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Gibbs state, Gibbs 
measure, non-ideal gas in $\reals^d$, multi-body interaction, 
uniqueness, existence, Dobrushin uniqueness condition
---------------0105211031825
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%%%%%%%%%%%%%  notations

\newcommand{\Z}{{\mathbb Z}}
\newcommand{\naturals}{{\mathbb N}}
\newcommand{\reals}{{\mathbb R}}
\newcommand{\mo}{{\rm I\!\!E}}
\newcommand{\pr}{{\rm I\!\!P}}
\newcommand{\given}{\, \big|\,}
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\def\cP{{\cal P}}
\def\cU{{\cal U}}
\def\ozna{c}
\newcommand{\sign}{{\mathrm sign}}
\newcommand{\diam}{{\rm diam}}
\newcommand{\half}{{\scriptstyle\frac{1}{2}}}
\def\Bphi{\overline{\Phi}}
\def\raz{{1\!\!{\rm I}}}
\def\bX{{\mathbf X}}
\def\bN{\ovl{N}}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{assertion}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\T}{\mathbf{T}}
\newcommand{\ds}{\,\nu({\rm d}\x)}
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\renewcommand{\epsilon}{\varepsilon}


\begin{document}

%%%%%%%%%%%%%  title
\title{Uniqueness of Gibbs State for Non-Ideal Gas in
${\bf R}^d$: the Case of Multibody Interaction}
\author{V. Belitsky${}^\dagger$\and E. A. Pechersky${}^\ddagger$
\thanks{The work was partly
supported by the Russian Foundation for Basic Research,
French-Russian Institute, CRDF, FAPESP, CNPq}}

\date{}
\maketitle
 \centerline{${}^\dagger${\footnotesize Instituto de Matem\'atica e Estat\'\i stica
 da Universidade de S\~ao Paulo,}}
 \centerline{\footnotesize  Rua do Mat\~ao,
 1010, CEP 05508-900, S\~ao Paulo, SP, Brazil}
 \centerline{${}^\ddagger$
{\footnotesize Institute of Information Transmission Problems,
Russian AS,}}
 \centerline{\footnotesize 19, Bolshoi Karetnyi, GSP-4,
Moscow 101447, Russia}
%\centerline{\footnotesize /tex/pech/draft-04-04-01.tex}
%%%%%%%%%%%%%%%%%  introd

\begin{abstract}
We study the question of existence and uniqueness of non-ideal gas
in ${\mathbb R}^d$ with multi-body interactions among its
particles. For each $k$-tuple of the gas particles, $2\leq k\leq
m_0<\infty$, their interaction is represented by a potential
function $\Phi_k$ of a finite range.

We introduce a {\em stabilizing} potential function $\Phi_{k_0}$,
such that $\Phi(x_1, \ldots, x_{k_0})$ grows sufficiently fast,
when diam$\{x_1, \ldots, x_{k_0}\}$ shrinks to $0$. Our results
hold under the assumption that  at least one of the potential
functions is stabilizing, that causes a sufficiently strong
repulsive force due to the stabilizing potential. We prove that
{\it (i)} for any temperature  there exists at least one Gibbs
field, and {\it (ii)} there exists exactly one Gibbs filed $\xi$
at low enough temperature, such that ${\mathbb E} e^{\chi
|\xi_V|}<\infty$ for any real $\chi>0$  and any small volume $V$.
The proofs use the criterion of the uniqueness of Gibbs field in
non-compact case developed in (\cite{DP}), and the technique
employed in (\cite{PZh}) for studying a gas with pair interaction.
\end{abstract}

\noindent{\sl Key words and phrases:}\ \ Gibbs state, Gibbs
measure, non-ideal gas in $\reals^d$, multi-body interaction,
uniqueness, existence, Dobrushin uniqueness condition.

\section{Introduction}\label{introd}
This work is the second part of our studies of the uniqueness
region of non-ideal classical gas. The first part of this work,
published in \cite{PZh}, was devoted to the case of pair
interactions between gas particles. In the present paper, we
study the case of multi-body interactions.

Our main goal is to use the technique of Dobrushin's type for the
study of continuous statistical physics models. It is a far
different  approach with respect to the analysis employed in the
works \cite{R} and \cite{G}. Besides, the approach we use gives
rather explicit conditions for the uniqueness that is also one of
the aims of this work.

The method we use here, as well as in \cite{PZh}, amounts to
checking the conditions of a general theorem from \cite{DP}.
These conditions split in two groups: 1) the compactness
conditions; 2) the contraction conditions. The contraction
conditions are close to those from R.~Dobrushin's work
\cite{D70}. The difference is that the Dobrushin's contraction
conditions from \cite{D70} is normally checked for total spin
space while that from \cite{DP} is checked here only on a compact
sub-space of the whole spin space, although the spin space of a
considered model may  be non-compact. This difference is an
essential advantage of the condition from \cite{DP} over that from
\cite{D70} since  it allows us to get uniform estimates that we
need for our argument and that would not available, if were
verified on a non-compact space. We note that the theorem from
\cite{DP} that we use has been proved for lattice models.
Therefore, to apply it to gas models in $\reals^d$, we have to
map the studied continuous model to an equivalent lattice model.
We do so via partitioning $\reals^d$ into small cubes and
considering all particle configurations in every cube as the spin
space at an appropriate site of $\Z^d$.

In this work, we follow the plan of \cite{PZh}. However, the
verification of the uniqueness conditions for a multi-body
interaction case, compared to that for a pair interaction one,
requires some extension of the conditions on potential functions,
and accordingly, some new computations. The reason of the changes
needed to treat the multi-body interaction case is due to the
follow simple fact. We recall that the condition we check is
physically speaking, a repulsiveness of the particles at small
distances. Assume that $\Phi_{m_0}, m_0>2$, is the potential
function giving the energy of $m_0$-particle interactions. Assume
also that $\Phi_{m_0}$ takes negative values, bounded from below.
Then $m_0$ particles attract each  other. If there are $N$
particles in a small volume then there exist $C_N^{m_0}$ groups,
each having $m_0$ particles and giving its contribution to the
attraction. A repulsive potential is needed. Let $\Phi_2$ be the
repulsive potential. Then, there are $C_N^2$ pairs of the
particles in the same volume, each one having repulsive energy.
The number $C_{N}^{2}$ of the pairs of particles and the number
$C_N^{m_0}$ of the groups of $m_0$ particles in a volume
containing $N$ particles,  have different asymptotic as
$N\rightarrow \infty$; namely, $C_{N}^{2}/C_N^{m_0}\rightarrow 0$.
To prevent a collapse of infinite number of particles in a bounded
volume caused by the predominance of the attraction, the repulsive
potential $\Phi_2$ must have a special property that would
compensate this difference. Namely, it must grow with
approximation \vv{is it good term?}of particle positions.
Therefore, we introduce a stabilizing potential function
$\Phi_{k_{0}}$ which gives the main repulsive energy of the model.
We study the case when the multi-bodiness  is bounded from above
by some integer $m_0$. As we shall see, in the case $k_0<m_0$, the
stabilization is ensured, when  the stabilizing potential
$\Phi_{k_0}$ has a singularity at the diagonal of
$\left(\reals^{d}\right)^{k_0}$ and grows sufficiently fast nearby
the diagonal. However, in the case $k_0=m_0$, it is enough that
$\Phi_{k_0}$ is large, although finite, at the diagonal.

Our conditions imply also the existence of at least one Gibbs
field for any temperature (Theorem  \ref{super}). It is worth to
remark that the existence is proved under the same conditions
that provide the uniqueness. This existence result was not proved
in \cite{PZh}.

In Section~2, we give the assumptions on the potential functions
and define the stabilizing potential function. Section~3 contains
the main results of the work.  We prove our results in Section~4.

%%%%%%%%%%%%%% model


\section{The model}\label{model}
The spin space $\cX$ is the set of all locally finite point
configurations on $\reals^d$.  The reference measure $\nu$ is the
non-normalized Poisson measure on $\cX$, that gives weight
$\frac{|V|^{N}}{N!}$ to the set of all configurations of $\cX$
having  $N$ particles in a bounded volume $V$. The detailed
descriptions of the measurable space $(\cX,\nu)$ is given in
\cite{PZh}, Section~1.1. Next we describe the assumptions on
potential functions $\Phi_k(x_1, \ldots, x_k):\,
\reals^{dk}\rightarrow \reals$, $k\geq 2$, determining
 of the system.
\begin{description}
\item[i] Finite group:
There exists an integer $m_0$ such that
$$\Phi_k(x_1, \ldots,
x_k)\equiv 0,\hbox{ for all }k>m_0,
$$
and
$$
\inf\Phi_{m_0}(x_1, \ldots, x_{m_0})< 0.
$$
\item[ii] Finite range:
Each potential function is of finite range, i.e., there exists a
constant $D>0$ such that for any $2\leq k\leq m_0$
\begin{align*}
\Phi_k(x_1, \ldots, x_k)=0, \hbox{ if }\diam\{x_1, \ldots, x_k\}>D.
\end{align*}
\item[iii] Symmetry: $\Phi_k(x_1, \ldots, x_k)=\Phi_k(x_{i_1}, \ldots,
x_{i_k})$, for any $k$, any $(x_1, \ldots, x_k)\in \reals^{dk}$, and any
permutation $(i_1, \ldots, i_k)$ of $(1, \ldots, k)$.
\item[iv] Translation invariance: $\Phi_k(x_1+z, \ldots, x_k+z)=
\Phi_k(x_1, \ldots, x_k)$, for any $k$, any\\ \noindent $(x_1,
\ldots, x_k)\in \reals^{dk}$, and any $z\in \reals^d$.
\item[v] Boundness from below: There exists a constant $M>0$ such that
$\Phi_k(x_1, \ldots, x_k)\geq-M$, for all $k$ and all $(x_1, \ldots,
x_k)\in \reals^{dk}$.
\item[vi] No hard core:  For every $k$,
the function $\Phi_k(x_1, \ldots, x_k)$ is continuous at all $(x_1,
\ldots, x_k)\in \reals^{dk}$, except maybe the case when
$x_1=x_2=\ldots=x_k$.
\end{description}
Since the functions $\Phi_k$ are continuous out every neighborhood
of the diagonals $\{x_1=...=x_k\}$ we shall assume that the constant
$M$ from \textbf{v} is such that for some $\delta_{0}>0$ and all $k$
$$
\sup_{\diam\{x_1, \ldots, x_k\}>\delta_0}\left| \Phi_k(x_1, \ldots,
x_k)\right|\leq M.
$$
A concrete value of $\delta_0$ will be chosen later, when we
introduce the stabilizing potential function.

As usual, the Hamiltonian of a finite configuration $\sigma\in\cX$
is \beq\label{hami} H(\sigma)=\sum_{k=2}^{m_0}\sum_{x_1, \ldots,
x_k\in \sigma}\Phi_k(x_1, \ldots, x_k). \eeq For two configurations
$\sigma$ and $\tau$ having no common particles, let
\beq\label{inen} F(\sigma,
\tau)=\sum_{k=2}^{m_0}\sum_{\stackrel{\stackrel {\scriptstyle
x_1\cdots x_r\in \sigma}{ y_1\cdots y_p\in \tau}}{p+r=k,\, p\geq
1,\, r\geq 1}} \Phi_{k}(x_1, \ldots, x_r, y_1, \ldots, y_p) \eeq be
the energy of the interaction between $\sigma$ and $\tau$.

Let $V\subseteq \reals^d$ and $\sigma\in \cX$ be such that $V$ is
bounded, $\sigma\subset V$, and let $\tau\subset V^c$ then the value
of the {\em conditional} Hamiltonian at $\sigma$ under the condition
$\tau$ is
\begin{equation}\label{ct}
H(\sigma\, |\, \tau)=H(\sigma)+F(\sigma, \tau).
\end{equation}
Observe that $F(\sigma, \tau)$ is finite even though the
configuration $\tau$ can be infinite because the potentials are of
finite range.
\bigskip



\textit{Remark }1. Due to assumption {\bf ii}, $F(\sigma, \tau)$
and, consequently, $H(\sigma\, |\,\tau)$ depends on $\tau$ through
solely its restriction to the $D$--neighborhood of $V$. With some
abuse of notations, we shall denote this restriction by the same
letter $\tau$ and $|\tau|$ will denote the number of particles in
this neighborhood. Further $\partial V$ denotes the
$D$--neighborhood of $V$.
\bigskip

The potential function $\Phi_{k_0}$ is said to be {\em stabilizing}
if
\begin{description}
\item[1] in the case $k_0<m_0$,
let $\delta_1>0$ and $A_{1}>0$  be such that
 \beq\label{delta}
\delta_{1}^{d(m_{0}-1)}e^{2\left(\frac{D}{\delta_1}+3\right)^d+1}M<
\frac{ A_{1}}{2^{m_{0}-k_{0}}k_{0}^{m_{0}}}
 \eeq
 then
 \beq\label{3t.0} \Phi_{k_0}(x_1, \ldots, x_{k_0})\geq
\frac{A_1}{r^{d(m_0-1)}}\hbox{ if } \diam\{x_1,\ldots, x_{k_0}\}=
r\leq \delta_1. \eeq
\item[2] in the case $k_0=m_0$, let $\delta_{2}>0$  and $A_{2}>0$
be such that
 \beq\label{3t.ab}
A_2\geq m_0^{m_0}M e^{2\left(\frac{D}{\delta_2}+3\right)^d+1}
 \eeq
 then
 \beq\label{3t.0a} \Phi_{k_0}(x_1, \ldots, x_{k_0})\geq A_2\hbox{ if }
\diam\{x_1,\ldots, x_{k_0}\}\leq \delta_2.
 \eeq
\end{description}


Further we shall use the notation
\begin{equation}\label{del}
\delta_{0}=\begin{cases}\delta_1,&\mbox{ if } k_0<m_0,\\
\delta_2,&\mbox{ if } k_0=m_0,
\end{cases}
\end{equation}
and
\begin{equation}\label{del1}
A_{0}=\begin{cases}A_1,&\mbox{ if } k_0<m_0,\\
A_2,&\mbox{ if } k_0=m_0.
\end{cases}
\end{equation}

To construct the specification by the conditional Hamiltonian
(\ref{ct}), the integral
 \beq\label{3t.1} \int_{\cX_V}\exp\{-\beta
H(\sigma\, |\, \tau)+\mu|\sigma|\}\nu(d\sigma),
 \eeq
 must be finite, where $\cX_V$ is the configuration set in $V$.
In (\ref{3t.1}) $V\subseteq \reals^d$ is a bounded volume and
$\mu\in \reals$. As we shall show the existence of the stabilizing
potential function provides the finiteness of (\ref{3t.1}) for
finite volumes $V$.

%%%%%%%%%%%%%%%%%%  main-results

\section{Main Results}\label{main-results}

\begin{theorem}\label{existsz} Let {\rm\textbf{i}--\textbf{vi}} be satisfied.
If there exists the stabilizing potential function, then for every
finite  volume  $V$  and every boundary condition $\tau$, the
integral (\ref{3t.1}) is finite.
\end{theorem}


As a consequence of Theorem \ref{existsz}, we can introduce a Gibbs
specification $\{P_{V, \tau}\, ;\, V\subseteq \reals^d, \, \tau\in
\cX_{V^c}\}$ such that the density $p_{V, \tau}$ of the measure
$P_{V, \tau}$ with the respect to $\nu$ is
 \beq\label{dens}
 p_{V,
\tau}(\sigma)=\frac{\exp\{-\beta H(\sigma\,
|\,\tau)+\mu|\sigma|\}} {\int_{\cX_V}\exp\{-\beta H(\sigma\,
|\,\tau)+\mu|\sigma|\}\nu(d\sigma)},
 \eeq
where $\mu\in \mathbb{R}$.

\begin{theorem}\label{super}
Let {\rm\textbf{i}--\textbf{vi}} be satisfied and  $\Phi_{k_0}$ be
the stabilizing potential function for some $k_{0}\leq m_{0}$.

Then for any $\mu\in \mathbb{R}$ and $\beta>0$ there exists at
least one Gibbs field corresponding to the specification
(\ref{dens}).
\end{theorem}


\begin{theorem}\label{uniq} Let the conditions {\rm\textbf{i}--\textbf{vi}}
be satisfied and  $\Phi_{k_0}$ be the stabilizing potential
function for some $k_{0}\leq m_{0}$. Then for every $\mu\in
\reals$ there exists $\beta(\mu)>0$ such that for any $\beta$,
$0\leq\beta\leq\beta(\mu)$, there exists the unique random field
$\xi$ corresponding to the Gibbs specification (\ref{dens}) and
satisfying the condition
\begin{equation}\label{ojid}
\mo e^{\chi |\xi_V|}<\infty
\end{equation}
for every $\chi>0$ and for $\diam(V)\leq \delta_{0}$, where
$\xi_{V}\in \cX_V$.
\end{theorem}

%%%%%%%%%%%%%%%%%%  proofs

\section{Proofs}\label{proofs}

\subsection{Proof of Theorem~\ref{existsz}.}
We start off introducing notations that will be used throughout. Let
\begin{equation}\label{nbar}
\bN:=\left\{\begin{array}{rl}k_0(1-1/\sqrt[d]{2})^{-d},&
\hbox{ if }k_0<m_0,\\
m_0, &\hbox{ if } k_0=m_0.\end{array}\right.
\end{equation}
and
\begin{equation}\label{aro}
\rho:=\frac{1}{2}\left(\frac{2\sqrt{d}D}{\delta_{0}}+3\right)^{-d}.
\end{equation}

 For a bounded volume $V$ and for a configuration $\tau\in
\cX_{V^c}$ ($|\tau|$ below should be understood in the sense of
Remark~1) we define
\begin{equation}\label{nt} N_{\tau}:=[\rho|\tau|]+1,
\end{equation}
and
\begin{equation}\label{nbart}
\bN_{\tau}:=\max\{\bN, N_{\tau}\}.
\end{equation}
We note that the choice of $\rho$ in (\ref{aro}) will be clear in
the Section~4.2.

Lemma~\ref{gvul} presented below will be frequently used in the
proofs  of Theorem~\ref{existsz},  Theorem~\ref{super} and
Theorem~\ref{uniq}.

\begin{lemma}\label{gvul} Let  potential functions satisfy
the assumptions of {\rm\textbf{i}--\textbf{vi}}, and let $k_0\leq
m_{0}$ be the index of its stabilizing potential function. Then
\newline {\bf (a)} for any finite $V\subset \reals^d$,
$\sigma=\emptyset$, and  for any $\tau\in \cX_{V^c}$, it holds
that $H(\sigma\, |\, \tau)=0$;
\newline {\bf (b)}
for any finite $V\subset \reals^d$, any $\sigma\in\cX_{V}$ with
$|\sigma|>0$, and any $\tau\in \cX_{V^c}$, it holds that
\begin{equation}\label{beza} H(\sigma\,
|\, \tau)\geq -M |\sigma|^{m_0}\exp\{1+|\tau|/|\sigma|\}
\end{equation}
{\bf (c)} for any $V\subset \reals^d$ satisfying
$\diam(V)<\delta_0$, for any $\tau\in \cX_{V^c}$ and any
$\sigma\in \cX_{V}$ satisfying $|\sigma|\geq \bN_{\tau}$, it
holds that
\begin{equation}\label{xop}
H(\sigma\, |\, \tau)\geq C_{0} |\sigma|^{m_0},
\end{equation}
where
\begin{equation}\label{ce} C_{0}=\begin{cases}
\frac{A_1}{2^{m_0-k_0}k_0^{m_0}\delta_{1}^{d(m_0-1)}}\,
-\, Me^{2\left(\frac{2\sqrt{d}D}{\delta_{1}}+3\right)^d+1},&\mbox{ if }k_{0}<m_{0}\\
\frac{A_2}{m_0^{m_0}}\, -\,
Me^{2\left(\frac{2\sqrt{d}D}{\delta_{2}}+3\right)^d+1},&\mbox{ if
}k_{0}=m_{0},
\end{cases}
\end{equation}
\end{lemma}


The following two lemmas are used in the prove of the lemma
\ref{gvul}.

\begin{lemma}\label{comb} For any integers $N$ and  $k$ such that $N\geq k$
it holds that
$$
N^k/k!\geq C_N^k\geq N^k/k^k.
$$
\end{lemma}

\noindent{\it Proof.}\ \ The left inequality is obvious.
$C_{N}^{k}=\frac{N^{k}}{k^{k}}$ for $N=k$. Rewrite the right
inequality as
 \beq\label{d2} \left(1-\frac{1}{N}\right)\ldots
\left(1-\frac{k-1}{N}\right)\geq \frac{k!}{k^k}.
 \eeq
 It is clear
that the left hand side of (\ref{d2}) is increasing with $N$. Since
(\ref{d2}) holds for $N=k$ then it holds for $N>k$. \hfill$\Box$

\begin{lemma}\label{vova}
For arbitrary integers $P\geq 1$ and $k\geq 2$ and  real $R\ge
\frac{k^{2}}{k-1}P$, let
\begin{equation*}
T(r_1,...,r_P)=\sum_{i=1}^P\mathbf{1}(r_i\ge k)r_i^k
\end{equation*}
be a function of $(r_1,...,r_P)\in\Delta$, where
$\Delta=\reals^P_+ \cap
\left\{(r_1,...,r_P):\:\sum_{i=1}^Pr_i=R\right\}$ and
$\mathbf{1}(A)$ is the indicator of $A$. Then
\begin{equation}\label{vo1}
\inf_\Delta T(r_1,...,r_P)=T\left(\frac{R}{P},...,\frac{R}{P}\right)
\end{equation}
\end{lemma}
\noindent {\it Proof}. It is clear that
\begin{equation}\label{vo2}
T\left(\frac{R}{P},...,\frac{R}{P}\right)=\left(\frac{R}{P}\right)^{k}P
=\inf\left\{T(r_1,...,r_P):\:\sum r_{i}=R, r_{i}\geq k\mbox{ for
all }i=1,...,P\right\}.
\end{equation}
Assume next that some of $r_{i}$ are less than $k$. Let us for
$i=P-p+1,...,P$ take $r_{i}<k$, where $p$ is an integer less than
$P$. Then
\begin{multline}\label{vo3}
T\left(\frac{R-kp}{P-p},...,\frac{R-kp}{P-p},r_{P-p+1},...,r_{P}\right)=
\left(\frac{R-kp}{P-p}\right)^{k}(P-p)
=\\\inf\left\{T(r_1,...,r_P):\:\sum r_{i}=R, r_{i}\geq k\mbox{ for
}i\leq P-p, r_{i}<k\mbox{ for }i>P-p\right\}.
\end{multline}
The value in (\ref{vo3}) is greater than one in (\ref{vo2}) since
\begin{equation*}
R\geq \frac{k^{2}}{k-1}P\geq
\frac{kp}{1-\left(1-\frac{p}{P}\right)^{1-\frac{1}{k}}}
\end{equation*}
for any $1\leq p\leq P-1$. \qed

\bigskip
\noindent{\it Proof of Lemma~\ref{gvul}}.

Item ({\bf a}) is evident.

Item (\textbf{b}). Let $V$, $\sigma\in \cX_V$, $|\sigma|\geq 1$,
and $\tau\in \cX_{V^c}$ be arbitrary. Then, using {\bf v} and
Lemma~\ref{comb} we obtain that for any $k=2,3,\ldots, m_0$,
\begin{gather}
\sum_{\stackrel{\scriptstyle \sigma^\prime\subseteq \sigma}
{|\sigma^\prime|=k}} \Phi_k(\sigma^\prime)\geq -MC_{|\sigma|}^k\geq
-M\frac{|\sigma|^k}{k!},\label{8t.1}\\
 \sum_{\stackrel{\scriptstyle \sigma^\prime\subseteq \sigma,\,
\tau^\prime\subseteq \tau} {|\sigma^\prime|+|\tau^\prime|=k,\hbox{
}|\sigma^\prime|\geq 1,\, |\tau^\prime|\geq 1}}\kern-2em
\Phi_k(\sigma^\prime\cup\tau^\prime) \geq -M\sum_{1\leq \ell\leq
k-1}C_{|\sigma|}^\ell C_{|\tau|}^{k-\ell}\geq -M\sum_{1\leq
\ell\leq k-1} \frac{|\sigma|^\ell}{\ell
!}\frac{|\tau|^{k-\ell}}{(k-\ell)!}.\label{8t.2}
\end{gather}
Thus
\begin{equation}\label{rew}
H(\sigma|\tau)\geq - M|\sigma|^{m_0}\left[
\sum_{k=2}^{m_0}\frac{1}{|\sigma|^{m_0-k}k!}+\sum_{m=1}^{m_0-1}
\left(\frac{|\tau|}{|\sigma|}\right)^m\frac{1}{m!}\sum_{k=m+1}^{m_0}
\frac{1}{|\sigma|^{m_0-k}(k-m)!}\right].
\end{equation}
Since $|\sigma|>0$ we obtain from (\ref{rew})
\begin{equation}\label{tr}
H(\sigma|\tau)\geq -M |\sigma|^{m_0}
e\left(1+\sum_{m=1}^{\infty}(m!)^{-1}(|\tau|/|\sigma|)^m\right)= -M
|\sigma|^{m_0}\exp\{1+|\tau|/|\sigma|\}
\end{equation}
that proves the assertion {(\bf b)} of the lemma.

We shall prove ({\bf c}) for the volume $V=G$, where $G$ is a cube
with its edge size $\delta_{0}/\sqrt d$.  It is clear that having
the result for $G$ we have it for any volume $V$ that can be
surrounded by a cube with its diameter less than or equal to
$\delta_{0}$. We prove  ({\bf c}) treating the cases $k_0<m_0$ and
$k_0=m_0$ separately.

Let $k_0<m_0$. Let us pick a sequence of integers $P=P_{N}$ that
satisfies the following two relations:
\begin{equation}\label{nap} \frac{1}{2k_0}N\leq P_{N}\leq
\frac{1}{k_0}N, \kern1em\hbox{ and }\kern1em P_{N}=\ell^d\hbox{ for
some }\ell\in \Z^+.
\end{equation}
The left hand side of the inequality holds for $N\geq \bN\equiv
k_0(1-1/\sqrt[d]{2})^{-d}$ (we could take a smaller  value of
$\bN$ and enlarge appropriately the coefficient 2 in the
denominator of the fraction in the left hand side of (\ref{nap})).
For every $\sigma$ we consider $G$ represented as a union of
$P_{|\sigma|}$ non-intersecting small cubes, each one with the
edge length
\begin{equation}\label{edge}
r=\frac{\delta_1}{\sqrt{d} P^{\frac{1}{d}}_{|\sigma|}}.
\end{equation}
Let   the small cubes of the partition of $G$ be enumerated from 1
to $P=P_{|\sigma|}$ and let $\sigma_i$ denote the sub-configuration
of $\sigma$ in the $i$-th cube. Then
 \beq\label{7t.1}
\sum_{\stackrel{\scriptstyle \sigma^\prime\subseteq
\sigma}{|\sigma^\prime|=k_0}}\Phi_{k_0}(\sigma^\prime)\geq
\sum_{i=1}^P\sum_{\stackrel{\scriptstyle \sigma^\prime\subseteq
\sigma_i} {|\sigma^\prime|=k_0}} \Phi_{k_0}(\sigma^\prime)\geq
\Bphi_{k_0}(r\sqrt{d})\sum_{i=1}^PC_{|\sigma_i|}^{k_0},
 \eeq
 where
 \beq\label{phibar} \Bphi_{k_0}(r\sqrt{d})=\inf\left\{\Phi_{k_0}(x_1,
\ldots, x_{k_0})\, :\, \diam\{x_1, \ldots, x_{k_0}\}\leq
r\sqrt{d}\right\}.
 \eeq
 We now apply Lemma~\ref{vova}, and after (\ref{3t.0}), (\ref{edge})
and the left hand side of (\ref{nap}) to obtain that the right hand
side of (\ref{7t.1}) is greater than
\begin{equation}\label{7t.2}
\Bphi_{k_0}(r\sqrt{d})\frac{1}{k_0^{k_0}}\frac{|\sigma|^{k_0}}{P^{k_0}}P
\geq \frac{A_1 |\sigma|^{k_0} P^{m_0-k_0}
}{{k_0}^{k_0}\delta_1^{d(m_0-1)}} \geq \frac{A_1 |\sigma|^{m_0}}
{2^{m_0-k_0}{k_0}^{m_0}\delta_1^{d(m_0-1)}}.
\end{equation}
Gathering (\ref{8t.1}), (\ref{8t.2}), (\ref{7t.1}) and
(\ref{7t.2}) and employing the arguments used for (\ref{tr}) give
the following relations
\begin{eqnarray}
H(\sigma\, |\,\tau)&\geq &
 \frac{A_1|\sigma|^{m_0}}{2^{m_0-k_0}k_0^{m_0}\delta_1^{d(m_0-1)}}
\, -\, M\sum_{k=2}^{m_0}\frac{|\sigma|^k}{k!}\,
-\,M\sum_{m=1}^{m_0-1}\frac{|\tau|^m}{m!}
\sum_{k=m+1}^{m_0}\frac{|\sigma|^{k-m}}{(k-m)!}\nonumber\\
&\geq & |\sigma|^{m_0}\left\{
\frac{A_1}{2^{m_0-k_0}k_0^{m_0}\delta_1^{d(m_0-1)}}
-M\exp\left\{1+\frac{|\tau|}{|\sigma|}\right\} \right\}  \label{mia}
\end{eqnarray}
Since $|\sigma|\geq \bN_{\tau}$ we can use (\ref{nt}), (\ref{aro})
and obvious estimates to conclude that
\begin{equation}\label{posi}
H(\sigma\, |\,\tau) \geq
|\sigma|^{m_0}\left(\frac{A_1}{2^{m_0-k_0}k_0^{m_0}\delta_{1}^{d(m_0-1)}}\,
-\, Me^{2\left(\frac{2\sqrt{d}D}{\delta_{1}}+3\right)^d+1}\right) =
C_{0}|\sigma|^{m_0},
\end{equation}
where $C_{0}$ is defined in (\ref{ce}).

We next prove ({\bf c}) in the case $k_0=m_0$. The procedure is
similar to that employed for the case $k_0<m_0$. However, now we
do not divide the cube $G$ into small ones. Therefore, the
counterpart of (\ref{7t.1}) and (\ref{7t.2}) are:
\begin{equation}\label{re}
\sum_{\stackrel{\scriptstyle \sigma^\prime\subseteq
\sigma}{|\sigma^\prime|=m_0}}\Phi_{m_0}(\sigma^\prime)\geq
\Bphi_{m_0}\left(\delta_2\right)C_{|\sigma|}^{m_0} \geq
A_2\frac{|\sigma|^{m_0}}{m_0^{m_0}},
\end{equation}
where $\Bphi$ has been defined in (\ref{phibar}). In the first
passage of (\ref{re}), we used that $|\sigma|\geq m_0$, since
$|\sigma|\geq \bN$, and in the last one, we used (\ref{3t.0a}) and
Lemma~\ref{comb}. The rest of the argument is identical to that for
the case $k_0<m_0$. We obtain
\begin{equation}\label{posit}
H(\sigma\, |\,\tau) \geq |\sigma|^{m_0}\left(\frac{A_2}{m_0^{m_0}}\,
-\, Me^{2\left(\frac{2\sqrt{d}D}{\delta_{2}}+3\right)^d+1}\right) =
C_{0}|\sigma|^{m_0},
\end{equation}
(see (\ref{ce})). \hfill$\Box$

\noindent{\it Remark 2}. Now we can prove the existence of the
integral (\ref{3t.1}) for  small volumes with diameter less than
$\delta_{0}$. For fixed $V$ with $\diam(V)\leq \delta_{0}$ and
$\tau$ on $V^{c}$, define $\cX_V^N:=\{\sigma\in \cX_V\, :\,
|\sigma|=N\},\; N\in \naturals$. Then, the partition function
(\ref{3t.1}) is
\begin{equation}\label{nado}
Z_{V, \tau}=\sum_{N=0}^\infty \int_{\cX_V^N}e^{-\beta H(\sigma\,|\,
\tau)+\mu|\sigma|}\nu(d\sigma).
\end{equation}
Using ({\bf c}) of Lemma~\ref{gvul}, each term with $N\geq
\bN_{\tau}$ may be estimated from above by $e^{\mu N-\beta
C_{0}N^{m_0}}\int_{\cX_V^N}\nu(d\sigma)= e^{\mu N-\beta
C_{0}N^{m_0}} |V|^{N}/N!$. Thus, (\ref{nado}) is a converging
series.

\medskip
We prove Theorem \ref{existsz} proving in the next lemma the
finiteness of (\ref{3t.1}) for any bounded volume $V$. Lemma
\ref{gvul} is essential in this proof.
\begin{lemma}\label{bigv}
For any bounded volume $V$ and the boundary configuration $\tau$ on
$V^{c}$ the following inequality
\begin{equation}\label{1m}
H(\sigma|\tau)\geq 0
\end{equation}
holds for $|\sigma|$ large enough.
\end{lemma}
\noindent\textit{Proof}. Represent $\mathbb{R}^{d}$ as the union
of small non-intersected cubes $G_{t}$ with the diagonal length
equal to $\delta_{0}$. We enumerate the small cubes by points $t$
of $\mathbb{Z}^{d}$. Let $V$ be a cube in $\mathbb{R}^{d}$ such
that it can be represented as the union of the small cubes
$G_{t},\;t\in U\subseteq\mathbb{Z}^{d}$. We assume also that
\begin{equation*}
U=\{t\in \mathbb{Z}^{d}:\: |t|\leq \ell\},
\end{equation*}
where $\ell$ is an integer. If $\sigma$ is a configuration on $V$
then $\sigma_{t},\;t\in U$, is the subconfiguration localized in
$G_{t}\subset V$. The neighborhood $\partial t$ of
$t\in\mathbb{Z}^{d}$ is
\begin{equation}\label{1d11}
\partial t=\{u:\:G_{u}\cap\partial G_{t}\neq\emptyset\},
\end{equation}
where $\partial G_{t}$ is taken in the sense of Remark 1. Further
we use the notation
\begin{equation}\label{1d12}
a=|\partial t|.
\end{equation}
It follows from (\ref{aro}) that $a\rho\leq \frac{1}{2}$.



Let $L_0=\max\{N_s:\:s\in U\}$, where $N_{s}=|\sigma_{s}|$, and
$L_{i}=L_{0}-i,\;i\geq 0$. Assume that $|\sigma|\geq \ovl{N}|U|$
then we consider a sequence $U_{i}$ of subsets in $U$:
\begin{equation*}
  U_{i}=\{t\in U:\: N_{t}=L_{i}\}.
\end{equation*}
For every $i$ we enumerate points in $U_{i}$ by the integers. For
$t\in U_{i}$ let $n_{i}(t)$ be the number of $t$. The sequence
$U^{j}_{i},\;j\geq 0$, of subsets of $U_{i}$ is determined as
\begin{equation*}
U_{i}^{j}=\{t\in U_{i}:\:n_{i}(t)>j\}.
\end{equation*}
Clear, $U_{i}^{0}=U_{i}$. We build next a sequence of
configurations $\sigma^{i,j}$  as the following.
\begin{equation*}\begin{array}{lll}
&\sigma^{0,0}&=\sigma, \\
&...&\\
\sigma_{G_t}^{i,j}=\begin{cases}\emptyset,&\mbox{ if }t\in \cup_{k=0}^{i-1}
U_k\cup (U_{i}\setminus U_{i}^{j}) \\
\sigma_{G_t}&\mbox{ otherwise }
\end{cases},
&\sigma^{i,j}&=\cup_{t\in U}\sigma_{G_{t}}^{i,j},\\
&...&
\end{array}\phantom{aaaaaaaaaaaa}
\end{equation*}
The sequence composition continues until $L_i>\ovl{N}$.

The following property of the sequence is important for the
estimate of $H(\sigma)$. Recall that
$H(\sigma_{G_t}|\sigma_{\partial G_t})=H(\sigma_{G_t})+ \sum_{u\in
\partial t}F(\sigma_t,\sigma_u)$, where $\partial t$ is defined in
(\ref{1d11}). The property satisfied for  any $t\in U_i^{j}$ is
\begin{equation}\label{1d1}
H(\sigma_{G_t}^{i,j}|\sigma_{\partial G_t}^{i,j})\geq
C_0L_i^{m_0}.
\end{equation}
This fact follows from Lemma (\ref{gvul})({\bf c})  since the
condition $|\sigma_{G_t}^{i,j}|>[\rho|\sigma_{\partial
G_t}^{i,j}|]+1$ holds if $t\in U_i^{j}$. Recall that
$|\sigma_{G_t}^{i,j}|=L_{i}>\bN$ if $j=n_{i}(t)$.

Let $W_i^{j}=\cup_{k=0}^{i-1}U_k\cup (U_{i}\setminus U_{i}^{j})$,
$W=\cup_iU_i$ and $\ovl{U}=\{t:\: N_t\leq \ovl{N}\}$. For
$T\subseteq U$ let $\wh{T}=\cup_{s\in T}G_{s}\subseteq V$. We
calculate next $H(\sigma_{\wh{W}}|\sigma_{\partial \wh{W}})$. We
have
\begin{multline}\label{1d2}
H(\sigma_{\wh{W}}|\sigma_{\partial \wh{W}})=\sum_{t\in
W}H(\sigma_{G_t})+\sum_{u,s\in W}F(\sigma_{G_u},\sigma_{G_s})
+\sum_{u\in W,s\in \partial W}F(\sigma_{G_u},\sigma_{G_s})=\\
\sum_{i\geq 0}\mathop{{\sum}^{o}}_{{n_{i}(t)\geq 1\atop t\in
U_i}}\left[ H(\sigma_{G_t}) +\sum_{u\in\partial t\setminus
W_i^{n_{i}(t)}} F(\sigma_{G_t},\sigma_{G_u}) \right],
\end{multline}
where $\mathop{{\sum}^{o}}$ means that the sum is taken over $t$
such that $n(t)$ is increasing. Remark now that
$\sum_{u\in\partial t\setminus {W}_i^{n_{i}(t)}}
F(\sigma_{G_t},\sigma_{G_u})=\sum_{u\in\partial t}
F(\sigma_{G_t}^{i,n_{i}(t)},\sigma_{G_u}^{i,n_{i}(t)})$. Thus
\begin{equation}\label{1d3}
H(\sigma_{\wh{W}}|\sigma_{\partial \wh{W}})= \sum_{i\geq
0}\sum_{t\in U_i}\left[ H(\sigma_{G_t}^{i,n_{i}(t)})
+\sum_{u\in\partial t}
F(\sigma_{G_t}^{i,n_{i}(t)},\sigma_{G_u}^{i,n_{i}(t)})
\right]\geq C_0\sum_iL_i^{m_0}|U_i|.
\end{equation}
Let $U^-=U\setminus (W\cup\partial W)$. If $U^-$ is not empty then
$N_t\leq \ovl{N}$ for any $t\in U^-$. We enumerate all sites from
$U^-$ in arbitrary order and denote  $U^-_k=\{t:\:n^{-}(t)\leq k
)\}$, where $n^{-}(t)$ is a number of $t$ in the enumeration. Using
the following relations for any $t,s \in U^-$:
\begin{eqnarray*}
H(\sigma_{G_t})\geq -Mm_0\ovl{N}^{m_0}\\
F(\sigma_{G_t},\sigma_{G_s})\geq -Mm_0\ovl{N}^{m_0}
\end{eqnarray*}
we obtain
\begin{multline}\label{2d1}
H(\sigma_{\wh{U}^-})=\mathop{{\sum}^{o}}_{t\in
U^-}\left[H(\sigma_{G_{t}})+ \sum_{s\in\partial t\setminus
(U^-_{n(t)-1}\cup W)}F(\sigma_{G_{t}},\sigma_{G_s})\right]
\geq-Mm_0|U^-|\ovl{N}^{m_0}(1+a).
\end{multline}
The total energy of the $\sigma$ can be estimated now
\begin{multline}\label{2d2}
H(\sigma_V)=H(\sigma_{\wh{W}}|\sigma_{\partial
\wh{W}})+H(\sigma_{\wh{U}^-}) \geq
C_0\sum_iL_i^{m_0}|U_i|-Mm_0|U^-|\ovl{N}^{m_0}(1+a)
\end{multline}
Now it is clear that $H(\sigma_{V})\geq 0$ if $|\sigma|$ is large.

The proof that we have works for the case $\tau=\emptyset$. It is
clear that for non-empty $\tau\in\cX_{V^{c}}$ the proof can be
done in the similar way. In this case the value of $\ovl{N}$
depends on $\tau$, namely,
$\ovl{N}=\max\{\ovl{N},N_{s},\;s\in\partial U\}$. \qed
\vspace{1cm}

\subsection{Proof of Theorem \ref{super} and \ref{uniq}}
The proofs are based on general theorems for the existence from
\cite{D70} and for the uniqueness  from \cite{DP} (the formulation
of the latter may be also found in \cite{PZh}). These theorems
work for lattice models, therefore, as in Subsection~4.1 we
partition $\reals^d$ into cubes $\{G_t,\, t\in \Z^d\}$ such that
the diagonal length of each cube is equal to $\delta_{0}$. All
cubes are equivalent in the sense that $G_{t}=G_{0}+t$. The spin
space $\bX$ is the set of particle configurations in $G_0$
endowed with a metric such that it becomes a Polish space (see
\cite{PZh}). The models, continuous and lattice, are equivalent.
In fact, $\bX$ becomes a Polish space if $G_{0}$ is a closed cube
and so $G_{t}$ are. Then adjacent cubes have common faces.
However, the lattice model with configurations on closed cubes is
still equivalent to the original continuous one (see details in
\cite{PZh}). Thus the existence and the uniqueness of the lattice
model implies the existence and the uniqueness of the continuous
one. Further $G_{0}$ means the closed cube.

The neighborhood of a site $t$ is described in (\ref{1d11}) (see
also (\ref{1d12})).

Further we say '$\x$ is a configuration in $G_{t}$' in the sense
that $\x$ is an element of $\bX$, or, in other words, $\x$ is a
configuration in $G_{0}$ which is assigned to the site
$t\in\mathbb{Z}^{d}$. We now define the specification of the
corresponding lattice model. Let $t\in \Z^d$ and $\ovl{\y}$ be a
configuration of the lattice model, $\ovl{\y}\in
\bX^{\Z^d\setminus\{t\}}$. The Hamiltonian $H(\x|\ovl{\y})$ is
then constructed as the energy of the interactions between the
particles from $\x\in G_{t}$ and between the particles from $\x$
and from $\ovl{\y}$. Then the distribution density with respect
to $\nu$ on $\bX$, which is, in fact, a distribution on the
configuration set in $G_t$, under the condition $\ovl{\y}$ is
 \beq\label{10t.1} P_{t,
\ovl{\y}}(\x)=Z^{-1}_{\ovl{\y}}\exp\left\{-\beta\, H(\x\, |\,
\ovl{\y})+\mu|\x| \right\}
 \eeq
(for details, see \cite{PZh}). It follows from Theorem
\ref{existsz} that we can define a distribution $P_{U,\ovl{\y}}$
on $\wh{U}=\cup_{s\in U}G_{s}$ for finite $U\subset\mathbb{Z}^{d}$
under a condition $\ovl{\y}$ out of $\wh{U}$.


\subsubsection{Proof of Theorem~\ref{super}}
In the first step we check the \textit{compactness} condition that
will be used to establish both the existence and the uniqueness.
Actually the compactness condition for the uniqueness from
\cite{DP} is slightly stronger than that for the existence from
\cite{D70}. In the next lemma we prove the stronger version of
the condition.

\begin{lemma}\label{compc} For any   $\chi>0$, $t\in \Z^d$ and
$\ovl{\y}\in \bX^{\Z^d\setminus\{t\}}$, there exist $C>0$ and
$c_u\geq 0, u\in \partial t$, such that
 \beq\label{10t.2} \mo_{\ovl{\y}} e^{\chi|\x|}=\int_{\bX}
e^{\chi |\x|}P_{t, \ovl{\y}}(d\x)\leq C+\sum_{u\in
\partial t}c_u e^{\chi|\y(u)|}
 \eeq
  and $\sum_{u\in \partial t}c_u$
is small enough.
\end{lemma}

\noindent \textit{Remark 3}. In order to prove the existence it
is sufficient that (\ref{10t.2}) holds with  $\sum_{u\in\partial
t}c_u<1$. However for the needs of the proof of uniqueness, the
sum in the above inequality must be less than a certain number
determined by some specific characteristics of the potential
functions (see \cite{DP}). Of course, this number is less than 1.

\bigskip

\noindent{\it Proof}.\ \ We represent the space
$\bX=\cup_{N=0}^\infty \bX^N$ as the union of the subspaces
$\bX^N$ such that  configurations $\x\in \bX^N$ have the fixed
particle number: $|\x|=N$. Then,
\begin{eqnarray}
\mo_{\ovl{\y}} e^{\chi|\x|}& = &
\sum_{N=0}^\infty e^{\chi N}\Pr\left(|\x|=N\right)\nonumber\\
& = &
\left(\sum_{N=0}^{N_{\ovl{\y}}}+\sum_{N=N_{\ovl{\y}}+1}^{\bN}+
\sum_{N=\bN+1}^\infty\right) e^{\chi N}\Pr\left(|\x|=N\right)\label{11t.1}\\
&=:& J_1+J_2+J_3\nonumber
\end{eqnarray}
with the convention that $J_2=0$, if $\bN\leq N_{\ovl{\y}}$.

We shall show that
 \beq \label{12t.1} J_1\leq C_1+\sum_{u\in
\partial t}c_ue^{\chi |{\oy}(u)|},\kern1em J_2\leq C^\prime_1,
\kern1em J_3\leq C_1^{\prime \prime},
 \eeq
  and $C_{1},\kern1em C^\prime_1,
\kern1em  C_1^{\prime \prime}$ are independent on $\ovl{\y}$.
Recall that $|\ovl{\y}|$ denotes the number of particles of
$\ovl{\y}$ outside of $G_t$ and inside $\cup_{r\in G_t}B_D(r)$.

First, we shall estimate $J_1$. Since $\chi>0$ the sum $J_{1}$ is
less than $e^{\chi N_{\oy}}\leq e^{\chi \rho|\oy|+\chi}$ (see
(\ref{nt})). Applying then the obvious inequality
 \beq\label{nera}
\prod_{i=1}^n x_i\leq \frac{1}{n}\sum_{i=1}^nx_i^n, \kern1em
 n\in \naturals,\kern1em  x_1,\ldots, x_n\geq 0
 \eeq
  to $e^{\chi \rho|\oy|}=\Pi_{i=1}^n e^{\chi \rho |\y(u)|}$,
where $\oy(u)$ is the part of $\oy$ in $G_u,\,u\in\partial t$, we
obtain
 \beq\label{sir} e^{\chi}\prod_{u\in \partial t}e^{\chi
\rho|\oy(u)|}\leq \frac{e^{\chi}}{a}\sum_{u\in
\partial t}e^{\chi
\rho a |\oy(u)|}\leq \frac{e^{\chi}}{2a}\sum_{u\in
\partial t}e^{2T}+\frac{e^{\chi}}{2a}\sum_{u\in \partial
t}e^{-2T}e^{2\chi\rho a|\oy(u)|}
 \eeq
for any $T>0$. Because of the convexity of the function $e^x$, we
obtain
 \beq\label{ota} J_1\leq e^{2T+\chi}/2+
e^{-2T+\chi}(1-2\rho a)/2+\sum_{u\in \partial t} e^{\chi
|\oy(u)|}\, e^{-2T+\chi}2\rho a.
 \eeq
The crucial point in the previous estimates is that $1-2\rho a\geq
0$ since
\begin{equation}\label{neig}
a\leq\left(\frac{2D\sqrt{d}}{\delta_{0}}+3\right)^{d}
=\frac{1}{2\rho},
\end{equation}
(see (\ref{aro})).

Denoting $C_1:= e^{2T+\chi}/2+ e^{-2T+\chi}(1-2\rho a)/2$ and
$c=c_u= e^{-2T+\chi}2\rho a$, we obtain the first inequality of
(\ref{12t.1}).

Note that both $C_1$ and $c_u$ do not depend on $\beta$, and that
we can make $\sum c_u=ca$ as small as desired by choosing $T$
appropriately large.

To estimate $J_{2}$ we use (\textbf{b}) of Lemma \ref{gvul}It
gives that
\begin{multline}\nonumber
J_{2}\leq\sum_{N=N_{\oy}+1}^{\bN}e^{(\chi+\mu)N}\exp\left\{\beta
MN^{m_{0}}e^{1+\frac{|\tau|}{N}}\right\}\frac{\delta^{dN}_{0}}{N!}\leq\\
\exp\left\{(\chi+\mu)\bN+\beta
M\bN^{m_{0}}e^{1+\frac{1}{\rho}}+\delta^{d}_{0}\right\}:=C_{1}^{\prime}.
\end{multline}
We assumed above that $N_{\oy}<\bN$, otherwise
$C_{1}^{\prime}=J_{2}=0$.

Finally, let us estimate $J_{3}$. For $|\x|>\bN$ we can apply
Lemma~\ref{gvul}, ({\bf c}), to conclude that $H(\x\, |\, \oy)\geq
0$. Hence, we have that
\begin{eqnarray}
J_3  & \leq & \sum_{N=\bN+1}^\infty \int_{\bX^N} e^{-\beta H(\x\,
|\,\oy)+(\chi+\mu)|\x|}\nu(d\x)\leq \sum_{N\geq
\bN+1}e^{(\mu+\chi)N}
\frac{(\delta^d_{0})^N}{N!}\nonumber\\
&\leq & \exp\{\delta_{0}^de^{\mu+\chi}\}-1=:C^{\prime\prime}_1.
\label{natan}
\end{eqnarray}
Remark that $C_{1}$ and $C_{1}^{\prime\prime}$ do not depend on
$\beta$. However,  $C_1^{\prime}$ depends on $\beta$ such that it
is not decreasing with $\beta$. We take
$C=C_{1}+C_1^{\prime}+C_1^{\prime\prime}$. \mbox{} \hfill$\Box$

\bigskip
The  Dobrushin's existence condition (see \cite{D70}) rests on the
lemma just proved and on continuity  of the specification with
respect to boundary configurations that will be established below.


\begin{lemma}\label{con}
Let \(\oy\) be a boundary configuration on
$\mathbb{Z}^d\backslash\{t\}$ and let \((\oy_n)\) be a sequence of
configurations on $\mathbb{Z}^d\backslash\{t\}$ that converges to
\(\oy\). Then the measures $P_{t,\oy_n}$ weakly converge to the
measure $P_{t,\oy}$.
\end{lemma}

\noindent{\it Proof.}\ Because the potential functions are
translation invariant, it is sufficient to treat solely the case
$t=0$. Therefore we omit the subscript $t$.

The weak convergence means that for any bounded continuous function
$\phi$ on $\bX$ the following convergence
 \beq\label{1L.1}
\int_{\bX}\phi(\x)P_{\oy_n}(d \x)\rightarrow
\int_{\bX}\phi(\x)P_{\oy}(d \x), \hbox { as } \oy_n\rightarrow \oy
 \eeq
holds. The convergence $\oy_n\to\oy$ means that
$\oy_n(u)\to\oy(u)$ for every $u\in \partial 0$. The convergence
$\oy_n(u)\rightarrow \oy(u)$ is understood in the sense of the
metric on $\bX$. According to this metric, $\oy_n(u)\rightarrow
\oy(u)$ implies the following. Let $B_\epsilon(y),\,y\in\oy(u)$,
be a $\varepsilon$--ball around $y$ such that no particles of
$\oy(u)$ belong to $B_\epsilon(y)\setminus\{y\}$. Then there
exists a sequence $y_n\in \oy_n(u)$ such that for $n$ large only
$y_n$ of $\oy_n(u)$ belongs to $B_\epsilon(y)$. It follows from
this fact and the continuity of the potential functions out of
diagonals that
$$
H(\x\, |\, \oy_n)\rightarrow  H(\x\, |\, \oy),
$$
almost everywhere with respect to Lesbegue measure in
$|G_{0}|^{|\x|}$. Hence,
 \beq\label{2L.1} \int_{\bX^N}\left| e^{-\beta H(\x\, |\,
\oy_n)}-e^{-\beta H(\x\, |\, \oy)}\right| \nu(d \x)\rightarrow 0,
\hbox{ for any }N\hbox{ as } \y_n\rightarrow \y.
 \eeq
Recall that $\x\in \bX^{N}$ if $|\x|=N$. It is not difficult to
obtain the inequality
\begin{multline} \label{2L.2}
R_n(\phi) =  \left|\int_{\bX}\phi(\x)P_{\oy_n}(d
\x)-\int_{\bX}\phi(\x)P_{\oy}(d \x)\right|\\
\leq  \sum_{N,
N^\prime}e^{\mu(N+N^\prime)}\int_{\bX^N}\int_{\bX^{N^\prime}}|\phi(\x)|\cdot
\left|e^{-\beta H(\x^\prime\, |\, \oy)-\beta H(\x\, |\, \oy_n)}-
e^{-\beta H(\x^\prime\, |\, \oy_n)-\beta H(\x\, |\,
\oy)}\right|\nu(d \x)\nu(d\x^{\prime})
\end{multline}
Let $\epsilon>0$ be small then we can find $N_\epsilon$ such that
the sum tail of the right hand side of (\ref{2L.2}) is less than or
equal to $\epsilon$:
$$
\sum_{N\geq N_\epsilon}\sum_{N^\prime\geq N_\epsilon}\leq\epsilon.
$$
Taking the limit along $n$ in (\ref{2L.2}) and using (\ref{2L.1}),
we obtain
$$
\lim_{n\to\infty} R_n(\phi)\leq \epsilon\qquad\mbox{\hfill\qed}
$$

\noindent{\it Proof of Theorem \ref{super}} is now a direct
consequence of Lemmas~\ref{compc} and \ref{con}.\qed.



\bigskip
\subsubsection{Proof of Theorem~\ref{uniq}}
In Lemma \ref{compc}, we proved the compactness condition of the
theorem from \cite{DP}. We next verify the \textit{contraction
condition} (see \cite{PZh}) that in our case, amounts to the
following requirement: If we take two different boundary
conditions $\oy_1$ and $\oy_2$ at a site $t\in \Z^d$, then the
variation distance $V(P_{t, \oy_1},P_{t, \oy_2})$ between two
distributions on ${\bf X}_t$ corresponding to $\oy_1$ and $\oy_2$
must be less than the discrete distance between $\oy_1$ and
$\oy_2$. This is the general Dobrushin condition of uniqueness,
which is here formulated for the case of variation distance
between measures. In our approach based on \cite{DP}, we verify
this condition only for a compact set of boundary configurations
$\oy$. According to \cite{DP} this compact set must be large
enough. We consider the compact sets of the type
$\{\sigma:\:e^{\chi|\sigma|}\leq K\}$. The compact set of this
kind is large if $K$ is large. There is no any estimate of $K$ in
\cite{DP}. Therefore we check the contraction condition for any
$K$. It gives us also the contraction condition with a large
compact set as it is necessary.
\begin{lemma}\label{l3}
For any positive integer $K_0$ there exists $\beta(K_0)>0$ such
that for any $\beta\leq\beta(K_0)$, any $t\in \mathbb Z^d$, and
any two configurations $\oy_1$ and $\oy_2$ from $\bX^{{\mathbb
Z}^d\setminus\{t\}}$ satisfying
 \beq\label{ogr}
\max_{u\in\partial t}\{|\oy_1(u)|,|\oy_2(u)|\}\le K_0,
 \eeq
 the following inequality holds:
\begin{equation}\label{V}
V(P_{t,\oy_1},P_{t,\oy_2})<\frac{1}{a}\sum_{u\in \partial
t}\raz_{\{\oy_1(u)\not =\oy_2(u)\}}.
\end{equation}
\end{lemma}
\noindent {\it Proof.}\ As in the lemma~\ref{con} we  perform all
our calculations for $t=0\in{\mathbb Z}^d$ and  omit $t$ from all
subscripts in this proof.

The following estimate for the variational distance may be obtained
straightforwardly. For any  $\oy_1, \oy_2\in\bX^{{\mathbb
Z}^d\setminus\{0\}}$,
\begin{eqnarray}
& &V(P_{\oy_1},P_{\oy_2}) = \frac{1}{2}\sum_{N=0}^{\infty}
\int_{\bX^N}
|p_{\oy_1}(\x)-p_{\oy_2}(\x)|\,\nu(d\x)\nonumber\\
& &\le \frac{1}{2}\sum_{N=0}^{\infty} e^{\mu N}\int_{\bX^N}
\Big|\frac{e^{-\beta H(\x|\oy_1)}}{Z_{\oy_1}}-
\frac{e^{-\beta H(\x|\oy_2)}}{Z_{\oy_2}}\Big|\ds\label{in2}\\
& &\leq \sum_{N=1}^{\infty} e^{\mu N}\int_{\bX^N} \big|e^{-\beta
H(\x|\oy_1)}-
e^{-\beta H(\x|\oy_2)}\big|\ds\nonumber\\
& &+\frac{1}{2}\sum_{N=1}^{\infty}\sum_{N'=1}^{\infty}
e^{\mu(N+N')}\int_{\bX^N} \int_{\bX^{N'}} \left|e^{-\beta
H(\x|\oy_1)-\beta H(\x'|\oy_2)}
-e^{-\beta H(\x|\oy_2)-\beta H(\x'|\oy_1)}\right|\times\nonumber\\
& & \kern18em \times \nu(d\x)\,\nu(d\x').\nonumber
\end{eqnarray}
where, to obtain the last inequality, we used that $Z_{\oy}\geq 1$
for any $\oy$.

First, we shall prove the lemma in the case
\begin{description}\item[A]
There exists $u_0\in \partial 0$  such that the particle
configuration $\oy_2(u_0)$
 is exactly the particle configuration $\oy_1(u_0)$  plus one ``extra''
particle whose coordinate will be denoted by $y_0$, and
$\oy_1(u)=\oy_2(u)$ for all $u\in\partial 0, u\ne u_0$.
\end{description}
Let
 \beq\label{razny} \Psi(\x):=H(\x|\oy_2)-H(\x|\oy_1).
  \eeq
  Plugging
(\ref{razny}) in (\ref{in2}), we get that
\begin{eqnarray}\label{11.3}
V\left(P_{\oy_1},P_{\oy_2}\right) & \leq & \sum_{N=1}^{\infty}e^{\mu
N}\int_{\bX^N} e^{-\beta H(\x|\oy_1)}\left|
1-\exp\{-\beta\Psi(\x)\}\right|
\ds\nonumber\\
& + &\sum_{N=1}^{\infty}\sum_{N'=1}^{\infty}
e^{\mu(N+N')}\int_{\bX^N}e^{-\beta H(\x|\oy_1)}\ds\label{11.3a}\\
& &\kern4em \times\int_{\bX^{N'}} e^{-\beta H(\x'|\oy_1)}\left|
1-\exp\{-\beta\Psi(\x')\}\right| \nu(d\x').\nonumber
\end{eqnarray}
Introducing
\begin{equation}\label{12.1}
Y_{\oy_1}(y_0)=\sum_{N'=1}^{\infty}\int_{\bX^{N'}} e^{-\beta
H(\x'|\oy_1)+\mu N'}\left| 1-\exp\{-\beta\Psi(\x')\}\right|
\nu(d\x'),
\end{equation}
we thus have from (\ref{11.3a}) that
 \beq\label{prieh}
V(P_{\oy_1},P_{\oy_2})\leq Y_{\oy_1}(y_0)Z_{\oy_1}.
 \eeq

Our programme is as follows: we shall found estimates on $Z_{\oy_1}$
and $Y_{\y_1}(y_0)$  that will show that, as $\beta\rightarrow 0$,
$Z_{\oy_1}$ remains bounded while $Y_{\oy_1}\rightarrow 0$. These
facts together with (\ref{prieh}) will imply the lemma assertion.

We now start an argument that will estimate $Z_{\oy_1}$. We note
that $|\oy_1|\leq aK_0$, according to (\ref{ogr}). This inequality,
together with the definitions of $N_{\oy_1}$ (see (\ref{nt})) and
the estimate (\ref{neig}) imply the inequality $N_{\oy_1}\leq K_0$
that will be used below. We represent the partition function in
the following form:
\begin{equation}\label{nac}
Z_{\oy_1} = 1+ \sum_{N=1}^{\bN_{\oy_1}}\int_{\bX^N}e^{-\beta
H(\x\, |\, \oy_1)+\mu |\x|} \nu(d\x)+
\sum_{N=\bN_{\oy_1}+1}^\infty\int_{\bX^N} e^{-\beta H(\x\, |\,
\oy_1)+\mu |\x|} \nu(d\x).
\end{equation}

Due to Lemma~\ref{gvul},  $H(\x\, |\,\oy_1)\geq
-MN^{m_0}e^{1+|\oy_1|/N}$ in each term of the first sum, while
$H(\x\, |\,\oy_1)\geq 0$ in each term of the second sum. We plug in
these estimates in (\ref{nac}) and obtain that
\begin{eqnarray}
Z_{\y_1} & \leq & 1+ \sum_{N=1}^{\max\{\bN, K_0\}} \exp\{\beta M
N^{m_0}e^{|\partial 0|K_0+1}+\mu N\} \frac{{\delta}^{dN}_{0}}{N!}+
\left(\exp\{\delta^d_{0} e^\mu\}-1\right)\label{last}
\end{eqnarray}


We now start an argument that will estimate $Y_{\oy_1}(y_0)$ from
above. It consists of two steps.

At the first step, we estimate
$\int_{\bX^{N}}|1-\exp\{-\beta\Psi(\x)\}|\nu(d\x)$. Let $L$ denote
the number of particles of $\oy_1$ in $\partial 0$ and let $y_0$ be
as determined in  {\bf A}. We define a set of multi-indices $R$ as
the collection of triples
\begin{equation}\label{6.1.1}
r=(k,\,\{i_1,i_2,\ldots,i_{k_1}\},\,\{j_1,\ldots,j_{k_2}\}),
\end{equation}
where the first element of each triple is an integer $k\in \{2,3,
\ldots, m_0\}$, the second one is a subset
$\{i_1,i_2,\ldots,i_{k_1}\}$ of $\{1, \ldots, N\}$, and the third
element is a subset $\{j_1,\ldots,j_{k_2}\}$ of $\{1, \ldots, L\}$
that satisfy the conditions $k_1\geq 1$, $k_2\geq 0$, and
$k_1+k_2=k-1$. We shall use the notation:
$$
\Phi_{k}((\x\oy)_{r})=\Phi_k(x_{i_1}, \ldots, x_{i_{k_1}}, y_0,
y_{j_1}, \ldots, y_{j_{k_2}}),
$$
where $r$ as in (\ref{6.1.1}), $x_{i_{1}},...,x_{i_{k_{1}}}\in \x$
and $ y_{j_{1}},...,y_{j_{k_{2}}}\in\cup_{u\in\partial
0}\oy_{1}(u)\setminus\{y_{0}\}$. Then (see (\ref{razny}))
\begin{equation}\label{vot}
\Psi(\x)= \sum_{r\in R}\Phi_k((\x\oy)_{r}),
\end{equation}
and
\begin{eqnarray}
\int_{\bX^{N}}\left| 1-\exp\{-\beta\Psi(\x)\}\right|\nu(d\x)
=\frac{1}{N!} \int_{{G}_0^{N}} \left|1-\exp\left\{-\beta \sum_{r\in
R}\Phi_k((\x\oy)_{r})\right\}\right| dx_1\ldots dx_N.\label{paw}
\end{eqnarray}
Using the inequality
\begin{equation}\label{2.0}
\left| 1-\exp\left\{a-\sum_{i=1}^n a_i\right\}\right|\le \left|
1-\exp\{a\}\right| +\sum_{i=1}^n\left| 1-\exp\{-a_i\}\right|,
\end{equation}
that holds for any $n\in \naturals$, and any $a, a_1, \ldots,
a_r\geq 0$, we get that the integral in the r.h.s. of (\ref{paw}) is
bounded from above  by the following expression:
\begin{eqnarray}
& &\int_{G_0^N}\left|1-\exp\left\{-\beta {\textstyle \sum_{r\in
R}}\Phi_k((\x\oy)_{r})  \raz_{\{ \Phi_k((\x\oy)_{r})< 0\}}(x_1,
\ldots, x_N)\right\}
               \right|dx_1\ldots dx_N\nonumber\\
                           &  &+
\sum_{r\in R}\int_{G_0^N}\left|1-\exp\left\{-\beta
\Phi_k((\x\oy)_{r})
   \raz_{\{ \Phi_k((\x\oy)_{r})\geq 0\}}(x_1, \ldots, x_N)\right\}
               \right|dx_1\ldots dx_N\nonumber\\
& & =:I_1+I_2,\label{rot}
\end{eqnarray}
The following inequality is obvious
\begin{multline*}
I_{1}\leq -\beta\int_{G_0^N}\sum_{r\in R}\Phi_k((\x\oy)_{r})
\raz_{\{ \Phi_k((\x\oy)_{r})< 0\}}(x_1, \ldots, x_N)\times\\
\exp\left\{-\beta {\textstyle \sum_{r\in
R}}\Phi_k((\x\oy)_{r})\raz_{\{ \Phi_k((\x\oy)_{r})< 0\}}(x_1,
\ldots, x_N)\right\}dx_1\ldots dx_N
\end{multline*}
Combining this estimate with the lemma \ref{gvul} (\textbf{b}) we
obtain that
\begin{equation}\label{okot}
I_1\leq \beta M m_0(N+L)^{m_0-1}\exp\{\beta M
m_0(N+L)^{m_0-1}\}(\delta_{0}/\sqrt{d})^{dN}.
\end{equation}

We next estimate $I_2$. The assumption {\bf v} on the potential
functions guarantees that when $u_1, \ldots, u_{k_1} \in {G_0}$
and $v_1, \ldots, v_{k_2}\in \ovl{\reals^d\setminus G_0}$ (the bar
means the closure) then
$\Phi_{k}(u_{1},...,u_{k_1},v_{1},...,v_{{k_2}})$  may take an
infinite value, only if $u_{1}=\ldots =u_{k_1}=v_{1}=\ldots
=v_{k_2}$. A necessary condition for that to occur, is that all
$u$'s belong to the boundary of $G_0$. Hence, the function
\begin{eqnarray}
& & F_k(u_1, \ldots, u_{k_1}):=\nonumber\\
& & \sup_{v_{1},...,v_{k_2}}
\left\{\left|1-e^{-\beta\Phi_{k}(u_{1},\ldots,
u_{k_1},v_{1},...,v_{k_2}) \raz_{\{\Phi_{k}(u_{1},\ldots,
u_{k_1},v_{1},...,v_{k_2})\geq 0\}}(u_1, \ldots, u_{k_1})}
\right|\,\right\}\label{samoe}
\end{eqnarray}
converges to $0$, as $\beta\rightarrow 0$, almost surely with
respect to Lebesgue measure on ${G_0}^{k_1}$. Since for any $r\in R$,
\begin{eqnarray}
& &\int_{G_0^N}\left|1-\exp\left\{-\beta \Phi_k((\x\oy)_{r})
   \raz_{\{ \Phi_k((\x\oy)_{r})\geq 0\}}(x_1, \ldots, x_N)\right\}
               \right|dx_1\ldots dx_N \nonumber\\
& &\leq\int_{G_0^{k_1}}F_k(x_1, \ldots, x_{k_1})dx_1\ldots
dx_{k_1}\, \int_{G_0^{N-k_1}}dx_{k_1+1}\ldots dx_{N},\label{te}
\end{eqnarray}
we have that for any $\varepsilon>0$ there exists
$\beta_{\varepsilon}>0$ such that for all $\beta\leq
\beta_{\epsilon}$
\begin{equation}\label{hava}
I_2\leq m_0(N+L)^{m_0-1}\epsilon\, \max\{1;
(\delta_{0}/\sqrt{d})^{dN}\}.
\end{equation}

We now combine  (\ref{okot}) and (\ref{hava}) and obtain that for
all $\beta\leq \beta_{\epsilon}$, the following estimate is valid:
\begin{eqnarray}
& &\int_{\bX^{N}}\left| 1-\exp\{-\beta\Psi(\x)\}\right|\ds
\nonumber\\
& &\leq \frac{1}{N!}\left[ \beta M
m_0(N+L)^{m_0-1}\exp\{\beta M m_0(N+L)^{m_0-1}\}
(\delta_{0}/\sqrt{d})^{dN}\right.\nonumber\\
& &\kern6em\left.+m_0(N+L)^{m_0-1}\epsilon \,
\max\{1; (\delta_{0}/\sqrt{d})^{dN}\}\right]\nonumber\\
& &\leq (\beta+\epsilon)\frac{1}{N!}\,\max\{1;
(\delta_{0}/\sqrt{d})^{dN}\}Mm_0(N+a|K_0)^{m_0-1}\nonumber\\
& &\kern10em\times\exp\{\beta Mm_0(N+aK_0)^{m_0-1}\}
\label{vmeste},
\end{eqnarray}
where we have used that $L\equiv |\oy_1|\leq aK_0$, as it follows
from (\ref{ogr}).

For the second step of the argument that estimates $Y_{\oy_1}(y_0)$
we define
\begin{equation}\label{novyi}
N^\ast:=\min\{N\,:\, N\geq \bN,\, N\geq K_0,\,\hbox{ and }
C_{0}N^{m_0}-Mm_0(N+|\partial 0|K_0)^{m_0-1}>N\}.
\end{equation}
Then (see (\ref{12.1}))
\begin{equation}\label{nf}
Y_{\oy_1}(y_0)=\left(\sum_{N=1}^{N^{*}}+\sum_{N=N^{*}+1}^\infty\right)
\int_{\bX_{N}} e^{-\beta H(\x|\oy_1)+\mu |\x|}\left|
1-\exp\{-\beta\Psi(\x)\}\right| \nu(d\x).
\end{equation}
We shall estimate $-H(\x\, |\, \oy_1)$ from above by
$M|\x|^{m_0}e^{|\oy_1|+1}$, when $|\x|\leq N^{*}$, and by
$-C_{0}|\x|^{m_0}$, when $|\x|> N^{*}$. The first estimate holds
directly by Lemma~\ref{gvul}, {(\bf b)}. The second estimate
follows from Lemma~\ref{gvul}, {(\bf c)} since $N^{*}\geq
K_{0}\geq N_{\oy_1}$. Plugging the estimates in (\ref{nf}), using
(\ref{vmeste}), and substituting $|\oy_1|$ by its maximal possible
value $aK_0$, we conclude that
\begin{eqnarray}\label{biaka}
Y_{\oy_1}(y_0)&\leq & (\beta+\epsilon)
Mm_0\\
& &\kern-6em\times\left[\sum_{N=1}^{N^{*}}
(\delta_{0}/\sqrt{d})^{dN}\frac{(N+aK_0)^{m_0-1}}{N!}\right.
\nonumber\\
&&\times\exp\left\{\beta N^{m_0}e^{aK_0+1}+\mu N+ \beta
Mm_0(N+aK_0)^{m_0-1}\right\}
\nonumber\\
& & \kern-6em+\kern-0.3em \sum_{N=N^{*}+1}^\infty
\kern-0.3em(\delta_{0}/\sqrt{d})^{dN}
\frac{(N+aK_0)^{m_0-1}}{N!}\nonumber\\&& \times\exp\left\{-\beta
C_{0}N^{m_0}+\mu N +\beta Mm_0(N+aK_0)^{m_0-1}\right\}
\biggr].\nonumber
\end{eqnarray}
It follows from the above expression and the fact that
$\varepsilon\to 0$ as $\beta\to 0$ that $Y_{\oy_1}(y_0)\rightarrow 0$
as $\beta\rightarrow 0$. Thus there exists $\beta(K_{0})$ such  that
 \beq\label{pochti}
V(P_{\oy_1}, P_{\oy_2})<\frac{1}{2a}, \hbox{ for any }\beta\leq
\beta(K_0), \hbox{ when }\oy_1, \oy_2 \hbox{ satisfy {\bf A}}.
 \eeq

Let now $\oy_1$ and $\oy_2$ be two arbitrary configurations that
satisfy the assumption (\ref{ogr}) of Lemma~\ref{l3}. Then, there is
a sequence $\oy^1, \oy^2, \ldots, \oy^j$ of at most $2K_0$
configurations such that each one satisfies (\ref{ogr}), and $\oy^i$
and $\oy^{i+1}$ differ at solely one particle (in the sense of {\bf
A}). Using the triangle inequality and the conclusion (\ref{pochti})
we complete the proof of Lemma~\ref{l3}.\hfill\qed


\bigskip
\noindent\textit{Proof of Theorem \ref{uniq}.} Lemmas~\ref{compc}
and \ref{l3} allow us to check the uniqueness conditions of
\cite{DP}. For any $\chi>0$ and any $\mu\in \mathbb{R}$ there
exists $\beta(\wh{K},\,\chi,\,\mu)>0$ such that for $\beta\leq
\beta(\wh{K},\,\chi,\,\mu)$ there exists only Gibbs field $\xi$
corresponding to the specification $P_{V,\tau}$ and $\mo
e^{\chi|\xi_{V}|}<\infty$ if $\diam(V)<\delta_{0}$. According to
the uniqueness theorem from \cite{DP} the value of $\wh{K}$
depends on $D,\;\delta_{0}$ and $C$ (see (\ref{10t.2})). We recall
that $C$ depends on $\beta$. Thus $\wh{K}$ depends on $\beta$.
However, $C$ can only increase with $\beta$ (see the proof of the
lemma \ref{compc}). Hence for any interval $[0,\beta_{0}]$ we can
choose a unique $C(\beta_{0})$ such that Lemma \ref{compc} holds
for $\beta\leq\beta_{0}$ with the same $C\equiv C(\beta_{0})$.
Then $\wh{K}\equiv \wh{K}(\beta_{0})$ is a constant when
$\beta\in[0,\beta_{0}]$. Finally, we obtain the uniqueness for
any $\beta\leq\min\{\beta_{0},\,
\beta(\wh{K}(\beta_{0}),\,\chi,\,\mu)\}$. Moreover we obtain the
uniqueness for
\begin{equation}\label{beta}
\beta\leq\beta(\chi,\mu)=\max_{\beta_{0}}\min\{\beta_{0},\,
\beta(\wh{K}(\beta_{0}),\,\chi,\,\mu)\}.
\end{equation}

Consider next two different positive numbers $\chi_{1}$ and
$\chi_{2}$. Let $\chi_{1}<\chi_{2}$. It follows from the previous
considerations that for
$\beta\leq\min\{\beta(\chi_{1},\mu),\beta(\chi_{2},\mu)\}$ there
exists an unique Gibbs field $\xi$ corresponding to the
specification. On other hand there are Gibbs fields $\xi^{1}$ and
$\xi^{2}$ which are unique in the regions
$\beta\leq\beta(\chi_{1},\mu)$ and $\beta\leq\beta(\chi_{2},\mu)$
accordingly. It is now clear that $\xi=\xi^{1}=\xi^{2}$ because of
the uniqueness. Hence $\xi$ is unique in the region
$\beta\leq\max\{\beta(\chi_{1},\mu),\beta(\chi_{2},\mu)\}$. It is
obvious that $\xi$ is unique in the region
\begin{equation}\label{beta1}
\beta\leq\beta(\mu)=\sup_{\chi}\{\beta(\chi,\mu)\}.\mbox{\qed}
\end{equation}




%%%%%%%%%%%%%  conclusion

\section{Conclusion Remarks} 1. The stabilizing function $\Phi_{k_0}$ has a
singularity the order of which is $d(m_0-1)$ if $k_0<m_0$. The
order of the singularity is independent on $k_0$. However, the
greater $k_{0}$ is the weaker the singularity is, because the
dimension of the domain of $\Phi_{k_{0}}$ increases with $k_{0}$.

\noindent 2. We require in this work that the stabilizing
potential $\Phi_{k_0}$ has its singularity near diagonal, if
$k_0<m_0$. It means that $\Phi_{k_0}(x_1,...,x_{k_0})$ takes large
values if the  co-ordinates $x_1,...,x_{k_0}$ of all $k_0$
particles pertain to  a small volume of $\mathbb R^d$, that is,
when $\diam\{x_1,...,x_{k_0}\}$ is small. However, our technics
permits to extend our results to the cases when $\Phi_{k_0}$ takes
large values,  when $l_0,\;l_0<k_0$, variables of
$\{x_1,...,x_{k_0}\}$ pertain to a small volume.

\noindent 3.   We can extend our results to the case when there
is a finite number of potential functions
$\Phi_{m_0+1},\Phi_{m_0+2},...$ all taking non-negative values.
This extension can be done with the same technics we use.



\section{Acknowledgment} V.~Belitsky thanks CNPq (grant
301637/91-1) and FAPESP (grant 99/11962-9). E.~Pechersky thanks
the Russian Foundation for Basic Research (grant 99-01-00003),
French-Russian Institute (grant 02-98), CRDF (grant RM1-2085),
FAPESP (grants  98/14795-3 and 00/06186-9) and MCT/FINEP/CNPq
(grant   41.96.0923.00). E.~Pechersky expresses his thanks to
Professors A. Galves  and P. Ferrari for warm hospitality during
his visits to the University of S\~ao Paulo.

\medskip
\begin{thebibliography}{99}

\bibitem{PZh} E.~Pechersky, Yu.~Zhukov, Uniqueness of Gibbs State for Non-Ideal Gas in
${\bf R}^d$: the case of pair potentials. {\it J.Stat.Phys.} V.97,
Nos.1/2 (1999), pp.145-172.

\bibitem{R} D.~Ruelle, Superstable Interactions in Classical Statistical
Mechanics. {\it  Comm. Math. Phys.} V.18 (1970), pp.127-159.

\bibitem{G} W.~Greenberg, Thermodinamical States of Classical
Systems. {\it Comm. Math. Phys.} V.22 (1971), pp. 269-279.

\bibitem{DP}  R.L. Dobrushin,  E.A. Pechersky,
A Criterion of the Uniqueness of Gibbs Fields in the Non-Compact
Case. {\it Lectures Notes in Mathematics.} Springer-Verlag,
V.~1021 (1982), pp.97-110.

\bibitem{D70} R.L. Dobrushin,
Prescribing a System of Random Variables by Conditional
Distributions, {\it Teor. Veroatn. Primen.} (transl. from the
Russian) V.~15 (1970), pp.458-486


\end{thebibliography}

\end{document}
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