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Lattice Spin Systems, Witten Laplacian, 
Lattice Greens function.
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\newcommand{\DATUM}{2003.09.08}            %                        %
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\markboth{\centerline{V. Bach and J. S. M{\o}ller}}%
         {\centerline{Correlation at low temperature: II.  Asymptotics}}
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%         {\centerline{Correlation at low temperature: II.\ Asymptotics}}
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\begin{document}
\bibliographystyle{amsplain}
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%
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\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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% \MRhref is called by the amsart/book/proc definition of \MR.
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