Content-Type: multipart/mixed; boundary="-------------0901211031755" This is a multi-part message in MIME format. ---------------0901211031755 Content-Type: text/plain; name="09-12.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-12.comments" AMS-Code: 74J20; 81Q30; 37K60; 35Q55; 70K70 ---------------0901211031755 Content-Type: text/plain; name="09-12.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-12.keywords" Equilibrium time-correlations, wave turbulence, kinetic theory, perturbation theory ---------------0901211031755 Content-Type: application/x-tex; name="nonlin.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nonlin.tex" %%%% Final: Jan 21, 2009 %%%%%%%%%%%% \documentclass[11pt,a4paper]{article} %\usepackage[active]{srcltx} %\usepackage[pdftex]{graphicx} %\usepackage{epstopdf} \usepackage{graphicx} \setlength{\oddsidemargin}{3 mm} \setlength{\evensidemargin}{3 mm} 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\newcommand{\rme}{{\rm e}} \newcommand{\rmd}{{\rm d}} \newcommand{\braket}[2]{\langle #1|#2\rangle} \renewcommand{\braket}[2]{ \left\langle #1 , #2\right\rangle} \newcommand{\FT}[1]{\hat{#1}} \newcommand{\IFT}[1]{\widetilde{#1}} % 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} % }}{\end{list}} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{assumption}[theorem]{Assumption} \newenvironment{proof}{\begin{trivlist}\item[]{\em Proof:}\/}{\hfill\mbox{$\Box$}\end{trivlist}} \numberwithin{equation}{section} \begin{document} \selectlanguage{english} \newcommand{\email}[1]{E-mail: \tt #1} \newcommand{\emailjani}{\email{jani.lukkarinen@helsinki.fi}} \newcommand{\addressjani}{\em University of Helsinki, Department of Mathematics and Statistics\\ \em P.O. Box 68, FI-00014 Helsingin yliopisto, Finland} \newcommand{\addressherbert}{\em Zentrum Mathematik, Technische Universit\"at M\"unchen, \\ \em Boltzmannstr. 3, D-85747 Garching, Germany} \newcommand{\emailherbert}{\email{spohn@ma.tum.de}} \title{Weakly nonlinear Schr\"{o}dinger equation with\\ random initial data} \author{Jani Lukkarinen\thanks{\emailjani}, Herbert Spohn\thanks{\emailherbert}\\[1em] $^*$\addressjani \\[1em] $^\dag$\addressherbert } \maketitle \begin{abstract} There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schr\"{o}dinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution $\psi_t(x)$ of the nonlinear Schr\"{o}dinger equation yields then a stochastic process stationary in $x\in\mathbb{Z}^d$ and $t\in \mathbb{R}$. If $\lambda$ denotes the strength of the nonlinearity, we prove that the space-time covariance of $\psi_t(x)$ has a limit as $\lambda\to 0$ for $t=\lambda^{-2}\tau$, with $\tau$ fixed and $|\tau|$ sufficiently small. The limit agrees with the prediction from kinetic theory. \end{abstract} \tableofcontents \section{Introduction}\label{sec:intro} The nonlinear Schr\"{o}dinger equation (NLS) governs the evolution of a complex valued wave field $\psi:\mathbb{R}\times\mathbb{Z}^d\to\mathbb{C}$ and reads \begin{equation}\label{eq:1.1} \textrm{i}\frac{\textrm{d}}{\textrm{d}t} \psi_t(x)=\sum_{y\in \mathbb{Z}^d}\alpha(x-y)\psi_t(y)+\lambda|\psi_t(x)|^2 \psi_t(x)\,. \end{equation} Here $\alpha(x)$ are the ``hopping amplitudes'' and we assume that they satisfy \begin{jlist} \item $\alpha:\mathbb{Z}^d\to \mathbb{R}$, $\alpha(x)=\alpha(-x)$. \item $\alpha$ has an exponentially decreasing upper bound. \end{jlist} We consider only the dispersive case $\lambda\geq 0$. Usually the NLS is studied in the continuum setting, where $\mathbb{Z}^d$ is replaced by $\mathbb{R}^d$ and the linear term is $\Delta\psi_t(x)$. It will become evident later on why for our purposes the spatial discretization is a necessity. The NLS is a Hamiltonian system. To see this, we define the canonical degrees of freedom $q_x,p_x\in \R$, $x\in\mathbb{Z}^d$, via $\psi(x)=(q_x+i p_x)/\sqrt{2}$. Their Hamiltonian function is obtained by substitution in \begin{equation}\label{eq:1.2} H(\psi)= \sum_{x,y\in\mathbb{Z}^d}\alpha(x-y)\psi(x)^\ast\psi(y) +\tfrac{1}{2}\lambda \sum_{x\in\mathbb{Z}^d} |\psi(x)|^4\,. \end{equation} It is easy to check that the corresponding equations of motion, \begin{equation}\label{eq:1.3} \frac{\textrm{d}}{\textrm{d}t}q_x=\frac{\partial}{\partial p_x} H\,,\quad \frac{\textrm{d}}{\textrm{d}t}p_x=-\frac{\partial}{\partial q_x} H\, , \end{equation} are identical to the NLS. In particular, we conclude that the energy is conserved, $H(\psi_t)=H(\psi_0)$ for all $t\in\mathbb{R}$. Also the $\ell_2$-norm is conserved, in this context also referred to as particle number $N$, \begin{equation}\label{eq:1.3a} N(\psi)= \sum_{x\in\mathbb{Z}^d}|\psi(x)|^2\,,\quad N(\psi_t)=N(\psi_0)\textrm{ for all } t\in\mathbb{R}\,. \end{equation} Because of energy conservation law, if $H(\psi_0)<\infty$, then the Cauchy problem for (\ref{eq:1.1}) has a unique global solution. We refer to \cite{SuSu99} for a more detailed information on the NLS. In this work we are interested in random initial data. From a statistical physics point of view a very natural choice is to take the initial $\psi$-field to be distributed according to a Gibbs measure for $H$ and $N$, which physically means that the wave field is in thermal equilibrium. Somewhat formally the Gibbs measure is defined through \begin{equation}\label{eq:1.4} \frac{1}{Z}\exp \big[-\beta\big(H(\psi)-\mu N(\psi)\big)\big]\prod_x \left[\rmd\bigl(\re \psi(x)\bigr) \rmd\bigl(\im \psi(x)\bigr)\right] . \end{equation} Here $\beta>0$ is the inverse temperature and $\mu\in\mathbb{R}$ the chemical potential. The partition function $Z$ is a constant chosen so that (\ref{eq:1.4}) is a probability measure. To properly define the Gibbs measure one has to restrict (\ref{eq:1.4}) to some finite box $\Lambda\subset\mathbb{Z}^d$, which yields a well-defined probability measure $\mathbb{P}^\lambda_{\beta,\mu,\Lambda}$ on $\mathbb{C}^{|\Lambda|}$. The Gibbs probability measure $\mathbb{P}^\lambda_{\beta,\mu}$ on $\mathbb{Z}^d$ is then obtained in the limit $\Lambda\nearrow\mathbb{Z}^d$. The existence of this limit is a well-studied problem \cite{LePr76}. If $\lambda$ is sufficiently small and $\mu$ sufficiently negative, then the Gibbs measure $\mathbb{P}^\lambda_{\beta,\mu}$ exists and has exponential mixing. Let us denote the corresponding expectation by $\mathbb{E}^\lambda_{\beta,\mu}$. The random field $\psi(x)$, $x\in\mathbb{Z}^d$, distributed according to $\mathbb{P}^\lambda_{\beta,\mu}$, is stationary with a rapid decay of correlations. It is also gauge invariant in the sense that $\psi(x)=\mathrm{e}^{i\theta}\psi(x)$ in distribution for any $\theta\in [0,2\pi]$. Of course, $\mathbb{P}^\lambda_{\beta,\mu}$-almost surely it holds $H(\psi)=\infty$ and $N(\psi)=\infty$. Thus one has to define solutions for the NLS with initial data of infinite energy. This has been accomplished for standard anharmonic Hamiltonian systems by Lanford, Lebowitz, and Lieb \cite{lll77}, who prove existence and uniqueness under a suitable growth condition at infinity for the initial data. Their argument can be extended to the Hamiltonian system (\ref{eq:1.3}). In particular, then $\psi_t(x)$ exists $\mathbb{P}^\lambda_{\beta,\mu}$-almost surely and $\mathbb{E}^\lambda_{\beta,\mu}\big(|\psi_t(x)|^2\big)<\infty$. Under $\mathbb{P}^\lambda_{\beta,\mu}$ the space-time random field $\psi_t(x)$, $x\in\mathbb{Z}^d$, $t\in\mathbb{R}$, is stationary in both arguments. Let us return for a moment to the issue of the continuum limit $(\varepsilon\mathbb{Z})^d$ with $\varepsilon\to 0$. To establish this limit for the Gibbs measure $\mathbb{P}^\lambda_{\beta,\mu}$ is one of the central goals of constructive quantum field theory \cite{glimmjaffe}. If the limit exists at all, the limiting random field will be rather singular at short distances. To construct solutions for the NLS with such singular initial data has never been attempted. As a general rule, it is not so wise to mix two difficulties and we stay with the lattice version for which all mathematical objects are well-defined. In the present context the most basic quantity is the stationary covariance \begin{equation}\label{eq:1.5} \mathbb{E}^\lambda_{\beta,\mu}\big(\psi_0(x_0)^\ast\psi_t(x)\big)= F^\lambda_2 (x-x_0,t)\,. \end{equation} $F^\lambda_2$ is well defined and one would like to know its qualitative dependence on $x,t$. For deterministic infinitely extended Hamiltonian systems, such as the NLS, establishing the qualitative behavior of equilibrium time correlations is known to be an extremely difficult problem with very few results available, despite intense efforts. For linear systems one has an explicit solution in Fourier space, see below. But already for completely integrable systems, like the Toda chain, not much is known about time correlations in thermal equilibrium. It is instructive to briefly discuss the linear case, $\lambda=0$, for which purpose we introduce Fourier transforms. For $f:\mathbb{Z}^d\to\mathbb{C}$ let us denote its Fourier transform by \begin{equation}\label{eq:1.6} \hat{f}(k)=\sum_{x\in\mathbb{Z}^d} f(x) \textrm{e}^{-\textrm{i} 2\pi k\cdot x}\,, \end{equation} $k\in\mathbb{R}$, and the inverse Fourier transform by \begin{equation}\label{eq:1.7} \tilde{g}(x)=\int_{\mathbb{T}^d} \textrm{d}k g(k) \textrm{e}^{\textrm{i} 2\pi k\cdot x} \end{equation} with $\mathbb{T}^d=[0,1]^d$, a parametrization of the $d$-dimensional torus. (We will use arithmetic relations on $\T^d$. These are defined using the arithmetic induced on the torus via its definition as equivalence classes $\R^d/\Z^d$, i.e., by using ``periodic boundary conditions''.) In particular, we set \begin{equation}\label{eq:1.8} \omega(k)=\hat{\alpha}(k)\,. \end{equation} $\omega$ is then the dispersion relation of our discretized Schr\"{o}dinger equation. It follows from the assumptions on $\alpha$ that \begin{jlist} \item $\omega:\mathbb{T}^d\to\mathbb{R}$ and its periodic extension is a real analytic function. \item $\omega(k)=\omega(-k)$. \end{jlist} In Fourier space the energy is given by \begin{align}\label{eq:1.9} & H(\psi)= \int_{\mathbb{T}^d} \textrm{d}k\, \omega(k)|\hat{\psi}(k)|^2 \nonumber\\ & \qquad + \tfrac{1}{2}\lambda \int_{(\mathbb{T}^{d})^4} \textrm{d}k_1 \textrm{d}k_2 \textrm{d}k_3 \textrm{d}k_4 \delta(k_1+k_2-k_3-k_4) \hat{\psi}(k_1)^\ast \hat{\psi}(k_2)^\ast \hat{\psi}(k_3)\hat{\psi}(k_4)\, , \end{align} where $\delta$ is a formal Dirac $\delta$-function, used here to simplify the notation for the convolution integral. Clearly, $H(\psi)\geq (\inf_k \omega(k)) N(\psi)$. The NLS after Fourier transform reads \begin{align}\label{eq:1.10} & \frac{\textrm{d}}{\textrm{d}t}\hat{\psi}_t(k_1)= -\textrm{i} \omega(k_1) \hat{\psi}_t(k_1) -\textrm{i} \lambda \int \textrm{d}k_2 \textrm{d}k_3 \textrm{d}k_4 \delta(k_1+k_2-k_3-k_4) \nonumber\\ & \qquad\times \hat{\psi}_t(k_2)^\ast\hat{\psi}_t(k_3)\psi_t(k_4)\,. \end{align} For $\lambda=0$, $\mathbb{P}^0_{\beta,\mu}$ is a Gaussian measure with mean zero and covariance \begin{equation}\label{eq:1.11} \mathbb{E}^0_{\beta,\mu} \big( \psi(0)^\ast \psi(x)\big)= F^0_2 (x,0)=\int_{\mathbb{T}^d} \textrm{d}k \big(\beta(\omega(k)-\mu)\big)^{-1} \textrm{e}^{\textrm{i} 2\pi k\cdot x}\,, \end{equation} provided $\mu<\inf_k \omega(k)$. Under our assumptions on $\omega$ the Gaussian field has exponential mixing. For the time-dependent equilibrium covariance one obtains \begin{equation}\label{eq:1.12} F^0_2(x,t)=\int_{\mathbb{T}^d} \textrm{d}k \big(\beta(\omega(k)-\mu)\big)^{-1} \textrm{e}^{\textrm{i} 2\pi k\cdot x} \textrm{e}^{-\textrm{i} \omega(k)t}\,. \end{equation} Clearly, $F^0_2(x,t)$ is a solution of the linear wave equation for exponentially localized initial data and thus spreads dispersively. If $\lambda> 0$, as general heuristics the nonlinearity should induce an exponential damping of $F^\lambda_2$. The physical picture is based on excitations of wave modes which interact weakly and are damped through collisions. Approximate theories have been developed in the context of phonon physics and wave turbulence, see e.g. \cite{Gu86,ZLF92}. To mathematically establish such a time-decay is completely out of reach, at present, whatever the choice of the nonlinear wave equation. To make some progress we will investigate here the regime of small nonlinearity, $\lambda\ll 1$. The idea is not to aim for results which are valid globally in time, but rather to consider the first time scale on which the effect of the nonlinearity becomes visible. For small $\lambda$ the rate of collision for two resonant waves is of order $\lambda^2$. Therefore, the nonlinearity is expected to show up on a time scale $\lambda^{-2}$. This suggests to study the limit \begin{equation}\label{eq:1.13} F^\lambda_2 (x,\lambda^{-2} t)\quad \textrm{as }\lambda\to 0\,. \end{equation} Note that the location $x$ is not scaled. For this limit to exist, one has to remove the oscillating phase resulting from (\ref{eq:1.12}), which on the speeded-up time scale is rapidly oscillating, of order $\lambda^{-2}$. In fact, a second rapidly oscillating phase of order $\lambda^{-1}$ will show up, which also has to be removed. Under suitable conditions on $\omega$, we will prove that $F^\lambda_2 (x,\lambda^{-2} t)$, with the removals just mentioned, has a limit for $\lambda\to 0$, at least for $|t|\leq t_0$ with some suitable $t_0>0$. The limit function indeed exhibits an exponential damping. A similar result has been obtained a long time ago for a system of hard spheres in equilibrium and at low density \cite{BLLS80}. There the small parameter is the density rather than the strength $\lambda$ of the nonlinearity. But the over-all philosophy is the same. To establish the decay of time-correlations in equilibrium at a fixed low density is an apparently very hard problem. Therefore, one looks for the first time scale on which the collisions between hard spheres have a visible effect. By fiat, hard spheres remain well localized in space, and on the time scale of interest only a finite number of collisions per particle are taken into account. In contrast, waves tend to delocalize through collisions. This is the reason why the problem under study has remained open. Our resolution uses techniques totally different from \cite{BLLS80}. The limit $\lambda\to 0$, $t=\lambda^{-2}\tau$ with $\tau$ fixed, together with a possible rescaling of space by a factor $\lambda^{-2}$, is called \textit{kinetic limit}, because the limit object is governed by a kinetic type transport equation. Formal derivations are discussed extensively in the literature, e.g., see \cite{janssen03,LN04}. On the mathematical side, Erd{\H o}s and Yau \cite{erdyau99} study in great detail the linear Schr\"{o}dinger equation with a random potential, extended to even longer time scales in \cite{erdyau05a,erdyau05b}. The discretized wave equation with a random index of refraction is covered in \cite{ls05}. For nonlinear wave equations the only related study is by Benedetto \textit{et al.\/} \cite{BCEP08} on the dynamics of weakly interacting quantum particles. They transform to multipoint Wigner functions, which leads to an expansion somewhat different from the one used here. We refer to \cite{ls08} for a comparison. As in our contribution, Benedetto \textit{et al}. have to analyze the asymptotics of high-dimensional oscillatory integrals. But in contrast, they have no control on the error term in the expansion. Before closing the introduction, we owe the reader some explanations why a seemingly perturbative result requires so many pages for its proof. From the solution to (\ref{eq:1.1}) one can regard $\psi_t(x)$ as some functional $\mathcal{F}_{x,t}$ of the initial field $\psi$, \begin{equation}\label{eq:1.14} \psi_t(x)=\mathcal{F}_{x,t}(\psi)\,. \end{equation} For given $t$ it depends only very little on those $\psi(y)$'s for which $|y-x|\gg t$. To make progress it seems necessary to first average the initial conditions over $\psi(x_0)^\ast\mathbb{P}^\lambda_{\beta,\mu}$ so that subsequently one can control the limit $\lambda\to 0$ with $t=\lambda^{-2}\tau$, $\tau>0$. Such an average can be accomplished by writing $\mathcal{F}_{x,t}$ as a power series in $\psi$, which is done through the Duhamel formula. For any $n\geq 1$ we write \begin{align}\label{eq:1.15} & \prod^n_{j=1} \textrm{e}^{\textrm{i} 2\pi \sigma_j \omega(k_j)t} \hat{\psi}_{t}(k_j,\sigma_j ) % \nonumber \\ & \quad = \prod^n_{j=1}\hat{\psi}_{0}(k_j,\sigma_j ) +\int^t_0 \textrm{d}s \frac{\textrm{d}}{\textrm{d}s}\prod^n_{j=1} \textrm{e}^{\textrm{i} 2\pi \sigma_j \omega(k_j)s}\hat{\psi}_{s}(k_j,\sigma_j)\,. \end{align} Here $\sigma_j \in \set{\pm 1}$ and $\hat{\psi}_t(k,1)=\hat{\psi}_t(k)$, $\hat{\psi}_{t}(k,-1)=\hat{\psi}_t(-k)^\ast$, $\hat{\psi}_0(k)=\hat{\psi}(k)$. Using the product rule and the equations of motion (\ref{eq:1.10}) yields a formula relating the $n$:th moment at time $t$ to the time-integral of a sum over $(n+2)$:th moments at time $s$. Iterating this equation leads to a (formal) series representation \begin{equation}\label{eq:1.16} \hat{\psi}_t(k)=\sum^\infty_{n=1} \mathcal{P}^n_{k,t} (\hat{\psi})\,, \end{equation} where $\mathcal{P}^n_{k,t}$ is a sum/integral over monomials of order $n$ in $\hat{\psi}$ and $\hat{\psi}^\ast$. Since each time-derivative increases the degree of the monomial by two, we have \begin{equation}\label{eq:1.17} \delta(k'-k)\sum_{x\in\mathbb{Z}^d} \textrm{e}^{-\textrm{i}2\pi k\cdot x} \mathbb{E}^\lambda_{\beta,\mu}\big(\psi(0)^\ast \psi_t(x)\big)=\sum^\infty_{n=0} \mathbb{E}^\lambda_{\beta,\mu}\big(\hat{\psi}(k')^\ast \mathcal{P}^{2n+1}_{k,t}(\hat{\psi})\big)\,. \end{equation} The first difficulty arises from the fact that the sum in (\ref{eq:1.17}) does not converge absolutely for any $t$. Very roughly, $\mathcal{P}^n_{k,t}$ is a sum of $n!$ terms of equal size. The iterated time-integration yields a factor $t^n/n!$. However, for the approximately Gaussian average the $n$:th moment grows also as $n!$. To be able to proceed one has to stop the series expansion at some large $N$ which depends on $\lambda$. A similar situation was encountered by Erd\H{o}s and Yau \cite{erdyau99} in their study of the Schr\"{o}dinger equation with a weak random potential. We will use the powerful Erd\H{o}s-Yau techniques as a guideline for handling the series in (\ref{eq:1.17}). The stopping of the series expansion will leave a remainder term containing the full original time-evolution. Erd\H{o}s and Yau control the error term in essence by unitarity of the time-evolution. For the NLS mere conservation of $N(\psi)$ will not suffice. Instead, we use stationarity of $\psi_t(x)$. In wave turbulence \cite{ZLF92} one is also interested in non-stationary initial measures, e.g., in Gaussian measures with a covariance different from $(\beta(\omega(k)-\mu))^{-1}$. For such initial data we have no idea how to control the error term, while other parts of our proof apply unaltered. The central difficulty resides in $\mathbb{E}^\lambda_{\beta,\mu}\big(\hat{\psi}(k')^\ast \mathcal{P}^{2n+1}_{k,t}(\hat{\psi})\big)$ which is a sum of rather explicit, but high-dimensional, dimension $n(1+3d)+d$, oscillatory integrals. On top, because of the $\delta$-function in (\ref{eq:1.10}), the integrand is restricted to a non-trivial linear subspace. In the limit $\lambda\to 0$, $t=\lambda^{-2}\tau$, $\tau>0$, only a few oscillatory integrals have a non-zero limit. Summing up these leading oscillatory integrals results in the anticipated exponential damping. The major task of our paper is to discover an iterative structure in all remaining oscillatory integrals, in a way which allows for an estimate in terms of a few basic ``motives''. Each of these subleading integrals is shown to contain at least one motive whose appearance leads to an extra fractional power of $\lambda$, thereby ensuring a zero limit. In Section \ref{sec:finite volume} we first give the mathematical definition of the above system in finite volume, and state in Section \ref{sec:model} the assumptions and main results. Their connection to kinetic theory is discussed in Section \ref{sec:link}. The proof of the main result is contained in the remaining sections: we derive a suitable time-dependent perturbation expansion in Section \ref{sec:graphs}, and develop a graphical language to describe the large, but finite, number of terms in the expansion in Section \ref{sec:diagrams}. The analysis of the oscillatory integrals in the expansion is contained in Sections \ref{sec:momdeltas}--\ref{sec:fullypaired}. More detailed outline of the technical structure of the proof can be found in Section \ref{sec:structureofproof}. The estimates are collected together and the limit of the non-zero terms is computed in Section \ref{sec:completion} where we complete the proof of the main theorem. In an Appendix, we show that the standard nearest neighbor couplings in $d\ge 4$ dimensions lead to harmonic interactions satisfying all assumptions of the main theorem. \bigskip \noindent {\it Acknowledgments.} We would like to thank L\'{a}szl\'{o} Erd\H{o}s and Horng-Tzer Yau for many illuminating discussions on the subject. The research of J.\ Lukkarinen was supported by the Academy of Finland. \section{Kinetic limit and main results}\label{sec:main} \subsection{Finite volume dynamics}\label{sec:finite volume} To properly define expectations such as (\ref{eq:1.5}), one has to go through a finite volume construction, which will be specified in this subsection. Let \begin{equation}\label{eq:1.1.1} L\ge 2\,,\quad \Lambda = \{0,1,\ldots,L-1\}^d\,, \end{equation} the dimension $d$ an arbitrary positive integer. We apply periodic boundary conditions on $\Lambda$, and let $[x]=x \bmod L\in \Lambda$ for all $x\in \mathbb{Z}^d$. Fourier transform of $f:\Lambda\to\mathbb{C}$ is denoted by $\hat{f} : \Lambda^* \to \mathbb{C}$, with the dual lattice $\Lambda^* = \{0,\frac{1}{L},\ldots,\frac{L-1}{L}\}^d$ and with \begin{equation}%\label{eq:} \hat{f}(k) = \sum_{x\in \Lambda} f(x) \textrm{e}^{-\textrm{i} 2\pi k \cdot x} \end{equation} for all $k\in \Lambda^*$ (or for all $k\in (\mathbb{Z}/L)^d$, which yields the periodic extension of $\hat{f}$). The inverse transform is given by \begin{equation}%\label{eq:} \tilde{g}(x) = \frac{1}{|\Lambda|} \sum_{k\in \Lambda^*} g(k) \textrm{e}^{\textrm{i} 2\pi k \cdot x}\,, \end{equation} where $|\Lambda|=L^d$. For all $x\in \Lambda$, it holds $\tilde{{\hat{f}}}(x)=f(x)$. The arithmetic operations on $\Lambda$ are done periodically, identifying it as a parametrization of $\mathbb{Z}_L^d$, the cyclic group of $L$ elements (for instance, for $x,y\in \Lambda$, we have then $x+y=[x+y]$ and $-x=[-x]$.) Similarly, $\Lambda^*$ is identified as a subset of the $d$-torus $\mathbb{T}^d$. We will use the short-hand notations \begin{equation}%\label{eq:} \int_{\Lambda^*} \textrm{d} k\, \cdots = \frac{1}{|\Lambda|} \sum_{k\in \Lambda^*} \cdots \, , \end{equation} and \begin{equation}\label{eq:2.5} \langle f,\psi\rangle=\sum_{x\in\Lambda} f(x)^\ast \psi(x)\, , \end{equation} as well as the similar but unrelated notation for ``regularized'' absolute values \begin{align} \sabs{x} =\sqrt{1+x^2}, \qquad \text{for all } x\in \R\, . \end{align} Let us also denote the limit $L\to \infty$ by $\Lambda\to\infty$. Let $\omega:\mathbb{T}^d\to\mathbb{R}$ be defined as in (\ref{eq:1.8}). For the finite volume, we introduce the periodized $\alpha_\Lambda$ through \begin{equation}%\label{eq:} \alpha_\Lambda(x) = \int_{\Lambda^*}\! \textrm{d} k \, \textrm{e}^{\textrm{i} 2\pi x \cdot k} \omega(k) = \frac{1}{|\Lambda|} \sum_{k\in \Lambda^*} \textrm{e}^{\textrm{i} 2\pi x\cdot k} \omega(k). \end{equation} Clearly $\alpha_\Lambda\in \mathbb{R}$ and $\alpha_\Lambda(-x)=\alpha_\Lambda(x)$ for all $x\in \Lambda$. After these preparations, we define the finite volume Hamiltonian for $\psi:\Lambda \to \mathbb{C}$ by \begin{align}%\label{eq:} & H_\Lambda(\psi) = \sum_{x,y\in \Lambda} \alpha_\Lambda(x-y) \psi(x)^* \psi(y) + \tfrac{1}{2} \lambda \sum_{x\in\Lambda} |\psi(x)|^4 \nonumber \\ & \quad = \int_{\Lambda^*} \textrm{d} k \, \omega(k) |\hat{\psi}(k)|^2 \nonumber \\ & \qquad + \tfrac{1 }{2}\lambda \int_{(\Lambda^*)^4}\!\!\! \textrm{d} k_1 \textrm{d} k_2 \textrm{d} k_3\textrm{d} k_4\, \delta_{\Lambda}(k_1+k_2-k_3-k_4) \hat{\psi}(k_1)^*\hat{\psi}(k_2)^*\hat{\psi}(k_3) \hat{\psi(}k_4)\,, \end{align} where $\lambda\ge 0$ and $\delta_{\Lambda}: (\mathbb{Z}/L)^d \to \mathbb{R}$ is the following discrete $\delta$-function \begin{equation}%\label{eq:} \delta_{\Lambda}(k) = |\Lambda| \mathbbm{1}(k \bmod 1 = 0) . \end{equation} Here $\1$ denotes a generic characteristic function: $\1(P)=1$, if the condition $P$ is true, and $\1(P)=0$ otherwise. $H_\Lambda(\psi)\ge c\|\psi\|^2_2$ for all $\psi$, with $c=\inf_k\omega(k)>-\infty$ and $\|\psi\|_2$ denoting the $\ell_2(\Lambda)$-norm. Introducing, as before, the canonical conjugate pair $q_x,p_x\in\R$, $\psi(x)=(q_x+\ci p_x)/\sqrt{2}$, and the evolution equations associated to $H_\Lambda$, we find that $\psi_t(x)$ satisfies the finite volume discrete NLS \begin{equation}\label{eq:dNLS} \textrm{i} \frac{\textrm{d} }{\textrm{d} t} \psi_t(x) = \sum_{y\in \Lambda} \alpha_\Lambda(x-y) \psi_t(y) %\nonumber \\ \quad + \lambda |\psi_t(x)|^2 \psi_t(x) \, . \end{equation} The Fourier-transform $\FT{\psi}_t(k)$ satisfies the evolution equation \begin{align}\label{eq:FTdNLS2} & \frac{\rmd }{\rmd t} \FT{\psi}_t(k_1) = -\ci \omega(k_1) \FT{\psi}_t(k_1) %\nonumber \\ & \qquad -\ci \lambda \int_{(\Lambda^*)^3}\!\! \rmd k_2 \rmd k_3\rmd k_4\, \delta_{\Lambda}(k_1+k_2-k_3-k_4) \FT{\psi}_t(k_2)^* \FT{\psi}_t(k_3) \FT{\psi}_t(k_4) . \end{align} The evolution equations have a continuously differentiable solution for all $t\in\R$ and for any given initial conditions $\psi_0\in \C^{\Lambda}$, which follows by a standard fixed point argument and the conservation laws stated below. The energy $H_\Lambda(\psi)$ is naturally conserved by the time-evolution. In addition, for all $x$, \begin{align}\label{eq:l2deriv} \frac{\rmd }{\rmd t} |\psi_t(x)|^2 = -\ci \sum_{y\in \Lambda} \alpha_\Lambda(x-y) \left(\psi_t(x)^* \psi_t(y)- \psi_t(y)^* \psi_t(x)\right) . \end{align} The right hand side sums to zero if we sum over all $x\in\Lambda$. Therefore, for $t\in\R$, \begin{align}\label{eq:normisconstant} \norm{\psi_t}_2^2=\sum_{x\in \Lambda} |\psi_t(x)|^2 = \norm{\psi_0}_2^2\, , \end{align} and thus also $\norm{\psi_t}_2$ is a constant of motion. The initial field is taken to be distributed according to the finite volume Gibbs measure as explained in the introduction. We assume $\beta,\mu$ to be fixed and drop the dependence on these parameters from the notation. Then the Gibbs measure is \begin{align}%\label{eq:} & \int_{\C^\Lambda} \mathbb{P}_{\Lambda}^\lambda(\rmd \psi) f(\psi) %\nonumber \\ & \quad = \frac{1}{Z_{\beta,\mu,\Lambda}^\lambda} \int_{(\R^2)^{\Lambda}} \prod_{x\in \Lambda} \left[\rmd(\re \psi(x))\, \rmd(\im \psi(x))\right] \rme^{-\beta (H_{\Lambda}(\psi)-\mu \norm{\psi}^2)} f(\psi) \, . \end{align} Expectation values with respect to the finite volume, perturbed measure $\mathbb{P}_{\Lambda}^\lambda$ are denoted by $\Elfin$. $\mathbb{P}_{\Lambda}^\lambda$ has a well defined infinite volume and zero coupling limit. The expectation values with respect to the corresponding infinite volume limit will be denoted by $\Elinf$. The zero coupling limit of these measures will be Gaussian and denoted by $\EG$. The covariance of the corresponding Gaussian measure has a Fourier transform \begin{align}%\label{eq:} W(k) = \FT{F}_2^0(k,0)= \frac{1}{\beta(\omega(k)-\mu)}\, . \end{align} Note that by the translation invariance of the finite volume Gibbs measure, there always exists a function $W^{\lambda}_\Lambda:\Lambda^*\to \C$ such that for all $k,k'\in \Lambda^*$, \begin{align}%\label{eq:} \E[\FT{\psi}(k)^* \FT{\psi}(k')] = \delta_\Lambda(k-k') W^{\lambda}_\Lambda(k)\, . \end{align} Since the energy and norm are conserved, the Gibbs measure is time stationary, in other words, for all $t$ and any integrable $f$ \begin{align}%\label{eq:} \Elfin[f(\psi_t)] = \Elfin[f(\psi_0)] . \end{align} In addition, since the dynamics and the Gibbs measure are invariant under periodic translations of $\Lambda$, under $\mathbb{P}^\lambda_\Lambda$ the stochastic process $(x,t)\mapsto \psi_t(x)$ is stationary jointly in space and time. \subsection{Main results}\label{sec:model} We have to impose two types of assumptions. Those in Assumption 2.2 are conditions on the dispersion relation $\omega$. Assumption 2.1 is concerned with a specific form of the clustering of the Gibbs measure. In each case we comment on their current status. \begin{assumption}[Equilibrium correlations]\label{th:Ainitcond} Let $\beta>0$ and $\mu < \inf_k\omega(k)$ be given. We take the initial conditions $\psi_0$ to be distributed according to the Gibbs measure $\mathbb{P}_{\Lambda}^\lambda$ which is assumed to be {\em $\ell_1$-clustering\/} in the following sense: We assume that there exists $\lambda_0 > 0$ and $c_0 > 0$, independent of $n$, such that for $0 <\lambda\le \lambda_0$ and all $n\ge 4$ one has the following bound for the fully truncated correlation functions (i.e., cumulants) \begin{align}\label{eq:l1clustering} \sup_{\Lambda,\sigma\in \set{\pm 1}^n} \sum_{x\in \Lambda^{n}} \delta_\Lambda(x_1) \Bigl|\Elfin\Bigl[\prod_{i=1}^n \psi(x_i,\sigma_i)\Bigr]^{\rm trunc}\Bigr| \le \lambda (c_0)^n n!\, , \end{align} where $\psi(x,1)=\psi(x)$, $\psi(x,-1)=\psi(x)^*$. We also assume a similar convergence of the two-point correlation function, \begin{equation}\label{eq:twopointlim} \limsup_{\Lambda\to \infty} \sum_{\norm{x}_\infty\le L/2} \left| \Elfin[\psi(0)^* \psi(x)] - \E^0[\psi(0)^* \psi(x)]\right| \le \lambda 2 (c_0)^2\,. \end{equation} \end{assumption} In the present proof for $d\ge 4$ we do not use the full strength of the bound in (\ref{eq:l1clustering}), namely, we could omit the prefactor $\lambda$. However, the prefactor could be needed in any proof which concerns $d\le 3$. Technically, Assumption 2.1 refers to the clustering of a weakly coupled massive two-com\-po\-nent $\lambda\phi^4$-theory. Such problems have a long tradition in equilibrium statistical mechanics and are handled through cluster expansions, e.g., see \cite{MM91}. Unfortunately, the result needed in our context does not seem to be available in the published literature. One reason is that often the harmonic coupling to the neighbors is used as expansion parameter, while for us the strength of the nonlinearity, $\lambda$, is small. A further difficulty is the required $n$-dependence of the above bound. The internal note \cite{APS} studies both (\ref{eq:l1clustering}) and (\ref{eq:twopointlim}). Very recently, Salmhofer \cite{Salm08} has established that the equilibrium correlations of lattice fermions interacting through a short range potential satisfy analogous assumptions. In fact, there are sign cancellations and his bound has no factor $n!$ as in (\ref{eq:l1clustering}). For the main theorem we will need properties of the linear dynamics, $\lambda=0$, which can be thought of as implicit conditions on $\omega$. \begin{assumption}[Dispersion relation]\label{th:disprelass} Suppose $d\ge 4$, and $\omega:\T^d\to\R$ satisfies all of the following: \begin{jlist}[(DR\thejlisti)] \item\label{it:DR1} The periodic extension of $\omega$ is real-analytic and $\omega(-k)=\omega(k)$. \item\label{it:DRdisp} ($\ell_3$-dispersivity). Let us consider the {\em free propagator\/} \begin{align}\label{eq:defptx} p_t(x) = \int_{\T^d} \!\rmd k\, \rme^{\ci 2\pi x\cdot k} \rme^{-\ci t \omega(k)} \, . \end{align} We assume that there are $C,\delta>0$ such that for all $t\in\R$, \begin{align}%\label{eq:} \norm{p_t}_3^3 = \sum_{x\in\Z^d} |p_t(x)|^3 \le C \sabs{t}^{-1-\delta} \, . \end{align} \item\label{it:DRinterf} (constructive interference). There exists a set $\Msing \subset \T^d$ consisting of a union of a finite number of closed, one-dimensional, smooth submanifolds, and a constant $C$ such that for all $t\in \R$, $k_0\in \T^d$, and $\sigma\in \set{\pm 1}$, \begin{align}%\label{eq:} \Bigl| \int_{\T^d}\!\rmd k\, \rme^{-\ci t (\omega(k)+\sigma \omega(k-k_0))}\Bigr| \le \frac{C\sabs{t}^{-1} }{d(k_0,\Msing)}\, . \end{align} where $d(k_0,\Msing)$ is the distance (with respect to the standard metric on the $d$-torus, $\R^d/\Z^d$) of $k_0$ from $\Msing$. \item\label{it:DRcrossing} (crossing bounds). Define for $t_0,t_1,t_2\in \R$, $u_1,u_2\in \T^d$, and $x\in \Z^d$, \begin{align}\label{eq:defp2tx} K(x;t_0,t_1,t_2,u_1,u_2) = \int_{\T^d} \!\rmd k\, \rme^{\ci 2\pi x\cdot k} \rme^{-\ci (t_0 \omega(k)+t_1 \omega(k+u_1)+ t_2 \omega(k+u_2))} \, . \end{align} We assume that there is a measurable function $\Fbcr:\T^d \times \R_+ \to [0,\infty]$ so that constants $0<\gamma\le 1$, $c_1,c_2$, for the following bounds can be found. \begin{enumerate} \item For any $u_i\in \T^d$, $\sigma_i\in \set{\pm 1}$, $i=1,2,3$, and $0<\beta\le 1$, the following bounds are satisfied: \begin{align} & \int_{-\infty}^\infty\! \rmd t\, \norm{p_{t}}_3^2 \int_{-\infty}^\infty\! \rmd s\, \rme^{-\beta |s|} \norm{K(t,\sigma_1 s,\sigma_2 s,u_1,u_2)}_3 \le \beta^{\gamma-1} \Fbcr(u_2-u_1;\beta) \, ; \label{eq:crossingest1c} \\ & \int_{-\infty}^\infty\! \rmd t \int_{-\infty}^\infty\! \rmd s\, \rme^{-\beta |s|} \prod_{i=1}^3 \norm{K(t,\sigma_i s,0,u_i,0)}_3 % \nonumber \\ & \quad \le \beta^{\gamma-1} \Fbcr(u_n;\beta),\quad \text{for any }n\in\set{1,2,3}\, . \label{eq:crossingest1b} \end{align} \item For all $0<\beta\le 1$ we have \begin{align}\label{eq:crossingest2a} & \int_{\T^d} \rmd k\, \Fbcr(k;\beta) \le c_1 \sabs{\ln \beta}^{c_2} , \end{align} and if also $u,k_0\in \T^d$, $\alpha\in \R$, $\sigma\in\set{\pm 1}$, and $n\in \set{1,2,3}$, and we denote $k=(k_1,k_2,k_0-k_1-k_2)$, then \begin{align}\label{eq:crossingest2} & \int_{(\T^d)^2} \rmd k_1\rmd k_2\, \Fbcr(k_n+u;\beta) \frac{1}{|\alpha-\Omega(k,\sigma)+\ci\beta|} %\nonumber \\ & \qquad\times \le c_1 \sabs{\ln \beta}^{1+c_2} , \end{align} where $\Omega:(\T^d)^3\times \set{\pm 1}\to \R$ is defined by \begin{align}\label{eq:defOmega} & \Omega(k,\sigma) = \omega(k_3)-\omega(k_1)+\sigma(\omega(k_2)-\omega(k_1+k_2+k_3))\, . \end{align} \end{enumerate} \end{jlist} \end{assumption} We prove in Appendix \ref{sec:appNN} that the nearest neighbor interactions satisfy all of the above assumptions for $d\ge 4$, at least with constants $\gamma=\frac{4}{7}$, $c_2=0$, and using the function \begin{align}\label{eq:defnnFcr} \Fbcr(u;\beta) = C \prod_{\nu=1}^d \frac{1}{|\sin (2 \pi u^\nu)|^{\frac{1}{7}}} \end{align} with a certain constant $C$ depending only on $d$ and $\omega$. Presumably a larger class of $\omega$'s could be covered, but this needs a separate investigation. We wish to inspect the decay of the space-time covariance on the kinetic time scale $t=\order{\lambda^{-2}}$. More precisely, given some test-functions $f,g\in \ell_2(\Z^d)$, with a compact support, we study the expectation of a quadratic form, \begin{align}%\label{eq:} \Elfin\bigl[\,\mean{f_\Lambda,\psi_0}^* \mean{g_\Lambda,\psi_{t/\vep}}\,\bigr]\, , \end{align} where $\vep=\lambda^2$, $f_\Lambda(x)= \sum_{n\in \Z^d} f(x+L n)$, and $g_\Lambda$ is obtained from $g$ similarly. Since we assume the test-functions to have a compact support, $f_\Lambda$ and $g_\Lambda$ are, in fact, independent of $\Lambda$ for all large enough lattice sizes. In addition, $\FT{f}_\Lambda(k)=\FT{f}(k)$, and $\FT{g}_\Lambda(k)=\FT{g}(k)$ for all $k\in \Lambda^*$. To get a finite limit, it will be necessary to cancel the rapidly oscillating factors. To this end, let us define \begin{align}%\label{eq:} \omla(k) = \omega(k) + \lambda R_0 \, , \end{align} where \begin{equation}\label{eq:defR0} R_0=R_0(\lambda,\Lambda)= 2 \Evepfin[|\psi_0(0)|^2]\,. \end{equation} Differentiating the expectation value and applying Assumption \ref{th:Ainitcond} shows that \begin{align}\label{eq:R0lim} \lim_{\Lambda\to\infty}R_0(\lambda,\Lambda) = 2\int_{\T^d}\rmd k\,W(k) \Bigl(1-2 \beta \lambda \int_{\T^d}\rmd k'\,W(k')^2\Bigr) + \order{\lambda^2}\, . \end{align} Then the task is to control the limit of the quadratic form \begin{align}\label{eq:cov} \Qlfin[g,f](t) = \Elfin\!\left[\mean{\FT{f},\FT{\psi}_0}^* \mean{\rme^{-\ci \omla t/\vep} \FT{g},\FT{\psi}_{t/\vep}}\right] , \quad \vep=\lambda^2\, . \end{align} \begin{theorem}\label{th:main} Consider the system described in Section \ref{sec:model} with an initial Gibbs measure satisfying Assumption \ref{th:Ainitcond} and a dispersion relation satisfying Assumption \ref{th:disprelass}. Then there is $t_0>0$ such that for all $|t|< t_0$, and for any $f,g\in \ell_2(\Z^d)$ with finite support, \begin{align}\label{eq:mainQlim} \lim_{\lambda\to 0} \limsup_{\Lambda\to \infty} \left| \Qlfin[g,f](t) - \int_{\T^d}\rmd k\, \FT{g}(k)^* \FT{f}(k) W(k) \rme^{-\Gamma_1(k)|t|-\ci t \Gamma_2(k)}\right| = 0\, , \end{align} where $\Gamma_j(k)$ are real, and $\Gamma(k)=\Gamma_1(k)+\ci \Gamma_2(k)$ is given by \begin{align}\label{eq:defGamma} & \Gamma(k_1) = -2 \int_0^\infty \!\rmd t \int_{(\T^d)^3} \rmd k_2 \rmd k_3 \rmd k_4 \delta(k_1+k_2-k_3-k_4) \nonumber \\ & \quad \times \rme^{\ci t (\omega_1+\omega_2-\omega_3-\omega_4)} \left( W_3 W_4 - W_2 W_4 - W_2 W_3 \right) \end{align} with $\omega_i = \omega(k_i)$, $W_i= W(k_i)$. \end{theorem} In fact, we expect that the infinite volume limit of $\Qlfin[g,f](t)$ exists, but since proving this would have been a diversion from our main results, we have stated the main theorem in a form which does not need this property. Clearly, if the limit does exist, then (\ref{eq:mainQlim}) implies the stronger result \begin{align}%\label{eq:mainQlim2} \lim_{\lambda\to 0} \lim_{\Lambda\to \infty} \Qlfin[g,f](t) = \int_{\T^d}\rmd k\, \FT{g}(k)^* \FT{f}(k) W(k) \rme^{-\Gamma_1(k)|t|-\ci t \Gamma_2(k)}\, . \end{align} We point out that $\Gamma_1(k)\ge 0$, as by explicit computation \begin{align}\label{eq:Gamma2} & \Gamma_1(k_1) = 2\pi W(k_1)^{-2} \int_{(\T^d)^3} \rmd k_2 \rmd k_3 \rmd k_4 \delta(k_1+k_2-k_3-k_4) \nonumber \\ & \qquad \times \delta(\omega_1+\omega_2-\omega_3-\omega_4) \prod_{i=1}^4 W(k_i) \, . \end{align} (We prove in Section \ref{sec:freeint} that the integral in (\ref{eq:defGamma}) and the positive measure in (\ref{eq:Gamma2}) are well-defined for any $\omega$ satisfying Assumption \ref{th:disprelass}.) If $\Gamma_{1}(k) > 0$, then the term $\exp[-\Gamma_{1}(k)|t|]$ yields the exponential damping in $|t|$, both forward and backwards in time, and if $\Gamma_{1}(k) \geq \gamma >0$ for all $k\in \T^d$, then on the kinetic scale the covariance has an exponential bound $\mathrm{e}^{-\gamma|t|}$. \subsection{Link to kinetic theory}\label{sec:link} To briefly explain the connection of our result to the kinetic theory for weakly nonlinear wave equations, we assume that the initial data $\psi(x)$, $x\in\mathbb{Z}^d$, are distributed according to a Gaussian measure, $\mathbb{P}_\mathrm{G}$, with mean zero and covariance \begin{equation}\label{1} \mathbb{E}_\mathrm{G}\big(\psi(y)^\ast\psi(x)\big)=\int_{\mathbb{T}^d} \rmd k\, h^0(k) \rme^{\ci 2\pi k\cdot(x-y)}\,,\quad \mathbb{E}_\mathrm{G}\big(\psi(y)\psi(x)\big)=0\,. \end{equation} $\mathbb{P}_\mathrm{G}$ is stationary under the $\lambda=0$ dynamics, but nonstationary for $\lambda>0$. Since translation and gauge invariance are preserved in time, necessarily \begin{equation}\label{2} \mathbb{E}_\mathrm{G}\big(\psi_t(y)^\ast\psi_t(x)\big)=\int_{\mathbb{T}^d} \rmd k\, h_\lambda(k,t) \rme^{\ci 2\pi k\cdot(x-y)}\,,\quad \mathbb{E}\big(\psi_t(y)\psi_t(x)\big)=0\,. \end{equation} The central claim of kinetic theory is the existence of the limit \begin{equation}\label{3} \lim_{\lambda\to 0} h_\lambda(k,\lambda^{-2}t)=h(k,t)\,, \end{equation} where $h(t)$ is the solution of the spatially homogeneous kinetic equation \begin{equation}\label{4} \frac{\partial}{\partial t} h(k,t)=\mathcal{C}\big(h(\cdot,t)\big)(k) \end{equation} with initial conditions $h(k,0)=h^0(k)$. The collision operator, $\mathcal{C}$, is given by \begin{align}\label{5} & \mathcal{C}(h)(k_1) = 4\pi\int_{(\T^{d})^3} \rmd k_2 \rmd k_3 \rmd k_4 \delta(k_1+k_2-k_3-k_4) \delta (\omega_1+\omega_2-\omega_3-\omega_4)\nonumber \\ & \qquad\qquad\qquad\quad \times\left(h_2 h_3 h_4 + h_1 h_3 h_4 - h_1 h_2 h_3 - h_1 h_2 h_4\right) \end{align} with $h_j$ shorthand for $h(k_j)$, $j=1,2,3,4$. The proof of the limit (\ref{3}) remains as mathematical challenge. For equilibrium expectations of the form $\mathbb{E}^\lambda_{\beta,\mu}(\psi^\ast_t \psi_t \psi^\ast_0 \psi_0)$ it is conjectured that they are governed by the linearization of (\ref{4}) at the stationary solution $h^\mathrm{eq}(k)= (\beta(\omega(k)-\mu))^{-1}$, c.f., \cite{spohn05}. In this paper we study $\mathbb{E}^\lambda_{\beta,\mu}(\psi^\ast_t \psi_0)$, for which only the linearization of the loss term, i.e., of $h_1 (h_3 h_4 - h_2 h_3 - h_2 h_4)$, at $h_1$ relative to $h^\mathrm{eq}$ is needed. In addition, only ``half'' of the energy conservation shows up: instead of \begin{equation}\label{6} \int_{-\infty}^\infty \rmd t\, \rme^{\ci t (\omega_1+ \omega_2 - \omega_3 - \omega_4)} = 2\pi\delta (\omega_1+\omega_2-\omega_3-\omega_4)\, , \end{equation} only \begin{equation}\label{7} \int^\infty_0 \rmd t\, \rme^{\ci t (\omega_1+ \omega_2 - \omega_3 - \omega_4)} \end{equation} appears in the definition of the decay rate for $\mathbb{E}^\lambda_{\beta,\mu}(\psi_t^\ast \psi_0)$, compare with Equation (\ref{eq:defGamma}). \subsection{Restriction to times $t>0$} \label{sec:firstpf} From now on we assume that Assumptions \ref{th:Ainitcond} and \ref{th:disprelass} are satisfied. We begin by showing that then it is sufficient to prove the theorem under the assumption $t>0$. For simplicity, let us denote $\E= \Elfin$ and $F_2= F^\lambda_2$, i.e., we define \begin{align}%\label{eq:} F_2(x,t) = \E[\psi_0(0)^* \psi_t(x)], \quad x\in \Lambda,\ t\in \R\,. \end{align} In order to study the infinite volume limit $\Lambda\to \Z^d$, we define the natural ``cell step function'' $\tdfloor{k}:\R^d\to\Lambda^*$ by setting $\tdfloor{k}_i$ equal to $\lfloor L (k_i \bmod 1)\rfloor /L$. Since $\tdfloor{k}$ is periodic, we can also identify it with a map $\T^d\to \Lambda^*$. Clearly, for any $F:\Lambda^* \to \C$ we can then apply the following obvious formula relating the discrete sum over $\Lambda^*$ and a Lebesgue integral: \begin{align}\label{eq:LamtoLeb} \int_{\Lambda^*}\rmd k\, F(k) = \int_{\T^d}\rmd k\, F(\tdfloor{k}) \, , \end{align} where $F(\tdfloor{k})$ is a piecewise constant ``step function'' on $\T^d$. Now if $F_\Lambda$ is any sequence of functions $\Lambda^* \to \C$ such that $F_\Lambda([k])$ converges on $\T^d$ to $F$, and $\sup_{\Lambda}\sup_{k\in\Lambda^*} |F_\Lambda(k)|<\infty$, then by dominated convergence, we have \begin{align}%\label{eq:} \lim_{\Lambda\to \infty} \int_{\Lambda^*}\rmd k\, F_\Lambda(k) = \int_{\T^d}\rmd k\, F(k) \, . \end{align} At $t=0$, $\FT{F}_2(k,0)=W^{\lambda}_\Lambda(k)$, for $k\in \Lambda^*$. Combined with (\ref{eq:cov}) for $t=0$ and the smoothness of $\FT{f},\FT{g}$, the following Lemma implies that \begin{align}%\label{eq:} \lim_{\lambda\to 0} \limsup_{\Lambda\to \infty}\left| \Qlfin[g,f](0) - \int_{\T^d}\rmd k\, \FT{g}(k)^* \FT{f}(k) W(k)\right|=0\, , \end{align} and, therefore, the main theorem holds at $t=0$. \begin{lemma}\label{th:unifW2} For all $0<\lambda\le \lambda_0$, \begin{align}%\label{eq:} \limsup_{\Lambda\to \infty}\sup_{k\in \T^d}\left| W^{\lambda}_\Lambda([k])-W(k)\right|\le 2c_0^2 \lambda \, . \end{align} \end{lemma} \begin{proof} Since our conditions on $\beta$ and $\mu$ imply that \begin{align}%\label{eq:} W(k) = \frac{1}{\beta(\omega(k)-\mu)} = \sum_{x\in \Z^d} \rme^{-\ci 2\pi x \cdot k} \EG\bigl[\psi(0)^*\psi(x)\bigr] \end{align} is smooth, its inverse Fourier transform $\IFT{W}(x)=\EG\bigl[\psi(0)^*\psi(x)\bigr]$ belongs to $\ell_1(\Z^d)$. Now for any $k\in \T^d$ \begin{align}%\label{eq:} & |W^{\lambda}_\Lambda([k])-W(k)| \le |W(k)-W([k])| \nonumber \\ & \qquad + \sum_{\norm{x}_\infty \ge L/2}\!\!\! |\IFT{W}(x)| + \sum_{\norm{x}_\infty \le L/2} \Bigl|\Elfin\bigl[\psi(0)^*\psi(x)\bigr] - \EG\bigl[\psi(0)^*\psi(x)\bigr] \Bigr| \, , \end{align} and the second part of Assumption \ref{th:Ainitcond} implies that the Lemma holds. \end{proof} The initial state is invariant under periodic translations of the lattice. Since the time evolution also commutes with these translations, we have \begin{align}%\label{eq:} \E[\psi_0(x_0)^* \psi_t(x)] = F_2(x-x_0,t)\, , \end{align} and thus for $k,k'\in \Lambda^*$, \begin{align}%\label{eq:} \E[\FT{\psi}_0(k')^* \FT{\psi}_t(k)] = \delta_{\Lambda}(k'-k) \FT{F}_2(k,t)\, . \end{align} Therefore, \begin{align}\label{eq:Qlformula} \Qlfin[g,f](t) = \int_{\Lambda^*}\rmd k\, \FT{g}(k)^* \FT{f}(k) \rme^{\ci \omla (k) t/\vep} \FT{F}_2(k,t/\vep)\, . \end{align} In addition, since the initial measure is stationary and the process fully translation invariant, we have \begin{align}\label{eq:C2inv} F_2(-x,-t)^* = \E[\psi_0(0) \psi_{-t}(-x)^*] = \E[\psi_0(x) \psi_{-t}(0)^*] = F_2(x,t) \, , \end{align} and thus \begin{align}\label{eq:FTC2inv} \FT{F}_2(k,-t)^* = \FT{F}_2(k,t) \, . \end{align} Applied to (\ref{eq:Qlformula}) this implies that, in fact, \begin{align}\label{eq:Qlinvrel} \Qlfin[g,f](-t)^* = \Qlfin[f,g](t) \, . \end{align} Let us assume that the main theorem has been proven for $t>0$. Then for any $-t_00$. This will be done in the following sections. \section{Duhamel expansion} \label{sec:graphs} From now on, let $d\ge 4$ and $t>0$ be given and fixed. We also denote $\E= \Elfin$, as before. We will now describe how we plan to do the expansion of the time-correlations into a sum over amplitudes determined by graphs. We begin from the Fourier transformed evolution equations, (\ref{eq:FTdNLS2}). Constructive interference turns out to be a problem for the perturbation expansion, and we have to treat the wave numbers near the ``singular'' manifold $\Msing$ differently from the rest. To this end, we introduce a cutoff function $\PFzero:(\T^d)^3\to [0,1]$ which is smooth, depends on $\lambda$, and is zero apart from a small neighborhood of $\Msing$. Given such a function let us denote $\PFone=1-\PFzero$. We will construct in Section \ref{sec:cutoff} a function $\PFzero$ and find a constant $\lambda'_0>0$ such that the following Proposition holds with the choice $b=\frac{3}{4}$. \begin{proposition}\label{th:PFcorr} Suppose $k\in (\T^d)^3$ and $0<\lambda<\lambda'_0$. Then for any pair of indices $i\ne j$, $i,j\in \set{1,2,3}$, all of the following hold: \begin{enumerate} \item If $k_i+k_j=0$, $\PFone(k)=0$ and $\PFzero(k)=1$. \item $0\le \PFone (k)\le C_1 \lambda^{-b} d(k_i+k_j,\Msing)$. \end{enumerate} In addition, $\PFone(k_3,k_2,k_1)=\PFone(k_1,k_2,k_3)$, $\PFzero(k_3,k_2,k_1)=\PFzero(k_1,k_2,k_3)$, and \begin{align}\label{eq:PFzeroineq} 0\le \PFzero (k)\le \sum_{i,j=1; i0$, the first two terms can be combined to get a formula similar to that given in Theorem 4.3 in \cite{ls05}, \begin{align}\label{eq:softpiold} & \rme^{-\kappa t} \prod_{i=1}^n \FT{a}_0(k_i,\sigma_i) + \kappa \int_0^t\!\rmd s\, \rme^{-(t-s)\kappa} \prod_{i=1}^n \FT{a}_s(k_i,\sigma_i) %\nonumber \\ & \quad = \kappa \int_0^\infty \!\rmd r\, \rme^{-r \kappa} \prod_{i=1}^n \FT{a}_{(t-r)_+}(k_i,\sigma_i) \end{align} with $(r)_+=r$, if $r\ge 0$, and $(r)_+=0$, if $r< 0$. We now iterate this formula for $N_0\ge 1$ times, using it \defem{only} in the term containing $\PFone$, the complement of the cutoff function. Then at each iteration step we get three new terms, one depending only on the initial field, $a_0$, one coming from the remainder of the partial time integration and one containing the cutoff function $\PFzero$. Explicitly, this yields for any $\kappa\in \R_+^{\set{0,1,\ldots,N_0-1}}$ an expansion \begin{align}\label{eq:mainaiter} & \FT{a}_t(k,\sigma) = \sum_{n=0}^{N_0-1} \mathcal{F}_n(t,k,\sigma,\kappa)[\FT{a}_0] + \sum_{n=0}^{N_0-1} \kappa_n \int_0^t\! \rmd s\, \mathcal{G}_{n}(s,t,k,\sigma,\kappa)[\FT{a}_s] \nonumber \\ & \quad + \sum_{n=1}^{N_0} \int_0^t\! \rmd s\, \mathcal{Z}_n(s,t,k,\sigma,\kappa)[\FT{a}_s] + \int_0^t\! \rmd s\, \mathcal{A}_{N_0}(s,t,k,\sigma,\kappa)[\FT{a}_s] . \end{align} Each of the functionals is a polynomial of $\FT{a}_s$, for some time $s$, and their structure can be encoded by diagrams whose construction is described next. For given $n, n'$, with $0\le n \le n'$, we first define the index sets $I_n=\set{1,2,\ldots,n}$ and $I_{n',n}=\set{n',n'+1,\ldots,n}$. For further use, let $m_0\ge 1$ denote the number of fields at the final time $t$ (in the above case of $\FT{a}_t$ we thus set $m_0=1$). Also, let $N\ge 0$ denote the total number of interactions, i.e., iterations of the Duhamel formula. A term with $N$ interactions has the total time $t$ divided into $N+1$ ``time-slices'' of length $s_i$, $i=0,1,\ldots,N$, labeled in their proper time-order (from bottom to top in the diagram). Associated with a time-slice $i$ there are total $m_{N-i}$ ``momentum integrals'' over $\Lambda^*$, where $m_n=m_0+2 n$. We label the momenta by $k_{i,j}$ and associate a line segment in the diagram to each of them. The interactions are denoted by an interaction vertex. Each interaction vertex thus contains a $\delta_\Lambda$-function which enforces the momenta below the vertex (belonging to an earlier time-slice) to sum up to the momenta above the vertex. The momenta not involved in an interaction are continued unchanged from one time-slice to the next. Thus a natural way of representing the line segments is to connect them into straight lines passing through several time-slices until they encounter an interaction vertex at which three such lines fuse into one new momentum line. For this reason, we will call the interactions \defem{fusions} from now on. To summarize the notations, the fusion number $1$, denoted by an interaction vertex $v_1$, happens after time $s_0$ which is the length of the time-slice number $0$, fusion $2$ happens after time $s_0+s_1$, where $s_1$ is the length of the time-slice $1$, etc. In general, fusion $i$ happens in the beginning of the slice $i$. For each time-slice $i\in I_{0,N}$ we label the momenta by $k_{i,j}$, $j=1,\ldots,m_{N-i}$. Similar labeling is used for the ``parity'' $\sigma_{i,j}\in \set{\pm 1}$. The structure of interactions is such that the parity of each line is uniquely determined by the parities of the final lines. In our diagrams, we use the order implicit in (\ref{eq:kappaDuh}): the parities of the fusing line-segments are required to appear in the order $(-1,\sigma,+1)$, and then the parity of the \defem{middle} line will be carried on to determine the parity resulting in a fusion. Figure \ref{fig:indexgr} illustrates these definitions. \begin{figure} \centering \myfigure{width=0.9\textwidth}{DiagramC} \caption{Two examples of interaction diagrams with $N=2$ interactions and $m_0=1$ final fields. In the left diagram we have indicated the notations used for time-slices and interaction vertices. In the right, the parity of each line is shown, assuming that the final line has parity $\sigma$, as well as some of the notations used for momenta associated with line-segments on each time-slice. The ``interaction history'' of the diagram is $\ell=(3,1)$ on the left and $\ell=(2,1)$ on the right.\label{fig:indexgr}} \end{figure} Let $\mathcal{I}_{n;m_0} = \defset{(i,j)}{0\le i\le n-1, 1\le j \le m_0+2(n-i)} \subset I_{0,n-1}\times I_{m_0+2n}$. Then the set $\mathcal{I}_{N;m_0}$ collects all index pairs associated with momentum line-segments, excluding the final time-slice with $i=N$. We also employ the shorthand notation $\mathcal{I}_{n}$ for $\mathcal{I}_{n;1}$. We use a vector $\ell$ to define the interaction history by collecting, for every time-slice with $i\ge 1$, the index of the new line formed in the fusion at the beginning of the slice. Then $\ell\in G_N$, with $G_N= I_{m_{N-1}}\times I_{m_{N-2}}\times \cdots \times I_{m_{0}}$, and let also $G_0=\emptyset$. By the earlier explained procedure, the indices in each time slice are matched so that the indices made vacant by the fusion are filled by shifting the indices following the fusion line down by two. This corresponds to labeling the momenta in each time-slice by counting them from left to right in the natural graphical representation of the interaction history, where the lines intersect only at interaction vertices. (See Figure \ref{fig:indexgr} for an illustration.) Explicitly, we have $\mathcal{F}_0(t,k,\sigma,\kappa)[\FT{a}]=\rme^{-\kappa_0 t} \FT{a}(k,\sigma)$ and, for $n> 0$, \begin{align}\label{eq:defFn} & \mathcal{F}_n(t,k_{n1},\sigma_{n1},\kappa)[\FT{a}] \nonumber \\ & = (-\ci \lambda)^n \sum_{\ell \in G_n} \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}_{n}}} \int_{(\Lambda^*)^{\mathcal{I}_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma;\Lambda) \prod_{j=1}^{m_0+2 n} \FT{a}(k_{0,j},\sigma_{0,j}) % \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] \nonumber \\ & \quad \times \int_{(\R_+)^{I_{0,n}}}\!\rmd s \, \delta\Bigl(t-\sum_{i=0}^{n} s_i\Bigr) \prod_{i=0}^{n} \rme^{-s_i \kappa_{n-i}} \prod_{i=1}^{n} \rme^{-\ci t_i(s) \Omega_{i-1;\ell_i}(k,\sigma)} \, , \end{align} where $t_i(s) = \sum_{j=0}^{i-1} s_j$ is the time needed to reach the beginning of the slice $i$, and \begin{align}%\label{eq:} & k_{i;j} = (k_{i,j},k_{i,j+1},k_{i,j+2})\in (\T^d)^3\, , \\ & \Omega_{i;j}(k,\sigma) = \Omega(k_{i;j},\sigma_{i+1,j}), \end{align} and $\Delta_{n,\ell}$ contains $\delta$-functions restricting the integrals over $k$ and $\sigma$ to coincide with the interaction history defined by $\ell$, as described above. Explicitly, \begin{align}%\label{eq:} &\Delta_{n,\ell}(k,\sigma;\Lambda) = \prod_{i=1}^{n} \Bigl\{ \prod_{j=1}^{\ell_i-1} \Bigl[ \delta_\Lambda(k_{i,j}-k_{i-1,j}) \1(\sigma_{i,j}=\sigma_{i-1,j}) \Bigr] \nonumber \\ & \qquad \times \delta_\Lambda\Bigl(k_{i,\ell_{i}}-\sum_{j=0}^{2} k_{i-1,\ell_i+j} \Bigr) \prod_{j=\ell_i+1}^{m_{n-i}} \Bigl[ \delta_\Lambda(k_{i,j}-k_{i-1,j+2}) \1(\sigma_{i,j}=\sigma_{i-1,j+2}) \Bigr] \nonumber \\ & \qquad \times \1(\sigma_{i-1,\ell_{i}}=-1) \1(\sigma_{i-1,\ell_{i}+1}=\sigma_{i,\ell_i}) \1(\sigma_{i-1,\ell_{i}+2}=+1) \Bigr\} \, . \end{align} The remaining terms are very similar. For $n\ge 1$, \begin{align}%\label{eq:} & \mathcal{A}_n(s_0,t,k_{n1},\sigma_{n1},\kappa)[\FT{a}] \nonumber \\ & = (-\ci \lambda)^n \sum_{\ell \in G_n} \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}_{n}}} \int_{(\Lambda^*)^{\mathcal{I}_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma;\Lambda) \prod_{j=1}^{m_0+2 n} \FT{a}(k_{0,j},\sigma_{0,j}) % \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] \nonumber \\ & \quad \times \int_{(\R_+)^{I_{n}}}\!\rmd s \, \delta\Bigl(t-s_0-\sum_{i=1}^{n} s_i\Bigr) \prod_{i=1}^{n} \rme^{-s_i \kappa_{n-i}} \prod_{i=1}^{n} \rme^{- \ci t_i(s) \Omega_{i-1;\ell_i}(k,\sigma)} \, . \end{align} We define $\mathcal{G}_{0}(s,t,k,\sigma,\kappa)[\FT{a}] =\rme^{-(t-s)\kappa_0}\FT{a}(k,\sigma)= \mathcal{F}_0(t-s,k,\sigma,\kappa)[\FT{a}]$, and for $n> 0$, \begin{align}\label{eq:defGnviaA} \mathcal{G}_{n}(s,t,k,\sigma,\kappa)[\FT{a}] =\int_0^{t-s}\!\rmd r\, \rme^{-r\kappa_n} \mathcal{A}_{n}(s+r,t,k,\sigma,\kappa)[\FT{a}], \end{align} and, finally, \begin{align}%\label{eq:} & \mathcal{Z}_n(s_0,t,k_{n1},\sigma_{n1},\kappa)[\FT{a}] = (-\ci \lambda)^n \sum_{\ell \in G_n} \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}_{n}}} \int_{(\Lambda^*)^{\mathcal{I}_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma;\Lambda) \nonumber \\ & \ \times \prod_{j=1}^{\ell_1-1} \FT{a}(k_{0,j},\sigma_{0,j}) \Ptrunc\Bigl[ \prod_{j=\ell_1}^{\ell_1+2} \FT{a}(k_{0,j},\sigma_{0,j}) \Bigr] \prod_{j=\ell_1+3}^{m_0+2 n} \FT{a}(k_{0,j},\sigma_{0,j}) % \nonumber \\ & \ \times \sigma_{1,\ell_1} \PFzero (k_{0;\ell_1}) \prod_{i=2}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] \nonumber \\ & \ \times \int_{(\R_+)^{I_{n}}}\!\rmd s \, \delta\Bigl(t-s_0-\sum_{i=1}^{n} s_i\Bigr) \prod_{i=1}^{n} \rme^{-s_i \kappa_{n-i}} \prod_{i=1}^{n} \rme^{-\ci t_i(s) \Omega_{i-1;\ell_i}(k,\sigma)} \, . \end{align} Using these definitions, the validity of (\ref{eq:mainaiter}) can be proven by induction in $N_0$, applying (\ref{eq:kappaDuh}) to $\mathcal{A}_{N_0}$ in (\ref{eq:mainaiter}). For later use, let us point out that the total oscillating phase factor in the above formulae can also be written as \begin{align}\label{eq:phase2} \prod_{i=1}^{n} \rme^{-\ci t_i(s) \Omega_i} = \prod_{j=0}^{n-1} \exp\Bigl[-\ci s_j \sum_{i=j+1}^n \Omega_i\Bigr], \quad \Omega_i = \Omega_i(\ell,k,\sigma) = \Omega_{i-1;\ell_i}(k,\sigma) \, . \end{align} Applying the expansion in (\ref{eq:Qwitha}) proves that the following result holds. \begin{proposition}\label{th:Qmainerr} For any $N_0\ge 1$ and for any choice of $\kappa\in \R_+^{I_{0,N_0-1}}$, we have \begin{align}\label{eq:pexpmainterm} & \Qlfin[g,f](t) = \Qmain + Q^{\rm err}_{\rm pti}+ Q^{\rm err}_{\rm cut} + Q^{\rm err}_{\rm amp} \, , \end{align} where \begin{align}\label{eq:Qmaindef} \Qmain = \int_{(\Lambda^*)^2}\!\! \rmd k\rmd k'\, \FT{g}(k)^* \FT{f}(-k') \sum_{n=0}^{N_0-1} \E\!\left[\FT{\psi}_0(k',-1)\mathcal{F}_n(t/\vep,k,1,\kappa)[\FT{\psi}_0] \right] \end{align} and the error terms are given by \begin{align}%\label{eq:pexpmainterm} & Q^{\rm err}_{\rm pti} = \sum_{n=0}^{N_0-1} \kappa_n \int_0^{t/\vep}\! \rmd s\, \E\Bigl[\mean{\FT{f},\FT{a}_0}^* \int_{\Lambda^*}\!\! \rmd k\, \FT{g}(k)^* \mathcal{G}_{n}(s,t/\vep,k,1,\kappa)[\FT{a}_s]\Bigr], \\ & Q^{\rm err}_{\rm cut} = \sum_{n=1}^{N_0} \int_0^{t/\vep}\! \rmd s\, \E\Bigl[\mean{\FT{f},\FT{a}_0}^* \int_{\Lambda^*}\!\! \rmd k\, \FT{g}(k)^* \mathcal{Z}_{n}(s,t/\vep,k,1,\kappa)[\FT{a}_s]\Bigr], \\ & Q^{\rm err}_{\rm amp} = \int_0^{t/\vep}\! \rmd s\, \E\Bigl[\mean{\FT{f},\FT{a}_0}^* \int_{\Lambda^*}\!\! \rmd k\, \FT{g}(k)^* \mathcal{A}_{N_0}(s,t/\vep,k,1,\kappa)[\FT{a}_s]\Bigr] \, . \end{align} \end{proposition} \subsection{Structure of the proof} \label{sec:structureofproof} We have now derived a time-evolution equation for arbitrary moments of the field, and constructed a related Duhamel expansion of our observable. Already at this stage we had to introduce certain additional structure compared to the standard Duhamel formula. Certain regions of wavenumbers are treated differently, in order to control ``bad'' constructive interference effects. In addition, we have introduced an artificial exponential decay for partial time integration which will be used to amplify decay estimates which are too weak to be used in the error estimates. The terms in this expansion either contain only finite moments of the initial fields, or after relying on stationarity of the initial measure, can be bounded by such moments. We will employ our assumption about the strong clustering properties of the initial measure to turn the moments into cumulants whose analysis in the Fourier-space will result only in additional ``Kirchhoff rules'' on the initial time-slice. The expectation values can then be expressed as a sum over graphs encoding the various possible momentum- and time-dependencies of the integrand. The construction of the graphs will be explained in Section \ref{sec:diagrams} We will then derive a certain, essentially unique, way to resolve all the momentum dependencies dictated by a graph, see Section \ref{sec:momdeltas}. After this, it will be a modest step to show that the limit $\Lambda\to \infty$ in essence corresponds to replacing the discrete sums over $\Lambda^*$ by integrals over $\T^d$. The resulting graphs can then be classified, in the spirit of \cite{erdyau99}, by identifying in most of them certain integrals with oscillating factors which produce additional decay compared to the leading graphs. Here the idea is first to identify all ``motives'' which make the phase factors to vanish identically in every second time-slice, while the remaining time-slices are forced to have a subkinetic length due to the oscillating phases. These correspond to immediate recollisions in the language of the earlier works, and repetitions of these motives yield the leading term graphs. Other graphs will be subleading either because they contain additional $k$-integrals, or because the $k$-integrals overlap in such a way that additional time-slices can be proven to have a subkinetic length. As before, the overlap needs to be controlled in several different fashions to find the appropriate mechanism for decay. This results in a classification of these graphs into partially paired, nested, and crossing graphs. The control of the three different types of remainder terms can be accomplished by slight modifications of the estimates used for the main term. The limit of the sum of the leading graphs is then shown to coincide with the expression given in the main theorem. The precise choice of expansion parameters, as well as a preliminary classification of the graphs, will be given in Section \ref{sec:classification}. After establishing the main technical lemmata in Section \ref{sec:lemmas}, we derive the various estimates in two parts. In Section \ref{sec:higherorder} we consider higher order effects and the infinite volume limit. Pairing graphs can only be treated after taking $\Lambda\to \infty$, and their analysis is given in Section \ref{sec:fullypaired}. Combined, the various estimates yield the result stated in Theorem \ref{th:main}, as is shown in Section \ref{sec:completion}. \section{Diagrammatic representation}\label{sec:diagrams} In this section, we derive diagrammatic representations related to the terms in Proposition \ref{th:Qmainerr}. For the main terms summing to $\Qmain$ the representation describes the value of the term, whereas for the error terms, the representation is a contribution to an upper bound of the term. The representations arise since we are able to derive upper bounds which depend only on moments of the initial fields. We first recall a standard result which relates moments to truncated correlation functions of the time zero fields. \subsection{Initial time clusters from a cumulant expansion} \label{sec:cumulants} Since $a_0(x)=\psi_0(x)$, the conditions for initial fields imply $\EG[\FT{a}_0(k,\sigma)]=0$, and \begin{align}%\label{eq:} \EG[\FT{a}_0(k,\sigma) \FT{a}_0(k',\sigma')] = \1(\sigma+\sigma'=0) \delta(k+k') W(k)\, , \end{align} for $\sigma',\sigma \in \set{\pm 1}$. However, this formula is correct only after taking the infinite volume and the weak coupling limit. Before taking these limits there will be corrections to the cumulants. We describe here how this works out explicitly. \begin{definition}\label{th:defPiI} For any finite, non-empty set $I$, let $\pi(I)$ denote the set of 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} Given $n\in \N$, we define for $k\in (\Lambda^*)^n$, $\sigma\in \set{\pm 1}^n$, the truncated correlation function in Fourier-space as \begin{align}%\label{eq:} C_n(k,\sigma;\lambda,\Lambda) := \sum_{x\in \Lambda^{n}} \delta_\Lambda(x_1) \rme^{-\ci 2\pi \sum_{i=1}^{n} x_i\cdot k_i} \Elfin\Bigl[\prod_{i=1}^n \psi(x_i,\sigma_i)\Bigr]^{\rm trunc} \, . \end{align} An immediate consequence of the gauge invariance of the measure is that if $\sum_{i=1}^n \sigma_i\ne 0$, then $C_n = 0$. In particular, all odd cumulant functions vanish. By Assumption \ref{th:Ainitcond}, apart from $n=2$, the functions for all other even $n$ have uniform bounds \begin{align}\label{eq:Cnbound} | C_n(k,\sigma;\lambda,\Lambda)| \le \lambda (c_0)^n n!\, . \end{align} For $n=2$, we have more explicitly \begin{align}%\label{eq:} C_2(k,\sigma;\lambda,\Lambda) = \sum_{x\in \Lambda} \rme^{-\ci 2\pi x \cdot k_2} \Elfin\!\left[\psi(0,\sigma_1)\psi(x,\sigma_2)\right] \, . \end{align} Thus, if $\sigma_1=\sigma_2$, $C_2=0$, and clearly also $C_2(k,(1,-1))=C_2(-k,(-1,1))^*$. A comparison with the definition of $W^{\lambda}_\Lambda$ shows that $C_2((k',k),(-1,1))=W^{\lambda}_\Lambda(k)$. Thus $C_2((k',k),(1,-1))=W^{\lambda}_\Lambda(-k)^*$ which is equal to $W^{\lambda}_\Lambda(k)$ since the initial Gibbs measure is invariant also under the reflection $x\to -x$. By Lemma \ref{th:unifW2}, there is a constant $c_0'$ such that $|W^{\lambda}_\Lambda(k)|\le c_0'$, which, apart from the factor $\lambda$, yields a generalization of the bound (\ref{eq:Cnbound}) for $n=2$. The cumulant functions are of interest since they allow expanding moments in terms of uniformly bounded functions via the following general result. \begin{lemma}\label{th:cumulants} For any index set $I$, and any $k\in (\T^d)^I$, $\sigma\in \set{\pm 1}^I$, \begin{align}%\label{eq:} & \Elfin\Bigl[\prod_{i\in I} \FT{\psi}(k_i,\sigma_i)\Bigr] %\nonumber \\ & \quad = \sum_{S\in \pi(I)} \prod_{A\in S} \Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} k_i\Bigr) C_{|A|}(k_A,\sigma_{\!A};\lambda,\Lambda) \Bigr] \, , \end{align} where the sum runs over all partitions $S$ of the index set $I$, and the shorthand notations $k_A$ and $\sigma_{\!A}$ refer to $(k_a)_{a\in A}\in (\T^d)^A$ and $(\sigma_a)_{a\in A}\in \set{\pm 1}^A$. \end{lemma} \begin{proof} Let $\E=\Elfin$. We need to study \begin{align}%\label{eq:} & \E\Bigl[\prod_{i\in I} \FT{\psi}(k_i,\sigma_i)\Bigr] = \sum_{x\in \Lambda^I} \rme^{-\ci 2\pi \sum_i x_i\cdot k_i} \E\Bigl[\prod_{i\in I} \psi(x_i,\sigma_i)\Bigr]\, , \end{align} where in terms of the cumulant generating function $\mathcal{G}_c[f]=\ln \E[\rme^{\ci \sum_{x,\sigma}f(x,\sigma) \psi(x,\sigma)}]$, \begin{align}%\label{eq:} & \E\Bigl[\prod_{i\in I} \psi(x_i,\sigma_i)\Bigr] = (-\ci)^{|I|} \Bigl[\prod_{i\in I} \partial_{f(x_i,\sigma_i)}\Bigr] \left.\rme^{\mathcal{G}_c[f]}\right|_{f=0} \nonumber \\ & \quad = \sum_{S\in \pi(I)} \prod_{A\in S} \Bigl[(-\ci)^{|A|} \prod_{i\in A} \partial_{f(x_i,\sigma_i)} \left.\mathcal{G}_c[f]\right|_{f=0}\Bigr] %\nonumber \\ & \quad = \sum_{S\in \pi(I)} \prod_{A\in S} \Elfin\Bigl[\prod_{i\in A} \psi(x_i,\sigma_i)\Bigr]^{\rm trunc} \, . % \mathcal{C}_{|A|}(x_A,\sigma_A)\, . \end{align} Since the measure is translation invariant, so are all of the truncated correlation functions (cumulants). Thus, if we choose an arbitrary order for the elements for $A\subset S$ and let $i_A$ denote the first element, we find \begin{align}%\label{eq:} & \E\Bigl[\prod_{i\in I} \FT{\psi}(k_i,\sigma_i)\Bigr] = \sum_{S\in \pi(I)} \prod_{A\in S} \Bigl[ \sum_{x\in \Lambda^A} \rme^{-\ci 2\pi \sum_{i\in A}x_i\cdot k_i} \E\Bigl[\prod_{i\in A} \psi(x_i-x_{i_A},\sigma_i)\Bigr]^{\rm trunc} \Bigr] %\mathcal{C}_{|A|}(x_A-x_{a_1},\sigma_A) \Bigr] \nonumber \\ & \quad = \sum_{S\in \pi(I)} \prod_{A\in S} \Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} k_i\Bigr) C_{|A|}(k_A,\sigma_A) \Bigr] \, , \end{align} which completes the proof of the Lemma. \end{proof} %\newpage \subsection{Main terms}\label{sec:maindiag} Using Lemma \ref{th:cumulants} in $\Qmain$ yields a high dimensional integral over the momenta, restricted to a certain subspace determined by $\Delta_{n,\ell}$ and the $\delta_\Lambda$-functions arising from the cumulant expansion. The restrictions can be encoded in a ``Feynman-diagram'', which is a planar graph where each edge corresponds to an independent momentum integral, and each vertex carries the appropriate $\delta_\Lambda$-function (in physics language, these can be interpreted as ``Kirchhoff rules'' applied at the vertex). The explicit integral expressions are given in the following proposition, and we will discuss their graphical representation in Section \ref{sec:momdeltas}. \begin{proposition}\label{th:main1st} For a given $N_0\ge 1$, \begin{align} \Qmain = \sum_{n=0}^{N_0-1} \sum_{\ell \in G_n} \sum_{S\in \pi(I_{0,2 n+1})} \mathcal{F}_n^{\rm ampl}(S,\ell,t/\vep,\kappa)\, , \end{align} where, setting $\mathcal{I}''_{n} =\mathcal{I}_{n} \cup \set{(n,1)}\cup \set{(0,0)}$, \begin{align}\label{eq:defFnampl} & \mathcal{F}_n^{\rm ampl}(S,\ell,t/\vep,\kappa) = (-\ci \lambda)^n \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}''_{n}}} \int_{(\Lambda^*)^{\mathcal{I}''_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma;\Lambda) \nonumber \\ & \quad \times \prod_{A\in S}\Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} k_{0,i}\Bigr) C_{|A|}(\sigma_{0,A},k_{0,A};\lambda,\Lambda) \Bigr] \nonumber \\ & \quad \times \1(\sigma_{n,1}=1)\1(\sigma_{0,0}=-1) \FT{g}(k_{n,1})^* \FT{f}(k_{n,1}) % \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] \nonumber \\ & \quad \times \int_{(\R_+)^{I_{0,n}}}\!\rmd s \, \delta\Bigl(\frac{t}{\vep}-\sum_{i=0}^{n} s_i\Bigr) \prod_{i=0}^{n} \rme^{-s_i \kappa_{n-i}} \prod_{m=0}^{n-1} \rme^{-\ci s_m \sum_{i=m+1}^{n}\Omega_{i-1;\ell_i}(k,\sigma)} \, . \end{align} \end{proposition} \begin{proof} The representation is a corollary of the results in Section \ref{sec:graphs}, after we relabel $k=k_{n1}$ and $k'=k_{00}$ and set $\sigma_{n1}=1$, $\sigma_{00}=-1$ in (\ref{eq:Qmaindef}). In the resulting formula the cluster momentum $\delta_\Lambda$-functions enforce $\sum_i k_{0,i}=0$. Combined with the interaction $\delta_\Lambda$-functions this implies $k_{0,0}=-k_{n,1}$ which we have used to simplify the final formula by changing the argument of $\FT{f}$. \end{proof} \begin{figure} \centering \myfigure{width=0.9\textwidth}{GrF2fig.eps} \caption{Two diagrams representing nonzero $\mathcal{F}_2^{\rm ampl}(S,\ell,t,\kappa)$ with interaction history $\ell=(2,1)$. The left one has clustering $S=\set{\set{0,4},\set{1,3},\set{2,5}}$ and it corresponds to a leading term. The right one has $S=\set{\set{0,4},\set{1,2,3,5}}$ and corresponds to a subleading term. The symbol ``$\ominus$'' denotes the root of the (here trivial) minus tree, and ``$\oplus$'' the root of the plus tree.\label{fig:maingraph}} \end{figure} Each choice of $n$, $S$, and $\ell$ corresponds to a unique diagram: we take the earlier discussed ``interaction diagrams'' (as in Fig.~\ref{fig:indexgr}), add a ``dummy'' placeholder vertex for each of the fields $\FT{a}_0$ at the bottom of the graph, and add a ``cluster'' vertex for each $A\in S$ with the appropriate connections to the placeholder vertices. Two simple examples are shown in Fig.~\ref{fig:maingraph}. We have also added a line from the $(0,0)$-placeholder vertex to the top line, for reasons which will become apparent shortly. For further use, we introduce here the concepts of ``plus'' and ``minus tree''. When all cluster vertices and their edges are removed, the diagram splits into two components which are graph-theoretically trees. The left tree (which here is a single edge connecting to the placeholder of the original $\FT{\psi}_0(k',-1)$) is called the minus tree, and the right tree is called the plus tree, for obvious reasons. The integral defining the corresponding amplitude can be constructed from a diagram by applying the following ``Feynman rules'': the parities of the two topmost lines are fixed to $-1$ on the left and $1$ on the right. The remaining parities can be computed going from top to bottom and at each interaction vertex continuing the parity unchanged in the middle line, and setting $-1$ on the left and $+1$ on the right. The cluster vertices do not affect parity nor momentum. To each edge in the diagram there is attached a momentum and they are related by Kirchhoff rules at the vertices: at a fusion vertex, the three momenta below need to sum to the single momentum above, and at a cluster vertex all momenta sum to zero. In addition, each fusion vertex carries a factor $-\ci \lambda \sigma \PFone$ ($\sigma$ is determined by the middle edge and the arguments of $\PFone$ by the edges below the vertex) and each cluster vertex a factor $C_{|A|}$ (with $\sigma$ and $k$ determined by the edges attached to the vertex). The total amplitude still needs to be multiplied by $\FT{g}(k_{n,1})^* \FT{f}(k_{n,1})$ and by the appropriate time-dependent factor, the integrand in the last line of (\ref{eq:defFnampl}), before integrating over $s$ and $k$. The time-dependent factor can also be written as \begin{align}\label{eq:defgammam} \prod_{m=0}^n \rme^{-\ci s_m \gamma(m)},\qquad \text{where}\quad \gamma(m) = \sum_{i=m+1}^n \Omega_i -\ci \kappa_{n-i}\, , \end{align} and we recall the notation $\Omega_i =\Omega_{i-1;\ell_i}(k,\sigma)$. The pure phase part for the time-slice $m$, i.e., $\rme^{-\ci s_m \re \gamma(m)}$, can also be read directly from the diagram: collect all edges which go through the time-slice $m$ and for each edge $e$ add a factor $\rme^{-\ci s_m \sigma_e \omega(k_e)}$. This follows from the following Lemma according to which inside any of the above amplitude integrals we have \begin{align} \sum_{i=m+1}^{n}\Omega_{i-1;\ell_i}(k,\sigma) = \sum_{j=1}^{2(n-m)+1} \sigma_{m,j} \omega(k_{m,j}) - \omega(k_{n1}) \, . \end{align} This yields the above mentioned factors when we follow the construction explained earlier; since $-\omega(k_{n1})=\sigma_{00}\omega(k_{00})$, and the corresponding edge intersects all time-slices of the diagram, also the last term comes out correctly. \begin{lemma}\label{th:omOmconv} Suppose $m_0=1$, and $n\ge 0$ is given. Then for any $\ell\in G_n$, for all $0\le m\le n$, and with $\sigma$ and $k$ such that $\Delta_{n,\ell}(k,\sigma;\Lambda)\ne 0$, \begin{align}\label{eq:convertphase} \sum_{j=1}^{2(n-m)+1} \sigma_{m,j} \omega(k_{m,j}) - \sum_{i=m+1}^{n} \Omega_{i-1;\ell_i}(k,\sigma) = \sigma_{n,1} \omega(k_{n1}) \, , \end{align} and \begin{align}\label{eq:convertphase2} \sum_{j=1}^{2(n-m)+1} \sigma_{m,j} = \sigma_{n,1}\, . \end{align} \end{lemma} \begin{proof} The proof goes via induction in $m$, starting from $m=n$ and proceeding to smaller values. The equation holds trivially for $m=n$, as the second sum is not present then. Assume that the equation holds for $m$, where $1\le m\le n$, and to complete the induction, we need to prove that the equation then holds for $m-1$. Since $k,\sigma$ is consistent with $\Delta_{n,\ell}$, we have \begin{align}%\label{eq:convertphase} & \sum_{j=1}^{2(n-m+1)+1} \sigma_{m-1,j} \omega(k_{m-1,j}) % \nonumber \\ & \quad = \sum_{j=0}^{2} \sigma_{m-1,\ell_{m}+j} \omega(k_{m-1,\ell_{m}+j}) + \sum_{j=1;j\ne \ell_{m}}^{2(n-m)+1} \sigma_{m,j} \omega(k_{m,j}) \, . \end{align} Here the first sum is equal to \begin{align}%\label{eq:} & -\omega(k_{m-1,\ell_{m}})+\sigma_{m,\ell_m} \omega(k_{m-1,\ell_{m}+1}) + \omega(k_{m-1,\ell_{m}+2}) % \nonumber \\ & \quad = \Omega_{m-1;\ell_{m}}(k,\sigma) + \sigma_{m,\ell_m} \omega(k_{m,\ell_m}) . \end{align} Thus by the induction assumption, \begin{align}%\label{eq:convertphase} & \sum_{j=1}^{2(n-m+1)+1} \sigma_{m-1,j} \omega(k_{m-1,j}) - \sum_{i=m}^{n} \Omega_{i-1;\ell_i}(k,\sigma) = \sigma_{n,1} \omega(k_{n1}) \, , \end{align} as was claimed in the Lemma. The proof of (\ref{eq:convertphase2}) is essentially identical, and we will skip it. \end{proof} \subsection{Error terms} \label{sec:errors} Each of the three error terms $Q^{\rm err}$ is a sum over terms of the type \begin{align}%\label{eq:} \int_0^{t/\vep}\! \rmd s\, \E\!\left[\mean{\FT{f},\FT{a}_0}^* F_s[\FT{a}_s]\right]\, , \end{align} where $F_s$ contains only a finite moment of the fields $\FT{a}_s$. We estimate it using the Schwarz inequality, \begin{align}%\label{eq:} &\Bigl|\int_0^{t/\vep}\! \rmd s\, \E\Bigl[\mean{\FT{f},\FT{a}_0}^* F_s[\FT{a}_s]\Bigr]\Bigr|^2 \le \Bigl(\int_0^{t/\vep}\! \rmd s\, \E\Bigl[\bigl|\mean{\FT{f},\FT{a}_0}\bigr| \, \bigl|F_s[\FT{a}_s]\bigr|\Bigr]\Bigr)^2 \nonumber \\ & \quad \le \frac{t}{\vep} \E[|\mean{\FT{f},\FT{a}_0}|^2]\, \int_0^{t/\vep}\! \rmd s\, \E\bigl[|F_s[\FT{a}_s]|^2\bigr] \, . \end{align} Here $\E[|\mean{\FT{f},\FT{a}_0}|^2]$ is uniformly bounded, since \begin{align}%\label{eq:} \E[|\mean{\FT{f},\FT{a}_0}|^2] = \int_{\Lambda^*} \rmd k\, |\FT{f}(k)|^2 W^{\lambda}_\Lambda(k)\, . \end{align} Thus \begin{align}%\label{eq:} \limsup_{\Lambda\to\infty} \frac{t}{\vep} \E[|\mean{\FT{f},\FT{a}_0}|^2] \le \lambda^{-2} t c'_0 \norm{f}_2^2\, . \end{align} Therefore, we need to aim at estimates for $\sup_{0\le s\le t \lambda^{-2}}\E\bigl[|F_s[\FT{a}_s]|^2\bigr]$ which decay faster than $\lambda^{4}$ in order to get a vanishing bound. Although the Gibbs measure is not stationary with respect to $\FT{a}_t$, it {\em is} stationary with respect to $\FT{\psi}_t$. The non-stationarity manifests itself only via an additional phase factor: \begin{align}\label{eq:anonstat} & \E\Bigl[\prod_{i\in I} \FT{a}_t(k_i,\sigma_i)\Bigr] = \prod_{i\in I} \rme^{\ci t \sigma_i \omla(k_i)} \E\Bigl[\prod_{i\in I} \FT{a}_0(k_i,\sigma_i)\Bigr] %\nonumber \\ & \quad = \rme^{\ci t \lambda R_0 \sum_i\sigma_i} \prod_{i\in I} \rme^{\ci t \sigma_i \omega(k_i)} \E\Bigl[\prod_{i\in I} \FT{a}_0(k_i,\sigma_i)\Bigr]\, . \end{align} The extra phase factor can always expressed in terms of the previously used $\Omega$-factors, employing Lemma \ref{th:omOmconv}. Applying the Lemma for $m=0$ implies that the phase factor generated by the non-stationarity of $\FT{a}$ can be resolved by using the equality \begin{align}\label{eq:s0phase} & \prod_{j=1}^{m_0+2 n} \FT{a}_s(k_{0,j},\sigma_{0,j}) % \nonumber \\ & \quad = \rme^{\ci s \sigma_{n,1} \omla(k_{n,1})} \prod_{i=1}^{n} \rme^{\ci s \Omega_{i-1;\ell_i}(k,\sigma)} \prod_{j=1}^{m_0+2 n} \FT{\psi}_s(k_{0,j},\sigma_{0,j})\, , \end{align} which will always hold inside the relevant integrals. The following lemma gives a recipe how the two simplex time-integrations resulting from the Schwarz inequality can be represented in terms of a single simplex time-integration. We begin by introducing the concept of interlacing of two sequences. \begin{definition} Let $n,n'\ge 0$ be integers. A map $J:I_{n+n'}\to \set{\pm 1}$ \defem{interlaces} $(n,n')$, if $|J^{\gets}(\set{+1})|=n$ and $|J^{\gets}(\set{-1})|=n'$. For any such $J$, we define further two maps $C_\pm:I_{0,n+n'}\to \N_0$ by setting for $\sigma\in\set{\pm 1}$, $i\in I_{0,n+n'}$, \begin{align}%\label{eq:} C_\sigma(i;J) = \sum_{j=1}^i \1(J(j)=\sigma)\, . \end{align} \end{definition} Thus $C_\sigma(0;J)=0$ and else $C_\sigma(i;J)= |J^{\gets}(\set{\sigma})\cap I_i|$. In addition, as $J$ interlaces $(n',n)$, clearly $C_+:I_{0,n+n'}\to I_{0,n}$ and $C_-:I_{0,n+n'}\to I_{0,n'}$ and both maps are increasing and onto. We claim that with these definitions the following representation Lemma holds, saving the proof of the Lemma until the end of this section. \begin{lemma}\label{th:recombinationlemma} Let $t>0$, $n,n'\ge 0$, and suppose $\gamma^+_i,\gamma^-_j\in\C$ are given for $i\in I_{0,n}$ and $j\in I_{0,n'}$. Then \begin{align}\label{eq:recombinationlemma} & \int_{(\R_+)^{I_{0,n}}}\!\rmd s \, \delta\Bigl(t-\sum_{i=0}^{n} s_i\Bigr) \prod_{i=0}^n \rme^{-\ci s_i \gamma^+_i} \times \int_{(\R_+)^{I_{0,n'}}}\!\rmd s' \, \delta\Bigl(t-\sum_{i=0}^{n'} s'_i\Bigr) \prod_{i=0}^{n'} \rme^{-\ci s'_i \gamma^-_i} \nonumber \\ & \quad = \sum_{J\text{ interlaces }(n,n')} \int_{(\R_+)^{I_{0,n+n'}}}\!\rmd r \, \delta\Bigl(t-\sum_{i=0}^{n+n'} r_i\Bigr) \prod_{i=0}^{n+n'} \rme^{-\ci r_i (\gamma^+_{C_+(i;J)}+\gamma^-_{C_-(i;J)})} . \end{align} \end{lemma} We recall that the $\delta$-functions in the above formula are a shorthand notation for restricting the integration to the standard simplex of size $t$. The exact definition is obtained by choosing an arbitrary index $i$ and ``integrating out'' the delta function with respect to $s_i$. It is an easy exercise to show that for the above exponentially bounded functions, an equivalent definition is obtained by replacing the $\delta$-function by a Gaussian approximation and then taking the variance of the Gaussian distribution to zero. (The latter property combined with Fubini's theorem allows for free manipulation of the order of integration.) Using the above observations, we can derive diagrammatic representations of the expectation values $\E\bigl[|F_s[\FT{a}_s]|^2\bigr]$ very similar to what was described in Section \ref{sec:maindiag}. We consider only the case of $\mathcal{A}_n$ in detail, the treatment of the remaining error terms is very similar, and we merely quote the results later. Let $\mathcal{I}'_{n} =\mathcal{I}_{n} \cup \set{(n,1)}= \defset{(i,j)}{0\le i\le n, 1\le j \le m_{n-i}}$. Then by (\ref{eq:s0phase}) and (\ref{eq:phase2}) we can write \begin{align}\label{eq:gAnprod} & \mean{\FT{g},\mathcal{A}_n(r,t,\cdot,1,\kappa)[\FT{a}_r]} \nonumber \\ & \quad = \sum_{\ell \in G_n} \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}'_{n}}} \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma;\Lambda) \1(\sigma_{n,1}=1) \FT{g}(k_{n,1})^* \rme^{\ci r \omla(k_{n,1})} \prod_{j=1}^{m_0+2 n} \FT{\psi}_r(k_{0,j},\sigma_{0,j}) \nonumber \\ & \qquad \times \prod_{i=1}^{n} \Bigl[-\ci \lambda \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] % \nonumber \\ & \qquad \times \int_{(\R_+)^{I_{n}}}\!\rmd s \, \delta\Bigl(t-r-\sum_{m=1}^{n} s_m\Bigr) \prod_{m=1}^{n} \rme^{-\ci s_m \gamma^+_m} \, , \end{align} where $\gamma^+_n=-\ci \kappa_0$ and for $1\le m\le n-1$, \begin{align} \gamma^+_m = \sum_{i=m+1}^n \Omega_i -\ci \kappa_{n-m}\, . \end{align} Now we can apply Lemma \ref{th:recombinationlemma} to study the expectation of the square. However, before taking the expectation value, we make a change of variables $\sigma'_{i,j}=-\sigma_{i,2 (n-i+1)-j}$, $k'_{i,j}=-k_{i,2 (n-i+1)-j}$, and $\ell'_i=2(n-i+1)-\ell_i$ in the complex conjugate (i.e., we swap the signs and invert the order on each time-slice). We also define $I=I_{2 m_n}=I_{2(2 n+1)}$ to give labels to the fields $\FT{\psi}_r$: we denote $K=(k'_{0,\cdot},k_{0,\cdot})\in (\T^d)^{I}$ and $o=(\sigma'_{0,\cdot},\sigma_{0,\cdot})\in \set{\pm 1}^{I}$, and thus, for instance, $K_{m_n+1}=k_{0,1}$. Applying Lemma \ref{th:recombinationlemma} and Proposition \ref{th:PFcorr} we then obtain \begin{align}\label{eq:def Aampl} & \E\Bigl[|\mean{\FT{g}, \mathcal{A}_n(s,t/\vep,\cdot,1,\kappa)[\FT{a}_s]}|^2\Bigr] \nonumber \\ & \quad = \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I)} \mathcal{A}_n^{\rm ampl}(S,J,\ell,\ell',t/\vep-s,\kappa)\, , \end{align} where the ``amplitudes'' are explicitly \begin{align}%\label{eq:} & \mathcal{A}_n^{\rm ampl} = (-\lambda^{2})^{n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k'\, \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n,\ell'}(k',\sigma';\Lambda) \nonumber \\ & \quad \times \prod_{A\in S}\Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) C_{|A|}(K_A,o_A;\lambda,\Lambda) \Bigr] % \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \sigma'_{i,\ell'_{i}} \PFone(-k'_{i-1;\ell'_i}) \Bigr] \nonumber \\ & \quad \times \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) \FT{g}(k_{n,1})^* \FT{g}(-k'_{n,1}) \nonumber \\ & \quad \times \rme^{\ci s (\omega(k_{n,1})-\omega(k'_{n,1}))} \int_{(\R_+)^{I_{0,2n-2}}}\!\rmd r \, \delta\Bigl(\frac{t}{\vep}-s-\sum_{i=0}^{2 n-2} r'_i\Bigr) % \nonumber \\ & \quad\quad \times \prod_{i=0}^{2n-2} \rme^{-\ci r'_i \left(\gamma^+_{C_+(i;J)+1}+\gamma^-_{C_-(i;J)+1}\right)} , \end{align} where $\gamma^+_m$ is defined as before, and since $\Omega(-(k_3,k_2,k_1),-\sigma)=-\Omega((k_1,k_2,k_3),\sigma)$, \begin{align}%\label{eq:} \gamma^-_m & = \sum_{i=m+1}^{n} \Omega_{i-1;\ell'_i}(k',\sigma')-\ci \kappa_{n-m} \, . \end{align} The cluster $\delta$-functions imply that $\sum_{i\in I} K_i = 0$. Applying the interaction $\delta$-functions iteratively in the direction of time then shows that the integrand is zero unless $k_{n,1}+k'_{n,1}=0$ (modulo 1). Therefore, $\omega(k_{n,1})=\omega(k'_{n,1})$, and the amplitude depends on $s$ and $t/\vep$ only via their difference $t/\vep-s$, as implied by the notation in (\ref{eq:def Aampl}). The final, somewhat simplified expression, for the amplitude function is thus \begin{align}\label{eq:Aamplsimp} & \mathcal{A}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa) = (-\lambda^{2})^{n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n,\ell'}(k',\sigma';\Lambda) \prod_{A\in S}\Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) C_{|A|}(o_A,K_A;\lambda,\Lambda) \Bigr] \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \sigma'_{i,\ell'_{i}} \PFone(-k'_{i-1;\ell'_i}) \Bigr] \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) |\FT{g}(k_{n,1})|^2 \nonumber \\ & \quad \times \int_{(\R_+)^{I_{2,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=2}^{2 n} r_i\Bigr) \prod_{i=2}^{2n} \rme^{-\ci r_i \gamma({i;J})}\, , \end{align} where, for $i=2,3,\ldots,2n$, we define \begin{align}\label{eq:giJ} & \gamma({i;J}) = \sum_{j=m+1}^{n} \Omega_{j-1;\ell_j}(k,\sigma) + \sum_{j=m'+1}^{n} \Omega_{j-1;\ell'_j}(k',\sigma') -\ci (\kappa_{n-m}+\kappa_{n-m'}) \nonumber \\ & = \sum_{j=1}^{2(n-m)+1}\!\! \sigma_{m,j} \omega(k_{m,j}) + \sum_{j=1}^{2(n-m')+1}\!\! \sigma'_{m',j} \omega(k'_{m',j}) -\ci (\kappa_{n-m}+\kappa_{n-m'})\, , \end{align} with $m=m(i)=C_+(i-2;J)+1$ and $m'=m'(i)=C_-(i-2;J)+1$. In particular, $\gamma(2 n;J)=\gamma^+_n + \gamma^-_n =-\ci 2\kappa_0$. We can now describe the integral (\ref{eq:Aamplsimp}) in the earlier defined diagrammatic scheme. To make the identification more direct, we have shifted the time-indices upwards by two: the idea is that the first two time-slice have zero length, i.e., they are \defem{amputated}. Formally, we could write the time-integral as \begin{align} \int_{(\R_+)^{I_{0,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=0}^{2 n} r_i\Bigr) \delta(r_0) \delta(r_1) \prod_{i=0}^{2n} \rme^{-\ci r_i \gamma(i;J)}\, . \end{align} Clearly, the result is independent of how we define $\gamma(0;J)$ and $\gamma(1;J)$. To make the identification between an amputated amplitude and the diagram unique, we arbitrarily require that in an amputated diagram the first fusion always happens in the minus tree and the second fusion in the plus tree. The construction of the phase factor of the time-integrand is then done using the same rules as before: for each time slice $i$, we collect all edges which go through the time-slice and for each edge $e$ add a factor $\rme^{-\ci r_i \sigma_e \omega(k_e)}$. Under the above amputation condition, we arrive this way to the integrand in (\ref{eq:Aamplsimp}). Compared to the Feynman rules explained for the main term, we have only one additional rule here: in the \defem{minus} tree, the sign inside the cutoff-function is swapped, i.e., there we use a factor $-\ci \lambda \sigma' \PFone(-k')$. Otherwise, the Feynman rules are identical, apart from the overall testfunction factor which is here $|\FT{g}(k_{n,1})|^2$. We have illustrated these definitions in Figure \ref{fig:ampdiag}. \begin{figure} \centering \myfigure{height=0.3\textheight}{GrAmpfig.eps} \caption{An amputated diagram representing a nonzero $\mathcal{A}_2^{\rm ampl}(S,J,\ell,\ell',s,\kappa)$ with a pairing $S=\set{\set{1,7},\set{2,10},\set{3,9},\set{4,8},\set{5,6}}$ and $\ell'=(2,1)$, $\ell=(3,1)$, $J=(+1,-1)$. The shading on the first two time-slices is used to denote the fact that these time-slices have zero length, as explained in the text. As before, the symbols ``$\oplus$'' and ``$\ominus$'' denote the roots of the plus and minus trees, respectively.\label{fig:ampdiag}} \end{figure} To complete the above derivation, we still need to prove the time-simplex Lemma. \begin{proofof}{Lemma \ref{th:recombinationlemma}} The Lemma is based on rearrangement of the time-integrations by iteratively splitting time-integrations into two independent parts. The splitting will depend on the relative order of the times accumulated from $s$ and $s'$, which is captured by the sum over $J$ on the right hand side. The value of $C_+(i;J)$ yields the index for the phase factor $\gamma^+$ which is ``active'' at the new time-slice obtained after the splitting. The proof below will be given mainly to show that the above definitions yield a correct description of the result. Suppose first that $n=0$. Then the first factor is $\rme^{-\ci t \gamma^+_0}$. On the other hand, the only admissible $J$ is then $J(i)=-1$ for all $i\in I_{n'}$, and thus $C_+(i;J)=0$ and $C_-(i;J)=i$ for all $i\in I_{0,n'}$. Therefore, (\ref{eq:recombinationlemma}) holds by inspection. A symmetrical argument applies to the case $n'=0$. Assume thus that $n'\ge 1$, and we will prove the rest by induction in $n$. The initial case $n=0$ was checked to hold above. Assume then that (\ref{eq:recombinationlemma}) holds for all $n\le N$, with $N\ge 0$, and consider the case $n=N+1\ge 1$. It is clear that both sides of (\ref{eq:recombinationlemma}) are continuous in $\gamma^+_{N+1}$, and thus it suffices to prove it assuming $\gamma^+_{N+1}\ne \gamma^+_{N}$. Let us first concentrate on the first factor. We change the integration variable from $s_{N}$ to $u=s_{N}+s_{N+1}$. This shows that \begin{align}%\label{eq:} & \int_{0}^\infty\! \rmd s_{N}\, \int_{0}^\infty\! \rmd s_{N+1}\, \delta\Bigl(t-\sum_{i=0}^{N+1} s_i\Bigr) \rme^{-\ci s_N \gamma^+_N -\ci s_{N+1} \gamma^+_{N+1}} \nonumber \\ & \quad = \int_{0}^\infty\! \rmd s_{N+1}\, \int_{s_{N+1}}^\infty\! \rmd u\, \delta\Bigl(t-u-\sum_{i=0}^{N-1} s_i\Bigr) \rme^{-\ci (u-s_{N+1}) \gamma^+_N -\ci s_{N+1} \gamma^+_{N+1}} \nonumber \\ & \quad = \int_{0}^\infty\! \rmd u\, \delta\Bigl(t-u-\sum_{i=0}^{N-1} s_i\Bigr) \rme^{-\ci u \gamma^+_{N}} \int_{0}^u\! \rmd s_{N+1}\, \rme^{-\ci s_{N+1} ( \gamma^+_{N+1}- \gamma^+_N ) } \nonumber \\ & \quad = \frac{\ci}{\gamma^+_{N+1}- \gamma^+_N} \int_{0}^\infty\! \rmd u\, \delta\Bigl(t-u-\sum_{i=0}^{N-1} s_i\Bigr) \Bigl(\rme^{-\ci u \gamma^+_{N+1}} -\rme^{-\ci u \gamma^+_{N}} \Bigr) . \end{align} The induction assumption can be applied to both terms separately, which proves that (\ref{eq:recombinationlemma}) is equal to \begin{align}%\label{eq:} & \frac{\ci}{\gamma^+_{N+1}- \gamma^+_N} \sum_{J\text{ interlaces }(N,n')} \int_{(\R_+)^{I_{0,N+n'}}}\!\rmd r \, \delta\Bigl(t-\sum_{i=0}^{N+n'} r_i\Bigr) \nonumber \\ & \qquad \times \Bigl( \prod_{i=0}^{N+n'} \left. \rme^{-\ci r_i (\gamma^+_{C_+(i;J)}+\gamma^-_{C_-(i;J)})} \right|_{\gamma^+_{N}\to\gamma^+_{N+1}} - \prod_{i=0}^{N+n'} \rme^{-\ci r_i (\gamma^+_{C_+(i;J)}+\gamma^-_{C_-(i;J)})} \Bigr) . \end{align} For a fixed $J$, let $j_0=\min \defset{i\in I_{0,N+n'}}{C_+(i;J)=N}$, i.e., $j_0$ denotes the last appearance of $+1$ in $J$. Then $N\le j_0\le N+n'$. The difference in the brackets can then be expressed as \begin{align}%\label{eq:} & \prod_{i=0}^{N+n'} \rme^{-\ci r_i\gamma^-_{C_-(i;J)}} \prod_{i=0}^{j_0-1} \rme^{-\ci r_i \gamma^+_{C_+(i;J)}} \Bigl( \rme^{-\ci \gamma^+_{N+1} \sum_{i=j_0}^{N+n'} r_i} - \rme^{-\ci \gamma^+_{N}\sum_{i=j_0}^{N+n'} r_i} \Bigr) \nonumber \\ & \quad = (-i) (\gamma^+_{N+1}- \gamma^+_N) \prod_{i=0}^{N+n'} \rme^{-\ci r_i\gamma^-_{C_-(i;J)}} \prod_{i=0}^{j_0-1} \rme^{-\ci r_i \gamma^+_{C_+(i;J)}} \nonumber \\ & \qquad \times \left. \int_0^{u}\! \rmd s\, \rme^{-\ci s \gamma^+_{N}} \rme^{-\ci (u-s)\gamma^+_{N+1}}\right|_{u=\sum_{i=j_0}^{N+n'}r_i} . \end{align} The final integral is split according to the position of $s$ in the sequence $(S_\ell)_{\ell=j_0,\ldots,N+n'+1}$, where $S_\ell =\sum_{i=j_0}^{\ell-1}r_i$. Explicitly, this yields \begin{align}%\label{eq:} \sum_{\ell =j_0}^{N+n'} \int_0^{\infty}\! \rmd s\, \1(S_{\ell}\le s \le S_{\ell+1}) \rme^{-\ci s \gamma^+_{N}} \rme^{-\ci (S_{N+n'+1}-s)\gamma^+_{N+1}} . \end{align} Given a map $J$ and $\ell\in \set{j_0(J),N+n'}$, we define a map $J'=J'_{\ell,J}:I_{N+1+n'}\to \set{\pm 1}$ by the rule \begin{align}%\label{eq:} J'(i) = \begin{cases} J(i),& \text{if }i\le \ell,\\ +1,& \text{if }i=\ell+1,\\ -1,& \text{if }i> \ell+1. \end{cases} \end{align} Obviously, $J'$ interlaces $(N+1,n')$, and for later use, let us point out that the maps $C_\pm(\cdot;J')$ then satisfy $C_\pm (i;J')=C_\pm(i;J)$ for $i\le \ell$ and $C_+ (i;J')=N+1$, $C_- (i;J')=C_-(i-1;J)$ for $i> \ell$. Conversely, if $J''$ is an arbitrary map interlacing $(N+1,n')$ then there are unique $\ell$ and $J$ such that $J''=J'_{\ell,J}$, determined by the choices $\ell=j_0(J'')$, and $J$ obtained from $J''$ by canceling $\ell$. Therefore, \begin{align}%\label{eq:} \sum_{J'\text{ interlaces }(N+1,n')} F(J') = \sum_{J\text{ interlaces }(N,n')} \sum_{\ell=j_0(J)}^{N+n'} F(J'_{\ell,J}). \end{align} Thus we only need to prove that the remaining integrals are equal, i.e., that the integral on the right hand side of (\ref{eq:recombinationlemma}) for $J\to J'=J'_{\ell,J}$ and $n\to N+1$, is equal to \begin{align}\label{eq:nearlythere} & \int_{(\R_+)^{I_{0,N+n'}}}\!\rmd r \, \delta\Bigl(t-\sum_{i=0}^{N+n'} r_i\Bigr) \prod_{i=0}^{N+n'} \rme^{-\ci r_i\gamma^-_{C_-(i;J)}} \prod_{i=0}^{j_0-1} \rme^{-\ci r_i \gamma^+_{C_+(i;J)}} \nonumber \\ & \quad \times \int_0^{\infty}\! \rmd s\, \1(S_{\ell}\le s \le S_{\ell+1}) \rme^{-\ci s \gamma^+_{N}} \rme^{-\ci (S_{N+n'+1}-s)\gamma^+_{N+1}} . \end{align} To see this, let us change the integration variables $(r_i,s)_i$ to $(r'_j)_j$ by using $r'_j=r_j$ for $j<\ell$, $r'_j=r_{j-1}$ for $\ell0$ with $n'\ne n$, and as such is not related to any of the present amplitudes. However, we use the more general graph to show that the scheme does not depend on the special relation between $n$ and $n'$.) Our aim is next to ``integrate out'' all the constraint $\delta$-functions. We do this by associating with every vertex a unique edge attached to it which we use for the integration. As long as we use each edge not more than once, this results in a complete resolution of the momentum constraints. The edges used in the integration of the $\delta$-functions are called \defem{integrated}, and the remaining edges are called \defem{free}. We use the notation $\edges'$ for the collection of integrated edges and $\fedges$ for the free edges. The following theorem shows that there is a way of achieving such a division of edges which respects their natural time-ordering. \begin{theorem}\label{th:intdeltas} Consider a momentum graph $\mathcal{G}$. There exists a complete integration of the momentum constraints, determined by a certain unique spanning tree of the graph, such that for any free edge $f$ all $k_e$ with $e\tau(v')$ and $f=\set{v,v'}$. \end{theorem} From now on, we assume that the momentum constraints are integrated out using the unique construction in Theorem \ref{th:intdeltas}. For any fusion vertex $v\in \Fverts$, we call the number of free edges in $\edges_-(v)$ the \defem{degree} of the fusion vertex, and denote this by $\deg v$. The following theorem summarizes how the integrated edges ending at an interaction vertex depend on its free momenta. \begin{proposition}\label{th:momatintv} The degree of a fusion vertex belongs to $\set{0,1,2}$. If $v\in\Iverts$ is a degree one interaction vertex, then $\edges_-(v)=\set{f,e,e'}$ where $f$ is a free edge, and $k_e=-k_f+G$, $k_{e'}=G'$, where $G$ and $G'$ are independent of $k_f$. If $v$ is a degree two interaction vertex, then $\edges_-(v)=\set{f,f',e}$ where $f,f'$ are free edges, and $k_e=-k_f-k_{f'}+G$, where $G$ is independent of $k_f$ and $k_{f'}$. \end{proposition} We will need other similar properties of the integrated momenta, to be given later in this section. However, let us first explain how the constraints are removed. \begin{proofof}{Theorem \ref{th:intdeltas} and Proposition \ref{th:momatintv}} We construct a spanning tree for $\graph$ which provides a recipe for integration of the vertex $\delta$-constraints and leads to the properties stated in Theorem \ref{th:intdeltas}. We first construct an unoriented tree $\Ttree=(\Tverts,\Tedges)$ from $\mathcal{G}$, and then define an oriented tree $\oTtree=(\Tverts,\oTedges)$ by assigning an orientation to each of the edges in $\Tedges$. Let $\Ttree\upn{0}=(\Tverts\upn{0},\Tedges\upn{0})$, with $\Tverts\upn{0}=\emptyset=\Tedges\upn{0}$. We go through all edges in $\edges$ in the opposite order they were created, i.e., decreasing with respect to their order. At the iteration step $l$, let $e$ denote the corresponding edge, and consider the previous graph $\Ttree\upn{l-1}$. If adding the edge $e$ to $\Ttree\upn{l-1}$ would create a loop, we define $\Ttree\upn{l}=\Ttree\upn{l-1}$. Otherwise, we define $\Ttree\upn{l}$ as the graph resulting from this addition, i.e., we define $\Tverts\upn{l}=\Tverts\upn{l-1}\cup e$, and $\Tedges\upn{l}=\Tedges\upn{l-1}\cup \set{e}$. Since in the first case necessarily $e\subset \Tverts^{\ell-1}$, we will always have $e\subset \Tverts^{\ell}$, and thus no vertex in $e$ can be lost in the iteration step. Let $\Ttree=(\Tverts,\Tedges)$ denote the graph obtained after the final iteration step. By construction, at each step $\Ttree\upn{\ell}$ is a forest, and thus so is $\Ttree$. Moreover, since $\graph$ is connected, $\Ttree$ is actually a tree. Since every vertex in $\verts$ is contained in some edge, we also have $\Tverts=\verts$. In addition, $\Tedges\subset \edges$, and every $e\in \edges\setminus\Tedges$ has the following property: adding it to $\Tedges$ would make a unique loop {\em composed out of edges $\set{e'}$ each of which satisfies $e'\ge e$,\/} and thus also $\taup (e')\le \taup (e)$. (The loop is unique since $\Ttree$ itself has no loops.) Next we create $\oTtree$ by assigning an orientation to the edges of $\Ttree$. We root the tree at $\rootv$. This is achieved by the following algorithm: we first note that for any vertex $v$ there is a unique path connecting it to $\rootv$. We orient the edges of the path so that it starts from $v$ and ends in $\rootv$. This is iterated for all vertices in the tree. Although it is possible that two different vertices share edges along the path, these edges are assigned the same orientation at all steps of the algorithm. (If two such paths share any vertex, then the paths must coincide past this vertex; otherwise there would be a loop in the graph.) This results in an oriented graph in which for every $v\in\verts\setminus\Rverts$ there is a {\em unique\/} edge $E(v)\in \edges(v)$ pointing {\em out\/} of the vertex. In addition, the map $E:\verts\setminus\Rverts\to \edges$ is one-to-one. Thus we can integrate all the momentum $\delta$-functions, by using the variable $k_{E(v)}$ for the $\delta$-function at the vertex $v\in\verts\setminus\Rverts$. We have depicted the oriented tree resulting from the graph of Figure \ref{fig:graphex1} in Figure \ref{fig:graphex2}. \begin{figure} \centering \myfigure{height=0.3\textheight}{Graph2} %\myfigure{width=0.8\textwidth}{Graph2} \caption{The oriented spanning tree $\oTtree$ corresponding to the graph $\mathcal{G}$ given in Fig.~\ref{fig:graphex1}. The edges in the complement of the tree have also been depicted by dashed lines. The enumeration $(e_\ell)$ of the edges corresponds to the one explained in the text; the spanning tree is constructed by adding the edges in the graph in {\em decreasing\/} order. \label{fig:graphex2}} \end{figure} After the above integration steps, all the constraints have been resolved, and the set of remaining integration variables will consist of $k_e$ with $e\in \edges\setminus \Tedges$. These are all free integration variables, and thus $\fedges:=\edges\setminus \Tedges$ is the set of free edges, and $\edges':=\Tedges$ is the set of integrated edges. Obviously, one has to add at least all edges attached to a cluster vertex before a loop can be created, and thus no such edge is free. Also, the addition of the last edge $e_0$ never creates a loop. All remaining edges end at a fusion vertex, and thus this is true also of all free edges. In order to conclude the proof of Theorem \ref{th:intdeltas}, we need to find out how the integrated momenta depend on the free ones. For later use, let us spell out also this fairly standard part in detail. For any $v\in \verts$, let $\fedges(v)$ collect the free edges attached to $v$, $\fedges(v)=\edges(v)\cap \fedges$. Let us also associate for any $v\in \verts\setminus\Rverts$ an ``edge parity'' mapping $\sigma_v:\edges(v)\to \set{-1,+1}$ defined by \begin{align}%\label{eq:} \sigma_v(e) = \begin{cases} +1, & \text{if } e\in \edges_+(v),\\ -1, & \text{if } e\in \edges_-(v). \end{cases} \end{align} \begin{lemma}\label{th:mesigns} If $e=\set{v,v'}\in \edges$ does not intersect $\Rverts$, then $m(e)=-\sigma_v(e)\sigma_{v'}(e)=1$. \end{lemma} \begin{proof} Assume first that $v\in \Cverts$. Then $\sigma_v(e)=1$ and $v'\in \Dverts$. Since $\edges(v')$ then contains only two elements, of which $e$ is created later, we have $\sigma_{v'}(e)=-1$, and thus $m(e)=1$. Assume then $e\cap \Cverts = \emptyset$. If $v\in \Dverts$, then $v'\in \Fverts$, and thus $e$ is the earlier of the two edges attached to $v$ and $\sigma_v(e)=1$. However, then also it must be one of the three later ones attached to $v'$, and thus $\sigma_v(e)=-1$. This implies $m(e)=1$. Since $m(e)$ is symmetric under the exchange of $v$ and $v'$, we can now assume that $e\cap \Cverts \cap\Dverts = \emptyset$. Then both $v,v'\in \Fverts$ and it follows from the construction that $\sigma_v(e) \sigma_{v'}(e) = -1$. This proves that for any edge $e$ with $e\cap \Rverts = \emptyset$, $m(e)=1$. \end{proof} Consider then an integrated variable $k_e$, with $e\in \edges'$. The edge $e$ has been assigned an orientation, say $e=(v_1,v_2)$, going from the vertex $v_1$ to the vertex $v_2$. Let $\mathcal{P}(v)$, $v\in \verts$, denote the collection of the vertices $v'$ for which there exists a path from $v'$ to $v$ in the {\em oriented\/} tree $\oTtree$. In particular, we include here the trivial case $v'=v$. We claim that then \begin{align}\label{eq:kesol} k_e = \sum_{v\in \mathcal{P}(v_1)} \sum_{f\in \fedges (v)} \left( - \sigma_{v_1}(e) \sigma_{v}(f) \right) k_{f} . \end{align} This can be proven by induction in a degree $j$ associated with an oriented edge $e=(v_1,v_2)\in \oTedges$: $j$ is defined as the maximum of the number of vertices in an oriented path from any leaf to $v_1$ (note that such paths always exist). For $j=1$, $v_1$ is itself a leaf, and thus $\mathcal{P}(v_1)=\set{v_1}$ and $\fedges (v_1)=\edges(v_1)\setminus\set{e}$. Also $v_1\not\in \Rverts$, since the edge $e_0$ must be oriented as $(v_{N+1},\rootv)$. Thus there is a $\delta$-function associated with $v_1$, and it enforces \begin{align}%\label{eq:} \sum_{e'\in \edges(v_1)} \sigma_{v_1}(e') k_{e'} =0. \end{align} The designated integration of this $\delta$-function yields, with $v=v_1$, \begin{align}%\label{eq:} k_e = -\sigma_v(e) \sum_{e'\in \edges(v)\setminus\set{e}} \sigma_{v}(e') k_{e'} = \sum_{e'\in \fedges (v)} ( -\sigma_v(e) \sigma_{v}(e')) k_{e'} , \end{align} and therefore (\ref{eq:kesol}) holds for $j=1$. Assume then that (\ref{eq:kesol}) holds for any edge up to degree $j\ge 1$, and suppose $e=(v_1,v_2)$ is an edge with a degree $j+1$. Again $v_1\not\in \Rverts$, and the corresponding $\delta$-function implies that, with $v=v_1$, \begin{align}%\label{eq:} k_e = \sum_{e'\in \edges(v)\setminus\set{e}} (-\sigma_{v}(e) \sigma_{v}(e')) k_{e'}. \end{align} In the sum, an edge $e'$ is either free or it must have a degree of at most $j$, as otherwise $e$ would have a degree of at least $j+2$. Thus by the induction assumption, \begin{align}\label{eq:kesol2} & k_e = \sum_{f\in \fedges (v)} (-\sigma_v(e) \sigma_{v}(f)) k_{f} + \sum_{e'\in \edges(v)\setminus\set{e}\setminus \fedges (v)} (-\sigma_v(e) \sigma_{v}(e')) \nonumber \\ & \qquad \times \sum_{v'\in \mathcal{P}(V_1(e'))} \sum_{f\in \fedges (v')} \left( -\sigma_{V_1(e')}(e') \sigma_{v'}(f) \right) k_{f}, \end{align} where $e'=(V_1(e'),v)$ and $V_1(e)$ denotes the first vertex of an oriented edge $e$. Therefore, \begin{align}%\label{eq:} & (-\sigma_v(e) \sigma_{v}(e')) \left( -\sigma_{V_1(e')}(e') \sigma_{v'}(f) \right) %\nonumber \\ & \quad = -\sigma_v(e) m(e') \sigma_{v'}(f) = -\sigma_v(e) \sigma_{v'}(f) \, . \end{align} Now (\ref{eq:kesol2}) can be checked to coincide with (\ref{eq:kesol}). This completes the induction step, and thus proves (\ref{eq:kesol}). Consider then a free integration variable corresponding to $f_0=\set{u,u'}\in\fedges$, where we can choose $\tau(u)>\tau(u')$. Then $f_0\cap \Rverts =\emptyset$, $\taup (f_0)=\tau(u)$, and $\sigma_u(f_0)=-1$, $\sigma_{u'}(f_0) = +1$. The unique oriented paths from $u$ and $u'$ to the root of the tree must coincide starting from a unique vertex $v_0$, which can {\em a priori\/} also be either $u$ or $u'$. In addition, the paths before $v_0$ cannot have any common vertices. Suppose $e=(v_1,v_2)$ belongs to the path from $u$ to $v_0$ in $\oTtree$. Then $k_e$ depends on $k_{f_0}$, and using (\ref{eq:kesol}) we find that $k_e=\sigma_{v_1}(e) k_{f_0} + \cdots$. Similarly, if $e$ belongs to the path from $u'$ to $v_0$, then $k_e=-\sigma_{v_1}(e) k_{f_0} + \cdots$. For any $e$ which comes after $v_0$ in the path, both terms will be present, and they cancel each other. Resorting to (\ref{eq:kesol}) thus proves that only those $k_e$ whose edges are contained in either of the paths $u\to v_0$ and $u'\to v_0$ depend on the free variable $k_{f_0}$. However, as these edges, together with $f_0$, would form a loop in $\mathcal{G}$, it follows from the construction of $\Ttree$ that for any such edge $e$ we have $e\ge f_0$. This proves that if $f\in \fedges$ and $e\in\edges$ with $ef_0$, we have $e\in \edges_-(u)$,and thus $k_e=-k_{f_0}+\cdots$. If $v_0= u$, there is a non-trivial path from $u'$ to $u$, and let $e=(v_1,u)$ be the last edge in that path. By $e>f_0$, again we then have $e\in \edges_-(u)$. Since $m(e)=1$, we find $k_e=-\sigma_{v_1}(e) k_{f_0} + \cdots=\sigma_{u}(e) k_{f_0} + \cdots= -k_{f_0} + \cdots$. Finally, consider the third edge $e'\in \edges_-(u)$. This cannot belong to either of the paths from $u\to v_0$ and $u'\to v_0$, and thus $k_{e'}$ is always independent of $k_{f_0}$. If $e'$ is integrated, the degree of $u$ is one, and we have proved the statement made in the Proposition. If $e'$ is a free edge, the degree of $u$ is two. If we then apply the above result to $e'$ instead of $f_0$, we can conclude that $k_e=-k_{e'}-k_{f_0}+\cdots$, where the remainder is independent of both $k_{e'}$ and $k_{f_0}$. (Note that $e$ is then the only integrated edge in $\edges_-(u)$, and must therefore contain both of the free variables.) We have thus proven both Theorem \ref{th:intdeltas} and Proposition \ref{th:momatintv}. \end{proofof} In the following, the term {\em oriented path} refers to a path in $\oTtree$. Without this clarifier, a path always refers to an unoriented path in a subgraph of $\mathcal{G}$. The following Lemma improves on (\ref{eq:kesol}) and yields the exact dependence of integrated momenta on the free ones. \begin{lemma}\label{th:kesol3} For any integrated edge $e=(v_1,v_2)\in \edges'$, let $P=\mathcal{P}(v_1)$ denote the collection of vertices such that there is an oriented path from the vertex to $v_1$. Then \begin{align}\label{eq:kesol3} & k_e = \sum_{v\in P} \sum_{f=\set{v,v_f}\in \fedges (v)} \1(v_f\not\in P) \left( - \sigma_{v_1}(e) \sigma_{v}(f) \right) k_{f} . \nonumber \\ & \quad = - \sigma_{v_1}(e) \sum_{f\in \fedges} \1(\exists v\in f\cap P\text{ and }f\cap P^c\ne \emptyset ) \sigma_{v}(f) k_{f} . \end{align} In addition, any $f=\set{v,v'}\in\edges$, such that $f\ne e$, $v\in P$, and $v'\not \in P$, is free. \end{lemma} \begin{proof} By (\ref{eq:kesol}) the result in (\ref{eq:kesol3}) holds without the characteristic function $\1(v_f\not\in \mathcal{P}(v_1))$. Consider thus $v,v'\in \mathcal{P}(v_1)$, $v'\ne v$, such that $f=\set{v,v'}$ is free. Then $\sigma_{v}(f)=-\sigma_{v'}(f)$, and thus $-\sigma_{v_1}(e) \sigma_{v}(f)-\sigma_{v_1}(e) \sigma_{v'}(f)=0$. Now sums over edges in $\fedges (v)$ and $\fedges (v')$ appear in (\ref{eq:kesol}), and thus the terms proportional to $k_f$ in these sums cancel each other. This proves (\ref{eq:kesol3}). To prove the last statement let us assume the converse. We suppose $f$ is not free, which implies that $f$ has a representative in $\oTtree$. Suppose first that it is $(v,v')$. There is a unique oriented path from $v'$ to the root of the corresponding oriented tree. Since $v'\not\in P$, the path does not contain $v_1$. This however is not possible because there is also an oriented path from $v$ to $v_1$ to the root (otherwise $\Ttree$ contains a loop). Therefore, we only need to consider $f=(v',v)$. Then there is an oriented path from $v$ to $v_1$, and thus also an oriented path from $v'$ to $v_1$. This contradicts $v'\not\in P$, and thus we can conclude that $f$ must be free. This completes the proof of the Lemma. \end{proof} \begin{corollary}\label{th:zerok} For any edge $e\in \edges$, there is a unique collection of free edges $\fedges_e$, and of $\sigma_{e,f}\in \set{\pm 1}$, $f\in\fedges_e$, such that \begin{align}%\label{eq:} k_e =\sum_{f\in \fedges_e} \sigma_{e,f} k_f\, . \end{align} In addition, $k_e$ is independent of all free momenta if and only if $k_e=0$. This is equivalent to $\fedges_e=\emptyset$, which is possible if and only if the number of connected components increases by one when the edge $e$ is removed from $\mathcal{G}$. \end{corollary} \begin{proof} If $e\in \fedges$, we choose $\fedges_e=\set{e}$ and $\sigma_{e,e}=1$. Otherwise, the existence part follows from the Lemma. Suppose there are two such expansions given by $\fedges_e$, $\sigma_{\cdot,e}$ and $\fedges'_e$, $\sigma'_{\cdot,e}$. If $\fedges'_e\ne \fedges_e$, the difference of the expansions would contain some free momenta with coefficients $\pm 1$, and if $\fedges'_e= \fedges_e$ but $\sigma'_{\cdot,e}\ne \sigma_{\cdot,e}$, some free momenta would appear in the difference with coefficients $\pm 2$. This proves that the expansion is unique. Obviously, $k_e$ is a constant if and only if $\fedges_e=\emptyset$, when $k_e=0$. If $e$ is free, then $\fedges_e$ is not empty. However, then the spanning tree is not affected by removal of $e$, and thus the number of connected components is not altered. Therefore, the Corollary holds then. Else $e=(v_1,v_2)$ is an integrated edge and we can apply Lemma \ref{th:kesol3}. Denote $P=\mathcal{P}(v_1)$, and suppose there is a path from $v_1$ to $v_2$ which does not contain $e$. In this case, removing $e$ from $\graph$ does not create any new components. Along this path there is an edge $f=\set{v,v'}$ such that $v\in P$ but $v'\not\in P$. Since $f\ne e$, by Lemma \ref{th:zerok}, $f\in \fedges$. This implies that $k_e$ depends on $k_f$, and is not uniformly zero, in accordance with the Corollary. Finally, assume that every path from $v_1$ to $v_2$ contains $e$. This implies that $v_1$ and $v_2$ belong to different components if $e$ is removed from $\graph$. However, then there still must be a path from any vertex to either $v_1$ or $v_2$, and the number of components is thus exactly two. Suppose $f=\set{v,v'}\in \fedges_e$, and choose $v\in P$, $v'\not \in P$. Following the oriented paths in the opposite direction, we obtain paths $v_1\to v$, $\rootv\to v_2$ which do not contain $e$. Since by assumption also the oriented path $v'\to \rootv$ does not contain $e$, we can construct a path $v_1\to v_2$ which avoids $e$. This contradicts the assumption, and thus now $\fedges_e=\emptyset$. This completes the proof of the Corollary. \end{proof} Since removing $e_0$ from $\graph$ isolates $\rootv$, the Corollary implies that always $k_{e_0}=0$, i.e., the sum of the top momenta of plus and minus trees is zero. We have already used this property in the derivation of the amplitudes. \begin{lemma}\label{th:nokkdiff} Suppose $f,f'$ are the two free edges ending at a degree two interaction vertex $v_0\in \Iverts$. Let $e=(v_1,v_2)\in \edges'$ be an integrated edge. Then $k_e=F_e(k_f,k_{f'})+G_e$ where $G_e$ is independent of $k_f,k_{f'}$ and $(k,k')\mapsto F_e(k,k')$ is one of the following seven functions: $0$, $\pm k$, $\pm k'$, $\pm (k+k')$. Let $v\in \Iverts$, and suppose $e,e'\in \edges_-(v)$, $e\ne e'$. Then $k_e+k_{e'}=F(k_f,k_{f'})+G$ where $G$ is independent of $k_f,k_{f'}$ and $(k,k')\mapsto F(k,k')$ is also one of the above seven functions. If $v=v_0$, the choice is reduced to one of the functions $-k$, $-k'$, and $k+k'$. \end{lemma} \begin{proof} There are $w,w'\in \Iverts\cup\Cverts$ such that $k_f=\set{v,w}$, $k_{f'}=\set{v,w'}$. Then $\sigma_w(f)=1=\sigma_{w'}(f')$ and $\sigma_v(f)=-1=\sigma_{v}(f')$. We express $k_e$ using (\ref{eq:kesol}), which shows that $k_e=F_e+G_e$ where $F_e = -\sigma_{v_1}(e)(-o_v (k_f+k_{f'})+o_{w} k_f +o_{w'} k_{f'})$, and $o_x$ is one, if there is an oriented path from the vertex $x$ to $v_1$, and zero otherwise. Checking all combinations produces the list of seven functions stated in the Lemma. Let us then consider the second statement. If $v=v_0$, then either one of the vertices is the unique integrated one, or $e,e'$ are equal to the free momenta $f,f'$. In the first case, we can apply Proposition \ref{th:momatintv} which shows that either $F=-k$ or $F=-k'$ will work, and in the second case we can choose $F=k+k'$. We can thus assume $v\ne v_0$. If both $e,e'$ are free, then $F=0$ works. If only one of the edges is free, then the previous result implies the existence of the decomposition. Thus we can assume that both edges are integrated, and $e=\set{v,w_1}$, $e'=\set{v,w'_1}$, where $w_1\ne w'_1$. Suppose first that $e$ points out from $v$, i.e., $e=(v,w_1)$. Then $e'$ points in, $e'=(w'_1,v)$, and any oriented path to $w'_1$ extends into an oriented path to $v$. Since $\sigma_{w'_1}(e')=1$ and $\sigma_{v}(e)=-1$, we have $k_e+k_{e'}=F_e + F_{e'} +G_e + G_{e'}$ with $F_e + F_{e'}=-(-o_v (k_f+k_{f'})+o_{w} k_f +o_{w'} k_{f'}) -o_v (k_f+k_{f'})+o_{w} k_f +o_{w'} k_{f'} -o'_v (k_f+k_{f'})+o'_{w} k_f +o'_{w'} k_{f'}$, where $o_x$ is one, if there is an oriented path from $x$ to $w'_1$, and $o'_x$ is one if there is an oriented path from $x$ to $v$ which does not go via $w'_1$. Thus $F_e + F_{e'}=-o'_v (k_f+k_{f'})+o'_{w} k_f +o'_{w'} k_{f'}$ is also of the stated form. In the remaining case, we can assume that both $e$ and $e'$ point into $v$: $e=(w_1,v)$, $e'=(w'_1,v)$. If there is an oriented path from a vertex $x$ to $w_1$, there cannot be an oriented path from $x$ to $w'_1$, and vice versa. Since, in addition, $\sigma_{w'_1}(e')=1=\sigma_{w_1}(e)$, we have $k_e+k_{e'}=F_e + F_{e'} +G_e + G_{e'}$ with $F_e + F_{e'}=-(-o_v (k_f+k_{f'})+o_{w} k_f +o_{w'} k_{f'})$ where $o_x$ is one if there is an oriented path from $x$ to either $w_1$ or $w'_1$, and zero otherwise. This completes the proof of the Lemma. \end{proof} \begin{proposition}\label{th:pairmom} Suppose $e, e'\in \edges$, $e\ne e'$, are such that $\fedges_e=\fedges_{e'}$. Then either $k_e=0=k_{e'}$ or removal of $e$ and $e'$ from $\graph$ splits it into exactly two components. \end{proposition} \begin{proof} If $\fedges_e=\fedges_{e'}=\emptyset$, then by Corollary \ref{th:zerok} $k_e=0=k_{e'}$, and the first alternative holds. If both $e$ and $e'$ are free, then $\fedges_e=\fedges_{e'}$ implies $e'=e$, which was not allowed. We can thus assume that $k_e,k_{e'}\ne 0$, and that at least one of $e,e'$ is integrated. Let $\graph''$ denote the graph obtained by removing $e$ and $e'$ from $\graph$. Suppose next that one edge is free, but the other is not. By symmetry, we can choose $e'$ to be the free one, and assume $e=(v_1,v_2)$. Let $P=\mathcal{P}(v_1)$. Since $e'\in \fedges_e$, $e'=\set{v,v'}$ where $v\in P$ and $v'\not\in P$. Removal of $e'$ does not change the number of components, since the spanning tree is not affected by it. We thus need to show that the subsequent removal of $e$ will split the component. Suppose $f=\set{w,w'}$ is an edge such that $w\in P$ and $w'\not\in P$. If $f\ne e$, $f$ is free by Lemma \ref{th:kesol3}, and thus $f\in \fedges_e$ which implies $f=e'$. Thus $e,e'$ are the only edges with this property, and every path from the component containing $P$ to the one containing $P^c$ must use either $e$ or $e'$. On the other hand, the vertices in $P$ (respectively $P^c$) can be connected without using $e$ or $e'$, and thus $\graph''$ has exactly two components. Thus we can assume that neither $e$ nor $e'$ is free. We identify $e'=(v'_1,v'_2)$ and $e=(v_1,v_2)$ in $\oTedges$. Let $P=\mathcal{P}(v_1)$, and $P'=\mathcal{P}(v'_1)$. Suppose first that the oriented path from $v_1$ to root contains $v_1'$. Then $P\subset P'$, and we claim that $P\cup (P')^c$ and $P'\setminus P$ span independent connected components in $\mathcal{G}''$. Suppose $f=\set{w,w'}$ is free and $w\in P' \setminus P$. If $w'\in P$, then $f\in \fedges_e\setminus \fedges_{e'}$, which is empty by assumption. Similarly, if $w'\not\in P'$, then $f\in \fedges_{e'}\setminus \fedges_{e}=\emptyset$. Thus $w'\in P' \setminus P$. This implies that cutting $e$ and $e'$ isolates $P' \setminus P$ from both $P$ and $(P')^c$. $P' \setminus P$ is not empty since it contains at least $v_1'$, and for any $w\in P'\setminus P$ there is an oriented path from $w$ to $v'_1$ which, by the above results, cannot contain $e$ nor (obviously) $e'$. Thus $P' \setminus P$ spans a connected component in $\mathcal{G}''$. Both $P$ and $(P')^c$ are connected in $\mathcal{G}''$ (note that every vertex in $(P')^c$ has a path to the root which does not go via $v_1'$). On the other hand, there must be a free edge connecting $P$ and $(P')^c$, which thus is different from $e$ or $e'$, as else $k_{e}=0$. Therefore, $P\cup (P')^c$ spans the second, and last, component in $\mathcal{G}''$. We have proven the result for the case the oriented path from $v_1$ to root contains $v'_1$ and obviously this also proves the result in the case if the path from $v'_1$ to root contains $v_1$. Thus we can assume that the oriented paths from $v_1$ and $v_1'$ to root are not contained in each other. This implies that $P\cap P'=\emptyset$. If there exists a path from $P$ to $(P\cup P')^c$ avoiding $e$, there is a free edge between these sets which thus belongs to $\fedges_{e}\setminus \fedges_{e'}=\emptyset$. Similarly, there cannot be any path from $P' $ to $(P\cup P')^c$ avoiding $e'$. Since $k_e\ne 0$, there must thus be a free edge from $P$ to $P'$. Therefore, $P\cup P'$ and $(P\cup P')^c$ span disjoint connected components in $\mathcal{G}''$. This completes the proof of the Proposition. \end{proof} \begin{lemma}\label{th:pairmom2} Suppose $e, e'\in \edges$, $e\ne e'$. Then $\fedges_e=\fedges_{e'}$ if and only if there is $\sigma\in \set{\pm 1}$ such that $k_e=\sigma k_{e'}$ independently of the free momenta. \end{lemma} \begin{proof} If $k_e=\sigma k_{e'}$, then by uniqueness of the representation in Corollary \ref{th:zerok} $\fedges_e=\fedges_{e'}$. If $e$ and $e'$ are both free, then $\fedges_e=\fedges_{e'}$ implies $e'=e$ which is not allowed. If one of them is free, say $e'$, then $\fedges_e=\fedges_{e'}$ and (\ref{eq:kesol3}) imply $k_e= \pm k_{e'}$. Thus we can assume that $e$ and $e'$ are not free, and set $e=(v_1,v_2)$, $e'=(v'_1,v'_2)$. We also denote $P=\mathcal{P}(v_1)$, $P'=\mathcal{P}(v'_1)$. Suppose first that $P\subset P'$. Then for any $f\in \fedges_e$ there are $v\in P$, $v'\not\in P$ such that $f=\set{v,v'}$. Since also $f\in \fedges_{e'}$ then necessarily $v\in P'$, $v'\not\in P'$, and the factor $\sigma_v(f)$ is the same in the representation (\ref{eq:kesol3}) both of $k_e$ and of $k_{e'}$. This implies that $k_{e'}=\sigma_{v'_1}(e')\sigma_{v_1}(e)k_e$, in accordance with the Corollary. By symmetry, the same results also holds if $P'\subset P$. If the oriented path from $v_1$ to the root is contained in the oriented path from $v'_1$ to the root, then $P\subset P'$, and if the path from $v'_1$ is contained in the path from $v_1$, then $P'\subset P$. Thus we can assume the converse, which clearly implies $P\cap P'=\emptyset$. Then if $f\in \fedges_e=\fedges_{e'}$, we have $f=\set{v,v'}$ where $v\in P$, $v'\not \in P$ and $v\not \in P'$, $v'\in P'$. Thus by Lemma \ref{th:mesigns} $\sigma_v(f)=-\sigma_{v'}(f)$, and we can conclude from (\ref{eq:kesol3}) that $k_{e'}=-\sigma_{v'_1}(e')\sigma_{v_1}(e)k_e$. This concludes the proof of the Lemma. \end{proof} \begin{corollary} \label{th:kkpl} Let $v\in \Iverts$, and suppose $e,e'\in \edges_-(v)$, $e\ne e'$. Then $k_e+k_{e'}$ is independent of all free momenta if and only if the initial time vertices at the bottom of the interaction trees starting from $e$ and $e'$ are isolated from the rest of the initial time vertices. In this case, $k_e+k_{e'}=0$. \end{corollary} \begin{proof} Suppose $k_e+k_{e'}$ is independent of all free momenta. By Corollary \ref{th:zerok} this is possible only if, in fact, $k_{e'}=-k_e$. In particular, then $\fedges_e=\fedges_{e'}$ and by Proposition \ref{th:pairmom} either $k_e=0=k_{e'}$, or removing $e,e'$ splits a connected component. Denote the set of initial time vertices at the bottom of the interaction subtree starting from $e$ ($e'$) by $D_e$ ($D_{e'}$). If $k_e=0$, then $k_e=0=k_{e'}$ which implies that $D_e$ and $D_{e'}$ are separately isolated from the rest of the initial time vertices, and the theorem holds. Otherwise, we can assume that removing $e,e'$ splits the graph into two components. Thus there can be no connection from $D_e\cup D_{e'}$ to its complement in $\Dverts$. This proves the ``only if'' part of the theorem. For the converse, suppose $D_e\cup D_{e'}$ are isolated from the rest of the initial time vertices. If there is no connection between $D_e$ and $D_{e'}$ then any path from $D_e$ to the root must go via $e$, which implies that $k_e=0$. Similarly, then $k_{e'}=0$, and thus also $k_e+k_{e'}=0$. If there is a connection between $D_e$ and $D_{e'}$, then the larger of the edges $e$, $e'$ is integrated, the other is free, and they sum to zero. This completes the proof of the theorem. \end{proof} \begin{corollary}\label{th:zeroisirr} If the momentum graph has an edge $e\ne e_0$ such that $k_e=0$ identically, $S$ contains a cluster with odd number of elements. \end{corollary} \begin{proof} Suppose there is an edge $e$ such that $k_e=0$ identically. If $e\in\edges_-(v)$ for some fusion vertex $v$, then the argument used in the proof of Corollary \ref{th:kkpl} shows that the subtree spanned by $e$ must have isolated clustering. This implies that the size of one of the clusters is odd. Since $e\ne e_0$, we can then assume that $e$ contains a cluster vertex. However, since every cluster has a size of at least two, removal of one such edge cannot split the graph. This contradicts $k_e=0$. \end{proof} The following theorem proves that the number of free momenta is independent of the choice of the spanning tree. It is a standard result and included here mainly for the sake of completeness. \begin{proposition}\label{th:loopsinv} Let $\Ttree_1=(\verts,\edges_1)$ and $\Ttree_2=(\verts,\edges_2)$ be spanning forests of a graph $\mathcal{G}=(\verts,\edges)$. Then $|\edges_2|=|\edges_1|$. \end{proposition} \begin{proof} We make the proof by induction in $|\edges_2\setminus \edges_1|$. If this number is zero, then $\edges_2\subset\edges_1$, and as $\edges_1$ cannot contain any loops, we have $\edges_2=\edges_1$, and the theorem holds. Make the induction assumption that the theorem holds up to $N\ge 0$. Consider $\edges_2$ such that $|\edges_2\setminus \edges_1|=N+1$. Then there is $f_0\in \edges_2\setminus \edges_1$. Adding $f_0$ to $\Ttree_1$ creates a unique loop. Let $f'_i$, $i=1,\ldots,n$, count the momenta along this loop which do not belong to $\edges_2$, i.e., which belong to $\edges_1\setminus \edges_2$. Then $n\ge 1$, as otherwise $\Ttree_2$ would contain a loop. On the other hand, adding one of $f'_i$ to $\Ttree_2$ also creates a unique loop. If none of these new loops contains $f_0$, then $\edges_2$ has a loop: if $f_0=\set{a,b}$, one can start from $a$, follow the first loop, and go around each $f'_i$ along the new loops, arriving finally to $b$. Thus we can assume that $f'\in \edges_1\setminus \edges_2$ is such that it belongs to the loop created by $f_0$ in $\Ttree_1$, and $f_0$ belongs to the loop created by $f'$ in $\Ttree_2$. We then set $\edges_3=(\edges_2\cup \set{f'})\setminus \set{f_0}$, and consider $\Ttree_3=(\verts,\edges_3)$. Since the removal of $f_0$ cuts the unique loop generated by $f'$ in $\edges_2$, $\Ttree_3$ has no loops, i.e., it is also a forest. If $g\in \edges_3^c$, then $g\ne f'$ and either $g\in \edges_2^c$ or $g=f_0$. Adding $f_0$ to $\Ttree_3$ creates a loop by construction. Else $g\in \edges_2^c$, and adding it to $\Ttree_2$ creates a unique loop. If $f_0$ is not along this loop, it is composed of edges in $\edges_3$, and adding $g$ to $\Ttree_3$ creates a loop. If $f_0$ is along this loop, we can avoid it by using the loop created by the addition of $f'$, and construct a loop out of edges in $\set{g}\cup\edges_3$. Thus $\Ttree_3$ is a spanning forest. Since $|\edges_3\setminus\edges_1|=N$, we can apply the induction assumption to it, which shows that $|\edges_2|=|\edges_3|=|\edges_1|$. This completes the induction step. \end{proof} \begin{proposition}\label{th:noffreemom} A momentum graph has exactly $2 N+2-|S|$ free momenta, where $N=n+n'$. \end{proposition} \begin{proof} We construct a second spanning tree by first going from top to bottom, then adding the edges containing the cluster vertices, going from left to right (this is exactly the opposite order in which the spanning tree was constructed before). Clearly, the spanning tree then contains all edges in the interaction tree, and exactly one edge per cluster in $S$ (the first edge connects the cluster vertex to the tree, but every further edge would create a loop). Thus there are altogether $\sum_{A\in S} (|A|-1)=2 N+2-|S|$ free edges attached to the cluster vertices. By Proposition \ref{th:loopsinv}, the number of free momenta is independent of the choice of the spanning tree, and thus the result holds also for the first spanning tree. \end{proof} \section{Expansion parameters and classification of graphs} \label{sec:classification} \begin{definition}[Expansion parameters]\label{th:defkappaetc} Let $\delta$ be a constant for which the dispersion relation $\omega$ satisfies the dispersion bound (DR3), and $\gamma$ be a constant for which the dispersion relation satisfies the crossing bounds in (DR4). We define \begin{align}\label{eq:defgammap} b= \frac{3}{4}, \quad \gamma' = \min (\frac{1}{4},2 \gamma,2\delta),\quad a_0 = \frac{\gamma'}{24},\quad\text{and}\quad b_0 =16 \Bigl( 3 + \frac{1}{a_0}\Bigr). \end{align} For any $\lambda>0$ let us then define \begin{align}\label{eq:chooseN0} \vep=\lambda^2\quad\text{and}\quad N_0(\lambda) = \max\Bigl(1,\Bigl\lfloor\,\frac{a_0 \, |\ln \lambda|}{\ln \sabs{\ln \lambda}}\, \Bigr\rfloor\Bigr) \, , \end{align} where $\lfloor x\rfloor$ denotes the integer part of $x\ge 0$. Let also, with $N_0=N_0(\lambda)$, \begin{align}\label{eq:defkappa} \kappa'(\lambda)=\lambda^2 N_0^{b_0}\qand \kappa_n(\lambda) = \begin{cases} 0,& 0\le n< N_0/2\, ,\\ \kappa'(\lambda), & N_0/2 \le n\le N_0\, . \end{cases} \end{align} \end{definition} The definition of $b$, associated with the removal of the singular manifold, coincides with the one given earlier in Section \ref{sec:graphs}. For this choice of parameters, in the limit $\lambda\to 0^+$ we have $N_0\to \infty$, $\max_n\kappa_n\to 0$, and \begin{align}\label{eq:N0limit1} \frac{N_0(\lambda) \ln \sabs{\ln\lambda}}{|\ln \lambda|} \to a_0\qand \frac{N_0(\lambda) \ln N_0(\lambda)}{|\ln \lambda|} \to a_0 \, . \end{align} If $c,c'>0$, $n_1,n_2,n_3\in \N_+$, and $p_1,p_2\in \R$ are some fixed given constants, then using $n!\le n^n$ we easily find that \begin{align}\label{eq:N0limit2} c^{N_0} \lambda^{p_1} N_0^{p_2 N_0+c'} ((n_1 N_0)!)^{n_2} \sabs{\ln \lambda}^{n_3 N_0+c'} \to 0\, , \end{align} as soon as the inequality $p_1- a_0( p_2+ n_1 n_2 +n_3)>0$ is satisfied. The decay is then actually powerlaw in $\lambda$, with the supremum of the power determined by the above difference. For instance, with our choices of $a_0,b_0$, we have up to a powerlaw $\lambda^2$ decay in \begin{align}\label{eq:N0lima} c^{N_0} \lambda^{-2} N_0^{-b_0\frac{1}{4} N_0+4 N_0+c'} (4 N_0)! \sabs{\ln \lambda}^{4 N_0+c'} \to 0\, , \end{align} and up to a powerlaw $\lambda^{\gamma' \! /2}$ decay in \begin{align}\label{eq:N0limb} c^{N_0} \lambda^{\gamma'} N_0^{4 N_0+c'} (4 N_0)! \sabs{\ln \lambda}^{4 N_0+c'} \to 0\, . \end{align} Consider a generic momentum graph, defined using parameters $(S,J,n,\ell,n',\ell')$. We integrate out all the momentum constraints using the spanning tree which respects the time-ordering, as explained in the previous section. We recall also the definition of a degree of an interaction vertex (we stress here that this concept is not a graph invariant, and thus depends on the way we have constructed the spanning tree). By Proposition \ref{th:momatintv}, the degree counts the number of free momenta ending at the vertex, and it belongs to $\set{0,1,2}$. The following terminology will be used from now on: \begin{definition} Consider a time slice $i\in I_{0,n+n'}$ in a momentum graph. If it has exactly zero length, it is called \defem{amputated}. If it ends in an interaction vertex of degree $1$ or $2$ it is called {\em short\/}. Otherwise, it is called {\em long\/}. \end{definition} By this definition, the time-slice $i=n+n'$ is always long. The graph is called \begin{description} \setlength{\itemsep}{0pt} \item[{\em irrelevant,}\/] if the amplitude corresponding to the graph is identically zero. Otherwise it is {\bf\em relevant.} \item[{\em pairing,}\/] if for every $A\in S$ we have $|A|=2$. Otherwise it is {\bf\em non-pairing.} \item[{\em higher order,}\/] if it is a relevant non-pairing graph. \item[{\em fully paired,}\/] if it is a pairing graph and has no interaction vertices of degree one. A pairing graph which is not fully paired is called {\bf\em partially paired.} \end{description} Clearly, if there is $A\in S$ such that $|A|$ is odd, the graph is irrelevant. \begin{figure} \centering \myfigure{height=0.7\textheight}{Leadmotiv} % \myfigure{width=0.9\textwidth}{Leadmotiv} \caption{Half of the leading motives: the first four depict ``gain motives'', and the lower six ``loss motives''. The ``truncated'' lines denote the places where the motive is attached to a graph with the appropriate parities shown next to the line. The dashed lines will always extend to the initial time vertices, i.e., these parts of the edges will stretch over several time slices when more vertices are added below the motive. The remaining leading motives can be obtained from the above by inverting the parities of all edges, and then inverting the order of the edges below the interaction vertices.\label{fig:leadmot}} \end{figure} Every fully paired graph has thus interaction vertices only of degree $0$ and $2$. Such graphs can be obtained for instance by iteration of the graph ``motives'' depicted in Figure \ref{fig:leadmot}. These motives are called {\em leading motives\/} or {\em immediate recollisions\/}. The latter name comes from the fact that the motive does not change the ``incoming'' momentum. The motives can be iteratively attached to a graph in two ways: the gain motives can replace any pairing cluster (the order of parities of the cluster determines which four of the total eight gain motives can be used: the first four in Figure \ref{fig:leadmot} are used for replacing $(-1,1)$-pairings), and the loss motives can be attached to any line with the correct parity (the bottom six in Figure \ref{fig:leadmot} can be attached to a line of parity $1$). Any graph which is obtained by such an iteration starting from the simple graph corresponding to $n=0=n'$ (a single loop) is called a leading term graph. A straightforward induction shows that a leading term graph is fully paired. Furthermore, the fully paired graphs are classified into three categories, depending on properties of the phase factors of long time-slices: a fully paired graph is called \begin{description} \setlength{\itemsep}{0pt} \item[{\em leading,}\/] if it is formed by iteration of leading motives. \item[{\em crossing or nested,}\/] otherwise. \end{description} The precise classifications require technical definitions, to be given in Section \ref{sec:fullypaired}. We only mention here that in a nested graph the first short time-slices (at the bottom of the graph) consist of leading motives ``nested'' inside another leading motive. (This explains the name, already used in \cite{erdyau99} for a similar construction, although it would be more precise to call our nested graphs as ones which {\em begin\/} with a nest.) \subsection{Iterative cluster scheme}\label{sec:iterclsuters} An important estimate of the magnitude of the amplitude associated with a momentum graph, is the total number of interaction vertices of the various degrees. We denote by $n_i$ the number of interaction vertices of degree $i$. Then for instance $n_0+n_1+n_2=n+n'$, and the following Lemma captures other basic relations between these numbers for relevant graphs. \begin{lemma}\label{th:ivdegrees} Consider a relevant graph, and let $r=N+1-|S|$, $N=n+n'$. Then $\Nnp\le r \le N$, where $\Nnp:=|\defset{A\in S}{|A|>2}|$ is the number of clusters which are not pairs, and $r=0$ if and only if $S$ is a pairing. In addition, $n_2-n_0=r$, $n_0=\frac{1}{2}(N-r-n_1)$, and $0\le n_1\le N$, $0\le n_0\le \lfloor\frac{N-r}{2}\rfloor$. \end{lemma} \begin{proof} The graph is relevant and thus has no odd clusters. Then a cluster in $S$ is either a pair, or has a size of at least $4$. Therefore, $2 N +2 = \sum_{A\in S}|A|= 2 |S| + \sum_{A\in S}(|A|-2) \ge 2|S|+ 2 \Nnp$. This implies $r\ge \Nnp$ and $r\le N$ is obvious. Also $r=0$ if and only if $S$ is a pairing. If there is a cluster which contains initial time vertices from both plus and minus trees, then adding the second edge ($e_1$) to the top fusion vertex $v_{N+1}$ creates a loop. Otherwise, the clusters of plus and minus trees are separated, which implies that both contain at least one odd cluster. Therefore, in a relevant graph exactly one free momenta is not attached to an interaction vertex. By Proposition \ref{th:noffreemom} thus $2 N+1-|S|=2 n_2+n_1$, and clearly also $N=n_0+n_1+n_2$. We substitute $|S|=N+1-r$, and find the stated formulae for $n_2$ and $n_0$. The upper bound for $n_0$ then arises, as $n_1\ge 0$ and $n_0$ needs to be an integer. \end{proof} We will also use the corresponding cumulative counters: we let $n_j(i)$, $j=0,1,2$, denote the number of interaction vertices of degree $j$ below and including $v_i$. \begin{proposition}\label{th:ivordering} Consider an interaction vertex $v_i$, $1\le i \le N$, in a relevant graph. Then always \begin{align}%\label{eq:} n_2(i)\le r + n_0(i)\qand n_0(i)\ge \frac{i-(n_1+r)}{2}\, , \end{align} where $r=N+1-|S|$ and $n_1=n_1(N)$, as in Lemma \ref{th:ivdegrees}. \end{proposition} The proof of Proposition \ref{th:ivordering} is based on one more construction related to the momentum graph, which we call the \defem{iterative cluster scheme}. Since the scheme will reappear later, let us first explain it in detail. Let us consider the evolution of the cluster structure while the spanning tree is being built. We define $S^{(0)}=S$ and let $S^{(i)}$ denote a clustering of the edges intersecting the time-slice $i$, induced by the following iterative procedure where the interaction vertices are added to the graph, one by one from bottom to top. The addition of the vertex $v_i$ will thus fuse the three edges in $\edges_-(v_i)$ into the one in $\edges_+(v_i)$. All the three ``old'' edges belong to some clusters in $S^{(i-1)}$ while the ``new'' edge does not appear there. We construct $S^{(i)}$ by first joining all clusters in $S^{(i-1)}$ which intersect $\edges_-(v_i)$, and then replacing the three edges by the unique new one in $\edges_+(v_i)$. The rest of the clusters are kept unchanged. If two of the three old edges belong to the same cluster in $S^{(i-1)}$, then adding the second one would create a loop in the construction of the spanning tree. Similarly, if all three edges go into the same cluster, then this creates two separate loops. Therefore, this also determines the degree of the added interaction vertex: if the vertex joins three separate old clusters, it has degree $0$, if it joins two clusters, it has degree $1$, and if all edges belong to the same previous cluster, it has degree $2$. By going through all the alternatives, we then find that in the iteration step the number of clusters changes as $|S^{(i)}|=|S^{(i-1)}|-2+\deg{v_i}$. In addition, the structure of the clustering is conserved, in the sense that each $S^{(i)}$ contains only even (and non-empty) clusters. \begin{proofof}{Proposition \ref{th:ivordering}} Consider the iterative cluster scheme, where for each $i$, the set $S^{(i)}$ is a partition of $2(N-i)+2$ elements. Since all clusters have size of at least two, we thus have $|S^{(i)}|\le N-i+1$. On the other hand, here \begin{align}%\label{eq:} |S^{(i)}| = |S|-\sum_{j=1}^i (2-\deg v_j) = |S|-2 i + n_1(i)+ 2 n_2(i)\, . \end{align} Since by construction $i=n_0(i)+n_1(i)+n_2(i)$, we have proven that \begin{align}%\label{eq:} n_2(i) \le N+1-|S|+i-n_1(i)-n_2(i)= r+n_0(i), \end{align} as claimed in the Proposition. But then also \begin{align}%\label{eq:} i\le 2 n_0(i)+n_1(i)+r\le 2 n_0(i)+n_1+r\, , \end{align} from which the second inequality follows. \end{proofof} \section{Main lemmata} \label{sec:lemmas} We have collected here the main technical tools and results to be used in the proof of the main estimates. \subsection{Construction of momentum cutoff functions} \label{sec:cutoff} We first explain the construction of the cutoff function $\PFzero$, and prove that it satisfies Proposition \ref{th:PFcorr} which was used in Section \ref{sec:graphs} in the derivation of the basic Duhamel formulae. We recall that $b=\frac{3}{4}$. Let $\Msing$ denote the singular manifold in Assumption \ref{th:disprelass}. Then there are $\Ns > 0$ and smooth closed one-dimensional submanifolds $M_j$, $j=1,\ldots,\Ns$, of $\T^d$ such that $\Msing=\bigcup_{j=1}^{\Ns} M_j$. Since the manifold $M_j$ is actually compact, for each $j$ there exists $\vep_j>0$ such that the map $k\mapsto d(k,M_j)$ is smooth in the neighborhood $U_j:= \defset{k\in \smash{\T^d}}{d(k,M_j)<\vep_j}$ of $M_j$. We define $\vep_0=\min_j \vep_j$, when $\vep_0>0$, and consider an arbitrary $\vep$ such that $0<\vep< \vep_0$. We recall here that $\Msing\ne \emptyset$ since at least $0\in \Msing$. We choose an arbitrary one-dimensional smooth ``step-function'' $\varphi$. Explicitly, we assume that $\varphi\in C^\infty(\R)$ is symmetric, $\varphi(-x)=\varphi(x)$, monotone on $[0,\infty)$, and $\varphi(x)=0$ for $|x|\ge 1$ and $\varphi(x)=1$ for $|x|\le \frac{1}{2}$. In particular, then $\varphi(0)=1$. We define further, for $0<\vep< \vep_0$, $j=1,\ldots,\Ns$, the functions $f^{j}:\T^d \to [0,1]$ by \begin{align}%\label{eq:} f^{j}(k;\vep) = \varphi\Bigl(\frac{d(k,M_j)}{\vep}\Bigr), \quad k\in\T^d\, . \end{align} Then $f^{j}(k;\vep) =0$ for all $d(k,M_j)\ge \frac{\vep}{\vep_j}\vep_j$, where $\frac{\vep}{\vep_j}<1$. Thus, by construction, $f^j$ is smooth on $\T^d$, and we can find a constant $C$ independent of $\vep$ such that $|\nabla f^j(k;\vep)|\le \frac{C}{\vep}$ for all $j,k,\vep$. In addition, we have $f^{j}(k;\vep)=1$ if $k\in M_j$. Next we construct $d$-dimensional cut-off functions. Let $\lambda'_0 = \min(1,\lambda_0,\vep_0^{1/b})$, and define for all $0<\lambda<\lambda'_0$ the functions $\Fone,\Fzero:\T^d\to \R$ by \begin{align}%\label{eq:} \Fone(k) = \prod_{j=1}^{\Ns} \left( 1-f^{j}(k;\lambda^b) \right), \quad \Fzero = 1-\Fone \, . \end{align} \begin{lemma}\label{th:Foneprop} There is a constant $C_1\ge 1$ such that for any $0<\lambda<\lambda'_0$, \begin{jlist} \item $0\le \Fone,\Fzero \le 1$. \item If $k\in \Msing$, then $\Fone (k)=0$ and $\Fzero (k)=1$. \item If $d(k,\Msing)\ge \lambda^b$, then $\Fone (k)=1$ and $\Fzero (k)=0$. \item $\Fone,\Fzero$ are smooth, and $|\nabla\Fone(k)|, |\nabla\Fzero(k)| \le C_1 \lambda^{-b}$, for all $k$. \item $0\le \Fone (k)\le C_1 \lambda^{-b} d(k,\Msing)$ for all $k\in \T^d$. \item There is a constant $C$ such that \begin{align}%\label{eq:} \int_{\T^d} \rmd k\, \Fzero(k) \le \int_{\T^d} \rmd k\, \1(d(k,\Msing)<\lambda^b) \le C \lambda^{b(d-1)} \, . \end{align} \end{jlist} \end{lemma} \begin{proof} The first four items follow from the above-mentioned properties of $f^j$. For the fifth item, fix $k$ and let $l=d(k,\Msing)$. If $l\ge \lambda^b$, then $\Fone(k)=1$, and the inequality holds trivially for any $C_1\ge 1$. If $l< \lambda^b$, then $l<(\lambda'_0)^b\le\vep_0$. Since $\Msing$ is a compact set, there are $j$ and $k'\in M_j$, such that $l=d(k,k')$. In addition, there is a smooth path $\gamma:[0,1]\to U_j$ from $k'$ to $k$ such that $d(k,k')=\int_0^1 \rmd s |\gamma'(s)|$. Then $\Fone(k)=\Fone(k)-\Fone(k')=\int_0^1 \rmd s \frac{\rmd}{\rmd s} \Fone(\gamma(s))$. Using the chain rule, and then applying the result in item 4, shows that $\Fone (k)\le C_1 \lambda^{-b} l$ also in this case. The last estimate follows by first estimating $\Fzero(k)\le \1(d(k,\Msing)<\lambda^b)$, and then using the compactness of the manifold and the fact that it has maximally codimension $d-1$. \end{proof} Now we are ready to define the $3 d$-dimensional cut-off functions introduced in Section \ref{sec:graphs}. We define $\PFone,\PFzero:(\T^d)^3 \to [0,1]$ by \begin{align}%\label{eq:} \PFone(k_1,k_2,k_3) = \Fone (k_1+k_2)\Fone (k_2+k_3) \Fone (k_3+k_1)\, , \quad \PFzero=1-\PFone . \end{align} \begin{proofof}{Proposition \ref{th:PFcorr}} Inequality (\ref{eq:PFzeroineq}) follows from the previous properties, since for any $0\le a_i\le 1$, $i=1,2,3$, it holds that $1-\prod_{i=1}^3 (1-a_i) \le a_1+a_2+a_3$. The other points are obvious corollaries of Lemma \ref{th:Foneprop}. \end{proofof} \subsection{From phases to resolvents} The following result generalizes the standard formula used in connection with time-dependent perturbation expansions. \begin{theorem}\label{th:resolvents} Let $I$ be a non-empty finite index set, assume $t>0$, and let $\gamma_i\in D$, $i\in I$, with $D\subset \C$ compact. Suppose $A$ is a non-empty subset of $I$. We choose an additional time index label $*$, i.e., assume $*\not\in I$, and let $A^{\rm c}=I\setminus A$, and $A'=A^{\rm c}\cup\set{*}$. Then for any path $\Gamma_D$ going once anticlockwise around $D$, we have \begin{align}\label{eq:phtores} &\int_{(\R_+)^{I}}\!\rmd s \, \delta\Bigl(t-\sum_{i\in I} s_i\Bigr) \prod_{i\in I}\rme^{-\ci \gamma_i s_i} \nonumber \\ & \quad = -\oint_{\Gamma_D} \frac{\rmd z}{2\pi} \int_{(\R_+)^{A'}}\!\rmd s \, \delta\Bigl(t-\sum_{i\in A'} s_i\Bigr) \prod_{i\in A'} \left.\rme^{-\ci \gamma_i s_i}\right|_{\gamma_*=z} \prod_{i\in A} \frac{\ci}{z-\gamma_i} \, . \end{align} \end{theorem} \begin{proof} Let us first consider the case $A=I$. Then $A'=\set{*}$, and by definition the ``$s$-integral'' on the right hand side yields a factor $\rme^{-\ci z t}$. Therefore, in this case the formula is equal to the standard formula (whose proof under the present assumptions can be found for instance from Lemma 4.9 in \cite{ls05}). If $A\ne I$, there is $i_0\in A^{\rm c}$. Resorting to the definition of the time-integration as an integral over a standard simplex, it is straightforward to prove that now \begin{align} & \int_{(\R_+)^{I}}\!\rmd s \, \delta\Bigl(t-\sum_{i\in I} s_i\Bigr) \prod_{i\in I}\rme^{-\ci \gamma_i s_i} \nonumber \\ & \quad = \int_0^t \!\rmd s_{i_0} \rme^{-\ci \gamma_{i_0} s_{i_0}} \Bigl[\int_{(\R_+)^{I'}}\!\rmd s \, \delta\Bigl(t-s_{i_0}-\sum_{i\in I'} s_i\Bigr) \prod_{i\in I'}\rme^{-\ci \gamma_i s_i} \Bigr] \, , \end{align} where $I'=I\setminus\set{i_0}$. Therefore, we can now perform an induction in the number of elements in $A^{\rm c}$, starting from $|A^{\rm c}|=0$. Applying the above formula, induction assumption, and then Fubini's theorem shows that (\ref{eq:phtores}) is valid for all $A$. \end{proof} \subsection{Cluster combinatorics} \begin{lemma}\label{th:clustercomb} There is a constant $c$ such that for all $N>0$, $0<\lambda<\lambda_0$, \begin{align}\label{eq:ccmbbound} \sum_{S\in\pi(I_{N})} % |A|>2\text{ for some }A\in S}} \prod_{A\in S} \sup_{\Lambda,k,\sigma}|C_{|A|}(k,\sigma;\lambda,\Lambda)| \le c^{N} N!\, . \end{align} If the sum is restricted to non-pairing $S$, then the bound can be improved by a factor of $\lambda$. \end{lemma} \begin{proof} Any $S$ which is non-pairing either has an odd cluster, or contains a cluster of size of at least four. If there is an odd cluster, the corresponding $C_{|A|}$ term is zero, and thus any positive bound works for them. We cancel all partitions containing a singlet, and use the bound in (\ref{eq:Cnbound}) for all clusters which are not pairs. As proven in Section \ref{sec:cumulants}, the constant can be adjusted so that for pairs we can use (\ref{eq:Cnbound}) without the factor of $\lambda$. Let $\pi'(I_N)$ consist of all partitions of $I_N$ which do not contain singlets, i.e., of $S\in \pi(I_N)$ such that $|A|\ge 2$ for all $A\in S$. Then \begin{align}%\label{eq:} & \sum_{\substack{S\in\pi(I_{N}),\\ S\text{ not a pairing}}} \prod_{A\in S} \sup_{k,\sigma}|C_{|A|}(k,\sigma;\lambda,\Lambda)| \le \lambda \sum_{S\in\pi'(I_{N})} \prod_{A\in S} \left( (c_0)^{|A|} |A|!\right) \end{align} and \begin{align}%\label{eq:} & \sum_{S\in\pi(I_{N})} \prod_{A\in S} \sup_{k,\sigma}|C_{|A|}(k,\sigma;\lambda,\Lambda)| \le \sum_{S\in\pi'(I_{N})} \prod_{A\in S} \left( (c_0)^{|A|} |A|!\right) \nonumber \\ & \quad \le (c_0)^{N} \sum_{m=1}^{\lfloor N/2\rfloor} \sum_{S\in\pi(I_{N})} \1(|S|=m) \prod_{A\in S} |A|! \, . \end{align} A combinatorial computation along the proof of Lemma C.4 in \cite{ls05} shows that \begin{align}%\label{eq:} & \sum_{S\in\pi(I_{N})} \1(|S|=m) \prod_{A\in S} |A|! = \frac{N!}{m!} \sum_{n\in \N_+^m} \1\Bigl(\sum_{j=1}^m n_j = N\Bigr) %\nonumber \\ & \quad = \frac{N!}{m!} \binom{N-1}{m-1} \le \frac{N!}{m!} (N-1)^{m-1}\, . % \frac{(2 n-1)!}{(m-1)! (2n -m)!} \end{align} The sum over $m$ from $1$ to $\infty$ of the last bound is bounded by $N! \rme^{N}$. This proves that (\ref{eq:ccmbbound}) holds with $c=c_0 \rme$. \end{proof} \subsection{Integrals over free momenta} \label{sec:freeint} \begin{proposition}\label{th:defendelta} Suppose the assumption (DR\ref{it:DRdisp}) holds with constants $C,\delta>0$, and assume $f \in \ell_1((\Z^d)^3)$. Then for all $s\in \R$, $k_0\in \T^d$, and $\sigma,\sigma'\in\set{\pm 1}$, \begin{align}\label{eq:defendelta} &\left| \int_{(\T^d)^2} \!\rmd k'\rmd k\, \rme^{\ci s (\omega(k) + \sigma' \omega(k')+ \sigma \omega(k_0-k-k'))} \FT{f}(k,k',k_0-k-k') \right| \le C\norm{f}_1 \sabs{s}^{-1-\delta}\, . \end{align} In particular, \begin{align}\label{eq:leadphasebnd} &\left| \int_{(\T^d)^2} \!\rmd k'\rmd k\, \rme^{\ci s (\omega(k) + \sigma' \omega(k')+ \sigma \omega(k_0-k-k'))} \right| \le C\sabs{s}^{-1-\delta}\, . \end{align} \end{proposition} In particular, the Proposition implies that $\Gamma(k_1)$ in (\ref{eq:defGamma}) is well defined. Adapting the proof of Proposition A.1 in \cite{ls05}, the Proposition also shows that our assumptions on the dispersion relation $\omega$ guarantee that the map \begin{align} F\mapsto \lim_{\beta\to 0^+} \int_{(\T^d)^2} \!\rmd k_2\rmd k_3\, \frac{\beta}{\pi} \frac{1}{(\omega_1+\omega_2-\omega_3-\omega_4)^2+\beta^2}F(k_2,k_3,k_1-k_2-k_3) \, , \end{align} where $\omega_4=\omega(k_1-k_2-k_3)$, defines for all $k_1\in \T^d$ a bounded positive Radon measure on $(\T^d)^3$ which we denote by $\rmd k_2\rmd k_3\rmd k_4\, \delta(k_1+k_2-k_3-k_4) \delta(\omega_1+\omega_2-\omega_3-\omega_4)$. In addition, if $F\in L^2((\T^d)^3)$ has summable Fourier-transform, we also have \begin{align} & \int_{(\T^d)^3} \!\rmd k_2\rmd k_3\rmd k_4\, \delta(k_1+k_2-k_3-k_4) %\nonumber \\ & \quad \times \delta(\omega_1+\omega_2-\omega_3-\omega_4) F(k_2,k_3,k_4) \nonumber \\ & \quad = \int_{-\infty}^{\infty}\!\frac{\rmd s}{2\pi} \left[ \int_{(\T^d)^3} \!\rmd k_2\rmd k_3\rmd k_4\, \delta(k_1+k_2-k_3-k_4) \rme^{\ci s (\omega_1+\omega_2-\omega_3-\omega_4)} %\nonumber \\ & \quad \times F(k_2,k_3,k_4) \right] \, . \end{align} This gives a precise meaning to the ``energy conservation'' $\delta$-function in (\ref{eq:Gamma2}), and proves the equality. We wish to stress here that this $\delta$-function is a non-trivial constraint, and can produce non-smooth behavior even for smooth dispersion relations. \begin{proofof}{Proposition \ref{th:defendelta}} Since $f \in \ell_1((\Z^d)^3)$, we have as an absolutely convergent sum, \begin{align} \FT{f}(k,k',k_0-k-k') = \sum_{x_1,x_2,x_3\in \Z^d} \rme^{-\ci 2\pi (k\cdot (x_1-x_3)+k'\cdot (x_2-x_3)+k_0\cdot x_3)} f(x_1,x_2,x_3)\, . \end{align} We insert this in the integrand and use Fubini's theorem to exchange the order of $x$-sum and $k,k'$-integrals. The resulting convolution integral over $k,k'$ can be expressed in terms of $p_t(x)$, which is the inverse Fourier-transform of $k\mapsto\rme^{-\ci t \omega(k)}$. This proves that \begin{align}\label{eq:convsplit} & \int_{(\T^d)^2} \!\rmd k'\rmd k\, \rme^{\ci s (\omega(k) + \sigma' \omega(k')+ \sigma \omega(k_0-k-k'))} \FT{f}(k,k',k_0-k-k') \nonumber \\ & \quad = \sum_{x_1,x_2,x_3\in \Z^d} f(x_1,x_2,x_3) \sum_{y\in \Z^d} \rme^{-\ci 2\pi k_0\cdot (y+x_3)} p_{-s}(y+x_3-x_1) p_{-\sigma' s}(y+x_3-x_2) p_{-\sigma s}(y)\, . \end{align} Thus by H\"{o}lder's inequality and the property $\norm{p_{-s}}_3 = \norm{p_{s}}_3$, its absolute value is bounded by $\norm{f}_1\norm{p_{s}}_3^3\le \norm{f}_1 C\sabs{s}^{-1-\delta}$. This proves (\ref{eq:defendelta}). Equation (\ref{eq:leadphasebnd}) follows then by applying the result to $f(x_1,x_2,x_3)= \prod_{i=1}^3 \1(x_i=0)$. \end{proofof} \begin{lemma}[Degree one vertex]\label{th:degoneest} For any $k_0\in \T^d$, $\alpha\in \R$, $|\beta|>0$, $0<\lambda\le \lambda'_0$, and $\sigma,\sigma'\in\set{\pm 1}$, \begin{align}\label{eq:degoneest} \int_{\T^d} \!\rmd k\, \frac{\Fone(\sigma' k_0)}{|\omega(k) + \sigma \omega(k_0-k)-\alpha +\ci \beta|} \le C\lambda^{-b} \sabs{\ln |\beta|}^2\, , \end{align} where $C$ depends only on $\omega$ and the basic cutoff function $\varphi$. \end{lemma} \begin{proof} The left hand side of (\ref{eq:degoneest}) does not depend on the sign of $\beta$, and thus it suffices to consider $\beta>0$. The result holds trivially for any $C\ge 1$ if $|\beta|\ge 1$. Furthermore, if we change the integration variable from $k$ to $k'=\sigma' k$, the left hand side becomes \begin{align}%\label{eq:} & \int_{\T^d} \!\rmd k'\, \frac{\Fone(\sigma' k_0)}{|\omega(\sigma' k') + \sigma \omega(\sigma'(\sigma' k_0-k'))-\alpha +\ci \beta|} % \nonumber \\ & \quad = \int_{\T^d} \!\rmd k\, \frac{\Fone(\sigma' k_0)}{|\omega(k) + \sigma \omega(\sigma' k_0-k)-\alpha +\ci \beta|} \, . \end{align} Thus it is enough to prove the theorem for $\sigma'=1$. Let us thus assume $0<\beta\le 1$, $\sigma'=1$. We apply Lemma \ref{th:Foneprop} to the left hand side, which proves that it is then bounded by \begin{align}%\label{eq:degoneest} C_1 \lambda^{-b} d(k_0,\Msing) \int_{\T^d} \!\rmd k\, \frac{1}{|\omega(k) + \sigma \omega(k_0-k)-\alpha+\ci \beta|} \,. \end{align} In particular, if $d(k_0,\Msing)=0$ the left hand side is zero, and the bound (\ref{eq:degoneest}) holds trivially. Let us thus assume $k_0\not\in \Msing$. By Lemma 4.21 in \cite{ls05}, for any real $r,\beta$, \begin{align}\label{eq:restophases} \frac{1}{|r+\ci \beta|} = \sabs{\ln\beta} \int_{-\infty}^{\infty} \!\rmd s\, \rme^{\ci s r} F(s;\beta) \end{align} where $F(s;\beta)\ge 0$ is such that $F(s;\beta)\le \rme^{-\beta|s|} + \1(|s|\le 1) \ln |s|^{-1}$. The bound is uniformly integrable in $s$. Applying this representation and then Fubini's theorem shows that \begin{align}%\label{eq:degoneest} & \int_{\T^d} \!\rmd k\, \frac{1}{|\omega(k) + \sigma \omega(k_0-k)-\alpha+\ci \beta|} \nonumber \\ & \quad \le \sabs{\ln \beta} \int_{-\infty}^{\infty} \!\rmd s\, F(s;\beta) \left| \int_{\T^d} \!\rmd k\, \rme^{\ci s (\omega(k) + \sigma \omega(k_0-k))} \right| \nonumber \\ & \quad \le \sabs{\ln \beta} \int_{-\infty}^{\infty} \!\rmd s\, F(s;\beta) \frac{C\sabs{s}^{-1} }{d(k_0,\Msing)} \nonumber \\ & \quad \le \sabs{\ln \beta}\frac{C}{d(k_0,\Msing)} \Bigl( \int_{-1}^{1} \!\rmd s\, (1+\ln |s|^{-1}) + 2 \int_{1}^{\infty} \!\rmd s\, \frac{1}{s} \rme^{-\beta s} \Bigr) \nonumber \\ & \quad \le \sabs{\ln \beta}^2\frac{C'}{d(k_0,\Msing)}\, , \end{align} where in the second inequality we have used assumption (DR\ref{it:DRinterf}). Collecting the estimates together yields the bound in (\ref{eq:degoneest}). \end{proof} \begin{lemma}[Degree two vertex]\label{th:degtwoest} For any $k_0\in \T^d$, $\alpha\in \R$, $|\beta|>0$, and $\sigma,\sigma'\in\set{\pm 1}$, \begin{align}\label{eq:degtwoest} \int_{(\T^d)^2} \!\rmd k'\rmd k\, \frac{1}{|\omega(k) + \sigma' \omega(k') + \sigma \omega(k_0-k-k')-\alpha +\ci \beta|} \le C \sabs{\ln |\beta|}\, , \end{align} where $C$ depends only on $\omega$. \end{lemma} \begin{proof} Again it suffices to consider $\beta>0$. We apply the same representation of the resolvent term as in the proof of the previous Lemma. This shows that \begin{align}%\label{eq:degtwoest} &\int_{(\T^d)^2} \!\rmd k'\rmd k\, \frac{1}{|\omega(k) + \sigma' \omega(k') + \sigma \omega(k_0-k-k')-\alpha +\ci \beta|} \nonumber \\ & \quad \le \sabs{\ln \beta} \int_{-\infty}^{\infty} \!\rmd s\, F(s;\beta) \left| \int_{(\T^d)^2} \!\rmd k'\rmd k\, \rme^{\ci s (\omega(k) + \sigma' \omega(k')+ \sigma \omega(k_0-k-k'))} \right|\, . \end{align} Here by Proposition \ref{th:defendelta} the absolute value has an $s$-integrable bound. Therefore, a constant $C$ for (\ref{eq:degtwoest}) can be found. \end{proof} \section{Partially paired and higher order graphs} \label{sec:higherorder} In this section we consider relevant graphs which are either higher order, when they necessarily contain a cluster $A'\in S$ with $|A'|\ge 4$, or they are pairing and contain an interaction vertex of degree one. We will show that the contribution of these graphs is negligible in all error terms and in the main term. In addition, the related estimates will suffice to prove that also all other contributions to the amputated and constructive interference error terms are negligible. We will use the notations introduced in the earlier sections, in particular, in Section \ref{sec:iterclsuters}. \begin{lemma}[Basic $\mathcal{A}$-estimate]\label{th:basicAest} There is a constant $C>0$ such that for any (amputated) momentum graph $\graph(S,J,n,\ell,n,\ell')$, $1\le n\le N_0$, and $s>0$ we have \begin{align}\label{eq:basicAest} & \limsup_{\Lambda\to \infty} \lambda^{2n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n,\ell'}(k',\sigma';\Lambda) \prod_{A\in S} \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) \prod_{i=1}^{n} \Bigl[ \PFone(k_{i-1;\ell_i}) \PFone(-k'_{i-1;\ell'_i}) \Bigr] \nonumber \\ & \quad \times \left| \int_{(\R_+)^{I_{2,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=2}^{2 n} r_i\Bigr) \prod_{i=2}^{2n} \rme^{-\ci r_i \gamma({i;J})} \right| \nonumber \\ & \le \rme^{s\lambda^2} \frac{(s\lambda^2)^{\tilde{n}_0-n'_0} }{(\tilde{n}_0-n'_0)!} \lambda^{2+\tilde{n}_2+(1-b)\tilde{n}_1-\tilde{n}_0} N_0^{-b_0 n'_0} C^{1+\tilde{n}_1+\tilde{n}_2} \sabs{\ln n} \sabs{\ln \lambda}^{1+\tilde{n}_2+2 \tilde{n}_1} \, , \end{align} where $\tilde{n}_j=n_j-n_j(2)$ denotes the number of non-amputated interaction vertices of degree $j$, and $n'_0\ge 0$ counts the number of degree zero interaction vertices $v_i$ with $2< i\le 2 n-N_0+1$. \end{lemma} In the above, if $n\ge (N_0+1)/2$, we have $n'_0=n_0(2 n-N_0+1)-n_0(2)$, and $n'_0=0$, otherwise. \begin{proof} We first perform the sums over $\sigma$ and $\sigma'$, which have only one non-zero term, the one with the appropriate propagation of parities. We resolve the momentum constraints as explained in Section \ref{sec:momdeltas}, i.e., we integrate out all the $\delta_\Lambda$-terms using the time-ordered spanning tree. We rewrite the remaining (free) $k,k'$-integrals as in (\ref{eq:LamtoLeb}) and thus convert all sums into Lebesgue integrals over step functions. Since the resulting integrand is uniformly bounded, using dominated convergence theorem we can take the limit $\Lambda\to \infty$ inside the integrals. This proves the existence of the limit, and the resulting formula is, in fact, identical to the one obtained by replacing in the left hand side of (\ref{eq:basicAest}) every $\Lambda^*$ by $\T^d$ and all $\delta_\Lambda$ by $\delta=\delta_{\T^d}$. However, we continue using the time-ordered resolution of momentum constraints also after the continuum limit $\Lambda\to\infty$ has been taken. There are total $N=2 n$ interaction vertices, and let $A_j$, $j=0,1,2$, denote the collection of time-slice indices $2\le i< N$ such that $\deg v_{i+1} =j$. Some of the sets can be empty, but they are disjoint and their union is $\set{2,3,\ldots,N -1}$. Let further $B=\defset{i\in A_0}{i\le N-N_0}$ (which can be empty). For every $i\in B$ we thus have $\lfloor i/2\rfloor \le n-(N_0/2)$. Set $\gamma_i=\gamma(i;J)$. Then $-2 \kappa'\le \im \gamma_i \le 0$ and $|\re\gamma_i|\le 2 N\norm{\omega}_\infty$ for all $i$. We can thus apply Lemma \ref{th:resolvents} with $A=\set{N}\cup A_1\cup A_2$, and using the path $\Gamma_N$ depicted in Figure \ref{fig:gnpath}. Since then $A'=\set{*}\cup A_0$, we find \begin{align}%\label{eq:} & \left| \int_{(\R_+)^{I_{2,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=2}^{2 n} r_i\Bigr) \prod_{i=2}^{2n} \rme^{-\ci r_i \gamma_i} \right| \nonumber \\ & \quad \le \oint_{\Gamma_N} \frac{|\rmd z|}{2\pi} \int_{(\R_+)^{A'}}\!\rmd r \, \delta\Bigl(s-\sum_{i\in A'} r_i\Bigr) \left|\rme^{-\ci r_* z}\right| \prod_{i\in A_0} \left|\rme^{-\ci r_i \gamma_i}\right| \prod_{i\in A} \frac{1}{|z-\gamma_i|} \, . \end{align} \begin{figure} \centering \myfigure{height=5cm}{Gnpath} \caption{Integration path $\Gamma_N$. Here $c_N=2 (N+1) \norm{\omega}_\infty$, $\beta=\lambda^2$, and the shaded area contains all possible values of $\gamma(i;J)$ for momentum graphs with $N$ interaction vertices. \label{fig:gnpath}} \end{figure} If $i\in B$, we have $\im (-\gamma_i)= \kappa_{n-m} + \kappa_{n-m'}$ with $m+m'= 2 + C_+(i-2;J) + C_-(i-2;J)=i$. Thus then $\min(m,m')\le \lfloor i/2\rfloor \le n-(N_0/2)$, and therefore, $\im (-\gamma_i) \ge \kappa_{n-\min(m,m')}=\kappa'=\lambda^2 N_0^{b_0}$. In general, $\im (-\gamma_i) \ge 0$, and we obtain the estimates \begin{align}\label{eq:intrtrick} & \int_{(\R_+)^{A'}}\!\rmd r \, \delta\Bigl(s-\sum_{i\in A'} r_i\Bigr) \left|\rme^{-\ci r_* z}\right| \prod_{i\in A_0} \left|\rme^{-\ci r_i \gamma_i}\right| \nonumber \\ & \quad \le \rme^{s (\im z)_+} \int_{(\R_+)^{B}}\!\rmd r \, \prod_{i\in B} \rme^{-\kappa' r_i} \int_{(\R_+)^{A'\setminus B}}\!\rmd r \, \delta\Bigl(s-\sum_{i\in B} r_i-\sum_{i\in A'\setminus B} r_i\Bigr) \nonumber \\ & \quad \le \rme^{s (\im z)_+} (\kappa')^{-|B|} \frac{s^{\tilde{n}}}{\tilde{n}!} \, , \end{align} where $(\cdot)_+$ was defined in (\ref{eq:softpiold}) and $\tilde{n}=|A'\setminus B|-1=|A_0\setminus B|=|A_0|-|B|=\tilde{n}_0-n'_0$. Since $\gamma_{2 n-2}=-\ci 2 \kappa_0=0$, this shows that \begin{align}%\label{eq:} & \left| \int_{(\R_+)^{I_{2,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=2}^{2 n} r_i\Bigr) \prod_{i=2}^{2n} \rme^{-\ci r_i \gamma_i} \right| \nonumber \\ & \quad \le (\kappa')^{-n'_0} \frac{s^{\tilde{n}_0-n'_0}}{(\tilde{n}_0-n'_0)!} \oint_{\Gamma_N} \frac{|\rmd z|}{2\pi} \frac{\rme^{s (\im z)_+}}{|z|} \prod_{i\in A_1\cup A_2} \frac{1}{|z-\gamma_i|} \, . \end{align} We then estimate $\PFone(k_{0;\ell_1})\PFone(-k'_{0;\ell_1}) \le 1$ to remove the dependence on the two ``amputated'' interaction vertices at the bottom of the interaction trees. If there is any free momenta associated with these vertices, they will be integrated over next, resulting in an irrelevant factor of $1$. Each of the resolvents in $\frac{1}{|z-\gamma_i|}$, $i\in A_1\cup A_2$, depends only on the free momenta associated with edges ending on a time-slice $i'\ge i$. Consider a degree one vertex in the plus-tree. By Proposition \ref{th:momatintv}, there is a permutation $\pi$ of $\set{0,1,2}$ such that $\tilde{k}=k_{i-1,\ell_i+\pi(1)}$ is the free momentum, and neither $k_{i-1,\ell_i+\pi(3)}$ nor $k_0=k_{i-1,\ell_i+\pi(2)}+k_{i-1,\ell_i+\pi(1)}$ depend on $\tilde{k}$. We then estimate $\PFone(k_{i-1;\ell_i}) \le \Fone (k_0)$. Analogously, for every degree one vertex in the minus tree, we can estimate $\PFone(-k'_{i-1;\ell'_i}) \le \Fone (-k'_0)$. We remove all remaining $\PFone$, which are thus attached to a degree zero or two vertex, by the trivial estimate, $\PFone\le 1$. After this we can use the estimates given in Lemmata \ref{th:degoneest} and \ref{th:degtwoest} to iterate through the free momentum integrals in the direction of time, i.e., from bottom to top in the graph. At each iteration step, only one resolvent factor depends on the corresponding free momenta, and the remaining free momenta only affect the value of ``$\alpha$'' in the resolvent factor. The estimates can be used with $\beta=\lambda^2$ for those $z\in \Gamma_N$ in the top horizontal part of the path; we can ignore the imaginary part of $\gamma_i$, since this is always negative, and thus would only increase the ``$\beta$'' in the Lemmas and lower the value of the resolvent factor. For the remaining $z$ we have $|z-\gamma_i|\ge 1$, and the upper bounds remain valid also for these values of $z$, after we adjust the constant so that $C\ge 1$. After the last iteration step, there is one free momentum integral left, provided that there are any free momenta attached to the top fusion vertex. However, since the remaining integrand is momentum-independent, this integral yields a trivial factor $1$, and can thus be ignored. The only remaining integral is over $z$. This we estimate by \begin{align}%\label{eq:} \oint_{\Gamma_N} \frac{|\rmd z|}{2\pi} \frac{\rme^{s (\im z)_+}}{|z|} \le C \rme^{s\lambda^2} \sabs{\ln N} \sabs{\ln \beta}\, , \end{align} where $C$ is a constant which depends only on $\norm{\omega}_\infty$. Collecting the estimates together yields the upper bound in (\ref{eq:basicAest}); the power of $\lambda$ arising from the estimates is $2 n-2(\tilde{n}_0-n'_0)-2 n'_0-b \tilde{n}_1$ which we have simplified to $2+\tilde{n}_2+(1-b)\tilde{n}_1-\tilde{n}_0$ using $2 n-2=\tilde{n}_0+\tilde{n}_1+\tilde{n}_2$. \end{proof} For the following result, let us recall the definition of the time-dependent exponents in a main term, $\gamma(i)$ in equation (\ref{eq:defgammam}). In the analysis of the partial integration error term, Section \ref{sec:partierr}, we will need a generalization of this phase factor to a case with interactions also in the minus tree. To this end we define then \begin{align}\label{eq:defgtilde} \tilde\gamma({i;J})=\gamma^+_j+\gamma^-_{j'},\quad\text{with}\quad j=C_+(i;J),\ j'=C_-(i;J)\, , \end{align} and thus $j,j'\in\set{0,1,\ldots,n}$. For the main term, with $n'=0$, we have $J(i)=+1$, for all $i$, and $\gamma(i)=\tilde\gamma({i;1})$. Although the functional dependence on the mapping $J$ is different between $\tilde{\gamma}$ and the amputated $\gamma$, in both cases the correct exponential can be read off from our momentum graphs by summing over $\sigma_e\omega(k_e)$, for all edges $e$ which intersect the corresponding time-slice. Therefore, we will make no distinction between the amputated and non-amputated exponentials and denote both by $\gamma(i;J)$. \begin{lemma}[Basic $\mathcal{F}$-estimate]\label{th:basicFest} There is a constant $C>0$ such that for any momentum graph $\graph(S,J,n,\ell,n',\ell')$, $n,n'\ge 0$, and $s>0$ we have \begin{align}\label{eq:basicFest} & \lim_{\Lambda\to \infty} \lambda^{n+n'} \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}'_{n}} } \sum_{\sigma'\in \set{\pm 1}^{\mathcal{I}'_{\smash[t]{n'}}} } % \nonumber \\ & \qquad \times \1(\sigma_{n,1}=1)\1(\sigma'_{n',1}=-1) \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{\smash[t]{n'}}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n',\ell'}(k',\sigma';\Lambda) \prod_{A\in S} \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) \prod_{i=1}^{n} \PFone(k_{i-1;\ell_i}) \prod_{i=1}^{n'} \PFone(-k'_{i-1;\ell'_i}) \nonumber \\ & \quad \times \left| \int_{(\R_+)^{I_{0,n+n'}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=0}^{n+n'} r_i\Bigr) \prod_{i=0}^{n+n'} \rme^{-\ci r_i \gamma({i;J})}\right| \nonumber \\ & \le \rme^{s\lambda^2} \frac{(s\lambda^2)^{n_0} }{(n_0)!} \lambda^{n_2+(1-b)n_1-n_0} C^{1+n_1+n_2} \sabs{\ln (n+n'+1)} \sabs{\ln \lambda}^{1+n_2+2 n_1} \, , \end{align} where $n_i$ denotes the number of interaction vertices of degree $i$. \end{lemma} \begin{proof} There are $N=n+n'$ interaction vertices, and we let $A_j=\defset{0\le i< N}{\deg v_{i+1} =j}$, $j=0,1,2$, and $B=\emptyset$. With these adjustments, we can derive the bound as in the proof of Lemma \ref{th:basicAest}. (Choosing $B=\emptyset$ implies $n'_0=0$ and the estimate thus ignores any additional decay arising from factors with $\im \gamma_i<0$.) The resulting power of $\lambda$ is $n+n'-2 n_0-b n_1$ which we have simplified using $n+n'=n_0+n_1+n_2$. \end{proof} \subsection{Amputated error term} \label{sec:proveerramp} \begin{proposition}\label{th:amperr} Suppose $t>0$ and $0<\lambda<\lambda'_0$ are given, and define $N_0$ and $\kappa$, as in Definition \ref{th:defkappaetc}. There is a constant $C>0$ depending only on $f$ and $g$, and $c>0$ depending only on $\omega$ such that \begin{align}\label{eq:amperr} & \limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm amp}[g,f](t)|^2 %\nonumber \\ & \quad \le C t^2 \rme^t \sabs{c t}^{N_0} N_0^{2 N_0+2} (4 N_0)! \sabs{\ln \lambda}^{4 N_0+2} \Bigl( \lambda + \lambda^{-2} N_0^{-b_0 N_0/4}\Bigr) \, , \end{align} as soon as $N_0(\lambda)\ge 56$. \end{proposition} Since $\gamma'\le 1$, we can conclude from (\ref{eq:N0lima}) and (\ref{eq:N0limb}) that then $\limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm amp}[g,f](t)|\to 0$ as $\lambda\to 0$. \begin{proof} By Proposition \ref{th:Qmainerr}, and according to the discussion in Section \ref{sec:errors}, we have \begin{align}%\label{eq:} & \limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm amp}[g,f](t)|^2 \le C\norm{f}_2^2 t^2 \lambda^{-4} \nonumber \\ & \quad \times \sup_{0\le s\le t\lambda^{-2}} \limsup_{\Lambda\to\infty} \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} |\mathcal{A}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa)|\, , \end{align} where $n=N_0$. We have here first applied dominated convergence to move $\limsup_{\Lambda\to\infty}$ inside the $s$-integral which was then estimated trivially. The bound for domination is contained in the following. The $\limsup$ can be bounded by first employing (\ref{eq:Aamplsimp}) and then Lemma \ref{eq:basicAest}. This yields the bound \begin{align}%\label{eq:Aamplsimp} & \norm{\FT{g}}_\infty^2 \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} \prod_{A\in S} \sup_{\Lambda,k,\sigma} |C_{|A|}(\sigma,k;\lambda,\Lambda)| \nonumber \\ & \qquad \times \rme^{s\lambda^2} \frac{(s\lambda^2)^{\tilde{n}_0-n'_0} }{(\tilde{n}_0-n'_0)!} \lambda^{2+\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1} n^{-b_0 n'_0} % \nonumber \\ & \qquad \times C^{1+\tilde{n}_1+\tilde{n}_2} \sabs{\ln n} \sabs{\ln \lambda}^{1+\tilde{n}_2+2 \tilde{n}_1} \nonumber \\ & \quad \le \norm{\FT{g}}_\infty^2 \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} \prod_{A\in S} \sup_{\Lambda,k,\sigma} |C_{|A|}(\sigma,k;\lambda,\Lambda)| \nonumber \\ & \qquad \times \rme^{t} \sabs{c t}^{n} \sabs{\ln \lambda}^{2 + 4 n}% \nonumber \\ & \quad \times \lambda^{2+\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1} n^{-b_0 n'_0} \end{align} where we have used $\tilde{n}_0\le n_0 \le n$ which is implied by Lemma \ref{th:ivdegrees}. We set $r=2n+1-|S|$ as in Proposition \ref{th:ivordering}, and note that here $\tilde{n}_j=n_j-n_j(2)\ge 0$ and $n'_0=n_0(n+1)-n_0(2)$. By Lemma \ref{th:ivdegrees}, $\tilde{n}_2-\tilde{n}_0= r+n_0(2)-n_2(2) \ge r-2$, and $\tilde{n}_1\ge n_1-2$. If $r+n_1\ge 24$, we thus have $\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1\ge r +\frac{1}{4} n_1 -3\ge 3$. In such cases we estimate $\lambda^{2+\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1} n^{-b_0 n'_0}\le \lambda^5$. If $r+n_1< 24$, we get from Proposition \ref{th:ivordering}, the estimates $n'_0\ge (n-n_1-r)/2-2$ and $\tilde{n}_2-\tilde{n}_0\ge 0$. Thus for any $n\ge 56$, we have $n'_0\ge (n/2)-14\ge n/4$, and therefore also $\lambda^{2+\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1} n^{-b_0 n'_0}\le \lambda^2 n^{-b_0 n/4}$. The number of terms is the sum over $J$ is less than $2^{2 n-2}$ and $\sum_{\ell\in G_n} 1 = ( 2n -1)!!\le (2 n)^n$. For the sum over $S$ we can apply Lemma \ref{th:clustercomb}. Collecting all the estimates together proves that (\ref{eq:amperr}) holds, after readjustment of the constants $c$ and $C$. \end{proof} \subsection{Constructive interference error terms} \label{sec:proveerrcut} \begin{proposition}\label{th:cuterr} Suppose $t>0$ and $0<\lambda<\lambda'_0$ are given, and define $N_0$ and $\kappa$, as in Definition \ref{th:defkappaetc}. There is a constant $C>0$ depending only on $f$ and $g$, and $c>0$ depending only on $\omega$ such that \begin{align}\label{eq:cuterr} & \limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm cut}[g,f](t)|^2 \le C t^2 \rme^t \sabs{c t}^{N_0} N_0^{2 N_0+4} (4 N_0)! \sabs{\ln \lambda}^{4 N_0+3} \lambda^{1/4} \,. \end{align} \end{proposition} Since $\gamma'\le \frac{1}{4}$, this estimate proves that $\limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm amp}[g,f](t)|\to 0$ as $\lambda\to 0$. \begin{proof} We again denote $\mathcal{I}'_{n} =\mathcal{I}_{n} \cup \set{(n,1)}= \defset{(i,j)}{0\le i\le n, 1\le j \le m_{n-i}}$, and define $I=I_{2 m_n}=I_{4 n +2}$ to give labels to the final $\FT{a}$, as before. By expanding the pairing truncations in $\Ptrunc$ to individual components, and then applying the cluster expansions in Lemma \ref{th:cumulants}, we find that the effect of the extra terms in the truncations is the cancellation of all those terms from the main cumulant expansion which contain one of the corresponding pairings. None of the other clusterings is affected. The remainder of the analysis is completely analogous to that used for $\mathcal{A}_{n}$ in Section \ref{sec:errors}, and it shows that \begin{align}%\label{eq:} &\E\Bigl[|\mean{\FT{g}, \mathcal{Z}_n(s,t/\vep,\cdot,+1,\kappa)[\FT{a}_s]}|^2\Bigr] \nonumber \\ & \quad = \sum_{J\text{ interlaces }(n-1,n-1)} \ \sum_{\ell,\ell' \in G_n}\ \sum_{S\in \pi(I_{4 n +2})} \mathcal{Z}_n^{\rm ampl}(S,J,\ell,\ell',t/\vep-s,\kappa)\, . \end{align} Here $\mathcal{Z}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa)=0$ for any partition $S$ which contains a pairing of any two edges attached to one of the ``truncated'' vertices (i.e., if there are $A\in S$ and $i\in \set{1,2}$ such that $|A|=2$ and $\edges_+(u)\subset\edges_-(v_i)$ for every $u\in A$). For all other $S$ we have \begin{align}\label{eq:Znampl} & \mathcal{Z}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa) = (-\lambda^{2})^{n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n,\ell'}(k',\sigma';\Lambda) \prod_{A\in S}\Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) C_{|A|}(o_A,K_A;\lambda,\Lambda) \Bigr] \nonumber \\ & \quad \times \sigma_{1,\ell_{1}} \PFzero(k_{0;\ell_1}) \sigma'_{1,\ell'_{1}} \PFzero(-k'_{0;\ell'_1}) \prod_{i=2}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \sigma'_{i,\ell'_{i}} \PFone(-k'_{i-1;\ell'_i}) \Bigr] \nonumber \\ & \quad \times \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) |\FT{g}(k_{n,1})|^2 \nonumber \\ & \quad \times \int_{(\R_+)^{I_{2,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=2}^{2 n} r_i\Bigr) \prod_{i=2}^{2n} \rme^{-\ci r_i \gamma({i;J})} \, , \end{align} where $\gamma({i;J})$ is given by (\ref{eq:giJ}). We use the Schwarz inequality in the sum over $n$, and then proceed as in the proof of Proposition \ref{th:amperr}. Then using Proposition \ref{th:Qmainerr} yields the estimate \begin{align}%\label{eq:} & \limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm cut}[g,f](t)|^2 \le N_0 C\norm{f}_2^2 t^2 \lambda^{-4} \sup_{0\le s\le t\lambda^{-2}} \nonumber \\ & \quad \times \sum_{n=1}^{N_0} \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} \limsup_{\Lambda\to\infty} |\mathcal{Z}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa)|\, . \end{align} Compared to the amputated error term, the terms in the sum have the following improved upper bounds whose proof will be given at the end of this section. \begin{lemma}[Basic $\mathcal{Z}$-estimate]\label{th:basicZest} If we change the term $\PFone(k_{0;\ell_1}) \PFone(-k'_{0;\ell'_1})$ on the left hand side of (\ref{eq:basicAest}) to $\PFzero(k_{0;\ell_1}) \PFzero(-k'_{0;\ell'_1})$, the estimate can either be improved by a factor of $C \sabs{\ln \lambda} \lambda^{z_0}$, $z_0=bd-2$, or the corresponding $\mathcal{Z}_n^{\rm ampl}$ is zero. If $\tilde{n}_1=0$, then the estimate is valid also with $z_0=(d-2) b$. If any of the amputated vertices has degree two, the estimate is valid with $z_0=(d-1) b$. \end{lemma} The $\limsup$-factor can then be bounded by first employing (\ref{eq:Znampl}) and then Lemma \ref{th:basicZest}. We neglect the extra decay provided by the partial time integration, and estimate $n^{-b_0 n'_0}\le 1$. Simplifying the expression somewhat along the lines used in the proof of Proposition \ref{th:amperr}, we thus have the following bound for the $\limsup$-term: \begin{align}%\label{eq:Aamplsimp} & \norm{\FT{g}}_\infty^2 \sum_{n=1}^{N_0} \sum_{J\text{ interlaces }(n-1,n-1)} \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} \prod_{A\in S} \sup_{\Lambda,k,\sigma} |C_{|A|}(\sigma,k;\lambda,\Lambda)| \nonumber \\ & \qquad \times \rme^{t} \sabs{c t}^{n} \sabs{\ln \lambda}^{3 + 4 n}% \nonumber \\ & \quad \times \lambda^{2+\tilde{n}_2-\tilde{n}_0+(1-b)\tilde{n}_1+db} \nonumber \\ & \qquad \times \1(\mathcal{Z}_n^{\rm ampl}\ne 0)\times\begin{cases} \lambda^{-b}, & \text{if }n_2(2)>0;\\ \lambda^{-2 b}, & \text{if }n_2(2)=0, \tilde{n}_1=0;\\ \lambda^{-2}, & \text{otherwise.}\\ \end{cases} \end{align} By Proposition \ref{th:ivordering}, here always $\tilde{n}_2-\tilde{n}_0\ge 0$ and $\tilde{n}_1\ge 0$. Thus if $n_2(2)>0$, the power of $\lambda$ can be bounded from above by $\lambda^{2+b(d-1)}\le \lambda^{4+\frac{1}{4}}$. We can thus assume that $n_2(2)=0$. Then $\tilde{n}_2-\tilde{n}_0=n_2-n_0-n_2(2)+n_0(2)=r+n_0(2)$. If also $n_0(2)=0$, we need to have $n_1(2)=2$, i.e., that both amputated interaction vertices have exactly one free momentum attached to them. By the iterative cluster scheme, this implies there is a cluster $A_0\in S$ such that exactly two of the edges in the first interaction vertex, the amputated minus vertex, connect to it. If $A_0$ is a pairing, then $\mathcal{Z}_n^{\rm ampl}=0$ by definition. Otherwise, $|A_0|\ge 4$, and thus then $r\ge 1$. Therefore, in all cases we can conclude that now either $\mathcal{Z}_n^{\rm ampl}=0$ or $\tilde{n}_2-\tilde{n}_0\ge 1$. Therefore, if $n_2(2)=0$ and $\tilde{n}_1=0$, the power of $\lambda$ can be bounded by $\lambda^{2+1+b(d-2)}\le \lambda^{4+\frac{1}{2}}\le \lambda^{4+\frac{1}{4}}$. If $n_2(2)=0$ and $\tilde{n}_1>0$, then the bound $\lambda^{2+1+1-b+db-2}=\lambda^{2+b(d-1)}\le \lambda^{4+\frac{1}{4}}$ can be used. Therefore, whatever the clustering, a bound $\lambda^{4+\frac{1}{4}}$ is always available. The rest of the sums can be bounded exactly as in the proof of Proposition \ref{th:amperr}, apart from the first sum over $n$ which yields an additional factor $N_0$. Collecting all the estimates together proves that (\ref{eq:cuterr}) holds. \end{proof} \begin{proofof}{Lemma \ref{th:basicZest}} The statement with $z_0=0$ is a corollary of the proof of Lemma \ref{th:basicAest}, since the estimate $\PFzero(k_{0;\ell_1}) \PFzero(-k'_{0;\ell'_1})\le 1$ allows to remove these terms at the right place in the proof. However, we can improve on the estimate by using the fact that $\PFzero(k_{0;\ell_1})$ enforces particular sums of momenta to lie very close to the singular manifold. Let us first consider the case were one of the amputated vertices has degree two. If this is the amputated minus vertex, we find from Proposition \ref{th:PFcorr} that \begin{align}%\label{eq:} &\PFzero(-k'_{0;\ell'_1}) \le \sum_{e_1,e_2 \in \edges_-(v_1);\ e_1< e_2} \1\!\left(d(-(k_{e_1}+k_{e_2}),\Msing)<\lambda^b\right)\, . \end{align} By Lemma \ref{th:nokkdiff}, for any choice of $e_1$, $e_2$ here $-(k_{e_1}+k_{e_2})$ depends on the free momenta $k,k'$ of $v_1$ either as $k$, $k'$, or $-(k+k')$. Thus when we first integrate over $k$ and then over $k'$, in one of the integrals we can apply Lemma \ref{th:Foneprop}, according to which \begin{align}\label{eq:Fzerovol} \sup_{k_0\in \T^d} \int_{\T^d} \rmd k\, \1\!\left(d(\pm k+k_0,\Msing)<\lambda^b\right) \le C \lambda^{b(d-1)} \, . \end{align} After this we can estimate $\PFzero(k_{0;\ell_1})\le 1$, and then continue as in the proof of Lemma \ref{th:basicAest}. If the amputated minus vertex does not have degree two, we estimate trivially $\PFzero(-k'_{0;\ell'_1})\le 1$ and integrate over any free momenta attached to it, which yields an irrelevant factor of one for the iterative bound. We then estimate the extra factor in the amputated plus vertex by \begin{align}\label{eq:ampv2est} &\PFzero(k_{0;\ell_1}) \le \sum_{e_1,e_2 \in \edges_-(v_2);\ e_1< e_2} \1\!\left(d(k_{e_1}+k_{e_2},\Msing)<\lambda^b\right) \end{align} If the amputated plus vertex, $v_2$, has degree two, we again gain a factor $C\lambda^{b(d-1)}$ from performing the two free integrations attached to it. After this the proof can proceed as in Lemma \ref{th:basicAest}. Thus if either of the amputated vertices has degree two, we have proven a gain by the stated factor with $z_0=b(d-1)$. This proves the last statement made in the Lemma. To prove the other two statements, it is sufficient to consider the term corresponding to some fixed pair $e_1< e_2$, $e_1,e_2\in \edges_-(v_2)$ in the bound (\ref{eq:ampv2est}). If $k_{e_1}+k_{e_2}$ is independent of all free momenta, then by Proposition \ref{th:kkpl}, the two initial time vertices belonging to $e_1\cup e_2$ must be isolated from the rest of the initial time vertices. This is possible only if they are paired, but then $\mathcal{Z}_n^{\rm ampl}=0$ by definition. Thus we can assume that there is some free momenta on which $k_{e_1}+k_{e_2}$ depends. Of the corresponding free edges, let $f_0$ denote the one added first (i.e., it is the maximum in the ordering of edges). We next estimate all $\PFone$-factors as in the proof of Lemma \ref{th:basicAest}, with one exception: if the fusion vertex at which $f_0$ ends is a degree two interaction vertex, we will need the corresponding $\PFone$-factor, and this is kept unchanged. Then we use the estimates in the proof of Lemma \ref{th:basicAest}, and iterate through the interaction vertices until the vertex at which $f_0$ ends is reached. If $f_0$ is attached to either an amputated interaction vertex or the top fusion vertex, then using (\ref{eq:Fzerovol}) we gain an improvement with $z_0=b(d-1)$, which is the best bound of all the three possibilities. If $f_0$ is attached to a degree one non-amputated interaction vertex, we first remove the corresponding ``resolvent'' factor using the trivial $L^\infty$ estimate and then apply (\ref{eq:Fzerovol}). Since in the proof of Lemma \ref{th:basicAest} this term would be estimated by $C \sabs{\ln \lambda}^2 \lambda^{-b}$, we gain an improvement by a factor of $C \lambda^{b-2+b(d-1)}$, as compared to the estimate in Lemma \ref{th:basicAest}. This yields the worst bound with $z_0=b d-2$. Otherwise, $f_0$ is attached to a non-amputated interaction vertex of degree two. Let the two free momenta be denoted by $k_1$ and $k_2$, and the third integrated momenta by $k_3$. In addition, denote $k_0=k_1+k_2+k_3$ which is independent of $k_1$ and $k_2$. By Lemma \ref{th:nokkdiff}, $k_{e_1}+k_{e_2}=\pm k_i+k_0'$, for some choice of sign and $i\in \set{1,2,3}$, where $k_0'$ is independent of $k_1$, $k_2$. Thus we need to consider \begin{align}\label{eq:degtwoest2} &\int_{(\T^d)^2} \!\rmd k_1\rmd k_2\, \1\!\left(d(\pm k_i+k_0',\Msing)<\lambda^b\right) \frac{\PFone(\pm(k_1,k_2,k_0-k_1-k_2))}{|\omega(k_1) + \sigma' \omega(k_2) + \sigma \omega(k_0-k_1-k_2)-\alpha +\ci \beta|}\, . \end{align} (The $\PFone$-factor is present here, since we did not estimate it trivially. $\PFone$ is also clearly invariant under permutation of its arguments.) If $i=1$, we estimate $\PFone\le \Fone(\pm(k_0-k_1))$, and change the integration variable $k_1$ to $k=\pm k_1+k_0'$. Then first applying Lemma \ref{th:degoneest} to the $k_2$ integral and then (\ref{eq:Fzerovol}) to the $k$ integral, we find that the integral is bounded by $C\sabs{\ln \beta}^2\lambda^{-b+b(d-1)}$. Analogous chance of variables can be done to show that the bound is valid also if $i=2$ or $i=3$. Thus we get an improvement by a factor of $C\sabs{\ln \beta}\lambda^{b(d-2)}$ compared to the estimate used in the proof of Lemma \ref{th:basicAest}. After one of the above estimates, we can finish the iteration of the interaction vertices, and complete the rest of the proof as in Lemma \ref{th:basicAest}. If $\tilde{n}_1=0$, then there are no non-amputated degree one vertices. Thus either of the two better estimates apply, and $z_0=b (d-2)> bd-2$ can be used. Otherwise, we need to resort to the worst estimate with $z_0=bd-2$. This completes the proof of the Lemma. \end{proofof} \subsection{Partial time-integration error terms} \label{sec:partierr} \begin{proposition}\label{th:ptierr1} Suppose $t>0$ and $0<\lambda<\lambda'_0$ are given, and define $N_0$ and $\kappa$, as in Definition \ref{th:defkappaetc}. There is a constant $C>0$ depending only on $f$ and $g$, and $c>0$ depending only on $\omega$ and $\lambda'_0$ such that \begin{align}\label{eq:ptierr1} & \limsup_{\Lambda\to\infty} | Q^{\rm err}_{\rm pti}[g,f](t)|^2 \le C t^2 \rme^t \sabs{c t}^{N_0} N_0^{2 N_0+5+2 b_0} (4 N_0)! \sabs{\ln \lambda}^{4 N_0+2} \lambda^{1/4} \nonumber \\ & \quad + C t^2 N_0^{2+2 b_0} \sup_{0\le s\le t\lambda^{-2}, N_0/2\le n < N_0} \sum_{J\text{ interlaces }(n,n)} \ \sum_{\ell,\ell' \in G_n}\ \sum_{S\in \pi(I_{4 n +2})} %\nonumber \\ & \qquad \times |\mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa)| \, , \end{align} where $\mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa)=0$, if the graph defined by $J,S,\ell,\ell'$ is not fully paired, and otherwise it is equal to \begin{align}\label{eq:defGnpairs} & \mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa) = (-\lambda^{2})^{n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \int_{(\T^d)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\T^d)^{\mathcal{I}'_{n}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma) \Delta_{n,\ell'}(k',\sigma') \prod_{A=\set{i,j}\in S} \Bigl[ \delta(K_i+K_j) \1(o_i=-o_j) W(K_i) \Bigr] \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \sigma'_{i,\ell'_{i}} \PFone(-k'_{i-1;\ell'_i}) \Bigr] \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) |\FT{g}(k_{n,1})|^2 \nonumber \\ & \quad \times \int_{(\R_+)^{I_{0,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=0}^{2 n} r_i\Bigr) \prod_{i=0}^{2n} \rme^{-\ci r_i \gamma({i;J})} \, . \end{align} \end{proposition} \begin{proof} The error term $\mathcal{G}_n$ was defined in (\ref{eq:defGnviaA}), where we can directly apply (\ref{eq:gAnprod}). Comparing the result to the definition of $\mathcal{F}_n$ in (\ref{eq:defFn}) shows that \begin{align}%\label{eq:} \mean{\FT{g},\mathcal{G}_{n}(s,t,\cdot,1,\kappa)[\FT{a}_s]} = \mean{\FT{g},\rme^{\ci s \omla} \mathcal{F}_n(t-s,\cdot,1,\kappa)[\FT{\psi}_s]} \, . \end{align} Thus in this case \begin{align}%\label{eq:} &\E\Bigl[|\mean{\FT{g}, \mathcal{G}_n(s,t/\vep,\cdot,1,\kappa)[\FT{a}_s]}|^2\Bigr] %\nonumber \\ & \quad = \E\Bigl[|\mean{ \rme^{-\ci s \omla} \FT{g}, \mathcal{F}_n(t/\vep-s,\cdot,1,\kappa)[\FT{\psi}_0]}|^2\Bigr] \nonumber \\ & \quad = \sum_{J\text{ interlaces }(n,n)} \ \sum_{\ell,\ell' \in G_n}\ \sum_{S\in \pi(I_{4 n +2})} \mathcal{G}_n^{\rm ampl}(S,J,\ell,\ell',t/\vep-s,\kappa)\, , \end{align} where \begin{align}\label{eq:defGnampl} & \mathcal{G}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa) = (-\lambda^{2})^{n} \sum_{\sigma,\sigma'\in \set{\pm 1}^{\mathcal{I}'_{n}} } \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k\, \int_{(\Lambda^*)^{\mathcal{I}'_{n}}} \!\rmd k'\, \nonumber \\ & \quad \times \Delta_{n,\ell}(k,\sigma;\Lambda) \Delta_{n,\ell'}(k',\sigma';\Lambda) \prod_{A\in S}\Bigl[ \delta_\Lambda\!\Bigl(\sum_{i\in A} K_i\Bigr) C_{|A|}(o_A,K_A;\lambda,\Lambda) \Bigr] \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \sigma'_{i,\ell'_{i}} \PFone(-k'_{i-1;\ell'_i}) \Bigr] \1(\sigma_{n,1}=1)\1(\sigma'_{n,1}=-1) |\FT{g}(k_{n,1})|^2 \nonumber \\ & \quad \times \int_{(\R_+)^{I_{0,2n}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=0}^{2 n} r_i\Bigr) \prod_{i=0}^{2n} \rme^{-\ci r_i \gamma({i;J})}\, . \end{align} Thus the amplitude differs from the ``amputated'' amplitudes $\mathcal{A}_n$ and $\mathcal{Z}_n$ by containing also the propagators associated with the first two interactions. We recall the discussion about the definition of the non-amputated exponentials $\gamma(i;J)$ in (\ref{eq:defgtilde}). As in the proof of Proposition \ref{th:cuterr}, we then find \begin{align}%\label{eq:} & \limsup_{\Lambda\to\infty} |Q^{\rm err}_{\rm pti}[g,f](t)|^2 \le N_0^2 (\kappa')^2 C\norm{f}_2^2 t^2 \lambda^{-4} \nonumber \\ & \quad \times \sup_{0\le s\le t\lambda^{-2}, N_0/2\le n \le N_0} \sum_{J\text{ interlaces }(n,n)} \ \sum_{\ell,\ell' \in G_n} \sum_{S\in \pi(I_{4 n+2})} \limsup_{\Lambda\to\infty} |\mathcal{G}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa)|\, . \end{align} For any graph which is not fully paired, we take absolute values up to the time-integrations and apply Lemma \ref{th:basicFest}. Using the notations of Lemma \ref{th:ivdegrees}, then \begin{align}%\label{eq:} & |\mathcal{G}_n^{\rm ampl}(S,J,\ell,\ell',s,\kappa)| %\nonumber \\ & \quad \le \norm{\FT{g}}_\infty^2 \rme^{t} \frac{t^{n_0} }{(n_0)!} \lambda^{r+(1-b) n_1} C^{2 n +1} \sabs{\ln \lambda}^{2+4 n} \prod_{A\in S} \sup_{\Lambda,k,\sigma} |C_{|A|}(\sigma,k;\lambda,\Lambda)|\, . \end{align} If the graph is higher order, $r\ge 1$, and if the graph is not fully paired, $n_1\ge 1$. Thus for both types of graphs, and trivially for irrelevant graphs, we can use a bound \begin{align}%\label{eq:} \lambda^{1/4} \norm{\FT{g}}_\infty^2 \rme^{t} \sabs{c t}^{n} \sabs{\ln \lambda}^{2+4 n} \prod_{A\in S} \sup_{\Lambda,k,\sigma} |C_{|A|}(\sigma,k;\lambda,\Lambda)|\, . \end{align} Consider then a fully paired graph. Then all clusters are pairings, with $C_2((k',k),(\sigma',\sigma))=\1(\sigma'+\sigma=0) W^\lambda_\Lambda(k)$. By Lemma \ref{th:unifW2}, $W^\lambda_\Lambda(k)= W(k)+\Delta$, where $\limsup_{\Lambda\to \infty}\sup_{k\in \T^d} |\Delta|\le 2 c_0^2 \lambda$. Thus for any finite index set $I$ we have \begin{align}%\label{eq:} \limsup_{\Lambda\to\infty} \Bigl|\prod_{i\in I} W^\lambda_\Lambda(k_i)-\prod_{i\in I} W(k_i)\Bigr|\le |I| C^{|I|-1} 2 c_0^2 \lambda \, , \end{align} where $C=2 c_0^2 \lambda'_0+\norm{W}_\infty<\infty$. (The statement can be proven by induction in $|I|$.) Thus if $S$ is a pairing, we have $|S|=2 n +1$ and by Lemma \ref{th:basicFest}, we can exchange in the definition of $\mathcal{G}_n^{\rm ampl}$ all $C_2$ terms by $W(K_i)\1(o_i=-o_j)$ with an error whose $\limsup$ is bounded by \begin{align}%\label{eq:} C n \norm{\FT{g}}_\infty^2 \rme^{t} \sabs{c t}^{n} \sabs{\ln \lambda}^{2+4 n} \lambda \, . \end{align} In the resulting formula, we first resolve all $\delta_\Lambda$-functions as explained before. The summations over the free momenta are then turned into Lebesgue integrals as explained in Section \ref{sec:firstpf}, with an integrand which is uniformly bounded and has a pointwise limit when $\Lambda\to \infty$. By dominated convergence we can thus take the limit $\Lambda\to \infty$ inside the integrals which shows that the limit is given by $\mathcal{G}_n^{\rm pairs}$ defined in (\ref{eq:defGnpairs}). Now collecting all the above bounds together, estimating the number of terms in the $J,\ell,\ell',S$ sums as before, readjusting $c$ and $C$ whenever necessary, and using $N_0^2 (\kappa')^2 \lambda^{-4}=N_0^{2+2 b_0}$, proves that (\ref{eq:ptierr1}) holds. \end{proof} \subsection{Main term} We recall the graphical representation of the main term, and the related notations, in particular, Proposition \ref{th:main1st} and the definition of $\gamma(m)$ in (\ref{eq:defgammam}). \begin{proposition}\label{th:main2nd} Suppose $t>0$ and $0<\lambda<\lambda'_0$ are given, and define $N_0$ and $\kappa$, as in Definition \ref{th:defkappaetc}. There is a constant $C>0$ depending only on $f$ and $g$, and $c>0$ depending only on $\omega$ and $\lambda'_0$ such that \begin{align}\label{eq:main2nd} & \limsup_{\Lambda\to\infty} |\Qmain [g,f](t)- \Qpairs [g,f](t)| %\nonumber \\ & \quad \le C \rme^t \sabs{c t}^{N_0} N_0^{N_0+4} (2 N_0)! \sabs{\ln \lambda}^{2 N_0+2} \lambda^{1/4} \, , \end{align} where $\Qpairs$ is defined by \begin{align}%\label{eq:} \Qpairs [g,f](t) = \sum_{n=0}^{N_0-1} \sum_{\ell \in G_n} \sum_{S\in \pi(I_{0,2 n+1})} \mathcal{F}_n^{\rm pairs}(S,\ell,t/\vep,\kappa) \end{align} with $\mathcal{F}_n^{\rm pairs}(S,\ell,t/\vep,\kappa)=0$, if the graph defined by $S,\ell$ is not fully paired, and otherwise it is equal to \begin{align}\label{eq:defFnpairs} &\mathcal{F}_n^{\rm pairs}(S,\ell,t/\vep,\kappa) = (-\ci \lambda)^n \sum_{\sigma\in \set{\pm 1}^{\mathcal{I}''_{n}}} \int_{(\T^d)^{\mathcal{I}''_{n}}} \!\rmd k\, \Delta_{n,\ell}(k,\sigma) \FT{g}(k_{n,1})^* \FT{f}(k_{n,1}) \nonumber \\ & \quad \times \1(\sigma_{n,1}=1)\1(\sigma_{0,0}=-1) \prod_{A=\set{i,j}\in S}\Bigl[ \delta(k_{0,i}+k_{0,j}) \1(\sigma_{0,i}=-\sigma_{0,j}) W(k_{0,i}) \Bigr] \nonumber \\ & \quad \times \prod_{i=1}^{n} \Bigl[ \sigma_{i,\ell_{i}} \PFone(k_{i-1;\ell_i}) \Bigr] % \nonumber \\ & \quad \times \int_{(\R_+)^{I_{0,n}}}\!\rmd r \, \delta\Bigl(t\lambda^{-2}-\sum_{i=0}^{n} r_i\Bigr) \prod_{m=0}^{n} \rme^{-\ci r_m \gamma(m)} \, . \end{align} \end{proposition} \begin{proof} We can apply Lemma \ref{th:basicFest} with $n'=0$ to estimate $|\mathcal{F}_n^{\rm ampl}(S,\ell,t/\vep,\kappa)|$. The steps of the proof are otherwise identical to those used in the proof of Proposition \ref{th:ptierr1}. To avoid repetition, we skip the rest of the details here. \end{proof} \section{Fully paired graphs} \label{sec:fullypaired} By the results proven in the previous section, only fully paired graphs remain to be estimated, with the corresponding amplitudes given by $\mathcal{G}_n^{\rm pairs}$ and $\mathcal{F}_n^{\rm pairs}$. In these terms, all sums over $\Lambda^*$ have already been replaced by integrals over $\T^d$, and we have changed the covariance function to its $\lambda\to 0$ limiting value. The related momentum graphs differ by the number of interactions in the minus tree: for $\mathcal{F}_n^{\rm pairs}$, we have $n'=0$, and for $\mathcal{G}_n^{\rm pairs}$, $n'=n$. For such relevant graphs, we have $r=0$ and $n_1=0$, and thus by Proposition \ref{th:ivordering} for any $1\le i\le n+n'$, \begin{align}%\label{eq:} n_2(i)\le n_0(i)\qand n_0(i)\ge \frac{i}{2}\, . \end{align} In addition, by Lemma \ref{th:ivdegrees} also $n_2=n_0=\frac{n+n'}{2}$. Since $n+n'$ must then be even this implies that any $\mathcal{F}_n^{\rm pairs}$ with odd $n$ is zero. Also necessarily $\deg v_1=0$ and $\deg v_{n+n'}=2$. Therefore, we can conclude that either the degrees of the interaction vertices form an alternating sequence $(0,2,0,2,\ldots,0,2)$, or this alternating behavior ends in two or more consecutive zeroes. Moreover, the first phase is always zero, since by Lemma \ref{th:omOmconv} for any relevant graph \begin{align}%\label{eq:} \re \gamma(0;J) = \sum_{i=1}^{2 n+1}\!\! \sigma_{0,i} \omega(k_{0,i}) +\sum_{i=1}^{2 n'+1}\!\! \sigma'_{0,i} \omega(k'_{0,i}) \end{align} and the pairing of momenta and parities on the initial time-slice implies that the terms cancel each other pairwise. For simplicity, let us now drop the dependence on $J$ from the notation, i.e., we denote $\gamma(m)$ instead of $\gamma(m;J)$ also for $\mathcal{G}_n^{\rm pairs}$. We recall that the time-slice $m1$ for some $j$. Then $j\ne 0$, $|A_j|>2$, and, since all clusters are pairs, any path from one edge of $A_j$ to another must then go via an interaction vertex $v'\ne v,v_i$. Since the double-loop of $v$ can only contain $v_i$, we must have $A'_{j_e}=\set{e}$ for all $e\in \edges_-(v)$. If $j_e\ne 0$ for all $e$, we can conclude that all of $A_j$ are pairings, and that the addition of $v_i$ and $v$ is equivalent to splitting of a pairing using a gain motive. Else $j_e=0$ for some $e$. Then the remaining two edges connect via a pairing to $v_i$, while the size of the third iterated cluster remains unaffected. This is equivalent to an addition of a loss motive to one of the edges in the third cluster. In both cases, the result is an immediate recollision, which thus leaves the $\gamma$-factors and $k$-dependence invariant. This implies that we can apply the induction assumption to the graph which is obtained by cutting out the leading motive of $v$ from the original graph, i.e., by removing the time-slices $i-1$ and $i$, all edges and vertices associated with the pairings used in the leading motive, and then repairing the graph by either adding the pairing previously formed by a gain term or by reconnecting the two ends previously joined by a loss motive. This shows that the statement holds for arbitrary $M\ge 0$. \end{proof} \begin{proofof}{Proposition \ref{th:leadingremains}} By assumption the graph is relevant, but none of its long time-slices depends on any of the double-loops. Then by Lemma \ref{th:immrecstart} all double-loops correspond to immediate recollisions, and if all recollisions are removed, a simple loop corresponding to $n=n'=0$ is left over. Thus the graph is leading, and as immediate recollisions preserve the phase factor, which is initially zero, all long time-slices are trivial. Conversely, if all long slices are trivial, they are zero independently of all free momenta. Therefore, also then the graph is leading. \end{proofof} \subsection{Crossing graphs} \begin{proposition}\label{th:crossingbnd} There is a constant $c_0$, which depends only on $\omega$, and $C$, which depends only on $\omega,f,g$, such that the amplitudes of all crossing graphs satisfy the bounds \begin{align}%\label{eq:} & |\mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa)| %\nonumber \\ & \quad \le C \lambda^{2 \gamma} \rme^{s\lambda^2} \sabs{c_0 s\lambda^2}^{n-1} \sabs{\ln \lambda}^{3+c_2+2 n}\, , \\ & |\mathcal{F}_n^{\rm pairs}(S,\ell,t \lambda^{-2},\kappa)| %\nonumber \\ & \quad \le C \lambda^{2 \gamma} \rme^{t} \sabs{c_0 t}^{n/2-1} \sabs{\ln \lambda}^{3+c_2+n} \, . \end{align} \end{proposition} \begin{proof} Both bounds can be derived simultaneously, if we consider a general crossing graph. Obviously, it also suffices to derive the bound merely for relevant graphs. By Lemma \ref{th:immrecstart}, then there is $i_2\in I_{2,N}$ such that $\deg v_{i_2}=2$ and every $1\le ii_2} \Omega_i \, . \end{align} By construction of the spanning tree, $\alpha_2$ cannot depend on $k_1,k_2$, nor on any other double-loop of $v_i$ with $i\le i_2$. We prove next that there is $\alpha_1$, which is also independent of all such double-loops, and $p\in\set{0,1}$ such that $a_1=-(1-p) \Omega_{i_2}+\alpha_1$. This implies \begin{align}%\label{eq:} \re \gamma(i_0-1) = p\Omega_{i_2} + \Omega_{i_0} + \alpha_1 + \alpha_2\, . \end{align} Then the vertex $v_{i_0}$ has to be either an $X$- or $T$-vertex for the double-loop of $v_{i_2}$. Otherwise, $\Omega_{i_0}$ does not depend on $k_1,k_2$, which would imply that either $\re \gamma(i_0-1)$ or $\re \gamma(i_0-1) - \Omega_{i_2}$ is independent of $k_1,k_2$ contradicting the assumption that $i_0-1$ propagates a crossing. Let us first consider $i$ such that $\deg v_i=2$ and $i_0+1\le ii_0-1$ is long, and we have $\re \gamma(i'-1)=\Omega_{i_2} + a_1 + \alpha_2$. Since this slice cannot propagate a crossing, we must have that either $a_1$ or $a_1+\Omega_{i_2}$ is independent of the double-loop of $v_{i_2}$. In the first case, we define $\alpha_1=a_1$ and $p=1$, and in the second we let $\alpha_1=a_1+\Omega_{i_2}$ and $p=0$. With these definitions, all of the previous claims holds. Set $m'=i_0-1$. We follow the iteration scheme used in the basic $\mathcal{F}$-estimate (Lemma \ref{th:basicFest}) with the following exceptions: we now have $A_1=\emptyset$, and we define $A=\set{m',2 n}\cup A_2$, i.e., we move the index $m'$ from $A_0$ to $A$. Since $|A_2|=|A_0|=N/2$, the integrated phase factor satisfies \begin{align}%\label{eq:} & \Bigl|\int_{(\R_+)^{I_{0,N}}}\!\rmd r \, \delta\Bigl(s-\sum_{i=0}^{N} r_i\Bigr) \prod_{i=0}^{N} \rme^{-\ci r_i \gamma(i)} \Bigr| % \nonumber \\ & \quad \le \frac{s^{N/2-1}}{(N/2-1)!} \oint_{\Gamma_N} \frac{|\rmd z|}{2\pi} \frac{\rme^{s (\im z)_+}}{|z|} \prod_{i\in \set{m'}\cup A_2} \frac{1}{|z-\gamma(i)|} \, . \end{align} Then we follow the iteration procedure until index $m'$ is reached. At that point we have to deal with the dependence of the factor $1/|z-\gamma(m')|$ on the various free momenta. If $z$ does not belong to the top of the integration path, we can estimate trivially $1/|z-\gamma(m')|\le 1$, and then complete the iterative estimate as in the proof of Lemma \ref{th:basicFest}. This yields an improvement of the upper bound by a full factor of $\lambda^2$. For those $z$ belonging to the top of the integration path, we have $z=\alpha+\ci \beta$ for some $|\alpha|\le 1+2 (N+1) \norm{\omega}_\infty$ and with $\beta=\lambda^2>0$. We next remove any dependence on $\kappa'$: since $\im \gamma(m)\le 0$ for all $m$, we can estimate all the remaining resolvent factors by \begin{align}%\label{eq:} & \frac{1}{|\alpha-\gamma(m)+\ci \beta|} \le \frac{1}{|\alpha-\re \gamma(m)+\ci \beta|} \, . \end{align} As we have shown above, $\re \gamma(m')$ is independent of any free momenta appearing before $k_1,k_2$. These can thus be estimated as before. Finally, we arrive at the $k_1,k_2$-integral, which is equal to \begin{align}\label{eq:crossTint} & \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \frac{1}{|\alpha-\alpha_2-\Omega_{i_2}+\ci\beta| |\alpha-\alpha_2-\alpha_1-p \Omega_{i_2} -\Omega_{i_0} +\ci\beta|} \, . \end{align} We represent both factors in terms of the oscillating integrals, using (\ref{eq:restophases}). Since all $\alpha$-terms above are independent of $k_1,k_2$, Fubini's theorem shows that the integral is bounded by \begin{align}\label{eq:crossTint2} & \sabs{\ln \beta}^2 \int_{\R^2}\!\! \rmd r\rmd s\, F(r;\beta) F(s;\beta) \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \rme^{-\ci (r+p s) \Omega_{i_2}-\ci s\Omega_{i_0}} \Bigr| \nonumber \\ & \quad \le 4 \sabs{\ln \beta}^2 \biggl( 1 + \int_{\R^2}\!\! \rmd r\rmd s\, \rme^{-\beta |s|} \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \rme^{-\ci (r+p s) \Omega_{i_2}-\ci s\Omega_{i_0}} \Bigr| \biggr) \, . \end{align} Suppose $v_{i_0}$ is an $T_j$-vertex. Then $\Omega_{i_0} = \pm \omega(k_j+u)\pm \omega(k_j+u') + \alpha'$ for some choice of the signs and for $\alpha'$ and $u,u'$ which are independent of $k_1,k_2$. The $j$-part of the double-loop goes through the vertex via two edges. If both edges $e,e'$ are $\edges_-(v_{i_0})$, we have $k_e=\sigma(k_j+u)$ and $k_{e'}=-\sigma (k_j+u')$ for some $\sigma\in \set{\pm 1}$, which implies $u'-u=-\sigma(k_e+k_{e'})$. Otherwise, the loop uses $\tilde{e} \in \edges_-$ and $\tilde{e}'\in \edges_+$, and then $k_{\tilde{e}}=\sigma(k_j+u)$ and $k_{\tilde{e}'}=\sigma (k_j+u')$, implying $u'-u=\sigma(k_{\tilde{e}'}-k_{\tilde{e}})= \sigma(k_{e}+k_{e'})$ where $e,e'$ are the remaining two edges in $\edges_-$. Thus by Lemma \ref{th:nokkdiff}, for any free momenta of a degree two vertex, $u'-u$ is either independent of the momenta, or depends on it by ``$\pm k_{j'}$'' for some $j'\in\set{1,2,3}$. In addition, if $u'-u$ is independent of all free momenta, then by Corollary \ref{th:kkpl} $k_e+k_{e'}=0$, and the corresponding graph is thus irrelevant. We change variable $r$ to $t=r+p s$, and estimate \begin{align}%\label{eq:} & \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \rme^{-\ci t \Omega_{i_2}-\ci s\Omega_{i_0}} \Bigr| %\nonumber \\ & \quad = \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \rme^{-\ci t (\pm \omega_1\pm \omega_2\pm \omega_3) -\ci s (\pm \omega(k_j+u)\pm \omega(k_j+u'))} \Bigr| \nonumber \\ & \quad \le \norm{p_{t}}_3^2 \norm{K(\pm t,\pm s,\pm s,u,u')}_3\, , \end{align} where we have used a convolution estimate similar to (\ref{eq:convsplit}). It is obvious from the definitions that not only $\norm{p_{-t}}_3=\norm{p_{t}}_3$, but also the norm of $K$ remains invariant under a swap of the signs of its time-arguments. Thus we can use this to change the first argument of $K$ to $t$. The resulting integral over $t,r$ is of a form given in (\ref{eq:crossingest1c}). Thus by Assumption (DR\ref{it:DRcrossing}) we find that (\ref{eq:crossTint2}) is bounded by \begin{align}%\label{eq:} 4 \sabs{\ln \beta}^2 \beta^{\gamma-1} ( 1 + \Fbcr(u'-u;\beta) ) \, . \end{align} As mentioned above, for a relevant graph, $u'-u$ must depend on some free momenta. We iterate the basic estimates, until the first such momenta appear. Since the dependence of $u'-u$ is of the form $\pm k_{j'}$, we can then apply the second part of Assumption (DR\ref{it:DRcrossing}). If $u'-u$ depends only on free momenta of the top fusion vertex, we use the first estimate, otherwise we use the second estimate. The remainder of the integrals can be iterated as in the basic estimate. Comparing the resulting bound to the basic estimate shows that we have gained an improvement by a factor of \begin{align}\label{eq:crossinggain} C \frac{1}{\sabs{s\beta}}\sabs{\ln \beta}^{c_2+1} \beta^\gamma\, . \end{align} This yields the bounds given in the Proposition, since for a $\mathcal{G}_n$-graph we have $N=2 n$ and for $\mathcal{F}_n$-graph $N=n$. We still need to consider the case where $v_{i_0}$ is an $X$-vertex. Then $\Omega_{i_0} = \sum_{i=1}^3 (\pm \omega(k_i+u_i)) + \alpha'$ for some choice of the signs and for $\alpha'$ and $u_i$, $i=1,2,3$, which are independent of $k_1,k_2$. Suppose $u_i=0$ for all $i$. Then also the fourth momentum is equal to $\pm k_0$. We can use the iterative cluster scheme as in the proof of Lemma \ref{th:immrecstart} to show that then $\re \gamma(i_0-1)$ would be independent of $k_1,k_2$, which is against the construction. For this, consider the addition of the degree zero vertex $v_{i_0}$ and let $A_i$, $i=1,2,3$, $A'$, and its partition $A'_i$, $i=0,1,2,3$, be defined as in Lemma \ref{th:immrecstart}. Since $v_{i_0}$ is not part of an immediate recollision, the next added vertex is either $v_{i_2}$ or has degree zero. In the latter case, there can be also more vertices before $i_2$ is added. Of these, we can ignore all immediate recollisions, since they leave the momenta and phase factors invariant. Thus we only need to consider the iterative cluster scheme when a number of degree zero vertices are added before $i_2$. Any such addition either leaves $A'$ invariant, or increases the number of edges in one of $A'_j$ by at least two. However, the argument used in Lemma \ref{th:immrecstart} implies that at the moment when $v_{i_2}$ is added, all of the sets $A'_i$ which intersect $\edges_-(v_{i_2})$ must be singlets, as else one of $u_i$ is not zero, or $v_{i_0}$ is not an $X$-vertex. Then none of $\Omega_{i}$, $i\in I'$, can depend on $k_1,k_2$, and thus we have $p=1$ above. On the other hand, $v_{i_0}$ is effectively a delayed recollision, and an explicit computation shows that $\Omega_{i_0}=-\Omega_{i_2}$, implying that $\re \gamma(i_0-1)$ is independent of $k_1,k_2$. Therefore, there is $j'=1,2,3$ such that $u_{j'}$ depends on some free momenta. We change variable $r$ to $t=r+p s$, and estimate \begin{align}%\label{eq:} & \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \rme^{-\ci t \Omega_{i_2}-\ci s\Omega_{i_0}} \Bigr| = \Bigl| \int_{(\T^d)^2}\!\! \rmd k_1\rmd k_2\, \prod_{i=1}^3 \rme^{-\ci (\pm t \omega(k_i)\pm s \omega(k_i+u_i))} \Bigr| \nonumber \\ & \quad \le \prod_{i=1}^3 \norm{K(t,\pm s,0,u_i,0)}_3\, , \end{align} where we have again used the invariance of $\norm{K}_3$ under reversal of its time-arguments. Employing the assumption (DR\ref{it:DRcrossing}) we thus find that (\ref{eq:crossTint2}) is then bounded by \begin{align}%\label{eq:} 4 \sabs{\ln \beta}^2 \beta^{\gamma-1} ( 1 + \Fbcr(u_{j'};\beta) ) \, . \end{align} By construction, $u_{j'}$ depends on some free momenta, and by Lemma \ref{th:nokkdiff} the dependence is of the earlier encountered form. Thus we can then conclude the rest of the estimate following the steps used for the $T$-vertex. This results in an improvement by a factor given in (\ref{eq:crossinggain}) compared to the basic estimate, and concludes the proof the Proposition. \end{proof} \subsection{Leading and nested graphs} For the leading and nested graphs we cannot take the absolute value of too many phase factors. In addition, the contribution from the immediate recollisions needs to be estimated more carefully. Let us thus consider a {\em relevant\/} graph which is either nested or leading. The momentum cut-offs have now fulfilled their purpose, and need to be removed. We use an iterative scheme to expand $\PFone = 1- \PFzero$ one by one, going through the interaction vertices $i'$ from the bottom to the top. At each step, we obtain two terms, one of which corresponds to replacing $\PFone \to 1$ in the iterated vertex. This term will be continued for the next iteration step. In the other term we take absolute values inside the $k$-integrals and estimate the phase factor using the iteration scheme of the basic estimate. We can then estimate the $\PFzero$-factor using Proposition \ref{th:PFcorr}: $\PFzero (\pm k)\le \sum_{e_ii_2} \Omega_i+\ci \im \gamma(i_2-1)$ is independent of the free momenta of $v_{i_2}$. In addition, $\zeta_{i_2}$ is also independent of free momenta of any other immediate recollisions. Thus for the first step where immediate recollisions are integrated out, we can take the corresponding exponential factors out of these integrals. If the recollision corresponds to a gain motive, adding it to a pairing corresponds to changing a factor $W(k_0)$ to \begin{align}%\label{eq:} -G_{s,\tau(\sigma)}[1,W,W,W](k_0) \rme^{-\ci s \zeta_{i_2}}\, , \end{align} where $s=s_{i_2-1}$ is the time-variable of the ``recollision'' time-slice and $\tau(\sigma)=(-\sigma,-1,\sigma,1)$, $\sigma$ being the parity of the part where the \defem{higher} of the two vertices is attached. Similarly, adding a loss motive to a line with parity $\sigma$ and momentum $k_0$ changes a factor $W(k_0)$ to \begin{align}%\label{eq:} \sigma \tau_{1,j} G_{s,\tau(\sigma)}[W,W_{j,1},W_{j,2},W_{j,3}](k_0) \rme^{-\ci s \zeta_{i_2}}\, , \end{align} where $j\in \set{1,2,3}$ and $W_{j,j}=1$, $W_{j,i}=W$ if $i\ne j$. The addition of the remaining immediate recollisions corresponds to similar modification of the integrand. However, an input function can then also be one of the previously generated $G_{s,\tau}$-factors in addition to the initial factors $W$. We need to control the time-integrability of these iterated terms. We use (\ref{eq:Gsest1}), which requires controlling the $\ell_3$-norm of $G_{s,\tau}$. For this, we note that for any $h\in \ell_1(\Z^d)$ and $t\in \R$, $x\in \Z^d$, \begin{align}%\label{eq:} \sum_{x\in \Z^d} |(U_t h)(x)|^3 \le \sum_{x\in \Z^d} \sum_{y\in (\Z^d)^3} \prod_{i=1}^3 (|p_t(x-y_i)|\, |h(y_i)|) \le \norm{p_t}_3^3 \norm{h}_1^3 \, , \end{align} and thus $\norm{U_t h}_3\le \norm{p_t}_3 \norm{h}_1$. Then Assumption (DR\ref{it:DRdisp}) provides decay in $t$. However, the estimate is useful only if $\ell_1$-norm of $h$ remains bounded, and this requires carefully separating the free evolution from the initial states; we note that even if $\norm{h}_1<\infty$, typically $\norm{U_t h}_1=\order{t^{p}}$ with $p \ge d/2$. Consider thus one of the factors obtained from the iteration of the leading motive integrals and let $f\in \ell_2$ denote its inverse Fourier-transform. Since $G_{s,\tau}$ is linear in all of its arguments, we can neglect the sign- and $\zeta$-factors in the estimation of the $\ell_3$-norm. However, we have to iteratively expand the first argument until either $1$ (gain motive) or $W$ (the initial pairing for a sequence of loss motives) is reached. Let $M\ge 1$ denote the number of iterations needed for this. Since both $1$ and $W$ have an inverse Fourier-transform in $\ell_1$, we conclude that the factor is then of the form \begin{align}%\label{eq:} \hat{f}(k) := \hat{h}_0(k) \prod_{m=1}^M \rme^{\ci \sigma_m s_m \omega(k_0)} \prod_{m=1}^M F_{m}(k) \, , \end{align} where $\sigma_m\in\set{\pm 1}$, $s_m\in \R$, are the appropriate parity- and time-variables, $\FT{h}_0\in\set{1,W}$, and \begin{align}%\label{eq:} & F_{m}(k_0) = \int_{(\T^d)^3}\!\! \rmd k_1 \rmd k_2\rmd k_3\, \delta(k_0-k_1-k_2-k_3) \prod_{i=1}^3 \left( \rme^{-\ci s_m \tau_{m,i} \omega(k_i)} \FT{f}_{m,i}(k_i) \right) \, , \end{align} where $\tau_{m,i}=\tau(\sigma_m)_i$ and all of the functions $\FT{f}_{m,i}$ are obtained from earlier iterations, and thus are one of $1$, $W$, or $G_{s,\tau}$. In any case, $h_0\in \ell_1(\Z^d)$. Now for any $t\in \R$, \begin{align}%\label{eq:} (U_t f)\hat{\;}(k) = (U_{t-\sum_{m=1}^M \sigma_m s_m}H)\hat{\;}(k), \quad \text{where } \FT{H}(k) = \hat{h}_0(k) \prod_{m=1}^M F_{m}(k) \, . \end{align} As $F_{m}(k_0) = \sum_{x\in \Z^d} \rme^{-\ci 2 \pi x\cdot k_0} \prod_{i=1}^3 (U_{\tau_{m,i} s_m } f_{m,i})(x)$, we have \begin{align}%\label{eq:} H(y) = \sum_{x\in (\Z^d)^M} h_0\Bigl(y-\sum_{m=1}^M x_m\Bigr) \prod_{i=1}^3 \prod_{m=1}^M (U_{\tau_{m,i} s_m } f_{m,i})(x_m)\, . \end{align} Therefore, $\norm{H}_1\le \norm{h_0}_1 \prod_{i=1}^3 \prod_{m=1}^M \norm{U_{\tau_{m,i} s_m } f_{m,i}}_3$. We conclude that \begin{align}\label{eq:Giterest} \norm{U_t f}_3 \le c_1 \norm{p_{t-\sum_{m=1}^M \sigma_m s_m}}_3 \prod_{m=1}^M \prod_{i=1}^3 \norm{U_{\tau_{m,i} s_m } f_{m,i}}_3 \, , \end{align} where $c_1=\max(1,\norm{\IFT{W}}_1)<\infty$. \begin{proposition}\label{th:leadingest} There are constants $c,c_0>0$, which depend only on $\omega$, and $C$ which depends only on $\omega,f,g$ such that for any leading graph \begin{align} & |\mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa)| \le C \lambda^{\gamma'} \rme^{s\lambda^2} \sabs{c s\lambda^2}^{n} \sabs{\ln \lambda}^{2+n} + \frac{C}{n!} (c_0 \lambda^2 s)^n \, , \label{eq:Gnleadb}\\ & \left|\left.\mathcal{F}_n^{\rm pairs}(S,\ell,t \lambda^{-2},\kappa) \right|_{\PFone \to 1} \right| \le \frac{C}{(n/2)!} (c_0 t)^{n/2} \, , \label{eq:Fnleadb1}\\ & \left|\mathcal{F}_n^{\rm pairs}(S,\ell,t \lambda^{-2},\kappa) -\left.\mathcal{F}_n^{\rm pairs}(S,\ell,t \lambda^{-2},\kappa) \right|_{\PFone \to 1}\right| \le C \lambda^{\gamma'} \rme^{t} \sabs{c t}^{n/2}\sabs{\ln \lambda}^{2+n/2} \label{eq:Fnleadb2} \, . \end{align} \end{proposition} \begin{proof} Consider a leading graph with $N$ interaction vertices. Then $N$ is even. The first term in (\ref{eq:Gnleadb}) and the bound in (\ref{eq:Fnleadb2}) arise from exchanging all $\PFone$ factors to $1$ and both follow from applying (\ref{eq:somebound}). In the remaining term, we leave the time-integrals unmodified, and perform first all $k$-integrals apart from the top fusion integral on which the original $\FT{f}$- and $\FT{g}$-factors depend. A leading graph consists of a sequence of $N/2$ leading motives. Since the leading motives preserve the phase, we thus have $\re \gamma(i)=0$ for all even $i$. In addition, for all odd $i$ we have $\Omega_{i} = -\Omega_{i+1}$. Therefore, in this case the total phase is \begin{align}\label{eq:leadingphases} \sum_{i=0}^N r_i \re \gamma(i) = \sum_{j=1}^N \Omega_j \sum_{i=0}^{j-1} r_i = \sum_{m=1}^{N/2} \Omega_{2m} r_{2 m-1} \, . \end{align} As explained above, performing the immediate recollision $k$-integrals results in an iterative application of $G_{r_i,\tau}$. We estimate the absolute value of the amplitude by taking the absolute value inside the time-integrals. For the outmost (i.e., last) application of $G_{r_i,\tau}$ corresponding to $m=N/2$ we use (\ref{eq:Gsest1}) and in the resulting bound we can iterate estimates (\ref{eq:Gsest1}) and (\ref{eq:Giterest}) further until only $\ell_1$-norms of $\tilde{W}$ remain. This shows that for each $m=1,2,\ldots,N/2$ there are three subsets $B_{m,i}$, $i=1,2,3$, of $I_{1,m-1}$ such that the $k$-integrated phase factor has a bound \begin{align}\label{eq:leadingphase} c_1^{3 N/2+1} \prod_{m=1}^{N/2} \prod_{i=1}^3 \norm{p_{\pm r_{2 m -1}-\sum_{j\in B_{m,i}} (\pm r_{2 j-1})}}_3 \, , \end{align} for some choice of signs. By H\"{o}lder's inequality and assumption (DR\ref{it:DRdisp}) there is a constant $c_0$ such that if this bound is integrated over all of $r_j$, $j$ odd, the result is bounded by $c_0^{N/2}$. (The integration over $r_{N-1}$ is performed first, and the rest are iterated until $r_{1}$ is reached.) We can apply an estimate similar to that used in (\ref{eq:intrtrick}) to separate the odd and even integrations and obtain an additional factor $s^{N/2}/(N/2)!$ from the even integrations. Collecting all the estimates together yields the bounds stated in the Proposition. \end{proof} \begin{proposition}\label{th:nestedbnd} There is a constant $c_0$, which depends only on $\omega$, and $C$, which depends only on $\omega,f,g$, such that the amplitudes of all nested graphs satisfy the bounds \begin{align}%\label{eq:} & |\mathcal{G}_n^{\rm pairs}(S,J,\ell,\ell',s,\kappa)| \le C \lambda^{\gamma'} \rme^{s\lambda^2} \sabs{c_0 s\lambda^2}^{n} \sabs{\ln \lambda}^{2+n}\, , \\ & |\mathcal{F}_n^{\rm pairs}(S,\ell,t \lambda^{-2},\kappa)| \le C \lambda^{\gamma'} \rme^{t} \sabs{c_0 t}^{n/2} \sabs{\ln \lambda}^{2+n/2} \, . \end{align} \end{proposition} \begin{proof} Consider a relevant nested graph. Let $i_2$ denote the index of the first degree two interaction vertex $v=v_{i_2}$ which is not an immediate recollision. By assumption, every long time-slice which depends on the double-loop of $v$ is nested inside the double-loop, and there is at least one such time-slice. Let $N_2$ collect the indices of these time-slices. In addition, applying Lemma \ref{th:immrecstart}, we can conclude that every double-loop before $i_2$ corresponds to an immediate recollision. Let $j_0=\min N_20$ and thus there is a time-slice $j_0-1$. If it is short, then $j_0-1>0$ and it belongs to an immediate recollision. This however leads to contradiction, since immediate recollisions preserve the phase factor, and thus $\re \gamma(j_0)=\re \gamma(j_0-2)$ implying that the slice $j_0-20$ denote a constant for which Proposition \ref{th:leadingest} holds. We choose $t_0 = (2^6 c_0)^{-1}>0$, when for all $0 4m $. Then $\kappa_{2 m-j}=0$ for all $j$, and only the $\delta$-function depends on $r_i$, for $i$ even. We change integration variables from $r$ to $(t,s)$ with $t_i=\lambda^2 r_{2 i}$, for $i=0,1,\ldots,m$, and $s_i=r_{2 i-1}$ for $i=1,\ldots,m$. Then the last line of (\ref{eq:Fnpairs2}) is equal to \begin{align}%\label{eq:} & \lambda^{-2 m} \int_{(\R_+)^{I_{m}}}\!\rmd s \prod_{j=1}^{m} \rme^{-\ci s_{j} \Omega_{2 j}} \int_{(\R_+)^{I_{0,m}}}\!\rmd t \, \delta\Bigl(t-\sum_{i=0}^{m} t_i - \lambda^{2}\sum_{i=1}^{m} s_i\Bigr) \nonumber \\ & \quad = \lambda^{-2 m} \int_{(\R_+)^{I_{m}}}\!\rmd s \prod_{j=1}^{m} \rme^{-\ci s_{j} \Omega_{2 j}} \1\Bigl(\sum_{i=1}^{m} s_i \le t \lambda^{-2}\Bigr) \frac{1}{m!} \Bigl(t-\lambda^{2}\sum_{i=1}^{m} s_i\Bigr)^m\, . \end{align} Next we integrate over the double-loop momenta. This leads to iterated applications of $G_{s_i,\tau_i}$ which yields a function $\tilde{G}(-k_{e_1},s;S,\ell)$. This function also has an $s$-integrable upper bound which is independent of $k$ and $\lambda$, and thus we can apply dominated convergence to take the $\lambda\to 0$ limit inside the $s$-integration. This proves that for a leading graph \begin{align}\label{eq:sumleading2} &\lim_{ \lambda \to 0 } \left.\mathcal{F}_n^{\rm pairs}(S,\ell,t\lambda^{-2},\kappa) \right|_{\PFone \to 1} % \nonumber \\ & \quad = (-1)^m \frac{t^m}{m!} \int_{\T^d}\rmd k\, \FT{g}(k)^* \FT{f}(k) \int_{(\R_+)^{I_{m}}}\!\rmd s \, \tilde{G}(k,s;S,\ell) \, . \end{align} We now sum over the leading graphs with $m$ motives, which is a finite sum and thus can be taken directly of $\tilde{G}(k,s;S,\ell)$. Every leading diagram in the sum in (\ref{eq:sumleading}) is obtained by iteratively adding $m$ leading motives into the single pairing graph. It is then easy to see that there is a one-to-one correspondence between leading graphs with $m$ motives and no interactions vertices in the minus tree, and graphs which are obtained by adding a leading motive to a pairing cluster of such a graph with $m-1$ motives so that the result does not contain any interaction vertices in the minus tree. Consider thus a graph with $m-1$ motives which could give rise to a leading diagram in the sum in (\ref{eq:sumleading}). This graph has $2 m -1$ pairing clusters, exactly one of which connects to the minus tree. In addition, if we add the leading motive to this special pairing, then the resulting graph also cannot contain any interaction vertices in the minus tree. This rules out all gain motives, and the 6 loss motives which connect on the left leg of the pairing cluster. Thus we get only 6 new terms from such an addition, those corresponding to adding a loss motive to the right leg of the pairing cluster. There are no such restrictions to motives added to any of the remaining $2m-2$ pairings since all the new interaction vertices then belong to the plus tree. Fix the graph and such a pairing, and let $\sigma$ denote the parity on the left leg of the pairing. We sum over all graphs obtained by adding a leading motive to this pairing. Then the iteration steps to obtain the corresponding $\tilde{G}(k,s;S,\ell)$ are equal apart from the first step which is of the form $\pm G_{s_1,\tau}(k_0)$. Computing the sum over all possibilities in the first iteration yields a contribution \begin{align}\label{eq:alllead} & \int_{(\T^d)^2}\!\! \rmd k_1 \rmd k_2\, \Bigl[ \rme^{-\ci s_1 \Omega(k,\sigma)} %\nonumber \\ & \qquad \times \left( -2 W_1 W_2 W_3+ 2 \sigma W_0 W_1 W_2 + 2 W_0 W_1 W_3- 2 \sigma W_0 W_2 W_3\right) \nonumber \\ & \quad +\rme^{-\ci s_1 \Omega(k,-\sigma)} \left( -2 W_1 W_2 W_3- 2 \sigma W_0 W_1 W_2 + 2 W_0 W_1 W_3+ 2 \sigma W_0 W_2 W_3\right) \Bigr]\, , \end{align} where $W_i=W(k_i)$ and $k_3=k_0-k_1-k_2$. Since $\Omega((k_1,k_2,k_3),-\sigma)=-\Omega((k_3,k_2,k_1),\sigma)$, we can make a change of integration variables $k_1\to k_3$ in the second term, which shows that (\ref{eq:alllead}) is equal to \begin{align}%\label{eq:alllead} & 2 \int_{(\T^d)^2}\!\! \rmd k_1 \rmd k_2\, \Bigl[ \rme^{-\ci s_1 \Omega(k,\sigma)} +\rme^{\ci s_1 \Omega(k,\sigma)} \Bigr] W_0 W_1 W_2 W_3 %\nonumber \\ & \qquad \times \left( -W_0^{-1}+ \sigma W_3^{-1} + W_2^{-1} - \sigma W_1^{-1}\right) \, . \end{align} Since $W(k)^{-1} = \beta(\omega(k)-\mu)$, the final factor in parenthesis is equal to $\beta \sigma \Omega(k,\sigma)$. This implies that if we integrate (\ref{eq:alllead}) over $s_1\in [0,M]$ for some $M>0$, then the result is equal to \begin{align}%\label{eq:} & 2 \beta\sigma \int_{-M}^M\! \rmd s_1 \int_{(\T^d)^2}\!\! \rmd k_1 \rmd k_2\, \Omega(k,\sigma) \rme^{-\ci s_1 \Omega(k,\sigma)} W_0 W_1 W_2 W_3 \nonumber \\ & \quad = 2 \beta\sigma \int_{(\T^d)^2}\!\! \rmd k_1 \rmd k_2\, \ci \Bigl[ \rme^{-\ci M \Omega(k,\sigma)} +\rme^{\ci M \Omega(k,\sigma)} \Bigr] W_0 W_1 W_2 W_3 \, , \end{align} which vanishes as $M\to \infty$. Therefore, the sum over such graphs in (\ref{eq:sumleading2}) is exactly zero, even though the individual terms can be non-vanishing. (The vanishing is not accidental: $W$ is a stationary solution of the corresponding nonlinear Boltzmann equation, (\ref{4}), and the above sum corresponds to the action of its collision operator $\mathcal{C}$ to $W$, which thus should be equal to zero. Compare with the discussion in Section \ref{sec:link}.) Therefore, in the sum over the relevant leading diagrams, only those terms can be non-zero which come from the application of a loss term to the right leg of the unique pairing which connects to the minus tree. Since this only changes the multiplicative factor associated with the pairing, we can iterate the above argument and conclude that the sum must be equal to $W(k) \Gamma(k)^m$, where $\Gamma(k)$ is the result from the sum over the 6 relevant loss terms. As above, this can be computed explicitly, and it is seen to be equal to $\Gamma(k)$ defined in (\ref{eq:defGamma}) after a change of variables and using the evenness of the functions $\omega$ and $W$. Collecting all the results together yields (\ref{eq:sumleading}). \end{proofof} \appendix \section{Nearest neighbor interactions}\label{sec:appNN} Let us consider here the dispersion relation $\omega$ defined by \begin{align}%\label{eq:} \omega(k):= c - \sum_{\nu=1}^d \cos p^\nu, \quad \text{with }p=2\pi k, \end{align} where $c\in \R$ is arbitrary. This clearly satisfies (DR1). We consider the function $K$ defined in (\ref{eq:defp2tx}). Then \begin{align}%\label{eq:} & K(x;t_0,t_1,t_2,\tfrac{1}{2\pi} q_1, \tfrac{1}{2\pi} q_2) = \rme^{-\ci c (t_0+t_1+t_2)} %\nonumber \\ & \qquad \times \prod_{\nu=1}^d \int_{0}^{2\pi} \!\frac{\rmd p}{2\pi}\, \rme^{\ci p x^\nu} \rme^{\ci (t_0 \cos p +t_1 \cos (p +q_1^\nu) + t_2 \cos (p +q_2^\nu))} \, . \end{align} Since \begin{align}%\label{eq:} & t_0 \cos p +t_1 \cos (p +q_1^\nu) + t_2 \cos (p +q_2^\nu) %\nonumber \\ & \quad = \re\!\left[ \rme^{\ci p} ( t_0 + t_1 \rme^{\ci q_1^\nu} + t_2 \rme^{\ci q_2^\nu} ) \right]\, , \end{align} there is $\varphi^\nu$, which does not depend on $p$, such that this is equal to \begin{align}%\label{eq:} R^\nu \cos(p+\varphi^\nu),\quad \text{with}\quad R^\nu = | t_0 + t_1 \rme^{\ci q_1^\nu} + t_2 \rme^{\ci q_2^\nu} |\, . \end{align} This proves that \begin{align}%\label{eq:} | K(x;t_0,t_1,t_2,\tfrac{1}{2\pi} q_1, \tfrac{1}{2\pi} q_2)| = \prod_{\nu=1}^d \left| \int_{0}^{2\pi} \!\frac{\rmd p}{2\pi}\, \rme^{\ci p x^\nu+\ci R^\nu \cos p}\right| \, , \end{align} and, therefore, \begin{align}%\label{eq:} \norm{K(t_0,t_1,t_2,\tfrac{1}{2\pi} q_1, \tfrac{1}{2\pi} q_2)}_3 \le \prod_{\nu=1}^d \norm{p^{(d=1)}_{R^\nu}}_3 \le C \prod_{\nu=1}^d \frac{1}{\sabs{R^\nu}^{\frac{1}{7}}}, \end{align} where we have applied a known bound for the $\ell_3$-norm of the one-dimensional propagator, following \cite{HoLa97, LaLu01}. We note that $p_t(x)=K(x;t,0,0,0,0)$ and thus the above bound shows that \begin{align}\label{eq:pt33bound} \norm{p_t}_3^3 \le C \sabs{t}^{-\frac{3 d}{7}} \le C \sabs{t}^{-1-\frac{2}{7}} \end{align} for all $d\ge 3$. Thus (DR2) is satisfied then. On the other hand, \begin{align}%\label{eq:} \Bigl| \int_{\T^d}\!\rmd k\, \rme^{-\ci t (\omega(k)+\sigma \omega(k-k_0))}\Bigr| = |K(0;t,\sigma t,0,-k_0,0)| \le C \prod_{\nu=1}^d \frac{1}{\sabs{R^\nu}^{\frac{1}{2}}}, \, \end{align} since $\cos p$ is a Morse function. Here $R^\nu = |t|\, | 1 + \sigma \rme^{-\ci 2 \pi k_0^\nu}| \ge |t| \,|\sin (2 \pi k_0^\nu )|$. Thus, given $k_0$, let $\nu_0$ denote the index corresponding to the {\em second\/} largest of the numbers $|\sin (2 \pi k_0^\nu )|$. (This might not be unique, but this is irrelevant for the following estimates.) Then we have \begin{align}%\label{eq:} \Bigl| \int_{\T^d}\!\rmd k\, \rme^{-\ci t (\omega(k)+\sigma \omega(k-k_0))}\Bigr| \le C \sabs{t}^{-1} \frac{1}{|\sin (2\pi k_0^{\nu_0})|} \le C \sabs{t}^{-1} \frac{1}{d(k_0^{\nu_0},\set{0,\frac{1}{2}})} \, . \end{align} Thus (DR3) holds if $d\ge 3$ and we choose $\Msing$ to consist of those $k$ for which all but one component belong to the set $\set{0,\frac{1}{2}}$. (This set is clearly a union of lines.) Therefore, we only need to check (DR4). Let us first consider (\ref{eq:crossingest1b}), where $t_0=t$, $t_1=\pm s$, $t_3 = 0$, $q_2=0$, and we have some fixed $n\in \set{1,2,3}$. Then $R^\nu = | t\pm s \rme^{\ci q_1^\nu}|$ which, by the triangle inequality, has a lower bound $| |t| -|s| |$. On the other hand, by inspecting only the imaginary part, we find that it also has a lower bound $|s| |\sin q_1^\nu|$. We use the second bound in the $n$:th factor in (\ref{eq:crossingest1b}) and the first bound in the remaining two factors. This shows that the left hand side of (\ref{eq:crossingest1b}) can be bounded by \begin{align} & C \int_{\R^2}\!\! \rmd t\rmd s\, \rme^{-\beta |s|} \sabs{ |t|-|s| }^{-\frac{8}{7}} |s|^{-\frac{4}{7}} \prod_{\nu=1}^4 \frac{1}{|\sin (2 \pi u_n^\nu)|^{\frac{1}{7}}} % \nonumber \\ & \quad \le C \beta^{\frac{4}{7}-1} \prod_{\nu=1}^d \frac{1}{|\sin (2 \pi u_n^\nu)|^{\frac{1}{7}}} \, , \end{align} where we have assumed $d\ge 4$. In the other inequality (\ref{eq:crossingest1c}), we need to consider $t_0=t$, $t_1=\pm s$, $t_3 = \pm s$. We apply the previous estimates which shows that the left hand side of (\ref{eq:crossingest1c}) is bounded by \begin{align}%\label{eq:} & C \int_{\R}\! \rmd t\, \sabs{t}^{-\frac{8}{7}} \int_{\R}\!\rmd s\, \rme^{-\beta |s|} \prod_{\nu=1}^4 \frac{1}{\sabs{R^\nu}^{\frac{1}{7}}} \end{align} where \begin{align}%\label{eq:} R^\nu = | t \pm s (\rme^{\ci q_1^\nu} \pm \rme^{\ci q_2^\nu})| \ge \left|\, |s|\, |1\pm \rme^{\ci (q_2^\nu-q_1^\nu)}|-|t|\,\right| \,. \end{align} Since here $|1\pm \rme^{\ci (q_2^\nu-q_1^\nu)}|\ge |\sin (q_2^\nu-q_1^\nu)|$ and $q_2-q_1=2\pi(u_2-u_1)$, this shows that (\ref{eq:crossingest1c}) can be bounded by $C \beta^{\frac{4}{7}-1} \prod_{\nu=1}^d |\sin (2 \pi (u_2-u_1)^\nu)|^{-\frac{1}{7}}$. Thus we now only need to check that the second item in (DR4) holds for $\Fbcr$ defined by (\ref{eq:defnnFcr}). Since $\Fbcr$ is independent of $\beta$ and obviously belongs to $L^1(\T^d)$, (\ref{eq:crossingest2a}) holds with $c_2=0$. For the second integral we need to estimate \begin{align}\label{eq:DR4nn2est} \left| \int_{(\T^d)^2} \rmd k_1\rmd k_2\, \Fbcr(k_1+u;\beta) \rme^{-\ci s (\sigma_1 \omega(k_1)+ \sigma_2 \omega(k_2)+ \sigma_3 \omega(k_1+k_2-k_0))}\right| \end{align} for any choice of the signs $\sigma\in \set{\pm 1}^3$. Since \begin{align}%\label{eq:} \cos(p_2) \pm \cos(p_1+p_2-p_0) = \cos(p_2+\varphi) | 1\pm \rme^{\ci(p_1^\nu-p_0^\nu)}| \end{align} for some $\varphi$ independent of $p_2$, we can bound (\ref{eq:DR4nn2est}) by \begin{align}%\label{eq:} C \sabs{s}^{-\frac{d}{2}} \prod_{\nu=1}^d \left[ \int_0^{2\pi} \rmd p_1^\nu\, |\sin(p_1^\nu +2\pi u^\nu)|^{-\frac{1}{7}} |\sin(p_1^\nu - 2\pi k_0^\nu)|^{-\frac{1}{2}} \right] \, . \end{align} Then an application of H\"{o}lder's inequality with the conjugate pair $(3,\frac{3}{2})$, reveals that the remaining integral is bounded uniformly in $u$ and $k_0$. Thus (\ref{eq:DR4nn2est}) is uniformly bounded by $C\sabs{s}^{-2}$ which is integrable in $s$. 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Sulem, \textit{The Nonlinear Schr\"{o}dinger Equation: Self-Focusing and Wave Collapse}, Springer, Berlin, 1999. \bibitem{ZLF92} V.~E.~Zakharov, V.~S.~L'vov, and G.~Falkovich, \textit{Kolmogorov Spectra of Turbulence I: Wave Turbulence}, Springer, Berlin, 1992. \end{thebibliography} \end{document} ---------------0901211031755 Content-Type: application/postscript; name="DiagramC.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="DiagramC.eps" %!PS-Adobe-2.0 EPSF-2.0 %%BoundingBox: 105 638 469 733 %%HiResBoundingBox: 106.000000 639.500000 467.500000 731.500000 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: dummy.dvi %%CreationDate: Wed Nov 26 10:59:05 2008 %%PageOrder: Ascend %%DocumentFonts: Times-Roman CMMI10 CMR7 CMMI7 CMSY10 CMR10 %%DocumentPaperSizes: a4 %%EndComments % EPSF created by ps2eps 1.64 %%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 %DVIPSSource: TeX output 2008.11.26:1059 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/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 %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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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 43 /plus put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /minus put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 59 /comma put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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. 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8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix 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2430 4995 l gs col0 s gr % Polyline n 1890 4590 m 1890 5940 l gs col0 s gr % Polyline n 1845 4590 m 1935 4590 l gs col0 s gr % Polyline n 1845 5940 m 1935 5940 l gs col0 s gr % Ellipse 15.000 slw n 4185 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3285 5040 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6750 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6750 5045 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 7.500 slw [15 60] 60 sd n 4905 5490 m 2565 5490 l gs col0 s gr [] 0 sd % Polyline [15 60] 60 sd n 4905 5940 m 2565 5940 l gs col0 s gr [] 0 sd % Polyline [15 60] 60 sd n 4905 5040 m 2565 5040 l gs col0 s gr [] 0 sd % Polyline [15 60] 60 sd n 4905 4590 m 2565 4590 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 3285 5040 m 3285 4590 l gs col0 s gr % Polyline 7.500 slw [15 60] 60 sd n 7920 5940 m 5580 5940 l gs col0 s gr [] 0 sd % Polyline [15 60] 60 sd n 7920 5040 m 5580 5040 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/Times-Roman-iso ff 190.50 scf sf 7605 6165 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 6255 6165 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 6705 6165 m gs 1 -1 sc (sig) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5805 6165 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 6390 4860 m gs 1 -1 sc (k21) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 6795 5355 m gs 1 -1 sc (sig) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 6795 4815 m gs 1 -1 sc (sig) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5490 5760 m gs 1 -1 sc (k01) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7695 5760 m gs 1 -1 sc (k05) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7560 5310 m gs 1 -1 sc (k13) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5715 5310 m gs 1 -1 sc (k11) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5985 5760 m gs 1 -1 sc (k02) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2115 5310 m gs 1 -1 sc (s1) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2115 4905 m gs 1 -1 sc (s2) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2115 5760 m gs 1 -1 sc (s0) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 4230 5400 m gs 1 -1 sc (v1) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 3375 4950 m gs 1 -1 sc (v2) col0 sh gr % here ends figure; pagefooter showpage %%Trailer %EOF %%EndDocument @endspecial 275 1121 a /End PSfrag 275 1121 a 275 -464 a /Hide PSfrag 275 -464 a -410 -408 a Ff(PSfrag)20 b(replacements)p -410 -377 685 4 v 275 -373 a /Unhide PSfrag 275 -373 a 198 -298 a { 198 -298 a 12 x Fe(v)238 -274 y Fd(1)198 -298 y } 0/Place PSfrag 198 -298 a 197 -198 a { 197 -198 a 12 x Fe(v)237 -174 y Fd(2)197 -198 y } 1/Place PSfrag 197 -198 a 198 -99 a { 198 -99 a 12 x Fe(s)237 -75 y Fd(0)198 -99 y } 2/Place PSfrag 198 -99 a 198 1 a { 198 1 a 12 x Fe(s)237 25 y Fd(1)198 1 y } 3/Place PSfrag 198 1 a 198 101 a { 198 101 a 12 x Fe(s)237 125 y Fd(2)198 101 y } 4/Place PSfrag 198 101 a 245 199 a { 245 199 a 25 x Fe(t)245 199 y } 5/Place PSfrag 245 199 a 141 283 a { 141 283 a 17 x Fe(k)184 312 y Fd(2)p Fc(;)p Fd(1)141 283 y } 6/Place PSfrag 141 283 a 141 383 a { 141 383 a 17 x Fe(k)184 412 y Fd(1)p Fc(;)p Fd(1)141 383 y } 7/Place PSfrag 141 383 a 141 482 a { 141 482 a 17 x Fe(k)184 511 y Fd(1)p Fc(;)p Fd(3)141 482 y } 8/Place PSfrag 141 482 a 141 582 a { 141 582 a 17 x Fe(k)184 611 y Fd(0)p Fc(;)p Fd(1)141 582 y } 9/Place PSfrag 141 582 a 141 682 a { 141 682 a 17 x Fe(k)184 711 y Fd(0)p Fc(;)p Fd(2)141 682 y } 10/Place PSfrag 141 682 a 141 781 a { 141 781 a 17 x Fe(k)184 810 y Fd(0)p Fc(;)p Fd(5)141 781 y } 11/Place PSfrag 141 781 a 225 904 a { 225 904 a 18 x Fe(\033)225 904 y } 12/Place PSfrag 225 904 a 210 994 a { 210 994 a 20 x Fb(\000)210 994 y } 13/Place PSfrag 210 994 a 210 1093 a { 210 1093 a 21 x Fa(+)210 1093 y } 14/Place PSfrag 210 1093 a eop end %%Trailer userdict /end-hook known{end-hook}if %%Trailer cleartomark countdictstack exch sub { end } repeat restore %%EOF ---------------0901211031755 Content-Type: application/postscript; 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Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% 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 %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, % W. Schmidt, P. Lehman", % version = "2.0", % date = "27nov06", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "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" % @} % % The 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, Windows ANSI including the euro symbol, % MacRoman, and some extra characters from Lucida. % % 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. % These are /dotlessj /ff /ffi /ffl. % % (4) hyphen appears twice for compatibility with both ASCII and Windows. % % (5) /Euro was assigned to 128, as in Windows ANSI % % (6) Missing characters from MacRoman encoding incorporated as follows: % % PostScript MacRoman TeXBase1 % -------------- -------------- -------------- % /notequal 173 0x16 % /infinity 176 0x17 % /lessequal 178 0x18 % /greaterequal 179 0x19 % /partialdiff 182 0x1A % /summation 183 0x1B % /product 184 0x1C % /pi 185 0x1D % /integral 186 0x81 % /Omega 189 0x8D % /radical 195 0x8E % /approxequal 197 0x8F % /Delta 198 0x9D % /lozenge 215 0x9E % /TeXBase1Encoding [ % 0x00 /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron % 0x10 /caron /dotlessi /dotlessj /ff /ffi /ffl /notequal /infinity /lessequal /greaterequal /partialdiff /summation /product /pi /grave /quotesingle % 0x20 /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 % 0x80 /Euro /integral /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /Omega /radical /approxequal % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /Delta /lozenge /Ydieresis % 0xA0 /.notdef /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron % 0xB0 /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 %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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 %%EndProcSet %%BeginProcSet: special.pro 0 0 %! 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 %%EndProcSet %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. 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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 49 /one put dup 50 /two put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 78 /N put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 12 /beta put dup 20 /kappa put dup 99 /c put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark TeXDict begin 39158280 55380996 1000 600 600 (dummy.dvi) @start /Fa 207[19 48[{}1 58.1154 /CMSY7 rf /Fb 205[42 42 49[{}2 83.022 /CMR10 rf /Fc 177[53 78[{}1 58.1154 /CMMI7 rf /Fd 156[36 78[48 7[47 12[{}3 83.022 /CMMI10 rf /Fe 139[23 32 28 1[42 1[42 65 23 4[42 28 37 1[37 1[37 13[46 2[46 80[{TeXBase1Encoding ReEncodeFont}14 83.022 /Times-Roman rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 /setpagedevice where { pop << /PageSize [595 842] >> setpagedevice } { /a4 where { pop a4 } if } ifelse %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 275 1420 a /PSfrag where{pop((om))[[0()1 0]]((bt))[[1()1 0]]((1))[[2()1 0]]((2kp))[[3()1 0]]4 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 275 1420 a @beginspecial 0 @llx 0 @lly 435 @urx 205 @ury 2834 @rwi @clip @setspecial %%BeginDocument: Gbpath_raw.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: Gbpath_raw.fig %%Creator: fig2dev Version 3.2 Patchlevel 5 %%CreationDate: Tue Dec 2 10:38:58 2008 %%For: jlukkari@mat-417 (Jani Lukkarinen,,,) %%BoundingBox: 0 0 435 205 %Magnification: 1.0000 %%EndComments %%BeginProlog /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % 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 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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 2475 m 1065 2715 l 1185 2715 l 1185 2475 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 6389 3855 m 6540 3855 l 6540 3795 l 6389 3795 l 6389 3795 l 6509 3825 l 6389 3855 l cp 5761 3795 m 5610 3795 l 5610 3855 l 5761 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 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Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% 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 %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, % W. Schmidt, P. Lehman", % version = "2.0", % date = "27nov06", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "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" % @} % % The 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, Windows ANSI including the euro symbol, % MacRoman, and some extra characters from Lucida. % % 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. % These are /dotlessj /ff /ffi /ffl. % % (4) hyphen appears twice for compatibility with both ASCII and Windows. % % (5) /Euro was assigned to 128, as in Windows ANSI % % (6) Missing characters from MacRoman encoding incorporated as follows: % % PostScript MacRoman TeXBase1 % -------------- -------------- -------------- % /notequal 173 0x16 % /infinity 176 0x17 % /lessequal 178 0x18 % /greaterequal 179 0x19 % /partialdiff 182 0x1A % /summation 183 0x1B % /product 184 0x1C % /pi 185 0x1D % /integral 186 0x81 % /Omega 189 0x8D % /radical 195 0x8E % /approxequal 197 0x8F % /Delta 198 0x9D % /lozenge 215 0x9E % /TeXBase1Encoding [ % 0x00 /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron % 0x10 /caron /dotlessi /dotlessj /ff /ffi /ffl /notequal /infinity /lessequal /greaterequal /partialdiff /summation /product /pi /grave /quotesingle % 0x20 /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 % 0x80 /Euro /integral /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /Omega /radical /approxequal % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /Delta /lozenge /Ydieresis % 0xA0 /.notdef /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron % 0xB0 /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 %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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 %%EndProcSet %%BeginProcSet: special.pro 0 0 %! 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 %%EndProcSet %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 102 /braceleft put dup 103 /braceright put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 59 /comma put dup 78 /N put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B B078D20B4B8A6D7F0300CF694E6956FF9C29C84FCC5C9E8890AA56B1BC60E868 DA8488AC4435E6B5CE34EA88E904D5C978514D7E476BF8971D419363125D4811 4D886EDDDCDDA8A6B0FDA5CF0603EA9FA5D4393BEBB26E1AB11C2D74FFA6FEE3 FAFBC6F05B801C1C3276B11080F5023902B56593F3F6B1F37997038F36B9E3AB 76C2E97E1F492D27A8E99F3E947A47166D0D0D063E4E6A9B535DC9F1BED129C5 123775D5D68787A58C93009FD5DA55B19511B95168C83429BD2D878207C39770 012318EA7AA39900C97B9D3859E3D0B04750B8390BF1F1BC29DC22BCAD50ECC6 A3C633D0937A59E859E5185AF9F56704708D5F1C50F78F43DFAC43C4E7DC9413 44CEFE43279AFD3C167C942889A352F2FF806C2FF8B3EB4908D50778AA58CFFC 4D1B14597A06A994ED8414BBE8B26E74D49F6CF54176B7297CDA112A69518050 01337CBA5478EB984CDD22020DAED9CA8311C33FBCC84177F5CE870E709FC608 D28B3A7208EFF72988C136142CE79B4E9C7B3FE588E9824ABC6F04D141E589B3 914A73A42801305439862414F893D5B6C327A7EE2730DEDE6A1597B09C258F05 261BC634F64C9F8477CD51634BA648FC70F659C90DC042C0D6B68CD1DF36D615 24F362B85A58D65A8E6DFD583EF9A79A428F2390A0B5398EEB78F4B5A89D9AD2 A517E0361749554ABD6547072398FFDD863E40501C316F28FDDF8B550FF8D663 9843D0BEA42289F85BD844891DB42EC7C51229D33EE7E83B1290404C799B8E8C 889787CDC0C51802EA1E0C63E6DE20980D3DD206F0536A88537C60244BA8D64B 5FAA26C67A5BAC9D35F1002999D0BFB71CCB0D74AB347366D7426CFE90FA93F0 40D39FF86D8316C77EFF6703B4D5F41F277DFAA2C71DA78ECC85A83D29955A10 719C8E7D0E5F9FF17B42571F1FB1A2369C626161829C346B437671F728FDB37A 01DCCF4397976DB18BFC2EF86CBDF07DE0549ED81E6A584803F9915EDE1F0031 6096A2FFB7417C54A2AF1FBACE79BA8C8EDE238630B6C5329A951CE2E979BAEF F372811A3D2AB56FC901285163C273E591BE38B3561B4AD4E9EBF3347D393C90 2BEF7F00389A1638E7ADD892668631813A55721B5C0BDAE7BCCB419F4A95C3F6 A87B1AEDB6708F4AFA8EB404D1EB1921822ADA6E2273D0E26E5298D0B83CEFCC CBD2D6B2B88894282A402B29447AA36BD448B9DC0DCE19785524B75513FF4C85 7D95F119EE860B9C3654BEE15381CDB3A9934F9F72C47A6AC7D22E97691754D9 32E9C6ACE3F96DB8134B05926F1283B67215C27ABC6D51F7CFDE9EE62A4737DD 8AD735AC1BE436D59896BC1DEAF919EB95FB1B9ECD4039BF32C12FA6CCF8DFD8 C1A9F585533F61472BAA47095411CB8B828D5197CC690D42486393A794EADEE9 242E6DE85F087CA75ABE9C8712B90E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 48 /zero put dup 49 /one put dup 50 /two put dup 67 /C put dup 70 /F put dup 82 /R put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 28 /tau put dup 78 /N put dup 117 /u put dup 118 /v put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8 42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8 40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837 B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53 9560176671A33FC55340CD04C2DE46FD812FC2711A85E2568A8DD2C5EBA154B6 630CEAAB56BEAD0D795596E34DAEB37993EBF354A3C9D89403E2EB950F27526E EB8F2375FDD05F69F1B0F622FF73B3A4062FB6ADA5A6257CFC50BBAAB94FE87C 7737D8D06D2EFB88F21EBAF957A1D60EF6B0F333906EE630039B3F87AB6D8D16 3D446E16384AB20208CE5A4413346BEE8187F3BD3D400990CD5D491C65D10FA1 88C6B0277619D3B4DFDF2A32F6E3204B96183A6AF819CFB34AC1F3B7F90F5072 DA3F163345C1BB72815E07237B25C63FA5E95B70679FE2AA9E1BCF4E22770E20 EEB29F7893B47296C6743BED8DC9BF852A40D94543EB08710BA1F979624CB163 64F0B61F2D582A53626ADB90F7740EAD3AF6876D35CAE3F4BBBDBD27519FE398 D1F753FCCDCE7921692A545C9BAC0D07E300597C1D9A6E13D66EDBEFB99C6D60 92CB072B49C0544A6E4ACD85705B4602BDF9267CACA1D6E9002FD1812C29A32D AFE2FF112922283F007464B83CF974FE1B7D096F1AAD52D234C6C3772A965263 34A761467D38148CB4909F48D4127B3CA08E11D4B7195996376D979C9A7A0033 7F485614481C8E0B8E09EC68C22A92DB8F3F15222DA352BD76866B904DB740AE 9C3AF9B1F9A720201D44714451A740CEDC6080993FBEAE994730489512AC1D7B 929A12CD4367EA173AA661A3D9580940513BE0F1ED7CB062D8766EB7897EDB37 0BB4DAD1329B04753C4A8811FDD857FA24E35988380580E11B9D9DA36AEA4D01 2DF421738A9C8EDF7D23A67AA3C59BA85376AF61F6E666F7C25DE7794D46BEB5 A0B0BD7069B4E23EE4273131226EC8A836B1E7B353BFF7A715FD30352B745042 FA854E992AFBB267C3DFAAEE9829E8D7FE177856A08ACF9A221FB0AF4351E679 3A3693E3E453AD5C7443E1F18C80A6948B5E386D8F0F2B023BE72185AD763373 9E69CC1646F94F6A5FA9F977D2D65F0EED5E39F1E11686FD1C68D135CCCF2153 65B7B18F9633B30233B80AEC3D6AA32E9B92EE935F8BA4BE65BAB78DB4FDA43A 0CBD5F68D046DB66706BE3B30AF6BCAD8C141B4BD63183B4444EFDEFE4F0432F 9B0110DC4D483303A42B618475A796C3099CAD79974C4E7C3D5E2ADDB89929F7 732EA82B4E9124AAAC014711151749178FE32945E55470CAC163F3BB4A690A16 D74B96BAC6EED8C743700AD5F148ECEBC9E6573EFB29EB6CAFA17C80FA10A88D 106214E919EC19BE92F42731E8D2D69EB6F73833CFC6285EBB7A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /minus put dup 86 /V put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 43 /plus put dup 48 /zero put dup 49 /one put dup 61 /equal put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 87100FB1FFC4C8AF409B2067537220E605DA0852CA49839E1386AF9D7A1A455F 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Polyline n 7650 3600 m 7650 4050 l gs col0 s gr % Polyline n 7650 3600 m 7653 3602 l 7660 3606 l 7673 3614 l 7691 3625 l 7714 3640 l 7742 3657 l 7772 3676 l 7803 3695 l 7833 3715 l 7862 3734 l 7889 3752 l 7914 3768 l 7936 3784 l 7956 3798 l 7973 3811 l 7988 3823 l 8002 3834 l 8014 3845 l 8025 3855 l 8039 3870 l 8050 3884 l 8060 3899 l 8069 3914 l 8076 3931 l 8082 3949 l 8087 3969 l 8091 3989 l 8095 4009 l 8097 4026 l 8099 4038 l 8100 4046 l 8100 4049 l 8100 4050 l gs col0 s gr % Polyline n 7650 3600 m 7647 3602 l 7640 3606 l 7627 3614 l 7609 3625 l 7586 3640 l 7558 3657 l 7528 3676 l 7497 3695 l 7467 3715 l 7438 3734 l 7411 3752 l 7386 3768 l 7364 3784 l 7344 3798 l 7327 3811 l 7312 3823 l 7298 3834 l 7286 3845 l 7275 3855 l 7261 3870 l 7250 3884 l 7240 3899 l 7231 3914 l 7224 3931 l 7218 3949 l 7213 3969 l 7209 3989 l 7205 4009 l 7203 4026 l 7201 4038 l 7200 4046 l 7200 4049 l 7200 4050 l gs col0 s gr % Polyline n 6750 1800 m 6752 1800 l 6756 1802 l 6763 1804 l 6775 1807 l 6791 1812 l 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4070 2988 l 4064 3021 l 4060 3053 l 4056 3083 l 4054 3107 l 4052 3126 l 4051 3139 l 4050 3147 l 4050 3150 l gs col0 s gr % Polyline n 6750 1800 m 6749 1798 l 6748 1794 l 6745 1788 l 6741 1777 l 6735 1764 l 6728 1747 l 6719 1728 l 6709 1707 l 6697 1685 l 6683 1661 l 6668 1637 l 6650 1613 l 6630 1589 l 6608 1564 l 6582 1539 l 6552 1514 l 6517 1488 l 6478 1464 l 6435 1440 l 6397 1422 l 6358 1407 l 6320 1393 l 6283 1382 l 6247 1373 l 6212 1366 l 6178 1360 l 6145 1356 l 6113 1352 l 6082 1350 l 6051 1348 l 6022 1347 l 5993 1347 l 5966 1347 l 5941 1347 l 5919 1347 l 5899 1348 l 5883 1348 l 5870 1349 l 5861 1349 l 5855 1350 l 5852 1350 l 5850 1350 l gs col0 s gr /Times-Roman-iso ff 180.00 scf sf 6885 1710 m gs 1 -1 sc (vn) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 7740 3510 m gs 1 -1 sc (v1) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9540 4095 m gs 1 -1 sc (VD) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9540 4545 m gs 1 -1 sc (VC) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3605 4477 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3960 4477 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3891 3044 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3420 4214 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3420 3952 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4230 3260 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3778 3260 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3944 4763 m gs 1 -1 sc (u12) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3906 3437 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9540 2520 m gs 1 -1 sc (VI) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 6030 900 m gs 1 -1 sc (vr) col0 sh gr % here ends figure; $F2psEnd rs showpage %%EndDocument @endspecial 275 1916 a /End PSfrag 275 1916 a 275 431 a /Hide PSfrag 275 431 a -410 487 a Fg(PSfrag)20 b(replacements)p -410 518 685 4 v 275 521 a /Unhide PSfrag 275 521 a 243 593 a { 243 593 a 211 614 a Ff(+)243 593 y } 0/Place PSfrag 243 593 a 243 693 a { 243 693 a 211 714 a Fe(\000)243 693 y } 1/Place PSfrag 243 693 a 77 794 a { 77 794 a 26 x Fd(\034)32 b Ff(=)23 b(0)77 794 y } 2/Place PSfrag 77 794 a 77 893 a { 77 893 a 27 x Fd(\034)32 b Ff(=)23 b(1)77 893 y } 3/Place PSfrag 77 893 a 42 991 a { 42 991 a 29 x Fd(\034)33 b Ff(=)23 b Fd(N)42 991 y } 4/Place PSfrag 42 991 a -101 1087 a { -101 1087 a 25 x Fd(\034)33 b Ff(=)23 b Fd(N)k Ff(+)18 b(1)-101 1087 y } 5/Place PSfrag -101 1087 a 182 1195 a { 182 1195 a 12 x Fd(v)222 1219 y Fc(R)182 1195 y } 6/Place PSfrag 182 1195 a 171 1294 a { 171 1294 a 12 x Fd(v)211 1318 y Fb(N)171 1294 y } 7/Place PSfrag 171 1294 a 197 1394 a { 197 1394 a 12 x Fd(v)237 1418 y Fc(1)197 1394 y } 8/Place PSfrag 197 1394 a 171 1483 a { 171 1483 a 22 x Fe(V)222 1517 y Fc(R)171 1483 y } 9/Place PSfrag 171 1483 a 177 1583 a { 177 1583 a 22 x Fe(V)228 1617 y Fc(F)177 1583 y } 10/Place PSfrag 177 1583 a 186 1682 a { 186 1682 a 22 x Fe(V)237 1716 y Fc(0)186 1682 y } 11/Place PSfrag 186 1682 a 172 1782 a { 172 1782 a 22 x Fe(V)223 1816 y Fc(C)172 1782 y } 12/Place PSfrag 172 1782 a 69 1884 a { 69 1884 a 3 x Fd(u)117 1902 y Fa(f)p Fc(1)p Fb(;)p Fc(2)p Fa(g)69 1884 y } 13/Place PSfrag 69 1884 a eop end %%Trailer userdict /end-hook known{end-hook}if %%Trailer cleartomark countdictstack exch sub { end } repeat restore %%EOF ---------------0901211031755 Content-Type: application/postscript; name="Graph2.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Graph2.eps" %!PS-Adobe-2.0 EPSF-2.0 %%BoundingBox: 107 540 406 734 %%HiResBoundingBox: 108.000000 541.000000 404.500000 732.500000 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: dummy.dvi %%CreationDate: Mon Dec 1 10:30:17 2008 %%PageOrder: Ascend %%DocumentFonts: Times-Roman CMMI10 CMR7 %%DocumentPaperSizes: a4 %%EndComments % EPSF created by ps2eps 1.64 %%BeginProlog save countdictstack mark newpath /showpage {} def /setpagedevice {pop} def 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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 %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% 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 %%EndProcSet %%BeginProcSet: 8r.enc 0 0 % File 8r.enc TeX Base 1 Encoding Revision 2.0 2002-10-30 % % @@psencodingfile@{ % author = "S. 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All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /minus put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 61 /equal put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 48 /prime put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 107 /k put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 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All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /minus put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 61 /equal put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 48 /prime put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 107 /k put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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6436 l 4860 6435 l gs col0 s gr % Polyline n 4860 6435 m 4862 6436 l 4868 6437 l 4878 6440 l 4894 6444 l 4915 6450 l 4942 6458 l 4974 6467 l 5011 6478 l 5052 6490 l 5095 6503 l 5139 6517 l 5184 6531 l 5227 6545 l 5269 6559 l 5309 6573 l 5347 6587 l 5382 6601 l 5415 6614 l 5445 6627 l 5473 6640 l 5498 6653 l 5521 6665 l 5542 6678 l 5562 6692 l 5580 6705 l 5596 6719 l 5611 6734 l 5628 6751 l 5643 6770 l 5657 6789 l 5669 6810 l 5680 6833 l 5690 6857 l 5700 6884 l 5708 6913 l 5716 6945 l 5723 6979 l 5729 7015 l 5735 7053 l 5740 7093 l 5745 7133 l 5748 7172 l 5752 7209 l 5754 7243 l 5756 7273 l 5758 7296 l 5759 7314 l 5760 7326 l 5760 7332 l 5760 7335 l gs col0 s gr % Polyline [60] 0 sd n 5760 7335 m 5760 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4860 7335 m 4860 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4410 7335 m 4410 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3960 7335 m 3960 7785 l gs col0 s gr [] 0 sd % Polyline n 4860 6885 m 4863 6886 l 4870 6889 l 4883 6894 l 4901 6902 l 4924 6912 l 4952 6924 l 4982 6937 l 5013 6950 l 5043 6963 l 5072 6977 l 5099 6989 l 5124 7001 l 5146 7011 l 5166 7021 l 5183 7031 l 5198 7040 l 5212 7048 l 5224 7057 l 5235 7065 l 5251 7079 l 5264 7093 l 5275 7109 l 5284 7126 l 5291 7145 l 5297 7167 l 5302 7189 l 5306 7210 l 5308 7228 l 5309 7239 l 5310 7244 l 5310 7245 l gs col0 s gr % Ellipse n 7920 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6570 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7920 6885 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7020 6435 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7020 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7470 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7245 7920 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7245 8100 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj n 7470 7785 m 7245 7920 l 7020 7785 l gs col0 s gr % Polyline n 7920 7785 m 7245 8100 l 6570 7785 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 8640 6435 m 6300 6435 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 6885 m 6300 6885 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 7335 m 6300 7335 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 7785 m 6300 7785 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 7920 6885 m 7920 7335 l gs col0 s gr % Polyline n 7020 6435 m 7020 6255 l gs col0 s gr % Polyline n 7110 6255 m 6930 6255 l gs col0 s gr % Polyline n 7020 6435 m 7020 7335 l gs col0 s gr % Polyline n 8460 7245 m 8280 7245 l gs col0 s gr % Polyline 2 slj n 7020 6435 m 7017 6437 l 7010 6441 l 6997 6448 l 6979 6458 l 6955 6472 l 6928 6488 l 6898 6506 l 6867 6525 l 6836 6545 l 6807 6565 l 6780 6584 l 6755 6603 l 6733 6622 l 6713 6640 l 6696 6658 l 6680 6676 l 6667 6694 l 6655 6714 l 6644 6734 l 6635 6751 l 6628 6770 l 6621 6789 l 6615 6810 l 6609 6833 l 6604 6857 l 6600 6884 l 6595 6913 l 6592 6945 l 6588 6979 l 6585 7015 l 6582 7053 l 6580 7093 l 6577 7133 l 6576 7172 l 6574 7209 l 6573 7243 l 6572 7273 l 6571 7296 l 6570 7314 l 6570 7326 l 6570 7332 l 6570 7335 l gs col0 s gr % Polyline [60] 0 sd n 6570 7335 m 6570 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 7920 7335 m 7920 7785 l gs col0 s gr [] 0 sd % Polyline n 7924 6881 m 7924 6879 l 7923 6874 l 7921 6866 l 7918 6854 l 7915 6838 l 7910 6821 l 7904 6803 l 7897 6785 l 7888 6767 l 7879 6750 l 7869 6734 l 7857 6720 l 7844 6707 l 7829 6694 l 7812 6682 l 7792 6671 l 7771 6659 l 7754 6651 l 7736 6644 l 7717 6635 l 7695 6627 l 7672 6619 l 7646 6610 l 7617 6601 l 7586 6591 l 7552 6580 l 7515 6569 l 7475 6558 l 7432 6546 l 7386 6533 l 7340 6520 l 7292 6507 l 7245 6494 l 7199 6482 l 7157 6471 l 7119 6461 l 7087 6452 l 7061 6446 l 7042 6441 l 7030 6438 l 7023 6436 l 7020 6435 l gs col0 s gr % Polyline [60] 0 sd n 7020 7335 m 7020 7785 l gs col0 s gr [] 0 sd % Polyline n 7920 6885 m 7923 6886 l 7930 6889 l 7943 6894 l 7961 6902 l 7984 6912 l 8012 6924 l 8042 6937 l 8073 6950 l 8103 6963 l 8132 6977 l 8159 6989 l 8184 7001 l 8206 7011 l 8226 7021 l 8243 7031 l 8258 7040 l 8272 7048 l 8284 7057 l 8295 7065 l 8311 7079 l 8324 7093 l 8335 7109 l 8344 7126 l 8351 7145 l 8357 7167 l 8362 7189 l 8366 7210 l 8368 7228 l 8369 7239 l 8370 7244 l 8370 7245 l gs col0 s gr % Polyline n 7466 7331 m 7466 7330 l 7466 7327 l 7467 7318 l 7468 7302 l 7470 7281 l 7473 7256 l 7477 7228 l 7481 7201 l 7486 7174 l 7491 7150 l 7498 7129 l 7505 7109 l 7513 7092 l 7523 7077 l 7534 7063 l 7546 7050 l 7558 7040 l 7570 7030 l 7584 7021 l 7600 7012 l 7618 7002 l 7638 6993 l 7661 6983 l 7686 6972 l 7714 6962 l 7744 6951 l 7775 6940 l 7807 6929 l 7838 6918 l 7866 6909 l 7890 6901 l 7909 6895 l 7921 6891 l 7929 6889 l 7932 6888 l gs col0 s gr % Polyline [60] 0 sd n 7470 7335 m 7470 7785 l gs col0 s gr [] 0 sd % Ellipse n 6570 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7920 9090 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7020 8640 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7020 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7470 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7245 10125 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7245 10305 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 8370 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj n 7470 9990 m 7245 10125 l 7020 9990 l gs col0 s gr % Polyline n 8370 9990 m 7245 10305 l 6570 9990 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 8640 8640 m 6300 8640 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 9090 m 6300 9090 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 9540 m 6300 9540 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 8640 9990 m 6300 9990 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 7920 9090 m 7920 9450 l gs col0 s gr % Polyline n 7020 8640 m 7020 8460 l gs col0 s gr % Polyline n 7110 8460 m 6930 8460 l gs col0 s gr % Polyline n 7020 8640 m 7020 9540 l gs col0 s gr % Polyline n 8010 9450 m 7830 9450 l gs col0 s gr % Polyline 2 slj n 7020 8640 m 7017 8642 l 7010 8646 l 6997 8653 l 6979 8663 l 6955 8677 l 6928 8693 l 6898 8711 l 6867 8730 l 6836 8750 l 6807 8770 l 6780 8789 l 6755 8808 l 6733 8827 l 6713 8845 l 6696 8863 l 6680 8881 l 6667 8899 l 6655 8919 l 6644 8939 l 6635 8956 l 6628 8975 l 6621 8994 l 6615 9015 l 6609 9038 l 6604 9062 l 6600 9089 l 6595 9118 l 6592 9150 l 6588 9184 l 6585 9220 l 6582 9258 l 6580 9298 l 6577 9338 l 6576 9377 l 6574 9414 l 6573 9448 l 6572 9478 l 6571 9501 l 6570 9519 l 6570 9531 l 6570 9537 l 6570 9540 l gs col0 s gr % Polyline [60] 0 sd n 6570 9540 m 6570 9990 l gs col0 s gr [] 0 sd % Polyline n 7924 9086 m 7924 9084 l 7923 9079 l 7921 9071 l 7918 9059 l 7915 9043 l 7910 9026 l 7904 9008 l 7897 8990 l 7888 8972 l 7879 8955 l 7869 8939 l 7857 8925 l 7844 8912 l 7829 8899 l 7812 8887 l 7792 8876 l 7771 8864 l 7754 8856 l 7736 8849 l 7717 8840 l 7695 8832 l 7672 8824 l 7646 8815 l 7617 8806 l 7586 8796 l 7552 8785 l 7515 8774 l 7475 8763 l 7432 8751 l 7386 8738 l 7340 8725 l 7292 8712 l 7245 8699 l 7199 8687 l 7157 8676 l 7119 8666 l 7087 8657 l 7061 8651 l 7042 8646 l 7030 8643 l 7023 8641 l 7020 8640 l gs col0 s gr % Polyline [60] 0 sd n 7020 9540 m 7020 9990 l gs col0 s gr [] 0 sd % Polyline n 7920 9090 m 7923 9091 l 7930 9094 l 7943 9099 l 7961 9107 l 7984 9117 l 8012 9129 l 8042 9142 l 8073 9155 l 8103 9168 l 8132 9182 l 8159 9194 l 8184 9206 l 8206 9216 l 8226 9226 l 8243 9236 l 8258 9245 l 8272 9253 l 8284 9262 l 8295 9270 l 8311 9284 l 8324 9298 l 8335 9314 l 8344 9331 l 8351 9350 l 8357 9372 l 8362 9394 l 8366 9415 l 8368 9433 l 8369 9444 l 8370 9449 l 8370 9450 l gs col0 s gr % Polyline n 7466 9536 m 7466 9535 l 7466 9532 l 7467 9523 l 7468 9507 l 7470 9486 l 7473 9461 l 7477 9433 l 7481 9406 l 7486 9379 l 7491 9355 l 7498 9334 l 7505 9314 l 7513 9297 l 7523 9282 l 7534 9268 l 7546 9255 l 7558 9245 l 7570 9235 l 7584 9226 l 7600 9217 l 7618 9207 l 7638 9198 l 7661 9188 l 7686 9177 l 7714 9167 l 7744 9156 l 7775 9145 l 7807 9134 l 7838 9123 l 7866 9114 l 7890 9106 l 7909 9100 l 7921 9096 l 7929 9094 l 7932 9093 l gs col0 s gr % Polyline [60] 0 sd n 7470 9540 m 7470 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 8370 9540 m 8370 9990 l gs col0 s gr [] 0 sd % Ellipse n 5310 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5760 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5760 2430 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6660 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7110 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7560 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7110 1980 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6210 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6435 3420 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6030 3645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6840 3645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj n 5310 3330 m 6030 3645 l 7110 3330 l gs col0 s gr % Polyline n 5760 3330 m 6840 3645 l 7560 3330 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 5040 1980 m 7830 1980 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 2430 m 7830 2430 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 2880 m 7830 2880 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 3330 m 7830 3330 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 5760 2430 m 5760 2880 l gs col0 s gr % Polyline n 5760 1800 m 5760 2430 l gs col0 s gr % Polyline n 5670 1800 m 5850 1800 l gs col0 s gr % Polyline n 7020 1800 m 7200 1800 l gs col0 s gr % Polyline n 6210 3330 m 6435 3420 l 6660 3330 l gs col0 s gr % Polyline 2 slj [60] 0 sd n 7110 2880 m 7110 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 6210 2880 m 6210 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5760 2880 m 5760 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5310 2880 m 5310 3330 l gs col0 s gr [] 0 sd % Polyline n 5760 2430 m 5763 2431 l 5770 2434 l 5783 2440 l 5801 2447 l 5824 2458 l 5852 2470 l 5882 2483 l 5913 2497 l 5943 2511 l 5972 2525 l 5999 2538 l 6024 2551 l 6046 2562 l 6066 2574 l 6083 2584 l 6098 2595 l 6112 2605 l 6124 2615 l 6135 2625 l 6147 2638 l 6158 2652 l 6167 2666 l 6175 2682 l 6181 2700 l 6187 2720 l 6193 2741 l 6197 2765 l 6201 2789 l 6204 2814 l 6206 2836 l 6208 2855 l 6209 2868 l 6210 2876 l 6210 2879 l 6210 2880 l gs col0 s gr % Polyline n 5310 2880 m 5310 2879 l 5310 2876 l 5311 2868 l 5312 2855 l 5314 2836 l 5316 2814 l 5319 2790 l 5323 2765 l 5328 2742 l 5333 2720 l 5339 2700 l 5346 2683 l 5354 2667 l 5364 2652 l 5375 2639 l 5387 2626 l 5398 2615 l 5411 2605 l 5424 2595 l 5440 2585 l 5458 2574 l 5478 2563 l 5501 2552 l 5526 2539 l 5554 2526 l 5584 2513 l 5615 2499 l 5647 2485 l 5678 2472 l 5706 2460 l 5730 2450 l 5749 2443 l 5761 2437 l 5769 2434 l 5772 2433 l gs col0 s gr % Polyline n 6660 2880 m 6660 2877 l 6660 2870 l 6661 2858 l 6661 2839 l 6662 2814 l 6663 2783 l 6664 2747 l 6666 2707 l 6668 2666 l 6670 2623 l 6673 2582 l 6676 2541 l 6679 2503 l 6683 2468 l 6686 2435 l 6691 2404 l 6695 2377 l 6701 2351 l 6706 2328 l 6712 2306 l 6719 2286 l 6727 2268 l 6735 2250 l 6746 2230 l 6758 2211 l 6772 2193 l 6787 2176 l 6804 2159 l 6824 2142 l 6846 2125 l 6871 2108 l 6898 2091 l 6927 2074 l 6957 2057 l 6988 2040 l 7018 2025 l 7046 2011 l 7069 1999 l 7087 1991 l 7100 1985 l 7107 1981 l 7110 1980 l gs col0 s gr % Polyline n 7110 2880 m 7110 1800 l gs col0 s gr % Polyline n 7560 2880 m 7560 2877 l 7560 2870 l 7559 2858 l 7559 2839 l 7558 2814 l 7557 2783 l 7556 2747 l 7554 2707 l 7552 2666 l 7550 2623 l 7547 2582 l 7544 2541 l 7541 2503 l 7537 2468 l 7534 2435 l 7529 2404 l 7525 2377 l 7519 2351 l 7514 2328 l 7508 2306 l 7501 2286 l 7493 2268 l 7485 2250 l 7474 2230 l 7462 2211 l 7448 2193 l 7433 2176 l 7416 2159 l 7396 2142 l 7374 2125 l 7349 2108 l 7322 2091 l 7293 2074 l 7263 2057 l 7232 2040 l 7202 2025 l 7174 2011 l 7151 1999 l 7133 1991 l 7120 1985 l 7113 1981 l 7110 1980 l gs col0 s gr % Polyline [60] 0 sd n 6660 2880 m 6660 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 7560 2880 m 7560 3330 l gs col0 s gr [] 0 sd % Ellipse n 5310 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5760 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5760 4590 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6660 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7110 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7560 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7110 4140 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6210 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6435 5580 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6435 5760 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6435 5940 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj 7.500 slw [15 68] 68 sd n 5040 4140 m 7830 4140 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 4590 m 7830 4590 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 5040 m 7830 5040 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 5040 5490 m 7830 5490 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 5760 4590 m 5760 5040 l gs col0 s gr % Polyline n 5760 3960 m 5760 4590 l gs col0 s gr % Polyline n 5670 3960 m 5850 3960 l gs col0 s gr % Polyline n 7020 3960 m 7200 3960 l gs col0 s gr % Polyline n 6210 5490 m 6435 5580 l 6660 5490 l gs col0 s gr % Polyline n 5760 5490 m 6435 5760 l 7110 5490 l gs col0 s gr % Polyline n 5310 5490 m 6435 5940 l 7560 5490 l gs col0 s gr % Polyline 2 slj [60] 0 sd n 7110 5040 m 7110 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 6210 5040 m 6210 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5760 5040 m 5760 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5310 5040 m 5310 5490 l gs col0 s gr [] 0 sd % Polyline n 5760 4590 m 5763 4591 l 5770 4594 l 5783 4600 l 5801 4607 l 5824 4618 l 5852 4630 l 5882 4643 l 5913 4657 l 5943 4671 l 5972 4685 l 5999 4698 l 6024 4711 l 6046 4722 l 6066 4734 l 6083 4744 l 6098 4755 l 6112 4765 l 6124 4775 l 6135 4785 l 6147 4798 l 6158 4812 l 6167 4826 l 6175 4842 l 6181 4860 l 6187 4880 l 6193 4901 l 6197 4925 l 6201 4949 l 6204 4974 l 6206 4996 l 6208 5015 l 6209 5028 l 6210 5036 l 6210 5039 l 6210 5040 l gs col0 s gr % Polyline n 5310 5040 m 5310 5039 l 5310 5036 l 5311 5028 l 5312 5015 l 5314 4996 l 5316 4974 l 5319 4950 l 5323 4925 l 5328 4902 l 5333 4880 l 5339 4860 l 5346 4843 l 5354 4827 l 5364 4812 l 5375 4799 l 5387 4786 l 5398 4775 l 5411 4765 l 5424 4755 l 5440 4745 l 5458 4734 l 5478 4723 l 5501 4712 l 5526 4699 l 5554 4686 l 5584 4673 l 5615 4659 l 5647 4645 l 5678 4632 l 5706 4620 l 5730 4610 l 5749 4603 l 5761 4597 l 5769 4594 l 5772 4593 l gs col0 s gr % Polyline n 6660 5040 m 6660 5037 l 6660 5030 l 6661 5018 l 6661 4999 l 6662 4974 l 6663 4943 l 6664 4907 l 6666 4867 l 6668 4826 l 6670 4783 l 6673 4742 l 6676 4701 l 6679 4663 l 6683 4628 l 6686 4595 l 6691 4564 l 6695 4537 l 6701 4511 l 6706 4488 l 6712 4466 l 6719 4446 l 6727 4428 l 6735 4410 l 6746 4390 l 6758 4371 l 6772 4353 l 6787 4336 l 6804 4319 l 6824 4302 l 6846 4285 l 6871 4268 l 6898 4251 l 6927 4234 l 6957 4217 l 6988 4200 l 7018 4185 l 7046 4171 l 7069 4159 l 7087 4151 l 7100 4145 l 7107 4141 l 7110 4140 l gs col0 s gr % Polyline n 7110 5040 m 7110 3960 l gs col0 s gr % Polyline n 7560 5040 m 7560 5037 l 7560 5030 l 7559 5018 l 7559 4999 l 7558 4974 l 7557 4943 l 7556 4907 l 7554 4867 l 7552 4826 l 7550 4783 l 7547 4742 l 7544 4701 l 7541 4663 l 7537 4628 l 7534 4595 l 7529 4564 l 7525 4537 l 7519 4511 l 7514 4488 l 7508 4466 l 7501 4446 l 7493 4428 l 7485 4410 l 7474 4390 l 7462 4371 l 7448 4353 l 7433 4336 l 7416 4319 l 7396 4302 l 7374 4285 l 7349 4268 l 7322 4251 l 7293 4234 l 7263 4217 l 7232 4200 l 7202 4185 l 7174 4171 l 7151 4159 l 7133 4151 l 7120 4145 l 7113 4141 l 7110 4140 l gs col0 s gr % Polyline [60] 0 sd n 6660 5040 m 6660 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 7560 5040 m 7560 5490 l gs col0 s gr [] 0 sd % Ellipse n 2160 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2610 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3510 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3960 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4410 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3060 3330 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3285 3420 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2880 3645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3690 3645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2160 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2610 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3510 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3960 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4410 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3060 5490 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3285 5580 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3285 5760 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3285 5940 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2610 1980 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3960 2430 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2610 4140 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3960 4590 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj n 2160 3330 m 2880 3645 l 3960 3330 l gs col0 s gr % Polyline n 2610 3330 m 3690 3645 l 4410 3330 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 1890 1980 m 4680 1980 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 2430 m 4680 2430 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 2880 m 4680 2880 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 3330 m 4680 3330 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 2610 2430 m 2610 2880 l gs col0 s gr % Polyline n 2610 1800 m 2610 2430 l gs col0 s gr % Polyline n 2520 1800 m 2700 1800 l gs col0 s gr % Polyline n 3870 1800 m 4050 1800 l gs col0 s gr % Polyline n 3060 3330 m 3285 3420 l 3510 3330 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 1890 4140 m 4680 4140 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 4590 m 4680 4590 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 5040 m 4680 5040 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1890 5490 m 4680 5490 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 2610 4590 m 2610 5040 l gs col0 s gr % Polyline n 2610 3960 m 2610 4590 l gs col0 s gr % Polyline n 2520 3960 m 2700 3960 l gs col0 s gr % Polyline n 3870 3960 m 4050 3960 l gs col0 s gr % Polyline n 3060 5490 m 3285 5580 l 3510 5490 l gs col0 s gr % Polyline n 2610 5490 m 3285 5760 l 3960 5490 l gs col0 s gr % Polyline n 2160 5490 m 3285 5940 l 4410 5490 l gs col0 s gr % Polyline 2 slj [60] 0 sd n 3960 2880 m 3960 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3060 2880 m 3060 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 2610 2880 m 2610 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 2160 2880 m 2160 3330 l gs col0 s gr [] 0 sd % Polyline n 3960 2880 m 3960 1800 l gs col0 s gr % Polyline [60] 0 sd n 3510 2880 m 3510 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4410 2880 m 4410 3330 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3960 5040 m 3960 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3060 5040 m 3060 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 2610 5040 m 2610 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 2160 5040 m 2160 5490 l gs col0 s gr [] 0 sd % Polyline n 3960 5040 m 3960 3960 l gs col0 s gr % Polyline [60] 0 sd n 3510 5040 m 3510 5490 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4410 5040 m 4410 5490 l gs col0 s gr [] 0 sd % Polyline n 3960 2430 m 3963 2431 l 3970 2434 l 3983 2440 l 4001 2447 l 4024 2458 l 4052 2470 l 4082 2483 l 4113 2497 l 4143 2511 l 4172 2525 l 4199 2538 l 4224 2551 l 4246 2562 l 4266 2574 l 4283 2584 l 4298 2595 l 4312 2605 l 4324 2615 l 4335 2625 l 4347 2638 l 4358 2652 l 4367 2666 l 4375 2682 l 4381 2700 l 4387 2720 l 4393 2741 l 4397 2765 l 4401 2789 l 4404 2814 l 4406 2836 l 4408 2855 l 4409 2868 l 4410 2876 l 4410 2879 l 4410 2880 l gs col0 s gr % Polyline n 3060 2880 m 3060 2877 l 3060 2870 l 3059 2858 l 3059 2839 l 3058 2814 l 3057 2783 l 3056 2747 l 3054 2707 l 3052 2666 l 3050 2623 l 3047 2582 l 3044 2541 l 3041 2503 l 3037 2468 l 3034 2435 l 3029 2404 l 3025 2377 l 3019 2351 l 3014 2328 l 3008 2306 l 3001 2286 l 2993 2268 l 2985 2250 l 2974 2230 l 2962 2211 l 2948 2193 l 2933 2176 l 2916 2159 l 2896 2142 l 2874 2125 l 2849 2108 l 2822 2091 l 2793 2074 l 2763 2057 l 2732 2040 l 2702 2025 l 2674 2011 l 2651 1999 l 2633 1991 l 2620 1985 l 2613 1981 l 2610 1980 l gs col0 s gr % Polyline n 2160 2880 m 2160 2877 l 2160 2870 l 2161 2858 l 2161 2839 l 2162 2814 l 2163 2783 l 2164 2747 l 2166 2707 l 2168 2666 l 2170 2623 l 2173 2582 l 2176 2541 l 2179 2503 l 2183 2468 l 2186 2435 l 2191 2404 l 2195 2377 l 2201 2351 l 2206 2328 l 2212 2306 l 2219 2286 l 2227 2268 l 2235 2250 l 2246 2230 l 2258 2211 l 2272 2193 l 2287 2176 l 2304 2159 l 2324 2142 l 2346 2125 l 2371 2108 l 2398 2091 l 2427 2074 l 2457 2057 l 2488 2040 l 2518 2025 l 2546 2011 l 2569 1999 l 2587 1991 l 2600 1985 l 2607 1981 l 2610 1980 l gs col0 s gr % Polyline n 3510 2880 m 3510 2879 l 3510 2876 l 3511 2868 l 3512 2855 l 3514 2836 l 3516 2814 l 3519 2790 l 3523 2765 l 3528 2742 l 3533 2720 l 3539 2700 l 3546 2683 l 3554 2667 l 3564 2652 l 3575 2639 l 3587 2626 l 3598 2615 l 3611 2605 l 3624 2595 l 3640 2585 l 3658 2574 l 3678 2563 l 3701 2552 l 3726 2539 l 3754 2526 l 3784 2513 l 3815 2499 l 3847 2485 l 3878 2472 l 3906 2460 l 3930 2450 l 3949 2443 l 3961 2437 l 3969 2434 l 3972 2433 l gs col0 s gr % Polyline n 2160 5040 m 2160 5037 l 2160 5030 l 2161 5018 l 2161 4999 l 2162 4974 l 2163 4943 l 2164 4907 l 2166 4867 l 2168 4826 l 2170 4783 l 2173 4742 l 2176 4701 l 2179 4663 l 2183 4628 l 2186 4595 l 2191 4564 l 2195 4537 l 2201 4511 l 2206 4488 l 2212 4466 l 2219 4446 l 2227 4428 l 2235 4410 l 2246 4390 l 2258 4371 l 2272 4353 l 2287 4336 l 2304 4319 l 2324 4302 l 2346 4285 l 2371 4268 l 2398 4251 l 2427 4234 l 2457 4217 l 2488 4200 l 2518 4185 l 2546 4171 l 2569 4159 l 2587 4151 l 2600 4145 l 2607 4141 l 2610 4140 l gs col0 s gr % Polyline n 3510 5040 m 3510 5039 l 3510 5036 l 3511 5028 l 3512 5015 l 3514 4996 l 3516 4974 l 3519 4950 l 3523 4925 l 3528 4902 l 3533 4880 l 3539 4860 l 3546 4843 l 3554 4827 l 3564 4812 l 3575 4799 l 3587 4786 l 3598 4775 l 3611 4765 l 3624 4755 l 3640 4745 l 3658 4734 l 3678 4723 l 3701 4712 l 3726 4699 l 3754 4686 l 3784 4673 l 3815 4659 l 3847 4645 l 3878 4632 l 3906 4620 l 3930 4610 l 3949 4603 l 3961 4597 l 3969 4594 l 3972 4593 l gs col0 s gr % Polyline n 3960 4590 m 3963 4591 l 3970 4594 l 3983 4600 l 4001 4607 l 4024 4618 l 4052 4630 l 4082 4643 l 4113 4657 l 4143 4671 l 4172 4685 l 4199 4698 l 4224 4711 l 4246 4722 l 4266 4734 l 4283 4744 l 4298 4755 l 4312 4765 l 4324 4775 l 4335 4785 l 4347 4798 l 4358 4812 l 4367 4826 l 4375 4842 l 4381 4860 l 4387 4880 l 4393 4901 l 4397 4925 l 4401 4949 l 4404 4974 l 4406 4996 l 4408 5015 l 4409 5028 l 4410 5036 l 4410 5039 l 4410 5040 l gs col0 s gr % Polyline n 3060 5040 m 3060 5037 l 3060 5030 l 3059 5018 l 3059 4999 l 3058 4974 l 3057 4943 l 3056 4907 l 3054 4867 l 3052 4826 l 3050 4783 l 3047 4742 l 3044 4701 l 3041 4663 l 3037 4628 l 3034 4595 l 3029 4564 l 3025 4537 l 3019 4511 l 3014 4488 l 3008 4466 l 3001 4446 l 2993 4428 l 2985 4410 l 2974 4390 l 2962 4371 l 2948 4353 l 2933 4336 l 2916 4319 l 2896 4302 l 2874 4285 l 2849 4268 l 2822 4251 l 2793 4234 l 2763 4217 l 2732 4200 l 2702 4185 l 2674 4171 l 2651 4159 l 2633 4151 l 2620 4145 l 2613 4141 l 2610 4140 l gs col0 s gr % Ellipse n 1350 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1800 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1800 9090 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2700 8640 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2700 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3150 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2250 10125 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2250 10305 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4860 9090 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4860 8645 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4410 10260 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5310 10260 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3960 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 4410 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5760 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 5310 9990 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1800 8055 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2700 8055 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1350 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1800 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 3150 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1800 6885 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2700 6435 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 2700 7785 64 64 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 0 slj n 1350 9990 m 2250 10305 l 3150 9990 l gs col0 s gr % Polyline n 1800 9990 m 2250 10125 l 2700 9990 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 1080 8640 m 3420 8640 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 9090 m 3420 9090 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 9540 m 3420 9540 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 9990 m 3420 9990 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 1800 9090 m 1800 9540 l gs col0 s gr % Polyline n 2160 9450 m 2340 9450 l gs col0 s gr % Polyline n 2700 8640 m 2700 8460 l gs col0 s gr % Polyline n 2610 8460 m 2790 8460 l gs col0 s gr % Polyline n 2700 8640 m 2700 9540 l gs col0 s gr % Polyline n 4860 8640 m 4860 8460 l gs col0 s gr % Polyline n 4860 9090 m 4860 8640 l gs col0 s gr % Polyline n 3960 9990 m 4410 10260 l 5310 9990 l gs col0 s gr % Polyline n 4410 9990 m 5310 10260 l 5760 9990 l gs col0 s gr % Polyline n 4860 9090 m 4860 9450 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 3690 8640 m 6030 8640 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 3690 9090 m 6030 9090 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 3690 9540 m 6030 9540 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 4770 8460 m 4950 8460 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 3690 9990 m 6030 9990 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 4770 9450 m 4950 9450 l gs col0 s gr % Polyline n 1350 7785 m 1800 8055 l 2700 7785 l gs col0 s gr % Polyline n 1800 7785 m 2700 8055 l 3150 7785 l gs col0 s gr % Polyline 7.500 slw [15 68] 68 sd n 1080 6435 m 3420 6435 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 6885 m 3420 6885 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 7335 m 3420 7335 l gs col0 s gr [] 0 sd % Polyline [15 68] 68 sd n 1080 7785 m 3420 7785 l gs col0 s gr [] 0 sd % Polyline 15.000 slw n 1800 6885 m 1800 7335 l gs col0 s gr % Polyline n 2160 7245 m 2340 7245 l gs col0 s gr % Polyline n 2700 6435 m 2700 6255 l gs col0 s gr % Polyline n 2610 6255 m 2790 6255 l gs col0 s gr % Polyline n 2700 6435 m 2700 7335 l gs col0 s gr % Polyline 2 slj n 2700 8640 m 2703 8642 l 2710 8646 l 2723 8653 l 2741 8663 l 2765 8677 l 2792 8693 l 2822 8711 l 2853 8730 l 2884 8750 l 2913 8770 l 2940 8789 l 2965 8808 l 2987 8827 l 3007 8845 l 3024 8863 l 3040 8881 l 3053 8899 l 3065 8919 l 3076 8939 l 3085 8956 l 3092 8975 l 3099 8994 l 3105 9015 l 3111 9038 l 3116 9062 l 3120 9089 l 3125 9118 l 3128 9150 l 3132 9184 l 3135 9220 l 3138 9258 l 3140 9298 l 3143 9338 l 3144 9377 l 3146 9414 l 3147 9448 l 3148 9478 l 3149 9501 l 3150 9519 l 3150 9531 l 3150 9537 l 3150 9540 l gs col0 s gr % Polyline [60] 0 sd n 3150 9540 m 3150 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1800 9540 m 1800 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 9540 m 1350 9990 l gs col0 s gr [] 0 sd % Polyline n 1796 9086 m 1796 9084 l 1797 9079 l 1799 9071 l 1802 9059 l 1805 9043 l 1810 9026 l 1816 9008 l 1823 8990 l 1832 8972 l 1841 8955 l 1851 8939 l 1863 8925 l 1876 8912 l 1891 8899 l 1908 8887 l 1928 8876 l 1949 8864 l 1966 8856 l 1984 8849 l 2003 8840 l 2025 8832 l 2048 8824 l 2074 8815 l 2103 8806 l 2134 8796 l 2168 8785 l 2205 8774 l 2245 8763 l 2288 8751 l 2334 8738 l 2380 8725 l 2428 8712 l 2475 8699 l 2521 8687 l 2563 8676 l 2601 8666 l 2633 8657 l 2659 8651 l 2678 8646 l 2690 8643 l 2697 8641 l 2700 8640 l gs col0 s gr % Polyline n 1800 9090 m 1803 9091 l 1810 9094 l 1823 9099 l 1841 9107 l 1864 9117 l 1892 9129 l 1922 9142 l 1953 9155 l 1983 9168 l 2012 9182 l 2039 9194 l 2064 9206 l 2086 9216 l 2106 9226 l 2123 9236 l 2138 9245 l 2152 9253 l 2164 9262 l 2175 9270 l 2191 9284 l 2204 9298 l 2215 9314 l 2224 9331 l 2231 9350 l 2237 9372 l 2242 9394 l 2246 9415 l 2248 9433 l 2249 9444 l 2250 9449 l 2250 9450 l gs col0 s gr % Polyline [60] 0 sd n 2700 9540 m 2700 9990 l gs col0 s gr [] 0 sd % Polyline n 4410 9540 m 4410 9539 l 4410 9536 l 4411 9527 l 4412 9511 l 4414 9490 l 4417 9465 l 4421 9437 l 4425 9410 l 4430 9383 l 4435 9359 l 4442 9338 l 4449 9318 l 4457 9301 l 4467 9286 l 4478 9272 l 4490 9259 l 4502 9249 l 4514 9239 l 4528 9230 l 4544 9221 l 4562 9211 l 4582 9202 l 4605 9192 l 4630 9181 l 4658 9171 l 4688 9160 l 4719 9149 l 4751 9138 l 4782 9127 l 4810 9118 l 4834 9110 l 4853 9104 l 4865 9100 l 4873 9098 l 4876 9097 l gs col0 s gr % Polyline n 3960 9540 m 3960 9537 l 3960 9531 l 3961 9521 l 3962 9504 l 3964 9482 l 3966 9455 l 3968 9424 l 3972 9389 l 3975 9353 l 3980 9316 l 3985 9279 l 3991 9243 l 3997 9209 l 4004 9178 l 4012 9148 l 4020 9121 l 4030 9096 l 4040 9072 l 4051 9050 l 4063 9030 l 4077 9011 l 4092 8993 l 4109 8975 l 4124 8961 l 4140 8947 l 4158 8933 l 4178 8919 l 4199 8905 l 4222 8892 l 4247 8878 l 4275 8864 l 4305 8849 l 4338 8835 l 4373 8819 l 4411 8804 l 4451 8787 l 4493 8771 l 4536 8754 l 4581 8738 l 4625 8722 l 4668 8706 l 4709 8692 l 4746 8679 l 4778 8668 l 4805 8659 l 4826 8651 l 4842 8646 l 4852 8643 l 4858 8641 l 4860 8640 l gs col0 s gr % Polyline n 4860 8640 m 4862 8641 l 4868 8642 l 4878 8645 l 4894 8649 l 4915 8655 l 4942 8663 l 4974 8672 l 5011 8683 l 5052 8695 l 5095 8708 l 5139 8722 l 5184 8736 l 5227 8750 l 5269 8764 l 5309 8778 l 5347 8792 l 5382 8806 l 5415 8819 l 5445 8832 l 5473 8845 l 5498 8858 l 5521 8870 l 5542 8883 l 5562 8897 l 5580 8910 l 5596 8924 l 5611 8939 l 5628 8956 l 5643 8975 l 5657 8994 l 5669 9015 l 5680 9038 l 5690 9062 l 5700 9089 l 5708 9118 l 5716 9150 l 5723 9184 l 5729 9220 l 5735 9258 l 5740 9298 l 5745 9338 l 5748 9377 l 5752 9414 l 5754 9448 l 5756 9478 l 5758 9501 l 5759 9519 l 5760 9531 l 5760 9537 l 5760 9540 l gs col0 s gr % Polyline [60] 0 sd n 5760 9540 m 5760 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 4410 9540 m 4410 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 3960 9540 m 3960 9990 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5310 9540 m 5310 9990 l gs col0 s gr [] 0 sd % Polyline n 4860 9090 m 4863 9091 l 4870 9094 l 4883 9100 l 4901 9107 l 4924 9118 l 4952 9130 l 4982 9143 l 5013 9157 l 5043 9171 l 5072 9185 l 5099 9198 l 5124 9211 l 5146 9222 l 5166 9234 l 5183 9244 l 5198 9255 l 5212 9265 l 5224 9275 l 5235 9285 l 5247 9298 l 5258 9312 l 5267 9326 l 5275 9342 l 5281 9360 l 5287 9380 l 5293 9401 l 5297 9425 l 5301 9449 l 5304 9474 l 5306 9496 l 5308 9515 l 5309 9528 l 5310 9536 l 5310 9539 l 5310 9540 l gs col0 s gr % Polyline n 2700 6435 m 2703 6437 l 2710 6441 l 2723 6448 l 2741 6458 l 2765 6472 l 2792 6488 l 2822 6506 l 2853 6525 l 2884 6545 l 2913 6565 l 2940 6584 l 2965 6603 l 2987 6622 l 3007 6640 l 3024 6658 l 3040 6676 l 3053 6694 l 3065 6714 l 3076 6734 l 3085 6751 l 3092 6770 l 3099 6789 l 3105 6810 l 3111 6833 l 3116 6857 l 3120 6884 l 3125 6913 l 3128 6945 l 3132 6979 l 3135 7015 l 3138 7053 l 3140 7093 l 3143 7133 l 3144 7172 l 3146 7209 l 3147 7243 l 3148 7273 l 3149 7296 l 3150 7314 l 3150 7326 l 3150 7332 l 3150 7335 l gs col0 s gr % Polyline [60] 0 sd n 3150 7335 m 3150 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1800 7335 m 1800 7785 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 1350 7335 m 1350 7785 l gs col0 s gr [] 0 sd % Polyline n 1796 6881 m 1796 6879 l 1797 6874 l 1799 6866 l 1802 6854 l 1805 6838 l 1810 6821 l 1816 6803 l 1823 6785 l 1832 6767 l 1841 6750 l 1851 6734 l 1863 6720 l 1876 6707 l 1891 6694 l 1908 6682 l 1928 6671 l 1949 6659 l 1966 6651 l 1984 6644 l 2003 6635 l 2025 6627 l 2048 6619 l 2074 6610 l 2103 6601 l 2134 6591 l 2168 6580 l 2205 6569 l 2245 6558 l 2288 6546 l 2334 6533 l 2380 6520 l 2428 6507 l 2475 6494 l 2521 6482 l 2563 6471 l 2601 6461 l 2633 6452 l 2659 6446 l 2678 6441 l 2690 6438 l 2697 6436 l 2700 6435 l gs col0 s gr % Polyline n 1800 6885 m 1803 6886 l 1810 6889 l 1823 6894 l 1841 6902 l 1864 6912 l 1892 6924 l 1922 6937 l 1953 6950 l 1983 6963 l 2012 6977 l 2039 6989 l 2064 7001 l 2086 7011 l 2106 7021 l 2123 7031 l 2138 7040 l 2152 7048 l 2164 7057 l 2175 7065 l 2191 7079 l 2204 7093 l 2215 7109 l 2224 7126 l 2231 7145 l 2237 7167 l 2242 7189 l 2246 7210 l 2248 7228 l 2249 7239 l 2250 7244 l 2250 7245 l gs col0 s gr % Polyline [60] 0 sd n 2700 7335 m 2700 7785 l gs col0 s gr [] 0 sd % Polyline n 1350 7335 m 1350 7334 l 1350 7331 l 1351 7323 l 1352 7310 l 1354 7291 l 1356 7269 l 1359 7244 l 1363 7220 l 1367 7196 l 1373 7175 l 1379 7155 l 1385 7137 l 1393 7121 l 1402 7107 l 1413 7093 l 1425 7080 l 1436 7070 l 1448 7060 l 1462 7050 l 1477 7039 l 1494 7029 l 1514 7017 l 1536 7006 l 1561 6993 l 1588 6980 l 1617 6966 l 1647 6952 l 1678 6938 l 1708 6925 l 1736 6913 l 1759 6902 l 1777 6895 l 1790 6889 l 1797 6886 l 1800 6885 l gs col0 s gr % Polyline n 1350 9540 m 1350 9539 l 1350 9536 l 1351 9528 l 1352 9515 l 1354 9496 l 1356 9474 l 1359 9449 l 1363 9425 l 1367 9401 l 1373 9380 l 1379 9360 l 1385 9342 l 1393 9326 l 1402 9312 l 1413 9298 l 1425 9285 l 1436 9275 l 1448 9265 l 1462 9255 l 1477 9244 l 1494 9234 l 1514 9222 l 1536 9211 l 1561 9198 l 1588 9185 l 1617 9171 l 1647 9157 l 1678 9143 l 1708 9130 l 1736 9118 l 1759 9107 l 1777 9100 l 1790 9094 l 1797 9091 l 1800 9090 l gs col0 s gr /Times-Roman-iso ff 190.50 scf sf 5040 6345 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5445 7245 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7200 6345 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 8460 7200 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 8010 9405 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7155 8550 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5895 1935 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7245 1935 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 5895 4095 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 7245 4095 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2745 1935 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 4095 1935 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2745 4095 m gs 1 -1 sc (-) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 4095 4095 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2835 8550 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2385 9405 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 4950 9405 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 4995 8550 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2835 6345 m gs 1 -1 sc (+) col0 sh gr /Times-Roman-iso ff 190.50 scf sf 2385 7200 m gs 1 -1 sc (+) col0 sh gr % here ends figure; pagefooter showpage %%Trailer %EOF %%EndDocument @endspecial 275 3796 a /End PSfrag 275 3796 a 275 3506 a /Hide PSfrag 275 3506 a -410 3562 a Fc(PSfrag)20 b(replacements)p -410 3593 685 4 v 275 3596 a /Unhide PSfrag 275 3596 a 210 3668 a { 210 3668 a 21 x Fb(\000)210 3668 y } 0/Place PSfrag 210 3668 a 210 3768 a { 210 3768 a 21 x Fa(+)210 3768 y } 1/Place PSfrag 210 3768 a eop end %%Trailer userdict /end-hook known{end-hook}if %%Trailer cleartomark countdictstack exch sub { end } repeat restore %%EOF ---------------0901211031755--