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\begin{document}
\begin{center}
{\Large \bf Exact large deviation bounds up to $T_c$ for the Ising\\ model
in two dimensions
}
%\footnote{To appear in the JSP}
\vskip 0.2in
{\em Dmitry Ioffe\\
%\footnote{Partially supported by NSF, under grant DMS 9112654}
Departement of Mathematics, Northwestern University\\
2033 Sheridan Rd., Evanston, IL 60208\\
e-mail: ioffe@math.nwu.edu
}
\end{center}
\begin{abstract}
We prove an upper large deviation bound for the block spin magnetization
in the 2D Ising model in the phase coexistence region. The precise rate
given by the Wulff construction is shown to hold true
for all $\beta >\ \beta_c$. Combined with the lower bounds derived in [I]
those results yield an exact second order large deviation theory up to
the critical temperature.
\vskip 0.1in
{\em Key words and phrases:} Large deviations, Ising model, Wulff construction,
FK percolation.
\end{abstract}
\vskip 0.1in
\vskip 0.1in
\section{Introduction.}
This work is a continuation of [I], where we derived precise lower large
deviation bounds for the block spin magnetization in the 2D Ising model in
the phase coexistence region up to the critical temperature. We, thereby,
refer to the latter paper as well as for the basic manuscripts [DKS] and [P]
for a more comprehensive discussion of the underlying problem. To set up
notations let ${\bf Z}^2$ be a 2D integer lattice, $d$ a lattice distance;
$d((x_1 , x_2),(y_1 ,y_2 ))=max\{ |x_1 - y_1 |, |y_1 - y_2 |\}\ $ and ${\bf
B}_N (x)$ a
lattice box ; $ {\bf B}_N (x) = \{y\in {\bf Z^2} : d(y,x)\leq N\}$. Also,
let $\Lambda (L) \subset {\bf Z^2}$ be a square
box of volume $L^2$ and let $P_{\Lambda ,+}^{\beta}$ to denote the Gibbs
measure
on $\Lambda$ with plus boundary conditions at inverse temperature $\beta$,
that is the Hamiltonian $H_{\Lambda ,+}$ of a spin configuration $\sigma$ on
${\{-1,1\}}^{\Lambda}$ is given by
$$- H_{\Lambda ,+}(\sigma )\ =\ \frac{1}{2}\sum_{\stackrel{}
{x,y\in\Lambda}}
\sigma_x\sigma_y\
+\ \sum_{\stackrel{}{x\in\Lambda ,y\in\Lambda^{c}}}\ \sigma_x ,$$
where the summation is over all (ordered) pairs of
nearest neighbours $x$ and $y$. The
probability distribution of $\sigma\in {\{-1,1\}}^{\Lambda}$ is defined, then,
via
$$
P_{\Lambda ,+}^{\beta}
(\sigma)\ =\ \frac{1}{Z_{\Lambda ,+}^{\beta}}e^{-\beta
H_{\Lambda ,+}(\sigma )}\ ,$$
where $Z_{\Lambda ,+}^{\beta}$ is the corresponding partition function.
In a similar fashion one can define $P_{\Lambda ,-}^{\beta}$. When
$L\to\infty$ the measures $P_{\Lambda, +}^{\beta}$ and $P_{\Lambda
,-}^{\beta}$ weakly converge to limits $P_{+}^{\beta}$ and $P_{-}^{\beta}$
respectively, both measures being ergodic on $\{1, -1\}^{{\bf Z}^2}$ [Pr]. Then
the critical value $\beta_c$, such that
$$ P_{+}^{\beta}= P_{-}^{\beta}\ for \ \beta\leq{\beta}_c\ \ and\ \
P_{+}^{\beta}\neq P_{-}^{\beta}\ for\ \beta > {\beta}_c $$
is well defined. In our particular case $\beta_c\in (0,\infty)$ and two
alternative definitinions of $\beta_c$ are the Cramer-Wannier relation
\begin{equation}
tanh(\beta_c )\ =\ e^{-2\beta_c }
\end{equation}
and the order parameter relation
\begin{equation}
\beta\ >\ \beta_c\ \ iff \ \ E_{+}^{\beta}\sigma_0 \ =\ m^{*}\ >\ 0 ,
\end{equation}
where $m^*$ is a spontaneous magnetization.
Define now the block-spin magnetization
$X_{\Lambda}$ on $\Lambda$ as
$$
X_{\Lambda}\ =\ \frac{1}{|\Lambda |} \sum_{x\in\Lambda } \sigma_x \ ,
$$
where $\sigma_x = \pm 1$ is the value of the spin at the site $x\in\Lambda$.
Since $P_{+}^{\beta}$ is ergodic, $X_{\Lambda}$ converges to $m^*$
$P_{+}^{\beta}$-a.s.. The purpose of this article is to derive a
corresponding large deviation theorem. More precisely we are going to find
the main term in the logarithmic asymptotics of
\begin{equation}
P_{\Lambda , +}^{\beta}\{X_{\Lambda}\ \in\ [a,b]\} .
\end{equation}
Such large deviations results have a very different flavour depending on
whether $[a,b]\cap [-m^* ,m^* ]$ is empty or not. In the former case
$(1.3)$ decays exponentially with the volume $|\Lambda |$ and the precise
nontrivial answer is provided within the framework of a more general
theory developed in [FO] and [O]. On the other hand, points in $[-m^* ,m^*
]$ correspond to Gibbs states which have zero specific relative entropy
with respect to $P_{+}^{\beta}$. Consequently the exponential decay of
$(1.3)$ in this region is at most of the surface order. It turns out that
the main mechanism responsible for large deviations
in the phase coexsistence region $[-m^*
,m^*]$ is that of the phase separation. Schonmann [S1] was the first to
derive nontrivial surface order bounds for large values of $\beta$, which
were subsequently extended to all $\beta >\beta_c $ in [CCS]. Those bounds
were, however, imprecise. Precise bounds, their relation to the phase
separation and many more remarkable results were recently obtained in the
extensive works of [DKS] and [P]. Both works, however, relied on the method
of cluster expansions. Consequently their results are valid only for
large values of $\beta$, although being almost exhaustive when they apply.
A weak version of their theorem states that for $m\in [-m^* ,m^* ]$
which is sufficiently close to $m^*$ and for $\beta >\beta_c$ large
enough,
\begin{equation}
\lim_{L\to\infty} \frac{1}{L}\log P_{\Lambda ,+}^{\beta}
\{X_{\Lambda}\ \leq\ m\}
\ =\ -2(\lambda_F\alpha (m))^{1/2}\ ,
\end{equation}
where
$\alpha (m)\ =\ (m^* -m)/(2m^* )$ and $\lambda_F$ is the volume
of the (unnormalized) Wulff shape, generated by the surface tension
function $F$ of the model.
In order to explain $(1.4)$ recall that the surface tension
(see [A],[P] for the definition and properties) is a
function $F:{\bf S}^1\to {\bf R}_+ $ and it's value $F(n)$ is
designed to measure the free energy of the $\pm$-interface in the direction
othogonal to the unit vector $n$. It is also known [P] that the
homogeneous extension of $F$,
$$
F(x)\ =\ |x|F(\frac{x}{|x|})
$$
is convex. Being such it is the support function of some (uniquely defined)
convex body $K_F$, which is called the Wulff shape. $K_F$ can be recovered
from $F$ via
$$
K_F\ =\ \{x\in{\bf R}^2\ :\ \ \leq\ F(n)\ \forall n\},
$$
where $<\bullet ,\bullet >$ above denotes the scalar product in ${\bf
R}^2$. By the definition, $\lambda_F = Vol(K_F )$. Yet, there is another
description of $K_F$ - a variational one. To this end
let $\gamma_F$ be the boundary of $K_F$; $\gamma_F =\partial K_F$, and
let ${\cal A}$ be the class of all closed rectifiable curves in
${\bf R}^2$ without selfintersections
. For $\gamma\in{\cal A}$ define the Wulff functional as
$$
{\cal W}_F (\gamma )\ =\ \int_{\gamma } F(n_s )ds\ ,
$$
where $n_s$ is the unit normal to $\gamma $ at $s$. Then,
$$
2\lambda_F\ =\ \min_{\stackrel{\gamma\in {\cal A}}{vol(\gamma )=
\lambda_F}} {\cal W}_F(\gamma )\ =\ {\cal W}_F (\gamma_F ) .
$$
Roughly speaking the least energy consuming way to emerse an island of
$"-"$ phase of the volume $\lambda_F$ into the sea of $"+"$ phase is to
pack it into the Wulff shape. Since the Wulff functional scales with
the length of the boundary,
$$
{\cal W}_F (a\gamma )\ =\ a{\cal W}_F (\gamma ) ,
$$
its minimum on simple Jordan curves of volume $v$ is attained on the
corresponding dilatation of the Wulff shape, $(v/\lambda_F )^{1/2}K_F$ and is
equal to $2(v\lambda_F )^{1/2}$. In particular the left hand side of
$(1.4)$ is nothing but the minimal value of the Wulff functional on shapes
of the volume $\alpha (m)$.
The meaninig of $(1.4)$ becomes more transparent now: the main contribution
to the event $\{X_{\Lambda}\leq m\}$ comes from the formation of the Wulff
drop of the $"-"$ phase of the relative volume $\alpha (m)$. In its turn
the choice of $\alpha (m)$ is based on the following identity:
$$
(1-\alpha (m))m^*\ -\ \alpha (m)m^*\ =\ m.
$$
In the present paper we establish $(1.4)$ for all values of $\beta
>\beta_c$. Half of the work was done in the earlier paper [I], where the
corresponding lower bound was verified. Ou main result here is the
following
\begin{thm}
Let $m\in [-m^* ,m^*]$ and $\beta >\beta_c$ .
Then,
\begin{equation}
\limsup_{L\to\infty} \frac{1}{L}\log P_{\Lambda ,+}^{\beta}
\{X_{\Lambda}\ \leq\ m\}\ \leq\ -2(\lambda_F \alpha (m))^{1/2}.
\end{equation}
\label{thm1}
\end{thm}
As in [I] our global strategy of the proof as well as some notations we
use is
inherited from [P] . The novel
feature is , however, that we've been able to avoid cluster
expansions while deriving all the necessary estimates in the phase
of small contours (see section 3). Our technical tools to do so are
those of percolation theory, the principal inspiration being
drawn from the works of [ACC] and [Pi]. The main technical idea is
to translate events stated in terms of small contours into the
events stated in terms of small FK
(Fortuin-Kasteleyn) clusters. In section 2 we
briefly review some relevant facts about dependent percolation (FK
measures) and prove our main large deviation bound for the FK
percolation. Section 3 is devoted to the basic reduction to the FK
language. The proof of Theorem 1.1 is concluded in section 4.
\section{FK percolation and an estimate on the density of small clusters.}
\setcounter{equation}{0}
Let $A\in{\bf Z}^2$ be a finite set of sites. The bond(edge) set of $A$
, ${\cal B} (A)$, is defined to be the set of all lattice bonds $$ with at
least one end belonging to $A$. Given $p\in (0,1)$ set ${\cal P}_{A,w}^{p}$
and ${\cal P}_{A,f}^p$ to denote the FK (random cluster) measures on bond
configurations $n\in\ {\cal D}_A\stackrel{def}{=}{\{0,1\}}^{{\cal B}(A)}$
with wired and free boundary conditions respectively. We refer to [ACCN]
for definitions and a comprehensive account of properties of the FK
measures as well as to an excellent review in a recent paper [Pi]. Since
${\cal D}_A$ has the natural partial order one may consider FKG
(Fortuin-Kasteleyn-Ginibre) properties
of ${\cal P}_{A,w}^{p}$ and ${\cal P}_{A,f}^{p}$. Also note that it is
possible to view
those measures as being defined on ${\cal D}_{{\bf Z}^2}$ by requiring all
the bonds $b\notin {\cal B}(A)$ to be open (i.e $n(b)=1$) or closed
($n(b)=0$) respectively. Below we list all the properties of random cluster
measures which we are going to use later on.
{\bf P1.}(FKG relations)[ACCN]. Let $\succ$ denote the FKG order
relation. Both ${\cal P}_{A,w}^{p}$ and ${\cal P}_{A,f}^{p}$ are FKG measures.
Moreover, if $A\subset B$, then
$$
{\cal P}_{A,w}^{p}
\ \succ\
{\cal P}_{B,w}^{p}\ \ and\ \
{\cal P}_{A,f}^{p}\ \prec\
{\cal P}_{B,f}^{p}.
$$
Also,
$$
\lim_{L\to\infty}
{\cal P}_{\Lambda (L),w}^{p}\ \stackrel{def}{=}
\ {\cal P}_{w}^{p}
$$
exists and is an FKG translation invariant measure on ${\cal D}_{{\bf
Z}^2}$.
\vskip 0.1in
{\bf P2.}(Relations to the Ising model)[ACCN],[ES]. Let $\Omega_A = {\{-1,1\}}^A$
be the set of spin configurations on $A$. Choose $p\ = \ e^{-2\beta}$ and
define ${\bf Q}_{A,+}$ to be a joint site-bond measure on $\Omega_A \times
{\cal D}_A$, given by
$$
{\bf Q}_{A,+}\{\sigma ,n \}\ =\
\frac{1}{K_{A,+}}\prod_{\stackrel{b=}{n(b)=0}} (1-p)
\prod_{\stackrel{b=}{n(b)=1}} p{\delta}_{\sigma_x =\sigma_y} ,
$$
where $K_{A,+}$ is the normalizing constant and all the spins on $\partial
A$ are set to be $+1$. Then the bond marginal of ${\bf Q}_{A,+}$ is
precisely
${\cal P}_{A,w}^{p}$
, whereas the site marginal is
$P_{A ,+}^{\beta}$
. Moreover for each site $x\in A$,
$$
E_{\Lambda ,+}^{\beta} \sigma_x\ =\
{\cal P}_{A,w}^{p}\{x\ \leftrightarrow\ \partial A\},
$$
where as usual $\{x\leftrightarrow \partial A\}$ denotes the event
that $x$ belongs to the open cluster of the boundary $\partial A$ (
$x$ is connected to $\partial A$ by open bonds). In
particular, for any $\beta >\beta_c$ we have,
$$
m^*\ =\
E_{+}^{\beta} \sigma_x\ =\
{\cal P}_{w}^{p}\{|C(x)|\ =\ \infty\},
$$
where $C(x)$ is the cluster of $x$ and $|C(x)|$ is its
cardinality.
\vskip 0.1in
{\bf P3.}(Duality relations)[CCS] Each bond $b$ of the direct
lattice is intersected by precisely one bond $b^*$ of the
dual lattice ${\bf Z}^2 + (1/2,1/2)$. Consequently, to a given
bond configuration $n\in {\cal D}_A$ one can correspond the dual
bond configuration $n^* \in {\cal D}_{A^*}$ (where $A^*\subset
{\bf Z}^2 + (1/2,1/2)$ ) via
$$
n^* (b^* )\ =\ 1\ -\ n(b).
$$
The above correspondence is obviously one to one. The measure
${\cal P}_{A,w}^{p}$
,therefore, induces some measure on ${\cal D}_{A^*}$ which happens
to be nothing but
${\cal P}_{A^*,f}^{p^*}$,
where $p^* = 1-p/(2-p)$. Note that if $p^* = 1-e^{-2\beta^*}$, then
$e^{-2\beta^*}\ =\ tanh(\beta )$. Consequently, dual FK measures
correspond to dual Ising models. Likewise the dual measure
induced by
${\cal P}_{A,f}^{p}$
on ${\cal D}_{A^*}$ is
${\cal P}_{A^*,w}^{p^*},$
where $p^*$ is as above.
\vskip 0.1in
{\bf P4.}(Strong FKG and decoupling)[K],[Pi] Let $B\subset {\cal B}
(A)$ and $\eta_1 ,\eta_2$ are two bond configurations on $B$ such
that $\eta_1\geq\eta_2$. Then,
$$
{\cal P}_{A,w}^{p}\{\bullet\ |\ n|_B =\eta_1\}\ \succ\
{\cal P}_{A,w}^{p}\{\bullet |\ n|_B =\eta_2\}.
$$
Assume now that $B\subset\ {\cal B}(A)$ doesn't contain sites
belonging to the boundary $\partial A$ and after the removal of
all sites attached to $B$, $A$ is splitted into two
disjoint components - the
inner componet $A_1$ and the outer component $A_2$; $A_2$ is
assumed to be connected to $\partial A$ by the bonds from
${\cal B}(A)\setminus B$. Then,
$$
{\cal P}_{A,w}^{p}\{n|_{{\cal B}(A_1 )}=\bullet\ | \ n|_B \equiv 0\}\ =\
{\cal P}_{A_1 ,f}^{p}\{\ \bullet\ \}
$$
and
$$
{\cal P}_{A,w}^{p}\{n|_{{\cal B}(A_1 )}=\bullet ;\ n|_{{\cal B}(A_2
)} = \bullet\ |\ n|_B \equiv 1\}\ =\
{\cal P}_{A_1 ,w}^{p}\{\bullet \}
{\cal P}_{A_2 ,w}^{p}\{\bullet \}.
$$
\vskip 0.1in
We are now in a position to prove our first large deviation result
about the infinite volume FK measure
${\cal P}_{w}^{p}$
. Let $p\ >\ p_c\ \stackrel{def}{=}\ 1-e^{-2\beta_c}$. Pick a
number $N$, $00$, such that,
$$
{\cal P}_{w}^{p}
\{|\frac{1}{|\Lambda |} \sum_{x\in\Lambda}\eta_x\ -\
d(N)|\ \geq\ \epsilon\ \}\ \leq\ exp\{\ -\frac{L^2}{N^2}c(\epsilon )\} .
$$
\end{lem}
Proof: The main observation is that the field $\{\eta_x \}$ has nice mixing
properties. Going along the lines of [S2] one can in principle use this to
derive the desired estimate. We'll, however, follow an elegant approach of
[Pi], based on renormalization and comparison with an independent site
percolation.
Note first of all that $\eta_x$ is a decreasing function of the bond
configuration $n$, consequently the field $\{\eta_x \}$ is positively
associated, i.e. it posseses the FKG property. We'll show next that
\begin{equation}
0\ \leq\ Cov(\eta_x ,\eta_y )\ \leq\ exp\{\ -c^{\prime}d(x,y)\}\ ,
\end{equation}
where $c^{\prime}$ is some constant. Let us consider $d(x,y)$ so large that
the boxes
${\bf B}_{2N}(x)$
and
${\bf B}_{2N}(y)$
are far apart.
Since,
$$
Cov(\eta_x ,\eta_y )\ =\
{\cal P}_{w}^{p}
\{\eta_x = 1;\ \eta_y = 1\}\ -\
{\cal P}_{w}^{p}\{\eta_x = 1\}
{\cal P}_{w}^{p}\{\eta_y = 1\}
$$
the nonnegativity part of $(2.1)$ is implied by the FKG property.
To derive the upper bound in $(2.1)$ we`ll split $
{\cal P}_{w}^{p}
\{\eta_x = 1;\ \eta_y = 1\}
$ according to whether
${\bf B}_{2N}(x)$
and
${\bf B}_{2N}(y)$
are separated by open bonds of a configuration $n$ or not. To be more
precise let us say that a closed contour (Jordan curve) $\gamma$ separates
${\bf B}_{2N}(x)$
from
${\bf B}_{2N}(y)$
if
${\bf B}_{2N}(y)\subseteq int(\gamma )$
and
${\bf B}_{2N}(x)\subseteq ext(\gamma )$. Set
$$
{\cal A}\ =\ \{n\in{\cal B}({\bf Z^2}):
\exists\ \gamma - closed\ contour\ of\ open\ bonds\ which\ separates\
{\bf B}_{2N}(x)\ from\
{\bf B}_{2N}(y)\
\}
$$
and
$$
{\cal A}_{\gamma}\ =\ \{n\in{\cal A}:
\gamma \ is\ the\ smallest\ contour\ of\ n\
which\ separates\
\ {\bf B}_{2N}(x)\ from\
{\bf B}_{2N}(y)\
\}
$$
Obviously, ${\cal A}$ is the disjoint union of ${\cal A}_{\gamma}$.
On the other hand ${\cal A}_{\gamma }\ =\ {\bf
1}_{\{n|_{\gamma}\equiv 1\}}\cap {\cal A}_{\gamma}^{\prime}$, where
the event ${\cal A}_{\gamma}^{\prime}$ depends only on bond
configurations inside $int(\gamma )$. Thus, by the strong FKG and
decoupling properties P4,
$$
{\cal P}_{w}^{p}
\{\eta_x = 1;\ \eta_y = 1;\ {\cal A}_{\gamma}\}\ =\
{\cal P}_{w}^{p}
\{\eta_x = 1\ |\ {\bf 1}_{\{n|_{\gamma}\equiv 1\}}\}
{\cal P}_{w}^{p}
\{{\cal A}_{\gamma} \ |\ \eta_y = 1\ \}
{\cal P}_{w}^{p}
\{\eta_y = 1\ \} \ \leq
$$
$$
{\cal P}_{w}^{p}
\{\eta_x = 1\ \}
{\cal P}_{w}^{p}
\{\eta_y = 1\ \}
{\cal P}_{w}^{p}
\{{\cal A}_{\gamma} \ |\ \eta_y = 1\ \}
$$
Consequently,
$$
{\cal P}_{w}^{p}
\{\eta_x = 1;\ \eta_y = 1;\ {\cal A}\}\ \leq\
{\cal P}_{w}^{p}
\{\eta_x = 1\ \}
{\cal P}_{w}^{p}
\{\eta_y = 1\ \}
{\cal P}_{w}^{p}
\{{\cal A} \ |\ \eta_y = 1\ \} \ \leq
$$
$$
\leq\ {\cal P}_{w}^{p}
\{\eta_x = 1\ \}
{\cal P}_{w}^{p}
\{\eta_y = 1\ \}
{\cal P}_{w}^{p}
\{{\cal A} \ \} .
$$
The argument above may look formal since the infinite volume measure
$
{\cal P}_{w}^{p}
$ is involved, but we can reiterate it for
${\cal P}_{\Lambda ,w}^{p}$
with $\Lambda \nearrow\infty$ to obtain the desired estimate.
We claim, furthermore, that
\begin{equation}
{\cal P}_{w}^{p}
\{{\cal A}^c\}\ \leq\ exp\{\ -c^{\prime}d(x,y)\}
\end{equation}
$(2.1)$, then, follows immediately.
The best way to see $(2.2)$
is to go to the dual model, the
pass being secured by the property P3. Indeed,
$$
{\cal P}_{w}^{p}\{{\cal A}^c\}\ =\
{\cal P}_{f}^{p^*}\{{\bf B}^{*}_{2N}(x)\ \leftrightarrow
\ {\bf B}^{*}_{2N}(y)\}\ ,
$$
where ${\bf B}^{*}_{2N}(x)(\ {\bf B}^{*}_{2N}(y))
$ is the corresponding dual box. But,
$$
{\cal P}_{f}^{p^*}\{{\bf B}^{*}_{2N}(x)\ \leftrightarrow
\ {\bf B}^{*}_{2N}(x)\}\ \leq\ 64N^2 \max_{u\in {\bf
B}^{*}_{2N}(x),\ v\in {\bf B}^{*}_{2N}(y)}
{\cal P}_{f}^{p^*}\{\ u\leftrightarrow v\ \}.
$$
However,
$$
{\cal P}_{f}^{p^*}\{\ u\leftrightarrow\ u\}\ =\ <\sigma_u ,\sigma_v
>_{\beta^*},
$$
where the latter quantity is the correlation function of the
infinite volume Ising Gibbs measure at the inverse dual temperature
$\beta^* <\beta_c$, which is known to decay exponentially with
$d(x,y)$ [P]. $(2.3)$, thereby, follows.
We`ll use a renormalization procedure to finish the proof of lemma 2.1.
Note that because of $(2.1)$,
$$
Var(\ \frac{1}{|{\bf B}_M (x)|}\sum_{y\in {\bf B}_M (x)}\eta_y\ )
$$
tends to zero as $M$ tends to $\infty$. Therefore, by the Chebyshev's
inequality,
$$
{\cal P}_{w}^{p}
\{\ |\frac{1}{|{\bf B}_M (x)|}\sum_{y\in {\bf B}_M (x)}\eta_y\ -\
\ d(N)|\ \geq\ \epsilon /2\}
$$
becomes arbitrary small provided only that $M$ is large enough. Set
$M\ =\ NK^2$ and choose $K$ in such a fashion that,
$
a)
$
$$\ {\cal P}_{w}^{p}
\{\ |\frac{1}{|{\bf B}_M|}\sum_{y\in {\bf B}_M (x)}\eta_y\ -
\ d(N)|\ \geq\ \epsilon /2\}\ \leq\ \epsilon /16
$$
\vskip 0.1in
$
b)\ K\ \geq\ 1/\epsilon\ and \ N(K^2\ +\ 2K)\ divides\ L
$
\vskip 0.1in
$
c)\ exp\{\ -c^{\prime}KN\}\ \leq\ \epsilon /16,
$
\vskip 0.1in
where the constant $c^{\prime}$ in $c)$ is to be specified below.
Let us define a sublattice ${\bf Z}_{K}^{2}\subset {\bf Z}^{2}$
with
the step $2N(K+K^2 )$, i.e. ${\bf Z}_{K}^{2}\ =\ 2N(K^2+K){\bf
Z}^2$. Consider a partition of the large box $\Lambda (L)$ into smaller
boxes $\{{\bf B}_M (u)\}_{u\in {\bf Z}_{K}^{2}\cap\Lambda}$ and the
corresponding corridors of the length $2KN$. The total number of small
boxes in $\Lambda (L)$ equals, then, to
$$
T\ \stackrel{def}{=}\ \frac{|\Lambda |}{4[N(K^2 +K )]^2} .
$$
Also condition $b)$ implies that the relative volume of the corridors is
bounded above by $\epsilon /4$. Consequently,
$$
\{\frac{1}{|\Lambda |}\sum_{x\in \Lambda}\eta_x\geq d(N)+\epsilon \}\
\subseteq\ \{\frac{1}{T}\sum_{u\in {\bf Z}^{2}_{K}\cap\Lambda}
\frac{1}{|{\bf B}_M|}
\sum_{x\in {\bf B}_M (u)}\eta_x\geq d(N)+3\epsilon /4\} .
$$
and
$$
\{\frac{1}{|\Lambda |}\sum_{x\in \Lambda}\eta_x\leq d(N)-\epsilon \}\
\subseteq\ \{\frac{1}{T}\sum_{u\in {\bf Z}^{2}_{K}\cap\Lambda}
\frac{1}{|{\bf B}_M|}
\sum_{x\in {\bf B}_M (u)}\eta_x\leq d(N)-3\epsilon /4\} .
$$
Now, at each site $u\in {\bf Z}_{K}^{2}$ define random variables
$Y_{u}^i ;\ i=1,2;$ as
$$
Y_{u}^1\ =\ {\bf 1}_{
\{\ \frac{1}{|{\bf B}_M |}\sum_{y\in {\bf B}_M (u)}\eta_y\ \geq
\ d(N)\ +\ \epsilon /2\}}.
$$
$$
Y_{u}^2\ =\ {\bf 1}_{
\{\ \frac{1}{|{\bf B}_M |}\sum_{y\in {\bf B}_M (u)}\eta_y\ \leq
\ d(N)\ -\ \epsilon /2\}}.
$$
By the property $a)$,
$
{\cal P}_{w}^{p}\{Y_{u}^i\ =\ 1\}\ \leq\ \epsilon /16
$; $i=1,2$.
We claim, moreover, that
\begin{equation}
{\cal P}_{w}^{p}
\{\ Y_{u}^i\ =\ 1\ | \Sigma_u \}\ \leq\ \epsilon /8;\ i=1,2\ ,
\end{equation}
where $\Sigma_u$ is the $\sigma-$algebra generated by the variables
$\{Y_v\}_{ v\neq u}.$
As in [Pi] it is easy to see that under $(2.3)$ the fields $\{Y_{u}^i\}$
are FKG dominated by the process of independent site percolation on
${\bf Z}^{2}_{K}$, $\{S_u\}$, with
$$
P\{\ S_u\ =\ 1\}\ =\ \epsilon /8\ =\ 1\ -\ P\{\ S_u\ =\ 0\} .
$$
Note that,
$$
\{\frac{1}{T}\sum_{u\in {\bf Z}^{2}_{K}\cap\Lambda}
\frac{1}{|{\bf B}_M |}\sum_{x\in {\bf B}_M (u)}\eta_x\geq d(N)+3\epsilon /4\}\
\subseteq\ \{\frac{1}{T} \sum_{u\in\Lambda\cap
{\bf Z}_{K}^{2}} Y_{u}^1\ \geq\ \epsilon /4\}
$$
and
$$
\{\frac{1}{T}\sum_{u\in {\bf Z}^{2}_{K}\cap\Lambda}
\frac{1}{|{\bf B}_M |}\sum_{x\in {\bf B}_M (u)}\eta_x\leq d(N)-3\epsilon /4\}\
\subseteq\ \{\frac{1}{T} \sum_{u\in\Lambda\cap
{\bf Z}_{K}^{2}} Y_{u}^2\ \geq\ \epsilon /4\}.
$$
However, for $i=1,2,$
$$
{\cal P}_{w}^{p}
\{\frac{1}{T} \sum_{u\in\Lambda\cap
{\bf Z}_{K}^{2}} Y_{u}^i\ \geq\ \epsilon /4\}\ \leq\
P
\{\frac{1}{T} \sum_{u\in\Lambda\cap
{\bf Z}_{K}^{2}} S_u\ \geq\ \epsilon /4\}\ \leq\
$$
$$
\leq\ exp\{-T\phi(\epsilon /4)\}\ =\ exp\{\
-\frac{L^2}{4[N(K^2 +K)]^2}\phi (\epsilon /4)\}\ ,
$$
where
$$
\phi (a)\ = alog(\frac{8a}{\epsilon} )\ +\
(1-a)log(\frac{8(1-a)}{8-\epsilon}
)$$
is the rate function which governs
large deviations for the Bernoulli i.i.d.-s with $p=\epsilon /8$. In
particular, $\phi (\epsilon /4)\geq k{\epsilon}^2$, where $k$ is a
positive constant. Since the choice of $K$ depended in essential
way only on $\epsilon$ this implies the claim of the lemma. Thus it
remains to prove $(2.3)$.
Set ${\bf B}_1\ =\ {\bf B}_{M+N}(u)$ and ${\bf B}_2\ =\ {\bf
B}_{M+(2K-1)N}(u)$. Note that the random variables $Y_{u}^i ;i=1,2;$ depend only on
bond configurations from ${\cal D}_{{\bf B}_1}$, whereas the rest of
${\{Y_{v}^i\}}_{i=1,2;v\neq u}$ depends only on bond configurations from
${\cal D}_{({\bf Z}^2\setminus{\bf B}_2 )}$. By the strong FKG and
decoupling properties P4,
$$
{\cal P}_{w}^{p}
\{\ Y_{u}^1 =1\ |\ \Sigma_u\}\ \leq\
{\cal P}_{w}^{p}
\{\ Y_{u}^1 =1\ |\ n|_{{\cal B}({\bf Z}^2\setminus{\bf B}_2 )}\equiv
0\}\ =\
{\cal P}_{{\bf B}_2 ,f}^{p}\{\ Y_{u}^1 =1\}
$$
Set ${\cal A}\ =\ \{n\in{\cal D}_{{\bf B}_2}:\ \exists\ closed\
contour\ of\ open\ bond\ lying\ in\ {\bf B}_2\setminus{\bf B}_1\}.$
${\cal A}$ is an increasing event. As it was done above one
can use the FKG property and
decoupling to show that
$$
{\cal P}_{{\bf B}_2,f}^{p}
\{Y_{u}^1\ =\ 1\ |\ {\cal A}\}
\ \leq\
{\cal P}_{{\bf B}_2 ,w}^{p}
\{Y_{u}^1\ =\ 1\}\ \leq
{\cal P}_{w}^{p}
\{Y_{u}^1\ =\ 1\}\ \leq\ \epsilon /16.
$$
Similarly,
$$
\epsilon /16\ \geq
{\cal P}_{w}^{p}
\{Y_{u}^2\ =\ 1\}\ \geq
{\cal P}_{w}^{p}
\{Y_{u}^2\ =\ 1\ |\ {\cal A}\}
{\cal P}_{w}^{p}
\{ {\cal A}\}\ \geq
{\cal P}_{w}^{p}
\{\ Y_{u}^2 =1\ |\ \Sigma_u\}
{\cal P}_{w}^{p}
\{ {\cal A}\}.
$$
On the other hand
$$
{\cal P}_{{\bf B}_2,f}^{p}
\{{\cal A}^c\}\ =\
{\cal P}_{{\bf B}^{*}_2 ,w}^{p^*}
\{\ \partial {\bf B}^{*}_{2}\ \leftrightarrow\ \partial{\bf B}^{*}_{1}\}\
\leq
$$
$$
\leq\ |\partial {\bf B}_{1}^{*}|\max_{z\in\partial {\bf
B}^{*}_{1}}
{\cal P}_{{\bf B}^{*}_2,w}^{p^*}
\{z\ \leftrightarrow\ \partial {\bf B}^{*}_{2}\}\ \leq
\ 9N(K^2 +1)\max_{z\in\partial{\bf B}^{*}_{1}}
E^{\beta^*}_{{\bf B}^{*}_{2},+}\sigma_z .
$$
Since $\beta^* <\beta_c$ there exists a constant
$c_1 =c_1 (\beta ) >0$ ([CCS],pp. 442,443), such that
$$
\max_{z\in\partial {\bf B}^{*}_{1}}
E^{\beta^*}_{{\bf B}^{*}_{2},+}\sigma_z \ \leq
\ exp\{\ -c_1 d(
\partial{\bf B}_{1}^{*},\partial{\bf B}^{*}_{2})\}\ \leq
\ exp\{\ -c_1 (2K-3)N\}.
$$
Consequently, for $KN$ sufficeinly large,
$$
{\cal P}_{{\bf B}_{2},f}^{p}
\{{\cal A}^c\}\ \leq\ exp\{\
-c_1 KN\}.
$$
As far as $Y_{u}^2$ is concerned we can proceed as above using
estimates on the infinite volume correlation function, which were
already employed in the verification of $(2.2)$. We, thereby,
obtain:
$$
{\cal P}_{w}^{p}
\{{\cal A}^c\}\ \leq\ exp\{\
-c_2 KN\},
$$
where $c_2$ is another positive constant.
To complete the proof of the lemma define the constant $c^{\prime}$
in $c)$ as $c^{\prime}=min(c_1 ,c_2 )$.
Below we'll use a slight generalization of lemma 2.1. Namely, given
$\alpha >0$ let $A$ be a set of sites, $A\subset\Lambda (L)$, such that
\begin{equation}
|A|\ \geq\ \alpha L^2 ,
\end{equation}
and let $\{\eta_x\}$ be as before.
\begin{lem}
There exists a function $c=c(\epsilon ,\alpha )>0$, such that,
$$
{\cal P}_{w}^{p} \{|\frac{1}{|A|} \sum_{x\in A}\eta_x\ -\
d(N) |\ \geq \epsilon\ \}\ \leq\ exp\{\ -\frac{L^2}{N^2}c(\epsilon ,
\alpha )\} .
$$
for all sets $A\subset \Lambda (L)$ which satisfy $(2.4)$.
\end{lem}
We omit the proof of this lemma since it essentially repeats the
proof of lemma 2.1.
\section{Basic reduction to the FK case.}
\setcounter{equation}{0}
We are using two parameters $\nu$ and $b$, $0<\nu **\ 0$. Then there exists a function $c=c(\alpha ,
\epsilon )>0$, such that for any $A\subset \Lambda (L)$ with $|A|\geq
\alpha L^2$,
\begin{equation}
P_{A,+}^{\beta ,b}
\{\ X_A\ \leq\ m^* -\epsilon \}\ \leq\ exp\{\ -cL^{2-4b}\}
\end{equation}
provided only that $L$ is large enough (i.e. $L\geq l(\beta ,\alpha
,\epsilon ))$. As usual, $X_A = 1/|A|\sum_{x\in A}\sigma_x$.
\end{lem}
{\bf Remark:} Since $b\in (0,1/4)$ $(3.1)$ yields a supersurface
order of decay in the phase of small contours.
\vskip 0.1in
Proof: Set ${\cal A}\ =\ \{\sigma :\ all\ contours\ of\ \sigma\
are\
small\}$. We`ll see in the next section that
\begin{equation}
P_{A,+}^{\beta}
\{{\cal A}\}\ \rightarrow\ 1\ as\ L\rightarrow\ \infty.
\end{equation}
Consequently,
$$
P_{A,+}^{\beta ,b}
\{\ X_A\ \leq\ m^* -\epsilon\}\ =\
P_{A,+}^{\beta}
\{\ X_A\ \leq\ m^* -\epsilon ;{\cal A}\}(1\ +\ o(1)).
$$
Let us consider joint site-bond configurations
$$
(\sigma ,n)\ \in\ \Omega_A \times {\cal D}_A ,
$$
as defined in the previous section. Also let ${\bf Q}_{A,+}$ be
the corresponding
joint site-bond measure. Since,
$$
({\cal A}\times{\cal D}_A )\cap supp({\bf Q}_{A,+})\ \subseteq\ \{(\sigma ,n
):\ all\ "-"\ FK\ clusters\ are\ small\} \ \stackrel{def}{=}\ {\cal N}_{-},
$$
we have,
\begin{equation}
P_{A,+}^{\beta }
\{\ X_A\leq m^* -\epsilon ;\ {\cal A}\}\ \leq
\ {\bf Q}_{A,+}\{X_A\leq m^* -\epsilon ;\ {\cal N}_{-}
\}.
\end{equation}
Set $\phi_A$ to denote the density of small FK clusters,
$$
\phi_A\ =\ \frac{1}{|A|}\sum_{x\in A} {\bf 1}_{\{|C(x)|\leq L^{2b}\}}
.
$$
Note that
$$
\{X_A\leq m^* -\epsilon ;\ {\cal N}_{-}\}\ \subseteq\
\{\phi_A\ \geq\ \delta\}
$$
where $\delta\ =\ (1-m^* +\epsilon )/2$. Indeed, the only place
where $"-"$ spins can appear if large $"-"$ clusters are prohibited
are small FK clusters.
Then in the absence of
large $"-"$ FK clusters the density of large $"+"$ FK clusters is
given by $1-\phi_A$. Consequently,
$$
{\bf Q}_{A,+}\{X_A\leq m^* -\epsilon ;\ {\cal N}_{-}
\}\ \leq\
$$
$$
\leq {\bf Q}_{A,+}\{\frac{1}{|A|}\sum_{x\in A} \sigma_x{\bf 1}_
{\{|C(x)|\leq L^{2b}\}}\ \leq\ -\epsilon /2\ |\ \phi_A\geq
\delta\}\ +\ {\bf Q}_{A,+}\{\ 1-\phi_A \leq\ m^*
- \epsilon /2\}\ \leq
$$
$$
\leq {\bf Q}_{A,+}\{\frac{1}{ |A|}\sum_{x\in A} \sigma_x{\bf 1}_
{\{|C(x)|\leq L^{2b}\}}\ \leq\ -\epsilon /2\ ;\ \phi_A\geq
\delta\}\ +\
{\cal P}_{A,w}^{p}\{\phi_A \geq\ 1-m^* +\epsilon /2\}\ =\ (I)\ +\
(II).
$$
In order to estimate $(II)$ note that the event $\{\phi_A\geq
1-m^* +\epsilon /2\}$ is decreasing. Therefore,
$$
(II)\ \leq\
{\cal P}_{w}^{p}\{\phi_A \geq\ 1-m^* +\epsilon /2\}.
$$
Furthermore, observe that
$$
\lim_{L\to\infty}{\cal P}_{w}^{p}\{|C(x)|\leq\ L^{2b}\}\ =\ 1-m^* .
$$
Therefore for large enough values of $L$, the ${\cal P}_{w}^{p}$
expectation of $\phi_A$ is less than $1-m^* +\epsilon /4$. Lemma
2.2 ,then, implies that for large enough values of $L$,
\begin{equation}
(II)\ \leq\ exp\{\ -cL^{2-4b}\} .
\end{equation}
In order to treat $(I)$ set
$$
{\phi}_{A}^{N}
\ =\ \frac{1}{|A|}\sum_{x\in A} {\bf 1}_{\{|C(x)|\leq N\}}
.
$$
Pick $N=N(\epsilon )$ so large that $d(N)\geq 1-m^* -\epsilon /16$.
Then, by the virtue of lemma 2.2,
$$
{\cal P}_{w}^{p}\{{\phi}_{A}^{N} \leq\ 1-m^* -\epsilon /8\}\
\leq\ e^{-c\frac{L^2}{N^2}}
$$
for $L$ large enough. One more application of lemma 2.2 then, reveals
that for $L$ sufficiently large,
$$
{\cal P}_{w}^{p}\{\
{\phi}_{A}-
{\phi}_{A}^{N}\ \geq\ \epsilon /4\}\ \leq\ e^{-cL^{2-4b}},
$$
where $c=c(\epsilon )$ is a positive constant. Consequently, $(I)$
can be estimated above as
$$
(I)\ \leq\
{\bf Q}_{A,+}\{\frac{1}{ |A|}\sum_{x\in A} \sigma_x{\bf 1}_
{\{|C(x)|\leq N\}}\ \leq\ -\epsilon /2\ ;\ {\phi}_{A}^{N}\geq
\delta /2\}\ +\ e^{-cL^{2-4b}} .
$$
Finally the ${\bf Q}_{A,+}$ term above
can be estimated by means of a
generalization of the usual Cramer's approach to large deviations of
normalized sums of i.i.d. random variables. Such an estimate may be
found in full details in [Pi], here we only sketch it for the sake
of completeness. For each bond configuration with ${\phi}_{A}^{N}
\geq \delta /2$
we have at least $\alpha \delta L^{2}/2N$ small clusters, which are
painted independently into $"+"$ or $"-"$ with probability $1/2$ each.
Thus the problem of the estimation of $(I)$ reduces to the following
situation:
Given independent random variables $Z_1 ,Z_2 ,...,Z_M$ with $M
\geq \frac{\alpha
\delta L^{2}}{2N}$ and
$$
P\{\ Z_k =\pm a_k\}\ =\ \frac{1}{2};\ \ 0\leq a_k\leq N,
$$
find an upper bound for
$$
P\{\ \frac{1}{M}\sum_{k=1}^{M} Z_k\ \geq\ \epsilon /4\}.
$$
Note that the log-moment generating function of $Z_k$;
$$
h_k (t)\ =\ log(Ee^{tZ_k })\ =\ h_B(a_k t)\ =\ log(\frac{e^{a_k
t}+e^{-a_k t}}{2}),
$$
where $h_B$ is the log-moment generating function of the $\pm 1$ Bernoulli
random variable. Note also that for $t> 0\ $ $h_B (at)$ is increasing in
$a\geq 0$. By the exponential Chebyshev's inequality,
$$
P\{\ \frac{1}{M}\sum_{k=1}^{M} Z_k\ \geq\ \epsilon /4\}\ \leq\
exp\{\ -M\sup_{t>0}\{t\epsilon /4\ -\ \frac{1}{M}\sum_{k=1}^{M}
h_k(t)\}\}.
$$
But,
$$
\sup_{t>0}\{t\epsilon /4\ -\ \frac{1}{M}\sum_{k=1}^{M}h_k (t)\}\ \geq
\sup_{t>0}\{t\epsilon /4\ -\ h_B(Nt)\}\ =\ I_B(\frac{\epsilon}{
4N}),
$$
where $I_B$ is the large devaition rate function for the $\pm 1$
Bernoulli i.i.d.-s. Obviously, there exist a constant $c>0$, such that
$$
I_B(\frac{\epsilon}{4N})\ \geq\ c\frac{\epsilon^2}{N^2}.
$$
Consequently,
$$
P\{\ \frac{1}{M}\sum_{k=1}^{M} Z_k\ \geq\ \epsilon /4\}\ \leq\
exp\{\ -c\frac{M\epsilon^2}{N^2}\}.
$$
Combined with $(3.4)$ the above estimate implies the claim of the lemma.
\section{Proof of the main result.}
\setcounter{equation}{0}
We define a droplet to be an ordered sequence of dual sites $S\ =\
(u_1,...,u_{N+1})$, with $u_1\equiv u_{N+1}$ such that:
\vskip 0.1in
$ a)\ L^{\nu}\ \leq\ d(u_k ,u_{k+1})\ \leq\ 2L^{\nu}; \ k=1,...,N.$
$ b)\ The\ polygonal\ curve\ P(S)\ through\ u_1 ,...,u_{N+1}\ is\ a\
simple\ Jordan\ curve.$
\vskip 0.1in
Given a contour $\gamma$, we say that $\gamma$ is compatible with a droplet
$S=\{u_1,...,u_{N+1}\}$,
$\gamma\sim S$, if each $u_k$ is at distance no more than $1/3$ (to include
rounded corners)
from $\gamma$ and $\rho (\gamma ,P(S))\ \leq\ L^{\nu}$, where $\rho$ is the
Hausdorff distance. We'll also say that $S$ is a large droplet if
$Vol(P(S))\ \geq\ L^{2b}/2$. It is not hard to see that for any large contour
$\gamma$ there exists a large droplet $S$, which is compatible with
$\gamma$.
A collection of droplets ${\bf S}\ =\ \{S_1 ,...,S_n \}$ is called an
arrangement of droplets if there exists a configuration $\sigma$ such that
for $\{\gamma_1 ,...,\gamma_n\}$- the only large contours of $\sigma$ we have
$$
S_k\ \sim\ \gamma_k ;\ k=1,...,n\ .
$$
If this is the case, we`ll say that ${\bf S}$ is compatible with $\sigma$;
$\sigma\sim{\bf S}$. Likewise a collection of large droplets ${\bf S}\ =\
\{S_1 ,...,S_n \}$ is called an arrangement of exterior droplets, if there
exists a configuration $\sigma$ such that for $\{\gamma_1 ,...,\gamma_n\}$ -
the only large exterior contours of $\sigma$ we have $S_k\sim\gamma_k ;\
k=1,...,n$. We denote this as ${\bf S}\stackrel{ext}{\sim}\sigma$.
For an arrangement ${\bf S}$
the value of the Wulff functional ${\cal W}_F ({\bf
S})$ is defined via
$$
{\cal W}_F ({\bf S})\ =\ \sum_{k=1}^{n}{\cal W}_F (P(S_k )) .
$$
With each set of arrangements $E$ one can associate an event
$$
{\cal E}\ =\ \{\sigma :\ \exists\ {\bf S}\in E\ such\ that\ {\bf
S}\sim\sigma\}.
$$
Similarly, with each set of exterior arrangements $E$ one can associate an
event ${\cal E}_{ext}$. Thus starting with the probability measure
$P_{A,+}^{\beta}$ on a space of configurations $\sigma\in \Omega_A$ we have
a well defined (subprobability) set function on a space of arrangements (
exterior arrangements) of droplets. Now a
key lemma of [P, Lemma 10.1] states that
given an arrangement ${\bf S}$,
\begin{equation}
P_{A,+}^{\beta}\{\sigma :\ \sigma\sim {\bf S}\}
\ \leq\ exp\{\ -
{\cal W}_F ({\bf S}) + cN({\bf S})\},
\end {equation}
where $c=c(\beta )$ is a constant and $N({\bf S})$ is the number of
vertices in all droplets of the arrangement ${\bf S}$. Though the
original proof of [P] makes use of cluster expansions one can easily
verify that that the claim in the form of $(4.1)$ remains true for
all values of $\beta >\beta_c$. Note that
\begin{equation}
N({\bf S})\ \leq\ \frac{1}{L^{\nu}}\sum_{k=1}^{n} |\partial P(S_k )|\ \leq
\ \frac{{\cal W}_F ({\bf S})}{L^{\nu}\inf_{n\in{\bf S}^1}F(n)}
\ =\ o({\cal W}_F({\bf S }).
\end{equation}
Following [P] one can use $(4.1)$ to derive the following estimate:
Let $A\subseteq \Lambda (L)$ and $\delta >\nu$, then
\begin{equation}
P_{A,+}^{\beta}\{{\bf S} :\ {\cal W}_F ({\bf S}
)\ \geq\ kL^{\delta}\}
\
\leq\ exp\{\ -kL^{\delta}(1-o(1))\} .
\end{equation}
Indeed, let us rewrite $(4.2)$ as
$$
P_{A,+}^{\beta}\{\sigma :\ \sigma\sim {\bf S}\}\ \leq
\ exp\{\ -{\cal W}_F ({\bf S})(1-o(1))\ -\ a(N({\bf S})\},
$$
where
$$
o(1)\ =\ \frac{cN({\bf S}}{{\cal W}_F ({\bf S})}\ =
\ \frac{a(N({\bf S})}{{\cal W}_F ({\bf S})}
$$
and $a=a(n)$ is some function yet to be determined. Now,
$$
P_{A,+}^{\beta}\{{\bf S} :\ {\cal W}_F ({\bf S}
)\ \geq\ kL^{\delta}\}\ =\
\sum_{n}
P_{A,+}^{\beta}\{{\bf S} :\ {\cal W}_F ({\bf S}
)\ \geq\ kL^{\delta}; \ N({\bf S})=n\}\ =
$$
$$
\ =\ \sum_{n}\sum_{\stackrel{{\bf S}: N({\bf S})=n}
{{\cal W}_F ({\bf S})\geq
kL^{\delta}}}
P_{A,+}^{\beta}\{\sigma :\ \sigma\sim {\bf S}\}\ \leq
$$
$$
exp\{\ -kL^{\delta}(1-o(1))\}\sum_{n}e^{-a(n)}\#\{{\bf S}:\ N({\bf
S})=n\}.
$$
But a very rough estimate on the number of droplet arrangements
with $n$ vertices, $\#\{{\bf S}:\ N({\bf S})=n\}$ yields:
$$
\#\{{\bf S}:\ N({\bf S})=n\}\ \leq\ (L^2 )^n n^n .
$$
Thus, if we pick $a(n) = 2nlogL\ +\ nlogn\ +\ n$, then
$$
\sum_{n}e^{-a(n)}\#\{{\bf S}:\ N({\bf S})=n\}\ \sum_{n}e^{-n}\ <\
\infty .
$$
It remains to verify that $a(N({\bf S}))/{\cal W}_F ({\bf S})=
o(1)$ on ${\bf S}$ such that ${\cal W}_F ({\bf S})\geq
kL^{\delta}$. But this is clear, since ${\cal W}_F
({\bf S})$ is always bounded above by $ cL^2$, and $(4.3)$
follows.
In particular, picking $\delta =b$ and $k$ the optimal constant in the
lattice isoperimetric inequality over $infF$, we see that, because
$\nu < b$,
$$
P_{A,+}^{\beta}\{\ \exists \ large\ \pm -contours\}\ \leq\
exp\{\ -cL^b (1-o(1))\},
$$
which implies $(3.2)$ of the previous section.
We now turn to the proof of Theorem 1.1. The test event for the upper bound
is $
{\cal E}^{m}_{ext}
$
, which is generated by the set of external arrangements
$$
E^{m}_{ext}\ =\ \{{\bf S}:\ {\cal W}_F ({\bf S})\ \geq\ 2(\alpha
(m)\lambda_F )^{1/2}L\} .
$$
For any $\delta>0$,
\begin{equation}
P_{\Lambda ,+}^{\beta}\{\ X_{\Lambda}\ \leq\ m\}\ \leq
\ P_{\Lambda ,+}^{\beta}\{\
{\cal E}^{m+\delta}_{ext}\}\ +\
P_{\Lambda ,+}^{\beta}\{\ X_{\Lambda}\ \leq\ m\ /\
({\cal E}^{m+\delta}_{ext})^c\}.
\end{equation}
The first term in the right hand side of $(4.4)$ can be estimated
above by means of $(4.3)$ with $\delta =1$\ and $k=2(\alpha (m+
\delta )\lambda_F )^{1/2}$ as
$$
\limsup_{L\to\infty}\frac{1}{L}logP_{\Lambda ,+}^{\beta}\{
{\cal E}^{m+\delta}_{ext}\}\ \leq\
-2(\alpha (m+\delta )\lambda_F )^{1/2} .
$$
Thus if we show that the second term in $(4.3)$ can be neglected,
i.e. it has a supersurface order of decay, the claim of the theorem
follows by letting $\delta$ tend to zero.
\begin{lem}
For any $\delta>0$,
$$
P_{\Lambda ,+}^{\beta}\{ X_{\Lambda}\ \leq\ m\ /\ ({\cal E}_{ext}^{m+
\delta})^c\}\ \leq\ exp\{\ -cL^{2-6b}\},
$$
where $c\ =\ c(m,\delta ,\beta )\ >\ 0.$
\end{lem}
{\bf Remark:} Since $2-4b>1$, the above lemma provides the required
supersurface estimate.
\vskip 0.1in
Let $\gamma_1 ,...,\gamma_n$ be a collection of external contours
in $\Lambda$. Set
$$
\Gamma (\gamma_1 ,...,\gamma_n )\ =\ \{\sigma :\ \gamma_1 ,...,
\gamma_n\ are\ the\ only\ external\ \pm -contours\ of\ \sigma\}.
$$
Note that if we remove sites adjacent to
$\gamma_1 ,...,\gamma_n$ then the remaining sites of $\Lambda$ will
be splitted into two
disjoint components; the external component ${\Lambda_{ext}}$ and the
internal component ${\Lambda}_{int}$. Moreover, if $\{\gamma_1
,...,
\gamma_n\}$ is compatible with some arrangement of external
droplets ${\bf S}\in (E^{m+\delta})^c$, then
\begin{equation}
\frac{1}{2}Vol(\Lambda\setminus (\Lambda_{int}\cup\Lambda_{ext}))\
\leq\
\sum_{k=1}^{n} |\gamma_k |\ \leq\ \frac{2L^{1+2\nu}(\alpha
(m+\delta )\lambda_F )^{1/2}}{\inf_{n\in{\bf S}^1}F(n)}\ \leq
\ cL^{1+2\nu}
\end{equation}
and
\begin{equation}
Vol({\Lambda}_{int})\ \leq\ L^2\alpha (m+\delta ) \ +\ o(L^2),
\end{equation}
where $c=c(m)$ is a constant. Consequently, for $L$ large enough,
$$
P_{\Lambda ,+}^{\beta}\{\ X_{\Lambda} \ \leq\ m\ /\ \Gamma (\gamma_1
,...,\gamma_n )\}\ \leq
$$
$$
\leq\ P_{\Lambda ,+}^{\beta}\{\ \frac{|\Lambda_{int}|}{|\Lambda
|}X_{\Lambda_{int}}\ +\ \frac{|\Lambda_{ext}|}{|\Lambda
|}X_{\Lambda_{ext}}\ \leq\ m\ +\ \delta /3\ /\ \Gamma (\gamma_1
,...,\gamma_n )\}\ \leq
$$
$$
P_{\Lambda_{ext},+}^{\beta ,b}\{\ X_{\Lambda_{ext}}\ \leq\ m^*\ -\delta
/3\}\ +\ {\bf 1}_{\{|\Lambda_{int}|\geq \delta L^2 /3\}}
P_{\Lambda_{int},-}^{\beta}\{\ X_{\Lambda_{int}}\ \leq\ -m^* -\delta /3\},
$$
where the first inequality follows by $(4.5)$.
By lemma 3.1 and in a view of $(4.6)$
the first term in the rightmost expression is
$o(e^{-cL^{2-4b}})$. Thus it remains to estimate the second term. We claim
that for all collections $\gamma_1 ,...,\gamma_n$ as above,
\begin{equation}
{\bf 1}_{\{|\Lambda_{int}|\geq \delta L^2 /3\}}P_{\Lambda_{int},+}^{\beta}
\{\ X_{\Lambda_{int}}\ \leq\ -m^*\ -\ \delta /3\}\ \leq\ exp\{\ -cL^2\},
\end{equation}
where $c=c(\delta ,\beta )>0$ is a constant which doesn't depend on a
particular collection. The claim of the lemma will, then, follow easily,
since
$$
P_{\Lambda ,+}^{\beta}\{\ X_{\Lambda}\ \leq\ m\ /\ ({\cal
E}_{ext}^{m+\delta})^c\}\ \leq\ \max_{\{\gamma_1 ,...,\gamma_n\}\sim{\bf S}\in
(E_{ext}^{m+\delta })^c} P_{\Lambda ,+}{\beta}\{\ X_{\Lambda }\ \leq\ m\ /\
\Gamma (\gamma_1 ,...,\gamma_n )\} .
$$
Proof of $(4.7)$: We'll establish a slightly more general result:
Let $A\subseteq\Lambda (L)$ with $|A|\geq \delta L^2$ and $|\partial A|\leq
L^{\rho};\ \rho <2$ and let $\epsilon >0$. Then there exists $ c=c(\epsilon ,
\delta ,\rho ,\beta )>0$ which doesn't depend on $A$, such that
\begin{equation}
P_{A,+}^{\beta}\{X_A\ >\ m^*\ +\ \epsilon\}\ \leq\ exp\{\ -cL^2\}.
\end{equation}
Fix a number $N$ and tile as much as possible of $A$ by disjoint boxes
${\bf B}_N (x_k )\subset A$. Set ${\bf B}=\cup{\bf B}_{N}(x_k )$. Clearly,
$$
\frac{1}{|A|}|A\setminus {\bf B}|\ \rightarrow\ 0\ as \
L\rightarrow\infty .
$$
Thus, for $L$ large enough,
$$
P_{A,+}^{\beta}\{\ X_A\ \geq\ m^* +\epsilon\}\ \leq\ P_{{\bf B},+}^{\beta}
\{\ X_{{\bf B}}\ \geq\ m^* +\epsilon /2\}.
$$
Setting $M=|{\bf B}|/4N^2 $, i.e. $M$ is the number of elementary boxes in
${\bf B}$, we see that
$$
\{\ X_{{\bf B}}\ \geq\ m^* +\epsilon\}\ \subseteq\ \{\ X_{{\bf B}_N} (x_k )\
\geq\ m^* +\epsilon /4\ for\ at\ least\ M\epsilon /4\ elementary\ boxes\}.
$$
To complete the proof of $(4.5)$ just choose $N$ so large, that
$$
P_{{\bf B}_{N}(0),+}^{\beta}\{\ X_{{\bf B}_{N}(0)}\ \geq\ m^*\ +\ \epsilon
/4\}\ \leq\ \epsilon /8 ,
$$
and use the renormalization technique already employed in section 2.
\vskip 0.2in
{\bf Acknowledgement:} My thanks go to Ago Pisztora for providing me
with a preliminary version of [Pi] as well as for several useful
discussions we had. I'm also grateful to
Roberto Schonmann for a very careful reading of the
preliminary draft and a number of valuable remarks he made, to Tom
Mountford who pointed out a mistake in the original proof of lemma 2.1
and to Bao Nguyen for the interest he
expressed in this work.
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\end{document}
%ENDBODY
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