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Kac potentials, Lebowitz-Penrose limit, 
Pirogov-Sinai theory
---------------0508101403131
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\title[\red{Phase transition induced by increasing the range of
interaction}]
{{Phase transition induced by increasing the range of
interaction in Potts Model}}

\vskip1cm


\vskip1cm

\author{T. Gobron}
\address{Thierry Gobron, CNRS-UMR 8089, Laboratoire
de physique th\'eorique et mod\'elisation, Universit\'e de
Cergy-Pontoise; 
2 avenue Adolphe Chauvin, Pontoise
95302 Cergy-Pontoise cedex} 
\email{gobron@ptm.u-cergy.fr}



\author{I. Merola}
\address{Immacolata Merola,
Dipartimento di matematica pura e applicata, Universit\`a
dell'Aquila,
 Italy}
\email{merola@univaq.it}
%\thanks{}
%\keywords{}
%\subjclass{}
%
\begin{document}

\vskip .5cm \noindent\vskip .5cm \noindent\vskip .5cm
\noindent


\begin{abstract}
We consider the $Q$-state  Potts model on $\mathbb Z^d$,
$Q\ge 3$, $d\ge 2$, with a Kac ferromagnetic interaction,
with scaling parameter $\ga$. {When $\ga \to 0$ the range  increases
as $\ga^{-1}$ and the ``strength" remains equal to $1$.}
We prove  the existence of a first order phase
transition for $\ga$ small enough (thus with potential  range finite).
The proof is obtained by a perturbation around mean-field
using the Pirogov-Sinai theory.
The result is valid in particular for $d=2$, $Q=3$, thus  providing an
example of
a system which undergoes a transition from second to first order phase
transition
when varying the   finite range of the interaction.

\keywords{Kac potentials, Lebowitz-Penrose limit,
Pirogov-Sinai theory}
\end{abstract}
\maketitle
%\newpage
\section{Introduction}

The Potts model is among the most interesting models in Statistical
Mechanics and beyond. Since its original description by Potts
as a simplified version of the clock model\cite{Potts}, it
has become an ever growing source of interest, in particular in the field of phase
transition. Started as the simplest generalization of the
Ising Model (classical spins with $Q$ values interacting
through alike/dislike interactions), it acquired a widest
meaning through the Kasteleyn-Fortuin representation
\cite{FK}, \cite{Pi}, %\cite{Grimmett},
which allows for both a
straightforward generalization to any real positive value of the
parameter $Q$ (random cluster model
\cite{Grimmett}-\cite{bollobas}), and a direct connection
between its partition function and the Tutte dichromatic
polynomial \cite{Tutte}-\cite{Sokal}, which have a central
meaning in large areas of graph theory. The first instance
gave rise to important connections with percolation theory
($Q\to 1$) and resistor networks ($Q\to 0$), while the
second emphasizes on the properties of the graph it is
defined on.


Though the Potts model is in general not solvable, it has
been known since the original work by Potts that it
undergoes an order-disorder phase transition.
Since that time, a lot of work has been dedicated to the
study of its critical properties, beginning with the
nature of the transition, which appears to depend both on
the number of spin values $Q$ and the dimension of the lattice.
In particular it has been proven
that the transition is first order for nearest neighbor
interactions in two dimensions for $Q> 4$ and continuous
for $Q\le 4$, \cite{baxter73}-\cite{baxter78}-\cite{Wu}.


Besides these exact results, an important guideline in the
rigorous analysis of the Potts model has been the mean
field theory \cite{Wu}, in which the
transition is continuous for $Q\le 2$ and first order for
$Q\ge 3$. The Potts model with nearest neighbor
interactions on $\Zz^d$ gets close to its mean field
approximation in two ways, namely in high dimensions, or for
large values of $Q$. Early attempts led to the heuristic
determination of a critical value $Q_c(d)$ beyond which the
transition become mean-field like, and in particular to a
few exact results $Q_c(2)=4$, $Q_c(4)=2$, $Q_c(6)=1$
\cite{Wu}. Rigorous results considered the large
values of $Q$ \cite{KotS}
or high dimension limit \cite{KS}%\cite{kesten-schonmann}
 while a more recent account gives a proof for $d$
sufficiently large and all $Q\ge 3$, \cite{Biskup-Chayes}.


Another possible approach consists in considering a larger
range of interaction. The known results include one
dimensional systems with $\frac{1}{r^s}$ interactions,
$1<s\le 2$. In \cite{ACCN}, the existence of a transition
has been proven in one dimension for $s=2$. Numerical
results \cite{RD} indicate that a transition from
continuous to first order behavior occurs when $s$ varies
across some $Q$-dependant critical value. Recently
\cite{b-chays-c} extended the previous rigorous results for
long range potentials by proving the existence of a first
order phase transition for potentials with the property of
``reflection positivity" and sufficient spread out. It
applies mainly to power law potentials for $d< 3$, and
exponentially decaying potentials for $ d\ge 3 $. In
particular this lead to a class of bidimensional systems
with $Q=3$ which undergo a first order phase transition in
contradistinction with the nearest neighbor case. This
result confirms the behavior suggested by the mean field
that the order of the transition may change with the range
of interaction.


\vskip .5cm In the present work, we consider long
but finite interaction potentials, and prove that for any
$Q\ge 3$, $d\ge 2$, such a system undergoes a first order
phase transition provided the range of interactions is long
enough. In particular we prove that the transition takes
place at a temperature close to the mean field critical
temperature and that there is
coexistence at the critical point between one disordered and $Q$ ordered phases.
As a byproduct, we obtain that for $d=2$, $Q=3$ there is a
change in the nature of the transition from second order
for nearest neighbor interaction to first order beyond some
finite value for the interaction range.


Our method relies on the derivation of a Peierls estimate
for the system of contours defined on top of coarse grained
configurations, where the ``mean-field states'' play the
role of the ground states. This method was first developed
in the context of Ising models with Kac interactions
\cite{CP},\cite{BZ1},\cite{BZ2}, and later included an
extended Pirogov-Sinai theory  \cite{Z} in order to cope
with non symmetrical systems, such as a continuous point
particle model \cite{lmp} and a quantum version of it
\cite{bkmp1}. See also \cite{errico-leip} for a thorough
account on this approach. Absence of symmetry between ordered and disordered
phases also lead us to use the Pirogov-Sinai theory.
This requires in turn a precise estimate of finite volume
corrections to the pressures which follows from an
extension of the Dobrushin uniqueness criterion
(\cite{dob}) introduced in \cite{bkmp2}.

\vskip .5cm  Our results suggests three possible
directions for further work: (1) the proof of the existence
of a transition from first to second order when increasing the
range of interactions. (2) the extension of
our results to the random
cluster model for all real positive values of $Q$.
(3) the study of the phase diagram in a neighborhood
of the critical temperature, applying the techniques developed in \cite{bmpz}.


\vskip .5cm \noindent

The paper is organized as follows: in Section 2, we define
the model and state our main result. In Section 3, we
explain the strategy of the proof and give some hints on
why such approach is successful. In section 4, we define
various notations and introduce the scales which
appear in the problem. In section 5, we define the
coarse grained configurations and prove a ``Lebowitz
Penrose theorem", introducing
a mean field functional. Section 6
contains the definition of contours and states the Peierls
bound for their weights.
This bound is proven in the remaining sections:
in section 7 we prove a factorization theorem for the
contour weights and introduce abstract contour models which
are at the core of the Pirogov-Sinai
in the Zahradn\`\i k approach, \cite{Z}.

In section 8 we identify the value of the inverse
temperature $\beta$
at which the order-disorder phase transition takes place.
Section 9 contains an
estimate for the finite volume corrections to the pressure,
which requires both a control on the decay of correlation
and a small deviation estimate. Section 10 gives a large
deviation estimate which provides the Peierls constant.
Five appendices are added at the end of the paper: Appendix
A contains a short review of the mean field theory for the
Potts model as well as the derivation of the properties
on which we rely in the rest of the paper.
Appendix B discuss the local
equilibrium properties which are used in various parts of
the paper.
In Appendix C,
we prove the existence of the pressure for the abstract
models introduced in section 7. In Appendix D, we prove a
lemma needed in section 8.  Finally, in Appendix E, we prove a lemma needed
in the proof of the small deviation estimate of section 9.

\vskip .5cm \noindent




\vskip 2.5cm \noindent
\section{Potts model with Kac interactions: definitions and main
{results.}}


We first define the $Q$-state Potts model on $\Zz^d$ with
long but finite interactions. In the following, $\Zz^d$ is
considered as naturally embedded in $\Rr^d$. 

For a fixed value of the scaling parameter $\ga
> 0$, we introduce the Kac potential $J_\ga$ defined on
$\Rr^d\times \Rr^d$ as
\begin{eqnarray}
\label{def:J} {J_\ga(x,y)}=\ga^{d}\mathcal J(\ga|x-y|)
\end{eqnarray}
where $\mathcal J(r)$ is a symmetric probability kernel
with range 1, differentiable with bounded derivative
and $|.|$ denotes the Euclidean norm on $\Rr^d$.

\par
Let denote by $a_q$, $q=1,\dots, Q$, the $Q$ different
``colors''. We denote by \blue{${\tilde\cXx}_o:=\{a_1,\dots
a_Q\}$ and by ${\tilde\cXx} :={\tilde\cXx}_o^{\Zz^d}$} the
space of configurations of the Q-state Potts model on
$\Zz^d$ and by ${\tilde\cXx}_\La={\tilde\cXx}_o^\La$ its
restriction to a finite subset $\La$ of $\Zz^d$. \red{The
color at the site $i$ is denoted by $\si(i)\in\tilde\cXx_o$
}

For any finite region $\La$ of $\Zz^d$, we define the
Potts-Kac Hamiltonian with boundary conditions
$\si_{\La^c}\in {\tilde\cXx}_{\La^c}$ as





\begin{eqnarray*}
\label{def:Kac-Ham-b}
{H_{\ga,\La}}(\si_\La|\si_{\La^c}):=-\frac{1}{2}\sumtwo{i,j\in
\La}{i\neq j}{J_\ga(i,j)}\Ii_{\{\si_\La(i)=\si_\La(j)\}}
-\sumtwo{i\in \La}{j\in
\La^c}{J_\ga(i,j)}\Ii_{\{\si_\La(i)=\si_{\La^c}(j)\}}
\end{eqnarray*}


The Gibbs measure  specifications are:
\begin{eqnarray}
\label{eq:gibbskac} \mu_{\ga,
\beta,\La}(\si_\La|\si_{\La^{c}})= \frac{e^{-\beta
{H_{\ga,\La}}(\si_\La|\si_{\La^{c}})}}{Z_{\ga,\beta,\La}(\si_{\La^{c}})}
 %\hskip1.6cm q\in \{d,1,\dots,Q\}
\end{eqnarray}
where $Z_{\ga,\beta,\La}(\si_{\La^{c}})$ is the partition
function
\begin{eqnarray*}
Z_{\ga,\beta,\La}(\si_{\La^c}):=\sum_{\si_\La}e^{-\beta
{H_{\ga,\La}}(\si_\La|\si_{\La^{c}})}
\end{eqnarray*}

\vskip .5cm \noindent In this paper, we will work with an
alternative formulation of the Potts model which is more
suited for our purpose and prepare for the introduction of
coarse graining: Let $\vec u_q$, $q=1,\cdots, Q$ the
vectors of $\Rr^Q$ with components $u_{q,k}=\delta_{q,k}$.
We define \blue{ $\Omega_o:= \{\vec u_1,\cdots,\vec u_Q\}$
and $\Omega :=\Omega_o^{\Zz^d}$}. Then, we associate to
each element $\si$ of $\tilde\Omega$, a vector configuration
$\vw$ of $\Omega$ as
\begin{eqnarray}
\vw(i)=\vec u_q \Longleftrightarrow \si(i)=a_q
\end{eqnarray}

In these notations, the Hamiltonian \eqref{def:Kac-Ham-b}
reads
\begin{eqnarray}
\label{def:Kac-Ham-c}
{H_{\ga,\La}}(\vw_\La|\vw_{\La^c}) =
-\frac{1}{2}\sumtwo{i,j\in \La}{i\neq j}{J_\ga(i,j)}
\vw_\La(i)\cdot\vw_\La(j) -\sumtwo{i\in \La}{j\in
\La^c}{J_\ga(i,j)} \vw_\La(i)\cdot\vw_{\La^c}(j)
\end{eqnarray}
where ``$\cdot$" denote the scalar product:  $\vec v\cdot
\vec v' :=\sum_{k=1}^{Q}v_k v_k'$. In these variables, the
Potts model appears as a superposition of $Q$ lattice
gases, with configurations $\w_q\in\{0,1\}^{\Zz^d}$,
interacting through hard core exclusion, $\sum _q
\w_q(i)=1$ for all $i\in\Zz^d$.




\vskip 1.5cm \noindent

The main result of this paper is stated in the following
theorem:
\begin{thm}
\label{thm:main} For any $d\le 2$ and $Q\ge 3$, there
exists $\bar\ga>0$ such that for any $\ga\in(0,\bar\ga)$,
there is a value of the inverse temperature $\bcg$ at which
there exist $Q+1$
distinct DLR measures.
\end{thm}
In particular there are $Q$ DLR measures
$\mu^{(q)}_{\ga,\beta}$, $q=1, \cdots,Q$, the ``ordered''
states, where one of $Q$
colors
 is dominant:
\begin{eqnarray}
\mu^{(q)}_{\ga,\beta}(\si_0=a_q)>\mu^{(q)}_{\ga,\beta}(\si_0=a_p)
\quad \forall p,q \in \{1, \cdots,Q\}
\end{eqnarray}
The remaining DLR measure is the ``disordered" state
$\mu^{(-1)}_{\ga,\beta}$ where all colors appear with the same
density:
\begin{eqnarray*}
\mu^{(-1)}_{\ga,\beta}(\si_0=a_q)=\frac{1}{Q}
\end{eqnarray*}


\setcounter{equation}{0} \vskip 2.5cm \noindent
\section{Idea of the proof and stability properties of the Free energy functional}
\label{sec:Stability}


The main stream of ideas which leads to the present proof originates in the work by Lebowitz
and Penrose \cite{LP} on the rigorous derivation of the Van der Waals theory of liquid-vapor transition
using the inverse range of interaction $\ga$ as a control parameter.

The first step consists in the introduction of a continuous approximation of the free energy
of the system by a ``mean field free energy functional''

\begin{eqnarray*}
{\cF_{\ga,\beta}(\vec\rho)}=\int dr ~
\Phi_\beta^{\mf}(\vec\rho) +\frac{1}{4}\int\int dx dy ~
J_\ga(x,y) (\vec\rho(y)-\vec \rho(x))^{2}
\end{eqnarray*}
defined for all $\vec \rho \in L^{\infty}(\Rr^{d},S_{Q})$, where
\begin{eqnarray}
S_Q = \bigl\{\vec\rho\in\Rr^Q_+,\sum_q \rho_q =1\bigr\}
\end{eqnarray}
The mean field free energy density $\phi_\beta^{\mf}$ is defined on $S_{Q}$ as
\begin{eqnarray*}
\phi_\beta^{\mf}(\vec
v)&:=&\frac{1}{2}(\vec v\cdot \vec v)-\frac{1}{\beta}\vec v\cdot\ln(\vec v)
\end{eqnarray*}
and finally
\begin{eqnarray*}
\Phi_\beta^{\mf}(\vec v)&:=&\phi_\beta^{\mf}(\vec
v)-\inf_{\vec v\in S_Q}\phi_\beta^{\mf}(\vec v)
\end{eqnarray*}



This approximation allows for a perturbative analysis
 that relies on the stability properties
of $\cF_{\ga,\beta}(\vec\rho)$.

The critical points
in $L^{\infty}(\Rr^{d},S_{Q})$ of $\cF_{\ga,\beta}(\cdot )$ identify with
the fixed points of the map
$\vec\fM=(\fM_1,\dots,\fM_Q)$ defined on
$L^{\infty}(\La,S_Q)$ as:
\begin{eqnarray}
\label{def:op} \vec \rho \to \vec\fM(\vec \rho)\hskip1cm
\mbox{ with }\hskip1cm\fM_k(\vec \rho) := \frac{e^{\beta
\cL_k(\vec \rho)}}{\sum _l e^{\beta \cL_l(\vec \rho)}}
\end{eqnarray}
where $\vec \cL(\vec\rho)(r)=\int
dy~J_\ga(x,y)\vec\rho(y)$. The stability property we are referring to is the fact that
the map $\vec\fM(\cdot)$ is a contraction in a neighborhood of each fixed point
relevant to our analysis.
These fixed points are the constant
vector functions, which take values in the set of the fixed points of the ``mean
field map"$\vec\fM^{\mf}(\cdot)$ defined on $S_Q$:
\begin{eqnarray*}
\fM^{\mf}_k(\vec v):=\frac{e^{\beta v_k}}{\sum_{l}e^{\beta
v_l}}
\end{eqnarray*}
This correspondence uses the fact that $\vL(\vec \rho)=\vec \rho$ for any
constant profile $\vec \rho$ on $\Rr^{d}$.
In appendix \ref{app:meanfield} it is shown that for
$\beta$ in an interval $(\beta_0,Q)$ %containing $\bcmf$
there are $Q+1$ solutions of the mean field equation in $S_Q$:
\begin{eqnarray}
\label{eq:mean field}
\vec v = \fM^{\mf}_k(\vec v)
\end{eqnarray}

 denoted by
$\vec\rho^{(\q)}\equiv \vec\rho^{(\q)}_\beta$, $\q\in
\{-1,1,\dots,Q\}$, which are the local
 minimizers of the mean field free
energy density function $\phi^{\mf}_\beta(\vec v)$ on $S_Q$
(Theorem \ref{thm:mflocalminimizers}). $Q$ of them are the ``ordered states'' $\vec\rho^{(q)}$, $q>0$,
in which one of the color dominates, and the remaining one is a ``disordered state'', $\vec\rho^{(-1)}$
in which all colors have the same density.
Around each of these minimizers,
the mean field map is a contraction (Theorem \ref{thm:mflocalstability}).
and the map $\vec\fM$ inherits the same property for $\beta$ close enough
to the mean field critical inverse temperature $\bcmf$
as shown in section \ref{subsect:decayofthecorrelations}.



The next step consists in the introduction of a coarse grained level of description,
at which one can introduce reference states in which the local empirical mean is
everywhere close to one of the constant minimizers of $\cF_{\ga,\beta}(\vec\rho)$.
Our result is then based on the derivation of Peierls bounds for a systems of
contours defined on a larger scale, as perturbations of these references states.

The absence of symmetry
between ordered and disordered phases lead to use the Pirogov-Sinai theory.
We follow the approach of Zahradn\`\i k \cite{Z}
(see also \cite{errico-leip}, \cite{lmp}, \cite{bkmp1})
and define abstract contours models with cut-off weights, supported by a
the restricted ensemble of configurations associated to a reference state.
The value of the inverse temperature at which the
pressures of these models are equal identifies with the critical value
at which the transition takes place. In fact, equality between these pressures
together with an accurate estimate on their finite volume corrections
lead to a Peierls estimate for the ``true model''.
This latter point is obtained by proving for the abstract models both the fast
decay of correlations and a small deviation estimate.
These proofs depend in a crucial way
on the stability properties
of the free energy functional $\cF_{\ga,\beta}(\vec\rho)$.
The value of the Peierls constant as well as the control over the various error terms
then follows from a large deviation estimate based on the continuous approximation
introduced above.


Technically, we first derive a factorization theorem (Theorem \ref{thm:Peierls-factoriz})
for the weight of a contour. The proof then splits into two parts: a large deviation estimate
gives an estimate for the free energy of a contour (Theorem \ref{thm:Peierls-3}) while the control on
the other term in the factorization formula follows from the analysis of the abstract contours
models (Theorem \ref{thm:Peierls-2}). This last theorem uses estimates on decay of correlations
and small deviation estimates for these models. Connection with the original model holds
when the cut-off weights %$\hat W^{\p}_\ga(\Ga;\vw)$
for the abstract models identify with
the true weights%$w^{\p}_{\abs,\beta}(\Ga;\vw)$. This is proven for a single value $\beta=\bcg$
%where we have $w^{\p}_{\abs,\beta}(\Ga;\vw)< e^{-\frac{\fK_{\ga}}{2} N_\Ga}$.
which occurs for a single value $\beta=\bcg$.

By these  three Theorem (\ref{thm:Peierls-factoriz},
\ref{thm:Peierls-3} and \ref{thm:Peierls-2}) we prove that a
Peierls bound holds at the critical temperature $\bcg$ for $\ga$ small enough
(Theorem \eqref{thm:Peierls-0}).
\red{A classical Peierls argument then proves} that, at $\beta=\bcg$, the
 $Q$ ordered phases coexist with the disordered one
(Theorem \eqref{thm:main}).



\vskip 1.5cm \noindent

\section{ Notations and scales}



{\em Notations}

We consider $\Zz^d$ as imbedded in  $\Rr^d$ and
frequently need to define similar functions both on $\Zz^d$
and on $\Rr^d$. In order to avoid confusion and duplication
of all notations, we will stick to the
convention that functions are defined on $\Zz^d$ or on
$\Rr^d$ according to which of the two following notations is used for their
arguments:
{\obeylines
$x$,$y$ are points in $\Rr^d$.
$i$, $j$ are points in $\Zz^d$.
}

Accordingly, for any subset  $\La\in\Rr^d$, we
will write $\{i\in\La\}$ for the set
$\{i\in\La\sqcap\Zz^d\}$.


\red{We denote by $|\La|$ the
volume of the set $\La$ and by $|\partial \La|$ its
surface.}

\noindent Furthermore, the $Q$ colors in the microscopic
model are denoted $\{a_1,\cdots,a_Q\}$, and we use indices
$p$ or $q$ in $\{1,\cdots, Q\}$ to denote them.
After coarse graining, these indices refer to the $Q$ ordered states
in which one color is dominant, while one extra color
$a_{-1}$ is introduced corresponding to a disordered state.
Thus we use indices $\p$ or $\q$ in $\{-1,1,\cdots, Q\}$ to designate the colors after
coarse graining. Correspondingly there are $Q+1$ local minimizers
in the mean field model which we denote by the symbol $\vec\rho^{~(\q)}\equiv \vec\rho^{~(\q)}_\beta$.
We will also denote by the same symbol the $Q+1$ density functions in
$L^\infty(\Zz^{d},S_Q)$ (respectively
$L^\infty(\Rr^{d},S_Q)$) constantly equal to
$\vec\rho^{~(\q)}$ on $\Zz^d$ (respectively $\Rr^d$).


We will use instead $k$ and $l$ as indices in $\{1,\cdots,Q\}$ to designate
the $k^{th}$ and $l^{th}$ coordinates of a vector in $S_Q$.
Finally, $m$ is an index in $\{1,\cdots,d\}$.
\vskip 1.5cm \noindent We equip  $\Rr^{d}$ with the
following two norms: let $x,y\in\Rr^d$
\begin{eqnarray}
\dist(x,y)&:=&\max_{m\in\{1,\dots,d\}} |x_m-y_m|
\label{def:dist}\\
|x-y|&:=&\sqrt{\sum_{m=1}^{d}(x_m-y_m)^2}
\label{def:modulo-sp}
\end{eqnarray}

\noindent We also introduce two norms on $S_Q$. For all $\vec v$ in $S_Q$, let
\begin{eqnarray}
\label{def:star} \|\vec v\|_{\star}&:=&\sup_{k\in
\{1,\dots,Q\}}|v_k|\\
|\vec v|&:=& \sqrt{\sum_{k=1}^{Q} v_k^2}
\end{eqnarray}

Finally, we denote by $\fJ_\ga$ the normalization constant
of the interaction kernel $J_\ga$ on $\Zz^d$:
\begin{eqnarray}
\label{def:Jb} \fJ_\ga:= \sum_{j\in \Zz^d
}{J_\ga(0,j)}\hskip1.5cm\lim_{\ga\to 0}\fJ_\ga=1
\end{eqnarray}


{\em Scales:}



In the sequel, we introduce three lengths $\ell_0$, $\ell_{-,\ga}$ and $\ell_{+,\ga}$  which grow as
powers of the potential range, respectively as $\ga^{-\frac{1}{2}}$, $\ga^{-(1-\alpha)}$ and $\ga^{-(1+\alpha)}$,
where $\alpha\in (0,\frac{1}{16 d})$. The bound $\frac{1}{16 d}$ will be 
requested in the course of the proofs in order to satisfy various requirements.
For $\ell = \ell_0, \ell_{-,\ga}, \ell_{+,\ga}$ we construct a partition $\cD^{\ell}$ of $\Rr^d$
in cubes of side length $\ell$.
In order that these three partitions are one coarser than the other, we define $\ell_0$,
(respectively $\ell_{-,\ga}$ and $\ell_{+,\ga}$)
as the largest power of $2$ smaller than $\ga^{-\frac{1}{2}}$,
(respectively $\ga^{-(1-\alpha)}$ and $\ga^{-(1+\alpha)}$). In other terms,
\begin{eqnarray}
\ell_0&=&2^{\left[\frac{1}{2}\frac{\ln\ga^{-1}}{\ln 2}\right]}\\
\ell_{\pm,\ga}&=&2^{(1\pm\alpha)\left[\frac{\ln\ga^{-1}}{\ln 2}\right]}
\end{eqnarray}
where $[\cdot]$ denotes the integer part
We will also consider $\ga$ small enough so that these lengths are ordered as
\begin{eqnarray}
1<\ell_0<\ell_{-,\ga}<\ga^{-1}<\ell_{+,\ga}
\end{eqnarray}

\vskip 2.5cm \noindent
\section{Coarse graining and mean
field functional} \label{sec:freeenergy} \vskip .5cm
\noindent


Let $\ell$ a large positive integer and
 $\cD^{\ell}$ a partition of $\Rr^{d}$ in cubes of size
 $\ell$. For all $x$ in $\Rr^d$,
we denote by $C_x^{\ell}$ the cube of $\cD^{\ell}$
containing $x$.
We define a coarse grained configuration on $\cD^{\ell}$
as follows: for each configuration $\vw\in{\Omega}$
and any $x\in \Rr^{d}$, we define the coarse grained
configuration (at scale $\ell$) as the Q-dimensional vector,
\begin{eqnarray}
\label{def:p-ell0}
\vw^{\,\ell}(x) = \ell^{-d} \sum_{i\in
C_x^{\ell}} {\vw}(i)
\end{eqnarray}
where we make use of our notational conventions.
The $k^{th}$ component $\w_k^{\ell}(x)$ is the
empirical density of color $a_k$ in $C_x^{\ell}$.
 Due to the underlying discretization, $\vw^{\ell}(x)$
takes values in the finite set $M_{\ell^d}^Q$
\begin{eqnarray*}
M_{\ell^d}^Q=\bigl\{(r_1,\cdots,r_Q); (\ell^d r_k)
 \in
\Nn, \sum_{k=1}^Q r_k=1 \bigr\}
\end{eqnarray*}
Let $\La$ a $\cD^{\ell}$-measurable subset of $\Rr^d$.
The set of coarse grained configuration in $\La$ is denoted
by ${\cXx}_\La^{\ell}$ and identifies with the set of
$\cD^{\ell}$-measurable functions on $\La$ with values in
$M_{\ell^d}^Q$.
We extend the discrete set $M_{\ell^d}^Q$ to the simplex $S_Q$ in $\Rr^Q$ defined as
\begin{eqnarray}
\label{def:SQ}
S_Q = \bigl\{\vec\rho\in\Rr^Q_+,\sum_q \rho_q =1\bigr\}
\end{eqnarray}
Thus all coarse grained configurations in ${\cXx}_\La^{\ell}$ are also elements of
$L^\infty(\La,S_Q)$. Conversely we will approximate any
function in $L^\infty(\La,S_Q)$ by a coarse grained
configuration. For any $\vec\rho\in L^\infty(\La,S_Q)$, we
will denote by $\vec\rho^{\,\ell}$ its
$\cD^{\ell}$-measurable approximation
\begin{eqnarray}
\label{def:rhol0}
\vec\rho^{\,\ell}(x) = \ell^{-d}
\int_{C_x^{\ell}} {\vec\rho(y)} dy
\end{eqnarray}
for all $x$ in $\La$, and by $[\vec\rho\,]^{\ell}$ the
only function  in ${\cXx}_\La^{\ell}$ such that
\begin{eqnarray}
\label{def:parteintera}
-\frac{1}{2\ell^d}< [\vec\rho\,]_k^{\ell}(x)-\rho_k^{\ell}(x)\le
\frac{1}{2\ell^d}
\end{eqnarray}
for all $k$ and all $x$ in $\La$.


For $\La$ a finite
$\cD^{\ell}$-measurable region of $\Rr^d$, we define the
mean field free energy functional on $L^\infty(\La,S_Q)$
as,
\begin{eqnarray}
\label{def:freeenergy}
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c}) =
V_{\ga,\La}(\vec\rho_\La|\vec\rho_{\La^c}) -\frac{1}{\beta}
I(\vec\rho_\La)
\end{eqnarray}
where $\vec\rho_{\La^c}\in L^\infty(\La^c,S_Q)$ defines the boundary conditions.
The two functionals $V_{\ga,\La}(\vec\rho_\La|\vec\rho_{\La^c})$
and $I(\vec\rho_\La)$ are respectively the energy and the
entropy of configuration $\vec\rho_\La$,
\begin{eqnarray*}
V_{{\ga,\La}}(\vec\rho_\La|\vec\rho_{\La^c}) =
-\frac{1}{2}\int_\La\int_\La {J_\ga(x,y)} \bigl(
\vec\rho_\La(x)\cdot\vec\rho_\La(y) \bigr) dx dy
-\int_\La\int_{\La^c} {J_\ga(x,y)} \bigl(
\vec\rho_\La(x)\cdot\vec\rho_{\La^c}(y) \bigr) dx dy
\end{eqnarray*}
\begin{eqnarray*}
I(\vec\rho_\La) = - \int_\La \sum_{q=1}^Q \rho_{\La,q}(x)
\ln (\rho_{\La,q}(x))   dx{=:- \int_\La\vec\rho_{\La}(x)
\cdot\ln \vec\rho_{\La}(x)}  dx
\end{eqnarray*}
For all $\cA\sqsubset{\cXx}_\La$ we define the constrained
partition function $Z_{\ga,\beta,\La}(\cA|\vw_{\La^c})$
\begin{eqnarray*}
Z_{\ga,\beta,\La}(\cA|\vw_{\La^c})= \sum_{\vw_\La\in \cA}
e^{-\beta {H_{\ga,\La}}(\vw_\La|\vw_{\La^c})}
\end{eqnarray*}
We now state a theorem relating constrained partition functions
and mean field free energy:
\begin{thm}
\label{thm:app} There exists a constant $c>0$ such that for
all $\ga>0$, $\ell \in (1,\ga^{-1})$ and all bounded
$\cD^{\ell}$-measurable region $\La$ of $\Rr^d$, the
following inequalities hold:\par For all subsets $\cA$ of
${\cXx}_\La^{\ell}$,
\begin{eqnarray*}
\ln Z_{\ga,\beta,\La}(\{\vw^{\,\ell}_\La\in
\cA\}|\vw_{\La^c}) +\beta \inf_{\vec\rho_\La\in \cA}
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw^{\,\ell}_{\La^c}) \le
\beta c \eps(\ga,\ell) |\La|
\end{eqnarray*}
and for all $\vec\rho_\La\in L^\infty(\La,S_Q)$,
\begin{eqnarray*}
\ln Z_{\ga,\beta,\La}(\{\vw_\La : \vw^{\,\ell}_\La=
[\vec\rho_\La]^{\ell}\}|\vw_{\La^c}) +\beta
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw^{\,\ell}_{\La^c})
\ge -\beta c \eps(\ga,\ell) |\La|
\end{eqnarray*}
where
\begin{eqnarray*}
\eps(\ga,\ell) = \ga \ell + \frac{\ln
\ell}{\ell^d}
\end{eqnarray*}
\end{thm}
\begin{proof}
We first estimate the difference between the energy
${H_{\ga,\La}}(\vw_\La|\vw_{\La^c})$ and its coarse grained
approximation
$V_{{\ga,\La}}(\vw^{\,\ell}_\La|\vw^{\,\ell}_{\La^c})$.
Given two cubes $C_1$ and $C_2$ of the partition
$\cD^{\ell}$, for any two points $i\in C_1$ and  $j\in
C_2$, we have
\begin{eqnarray*}
|{J_\ga(i,j)} - \frac{1}{\ell^{2 d}}\int_{C_1\times
C_2} {J_\ga(x,y)} ~dx ~dy| \le 2\sqrt{d} \|\nabla \cJ\|_\infty
\ga^{d+1} \ell \Ii_{d(C1,C2)\le\ga^{-1}}
\end{eqnarray*}
where $d(C1,C2)=\inf_{x\in C_1,y\in C_2} |x-y|$.

Hence
\begin{eqnarray*}
&&\big|{H_{\ga,\La}}(\vw_\La|\vw_{\La^c}) -
V_{{\ga,\La}}(\vw^{\,\ell}_\La|\vw^{\,\ell}_{\La^c})\big|
\\
&&\hskip1cm\le
\sumtwo{C_1\in\cD^{\ell}_\La}{C_2\in\cD^{\ell}}
\big|\sumtwo{i\in C_1\cap\Zz^d}{j\in C_2\cap\Zz^d}
{J_\ga(i,j)}\vw_\La(i)\cdot \vw_\La(j)
 -\int_{C_1\times C_2} {J_\ga(x,y)}
\vw^{\,\ell}_\La(x)\cdot \vw^{\,\ell}_\La(y)\big|\\
&&\hskip1cm\le
\sumtwo{C_1\in\cD^{\ell}_\La}{C_2\in\cD^{\ell}}
\big|\sumtwo{i\in C_1\cap\Zz^d}{j\in C_2\cap\Zz^d}
\Big({J_\ga(i,j)}
 -\frac{1}{\ell^{2d}}\int_{C_1\times C_2} {J_\ga(x,y)}~ dx
~dy\Big)\vw_\La(i)\cdot \vw_\La(j)
 \big|\\
&&\hskip1cm\le 2\sqrt{d} \|\nabla \cJ\|_\infty \ga^{d+1} \ell
\sum_{C_1, C_2} |C_1| |C_2| \Ii_{d(C_1,C_2)\le\ga^{-1}}\\
&&\hskip1cm\le c_d \ga \ell |\La|
\end{eqnarray*}
where $c_d$ is a constant independent on $\ga$,
$\ell\le\ga^{-1}$ and $\La$
\begin{eqnarray}
\label{def:cd}
c_d=2^{d+1}\sqrt{d}\|\nabla \cJ\|_\infty
\end{eqnarray}
Thus for any profile $\vec\rho_\La$ in
${\cXx}_\La^{\ell}$, we have
\begin{eqnarray*}
\big| \ln
\frac{Z_{\ga,\beta,\La}(\{\vw_\La:\vw^{\ell}_\La=\vec\rho_\La\}
|\vw_{\La^c})} {|\{\vw_\La:
{\vw^{~\ell}}_\La=\vec\rho_\La \}| e^{-\beta
V_{{\ga,\La}}(\vec\rho_\La|\vw^{\ell}_{\La^c})}}\big|\le
c_d \ga \ell |\La|
\end{eqnarray*}
The cardinality of $\{\vw_\La: \vw^{\ell}_\La=\vec\rho_\La
\}$ can be related to the entropy of $\vec\rho_\La$. The
error bounds for the Stirling formula
\begin{eqnarray}
\frac{1}{12N+1}\le
\frac{\ln(N!)}{N(\ln(N)-1)+\ln(\sqrt{2\pi N})}\le
\frac{1}{12N}
\end{eqnarray}
leads to the following estimate
\begin{eqnarray*}
\big|\ln |\{\vw_\La: {\vw^{~\ell}}_\La=\vec\rho_\La \}| -
I(\vec\rho_\La)\big| \le d Q |\La| \frac{\log
\ell}{\ell^d}
\end{eqnarray*}
one gets
\begin{eqnarray*}
\big| \ln Z_{\ga,\beta,\La}(\{\vw_\La:
\vw^{\,\ell}_\La=\vec\rho_\La\}|\vw_{\La^c}) +\beta
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw^{\,\ell}_{\La^c})\big|
\le \beta c_d \ga \ell |\La| + Q d |\La| \frac{\log
\ell}{\ell^d}
\end{eqnarray*}
Now an easy upper bound for the cardinality of
${\cXx}_\La^{\ell}$ gives for all
$\cA\sqsubset{\cXx}_\La^{\ell}$
\begin{eqnarray*}
\ln|\cA|\le \ln |{\cXx}_\La^{\ell}| \le {Q
d\frac{|\La|}{\ell^d}}\log \ell
\end{eqnarray*}
where $|\cA|$ denotes the cardinality of the set $\cA$.
Combining these two last inequalities gives the first part
of Theorem  \ref{thm:app} with $c=\max(c_d,\frac{2Qd}{\beta})$. On the other side, for all
$\vec\rho\in L^\infty(\La,S_Q)$, one has
\begin{eqnarray*}
\log Z_{\ga,\beta,\La}
(\vw^{\,\ell}_\La=[\vec\rho_\La\,]^{\ell}|\vw_{\La^c}) \ge
-\beta
F_{{\ga,\beta,\La}}([\vec\rho]^{\ell}|\vw^{\,\ell}_{\La^c})
-c \eps(\ga,\ell) |\La|
\end{eqnarray*}
Now using
\begin{eqnarray*}
|V_{{\ga,\La}}(\vec\rho_\La|\vw_{\La^c}^{\,\ell})
-V_{{\ga,\La}}(\vec\rho_\La^{\,\ell}|
\vw_{\La^c}^{\,\ell})|\le c_d \ga \ell |\La|
\end{eqnarray*}
and the concavity of the entropy, one gets
\begin{eqnarray*}
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw_{\La^c}^{\,\ell})
\ge F_{{\ga,\beta,\La}}(\vec\rho_\La^{\,\ell}|
\vw_{\La^c}^{\,\ell}) - c_d \ga \ell |\La|
\end{eqnarray*}
Furthermore, approximating $\vec\rho_\La^{\,\ell}$ by
$[\vec\rho_\La\,]^{\ell}$ gives
\begin{eqnarray*}
F_{{\ga,\beta,\La}}(\vec\rho_\La^{\,\ell}|
\vw_{\La^c}^{\,\ell})\ge
F_{{\ga,\beta,\La}}([\vec\rho_\La\,]^{\ell}|
\vw_{\La^c}^{\,\ell}) - d Q \frac{|\La|}{\ell^d}
\end{eqnarray*}
we thus get:
\begin{eqnarray*}
\log Z_{\ga,\beta,\La}
(\vw^{\,\ell}_\La=[\vec\rho\,]^{\ell}|\vw_{\La^c}) \ge
-\beta
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw^{\,\ell}_{\La^c}) - \beta c
\eps(\ga,\ell) |\La|
\end{eqnarray*}
which gives the second part of the theorem with the same constant as before.
\end{proof}

We will use theorem \ref{thm:app} mostly in the following weaker form:
\begin{coro}
There exists a constant $c>0$ such that for all $\ga>0$,
$\ell$ an integer in $(1,\ga^{-1})$ and all bounded
$\cD^{\ell}$-measurable region $\La$ of $\Rr^d$,
\begin{eqnarray*}
\big|\log Z_{\ga,\beta,\La} (\vw_{\La^c}) +\beta
\inf_{\vec\rho_\La}
 F_{{\ga,\beta,\La}}(\vec\rho_\La|\vw^{\,\ell}_{\La^c}) \big| \le \beta c \eps(\ga,\ell)
|\La|
\end{eqnarray*}
\end{coro}
Theorem \ref{thm:app} leads also to the Lebowitz-Penrose
limit for the Potts model
\begin{thm}{[Lebowitz-Penrose]}
\label{thm:L-P}
There exists the limit
\begin{eqnarray*}
\lim_{\ga\to\infty} \lim_{\La\to\Rr^d} \frac{\log
Z_{\ga,\beta,\La}}{|\La|}={P^{\mf}_\beta}
\end{eqnarray*}
where $P^{\mf}_\beta$ is the mean field pressure.
\end{thm}
\vskip .5cm \noindent

\begin{proof}
The free energy functional on $L^\infty(\La,S_Q)$ with
boundary conditions $\vec\rho_{\La^c}$ can be rewritten as
\begin{eqnarray*}
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c})=
\cF_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c})
-\frac{1}{2}\int_\La\int_{\La^c} {J_\ga(x,y)}
{|\vec\rho_{\La^c}(y)|^2} dx dy
\end{eqnarray*}
where
\begin{eqnarray*}
\label{proof:lp:2}
\cF_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c}) =&&
\int_\La \phi^{{\mf}}(\vec\rho_\La(x)) dx
+\frac{1}{4}\int_\La\int_\La {J_\ga(x,y)}
|\vec\rho_{\La}(x)-\vec\rho_{\La}(y)|^2 dx dy
\\
&&+\frac{1}{2}\int_\La\int_{{\La^c}} {J_\ga(x,y)}
|\vec\rho_{\La}(x)-\vec\rho_{\La^c}(y)|^2 dx dy
\end{eqnarray*}
with $\phi^{\mf}_\beta(\vec\rho)$ the mean field free
energy density on $S_Q$ \eqref{def:mean-field-free-energy}.
We have clearly
\begin{eqnarray*}
\inf_{\vec\rho_\La\in L^\infty(\La,S_Q)}
\cF_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c}) \ge
\inf_{\vec\rho_\La\in L^\infty(\La,S_Q)}  \int_\La
\phi^{\mf}_\beta(\vec\rho_\La(x)) dx
\end{eqnarray*}
which gives a lower bound for the free energy as
\begin{eqnarray*}
\inf_{\vec\rho_\La\in L^\infty(\La,S_Q)}
F_{{\ga,\beta,\La}}(\vec\rho_\La|\vec\rho_{\La^c}) \ge
-{P^{\mf}_\beta} |\La| -c\ga^{-1} |\partial\La|
\end{eqnarray*}
with
\begin{eqnarray}
P^{\mf}_\beta=-\inf_{\vec v\in S_Q}\phi^{\mf}_\beta(\vec v)
\end{eqnarray}
{}From \ref{thm:app} with $\ell=\ga^{-\frac{1}{2}}$, one gets
\begin{eqnarray*}
\frac{\log Z_{\ga,\beta,\La}}{\beta |\La|}\le
{P^{\mf}_\beta} + c \eps(\ga,\ga^{-1/2}) + \ga^{-1}
\frac{|\partial\La|}{|\La|}
\end{eqnarray*}
and hence
\begin{eqnarray*}
\liminf_{\ga\to 0}\liminf_{\La\nearrow\Rr^d} \frac{\log
Z_{\ga,\beta,\La}}{\beta |\La|} \le {P^{\mf}_\beta}
\end{eqnarray*}
where the limit $\liminf_{\La\nearrow\Rr^d}$ is taken on a sequence of Van Hove subsets of $\Rr^d$.
On the other side, writing \eqref{proof:lp:2} for
$\vec\rho_\La =\vec\rho^{\p} \Ii_\La$, where
$\vec\rho^{\p}$ is the absolute minimizer of $\phi^{\mf}_\beta$, we
get
\begin{eqnarray*}
F_{{\ga,\beta,\La}}(\vec\rho^{\,*}
\Ii_\La|\vec\rho_{\La^c}) \le
\cF_{{\ga,\beta,\La}}(\vec\rho^{\,*}
\Ii_\La|\vec\rho_{\La^c}) \le -{P^{\mf}_\beta} |\La|
+c\ga^{-1} |\partial\La|
\end{eqnarray*}
From the second inequality in \ref{thm:app} one gets
\begin{eqnarray*}
\frac{\log Z_{\ga,\beta,\La}}{\beta |\La|}\ge
{P^{\mf}_\beta} - c \eps(\ga,\ga^{-1/2}) + \ga^{-1}
\frac{|\partial\La|}{|\La|}
\end{eqnarray*}
and finally
\begin{eqnarray*}
\limsup_{\ga\to 0}\red{\limsup_{\La\nearrow \Rr^d}} \frac{\log
Z_{\ga,\beta,\La}}{\beta |\La|} \ge {P^{\mf}_\beta}
\end{eqnarray*}
\end{proof}


\vskip .5cm \noindent


%xxxxxxxx
\vskip2cm
\setcounter{equation}{0}
\section{Contours}
\label{sec:contours}


We first define three {\em phase indicators}:~ $\eta$,
$\theta$, $\Theta$.
These indicators are block spin variables which take
value in $\{0,a_{\ds},a_1,\dots, a_Q\}$, ($a_{\ds}$ a new
``color" which will sign the disordered state). We will define them both on configurations
in $\Om$ and on functions in $L^\infty(\Rr^d,S_Q)$ or in $L^\infty(\Zz^d,S_Q)$,
using the same notation, as
follows:
Let $C_x^{\ell_{-,\ga}}$ the block of the partition $\cD^{\ell_{-,\ga}}$
of size $\ell_{-,\ga}$ %and centered in $x\in \ell_{-,\ga}\Zz^d$
that contains the point $x\in \Rr^{d}$.

\begin{eqnarray}
\label{def:eta} \eta_x(\vec s):=
  \begin{cases}
    a_{{\p}} & \text{if }~
    \dis{\|{\vec s^{\,\ell_{-,\ga}}(x)}-
    \red{\vec \rho_{\beta}^{~(\p)}}\|_{\star}< \ga^{a}}, \quad
    {\p\in\{{\ds},1,\dots, Q\}}\\
    \\
    0 & \text{otherwise}.
  \end{cases}
\end{eqnarray}
where $\vec s^{\,\ell_{-,\ga}}$ is a $\cD^{\ell_{-,\ga}}$-measurable
function defined either by \eqref{def:p-ell0} or by \eqref{def:rhol0}
and $\vec\rho^{~(\q)}$ are the local minimizers in $S_Q$ of the
mean field free energy density.

The value of the exponent $a >0$ defines the ``accuracy
parameter'' $\ga^{a}$ in the right hand side of \eqref{def:eta}.
Our proof will require in addition that $a < \alpha$.

Let then denote by { $C^{\ell_{+,\ga}}_y$} the cube of the partition
$\cD^{\ell_{+,\ga}}$ that contains the point $y\in\Rr^{d}$. We define
\begin{eqnarray}
\label{def:theta} \theta_x(\vec s):=
  \begin{cases}
    \eta_x(\vec s) & \text{if }\eta_y(\vec s)= a_{\p} \quad\forall y\in
{C_x^{\ell_{+,\ga}}}, \\
    0 & \text{otherwise}.
  \end{cases}
\end{eqnarray}
\begin{eqnarray}
\label{def:Theta} \Theta_x(\vec s)=\theta_x(\vec s)
\prod_{{C^{\ell_{+,\ga}}_y:C^{\ell_{+,\ga}}_y\sim
C^{\ell_{+,\ga}}_x}} \Ii_{\theta_x(\vec s)=\theta_y(\vec s)}
\end{eqnarray}
\noindent
The set $\{C^{\ell_{+,\ga}}_y: C^{\ell_{+,\ga}}_y\sim C^{\ell_{+,\ga}}_x\}$ is the set of cubes
in $\cD^{\ell_{+,\ga}}$ $*$-connected with $C^{\ell_{+,\ga}}_x$, i.e. :
$\overline{C^{\ell_{+,\ga}}_y}\sqcap\overline{C^{\ell_{+,\ga}}_x}\neq \emptyset$.

\vskip .5cm \noindent Notice that, by definition,
if $\Theta_x(\vec s)=a_{\p}$ and $\Theta_y(\vec s)=a_{\q}$ and $\p \ne \q$, then
$\dist(x,y)>2\ell_{+,\ga}$
%{ $\{x:
%\Theta_x(\vec s)=a_{\p'} \}\sqcap \{x: \Theta_x(\vec s)=a_{\p} \}= \emptyset $, if
%$\p\ne \p'$, $\p,\p' \in \{{\ds},1,\dots, Q\}$; }~
Thus the
regions  $\{x:\Theta_x(\vec s)=a_{\p}\}$ and  $\{x:\Theta_x(\vec s)=a_{\q}\}$
will be separated by a domain which form the support for the {\em contours}
which we now define with respect to a
configurations $\vw$. A similar definition with respect to a continuous profile
$\vec \rho$ will also hold.


\begin{defin}
\label{def:contours} A contour
$\Ga\equiv(\ssp(\Ga),{\eta_\Ga})$ for a configuration
$\vw$ in $\Om$ , is specified by
%\notes{o \fbox{$\eta_\ga$ ?????}.}
a couple $(\ssp(\Ga),{\eta_\Ga})$ where $\ssp(\Ga)$ is one
of the maximal connected component of the subset of $\Rr^d$
$\{x:\Theta_x(\vw)=0\}$ and {$\eta_{\Ga}$} is the coarse grained configuration on $\ssp(\Ga)$
on the scale $\ell_{\ga,-}$, {$\eta_{\Ga}\equiv\{\eta_x(\vw)\}_{x\in \ssp(\Ga)}$.}
\end{defin}

\vskip .5cm \noindent

We will prove that a Peierls bound for these contours holds
for a range of values in $\beta$. Before stating our precise
result in a Theorem, we need to introduce some notations
related to the notion of contour:

\vskip .5cm \noindent
\centerline{\em Notations }
\vskip .5cm \noindent


For any set $B\sqsubset \Rr^{d}$, $r\in \Rr$, we define
\begin{eqnarray}
\delta_{\ins}^r[B]&:=&\{x\in B: \dist(x,B^c)\leq r\}\\
\delta_{\out}^r[B]&:=&\{x\in B^{c}: \dist(x,B)\leq r\}
\end{eqnarray}
where $\dist(x,B)=\inf_{y\in B} \dist(x,y)$.% as in \eqref{def:dist}.


We denote by $|\Ga|\equiv|\ssp(\Ga)|$ the volume of the region
$\ssp(\Ga)$. For any bounded contour ($|\Ga|<\infty$), we denote
by $\Ext(\Ga)$ the (unique) unbounded connected component of $\ssp(\Ga)^{c}$ and  by $\{\Int_{i}(\Ga)\}$,
$i\in I$ the collection of its bounded connected components:
\begin{eqnarray}
\ssp(\Ga)^{c}=\Ext(\Ga)\sqcup_{i\in I}\Int_{i}(\Ga)
\end{eqnarray}
By definition of a contour, for any point $x$ in $\delta_{\out}^{\ell_{+,\ga}}[\ssp(\Ga)]$, $\Theta_x\ne 0$
and we define
\begin{eqnarray}
A^{\q}\equiv A^{\q}(\Ga):=\{x\in  \delta_{\out}^{\ell_{+,\ga}}[\ssp(\Ga)]: \Theta_x= a_q \}
\qquad ; \qquad A(\Ga):= \bigsqcup_{\q} A^{\q} = \delta_{\out}^{\ell_{+,\ga}}[\ssp(\Ga)]
\end{eqnarray}
Note that $A^{\q}$ depends only on $\Ga$.
Correlatively, for each connected component of $\ssp(\Ga)^{c}$, $\Int_i(\Ga)$ (respectively $\Ext(\Ga)$)
the phase indicator $\Theta$ is constant and nonzero on $\delta_{\ins}^{\ell_{+,\ga}}[\Int_i(\Ga)]$
(respectively on $\delta_{\ins}^{\ell_{+,\ga}}[\Ext(\Ga)]$). Thus we write
\begin{eqnarray}
I^{\q}(\Ga):=\{i: \Int_i(\Ga)\sqcap A^{\q}\neq \emptyset; \}
\qquad ; \qquad \Int^{\q}(\Ga):= \bigsqcup_{i\in I^{\q}(\Ga)}\Int_i(\Ga)
\end{eqnarray}
We call $\Ga$ a ``$\p$-contour'' if $\Ext(\Ga)\sqcap A^{\p}\neq \emptyset$
%if  $\theta_x=a_{\q}$  $\forall x\in A\sqcap \Ext(\Ga)$.
and define
\begin{eqnarray}
c(\Ga):=\ssp(\Ga)\sqcup_{i\in I}\Int_{i}(\Ga)
\end{eqnarray}
\vskip .5cm \noindent

\centerline{\em Weights of contours }

\vskip .5cm \noindent
Let denote by $\cE(\Ga,\p)$ the event that $\Ga$ is a $\p$-contour:
\begin{eqnarray*}
& &\cE(\Ga,\p):= \left\{\vw: \eta_x(\vw)=\eta_\Ga ~~\forall~ x\in
\ssp(\Ga)~~; ~~\red{ \linetwo{\Theta_x(\vw)=a_{\q} ~~\forall~  x\in
A^{\q}(\Ga)~\forall \q\in \{{\ds},1,\dots, Q\}}{\Theta_x(\vw)=a_{\p} ~~\forall~  x\in
A(\Ga)\sqcap \Ext(\Ga)}}\right\}
\end{eqnarray*}
and by $\cE(\not\Ga,\p)$ the event that the phase $\p$ extends on $\ssp(\Ga)\sqcup A(\Ga)$
\begin{eqnarray*}
& &\cE(\not\Ga,\p):= \{\vw: \Theta_x(\vw)=a_{\p} ~~\forall~
x\in \ssp(\Ga)\sqcup A(\Ga)\}
\end{eqnarray*}

\vskip .5cm \noindent
We then define the { weight} of a
$\p$-contour as follows:
\begin{eqnarray}
\label{weight} w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}}):=
\frac{\mu_{\ga,\beta,c(\Ga)\setminus
\Int^{\p}(\Ga)}(\cE(\Ga,\p)|{\vw_{A^{\p}}})}{\mu_{\ga,\beta,c(\Ga)\setminus
\Int^{\p}(\Ga)}(\cE(\not\Ga,\p)|{\vw_{A^{\p}}})}
%\frac{Z_{R^-(\Ga)}(\Ga|\bar\si)}{Z_{R^-(\Ga)}(\not
%\Ga|\bar\si)}
\end{eqnarray}

\vskip 1.5cm \noindent

\centerline{\em Peierls bounds }
\vskip .5cm \noindent

We say that the ${\p}$-contours satisfy a Peierls bound if
there exists {a constant $\fK >0$} such that for any ${\p}$-contour $\Ga$:
\begin{eqnarray}
\label{eq:standard} w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}})\le
e^{-{\fK } ~N_\Ga}
\end{eqnarray}
where $N_\Ga$ is the number of cubes of
$\cD^{\ell_{+,\ga}}$ in $\ssp(\Ga)$:
\begin{eqnarray*}
N_\Ga:= \frac{|\Ga|}{|C_0^{\ell_{+,\ga}}|}
\end{eqnarray*}

\begin{thm}
\label{thm:standard} \red{If there is $\fK_\ga $ large enough such that,
for any $\p\in\{-1,1,\cdots,Q\}$, the weights
$w^{\p}_\ga(\Ga;\vw_{A^{\p}})$ satisfy  the Peierls bound
\eqref{eq:standard}
then
%, for these values of the parameters, there is a phase transition, in the sense that
there exist $Q+1$  distinct DLR measures and:% in particular
\begin{eqnarray*}
&&\mu^{(q)}_{\ga,\beta}(\Theta=a_{\ds})\neq
%\mu^{(q)}_{\ga,\beta}(\Theta=a_{\ds})\neq
\mu^{(\ds)}_{\ga,\beta}(\Theta=a_{\ds}) \hskip1cm \forall
q>0%q\in\{1,\dots,Q\}
\\
&&\mu^{(q)}_{\ga,\beta}(\Theta=a_{\ds})\le e^{-(\fK_\ga -b)}
\hskip2.5cm \forall q>0%q\in\{1,\dots,Q\}~~
\end{eqnarray*}
with $b $ a positive
constant.
}\end{thm}

The proof of Theorem \ref{thm:main} follows from \blue{Theorem \ref{thm:standard}},
 showing that \blue{there exist a value of
$\beta=\beta_c(Q,\ga)$ and a constant $\fK:=\fK_\ga (\ga,\beta)$} such that the bounds \eqref{eq:standard}
hold.

\red{The proof of Theorem \ref{thm:standard} uses the well known Peierls
argument and is omitted.}

\vskip .5cm \noindent
The key estimate is stated in the following Theorem:
\begin{thm}
   \label{thm:Peierls-0}
there
 exists a value $\bar \ga $ such that for any $\ga<\bar\ga$ there exists $\bcg$:
       \begin{equation}
       \label{eq:key}
       w^{\p}_{\ga,\bcg}(\Ga;\vw_{A^{p}})\le \exp\{-\fK_\ga |N_\Ga|\}\hskip1cm
  \text{with }~~    \fK_\ga= c_p \bcg\ga^{2a}
|C^{\ell_{-,\ga}}|
       \end{equation}
       with $c_d$ a constant depending only on $Q$ and on the dimension of the space,
%which value is given in \eqref{def:cd}
\begin{equation}
c_p=\frac{1}{3^{d+1}}(Q-\bcmf) > 0
\end{equation}

\end{thm}
%xxxxxxxxxxxxxxxxxxxxxxxxx
%\newpage
\vskip 2.5cm \noindent
\section{Peierls estimates}
\vskip .5cm \noindent


\red{Let $\cX^{\q}:=\{\vec s: \Theta_x(\vec s)=a_{\q}~\forall x\in
\Rr^{d}\}$, defined for  $\vec s$ that can be either in $\Om$, or in
$ L^{\infty}(\Rr^d,S_Q)$ or in $L^{\infty}(\Zz^d,S_Q)$}.


For any $\q\in \{\ds,1,\dots,Q\}$, and any
$\vec s\in\cX^{\q}:=\{\vec s\in L^{\infty}(\Zz^d,S_Q): \Theta_x(\vec s)=a_{\q}~\forall x\in
\Rr^{d}\}$, we define the ``diluted partition
function"
$Z^{\q}_{\ga,\beta,\La}(\vec s_{\La^c})$, as

\begin{eqnarray}
\label{def:diluted-true}
Z^{\q}_{\ga,\beta,\La}(\vec s_{\La^{c}})&:=&
Z_{\ga,\beta,\La}
(\{\vw_\La:\theta_x(\vw_\La)=a_{\q} \quad\forall x\in
\delta_{\ins}^{\ell_{+,\ga}}[\La]\}|\vec s_{\La^{c}})
\end{eqnarray}
where the definition of the Hamiltonian \eqref{def:Kac-Ham-c} have been naturally
extended to boundary conditions in $L^{\infty}(\Zz^d,S_Q)$.

Let $\Ga$ a $\p$-contour and
\begin{eqnarray}
\label{def:gp}
\hG^{\p}:=\ssp(\Ga)\bigsqcup_{\q\ne \p} A^{\q}
\end{eqnarray}

\vskip .5cm \noindent
\begin{thm}
   \label{thm:Peierls-factoriz}
  \red{ There are $\bar \ga,
   \bar b$ and a constant $c$ such that for
all $\ga<\bar \ga, |\beta-\bcmf|<\bar b $}
%and all $\beta$ such that \red{$|\beta-\bcmf|<\ga^{1/2}$}
:
       \begin{eqnarray}
       \label{eq:Peierls-factoriz}
       w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}})&\le &
       \frac{\prod_{\q\ne \p} Z^{\q}_{\ga,\beta,\Int^{q}(\Ga)\setminus A^{\q}}
       ({\vec\rho^{\q}})}{
       \prod_{\q\ne \p} Z^{\p}_{\ga,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}(\vec\rho^{\p})}
       ~~\exp\{c\ga^{\frac{1}{2}}|\Ga|\}
       \\ \nn
       &&\hskip-1cm
       \cdot\exp\bigg\{-\beta\bigg[\inftwo{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=\eta_\Ga}{\vec\rho_{A^{\q}}=\vec\rho^{~\q}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^{\p}})
-\inftwo{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=a^{p}}{\vec\rho_{A^{\q}}=\vec\rho^{~\p}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^p})
       \bigg]\bigg\}
       \end{eqnarray}
\end{thm}
\vskip .5cm \noindent



\begin{proof}[Proof of Theorem \ref{thm:Peierls-factoriz}]
\label{proofofPeierls-factoriz}

The proof of this theorem is a straightforward  consequence
of the analysis of section \ref{sec:freeenergy} and appendix
\ref{app:dinamica}.
We recall the definition of the weight of a $\p$-contour given in \eqref{weight}:
\begin{eqnarray*}
w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}}):=
\frac{\mu_{\ga,\beta,c(\Ga)\setminus
\Int^{\p}(\Ga)}(\cE(\Ga,\p)|{\vw_{A^{\p}}})}{\mu_{\ga,\beta,c(\Ga)\setminus
\Int^{\p}(\Ga)}(\cE(\not\Ga,\p)|{\vw_{A^{\p}}})}
\end{eqnarray*}
For each set $A^{\q}$, $\q\ne\p$, we denote by:

\begin{eqnarray}
\label{def: hatAq}
\hat A^{\q}&:=& A^{\q}\sqcup\delta_{\out}^{{\ell_{+,\ga}}}[A^{\q}]\\
\label{def: checkAq}
\check A^{\q}&:=& A^{\q}\sqcup\delta_{\out}^{{\ell_{+,\ga}/2}}[A^{\q}]\\
\label{def: Siq}
%\Si^{\q}:=\hat A^{\q}\setminus \check A^{\q}&\qquad& \bar\Si^{\q}:=\check A^{\q}\setminus A^{\q}\\
%\label{def: Siqi}
%\Si^{\q,i}&:=& \Si^{\q}\sqcap \ssp(\Ga)\hskip1cm;\hskip1cm
\Si^{\q,e}&:=&\bigl(\hat A^{\q}\setminus \check A^{\q}\bigr)\sqcap \ssp(\Ga)^c\\
%\bar\Si^{\q,i}&:=& \bar\Si^{\q}\sqcap \ssp(\Ga)\hskip1cm;\hskip1cm
\bar\Si^{\q,e}&:=&\bigl(\check A^{\q}\setminus A^{\q}\bigr)\sqcap \ssp(\Ga)^c\\
\Si^{e}&:=&\sqcup_{\q\ne\p}\Si^{\q,e}
\end{eqnarray}

and also:
\begin{eqnarray*}
& &\Delta^{\p}:=\ssp(\Ga)\sqcup_{\q\ne\p}\check A^{\q}
\\
& &V^{\q}:=\Int^{\q}(\Ga)\setminus \hat A^{\q}
\end{eqnarray*}

We write:
\begin{eqnarray*}
w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}})&:=&
\frac{\dis{
\sum_{\vw_{\Si^{e}}:\vw_{\Si^{\q,e}}\in\cX^{\q}}e^{-\beta \sum_{\q\ne \p} H_\ga(\vw_{\Si^{\q,e}})}
Z_{\beta,\Delta^{\p}}(\cE(\Ga)|\vw_{\Si^{e}};\vw_{A^{\p}})
\prod_{\q\ne\p}
Z^{\q}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})
}}{\dis{\sum_{\vw_{\Si^{e}}\in\cX^{\p}}e^{-\beta H_\ga(\vw_{\Si^{e}})}
Z_{\beta,\Delta^{\p}}(\cE(\not\Ga)|\vw_{\Si^{e}};\vw_{A^{\p}})
\prod_{\q\ne\p}
Z^{\p}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})}}
\end{eqnarray*}



\vskip .5cm \noindent
By Theorem \ref{thm:app} with $\ell=\ell_0$, we have:
\begin{eqnarray}
\label{prf:app1}
& &w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}})\le e^{c\ga\ell_0|\Ga|}\cdot\\
&&\hskip.3cm
\frac{\dis{
\sum_{\vw_{\Si^{e}}:\vw_{\Si^{\q,e}}\in\cX^{\q}}e^{-\beta \sum_{\q\ne \p} H_\ga(\vw_{\Si^{\q,e}})}
\exp\{-\beta\inftwo{\rho: \eta(\rho)=\eta_\Ga ~r\in\ssp(\Ga)}{~\eta(\rho)=a_{\q} ~r\in\check A^{\q}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};
\vw^{\ell_0}_{A^{\p}})\}
\prod_{\q\ne\p}
Z^{\q}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})
}}{\dis{\sum_{\vw_{\Si^{e}}\in\cX^{\p}}e^{-\beta H_\ga(\vw_{\Si^{e}})}
\exp\{-\beta\inf_{\rho: \eta(\rho)=a_{\p}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};
\vw^{\ell_0}_{A^{\p}})\}
\prod_{\q\ne\p}
Z^{\p}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})}}
\nn\end{eqnarray}
Using Corollary \ref{corol:dimamica} with $\La=\check A^{\q}$, we can find a lower bound
for the free energy term in the numerator by considering density profiles $\vec\rho$ identically
equal to $\vec\rho^{\q}$ on $A^{\q}$,at the expense of a small error term; we have:
\begin{eqnarray*}
& &\inftwo{\rho: \eta(\rho)=\eta_\Ga ~r\in\ssp(\Ga)}{~\eta(\rho)=a_{\q} ~r\in\check A^{\q}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw^{\ell_0}_{A^{\p}})\ge
\infthree{\rho: \eta(\rho)=\eta_\Ga \; r\in\ssp(\Ga)}{~\eta(\rho)=a_{\q} \; r\in\check A^{\q}}
{\vec\rho\equiv\vec\rho^{\q} \;  r\in  A^{\q}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw^{\ell_0}_{A^{\p}})-c_\om e^{-\om \ga\ell_{+,\ga}/2}
|\check A|\\
& &=\inftwo{\rho: \eta(\rho)=\eta_\Ga \; r\in\ssp(\Ga)}{\vec\rho\equiv\vec\rho^{\q} \;  r\in  A^{\q}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})
 +\sum_{\q}\inf_{\rho: \eta(\rho)=a_{\q}}
F_{\ga,\beta,\bar\Si^{\q,e}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vec\rho^{q}_{A^{\q}})
-c_\om e^{-\om \ga\ell_{+,\ga}/2}|\Ga|\\
& &\ge\inftwo{\rho: \eta(\rho)=\eta_\Ga \; r\in\ssp(\Ga)}{\vec\rho\equiv\vec\rho^{\q} \;  r\in  A^{\q}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})
 -\frac{1}{\beta}\sum_{\q}\log\bigl(Z_{\beta,\bar\Si^{\q,e}}(\vw_{\Si^{e}};\vec\rho^{q}_{A^{\q}})\bigr)
- \bigl(c_\om e^{-\om \ga\ell_{+,\ga}/2} +c \ga\ell_0\bigr)|\Ga|
\end{eqnarray*}
The free energy term in the denominator of \eqref{prf:app1} can be directly bounded from above, as
\begin{eqnarray*}
& &\inf_{\rho: \eta(\rho)=a_{\p}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw^{\ell_0}_{A^{\p}})
\le
\inftwo{\rho: \eta(\rho)=a_{\p}}{\vec\rho\equiv\vec\rho^{\q} \;  r\in  A^{\q}}
F_{\ga,\beta,\Delta^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw^{\ell_0}_{A^{\p}})
\\
& &=\inf_{\rho: \eta(\rho)=a_{\p}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})
 +\sum_{\q}\inf_{\rho: \eta(\rho)=a_{\p}}
F_{\ga,\beta,\bar\Si^{\q,e}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vec\rho^{\p}_{A^{\q}})
\\
& &\le
\inf_{\rho: \eta(\rho)=a_{\p}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})
 -\frac{1}{\beta}\sum_{\q}\log\bigl(Z_{\beta,\bar\Si^{\q,e}}(\vw_{\Si^{e}};\vec\rho^{\p}_{A^{\q}})\bigr)
 +c \ga\ell_0|\Ga|
\end{eqnarray*}
Inserting both estimates in \eqref{prf:app1} we get the following factorization:
\begin{eqnarray*}
& &w^{\p}_{\ga,\beta}(\Ga;\vw_{A^{\p}})\\
& &\le e^{\beta(3 c\ga\ell_0+ c_\om e^{-\om \ga\ell_{+,\ga}/2})|\Ga|}\quad
\frac{\dis{
\sum_{\vw_{\Si^{e}}:\vw_{\Si^{\q,e}}\in\cX^{\q}} \prod_{\q\ne\p} e^{-\beta H_\ga(\vw_{\Si^{\q,e}})}
Z_{\beta,\bar\Si^{\q,e}}(\vw_{\Si^{e}};\vec\rho^{q}_{A^{\q}})
Z^{\q}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})}}
{\dis{
\sum_{\vw_{\Si^{e}}\in\cX^{\p}} \prod_{\q\ne\p} e^{-\beta H_\ga(\vw_{\Si^{\q,e}})}
Z_{\beta,\bar\Si^{\q,e}}(\vw_{\Si^{e}};\vec\rho^{\p}_{A^{\q}})
Z^{\p}_{\beta,V^{\q}}(\vw_{\Si^{\q,e}})
}}
\\& &\qquad\times
\exp\bigl\{-\beta \bigl(\inftwo{\rho: \eta(\rho)=\eta_\Ga \; r\in\ssp(\Ga)}{\vec\rho\equiv\vec\rho^{\q} \;  r\in  A^{\q}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})- \inf_{\rho: \eta(\rho)=a^{p}}
F_{\ga,\beta,G^{\p}}(\vec\rho|\vw^{\ell_0}_{\Si^{e}};\vw_{A^{\p}})\bigr)\bigr\}
\\
& & = e^{3 \beta c\ga\ell_0|\Ga|+ \beta c_\om e^{-\om \ga\ell_{+,\ga}/2}}\quad
       \frac{\prod_{\q\ne \p} Z^{\q}_{\ga,\beta,\Int^{q}(\Ga)\setminus A^{\q}}
       ({\vec\rho^{\q}})}{
       \prod_{\q\ne \p} Z^{\p}_{\ga,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}(\vec\rho^{\p})}
       \\ \nn
       &&\hskip-1cm
       \cdot\exp\bigg\{-\beta\bigg[\inftwo{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=\eta_\Ga}{\vec\rho_{A^{\q}}=\vec\rho^{~\q}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^{\p}})
-\inftwo{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=a^{p}}{\vec\rho_{A^{\q}}=\vec\rho^{~\p}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^p})
       \bigg]\bigg\}
\end{eqnarray*}

\end{proof}




%{sec:proof-Peierls-factoriz}.
Using Theorem \ref{thm:Peierls-factoriz},
the proof of the Theorem \ref{thm:Peierls-0}
follows from estimates %will be then obtained by an  estimate
on the terms appearing in \eqref{eq:Peierls-factoriz},
which are provided by two theorems, \ref{thm:Peierls-3} and
\ref{thm:Peierls-2}.


%The bound on the last
%factor is postponed to section \ref{section:largedeviation}
%and it will be the smaller factor
%%main gain in removing the contour,
%while the first factor will come out to be sufficiently close to one
%(for a suitable value of $\beta$).
%%to not destroy the gain due to the last .....


\vskip .5cm \noindent Notice that the first factor in
\eqref{eq:Peierls-factoriz}:
\begin{eqnarray*}
\frac{\prod_{\q\ne \p}
Z^{\q}_{\ga,\beta,\Int^{\q}(\Ga)\setminus
A^{\q}}(\vec\rho^{(\q)})}{
       \prod_{\q\ne \p} Z^{\p}_{\ga,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}(\vec\rho^{(\p)})}
\end{eqnarray*}
would be identically equal to $1$ in presence of symmetry
between all the phases. In a non-symmetric case this factor
is close to one only for special values of the parameters.
Our case is somehow an intermediate one since there is a
symmetry (by color permutation) relating the ordered
phases, and we need only to distinguish between ordered
($\q>0$) and disordered ($\q<0$) phases. The proof of the
Peierls bound requires an accurate estimate of the
partition functions in this ratio, which means to go beyond
the bulk contribution and get also some control over
boundary terms.



We now state the two Theorems which allow us to control the
terms appearing in the factorization formula of Theorem
\ref{thm:Peierls-factoriz}:
\begin{thm}
\label{thm:Peierls-3} For all $\beta$ such that
$|\beta-\bcmf|\le c_b\ga^{1/2}$
\begin{eqnarray*}
& &
%\bigg[
\inftwo{\red{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=\eta_\Ga}}{\red{\vec\rho_{A^{\q}}=\vec\rho^{\q}}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^p})
-\inftwo{\red{\vec\rho_{\hG^{\p}}:
       \eta(\vec\rho_{\hG^{\p}})=a_{\p}}}{\red{\vec\rho_{A^{\q}}=\vec\rho^{\p}}}
       F_{\ga,\beta,\hG^{\p}}
       (\vec\rho_{\hG^{\p}}|\vw^{(\ell_0)}_{A^p})
%       \bigg]
\\\nn
&&\hskip5.3cm
\red{\ge c_f \ga^{2a}\ell_{-,\ga}^{d}|\Ga|}
       +\sum_{\q\ne\p}\cI_{A^{\q}}(|\vec\rho_{\q}|^{\;2}-|\vec\rho_{\p}|^{\;2})
\end{eqnarray*}
where
\begin{eqnarray*}
\cI_{A^{\q}} = \frac{1}{2} \int_{A^{\q}}\int_{(\Int^{\q}(\Ga)\setminus A^{\q})} J_\ga(x,y) dx dy
\end{eqnarray*}
and the constant $c_b$ is fixed by theorem \ref{thm:equalitypresure} and $c_f$ reads
\begin{eqnarray}
c_f=\frac{1}{3^{d+1}}(\frac{Q}{\beta}-1)
\end{eqnarray}
\end{thm}
\vskip .5cm \noindent
\begin{thm}
   \label{thm:Peierls-2}
There is $\bar \ga$, and a constant $\kappa_1$ such that
for any $\ga<\bar \ga$, there is a value of $\beta$, $\bcg$:
\begin{equation}
       \label{eq:Peierls-2}
\left|\ln\bigl(
       \frac{Z^{(q)}_{\ga,\red{\bcg},\La}(\vec\rho^{(q)}) e^{\cI_{A^{\q}}|\vec\rho^{(q)}|^{\;2}}}{
        Z^{(-1)}_{\ga,\red{\bcg},\La}(\vec\rho^{(-1)})e^{\cI_{A^{\q}}|\vec\rho^{(-1)}|^{\;2}}}
        \bigr)\right|\le \kappa_1\ga^{\red{1/8}}|\Ga|\hskip3cm \forall q>0
       \end{equation}

       \end{thm}

The proof of Theorem \ref{thm:Peierls-3} is postponed until
Section \ref{section:largedeviation}, while
Theorem \ref{thm:Peierls-2} will be proved in the next sections.

In order to prove Theorem \ref{thm:Peierls-2}
we will introduce $Q+1$ ``abstract contour models"
and estimate the ratio between their finite volume  partition functions.
The relation between  the partition functions of the
``abstract contour models" and  the corresponding diluted partition functions of the
``true model" (see \ref{def:diluted-true}), will
be stated for suitable values of $\beta$ in Theorem \ref{thm:ab-true} below.

We will prove that there exists a value of $\beta$, $\bcg$, so that
the pressures of these abstract systems are all equal.
For this value of the temperature we will be also able to get an estimate
on the ``finite volume correction
to these pressures", hence on the value of finite volume partition functions
of the abstract models.
This result is obtained by estimates
on the decay of the correlations and a control on the probability of
``small deviations"
for the measures of these abstract contour models.


\centerline{\em Abstract contour models}


\vskip .5cm \noindent For any $\p$
we define:
    \begin{eqnarray}
    &&
\mathcal X^{\p} :=\{ \vw: \theta_x(\vw)\equiv a_{\p}, \,
x\in \mathbb Z^d\}
       \label{z4.4}
       \\
         \label{z4.2}
&&\{\Ga\}^{\p}   := \text{ the collection of all $\p$ contours
s.t $|\ssp(\Ga)|<\infty$ }
\\&&
\mathcal B^{\p} := \big\{ \und \Ga=(\Ga_1,..,\Ga_n): n \in \mathbb
N_+,\; \Ga_i \in \{\Ga\}^{\p},\,
 \overline{\text{sp}(\Ga_i)}  \sqcap  \overline{\text{sp}(\Ga_j)} = \emptyset ,\; i\ne j \big\}
         \label{z4.3}
       \\
       &&
\mathcal B^{\Ext,\p} :=\{\und\Ga\in \mathcal B^{\p}:
\ssp(\Ga)\sqcap\Int(\Ga')=\emptyset ~~\forall \Ga,\Ga'\in \und\Ga\}
    \end{eqnarray}
   \red{where $\overline{A}$ denotes the closure of the set $A$.}
    \vskip .5cm \noindent
$\{\Ga\}^{\p}_\La$, $\mathcal B^{\p}_\La$ and $\mathcal B_\La^{\Ext,\p}$ are defined
similarly, but with the condition that all contours should have
spatial support in $\La$.
%$\und\Ga^{\Ext,\q}_\La :=\{\Ga\in \mathcal B^{\q}_\La:
%\Ext(\Ga)\sqcap\Int(\Ga')=\emptyset \forall \Ga'\in \mathcal
%B^{\q}_\La\}$.



\vskip .7cm \noindent Abstract contour models are systems defined on
the product spaces $\cX^{\p}\times \cB^{\p}$. The partition functions read:
%$p\in \{o,d\}$:
\begin{eqnarray}
\label{z4.7} \red{Z_{\abs,\beta,\La}^{\p}(\und\vw_{\La^c})}=
\sum_{\und\Ga\in\cB^{\p}_{\La}}
\sum_{\vw_{\La}\in\cX^{\p}_{\La}}\prod_{\Ga\in \und\Ga
}W^{\p}(\Ga;\vw)e^{-\beta
H_{\ga,\La}(\vw_{\La}|\und\vw_{\La^c})}
\end{eqnarray}
%where ``abs'' stands for ``abstract" and
where the weights $W^{\p}(\Ga;\vw)$ are assigned with an
arbitrary rule and may depend on $\vw\in \cX^{\p} $, but
only through its restriction on $\ssp(\Ga)\bigsqcup_{\p}
A^{\p}$. The abstract contours are given by specifying
$(\ssp(\Ga),\eta_{\Ga})$, where $\eta_{\Ga}$ is an arbitrary $\cD^{\ell_{-,\ga}}$-measurable function
with value in $\{0,a_{-1},a_1,\cdots,a_Q\}$,
%a set of
%value of the variables $\eta_{x};~ x\in \ssp(\Ga),
%~C^{\ell_{-,\ga}}_x\in\cD^{\ell_{-,\ga}}$, that are
%specified arbitrarily, with the only conditions to have
with
a constant value different from zero on each connected component of the
boundaries $\delta_{\ins}^{\ell_{+,\ga}}[\ssp(\Ga)]$, and
compatible with the condition $\Ga\in \{\Ga\}^{\p}$ .


\vskip .5cm \noindent
Then, for any $\und\vw \in \mathcal X^{\p}$, we introduce
the ``dilute'', finite volume Gibbs measures on
$\cX^{\p}_{\La} \times \mathcal B^{\p}_\La$, $\La$ a
bounded $\mathcal D^{\ell_{+,\ga}}$ measurable region, by
setting
%       \begin{equation}
%         \label{z4.6}
%\mu_{\abs,\La}^{\p}(d\vw',\und \Ga|\und\vw_{\La^{c}}) :=
%\frac{{\bf 1}_{\vw' \in
%\cX^{\p}}}{Z_{\abs,\beta,\La}^{\p}(\und\vw_{\La^{c}})}
%%W^{\p}(\und\Ga,\vw')
%\prod_{\Ga \in \und\Ga} W^{\p}(\Ga,\vw')
%e^{-H_{\ga,\La}( \vw'_\La |
%\und\vw_{\La^c})} \delta( \vw'_{\La^c}- \und\vw_{\La^c})d
%\vw'_{\La^c}
%        \end{equation}
       \begin{equation}
         \label{z4.6}
\mu_{\abs,\La}^{\p}(\vw',\und \Ga|\und\vw_{\La^{c}}) :=
\frac{{\bf 1}_{\vw' \in
\cX^{\p}}}{Z_{\abs,\beta,\La}^{\p}(\und\vw_{\La^{c}})}
\prod_{\Ga \in \und\Ga} W^{\p}(\Ga,\vw')
e^{-H_{\ga,\La}( \vw'_\La |
\und\vw_{\La^c})}
        \end{equation}

 % and
%      \begin{equation}
%      \label{z4.7}
%Z_{\abs,\La}^p( \bar\si_\La^{c})=  \sum_{\und \Ga \in \mathcal
%B^{q}_\La} \sum_{{\si'\in \cX_\La^{p}}}
%W^p(\und\Ga,\si) e^{-H_{\ga,\La}(  \si'_\La |
%\si_{\La^c})}\
%    \end{equation}
%By $H_{\ga,\La}(s'_\La|s_{\La^c})$ we denote
%the conditional energy, formally defined as $H(s'_\La
%s_{\La^c}) - H(s_{\La^c})$.


\vskip.5cm

{\it Remarks.}  Notice that the elements of a pair
$(\vw,\und \Ga)$ in an abstract model are totally
unrelated, in fact the configuration $\vw$ is in $\mathcal
X^{\p}$ and therefore
has no contour. %(the weight of a contour may  depend on $\si$, but not its existence),


\vskip 1.5cm %\noindent
 We will consider the $Q+1$ ``abstract contour models" , for any $\p\in\{\ds,1,\dots,Q\}$,
  defined by the following choice of ``cut-off weights" $W^{\p}(\Ga|\vw)$,
 \begin{eqnarray}
 \label{cutoff-weight}
\hat W^{\p}_\ga(\Ga;\vw):=\min
\{w^{\p}_{\abs,\beta}(\Ga;\vw), e^{-\frac{\fK_{\ga}}{2} N_\Ga}\}
\hskip1cm %\hat\fK =\frac{1}{2}c\beta \ga^{2a}|C^{\ell_{-,\ga}}|
 \end{eqnarray}
 \blue{where $\fK_\ga$ is given by \eqref{eq:key} }
 and   $w^{\p}_{\abs,\beta}(\Ga;\vw)$ are defined  by an inductive way on the volume, as:
% is given by:% \eqref{weight}.

\begin{equation}
\label{def:w-abs} w^{\p}_{\abs,\beta}(\Ga;\vw_{A^{\p}}):= \frac{
\sum_{\vw_{\hG^{\p}}}e^{-\beta
H_{\ga,\hG^{\p}}(\vw_{\hG^{\p}}|\vw_{A^{\p}})}
\Ii_{\eta(\vw_{\ssp(\Ga)})=\eta_\Ga}\prod_{\q\ne\p}\Ii_{\eta_{\vw_{A^{\q}}}=a_{\q}}
 Z_{\abs,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}^{\q}(\vw_{A^{\q}})}
{\sum_{\vw_{\hG^{\p}}}e^{-\beta
H_{\ga,\hG^{\p}}(\vw_{\hG^{\p}}|\vw_{A^{\p}})}
\Ii_{\eta(\vw_{\hG^{\p}})=a_{\p}}\prod_{\q\ne\p}
 Z_{\abs,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}^{\q}(\vw_{A^{\q}})}
\end{equation}
 %Since $\frac{\fK}{2}<\fK$,  if $w^{\q}(\Ga|\si)$ satisfies
% \eqref{eq:key}, $\hat W^{\p}_\ga(\Ga;\si)=w^{\q}(\Ga|\si)<e^{-\frac{\fK}{2} N_\Ga}$.

We point out   that
%Notice that
the sum $\sum_{\vw_{\hG^{\p}}}$ in the
definition \eqref{def:w-abs}, that is a sum on
the whole space $\Om_{\hG^{\p}}$,
%$\vw_{\hG^{\p}}\in \Om_{\hG^{\p}}$ and only defines the
enter only in the definition of the
weights,
without any relations with  the configurations
$\vw\in\cX^{\p}$ of the abstract model that we are
considering.


Recalling the definition of diluted partition function \eqref{def:diluted-true}
we state the following theorem:
\vskip .5cm \noindent
\begin{thm}
   \label{thm:ab-true}
If for any $\p$ and any $\p$-contour $\Ga$ the weights
$\hat W^{\p}_\ga(\Ga;\vw)$, defined by
\eqref{cutoff-weight},
satisfy %(for any $\p$):
       \begin{equation}
       \label{eq:ab-true1}
       \hat W^{\p}_\ga(\Ga;\vw)< e^{-\frac{\fK_\ga}{2} N_\Ga}
       \end{equation}

then %(for any $\p$):
\begin{equation}
       \label{eq:ab-true2}
W^{\p}_\ga(\Ga;\vw)= w^{\p}_{\ga,\beta}(\Ga;\vw)\hskip1cm
;\hskip1cm
       Z_{\abs,\beta,\La}^{\p}( \und\vw_\La^{c})= Z_{\beta,\La}^{\p}( \und\vw_\La^{c})
       \end{equation}
\end{thm}

\vskip .5cm \noindent
\begin{proof}
 We first rewrite the  diluted partition function of the ``true model",
$Z_{\beta,\La}^{\p}(\und\vw_{\La^c})$, as

\begin{eqnarray*}
Z_{\beta,\La}^{\p}(\und\vw_{\La^c})=
\sum_{\und\Ga\in\cB^{\Ext,\p}_\La}
\sum_{\vw_{R(\und\Ga)}\in\cX^{\p}}\prod_{\Ga\in
\und\Ga }w^{\p}_{\ga,\beta}(\Ga;\vw)e^{-\beta
H_{\ga,\La}(\vw_{R(\und\Ga)}|\und\vw_{\La^c})}
\blue{\prod_{\q}\prod_{i\in I_{\q}}
 Z_{\abs,\beta,\Int_i^{\q}(\Ga)\setminus A^{\q}(\Ga)}^{\p}(\vw_{R(\und\Ga)})}
 %\prod..
\end{eqnarray*}
where
%$R(\und\Ga):=\La\sqcap[\Ext(\und\Ga)\sqcup\ssp(\und\Ga)\sqcup A(\Ga)]$,
%$\Ext(\und\Ga):=\sqcap_{\Ga\in \und\Ga} \Ext(\Ga)$.
\begin{eqnarray*}
R(\und\Ga):=\La\setminus\bigl(\sqcup_{\Ga\in\und\Ga}\sqcup_{i\in I_\Ga} (\Int_i^{\q}(\Ga)\setminus A^{\q}(\Ga))\bigr)
\end{eqnarray*}
\red{Proceeding by iteration on $Z_{\abs,\beta,\Int^{\q}(\Ga)\setminus A^{\q}}^{\p}(\vw_{R(\und\Ga)})$
one gets:}
\begin{eqnarray*}
Z_{\beta,\La}^{\p}(\und\vw_{\La^c})= \sum_{\und\Ga\in
\cB^{\p}_\La}
\sum_{\vw_{\La}\in\cX^{\q}_{\La}}\prod_{\Ga\in \und\Ga
}w^{\p}_{\ga,\beta}(\Ga;\vw)e^{-\beta
H_{\ga,\La}(\vw_{\La}|\und\vw_{\La^c})}
\end{eqnarray*}
\vskip.5cm \noindent
The proof of the theorem \ref{thm:ab-true} is then obtained  by induction on the volume
(see for instance \cite{errico-leip}, or \cite{bkmp2}).

\blue{Let $\Delta_n$ a sequence of increasing $\cD^{\ell_{+,\ga}}$-measurable regions with
$\Delta_1: |\Delta_1|=3^{d}|C^{\ell_{+,\ga}}|$ (the minimal size for the support of a contour);
and $\Delta_{n+1}:=\delta_{\out}^{2\ell_{+,\ga}}[\Delta_n]$. Let $\cT_n:=\{
\Ga\in \cB^{\q}: |\Int_i(\Ga)|\le|\Delta_n| \forall i\}$.
Suppose then that for any $  \Ga\in\cT_1$,
the weights  $\hat W^{\p}_\ga(\Ga;\vw)$, defined by \eqref{cutoff-weight},
satisfy %(for any $\p$)
the hypothesis \eqref{eq:ab-true1}, that means $\hat W^{\p}_\ga(\Ga;\vw)=
w^{\p}_{\abs,\beta}(\Ga;\vw)$ (see \eqref{cutoff-weight}). Since contours in
$\cT_1$ cannot contain any smaller contour in their interior, it follows that:
%for any $\Ga\in \cT_1$:
\begin{equation*}
Z^{\q}_{\abs,\beta,\Int(\Ga)\setminus
A^{\q}}(\vw_{A^{\q}})= Z^{\q}_{\beta,\Int(\Ga)\setminus
A^{\q}}(\vw_{A^{\q}})
\hskip3cm
\forall ~~\Ga\in \cT_1
\end{equation*}
 and then, recalling \eqref{def:w-abs},
$w^{\p}_{\abs,\beta}(\Ga;\vw)=
w^{\p}_{\ga,\beta}(\Ga;\vw)$.
As a consequence the diluted partition functions on $\Delta_2$ are equal :
\begin{equation*} 
Z^{\q}_{\abs,\beta,\Delta_2}=
Z^{\q}_{\beta,\Delta_2}
\end{equation*}
because all contours in
$\cB^{\p}_{\Delta_2\setminus
\delta_{\ins}^{\ell_{+,\ga}}[\La_1]}$ are in $\cT_1$.
This equality induces, via \eqref{def:w-abs},
equality of the weights $w^{\p}_{\abs,\beta}(\Ga;\vw)=
w^{\p}_{\ga,\beta}(\Ga;\vw)$ for contours in $\cT_2$.
The iteration of this argument proves the
Theorem.}

\end{proof}



An immediate  corollary  of Theorem \ref{thm:ab-true} is:
\begin{corol}
\label{corol:key}
If for any $\q$, any $\q$-contour $\Ga$ satisfies:

\begin{eqnarray}
\label{eq:c-key} w^{\q}_{\abs,\beta}(\Ga;\vw)<e^{-\fK_\ga
N_{\Ga}}
\end{eqnarray}
then  the Peierls bound \eqref{eq:key} holds for any $\q$ contour.
\end{corol}

In fact in this case \eqref{eq:ab-true1} holds, and then
$w^{\q}_{\ga,\beta}(\Ga;\vw)=w^{\q}_{\abs,\beta}(\Ga;\vw)<e^{-\fK_\ga
N_{\Ga}}$.
\red{ The proof of the phase transition is then obtained by
proving the existence of  a value of $\beta$ such that
\eqref{eq:c-key} holds. }
\vskip .5cm \noindent

A statement similar to Theorem \ref {thm:Peierls-factoriz} holds analogously
for the weights
$w^{\q}_{\abs,\beta}(\Ga;\vw)$ defined in
\eqref{def:w-abs}. Hence
the proof of \eqref{eq:c-key} will follow by
Theorem \ref{thm:Peierls-3} and
 from the analogous of Theorem \ref{thm:Peierls-2} for the
 abstract partition functions,
that also implies, via Theorem \ref{thm:ab-true}, the
validity of \eqref{eq:Peierls-2} for the dilute partition functions.

\red{
The proof of the version of Theorem \ref{thm:Peierls-2} for abstract
partition functions  requires estimates on the pressures of the
abstract models and their finite volume corrections that will given respectively in sections
\ref{sec:equalitypresure} and \ref{sect:finitevolumecorrections}. The proof of
Theorem \ref{thm:Peierls-3}
is given in  section \ref{section:largedeviation}.}

\vskip 2.5cm \noindent

\red{Working with the abstract models has many advantages, since Peierls bounds
hold by definition \eqref{cutoff-weight}. We will show that
Peierls bounds for the cut-off weights $\hat W^{\p}_\ga(\Ga;\vw)$
together with the stability properties discussed in section \ref{sec:Stability}
allow to prove uniqueness
of the Gibbs measure $\mu^{\q}_{\abs,\beta}$ of the abstract models.
From a technical point of view the we will take advantage of \eqref{cutoff-weight}
making a partial cluster expansion of the partition function that gives
an extra interaction,
an ``effettive hamiltonian " $\cH^{\p}_{\ga,\La}$, with infinite range but fast decay.
}
\red{In this way the space of the configurations can be restricted to $\cX^{\p}$} while
the ``effective Hamiltonian"
$\cH^{\p}_{\ga,\La}(\vw)$ takes into account the sum
over contours as a Gibbs factor,
     \begin{equation}
       \label{8.1a.4.1}
\sum_{\und \Ga \in \mathcal B^{\p}_\La}\hat
W^{\p}_\ga(\und\Ga;\vw)= e^{-\beta
\cH^{\p}_{\ga,\La}(\vw_\La)}
    \end{equation}
The validity of the cluster expansion leading to the Hamiltonians $\cH^{\p}_{\ga,\La}(\vw)$
follows here by the Peierls bound that the abstract weights satisfy by
definition \eqref{cutoff-weight} (for $\ga$ small enough):
  \begin{thm}
  \label{thmz4.1}
If the weights $W^{\p}(\Ga,\vw)$  satisfy the Peierls
bounds with a constant $\fK_\ga$ large enough
%(in particular
%$\fK> 2 b$, $b$ as in \eqref{z4.11b} below)
then, for any
bounded, $\cD^{\ell_{+,\ga}}$-measurable region $\La$ and any
$\vw \in \mathcal X^{\p}$,
\begin{equation}
      \label{z4.7b}
Z_{\abs,\beta,\La}^{\p}(\und\vw_{\La^{c}})=  \sum_{\und \Ga
\in \mathcal B^{\p}_\La} \sum_{{\vw'\in \cX_\La^{\p}}}
\hat W^{\p}_\ga(\und\Ga,;\vw') e^{-H_{\ga,\La}(  \vw'_\La
| \vw_{\La^c})}= \sum_{{\vw'\in \cX_\La^{\p}}}
 e^{-\tilde H_{\ga,\La}^{\p}(\vw'_\La |
\vw_{\La^c})}
    \end{equation}
 \begin{eqnarray}
 \label{def:tildeH}
\tilde H_{\ga,\La}^{\p}(\vw'_\La |
\vw_{\La^c}):=H_{\ga,\La}(\vw'_\La | \vw_{\La^c}) +
\cH^{\p}_{\ga,\La}(\vw'_\La)
\end{eqnarray}

\begin{eqnarray}
\label{def:cH}
\cH^{\p}_{\ga,\La}(\vw)=\sum_{\Delta\sqsubset
\La}U^{\p}_\Delta(\vw)
\end{eqnarray}
where the potentials $U^{\p}_\Delta( \vw_{\Delta})$ are defined by \eqref{z4.15} below
and satisfy:

      \begin{eqnarray}
          \label{z4.11a}
& &U^{\p}_\Delta=0 ~~~~\mbox{if  $\Delta$ is not connected}
\\
  \label{z4.11b}
& &\|U^{\p}_\Delta(\cdot)\|_\infty= \sup_{ \vw_\Delta}
|U^{\p}_\Delta( \vw_{\Delta})| \le e^{-(\frac{\fK_\ga}{2} -b) N_{\Delta}}
    \end{eqnarray}
where $N_\Delta$ is the number of $\mathcal D^{\ell_{+,\ga}}$-cubes in
$\Delta$; $b>0$ a dimension dependent constant, such that
$\dis{\sum_{\Delta \ni x} e^{-b N_\Delta} \leq 1}$.



The  Gibbs measures relative to $\tilde
H_{\ga,\La}^{\p}(\vw'_\La |\vw_{\La^c})$ on $\cX^{\p}$,

\begin{eqnarray}
\label{absspec}
\tilde\mu_{\abs,\La}^{\p}(\vw_\La|\und\vw_{\La^{c}}):=
\frac{e^{-\beta\tilde H_{\ga,\La}^{\p}(\vw_\La
|\vw_{\La^c})}}{Z_{\abs,\beta,\La}^{\p}(\und\vw_{\La^{c}})}
\end{eqnarray}
 are  the marginals
on $\cX^{\p}$ of the measures
$\mu_{\abs,\La}^{\p}(\vw,\und \Ga|\und\vw_{\La^{c}})$.
  \end{thm}

  \vskip.5cm

{\bf Proof.} \eqref{z4.7b} follows from  \eqref{z4.7} and
\eqref{def:tildeH},  by
setting
      \begin{equation}
         \label{z4.13}
 e^{-\cH^{\p}_\La( \vw_\La)}=
  \sum_{\und\Ga\in \mathcal B^{+}_\La} W^{\p}(\und\Ga,\vw_\La)
     \end{equation}
To prove the remaining statements we use a cluster
expansion to express the energy $\cH^{\p}_\La(\vw_\La)$ in terms
of a sum of weights of polymers, which will then identify
the many-body potentials $U^{\p}_\Delta(\vw_\Delta)$.

Polymers are functions $I:\{\Ga\}^{\p}\to \mathbb N$ such that
the collection $\{\Ga: I(\Ga)>0\}$ is finite and connected,
where  two elements $\Ga$ and $\Ga'$ in $\{\Ga\}^{\p}$ are
connected if $\overline{\ssp(\Ga)}  \sqcap
\overline{\ssp(\Ga')} \ne \emptyset$. Denote by
$\mathcal P^{\p}$ the collection of all polymers and by
$\mathcal P^{\p}_\La$ those made by contours in $\{
\Ga\}^{\p}_\La$.  It then follows from Kotecki and Preis,
\cite{KP}, that,  if
the Peierls constant
is large enough,
there are numbers $\varpi(I,\und s)$,
such that
      \begin{equation}
         \label{z4.14}
\ln \sum_{\und\Ga\in \,\mathcal B^{\p}_\La}
W^{\p}(\und\Ga,\vw) = \sum_{I\in\, \mathcal P^{\p}_\La}
 \varpi^{\p} (I,\vw)
    \end{equation}
Calling $\dis{\ssp(I)= \bigsqcup_{\Ga: I(\Ga) >0}
\ssp(\Ga)}$, we then set
      \begin{equation}
         \label{z4.15}
-U^{\p}_\Delta( \vw_\Delta) = \sum_{I\in\, \mathcal P^{\p}_\La,\,
\ssp(I)= \Delta}
  \varpi^{\p}(I,\vw_\Delta)
    \end{equation}
\eqref{z4.11b} follows from \eqref{z4.15} and  \cite{KP}.

The proof of \eqref{absspec} follows from \eqref{z4.6} in a similar way. Theorem
\ref{thmz4.1} is proved.

\qed

 \vskip.5cm \noindent

\vskip2cm
\setcounter{equation}{0}
\section{Equality of the pressures}
\label{sec:equalitypresure}



Since all the measures with $\q>0$ are equivalent (equal up to a permutation of the indices),
 \eqref{z4.7}-\eqref{cutoff-weight} define actually only two  distinct ``abstract models",
the ordered and the disordered one,
relative  respectively to values $\q>0$, $\q<0$.
In the next two sections we will use the superscripts $\pm$ instead of $\p$ in order
to distinguish only between disordered and ordered phases.
\vskip .7cm \noindent


{
Let denote by
\begin{equation*}
P^\pm_{\abs,\La,\ga,\beta}(\und\vw_{\La^c}):=\frac{1}{\beta |\La|}\ln
Z_{\abs,\beta,\La}^{\pm}(\und\vw_{\La^c})\end{equation*} the ``finite
volume pressures".
%The limit \eqref{def:P-abs} exists on the
%sequence of Van Hove sets.\blue{(see section \ref{sec:equalitypresure})}
\vskip .5cm
The proof of Theorem \ref{thm:Peierls-2} requires the
proof of the
following theorem to control the bulk contribution to the ratio
in \eqref{eq:Peierls-2}:}

\begin{thm}
   \label{thm:equalitypresure}
   {Let $\{\La_n\}$ a sequence of sets in $\Rr^d$ of side $2^n\ell_{+,\ga}$.
  There exist the two limits:
\begin{eqnarray}
\label{def:P-abs} P^\pm_{\abs,\ga,\beta} :=
\lim_{n\to\infty}\frac{1}{\beta |\La_n|}\ln
P^\pm_{\abs,\La,\ga,\beta}(\vec\rho^{\pm}_{\beta})
\end{eqnarray}
that are continuous in $\beta$, moreover there are
constants} $c_b$, $\bar \ga$ s.t. for any $\ga<\bar \ga$
there is a value of $\beta$, $\bcg$, s.t.
       \begin{equation}
       \label{eq:abs1}
P^+_{\abs,\ga,\bcg}=P^-_{\abs,\ga,\bcg}\hskip1cm
 |\bcmf-\bcg|<c_b\ga^{1/2}
    \end{equation}


\end{thm}


\vskip .5cm \noindent

\blue{The proof of the Theorem \ref{thm:equalitypresure}
relies on the  well known mean field result on the
existence of a first order phase transition for any $Q>2$,
(see \cite{Wu}), that we recall for completeness in appendix
\ref{app:meanfield}. In particular we prove that }
there exists an inverse temperature $\bcmf$ such that the
mean field free energy function satisfies:

\begin{eqnarray}\
\label{eq:equalitymeanfiel}
\phi_{\bcmf}(\vec\rho^{+}_{\bcmf})=\phi_{\bcmf}(\vec\rho^{-}_{\bcmf})=
\inf_{\red{\vec\rho\in S_Q}}\phi_{\bcmf}(\vec\rho)
\end{eqnarray}
and
 \begin{eqnarray}
 \label{eq:lemmaa.2}
\frac{d}{d\beta}\left[\phi_\beta^{\mf}(\vec\rho^{~+}_\beta)-
\phi_\beta^{\mf}(\vec\rho^{~-}_\beta)\right]
\bigg|_{\beta=\bcmf}\neq 0
 \end{eqnarray}
\vskip 1.5cm \noindent

\begin{proof}[Proof of Theorem \ref{thm:equalitypresure}]

The proof of existence of the two pressures
$P^\pm_{\abs,\ga,\bcg}$ and their continuity in $\beta$ is
given in Appendix \ref{app:exist-pressure}, while the proof
of
\eqref{eq:abs1} is an immediate consequence the following
lemma :
\begin{lemma}
\label{lemma:a}
There are constants $\kappa$ and  $\bar \ga$ such that for any $\ga<\bar \ga$ and
\red{for any $\beta: |\beta-\bcmf|\le \ga^{2a}$}:

\vskip .5cm \noindent

\begin{eqnarray}
 \label{eq:lemmaa.1}
|P^{\pm}_{\abs,\ga,\beta}+\phi_\beta^{\mf}(\vec\rho^{~\pm}_\beta)|
\leq\frac{\kappa}{\beta}\ga^{1/2}
 \end{eqnarray}
\end{lemma}
that implies:
\begin{eqnarray}
\label{eq:starstar}
|P^{+}_{\abs,\ga,\beta}-P^{-}_{\abs,\ga,\beta}-(\phi_\beta^{\mf}(\vec\rho^{~-})-\phi_\beta^{\mf}(\vec\rho^{~+}))|<
2\frac{\kappa}{\beta}\ga^{1/2}
\end{eqnarray}

\eqref{eq:starstar}, \eqref{eq:lemmaa.2},
\eqref{eq:equalitymeanfiel}, and the continuity in $\beta$
of the pressures prove \eqref{eq:abs1} and complete the
proof of the Theorem  \ref{thm:equalitypresure}
\end{proof}
\vskip .5cm \noindent

\begin{proof}[proof of Lemma \ref{lemma:a}]


The proof of \eqref{eq:lemmaa.1}  could be  obtained as a
byproduct of a more detailed analysis contained in the next
section but since a direct proof  is quite shorter we sketch it here.
We first prove an upper bound for $P^{\pm}_{\abs,\ga,\beta}$.


\begin{eqnarray*}
P^{\pm}_{\abs,\ga,\beta}&=&\lim_{n\to\infty}\frac{1}{\beta |\La_n|}\ln
Z_{\abs,\beta,\La}^{\pm}(\vec\rho^{\pm}_{\beta})
\end{eqnarray*}
and denoting by $\hat Z_{\abs,\beta,\La}^{\pm}(\vec\rho^{\pm}_{\beta})$ the abstract partition function
with interactions $H_{\ga,\La}(\vw_\La|\vw_{\La^c})$, we get:
\begin{eqnarray*}
P^{\pm}_{\abs,\ga,\beta}& \le&\lim_{n\to\infty}\frac{1}{\beta |\La_n|}\ln
\hat Z_{\abs,\beta,\La}^{\pm}(\vec\rho^{\pm}_{\beta})+\sup_{\vw\in \cX^{\pm}}
\sum_{\Delta:\Delta\ni 0} |U^\pm_\Delta(\vw)|
\end{eqnarray*}
By \eqref{z4.11b} the last term is bounded as
$e^{-\frac{\fK_\ga}{2}+2b}$   and by
 Theorem \ref{thm:app} we have:

\begin{eqnarray*}
P^{\pm}_{\abs,\ga,\beta}\le -\lim_{n\to \infty}\inf_{\vec\rho\in \cX_{\La_n}}
\frac{F_{\beta,\ga,\La_n}(\vec\rho|\vec\rho^{\pm}_\beta)}{\beta|\La_{n}|}+c_d \ga^{1/2}
\end{eqnarray*}
we postpone at the end of this section the proof of the
following bound that follows by the concavity of the
entropy:
\begin{eqnarray}
\label{eq:star1}
P^{\pm}_{\abs,\ga,\beta}\le c_d\ga^{1/2} -\lim_{n\to \infty}\inf_{\vec\rho\in \cX_{\La_n}}
\frac{1}{|\La_{n}|}\int_{\La_{n}}\phi_{\beta}
\bigg(J_{\ga}*(\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c})\bigg)dr
\end{eqnarray}


\noindent where $\Ii_A:= \Ii_{\{x\in A\}}$.
Let
    \begin{eqnarray*}
J_\ga^{(\ell_{-,\ga})}(x,y) := \frac{1}{(\ell_{-,\ga})^d}\int_{z\in C^{\ell_{-,\ga}}_y} J_\ga(x,z)
     \end{eqnarray*}
     where $C^{\ell_{-,\ga}}_y$ is the cube of the partition  $\cD^{\ell_{-,\ga}}$ containing
     the point $y$. Then for any $\vec s\in \cX^{\pm}$:

    \begin{eqnarray*}
& &\|J_\ga * \vec s(r)-\vec\rho^{\pm}\|_\star\le \|\int
J_\ga^{(\ell_{-,\ga})}(r,r')(\vec s(r')-\vec \rho^{\pm})\|_\star
\\
& &\hskip5cm+\|\int
[J_\ga^{(\ell_{-,\ga})}(r,r')-J_\ga(r,r)](\vec s(r')-\vec \rho^{\pm})\|_\star
     \end{eqnarray*}

By the assumptions on $J_\ga$ (see  \eqref{def:J}-\eqref{def:Jb})
 the second term is bounded as :
\begin{eqnarray*}
\|\int
[J_\ga^{(\ell_{-,\ga})}(r,r')-J_\ga(r,r')]
(\vec s(r')-\vec \rho^{\pm})\|_\star\le \kappa_3 \ga\ell_{-,\ga}=\kappa_3\ga^{\al}
\end{eqnarray*}

while the first term, since $\vec s\in \cX\pm$ and $J^{(\ell_{-,\ga})}(r, r')$ is
constant w.r.t. the second variable in  each cube of $ \cD^{\ell_{-,\ga}}$:
\begin{eqnarray*}
 \|\int
J_\ga^{(\ell_{-,\ga})}(r,r')(\vec s(r')-\vec \rho^{\pm})\| _{\star}\le
\ga^a %\sum_{j}  J_\ga(i,j)=\ga^{a}\fJ_\ga
\end{eqnarray*}
going back to \eqref{eq:star1} we have:

\begin{eqnarray}
\label{eq:star2}
P^{\pm}_{\abs,\ga,\beta}\le c\ga^{1/2} -
\inf_{\|\vec s-\vec\rho^\pm\|_\star <\ga^{a}+\kappa_3\ga^{\al} }
\phi_{\beta}(\vec s)
\end{eqnarray}

 In appendix \ref{app:meanfield}, is shown that
$\inf_{\vec s}\phi_{\bcmf}(\vec s)= \phi_{\bcmf}(\vec
\rho^{\q}_{\bcmf})$, for any $\q\in\{-1,1,\dots Q\}$.
\red{By continuity  for $|\beta-\bcmf|<\ga^{2a}$, $\ga$
small enough $\phi_{\beta}(\vec s)$
%$\rho^{\q}$, $\q\in\{-1,1,\dots Q\}$
has   $Q+1$ local minima $\vec \rho^{\q}_{\beta}$ s.t.
$|\vec \rho^{\q}_{\beta}-\vec
\rho^{\q}_{\bcmf}|<c|\beta-\bcmf|<c \ga^{2a}$. The are respectively then
absolute minimizers  in the sets $\|\vec s-\vec\rho^\pm\|_\star
<\ga^{a}+\kappa_3\ga^{\al}$.} We then get:


\begin{eqnarray}
\label{eq:star3}
P^{\pm}_{\abs,\ga,\beta}\le c\ga^{1/2} -
\phi_{\beta}(\vec \rho^{\pm}_{\beta})
\end{eqnarray}



We now prove a lower bound for the pressures.
By Theorem \ref{thm:app} with $\ell=\ell_0$, we have for any $\vec\rho^{(n)}\in \cX^{\pm}_{\La_n}$:

\begin{eqnarray*}
P^{\pm}_{\abs,\ga,\beta}\ge -\lim_{n\to \infty}
\frac{F_{\beta,\ga,\La_n}(\vec\rho^{(n)}|\vec\rho^{\pm}_\beta)}{\beta|\La_{n}|}-c\ga^{1/2}
\end{eqnarray*}

and for $\ga$ small enough the
same argument as before shows that $\rho^{\pm}_{\beta}\Ii_{\La_n}\in\cX^{\pm}_{\La_n} $. We then get:
\begin{eqnarray}
\label{eq:star4}
P^{\pm}_{\abs,\ga,\beta}\ge -c\ga^{1/2} -
\phi_{\beta}(\vec \rho^{\pm}_{\beta})
\end{eqnarray}

\end{proof}

\vskip .5cm \noindent
Proof of \eqref{eq:star1}:

\begin{eqnarray*}
F_{\beta,\ga,\La_n}(\vec\rho|\vec\rho^{\pm}_\beta)&=&
F_{\beta,\ga}(\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c})-F_{\beta,\ga}(\vec\rho^{\pm}_\beta \Ii_{\La_n^c})
\\& =&\int_{\La_{n}\sqcup \delta_{\out}^{\ga^{-1}}[\La_{n}]}\phi_{\beta}
\big({\st{ J_\ga*(\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c})}}\big)dr\\
&& +\frac{1}{\beta}\int_{\La_{n}\sqcup \delta_{\out}^{\ga^{-1}}[\La_{n}]}
\bigg[I\big({\st{J_\ga*(\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c})}}\big)-
J_\ga*I({\st{\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c}}})\bigg]dr
\\&&
-\int_{\delta_{\out}^{\ga^{-1}}[\La_{n}]}\phi_{\beta}
\big({\st J_\ga*(\vec\rho^{\pm}_\beta \Ii_{\La_n^c})}\big)dr-\frac{1}{\beta}
\int_{\delta_{\out}^{\ga^{-1}}[\La_{n}]}
\bigg[I({\st J_\ga*\vec\rho^{\pm}_\beta \Ii_{\La_n^c}})-
J_\ga*I({\st\vec\rho^{\pm}_\beta \Ii_{\La_n^c}}) \bigg]dr
\end{eqnarray*}

where we have used the fact that $\phi_{\beta}(0)=I(0)=0$. By concavity of
 $I(\cdot)$ the second term is non negative and since $|\phi_{\beta}(\vec s)|$ is bounded in
 $\vec s\in S_Q$ we have:

 \begin{eqnarray*}
F_{\beta,\ga,\La_n}(\vec\rho|\vec\rho^{\pm}_\beta)\ge
\int_{\La_{n}}\phi_{\beta}
\big({\st{ J_\ga*(\vec\rho\Ii_{\La_n}+\vec\rho^{\pm}_\beta \Ii_{\La_n^c})}}\big)dr-
c|\delta_{\out}^{\ga^{-1}}[\La_n]|
 \end{eqnarray*}
\vskip 2cm

\section{finite volume corrections to the pressure}
\label{sect:finitevolumecorrections}
\setcounter{equation}{0}


The proof of the Theorem \ref{thm:Peierls-2}, is obtained  by proving the same bound
as in \eqref{eq:Peierls-2} for the abstract partition functions:


\begin{eqnarray}
\label{eq:Peierls-2-abs}
       |\log\bigl(
       \frac{Z^{+}_{\abs,\beta,\La}(\vec\rho^{(+1)}) e^{\cI_{A^{\q}}|\vec\rho^{(+1)}|^{\;2}}}{
        Z^{-}_{\abs,\beta,\La}(\vec\rho^{(-1)})e^{\cI_{A^{\q}}|\vec\rho^{(-1)}|^{\;2}}}
        \bigr)|\le
       \kappa_1\ga^{\red{1/8}}|\Ga|
\end{eqnarray}
for $|\beta-\bcmf|<c_b\ga^{1/2}$ and the conclusion,
at the inverse temperature $\bcg$, comes from
\eqref{eq:ab-true2} once proved \eqref{eq:ab-true1}.


\vskip .5cm \noindent In the regime when Theorem
\ref{thm:equalitypresure} holds, the proof of
\eqref{eq:Peierls-2-abs} requires a control of the finite
volume corrections to the pressure.

We define $R^{\pm}_{\abs,\La}$  the finite volume correction to
the thermodynamical pressure
$P^{\pm}_{\abs,\ga}$ %defined in \eqref{def:P-abs}:
as:
     \begin{equation*}
R^{\pm}_{\abs,\La} :=\log \blue{Z^{\pm}_{\abs,\beta,\La}}(\und\vw)
-\beta |\La|P^{\pm}_{\abs,\ga} \quad \quad \quad  \lim_{\La  \nearrow \mathbb Z^d}
  \frac {R^{\pm}_{\abs,\La}}{|\La|}=0
    \end{equation*}
    %%     \begin{equation*}
% \log Z_{\beta,\La} =\beta |\La|P_\beta +R_{\beta,\La},\quad  \lim_{\La  \nearrow \mathbb Z^d}
%  \frac {R_{\beta,\La}}{|\La|}=0
%    \end{equation*}

In this section we will study  the partition functions
$Z^{\pm}_{\abs,\beta, \La}(\und\vw)$, with $\und\vw\in \cX^{\pm}$
an ordered-disordered  boundary condition, %(the ordered boundary condition
%is in  one of the $Q$ possible sets $\cX^{q}$, $q\in \{1,\dots,q\}$),
$\La$ a $\mathcal
D^{(\ell_{+,\ga})}$ bounded region, and prove the following
theorem.\vskip1cm

\begin{thm}
   \label{thm:surfacecorrections}

 There is a constant $c>0$ so that
      \begin{eqnarray}
 &&
\Big|\ln Z^{\pm}_{\abs,\beta, \La}(\und\vw) - \{\beta
|\La|P^{\pm}_{\abs,\ga} + \frac \beta 2
\sumtwo{i\in\La}{j\in \La^c} J_\ga(i,j)\, (\vec\rho^{\pm}\cdot
\vec\rho^{\pm})\} \Big| \nn\\&&\hskip1cm\le \beta
\sum_{i\in\La}| \sum_{j\in \La^c} J_\ga(i,j)\,(
\und\vw(i)-\vec\rho^{\pm})| + c \ga^{1/8}|\delta_{\rm
in}^{\ell_{+,\ga}}[\La]|
        \label{8.0.0.1}
    \end{eqnarray}
\end{thm}


The proof of Theorem \ref{thm:surfacecorrections} is obtained in two steps,
the first one is a bound on the decay of correlations, that allows to control
the contribution of the bulk to the surface corrections, and the second step
is a small deviation estimate to control the contribution coming from regions near the boundary.

\vskip .5cm \noindent
Notice that if the boundary condition $\und\vw$ satisfy:
\begin{eqnarray}
\label{eq:smalldev0} {\|\sum_{j\in \La^c} J_\ga(i,j)\,
(\und\vw(j)-\vec\rho^{\pm})\|}< c \ga^{1/8}|\delta_{\rm
in}^{\ell_{+,\ga}}[\La]|
\end{eqnarray}

then the leading contribution to $R^{\pm}_{\abs,\La}$ is the finite volume correction
to the mean field pressure (with $\rho^{\pm}$ b.c.) %in the  approximation:
\begin{eqnarray}
\label{def:Rmf}
R_{\ga,\La}^{\mf,{\pm}}:=\frac{\beta}{2}\sumtwo{i\in\La}{j\in \La^c} J_\ga(i,j)\,
(\vec\rho^{\pm}\cdot
\vec\rho^{\pm})
\end{eqnarray}
 \vskip.5cm


Since the proof of the bounds in \ref{eq:Peierls-2-abs}
only requires to consider ``perfect boundary conditions"(
through Theorem \ref{thm:Peierls-factoriz}) we restrict the analysis to this case.
In the sequel,  we will consider boundary conditions
$\und\vw = \vec\rho^{\pm}$ and denote by:
\begin{eqnarray*}
P^{\pm}_{\abs,\ga,\La}:= \ln  Z^{\pm}_{\abs,\beta,\La}(\vec\rho^{\pm})
\end{eqnarray*}

\vskip .5cm \noindent

Following Dobrushin \cite{dob} we write the pressures of the abstract
models  in terms of correlations functions introducing the
interpolating Hamiltonians $h^{\pm}_u(\vw)$, $u\in [0,1]$
 defined as follows: let $u\in [0,1]$, and:

\begin{eqnarray*}
h^{\pm}_u(\vw):=u \tilde H^{\pm}(\vw|\vec\rho^{\pm})+(1-u)\fH_{\ga}^{\pm}(\vw) \hskip1.6cm
\end{eqnarray*}
where $\fH^{\pm}$ are the one body ``mean field"
Hamiltonians,
\begin{eqnarray}
\label{def:H0}
\fH_{\ga,\La}^{\pm}(\vw)
:=-
\sum_{i\in \La}%J_\ga(x,y)(\vw_x(\si_x)\cdot \vec\rho^{q}(y))
(\vvw_i\cdot \vec\cL_{\ga}(\vec\rho^{\pm};i))+ H_{\ga,\La}(\vec\rho^{\pm}|\vec\rho^{\pm})
\end{eqnarray}
\red{with
$\vL_{\ga}(\vw;i)$ the ``convolution":}

\begin{eqnarray}
\label{eq:vec-L}
\vL_{\ga}(\vw;i):=\sum_{j\ne i}J_{\ga}(i,j)\vw(j)
\end{eqnarray}
and
\begin{equation}
\label{def:deltap}
\vvw:=\vw-\vec\rho^{\pm}
\end{equation}

\vskip .5cm \noindent

\blue{The choice of the one body hamiltonians
$\fH_{\ga,\La}^{\pm}(\vw)$ is done in such a way
 that the expectations
of the densities  with respect to the corresponding  Gibbs measures
converges to the mean field values:
\begin{eqnarray}\label{MFE}
 \lim_{\ga\to 0}\langle\w_q\rangle_{\fH_{\ga,\La}^{\pm}}=\lim_{\ga\to 0}\frac{e^{\beta \cL_q(\rho^{\pm})}}
 {\sum_ie^{\beta \cL_i(\rho^{\pm})}}
 =\rho_q^{\pm}
~~~~~~~~~~~  q\in\{1,\dots,Q\}
\end{eqnarray}
The additive constant appearing in \eqref{def:H0} is added to get the mean
field pressure in the limit when $\ga$ goes to $0$:
}

\begin{eqnarray}
\label{P-abs-0}
P^{\pm}_{\abs,\ga,0}&:=&\lim_{\La\nearrow \infty}\frac{1}{\beta|\La|}\ln
 Z^{\pm}_{\abs,\beta,\La;0}
 \\
 &\equiv& \frac{1}{\beta|C_0|}
\ln \sum_{\vw\in\cX^\pm_{C_0}} \prod_{i\in C_0} e^{\beta
\big(\vvw_i\cdot
\vec\cL(\rho^{\pm};i)\big)}+\frac{1}{2}\fJ_\ga
(\vec\rho^{\pm}\cdot \vec\rho^{\pm}) \nn
\end{eqnarray}
where $C_0$ the cube of the partition $\cD^{\ell_{+,\ga}}$ that contains $0$ and
\begin{eqnarray*}
\fJ_\ga:=\sum_j J_{\ga}(i,j)
\end{eqnarray*}

The pressure relative to the
 one body interaction $\fH_{\ga,\La}^{\pm}(\vw)$
 represents
 the first order approximation of $P^{\pm}_{\abs,\ga}$.

The proof of \eqref{P-abs-0} and the proof of the convergence of the pressures
$P^{\pm}_{\abs,\ga,0}$ to the mean field free energy is given
in Appendix \ref{app:meanfieldpressurelimit}. Estimates on the finite volume corrections
to the pressures $P^{\pm}_{\abs,\ga,0}$
are  also  given in Appendix \ref{app:meanfieldpressurelimit}.

%$P^{\abs,0}_{\ga,\beta,{\pm}} $  differ from the mean field ones $P^{\mf}_{\beta,{\pm}}$,
%because of the restriction of the sum on $\si$ on the restricted ensemble.
%(Notice that the condition to be in the restricted ensemble depends on $\ga$
%(through the size of the cube of the partition)).

\vskip 1.5cm \noindent Let denote by
$\tilde \mu^{\pm}_{\abs,\La;u}$ the [finite volume] Gibbs
measure with hamiltonian $h^{\pm}_u(\vw)$
on  $\cX^{\pm}_\La$.
Following Dobrushin \cite{dob},  we write the ``finite
volume pressure" in terms of correlation functions:
     \begin{eqnarray}
%&&
P^{\pm}_{\abs,\ga,\La }:=&& \frac{1}{\beta|\La|}\ln \tittti{Z}{\tilde Z}_{\abs,\beta,\La;1}^{\pm}( \vec\rho^{\pm}) \nn\\
=&&
\frac{1}{\beta|\La|}\ln  Z_{\abs,\beta,\La;0}^{\pm}
- \frac{1}{|\La|}\int_0^1 du\; \left\langle \tilde H^{\pm}_\La( \vw'_\La|\vec\rho^{\pm})-
 \fH^{\pm}_{\La}( \vw'_\La) \right\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}
          \label{interpol}
    \end{eqnarray}
%    for any $q=d,1,2,\dots Q$.
    %, since by symmetry the
%    pressures of the models $i=1,2,\dots Q$ are all the
%    same, we denote by the index $(o)$ (ordered) the pressure
%    relative to the models $i=1,2,\dots Q$ and by the
%    index $(d)$ (disordered) the case $i=d$


\vskip .5cm \noindent
\begin{prop}
\label{prop:1}
The exist \red{$g_{\ga,i,\La}, ~g_{\ga,i}: \Zz^{d}\to\Rr$}:
%\begin{eqnarray*}
%\lim_{\La\nearrow \Rr^{d}}\frac{1}{|\La|}\left[\tilde H^{q}_\La( \si'_\La|\vec\rho^{\pm})-
% \fH^{q}_{\La}( \si'_\La)\right]=g_{\ga,0}
%\end{eqnarray*}
\begin{eqnarray}
\label{eq:prop1-a}
& &\left[\tilde H^{\pm}_\La( \vw'_\La|\vec\rho^{\pm})-
\fH^{\pm}_{\La}( \vw'_\La)\right]= \sum_{i\in\La}g_{\ga,i,\La}
\\
\label{eq:prop1-b}
& &\lim_{\La\nearrow \Rr^{d}}g_{\ga,i,\La}=g_{\ga,i}
\end{eqnarray}
\end{prop}

\vskip .5cm \noindent
\begin{proof}[Proof] %of proposition \ref{prop:1}]

%{\notes{\red{
%\begin{tabular}{|l|} \hline \\
%ho cambiato \\il segno\\controlli ?     \\
%     \\ \hline
%\end{tabular}}
%.}}
\begin{eqnarray*}
\tilde H^{\pm}_\La( \vw_\La|\vec\rho^{\pm})-
 \fH^{\pm}_{\La}( \vw_\La) \tittti{=+}{=-}\frac{1}{2}
 \sum_{i\in\La}\vvw_i\sumtwo{j\in\La}{j\ne i}J_\ga(i,j)\vvw_j+ \cH^{\pm}_{\ga,\La}(\vw_\La)
 %\\& &\hskip3cm+(\vec\rho^{\pm}\cdot \vec\rho^{\pm})\sum_{x\in\La}\sum_{y\in \La^c}J_\ga(x,y)
% \\
%& =&-\frac{1}{2}
% \sum_{x\in\La}\vvw_x\sum_{y:y\ne x}J_\ga(x,y)\vvw_y+ \cH^{\pm}_{\ga,\La}(\si_\La)
% +(\vec\rho^{\pm}\cdot \vec\rho^{\pm})\sum_{x\in\La}\sum_{y\in \La^c}J_\ga(x,y)
\end{eqnarray*}
%because $\vvw(y)\equiv \vec 0$ for $y\in \La^c$.
Recalling \eqref{def:cH}:
\begin{eqnarray*}
\cH^{\pm}_{\La}(\vw)=\sum_{\Delta\sqsubset \La}U^{\pm}_\Delta(\vw)=\sum_{i\in \La}
\sumtwo{\Delta\sqsubset\La}{\Delta\ni i} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)
\end{eqnarray*}
Defining:
\begin{eqnarray}
\label{def:ggaLa}
g_{\ga,i,\La}(\vw)\tittti{:=+}{:=-}\frac{1}{2}\vvw_i\sumtwo{j\in\La:}{j\ne i}J_\ga(i,j)\vvw_j
   % +(\vec\rho^{\pm}\cdot \vec\rho^{\pm})\sum_{j\in \La^c}J_\ga(i,j)
    +\sumtwo{\Delta\ni i}{\Delta\sqsubset \La} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)
%    +(\vec\rho^{\pm}\cdot \vec\rho^{\pm})\sum_{i\in\La}\sum_{j\in \La^c}J_\ga(i,j)
\end{eqnarray}
we can write:
\begin{eqnarray*}
\tilde H^{\pm}_\La( \vw_\La|\vec\rho^{\pm})-
 \fH^{\pm}_{\La}( \vw_\La) =\sum_{i\in\La}
g_{\ga,i,\La}(\vw)
\end{eqnarray*}

\vskip .5cm \noindent
Recalling \eqref{z4.11b}, the  limit \eqref{eq:prop1-b} exists:
\begin{eqnarray}
\label{def: gx}
\lim_{\La\nearrow \Rr^{d}}g_{\ga,i,\La}(\vw)=
-\frac{1}{2}\vvw_i\sum_{j:j\ne i}J_\ga(i,j)\vvw_j
+\sum_{\Delta\ni i} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)=:g_{\ga,i}(\vw)
\end{eqnarray}
\end{proof}
Suppose now that the Gibbs measure $\tilde \mu^{\pm}_{\abs,\La;u}$
has a unique limit when $\La\nearrow \Rr^d$ that we
denoted by $\tilde \mu^{\pm}_{\abs;u}$,
%$\langle \cdot\rangle_{\tilde \mu^{\pm}_{\abs;u}}$
%the expectation w.r.t. this measure
\blue{then taking the limit of \eqref{interpol},   we would have :}
\begin{eqnarray}
\label{eq:press-abst}
P^{\pm}_{\abs,\ga} =
P^{\pm}_{\abs,\ga,0}
- \int_0^1 du\;
%\int_{\mathcal{X}^{q}}
 \left\langle g_{\ga,0} \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}
    \end{eqnarray}


and we can rewrite \eqref{interpol} as:
\begin{eqnarray*}
&&P^{\pm}_{\abs,\ga,\La }=P^{\pm}_{\abs,\ga}+
\left[\frac{1}{\beta |\La|}\ln  Z_{\abs,\beta,\La;0}^{\pm}-P^{\pm}_{\abs,\ga,0} \right]
\\
&&\hskip3cm
- \int_0^1 du\;
%%\int_{\mathcal{X}^{q}}
 \bigg[\frac{1}{|\La|}\sum_{i\in\La}\left\langle g_{\ga, i,\La}
 \right\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}
-
 \left\langle g_{\ga,0} \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}\bigg]
 \end{eqnarray*}
\vskip .5cm \noindent

The finite volume corrections are then given by:

\begin{eqnarray}
\label{eq:surf-corr}
R_{\abs,\La}^{\pm}=R_{\abs,\La,0}^{\pm}-\beta \int_0^1 du\;\sum_{i\in\La}
\left[\left\langle g_{\ga, i,\La}
\right\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}-
 \left\langle g_{\ga,i} \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}\right]
\end{eqnarray}
where
\begin{equation*}
R_{\abs,\La,0}^{\pm}:=\ln Z_{\abs,\beta,\La;0}^{\pm}- \beta |\La|P^{\pm}_{\abs,\ga,0}
\end{equation*}

In appendix \ref{app:meanfieldpressurelimit} it is proven that:
%{\notes{\red{
%\begin{tabular}{|l|} \hline \\
%check: \\it is proved ?     \\
%     \\ \hline
%\end{tabular}}
%.}}

\begin{eqnarray*}
R_{\abs,\La,0}^{\pm}=R_{\ga,\La}^{\mf,{\pm}}%\blue{=\eqref{def:Rmf}}
\end{eqnarray*}
and then the proof of the Theorem \ref{thm:surfacecorrections} follows by estimating the
remaining terms in \eqref{eq:surf-corr}. In the next subsections we will prove that
there are positive constant $c, \om$ such that:

\begin{eqnarray}
\label{eq:decay}
&&\left|
\left\langle g_{\ga, i,\La} \right\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}- \left\langle g_{\ga,i} \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}
 \right|\le c~e^{-\om\ga\dist(i,\La)}
 \\
 \label{eq:smalldev}
&&\hskip-.3cm\sum_{i\in \delta_{\ins}^{\ell_{+,\ga}}[\La]}
\{|\langle g_{\ga,i}\rangle_{\tilde \mu^{\pm}_{\abs;u}}|
+|\langle g_{\ga,i,\La}\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}|\}
\le c\ga^{1/8}|\delta_{\ins}^{\ell_{+,\ga}}[\La]|
\end{eqnarray}

\vskip .5cm \noindent
\blue{The proof of \eqref{eq:decay} and the existence of the limit of
$\tilde \mu^{\pm}_{\abs,\La;u}$
%has a unique limit
when $\La\nearrow \Rr^d$ is given in the
next subsection as a consequence
of the exponentially decay of the correlations, while the estimate
\eqref{eq:smalldev} is a consequence of the small deviations estimates
proved in the last subsection.}
\vskip 1.5cm \noindent



\subsection{Decay of the correlations}
\label{subsect:decayofthecorrelations}



In this subsection we prove the following Theorem:

\begin{thm}

 \label{thmz5.1}

%Under the  assumptions............
There are $c$ and $\om$ positive so that for any bounded sets $\La$ and $\Delta$, $\La$
$\cD^{\ell_{+,\ga}}$-measurable, and any
$\und \vw \in \cX^{({\pm})}$, there is a coupling
$Q_u$ of $\tilde\mu^{({\pm})}_{\abs,\La;u}(\cdot|\und \vw)$
and $\tilde \mu^{({\pm})}_{\abs;u}(\cdot)$ such that
     \begin{equation}
          \label{3.K19.5}
Q_u(\vw_\Delta \ne \vw'_\Delta) \le c |\Delta| e^{-  \om\ga
\text{dist}(\Delta,\La^c )}
    \end{equation}


    \end{thm}

It follows that:

\begin{corol}
\label{cor:mu-limit} For any $u\in[0,1]$ there is a unique
DLR measure $\tilde \mu^{\pm}_{\abs;u}$ with
hamiltonian $h^{\pm}_u$ and for any local function $f$ with
support in $\Delta$ :
\begin{eqnarray*}
\lim_{\La\nearrow \Zz^{d}} \langle f \rangle_{\tilde
\mu^{\pm}_{\abs,\La;u}}= \langle f \rangle_{\tilde
\mu^{\pm}_{\abs;u}}
\end{eqnarray*}
and, for any  $\Delta\sqsubset \La$,  there are positive
constants $c, \om$:
\begin{eqnarray*}
\left| \left\langle f \right\rangle_{\tilde
\mu^{\pm}_{\abs,\La;u}}-
 \left\langle f \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}
 \right|\le c~|\Delta|\sup_{x\in\Delta}\{f(x)\}~~e^{-\om\ga\dist(\Delta,\La^c)}
\end{eqnarray*}
\end{corol}

\vskip 1.5cm \noindent

Corollary \ref{cor:mu-limit} applied
to our case, proves inequality \eqref{eq:decay}:
\begin{eqnarray*}
\left| \left\langle g_{\ga, x,\La} \right\rangle_{\tilde
\mu^{\pm}_{\abs,\La;u}}-
 \left\langle g_{\ga,x} \right\rangle_{\tilde \mu^{\pm}_{\abs;u}}
 \right|\le \kappa_4~e^{-\om\ga\dist(x,\La^c)}
\end{eqnarray*}
\vskip .5cm \noindent

\begin{proof}
The proof of Theorem
\ref{thmz5.1} requires an extension of the Dobrushin high
temperature uniqueness Theorem. In fact the
classical Dobrushin Theorem does not apply to this case because of the restriction
to the set $\cX^{\pm}$, that corresponds to %introduce
an hard-core interaction.

We recall  that the classical
Dobrushin uniqueness criterium  requires that
 there is a function $r_{ij}: \Zz^{d}\times \Zz^{d}\to\Rr^{+}$, $r_{ii}=0$
 such that for all $i\in\Zz^{d}$: and all $\vw^{(1)},\vw^{(2)}\in \cX^{\pm}$:

\begin{eqnarray}
& &R_{i}(\mu_{i}(\cdot|\vw^{(1)}_{i^{c}}), \mu_{i}(\cdot|\vw^{(2)}_{i^c}))\le
\sum_{j\ne i}r_{ij}
~\dist_j(\vw_j^{(1)},\vw_j^{(2)})
\label{eq:dob-a}
\\
&&\sup_{j}\sum_{j\ne i}r_{ij}<\delta\hskip3cm 0<\delta <1
\label{eq:dob-b}
\end{eqnarray}
%where $d$
where $R_{i}(\mu_{i}(\cdot|\vw^{(1)}_{i^{c}}), \mu_{i}(\cdot|\vw^{(2)}_{i^c}))$
is the Vaserstein distance between the two single spin measures
$\mu_{i}(\cdot|\vw^{(1)}_{i^{c}}), \mu_{i}(\cdot|\vw^{(2)}_{i^c})$.

In our case this criterium cannot be satisfied uniformly in $\vw_{i^c}^{(1)},\vw_{i^c}^{(2)}$
 because there is a subset of configurations $\und\vw_{i^c}$ that are ``critical"
in the sense that the conditional measures $\mu^{\p}_{\abs,i;u}(\cdot|\und\vw_{i^c})$
have support only a strict subset of $\Om_i$ (the subset of values of $\vw_x$
for which $\und\vw_y\Ii_{y\ne x}+ \vw_x\Ii_{y=x}$ is in $\cX^{\pm}$), while the
Vaserstein distance between measures with different supports is not small.



In reference \cite{bkmp2}, the Dobrushin
uniqueness  criterium is extended to the case when the ``classical Dobrushin condition"
 is not satisfied uniformly  in the boundary conditions,
but only for \blue{``most of the configurations".} Uniqueness of the measure can be
proven here along the same line as in the original Dobrushin proof,
using a criterium analogous to \eqref{eq:dob-a}-\eqref{eq:dob-b}, where
the single spin measures are replaced by measures on $\Om_{C_n}$, with $C_n$ a cube
of a partition $\cD^{\ell}$ of $Z^d$.
With such a choice, the uniqueness Dobrushin criterium would read
as:

\begin{eqnarray}
& &
R_{C_n}(\mu_C(\cdot|\vw_{C_n^c}^{(1)}), \mu_{C_n}(\cdot|\vw_{C_n^c}^{(2)}))\le
\sum_{m\in (\ell\Zz)^d}r_{nm}
\dist(\vw_{C_m}^{(1)},\vw_{C_m}^{(2)})
\label{eq:dob-m-a}
\\
&&\sup_{n}\sum_{m\ne n}r_{nm}<\delta \hskip3cm 0<\delta <1
\label{eq:dob-m-b}
\end{eqnarray}
with $r_{nm}: (\ell\Zz)^{d}\times (\ell\Zz)^{d}\to\Rr^{+}$, $r_{nn}\equiv 0$. %and $\dist$.

\vskip .5cm \noindent

Here we use an alternative  line of proof \cite{bkmp2} inspired by \cite{gross}
and based on the Kantorovich-Rubinstein
 duality in order to avoid measurability problems.
\red{This formulation has the advantage of being less restrictive and in particular
the [small] probability of ``bad configurations"  can be  ``adsorbed" in $r_{nm}$,
but the nice situation of having an easily checkable criterium is lost.
(In the context of spin systems, the Vaserstein distance
between single spin measures can be explicitly calculated). }

In reference \cite{bkmp2}, a set of
five assumptions on the measures is provided which
validity imply \eqref{eq:dob-m-a}-\eqref{eq:dob-m-b} and can be proven more easily
that a direct check of the criterium.
Theorem \ref{thmz5.1} will be proved through the demonstration
that
these five assumptions hold
for the two abstract models $\tilde \mu^{({\pm})}_{\abs;u}(d\vw')$.

In our context, recalling the definition of $\cX^{\pm}$, the ``natural" choice for the partition
in \eqref{eq:dob-m-a} is $\cD^{\ell_{-,\ga}}$ and the five assumptions stated
below will imply the validity of the criterium \eqref{eq:dob-m-a}-\eqref{eq:dob-m-b}
with $\ell=\ell_{-,\ga}$.

\vskip .5cm \noindent
In the sequels we prove assumptions 1 and 2 which require a specific treatment in the present case.
Assumptions 3-5
can be verified along the same lines as in the
Ising case (section 9 of \cite{bkmp2}) and will be stated here without proofs.


\vskip .5cm \noindent
\subsubsection{First assumption:}
\label{subsec:first-assumption}
\vskip .5cm \noindent

The first assumption requires to prove that for any
$i\in \Zz^d$ there is a measurable set
$G^{\pm}_i\sqsubset \cX^{\pm}$ %determined only by the variables
%measurable w.r.t.
depending only on $\{\und\vw_j,\; j\in C^{\ell_{-,\ga}}_i\setminus i\}$, such that there exists
$b(i,j)$ with the following properties:
\begin{eqnarray}
\label{firstassumption}
&&b(i,j)\ge 0 \;\; ;\;\; b(i,i)=0 \nonumber\\
&&\sup_{i\in\Zz^d}\sum_{j\in C^{\ell_{-,\ga}}_i} b(i,j)<\delta< 1 \\
&& \cR(\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw),\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw'))\le
\sum_j b(i,j)\; \dist(\und\vw_j,\und\vw_j') \hskip1.6cm \text{for any $\und\vw,\und\vw'\in G_i$}
\nonumber
\end{eqnarray}
where $\dist(\vw_j,\vw_j')$ is a distance defined on the configuration space and\linebreak
$\cR(\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw),\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw'))$ is the
associated Vaserstein distance.
Here, we consider the  following distance between configurations:


\begin{eqnarray}
\label{def:dist2}
\dist(\vw_i,\vw'_i):=
%\frac{1}{2}\sum_{ik}\left|\Ii_{\vw_i=a_i}-\Ii_{\vw'_i=a_k}\right|\\\equiv
\frac{1}{2}\sum_i|\w_i(i)-\w'_i(i)|
\end{eqnarray}
 and define  $G_i$ as :
\begin{eqnarray}
\label{Gx}
G^{\pm}_i:=\{\vw\in \cX^{({\pm})}:\vw^{(i,q)}\in \cX^{({\pm})} \forall q\}
\end{eqnarray}
 where we have denoted
\begin{eqnarray}
\hskip1cm \vw^{(i,q)}_j=
  \begin{cases}
    \vw_j & \text{$j\ne i$}, \\
    \vec u_q & \text{$j=i$}.
  \end{cases}
\end{eqnarray}


\vskip .5cm \noindent
{\bf Remark:}
$G^{\pm}_i$ is the set of configurations $\vw$ which belong to
$\cX^{({\pm})}$ independently of the value of
$\vw_i$ and is measurable on $\vw_{i^c}$.
When $\und\vw,\und\vw'$ are not in $G^{\pm}_i$, the probability
measures for  $\vw_i$,
have support on a strict subset of $\Om$
and the Vaserstein distance can be larger than the bound in \eqref{firstassumption}.
\vskip .5cm \noindent

\begin{thm}

There are $\bar \ga,  \varsigma$, so that for
$\ga<\bar\ga$, and for any $\beta:~|\beta-\bcmf|<c_b \ga^{\frac{1}{2}}$, there exists $b(i,j)$ satisfying the relations
\eqref{firstassumption} with  $G_i$ as in \eqref{Gx}.
$b(i,j)$ has the expression:
\begin{eqnarray*}
b(i,j)=r\left[J_{\ga}({i,j})+ e^{-(\frac{\fK_\ga}{2} -2b)N_\ga(i,j)}\right]
\end{eqnarray*}
where $r<1$, $\fK_\ga $ and $b$ as in \eqref{z4.11b} and  $N_\ga(i,j)$:
\begin{eqnarray*}
N_\ga(i,j)=1+\text{integer part}\left[\frac{|i-j|}{\ell_{+,\ga}}\right]
\end{eqnarray*}
\end{thm}

\begin{proof}
The Vaserstein distance between the two measures $\mu_{i}(\vw_i|\vw_1),\mu_i(\vw_i|\vw_2)$
is defined as

\begin{eqnarray*}
\cR(\mu_{i}(\vw_i|\vw_1),\mu_i(\vw_i|\vw_2)):= \inf_{\mathcal q}E_{\mathcal q}(\dist(\vw_i,\vw'_i))
\end{eqnarray*}
where the infimum is taken over all couplings between
$\mu_{i}(\vw_i|\vw_1),\mu_i(\vw_i|\vw_2)$.
Recalling \eqref{def:dist},
the infimum is realized on the couplings which have the maximal mass on the diagonal
\begin{eqnarray*}
\mathcal q(\vw_i=\vw'_i=\vec u_q)=\min\{\mu_i(\vec u_q|\vw_1),\mu_i(\vec u_q|\vw_2)\}
\end{eqnarray*}
Let { $\mathcal q$} be such a coupling
\red{for $\mu_{\abs,i;u}(\vw_i|\vw_1),\mu_{\abs,i;u}(\vw_i|\vw_2)$.} We have:
\begin{eqnarray*}
\cR(\mu_{\abs,i;u}(\vw_i|\vw_1),\mu_{\abs,i;u}(\vw_i|\vw_2))&=&
E_{\mathcal q}\bigg(\dist(\vw_i,\vw'_i)\bigg)
=\mathcal q(\vw_i\ne\vw'_i)\\
&=&1-\sum_{q}\min\{\mu_{\abs,i;u}(\vec u_q|\vw_1),\mu_{\abs,i;u}(\vec u_q|\vw_2)\}
\\
&=&\frac{1}{2}\sum_{q}\bigl(\mu_{\abs,i;u}(\vec u_q|\vw_1)+\mu_{\abs,i;u}(\vec u_q|\vw_2)\\
& &\hskip3cm-2\min\{\mu_{\abs,i;u}(\vec u_q|\vw_1),\mu_{\abs,i;u}(\vec u_q|\vw_2)\}\bigr)\\
&=& \frac{1}{2}\sum_{q}|\mu_{\abs,i;u}(\vec u_q|\vw_1)-\mu_{\abs,i;u}(\vec u_q|\vw_2)|
\end{eqnarray*}

Since $\und\vw$ and $\und\vw'$ are in $G_i$, the constraint of being in $\cX^{\pm}$ does
not affect the values of $\vw_i$, and the conditional measures
$\mu^{\pm}_{\abs,i;\red{u}}(\cdot|\vw)$
are given by \red{the Gibbs conditional measures}. Using \eqref{absspec}, we get,

\begin{eqnarray*}
\cR(\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw),\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw'))&=&
\frac{1}{2}\sum_{q}\left|\frac{e^{\beta k^{\pm;\red{u}}_q(\und\vw)}}{\sum_{p=1}^{Q}
e^{\beta k^{\pm;\red{u}}_{p}(\und\vw)}}-
\frac{e^{\beta k^{\pm;\red{u}}_{q}(\und\vw')}}{\sum_{p=1}^{Q}e^{\beta k^{\pm;\red{u}}_{p}(\und\vw')}}\right|
\\
&=&\frac{1}{2}\sum_{q}|g_q(\vec k^{\pm;\red{u}}(\und\vw))-g_q(\vec k^{\pm;\red{u}}(\und\vw'))|
\end{eqnarray*}
Where $\vec g(\cdot)$ is defined in \eqref{mf:g} and
$\vec k^{\pm;\red{u}}_i(\und\vw)\equiv \vec k^{\pm;\red{u}}_i(\und\vw_{i^c})$
is the vector
\begin{eqnarray*}
\vec k^{\pm}_i(\und\vw)&:=&
u \vL_{\ga}(\vw;i) + (1-u) \vL_{\ga}(\vec \rho^\pm;i) - u
\sum_q\bigl(\cH^{\pm}(\und\vw^{(i,q)}) -\cH^{\pm}(\und\vw_{i^c})\bigr) \vec u_q\nn\\
&=& u \sum_{j\ne i}J_{\ga}({i,j})\vw_j +(1-u) \sum_{j\ne i}J_{\ga}({i,j})\vec\rho^\pm_j
- u \sum_q \sum_{\Delta\ni i}U^{\pm}_\Delta(\vw^{(i,q)}) \vec u_q
\end{eqnarray*}

Using \eqref{z4.11b}, we have:
\begin{eqnarray}
\label{eq:vec-k}
\lim_{\ga\to 0}\sup_{\und\vw\in G_i}
\|\vec k^{\pm}(\und\vw)-\vec \rho^{\pm}\|\le {\lim_{\ga\to 0}}\left(c\ga^a+\sum_{\Delta\ni
i}Q\|U^{\pm}_\Delta(\cdot)\|_{\infty}\right)=0
\end{eqnarray}
Hence , for $\ga$ small enough, we can use theorem \ref{thm:mflocalstability} and get

\begin{eqnarray*}
\cR\bigg(\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw),\mu^{\pm}_{\abs,i;\red{u}}(\vw_i|\und\vw')\bigg)
&\le&\frac{1}{2} (1-\frac{1}{2Q})
\sum_q|k^{\pm;\red{u}}_q(\und\vw)-k^{\pm;\red{u}}_q(\und\vw')|
\end{eqnarray*}
On the other hand, we have the following bound for all $\Delta\ni x$
    \begin{equation*}
 |U^{\pm}_\Delta(\vw^{(i,q)})-
 U^{\pm}_\Delta(\vw'^{(i,q)})| \le
2\|U^{\pm}_\Delta(\cdot)\|_{\infty} \sum_{j\in \Delta\setminus
i}\dist(\und\vw_j,\und\vw'_j)
      \end{equation*}
and thus

\begin{eqnarray}
\label{k-dist}
\sum_q|k^{\pm;\red{u}}_q(\und\vw)-k^{\pm;\red{u}}_q(\und\vw')|\le 2
\sum_{j\ne i} \left(\red{u}J_{\ga}(i,j)+\sum_{\Delta \ni i,j}
Q\|U^{\pm}_\Delta(\cdot)\|_{\infty}\right)\;\;\dist(\und\vw_j,\und\vw'_j)
\end{eqnarray}

Recalling \eqref{z4.11b} we
bound the last term in \eqref{k-dist}
as:
    \begin{equation}
       \label{6.8b}
  \sum_{\Delta\ni i,j}\|U^{\pm}_\Delta(\cdot)\|_{\infty} \le   \sum_{\Delta\ni
i,j} e^{-(\frac{\fK_\ga}{2} -b)N_{\Delta}} \le e^{-(\frac{\fK_\ga}{2} -2b) N_\ga(i,j)}
\sum_{\Delta\ni i} e^{-bN_{\Delta}}\le e^{-(\frac{\fK_\ga}{2} -2b) N_\ga(i,j)}
     \end{equation}
Since
$\fK_\ga = c \ga^{-(1-\alpha)d + 2a}$,
\begin{eqnarray*}
\lim_{\ga\to 0}  \sum_{\Delta\ni
i,j}Q\|U^{\pm}_\Delta(\cdot)\|_{\infty}=0
\end{eqnarray*}

hence, we take for all $u$,

\begin{eqnarray}
\label{eq:b-xy}
b(i,j):=(1-\frac{1}{2Q}) \left(\red{u}J_{\ga}(i,j)+e^{-(\frac{\fK_\ga}{2} -2b) N_\ga(i,j)}\right)
\end{eqnarray}
We have
\begin{eqnarray}
\label{eq:blim}
\lim_{\ga\to 0}\sup_{i\in \mathbb Z^d} \sum_{j\in
C_i^{\ell_{-,\ga}}\setminus i} b(i,j)=0
\end{eqnarray}
so that \eqref{firstassumption} holds for all $\ga$ small enough.
\end{proof}

\vskip .5cm \noindent
\subsubsection{Second Assumption:}
\vskip .5cm \noindent


The second assumption is a bound on the probability of ${G_i^{\pm}}^c$, where $G^{\pm}_i$
is defined in
\eqref{Gx}.
Here we deal with a bounded state space and the assumption may be written as:
\begin{eqnarray}
\label{assumpt2}
\sup_{ \vw^*\in \cX^{\pm}}\mu^{({\pm})}_{\red{\abs,D,u}}
({G_i^{\pm}}^{c}|\vw^{~*})\le
 e^{-c\ga^{2a}\ell^{d}_{-,\ga}}
\end{eqnarray}
$c$ a positive constant and $D=C^{\ell_{-,\ga}}_i$.
The proof of \eqref{assumpt2} requires a Peierls estimate inside the
restricted set $\cX^{\pm}$.
Let $\zeta'>0$, we define the  ``bad" set
\begin{eqnarray*}
\cAA_i :=\{\vw\in \cX^{({\pm})}: \|\vw^{(\ell_{-,\ga})}(i)- \vec\rho^{\pm}\|_{\star}>(1-\zeta')\ga^a\}
\end{eqnarray*}
Notice that for any $\zeta'>0$ and  $\ga$ small enough, we have for all $i$
\begin{eqnarray*}
{G^\pm}^c_i\sqsubset \cAA_i
\end{eqnarray*}
so that $\cAA_i$ does not depend on $\vw_i$, and we have
\begin{eqnarray*}
\sup_{\vw^{~*}\in \cX^{\pm}}\mu^{({\pm})}_{\red{\abs,D,u}}%{\ga,\beta,D}
({G_i^{\pm}}^{c}|\vw^{~*})\le
\sup_{ \vw^{~*}\in \cX^{\pm}}\mu^{({\pm})}_{\red{\abs,D,u}}%{\ga,\beta,D}
(\cAA_i|\vw^{~*})
\end{eqnarray*}

\vskip 1.5cm \noindent

\red{Let
\begin{eqnarray*}
{F_{\ga,D,u}}&:=&-\frac{u}{2}\int_{D\times D}dr~dr'~~
J_\ga(r,r')\vec\rho_{D}(r)\cdot \vec\rho_{D}(r')-(1-u)\int_{D}dr~~
\vec\rho_{D}(r)
\cdot\vec \rho^{\pm}
\\
& &-\int_{D\times D^c}dr~~
J_\ga(r,r')\vec\rho_{D}(r)
\cdot \vec\rho_{D^{c}}(r')+\frac{1}{\beta}\int_{D} dr
\vec\rho_D(r)\cdot\ln\vec\rho_D(r)
\end{eqnarray*}}
The proof of the second assumption is thus based on the following proposition whose
proof is given at the end of subsection \ref{subsection:smalldeviation}:
\begin{prop}
\label{prop:smalldeviation}
There is a constant $c>0$ so that for all  $\ga$ small enough, any $x$:
\begin{eqnarray}
\label{eq:smalldev01}
&& \hskip-1cm \sup_{\vw^{*}\in\cX^{({\pm})}}\ln \; \mu_{\red{\abs,D,u}}
(\cAA|\vw^{~*})
\le
-\hskip-.3cm\inf_{\vec \rho_{D^{c}}\in
\cX_{D^{c}}^{\pm}}
\Bigg(\inftwo{\vec\rho_{D} \in \cX_{D}^{\pm},}{
\left|\vec\rho^{(\ell_{-,\ga})}(r) - \vec\rho^{\pm}\right|
>(1-\zeta')\zeta}
\hskip-.7cm \red{F_{\ga,D,u}}(\vec\rho_{D}|\vec \rho_{D^{c}})
 \nonumber\\&&\hskip3cm
-\inf_{\vec\rho_D \in \cX_{D}^{\pm}}
\red{F_{\ga,D,u}}(\vec\rho_{D}|\vec \rho_{D^{c}})\Bigg)
 + c \ga^{1/2} \ell_{-,\ga}^d
\end{eqnarray}
\end{prop}

where, by an abuse of notation, we have denoted by the same symbol $\cX^{\pm}$,
the restricted ensemble:
         \begin{eqnarray}
 \cX^{\pm}:= \Big\{ \vec\rho\in L^\infty(\Rr^d,S_Q):\|\vec \rho^{(\ell_{-,\ga})}(r)
 - \vec\rho^{\pm}\|_{\star} \le \ga^a, ~~\forall r\in \Rr^d\Big\}
       \label{7.3}
     \end{eqnarray}
and denote by $\cX^{\pm}_\La$ the above expression \eqref{7.3}
when the constrain is imposed on
$\La$, with   $\La \sqsubset \Rr^d$, a $\cD^{(\ell_{-,\ga})}$-measurable set.




\vskip 1.5cm \noindent
\begin{proof}
Using a result similar to Theorem \ref{thm:app} but with a slightly different functional,
one gets for all $\vw^{*}\in\cX^{({\pm})}$
\begin{eqnarray*}
&&\ln \; \mu_{\red{\abs,D,u}}(\cAA|\vw^{~*})
= \ln Z_{\red{\abs,D,u}}(\cAA|\vw^{~*})- \ln Z_{\red{\abs,D,u}}(\vw^{~*})\\
& &\le
\inftwo{\vec\rho_{D} \in \cX_{D}^{\pm},}{
\left|\vec\rho^{(\ell_{-,\ga})}(r) - \vec\rho^{\pm}\right|
>(1-\zeta')\zeta}
\hskip-.7cm \red{F_{\ga,D,u}}(\vec\rho_{D}|(\vw^{~*})^{(\ell_0)})
-\inf_{\vec\rho_D \in \cX_{D}^{\pm}}
\red{F_{\ga,D,u}}(\vec\rho_{D}|(\vw^{~*})^{(\ell_0)})
 + c \ga^{-1} \ell_0 |D|\\
 & &\le
-\inf_{\vec \rho_{D^{c}}\in
\cX_{D^{c}}^{\pm}}
\Bigg(\inftwo{\vec\rho_{D} \in \cX_{D}^{\pm},}{
\left|\vec\rho^{(\ell_{-,\ga})}(r) - \vec\rho^{\pm}\right|
>(1-\zeta')\zeta}
\hskip-.7cm \red{F_{\ga,D,u}}(\vec\rho_{D}|\vec \rho_{D^{c}})
-\inf_{\vec\rho_D \in \cX_{D}^{\pm}}
\red{F_{\ga,D,u}}(\vec\rho_{D}|\vec \rho_{D^{c}})\Bigg)
 + c \ga^{1/2} \ell_{-,\ga}^d
\end{eqnarray*}

\end{proof}


\vskip 1.5cm \noindent

The proof of the validity of the second assumption \eqref{assumpt2} follows from an estimate for the r.h.s of
\eqref{eq:smalldev0}. We define:


\begin{eqnarray*}
F^0_{\red{\ga,D,u}}(\vec\rho_{D}|
\vec\rho_{D^{c}})
:=
-\int_{D}dr~~
\left(
\red{\vec\cL^{u}(r,\vec\rho_{D^{c}})}
\cdot \vec\rho_{D}\red{-
}\frac{1}{\beta}\vec\rho_D\cdot\ln\vec\rho_D\right)
\end{eqnarray*}
where \red{$\vec\cL^u(r,\vec\rho_{D^{c}})$} is the external field:
\begin{eqnarray}
\label{eq:hi}
\vec\cL^{u}(r,\vec\rho_{D^{c}}):= (1-u)\vec \rho^{\pm}+ u\int_{D^{c}}
J_\ga(r,r')\vec\rho_{D^{c}}(r')dr'
\end{eqnarray}

Since $F^0_{\ga,\beta,D}(\vec\rho_{D}|
\vec\rho_{ D^{c}   })$ differs from
 $F_{\ga,\beta,D}(\vec\rho_{D}|
\vec\rho_{ D^{c}   })$ by the self interaction
energy, %i.e.\ the first term on the r.h.s.\ of \eqref{}
which is bounded
proportionally to $|D|^2$,
there is a constant $c'>0$ such that:
\begin{eqnarray}
\label{eq:smalldev1}
|F_{\red{\ga,D,u}}(\vec\rho_{D}|
\vec\rho_{ D^{c}})-F^0_{\red{\ga,D,u}}(\vec\rho_{D}|
\vec\rho_{D^{c}})|\le c'\red{u}\ga^{d}|D|^2
\end{eqnarray}


 $F^0_{\ga,D}(\vec\rho_{D}|
\vec\rho_{D^{c}})$ is a convex functional on $L^\infty(D,S_Q)$,
and has thus a unique minimizer that we denote by
$\vec\rho^{~*}(r;\vec\rho_{D^c})\equiv \vec\rho^{~*}(r)$, whose components are  given by:
\begin{eqnarray}
\label{barmD}
\rho^{~*}_k(r)=\frac{e^{\beta
\red{\cL^u_k(r,\vec\rho_{D^{c}})}}}
{\sum_l e^{\beta \red{\cL^u_l(r,\vec\rho_{D^{c}})}}}
\end{eqnarray}
We need to evaluate the difference:
\begin{eqnarray*}
& &F^0_{\red{\ga,D,u}}(\vec\rho_{D}|\vec\rho_{D^c})-F^0_{\red{\ga,D,u}}(\vec\rho^{~*}_D|\vec\rho_{D^c})=\\
& &\hskip2cm\int_D dr'
\left(- \vec\rho_{D}(r') \cdot \red{\vec\cL^{u}}(r,\vec\rho_{D^c})+\frac{1}{\beta}
\vec\rho_{D}(r')\cdot\ln \vec\rho_{D}(r')\right)-
\\
& &\hskip3cm\left(-\vec\rho^{~*}_{D}(r')\cdot\red{\vec\cL^{u}}(r,\vec\rho_{D^c})+\frac{1}{\beta}
\vec\rho^{~*}_{D}(r')\cdot\ln\vec\rho^{~*}_{D}(r')\right)
\end{eqnarray*}
Using \eqref{barmD}, we write $\red{\vec\cL^{u}}(r,\vec\rho_{D^c})$ in terms of $\vec\rho^{~*}$ and get:
\begin{eqnarray}
\label{relativentropy}
& &F^0_{\red{\ga,D,u}}(\vec\rho_{D}|\vec\rho_{D^c})-F^0_{\red{\ga,D,u}}(\vec\rho^{~*}_D|\vec\rho_{D^c})=
 \frac{1}{\beta}\int_D dr'
\sum_{k=1}^{Q}\rho_{k}(r') \ln\frac{\rho_{k}(r')}{\rho^{~*}_{k}(r')}
\end{eqnarray}
%The function inside the integral
%\begin{equation}
%\label{def:rel-entropy}
%f(\vec z):= \sum_{q=1}^Q z_q\ln\frac{z_q}{d_q}
%\end{equation}
%is positive
%(on the space $\sum_q z_q=1)$ and convex with a (unique)
%minimum  $\vec z^{~*}=\vec d=(d_1,\dots,d_Q)$, moreover  any second derivative
%$\frac{\partial^{2} f(\vec z)}{\partial z_q^2}>1$ , (notice that
%the matrix of second derivative is diagonal and positive
%defined in any point $\vec z$). Then:
%\begin{eqnarray}
%\label{eq:rel-entropy}
%f(\vec d+ \vec z)>\frac{1}{2}\sum_q z_q^2
%\end{eqnarray}
%\begin{proof}[Proof of \eqref{eq:rel-entropy}]
%
%
%In fact, let $t,s\in \Rr$,since $\nabla f|_d=0$ and :
%\begin{eqnarray*}
%f(\vec d+\vec z)&=&f(\vec d)+\int_0^{1}dt \int_0^t ds \frac{d^2}{ds^2}f(\vec d+s \vec z) \\
%\\
%& =& \int_0^{1}dt \int_0^t ds \frac{d}{ds} \sum_i \frac{\partial }{\partial x_i}
%f(\vec x)|_{\vec x=\vec d+s\vec z}
%\;\; z_i\\
%& =& \int_0^{1}dt \int_0^t ds  \sum_i \frac{\partial^2 }{\partial x_i^2}
%f(\vec x)|_{\vec x=\vec d+s\vec z}
%\;\; z_i^2\\
%\end{eqnarray*}
%because  $\frac{\partial }{\partial x_i}f(\vec x)$ depends only on $x_i$. Since
%$\frac{\partial^2 }{\partial x_i^2}f(\vec x)\ge 1$ uniformly in $i,x$, then:
%\begin{eqnarray*}
%f(\vec d+\vec z)&\ge&\sum_i z_i^2 \int_0^{1}dt \int_0^t ds=\frac{1}{2}\sum_iz_i^{2}
%\end{eqnarray*}
%\end{proof}
%\vskip .5cm \noindent
%\eqref{eq:rel-entropy} %In these notations $z_i$ are the increments,
% applied to \eqref{relativentropy},  with $z_i = \rho_i-\rho^{*}_i$, gives
Thus by the Kullback-Leibler inequality, one gets
\begin{eqnarray*}
\label{relativentropy2}
F^0_{\red{\ga,D,u}}(\vec\rho_{D}|\vec\rho_{D^c})-F^0_{\red{\ga,D,u}}(\vec\rho^{~*}_D|\vec\rho_{D^c})%&=&
%\frac{1}{\beta}\int_D dr'
%\sum_{i=1}^Q m^{(i)}_{D}(r') \ln\frac{m^{(i)}_{D}(r')}{\bar m^{(i)}_{D}(r')}\\
& \ge &\frac{1}{2\beta}\int_D dr'|
 \vec\rho_{D}(r')- \vec\rho^{~*}_{D}(r')|^{2}
\end{eqnarray*}

\vskip .5cm \noindent
We claim that there is $\eps>0$ such that,  for any
$\vec\rho_{D^{c}}$ and for $\ga>0$ small enough,
\begin{eqnarray}
\label{eq:smalldev2}
\|\vec \rho^{~*}- \vec \rho^{\pm}\|_{\star}\le (1-\eps)\ga^a
\end{eqnarray}
Using Cauchy-Schwartz inequality, we thus get (taking $\zeta'< \eps$):
\begin{eqnarray}
\label{eq:smalldev3}
F^0_{\red{\ga,D,u}}(\vec\rho_{D}|\vec\rho_{D^c})-F^0_{\red{\ga,D,u}}(\vec\rho^{~*}_D|\vec\rho_{D^c})
\ge \frac{1}{2\beta}|D|(\eps-\zeta')^2\ga^{2a}
\end{eqnarray}

\vskip .5cm \noindent
Postponing the proof of \eqref{eq:smalldev2},
 we get the following bound by using proposition
\ref{prop:smalldeviation} together with \eqref{eq:smalldev1} and \eqref{eq:smalldev3}:

\begin{eqnarray*}
\sup_{\vw^{*}\in\cX^{({\pm})}}\ln \; \mu_{\red{\abs,D,u}}(\cAA_x|\vw^{~*})&\le&
\exp\left\{-\left(
\frac{1}{2\beta}|D|(\eps-\zeta')^2\ga^{2a}-c'\ga^{d}|D|^2 \right)+
c\ga^{1/2}\ell_{-,d}^{d}\right\}
\\
&=&\exp\left\{-\left(
\frac{1}{2\beta}(\eps-\zeta')^2\ga^{2a}-c'\ga^{d\al} +c\ga^{1/2}\right)\ell_{-,d}^{d}\right\}
\end{eqnarray*}
%Recall that $\ell_{-,\ga}=\ga^{-1+\al }$, $|D|=\ga^{\al d}$,
%$\beta\sim\beta_c<Q$ (cfr \eqref{betacQ})
Assumption \eqref{assumpt2} is then proven.
We now turn to the proof of \eqref{eq:smalldev2}:


\vskip .5cm \noindent
In order to prove \eqref{eq:smalldev2}, we show that there is $b\in (0,1)$:
\begin{eqnarray}
\label{eq:star-L}
|\rho^{~*}_k(r)-  \rho^{\pm}_k|\le b |\cL^u_k(r)- \rho^{\pm}_k|
\end{eqnarray}
while $\|\vec\cL^u(r)- \vec\rho^{\pm}\|_{\star}$ is small and close enough to $\ga^a$.

We first prove that $| \cL^u_k(r)-\rho_k^{\pm}|$ is small enough.
We define

     \begin{equation*}
     J^{(\ell_{-,\ga})}_{\ga}(r,\lng r' \rng):= \frac{1}{|C^{\ell_{-,\ga}}_{r'}|}\int J_{\ga}(r,r'')dr''
    \end{equation*}
For $\ga$ small enough, by hypothesis on $\mathcal J$ (see \eqref{def:J})

    \begin{equation}
      \label{p4.3.1.5}
\big| J_{\ga}(r,r')-J^{(\ell_{-,\ga})}_{\ga}(r,\lng r' \rng)\big| \le
 \sqrt{d} \|\nabla \cJ\|_\infty\ga^{d+1}\ell_{-,\ga} \text{\bf
1}_{\{|r-r'|\le 2\ga^{-1}\}}
     \end{equation}
Then, for any $\vec\rho\in L^\infty(\mathbb
R_+^d,S_Q)$, there is a constant $c_d=2^d  \sqrt{d} \|\nabla \cJ\|_\infty$
%\marginpar{\blue{\maltese}.}
    \begin{eqnarray*}
\big\| J_\ga* \vec\rho - J_{\ga}^{(\ell_{-,\ga})}* \vec\rho \big\|
%&\leq & \red{c'_d~ \frac{\ga^{-d}}{(\ell_{-,\ga})^{d}} (\ell_{-,\ga})^{d} \ga^{d+1}\ell_{-,\ga}}
%\\
&\leq& {c_d~  \ga\ell_{-,\ga}= c_d\ga^{\al}}
    \end{eqnarray*}

Using the fact that $\vec\rho_{D^c}\in \cX^\pm_{D^c}$, we write:

\begin{eqnarray*}
\big\|\vec\cL^u(r) - \vec\rho^{\pm} \big\|
&=&u\big\|\int_{D^c}J_\ga(r,r') \vec\rho_{D^c}(r') dr' - \vec\rho^{\pm} \big\|\\
&\le&\big\|\int_{D^c}\bigl(J_\ga(r,r')-J_{\ga}^{(\ell_{-,\ga})}(r,r')\bigr) \vec\rho_{D^c}(r') dr' \big\|
\\
& & \hskip1cm+\big\| \int_{D^c}J_{\ga}^{(\ell_{-,\ga})}(r,r') \bigl(\vec\rho_{D^c}(r')-
\vec\rho^{\pm}  \bigr)dr'\big\|
+\big\|\int_{D}J_{\ga}^{(\ell_{-,\ga})}(r,r') \vec\rho^{\pm} dr' \big\|\\
&\le& c_d\ga^{\al} +\ga^a + c'_d \ga^{\al d}
\end{eqnarray*}
with $c_d'=\|\cJ\|_\infty$.
Recalling \eqref{barmD}, we have
\begin{eqnarray}
\big\|\vec\rho^{~*}_{D}(r)-  \vec\rho^{\pm}\big\|=\big\|\vec g(\vec\cL^u(r)) - \vec g(\vec\rho^{\pm}) \big\|
\end{eqnarray}
Applying Theorem \ref{thm:mflocalstability}, we get
\begin{eqnarray*}
\big\|\vec\rho^{~*}_{D}(r)-  \vec\rho^{\pm}\big\|&\le&
(1-\frac{1}{2Q})\bigg(\ga^{a}+c_d\ga^{\al}+c'_d\ga^{\al d}\bigg)
\\
&\le& (1-\frac{1}{4Q})\ga^{a}-\bigg(\frac{1}{4Q}\ga^{a}-(1-\frac{1}{2Q})(c_d\ga^{\al}+c_d'\ga^{\al d})\bigg)
\\
& \le& \ga^{a}(1-\eps)
\end{eqnarray*}
for $\ga$ small enough and $a<\al$, taking $\eps=1-\frac{1}{4Q}$.


\vskip .5cm \noindent
\subsubsection{Third Assumption:}
\vskip .5cm \noindent

Let
\begin{eqnarray*}
\cB(n):=\{m\in (\ell_{-,\ga}\Zz)^d: \dist(C^{\ell_{-,\ga}}_n,C^{\ell_{-,\ga}}_m)\le\ga^{-1} \}
\end{eqnarray*}
there is $\tilde r_{nm}: (\ell_{-,\ga}\Zz)^{d}\times (\ell_{-,\ga}\Zz)^{d}\to\Rr^{+}$
s.t. for any $\vw^{1},\vw^{2}:\vw^{1}(j)=\vw^{2}(j)~~\forall j\in [\Zz^d\sqcap \cB(n)]\setminus i $,
it holds that:
\begin{eqnarray}
& &
R_{C_n}(\mu_C(\cdot|\vw_{C_n^c}^{(1)}), \mu_{C_n}(\cdot|\vw_{C_n^c}^{(2)}))\le
\sum_{m\in (\ell_{-,\ga}\Zz)^d\setminus \cB(n)}%{m\notin \cB(n)}
\tilde r_{nm}
\dist(\vw_{C_m}^{(1)},\vw_{C_m}^{(2)})
\label{eq:a3-a}
\\
&&\sup_{n}\sum_{m\notin \cB(n)} \tilde r_{nm}<1
\label{eq:a3-b}
\end{eqnarray}

This is a condition on the tail of the interaction, which in our case
is satisfied because of the exponential decay of the interaction $h_u^{\pm}$
due to the Peierls estimates.

\vskip .5cm \noindent
\subsubsection{Fourth Assumption:}
\vskip .5cm \noindent

Let $r(n,m): (\ell_{-,\ga}\Zz)^{d}\times (\ell_{-,\ga}\Zz)^{d}\to\Rr^{+}$ %, $r_{nn}\equiv 0$
defined as follows:

\begin{eqnarray*}
& &r(n,m):=
\begin{cases}
    0 & \text{if  }m=n\\
    \tilde r_{nm} & \text{if } m\notin \cB(n)\\
 \tilde r_{nm}^*    &
    \text{if } m\in \cB(n)\setminus n
  \end{cases}
\end{eqnarray*}
where:
\begin{eqnarray*}
\tilde r_{nm}^*:=\dis{\sup_{j\in C^{\ell_{-,\ga}}_m}\sum_{i\in C^{\ell_{-,\ga}}_n}\sum_{k>0}
    b_{C^{\ell_{-,\ga}}_n}^{k}(i,j)+2e^{-c\ga^{2a}\ell^{d}_{-,\ga}} \bigg(\ell_{-,\ga}^d+
    \sum_{i,i'\in C^{\ell_{-,\ga}}_n}\sum_{k>0}b_{C^{\ell_{-,\ga}}_n}^{k}(i,i')\bigg)}
\end{eqnarray*}
with
\begin{eqnarray*}
b_{C^{\ell_{-,\ga}}_n}(i,j):=b(i,j)\Ii_{i\in C_{n}} %~~~\text{and } ~~~
\end{eqnarray*}
and $b_{C^{\ell_{-,\ga}}_n}^{k}(i,j)$ is the $k$-th convolution of $b_{C^{\ell_{-,\ga}}_n}(i,j)$.
Assumption $4$ then states:
\begin{eqnarray}
\label{eq:as-4}
\sup_{n}\sum_{m}r(n,m)\le\delta     \hskip3cm 0<\delta<1
\end{eqnarray}


\vskip .5cm \noindent
\subsubsection{Fifth Assumption:}
\vskip .5cm \noindent

There is a metric $\psi(n,m)$ on $(\ell_{-,\ga}\Zz)^d$ s.t. $\psi(n,m)\ge d_0>0$ for all
$n\ne m\in (\ell_{-,\ga}\Zz)^d$, and

\begin{eqnarray}
\label{eq:as-5}
\sup_{n}\left[\delta e^{\sup_{j\in\cB(n)\psi(n,m)}}+
\sum_{m\notin \cB(n)}r(n,m)e^{\psi(n,m)}\right]<1
\end{eqnarray}
for $\delta$ as in \eqref{eq:as-4}.




\vskip .5cm \noindent
\subsubsection{Conclusion of subsection \eqref{subsect:decayofthecorrelations}}
\vskip .5cm \noindent

We have proved that the assumptions $1$, $2$, $3$, $4$ hold
for the two abstract models, which imply uniqueness
of the measure.
Moreover assumption $5$ holds and implies exponential decay of
the correlation of the measures $\tilde \mu^{\pm}_{\abs,u}$. see \cite{bkmp2}.
\vskip .3cm
Hence the validity of assumptions $1$, $2$, $3$, $4$ $5$ imply
Theorem \ref{thmz5.1}
and Corollary \ref{cor:mu-limit}
\end{proof}


\vskip 2.5cm \noindent
\subsection{Small deviation estimates}
\label{subsection:smalldeviation}
\vskip .5cm \noindent

In this subsection we prove the estimate \eqref{eq:smalldev}
\begin{eqnarray*}
\int_0^1 du\sum_{i \in A}
\{|\langle g_{\ga,i}\rangle_{\tilde \mu^{\pm}_{\abs;u}}|
+|\langle g_{\ga,i,\La}\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}|\}
\le c\ga^{1/8}|\delta_{\ins}^{\ell_{+,\ga}}[\La]|
\end{eqnarray*}
Let $A:=\delta_{\ins}^{\ell_{+,\ga}}[\La]$ and define the set $S^{\pm}(\vw)$ as:
\begin{eqnarray}
\label{def: S}
S^{\pm}(\vw):= \{i\in\check A: \|\vw^{(\ell_0)}(i)-\vec \rho^{\pm}\|_{\star}\ge \ga^{1/8}\}
\end{eqnarray}
where
\begin{eqnarray}
\label{def: hatA}
\check A:= A\sqcup\delta_{\out}^{\ga^{-1}}[A]
\end{eqnarray}
We first prove the following bound:
\begin{eqnarray}
\label{eq:sm-dev-1}
&&\int_0^1 du\sum_{i\in A}
\{|\langle g_{\ga,i}\rangle_{\tilde \mu^{\pm}_{\abs;u}}|
+|\langle g_{\ga,i,\La}\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}|\}
\\
&&\hskip1cm\le \fJ_\ga\int_0^1~ du
\left[\langle \Ii_{|S^\pm|\ge\ga^{1/8}|A|}\rangle_{\tilde \mu^{\pm}_{\abs;u}}
+\langle \Ii_{|S^\pm|\ge\ga^{1/8}|A|}\rangle_{\tilde \mu^{\pm}_{\abs,\La;u}}\right]
%\\
%&&\hskip1.5cm
+ c(\ga^{1/2}|A|+\ga^{1/8}|A|)\nn
\end{eqnarray}

\vskip .5cm \noindent
\begin{proof}[proof of \eqref{eq:sm-dev-1}]


\vskip .5cm \noindent
We recall the definition of $g_{\ga,i,\La}(\vw)$ \eqref{def:ggaLa} and $g_{\ga,i}(\vw)$
\eqref{def: gx}:
\begin{eqnarray*}
g_{\ga,i,\La}(\vw)&:=&-\frac{1}{2}\vvw_i\sumtwo{j\in\La}{j\ne i}J_\ga(i,j)\vvw_j
    +\sumtwo{\Delta\ni i}{\Delta\sqsubset \La} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)
\\
g_{\ga,i}(\vw)&:=&\lim_{\La\nearrow \Rr^{d}}g_{\ga,i,\La}(\vw)=
-\frac{1}{2}\vvw_i\sum_{j\ne i}J_\ga(i,j)\vvw_j
+\sum_{\Delta\ni i} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)
\end{eqnarray*}

\vskip .5cm \noindent

By \eqref{z4.11b},
\begin{eqnarray*}
\dis{\sum_{\Delta\ni i} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)\le
e^{-(\frac{\fK_\ga}{2} -b)}}
\end{eqnarray*}
Hence, we have:
\begin{eqnarray*}
&&\sum_{i\in A}\big| g_{\ga,i}(\vw)\big|\le
\frac{1}{2}\sum_{i\in A}\big|\vvw_i\sum_{j\ne i}J^{(\ell_0)}_\ga(i,j)\vvw_j\big| + c \ga\ell_0|A|\\
&&\le \sum_{i\in A}\sum_{j\in\check A}J^{(\ell_0)}_\ga(i,j)\|\vvw_j^{(\ell_0)}\|_\star +c \ga^{1/2}|A|\\
&&\le \fJ_\ga \sum_{j\in\check A}\|\vvw_j^{(\ell_0)}\|_\star +c \ga^{1/2}|A|\\
&&\le \fJ_\ga |S^\pm(\vw)|+ c(\ga^{1/2}|A|+\ga^{1/8}|A|)
%+\sum_{\Delta\ni x} \frac{1}{|\Delta|}U^{\pm}_\Delta(\vw)
\end{eqnarray*}
where we have defined
\begin{eqnarray*}
\blue{J^{(\ell_0)}_\ga(i,\lng j \rng)}:=\frac{1}{|C^{\ell_0}|}
\sum_{j'\in C_j^{\ell_0}} J_\ga(i,j)
\end{eqnarray*}
and used that by definition \eqref{def: S},
$\|\vvw_j\|_{\star}\le \ga^{1/8}$  for all $j\notin S$.
A similar estimate for $g_{\ga,i,\La}$ holds and \eqref{eq:sm-dev-1} follows.
\end{proof}

\vskip .5cm \noindent
Since the two terms in the integral in the right hand side of \eqref{eq:sm-dev-1} are very similar, we give
the derivation of an estimate for the first term only.
%Using the results of section \ref{sec:freeenergy} on the approximation to the continuum.
We prove the following
\begin{eqnarray}
\label{eq:bound8}
\langle \Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}\rangle_{\tilde \mu^{\pm}_{\abs;u}}
\le e^{-c\ga^{3/8}|A|}
\end{eqnarray}

\begin{proof}[proof of \eqref{eq:bound8}]

\vskip .5cm \noindent

Let

\begin{eqnarray*}
\hat A:=A\sqcup \delta_{\out}^{\ell_{+,\ga}}[A]
\end{eqnarray*}
\begin{eqnarray}
\label{eq:proto8}
\langle \Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}\rangle_{\tilde \mu^{\pm}_{\abs;u}}
=\langle
\frac{\dis{\hat \mu^{\pm}_{\abs;\hat A,u}(\Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}
e^{-\beta\sum_{\Delta:\Delta\sqcap \hat A\not=\varnothing} U^\pm_\Delta(\vw)}|\vw_{\hat A^c})}}
{\dis{\hat \mu^{\pm}_{\abs;\hat A,u}(
e^{-\beta\sum_{\Delta:\Delta\sqcap \hat A\not=\varnothing} U^\pm_\Delta(\vw)}|\vw_{\hat A^c})}}
\rangle_{\tilde \mu^{\pm}_{\abs;u}}
\end{eqnarray}
where $\hat \mu^{\pm}_{\abs;\hat A,u}$ is the measure on $\cX^{\pm}_{\hat A}$
associated to the finite range interpolating Hamiltonian $\hat h^{\pm}_u(\vw_{\hat A}|\vw_{\hat A^c})$
\begin{eqnarray*}
\hat h^{\pm}_u(\vw_{\hat A}|\vw_{\hat A^c}):=u H_{\ga,\hat A}(\vw_{\hat A}|\vw_{\hat A^c})
+(1-u)\fH_{\ga,\hat A}^{\pm}(\vw)
\end{eqnarray*}
Recalling \eqref{z4.11b}, we get:
\begin{eqnarray}
\langle \Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}\rangle_{\tilde \mu^{\pm}_{\abs;u}}
\le\langle
\hat \mu^{\pm}_{\abs;\hat A,u}(\Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}|\vw_{\hat A^c})
\rangle_{\tilde \mu^{\pm}_{\abs;u}}\times e^{\beta|\hat A|e^{-(\fK_\ga -4b)}}
\end{eqnarray}
we write
\begin{eqnarray}
\hat \mu^{\pm}_{\abs;\hat A,u}(\Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}|\vw_{\hat A^c})
=\frac{\dis{\hat Z_{\abs,\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c})}}
{\dis{\hat Z_{\abs,\beta,\hat A;u}^{\pm}(\vw_{\hat A^c})}}
\end{eqnarray}
where
\begin{eqnarray*}
\hat Z_{\abs,\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c}):=
&&\sum_{\vw_{\hat A}\in \cX^{\pm}_{\hat A}} \Ii_{|S^{\pm}|\ge\ga^{1/8}|A|}
e^{-\beta \hat h^{\pm}_u(\vw_{\hat A}|\vw_{\hat A^c})}\\
\hat Z_{\abs,\beta,\hat A;u}^{\pm}(\vw_{\hat A^c}):=
&&\sum_{\vw_{\hat A}\in \cX^{\pm}_{\hat A}}
e^{-\beta \hat h^{\pm}_u(\vw_{\hat A}|\vw_{\hat A^c})}
\end{eqnarray*}
Now the partition function $\hat Z_{\abs,\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c})$
can be estimated using an approximation to the continuum. Defining
\begin{eqnarray*}
&&F_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c}):=
-\frac{1}{2}\int_{\hat A\times \hat A}J_\ga(r,r')
\big( u \vw_{\hat A}(r)\cdot\vw_{\hat A}(r') +(1-u) \vec\rho^{\pm}\cdot \vec\rho^{\pm}\big)dr dr'\\
&&\hskip3cm-\int_{\hat A\times \hat A^c}J_\ga(r,r')
\big(u \vw_{\hat A}(r)\cdot\vw^{(\ell_0)}_{\hat A^c}(r') +(1-u) \vec\rho^{\pm}\cdot \vec\rho^{\pm}\big) dr dr'\\
&&\hskip3cm-(1-u)\int_{\hat A}\big(\vw_{\hat A}(r)\cdot\vec\rho^{\pm}\big) dr
+\frac{1}{\beta}\int_{\hat A}\vw_{\hat A}(r)\cdot\ln\vw_{\hat A}(r) dr
\end{eqnarray*}
and
\begin{eqnarray*}
\fZ^{\pm}:=\left\{\vw_{\hat A}\in \cXx_\La^{\ell_0}: \vw_{\hat A}(r)=\vw^{(\ell_0)}(r)~;~
\eta(\vw;r)=a_{\pm}~~ r\in \hat A~;~\int_{\check A}
\Ii_{\|\vw_{\hat A}-\vec\rho^{\pm}\|_{\star}\ge\ga^{1/8}}\ge\ga^{1/8}|A| \right\}
\end{eqnarray*}
A result similar to theorem \ref{thm:app} holds for the above functional and leads to
\begin{eqnarray}
\label{eq:sm-dev-2}
\ln\hat Z_{\abs,\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c})\le-\beta\inf_{\vw_{\hat A}\in\fZ^{\pm}}
F_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})+c\ga^{1/2}|\hat A|
\end{eqnarray}
where $c$ is a  constant independent on $u$.



We are then reduced to study the variational problem in \eqref{eq:sm-dev-2}
for $F_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})$ on  $\fZ^{\pm}$.
We define the ``excess free energy functional":
\begin{eqnarray}
\label{def:F-eff-u}
& &\cF^{\eff}_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c}):=
F_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})+\fA
\\
& &\fA:=-\inf_{\vec v: \|\vec v\|=1}\hskip-.3cm\phi^{\mf}_{u}(\vec v)|\hat A|
+\frac{u}{2}\int_{\hat A\times \hat A^c}\hskip-.73cmJ_\ga(r,r')\big(\vw^{(\ell_0)}_{\hat A^c}(r')\cdot
\vw^{(\ell_0)}_{\hat A^c}(r')\big)\; dr\ dr'
%\red{ + ....}
\\
\label{def:fmfu}
& &\hskip-.73cm\phi^{\mf,\pm}_{u}(\vec v):=-u\frac{(\vec v\cdot\vec v )}{2}-
(1-u)(\vec v\cdot \rho^{\pm})+\frac{1}{\beta}
(\vec v\cdot \ln\vec v)
\end{eqnarray}
$\cF^{\eff}_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c}) $ is positive and
 differs from $F_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})$
by a constant \red{(hence has the same minimizers and its minimum is finite)}

\vskip .5cm \noindent
Denoting by:
\begin{eqnarray}
\label{def:Fmfu}
\Phi^{\eff,\pm}_u(\vw_{\hat A}):=\phi^{\mf,\pm}_u(\vw_{\hat A})-\inf_{\vec v: \|\vec v\|=1}\phi^{\mf,\pm}_{u}(\vec v)
\end{eqnarray}
$\cF^{\eff}_{\ga,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})$ can be rewritten then as:
\begin{eqnarray*}
& &\cF^{\eff}_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})=
\int_{\hat A} \Phi^{\eff,\pm}_u(\vw_{\hat A})
+\frac{u}{4}\int_{\hat A\times \mathbb{B}}J_\ga(r,r')[\vw_{\hat A}(r)-\vw_{\hat A}(r')]^{2}
\\
& &\hskip7cm+\frac{u}{2}\int_{\hat A\times \hat A^c}
J_\ga(r,r')[\vw_{\hat A}(r)-\vw^{(\ell_0)}_{\hat A^c}(r')]^{2}
\end{eqnarray*}

\vskip .5cm \noindent
The analysis in  appendix \ref{app:dinamica}, see corollary \ref{corol:dimamica},  proves
that there are positive constants $\om, c_\om$, so that for any
\red{$\vw\in \fZ^{\pm}$} there is $\vec\psi_{\hat A}$:
\begin{eqnarray}
\label{eq:dinamica}
\cF^{\eff}_{\ga,\hat A,u}(\vw_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})\ge \cF^{\eff}_{\ga,\hat A,u}
(\vec \psi_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})-
c_\om e^{-\om\ga\ell_{+,\ga}/4}|\hat A|
\end{eqnarray}
where $\vec\Psi_{\hat A}$ has the following properties:
\begin{eqnarray}
& &\eta(\vec \psi_{\hat A};r)=a_{\pm} ~~~\forall r\in \hat A
\nn \\
& & \vec \psi_{\hat A}(r) =\vec \rho^{\pm} ~~~\forall r\in \blue{\Si:=}
\blue{\big(\hat A\setminus A\big)\setminus \delta_{\ins}^{\ell_{+,\ga}/4}\big(\hat A\setminus A\big)}
\label{def: Si} \\
& & \vec \psi_{\hat A}(r)=\vw(r) ~~~\forall r\in \check A % = \eqref{def: hatA}
 \nn\\
& &
\sup_q|\psi_{\hat A,q}(r)-\rho_q^{\pm}|\le(1-\kappa_0)\ga^{a} ~~~
\forall r\in\delta_{\ins}^{\ell_{+,\ga}/4}[\hat A]
~~~~~~~~~~~~~~~~\kappa_0>0 \nn
\end{eqnarray}
       % \begin{figure}[h]%h=here t=top b=bottom p=Page of floats
%        \centering
%        \resizebox{7cm}{!}
%        {{\includegraphics{graf1.eps}}}
%        \caption{}
%        \label{fig:}
%        \end{figure}


We can write:

\begin{eqnarray}
\label{eq:u1-0}
\ln\hat Z_{\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c})\le -\beta \inf_{\vec\psi_{\hat A}\in\cB^0}
\cF^{\eff}_{\ga,\hat A,u}(\vec\psi_{\hat A}|\vw_{\hat A^c}^{(\ell_0)})+c\ga\ell_0|\hat A|
+\beta c_\om e^{-\om\ga\ell_{+,\ga}/4}|\hat A|+\fA\hskip.73cm
\end{eqnarray}
where

\begin{eqnarray}
\label{def: B0}
&&\hskip-1.6cm\cB^{0}:=\{\vec\psi_{\hat A}: \sup_q|\psi_{\hat A,q}-\rho_q^{\pm}|\le(1-\kappa_0)\ga^{a},
~r\in\delta_{\ins}^{\ell_{+,\ga/4}}[\hat A]; ~~ \eta(\vec\psi_{\hat A};r)=a_{\pm},
~r\in \hat A;\\ \nn
&&\hskip-.5cm \red{\vec\psi_{\hat A}=\vec\rho^{\pm} ~ r\in \Si}; ~~
 \vec\psi_{\hat A}(r)=\vec\psi_{\hat A}^{(\ell_0)}(r), \red{~r\in \check A;} ~~
 \int_{\check A}\Ii_{\blue{\| \vec\psi_{\hat A,q}-\vec\rho^{\pm}\|_{\star}}\ge \ga^{1/8}}>\ga^{1/8}|A|\}
\end{eqnarray}


In appendix \ref{app:u1} it is proved that for any $\psi_{\hat A}\in\cB^0$ there is
$\vec\psi^{*}_{\hat A}:$
\begin{eqnarray}
\label{eq:u1-a}
\vec \psi^*_{\hat A}=
  \begin{cases}
    \vec \psi_{\hat A} & \text{on}~~\delta_{\ins}^{\ell_{+,\ga}/4}[\hat A], \\
    \vec\rho^{\pm} & \text{elsewhere}.
  \end{cases}
\end{eqnarray}
so that:
\begin{eqnarray}
\label{eq:u1-b}
\cF^{\eff}_{\ga,\hat A,u}(\vec\psi_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})\ge
\cF^{\eff}_{\ga,\hat A,u}(\vec\psi^*_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})+c\ga^{1/4}(\ga^{1/8}|A|)
\end{eqnarray}
Let $[\psi^{*}]^{\ell_0}$ defined as in \eqref{def:parteintera}.
By theorem \ref{thm:app} we then have:

\begin{eqnarray*}
-\beta F_{\ga,\hat A,u}(\vec\psi^{~*}_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})\le
\ln \hat Z_{\beta,\hat A;u}\big(\{\vw_{\hat A}^{\ell_0}=[\vec\psi^{~*}_{\hat A}]^{\ell_0}\}\big|\vw_{\hat A^c}\big)+\ga\ell_0|\hat A|
\end{eqnarray*}
and since by definition :
%\notes{definire le norme!.}
\begin{eqnarray*}
\|[\vec\psi^{~*}_{\hat A}]^{\ell_0}-\vec\psi^{~*}_{\hat A}\|_{\star}<\frac{1}{2} \ell_0^{-d}
\end{eqnarray*}
we have:

\begin{eqnarray*}
\|\red{[\vec\psi^{~*}_{\hat A}]_q^{\ell_0}}-\vec\rho^{~\pm}\|_{\star}<\frac{1}{2}\ell_0^{-d}+
(1-\kappa_0)\ga^{a}
\end{eqnarray*}
\blue{so that the set $\{\vw_{\hat A}^{\ell_0}=[\vec\psi^{~*}_{\hat A}]^{\ell_0}\}\in \cX^{\pm}$ and
}
\begin{eqnarray*}
-\beta F_{\ga,\hat A,u}(\vec\psi^{~*}_{\hat A}|\vw^{(\ell_0)}_{\hat A^c})\le
\ln \hat Z^\pm_{\abs,\beta,\hat A;u}\big(\vw_{\hat A^c}\big)+\ga\ell_0|\hat A|
\end{eqnarray*}

By \eqref{def:F-eff-u} and \eqref{eq:u1-0}
\begin{eqnarray*}
\ln\hat Z_{\abs,\beta,\hat A;u}^{\pm}(S^{\pm}|\vw_{\hat A^c})\le
\ln \hat Z_{\abs,\beta,\hat A;u}^\pm\big(\vw_{\hat A^c}\big)
+\beta c_\om e^{-\om\ga\ell_{+,\ga}/4}|\hat A|-c\ga^{1/4}(\ga^{1/8}|A|)+c'\ga\ell_0|\hat A|
\end{eqnarray*}
Inserting this inequality in \eqref{eq:proto8}, we get \eqref{eq:bound8} for $\ga$ small enough.

\end{proof}

\vskip .5cm \noindent
\setcounter{equation}{0}
\section{Large deviation estimates}
\label{section:largedeviation}



Let $\Ga$ a $\p$-contour. Recalling the definition \eqref{def:gp} of $\hG^{\p}$,
\begin{eqnarray*}
\hG^{\p}:=\ssp(\Ga)\bigsqcup_{\q\ne \p} A^{\q}
\end{eqnarray*}
we define $\cR(\Ga)$ and  $\cR(\not\Ga)$
as the two subsets of $L^\infty(\hG^{\p},S_Q)$ such that
\begin{eqnarray*}
\cR(\Ga):= \{\vec\rho\in L^\infty(\hG^{\p},S_Q):
\eta_x(\vec\rho)=\eta_\Ga(x)\; \forall x\in \ssp(\Ga);\quad \vec
\rho(x)= \vec\rho^{\,\q} \;\forall x\in A^{\q}\;\forall \q\ne\p\}
\end{eqnarray*}
\begin{eqnarray*}
\cR(\not\Ga):= \{\vec\rho\in L^\infty(\hG^{\p},S_Q):
\eta(\vec\rho,x)=a_{\p}\; \forall x\in \Ga;\quad
\rho (x)= \vec\rho^{\,\p} \;\forall x\in \bigsqcup_{\q\ne\p} A^{\q}\}
\end{eqnarray*}


We prove the following:

\begin{thm}
%\label{thm:Peierls-3}
There exists $\bar\ga>0$ and a constant $c_f>0$ such that for all $\ga\le\bar\ga$
and all $\beta$ such that $|\beta-\bcmf|\le c_b \ga^{\frac{1}{2}}$,
the following large deviation estimate holds:
\begin{eqnarray*}
\inf_{\vec\rho\in \cR(\Ga)}
F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})
- \inf_{\vec\rho\in \cR(\not\Ga)}
F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})
\ge
 c_f |\Ga| \ga^{2a}\bigl(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d
+\sum_{\q\ne \p} \cI_{A^{\q}}(\vec \rho_{\q}^2 -\vec \rho_{\p}^2)
\end{eqnarray*}
where
\begin{eqnarray*}
\cI_{A^{\q}} = \frac{1}{2} \int_{A^{\q}}\int_{(\Int^{\q}(\Ga)\setminus A^{\q})} J_\ga(x,y) dx dy
\end{eqnarray*}
\end{thm}
\vskip .5cm \noindent
\begin{proof}

For each $\cD^{\ell_{+,\ga}}$-measurable cube $C$
in $\ssp(\Ga)$,
at least one of the two following events occurs (recall definition \eqref{def:Theta}):

$(a)$ there is $x$ in $\Ga$ such that $C_x^{\ell_{+,\ga}}\sim C$ and
$\eta_x^{\ell_{-,\ga}}=0$.\par

$(b)$ there are $x_1$, $x_2$ in  $\Ga$ such that  $C_{x_1}^{\ell_{+,\ga}}\sim C$,
$C_{x_2}^{\ell_{+,\ga}}\sim C$, $C_{x_1}^{\ell_{-,\ga}}\sim C_{x_2}^{\ell_{-,\ga}}$ and
\tiva{
$\eta_{x_1}\not=0$,$\eta_{x_2}\not=0$,$\eta_{x_1}\not=\eta_{x_2}$}
{$\eta_{x_1}^{\ell_{-,\ga}}\not=0$,$\eta_{x_2}^{\ell_{-,\ga}}\not=0$,$\eta_{x_1}^{\ell_{-,\ga}}\not=\eta_{x_2}^{\ell_{-,\ga}}$}.\par

Let $N_\Ga$ the number of cubes of
$\cD^{\ell_{+,\ga}}$ contained in  $\ssp(\Ga)$. Since a single event
can be associated to at most $3^d$ cubes, there are
at least $3^{-d} N_\Ga$ distinct and nonintersecting
events.
Let consider a maximal family of nonintersecting events and
denote by $(x_0^i)_{i\in I_a}$, respectively
$(x_1^j,x_2^j)_{j\in I_b}$  a set of points (respectively a
set of pairs of points) characterizing the events of type
$a$ (respectively of type $b$). We also define
\begin{eqnarray}
&&B_a=\bigsqcup_{i\in I_a} C_{x_0^i}^{\ell_{-,\ga}}\\
&&B_b=\bigsqcup_{j\in I_b} (C_{x_1^j}^{\ell_{-,\ga}}\sqcup C_{x_2^j}^{\ell_{-,\ga}})
\end{eqnarray}

We have
\begin{eqnarray}
|I_a| +|I_b|\ge \frac{1}{3^d} N_\Ga
\end{eqnarray}
\begin{eqnarray}
|B_a| +|B_b|\ge \frac{1}{3^d} \bigl(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d|\Ga|
\end{eqnarray}

We separate $\hG^{\p}$ in two components $\hG^{\p}=\hG_1 \sqcup \hG_2$,
\begin{eqnarray}
\hG_1 &=& \delta_{\out}^{\ell_{+,\ga}}(A^{\p})\sqcap \ssp(\Ga)\\
\hG_2  &=& \hG^{\p} \setminus \hG_1
\end{eqnarray}
and write for all $\vec \rho$ in $\cR(\Ga)$
\begin{eqnarray}
F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})=
F_{\ga,\beta,\hG_1}(\vec\rho_{\hG_1}|\vw^{(\ell_0)}_{A^{\p}})
+F_{\ga,\beta,\hG_2}(\vec\rho_{\hG_2}|\vec\rho_{\hG_1})
\end{eqnarray}
We first apply corollary \ref{corol:dimamica} with $\La =
\hG_1\setminus \delta_{\ins}^{\ga^{-1}}(\hG_1)$,
$\rho^*=\rho$ and $u=0$. Thus there are positive constants
$\omega$ and $c_\omega$ and there exists $\psi\in\cR(\Ga)$
such that
\begin{eqnarray}
F_{\ga,\beta,\hG_1}(\vec\rho_{\hG_1}|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG_1}(\vec\psi_{\hG_1}|\vw^{(\ell_0)}_{A^{\p}})
- c_\omega |\hG_1|e^{-\omega \ell_{+,\ga}/4}
\end{eqnarray}
with
\begin{eqnarray}
\label{ld-psiG1}
&&\vec\psi_{\hG_1} =
  \begin{cases}
    \vec\rho_{\hG_1} & \text{ on } \delta_{\ins}^{\ga^{-1}}(\hG_1), \\
    \vec\rho^{\p} & \text{ on } \hG_1\setminus \delta_{\ins}^{\ell_{+,\ga}/4}(\hG_1)
  \end{cases}
\end{eqnarray}
and
\begin{eqnarray}
F_{\ga,\beta,\hG_2}(\vec\rho_{\hG_2}|\vec\rho_{\hG_1}) =
F_{\ga,\beta,\hG_2}(\vec\rho_{\hG_2}|\vec\psi_{\hG_1})
\end{eqnarray}
Thus we have
\begin{eqnarray}
\label{ld-st1}
F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG^{\p}}(\vec\psi|\vw^{(\ell_0)}_{A^{\p}})
-c_\omega |\hG_1|e^{-\omega \ell_{+,\ga}/4}
\end{eqnarray}
For $i\in I_a$, we write $C_i = C_{x_i}^{\ell_{-,\ga}}$ as a shorthand notation,
and define the function $\vec\psi^0$ on $\hG^{\p}$ as
\begin{eqnarray}
\vec\psi^0(r)=
  \begin{cases}
    \dis{\frac{1}{|C_i|}\int_{C_r^{\ell_{-,\ga}}} \vec\psi(r') dr'} & \text{ if } r\in B_a, \\
    \psi(r)  & \text{otherwise}.
  \end{cases}
\end{eqnarray}
From the definition of a contour, it follows that $\dist(B_a,\ssp(\Ga)^c)\ge\ga^{-1}$.
Thus the energy terms in the free energies of $\vec\psi$ and $\vec\psi^0$ differ by
the quantity
\begin{eqnarray}
&&|U_{\ga,\hG^{\p}}(\vec\psi|\vw^{(\ell_0)}_{A^{\p}})
-U_{\ga,\hG^{\p}}(\vec\psi^0|\vw^{(\ell_0)}_{A^{\p}})|\nn\\
&&=\frac{1}{2}\bigl|\int_{\ssp(\Ga)\times\ssp(\Ga)} J_\ga(x,y)
\vec\psi(x)\cdot\vec\psi(y) dx dy - \int_{\ssp(\Ga)\times\ssp(\Ga)}
J_\ga(x,y) \vec\psi^0(x)\cdot\vec\psi^0(y) dx dy\bigr|\nn\\
&&=\frac{1}{2}\bigl|\int_{\ssp(\Ga)\times\ssp(\Ga)} J_\ga(x,y)
(\vec\psi(x)+\vec\psi^0(x))\cdot(\vec\psi(y)-\vec\psi^0(y))
dx dy\bigr|\nn\\
&&=\frac{1}{2}\sum_{i\in I_a} \bigl|\frac{1}{|C_i|}
\int_{\ssp(\Ga)\times C_i} dx dy J_\ga(x,y)
(\vec\psi(x)+\vec\psi^0(x)) \cdot \int_{C_i} dy'
(\vec\psi(y)-\vec\psi(y'))\bigr|\nn\\
&&=\frac{1}{2}\sum_{i\in I_a} \bigl|\frac{1}{|C_i|}
\int_{\ssp(\Ga)\times C_i} dx dy (\vec\psi(x)+\vec\psi^0(x))
\cdot \vec\psi(y) \int_{C_i} dy'
(J_\ga(x,y)- J_\ga(x,y'))\bigr|\nn\\
&&\le\frac{1}{2}\sum_{i\in I_a} \frac{1}{|C_i|}
\int_{\ssp(\Ga)\times C_i} dx dy |(\vec\psi(x)+\vec\psi^0(x))
\cdot \vec\psi(y)| \int_{C_i} dy'
|J_\ga(x,y)- J_\ga(x,y')|\nn\\
&&\le 2^d\sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty
|B_a|\nn
\end{eqnarray}
where in the last inequality we used
\begin{eqnarray}
|(\vec\psi(x)+\vec\psi^0(x)) \cdot \vec\psi(y)|\le 2
\end{eqnarray}
and
\begin{eqnarray}
|J_\ga(x,y)- J_\ga(x,y')|\le \sqrt{d} \ell_{-,\ga}
\ga^{d+1} \|\nabla \cJ\|_\infty {\bf 1}_{\{|x-y|\le
2\ga^{-1}\}}
\end{eqnarray}
that for all $y$, $y'$ in $C_i$.
By concavity of the entropy, $I(\vec\psi)\le I(\vec\psi^0)$
and we get
\begin{eqnarray}
\label{ld-st2}
F_{\ga,\beta,\hG^{\p}}(\vec\psi|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG^{\p}}(\vec\psi^0|\vw^{(\ell_0)}_{A^{\p}})
-2^d\sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty |B_a|
\end{eqnarray}
We look for a lower bound of the free energy of
$\vec\psi^0$.
We again divide $\hG^{\p}$ in two parts
$\hG^{\p}=\hG_1'\sqcup\hG_2'$ where
\begin{eqnarray}
&&\hG_1'=\delta_{\out}^{\ell_{+,\ga}/2}(A^{\p})\sqcap\ssp(\Ga)\\
&&\hG_2'= \hG\setminus \hG_1'
\end{eqnarray}
We have
\begin{eqnarray}
F_{\ga,\beta,\hG^{\p}}(\vec\psi^0|\vw^{(\ell_0)}_{A^{\p}})=
F_{\ga,\beta,\hG_1'}(\vec\psi^0_{\hG_1'}|\vw^{(\ell_0)}_{A^{\p}})
+F_{\ga,\beta,\hG_2'}(\vec\psi^0_{\hG_2'}|\vec\psi^0_{\hG_1'})
\end{eqnarray}
We write the second term as:
\begin{eqnarray}
&&F_{\ga,\beta,\hG_2'}(\vec\psi^0_{\hG_2'}|\vec\psi^0_{\hG_1'})=
\int_{\hG_2'} dx \phi^{mf}_{\beta}(\vec\psi^0(x))
+\frac{1}{4}\int_{{\hG_2'}\times{\hG_2'}} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2 \nn\\
&&-\int_{{\hG_2'}\times{\hG_1'}} dx dy J_\ga(x,y) (\vec\psi^0(x)\cdot\vec\psi^0(y))
+\frac{1}{2}\int_{{\hG_2'}\times{\hG_2'^c}} dx dy J_\ga(x,y) |\vec\psi^0(x)|^2
\end{eqnarray}
The last two terms can be calculated since,
by \eqref{ld-psiG1}, $\vec\psi^0$ is constant and equal to $\vec\rho^{\p}$ on both
$\delta_{\out}^{\ga^{-1}}(\hG_1')\sqcap \hG_2'$ and $\hG_1'\sqcap\delta_{\out}^{\ga^{-1}}(\hG_2')$,
and equal to $\vec\rho^{\q}$ on $A^{\q}$, $\q\ne \p$. We get:
\begin{eqnarray}
F_{\ga,\beta,\hG_2'}(\vec\psi^0_{\hG_2'}|\vec\psi^0_{\hG_1'})=
\int_{\hG_2'} dx \phi^{\mf}_{\beta}(\vec\psi^0(x))
+\frac{1}{4}\int_{{\hG_2'}\times{\hG_2'}} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2 \nn\\
+\frac{1}{2}\sum_{\q\ne\p}\int_{{\hG_2'}\times (\Int^{\q}(\Ga)\setminus A^{\q})} dx dy J_\ga(x,y) |\vec\rho^{\q}|^2\nn
-\frac{1}{2}\int_{{\hG_2'}\times {\hG_1'}} dx dy J_\ga(x,y) |\vec\rho^{\p}|^2
\end{eqnarray}
Since $B_b\sqsubset \hG_2'$, the second term can be bounded by the contributions of the $b$-events.
We have the following estimate:
\begin{eqnarray}
&&\frac{1}{4}\int_{\hG_2'\times\hG_2'} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2 \nn\\
&&\ge \frac{1}{4}\sum_{j\in I_b}  \bigl(\int_{\hG_2'\times C_j}
dx dy J_\ga(x,y) |\vec\psi^0(x)-\vec\psi^0(y)|^2+
\int_{\hG_2'\times C_j'} dx dy' J_\ga(x,y')
|\vec\psi^0(x)-\vec\psi^0(y')|^2\bigr)\nn\\
&&=\frac{1}{4}\sum_{j\in I_b} \frac{1}{|C_j|}\int_{\hG_2'}
dx \int_{C_j\times C_j'} dy dy' \bigl(J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2+  J_\ga(x,y')
|\vec\psi^0(x)-\vec\psi^0(y')|^2\bigr)\nn
\end{eqnarray}
where we used the notations $C_j=C_{x_1^j}^{\ell_{-,\ga}}$,
$C_j'=C_{x_2^j}^{\ell_{-,\ga}}$ for all $j \in I_b$.
Using now the fact that
for all $(y,y')$ in $C_j\times C_j'$,
\begin{eqnarray}
\bigl|J_\ga(x,y)- J_\ga(x,y')\bigr|
\le 2\sqrt{d} \ell_{-,\ga} \ga^{d+1} \|\nabla \cJ\|_\infty
{\bf 1}_{\{|x-y|\le2\ga^{-1}\}}
\end{eqnarray}
we get
\begin{eqnarray}
&&\frac{1}{4}\int_{\hG_2'\times\hG_2'} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2 \\
&&\ge\frac{1}{4}\sum_{j\in I_b} \int_{\hG_2'} dx
\frac{1}{2|C_j|}\cdot \\
& &\hskip1cm
\int_{C_j\times C_j'} dy dy' (J_\ga(x,y)+J_\ga(x,y'))
\bigl(|\vec\psi^0(x)-\vec\psi^0(y)|^2+|\vec\psi^0(x)-\vec\psi^0(y')|^2\bigr)\nn\\
&&\hskip1cm-2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty |B_b|\nn
\end{eqnarray}

Now using successively the inequality
\begin{eqnarray}
|a-b|^2+|a-c|^2\ge\frac{1}{2}|b-c|^2
\end{eqnarray}
and $\dist(B_b,\hG_2'^c)\ge\ga^{-1}$, we can sum over the $x$ variable to obtain

\begin{eqnarray}
&&\frac{1}{4}\int_{\hG_2'\times\hG_2'} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2 \\
&&\ge\frac{1}{8}\sum_{j\in I_b} \int_{\hG_2'} dx
\frac{1}{2|C_j|}
\int_{C_j\times C_j'} dy dy' (J_\ga(x,y)+J_\ga(x,y'))|\vec\psi^0(y)-\vec\psi^0(y')|^2\nn\\
&&-2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty |B_b|\nn\\
&&\ge\frac{1}{8}\sum_{j\in I_b} \frac{1}{|C_j|}
\int_{C_j\times C_j'} dy dy'
|\vec\psi^0(y)-\vec\psi^0(y')|^2
 -2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty
|B_b|\nn
\end{eqnarray}
We finally get
\begin{eqnarray}
\label{ld-st3}
&&\frac{1}{4}\int_{\hG_2'\times\hG_2'} dx dy J_\ga(x,y)
|\vec\psi^0(x)-\vec\psi^0(y)|^2\nn\\
&&\ge\frac{1}{8}\sum_{j\in I_b} \frac{1}{|C_j|}
\bigl|\int_{C_j} dy \vec\psi^0(y)- \int_{C_j'}
dy'\vec\psi^0(y')\bigr|^2
 -2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty
|B_b|\nn
\end{eqnarray}
Now collecting all estimates \eqref{ld-st1},\eqref{ld-st2} and \eqref{ld-st3},
we get for all $\vec\rho$ in
$\cR(\Ga)$:
\begin{eqnarray}
&&F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG_1'}(\vec\psi^0_{\hG_1'}|\vw^{(\ell_0)}_{A^{\p}})
+ \sum_{\q\ne\p}\cI_{A^{\q}}|\vec\rho^q|^2\nn
-\frac{1}{2}\int_{{\hG_2'}\times {\hG_1'}} dx dy J_\ga(x,y) |\vec\rho^{\p}|^2\\
&&+ \int_{\hG_2'} dx \phi_\beta^{\mf}(\vec\psi^0(x)) +
\frac{1}{8}\sum_{j\in I_b} \frac{1}{|C_j|} \bigl|\int_{C_j}
dy' \vec\psi^0(y')- \int_{C_j'} dy''\vec\psi^0(y'')\bigr|^2\nn\\
&&-c_\omega|\hG_1|e^{-\omega \ell_{+,\ga}/4}
 -2^d\sqrt{d}\ell_{-,\ga} \ga \|\nabla \cJ\|_\infty |B_a|
-2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty
|B_b|\nn
\end{eqnarray}
Using the definitions of $B_a$ and $B_b$ and the results of appendix A, we have
for all $\beta$ such that $\beta-\bcmf\le c_b \ga^{\frac{1}{2}}$,
\begin{eqnarray}
\phi^{mf}_{\beta}(\vec\psi^0(x))\ge \phi^{mf}_{\beta}(\vec\rho^{\p})
+ (\frac{Q}{\beta}-1) \ga^{2a}- c \ga^{\frac{1}{2}}
\end{eqnarray}
for all $x$ in $B_a$ and
\begin{eqnarray}
\frac{1}{|C_j|^2} \bigl|\int_{C_j} dy \vec\psi^0(y)-
\int_{C_j'} dy'\vec\psi^0(y')\bigr|^2 \ge (\frac{Q-1}{Q}-
2\ga^a)^2
\end{eqnarray}
on all $j$ in $I_b$
Hence
\begin{eqnarray}
&&F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG_1'}(\vec\psi|\vw^{(\ell_0)}_{A^{\p}})
+ |\hG_2'|  \phi^{\mf}_{\beta}(\vec\rho^{\p})
+ \sum_{\q\ne\p}\cI_{A^{\q}} |\vec\rho^q|^2\nn
-\frac{1}{2}\int_{{\hG_2'}\times {\hG_1'}} dx dy J_\ga(x,y) |\vec\rho^{\p}|^2\\
&&+(\frac{Q}{\beta}-1) \ga^{2a} |B_a|
 +\frac{1}{4}(\frac{Q-1}{2Q}-\ga^a)^2 |B_b|\nn\\
  &&-c_\omega|\hG_1|e^{-\omega \ell_{+,\ga}/4}
 -2^d\sqrt{d}\ell_{-,\ga} \ga \|\nabla \cJ\|_\infty |B_a|- c\ga^{\frac{1}{2}}
-2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty
|B_b|\nn
\end{eqnarray}
Now for $a<\min(\frac{1}{4},\frac{\alpha}{2})$, we may choose $\ga$ so that the last three error terms are a fraction of
the respective gain terms. In addition, since we have both $|B_a|+|B_b| \ge
\frac{1}{3^d}\big(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d|\Ga|$
and $\hG_1\sqsubset\Ga$, the remaining error term can be also compensated for $\ga$ small enough.
We define $\bar\ga$ as the largest
value of  $\ga$ such that the following inequalities hold
simultaneously
\begin{eqnarray}
&&2^{d-1} \sqrt{d} \ell_{-,\ga} \ga \|\nabla \cJ\|_\infty\le
\frac{1}{3} (\frac{Q-1}{2Q}-\ga^a)^2\nn\\
&& 2^d\sqrt{d}\ell_{-,\ga} \ga \|\nabla \cJ\|_\infty \le
\frac{1}{3} (\frac{Q}{\beta}-1) \ga^{2a}\nn\\
&&c_\omega e^{-\omega \ell_{+,\ga}/4} \le
\frac{1}{3} (\frac{Q}{\beta}-1) \ga^{2a}\nn\\
&&\frac{1}{4}(\frac{Q-1}{2Q}-\ga^a)^2 \ge \frac{1}{3}
(\frac{Q}{\beta}-1) \ga^{2a}\nn
\end{eqnarray}
For all $\ga\le\bar\ga$, we have
\begin{eqnarray}
&&F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG_1'}(\vec\psi|\vw^{(\ell_0)}_{A^{\p}})
 + |\hG_2'|  \phi^{mf}_{\beta}(\vec\rho^{\p})
 -\frac{1}{2}\int_{{\hG_2'}\times {\hG_1'}} dx dy J_\ga(x,y) |\vec\rho^{\p}|^2\nn\\
&&+ \sum_{\q\ne\p}\cI_{A^{\q}}|\vec\rho^q|^2+
\frac{1}{3^{d+1}}(\frac{Q}{\beta}-1) \ga^{2a}
\bigl(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d|\Ga|
\end{eqnarray}
Now consider the function $\vec\varphi$ in
$L^\infty(\hG^{\p},S_Q)$ defined as
\begin{eqnarray}
\vec\varphi(r) =
  \begin{cases}
    \vec\psi(r) & r\in\hG_1', \\
    \vec\rho^{\p} & r\in\hG_2'.
  \end{cases}
\end{eqnarray}
$\vec\varphi(r)$ belongs clearly to $\cR(\not\Ga)$, and
since $\vec\varphi(r)=\vec\rho^{\p}$ on
$\hG_1'\sqcap\delta_{\out}^{\ga^{-1}}(\hG_2')$, its
free energy reads
\begin{eqnarray}
&&F_{\ga,\beta,\hG^{\p}}(\vec\varphi|\vw^{(\ell_0)}_{A^{\p}})=
F_{\ga,\beta,\hG_1'}(\vec\varphi|\vw^{(\ell_0)}_{A^{\p}})\nn\\
&&+ |\hG_2'|\phi^{mf}_{\beta}(\vec\rho^{\p})
-\frac{1}{2}\int_{{\hG_2'}\times {\hG_1'}} dx dy J_\ga(x,y) |\vec\rho^{\p}|^2
+\sum_{\q\ne\p}\cI_{A^{\q}}|\vec\rho^p|^2\nn
\end{eqnarray}
Hence we
have for all $\vec\rho$ in $\cR(\Ga)$,
\begin{eqnarray}
\label{ld-st4}
&&F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
F_{\ga,\beta,\hG^{\p}}(\vec\varphi|\vw^{(\ell_0)}_{A^{\p}})\nn\\
&&+\sum_{\q\ne\p}\cI_{A^{\q}}( |\vec\rho^{\q}|^2 -
|\vec\rho^{\p}|^2) + \frac{1}{3^{d+1}}(\frac{Q}{\beta}-1) \ga^{2a}
\bigl(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d|\Ga|
\end{eqnarray}
and since $\vec\varphi$ is in $\cR(\not\Ga)$,
\begin{eqnarray}
F_{\ga,\beta,\hG^{\p}}(\vec\varphi|\vw^{(\ell_0)}_{A^{\p}})\ge
\inf_{\rho\in \cR(\not\Ga)} F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})
\end{eqnarray}
Taking now the infimum over $\cR(\Ga)$ in \eqref{ld-st4}, we get for all $\ga\le\bar\ga$,
\begin{eqnarray}
&&\inf_{\vec\rho\in\cR(\Ga)}F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\ge
\inf_{\rho\in \cR(\not\Ga)} F_{\ga,\beta,\hG^{\p}}(\vec\rho|\vw^{(\ell_0)}_{A^{\p}})\nn\\
&&+\sum_{\q\ne\p}\cI_{A^{\q}}( |\vec\rho^{\q}|^2 -
|\vec\rho^{\p}|^2) + c_f
\ga^{2a}
\bigl(\frac{\ell_{-,\ga}}{\ell_{+,\ga}}\bigr)^d|\Ga|
\end{eqnarray}
with
\begin{eqnarray}
c_f=\frac{1}{3^{d+1}}(\frac{Q}{\beta}-1)
\end{eqnarray}
\end{proof}



\vskip 2.5cm \noindent
\centerline{\bf Acknowledgments }

We thank Marzio Cassandro, Roberto Fernandez, Joel Lebowitz and
Errico Presutti for many helpful discussions. T.G. thanks the warm hospitality
of the Mathematics departments of Universities of Rome 2 and l'Aquila where most of
the work has been done.

\vskip 2.5cm \noindent
%\newpage
\appendix


%=======================================================================================

\vskip 2.5cm \noindent
\section{Mean Field Model}
\setcounter{equation}{0}
\label{app:meanfield} In this appendix, we review briefly
the mean field theory of the Potts model
\cite{Wu}, and derive the various quantities
needed in the rest of the paper.



We consider a $Q$-state Potts model defined on a complete
graph with $N$ sites and derive its behavior in the large
$N$ limit. A variable $\sigma_i$, $\sigma_i\in
[1,\cdots,Q]$, is attached to each site $i$ of the graph so
that the space of configurations is
$\Omega_N=[1,\cdots,Q]^N$. The mean field
Hamiltonian on $\Omega_N$ is:

\begin{eqnarray*}
H^{\mf}(\si):=-\frac{1}{2 N}\sum_{i\ne j}
\Ii_{\{\si_i=\si_j\}}
\end{eqnarray*}
$H^{\mf}$ is invariant under any permutation of sites so
that its value on a given configuration depends only on the
number of sites with color $q$, says $N_q$,
$q\in[1,\cdots,Q]$. The partition function of the model is thus:

\begin{eqnarray} Z_{N,\beta}=\sum_{\{N_q\}:\sum N_q
=N}\frac{N!}{\prod_q N_q!} e^{\frac{\beta}{2N}\sum_q N_q
(N_q -1)}
\end{eqnarray} For $N$ large, $Z_{N,\beta}$ is
dominated by the configurations which realize the minimum
of the free energy density $\phi^{\mf}_\beta$ defined on
$S_Q=\{\vec\rho\in\Rr_+^Q,\sum_q\rho_q=1\}$ as
\begin{eqnarray}
\label{def:mean-field-free-energy}
\phi^{\mf}_\beta(\vec\rho) = -\frac{1}{2}\sum_q \rho_q^2
+\frac{1}{\beta} \sum_q\rho_q \ln (\rho_q)
\end{eqnarray}
with the correspondence $\rho_q=\frac{N_q}{N}$.
In fact, for our present purpose, we are also interested in all local
minimizers of $\phi^{\mf}_\beta$, which appear to be of two kinds:
one ``disordered'' (or uniform) state in which all colors have the same density
and $Q$ degenerated ``ordered'' (or colored) states in which one color dominates.
As can be expected, the first one exists for small values of $\beta$,
while the other $Q$ exist only for $\beta$ large enough. In addition,
there is a critical value of $\beta$ which determines which kind of local minimizer
is the actual absolute minimizer for $\phi^{\mf}_\beta$. We make these statements
precise in the two following theorems. We first characterize all local minimizers:
\begin{thm}
   \label{thm:mflocalminimizers}
For all $Q>2$, there exists $\beta_0<Q$ such that the mean field free energy $\phi^{\mf}_\beta$
\eqref{def:mean-field-free-energy} on $S_Q$ has exactly:
{\obeylines
$\bullet$ 1 local minimizer for $\beta\le\beta_0$,
$\bullet$ $Q+1$ local minimizers for $\beta_0<\beta<Q$, and
$\bullet$ $Q$ local minimizers for $\beta\ge Q$.
}
\noindent These minimizers are of two kinds and characterized as follows:
\begin{itemize}
  \item
  For all $\beta<Q$, the uniform state $\vec\rho^{(-1)}$, with components
  \begin{eqnarray}
  \label{mf:rhod}
\rho^{(-1)}_q=\frac{1}{Q} \quad\hbox{ for all } q\in\{1,\cdots,Q\}
  \end{eqnarray}
  and free energy
  \begin{eqnarray}
  \label{mfphi(rhod)}
\phi^{\mf}_\beta(\vec\rho^{(-1)})=\frac{-1}{2Q}-\frac{1}{\beta}\log(Q)
  \end{eqnarray}
  \item
  For all $\beta>\beta_0$, $Q$ colored states $\vec\rho^{(p)}\equiv\vec\rho^{(p)}_\beta$, $p\in\{1,\cdots,Q\}$,
  with components
  \begin{eqnarray}
  \label{mf:rhoo}
\rho^{(p)}_q=
  \begin{cases}
    \rho_A & \text{ if } q=p, \\
    \rho_B & \text{otherwise}.
  \end{cases}
  \end{eqnarray}
where $(\rho_A,\rho_B)$, $\rho_A>\rho_B$, is the solution of the set of equations
\begin{eqnarray}
\label{mf:defrhoo1}
\frac{\log(\rho_A)-\log(\rho_B)}{\rho_A-\rho_B}&=& \beta\\
\label{mf:defrhoo2}
\rho_A+ (Q-1)\rho_B &=& 1
\end{eqnarray}
which verify
\begin{eqnarray}
\label{mf:defrhoo3}
Q\beta \rho_A \rho_B <1
\end{eqnarray}
These $Q$ states are degenerate and have free energy
\begin{eqnarray}
\label{mf:phi(rhoo)}
\phi^{\mf}_\beta(\vec\rho^{(p)})=\frac{-1}{2}Q\rho_A\rho_B+\frac{1}{2\beta}\log(\rho_A\rho_B)
\end{eqnarray}
\end{itemize}
\end{thm}
The mean field first order transition is described in the following
\begin{thm}
   \label{thm:mfabsoluteminimizer}
For all $Q>2$, there exists a critical value of $\beta$, in $(\beta_0,Q)$,
\begin{equation}
\label{mf:bcmf}
\bcmf\equiv \frac{2(Q-1)}{Q-2}\log(Q-1)
\end{equation}
such that $\phi^{\mf}_\beta$ has:
{\obeylines
$\bullet$ 1 minimizer $\vec\rho^{(-1)}$ for all $\beta<\bcmf$;
$\bullet$ $Q+1$ minimizers $\vec\rho^{(\p)}$, $\p\in\{-1,1,\cdots,Q\}$ for $\beta=\bcmf$;
$\bullet$ $Q$ minimizers $\vec\rho^{(p)}$, $p\in\{1,\cdots,Q\}$ for $\beta>\bcmf$.
}
\end{thm}



Finally, the relevance of local minimizers in our problem arises from their local stability, which is stated
in the following theorem:
\begin{thm}
   \label{thm:mflocalstability}
For all $Q>2$ and all $\beta$ in $(\beta_0,Q)$, the map
       \begin{equation}
\vec\rho \rightarrow \vec g(\vec\rho)
       \end{equation}
       where
       \begin{eqnarray}\dis
       \label{mf:g}
g_q(\vec\rho)=\frac{\exp(\beta\rho_q)}{\sum_{p=1}^Q\exp(\beta\rho_p)}
       \end{eqnarray}
is a contraction around the $Q+1$ local minimizers $\vec\rho^{(\p)}$ of the mean field free energy
$\phi^{\mf}_\beta$. In particular,
\begin{eqnarray}
\sup_{q,q'} \bigl|\frac{\partial g_q(\vec\rho)}{\partial\rho_{q'}}\bigr|\le 1-\frac{1}{2Q}
\end{eqnarray}
for all $\vec\rho$ such that $\sup_q|\rho_q -\rho_q^{\p}|\le \frac{1}{4\beta^2Q^2}$ for some $\p$.

\end{thm}
\vskip2.0truecm


\begin{proof}{\bf of Theorem \ref{thm:mflocalminimizers}}:


We consider the variational
problem for $\phi^{\mf}_\beta(\vec\rho)$.
\begin{eqnarray*}
\inf_{\vec \rho: \sum_{q=1}^Q\rho_q=1}\phi^{\mf}_\beta(\vec
\rho)= {\min_{\vec\rho,\la}}^{\loc}\left[\phi^{\mf}_\beta(\vec
\rho)+\la\left(\sum_{j=q}^Q\rho_q-1\right)\right]
\end{eqnarray*}
where $\lambda$ is a Lagrange parameter associated to the
constraint $\sum_q \rho_q=1$. Since the gradient of the
free energy points inward the simplex $S_Q$, the (local) minima
cannot stay on the boundary of $S_Q$ and are thus solutions
of the set of equations:
\begin{eqnarray}
\label{mf:criticalpoints}
\frac{\partial \phi^{\mf}_\beta(\vec \rho)}{\partial \rho_q}
+\la = 0,\qquad
q=1,\cdots, Q
\end{eqnarray}
together with the condition
\begin{eqnarray}
\label{mf:minima}
\sum_{q1,q2 =1}^Q\frac{\partial^2 \phi^{\mf}_\beta(\vec \rho)}{\partial \rho_{q_1}\rho_{q_2}} x_{q_1}x_{q_2} \ge 0
\end{eqnarray}
for all $\vec x\in\Rr^Q$ such that $\sum_{q=1}^Q x_q =0$.

Explicitly the first derivatives of the free energy read
\begin{eqnarray}
\frac{\partial \phi^{\mf}_\beta(\vec \rho)}{\partial \rho_q}
\equiv 1-\rho_q+\frac{1}{\beta}(\ln\rho_q+1)
\end{eqnarray}
while the Hessian matrix of $\phi^{\mf}_\beta(\vec \rho)$ is diagonal and
\begin{eqnarray}
\label{mf:hessian}
\frac{\partial^2 \phi^{\mf}_\beta(\vec \rho)}{\partial \rho_{q_1}\rho_{q_2}}
=\delta_{q_1,q_2} \bigl(-1 +\frac{1}{\beta \rho_{q_1}}\bigr)
\end{eqnarray}
As a function of $\rho_q$ alone, $\frac{\partial
\phi^{\mf}_\beta(\vec \rho)}{\partial \rho_q}$ is a strictly
concave $C^\infty$ function and hence cannot take the same value more than twice.
Thus there are two kind of solutions for \eqref{mf:criticalpoints},
depending on whether $\rho_q$ takes one or two values.


The first case correspond to a ``disordered'' solution
$\vec\rho^{(-1)}$ in which each color has the same density:
\begin{eqnarray}
\rho^{(-1)}_q &=& \frac{1}{Q}\qquad \hbox{for all } q=1,\cdots,Q\\
\phi^{\mf}_\beta(\vec\rho^{(-1)})&=&\frac{-1}{2Q}-\frac{1}{\beta}\ln(Q)
\end{eqnarray}
Using \eqref{mf:hessian}, $\vec\rho^{(-1)}$ is a local minimum of $\phi^{\mf}_\beta(\vec
\rho)$ if and only if $\beta< Q$\vskip .5cm


In the second case, let $\vec\rho$ a vector in $S_Q$ which components takes two values,
 says $\rho_A$ and $\rho_B$ with $\rho_A>\rho_B$, and let $n$, $0<n<Q$, the number of components
 equal to $\rho_A$ (the remaining $Q-n$ are thus equal to $\rho_B$).
According to \eqref{mf:criticalpoints}, $\vec\rho$ is a critical point
for $\phi^{\mf}_\beta(\vec \rho)$ in $S_Q$ whenever the constraint is
satisfied and both $\rho_A$ and $\rho_B$ are associated to
the same value of the Lagrange parameter:
\begin{eqnarray}
\label{os1}
&& n\rho_A+(Q-n)\rho_B=1\\
\label{os2}
 &&\lambda=\rho_A-1-\frac{1}{\beta}(\ln\rho_A+1)=\rho_B-1
-\frac{1}{\beta}(\ln\rho_B+1)
\end{eqnarray}
However, all such points are not local minima: equation \eqref{os2} implies the relation \eqref{mf:defrhoo1}
between $\beta$,
$\rho_A$ and $\rho_B$,
\begin{eqnarray}
\label{mf:betaAB}
\beta=\frac{\ln \rho_A -\ln \rho_B}{\rho_A - \rho_B}
\end{eqnarray}
and by concavity of the logarithmic function, we have necessarily
\begin{eqnarray}
\frac{1}{\rho_A}<\beta<\frac{1}{\rho_B}
\end{eqnarray}
Thus the second derivative of $\phi^{\mf}_\beta(\vec \rho)$ is negative in each direction in which
$\rho_q=\rho_A$. This is obviously not compatible with the condition \eqref{mf:minima} for $n\ge 2$.
In the case $n=1$, the constraint \eqref{os2} reduces to \eqref{mf:defrhoo2}:
\begin{eqnarray}
\label{mf:QAB}
\rho_A+(Q-1)\rho_B=1
\end{eqnarray}
while the condition \eqref{mf:minima} can be made explicit using \eqref{mf:hessian}:
\begin{eqnarray*}
\bigl(-1 +\frac{1}{\beta \rho_A}\bigr)+\frac{1}{Q-1}\bigl(-1 +\frac{1}{\beta \rho_B}\bigr)\ge 0
\end{eqnarray*}
or equivalently using the constraint \eqref{mf:QAB}, one gets \eqref{mf:defrhoo3}:
\begin{eqnarray}
\label{mf:stabilite}
\beta Q\rho_A\rho_B\le 1
\end{eqnarray}
We need to find out all triples $(\beta,\rho_A,\rho_B)$ which solve simultaneously
\eqref{mf:betaAB}, \eqref{mf:QAB} and \eqref{mf:stabilite}. For $\rho_A$ in $(\frac{1}{Q},1)$,
we consider the function $\tilde\beta(\rho_A)$ as
\begin{eqnarray}
\label{mf:beta(rhoA)}
\tilde\beta(\rho_A)=\frac{\ln \rho_A -\ln \rho_B}{\rho_A - \rho_B}
\end{eqnarray}
where $\rho_B$ is taken implicitly as a function of $\rho_A$, through \eqref{mf:QAB}
%\begin{eqnarray}
%\rho_B:=\frac{1-\rho_A}{Q-1}
%\end{eqnarray}
Its first derivative reads
\begin{eqnarray}
\frac{\partial \tilde\beta(\rho_A)}{\partial \rho_A}=\frac{1}{(Q-1)(\rho_A-\rho_B)}
\bigl[\frac{1}{\rho_A \rho_B} -\beta Q\bigr]
\end{eqnarray}
while the second derivative can be cast in the form
\begin{eqnarray}
\frac{\partial^2 \tilde\beta(\rho_A)}{\partial \rho_A^2}=
\frac{\varphi(\rho_A)-\varphi(\rho_B)}{(Q-1)^2(\rho_A-\rho_B)^3}
\end{eqnarray}
where
\begin{eqnarray}
\varphi(\rho)=2Q^2\log(\rho)+\frac{4Q}{\rho}-\frac{1}{\rho^2}
\end{eqnarray}
One can check that the function $\varphi(\rho)$ is strictly increasing and thus the second derivative
of $\tilde\beta(\rho_A)$ is always positive. It follows that
there exists a unique value $\rho_0$ in $(\frac{1}{Q},1)$ so that the first derivative is zero
in $\rho_0$, strictly negative for $\rho<\rho_0$ and strictly positive for $\rho> \rho_0$.
We define $\beta_0\equiv\tilde\beta(\rho_0)$.
Since equation \eqref{mf:betaAB} is equivalent to $\beta=\tilde\beta(\rho_A)$, solutions to \eqref{mf:betaAB}
will exist only for $\beta$ in the image of $\tilde\beta(\cdot)$, and thus for $\beta\ge\beta_0$.
Now the condition for a local minimum \eqref{mf:stabilite} is equivalent
to $\frac{\partial \tilde\beta(\rho_A)}{\partial \rho_A}\ge 0$ and thus to $\rho_A\ge\rho_0$.
Furthermore the function $\tilde\beta(\cdot)$ is invertible from  $(\rho_0,1)$ onto $(\beta_0,+\infty)$ and
therefore, for all $\beta>\beta_0$ there is a unique couple $(\rho_A,\rho_B)$ for which the vectors \eqref{mf:rhoo}
are $Q$ local minima, and there is no ``colored'' solutions for $\beta<\beta_0$.
Whenever they exist, those minima are degenerate and their mean field free energy is given by \eqref{mf:phi(rhoo)}.
We postpone the proof that $\beta_0<Q$ at the end of the proof of the next theorem.
\end{proof}
\begin{proof}{\bf of Theorem \ref{thm:mfabsoluteminimizer}}
For $\beta$ in $(\beta_0,Q)$, we consider the difference of free energy between ordered
and disordered local minima
\begin{eqnarray}
\label{mf:diff-freeenergy}
\Delta(\phi^{\mf}_\beta)\equiv
\phi^{\mf}_\beta(\vec\rho^{(p)})-\phi^{\mf}_\beta(\vec\rho^{(-1)})=
\frac{1-Q^2\rho_A\rho_B}{2Q}+\frac{1}{2\beta}\log(Q^2\rho_A\rho_B)
\end{eqnarray}
We first note that $\beta\Delta(\phi^{\mf}_\beta)$ is a strictly decreasing function
of $\beta$: using \eqref{mf:beta(rhoA)}, we have for $\beta$ in $(\beta_0,Q)$,
\begin{eqnarray}
\label{mf:Delta}
\frac{\partial}{\partial\beta}\bigl(\beta\Delta(\phi^{\mf}_\beta)\bigr)=
\bigr(\frac{\partial\tilde\beta}{\partial\rho_A}\bigr)^{-1}
\frac{\partial\tilde\beta\Delta(\phi^{\mf}_{\tilde\beta})}{\partial\rho_A}=
-\frac{(1-Q\rho_A)^2}{2Q(Q-1)}<0
\end{eqnarray}
Furthermore, $\Delta(\phi^{\mf}_\beta)$ has one zero for $Q^2\rho_A\rho_B=1$
or equivalently
\begin{eqnarray}
\rho_A=\frac{Q-1}{Q}
\end{eqnarray}
This zero is thus necessarily unique and defines a critical value of $\beta$
\begin{eqnarray}
\bcmf=\tilde\beta(\frac{Q-1}{Q})=\frac{2(Q-1)}{Q-2}\log(Q-1)
\end{eqnarray}
We complete the proof by showing that \eqref{mf:stabilite} holds at 
$\beta=\bcmf$.
We have for all $Q>2$:
\begin{eqnarray}
\label{mf:bcmfstab}
Q\bcmf\rho_A\rho_B = \frac{2(Q-1)}{Q(Q-2)}\log(Q-1)=
\frac{\log(Q-1)}{\sinh(\log(Q-1))}<1
\end{eqnarray}
Thus $\bcmf>\beta_0$. On the other hand, since $\rho_A\rho_B=Q^{-2}$
at $\beta=\bcmf$, \eqref{mf:bcmfstab}  proves also that 
$\bcmf<Q$, and thus $\beta_0<\bcmf<Q$. This also complete
 the proof of theorem \ref{thm:mflocalminimizers}.

\end{proof}
\begin{proof}{\bf of Theorem \ref{thm:mflocalstability}}

Let $\beta$ be in the interval $(\beta_0,Q)$ and consider the map $\vec\rho\rightarrow\vec g(\vec\rho)$
defined in \eqref{mf:g}. We have
\begin{eqnarray}
\label{mf:dg}
\frac{\partial g_q(\vec\rho)}{\partial\rho_{q'}}=\beta g_q(\vec\rho)\bigl(\delta_{q,q'}-g_{q'}(\vec\rho)\bigr)
\end{eqnarray}
and since $0\le g_q(\vec\rho)\le 1$, we have a first bound uniform in $\vec\rho$:
\begin{eqnarray}
\label{mf:uniformbounddg}
\sup_{q,q'} \bigl|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho)\bigr|\le \beta
\end{eqnarray}
On the other hand, from \eqref{mf:dg}, one can also write
\begin{eqnarray}
\label{mf:bounddg1}
\sup_{q,q'}|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho)|\le
\sup_{q,q'}|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho^{\p})|+
\sup_{q,q'}|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho)-\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho^{\p})|
\end{eqnarray}
The first term can be bounded by
\begin{eqnarray}
\sup_{q,q'} |\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho^{\p})|=
\beta \sup_{q,q'} |\rho^{\p}_q (\delta_{q,q'}-\rho^{\p}_{q'})|\le 1-\frac{1}{Q}
\end{eqnarray}
where the inequality follows from \eqref{mf:rhod} and $\beta\le Q$ for $\p=-1$,
and from \eqref{mf:defrhoo3} and \eqref{mf:defrhoo2} for $\p>0$.
The second term in \eqref{mf:bounddg1} can be bounded using \eqref{mf:uniformbounddg}
as
\begin{eqnarray}
\sup_{q,q'}
|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho)-\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho^{\p})|\le
2\beta\sup_{q}| g_q(\vec\rho)- g_q(\vec\rho^{\p})|\le 2Q\beta^2 \sup_{q}|\rho_q- \rho^{\p}_q|
\end{eqnarray}
Thus for all $\rho$ such that $\sup_{q}|\rho_q- \rho^{\p}_q|\le\frac{1}{4\beta^2Q^2}$ for some $\p$
one gets
\begin{eqnarray}
\sup_{q,q'}|\frac{\partial g_q}{\partial\rho_{q'}}(\vec\rho)|\le 1-\frac{1}{2Q}
\end{eqnarray}
\end{proof}

We conclude this appendix by a proof of \eqref{eq:lemmaa.2}:


From equation \eqref{mf:Delta} and the definition of $\bcmf$
\eqref{mf:bcmf}, one gets explicitly:

\begin{eqnarray}
\frac{d}{d\beta}\left[P^{-}_{\mf,\beta}-P^{+}_{\mf,\beta}\right]\Big|_{\beta=\bcmf}
&=&\frac{1}{\bcmf}\frac{\partial}{\partial \beta} \bigl(\beta\Delta(\phi^\mf_\beta)\bigr)|_{\beta=\bcmf}\nn\\
&=&-\frac{(Q-2)^2}{2Q(Q-1)\bcmf}<0
\end{eqnarray}
%=======================================================================================

\vskip 2.5cm \noindent
\section{Local equilibrium}
\setcounter{equation}{0}
\label{app:dinamica}


The main result of this appendix is the proof that, for suitable values of the temperature,
if a density profile is in a neighborhood of an equilibrium value in a region
$\La\sqcup\delta_{\out}^{\ga^{-1}}[\La]$, then it can be made closer to equilibrium inside $\La$
at an exponential rate from its boundary, decreasing the free energy.
This result is essentially due to the stability properties of the free energy functionals
discussed in section \ref{sec:Stability}, namely the contraction property  of the map
 $\vec\fM$ around
 % its fixed points.%, when $\beta$ is close to $\beta_c$
its constant fixed point.
The precise result is stated in  the Theorem \ref{thm:dimamica}
 below. The proof follows the lines developed in \cite{errico-leip},(see also \cite{bkmp1})
 for Ising model and continuum particle models, and we will stress here only the points
 specific for our model while we will only  sketch    the points that are
 quite analogous to the other cases.

Without lost of generality we study the local equilibrium around the phase
$``\ \p\ "$, $\p\in \{-1,1,\dots Q\}$.
Let $\La$ a bounded $\cD^{\ell_{+,\ga}}$-measurable region,
and $\eta^{\zeta}_x(\vec\rho)\in L^\infty(\Rr^d,S_Q)$ defined analogously
as $\eta_x(\vec\rho)$  in \eqref{def:eta},
but with an accuracy parameter, denoted by $\zeta$, that here we leave free
\begin{eqnarray*}
\eta^{\zeta}_x(\vec\rho)=
\begin{cases}
    a_{{\p}} & \text{if }~
    \dis{\|{\vec \rho^{\ell_{-,\ga}}(x)}-
    \red{\vec \rho_{\beta}^{~(\p)}}\|_{\star}< \zeta}
    \\
    0 & \text{otherwise}.
  \end{cases}
\end{eqnarray*}


\begin{equation*}
\cM_{\La,\zeta}^{\p}:=\{ \vec\rho\in
L^{\infty}(R^{d},S_Q): \red{\eta^{\zeta}_x(\vec\rho)=a_{\p}} ~~
\forall x\in\La\sqcup\delta^{\ga^{-1}}_{\out}[\La ]\}
\end{equation*}
and for any $\vec\rho^{~*}\in \cM_{\La}^{\p}$, we define
\begin{equation*}
\cX^{\p}_{\La,\vec\rho^{~*}}:=
    \{\vec\rho\in M_{\La,\zeta}^{\p}: \vec\rho_{\La^{c}}(x)
    \equiv \vec\rho^{~*}_{\La^{c}}(x) \}
\end{equation*}



\begin{thm}
   \label{thm:dimamica}
There are positive constant $\zeta_0$, $\om$, $c_\om$
so that for any \red{$\zeta<\zeta_0$,
$ \ga^\al<\kappa_0\zeta $} and any $\vec\rho^{~*}\in M_{\La,\zeta}^{\p}$, s.t.
\begin{itemize}
  \item there is a unique $\vec\psi\in \cX_{\La,\vec\rho^{~*}}$ s.t.:
   \begin{equation}
       \inf_{\vec\rho\in \cX^{\p}_{\La,\vec\rho^{~*}}}\cF_{\ga,u,\La}(\vec\rho|\vec\rho^{~*}_{\La^c})
       =\cF_{\ga,u,\La}(\vec\psi|\vec\rho^{~*}_{\La^c})
       \end{equation}
  \item $\vec\psi$ is the unique solution of the mean field equation and has
  the following properties:
\begin{itemize}
  \item[*] \red{$\vec\psi_\La\in C^{\infty}(\La,M_{\La,(1-\kappa_0)\zeta}^{\p})$, \;\;\;
  $\sup_{r\in\La}\|\nabla \psi_{\La} (r)\|_{\star}\le \beta\|\nabla J_\ga\|_1$}

\vskip .5cm \noindent
  \item[*] \begin{equation*}\|\vec\psi_{\La}(r)-\vec \rho^{\p}\|
  \le c_\om e^{-\om\; \dist(r,\La^c_{\ne})}\end{equation*}
 where
$\La^c_{\ne}:= \{r\in \La^c: \dist(r,\La)\le\ga^{-1}; \vec\rho^{~*}(r)\ne\vec\rho^{\p}\}$

\vskip .5cm \noindent

\end{itemize}

   \item If $\vec\psi, \vec\phi$ are minimizers resp. in
  $\cX^{\p}_{\La,\vec\rho_{1}}, \cX^{\p}_{\La,\vec\rho_{2}}$ then:
  \begin{equation*}\|\vec\psi_{\La}(r)-\vec \phi_{\La}(r)\|
  \le c_\om e^{-\om\dist(r,\La^c_{1,2,\ne})}\end{equation*} where
$\La^c_{1,2,\ne}:= \{r\in \La^c: \dist(r,\La)\le\ga^{-1}; \vec\rho_{1}(r)\ne\vec\rho_{2}(r)\}$

\end{itemize}



\end{thm}



\vskip .5cm \noindent


The proof of Theorem \ref{thm:dimamica} is obtained by defining a dynamic
$\vec T^{u,\La,\p}_t$ on
$L^{\infty}(\Rr^{d},S_Q)$, and
proving that this dynamic maps $M_{\La,\zeta}^{\p}$
into itself and that it is dissipative for the free energy $\cF_{\ga,u,\La}$.
The minimizer
$\vec \psi$ is then obtained as the limit point of the orbit
$\vec T^{u,\La,\p}_t$ as $t\to\infty$.

\red{Following \cite{errico-leip}} we define
an opportune dynamic $\vec T^{u,\La,\p}_t$ (suitable for our model)
that has the properties
that allow to conclude as in reference \cite{errico-leip}.

The essential point in the proof of the Theorem
\ref{thm:dimamica} is the contraction property of the map
$\vec\fM(\cdot)$ discussed in section \ref{sec:Stability},
that extends to the map $\vec \fM^{(u,\p)}(\cdot)$
parameterized by $u$, $u\in[0,1]$, defined as follows:

\begin{eqnarray*}
& & \fM^{(u,\p)}_q(\vec \rho)(r):=
\frac{e^{\beta \cL^{u,\p}_q(\vec\rho;r)}}{\sum _{q'} e^{\beta \cL_{q'}(\vec \rho;r)}}\\
& &\cL^{u,\p}_q(\vec\rho;r):=u\int dr'~J_\ga(r,r')\rho_q(r')+(1-u)\; \rho^{\p}_q\\
\end{eqnarray*}
 We state here a lemma  which proof is postponed at the end of this appendix :
\begin{lemma}
   \label{lem:dinam}
There are  $\zeta'_0$ \tiva{}{($\zeta_0'<\left(\frac{1}{2\beta Q}\right)^2$)} and $\kappa_0$ positive, so that for any $\zeta<\zeta'_0$
and $\ga^{\al}<\kappa_0\zeta$, any bounded $\cD^{\ell_{-,\ga}}$-measurable region
$\La$, $r\in \La$, $\vec\rho\in M_{\La,\zeta}^{\p}$

       \begin{eqnarray}
       \label{eq:lem-dinam-1}
       &&\sup_{r\in\La}\|\vL^{u,\p}(\vec\rho)(r)-\vec\rho^{~\p}(r)\|_{\red{\star}}\le {u(1+c_d\kappa_0)}\zeta<2\zeta\\
       \label{eq:lem-dinam-2}
&&\sup_{r\in\La}\|\vec\fM^{(u,\p)}(\vec\rho)(r)-\vec\rho^{~\p}(r)\|_{\red{\star}}\le u(1-\kappa_0)\zeta
       \end{eqnarray}

\end{lemma}

\vskip .5cm \noindent

We then define a  dynamic given by the semigroup $\vec T^{u,\La,\p}_{t}(\cdot)$, $t\geq 0$ on
$L^{\infty}(\La,S_Q)$,
\begin{eqnarray}
\label{def:dynamic}
\vec T^{u,\La,\p}_t(\vec \rho):=\big(T^{u,\La,\p}_{1,t}(\rho_1),\dots,
T^{u,\La,\p}_{Q,t}(\rho_Q))\hskip3cm
\mbox{for any $\vec \rho \in L^{\infty}(\Rr^{d},S_Q)$}
\end{eqnarray}
where $T^{u,\La,\p}_{q,t}(\rho_i^*)$ are solutions of the Cauchy problem:
\begin{eqnarray}
\label{def:dynamics}
\left\{\ba{lr}
\dis{\frac{d\rho^{\La}_q(r,t)}{dt}=-\rho^{\La}_q(r,t)+\fM_q^{(u,\p)}(\vec \rho^{\La})}
&  \mbox{for any $q=1,\dots, Q$}
\\
\\
\dis{\vec\rho^{\La}(r,t)=\vec\rho^{~*}(r,t)}& (r,t)\in\left[\La^c\times \{t\ge 0\}\right]
\sqcup[\Rr^{d}\times\{t=0\}]
\ea\right.
\end{eqnarray}

 Existence, uniqueness and continuity w.r.t. the initial datum of the solution follows
by the continuity and the Lipschitz property of the r.h.s. of equation \eqref{def:dynamics}

Notice that $\vec T^{u,\La}_t(\cdot)$ maps $L^{\infty}(\Rr^{d},S_Q)$ in itself,
and has as a  fixed point
$\vec\rho^{\p}$. \red{We next  prove} the following properties:



\vskip .5cm \noindent
\begin{enumerate}
  \item %\begin{eqnarray*}
$\vec T^{u,\La,\p}_t(M_{\La,\zeta}^{\p})\sqsubset M_{\La,\zeta}^{\p}$ for any $t\ge 0$

\vskip .5cm \noindent
  \item For any $\vec\rho_0\in L^{\infty}(\Rr^d,S_Q)$  and  $\La$ a Borel set,
   $T^{u,\La,\p}_{q,t}(\vec\rho_0)$, $q=1,\dots, Q$,
  converges by subsequences  as $t\to \infty$ to  functions
  $v_q$ that are bounded in $\La$ and with $\nabla v_q$ bounded in $\La$.
  The limit points are solutions
  of \eqref{eq:staz} below.

\vskip .5cm \noindent
  \item $\cF_{\ga,u,\La}(\vec T^{u,\La,\p}_t(\vec\rho_0))$ decrease with $t$,
  strictly unless $\vec\rho_0$
  is stationary, in which case satisfies:% for any $q=1,\dots, Q$:
\begin{eqnarray}
\label{eq:staz}
\rho_{0,q}=\frac{\dis{e^{\beta\cL^u_q(\vec\rho_0(r,t))}}}
{\dis{\sum_{q'=1}^{Q}e^{\beta\cL^u_{q'}(\vec\rho_0(r,t))}}}
\hskip3cm \forall q=1,\dots Q ~~~~~\forall r\in \La
\end{eqnarray}

\vskip .5cm \noindent
    \item  As a consequence of the  property $(3)$, for any
    $\vec\rho^{~*}(r)\in M_{\La,\zeta}^{\p}$,
    the minimizers of $\cF_{\ga,u,\La}(\cdot)$ in $\cX_{\La,\vec\rho^{~*}}:=
    \{\vec\rho\in M_{\La,\zeta}^{\p}: \vec\rho_{\La^{c}}(r)
    \equiv \vec\rho^{~*}_{\La^{c}}(r) \}$ are solutions of \eqref{eq:staz}:
    $\vec\rho=\vec\fM^{u,\p}(\vec\rho)$. By the
    contraction property of the map $\vec\fM^{u,\p}(\cdot)$ %\red{around .....}
    we get uniqueness of the minimizer.
\end{enumerate}


\vskip 1.5cm \noindent
{\em Proof of the properties 1,2,3}
\begin{enumerate}


\item
Clearly we have
$\vec T^{u,\La,\p}_{t}(S_Q)\sqsubset S_Q$.
To prove the first point, %propriety,
let then $\tau>0$, $\vec\rho_0\in M_{\La,\zeta}^{\p}$ and
\begin{eqnarray*}
\cX_{\tau,\vec\rho_0}:=\{\vec\rho\in L^{\infty}(\Rr^{d}\times[0,\tau];S_{Q}):
\vec\rho(r,t)=\vec\rho_0(r)
~~(r,t)\in\big[ \Rr^{d}\times 0\big]\sqcup\big[\La^c\times[0,\tau]\big] \}
\end{eqnarray*}

Let  $\hat\Omega^{\p}(\cdot)$ the map from $\cX_{\tau,\vec\rho_0}$ into itself defined
for any $t\in[0,\tau]$, $r\in \La$ as

\begin{eqnarray*}
\vec\Omega^{\p}(\vec\rho)(r,t)=e^{-t}\vec\rho_0(r)+\int_{0}^{t} e^{-(t-s)}
\vec\fM^{(u,\p)}(\vec\rho) ~ds
\end{eqnarray*}
if $\tau$ is small enough $\vec \Omega^{\p}$ is a contraction and its fixed point is the solution
of \eqref{def:dynamics},
$T^{u,\La,\p}_{t}(\vec\rho_0)$, $t\in [0,\tau]$. By \eqref{eq:lem-dinam-2} the set:
\begin{eqnarray*}
\{\vec v\in \cX_{\tau,\vec\rho_0}: \vec v(\cdot ,t)\in M_{\La,\zeta}^{\p}~ \forall t\in [0,\tau]\}
\end{eqnarray*}
is invariant under the map $\hat\Omega$, and since it is closed, it contains the fixed point of
$\hat\Omega$. By induction the statement can be extended fo any $t$: $T^{u,\La,\p}_{t}(\vec\rho_0)
\sqsubset M_{\La,\zeta}^{\p}$, $t\ge 0$.

\vskip .5cm \noindent
\item
Convergence on subsequences follows by Ascoli-Arzel\`a theorem, after having written the
integral expression of\; \blue{the evolution} \eqref{def:dynamics} and observed
$T^{u,\La,\p}_{q,t}(\vec \rho)-e^{-t}\rho_q$
is bounded with bounded gradient.

\vskip .5cm \noindent

 \item The decreasing of the free energy functional, follows by observing that:
 \begin{eqnarray*}
& &\frac{d}{dt}\cF_{\ga,u,\La}([\vec T^{u,\La,\p}_t(\vec\rho)] \; |\; \vec\rho_{\La^c})<0
\end{eqnarray*}
an explicit calculation gives in fact :
\begin{eqnarray*}
&&%\hskip1cm
\frac{d}{dt}\cF_{\ga,u,\La}([\vec T^{u,\La,\p}_t(\vec\rho)] \; |\; \vec\rho_{\La^c}=
\sum_q\int_{\La}dr~\bigg(-\cL_q^{u}(\vec T^{u,\La,\p}_t(\vec\rho))+\frac{1}{\beta}
(1+\ln T^{u,\La,\p}_{q,t}(\vec\rho))\bigg)
\\
& &\hskip6cm \cdot
 \bigg(-T^{u,\La,\p}_{q,t}(\vec\rho)+\fM^{(u,\p)}_q(\vec T^{u,\La,\p}_t(\vec\rho))\bigg)
\\
&&\hskip1cm=
\sum_q\int_{\La} dr~\frac{1}{\beta}\bigg(\ln\frac{T^{u,\La,\p}_{q,t}(\vec\rho)}{\fM^{(u,\p)}_q
(\vec T^{u,\La,\p}_t(\vec\rho))}
+1-\ln\sum_{q'} e^{\beta\cL^{u}_{q'}(\vec T^{u,\La,\p}_t(\vec\rho))})\bigg)
\\ & &\hskip6cm \cdot
\bigg(-T^{u,\La,\p}_{q,t}(\vec\rho)+\fM^{(u,\p)}_q(\vec T^{u,\La,\p}_t(\vec\rho))\bigg)
\end{eqnarray*}
\begin{eqnarray*}
&&\hskip1cm =
\frac{1}{\beta}\sum_q\int_{\La} dr~\bigg(\ln\frac{T^{u,\La,\p}_{q,t}(\vec\rho)}{\fM^{(u,\p)}_q(\vec T^{u,\La}_t(\vec\rho))}\bigg)
\bigg(-T^{u,\La,\p}_{q,t}(\vec\rho)+\fM^{(u,\p)}_q(\vec T^{u,\La,\p}_t(\vec\rho))\bigg)\\& &\hskip6cm+
\frac{K}{\beta}\sum_q\int_{\La} dr~%(1+\ln\sum_{q'} e^{\beta\cL^{u}_{q'}(\vec\rho)})
\bigg(- T^{u,\La,\p}_{q,t}(\vec\rho)+\fM^{(u,\p)}_q(\vec T^{u,\La,\p}_t(\vec\rho))\bigg)
 \end{eqnarray*}
 with $K=(1-\ln\sum_{q'} e^{\beta\cL^{u}_{q'}(\vec\rho)})$. By normalization condition,
 last term is null, while
 the first one is negative (in fact if the first factor  inside the integral is positive,
 the last one is negative and viceversa). We denote by:
 \begin{eqnarray*}
\cD^{u,\p}_{\La}(\vec \rho):=\frac{1}{\beta}\sum_i \bigg(\ln\frac{\rho_i}{\fM^{(u,\p)}_i(\vec\rho)}\bigg)
\bigg(-\rho_i+\fM^{(u,\p)}_i(\vec\rho)\bigg)
 \end{eqnarray*}

$\cD^{u,\p}_{\La}(\vec \rho)=0$  if and only if $\vec\rho$ satisfies the equation:
\begin{eqnarray*}
\rho_q=\fM^{(u,\p)}_q(\vec\rho) \hskip3cm \forall q=1,\dots,Q
\end{eqnarray*}
\blue{To conclude we need
a lower bound on $\vec T^{u,\La,\p}_t(\vec\rho)$ that assure that the dynamic is always well
defined for any time $t>0$.
(notice  that $\cD^{u,\p}_{\La}(\vec \rho) $ diverges if one of the coordinates
$\rho_q(t)$  becomes null.)}

\begin{lemma}
   \label{lem:bound}
Let $\vec\rho\in L^{\infty}(\Rr^{d},S_Q)$, $\La$ a Borel set in $\Rr^{d}$, $\vec\rho^{\La}(\cdot,t)
=\vec T^{u,\La,\p}_t(m)$
and $v_q^{0}:=\dis{\inf_{r'\in\La}}\rho_q^{\La}(r')$.
Then for all $r\in\La$ and $t\ge 0$:
       \begin{equation}
       \label{eq:lembound}
       \rho_q(r,t)\ge\big[ v_q^0-c_q(u,\beta)\big]e^{-t}+c_q(u,\beta)
       \end{equation}
where $c_q(u,\beta)=\frac{1}{e^{\beta u}}
\frac{e^{(1-u)\beta\rho_q^{\p}}}{\sum_{q'}e^{(1-u)\beta\rho_{q'}^{\p}}} \ge
\frac{1}{Q e^{\beta [u+(1-u)]}}\ge\frac{1}{Q e^{\beta }}$
\end{lemma}

\vskip .5cm \noindent
\begin{proof}
 For any $q=1,\dots,Q$, let  $v_q(t)$ the solutions of the Cauchy problems:
\begin{eqnarray}
\label{def:lemboundproof1}
\left\{\ba{lr}
\dis{\frac{dv_q(t)}{dt}=-v_q(t)+\frac{1}{e^{\beta u}}c_q(u,\beta)}
\\
\dis{\vec v(t)=\vec v^{0}}
\ea\right.
\end{eqnarray}
Let $w_q(r,t):=v_q(t)\Ii_{r\in\La}+\vec \rho^*(r,t) \Ii_{r\in \La^{c}}$,
Since $-w_q(r,t)+\frac{1}{e^{\beta u}}c_q(u,\beta)\le -w_q(r,t)+\fM^{(u,\p)}_q(\vec w)$ uniformly in
$r$ and
 $w_q\in [0,1]$.{By Gronwall Lemma }, $\rho_q(r,t)\ge w_q(r,t)$
 for any $r\in \Rr^d,t>0$, and it
 is strictly positive for any $t>s>0$.

\end{proof}
 By Lemma \ref{lem:bound}  for any  $s>0$, the functions
$[ T_{q,t}^{u,\La,\p}(\vec \rho)]_{\La}$, $t\ge s$ , for any $q\in [1,\dots, Q]$ are
 bounded away from $0$, so that:
 \begin{eqnarray*}
\cF_{\ga,u,\La}([\vec T_{t}^{u,\La,\p}(\vec \rho)]_{\La}| \vec\rho^{~*})-
\cF_{\ga,u,\La}([\vec T_{s}^{u,\La,\p}(\vec \rho)]_{\La}| \vec\rho^{~*})=
\int_s^{t}\cD^{u,\p}_\La(\vec T_{t'}^{u,\La,\p}(\vec \rho))dt'
 \end{eqnarray*}
 Since  $\cD^{u,\p}_\La(\vec T_{s}^{u,\La,\p}(\vec \rho))$ is monotone in $s$,
by  the Lebesgue dominated convergence theorem, the limit $s\to 0$ exists, and we get:

\begin{eqnarray*}
\cF_{\ga,u,\La}([\vec T_{t}^{u,\La,\p}(\vec \rho)]_{\La}| \vec\rho^{~*})-
\cF_{\ga,u,\La}(\vec \rho_{\La}| \vec\rho^{~*})=
\int_0^{t} \cD^{u,\p}_\La(\vec T_{t'}^{u,\La,\p}(\vec \rho))dt'
 \end{eqnarray*}

 \end{enumerate}
\vskip .5cm \noindent
We omit the proof of  the following theorem that follows by previous analysis:
\begin{thm}
   \label{thm:}
Let $\vec\rho\in \red{L^{\infty}(\Rr^{d} ,S_Q)}$ and $\La$ a  bounded, Borel set. Then, any limit
point of $\vec T_t^{\La}(\vec \rho)$ satisfies \eqref{eq:staz} and for any $\vec\rho^{~*}\in
\red{L^{\infty}(\La^{c} ,S_Q)}$ there is $\vec v\in \red{C^{1}(\La ,S_Q)}$,
\red{with $\nabla v_x$ bounded for any $x$ in $\La$} s.t.
       \begin{equation*}
     \cF_{\ga,u,\La}(\vec v|\vec\rho^{~*}_{\La^{c}})
     \le\cF_{\ga,u,\La}(\vec \rho^{~*}_{\La}|\vec\rho^{~*}_{\La^{c}})
       \end{equation*}
\end{thm}
\vskip .5cm \noindent


As a corollary of the Theorem \ref{thm:dimamica} we have the following result used in Subsection
\ref{subsection:smalldeviation}, and  \ref{section:largedeviation}
\begin{corol}
   \label{corol:dimamica}
There are positive constants $\om$ and $c'_\om$ so that for any
\red{$u\in [0,1]$} and any $\vec\rho^{~*}\in
M_{\La,\zeta,\ell}^{\p}$
there is $\vec\psi^{(u)}\in \cX_{\La,\vec\rho^{~*}}$
with the following properties:
\begin{eqnarray}
\ba{lr}
\vec\psi^{(u)}(r)=\vec\rho^{~*}(r) & \hskip3cm r\in\La^{c}\sqcup\delta^{\ga^{-1}}_{\ins}[\La]
\\
\vec \psi^{(u)}(r)=\vec \rho^{\p}(r)& r\in\La\setminus \delta_{\ins}^{\ell_{+,\ga}/4}[\La]
 \\
\cF_{\ga,u,\la}(\vec \psi^{(u)})\le \cF_{\ga,u,\la}(\vec\rho^{~*})+c_\om|\La|e^{-\om\ell_{+,\ga}/4}&
\ea
\end{eqnarray}

\end{corol}

\vskip .5cm \noindent
We conclude this appendix by giving the proof of the Lemma \ref{lem:dinam}

\vskip .5cm \noindent
\begin{proof}[Proof of Lemma \ref{lem:dinam}:]
\vskip .5cm \noindent
In order to prove the first statement, we define a $\cD^{(\ell_{-,\ga})}$- measurable approximation
of the interaction kernel as:
\begin{eqnarray}
J_\ga^{(\ell_{-,\ga})} (x,\lng y\rng ) = \frac{1}{|C^{(\ell_{-,\ga})}|}\int_{y'\in C_y^{(\ell_{-,\ga})}} J_\ga (x,y') dy'
\end{eqnarray}
We have for $\ell_{-,\ga}=\ga^{-1+\al}\ll \ga^{-1}$,
\begin{eqnarray}
|J_\ga^{(\ell_{-,\ga})} (x,\lng y\rng) - J_\ga (x,y)| &&\le
\sup_{y'\in C_y^{(\ell_{-,\ga})}}  |J_\ga^{(\ell_{-,\ga})} (x,\red{\lng y\rng}) - J_\ga (x,\red{y'})|\\
 &&\le \sqrt{d}~ \ell_{-,\ga} \ga^{d+1} \|\nabla J\|_\infty {\bf 1}_{\{|x-y|\}\le 2 \ga^{-1}}
\end{eqnarray}
Using this result, we can write for all $r$ and all $q$,
\begin{eqnarray*}
&&|\cL^{u}_q(\vec\rho)(r)-\rho_q^{~\p}| =
u \left| \int dr'~J_\ga(r,r')(\rho_q(r') - \rho^{\p}_q) \right|
\\ && \le
u \left| \int dr'~J^{(\ell_{-,\ga})}_\ga(r,\lng r'\rng)(\rho_q(r') - \rho^{\p}_q) \right| +
u \left| \int dr'~(J_\ga(r,r') - J^{(\ell_{-,\ga})}_\ga(r,\lng r'\rng))(\rho_q(r') -
\rho^{\p}_q) \right|
\\
&&\le
u \bigl( \sum_{j\in (\ell_{-,\ga}\Zz)^d} J^{(\ell_{-,\ga})}_\ga(r,\lng x \rng)
\int_{r'\in C_j^{(\ell_{-,\ga})}} |\rho_q(r') - \rho^{\p}_q| dr'
+ 2^d \sqrt{d} \|\nabla J\|_\infty \ga \ell_{-,\ga} \bigr)\\
&& \le u (1 + c_d \kappa_0 ) \zeta
\end{eqnarray*}
for $\ga^{\al}\le \kappa_0 \zeta $ and $c_d = 2^d \sqrt{d} \|\nabla J\|_\infty$.
Hence
\begin{eqnarray}
\sup_{r}\|\vL^{u}(\vec\rho)-\vec\rho^{~\p}\|_\star \le u (1 + c_d \kappa_0 ) \zeta \le 2  \zeta
\end{eqnarray}
for $\kappa_0$ small enough.

In order to prove \eqref{eq:lem-dinam-2},
we take $\zeta_0'$ small enough so that theorem \ref{thm:mflocalstability} holds
\blue{( for example $\zeta_0'<\left(\frac{1}{2\beta Q}\right)^2$)}
for all $\zeta<\zeta_0'$.
We get for any $r\in \La$
\begin{eqnarray}
\|\vec\fM^{(u,\p)}(\vec\rho)(r)-\vec\rho^{~\p}(r)\|_{\star}&=&
\|\vec g(\vL^{u}(\vec\rho))(r)-\vec g(\vec\rho^{~\p})(r)\|_{\star}\nn \\
&\le& (1-\frac{1}{2Q}) \|\vL^{u}(\vec\rho)(r)-\vec\rho^{~\p}(r)\|_\star \nn \\
&\le&  (1-\frac{1}{2Q}) u (1 + c_d \kappa_0 ) \zeta \le u (1 - \kappa_0 ) \zeta
\end{eqnarray}
having chosen  $\kappa_0$ so small that
\begin{eqnarray}
\kappa_0 \le \frac{1}{2Q(1+ c_d)}
\end{eqnarray}
and \eqref{eq:lem-dinam-2}  holds.
\end{proof}
%============================================================================================================


\vskip 2.5cm \noindent
\setcounter{equation}{0}
\section{Existence of the pressures $P^{\pm}_{\abs,\ga,\beta}$ of the abstract models }
\label{app:exist-pressure}

Let $\{\La_n\}$ a sequence of sets of side
$2^n\ell_{+,\ga}$ and
\begin{eqnarray*}
D_{\ga,\beta}(n):= \frac{\ln
Z^{\pm}_{\abs,\La_n\ga,\beta}(\vec\rho^{\pm}_{\beta})}{\beta|\La_n|}-\frac{\ln
Z^{\pm}_{\abs,\La_{n-1},\ga,\beta}(\vec\rho^{\pm}_{\beta})}{\beta|\La_{n-1}|}
\end{eqnarray*}

The proof of existence and continuity in $\beta$ o the
abstract pressures follows by the continuity in $\beta$ of
$D_{\ga,\beta}(n)$ and by proving that there is a constant
$\kappa_7$:
\begin{eqnarray}
\label{eq:dn} |D_{\ga,\beta}(n)|\le \kappa_7 2^{-n}
\end{eqnarray}

\vskip .5cm \noindent

\begin{proof}

\vskip .5cm \noindent
Decomposing $\La_n$ into cubes $\La_{n-1}(k)$, $k=1,\dots,2^d$,
 since the interaction energy  is bounded uniformly in $\vw$ and
 recalling \eqref{z4.11b}, we have:
\begin{eqnarray}
\label{eq:ab-1}
&&\ln Z^{\pm}_{\abs,\La_n\ga,\beta}(\vec\rho^{\pm}_{\beta})\ge
\\
&&\hskip1cm2^{d}\ln Z^{\pm}_{\abs,\La_{n-1}\ga,\beta}(\vec\rho^{\pm}_{\beta})-c\ga^{-1}
(2^n\ell_{+,\ga})^{d-1}-2d(2^{n-1})^{d-1}e^{-(\fK_{\ga}-2b)}\nn
\end{eqnarray}

where, denoting by
$[\delta\La_n^{0}]:=\sqcup_{k=1}^{2^d}
\delta_{in}^{\ell_{+,\ga}}[\La_{n-1}(k)] $, we have used the estimate
\begin{eqnarray*}
|\sum_{\Delta\sqsubset\La_n}U_{\Delta}(\vw)-\sum_k
\sum_{\Delta\sqsubset\La_{n-1}(k)}U_{\Delta}(\vw)|=
\sum_{\Delta: \Delta
\sqcap[\delta\La_n^{0}]\ne \emptyset}\|U_{\Delta}(\vw)\|_{\infty}
\le 2d(2^{n-1})^{d-1}e^{-(\fK_{\ga}-2b)}
\end{eqnarray*}

\vskip .5cm \noindent
\eqref{eq:ab-1} gives:
\begin{eqnarray*}
D_{\ga,\beta}(n)\ge -c2^{-n}
\end{eqnarray*}
the same arguments  give also the upper bound:
\begin{eqnarray*}
D_{\ga,\beta}(n)\le -c2^{-n}
\end{eqnarray*}
\end{proof}

%=======================================================================================

\vskip 2.5cm \noindent
\section{Existence of the pressure $P^{\pm}_{\abs,\ga;0}$  and surface correction.}
\label{app:meanfieldpressurelimit}
\setcounter{equation}{0}
In this appendix  we prove the following  Lemma:

\begin{lemma}
   \label{lem:0pressure}
There exists a constant $c_d$ such that for $\ga$ small enough
       \begin{eqnarray}
       \label{eq:lem0pressure}
|P^{\pm}_{\abs,\ga;0} + \phi^{\mf}_\beta(\rho^{\pm})|\le c_d \ga
\end{eqnarray}
where $P^{\pm}_{\abs,\ga;0}$ is defined in \eqref{def:H0} and \eqref{P-abs-0}, and
\begin{eqnarray}
       \label{eq:lem0correction}
R^{\pm}_{\abs,\La;0}=R^{\mf}_{\ga,\La}:=\frac{\beta}{2}\sumtwo{x\in \La}{y\in\La^c}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})
       \end{eqnarray}

\end{lemma}




\vskip .5cm \noindent
\begin{proof}
{\ }
\vskip .5cm \noindent
We denote by $C_0\equiv C_0^{\ell_{+,\ga}}$,
the cube of the partition $\cD^{\ell_{+,\ga}}$ that contains the point $0$,
\begin{eqnarray}
\ln  Z^{\pm}_{\abs,\beta,\La;0}&=&  \ln \{\sum_{\vw_{\La}} \prod_{x\in \La}
e^{\beta \big((\vw_x-\rho^{\pm})\cdot \vec\cL(\rho^{\pm})\big)}
\Ii_{\vw\in\cX^{\pm}_{C_0}}\}
-\beta  H_{\ga,\La}(\vec\rho^{\pm}_\La|\vec\rho^{\pm}_{\La^c})
\nn \\
& =&\ln \{\sum_{\vw_{\La}} \prod_{x\in \La}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}\Ii_{\vw\in\cX^{\pm}_{C_0}}\}
-\beta |\La|\fJ_\ga( \rho^{\pm}\cdot \vec\rho^{\pm})
-\beta H_{\ga,\La}(\vec\rho^{\pm}_\La|\vec\rho^{\pm}_{\La^c})
\nn \\
&=&\frac{|\La|}{|C_0|} \ln \{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}\Ii_{\vw\in\cX^{\pm}_{C_0}}\}
-\frac{\beta|\La|}{2}\fJ_\ga( \rho^{\pm}\cdot \vec\rho^{\pm})\label{P-abs-0a}
\\ \nn
& &\hskip7cm+\frac{\beta|}{2}\sumtwo{x\in\La}{y\in \La^c}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})
\end{eqnarray}
where we recall
$H_{\ga,\La}(\vec\rho^{\pm}_\La|\vec\rho^{\pm}_{\La^c})=-\frac{1}{2}\sumtwo{x\in\La}{y\in \La}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})-\sumtwo{x\in\La}{y\in \La^c}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})=-\frac{1}{2}\sumtwo{x\in\La}{y\in \Zz^{d}}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})-\frac{1}{2}\sumtwo{x\in\La}{y\in \La^c}J_\ga(x,y)
(\vec\rho^{\pm}\cdot\vec\rho^{\pm})$.
Then we get:

\begin{eqnarray}
P^{\pm}_{\abs,\ga,0}&:=&\lim_{\La\nearrow \infty}\frac{1}{\beta|\La|}\ln
 Z^{\pm}_{\abs,\beta,\La;0}=\lim_{\La\nearrow \infty}\frac{1}{\beta|\La|}\ln
\{\sum_{\vw_{C_0}} \prod_{x\in C_0} e^{-\beta\fH_{\ga,\La}^{\pm}(\vw)}
\Ii_{\vw\in\cX^{\pm}_{C_0}}\}
\nn \\\label{P-abs-0b}
&\equiv& \frac{1}{\beta|C_0|}
\ln \{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}\Ii_{\vw\in\cX^{\pm}_{C_0}}\}-
\frac{1}{2}\fJ_\ga
(\vec\rho^{\pm}\cdot \vec\rho^{\pm})
\end{eqnarray}

A comparison of \eqref{P-abs-0a} and \eqref{P-abs-0b} gives directly \eqref{eq:lem0correction}.

\vskip .5cm \noindent
We now prove \eqref{eq:lem0pressure}



Let consider the first term of \eqref{P-abs-0b}:
\begin{eqnarray*}
&&\frac{1}{\beta|C_0|}
\ln \{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}\Ii_{\vw\in\cX^{\pm}_{C_0}}\}
\hskip7cm\\
&&\hskip3cm=
\frac{1}{\beta|C_0|}
\ln G_{{\pm},\ga;0}[\Ii_{\vw\in\cX^{\pm}_{C_0}}]+\frac{1}{\beta|C_0|}\ln
\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}\\
&&\hskip3cm=
\frac{1}{\beta|C_0|}
\ln G_{{\pm},\ga;0}[\Ii_{\vw\in\cX^{\pm}_{C_0}}]+
  \frac{1}{\beta}\ln\sum_{i=1,Q}e^{\beta\fJ_\ga \rho^{\pm}_i}
\end{eqnarray*}
where $G_{{\pm},\ga;0}[\Ii_{\vw\in\cX^{\pm}_{C_0}}]$ is the  probability of the event
$\Ii_{\vw\in\cX^{\pm}_{C_0}}$ w.r.t. the Gibbs measure specified by:
\begin{eqnarray*}
\mu^{\pm}_{\ga;0}:=
\frac{e^{\beta\sum_x\big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}}{Z^{\pm}_{\ga;0}}
\end{eqnarray*}

Postponing the proof that the first term is negligible, as
\begin{eqnarray}
\label{app2:gpm}
%\lim_{\ga\to 0}
G_{{\pm},\ga;0}[\Ii_{\vw\in\cX^{\pm}_{C_0}}]>1-Ce^{-c\ga^{2a}|C^{\ell_{-,\ga}}|}
\end{eqnarray}
we consider the second term. We have
\begin{eqnarray*}
|\frac{1}{\beta}\ln\sum_{i=1,Q}e^{\beta\fJ_\ga \rho^{\pm}_i} -
\frac{1}{\beta}\ln\sum_{i=1,Q}e^{\beta\rho^{\pm}_i}|\le 2^d \sqrt{d} \|\nabla\cJ\|_\infty \ga
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{1}{\beta}\ln\sum_{i=1,Q}e^{\beta\rho^{\pm}_i}
&&=\frac{1}{\beta}\sum_j\rho^{\pm}_j\ln\sum_{i=1,Q}e^{\beta\rho^{\pm}_i}
\\
  &&=
  \frac{1}{\beta}\sum_j \rho^{\pm}_j\left[-\ln
  \rho^{\pm}_j+
  {\beta \rho^{\pm}_j}\right]
=-\phi^{\mf}_\beta(\vec\rho^{\pm})
\end{eqnarray*}
Going back to \eqref{P-abs-0b}, and using \eqref{app2:gpm}, we get

\begin{eqnarray*}
|P^{\pm}_{\abs,\ga,0} +\phi^{\mf}_\beta(\vec\rho^{\pm})| \le c_d \ga +Ce^{-c\ga^{2a}|C^{\ell_{-,\ga}}|}
\le c_d'\ga
\end{eqnarray*}
for $\ga$ small enough.
\vskip 1.5cm \noindent
To conclude we are then left with the proof of \eqref{app2:gpm}.
Since the derivation is similar (and simpler) to what is done in section \ref{section:largedeviation}
but with a different free energy functional, we just sketch the proof here. We consider the
one body functional $\cF_{\beta,C_0}^{(1)}$
on $C_0$,
\begin{eqnarray}
\cF_{\beta,C_0}^{(1)}(\vec\rho_{C_0})=-\frac{1}{2}\int_{C_0} (\vec\rho^\pm \cdot \vec\rho_{C_0}(x)) dx
 -\frac{1}{\beta} \int_{C_0} I(\vec\rho_{C_0}(x))dx
\end{eqnarray}
Using a result similar to \ref{thm:app} leads to an estimate valid for
$\ga$ small enough of $G_{{\pm},\ga;0}[\Ii_{\vw\notin\cX^{\pm}_{C_0}}]$ in terms of the
functional $\cF_{\beta,C_0}^{(1)}$, as
\begin{eqnarray*}
G_{{\pm},\ga;0}[\Ii_{\vw\notin\cX^{\pm}_{C_0}}]&=&
\frac{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}
\Ii_{\vw\notin\cX^{\pm}_{C_0^+}}
}{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \big(\vw_x\cdot \vec\cL(\rho^{\pm})\big)}}
\\
&=& \frac{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \fJ_\ga\big(\vw_x\cdot \vec\rho^{\pm}\big)}
\Ii_{\vw\notin\cX^{\pm}_{C_0}}
}{\sum_{\vw_{C_0}} \prod_{x\in C_0}
e^{\beta \fJ_\ga\big(\vw_x\cdot \vec\rho^{\pm}\big)}}
\\
&\approx&
\exp\bigl\{-\beta\bigl(\inf_{\vec\rho_{C_0}\not\in\cX^{\pm}_{C_0}}\cF_{\beta,C_0}^{(1)}(\vec\rho_{C_0})
-\inf_{\vec\rho_{C_0}}\cF_{\beta,C_0}^{(1)}(\vec\rho_{C_0})\bigr)\bigr\}
\end{eqnarray*}
Following the analysis of section \ref{section:largedeviation}, one gets a similar estimate for
the large deviation cost as
\begin{eqnarray}
\inf_{\vec\rho_{C_0}\not\in\cX^{\pm}_{C_0}}\cF_{\beta,C_0}^{(1)}(\vec\rho_{C_0})
-\inf_{\vec\rho_{C_0}}\cF_{\beta,C_0}^{(1)}(\vec\rho_{C_0})\ge c\ga^{2a}|C^{\ell_{\ga,-}}|
\end{eqnarray}
for some constant $c$ and $\ga$ small enough. The estimate \eqref{app2:gpm} then follows.
\end{proof}

%=======================================================================================


\setcounter{equation}{0}
\section{Proof of \eqref{eq:u1-a}-\eqref{eq:u1-b}}%------------------------------------------------------------------
\label{app:u1}


In this appendix we prove that for any $\psi_B\in\cB^0$ there is
$\vec\psi^{*}_B:$
\begin{eqnarray}
\label{eq:u1-aa}
\vec \psi^*_B=
  \begin{cases}
    \vec \psi_B & \text{on}~~\delta_{\ins}^{\ell_{+,\ga}/4}[B], \\
    \vec\rho^{\pm} & \text{elsewhere}.
  \end{cases}
\end{eqnarray}
so that:
\begin{eqnarray}
\label{eq:u1-ab}
\cF^{\eff}_{\ga,B,u}(\vec\psi_B|\vw^{(\ell_0)}_{B^c})\ge
\cF^{\eff}_{\ga,B,u}(\vec\psi^*_B|\vw^{(\ell_0)}_{B^c})+c\ga^{1/4}(\ga^{1/8}|A|)
\end{eqnarray}


\vskip .5cm \noindent
The proof is analogous to the case  of the Ising model widely analyzed in \cite{errico-leip}
to which we refer for details. A sketchy version is reported here for completeness.

\vskip .5cm \noindent
Let $\Si$ as in \eqref{def: Si},
and
\begin{equation*}
\Delta=
A\sqcup\delta_{\out}^{\ell_{+,\ga}/4}[A]
\end{equation*}

Then, recalling that the interaction term appearing in the excess of free energy
is always positive, we get a lower bound by neglecting the interaction between $\Delta$ and  $B\setminus \Delta$:

\begin{eqnarray*}
\cF^{\eff}_{\ga,B,u}(\vec\psi_B|\vw^{(\ell_0)}_{B^c})\ge
\cF^{\eff}_{\ga,B\setminus \Delta,u}(\vec\psi_{B\setminus \Delta}|\vw^{(\ell_0)}_{B^c})+
 \cF^{\eff}_{\ga,\Delta, u}(\vec\psi_\Delta)
\end{eqnarray*}

where, for any sets $D, F\sqsubset \Rr^{d}$:

\begin{eqnarray*}
& &\cF^{\eff}_{\ga,D, u}(\vec\psi_D):= \int_{D} \Phi^{\eff,\pm}_u(\vec\psi_D)
+\frac{u}{4}\int_{D\times D}J_\ga(r,r')[\vec\psi_D(r)-\vec\psi_D(r')]^{2}
\\
& & \cF^{\eff}_{\ga,D, u}(\vec\psi_D| \vw_F):=
\int_{D} \Phi^{\eff,\pm}_u(\vec\psi_D)
+\frac{u}{4}\int_{D\times D}J_\ga(r,r')[\vec\psi_D(r)-\vec\psi_D(r')]^{2}
\\
& &\hskip7cm+\frac{u}{2}\int_{D\times F}
J_\ga(r,r')[\vec\psi_D(r)-\vw_F(r')]^{2}
\end{eqnarray*}

\vskip .5cm \noindent
For any $\psi_{B}\in \cB^{0}$ (\eqref{def: B0}), since
$\vec\psi_B^*(r)\equiv\vec\psi_B(r)$ on $r\in \delta_{\ins}^{\ell_{+,\ga}/4}[B]\sqcup \Si$,
and $\vec\psi_B(r)=\vec\rho^{\pm}$ on $r\in \Si$, we have that:
\begin{eqnarray*}
\cF^{\eff}_{\ga,B\setminus \Delta,u}
(\vec\psi_{B\setminus \Delta}|\vw^{(\ell_0)}_{B^c})=\cF^{\eff}_{\ga,B,u}
(\vec\psi^*_{B}|\vw^{(\ell_0)}_{B^c})
\end{eqnarray*}
in fact the distance between the sets
$\delta_{\ins}^{\ell_{+,\ga}/4}[B]$ and $\Delta$ is larger than $\ga^{-1}$ and
\linebreak ${\cF_{\ga,\Delta,u}(\vec\rho^{\pm})\equiv 0}$:


\vskip .5cm \noindent
Hence we need to prove that for any $\vec\psi_B\in\cB^{0}$:
\begin{eqnarray}
\label{eq:boundDelta}
 \cF^{\eff}_{\ga,\Delta,u}(\vec\psi_\Delta)\ge c\ga^{1/4}\left(\ga^{1/8}|A|\right)
\end{eqnarray}
\vskip .5cm \noindent
It is convenient here to fix a specific color $\p$ instead of distinguish
only disordered and ordered configurations.%$\pm$ suffice.


 We then denote by
\begin{eqnarray*}
\cS^{\p}:=\{r\in\hat A:\| \vec\psi_{B}(r)-\vec\rho^{\p}\|_{\star}\ge \ga^{1/8} \}
\end{eqnarray*}
that can be written as the  sum of two sets $\cS_0$,  $\cS_1$:

\begin{eqnarray*}
\cS_0&:=&\{r\in \cS: \| \vec\psi_{B}(r)-\vec\rho^{\p}\|_{\star}\ge \ga^{1/8} ~~\forall \q=-1,1,\dots,Q\}
\\
\cS_1&:=&\{r\in \cS:  \exists \q\ne {\p}: \|\vec\psi_B(r)-\vec\rho^{\q}\|_{\star}\le \ga^{1/8} \}
\end{eqnarray*}


Recalling the definition of $\Phi^{\eff,\p}_u(\vec v)$ in \eqref{def:Fmfu},
 and \eqref{def:F-eff-u}-\eqref{def:fmfu},
we will prove that
there are positive constants, $c_0, c_1$, $c_2$, so that:
\begin{eqnarray}
\label{eq:s0}
& &\int_{\cS_0}    \Phi^{\eff,\p}_u(\vec\psi(r))\ge c_0\ga^{1/4}|\cS_0|
\\
\label{eq:s1}
& &\int_{\cS_1}   \Phi^{\eff,\p}_u(\vec\psi(r))\ge (c_1u+c_2(1-u))|\cS_1|%(\rho_A-\rho_B)^2
\end{eqnarray}

\vskip .5cm \noindent
\eqref{eq:s0} and \eqref{eq:s1} prove \eqref{eq:boundDelta}


\vskip 1.5cm \noindent
\begin{proof}[Proof of \eqref{eq:s0}] %\red{for any $\beta$?????}
\eqref{eq:s0}  follows from the bound immediately obtained by the  explicit expression of
$\Phi^{\eff,\p}_u(\vec v)$: %There is a constant $c_0$:

\begin{eqnarray*}
\inf_{u\in(0,1)}\inf_{\{\vec v:\|\vec v-\rho^{\q}\|_{\star}\ge \ga^{1/8} \forall \q\}}
\Phi^{\eff,\p}_u(\vec v)\ge c_0\ga^{1/4}
\end{eqnarray*}
$c_0$ a suitable constant
\end{proof}


\begin{proof}[Proof of \eqref{eq:s1}]
Suppose $\|\vec v-\vec\rho^{\q}\|_{\star}\le \ga^{1/8}$ for some  $\q\ne {\p}$.
We will prove separately two bounds:

\begin{eqnarray}
\label{eq:s1-a}
& &\int_{\cS_1}   \Phi^{\eff,\p}_u(\vec\psi(r))\ge 2c_2(1-u)|\cS_1|
\\
\label{eq:s1-b}
& &\int_{\cS_1}   \Phi^{\eff,\p}_u(\vec\psi(r))\ge 2 c_1 u|\cS_1|
\end{eqnarray}
that together give  \eqref{eq:s1}


Proof of \eqref{eq:s1-a}
\begin{eqnarray}
\label{eq:bound-s1-a} \Phi^{\eff,\p}_u(\vec v)\ge
(1-u)\bigg[-\vec\rho^{\p} \cdot(\vec
v-\vec\rho^{\p})+\frac{1}{\beta} \left[\vec v\ln\vec v-
\vec\rho^{\p}\ln\vec\rho^{\p}\right]\bigg]\ge 0
\end{eqnarray}


Since  $\vec\rho^{\p}$ is a  solution of the mean field equations
$\vec\rho^{\p}=\frac{ e^{\beta\vec\rho^{\p}}}{\sum_i e^{\beta\rho_i^{\p}}}$,
it satisfies:
\begin{eqnarray*}
\beta\rho_i^{\p}=\ln\rho_i^{\p}-\ln C
\end{eqnarray*}
with $C= \sum_i e^{\beta\rho_i^{\p}}$
and the square parenthesis in  r.h.s. of
\eqref{eq:bound-s1-a}, can be rewritten as:

\begin{eqnarray*}
&&\bigg[-\vec\rho^{\p} \cdot(\vec
v-\vec\rho^{\p})+\frac{1}{\beta} \left[\vec v\ln\vec v-
\vec\rho^{\p}\ln\vec\rho^{\p}\right]\bigg]
\\
&&\hskip1cm=-\frac{1}{\beta}\sum_i
(v_i-\rho_i^{\p})\ln\rho_i^p+\frac{1}{\beta}\sum_i
(v_i-\rho_i^{\p})\ln C^{-1}+\frac{1}{\beta} \left[\vec
v\ln\vec v- \vec\rho^{\p}\ln\vec\rho^{\p}\right]
\\
&&\hskip1cm=-\frac{1}{\beta}\sum_i
(v_i-\rho_i^{\p})\ln\rho_i^p+\frac{1}{\beta} \left[\vec
v\ln\vec v- \vec\rho^{\p}\ln\vec\rho^{\p}\right]
\end{eqnarray*}
where in the last equality we used the fact that $\sum_i
v_i=\sum_i\rho_i^{\p}=1$. We then have:

\begin{eqnarray*}
\Phi^{\eff,\p}_u(\vec v)\ge \frac{(1-u)}{\beta}~\vec v\ln
\frac{\vec v}{\vec\rho^{\p}}
\end{eqnarray*}
and by Kullback-Leibler inequality:

\begin{eqnarray*}
\Phi^{\eff,\p}_u(\vec v)&\ge &\frac{(1-u)}{2\beta}(\vec v-\vec\rho^{\p})^{2}
\\
&\ge&\frac{(1-u)}{2\beta}
\left[(\vec \rho^{\q}-\vec\rho^{\p})^{2}-\ga^{1/4}\right]
\end{eqnarray*}
We consider separately  the case when $\q=-1$, $\p>0$ (or viceversa)
and the case where both $\q,\p$ are positive.

In the first case:
\red{\begin{eqnarray*}
\Phi^{\eff,\p}_u(\vec v)
&\ge&\frac{(1-u)}{2\beta}\left[\left(\rho_A-\frac{1}{Q}\right)^{2}
+\left(\rho_B-\frac{1}{Q}\right)^{2}(Q-1)-\ga^{1/4}\right]
\\
&\ge&\frac{(1-u)}{2\beta}\left[\frac{Q(1-2/Q)^{2}}{(Q-1)}-\ga^{1/4}\right]
\end{eqnarray*}}
If both $\q,\p$ are positive
\begin{eqnarray*}
\Phi^{\eff,\p}_u(\vec v)
&\ge&\frac{(1-u)}{2\beta}\left[2\left(\rho_A-\rho_B\right)^{2}-\ga^{1/4}\right]
\\
&\ge&
\frac{(1-u)}{2\beta}\left[\left(\frac{Q(1-2/Q)}{(Q-1)}\right)^{2}-\ga^{1/4}\right]
\end{eqnarray*}

Finally, for $\ga$ small enough:
\begin{eqnarray*}
\Phi^{\eff,\p}_u(\vec v)
&\ge&\frac{(1-u)}{20\beta}
\end{eqnarray*}
We now prove \eqref{eq:s1-b}. Let $r\in \cS_1$
\begin{eqnarray*}
\int~~dr'
J_\ga(r,r')\left(\vec\psi_B(r)-\vec\psi_B(r')\right)^{2}
\ge\int~~dr'
J^{(\ell_{-,\ga})}_\ga(r,\lng r' \rng)\left(\vec\psi_B(r)-\vec\psi_B(r')\right)^{2}
-c\ga\ell_{-,\ga}
\end{eqnarray*}

where
\begin{eqnarray*}
J^{(\ell_{-,\ga})}_\ga(r,\lng r' \rng):=\frac{1}{|C^{\ell_{-,\ga}}|}
\int_{C_{r'}^{\ell_{-,\ga}}}
J_\ga(r,r'') dr''
\end{eqnarray*}
and it is constant on the cubes of the partition
$\cD^{\ell_{-,\ga}}$. By Cauchy-Schwartz inequality:%,
\begin{eqnarray*}
&&\frac{1}{|C^{\ell_{-,\ga}}|}\int_{C_{r'}^{\ell_{-,\ga}}}~~dr'
J^{(\ell_{-,\ga})}_\ga(r,\lng r'\rng)\left(\vec\psi_B(r)-\vec\psi_B(r')\right)^{2}
\\
&&\hskip1cm\ge
J^{(\ell_{-,\ga})}_\ga(r,\lng r' \rng)\left(\vec\psi_B(r)-\frac{1}{|C^{\ell_{-,\ga}}|}
\int_{C_{r'}^{\ell_{-,\ga}}}\vec\psi_B(r'')~~dr''\right)^{2}
\\
&&\hskip1cm\ge J^{(\ell_{-,\ga})}_\ga(r,\lng r' \rng)
\bigg(\left(\vec\rho^{\q}-\vec\rho^{\p}\right)^2-\ga^{1/4}-\ga^{2a}\bigg)
\end{eqnarray*}
if both $\q, \p>0$,
$\left(\vec\rho^{\q}-\vec\rho^{\p}\right)^2=2(\rho_A-\rho_B)^2=2
\left(\frac{Q(1-{2}/{Q})}{Q-1}\right)^2$. While if $\q$ or
$\p$ is equal to $-1$:
$\left(\vec\rho^{\q}-\vec\rho^{\p}\right)^2=2(1-2/Q)^{2}$
Then:
\begin{eqnarray*}
\int~~dr'
J_\ga(r,r')\left(\vec\psi_B(r)-\vec\psi_B(r')\right)^{2}\ge
\frac{1}{5}-c\ga\ell_{-,\ga}-\ga^{1/4}-\ga^{2a}
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{u}{4}\int_{\cS_1}~~dr~dr'
J_\ga(r,r')\left(\vec\psi_B(r)-\vec\psi_B(r')\right)^{2}\ge
\left(\frac{1}{30}\right)u|\cS_1|
\end{eqnarray*}
\end{proof}




%===================================================================================
\bibliographystyle{amsalpha}
\begin{thebibliography}{A}

\bibitem{ACCN} M. Aizenman, J. T. Chayes, L Chayes, C. M. Newman:
{\it Discontinuity of the magnetization in one-dimensional
$\frac{1}{|x-y|^2}$ Ising and Potts models},
{ J. Stat. Phys.} {\bf 50}  1-40, (1988)

\vskip .5cm
\bibitem{bkmp1}F. Baffioni, T. Kuna, I. Merola, E. Presutti:
{\it A liquid vapor  phase transition in quantum statistical
mechanics},
{ Submitted to Memoirs AMS}~(2004)

\vskip.5cm
\bibitem{bkmp2}F. Baffioni, T. Kuna, I. Merola, E. Presutti:
{\it A relativized Dobrushin uniqueness condition and
applications to Pirogov-Sinai models},
{ Submitted to Memoirs AMS}~(2004)

\vskip .5cm
\bibitem{baxter73} R. J. Baxter:
{\it Potts model at the critical temperature},
{ J. Phys. C }{\bf 6}, L445 (1973) \vskip .5cm

\vskip .5cm
\bibitem{baxter78} R. J. Baxter, H. N. V. Temperley, S. E. Ashley:
{\it Triangular Potts model at its transition temperature, and related models},
{ Proc. Roy. Soc. London, Ser. A} {\bf 358}, 535 (1978)

\vskip .5cm
\bibitem{baxter} R. J. Baxter:
 { Exactly solved sodels in statistical mechanics},
Academic Press, London (1982).

\vskip .5cm
\bibitem{Biskup-Chayes} M.Biskup, L. Chayes:
 {\it Rigourous analysis of discontinous phase transitions via mean field bounds},
 { Commun. Math. Phys.} {\bf 238}, 53--93 (2003)

\vskip .5cm
\bibitem{b-chays-c} M.Biskup, L. Chayes, N.Crawford:
{\it Mean-field driven first order phase transitions in systems with long-range interactions},
{ Submitted to J. Stat. Phys.}(2005)

\vskip .5cm
\bibitem{bollobas} B. Bollobas, G. Grimmett, S.Janson:
{\it The random-cluster model on the complete graph},
{ Probab. Theory Related Fields} {\bf 104}, 283--317 (1996)


\vskip .5cm \bibitem{bmpz} A. Bovier, I. Merola, E. Presutti, M. Zahradn\`ik:
{\it On the Gibbs phase rule in the Pirogov-Sinai regime},
{ J. Stat. Phys.} {\bf 114}, 1235--1267 (2004)

\vskip .5cm
 \bibitem{BZ1} A. Bovier, M. Zahradn\`ik:
{\it The low-temperature phase of Kac-Ising models},
{ J. Stat. Phys.} {\bf 87}, 311--332 (1997)

\vskip .5cm
\bibitem{BZ2} A. Bovier,M. Zahradn\`ik:
{\it Cluster expansions and Pirogov-Sinai theory for long range spin systems},
{ Markov Process and Related Fields} {\bf 8}, 443--478 (2002)

\vskip .5cm
\bibitem{bkls} J. Bricmont, H. Kesten, J. L. Lebowitz, R. H. Schonmann:
{\it A note in the Ising model in high dimensions},
{ Comm. Math. Phys.} {\bf 122}, 597--607 (1989)

\vskip .5cm
\bibitem{CP} M. Cassandro, E. Presutti:
{\it Phase transitions in Ising systems with long but finite range},
{ Markov Processes and Related Fields} {\bf 2}, 241--262 (1996)

\vskip .5cm
\bibitem{CP01} R. Cerf, A. Pisztora:
{\it Phase coexistence in Ising, Potts and percolation models},
{ Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques}
{\bf 37}, 643-724 (2001)

\vskip .5cm
\bibitem{dob} R. L. Dobrushin:
{\it Prescribing a system of random variables by conditional distributions},
{ Th. Prob. Appl.} {\bf 15}, 458--486 (1970)

\vskip .5cm
\bibitem {D1} R. L.Dobrushin:
{\it Existence of phase transition in two and three dimensional Ising models},
{ Th. Prob. Appl.} {\bf 10}, 193-313 (1965)

\vskip .5cm
\bibitem {D3} R. L. Dobrushin:
{\it Estimates of Semi-invariants for the Ising Model at Low Temperatures},
{ Topics in statistical and theoretical physics}, { Amer. Math. Soc. Transl.}
{\bf 177}, 59--81 (1996)

\vskip .5cm
\bibitem {D4} R. L. Dobrushin, E. A. Pecherski:
{\it A criterion of the uniqueness of Gibbsian field in the non compact case},
 { Lecture Notes in Mathematics} {\bf 1021}, 97--110 (1982)

\vskip .5cm
\bibitem{ds2} R. L. Dobrushin, S. B. Shlosman:
{\it Completely analytical interactions: constructive description},
{ J. Stat. Phys.} {\bf 46}, 983--1014 (1987)

\vskip .5cm
\bibitem{FK} C. M. Fortuin, P. W. Kasteleyn:
{\it On the random cluster model I: introduction and relation to other models},
{ Physica} {\bf 57}, 536--564 (1972)

\vskip .5cm
\bibitem{Grimmett} G. R. Grimmett:
{\it The stochastic random cluster process and the uniqueness of random cluster measures},
{ Ann. Prob.} {\bf 23}, 1461--1510 (1995).

\vskip .5cm
\bibitem{Grimm} G. R. Grimmett:
{ Percolation},
 Springer-Verlag, Berlin (1989)

\vskip .5cm
\bibitem{Grimm-a} G. R. Grimmett:
{\it The random-cluster model},
{ Probability, Statistics and Optimization},
F.P.Kelly ed., John Wiley \& Sons, Chichester, 49--63 (1994)

\vskip .5cm
\bibitem{Grimm-b} G. R. Grimmett: {\it Percolative problems},
{ Probability and phase transition}, G.R. Grimmett ed., Kluwer, Dordrecht, 69--86 (1994)

\vskip .5cm

\bibitem{gross} L. Gross: {\it Decay of correlations in classical
lattice models at hight temperature}, { Commun.Math.Phys.}
{\bf 68} 9--27 (1979).

\vskip .5cm
\bibitem {KUH} M. Kac, G. Uhlenbeck and P. C. Hemmer:
{\it On the Van der Waals Theory of Vapor-Liquid equilibrium}
{ J. Mat. Phys.}{\bf 4}, 216--228, 229--247 (1963); { J.
Mat. Phys.}{\bf 5}, 60--74 (1964)

\vskip .5cm
\bibitem{KS} H. Kesten, R. H. Schonmann:
 {\it Behavior in large dimension of the Potts and Heisenberg models},
{ Reviews in Math. Phys.} {\bf 1}, 147--182 (1990)

\vskip .5cm
\bibitem{KotS} R. Koteck\'y, S. B. Shlosman:
{\it First order phase transition in large entropy lattice models},
{ Commun. Math. Phys.} {\bf 83}, 493--515 (1982)

\vskip .5cm
\bibitem{KP} R. Koteck\'y, D. Preis:  {\it Cluster
expansion for abstract polymer models} { Comm. Math.
Phys.}, {\bf 103}, 491--498 (1986)

\vskip .5cm
\bibitem{holl}
F. den Hollander, J. E. Steif: {\it Mixing properties of the
generalized $T,T\sp {-1}$-process}, { J. Anal. Math.}
{\bf 72} 165--202  (1997)

\vskip .5cm
\bibitem{LM} J. L. Lebowitz, A. Mazel:~
{\it On the uniqueness of Gibbs states in the Pigorov-Sinai
theory}, { Comm. Math. Phys.} {\bf 189}, 311--321
(1997)

\vskip .5cm
\bibitem{lmp} J. L. Lebowitz, A. E. Mazel and E. Presutti: {\it Liquid-vapour phase
transitions for systems with finite range interactions},
{ J. Statist. Phys.} {\bf 94}, 955--1025 (1999)
\vskip .5cm

\bibitem {LMP2}
J. L. Lebowitz, A. E. Mazel, E. Presutti:~ {\it Rigorous proof of a
liquid-vapour phase transition in a continuum particle
system} { Phys. Rev. Letters} {\bf 80}, 4701 (1998)

\vskip .5cm

\bibitem {LP} J. L. Lebowitz and O. Penrose:
{\it Rigorous treatment of the Van der Waals Maxwell theory of
the liquid-vapor transition}  { J. Mat. Phys.} {\bf 7},
98--113 (1966)

\vskip .5cm
\bibitem{ps} S.A. Pirogov, Ya. G. Sinai: (i) { Teor. Mat. Fiz.}
{\bf 25}, 358--369, (1975), in Russian; English
translation: {\it Phase diagrams of classical lattice
systems}, {Theor. Math. Phys.} {\bf 25} 1185--1192, (1975).
(ii) { Teor. Mat. Fiz.} {\bf 26}, 61--76, (1976), in Russian;
English translation: {\it Phase diagrams of classical
lattice systems. Continuation}, {Theor. Math. Phys.} {\bf
26} 39--49, (1976)

\vskip .5cm
\bibitem{Pi} A. Pisztora :
  {\it Surface order large deviations of Ising, Potts and percolation models},
{ Probab. Theory Related Fields} {\bf 104}, 427--466 (1996)

\vskip .5cm

\bibitem{Pisztora2} A. Pisztora :
 {\it Lectures at the IHP},
 Paris, June (1998) (unpublished)


\vskip .5cm
\bibitem{Potts} R. Potts:
{\it Some generalyzed order-disorder transformations}
{ Proc. Camb. Phil. Soc.} {\bf48}, 106--109 (1952)
\vskip .5cm

\bibitem{errico-leip} E. Presutti: {\it Scaling limits in statistical  mechanics \&
microstructures in continuum mechanics.} ~{ In preparation} (2005)


\vskip .5cm

\bibitem{RD} S. Reynal, H. T. Diep:~
{\it Reexamination of the long-range Potts model: A multicanonical approach}
{ Phys. Rev E}{\bf 69} 026109 (2004)

\vskip .5cm

\bibitem{Sokal} A. D. Sokal: {\it Chromatic polynomials, Potts models and all that},
{ Physica A279}, 324--332 (2000)

\vskip .5cm
\bibitem{suzuki} M. Suzuki, M. E. Fisher:
{\it Zeros of the partition function for the Heisenberg, ferroelectric and general Ising models}
 { J. Math. Phys.} {\bf12}, 235--246 (1971)

\vskip .5cm
\bibitem{Tutte} W. T.  Tutte:
{\it On dichromatic polynomials},
{ J. Comb. Theory} {\bf 2}, 301--320 (1967)

\vskip .5cm

\bibitem{Wu} F. Y. Wu:~ {\it The Potts model}, { R. Mod. Phys.} {\bf 54}, 235--268 (1982)


\vskip .5cm

\bibitem{Z}
 M. Zahradn\`\i k: {\it A short course on the Pirogov-Sinai theory},
 { Rend. Mat. Appl} {\bf 18}, 411--486 (1998)

\end{thebibliography}
\end{document}
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