begin{align}%\label{eq:} & g(k;w) = \int_{\T^3}\!\! \rmd k' v(k,k') \frac{\ci}{w-\ho(k')} v(k',k) . \end{align} \end{definition} Explicitly, the $\sigma_1 \sigma_2$-component of $g$ is then given by \begin{align}\label{eq:gcomponents} \sum_{\sigma'=\pm 1} \int_{\T^3}\! \rmd k' \frac{\ci}{w-\sigma'\omega(k')} \frac{\omega(k)+\sigma_1 \sigma'\omega(k')}{2} \frac{\omega(k)+\sigma_2 \sigma'\omega(k')}{2} . \end{align} We need to study this function in fairly great detail, and for this we will also need certain properties of the level set measures of $\omega$, derived in Appendix \ref{sec:appBoltzmann}. \begin{lemma}\label{th:gateint} As a function of $k$, $g(k;w)$ is a second order polynomial in $\omega(k)$, with coefficients uniformly bounded for all $w\in \gpath+\ci \kappa$ and $\beta,\kappa>0$, with $\kappa+\beta\le 1$. In addition, the following limit converges for all $k\in \T^3$ and $\sigma=\pm 1$: \begin{align} \Theta_\sigma(k) = \lim_{\beta\to 0^+} g_{\sigma\sigma}(k;\sigma \omega(k)+\ci \beta). \end{align} The functions $\Theta_\sigma$ are H\"{o}lder continuous with exponent $\frac{1}{2}$, $\Theta_- = \Theta_+^*$, \begin{align}\label{eq:reTheta} \re \Theta_+(k) = \pi \omega(k)^2 \int_{\T^3} \!\rmd k' \, \delta(\omega(k')-\omega(k)) = \frac{1}{2} \nu_k(\T^3), \end{align} for all $k\in\T^3$, and there are constants $c'_i$, $i=1,2,3$, such that for all $\beta,\kappa$ as above, $k,k'\in \T^3$, $\sigma=\pm 1$, and $w\in\gpath+\ci \kappa$, \begin{align}\label{eq:gtotheta} & \left|g_{\sigma\sigma}(k';w)- \Theta_\sigma(k) \right| % \nonumber \\ & \quad \le c'_1 |\omega(k')-\omega(k)| + \frac{c'_2}{\sqrt{\beta}} |w-\sigma\omega(k')| + c'_3 \sqrt{\beta+\kappa} . \end{align} \end{lemma} %\begin{proofof}{Lemma \ref{th:gateint}} \begin{proof} By (\ref{eq:gcomponents}) and Lemma \ref{th:morseprop}:\ref{it:mpure}, $k\mapsto g_{\sigma_1\sigma_2}(k;w)$ is a second order polynomial in $\omega(k)$ with uniformly bounded coefficients. In particular, also \begin{align}\label{eq:defbarg} & \bar{g} = \sup_{\beta,\kappa} \sup_{z\in \gpath, k\in \T^3} \left\Vert g(k;z+\ci\kappa)\right\Vert <\infty. \end{align} To study the limit of small $\beta$, we apply to (\ref{eq:gcomponents}) the following equality, which is valid for all $\alpha\in\R$, $\beta\ge\beta'>0$, \begin{align} \frac{\ci}{\alpha+\ci \beta'} - \frac{\ci}{\alpha+\ci \beta} = \int_{\beta'}^{\beta} \rmd \lambda\, \frac{1}{(\alpha+\ci \lambda)^2}. \end{align} Then by Lemma \ref{th:morseprop}, there is $c''$ such that for $0<\beta'<\beta\le 1$, $\sigma=\pm 1$, and $k\in\T^3$, \begin{align} & \left|g_{\sigma\sigma}(k;\sigma\omega(k)+\ci \beta')- g_{\sigma\sigma}(k;\sigma\omega(k)+\ci \beta)\right| % \\ & \quad \le \frac{c''}{2} \int_{\beta'}^\beta\rmd \lambda\, \lambda^{-1/2} \le c''\sqrt{\beta}. \end{align} This proves that the limits $\Theta_\sigma(k)$ exist for all $k$ and $\sigma$, and that \begin{align}\label{eq:Thetaest} & \sup_{k,\sigma}\left|\Theta_\sigma(k)- g_{\sigma\sigma}(k;\sigma\omega(k)+\ci \beta)\right| \le c''\sqrt{\beta}. \end{align} As $g_{--}(k;-\omega(k)+\ci \beta) = g_{++}(k;\omega(k)+\ci \beta)^*$, we have then $\Theta_- = \Theta_+^*$. By the same Lemma, there is a constant $c'_1$ such that \begin{align} \sup_{w\in\gpath+\ci\kappa,\sigma=\pm 1} \left|g_{\sigma\sigma}(k';w) - g_{\sigma\sigma}(k;w) \right| \le \left|\omega(k)-\omega(k')\right| c'_1. \end{align} Suppose for a moment that $0<\beta\le 1$ and $\alpha\in\R$ satisfies $|\alpha|\le \ommax+1$. Since for all $a,a'\in \R$, \begin{align} & \frac{1}{ a'+\ci \beta} - \frac{1}{ a+\ci \beta} %\\ & \quad = = \int_0^1\!\! \rmd\lambda \frac{a-a'}{(a+ \lambda(a'-a)+\ci \beta)^2} \end{align} we can apply Lemma \ref{th:morseprop}:\ref{it:mpure} with $n=2$ and conclude that there is a constant $c''_2$, depending only on the function $\omega$, such that \begin{align}\label{eq:gvaryzbound} & \left| g_{\sigma\sigma}(k;\alpha+\ci \beta)- g_{\sigma\sigma}(k;\sigma\omega(k)+\ci \beta) \right| \le c''_2 |\alpha-\sigma\omega(k)| \beta^{-1/2} \nonumber \\ & \quad \le c''_2 |\alpha+\ci \beta-\sigma\omega(k)| \beta^{-1/2}+ c''_2 \beta^{1/2}. \end{align} Let then $\kappa,\beta$ be as in the assumptions of the Lemma. Then for $w\in\gpath+\ci \kappa$, either $w$ is of the already considered form, or $|w-\sigma\omega(k)|\ge 1$. But in the latter case $\left| g_{\sigma\sigma}(k;w)- g_{\sigma\sigma}(k;\sigma\omega(k)+\ci \beta)\right| \le 2\bar{g} |w-\sigma\omega(k)| \beta^{-1/2}$, and the inequalities proven so far imply the inequality (\ref{eq:gtotheta}). For the continuity of $\Theta_\sigma$, let $h$ be such that $|h|<1$, and define $\beta=|h|$. Then % by (\ref{eq:gtotheta}) \begin{align} & \left|\Theta_\sigma(k+h)-\Theta_\sigma(k) \right| \le (c'_2+c'_3) \sqrt{\beta} + \left|\Theta_\sigma(k+h)-g_{\sigma\sigma}(k;\sigma\omega(k)+\ci\beta) \right| \nonumber \\ & \quad \le 2 (c'_2+c'_3)\sqrt{\beta} + (c'_1+\frac{c'_2}{\sqrt{\beta}}) |\omega(k+h)-\omega(k)| . \end{align} Since $\omega$ is smooth, there thus is a constant $c$ such that \begin{align} \left|\Theta_\sigma(k+h)-\Theta_\sigma(k) \right|\le c \beta^{1/2}= c |h|^{1/2} \end{align} which proves that the function is H\"{o}lder-continuous with an exponent $1/2$. Finally, to prove (\ref{eq:reTheta}) note that $\re \Theta_+(k)$ is the $\beta\to 0^+$ limit of \begin{align} \sum_{\sigma'=\pm 1} \int_{\T^3} \rmd k'\! \frac{\beta}{ (\omega(k)-\sigma'\omega(k'))^2+\beta^2} \left(\frac{\omega(k)+\sigma'\omega(k')}{2}\right)^2 . \end{align} As $|\omega(k)|\ge \ommin > 0$, the $\sigma'=-1$ term is $\order{\beta}$, and for the $\sigma'=+1$ we use \begin{align} \frac{\omega(k)+\omega(k')}{2} = \omega(k) + \frac{\omega(k')-\omega(k)}{2} \end{align} to expand the square. The term having $(\omega(k')-\omega(k))^2$ is $\order{\beta}$, and the term with $\omega(k')-\omega(k)$ vanishes by dominated convergence, justifiable by the estimate (\ref{eq:bxunif}). The only non-vanishing term is \begin{align} \omega(k)^2 \int_{\T^3}\!\rmd k'\, \frac{\beta}{(\omega(k)-\omega(k'))^2+\beta^2} \end{align} which converges to the middle formula in (\ref{eq:reTheta}). The last equality follows then from Proposition \ref{th:crosssectprop}:\ref{it:movecontel}. \end{proof} \begin{proofof}{Lemma \ref{th:nestedest}} We call a pairing $P\in S$ {\em nesting\/} if $\min P > \bar{N}_1+1$ or $\max P<\bar{N}_1+1$, and there is $P'\in S$ such that $\min P<\min P'<\max P'<\max P$ -- the first condition is to exclude nests going over $\bar{N}_1+1$ which will contribute to the main term. A nesting is called {\em minimal\/} if the nest contains only gates, i.e., $\min P 1$ such that $\sigma_{j_0}\ne \sigma_1$. Then, by using $1/(ab) = (1/a-1/b)/(b-a)$ and $\ommin >0$, we find that \begin{align}\label{eq:oneoverab} & \left|\frac{1}{w + \ci \kappa_{i_1+1} -\tau \sigma_1 \omega(k)} \frac{1}{w + \ci \kappa_{i_1+2 j_0-1} -\tau \sigma_{j_0} \omega(k)} \right| \nonumber \\ & \quad \le \frac{1}{2 \ommin} \Bigl( \frac{1}{|w + \ci \kappa_{i_1+1} -\tau \sigma_1 \omega(k)|} + \frac{1}{|w + \ci \kappa_{i_1+2 j_0-1} -\tau \sigma_{j_0} \omega(k)|} \Bigr) . \end{align} But, since $|g|,|v|\le c_1$, we can use the trivial bound for the rest of the terms, and then bound the remaining integral by Lemma \ref{th:morseprop}:\ref{it:mk1}. This shows that the absolute value of the summand is bounded by \begin{align}%\label{eq:} & \frac{c_1^{m+2}}{2\ommin} \beta^{1-m} 24 \comega \sabs{\ln \beta} . \end{align} Since there are less than $2^{m+1}$ such terms, we have proven that \begin{align}%\label{eq:} & \norm{G'_m} \le 2 \max_{\sigma',\sigma}{|(G'_m)_{\sigma'\sigma}|} % \nonumber \\ & \quad \le (2 c_1)^{m+2} 12 \comega \beta^{\frac{1}{2}-m} \Bigl( \frac{2}{\ommin} \beta^{\frac{1}{2}} \sabs{\ln \beta} + (m+2)^{d_1} \Bigr). \end{align} Here $\beta^{\frac{1}{2}} \sabs{\ln \beta} \le 1$ since $0<\beta\le 1$, and we find that there are constants $c'_1$, $c'_2$, which depend only on $\omega$, such that \begin{align}\label{eq:xxb1} & \norm{G'_m} \le c'_1 (c'_2)^{m+1} (m+2)^{d_1} \beta^{\frac{1}{2}-m} . \end{align} This upper bound is a constant, and we can take it out of all of the remaining integrals. Estimating the remainder by Corollary \ref{th:genbasicb} yields a bound which is the basic bound times \begin{align}\label{eq:xxb2} (2\ommax)^{-2(m+1)} \beta^m (24 \comega\sabs{\ln \beta})^{-m-1} \end{align} since there are $m+1$ missing $L^1$-norms, $m$ missing $L^\infty$-norms and $2(m+1)$ missing bounds for $v$. To finish the proof of the Lemma, we multiply (\ref{eq:xxb1}) with (\ref{eq:xxb2}), and then use $2 m+2\le \bar{N}$ and $\beta\le 1/\sabs{t}$. \end{proofof} \section{Simple partitions}\label{sec:simple} \subsection{General bound (proof of Lemma \ref{th:simpleest})} \label{sec:gensimple} Clearly, every $S_m(n,n')$, which has $m$, $n$ and $n'$ such that $m+2\sum_{j=1}^{m+1} n_j = \bar{N}_1$ and $m+2\sum_{j=1}^{m+1} n'_j = \bar{N}_2$, belongs to $\pi(I_{\bar{N}_1,\bar{N}_2})$ and is simple. Let us next prove that also the converse holds. Suppose $S$ is simple. If $A\in S$ is such that $\max A<\bar{N}_1+1$, then $A$ must be a gate since otherwise it would form either a nest or crossing for some $A(n)$ with $\min A\bar{N}_1+1$, then $A$ must also be a gate. The remaining pairings form a subset $S'=\defset{A\in S}{\min A<\bar{N}_1+1<\max A}$, and let $m=|S'|$. If $S'$ is empty, $S=S_0(\bar{N}_1/2,\bar{N}_2/2)$. Otherwise, let us order $A'\in S'$ into a sequence such that $\min A'_j<\min A'_{j+1}$ for all $j=1,\ldots,m$. Then $\max A'_j>\max A'_{j+1}$ for all $j$, since otherwise $(A'_j, A'_{j+1})$ is crossing. Let also $A'_0=\set{0,\bar{N}+2}$, $A'_{m+1}=\set{\bar{N}_1+1}$, and define, for $i=0,\ldots,m$, $n_i$ as the number of gates $A$ with $\min A'_i< \min A <\min A'_{i+1}$ and $n'_i$ as the number of gates $A$ with $\max A'_i> \min A >\max A'_{i+1}$. Then $S=S_m(n,n')$, with $m$, $n$, $n'$ satisfying the condition given in the Lemma. We have begun indexing the components of $n$ and $n'$ from $0$, not from $1$ -- this will become convenient later. Consider then $\Kpart(t, S;(N',N_2),(N',N_1),\vep,\kappa,p,f)$ corresponding to such $S$. First we integrate out the gates, each yielding a factor $\pm g$, as before. The remaining free indices, if any, are $\eta_{r_j}$ with $r_j = \min A'_j=\sum_{j'=0}^{j-1} (2 n_{j'}+1)$, for $j=1,\ldots,m$. We then make a change of variables to \begin{align}%\label{eq:} k''_j = \sum_{j'=0}^{j-1} \eta_{\ell_{j'}} - \frac{1}{2} p \end{align} where $j=1,\ldots,m+1$. This implies that for all $\ell\le \bar{N}_1$, which are not inside a gate, $k_\ell = k''_j + p/2$ for some $j$, and for all $\ell> \bar{N}_1+1$, which are not inside a gate, $k_\ell = k''_j - p/2$ for some $j$. For an explicit example, see Fig.\ \ref{fig:gensimple}. To write the result in a convenient form, let us define $r'_j = \max A'_j=\bar{N}+2-\sum_{j'=0}^{j-1} (2 n'_{j'}+1)$, for $j=1,\ldots,m$, and $r_0=0$, $r'_{0}=\bar{N} +2$, and let then $\kappa_{j,i}=\kappa_{r_j+1+i}$ and $\kappa'_{j,i}=\kappa_{r'_j-i}$ for any appropriate choice of indices $j,i$. Dropping the double-primes, and using the short-hand notations \begin{align}%\label{eq:} k^{\pm}_j = k_j \pm \frac{1}{2} p, \end{align} we obtain the following representation for $\Kpart$ \begin{align}\label{eq:notsosimplesum2} & \vep^{m} \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \oint_\gpath \frac{\rmd z}{2\pi} \oint_\gpath \frac{\rmd z'}{2\pi} \, \rme^{-\ci t (z+z')} \FT{\psi}^\vep(k_0^-)\cdot \nonumber \\ & \quad \Bigl\{\prod_{j=m}^0 \Bigl[ \frac{\ci}{z'+\ci\kappa'_{j,0}+H(k_j^-)} \prod_{i=n'_j}^{1} \Bigl(g(k_j^- ;-z'-\ci\kappa'_{j,2 i-1}) \frac{\ci \vep }{z'+\ci\kappa'_{j,2 i}+H(k_j^-)} \Bigr) \nonumber \\ & \quad \times v(k_j^-,k_{j+1}^-)^{\1(j\ne m)} \Bigr] f(k_m) \prod_{j=0}^m \Bigl[ v(k_{j+1}^+,k_j^+)^{\1(j\ne m)} \nonumber \\ & \quad \prod_{i=1}^{n_j} \Bigl(\frac{-\ci \vep }{z+\ci\kappa_{j,2 i}-H(k_j^+)} g(k_j^+ ;z+\ci\kappa_{j,2 i-1}) \Bigr) \frac{\ci}{z+\ci\kappa_{j,0}-H(k_j^+)} \Bigr] \FT{\psi}^\vep(k_0^+) \Bigr\}. \end{align} where the index set $I'_m=\set{0,\ldots,m}$, we have defined $M^0=\1$ for all matrices $M$, and we have used the equality \begin{align}%\label{eq:} (-\ci)^{\bar{N}_2-\bar{N}_1} \vep^{(\bar{N}_1+\bar{N}_2)/2} = \vep^m \prod_{j=0}^{m} \left((-\vep)^{n_j} (-\vep)^{n'_j} \right). \end{align} The following Lemma, whose proof we postpone for the moment, is used also in the computation of the limit of the main term. \begin{lemma}\label{th:maindiff} \begin{align}\label{eq:simplebound4} & \Bigl| \Kpart(t, S_m(n,n');(N',N_2),(N',N_1),\vep,\kappa,p,f) \nonumber \\ & \qquad - \Kpmain(t, S_m(n,n'),N',\vep,\kappa,p,f) \Bigr| \le c' \norm{f}_\infty \Emax (c \vep \sabs{t})^{\!\frac{\bar{N}}{2}} \nonumber \\ & \quad \times \Bigsabs{\ln \frac{\sabs{\vep t}}{\vep}}^{3} (\bar{N}+1) \Bigl(1 + c_1 \frac{|p|}{\sqrt{\vep}} + c_2 \frac{\kappa}{\vep} + c_3 \sqrt{1+\kappa/\vep}\Bigr) \frac{ \sabs{\vep t}}{\sabs{t}^{1/2}}. \end{align} where \begin{align}\label{eq:defKmain} & \Kpmain(t, S_m(n,n'),N',\vep,\kappa,p,f) = \sum_{\sigma',\sigma\in \set{\pm 1}} \vep^{m} \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \FT{\psi}^\vep_{\sigma'}(k_0^-)^* \FT{\psi}^\vep_{\sigma}(k_0^+) \nonumber \\ & \quad \times f_{\sigma'\sigma}(k_m) (\sigma'\sigma)^{m} \prod_{j=1}^{m} \omega(k_{j})^2 %\nonumber \\ & \quad (-\vep \Theta_\sigma(k_0))^{\sum_{j=0}^m n_j} (-\vep \Theta_{-\sigma'}(k_0))^{\sum_{j=0}^m n'_j} \nonumber \\ & \quad \times K_{(\bar{N}_2+m)/2+1}(t,w'_{\sigma'}(k^-)) K_{(\bar{N}_1+m)/2+1}(t,w_{\sigma}(k^+)) \end{align} with \begin{align}%\label{eq:} (w_{\sigma}(k))_{j,i} = \sigma \omega(k_j)-\ci \kappa_{j,2 i}\qand (w'_{\sigma}(k))_{j,i} = -\sigma \omega(k_j)-\ci \kappa'_{j,2 i}. \end{align} \end{lemma} Then we can finish the proof by using the following upper bound for $|\Kpmain|$ % above \begin{align}\label{eq:kpmainbnd} &\norm{f}_\infty \ommax^{2m} \bar{\Theta}^{\bar{N}/2-m} \vep^{\bar{N}/2} % \nonumber \\ & \quad \times \Bigl[ \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \norm{\FT{\psi}^\vep(k_0)}^2 |K_{N'_1}(t,w_{+}(k))|^2 \nonumber \\ & \qquad \times \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \norm{\FT{\psi}^\vep(k_0)}^2 | K_{N'_2}(t,w'_{+}(k))|^2 \Bigr]^{\frac{1}{2}} \end{align} where $N'_1=(\bar{N}_1+m)/2+1$, $N'_2=(\bar{N}_2+m)/2+1$, and $\bar{\Theta} = \sup_{k}|\Theta_+(k)|<\infty$. To get the bound we have applied the Schwarz inequality, then shifted all integration variables by $\pm p/2$ and, finally, used $|K_{N}(t,w_{-}(k))|=|K_{N}(t,w_{+}(k))|$. If $m=0$, we apply Lemma \ref{th:Knprop}:\ref{it:Knbound} to find that the square root in (\ref{eq:kpmainbnd}) is bounded by $\Emax t^{\bar{N}/2}((\bar{N}_1/2)! (\bar{N}_2/2)!)^{-1/2}$. Therefore, for $m=0$, \begin{align}\label{eq:Kpmb1} & |\Kpmain(t, S_m(n,n'),N',\vep,\kappa,p,f)|\le \frac{ \Emax \norm{f}_\infty (c \vep t)^{\bar{N}/2}}{ [((\bar{N}_1+m)/2)!((\bar{N}_2+m)/2)!]^{1/2}} \end{align} with $c=\ommax^2 \bar{\Theta}$. Let then $m>1$. Denoting $N=N'_1$, we need to inspect \begin{align}%\label{eq:} & |K_{N}(t,w_{+}(k))|^2 = \int_{\R_+^{N}}\! \rmd s \, \delta\Bigl(t-\sum_{\ell=1}^{N} s_\ell\Bigr) \int_{\R_+^{N}}\! \rmd s' \, \delta\Bigl(t-\sum_{\ell=1}^{N} s'_\ell\Bigr) \nonumber \\ & \qquad \times \prod_{\ell=1}^N \rme^{-\ci (s_\ell-s'_\ell) \omega(k_{j(\ell)})} \prod_{\ell=1}^N \rme^{-\kappa_{j(\ell),i(\ell)} (s_\ell+s'_\ell)}. \end{align} where $(j(\ell),i(\ell))$ define the natural index mapping from $I_N$ to allowed $(j,i)$ such that $1\mapsto (0,0)$, $r_1+1\mapsto (1,0)$, etc. Then we use Fubini's theorem to integrate out $k_j$ with $j\ge 1$, and estimate the integral by (DR\ref{it:suffdisp}). This shows that \begin{align}\label{eq:KL2est} &\int_{(\T^3)^{I_m}} \!\!\rmd k\, |K_{N}(t,w_{+}(k))|^2 \le \comega^m \int_{\R_+^{N}}\! \rmd s \, \delta\Bigl(t-\sum_{\ell=1}^{N} s_\ell\Bigr) \int_{\R_+^{N}}\! \rmd s' \, \delta\Bigl(t-\sum_{\ell=1}^{N} s'_\ell\Bigr) \nonumber \\ & \qquad \prod_{j=1}^m \Bigl\langle \sum_{i=0}^{n_j}(s_{\ell(j,i)}-s'_{\ell(j,i)})\Bigr\rangle^{-\frac{3}{2}}. \end{align} Let us next define \begin{align}\label{eq:stoab} a_\ell = \vep\frac{s_\ell+s'_\ell}{2}\qand b_\ell = s_\ell - s'_\ell. \end{align} when $s_\ell=a_\ell/\vep +\frac{1}{2} b_\ell$, $s'_\ell = a_\ell/\vep -\frac{1}{2} b_\ell$. If we first resolve the delta-functions by integrating out $s_1$ and $s'_1$, and then make the above change of variables, the Jacobian is $\vep^{-(N-1)}$, and we find that the right hand side of (\ref{eq:KL2est}) is equal to \begin{align} & \vep^{-(N-1)} \comega^m \int_{\R_+^{N}}\!\!\! \rmd a\, \delta\Bigl(\vep t-\sum_{\ell=1}^{N} a_\ell\Bigr) \nonumber \\ & \qquad \times \int_{\R^{I_N\setm\set{1}}}\!\!\! \rmd b \, \1\Bigl(\Bigl|\sum_{\ell=2}^{N}b_\ell \Bigr| \le 2\frac{a_1}{\vep} \Bigr) \prod_{\ell=2}^{N} \1\!\left(|b_\ell|\le 2 \frac{a_\ell}{\vep}\right) \prod_{j=1}^m \Bigl\langle \sum_{i=0}^{n_j} b_{\ell(j,i)}\Bigr\rangle^{-\frac{3}{2}}. \end{align} Here we use the trivial bound $\1(\cdot)\le 1$ to remove the characteristic functions containing $a_{r_j+1}$ for $j=0,1,\ldots,m$, and estimate the integrals over $b_{r_j+1}$, $j=1,\ldots,m$, by the bound (\ref{eq:int0bounds}). Then we can integrate the remaining $N-1-m$ integrals over $b_\ell$, use the bounds $a_\ell\le \vep t$, and then finally estimate the $a$-integral by Lemma \ref{th:Knprop}:\ref{it:Knbound}. This shows that \begin{align}%\label{eq:KL2est} &\int_{(\T^3)^{I_m}} \!\!\rmd k\, |K_{N}(t,w_{+}(k))|^2 \le \vep^{-(N-1)}\comega^m 6^m (4 t)^{N-1-m} \frac{ (\vep t)^{N-1}}{(N-1)!} \nonumber \\ & \quad \le \frac{(6\comega t)^{\bar{N}_1}}{((\bar{N}_1+m)/2)!}. \end{align} where we used $N-1=(\bar{N}_1+m)/2$. Since the above argument works for any partition $n$ and $\kappa_{i,j}$, we can now also conclude that \begin{align}%\label{eq:KL2est} &\int_{(\T^3)^{I_m}} \!\!\rmd k\, |K_{N'_2}(t,w'_{+}(k))|^2 \le \frac{(6\comega t)^{\bar{N}_2}}{((\bar{N}_2+m)/2)!}. \end{align} Therefore (\ref{eq:Kpmb1}) is valid also in this case for $c=6 \comega\ommax^2 \bar{\Theta}$ which is larger than the $c$ for the $m=0$ case. Combined with Lemma \ref{th:maindiff} we obtain (\ref{eq:simplebound}) and this finishes the proof of Lemma \ref{th:simpleest}. We still need to prove Lemma \ref{th:maindiff}. This will be based on the following result which shows that removing any of the denominators improves the estimate: \begin{lemma}\label{th:missingpropag} For any $\sigma_{j,i}, \sigma'_{j,i}= \pm 1$, the following integral \begin{align}\label{eq:absval} & \int_{(\T^3)^{I'_m}} \!\!\rmd k\, |\FT{\psi}^\vep_{\sigma'_{0,0}}\!(k_0^-)| |\FT{\psi}^\vep_{\sigma_{0,0}^{\phantom a}}\!(k_0^+)| %\nonumber \\ & \qquad \times \oint_\gpath \frac{|\rmd z|}{2\pi} \oint_\gpath \frac{|\rmd z'|}{2\pi} \nonumber \\ & \qquad \times \prod_{j=0}^{m} \Bigl( \prod_{i=0}^{n_j} \frac{1}{|z+\ci \kappa_{j,2 i}-\sigma_{j,i} \omega(k^+_j)|} \prod_{i=0}^{n'_j} \frac{1}{|z'+\ci \kappa'_{j,2 i}+\sigma'_{j,i} \omega(k^-_j)|} \Bigr). \end{align} with any $0<\beta\le 1-\kappa$ is bounded by \begin{align}\label{eq:sbound0} \Emax |\gpath|^2 \sabs{\ln \beta}^2 (3 \comega)^m \beta^{-m-\sum_j(n_j+n'_j)}. \end{align} If $m+\sum_j n_j > 0$ and the integrand is multiplied by $|z+\ci \kappa_{j,2 i}-\sigma_{j,i} \omega(k^+_j)|$ for some pair of indices $j\in I'_m$, $i\in\set{0,\ldots,n_j}$, then the integral has an upper bound which is given by (\ref{eq:sbound0}) times \begin{align}\label{eq:sbound1} 4\beta \sabs{\ln\beta}. \end{align} The same is true whenever $m+\sum_j n'_j > 0$, and the integrand is multiplied by $|z'+\ci \kappa'_{j,2 i}+\sigma'_{j,i} \omega(k^-_j)|$ for some pair of indices $j\in I'_m$, $i\in\set{0,\ldots,n'_j}$. \end{lemma} \begin{proof} Using Lemma \ref{th:morseprop}:\ref{it:m0}, we find an upper bound \begin{align} & \beta^{-\sum_j (n_j +n'_j)} \int_{(\T^3)^{I'_m}} \!\!\rmd k\, |\FT{\psi}^\vep_{\sigma_{0,0}^{\phantom a}}\!(k_0^+)| |\FT{\psi}^\vep_{\sigma'_{0,0}}\!(k_0^-)| %\nonumber \\ & \qquad \times \oint_\gpath \frac{|\rmd z|}{2\pi} \oint_\gpath \frac{|\rmd z'|}{2\pi} \nonumber \\ & \qquad \times \prod_{j=0}^{m} \Bigl( \frac{1}{|z+\ci \kappa_{j,0}-\sigma_{j,0} \omega(k^+_j)|} \frac{1}{|z'+\ci \kappa'_{j,0}+\sigma'_{j,0} \omega(k^-_j)|} \Bigr). \end{align} We estimate the $k_j$ integrals for $j=1,\ldots,m$ by \begin{align} & \int_{\T^3} \rmd k_j \frac{1}{|z+\ci \kappa_{j,0}-\sigma_{j,0} \omega(k^+_j)|} \frac{1}{|z'+\ci \kappa'_{j,0}+\sigma'_{j,0} \omega(k^-_j)|} %\\ & \quad \le \frac{3 \comega}{\beta} \end{align} which follows from the Schwarz inequality and Lemma \ref{th:morseprop}:\ref{it:mk2}. Then we estimate $z$ and $z'$-integrals by Lemma \ref{th:morseprop}:\ref{it:mz1}, after which the remaining $k_0$-integral can be bound by the Schwarz inequality. This proves (\ref{eq:sbound0}). Assume then that $m+\sum_j n_j > 0$, for some index pair $j,i$. If $n_j>0$, the only change needed to be made to the above steps is to retain one of the remaining factors depending on $k_j$. This will yield a bound which is better than (\ref{eq:sbound0}) by a full factor of $\beta$. If $n_j=0$, we necessarily have $m>0$. If $j=0$, let $j'=1$, otherwise let $j'=j$. We use the trivial estimate for all terms with $i >0$, and estimate also the remaining factors independent of $k_0$ and $k_{j'}$ as before. Then we can apply Lemma \ref{th:morseprop} to estimate the remaining integrals in the following order: first the $z$-integral, then the $k_{j'}$-integral, $z'$-integral, and finally $k_0$-integral. This yields a bound which is (\ref{eq:sbound0}) times $4\beta \sabs{\ln\beta}\ge \beta$. The remaining case, where a $z'$-factor is cancelled instead of a $z$-factor follows by identical reasoning. \end{proof} \begin{proofof}{Lemma \ref{th:maindiff}} Let us begin by writing the $2\times 2$ matrix product in (\ref{eq:notsosimplesum2}) in component form, and let $\sigma_{j,i }$ and $\sigma'_{j,i }$ denote the component attached to the factor with $\kappa_{j,i}$, respectively $\kappa'_{j,i}$. We also use $\beta=(2\sabs{t/2})^{-1}$, as before. For the absolute value of any term in the resulting sum over $\sigma'$ and $\sigma$ we then have an upper bound: \begin{align}%\label{eq:} \norm{f}_\infty \vep^{\frac{\bar{N}}{2}} \bar{g}^{\frac{\bar{N}}{2}-m} \ommax^{2 m} \rme^2 \times \text{(\ref{eq:absval})} \end{align} where $\bar{g}$ is the finite constant in (\ref{eq:defbarg}), for which also $\sup_k |\Theta_+(k)|\le \bar{g}$. Suppose that there is an index pair $(j,i)$ such that $\sigma_{j,i}=-\sigma_{0,0}$. Then we take the absolute value inside the integrals where, similarly to (\ref{eq:oneoverab}), we apply the inequality \begin{align}%\label{eq:} & \frac{1}{|z+\ci \kappa_{j,2 i}-\sigma_{j,i} \omega(k^+_j)|} \frac{1}{|z+\ci \kappa_{0,0}-\sigma_{0,0} \omega(k^+_0)|} \nonumber \\ & \quad \le \frac{1}{2 \ommin} \Bigl( \frac{1}{|z+\ci \kappa_{j,2 i}-\sigma_{j,i} \omega(k^+_j)|} + \frac{1}{|z+\ci \kappa_{0,0}-\sigma_{0,0} \omega(k^+_0)|} \Bigr) . \end{align} Since $(j,i)\ne (0,0)$, we must have $\bar{N}_1>0$, and we can apply Lemma \ref{th:missingpropag}. This yields an upper bound $\frac{2}{\ommin}\beta \sabs{\ln\beta}$ times \begin{align}\label{eq:finalupperb} c'' \norm{f}_\infty \Emax (c \sabs{\vep t})^{\bar{N}/2} \sabs{\ln \frac{\sabs{\vep t}}{\vep}}^{2} \end{align} where $c=\sabs{3 \comega \ommax^2} \sabs{\bar{g}}$ and $c''= \rme^2 4 (2 \ommax+5)^2$ depend only on $\omega$. The same estimate is valid also whenever there is an index pair $(j,i)$ such that $\sigma'_{j,i}=-\sigma'_{0,0}$. Therefore, the sum over all those sign combinations which do not have constant $\sigma$ and $\sigma'$ is bounded by (\ref{eq:finalupperb}) times \begin{align}\label{eq:firstextra} %2^{\bar{N}+4}\sabs{t}^{-1} \sabs{\ln \frac{\sabs{\vep t}}{\vep}} . 2^{\bar{N}}\frac{2^3}{\ommin} \beta \sabs{\ln \beta}. \end{align} Thus we have proven that up to such an error, $\Kpart$ is equal to \begin{align}\label{eq:goingtoKmain} & \sum_{\sigma',\sigma\in \set{\pm 1}} \vep^{m} \int_{(\T^3)^{I'_m}} \!\!\rmd k\, f_{\sigma'\sigma}(k_m) \FT{\psi}^\vep_{\sigma'}(k_0^-)^* \FT{\psi}^\vep_{\sigma}(k_0^+) \oint_\gpath \frac{\rmd z}{2\pi} \oint_\gpath \frac{\rmd z'}{2\pi} \, \rme^{-\ci t (z+z')} \nonumber \\ & \qquad \times \prod_{j=0}^m \Bigl[ \frac{\ci}{z'+\ci\kappa'_{j,0}+\sigma'\omega(k_j^-)} \prod_{i=1}^{n'_j} \Bigl( \frac{\ci \vep g_{\sigma'\sigma'}(k_j^- ;-z'-\ci\kappa'_{j,2 i-1}) }{z'+\ci\kappa'_{j,2 i}+\sigma'\omega(k_j^-)} \Bigr) \nonumber \\ & \qquad \times \frac{\ci}{z+\ci\kappa_{j,0}-\sigma\omega(k_j^+)} \prod_{i=1}^{n_j} \Bigl(\frac{-\ci \vep g_{\sigma\sigma}(k_j^+ ;z+\ci\kappa_{j,2 i-1})}{ z+\ci\kappa_{j,2 i}-\sigma\omega(k_j^+)} \Bigr) \Bigr] \nonumber \\ & \qquad \times \prod_{j=0}^{m-1} \Bigl[ v_{\sigma'\sigma'}(k_j^-,k_{j+1}^-) v_{\sigma\sigma}(k_{j+1}^+,k_j^+) \Bigr] . \end{align} If $\bar{N}=0$, then this formula is equal to (\ref{eq:defKmain}). Otherwise, we can express the two $K$ factors in (\ref{eq:defKmain}) as integrals over $z'$ and $z$. This yields a formula which would be equal to (\ref{eq:goingtoKmain}) if we could change each $v_{\sigma\sigma}$ to $\sigma \omega$, and each $g_{\sigma\sigma}$ to $\Theta_\sigma$ there. However, we can do these changes one by one and compute an upper bound for $|\mbox{(\ref{eq:goingtoKmain})}-\Kpmain|$ using the following estimates: \begin{align}%\label{eq:} & \left|v_{\sigma\sigma}(k_{j+1}^+,k_{j}^+)-\sigma \omega(k_{j+1})\right| %\nonumber \\ & \quad \le |\omega(k^+_{j+1})-\omega(k_{j+1})| + \frac{1}{2} |\kappa_{j,0}-\kappa_{j+1,0}| \nonumber \\ & \qquad + \frac{1}{2}|z+\ci\kappa_{j,0}-\sigma \omega(k^+_{j})| + \frac{1}{2}|z+\ci\kappa_{j+1,0}-\sigma \omega(k^+_{j+1})| \nonumber \\ & \quad \le \frac{\norm{\nabla \omega}_\infty}{2} |p| + \frac{1}{2} \kappa + \frac{1}{2}|z+\ci\kappa_{j,0}-\sigma \omega(k^+_{j})| + \frac{1}{2}|z+\ci\kappa_{j+1,0}-\sigma \omega(k^+_{j+1})|, \end{align} and a similar estimate for $|v_{\sigma'\sigma'}(k_{j}^-,k_{j+1}^-)-\sigma'\omega(k_{j+1})|$. By Lemma \ref{th:gateint}, \begin{align}%\label{eq:} & \left|g_{\sigma\sigma}(k_j^+ ;z+\ci\kappa_{j,2 i-1})-\Theta_{\sigma}(k_0)\right| \le c'_1 |\omega(k_0^+)-\omega(k_0)| \nonumber \\ & \qquad + c'_1 |z+\ci \kappa_{0,0}-\sigma \omega(k^+_0)| + (c'_1+c'_2/\beta^{1/2}) |z+\ci \kappa_{j,2 i}-\sigma \omega(k^+_j)| \nonumber \\ & \qquad + c'_1 |\kappa_{j,2 i}-\kappa_{0,0}| + c'_2 |\kappa_{j,2 i}-\kappa_{j,2 i-1}|/\beta^{1/2} +c'_3\sqrt{\beta+\kappa} \nonumber \\ & \quad \le c''_1 |z+\ci \kappa_{0,0}-\sigma \omega(k^+_0)| + \frac{c''_2}{\beta^{1/2}} |z+\ci \kappa_{j,2 i}-\sigma \omega(k^+_j)| \nonumber \\ & \qquad + \frac{c''_2}{\beta^{1/2}} \kappa + c''_3 \sqrt{\beta+\kappa} + c''_4 |p| \end{align} where all the constants $c''_i$ depend only on $\omega$. Since $-g_{\sigma\sigma}(k;-w)=g_{-\sigma,-\sigma}(k;w)$, a similar bound is valid also for $|-g_{\sigma'\sigma'}(k_j^- ;-z-\ci\kappa'_{j,2 i-1}) -\Theta_{-\sigma'}(k_0)|$. We have to iterate $2 m$ times the change of $v$ and $\bar{N}/2- m$ times the change of $g$. Collecting the estimates, and applying Lemma \ref{th:missingpropag} when needed, shows that $|\mbox{(\ref{eq:goingtoKmain})}-\Kpmain|$ is bounded by (\ref{eq:finalupperb}) times \begin{align}\label{eq:finalextra} \bar{N} \sqrt{\beta} \Bigl(c_1 \frac{|p|}{\sqrt{\beta}} + c_2 \sabs{\ln \beta} + c_3 \frac{\kappa}{\beta} + c_4 \sqrt{1+\kappa/\beta}\Bigr) \end{align} where the constants $c_i$ depend only on $\omega$. Then the terms containing $\beta$ can be bounded from above as before and, together with (\ref{eq:firstextra}) and after a redefinition of the constants, this proves (\ref{eq:simplebound4}). \end{proofof} \subsection{Convergence of the main term} \label{sec:maintermconv} It will be enough to study the limit of a sum of functions $\Kpmain$ defined in (\ref{eq:kpmainbnd}), more precisely, the limit of \begin{align}\label{eq:Kpmainbegin} & \sum_{N_1,N_2=0}^{N_0(\vep)-1} \sum_{m=0}^{N_0(\vep)-1}\!\! \sum_{n,n'\in \N^{I'_m}} \1\Bigl(N_1=m+2\sum_{j=0}^{m} n_j\Bigr) \1\Bigl(N_2=m+2\sum_{j=0}^{m} n'_j\Bigr) \nonumber \\ & \qquad \times \Kpmain(\tmacro/\vep, S_m(n,n'),0,\vep,0,\vep\pmacro,e_{\nmacro}P_{++}) . \end{align} To see this, first note that the difference between this and \begin{align}\label{eq:sumsimplie} \sum_{N_1,N_2=0}^{N_0-1} \sum_{\substack{S\in\pi(I_{N_1,N_2}),\\ S \text{ simple}}} \Kpart (\tmacro /\vep,S;(0,N_2),(0,N_1),\vep,0,\vep\pmacro,e_{\nmacro}P_{++}) \end{align} is by Lemma \ref{th:maindiff} bounded by \begin{align}%\label{eq:simplebound4} & N_0^2 c' \Emax (c \sabs{\tmacro})^{\!\frac{\bar{N}}{2}} % \nonumber \\ & \quad \times \Bigsabs{\ln \frac{\sabs{\tmacro}}{\vep}}^{3} (\bar{N}+1) (1 + c_3+ c_1 \sqrt{\vep}|\pmacro|) \frac{ \sabs{\tmacro}}{\sabs{\tmacro/\vep}^{1/2}} \end{align} for some constants $c,c'$ and $\bar{N}=2(N_0-1)$. The bound goes to zero as $\vep\to 0$. Secondly, by Corollaries \ref{th:EFF} and \ref{th:nonsimple}, for all $0<\vep\le \vep'$ and $N_1+N_2>0$, the difference between (\ref{eq:sumsimplie}) and $\Fmain(\pmacro,\nmacro,\tmacro) $ is bounded by \begin{align}\label{eq:nonsimplebound2a} & \sum_{N_1,N_2=0}^{N_0-1} \sum_{\substack{S\in\pi(I_{N_1,N_2}),\\ S \text{ not simple}}} \prod_{A\in S} |C_{|A|}| \, |\Kpart (\tmacro/\vep,S; (0,N_2),(0,N_1),\vep,0,\vep\pmacro,e_{\nmacro}P_{++}) | \nonumber \\ & \quad \le N_0^2 c' \Emax \left(c\sabs{\tmacro}\right)^{\!\frac{\bar{N}}{2}} \bar{N}! \Bigsabs{\ln \frac{\sabs{\tmacro}}{\vep}}^{\bar{N}+\max(2,d_2)} \bar{N}^{d_1} \Bigl(\frac{\sabs{\tmacro}}{\sabs{\tmacro /\vep}}\Bigr)^{\gamma'}. \end{align} for some constants $c,c'$ and $\bar{N}=2(N_0-1)$. Also this bound vanishes as $\vep\to 0$, and it is thus sufficient to study the limit of (\ref{eq:Kpmainbegin}). For any $n\in \N$, \begin{align}%\label{eq:} 1 = \sum_{R\in \N} \1(R=n) = \sum_{R\in \N} \int_0^1\rmd \varphi\, \rme^{\ci 2 \pi \varphi (R-n)}. \end{align} We insert this identity twice into (\ref{eq:Kpmainbegin}), with $n=\sum_j n_j$ and with $n=\sum_j n'_j$. Then we can perform the sums over $N_1$ and $N_2$. We express the two $K$-factors again as integrals over $z$ and $z'$, but this time choosing $\beta = 2 \vep \bar{g}$, with $\bar{g}$ defined in (\ref{eq:defbarg}). This shows that (\ref{eq:Kpmainbegin}) equals \begin{align}\label{eq:onlyafewmore} & \sum_{m=0}^{N_0-1} \sum_{n,n'\in \N^{m+1}} \sum_{R \in \N^2} \1(m+2 R_1\le N_0-1)\1(m+2 R_2\le N_0-1) \nonumber \\ & \quad \times \int_{[0,1]^2} \rmd \varphi \, \rme^{\ci 2 \pi R\cdot \varphi} \vep^{m} \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \FT{\psi}^\vep_{+}(k_0^-)^* \FT{\psi}^\vep_{+}(k_0^+) \rme^{\ci 2\pi \nmacro\cdot k_m} \prod_{j=1}^{m} \omega(k_{j})^2 %\nonumber \\ & \quad \nonumber \\ & \quad \times (-\vep \Theta_{+}(k_0)\rme^{-\ci 2\pi \varphi_1})^{\sum_{j=0}^m n_j} (-\vep \Theta_{-}(k_0)\rme^{-\ci 2\pi \varphi_2})^{\sum_{j=0}^m n'_j} \nonumber \\ & \quad \times \oint_\gpath \frac{\rmd z}{2\pi} \oint_\gpath \frac{\rmd z'}{2\pi} \, \rme^{-\ci \frac{\tmacro}{\vep} (z+z')} % \nonumber \\ & \times \prod_{j=0}^{m} \Bigl[ \Bigl( \frac{\ci}{z-\omega(k^+_j)} \Bigr)^{n_j+1} \Bigl(\frac{\ci}{z'+\omega(k^-_j)} \Bigr)^{n'_j+1}\Bigr] \end{align} where $k^{\pm}_j = k_j \pm \frac{1}{2}\vep \pmacro $. By our choice of $\beta$, we have for all $\vep\le (2\bar{g})^{-1}$ and $z\in\gpath$, $|\vep\Theta(k_0)|/|z\pm\omega(k)|\le 1/2$. This implies that for such $\vep$ the sums over $n$ and $n'$ are absolutely summable, and we can use Fubini's theorem to perform them first. As for any $a,b\in \C$ such that $|b|<|a|$, \begin{align}%\label{eq:} \sum_{n\in \N} \frac{1}{a} \left(\frac{b}{a}\right)^n = \frac{1}{a-b}, \end{align} we find that the last two lines of (\ref{eq:onlyafewmore}), summed over $n$ and $n'$, become \begin{align} & \oint_\gpath\! \frac{\rmd z}{2\pi} \oint_\gpath\! \frac{\rmd z'}{2\pi} \, \rme^{-\ci \frac{\tmacro}{\vep} (z+z')} \prod_{j=0}^{m} \Bigl( \frac{\ci}{z-\omega(k^+_j)+\ci\vep \Theta_+(k_0)\rme^{-\ci 2\pi \varphi_1}} \nonumber \\ & \qquad \times \frac{\ci}{z'+\omega(k^-_j)+\ci\vep \Theta_-(k_0)\rme^{-\ci 2\pi \varphi_2}} \Bigr) . \end{align} We use here Theorem \ref{th:Knprop} to evaluate the $z$ and $z'$ integrals and insert the result in (\ref{eq:onlyafewmore}). For any $a\in \C$, and $R\in \N$, \begin{align} \int_0^1\rmd \varphi\, \rme^{\ci 2 \pi \varphi R} \exp\!\left( a\rme^{-\ci 2\pi \varphi}\right) = \frac{a^R}{R!}, \end{align} which can be proven, e.g., by a series expansion. Using this to evaluate the $\varphi_1$ and $\varphi_2$ integrals, we arrive at the following expression for (\ref{eq:onlyafewmore}) \begin{align}\label{eq:nearly2} & \sum_{m=0}^{N_0-1} \sum_{R \in \N^2} \1(m+2 R_1\le N_0-1)\1(m+2 R_2\le N_0-1) \nonumber \\ & \times \int_{(\T^3)^{I'_m}} \!\!\rmd k\, \FT{\psi}^\vep_{+}(k_0^-)^* \FT{\psi}^\vep_{+}(k_0^+) \prod_{j=1}^{m} \omega(k_{j})^2 \rme^{\ci 2\pi \nmacro\cdot k_m} \frac{(-\tmacro\Theta_+(k_0))^{R_1}}{R_1!} \frac{(-\tmacro\Theta_-(k_0))^{R_2}}{R_2!} \nonumber \\ & \times \vep^{m} \int_{\R_+^{I'_m}}\! \rmd s' \int_{\R_+^{I'_m}}\! \rmd s \, \delta\Bigl(\frac{\tmacro}{\vep}-\sum_{j=0}^{m} s_j\Bigr) \delta\Bigl(\frac{\tmacro}{\vep}-\sum_{j=0}^{m} s'_j\Bigr) %\\ & \qquad \times \rme^{-\ci \sum\limits_{j=0}^{m} \left[ s_j \omega(k^+_j) - s'_j \omega(k^-_j) \right]} . \end{align} The $m=0$ term in the sum is equal to \begin{align} & \sum_{R \in \N^2} \1(2 R_1\le N_0-1)\1(2 R_2\le N_0-1) \int_{\T^3} \!\!\rmd k_0\, \FT{\psi}^\vep_{+}(k_0^-)^* \FT{\psi}^\vep_{+}(k_0^+) \nonumber \\ & \qquad \times \frac{(-\tmacro\Theta_+(k_0))^{R_1}}{R_1!} \frac{(-\tmacro\Theta_-(k_0))^{R_2}}{R_2!} \rme^{\ci 2\pi \nmacro\cdot k_0} \rme^{-\ci \tmacro ( \omega(k^+_0) - \omega(k^-_0) )/\vep}. \end{align} For all $k_0$, \begin{align}\label{eq:eme} & \Bigl| \rme^{-\ci \tmacro \bigl( \frac{\omega(k^+_0) - \omega(k^-_0) }{\vep} - \pmacro\cdot \nabla \omega(k_0)\bigr)} -1 \Bigr| \le \vep\frac{1}{4}\tmacro \bar{p}^2 \norm{D^2\omega}_\infty \end{align} which allows replacing the last exponential by $\rme^{-\ci \tmacro p\cdot \nabla \omega(k_0)}$ with an error which vanishes in the $\vep \to 0$ limit. The remaining integrand is $\vep$-independent, apart from the $\FT{\psi}^\vep$ factors. Dominated convergence can be applied to take the $\vep\to 0$ limit inside the $R$-sum, where $N_0(\vep)\to \infty$ and, as $\Theta$ is continuous by Lemma \ref{th:gateint}, we can apply Lemma \ref{th:winitlim} and obtain the limit \begin{align} & \sum_{R \in \N^2} \int_{\T^3}\! \mu_{0}(\rmd x\, \rmd k) \rme^{-\ci 2 \pi (\pmacro\cdot x-\nmacro\cdot k)} \rme^{-\ci \tmacro \pmacro\cdot \nabla \omega(k)} \frac{(-\tmacro \Theta_+(k))^{R_1}}{R_1!} \frac{(-\tmacro \Theta_-(k))^{R_2}}{R_2!} \nonumber \\ \quad & = \int_{\T^3}\! \mu_{0}(\rmd x\, \rmd k) \rme^{-\ci 2 \pi (\pmacro\cdot x-\nmacro\cdot k)} \rme^{-\ci \tmacro \pmacro\cdot \nabla \omega(k)} \rme^{-2 \tmacro \re \Theta_+\!(k)}. \end{align} The equality follows from Fubini's theorem, which allows swapping the $R$-sum and the $k$-integral, and then using $\Theta_-=\Theta_+^*$. Consider then the remaining case $m\ge 1$. We make the same change of variables as in (\ref{eq:stoab}), \begin{align}%\label{eq:stoab2} r_j = \vep\frac{s_j+s'_j}{2}\qand b_j = s_j - s'_j \end{align} when $s_j=r_j/\vep +\frac{1}{2} b_j$, $s'_j = r_j/\vep -\frac{1}{2} b_j$. The Jacobian is now $\vep^{-m}$, which cancels the remaining $\vep$-factors, and the last line of (\ref{eq:nearly2}) becomes \begin{align} & \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int_{\R^{I_m}}\!\!\! \rmd b \, \1\Bigl(\Bigl|\sum_{j=1}^{m}b_j \Bigr| \le 2\frac{r_0}{\vep} \Bigr) \prod_{j=1}^{m} \1\!\left(|b_j|\le 2 \frac{r_j}{\vep}\right) \nonumber \\ & \qquad \times \rme^{\ci \frac{\omega(k^+_0) + \omega(k^-_0)}{2} \sum_{j=1}^m b_j } \prod_{j=1}^{m}\rme^{-\ci b_j \frac{\omega(k^+_j) + \omega(k^-_j)}{2}} \prod_{j=0}^{m}\rme^{-\ci r_j \frac{ \omega(k^+_j) - \omega(k^-_j)}{\vep}} . \end{align} For any $\vep>0$ the integration region over $(r,b)$ is bounded which allows using Fubini's theorem and performing first the $k$ integrals. Therefore, for $m\ge 1$ the summand in (\ref{eq:nearly2}) is equal to \begin{align}\label{eq:almostthere} & \sum_{R \in \N^2} \1(m+2 R_1\le N_0-1)\1(m+2 R_2\le N_0-1) \nonumber \\ & \quad \times \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int_{\R^{I_m}}\!\!\! \rmd b \, \1\Bigl(\Bigl|\sum_{j=1}^{m}b_j \Bigr| \le 2\frac{r_0}{\vep} \Bigr) \prod_{j=1}^{m} \1\!\left(|b_j|\le 2 \frac{r_j}{\vep}\right) \nonumber \\ & \quad \times \int_{\T^3} \!\rmd k_0\, \FT{\psi}^\vep_{+}(k_0^-)^* \FT{\psi}^\vep_{+}(k_0^+) \frac{(-\tmacro\Theta_+(k_0))^{R_1}}{R_1!} \frac{(-\tmacro\Theta_-(k_0))^{R_2}}{R_2!} \nonumber \\ & \quad\times \rme^{\ci \frac{\omega(k^+_0) + \omega(k^-_0)}{2} \sum_{j=1}^m b_j } \rme^{-\ci r_0 \frac{ \omega(k^+_0) - \omega(k^-_0)}{\vep}} \int_{(\T^3)^{I_m}} \!\!\rmd k\, \rme^{\ci 2\pi \nmacro\cdot k_m} \nonumber \\ & \quad\times \prod_{j=1}^{m} \Bigl( \omega(k_{j})^2 \rme^{-\ci b_j \frac{\omega(k^+_j) + \omega(k^-_j)-2 \omega(k_j)}{2}} \rme^{-\ci r_j \frac{ \omega(k^+_j) - \omega(k^-_j)}{\vep}} \rme^{-\ci b_j \omega(k_j)} \Bigr). \end{align} For any multi-index $\alpha$, differentiation with respect to $k$ satisfies \begin{align}%\label{eq:} D^{\alpha}\! \left(\omega(k^+) - \omega(k^-)\right) = D^{\alpha} \omega(k^+) - D^{\alpha}\omega(k^-) \end{align} which, by the smoothness of $\omega$, is bounded by $\vep \bar{p}\norm{\omega}_{\infty,|\alpha|+1}$. When multiplied with $r_j/\vep$, the bound remains uniformly bounded in $\vep$. Similarly, \begin{align} & D^{\alpha} \!\left(\omega(k^+) - \omega(k) + \omega(k^-) - \omega(k) \right) \nonumber \\ &\quad = D^{\alpha} \omega(k^+) - D^{\alpha} \omega(k) + D^{\alpha}\omega(k^-) - D^{\alpha}\omega(k) \end{align} is bounded by $\frac{1}{4}\vep^2\pmacro^2 \norm{\omega}_{\infty,|\alpha|+2}$, and thus, when multiplied by $b_j$, it is bounded by $c_2\tmacro \vep \pmacro^2$ where $c_2$ is a constant independent of $\vep$. Applying (DR\ref{it:suffdisp}) we thus find that there is a constant $c'$ such that the $k_j$ integral in (\ref{eq:almostthere}) is for any $j=1,\ldots,m$ bounded by $c'/\sabs{b_j}^{3/2}$ which is integrable over $b_j$. Therefore, by a similar argument as in the $m=0$ case, we can now use \begin{align} &\rme^{\ci \frac{\omega(k^+_0) + \omega(k^-_0)}{2} \sum_{j=1}^m b_j } \rme^{-\ci r_0 \frac{ \omega(k^+_0) - \omega(k^-_0)}{\vep}} %\\ & \quad = \rme^{\ci \omega(k_0) \sum_j b_j } \rme^{-\ci r_0 \pmacro \cdot\nabla \omega(k_0)} +\order{\vep} \end{align} to remove the $\vep$-dependence from this term. %(For this, note that $|\sum_j b_j|\le 2 \tmacro/\vep$.) Let us then consider the sum over all $m=1,\ldots,N_0-1$. We apply the above bounds to justify using dominated convergence to take the $\vep\to 0$ limit up to inside the $b$-integrals (for the sum over $m$, note that due to the $r$-integral each term has an upper bound of the type $(c \tmacro)^m/m!$). Applying Lemma \ref{th:winitlim}, we then find that the sum over these $m$ converges to \begin{align} & \sum_{m=1}^\infty \sum_{R \in \N^2} \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int_{\R^{I_m}}\!\!\! \rmd b \, %\nonumber \\ & \quad \times \prod_{j=1}^{m-1} \int_{\T^3} \!\!\rmd k_j\, \omega(k_{j})^2 \rme^{-\ci b_j \omega(k_j)} \rme^{-\ci r_j \pmacro\cdot \nabla \omega(k_j)} \nonumber \\ & \quad \times \int_{\T^3} \!\!\rmd k_m\, \rme^{\ci 2\pi \nmacro\cdot k_m} \omega(k_{m})^2 \rme^{-\ci b_m \omega(k_m)} \rme^{-\ci r_m \pmacro\cdot \nabla \omega(k_m)} \int \mu_{0}(\rmd x\, \rmd k_0) \rme^{-\ci 2 \pi \pmacro\cdot x} \nonumber \\ & \quad\times \rme^{\ci \omega(k_0) \sum_j b_j } \rme^{-\ci r_0 \pmacro \cdot\nabla \omega(k_0)} \frac{(-\tmacro\Theta_+(k_0))^{R_1}}{R_1!} \frac{(-\tmacro\Theta_-(k_0))^{R_2}}{R_2!}. \end{align} Since $\mu_0$ is a bounded Borel measure, we can apply Fubini's theorem here to reorder the integrals so that we first perform the $k_j$ integrals for $j=1,\ldots,m$, then the sum over $R$, then all $b$-integrals, and finally the $\mu_0$ integral. This shows that the above sum is equal to \begin{align} & \sum_{m=1}^\infty \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int \mu_{0}(\rmd x\, \rmd k_0) \rme^{-\ci 2 \pi \pmacro\cdot x} %\nonumber \\ & \qquad\times \rme^{-\ci r_0 \pmacro \cdot\nabla \omega(k_0)} \rme^{-\tmacro 2 \re \Theta_+(k_0)} \nonumber \\ & \quad \times \prod_{j=1}^{m-1} \int_{-\infty}^\infty\! \rmd b_j \, \int_{\T^3} \!\!\rmd k_j\, \omega(k_{j})^2 \rme^{-\ci b_j (\omega(k_j)-\omega(k_0))} \rme^{-\ci r_j \pmacro\cdot \nabla \omega(k_j)} \nonumber \\ & \quad \times \int_{-\infty}^\infty\! \rmd b_m \, \int_{\T^3} \!\!\rmd k_m\, \rme^{\ci 2\pi \nmacro\cdot k_m} \omega(k_{m})^2 \rme^{-\ci b_m (\omega(k_m)-\omega(k_0))} \rme^{-\ci r_m \pmacro\cdot \nabla \omega(k_m)} . \end{align} Using the equation (\ref{eq:smoothdelta}) in Proposition \ref{th:defcrosssect}, and (\ref{eq:reTheta}) in Lemma \ref{th:gateint}, we obtain, by collecting all the results proven in this section, \begin{align}\label{eq:simplelim} & \lim_{\vep\to 0}\Fmain(\pmacro,\nmacro,\tmacro) \nonumber \\ & \quad = \sum_{m=0}^\infty \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int \mu_{0}(\rmd x\, \rmd k_0) %\nonumber \\ & \qquad\times \rme^{-\ci r_0 \pmacro \cdot\nabla \omega(k_0)} \rme^{-\tmacro \sigma(k_0)} \nonumber \\ & \qquad \times \prod_{j=1}^{m-1} \int_{\T^3} \!\nu_{k_0}(\rmd k_j) \rme^{-\ci r_j \pmacro\cdot \nabla \omega(k_j)} %\nonumber \\ & \quad \times \int_{\T^3} \!\nu_{k_0}(\rmd k_m) \rme^{-\ci r_m \pmacro\cdot \nabla \omega(k_m)} \rme^{-\ci 2 \pi (\pmacro\cdot x- \nmacro\cdot k_m)} \nonumber \\ & \quad = \sum_{m=0}^\infty \int_{\R_+^{I'_m}}\!\!\! \rmd r\, \delta\Bigl(\tmacro-\sum_{j=0}^{m} r_j\Bigr) \int \mu_{0}(\rmd x\, \rmd k_0) %\nonumber \\ & \qquad\times \int_{\T^3} \!\nu_{k_0}(\rmd k_1)\cdots \int_{\T^3} \!\nu_{k_m-1}(\rmd k_m)\, \nonumber \\ & \qquad \times \prod_{j=0}^m \rme^{-r_j (\sigma(k_j)+\ci \pmacro\cdot \nabla \omega(k_j))} %\nonumber \\ & \quad \times \rme^{-\ci 2 \pi (\pmacro\cdot x- \nmacro\cdot k_m)} \end{align} where $\sigma(k)=\nu_k(\T^3)$ is the total collision rate, and we used Proposition \ref{th:crosssectprop} to derive the second equality. The final form is a Dyson series solution to the characteristic function of the Boltzmann equation (\ref{eq:Btransporteq}) at time $\tmacro$ with the required initial conditions. This proves that (\ref{eq:Fveptomut2}) holds and concludes the proof of the main theorem. \section{Dispersion relation} \label{sec:dispersion} To make the main theorem, Theorem \ref{th:main}, a nonempty statement, we still have to discuss how the assumptions (DR\ref{it:DC1}) -- (DR\ref{it:crossing}) could be verified for a given dispersion relation $\omega$. We will also give two explicit examples of elastic couplings which satisfy the conditions. The bound (\ref{eq:suffdisp}) follows immediately by standard stationary phase methods in case $\omega$ is a Morse function, i.e., if $\omega$ has only isolated, non-degenerate critical points. For instance, one can then use a partition of unity to isolate the critical points and then apply Theorem 7.7.5.\ in \cite{horm:PDE1} which proves the validity of the bound with $d_1=4$. The suppression of crossings, (DR\ref{it:crossing}), is much harder to verify. It has been shown to be valid for the function $\sum_{\nu=1}^3 2(1-\cos(2 \pi k^{\nu}))$ in \cite{chen03} with $\gamma=1/5$ and $d_2=2$ and, independently, in \cite{erdyau04} with $\gamma=1/4$ and $d_2=6$. Therefore, $k\mapsto \omega_0^2+\sum_{\nu=1}^3 2(1-\cos(2 \pi k^{\nu}))$ is a Morse function satisfying (DR\ref{it:DC1}) -- (DR\ref{it:crossing}) for any $\omega_0>0$. We prove in Proposition \ref{th:crossing1}, that the taking of the square root, which is necessary to get the dispersion relation from the Fourier transform of the elastic couplings, very generally preserves the Assumptions \ref{th:disprelass}. In particular, this is then true for \begin{align}\label{eq:nndisp} \omega_{\text{nn}}(k) = \Bigl[\omega_0^2 + \sum_{\nu=1}^3 2(1-\cos(2 \pi k^{\nu}))\Bigr]^{\frac{1}{2}} \end{align} whenever $\omega_0>0$. Both $\omega_{\text{nn}}(k)$, and $\omega_{\text{nn}}(k)^2$ are dispersion relations of simple lattice systems. $\omega_{\text{nn}}$ corresponds to the nearest neighbour elastic couplings, $\alpha(0)=\omega_0^2+6$, $\alpha(y)=-1$ for $|y|=1$, and $\alpha(y)=0$ otherwise, while $\omega_{\text{nn}}^2$ corresponds to $\alpha(0)=(\omega_0^2+6)^2+6$, $\alpha(y)=-2(\omega_0^2+6)$ for $|y|=1$, $\alpha(y)=2$ for $|y|=\sqrt{2}$, $\alpha(y)=1$ for $|y|=2$, and $\alpha(y)=0$ otherwise. \begin{proposition}\label{th:crossing1} If $\omega$ is a Morse function which satisfies all of the Assumptions \ref{th:disprelass}, then $\sqrt{\omega}$ is also a Morse function satisfying them with the same value for the parameter $\gamma$. \end{proposition} %\begin{proofof}{Remark \ref{th:crossing1}} \begin{proof} Since $\omega\ge \ommin>0$, the function $g(k)=\sqrt{\omega(k)}$ is well-defined and smooth. The assumptions also immediately imply that $g$ is symmetric and $g\ge\sqrt{\ommin}>0$, and thus $g$ satisfies (DR\ref{it:DC1}) and (DR\ref{it:DC2}). As also \begin{align} D g(k) = \frac{1}{2 g(k)} D \omega(k), \end{align} the critical points of $g$ and $\omega$ coincide, and if $k_0$ is a critical point, \begin{align} D^2 g(k_0) = \frac{1}{2 g(k_0)} D^2 \omega(k_0) - \frac{1}{2 g(k_0)^2} D\omega(k_0)\otimes D\omega(k_0) = \frac{1}{2 g(k_0)} D^2 \omega(k_0) \end{align} which is non-degenerate since $\omega$ is a Morse function. This proves that $g$ is a Morse function, which implies that assumption (DR\ref{it:suffdisp}) holds. Then we only need to check the crossing estimate. If $|\alpha_1|\le \sqrt{\ommin}/2$, we can prove (\ref{eq:crossingest}) for the function $g$ using the trivial bound \begin{align} |\alpha_1-\sigma_1 g(k_1)+\ci\beta|\ge g(k_1)-|\alpha_1|\ge \sqrt{\ommin}/2 \end{align} and evaluation of the remaining integrals by Lemma \ref{th:morseprop}:\ref{it:mz1}. This yields a bound $\order{\sabs{\ln \beta}^2}$. If $|\alpha_1|\ge 2\sqrt{\bar{\omega}}$, we get the same result using the bound $ |\alpha_1-\sigma_1 g(k_1)+\ci\beta|\ge \sqrt{\bar{\omega}}$. If either $|\alpha_i|\le \sqrt{\ommin}/2$ or $|\alpha_i|\ge 2\sqrt{\bar{\omega}}$, for $i=2$, or $i=3$, we get similarly a bound $\order{\sabs{\ln \beta}^2}$. Let us then assume that $\sqrt{\ommin}/2\le |\alpha_i|\le 2\sqrt{\bar{\omega}}$ for all $i=1,2,3$. Then we can apply the following bound to all of the three fractions in the integrand, \begin{align} & \frac{1}{|\alpha-\sigma g(k)+\ci\beta|}\le \left|\frac{\alpha+\ci\beta+\sigma g(k)}{ (\alpha+\ci\beta)^2-g(k)^2}\right|\le \frac{3\sqrt{\bar{\omega}}+1}{ \left| \alpha^2-\beta^2-\omega(k)+\ci 2\alpha\beta\right|} \nonumber \\ & \quad \le \frac{3\sqrt{\bar{\omega}}+1}{ \left| \alpha^2-\beta^2-\omega(k)+\ci \sqrt{\ommin}\beta\right|} . \end{align} This allows using the crossing bound of $\omega$ to prove that of $g$. \end{proof} Finally, let us give a result which could become useful if one needs to check whether a given dispersion relation satisfies the crossing condition. We will show that Assumption (DR\ref{it:crossing}) can also be replaced by the following one which should be more accessible to stationary phase methods. \begin{assumption}[DR\ref{it:crossing}'] \label{th:crossing2a} Assume that there are constants $c'_2>0$, $0<\gamma\le 1$ and $d'_2\in\N$ such that for all $0< \beta \le 1$, using $k_3=k_1-k_2+u$, \begin{align}\label{eq:crossingcond2} \sup_{u\in \T^3} \int_{\R^3} \!\rmd s\, \rme^{-\beta |s|} \left| \int_{(\T^3)^2} \rmd k_1\rmd k_2\, \rme^{-\ci \sum_{i=1}^3 s_i \omega(k_i)}\right| \le c_2' \beta^{\gamma-1} \sabs{\ln \beta}^{d'_2} . \end{align} \end{assumption} \begin{proposition}\label{th:crossing2} Let the assumption \ref{th:crossing2a} be satisfied. Then there is constant $c_2$ such the assumption (DR\ref{it:crossing}) holds for this $\gamma$ and for $d_2=3+d_2'$. \end{proposition} %\begin{proofof}{Remark \ref{th:crossing2}} \begin{proof} By Lemma \ref{th:absest}, one has for any $a\in\R$, \begin{align} \frac{1}{|a+\ci \beta|} = \int_{-\infty}^\infty\!\! \rmd s \, \rme^{\ci s a} f(\beta |s|) \end{align} with \begin{align} 0\le f(\beta |s|) \le \sabs{\ln\beta} \rme^{-\beta |s|} + \1(|s|\le 1)\ln |s|^{-1} . \end{align} We use this to evaluate the left hand side of (\ref{eq:crossingest}) and then Fubini's theorem to swap the order of the $s$- and $k$-integrals. We then split the integration region $\R^3$ over the $s$-variables into two parts: $\norm{s}_\infty \le 1$ and $\norm{s}_\infty > 1$. %Since $\int_{-1}^1\rmd x\ln x^{-1} = 2$, The first integration region yields a value bounded by a constant times $\sabs{\ln \beta}^3$. For the second region we use $\sum_{i=1}^3 |s_j| \ge |s|$ combined with the estimate (\ref{eq:crossingcond2}), and obtain a bound which proves the stated result. \end{proof} \section{Energy transport for harmonic lattice dynamics} \label{sec:classical} We return to the lattice dynamics in Section \ref{sec:model} with the goal of reading off from the main theorem the implications on energy transport in the kinetic scaling limit. Let us first consider a fixed realization of the random masses % given by $\xi$, and a state $(q,v)$ with a finite energy: $E(q,v;\xi)<\infty$ with $E$ defined in (\ref{eq:defHam}). An energy density is a function $E(x;q,v,\xi)$ such that $\int_{\R^3}\! \rmd x \, E(x;q,v,\xi) = E(q,v;\xi)$. In general, there are many ways to divide up the energy into local pieces. However, there is one particularly convenient choice in our case: we define the energy density at a scale $\vep^{-1}>0$ as the random distribution $\Edens^\vep[q,v]$ with \begin{align}%\label{eq:} \mean{f,\Edens^\vep[q,v]} = \sum_{y\in\Z^3} f(\vep y)^* \frac{1}{2} \Bigl( (1+\sqrt{\vep}\, \xi_y)^{-2} v^2_y + |(\Omega q)_{y}|^2 \Bigr) \end{align} for any test function $f\in \cals(\R^3)$. Here $\Omega$ is the convolution operator defined in (\ref{eq:defOm}). Since it is assumed that $E(q,v;\xi)<\infty$, one has $\Omega q\in \ell_2$. This implies that the above formula makes sense for any $f\in C^\infty(\R^3)\cap L^\infty(\R^3)$ and that $\mean{1,\Edens^\vep[q,v]} = E(q,v)$. This particular choice for energy density is appealing since it is a positive distribution for any choice of $(q,v)$ -- in fact, when divided by the total energy, it defines a probability measure on $\R^3$. Let us then consider some initial conditions $q(0)=q^\vep_0$ and $v(0)=v^\vep_0$ with a bounded unperturbed energy, i.e., with \begin{align}%\label{eq:} \sup_\vep \left. E(q_0^\vep,v_0^\vep)\right|_{\xi=0} <\infty. \end{align} We define further $\psi^\vep\in\hilb$ by \begin{align}\label{eq:defPsiinit} \psi^\vep_{\sigma,y} = \frac{1}{2} \left( (\Omega q_0^\vep)_y + \ci \sigma (v_0^\vep)_y \right),%\qand \psi_-^\vep = (\psi_+^\vep)^* \end{align} which differs from $\psi(0)$ defined in (\ref{eq:defPsi}) by omission of the random perturbations. This omission will lead to errors which are uniformly of order $\sqrt{\vep}$: The mechanical energy density and the Wigner function of $\psi^\vep$ evolved according to (\ref{eq:hilbevol}) are related by \begin{align}%\label{eq:} \Edens^\vep[q(t),v(t)](x) = 2\int_{\T^3}\!\rmd k\, W_{++}^\vep[\rme^{-\ci t H_\vep} \psi^\vep](x,k) + \order{\smash{\vep^{\frac{1}{2}}}}. \end{align} More precisely, if $f\in \cals(\R^3)$, we define $J(x,k)=f(x)$ as a test-function in $\cals(\R^3\times\T^3)$, and then for all $t\in \R$ and all sufficiently small $\vep$, \begin{align}\label{eq:initcerror} & \bigl|\mean{f,\Edens^\vep[q(t),v(t)]} - 2\mean{J, W_{++}^\vep[\rme^{-\ci t H_\vep} \psi^\vep]} \bigr| %\nonumber \\ & \quad \le c \norm{f}_{4,\infty} \norm{\psi^\vep}^2 \sqrt{\vep} \end{align} where $c$ is a constant which depends only on $\bar{\xi}$. To prove (\ref{eq:initcerror}), first note that, if $\psi(t)$ is defined by (\ref{eq:defPsi}), then by unitarity \begin{align}%\label{eq:} \norm{\psi(t)-\rme^{-\ci t H_\vep}\psi^\vep} = \norm{\psi(0)-\psi^\vep} \le \sqrt{2\vep} \frac{\bar{\xi}}{1-\sqrt{\vep}\bar{\xi}} \norm{v^\vep_0}. \end{align} Therefore, using (\ref{eq:JW2}) and $\norm{v^\vep_0}^2\le 2\norm{\psi^\vep}^2$, there is a constant $c'$ such that \begin{align}\label{eq:Jdiff2} \bigl|\mean{J,W_{++}^\vep[\psi(t)]}- \mean{J,W_{++}^\vep[\rme^{-\ci t H_\vep}\psi^\vep]} \bigr| \le c' \norm{f}_{4,\infty} \norm{\psi^\vep}^2 (2 \sqrt{\vep} b + \vep b^2), \end{align} where $b=2 \bar{\xi}/(1-\sqrt{\vep}\bar{\xi})$ which goes to $2 \bar{\xi}$ when $\vep\to 0$. On the other hand, since $J$ does not depend on $k$, we obtain directly from the definition (\ref{eq:defEW}) \begin{align}%\label{eq:} \mean{J,W_{++}^\vep[\psi(t)]} = \sum_{y\in\Z^3} f(\vep y)^* |\psi(t)_{+,y}|^2 = \frac{1}{2}\mean{f,\Edens^\vep[q(t),v(t)]}. \end{align} Therefore, (\ref{eq:Jdiff2}) implies (\ref{eq:initcerror}) for all sufficiently small $\vep$. The following result establishes that the time-evolved, disorder-averaged energy density of the harmonic lattice dynamics in the kinetic scaling limit can be obtained by solving the linear Boltzmann equation and then integrating out the $k$-variable. \begin{corollary} Consider the lattice dynamics (\ref{eq:defDyn}) with initial conditions $q_0^\vep$, $v_0^\vep$, and let $\psi^\vep$ be defined as in (\ref{eq:defPsiinit}). Assume that the initial conditions are independent of $\xi$, and the family $(\psi^\vep)$ satisfies the assumptions (IC\ref{it:I1}) -- (IC\ref{it:I3}), and suppose that the elastic couplings satisfy (E\ref{it:EC0}) -- (E\ref{it:EC3}) and have a dispersion relation which satisfies (DR\ref{it:suffdisp}) and (DR\ref{it:crossing}). Then there is a family of bounded positive measures $\mu_t$, $t\ge 0$, on $\R^3\times \T^3$ which satisfy the Boltzmann equation (\ref{eq:Btransporteq}), such that for any $f\in \cals(\R^3)$ and $t\ge 0$ \begin{align}%\label{eq:} \lim_{\vep\to 0}\E[\mean{f,\Edens^\vep[q(t/\vep),v(t/\vep)]}]= 2 \int_{\R^3\times \T^3} \!\!\mu_t(\rmd x \,\rmd k)\, f(x)^*. \end{align} \end{corollary} \begin{proof} Since (E\ref{it:EC0}) -- (E\ref{it:EC3}) imply (DR\ref{it:DC1}) and (DR\ref{it:DC2}) we can now apply Theorem \ref{th:main} to compute the limit of $\mean{J,W_{++}^\vep[\rme^{-\ci t H_\vep}\psi^\vep]}$ for all $J$ of the form $J(x,k)=f(x)$, $f\in\cals$. Together with (\ref{eq:initcerror}) this proves the corollary. \end{proof} In the previous section we have already given examples of elastic couplings which satisfy the assumptions of the Corollary. The assumptions on the initial conditions can be satisfied, for instance, by using the following two standard examples of Wigner functions in the semi-classical limit: \begin{enumerate} \item $\vep$-independent $\psi\in\hilb$: Then we have the weak-$*$ limit \begin{align} \lim_{\vep\to 0^+} W^\vep[\psi](x,k) =\delta(x)\, \FT{\psi}(k)^*\otimes \FT{\psi}(k). \end{align} \item WKB-type $\psi^\vep\in\hilb$: For some given $h,S\in\cals(\R^3)$, $S$ real, define \begin{align} \psi^\vep_{+,y} = \vep^{3/2} h(\vep y) \rme^{\ci S(\vep y)/\vep}, \end{align} and $\psi^\vep_{-,y} = (\psi^\vep_{+,y})^*$. Then for both $\sigma= \pm 1$, \begin{align} \lim_{\vep\to 0^+} W_{\sigma\sigma}^\vep(x,k) = |h(x)|^2 \delta\!\left(k-\left[\frac{\sigma}{2 \pi}\nabla S(x)\right]\right). \end{align} where $[\cdot]$ denotes the natural injection of $\R^3$ to $\T^3$ defined by removal of the integer part. %which for our chosen parameterization of $\T^3$ reads %$[x] = (x+x_0)\ {\rm mod}\ \Z^3-x_0$ where $x_0=(1,1,1)/2$. The off-diagonal components $W^\vep_{+-}$ and $W^\vep_{-+}$ do not necessarily have a weak-$*$ limit as $\vep\to 0$. Note that the normalization has been chosen so that $\sup_\vep\norm{\psi^\vep} < \infty$. \end{enumerate} Given such $\psi^\vep$, initial positions and velocities of the particles are obtained from \begin{align} q_0^\vep = \Omega^{-1}(2\, \re \psi^\vep_+), \quad\text{and}\quad v_0^\vep = 2\,\im \psi^\vep_+ . \end{align} \appendix \section{Definition of the collision operator} \label{sec:appBoltzmann} For writing down the collision term in the Boltzmann equation, we need to know that our assumptions yield ``energy-level'' measures which are sufficiently regular. This is the content of the following two propositions: \begin{proposition}\label{th:defcrosssect} Let $\omega:\T^3\to \R$ be measurable and assume it satisfies (DR\ref{it:suffdisp}). Then for all $\alpha\in \R$, the mapping \begin{align}\label{eq:deltadef} & C(\T^3)\ni f \mapsto \lim_{\beta\to 0^+} \int_{\T^3}\!\rmd k\, \frac{\beta}{\pi} \frac{1}{(\alpha-\omega(k))^2+\beta^2} f(k) \end{align} defines a positive bounded Borel measure which we denote by $\rmd k\, \delta(\alpha-\omega(k))$. In addition, for all $f\in C^{\infty}(\T^3)$, \begin{align}\label{eq:smoothdelta} \int_{-\infty}^\infty \rmd s \left( \int_{\T^3} \! \rmd k\, f(k) \rme^{\ci s (\alpha-\omega(k))} \right) = 2\pi \int_{\T^3} \! \rmd k\, \delta(\alpha-\omega(k)) f(k) . \end{align} \end{proposition} %\begin{proofof}{Proposition \ref{th:defcrosssect}} \begin{proof} Let us consider the family of linear mappings $\Lambda_{\alpha,\beta}\in C(\T^3)^*$ defined by the formula \begin{align}\label{eq:defLambda} \Lambda_{\alpha,\beta}[f] = \int_{\T^3}\!\rmd k\, \frac{\beta}{\pi} \frac{1}{(\alpha-\omega(k))^2+\beta^2} f(k) \end{align} for all $0<\beta\le 1$. Then \begin{align} \norm{\Lambda_{\alpha,\beta}} \le \int_{\T^3}\!\rmd k\, \frac{\beta}{\pi} \frac{1}{(\alpha-\omega(k))^2+\beta^2} \end{align} and we shall soon prove that the integral has an upper bound $\comega$. Therefore, the family is equicontinuous. The set of smooth functions is dense in $C(\T^3)$, and if we can prove that the limit $\beta\to 0^+$ exists for all smooth functions, it follows that the limit in fact exists in all of $C(\T^3)$, and the limit functional belongs to $C(\T^3)^*$ with a norm bounded by $\comega$. The limit is positive for any positive $f$, implying that the limit functional is positive, and thus is determined by a unique regular positive Borel measure on $\T^3$, bounded by $\comega$. Suppose thus that $f\in C^{\infty}(\T^3)$. By (\ref{eq:deltaappr}), \begin{align} \Lambda_{\alpha,\beta}[f] = \int_{-\infty}^\infty \frac{ \rmd s}{2\pi} \rme^{\ci s \alpha-\beta |s|} \Bigl( \int_{\T^3} \! \rmd k\, f(k) \rme^{-\ci s \omega(k)} \Bigr). \end{align} However, the dispersion bound then also provides a bound for dominated convergence theorem, which implies that the limit in (\ref{eq:deltadef}) exists and is equal to \begin{align} \int_{-\infty}^\infty \frac{\rmd s}{2\pi} \int_{\T^3} \! \rmd k\, f(k) \rme^{\ci s(\alpha-\omega(k))}. \end{align} In addition, we have then also \begin{align}\label{eq:betaintbound} \int_{\T^3}\!\rmd k\, \frac{\beta}{\pi} \frac{1}{(\alpha-\omega(k))^2+\beta^2} \le \int_{-\infty}^\infty \frac{\rmd s}{2\pi} \frac{\comega}{\sabs{s}^{3/2}} \le \comega. \end{align} Therefore, we can conclude that the limit defines a bounded positive Borel measure, such that for any smooth function $f$ equation (\ref{eq:smoothdelta}) holds. \end{proof} For the next result, we also need to require continuity of $\omega$. \begin{proposition}\label{th:crosssectprop} %\begin{definition} Let $\omega$ satisfy the assumptions of Proposition \ref{th:defcrosssect} and let \begin{align}\label{eq:defnuk2} \nu_k(\rmd k') = \rmd k' \delta(\omega(k)-\omega(k')) 2\pi \omega(k')^2, \quad k\in\T^3. \end{align} If $\omega$ is continuous, then all of the following statements are true: \begin{enumerate} \item\label{it:contin} For any $g\in C(\T^3)$, the functions $\R\ni \alpha \mapsto \int \rmd k' \delta(\alpha-\omega(k')) g(k')$ and $\T^3\ni k \mapsto \int \nu_k(\rmd k') g(k')$ are continuous. \item\label{it:gsymm} For any $g\in C(\T^3\times \T^3)$ \begin{align} \int_{\T^3}\!\rmd k \left( \int_{\T^3} \nu_k(\rmd k') g(k,k')\right) = \int_{\T^3}\!\rmd k' \left( \int_{\T^3} \nu_{k'}(\rmd k) g(k,k')\right). \end{align} \item\label{it:movecontel} If $f\in C(\R)$, then for all $k\in \T^3$, \begin{align}\label{eq:fswap} & \rmd k'\delta(\omega(k)-\omega(k'))f(\omega(k)) = \rmd k'\delta(\omega(k)-\omega(k')) f(\omega(k')) . \end{align} \end{enumerate} \end{proposition} %\begin{proofof}{Proposition \ref{th:crosssectprop}} \begin{proof} For $g\in C(\T^3)$ and $0<\beta\le 1$, let $h_\beta(\alpha;g)=\Lambda_{\alpha,\beta}[g]$ with $\Lambda$ defined in (\ref{eq:defLambda}). Then by Proposition \ref{th:defcrosssect} the limit $h_0(\alpha;g)=\lim_{\beta\to 0^+} h_\beta(\alpha;g)$ exists. To prove item \ref{it:contin}, we only need to prove that $h_0$ is continuous: this is equal to the first statement and also implies the second, as then $k\mapsto h_0(\omega(k);\omega^2 g)\in C(\T^3)$. Since for any $x\in \R$, \begin{align}\label{eq:bxunif} \frac{\beta |x|}{x^2+\beta^2} \le \frac{1}{2}. \end{align} we get from (\ref{eq:betaintbound}), \begin{align} |h_\beta(\alpha')-h_\beta(\alpha)| \le \frac{|\alpha'-\alpha|}{\beta} \comega \norm{g}_\infty. \end{align} Taking $\beta$ sufficiently small then allows us to conclude that $h_0$ is continuous. The proof of \ref{it:gsymm} is a straightforward application of the dominated convergence and Fubini's theorems, with the necessary bounds provided by (\ref{eq:betaintbound}). To prove item \ref{it:movecontel}, let us first assume that $f$ is smooth. Then \begin{align}\label{eq:dfbound} |f(\omega(k'))-f(\omega(k))|\le |\omega(k')-\omega(k)| \sup_{|x|\le \ommax} |f'(x)|, \end{align} where $\ommax=\sup_k|\omega(k)|<\infty$ since $\omega$ is continuous. Thus for all $g\in C(\T^3)$, \begin{align}\label{eq:vara1} & \int \rmd k'\delta(\omega(k)-\omega(k')) ( f(\omega(k)) - f(\omega(k')) ) g(k') \nonumber \\ & \quad = \lim_{\beta\to 0^+} \int_{\T^3}\!\rmd k'\, \frac{\beta}{\pi} \frac{1}{(\omega(k)-\omega(k'))^2+\beta^2} ( f(\omega(k)) - f(\omega(k')) ) g(k') =0, \end{align} as (\ref{eq:bxunif}) and (\ref{eq:dfbound}) allow applying dominated convergence theorem to take the limit inside the $k'$-integral. However, since the left hand side of (\ref{eq:vara1}) is continuous in $f$ in the $\sup$-norm, this implies that (\ref{eq:vara1}) holds also for all continuous $f$. This proves (\ref{eq:fswap}). \end{proof} \section{Lattice Wigner transform} \label{sec:appWigner} It will be convenient for us to generalize the definition of the Wigner transform slightly, and consider also Wigner transforms of probability measures. Since the following results do not depend on the specific model of our study and can be of use in later work, we state the results in greater generality than what was assumed for the main theorem. In particular, we consider here the Wigner transform in any dimension $d\in \N_+$ and with arbitrary number of components $N\in \N_+$. Using our conventions for Fourier transform, the Wigner transform of $\psi\in L^2(\R^d)$ would be defined as the function \begin{align}\label{eq:L2Wigner} \R^{3}\times\R^{3} \ni (x,k) \mapsto \int_{\R^d}\! \!\rmd p\, \rme^{-\ci 2\pi x\cdot p} \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr)^* \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr) \end{align} where $\FT{\psi}$ is the Fourier transform of $\psi$ -- this is often also called the Wigner function. Most of the properties listed below have then been proven in \cite{gerard97}, but Wigner transforms of lattice systems have not been so widely discussed. We are aware only of \cite{macia04,mielke05}. In \cite{mielke05} the approach is to consider $\FT{\psi}$ as a function in $L^2(\R^d)$ by setting $\FT{\psi}(k)=0$ for $k$ not in the fundamental Brillouin zone. One can then apply the standard results valid for wave functions on $\R^d$. In \cite{macia04}, the discrete Wigner transform is defined as a distribution, similarly to what we have done here. Similar proposals have been made in the context of studying semi-classical limits of the Schr\"{o}dinger equation in a periodic potential, see for instance \cite{bal99,gerard97,SP04}. We find it convenient to define the Wigner transform as a distribution which, for $\psi\in L^2(\R^d)$, would correspond to using (\ref{eq:L2Wigner}) as an integral kernel. \begin{definition}\label{th:genWigner} Let $\nu$ be a Borel probability measure on $\ell_2(\Z^d,\C^N)$ equipped with its weak topology, and let $\E_\nu$ denote the expectation value with respect to $\nu(\rmd \psi)$. Whenever $\E_\nu\!\left[\norm{\psi}^2\right] <\infty$, we define for any $\vep>0$ the Wigner transform $W_\nu^\vep$ of $\nu$ at the scale $\vep^{-1}$ via \begin{align}\label{eq:defEW} \mean{J,\wvep_\nu}% \nonumber \\ & \quad = \sum_{y',y\in\Z^d} \sum_{i',i=1}^N \E_\nu\!\!\left[\psi_{i',y'}^* \psi_{i,y} \right] \int_{\T^d}\! \rmd k\, \rme^{\ci 2\pi k\cdot (y'-y)} J_{i',i}\Bigl(\vep\frac{y'+y}{2},k\Bigr)^*. \end{align} where $J\in \cals(\R^d\times \T^d,\M_N)$. \end{definition} This definition includes the deterministic case, where $\nu$ is the Dirac measure $\nu=\delta_{\phi}$ for any $\phi\in \ell_2(\Z^d,\C^N)$. In this case $\wvep_\nu $ is the Wigner transform of the vector $\phi$, denoted by $\wvep[\phi]$. The topology of $\cals(\R^d\times \T^d, \M_N)$ is defined as usual, via a countable family of seminorms (see e.g.\ \cite{gelfandII}). The next theorem proves that, under the above assumptions, $W_\nu^\vep$ is a tempered distribution, and it lists some of their general properties. In particular, item (b) establishes that this definition coincides with the one given in Eq.~(\ref{eq:Wdef}). \begin{theorem}\label{th:cwvep} Under the assumptions of Definition \ref{th:genWigner}, $W_\nu^\vep\in \cals'(\R^d\times \T^d,\M_N)$, and for every test-function $J$, \begin{align}\label{eq:meanlast} \mean{J,W_\nu^\vep} = \E_\nu\!\left[\mean{J,W^\vep[\psi]} \right]. \end{align} Furthermore, denoting $\hilb=\ell_2(\Z^d,\C^N)$ and for arbitrary $J\in \cals(\R^d\times \T^d,\M_N)$, the following properties hold: \begin{jlist}[(\alph{jlisti})] \item There is a bounded operator $\cwvep [J]$ and a constant $c$, depending only on the dimensions $d$ and $N$, such that for all $\psi\in \hilb$ \begin{gather}\label{eq:JW2} \mean{J,\wvep [\psi]} = \braket{\psi}{\cwvep [J]\psi} \qquad \text{with}\qquad \norm{\cwvep[J]} \le c \norm{J}_{d+1,\infty} . % \schnorm{J}{d+1} . \end{gather} In addition, for the same constant $c$ as above, \begin{gather}\label{eq:Wnubound} \left|\mean{J,W_\nu^\vep}\right| \le c \norm{J}_{d+1,\infty} \E_\nu\!\!\left[\norm{\psi}^2\right]. \end{gather} \item\label{it:GW2} For any $\psi\in \hilb$, \begin{align}\label{eq:WJFourier} & \mean{J,\wvep [\psi]} = \int_{\R^d}\! \!\rmd p \int_{\T^d}\! \!\rmd k\, \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr) \cdot \FT{J}(p,k)^* \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr) \end{align} where $\FT{J}$ is the Fourier transform of $J$ in the first variable, as in (\ref{eq:defFT1J}). \end{jlist} \end{theorem} \begin{proof} Consider $\vep>0$ and an arbitrary test-function $J$. Define component-wise the operator $\cwvep [J]$ by \begin{align} \cwvep [J](i',y';i,y) = \int_{\T^d}\! \rmd k\, \rme^{\ci 2\pi k\cdot (y'-y)} J_{i',i}\Bigl(\vep\frac{y'+y}{2},k\Bigr)^*. \end{align} By partial integration in $k$ we find that there is a constant $c'<\infty$, depending only on $d$, such that \begin{align}\label{eq:Jschbound} \left| \cwvep [J](i',y';i,y)\right| \le \frac{c'}{\sabs{y'-y}^{d+1}} \norm{J}_{d+1,\infty} . \end{align} Therefore, for all $\phi,\psi\in \hilb$, \begin{align}%\label{eq:WJbound} & \sum_{y',y\in\Z^d} \sum_{i',i=1}^N \left|\phi_{i',y'}^* \psi_{i,y} \cwvep [J](i',y';i,y)\right| \ %\nonumber \\ & \quad \le \norm{J}_{d+1,\infty} \norm{\phi} \norm{\psi} \sum_{n\in\Z^d} \frac{c' N}{\sabs{n}^{d+1}}. \end{align} Let us denote the result from the sum over $n$ by $c$. Then $c$ is finite and depends only on $d$ and $N$, and we have proven that $\norm{\cwvep[J]} \le c \norm{J}_{d+1,\infty}$. By the definition (\ref{eq:defEW}), $\mean{J,\wvep[\psi]} = \braket{\psi}{\cwvep [J]\psi}$, and (\ref{eq:JW2}) is valid. Under the assumptions made on $\nu$, $\psi_{i',y'}^* \psi_{i,y}$ is measurable, and the mean of its absolute value is bounded by $\E_\nu[|\psi_{i',y'}|^2]^{1/2}\E_\nu[|\psi_{i,y}|^2]^{1/2}$ by the Schwarz inequality. An application of (\ref{eq:JW2}) shows that the sum in (\ref{eq:defEW}) is absolutely summable, and it is bounded by $c \norm{J}_{d+1,\infty} \E_\nu\!\!\left[\norm{\psi}^2\right]<\infty$. Therefore, $\mean{J,\wvep_\nu}$ is well-defined, the inequality (\ref{eq:Wnubound}) is satisfied, and by Fubini's theorem (\ref{eq:meanlast}) holds. The mapping $\wvep_\nu$ is linear and, as $\norm{J}_{d+1,\infty}$ is bounded from above by one of the semi-norms defining the topology of $\cals$, (\ref{eq:Wnubound}) implies that $\wvep_\nu$ is a tempered distribution. Now we only need to prove the item (b). By (\ref{eq:JW2}) both sides of the equality (\ref{eq:WJFourier}) are continuous in $\psi$, and thus it is enough to prove it for $\psi\in\hilb$ which have a compact support. However, then we can first use \begin{align} J\Bigl(\vep\frac{y'+y}{2},k\Bigr)^* = \int_{\R^d} \!\rmd p\, \rme^{-\ci 2\pi (\vep p/2)\cdot (y'+y)} \FT{J}(p,k)^* \end{align} in the definition (\ref{eq:defEW}) and perform the finite sums over $y$ and $y'$. This yields (\ref{eq:WJFourier}) after changing the order of $p$ and $k$ integral, which is possible by the integrability of $p\mapsto\sup_k\norm{\FT{J}(p,k)}$. \end{proof} Let us next investigate properties of limit points of a sequence of Wigner transforms when $\vep\to 0$. For simplicity, we shall do this only in the case $N=1$, but for arbitrary $d$. In most cases, it is sufficient to study the limit of the Fourier transforms of the Wigner distributions. Explicitly, let $\ell_2=\ell_2(\Z^d)$, let $\nu$ be a probability measure on $\ell_2$ satisfying the assumptions of Definition \ref{th:genWigner}, let $W_\nu^\vep$ denote its Wigner transform for some $\vep>0$, and define the function $F^\vep_\nu:\R^d\times \Z^d\to \C$ by the formula \begin{align}\label{eq:defFvep} F^\vep_\nu(p,n) = \sum_{y\in \Z^d} \E_\nu\!\!\left[\psi_{y-n}^* \psi_{y} \right] \rme^{-\ci 2\pi \vep p \cdot (y-n/2)} . \end{align} \begin{proposition}\label{th:Fnuprop} For $\nu$, $F^\vep_\nu$ and $W^\vep_\nu$ defined as above, all of the following hold: \begin{jlist}[(\alph{jlisti})] \item $|F^\vep_\nu(p,n)|\le F^\vep_\nu(0,0)=\E_\nu[\norm{\psi}^2]$. \item $F^\vep_\nu$ is the Fourier transform of $W^\vep_\nu$: for all $J\in \cals(\R^d\times \T^d)$, \begin{align}\label{eq:FTWnu} \mean{J,W_\nu^\vep} = \sum_{n\in\Z^d} \int_{\R^d}\! \!\rmd p \, \mathcal{J}(p,n)^* F^\vep_\nu(p,n) \end{align} where $\mathcal{J}$ is the Fourier transform of $J$ in both variables. \item For all $p\in \R^d$ and $n\in \Z^d$, \begin{align}\label{eq:FvepFourier} & F^\vep_\nu(p,n) = \E_{\nu}\!\!\left[ \int_{\T^d}\! \!\rmd k\, \rme^{\ci 2 \pi n\cdot k} \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr)\right] . \end{align} \end{jlist} \end{proposition} \begin{proof} That $F^\vep_\nu(p,n)$ is well-defined, and the bounds in (a) follow as in the proof of Theorem \ref{th:cwvep}. (b) follows directly from Fubini's theorem, since performing the integral over $k$ yields $\1(y'=y-n)$. The proof of (c) is similar to that of (b) in Theorem \ref{th:cwvep}. First prove the result for $\nu=\delta_\phi$ with $\phi$ having a compact support, then extend it to all $\phi\in \ell_2$ by continuity, and finally use Fubini's theorem to change the order of the sum and the expectation value in (\ref{eq:defFvep}). \end{proof} In both of the following theorems, let $I=(\vep_k)$, $k=1,2,\ldots$, be a sequence in $(0,\infty)$ such that $\vep_k\to 0$ when $k\to\infty$. For notational simplicity, we will again denote the limits of the type $\lim_{k\to\infty} f(\vep_k)$ by $\lim_{\vep\to 0} f(\vep)$. Let $(\nu^\vep)_{\vep\in I}$ be a family of probability measures on $\ell_2$ satisfying the assumptions of Definition \ref{th:genWigner}, and denote also $\E^\vep = \E_{\nu^\vep}$, $\owvep = W_{\nu^\vep}^\vep$ and $F^\vep =F_{\nu^\vep}^\vep$. \begin{theorem}\label{th:weakimpliesborel} If $\owvep \to \owl$ in the weak-$*$ topology and \begin{align}\label{eq:Emaxass} \sup_{\vep \in I} \E^{\vep}\!\left[\norm{\psi}^2\right] <\infty, \end{align} then there is a unique bounded positive Borel measure $\mu$ on $\R^d\times \T^d$ such that for all test functions $J$ \begin{align}\label{eq:Wismeas} \mean{J,\owl} = \int_{\R^d\times \T^d} \!\!\mu(\rmd x \,\rmd k)\, J(x,k)^* \end{align} and $\mu$ is bounded by $\sup_{\vep \in I}\E^{\vep}\!\left[\norm{\psi}^2\right]$. If, in addition, the family $(\nu_\vep)_\vep$ is tight on the scale $\vep^{-1}$, in the sense that \begin{align}\label{eq:mthightness} \limsup_{\vep\to 0} \sum_{|y|> R/\vep} \E^\vep\!\!\left[|\psi_{y}|^2\right] \to 0 \text{, when }R\to \infty, \end{align} then $F^\vep$ converges to the characteristic function of $\mu$: For all $p\in \R^d$ and $n\in \Z^d$, \begin{align}\label{eq:Fvepchar} \lim_{\vep\to 0} F^\vep(p,n) = \int_{\R^d\times \T^d} \!\!\mu(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi (p\cdot x- n\cdot k)}. \end{align} \end{theorem} \begin{proof} We start by proving that \begin{align}\label{eq:owlposit} \mean{|J|^2,\owl}\ge 0,\qquad \text{for all }J\in \cals . \end{align} Since $\R^d\times \Z^d$ is a locally compact Abelian group, it then follows from the Bochner-Schwartz theorem \cite{warz68} that there is a unique tempered positive Borel measure $\mu$ such that (\ref{eq:Wismeas}) holds. Let $J\in \cals$. Then for all $k\in \T^d$, and $y,n\in \Z^d$, \begin{align} J\Bigl(\vep y + \vep \frac{n}{2},k\Bigr) - J(\vep y,k) = \frac{\vep}{2} \int_0^1\! \rmd s \, n\cdot \nabla_{\! 1} J\Bigl(\vep y + s \vep \frac{n}{2},k\Bigr) . \end{align} Therefore, for any $y',y\in \Z^d$, there is $c$, depending only on $d$, such that \begin{align} & \Bigl| \int_{\T^d}\! \rmd k\, \rme^{\ci 2\pi k\cdot (y'-y)} \Bigl( \Bigl|J\Bigl(\vep\frac{y'+y}{2},k\Bigr)\Bigr|^2 - J(\vep y',k)^* J(\vep y,k) \Bigr) \Bigr| \nonumber \\ & \quad \le \vep \frac{c}{\sabs{y'-y}^{d+1}} \norm{J}_{d+2,\infty}^2, \end{align} which can be proven, e.g., by $d+2$ partial integrations over $k_i$ with $i$ chosen so that $|y'_i-y_i|$ is at maximum. Proceeding as in the proof of Theorem \ref{th:cwvep}, we find \begin{align} & \mean{|J|^2,\owvep} = \E^\vep\Bigl[ \int_{\T^d}\!\rmd k \Bigl| \sum_{y\in\Z^d} J(\vep y,k) \rme^{-\ci 2\pi k\cdot y} \psi_y \Bigr|^2 \Bigr] + \order{\vep}. \end{align} This implies that $\mean{|J|^2,\owl} = \lim_{\vep\to 0}\mean{|J|^2,\owvep} \ge 0$, and proves (\ref{eq:owlposit}). We still need to prove that $\mu$ is bounded. For this, let \begin{align}\label{eq:defJlambda} J_{\lambda,p,n}(x,k) = \rme^{-\lambda^2 x^2+\ci 2\pi (p\cdot x-n\cdot k)}\text{ for }\lambda>0. \end{align} Then \begin{align}\label{eq:Jlambda} & \mean{J_{\lambda,p,n},\owvep} = \sum_{y\in \Z^d} \rme^{-\lambda^2 \vep^2 (y-n/2)^2} \E^\vep\!\!\left[\psi_{y-n}^* \psi_{y} \right] \rme^{-\ci 2\pi \vep p \cdot (y-n/2)} , \end{align} and thus $|\mean{J_{\lambda,p,n},\owvep} |\le\E^\vep[\norm{\psi}^2]$. On the other hand, by monotone convergence $\mu(\R^d\times \T^d) = \lim_{\lambda\to 0} \int\mu(\rmd x \,\rmd k)\, \rme^{-\lambda^2 x^2} = \lim_{\lambda\to 0} \mean{J_{\lambda,0,0},\owl}$, and we can infer that $\mu$ is bounded by $\sup_{\vep} \E^\vep[\norm{\psi}^2]<\infty$. Let us then assume that also (\ref{eq:mthightness}) holds, and consider any fixed $p\in \R^d$ and $n\in \Z^d$. By (\ref{eq:Jlambda}), \begin{align}\label{eq:deltaFv} & \left| \mean{J_{\lambda,p,n},\owvep} - F^\vep(p,n)\right| \le \sum_{y\in \Z^d} \left|1-\rme^{-\lambda^2 \vep^2 (y-n/2)^2}\right| \E^\vep\!\left[|\psi_{y-n}||\psi_{y}| \right]. \end{align} Then for any $R>0$ and $\vep \le 2/|n|$ we have \begin{align} & \sum_{y\in \Z^d} \left(1-\rme^{-\lambda^2 \vep^2 (y\pm n/2)^2}\right) \E^\vep\!\!\left[|\psi_{y}|^2 \right] %\nonumber \\ & \quad \le \lambda^2 (R+1)^2 \E^\vep[\norm{\psi}^2] + \sum_{|y|>R/\vep} \!\!\! \E^\vep\!\!\left[|\psi_{y}|^2 \right], \end{align} which can be applied in (\ref{eq:deltaFv}) yielding \begin{align}%\label{eq:deltaFv} & \left| \mean{J_{\lambda,p,n},\owvep} - F^\vep(p,n)\right| \le \lambda^2 (R+1)^2 \E^\vep[\norm{\psi}^2] + \sum_{|y|>R/\vep} \E^\vep\!\!\left[|\psi_{y}|^2 \right]. \end{align} Let $F$ denote the characteristic function of $\mu$, defined by the right hand side of (\ref{eq:Fvepchar}). By dominated convergence, $F(p,n) = \lim_{\lambda\to 0}(\lim_{\vep\to 0} \mean{J_{\lambda,p,n},\owvep})$, and thus for all $R>0$, \begin{align}%\label{eq:deltaFv} & \limsup_{\vep\to 0} \left| F(p,n) - F^\vep(p,n)\right| \le \limsup_{\vep\to 0} \sum_{|y|>R/\vep} \E^\vep\!\!\left[|\psi_{y}|^2 \right]. \end{align} We take here $R\to\infty$, when the tightness assumption implies that (\ref{eq:Fvepchar}) holds. \end{proof} For the converse of this theorem, we do not even need to require tightness. The main part of the statement can be summarized as follows: If the Fourier transforms $F^\vep$ converge pointwise almost everywhere, then the corresponding Wigner transforms converge to a measure whose characteristic function coincides almost everywhere with $\lim_{\vep\to 0}F^\vep$. \begin{theorem}\label{th:FimpliesWweak} Let the family $(\nu^\vep)$ satisfy Eq.~(\ref{eq:Emaxass}), and assume that for all $n\in \Z^d$ and almost every $p\in \R^d$, the limit $\lim_{\vep\to 0} F^\vep(p,n)$ exists. Then $\owvep \to \owl$ in the weak-$*$ topology when $\vep\to 0$, and $\owl$ is given by a bounded positive Borel measure $\mu$ such that almost everywhere \begin{align}\label{eq:Fveptomuch} \lim_{\vep\to 0} F^\vep(p,n) = \int_{\R^d\times \T^d} \!\!\mu(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi (p\cdot x- n\cdot k)}. \end{align} In addition, for almost every $p\in \R^d$, in particular for every $p$ for which (\ref{eq:Fveptomuch}) holds for all $n$, we have for any $f\in C(\T^d)$ \begin{align}\label{eq:Fveptomucont} \lim_{\vep\to 0} \E^\vep\!\!\left[ \int_{\T^d} \!\!\rmd k f(k)\, \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr) \right] = \int_{\R^d\times \T^d}\!\! \!\!\mu(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi p\cdot x} f(k) . \end{align} \end{theorem} \begin{proof} Let us define $F^0(p,n)=\lim_{\vep\to 0} F^\vep(p,n)$ for every $p$, $n$ for which the limit exists and $0$ elsewhere. By Proposition \ref{th:Fnuprop}, items (a) and (b), we can apply dominated convergence to the equation (\ref{eq:FTWnu}) which proves that for any test-function $J$, with Fourier transform $\mathcal{J}$, \begin{align}\label{eq:F0eq} \lim_{\vep\to 0} \mean{J,\owvep} = \sum_{n\in\Z^d} \int_{\R^d}\! \!\rmd p \, \mathcal{J}(p,n)^* F^0(p,n) . \end{align} By the Banach-Steinhaus theorem, then there is $\owl\in \cals'$ such that $\lim_{\vep\to 0}\owvep = \owl$ in the weak-$*$ topology. Therefore, we can apply Theorem \ref{th:weakimpliesborel} and conclude that there is a bounded positive Borel measure such that (\ref{eq:Wismeas}) holds. Let $F$ be the characteristic function of $\mu$. Then by Fubini's theorem, we have for all test-functions $J$ \begin{align} \mean{J,\owl} = \sum_{n\in\Z^d} \int_{\R^d}\! \!\rmd p \, \mathcal{J}(p,n)^* F(p,n) . \end{align} As this needs to be equal to (\ref{eq:F0eq}), for all $n$ there is $A_n\subset \R^d$ with Lebesgue measure zero such that $F(p,n)=F^0(p,n)$ for $p\not\in A_n$. Then also $A=\cup_n A_n$ has Lebesgue measure zero, and we have proven (\ref{eq:Fveptomuch}). For the final result, consider any $n\in \Z^d$ and $p\not\in A$. As for any $\vep>0$, $|\E^\vep[ \int \rmd k f(k) \FT{\psi}(k-\vep p/2)^* \FT{\psi}(k+\vep p/2)]| \le \norm{f}_\infty \E^\vep[\norm{\psi}^2]$, and smooth functions are dense in $C(\T^d)$, it is enough to prove (\ref{eq:Fveptomucont}) for all smooth $f$. Let thus $f\in C^{\infty}(\T^d)$ and let $\IFT{f}$ denote its Fourier transform. Then by Fubini's theorem and Proposition \ref{th:Fnuprop}, item (c), \begin{align}%\label{eq:Fveptomucont} \E^\vep\!\!\left[ \int_{\T^d} \!\!\rmd k f(k)\, \FT{\psi}\Bigl(k-\frac{1}{2}\vep p\Bigr)^* \FT{\psi}\Bigl(k+\frac{1}{2}\vep p\Bigr) \right] = \sum_{n\in Z^d} \IFT{f}(n) F^\vep(p,-n). \end{align} Using the dominated convergence theorem and the assumption $p\not\in A$, we find that, when $\vep\to 0$, this converges to \begin{align} \sum_{n\in Z^d} \IFT{f}(n) F(p,-n)= \int_{\R^d\times \T^d}\!\! \!\!\mu(\rmd x \,\rmd k)\, \rme^{-\ci 2 \pi p\cdot x} f(k). \end{align} This finishes the proof of the theorem. \end{proof} \section{Cumulant bounds} \label{sec:appComb} Let $\nu$ denote the distribution of $\xi_0$, which, by assumption, has zero mean, unit variance, and whose support is bounded by $\bar{\xi}$. Let $g_m(z) = \int \nu(\rmd x) \exp(\ci x z)$ and $g_c(z)=\ln g_m(z)$ denote its moment and cumulant generating functions, respectively. The cumulants $C_n$ of $\nu$ are then defined by the formula $C_n = (-\ci)^n g_c^{(n)}(0)$, for $n\in \N_+$. \begin{lemma}\label{th:cumulb} $C_1=0$ and $C_2=1$, and for all $n> 2$, \begin{equation} \label{eq:bcumul} |C_n| \le 3 \bar{\xi}^n n!. \end{equation} \end{lemma} \begin{proof} Since $\nu$ has a zero mean and unit variance, $C_1=0$ and $C_2=1$. Due to the compact range, the generating function $g_m$ is analytic near origin, and since $g_m(0)=1$, so is then the cumulant generating function $g_c=\ln g_m$. On the other hand, for $|z|\le 1/\bar{\xi}$ we have by Jensen's inequality \begin{align} & \re g_m(z) = \int_{[-\bar{\xi},\bar{\xi}]} \nu(\rmd x)\, \rme^{-x \im z}\cos\left(x\re z\right) \nonumber \\ & \quad \ge \cos(1) \int \nu(\rmd x)\, \rme^{-x \im z} \ge \cos(1) \exp\Bigl[-\im z \int \nu(\rmd x)\, x \Bigr] = \cos(1)>0. \end{align} Thus $g_c(z)$ is analytic in an open set containing the closed disc $|z|\le 1/\bar{\xi}$, and inside the disc $|g_c(z)|\le |\ln|g_m(z)||+|\arg g_m(z)| \le 1 + \pi/2<3$, since now $\cos(1)\le |g_m(z)| \le \rme^1$. Then the Cauchy estimates for derivatives yield (\ref{eq:bcumul}). \end{proof} \begin{definition}\label{th:defPiI} For any $N\in \N_+$, let $I_N=\set{1,\ldots,N}$, and define $I_0=\emptyset$. For any finite, non-empty set $I$, let $\pi(I)$ denote the set of all its partitions: $S\in \pi(I)$ if and only if $S \subset \mathcal{P}(I)$ such that each $A\in S$ is non-empty, $\cup_{A\in S} A = I$, and if $A,A'\in S$ with $A'\ne A$ then $A'\cap A=\emptyset$. In addition, we define $\pi(\emptyset)=\set{\emptyset}$. \end{definition} \begin{lemma}[Moments to cumulants formula]\label{th:momtocum} Let $N\in\N_+$ and $I=I_N$. Then for any mapping $i:I\to \Z^d$ \begin{align}\label{eq:momtocum} \E\Bigl[\prod_{\ell=1}^N \xi_{i_\ell}\Bigr] & = \sum_{S\in\pi(I)} \prod_{A\in S} \Bigl[ C_{|A|} \sum_{y\in \Z^{3}} \prod_{\ell\in A} \delta_{i_\ell,y} \Bigr] . \end{align} \end{lemma} \begin{proof} Proof is by induction in $N$. The formula is clearly true for $N=1$. For the induction step, let us assume it is true for all values less than a given $N>1$. Let us first define the finite set $X=i(I) \subset \Z^3$, when \begin{align} \E\Bigl[\prod_{\ell=1}^N \xi_{i_\ell}\Bigr] & = \prod_{\ell=1}^N \Bigl(-\ci \frac{\partial}{\partial z_{i_\ell}}\Bigr) \E\Bigl[\rme^{\ci \sum_{x\in X}\! \xi_x z_x}\Bigr]_{z=0} . \end{align} By the assumed independence of $(\xi_y)$, we get for all $z\in \R^X$ in a sufficiently small neighbourhood of zero, \begin{align} & \prod_{\ell=1}^N \Bigl(-\ci \frac{\partial}{\partial z_{i_\ell}}\Bigr) \E\Bigl[\rme^{\ci \sum_{x\in X}\! \xi_x z_x}\Bigr] % \\ & \quad = \prod_{\ell=1}^N \Bigl(-\ci \frac{\partial}{\partial z_{i_\ell}}\Bigr) \Bigl[\prod_{x\in X} g_m(z_{x})\Bigr] \nonumber \\ & \quad = \prod_{\ell=1}^{N-1} \Bigl(-\ci \frac{\partial}{\partial z_{i_\ell}}\Bigr) \Bigl[-\ci \partial_{z_{i_N}} \exp\Bigl(\sum_{x\in X} g_c(z_{x})\Bigr)\Bigr] \nonumber \\ & \quad = \prod_{\ell=1}^{N-1} \Bigl(-\ci \frac{\partial}{\partial z_{i_\ell}}\Bigr) \Bigl\{\bigl(-\ci g'_c(z_{i_N})\bigr) \E\Bigl[\rme^{\ci \sum_{x\in X}\! \xi_x z_x}\Bigr]\Bigr\} . \end{align} By induction, we then can prove that this is equal to \begin{align} \sum_{B\subset\set{1,\ldots,N-1}} \Bigl\{ \Bigl[\prod_{\ell\in B\cup\set{N}} \!\! \bigl(-\ci \partial_{z_{i_\ell}}\bigr)\Bigr] g_c(z_{i_N}) \Bigl[\prod_{\ell\in B^c} \bigl(-\ci \partial_{z_{i_\ell}}\bigr)\Bigr] \E\Bigl[\rme^{\ci \sum_{x\in X}\! \xi_x z_x}\Bigr]\Bigr\}. \end{align} We evaluate this at $z=0$, showing \begin{align} & \E\Bigl[\prod_{\ell=1}^N \xi_{i_\ell}\Bigr] %\\ & \quad = \sum_{B\subset\set{1,\ldots,N-1}} \Bigl\{ C_{|B|+1} \sum_{y\in\Z^3} \prod_{\ell\in B\cup\set{N}} \delta_{i_\ell,y} \Bigr\} \E\Bigl[ \prod_{\ell\in B^c} \xi_{i_\ell}\Bigr] \end{align} where an application of the induction assumption yields (\ref{eq:momtocum}). \end{proof} Finally, we need the following bound: \begin{lemma}\label{th:highosum} Let $I$ be an index set with $|I|=N$, let $M$ be an integer such that $2\le M< N$, and define $a=2\bar{\xi}(3\bar{\xi}^2+1)$. Then for any $0\le r \le 1/a$, \begin{align} \sum_{\substack{S\in\pi(I),\\ |A|>M\text{ for some }A\in S}} r^{N-2 |S|} \prod_{A\in S} \left|C_{|A|} \right| \le N!\, (a r)^{M-\1(N-M\text{ is odd})}. \end{align} \end{lemma} \begin{proof} We begin by \begin{align} \sum_{\substack{S\in\pi(I),\\ |A|>M\text{ for some }A\in S}} r^{N-2 |S|} \prod_{A\in S} \left|C_{|A|} \right| = \sum_{m=1}^N r^{N-2 m} \sum_{\substack{S\in\pi(I): |S|=m,\\ |A|>M\text{ for some }A\in S}} \prod_{A\in S} \left|C_{|A|} \right| \end{align} and focus on the second sum. For any finite, non-empty set $S$, let $\text{Ind}_S$ denote the set of indexings of $S$, i.e., it is the collection of all bijections $\set{1,\ldots,|S|} \to S$. As this set always has $|S|!$ elements and the summand is indexing invariant, we get \begin{align} & \sum_{\substack{S\in\pi(I): |S|=m,\\ |A|>M\text{ for some }A\in S}} \prod_{A\in S} \left|C_{|A|} \right| = \frac{1}{m!} \sum_{\substack{S\in\pi(I): |S|=m,\\ |A|>M\text{ for some }A\in S}} \sum_{P\in \text{Ind}_S} \prod_{j=1}^m \left|C_{|P(j)|} \right| \nonumber \\ & \quad = \frac{1}{m!} \sum_{\substack{S\in\pi(I): |S|=m,\\ |A|>M\text{ for some }A\in S}} \sum_{P\in \text{Ind}_S} \sum_{n\in \N_+^m} \prod_{j=1}^m \1(|P(j)|=n_j) \prod_{j=1}^m \left|C_{n_j} \right| \nonumber \\ & \quad = \frac{1}{m!} \sum_{n\in \N_+^m} \1\Bigl(\sum_j n_j = N\Bigr) \1(\exists j: n_j>M) \prod_{j=1}^m \left|C_{n_j} \right| \nonumber \\ & \qquad \times \sum_{S\in\pi(I): |S|=m} \sum_{P\in \text{Ind}_S} \1(\forall j: |P(j)|=n_j) \label{eq:sumSAM} \end{align} Every pair $S$, $P$ defines, by the formula $S_j=P(j)$, a sequence $(S_1,\ldots,S_m)$ which is a collection of non-empty, disjoint sets which partition $I$. Conversely, to every such sequence corresponds a unique pair $S$ and $P\in \text{Ind}_S$. On the other hand, since the number of such sequences, which also satisfy $n_j=|S_j|$ for all $j$, is exactly \begin{align} \prod_{j=1}^m \binom{N-\sum_{j'=1}^{j-1} n_{j'}}{n_j} = \frac{N!}{\prod_{j=1}^m n_j!}, \end{align} we have now proven that (\ref{eq:sumSAM}) is equal to \begin{align} &\frac{N!}{m!} \sum_{n\in \N_+^m} \1\Bigl(\sum_j n_j = N\Bigr) \1(\exists j:n_j>M) \prod_{j=1}^m \frac{\left|C_{n_j} \right|}{n_j!} . \end{align} Since $C_1=0$, the summand in is zero, unless $n_j \ge 2$ for all $j$. Taking also into account that $C_2=1$, we get from Lemma \ref{th:cumulb} \begin{align}\label{eq:onlycumul} & \sum_{\substack{S\in\pi(I): |S|=m,\\ |A|>M\text{ for some }A\in S}} \prod_{A\in S} \left|C_{|A|} \right| \nonumber \\ & \quad \le \frac{N!}{m!} \sum_{\substack{n\in \N_+^m\\n_j\ge 2}} \1\Bigl(\sum_j n_j = N\Bigr) \1(\exists j: n_j>M) \prod_{j:n_j>2} \left(3 \bar{\xi}^{n_j}\right) . \end{align} Let us denote $k=|\set{j:n_j=2}|$, when the second condition implies $1\le k< m$. Then, by shifting each $n_j$ by $2$ and by using the permutation invariance of the summand, the right hand side of (\ref{eq:onlycumul}) can also be written as \begin{align} & \frac{N!}{m!} \sum_{k=0}^{m-1} \binom{m}{k} \sum_{n\in \N_+^{m-k}} \1\Bigl(\sum_j n_j = N-2 m\Bigr) \1(\exists j: n_j>M-2) 3^{m-k} \bar{\xi}^{N-2 k} \nonumber \\ & \ \le \frac{N!}{m!} \sum_{k=0}^{m-1} \binom{m}{k} (m-k) 3^{m-k} \bar{\xi}^{N-2 k} \sum_{n'\in \N_+^{m-k}} \!\!\1\Bigl(\sum_j n'_j = N-2 m-M+2\Bigr) \end{align} where we have estimated $\1(\exists j: n_j>M-2) \le \sum_{j=1}^{m-k} \1(n_j>M-2)$. Now for any $m'\ge k'\ge 1$, \begin{align} \sum_{n\in \N_+^{k'}} \1\Bigl(\sum_j n_j = m'\Bigr) = \binom{m'-1}{k'-1}, \end{align} and, if $m'M\text{ for some }A\in S}} r^{N-2 |S|} \prod_{A\in S} \left|C_{|A|} \right| \le N! \sum_{m=1}^{\bar{n}} r^{N-2 m} \nonumber \\ & \quad \times \sum_{k=\max(0,3m-N+M-2)}^{m-1} \frac{m-k}{m!}\binom{m}{k} 3^{m-k} \bar{\xi}^{N-2 k} \binom{N-2m-M+1}{m-k-1}. \end{align} where $\bar{n}=\lfloor \frac{N-M+1}{2} \rfloor$, so that $N=2 \bar{n} +M -\sigma$ with $\sigma=\1(N-M\text{ is odd})$. Using the new summation variable $k'=m-1-k$ and estimating $\frac{m-k}{m!} \binom{m}{k} \le 1$, we obtain a new upper bound \begin{align} & N! \sum_{m=1}^{\bar{n}} (\bar{\xi} r)^{N-2 m} \sum_{k'=0}^{\min(m-1,N-2m-M+1)} \binom{N-2m-M+1}{k'} (3 \bar{\xi}^{2})^{k'+1} \nonumber \\ & \quad \le N! \sum_{m=1}^{\bar{n}} (\bar{\xi} r)^{N-2 m} 3 \bar{\xi}^{2} (3 \bar{\xi}^{2}+1)^{N-2m-M+1} \nonumber \\ & \quad = N! \frac{3 \bar{\xi}^{2}}{(3 \bar{\xi}^{2}+1)^{M-1}} \sum_{m'=0}^{\bar{n}-1} (a r/2)^{2m'+M-\sigma} \le N! (a r/2)^{M-\sigma} \frac{1}{1-(ar/2)^2} \nonumber \\ & \quad \le N! (a r)^{M-\sigma} \end{align} where we have used the assumption $a r \le 1$. \end{proof} \begin{thebibliography}{10} \bibitem{bal99} \textsc{Bal, G., Fannjiang, A., Papanicolaou, G., Ryzhik, L.}: Radiative transport in a periodic structure. \newblock \textit{J. Stat. 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Anal.} \textbf{36}, 347--383 (2004) \bibitem{mielke05} \textsc{Mielke, A.}: Macroscopic behavior of microscopic oscillations in harmonic lattices via {W}igner-{H}usimi transforms. {WIAS} preprint No.\ 1027 (2005) \bibitem{Rudin:FA} \textsc{Rudin, W.}: \textit{Functional Analysis}. \newblock Tata McGraw-Hill, New Delhi, 1974 \bibitem{ryzhik96} \textsc{Ryzhik, L., Papanicolaou, G., Keller, J.B.}: Transport equations for elastic and other waves in random media. \newblock Wave Motion \textbf{24}, 327--370 (1996) \bibitem{spohn05} \textsc{Spohn, H.}: The phonon {B}oltzmann equation, properties and link to weakly anharmonic lattice dynamics. Preprint {\tt ar{X}iv.org: math-ph/0505025} (2005) \bibitem{SP04} \textsc{Teufel, S., Panati, G.}: Propagation of {W}igner functions for the {S}chr{\"{o}}dinger equation with a perturbed periodic potential. \newblock In: \textit{Multiscale Methods in Quantum Mechanics}, (Ph. Blanchard, G. Dell'Antonio, eds.), Birkh{\"{a}}user, Boston, 2004 \bibitem{warz68} \textsc{Wawrzy{\'{n}}czyk, A.}: On tempered distributions and {B}ochner-{S}chwartz theorem on arbitrary locally compact {A}belian groups. \newblock \textit{Colloq. Math.} \textbf{19}, 305--318 (1968) \end{thebibliography} % \newcommand{\utildir}[1]{../../texstuff/#1} % \bibliographystyle{\utildir{fenno}} % % \bibliographystyle{\utildir{abbrv}} % \bibliography{\utildir{myabbr},\utildir{mrabbrev},\utildir{allrefs}} % \end{document} \end{document} ---------------0505310851255 Content-Type: application/postscript; name="Gbpath.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Gbpath.eps" %!PS-Adobe-2.0 EPSF-2.0 %%BoundingBox: 105 598 388 733 % EPSF created by ps2eps 1.47 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: dummy.dvi %%PageOrder: Ascend %%DocumentFonts: Times-Roman CMMI10 CMR10 CMMI12 %%DocumentPaperSizes: a4 %%EndComments %%BeginProlog save countdictstack mark newpath /showpage {} def /setpagedevice {pop} def %%EndProlog %%Page 1 1 %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -q dummy.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.05.17:1540 %! 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TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %!PS-AdobeFont-1.1: CMMI12 1.100 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DCC57F7903449E07914FBE9E67 3C15C2153C061EB541F66C11E7EE77D5D77C0B11E1AC55101DA976CCACAB6993 EED1406FBB7FF30EAC9E90B90B2AF4EC7C273CA32F11A5C1426FF641B4A2FB2F 4E68635C93DB835737567FAF8471CBC05078DCD4E40E25A2F4E5AF46C234CF59 2A1CE8F39E1BA1B2A594355637E474167EAD4D97D51AF0A899B44387E1FD933A 323AFDA6BA740534A510B4705C0A15647AFBF3E53A82BF320DD96753639BE49C 2F79A1988863EF977B800C9DB5B42039C23EB86953713F730E03EA22FF7BB2C1 D97D33FD77B1BDCC2A60B12CF7805CFC90C5B914C0F30A673DF9587F93E47CEA 5932DD1930560C4F0D97547BCD805D6D854455B13A4D7382A22F562D7C55041F 0FD294BDAA1834820F894265A667E5C97D95FF152531EF97258F56374502865D A1E7C0C5FB7C6FB7D3C43FEB3431095A59FBF6F61CEC6D6DEE09F4EB0FD70D77 2A8B0A4984C6120293F6B947944BE23259F6EB64303D627353163B6505FC8A60 00681F7A3968B6CBB49E0420A691258F5E7B07B417157803FCBE9B9FB1F80FD8 CA0DA1186446DD565542BCCC7D339A1EB34C7F49246E8D72E987EB477C6DB757 99AF86CEBCD7605C487A00CD2CD093098182DC57B20D78ECE0BECF3A0BF88EBA C866DB19F34BBBED6634AFC0F08D2AFB2A92578A6F8B4ADCD6594737FF6EED7D 5B536DA9E3E2CADB40DB7C600EA4D100D33C3B92B1CF857E012C4EB370BA8295 55B50047CC8911C98FE1A7BA6CDEA82D34476286E7107DC363D9E871D7CA538E CFB278CE789B783647916E244525A790278816F2CB2A7D8327E57E46983195B6 797EFE7972A11670AF013EA6FBA8207521B8AC6708438622161E5675A380AA87 F3BA6A84EDA88BA0462F14EAD0D590199C622FE5B9FDEAF094C8C96324CF3D5B D9D578A993C83292EE107908EAF32A6FA609F6607BBC62055C7F38108F42267F 59F2E80F2FFC6674691668CD3B3A1D354880903C97A05E60B07C82F703E175EA E9A81B81D3037FDFCD699CC78FE5D21665C2DCF541A85641AD51E48FFDC53DB5 59B86C9497648C8518CBCCFA8D015DD0557D208B13673B82E03EFBB4EF1F17E6 32C9622AF391FE7E7CB865EE4C5703253B65C7F0BBE6E9E0739D3C4E64408A46 648319C2F74B09B494AB5EA6AE1EC1B61F13AC16F7242C18506080BC215AB5BC 05FFFA3A02A1A2C14E107B6749205E3928EB7B163F001EFB760AFFCF46C617E0 6118544283541CC356CAE83239994BD4238153A7F2C2B862F6C082EF33138778 570CE37F7B8464B6EB9470741EBFBFBC 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %!PS-AdobeFont-1.1: CMR10 1.00B % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec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cleartomark %!PS-AdobeFont-1.1: CMMI10 1.100 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % right45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 20 20 true [ 20 0 0 -20 0 20 ] {<0040100080200100400200800401000802001004 0020080040100080200000401000802001004002 0080040100080200100400200800401000802000>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P5 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % % % here starts figure with depth 50 % Polyline 7.500 slw gs clippath 6720 2715 m 6780 2715 l 6780 2564 l 6750 2684 l 6720 2564 l cp 6780 1785 m 6720 1785 l 6720 1936 l 6750 1816 l 6780 1936 l cp eoclip n 6750 1800 m 6750 2700 l gs col0 s gr gr % arrowhead n 6780 1936 m 6750 1816 l 6720 1936 l 6780 1936 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 6720 2564 m 6750 2684 l 6780 2564 l 6720 2564 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 6720 1815 m 6780 1815 l 6780 1731 l 6750 1791 l 6720 1731 l cp 6780 1560 m 6720 1560 l 6720 1644 l 6750 1584 l 6780 1644 l cp eoclip n 6750 1575 m 6750 1800 l gs col0 s gr gr % arrowhead n 6780 1644 m 6750 1584 l 6720 1644 l 6780 1644 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 6720 1731 m 6750 1791 l 6780 1731 l 6720 1731 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 6720 3615 m 6780 3615 l 6780 3464 l 6750 3584 l 6720 3464 l cp 6780 2685 m 6720 2685 l 6720 2836 l 6750 2716 l 6780 2836 l cp eoclip n 6750 2700 m 6750 3600 l gs col0 s gr gr % arrowhead n 6780 2836 m 6750 2716 l 6720 2836 l 6780 2836 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 6720 3464 m 6750 3584 l 6780 3464 l 6720 3464 l cp gs 0.00 setgray ef gr col0 s % Polyline n 4275 1800 m 5625 1800 l 5625 2700 l 4275 2700 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 285.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 2025 1800 m 3375 1800 l 3375 2700 l 2025 2700 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 135.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline gs clippath 4035 1515 m 4035 1635 l 4275 1635 l 4065 1575 l 4275 1515 l cp eoclip n 4635 1575 m 4050 1575 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 270.00 105.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 4275 1515 m 4065 1575 l 4275 1635 l col0 s % Polyline gs clippath 6585 2505 m 6465 2505 l 6465 2745 l 6525 2535 l 6585 2745 l cp eoclip n 6525 3105 m 6525 2520 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 435.00 168.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 6585 2745 m 6525 2535 l 6465 2745 l col0 s % Polyline gs clippath 3615 3660 m 3615 3540 l 3375 3540 l 3585 3600 l 3375 3660 l cp eoclip n 3015 3600 m 3600 3600 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 201.00 240.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 3375 3660 m 3585 3600 l 3375 3540 l col0 s % Polyline gs clippath 1065 2715 m 1185 2715 l 1185 2475 l 1125 2685 l 1065 2475 l cp eoclip n 1125 2115 m 1125 2700 l gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 75.00 141.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 1065 2475 m 1125 2685 l 1185 2475 l col0 s % Polyline gs clippath 6540 3855 m 6540 3795 l 6389 3795 l 6509 3825 l 6389 3855 l cp 5610 3795 m 5610 3855 l 5761 3855 l 5641 3825 l 5761 3795 l cp eoclip n 5625 3825 m 6525 3825 l gs col0 s gr gr % arrowhead n 5761 3795 m 5641 3825 l 5761 3855 l 5761 3795 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 6389 3855 m 6509 3825 l 6389 3795 l 6389 3855 l cp gs 0.00 setgray ef gr col0 s % Polyline n 1125 1575 m 6525 1575 l 6525 3600 l 1125 3600 l cp gs col0 s gr % Polyline gs clippath 3840 3855 m 3840 3795 l 3723 3795 l 3813 3825 l 3723 3855 l cp 3360 3795 m 3360 3855 l 3477 3855 l 3387 3825 l 3477 3795 l cp eoclip n 3375 3825 m 3825 3825 l gs col0 s gr gr % arrowhead n 3477 3795 m 3387 3825 l 3477 3855 l 3477 3795 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 3723 3855 m 3813 3825 l 3723 3795 l 3723 3855 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5640 3855 m 5640 3795 l 5489 3795 l 5609 3825 l 5489 3855 l cp 3810 3795 m 3810 3855 l 3961 3855 l 3841 3825 l 3961 3795 l cp eoclip n 3825 3825 m 5625 3825 l gs col0 s gr gr % arrowhead n 3961 3795 m 3841 3825 l 3961 3855 l 3961 3795 l cp gs 0.00 setgray ef gr col0 s % arrowhead n 5489 3855 m 5609 3825 l 5489 3795 l 5489 3855 l cp gs 0.00 setgray ef gr col0 s % Polyline 15.000 slw gs clippath 7215 1845 m 7215 1755 l 6952 1755 l 7162 1800 l 6952 1845 l cp eoclip n 450 1800 m 7200 1800 l gs col0 s gr gr % arrowhead n 6952 1845 m 7162 1800 l 6952 1755 l 6952 1845 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3870 885 m 3780 885 l 3780 1148 l 3825 938 l 3870 1148 l cp eoclip n 3825 4050 m 3825 900 l gs col0 s gr gr % arrowhead n 3870 1148 m 3825 938 l 3780 1148 l 3870 1148 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 180.00 scf sf 6840 2250 m gs 1 -1 sc (\(1\)) col0 sh gr /Times-Roman ff 180.00 scf sf 6840 3150 m gs 1 -1 sc (\(1\)) col0 sh gr /Times-Roman ff 180.00 scf sf 6795 1665 m gs 1 -1 sc (\(bt\)) col0 sh gr /Times-Roman ff 180.00 scf sf 4500 4050 m gs 1 -1 sc (\(om\)) col0 sh gr /Times-Roman ff 180.00 scf sf 5940 4050 m gs 1 -1 sc (\(1\)) col0 sh gr /Times-Roman ff 180.00 scf sf 3420 4050 m gs 1 -1 sc (\(O0\)) col0 sh gr % here ends figure; $F2psEnd rs end showpage @endspecial 275 1428 a /End PSfrag 275 1428 a 275 839 a /Hide PSfrag 275 839 a -410 895 a Fe(PSfrag)20 b(replacements)p -410 926 685 4 v 275 929 a /Unhide PSfrag 275 929 a 257 1011 a { 257 1011 a 239 1029 a Fd(c)257 1011 y } 0/Place PSfrag 257 1011 a 202 1105 a { 202 1105 a 129 1117 a Fd(!)181 1129 y Fc(min)202 1105 y } 1/Place PSfrag 202 1105 a 249 1191 a { 249 1191 a 223 1212 a Fd(\014)249 1191 y } 2/Place PSfrag 249 1191 a 254 1301 a { 254 1301 a 233 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putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end userdict begin /PSfragLib 90 dict def /PSfragDict 6 dict def /PSfrag { PSfragLib begin load exec end } bind def end PSfragLib begin /RO /readonly load def /CP /currentpoint load def /CM /currentmatrix load def /B { bind RO def } bind def /X { exch def } B /MD { { X } forall } B /OE { end exec PSfragLib begin } B /S false def /tstr 8 string def /islev2 { languagelevel } stopped { false } { 2 ge } ifelse def [ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall sM currentmatrix RO pop dM defaultmatrix RO idM invertmatrix RO pop srcFM identmatrix pop /Hide { gsave { CP } stopped not newpath clip { moveto } if } B /Unhide { { CP } stopped not grestore { moveto } if } B /setrepl islev2 {{ /glob currentglobal def true setglobal array astore globaldict exch /PSfrags exch put glob setglobal }} {{ array astore /PSfrags X }} ifelse B /getrepl islev2 {{ globaldict /PSfrags get aload length }} {{ PSfrags aload length }} ifelse B /convert { /src X src length string /c 0 def src length { dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def } repeat } B /Begin { /saver save def srcFM exch 3 exch put 0 ne /debugMode X 0 setrepl dup /S exch dict def { S 3 1 roll exch convert exch put } repeat srcM CM dup invertmatrix pop mark { currentdict { end } stopped { pop exit } if } loop PSfragDict counttomark { begin } repeat pop } B /End { mark { currentdict end dup PSfragDict eq { pop exit } if } loop counttomark { begin } repeat pop getrepl saver restore 7 idiv dup /S exch dict def { 6 array astore /mtrx X tstr cvs /K X S K [ S K known { S K get aload pop } if mtrx ] put } repeat } B /Place { tstr cvs /K X S K known { bind /proc X tM CM pop CP /cY X /cX X 0 0 transform idtransform neg /aY X neg /aX X S K get dup length /maxiter X /iter 1 def { iter maxiter ne { /saver save def } if tM setmatrix aX aY translate [ exch aload pop idtransform ] concat cX neg cY neg translate cX cY moveto /proc load OE iter maxiter ne { saver restore /iter iter 1 add def } if } forall /noXY { CP /cY X /cX X } stopped def tM setmatrix noXY { newpath } { cX cY moveto } ifelse } { Hide OE Unhide } ifelse } B /normalize { 2 index dup mul 2 index dup mul add sqrt div dup 4 -1 roll exch mul 3 1 roll mul } B /replace { aload pop MD CP /bY X /lX X gsave sM setmatrix str stringwidth abs exch abs add dup 0 eq { pop } { 360 exch div dup scale } ifelse lX neg bY neg translate newpath lX bY moveto str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall flattenpath pathbbox [ /uY /uX /lY /lX ] MD CP grestore moveto currentfont /FontMatrix get dstFM copy dup 0 get 0 lt { uX lX /uX X /lX X } if 3 get 0 lt { uY lY /uY X /lY X } if /cX uX lX add 0.5 mul def /cY uY lY add 0.5 mul def debugMode { gsave 0 setgray 1 setlinewidth lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath lX bY moveto uX bY lineto lX cY moveto uX cY lineto cX lY moveto cX uY lineto stroke grestore } if dstFM dup invertmatrix dstM CM srcM 2 { dstM concatmatrix } repeat pop getrepl /temp X S str convert get { aload pop [ /rot /scl /loc /K ] MD /aX cX def /aY cY def loc { dup 66 eq { /aY bY def } { % B dup 98 eq { /aY lY def } { % b dup 108 eq { /aX lX def } { % l dup 114 eq { /aX uX def } { % r dup 116 eq { /aY uY def } % t if } ifelse } ifelse } ifelse } ifelse pop } forall K srcFM rot tM rotate dstM 2 { tM concatmatrix } repeat aload pop pop pop 2 { scl normalize 4 2 roll } repeat aX aY transform /temp temp 7 add def } forall temp setrepl } B /Rif { S 3 index convert known { pop replace } { exch pop OE } ifelse } B /XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B /xs { pop } XA XC /xks { /kern load OE } XA /kern XC /xas { pop ax ay rmoveto } XA /ay /ax XC /xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC /xaws { ax ay rmoveto c eq { cx cy rmoveto } if } XA /ay /ax /c /cy /cx XC /raws { xaws { awidthshow } Rif } B /rws { xws { widthshow } Rif } B /rks { xks { kshow } Rif } B /ras { xas { ashow } Rif } B /rs { xs { show } Rif } B /rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B PSfragDict begin islev2 not { /restore { /rrs PSfrag } B } if /show { /rs PSfrag } B /kshow { /rks PSfrag } B /ashow { /ras PSfrag } B /widthshow { /rws PSfrag } B /awidthshow { /raws PSfrag } B end PSfragDict RO pop end % File 8r.enc as of 2002-03-12 for PSNFSS 9 % % This is the encoding vector for Type1 and TrueType fonts to be used % with TeX. This file is part of the PSNFSS bundle, version 9 % % Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % % (6) /Euro is assigned to 128, as in Windows ANSI % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /Euro /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 % /TeXbbad153fEncoding [ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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This file is part of the PSNFSS bundle, version 9 % % Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % % (6) /Euro is assigned to 128, as in Windows ANSI % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /Euro /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmmi10 cmmi12 cmmi5 cmmi6 cmmi7 cmmi8 cmmi9 cmmib10 % /TeXaae443f0Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /omega /epsilon /theta1 /pi1 /rho1 /sigma1 /phi1 /arrowlefttophalf /arrowleftbothalf /arrowrighttophalf /arrowrightbothalf /arrowhookleft /arrowhookright /triangleright /triangleleft /zerooldstyle /oneoldstyle /twooldstyle /threeoldstyle /fouroldstyle /fiveoldstyle /sixoldstyle /sevenoldstyle /eightoldstyle /nineoldstyle /period /comma /less /slash /greater /star /partialdiff /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /flat /natural /sharp /slurbelow /slurabove /lscript /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /dotlessi /dotlessj /weierstrass /vector /tie /psi /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /alpha /beta /gamma /delta /epsilon1 /zeta /eta /theta /iota /kappa /lambda /mu /nu /xi /pi /rho /sigma /tau /upsilon /phi /chi /psi /tie /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmb10 cmbx10 cmbx12 cmbx5 cmbx6 cmbx7 cmbx8 cmbx9 cmbxsl10 % cmdunh10 cmr10 cmr12 cmr17cmr6 cmr7 cmr8 cmr9 cmsl10 cmsl12 cmsl8 % cmsl9 cmss10cmss12 cmss17 cmss8 cmss9 cmssbx10 cmssdc10 cmssi10 % cmssi12 cmssi17 cmssi8cmssi9 cmssq8 cmssqi8 cmvtt10 % /TeXf7b6d320Encoding [ /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /exclam /quotedblright /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /exclamdown /equal /questiondown /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /quotedblleft /bracketright /circumflex /dotaccent /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /endash /emdash /hungarumlaut /tilde /dieresis /suppress /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /Gamma /Delta /Theta /Lambda /Xi /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl /dotlessi /dotlessj /grave /acute /caron /breve /macron /ring /cedilla /germandbls /ae /oe /oslash /AE /OE /Oslash /suppress /dieresis /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def % Thomas Esser, Dec 2002. public domain % % Encoding for: % cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9 % /TeXbbad153fEncoding [ /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast /arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup /arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional /prime /infinity /element /owner /triangle /triangleinv /negationslash /mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur /latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /union /intersection /unionmulti /logicaland /logicalor /turnstileleft /turnstileright /floorleft /floorright /ceilingleft /ceilingright /braceleft /braceright /angbracketleft /angbracketright /bar /bardbl /arrowbothv /arrowdblbothv /backslash /wreathproduct /radical /coproduct /nabla /integral /unionsq /intersectionsq /subsetsqequal /supersetsqequal /section /dagger /daggerdbl /paragraph /club /diamond /heart /spade /arrowleft /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /minus /periodcentered /multiply /asteriskmath /divide /diamondmath /plusminus /minusplus /circleplus /circleminus /.notdef /.notdef /circlemultiply /circledivide /circledot /circlecopyrt /openbullet /bullet /equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal /greaterequal /precedesequal /followsequal /similar /approxequal /propersubset /propersuperset /lessmuch /greatermuch /precedes /follows /arrowleft /spade /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] def %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def @MacSetUp}N/doclip{psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N} N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N /setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{ pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B /rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %!PS-AdobeFont-1.1: CMSY7 1.0 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %!PS-AdobeFont-1.1: CMR7 1.0 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % % % here starts figure with depth 50 % Polyline 7.500 slw n 855 1305 m 945 1395 l gs col0 s gr % Polyline n 855 1395 m 945 1305 l gs col0 s gr % Polyline n 1305 1305 m 1395 1395 l gs col0 s gr % Polyline n 1305 1395 m 1395 1305 l gs col0 s gr % Polyline n 1755 1305 m 1845 1395 l gs col0 s gr % Polyline n 1755 1395 m 1845 1305 l gs col0 s gr % Polyline n 2655 1305 m 2745 1395 l gs col0 s gr % Polyline n 2655 1395 m 2745 1305 l gs col0 s gr % Polyline n 3105 1305 m 3195 1395 l gs col0 s gr % Polyline n 3105 1395 m 3195 1305 l gs col0 s gr % Polyline n 3555 1305 m 3645 1395 l gs col0 s gr % Polyline n 3555 1395 m 3645 1305 l gs col0 s gr % Polyline n 4005 1305 m 4095 1395 l gs col0 s gr % Polyline n 4005 1395 m 4095 1305 l gs col0 s gr % Polyline n 4455 1305 m 4545 1395 l gs col0 s gr % Polyline n 4455 1395 m 4545 1305 l gs col0 s gr % Polyline n 4905 1320 m 4995 1380 l gs col0 s gr % Polyline n 4905 1380 m 4995 1320 l gs col0 s gr % Polyline n 4950 1305 m 4950 1395 l gs col0 s gr % Ellipse n 450 1350 90 90 0 360 DrawEllipse gs col0 s gr % Ellipse n 4950 1350 90 90 0 360 DrawEllipse gs col0 s gr % Polyline n 2205 1305 m 2295 1305 l 2295 1395 l 2205 1395 l cp gs col0 s gr % Polyline n 549 1350 m 1350 1350 l gs col0 s gr % Polyline [60] 0 sd n 1350 1350 m 2205 1350 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 2295 1350 m 3150 1350 l gs col0 s gr [] 0 sd % Polyline n 3150 1350 m 4860 1350 l gs col0 s gr % Polyline [15 37] 37 sd n 900 1350 m 900 900 l 4050 900 l 4050 1350 l gs col0 s gr [] 0 sd % Polyline [15 37] 37 sd n 1350 1350 m 1350 1035 l 3600 1035 l 3600 1350 l gs col0 s gr [] 0 sd % Polyline [15 37] 37 sd n 2700 1350 m 2700 1215 l 3150 1215 l 3150 1350 l gs col0 s gr [] 0 sd % Polyline [15 37] 37 sd n 1800 1350 m 1800 1125 l 4500 1125 l 4500 1350 l gs col0 s gr [] 0 sd /Times-Roman ff 180.00 scf sf 2205 1575 m gs 1 -1 sc (4) col0 sh gr /Times-Roman ff 180.00 scf sf 2655 1575 m gs 1 -1 sc (5) col0 sh gr /Times-Roman ff 180.00 scf sf 3555 1575 m gs 1 -1 sc (7) col0 sh gr /Times-Roman ff 180.00 scf sf 4005 1575 m gs 1 -1 sc (8) col0 sh gr /Times-Roman ff 180.00 scf sf 4455 1575 m gs 1 -1 sc (9) col0 sh gr /Times-Roman ff 180.00 scf sf 3105 1575 m gs 1 -1 sc (6) col0 sh gr /Times-Roman ff 180.00 scf sf 1755 1575 m gs 1 -1 sc (3) col0 sh gr /Times-Roman ff 180.00 scf sf 855 1575 m gs 1 -1 sc (1) col0 sh gr /Times-Roman ff 180.00 scf sf 1305 1575 m gs 1 -1 sc (2) col0 sh gr % Polyline 2 slc n 2250 1620 m 2250 1935 l gs col0 s gr % Polyline gs clippath 2580 1695 m 2580 1635 l 2428 1635 l 2548 1665 l 2428 1695 l cp 1965 1635 m 1965 1695 l 2117 1695 l 1997 1665 l 2117 1635 l cp eoclip n 1980 1665 m 2565 1665 l gs col0 s gr gr % arrowhead 0 slc n 2117 1635 m 1997 1665 l 2117 1695 l col0 s % arrowhead n 2428 1695 m 2548 1665 l 2428 1635 l col0 s /Times-Roman ff 180.00 scf sf 585 1710 m gs 1 -1 sc (k1) col0 sh gr /Times-Roman ff 180.00 scf sf 1035 1710 m gs 1 -1 sc (k2) col0 sh gr /Times-Roman ff 180.00 scf sf 2025 1845 m gs 1 -1 sc (\(z\)) col0 sh gr /Times-Roman ff 180.00 scf sf 2340 1845 m gs 1 -1 sc (\(zp\)) col0 sh gr % here ends figure; $F2psEnd rs showpage @endspecial 275 842 a /End PSfrag 275 842 a 275 -544 a /Hide PSfrag 275 -544 a -410 -488 a Fe(PSfrag)20 b(replacements)p -410 -457 685 4 v 275 -453 a /Unhide PSfrag 275 -453 a 256 -379 a { 256 -379 a 237 -354 a Fd(1)256 -379 y } 0/Place PSfrag 256 -379 a 256 -279 a { 256 -279 a 237 -254 a Fd(2)256 -279 y } 1/Place PSfrag 256 -279 a 256 -180 a { 256 -180 a 237 -155 a Fd(3)256 -180 y } 2/Place PSfrag 256 -180 a 256 -80 a { 256 -80 a 237 -55 a Fd(4)256 -80 y } 3/Place PSfrag 256 -80 a 256 19 a { 256 19 a 237 44 a Fd(5)256 19 y } 4/Place PSfrag 256 19 a 256 119 a { 256 119 a 237 144 a Fd(6)256 119 y } 5/Place PSfrag 256 119 a 256 219 a { 256 219 a 237 243 a Fd(7)256 219 y } 6/Place PSfrag 256 219 a 256 318 a { 256 318 a 237 343 a Fd(8)256 318 y } 7/Place PSfrag 256 318 a 256 417 a { 256 417 a 237 442 a Fd(9)256 417 y } 8/Place PSfrag 256 417 a 235 508 a { 235 508 a 195 531 a Fc(k)238 543 y Fb(1)235 508 y } 9/Place PSfrag 235 508 a 235 608 a { 235 608 a 195 630 a Fc(k)238 642 y Fb(2)235 608 y } 10/Place PSfrag 235 608 a 221 711 a { 221 711 a 167 742 a Fc(z)221 711 y } 11/Place PSfrag 221 711 a 242 811 a { 242 811 a 209 842 a Fc(z)252 812 y Fa(0)242 811 y } 12/Place PSfrag 242 811 a eop end userdict /end-hook known{end-hook}if %%Trailer cleartomark countdictstack exch sub { end } repeat restore %%EOF ---------------0505310851255--