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Ising model, wetting, pinning, surface tension, Wulff, Winterbottom, 
droplet, large deviations
---------------9810020448641
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\begin{document}

\noindent{\Large \bf
Macroscopic Description of Phase Separation\\ in the 2D Ising Model\footnote{To
appear in the proceedings of the conference {\em Mathematical Results in Statistical
Mechanics}, held in Marseille, July 27-31 1998.}
\normalsize \rm}

\vspace{24pt}

\noindent{\bf  Charles-Edouard PFISTER$^{1}$ and \ul{Yvan VELENIK}$^{2}$}

\vspace{12pt}
\it
\noindent
$^{(1)}$ D\'epartement de Math\'ematiques, EPF-L, CH-1015 Lausanne,
Switzerland\\
\phantom{$^{(1)}$} e-mail: {\tt charles.pfister@dma.epfl.ch}
\newline
$^{(2)}$ Fachbereich Mathematik, Sekr. MA 7-4, TU-Berlin, Stra\ss e des 17. Juni
136,\\
\phantom{$^{(2)}$} D-10623 Berlin, Germany\\
\phantom{$^{(2)}$} e-mail: {\tt velenik@math.tu-berlin.de}
\rm

\vspace{24pt}
\footnotesize
\begin{quote}
{\bf Abstract:} We review recent results about the macroscopic description of
phase separation in the 2D Ising model, with special emphasis on boundary
effects and related surface phase transitions. In particular, after having
recalled some facts about the wetting transition, we describe two situations in
which this transition has major consequences at the macroscopic scale. We also
briefly describe a more general situation for which it is possible to derive
the thermodynamical variational problem characterizing the interfaces of the
equilibrium state.

\vspace{3pt}

{\bf Keywords:} Phase separation, Wulff droplet, Winterbottom droplet, interface
pinning, wetting transition.

\end{quote}
\normalsize

\vspace{12pt}

%
\section{Introduction}\label{s_intro}
\setcounter{equation}{0}
%
The aim of these notes is to give a non-technical account of recent results
about the macroscopic description of phase separation in the 2D Ising model.
They are based on a series of works by the authors
\cite{PVHPA,PVPTRF,PVCMP,V}.

\bigskip
Consider some two-dimensional container $Q$, filled with some substance in the
phase coexistence regime (we suppose to simplify that there are only two
phases) at some fixed temperature. Macroscopically, the states can be described
by a family of curves $\ul\cC$, the {\em interfaces}, which are the boundaries
of the regions occupied by the two phases. These families may have to satisfy
some constraints, see below. The problem addressed here is how to determine the
macroscopic geometry of the phase separation, i.e. to characterize the
family of interfaces of the equilibrium state.  Let us first consider the answer to this
question which is provided by Thermodynamics (or rather, here, Thermostatics).

To do this, Thermodynamics postulates the existence of a functional on these
families of curves, the {\em surface free energy}, the minimum (or minima) of
which is attained on the interfaces of the equilibrium state (or states).

In the following, it will be important to distinguish between two contributions
to this surface free energy: The {\em surface tension} is the contribution
coming from the presence of an interface between the two phases, an
infinitesimal part of interface of length $\dd l$, with normal $\ul n$,
contributing $\tau(\ul n)\dd l$. The {\em wall free energy} is the contribution
coming from the interaction of an interface with the walls of the container, an
infinitesimal segment of length $\dd l$ at some point $x$ along the boundary of
$Q$ contributing $\taubd(x)\dd l$. Physically, the wall free energy will depend
on the chemical structure of the wall and may favor the presence of one of the
two phases in its vicinity. The surface free energy of a family $\ul C$ can
then be written as
$$
\cT(\ul\cC) = \sum_{\cC\in\ul\cC} \cT_1(\cC){\rm\ with\ \ } \cT_1(\cC)=\int_\cC
\tau(\ul n_s) \dd s + \int_{\cC\cap \bd Q} (\taubd(x_s) - \tau(\ul n_s)) \dd
s\,.
$$
Since this functional is positive, if there were no constraints the solution of
the variational problem would always be the empty set. Hence, it is necessary
to force the presence of interfaces in the system; this can be done in several
ways, and we will be interested in the following two: 1) Boundary
conditions: the endpoints of the curves of an admissible family are given by a
prescribed set of points of $\bd Q$; 2) Volume constraint: a family is
admissible if the volume of the two phases it determines takes some prescribed
value.

\bigskip
Our aim now is to show how the above description can be understood from the
point of view of statistical mechanics, and in particular how this variational
problem can be derived.
%
\section{Some definitions}\label{s_def}
\setcounter{equation}{0}
%
Let $L$ be a strictly positive integer; we define
%
\begin{equation}
\Lambda_L = [-L,L]^2\cap\bZ^2\,,\quad\quad
\bd\Lambda_L=\{x\in\Lambda_L\,|\,\exists y\not\in\Lambda_L,\,\|x-y\|_1=1\}\,.
\end{equation}
%
A boundary condition is an element $\eta\in\{-1,1\}^{\bd\Lambda_L}$\,. A
configuration $\sigma\in\{-1,1\}^{\Lambda_L}$ is $\eta$-compatible is
$\sigma(x)=\eta(x)$ for all $x\in\bd\Lambda_L$.\\
The Hamiltonian in $\Lambda_L$ with $\eta$-boundary conditions (b.c.) is defined by
%
\begin{equation}
H^\eta_{\Lambda_L}(\sigma) = 
\cases{\displaystyle -\sum_{\bk{t,t'}\subset\Lambda_L}
J(t,t')\sigma(t)\sigma(t') & if 
$\sigma$ is $\eta$-compatible;\cr
+\infty & otherwise,\cr}
\end{equation}
%
where $\bk{t,t'}$ denotes nearest neighbors and $J(t,t')$ are real numbers.
The Gibbs measure in $\Lambda_L$ with $\eta$-b.c. at inverse temperature
$\beta\geq 0$ is defined by
%
\begin{equation}
\mu^\eta_{\Lambda_L}(\sigma) = \frac 1{Z^\eta_{\Lambda_L}} \exp\{-\beta
H^\eta_{\Lambda_L}(\sigma)\}\,.
\end{equation}
%
In the special cases of $+$-b.c. ($\eta\equiv 1$), we write
$\mu^+_{\Lambda_L}$, $Z^+_{\Lambda_L}$, ..., and similarly for $-$-b.c.
($\eta\equiv -1$).

Let $J(t,t')\equiv 1$; the $+$-phase is described by the measure $\mu^+ =
\lim_{L\ra\infty} \mu^+_{\Lambda_L}$; the $-$-phase is defined similarly.

\medskip
Let $\ul n$ be a unit vector, and $\cD_{\ul n}$ be the straight line through the
origin with normal $\ul n$. We denote by $D_{\ul n}$ the length of the segment
$\cD_{\ul n}\cap [-1,1]^2$, and define the following b.c.
%
\begin{equation}
\eta_{\ul n} (x) = {\rm sign}(x\cdot \ul n)\,,
\end{equation}
%
where ${\rm sign}(0)=1$. Let $J(t,t')\equiv 1$. The {\em surface tension} in the
direction $\ul n$ is defined by
%
\begin{equation}
\tau(\ul n;\beta) = -\lim_{L\ra\infty} \frac 1{LD_{\ul n}} \ln\frac {Z^{\eta_{\ul
n}}_{\Lambda_L}}{Z^+_{\Lambda_L}}\,.
\end{equation}
%

\medskip
Let now
%
\begin{equation}\label{eq_betah}
J(t,t') = \cases {
	h & if $t_2=-L$ or $t'_2=-L$\cr
	1 & otherwise,
}
\quad{\rm and}\quad \eta_\pm (x) = \cases{
	-1 & if $x_2=-L$,\cr
	 1 & otherwise.
}
\end{equation}
%
The {\em wall free energy} is defined by
%
\begin{equation}
\taubd(\beta,h) = -\lim_{L\ra\infty} \frac 1{2L+1} \ln\frac
{Z^{\eta_\pm}_{\Lambda_L}}{Z^+_{\Lambda_L}}\,.
\end{equation}
%
Let us write $\tau^*(\beta) = \tau((0,1);\beta)$. It was proved in \cite{FP}
that $|\taubd(\beta,h)|\leq \tau^*(\beta)$, for all values of $\beta$ and $h$.
%
\section{Wetting transition: microscopic point of view}\label{s_microwetting}
\setcounter{equation}{0}
%
The structure of the wall of a container, which is modelized in our model by the
coupling constants between the spins in $\Lambda_L\setminus\bd\Lambda_L$ and those of
$\bd\Lambda_L$, can have major effects on the behavior
of the system, even deep inside the box, as will be shown later. What we will
see is that it can even induce {\em boundary phase transitions}, i.e. a
dramatic change of behavior of the system resulting from a smooth change at the
boundary. The prototypical example of such a phenomenon in our model is the {\em
wetting transition}.  In this section, we recall the microscopic description of
this transition. We refer to the original works \cite{A,FP} for details (see also
\cite{PVHPA}).

We consider the following choice of boundary coupling constants: Let $h\in\bR$,
and $a=\pm 1$; then we choose $h(e)=h$ if $e=\bk{t,t'}$ with $t(2)=1$,
$t'(2)=0$ (or vice-versa), and $h(e)=a$ otherwise. We refer to $h$ as the {\em
boundary magnetic field} and $a$ as the boundary condition.

%
\begin{figure}[t]
 \centerline{\psfig{figure=wettingmicro.ps,height=40mm}}
 \caption[]{\captionsize Schematic representation of the regimes of partial and
 complete wetting of the wall by the $+$-phase. When $\hw(\beta)>h>0$, the
 phase separation line sticks to the wall: the height of its excursions away
 from the wall has bounded expectation (in fact even exponential moments)
 uniformly in the size of the system (left); this is the regime of partial
 wetting. When $h\geq \hw(\beta)$, the phase separation line takes off and
 fluctuates at a distance of order $\sqrt L$ away from the wall; this is the
 regime of complete wetting (right).} \label{fig_wettingmicro}
\end{figure}
%
%
\begin{figure}[t]
 \centerline{\psfig{figure=phdiag.ps,height=50mm}}
  \figtext{
	\writefig	7.1	4.9	{$T$}
	\writefig	7.75	4.25	{$T_c$} 
	\writefig	11.9	0.1	{$h$}
	\writefig	10.6	0.1	{$1$} 
	\writefig	4.2	0.1	{$-1$}
	\writefig	6.92	2.5	{Partial}
	\writefig	6.9	2.0	{Wetting}
  }
 \caption{\captionsize The phase diagram of the wetting transition. The shaded area
 corresponds to the regime of partial wetting; it is characterized by
 $|\taubd(\beta,h)|<\tau^*(\beta)$ or, equivalently, by non-uniqueness of
 the surface Gibbs state.}
 \label{fig_phdiag}
\end{figure}
%
Let $\Lambda'_L=\Lambda+(0,L)$ and $\bL=\{x\in\bZ^2\,|\, x_2\geq 0\}$; all
definitions done for $\Lambda_L$ are straightforwardly extended to
$\Lambda'_L$. Let
$\mu_\bL^{a,\beta,h}=\lim_{L\rightarrow\infty}\mu_{\Lambda'_L}^{a,\beta,h}$;
such a limit is called a {\em surface Gibbs state}. The question here is the
following: does the phase in the vicinity of the wall depend on the boundary
condition $a$? In other words, does $\mu_\bL^{a,\beta,h}$ actually depend on
$a$? It turns out that the answer depends on the value of $h$; more precisely,
Fr\"ohlich and Pfister proved the following
%
\begin{thm}\cite{FP}
For any value of $\beta$ and $h$, $|\taubd(\beta,h)| \leq \tau^*(\beta)$.
Moreover, $|\taubd(\beta,h)| = \tau^*(\beta)$ if and only if
$\mu_\bL^{+,\beta,h}=\mu_\bL^{-,\beta,h}$.
\end{thm}
%
This result provides a thermodynamical characterization of the wetting
transition in terms of the surface tension and wall free energy; this
characterization is known as Cahn's criterion in the physical literature.\\
This result can be understood heuristically. Suppose, without loss of
generality, that $h\geq 0$. Then if $a=1$, the magnetization near the wall will
be positive. If $a=-1$, then there will be competition between attraction of
the wall (when $h<1$ there is an energetic gain in putting the interface along
the wall) and entropic repulsion (close to the wall the phase separation line
cannot fluctuate as much as far from it). Indeed, when $\taubd<\tau^*$, the
attraction of the wall wins, and the phase separation line stays very close to
it and visits it very often; when $\taubd=\tau^*$, entropic repulsion dominates,
meaning that the phase separation line takes off and fluctuates far away from
the wall (in fact at a distance of order $O(L^{1/2})$, see \cite{A,D}). In
the first situation, the phase in the bulk will extend up to the bottom wall,
and consequently the magnetization near the wall will be negative. In the
second situation, however, no information from the bulk can reach the bottom
wall and the phase in its vicinity will have positive magnetization.

\smallskip
The phase transition line in the (positive $h$ part of the) phase diagram can be
parameterized by $\hw(\beta) =
\inf\{h\in\bR\,|\,\taubd(\beta,h)=\tau^*(\beta)\}$, as was shown in \cite{FP};
the complete line can be obtained by symmetry.
%
\section{Macroscopic manifestation of the wetting transition}\label{s_macrowetting}
\setcounter{equation}{0}
%
The wetting transition as described in the previous section cannot be observed
macroscopically: in both regimes, the interface lies, in the continuum limit,
on the bottom wall, since it is never repelled to a macroscopic distance. In
this section, we present two versions of the Ising model in which the wetting
transition can be studied macroscopically; these two versions correspond,
respectively, to forcing the presence of interfaces inside the system by
boundary conditions, and by a volume constraint.
%
\subsection{Grand-canonical ensemble}\label{ss_GCan}
%
This is the simplest modification of the settings of Section
\ref{s_microwetting}. To be able to see the transition, we will raise the
endpoints of the interface at some macroscopic heights along the vertical walls 
of the box.

The coupling constants are given by \req{eq_betah} with $h>0$\footnote{The
case $h\leq 0$ can be treated in exactly the same way, but is not as
interesting.}. Let $a,b \in (-1,1)$; the boundary condition is given by
%
\begin{equation}
\eta_{ab}(x) = \cases{
1  & if $x_2=L$,\cr
   & or $x_1=-L$ and $aL\leq x_2\leq L$,\cr
   & or $x_1=L$  and $bL\leq x_2\leq L$,\cr
-1 & otherwise.
}
\end{equation}
%

\medskip
Let $Q=[-1,1]^2$, and $A=(-1,a)$, $B=(1,b)$; we denote by $\Omega$ the set of
all rectifiable curves inside $Q$ with endpoints $A$ and $B$. On this set we
define the following (surface free energy) functional,
%
\begin{equation}\label{eq_cT}
{\cT}(\cC; \beta,h) = \int_\cC \tau(\ul n_s;\beta) \dd s + |\cC\cap w_Q|
(\taubd(\beta,h)-\tau^*(\beta))\,,
\end{equation}
%
where $w_Q=\{x\in Q\,|\, x_2=-1\}$ and $|\cC\cap w_Q|$ is the length of the
portion of $\cC$ in contact with the wall $w_Q$.

It is not difficult to solve the corresponding thermodynamical variational
problem (see \cite{PVCMP} for details); the result is given in Proposition
\ref{pro_vpgc} below.

Let $\cD$ be the straight line from $A$ to $B$ and $\cW$ be the curve composed
of  three straight line segments:  from $A$ to a point $P_1\in w_Q$ ,  from
$P_1$ to $P_2\in w_Q$, and from $P_2$ to $B$. The points $P_1$ resp. $P_2$ are
such that the angles between the first segment and the wall resp. between the
last segment and the wall are equal to $\theta_Y\in [0,\pi/2]$, which is
solution of the Herring-Young
equation\footnote{$\tau(\theta)\equiv\tau((\cos\theta,\sin\theta))$.}
%
\begin{equation}
\cos\theta_Y \, \tau(\theta_Y) - \sin\theta_Y \, \tau'(\theta_Y) =
\taubd\,.\label{Young}
\end{equation}
%
$\cW$ is a simple curve in $Q$ if and only if $\theta_Y\in [\arctan
\frac{a+b}{2},\pi/2)$.
%
\begin{pro}\cite{PVCMP}\label{pro_vpgc}
Let $\theta_Y$ be the solution of the Herring-Young equation 
\req{Young}.
%
Let $\cM_{\cT}$ be the set of curves minimizing $\cT$.
%
\begin{enumerate}
\item
If $\tan\theta_Y\leq \frac{a+b}{2}$, then $\cM_{\cT}=\{\cD\}$.
\item
If $\pi/2>\theta_Y>\arctan(\frac{a+b}{2})$, then $\cM_{\cT}=\{\cD\}$ if
${\cT}(\cD)<{\cT}(\cW)$, $\cM_{\cT}=\{\cW\}$ if
${\cT}(\cD)>{\cT}(\cW)$ and $\cM_{\cT}=\{\cD,\cW\}$ if ${\cT}(\cD)={\cT}(\cW)$.
\end{enumerate}
%
\end{pro}
%
%
\begin{figure}[t]
 \centerline{\psfig{figure=hcpin.ps,height=60mm}}
  \figtext{
        \writefig       4.31     5.9     {\footnotesize $h$}
        \writefig       4.22    4.46    {\footnotesize $0.8$}
        \writefig       4.22     1.60    {\footnotesize $0.2$}
        \writefig       9.31     0.37     {\footnotesize $T_c$}
        \writefig       10.08    0.37     {\footnotesize $T$}
        \writefig       6.41     4.67     {\footnotesize $i$}
        \writefig       6.41     4.27     {\footnotesize $ii$}
        \writefig       6.41     3.65    {\footnotesize $iii$}
        \writefig       6.41     1.71    {\footnotesize $iv$}
  }
\caption{\captionsize A sequence of phase-transition lines for various values of $a$
and $b$, separating the phase in which the interface is a straight line and the
phase in which it is pinned to the wall. The shaded area corresponds to the
value of $(T,h)$ so that $\taubd(\beta,h)<\tau^*(\beta)$ (i.e. it's a part of
the phase diagram of Fig.~\ref{fig_phdiag}). Observe that the system in case i)
exhibits reentrance (see also Fig. \ref{fig_reentrance}).} \label{fig_pindiag}
\end{figure}
%
So Thermodynamics predicts that the interface of the equilibrium state should
be given either by $\cD$ or by $\cW$, depending on the values of $a$, $b$,
$\beta$ and $h$. We show now that it is indeed possible to derive this starting
from statistical mechanics. To state the result, we need to define some
microscopic version of these two curves. Let $C>0$; we define ($\dd_2$ being
the Euclidean metric)\footnote{In fact, the sets given here are not optimal;
those given in \cite{PVCMP} are slightly better, but more complicated to
describe.}\footnote{If $\cV\subset\bR^2$, then $L\cV=\{Lx,\,x\in\cV\}$.}
%
\begin{eqnarray}
{\cD}^C_L &=& \{t\in\Lambda_L\,|\, \dd_2(t,L\cD)<C\sqrt{L\log L}\}\,,\\
{\cW}^C_L &=& \{t\in\Lambda_L\,|\, \dd_2(t,L(\cW\setminus w_Q) < C\sqrt{L\log L} {\rm\
or\ }
\dd_2(t,L(\cW\cap w_Q))<C\log L)\}\,,\\
{\bd Q}^C_L &=& \{t\in\Lambda_L\,|\, \dd_2(t, \bZ^2\setminus\Lambda_L)< C\log L\}.
\end{eqnarray}
%
Let $\beta>\beta_c$ and $h$ such that $\cT(\cD)\neq\cT(\cW)$. If
$\cM_\cT=\{\cD\}$, then we define $\Lambda^+_L$ as the component of
$\Lambda_L\setminus (\cD^C_L\cup\bd Q^C_L)$ in contact with $+$-b.c., and
$\Lambda^-_L$ as the component in contact with $-$-b.c.. If $\cM_\cT=\{\cW\}$,
we make the corresponding definition, but this time there are two components
$\Lambda^-_{L,1}$ and $\Lambda^-_{L,2}$ in contact with $-$-b.c..\\
The following statement can be proved using the results of \cite{PVCMP}.
%
\begin{thm}\label{thm_pinning}\cite{PVCMP}
Let $\beta>\beta_c$ and $h>0$. There exist $C>0$ and $L_0$ such that, if
$\cM_\cT\neq\{\cD,\cW\}$, then for any $A\subset\Lambda^+_L$,
$$
|\bk{\sigma_A}^{\eta_{ab}}_{\Lambda_L}-\bk{\sigma_A}^+| <
L^{-\cO(C)}\,,\quad\quad \forall L\geq L_0\,.
$$
The corresponding statement for $A\subset\Lambda^-_L$ (if $\cM_\cT=\{\cD\}$), or
$A\subset\Lambda^-_{L,i}$ (i=1,2) (if $\cM_\cT=\{\cW\}$) also holds.
%
\end{thm}
%
This result shows that the set $\cD^C_L$ or $\cW^C_L$ corresponding to the
solution of the thermodynamical variational problem does indeed play the role
of a macroscopic interface, in the sense that it separates regions occupied by
$+$ and $-$-phases (notice also that it converges to the corresponding solution
of the variational problem as $L\ra\infty$). This is therefore a precise
derivation of the thermodynamical description in this case. In fact, even more
can be proved, namely that for any rectifiable, simple curve $\cC$ in Q with
endpoints $A$ and $B$, the probability that the interface is ``close'' to $\cC$
is roughly given by $\exp\{-(\cT(\ul\cC)-\cT^*)\,L\}$, where $\cT^*$ is the
minimum of the functional $\cT$ on such curves. This shows that the surface
free energy is the rate-function for the large deviations of the interface, in
a way completely similar to the role played by the bulk thermodynamic
potentials in the case of large deviations of bulk quantities.
%
\begin{figure}[t]
 \centerline{\psfig{figure=pd11.ps,height=50mm}\hspace{1cm}
 \psfig{figure=pd112.ps,height=50mm}}
  \figtext{
        \writefig       1.38    2.22     {\footnotesize $0.80$}
        \writefig       1.38    0.48     {\footnotesize $0.78$}
        \writefig       1.38    4.02     {\footnotesize $0.82$}
        \writefig       6.38    0.27     {\footnotesize $1$}
        \writefig       6.78    0.27     {\footnotesize $T$}
        \writefig       1.83    5.07     {\footnotesize $h$}
        \writefig       7.31    2.22     {\footnotesize $0.80$}
        \writefig       7.31    0.48     {\footnotesize $0.78$}
        \writefig       7.31    4.02     {\footnotesize $0.82$}
        \writefig       12.31   0.27     {\footnotesize $1$}
        \writefig       12.71   0.27     {\footnotesize $T$}
        \writefig       7.76    5.07     {\footnotesize $h$}
  }
\caption{\captionsize This figure shows part of the phase diagram for $a=-0.8,
b=-0.8$ (left), and $a=-0.8,b=-0.76$ (right). For values of the parameters $T$ and
$h$ below these curves the interface is pinned, while it is a straight line
above these curves. Increasing the temperature along the dashed lines, we see
that the system exhibits reentrance.}
\label{fig_reentrance}
\end{figure}
%

\medskip
Since we are in the 2D Ising model, it is in fact possible to compute
explicitly the corresponding phase diagrams. Figure~\ref{fig_pindiag} shows
the phase diagrams obtained for various choices of $a$ and $b$. An interesting
feature is that, for some values of $a$ and $b$, the system exhibits reentrance
(see Fig.~\ref{fig_reentrance}).


%
\subsection{Canonical ensemble}\label{ss_Can}
%
We describe now another situation in which the wetting transition occurring on
the microscopic scale results in a transition at the macroscopic scale. We do
not induce the presence of interfaces by a suitable choice of boundary
conditions, but rather by a more subtle mechanism, namely fixing the total
amount of magnetization. It turns out that the system reacts to such a
constraint by spontaneously segregating the two phases. Previous results on this
kind of problems, but neglecting boundary effects, are \cite{DKS,P,I}; see also
\cite{DKS,IS} for a detailed description of the {\em microscopic} properties of the
canonical phase.

The coupling constant are given as before by \req{eq_betah}, with $h\in\bR$
(here $h$ can be positive or negative).  We consider $+$-b.c.. Let
$m^*(\beta)=\bk{\sigma(0)}^+$ be the spontaneous magnetization, and choose $m$
such that $|m|<m^*(\beta)$. We can now define the canonical states at finite
volume. Let $c=1/4-\delta>0$, with $\delta>0 $. We introduce the event
%
\begin{equation}
A(m;c):=\{\sigma:\,|\,\sum_{t\in\Lambda_L}\sigma(t)-m|\Lambda_L|\,| \leq
|\Lambda_L|\cdot L^{-c}\}\,.
\end{equation}
%
We want to study the typical macroscopic configurations under the measure
$\mu^+_{\Lambda_L}(\,\cdot\,|\,A(m;c))$, to which we refer as the {\em
canonical state}. Let us first briefly recall what is expected on
thermodynamical grounds. Let $Q=[-1,1]^2$, and for any family $\cC$ of
rectifiable closed curves in $Q$ define
%
\begin{equation}
\ul\cT(\ul\cC) = \sum_{\cC\in\ul\cC}\cT(\cC)\,,
\end{equation}
%
with $\cT(\cC)$ defined as in \req{eq_cT}. Finally, let $V(m)=|Q|(m^*-m)/2m^*$.
The corresponding thermodynamical variational problem is\\
%
{\bf Variational Problem:} Among all subsets $V$ of $Q$ of volume $V(m)$ and with
rectifiable boundary, find those minimizing $\ul\cT(\bd V)$.
%
The idea is that if the $-$-phase occupies such a set, while the $+$-phase occupies
its complement in $Q$, the expected magnetization should be close to
$m^*(|Q|-V(m))-m^*V(m)=|Q|m$ and should therefore satisfy the constraint.

The solution of this variational problem is well known, at least if we ignore
the constraint $V \subset Q$ (see \cite{PVHPA} for more details):
%
\begin{itemize}
\item If $\taubd(\beta,h)=\tau^*(\beta)$, then the solution is given by a
convex set, not touching the wall, whose shape can be obtained through the
Wulff construction;
\item If $\taubd(\beta,h)<|\tau^*(\beta)|$, then the solution is given by a
convex body attached to the wall, whose shape is given by the Winterbottom
construction;
\item If $\taubd(\beta,h)=-\tau^*(\beta)$, then the solution is degenerate, in
the sense that any minimizing sequence consists of unbounded sets (for example a
sequence of rectangle attached to the wall with increasingly larger base and fixed
volume).
\end{itemize}
%
These three situations are referred to, respectively, as: complete drying;
partial wetting (partial drying); complete wetting.

% 
\begin{figure}[t]
\centerline{
\psfig{figure=droplet.ps,height=50mm}\hspace{1cm}
\psfig{figure=bigdroplet.ps,height=50mm}}
\caption{\captionsize {\bf Left:} A sequence of droplets for decreasing values of the
boundary magnetic field. The first picture corresponds to $h\geq\hw(\beta)$;
the droplet has the Wulff shape and ``floats'' inside the box. As soon as $h$
becomes smaller than $\hw(\beta)$, the droplet starts to touch the wall.
Further decrease of $h$ makes it spread until it touches both vertical sides of
the box. Further reduction of the magnetic field does not modify the shape of
the droplet, but makes it unstable in the sense that the removal of the
vertical walls would result in a spreading of the droplet (for example, the
dashed line in the last picture shows part of the droplet which would be
obtained by removing the walls for some $0>h>-h^*$. {\bf Right:} A sequence of big
droplets in a tube for decreasing values of the magnetic field. The droplets
have neither Wulff, nor Winterbottom shape; however Theorem~\ref{thm_droplet}
still holds.}
\label{fig_droplets}
\end{figure}
%
Of course, the constraint $V\subset Q$ may prevent the system from reaching the
true minimum of the variational problem; in particular, the third case will
never be observed. The solution to the constrained variational problem is much
more complicated \cite{KP}; see Fig.~\ref{fig_droplets}
for illustrations.
%

\medskip
We want to show that the typical macroscopic configurations of the model in the
canonical state are close in some sense to the minimum of the (constrained)
variational problem described above. Using the fact that the infimum of the
variational problem is necessarily taken on a unique convex body (by convexity
of the surface tension), it is possible to prove such a claim in complete
generality \cite{PVPTRF}; one of the strength of our method is that we don't
need to know what the solution explicitly is, neither require informations on
its  stability properties.

To define the notion of closeness, we need to be more careful than in the previous
section. Indeed in the present case the minimum of the variational problem is
typically taken on an uncountable set (since any translate of one solution
staying inside the box is also a solution). We therefore use a somewhat weaker
notion of proximity as before, which we describe now.

\medskip
\noindent
Let $C\subset\bZ^2$; {\bf the empirical magnetization in} $C$ is
%
%\begin{equation} 
$m_{C}(\sigma):={1\over |C|}\sum_{t\in C}\sigma(t)\,.
$
%\end{equation}
%

\noindent
Let $0<a<1$; we introduce a grid $\cL(a)$ in $\Lambda_L$ made of cells which
are translates of the square box 
%
\begin{equation}
B(0;[L^{a}])=\{t\in\bR^2:\,|t(j)|\leq [L^{a}] \;,\;
j=1,2\}\,. 
\end{equation}
%
The value  of $a$ is  close to $1$. In most of the cells the empirical
magnetization is close to $m^*$ or $-m^*$ with high probability. For each cell
of the grid $\cL(a)$  we compute the empirical magnetization $m_C(\sigma)$.
Then we scale all lengths by $1/L$, so that after scaling the box $\Lambda_L$ 
is the rectangle $Q$. For each $\sigma$ we define a magnetization profile
$\rho_{L}(x;\sigma)$ on $Q$ by
%
%\begin{equation}
$
\rho_{L}(x;\sigma):=m_{C}(\sigma)
%\; \hbox{if $Lx\in C$}\, 
$
%\end{equation}
%
if $Lx\in C$, where $Lx$ is the point $x\in Q$ scaled by $L$ and $C$ a cell
of the grid $\cL(a)$.\\
The set of macroscopic droplets at equilibrium is 
%
\begin{equation}
\cD(m):=\{\,V\subset Q:\,|V|=V(m)\,,\, \cT(\bd V) {\rm\ is\ minimum}\,\}\,. 
\end{equation}
%
For each $V\in\cD(m)$ we have a magnetization profile, 
%
\begin{equation}
\rho_{V}(x):=\cases{m^*& if $x\in Q\setminus V$,\cr -m^*&
$x\in V$.\cr} 
\end{equation}
%
Let $f$ be a real--valued function on $Q$;
we set
% 
\begin{equation}
d_1(f,\cD(m)):=\inf_{V\in\cD(m)}\int_{Q}dx\,|\,f(x)-\rho_{V}(x)\,|\,.
\end{equation}
%
The result is
%
\begin{thm}\cite{PVPTRF}\label{thm_droplet} Let $\beta>\beta_c$, $h\in\bR$,
$-m^*<m<m^*$, $c=1/4-\delta>0$.
There exists a positive function $\overline{\varepsilon}(L)$ such that
$\lim_{L\ra\infty}\overline{\varepsilon}(L)=0$ and for $L$ large
enough
%
$$
\mu^+_{\Lambda_L}\bigl(\{\,d_1(\rho_{L}(\,\cdot\,;\sigma),\cD(m))\leq
\overline{\varepsilon}(L)\,\}\,|\,A(m;c)\bigr)\geq
1-\exp\{-O(L^{\kappa})\}\,. 
$$
\end{thm}
%
The thermodynamical description has then again been derived, since this result
proves that typical macroscopic configurations are close (in some $L^1$ sense)
to the solution of the variational problem (which we don't need to know!). In
particular, the complete drying/partial drying phase transition in this case is
observable macroscopically (and occurs at exactly the same value of the
boundary magnetic field as in the wetting transition described in
Section~\ref{s_microwetting}).

\bigskip
Because of the finite size of the box, it is not possible to observe the
partial wetting/complete wetting transition; indeed, the droplet always
completely covers the bottom wall before the magnetic field reaches
$-\hw(\beta)$. However, if instead of looking at macroscopic droplets we look
at mesoscopic droplets, this transition can be recovered. This corresponds to
choosing $m=m^*-CL^{-\nu}$, for some $C>0$ and small enough $\nu>0$. Then a
result analogous to the above can be proved \cite{V}. In particular, as long as
$h>-\hw(\beta)$, the equilibrium droplets will always have the Wulff, or the
Winterbottom shape (depending on $h$), as soon as $L$ is large enough. Indeed,
the volume of the corresponding subset of $Q$ is of order $L^{-\nu}$ and
therefore the unconstrained solution always fits inside the box if $L$ is large
enough. However, if $h\leq -\hw(\beta)$, the equilibrium ``droplet'' completely
covers the bottom wall for {\em arbitrarily large $L$}! Since its volume
vanishes in the limit, the height of the droplet goes to zero; in this case,
the limiting shape is a film along the bottom wall.

Considering this dramatic effect of the partial wetting/complete wetting
transition on the equilibrium droplet, one might expect that this transition
must also have non-trivial consequences on the probability of large moderate
deviations. This is indeed the case; it can be shown that\footnote{Exact
expression of these probabilities at leading order in $L$ can be computed, see
\cite{V}.}
%
\begin{equation}\label{eq_lmod}
\mu^+_{\Lambda_L}(A(m;c)) = \cases{
e^{-\cO(L^{1-\frac\nu 2})} & if $h\geq -\hw(\beta)$,\cr
e^{-\cO(L^{1-2\nu})} & if $h< -\hw(\beta)$.\cr
}
\end{equation}
%
Therefore, this phase transition changes the scale of the large moderate
deviations. The results proven in \cite{V} have been obtained for small enough
$\nu$, but are expected to hold for all $\nu<2/3$ if $h\geq-\hw(\beta)$ (this
could be proven combining the argument in \cite{V} with the local limit theorem
of \cite{IS}), and for $\nu<1/2$ if $h<-\hw(\beta)$ (even though this would
be more difficult to prove). As $\nu$ becomes close to $1/2$, the exponent in
\req{eq_lmod} goes to zero. This should not be surprising since in the complete
wetting regime, even without imposing a deviation of the magnetization, the
phase separation line is repelled away from the wall and makes fluctuations of
order $L^{1/2}$, as can be shown at low temperature (since this line converges
to the Brownian excursion \cite{D} for which this result is known). Therefore,
in the complete wetting regime, fluctuations of magnetization are not dominated
by bulk fluctuations, which are of order $L$, but by the fluctuations of the
phase separation line. This implies that {\em typical} fluctuations of
magnetization in the complete wetting regime should be of order $L^{3/2}$, which
is just $L^{2-\nu}$ for $\nu=1/2$.
%
\section{A more general situation}\label{s_general}
\setcounter{equation}{0}
%
In this last section, we describe briefly a ``general'' situation for which it
is possible to derive the macroscopic description provided by Thermodynamics;
the walls of the container can now have a rather complicated structure (they
can be split into an arbitrary number of macroscopic pieces on each of which
a different boundary magnetic field is acting) and the number of interfaces can be
as large as desired.

More precisely, Let $Q=[-1,1]^2$, and let $M$ and $N$ be
two strictly positive integers. We consider a partition of $\bd Q$ into $N$
disjoint, connected components, $\bd Q=\bigcup_{k=1}^N\Delta_k$. Moreover, let
$\cB=\{x_1,\dots,x_{2M}\}$ be $2M$ distinct points of $\bd Q$. The macroscopic
problem we consider now is the following: Find the equilibrium state of a binary
mixture in the box $Q$, with wall free energies given by $\taubd^k$ on $\Delta_k$,
under the constraint that there are $M$ interfaces with endpoints given by the
set $\cB$. The corresponding variational problem is then given by\\
{\bf Variational problem:} Among all families $\ul\cC$ of M open, rectifiable simple
curves in $Q$ with endpoint given by the set $\cB$, find those minimizing the
total free energy $\cT(\ul\cC)=\sum_{k=1^M}\cT(\cC)$, where
%
\begin{equation}
\cT(\cC) = \int_\cC \tau(\ul n_s) \dd s + \sum_{k=1}^N
(\taubd^k-\tau^*)|\Delta_k\cap\cC|\,.
\end{equation}
%
Of course, in such generality not much can be said about the solution, except
that it is taken on families of polygonal lines (by convexity of the surface
tension). It is however again possible to derive, as before this variational
problem from statistical mechanics for the corresponding Ising model. The result
is completely similar to Theorem \ref{thm_pinning}.
%
\begin{thebibliography}{99}

\bibitem{A}
D.B.\ Abraham,
{\em Solvable model with a roughening transition for a planar Ising ferromagnet\/},
Phys. Rev. Lett. {\bf 44}, 1165--1168 (1980).

\bibitem{D}
R.L.\ Dobrushin,
{\em A statistical behavior of shapes of boundaries of phases\/},
in ``Phase Transitions: Mathematics, Physics, Biology...'', ed. R.\ Koteck\'y,
World Scientific, Singapore, 1993.

\bibitem{DKS}
R.L.\ Dobrushin, R.\ Koteck\'y, S.\ Shlosman,
{\em Wulff construction: a global shape from local interaction\/}, AMS
translations series, 1992.

\bibitem{FP}
J.\ Fr\"ohlich, C.-E.\ Pfister,
{\em Semi--infinite Ising model I. Thermodynamic functions and phase diagram in
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{\em Semi--infinite Ising
model II. The wetting and layering transitions\/}, Commun. Math. Phys. {\bf 112},
51--74 (1987).

\bibitem{I}
D.\ Ioffe,
{\em Large deviations for the 2D Ising model: a lower bound without cluster
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the Ising model in two dimensions\/}, Prob. Th. Rel. Fields. {\bf 102},
313--330  (1995).

\bibitem{IS}
D.\ Ioffe, R.\ Schonmann,
{\em Dobrushin-Koteck\'y-Shlosman theory up to the critical temperature\/},
preprint, 1997.

\bibitem{KP}
R.\ Koteck\'y, C.-E.\ Pfister,
{\em Equilibrium shapes of crystals attached to walls\/},
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\bibitem{P}
C.-E.\ Pfister,
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model\/},
Helv. Phys. Acta {\bf 64}, 953--1054 (1991).

\bibitem{PVHPA}
C.-E.\ Pfister, Y.\ Velenik,
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\bibitem{PVPTRF}
C.-E.\ Pfister, Y.\ Velenik, 
{\em Large deviations and continuum limit in the 2D Ising model\/},
Prob. Th. Rel. Fields {\bf 109}, 435--506 (1997).

\bibitem{PVCMP}
C.-E.\ Pfister, Y.\ Velenik, 
{\em Interface, surface tension and reentrant pinning transition in the 2D Ising
model\/},
preprint, 1998.

\bibitem{V}
Y.\ Velenik,
{\em Phase separation as a large deviations 
problem: a microscopic derivation of surface thermodynamics for some 2D 
spin systems\/},
Th\`ese {\bf 1712} EPF-L, 1997; available electronically from the author.
\end{thebibliography}
%
\end{document}

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/tnt {dup dup currentrgbcolor
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/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
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/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def
%%EndProlog

$F2psBegin
10 setmiterlimit
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 0.04500 0.04500 sc
%%Page: 1 1
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
7.500 slw
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% Polyline
30.000 slw
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% Polyline
7.500 slw
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% Polyline
30.000 slw
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% Polyline
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% Polyline
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% Polyline
n 1725 5400 m 2175 5400 l gs col0 s gr 
$F2psEnd
rs
showpage

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