Content-Type: multipart/mixed; boundary="-------------0309081023611" This is a multi-part message in MIME format. ---------------0309081023611 Content-Type: text/plain; name="03-408.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-408.keywords" Lattice Spin Systems, Witten Laplacian, Lattice Greens function. ---------------0309081023611 Content-Type: application/x-tex; name="BM-II-final.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BM-II-final.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[a4paper,12pt,twoside]{article} % %\setlength{\textwidth}{27pc} %JSP %\setlength{\textlength}{43pc} %JSP % \usepackage{amsfonts}% \usepackage{times} \usepackage{amsthm} \usepackage{epsfig} % %\usepackage{theorem} %JSP %\theorembodyfont{\upshape} %JSP % % % %\typeout{TransFig: figure text in LaTeX.} \typeout{TransFig: %figures in PostScript.} % \def\bbbone{{\mathchoice {\rm 1\mskip-4mu 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The section % {\thesection.\arabic{equation}} % number will then be displayed % \newcommand{\secct}[1]{\section{#1}% % as a roman number, e.g., IV. for % \setcounter{equation}{0}} % the fourth section % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{hypothesis}{Hypothesis} \newtheorem{condition}{Condition}[section] \newtheorem{theorem}{Theorem}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{lemma}[theorem]{Lemma} % These Theoremlike environm.% \newtheorem{corollary}[theorem]{Corollary} % are counted according to % \newtheorem{definition}[theorem]{Definition} % the section where they % \newtheorem{remark}[theorem]{Remark} % appear. % \newtheorem{proposition}[theorem]{Proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theoremstyle{plain} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\DATUM}{2003.09.08} % % \pagestyle{myheadings} % Date and Page Headings % \markboth{\centerline{V. Bach and J. S. M{\o}ller}}% {\centerline{Correlation at low temperature: II. Asymptotics}} %\markboth{\centerline{V. Bach and J. S. M{\o}ller}}% % {\centerline{Correlation at low temperature: II.\ Asymptotics}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \bibliographystyle{amsplain} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \setcounter{page}{0} \thispagestyle{empty} \title{Correlation at low temperature:\\ II.~Asymptotics} \author{ Volker Bach and Jacob Schach M{\o}ller\footnote{Supported in parts by Carlsbergfondet and by a Marie Curie Individual Fellowship.} \\ FB Mathematik (17) \\ Johannes Gutenberg Universit{\"a}t\\ 55099 Mainz\\ Germany\\ {\em email:} vbach@mathematik.uni-mainz.de \\ {\em email:} jacob@mathematik.uni-mainz.de } \date{\DATUM} \maketitle \begin{abstract} The present paper is a continuation of \cite{BachMoeller2003a}, where the truncated two-point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature, under a weak coupling assumption. In this paper we compute the asymptotics of the correlation function as the temperature goes to zero. This paper thus extends \cite{BachJeckoSjoestrand2000} in two directions: The Hamiltonian function is allowed to have several local minima other than a unique global minimum, and we do not require translation invariance of the Hamiltonian function. We are in particular able to handle spin systems on a general lattice. \vspace{2mm} \noindent {\bf MSC:} 82B05, 82B20, 81Q20. \vspace{2mm} \noindent {\bf Keywords:} Lattice Spin Systems, Witten Laplacian, Lattice Greens function. \end{abstract} \thispagestyle{empty} \newpage \setcounter{page}{1} \tableofcontents \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \secct{Introduction and Results} \label{PartI} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Introduction} \label{SecI.1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\Lambda$ be a finite set. The reader should think of $\Lambda$ as an element of an infinite family of sets $\Gamma = \{\Lambda\}$ ordered by inclusion. Constants appearing in this paper which neither depend on a particular point $i \in \Lambda$ nor on the choice for $\Lambda \in \Gamma$ are called \emph{universal}. Given a Hamiltonian function $H = H_\Lambda: \RR^\Lambda \to \RR$, we define the associated Gibbs measure at inverse temperature $\beta>0$ by % \begin{equation} \label{Gibbs} d\mu_\Lambda^\beta(x) \ := \ e^{-2\beta H(x)} \, \frac{d^\Lambda x}{\cZ_\beta} \period \end{equation} % Here $x = (x_i)_{i \in \Lambda} \in \RR^\Lambda := \oplus_{i\in\Lambda} \RR = \{x:\Lambda\to\RR\}$, and $d^\Lambda x$ is the Lebesgue measure on $\RR^\Lambda$. The constant $\cZ_\beta = \int_{\RR^\Lambda}e^{-2\beta H(x)}d^\Lambda x$ is a normalization constant chosen so that $d\mu_\Lambda^\beta$ is a probability measure. We assume the Hamiltonian $H$ to be a sum of single-spin potentials $\{f_i\}_{i \in \Lambda}$ and pair-interactions $\{w_{ij}\}_{i,j \in \Lambda}$ of the form % \begin{equation} \label{Gibbs2} H(x) \ = \ \sum_{i \in \Lambda} f_i(x_i) \; + \; \alpha \sum_{i,j \in \Lambda, i \neq j} w_{ij}(x_i,x_j)\period \end{equation} % We stress that the $f_i$'s and $w_{ij}$'s may have an explicit $\Lambda$-dependence. The coupling constant $\alpha$ is assumed to be small and positive. The $f_i$'s should have a unique global minimum at $x_i=0$ and the interaction term should be ferromagnetic at $0$. See Hypotheses~\ref{H-1}, \ref{H-2}, and~\ref{H-3} below for a precise formulation. The object to be studied in this paper is the truncated two-point correlation function given by % \begin{equation} \label{Gibbs3} \EE^T_\beta(x_i\, ;\,x_j) \ := \ \EE_\beta(x_i\,x_j) \; - \; \EE_\beta(x_i)\,\EE_\beta(x_j)\comma \end{equation} % where $\EE_\beta(\cdot)$ denotes expectation value with respect to the Gibbs measure, $\EE_\beta(u) := \int_{\RR^\Lambda} u\,d\mu_\Lambda^\beta$, for a polynomially bounded observable $u:\RR^\Lambda \to \RR$. In \cite{BachMoeller2003a} we assumed the interactions to decay exponentially fast. More precisely, we assumed the existence of a (universal) metric $d$ on $\Lambda$ such that $w_{ij}$ is bounded by $e^{-d(i,j)}$, in a suitable sense. Under these assumptions, it was shown in \cite[Theorem~I.1]{BachMoeller2003a} that, for small $\alpha >0$, large $\beta < \infty$, and any $\eta >0$, the correlations are bounded by % \begin{equation} \label{Gibbs4} \big| \EE^T_\beta(x_i\, ;\,x_j) \big| \ \leq \ \frac{1+ C\eta}{2 \, \beta \, \lmin} \; e^{-(1-\eta)d(i,j)} \comma \end{equation} % where $\lmin = \inf\sigma[H''(0)] > 1/C >0$ is the lowest eigenvalue of the Hessian at $x=0$, and $C < \infty$ is a universal constant. The purpose of the present paper is to sharpen this result, in particular, to give upper \emph{and} lower bounds on the correlations that agree asymptotically in the low temperature limit $\beta \to 0$. To this end, we replace the metric $d$, which in \cite{BachMoeller2003a} was assumed to be given a priori, by the logarithm of the resolvent of the Hessian of $H$ at $x=0$, i.e., $\lmin^{-1} e^{-\rho(i,j)} \to \{ H''(0)^{-1} \}_{ij}$. That is, all decay properties derived in this paper are to be compared to the decay of $H''(0)^{-1}$. Under a finite range assumption and a ferro-magnetic assumption on the interaction, but no assumption of translation invariance or any other geometric structure of the lattice, we improve in the present paper the estimate (\ref{Gibbs4}) to % \begin{equation} \label{Gibbs5} \frac{2 \, [1 - \tau(\beta)]}{\beta} \, \big\{ H''(0)^{-1} \big\}_{ij}^{1+\tau(\beta)} \ \leq \ \EE^T_\beta(x_i\, ;\,x_j) \ \leq \ \frac{2 \, [1 + \tau(\beta)]}{\beta} \, \big\{ H''(0)^{-1} \big\}_{ij}^{1-\tau(\beta)} \comma \end{equation} % where $\tau(\beta) = C \beta^{-1/2}$, and $C < \infty$ is a universal constant. The precise formulation of this main result is given in Theorem~\ref{asymptotics}. The correlation asymptotics have been derived in a form similar to (\ref{Gibbs5}) in \cite{BachJeckoSjoestrand2000}. The assumptions in \cite{BachJeckoSjoestrand2000} were, however, more stringent than those used here (namely, the $f_i$'s were forced to have only one critical point) and ruled out various important natural examples for $H$, like an Ising-ferromagnet in a uniform, non-zero, external magnetic field. In view of deriving the correlation asymptotics, rather than mere exponential bounds of type~(\ref{Gibbs4}), the assumption of translation invariance was crucial in \cite{BachJeckoSjoestrand2000}, while no such requirement is necessary in the present paper. In fact, one of the novelties of our approach is based on the observation, that the assumption of existence of an a-priori metric is obsolete because the Hessian $H''(0)$ of $H$ at $x=0$ \emph{defines} a metric $\rho$ on $\Lambda$ which yields the correlation length, see Theorem~\ref{asymptotics}. We approach the problem via a representation formula, see Theorem~\ref{correlations}, which expresses the truncated two-point correlations functions in terms of matrix elements of the resolvent of a so-called Witten Laplacian (restricted to $1$-forms). See Subsect.~\ref{SecI.2}. For a more thorough discussion of the Witten Laplacian techniques used here, and of related works, we refer the reader to the introduction to our first paper on the subject \cite{BachMoeller2003a}. In the remaining part of this subsection we mention some recent works, and one application which was not discussed in our previous paper. A number of works, starting with a paper by Naddaf and Spencer \cite{NaddafSpencer1997}, uses semiclassical analysis of the Witten Laplacian on $1$-forms to construct the continuum limit of some massless spin models. Here the lattice spacing plays the role of a semiclassical parameter. We refer the reader to the recent paper by Conlon \cite{Conlon2003}, and references therein, for further material related to this approach. In a work \cite{MatteMoeller2003} of Matte and the second author, the main technical form bound of \cite{BachJeckoSjoestrand2000} is used to show that the usual semiclassical picture, of the low-lying spectrum of a Schr{\"o}dinger operator with convex potential, persists in the thermodynamic limit. After a ground state transform the Witten Laplacian on $0$-forms takes the form $-\beta^{-2}\Delta + \nabla H\cdot \nabla$ as an operator on $L^2(\RR^\Lambda;\exp(-\beta H)d^\Lambda x)$. In this form it appears often in the theory of kinetic equations, and was studied by H{\'e}reau and Nier who obtained bounds on the rate of convergence to equilibrium for the Fokker-Planck equation. See \cite{HereauNier2002}, and references therein. In this connection no uniformity in $\Lambda$ is sought for. Helffer and Nier have recently obtained in \cite{HelfferNier2003} delicate conditions under which a Poincar{\'e} inequality holds for the Gibbs measure (\ref{Gibbs}). They approach the problem by proving the stronger statement that the Witten Laplacian on $0$-forms has compact resolvent. The Poincar{\'e} inequalities thus obtained are not uniform in the cardinality of $\Lambda$. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Correlation Asymptotics} \label{SecI.3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This subsection is devoted to a presentation of the main result, Theorem~\ref{asymptotics}, below. We begin by formulating the hypotheses under which we work. \subsubsection{Hypotheses on the Hamiltonian} % We start with the hypothesis on the self-energies $f_j$. % \begin{hypothesis} \label{H-1} For any $j \in \Lambda$, zero is the unique, non-degenerate minimum of $f_j\in C^2(\RR;\RR)$, attained at $t=0$, i.e., $f_j(0) = 0$, $f_j''(0) >0$, and $f_j(t) >0$, whenever $t \neq 0$. Moreover, there exist universal constants $0 < c_f \leq 1 \leq C_f < \infty$, $R_f < \infty$, such that % \begin{eqnarray} \label{eq-I-1-2} & c_f \ \leq \ f_j''(0) \ \leq \ C_f \comma & \\ \label{eq-I-1-2-1} \forall t \neq 0 : & f_j'(t) \; = \; 0 \ \Rightarrow \ f_j(t) \; \geq \; c_f \comma & \\ \label{eq-I-1-3} \forall t \in \RR : & | f_j''(t) - f_j''(0) | \ \leq \ C_f \big( | f_j'(t) | \, + \, \min\{1, |t|\} \big) \comma & \\ \label{eq-I-1-4} \forall |t| \geq R_f : & | f_j'(t) | \ \geq \ c_f \: \max\big\{ | f_k'(s)| \; \big| \ |s| \leq |t| \comma \ k \in \Lambda \big\} & \end{eqnarray} % for all $j \in \Lambda$. \end{hypothesis} % For the formulation of the hypotheses on the interactions $w_{ij}$, where $i \neq j$, it is convenient to use the following notation for the partial derivatives of $w_{ij}$, % \begin{eqnarray} \label{eq-I-1-5.6} & \partial_1 w_{ij}(x_i,x_j) \ := \ \frac{\partial w_{ij}}{\partial x_i} (x_i,x_j) \comma \mss \partial_2 w_{ij}(x_i,x_j) \ := \ \frac{\partial w_{ij}}{\partial x_j} (x_i,x_j) \comma & \\ \label{eq-I-1-5.7} & \partial_1^2 w_{ij}(x_i,x_j) \ := \ \frac{\partial^2 w_{ij}}{\partial x_i^2} (x_i,x_j) \comma \mss \partial_2^2 w_{ij}(x_i,x_j) \ := \ \frac{\partial^2 w_{ij}}{\partial x_j^2} (x_i,x_j) \comma & \\ \label{eq-I-1-5.71} & \mand \partial_{12}^2 w_{ij}(x_i,x_j) \ := \ \frac{\partial^2 w_{ij}}{\partial x_i \partial x_j} (x_i,x_j) \period & \end{eqnarray} % We introduce two symmetric matrices $\ua = (a_{ij})_{i,j \in \Lambda}$, $\us = (s_{ij})_{i,j \in \Lambda}$ by % \begin{eqnarray} \label{eq-I-1-5.8} a_{ij} & := & - \partial_{12}^2 w_{ij}(0,0) \comma \\ \label{eq-I-1-5.9} s_{ij} & := & \one[ a_{ij} \neq 0 ] \period \end{eqnarray} % % \begin{hypothesis}\label{H-2} There exist universal constants $C_a, C_s < \infty$ such that the two symmetric matrices $\ua = (a_{ij})_{i,j \in \Lambda}$ and $\us = (s_{ij})_{i,j \in \Lambda}$ possess the following properties: % \begin{eqnarray} \label{eq-I-1-9.1a} \forall i,j \in \Lambda: \hspace{12mm} a_{ij} & \geq & 0 \comma \\ \label{eq-I-1-9} \max_{i \in \Lambda} \sum_{j \in \Lambda} a_{ij} & \leq & C_a \comma \\ \label{eq-I-1-5.12} \max_{i \in \Lambda} \sum_{j \in \Lambda} s_{ij} & \leq & C_s \period \end{eqnarray} % \end{hypothesis} % We remark that (\ref{eq-I-1-9.1a}) is a ferromagnetic property. Eq.~(\ref{eq-I-1-5.12}) can be viewed as a finite range condition on the Hessian of $H$ at $x=0$. Namely, given $i \in \Lambda$, the number of nearest-neighbor sites of $i$ is $\sum_{j \in \Lambda} s_{ij}$. Condition~(\ref{eq-I-1-5.12}) requires these to be uniformly bounded in $i \in \Lambda$. The second hypothesis on the interactions $w_{ij}$ is formulated with the aid of the following functions, % \begin{eqnarray} \label{eq-I-1-5.5} h_j(s) & := & \min\big\{ |f_j(s)|, |f_j(s)|^{\frac{1}{2}} \big\} \; + \; \min\{ 1, |s| \} \comma \end{eqnarray} % which we introduce for all $j \in \Lambda$. % \begin{hypothesis} \label{H-3} For all $i,j \in \Lambda$, the pair interaction functions $w_{ij} \in C^2( \RR \times \RR; \RR)$ vanish on-site and at the origin, i.e., $w_{ii}(s,t) \equiv 0$ and $w_{ij}(0,0)=0$. Furthermore, for $i,j \in \Lambda$, there exists a universal number $C_w$, such that % \begin{eqnarray} \label{eq-I-1-6} \nonumber & & \hspace{-4mm} \big| \partial_1 w_{ij}(x_i,x_j) \big| + \big| \partial_2 w_{ij}(x_i,x_j) \big| + \Big| \partial_1^2 w_{ij}(x_i,x_j) - \partial_1^2 w_{ij}(0,0) \Big| \\ \nonumber & & \hspace{-4mm} + \Big| \partial_2^2 w_{ij}(x_i,x_j) - \partial_2^2 w_{ij}(0,0) \Big| + \Big| \partial_{12}^2 w_{ij}(x_i,x_j) - \partial_{12}^2 w_{ij}(0,0) \Big| \\[2mm] & & \hspace{4mm} \leq \ C_w \, a_{ij} \, \big( h_{i}(x_i) + h_{j}(x_j)\big)\comma \end{eqnarray} % and % \begin{equation} \label{eq-I-1-7} \big| \partial_1^2 w_{ij}(0,0) \big| + \big| \partial_2^2 w_{ij}(0,0) \big| \ \leq \ C_w \, a_{ij} \period \end{equation} % \end{hypothesis} % All (derived) universal constants appearing in the paper depend only on ingredients through the universal constants which appear in the relevant hypotheses, i.e., $c_f$, $C_f$, $R_f$, $C_a$, $C_s$, and $C_w$ from Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. By \cite[Lemma B.1]{BachMoeller2003a}, Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3} insure the well-definedness of the Gibbs measure (\ref{Gibbs}), for small $\alpha$. Moreover, polynomially bounded, measurable observables $u : \RR^\Lambda\to \CC$ are integrable. \subsubsection{The Main Result} For the formulation of the main result we introduce the real symmetric $\Lambda\times\Lambda$ transition matrix $T = \{T_{ij}\}_{i, j \in \Lambda}$ by % \begin{equation}\label{Tij} T_{ij} \ := \ \frac{- \partial_{12}^2 w_{ij}(0,0)} {\sqrt{H_{ii}^\dprime(0) \, H_{jj}^\dprime(0) \,}} \ \leq \ 0 \comma \end{equation} % for all $i, j \in \Lambda$. Here we use that, due to Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}, we have $c_f - \alpha C_a \leq H_{ii}^\dprime(0) \leq C_f + \alpha C_a$ and hence $0 < c_f/2 \leq H_{ii}^\dprime(0) \leq 2C_f$, for $0 < \alpha \leq c_f / (2C_a)$. Note that $T_{ii} = 0$, for all $i \in \Lambda$. Further note that % \begin{equation}\label{Tij-1} H_{ij}^\dprime(0) \ = \ H_{ii}^\dprime(0)^{1/2} \, \big\{ \one - \alpha T \big\}_{ij} \, H_{jj}^\dprime(0)^{1/2} \comma \end{equation} % and a Neumann series expansion shows that $\{ H''(0)^{-1} \}_{ij} \geq 0$. If $i$ and $j$ belong to the same connected component of $\Lambda$, i.e., $d_a(i,j) < \infty$ (see Eq.~(\ref{eq-I-1-9.2}) and below), then we even have strict positivity of the matrix elements $\{ H''(0)^{-1} \}_{ij} > 0$. We may hence define a map $\rho: \Lambda \times \Lambda \to [0,\infty]$ by % \begin{equation}\label{Tij-2} \exp\big[ - \rho(i,j) \big] \ := \ \frac{ \{ H''(0)^{-1} \}_{ij} } { \{ H''(0)^{-1} \}_{ii}^{1/2} \, \{ H''(0)^{-1} \}_{jj}^{1/2} } \period \end{equation} % We are now in position formulate the main result of this paper. % \begin{theorem}\label{asymptotics} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist universal constants $C < \infty$, $\beta_0 < \infty$ and $\alpha_0 >0$ such that, for any $0 \leq \alpha \leq\alpha_0$ and any $\beta_0 \leq \beta < \infty$, we have % %\begin{eqnarray} \label{eq-asymptotics-1} %\lefteqn{ %\Big(\frac{2\, [1 - \tau(\beta)]}% %{ \beta \, H_{ii}''(0)^{1/2} \, H_{jj}''(0)^{1/2} } \Big) \, \Big\{ %\Big( \one \, - \, \frac{\alpha}{1+ \tau(\beta)} \, T \Big)^{-1} %\Big\}_{ij} %\ \; \leq \ \; %} %\\ \nonumber %& & \hspace{5mm} %\EE^T_\beta(x_i\, ; \,x_j) %\ \; \leq \ \; %\Big(\frac{2\, [1 - \tau(\beta)]}% %{ \beta \, H_{ii}''(0)^{1/2} \, H_{jj}''(0)^{1/2} } \Big) \, \Big\{ %\Big( \one \, - \, \frac{\alpha}{1- \tau(\beta)} \, T \Big)^{-1} %\Big\}_{ij} %\end{eqnarray} % %and % \begin{equation} \label{eq-asymptotics-2} \frac{2\, [1 - \tau(\beta)]}{\beta} \, \big\{ H''(0)^{-1} \big\}_{ij}^{1+\tau(\beta)} \ \leq \ \EE^T_\beta(x_i\, ;\,x_j) \ \leq \ \frac{2\, [1 - \tau(\beta)]}{\beta} \, \big\{ H''(0)^{-1} \big\}_{ij}^{1-\tau(\beta)} \period \end{equation} % where $\tau(\beta) := C \beta^{-1/2}$. Moreover, $\rho: \Lambda \times \Lambda \to [0,\infty]$ defined by (\ref{Tij-2}) is a metric on $\Lambda$, and % \begin{eqnarray} \label{eq-asymptotics-3} \lefteqn{ \frac{2\, [1 - \tau(\beta)]}{\beta} \, \exp\big[ -(1+\tau(\beta)) \, \rho(i,j) \big] \ \leq \ } \\ \nonumber \EE^T_\beta(x_i\, ;\,x_j) & \leq & \frac{2\, [1 - \tau(\beta)]}{\beta} \, \exp\big[ -(1+\tau(\beta)) \, \rho(i,j) \big] \period \end{eqnarray} % \end{theorem} % We stress that we impose no translation invariance assumption on the Hamiltonian, which was crucial for the method used in \cite{Sjoestrand1997}, \cite{BachJeckoSjoestrand2000}, and \cite{Sjoestrand2000}. Theorem~\ref{asymptotics} reduces the problem of studying the correlation function at low temperature to that of the study of resolvents of transition matrices. This can be viewed as a problem related to random walks on infinite graphs, or more precisely, Ornstein-Zernike theory. We refer the interested reader to Sj{\"o}strand \cite[Sect.~5]{Sjoestrand1997}, Campanino, Ioffe, and Velenik \cite{CampaninoIoffeVelenik2003b} (and references therein), and the monographs by Spitzer \cite{Spitzer1976} (translation invariant random walks on $Z^d$) and Woess \cite{Woess2000} (general theory). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{A Recollection of Earlier Results} \label{SecI.2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this Subsection we recall the ingredients and results from \cite{BachMoeller2003a} which are used here. They hold under Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and a seemingly additional requirement of existence of an a-priori metric $d$ on $\Lambda$ satisfying the summability condition % \begin{equation} \label{eq-summ-cond-d} \max_i\,\sum_j e^{-d(i,j)} \ \leq \ C_d \hspace{4mm} \mbox{and} \hspace{4mm} \max_i\,\sum_j e^{d(i,j)} \, a_{ij} \ \leq \ C_d \comma \end{equation} % for a universal constant $C_d < \infty$. For the construction of $d$ from Hypothesis~\ref{H-2}, we introduce the set of \emph{nearest-neighbour bonds} % \begin{equation}\label{eq-I-1-9.21} \cB_a \ := \ \big\{ (i,j) \in \Lambda \times \Lambda \; \big| \ a_{ij} \neq 0 \big\} \ = \ \big\{ (i,j) \in \Lambda \times \Lambda \; \big| \ s_{ij} = 1 \big\} \period \end{equation} % Given two points $i, j \in \Lambda$, a nonempty, finite collection of nearest-neighbour bonds of the form $\gamma = \{(i_0,i_1), (i_1,i_2), \dots, (i_{n-1},i_n)\} \subseteq \cB_a$, with $i_0 = i$, $i_n = j$, and $i_k \neq i_{k+1}$, is called a \emph{path} from $i$ to $j$ and is denoted $\gamma:i \to j$. The number $|\gamma| := n$ of bonds $b = (i_k,i_{k+1}) \in \cB_a$ in the path is referred to as its \emph{length}. The collection of all paths from $i$ to $j$ is denoted $\Gamma(i,j)$. Note that $\gamma = \emptyset$ is not a path, and there is no path of zero length. Further note that $(i,i) \notin \cB_a$ is neither a nearest-neighbour bond, nor is $\{(i,i)\} \notin \Gamma(i,i)$ a path. A metric $d_a$ is defined as the canonical metric of the graph $(\Lambda, \cB_a)$. So, given two points $i, j \in \Lambda$, their distance with respect to the metric $d_a$ is defined to be the minimal length of all paths linking $i$ and $j$, i.e., % \begin{equation}\label{eq-I-1-9.2} d_a(i,j) \ := \ \min\big\{ |\gamma| \; \big| \ \gamma \in \Gamma(i,j) \big\} \comma \end{equation} % $d_a(i,i) := 0$, and $d_a(i,j) := \infty$, if no such path exists. If $d_a(i,j) = 1$ then $(i,j) \in \cB_a$ and $i$ and $j$ are called \emph{nearest neighbours}. A sufficiently large multiple of $d_a$ then possesses the desired summability properties % \begin{lemma} \label{lem-summ-cond-dw} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and define a metric $d := \ln(2C_s) \, d_a$ on $\Lambda$. Then % \begin{equation} \label{eq-summ-cond-dw} \max_i\,\sum_j e^{-d(i,j)} \ \leq \ 1 \hspace{4mm} \mbox{and} \hspace{4mm} \max_i\,\sum_j e^{d(i,j)} \, a_{ij} \ \leq \ 2 \, C_s^2 \period \end{equation} % \end{lemma} % \Proof To derive the first estimate in (\ref{eq-summ-cond-dw}), we remark that, given a fixed site $i \in \Lambda$, the number of sites $j \in \Lambda$, which are at distance $n = d_a(i,j)$ of $i$, is bounded by $C_s^n$, due to Hypothesis~\ref{H-2}. Therefore, % \begin{equation} \label{eq-summ-cond-nu} \sum_j e^{-d(i,j)} \ \leq \ \sum_{n=1}^\infty C_s^n \: e^{-\ln(2C_s)} \ = \ 1 \period \end{equation} % The second estimate in (\ref{eq-summ-cond-dw}) is a trivial consequence of Hypothesis~\ref{H-2} which implies that $d_a(i,j) = 1$, whenever $a_{ij} > 0$ does not vanish. \QED \subsubsection{Modified Single-Spin Potentials} We first introduce modifications $g_i$'s of the $f_i$'s, which coincide with the $f_i$'s near $0$ and, as the main point, differ from the $f_i$'s by having no local minima away from $0$. They were constructed in \cite[Lemma 1.2]{BachMoeller2003a}, and we present them in the following lemma, leaving out those properties not needed here. % \begin{lemma} \label{lem-I-1} Assume Hypothesis~\ref{H-1}. There exist universal numbers $\qmax >0$, % \begin{equation} \label{eq-I-1-9.1b} 0 \ < \ \Ri_0 \ < \ \Ri_1 \ < \ \Ro_0 \ < \ \Ro_1 \ < \ \infty \comma \end{equation} % to which we associate the unions of intervals % \begin{equation} \label{eq-I-1-10.1} \begin{array}{cc} I_0 \ := \ [-\Ri_0 \; , \; \Ri_0] \comma & \Ii \ := \ (-\Ri_1 , -\Ri_0) \cup (\Ri_0 , \Ri_1) \comma \\ \Io \ := \ (-\infty , -\Ro_0) \cup (\Ro_0 , \infty) \comma & I_\infty \ := \ (-\infty , -\Ro_1) \cup (\Ro_1 , \infty) \comma \end{array} \end{equation} % and functions % \begin{eqnarray} \label{eq-I-1-10.55} \nonumber & \qi_j \: \in \: C^2\big( \Ii \, ; \, [0, \qmax] \big) \hspace{3mm} \comma \hspace{4mm} \qo_j \: \in \: C^2\big( \Io \, ; \, [0, \qmax] \big) \comma & \\ & \mbox{and} \hspace{10mm} q_j \; := \; \qi_j + \qo_j \: \in \: C^2( \RR \, , \, [0, \qmax] \big) \comma & \end{eqnarray} % possessing the following properties: % \begin{itemize} \item[(i)] On $I_\infty$, we have $\qo_j \equiv \qmax$. \item[(ii)] The functions % \begin{equation} \label{eq-I-1-10.6} g_j \; := \; f_j - q_j \hspace{2mm} \mbox{and} \hspace{2mm} \gi_j \; := \; f_j - \qi_j \end{equation} % are nonnegative and have a unique critical point at $t =0$. \item[(iii)] There exist universal constants $c_g >0$ and $C_g<\infty$ such that we have $\sup_{t} |q_j^\prime(t)| < C_g$, and for all $t \in \RR$ \begin{equation}\label{new-est-on-g} c_g \,\min\{ 1, |t| \}\ \leq \ \sgn(t) \,g_j'(t) \ \leq \ C_g\,e^{C_g |t|}\period \end{equation} \end{itemize} % \end{lemma} % \subsubsection{Semiclassical Localization Estimates} The second result we invoke in this paper is a semiclassical localization estimate deriving from \cite[Theorem 1.6]{BachMoeller2003a}, which we use to argue that the twists we introduce into the Witten Laplacian only give rise to small corrections in the low temperature limit. It holds under Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and the existence of a metric $d$ on $\Lambda$ satisfying the summability condition (\ref{eq-summ-cond-d}), which is insured by Lemma~\ref{lem-summ-cond-dw}. % \begin{theorem} \label{LocofZ1} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}, and let $R >0$ be a fixed universal number. Then there exist universal constants $\alpha_0, \delta >0$, and $\beta_0 < \infty$ such that, for all $k \in \Lambda$, $0 \leq \alpha \leq \alpha_0$, and $\beta > \beta_0$, we have % \begin{equation} \label{semiclassicalest.1a} \int_{ |x_k| \geq R} e^{ - 2 \beta H(x) } \, \frac{d^\Lambda x}{\cZ_\beta} \ \leq \ e^{ - \beta \, \delta} \period \end{equation} % \end{theorem} % We remark that, for a given lattice site $k \in \Lambda$, the Hamiltonian function $\tH_k(x) := H(x) - q_k(x_k)$ also satifies Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. Moreover, $H(x) \equiv \tH_k(x)$ on $\{x \in \RR^\Lambda: |x_k| \leq \Ri_0 \}$. Thus, an application of Theorem~\ref{LocofZ1} to the metric $d$ and the Hamiltonian function $\tH_k(x)$ yields the following corollary. % \begin{corollary} \label{LocofZ2} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. Then there exist universal constants $\alpha_0, \delta >0$, and $\beta_0 < \infty$ such that, for all $k \in \Lambda$, $0 \leq \alpha \leq \alpha_0$, and $\beta > \beta_0$, we have % \begin{equation} \label{semiclassicalest.1b} \Big| \int e^{ - 2 \beta [H(x) - q_k(x)]} \, \frac{d^\Lambda x}{\cZ_\beta} \; - \; 1 \Big| \ \leq \ e^{ - \beta \, \delta} \period \end{equation} % \end{corollary} % It is important to notice that Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3} and, hence, also Corollary~\ref{LocofZ2} may fail to hold for the Hamiltonian function $H(x) - 2q_k(x_k)$. This fact plays a certain role for our choice of the projection $p$ in Eq.~(\ref{Feshbachprojection}). \subsubsection{Twisted de Rham Complex} The following is a brief recollection of the remaining part of \cite[Section 1]{BachMoeller2003a}, and we refer the reader to \cite[Section 1 and Appendix B]{BachMoeller2003a} for details. We use the summation convention $\sum_i (\cdot) := \sum_{i \in \Lambda} (\cdot)$, $\sum_{i,j} (\cdot) := \sum_{(i,j) \in \Lambda^2} (\cdot)$, $\sum_{i \neq j} (\cdot) := \sum_{(i,j) \in \Lambda^2 \setminus \{(i,i) | i \in \Lambda \} } (\cdot)$, and $\sum_{i (\neq j)} (\cdot) := \sum_{i \in \Lambda \setminus \{j\}} (\cdot)$. Next, we introduce the fermionic Fock space over $\CC^\Lambda$, % \begin{equation} \label{Fock-space} \cF \ \equiv \ \cF(\CC^\Lambda) \ := \ \oplus_{n=0}^\infty \, \cF^{(n)} \comma \hspace{10mm} \cF^{(n)} \ := \ \big(\CC^\Lambda \big)^{\otimes_a n} \comma \end{equation} % where $\otimes_a n$ denotes the $n$-fold antisymmetric tensor product, and $\cF^{(0)} \simeq \CC \, \Om$ is a one-dimensional subspace spanned by the normalized vacuum vector $\Om$. The standard annihilation and creation operators $\{a_i , a_i^* \}_{i\in\Lambda}$ represent the canonical anticommutation relations (CAR); $\forall i,j \in \Lambda$: % \begin{equation} \label{commrel} \{a_i , a_j\} \ = \ \{a_i^* , a_j^*\} \ = \ 0 \comma \hspace{4mm} \{a_i^* , a_j\} \ = \ \delta_{ij} \comma \hspace{2mm}\mbox{and}\hspace{4mm} a_i\Omega \ = \ 0 \comma \end{equation} % on $\cF$, where $\{A,B\} = AB+BA$, and $\delta_{ij}$ is the Kronecker delta. % The Hilbert space of forms over $\RR^\Lambda$ is the tensor product % \begin{equation} \label{Hilbertspace} \cH \ := \ L^2(\RR^\Lambda) \, \otimes \, \cF \ = \ \oplus_{n=0}^\infty \cH^{(n)} \comma \hspace{5mm} \cH^{(n)} \ = \ L^2(\RR^\Lambda)\,\otimes \, \cF^{(n)} \period \end{equation} % We introduce a (multiple of the) standard exterior derivative on $\cH$, for $\beta>0$, and its adjoint % \begin{equation} \label{extder} d \ := \ \sum_{i} \frac{1}{\beta}\partial_i \, \otimes \, a_i^* \hspace{3mm}\mbox{and}\hspace{3mm} d^* \ = \ -\sum_{i} \frac{1}{\beta} \partial_i \, \otimes \, a_i \comma \end{equation} % where $\partial_i$ is shorthand for $\frac{\partial}{\partial x_i}$. Note that $d^2 = (d^*)^2 = 0 $. The Hodge Laplacian associated to this exterior derivative is $dd^*+d^*d = (d+d^*)^2$. To an operator $T$ on $\cH^{(1)}$, we associate its second quantization $d\Gamma(T):\cH\to\cH$ by the standard formula, i.e., if $T$ is represented by a matrix $(T_{ij})_{i,j \in \Lambda}$ whose entries take their values in operators on $L^2(\RR^\Lambda)$ then % \begin{equation} \label{secquant1} d\Gamma(T) \ := \ \sum_{i,j} T_{ij} \, \otimes \, a_i^* \, a_j \period \end{equation} % In the present paper we second-quantize only operators whose entries are semi-bounded, self-adjoint operators on $L^2(\RR^\Lambda)$. We remark that, if $\cap_{i,j} \cD(T_{ij})$ is a core for all the $T_{ij}$'s, then $T$ and $d\Gamma(T)$ are essentially self-adjoint and semi-bounded on $\{\cap_{i,j} \cD(T_{ij})\}\otimes \CC^\Lambda$ and $\{\cap_{i,j} \cD(T_{ij})\}\otimes \cF$ respectively. If $T$ is bounded then $d\Gamma(T)$ is also bounded. In particular, let $Q$, $\Qio$, and $\Qo$ denote the matrix-valued functions with entries % \begin{eqnarray} \label{Qmatrices1} & \Qio_{ij} (x) \ = \ \delta_{ij} \, \qi_i(x_i) \comma \hspace{4mm} \Qo_{ij} (x) \ = \ \delta_{ij} \, \qo_i(x_i) & \\ \label{Qmatrices2} & \hspace{3mm}\mbox{and}\hspace{3mm} Q_{ij}(x) \ = \ \Qio_{ij} (x) \; + \; \Qo_{ij} (x) \period & \end{eqnarray} % Here $x \equiv (x_i)_{i \in \Lambda} \in \RR^\Lambda$, and $\qi_i$ and $\qo_i$ are introduced in (\ref{eq-I-1-10.55}). We frequently omit the argument and write $\qi_i := \qi_i(x_i)$ and $\qo_i := \qo_i(x_i)$, et cetera. Then their second quantization is given by % \begin{equation} \label{secquant2} d\Gamma(Q^\#) \ = \ \sum_{j} q_j^\# \otimes \, a_j^* \, a_j \comma \end{equation} % where $Q^\#$ denotes $Q$, $\Qio$, or $\Qo$, and $q_j^\#$ denotes $q_j$, $\qi_j$, or $\qo_j$, respectively. Now, we introduce the twisted exterior derivative % \begin{equation} \label{dHQ} d_{H,Q} \ := \ e^{-\beta(H-d\Gamma(Q))} \: d \: e^{\beta(H-d\Gamma(Q))} \end{equation} % on $C_0^\infty(\RR^\Lambda) \otimes \cF$, which is a core for $d_{H,Q}$. We denote its closure by the same symbol. Here $H \equiv H \otimes \one$ is considered a multiplication operator on $\cH$. The \emph{twisted Dirac operator} is defined as the sum of the twisted exterior derivative and its adjoint, % \begin{equation} \label{Dirac} d_{H,Q} \; + \; d_{H,Q}^* \period \end{equation} % By construction it is clear that $d_{H,Q}^2 = (d_{H,Q}^*)^2 = 0$. Thus the square of the twisted Dirac operator is the associated Hodge Laplacian % \begin{equation} \label{Laplace} \Delta_{H,Q} \ := \ \big(d_{H,Q} \, + \, d_{H,Q} \big)^2 \ = \ d_{H,Q} \, d_{H,Q}^* \; + \; d_{H,Q}^* \, d_{H,Q} \comma \end{equation} % which we call the \emph{twisted Witten Laplacian}. Similar to the situation in \cite{BachMoeller2003a}, we have that $C_0^\infty(\RR^\Lambda) \otimes \cF$ is a form-core for $\Delta_{H,Q}$. We write $\Delta_{H,Q}^{(n)}$ for the restriction of $\Delta_{H,Q}$ to $\cH^{(n)}$. We recall \cite[Theorem 1.4]{BachMoeller2003a} % \begin{theorem} \label{thm-kernelofDelta} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. % \begin{itemize} \item[(i)] There exist $\alpha_0 > 0$ and $\beta_0 < \infty$ such that, for $0 \leq \alpha < \alpha_0$ and $\beta > \beta_0$, % \begin{equation} \label{eq-kernelofDelta} \cern\{\Delta_{H,Q}^{(0)}\} \ = \ \CC \, e^{-\beta H}\hspace{2mm}\mbox{and}\hspace{2mm} \cern\{\Delta_{H,Q}^{(1)}\} \ = \ \{0\}\period \end{equation} % \item[(ii)] If all the $f_i$'s and $w_{ij}$'s are $C^\infty$ in (not necessarily universal) neighbourhoods of $0$, then there exists a universal constant $\alpha_0>0$ such that for $0\leq \alpha <\alpha_0$ and all $\beta>0$, % \begin{equation}\label{eq-kernelofDelta2} \cern\{\Delta_{H,Q}\} \ = \ \CC \, e^{-\beta H}\,\otimes\,\Omega\period \end{equation} % \end{itemize} % \end{theorem} % In \cite{BachMoeller2003a}, the following representation of the correlations has been derived from Theorem~\ref{thm-kernelofDelta}, which is of key importance to our analysis. % \begin{theorem} \label{correlations} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist $\alpha_0 > 0$ and $\beta_0 < \infty$ such that, for $0 \leq \alpha < \alpha_0$ and $\beta > \beta_0$, % \begin{equation} \label{eq-correlations} \EE^T_\beta(x_i\, ; \,x_j) \ = \ \beta^{-2} \, \cZ_\beta^{-1} \, \Big\la e^{-\beta(H-q_i)} \otimes e_i \, \Big| \; \big( \Delta_{H,Q}^{(1)} \big)^{-1} \, e^{-\beta(H-q_j)} \otimes e_j \Big\ra \period \end{equation} % \end{theorem} % For $Q =0$, (\ref{eq-correlations}) has first been observed (implicitly) by Helffer and Sj\"ostrand in \cite{HelfferSjoestrand1994} and, more explicitly, by Sj\"ostrand in \cite{Sjoestrand1997}. We have also the following important supersymmetric property. % \begin{theorem} \label{susy} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist $\alpha_0 > 0$ and $\beta_0 < \infty$ such that, for $0 \leq \alpha < \alpha_0$ and $\beta > \beta_0$, % \begin{equation} \label{eq-susy} \sigma(\Delta_{H,Q}^{(0)}) \setminus \{0\} \ \subseteq \ \sigma(\Delta_{H,Q}^{(1)}) \period \end{equation} % \end{theorem} % \Proof The proof of Theorem~\ref{susy} in \cite{Sjoestrand1997} assumes the discreteness of the spectrum of $\Delta_{H,Q}$. Here we refer instead to an abstract result of Johnsen, cf.~\cite[Theorem 3.1]{Johnsen2000}. We apply this result to $H = \cH^{(0)}$, $H_1 = \cH^{(1)}$, $T = d_{H,Q |\cH^{(0)}}$, and $F_1 = \overline{\Ran{T}}$. We verify one of the five equivalent conditions in Johnsen's theorem, namely condition (iv). It requires that ${TT^*}_{|F_1}$ has closed range and does not have $0$ in its spectrum. But this follows directly from the estimate ${TT^*}_{|F_1} \geq \Delta^{(1)}_{H,Q|F_1}$ and (\ref{eq-I-specbound}) below. A similar argument was used in \cite{MatteMoeller2003} (for the case $Q=0$). \QED \subsubsection{Explicit Expressions} We now give more explicit formulas for the objects introduced above. First, we introduce some exponential weights, % \begin{equation} \label{Thetas} \Thio_i \; := \; e^{\beta \qi_i} \comma \hspace{4mm} \Tho_i \; := \; e^{\beta \qo_i} \hspace{3mm} \mbox{and} \hspace{3mm} \Th_i \; :=\; e^{\beta q_i} \; =\; \Thio_i\,\Tho_i \comma \end{equation} % and twisted derivatives, together with their adjoints, % \begin{equation} \label{Zs} Z_i(H) \ := \ \frac{1}{\beta}\,\partial_i \; + \; H_i' \hspace{3mm} \mbox{and} \hspace{3mm} Z_i^*(H) \ = \ -\frac{1}{\beta}\,\partial_i \; + \; H_i' \period \end{equation} % We compute, using (\ref{commrel}) and (\ref{extder})--(\ref{dHQ}), % \begin{equation} \label{ds} d_{H,Q}\; =\; \sum_{i} \Th_i \,Z_i(H)\,\otimes\, a_i^* \hspace{3mm}\mbox{and}\hspace{3mm} d_{H,Q}^* \; =\; \sum_{i} Z_i^*(H)\,\Th_i\,\otimes \,a_i\period \end{equation} % Note the intertwining relations % \begin{equation} \label{intertwining} \Th_i \, Z_i(H) \ = \ Z_i(G) \, \Th_i \hspace{6mm}\mbox{and}\hspace{6mm} \Thi_i \, Z_i(H) \ = \ Z_i(\Gi) \, \Thi_i \comma \end{equation} % where % \begin{eqnarray} \label{Gs} G & := & H \: - \: \sum_i q_i \ = \ \sum_i g_i \: + \: \alpha \sum_{i,j} w_{ij} \comma \\ \label{Ginns} \Gi & := & H \: - \: \sum_i \qi_i \ = \ \sum_i \gi_i \: + \: \alpha \sum_{i,j} w_{ij} \period \end{eqnarray} % The intertwining relations (\ref{intertwining}) were of key importance to our analysis in \cite{BachMoeller2003a}, as they allowed us to pass from $H$ to a new Hamiltonian function $G$ which agrees with $H$ at zero, but yet has no critical points other than zero. Note that $\cH^{(0)} = L^2(\RR^\Lambda) \otimes \cF^{(0)}$ can be identified with $L^2(\RR^\Lambda)$, and $\cH^{(1)}$ with $L^2(\RR^\Lambda) \otimes \CC^\Lambda$. We will use the same notation, $\Delta_{H,Q}^{(0)}$ and $\Delta_{H,Q}^{(1)}$, for both representations. The fully twisted Witten Laplacian is of the form (cf. (\ref{commrel}), (\ref{Laplace}), (\ref{ds}) and (\ref{intertwining})) % \begin{eqnarray} \label{Deltafull} \Delta_{H,Q} & = & \sum_{i,j} \Th_i \, Z_i(H) \, Z_j^*(H) \, \Th_j \otimes a_i^* \,a_j \; + \; Z_j^*(H) \, \Th_j \, \Th_i \, Z_i(H) \otimes a_j \, a_i^* \nonumber \\[1mm] & = & \sum_{j} \Big\{ \Th_j \, Z_j(H) \, Z_j^*(H) \, \Th_j \otimes a_j^* \, a_j \; + \; \Th_j \, Z_j^*(G) \, Z_j(G) \, \Th_j \otimes a_j \, a_j^* \Big\} \nonumber \\[1mm] & & \hspace{20mm} + \; \sum_{i \neq j} \big[ \Th_i \, Z_i(H) \: , \: Z_j^*(H) \, \Th_j \big] \, \otimes \, a_i^* \, a_j \comma \nonumber \\[1mm] & = & \sum_{j} \Big\{ \Th_j \, Z_j(H) \, Z_j^*(H) \, \Th_j \otimes a_j^* \, a_j \; + \; \Th_j \, Z_j^*(G) \, Z_j(G) \, \Th_j \otimes a_j \, a_j^* \Big\} \nonumber \\[1mm] & & \hspace{20mm} + \; \frac{2}{\beta} \sum_{i \neq j} \Th_i \, \Th_j \, H_{ij}''(x) \, \otimes \, a_i^* \, a_j \comma \end{eqnarray} % where we used that $[ Z_i(H) , \Th_j ] = 0$ and $[ Z_i(H) , Z_j^*(H) ] = 2 \beta^{-1} H_{ij}''$, for $i\neq j$, and we denote $H_{ij}'' := \partial_i \partial_j H$. Restricting (\ref{Deltafull}) to $\cH^{(0)}$ and $\cH^{(1)}$, we arrive at the twisted Witten Laplacian on $0$- and $1$-forms, % \begin{eqnarray} \label{Delta0} \Delta_{H,Q}^{(0)} & = & \sum_{i} \Th_i \, Z_i^*(G) \, Z_i(G) \, \Th_i \comma \\\label{Delta1} \Delta_{H,Q}^{(1)} & = & \sum_{j} \Big\{ \Th_j\,Z_j(H)\,Z_j^*(H)\,\Th_j \; + \; \sum_{k(\neq j)} \Th_k \, Z_k^*(G) \, Z_k(G) \, \Th_k \Big\} \,\otimes\, E_{jj} \nonumber \\[1mm] & & + \; \frac{2}{\beta} \sum_{j \neq k} \Th_j \, \Th_k \, H_{jk}'' \, \otimes \, E_{jk}\period \end{eqnarray} \subsubsection{Comparison Operator and Perturbation} We now recall the comparison operator which was shown in \cite{BachMoeller2003a} to approximate $\Delta_{H,Q}$ at low temperature in a form sense % \begin{equation} \label{eq-I-4-4} A_{H,Q} \ := \ \sum_{j} \Big\{ \Ao_{j} \otimes a_j^* \, a_j \: + \: A_{j} \otimes a_j \, a_j^* \Big\} + \; \frac{2}{\beta} \, \sum_{i,j} H_{ij}''(0) \, \otimes \, a_i^* \, a_j \comma \end{equation} % where % \begin{eqnarray} \label{eq-I-4-5} \Ao_{j} & := & \Th_j \, Z_j(H) \, Z_j^*(H) \, \Th_j \: - \: 2 \beta^{-1} \, \Tho_j^2 \, \Gio_{jj}''(x) \comma \\[2mm] \label{eq-I-4-6} A_{j} & := & \Th_j \, Z_j^*(G) \, Z_j(G) \, \Th_j \period \end{eqnarray} % Furthermore, we define % \begin{eqnarray} \label{eq-I-4-7} W_{H,Q}(x) & := & \Wdiag(x) \; + \; \Woff(x) \comma \\ \label{eq-I-4-8} \Wdiag(x) & := & \frac{2}{\beta} \sum_{j} \Big(\Tho_j^2 \, \Gio_{jj}''(x) \, - \, H_{jj}''(0) \Big) \, \otimes \, a_j^* a_j \comma \\ \label{eq-I-4-9} \Woff(x) & := & \frac{2}{\beta} \sum_{i \neq j} \Big( \Th_i \Th_j \, H_{ij}''(x) \, - \, H_{ij}''(0) \Big) \, \otimes \, a_i^* a_j \\ \nonumber & = & \frac{2 \, \alpha}{\beta} \sum_{i \neq j} \Big( \Th_i \Th_j \, \partial_{12}^2 w_{ij}(x_i,x_j) \, - \, \partial_{12}^2 w_{ij}(0,0) \Big) \, \otimes \, a_i^* a_j \comma \end{eqnarray} % and we observe the decomposition identity % \begin{equation} \label{eq-I-4-10} \Delta_{H,Q} \ = \ A_{H,Q} \; + \; W_{H,Q} \period \end{equation} % The argument ``$x$'' in (\ref{eq-I-4-7})--(\ref{eq-I-4-9}) indicates that these operators act as matrix-valued multiplication operators and contain no differential operator. We frequently omit to display $x$. The restrictions of $A_{H,Q}$, $W_{H,Q}$, $\Wdiag$, and $\Woff$ onto $\cH^{(0)}$ are given by % \begin{eqnarray} \label{eq-I-4-11} A_{H,Q}^{(0)} & = & \sum_{j} A_{j} \ = \ \sum_{j} \Th_j \, Z_j^*(G) \, Z_j(G) \, \Th_j \comma \\ \label{eq-I-4-10.01} W_{H,Q}^{(0)} & = & \Wdiag^{(0)} \ \; = \ \; \Woff^{(0)} \ \; = \ \; 0 \period \end{eqnarray} % Before we write down the restrictions of $A_{H,Q}$, $W_{H,Q}$, $\Wdiag$, and $\Woff$ onto $\cH^{(1)} \simeq \cH^{(0)} \otimes \CC^\Lambda$, we note that we may view any operator on $\cH^{(1)}$ as a $\Lambda \times \Lambda$-matrix with entries in the operators on $\cH^{(0)}$. More specifically, given an operator $X$ on $\cH^{(1)}$, we denote by $( \{X\}_{ij} )_{i,j \in \Lambda}$ the unique family of operators on $\cH^{(0)}$ such that % \begin{equation} \label{eq-I-4-10.02} X \ = \ \sum_{i,j} \{X\}_{ij} \otimes E_{ij} \period \end{equation} % Equipped with this notation, we find % \begin{eqnarray} \\ \label{eq-I-4-12} A_{H,Q}^{(1)} & = & \sum_{j} \tA_j \otimes E_{jj} \; + \; \frac{2}{\beta} \, \sum_{i, j} H_{ij}''(0) \otimes E_{ij} \comma \\ \label{eq-I-4-12.1} \tA_j & := & \Ao_j \; + \: \sum_{k (\neq j)} A_k \comma \\[1mm] \label{eq-I-4-10.1} \big\{ W_{H,Q}^{(1)} \big\}_{ij} & = & \delta_{ij} \, \big\{ \Wdiag^{(1)} \big\}_{jj} \; + \; (1-\delta_{ij}) \, \big\{ \Woff^{(1)} \big\}_{ij} \comma \\ \label{eq-I-4-10.2} \big\{ \Wdiag^{(1)} \big\}_{jj} & = & \frac{2}{\beta} \Big(\Tho_j^2 \, \Gio_{jj}''(x) \, - \, H_{jj}''(0) \Big) \comma \\ \label{eq-I-4-9.1} \big\{ \Woff^{(1)} \big\}_{ij} & := & \frac{2}{\beta} \Big( \Th_i \Th_j \, H_{ij}''(x) \, - \, H_{ij}''(0) \Big) \period \end{eqnarray} % \subsubsection{Form Bounds on the Perturbation} The main technical result in \cite{BachMoeller2003a} is Theorem~2.1 which we quote in a special case ($\kappa =0$, see \cite[Eq.~(II.2)]{BachMoeller2003a}). % \begin{theorem} \label{thm-II-0-1} Assume Hypotheses~\ref{H-1}, \ref{H-2} and \ref{H-3}. There exist universal constants $C < \infty$, $\alpha_0 >0$, and $\beta_0 < \infty$ such that, for all $0 \leq \alpha \leq \alpha_0$ and all $\beta > \beta_0$, we have % \begin{equation} \label{eq-II-0-2} \pm W_{H,Q}^{(1)} \ \leq \ \frac{C}{\beta^{1/2}} \, A_{H,Q}^{(1)} \comma \end{equation} % in the sense of quadratic forms. \end{theorem} % Note that, as a consequence, for all $0 \leq \alpha \leq \alpha_0$ and all $\beta > \beta_0$, we have % \begin{equation}\label{eq-I-specbound} \Delta^{(1)}_{H,Q} \ \geq \ \frac{2(1-\beta^{\frac12}C)\lmin}{\beta}\,\one \comma \end{equation} % where the lowest eigenvalue $\lmin := \inf\sigma(H^\dprime(0)) > 0$ of the Hessian of $H$ at $x=0$ is strictly positive, for small $\alpha$. It turns out that Theorem~\ref{thm-II-0-1} is not precise enough for the derivation of the correlation asymptotics. What we really need is the following bound whose proof, sketched below, is based on the constructions in \cite{BachMoeller2003a}. % \begin{theorem} \label{thm-II-0-2} Assume Hypotheses~\ref{H-1}, \ref{H-2} and \ref{H-3}. There exist universal constants $C < \infty$, $\alpha_0 > 0$, and $\beta_0 < \infty$ such that, for all $0 \leq \alpha \leq \alpha_0$, $\beta > \beta_0$, and all $i,j \in \Lambda$, we have % \begin{equation} \label{eq-II-0-2.1} \Big\| \big( \tA_i + \beta^{-1} \big)^{-1/2} \: \big\{ W_{H,Q}^{(1)} \big\}_{ij} \: \big( \tA_j + \beta^{-1} \big)^{-1/2} \Big\| \ \leq \ \frac{C}{\beta^{1/2}} \Big( \delta_{ij} \, + \, \alpha \, a_{ij} \Big) \comma \end{equation} % where the norm is for bounded operators on $\cH^{(0)}$. \end{theorem} % For the derivation of Theorem~\ref{thm-II-0-2}, we use the matrix $\us := (s_{ij})_{i,j \in \Lambda}$, defined in (\ref{eq-I-1-5.9}), and the functions % \begin{equation} \label{eq-II-0-6} J_j[\us] \ := \ \Tho_j^2 \, |\gi_j'| \: + \: \sum_{k (\neq j)} s_{jk} \, \Th_k^2 \, |g_k'| \period \end{equation} % The result of the estimates in \cite[Eqs.~(II.14)--(II.27)]{BachMoeller2003a} can be rephrased as follows, % \begin{lemma} \label{lem-II-0-3} Assume Hypotheses~\ref{H-1}, \ref{H-2} and \ref{H-3}. There exist universal constants $C < \infty$, $\alpha_0 >0$, and $\beta_0 < \infty$ such that, for all $0 \leq \alpha \leq \alpha_0$, all $\beta > \beta_0$, and all $j \in \Lambda$, we have % \begin{equation} \label{eq-II-0-6.1} J_j[\us] \ \leq \ \frac{C}{\beta^{1/2}} \, \big( \tA_j + \beta^{-1} \big) \comma \end{equation} % in the sense of quadratic forms. \end{lemma} % Next, we recall the statement of \cite[Lemma~II.5]{BachMoeller2003a}. % \begin{lemma} \label{lem-II-0-4} Assume Hypotheses~\ref{H-1}, \ref{H-2} and \ref{H-3}. For some universal constant $C < \infty$, we have % \begin{eqnarray} \label{eq-II-2-5.1} \big| \Tho_j^2 (x_j) \Gio_{jj}''(x) \: - \: H_{jj}''(0) \big| & \leq & C \; J_j[\us](x) \comma \\ \label{eq-II-2-5.2} \big| \Th_j^2 (x_j) G_{jj}''(x) \: - \: H_{jj}''(0) \big| & \leq & C \; J_j[\us](x) \period \end{eqnarray} % \end{lemma} % \Proof In \cite[Lemma~II.5]{BachMoeller2003a}, the left sides of (\ref{eq-II-2-5.1}) and (\ref{eq-II-2-5.2}) are bounded by $C' J_j[\ua](x)$. To bound this quantity by $C J_j[\us](x)$, we additionally observe that $a_{ij} \leq C_a s_{ij}$, which implies $J_j[\ua](x) \leq (1+C_a) J_j[\us](x)$, for all $x \in \RR^\Lambda$. \QED Note that Lemma~\ref{lem-II-0-4} implies that % \begin{equation} \label{eq-II-0-2-5.01} \big\| J_j[\us]^{-1/2} \: \big\{ \Wdiag^{(1)} \big\}_{jj} \: J_j[\us]^{-1/2} \big\| \ \leq \ \frac{C}{\beta^{1/2}} \comma \end{equation} % for some universal $C < \infty$ and all $j \in \Lambda$. Next, we consider $\Woff^{(1)}$. % \begin{lemma} \label{lem-II-0-5} Assume Hypotheses~\ref{H-1}, \ref{H-2} and \ref{H-3}. For some universal constant $C < \infty$ and all $i,j \in \Lambda$, we have % \begin{equation} \label{eq-II-0-2-5.1} \big\| J_i[\us]^{-1/2} \: \big\{ \Woff^{(1)} \big\}_{ij} \: J_j[\us]^{-1/2} \big\| \ \leq \ \frac{C \, \alpha }{\beta^{1/2}} \, a_{ij} \period \end{equation} % \end{lemma} % \Proof As in \cite[Eq.~(II.36)]{BachMoeller2003a}, we have % \begin{eqnarray} \label{eq-II-1-5.1} \lefteqn{ \big| \Th_i \Th_j \, \partial_{12}^2 w_{ij}(x) \, - \, \partial_{12}^2 w_{ij}(0) \big| } % \\ \nonumber & \leq & C \, a_{ij} \, \Big( \Th_i |g_i'| + \Th_j |g_j'| + \Th_i \Th_j |g_i'|^{1/2} |g_j'|^{1/2} \Big) \comma \end{eqnarray} % for all $i \neq j$. Now, we use % \begin{equation} \label{eq-II-0-1-5.2} a_{ij} \ = \ a_{ij} \, s_{ij}^{1/2} \ = \ a_{ij} \, s_{ij}\comma \end{equation} % $s_{ij} = s_{ji}$, and $|g_j^\prime|\leq c_g^{-1}|\hat{g}_j^\prime|$ (cf. \cite[Eq.~(1.30)]{BachMoeller2003a}), which yield % \begin{eqnarray} \label{eq-II-1-5.3} a_{ij} \, \Th_i |g_i'| & = & a_{ij} \, \big( |g_i'| \big)^{1/2} \, \big( s_{ji} \Th_i^2 |g_i'| \big)^{1/2} \ \leq \ c_g^{-\frac12}\, a_{ij} \, J_i[\us]^{1/2} \, J_j[\us]^{1/2} \comma \hspace{6mm} \\ \label{eq-II-1-5.4} a_{ij} \, \Th_j |g_j'| & = & a_{ij} \, \big( s_{ij} \Th_j^2 |g_j'| \big)^{1/2} \, \big( |g_j'| \big)^{1/2} \ \leq \ c_g^{-\frac12}\, a_{ij} \, J_i[\us]^{1/2} \, J_j[\us]^{1/2} \comma \hspace{6mm} \end{eqnarray} % and % \begin{eqnarray} \label{eq-II-1-5.5} a_{ij} \, \Th_i \Th_j |g_i'|^{1/2} |g_j'|^{1/2} & = & a_{ij} \, \big( s_{ij} \Th_j^2 |g_j'| \big)^{1/2} \, \big( s_{ji} \Th_i^2 |g_i'| \big)^{1/2} \nonumber \\ & \leq & c_g^{-\frac12}\, a_{ij} \, J_i[\us]^{1/2} \, J_j[\us]^{1/2} \period \end{eqnarray} \QED \noindent\emph{Proof of Theorem~\ref{thm-II-0-2}: } The asserted estimate (\ref{eq-II-0-2.1}) follows directly from combining Eq.~(\ref{eq-II-0-2-5.01}), Lemma~\ref{lem-II-0-5}, and Lemma~\ref{lem-II-0-3}. \QED \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \secct{Greens Function Estimates} \label{PartII} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section, we study the stability of the Greens function $(\one- \alpha T)^{-1}$ under two types of perturbations: $T \to e^g T$ and $T\to T + gY$, where $g$ is a small parameter, $Y$ is a matrix with no definite sign, and $T$ is the real symmetric $\Lambda\times\Lambda$ transition matrix defined in Eq.~(\ref{Tij}), % \begin{equation} \label{eq-II-a-2} \forall \, i,j \in \Lambda \; : \ \ \ \ \ \ \ T_{ij} \ := \ \frac{-\partial_{12}^2 w_{ij}(0,0)} {\sqrt{H_{ii}^\dprime(0)\,H_{jj}^\dprime(0)}} \period \end{equation} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Stability under Perturbations of the Form $T\to T + gY$} \label{SecII.1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Our first goal is the derivation of estimates on the matrix elements of resolvent of the form % \begin{equation} \label{eq-II-a-1} \Res[\alpha T, g , \vth] \ := \ \Big( \one \, - \, \alpha T \, - \, g \big(\one - \vth \alpha T \big)^{-1} \Big)^{-1} \comma \end{equation} % where $1/2 < \vth < 1$. Note that $(\one - \vth \alpha T )^{-1}$ decays faster than $(\one - \alpha T )^{-1}$. Hence we expect the decay of $\Res[\alpha T, g , \vth]$ to be dominated by the decay of $(\one - \alpha T )^{-1}$. We quantify this by deriving nontrivial upper and lower bounds on the matrix elements of $\Res[\alpha T, g ,\vth]$. % \begin{lemma} \label{lem-II-a-1} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and that $0 \leq \alpha C_a \leq 1/4$. Let $1/2 < \vth < 1$, and set $\kappa(\vth) := 12 (1-\vth)^{-3}$. Then, for all $|g| \leq (1-\vth)^3/24$, % \begin{eqnarray} \label{eq-II-a-2.1} \lefteqn{ \big(1 \, - \, \kappa(\vth) \, |g| \big) \, \Big\{ \Big( \one \, - \, \frac{\alpha}{1+2(1-\vth)^{-1}|g|} \, T \Big)^{-1} \Big\}_{ij} \ \leq \ \Res[\alpha T, g , \vth]_{ij} } \nonumber \\ & \hspace{15mm} \leq & \big(1 \, + \, \kappa(\vth) \, |g| \big) \, \Big\{ \Big( \one \, - \, \frac{\alpha}{1-2(1-\vth)^{-1}|g|} \, T \Big)^{-1} \Big\}_{ij} \comma \end{eqnarray} % for all $i,j \in \Lambda$. \end{lemma} % \Proof For the proof of (\ref{eq-II-a-2.1}), it is convenient to replace $\alpha T$ by a complex variable and consider the complex rational function $f(z) := \Res[z , g , \vth]$ given by % \begin{eqnarray} \label{eq-II-a-3} f(z) & := & \big( 1 \, - \, z \, - \, g (1 - \vth z)^{-1} \big)^{-1} \ = \ \frac{1 - \vth z}{\vth} \, \Big( z^2 \, - \, \frac{1+\vth}{\vth} z \, + \, \frac{1-g}{\vth} \Big)^{-1} \nonumber \\ & = & \Big( \frac{1 - \vth z}{\vth} \Big) \, \Big( \frac{1}{\zeta_+ - z} \Big) \, \Big( \frac{1}{\zeta_- - z} \Big) \comma \end{eqnarray} % where % \begin{equation} \label{eq-II-a-4} \zeta_\pm \ = \ \frac{1}{2\vth} \, \Big( 1+\vth \, \pm \, (1-\vth) \, \sqrt{1 \, + \, 4 \vth (1-\vth)^{-2} \, g } \Big) \period \end{equation} % Note that the condition $|g| \leq (1-\vth)^3/24 \leq (1-\vth)^2/(8 \vth)$ insures that $\zeta_\pm \in \RR$ are real and that % \begin{equation} \label{eq-II-a-5} 1 \, - \, 4 \vth (1-\vth)^{-2} \, |g| \ \leq \ \sqrt{1 + 4 \vth (1-\vth)^{-2} \, g } \ \leq \ 1 \, + \, 4 \vth (1-\vth)^{-2} \, |g| \comma \end{equation} % which, together with $|g| \leq (1-\vth)^3/24 \leq (1-\vth)/4$, implies the following bounds, % \begin{eqnarray} \label{eq-II-a-6} & \frac{1-\vth}{\vth} \, \Big( 1 \, - \, \frac{4 \vth |g|}{(1-\vth)^2} \Big) \ \leq \ \zeta_+ - \zeta_- \ \leq \ \frac{1-\vth}{\vth} \, \Big( 1 \, + \, \frac{4 \vth |g|}{(1-\vth)^2} \Big) \comma & \\ \label{eq-II-a-7} & \frac{1}{\vth} \, - \, \frac{2 \vth |g|}{(1-\vth)^2} \ \leq \ \zeta_+ \ \leq \ \frac{1}{\vth} \, + \, \frac{2 \vth |g|}{(1-\vth)^2} \comma & \\ \label{eq-II-a-8} & \frac{1}{2} \ \leq \ 1 \, - \, \frac{2 |g|}{1-\vth} \ \leq \ \zeta_- \ \leq \ 1 \, + \, \frac{2 |g|}{1-\vth} \leq \ \frac{3}{2} \comma \\ \label{eq-II-a-9} & \frac{1}{\vth} \, - \, 1 \, - \, \frac{4 |g|}{\vth (1-\vth)} \ \leq \ \frac{1 - \vth \zeta_-}{\vth \zeta_-} \ \leq \ \frac{1}{\vth} \, - \, 1 \, + \, \frac{4 |g|}{\vth (1-\vth)} \comma \\ \label{eq-II-a-9a} & |1 - \vth \zeta_+| \ \leq \ \frac{2 \vth^2 |g|}{(1-\vth)^2} \ \leq \ \frac{1}{12} \period \end{eqnarray} % After some algebra, we arrive at % \begin{equation} \label{eq-II-a-10} (\zeta_+ - \zeta_-) \, f(z) \ = \ \Big( \frac{1 - \vth \zeta_-}{\vth \zeta_-} \Big) \, \Big( 1 - \frac{z}{\zeta_-} \Big)^{-1} \; - \; \Big( \frac{1 - \vth \zeta_+}{\vth \zeta_+} \Big) \, \Big( 1 - \frac{z}{\zeta_+} \Big)^{-1} \period \end{equation} % Note that (\ref{eq-II-a-10}) yields an identity for the matrix $\Res[\alpha T, g , \vth]$ by substituting the matrix $\alpha T$ for $z$ on its right side, invoking functional calculus (or comparing norm-convergent power series). We now bound the matrix elements of the resolvents $(1 - \zeta_\pm^{-1} \alpha T)^{-1}$ which then appear. Since $\zeta_+ > \zeta_-$, by (\ref{eq-II-a-6}), we have, for $\alpha >0$ sufficiently small, that % \begin{equation} \label{eq-II-a-11} 0 \ \leq \ \Big\{ \big( 1 \, - \, \zeta_+^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \ \leq \ \Big\{ \big( 1 \, - \, \zeta_-^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \end{equation} % from which we deduce that % \begin{equation} \label{eq-II-a-12} \Big| \Big( \frac{1 - \vth \zeta_+}{\vth \zeta_+} \Big) \, \Big\{ \big( 1 \, - \, \zeta_+^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \Big| \ \leq \ \frac{4 \vth^2 |g|}{(1-\vth)^2} \, \Big\{ \big( 1 \, - \, \zeta_-^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \end{equation} % additionally taking (\ref{eq-II-a-7}) and (\ref{eq-II-a-9a}) into account. Inserting Eqs.~(\ref{eq-II-a-12}) and (\ref{eq-II-a-9}) into (\ref{eq-II-a-10}), we conclude that % \begin{eqnarray} \label{eq-II-a-13} \lefteqn{ (\zeta_+ - \zeta_-)^{-1} \, \Big( \frac{1}{\vth} \, - \, 1 \, - \, \frac{8|g|}{\vth (1-\vth)^2} \Big) \, \Big\{ \big( 1 \, - \, \zeta_-^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \ \leq \ \Res[\alpha T, g , \vth]_{ij} } \nonumber \\ & \hspace{7mm} \leq & (\zeta_+ - \zeta_-)^{-1} \, \Big( \frac{1}{\vth} \, - \, 1 \, + \, \frac{8|g|}{\vth (1-\vth)^2} \Big) \, \Big\{ \big( 1 \, - \, \zeta_-^{-1} \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \hspace{10mm} \end{eqnarray} % which, together with (\ref{eq-II-a-6}), yields the claim. \QED Lemma~\ref{lem-II-a-1} is an important input for the proof of the following lemma, because it insures the positivity of the matrix elements of $\Res[\alpha T, g , \vth]$. % \begin{lemma} \label{lem-II-a-2} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and that $0 \leq \alpha C_a \leq 1/4$. Let $1/2 < \vth < 1$, set $\kappa(\vth) := 12 (1-\vth)^{-3}$, and assume $|g| \leq (1-\vth)^3/24$. Suppose that $Y$ is a real $\Lambda \times \Lambda$ matrix obeying % \begin{equation} \label{eq-II-a-14} | Y_{ij} | \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{equation} % Then $\one - \alpha T - gY$ is invertible, and the matrix elements of its resolvent fulfill the following estimates. % \begin{equation} \label{eq-II-a-15} \Res[\alpha T, -|g| , \vth]_{ij} \ \leq \ \Big\{ \big( \one \, - \, \alpha T \, - \, g Y \big)^{-1} \Big\}_{ij} \ \leq \ \Res[\alpha T, |g| , \vth]_{ij} \comma \end{equation} % for all $i,j \in \Lambda$. \end{lemma} % \Proof First, we expand the inverse of $\one - \alpha T - gY$ in a Neumann series and use the upper bound on $Y_{ij}$ in (\ref{eq-II-a-14}) to obtain the upper bound asserted in (\ref{eq-II-a-15}), % \begin{eqnarray} \label{eq-II-a-16} \lefteqn{ \Big\{ \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} \ = \ \sum_{n=0}^\infty \Big\{ \big( \one - \alpha T \big)^{-1} \, \Big[ g Y \, \big( \one - \alpha T \big)^{-1} \Big]^n \Big\}_{ij} } \nonumber \\ & \hspace{20mm} \leq & \sum_{n=0}^\infty \Big\{ \big( \one - \alpha T \big)^{-1} \, \Big[ g \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \, \big( \one - \alpha T \big)^{-1} \Big]^n \Big\}_{ij} \nonumber \\ & \hspace{20mm} = & \Res[\alpha T, |g| , \vth]_{ij} \period \end{eqnarray} % The lower bound in (\ref{eq-II-a-15}) follows similarly from a Neumann series, % \begin{eqnarray} \label{eq-II-a-17} \lefteqn{ \Big\{ \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} } \nonumber \\ & = & \sum_{n=0}^\infty \Big\{ \Res[\alpha T, -|g| , \vth] \, \Big[ g \Big( \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \, - \, Y \Big) \, \Res[\alpha T, -|g| , \vth] \Big]^n \Big\}_{ij} \nonumber \\ & \geq & \Res[\alpha T, -|g| , \vth]_{ij} \comma \end{eqnarray} % retaining from the series only the term corresponding to $n=0$. Here we use the positivity of the matrix elements of $(\one - \vth \alpha T)^{-1} - Y$, following from (\ref{eq-II-a-14}), as well as, the positivity of the matrix elements of $\Res[\alpha T, -|g| , \vth]$, which follows from Lemma~\ref{lem-II-a-2}. \QED Putting together Lemma~\ref{lem-II-a-2} and Lemma~\ref{lem-II-a-1}, we arrive at the first main result of this subsection. % \begin{theorem} \label{thm-II-a-3} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and that $0 \leq \alpha C_a \leq 1/4$. Let $1/2 < \vth < 1$, set $\kappa(\vth) := 12 (1-\vth)^{-3}$, and assume $|g| \leq (1-\vth)^3/24$. Suppose that $Y$ is a real $\Lambda \times \Lambda$ matrix obeying % \begin{equation} \label{eq-II-a-18} | Y_{ij} | \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{equation} % Then $\one - \alpha T - gY$ is invertible, and the matrix elements of its resolvent fulfill the following estimates. % \begin{eqnarray} \label{eq-II-a-19} \lefteqn{ \big(1 - \kappa(\vth) |g| \big) \Big\{ \Big( \one - \frac{\alpha \, T}{1+2(1-\vth)^{-1}|g|} \Big)^{-1} \Big\}_{ij} \ \leq \ \Big\{ \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} } \nonumber \\ & \hspace{20mm} \leq & \big(1 + \kappa(\vth) |g| \big) \Big\{ \Big( \one - \frac{\alpha \, T}{1-2(1-\vth)^{-1}|g|} \Big)^{-1} \Big\}_{ij} \comma \hspace{10mm} \end{eqnarray} % for all $i,j \in \Lambda$. \end{theorem} % For the derivation of the correlation asymptotics we actually need a more refined version of the Neumann series expansion in (\ref{eq-II-a-17}) which yields the following estimate. % \begin{theorem} \label{thm-II-a-4} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and that $0 \leq \alpha C_a \leq 1/4$. Let $1/2 < \vth < 1$, set $\kappa(\vth) := 12 (1-\vth)^{-3}$, and assume $|g| \leq (1-\vth)^3/48$. Suppose that $S$ and $Y$ are real $\Lambda \times \Lambda$ matrices obeying % \begin{equation} \label{eq-II-a-20} | S_{ij} | \, , \: | Y_{ij} | \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{equation} % Then we have the following estimate, % \begin{equation} \label{eq-II-a-21} \Big| \Big\{ S \, \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} \Big| \ \leq \ |g|^{-1} \, \Big\{ \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} \comma \end{equation} % for all $i,j \in \Lambda$. \end{theorem} % \Proof Clearly, we may assume without loss of generality that $g \geq 0$. Due to Assumption~(\ref{eq-II-a-20}), we have that % \begin{equation} \label{eq-II-a-22} | S_{ij} | \ \leq \ X_{ij} \ := \ \Big\{ 2 \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \: - \: Y \Big\}_{ij} \period \end{equation} % Moreover, $(\one - \alpha T - g Y )^{-1}$ has nonnegative matrix elements, by Theorem~\ref{thm-II-a-3} and Lemma~\ref{lem-II-a-2}. The latter also implies that $\Res[\alpha T, -2g, \vth]_{ij} \geq \delta_{ij}$. Therefore, % \begin{eqnarray} \label{eq-II-a-23} \lefteqn{ \Big| \Big\{ S \, \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} \Big| \ \leq \ \Big\{ X \, \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} } \nonumber \\ & = & \Big\{ X \, \big( \one - \alpha T + 2 g ( 1 - \vth \alpha T)^{-1} - g X \big)^{-1} \Big\}_{ij} \nonumber \\ & = & \frac{1}{g} \, \Big\{ \sum_{n=1}^\infty \big( gX \, \Res[\alpha T, -2g, \vth] \big)^n \Big\}_{ij} \nonumber \\ & \leq & \frac{1}{g} \, \Big\{ \sum_{n=0}^\infty \Res[\alpha T, -2g, \vth] \, \big( gX \, \Res[\alpha T, -2g, \vth] \big)^n \Big\}_{ij} \nonumber \\ & = & \frac{1}{g} \, \Big\{ \big( \one - \alpha T - g Y \big)^{-1} \Big\}_{ij} \comma \end{eqnarray} % proving the claim. \QED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Stability under Perturbations of the Form $T\to e^g T$} \label{SecII.2} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this subsection we study stability of the correlations under scaling of the transition matrix $T$ defined in (\ref{Tij}) (see also (\ref{eq-II-a-2})). We begin with a log-convexity estimate, which is similar to \cite[(VI.42)]{BachJeckoSjoestrand2000}. % \begin{lemma}\label{lem-III-logcon} Suppose Hypothesis~\ref{H-2} and that $\alpha C_a \leq 1/4$. Then there exists a universal constant $g_0>0$ such that, for $g_1, g_2 \in [-g_0, g_0]$, $0 \leq \kappa \leq 1$,and all $i,j \in \Lambda$, we have % \begin{eqnarray}\label{logconvexity} \lefteqn{ \big\{ \big( \one \, - \, e^{\kappa g_1 + (1-\kappa) g_2} \, \alpha T \big)^{-1} \big\}_{ij} \ \leq \ } \\ \nonumber & & \Big( \big\{ \big( \one \, - \, e^{g_1} \, \alpha T \big)^{-1} \big\}_{ij} \Big)^\kappa \; \Big( \big\{ \big( \one \, - \, e^{g_2} \, \alpha T \big)^{-1} \big\}_{ij} \Big)^{1-\kappa} \period \end{eqnarray} % \end{lemma} % \Proof Note that all resolvent can be expanded in norm-convergent Neumann series thanks to the condition $\alpha C_a \leq 1/4$. Then we obtain % \begin{eqnarray} \label{eq-II-b-1} \big\{ \big( \one \, - \, e^{\kappa g_1 + (1-\kappa) g_2} \, \alpha T \big)^{-1} \big\}_{ij} & = & \sum_{n=0}^\infty \big( e^{g_1 n} \, \{T^n\}_{ij} \big)^\kappa \big( e^{g_2 n} \, \{T^n\}_{ij} \big)^{1-\kappa} \\ \nonumber & \leq & \Big( \sum_{n=0}^\infty e^{g_1 n} \, \{T^n\}_{ij} \Big)^\kappa \: \Big( \sum_{n=0}^\infty e^{g_2 n} \, \{T^n\}_{ij} \Big)^{1-\kappa} \comma \end{eqnarray} % using H{\"o}lder's inequality. \QED We also use the following elementary result % \begin{lemma} \label{lem-III-Jensen} Let $f:[0,\infty)\to \RR$ with $f(0) = 0$. If $f$ is convex then % \begin{equation} \label{JensenConvex} \forall a,b \geq 0 \, : \ \ f(a\,+\,b)\ \geq \ f(a) \; + \; f(b) \comma \end{equation} % and if $f$ is concave then % \begin{equation} \label{JensenConcave} \forall a,b \geq 0 \, : \ \ f(a\,+\,b)\ \leq \ f(a) \; + \; f(b) \period \end{equation} % \end{lemma} % \Proof We consider only the convex case. The concave case follows from replacing $f$ by $-f$. We can assume without loss of generality that $0\leq a \leq b$ and that $b > 0$. Writing % \begin{equation} \label{eq-II-b-2} a \ = \ \frac{a}{b} \cdot b \: + \: \frac{b-a}{b} \cdot 0 \hspace{4mm} \mbox{and} \hspace{4mm} b \ = \ \frac{b}{a+b}\cdot (a+b) \: + \: \frac{a}{a+b}\cdot 0 \comma \end{equation} % the convexity of $f$ implies that % \begin{equation} \label{eq-II-b-3} f(a) \: + \: f(b) \ \leq \ \Big( 1 \, + \, \frac{a}{b} \Big) \, f(b) \ \leq \ \Big( 1 \, + \, \frac{a}{b} \Big) \, \frac{b}{a+b} \, f(a+b) \ = \ f(a+b) \period \end{equation} % This completes the proof. \QED We now turn to the main result of this subsection % \begin{theorem} Suppose Hypothesis~\ref{H-2} and $\alpha C_a \leq 1/4$. Then there exist universal constants $g_0 > 0$ and $C < \infty$, such that, for $0 \leq g \leq g_0$, we have % \begin{eqnarray}\label{eq-III-pert-1} \big\{ ( \one - e^{g} \alpha T )^{-1} \big\}_{ij} & \leq & (1 + C g) \, \big\{ ( \one - \alpha T )^{-1} \big\}_{ij}^{1-Cg} \comma \\ \label{eq-III-pert-2} (1 - C g) \, \big\{ ( \one - \alpha T )^{-1} \big\}_{ij}^{1+Cg} & \leq & \big\{ ( \one - e^{-g} \alpha T )^{-1} \big\}_{ij} \period \end{eqnarray} % Assume additionally that either $T_{ij}=0$ or $T_{ij} \geq \sigma_T$, for some universal constant $\sigma_T > 0$. Then there exist universal constants $g_0 > 0$ and $C_1 < \infty$, such that, for $0 \leq g \leq g_0$, we have % \begin{eqnarray}\label{eq-III-pert-3} \big\{ ( \one - \alpha T )^{-1} \big\}_{ij}^{1- \ln[1/(\alpha \sigma_T)] g} & \leq & \big\{ ( \one - e^{g} \alpha T )^{-1} \big\}_{ij} \comma \\ \label{eq-III-pert-4} \big\{ ( \one - e^{-g} \alpha T )^{-1} \big\}_{ij} & \leq & \big\{ ( \one - \alpha T )^{-1} \big\}_{ij}^{1+ \ln[1/(\alpha \sigma_T)] g} \period \end{eqnarray} % \end{theorem} % \textbf{Remark.} In applications we will use this theorem with $e^{g}$ replaced by $1+{g}$ and $e^{-{g}}$ replaced by $1-{g}$. The corresponding estimates are clearly equivalent (but with different $g_0$'s). Also notice that $\alpha \sigma_T \leq \alpha C_a \leq 1/4$. \Proof An application of Lemma~\ref{lem-III-logcon} with $\kappa := g/g_0$, $g_1 := g_0$ (so that $\kappa g_1 = g$), and $g_2 = 0$ yields % \begin{eqnarray} \label{eq-II-b-4} \big\{ (\one - e^g \alpha T )^{-1} \big\}_{ij} & \leq & \Big( \big\{ (\one - e^{g_0} \alpha T )^{-1} \big\}_{ij} \Big)^{g/g_0} \: \Big( \big\{ (\one - \alpha T )^{-1} \big\}_{ij} \Big)^{1 - g/g_0} \nonumber \\ & \leq & \big( 1 - e^{g_0} \alpha C_a )^{-g/g_0} \: \Big( \big\{ (\one - \alpha T )^{-1} \big\}_{ij} \Big)^{1 - g/g_0} \comma \end{eqnarray} % and hence (\ref{eq-III-pert-1}). Eqn.~(\ref{eq-II-b-4}) also implies (\ref{eq-III-pert-2}) upon the substitution $\alpha \to \alpha' := e^g \alpha$. To prove (\ref{eq-III-pert-4}), we consider $b = (i,j) \in \cB_a$, i.e., $T_{ij} \neq 0$. Then, by assumption, $\alpha T_{ij} \geq \alpha \sigma_T$, and we conclude that % \begin{equation} \label{eq-II-b-5} e^{-g} \, \alpha \, T_{ij} \ \leq \ (\alpha T_{ij})^{1 + \ln[1/(\alpha \sigma_T)] g} \period \end{equation} % Next we (repeatedly) use Lemma~\ref{lem-III-Jensen} with the convex function $t \mapsto t^{1+ \ln[1/(\alpha \sigma_T)] g}$, which vanishes at $0$, where $\ln[1/(\alpha \sigma_T)] > 0$. An expansion of the resolvent matrix elements in terms of paths thus gives the desired estimate, % \begin{eqnarray} \label{eq-II-b-6} \lefteqn{ \big\{ (\one - e^{-{g}} \alpha T)^{-1} \big\}_{ij} } \\ \nonumber & = & \delta_{ij} \, + \, \sum_{\gamma \in \Gamma(i,j)} \prod_{b \in \gamma} e^{-{g}}\, \alpha \, T_b \ \leq \ \delta_{ij} \, + \, \sum_{\gamma \in \Gamma(i,j)} \Big( \prod_{b \in \gamma} \alpha \, T_b \Big)^{1+\ln[1/(\alpha \sigma_T)] g} \\ \nonumber & \leq & \Big( \delta_{ij} \, + \, \sum_{\gamma \in \Gamma(i,j)} \prod_{b \in \gamma} \alpha \, T_b \Big)^{1+\ln[1/(\alpha \sigma_T)] g} \ = \ \Big( \big\{ (\one - \alpha T)^{-1} \big\}_{ij} \Big)^{1+\ln[1/(\alpha \sigma_T)] g} \comma \end{eqnarray} % additionally taking into account that $i \neq j$, since $T_{ij} \neq 0$. The remaining inequality (\ref{eq-III-pert-3}) is proved analogously, using the concave function $t \mapsto t^{1-C_1 g}$. \QED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Greens Functions and Associated Metrics} \label{SecII.3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % In this section, we study resolvents $(\one - \alpha T)^{-1}$, where the transition matrix $T$ is defined in (\ref{Tij}), and $\alpha \leq 0$ is sufficiently small such that $\alpha C_a \leq 1/4$. Since $(\one - \alpha T)^{-1}$ has positive matrix elements, so % \begin{equation} \label{eq-II-c-2} \exp[ -\rho(i,j) ] \ := \ \frac{ \big\{ (\one - \alpha T)^{-1} \big\}_{ij} }% { \big\{ (\one - \alpha T)^{-1} \big\}_{ii}^{1/2} \: \big\{ (\one - \alpha T)^{-1} \big\}_{jj}^{1/2} } \end{equation} % defines a function $\rho: \Lambda \times \Lambda \to [0,\infty]$. It is a remarkable fact that this function is actually a metric on $\Lambda$. % \begin{theorem} \label{thm-II-c-1} Assume Hypotheses~\ref{H-1}, \ref{H-2}, \ref{H-3}, and that $0 \leq \alpha C_a \leq 1/4$. Then $\rho: \Lambda \times \Lambda \to [0,\infty]$ is a metric on $\Lambda$. \end{theorem} % \Proof For the proof, we denote $\Res := (\one - \alpha T)^{-1}$. Its matrix elements are nonnegative, and $\Res$ is positive, as a quadratic form. The symmetry $\rho(i,j) = \rho(j,i)$ is trivial, since $T$ and $Y$ are symmetric. Since $\Res$ is positive, as a quadratic form, % \begin{equation}\label{eq-III-rhoT-1} \Res_{ij} \ = \ \la \Res^{1/2} e_i | \Res^{1/2} e_j \ra \ \leq \ \| \Res^{1/2} e_i \| \; \| \Res^{1/2} e_j \| \ = \ \Res_{ii} \, \Res_{jj} \comma \end{equation} % with equality iff $\Res^{1/2} e_i$ and $\Res^{1/2} e_j$ are parallel, which is equivalent to $e_i$ and $e_j$ being parallel, i.e., $i=j$. Therefore, $\rho \geq 0$, and $\rho(i,j) = 0$ iff $i=j$. As for the triangle inequality % \begin{equation}\label{triangle1} \rho(i,j) \ \leq \ \rho(i,k) \: + \: \rho(k,j) \comma \end{equation} % which is equivalent to % \begin{equation}\label{triangle2} \Res_{ik} \, \Res_{kj} \ \leq \ \Res_{ij} \, \Res_{kk} \comma \end{equation} % we note that it is sufficient to consider the case where $i, j, k \in \Lambda$ are three different points. Expanding in a Neumann series, we see that $T_{ij} \geq 0$ implies that $\Res_{kk} \geq \{\one\}_{kk} = 1$. Moreover, we may expand $\Res_{ij}$ as a sum over all paths $\gamma$ from $i$ to $j$, % \begin{equation} \label{triangle8} \Res_{ij} \ = \ \delta_{ij} \: + \: \sum_{\gamma \in \Gamma(i,j)} \prod_{b \in \gamma} T_b \period \end{equation} % We recall that a path $\gamma$ is a (ordered) nonempty, finite collection of nearest-neighbour bonds of the form $\gamma = \{(i_0,i_1), (i_1,i_2), \dots, (i_{n-1},i_n)\} \subseteq \cB_a$, with $i_0 = i$ and $i_n = j$. The collection of all paths from $i$ to $j$ is denoted $\Gamma(i,j)$. We further introduce the set $\Gamma'(i,j) \subset \Gamma(i,j)$ of paths from $i$ to $j$ which do not visit $j$ in between. So, if $\gamma = \{(i_0,i_1), (i_1,i_2), \dots, (i_{n-1},i_n)\} \in \Gamma'(i,j)$, then $i_0 = i$, $i_n = j$, and $i_1 \neq j, \ldots, i_{n-1} \neq j$. We define the concatenation $\circ : \Gamma(i,j) \times \Gamma(j,k) \to \Gamma(i,k)$ of two paths in the obvious way, i.e., $\gamma_1 \circ \gamma_2 := (b_1, \ldots, b_{m+n})$, for $\gamma_1 = (b_1, \ldots, b_m) \in \Gamma(i,j)$ and $\gamma_2 = (b_{m+1}, \ldots, b_{m+n}) \in \Gamma(j,k)$. Given two points $i,j \in \Lambda$, we observe the following disjoint decomposition identity, % \begin{equation} \label{triangle8d} \Gamma(i,j) \ = \ \Gamma'(i,j) \: \cup \: \Gamma'(i,j) \circ \Gamma(j,j) \period \end{equation} % Thus, defining a Greens function $\Res'$ by % \begin{equation} \label{triangle8e} \Res_{ij}' \ := \ \sum_{\gamma' \in \Gamma'(i,j)} \prod_{b' \in \gamma'} T_{b'} \comma \end{equation} % we have the following identity % \begin{equation} \label{triangle8g} \Res_{ik} \ = \ \delta_{ik} \: + \: \Res_{ik}' \: + \: \Res_{ik}' \, ( \Res_{kk} - 1 ) \ = \ \delta_{ik} \: + \: \Res_{ik}' \, \Res_{kk} \comma \end{equation} % for all $i,k \in \Lambda$. Now suppose that $i,j,k \in \Lambda$ are three different point in the lattice. Then, the concatenation $\circ$, viewed as a map % \begin{equation} \label{triangle8i} \circ: \; \Gamma'(i,k) \times \Gamma(k,j) \to \Gamma(i,j) \comma \end{equation} % is injective. This implies that % \begin{equation} \label{triangle8j} \Res_{ik}' \, \Res_{kj} \ \leq \ \Res_{ij} \period \end{equation} % Therefore, using (\ref{triangle8g}), we observe that % \begin{equation} \label{triangle8k} \Res_{ik} \, \Res_{kj} \ = \ \Res_{ik}' \, \Res_{kk} \, \Res_{kj} \ \leq \ \Res_{ij} \, \Res_{kk} \comma \end{equation} % which proves the triangle inequality (\ref{triangle2}). \QED \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \secct{Spectral Separation and Improved Decay} \label{PartIII} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The Feshbach projection method} \label{SecIII.1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Our analysis of the correlation asymptotics is built upon the Feshbach map associated to the projection % \begin{equation} \label{Feshbachprojection} P \ := \ p \otimes \one \comma \hspace{4mm} \mbox{where} \hspace{4mm} p \ := \ \cZ_\beta^{-1} \; \big| e^{-\beta H} \big\ra \big\la e^{-\beta H} \big| \end{equation} % is the rank-1 projection onto $e^{-\beta H} \in L^2(\RR^\Lambda)$. We write $\bp := \one - p$ and $\bP := \one - P = \bp \otimes \one$. The Feshbach operator $\cF_P(\Delta_{H,Q})$ is defined to be the image of $\Delta_{H,Q}$ under the Feshbach map, see \cite{BachFroehlichSigal1998b,DerezinskiJaksic2001,BachJeckoSjoestrand2000,BachChenFroehlichSigal2003} for a detailed description of the Feshbach map and its properties. The Feshbach operator $\cF_P := \cF_P(\Delta_{H,Q}): \Ran P \to \Ran P$ is now defined as % \begin{equation}\label{Feshbachop} \cF_P \ = \ P \, \Delta_{H,Q}^{(1)} \, P \; - \; \DPP^*\,\big(\bDelta_{H,Q}^{(1)}\big)^{-1}\, \DPP \comma \end{equation} % where we write $\ol{\Delta}_{H,Q}^{(1)} := \bP \Delta_{H,Q}^{(1)} \bP$ and use the overlap operator $\DPP: \Ran P\to \Ran\bP$ defined by % \begin{equation} \DPP \ := \ \bP\,\Delta_{H,Q}^{(1)} \, P \period \end{equation} % Note that $\ol{\Delta}_{H,Q}^{(1)} \geq (2-C\beta^{-1/2})\beta^{-1} \one > 0$ is bounded invertible on $\Ran \bP$ and that $\DPP$ is bounded. Hence $\cF_P$ is well-defined. One of the crucial properties of the Feshbach map is its isospectrality. That is, $\cF_P$ is invertible on $\Ran P$ if and only if $\Delta^{(1)}_{H,Q}$ is invertible on $\cH^{(1)}$. In this case, we have % \begin{eqnarray}\label{Feshbachreduction} (\Delta_{H,Q}^{(1)})^{-1} & = & \Big( P \; - \; \bP\,\big(\bDelta_{H,Q}^{(1)}\big)^{-1}\,\DPP\Big) \, \cF_P^{-1} \, \Big(P \; - \; \DPP^*\,\big(\bDelta_{H,Q}^{(1)}\big)^{-1}\, \bP\Big) \nonumber \\ & & + \; \bP \, \big( \bDelta_{H,Q}^{(1)}\big)^{-1}\, \bP \period \end{eqnarray} % The rest of this subsection is devoted to properties of the projection $P$, in relation to $A_{H,Q}$. We recall that, for $j \in \Lambda$, % \begin{eqnarray} \label{eq-III-1.21} & \tA_j \ = \ \Ao_j + \sum_{k(\neq j)} A_k \comma & \\ \label{eq-III-1.22} & \Ao_j \ = \ \Th_j Z_j(H) Z_j^*(H) \Th_j - \frac{2}{\beta} \Tho_j^2 \Gio_{jj}''(x) \comma \ \ \ A_j \ = \ \Th_j Z_j^*(G) Z_j(G) \Th_j \period & \hspace{6mm} \end{eqnarray} % On $\Ran p$, we have the following upper bound, % \begin{lemma}\label{III-lem-1} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist universal constants $C, \alpha_0 >0 $, and $\beta_0 < \infty$ such that, for all $j \in \Lambda$, $0 \leq \alpha < \alpha_0$, and $\beta> \beta_0$, we have % \begin{equation} \label{eq-III-1.23} \big\| p \, \tA_j \, p \big\| \ \leq \ \frac{C}{\beta^{3/2}} \period \end{equation} % \end{lemma} % \noindent\textbf{Remark.} By virtue of Theorem~\ref{LocofZ1}, we could have chosen $p$ to be $j$-dependent projections onto vectors of the form $e^{-\beta(H+r_j)}$, for a large class of $r_j$'s. The choice (\ref{Feshbachprojection}) seems the most convenient here. We note that the most desirable choice, $r_j = -q_j$, may cause Lemma~\ref{III-lem-1} to be false. The reason being that the expression $p \Ao_j p$ would contain (the square of) a term of the form $e^{-\beta(H-2q_j)}$, and the Hamiltonian $H-2q_j$ does not in general localize at $0$. \Proof We first remark that, for $j\in\Lambda$, we have % \begin{equation}\label{eq-III-1.3} \Big(\sum_{k(\neq j)} A_k\Big) \, e^{-\beta H} \ = \ \sum_{k(\neq j)} \Theta_k\, Z_k^*(G)\,Z_k(G)\,\Theta_k\, e^{-\beta H} \ = \ 0 \comma \end{equation} % due to (\ref{eq-I-4-5}), (\ref{eq-I-4-6}), and (\ref{eq-I-4-12}). Next, a short computation yields % \begin{eqnarray}\label{eq-III-prelem1} \Ao_j & = & \Theta_j \, Z_j(H) \, Z_j^*(H) \, \Theta_j \: - \: \frac{2}{\beta} \, \Tho_j^2 \, \Gio_{jj}'' \\ \nonumber & = & \Theta_j \, Z_j^*(H) \, Z_j(H) \, \Theta_j \: + \: \frac{2}{\beta} \big(\Th_j^2 \, G_{jj}'' \: - \: \Tho_j^2 \, \Gio_{jj}''\big) \: + \: \frac{2}{\beta} \Th_j^2 \, q_j'' \period \end{eqnarray} % By Lemmata~\ref{lem-II-0-3} and \ref{lem-II-0-4}, we have % \begin{equation} \label{eq-III-0-6.1} \Big\| \frac{2}{\beta} \big( \Th_j^2 \, G_{jj}'' \: - \: \Tho_j^2 \, \Gio_{jj}''\big) \Big\| \ \leq \ \frac{C}{\beta^{1/2}} \, \big( \tA_j + \beta^{-1} \big) \comma \end{equation} % in the sense of quadratic forms, for some universal $C<\infty$. Inserting (\ref{eq-III-0-6.1}) into (\ref{eq-III-prelem1}) and sandwiching with $e^{-\beta H}$, we thus obtain, for $\beta<\infty$ sufficiently large, % \begin{eqnarray}\label{eq-III-prelem2} \lefteqn{ \big\la e^{-\beta H} \big| \; \tA_j \, e^{-\beta H} \big\ra } \nonumber \\ & \leq & 2 \, \big\la e^{-\beta H} \big| \; \Theta_j \, Z_j(H) \, Z_j^*(H) \, \Theta_j \, e^{-\beta H} \big\ra \: + \: \frac{4}{\beta} \, \big\| \Th_j \, |q_j''|^{1/2} e^{-\beta H} \big\|^2 \: + \: \frac{C\, \cZ_\beta}{\beta^{3/2}} \nonumber \\ & = & 2 \big\| \Th_j \, q_j' \, e^{-\beta H} \big\|^2 \: + \: \frac{4}{\beta} \, \big\| \Th_j \, |q_j''|^{1/2} e^{-\beta H} \big\|^2 \: + \: \frac{C \, \cZ_\beta}{\beta^{3/2}} \nonumber \\ & \leq & C \Big( \frac{\, \cZ_\beta}{\beta^{3/2}} \: + \: \int_{|x_j| \geq \Ri_0} e^{- 2 \beta (H - q_j)} \, d^\Lambda x \Big) \comma \end{eqnarray} % using that $\|q_j'\|_\infty, \|q_j''\|_\infty \leq C'$ are bounded by some universal constant $C'<\infty$ and vanish on $[-\Ri_0, \Ri_0]$. According to Theorem~\ref{LocofZ2}, the integral on the right side of (\ref{eq-III-prelem2}) is bounded by $\cZ_\beta \, e^{-\delta\beta}$, for some universal $\delta >0$. This yields the asserted estimate. \QED On $\Ran \bp$, we have the following complementary lower bound, % \begin{lemma} \label{III-lem-1.2} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist universal constants $\alpha_0 >0 $, and $C, \beta_0 < \infty$ such that, for all $j \in \Lambda$, $0 \leq \alpha < \alpha_0$, and $\beta> \beta_0$, we have % \begin{equation} \label{eq-III-a-1} \bp \, \tA_j \, \bp \ \geq \ \big( 1 - C \beta^{-\frac12} \big) \, \frac{2 \, \lmin}{\beta} \, \bp \period \end{equation} % \end{lemma} % \Proof We first pick a smooth characteristic function $\chi \in C_0^\infty( \RR_0^+; [0,1])$ on the interval $[0, \Ri_0)$, such that $\chi \equiv 1$ on $[0, \Ri_0/2)$, $\chi \equiv 0$ on $[\Ri_0, \infty)$, and $\bchi := \sqrt{1 - \chi^2} \in C^\infty$ is smooth, as well. We denote $\chi_j := \chi(|x_j|)$. The IMS localization formula reads % \begin{equation} \label{eq-III-a-2} \tA_j \ = \ \chi_j \, \tA_j \, \chi_j \: + \: \bchi_j \, \tA_j \, \bchi_j \: - \: \beta^{-2} \big( (\chi_j')^2 + (\bchi_j')^2 \big) \period \end{equation} % Note that, by Lemma~\ref{lem-II-0-3}, % \begin{equation} \label{eq-III-a-3} \frac{C}{\beta^{1/2}} \big( \tA_j \, + \, \beta^{-1} \big) \ \geq \ J_j[\us] \ \geq \ |\gi_j'| \comma \end{equation} % for some universal constant $C < \infty$. Furthermore, Eq.~(\ref{new-est-on-g}) yields that $|\gi_j'(x_j)| \geq c_g \,\min\{ 1, |\Ri_0|/2 \}$, for $|x_j| \geq |\Ri_0|/2$. Thus, for some universal constant $c > 0$ and $\beta < \infty$ sufficiently large, we have that % \begin{equation} \label{eq-III-a-4} \bchi_j \, \tA_j \, \bchi_j \ \geq \ \frac{c}{\beta^{1/2}} \, \bchi_j^2 \period \end{equation} % Next, the supersymmetric property (\ref{susy}) implies that % \begin{equation} \label{eq-III-a-5} \Delta_{H,Q}^{(0)} \ \geq \ \big( 1 - C \beta^{-\frac12} \big) \, \frac{2 \, \lmin}{\beta} \, \bp \comma \end{equation} % for some universal $C < \infty$. Moreover, $\chi_j \tA_j \chi_j = \chi_j \Delta_{H,Q}^{(0)} \chi_j$, and hence we have the lower bound % \begin{equation} \label{eq-III-a-6} \chi_j \, \tA_j \, \chi_j \ \geq \ \big( 1 - C \beta^{-\frac12} \big) \, \frac{2 \, \lmin}{\beta} \, \big( \chi_j^2 \: - \: \chi_j \, p \, \chi_j \big) \period \end{equation} % Putting together (\ref{eq-III-a-2})--(\ref{eq-III-a-6}), we have that % \begin{eqnarray} \label{eq-III-a-7} \tA_j & \geq & \big( 1 - C \beta^{-\frac12} \big) \, \frac{2 \, \lmin}{\beta} \, \chi_j^2 \: + \: \frac{c}{\beta^{1/2}} \, \bchi_j^2 \: - \: \frac{2 \, \lmin}{\beta} \, \chi_j \, p \, \chi_j \: - \: \frac{C}{\beta^2} \nonumber \\ & \geq & \frac{2 \, \lmin}{\beta} \, \Big\{ 1 \: - \: \frac{C'}{\beta^{1/2}} \: - \: \chi_j \, p \, \chi_j \Big\} \comma \end{eqnarray} % for universal $c>0$, $C, C' < \infty$ and $\beta < \infty$ sufficiently large. Sandwiching Eq.~(\ref{eq-III-a-7}) with $\bp$, we arrive at % \begin{equation} \label{eq-III-a-8} \bp \, \tA_j \, \bp \ \geq \ \frac{2 \, \lmin}{\beta} \, \Big\{ 1 - \frac{C'}{\beta^{1/2}} - \| \bp \, \chi_j \, p \, \chi_j \, \bp \| \Big\} \, \bp \period \end{equation} % Now, observe that due to $\bp \, p = p \, \bp = 0$, % \begin{equation} \label{eq-III-a-9} \bp \, \chi_j \, p \, \chi_j \, \bp \ = \ \bp \, (1-\chi_j) \, p \, (1-\chi_j) \, \bp \period \end{equation} % Since $1-\chi_j$ vanishes on $[-\Ri_0/2, \Ri_0/2]$, there exist a universal $\delta >0$ such that % \begin{eqnarray} \label{eq-III-a-10} \| \bp \, \chi_j \, p \, \chi_j \, \bp \| & \leq & \| (1-\chi_j) \, p \, (1-\chi_j) \| \ \; = \ \; \cZ_\beta^{-1} \, \big\| (1-\chi_j) \, e^{-\beta H} \big\|^2 \nonumber \\ & \leq & \int_{|x_j| \geq \Ri_0/2} e^{- 2 \beta H} \, \frac{d^\Lambda x}{\cZ_\beta} \ \; \leq \ \; e^{-\delta \beta} \comma \end{eqnarray} % according to Theorem~\ref{LocofZ1}. We finally obtain the asserted estimate (\ref{eq-III-a-1}) for sufficiently large $\beta < \infty$ by inserting (\ref{eq-III-a-10}) into (\ref{eq-III-a-8}). \QED The following Theorem is an immediate consequence of Lemma~\ref{III-lem-1.2} and Theorem~\ref{thm-II-0-1}. % \begin{theorem} \label{specsepestim} Assume Hypotheses~\ref{H-1}, \ref{H-2} and~\ref{H-3}. We have the following \emph{spectral separation} estimates: There exist universal constants $\alpha_0>0$, $\beta_0<\infty$, and $C<\infty$ such that, for any $0 \leq \alpha < \alpha_0$ and $\beta > \beta_0$, we have % \begin{eqnarray} \label{eq-III-a-11} \bP \, A^{(1)}_{H,Q} \, \bP & \geq & \big( 1 \, - \, C \beta^{-\frac12} \big) \, \frac{4 \, \lmin}{\beta} \, \bP \comma \\ \label{eq-III-a-12} \bP \, \Delta^{(1)}_{H,Q} \, \bP & \geq & \big( 1 \, - \, C \beta^{-\frac12} \big) \, \frac{4 \, \lmin}{\beta} \, \bP \period \end{eqnarray} % \end{theorem} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Improved decay} \label{SecIII.3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We begin by introducing some notation. Let % \begin{eqnarray} \label{AtoD} D & := & \sum_j D_j \otimes E_{jj} \ := \ \Hdiag^{-\frac12} \, \Big( \sum_j \tA_j \otimes E_{jj} \Big) \, \Hdiag^{-\frac12} \comma \\ \label{Bone0} B_0 & := & D \: + \: \frac2{\beta} \, \one \comma \\ \label{Bone} B & := & \Hdiag^{-\frac12} \, A_{H,Q}^{(1)} \, \Hdiag^{-\frac12} %\nonumber \\ %& = & \ = \ D \: + \: \frac2{\beta}(\one \, - \, \alpha \, T) \ = \ B_0 \: - \: \frac{2 \alpha}{\beta} \, T \comma \\ \label{AtoD.2} V & := & \Hdiag^{-\frac12} \, W_{H,Q}^{(1)} \, \Hdiag^{-\frac12} \comma \end{eqnarray} % where $\Hdiag$ is the diagonal matrix given by % \begin{equation}\label{DH} \Hdiag \ := \ \sum_j H_{jj}''(0) \otimes E_{jj} \period \end{equation} % We frequently use without further comment that $(c_f - \alpha C_a) \one \leq \Hdiag \leq (C_f + \alpha C_a) \one$ is bounded above and below by universal constants, for small $\alpha >0$. Furthermore, $D_j := \{D\}_{jj} = H_{jj}''(0)^{-1} \, \tA_j$, and $T$ is the matrix given by (\ref{Tij}). We also introduce $\bA_{H,Q}^{(1)} := \bP A_{H,Q}^{(1)}\bP$, $\bB := \bP B \bP$, $\bB_0 := \bP B_0 \bP$, $\bD := \bP D \bP$, $\bD_j := \bp D_j \bp$, $\bV := \bP V \bP$, and $\bT := \bp \otimes T$. We observe that % \begin{eqnarray} \label{eq-III-a-12.1} \big( \bDelta_{H,Q}^{(1)} \big)^{-1} & = & \Hdiag^{-\frac12} \, (\bB + \bV)^{-1} \, \Hdiag^{-\frac12} \period \end{eqnarray} % Furthermore, we observe that, as a consequence of Lemma~\ref{specsepestim}, we have % \begin{equation} \label{eq-III-a-12.2} \bB_0 \ \geq \ \frac{2}{\vth \, \beta} \, \bP \comma \end{equation} % where % \begin{equation} \label{eq-III-a-13} 1 \ < \ \frac{1}{\vth} \ := \ 1 + \frac{\lmin}{2 \, C_f} \ \leq \ 2 \comma \end{equation} % for $\alpha>0$ sufficiently small and $\beta < \infty$ sufficiently large. We then have the following decay estimate. % \begin{lemma} \label{DADdecay} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}, and define $1/2 < \vth < 1$ by (\ref{eq-III-a-13}). Then % \begin{equation} \label{bBonedecay} \Big\| \big\{ \bB_0^{\, 1/2} \, \bB^{\, -1} \, \bB_0^{\, 1/2} \big\}_{ij} \Big\| \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \end{equation} % where $\| \cdot \|$ is the operator norm on $\cB(\cH^{(0)})$, see (\ref{eq-I-4-10.02}). \end{lemma} % \Proof Expanding the inverse of $\bB$ in a Neumann series, we obtain from (\ref{Bone}) that % \begin{equation} \label{eq-III-a-14} \bB^{\, -1} \ := \ \sum_{n=0}^\infty \bB_0^{\, -1} \Big\{ \frac{2\alpha}{\beta} \, \bT \, \bB_0^{\, -1} \Big\}^n \, \bP \period \end{equation} % Therefore, we have % \begin{equation} \label{eq-III-a-15} \Big\| \big\{ \bB_0^{\, 1/2} \, \bB^{\, -1} \, \bB_0^{\, 1/2} \big\}_{ij} \Big\| \ \leq \ \sum_{n=0}^\infty \big\{ M^n \big\}_{ij} \comma \end{equation} % where % \begin{eqnarray} \label{eq-III-a-16} M_{ij} & := & \Big\| \big\{ \bB_0^{\, -1/2} \: \frac{2\alpha}{\beta} \, \bT \: \bB_0^{\, -1/2} \big\}_{ij} \Big\| \\ \nonumber & \leq & \frac{2\alpha}{\beta} \, T_{ij} \, \big\| \bB_0^{\, -1} \big\| \ \leq \ \frac{2\alpha}{\beta} \, T_{ij} \, \Big( \frac{2}{\vth \, \beta} \Big)^{-1} \ = \ \vth \, \alpha \, T_{ij} \comma \end{eqnarray} % which, inserted into (\ref{eq-III-a-15}), yields the Neumann series for the inverse of the matrix $\one - \vth \alpha T$. \QED Next, we note the following consequence of Theorem~\ref{thm-II-0-2} and Lemma~\ref{III-lem-1}. % \begin{lemma} \label{DWDdecay} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}. There exist universal constants $C < \infty$, $\alpha_0 > 0$, and $\beta_0 < \infty$ such that, for all $0 \leq \alpha \leq \alpha_0$, $\beta > \beta_0$, and all $i,j \in \Lambda$, we have % \begin{eqnarray} \label{eq-DWDdecay} \Big\| \big\{ \bB_0^{\, -1/2} \: \bV \: \bB_0^{\, -1/2} \big\}_{ij} \Big\| & \leq & \frac{C}{\beta^{1/2}} \Big( \delta_{ij} \, + \, \alpha \, T_{ij} \Big) \comma \\ \label{eq-DPPdecay} \Big\| \big\{ \Hdiag^{-1/2} \, \bB_0^{\, -1/2} \: \DPP \big\}_{ij} \, p \Big\| & \leq & \frac{C}{\beta^{3/4}} \Big( \delta_{ij} \, + \, \alpha \, T_{ij} \Big) \period \end{eqnarray} % \end{lemma} % \Proof We first remark that due to Theorem~\ref{thm-II-0-2}, there exist a universal number $C' < \infty$, such that % \begin{equation} \label{eq-III-a-17} \Big\| \big\{ \bB_0^{\, -1/2} \: \bV \: \bB_0^{\, -1/2} \big\}_{ij} \Big\| \ \leq \ M_i \, \frac{C'}{\beta^{1/2}} \Big( \delta_{ij} \, + \, \alpha \, a_{ij} \Big) \, M_j \comma \end{equation} % \begin{equation} \label{eq-III-a-18} \Big\| \big\{ \bB_0^{\, -1/2} \: \bP V P \big\}_{ij} \Big\| \ \leq \ M_i \, \frac{C'}{\beta^{1/2}} \Big( \delta_{ij} \, + \, \alpha \, a_{ij} \Big) \, \Big\| \big( \tA_j + \beta^{-1} \big)^{1/2} \, p \Big\| \comma \end{equation} % and % \begin{equation} \label{eq-III-a-19} \Big\| \big\{ \bB_0^{\, -1/2} \: \bP \, B \, P \big\}_{ij} \Big\| \ = \ \delta_{ij} \, \Big\| \big( \bD_j + 2\beta^{-1} \big)^{-1/2} \, \bp \, \tA_j \, p \Big\| \ \leq \ \delta_{ij} \, M_j \, \Big\| \tA_j^{1/2} \, p \Big\| \comma \end{equation} % where % \begin{eqnarray} \label{eq-III-a-18.1} M_j & := & \Big\| \big( \tA_j + \beta^{-1} \big)^{1/2} \, \bp \, \big( \bD_j + 2\beta^{-1} \bp \big)^{-1/2} \Big\| \nonumber \\ & = & \Big\| \big( \bD_j + 2\beta^{-1} \bp \big)^{-1/2} \, \big( \bp \, \tA_j \, \bp + \beta^{-1} \bp \big) \, \big( \bD_j + 2\beta^{-1} \bp \big)^{-1/2} \Big\|^{1/2} \nonumber \\ & \leq & C'' \comma \end{eqnarray} % since $\tA_j \leq (C'')^2 D_j$, for some universal number $C'' < \infty$. Finally, we invoke Lemma~\ref{III-lem-1} to arrive at % \begin{equation} \label{eq-III-a-21} \big\| \tA_j^{1/2} \, p \| \ = \ \big\| p \, \tA_j \, p \|^{1/2} \ \leq \ \frac{C'''}{\beta^{3/4}} \comma \end{equation} % for some universal constant $C''' < \infty$. We conclude by remarking that $T_{ij} = H_{ii}''(0)^{-1/2} a_{ij} H_{jj}''(0)^{-1/2}$ is bounded above and below by universal multiples of $a_{ij}$. \QED We now come to proving the main result of this subsection: the fast decay of all terms but the main term in $\cF_P$. % \begin{theorem} \label{improveddecay} Assume Hypotheses~\ref{H-1}, \ref{H-2}, and \ref{H-3}, and let $\vth < \vth' < 1$ be a universal number, where $\vth^{-1} := 1 + (\lmin/2 C_f)$ is defined in (\ref{eq-III-a-13}). Then there exist universal constants $C < \infty$, $\alpha_0 > 0$, and $\beta_0 < \infty$ such that, for all $0 \leq \alpha \leq \alpha_0$, $\beta > \beta_0$, and all $i,j \in \Lambda$, we have % \begin{eqnarray} \label{eq-III-a-22} \Big\| \bp \, \big\{ \bDelta_{H,Q}^{-1} \big\}_{ij} \, \bp \Big\| & \leq & C \, \beta \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \\ \label{eq-III-a-23.1} \Big\| \bp \, \big\{ \bDelta_{H,Q}^{-1} \, \DPP \big\}_{ij} \, p \Big\| & \leq & \frac{C}{\beta^{1/4}} \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \\ \label{eq-III-a-23.2} \Big\| p \, \big\{ \DPP^* \, \bDelta_{H,Q}^{-1} \big\}_{ij} \, \bp \Big\| & \leq & \frac{C}{\beta^{1/4}} \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \\ \label{eq-III-a-24} \Big\| p \, \big\{ \DPP^* \, \bDelta_{H,Q}^{-1} \, \DPP \big\}_{ij} \, p \Big\| & \leq & \frac{C}{\beta^{3/2}} \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{eqnarray} % \end{theorem} % \Proof We only derive Estimate~(\ref{eq-III-a-24}). The derivations of Estimates~(\ref{eq-III-a-22})--(\ref{eq-III-a-23.2}) are similar. We first observe that due to (\ref{eq-III-a-12.1}), % \begin{eqnarray} \label{eq-III-a-25} \lefteqn{ \big\{ \DPP^* \, \bDelta_{H,Q}^{-1} \, \DPP \big\}_{ij} \ = \ \sum_{k,\ell} \big\{ \DPP^* \, \bB_0^{\, -1/2} \, \Hdiag^{-\frac12} \big\}_{ik} } \\ \nonumber & & \Big\{ \big( \bB_0^{\, -1/2} \, \bB \, \bB_0^{\, -1/2} \: + \: \bB_0^{\, -1/2} \, \bV \, \bB_0^{\, -1/2} \big)^{-1} \Big\}_{k \ell} \; \big\{ \Hdiag^{-\frac12} \, \bB_0^{\, -1/2} \, \DPP \big\}_{\ell j} \period \end{eqnarray} % A Neumann series expansion, (\ref{bBonedecay}), and (\ref{eq-DWDdecay}) yield % \begin{eqnarray} \label{eq-III-a-26} \lefteqn{ \Big\| \bp \, \Big\{ \big( \bB_0^{\, -1/2} \, \bB \, \bB_0^{\, -1/2} \: + \: \bB_0^{\, -1/2} \, \bV \, \bB_0^{\, -1/2} \big)^{-1} \Big\}_{k \ell} \, \bp \Big\| \ \leq \ } \\ \nonumber & & \bigg\{ \sum_{n=0}^\infty \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \: \Big[ \frac{C}{\beta^{1/2}} \, ( \one \, + \, \alpha T ) \: \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big]^{n} \bigg\}_{k \ell} \period \end{eqnarray} % Inserting this estimate and (\ref{eq-DPPdecay}) into (\ref{eq-III-a-25}), we hence obtain % \begin{eqnarray} \label{eq-III-a-27} \lefteqn{ \Big\| p \, \big\{ \DPP^* \, \bDelta_{H,Q}^{-1} \, \DPP \big\}_{ij} \, p \Big\| } \\ \nonumber & \leq & \frac{C}{\beta^{1/2}} \, \bigg\{ \sum_{n=1}^\infty \Big[ \frac{C}{\beta^{1/2}} \, ( \one \, + \, \alpha T ) \: \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big]^{n} \: \frac{C}{\beta^{1/2}} \, ( \one \, + \, \alpha T ) \bigg\}_{ij} \\ \nonumber & \leq & \frac{C \, b^2}{\beta^{3/2}} \, \bigg\{ \sum_{n=1}^\infty \Big[ \frac{C}{b} \, ( \one \, + \, \alpha T ) \: \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big]^{n} \: \frac{C}{b} \, ( \one \, + \, \alpha T ) \bigg\}_{ij} \comma \end{eqnarray} % for any $b^2 < \beta$, additionally using that all matrices involved have only nonnegative matrix elements. (At this point we would additionally make use of the trivial bound $\| \{ \Hdiag^{-1/2} \bB_0^{\, -1/2} \}_{ij} \bp \| \leq C \beta^{1/2} \delta_{ij}$ to derive (\ref{eq-III-a-22})--(\ref{eq-III-a-23.2}).) We further observe that, % \begin{equation} \label{eq-III-a-28} \delta_{ij} \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \end{equation} % for all $i,j \in \Lambda$. Thus, for any matrix $M$ with only nonnegative matrix elements $M_{ij} \geq 0$, we have that % \begin{equation} \label{eq-III-a-29} M_{ij} \ \leq \ \Big\{ \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \: M \: \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{equation} % Applying this to the right side of (\ref{eq-III-a-27}), we arrive at % \begin{eqnarray} \label{eq-III-a-30} \lefteqn{ \Big\| p \, \big\{ \DPP^* \, \bDelta_{H,Q}^{-1} \, \DPP \big\}_{ij} \, p \Big\| } \\ \nonumber & \leq & \frac{C \, b^2}{\beta^{3/2}} \, \bigg\{ \sum_{n=0}^\infty \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \: \Big[ \frac{C}{b} \, ( \one \, + \, \alpha T ) \: \big( \one \, - \, \vth \, \alpha \, T \big)^{-1} \Big]^{n} \bigg\}_{ij} \\ \nonumber & = & \frac{C \, b^2}{\beta^{3/2}} \, \bigg\{ \Big( (1- Cb^{-1}) \, \one \: - \: \alpha \, \big( \vth + Cb^{-1} \big) \, T \Big)^{-1} \bigg\}_{ij} \\ \nonumber & = & \frac{C \, b^2}{(1 - Cb^{-1}) \, \beta^{3/2}} \, \bigg\{ \Big( \one \: - \: \alpha \, \frac{ \vth + Cb^{-1} }{ 1 - Cb^{-1} } \, T \Big)^{-1} \bigg\}_{ij} \period \end{eqnarray} % The claim now follows from choosing a sufficiently large, universal number $b < \infty$ such that % \begin{equation} \label{eq-III-a-31} \vth \: + \: Cb^{-1} \ \leq \ (1 - Cb^{-1}) \, \vth' \period \end{equation} % \QED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of the main theorem} \label{SecIII.4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this subsection we give a proof of our main result Theorem~\ref{asymptotics}. This proof differs in an essential way from the corresponding proofs given in \cite{Sjoestrand1997} and \cite{BachJeckoSjoestrand2000}, by not relying on translation invariance. We recall that translation invariance was used in order to diagonalize, using the Fourier transform, and then do estimates in momentum space. We first present the correlation formula (\ref{eq-correlations}) with the resolvent of the Witten Laplacian expanded using the Feshbach reduction formula (\ref{Feshbachreduction}). Abbreviating % \begin{eqnarray}\label{SP} \cS_P & := & \bP\,\big(\bDelta_{H,Q}^{(1)}\big)^{-1}\,\DPP \ :\; \Ran{P}\; \to\; \cH^{(1)} \comma \\ \label{eq-III-a-31.1} \phi_j & := & \cZ_\beta^{-1/2} \: e^{-\beta(H-q_j)} \otimes e_j \comma \end{eqnarray} % for $j \in \Lambda$, we can write % \begin{eqnarray}\label{corformulaP} \beta^2 \, \sqrt{H_{ii}''(0) \, H_{jj}''(0) \, } \, \EE^T_\beta(x_i\, ; \,x_j) & = & \Big\la \phi_i \, \Big| \, \Hdiag^{\frac12} \, (P - \cS_P) \, \cF_P^{-1} \, (P - \cS_P^*) \, \Hdiag^{\frac12} \, \phi_j \Big\ra \nonumber \\ & & + \, \Big\la \phi_i \, \Big| \, \Hdiag^{\frac12} \, \bP \, \big( \bDelta_{H,Q}^{(1)} \big)^{-1} \, \bP \, \Hdiag^{\frac12} \, \phi_j \Big\ra \period \end{eqnarray} % Let $U: \Ran P \to \CC^\Lambda$ be given by % \begin{equation}\label{Umap} U \, \psi \ := \ \|e^{-\beta H}\|_{\cH^{(0)}}^{-1} \sum_k \la e^{-\beta H} \otimes e_k | \psi \ra \, e_k \period \end{equation} % Clearly $U^*U = \one_{\Ran{P}}$ and $UU^* = \one_{\RR^\Lambda}$. After conjugation with this unitary map, the Feshbach operator $\cF_P$ becomes a $\Lambda \times \Lambda$ matrix % \begin{equation} \label{UFPU} \cF_P \ =: \ \frac2{\beta} \, U^* \, \Hdiag^{\frac12} \, F \, \Hdiag^{\frac12} \, U \, \comma \end{equation} % where % \begin{eqnarray} \label{eq-III-a-32} F & = & \one \, - \, \alpha \, T \, - \, \beta^{-1/2} \, Y \comma \\ \label{eq-III-a-33} Y & := & \frac{\beta^{3/2}}{2} \, U \, P \, ( D + V ) \, P \, U^* \\ \nonumber & & - \; \frac{\beta^{3/2}}{2} \, U \, P \, \Hdiag^{-\frac12} \, \DPP^* \, \big( \bDelta_{H,Q}^{(1)} \big)^{-1} \, \DPP \, \Hdiag^{-\frac12} \, P \, U^* \period \end{eqnarray} % To estimate the matrix elements of $Y$, we observe that Lemma~\ref{III-lem-1} implies % \begin{equation} \label{eq-III-a-32.1} \la e_i | \; U \, P \, D \, P \, U^* \, e_j \ra \ \leq \ C \, \delta_{ij} \, \big\| p \, \tA_j \, p \big\| \ \leq \ \frac{C'}{\beta^{3/2}} \, \delta_{ij} \comma \end{equation} % for some universal constants $C,C' < \infty$. Due to Theorem~\ref{thm-II-0-2}, we have that % \begin{eqnarray} \label{eq-III-a-32.2} \la e_i | \; U \, P \, V \, P \, U^* \, e_j \ra & \leq & \frac{C''}{\beta^{1/2}} \Big( \delta_{ij} + \alpha T_{ij} \Big) \big\| (\tA_i + \beta^{-1})^{\frac12} \, p \big\| \, \big\| (\tA_j + \beta^{-1})^{\frac12} \, p \big\| \nonumber \\ & \leq & \frac{C'''}{\beta^{3/2}} \, \Big( \delta_{ij} + \alpha T_{ij} \Big) \comma \end{eqnarray} % for some universal constants $C'',C''' < \infty$. Estimates~(\ref{eq-III-a-32.1}), (\ref{eq-III-a-32.2}), and (\ref{eq-III-a-24}) hence imply that there exist universal constants $C < \infty$ and $\vth < \vth' < 1$ such that % \begin{equation} \label{eq-III-a-32.3} |Y_{ij}| \ \leq \ C \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{equation} % Next, we introduce the $\Lambda \times \Lambda$ matrices $\eps$, $S$, and $\bR$ by % \begin{eqnarray} \label{eq-III-a-32.4} \eps_{ij} & := & \delta_{ij} \, \big( \la \phi_j | \, P \, \phi_j \ra^{1/2} \, - \, 1 \big) \comma \\ \label{eq-III-a-34} S_{ij} & := & \Big\la \phi_i \, \Big| \, \Hdiag^{\frac12} \, \cS_P \, \Hdiag^{\frac12} \, U^* \, e_j \Big\ra \comma \\ \label{eq-III-a-35} \bR_{ij} & := & \Big\la \phi_i \, \Big| \, \Hdiag^{\frac12} \, \bP \, \big( \bDelta_{H,Q}^{(1)} \big)^{-1} \, \bP \, \Hdiag^{\frac12} \, \phi_j \Big\ra \end{eqnarray} % and obtain % \begin{equation} \label{eq-III-a-36} \frac{\beta}{2} \, \sqrt{H_{ii}''(0) \, H_{jj}''(0) \, } \, \EE^T_\beta(x_i\, ; \,x_j) \ = \ \big\{ (\one + \eps - S) \, F^{-1} \, (\one + \eps^* - S^*) \: + \: \bR \big\}_{ij} \period \end{equation} % We observe that due to Corollary~\ref{LocofZ2}, there exists a universal $\delta >0$ such that, for all $j \in \Lambda$, % \begin{eqnarray} \label{eq-III-a-37} \big| \la \phi_j | \phi_j \ra \, - \, 1 \big| & \leq & e^{-2\delta\beta} \comma \\ \label{eq-III-a-38} \big| \la \phi_j | \, P \, \phi_j \ra \, - \, 1 \big| & \leq & e^{-2\delta\beta} \comma \\ \label{eq-III-a-39} \la \phi_j | \, \bP \, \phi_j \ra & \leq & e^{-2\delta\beta} \comma \end{eqnarray} % provided $\alpha >0$ is sufficiently small and $\beta < \infty$ is sufficiently large. We recall from (\ref{eq-III-a-13}) that $\vth^{-1} := 1 + (\lmin/2 C_f)$, and we introduce a universal number $1 > \vth' > \vth$ by $(\vth')^{-1} := 1 + (\lmin/4 C_f)$. Now, Theorem~\ref{improveddecay} and (\ref{eq-III-a-39}) imply that, for $\alpha >0$ sufficiently small and $\beta < \infty$ sufficiently large, there exists a universal $C < \infty$, such that, for all $i,j \in \Lambda$, % \begin{eqnarray} \label{eq-III-a-40} |S_{ij}| & \leq & \frac{C \, e^{-\delta\beta} }{\beta^{1/4}} \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \comma \\ \label{eq-III-a-41} |\bR_{ij}| & \leq & C \, \beta \, e^{-2\delta\beta} \, \Big\{ \big( \one \, - \, \vth' \, \alpha \, T \big)^{-1} \Big\}_{ij} \period \end{eqnarray} % Moreover, (\ref{eq-III-a-38}) directly yields that $|\eps_{ij}| \leq \delta_{ij} e^{-\delta\beta}$. Thus, applying Theorems~\ref{thm-II-a-3} and \ref{thm-II-a-4}, using (\ref{eq-III-a-32.3}), we have % \begin{equation} \label{eq-III-a-42} \Big| \big\{ (\eps - S) F^{-1} \, + \, F^{-1} (\eps^* - S^*) \, + \, (\eps - S) F^{-1} (\eps^* - S^*) \, + \, \bR \big\}_{ij} \Big| \ \leq \ e^{-\delta\beta} \: \big\{ F^{-1} \big\}_{ij} \comma \end{equation} % for $\beta < \infty$ sufficiently large, which, inserted into (\ref{eq-III-a-36}), yields % \begin{eqnarray} \label{eq-III-a-43} \big( 1 \, - \, C \, e^{-\delta\beta} \big) \, \big\{ F^{-1} \big\}_{ij} & \leq & \frac{\beta}{2} \, \sqrt{H_{ii}''(0) \, H_{jj}''(0) \, } \, \EE^T_\beta(x_i\, ; \,x_j) \nonumber \\ & \leq & \big( 1 \, + \, C \, e^{-\delta\beta} \big) \, \big\{ F^{-1} \big\}_{ij} \comma \end{eqnarray} % for all $i,j \in \Lambda$. Now applying again Theorem~\ref{thm-II-a-3}, we arrive at % \begin{eqnarray} \label{eq-asymptotics-1a} \lefteqn{ \Big(1 \, - \, \frac{C'}{\sqrt{\beta} } \Big) \, \Big\{ \Big( \one \, - \, \frac{\alpha}{1+C' \, \beta^{-1/2}} \, T \Big)^{-1} \Big\}_{ij} \ \leq \ \frac{\beta}{2} \, \sqrt{H_{ii}''(0) \, H_{jj}''(0) \, } \, \EE^T_\beta(x_i\, ; \,x_j) } \nonumber \\ & \hspace{30mm} \leq & \Big(1 \, - \, \frac{C'}{\sqrt{\beta} } \Big) \, \Big\{ \Big( \one \, - \, \frac{\alpha}{1-C' \, \beta^{-1/2}} \, T \Big)^{-1} \Big\}_{ij} \comma \hspace{20mm} \end{eqnarray} % where $C' < \infty$ is universal. The first assertion (\ref{eq-asymptotics-2}) of the main theorem~\ref{asymptotics} then results from an additional application of the Greens function estimates (\ref{eq-III-pert-1}) and (\ref{eq-III-pert-2}). The fact that $\rho$ is a metric is proved in Theorem~\ref{thm-II-c-1}, and, finally, the second claim (\ref{eq-asymptotics-3}) is a transcription of (\ref{eq-asymptotics-2}). \QED {\bf Acknowledgement: } The authors would like to thank B.~Helffer and O.~Matte for discussions, T.~Bodineau for bringing Ornstein-Zernike theory to our attention, and the second author would like to thank Dokuz Eyl{\"u}l University for hospitality. %\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\bibliography{/home/vbach/BIB/volle} %\end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{10} \bibitem{BachChenFroehlichSigal2003} V.~Bach, T.~Chen, J.~Fr{\"{o}}hlich, and I.~M. Sigal, \emph{Smooth {F}eshbach map and operator-theoretic renormalization group methods}, J.~Funct.~Anal. (in print) (2003). \bibitem{BachFroehlichSigal1998b} V.~Bach, J.~Fr{\"{o}}hlich, and I.~M. Sigal, \emph{Renormalization group analysis of spectral problems in quantum field theory}, Adv.~in Math.~ \textbf{137} (1998), 205--298. \bibitem{BachJeckoSjoestrand2000} V.~Bach, T.~Jecko, and J.~Sj{\"{o}}strand, \emph{Correlation asymptotics of classical lattice spin systems with nonconvex {H}amilton function at low temperature}, Ann.~Henri Poincar\'e \textbf{1} (2000), 59--100. \bibitem{BachMoeller2003a} V.~Bach and J.~M{\o}ller, \emph{Correlation at low temperature: {I}. exponential decay}, J.~Funct.~Anal. \textbf{203} (2003), 93--149. \bibitem{CampaninoIoffeVelenik2003b} M.~Campanino, D.~Ioffe, and Y.~Velenik, \emph{Rigorous nonperturbative {O}rnstein-{Z}ernike theory for {I}sing ferromagnets}, Europhys. Lett. \textbf{62} (2003), 182--188. \bibitem{Conlon2003} J.~G. Conlon, \emph{{PDE} with random coefficients and {E}uclidean field theory}, preprint (2003). \bibitem{DerezinskiJaksic2001} J.~Derezinski and V.~Jaksic, \emph{Spectral theory of {P}auli-{F}ierz operators}, J.~Funct. Anal. \textbf{180} (2001), no.~2, 243--327. \bibitem{HelfferNier2003} B.~Helffer and F.~Nier, \emph{Criteria to the {P}oincar{\'e} inequality associated with {D}irichlet forms in {$\RR^d$}, $d\geq 2$}, Int. Math. Res. Not. (2003), 1199--1223. \bibitem{HelfferSjoestrand1994} B.~Helffer and J.~Sj{\"o}strand, \emph{On the correlations for {K}ac like models in the convex case}, J.~Stat.~Phys. \textbf{74} (1994), 349--369. \bibitem{HereauNier2002} F.~H{\'e}reau and F.~Nier, \emph{Isotropic hypoellipticity and trends to equilibrium for the {F}okker-{P}lanck equation with high degree potential}, preprint (2002). \bibitem{Johnsen2000} J.~Johnsen, \emph{On the spectral properties of {W}itten-{L}aplacians, their range projections and {B}rascamp-{L}ieb's inequality}, Integr.~Eq.~Oper.~Theor. \textbf{36} (2000), no.~3, 288--324. \bibitem{MatteMoeller2003} O.~Matte and J.~S. M{\o}ller, \emph{On the spectrum of semiclassical {W}itten {L}aplacians and {S}chr{\"o}dinger operators in large dimensions}, In preparation (2003). \bibitem{NaddafSpencer1997} A.~Naddaf and T.~Spencer, \emph{On homogenization and scaling limit of some gradient perturbations of a massless free field}, Commun.~Math.~Phys. \textbf{183} (1997), no.~1, 55--84. \bibitem{Sjoestrand1997} J.~Sj{\"o}strand, \emph{Correlation asymtotics and {W}itten {L}aplacians}, St.~Petersburg Math.~J.~(AMS) \textbf{8} (1997), 123--147, Original in Algebra i Analiz 8 (1996), no.\ 1, pp.~160--191. \bibitem{Sjoestrand2000} \bysame, \emph{Complete asymptotics for correlations of {L}aplace integrals in the semi-classical limit}, M{\'e}m.~Soc.~Math.~Fr.~(N.S.) \textbf{83} (2000), iv+104~pp. \bibitem{Spitzer1976} F.~Spitzer, \emph{Principles of random walk}, Graduate Texts in Mathematics, no.~30, Springer-Verlag, New York, 1976. \bibitem{Woess2000} W.~Woess, \emph{Random walks on infinite graphs and groups}, 1 ed., Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. \end{thebibliography} \end{document} ---------------0309081023611--