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\null\vskip 3cm
 
\centerline {\bf ON THE GIBBS STATES FOR ONE-DIMENSIONAL LATTICE}
\centerline{\bf BOSON SYSTEMS WITH A LONG-RANGE INTERACTION}
 
\vskip 10truemm
 
\centerline{\bf E. Olivieri$^{1,2}$, P. Picco$^{1,2,6}$ and Yu.M.
Suhov$^{2,3,4,5}$}
 
\vskip 10truemm
 
 
{\leftskip=2.5cm\rightskip=2cm\n\eightrm
{\bf Abstract} : We consider an infinite chain of interacting quantum
(anharmonic) oscillators. The pair potential for the oscillators at 
lattice distance $d$ is proportional to $(d^2(\log(d+1))F(d))^{-1}$ where
${\displaystyle{\Sigma_{r\in{\bf Z}}(rF(r))^{-1}<\infty}}$. 
We prove that for any value of the
inverse temperature $\beta>0$ there exists a limiting Gibbs state which is
translationally invariant and ergodic. Furthermore, it is analytic in a 
natural sense. This shows
the absence of phase transitions in the systems under consideration for any
value of the thermodynamic parameters.
 
\vskip 10truemm
 
\n February 1992
 
 
\vskip 15truemm
 
\n 1\quad Dipartimento di Matematica, Universit\'a di Roma ``Tor Vergata'', 
$\;$Roma ITALIA.CNR-GNFM.
 
\n 2\quad Centre de Physique Th\'eorique, CNRS-Luminy, Marseille FRANCE.
 
\n 3\quad Institute for Problems of Information Transmission, The Russian 
Academy of Sciences, Moscow RUSSIA.
 
\n 4\quad Dipartimento di Matematica "Guido Castelnuovo", Universit\'a degli
Studi di Roma "La Sapienza", Roma ITALIA.
 
\n 5\quad Statistical Laboratory, Department of Pure Mathematics and 
Mathematical Statistics, University of Cambridge, Cambridge ENGLAND UK. 
 
\n 6\quad Courant Institute of Mathematical Sciences, New York, USA 
 
 
\par}
 
 
 
 
 
\vfill\eject
 
\n {\bf 0. INTRODUCTION}
\m
One-dimensional systems of statistical mechanics, both classical and quantum,
are believed not to exhibit phase transitions provided that the interaction
between particles decreases fast enough with the distance. A border case is
the inverse square power interaction: classical one-dimensional systems
with that type of interaction were investigated in [FS].
Quantum systems are more difficult to study; even for relatively simple
classes of systems (spins on a one-dimensional lattice or one-dimensional
particle systems with a fermion-type interaction) the rigorous proof of the
absence of phase transitions requires sophisticated techniques.
 
In this paper we investigate a class of one-dimensional quantum lattice
boson systems (chains of quantum anharmonic 
oscillators) with long-range interaction
potentials that decrease slightly faster than the inverse square power of the
lattice distance. The main technical tools to use are the Wiener integral
representation [Gi] and the cluster expansions which, in the one-dimensional
classical situation, was elaborated in [C.C.O.]. The absence of phase
transitions is expressed here in the following terms: for any  value of the
inverse temperature $\beta >0$ there exists a limiting Gibbs state which is
translation-invariant and ergodic. Moreover, this state is analytic, in
terms of the self-interaction and two-body interaction potentials, in the
sense that the expectation values of certain observables admit an
analytic continuation to a complex domain containing part of the real
axis.
 
Our method may be considered as alternative to the one used in [P1] and 
[P2]. We aim to extend our results to one-dimensional continuous
quantum  systems in a separate publication.
 
The paper is organized as follows. In Section 1 we formulate our main results
(Theorems 1,2) and introduce basic objects for future use. In Section
2 the proofs are accomplished. An  Appendix contains the proof
of superstability estimates which are not related to the
one-dimensional structure of systems under consideration.
\b
\n {\bf 1. PRELIMINARIES, RESULTS AND TECHNICAL TOOLS}
\m
A Hilbert space ${\cal H}_j$ identified as $L_{2}({\bf R})$
is associated with any site $j$ of the one-dimensional lattice $\bf Z$.
By ${\cal B}_j$ we denote the $C^{\star}$-algebra of the bounded operators in
${\cal H}_j$. Given a finite set $\L\subset \bf Z$, we identify a 
Hilbert space ${\cal H}_\L$ with $\;L_2({\bf R}^J)\;$ (which is nothing but
the tensor product $\;\otimes_{j\in
\L}{\cal H}_j\;$) and denote by ${\cal B}_\L$ the $C^{\star}$-algebra of
the bounded operators in ${\cal H}_\L$. The inductive limit $\lim
_{\L\nearrow \bf Z} {\cal B}_\L$ is denoted as ${\cal B}$; this is
the $^{\star}$-algebra of local observables of our system. Its completion in
the operator norm is the $C^{\star}$-algebra $\cal B$ of quasilocal
observables. In the sequel we do not distinguish between the operators in
${\cal H}_\L$ and the corresponding elements of $\cal B$.
 
The action of the space translation group $S_{y}, y\in \bf Z,$ on $\cal
B$ is defined in the standard way.
 
By $q_j$ and $p_j$ we denote the position and momentum operators in ${\cal
H}_j$ (or the corresponding operators in ${\cal H}_\L$ with $\L\ni
 j:$)
$$q_{j}\;f(x_{j}) = x_{j}f(x_{j}),\quad\quad p_{j}\;f(x_{j}) = - i {d
\over dx_j}f(x_j).$$
The Hamiltonian (the operator of the energy) $\;H_\L\;$ of the system in a
finite `volume' $\;\L\;$ is the self-adjoint operator in $\;{\cal H}_\L\;$ 
$$H_{\L} \  = \  K_\L \;+ \; U_\L.\eqno (1.1)$$ 
Here $K_\L \; $ is the kinetic part and $\; U_\L \;$ the
potential part: 
$$ K_\L\; = \; {1\over 2} \sum _{j \in \L} {p_j}^2
,\eqno (1.2)$$ 
and
$$\  U_\L \;=\;U_{\L,0} \;+\;U_{\L,1}, \eqno (1.3) $$
where $\  U_{\L,0} \ $ is the self-interaction energy and $\
U_{\L,1}\  $ is the two-body interaction energy
$$U_{\L,0} \;= \; \sum _{j \in \L}
\Phi(\;q_j\;),\quad U_{\L,1} \;=\;{1\over 2} 
\sum _{j,j^\prime \in \L:j\ne
j^\prime} \Psi _{\vert j - j^{\prime}\vert}(\;q_{j},q_{j^\prime}\;). \eqno
(1.4) $$
 
Here,  $\Phi : {\bf R}\rightarrow {\bf R}$ (a self-interaction
potential) and $ \Psi_{d} : {\bf R}\times {\bf R}\rightarrow {\bf R}$
 (a two-body interaction potential, at distance $d$), $\;d\in
{\bf Z}^+\;$, are $C^2$-functions, $\;\Psi_{d}\;$ is symmetric.
[We use the same symbols for denoting functions (of real variables)
and the corresponding multiplication operators.] We list below the
conditions that are imposed on the interaction potentials.
 
(I)The function $\;\Psi_d (x,y)\;$ obeys  
$$\vert\Psi_{d}(x,y)\vert\;\le{{(\vert x\vert+1)\;
(\vert y\vert +1)}\over{d^{2}F(d)\log\;(d+1)}},\  x,y \in \bf R,
\eqno (1.5)$$ 
where $\;F\;$ is a monotone function $\;\bf Z^+\rightarrow\bf R^+\;$ 
with 
$$\sum _{r\in
\bf Z} (r F(r))^{-1} <\infty .\eqno (1.6)$$
 
In addition, we suppose that
\m
 
(II) there exists $\;r^0>0\;$ such that for any finite
$\;\L\subset\bf Z$, $\;x_\L=(x_j,j\in\L)\in{\bf
R}^\L,\;$ and positive integer $\;r\geq r^0\;$ 
$$U^{(\leq r )}_{\L}(x_{\L})\ge\;\sum_{j\in\L}(c_1x_{j}^{2} -c_{2} ),
\eqno (1.7)$$ 
where $\;c_{1}>0\;$ and $\;c_{2}\in\bf R\;$ are
constants. Here
$$U_\L^{(\leq
r)}(x_\L )={1\over 2}\sum_{j,j'\in\L :|j-j'|\leq r}\Psi_{|j-j'|}
(x_j,x_{j'}) + \sum_{j \in \L} \Phi (x_j) .\eqno (1.8)$$ 
The bound (1.7), with 
$\;r\;$ greater than the length of  $\;\L,\;$ is usually called a
superstability condition.
 
At a certain stage we shall need a similar condition for the 
derivatives:
 
(III)The functions $\;\displaystyle{\Phi^{(\mu )}(x)=
{{\partial^\mu}\over{\partial x^\mu}}\Phi (x)}\;$ and 
$\;\displaystyle{\Psi_d^{(\mu )}(x,y)=
{{\partial^\mu}\over{\partial x^\mu}}\Psi_d
(x,y)}$, $\mu=1,2$, satisfy the bounds  
$$|\Psi^{(\mu )}_d (x,y)|\;\le {{(|x|+1)\;(|y|+1)}\over{d^2F(d)\log
(d+1)}},\;x,y\in{\bf R},\;\mu=1,2,\eqno (1.9)$$
and
$$|\Phi^{(\mu )}(x)|\leq \exp(\tilde c_1x^2-\tilde c_2)
\;x\in{\bf R},\;\mu=1,2,\eqno (1.10)$$
where $\;\tilde c_1>0\;$ and $\;\tilde c_2\in{\bf R}\;$ are some constants.
 
Examples of potentials $\;\Phi\;$ and $\;\Psi_d\;$satisfying the 
conditions stated are easily provided by `polynomial' interactions.
 
A Gibbs state in a finite volume $\L \subset \bf Z$ is defined by
$$\phi_{\L}(a)\ =\ \hbox{tr}\;(a\rho_{\L}),\  a\in{\cal
B}_\L, \eqno (1.11)$$
where $\rho_\L$ is the density matrix
$$ \rho_{\L} \  =
\  \Xi_{\L}^{-1} \exp (-\beta H_{\L}), \eqno (1.12)$$
$\Xi_\L$ is the
partition function
$$\Xi_\L \  =\  \hbox{tr} \  \exp (- \beta
H_{\L}) \eqno (1.13)$$
and $\;\beta > 0\;$ is the inverse temperature of the
system. The existence of a state $\;\phi_\L\;$ is guaranteed by
 
\n {\bf Proposition 1.1}
\m
{\it Under the condition (II), 
for any $\ \beta >0\;$ operator $\;\exp (-\beta H_\L)\;$ is
of trace class.}
\b
 
Being of trace class, the operator $\;\rho_\L\;$ is determined by its 
integral kernel $\;k_\L(x_\L,y_\L):\;$
$$\rho_\L f(x_\L)=\int_{{\bf R}^\L}dy_\L k_\L(x_\L,y_\L)f(y_\L),\;f\in 
L_2({\bf R}^\L),\;x_\L,y_\L\in{\bf R}^\L,\eqno (1.14)$$
where $\;dy_\L\;$ denotes the Lebesgue measure on $\;{\bf R}^\L.\;$
 
The quantity $\lambda_\L\;=\;-1/ \vert \L\vert \;\ln \;\hbox
{tr}
\;\exp (-\beta H_\L)$ gives the free energy per lattice site in
the volume $\L\;(\;\vert\L\vert$ denotes the number of 
lattice sites in $\L$).
 
Actually, it is of interest to consider  $\;\phi_\L(a)\;$ also for  
some unbounded operators 
\break
\n$a\;$ in $\;{\cal H}_\L$ (this is possible when $\;a\rho_\L\;$ 
is a trace class operator). Good examples 
are the position and the momentum operators $\;q_j$, $p_j$, $j\in\L\;$ and 
functions of them
such as the Hamiltonian $\;H_{\L}\;$ and its kinetic and potential 
parts, $\;K_{\L}\;$ and $\;U_{\L}.\;$ It is also interesting 
to consider the Hamiltonian $\;H_{J}\;$ for a `sub-system' 
in a smaller volume $\;J\subseteq\L$ as well as its kinetic and 
potential parts. Finally, we can take a `relative' potential energy 
operator $\;U_{J\;|\;\L\setminus J}$ = $U_\L-U_{J}$. 
 
In general, we consider operators of the form 
$\;{\cal F}_J$ = ${\cal F}_J(q_j,j\in J)\;$ where 
$\;{\cal F}_J:{\bf R}^J\to{\bf R}\;$ 
is a measurable function such that
$$|{\cal F}_J(x_J)|\leq
\exp [\sum_{j\in J}(\bar c_1x^2-\bar c_2)],\eqno (1.15)$$
where $\;\bar c_1<c_1\;$ and $\bar c_2\in\bf R\;$ (cf (1.7), 
(1.10)). We can also treat a more general case where $\;{\cal F}\;$
is a function of an infinite number of variables, but its `essential' 
dependence is upon variables associated with a finite set $\;J\subseteq
{\bf Z}.\;$ An example of this kind is $\;U_{J\;|\;{\bf Z}\setminus J}$,
the potential energy of a subsystem in $\;J\;$ relative to the whole 
exterior $\;{\bf Z}\setminus J$. See Theorems 1 and 2 below. To avoid
technically complicated constructions, we omit a formal 
general set-up related to functions of that type; 
the interested reader may reconstruct 
it by following the example mentioned. 
 
The formalism introduced so far is indeed insensitive to the dimensionality of
the system: we can replace the lattice $\;{\bf Z}\;$ by its multi-dimensional 
analogue $\;{\bf Z}^\nu ,\;\nu\geq 1,$ and the single-oscillator phase 
space $\;L_2({\bf R})\;$ by $\;L_2({\bf R}^k),\;k\geq 1.\;$ Also the 
potentials may not be translation-invariant, in which case the 
self-interaction will be described by a family $\;\{\Phi_j,j\in\bf Z\}\;$ 
and the two-body interaction by a family $\;\{\Psi_{j,j'},j,j'\in\bf Z\}.\;$ 
This type of models is called a general system of (quantum) oscillators (later
on, we make this definition precise). As it was pointed out earlier, the main 
results of this paper are formulated for the one-dimensional case 
($\;\nu\;=\;1$) and translation-invariant interaction potentials. However,
some auxiliary assertions (see e.g. Lemma 3 below) are of interest for 
purposes outside this paper and are stated in a general situation. 
 
We are interested in studying the limits of$\; \phi_\L\;$and $\;
\lambda_\L\;$when $\; \L \nearrow \bf Z$.
\b
\n {\bf Theorem 1}
\m
{\it Suppose that the conditions (I) and (II) are fulfilled. Then,
for any $\beta>0$,
 
(a) there exists the limit
$$\lambda\ =\lim_{\L\nearrow\bf Z}\ \lambda_\L\eqno (1.16) $$
(the free energy of the infinite system),
 
(b) there exists the $\; w^{\star}-$limit
$$ \phi\ =\ \lim_{\L\nearrow \bf Z}\ \phi_\L \eqno (1.17) $$
which defines a state $\ \phi\ $ on $\;C^{{\star}}$-algebra $\
\cal B\;.$
 
Furthermore, the state $\ \phi\ $ is locally normal
(i.e. is given by a family $\ (\rho^{(J)})\ $ of local density
matrices), translationally invariant and has the following mixing property:
$$ \lim _{u\nearrow \pm
\infty }\phi (a_{1}\;S_{u}a_{2})\  =\  \phi (a_{1})\  \phi
(a_{2}). \eqno (1.18) $$
Hence, $\;\phi\;$is ergodic, i.e., $\  \phi \  $ gives an
extreme point of the set of translationally invariant states  on the
$C^{{\star}}$-algebra $\  \cal B $.
For any finite $\  J \subset \bf Z, \ $ the operator $\  \rho^{(J)}
\  $acts in $\  {\cal H }_{J}; \  $it is positively defined,
of trace class and with ${\rm {tr}}\;\rho^{(J)}\;=
\;1.$ 
 
(c) The functionals  $\phi ({\cal F}_{J})\;$ (defined as 
{\rm{tr}} $\;({\cal F}_{J})\rho^{(J)})\;$
are finite for any finite $\;J\subset\bf Z\;$
and any measurable function $\;{\cal F}_J\; :{\bf R}^{J}
\rightarrow \bf R \ $ which obeys (1.15). In addition, they 
coincide with the limits
$$\lim_{\L\nearrow\bf Z}\phi_\L({\cal F}_J).\eqno (1.19)$$
In particular, this is true for
$\;{\cal F}_J=U_J.\;$ Moreover, there exists a finite limit  
$$\phi (U_{J\;|\;{\bf Z}\setminus J})=\lim_{\tilde\L\nearrow{\bf Z}}
\phi (U_{J\;|\;\tilde\L\setminus J})=\lim_{\L\nearrow{\bf Z}}\phi_\L
(U_{J\;|\;\L\setminus J}).\eqno (1.20)$$
 
(d) If, in addition, condition (III) is valid,
then, for any finite $\;J\subset{\bf Z},\;$ 
the integral kernel $\;k^{(J)}\;$ of operator $\;\rho^{(J)}\;$ 
is a $\;C^2$-function of arguments $\; 
x_J,\;{y}_J\;\in{\bf R}^J.\;$ Furthermore, 
$\;\phi (p^2_{l})\;$ 
(defined as {\rm {tr}}$\;(p^2_{l}\rho^{(\{l\})})\;$) is finite  
and coincides with the limit}
$$\lim_{\L\nearrow\bf Z}\phi_\L(p^2_l).\eqno (1.21)
$$
 
{\bf Remarks.} 1. The kernels $\;k^{(J)}\;$ are given
by the formula (cf. (1.14))
$$\rho^{(J)}f(x_J)=\int_{{\bf R}^J}dy_Jk^{(J)}
(x_J,y_J)f(y_J),\;f\in L_2({\bf R}^J),\;x_J,y_J
\in{\bf R}^J,\eqno (1.22)$$
where $\;dy_J\;$ denotes, as before, the Lebesgue 
measure on $\;{\bf R}^J;\;$
these kernels are formally determined almost everywhere with respect to
this measure. Speaking of their smoothness property, we have in mind 
their variants that are determined everywhere on $\;{\bf R}^J$. The same 
is true for the analyticity property .  
 
In fact, this remark holds for every kernel we deal with in the sequel, and 
for every property that is stated for any $\;x_J$, $y_J$  $\in{\bf R}^J$.
 
2. In view of the translation-invariance property of $\;\phi\;$,
$\phi (p^2_l)\;$ does not depend on $\;l\;$ and 
also $\;\phi (U_{S_uJ})\;$ and 
$\;\phi (U_{S_uJ|{\bf Z}\setminus S_uJ})\;$ do not depend on $\;u.\;$
\b
 
The following theorem expresses the analyticity properties of the limiting
state $\;\phi\;$ constructed in Theorem 1.
\b
\n {\bf Theorem 2}
\m
{\it Let the functions $\;\Phi (\;\cdot\;,z_0)\;$ and $\;
\Psi_{d}(\cdot\;,z_1)\;$ depend on parameters $\;z_{l}\in{\bf
C},\;l\;=\;0,1,\;$ in the following way:
$$\Phi(\;\cdot\;,z_{0}\;)\;=\;z_{0}\;\Phi ,
\;\Psi_{d}(\;\cdot\;,z_{1}\;)\;=\;z_{1}\;\Psi_{d},\eqno (1.23) $$
where $\;\Phi\;$ and $\;\Psi_d\;$ satisfy conditions (I) and (II). 
Then, for any $\;\beta > 0,\;$ 
 
(a) The free energy $\;\lambda\;$ is a real analytic function of
variables $\;z_0,z_1\;$ in the \break region $\;{\cal V}=\{z_0\in{\bf
R}^{+},z_1\in[0,z_0]\}\;$ which has an analytic continuation in a
complex domain in $\;{\bf C}^{2}\;$ containing $\;{\cal V}$,
 
(b) For any finite $\;J\;$ and any $\;x_J,y_J\in{\bf R}^J,\;$ 
the same assertion holds for the kernels $\;k^{(J)}(x_J,y_J).\;$
Furthermore the same is true for 
$\;\phi ({\cal F}_{J})\;$ where $\;F_{J}\;
$ is as in Theorem 1 (in particular, for $\;F_J=U_J\;$). Finally,
the same is true for  $\;\phi (U_{J\;|\;{\bf Z}\setminus J}).\;$ 
 
(c) For any finite $\;J\;$ and a Hilbert-Schmidt operator 
$\;a\in{\cal B}_J,\;$  $\;\phi (a)\;$ admits an analytic 
continuation in a complex domain of $\;{\bf C}^2\;$ containing
$\;{\cal V}.$
 
(d) If, in addition, the condition (III) is valid, then 
$\;\phi (p_l^2)\;$ is also a real-analytic function of
$\;z_0$, $z_1$  $\in{\cal V}\;$ which admits an analytic 
continuation in a complex domain containing $\;\cal V.\;$ 
Moreover, for any finite $\;J\;$ and any $\;x_J$, $y_J$  $\in
{\bf R}^J,\;$ 
$\;\displaystyle{{{\partial^\mu}\over{\partial x^\mu}}} 
k^{(J)}(x_J,y_J)\;$ and $\;\displaystyle{{{\partial^\mu}\over{\partial
y^\mu}}}k^{(J)}(x_J,y_J)$, $\mu =1,2,\;$ are real-analytic functions 
which again admit an  analytic continuation in a complex 
domains of the same kind as before.} 
\b 
 
{\bf Remarks.} 1. the variables $\;z_{0}\;$ and $\;z_1\;$ are
subject to the restriction Re $z_0>0,\;\;0\leq$ Re $z_1\leq\;$
Re $z_0\;$ in order to preserve the superstability condition for
Im $z_0=$Im $z_1=0$.
 
2. Of course, one can admit a more general form of dependence of the
potentials $ \; \Phi \; $ and $ \; \Psi_d \; $ on the variables $ \;
z_{l} \; $ (with the same kind of restrictions as 
in the previous remark).
We have chosen the form of (1.23) for the sake of simplicity of 
the exposition.
 
3. Combining the results of this paper with those from [P2], one can also 
prove a theorem establishing a (weak) KMS property of the limiting state 
$\;\phi.\;$
 
4. The complex domain of  analyticity of  $\;\phi (\cdot )\;$ 
in the assertions (b) and (c) of  Theorem 2 depends on the operator in the 
argument of $\;\phi.\;$ The same is true for the assertion (d) where the
domain of analyticity of $\;k^{(J)}(x_J,y_J)\;$ depends on $\;x_J,y_J.\;$ 
However, under some extra conditions controlling the increasing of 
$\;{\cal F}_J\;$ or the decreassing of  the kernel of a Hilbert-Schmidt 
operator $\;a,\;$ this domain may be chosen independently on
$\;{\cal F}_J\;$ or on$\;a.\;$ Similarly, the analyticity domain of 
$\;k^{(J)}(x_J,y_J)\;$ may be chosen independently on $\;x_J,y_J\;$ 
running over any given  compact domain in $\;{\bf R}^J.\;$ In any 
case, a `width' (in `imaginary
directions') of the  complex analyticity domain varies with $\;z_0\;$
and $\;z_1\;$ and in general  tends to zero as $\;z_0\to\infty .$
 
In the sequel, we write 
$$z_0=1+w_0,\;z_1=1+w_1,\eqno (1.24)$$
incorporating in the potentials $\;\Phi\;$ 
and $\;\Psi_d\;$ `unperturbed' values
belonging to $\;\cal V\;$ and treating $\;w_0\;$ and $\;w_1\;$ as a small
complex perturbations.
 
5. As it was noted before, the condition (III) involving derivatives 
of functions $\Phi\;$ and $\;\Psi_d\;$
is used only for proving the assertions concerning the functional 
$\;\phi (p^\mu_l).\;$ 
\b
 
We  now introduce some basic technical tools and mention preliminary
facts which will be repeatedly used below. The main statement of
Theorem 1, the existence of a limiting state $\;\phi\;$ (see assertion (b)), 
is a direct corollary of  the following fact.
For any finite $\; J \subset \bf Z \;$  the limit
$$ \lim_{\L\nearrow\bf Z}\;\rho_{\L}^{(J)}\;=
\;\rho^{(J)}\eqno (1.25)$$
exits in the trace norm in $\  {\cal H}_J$. Here $\
\rho_{\L}^{(J)}\  $ is the density matrix for the restriction of the
state $\  \phi_{\L}\  $ to the $C^{\star}$-algebra $\  {\cal
B}_{J}$:
$$ \rho _{\L}^{(J)} \  = \  \hbox {tr}_{{\cal H}_{\L \setminus
J}} \rho _{\L} . \eqno (1.26)$$
 
By using  Lemma 1 from [S2], one reduces the problem to prove that
the limit (1.25) holds in the Hilbert-Schmidt norm in $\;{\cal
H}_{J}.\;$ It is convenient to pass to the integral kernels
$\;k_{\L}^{(J)}(x_J,y_J),\;
x_J$, ${y}_J$  $\in{\bf R}^{J},\;$ of 
the operators $\;\rho _{\L}^{(J)},\;J\subset\L,\;$ which
are given by
 
$$ \rho_{\L}^{(J)}f(x_J)\;=\;\int_{{\bf R}^{J}}
d{y}_J\;k_{\L}^{(J)}(x_J,
y_J)\;f(y_J).\eqno (1.27)$$
In terms of the kernels $\  k _{\L}^{(J)} ( x_J,
y_J ) \ $ the Hilbert-Schmidt norm convergence means
that
 
$$\lim_{\L\nearrow{\bf Z}}\int_{{\bf R}^J\times{\bf
R}^J}dx_J\times dy_J\;
(\  k_{\L}^{(J)}(x_J,y_J )\;-\;
k^{(J)}( x_J, y_J )\ )^2\  =\ 0. \eqno (1.28)$$
Here $\ k^{(J)}( x_J,{y}_J)
\ $ is a limiting kernel that defines the limiting density matrix
$\  \rho^{(J)} \  $ in the same way as in (1.27).
By the Lebesgue's dominated convergence theorem, it is enough to check that
the kernels $\  k_{\L}^{(J)} \  $ satisfy, uniformly in $\ \L
\supset J,\  $ a bound
$$0\;\le \; k_{\L}^{(J)} ( x_J,  {y}_J)\;\le
 \; k_{{\star}}^{(J)} ( x_J, {y}_J ), \
x_J, {y}_J \; \in {\bf R}^{J}, \eqno (1.29)$$
with $\  k_{{\star}}^{(J)} \in L_{2} ( {{\bf R}^{J}}\times
{{\bf R}^{J}} ) \  $ and that the following pointwise
convergence takes place:
 
$$\lim_{\L \nearrow Z} k_{\L}^{(J)} ( x_J,{
y}_J )\  = \  k^{(J)} ( x_J,
y_J ), \  x_J,{y}_J \in {\bf
R}^{J}. \eqno (1.30)$$
 
The translation invariance of the limiting state  $\  \phi \  $
follows from the equality for the kernels $ \  k^{(J)}$:
$$ k^{(J)}(x_J,{y}_J)\  = \
k^{(S_{u}J)} ( S_{u} x_J, S_{u} {y}_J ),\;
x_J,{y}_J \in {\bf R}^J, \; u \in {\bf
Z}, \eqno (1.31)$$
where $\  S_{u}J \; = \; (j:\; j-u \; \in J)\  $ and $\
S_{u}{ z}_J \  $ denotes, for $\  { z}_J \;=
\; (z_{j}, \; j \in J) \; \in \; {\bf R}^{J},\  $ the
element of $\  {\bf R}^{S_{u}J}\  $ given by
$$S_{u}{ z}_J\;=\;(z^{\prime}_{j^{\prime}}, \; j^{\prime}
\in S_{u} J ) \  \hbox {with}\  z^{\prime}_{j^{\prime}} \; = \;
z_{j'-u}.$$
 
The proof of the mixing property (1.18) proceeds in a similar way. Here
the problem is reduced to proving the following relation for the limiting
kernels $\  k^{(J)}$:
$$\lim_{u\rightarrow\infty}k^{(J^{(1)}\cup S_{u}J^{(2)})}({
x}_{J^{(1)}}\vee S_{u}{x}_{J^{(2)}},\;{
y}_{J^{(1)}}\vee S_{u}{y}_{J^{(2)}})\  =$$
$$=\  k^{(J^{(1)})}({x}_{J^{(1)}},{
y}_{J^{(1)}})\;k^{(J^{(2)})}({x}_{J^{(2)}},{
y}_{J^{(2)}}),\eqno (1.32)$$
 
$$x_{J^{(1)}},{y}_{J^{(1)}} \in{\bf R}^{J^{(1)}},
\;{x}_{J^{(2)}},{y}_{J^{(2)}}\in
{\bf R}^{J^{(2)}}. $$
Symbol $\ \vee\ $ indicates the operation of
``glueing" configurations over non-intersecting volumes on $\  {\bf Z}.
\ $
 
The ergodicity of the limiting state $\  \phi \  $ follows from
the mixing property by virtue of general theorems (see e.g. [Ru1])
 
Let us now comment on the existence of  $\;\phi ({\cal F}_{J}
)\;$ (see assertion (c) of Theorem 1). Without loss of generality,
we can assume that the function $\;{\cal F}_J\;$ is non-negative. We can 
write 
$$\phi_\L({\cal F}_J)=\int_{{\bf R}^J}dx_Jk_\L^{(J)}
(x_J,x_J){\cal F}_J(x_J);\eqno (1.33)$$
a similar equality holds for $\;\phi ({\cal F}_J).\;$
Under the condition (1.15) we will 
establish the estimate
$$k_\L^{(J)}(x_J,x_J){\cal F}_J(x_J)\leq 
\exp [-\sum_{j\in J}
({\u c}_1x_j^2-{\u c}_2)]\eqno (1.34)$$
for some constants $\;{\u c}_1>0\;$ and $\;{\u c}_2
\in{\bf R}\;$ depending on $\;{\cal F}_J,\;$ but not on 
$\;x_J,y_J\in{\bf R}^J.\;$ The existence of the pointwise limit in (1.30) 
and the Lebesgue's dominated convergence Theorem will then imply the 
convergence to a finite limit in (1.19) and the coincidence 
of the limiting value with $\;\phi ({\cal F}_J)$.
 
In a similar way  one can  prove the existence of the  limits 
in (1.20) and  their coincidence with 
$\;\phi (U_{J\;|\;{\bf Z}\setminus J}).\;$ We omit a detailed argument 
since it is the same 
as in the case of $\;\phi_\L (p^2_l)$.
 
Finally, the smoothness of the limiting kernels $\;k^{(J)}\;$ and the 
existence of  $\;\phi (p_l^2)\;$ (see the assertion (d)
of Theorem 1) is established as follows. For any finite $\;\L\subset{\bf Z}\;$
and $\;J\subseteq\L\;$ the kernel $\;k_\L^{(J)}\;$ is a $\;C^2$-function of
the variables $\;x_J,y_J\;$ and it indeed converges to a limit, 
as $\;\L\nearrow{\bf Z},\;$
together with its derivatives
$${{\partial^\mu}\over{\partial x_j^\mu}}k_\L^{(J)}(x_J,y_J)\quad
{\rm{and}}\quad{{\partial^\mu}\over{\partial y_j^\mu}}k_\L^{(J)}(x_J,y_J),
\;j\in J,\;x_J,y_J\in{\bf R}^J,\;\mu =1,2.\eqno (1.35)$$
Moreover, the convergence is uniform in $\;x_J,y_J\;$ running over a compact 
set in $\;{\bf R}^J.\;$ 
The Fubini's theorem then implies that the limiting kernel $\;k^{(J)}\;$ is a 
$\;C^2$-function of $\;x_J,y_J\in{\bf R}^J\;$ and that the limits of 
the derivatives coincide with the derivatives of the limiting kernel. 
 
In addition, we establish a bound: for any finite $\;\L\subset\bf Z\;$ and $\;
J\subseteq\L,\;$ 
$$\vert{{\partial^\mu}\over{\partial x^\mu}}k^{(J)}_\L(x_J,y_J)
\lceil_{x_J=y_J}\vert\leq\exp [-\sum_{j\in J}(\tilde c_1x_j^2
-\tilde c_2)],\eqno (1.36)$$
where, for a fixed $\;J,\;$ constants $\;\tilde c_1>0\;$ 
and $\;\tilde c_2\in\bf R\;$ do not depend on $\;
\L\;$ and $\;x_J,y_J\in\bf R.\;$ We then
write 
$\;\phi_\L(p_l^2)\;$ as an integral
$$\phi_\L(p_l^2) \;= \; \int_{\bf R} dx_l(
{\partial^{2}\over\partial x_l^{2}} k_\L^{(\{l\})} (x_l,
y_l) \lceil _{x_l=y_l});\eqno (1.37)$$  
a similar representation holds also for  $\;\phi (p^2_l).\;$
After the estimate (1.36), the existence of a finite limit of the quantity 
(1.37) and its coincidence with
$\;\phi_(p^2_l)\;$ follows, as before, from the convergence of the 
derivatives (1.35) and the Lebesgue's dominated 
convergence theorem.\footnote*
{As a byproduct of this argument, we get that $\;\phi (p_l)\;$
is finite (and equals zero).}
 
A key role in the proof of the assertions stated is played by the Wiener
integral representations for the partition function $\  \Xi_\L
\ $ and the kernels $\  k_\L^{(J)},\;J\subset\L,\  $
which follow from the Feynman-Kac formula for the integral kernel
$\  e_\L\ $ of operator
$\  \exp (-{\beta} H_\L )\ $ (see [Gi]):
$$e_\L ({x}_{\L},{y}_{\L})\;=\;\int_{{\bf W}_{
x_{\L},{y}_{\L}}^{(\beta )}} dP_{x_{\L},
y_{\L}}^{(\beta )}({\omega}_{\L})\;\exp\;(-V_\L ({
\omega}_{\L})),\quad {x}_{\L},{y}_{\L}\in{\bf R}^\L.\eqno (1.38)$$
 
Here, the space $\;{\bf W}_{{x}_{\L},{y}_{\L}}^{(\beta )}
\  $ is the Cartesian product
$$\times _{j \in \L} {\bf W}_{x_j,y_j}^{(\beta )},$$
where $\  {x}_{\L}\;=\;(x_j,j\in \L),\  {
y}_{\L}\;=\;(y_j,j\in\L),\  $ and $\  {\bf W}_{x,y}^{(\beta
)},\;x,y\in{\bf R},\  $ is defined as the set of
continuous functions (paths) $\  \omega : \; [0, \beta ]
\rightarrow {\bf R} \  $ with $ \  \omega (0) \; = \; x, \;
\omega ( \beta ) \; = \; y ,\  $ which is endowed with the 
standard Borel space structure. Furthermore, a measure $\
P_{{x}_{\L},{y}_{\L}}^{(\beta )}\qquad $ is the product
$$\times_{j\in\L}P_{x_j,y_j}^{(\beta )}$$
where $\  P_{x,y}^{(\beta )}\ $ is the  non-normalized
conditioned Wiener measure  on $\  {\bf 
W}_{x,y}^{(\beta )}\ $ (this means that $\  P_{x,y}^{(\beta )}(
{\bf W}_{x,y}^{(\beta )})\;=\;(2\pi\beta )^{-1/2}\;\exp\;[-
1/2\beta (x-y)^2)]).\ $ Finally,$\  V ({
\omega}_{\L}) \  $ is the `potential energy' of the path `configuration'
$\  {\omega}_{\L}\;=\;(\omega_j,\;j\in\L )\in{\bf
W}_{{x}_{\L},{y}_{\L}}^{(\beta )}.\ $ In analogy with
(1.2b,c) we set
$$V_\L (\;{ \omega}_{\L}\;) \;= \;V_{\L ,0}(\;{
\omega}_{\L}\;)\;+\;V_{\L ,1}(\;{ \omega}_{\L}\;),\eqno (1.39)$$
where
$$V_0(\;{ \omega}_{\L}\;) \;=\;\sum_{j\in \L}
\varphi(\;\omega_j\;),\qquad\;V_1(\;{
\omega}_{\L}\;)\;=\;{1\over 2}\sum_{j,j^\prime\in\L :j\ne
j^\prime}\;\psi_{|j-j'|} (\;\omega_j,\omega_{j^\prime}\;),
\eqno(1.40)$$ 
$$\varphi (\;\omega\;)\;=\;\int _0^{\beta} \Phi
(\;\omega (t)\;)dt, \quad\quad\psi_d
(\omega ,\omega')\;=\;\int_0^{\beta} \Psi_d
(\omega (t),\omega '(t)\;)dt\eqno (1.41) $$
 
For the sake of simplicity, we omit the index $\  (\beta )\  $ from the 
notations. We also identify the spaces $\  {\bf W}_{x,x},\;x\in
{\bf R},\ $ with a single space $\  {\bf W}\;=\;{\bf W}_{0,0} \
$ by means of the mapping $ \  \omega \hbox{\mm !}
\omega\;+\; x.\  $ Measures $\  P_{x,x}\  $ and
$\  P=P_{0,0} \ $ are transformed thereby into each other. A
measure space $\  ({\bf W}_{{x}^\L,{x}^\L},P_{{x}^\L,{
x}^\L})\  $ will be identified with the product-space $\  
({\bf W}^\L,P^\L).
\  $ Sometimes it will also be convenient to use the map $\ {\bf W}_{x,y}
\hbox{\mm !} {\bf W}_{0,0} \  $ given by $\  \omega
\hbox{\mm !} \omega \; + \; L_{x,y} \ $ where $\  L_{x,y}
\  $ is the linear function $\  L_{x,y} (t)\;=\;
x+t\beta^{-1}(y-x).\  $ The measure $\  P_{x,y} \ $ is
transformed thereby into $\  \exp\;(-1/2\beta (x-y)^2) P_{0,0}.\  $
The product-space $\;{\bf W}_{x_J,y_J}\;$ is transformed into $\;
{\bf W}^J\;$ by a vector analogue of this construction where the 
function $\;\bar L_{x_J,y_J}(t)=x_J+t\beta^{-1}(y_J-x_J)\;$ is used.
 
It is easy to check that, under our conditions on functions
$\Phi\ $ and $\  \Psi_d,\  $ the kernel $\  e_\L \ $
is a continuous function of variables $\  {x}_{\L} \ $ and
$\  {y}_{\L}.\  $ According to the Mercer's theorem and our previous 
arguments, we can
write the following formulas for the partition function $\  \Xi_\L
\ $ and the kernels $\  k_\L^{(J)}:$
$$\Xi_\L\;=\;\int_{{\bf R}^\L\times {{\bf W}^\L}}d{ u}_{\L}
dP^\L({\omega}_{\L})\exp (-V({\omega}_{\L}+{u}_{\L}))\eqno (1.42)$$
and
$$ k_\L^{(J)}(x_J, {y}_J) \; = \;
(\Xi_\L)^{-1} \; \Xi_\L^{(J)}(x_J,{y}_J),\quad 
x_J,y_J\in {\bf R}^J,\eqno (1.43)$$
where
$$\Xi_\L^{(J)}(x_J,y_J)\;=\;\exp [-1/2\beta\sum_{j\in J}(x_j-y_j)^2]
\int_{{\bf R}^{\L\setminus J}\times
{\bf W}^{\L\setminus J}}du_{\L\setminus J}dP^{\L\setminus J}(
\omega_{\L\setminus J})\;
\times $$
$$\int_{{\bf W}^{J}}
dP^{J}(\omega_J)\exp
(-V_\L (({\omega}_{\L\setminus J}+{u}_{\L\setminus J}
)\vee ({\omega}_J+\bar L_{x_J,y_J})).\eqno (1.44) $$
Here $\;\omega_{\L}\;+\;{u}_{\L}\;$ is the collection of the 
shifted trajectories 
$\;(\omega_j+u_j,j
\in \L),\;$ and $\;du_{\L}\;$ is the Lebesgue measure on $\;{\bf
R}^{\mid\L\mid}.\;$ 
The symbol $\  \vee\  $ has the same meaning as in (1.32).
 
We give at this point a scheme of the proof of  Proposition 1.1
and Theorems 1 and 2 (taking into account the intermediary
assertions stated so far in the course of exposition).
 
Using formula (1.38), we reduce the problem of proving 
Proposition 1.1 to check that the integral in the RHS is finite
for any value of $\  \beta > 0\  $. This is a straight-\break
forward (though
tedious) calculation based on the superstability condition (1.7).
Cf. the bound  (2.32) in Section 2 of this paper. 
 
The above arguments show that the part ($b$) of Theorem 1
follows from  Lemma 1 (see below). Furthermore, 
the proof of the part ($a$) is contained in the proof
of this lemma.
\b 
\n{\bf Lemma 1}
\m
{\it Assume that the interaction potentials $\;\Phi\;$ and $\;\Psi_d\;$ 
satisfy the conditions (I) and (II). Let the kernels $\;k^{(J)}_{\L}\;$
be given by (1.43). Then the pointwise limit (1.30) exits and the
limiting functions $k^{(J)}$
obey (1.31), (1.32). Moreover, the kernels $\;k^{(J)}_{\L}\;$
(and hence also the limiting kernels
$\;k^{(J)}\;$ ) satisfy the bound
(1.29) uniformly in $\L$ with a function $k^{(J)}_*\ $ 
that has the properties listed above. 
 
For any function $\;{\cal F}_J\;$ obeying (1.15) the bound (1.34) 
is fulfilled.
 
If, in addition, the condition (III) is valid, then the 
kernels $\;k_\L^{(J)}\;$ are of class $\;C^2\;$ in  the
variables $\;x_J,y_J\;$ 
and they converge to limits together with their derivatives (1.35).
Moreover, this convergence is uniform in $\;x_J,y_J\;$ 
running over a compact set in $\;{\bf R}^J.\;$ Finally, 
the bound (1.36) holds.}
\b
 
Theorem 2 follows from  Theorem 1 and  Lemma 2:
\b 
\n{\bf Lemma 2}
\m
 
{\it Assume that the potentials $\Phi$ and
 $\Psi_d$, $d\geq 1$, depend on the
parameters $z_0,\ z_1$ as indicated in (1.23).
Under  the conditions (I) and (II), for any $\;\beta\;$ 
there exist neighborhoods 
${\cal O}_0\ ,{\cal O}_1$ of the
origin in ${\bf C}$ such that, for any finite $\;\L\subset\bf Z$, 
$\;\lambda_\L\;$  admits an analytic continuation in 
$\;w_\ell\in{\cal O}_\ell\;$ and
$\ |\lambda_\L|\  $ is bounded uniformly in 
$\;\L\;$ and $\;w_{\ell}\in{\cal O}_{\ell}\ ,\ \ell=0,1.$ Furthermore, as
$\;\L\nearrow\bf Z,\;$ the analytic functions $\;\lambda_\L\;$ converge, 
uniformly in $\;{\cal O}_0\times{\cal O}_1,\;$ to a limit which is 
again an analytic function in
$\;w_\ell\in\cal O_\ell ,$ $\ell =0,1.$
 
Similarly, for any $\;\beta>0,\;$ any finite $\;J\subset{\bf Z}\;$ 
and any $\;x_J,y_J\in{\bf R}^J,\;$ there exist neighborhoods 
$\;{\cal O}_0\;$ and $\;{\cal O}_1\;$ of the origin 
in $\;\bf C\;$ such that for any finite $\;\L\supseteq J\;$ 
the kernel $\;k_\L^{(J)}(x_J,y_J)\;$ admits an  analytic continuation 
in a domain $\;{\cal O}_0\times{\cal O}_1\;$ and $\;|k_\L^{(J)}(x_J,y_J)|\;$ 
is bounded uniformly in $\;\L\;$ and $\;w_\ell\in{\cal O}_\ell$, $\ell=0,1$.
Furthermore, as $\;\L\nearrow{\bf Z},\;$ the analytic functions
$\;k_\L^{(J)}(x_J,y_J)\;$ converge, uniformly in 
$\;{\cal O}_0\times{\cal O}_1,\;$ to a limit which is again an analytic 
function in $\;w_\ell\in{\cal O}_\ell$, $\ell=0,1$. If $\;x_J,y_J\;$ run 
over a compact set in $\;{\bf R}^J,\;$ then the neighborhoods 
$\;{\cal O}_\ell\;$ may be chosen independently on $\;x_J,y_J\;$
and the convergence is uniform in $\;x_J,y_J.\;$
 
Moreover, similar assertions hold, for any $\;\beta>0\;$ and finite 
$\;J\subset\bf Z,\;$ for  $\;\phi_\L(a)\;$ where $\;a\;$ is
an operator in ${\cal H}_J$ of the kind considered
in assertions (b) and (c) of  Theorem 2. 
 
If, in addition, condition (III) is valid, then
the same assertions hold for the derivatives
(1.35) and  $\;\phi_\L(p^2_l)\;$ given by (1.37).}
\medskip
 
The proof of  Lemmas 1 and 2  are similar and are
given in Section 2. 
 
In the following we shall use the notation
$${\bf S}={\bf R}\times {\bf W}\quad{\rm{and}}\quad
s=(x,\omega )\in{\bf S}\eqno (1.45)$$
as well as 
$$\bar{\bf S}={\bf R}\times{\bf R}\times{\bf W}\quad{\rm{and}}\quad\bar s=(x,
y,\omega )\in\bar{\bf S}.\eqno (1.46)$$
Of course, $\;\bf S\;$ may be identified as a `diagonal part' of $\;\bar
{\bf S};\;$ in the sequel we use this fact without mentioning it. The 
spaces $\;\bf S\;$ and $\;\bar{\bf S}\;$ are provided with the norms 
$$\parallel s\parallel_r=(\int^{\beta}_{0} |s(t)|^rdt)^{1\over r},\;r=1,2,
\eqno (1.47)$$
and
$$\parallel\bar s\parallel_r=(\int_0^\beta |\bar s(t)|^rdt)^{1
\over r},\;r=1,2,\eqno (1.48)$$
where
$$s(t)=x+\omega (t),\;\bar s(t)=L_{x,y}(t)+\omega (t).$$
 
Given a finite $\;J\subset{\bf Z}\;$, we denote  
$$s_{J}=(s_j,j\in J)\in{\bf S}^{J}(=
{\bf R}^J\times {\bf W}^J),\;s_j=(x_j;\omega_j)\in{\bf S},\eqno (1.49)$$
or, equivalently, 
$$s_J=(x_J;\omega_J),\;x_J\in{\bf R}^J,\; \omega_J\in{\bf W}^J,$$
and
$$\bar s_J=(\bar s_j,j\in J)\in{\bar{\bf S}}^J(={\bf R}^J\times{\bf R}^J
\times{\bf W}^J),\;\bar s_j=(x_j,y_j;\omega_j)\in\bar{\bf S},\eqno (1.50)$$
or, equivalently,
$$\bar s_J=(x_J,y_J;\omega_J ),\;x_J,y_J\in{\bf R}^J,\;\omega_J\in{\bf W}^J.$$
As before, $\ s_J\  $ and $\  {\bar s}_J\  $ are called path configurations 
over $\  J.$
 
We also denote by $\;ds_J\;$ the measure $\;du_JdP(\omega_J)\;$
on ${\bf S}^J$ and use, as before, the notation $\;s_J\vee s_{J'}\;$ (and  
also $\;s_J\vee\bar s_{J'},\;\bar s_J\vee s_{J'},\;$ etc.), 
$\;J\;$ and $\;J'\;$ being non-intersecting 
finite subsets of $\;\bf Z,\;$ for the operation of ``glueing''
path configurations. 
The space-translations $\ S_u,\ u\in{\bf Z},\  $
act on  path configurations in a 
natural way: they map $\  {\bf S}^J\  $ onto $\  {\bf S}^{S_uJ}\  $ and 
transform the measure $\  du_JdP(\omega_J)\  $ to 
$\ du_{S_uJ}dP(\omega_{S_uJ}).\ $
We can then 
use the notation $\;\varphi (s)$, $\psi_{|j-j'|}(s_j,s_j')$, 
$\;V_\L(s_\L )\;$ and $\;V_\L (s_{\L \setminus J}\vee\bar s_J)\;$ 
and define: 
$$k^{(J)}_{\L}(\bar s_{J})=(\Xi_\L )^{-1}\Xi_\L^{(J
)}(\bar s_{J})\eqno (1.51)$$
where
$$\Xi_\L^{(J)}(\bar s_J)=\int_{{\bf
S}^{\L\backslash J}}ds_{\L\backslash J}\;\exp\;(-
V_\L (\bar s_J\vee s_{\L\setminus J})) .\eqno (1.52)$$
This allows us to write:
$$k_\L^{(J)}(x_J,y_J)=\exp [-1/2\beta\sum_{j\in J}(x_j-y_j)^2]\int_{{\bf W}^J}
dP^J(\omega_J)k_\L^{(J)}(\bar s_J)\eqno (1.53)$$
with $\;\omega_J=(\omega_j)_{j\in J}\;$ and $\;\bar s_J=(\bar s_j)_{j\in J}\;$
where $\;\bar s_j=(x_j,y_j;\omega_j).\;$
 
We are now going to write down formulas for the derivatives
(1.35). For the sake of brevity, we restrict ourselves to the case
of the derivatives $\;\displaystyle{{\partial^\mu}\over{\partial x^\mu_j}};\;$
obvious modifications needed to cover the case of $\;\displaystyle
{{\partial^\mu}\over{\partial y^\mu_j}}\;$ may easily be done by the reader.
Introducing the notation  
$$\varphi^{(\mu )}(\bar s)=\int_0^\beta dt(1-\beta^{-1}t)^\mu
\Phi^{(\mu )}(\bar s(t)),\;\mu=1,2,\;\bar s\in\bar{\bf S},\eqno (1.54)$$ 
$$\psi_d^{(\mu )}(\bar s,\bar s')=\int_0^\beta dt(1-\beta^{-1}t)^\mu
\Psi_d(\bar s(t),\bar s'(t)),\;\mu=1,2,
\;\bar s,\bar s'\in\bar{\bf S},\eqno (1.55)$$ 
and 
$$(V_\L)^{(\mu )}_j(\bar s_\L)=\varphi^{(\mu )} (\bar s_j)
+\sum_{j'\in\L:j'\neq j}\psi^{(\mu )}_{|j-j'|}
(\bar s_j,\bar s_{j'}),\;\bar s_\L=(\bar s_{\tilde j},
\tilde j\in\L),\eqno (1.56)$$ 
we can write, by using (1.42) - (1.44) and (1.51) - (1.53), 
$${\partial\over{\partial x_j}}k_\L^{(J)}(x_J,y_J)=-
(\Xi_\L)^{-1}[{1\over\beta}(x_j-y_j)\;\Xi_\L(x_J,y_J)+
(\Xi_\L^{(J)}(x_J,y_J))^{(1)}_j],\eqno (1.57)$$ 
where
$$\eqalignno{(&\Xi_\L^{(J)}(x_J,y_J))^{(1)}_j=\exp[-1/2\beta
\sum_{\tilde j\in J}(x_{\tilde j}-y_{\tilde j})^2]
\int_{{\bf W}^J}dP^J(\omega_J)\times\cr
&\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}(V_\L)^{(\mu )}_j
(\bar s_J\vee s_{\L\setminus J})\exp [-V_\L(\bar s_J\vee 
s_{\L\setminus J})],\;\bar s_J=(x_J,y_J;\omega_J ),&(1.58)\cr}$$
and
$$\eqalignno{{{\partial^2}\over{\partial x^2_j}}k_\L^{(J)}(x_J,y_J)=
&(\Xi_\L)^{-1}[({1\over{\beta^2}}(x_j-y_j)^2-{1\over\beta})\Xi_\L
(x_J,y_J)+\cr&+{2\over\beta}(x_j-y_j)\;(\Xi_\L^{(J)}(x_J,y_J))^{(1)}_j-
(\Xi_\L^{(J)}(x_J,y_J))^{(2)}_j],&(1.59)\cr}$$
where 
$$\eqalignno{(\Xi_\L^{(J)}(x_J,y_J))^{(2)}_j&=\exp [-1/2\beta
\sum_{\tilde j\in J}(x_{\tilde j}-y_{\tilde j})^2]\times\cr
&\int_{{\bf W}^J}dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}
ds_{\L\setminus J}\{[-(V_\L)^{(1)}_j(\bar s_J
\vee s_{\L\setminus J})]^2+\cr+(V_\L)^{(2)}_j
(\bar s_J\vee&s_{\L\setminus J})\}\exp\;[-V_\L
(\bar s_J\vee s_{\L\setminus J})],\;\bar s_J
=(x_J,y_J;\omega_J ).&(1.60)\cr}$$
 
The idea that we follow in the sequel (see the end of Section 2) is to 
treat separately the addends in the square brackets 
$\;[\dots ]\;$ in (1.57) and (1.59) and, in the case of (1.60), 
the single addends in the braces $\;\{\dots \}\;$ in the 
integrand. Furthermore, those addends are decomposed into series, 
according to (1.56), and we deal with a single term in such 
a series. In fact, at a certain point we need to consider separately 
the `positive' and 
`negative' parts of these terms meaning merely the integrals
of the positive and the negative parts of the corresponding
integrand. For example, the positive part of a term in the RHS of (1.60),
which, after decomposing quantity $\;(V_\L)^{(2)}_j(\bar s_J
\vee s_{\L\setminus J}),\;$ corresponds to $\;\psi_{|j-j'|}
(\bar s_j,\bar s_{j'}),\;$ is 
$$\eqalignno{\exp\;[-1/2\beta\sum_{\tilde j\in J}(x_{\tilde j}
-y_{\tilde j})^2]&\int_{{\bf W}^J}
dP^J(\omega_J )\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
[(V_\L)^{(1)}_j(\bar s_J\vee s_{\L\setminus J})]^2\times\cr&\exp\;
[-V_\L(\bar s_J\vee s_{\L\setminus J})],\;
\bar s_J=(x_J,y_J;\omega_J).&(1.61)\cr}$$
We use, for the positive and negative parts of such a term, 
a conventional notation
$[(\Xi_\L^{(J)}(x_J,y_J))^{(\mu )}_j]_\pm^{\rm{single}}.$ 
 
We conclude this section with a lemma containing the basic probability estimate
for a general system of oscillators. As it was noted before, a general system
of oscillators may be considered on a multi-dimensional lattice ${\bf
Z}^{\nu}\;$ and have $L_2({\bf R}^{l}\;$
as a single-particle phase space, $ \nu,\, l \geq 1$.We now make this
concept precise. 
In  Lemma 3 below, by 
a general system of quantum oscillators (in a finite volume 
$\L\subset{\bf Z}^\nu\;$) we  
mean a probability distribution on the path configuration space 
$\;\bar{\bf S}^\L\;$ (or on its subset such as $\;{\bf S}^\L\;$ or 
$\;{\bf W}^J\times{\bf S}^{\L\setminus J}\;$ where $\;J\subseteq\L\;$).
By a  path we now mean a multi-dimensional Wiener trajectory
$\;\omega$ :  $[0,\beta ]\to{\bf R}^l;\;$ all objects introduced
so far are  extended to this case without difficulties.
 
The structure of a probability distribution  is motivated 
for example by the
formula (1.51). More precisely, such a measure is determined by a normalizing
denominator which has the form of an integral, over a subset of 
$\;\bar{\bf S}^\L,\;$ with a non-negative integrand. In other words, we follow 
the definition of a Gibbs measure in classical Statistical Mechanics, in a
situation where the role of a `spin' is played by a path.
 
In these terms, the denominator $\;\Xi_\L\;$ determines the
`original' measure on  $\;{\bf S}^\L\;$: 
$${1 \over \Xi_{\Lambda}} du_{\Lambda}\,dP^{\Lambda}(\omega_{\Lambda})
\exp [-V(\omega_{\Lambda} + u_{\Lambda})]
$$
 Other
examples of interest are measures on $\;{\bf W}^J\times{\bf
S}^{\L\setminus J}\;$ determined by the denominators
$\;\Xi_\L^{(J)}(x_J,y_J)\;$ and $\;\pm [(\Xi_\L^{(J)}(x_J,y_J))^{(\mu
)}_j]_\pm^{\rm{single}}$,  $x_j,y_J\in{\bf R}^J,$ $\mu=1,2,\;$ 
provided that they do not vanish, of course 
(in the case $\;l>1\;$ where
derivatives are replaced with gradients, we treat  separately
different components of the corresponding vector- and 
tensor-functions).
 
Furthermore, given a non-negative function $\;{\cal E}_J:({\bf S}^l)^J\to{\bf
R}^+\;$ and a Hilbert-Schmidt operator $\;a\;$ in 
$\;{\cal H}_J(=L_2(({\bf R}^l)^J))\;$ with a non-negative kernel $\;{\cal A}:$ 
$({\bf R}^l)^J\times ({\bf R}^l)^J$ $\to{\bf R}^+,\;$ we can speak of measures 
on $\;{\bf S}^\L\;$ determined by the denominators $\;\Xi_\L({\cal E}_J)\;$ and
$\;\Xi_\L(a)\;$ given by
$$\Xi_\L ({\cal E}_J)=\int_{{\bf S}^\L}ds_\L{\cal F}(s_\L)
\exp [-V_\L (s_\L )]\eqno (1.62)$$
and
$$\Xi_\L (a)=\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
\int_{\bar{\bf S}^J}d\bar s_J {\cal A}(y_J,x_J)\exp [-V_\L(\bar s_J\vee 
s_{\L\setminus J})],\;\bar s_J=(x_J,y_J;\omega_J).\eqno (1.63)$$
 
In a general case (of complex-valued functions and kernels), we can deal
with the positive and negative restrictions (of both positive and negative
parts) and consider the corresponding probability measures on $\;{\bf S}^\L.$
 
Furthermore, an `energy' $\;V_\L(\bar s_\L )\;$ of a path configuration 
$\;\bar s_\L\in{\bf S}^\L\;$ may be generated by a non-translation-invariant,
multi-body interaction which is described by a family $\;\{\Phi_j,
j\in{\bf Z}^\nu\}\;$ of the self-interaction potentials with
$$\sup_{j\in{\bf Z}^\nu}\sup_{x\in{\bf R}^l:||x||\leq 1}|
\Phi_j(x)|\leq\u c<\infty ,\eqno (1.64)$$
and a family 
$\;\{\Psi_B,B\subset{\bf Z}^\nu\}\;$ of the  interaction potentials. 
Here, $\;\Psi_B\;$ describes the contribution, into the potential energy, 
of a subsystem of oscillators over a finite set $\;B\subset{\bf Z}^\nu.\;$
We assume that $\;\Psi_B\;$ is a function $\;({\bf R}^l)^B\to{\bf R}.\;$
Actually, for our purposes it is sufficient to assume that $\;\Psi_B\;$
is non-zero for a finite collection of sets $\;B\;$ with $\;|B|\geq 2.\;$ 
As for $\;B\;$ with $\;|B|=2,\;$ we assume that 
$$|\Psi_{\{j,j'\}}(x_j,x_{j'})|\leq \hat c(||x_j||+1)\;(||x_{j'}||+1)
\;|j-j'|^{-(\nu +\eta )},\eqno (1.65)$$
where $\;\hat c\;$ and $\;\eta\;$ are positive constants and $\;|j-j'|\;$
is the Euclidean distance between $\;j\;$ and $\;j'$.  
The condition (1.7) is preserved (with obvious modifications in
the corresponding definitions).
 
We use the notation 
$\;{\bf m}_\L\;$ for a measure of one of the types previously considered.
In analogy with (1.51), the `local' density in a volume $\;J\subset\L\;$ is 
denoted by $\;k_{{\bf m}_\L}^{(J)}(\bar s_J)$, $\bar s_J\in\bar{\bf
S}^J.$
 
\b
\n{\bf Lemma 3}
\m
{\it Consider a general system of quantum oscillators, as specified above.
Then for any $\;c^{\star}_1\in (0,c_1)\;$ (see (1.7)) there exists a 
constant $\;c^\star_2\in{\bf R}$
(depending on a measure $\;{\bf m}_\L\;$) such that for any finite 
$\;\L\subset{\bf Z}\;$ and $\;J\subset\L\;$ 
and any $\;\bar s_J=(\bar s_j,j\in J)\in\bar{\bf S}^J$
$$k^{(J)}_{{\bf m}_{\L}}(\bar s_J)\;\leq\;\exp\;[-\sum_{j\in J}^{}(c^{\star}_1
\parallel
\bar s_j\parallel^2_2-c^\star_2)].\eqno (1.66)$$
The constant $\;c^\star_2\;$ may be chosen to be uniform for 
measures with the same form of the denominator provided that, in the RHS of 
(1.7), (1.64) and (1.65), $\;c_1\;$ and $\;\eta\;$ are separated from zero and  
$\;c_2$, $\u c\;$ and $\;\hat c\;$ are varying in compact sets. 
In the case of the measures 
on $\;{\bf W}^J\times{\bf
S}^{\L\setminus J}\;$ with the denominators
$\;\Xi_\L^{(J)}(x_J,y_J)\;$ and $\;\pm [(\Xi_\L^{(J)}(x_J,y_J))^{(\mu
)}_j]_\pm^{\rm{single}}$,  $x_j,y_J\in{\bf R}^J,$ $\mu=1,2,\;$ (and fixed 
interaction potentials) constants $\;c^\star_1\;$ and $\;c^\star_2\;$ may be
chosen uniformly for $\;x_J,y_J\in\bf O\;$ where $\;{\bf O}\subset 
({\bf R}^k)^J\;$ is a compact set.}
 
\b
The proof of  Lemma 3 is carried out in the Appendix.
\b
\n {\bf 2. PROOF OF LEMMAS 1 AND 2}
 
\m
\n The proof is based on methods developed in [C.C.O.]. For
completeness we reproduce
a construction used in [C.C.O.] to transform our ``ensemble'' of
interacting paths into a polymer system. We start by  
treating the partition function $\Xi_{\L}$ for the unperturbed
Hamiltonian $H$ (that is, for $w_0=w_1=0$ in (1.24)). Let $L,\ n,\ p$
be positive integers and $\  \L(=\L_{L,n,p})\subset{\bf Z}\  $ 
be an interval of length $\;\vert\L\vert =(2p+1)L+2pnL,\;$     
centred at the origin.\footnote{$^*$}{The term `interval' and 
notation $\;[\alpha ,\alpha']\;$ are used here and below for
intervals on the lattice $\;\bf Z.\;$ By centred at the origin we
mean that the origin coincides with the rightmost point of the A-block
$\;A_0$ (see below).} Interval $\;\L\;$ is decomposed into pair-wise
disjoint  consecutive intervals, or blocks, $\ A_i\  $ and $\ B_i\  $
(alternatively  called blocks of type A and B, or briefly, A- and
B-blocks): $$\L_p=A_{-p}\cup B_{-p}\cup A_{-p+1}\dots \cup B_{-1}\cup
A_0\cup B_0\dots \cup B_{p-1}\cup A_p$$ where $\vert A_i\vert = L,\
\vert B_i\vert = nL.\  $  Furthermore, for any $\;i=-p,\dots ,p-1\;$
we decompose the  B-block $B_i$ into $n$ consecutive intervals
(blocks) of length $L$ : $$B_i=\bigcup_{k=1}^{n} B_{i,k},\qquad \vert
B_{i,k}\vert = L.$$ The blocks $\  A_j\  $ and $\  B_{j,k}\  $ are
sometimes called elementary.
 
The block partition of volume $\L$ allows us to write a path
configuration  $s_{\L}\in{\bf S}^{\L}$ as a sequence
$$(s_{A_{-p}},s_{B_{-p}},...,s_{B_p},s_{A_p})$$
and furthermore a path configuration $\;s_{B_j}\in{\bf S}^{B_j}\;$ as 
$$(s_{B_{j,1}},...,s_{B_{j,n}})\;$$
and to use the notation introduced so far. In particular, given an
integer $\;L>0,\;$ the potential energy of a collection $\;s_\L\;$
may be written as the sum
$$V^{\leq L}(s_{\L})+V^{> L}
(s_{\L})\eqno (2.1a)$$
(subscript $\;\L\;$ is omitted for the sake of simplicity). Here
$\;V^{\le L}\;$ includes the self-interaction energy and the energy
of two-body interaction on distances $\ \leq L\;:$ $$V^{\leq
L}(s_{\L})=V_0(s_{\L})+V_1^{\leq L}(s_{\L}),\eqno (2.1b)$$
 
$$V_1^{\leq L}(s_{\L})={1\over 2}\sum_{i,i^\prime\in\L:i\ne i'}
\psi^{ \leq L}_{\vert i-i^\prime\vert}( s_i,s_{i^\prime})\eqno (2.1c) $$
and $\ V^{>L}(s_{\L})\ $ is the remaining part of the energy
containing the long-range two-body interaction terms:
$$V^{>L}(s_{\L})={1\over 2}\sum_{i,i^\prime\in\L}\psi^{> L}_{\vert
i-i^\prime\vert}(s_i,s_{i^\prime}),\eqno (2.1d) $$
where (cf. (1.41)), for $\  s,s^\prime\in{\bf S},\;$
$$\psi^{\leq L}_d (s,s^\prime )=\psi_d(s,s^\prime ),\ \hbox
{if}\ d\leq L,\eqno (2.1e)$$
$$\psi^{\leq L}_d(s,s^\prime )=0,\  \hbox {if} \  d > L,\eqno (2.1f)$$
and
$$\psi^{>L}_d(s,s^\prime )=\psi_d
( s,s^\prime)-\psi (s,s^\prime ).\eqno (2.1g)$$
 
We call $\  V^{\leq L}\ $ a short-range part of the interaction
energy. Notice
that in the cut-off interaction picture the non-adjacent elementary blocks
do not interact.
 
Let us now consider the long-range part of the interaction.
Given two blocks $D$ and $D'$ (not necessarily distinct), we set:
$$W^{>L}(s_D,s_{D'})=\sum_{i\in D}\;
\sum_{i'\in D'}\; \psi^{>L}_{|i-i'|}(s_i,s_{i'})\eqno (2.2a)$$
For a pair $\;C=\{D,D'\}\;$ we then write 
$$                                                    
W^{>L}(s_C)\;=\;W^{>L}(s_D,s_{D'})
\eqno (2.2b)$$
We denote by ${\cal C}_{\L}$ the collection 
of block pairs $C$ of the following form:
$$\eqalign{&C=\{A_j,A_{j'}\},\quad\quad -p\leq j,j'\leq p,\quad j\not =
j-1,j,j+1,\cr
&C=\{A_j,B_{j'}\},\quad\quad -p\leq j\leq p,\quad -p\leq j'\leq
p-1,\quad j'\not = j,j-1,\cr
&C=\{B_j,B_{j'}\},\quad\quad -p\leq j,j'\leq p.\cr}$$
 
Furthermore, given a block `triplet' $\;\{A_j,B_j,A_{j+1}\},\;$ we
set
$$W^{>L}(s_{A_j},s_{B_j},s_{A_{j+1}})\;=\;W^{>L}(A_j,A_{j+1})\;+$$
$$\;W^{>L}(s_{A_j},s_{B_j})+W^{>L}(s_{B_j},s_{B_j})+W^{>L}
(s_{B_j},s_{A_{j+1}}). \eqno (2.2c)$$ We then denote by ${\cal
P}_{\L}$ the collection of triplets  of the form $C=\{A_j,B_j,A_{j+1
}\},\ -p\leq j<p\;$.
 
With this notation we can write
$$V^{>L}(s_{\L})=\sum_{C\in{\cal C}_\L\cup{\cal P}_\L}
W^{>L}(s_C)\eqno (2.3)$$
Therefore, calling $\varrho_C=\exp\  [-W^{>L}(s_C)]-1$, we get
$$\exp\;[-V^{>L}(s_{\L})]\;=\;\prod_{C\in{\cal C}_{\L}\cup{\cal P}_{\L}}\;
(1+\varrho_C(s_C))\;
=\;1+\sum_{\Gamma\subset{\cal C}_{\L}\cup{\cal P}_{\L}}\;\prod_{C\in\Gamma}
\varrho_C(s_C).\eqno (2.4)$$
 
To give an  idea of the proof, let us consider the term 
corresponding to unity in (2.4). We first introduce a cut-off on
the path configurations 
$\;s_k\;$ for $\;k\in A_j,\;j\in -p,...,p.\;$ That is, we write:
$$\eqalignno{
1&=\prod_{j=-p}^{p}(\left [\chi_{_M}(s_{A_j})\chi_{_M}(s_{A_{j+1}})\right ]+
\left [1-\chi_{_M}(s_{A_j})\chi_M(s_{A_{j+1}})\right ])\cr
&=\prod_{j=-p}^{p}\chi_{_M}(s_{A_j})\chi_{_M}(s_{A_{j+1}})\cr
&\hphantom{=}+\sum_{Y\subset\{-p,...,p\}}\;\prod_{k\in
Y^c}\;\chi_{_M}(s_{A_k})\chi_{_M}(s_{A_{k+1}})
\prod_{k'\in Y}\;\left (1-\chi_{_M}(s_{A_{k'}})\chi_{_M}(s_{A_{k'+1}})\right )
&(2.5)\cr}$$
where $\ \chi_{_M}\ $ denotes an indicator function 
of a set of path configuration 
that will be defined later.
 
Let us consider the following quantity :
$$\exp\;[-V^{\leq
L}(s_{\L})]\;\prod_{j=-p}^{p}\chi_M (s_{A_j})\chi_M (s_{A_{j+1}}).\eqno
(2.6)$$ 
it can be written as
$$\exp [-{1\over 2}V^{\leq L}_{A_{-p}}(s_{A_{-p}})]
\prod_{j=-p}^{p}\chi_M(s_{A_j}){\bf T}(s_{A_j},
s_{B_j},s_{A_{j+1}})\chi_M(s_{A_{j+1}})\time
\eqno(2.7)
$$
$$\exp [-{1\over 2}V^{\leq L}_{A_p}(s_{A_p})],$$ 
where
$$\eqalignno{&{\bf T}(s_{A_j};s_{B_j};s_{A_{j+1}}) =\cr
&T(s_{A_{j}};s_{B_{j,1}})\prod_{k=1}^{n-1}T(s_{B_{j,k}}; s_{B_{j,k+1}})
T(s_{B_{j,n}};s_{A_{j+1}})&(2.8)\cr}$$
 
\n and
$$T(s_D;s_{D'})=\exp\;[{1\over 2}
V^{\leq L}(s_D)-V^{\leq L}(s_D\vee s_{D'}) 
+ {1\over 2} V^{\leq L}(s_{D'})].\eqno (2.9)$$
 
Let ${\bb T} ( = {\bb T}_L ) \ $ denote the linear integral
operator generated by the kernel $\  T \ $. This operator acts in a
space of functions on  $\  {\bf S}^{J_L}\ $ (this may
be 
$\  C({\bf S}^{J_L})\ $ \break or $\  L_1({\bf S}^{J_L},ds_{J_L}),\ $ 
or $\;L_2({\bf S}^{J_L},ds_{J_L})\;$) where $\  J_L\  $ is the single 
lattice interval of length 
$\  L \  $ (it is convenient to set $\  J_L
=[0,L-1]\ $ and write $\  {\bf S}^L\  $ instead of $\  {\bf S}^{J_L}\  $ and 
$\  s^L\  $ instead of $\  s_{J_L}\  $). The operator $\  \bb T
\  $ transforms a non-negative function into a positive one and is
compact. According to the Krein-Rutman Theorem [K.Ru.] (in
either version), it has a  unique positive  eigenfunction 
$\  v  ( =
v_L )\ $ . The corresponding
eigenvalue $\ \gamma ( = \gamma_L ) \  $ is not degenerate and gives
the maximal point of the spectrum of $\  \bb T. \ $ Finally, the width
of the gap between $\ \gamma \  $ and the remainder of the spectrum is
positive.
 
A similar assertion holds for the adjoint operator $\ \bb
T^{{\star}};\ $ its extremal eigenvector is denoted by $\  v^{{\star}}
( = v^{{\star}}_L ) \ $ (the
corresponding eigenvalue $\ \gamma^{{\star}}\ $ coincides with $\ 
\gamma  ).\ $ We normalize $\ v \ $ and $\  v^{{\star}} \  $ in such a
way that $$\langle v,v^\star\rangle = 1.\eqno (2.10)$$ 
Here and below $\;\langle \cdot ,\cdot \rangle\;$ denotes the scalar product 
in $\;L_2({\bf S}^L,ds^L):$
$$\langle v,v^\star\rangle = \int_{{\bf S}^L}
ds^Lv(s^L)v^{\star}(s^L).$$
By using space-translations $\  S_u\  $ we can define the ``shifted'' 
functions $\  v(s_{A_j})\  $ and $\ v^*(s_{A_j}),$ $j=-p,\cdots ,p;\  $ for 
these functions the 
relation (2.10) will hold for any $\  j$.
 
Now let ${\cal G}_{\L}$ denote 
the family of pairs $\{A_i,A_{i+1}\},\;-p\leq i\leq
p-1.\;$ If $\;C$ = $\{A_i,A_{i+1}\}$  $\in{\cal G}_\L,$ we define
$$\rho^2_C(s_C)={{T^{(n)}(s_{A_j};s_{A_{j+1}})}\over
{\gamma^nv(s_{A_j})v^{\star}(s_{A_{j+1}})}}-1.\eqno (2.11)$$ 
Here, for $m\geq
2,\ T^{(m)}(s,s')$ is defined iteratively by : $$T^{(m)}(s;s')=\int
ds''\ T^{(m-1)}(s;s'')T(s'';\ s')$$ where $\;s,\ s'\;$ and $\;s''\;$
stand for path configurations  over appropriate intervals (e.g. for 
$s_{[0,L-1]},\;s_{[mL,(m+1)L-1]}\;$ and $\;s_{[(m-1)L+1,mL]}\;$,
respectively).
 
Returning to (2.6), we can write 
$$\int ds_\L\exp\;[-V^{\leq L}(s_\L
)]\;\prod_{j=-p}^p\chi_M(s_{A_j})\;=$$
$$ \gamma^{2pn}\int ds_{A_{-p}}
\times\dots\times ds_{A_p}
\prod_{j=-p}^{p-1}[v(s_{A_j})v^{\star}(s_{A_{j+1}})\times$$
$$\chi_M(s_{A_j})\chi_M(s_{A_{j+1}})]\exp [-{1\over
2}V_{A_{-p}}(s_{A_{-p}}) -{1\over 2}V_{A_p}(s_{A_p})]\times$$
$$(1\;+\;\sum_{{\Delta\subset{\cal G}_{\L}\atop
\Delta\not=\emptyset}} \;\prod_{C\in \Delta}\varrho^2_C(s_C)).\eqno
(2.12)$$
 
We can now specify the reference system around which we 
perform a perturbative expansion. This system is
formed by a family of independent paths configurations over A-blocks. The 
partition function of this system is precisely the term corresponding  
to unity in the RHS of (2.12).
 
Let us now explain how we make this expansion.
For $\Gamma\subset{\cal C}_{\L}\cup {\cal P}_{\L}$ we define
$${\bf B}(\Gamma )=\{i:B_i\in\bigcup_{C\in\Gamma}C\}\eqno (2.13)$$
and
$${\bf B}^c(\Gamma )=[-p,p]\backslash{\bf B}(\Gamma ).$$
 
We start the argument by writing:
$$\int ds_{\L}\;\exp\;[-V(s_\L )]
=\gamma^{2np}\sum_{\Gamma\subset
{\cal C}_{\L}\cup {\cal P}_{\L}}$$
$$\int ds_{\L\backslash\cup_{i\in{\bf B}^c(\Gamma
)}B_i}\prod_{i\in{\bf B}(\Gamma )}{{\bf T}(s_{A_i};s_{B_i};
s_{A_{i+1}})\over
\gamma^n}\prod_{C\in \Gamma}\varrho_C(s_C)\times$$
$$\prod_{j\in{\bf B}^c(\Gamma )}
[{T^{(n)}(s_{A_j};s_{A_{j+1}})\over\gamma^n}]\exp [-{1\over
2}V_{A_{-p}}(s_{A_{-p}})]\exp [-{1\over 2}V_{A_p}
(s_{A_p})]\eqno (2.14)$$
We have used here the fact that, if there is no coupling between a 
B-block and anything else coming from the $\varrho_C$-terms, 
we can perform the path configuration integration 
over this block. We finally get the region $\displaystyle
\bigcup_{i\in{\bf B}^c(\Gamma )}B_i$ where we deal with the $n$th 
iterates $\  {\bb T}^n\  $ of  
operator $\;\bb T\;.$
 
The next step is to analyse the terms 
$T^{(n)}(s_{A_j};s_{A_{j+1}})$ for $j\in{\bf B}^c(\Gamma )$
according to whether or not the path configuration $\;s_{A_j}\;$ belongs to 
a subset of ${\bf S}^{A_j}\  $ where $\;\chi_M\;=\;1.\;$ 
As before, we write:
$$\eqalignno{1&=\prod_{j\in{\bf B}^c(\Gamma )}\left
([\chi_M(s_{A_j})\chi_M(s_{A_{j+1}})]+
[1-\chi_M(s_{A_j})\chi_M(s_{A_{j+1}})]\right )\cr
&=\sum_{I\subset{\cal
B}^c(\Gamma )}\prod_{i\in I}\chi_M(s_{A_i})\chi_M(s_{A_{i+1}}) \prod_{j\in{\cal
B}^c(\Gamma )\backslash I}(1-\chi_M(s_{A_j})\chi_M(s_{A_{j+1}}))&(2.15)
\cr}$$
 
and then 
$$\prod_{j\in{\bf B}^c(\Gamma )}
\ {T^{(n)}(s_{A_j};s_{A_{j+1}})\over \gamma^n} =$$
$$=\sum_{I\subset{\bf B}^c(\Gamma )}\prod_{i\in
I}{T^{(n)}(s_{A_i};s_{A_{i+1}})\over\gamma^n}\chi_M(s_{A_i})
\chi_M(s_{A_{i+1}})\times$$
$$\prod_{j\in{\bf B}^c(\Gamma )\backslash
I}{T^{(n)}(s_{A_j};s_{A_{j+1}})\over \gamma^n}
[1-\chi_M(s_{A_j})\chi_M(s_{A_{j+1}})]\eqno (2.16)$$
 
For the terms with $i\in I$ we write :
$${T^{(n)}(s_{A_i};s_{A_{i+1}})\over \gamma^n}=\left
({T^{(n)}(s_{A_i};s_{A_{i+1}})\over\gamma^nv
(s_{A_i})v^{\star}(s_{A_{i+1}})}-1+1\right )v (s_{A_i})v^{\star}(s_{A_{i+1}})$$
$$= (1+\varrho^2_C(s_C))v
(s_{A_i})v^{\star}(s_{A_{i+1}})\eqno (2.17)$$
where $\;C=\{A_i,A_{i+1}\}\;$.  
Therefore, if we identify our pair $C=\{A_i,A_{i+1}\}$ with site
$\;i,\;$ we get:
$$\prod_{i\in I}{T^{(n)}(s_{A_i};s_{A_{i+1}})\over
\gamma^n}\chi_M(s_{A_i})\chi_M(s_{A_{i+1}}) =$$
$$\sum_{Y\subset I}\prod_{C\in
Y}\varrho^2_C(s_C)\prod_{i\in I}v
(s_{A_i})\chi_M(s_{A_i})v^{\star}(s_{A_{i+1}})\chi_M(s_{A_{i+1}})$$
 
We then collect the product-terms 
$$\prod_{i\in{\bf B}^c\backslash I}
(1-\chi_M(s_{A_j})\chi_M(s_{A_{j+1}}))$$ 
calling them 
$$\prod_{C\in{\bf B}^c(\Gamma )\backslash I}\varrho^1_C(s_C).$$ 
 
As a result, we get
$$\prod_{j\in{\bf B}^c(\Gamma )}{T^{(n)}(s_{A_j};s_{A_{j+1}})\over\gamma^n}=
\sum_{I\subset{\bf B}^c(\Gamma )}\sum_{Y\subset I}
\prod_{C\in Y}\varrho^2_C(s_C)$$
$$\prod_{C\in{\bf B}^c(\Gamma )\backslash I}
\varrho^1_C(s_C)\prod_{i\in
I}v(s_{A_i})\chi_M(s_{A_i})v^{\star}(s_{A_{i+1}})\chi_M(s_{A_{i+1}})$$
$$\prod_{j\in{\bf B}^c(\Gamma )\backslash I}[{T^{(n)}(s_{A_j};s_{A_{j+1}})\over
\gamma^{n}}]\eqno (2.18)$$
 
Let us note that, given $I\subset{\bf B}^c(\Gamma )\;$ and $\;Y\subset
I,\;$ if a pair $C=\{A_j,A_{j+1}\}$ appears as an index in  
a term $\varrho^2_C(s_C)\;$ with $\;C\in Y\subset I,\;$ it 
cannot appear simultaneously as an index of $\  \varrho^1\  $
(in which case $\  C\  $ would have to belong to 
$\  {\bf B}^c(\Gamma )\backslash I$ ).
 
Collecting all the previous expressions, we get:
$$\int ds_{\L}\exp [-V(s_{\L})]= \gamma^{2np}\sum_{\Gamma
\subset{\cal C}_{\L}\cup{\cal P}_{\L}} \sum_{I
\subset{\bf B}^c (\Gamma )}\sum_{Y\subset I}$$
$$\displaystyle\int ds_{\L\backslash
\bigcup_{i\in{\bf B}^c(\Gamma )}B_i}\exp [-{1\over 2}V_{A_{-p}}
(s_{A_{-p}})-{1\over 2}V_{A_p}(s_{A_p})]\times\eqno (2.19)$$
$$\prod_{i\in{\bf B}^c(\Gamma
)}{{\bf T}(s_{A_i};s_{B_i};s_{A_{i+1}})\over \gamma^n}\prod_{C\in
\Gamma} \varrho_C(s_C)\prod_{C\in Y}\varrho^2_C(s_C)\prod_{C\in{\bf
B}^c(\Gamma )\backslash I} \varrho^1_C(s_C)\times$$
$$\prod_{j\in{\bf B}^c(\Gamma )\backslash
I}{T^{n}(s_{A_j};s_{A_{j+1}})\over \gamma^n} \prod_{i\in
I}v(s_{A_i})\chi_M(s_{A_i})v^{\star}(s_{A_{i+1}})\chi_M(s_{A_{i+1}}).$$ 
The term
$\displaystyle \prod_{C\in Y}\varrho^2_C(s_C) \prod_{C\in{\bf B}^c(\Gamma
)\setminus I}\varrho^1_C(s_C)$ can be seen as associated to a pair
$\  (\Gamma_3,\Gamma_4)\  $ where $\  \Gamma_3,\Gamma_4\subset
{\cal G}_{\L},$ $\Gamma_3\cap\Gamma_4=\phi$:
 
$$\prod_{C\in \Gamma_3}\varrho^1_C(s_C)\prod_{C\in
\Gamma_4}\varrho^2_C(s_C)\eqno (2.20)$$ 
(the principle of the notation will be clear below).
 
We now want to express the partition function of the original system as the 
one of a polymer gas where the interaction is  a 
hard-core exclusion. Given $\  C\in{\cal C}_{\L}\cup{\cal
P}_{\L}\cup{\cal G}_{\L}\  $  (that is, $\  C\  $ is either a pair of
blocks of one of the types  indicated before or a triplet $\ 
(A_i,B_i,A_{i+1})$), we define the support of $\ C,\  $ denoted by
$\widehat C$, as:  $$\widehat C =(\bigcup_{i:A_i\in C}A_i)\cup 
(\bigcup_{i:B_i\subset C}B_i)\cup (\bigcup_{i:B_i\hbox{ or }B_{i-1}\subset
C}A_i)\eqno (2.21)$$ 
Note that in this way we have added the two neighbor A-blocks 
to any B-block which appears in $\ C$.
 
Now let us consider a quadruple $R=(\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4)$
with $\Gamma_1\subset{\cal C}_{\L},$ $\Gamma_2\subset{\cal 
P}_{\L}\  $ and $\  \Gamma_3,\Gamma_4\subset {\cal
G}_{\L}$. A quadruple $\  R\  $ is called admissible (which means 
that it could appear in the expression (2.19) as a particular term in   
the sum $\  \sum_{\Gamma}\sum_{I}\sum_{Y}\  $), if $\  \Gamma_3\cap\Gamma_4=\phi
\  $ and moreover if, for any block $\;
B_i\;$ that enters some pair or triplet 
$\ C\in\Gamma_1\cup\Gamma_2,\  $ the pair of the two 
neighbor-blocks $\  A_i,A_{i+1}\  $ does not belong to
$\Gamma_3\cup\Gamma_4$. The last condition comes from the fact that, 
by construction, a pair $\;\{A_i,A_{i+1}\}\;$
appears only in association with the intermediary B-block 
$\ B_i$. Hence, if $\;B_i\in C\;$ where $\;C\;$ is from
$\;\Gamma_1\cup\Gamma_2\;$ then pair $\;\{A_i,A_{i+1}\}\;$
can appear as an index neither in a term $\varrho^2_C(S_C)$
with $C\in Y\subset  I\subset{\bf B}^c(\Gamma )$ nor in a 
term $\varrho^1_C(S_C)$ with $C\in {\bf B}^c(\Gamma
)\backslash I$ (recall that we identify  $\;\{A_i,A_{i+1}\}\;$ with
site $\;i$).
 
If the block pairs or triplets, $\ C_1\ $ and $\ C_2,\ $ belong to 
$\;\cup_{i=1}^{4}\Gamma_i,\;$
we say that $\ C_1\ $ and $\ C_2\ $ are connected if 
$\;\widehat C_1\cap \widehat C_2\not =\emptyset.\;$
An admissible $\;R=(\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4 )\;$ is called a polymer
if, for any $C$ and $C'\in\cup_{i=1}^{4}\Gamma_i,\;$ there exists a
sequence $\;C_j,\;j=1,\dots,k,\;$ such that any $\;C_j\in
\cup_{i=1}^{4}\Gamma_i,$ $C_1=C,$ $C_k=C'\  $ and $\ C_j$ and $C_{j+1}$ 
are connected for $1\leq j\leq k-1$.
 
If $\ R=(\Gamma_1,\Gamma_2,
\Gamma_3,\Gamma_4 )\ $ is a polymer, we let $\;{\bf I}(R)\;$ be the set 
of those values $\;i\;$ for which either the corresponding B-block  
$\;B_i\;$ enters some $\;C\in\Gamma_1\cup\Gamma_2,\;$ or a pair 
$\;\{A_i,A_{i+1}\}\in\Gamma_3.\;$ Furthermore, $\;{\bf J}(R)\;$  
denotes the set of those values $\;i\;$ for which the corresponding 
A-block $\;A_i\;$ enters some $\;C\in\Gamma_1\cup\Gamma_4\;$. We
define the support of $\;R\;$ by
$${\widehat R=(\bigcup_{i\in{\bf I}(R)}(A_i\cup B_i\cup A_{i+1}))
\cup (\bigcup_{j\in{\bf J}(R)}A_j)}.\eqno (2.22)$$
 
For any polymer $\;R\;$ it is easy to see that the support 
$\widehat R\;$ can be decomposed into the 
union of pair-wise disjoint intervals:
$$\displaystyle{\widehat R=\bigcup_{i=1}^KG_i.}$$
Here $\;K=K(R)\;$ and $G_i$ may be either an A-block $\;A_j\;$ (in 
which case $\;j\in{\bf J}(R)\;$ and $\;A_j\;$ does not enter any triplet 
$\;\{A_i,B_i,A_{i+1}\}$ with $\;i\in{\bf I}(R)\;$) or a union corresponding 
to a chain of consecutive triplets:
$$G_i=A_{l_i}\cup B_{l_i}\cup A_{l_i+1}\cup\dots\cup B_{l_i+m_i}\cup
A_{l_i+m_i+1}.$$
 
Let us define the probability measures 
on $\;{\bf S}^{\widehat R}$:
$$\displaystyle\mu_R(ds_{\widehat
R})=\prod_{i=1}^{K}\mu_{G_i}(ds_{G_i}).\eqno (2.23)$$ 
Here, if $\;G=A_l,\;$ then  
$${d\mu_G(s_G)\over ds_G}={u_l^\star (s_{A_l})u_l(s_{A_l})
\over{\cal N}_G},\eqno (2.24)$$
and, if $\;G=A_l\cup B_l\cup A_{l+1}\cup\dots\cup B_{l+m}
\cup A_{l+m+1},\;$ then 
$${d\mu_G(s_G)\over ds_G}=$$
$${u^\star_l
(s_{A_l}){\bf T}(s_{A_l};s_{B_l};s_{A_{l+1}})\cdots
{\bf T}(s_{A_{l+m}};s_{B_{l+m}};s_{A_{l+m+1}})
u_{l+m+1}(s_{A_{l+m+1}})\over\gamma^{m(n+1)}{\cal N}_G}\eqno (2.25)$$
where 
$$u_\ell^\star=v^{\star}\chi_M,\quad{\rm{if}}\;\;-p+1\leq \ell\leq p,$$
$$u_\ell=v\chi_M,\quad{\rm{if}}\;\;-p\leq \ell\leq p-1,$$
$$u_\ell^\star (s_{A_\ell})= 
\exp [-{1\over 2}V_{A_\ell}(s_{A_\ell})]
=u_\ell (s_{A_\ell}),\quad{\rm{if}}\;\;\ell =-p\;{\rm{ or }}\;p,$$
and $\;{\cal N}_G\;$ is the normalization to have a probability
measure.
 
Next, we assign to a polymer $\;R\;$ its (not necessary 
positive) ``fugacity''. First, we set:
$${\widetilde\zeta}(R)=\hbox{$\int$}\mu_R(ds_{\widehat
R})\prod_{C\in\Gamma_1\cup\Gamma_2}\varrho^0_C(s_C)\  \prod_{C\in\Gamma_3}
\varrho^1_C(s_C)\  \prod_{C\in\Gamma_4}\varrho_C^2(s_C).\eqno (2.26)$$
where $\varrho^0_C \equiv \varrho_C$ see eq. (2.4).
One can then check that the partition function $\;\Xi_\L\;$ 
may be written in the following way:
$$\Xi_{\L}=\gamma^{2pn}\langle v\exp [-{1\over 2}V_{A_{-p}}]  
,\chi_M\rangle\langle v^\star\exp [-{1\over 2}V_{A_p}],\chi_M\rangle (\langle 
v^\star v,\chi_M\rangle )^{2p-1}\times$$
$$\left[ 1+\sum_{k\geq 1}\quad\sum_{{R_1,\dots ,R_k:\atop {\widehat
R}_i\subset\L ,1\leq i\leq k,}\atop{\widehat R}_i\cap{\widehat
R}_{i'}=\emptyset,1\leq i<i'\leq k}\prod^k_{i=1}
{\bar{\zeta}}(R_j)\right].\eqno (2.27)$$ 
Here the internal sum is taken over all (unordered) collections of
polymers for which the conditions indicated are fulfilled and  
$$\bar\zeta (R)={{\tilde\zeta (R){\cal N}_R}\over{(\langle
vv^{\star},\chi_M \rangle )^{\sharp \{i\ :\ i\not = -p,+p;\ 
A_i\subseteq\widehat R\}}}}\times$$ 
$${1\over\left [(\langle v\exp
[-{1\over 2}V_{A_{-p}}],\chi_M \rangle
)^{\widehat\delta ({-p,\widehat R}})(\langle  v^\star\exp [-{1\over
2}V_{A_p}],\chi_M\rangle )^{\widehat\delta ({p, \widehat R}})\right
]}\eqno (2.28)$$ 
where $\;\widehat\delta ({l,\widehat R})=1\;$ if
$\;A_l\subseteq\widehat R\;$  and $\;\widehat\delta ({l,\widehat
R})=0,\;$ otherwise.
 
The term in the square bracket in the RHS of (2.27) can be interpreted as
the partition function of a polymer system with a hard-core interaction and
fugacity $\;{\bar{\zeta}}$. The product in front of this term is the partition 
function of the system of non-interacting A-blocks. 
 
Notice that the fugacity $\;\bar\zeta (R)\;$ depends in general on $\;p\;$
(this is the case of those polymers that include the border blocks $\;A_{\pm p}
).\;$ However, this dependence is rather weak and does not affect the 
argument used.
 
Furthermore, introducing parameters $\;w_0\;$ and 
$\;w_1\;$ as specified in (1.24), one can produce a
`full' polymer expansion for $\;\Xi_\L\;$ 
where the complex terms are taken into account. This
leads to a more complicated definition of a polymer, which nevertheless is 
based on the same kind of ideas . The rigorous 
scheme repeats the one from [C.C.O.] and we will not go into
technical details. The corresponding complex fugacity of a polymer 
is again denoted by $\;\bar
\zeta (R)\;$; it contains, apart from the factors $\;\varrho^i,\;i=0,1,2,\;$ 
some new terms $\;\varrho^3\;$ that are  complex analogues 
of $\;\varrho^0.\;$ See [C.C.O.] for further details. 
 
In order to control our cluster expansion we need to estimate the fugacity 
of a polymer. We start by studying the contribution coming from the terms 
$\;\varrho^2_C(s_C)\;$. Let us first specify the indicator functions 
$\;\chi_M.\;$ The parameter $\;M\;$ runs over $\;{\bf R}_+,\;$ the positive 
half-axis. The function $\;\chi_M(s_{A_i}),$ 
$i=-p,\dots ,p,\;$ is defined as the  
space-translation of a function $\;\chi_M(s^L),$ $s^L\;\in
{\bf S}^L.\;$ The latter is the indicator of the set
$${\bf S}^L_M\;=\;\{s^L=(s_j,j\in [0,L-1])\in{\bf S}^L\;:\;\Vert s_j\Vert_2
\;\leq\;M(1+r(j))\}\eqno (2.29)$$
where the norm $\;\Vert\cdot\Vert_2\;$ is defined in (1.48) and 
$$r(j)\;=\;{\rm{min}}\;[\;\log (1+j),\log (L-j)\;].$$
 
Given a positive integer $\;N\;,$ let $\;E(=E_N)\;$ 
be the  probability density on ${\bf S}^{[0,NL-1]}\;$ (with respect to
the  measure $\;ds_{[0,NL-1]}\;$) of the form 
$$E(s_{[0,NL-1]})={u^\star 
(s^{(1)})T(s^{(1)};s^{(2)})\dots
T(s^{(N-1)};s^{(N)})u(s^{(N)})\over{\cal N}_{[0,NL-1]}}.\eqno (2.30)$$
Here, the path configuration $\;s^{(i)}$ $\in{\bf S}^{[(i-1)L,iL-1]}\;$ 
is the restriction of 
$\;s_{[0,NL-1]}\;$ to the interval $\;[(i-1)L,iL-1],\;$ $i=1,\dots ,N.\;$
Furthermore, the function $\;u^\star (s^{(1)})\;$ is 
either \break
$\;v^\star (s^{(1)})\;$ or 
$\;\exp [-{1\over 2}V_{[0,L-1]}(s^{(1)})]\;$ and $\;u(s^{(N)})\;$ is either 
$\;v(s^{(N)})\;$ or \break
$\;\exp [-{1\over 2}V_{[(N-1)L,NL-1]}(s^{(N)})]\;$
(all the combinations are possible), cf. (2.23). \break
Finally, 
$\;{\cal N}_{[0,NL-1]}\;$ is, as before, the normalization to have a
probability measure. 
 
Given $\;J\subseteq [0,NL-1],\;$ we set, in analogy with (1.51): 
$$k_E^{(J)}(s_J)=\int_{{\bf S}^{[0,NL-1]\setminus J}}ds_{[0,NL-1]\setminus
J}E(s_{[0,NL-1]\setminus J}\vee s_J).\eqno (2.31)$$ 
By  Lemma 3, we get an estimate: for any $\;J\subset [0,NL-1],\;$
$$k_E^{(J)}(s_J)\leq\exp\;[-\sum_{j\in\Delta}(c^{\star}\Vert
s_j\Vert^2_2-\delta )].\eqno (2.32)$$
In fact, the probability density $\;E_{[0,NL-1]}\;$ is either of a
form  assumed in  Lemma 3 or the limit of those densities. 
The constants 
$\;c^\star >0\;$ and $\;\delta\;$ do not depend on $\;N\;$ and $\;L\;.\;$ 
 
Moreover, for the probability measure $\;\mu_E\;$ on ${\bf S}^{[0,NL-1]}\;$ 
with a density $\;E\;$ of the form (2.30) and 
for any $\;j=0,\dots N-1,\;$ we have:
$$\mu_E(\{s_{[0,NL-1]}:\;\chi_M(s^{(j)})=0\})\leq 
{\bar c}_1e^{-{\bar c}_2M^2};\eqno (2.33)$$ 
the constants $\;\bar c_1\;$ and $\;\bar c_2\;$ 
again do not depend on $\;N\;$ 
and $\;L\;.$ 
 
Indeed, the bound (2.33) follows easily from the definition (2.29) -
(2.31),the  bound  (2.32) and the estimate: 
$$\int_{\{s\in{\bf S}:\Vert s\Vert_2\geq y\}}ds\exp [-
c^{\star}\Vert s\Vert_2^2]\leq \exp [-(c^\star -\epsilon )y^2]\int_{\bf S}ds
\exp [-\epsilon\Vert s\Vert^2_2]$$
provided that we are able to prove that the integral in the RHS is finite for
any $\;\epsilon >0.\;$ 
 
The last assertion may be proved in the following way. Given
$\;s=(x,\omega  ),\;$ we use the definition of the norm $\;\Vert
s\Vert^2_2\;$ and the Schwartz inequality to write the bound 
$$\Vert s\Vert_2^2\;\geq \;\beta x^2+2x\int_0^\beta dt
\omega (t)+\beta^{-1}
(\int_0^\beta dt\omega (t))^2$$
which implies 
$$\int_{\bf S} ds\exp (-\epsilon\Vert s\Vert^2_2)\leq \int_{\bf R}dx
\int_{\bf W}P(d\omega )\exp [-\epsilon\beta^{-1}
(\beta x+\int_0^\beta dt\omega (t))^2].$$
Performing in the RHS the integration in the variable $\;x,\;$ we get the 
desired result.
 
The following theorem is the crucial ingredient to control the 
terms $\;\varrho^2_C(s_C)\;$:
 
\b
 
\n{\bf Theorem 2.1}
 
\m
 
{\it There exist positive integers $\;L_0\;$ and positive constants $\;\bar c_3,
\;\bar c_4\;$ such that for any $\;L>L_0\;$ and $\;M>1\;$ and for any} $\;
s,\ s'\in{\bf S}_M$
$$\left |{T^n(s;s')\over\gamma^nv(s)v^{\star}(s')}-1\right |\leq\exp(\bar
C_1M^2-n\ e^{-\bar C_2M^2})\eqno (2.34)$$
 
 
Since the proof of Theorem 2.1 is similar to the one of Theorem 2.1
of [C.C.O], the estimate (2.33) playing the role of the bound (2.4)
of [C.C.O], we omit it.
For estimating the terms $\;\varrho^0\;$ and, in a complex domain, 
the terms
$\;\varrho^3,\;$ one again proceeds in a way similar to 
[C.C.O.]. It is convenient to summarize all the bounds in 
Proposition 2.2 below.  In this assertion, the condition that 
$\;M,\;n\,$ and $\;L\;$ are  large enough, $\;w_0\;$ and
$\;w_1\;$ are small enough means  the following. First, we impose the
restriction $\;M\geq M^0\;$ and $\;n\geq n^0\;$ where $\;M^0\;$ and
$\;n^0\;$ depend only on $\;\beta\;$ and the potentials $\;\Phi\;$
and $\;\Psi_d,\;d\geq 1.\;$ Then, for $\;M\;$ and  $\;n\;$ satisfying
these conditions, we take $\;L\geq L^0\;$ where $\;L^0= L^0(M,n).\;$
Finally, for $\;M,\;n\;$ and $\;L\;$ that satisfy the above 
restrictions, we consider $\;w_0\;$ and $\;w_1\;$ with max
$[|w_0|,|w_1|] \leq w^0\;$ where $\;w^0=w^0(M,n,L).\;$ The claim is
that we can guarantee a proper choice of the thresholds
$\;M^0,\;n^0,\;L^0\;$ and $\;w^0\;$ (in bounds (2.40) and (2.44) -
(2.46) below these thresholds depend on a  parameter $\;\sigma\;$).
Note that all the estimates that follow are claimed to be uniform in
$\;p.\;$
 
\b
 
\n{\bf Proposition 2.2}
 
\m
 
{\it Given $\;M,\;n\;$ and $\;L\;$ large enough and $\;w_0\;$ and $\;w_1\;$
small enough, the (complex) polymer fugacity $\;\bar\zeta (R)\;$ satisfies 
the bound   
$$\mid\bar\zeta(R)\mid\leq\;{\prod\atop
C\in\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4}{\widehat g}_C.\eqno (2.35)$$
Here, for $\;C=\{D,D'\}\in{\cal C}_\L,\;$
$${\widehat g}_C={\rm{max}}\{{12M^2n^2(1+w^2)\log(nL+1)\over{r_C  
F(r_CL)}},$$
$$6\bar c\exp\left
[-c^{\star}{M^2\over 8}\log(r_C+1)\right]\},\eqno (2.36)$$
where $\;r_C\;$ is the total number of blocks of the both types, {\rm
A} and  {\rm B}, situated between $\;D\;$ and
$\;D'\;$, whereas for $\;C\in{\cal P}_\L\cup{\cal G}_\L,$
$${\widehat g}_C(M,n,L,w)={\rm{max}}\{100M^2\log(nL+1)n\left[{1\over
\log(L+1)F(L)}+wL\right]+$$
$$100nLw,\;\;6\bar c\exp (-{c^\star\over 48}M^2),\;\; 
\exp (\bar c_3M^2-ne^{-\bar c_4M^2})\}.\eqno (2.37)$$
Constant $\;c^\star\;$ comes from Lemma 3, and $\;\bar c_3,\;\bar
c_4\;$  from Theorem 2.1 and} $\;w=$ max $[w_0,w_1].\;$  
\b 
The proof of Proposition 2.2 is similar to the one of 
Lemma 3.1 of
[C.C.O.]. In the proof one uses  Theorem 2.1 together with (2.33) 
and the following fact:
 
$$\mid\psi_d(s_i,s_j)\mid\leq{{\langle\mid s_i\mid ,\ \mid s_j\mid
\rangle}\over{(i-j)^2\log (1+\mid i-j\mid)F(\mid i-j\mid)}}\eqno(2.38)$$
where 
$$\eqalignno{\langle\mid s_i\mid ,
\ \mid s_j\mid\rangle&\equiv\int_0^{\beta}|x_i+w_i(t)|\
|x_j+w_j(t)|dt\cr &\leq{1\over 2}(\Vert s_i\Vert^2_2+\Vert s_j\Vert^2_2)
&(2.39)\cr}$$
As before, we omit the details referring the reader to [C.C.O.].
 
It now can be checked (cf. [C.C.O.], Proposition 2.2) 
that for $\;M\;$ and $\;L\;$ 
large enough we have, for some constant $\;{\cal N}>0$:
 
$$\eqalignno{
&{3\over 4}<(v\tilde v^{\star},\chi_{\bar
M})\leq(vv^{\star},\chi_M)\leq 1,\cr
&{3\over 4}{\cal N}^{-1}<\left({v\
e^{-h/2}\over\sqrt{\gamma}},\chi_M\right)\leq{\cal N},\cr
&{\cal N}_R\leq{\cal N}^2.\cr}$$
 
\n Furthermore, for every polymer $R$ the following inequality holds:
 
$$\mid R\mid\leq
3{\sharp(\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4)}$$
 
\n where $\mid R\mid$ denotes the number of blocks of type $A$ or $B$ contained
in $\widehat R$. It then follows that, given $\sigma\in [{1\over 2},1),$ 
we can choose $n,M$ and $\;L\;$ large enough and $\;w_0\;$ and $\;w_1\;$
small enough so that, for every polymer $R=(\Gamma_1,\Gamma_2,\Gamma_3,
\Gamma_4)$, the complex fugacity $\;\bar\zeta (R)\;$ admits the bound:
$$\mid\bar\zeta(R)\mid\leq\sigma^{\mid R\mid}\prod_
{C\in\Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4}g_c\eqno (2.40)$$
\n where 
$$g_c=2^3{\cal N}^4\hat g_c.\eqno (2.41)$$
The bound (2.40) is the basic ingredient for proving the convergence of the
cluster expansion. More precisely, proceeding as in [C.C.O.] 
Lemma 1, we get 
 
\b
 
\n{\bf Proposition 2.3}
 
\m
 
{\it Let $\;\tilde\Xi_\L\;$ denote the polymer partition function 
figuring in the square brackets
in the RHS of (2.27) (for the complex perturbation of the Hamiltonian):
$${\widetilde\Xi}_{\L}=1+\sum_{k\geq 1}\;\sum_{{R_1,\dots ,R_k:\atop{\widehat
R}_i\subseteq\L,1\leq i\leq k,}\atop{\widehat R}_i\cap{\widehat R}_{i'}
=\emptyset ,1\leq i<i'
\leq k}\prod^k_{j=1}\bar{\zeta}(R_j)\eqno (2.42)$$
and  $\;\kappa\;$ be given by 
$$\kappa\;=\;4\hat g_{\{A_0,A_1\}}+\sum_{C\in{\cal C}_A\cup{\cal
P}_{\L}}g_C\eqno (2.43)$$
(see (2.36), (2.37) and (2.40)).
Then, given $\sigma\in [{1\over 2},1),\;$ we can choose $\;n,\;L\;$ and 
$\;M\;$ large enough and $\;w_0\;$ and $\;w_1\;$ small enough so that 
estimate (2.40) is fulfilled and the following bounds hold:
 
\m
 
$$\exp \kappa<{1\over\sqrt{\sigma}(2-\sqrt{\sigma})},\eqno (2.44)$$
 
$$\sup_{V\subset\{A_i,i\in[-p_1,\dots,p]\}\atop\hphantom{V}\cup\{B_i,i
\in [-p_1,\dots,p]\}}\quad\sum_{R:V\subseteq\hat R\subseteq\L}\mid\bar\zeta 
(R)\mid\leq\sigma\kappa{[1-(e^\kappa-1)]\over 1+\sigma^2e^\kappa-2\sigma 
e^\kappa}\equiv G(\kappa,\sigma ),\eqno (2.45)$$
and for any polymer $\;R\;$  
\m
 
$$\eqalignno{
\sum_{k\geq 1}&\quad\sum_{R_1,\dots ,R_k\atop R_1=R} 
\mid\varrho (R_1,\dots ,R_k)\mid\prod^k_{i=2}\mid\bar{\zeta}(R_i)\mid\cr
&\leq \bar{\zeta}(R){\exp\left [G(\kappa,\sqrt{\sigma})\mid
R\mid\right]\over1-\sqrt{\sigma}\exp
[G(\kappa,\sqrt{\sigma})]}&(2.46)\cr}$$
 
\n Here $\;\varrho\;$ denotes the standard M\"obius function: 
 
$$\varrho (R_1,\dots ,R_k)={1\over k!}\sum_{{\bf g}\in{\bf G}(R_1
,\dots ,R_k)}(-1)^{\sharp(\sixrm\hbox{edges in $\bf g$})}\eqno(2.47)$$
 
\n where $\;{\bf G}(R_1,\dots ,R_k)\;$ stands for the 
set of all connected subgraphs of the graph with 
$\;k\;$ vertices $\{1,$ ... $,k\}$ and with the edges 
corresponding to those pairs $\;(i,j)\;$ for which ${\widehat R}_i$ 
$\cap$ ${\widehat R}_j$ $\not =$ $\emptyset$ (the sum in (2.47) equals 
zero if $\;{\bf G}(R_1,\dots ,R_k)\;$ is empty and one if $\;k=1$).}
\b 
\m
 
By using Proposition 2.3, we can control the standard cluster representation  
$${\widetilde\Xi}_{\L}=\exp\;[\;\sum_{k\geq 1}\;\;\sum_{R_1,\dots ,R_k:
\atop{\widehat R}_i\subseteq\L,1\leq i\leq k}\;\;
\varrho (R_1,\dots ,R_k)
\prod^k_{j=1}\bar{\zeta}(R_j)]\eqno (2.48)$$
which follows from (2.42). 
 
Now we can give  the proof of Lemmas 1 and 2.
The expansion 
argument provided so far allows us to prove the assertion of  Lemma 2 
about the analytic continuation of $\;\lambda_\L\;$ and the
boundedness of $\;|\lambda_\L|$. Furthermore, we are able to prove
the uniform convergence of the analytic functions 
$$\lim_{p\to\infty}\lambda_\L=\lambda,$$ 
while 
parameters $\;M,n,L\;$ and $\;w_0,w_1\;$ are kept fixed (but chosen
either large enough or small enough, respectively, in the sense  that
was specified above). 
 
The analysis of the  kernels
$\;k_\L^{(J)}\;$ and $\;\phi_\L (\cdot )\;$ for the operators 
that are specified in the formulation of  Lemma 2 is based on 
similar expansions of quantities $\;\Xi_\L^{(J)}(x_J,y_J)$, $x_J,y_J
\in{\bf R}^J\;$. 
 
Let us start by discussing the limit relation (1.30). We again assume
that $\;\L\;$ is of the form specified above. [This assumption
will soon be dropped.] Let us  suppose for definiteness that $\;J\;$
is a subset of B-block $\;B_0.\;$ Proceeding in the same way as
before, we can write the following representation for the quantity
under consideration: 
$$\eqalignno{\Xi^{(J)}_\L(x_J,y_J)&=\gamma^{(2p-1)n}\langle v\exp 
[-{1\over 2}V_{A_{-p}}],\chi_M \rangle\langle v^\star\exp [-{1\over
2}V_{A_p}],\chi_M\rangle (\langle v^\star v,\chi_M\rangle )^{2p-2}
\times\cr
&\int\;ds_{A_0}\;ds_{A_1}\;ds_{B_0\setminus J}d\omega_J{\bf
T}(s_{A_0};s_{B_0\setminus J}\vee\bar s_J;s_{A_1})\times\cr &\left[
1+\sum_{k\geq 1}\quad\sum_{{R_1,\dots ,R_k:\atop{\widehat
R}_i\subseteq\L ,1\leq i\leq k,}\atop{\widehat R}_i\cap{\widehat
R}_{i'}=\emptyset,1\leq i<i' \leq k}\;\prod_{j=1}^k\bar\zeta
(R_j;x_J,y_J)\right].&(2.49)\cr}$$ 
Here $\;\bar s_J=(x_J,y_J;\omega_J)\;$ and the quantity $\;{\bf
T}(s_{A_0};s_{B_0\setminus J}\vee\bar s_J;s_{A_1})\;$ is defined by
formulas similar to (2.8), (2.9). The sum in the
square brackets has the same nature as before (cf. (2.27)). The
definition of a polymer is again the same and the definition of the
polymer fugacity  $\;\bar\zeta (R;x_J,y_J)\;$ follows a similar idea
(cf. (2.23) - (2.26) and (2.28)). This fugacity now depends, in
general, not only on $\;p,\;$ but also
on $\;x_J,y_J,\;$ but as before, this  dependence is rather
weak: it affects only the polymers $\;R\;$ for which $\;{\widehat
R}\cap (A_{-p}\cup A_0\cup B_0\cup A_1\cup A_p)\ne\emptyset .\;$ 
 
Finally, introducing the parameters $\;w_0\;$ and $\;w_1\;$ 
as specified 
in (1.24), we are able to produce a full polymer 
expansion for 
$\;\Xi_\L(x_J,y_J)\;$ with complex terms. We again denote the
corresponding complex fugacity by $\;\bar\zeta (R;x_J,y_J).\;$
 
Our final aim is the same as above: we want to control the cluster 
representation
$$\widetilde\Xi_\L(x_J,y_J)\;=\;\exp\;[\sum_{k\geq
1}\;\sum_{R_1,\dots ,R_k: \atop{\widehat R}_i\subseteq\L,1\leq i\leq
k}\varrho (R_1,\dots ,R_k)) \prod_{j=1}^k\bar\zeta
(R_j;x_J,y_J)]\eqno (2.50)$$  
for a polymer partition function
$\;\widetilde\Xi_\L(x_J,y_J)\;$ given by  the term in the square
brackets in the RHS of (2.49). 
 
An analysis of the scheme of estimating the various terms contributing
to $\;\bar\zeta (R;x_J,y_J)\;$ shows no major
differences with the above construction. There are two changes
worth to note. First, the threshold values $\;M_0,n_0,L_0\;$ and
$\;w_0,w_1\;$ depend in general on $\;x_J\;$ and $\;y_J$. 
Second, we use the assertion of  Lemma 3 for both cases, as for
the denominator $\;\Xi_\L\;$ as for $\;\Xi_\L^{(J)}(x_J,y_J),\;$ because
the measures $\;\mu_G\;$ figuring in the definition of the polymer
fugacity either have denominators of this type or are limits of 
those measures. 
 
After performing the necessary estimates we arrive at assertions
that are similar to Propositions 2.2 and 2.3 and give a desired 
control of the convergence in representation (2.50). Having the  
control of both expansions, (2.48) and (2.50) we can proceed in a
standard way and guarantee the existence of the limit
$$\lim_{p\to\infty} k^{(J)}_\L(x_J,y_J)=k^{(J)}(x_J,y_J),\eqno
(2.51)$$
while the parameters $\;M,n,L\;$ and $\;w_0,w_1\;$ are, as before, kept
fixed (but again chosen large enough and small enough,
respectively). More precisely, we choose values of these parameters
so that the assertions of Propositions 2.2 and 2.3 and their
analogues for $\;\bar\zeta (R,x_J,y_J)\;$ hold and then perform the
limit $\;p\to\infty$. The same scheme is used in an argument that
follows.
 
In addition to (2.51), we get a representation of the limiting
kernel $\;k^{(J)}(x_J,y_J)\;$ in the form
$$k^{(J)}(x_J,y_J)=\exp\;[\;\sum_{k\geq 1}\;\sum_{{R_1,...,R_k:\atop
\hat R_i\cap (A_0\cup B_0\cup A_1)\neq\emptyset}\atop{\rm
for}\;{\rm some}\;i\in\{1,...k\}}\rho (R_1,...R_k)\times$$
$$\times (\prod_{j=1}^k\bar\zeta (R_j;x_J,y_J)-\prod_{j=1}^k\bar\zeta
(R_j))].\eqno (2.52)$$
Note that, for the polymers figuring in the expansion in the RHS of
(2.52), both fugacities $\bar\zeta\;$ and $\;\bar\zeta (R;x_J,y_J)\;$
do not depend on $\;p.$
 
Let us now comment on how to extend the limiting relation (2.51) to
a general case $\;\L\nearrow\bf Z$. Given an interval
$\;\L\subset{\bf Z},\;$ we can `fill' it with our A- and B-blocks 
as indicated (with $\;\displaystyle{p=p_\L=[{{|\L |-L}\over{2(n+1)}}]}$), 
with the proviso that the `border' blocks $\;A_{\pm p}\;$ have, in general,
a greater length (for example, we can include, in each of these 
blocks, half of the rest length $\;|\L |-L(2p(n+1)+1).\;$ Then we
can proceed in the same way as before, estimating the difference
between $\;k^{(J)}_\L(x_J,y_J)\;$ and $\;k^{(J)}(x_J,y_J)\;$ in
terms of $\;p_\L$. While $\;\L\nearrow\bf Z$,  $\;p_\L\;$
tends to infinity, which guarantees the convergence in (1.30).
 
In fact, we are able to do more. By using the argument developed,
we can prove that, given a finite $\;J\subset{\bf Z}\;$ and $\;
x_J,y_J\in{\bf R}^J,\;$ the kernel $\;k_\L^{(J)}(x_J,y_J)\;$ admits an
analytic continuation, in the variables $\;w_0,w_1,\;$ to $\;{\cal
O}_0\times{\cal O}_1\;$ where $\;{\cal O}_\ell$,
$\ell=0,1$, is a neighborhood of the origin in $\;\bf C\;$ (which
depends, in general, on $\;x_J,y_J$, but not on $\;\L\subseteq J$.
Furthermore, these analytic functions converge, uniformly in
$\;(w_0,w_1)\in{\cal O}_0\times{\cal O}_1$, to a limit, as
$\;\L\nearrow{\bf Z}$. The limiting function is nothing but the
analytic continuation of  $\;k^{(J)}(x_J,y_J)$.
 
Moreover, the formulas determining fugacities $\;\bar\zeta (R;x_J,y_J)\;$
show that $\;{\cal O}_0\;$ and $\;{\cal O}_1\;$ may be chosen independently
on $\;x_J,y_J\;$ and the series under consideration converge uniformly in 
$\;x_J,y_J\;$ provided that these variables run over a compact set in 
$\;{\bf R}^J.\;$ 
 
The same argument is used for proving the analyticity, boundedness
and convergence of  $\;\phi_\L(a)\;$ where $\;a\;$ is an operator
of the kind considered in the assertions ($b$) and ($c$) of Theorem 2. To 
avoid a repetition, we omit the detailed argument. The reader can 
reconstruct it from the corresponding discussion below related to 
$\;\phi_\L(p^2_l)\;$ (which is, apparently, the most difficult 
case from technical point of view). 
 
A cluster expansion argument is used also for proving the mixing
property (1.32). Here, the main construction has to be slightly
modified. Namely, for $\;u\;$ large enough,\break
$\;(x_{J^{(1)}},y_{J^{(1)}})\;$ and
$\;(x_{S_uJ^{(2)}},y_{S_uJ^{(2)}})\;$ are associated with
non-adjacent blocks. If $\;J^{(1)}\;$ and $\;S_uJ^{(2)}\;$ belong
to, say, blocks $\;B_{j_1}\;$ and $\;B_{j_2}$, respectively, then,
in a cluster representation for 
$$k^{(J^{(1)}\cup S_uJ^{(2)})}(x_{J^{(1)}}\vee S_u
x_{J^{(2)}},
y_{J^{(1)}}\vee S_u y_{J^{(2)}})$$
(which is similar to (2.51)), the condition 
$$\hat R_i\cap (A_{j_1}\cup B_{j_1}\cup A_{j_1+1}\cup A_{j_2}\cup
B_{j_2}\cup A_{j_2+1})\neq\emptyset$$
becomes `almost equivalent' to one of the mutually disjoint
conditions
$${\widehat R}_i\cap (A_{j_1}\cup B_{j_1}\cup A_{j_1+1})\neq\emptyset,\;
\;{\rm but}\;\;{\widehat R}_i\cap (A_{j_2}\cup B_{j_2}\cup
A_{j_2+1})=\emptyset ,$$
or
$${\widehat R}_i\cap (A_{j_2}\cup B_{j_2}\cup A_{j_2+1})\neq\emptyset,\;
\;{\rm but}\;\;{\widehat R}_i\cap (A_{j_1}\cup B_{j_1}\cup 
A_{j_1+1})=\emptyset .$$
 
More precisely, the polymers $\;R_i\;$ whose supports have a non-empty
intersection with both $\;A_{j_1}\cup B_{j_1}\cup A_{j_1+1}\;$ and
$\;A_{j_2}\cup B_{j_2}\cup A_{j_2+1}\;$ give a contribution that
vanishes as $\;|j_1-j_2|\to\infty$. This leads to relation (1.32).
 
The bound (1.29) (in the real domain) follows from the assertion of 
 Lemma 3 (see (1.66)), for the case of a measure $\;{\bf m}_\L\;$
with the denominator $\;\Xi_\L$. Indeed, by integrating the RHS 
of (1.66) in $\;P^J(d\omega_J)$, while $\;x_J,y_J\;$ remain fixed,
gives a function $\;k^{(J)}_*(x_J,y_J)\;$ with the desired
properties. The last fact may be proved by repeating 
the argument used for establishing (2.33).
 
To prove the bound (1.34), we proceed as follows. First, we write
$$k^{(J)}_\L(x_J,x_J){\cal F}_J(x_J)=\left[{{\Xi_\L({\cal F}_J)}\over
{\Xi_\L}}\right]\left[{{\Xi^{(J)}_\L(x_J,x_J){\cal F}_J(x_J)}\over{\Xi_\L
({\cal F}_J)}}\right]$$
(cf. (1.62)). [Recall that the function $\;{\cal F}_J\;$ is assumed to 
be non-negative.] Repeating the scheme elaborated above, we can 
prove the convergence to a finite limit, 
as $\;\L\nearrow\bf Z$, of the ratio $\;\displaystyle{{{\Xi_\L
({\cal F}_J)}\over{\Xi_\L}}}.\;$ Hence. this ratio is bounded 
uniformly in $\;\L$. The remaining ratio
$${{\Xi^{(J)}(x_J,x_J){\cal F}_J(x_J)}\over{\Xi_\L({\cal F}_J)}}$$
does not exceed, in view of Lemma 3, the RHS of (1.66). This gives the 
desired result.
 
A bit more sophisticated reasoning is used for proving the rest 
parts of  Lemmas 1 and Lemma 2. We begin the discussion with 
proving the analyticity, boundedness and convergence
of the derivatives (1.35). Again to avoid a repetition, let us 
consider the second derivative only.
 
The starting point is formula (1.59). We see that the problem is 
reduced to prove that the ratio of a single addend 
in the parentheses and the denominator $\;\Xi_\L\;$ possesses 
the aforementioned properties. For definiteness, consider
the ratio $\;\displaystyle{{{\Xi_\L^{(J)}(x_J,y_J))^{(2)}_j}
\over{\Xi_\L}}}.\;$ As we noted before, we can
consider separately each addend arising in the braces $\;\{\dots \}\;$
in the RHS; we again
confine the consideration to one of them, say, the quantity
$$\eqalign{\int_{{\bf W}^J}
dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
(V_\L)^{(2)}&(\bar s_J\vee s_{\L\setminus J})\exp
[-V_\L(\bar s_J\vee s_{\L\setminus J})],\cr &\;\bar s_J=(x_J,y_J;
\omega_J)\cr}\eqno (2.53)$$
(we omitted non-essential factor $\;\exp[-1/2\beta\displaystyle
{\sum_{i\in J}(x_j-y_j)^2}]\;$). Furthermore, we expand (2.53) into 
the sum
$$\eqalign{&(\Xi_\L)^{-1}\int_{{\bf W}^J}
dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
\varphi^{(2)}(s_j)\exp
[-V_\L(\bar s_J\vee s_{\L\setminus J})]+\cr
&(\Xi_\L)^{-1}\sum_{j'\in\L:j'\ne j}\int_{{\bf W}^J}
dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
\psi_{|j'-j|}^{(2)}(\bar s_j,\bar s_{j'})\exp
[-V_\L(\bar s_J\vee s_{\L\setminus J})],\cr}\eqno (2.54)$$
where 
$\;\bar s_j=(x_j,y_j;
\omega_j),\;$ and $\;\bar s_{j'}=(x_{j'},y_{j'}\omega_{j'})\;$ for $\;
j'\in J\;$ and $\;\bar s_{j'}=s_{j'}=(x_{j'};\omega_{j'})\;$ for $\;
j'\in\L\setminus J.$
 
We claim that a single addend in (2.54) admits an analytic 
continuation, in the variables $\;w_0$, $w_1,\;$ to a complex domain, of the 
same type as before, and 
 
(i) this domain may be chosen independently of an addend, and also of 
$\;\L\supseteq J\;$ and of $\;x_J$, $\;y_J\;$ running over a compact set 
$\;{\bf O}\subset{\bf R}^J,\;$
 
(ii) the whole series (2.54) converges uniformly in $\;w_0$, $w_1\;$ varying 
within this domain and uniformly in $\;\L\supseteq J\;$ and in 
the variables
$\;x_J$, $y_J\in{\bf O}.\;$
 
\noindent Furthermore, as $\;\L\nearrow\bf Z$, each addend tends to a limit, 
uniformly in $\;w_0$, $w_1$, $x_J\;$ and $\;y_J\;$ in the same sense as above,
and the limiting (analytic) functions form a series that converges, in the 
complex domain under consideration, again uniformly in the same sense. 
 
To verify these statements, it is convenient to take a single addend 
in the `positive'
and `negative' parts as was explained before and treat them separately.
Again for definiteness, let us consider the term corresponding to a 
fixed $\;j'\not\in J\;$ and take its positive part only:
$${{[(\Xi_\L^{(J)}(x_J,y_J))^{(2)}_{j,j'}]_+}\over{\Xi_\L}}\eqno (2.55)$$
where
$$\eqalignno{[(\Xi_\L^{(J)}(x_J,y_J))^{(2)}_{j,j'}]_+=
&\int_{{\bf W}^J}
dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
[\psi_{|j'-j|}^{(2)}(\bar s_j,s_{j'})]_+\times\cr &\exp
[-V_\L(\bar s_J\vee s_{\L\setminus J})],\;\;\bar s_J=(x_J,y_J;
\omega_J)&(2.56)\cr}$$
 
For the sake of simplicity, we can assume that $\;j=0$, $J=\{0\}\;$ 
and omit the subscripts $\;j$,  
$J\;$ and $\;+\;$ and the superscripts $\;(2)\;$ and 
$\;(J)\;$ from the notation,
writing $\;\Xi_L(x,y)_{j'}\;$ instead of $\;[(\Xi_\L^{(J)}(x_J,
y_J)^{(2)}_{j,j'}]_+.$
 
The analysis of the term (2.55) proceeds in a way similar to that 
for $\;\displaystyle{{{\Xi^{(J)}_\L(x_J,y_J)}\over{\Xi_\L}}.}\;$
We can write for $\;\Xi_\L(x,y)_{j'}\;$ a representation similar
to (2.49). The point is that the threshold values $\;M^0$, $N^0$,
$L^0\;$ and $\;w_0$, $w_1\;$ can be chosen independently on $\;j'\;$
and on $\;x_J$, $y_J\;$ running over $\;\bf O$. Therefore, the series 
that arises  have a radius of convergence that does not depend 
on $\;x_J,y_J\in\bf O.\;$ Furthermore, by virtue of (1.5), the sum may 
be estimated by a quantity 
$$const\;(d^2\log(d+1)F(d))^{-1}$$
which guarantees the uniform convergence of the series (2.54).  
 
Similar reasoning is used for establishing the bound (1.36). Let us again 
discuss the case $\;\mu =2\;$ and consider the contribution from $\;
(\Xi_\L^{(J)}(x_J,y_J))_j^{(2)}.\;$ As before, we decompose it 
according to (1.60). For diversity, we will consider the term 
$\;[(V_\L)^{(1)}_j(s_J\vee s_{\L\setminus J})]^2$. A single term
in the corresponding series is related either to $\;\varphi^{(1)}
(s_j)^2,\;$ or $\;\varphi^{(1)}(s_j)\psi_{|j-j'|}^{(1)}(s_j,s_{j'})$, 
or $\;\psi_{|j-j'|}^{(1)}(s_j,s_{j'})\psi_{|j-j''|}^{(1)}(s_j,s_{j''})$,
$j',j''\in\L\setminus\{j\}.\;$ For definiteness, consider a term 
that corresponds to 
$\;\psi_{|j-j'|}^{(1)}(s_j,s_{j'})\psi_{|j-j''|}^{(1)}(s_j,s_{j''})$, 
$j',j''\in\L\setminus J.\;$ 
 
Such a term obviously does not exceed the ratio 
$${{[(\Xi_\L^{(J)}(x_J,x_J))^{(1)}_{j;j',j''}]^+}\over{\Xi_\L}}$$
where
$$\eqalignno{&[(\Xi_\L^{(J)}(x_J,x_J))^{(1)}_{j;j',j''}]^+=
\int_{{\bf W}^J}
dP^J(\omega_J)\int_{{\bf S}^{\L\setminus J}}ds_{\L\setminus J}
|\psi_{|j-j'|}^{(1)}(s_j,s_{j'})|\times\cr &|\psi_{|j-j''|}(s_j,s_{j''})|
\exp [-V_\L(s_J\vee s_{\L\setminus J})],\;s_J=(x_J;
\omega_J)&(2.57)\cr}$$
 
As before, we assume for simplicity that $\;j=0$ and  $J=\{0\}\;$ and omit 
excessive indeces from the notation. According to (1.5) and 
(1.9), (2.57) is less than or equal to 
$$\eqalignno{[|j'|^2&\log\;(|j'|+1)F(|j'|)]^{-1}[|j''|^2\log\;(|j''|+1)
F(|j''|)]^{-1}\times\cr 
&\int_{\bf W}
dP(\omega_J)\int_{{\bf S}^{\L\setminus\{0\}}}ds_{\L\setminus\{0\}}
(||s_0||_1+1)^2(||s_{j'}||_1+1)\cr&(||s_{j''}||_1+1)
\exp [-V_\L(s_0\vee s_{\L\setminus\{0\}})],\;s_0=(x_0;
\omega_0)&(2.58)\cr}$$
 
Hence, all we need is to prove the bound
$${{\tilde\Xi_\L(x)_{j',j''}}\over{\Xi_\L}}\leq\exp\;[-\tilde c_1(x^2-
\tilde c_2)]\eqno (2.59)$$
with constants $\;\tilde c_1>0\;$ and $\;\tilde c_2\in\bf R\;$ 
independent on $\;j'$, $j''\;$ and $\;\L.\;$ Here 
$$\eqalignno{{{\tilde\Xi_\L(x)_{j',j''}}\over{\Xi_\L}}=&
\int_{\bf W}
dP(\omega_J)\int_{{\bf S}^{\L\setminus\{0\}}}ds_{\L\setminus\{0\}}
(||s_0||_1+1)^2\times\cr
&(||s_{j'}||_1+1)(||s_{j''}||_1+1)
\exp [-V_\L(s_0\vee s_{\L\setminus\{0\}})].&(2.60)\cr}$$
This may be achieved by using the assertion of  Lemma 3 
(with the integration in $\;dP(\omega_0)\;$ over $\;\bf W\;$) 
for a measure with the denominator $\;\Xi_\L({\cal E}_{\widetilde J})\;$ 
where
$\;\widetilde J$ = $\{0,j',j''\}\;$ and
$${\cal E}_{\widetilde J}=(||s_0||_1+1)^2(||s_{j'}||_1+1)
(||s_{j''}||_1+1).$$
 
\vfill\eject
\n{\bf APPENDIX}
 
\m\m
 
\n We prove  Lemma 3 by using an argument which is similar to 
the one from 
[R2] and [R3]. To make the exposition easier we will use, wherever possible, 
the notations from [R3], or a close one. 
Let us start by proving the assertion 
of Lemma 3 in the simplest case of the measure with the denominator $\;\Xi_\L$.
 As in [R2] and [R3], we deal with a sequence of volumes (cubes) $\;[\;q\;]$, 
$q=1,2,$..., where 
$$[\;q\;]\;=\;\{j=(j_1,...,j_\nu)\in{\bf Z}^\nu :\;|j_i|\leq l_q\}$$
and a sequence of positive integers $\;l_q\;$ is chosen so that 
$|l_{q+1}/l_q$ $-$ $(1+2\alpha )|$ $<$ $\alpha\;$ where $\;\alpha>0\;$ is 
a constant. The volume of $\;[\;q\;]\;$ is denoted by $\;v_q\;$ : $v_q$ = 
$(2l_q+1)^\nu$. We denote by $\;||\bar s||\;$ the norm (1.48) with $\;r=2.\;$
The  key technical assertion is the following proposition (cf. Proposition 2.1 
from [R2] and [R3]):
\m\m
 
{\bf Proposition A.1} {\it Under the conditions of Lemma 3, 
for any $\;\epsilon >0\;$ and $\;C\geq 0,\;$ there exists $\;\alpha^0>0\;$ 
such that for any $\;\alpha\in (0,\alpha^0)\;$ one can 
find $\;P\geq 1\;$ and a monotone 
increasing sequence $\;\varpi_q$, $q=P,P+1,$..., with $\;\varpi_q\geq 1\;$ 
and $\;{\displaystyle\lim_{q\to\infty}\varpi_q=\infty}\;$ 
such that the following holds: Let $\;\L\subset{\bf Z}^\nu\;$ be a finite set 
and $\;\bar s_\L=(\bar s_j)\;$ be a path configuration from 
$\;\bar{\bf S}^\L$. Suppose that $\;q\geq P\;$ is the largest
integer for which $\;{\displaystyle\sum_{j\in [\;q\;]\cap\L}||\bar s_j||^2}$ 
$\geq$ $\varpi_qv_q$. Then
$$\eqalignno{\sum_{j\in [\;q+1\;]\cap\L}C+\sum_{j\in [\;q+1\;]\cap\L}
\;\sum_{j'\in\L\setminus [\;q+1\;]}
&|\Psi_{\{j,j'\}}|{1\over 2}(||\bar s_j||^2+||\bar s_{j'}||^2)\leq
\cr\leq\epsilon&\sum_{j\in
[\;q+1\;]\cap\L}||s_j||^2.&(A.1)\cr}$$
Moreover, if $\;\epsilon\;$ and $\;C\;$ and  $\;c_1$, $c_2$, 
$\u c$, $\hat c\;$ and $\;\eta\;$ figuring in (1.7), (1.64)
and (1.65) are varying within compact sets (in the case 
of $\;\epsilon$, $\;c_1$, $\hat c\;$
and $\;\eta\;$--- separated from $\;0\;$), 
then $\;\alpha^0\;$ and --- for any $\;\alpha\in(0,\alpha^0)\;$---$\;P\;$ and 
$\{\varpi_q,q\geq P\}\;$ may be chosen independently on these values.}
\m\m
 
We omit the proof of  Proposition A.1: 
it repeats that of  Proposition 2.1
of[R2]. In what follows we fix  $\;\epsilon\in (0,c_1/3),\;$ 
$\;C=c_2$ and fix $\;\alpha\in (0,\alpha^0)\;$. Given 
$\;\L\subset{\bf Z}^\nu$, denote by $\;\Re_0\;$ the set of 
the path configurations $\;\bar s_\L\in\bar{\bf S}^\L\;$ for which 
$\;{\displaystyle\sum_{j\in [\;q\;]\cap\L}||\bar s_j||^2}$ $\leq$ 
$\varpi_qv_q\;$ for any $\;q\geq P$ and by $\;\Re_q$, $q\geq P$, the set of
the path configurations $\;\bar s_\L$ for which $\;q\;$ is the largest integer
$\;\geq P\;$ with ${\displaystyle\sum_{j\in [\;q\;]\cap\L}||\bar s_j||^2}$
$>\varpi_qv_q.$ 
 
An important corollary of  Proposition A.1 is
\m
{\bf Proposition A.2} (a) {\it Let a path configuration $\;\bar s_\L
\in\Re_q\;$ where $\;q\geq P.\;$ Then 
$$\eqalignno{&-V_\L(\bar s_\L)+V_{\L\cap [\;q+1\;]}(\bar s_{\L\cap [\;q+1\;]})
\leq\cr\leq&(-c_1+3\epsilon )\sum_{j\in [\;q+1\;]}
||s_j||^2-C'\varpi_{q+1}v_{q+1}&(A.2)\cr}$$ 
where the constant $\;C'>0\;$ does not depend on} $\;\L$. 
 
(b) {\it Let $\;\bar s_\L\in\Re_0.\;$ Then, for any $\;j\in\L$, 
$$-V_\L(\bar s_\L)+\varphi_j(\bar s_j)\leq D', \eqno (A.3)$$
where $\;\varphi_j(\bar s)=\int_0^\beta dt\Phi_j(\omega (t)+L_{x,y}(t))$,
$\bar s=(x,y;\omega )\;$ (cf. (1.41)) and  the constant $\;D'\;$ 
does not depend on $\;\L$. 
 
The constants, $\;C'\;$ and $\;D',\;$ possess the uniformity property 
stated in  Proposition A.1}. 
\m 
The proof of  Proposition A.2 is similar 
to the one of  Proposition 2.5 of 
[R2] (combined with the proof of the bound (2.29) from [R2]) 
and we again omit
it . Notice  that all constants figuring in the various estimates below
possess the  uniformity property.
 
The partition of $\;\bar{\bf S}^\L\;$ into sets $\;\Re_0$ and 
$\;{\displaystyle{\bigcup_{q\geq P}}\Re_q}\;$ generates, 
for any $\;J\subset\L\;$ 
and $\;\bar s_J$ = $(\bar s_j$, $j\in J)\in\bar{\bf S}^J,\;$ 
the corresponding expansion 
$\;k_{{\bf m}_\L}^{(J)}(\bar s_J)$ = $k'(\bar s_J)$ $+$ $k''(\bar s_J).\;$
As in [R3], we are going to prove later that 
$$k'(\bar s_J)\leq C''\exp\;[\;(\bar\Psi -c_1)||\bar s_j||^2\;]\;
k_{{\bf m}_\L}^{(J\setminus\{j\})}(\bar s_{J\setminus\{j\}})\eqno (A.4)$$
for any $\;j\in\L\;$ and
$$\eqalignno{k''(\bar s_J)\;\leq\;\sum_{q\geq P}\exp &
\;[\;-C'\varpi_{q+1}v_{q+1}
+D''v_{q+1}-
(c_1-3\epsilon )\sum_{j\in [\;q+1\;]\cap J}||\bar s_j||^2\;]\times\cr
&\times k_{{\bf m}_\L}^{(J\setminus [\;q+1\;])}
(\bar s_{J\setminus [\;q+1\;]}),&(A.5)\cr}$$
where $\;C''\,>0\;$ and $\;D''\in{\bf R}\;$ are constants independent on 
$\;\L\;$ and $\;J\subseteq\L\;$ and
$$\bar\Psi\;=\;\sup_{j\in{\bf Z}^\nu}
\sum_{j'\in{\bf Z}^\nu :j'\neq j}|\Psi_{\{j,j'\}}|$$  
(cf. (2) and (3) in [R3]). 
 
Having proved (A.4) and (A.5), we can establish, by using an induction on
the card of $\;J\;$ (cf. [R3]), that
$$k^{(J)}_{{\bf m}_\L}(\bar s_J)\;\leq\;\exp\;[\;\sum_{j\in J}(E||\bar s_j||^2
+F)]\eqno (A.4)$$
for some constants $\;E,\,F\in{\bf R}\;$ (cf. (4) in [R3]). The next step 
is to check that indeed a stronger inequality holds:
$$k^{(J)}_{{\bf m}_\L}(\bar s_J)\;\leq\;\exp\;(\;\sum_{j\in J}
\;[(-c_1+3\epsilon )
||\bar s_j||^2+\delta\;]),\eqno (A.7)$$
where $\;\delta >0\;$ again does not depend on $\;\L$, $J\subset\L\;$ 
(cf. (5) in [R3]). The assertion of Lemma 3 then 
follows with $\;c^\star_1=c_1-3\epsilon$, $c^\star_2=\delta$. 
 
The choice of $\;\delta\;$ is: $\delta =(E+c_1-3\epsilon )\varpi_Pv_P+F\;$ 
(precisely as in [R3]). If $\;||\bar s_j||^2\leq\varpi_Pv_P\;$ for any $\;j
\in J$, (A.7) follows from (A.4). Therefore, we can assume that 
$\;||\bar s_j||^2>\varpi_Pv_P\;$ for some $\;j\in J$. Then $\;k'(\bar s_J)$
= $0\;$ and $\;k^{(J)}_{{\bf m}_\L}(\bar s_J)$ = $k''(\bar s_J)$. 
 
By using (A.5) and an induction on the card of $J$, 
we can write in this case (cf.
[R3]):
$$\eqalign{k^{(J)}_{{\bf m}_\L}&(\bar s_J)\;\leq\;\exp\;\left[\;\sum_{j\in J}
(-c_1+3\epsilon )
||\bar s_j||^2\;\right]\times\cr&\times\sum_{q\geq P}\exp\;
[-C'''\varpi_{q+1}v_{q+1}+
D''v_{q+1}+\delta\;{\rm card}\;(J\setminus [\;q+1\;])\;]\leq\cr
&\leq\;\exp\;[\;-\sum_{j\in J}(c_1-3\epsilon )
||\bar s_j||^2+\delta\;{\rm card}\;
(J\setminus [\;P+1\;])\;]\leq\cr
&\leq\;\exp\;[\;-\sum_{j\in J}(c_1-3\epsilon )||\bar s_j||^2
+\delta\;{\rm card}\;J\;]\cr}$$
which finishes the proof of Lemma 3.
 
It remains to check the bounds (A.4) and (A.5). 
The reasoning is again similar to 
[R3] (cf. Appendix in [R3]). We write
$$\eqalignno{k'(\bar s_J)=&\Xi_\L^{-1}
\int_{{\bf S}^{\L\setminus J}}
ds_{\L\setminus J}\chi_{\Re_0} (\bar s_J
\vee s_{\L\setminus J})\;\exp\;[-V_\L(\bar s_J\vee s_{\L\setminus J})\;]\leq\cr
\leq&\exp\;[-V_J(\bar s_J)]\;\Xi_\L^{-1}
\int_{{\bf S}^{\L\setminus J}}
ds_{\L\setminus J}\chi_{\Re_0} (\bar s_J
\vee s_{\L\setminus J})\;\exp\;[-V_\L
(s'_J\vee s_{\L\setminus J})]\times\cr
&\times\exp\;\left[{1\over 2} (||\bar s_j||^2+||s'_j||^2)\bar\Psi +2D'\right].
&(A.8)\cr}$$
Bound (A.8) follows from Proposition A.2 (b).
 
Now pick a subset $\;{\bf S}_0$ = $\{s\in{\bf S}:||s||^2\leq 1\}$, then for 
any finite $\;\widetilde\L\subset{\bf Z}^\nu$ 
$$\int_{{\bf S}_0^{\widetilde\L}}ds_{\widetilde\L}\exp\;[-V_{\widetilde\L}
(s_{\widetilde\L})]\geq\lambda^{-{\rm card}\;\widetilde\L},\eqno (A.9)$$
where $\;\lambda >0$ (see 
below). Then the RHS of (A.8) is
$$\eqalignno{\leq&\lambda e^{2D'}\exp\;\left[-c_1||\bar s_j||^2+c_2+
{1\over 2}\bar\Psi ||\bar s_j||^2+{1\over 2}\bar\Psi\right]\times\cr
&\times\Xi_\L^{-1}\int_{{\bf S}^{(\L\setminus J)\cup\{j\}}}
ds_{(\L\setminus J)\cup\{j\}}'\exp\;[-V_\L(\bar s_{J\setminus\{j\}}\vee
s'_{(\L\setminus J)\cup\{j\}})]\leq\cr
\leq&\;C''\exp\;[(\bar\Psi-c_1)||\bar s_j||^2]\;
k^{(J\setminus\{j\})}_{{\bf m}_\L}(\bar s_{J\setminus\{j\}}),&(A.10)\cr}$$
which proves (A.4).
 
Furthermore,
$$\eqalign{k''(\bar s_J)=&\sum_{q\geq P}\Xi_\L^{-1}
\int_{{\bf S}^{\L\setminus J}}
ds_{\L\setminus J}\chi_{\Re_q} (\bar s_J
\vee s_{\L\setminus J})\;\exp\;[-V_\L(\bar s_J\vee s_{\L\setminus J})\;]\leq\cr
\leq\sum_{q\geq P}\Xi_\L^{-1}&
\int_{{\bf S}^{\L\setminus J}}
ds_{\L\setminus J}\chi_{\Re_q} (\bar s_J
\vee s_{\L\setminus J})\;\exp\;\left[\;\sum_{j\in [\;q+1\;]\cap\L}
(-c_1||\bar s_j||^2+c_2)\right]\times\cr
&\times\exp\;\left[\;\sum_{j\in [\;q+1\;]\cap\L}\;
\sum_{j'\in\L\setminus [\;q+1\;]}
|\Psi_{\{j,j'\}}|{1\over 2}(||\bar s_j||^2+||\bar s_{j'}||^2)\right]\times\cr
&\times\exp\;\left[\;\sum_{j\in [\;q+1\;]\cap\L}\;
\sum_{j'\in\L\setminus [\;q+1\;]}
|\Psi_{\{j,j'\}}|{1\over 2}(||s'_j||^2+||\bar s_{j'}||^2)\right]\times\cr}$$
\m
$$\times\exp\;[-V_\L(\bar s_{J\setminus 
[\;q+1\;]}\vee s_{(\L\setminus J)
\setminus [\;q+1\;]}\vee s'_{[\;q+1\;]\cap\L})+V_{[\;q+1\;]\cap\L}
(s'_{[\;q+1\;]\cap\L})\;];\eqno (A.11)$$
as before, bound (A.11) holds for any (sequence of) path configurations 
$\;s'_{[\;q+1\;]\cap\L}$ $\in
{\bf S}^{[\;q+1\;]\cap\L}$, $q\geq P$. By using Proposition A.2 (a), we 
continue (A.11) by estimating the RHS as
$$\eqalignno{
\leq&\sum_{q\geq P}\exp\;\left[\;\sum_{j\in [\;q+1\;]\cap J}(-c_1+3\epsilon )
||\bar s_j||^2-C'\varpi_{q+1}v_{q+1}\right]\times\cr
&\times\left[\lambda\;\exp\;\left({1\over 2}\bar\Psi\right)
\right]^{{\rm card}\;([\;q+1\;]\cap\L)}\times\cr
\times\Xi_\L^{-1}&\int_{{\bf S}^{\L\setminus (J\setminus [\;q+1\;])}}
ds'_{\L\setminus (J\setminus [\;q+1\;])}\exp\;[-V_\L
(\bar s_{J\setminus [\;q+1\;]}
\vee s'_{\L\setminus (J\setminus [\;q+1\;])})]\leq\cr
\leq&\sum_{q\geq P}\exp\;\left[\sum_{j\in [\;q+1\;]\cap J}(-c_1+3\epsilon )
||\bar s_j||^2-C'\varpi_{q+1}v_{q+1}+D''v_{q+1}\right]\times\cr
&\times k^{(J\setminus [\;q+1\;])}_{{\bf m}_\L}(\bar s_{J\setminus [\;q+1\;]}),
&(A.12)\cr}$$\m
\n which proves (A.5).
 
To prove (A.9), we observe (as in [R3]) that 
$$V_{\widetilde\L}(\bar s_{\widetilde\L})\leq\sum_{j\in\widetilde\L}
\varphi_j(\bar s_j)+\bar\Psi\sum_{j\in\widetilde\L}||\bar s_j||^2$$
and therefore
$$\eqalign{&\int_{{\bf S}_0^{\widetilde\L}}ds_{\widetilde\L}
\exp\;[-V_{\widetilde\L}(s_{\widetilde\L})]\geq\cr
&\leq\prod_{j\in\widetilde\L}\int_{{\bf S}_0}ds
\exp\;[-\varphi_j(s)-\bar\Psi ||s||^2]\geq\cr
&\geq\lambda^{{\rm card}\;\widetilde\L}\cr}$$
where
$$\lambda=\inf_j\int_{{\bf S}_0}ds\exp\;[-\varphi_j(s)-
\bar\Psi||s||^2>0;$$
the last estimate follows directly from the properties of the 
Wiener measure and the conditions imposed on potentials $\;\Phi_j$.
 
The proof of the assertion of  Lemma 3 for other types of measures
does not differ from the case of the measure with the denominator
$\;\Xi_\L.\;$ In fact what matters is the system of bounds (1.7),
(1.64) and (1.65). The extension of the proof provided to 
other cases is immediate and we leave it to the reader.
 
 
\vfill\eject
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\end