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\noindent{ \normalsize\bf Acknowledgements\par}} \vskip13pt}

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\begin{center}
{\Large \bf On the  Wulff Construction as a Problem  \\[2mm] 
of Equivalence of Statistical Ensembles}\\[5mm]
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
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\footnote{Preprint CPT-93/P.2952}
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\footnote{ To appear in 
           ``Micro, Meso and Macroscopic Approaches in Physics'',
              edited by A. Verbeure, Plenum Press, New York (1993)}
\renewcommand{\thefootnote}{\arabic{footnote}}
{\bf Salvador  MIRACLE-SOLE}\footnote{ {\it Email address\/}:
                                      miracle@cptsu2.univ-mrs.fr} 
{\bf and Jean RUIZ}\footnote{ {\it Email address\/}:
                                      ruiz@cptsu2.univ-mrs.fr}\\[4mm]
{\small \sc  Centre de Physique Th\'eorique, CNRS Luminy Case 907,\\
F-13288 Marseille Cedex 9, France}
\end{center}
\vspace{35pt plus 12pt}


In this note, the statistical mechanics of SOS (solid-on-solid) 
1-dimensio-
\hfill\break
nal models 
under the global constraint of having a specified area 
between the interface and the horizontal axis, is studied.
We prove the existence of the thermodynamic limits and 
the equivalence of the corresponding statistical ensembles. 
This gives a simple alternative microscopic proof of
the validity of Wulff construction for such models,
first established in \cite{DDR}.  

We consider the SOS model defined as follows:
To each site $i$ of the lattice $\relatif$ an integer variable $h_i$
is assigned which represents the height of the interface at this site.
The energy 
$ H_N ({ \confh }) $
of a configuration 
${\confh} = \{ h_0,h_1,...,h_N \} $,
in the box $0 \le i \le N$, of length $N$,
is equal to the length of the corresponding 
interface
\beq 
H_N ({\confh}) = \sum _{i=1}^N ( 1 + |h_i - h_{i-1}|) 
\label{1} 
\eeq
Its weight, at the inverse temperature $\beta $,
is proportional to the Boltzmann factor \  
$ \exp [ - \beta H({\confh})] $.
More general Hamiltonians of the form
$H=\sum P(|h_i - h_{i-1}|)$,
where $P$ is a strictly increasing function such that
$P(x)\geq |x|$, when $x\to \infty$, can be treated in the same way.
The proofs extend also to the case of continuous height variables.

We introduce the Gibbs ensemble
which consits of all configurations, 
in the box of length $N$,
with specified boundary conditions
$h_0=0$ and $h_N=Y$.
The associated partition function is given by
\beq 
Z_1 (N,Y) = \sum _{\confh} e^{- \beta H({\confh})}
\ \delta (h_0) \delta (h_N - Y) 
\label{2}
\eeq
where the sum runs over all configurations in the box
and $\delta (t) $ is the dicrete Dirac delta
($\delta (t) = 1 $ if $t=0$ and $\delta (t) = 0 $ 
otherwise).
We define the corresponding free energy per site as
the limit
\beq 
\tau _p (y) = \lim _{N \to \infty } -{ 1 \over {\beta N} }
\ln Z_1 (N,yN) 
\label{3} 
\eeq
where $y = - \tan \theta $, the slope of the interface,
is a real number.
This free energy is called the projected surface tension.
The surface tension,
which represents the interfacial free energy per unit length
of the mean interface, 
is
\beq 
\tau (\theta ) = \cos \theta \  \tau _p (- \tan \theta ) 
\label{4} 
\eeq

We introduce also a second Gibbs ensemble,
which is conjugate to the previous ensemble,
and whose partition function,
in the box of length $N$,
is given by
\beq 
Z_2 (N,x) 
= \sum _{\confh} e^{- \beta H({\confh})}
e^{\beta x h_N}
\ \delta (h_0) 
\label{5} 
\eeq
where 
$x \in \reel $
replaces as a thermodynamic parameter the slope $y$
and $h_0=0$. 
We define the associated free energy as
\beq 
\varphi (x) = \lim _{N \to \infty } -{ 1 \over {\beta N} }
\ln Z_2 (N,x) 
\label{6} 
\eeq

\medskip

{\bf Theorem 1.}
\it
Limits (\ref{3}) and (\ref{6}), which define the above free energies,
exist.
The first, $\tau _p$, 
is a  convex even  function of $y$.
The second, $\varphi $,
is a concave even  function of $x$.
Moreover, $\tau _p$ and $ -\varphi $ are conjugate
convex functions, i.e.,
they are related by the Legendre transformations
\beq
\begin{array}{rcl}
- \varphi (x) 
&=&
\displaystyle \sup_y \  [ x y - \tau_p (y) ] 
\\
\tau _p (y)
&=&
\displaystyle \sup_x \  [ x y + \varphi (x) ] 
\end{array}
\label{7}
\eeq
\rm


\noindent
{\it Proof.\/}
The validity of the above statements is well known.
See for instance \cite{DKS,MMR}
for a proof of these results
in a more general setting.
$\Box$



The convexity of $\tau _p$ is equivalent to the fact that
the surface tension $\tau $ satisfies 
a stability condition called
the triangular inequality \cite{DKS,MMR}.
Relations (\ref{7}) between the free energies express
the thermodynamic equivalence of the two ensembles (\ref{2}) and (\ref{5}).

These relations imply that the curve
$z = \varphi (x)$ gives,
according to the Wulff construction 
or its modern equivalent the Andreev construction,
the equilibrium shape of the crystal
associated to our system \cite{MMR}.

The function $\varphi (x)$ defined by (\ref{6}) is easily computed
by summing a geometrical serie. 
One introduces the difference variables
\beq 
n_i = h_{i-1} - h_i 
\label{8} 
\eeq
for $i = 1,...,N $, so that
the partition function factorizes
and one obtains
\beq 
\varphi (x) = 1
- \beta ^{-1} \ln \sum _{n\in \relatif } 
e^{ - \beta |n| + \beta x n} 
\label{9} 
\eeq
The explicit form of this function is
\beq 
\varphi (x) = 
1
- \beta^{-1} \ln { {\sinh \beta} \over {\cosh \beta - \cosh \beta x} } 
\label{10} 
\eeq 
if $-1 < x < 1$ and $ \varphi (x) = - \infty $
otherwise.



We next define two new Gibbs ensembles for the system under 
consideration.
In the first of these ensembles we consider the configurations
such that $h_N = 0$, which have  a specified height 
at the origin $h_0 = M$
and which have a specified volume $V$
between the interface and the horizontal axis,
this volume being counted negatively for negative 
heights:
\beq 
V = V({\confh}) 
\equiv \sum _{i=0}^N h_i 
\label{11} 
\eeq
The corresponding partition function is
\beq 
Z_3 (N,V,M) = \sum _{\confh} e^{- \beta H({\confh})}
\ \delta (h_N) \delta (V({\confh}) - V)
\delta (h_0 - M) 
\label{12} 
\eeq
The second of these ensembles is the conjugate ensemble of the first
one.
Its partition function is given by
\beq 
Z_4 (N,u,\mu) = \sum _{\confh} e^{- \beta H({\confh})}
\ e^{\beta u (V({\confh})/N) + \beta \mu h_0 }
\ \delta (h_N) 
\label{13} 
\eeq
where $u \in \reel $ and $\mu \in \reel $
are the conjugate variables.
Our next step will be to prove the existence of the
thermodynamic limit for these ensembles and their
equivalence in this limit.
We shall take 
$N=2^n$, $n \in \naturel$.
In (\ref{12}), $V$ and $M$ must be understood
as their integer part when they do not belong to $\relatif$.

\medskip

{\bf Theorem 2.} 
\it The following limits exist
\beqn 
\psi _3 (v,m) 
&=& 
\lim _{N \to \infty } -{ 1 \over {\beta N} }
\  \ln Z_3 (N, v N^2 , mN) 
\label{14} 
\\
\psi _4 (u,\mu) &=& \lim _{N \to \infty } -{ 1 \over {\beta N} }
\  \ln Z_4 (N, u,\mu ) 
\label{15} 
\eeqn
They define the free energies per site associated to the considered 
ensembles as, respectively,  
convex and concave functions of their variables.
Moreover, $\psi _3$ and $- \psi _4$ are conjugate convex functions:
\beq
\begin{array}{rcl}
- \psi_4 (u,\mu)  
&=&
\displaystyle \sup_{v,m} \  [ u v + \mu m - \psi_3 (v,m) ] 
\\
 \psi_3 (v, m) 
&=&
\displaystyle \sup_{u,\mu} \  [ u v + \mu m + \psi_4 (u,\mu) ] 
\end{array}
\label{16}
\eeq
\rm



\noindent
{\it Proof.\/}
The crucial observation is the subadditivity property
given in Lemma 1 below. Then we  addapt
known arguments \cite{R,GMS} in the theory of the
thermodynamic limit.
A more detailed discussion is given in the Appendix.
$\Box$

\medskip

{\bf Lemma 1.} 
\it 
The partition function
$Z_3$
satisfies the subadditivity property
\beq 
Z_3 ( 2N, 2(V' + V''), M' + M'' ) \ge 
Z_3 ( N, V', M')\  Z_3 ( N, V'', M'') \  
e^{- 2 \beta |M''| / (2N-1) } 
\label{17} 
\eeq
\rm 



\noindent 
{\it Proof.\/}
In order to prove this property we
associate a configuration ${\confh}$
of the first system in the box of length $2N$,
to a pair of configurations ${\confh}'$ and ${\confh}''$
of the system in  a box of length $N$, as follows
\beq
\begin{array}{rclr}
h_{2i}
&=&
\displaystyle h'_{i} + h''_{i} \; , 
& i=0,\dots , N
\\
h_{2i-1}
&=&
\displaystyle h'_{i-1 } + h''_{i} \; , 
& i=1,\dots , N 
\end{array}
\label{18}
\eeq
Then 
$h_{2N} = h'_{N} + h''_{N} = 0$,  
$h_{0} = h'_{0} + h''_{0} = M' + M''$ 
and
\beqn
V({\confh})
&=&
2 \sum_{i=1}^N h'_i 
+ \sum_{i=0}^N h''_i 
+ \sum_{i=1}^N h''_i 
\nonumber
\\
& =& 2\  [ V({\confh'}) + V({\confh''}) ] - M'' 
\nonumber
\eeqn
This shows that the configuration $\confh$ belongs to
$ Z_3 ( 2N, 2(V' + V'') - M'', M' + M'') $.  
Since $ H_{2N}({\confh}) = H_N ({\confh'}) + H_N ({\confh''})  $,
because 
$n_{2i} =n'_i$ 
and 
$n_{2i-1} =n''_i$ 
for $i=1,...,N-1 $, as follows from (\ref{18}),
we get 
$$
Z_3 ( N, V', M')\  Z_3 ( N, V'', M'') 
\leq 
Z_3 ( 2N, 2(V' + V'') - M'', M' + M'' )
$$
Then we use the change of variables
$\tilde{h}_i = h_i + [M''/(2N-1)]$
for $i=1,\dots,2N-1$,
$\tilde{h}_0 = h_0$, 
$\tilde{h}_{2N} = h_{2N}=0$ 
which gives
$$
Z_3 ( 2N, V - M'', M )
\leq
e^{ 2 \beta |M''| / (2N-1) }
\ Z_3 ( 2N, V , M )
$$
to conclude the proof.   
$\Box$

The subadditivity property and thus also Theorem 2  are 
satisfied when the height variables $h_i$ are restricted to take 
non negative values. 

\medskip

\bf Theorem 3.
\it
The functions $\psi _3$ and $\psi _4$
can be expressed in terms of the functions
$\varphi$ and $\tau_p$
as follows
\beqn
\psi _3 (v,m) 
&=&
\frac{1}{u_0}    
\int _{\mu_0} ^{\mu_0 + u_0}  \tau_p (\varphi' (x)) dx 
\label{19}
\\ 
\psi _4 (u,\mu) 
&=&
{1 \over u} \int _0 ^u \varphi (x + \mu) dx 
\label{20} 
\eeqn
if 
$u_0$ and $\mu_0$ satisfy
\beqn
 {1 \over {u_0^2} } 
\int _0 ^{u_0} \varphi (x+ \mu_0) dx 
- {1 \over {u_0} } \varphi ( \mu_0 + u_0 ) 
&=&
v
\label{21}
\\
 {1 \over u_0 } 
[\varphi (\mu_0 ) - \varphi ( \mu_0 + u_0) ]
&=&
m
\label{22}
\eeqn
\rm



\noindent
{\it Proof.\/}
We consider again the difference variables
(\ref{8}) and observe that
$$
V({\confh}) 
= \sum _{i=0}^N h_i 
= \sum _{i=1}^N \; i n _i 
$$
therefore
$$ 
Z_4 (N,u,\mu) 
= \prod _{i=1} ^N \Big( \sum _{n_i \in \relatif }
e^{ - \beta |n_i| + \beta ( u / N ) i n_i + \beta \mu n_i} \Big) 
$$
Taking expression (\ref{9}) into account it follows
$$
Z_4 (N,u, \mu ) 
= \exp \Big( - \beta \sum _{i=1} ^N
\varphi \big( { u \over N} i + \mu \big) \Big) 
$$
and
$$
\psi _4 (u,\mu)
= 
\lim _{N \to \infty} {1 \over N} 
\sum _{i=1}^N \varphi \big( { u \over N} i + \mu \big) 
=
\lim _{N \to \infty} {1 \over u} 
\sum _{i=1}^N {u \over N} 
\varphi \big( { u \over N} i + \mu \big) 
$$
which implies expression (\ref{20}) in the Theorem.



The function $\psi _3$ is determined by the Legendre transform
(\ref{16}).
The supremum over $u,\mu$ is obtained for the value $ u_0, \mu_0 $
for which the partial derivatives of the right hand side are zero:
$ v + (\partial \psi _4/ \partial u)  ( u_0,\mu_0 ) = 0, \; 
m + (\partial \psi _4/ \partial \mu)  ( u_0,\mu_0 ) = 0  $. 
That is, for $ u_0, \mu_0 $  which satisfy (\ref{21}) and (\ref{22}).

Then, from (\ref{16}), (\ref{20}), (\ref{21}) and (\ref{22}), we get
\beq
\psi _3 (v,m) 
= 
2 \psi_4 (u_0, \mu_0) 
-
\frac{1}{u_0}
[(\mu_0 + u_0) \varphi(\mu_0 + u_0)- \mu_0 \varphi(\mu_0)]
\label{23} 
\eeq
The right hand side of  (\ref{23}) represents twice the 
area of the sector $OBC$  in Fig.\thinspace 1 divided by $u_0$.


But, it is a known property 
in the Wulff construction,
that twice this area 
is equal to the integral in (\ref{19}).
Indeed, by using the relation (\ref{7})
under the form
$$
\varphi (x) 
= 
x \varphi'(x) + \tau _p (\varphi' (x)) 
$$
in (\ref{20}) and integrating by parts $x \varphi'(x)$, we get
$$
 2 \psi _4 (u_0,\mu_0) 
=   
\frac{1}{u_0} 
\int _{\mu_0} ^{\mu_0 + u_0}  \tau_p (\varphi' (x)) dx 
+
\frac{1}{u_0}
[(\mu_0 + u_0) \varphi(\mu_0 + u_0)- \mu_0 \varphi(\mu_0)]
$$
which together with (\ref{23}) implies the expression (\ref{19}) 
in the Theorem.
$\Box$

To interpret these relations, let us observe that the right hand side of 
(\ref{21}) represents the area $ABC$, in Fig.\t 1,
divided by $\overline{AC}^2$.
Therefore, the values $u_0$, $\mu_0$, which solve   
(\ref{21}) and (\ref{22}), 
are obtained when this area is equal to $v$, with the condition 
coming from (\ref{22}), that the slope 
$\overline{AB}/\overline{AC}$, is equal to $m$.
Then, according to (\ref{19}), the free energy $\psi_3(v,m)$
is equal to the integral of the surface tension along the arc
$BC$, of the curve $z=\varphi(x)$, divided by the same scaling factor 
$\overline{BC}=u_0$.



We conclude that, for large $N$, the configurations 
of the SOS model, with a prescribed area $vN^2$,
follow a well defined mean profile,
the macroscopic profile given by the Wulff construction,
with very small fluctuations.
This follows from the fact that the probability of the configurations
which deviate macroscopically from the mean profile
is  zero in the thermodynamic limit. 
The free energy associated to the configurations which satisfy 
the conditions above, and moreover, are constrained to 
pass through a given point not belonging to the mean profile, 
can be computed with the help of Theorem~3. 
The corresponding probabilities 
decay exponentially as $N\to \infty$,
as a concequence of the usual large deviations theory
in statistical mechanics \cite{RL}.  


\vskip35pt plus12pt
\noindent { \Large\bf Appendix \par } 
\vskip13pt

 
We give here a more detailed discussion of the proof of Theorem 2.
We define:
$$
f_n(v,m)
=
-{ 1 \over {\beta 2^n} }
\ln Z_3 (2^n,  2^{2n} v, 2^n m )
$$
For $v$ and $m$ of the form $2^{-q}p$, 
the subadditivity property with $N=2^n$, 
$V'=V''=vN^2$ and 
$M'=M''=mN$ 
implies that $f_n$ is
a decreasing sequence: $f_{n+1}(v,m) \leq f_n(v,m)$. Since this sequence is

bounded from below its limit exits when n tends to infinity
and coincide with
\beq
\psi_3(v,m) =
\inf _N \left[ -{ 1 \over {\beta N} } \ln Z_3 (N, v N^2,mN )\right]
\label{A.1}
\eeq
Indeed
$$
Z_3 (N,V,M) 
\leq 
\sum _{\confh} e^{- \beta H({\confh})}
\delta (h_N)  
$$
The right hand side of the above expression is easily computed 
by inroducing the 
difference variables (\ref{8}) and we get that $f_n$ 
is bounded from below by
$1 + (1/\beta) \ln \tanh (\beta / 2)$.
Let us notice that one can obtain a lower bound to $Z_3(N,vN^2,mN)$ 
by restricting
the summation to the configuration such that 
$(N-1)h_i = v N^2 - mN$ for $i=1,\dots,N-1$.
This gives that $f_n$ is bounded from above by
$1 +|v-m| + |v|$.


To prove that $\psi_3$ is convex, we notice that the 
subadditivity inequality (\ref{17}) with 
$N=2^n$, $V'=v_1 N^2$,  $V''=v_2 N^2$ and $M'=M''=mN^2$ gives:
$$
\psi_3 \Big( {1\over 2} (v_1 +v_2),m \Big)\leq 
{1\over 2} \psi_3(v_1,m)
+ {1\over 2} \psi_3(v_2,m)  
$$
which applied iteratively implies:
$$
\psi_3 \Big( \alpha v_1 + (1- \alpha) v_2,m \Big)\leq 
\alpha  \psi_3(v_1,m)
+ (1 -\alpha) \psi_3(v_2,m)
$$
for $\alpha$ of the form $2^{-q}p$ and $0\leq \alpha \leq 1$.
For such $\alpha$ we obtain analogously
$$
\psi_3 \Big( v ,  \alpha m_1 + (1- \alpha) m_2 \Big)\leq 
\alpha  \psi_3(v,m_1)
+ (1 -\alpha) \psi_3(v,m_2)
$$
by applying the subadditivity inequality (\ref{17}) with
$N=2^n$, $M'=m_1 N^2$,  $M''=m_2 N^2$ and $V'=V''=vN^2$.
Since $\psi _3$ is bounded, it follows, cf. \cite{R}, that
$\psi _3$ can be extended to a convex Lipshitz continuous
function of the real variables $v$ and $m$.

To prove the  the existence of the  limit (\ref{15})
and relations (\ref{16}),
we introduce 
$$
Z_4^+(N,u,\mu)
=\sup_{V,M \in \relatif } 
\left[ e^{\beta u  (V/N) + \beta \mu M } 
\  Z_3(N,V,M)
 \right]  
$$
and proceed, as in the Appendix of \cite{GMS}, 
to study the thermodynamic limit for this quantity.
We introduce the convex function
$$
\psi^{\ast}_4 (u,\mu) = \sup_{v,m} \  [uv+\mu m -\psi_3 (v,m)] 
$$
According to (\ref{A.1}), we 
have 
$$
e^{\beta u (V/N) + \beta \mu M} \  Z_3(N,V,M)
\leq  
e^{\beta N [uv+\mu m - \psi_3(v,m)]}
$$
for all 
$ V, M \in \relatif$,
so that
\beq
Z_4^+(N,u,\mu) \;
\leq \; 
e^{\beta N \psi_4^{\ast}(u,\mu)}
\label{A.2}
\eeq
On the other hand for any 
$\delta>0$ 
and sufficiently large $N$ one can find,
$V = v N^2$ and $M=mN$ 
such that 
\beqn
e^{\beta u (V/N)+\beta \mu M} \;  Z_3(N,V,M)
&=& 
e^{\beta N [uv+\mu m - \psi_3(v,m)]}
\; e^{\beta N \psi_3(v,m)}
\; Z_3(N,V,M)
\nonumber
\\
&\geq &
e^{\beta N [\psi_4^{\ast}(u,\mu)- \delta]}
\nonumber
\eeqn
and 
therefore
\beq
Z_4^+(N,u,\mu) \;
\geq \;
e^{\beta N [\psi_4^{\ast}(u,\mu)- \delta]}  
\label{A.3}
\eeq
Inequalities 
(\ref{A.2}) and (\ref{A.3}) imply that 
\beq
\lim_{N \rightarrow \infty}\; 
-{1\over \beta N} \; \ln Z_4^+(N,u,\mu) \;
=\; - \psi_4^{\ast}(u,\mu)  
\label{A.4}
\eeq
We shall now prove that the thermodynamic limit (\ref{15}) exits 
and gives the same 
quantity  following the argument of Theorem 2
in \cite{GMS}. 
First, we notice that 
$$
Z_4(N,u,\mu)
=\sum_{V,M \in \relatif }  
e^{\beta u  (V/N) + \beta \mu M } \  Z_3(N,V,M)     
$$
which implies 
\beq
Z_4^+(N,u,\mu)
\leq 
Z_4(N,u,\mu)
\label{A.5} 
\eeq
Moreover, the inequality
$$
Z_3(N,V,M)  
\leq 
e^{-\beta {\bar u} (V/N)-\beta {\bar {\mu}} M} 
\; Z_4^+(N,{\bar u},{\bar {\mu}}) 
$$
used with ${\bar u}=u'$ and ${\bar u}=u''$,
${\bar {\mu}}=\mu'$ and ${\bar {\mu }}=\mu''$ 
gives for any $u''<u<u'$ and any $\mu''<\mu < \mu'$ :
\beqn
Z_4(N,u,\mu) 
& \leq &
Z^{\ast}
\sum_{V\geq 0}  e^{\beta (u-u')(V/N)}
\sum_{M\geq 0} e^{\beta (\mu-\mu')M}
\nonumber
\\
&+&
Z^{\ast}
\sum_{V\leq 0}  e^{\beta (u''-u)(V/N)}
\sum_{M\geq 0} e^{\beta (\mu-\mu')M}
\nonumber
\\
&+&
Z^{\ast}
\sum_{V\geq 0}  e^{\beta (u-u')(V/N)}
\sum_{M\leq 0} e^{\beta (\mu''-\mu)M}
\label{A.6}
\\
&+&
Z^{\ast}
\sum_{V\leq 0}  e^{\beta (u''-u)(V/N)}
\sum_{M\leq 0} e^{\beta (\mu''-\mu)M}
\nonumber
\\
&\leq&
\frac{  4 Z^{\ast}  }{  (1-e^{-\beta (\Delta u/N)})  
(1-e^{-\beta (\Delta \mu)}) }
\nonumber
\eeqn     
where 
\beqn
Z^{\ast}
&=& \sup 
\big\{
 Z_4^+(N,u',\mu'), 
Z_4^+(N,u',\mu'') , 
Z_4^+(N,u'',\mu') , 
Z_4^+(N,u'',\mu'')  
\big\}
\nonumber
\\ 
 \Delta u &=& \min \{ u'-u,u-u''\} >0
\nonumber
\\ 
\Delta u &=& \min \{ \mu'-\mu,\mu-\mu'' \} >0
\nonumber
\eeqn
By referring to (\ref{A.4}), and to the continuity of $\psi^{\ast}_4$, the 

inequalities (\ref{A.5}) and (\ref{A.6}) imply (\ref{15}) and (\ref{16}).

\acknowledgements
The authors thank Mons University, 
where part of this work was done, 
for warm hospitality
and acknowledge the NATO and the ERASMUS project for financial support.

\begin{thebibliography}{9}
\bibitem{DDR}  
J.\t De Coninck, F.\t Dunlop, and V.\t Rivasseau,
On the microscopic validity of the Wulff construction
and of the generalized Young equation,
{\em Commun.\t Math.\t Phys\t.}  
{\bf 121}, 401--419 (1989).
\bibitem{DKS}
R.\t L.\t Dobrushin, R.\t Koteck\'y and S.\t B.\t Shlosman,
``Wulff Construction: A Global Shape from Local Interactions,''
Am.\t Math.\t Soc.\t, Providence, RI (1992).
\bibitem{MMR} 
A.\t Messager, S.\t Miracle-Sol\'e and J.\t Ruiz, 
Convexity properties of the surface tension and 
equilibrium crystals,
{\em J.\t Stat.\t Phys.\t} {\bf 67}, 449--470 (1992).

\bibitem{R}  
D.\t Ruelle,
`` Statistical Mechanics: Rigorous Results,''
Benjamin, New York Amsterdam (1969).


\bibitem{GMS}   
L.\t Galgani, L.\t Manzoni, and A.\t Scotti,
Asymptotic equivalence of equilibrium ensembles
of classical statistical mechanics,
{\em J.\t Math.\t Phys\t.}  {\bf12}, 933--935 (1971).

\bibitem{RL}   
D.\t Ruelle,
``Hasard et Chaos'' (chap.\t 19),
Odile Jacob, Paris (1991).  
O.\t Landford,
Entropy and equilibrium states in classical statistical mechanics,
{\it in\/}: ``Statistical Mechanics and Mathematical Problems,''
A.\t Lenard, ed.\t , Springer, Berlin (1973). 
\end{thebibliography}


\newpage

\centerline{\bf Figure Captions}

Figure 1. Graphical interpretation of Theorem 3.  

\end{document}

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/endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /divide
/lozenge
/ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright
/fi /fl
/daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand
/Acircumflex /Ecircumflex /Agrave
/Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute
/Ocircumflex
/apple /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde
/macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron
MacEncoding 128 128 getinterval astore pop
/local  [/exch load  /def load] cvx  def 

/copydict
{/newfontdict exch def
{exch dup /FID ne 
{exch newfontdict 3 1 roll put}
{pop pop} ifelse} forall
newfontdict
}bdef

/makename
{1 index length 
/prefixlength exch def
dup length prefixlength add string
dup prefixlength 4 -1 roll putinterval
dup 0 4 -1 roll putinterval
}bdef

/coordinatefont
{dup (|______) exch makename cvn 
/newname exch def 
FontDirectory newname known
{pop}
{cvn findfont dup
maxlength dict copydict 
dup /fontname known
{dup /fontname newname put} if
dup /Encoding MacEncoding put
newname exch definefont pop
} ifelse
} bdef
 
%-----------------------------------
%Encode PS Fonts to match Mac Fonts
(Symbol) coordinatefont
(Times-Roman) coordinatefont
%-----------------------------------

%----- Begin Main Program -----%

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grestore

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/|______Symbol findfont 14 scalefont setfont
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grestore

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/|______Symbol findfont 14 scalefont setfont
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/|______Times-Roman findfont 14 scalefont setfont
(+u) show
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/|______Symbol findfont 14 scalefont setfont
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(\(x\)) show
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grestore

gsave  %Grouped Object
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grestore
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gsave  %Line
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gsave  %Line
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gsave  %Line
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%------ End Main Program ------%

showpage end
vmstate restore

%%Trailer
%%DocumentFonts: Symbol
%%+ Times-Roman

%%Pages:1


ENDBODY
