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32 pages, 0 figures
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nonequilibrium statistical mechanics, quantum dynamics, large deviations
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%\author{Wojciech De Roeck\thanks{Postdoctoral fellow FWO-Flanders, Department of Mathematics, Harvard University, Cambridge, USA. On leave from  Instituut voor Theoretische Fysica, K.U.Leuven, Belgium,   email: {\tt
% wojciech.deroeck@fys.kuleuven.be}}}



%\title{Large deviations for currents in the spin-boson model}


\begin{document}

\begin{center}
\large{ \bf{ Large deviations for currents in the spin-boson model}} \\
\vspace{15pt} \normalsize



{\bf Wojciech De Roeck}\footnote{Postdoctoral Fellow FWO-Flanders,
email: {\tt
 wojciech.deroeck@fys.kuleuven.be}}\\
Instituut voor Theoretische Fysica, K.U.Leuven, Belgium\\
Department of Mathematics, Harvard University, Cambridge, USA
\\

\end{center}

\vspace{20pt} \footnotesize \noindent {\bf Abstract: } We consider
a finite quantum system coupled to quasifree thermal reservoirs at
different temperatures. Under the assumptions of small coupling
and exponential decay of the reservoir correlation function, the
large deviation generating function is shown to be analytic on a
compact set. Our method is different from the spectral deformation
technique which was introduced recently in the study of
spin-boson-like models. As a corollary, we derive the
Gallavotti-Cohen fluctuation relation for the entropy production.


\vspace{5pt} \footnotesize \noindent {\bf KEY WORDS:} weak
coupling limit, large deviations, nonequilibrium statistical
mechanics, cluster expansions, spin-boson model \vspace{20pt}
\normalsize
\section{Introduction}
\subsection{Large deviations}\label{sec: large deviations}
The theory of large deviations lies at the heart of classical
equilibrium statistical mechanics. Indeed thermodynamical
potentials like e.g.\ the free energy can be viewed as large
deviation generating functions. In the quest for a general theory
of nonequilibrium statistical mechanics, it is hence natural to
take large deviations as a starting point. Recently,
 some remarkable results have been obtained with that strategy, see e.g.\ \cite{maesnetocnyminent} and \cite{bertinidesolegabrielli}.
 A nonequilibrium large deviation result which has received huge attention in the last decade is the Gallavotti-Cohen
  fluctuation theorem \cite{gallavotticohen95prl,gallavotticohen2,evanscohen}, which states a symmetry in the
 fluctuations of entropy production.

What are large deviations? Assume that we have a family of
measures $\bbP_{ t \in \bbR^+ }$ and corresponding random
variables $A_{t}$ taking values in $\bbR^d$. Heuristically, the
family $A_t$ satisfies a LDP (large deviation principle) iff.\ ,
 \beq \bbP_t(A_t \approx \alpha) \sim   \e^{-t I(\alpha)},\qquad \alpha \in \bbR^d, t \uparrow
\infty \eeq for some  function $I(\alpha)$ which is called the
\emph{rate function}. A precise definition (see
\cite{dembozeitouni} for further details) states that there exists
a  lower-semicontinuous convex function $I: \bbR^d \to \bbR^+$
satisfying
\begin{eqnarray}
\limsup_{t \nearrow \infty}  \frac{1}{t} \log \bbP_t [ A_t \in  E] &\leq& -\inf_{\alpha \in E} I(\alpha) ,\qquad      E\quad  \textrm{closed}\label{ldp1}\\
\liminf_{t \nearrow \infty}  \frac{1}{t} \log \bbP_t [ A_t \in  E]
&\geq& -\inf_{\alpha \in E} I(\alpha) ,\qquad      E \quad
\textrm{open} \label{ldp2}
\end{eqnarray}

Define the large deviation generating function on $\bbR^d$ \beq
\label{def: intro generating function} F(\ka):= \lim_{t \uparrow
\infty}\frac{1}{t}\log\int \d \bbP_t (\alpha ) \e^{-t (\ka |
\alpha)} \eeq where $(\cdot|\cdot)$ is the canonical scalar
product on $\bbR^d$.

 The useful Gartner-Ellis theorem  states that if $F(\ka)$ exists and is differentiable, then
(\ref{ldp1}, \ref{ldp2}) hold with $I$ being the Legendre
transform of $F$. By a theorem of Bryc \cite{bryc}, analyticity of
$F$ in a neighbourhood of $\ka=0$ implies that the variables $A_t$
satisfy a central limit theorem, with covariance matrix given by
the Hessian of $F$ in $\ka=0$.

We now specialize the setup to nonequilibrium physics. We consider
the spin-boson model with several quasifree boson reservoirs
(indexed by $k \in \caK$) at inverse temperatures $\be_k$,
interacting through a small system (the `spin').  Let $t$ be time
and let $A_{k,t}$ be the time-averaged integrated heat flow up to
time $t$ into reservoir $k$. By $F_A$ we denote the large
deviation generating function corresponding to $A_{k,t}$ as
outlined above. Let $E_t:= \sum_{k \in \caK}\be_k A_{k,t}$, which
is interpreted as the time-averaged entropy production up to time
$t$ and let $I_{E},F_{E}$ be the rate function/generating function
associated to $E_t$. The Gallavotti-Cohen fluctuation theorem
states \beq I_E(\al)-I_E(-\al)=-\al ,\qquad \textrm{or,
equivalently}\qquad F_E(\ka)=F_E(1-\ka) \eeq under very general
conditions. It is easy to derive heuristically, see e.g.\
\cite{jarzynskiquantum} for a quantum version. The technical work
lies in establishing the LDP.


In the present paper, we establish analyticity of the large
deviation generating function $F_A$ (corresponding to the
variables $A_{k,t}$) on a compact set (Our method does not extend
to the full $\bbR^{|\caK|}$). This is described in Theorem
\ref{thm: 1}.  As a consequence, one has also analyticity of $F_E$
and we reproduce the Gallavotti-Cohen fluctuation theorem
(Corollary \ref{cor: gc}). For completeness, we state also the
central limit theorem (Corollary \ref{cor: central limit}).

In sharp contrast to classical lattice systems, where the existence of the generating function follows from quite general considerations, see \cite{simon},
 there is no standard quantum theory of large
 deviations yet, not even in equilibrium. However, some results have been obtained in \cite{gallavottilebowitzmastropietro,lebowitzlencispohn,netocnyredig,reybelletld} and
 recently in \cite{aboesalem}.





\subsection{Technique: classical polymer model}
A large part of the interest of the present paper resides  in the
technique of the proof. In the past years, quite some work has
been devoted to spin-boson and spin-fermion models. We mention
early work \cite{hubnerspohnspectral},\cite{bachfrohlichreturn},
the
 series of papers by Jak${\check {\mathrm s}}$i\'{c} and Pillet, e.g.\
\cite{jaksicpillet3,jaksicpillet4} and work on the Green-Kubo
relations \cite{jaksicogata}.  The elegant technique which was
developed for these models consists in identifying an operator
whose spectrum contains information on the approach to
equilibrium. This spectrum is then studied by complex
deformations. A drawback is that the conditions imposed on the
coupling between the small system and the quasi-free bath are
typically very restrictive, e.g.\ in \cite{jaksicpillet4} the
coupling function is required to be analytic in a strip, see
however \cite{derezinskijaksicspectral},
\cite{derezinskijaksicreturn} and \cite{merklicommutators} for an
approach which requires much weaker conditions.


We propose a different technique which reduces the time-evolution
of the coupled system to a classical polymer model with complex
polymer weights. This can be done by exploiting the weak coupling
limit in which the evolution of the small system becomes Markovian
and dissipative (see \cite{davies1} for the original rigorous
paper and \cite{lebowitzspohn1} or \cite{derezinskifruboesreview}
for a review.) The polymers represent excitations around the
Markovian dynamics, an idea which is inspired by
\cite{maesnetocnyspacetime}.

 (On the other  hand, the above-mentioned
spectral approach also exploits the weak coupling limit, but on
the level of the resolvent. Hence, one could call our technique
the dynamic counterpart to the spectral approach.)


The drawback of our technique is its focus on the small system,
and, most importantly, that it is much less elegant than the
spectral approach. A possible advantage of our technique is its
conceptual simplicity and its robustness. For example, it is
straightforward to prove and quantify the approach to a stationary
state and decay of correlations in the small system under
conditions comparable to those in \cite{derezinskijaksicreturn}.
This will be done in a future paper.

In this paper, we however assume quite strong conditions, namely
exponential decay of the reservoir correlation functions (see
Assumption \ref{ass: 1}). These strong conditions are necessary to
control the cluster expansion which proves analyticity of the
large deviation generating function.  This means however that
 our conditions are almost identical to those in
\cite{jaksicpillet4}.






\subsection{What are current fluctuations ?}\label{sec: finite}

The question what quantum current fluctuations are, and how one
should generalize the Gallavotti-Cohen theorem, has received quite
some attention in the statistical physics community, see
\cite{derezinskideroekmaes} for a discussion of our viewpoint and
a list of references. For the sake of consistency, we outline our
approach and motivation.

 We consider finite system where all
questions can be answered and then we take the thermodynamical
limit of interesting quantities, \emph{in casu} the Laplace
transform of the probability distribution of energy transport.

Fix a finite-dimensional Hilbert space $\caH_\sys$ with
self-adjoint Hamiltonian $H_\sys$ and let $\caK$ be a finite set
which indexes the heat reservoirs at inverse temperatures
$\be_{k\in \caK}
>0$. To each $k \in \caK$, we associate (the superscript $n \in
\bbN$ indicates that the thermodynamic limit has not yet been
taken) \ben

\item{The  self-adjoint coupling operator $V_k \in \caB(\caH_\sys)$.}
\item{A finite-dimensional one-particle Hilbert space $\frh_{k}^n$ and its bosonical second quantization $\fock (\frh_k^n)$. For $g \in \frh_{k}^n$, let  $a^*(g) / a(g)$ be creation/annihilation operators on $\fock (\frh_k^n)$, satisfying
\beq  [a(g),a^*(g')]=\langle g,g' \rangle_{\frh_{k}^n}, \qquad   g,g' \in \frh_{k}^n  \eeq}
\item{A self-adjoint one-particle Hamiltonian $h_{k}^n \in \caB(\frh_{k}^n)$ with corresponding second quantization $H_{\res_k}^n:=\d \Ga (h_{k}^n)$ and a coupling function $f_{k}^n \in \frh_{k}^n$.
}
\item{A Gibbs state $\rho_{k,\be_k}^n$ on $\caB(\fock (\frh_k^n))$ at inverse temperature $\be_k$
\beq \rho_{k,\be_k}^n\left[ \cdot \right] =
\frac{1}{Z^n_{k}(\be_k)}\Tr \left[ \e^{-\be_k  H_{\res_k}^n} \cdot
\right], \qquad  Z^n_{k}(\be_k)= \Tr \left[ \e^{-\be_k
H_{\res_k}^n } \right]   \eeq }

\een


We define the total interacting Hamiltonian on $\caH_\sys
\otimes_{k \in \caK} \fock (\frh_k^n) $ as \begin{eqnarray}
H_{\la}^n &=& H_\sys + \sum_{k \in \caK} H_{\res_k}^n + \la\sum_{k
\in \caK}  V_k \otimes  \left( a^*(f_{k}^n) +  a(f_{k}^n) \right )
\end{eqnarray} from which we construct the unitary time-evolution
\beq U_{t}^{\la,n} := \e^{-\i t H_{\la}^n } \eeq We take as
initial state \beq \label{def: initial state finite} \rho_\sys
\otimes \rho_\res^n := \rho_\sys \otimes [\otimes_{k\in \caK}
\rho_{k,\be_k}^n]\eeq Note that the $H_{\res_k}^n$  mutually
commute and that they have discrete spectrum. Denote by $x \in
\bbR^{|\caK|}$ (with components $x_{k \in \caK}$) elements of $
\mathop{\times}\limits_{k \in \caK} \sp H_{\res_k}^n$ and let $
P_x^n $  be the spectral projection on  $(x_k)_{k \in \caK}$
corresponding to $(H_{\res_k}^n)_{k \in \caK}$.



Inspired by physics, we define the
probability to measure an energy increase of resp.\ $y_k$ in
reservoir $k$ after time $t$ and starting from the state \eqref{def: initial state finite}, as (we group $y_{k \in \caK}$ into $y \in
\bbR^{|\caK|}$) \beq  \label{def: finite measure}\bbP_{\rho_\sys,t,\la}^n
(y):=\mathop{\sum}\limits_{\left.\begin{array}{c} x,x' \in \mathop{\times}\limits_{k \in \caK} \sp
H_{\res_k}^n \\
x'_k-x_k=y_k\end{array}\right.  } \rho_\sys \otimes \rho_\res^n
\left[ P_x^n U_{-t}^{\la,n} P_{x'}^n U_{t}^{\la,n}P_x^n
\right]\eeq where $\bbP_{\la,t,\rho_\sys}^n (y)$ is set to zero
when the sum on the RHS is empty. The physical idea behind this
formula is clear: measure the energies $H_{\res_k}^n$ in the state
$\rho_\sys \otimes \rho_\res^n$  (thereby reducing the reservoirs
to $P^n_x$), then switch on the time evolution $U^{\la,n}_{t}$,
finally measure again (reducing the reservoirs to $P^n_{x'}$).


We use that the initial state commutes with the projections $P_x$,
i.e.\ \beq \label{def: commuting state}\rho_\sys \otimes
\rho_\res^n [P_x \,\cdot\,]=\rho_\sys \otimes \rho_\res^n
[\,\cdot\, P_x ], \eeq
 to rewrite the Laplace transform of the measure \eqref{def: finite measure} as
 \beq
\label{alternative analogue2} \sum_{y \in
\bbR^{|\caK|}}\e^{-\sum_{k \in \caK} \ka_k y_k}
\bbP_{\rho_\sys,t,\la}^n\left( y \right)= \rho_\sys \otimes
\rho_\res^n \left[ Y_{-\ka}^n U^{\la,n}_{-t} Y_\ka^n U^{\la,n}_{t}
\right].
 \eeq
 where $\ka \in \bbR^{|\caK|}$ and $Y_\ka^n =\e^{- \sum_{k \in \caK} \ka_k H_{\res_k}^n   }
 $.

 The
infinite-volume analogue of \eqref{alternative analogue2},
to be defined through Lemma \eqref{lem: definition W}, is the subject of our main result: Theorem \ref{thm: 1}.











\subsection{Notation}\label{sec: notation}

For an indexed family of operators $J_{i=1}^m$, we assume the
ordering \beq \label{convention product} \prod_{i =1}^m J_i := J_m
J_{m-1} \ldots J_{2} J_{1} \eeq For a Hilbert space $\frh$, we
denote its symmetric (bosonical) second quantization by \beq \fock
(\frh):= \mathop{\oplus}\limits_{n \in \bbN}
\otimes_{\mathrm{s}}^n \frh \eeq where $\otimes_{\mathrm{s}}^n
\frh$ is the symmetrized $n$'th tensor power of $\frh$.  We define
the `vacuum vector' $\Psi \in \fock (\frh)$ as the unit vector in
$\bbC \sim \otimes_{\mathrm{s}}^{0} \frh$.

%For an operator $A$ and a subset $S \in \sp A$ we write $1_{S}(A)$
%for the spectral projection of $A$ on $S$.

When dealing with tensor products of Hilbert spaces, we will often
abbreviate $A\otimes 1$ and $1 \otimes B$ by respectively $A$ and
$B$.

If $\rho$  is a state (positive normalized functional) on a
$\caC^*$-algebra $\caA$, we use the notation $\rho [R]$ with $R
\in \caA$ for the value of the functional $\rho$ in $R$. If $\caA$
is finite-dimensional, we write $\tilde{\rho} \in \caA$ for the
associated density matrix, i.e.\ $\rho(R)= \Tr [\tilde{\rho} R]$.

If $A$ is a measurable subset of a measure space, we  write $
\mathrm{Ind}[A]$ for its characteristic function. (`Ind' from
`Indicator').

For $\ka \in \bbC^d$, we let $\re \ka, \im \ka \in \bbR^d$ be defined by
$(\re \ka)_i=\re \ka_i , (\im \ka)_i=\im \ka_i $ for $i=1,\ldots,d$.


\subsection{Outline}\label{sec: outline}

In Sections \ref{sec: zero temp} and \ref{sec: finite temp}, we state the model, which is the infinite-volume limit of the setup presented in Section \ref{sec: finite}.
The main result, Theorem \ref{thm: 1}, comes in Section \ref{sec: result}, some corollaries are given in Section \ref{sec: corollaries}.
An alternative way of reading the paper is by starting with Proposition \ref{prop: finite volume}, which builds on Section \ref{sec: finite} and allows to go immediately to the main result.

The proof is spread over Sections \ref{sec: proof of main theorem} and \ref{sec: proof of lemmas}.
In Section \ref{sec: proof of main theorem}, the general idea of the proof is given,
 necessary lemma's are stated and the connections between those lemma's are indicated.
In the technical Section \ref{sec: proof of lemmas}, all necessary
lemma's from Section \ref{sec: proof of main theorem} are proven.
Proofs of the corollaries are also  contained in Section \ref{sec:
proof of lemmas}.



%Define the complex domain (for $\delta>1$), $I_{D,\de}$
%as
% \beq I_{D,\de}:=\{ z \in \bbC^{|\caK|},  \mathrm{Re}\,   z \in D, \,  \|\mathrm{Im}\,    \ka\|
%< \delta \} \eeq

\section{Model and results}\label{sec: model}

\subsection{Zero-temperature Hamiltonian} \label{sec: zero temp}

Let $\caH_\sys$ be a finite-dimensional  Hilbert space with a
self-adjoint Hamiltonian $H_\sys$.
 To each reservoir $k \in \caK$, we associate
 \ben
\item{The self-adjoint
coupling operator $V_k \in  \caB(\caH_\sys)$ and an inverse temperature $\be_k > 0$.}
\item{The one-particle Hilbert space $\frh_k:= L^2(\bbR^+, \frl_k)$ for some separable Hilbert space
$\frl_k$ and its bosonical second quantization $\fock(\frh_k)$.
For $g \in \frh_k$, let  $a^*(g)/a(g)$ be creation/annihilation
operators on $ \fock(\frh_k) $ (see e.g.\ \cite{derzinski1}),
satisfying the canonical commutation relations. \beq
[a(g),a^*(g')]= \langle g,g' \rangle_{\frh_k},\qquad g,g' \in
\frh_k \eeq  }
\item{The coupling function $f_k \in \frh_k$ and the self-adjoint one-particle Hamiltonian $h_k$, acting
on $\frh_k=L^2(\bbR^+,\frl_k)$ as  multiplication by the variable
$x \in \bbR^+$, i.e.\ $(h_kg)(x)=x g(x)$. }
 \een
The total Hamiltonian with coupling constant $\la \in \bbR$,  \beq
H_\la= H_\sys + \sum_{k \in \caK} \d \Ga (h_k) +\la \sum_{k \in
\caK} V_k \otimes (a^*(f_k) +  a(f_k)) \eeq is formally defined on
a dense subspace of $\caH_\sys \otimes_{k \in \caK} \fock(\frh_k)$
(see Section \ref{sec: existence dynamics}).


\subsection{Finite-temperature dynamics} \label{sec: finite temp}


 Define the Hilbert spaces
\begin{eqnarray}
\frh &:=&\mathop{\oplus}\limits_{k \in \caK} (\frh_k \oplus \frh_k) \\
  \caH:&=&
\caH_\sys \otimes \fock (\frh)  \end{eqnarray} and  self-adjoint
operators
\begin{eqnarray}
l_k := &(-h_k) \oplus h_k  \quad &\textrm{on} \quad  \frh_k \oplus
\frh_k    \label{def: space lk}
\\
l :=&\mathop{\oplus} \limits_{k \in \caK} l_k  \quad  &
\textrm{on} \quad  \frh   \label{def: space l}
\end{eqnarray}

Identify \beq \frh_k \oplus \frh_k \sim L^2(\bbR^-,\frl_k) \oplus
L^2(\bbR^+,\frl_k) \sim L^2(\bbR,\frl_k) \eeq Remark that $l_k$,
as defined by \eqref{def: space lk}, acts on $L^2(\bbR,\frl_k)$ as
$(l_k g)(x)=x g(x)$.  Define the effective coupling functions
$\breve{f}_k : \bbR \to \frl_k$ by
 \beq \label{def: effective coupling} \breve{f}_k(x):=\left\{ \begin{array}{ll}
 \frac{1}{\sqrt{1-e^{-\beta_k x}}} f_k(x)  & x>0  \\[2mm]
 \frac{1}{\sqrt{e^{-\beta_k x}-1}}
\overline{f_k(-x)}  & x <0
\end{array} \right. \eeq
where  we equipped the Hilbert spaces  $\frl_k$ with a complex
conjugation $\frl_k \ni v \to \overline{v}$. By the forthcoming
Assumption \ref{ass: 1}, we have  $\breve{f}_k \in
L^2(\bbR,\frl_k)$.

By the natural embedding of $\frh_k \oplus \frh_k$ into $\frh$, we
will sometimes consider $\breve{f}_k$ as an element of $\frh$.
 We are ready
to write the formal \emph{semistandard Liouvillian}  on $\caH$
with coupling constant $\la \in \bbR$ \beq \label{def:
semiliouville}L_\la =H_\sys + \d \Ga (l)+ \la \sum_{k \in \caK}
V_k\otimes (a^*(\breve{f}_k)+ a(\breve{f}_k) )
 \eeq
This operator can easily be constructed rigorously as a
self-adjoint operator on a dense subspace of $\caH$, see Section
\ref{sec: existence dynamics}. Let $\rho_\sys$ be an arbitrary
state (see Section \ref{sec: notation}) on $\caB(\caH_\sys)$ and
let $\rho_\res$ be the vacuum state on $\caB(\fock (\frh))$, i.e.\
\beq \rho_\res \left[ \cdot \right]= \langle \Psi, \cdot   \Psi
\rangle \eeq where $\Psi$ is the vacuum vector in $\fock (\frh)$.
Our initial state on $\caB(\caH)$ will be $ \rho_\sys \otimes
\rho_\res $.

The relevance of the functions $\breve{f}_k$ and the Liouvillian
\eqref{def: semiliouville} in combination with the vacuum state
$\rho_\res$ is that \eqref{def: semiliouville} generates a
$W^*$-dynamics on an appropriate subalgebra (the Araki-Woods
$W^*$-algebra) of $\caB(\caH)$, on which $\rho_\res$ represents a
product of thermal states (at least on the reservoir part). This
elegant theory of Araki-Woods representations is extensively
discussed in the literature, see \cite{derzinski1}. We don't
pursue it here since it is not necessary to state the result. In
fact, we could even completely ignore the above setup and state
our result starting from finite-volume quantum systems. This is
illustrated in Proposition \ref{prop: finite volume}.


Define the following unitaries for $ t \in \bbR, \ka \in (\i
\bbR)^{|\caK|}$, \beq U_{t}^{\la}=\e^{-\i t L_\la },\qquad Y_\ka =
\e^{-\sum_{k \in \caK} \ka_k \d \Ga (l_k)} \eeq
%
%
We are ready to define our object of interest.
%
%
\begin{lemma}\label{lem: definition W}
 Assume there is a $D \subset \bbR^{|\caK|}$ with $0 \in D$ such that for $\ka
\in D $ and $k \in \caK$, \beq \label{eq: ass laplace} \langle \breve{f}_k, \e^{-\ka_k l_k}
 \breve{f}_k \rangle_{\frh} < \infty \eeq Than the functions \beq \label{def: functions analytic cont}\ka \longrightarrow
\left\{\begin{array}{l} \rho_\sys \otimes \rho_\res \left[
Y_{-\ka} U_{-t}^{\la} Y_\ka U_{t}^{\la} \right] \\[2mm]  \rho_\sys
\otimes \rho_\res \left[ U_{-s}^{\la} Y_{-\ka} U_{-t}^{\la} Y_\ka
U_{t}^{\la} U_{s}^{\la} \right] \end{array} \right.
 \eeq  have an analytical continuation from
 $(\i \bbR)^{|\caK|}$ into $D^\circ+(\i \bbR)^{|\caK|}$.
\end{lemma}


\subsection{Result}\label{sec: result}

To state our assumptions, we need to introduce some terminology.
For $e \in \sp H_\sys$, let $1_e(H_\sys)$ be the spectral
projection on $e$, corresponding to $H_\sys$. Let $ \caF:= \sp H_\sys -\sp
H_\sys $ and  define for $\om \in \caF$ and $V_k$ as in Section \ref{sec: zero temp},
 \beq V_{\om,k} := \sum_{e-e'=\om} 1_e(H_\sys) V_k
1_{e'}(H_\sys)\eeq
Denote for a subset $\caC \subset
\caB(\caH_\sys)$, the commutant $\caC'$ by \beq
 \caC' =\big\{ B \in \caB(\caH_\sys)\,\big | \,  \forall C \in \caC: \, [C,B]=0
 \big\} \eeq
%
%
%
%
 The first assumption expresses that the coupling is sufficiently effective.
\begin{assumption}\label{ass: wc}
\emph{ The functions $\breve{f}_k$ as defined in \eqref{def:
effective coupling} are continuous in $\caF$ and  \beq \left\{
V_{\om,k}  \, \big|\,   k \in \caK,\om \in \caF, \, \|
\breve{f}_k(\om) \| \neq 0  \right\} ' = \bbC 1\eeq }
\end{assumption}

The second assumption requires the reservoir correlation functions
to decay exponentially.


\begin{assumption}\label{ass: 1}
 \emph{The bound \eqref{eq: ass laplace} holds for a compact $D \subset \bbR^{|\caK|}$ with $0 \in D$. There
are constants $\al,c>0$ such that for $\ka \in D$, $k \in \caK$
and $t \in \bbR$, \beq \left|\langle \breve{f}_k, \e^{(-\ka_k-\i
t) l_k}
 \breve{f}_k \rangle_{\frh} \right|
< c \e^{-\alpha |t|} \eeq }
\end{assumption}





Our main result establishes existence and analyticity of the
large-deviation generating function:
\begin{theorem}\label{thm: 1}
Assume Assumptions \ref{ass: wc} and \ref{ass: 1}. There is a
$\la_0>0 $ such that for $ |\la| \leq |\la_0| $ and $\ka \in D$,
\beq F(\ka,\la):= \lim_{t \uparrow \infty} \frac{1}{t}\log
\rho_\sys \otimes \rho_\res \left[  Y_{-\ka} U_{-t}^{\la} Y_\ka
U_{t}^{\la} \right] \eeq exists and is independent of $\rho_\sys$.
The function  $D^\circ \ni \ka   \mapsto F(\ka,\la)$ is
real-analytic.


\end{theorem}

The previous result assumes a specific (decoupled) initial state
$\rho_\sys \otimes \rho_\res$. The result is however still valid,
when starting from an initial state, which has evolved from the
decoupled state $\rho_\sys \otimes \rho_\res$.
\begin{theorem}\label{thm: 2}
Assume the conditions of Theorem \ref{thm: 1}, then for all $s
\leq 0$, \beq F(\ka,\la)= \lim_{t \uparrow \infty} \frac{1}{t}\log
\rho_\sys \otimes \rho_\res \left[ U_{-s}^{\la} Y_{-\ka}
U_{-t}^{\la} Y_\ka U_{t}^{\la} U_{s}^{\la} \right] \eeq
\end{theorem}
\begin{vetremark}
From the proof, it follows that \beq F(\ka,\la)= \la^{2}e_\ka +
o(\la^{2}), \qquad \la \rightarrow 0 \eeq where $e_\ka$ is the
generating function obtained by starting from the weak coupling
limit \beq e_\ka:= \lim_{t \uparrow \infty} \lim_{\la \downarrow
0} \frac{1}{  t}\log \rho_\sys \otimes \rho_\res \left[ Y_{-\ka}
U_{- \lakl t}^{\la} Y_\ka U_{\lakl t}^{\la}  \right] \eeq (see
\cite{deroeckmaesfluct,derezinskideroekmaes} for a discussion).
Theorem \ref{thm: 1} hence implies that one can interchange the
limits $\la \searrow 0$ and $t \nearrow \infty$.
\end{vetremark}

\begin{vetremark}

Theorem \ref{thm: 2}, although mathematically more appealing than
Theorem \ref{thm: 1},
 does not have a clear physical meaning. This is because  states like
 \beq
\rho_\sys \otimes \rho_\res \left[U_{-s}^{\la}\cdot U_{s}^{\la} \right]
 \eeq
 in general do not commute\footnote{in the sense of \eqref{def: commuting state}} with functions of $\d \Ga (l_k)$,
 and hence one cannot repeat the construction of Section \ref{sec:
 finite}. For the same reason, we do not attempt to redo Theorem
 \ref{thm: 1}, starting from the NESS
 (nonequilibrium steady state), which is not considered in the present article.
\end{vetremark}

\begin{vetremark}\label{rem: bochner}
By Bochner's theorem, there is a nonnegative Borel measure
$\bbP_{\rho_\sys,t,\la}$ on $\bbR^{|\caK|}$ such that
 \beq \rho_\sys \otimes
\rho_\res [ Y_{-\i \ka} U^\la_{-t} Y_{\i \ka} U^\la_t  ] =
\int_{\bbR^{|\caK|}} \d \bbP_{\rho_\sys,t,\la}(y)  \e^{-\i \sum_{k
\in \caK}\ka_k y_k }
 \eeq
for $ \ka \in  \bbR^{|\caK|}$. Plugging in $\ka=0$, one sees that
$\bbP_{\rho_\sys,t,\la}$ is a probability measure. The variables
$y_{k \in \caK}$ represent the total heat flow into the $k$'th
reservoir, rather than the time-averaged heat flow, as in Section
\ref{sec: large deviations}. However, by the factor $t$ in the
exponent of \eqref{def: intro generating function}, the generating
function $F(\ka,\la)$ in Theorem \ref{thm: 1} is the exact
analogue of that in Section \ref{sec: large deviations}.
\end{vetremark}

\begin{vetremark}
If $\breve{f}_k \in L^2(\bbR,\frl_k)$, then one checks
straightforwardly that the bound \eqref{eq: ass laplace} is
automatically satisfied for $0\leq \ka_k \leq \be_k$.
Alternatively, one could construct the functions \eqref{def:
functions analytic cont} for these values of $\ka$ by using the
KMS-condition.
\end{vetremark}

\begin{vetremark}
Assumption \ref{ass: 1} cannot hold for $D=\bbR^{|\caK|}$, for that would imply that
\beq
\bbC \ni \ka \to\langle \breve{f}_k, \e^{-\ka_k l_k}
 \breve{f}_k \rangle_{\frh_k \oplus \frh_k}
\eeq
is analytic and bounded, hence constant.
\end{vetremark}




\subsection{Corollaries}\label{sec: corollaries}

We describe some straightforward, though physically relevant
corollaries to our main result.

It is convenient to introduce an additional assumption, which
expresses that the model has an additional symmetry (traditionally
this symmetry is time-reversal invariance)
\begin{assumption}\label{ass: timereversal}
There is an  antiunitary operator $\Theta$ on $\caH$, satisfying
\beq \Theta^2=1,\qquad  \Theta \Psi=\Psi,\qquad    \Theta L_\la
\Theta =L_\la , \qquad   \Theta \d \Ga(l_k)  \Theta  =\d \Ga(l_k)
\eeq
\end{assumption}
One derives the analog of the Gallavotti-Cohen fluctuation theorem
\begin{corollary} \label{cor: gc}
Assume the conditions of Theorem \ref{thm: 1} and  Assumption \ref{ass: timereversal}.  Define for $\zeta \in \bbR$, the vector
$\ka(\zeta) \in \bbR^{|\caK|}$ by \beq \label{def: kazeta} \ka_k(\zeta):=
\zeta \beta_k \eeq
Then, for $\zeta$ such that $\ka(\zeta),\ka( 1-\zeta) \in D$, \beq \label{gc equality} F(\ka(\zeta),\la)=F(\ka(1-\zeta),\la) \eeq
\end{corollary}

By standard methods, the large deviation principle with analytic
rate function yields a central limit theorem.
\begin{corollary} \label{cor: central limit}
Assume  the conditions of Theorem \ref{thm: 1} for a domain $D
\subset \bbR^{|\caK|}$ such that $0 \in D^\circ$. Let the
symmetric $|\caK|\times |\caK|$- matrix $M$ (covariance matrix) be
defined by
%
\beq M_{k,k'}= \frac{\partial^2}{\partial \ka_k \partial \ka_{k'}}
F(\ka,\la) \Big|_{\ka=0},   \eeq Recall the measure
 $\bbP_{\rho_\sys,t,\la}$ from Remark \ref{rem: bochner} and let
 $\bbE_{\rho_\sys,t,\la}$ be the associated expectation. Define
 the random variables
 \beq
Y_k^t:=  \frac{y_k - \bbE_{\rho_\sys,t,\la}[y_k]}{\sqrt{t}}
 \eeq
Then for $\ka \in \bbR^{|\caK|}$,   \beq \lim_{t \uparrow \infty}
  \bbE_{\rho_\sys,t,\la} \left[ \exp{\i \sum_{k \in \caK} \ka_k
Y_k^t } \right] =\exp{-\frac{1}{2}\left(\sum_{k,k' \in \caK} \ka_k
M_{k,k'} \ka_{k'} \right)} \eeq
\end{corollary}

The next proposition relates $\rho_\sys \otimes \rho_\res \left[
Y_{-\ka} U_{-t}^{\la} Y_\ka U_{t}^{\la}  \right]$ to finite-volume
quantities.
 It refers to notation introduced in Section \ref{sec: finite}.
Identify the $V_{k \in \caK},H_\sys \in \caB(\caH_\sys)$ and $\be_k > 0$
from Sections \ref{sec: finite} and \ref{sec: zero temp}.
%
%
\begin{proposition}\label{prop: finite volume}
Assume there is a $D \subset \bbR^{|\caK|}$  as in Lemma \ref{lem: definition W}.
 Assume that for all $\ka \in D $, $t \in \bbR$, $k \in \caK$ and for all choices $\zeta,\zeta' \in \{0,\ka_k \}$,
\begin{eqnarray} \label{ass: finite2a}  \rho_{k,\be_k}^n\left[ a^*( \e^{(-\i t+\zeta) h_{k}^n}f_{k}^n) a( \e^{-\overline{\zeta'} h_{k}^n} f_{k}^n)\right]
&\mathop{\longrightarrow}\limits_{n\uparrow \infty}& \langle f_k, (\e^{\be_k
h_{k}}-1)^{-1}\e^{(-\i t +\zeta-\zeta' )h_{k}} f_k \rangle_{\frh_k} \\
\label{ass: finite2b}
\rho_{k,\be_k}^n\left[ a(\e^{(-\i t-\overline{\zeta}) h_{k}^n} f_{k}^n) a^*( \e^{\zeta' h_{k}^n}
f_{k}^n)\right]& \mathop{\longrightarrow}\limits_{n\uparrow \infty} & \langle f_k, (1-\e^{-\be_k h_{k}})^{-1}\e^{(\i t -\zeta+\zeta' ) h_{k}} f_k \rangle_{\frh_k}.
\end{eqnarray}
 Then,
 \beq \label{eq: convergence finite}
\lim_{n \uparrow \infty}\rho_\sys\otimes \rho_\res^n \left[
Y_{-\ka}^n U^{\la,n}_{-t} Y_\ka^n U^{\la,n}_{t} \right]= \rho_\sys
\otimes \rho_\res \left[ Y_{-\ka} U_{-t}^{\la} Y_\ka U_{t}^{\la}
 \right] \eeq
\end{proposition}




\section{Proof of main theorem} \label{sec: proof of main theorem}



\subsection{The ``path space"} \label{sec: guichardet}
For a  (possibly unbounded) interval $I$, we write $I_2:=\{ \eta
\subset I \, | |\eta|=2 \} $ for the set of unordered pairs in
$I$. ($|\eta|$ is the number of elements in the set $\eta$.)
Introduce the set $\dsig_{I}$ as
 \beq  \dsig_I:= \mathop{\cup}\limits_{n \in \bbN}\dsig_{I,n},\qquad \dsig_{I,n} :=  \left\{ \dsi \subset (I_2 \times \{\mathrm{a},\mathrm{b} \})  \, \big| \,  |\dsi| =n \right\} \eeq
%
 We will
write
 $\dsig$ to denote $\dsig_{(-\infty,\infty)}$.
%
%%
%For $A,B \subset \dsig$, we define the product $A \times_\dsi B $ by
%\beq
%\dsi \in A \times_\dsi B \Leftrightarrow \left(\exists \dsi_1 \in A,\dsi_2 \in B: \dsi=  \dsi_1 \cup \dsi_2 \right)
%\eeq
%
%
For an interval $I$ and $\dsi \in \dsig$ define $\dsi_I \in
\dsig_I$ as \beq \dsi_I := \dsi \cap (I_2 \times
\{\mathrm{a},\mathrm{b} \}) \eeq
%
For intervals $I_1,I_2$ such that $I_1 \cup I_2$ is a disconnected
set, we define \beq  \dsi_{I_1 \cup I_2}:= \dsi_{I_1} \cup
\dsi_{I_2}, \qquad  \dsig_{I_1 \cup I_2} := \{\dsi \cup \dsi' \, |
\dsi \in \dsig_{I_1}, \dsi'\in \dsig_{I_2}\} \eeq Note however
that if $I:=I_1 \cup I_2$ is connected (hence an interval),
$\{\dsi \cup \dsi' \, | \dsi \in \dsig_{I_1}, \dsi'\in
\dsig_{I_2}\}$ is in general a strict subset of $\dsig_I$.
%
%
We use a special notation for the singletons in $\dsig_I$, \beq
\basig_I:= \dsig_{I,1}\qquad  \textrm{with elements}\qquad  \basi
\in \basig_I\eeq
%
%
and we slightly abuse notation by writing $\basi \in \dsi$
whenever $\basi \in \basig$ and $\dsi \cap \basi=\basi$. (The
`abuse' lies in confusing a singleton set with its only element)

 Define  $\varphi_1 : \bbR \times \bbR \times
\{\mathrm{a},\mathrm{b} \} \mapsto \basig $ by putting
$\varphi_1(s_1,t_i,\ell_1):=\{ (\{s_1,t_1 \},\ell_1) \}$ for
$s_1,t_1 \in \bbR$ and $\ell_1 \in \{\mathrm{a},\mathrm{b} \}$.
More generally, for \beq (s_1,t_1,\ell_1; \ldots; s_n,t_n,\ell_n)
\in  (\bbR \times \bbR \times \{\mathrm{a},\mathrm{b} \})^n, \eeq
we define \beq \varphi_n( s_1,t_1,\ell_1; \ldots; s_n,t_n,\ell_n)
:= \mathop{\cup}\limits_{i=1}^n \varphi_1( s_i,t_i,\ell_i ) \in
\dsig_n . \eeq

We now make $\dsig$ into a measure space.
 Let $\mu$ be the
(Lesbegue $\times$ Lesbegue $\times$ counting)- measure on
$\bbR\times \bbR \times \{\mathrm{a},\mathrm{b} \}$ and let
$\mu_n$ be its $n$-fold product on $(\bbR \times \bbR \times
\{\mathrm{a},\mathrm{b} \})^n$. Let $\caA$ be the $\sigma$-algebra
on $\dsig$ generated by the sets \beq \mathop{\bigcup}\limits_{n
\geq 1} \{ \varphi_n (A_n)\, | \, A_n \, \textrm{is a Borel subset
of } (\bbR\times \bbR \times \{\mathrm{a},\mathrm{b} \})^n \} \eeq
and finally we define the measure space $(\dsig,\caA,\mu_\dsi)$ by
putting, with $A_n$ as above,
\begin{eqnarray}
               \mu_\dsi(\emptyset)              & = &             1\\
               \mu_\dsi( \varphi_n (A_n))       & = &            \frac{1}{2^n n!} \mu_n \left(
               \varphi_n^{-1}\varphi_n (A_n) \right)
\end{eqnarray}

For convenience, we choose one element of
$\varphi_{|\dsi|}^{-1}(\dsi)$ to paramatrize $\dsi$, namely the
element satisfying $s_i \leq t_i$ for  $i=1,\ldots,|\dsi|$ and
$s_i\leq s_{i+1}$ for all $i=1,\ldots,|\dsi|-1$. The
$s_i,t_i,\ell_i$ thus determined will be indicated as
$s_i(\dsi),t_i(\dsi),\ell_i(\dsi)$ and simply
$s(\basi),t(\basi),\ell(\basi)$ for $\basi \in \basig$.
 We will
abbreviate $\d \mu_\dsi (\dsi)$ as $\d \dsi$. Note that \beq \d
\dsi =   \prod_{i=1}^{|\dsi|} \d s_i(\dsi) \d t_i(\dsi) \d
\ell_i(\dsi) \eeq where $\d \ell_i(\dsi)$ refers to the counting
measure on $\{\mathrm{a},\mathrm{b} \}$.


We define the \emph{support} of $\dsi \in \dsig$ as
\beq   \supp
(\dsi \in \dsig) = \mathop{\cup}\limits_{\basi \in \dsi } \supp (\basi),\qquad \supp (\basi \in \basig)=[s(\basi),t(\basi)],
\eeq





%Throughout the proof, we take a compact $D_c \subset D$ such that $0 \in D_c^\circ$.
%The statement for general $D$ as in \ref{thm: 1} and \ref{thm: 2} naturally follows.








\subsection{Perturbation expansion: the Duhamel series} \label{sec: duhamel}

\subsubsection{Existence of dynamics}\label{sec: existence dynamics}

Let $P_n$ be the projector on $\caH_\sys \otimes
\otimes_{\mathrm{s}}^n \frh$, the $n$-particle subspace of
$\caH_\sys \otimes \fock(\frh) \subset \caH$. Define the subspace
$\caD \subset \caH_\sys \otimes \fock(\frh)$ as  \beq \psi \in
\caD \Leftrightarrow \exists c>0: \, \| P_n \psi \| \leq
\frac{c^n}{\sqrt{n!}}, \, n \in \bbN \eeq Fix $\ka \in
\bbC^{|\caK|}$, let $L_0$ be as in \eqref{def: semiliouville} with
$\la=0$, and define \beq A_{\ka}(t):=  \e^{\i t L_0} \left(\sum_{k
\in \caK} V_k \otimes(a( \e^{
 -\overline{\ka_k} l_k}  \breve{f}_k) + a^*(\e^{
 \ka_k l_k} \breve{f}_k)) \right) \e^{-\i t L_0} \eeq
The following lemma is Theorem 6.1 in
\cite{derezinskideroeck2}.
\begin{lemma}\label{lem: existence dynamics}
  For $\psi \in
\caD$ and $\ka \in (\i \bbR)^{|\caK|} $, the series \beq
\label{eq: proof dyson} \e^{-\i t (L_0 +\la A_{\ka}(0)) }\psi :=
\e^{-\i t L_0}\sum_{n\geq 0} (-\i \la)^n \int_{0\leq t_1\leq
\ldots \leq t_n \leq t} \d t_1 \ldots \d t_n A_{\ka}(t_n) \ldots
A_{\ka}(t_1) \psi \eeq is absolutely convergent and $\e^{-\i t
(L_0 +\la A_{\ka}(0)) }\psi \in \caD$. The one-parameter family $t
\to \e^{-\i t (L_0 +\la A_{\ka}(0)) }$
 extends to a strongly continuous unitary group on $\caH$.
\end{lemma}
We recognize $ L_\la= L_0+\la A_{0}(0) $  with $L_\la$ as in
\eqref{def: semiliouville}.  This proves the existence of the dynamics $U_t^\la$.


The pull-through formula yields for $\ka \in (\i \bbR)^{|\caK|}$,
 \beq Y_{-\ka} U^\la_{-t} Y_{\ka} =   \e^{ \i t (L_0+ \la A_\ka(0))}.  \eeq
 Recall the vacuum vector $\Psi$ and remark that
obviously $\caH_\sys \otimes \Psi\in \caD$. Hence one can expand
the functions \eqref{def: functions analytic cont} for $\ka \in (\i \bbR)^{|\caK|}$  in a series involving
$A_{0}(u)$ and $ A_{\ka }(u)$ for $\tinit \leq u\leq \tfinal
$. Splitting the interaction by $V_k= \sum_{\om\in \caF} V_{\om,k}$ and using Wick's theorem,
 one arrives at  the equality \eqref{eq: W1}.

Note that Lemma \ref{lem: existence dynamics} can be extended to
some values of $\ka \in \bbC^{|\caK|}\setminus (\i\bbR)^{|\caK|}$
(sacrificing of course boundedness and unitarity of \eqref{eq:
proof dyson})
 and hence one could construct the functions \eqref{def: functions analytic cont} in a more direct way.
However, analytical continuation works on a larger domain.

\subsubsection{The Duhamel series as $\caB(\caB(\caH_\sys))$-valued integral over $\dsig$} \label{sec: duhamel as integral}



Let, for  $n \in \bbN$, $Pair(2n)$ be the set of all partitions,
(denoted by $\pi$), of $\{1,\ldots, 2n \}$ into $n$ unordered
pairs $\{p,q\}$ with $p < q$. That is, $\pi \subset
2^{\{1,\ldots,2n\}}$ belongs to $\mathrm{Pair}(2n)$ iff. \beq \pi
= \{ \{p_1,q_1\}, \{p_2,q_2\},\ldots, \{p_n,q_n\}  \}, \qquad
\mathop{\cup}\limits_{i=1}^n \{p_i,q_i\}= \{1,\ldots, 2n \}  \eeq
with the convention that $ p_i <q_i$ for $i=1,\ldots,n$.

Fix $\tfinal \geq 0\geq \tinit $ and let $\caX:= [\tinit,\tfinal]
\times \caF \times \caK$.

We denote by `$x$' elements of $\caX^n$ for any $n \in \bbN_0$
with components $(u_i,\om_i ,k_i) \in [\tinit,\tfinal]
\times\caF\times\caK,\, i=1,\ldots,n$.
%
The measure $\d^n x$ abbreviates
$\prod_{i=1}^n  \d u_i  \d \om_i \d k_i$ where the last two differentials $ \d \om_i \d k_i$ refer to the counting measure on resp. $ \caF , \caK$.
%
%
%          LEMMA DYSON
%

We have now the necessary notation to write an expansion for the
functions  \eqref{def: functions analytic cont}. For an arbitrary
state $\rho_\sys$ and $\la \in \bbR$;
\begin{eqnarray}\label{eq: W1}
& \rho_\sys \otimes \rho_\res \left[ U_{-\lakl \tinit}^\la
Y_{-\ka} U_{-\lakl \tfinal}^\la
Y_\ka U_{\lakl \tfinal}^\la U_{\lakl \tinit}^\la \right]= & \\
&\mathop{\sum}\limits_{n \in 2\bbN} \la^{-n}
\mathop{\sum}\limits_{m=0}\limits^{n} \i^{n-m} \int_{\caX^n}
\d^n x \, \mathrm{Ind}[ u_1 < \ldots <
u_{m}]\,\, \mathrm{Ind}[  u_{m+1} > \ldots > u_n] \,
&  \nonumber \\[3mm]
 & \Tr \left[ \left( \prod_{i=1}^m
 V_{\omega_i,k_i} \right) \, \, \tilde{\rho}_\sys \, \,   \left(\prod_{i=1}^{n-m} V_{\omega_{n+1-i},k_{n+1-i}}\right)  \right] & \nonumber\\[3mm]
&   \mathop{\sum}\limits_{\pi \in Pair(n)}
\mathop{\prod}\limits_{\{p,q\} \in \pi }
  \quad  Q_{\ka,\la}(x, \{p,q\},m)
    &
 \nonumber
\end{eqnarray}
where we abbreviated\footnote{Remark that we have written $l$
instead of $l_k$, which is justified since $f_k$ is embedded into
$\frh$.}
\begin{eqnarray}&   Q_{\ka,\la}(x,\{p,q\},m)= \de_{k_p,k_q}
 &  \nonumber \\&
  \left\langle  \e^{ -\overline{\zeta(x,q,m)}l} \e^{-\i  \lakl u_q(l + \om_q)}  \breve{f}_{k_q}, \quad
     \e^{ \zeta(x,p,m)l}  \e^{-\i  \lakl u_p(l - \om_p)}  \,  \breve{f}_{k_p}\right\rangle_{\frh}  \label{def: Q}&
\end{eqnarray}
with, for $x \in \caX^n$ and $j,m \in \{1,\ldots,n \}$,
\beq \zeta (x,j,m) := \left\{
\begin{array}{lll}
 \ka_{k_j} & \textrm{if}  \quad (   0<u_j  <
 \tfinal \quad & \textrm{and} \quad  j > m ) \\
 0 & \textrm{otherwise} &
\end{array} \right. \eeq


For $\ka \in (\i \bbR)^{|\caK|}$, the equality \eqref{eq: W1} follows from Section \ref{sec: existence dynamics}. In Section \ref{sec: proof of expansions}, we check that it can be analytically continued and hence, from now on, we assume  \eqref{eq: W1} for $\ka  \in D+ (\i \bbR)^{|\caK|}$.



We now connect the concepts and notation of Section \ref{sec:
guichardet} with those used in \eqref{eq: W1}.
%
%a $x \in \caX^{2n}$ can be associated to some elements $\dsi \in \dsig_n$. We write
%\beq
%\dsi \sim x \Leftrightarrow \{u_1,\ldots,u_n  \} = \cup_{\basi \in \dsi} \{s(\basi),t(\basi)\}
%\eeq
%
First we remark that a pair $(x,\pi) ,x \in \caX^{2n}, \pi \in Pair(2n)$ is associated to $2^{n}$ elements of $\dsig_n$, we write
\beq
\dsi \sim (x,\pi) \Leftrightarrow \forall \{p,q \}\in \pi, \exists \basi \in \dsi: \,   \{u_{p},u_{q}  \} =  \{s(\basi),t(\basi)\}
\eeq


%and $\pi Pair(2n)$ determine couple of  elements $\dsi \in \dsig_n$.
%Indeed, for $u_p,u_q$ (components of $x$) such that $\{p,q \} \in \pi$, we have
%\beq \{ \{u_p,u_q \}, \textrm{a} \}, \{ \{u_p,u_q \}, \textrm{b} \} \in \basig \eeq
%For a $\dsi \in \dsig$, we can make an element of $Pair(2|\dsi|)$ by letting
%$p_i := s_i(\dsi)$ and $q_i:=t_i(\dsi) $ for $i=1,\ldots,|\dsi|$. We denote this element by $\mathrm{Pr}_{\bbR_2}(\dsi)$.


 Let $(\de_\ep)_{\ep>0}$ be a
family of $C^{\infty}$ functions with compact support on $\bbR$
such that \beq \int_{\bbR}\d x \de_\ep (x)=2,\qquad
\de_\ep(x)=\de_\ep(-x)\geq 0 \eeq and for all bounded, measurable functions
$f$ on $\bbR$, which are right-continuous at $0$, \beq \lim_{\ep
\downarrow 0}\int_{\bbR^+} \d x f(x) \de_\ep (x)=\lim_{x
\downarrow 0} f(x) \eeq
%
Define, analogously to \eqref{def: Q},
\begin{eqnarray}&   Q^{\mathrm{a}}_{\ka,\la}(x,\{p,q\},m)= \de_{k_p,k_q} \de_{\om_p,-\om_q}
\int_{\bbR^+} \d u & \nonumber\\&  \left\langle  \e^{-\overline{\zeta(x,q,m)}l}
\breve{f}_{k_q},\,  \e^{-\i \lakl u (l-\om_p)
\mathrm{sign}(u_p-u_q)   }  \e^{ \zeta(x,p,m)l}  \,
\breve{f}_{k_p} \right\rangle_{\frh}  \label{def: Qa}&
\end{eqnarray}
which exists by Assumption \ref{ass: 1} (since $\| h\|_1 <\infty$ in the notation of Section \ref{sec: proof of expansions}).
For $\dsi \in \dsig_{[\tinit,\tfinal]}$ and $S \in \caB(\caH_\sys)$, and writing $n:= 2 |\dsi|$, we put
\begin{eqnarray}\label{eq: Y1} &W^\ep_{\ka,\la}(\dsi) S:= & \\
& \mathop{\sum}\limits_{\footnotesize{\left.\begin{array}{c} x \in \caX^{n},\pi \in Pair(n)\\ \dsi \sim (x,\pi)  \end{array}\right.}} \la^{-n}
\mathop{\sum}\limits_{m=0}\limits^{n} \i^{n-m}
 \, \mathrm{Ind}[ u_1 < \ldots <
u_{m}]\, \mathrm{Ind}[  u_{m+1} > \ldots > u_n]
&  \nonumber \\[1mm]
%
%
 & \Tr \left[ \left( \prod_{i=1}^m
 V_{\omega_i,k_i} \right) \, \, S \, \,   \left(\prod_{i=1}^{n-m} V_{\omega_{n+1-i},k_{n+1-i}}\right)  \right] & \nonumber\\[3mm]
 %
 %
&
\mathop{\prod}\limits_{\{p,q\} \in \pi }
  \Big(  \de_{\ell(\basi),\mathrm{a}}\,
 \big[\de_{\ep} ( \frac{t(\basi)-  s(\basi)}{\la^2})Q^{\mathrm{a}}_{\ka,\la}(x,\{p,q\},m) \big] \qquad \qquad \qquad \qquad &\nonumber\\
& \qquad \qquad \qquad \qquad  + \de_{\ell(\basi),\mathrm{b}} \,\big[Q_{\ka,\la}(x,\{p,q\},m)- \de_{\ep} ( \frac{t(\basi)-  s(\basi)}{\la^2})) Q^{\mathrm{a}}_{\ka,\la}(x,\{p,q\},m) \big]\Big)  & \nonumber
\end{eqnarray}
where the $\basi$ in the last two lines is the unique element $\basi \in \dsi$ such that $\{u_{p},u_{q}  \} =  \{s(\basi),t(\basi)\}$.
%$ \beq  g_\ka =
%\sum_{k \in \caK} \frac{f_k(x)\e^{\ka_k x}}{\sqrt{\e^{\be_k x}-1}}
%1_{x>0} + \frac{f_k(-x) \e^{\ka_k x}}{1-\sqrt{\e^{\be_k x}}}
%1_{x<0} \eeq and let $g := g_{\ka=0}$.

We state two important properties of the maps $W_{\ka,\la}^\ep (\dsi)$,
\beq \label{prop: factorization}
\max \supp (\dsi) \leq \min \supp (\dsi') \Longrightarrow  W_{\ka,\la}^\ep (\dsi \cup \dsi')=   W_{\ka,\la}^\ep (\dsi') W_{\ka,\la}^\ep (\dsi)
\eeq
\beq \label{eq: equality W=Y}  \textrm{LHS of \eqref{eq: W1}}= \int_{\dsig_{[\tinit,\tfinal]}} \d \dsi W^{\ep}_{\ka,\la}(\dsi)
\tilde{\rho}_\sys    \eeq
which follow from \eqref{eq: W1} and \eqref{eq: Y1}. (Note that \eqref{eq: equality W=Y} holds for all $\ep>0$.)

One can see in \eqref{eq: Y1}, the purpose of the coordinates
$\ell(\basi)=\mathrm{a},\mathrm{b}$. Each $\basi$  determines a
pair of time-instants (and hence interaction terms):
$s(\basi),t(\basi)$ between which a pairing occurs. If
$\ell(\basi)=\mathrm{a}$, these time-instants are constrained to
coincide (as $\ep \searrow 0$) and to have opposite values of $\om
\in \caF$. In  $\ell(\basi)=\mathrm{b}$, we collect the rest of
the term corresponding to this pairing. The reader who is familiar
with the \emph{weak coupling limit} from \cite{davies1},
recognizes the $\mathrm{a}$-terms as the ones contributing in the
limit $\la \searrow 0$.

\subsection{Time-discretized  polymers} \label{sec: time-discretized polymers}
The following construction depends on a time scale $\tau$. In
order to keep the notation manageable, we will drop this
time-scale from most of the notation.
\subsubsection{Primary polymers}\label{sec: primary polymers}
%
%
Let $j \in \bbZ$ and $\spo \subset \bbZ $ and put
 \beq I_j:= [\tau
j,\tau(j+1)], \qquad  I_\spo:= \cup_{j \in \spo} I_j \eeq
%
 Define a partial ordering  on $ \bbZ$ by \beq j
\prec j' \Leftrightarrow j \leq j'+2 \eeq The same symbol will be
used for subsets of $\spo,\spo' \subset  \bbZ$;  \beq \label{def:
order on subsets}\spo
 \prec \spo' \Leftrightarrow (\forall j \in \spo, \forall j' \in \spo': \ j\prec j')
\eeq Introduce also the compatibility relation  $\sim$ by \beq
\spo \sim \spo' \Leftrightarrow ( \spo \prec \spo' \quad
\textrm{or}\quad \spo' \prec \spo)\eeq

We define the set of \emph{primary polymers} $ \setfpo \subset 2^{
\bbZ}$. A $\fpo \in 2^{ \bbZ}$ belongs to $\setfpo$ iff.\ \ben
\item{For all $\fpo',\fpo'' \in 2^{ \bbZ}$ such that $\fpo =
\fpo' \cup \fpo''$: \beq
 \fpo' \sim \fpo'' \Longrightarrow \left(\fpo'=\emptyset
\quad \textrm{or}\quad \fpo''=\emptyset. \right)
 \eeq }
\item{ If $j \in \fpo$ and $j<-1$, then  $ j+1 \in \fpo$.} \een


 We now  construct a mapping
$ K: \dsig \to 2^{ \bbZ}$.
\begin{definition}
\emph{Let $M(\dsi) \subset  \bbZ$ be the union of all sets $\fpo
\subset \bbZ$ satisfying
 \beq \dsi_{I_\fpo}=\dsi \setminus \dsi_{\bbR \setminus
I_\fpo} \, \textrm{ and } \, \forall \basi \in \dsi_{I{\fpo}}:\, \ell(\basi)=\mathrm{a} \eeq
 Define $K(\dsi):= \bbZ \setminus M(\dsi)$ }
\end{definition}
We also construct operators on $\caB(\caH_\sys)$ indexed by $\fpo
\in 2^{ \bbZ}$ ;
 \beq \label{def: Wfpo} W_{\ka,\la}(\fpo)= \lim_{\ep \downarrow 0} \int_{K(\dsi)=\fpo} \d \dsi W^\ep_{\ka,\la}(\dsi) \eeq
%
See Section \ref{sec: proof of expansions} for well-definedness of
\eqref{def: Wfpo}.  The following lemma is our main technical tool
%
\begin{lemma}\label{lem: small exci}
Assume assumption \ref{ass: 1}.  There is a $c(\la) $  such that
for $\ka \in \{z \in \bbC |\, \re z \in D, |\im z| \leq 1   \}$
and for  $\fpo \in \setfpo$, \beq \| W_{\ka,\la}(\fpo) \| \leq
c(\la)^{|I_\fpo|} ,\qquad c(\la)
\mathop{\longrightarrow}\limits_{\la \searrow 0 } 0
 \eeq
% \item{ For each $\fpo \in
% \setfpo$, the function \beq  I_{D,1}  \ni  \ka \mapsto
% W_{\ka,\la}(\fpo) \eeq is analytic.}
\end{lemma}


\subsubsection{The reference dynamics} \label{sec: reference dynamics}
Recall the notation of Section \ref{sec: duhamel}, assume $\|h\|_1<\infty$ (see Section \ref{sec: proof of expansions}) and put  \beq \label{def: c}
c_\ka (\omega,k):=  \int_{\bbR^+} \d u \, \langle \breve{f}_k ,
\e^{-\i u (l_k-\om)} \e^{-\ka_k l_k} \breve{f}_k \rangle_{\frh_k
\oplus \frh_k} \eeq
%
%
%
Remark that, if $\breve{f}_k$ is continuous in $\caF$,  \beq \mathrm{Re}\,
c_\ka (\omega,k)= \| \breve{f}_k(\om) \|_{\frl_k}^2 \e^{-\ka_k \om}
\geq 0 \eeq and define for $S \in \caB(\caH_\sys)$:
%
%
%
 \beq \label{def: Lkappa}\caL_\ka
(S) := \sum_{\om \in \caF,k \in \caK} \left( 2 \mathrm{Re}\,   c_\ka(\om,k) V_{\om,k} S
V^*_{\om,k} - c_0(\om,k) S V^*_{\om,k} V_{\om,k} - \overline{c_0(\om,k)}
V^*_{\om,k} V_{\om,k} S \right) \eeq For $\ka \in \bbR^{|\caK|}$, the family $\e^{t \caL_{\ka}},
t \geq 0$ is a semigroup of completely positive maps on
$\caB(\caH_\sys)$, see e.g.\ \cite{lindblad}, or Theorem 3.4 in \cite{derezinskideroeck3}.  For $\ka=0$, it is
also trace-preserving. Its relevance  lies in the following
lemma's.
%
%
%
\begin{lemma}\label{lem: artificial wc}
Assume the bound \eqref{eq: ass laplace}, continuity of
$\breve{f}_k$ in $\caF$ and $\|h\|_1 <\infty$, then for $I \subset
\bbR^+$ \beq \label{eq: convergence reference}\lim_{\ep \searrow 0} \mathop{\int}\limits_{\left.
\footnotesize{
\begin{array}{c}\dsi \in \dsig_I \\
\ell_{1,\ldots,|\dsi|}(\dsi)= \mathrm{a}\end{array}} \right. }  \d \dsi
W_{\ka,\la}^\ep(\dsi)= \e^{|I |\caL_\ka} \eeq
\end{lemma}

\begin{lemma}\label{lem: frobenius}
Assume Assumption \ref{ass: wc} and the conditions of Lemma \ref{lem: artificial wc}.
For a compact  $D \in \bbR^{|\caK|}$, there are  $\de,g
>0$ such that for $\ka \in \{z \in \bbC |\, \re z \in D, |\im z| \leq \de   \}$, $\caL_\ka$ has a simple
eigenvalue $e_\ka$ and \beq    \inf  \big\{\mathrm{Re}\,   (e_\ka- x) \,
\big|\, x \in \sp \caL_\ka \setminus \{ e_\ka \} \big\} \geq
g \eeq The eigenvector $Z_\ka$  corresponding to $e_\ka$
can be chosen strictly positive.
\end{lemma}
%
We adopt the normalization $\Tr[ Z^*_\ka Z_\ka]=1 $. Let $P_{\ka}$
be the projection on $Z_\ka$ w.r.t.\ the scalar product given by
$\Tr[\cdot]$, i.e.\ \beq P_{\ka} S= Z_\ka \Tr\left[ Z_\ka^* S
\right],\qquad   S \in \caB(\caH_\sys)  \eeq and let $
R_\ka:=1-P_\ka $. Ordinary perturbation theory yields analyticity
of $\ka \to e_\ka,P_\ka$.



\subsubsection{Secondary polymers}\label{sec: secondary polymers}

It is good to pause now and justify  some concepts from the previous sections.


We  make some preliminary definitions, let for $\fpo \in \setfpo$,
  \beq \label{def: tilde primary polymers} \tilde{W}_{\ka,\la}(\fpo)=
\e^{-|I_\fpo| \caL_\ka}W_{\ka,\la}(\fpo) \eeq and  for $\tinit\leq
u <0$,  \beq   \tilde{\rho}_{\sys,u}:=
\int_{\dsig_{[\tinit,{u}]}} \d \dsi
W^{\ep}_{\ka,\la}(\dsi)\tilde{\rho}_\sys \eeq Remark that, by  the same
reasoning as the one leading to \eqref{eq: equality W=Y},
 \beq  \label{eq: resum initial} \Tr
\left[\tilde{\rho}_{\sys,u} S \right] = \rho_\sys \otimes \rho_\res \left [
U^\la_{\lakl(\tinit-u)} (S \otimes 1) U^\la_{\lakl(u-\tinit)}  \right]
\eeq In particular, $\Tr[\tilde{\rho}_{\sys,u}]=1$ and $\tilde{\rho}_{\sys,u}$ is a density matrix corresponding to a state $\rho_{\sys,u}$, which
is independent of $\ka$ and $\ep$.






Let $\tinit \leq 0 \leq \tfinal$ be such that  $\tau^{-1} \tinit,
\tau^{-1} \tfinal \in  \bbZ$ (this is a restriction on which we
will comment later). We write $\La:= \bbZ \cap
[\tau^{-1}\tinit,\tau^{-1}\tfinal-1]$. For an ordered sequence of
primary polymers $\fpo_1\prec\ldots \prec\fpo_n$ in $\setfpo \cap
2^{\La} $, we introduce \beq
%
 u_i= \left\{\begin{array}{ll}
%
\min \{\inf I_{\fpo_1},0 \} \quad  &\textrm{if}\quad i=0 \\
%
 \inf I_{\fpo_i} \quad  &\textrm{if}\quad  1\leq i \leq n \\
%
  \tfinal  \quad  &\textrm{if} \quad i =n+1
%
 \end{array}\right. \eeq
%
%
Starting from \eqref{eq: equality W=Y} and \eqref{def: Wfpo},
realizing that for every $\fpo \in 2^{ \bbZ}$, there is a unique
sequence $\fpo_1\prec\ldots \prec\fpo_n$ such that $\cup_{i=1}^n
\fpo_i=\fpo$, and using  Lemma \ref{lem: artificial wc} and
\eqref{eq: resum initial}, one derives
%
\begin{eqnarray}
&\rho_\sys \otimes \rho_\res \left[ U_{-\lakl \tinit}^\la
Y_{-\ka} U_{-\lakl \tfinal}^\la
Y_\ka U_{\lakl \tfinal}^\la U_{\lakl \tinit}^\la \right] = \Tr[ \e^{\tfinal \caL_\ka} \tilde{\rho}_{\sys,0}]&   \nonumber\\[2mm]
+& \mathop{\sum}\limits_{n \geq 1} \mathop{\sum}\limits_{\footnotesize{\left.\begin{array}{c} \fpo_1\prec\ldots \prec\fpo_n  \\ \fpo_{1,\ldots,n}\in \setfpo \cap 2^{\La} \end{array}\right. }}
\Tr \left[   \mathop{\prod}\limits_{i=1}^{n}
\e^{(u_{i+1}-u_{i}) \caL_\ka}   \tilde{W}_{\ka,\la}(\fpo_i) \, \times \, \e^{(u_1-u_0) \caL_\ka}
 \tilde{\rho}_{\sys,u}\right] & \label{def: first polymer expansion}
\end{eqnarray}
%
%
%
The trick is now to use the ergodicity of $\e^{t \caL_\ka}$, as
established by Lemma \ref{lem: frobenius}, to write the expansion
\eqref{def: first polymer expansion} as a product over independent
\emph{secondary polymers}, which will be collections of $\fpo \in
\setfpo$. To achieve this, we insert $1=P_\ka +R_\ka$ at times
$u_{i=0,\ldots,n+1}$. Consecutive primary polymers separated by
$R_\ka$ belong to the same secondary polymer. This is formalized
in the remainder of this Section.

We add two abstract elements, $-\varsigma$  and $\varsigma$ to $
\bbZ$ and we extend the partial ordering $\prec$ to $\bbZ \cup
\{-\varsigma,\varsigma \}$ by putting \beq       -\varsigma \nprec
-\varsigma,\qquad \varsigma \nprec \varsigma,\qquad \forall j \in
 \bbZ:\,  -\varsigma \prec j \prec \varsigma     \eeq The
extension to subsets of  $ \bbZ$ is done as in \eqref{def: order
on subsets}

Recall  $\La$ a defined above. We define the set of secondary polymers $\setspo_{\La} :=
2^{\La  \cup \{ -\varsigma,\varsigma \} } \setminus \{ \emptyset \} $.  Remark that every $\spo \in \setspo_{\La} $ has a
unique decomposition
%
%
\beq \label{parameters spo}  \spo=  \ep_- \cup \bigcup_{i=1}^n \fpo_i \cup \ep_+  \eeq
%
where  $\fpo_{1,\ldots,n} \in \setfpo $, $\fpo_1 \prec \ldots \prec \fpo_n$ and  $\ep_-=\emptyset, \{-\varsigma \} $ and $\ep_+=\emptyset,\{\varsigma\}$.
%
%
%
%
We  construct  \emph{polymer weights} $w^\La_{\ka,\la}(\spo)$ for $\spo \in
\setspo_\La$ using the parametrization \eqref{parameters spo}
\begin{eqnarray} &&  w^\La_{\ka,\la}(\spo=  \ep_- \cup \bigcup_{i=1}^n \fpo_i \cup \ep_+)
 =   \Tr \Bigg[W_{\ka}^{+}(\ep_+)   \times   \e^{(u_{n+1}-u_n)(\caL_\ka-e_\ka)}     \times  \nonumber\\ [1mm]
&&
\mathop{\prod}\limits_{i=2}^{n} \tilde{W}_{\ka,\la}(\fpo_i)R_\ka
\e^{(u_i-u_{i-1})(\caL_\ka-e_\ka)}  \times
  \tilde{W}_{\ka,\la}(\fpo_1) \e^{(u_1-u_0)(\caL_\ka-e_\ka)}   \times   W_{\ka}^{-}(\ep_-) \rho_{\sys,u_0  }  \Bigg] \label{polymer weight}
%
\end{eqnarray}
where
\beq
W_{\ka}^{+}(\ep_+)= \frac{1}{\Tr[Z_\ka]}\left\{ \begin{array}{ll}P_\ka \quad
\textrm{if} \quad \ep_+=\emptyset
\\R_\ka \quad
\textrm{if} \quad \ep_+=\{\varsigma\}
\end{array}\right., \qquad  W_{\ka}^{-}(\ep_-)= \frac{1}{\Tr[Z^*_\ka \tilde{\rho}_{\sys,0}]}\left\{\begin{array}{ll}P_\ka \quad
\textrm{if} \quad \ep_-=\emptyset
\\R_\ka \quad
\textrm{if} \quad \ep_-=\{-\varsigma\}
\end{array}\right.
\eeq
Remark that both $\Tr[Z_\ka],\Tr[Z^*_\ka \tilde{\rho}_{\sys,0}] >0$ by Lemma \ref{lem: frobenius}. Call
\beq b_\ka:= \Tr[Z_\ka]\Tr[Z^*_\ka \tilde{\rho}_{\sys,0}]  \eeq
For $n=0$, the definition \eqref{polymer weight} reduces to
\begin{eqnarray}w_{\ka,\la}^\La(\ep_- \cup \ep_+)&=& b_\ka^{-1}
\Tr [W_{\ka}^{+}(\ep_+)  \e^{\tfinal (\caL_{\ka}-e_\ka) } W_{\ka}^{-}(\ep_-)\tilde{\rho}_{\sys,0}]\nonumber\\
 &=& \de_{\ep_-,\emptyset}\de_{\ep_+,\emptyset}+  \de_{\ep_-,\{-\varsigma\}}\de_{\ep_+,\{\varsigma\}} b_\ka^{-1} \Tr [R_\ka  \e^{\tfinal (\caL_{\ka}-e_\ka) }R_\ka\tilde{\rho}_{\sys,0}]  \label{vanishing polymer weights}
\end{eqnarray}
which follows from $P_\ka R_\ka= R_\ka P_\ka=0$.

%
Note that the weights $w_{\ka,\la}^\La(\spo)$ depend weakly on $\rho_\sys$ (also via $\rho_{\sys,0}$).
However, for $\spo$ not containing $-\varsigma$ and $j<0$,
this dependence vanishes since  $(\Tr[Z^*_\ka \tilde{\rho}_{\sys,0}])^{-1} P_\ka \rho_{\sys,0}= Z_\ka$.
%
%
%
We finally write the polymer model starting from \eqref{def: first polymer expansion} and using the polymer weights \eqref{polymer weight},
\beq \label{def: polymer model}
\textrm{LHS of \eqref{def: first polymer expansion}} = \e^{\tfinal e_\ka} b_\ka \Big( 1+
\sum_{n \geq 1}
\mathop{\sum}\limits_{\footnotesize{\left.\begin{array}{c}\spo_1 \prec\ldots \prec\spo_n\\ \spo_{1,\ldots,n} \in \setspo_\La \\
\ \end{array}\right.}}
\prod_{i=1}^n w_{\ka,\la}^\La(\spo_i) \Big) \eeq


Note that the `$1$' corresponds to  $\spo=\emptyset$, which has
been excluded from $\setspo_\La$ but has weight $1$ by
\eqref{vanishing polymer weights}.




\subsection{Cluster expansion}\label{sec: cluster}

The expression \eqref{def: polymer model}  can be used to set up a
cluster expansion for the $\log$ of the LHS. We follow closely the
presentation of  cluster expansions in \cite{ueltschi}.

Let $\caC_n$ be the set of all (unoriented) connected graphs with $n$ vertices.
For a finite sequence $\spo_1,\ldots,\spo_n$ in  $\setspo_\La$, introduce the function
\beq
\phi(\spo_1,\ldots,\spo_n)= \frac{1}{n!}\left\{\begin{array}{ll} 1 \quad &\textrm{if}\quad n=1 \\[1mm]
\mathop{\sum}\limits_{G \in \caC_n} \mathop{\prod}\limits_{(i,j) \in G}  \mathrm{Ind}[\spo_i \nsim \spo_j] \quad &\textrm{if}\quad n>1   \end{array}\right.
\eeq
where $\prod_{(i,j) \in G}$ denotes the product over all edges $(i,j)$ of the graph $G$.


By standard combinatorics \cite{ueltschi}, \eqref{def: polymer model} yields
 \beq \label{def: cluster expansion}\log (\textrm{LHS of \eqref{def: first polymer expansion}})
=\tfinal e_\ka +\log{b_\ka} + \sum_{n\geq 1}  \sum_{\spo_1,\ldots,\spo_n \in  \setspo_\La}  \phi(\spo_1,\ldots,\spo_n)\prod_{i=1}^n   w^\La_{\ka,\la}(\spo_i)  \eeq
The sum on the RHS runs over sequences in $\setspo_\La$ with $n$ elements. As $\setspo_\La$ is finite, there is no
convergence issue.


To control the expansion \eqref{def: cluster expansion} as $\tfinal \nearrow \infty$ (recall that $\La$ also depends on $\tfinal$),
we apply a standard Theorem  which relies on the Kotecky-Preiss criterion
\cite{koteckypreiss}.


\begin{lemma}[Kotecky-Preiss criterion]\label{lem: kotecky}
Assume Assumptions \ref{ass: wc} and \ref{ass: 1}, and let
 $\de>0$ be as in Lemma \ref{lem: frobenius}. There is a
 $\la_0$ such that for $|\la |\leq \la_0$, $\ka \in \{ z \in \bbC^{|\caK|}  |\, \re \ka \in D, |\im \ka| \leq \delta \}$, and $\tfinal$ (and hence $\La$) large enough, there is a nonnegative function $\clustera$ on $\setspo_\La$, such that
 \ben

\item{
 \beq \label{kp bound1}
\sum_{\footnotesize{ \left. \begin{array}{c} \spo' \in \setspo_\La, \spo' \nsim \spo
\end{array} \right. }} |w^\La_{\ka,\la} (\spo')| \e^{\clustera(\spo')} < \clustera(\spo)
  \eeq
  }

\item{
 \beq
\sum_{\footnotesize{ \left. \begin{array}{c} \spo \in \setspo_\La
\end{array} \right. }}  |w^\La_{\ka,\la} (\spo)|  \e^{\clustera(\spo)} < \infty
  \eeq
 }
\een
Moreover, for each $\spo \in \setspo_\La$, $w^\La_{\ka,\la} (\spo)$ is analytic in $\ka \in \{ z \in \bbC^{|\caK|}  |\, \re \ka \in D, |\im \ka| \leq \delta \}$.

 \end{lemma}



By applying Theorem $1$ in \cite{ueltschi}, we conclude from
\eqref{def: polymer model} and Lemma \ref{lem: kotecky}, that
for every $\spo \in \setspo_\La$, the following bound holds
\beq \label{clusterbound}
v_{\ka,\la}^\La(\spo):=\sum_{n\geq 1} \sum_{\footnotesize{\left.\begin{array}{c}\spo_1,\ldots,\spo_n \in  \setspo_\La \\\exists i \in \{1,\ldots,n\}: \spo_i=\spo  \end{array}\right.}}  \phi(\spo_1,\ldots,\spo_n)
\prod_{i=1}^n  |  w^\La_{\ka,\la}(\spo_i)|  \leq \e^{\clustera(\spo)}w(\spo)
\eeq

Using first \eqref{clusterbound} and then \eqref{kp bound1},
\begin{eqnarray}
&& \sum_{n \geq 1}\sum_{\spo_1,\ldots,\spo_n \in  \setspo_\La}  \phi(\spo_1,\ldots,\spo_n)\prod_{i=1}^n   w^\La_{\ka,\la}(\spo_i)
\leq \sum_{j \in \La \cup \{ -\varsigma,\varsigma\} } \sum_{  \spo \in \setspo_{\La}, \spo \nsim \{j\}} v_{\ka,\la}^\La(\spo) \\
&\leq& \sum_{j \in \La \cup \{ -\varsigma,\varsigma\} } \sum_{ \spo \in \setspo_{\La}, \spo \nsim \{j\} } w_{\ka,\la}^\La(\spo) \e^{\clustera(\spo)}
\leq \sum_{j \in \La \cup \{ -\varsigma,\varsigma\} } a_{\La}(\{ j \}) \label{bound logaritm4}
\end{eqnarray}

By the proof of Lemma \ref{lem: kotecky}, we see that  $\clustera(\spo)$ on $\spo= \{j \}$ with $j \in \La \cup \{ -\varsigma,\varsigma\} $ is constant and independent of $\La$.
Hence, upon dividing by $\tfinal$, \eqref{bound logaritm4} is bounded.

We define the set of \emph{bulk polymers} $\setspobu \subset
\setspo_{\bbZ} $ as $\setspobu:= 2^{ \bbN}$. Remark that for each
$\spo \in \setspobu$, $\spo + m \in \setspobu$ for $m \in \bbN$.
For $\spo \in \setspobu \cap \setspo_\La$, we write simply
$w_{\ka,\la}(\spo):=w_{\ka,\la}^\La(\spo) $ (which is independent
of $\rho_\sys$, see  above \eqref{def: polymer model}). Define for
a sequence of polymers $\spo_1,\ldots,\spo_n$ in $\setspobu$. \beq
\eta_N(\spo_1,\ldots,\spo_n):=  \sum_{m=0}^{N-1}
\phi(\spo_1,\ldots,\spo_n) \prod_{i=1}^n w_{\ka,\la}(\spo_i+ m)
\eeq and notice that one could as well replace
$\phi(\spo_1,\ldots,\spo_n)$ by $\phi(\spo_1+ m,\ldots,\spo_n+
m)$.

We parameterize $\tfinal = N\tau$ (and hence $\La$) with $N \in
\bbN$. Analogous to \eqref{bound logaritm4},
\begin{eqnarray}
 &\Bigg| \mathop{\sum}\limits_{n\geq 1}  \mathop{\sum}\limits_{\spo_1,\ldots,\spo_n \in  \setspo_{\La}}  \phi(\spo_1,\ldots,\spo_n)\mathop{\prod}\limits_{i=1}^n   w^{\La}_{\ka,\la}(\spo_i)  -\mathop{\sum}\limits_{n\geq 1}
\mathop{\sum}\limits_{\footnotesize{\left.\begin{array}{cc} \spo_1,\ldots,\spo_n \in \setspobu \\ 0 \in \cup_{i=1}^n \spo_i \end{array}\right. }} \eta_{N}(\spo_1,\ldots,\spo_n)\Bigg|&  \nonumber \\
\leq &   \mathop{\sum}\limits_{j = -\varsigma,\varsigma,0,\max \La} \qquad \mathop{\sum}\limits_{ \spo \in \setspo_{\La}, \spo \nsim \{j\} }   v_{\ka,\la}^\La(\spo)  \leq   \mathop{\sum}\limits_{j = -\varsigma,\varsigma,0,\max \La} a_{\La}(\{ j \})   & \label{vanishing boundary}
\end{eqnarray}
and \eqref{vanishing boundary} vanishes upon dividing by $\tfinal=\tau N$ as $N \nearrow \infty$.

Theorems \ref{thm: 1} and \ref{thm: 2} contain a limit $\tfinal \uparrow \infty$, whereas, as already mentioned, we restrict ourselves to $\tfinal=N\tau,N \in \bbN$ and hence we take the limit along a subsequence.
It is easily seen that the general results can be obtained by  modifying
 slightly the intervals $I_j$ for $j= \tau^{-1}\tinit,\tau^{-1}\tfinal-1$ (such that $\cup_{j \in \La} I_j=[\tinit,\tfinal]$ still holds)
 and hence also modifying slightly the polymer weights of
 polymers containing $j= \tau^{-1}\tinit$ or $j= \tau^{-1} \tfinal-1$.
  Indeed, by \eqref{vanishing boundary}, these `boundary polymers' do not contribute to $F(\ka,\la)$.

That taken into account, and using \eqref{def: cluster expansion}
and \eqref{vanishing boundary}, Theorems \ref{thm: 1} and
\ref{thm: 2} will follow once we prove that \beq \label{cesaro1}
F(\ka,\la) = \la^{2} e_\ka+  \la^{2} \lim_{N \nearrow
+\infty}\sum_{\footnotesize{\left.\begin{array}{cc}
\spo_1,\ldots,\spo_n \in \setspobu \\ 0 \in \cup_{i=1}^n \spo_i
\end{array}\right. }} \frac{1}{\tau
N}\eta_{N}(\spo_1,\ldots,\spo_n) \eeq exists.
 Indeed, by the bound \eqref{bound logaritm4},  analyticity follows from the analyticity of
$w^\La_{\ka,\la}(\spo),\spo \in \setspo_\La$ and the Vitali convergence theorem.

Again by the bound \eqref{bound logaritm4}, existence of \eqref{cesaro1} will  follow once we prove
that for each sequence $\spo_1,\ldots,\spo_n$, the limit $\lim_{N \nearrow +\infty} \frac{1}{N}\eta_N(\spo_1,\ldots,\spo_n)$ exists. For simplicity, we restrict to sequences with one element  $\spo_1 \in \setspo_\La$ and to
$\spo_1= \emptyset \cup \fpo \cup \emptyset $ ($\fpo \in \setfpo$) in the parametrization \eqref{parameters spo}. the general case is treated  analogously).
%
%

For $\dsi \in \dsig$ and $ q \in \bbR$,  we denote by $\dsi+q$ the
element in $\dsig$ with coordinates
$s_i(\dsi)+q,t_i(\dsi)+q,\ell_i(\dsi), i=1,\ldots,|\dsi|$. Since
$K(\dsi)=\fpo+ m \Leftrightarrow  K(\dsi-\tau m)=\fpo$ for $m \in
\bbN$, \beq
 \frac{1}{N} \eta_N (\spo_1):=   \frac{1}{N} b_\ka^{-1} \sum_{m=0}^{N-1} \lim_{\ep \searrow 0} \int_{K^{-1}(\fpo)} \d \dsi \, \Tr[P_\ka W_{\ka,\la}^\ep(\dsi+\tau m ) P_\ka \rho_{\sys,0}]
\eeq

By inspecting \eqref{eq: Y1}, one sees that for $\supp \dsi \subset \bbR^{+}$, $W^\ep_{\ka,\la}(\dsi)$
differs from $W^\ep_{\ka,\la}(\dsi+q)$ only
by (a product of) periodic factors $\e^{\i \lakl \om q}, \om \in \caF$ and from this it easily follows that
\beq
\lim_{N\uparrow +\infty}   \frac{1}{N}\sum_{m=0}^{N-1} \Tr[P_\ka W_{\ka,\la}^\ep(\dsi+\tau m) P_\ka \rho_{\sys,0}]
\eeq
exists for all $\dsi$. Existence of $\lim_{N\uparrow +\infty}  \frac{1}{N}\eta_N (\spo_1)$ follows hence by  dominated convergence  from some $m$-independent bound, for example (see \eqref{bound on series2})
\beq\|\int_{K^{-1}(\fpo)} \d \dsi W_{\ka,\la}^\ep(\dsi+\tau m) \| \leq  \e^{\|G \|_1 |I_{\fpo}|} \eeq











\section{Proofs of various lemma's}\label{sec: proof of lemmas}

\subsection{Proof of Lemma \ref{lem: definition W}, Proposition \ref{prop: finite volume} and well-definedness of \eqref{def: Wfpo} } \label{sec: proof of expansions}

Remark that for  $g \in L^1(\bbR^+)$, the function \beq z \to
\int_{\bbR^+} g(x) \e^{z x}  \eeq  has an analytical continuation
from $\mathrm{Re}\,   z=0$ to  $ \mathrm{Re}\,   z < 0 $. Hence,
the bound \eqref{eq: ass laplace} implies that $\ka \to
Q_{\ka,\la}(x, \{p,q\},m)   $ (with the notation of Section
\ref{sec: duhamel as integral}) is analytic on $D^{\circ}+(\i
\bbR)^{|\caK|}$  and
 \beq \label{def: h} h(t):=\mathop{\sup}\limits_{\om \in \caF,k \in \caK}\,\,\mathop{\sup}\limits_{\re   \ka \in D, | \im    \ka| \leq 1 } |\langle \breve{f}_k, \e^{-\ka_k l -\i t (l+\om)} \breve{f}_k \rangle |
 < \infty \eeq
Also,
\begin{eqnarray}
\left| Q_{\ka,\la}(x, \{p,q\},m) \right|  &\leq &   h(|u_q-u_p|  ) \label{eq:
estimate G}
\end{eqnarray}
%
%
%
Using  \eqref{eq: Y1}, \eqref{eq: equality W=Y},the estimates
\eqref{eq: estimate G}, $\|\tilde{\rho}_\sys \| \leq 1$ and putting $\| V\|:=\max_{k \in \caK} \|V_k\|$, we get
\begin{eqnarray} \| \int_{\dsig_{I}} \d \dsi W_{\ka,\la}^\ep(\dsi) \tilde{\rho}_\sys \| &\leq&  \int_{\dsig_{I}} \d
\dsi     \la^{-2|\dsi|} \| V\|^{2 |\dsi|}  \mathop{\prod}\limits_{\basi \in \dsi}  (1/2) \, 4
|\caK||\caF|^2
   h(\lakl t(\basi)-\lakl s(\basi))
 \nonumber \\
&\leq& \sum_{n \in \bbN}    \left(4 |\caK||\caF|^2\| V\|^2
\int_{ I} \lakl \d u h( \lakl u)  \right)^n  \int_{s_n
\geq \ldots \geq s_1} \prod_{i=1}^n \d s_i \nonumber \\
& \leq &    \exp{ \left( 4 |\caK||\caF|^2 \|V\|^2 |I|
\int_{\lakl I}  \d u h( u)  \right) } \label{bound on series}
 \end{eqnarray}
 The prefactors $ 4 |\caK||\caF|^2$ in the first inequality account for the fact that for each $\dsi$,
there are $(4 |\caK|^2|\caF|^2)^{|\dsi|}$ pairs $x \in \caX^{2|\dsi|}, \pi \in Pair(2|\dsi|)$ such that
$\dsi \sim (x,\pi)$ (The $|\caK|^2$ turns into $|\caK|$ because of the factors $\de_{k_p,k_q}$).
The factor $1/2$ appears to avoid double counting, since  $4 |\caK||\caF|^2
   h(\lakl t(\basi)-\lakl s(\basi))$ bounds the sum over $\ell(\basi)=\mathrm{a},\mathrm{b}$.

 Since the bound \eqref{bound on series} doesnot depend on  $\ka$,
 analyticity of \eqref{eq: equality W=Y} and Lemma \ref{lem: definition W}
follow from the Vitali convergence theorem.




To show Proposition \ref{prop: finite volume}, one expands the LHS
of \eqref{eq: convergence finite} just as in \eqref{eq: W1}. Note
that one does not need a bound like \eqref{eq: ass laplace} since
$\frh_k^n$ is finite-dimensional.  By (\ref{ass: finite2a},
\ref{ass: finite2b}),the expansion converges term by term to the
infinite-volume expansion \eqref{eq: W1}.
 Remark that for large enough $n$ and fixed $\ka$, the expressions on the LHS of (\ref{ass: finite2a},\ref{ass: finite2b}) are   bounded by some $C$, uniformly in $t \in \bbR$.
The total series is hence  dominated similarly to \eqref{bound on
series} with $C$ replacing $h(\lakl t(\basi)-\lakl s(\basi))$.




We now turn to expression \eqref{def: Wfpo}, establishing some notation that will be used in further proofs.  By Assumption \ref{ass: 1}, we have $\|h\|_1 = \int_{\bbR^+} \d u
h(u) <\infty $.
 Let for $u \geq 0$,
\beq
 G_\ep(u):=  4 |\caK||\caF|^2\|V \|^2    \left( 2 h(u)  + \|h \|_1  \de_\ep( u)  \right) \geq 0
\eeq and remark $\int_{\bbR^+} \d u  G_\ep(u) =: \|G\|_1 < \infty$ (independent of $\ep,\la$). Put \beq \label{def: H} H^\ep_{\la}(\dsi)=\prod_{\basi \in
\dsi} \lakl G_\ep(\lakl t(\basi)-\lakl s(\basi)) \eeq
Let  $A
\subset \dsig_I$, then by \eqref{def: H} and  analogous to
\eqref{bound on series}, \beq \label{bound on series2} \|
\int_{A} \d \dsi W^\ep_{\ka,\la}(\dsi) \| \leq   \int_{A} \d \dsi
H^{\ep}_\la(\dsi)  \leq \e^{I  \| G \|_1}
 \eeq
%
 Call \beq
\caV:=\left\{ \dsi \in \dsig | \exists \basi \in \dsi :
s(\basi)=t(\basi) \right\} \eeq the set on which (products of)
$\de_\ep$ becomes singular and let $\d_\caV \dsi$ stand for the
measure induced by $\d \dsi$ on (the submanifold) $\caV$.
 By the  bound \eqref{bound on series2}, one sees  that $\int_{A } \d \dsi W^{\ep}_{\ka,\la}(\dsi)$ has a well-defined limit as $\ep \searrow 0$ whenever
 \beq \int_\caV \d_\caV \dsi \, \mathrm{Ind}[\dsi \in \partial A]=0 \eeq
 Since $A:= K^{-1}(\fpo)$ for $\fpo \in \setfpo$  satisfies this condition, the expression \eqref{def: Wfpo} is well-defined. Its analyticity in $\ka$ follows analogously to analyticity of \eqref{eq: equality W=Y},
using the bound \eqref{bound on series2} and analyticity of $Q^{\mathrm{a}}_{\ka,\la}(x, \{p,q\},m)$.





\subsection{Estimates to be used in Section \ref{sec: proof of small exci}}\label{sec: a priori estimates}






Assume that $\int_{\bbR^+} \d u \,uh(u) <\infty$. Let $I$ be a
union of intervals and $a \in \bbR$. For \eqref{eq: lestimate b},
we assume that $0 \notin I^\circ$. Let for $j \in \bbZ$ and $\dsi
\in \dsig$, \beq A_j(\dsi):= \mathrm{Ind} \left[ \footnotesize{
\begin{array}{c}  \exists  \basi,\basi' \in \dsi:\,\supp \basi
\cap \supp \basi'\neq \emptyset, \\ (\supp \basi \cup \supp
\basi') \cap I_j \neq \emptyset  \end{array}}\right], \qquad
B_j(\dsi) := \mathrm{Ind} \left[\footnotesize{\begin{array}{c}
\exists  \basi \in \dsi:\,   \supp \basi \cap   \partial I_j \neq
\emptyset \end{array} }\right] \eeq
 There are
$c_1,c_2,c_3>0$, such that
 \begin{eqnarray}
&\mathop{\int}_{\dsig_I}  \d \dsi H^{\ep}_{\la}(\dsi )  A_j(\dsi)  \leq \la^2 c_1 \tau  \exp{( |I|\| G\|_1 )} &\label{eq: lestimate cross}
\\
& \mathop{\int}_{\dsig_I}  \d \dsi H^{\ep}_{\la}(\dsi )  B_j(\dsi) \leq  \la^2 c_2 \exp{( |I|\| G\|_1 )}& \label{eq: lestimate pin}
\\
& \lim_{\ep \downarrow 0} \Big\|  \mathop{\int}\limits_{\footnotesize{\left.\begin{array}{c} \dsi \in \dsig_{I,n}, \ell(\basi)=b  \\
%
\forall \basi,\basi' \in \dsi: \supp \basi \cap \supp \basi'=\emptyset \end{array} \right.}} \d \dsi
%
W^{\ep}_{\ka,\la}(\dsi) \Big\| \leq  (\la^2 c_3)^n    & \label{eq: lestimate b} \end{eqnarray}
\begin{proof}
We start with \eqref{eq: lestimate b}. Recall the notation of Section \ref{sec: duhamel as integral}. By the substitution $u_q
\rightarrow v, u_p-u_q \rightarrow u$,
\begin{eqnarray*}
&&\lim_{\ep \downarrow 0}\left|\int_{u_p > u_q \in I} \d u_p \d u_q Q_{\ka,\la}(x,\{p,q\},m)-Q^\mathrm{a}_{\ka,\la}(x,\{p,q\},m)\de_\ep(\frac{|u_p-u_q|}{\la^2})\right|
\\  &\leq&  \de_{\om_p,-\om_q}  \int_I \d v \int_{| I|-v}^\infty \d u   h(\lakl u) + (1-\de_{\om_p,-\om_q})\la^2\left|\int_I \d v \int_{0}^{| I|-v} \d
u  h(\lakl u) \e^{-\i \lakl v (\om_p+\om_q) }\right| \\
&\leq &   \de_{\om_p,-\om_q}  \left( \int_I \d u \, u h(\lakl u)+ |I|
\int_{| I|}^\infty \d u h(\lakl u) \right)
 +  \la^2 (1-\de_{\om_p,-\om_q})
    \int_{0}^{| I|} \d u    h (\lakl
u)\frac{ 2 \la^{2} }{\om_p+\om_q}  \
\end{eqnarray*}
%
(The case $u_q>u_p $ is bounded identically.) An overall
factor $\la^4$ can be extracted and we conclude
\begin{eqnarray*}
 \lim_{\ep \downarrow 0} \|\mathop{\int}\limits_{\basi \in \basig_{I},
\ell(\basi)=b} \d \basi W^{\ep}_{\ka,\la}(\basi) \| \leq  \la^2
4|\caK|\|V\|^2 \left(   |\caF| \int_{\bbR^+} d u \, u h(u)
 +  2 \|h \|_1 \sum_{\om \neq \om' \in \caF}
|\om-\om'|^{-1} \right) =: \la^2  c_{3}
\end{eqnarray*}
which yields \eqref{eq: lestimate b} by  using the property \eqref{prop: factorization}.

To show \eqref{eq: lestimate cross}, we have
\beq \label{proof: estimate cross}
\mathop{\int}_{\dsig_I}  \d \dsi H^{\ep}_{\la}(\dsi )  A_j(\dsi) \leq  \mathop{\int}\limits_{\basi,\basi' \in \basig_I}\d \basi \d \basi' H^{\ep}_{\la}(\basi \cup \basi')  A_j(\basi \cup \basi') \mathop{\int}_{\dsig_I}  \d \dsi H^{\ep}_{\la}(\dsi )
\eeq
and the second integral on the RHS is smaller than $\e^{|I|\|G\|_1}$ by \eqref{bound on series2}. For the first integral in \eqref{proof: estimate cross},
 we assume without loss of
generality that $s(\basi)<s(\basi')$. We assume that $A_j=1$ which
implies that at least one of
$s(\basi),t(\basi),s(\basi'),t(\basi')$ lie in $I_j$. We assume
that $s(\basi) \in I_j$, the other possibilities give an identical
bound (hence the factor $1/4$ on the last line).
\begin{eqnarray}
&& \mathop{\int}\limits_{\basi,\basi' \in \basig, s(\basi) \in I_j, s(\basi)< s(\basi')<t(\basi)  } \d \basi' \d \basi    H^{\ep}_\la(\basi \cup \basi')   \\
&\leq&  \| G\|_1  \mathop{\int}\limits_{\basi \in \basig, s(\basi) \in I_j,s(\basi)< s(\basi')<t(\basi)  } \d \basi  \d s(\basi')     \lakl  G^{\ep}(\lakl (t(\basi)-s(\basi)))  \\
&\leq&  \| G\|_1  \mathop{\int}\limits_{\basi \in \basig,s(\basi) \in I_j } \d \basi      (t(\basi)-s(\basi))    \lakl  G^{\ep}(\lakl (t(\basi)-s(\basi)))   \\
&\leq&  \la^2  \| G\|_1 |I_j|    \int_{\bbR^+} \d u  \, u
G^{\ep}(u) =: (1/4) \la^2 c_1 \tau
\end{eqnarray}
where we used  $|I_j|=\tau$.
The integral on the LHS of \eqref{eq: lestimate pin} is evaluated
analogously (though simpler), yielding the bound $\la^2
\int_{\bbR^+} \d u \,  u  G^{\ep}(u)=:   \la^2 c_2$.

\end{proof}

\subsection{Proof of Lemma \ref{lem: small exci}}\label{sec: proof of small exci}


For each $\fpo \in \setfpo $ and $\dsi \in K^{-1}(\fpo)$
we define a partition $ K_a(\dsi) \cup K_b(\dsi) = \fpo$
\begin{definition}\emph{
A $j \in \bbZ$ belongs to $ K_a(\dsi) \subset 2^{\bbZ} $ iff.\
there is a $J \subset I_j$ such that \beq \dsi_{J} = \dsi
\setminus \dsi_{\bbR \setminus J}, \qquad  \forall \basi \in
\dsi_{J}:\ell(\basi)=\textrm{b},\qquad \forall \basi,\basi' \in
\dsi_J,\supp \basi \cap \supp \basi'=\emptyset   \eeq
%
%
%\ben
%\item{  \beq     \eeq}
%\item{ $$ and  $\forall \basi,\basi' \in \dsi_J $, $\supp \basi \cap \supp \basi'=\emptyset $. }
%\een
%
%
 Let for $j \in K_a(\dsi)$, $J_j \subset I_j$ be the union of
all  intervals with the above property and put \beq \dsi_a(\dsi):=
\cup_{j \in K_a(\dsi) } \dsi_{J_j} \eeq
  Let
$K_b(\dsi):=K(\dsi) \setminus K_a(\dsi)$ and define $\dsi_l(\dsi)
\subset \dsi$ \beq \dsi_l(\dsi):= \left\{ \basi \in \dsi \setminus \dsi_a(\dsi) \, \big| \,
|\supp \basi| >\tau  \right\} \eeq
}
\end{definition}

We now estimate
\begin{eqnarray}
&& \| \mathop{\int}\limits_{K_{a,b}(\dsi)=\fpo_{a,b}} \d \dsi
W^{\ep}_{\ka,\la}(\dsi) \|    \leq \mathop{\int}\limits_{\footnotesize{\left.\begin{array}{c}\dsi_l(\dsi )=\dsi
  \end{array} \right.}}
 \d \dsi  H^{\ep}_\la(\dsi )          \label{long
 estimate1} \\
%
%
  && \label{monster2}
%
%
%
%
\mathop{\int}\limits_{\footnotesize{\left.\begin{array}{c}  K_{a}(\dsi' \cup \dsi)=\emptyset, \fpo_b \subset K_b (\dsi' \cup \dsi) \subset \fpo  \\ \dsi_l( \dsi' \cup \dsi )=\dsi  \end{array} \right.}}  \d \dsi'  H^{\ep}_\la(\dsi') \\
%
 %
&    &  \Big\|\mathop{\int}
\limits_{\footnotesize{\left.\begin{array}{c} \dsi_a(\dsi \cup \dsi' \cup \dsi'')=\dsi''  \\
 K_a(\dsi \cup \dsi' \cup \dsi'')= \fpo_a \end{array} \right.}} \d
\dsi'' W^{\ep}_{\ka,\la}(\dsi'') \Big\|  \label{long estimate3}
\end{eqnarray}
By \eqref{eq: lestimate b} and for $\la$ small enough and $c_a =2
c_3$, \beq \label{eq: estimate a} \lim_{\ep \searrow
0}\Big\|\mathop{\int}
\limits_{\footnotesize{\left.\begin{array}{c} \dsi_a(\dsi \cup \dsi' \cup \dsi'')=\dsi''  \\
 K_a(\dsi \cup \dsi' \cup \dsi'')= \fpo_a \end{array} \right.}} \d
\dsi'' W^{\ep}_{\ka,\la}(\dsi'') \Big\|
%
 \leq (\sum_{n=1}^\infty (\la^2 c_3)^n )^{|\fpo_a|} \leq
 (c_a \la^{2})^{|\fpo_a|}
\eeq

%

In  \eqref{monster2}, all $j \in  \fpo_b$ such that $I_j \cap
\supp\dsi=\emptyset$ necessarily satisfy $C_j(\dsi'):=A_j(\dsi')
\vee B_j(\dsi')=1$ where $x\vee y :=\max{(x,y)}$, hence \beq
\label{monster22}
 \mathop{\int}\limits_{\footnotesize{\left.\begin{array}{c}
K_{a}(\dsi' \cup \dsi)=\emptyset, \fpo_b \subset K_b (\dsi' \cup
\dsi) \subset \fpo  \\ \dsi_l( \dsi' \cup \dsi )=\dsi  \end{array}
\right.}}  \d \dsi'  H^{\ep}_\la (\dsi') \leq
\int_{\dsig_{I_{\fpo}}  } \d_\tau \dsi' H_\la^\ep (\dsi')
 \mathop{\prod}\limits_{\footnotesize{\left.\begin{array}{c}j \in  \fpo_b\\
 \supp (\dsi) \cap I_j= \emptyset \end{array}\right. }} C_j(\dsi')
\eeq where we have exploited $ \dsi_l(\dsi')=\emptyset$ to modify
the measure as
 \beq \d_\tau \dsi':= \mathrm{Ind}[\forall \basi' \in \dsi': |\supp \basi' |<\tau ] \, \d \dsi'
. \eeq Remark that $\dsi$ in $\eqref{monster22}$  intersects at
most $(3/\tau) |\supp (\dsi)|$ elements $j \in \fpo_b$ and hence
the product in \eqref{monster22} contains at least $|\fpo_b|-\min
\{(3/\tau) |\supp (\dsi)|,|\fpo_b| \}$ elements. For $ n \geq 3$,
one easily establishes  \beq \label{eq: factorization}\int_\dsig
\d_\tau \dsi' C_j(\dsi') C_{j+ n}(\dsi') = \int_\dsig \d_\tau
\dsi' C_j(\dsi') \int_\dsig \d_\tau \dsi' C_{j+ n}(\dsi')
 \eeq
 By keeping hence only each third element in the product in \eqref{monster22} ,using  the
 estimates \eqref{eq: lestimate cross} and \eqref{eq: lestimate
 pin} and the factorization \eqref{eq: factorization},
\beq \label{monster23}
  \textrm{RHS \eqref{monster22}} \leq  (c'(\la))^{|\fpo_b|-\min \{(3/\tau) |\supp
(\dsi)|,|\fpo_b | \}} \left(\e^{ |I_{\fpo}|\|G\|_1 } \right) \eeq
 with $c'(\la)=\sqrt[3]{(c_1 \tau +c_2)\la^2}$.

We replace \eqref{monster2} by the RHS of \eqref{monster23} (omitting for the moment the number $\e^{
|I_{\fpo}|\|G\|_1 }$) and we perform the
first integral in \eqref{long  estimate1}: By Assumption \ref{ass:
1} , there is a $c''>0$ such that
 \beq
 | h(t ) | \leq  c'' \e^{-\al |t|}
\eeq Introduce $q_i:=t_i(\dsi)-s_i(\dsi)$ and
$q:=\sum_{i=1}^{|\dsi|}q_i \geq |\supp \dsi |$,
\begin{eqnarray}
&&\mathop{\int}\limits_{\footnotesize{\left.\begin{array}{c}\dsi_l(\dsi
)=\dsi,
 K_b(\dsi ) \subset \fpo_b \end{array} \right.}}
 \d \dsi  H^{\ep}_\la (\dsi ) c'(\la)^{|\fpo_b|-\min \{(3/\tau) |\supp
(\dsi)|,|\fpo_b | \}} \\
 %
 &&\leq   c'(\la)^{|\fpo_b|} \sum_{n \geq 0} \frac{ (c''|I_{\fpo_b}|)^n}{n!}\int_{(I_{\fpo_b})^n} \d q_1 \ldots \d q_n   \e^{-\al \lakl  q} c'(\la)^{-\min\{|\fpo_b|, (3/\tau)q \}} \\
%
&&\leq c'(\la)^{|\fpo_b|} \sum_{n \geq 0} \frac{ (c''|I_{\fpo_b}|)^n}{n!} \int_{\bbR^+} \d q
 \frac{q^{n-1}}{(n-1)!}
     \e^{-\al \lakl  q}
c'(\la)^{-\min\{|\fpo_b|, (3/\tau)q\}} \label{proof Kb}
%
\end{eqnarray}
The first inequality follows by integrating over $s_i(\dsi)$, the
second  by integrating over $q_{1,\ldots,n}$ with the constraint
$q:=\sum_{i=1}^{n}q_i$. By a straightforward computation, for
small enough $\la$, the expression \eqref{proof Kb} is bounded by
some function $c_b(\la)^{|\fpo_b|}$ with $c_b(\la) \searrow 0$ as
$\la \searrow 0$.
 Combining this with the bounds \eqref{eq: estimate a} and \eqref{bound on
 series2}, we get
\begin{eqnarray}
&& \lim_{\ep \searrow 0}\sum_{\fpo_{a}\cup \fpo_b=\fpo} \|\int_{K_{a,b}(\dsi)=\fpo_{a,b}} W^{\ep}_{\ka,\la}(\dsi) \| \\
&\leq& \sum_{\fpo_{a}\cup \fpo_b=\fpo} \e^{\|G \|_1 |I_\fpo| }
c_a(\la)^{|\fpo_a|}c_b(\la)^{|\fpo_b|}
\leq \left( e^{\|G \|_1}(c_a(\la)+ c_b(\la))^{\tau^{-1}}\right)^{|I_\fpo|}
\end{eqnarray}
where the sum is over partitions of $\fpo$ into $\fpo=\fpo_a \cup \fpo_b$ and we have used the binomial theorem. This proves Lemma \ref{lem: small exci}.



\subsection{Proof of Lemma \ref{lem: artificial wc} and Lemma \ref{lem: frobenius}}

For $\basi \in \basig_I$ such that $\ell(\basi)= \mathrm{a}$. \beq
\lim_{\ep \downarrow 0}    W_{\ka,\la}^\ep(\basi)= \left\{\begin{array}{ll} \caL_\ka & \textrm{if}\quad s(\basi) >0 \\
\caL_0& \textrm{if}\quad s(\basi) <0 \end{array}\right. \eeq which
follows by comparing \eqref{def: Qa} and \eqref{def: c}.
Similarly,  choosing now $\supp\dsi \subset \bbR^+$, \beq
\ell(\dsi)_{1,\ldots,|\dsi|}=\textrm{a} \Rightarrow \lim_{\ep
\downarrow 0}    W_{\ka,\la}^\ep(\dsi)= (\caL_\ka)^{|\dsi|} \eeq
After integrating over $s_{1,\ldots,|\dsi|}$ (which yields a
factor $\frac{I^{|\dsi|}}{|\dsi|!}$) and summing over $|\dsi|$,
Lemma \ref{lem: artificial wc} follows by dominated convergence,
using the bounds  \eqref{bound on series2} for the LHS of
\eqref{eq: convergence reference}.



Lemma \ref{lem: frobenius} follows from a Perron-Frobenius theorem
for completely positive maps: From  \cite{evanshoegh}, one easily
derives (see the Appendix of \cite{deroeckmaesfluct} for more
precise references)
\begin{theorem}
Let $Q$ be a completely positive map on a finite dimensional
$C^*-$ algebra $\caA$ and assume that for all strictly positive
(i.e.\ positive and invertible) $S,S' \in \caA$, \beq \label{cond:
irreducible} \Tr \left[ S Q S'  \right] >0  \eeq Then, $Q$ has a
strictly positive simple eigenvalue $e$  such that, for all
eigenvalues $e' \neq e$, one has $|e'|< e$.  The eigenvector $Z$,
corresponding to $e$ can be chosen strictly positive.
\end{theorem}
By a well-known theorem of Frigerio \cite{frigerio}, Assumption
\ref{ass: wc} implies the property \eqref{cond: irreducible}.






\subsection{Proof of Lemma \ref{lem: kotecky}}
For $\spo \in \setspo_\La$ and with the conventions $\max \emptyset=0, \min\emptyset= \max \La$, we put
\begin{eqnarray} \min \spo &:=&
\left\{\begin{array}{ll}  \min \{\min(\spo\setminus \{-\varsigma,\varsigma  \}),0\}  \quad &\textrm{if}
\quad -\varsigma \in \spo  \\
 \min(\spo\setminus \{\varsigma  \}) \quad &\textrm{otherwise}
 \end{array}\right.
 %
\\
 %
\max \spo &:=& \left\{\begin{array}{ll} \max \La=
\tau^{-1}\tfinal-1 \quad &\textrm{if}
\quad \varsigma \in \spo \\
\max (\spo\setminus \{-\varsigma  \}) \quad &\textrm{otherwise}
 \end{array}\right. \\
d (\spo)&:= &\max \spo
-\min \spo +1
\end{eqnarray}
%
%
Starting from \eqref{polymer weight}, using the estimates
 $\|W_{\ka}^{-}(\ep_-) \| \times \|W_{\ka}^{+}(\ep_+) \| \leq b_\ka^{-1}$ and Lemma \ref{lem: frobenius}, we get
 \beq w_{\ka,\la}^\La(\spo) \leq \e^{-\tau d(\spo)g
}   \frac{\dim \caH_\sys}{b_\ka}   \prod_{i=1}^n \|\tilde{W}_{\ka,\la}(\fpo_i) \|\eeq
%
%
Remark that by Lemma \ref{lem: small exci}, definition \eqref{def:
tilde primary polymers} and the simple bound $\|\e^{-t
|\caL_\ka|}\| \leq \e^{t \|\caL_\ka \|}$, \beq  \label{def: A}
A(\la):= \sup_{j \in  \bbZ } \sum_{\fpo \in \setfpo, \min
\fpo=j}\| \tilde{W}_{\ka,\la}(\fpo) \|  \rightarrow 0, \qquad \la
\searrow 0  \eeq
%
Using that $\spo \in \setspo_\La \setminus \{
\{-\varsigma,\varsigma\} \}$ either has $n \geq 1$ (in the
parametrization \eqref{parameters spo}) or
$w_{\ka,\la}^\La(\spo)=0$ (as is clear in \eqref{vanishing polymer
weights}), one calculates
\begin{eqnarray}
\sup_{j \in \La } \sum_{\footnotesize{\left.\begin{array}{c}d(\spo)=d, \min \spo =j \\
\spo \in \setspo_\La \setminus  \{ \{-\varsigma,\varsigma\} \}
\end{array}\right. }} w_{\ka,\la}^\La(\spo)& \leq &  \frac{\dim \caH_\sys}{b_\ka} \e^{-\tau d g } \sup_{j \in \La } \sum_{\footnotesize{\left.\begin{array}{c}d(\spo)=d, \min \spo =j \\
\spo \in \setspo_\La \setminus  \{ \{-\varsigma,\varsigma\} \} \end{array}\right. }}
\prod_{i=1}^n \|
\tilde{W}_{\ka,\la}(\fpo_i) \| \nonumber \\
& \leq &  \frac{4\dim \caH_\sys}{b_\ka} \e^{-\tau d g} \sum_{n \in \bbN_0}  \frac{(d)^n}{n !}    A(\la)^n    \nonumber   \\
& \leq & \frac{4\dim \caH_\sys}{b_\ka} ( \e^{  d A(\la)}-1)  \e^{- \tau d g } \label{proof preiss1}
\end{eqnarray}
%%
%
To obtain the second inequality,
rewrite the sum over $\spo$ as a sum over
 the minima of $\fpo_{i=1,\ldots,n}$ (which lie necessarily between $\min \spo$ and $\min \spo+d$) and a sum over all possible $\fpo_i$ with fixed minima (as in \eqref{def: A}).
The factor $4$ counts the possibilities for $\ep_-,\ep_+$.



Choose $\clustera(\spo):=a d(\spo)$ with $a>0$.
\begin{eqnarray}
&& \sum_{\spo' \in \setspo_\La,\spo' \nsim \spo}  |w_{\ka,\la}^\La  (\spo')| \e^{a d(\spo')} \\
%
&\leq&
%
 \sum_{d=1}^{\infty} (d +d (\spo)) \e^{a d} \sup_{j \in \La } \sum_{\footnotesize{\left.\begin{array}{c}d(\spo')=d, \min \spo' =j \\
\spo \in \setspo_\La \setminus  \{ \{-\varsigma,\varsigma\} \}
\end{array}\right. }}w_{\ka,\la}^\La(\spo') +
  \frac{\dim \caH_\sys}{b_\ka} \e^{- (\tau g-a)  d(\{-\varsigma,\varsigma \}) } \label{proof preiss2} \\
& \leq&
\frac{\dim \caH_\sys}{b_\ka}
%
  \sum_{d=1}^{\infty} (d +d (\spo)) ( \e^{  d A(g,\la)}-1)
\e^{- d (\tau g - a ) }  + \frac{\dim \caH_\sys}{b_\ka} \e^{- (\tau g-a)  \tfinal }  \label{proof kp}
\end{eqnarray}
To get the first inequality, isolate  $\spo'= \{-\varsigma,\varsigma \}$ in the sum over $\setspo_\La$.
The factor $d(\spo)+d$ in \eqref{proof preiss2} is the number of possible values for $\min  \spo'$. The second inequality follows by \eqref{proof preiss1}.

 To get Statement (1) of Lemma \ref{lem: kotecky}, it suffices e.g.\ to
 take $a=(1/2) g \tau$. Indeed, the last term in \eqref{proof kp} is made arbitrarily small by choosing $\tfinal$ large enough,
 the first term is controlled since for e.g.\ $A(\la) < (1/4) \tau g$,  the exponent $\tau g -a -A(\la) \geq (1/4) \tau g >0$  and hence terms with large $d$ decay exponentially.
 The terms with small $d$  are controlled by $\e^{d A(\la)}-1$ which is made arbitrarily small by reducing $\la$.

Statement (2) of Lemma \ref{lem: kotecky} is obvious since
$\setspo_\La$ is a finite set. The analyticity (in $\ka$) of the
weights $w_{\ka,\la}^\La  (\spo)$ is a straightforward consequence
of the analyticity of \eqref{def: Wfpo}, which is proven in
Section \ref{sec: proof of expansions}, and analyticity of $\ka
\to \caL_\ka, \e_\ka$.


\subsection{Proof of Corollaries \ref{cor: gc} and \ref{cor: central limit}}


We prove Corollary \ref{cor: gc}. Let $\rho_\sys$ be the trace
state on $\caH_\sys$, i.e.\ $\rho_\sys [S]= (1/\dim \caH_\sys)
\Tr[S]$.  Let $\ka(\zeta)$ be as in \eqref{def: kazeta} for $\zeta
\in \bbC$, and let $F,F'$ be finite products of $A_0(u),u \in
\bbR$, as introduced in Section \ref{sec: existence dynamics}. One
easily checks (or consults \cite{derzinski1}) that  \beq \rho_\sys
\otimes \rho_\res [ Y_{-\ka(\i t)}F Y_{\ka(\i t)} F'  ]= \rho_\sys
\otimes \rho_\res [ F' Y_{-\ka(\i t-1)} F Y_{\ka(\i t-1)} ] \eeq
which expresses that the state $\rho_\sys \otimes \rho_\res$ is a
KMS-state at temperature $1$ for the dynamics given by $
Y_{-\ka(\i t)} \cdot Y_{\ka(\i t)}$ on the appropriate Araki-Woods
$W^*$-algebra.

 Since on the domain $\caD$, $U_t^\la$ is a sum of such terms $F$ (see \eqref{eq: proof dyson}), one gets
 \beq  \rho_\sys \otimes \rho_\res [
Y_{-\ka(\i t)} U_{-t}^\la Y_{\ka(\i t)} U_t^\la  ]= \rho_\sys \otimes
\rho_\res [ U_t^\la Y_{-\ka(\i t-1)} U_{-t}^\la Y_{\ka(\i t-1)} ] \eeq
%
Upon inserting $\Theta \Theta=1$ and  using $\Theta Y_{\ka} \Theta=
Y_{\overline{\ka}} $, $\Theta U_t^\la \Theta= U_{-t}^\la $ and
 \beq \rho_\sys
\otimes \rho_\res [\cdot] =\rho_\sys \otimes \rho_\res [\Theta
\cdot \Theta], \qquad  \rho_\sys
\otimes \rho_\res [\cdot] =\rho_\sys \otimes \rho_\res [Y_{-\ka}
\cdot Y_{\ka}], \eeq one obtains by analytic continuation
\beq \label{exact gc}\rho_\sys \otimes \rho_\res [
Y_{-\ka(\zeta)} U_{-t}^\la Y_{\ka(\zeta)} U_t^\la  ]= \rho_\sys \otimes \rho_\res [
Y_{-\ka(1-\zeta)} U_{-t}^\la Y_{\ka(1-\zeta)} U_t^\la  ] \eeq
from which the Corollary \ref{cor: gc} follows by Theorem \ref{thm: 1}.

Corollary \ref{cor: gc} follows immediately from a theorem by Bryc
\cite{bryc}.



\section*{Acknowledgments}
The author has benefited from good discussions with H.-T. Yau, C.
Maes, J. Bricmont, J. Derezi\'{n}ski and C-A. Pillet. The
financial support of the FWO-Flanders is greatly acknowledged.




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