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Quantum Anharmonic Oscillators, Quantum Effects, Suppression of the Critical Fluctuations
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\documentclass[11pt]{amsproc}
\usepackage{srcltx}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\begin{document}

\title{Quantum Effects in Lattice Models
of Vector Anharmonic Oscillators  }


\author{Yuri Kozitsky}

\address{Institute of Mathematics, Marie Curie-Sklodowska University, Lublin PL 20-31, Poland }

\address{Institute for Condensed Matter Physics. Lviv, Ukraine}

\email{{\sf jkozi@golem.umcs.lublin.pl}}
\thanks{Dedicated to Sergio Albeverio on the occasion of his 60th birthday}
\thanks{This research was supported in part by the Polish Scientific Research Committee, Grant KBN 2 PO3A 02915}


\subjclass{82B10; 82B27}

\begin{abstract}

A lattice model of $d$-dimensional quantum anharmonic oscillators with a
polynomial anharmonicity and a ferroelectric interaction is considered. For
arbitrary $d\in {I\!\! N}$, it is shown that the critical fluctuations of the
position operator, peculiar to the critical point, are suppressed at all
temperatures provided the particles are "strongly quantum". The latter means
that for the corresponding one-dimensional model, the smallest distance between
the energy levels of the isolated oscillator is large enough or that the reduced
mass of the oscillator is less than some threshold value depending on the
anharmonicity parameters. These results are the extension to the vector case of
the statements obtained recently in \cite{[00]} and \cite{[0]} regarding such
effects in similar lattice models.

\end{abstract}

\maketitle




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%%%%%%%%%%%% 1. Kapitel %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Posing of the problem and results.}
The results described in this note were obtained in the framework of the
approach in quantum statistical mechanics initiated by now classical paper \cite{[1]}
published by Sergio Albeverio and Raphael H\o egh--Krohn almost 25 years ago.

We consider a countable system of interacting quantum particles
(oscillators) with the reduced mass $m=\mu/\hbar^2$, performing $d$-dimensional
oscillations around their equilibrium positions, the set of which is a lattice
${I\!\! L}$, e.g. $ {I\!\! L}
= {Z\!\!\! Z}^\nu $ with certain $\nu\in {I\!\! N}$. The dynamics
of the oscillations around the position $l$ is described by means
of the momentum and displacement operators $\{ p_l , q_l \}$ densely
defined on the complex Hilbert space ${\mathcal H}_l= L^{2} ({I\!\! R}_l^d, dx_l
)$. The whole system is described by the formal Hamiltonian
\begin{equation}
\label{1.1}
H = \frac{1}{2} \sum_{ l,l' \in {I\!\! L} } d_{ll'} (q_{l}, q_{l'}) + \sum_{l
\in {I\!\! L}}H_l;
\end{equation}
\begin{equation}
\label{1.2}
H_l = \frac{p_{l}^{2}}{ 2m} + V(q_l ) , \ \ p_{l}^{2} \ \stackrel{\rm def }{=} \
(p_l , p_l ),
\end{equation}
%
where $(\ . \ , \ . \ )$ stands for the scalar product in ${I\!\! R}^d $
and $d_{ll'}$ form the dynamical matrix.
The one-particle potential $V$ has the form
\begin{equation}
\label{new1}
V(x) = v((x,x)), \ \ v(\xi)= \frac{1}{2}a\xi + \sum_{s=2}^{r}b_s \xi^s , \ \
a\in {I\!\! R},
\ b_s
\geq 0, \ b_r >0 .
\end{equation}
The formal Hamiltonian cannot be
defined directly and used is "represented" by its local versions
-- indexed by finite subsets $\Lambda
\subset{I\!\! L}$ local Hamiltonians $H_{\Lambda}$. For such $\Lambda$, we set $
{\mathcal H}_{\Lambda}=\bigotimes_{l \in \Lambda} {\mathcal H}_l$. Then each $
H_{\Lambda}$ is defined as an essentially self-adjoint lower bounded operator
acting in the Hilbert space $ {\mathcal H}_{\Lambda} $. For our purposes, it is
enough to consider these Hamiltonians indexed by boxes $\Lambda = \{ l = (l_1, \
\dots, \  l_\nu )
\ \vert
\ l_j^0 \leq l_j \leq l_j^1, \ j = 1, \ \dots, \nu ; \ l_j^0 < l_j^1,
\
l_j^0, l_j^1 \in {Z\!\!\! Z} \ \}$ with the periodic conditions on the
boundaries of $\Lambda$. Let ${\mathcal L}$ stand for the family of all such
boxes. For a box $\Lambda$, let ${\mathcal L} (\Lambda)$ denote the partition of
${I\!\! L}$ by the boxes which are obtained as the translations of $\Lambda$. We
also denote by ${\mathcal G} $ the group of all translations of ${I\!\! L}$. Let
${\mathcal G} (\Lambda)\subset {\mathcal G}$ be the subgroup consisting of the
translations which generate the partition ${\mathcal L} (\Lambda)$. This means $
{\mathcal L} (\Lambda) =
\{ t(\Lambda), \ t \in {\mathcal G} (\Lambda) \}$, where $ t(\Lambda)= \{ t(l),\
l\in\Lambda \}$. Then for a box $\Lambda$, the dynamical matrix $
(d^{\Lambda}_{ll'})_{l,l' \in \Lambda}$ corresponding to the periodic conditions
on the boundary of $\Lambda$ is defined as follows
\begin{equation}
\label{new}
d^{\Lambda}_{ll'} = \min \{ d_{lt(l')} \ : \ t\in {\mathcal G} (\Lambda) \}.
\end{equation}
Thereafter, the dynamical matrices $(d_{ll'})_{l,l'\in {I\!\! L}}$ and $
(d^{\Lambda}_{ll'})_{l,l' \in \Lambda}$ are supposed to have the following
properties
%
\begin{description}
\item[(D1)] $ d_{l l'}$ is invariant under translations on ${I\!\! L}$;
\item[(D2)] $d^{\Lambda}_{l l'} \leq 0$, $d_{l l'} \leq 0$ (ferroelectricity),
\
  \ $d^{\Lambda}_{ll} = d_{ll} = 0$;
\item[(D3)]  the series $\sum_{l' \in {I\!\! L}} d_{l l'}$ converges
 for all $l \in {I\!\! L}$.
\end{description}
The mentioned local Hamiltonian  $H_{\Lambda}$ corresponding to the periodic
boundary conditions is
%
\begin{equation}
\label{1.3}
H_{\Lambda} = \frac{1}{2} \sum_{l,l' \in \Lambda } d^{\Lambda}_{ll'} (q_l
,q_{l'} ) +
\sum_{l \in {\Lambda}}H_l .
\end{equation}
For every $\Lambda \in {\mathcal L}$ and the inverse temperature
 $\beta = T^{-1}$,
the periodic Gibbs state in $\Lambda$ is defined as a positive functional on
appropriate $C^{*}$-algebra of observables as follows:
%
\begin{equation}
\label{1.7}
\gamma_{\beta , \Lambda} (A) = \frac{{\rm trace} (A e^{-\beta H_\Lambda})}
{{\rm trace}( e^{-\beta H_\Lambda } )} .
\end{equation}
%
By means of this state one can define temperature Green functions. For $\{
\tau_0 ,
\tau_1 ,\ldots , \tau_n \}$, $\tau_i \leq
\tau_{i+1}$, $\tau_i \in I_\beta := [0,\beta]$, and appropriate operators $\{
A_0,A_1 , \ldots , A_n \}$, the temperature Green function is
\begin{eqnarray}
\label{1.8}
&  & \Gamma^{ \beta , \Lambda}_{ A_0 ; A_1 ; \ldots ; A_n} (\tau_0 , \tau_1 ,
\ldots , \tau_n ) \\
 & = & \gamma_{\beta , \Lambda} \left( A_{0}e^{
     -(\tau_1 -
       \tau_0)H_{\Lambda}}
      A_{1}e^{ -(\tau_2 - \tau_1 )H_{\Lambda}} \ldots
 A_n e^{ -(-\tau_n + \tau_0 )H_{\Lambda}} \right). \nonumber
\end{eqnarray}
These functions can be obtained by using the analytic continuation of the usual
Green functions corresponding to the unitary time evolution, see \cite{[1]},
\cite{[7]}. Considering
the limits $\Lambda \nearrow {I\!\! L}$  we will use collections  of
boxes ${\mathcal C} $, which are countable and completely ordered by inclusion.
Our intention is to show that the large fluctuations of the
displacements of particles in the model considered
are absent (suppressed) at any temperature if these
particles are "strongly quantum". The precise meaning of this will be clarified
below. The first results in this direction (on the "physical level" of
strictness) were published in \cite{[9]}. Since that time the suppression of
critical phenomena in lattice systems by quantum effects has been studied
theoretically and experimentally in various physical publications (see
\cite{[5]}, \cite{[S]}, \cite{[12]}. The first mathematically strict results for
the models with unbounded operators were obtained in \cite{[00]}, \cite{[0]}.
The research described in this note is a direct extension to the vector case of
the statements from the latter paper.

Thus we restrict ourselves to the consideration of the operators which describe
the mentioned fluctuations, namely
\begin{equation}
\label{1.10}
A^{(k)} (\Lambda) =
 \frac{1}{\sqrt{ \vert \Lambda \vert} }  \sum_{l \in \Lambda} q_l^{(k)} ,
\ \ \ k=1,2, \dots , d .
\end{equation}
We set
\begin{equation}
\label{new2}
\Gamma_{\Lambda}^{(k)}(\tau, \tau' ) =
\Gamma_{A^{(k)}(\Lambda ); A^{(k)}(\Lambda)}^{\beta , \Lambda} (\tau , \tau') .
\end{equation}
The $O(d)$-symmetry of the model implies that such functions with different $k$
coincide with each other. Further, we set
\begin{equation}
\label{1.13}
U_\Lambda  =  \sum_{k=1}^{d} U_\Lambda^{(k)}, \  \  \ \  U_\Lambda^{(k)}  =
\frac{1}{ \beta} \int\int_{I^2_\beta}\Gamma_\Lambda ^{(k)} (\tau ,
\tau ') d\tau d\tau' .
\end{equation}
This parameter $U_\Lambda$ describes, in some sense, variance of the
fluctuations. It depends on $\beta$ and is known as local (for $\vert \Lambda
\vert < \infty$) static ferroelectric susceptibility (see \cite{[5]}, \cite{[S]}).
\begin{definition}
\label{1df}
For given inverse temperature $\beta$ and arbitrary choice of the sequence of
boxes ${\mathcal C}$, let the sequence $\{ U_\Lambda ,\Lambda \in {\mathcal C}
\}$ be bounded. Then the critical fluctuations of the displacements of particles
are absent at this temperature.
\end{definition}
In case this sequence is convergent, one may establish analyticity properties
for thermodynamic potentials describing this model.
Now we are in a position to state our main result.
\begin{theorem}
\label{1tm}
For any positive integer $d$, there exists a positive threshold value of the
reduced mass $m_* $ such that for all $m<m_* $, the critical fluctuations of the
displacement of particles in the $d$-dimensional model described above are
absent at all temperatures.
\end{theorem}

The proof of this statement is based upon the comparison of our vector model
with the model of one--dimensional quantum oscillators described by the
following local Hamiltonians
\begin{eqnarray}
\label{new3}
\tilde{H}_\Lambda & = & \frac{1}{2}\sum_{l,l' \in\Lambda} d_{ll'}^{\Lambda}
q_l^{(1)} q_{l'}^{(1)} +\sum_{l \in \Lambda}\tilde{H}_l , \\
\label{new33}
\tilde{H}_l & = & \frac{1}{2m} \left(p_l^{(1)}\right)^2 + v\left((q_l^{(1)})^2 \right),
\end{eqnarray}
where the canonical operators $ p_l^{(1)} , q_l^{(1)} $ act in the space $L^2
({I\!\! R}_l , dx_l )$. They can be viewed as the components of the momentum and
displacement operators introduced above. In the same sense we consider the
operators $A^{(1)} (\Lambda)$. The other parameters of the Hamiltonians
$\tilde{H}_\Lambda$, $\tilde{H}_l $ are the same as in (\ref{1.3}), (\ref{1.2}),
and (\ref{new1}). Let the state $\tilde{\gamma}_{\beta,\Lambda}$ be defined by
(\ref{1.7}) with $\tilde{H}_\Lambda $ instead of $H_\Lambda$. For $0
\leq \tau \leq \tau' \leq \beta$, we set
\begin{eqnarray}
\label{new4}
\tilde{\Gamma}^{(1)}_\Lambda (\tau , \tau' ) & = & \tilde{\gamma}_{\beta , \Lambda}
\left(A^{(1)} (\Lambda )e^{-(\tau' - \tau )\tilde{H}_\Lambda} A^{(1)} (\Lambda )
e^{-(\tau - \tau' )\tilde{H}_\Lambda} \right), \\
\label{new44}
\tilde{U}^{(1)}_\Lambda & = & \frac{1}{ \beta} \int\int_{I^2_\beta}
\tilde{\Gamma}_\Lambda ^{(k)} (\tau ,\tau ') d\tau d\tau' .
\end{eqnarray}
\begin{lemma}
\label{1lm}
For every $\beta $, finite $\Lambda$, and any $k= 1,2, \dots , d$
\begin{equation}
\label{new5}
U^{(k)}_\Lambda \leq \tilde{U}^{(1)}_\Lambda .
\end{equation}
\end{lemma}
In what follows, in order to have the critical $d$-dimensional fluctuations
suppressed it is enough to suppress these fluctuations at the same temperature
in the corresponding one--dimensional model. This conclusion is evident from the
physical point of view but its mathematical justification was impossible so far,
mainly due of the fact that the tool of proving here -- Griffiths and Lebowitz
inequalities -- works only in one dimension (although the Griffiths inequality
holds also for $d=2$, it is not enough to obtain such effective estimates
like that described below in Proposition \ref{1.1lem}, which was
proved in \cite{[0]}). Having the bound (\ref{new5}) we may use the results
obtained in \cite{[0]} for the one--dimensional quantum oscillators
described by the local Hamiltonians (\ref{new3}).

The dynamical matrix $ (d_{ll'})$ can be defined as a linear bounded operator
$D$ on the Hilbert space $l^2({I\!\! L})$. Since it is translation invariant
(D1) and ferroelectric (D2), the series mentioned in (D3) does not depend on
$l$. Thus
\begin{equation}
\label{1.16}
 \| D \| = - \sum_{l'\in {I\!\! L}} d_{ll'} .
\end{equation}
Define
\begin{equation}
\label{1.17}
T = \tilde{H}_l + \| D \| \left(q^{(1)}_l\right)^2, \ \ l\in {I\!\! L},
\end{equation}
where $\tilde{H}_l$ is given by (\ref{new33}). The operator $T$ has purely
discrete nondegenerate spectrum. We set
%
\begin{equation}
\label{1.5}
T\psi_n = E_n \psi_n,
 \ \  n \in {Z\!\!\! Z}_{+},
\end{equation}
%
where the eigenvalues $E_n$ are numbered in increasing order. Denote
%
\begin{equation}
\label{1.6}
\Delta = \min \{ E_{n+1} - E_n \ \vert \ n \in {Z\!\!\! Z}_{+} \}.
\end{equation}
\begin{proposition}{\bf \cite{[0]}}
\label{1.1lem}
Let the mass $m$, the spectral parameter $\Delta$, and the interaction parameter
$\| D \| $  satisfy the condition
\begin{equation}
\label{1.18}
m\Delta^2  >    2 \| D \| .
\end{equation}
Then for every $\beta > 0$ and arbitrary $\Lambda \in {\mathcal L}$, the
following estimate holds
\begin{equation}
\label{1.19}
\tilde{U}^{(1)}_\Lambda \leq \frac{1}{ m \Delta^2 - 2 \| D \| }.
\end{equation}

\end{proposition}
Thus in order to have the critical one-dimensional fluctuations
suppressed for all temperatures it is enough to satisfy (\ref{1.18}). In this
case both one-dimensional and $d$-dimensional models may be called "strongly
quantum" and the parameter appeared in the left hand side of (\ref{1.18}) may be
used to characterize this property quantitatively. It can be set large by means
of strong hydrostatic pressure (see e.g. \cite{[S]} or \cite{[12]}). For the
harmonic oscillator (i.e. if in (\ref{new1}) $a>0$ and all $b_s= 0$), this
parameter does not depend on $m$ (one has $m\Delta^2= a$ which is growing with
pressure), but for the anharmonic oscillator described by the Hamiltonian
(\ref{new33}), (\ref{new1}) with $r\geq 2$, it does depend.
\begin{proposition}{\bf \cite{[0]}}
\label{1.2lem}
For the parameter $m\Delta^2 $ defined by (\ref{1.6}), (\ref{1.5}),
(\ref{1.17}), (\ref{new33}), and (\ref{new1}), there exists $K>0$ such that
\begin{equation}
\label{new6}
\lim_{m\rightarrow 0+} m^{(r-1)/(r+1)}\left(m\Delta^2 \right) = K.
\end{equation}
Therefore, there exists $m_* >0$ such that for $m< m_* $, the condition
(\ref{1.18}) is satisfied.
\end{proposition}
The proof of Theorem \ref{1tm} easily follows from the latter three statements.
The above results may be extended to the models with dynamical matrices, which
are not translation invariant. For such matrices, we write $\hat{D}
\leq D$ if for their elements one has $\hat{d}_{ll'} \leq d_{ll' }$ for
every pair $l,l'
\in {I\!\! L}$.
\begin{corollary}
\label{1co}
For the model described by the Hamiltonian (\ref{1.1}) - (\ref{new1}), let
conditions (D2), (D3) be satisfied. Let also there exist a dynamical matrix
$\hat{D}$ obeying (D1) - (D3) and such that $\hat{D}\leq D$. Then Theorem
\ref{1tm} holds for this model as well.
\end{corollary}
The proof of this statement follows from the Griffiths inequality, that will be
explained in the next section. It is also worth to mention the following
property of the one-dimensional models, relevant to the problems discussed here
(see \cite{[AKK]}, \cite{[AKKR1]}).
\begin{proposition}
\label{1.3lem}
The local susceptibility $\tilde{U}^{(1)}_\Lambda $ defined by (\ref{new44}) is
a monotone increasing continuous function of the reduced mass $m$.
\end{proposition}

\section{The Proofs}

For given inverse temperature $\beta$, let
\[
C_\beta = \{ \omega \in C(I_\beta\rightarrow {I\!\! R}^d ) \vert \omega (0) =
\omega (\beta )\},
\]
and
\[
\Omega_{\beta , \Lambda} = \{ \omega_\Lambda = (\omega_l )_{l\in \Lambda} \vert
\omega_l \in C_\beta \}, \ \ \  \Lambda \in {\mathcal L} .
\]
Since $\Lambda $ is finite, $\Omega_{\beta , \Lambda}$ may be equipped with the
usual Banach space structure. Let ${\mathcal B}_{\beta , \Lambda}$ stand for the
Borel $\sigma$--algebra of its subsets. The space $\Omega_{\beta ,
\Lambda}$ may be embedded densely into the following real Hilbert space
\[
{\mathcal X}_{\beta , \Lambda} = \bigotimes_{l\in \Lambda} {\mathcal X}_l ,
\ \
\ {\mathcal X}_l \cong {\mathcal X} = L^2 (I_\beta \rightarrow {I\!\! R}^d ) .
\]
The scalar product in the space $ {\mathcal X} $ is
\begin{equation}
\label{na2}
<\omega , \omega' > = \int_{I_\beta } (\omega (\tau ) , \omega' (\tau ) ) d\tau
.
\end{equation}
Consider the following strictly positive trace class operator on ${\mathcal X}$
\begin{equation}
\label{na3}
S = \left( - m \Delta_\beta + 1 \right)^{-1} {\bf 1 },
\end{equation}
where $\Delta_\beta $ is the Laplace operator in $L^2 (I_\beta )$, $m$ is the
reduced mass, and ${\bf 1}$ is the identity operator in ${I\!\! R}^d$. The
operator $S$ determines on ${\mathcal X}$ an isotropic Gaussian measure
$\chi_\beta $ having the following Laplace transform
\[
\int_{{\mathcal X}} \exp \left(<\varphi, \omega > \right) \chi_\beta (d\omega ) =
\exp \left(<S \varphi , \varphi >\right).
\]
One has $\chi_\beta (C_\beta ) =1$ (see e.g. \cite{[7]}), thus we can define on
${\mathcal X}$ the measure $\sigma_\beta $ which is supported on $C_\beta $ and
has there the following representation
\begin{equation}
\label{na4}
\sigma_\beta (d\omega ) = \frac{1}{Y_\beta } \exp \left( -\int_{I_\beta }
 V(\omega (\tau ) ) d\tau )\right) \chi_\beta (d\omega ),
\end{equation}
where $V$ is given by (\ref{new1}) and $Y_\beta $ is the normalization constant.

In the framework of the approach mentioned at the beginning of this note, the
states in the boxes $\Lambda$ are defined by means of Euclidean Gibbs measures.
The periodic Euclidean Gibbs measure which corresponds to the $d$--dimensional
model we consider in this work is the following probability measure on the
Hilbert space ${\mathcal X}_{\beta ,
\Lambda}$
\begin{equation}
\label{na5}
\mu_{\beta, \Lambda} (d\omega_\Lambda ) = \frac{1}{Z_{\beta , \Lambda }} \exp\left\{
\frac{1}{2} \sum_{l,l' \in \Lambda}d_{ll'}^{\Lambda} <\omega_l , \omega_{l'} >
\right\}\bigotimes_{l\in \Lambda} \sigma_\beta (d\omega_l ) ,
\end{equation}
where $Z_{\beta , \Lambda}$ is the normalization constant and the periodic
dynamical matrix $(d^{\Lambda}_{ll'} )_{l,l' \in \Lambda}$ is defined by
(\ref{new}). Just in the same way one introduces the objects which describe the
one-dimensional version of the model considered here. We denote them by
$\tilde{C}_\beta $, $\tilde{\mathcal X}$, $\tilde{\Omega}_{\beta ,\Lambda}$,
$\tilde{\mathcal X}_{\beta ,\Lambda}$,
 $\tilde{S}_\beta $, $\tilde{\chi}_\beta $.
Obviously
\begin{equation}
\label{a1}
C_\beta = \tilde{C}_\beta^{d} , \ \ {\mathcal X} = \tilde{{\mathcal X}}^d , \ \
\Omega_{\beta , \Lambda} = \tilde{\Omega}^{d}_{\beta , \Lambda} ,
\end{equation}
and
\begin{equation}
\label{a2}
\gamma_\beta (d\omega_l ) = \bigotimes_{k=1}^{d}\tilde{\gamma}_\beta (d\omega_l^{(k)} ) .
\end{equation}
 Then we set
\begin{equation}
\label{na6}
\tilde{\sigma}_\beta (d\omega^{(1)} ) = \frac{1}{\tilde{Y}_\beta }
 \exp \left\{ -\int_{I_\beta }
 v\left((\omega^{(1)} (\tau ))^2 \right) d\tau )\right\} \tilde{\chi}_\beta
(d\omega^{(1)} ),
\end{equation}
where $\tilde{Y}_\beta$ is the normalization constant and $v$ is given by
(\ref{new1}). Thus the periodic Euclidean Gibbs measure of the one-dimensional
model is
\begin{equation}
\label{na7}
\tilde{\mu}_{\beta , \Lambda} (d\omega^{(1)}_\Lambda ) =
\frac{1}{\tilde{Z}_{\beta , \Lambda }}
\exp\left\{\frac{1}{2} \sum_{l,l' \in \Lambda}d_{ll'}^{\Lambda}
\int_{I_\beta }\omega_l^{(1)}(\tau)  \omega_{l'}^{(1)} (\tau )d\tau
\right\}\bigotimes_{l\in \Lambda} \tilde{\sigma}_\beta (d\omega_l^{(1)} ) .
\end{equation}
In our approach the Green functions (\ref{1.8}),
(\ref{new2}), (\ref{new4}) are written as expectations with respect to the
above introduced Euclidean Gibbs measures. That is
\begin{eqnarray}
\label{na8}
{\ \  \ \ \ } \Gamma^{(k)}_{\Lambda} (\tau , \tau' ) & = & \frac{1}{\vert
\Lambda \vert} \sum_{l,l' \in \Lambda} {I\!\! E}_{\mu_{\beta , \Lambda}}
\left\{\omega_l^{(k)} (\tau )
\omega_{l'}^{(k)} (\tau' )\right\}, \ \ k = 1, 2, \dots , d , \\
\label{na9}
{\ \  \ \ \ } \tilde{\Gamma}^{(1)}_{\Lambda} (\tau , \tau' ) & = &
\frac{1}{\vert
\Lambda \vert}
\sum_{l,l' \in \Lambda} {I\!\! E}_{\tilde{\mu}_{\beta , \Lambda}}
\left\{\omega_l^{(1)} (\tau )
\omega_{l'}^{(1)} (\tau' )\right\}.
\end{eqnarray}
{\bf Proof of Lemma \ref{1lm}.} The proof is based on an infinite dimensional
version of the Griffiths-Kelly-Sherman (GKS) inequality (see e.g. \cite{[10]}
p..119-124). A detailed study of this and other correlation inequalities for
Euclidean Gibbs measures will be given in our forthcoming paper \cite{[AKKR1]}.
Let us consider
\begin{eqnarray*}
\label{na10}
 {I\!\! E}_{\mu_{\beta , \Lambda}}\left\{\omega_l^{(k)} (\tau )
\omega_{l'}^{(k)} (\tau' )\right\}  =
\int_{{\mathcal X}_{\beta , \Lambda}}\omega_l^{(k)} (\tau )
\omega_{l'}^{(k)} (\tau' )\mu_{\beta , \Lambda} (d \omega_{\Lambda})=
\qquad &  &
\\
    \frac{1}{Z_{\beta , \Lambda} Y_\beta^{\vert \Lambda\vert}}
\int_{{\Omega}_{\beta , \Lambda}}\omega_l^{(k)} (\tau )
\omega_{l'}^{(k)} (\tau' )\exp\{ -\frac{1}{2}
\sum_{\lambda,\lambda' \in \Lambda} d_{\lambda \lambda'}^{\Lambda}
\sum_{k=1}^{d} \int_{I_\beta } \omega_l^{(k)}(t)
\omega_{\lambda'}^{(k)} (t) dt- &  &  \nonumber \\
 \frac{a}{2}  \sum_{\lambda\in \Lambda}  \sum_{k=1}^{d}
\int_{ I_\beta }\left(\omega_{\lambda}^{(k)} (t) \right)^2 dt
-\sum_{\lambda \in \Lambda}\sum_{s=1}^{r} b_s \int_{I_\beta}
\left(\sum_{k=1}^{d} (\omega_{\lambda}^{(k)} (t))^2 \right)^{s} dt \}
\bigotimes_{\lambda\in \Lambda}
\chi(d\omega_{\lambda} ). & & \nonumber
\end{eqnarray*}
For $\eta \in [0,1]$, we set
\begin{equation}
\label{na11}
 \Xi_{l,l' }^{(k)} (\eta \vert \tau, \tau' ) =
{I\!\! E}_{\rho_\eta }\left\{
\omega_l^{(k)}(\tau )\omega_{l'}^{(k)}(\tau') \right\},
\end{equation}
where the expectation is taken with respect to such measure on
$\tilde{\Omega}_{\beta , \Lambda} $
\begin{eqnarray}
\label{nb11}
\rho_\eta (d\omega_{\Lambda }^{(k)})& =
&  \frac{1}{\Upsilon^{(k)} (\eta )}
 \exp\{ -\frac{1}{2}
\sum_{\lambda ,\lambda' \in \Lambda} d_{\lambda \lambda'}^{\Lambda}\int_{I_\beta }
\omega_{\lambda}^{(k)}(t)
\omega_{\lambda'}^{(k)} (t) dt \nonumber  \\
&-  & \frac{a}{2}  \sum_{\lambda\in \Lambda}  \int_{ I_\beta }
\left(\omega_{\lambda}^{(k)} (t)\right)^2 dt
-\sum_{\lambda\in \Lambda}\sum_{s=1}^{r} b_s \int_{I_\beta}
\left( \omega_{\lambda}^{(k)} (t) \right)^{2s} dt     \\
&- & \eta \sum_{\lambda\in \Lambda} \sum_{n=1}^{r-1}\int_{I_\beta}\left[
\left(\omega_{\lambda}^{(k)} (t)\right)^{2n} P_{\lambda ,n}^{(k)} (t)\right] dt \}
 \bigotimes_{\lambda\in\Lambda} \tilde{\gamma}(d\omega_{\lambda}^{(k)} ).\nonumber
\end{eqnarray}
Here
\begin{eqnarray}
\label{na12}
P_{\lambda ,n}^{(k)} & := & \sum_{s=n+1}^{r} b_s
\left(\sum_{k' \neq k}(\omega_{\lambda}^{(k')}
(t))^2 \right)^{s-n} \geq 0 , \\
 \Upsilon^{(k)} (\eta) & := &
\int_{\tilde{\Omega}_{\beta , \Lambda} }
\exp\{ -\frac{1}{2}
\sum_{\lambda, \lambda' \in \Lambda} d_{\lambda \lambda'}^{\Lambda}\int_{I_\beta }
\omega_{\lambda}^{(k)}(t)
\omega_{\lambda'}^{(k)} (t) dt  \nonumber   \\
& - & \frac{a}{2}  \sum_{\lambda\in \Lambda}  \int_{ I_\beta }
\left(\omega_{\lambda}^{(k)} (t)\right)^2 dt
-\sum_{\lambda\in \Lambda}\sum_{s=1}^{r} b_s \int_{I_\beta}
\left( \omega_{\lambda}^{(k)} (t) \right)^{2s} dt \nonumber   \\
&- & \eta \sum_{\lambda\in \Lambda} \sum_{n=1}^{r-1}\int_{I_\beta}\left\{
\left(\omega_{\lambda}^{(k)} (t)\right)^{2n} P_{\lambda ,n}^{(k)} (t)\right\} dt \}
 \bigotimes_{\lambda\in\Lambda} \tilde{\gamma}(d\omega_{\lambda}^{(k)} ). \nonumber
\end{eqnarray}
Taking into account (\ref{a1}) and (\ref{a2}) one obtains from
(\ref{na11})--(\ref{na12})
\begin{eqnarray}
\label{na13}
{\quad}   {I\!\! E}_{\mu_{\beta , \Lambda}}\left\{\omega_l^{(k)} (\tau )
\omega_{l'}^{(k)} (\tau' )\right\}  =
    \frac{1}{ Z_{\beta , \Lambda} Y_\beta^{\vert \Lambda \vert} }
\int_{\tilde{\Omega}_{\beta , \Lambda}^{d-1}}\Xi_{ll'}^{(k)} (1 \vert \tau, \tau')
\Upsilon^{(k)} (1) & & \\
\exp\{ -\frac{1}{2}
\sum_{\lambda, \lambda' \in \Lambda} d_{\lambda \lambda'}^{\Lambda}
\sum_{k' \neq k} \int_{I_\beta } \omega_{\lambda}^{(k')}(t)
\omega_{\lambda'}^{(k')} (t) dt  -
  \frac{a}{2}  \sum_{\lambda\in \Lambda}  \sum_{k' \neq k}
\int_{ I_\beta }\left(\omega_{\lambda}^{(k')} (t) \right)^2 dt & & \nonumber \\
-\sum_{\lambda\in \Lambda}\sum_{s=1}^{r} b_s \int_{I_\beta}
\left(\sum_{k' \neq k} (\omega_{\lambda}^{(k')} (t))^2 \right)^{s} dt \}
\bigotimes_{\lambda\in \Lambda}
\bigotimes_{k' \neq k}\tilde{\chi}(d\omega_{\lambda}^{(k')} ). \nonumber
\end{eqnarray}
It is not difficult to check that for any choice of $k$, $l,l' $, and
$\tau,\tau'$, $\Xi_{l,l'}^{(k)} (\eta \vert \tau, \tau') $ given by (\ref{na11})
is a continuous function of $\eta$  on $[0,1]$, which is differentiable on $(0,
1)$, where its derivative is
\begin{eqnarray}
\label{na14}
 &  & \frac{\partial}{\partial \eta}\Xi_{l,l'}^{(k)} (\eta \vert \tau, \tau')  =
- \sum_{\lambda \in \Lambda}\sum_{n=1}^{r-1} \int_{I_\beta }P_{\lambda , n}^{(k)}
(t) \\ & & \{{I\!\! E}_{\rho_\eta} \left[\left(\omega_\lambda^{(k)}
(t)\right)^{2n}
\omega_l^{(k)} (\tau) \omega_{l'}^{(k)} (\tau' ) \right]  \nonumber \\
   & - &
{I\!\! E}_{\rho_\eta} \left[\left(\omega_\lambda^{(k)} (t)\right)^{2n}\right]
{I\!\! E}_{\rho_\eta} \left[\omega_l^{(k)} (\tau) \omega_{l'}^{(k)} (\tau' )
\right]\} dt. \nonumber
\end{eqnarray}
The measure $\rho_\eta $ is such that, for every $\eta \in [0,1]$ and for any
value of $P_{\lambda ,n}^{(k)} (t) \geq 0$ (\ref{na12}), its moments obey the
GKS-inequality
\[
{I\!\! E}_{\rho_\eta} \left[\left(\omega_\lambda^{(k)}
(t)\right)^{2n}
\omega_l^{(k)} (\tau) \omega_{l'}^{(k)} (\tau' ) \right] \geq
{I\!\! E}_{\rho_\eta} \left[\left(\omega_\lambda^{(k)} (t)\right)^{2n}\right]
{I\!\! E}_{\rho_\eta} \left[\omega_l^{(k)} (\tau) \omega_{l'}^{(k)} (\tau' )
\right],
\]
for every $\tau , \tau', $ and $t$. Taking into account that $P_{\lambda
,n}^{(k)} (t) \geq 0$ one deduces from (\ref{na14})
\begin{eqnarray*}
\Xi_{l,l'}^{(k)} (1 \vert \tau, \tau')  \leq  \Xi_{l,l'}^{(k)} (0 \vert \tau, \tau')
={I\!\! E}_{\tilde{\mu}_{\beta, \Lambda}} \left\{\omega_l^{(k)} (\tau)
\omega_{l'}^{(k)} (\tau' )\right\}.
\end{eqnarray*}
We use the latter estimate in (\ref{na13}) and obtain for every $\tau ,
\tau' \in [0,\beta ]$
\begin{eqnarray*}
  {I\!\! E}_{\mu_{\beta , \Lambda}}\left\{\omega_l^{(1)} (\tau )
\omega_{l'}^{(1)} (\tau' )\right\}  \leq {I\!\! E}_{\tilde{\mu}_{\beta, \Lambda}}
\left\{\omega_l^{(1)} (\tau) \omega_{l'}^{(1)} (\tau' )\right\}
    \int_{\tilde{\Omega}_{\beta , \Lambda}^{d-1}}
\frac{\Upsilon^{(1)} (1)}{ Z_{\beta , \Lambda} Y_{\beta}^{\vert \Lambda \vert} }
{\ \ \qquad} & & \\
 \exp\{ -\frac{1}{2}
\sum_{\lambda ,\lambda' \in \Lambda} d_{\lambda \lambda'}^{\Lambda}
\sum_{k =2}^{d} \int_{I_\beta } \omega_{\lambda}^{(k)}(t)
\omega_{\lambda'}^{(k)} (t) dt  -
  \frac{a}{2}  \sum_{\lambda\in \Lambda}  \sum_{k=2}^{d }
\int_{ I_\beta }\left(\omega_{\lambda}^{(k)} (t) \right)^2 dt - {\qquad} & &  \\
\sum_{\lambda\in \Lambda}\sum_{s=1}^{r} b_s \int_{I_\beta}
\left(\sum_{k=2}^{d } (\omega_{\lambda}^{(k)} (t))^2 \right)^{s} dt \}
\bigotimes_{\lambda\in \Lambda}
\bigotimes_{k=2}^{d}\tilde{\chi}(d\omega_{\lambda}^{(k)} )
= {I\!\! E}_{\tilde{\mu}_{\beta, \Lambda}} \left\{\omega_l^{(1)} (\tau)
\omega_{l'}^{(1)} (\tau' )\right\}. & &
\end{eqnarray*}
Now we use the representation (\ref{na8}), (\ref{na9}) and the definitions
(\ref{1.13}), (\ref{new44}) and obtain the estimate (\ref{new5}). \Kasten







\section{Acknowledgments} It is a honor and pleasure to collaborate with
Sergio Albeverio that is gratefully acknowledged by the author.



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