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Wulff construction, Ising model, phase separation, surface tension
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\markboth{T. Bodineau}{Wulff construction}
\title{The Wulff construction\\
in three and more dimensions.}
\author{T. Bodineau}
\date{December 27, 1998}
\begin{document}
\maketitle
\begin{center}
CNRS - UMR 7599,\\
Universit\'e Paris 7,\\
D\'epartement de Math\'ematiques, Case 7012,\\
2 place Jussieu, F-75251 Paris, France
\end{center}
\begin{abstract}
In this paper we prove the Wulff construction in three and more dimensions
for an Ising model with nearest neighbor interaction.
\end{abstract}
\section{Introduction}
The problem of phase separation for two dimensional Ising model and the study
of the equilibrium shape of crystals (Wulff shape) has been initiated by
Dobrushin, Kotecky and Shlosman \cite{DKS}.
Among other things, they proved that if at very low temperatures we decrease the
averaged magnetization in the
$+$ pure phase, we observe the creation of a macroscopic droplet of the $-$ phase
which has a deterministic shape on the macroscopic scale.
The proof has been first simplified by Pfister \cite{Pfister} and then extended
to the
whole of the phase transition region by Ioffe \cite{I1}, \cite{I2} (see also
\cite{SS} and \cite{PfisterVel}).
Recently, Ioffe and Schonmann \cite{IS} have completed the DKS theory up to
the critical
temperature and greatly simplified the original proofs.
Moderate deviations in the exact canonical ensemble are also studied in \cite{IS}.\\
In two dimensions, the proofs have been based on duality arguments and
on a coarse graining procedure (skeleton).
These arguments do not seem to apply
in higher dimensions.
For more than two dimensions, an alternative procedure has been proposed
by Alberti, Bellettini, Cassandro and Presutti
\cite{ABCP}, \cite{BCP} for Ising systems with Kac potentials.
They rephrase the whole problem in terms of $L^1$ theory
and prove large deviations for the appearance of a droplet of the minority
phase in a scaling limit when the size of the domain diverges not much faster
than the range of the Kac potentials.
This amounts to a weak large deviation principle which is obtained by proving
$\G$-convergence of a functional associated to the spins
system \cite{ABCP}.
A large deviation principle has then been proved via a tightness property
\cite{BCP}.
Their approach has been generalized by Benois, Bodineau, Butta
and Presutti \cite{BBBP}, \cite{BBP} by taking first the thermodynamic limit
and then letting the range of interaction go to infinity.
The first paper \cite{BBBP} was devoted to the proof of a weak large deviation
principle for the
macroscopic magnetization which is equivalent to the computation of
surface tension.
The main idea has been to introduce a coarse graining in order to use
the $L^1$ setting.
Namely, events in $L^1$ were related to mesoscopic quantities by
an argument which we will refer to later as minimal section argument.
An exact expression of surface tension was difficult to
recover from coarse grained estimates and surface tension
was only derived in the Kac limit, i.e.
when the range of interactions tends to infinity.
The second step \cite{BBP} consisted of proving a
tightness property by using the compactness in $L^1$ of
the set of functions of bounded variation with finite
perimeter.
Wulff construction for three dimensional
independent percolation has been proven by Cerf \cite{Cerf}
using a procedure similar to the one of \cite{BBBP} and a novel
definition of surface tension.
In this case, the dependence on boundary conditions is
weaker and, the minimal section argument enables to prove
directly a weak large deviation principle by using this appropriate definition
of surface tension.
As percolation occurs in an infinite volume, there is an extra
difficulty and different compactness arguments have been required.\\
In this paper we proceed as in \cite{BBBP}. The main difficulty is
to recover surface tension from a constraint on the averaged magnetization.
The surface tension is defined as $\log \big( {Z^+ \over Z^{+,-}} \big)$ where
the partition functions are computed with + boundary conditions and with mixed
boundary conditions ($+$ at the top and $-$ at the bottom), see for instance
the paper of Messager, Miracle-Sol\'e and Ruiz \cite{miracle}.
To use directly this definition, one would have to find in the bulk
surfaces of + spins or of $-$ spins which in fact may not exist.
A way to circumvent this problem is to prove that
surface tension can be produced by averaging the boundary
conditions, choosing the spins with respect to the $+$ pure
phase and to the $-$ pure phase.
For Ising model with nearest neighbor interaction, the coarse
graining developed by Pisztora \cite{Pisztora1} will play
an analogous role to the one used for Kac model.
Pisztora's coarse graining is one of the most profound and powerful
technique for the study of the Ising (Potts) model, it provides an accurate
description of the Ising model in a non perturbative regime up to a temperature
$\hat T_c$ which is conjectured to agree with the critical temperature.
In the following, we will mention which of our results hold up to $\hat T_c$.
As Pisztora's coarse graining is defined via the FK representation,
several quantities need to be rewritten in terms of the FK representation.
In particular, our approach to the surface tension (Section 4)
is built upon the FK representation and, is motivated by the corresponding construction
in \cite{Cerf}.
This is a key to obtain precise surface order estimates on the logarithmic scale.
This is also the only point at which we refer to \cite{Cerf}, the core philosophy
of our proof is based on the renormalization ideas of \cite{BBBP} and \cite{BBP},
including the appropriate setup of the geometric measure theory.
The coarse graining schemes of the latter works, however, depend on
specific properties of Kac potentials and, one of our main technical tasks
here is to develop a relevant modification of these renormalization
procedures in the nearest neighbor context.
% The use of FK measure is mainly motivated by our need of a coarse graining.
A step further in the understanding of the surface tension will be to prove
a phase separation theorem for Kac model with finite range interactions by
using a coarse graining defined only in terms of the Gibbs measure \cite{Bod1}.
\newpage
After introducing the main notation, we state in Section 2 the results and an
overview of the paper (see subsection 2.3).\\
\noindent
{\bf Acknowledgments :}
I am deeply indebted to both D. Ioffe and E. Presutti who have been involved at every stage
of this work. This paper would not have been possible without their invaluable suggestions,
advice and support.
I am also very grateful to A. Chambolle for explaining many results of geometric measure theory
and providing advice and references.
My gratitude also goes to G. Bellettini and F. Comets for our very useful discussions.
Finally, I acknowledge the kind hospitality of the Dipartimento di Mathematica di Roma Tor Vergata,
where part of this work was completed.
\section{Notation and results}
For simplicity, notation and results are stated in
three dimensions, but they are valid for any dimension
larger or equal to three.
\subsection{Notation}
We introduce the following norms on $\Z^3$
\begin{eqnarray*}
\forall x \in \Z^3, \qquad
\| x \|_1 = \sum_{i=1}^3 |x_i| \qquad {\rm{and}}
\qquad \| x \|_2 = \sqrt{\sum_{i=1}^3 |x_i|^2}.
\end{eqnarray*}
Two vertices $x,y$ in $\Z^3$ are nearest neighbors
if $\| x -y \|_1 \le 1$ and we denote it by $x \sim y$.
For any finite subset $\L$ of $\Z^3$, we define its boundary by
$$
\partial \L = \{ x \in \L^c \; | \; y \in \L, \ \ x \sim y \},
$$
and denote its cardinality by $|\L|$.
We consider the Ising model on $\Z^3$ with nearest neighbor interaction.
Each spin $\s_i$, attached at the lattice site $i$ in $\Z^3$, can take
values $\pm 1$.
For any integer $N$, we set $\D_N = \{ 1, N \}^3$ and denote the space of
configurations in $\D_N$ by $\Sigma_{\D_N} = \{ \pm 1 \}^{\D_N}$.
Let $\s_{\D_N}$ be the spin configuration restricted to $\D_N$.
We introduce the Hamiltonian associated to $\s_{\D_N}$ with boundary conditions
$\s_{\partial \D_N}$
\begin{eqnarray*}
H( \s_{\D_N} \, | \, \s_{\partial \D_N} ) =
- {1 \over 2} \sum_{ i \sim j \atop i,j \in \D_N} \s_i \s_j
- \sum_{ i \sim j \atop i \in \D_N, j \in \partial \D_N} \s_i \s_j.
\end{eqnarray*}
The Gibbs measure on $\Sigma_{\D_N}$ at inverse temperature $\b > 0$ is
\begin{eqnarray*}
\label{Gibbs measure}
\mu_{\b, \D_N} ( \s_{\D_N} \, | \, \s_{\partial \D_N} ) =
{1 \over Z_\b (\s_{\partial \D_N}) }
\exp \big( - \b H( \s_{\D_N} \, | \, \s_{\partial \D_N} ) \big),
\end{eqnarray*}
where the partition function $Z_\b (\s_{\partial \D_N})$ is the normalizing
factor.
When the boundary conditions $\s_{\partial \D_N}$ are identically equal to 1,
we simply write $\mu_{\b, \D_N}^+$.
There is a critical value $\b_c >0$ and for all $\b$ larger than $\b_c$, there
exists $m_\b>0$ such that
%If $\b$ is larger than some critical value $\b_c >0$, there is a phase
%transition. This implies
\begin{eqnarray*}
\lim_{N \to \infty} \mu^+_{\b, \D_N} (\s_0) = m_\b > 0.
\end{eqnarray*}
The Gibbs measure $\mu^+_\b$ on $\Sigma_{\Z^3}$ obtained by taking the
thermodynamic limit of $\mu_{\b, \D_N}^+$ is called the + pure phase and
$m_\b$ is the equilibrium value of the magnetization.
\subsection{Surface tension}
Let us recall the definition of surface tension and related
results which can be found in \cite{miracle}.
The set of unit vectors in $\R^3$ is denoted by $\SS^2$.
We fix $\vec{n}$ a vector in $\SS^2$ and $\vec{e_1},\vec{e_2}$ two vectors
orthogonal to $\vec{n}$.
Let $h$ be a positive constant and $N \to f(N)$ a positive
function which diverges as $N$ goes to infinity.
For any integer $N$, we denote
by $\bar \L (h N, h N, f(N))$ the parallelepiped of $\R^3$
centered at 0 with faces parallel to the axis $(\vec{e_1},\vec{e_2},\vec{n})$
such that the lengths of the sides parallel
to $(\vec{e_1},\vec{e_2})$ are $h N,h N$ and the ones parallel to $\vec{n}$ is
$f(N)$.
%When there is no risk of confusion, we simply write $\bar
%\L_N$.
We introduce $\L_N$ the set of vertices
$\bar \L (h N, h N, f(N)) \cap \Z^3$.
The boundary $\partial \L_N$ is split into 2 sets
\begin{eqnarray*}
\partial^+ \L_N & = & \{ i \in \partial \L_N \; | \; \vec{i}.\vec{n} \ge 0\},\\
\partial^- \L_N & = & \{ i \in \partial \L_N \; | \; \vec{i}.\vec{n} < 0\}.
\end{eqnarray*}
We call $\partial^+ \L_N$ the upper and $\partial^- \L_N$ the lower part
of $\partial \L_N$.
We fix the boundary conditions outside $\L_N$ to be equal to
1 on $\partial^+ \L_N$ and to $-1$ on $\partial^- \L_N$.
The corresponding partition function on $\L_N$ is denoted by $Z^{+,-}_{\L_N}$.
\begin{defi}
\label{tau}
The surface tension in the direction $\vec{n} \in \SS^2$ is defined by
\begin{eqnarray*}
\t(\vec{n}) = \lim_{N \to \infty} \; - {1 \over h^2 N^2}
\log { Z^{+,-}_{\L_N} \over Z^+_{\L_N}}.
\end{eqnarray*}
\end{defi}
\noindent
The surface tension defined above coincides with the one defined
in \cite{miracle} (see Appendix 8.1).
It depends neither on $h$ nor on $f$ as proven in \cite{miracle} (Theorem 2).
Let us extend $\t$ by homogeneity
\begin{eqnarray}
\label{tau tilde}
\forall \vec{v} \in \R^3 - \{0\}, \quad
\tilde \t(\vec{v}) = \| \vec{v} \|_2 \, \t
\left( { \vec{v} \over \| \vec{v} \|_2} \right)
\qquad \rm{and} \qquad \tilde \t(0) = 0.
\end{eqnarray}
The pyramidal inequality proven in Theorem 3 of \cite{miracle} ensures
that $\tilde \t$ is convex.
As $\tilde \t$ is locally bounded and convex, it is continuous.
It was proven by Lebowitz and Pfister \cite{lebPfister} that for all
$\b$ larger than $\b_c$, the surface tension $\t(\vec{n}_0)$ in the direction
$\vec{n}_0 = (1,0,0)$ is positive.
From the symmetries and the convexity of $\t$, we check that $\t$
is uniformly positive on $\SS^2$.\\
The spin configuration $\s$ should be seen as a microscopic representation of the
system. The macroscopic state of the system is instead determined by the value of
an order parameter (the averaged magnetization) which specifies the phase
of the system.
As $\b$ is fixed, it is convenient to replace the order parameter
%and the two equilibrium phases
by a parameter $u$ with values $\pm 1$.
We suppose that the macroscopic region of $\R^3$ where our system is confined is
$\T = [0,1]^3$.
We denote by $\BV$ the set of functions of bounded variation in
$\T$ with values $\pm 1$ (see \cite{EG} for a review).
The fact that $u_r = 1$ for some $r$ in $\T$ means that locally at $r$ the system
is in equilibrium in the phase $+ m_\b$.
The precise correspondence between $\s$ and functions on $\T$ is described in
Section 3, where we approximate $\s$ by a coarse graining procedure, introducing
a mesoscopic scale.
For all $u$ in $\BV$, we denote by $\partial u$ the boundary of the
set $\{ u = -1\}$. If the set $\partial u$ has finite perimeter, there
exists a set $\partial^* u$, called the reduced boundary, such that
one can define in each point $x$ of $\partial^* u$ the outer normal
denoted by $\vec{n_x}$.
Let us introduce the functional $\F$ on
$L^1(\T,[-{1 \over m_\b},{1 \over m_\b}])$
\begin{eqnarray}
\label{functional F}
\forall u \in \BV, \qquad
\F (u) = \int_{\partial^* u} \t(\vec{n_x}) d \H_x,
\end{eqnarray}
where $d \H$ is the 2 dimensional Hausdorff measure in $\R^3$.
If $u$ is in $L^1(\T,[-{1 \over m_\b},{1 \over m_\b}])$
but not in $\BV$ then we set $\F(u) = \infty$.
To any subset $A$ of $\T$, we associate the function
$\1_A = 1_{A^c} - 1_A$ and simply write $\F(A)=\F(\1_A)$.
An important property is the lower semi-continuity of $\F$ with respect to
$L^1$ convergence.
As $\tilde \t$ is convex (see (\ref{tau tilde})), the lower semi-continuity
is a consequence of a result by Ambrosio and Braides (see \cite{Ambrosio}
Theorem 2.1 and example 2.8).\\
The equilibrium crystal shape $\W_m$, called Wulff shape, is a solution
of the following isoperimetric problem
\begin{eqnarray}
\label{variational}
\min \big\{ \F(u) \ \big| \ u \in \BV, \qquad m_\b \int_\T u_r \, dr \le m \big\},
\end{eqnarray}
where $m$ belongs to $]m^*,m_\b[$.
We will restrict the
parameter $m$ so that, for $m$ in $]m^*,m_\b[$ the minimizers
of the variational problems in $\T$ and $\R^3$ are the same.
This enables us to avoid boundary problems.
The shape $\W_m$ can be explicitly constructed (the Wulff construction) by
dilating the set
\begin{eqnarray*}
\W = \bigcap_{\vec{n} \in \SS^2} \, \big\{ x \in \R^3; \qquad
\vec{x}. \vec{n} \le \t(\vec{n}) \}
\end{eqnarray*}
in order to satisfy the volume constraint $m_\b \int_\T \1_{\W_m}(r) \, dr = m$.
As $\W_m = \lambda_m \, \W$, one has $\F(\W_m) = \lambda_m \, \F(\W)$.
Thus $\F(\W_m)$ is continuous with respect to $m$.
Taylor \cite{Taylor} proved that $\W_m$ is a closed convex surface
and that all
other minimizers of (\ref{variational}) are deduced from $\W_m$ by shifts.
In the following, we suppose that $\W_m$ is centered in
$({1 \over 2}, {1 \over 2}, {1 \over 2})$.
\subsection{Heuristics and Results}
The total magnetization
${1 \over N^3} \sum_{i \in \D_N} \s_i$ will be
denoted by $\M_{\D_N}$.
A shift of the magnetization from its equilibrium value leads to large
deviations controlled by a surface order
\begin{theo}
\label{theo 1}
There is $\b_0$ positive such that for any $\b$ larger than $\b_0$ and
$m$ in $]m^*,m_\b[$
\begin{eqnarray*}
\lim_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N} \big( \M_{\D_N} \le m \big) = - \F(\W_m ),
\end{eqnarray*}
where $m^*$ and $\W_m$ were defined in (\ref{variational}).
\end{theo}
More precisely, a phase separation occurs on the macroscopic level.
In order to describe it, we introduce an intermediate scale
called mesoscopic : the magnetization is locally averaged on boxes
of size $N^\a$ with $\a \in ]0,1[$ ($1 \ll N^\a \ll N$).
For any integer $L$, we define the sub-lattice
\begin{eqnarray}
\label{lattice}
{\cal L}_L = \left\{ \left( L x_1 + { L \over 2}, L x_2 + { L \over 2},
L x_3 + { L \over 2} \right) \; \big| \; x =(x_1,x_2,x_3) \in \Z^3 \right\}.
\end{eqnarray}
We introduce also $B(x,L)$ the box of length $L$ centered in $x$ in
${\cal L}_L$
\begin{eqnarray}
\label{box}
B(x,L) = \big\{ y \in \Z^3 \; | \quad \forall i \in \{1,2,3\}, \quad
- {L \over 2} < y_i - x_i \le {L \over 2} \big\}.
\end{eqnarray}
To simplify the notation, we fix $\a$ in $\Q \, \cap ]0,1[$ and suppose
from now that $N$ is of the form $2^{\a^{-1} \, k}$, where $k$ and
${k \over \a}$ are integers.
The set $\D_N$ is partitioned into boxes $B(x,N^\a)$ of side length $N^\a$
centered in $x$ in ${\cal L}_{N^\a}$.
For general values of $N$, one would need to partition $\D_N$ with boxes which
may have different sizes. This is a standard
technique and we refer the reader to Pisztora \cite{Pisztora1}.
The local magnetization $\MM$ is a piecewise constant function on $\T$
\begin{eqnarray*}
\forall r \in \T, \qquad \MM_r = {1 \over N^{3\a}}
\sum_{j \in B(x,N^\a)} \s_j \qquad
\rm{if} \ \ \forall i, \quad
- {L \over 2} < N r_i - x_i \le {L \over 2}.
\end{eqnarray*}
As explained in the introduction, it is convenient to formulate the
problem of phase separation in terms of $L^1$ theory.
For any function $u$ in $L^1( \T, [- {1 \over m_\b}, {1 \over m_\b}])$,
we denote by $\V(u,\d)$ the $\d$-neighborhood of $u$
\begin{eqnarray*}
\V(u,\d) = \big\{ v \in L^1 \big( \T, [- {1 \over m_\b}, {1 \over m_\b}] \big)
\ \big|
\qquad \int |u_r - v_r| \, dr \le \d \big\}.
\end{eqnarray*}
We can now state a theorem on phase separation which says that for
$\b$ large enough with $\mu^+_{\b,\D_N} \big( \, . \; \big| \, \M_{\D_N} \le m
\big)$-probability converging to 1, the function
$\MM$ is close to some translate of the Wulff shape
$m_\b \1_{\W_m}$.
\begin{theo}
\label{theo 2}
There is $\b_0$ positive such that for any $\b$ larger than $\b_0$ and
$m$ in $]m^*,m_\b[$
\begin{eqnarray*}
\forall \d >0, \qquad
\lim_{N \to \infty} \; \mu^+_{\b,\D_N} \big(
{\MM \over m_\b} \in \bigcup_{r \in \T'} \V(\1_{\W_m + r},\d)
\; \big| \, \M_{\D_N} \le m \big) = 1,
\end{eqnarray*}
where $m^*$, $\W_m$ were defined in (\ref{variational}) and
$\T'= \{ r \in \T \ | \ \W_m + r \subset \T \}$.
\end{theo}
\noindent
This result is far less sharp than those obtained in the 2 dimensional
case (see \cite{IS}).\\
We will follow the scheme of \cite{BBBP} and deduce Theorems
\ref{theo 1} and \ref{theo 2} from the following statements.
\begin{prop}
\label{prop 2}
Let $\b$ be large enough. Then for all $u$ in $\BV$ such that
$\F(u)$ is finite
\begin{eqnarray*}
\lim_{\d \to 0} \limsup_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N}
\big( {\MM \over m_\b} \in \V(u,\d) \big) \le - \F(u).
\end{eqnarray*}
\end{prop}
\begin{prop}
\label{prop 1}
For $\b$ large enough and $m$ in $]m^*,m_\b[$
\begin{eqnarray*}
\lim_{\d \to 0} \liminf_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N}
\big( {\MM \over m_\b} \in \V(\1_{\W_m},\d) \big)
\ge - \F(\W_m ),
\end{eqnarray*}
where $m^*$ and $\W_m$ were defined in (\ref{variational}).
\end{prop}
Since Proposition \ref{prop 2} is only a weak large deviation principle,
we need to strengthen it by proving an exponential tightness property which
is similar to the one in \cite{BBP}.
For any $a$ positive, the set
$$
K_a = \big\{ u \in \BV \; |\quad \F(u) \le a \big\}
$$
is compact with respect
to convergence in measure : As $\F$ is lower semi-continuous, $K_a$ is
closed and as the surface tension $\t$ is larger than a positive
constant $\t_0$, the set $K_a$ is included
in the compact set of functions of bounded variation in $\T$
with perimeter
smaller than ${a \over \t_0}$ (see \cite{EG} Section 5.2.3).
\begin{prop}
\label{prop 3}
We fix $\b$ large enough. Then there exists a constant $C_\b$ such
that for all $a$ and $\d$ positive
\begin{eqnarray*}
\limsup_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N}
\big( {\MM \over m_\b} \in \V(K_a,\d)^c \big)
\le - C_\b \, a,
\end{eqnarray*}
where $\V(K_a,\d)$ is the $\d$-neighborhood of $K_a$ in
$L^1( \T, [-{1 \over m_\b},{1 \over m_\b}])$.
\end{prop}
The proofs of Theorems \ref{theo 1} and \ref{theo 2} are based
on well known
large deviations arguments (see \cite{DS}). For completeness we prove
Theorem \ref{theo 1}, the proof of Theorem \ref{theo 2} is similar
\noindent
{\it Proof of Theorem \ref{theo 1}}.
In order to prove the upper bound, we fix $\d$ positive and split
the closed set
$$
F = \big\{ u \in L^1 \big( \T, [-{1 \over m_\b},{1 \over m_\b}] \big)
\ \big| \quad m_\b \,\int_\T u_r \, dr \le m \big\}
$$
into 2 sets
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in F \big) \le
\mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in F \cap
\V(K_a,\d) \big)
+ \mu^+_{\b,\D_N} \big({\MM \over m_\b} \in
\V(K_a,\d)^c \big).
\end{eqnarray*}
We choose $a$ such that $C_\b a$ is much larger than $\F (\W_m)$, then
Proposition \ref{prop 3} enables us to bound the last term in the RHS.
Let us fix $\e$ positive.
Since $K_a$ is compact, we cover it with a finite number $\l$
of neighborhoods $\V(u_i,\e_i)$, where each $\e_i$ belongs to
$]0,\e]$ and is chosen such that Proposition \ref{prop 2} implies
\begin{eqnarray*}
\limsup_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in \V(u_i,\e_i) \big)
\le - \F(u_i) + \e.
\end{eqnarray*}
For $\d$ small enough, we cover $F \cap \V(K_a,\d)$ with the above neighborhoods
which intersect $F$.
Since $u_i$ belongs to $\V(F,\e)$, we get from Lemma 2.1.2 of \cite{DS}
\begin{eqnarray*}
\lim_{\d \to 0} \limsup_{N \to \infty} {1 \over N^2}
\log \mu^+_{\b,\D_N} \big({\MM \over m_\b}
\in F \cap \V(K_a,\d) \big)
\le - \lim_{\e \to 0} \inf_{u \in \V(F,\e)} \F(u)
\le - \inf_{u \in F} \F(u).
\end{eqnarray*}
As $\W_m$ minimizes the variational problem
(\ref{variational}), the upper bound holds.
\vskip.2cm
To prove the lower bound, we fix $\e$ positive and check that
$\{ {\MM \over m_\b} \in \V(\1_{\W_{m - \e}},\d) \}$ is
included in $\{ \M_{\D_N} \le m\}$ for $\d$ small enough.
Proposition \ref{prop 1} implies
\begin{eqnarray*}
\lim_{\d \to 0} \liminf_{N \to \infty} \; {1 \over N^2}
\log \mu^+_{\b,\D_N}
\big( {\MM \over m_\b} \in \V(\1_{\W_{m-\e} },\d) \big) \ge - \F(\W_{m-\e}).
\end{eqnarray*}
Letting $\e$ go to 0, we complete Theorem \ref{theo 1}.\qed
\vskip.5cm
Let us comment on Propositions \ref{prop 1}, \ref{prop 2} and \ref{prop 3}.
A shift of the averaged magnetization can be realized by 2 competing effects.
The first one, which consists of producing a large droplet of $-$ inside the bulk,
is controlled by surface tension (Propositions \ref{prop 1} and \ref{prop 2}).
The second one consists of increasing homogeneously the number of small $-$ contours.
This requires a lot of energy, but may be favored by entropy.
This effect is ruled out by Proposition \ref{prop 3} which is a combination of
an estimate in the phase of small contours with a Peierls type estimate for large contours.
In fact, the underlying phenomena are more subtle and
it was shown by \cite{IS} in the case of dimension 2,
that on the level of moderate deviations the second effect may
be the most important.
We start by defining
a coarse graining on the mesoscopic scale which keeps more details of the
microscopic structure than $\MM$.
This is done, in Section 3, via the FK representation by using Pisztora's results
\cite{Pisztora1}.
This coarse graining procedure imposes that $\b$ is larger than a critical value
$\tilde \b_c$ related to the slab percolation threshold (see \cite{Pisztora1}) and to
condition (\ref{Theta}). It is conjectured that $\tilde \b_c$ equals the critical value
$\b_c$.
In Section 4, motivated by \cite{Cerf}, we use an alternative definition of
surface tension in terms of the FK representation.
We prove the equivalence of several expressions for surface
tension which will enable us to compare different boundary
conditions.
In Section 5, Proposition \ref{prop 2} is proven along the
lines of the argument developed in \cite{BBBP}.
It states that the most likely configurations in
$\big\{ {\MM \over m_\b} \in \V(u,\d) \big\}$ are those for which
the $+$ and $-$ phases coexist along the boundary of
$\partial u$, this coexistence induces deviations proportional
to a surface order.
The $L^1$ constraint $\big\{ {\MM \over m_\b} \in \V(u,\d) \big\}$
imposed on the magnetization is not strong enough to localize
the interface close to $\partial u$ : there might be
mesoscopic fingers of one phase percolating into the other.
Following \cite{BBBP}, we prove by the minimal section argument
that one can chop off these fingers without changing too much
the probability of the event.
The renormalization is an essential feature in the previous procedure.
Once the interface is localized on the mesoscopic level,
the main problem is to identify surface tension.
Note that in the case of percolation \cite{Cerf}, the minimal
section argument enables to cut the microscopic fingers which
connect the domains separated by $\partial u$.
Therefore, one can identify the surface tension factor, because
for independent percolation it is defined as the probability
that no cluster connects one domain to the other.
In the case of spin systems, one would need to find a microscopic surface
of $+$ spins on one side of $\partial u$ and
another one of $-$ spins on the other side in order
to use directly the definition of surface tension.
This would seem difficult to achieve because mesoscopic contours enable only
to control the averaged magnetization and do not ensure the existence of such
microscopic surfaces. We proceed differently and use an alternative definition
of surface tension in terms of the FK measure. This requires $\b$ to be large.
Proposition \ref{prop 1} is proven in Section 6 under the condition that
$\b$ is larger than $\tilde \b_c$. The coarse graining is only
useful to get the lower bound up to $\tilde \b_c$ and it
could be avoided
if one considers only $\b$ large, in which case, a direct proof without using
the FK representation is possible.
Section 7 is devoted to the proof of Proposition \ref{prop 3}.
Besides its probabilistic interpretation, i.e. the proof of an
exponential tightness property, Proposition \ref{prop 3}
deals with a physical phenomenon of a different nature than the
surface tension : it states that the occurrence of many small
contours is unlikely.
The production of surface tension supposes a balance between
energy and entropy.
It is a general feature of DKS theory that energy is the
dominant factor which rules out the occurrence of small contours.
The techniques developed in \cite{SS} and \cite{IS}
to control the phase of small contours, for the two
dimensional Ising model, are robust enough
to be extended to higher dimensions provided Peierls estimate holds. This observation
was used in \cite{BBP}.
For $\b$ large enough, one could have proceed as in \cite{BBP} and worked only with
the Gibbs measure.
We use an alternative approach borrowed to \cite{I2} and deduce directly estimates
on the phase of small contours
from Pisztora's results \cite{Pisztora1}.
Thus Proposition \ref{prop 3} holds as soon as $\b$ is larger
than $\tilde \b_c$.
As noticed in the papers on Ising model with Kac potentials, the strategy described
above can be applied in any dimension larger or equal to three.
As a final remark, we stress the fact that the above results could be easily extended
to prove a large deviation principle for the measures $\mu^+_{\b,\D_N}$ with action
functional $\F$. This setting was developed in \cite{BBP} and also used for percolation
\cite{Cerf}.
This requires a modification of Proposition \ref{prop 1}
which is described in remark 8.3 at the end
of subsection 8.3.
\section{Coarse graining and mesoscopic scale}
\subsection{The FK representation}
We describe now the FK representation of Ising model. For
a review of FK measures, we refer the reader
to \cite{Pisztora1}, \cite{Grimmett} and \cite{ACCN}.\\
The set of edges is $\E = \big\{ \{x,y\} \; | \; x \sim y \big\}$.
For bond percolation, the configurations $\o$ belong to
$\OM = \{ 0, 1 \}^\E$.
An edge $b$ in $\E$ is open if $\o_b =1$ and closed otherwise.
To any subset $\L$ of $\Z^3$ and $\pi$ included in $\partial \L$, we associate a set of edges
\begin{eqnarray*}
[\L]_e^\pi = \big\{ \{x,y\} \; | \; x \sim y, \ x \in \L, \ y \in \L \cup \pi
\big\},
\end{eqnarray*}
and the space of configurations in $\L$ is $\OM_\L^\pi = \{ 0, 1\}^{[\L]^\pi_e}$.
Let $\o$ be a configuration in $\OM$,
an open path $(x_1, \dots ,x_n)$ is a finite sequence of distinct
nearest neighbors $x_1, \dots ,x_n$ such that on each edge
$\o_{\{x_i , x_{i+1} \}} = 1$.
We write $\{ A \lra B \}$ for the event such that there exists an open
path joining a site of $A$ to one of $B$.
A $*$-connected path $(x_1, \dots ,x_n)$ is a finite sequence of distinct
vertices such that $\|x_k - x_{k +1} \|_2$ is smaller than $\sqrt{3}$
for all $k$.
The connected components of the set of open edges of $\o$ are called
$\o$-clusters. The $\o$-cluster associated to the site $i$ is
denoted by $C_i(\o)$.\\
%or simply $C_i$ when there is no possible confusion.
Let us now describe the FK representation of the Ising model (see
Edwards and Sokal \cite{ES}).
Let $\L$ be a finite subset in $\Z^3$ and $\pi$ a subset of $\partial \L$.
The first step is to introduce a measure on $\OM_\L^\pi$.
A vertex $x$ of $\L$ is called $\pi$-wired if it
is connected by an open path to $\pi$.
We call $\pi$-clusters the clusters defined with respect to the boundary
condition $\pi$ : a $\pi$-cluster is a connected set of open edges in
$\OM_\L^\pi$ and we identify to be the same cluster all the clusters
which are $\pi$-wired, i.e. connected to $\pi$.
For a given $p$ in $[0,1]$, we define the FK measure on $\OM_\L^\pi$ with
boundary conditions $\pi$ by
\begin{eqnarray*}
\P^{\pi,p}_{\L} (\o) = {1 \over Z_{\L}^{\pi,p}}
\left( \prod_{b \in [\L]_e^\pi} ( 1 - p)^{1 - \o_b} p^{\o_b} \right)
2^{c^\pi (\o)},
\end{eqnarray*}
where $Z_{\L}^{\pi,p}$ is a normalization factor and $c^\pi (\o)$
is the number of clusters which are not $\pi$-wired.
If $\pi = \partial \L$ then the boundary conditions are said to be
wired and the corresponding FK measure on $\OM^{\rm w}_{\L}$
is denoted by $\P^{\rm w,p}_{\L}$.
If $\pi = \O$, we write $\P^{\rm f,p}_{\L}$ for the measure on $\OM^{\rm f}_{\L}$.
For any subset $\D$ of $\L$, we denote by $\F_\L^\D$ the $\s$-field generated by
finite dimensional cylinders associated with configurations in
$\OM^{\rm w}_{\L} / \OM^{\rm f}_{\D}$, then strong FKG property (see \cite{Pisztora1})
implies that for every increasing function $g$ supported by $ \OM^{\rm f}_{\D}$
\begin{eqnarray}
\label{FKG 1}
\P^{\rm w,p}_{\L}-{\rm a.s}, \qquad
\P^{\rm f,p}_{\D} (g) \le \P^{\rm w,p}_{\L} (g \, |\, \F_\L^\D) \le \P^{\rm w,p}_{\D} (g).
\end{eqnarray}
In particular, one has
\begin{eqnarray}
\label{FKG 2}
\P^{\rm f,p}_{\D} (g) \le \P^{\rm f,p}_{\L} (g) \le
\P^{\rm w,p}_{\L} (g) \le \P^{\rm w,p}_{\D} (g).
\end{eqnarray}
In order to recover the Gibbs measure $\mu^+_{\b,\L}$, we fix the
percolation parameter $p_\b = 1 - \exp(-\b)$ and
generate the edges configuration $\o$ in $\OM_\L^{\rm w}$ according to
the measure $\P^{\rm w,p_\b}_{\L}$.
Given $\o$, we associate to the wired cluster the sign +1 and equip
randomly each $\o$-cluster with a color $\pm 1$ with probability
${1 \over 2}$ independently from the others.
This amounts to introduce the measure $P_\L^\o$ on $\{-1,1\}^\L$
such that the spin $\s_i =1$ if $C_i(\o)$ is $\pi$-wired and to
be the chosen color of $C_i(\o)$ otherwise.
The Gibbs measure $\mu^+_{\b,\L}$ can be viewed as the first marginal
of the coupled measure $P_\L^\o (\s) \P_\L^{\rm w,p_\b}(\o)$
\begin{eqnarray*}
\forall \s_\L \in \Sigma_\L, \qquad
\mu^+_{\b,\L}(\s_\L) = \int_{\OM_\L^{\rm w}} \, P_\L^\o (\s) \P_\L^{\rm w,p_\b}(d\o).
\end{eqnarray*}
By abuse of notation, the joint measure will be also denoted by $\mu^+_{\b,\L}$.
As a consequence of this representation one has
\begin{eqnarray*}
m_\b = \lim_{N \to \infty} \mu^+_{\b,\D_N}(\s_0) =
\lim_{N \to \infty} \P_{\D_N}^{\rm w,p_\b}(\{ 0 \lra \partial \D_N \} )
= \Theta.
\end{eqnarray*}
In the following, we use $m_\b$ or $\Theta$ depending on the context.
In Theorems \ref{theo 1} and \ref{theo 2}, we consider only the case $\b$ large.
The first reason to do so is to satisfy the hypothesis of Theorem 5.3
of \cite{Grimmett} which implies that for $\b$ large enough
\begin{eqnarray}
\label{Theta}
\lim_{N \to \infty} \P_{\D_N}^{\rm f,p_\b}(\{ 0 \lra \partial \D_N \} )
= \lim_{N \to \infty} \P_{\D_N}^{\rm w,p_\b}(\{ 0 \lra \partial \D_N \} )
= \Theta.
\end{eqnarray}
Throughout the paper we suppose that (\ref{Theta}) holds.
The assumption $\b$ large will also be useful for technical reasons
in the proof of Lemma \ref{lem step 3}.
\subsection{Coarse graining}
We recall the renormalization procedure introduced by Pisztora
\cite{Pisztora1}, \cite{DP} for the FK measure.
For our purposes, it is preferable to use an alternative construction of the coarse
graining \cite{Pisztora2}.
The results of this section hold for $\b$ larger than $\hat \b_c$, where $\hat \b_c$
was defined in \cite{Pisztora1} in terms of slab percolation threshold.
Let $\tilde \b_c$ be the smallest value such that
(\ref{Theta}) is satisfied and $\tilde \b_c \ge \hat \b_c$.
It is conjectured that $\tilde \b_c$ coincides with the critical
value $\b_c$.\\
Let $\g = {1 \over 9}$ and $\a = {1 \over 3} + {\g \over 9}$.
In fact $\g$ could be any positive parameter small enough.
As in subsection 2.3,
we partition the domain $\D_N = \{ 1,N \}^3$ into disjoint boxes
$B(x,N^\a)$ of length $N^\a$ centered in $x$ in ${\cal L}_{N^\a}$
(see (\ref{lattice}) and (\ref{box})).
For each $x$ in ${\cal L}_{N^\a}$, we consider also the bigger
box $B(x,{5 \over 4}N^\a)$ containing $B(x,N^\a)$.
Note that if $x$ and $y$ are $*$-neighbors in ${\cal L}_{N^\a}$
the boxes $B(x,{5 \over 4}N^\a)$ and $B(y,{5 \over 4}N^\a)$ overlap.
Following \cite{Pisztora1}, we introduce events which occur on the box
$B(x,{5 \over 4}N^\a)$ for each $x$ in ${\cal L}_{N^\a}$
\begin{eqnarray*}
U_x = \left\{ \o \in \OM^{\rm w}_{\D_N} \; \big| \; \text{there is a unique crossing
cluster $C^*$ in $B(x,{5 \over 4} N^\a)$} \right\}.
\end{eqnarray*}
A crossing cluster is a cluster which intersects all the faces of the box.
\begin{eqnarray*}
R_x & = & U_x \bigcap \left\{ \o \in \OM^{\rm w}_{\D_N} \; \big| \;
\text{every open path in $B(x,{5 \over 4}N^\a)$ with diameter} \right.\\
& & \qquad \qquad \qquad \qquad
\text{larger than $N^\g$ is contained in $C^*$} \Big\},
\end{eqnarray*}
where the diameter of a subset $A$ of $\Z^3$ is $\sup_{x,y \in A} \|x - y\|_1$.
We also define
\begin{eqnarray*}
O_x & = & R_x \bigcap \Big\{ \o \in \OM^{\rm w}_{\D_N} \; \big| \;
\text{$C^*$ crosses every sub-box of side length $N^\g$}\\
& & \qquad \qquad \qquad \qquad \qquad
\text{ contained in $B(x,{5 \over 4}N^\a)$} \Big\}.
\end{eqnarray*}
Finally, we consider an event which imposes that the density of the crossing
cluster is close to $\Theta$ (see (\ref{Theta})) in $B(x,N^\a)$ with accuracy
$\z >0$
\begin{eqnarray*}
V_x^\z = U_x \bigcap \big\{ \o \in \OM^{\rm w}_{\D_N} \; \big| \;
| C^*| \in [\Theta -\z, \Theta + \z] \, |B(x,N^\a)| \big\}.
\end{eqnarray*}
In the following, parameters $\a,\g$ will be fixed, therefore we omit
the dependence on these parameters in notation.
We will only consider different coarse graining for different values of $\z$.\\
Each box $B(x,N^\a)$ is labelled by the random variable $Y_x^\z(\o)$ depending
only on the configuration $\o$ in $\OM^{\rm w}_{\D_N}$
\begin{eqnarray*}
Y_x^\z (\o) & = & 1 \qquad \text{if} \qquad \o \in O_x \cap V_x^\z,\\
Y_x^\z (\o) & = & 0 \qquad \text{otherwise}.
\end{eqnarray*}
Let $\{x_1,\dots,x_\l \}$ be vertices in ${\cal L}_{N^\a}$ not $*$-neighbors of
$x$, then \cite{Pisztora1} implies that there is an integer $N_\z$ such that
\begin{eqnarray*}
\forall N \ge N_\z, \qquad
\P^{\rm w,p_\b}_{\D_N} (Y_x^\z = 0 \ | \ Y_{x_1}^\z, \dots, Y^\z_{x_\l} ) \le
\exp( - c N^\g) + \exp (- c_\z ' N^\a),
\end{eqnarray*}
where $c_\z '$ depends only on $\z$ and $c$ is a constant.
From \cite{LSS} (Theorem 1.3), we deduce that for $N$ large enough, the random
variables $\{Y_x^\z\}$ are dominated by a Bernoulli product measure $\pi_{\rho_N}$
\begin{eqnarray}
\label{domination Y}
\pi_{\rho_N} ( X = 0) =
\rho_N \le \exp( - c_\z N^\g),
\end{eqnarray}
where $c_\z$ is a positive constant. A similar result was already stated in
\cite{Pisztora1}.\\
The random variables $Y^\z$ are only related to $\o$, therefore the
next step is to define a family of random variables which depend on $(\s,\o)$.
We denote by $\M_x$ the averaged magnetization in the box $B(x,N^\a)$
\begin{eqnarray}
\label{aimantation}
\M_x = \MM_{{x \over N}} = {1 \over N^{3\a} } \sum_{i \in B(x,N^\a)} \s_i.
% \qquad \text{and} \qquad \e_x = \text{sign}(\M_x).
\end{eqnarray}
Pisztora's results \cite{Pisztora1} give a control of the deviation
of the averaged magnetization from its equilibrium values $\pm m_\b$ in
the boxes $B(x,N^\a)$.
If $Y_x^\z = 1$, this deviation comes from the random coloring of the small
clusters (those of diameter less than $N^\g$) included in $B(x,N^\a)$ :
this random coloring is independent of the boxes around
$B(x,N^\a)$.
Let $\z$ be positive and
define the new random variables $\{ Z_x^\z \}$ which depend on the joint
law of $(\s,\o)$
\begin{eqnarray*}
Z_x^\z (\s,\o) & = & {\rm{sign}}(C^*) \qquad {\text{if}}
\qquad Y_x^\z (\o) =1
\ \ {\rm{and}} \ \ | \M_{x} - {\rm{sign}}(C^*) \, m_\b |
< 2 \z,\\
Z_x^\z (\s,\o) & = & 0 \qquad \qquad \quad \text{otherwise}.
\end{eqnarray*}
Combining results of \cite{Pisztora1} and \cite{LSS}, we check that
there is $N_\z$ such that for all $N$ larger than $N_\z$ the
random variables $\{ |Z_x^\z| \}$, taking values in $\{0,1\}$,
are dominated by a Bernoulli product measure $\pi_{\rho_N '}$
\begin{eqnarray}
\label{domination Z}
\pi_{\rho_N '} ( X = 0) = \rho_N ' \le \exp( - c_\z N^\g),
\end{eqnarray}
where $c_\z$ is a positive constant depending only on $\z$.
Since the setting is different from \cite{Pisztora1}, we sketch
the proof in Appendix 8.2.
\subsection{Mesoscopic scale}
In subsection 2.3, we already used a homogenization procedure on the
mesoscopic scale $N^\a$. We introduce now a different mesoscopic representation
which takes into account more details of the microscopic structure.\\
For a given $\z$ positive, we associate to any configuration $(\s,\o)$ in
$\Sigma_{\D_N} \times \OM^{\rm w}_{\D_N}$ the piecewise constant function $T^\z$ on $\T$
\begin{eqnarray}
\label{def T}
\forall r \in \T, \qquad T^\z_r (\s,\o) = Z^\z_x (\s,\o) \qquad \quad
{\text{if}} \ \ \forall i, \quad
-{L \over 2} < Nr_i -x_i \le {L \over 2}.
\end{eqnarray}
If $(\s,\o)$ is close to an equilibrium phase on a mesoscopic scale
then $T^\z$ has the sign of this phase.
The 2 pure phases are represented by functions $T^\z$ constantly equal to
1 or $-1$. From (\ref{domination Z}), one knows that for $\b$ larger than
$\tilde \b_c$
\begin{eqnarray*}
\lim_{N \to \infty} \mu^+_{\b,\D_N}
(\{ T^\z_r = 1, \qquad \forall r \in \T \}) = 1.
\end{eqnarray*}
The next lemma proves that a knowledge of the asymptotic of $T^\z$
is sufficient to control the local magnetization $\MM$.
Therefore to prove Propositions \ref{prop 2}, \ref{prop 1} and
\ref{prop 3}, it will be enough to replace $\MM$ by $T^\z$.
The accuracy of the approximation depends on
the parameter $\z$ which controls the coarse graining.
\begin{lem}
\label{lem magnetization}
For any $\d$ positive, we set
$\z = {1 \over 4} \d$, then
\begin{eqnarray*}
\lim_{N \to \infty} {1 \over N^2} \log
\mu_{\b, \D_N}^+ \left( \int_\T \big| m_\b T_r^\z - \MM_r \big| \, dr
\ge \d \right) = - \infty.
\end{eqnarray*}
\end{lem}
\begin{pf}
One has
\begin{eqnarray*}
\int_\T \big| m_\b T_r^\z - \MM_r \big| \, dr \le
\left( {N^\a \over N} \right)^3 \sum_{B(x,N^\a)} \big| m_\b Z^\z_x - \M_x|,
\end{eqnarray*}
this implies
\begin{eqnarray*}
\int_\T \big| m_\b T_r^\z - \MM_r \big| \, dr \le
\left( {N^\a \over N} \right)^3 \sum_{B(x,N^\a)} 1_{Z^\z_x = 0} + 2 \z.
\end{eqnarray*}
Since $\z$ is small enough
\begin{eqnarray*}
\mu_{\b,\D_N}^+ \left( \int_\T \big| m_\b T_r^\z - \MM_r \big|
\, dr \ge \d \right)
\le \mu_{\b, \D_N}^+ \left( \# \{Z_x^\z = 0 \} \ge {\d \over 2}
N^{3(1-\a)} \right),
\end{eqnarray*}
where $\# \{Z_x^\z = 0 \} $ is the number of boxes with label 0.
Therefore the lemma above will be a consequence of
\begin{lem}
\label{lem blocks0}
For any $\d$ and $\z$ positive
\begin{eqnarray*}
\lim_{N \to \infty} {1 \over N^2} \log \mu_{\b,\D_N}^+
\big( \# \{Z_x^\z = 0 \} \ge \d N^{3(1-\a)} \big)
= - \infty.
\end{eqnarray*}
\end{lem}
\begin{pf}
One has
\begin{eqnarray*}
\mu_{\b,\D_N}^+ \big( \# \{ Z^\z_x = 0 \} \ge \d N^{3(1-\a)} \big)
\le \sum_{k = \d N^{3(1-\a)}}^{N^{3(1-\a)}}
\mu_{\b,\D_N}^+ \big( \# \{ Z^\z_x = 0 \} = k \big).
\end{eqnarray*}
The random variables $|Z^\z_x|$ are dominated by independent variables
(\ref{domination Z}), thus for $N$ large enough
\begin{eqnarray*}
\mu_{\b,\D_N}^+ \big( \# \{ Z^\z_x = 0 \} \ge \d N^{3(1-\a)} \big)
\le 2^{N^{3(1-\a)}} \exp( - c_\z \d N^{3(1-\a) + \g} ).
\end{eqnarray*}
This implies
\begin{eqnarray*}
\mu_{\b,\D_N}^+ \big( \# \{ Z^\z_x = 0 \} \ge \d N^{3(1-\a)} \big)
\le
\exp \big( \ln 2 \, {N^{3(1-\a)}} - c_\z \d N^{2 + {2 \over 3} \g}
\big).
\end{eqnarray*}
As $3(1 -\a) < 2$, the entropic factor is negligible and
the Lemma follows. \end{pf} \end{pf}
\section{Surface tension}
As explained in the introduction, the main problem to derive Wulff
construction is to recover surface tension from general
boundary conditions.
In this section we rewrite the surface tension in terms of the FK
measure and prove that this new expression depends weakly on boundary
conditions.
This expression is reminiscent to the one introduced by Cerf \cite{Cerf} in
the context of percolation.
We keep notation of subsection 2.2.
Throughout this section, we fix the direction $\vec{n}$
and without loss of generality, we set $h = 1$.
We also suppose that ${f(N) \over \log(N)}$ diverges to
infinity as $N$ goes to infinity.
\subsection{First step}
The next lemma will be useful to prove Proposition \ref{prop 1}
\begin{lem}
\label{lem step 1}
Let $\{ \partial^+ \L_N \nlra \partial^- \L_N \}$ be the event such that
there is no open path inside $[\L_N]^{\rm w}_e$ joining $\partial^+ \L_N$
to $\partial^- \L_N$. Then
\begin{eqnarray}
\t(\vec{n}) = \lim_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{w}, p_\b}_{\L_N} \big( \{ \partial^+ \L_N \nlra \partial^- \L_N \} \big).
\end{eqnarray}
\end{lem}
Note that the event $\{ \partial^+ \L_N \nlra \partial^- \L_N \}$
takes only into account the paths inside $\L_N$ and not the identification
produced by wired boundary conditions.
\begin{pf}
We rewrite the quantities in terms of the FK representation.
A well known argument implies that for $p_\b = 1 - \exp(-\b)$
\begin{eqnarray*}
Z^+_{\L_N} = \sum_{\s \in \Sigma_{\L_N}} \prod_{ \in [\L_N]_e^{\rm w}}
\exp \big( \b (\d_{\s_x, \s_y} -1) \big)
= \sum_{\o \in \OM_{\L_N}^{\rm w}} \prod_{b \in [\L_N]_e^{\rm w}}
( 1 - p_\b)^{1 - \o_b} p_\b^{\o_b}
2^{c^{\rm w}(\o)},
\end{eqnarray*}
where $c^{\rm w}(\o)$ is the number of clusters which are not wired.\\
We prove now an equivalent formula for
\begin{eqnarray*}
Z^{+,-}_{\L_N} = \sum_{\s \in \Sigma_{\L_N}} \prod_{ \in [\L_N]_e^{\rm w}}
\exp \big( \b (\d_{\s_x, \s_y} -1) \big),
\end{eqnarray*}
where boundary conditions are equal to 1 on $\partial \L^+_N$ and to $-1$ on
$\partial \L^-_N$.
We get
\begin{eqnarray*}
Z^{+,-}_{\L_N} = \sum_{\s \in \Sigma_{\L_N}} \prod_{ \in [\L_N]_e^{\rm w}}
\big( 1 - p_\b + p_\b \, \d_{\s_x, \s_y} \big),
\end{eqnarray*}
this gives
\begin{eqnarray*}
Z^{+,-}_{\L_N} = \sum_{\s \in \Sigma_{\L_N}} \sum_{\o \in \OM_{\L_N}^{\rm w}}
\prod_{b} ( 1 - p_\b)^{1 - \o_b} p_\b^{\o_b}
\prod_{b = \atop \o_b =1} \d_{\s_x, \s_y}.
\end{eqnarray*}
Therefore
\begin{eqnarray*}
Z^{+,-}_{\L_N} = \sum_{\o \in \OM_{\L_N}^{\rm w}} \prod_{b}
( 1 - p_\b)^{1 - \o_b} p_\b^{\o_b}
\sum_{\s \in \Sigma_{\L_N}} \prod_{b = \atop \o_b =1} \d_{\s_x, \s_y}.
\end{eqnarray*}
The boundary conditions imply that configurations $\o$ containing a path joining
$\partial^+ \L_N$ to $\partial^- \L_N$ are not taken into account.
We keep the definition of wired boundary conditions
identifying all the clusters which touch the boundary
$\partial \L_N$
\begin{eqnarray*}
Z^{+,-}_{\L_N} = \sum_{\o \in \OM^{\rm w}_{\L_N}}
1_{\{ \partial^+ \L_N
{\leftrightarrow \hskip -5.5pt \mid \hskip 5.5pt}
\partial^- \L_N \}}(\o) \prod_{b}
( 1 - p_\b)^{1 - \o_b} p_\b^{\o_b} 2^{c^{\rm w}(\o)}.
\end{eqnarray*}
Taking the ratio ${Z^{+,-}_{\L_N} \over Z^{+}_{\L_N}}$, we recover
$\P^{\rm{w}, p_\b}_{\L_N} \big( \{ \partial^+ \L_N \nlra \partial^- \L_N \} \big)$.
\end{pf}
\subsection{Second step}
In the following, we denote by $\L_N ' = \bar \L(N,N,{1 \over 2} f(N) ) \cap \Z^3$
the parallelepiped included in $\L_N$.
\begin{lem}
\label{lem step 2}
One has
\begin{eqnarray*}
\t(\vec{n}) = \lim_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{w}, p_\b}_{\L_N}
\big( \{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \} \big).
\end{eqnarray*}
\end{lem}
\begin{pf}
By definition $\{ \partial^+ \L_N ' \nlra \partial^- \L_N '
\}$ is included in
$\{ \partial^+ \L_N \nlra \partial^- \L_N \}$.
Therefore, Lemma \ref{lem step 1} implies
\begin{eqnarray*}
\t(\vec{n}) \le \liminf_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{w}, p_\b}_{\L_N}
\big( \{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \} \big).
\end{eqnarray*}
Let us prove the reverse inequality.
The event $\{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \}$
is decreasing and supported by $[\L_N ']_e^{\rm w}$.
Thus (\ref{FKG 2}) gives
\begin{eqnarray}
\label{surf 1}
\P^{\rm{w}, p_\b}_{\L_N}
\big( \{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \} \big)
\ge
\P^{\rm{w}, p_\b}_{\L_N '} \big( \{ \partial^+ {\L_N '} \nlra
\partial^- {\L_N '} \} \big).
\end{eqnarray}
Since surface tension does not depend on the function $f$, Lemma
\ref{lem step 1} implies
\begin{eqnarray*}
\t(\vec{n}) = \lim_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{w}, p_\b}_{\L_N '} \big( \{ \partial^+ {\L_N '} \nlra \partial^- {\L_N '} \} \big).
\end{eqnarray*}
Thus using (\ref{surf 1}), the lemma is proven.
\end{pf}
\subsection{Third step}
Let $\P^{\rm{f},\rm{w}}_{\L_N}$ be the FK measure with wired boundary
conditions on the sides of $\L_N$ parallel to $\vec{n}$ and free on the
sides orthogonal to $\vec{n}$.
\begin{lem}
\label{lem step 3}
There is a constant $\b_0$ independent of $\vec{n}$ and $f$ such
that for any $\b$ larger than $\b_0$
\begin{eqnarray*}
\t(\vec{n}) = \lim_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{f},\rm{w}}_{\L_N}
\big( \{ \partial^+ \L_N ' \nlra \partial^- \L_N '
\} \big).
\end{eqnarray*}
\end{lem}
\begin{pf}
The event $\{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \}$
is denoted by $S_N$.
Applying (\ref{FKG 1}), one observes that
$\P^{\rm{f},\rm{w}}_{\L_N} (S_N) \ge \P^{\rm w, p_\b}_{\L_N} (S_N)$ so that
Lemma \ref{lem step 2} implies
\begin{eqnarray}
\t(\vec{n}) \ge \limsup_{N \to \infty} \, - {1 \over N^2}
\log \P^{\rm{f},\rm{w}}_{\L_N} (S_N).
\end{eqnarray}
To prove the reverse inequality, we introduce the slabs
$\Sl^+$ and $\Sl^-$ in $\L_N$
\begin{eqnarray*}
\Sl^+ & = &
\Big( \bar \L \big( N,N, {1 \over 10} f(N) \big) \, + \,
{3 \over 8} f(N) \vec{n} \Big) \cap \Z^3, \\
\Sl^- & = &
\Big( \bar \L \big( N,N, {1 \over 10} f(N) \big) \, - \,
{3 \over 8} f(N) \vec{n} \Big) \cap \Z^3.
\end{eqnarray*}
For any $\o$ in $\OM_{\L_N}^{\rm w}$, we call a vertex $x$ white if $\o_b =1$
for all edge $b$ incident with $x$ and black otherwise.
Let $A^+_N$ (resp $A^-_N$) be the event such that there is
a surface of white vertices which crosses the slab $\Sl^+$
(resp $\Sl^-$) and separates the two sides of the slab orthogonal to $\vec{n}$.
Equivalently, one can define ${A^+_N}^c$ as the set of configurations
$\o_{\L_N}$ which contain a $*$-connected path of black vertices intersecting
the 2 sides of $\Sl^+$ orthogonal to $\vec{n}$.\\
One has
\begin{eqnarray}
\label{step 3.1}
\P^{\rm{f},\rm{w}}_{\L_N} (S_N) = \P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap A^+_N \cap A^-_N \big) +
\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap ( {A^+_N} \cap {A^-_N} )^c \big).
\end{eqnarray}
First we estimate the last term in the RHS. It is enough to prove an upper
bound for $\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap {A^+_N}^c \big)$.
The events $S_N$ and ${A^+_N}^c$ have distinct supports, so that we can take the
conditional expectation with respect to $\partial \o$, the configuration outside
$\Sl^+$
\begin{eqnarray*}
\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap {A^+_N}^c \big) =
\P^{\rm{f},\rm{w}}_{\L_N} \Big( S_N \, \P^{\partial \o}_{\Sl^+} ({A^+_N}^c) \Big).
\end{eqnarray*}
Since ${A^+_N}^c$ is decreasing, (\ref{FKG 1}) implies
$$
\P^{\partial \o}_{\Sl^+} ({A^+_N}^c) \le
\P^{\rm f, p_\b}_{\Sl^+} ({A^+_N}^c),
$$
where the free boundary conditions are outside the domain $[\Sl^+]^{\rm w}_e$.
In order to control this term, we use a Peierls argument (see \cite{Grimmett}
p. 1486).
By the comparison result of Aizenman, Chayes, Chayes and Newman \cite{ACCN},
the above probability is bounded by the
percolation (product) measure $\bar \P_{\Sl^+}^{p_\b '}$ with
$p_\b ' = { p_\b \over p_\b + (1 -p_\b)^2 }$
\begin{eqnarray*}
\P^{\rm f,p_\b}_{\Sl^+} ({A^+_N}^c) \le \bar \P_{\Sl^+}^{\rm p_\b '} ({A^+_N}^c).
\end{eqnarray*}
We choose $\b$ large enough so that $p_\b '$ is close to 1.
Then Peierls estimate holds and there is a constant $c >0$ such that
the probability that a $*$-connected black path joins 2 vertices $x$
and $y$ on both sides of $\Sl^+$ is less than $\exp( - {c \over 10} f(N))$.
This comes from the fact that the length of such a path is at least
${1 \over 10} f(N)$
\begin{eqnarray*}
\bar \P_{\Sl^+}^{\rm p_\b '} ({A^+_N}^c) \le N^2 \exp \big(- {c \over 10} f(N)\big).
\end{eqnarray*}
One finally obtains
\begin{eqnarray}
\label{step 3.2}
\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap {A^+_N}^c \big) \le
N^2 \exp \big( - {c \over 10} f(N) \big) \P^{\rm{f},\rm{w}}_{\L_N} ( S_N).
\end{eqnarray}
We turn now to the estimate of
$\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap A^+_N \cap A^-_N \big)$.
For a given $\o$ in $A^+_N$, we are going to define the surface $\S^+(\o)$
of white vertices which is the closest to the ``upper'' side of $\Sl^+$.
First we construct the black set $\B^+(\o)$ as follows :
$\B^+(\o)$ contains the vertices in the ``upper'' side of
$\partial \Sl^+$, i.e. the vertices at distance less than 2 of the
hyperplan parallel to $(\vec{e_1},\vec{e_2})$ and centered in
${17 \over 40} f(N) \vec{n}$.
Furthermore $\B^+(\o)$ contains all the
black vertices linked by a $*$-connected path of black vertices
to the boundary of the ``upper'' side of $\Sl^+$.
A vertex $x$ is in $\S^+(\o)$ if it belongs to the boundary of $\B^+(\o)$
and if there is a path of vertices joining $x$ to 0 without crossing
$\B^+(\o)$. By construction the vertices in $\S^+(\o)$ are white.
In the same way we define $\S^-(\o)$ as the surface of white vertices which is
the closest to the ``lower'' side of $\Sl^-$, i.e. to the set of
the vertices at distance less than 2 of the
hyperplan parallel to $(\vec{e_1},\vec{e_2})$ and centered in
$- {17 \over 40} f(N) \vec{n}$.
The region between the surfaces $\S^+$ and $\S^-$ is denoted by
$\S_N$ and by construction $\L_N '$ is included in $\S_N$.
Therefore we can consider the conditional expectation of $S_N$
with respect to the configurations outside $\S_N$ (the measurability
is discussed in \cite{Grimmett} p. 1487).
Since $\S^+$ and $\S^-$ contain only white vertices, one gets
\begin{eqnarray*}
\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap A^+_N \cap A^-_N \big) =
\P^{\rm{f},\rm{w}}_{\L_N} \big( A^+_N \cap A^-_N \
\P^{\rm w, p_\b}_{\S_N} ( S_N) \big).
\end{eqnarray*}
The event $S_N$ is decreasing, thus strong FKG property (\ref{FKG 2}) implies
\begin{eqnarray}
\label{step 3.3}
\P^{\rm{f},\rm{w}}_{\L_N} \big( S_N \cap A^+_N \cap A^-_N \big) \le
\P^{\rm{f},\rm{w}}_{\L_N} \big( A^+_N \cap A^-_N \big) \,
\P^{\rm w, p_\b}_{\L_N} ( S_N) \le \P^{\rm w, p_\b}_{\L_N} ( S_N).
\end{eqnarray}
Combining (\ref{step 3.1}), (\ref{step 3.2}) and (\ref{step 3.3}),
we obtain
\begin{eqnarray*}
\P^{\rm{f},\rm{w}}_{\L_N} (S_N) \le \P^{\rm w, p_\b}_{\L_N} ( S_N)
+ N^2 \exp \big( - {c \over 10} f(N) \big) \P^{\rm{f},\rm{w}}_{\L_N} (S_N).
\end{eqnarray*}
Applying Lemma \ref{lem step 2}, we get
\begin{eqnarray*}
\liminf_{N \to \infty} \, - {1 \over N^2} \log \P^{\rm{f},\rm{w}}_{\L_N} (S_N)
\ge \t(\vec{n}) .
\end{eqnarray*}
The Lemma is completed. \end{pf}
\subsection{Fourth step}
Now, we will modify the boundary conditions and prove that the
surface tension remains unchanged.
The following lemma will be important in the proof of Proposition \ref{prop 2}.
It requires the assumption $\b$ large.
We denote by $\partial^\top \L_N '$ (resp $\partial^\bot \L_N '$) the face of
$\partial^+ \L_N '$ (resp $\partial^- \L_N '$) orthogonal to $\vec{n}$.
Let $\{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \}$ be the event such that
there is no open path inside $[\L_N ']^{\rm w}_e$ connecting $\partial^\top \L_N '$
to $\partial^\bot \L_N '$.
Finally, we set
$$
\d = \limsup_{N \to \infty} {f(N) \over N},
$$
and suppose that $\d$ is finite.
\begin{lem}
\label{lem step 4}
There is a constant $\b_0$ independent of $\vec{n}$ and $f$ such
that for any $\b$ larger than $\b_0$
\begin{eqnarray*}
\limsup_{N \to \infty} \, {1 \over N^2}
\log \left( \sup_\pi \; \P^{\pi,\rm p_\b}_{\L_N}
\big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N '
\} \big) \right) \le - \t(\vec{n}) + c_\b \d,
\end{eqnarray*}
where the constant $c_\b$ depends only on $\b$. The above inequality holds
uniformly over the boundary conditions $\pi$ outside $[\L_N]^{\rm w}_e$.
\end{lem}
\noindent
\begin{pf}
As $\{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \}$ is decreasing,
strong FKG property (\ref{FKG 1}) implies
\begin{eqnarray*}
\sup_{\pi} \P^{\pi,\rm p_\b}_{\L_N}
\big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \} \big)
\le
\P^{\rm f, p_\b}_{\L_N} \big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \} \big),
\end{eqnarray*}
where the free boundary conditions are outside $[\L_N]^{\rm w}_e$.
Note also that
\begin{eqnarray}
\label{lem 4 1}
\P^{f,\rm p_\b}_{\L_N} \big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \} \big)
\le 2^{4 f(N) N} \;
\P^{\rm f, w}_{\L_N} \big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \} \big).
\end{eqnarray}
We fix a configuration $\o$ in $\{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \}$.
The inner boundary of $\L_N$ is defined by
$$
\partial^* \L_N =
\{ x \in \L_N \ | \ \exists y \not\in \L_N, \ y \sim x\}.
$$
For any vertex $x$ on the sides of $\partial^* \L_N$
parallel to $\vec{n}$, we modify the edges of $\o$ incident with $x$ into closed edges
and denote by $\bar \o$ the new configuration.
By construction $\bar \o$ belongs to $\{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \}$.
Noticing that the number of edges which have been modified is smaller than $50 f(N) N$
and using (\ref{lem 4 1}), one has
\begin{eqnarray*}
\P^{\rm f, p_\b}_{\L_N} \big( \{ \partial^\top \L_N ' \nlra \partial^\bot \L_N ' \} \big)
\le \exp( c_\b f(N) N ) \;
\P^{\rm f, w}_{\L_N} \big( \{ \partial^+ \L_N ' \nlra \partial^- \L_N ' \} \big),
\end{eqnarray*}
where $c_\b$ depends only on $\b$.
Using Lemma \ref{lem step 3}, we complete the proof. \end{pf}
\section{Upper bound : Proposition \ref{prop 2}}
Throughout this Section, we fix $u$ in $\BV$ such that $\F(u)$ is finite.
We split the proof into 3 steps.
\subsection{Approximation}
First we suppose that $\partial u$ is included in the interior of $\T$.
The general case will be treated in subsection 5.3.
We approximate the boundary of $u$ with a finite number of parallelepipeds.
Similar Theorems were already stated in \cite{ABCP} and \cite{Cerf}.
The following result is proven in Appendix 8.3.
\begin{theo}
\label{theo ABCP}
For any $\d$ positive, there exists $h$ positive such that there are $\l$ disjoint
parallelepipeds $R^1, \dots, R^{\l}$ included in $\T$ with cubic basis $B^1, \dots, B^\l$ of
size $h$ and height $\d h$.
The basis $B^i$ divides $R^i$ in 2 parallelepipeds $R^{i,+}$ and $R^{i,-}$
and we denote by $\vec{n}_i$ the normal to $B^i$.
Furthermore, the parallelepipeds satisfy the following properties
\begin{eqnarray*}
\int_{R^i} | \chi_{R^i}(r) - u(r)| \, dr \le \d \, \vol(R^i) \quad
{\rm{and}} \quad
\Big| \sum_{i = 1}^{\l} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\F(u) \Big| \le \d,
\end{eqnarray*}
where $\X_{R^i} = 1_{R^{i,+}} - 1_{R^{i,-}}$ and the volume of $R^i$ is
${\vol(R^i)} = \d h^3$.
The area $\int_{B^i} d \H_x$ of $B^i$ is $h^2$.
\end{theo}
We fix $\d$ positive. The approximation procedure implies
\begin{eqnarray*}
\lim_{\d ' \to 0} \limsup_{N \to \infty} { 1 \over N^2} &
\log & \mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in
\V(u,\d ') \big) \le\\
& & \qquad
\limsup_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in
\bigcap_{i =1}^{\l} \, \V(R^i ,2 \d \vol(R^i)) \big), \nonumber
\end{eqnarray*}
where the $\e$-neighborhood of $R^i$ is
\begin{eqnarray*}
\V(R^i, \e) = \left\{ v \in L^1 \big( \T,[-{1 \over m_\b},{1 \over m_\b}] \big)
\ \big| \quad \int_{R^i} | v(r) - \X_{R^i}(r) | \, dr \le \e \right\}.
\end{eqnarray*}
According to Lemma \ref{lem magnetization}, there is $\z$
small enough, depending on $\d$ and $h$, such that
\begin{eqnarray*}
\label{approximation 1}
\limsup_{N \to \infty} { 1 \over N^2}
& \log & \mu^+_{\b,\D_N} \big( {\MM \over m_\b} \in
\bigcap_{i =1}^{\l} \, \V(R^i ,2 \d \vol(R^i)) \big) \le \\
& & \qquad \limsup_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N} \big( T^\z \in
\bigcap_{i =1}^{\l} \, \V(R^i ,3 \d \vol(R^i)) \big).
\end{eqnarray*}
Therefore to prove Proposition \ref{prop 2}, it is enough to show that
\begin{eqnarray*}
\limsup_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N} \big( T^\z \in
\bigcap_{i =1}^{\l} \, \V(R^i ,3 \d \vol(R^i)) \big)
\le - \F(u) + C_{\b,u} \d,
\end{eqnarray*}
where the constant $C_{\b,u}$ depends only on $\b$ and $u$.
Each box can be labelled by 3 values $0,\pm1$, thus
the number of configurations $T^\z$ is less than
$3^{N^{3(1 -\a)}}$. As $3(1 -\a) < 2$, this term has no
entropic effect.
Thus it remains to check
\begin{eqnarray}
\label{upper 0}
\limsup_{N \to \infty} { 1 \over N^2}
\log \left(
\; \sup_{T^\z \in \, {\cal U}^\d} \;
\mu^+_{\b,\D_N} \big( \{ T^\z \} \big) \right)
\le - \F(u) + C_{\b,u} \d,
\end{eqnarray}
where ${\cal U}^\d$ denotes $\bigcap_{i =1}^{\l} \V(R^i ,3 \d \vol(R^i))$ and
$\{ T^\z \}$ is the set of configurations $(\s,\o)$ which realize $T^\z$.
\subsection{Minimal section argument}
The microscopic domain associated to $R^i$ is $R^i_N = N R^i \cap \D_N$.
We also set $R^{i,+}_N = N R^{i,+} \cap \D_N$ and
$R^{i,-}_N = R^i_N / R^{i,+}_N$.
Let ${\Bbb L}^i_N$ be the subset of boxes $B(x,N^\a)$ intersecting $R^i_N$.
The number of boxes intersecting $R^{i,+}_N$ is
\begin{eqnarray}
\label{NN}
\NN_N^{i,+} = N^{3(1-\a)} \vol(R^i) \left( {1 \over 2} + o(N^{2(1-\a)}) \right),
\end{eqnarray}
where the error term $o(N^{2(1-\a)})$ goes to 0 as $N$ increases.
This error is due to the fact that the partition may not be exact on the sides of $R^{i,+}_N$.
A similar estimate holds for $\NN_N^{i,-}$, the number of boxes intersecting $R^{i,-}_N$.
Let us fix $T^\z$ in $\bigcap_{i =1}^{\l} \V(R^i ,3 \d \vol(R^i))$.
To any configuration $(\s,\o)$ in $\{ T^\z\}$, we associate the set of {\it bad}
boxes which are the boxes in ${\Bbb L}^i_N$ labelled by $Z^\z_x =0$ and the ones
intersecting $R^{i,+}_N$ (resp $R^{i,-}_N$) labelled by $Z^\z_x =-1$ (resp $Z^\z_x =1$).\\
We will now use the $L^1$-constraint to derive bounds on the number
of {\it bad} boxes.
The number of boxes in ${\Bbb L}^i_N$ not included in $R^i_N$
is smaller than $50 h^2 N^{2(1 -\a)}$, therefore
\begin{eqnarray*}
\big| N^{3(1-\a)} \int_{R^{i,+}} |T^\z_r - 1| \, dr -
\sum_{B(x,N^\a) \cap R^{i,+}_N \not = \O} | Z^\z_x -1 | \big|
\le 100 h^2 N^{2(1 -\a)}.
\end{eqnarray*}
Since $T^\z$ belongs to $\V(R^i ,3 \d \vol(R^i))$, one gets from (\ref{NN})
that for $N$ large enough the number of {\it bad} boxes in $R^{i,+}_N$ is
smaller than $10 \d \NN_N^{i,+}$.
In the same way, we check that the number of {\it bad} boxes in $R^{i,-}_N$ is
smaller than $10 \d \NN_N^{i,-}$.
Let ${R^i} \, '$ be the parallelepiped included in $R^i$ with basis
$B^i$ and height ${\d \over 2} h$.
Its microscopic counterpart is ${R^i_N}'$.
We will apply the minimal section argument introduced in
\cite{BBBP} and relate the expectation of $\{ T^\z\}$ to the one of
$\bigcap_{i =1}^\l \{ \partial^\top {R^i_N} ' \nlra \partial^\bot {R^i_N} '\}$.\\
For any integer $k$, we set $B^{i,k} = B^i + 10 {k \over N^{1-\a} } \vec{n}_i$.
Let $B^{i,k}_N$ be the microscopic subset of ${R^i_N} '$ associated
to $B^{i,k}$
\begin{eqnarray*}
B^{i,k}_N = \big\{ j \in {R^i_N} ' \ | \
\exists r \in B^{i,k}, \qquad \|j - Nr\|_1 \le 10 \big\}.
\end{eqnarray*}
We define $\B_i^k$ as the smallest connected set of mesoscopic boxes containing
\begin{eqnarray*}
\big\{ B(y,N^\a) \in {\Bbb{L}}^i_N \ | \qquad B(y,N^\a) \cap B^{i,k}_N \ne \O \big\}.
\end{eqnarray*}
By construction the $\B_i^k$ are disjoint surfaces of boxes.
For $k$ positive, let $n_i^+(k)$ be the number of {\it bad} boxes in
$\B_i^k$ and define
\begin{eqnarray*}
n^+_i = \min \big\{ n_i^+(k); \qquad 0< k < {\d h \over 30} N^{1-\a} \big\}.
\end{eqnarray*}
Call $k^+$ the smallest location where the minimum is achieved and define
the minimal section as $\B_i^{k^+}$.
For $k$ non positive, we denote by $\B_i^{k^-}$ the minimal section in ${R^{i,-}_N}$
and $n_i^-$ the number of {\it bad} boxes in $\B_i^{k^-}$.\\
For any configuration $(\s,\o)$ in $\{ T^\z\}$, we will check that
the total number of {\it bad} boxes is bounded by
\begin{eqnarray}
\label{upper 3}
\sum_{i =1}^\l n^+_i + n_i^- \le C_u \d N^{2(1 -\a)},
\end{eqnarray}
where $C_u$ is a constant depending only on $u$.
By definition, one has
\begin{eqnarray*}
{\d h \over 30} N^{1-\a} \, n^+_i \le
\sum_{B(x,N^\a) \cap R^{i,+}_N \not = \O}
1_{Z^\z_x \not = 1} \le 10 \d \NN_N^{i,+}.
\end{eqnarray*}
For $N$ large enough, (\ref{NN}) implies that
$n_i^+ \le 10^3 \d h^2 N^{2(1-\a)}$.
Note that $h^2$ is in fact the area of $B^i$, therefore the approximation
procedure implies that $\l h^2$ is bounded by a constant
depending on the perimeter of $\partial u$.
Thus (\ref{upper 3}) holds.\\
We are now going to use all the previous estimates.
We define
\begin{eqnarray*}
\A = \big\{ \o \in \OM^{\rm w}_{\D_N} \ \big| \quad
\exists \s \ {\rm such \ that} \ (\s,\o) \in \{ T^\z \} \big\}.
\end{eqnarray*}
Any configuration $\o$ in $\A$ will be mapped into $\bar \o$ by the following procedure.
For any {\it bad} box $B(x,N^\a)$ in the minimal sections,
we change the open edges of $\o$ located on the
sides of the box $B(x,{5 \over 4} N^\a)$ into closed edges.
The new configuration $\bar \o$ belongs to
$\{ \partial^\top {R^i_N} ' \nlra \partial^\bot {R^i_N} ' \}$,
because any open path of $\o$ which joins $\partial^\bot {R^i_N} '$ to
$\partial^\top {R^i_N} '$ intersects at least one of the minimal
section on a {\it bad} block and therefore is cut by the above procedure.
Let ${\bf C}$ be an open path of $\o$ joining
$\partial^\top {R^i_N} '$ to $\partial^\bot {R^i_N} '$ and suppose
that ${\bf C}$ crosses the minimal sections without intersecting a
{\it bad} box.
Then ${\bf C}$ intersects the boxes $B(x^+,N^\a)$ and $B(x^-,N^\a)$ in $\B_i^{k^+}$
and $\B_i^{k^-}$ with labels $Y^\z_{x^+} = Y^\z_{x^-} =1$.
This would imply that the crossing clusters of $B(x^+,N^\a)$ and $B(x^-,N^\a)$
are connected to ${\bf C}$, so that $Z^\z_{x^+} = Z^\z_{x^-}$.
Therefore one of these boxes has to be a {\it bad} box.
Around the {\it bad} boxes, we change at most $20 (n^+_i + n^-_i) N^{2\a}$ edges.
From (\ref{upper 3}) the total number of edges involved in the previous procedure is bounded
by $100 C_u \d N^2$. Therefore we get
\begin{eqnarray*}
\mu^+_{\b,\D_N} ( \{ T^\z \} ) \le
\P^{\rm{w},p_\b}_{\D_N} \big( \A \big) \le
\exp \big( C_{\b,u} \d N^2 \big)
\P^{\rm{w},p_\b}_{\D_N} \big(
\bigcap_{i =1}^\l \{ \partial^\top {R^i_N} ' \nlra
\partial^\bot {R^i_N} ' \} \big),
\end{eqnarray*}
where the constant $ C_{\b,u}$ depends only on $\b$ and $u$.
Conditioning outside each domain $R^i_N$ and using Lemma \ref{lem step 4}, we derive
\begin{eqnarray*}
\limsup_{N \to \infty} \; {1 \over N^2}
\log \left(
\; \sup_{T^\z \in \, {\cal U}^\d} \;
\P^{\rm{w},p_\b}_{\D_N} \big( \{ T^\z \} \big) \right)
\le
- \sum_{i =1}^\l \int_{B_i} \t(\vec{n}_i) \, d \H_x + C_{\b,u} \, \d
+ c_\b \l h^2 \, \d ,
\end{eqnarray*}
where $c_\b$ was defined in Lemma \ref{lem step 4}.
Noticing that $\l h^2$ is bounded in terms of the perimeter of $u$ and
using Theorem \ref{theo ABCP}, we derive (\ref{upper 0}).
\subsection{Boundary conditions}
Let $U$ be the intersection of the reduced boundary $\partial^* u$ and of $\partial \T$.
Suppose that $U$ has a positive 2 dimensional Hausdorff measure.
In this case we cannot approximate the surface $U$ as in Theorem \ref{theo ABCP}
with parallelepipeds included in $\T$.
We state a variant of Theorem \ref{theo ABCP} proven in Appendix 8.3
\begin{theo}
\label{theo ABCP 1}
For any $\d$ positive, there exist $h$ positive and $\l$ disjoint squares $B^1, \dots, B^\l$
in $\partial \T$ of size $h$ and normal $\vec{n}_i$ such that
\begin{eqnarray*}
\Big| \sum_{i = 1}^{\l} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\int_{U} \t(\vec{n}_x) \, d \H_x \Big| \le \d.
\end{eqnarray*}
Furthermore, there are $\l$ disjoint parallelepipeds $R^1, \dots, R^{\l}$ included in $\T$
such that one of the face of $R^i$ is $B^i$ and the height of $R^i$ is $\d h$.
The parallelepipeds also satisfy
\begin{eqnarray*}
\forall i \le \l, \qquad
\int_{R^i} | 1 + u_r | \, dr \le \d \, \vol(R^i).
\end{eqnarray*}
\end{theo}
The proof of the upper bound is based on local estimates in each
parallelepiped, thus we will simply explain how to adapt the previous
proof to obtain
\begin{eqnarray}
\label{upper 15}
\limsup_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N} \big( T^\z \in
\bigcap_{i =1}^{\l} \, \V(R^i , \d \vol(R^i)) \big)
\le - \int_{U} \t(\vec{n}_x) \, d \H_x + C_{\b,u} \d,
\end{eqnarray}
where $C_{\b,u}$ is a constant and $\V(R^i , \d \vol(R^i))$ is
\begin{eqnarray*}
\V(R^i, \d \vol(R^i)) =
\left\{ v \in L^1 \big( \T,[-{1 \over m_\b},{1 \over m_\b}] \big)
\ \big| \quad \int_{R^i} | v_r + 1 | \, dr \le \d \vol(R^i) \right\}.
\end{eqnarray*}
Combining estimates (\ref{upper 0}) and (\ref{upper 15}), one derives Proposition
\ref{prop 2} for any function of bounded variation $u$ in $\T$.\\
Let $R^i_N$ be the microscopic set associated to $R^i$.
The set of {\it bad} boxes is the set of boxes $B(x,N^\a)$ intersecting
$R^i_N$ and labelled by 0 or 1.
Using the $L^1$ constraint, we see that the number of {\it bad} boxes is smaller than
$10 \d^2 h^3 N^{3(1 -\a)}$.
Let ${R^i} \, '$ be the parallelepiped included in $R^i$ with height ${\d h \over 2}$
and such that one of its faces is $B^i$.
The previous argument implies that
the minimal section contains less than $100 \d h^2 N^{2(1 -\a)}$ bad boxes.
Therefore we can cut the wired open paths which cross the minimal section and obtain
for $N$ large enough
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big( T^\z \in
\bigcap_{i =1}^{\l} \, \V(R^i , \d \vol(R^i)) \big) \le
\exp \big( C_{\b,u} \d N^2 \big)
{\P}^{\rm w}_{\D_N} \big( \bigcap_{i =1}^\l \{ \partial^\top {R^i_N} ' \nlra
\partial^\bot {R^i_N} ' \} \big).
\end{eqnarray*}
Let ${\bar R}^i$ be the union of $R^i$ and of the parallelepiped in $\T^c$ with height
${\d h \over 2}$ and one of its faces is equal to $B^i$.
Denote by ${\bar R}^i_N$ the corresponding microscopic domain, then by inequality
(\ref{FKG 1}) one has for any boundary condition $\pi$ outside ${\bar R}^i_N$
\begin{eqnarray*}
{\bar \P}^{\pi , \rm w}_{R^i_N} \big( \{ \partial^\top {R^i_N} ' \nlra
\partial^\bot {R^i_N} ' \} \big)
\le
{\P}^{\rm f, p_\b}_{{\bar R}^i_N} \big( \{ \partial^\top {R^i_N} ' \nlra
\partial^\bot {R^i_N} ' \} \big),
\end{eqnarray*}
where ${\bar \P}^{\pi, \rm w}_{R^i_N}$ is the gibbs measure with wired boundary conditions
on the face of $R^i_N$ which coincides with $\partial \D_N$ and $\pi$ otherwise.
This enables us to apply Lemma \ref{lem step 4} and to recover (\ref{upper 15}).
\section{Lower Bound : Proposition \ref{prop 1}}
The proof rests only on Lemma \ref{lem step 1}, therefore Proposition \ref{prop 1}
holds as soon as $\b$ is larger than ${\tilde \b}_c$ (see subsection 3.2).
The proof is divided into 2 steps.
\subsection{Approximation procedure}
We first state an approximation theorem which will be proven in Appendix 8.3.
We call polyhedral set, a set which has a boundary included in the union of
a finite number of hyperplans.
\begin{theo}
\label{theo approx}
For any $\d$ positive, there exists a polyhedral set $W$ such that
$$
\1_W \in \V( \1_{\W_m}, {\d \over 3})
\qquad {\rm and} \qquad
\big| \F(W)-\F(\W_m) \big| \le {\d \over 2}.
$$
For any $h$ small enough there are $\l$ disjoint cubes
$R^1, \dots, R^{\l}$ of size $h$ with basis $B^1, \dots, B^{\l}$ included in $\partial W$.
Furthermore, the squares $B^1, \dots, B^{\l}$ cover $\partial W$ up to a set of measure
less than $\d$ denoted by $U^\d =\partial W / \bigcup_{i =1}^\l B^i$ and they satisfy
\begin{eqnarray*}
\Big| \sum_{i = 1}^{\l} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\F(\W_m) \Big| \le \d,
\end{eqnarray*}
where the normal to $B^i$ is denoted by $\vec{n}_i$.
\end{theo}
We fix $\d$ positive and choose a set $W$ approximating $\W_m$, then
\begin{eqnarray*}
\big\{ {\MM \over m_\b} \in \V( T^\z , {\d \over 3}) \big\}
\bigcap
\big\{ T^{\z} \in \V(\1_{W},{\d \over 3}) \big\}
\subset
\big\{ {\MM \over m_\b} \in \V(\1_{\W_m},{\d}) \big\}.
\end{eqnarray*}
Lemma \ref{lem magnetization} implies that there exists $\z$ such that
the event $\big\{ {\MM \over m_\b} \not\in \V(T^{\z},{\d \over 3}) \big\}$
has a probability which vanishes exponentially fast, therefore
\begin{eqnarray*}
\label{lower 1}
\liminf_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N}
\big( {\MM \over m_\b} \in \V(\1_{\W_m},\d) \big)
\ge
\liminf_{N \to \infty} { 1 \over N^2}
\log \mu^+_{\b,\D_N} \big( T^{\z} \in \V(\1_{W},{\d \over 3}) \big).
\end{eqnarray*}
It remains to find a lower bound for the term in the RHS.
For any $\e$ positive, we construct a shell around $\partial W$ which splits $\T$ into
2 domains
$$
\S_\e = \{r \in \T \ | \ \text{dist}(r,\partial W) \le \e \}.
$$
We set
\begin{eqnarray*}
W^+_\e = \{r \in W^c \ | \ \text{dist}(r,\partial W) \ge \e \}
\quad \text{and} \quad
W^-_\e = \{r \in W \ | \ \text{dist}(r,\partial W) \ge \e \}.
\end{eqnarray*}
Let $W^\pm_{\e,N}$ be the set of mesoscopic boxes included in $N W^\pm_\e \cap \D_N$.
We fix $\e$ such that the volume of $\S_\e$ is smaller than ${\d \over 10}$
and choose $h$ smaller than ${\e \over 2}$.
\subsection{Exponential bound}
The microscopic domain associated to the cube $R^i$ is denoted by $R^i_N = N R^i \cap \D_N$.
We set $\A_N = \bigcap_{i =1}^\l \{ \partial^+ R^i_N \nlra \partial^- R^i_N \}$.
The microscopic domain
$$
U_N^\d = \big\{ x \in \D_N \ \big| \ \exists r \in U^\d, \
\|x - N r \|_1 \le 10 \big\}
$$
is an enlargement of the surface $U^\d$ defined in the approximation procedure.
We introduce $\B_N$, the set of configurations with closed edges in $U_N^\d$.
Hypotheses on $U^\d$ imply that $\B_N$ is supported by at most $10 \d N^2$
edges.\\
We decompose $\A_N \cap \B_N$ into 2 disjoint sets
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \big) & = &
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \cap
\{ \forall B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \quad |Z_x^\z| = 1 \} \big)\\
&+&
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \cap
\{ \exists B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \quad Z_x^\z =0 \} \big).
\end{eqnarray*}
We first estimate the last term in the RHS. By definition the events
$\A_N \cap \B_N$ and
$\{ (\s,\o) \ | \ \exists B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \ Z_x^\z = 0 \}$
have disjoint supports. Taking the conditional
expectation with respect to $\A_N \cap \B_N$ and using the stochastic domination
(\ref{domination Z}) (see also remark 8.1), we get
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big(
\{ \exists B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \quad Z_x^\z = 0 \}
\; \big| \; \A_N \cap \B_N \big) \le N^{3(1-\a)} \exp(- c_\z N^\g).
\end{eqnarray*}
Therefore, for $N$ large enough
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \big) \le 2
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \cap
\{ \forall B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \quad |Z_x^\z| = 1 \} \big).
\end{eqnarray*}
By construction, no configuration $\o$ of $\A_N \cap \B_N$ contains an open path joining
the 2 connected components of $\D_N / \big( \bigcup_{i=1}^\l R^i_N \bigcup U_N^\d \big)$.
Therefore any configuration in $\A_N \cap \B_N \cap
\{ \forall B(x,N^\a) \subset W^+_{\e,N} \cup W^-_{\e,N}; \quad |Z_x^\z| = 1 \}$ contains
2 disconnected microscopic crossing clusters.
The cluster connected to $\partial \D_N$ is denoted by $C^+$ and the other
one by $C^-$.
The wired constraint imposes the sign 1 to $C^+$. With probability
${1 \over 2}$ we choose the sign of $C^-$ to be $-1$.
We define the event
\begin{align*}
\begin{split}
\C_N (\s,\o) = \A_N \bigcap \B_N \bigcap
&\{ \forall B(x,N^\a) \subset W^+_{\e,N}; \quad Z_x^\z = 1 \}\\
& \quad \bigcap \{ \forall B(x,N^\a) \subset W^-_{\e,N}; \quad Z_x^\z = -1 \}.
\end{split}
\end{align*}
then
$\mu^+_{\b,\D_N} (\A_N \cap \B_N ) \le 4 \mu^+_{\b,\D_N} (\C_N)$.
Thus for any configuration $(\s,\o)$ in $\C_N$,
the function $T^\z (\s,\o)$ is equal to 1 on $W^+_\e$ and to $-1$ on $W^-_\e$
(see (\ref{def T})).
Since the volume of $\S_\e$ is less than ${\d \over 10}$, we have
\begin{eqnarray*}
\C_N \subset
\big\{ (\s,\o) \ \big| \ T^{\z} (\s,\o) \in \V(\1_W, {\d \over 3}) \big\},
\end{eqnarray*}
this leads to
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big(
T^{\z} (\s,\o) \in \V(\1_W, {\d \over 3}) \big) \ge
{1 \over 4}
\mu^+_{\b,\D_N} \big(\A_N \cap \B_N \big).
\end{eqnarray*}
As $\A_N \cap \B_N$ depends only on the variable $\o$, we replace the coupled measure by
the FK measure
$\P^{\rm{w}, p_\b}_{\D_N}$.
The support of $\B_N$ contains less than $10 \d N^2$ edges, so that
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big( T^\z \in \V(\1_W, {\d \over 3}) \big) \ge {1 \over 4} \exp( - c_\b \d N^2)
\P^{\rm{w}, p_\b}_{\D_N} \big( \bigcap_{i =1}^\l \{ \partial^+ R^i_N \nlra \partial^- R^i_N \} \big),
\end{eqnarray*}
where $c_\b$ is a constant depending on $\b$.
The events $\{ \partial^+ R^i_N \nlra \partial^- R^i_N \}$ occur on disjoint supports,
taking the conditional
expectation with respect to the configuration $\partial \o_i$ outside $R^i_N$, we have
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big( T^\z \in \V(\1_W,{\d \over 3}) \big) \ge
{ \exp( - c_\b \d N^2) \over 4} \P^{\rm{w}, p_\b}_{\D_N} \big( \prod_{i =1}^\l
\P^{\partial \o_i}_{R^i_N} (\{ \partial^+ R^i_N \nlra \partial^- R^i_N \}) \big).
\end{eqnarray*}
Since the events $\{ \partial^+ R^i_N \nlra \partial^- R^i_N \}$ are non increasing,
(\ref{FKG 1}) implies
\begin{eqnarray*}
\mu^+_{\b,\D_N} \big( T^\z \in \V(\1_W,{\d \over 3}) \big) \ge
{\exp( - c_\b \d N^2) \over 4} \prod_{i =1}^\l
\P^{\rm w, p_\b}_{R^i_N} \big(\{ \partial^+ R^i_N \nlra \partial^-
R^i_N \} \big).
\end{eqnarray*}
Taking the limit as $N$ goes to infinity and using Lemma \ref{lem step 1},
we obtain
\begin{eqnarray*}
\liminf_{N \to \infty} { 1 \over N^2} \log
\mu^+_{\b,\D_N} \big( T^\z \in \V(\1_W, {\d \over 3}) \big) \ge
- \sum_{i =1}^\l \int_{B^i} \t (\vec{n}_i) d \H_x - c_\b \d.
\end{eqnarray*}
The proof is completed by letting $\d$ go to 0. \qed
\section{Exponential tightness : Proposition \ref{prop 3}}
In view of Lemma \ref{lem magnetization}, Proposition \ref{prop 3} will be a consequence of
\begin{lem}
Let $\b$ be larger than $\tilde \b_c$ (see subsection 3.2), then there is $C_\b$ such
that for any $a$ positive
\begin{eqnarray*}
\forall \z,\d>0, \qquad
\limsup_{N \to \infty} {1 \over N^2} \log \mu_{\b, \D_N}^+
\big( T^\z \in \V(K_a,\d)^c \big)
\le - C_\b \, a.
\end{eqnarray*}
\end{lem}
A high density of small contours can be interpreted on the mesoscopic scale
as a high density of random variables $Z^\z =0$. Such events are ruled out by
Lemma \ref{lem blocks0}.
If $T^\z$ belongs to $\V(K_a,\d)^c$ and does not contain many boxes labelled by
$Z^\z =0$, then there are mainly mesoscopic sets of constant sign.
By definition of the variables $Z^\z$, two sets of different signs are disconnected
on the microscopic level
and therefore are separated by a layer of boxes with label $Z^\z =0$.
The problem is that the probability of the event $\{Z^\z_x =0\}$ may
only be of the order of $\exp( -N^\g)$ which is much higher than the expected
surface order $\exp( -N^{2\a})$.
Thus we introduce another coarse graining in order to recover the surface order.
The scheme of the following proof was suggested by D. Ioffe.
\begin{pf}
As noticed in subsection 5.2, the number of configurations $T^\z$ is less than
$3^{N^{3(1 -\a)}}$. Thus it is enough to fix $T^\z$ in $\V(K_a,\d)^c$ and
to estimate $ \mu_{\b,\D_N}^+ ( \{ T^\z \} )$.
Lemma \ref{lem blocks0} enables us to consider only
configurations $T^\z$ with a number of boxes labelled by 0 smaller
than $\d N^{3(1-\a)}$. This amounts to say that
\begin{eqnarray}
\label{7.1}
\int_\T \, 1_{T^\z_r = 0} \, dr \le \d.
\end{eqnarray}
Each realization of $T^{\zeta}$ splits $\T$ into $\T = \T_{+} \cup \T_{-} \cup \T_{0}$,
where $T^\z$ is constantly equal to $\pm 1$ on $\T_{\pm 1}$ and to 0 on $\T_0$.
From (\ref{7.1}), the measure of $\T_{0}$ is smaller than $\d$.
The microscopic counterparts of ${\T}_{+}$ and
${\T}_{-}$ will be denoted by $\D_{N,+}$ and $\D_{N,-}$.
Moreover as $T^\z$ belongs to $\V(K_a,\d)^c$, for any regular set $A$ of $\T$
such that ${\T}_{-} \subset A \subset {\T} \setminus {\T}_{+} $
\begin{eqnarray}
\label{7 2}
\int_{\partial A} \, d \H_x \ge a,
\end{eqnarray}
where $\int_{\partial A} \, d \H_x$ is the perimeter of $\partial A$.
Note that for each configuration in ${\cal V} (K_a,\d )^c$ the set
$\T_-$ is not empty.\\
Let $L$ be an integer large enough
which divides $N^{\a}$ and is independent of $N$.
We partition $\D_N$ into boxes $B(i,L)$ where $i$ is in
${\cal L}_L$. We also introduce the boxes $B(i,{5 \over 4}L)$ and
following subsection 3.2, define a coarse graining on the scale $L$.
Let $\{ y_i\}$ be the family of
random variables equal to 1 if the event $O_i$
(for the box $B(i,{5 \over 4}L)$) is satisfied and 0 otherwise.
We define the microscopic set $\tilde{A}_N$ as the union of $\D_{N,-}$
and of the boxes $B(i,L)$ labelled by 1 such that there is a $*$-connected
path of boxes $B(j,L)$ labelled by 1 joining $B(i,L)$ to $\D_{N,-}$.
For any configuration $(\s,\o)$ in $\{ T^\z \}$
there is no microscopic path connecting $\D_{N,+} \cup \partial \D_N$ to $\D_{N,-}$.
Therefore there is no $*$-connected path of boxes $B(j,L)$ labelled by 1
connecting $\D_{N,+} \cup \partial \D_N$ to $\D_{N,-}$.
This implies that
$$
\D_{N,-}\subset\tilde{A}_N\subset \D_N\setminus\D_{N,+}.
$$
We define also $G_N =\D_N\setminus\tilde{A}_N$. Let
$G_N ~=~ \bigcup_{i =1}^\l G_{N,i}$
be the decomposition of $G_N$ into maximal connected components composed
of boxes $B(j,L)$.
The components $G_{N,i}$ such that $|G_{N,i}|0$ and $\s_i =-1$ otherwise.
The boundary conditions above $D_M$ are equal to $1$ and to $-1$ below.
This definition coincides with definition \ref{tau}. This is straightforward from the
argument developed by \cite{miracle} (Theorem 2). We recall this argument for completeness.
First, we show that
\begin{eqnarray}
\label{surface 1}
\t '(\vec{n}) \le \liminf_{N \to \infty} \; - \log { Z^{+,-}_{\L_N} \over Z^+_{\L_N}}.
\end{eqnarray}
Let $M, N$ be 2 integers such that $N \ll M$. We shall write
$$
F_M ' = - \log { Z^{+,-}_{\D_M} \over Z^+_{\D_M}}
\qquad \text{and} \qquad
F(\L_N) = - \log { Z^{+,-}_{\L_N} \over Z^+_{\L_N}},
$$
where the parallelepiped $\L_N$ was introduced in definition \ref{tau}.
We tile $D_M$ with $k_M$ squares $B^i$ of side length $h N$ and at distance 10 from each others.
These squares are chosen such that the area of $D_M$ not covered by
$\bigcup_{i =1}^{k_M} B^i$ is smaller than $C(M f(N) + k_M h N)$,
where $C$ is some positive constant independent of $M$ and $N$.
If the center of $B^i$ does not coincide with a site on the lattice, we translate $B^i$ at a distance
smaller than 1 such that the center of its translate ${B^i} '$ belongs to $\Z^3$.
Let $\L^i_N$ be the parallelepiped of basis ${B^i} '$ deduced from $\L_N$ by translation.
There is a choice of the squares $B^i$ such that all the parallelepipeds $\L^i_N$
are included in $\D_M$. Note that $F(\L_N^i) = F(\L_N)$.
Inequality (C.2) of \cite{miracle} implies
\begin{eqnarray*}
F_M ' \le \sum_{i =1}^{k_M} F(\L_N^i) + K C \big( M f(N) + k_M h N \big),
\end{eqnarray*}
where $K$ is some positive constant.
%The error $K$ comes from the fact that the centers of
%$\L_N^i$ do not necessarily coincide with a site on the lattice.
Since $|S_M - k_M h^2 N^2| \le C(M f(N) + k_M h N)$, one gets as $M$ goes to infinity
\begin{eqnarray*}
\t '(\vec{n}) \le {1 \over h^2 N^2} F(\L_N) + {K C \over h N}.
\end{eqnarray*}
Letting $N$ go to infinity, we obtain (\ref{surface 1}).
The reverse inequality
\begin{eqnarray}
\label{surface 2}
\t '(\vec{n}) \ge \limsup_{N \to \infty} \; - \log { Z^{+,-}_{\L_N} \over Z^+_{\L_N}}
\end{eqnarray}
is derived in the
same way by choosing $M \ll N$ and by partitioning the basis of $\L_N$
with translates of $D_M$.
Combining (\ref{surface 1}) and (\ref{surface 2}), we see that
$\t(\vec{n}) = \t ' (\vec{n})$.
\subsection{Stochastic domination}
The following result is a consequence of \cite{I2} and of the proof of Theorem 1.1 of
\cite{Pisztora1}, we sketch the proof for completeness
\begin{theo}
For any $\z$ positive, there is $N_\z$ such that for all $N$ larger than $N_\z$,
the family of random variables $|Z^\z_x |$ is dominated by a product Bernoulli
measure $\pi_{\rho_N '}$ (see (\ref{domination Z})).
\end{theo}
\begin{pf}
According to \cite{LSS}, it is enough to check that
\begin{eqnarray*}
\mu^+_{\b,\D_N} \left( |Z^\z_x| = 0 \ \big| \ |Z^\z_{x_1}| = \e_1, \dots,
|Z^\z_{x_\l}| = \e_\l \right) \le \exp( - c_\z N^{\g}),
\end{eqnarray*}
where the vertices $\{x_1, \dots , x_\l \}$ are not $*$-neighbors of $x$
in ${\cal L}_{N^\a}$ and each $\e_i$ belongs to $\{0,1\}$.
Using notation from subsection 3.1, one has
\begin{eqnarray*}
& &\mu^+_{\b,\D_N} \big( |Z^\z_x| = 0, |Z^\z_{x_1}| = \e_1, \dots,
|Z^\z_{x_\l}| = \e_\l \big)
\le \\
& & \quad \P^{\rm{w},p_\b}_{\D_N} \big( Y^\z_x = 1; \
P^\o_{\D_N} \big( |\M_x - {\rm{sign}}(C^*) m_\b | > 2 \z,
|Z^\z_{x_1}| = \e_1, \dots, |Z^\z_{x_\l}| = \e_\l \big) \big)\\
& & \qquad +
\P^{\rm{w},p_\b}_{\D_N} \big( Y^\z_x = 0,
Y^\z_{x_1} = \e_1, \dots, Y^\z_{x_\l} = \e_\l; \
P^\o_{\D_N} \big( |Z^\z_{x_1}| = \e_1, \dots, |Z^\z_{x_\l}| = \e_\l \big) \big).
\end{eqnarray*}
Because of stochastic domination (\ref{domination Y}), the last term in the RHS
is bounded by
$$
\exp( - c_\z N^{\g}) \, \mu^+_{\b,\D_N} \big( |Z^\z_{x_1}| = \e_1, \dots,
|Z^\z_{x_\l}| = \e_\l \big).
$$
In order to control the other term, we have to take into account the deviations
occurring from the random coloring of the small clusters, i.e. those of diameter
less than $N^\g$.
We enumerate the small clusters $C_1,\dots, C_{k_0}$ included in $B(x,N^\a)$.
Their cardinals are denoted $c_1, \dots , c_{k_0}$
and their signs $s_1, \dots, s_{k_0}$.
The random variables $s_1, \dots, s_{k_0}$ are iid Bernoulli.
For $N$ large enough, one has
$$
\big| N^{3 \a} \M_x - {\rm{sign}} (C^*) |C^*| - \sum_{i =1}^{k_0} s_i c_i \big|
< {\z \over 2} N^{3 \a}.
$$
This comes from the fact that all the clusters intersecting the boundary of $B(x,N^\a)$
and distinct from $C^*$ have length smaller than $N^\g$ when $Y^\z_x =1$.
Thus the total magnetization produced by these clusters is less than
$6 N^{2 \a + \g}$ and does not contribute.
By definition of the event $V^\z_x$, the unique crossing cluster $C^*$ in
$B(x,N^\a)$ satisfies $| m_\b N^{3 \a}- |C^*| \, | \le \z N^{3\a}$.
Therefore, we just need to prove large deviations for
$P^\o_{\D_N} \big( | \sum_{i =1}^{k_0} s_i c_i | > {\z \over 2} N^{3 \a} \big)$,
for configurations $\o$ which satisfy $Y^\z_x (\o) =1$.
By symmetry, it is enough to bound
$P^\o_{\D_N} \big( \sum_{i =1}^{k_0} s_i c_i > {\z \over 2} k_o \big)$, with
$k_0 \ge {\z \over 2} N^{3 (\a - \g)}$ (note that $k_0$ is always smaller than
$N^{3 \a}$).
We follow the argument of \cite{I2} (p. 325).
For all $t$ positive, Chebyshev's inequality implies
\begin{eqnarray*}
P^\o_{\D_N} \big( \sum_{i =1}^{k_0} s_i c_i > {\z \over 2} k_o \big)
\le \exp \big( - t {\z k_0 \over 2}\big) \prod_{i =1}^{k_0}
P^\o_{\D_N} \big( \exp( t c_i s_i) \big).
\end{eqnarray*}
As each $c_i$ is smaller than $N^{3 \g}$, one has
%As each $c_i$ is smaller than $N^{3 \g}$, Jensen's inequality implies
%(see \cite{Pisztora1} lemma 5.3)
\begin{eqnarray*}
{ 1 \over k_0} \log
P^\o_{\D_N} \big( \sum_{i =1}^{k_0} s_i c_i > {\z \over 2} k_o \big)
\le - t {\z \over 2} + {1 \over k_0} \sum_{i =1}^{k_0} \log \cosh ( t c_i )
\le - t {\z \over 2} + \log \cosh ( t N^{3 \g} ),
\end{eqnarray*}
Let $\L^*$ be the Legendre-Laplace transform of the Bernoulli measure ${1 \over 2} (\d_1 + \d_{-1})$.
There exists $c_\z$ positive such that for $N$ large enough
\begin{eqnarray*}
{ 1 \over k_0} \log
P^\o_{\D_N} \big( \sum_{i =1}^{k_0} s_i c_i > {\z \over 2} k_o \big)
\le - \L^* \big( {\z \over 2 N^{3 \g} } \big) \le - {c_\z \over N^{6\g}}.
\end{eqnarray*}
Since $k_0 \ge {\z \over 2} N^{3 (\a - \g)}$, this leads to
\begin{eqnarray*}
P^\o_{\D_N} \big( \sum_{i =1}^{k_0} s_i c_i > {\z \over 2} k_o \big)
\le \exp \big( - c_\z {\z \over 2} N^{3 \a - 9\g} \big).
\end{eqnarray*}
As $\g = {1 \over 9}$, we obtain the expected upper bound. \end{pf}
\vskip.2cm
\noindent
{\bf Remark 8.1.}
Let $\A$ be a subset of $\OM^{\rm w}_{\D_N}$ with support disjoint from
the box $B(x,{5 \over 4} N^\a)$. Then the following holds for $N$ large enough
\begin{eqnarray*}
\mu^+_{\b,\D_N} \left( |Z^\z_x| = 0 \ \big| \ \A \right) \le \exp( - c_\z N^{\g}).
\end{eqnarray*}
This is straightforward from the previous arguments.
From \cite{Pisztora1}, we know that $\P^{\o,p_\b}_{\D_N} ( Y^\z_x = 0 )$
vanishes exponentially fast for arbitrary boundary conditions $\o$ outside
the box $B(x,{5 \over 4} N^\a)$.
Furthermore, if the magnetization differs from its equilibrium values $\pm m_\b$,
the deviation occurs from the random coloring of small clusters independent of $\A$.
\subsection{Approximation}
Before proving Theorems \ref{theo ABCP} and \ref{theo ABCP 1}, let us recall
some basic notions of geometric measure Theory.
Throughout this section, we fix $u$ in $\BV$ such that $\F(u) < \infty$ and $\d$ in
$]0,1]$.
As $\t$ is bounded, the perimeter of $\partial u$, which is
$\int_{\partial^* u} \, d \H_x$, is also finite.
The ball of radius $r$ centered in $y$ will be denoted by $B(y,r)$.
For $y$ in $\partial^* u$, we introduce the half-spaces
\begin{eqnarray*}
H^+(y) & = & \big\{ z \in \R^3 \ | \qquad \vec{n}_y . (z -y) \ge 0 \big\},\\
H^-(y) & = & \big\{ z \in \R^3 \ | \qquad \vec{n}_y . (z -y) \le 0 \big\},
\end{eqnarray*}
where $\vec{n}_y$ is the normal to $\partial^* u$ in $y$.
Let $H(y)$ be the hyperplan $H^+(y) \cap H^-(y)$.
%Finally, for any subset $A$ of $\R^3$ and any positive $\e$, we denote by
%${\cal E}(A,\e)$ its $\| \, .\, \|_1$-neighborhood
%\begin{eqnarray*}
%{\cal E}(A,\e) = \{ z \in \R^3 \ | \quad {\rm{dist}}(z,A) \le \e \}.
%\end{eqnarray*}
We fix $\z$ positive.
According to Theorem 2 (p. 205) of \cite{EG}, the reduced boundary $\partial^* u$
equals $\bigcup_{i=1}^n K_i \cup N$ where the 2 dimensional Hausdorff measure of $N$ is
less than $\z$ and each $K_i$ is a compact subset of a $C^1$-hypersurface $S_i$.
For all $x$ in $K_i$, the normal $\vec{n}_x$ is also normal to $S_i$ and there
is $r_0 > 0$ such that uniformly on $K_i$
\begin{eqnarray}
\label{appendice approx 1}
\forall i, \; \forall r \le r_0, \ \forall y \in K_i, \qquad
{\vol} \big( B(y,r) \cap \{u = -1\} \cap H^+(y) \big) & < & \z r^3,\\
{\vol} \big( B(y,r) \cap \{u = +1\} \cap H^-(y) \big) & < & \z r^3. \nonumber
\end{eqnarray}
In the decomposition of $\partial^* u$, one can choose each set $K_i$ such that
it is either included in $\partial \T$ or at a positive distance from $\partial \T$.
\vskip.2cm
\noindent
{\it Proof of Theorem \ref{theo ABCP}}
We first approximate the compact sets $K_i$ which do not touch the boundary.
The following construction is the same for each $S_i$ so it is enough to present it for one
$S_i$, that we shall denote by $S$ (with corresponding $K \subset S$).
As $S$ is $C^1$, we can find $M$ pairwise disjoint open subsets
$\Sigma_1, \dots, \Sigma_M$ of $S$ which cover $S$ up to a set of measure less
than $\z$ and such that each $\Sigma_i$ is congruent to the graph of a real function
$f_i : U_i \to \R$ of class $C^1$, where $U_i$ is a bounded open set of
$\R^2$ and $f_i$ satisfies the bound $| \nabla f_i | \le \z$.
To any $x$ in $U_i$, we associate the point $g_i(x) = (x,f_i(x))$ of $S$.
Let $K_i$ be the compact subset of $U_i$ such that $g_i(K_i) = K \cap \Sigma_i$.
We choose $h$ in $]0,{ r_0 \over 10}[$ (see (\ref{appendice approx 1})) arbitrarily small
and cover $U_i$ with pairwise disjoint
cubes $C^j \subset U_i$ of side $h$ up to a set of measure less than ${\z \over M}$.
For each cube $C^j$ centered in $x_j$ and intersecting $K_i$,
we denote by $B^j$ the translate of $C^j$ centered in $g_i(x_j)$.
The parallelepiped $R^j$ is defined as $R^{j,+} \cup R^{j,-}$, where both parallelepipeds
$R^{j,+}$ and $R^{j,-}$ have a common face $B^j$ and height ${\d \over 2} h$
(one above and the other below $B^j$).
Let $y$ be in $C^j \cap K_i$.
The parallelepiped $R^j$ is included in the ball $B( g_i(y), 10 h)$.
As $| \nabla f_i | \le \z$, the intersection of the hyperplan
$H( g_i(y))$ and $R^j$ is contained in
$\{ z \in \R^3 \ | \quad {\rm{dist}}(z,B^j) \le 2 \z h \}$.
Therefore (\ref{appendice approx 1}) implies that
\begin{eqnarray}
\label{approx 1}
\int_{R^j} | \chi_{R^j}(r) - u(r)| \, dr \le 2 \z h^3 + 10^3 \z h^3
\le 10^4 {\z \over \d} {\vol} (R^i).
\end{eqnarray}
The upper bound of $\t$ is denoted by $\| \t \|_\infty$. It remains to check that
\begin{eqnarray}
\label{approx 2}
\Big| \sum_{i = 1}^{k} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\int_{K} \t(\vec{n}_x) \, d \H_x \Big| \le \| \t \|_\infty \, C_K \, \z,
\end{eqnarray}
where $\vec{n}_i$ is the normal to $B^i$ and $C_K$ depends on
the Hausdorff measure of $K$.
Let $\C^i$ be the union of cubes $C^j$ which intersect $K_i$, then for $h$ small
enough the measure of $\C^i \Delta K_i$ (the symmetric difference) is smaller than
${\z \over M}$ and one has
\begin{eqnarray*}
\Big| \int_{g_i(\C^i)} \t(\vec{n '}_x) \, d \H_x -
\int_{g_i(K_i)} \t(\vec{n}_x) \, d \H_x \Big| \le {\z \| \t \|_\infty \over M},
\end{eqnarray*}
where $\vec{n '}$ is the normal vector to the surface $S$ which coincides with
$\vec{n}$ on $K$.
The normal $\vec{n '}$ is uniformly continuous on any compact. Therefore
for $h$ small enough, the following holds for any cube $C^j$ in $U_i$
\begin{eqnarray*}
\forall x,y \in C^j, \qquad
\big| \t( \vec{n '}_{g_i(x)} ) - \t( \vec{n '}_{g_i(y)} ) \big| \le \z.
\end{eqnarray*}
Using the fact that $| \nabla f_i | \le \z$ on each $U_i$, we derive
(\ref{approx 2}).
Let us go back to the previous notation and denote by $B^1, \dots B^\l$ the collection
of sets which approximate the union of sets $K_i$ which are not in $\partial^* u$.
We set also $U = \partial^* u \bigcap \partial \T$.
As $\t$ is bounded
\begin{eqnarray*}
\int_N \t (\vec{n}_x) \, d \H_x \le \| \t \|_\infty \z.
\end{eqnarray*}
We deduce from (\ref{approx 2}) that
\begin{eqnarray}
\label{approx 3}
\Big| \sum_{i = 1}^{\l} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\int_{\partial^* u / U} \t(\vec{n}_x) \, d \H_x \Big| \le C_u \, \z,
\end{eqnarray}
where $C_u$ depends only on the perimeter of $u$.
From (\ref{approx 1}) and (\ref{approx 3}), we derive Theorem \ref{theo ABCP}
for $\z$ small enough. \qed
\vskip.2cm
\noindent
{\it Proof of Theorem \ref{theo ABCP 1}}
We are going now to approximate the compact sets $K_i$ included in $U = \partial \T
\cap \partial^* u$.
One can also suppose that each $K_i$ is included in one face of $\partial \T$.
Note that $\H$-almost surely for $y$ in $K_i$, the normal $\vec{n}_y$ is orthogonal
to $\partial \T$.
For a given $\z$ positive and $h$ in $]0, {r_0 \over 10}[$ small enough,
there is a covering of $K_i$ with pairwise
disjoint cubes $B^j \subset \T$ of size $h$ up to a set of measure less than $\z$.
We denote by $R^j$ the parallelepiped in $\T$ with one face equal to
$B^j$ and height $\d h$.
Let $y$ be in $B^j \cap K_i$.
The parallelepiped $R^j$ is included in the ball $B( g_i(y), 10 h)$ and
(\ref{appendice approx 1}) implies
\begin{eqnarray}
\label{approx 4}
\int_{R^j} | 1 + u(r)| \, dr \le 10^3 \z h^3
\le 10^3 {\z \over \d} {\vol} (R^j).
\end{eqnarray}
Furthermore, for $h$ small enough
\begin{eqnarray}
\label{approx 6}
\Big| \sum_{j = 1}^{k} \int_{B^j} \t(\vec{n}_j) \, d \H_x -
\int_U \t(\vec{n}_x) \, d \H_x \Big|
\le \| \t \|_\infty \, \z,
\end{eqnarray}
where $\vec{n}_i$ is the normal to $B^i$.
Combining (\ref{approx 4}) and (\ref{approx 6}), we conclude the proof. \qed
\vskip.2cm
One could have also modified the proof of Lemma 6.4 \cite{Cerf} and replaced the
approximation in terms of balls, by cubes.
\vskip.2cm
\noindent
{\it Proof of Theorem \ref{theo approx}}
Theorem \ref{theo approx} can be viewed as a consequence of a general approximation
procedure developed by Alberti and Bellettini \cite{AlBe}.
We briefly recall their proof and refer the reader to \cite{AlBe} for details.
Let $u$ be a function of bounded variation, then
general results of measure Theory imply the existence of a sequence $\{u_n\}$ of
polyhedral functions
converging to $u$ in $L^1(\T)$ and such that the vectors measure of the partial first
derivatives $D u_n$ converge weakly to $D u$ and also that the perimeters of
$\partial u_n$ converge to the one of $\partial u$.
Since $\t$ is continuous, a Theorem of Reshetnyak (see \cite{LM}) implies that $\F(u_n)$
converge to $\F(u)$.
Therefore for any $\d$ positive, there exists a polyhedral set $W$ such that
$$
\1_W \in \V ( \1_{\W_m}, {\d \over 3}) \qquad
{\rm and} \qquad
\big| \F(W) - \F(\W_m) \big| \le {\d \over 2}.
$$
For any $h$ small enough, we approximate the polyhedral set $W$ with disjoint cubes $R^1,\dots,R^\l$
of size $h$ and basis $B^1, \dots B^\l$.
The set $\partial W / \bigcup_{i =1}^\l B^i$ has arbitrarily small area and is
denoted by $U^\d$. As $\t$ is bounded, one has
\begin{eqnarray*}
\Big| \sum_{i = 1}^{\l} \int_{B^i} \t(\vec{n}_i) \, d \H_x -
\F(W) \Big| \le \|\t \|_\infty \, \int_{U^\d} \, d \H_x \le \|\t \|_\infty \d.
\end{eqnarray*}
This concludes the Theorem. \qed
\vskip.2cm
\noindent
{\bf Remark 8.2.}
Since we are only interested in approximating the Wulff shape, one could have
also used Aleksandrov's Theorem (see \cite{EG}) which ensures that the boundary
of a convex function has almost surely a second derivative.
\vskip.2cm
\noindent
{\bf Remark 8.3.}
As explained previously, Theorem \ref{theo approx} holds for any function $u$ of
bounded variation.
Thus following the arguments developed
in Section 6, one can prove the lower bound (Proposition \ref{prop 1})
for any function $u$ of bounded variation.
This implies that a large deviation principle for the measures $\mu_{\b,\D_N}^+$ holds.
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\end{document}
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