\documentstyle[12pt]{article}
\setlength{\textheight}{8.9in}
\setlength{\textwidth}{6.2in}
\topmargin= -1.2cm \hoffset -1cm \raggedbottom
\renewcommand{\baselinestretch}{1.0}
\newcommand{\be}{\begin{equation}}
\newcommand{\en}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand{\bee}{\begin{enumerate}}
\newcommand{\ene}{\end{enumerate}}
\newcommand{\non}{\nonumber}
\newcommand{\no}{\noindent}
\newcommand{\vs}{\vspace}
\newcommand{\hs}{\hspace}
\newcommand{\e}{\'{e}}
\newcommand{\D}{\dagger}
\newcommand{\ef}{\`{e}}
\newcommand{\bC}{{\bf C}}
\newcommand{\Bbb}{\bf}
\newcommand{\p}{\partial}
\newcommand{\ha}{{1\over 2}}
\newcommand{\un}{\underline}
\newcommand{\var}{\varphi}
\newtheorem{Th}{Theorem}
\newtheorem{Def}{D\'efinition}
\newtheorem{Lem}{Lemma}
\newtheorem{Pro}{Proposition}
\pagestyle{plain}
\title{{\bf Renormalization Group Pathologies
and the Definition of Gibbs States}}
\author{J.Bricmont\thanks{Supported by EC grant CHRX-CT93-0411}\\ UCL, Physique
Th\'eorique, B-1348, Louvain-la-Neuve, Belgium\\bricmont@fyma.ucl.ac.be\and
A.Kupiainen\thanks{Supported by NSF grant DMS-9205296 and EC grant
CHRX-CT93-0411} \\ Helsinki University, Department of Mathematics,
\\ Helsinki 00014,
Finland\\ajkupiai@cc.helsinki.fi\and
R. Lefevere\\ UCL,
Physique Th\'eorique, B-1348, Louvain-la-Neuve,
Belgium\\lefevere@fyma.ucl.ac.be\\
\vs*{15 mm}\\
{\it Dedicated to the memory of Roland L. Dobrushin}}
\date{}
\begin{document}
\maketitle \begin{abstract}
We show that the so-called Renormalization Group
pathologies in low temperature Ising models
are due to the fact that the renormalized Hamiltonian
is defined only almost everywhere (with respect to
the renormalized Gibbs measures). We construct this renormalized Hamiltonian
using a Renormalization Group method developed for random systems and we show
that the pathologies are analogous to Griffiths' singularities.
\end{abstract}
\newpage
\section{Introduction.}
\setcounter{equation}{0}
The Renormalization Group
(RG) has been one of the most useful tools of theoretical physics during the
past
decades. It has led to an understanding of
universality in the theory of
critical phenomena and of the divergences in quantum field theories. It has also
provided a nonperturbative calculational framework as well as the basis of a
rigorous mathematical understanding of these theories.
Even though the RG was primarily devised for the study of (approximatively)
scale invariant situations such as statistical mechanical models at the critical
point, it was found useful in the mathematical analysis of problems that were
not critical but that nevertheless were "multiscale": for example, first order
phase transitions in regular \cite{GKK} and disordered \cite{BK} spin systems.
The spin variables
give an appropriate representation of these systems at and above
the critical point; however,
at low temperatures, these models are
most naturally
expressed in terms of contours (domain
walls) that separate the different ground states. To apply the RG method, one
inductively sums over the small scale contours,
producing an effective theory
for the larger scale contours.
However, the real power of the RG both
theoretically and
in most applications has been to realize it as a map between Hamiltonians, and
the latter are usually expressed in terms of the spin variables. So, one would
like to be define
rigorously such a map, but this program
has met some
difficulties.
It was observed in simulations \cite{hashas} that the RG transformation seems,
in some sense, "discontinuous" as a map between spin Hamiltonians.
These observations led subsequently to a rather extensive discussion
of the so-called ``pathologies" of
Renormalization Group Transformations (RGT):
van Enter, Fernandez and
Sokal have shown \cite{VFS1, VFS}
that, first of all, the
RG transformation is
not really discontinuous.
But they also show, using results of
Griffiths and Pearce
\cite{GP1,GP2} and of Israel
\cite{Is}, that, roughly speaking, there does not exist a renormalized
Hamiltonian
for many RGT applied to Ising-like models at low temperatures and in some cases
even at high temperatures (in particular in a large external field, see
\cite{vE96, vEFK_JSP}).
More precisely,
van Enter, Fernandez and
Sokal
consider various real-space RGT (block spin, majority vote, decimation) that can
be easily and rigorously defined as maps acting on measures (i.e. on
probability distributions of the infinite volume spin system). The problem
occurs when one tries to rewrite this map in terms of Hamiltonians.
Hamiltonians are usually
expressed as sums of ($n$-body) interactions of the spins that have sufficient
decay properties so as to define infinite volume Gibbs measures. If we start
with a Gibbs measure $\mu$ corresponding to a given Hamiltonian $H$, one can
easily define the renormalized measure $\mu'$. The problem then is to
reconstruct a renormalized Hamiltonian $H'$ (i.e. a set of interactions)
for which $\mu'$ is a Gibbs
measure. Although this is trivial in finite volume, it is not so in the
thermodynamic limit, and it is shown in
\cite{VFS} that, in many cases at low temperatures, even if $H$ contains only
nearest-neighbour interactions,
there is no absolutely
summable interaction (defined in (2.\ref{2.6}) below) giving rise to a
Hamiltonian
$H'$ for which $\mu'$ is a Gibbs
measure.
What is one supposed to think about these pathologies? {}From a practical point
of view, we understand the reason why there {\it might} be problems: one is
using the wrong variables, i.e. the spin variables rather than the contours
variables.
The fact that the usefulness of the RG method depends crucially on choosing the
right variables has been
known for a long time. The ``good" variables should be such that a single RG
transformation, which can be interpreted as solving the statistical mechanics of
the small scale variables with the large ones kept fixed, should be
``noncritical" i.e. should be away from the parameter regions where phase
transitions occur. This is true in particular in the low temperature
region if one uses the contour variables.
In all the cases where pathologies were found, they were due to the fact that a
single RG transformation involves a system that has a phase transition for some
fixed values of the large scale variables.
However, from the theoretical
point of view, we believe that it is interesting to see just how pathological
the renormalized Hamiltonians are. This question is related to
another one, of independent interest: when is a measure Gibbsian for some
Hamiltonian? For example, Schonmann showed
\cite{sch87} that, when one projects
a Gibbs measure (at low temperatures)
to the spins attached to a
lattice of lower dimension, the resulting measure is not, in general, Gibbsian.
There has been an extensive investigation of this problem of pathologies and
Gibbsianness. Martinelli and Olivieri \cite{MO1, MO2}
have shown that, in a non-zero external
field, the pathologies disappear
after sufficiently many decimations
. Fernandez and Pfister \cite{FP} study
the set
of configurations that are responsible for those pathologies. They give criteria
which hold in particular in a non-zero external field, and which imply that this
set
is of zero measure with respect to the
renormalized measures. Following the work of Kennedy \cite{ken92}, several
authors
\cite{Ke, benmaroli, ciroli96}
analyze the absence of pathologies
near the critical point.
Our goal in this paper is to further clarify the situation: following an idea of
Dobrushin \cite{Do}, we prove that, for several examples considered in
\cite{VFS}, the
renormalized Hamiltonian actually exists, but the corresponding interaction
satisfies a weaker summability condition than the one used
in \cite{VFS}. Our condition ((2.\ref{2.9}) below) is however
sufficient to define Gibbs
measures, in a way that is similar
to the one used before for ``unbounded spins". Thus in a sense, the pathologies
are not there in the end, and the renormalized measures
are Gibbsian. However, it turns out that the renormalized
Hamiltonian is only defined almost everywhere with respect
to the renormalized measure: it becomes "pathological"
on a set of measure zero that in particular includes
the configurations used in \cite{VFS} to exhibit
the pathologies. Our result is similar to the one of Maes and Vande Velde
\cite{maevel96}
on the Schonmann example of the
measures projected on a
lattice of lower dimension, except that,
in our case, we show that the two renormalized
states are Gibbsian with respect to the same
Hamiltonian (while this question is left open in \cite{maevel96}).
However, as pointed out to us by
A. Sokal, with our definition of Gibbs states,
the interaction defining the Hamiltonian is not unique,
while it is essentially unique (up to physical equivalence)
with the usual definition.
Of course, we do not claim
to justify the RG method in general, and, besides, our results
do not hold
near the critical point.
We do not even prove that, upon iteration, the RGT drives
the Hamiltonians to a
trivial fixed point, although this can probably be done, in some of the examples
discussed below. But we do clarify the nature of the ``pathologies".
Our proof is based on the following idea: we consider the spins distributed by
the renormalized measures as random external fields acting on the original
system, and we apply the methods of \cite{BK} to construct the renormalized
Hamiltonian. So, we use the Renormalization Group in order to justify the
Renormalization Group.
As in all random systems, there is a set of measure zero of
``bad" configurations
of the random fields, for which ``typical" results (e.g. decay of correlations)
do not hold, and
which are responsible
for the Griffiths'
singularities (\cite{Gr}).
We shall see below
that the pathologies used in \cite{VFS}
are actually due to those
bad configurations. But, once one excludes this set of measure zero, the
renormalized system has a nice Hamiltonian, with rapidly decaying interactions.
\section{Results.}
\setcounter{equation}{0}
We consider the nearest-neighbour Ising model on ${\Bbb Z}^d$, $d \geq 2$, at
$\beta$ large,
for simplicity. To each $i
\in {\Bbb Z}^d$, we associate a variable $\sigma_i
\in \{ -1,+1\}$,
and the (formal)
Hamiltonian is
\be
-\beta H = \beta \sum_{\langle ij
\rangle} (\sigma_i \sigma_j-1) \label{2.1} \en
where $\langle ij \rangle$ denotes
a nearest-neighbour pair and $\beta$
is the inverse temperature.
At low temperatures,
there are two extremal translation invariant
Gibbs measures corresponding to (1),
$\mu_+$ and $\mu_-$. To define our RGT,
let ${\cal L} = (b {\Bbb Z})^d$, $ b \in {\Bbb N}$, $ b \geq 2$ and cover ${\Bbb
Z}^d$ with disjoint {\it b-boxes} $B_x = B_0 + x,$ $ x \in
{\cal L}$ where $B_0$ is a box of size
$b$ centered around 0. Associate to each $x \in {\cal L}$ a variable $s_x \in \{
-1,+1\}$, denote by $\sigma_A$ an element of $ \{-1 , +1\}^A$, for $A \subset
{\Bbb Z}^d$, $|A | < \infty$, and
introduce the probability kernels
$$T_x = T(\sigma_{ B_x},s_x)$$
for $x
\in {\cal L}$, i.e. $T_x$ satisfies
\ba
&1)& T ( \sigma_{ B_x},s_x)\geq 0\nonumber\\ &2)& \sum_{s_x}
T(\sigma_{ B_x},s_x) = 1 \label{2.2}
\ea
For any measure $\mu$ on $ \{-1,+1\}^{{\Bbb Z}^d}$, we denote by $\mu(\sigma_A)$
the probability of the configuration $\sigma_A$.
\vspace*{2mm}
{\bf Definition.}
Given a measure $\mu$ on $ \{-1,+1\}^{{\Bbb Z}^d}$, the {\it renormalized
measure} $\mu'$ on $\Omega =\{-1,+1\}^{\cal L}$ is defined by:
\be
\mu' (s_A) = \sum_{\sigma_{B_A}}
\mu ( \sigma_{B_A})\prod_{x\in A} T
(\sigma_{ B_x},s_x)
\label{2.3}
\en
where $B_A=\cup_{x\in A}B_x$, $A \subset {\cal L}$, $|A | < \infty$, and $s_A
\in \Omega_A = \{-1 , +1\}^A$. \\
\vspace*{2mm}
It is easy to check, using 1) and 2),
that $\mu'$ is a measure. We shall call the spins $\sigma_i$ the {\it internal}
spins and the spins $s_x$ the {\it external} ones (they are also
sometimes called the block spins).
We shall need two other conditions on $T$: we assume that $T$ is symmetric:
\be
T(\sigma_{ B_x} , s_x) =
T (-\sigma_{ B_x}
, -s_x)
\label{2.4}
\en
and that
\be
0 \leq T (\sigma_{ B_x},s_x) \leq e^{-\beta} \label{2.5}
\en
if $\sigma_i \neq s_x \;\;\; \forall
i \in B_x$.
The condition (\ref{2.5}) means that there is a
coupling which tends to align $s_x$ and the spins in the block $B_x$.
It would be more
natural to have, instead of
(\ref{2.5}),
$0 \le T \le \epsilon$ (with $\epsilon$
independent of $\beta$ but small enough). However, assuming
(\ref{2.5}) simplifies the proofs.
Note that (\ref{2.2}, \ref{2.4}, \ref{2.5}) imply that \be
\overline T \equiv T(\{\sigma_i = +1\}_{i\in B_x} , +1) = T (\{\sigma_i =
-1\}_{i\in B_x}
, -1)
\geq 1- e^{-\beta}\geq \frac{1}{2}
\label{2.5b}
\en
for $\beta$ large.
The usual transformations, discussed
in \cite{VFS}, sect. 3.1.2, like decimation or majority rule, obviously satisfy
(\ref{2.4},\ref{2.5}). The Kadanoff transformation satisfies (\ref{2.5}) for $p$
large. Our results could be extended to the block spin transformations (where
$s_x$ does not belong to $\{-1 , +1\}$).
Let us now summarize the main result
of \cite{VFS}. Consider {\it interactions} $\Phi = (\Phi_X)$, which are families
of functions
$$
\Phi_X : \Omega_X \to {\Bbb R},
$$
indexed by $X \subset {\cal L}$, $|X| < \infty$. Assume that $\Phi$ is
\bee
\item[a)] translation invariant:
$$
\Phi_X = \Phi_{X+x} \;\;\forall x \in {\cal L} $$
\item[b)] uniformly absolutely summable: \be
\sum_{X \ni 0} \| \Phi_X \|_\infty < \infty \label{2.6}
\en
\ene
This set of interactions obviously forms a
Banach space, with the norm
(\ref{2.6}) (note that our terminology differs slightly from the one
of \cite{VFS}: we add the word ``uniformly" to underline the difference with
respect to condition (\ref{2.9}) below) .
Then, one defines, $\forall V \subset {\cal L}$, $|V| < \infty$, the Hamiltonian
\be
H (s_V | \bar s_{V^c}) = -\sum_{X \cap V \neq \emptyset} \Phi_X (s_{X \cap V}
\vee
\bar s_{X \cap V^c})
\label{2.7}
\en
where $s_V \in \Omega_V $,$ \bar s_{V^c}$ is the restriction to $V^c$ of $\bar s
\in \Omega$
and, for
$X
\cap Y =
\emptyset$, $s_X \vee s_Y$ denotes the obvious configuration in $\Omega_{X\cup
Y}$.
The condition (\ref{2.6}) implies that $H$ is a {\em continuous}\/
function of $s_V$ and $\bar s_{V^c}$ (in the product topology),
and that
\be
\pi^\Phi_V(s_V | \bar s_{V^c}) =
Z^{-1} (\bar s_{V^c}) \exp (-H (s_V | \bar s_{V^c})) \en
(where $Z^{-1} (\bar s_{V^c})$ is the obvious normalization factor)
defines a quasilocal specification in the sense of \cite{VFS}.
\vspace*{3mm}
{\bf Definition.}
$\mu$ is a {\it Gibbs measure} for $\Phi$ if the
conditional
probabilities satisfy,
$\forall V \subset {\cal L}$, $|V|$ finite,
$\forall s_V \in \Omega_V$,
\be
\mu (s_V | \bar s_{V^c}) = \pi^\Phi_V(s_V | \bar s_{V^c}) \label{2.8}
\en
$\mu$ a.e.
\vspace*{3mm}
Note that
we left out the
inverse temperature $\beta$. When we refer
to $\beta$ below, we mean the inverse
temperature of the original Ising model (\ref{2.1}), before acting with the RGT.
The main result of \cite{VFS} is that, for a variety of RGT, there is no
interaction satisfying a) and b) above
for which $\mu'_+$ or $\mu'_-$
are Gibbs measures.
However, we observe that, in order to define
$H (s_V | \bar s_{V^c})$ and
$\pi^\Phi_V(s_V | \bar s_{V^c})$,
it is not necessary to assume
(\ref{2.6}); it is enough to assume the existence
of a tail set $\overline{\Omega} \subset \Omega$ on which the following {\em
pointwise}\/ bounds hold: \bee
\item[b')] $\overline \Omega$-pointwise absolutely summable: \be
\sum_{X \ni x} |
\Phi_X (s_{X }) | <
\infty\;\;\; \forall x \in {\cal L}, \forall s \in {\overline \Omega}.
\label{2.9}
\en
\ene
We shall therefore enlarge the class of ``allowed" interactions by
dropping the condition (\ref{2.6})
and assuming (\ref{2.9}) instead.
Actually, a similar setup was
already used in the theory of ``unbounded spins" with infinite range
interactions, see \cite{Ge}.
With this condition, we can define
the specification $\pi^{\Phi,\overline{\Omega}}$ by
\be
\pi^{\Phi,\overline{\Omega}}_V(s_V | \bar s_{V^c}) \;=\; \cases{ Z^{-1} (\bar
s_{V^c}) \exp (-H (s_V | \bar s_{V^c}))
& for $\bar s_{V^c} \in \overline{\Omega}$ \cr 0 & for $\bar s_{V^c} \not\in
\overline{\Omega}$ \cr }
\en
and then define Gibbs measures for
the pair $\Phi,\overline{\Omega}$ as follows:
\vspace*{3mm}
{\bf Definition.}
Given a tail set $\overline{\Omega} \subset \Omega$,
$\mu$ is a {\it Gibbs measure} for the pair
($\Phi$, $\overline{\Omega}$) if $\mu (\overline \Omega) =1$,
and there exists a version of the
conditional
probabilities that satisfy,
$\forall V \subset {\cal L}$, $|V|$ finite, $\forall s_V \in \Omega_V$,
\be
\mu (s_V | \bar s_{V^c}) =
\pi^{\Phi,\overline{\Omega}}_V(s_V | \bar s_{V^c})
\label{2.80}
\en
$\forall \bar s \in \overline{\Omega}.$
\vspace*{3mm}
Since conditional probabilities are defined almost everywhere,
this definition is very similar to the usual one.
However, when condition (\ref{2.6}) holds, the conditional
probabilities can be extended everywhere, and are continuous,
which is not the case here.
We can now state our main result:
\begin{Th}
Under assumptions (\ref{2.4}, \ref{2.5}) on $T$, and for $\beta$ large enough,
there exist disjoint translation-invariant (hence, tail) sets $\overline
\Omega_+, \overline \Omega_- \subset \Omega$ such that $\mu'_+ (\overline
\Omega_+ )=\mu'_- (\overline \Omega_- )=1$ and
an interaction
$\Phi$ satisfying a) and b') with $\overline \Omega
=\overline \Omega_+\cup \overline \Omega_-$ such that
$\mu'_+$ and $\mu'_-$ are
Gibbs measures for the pair ($\Phi$, $\overline \Omega$).
\end{Th}
\vspace*{5mm}
\par\noindent
{\bf Remarks.}
1. It should be emphasized that we have the {\it same} interaction for both
$\mu'_+$ and $\mu'_-$ (which, moreover, has stronger decay
properties than (\ref{2.9}), see (3.\ref{3.6}) below). Thus, the result is
different from the one of Maes and Vande Velde \cite{MV} on
the projected Gibbs measure.
2. One can distinguish different type of models where ``pathologies" occur.
Our framework
is the one where the pathologies are the weakest. In the case
of \cite{MV}, it is an open question whether
one can take the {\em same}\/ interaction for
$\mu'_+$ and $\mu'_-$.
But if one combines projection with
enough decimation, as in \cite{lorvel94},
then one knows that each of the resulting states
is Gibbsian
(in the strongest sense, i.e. with
interactions satisfying (\ref{2.6})), but for different
interactions. This in turn implies that non-trivial
convex combinations of these states are not quasilocal
everywhere, see \cite{loren95}, where other examples of
``robust" non-Gibbsianness can be found.
3. Note that in the theory of ``unbounded spins" with long range interactions, a
set $\overline \Omega$ of ``allowed" configurations has to be introduced, where
a bound like (\ref{2.9}) holds \cite{Ge}. But here, of course, contrary to the
unbounded spins models,
each $\| \Phi_X\|_\infty$ is finite. We even have the bound $\| \Phi_X\|_\infty
\leq C\beta |X|$, see (3.\ref{3.5}) below.
4. The set $\overline \Omega =\overline \Omega_+\cup \overline \Omega_-$ is not
``nice" topologically: e.g. it has an empty interior (in the usual product
topology). Besides, our effective potentials do not belong to a natural Banach
space like the one defined by (\ref{2.6}). However, this underlines the fact
that the concept of Gibbs measure is a measure - theoretic notion and
the latter often do not match with topological notions.
5. We can regard $\{ s_x\}$ as a set of quenched external fields coupled to the
spins $\sigma$ by the probability kernels $T$. The distribution of $\{s_x\}$ is
given by
$\mu'_+$ or $\mu'_-$. Then, as in most disordered systems, there is a set of
``good" configurations of the random fields ($\overline \Omega$ here) for which
the system with the spins $\sigma$ has good clustering properties. And,
implicitely, we shall use the latter to construct our Hamiltonian. This is why
part of the proof below uses the techniques of \cite{BK,GKK}. Of course, there
are also ``bad"
configurations
of the random fields for which the $\sigma$ spins do not have good clustering
properties, and those are essentially the ones used in \cite{VFS} to prove
thatthere is no absolutely summable potential for which $\mu'_+$, $\mu'_-$ are
Gibbsian.
6. To illustrate the role of the set $\overline \Omega$,
consider the (trivial)
case,
where $b=1$, and $T=\delta (\sigma_i - s_x)$ with $i=x$, i.e. the
``renormalized" system is identical to the original one (this example
was suggested to us by A. Sokal). Then $\overline \Omega$,
as constructed in our proof, will be the set of configurations such that all the
(usual) Ising contours are finite and each site is surrounded by at most a
finite number of contours. When $X$ = a contour $\gamma$ (considered as a
suitable set of sites), we let \be
\Phi_X (s_X) = 2\beta |\gamma|
\label{2.10}
\en
for $s_X$ = a configuration making $\gamma$ a contour, and $\Phi_X (s_X)= 0$
otherwise. Obviously, this
$\Phi$ satisfies (\ref{2.9}) but not (\ref{2.6}). One can write $\overline
\Omega =\overline \Omega_+ \cup \overline \Omega_-$, according to the values of
the spins in the infinite connected
component of the complement of the
contours. It is easy to see that $\mu^+$, $\mu^-$ are indeed, at low
temperatures, Gibbs measures (in the sense considered here) for this new
interaction:
a Peierls argument shows that $\mu^+ (\overline \Omega_+) = \mu^- (\overline
\Omega_-)
=1$, and for $s \in
\overline \Omega$ the (formal) Hamiltonian (2.\ref{2.1}) is
$\beta H = 2 \beta \sum_\gamma |\gamma|$. Actually, we
shall prove the Theorem by using a kind of perturbative
analysis around this example. Of course, in this example one could alternatively
take
$\overline{\Omega} = \Omega$ and $\Phi =$ the original
nearest-neighbor interaction; this shows the
{\em nonuniqueness}\/ of the
pair $(\Phi,\overline{\Omega})$ in our generalized
Gibbs-measure framework.
\section{Outline of the proof.}
\setcounter{equation}{0}
\subsection{The main Propositions.}
We shall construct the interaction $\Phi$ inductively. We shall now give the
strategy and indicate the different steps of the proof.
Consider $\mu'_+$; using (2.\ref{2.3}), one sees that the conditional
probabilities
$\mu_+'(s_V | \bar s_{V^c})$ can be obtained through the following limit, if it
exists:
\be
\mu_+' (s_V | \bar s_{V^c}) = \lim_{ \Lambda_2 \uparrow {\cal L}}
\lim_{\Lambda_1 \uparrow {\Bbb Z}^d} \;
\frac{
Z^+_{\Lambda_1}(s_V \vee \bar s_{V^c \cap{\Lambda_2}})} {\sum_{ \tilde s_V}
Z^+_{\Lambda_1}(\tilde s_V \vee \bar s_{V^c \cap{\Lambda_2}})}. \label{3.1}
\en
Where
\be
Z^+_{\Lambda_1} (s_{{\Lambda_2}}) =
\sum_{\sigma_{\Lambda_1}} \; \prod_{x\in \Lambda_2} T_x e^{-\beta
H(\sigma_{\Lambda_1} |+)}
\label{3.3}
\en
where $H(\sigma_{\Lambda_1}
|+)$ is defined as in
(2.\ref{2.1}), but with the sum restricted to $i\in \Lambda_1$, with
$\sigma_j=+1$, $\forall j\in \Lambda^c_1$. The conditional probabilities
$\mu_-'(s_V | \bar s_{V^c})$ can be obtained by similar formulas, with $+$
replaced by $-$.
We shall prove
\begin{Pro}
Under assumptions (2.\ref{2.4}, 2.\ref{2.5}),
there exists, for all $ \eta >0$,
a $\bar \beta <\infty$,
such that, $\forall \beta \geq \bar \beta $ in (2.\ref{2.1}), (2.\ref{2.5}),
there exists a set $\overline \Omega \subset \Omega$, $\overline \Omega =
\overline \Omega_+ \cup \overline \Omega_-$, and an interaction $\Phi$,
such that,
$\forall s^1, s^2 \in \overline \Omega_+$, $\forall V \subset {\cal L}$, $|V| <
\infty$
\be
\lim_{ \Lambda_2 \uparrow {\cal L}}
\lim_{\Lambda_1 \uparrow {\Bbb Z}^d}
\;\frac{Z^+_{\Lambda_1}(s^1_{\Lambda_2})} {Z^+_{\Lambda_1}(s^2_{\Lambda_2})} =
\exp \biggl( \sum_{X \cap V \neq \emptyset} \Bigl(\Phi_X (s^1_X) - \Phi_X
(s^2_X)\Bigr) \biggr) \label{3.4}
\en
if $s^1_x = s^2_x \;\; \forall x \not \in V$. The functions $\Phi_X $ satisfy:
\be
\| \Phi_X \|_\infty \leq
c \beta |X| \label{3.5}
\en
for some $c < \infty$, and, $\forall x \in {\cal L}$, $\forall s\in \overline
\Omega$,
\be
\sum_{X \ni x} | \Phi_X (s_X) | \exp
(d(X)^{1-\eta}) ={ C}(x,s)<\infty
\label{3.6}
\en
where $d(X)$ = $\mbox{\em diam} (X)$.
A formula similar to
(\ref{3.4}) holds with $+$ replaced by $-$ for all $ s^1$, $s^2 \in \overline
\Omega_-$. \end{Pro}
\vspace*{5mm}
\par\noindent
{\bf Remark.}
The factor $\exp (d(X)^{1-\eta})$ is not optimal; we could replace it by $\exp
(d(X)^{1-\eta}+ |X|^{1-\eta})$; but we expect
$|\Phi_X (s_X) |$ to
decay as $\exp (-d(X)-|X|)$.
\vspace*{5mm}
\begin{Pro}
$\mu'_+ (\overline \Omega_+) = \mu'_- (\overline \Omega_-) = 1.$ \end{Pro}
Clearly, (1) and these two Propositions imply Theorem 1.
\vspace*{5mm}
\par\noindent
{\bf Remark.}
The set
$\overline \Omega$ will be of measure zero for Gibbs measures
which are not convex combinations of
$\mu'_+$ and $\mu'_-$,
such as the non-translation-invariant Gibbs measures with interfaces
that exist for $d \geq 3$ \cite{Dob}. An open question is to find
another set $\tilde \Omega$, of $\mu'$ measure one, for
the renormalized measure corresponding to a non-translation
invariant Gibbs measure $\mu$ (for $d \geq 3$), and an interaction
with respect to which
$\mu'$ is Gibbsian.
\subsection{The contour representation.}
To prove these propositions, we shall use a ``contour", or ``polymer"
representation of
$Z^{+}_{\Lambda_1} (s_{\Lambda_2})$. But, since we regard the external spins as
random fields acting on the internal ones, we shall first define the sets where
the external spins are ``bad", namely where they change sign and exert opposite
influences on the internal spins. Let
\be
D(s)=\bigcup\{ B_x| x \in {\cal L},\; \exists y \in {\cal L} , | x-y | = b, s_x
\neq s_y \}\label{3.7}
\en
where we use throughout the paper:
\be
|x| = \max_{i=1,\cdots,d} |x_i|.\label{3.8} \en
$D(s)$ is determined by the set of (ordinary) contours of the configuration $s$.
Define also
\be
D_{\Lambda_2}\equiv D^+(s_{\Lambda_2})
\label{3.9}
\en
with
\be
D^+(s_{\Lambda_2}) =
\bigcup\{ B_x| x \in {\Lambda_2}, \;
\exists \; y \in \Lambda_2
, | x-y | = b, s_x \neq s_y \}
\bigcup\{ B_x| x \in {\partial\Lambda_2}, s_x=-1\} \label{3.10}
\en
where \be
\partial\Lambda_2= \{x \in {\Lambda_2}, d(x,\Lambda^c_2) =b\}. \label{3.10a}
\en
($d$ is the distance corresponding to ({\ref{3.8}})). $D^-(s_{\Lambda_2})$ is
defined similarly, with $s_x=+1$ instead of $s_x=-1$.
Now, we introduce the ``contours" of the internal spins: let, for each term in
({\ref{3.3}})
\be
\underline \Gamma ({\sigma}_{\Lambda_1}|+) = \bigcup\{ B_x \subset \Lambda_1|
\forall i \in B_x, \; {\sigma}_i
\neq s_x \,\; \mbox{or}\; \exists {\langle ij \rangle}, i\in B_x \;{\sigma}_i
\neq {\sigma}_j \}\bigcup D_{\Lambda_2} \label{3.11}
\en
where ${\sigma}_i = +1$ for $i \not \in \Lambda_1$.
So $\underline \Gamma$ includes the boxes $B_x$ where all the internal spins
differ from $s_x$, and
the boxes intersected
by the usual contours of the configuration ${\sigma}_{\Lambda_1}$, plus all the
sites in $D_{\Lambda_2}$. Of course, these sets are not disjoint: in a box $B_x$
belonging to $D_{\Lambda_2}$, $x\notin \partial \Lambda_2$, either all internal
spins in
$B_x$ differ from $s_x$ or all internal spins differ from $s_y \neq s_x$ in some
box $B_y$ with $|x-y|= b$ or there is a pair $\langle ij \rangle$ with
${\sigma}_i \neq {\sigma}_j$ in $B_x\cup B_y$. We include $D_{\Lambda_2}$ in the
contours, and we coarse-grain them into $b$-boxes for convenience.
Note that in all boxes $B_x$ not contained in $\underline \Gamma$, the internal
spins are constant (and, in $\Lambda_2$, are equal to the external ones).
So, one may decompose ${\underline \Gamma} = \bigcup {\underline \gamma}$ into
connected components (a subset $Y$ of
${\Bbb Z}^d$ is {\it connected} if any two points of $Y$ can be joined by a path
$(i_\alpha)$,
with $|i_\alpha-i_{\alpha+1}|=1$), and one may define contours as pairs $\gamma=
( \underline \gamma, \sigma (\gamma))$
where $ \underline \gamma$ is the {\it support} of the contour, and $\sigma
(\gamma)$ is a configuration $\{\sigma_i(\gamma)\}_{i\in \underline \gamma^c}$,
$\sigma_i(\gamma)= +1$
or $-1$, defined on the
complement of $
\underline \gamma$, which is constant on the connected components of
$\underline \gamma^c$ (this notion of contour will be slightly generalized
below).
\vs{3mm}
{\bf Definition.} A family of contours
$\Gamma$ is
{\it compatible} if the supports
of the contours are
mutually disjoint:
$\underline \gamma_1 \cap \underline \gamma_2= \emptyset$,
and if their signs match and agree with the boundary conditions on $\Lambda_1$.
\vs{3mm}
So, the notion of compatibility is as for the usual Ising contours, and if
$\Gamma$ is compatible, $\sigma_i(\Gamma)$ is unambiguously defined, $\forall i
\in \underline \Gamma^c$.
\vs{3mm}
{\bf Definition.} A family of contours
$\Gamma$ is
{\it s-compatible} if $\Gamma$ is compatible and, moreover,
$$\sigma_x(\Gamma) =s_x\;\;\; \forall x\in (\Lambda_2\cap {\cal L})\backslash
\underline \Gamma$$
\vs{3mm}
The notion of $s$-compatibility imposes a constraint due to the external spins.
For example, if all the external spins have value $+1$, a single (Ising) contour
surrounding $\Lambda_2$, with $+$ spins outside and $-$ spins inside, is
compatible but is not $s$-compatible.
One may write: \be
Z^{+}_{\Lambda_1} (s_{{\Lambda_2}}) =
({\overline T})^{|{{\Lambda_2}}|}
\sum_{\Gamma\supset D_{\Lambda_2} }\rho (\Gamma) \label{3.12}
\en
where the sum runs over $s$-compatible
families
of contours with $\underline
\Gamma\subset \Lambda_1$,
$\rho (\Gamma) = \prod_{\gamma \in \Gamma} \rho (\gamma)$ with $\rho(\gamma)=0$
if $\underline\gamma$ does not contain the connected components of
$D_{\Lambda_2}$ that it intersects and if $\sigma_x(\Gamma) \neq s_x$, for some
$x$ with $B_x$ adjacent to $\underline\gamma$; it equals \be
\rho(\gamma) = \sum^{\hspace{7mm}{\star}}_{{ \sigma}_{\underline\gamma}}
\prod_{x\in
\underline\gamma\cap {\cal L}}
\frac{T_x}{\overline T}
\exp(-\beta
H
({\sigma}_{\underline\gamma}|
{\sigma}(\gamma))\label{3.13}
\en otherwise.
Here $H
({\sigma}_{\underline\gamma}|
\sigma(\gamma))$ is defined in the same way as (2.\ref{2.7}) but with the
Hamiltonian (2.\ref{2.1}); $ \sigma_i(\gamma)$, $i\notin {\underline\gamma}$, is
fixed by the signs associated to the complement of $\underline\gamma$, and the
sum $\sum^{\star}$
runs over spin configurations
${\sigma}_{\underline\gamma}$ such that
$\gamma$ is a contour of the configuration
${\sigma}_{\underline\gamma}\vee\sigma(\gamma)$. (and
$\rho(\gamma)=0$ if the sum is empty);
$\rho(\gamma)$ is a function of the external spins $\{s_x| x\in \Lambda_2, d(x,
\underline\gamma\cap {\cal L})\leq 2 b\}$, since it vanishes unless
$\underline\gamma$ contains the connected components of $D_{\Lambda_2}$ that it
intersects (observe that the property, for a set $D_i$, to be a connected
component of $D_{\Lambda_2}$ depends
on $\{s_x| x\in \Lambda_2, d(x, D_i\cap {\cal L})\leq 2 b\}$). The fact that we
sum in (\ref{3.12}) over $s$-compatible families introduces a global constraint
on the set of contours which will be characterized
explicitly in Lemma 4.1 below.
It is easy to see that:
\be
0 \leq \rho (\gamma) \leq \exp (-\beta_0 |{\underline\gamma}\backslash
(D_{\Lambda_2}\cap \partial \Lambda_2) |) \label{3.15}
\en
where
$\beta_0=\beta_0(b, \beta)$ depends on the choice of $b$ in the definition of
${\cal L}$ and goes to infinity as $\beta$ in (2.\ref{2.1}) goes to infinity. To
prove (\ref{3.15}), observe that one gets a factor $e^{-2\beta}$ from the
Hamiltonian
(2.\ref{2.1}) for each pair
$\langle ij\rangle$ with $\sigma_i \neq \sigma_j$, and a factor $e^{-\beta}$ for
each box $B_x$ such that $\forall i \in B_x$, ${\sigma}_i \neq s_x $, from our
assumption (2.\ref{2.5}) on the probability kernels
$T_x$;
for the boxes $B_x$ in
$D_{\Lambda_2}$, but with $x\notin \partial \Lambda_2$, we use the observation
made
above that in or near each such box,
either there is a pair
$\langle ij\rangle$ with $\sigma_i \neq \sigma_j$
or all the internal spins differs from
the external one.
Finally, using the lower bound (2.\ref{2.5b}) on ${\overline T}$, we get
(\ref{3.15}) for
\be
\beta_0= c\beta,
\label{3.16}
\en
with some $c>0$ (which has to be taken
small enough because, in the above argument, we implicitely assigned the same
factor
$e^{-2\beta}$ or $ e^{-\beta}$
to different sites of $\underline\gamma$).
\subsection{Renormalization.}
Let us now introduce the coarse-grained description of the system on which our
inductive scheme is based.
Let $L > b$ be some odd integer (which will be taken large enough below). Divide
${\Bbb Z}^d$ into disjoint $L - $ boxes $\{ i |
\; | i - L x | <
\frac{L}{2}\}$ where $x \in {\Bbb Z}^d$, i.e. each $i \in {\Bbb Z}^d$ can be
written as $i=L x + j$ with $x \in {\Bbb Z}^d$ and $|j_\mu | <
\frac{L}{2}, \mu = 1, \cdots , d$ (here and below, we use the letters $x, y$ to
denote sites in the new lattices ${\Bbb Z}^d$). We define
$[L^{-1} i] = x$ and the $L-$box
of sites $i$ such that $[L^{-1} i] = x$
is denoted by $Lx$. Also for a set
$Y \subset{\Bbb Z}^d ,
[\frac{Y}{L}] = \{[L^{-1} i] | i \in Y\}$ while $LY = \cup \{ L x | x \in Y\}$.
We use a similar notation for all scales $L^n , n = 1,2,\cdots$.
We shall now describe $D_{\Lambda_2}$ and $D(s)$ on these different
coarse-grained scales.
Let us
introduce the random variables $N^n_x=N^n_x(s)$, $n=0,1,2\cdots$, defined
inductively as follows:
\ba
N^0_x &=& 2d \;\;\;\;\; \mbox{if} \;\;\;\; x \in { D}(s) \nonumber\\
N^0_x &=& 0 \;\;\;\;\;\; \mbox{otherwise.} \label{3.17}
\ea
\be
N^{n+1}_{x'} = L^{-1+\eta} \sum_{y \in
Lx' \backslash {\cal D}^n(s)} N^n_y
\label{3.17a}
\en
where
\ba
{\cal D}^n (s)=\{D_i | D_i\;
\mbox{is a connected component of}\; D^n(s),\; | D_i | \leq L^\alpha,
N^n (D_i) \leq L^{-3\alpha}
\}
\label{3.17b}
\ea
with $\eta$ as in Proposition 1, $\alpha =\frac{\eta}{4}$, $D^0(s)=D(s)$;
\be
N^n (Y) =\sum_{x \in Y} N^n_x,
\en
for $Y\subset{\Bbb Z}^d$,
and
\be
D^{n+1}(s) = \overline {[L^{-1} (D^n(s) \backslash {\cal D}^n(s))]},
\label{3.18}
\en
where, for a
set $Y\subset {\Bbb Z}^d$, we write:
\be
\overline Y= \{ i | d(i,Y)
\leq 1\}.
\label{3.14}
\en
It is easy to see
inductively that
\be
N_x^n =0 \;\;\mbox{if}\;\; x\notin D^n(s) \label{3.18a}
\en
and
\be
D^n(s) \subset \overline {\{x| N_x^n \neq 0\}} \label{3.18b}
\en
We define also variables $N_{x,\Lambda_2}^n$ and sets $D_{\Lambda_2}^{n}$,
${\cal D}_{\Lambda_2}^{n}$ $n=2,\cdots$ by the same formulas (\ref{3.17a},
\ref{3.18}), but starting with $D_{\Lambda_2}$ instead of $D(s)$ in
(\ref{3.17}).
Then,
we define, $\forall x \in {\Bbb Z}^d$,
$$
\Omega_x = \{ s \in \Omega | \exists n(x) \;\; \forall n \geq n (x) , N^n_{x} =
0 \}
$$
and
\be
\overline\Omega = \bigcap_{x \in {\Bbb Z}^d} \Omega_x \label{3.19}
\en
To understand intuitively the meaning of $\overline\Omega$, observe that
iterating the operation (\ref{3.18}) removes, at each step, the
``small" connected components
of $D(s)$, and ``glues" or ``blocks" together the ``large" ones that are not too
far from each other. Then, the configurations in $\overline\Omega$ are those for
which this operation ends, after finitely many steps, for each sequence of
$L^n$-boxes labelled by a given site of ${\Bbb Z}^d$. Note that in the
configurations used in \cite{VFS} to construct ``counterexamples", $D(s)$ covers
an infinite connected subset of the lattice (and obviously $N^n_{0}\neq 0$, for
all large enough $n$'s).
Now we can formulate the inductive representation for the partition function
which will be used to prove Proposition 1. We shall need a somewhat more
general notion of contour: here and below, a {\it contour} on scale $n$ will be
a triple
$\gamma$ =
($\underline \gamma$, $\hat \gamma$, $\sigma(\gamma)$) where $\underline\gamma$
is a connected subset of ${\Bbb Z}^d$, $\hat \gamma \subset
\underline
\gamma$ ($\hat \gamma$ is not necessarily connected) and $\sigma(\gamma)$ is a
collection of signs $\sigma_x(\gamma)$, $x\in {\Bbb Z}^d$ on the complement of
$\underline\gamma$ which
are constant on the
connected components of the complement of
$\underline\gamma$. However, when
$\underline\gamma=D_i$, for $D_i$ a connected
component of $D_{\Lambda_2}^n$,
we shall have $ \underline\gamma = \hat\gamma$, and we shall simply denote
$\gamma$ by $D_i$. For those contours,
on scale $n=0$, the signs $\sigma(D_i)$
coincide with the values of
the external spins in $D_i^c$. We shall
define below (at the end of the proof of Proposition 3)
``renormalized" values
$s_x^n$ of the external spins,
for each
$x\in [L^{-n} \Lambda_2]\backslash D^n_{\Lambda_2}$. On scale $n$, the signs
$\sigma(D_i)$
will also coincide with the values $s^n$ of the external spins in $D_i^c$.
As before, a
set of contours $\Gamma$ is {\it compatible} if
$\underline{\gamma}_1 \cap \underline{\gamma}_2= \emptyset$, $\forall
\gamma_1,\; \gamma_2 \in \Gamma$, and if the signs match (among the contours and
with the boundary conditions on $\Lambda_1$). It is {\it s-compatible}, on scale
$n$,
if, moreover,
$\sigma_x(\Gamma)=s^n_x$, $\forall x \in {\Bbb Z}^d\backslash\underline\Gamma$.
We shall derive inductively the following representation for the partition
function (\ref{3.12}):
\be
Z^+_{\Lambda_1} (s_{\Lambda_2}) =
e^{f^n_{+,\Lambda_1}(s_{\Lambda_2})
}\exp (\sum_{X \subset \Lambda_2}
\Phi^n_{X} (s_X))
\widetilde Z^+_{\Lambda_1} (s_{\Lambda_2}) \label{3.20}
\en
where $f^n_{+,\Lambda_1}(s_{\Lambda_2})$ corresponds to a ``bulk" free energy.
Since in Proposition 1, we study a ratio of partition functions, we need only to
bound
the difference between
two $f^n_{+,\Lambda_1}(s_{\Lambda_2})$, for different $s_{\Lambda_2}$'s, and
this is done in (\ref{3.26a}) below. $\Phi^n_{X}$ will converge, as $n\to
\infty$, to the interactions $\Phi_{X}$, while
\be
\widetilde Z^+_{\Lambda_1}
(s_{\Lambda_2}) = \sum_{\Gamma} \rho^n
(\Gamma) \exp (W^n({\Gamma}))
\label{3.21}
\en
where the sum runs over
$s^n$-compatible sets of contours,
with
$\underline{\Gamma} \equiv
\cup_{\gamma \in \Gamma} \underline{\gamma}\subset [L^{-n} \Lambda_1]$
and the constraint
$\widehat{\Gamma} \equiv \cup_{\gamma \in \Gamma} \widehat{\gamma}\supset
D_{\Lambda_2}^n$;
\be
\rho^n (\Gamma) = \prod \rho^n (\gamma). \label{3.23}
\en
$\rho^n (\gamma)$, is, for $n\geq1$, a function of $\{ s_x | x
\in L^n \overline{\underline \gamma} \cap\Lambda_2\}$ while
\be
W^n ({\Gamma})= \sum_{Y\subset [L^{-n} \Lambda_1]} \Psi^n (Y, \Gamma),
\label{3.22}
\en
where
$\Psi^n (Y,\Gamma)$ is, for $n\geq1$, a function of
$\{ s_x | x \in L^n \overline Y\cap \Lambda_2\}$. We shall prove that,
eventually,
$\widetilde Z^+_{\Lambda_1}
(s_{\Lambda_2}) \to 1$ as $\Lambda_2 \uparrow {\cal L}$. Proposition 1 will then
follow easily from such a representation.
In the Proposition below, we collect the bounds satisfied by $\rho^n (\gamma)$,
$\Psi^n (Y,\Gamma)$ and $f^n_{+,\Lambda_1}(s_{\Lambda_2})$:
\begin{Pro}
Under the hypotheses of Proposition 1, $\forall s_{\Lambda_2}\in
\Omega_{\Lambda_2}$, and for $n$ such that $|[L^{-n}\Lambda_2]|\geq L^d$,
(\ref{3.20}, \ref{3.21}) hold,
where $\Phi^n_{X}$ satisfies
(3.\ref{3.5}, 3.\ref{3.6}) uniformly in $n$, \be
0 \leq \rho^n (\gamma) \leq \exp
(\beta_n k_n N_{\Lambda_2}^n (\underline \gamma) - \beta_n |\underline{\gamma}
\backslash D^n_{\Lambda_2} |) \label{3.24}
\en
and, for each connected component $D_i$ of $D^n_{\Lambda_2}$,
\be
\rho^n (D_i) \geq \exp(- \beta_n k_n N_{\Lambda_2}^n (D_i)). \label{3.25}
\en
Moreover,
\be
| \Psi^n (Y, \Gamma)| \leq e^{-\beta_n |Y|}; \label{3.26}
\en
$\Psi^n (Y, \Gamma)$ depends on $\Gamma$ only through $\Gamma\cap Y\equiv
\{\gamma |\underline{\gamma}\cap Y \neq \emptyset\}$ and $\Psi^n (Y, \Gamma)= 0$
unless $Y$ is connected and
$\underline \Gamma \cap Y \neq \emptyset$.
Finally, $\forall s^1_{\Lambda_2}, s^2_{\Lambda_2} \in \Omega_{\Lambda_2}$,
with $s_x^1=s_x^2$, $\forall x\in V$, and for $n$ such that\\
$d([L^{-n}V],[L^{-n}\Lambda_2]^c)\geq L $, \be
|f^n_{+, \Lambda_1}(s^1_{\Lambda_2})-
f^n_{+, \Lambda_1}(s^2_{\Lambda_2})| \leq C|V| \exp(- (d(V,
\Lambda^c_2))^{1-2\eta})
\label{3.26a}
\en
for some $C<\infty$, and
\ba
\beta_n &=& L^{(1-\eta)n} \beta_0 \nonumber \\ k_n &=& k_0-L^{-n}
\label{3.27}
\ea
where $k_0<\infty$, $\beta_0=c\beta$ and $\eta = 4 \alpha$.
Similar formulas and bounds hold for $Z^-_{\Lambda_1}$, with
a function $f^n_{-, \Lambda_1}$ satisfying (\ref{3.26a}), and $D_{\Lambda_2}$
(=$D^+(s_{\Lambda_2})$) replaced by $D^-(s_{\Lambda_2})$.
\end{Pro}
{\bf Remarks.}
1. We shall see in the proof (eq.(4.\ref{4.11}) below) that, at each scale,
there are two contributions to $\Phi^n_X$. One is given by $\ln \rho^n (D_i) $,
and is similar to the contour energy in (2.\ref{2.10}). Theother contribution
comes from the sum over
the internal spins that introduces ``interactions" between the contours of the
external spins, and gives rise to the last term in (4.\ref{4.11}).
2. The restriction on $n$ in the Proposition is technical: for larger $n$'s, all
the statements would remain true, except that $\beta_n$ would no longer increase
as in (\ref{3.27}). However, in the limit $\Lambda_2 \uparrow {\cal L}$, the
largest value of $n$ to which the Proposition applies increases to infinity.
3. In the proofs, we shall denote by $c$ or $C$ a generic constant that
depends only on the lattice dimension or on $b$, but not on the choice of $L$.
This constant may vary from place to place. We shall use $C(L)$ to denote a
generic constant that may depend also on $L$. We shall assume that $L$ is chosen
large enough (given $\eta =4\alpha$ in Proposition 1) so that inequalities like
$C\leq L^\alpha$ can be used. Besides, we shall assume that $\beta_0 =c\beta$
(see (3.\ref{3.16})) is large enough,
so that inequalities like
$C(L)\leq \beta_0$ can be used.
\section{Proofs}
\setcounter{equation}{0}
{\bf Proof of Proposition 3}
\vspace{3mm}
The proof will be made for the $+$ boundary conditions. The proof for the $-$
boundary condition is similar, but we shall indicate which quantities may depend
on the boundary conditions. Although the proof is rather technical, the main
idea is quite simple: we cannot take directly the logarithm of $Z^+_{\Lambda_1}
(s_{\Lambda_2})$, wherever $s_{\Lambda_2}$ is not constant, because the change
of signs of $s_{\Lambda_2}$ in
${ D}_{\Lambda_2}$ introduces constraints in that
partition function, i.e. it
forces the presence of contours. We define a sort of local partition function,
$\rho ({\cal D}_{\Lambda_2})$ (see (\ref{4.1}) below), containing the contours
that are constrained only by the ``small" connected components of ${
D}_{\Lambda_2}$, and factor it out of the sum. Then the sum over the contours
that do not intersect $(D_{\Lambda_2} \backslash {\cal
D}_{\Lambda_2})$ can be exponentiated via the usual polymer formalism. The
exponent is divided into three parts: the terms that are not inside $\Lambda_2$
contribute to $f^n_{+,\Lambda_1}(s_{\Lambda_2})$
(see (\ref{4.12a})), those that are inside $\Lambda_2$ but do not depend
on the remaining contours
(i.e. those that intersect
$(D_{\Lambda_2} \backslash {\cal
D}_{\Lambda_2})$) contribute to $\Phi^n (X)$
(see (\ref{4.11})),
and finally those that depend on the remaining contours contribute to
$\Psi^n (Y,\Gamma)$ (see (\ref{4.12}, \ref{4.8a})). Then, we ``block" the
remaining contours and iterate the operation. By definition of $\overline
\Omega$, eventually
$D_{\Lambda_2}$ becomes empty, there are no constraints left
and ${\tilde Z}^+_{\Lambda_1} (s_{\Lambda_2})$ converges to $1$ (see
(\ref{4.250})).
Turning to the proof, we see that, for $n = 0$, (3.\ref{3.20}) and
(3.\ref{3.21}) follow from (3.\ref{3.12}) with $\widehat \Gamma =\underline
\Gamma$,
$\Phi^0_X = 0$,
$\Psi^0 (Y, \Gamma) = 0$, $f^0_{+,\Lambda_1}(s_{\Lambda_2})= |\Lambda_2| \ln
\overline T$ (which is independent of $s_{\Lambda_2}$, so that (3.\ref{3.26a})
holds trivially)
and
$\rho^0 (\gamma) = \rho (\gamma)$. The bounds on $\rho (\gamma)$ will be
discussed later (see proof of Lemma 5).
Now assume that the Proposition holds for $n$, and let us prove it for $n+1$. We
shall delete the indices $n$, $n+1$ and denote
by a prime the scale $n+1$.
Let
\be
\rho ({\cal D}_{\Lambda_2}) = \prod_{D_i \in {\cal D}_{\Lambda_2}} \rho (D_i)
\label{4.1}
\en
and write (3.\ref{3.21}) as:
\be
\widetilde Z = \rho ({\cal D}_{\Lambda_2}) \sum_{\Gamma}
\prod_{\gamma \in \Gamma} \overline\rho
(\gamma) \exp (W({\Gamma}))
\label{4.2}
\en
$W({\Gamma})$ was defined in (3.\ref{3.22}), and
\be
\overline\rho (\gamma) = \frac{\rho (\gamma)}{\prod_i \rho (D_i)} \label{4.3}
\en
where the product runs over $D_i \subset {\underline{\gamma}}$, $D_i \in {\cal
D}_{\Lambda_2}$. In (\ref{4.2}), we use the fact that $\underline{\Gamma}
\supset D_{\Lambda_2}\supset {\cal D}_{\Lambda_2}$.
A contour $\gamma$ is {\em small} if,
\be
V(\gamma)\not\supset [L^{-n} \Lambda_2]
\label{4.4a}
\en
and
\be
\underline \gamma \cap (D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}) =
\emptyset
\label{4.4b}
\en
where $V(\gamma)$ is the complement of the infinite connected component of
${\Bbb Z}^d \backslash \underline \gamma$. A contour is {\it large} otherwise.
Note that, unlike in \cite{BK}, \cite{GKK}, the notion of small contour does not
refer to the size of $\underline \gamma$,
but, basically, to the subset
of $D_{\Lambda_2}$
intersected by $\underline \gamma $. It is convenient to include
in the large contours those for which
$V(\gamma)\supset [L^{-n} \Lambda_2] $. Indeed, as we shall see in Lemma 1
below,
the global constraints on families of contours due to the fact that they have to
be
$s$-compatible can be expressed
entirely in terms of those contours.
This inclusion is, however, what
limits the values of $n$ in Proposition 3: when $|[L^{-n} \Lambda_2] |$ becomes
too small, all the contours are ``large" and the iteration
stops.
As we shall see, the bounds
(3.\ref{3.24}, 3.\ref{3.25}) are sufficient to control the sum over the small
contours (see Lemma 3 below). So, we rewrite the sum in (\ref{4.2}) as
\be
\sum^{\hspace{7mm}{\ell}}_{ \widehat{\Gamma}_1 \supset D_{\Lambda_2} \backslash
{\cal D}_{\Lambda_2}} \overline \rho (\Gamma_1) \exp (W({\Gamma_1}))
\sum^{\hspace{7mm}{s}}_{
\Gamma_2} \overline
\rho ( \Gamma_2) \exp (W({\Gamma_1}, {\Gamma_2})) \label{4.5}
\en
$\sum_{\Gamma_1}$ runs over all families of large contours such that $\widehat
\Gamma_1 \supset (D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2})$ and
$\sum_{\Gamma_2}$ runs
over
the set $C_s (\Gamma_1)$ of
families of small contours $\Gamma_2$ such that $\Gamma_1 \cup \Gamma_2$ is
$s$-compatible
and such that $\widehat \Gamma_1 \cup \widehat \Gamma_2 \supset D_{\Lambda_2} $.
If $C_s
(\Gamma_1) = \emptyset$, then
$\sum_{\Gamma_2}$ = 0.
Finally,
\be
W({\Gamma_1}, {\Gamma_2}) = W({\Gamma_1}\cup {\Gamma_2}) -W({\Gamma_1})=
\sum_Y \Psi (Y, {\Gamma_1}\cup {\Gamma_2} ) - \sum_Y\Psi (Y, {\Gamma_1})
\equiv \sum_Y
\Psi (Y, {\Gamma_1}, {\Gamma_2})
\label{4.6}
\en
where the sums run over $Y\subset [L^{-n}\Lambda_1]$, and the last sum runs over
$ Y \cap
{\underline \Gamma}_2 \neq \emptyset$ because $\Psi (Y,\Gamma)$ depends on
$\Gamma$ only through
$\Gamma \cap Y$.
Let us first characterize explicitely the constraint that the families of
contours have to be $s$-compatible. For that, we define $$Out (\Gamma)=\{\gamma
\in \Gamma | V(\gamma) \supset [L^{-n} \Lambda_2]\}.$$
\begin{Lem}
A compatible family of contours
$\Gamma$ such that $\widehat{\Gamma}
\supset D_{\Lambda_2}$ is $s$-compatible if and only if $Out(\Gamma)\cup
D_{\Lambda_2}$
is $s$-compatible.
\end{Lem}
The proof of this Lemma and of the other ones is given in the Appendix.
Using this Lemma, one may
characterize the families of contours that enter the sum
$\displaystyle{\sum^{\hspace{7mm}{s}}_{\Gamma_2}}$:
\begin{Lem}
If $\Gamma_1$ is
a family of contours such that
$\widehat{\Gamma}_1
\supset D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}$ and
$C_s(\Gamma_1)\neq\emptyset$, then
$C_s
(\Gamma_1)$ is the set of families of small contours $\Gamma_2$ such
that:
\ba
&1)& \underline{\Gamma}_2 \cap \underline{\Gamma}_1 = \emptyset \\ &2)&
\mbox{The signs of the contours in}\; \Gamma_2 \cup \Gamma_1 \; \mbox{match} \\
&& \mbox{among themselves and with the
boundary conditions on}\; \Lambda_1\\
&3)& {\widehat \Gamma_2} \supset ({\cal D}_{\Lambda_2} \backslash
{\widehat\Gamma}_1). \ea
\end{Lem}
We shall show that
the sum
$\displaystyle{\sum^{\hspace{7mm}{s}}_{\Gamma_2}}$
can
be exponentiated using this Lemma,
the bounds (3.\ref{3.24}, 3.\ref{3.25}) and the standard polymer formalism:
\begin{Lem}
\ba
\sum^{\hspace{7mm}{s}}_{
\Gamma_2} \overline
\rho ( \Gamma_2) \exp (W({\Gamma_1}, {\Gamma_2}))
= \exp (\sum_{Y\subset [L^{-n}\Lambda_1] } \varphi^+ (Y, {\Gamma}_1))
\label{4.7}
\ea
where $\varphi^+ (Y, {\Gamma}_1) $
is a function of $\{s_x, x \in L^n \overline{Y}\cap \Lambda_2\}$, for $n\geq 1$,
and of $\{s_x| x \in \Lambda_2, d(x, Y) \leq 2 b\}$ for $n=0$; $\varphi^+ (Y,
{\Gamma}_1)
$ depends on $\Gamma_1$ only through $\Gamma_1 \cap Y$. In particular,
$\varphi^+ (Y,
{\Gamma}_1)=\varphi^+ (Y,
\emptyset)$ if ${\underline \Gamma}_1 \cap Y = \emptyset$, and we denote it by
$\varphi^+ (Y) $ in that case.
Moreover,
$\varphi^+ (Y, {\Gamma}_1) $
satisfies the bound:
\be
|\varphi^+ (Y, {\Gamma}_1)| \leq \exp (-\beta L^{-2\alpha} |Y|) \label{4.8}
\en
and $\varphi^+ (Y, {\Gamma}_1) = 0$ unless $Y$ is connected. Finally, one may
define $\varphi^- (Y, {\Gamma}_1)$ with $-$ boundary conditions, and we have
$\varphi^+ (Y, {\Gamma}_1)=\varphi^- (Y, {\Gamma}_1)$, if $L^n\overline Y\subset
\Lambda_2$.
\end{Lem}
Now, insert (\ref{4.7}) in (\ref{4.5}). We write \bea
\sum_{Y\subset [L^{-n}\Lambda_1]}
\varphi^+ (Y, {\Gamma}_1)
&=& \sum_{Y\subset [L^{-n}\Lambda_1]} \varphi^+ (Y) + \sum_{Y\subset
[L^{-n}\Lambda_1]}(\varphi^+ (Y, {\Gamma}_1)- \varphi^+ (Y))
\chi(Y \cap {\underline{\Gamma}_1} \neq \emptyset) \nonumber\\ &=&\sum_{Y\subset
[L^{-n}\Lambda_1]} \varphi^+ (Y) +
\sum_{Y\subset
[L^{-n}\Lambda_1]}{\tilde\varphi^+} (Y, {\Gamma}_1) \eea
where
\be
{\tilde\varphi^+} (Y, {\Gamma}_1)=
(\varphi^+ (Y, {\Gamma}_1)-
\varphi^+ (Y))
\chi(Y \cap {\underline{\Gamma}_1} \neq \emptyset). \label{4.8a}
\en
We get, using (\ref{4.2}, \ref{4.5})
and writing $\Gamma$ for $\Gamma_1$,
\be
\widetilde Z = \rho ({\cal D}_{\Lambda_2}) \exp(\sum_{Y\subset
[L^{-n}\Lambda_1]} \varphi^+ (Y)) \sum^{\hspace{7mm}{\ell}}_{\widehat{\Gamma}
\supset D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}}\overline\rho (\Gamma)
\exp (W({\Gamma}) + \sum_{Y\subset
[L^{-n}\Lambda_1]} \tilde\varphi^+ (Y, {\Gamma})) \label{4.9}
\en
Write:
\ba
\sum_{Y\subset [L^{-n}\Lambda_1]} \varphi^+ (Y)= &&\sum_{ X \subset
\Lambda_2}\sum_{Y}
\varphi^+ (Y) \chi (L^n \overline Y=X)\nonumber\\ &+&
\sum_{Y\subset [L^{-n}\Lambda_1] }
\varphi^+ (Y) \chi (L^n \overline Y\cap
\Lambda^c_2 \neq \emptyset)
\ea
and a similar formula for $\ln\rho ({\cal D}_{\Lambda_2})$. Then,
using (3.\ref{3.20}), we get:
\be
Z^{+}_{\Lambda_1} (s_{{\Lambda_2}}) =
e^{f'_{+,\Lambda_1}(s_{\Lambda_2})}
\exp (\sum_{X \subset
\Lambda_2} \Phi'_{X}(s_X))
\sum^{\hspace{7mm}{\ell}}_{\widehat{\Gamma}
\supset D_{\Lambda_2} \backslash
{\cal D}_{\Lambda_2}}\overline\rho (\Gamma) \exp (\widetilde W (\Gamma))
\label{4.10}
\en
where, for $n\geq 1$, and $ X \subset
\Lambda_2$,
\ba
\Phi'_{X} = \Phi_{X} +
\sum_{D_i \in {\cal D}_{\Lambda_2}} \ln
\rho (D_i)
\chi (L^n {\overline D_i}
= X)
+ \sum_Y \varphi^+ (Y) \chi (L^n \overline Y = X) \label{4.11}
\ea
and, using (3.\ref{3.22}),
\ba
\widetilde W ({\Gamma})
=\sum_{Y\subset [L^{-n}\Lambda_1] }
\Bigl(\Psi (Y, \Gamma) + {\tilde\varphi^+} (Y, \Gamma) \Bigr) \label{4.12}
\ea
while
\ba
f'_{+,\Lambda_1}(s_{\Lambda_2})&=&
f_{+,\Lambda_1}(s_{\Lambda_2})+
\sum_{Y\subset [L^{-n}\Lambda_1]}
\varphi^+ (Y) \chi (L^n \overline Y \cap \Lambda^c_2 \neq \emptyset) \nonumber\\
&+& \sum_{D_i \in {\cal D}_{\Lambda_2}} \ln \rho (D_i)
\chi (L^n {\overline D_i}\cap
\Lambda^c_2 \neq \emptyset)
\label{4.12a}
\ea
For $n=0$, one modifies (\ref{4.11}, \ref{4.12a}) by replacing $L^n
{\overline D_i} $ by $\{x| d (x,D_i) \leq 2b\}$, and $L^n
\overline Y$ by $\{ x| d(x,Y) \leq 2b\}$. Note that all the terms in
(\ref{4.12}) vanish unless $Y\cap \underline{\Gamma}\neq \emptyset$ (using
Proposition 3 and (\ref{4.8a})).
\begin{Lem}
The functions $\Phi^n_{X}$, defined inductively by (\ref{4.11}), are functions
of $\{s_x\}_{x\in X}$, are independent of the boundary conditions on
$\Lambda_1$, and
satisfy the bounds in Proposition 1
uniformly in $n$. The limit
$\lim_{n\to \infty}\Phi^n_{X}=\Phi_{X}$ exists. Moreover,
$f^n_{+,\Lambda_1}(s_{\Lambda_2})$, defined inductively by (\ref{4.12a}), and
$f^n_{-,\Lambda_1}(s_{\Lambda_2})$ defined similarly, satisfy (3.\ref{3.26a}).
\end{Lem}
Now we shall ``block" the terms in
\be
\sum^{\hspace{7mm}{\ell}}_{\widehat{\Gamma} \supset D_{\Lambda_2} \backslash
{\cal D}_{\Lambda_2}} \overline\rho (\Gamma) \exp (\widetilde W (\Gamma))
\label{4.13}
\en
in order to obtain the representation
(3.\ref{3.21}) for $\tilde Z$ on the next scale. Let, for each term in
(\ref{4.13}),
${\widehat\Gamma}' = [L^{-1} {\underline{\Gamma}}] \bigcup D'_{\Lambda_2}$ (note
that $D'_{\Lambda_2}$ is not necessarily included in $[L^{-1}
{\underline{\Gamma}}]$, because of the
bar in (3.\ref{3.18})) and decompose
${\widehat\Gamma}'$ into connected components: ${\widehat\Gamma}'= \bigcup_i
{\hat \gamma_i}'$. Write also
\be
\widetilde W (\Gamma) = \sum^{\hspace{7mm}{\ell}}_{Y\subset [L^{-n}\Lambda_1]
}(U
(Y, \Gamma) +
\widetilde \Psi (Y, \widehat\Gamma')) +
\sum_{i} E(\hat\gamma_i', \underline \Gamma \cap \hat\gamma_i' ) \label{4.14}
\en
where,
in
$\sum^{\ell}$,
we sum only over $Y$ with $d(Y) \geq \frac{L}{4}$, and we define
\be
U (Y, \Gamma) = \Psi (Y,\Gamma) + {\tilde\varphi^+} (Y, \Gamma)
- \min_{{\Gamma}} (\Psi (Y,\Gamma) + {\tilde\varphi^+} (Y, \Gamma) ),
\label{4.16}
\en
and
\be
\widetilde \Psi (Y, {\widehat \Gamma}') = \min_{{\Gamma}} (\Psi (Y,\Gamma) +
{\tilde\varphi^+} (Y, \Gamma) ) \label{4.17}
\en
where $\min_{{\Gamma}}$ is taken over all ${\Gamma}$ such that $[L^{-1}
{\underline{\Gamma}}]\bigcup D'_{\Lambda_2} = {\widehat \Gamma}'$. Finally,
\be
E(\hat\gamma',\underline\Gamma\cap \hat\gamma') = \sum_{Y\cap L
{\hat\gamma'}\neq
\emptyset} (\Psi (Y, \Gamma) + {\tilde\varphi^+} (Y, \Gamma)) \label{4.15}
\en
where all the terms satisfy $d(Y) < \frac{L}{4}$. Note that those $Y$'s can
intersect
$L{\hat\gamma'}$ for at most one $\hat\gamma'$, since, for disconnected sets
$Y_1$, $Y_2$ in ${\Bbb Z}^d$,
$d(LY_1, LY_2) \geq L$.
Let, for $Y' \subset [L^{-(n+1)}\Lambda_1]$, \be
\Psi' (Y', {\widehat \Gamma'}) = \sum^{\hspace{7mm}{\ell}}_{[L^{-1} Y] = Y'}
\widetilde
\Psi (Y, {\widehat \Gamma'})
\label{4.18}
\en
We get
\be
(\ref{4.13}) = \sum_{{\widehat \Gamma}' \supset D'_{\Lambda_2}} \exp (\sum_{Y'
\subset [L^{-(n+1)}\Lambda_1]} \Psi' (Y', {\widehat \Gamma'}))
\sum^{\hspace{7mm}\ell}_{[L^{-1}
{\underline{\Gamma}}]\cup D'_{\Lambda_2}= {\widehat \Gamma}'} \overline\rho
(\Gamma) \exp (\sum_{i}
E(\hat\gamma'_i,\underline\Gamma\cap \hat\gamma'_i)+
\sum^{\hspace{7mm}{\ell}}_{Y } U (Y,
\Gamma)).
\label{4.19}
\en
We need to do a Mayer expansion on $\exp (\sum^\ell_{Y } U (Y,\Gamma))$ in order
to factorize the sum over $\Gamma$ in (\ref{4.19}). We note, for further use,
that, by
(\ref{4.16}),
$U (Y, \Gamma) \geq 0$. We write
\ba
\exp (\sum^{\hspace{7mm}{\ell}}_Y U (Y, \Gamma)) = \prod_Y (e^{U (Y, \Gamma)} -1
+1) =
\sum_{{\cal Y}} \prod_{Y \in {\cal Y}} (e^{U (Y, \Gamma)}-1)
\equiv \sum_{\cal Y}
V ({\cal Y}, \Gamma),
\label{4.20}
\ea
where $V ({\cal Y}, \Gamma)\geq 0$.
Insert (\ref{4.20}) in (\ref{4.19}). We can write the first
$\sum^\ell$ in (\ref{4.19}) as:
\be
\sum^{\hspace{7mm}\ell}_{[L^{-1}
{\underline{\Gamma}}]\cup D'_{\Lambda_2}={\widehat \Gamma}'} \overline\rho
(\Gamma)
\exp (\sum_{i}
E(\hat\gamma'_i,\underline\Gamma\cap \hat\gamma'_i))\sum_{\cal Y} V ({\cal Y},
\Gamma)=
\sum_{\underline\Gamma'\supset {\widehat \Gamma'}}\prod_{\gamma'}\rho' (\gamma')
\label{4.21}
\en
where
$\underline\Gamma'= {\widehat \Gamma}' \cup [L^{-1}{\cal Y}] = \cup
{\underline{\gamma}'}$ is decomposed into connected components and
\be
\rho' (\gamma') =
\sum_{(\Gamma,{\cal Y})}^{\hspace{7mm}\ell} \overline\rho (\Gamma) V ({\cal Y},
\Gamma) \exp
(\sum_{i} E(\hat\gamma'_i,\underline\Gamma\cap \hat\gamma'_i)) \label{4.22}
\en
where the sum runs over
$(\Gamma,{\cal Y})$ such that
\be
[L^{-1} \underline{\Gamma}
]\cup (D'_{\Lambda_2}\cap {\underline{\gamma}}') ={\widehat \Gamma}' \cap
{\underline{\gamma}}',
\label{4.23}
\en
\be
[L^{-1} (\underline{\Gamma}
\cup {\cal Y})]\cup (D'_{\Lambda_2}\cap {\underline{\gamma}}') =
{\underline{\gamma}}',
\label{4.24}
\en
and such that the signs $\{\sigma (\gamma)\}_{\gamma \in \Gamma}$ are the same
for all the terms in the sum. Note that all the terms in (\ref{4.22}) are
positive by (\ref{4.16}).
The factorization of the sum in (\ref{4.21}) holds because $U(Y, \Gamma)$
depends on $\Gamma$ only through $\Gamma \cap Y$, and all the terms
in (\ref{4.15}) intersect only one $\tilde{\gamma}'$. We define $\gamma'
=(\underline{\gamma'}, \hat\Gamma'\cap \underline{\gamma'}, \sigma(\gamma'))$,
where $\sigma(\gamma')$ is determined by the common signs of $\{\sigma
(\gamma)\}_{\gamma \in \Gamma}$, and \be
\Psi' (Y',\Gamma') = \Psi' (Y',\widehat \Gamma'). \label{4.25}
\en
Finally,
we define $s'_{x'}$
for each
$x'\in [L^{-n-1} \Lambda_2]\backslash D'_{\Lambda_2}$, as the (constant) value
of $s_y$, for $y\in Lx'\backslash (\cup_{D\subset{\cal D}_{\Lambda_2}}V(D))$. To
see that $s_y$ is constant, observe that $Lx' \cap ({ D}_{\Lambda_2}\backslash
{\cal D}_{\Lambda_2})=\emptyset$ since $x'\notin D'_{\Lambda_2}$, and to see
that the set $Lx'\backslash
(\cup_{D\subset{\cal D}_{\Lambda_2}}V(D))$ is not empty, notice that
$|V(D)|\leq cL^{d\alpha}< |Lx|=L^d$, for $D\subset {\cal D}^n_{\Lambda_2}$ and
$\alpha$ small. So, $s'_{x'}$ is well-defined.
With this definition of $s'$ and of $\sigma(\gamma')$, we see that the sum
(\ref{4.21}) runs over $s'$-compatible families of contours.
Inserting
(\ref{4.21}) in (\ref{4.19}), and combining (\ref{4.10}, \ref{4.19}) we get
(3.\ref{3.20}, 3.\ref{3.21}) on the
next scale
and the proof of Proposition 3 is finished with:
\begin{Lem}
$\Psi' (Y', \Gamma')$ defined by (\ref{4.18}, \ref{4.25}) and $\rho' (\gamma')$
defined by (\ref{4.22}), satisfy the claims of Proposition 3, for $\beta' =
L^{1-4\alpha} \beta $.
\end{Lem}
{\bf Remark}
Before proving Proposition 1, let us characterize $\overline{\Omega}_+$ and
$\overline{\Omega}_-$. It is easy to see that, if $s \in \overline{\Omega}$,
each connected component $D_i$ of $D(s)$ is finite. Moreover, for each $x \in
{\cal
L}$, there are at most a finite number of $D_i$'s with $x \in V (D_i)$, (where
$V(D_i) =$ the complement
of the infinite connected
component of
${\Bbb Z}^d \backslash D_i$).
Indeed, otherwise,
$\forall n$, there
exists $ m\geq n$ and a connected component $D_i$ of $D^m(s)$ such that $x \in
V(D_i)$,
$|D_i| \geq L^\alpha$,
$D_i \cap L^2 \{x\} \neq \emptyset$,
where $L^2 \{x\}$ is
the cube of size $L^2$ centered at $x$ (indeed, if this last condition is not
satisfied for some $m$, it will hold for a larger $m$, because of the
``blocking" in (3.\ref{3.18})). This would imply (by 3.\ref{3.18b}) that,
$\forall n$, there exists $ m\geq n$ and $y\in {\Bbb Z}^d $, with $|x-y|\leq L^2
+1$, and $N_y^m \neq 0$;
but since there is a finite number of such $y$'s,
this in turn means
that $s\notin \overline \Omega$.
So, combining these two
facts, we see that,
for each $s \in \overline{\Omega}$, there exists a unique infinite $b$-connected
set (a subset $Y$ of ${\cal L}$ is $b$-{\it connected} if any two points of $Y$
can be joined by a path $(x_i)$, $x_i\in {\cal L}$ with $|x_i-x_{i+1}|=b$) where
$s_x$ is of a given sign and this sign defines $\overline{\Omega}_+$ and
$\overline{\Omega}_-$.
Now we can give the
\vspace{3mm}
{\bf Proof of Proposition 1}
Let us apply Proposition 3 up to the largest $n$ such that
$|[L^{-n}\Lambda_2]|\geq L^d $. For that $n$, $[L^{-n}\Lambda_2]\subset L^2
\{0\}$, where $L^2 \{0\}$ is the cube of size $L^2$ centered at $0$. Since
$\Lambda_2 \uparrow {\cal L}$, $n\to \infty$. So, we have $[L^{-n} V] =\{0\}$,
for $n$ (i.e. $\Lambda_2$)
large, since $V$ is
fixed. Therefore, we have, for $n$ as above, $d([L^{-n}V],
[L^{-n}\Lambda_2]^c)\geq L$ and we can use (3.\ref{3.26a}).
Then, given the bounds on $\Phi_X^n$ and (3.\ref{3.26a}), it is enough to show
that
\be
\widetilde Z^+_{\Lambda_1}
(s_{\Lambda_2}) =1 + {\cal O}(e^{-c\beta_n}). \label{4.250}
\en
for $s\in \overline{\Omega}_+$.
We claim that, in that case, for $s\in \overline{\Omega}_+$ and $n$ large
enough,
\be
D^n_{\Lambda_2}=\emptyset
\label{4.2500}
\en
(which implies, by (3.\ref{3.18a}), $N^n_{\Lambda_2,x} = 0$, $ \forall x$).
Postponing the proof of (\ref{4.2500}), and using the representation
(3.\ref{3.21}), where now $\Gamma=\emptyset$, $\rho(\emptyset)=1$,
$W(\emptyset)=0$, enters the sum since
$D^n_{\Lambda_2}=\emptyset$,
we get:
\be
\widetilde Z^+_{\Lambda_1}
(s_{\Lambda_2}) =1 +\sum_{\Gamma\neq \emptyset} \rho^n (\Gamma) \exp
(W^n({\Gamma}))
\en
where, for each $\gamma$ in the sum,
$V(\gamma) \cap [L^{-n}\Lambda_2] \neq \emptyset$ (all other $\gamma$'s were
small on the first scale, see (\ref{4.4a}, \ref{4.4b})).
Now, we use the bound
(3.\ref{3.24}), with $D^n_{\Lambda_2}=\emptyset$, and $N^n_{\Lambda_2,x} = 0$,
$\forall
x$.
We use also (3.\ref{3.22}) and
$$
|W^n({\Gamma})| \leq c e^{-\beta_n}
|\underline \Gamma|,
$$
which follows from (3.\ref{3.26}), and the fact that $\Psi^n(Y, \Gamma)=0$
unless $Y$ is connected and
$Y\cap \underline \Gamma\neq \emptyset$, to get
\be
\sum_{\Gamma\neq \emptyset} \rho^n
(\Gamma) \exp (W^n({\Gamma}))
\leq \sum_{\Gamma\neq \emptyset} \exp(-\beta_n|\underline\Gamma|/2).
\label{4.251}
\en
Since for each $\gamma$ in the previous sum, $V(\gamma) \cap [L^{-n}\Lambda_2]
\neq \emptyset$, and since
$|[L^{-n}\Lambda_2]|\leq L^{2d}$, for
$n$ as above ($[L^{-n}\Lambda_2]\subset L^2 \{0\}$), we have, \be
(\ref{4.251})
\leq \sum_{k\geq 1}\Bigl(\sum_{V(\gamma)\cap [L^{-n}\Lambda_2] \neq \emptyset}
\exp(-\beta_n|\underline\gamma|/2)\Bigr)^k \leq\sum_{k\geq 1}
(e^{-c\beta_n}L^{2d})^k, \en
which proves (\ref{4.250}).
We are left with the proof of (\ref{4.2500}). Now observe that (on scale $n=0$),
if $D_i$ is a connected component
of $D_{\Lambda_2} =D^+ (s_{\Lambda_2})$
(or of $D^- (s_{\Lambda_2}))$ such that $D_i \cap \partial \Lambda_2 =
\emptyset$, then $D_i \subset D(s)$. Besides, if $s \in \overline{\Omega}_+$ and
$s_x = -1$, $x \in \partial \Lambda_2$ then $x \in V(D_i)$ for some connected
$D_i \subset
D(s)$.
Hence, if $s \in
\overline{\Omega}_+$,
$$
D^+ (s_{\Lambda_2}) \subset \bigcup
\{V(D_i) | D_i \subset D(s), V (D_i) \cap \Lambda_2
\neq \emptyset \}.$$
This implies inductively
that
\be
D_{\Lambda_2}^n \subset \bigcup
\{V(D_i) | D_i \subset D^n(s), V (D_i) \cap [L^{-n} \Lambda_2]
\neq \emptyset \}.
\label{4.252}
\en
On the other hand,
if $s \in
\overline{\Omega}_+$,
there exists $n$ such that, $\forall m \geq n$, \be
D^m (s) \cap L^2 \{0\} = \emptyset.
\label{4.25a}
\en
This implies that
\be
V(D^n(s))\cap L^2 \{0\} = \cup_{D_i \subset D^n(s)}V(D_i) \cap L^2 \{0\}
=\emptyset.
\label{4.253}
\en
Indeed, if (\ref{4.253}) does not hold, and $D^n (s) \cap L^2 \{0\} =
\emptyset$, then, because of the blocking in (3.\ref{3.18}),
$D^m (s) \cap L^2 \{0\} \neq \emptyset$, for some $m\geq n$. By taking
$\Lambda_2$ large, we may assume that this $n$ is the one chosen at the
beginning of the proof, in particular that $[L^{-n} \Lambda_2] \subset L^2 \{0\}
$. Combining this fact and (\ref{4.252}, \ref{4.253}), we get
$D^n_{\Lambda_2} = \emptyset$ i.e. (\ref{4.2500}).
\vspace{3mm}
To prove Proposition 2 and Lemma 5, we need the following Lemma. \begin{Lem}
Let $D_i$ be a connected component of $D^n$. Then \be
\sqrt{\beta_0} L^{\alpha n/2} | D_i| \leq \beta_n N^n (D_i) \leq 2d\beta_0
L^{nd} | D_i|.
\label{4.25b}
\en
A similar bound holds for $D_{\Lambda_2}^n$, $N_{\Lambda_2}^n (D_i)$. \end{Lem}
\vspace{3mm}
Now we shall prove that the configurations in $\overline\Omega_+$,
$\overline\Omega_-$ are typical for
$\mu'_+$, $\mu'_-$:
\vspace{3mm}
{\bf Proof of Proposition 2}
\vspace{3mm}
It is enough to show, see (3.\ref{3.19}), that, for any $x \in {\Bbb Z}^d$, \be
\mu'_+ (\Omega^c_x) = \mu'_- (\Omega^{c}_x) = 0, \label{4.26}
\en
because, by (3.\ref{3.15}) and a simple Peierls argument, one sees that the
probability with respect to $\mu'_+$ (resp. $\mu'_-$)
of an infinite $b$-connected set of $-$ (resp. $+$) spins is zero, hence
$\mu'_+ (\overline \Omega_-) = \mu'_- (\overline \Omega_+) = 0$. We shall prove
that, $\forall A \subset{\Bbb Z}^d$, and $\forall \{N^n_x \in L^{-(1-\eta)n}
{\Bbb N} \}_{x\in A}$,
\be
\mu'_+ (\prod_{x\in A}\chi_{N^n_x} \chi_A) \leq \exp (-\widetilde \beta_n N^n
(A))
\label{4.27}
\en
where $\chi_{N^n_x}$ means that the random variable $N^n_x(s)$ defined by
(3.\ref{3.17}, 3.\ref{3.17a}) takes the value $N^n_x$ (by (3.\ref{3.17a}),
$N^n_x
\in L^{-(1-\eta)n} {\Bbb N}$),
$\chi_A$ is the
indicator function of the event: ``$A$ is a union of connected components of
$D^n(s)$", and
\be
\widetilde \beta_n =c
L^{-\alpha n/4}
\beta_n.
\label{4.27b}
\en
Then (\ref{4.26}) follows from
\ba
&&\mu'_+ (\Omega^c_x) \leq \lim_{N\to\infty} \sum^\infty_{n=N} \mu'_+
(N^n_{x} \neq 0) \nonumber \\
&&\leq \lim_{N\to\infty} \sum^\infty_{n=N} \sum_{A\ni x}^{\hspace{7mm}{c}}
\exp(-c\sqrt{\beta_0} L^{\alpha n/4}|A|) \leq \lim_{N\to\infty} C
\sum^\infty_{n=N} \exp(-c\sqrt{\beta_0} L^{\alpha n/4})
= 0
\ea
where $\sum^c$ runs over connected sets, and, in the second inequality, we use
(\ref{4.27}) and the lower bound in (\ref{4.25b}). For $\mu'_-$, we can use the
symmetry
$\mu'_+ (\Omega^c_x) = \mu'_- (\Omega^c_x)$, which follows from (2.\ref{2.4}).
To prove (\ref{4.27}), consider first $n=0$. Using (3.\ref{3.17}), it is enough
to show:
\be
\mu'_+ (A \subset D(s))\leq \exp(-c\beta_0 |A|) \label{4.27a}
\en
We have, by definition
(3.\ref{3.7}) of $D(s)$,
\be
\mu'_+ (A \subset D(s)) = \lim_{\Lambda_2 \uparrow {\cal L}} \lim_{\Lambda_1
\uparrow {\Bbb Z}^d}
\sum^{\hspace{7mm}{A}}
\;
\frac{Z^+_{\Lambda_1} (\{s_x| d(x,A)\leq b\})}{Z^+_{\Lambda_1}} \label{4.28}
\en
where the sum runs over $\{s_x| d(x,A)\leq b\}$ such that, $\forall x \in A$, $
\exists y\in {\cal L}, \; |x-y| = b$ and $s_x \neq s_y$. Using the
representation (3.\ref{3.12}) in the numerator
of (\ref{4.28}), we get:
\be
(\overline T)^{-| \Lambda_2|} \sum^{\hspace{7mm}{A}} Z^+_{\Lambda_1} (\{s_x|
d(x,A)\leq b\})
\leq \sum_{ {\underline\Gamma}
\supset B_A} \rho (\Gamma)=
\sum^{\hspace{7mm}{1}}_{ {\underline\Gamma_1} \supset B_A}
\rho (\Gamma_1)\sum^{\hspace{7mm}{\Gamma_1}} _{\Gamma_2}\rho (\Gamma_2)
\label{4.29}
\en
where the first sum runs over $\Gamma_1$ such that $\underline\gamma\cap B_A
\neq \emptyset$, $\forall \gamma \in \Gamma_1$, with $B_A=\cup_{x\in A} B_x$,
and the second sum runs over
$\underline\Gamma_2
\cap B_A =\emptyset$, $\underline\Gamma_2 \cap \underline\Gamma_1 =\emptyset$
Now, for any $\Gamma_1$ in (\ref{4.29}), $(\overline T)^{-|\Lambda_2|}
Z^+_{\Lambda_1} \geq \sum_{\Gamma_2}^{\Gamma_1}\rho (\Gamma_2)$. Inserting this
and (\ref{4.29}) in (\ref{4.28}), and using
the bound (3.\ref{3.15}), where we can put
$\exp(-\beta_0|\underline\gamma|)$
in the RHS (by taking $\Lambda_2$ large enough), we get
\ba
\mu'_+ (A \subset D(s))& \leq& \sum^{\hspace{7mm}{1}}_{ {\underline\Gamma_1}
\supset B_A} \exp (- \beta_0 |\underline\Gamma_1|)\nonumber \\ &\leq&
\exp (- \beta_0 |A|/2) \sum^{\hspace{7mm}{1}}_{ {\underline\Gamma_1} \supset
B_A} \exp (- \beta_0 |\underline\Gamma_1|/2) \nonumber\\ &\leq &
\exp (- \beta_0 |A|/2)\exp (ce^{- \beta_0/2} |A|) \nonumber\\ &\leq & \exp (-
c\beta_0|A|)
\label{4.30}
\ea
for $\beta_0$ large enough, using ${\underline\Gamma_1} \supset B_A$
in the first inequality, and the fact that each (connected) $\underline\gamma$
in $\underline\Gamma_1$ contains a box in $B_A$ in the
second inequality, which implies
\be
\sum^{\hspace{7mm}{1}}_{\underline\Gamma_1 \supset B_A} \exp (-\beta_0
|\underline\Gamma_1|/2)
\leq
\prod_{i\in B_A} (1+\sum_{\underline\gamma \ni i} e^{-\beta_0
|\underline\gamma|/2}) \leq \exp (c'e^{-\beta_0/2} |B_A|)\leq \exp
(ce^{-\beta_0/2} |A|) \label{4.31}
\en
with $c=c'b^d$.
This proves (\ref{4.27a}), i.e. (\ref{4.27}), for $n=0$.
Then we proceed inductively: we have, by (3.\ref{3.17a}, 3.\ref{3.18}), \be
\mu'_+ (\prod_{x'\in A'}\chi_{N'_{x'}} \chi_{A'}) \leq \sum_{\overline{[L^{-1}
A]} = A',\{N_y\}}
\mu'_+ (\prod_{y\in A}\chi_{N_y}\chi_{A}) \prod_{x' \in A'} \chi (L^{-1+\eta}
\sum_{y \in L x' \cap A} N_y =
N'_{x'})
\label{4.32}
\en
Since
$A$ is a union of connected components
of $D^n(s)$, using (\ref{4.27}) on scale $n$ and Lemma 6, we have:
\be
(\ref{4.32}) \leq
\sum^{\hspace{7mm}{*}}_{A,\{N_y\}} e^{-\tilde \beta N(A)} \prod_{x' \in A'} \chi
(L^{-1+\eta} \sum_{y \in L x' \cap A} N_y = N'_{x'}) \label{4.33}
\en
where $\sum^*$
runs over $A,\{N_y\}$ such that $\tilde
\beta N(A) \geq c\sqrt{\beta_0} L^{\alpha n/4} |A|$ (see (\ref{4.25b},
\ref{4.27b})). Since $N_y \in L^{-(1-\eta)n} {\Bbb N}$, the sum over
$N_y , y
\in Lx'\cap A$ has at most $L^{(1-\eta)(n+1)} N'_{x'}$ terms. Note
also that, by (3.\ref{3.17a}, 3.\ref{3.27}), $\beta N (A) = \beta' N' (A')$,
i.e., again by (\ref{4.27b}),
$\tilde \beta N (A) \geq
\tilde
\beta'L^{\alpha/4} N' (A').$
So, using $\tilde \beta N (A) \geq \frac{\tilde \beta'}{2} L^{\alpha/4} N' (A')+
\sqrt{\beta_0} L^{\frac{\alpha n}{4}} \frac{|A|}{2}$, we have: \be
(\ref{4.33}) \leq \exp \Bigl( - \frac{\tilde \beta'}{2} L^{\alpha/4} N'
(A')\Bigr) (\prod_{x' \in A'} N'_{x'})^{L^d} \sum_{\overline{[L^{-1} A]} = A'}
L^{(1-\eta)(n+1)|A|} \exp (-\sqrt{\beta_0} L^{\frac{\alpha n}{4}}
\frac{|A|}{2}); \label{4.34}
\en
for $\beta_0$ and $L$ large, the sum over $A$ is bounded by ${\cal O} (1)$, and
we get:
\be
(\ref{4.34}) \leq \exp (-\tilde \beta' N' (A')), \label{4.35}
\en
which proves (\ref{4.27}) on scale $n+1$.
\vspace*{15mm}
\par\noindent
{\Large{\bf Appendix: Proof of the Lemmas}} \setcounter{equation}{0}
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 1}
\vspace{3mm}
First,
note
that a compatible family $ \Gamma$
that contains
an $s$-compatible family $\tilde \Gamma$ is trivially $s$-compatible:
each $x\in \underline \Gamma^c \cap {\cal L}$ belongs also to $\underline{\tilde
\Gamma}^c$ and we have $\sigma_x
(\tilde \Gamma) =s_x$ (because
$\tilde \Gamma$
is $s$-compatible) and
$\sigma_x
(\tilde \Gamma)=\sigma_x
( \Gamma)$ (because $\Gamma$ is compatible and $\tilde \Gamma \subset \Gamma$).
Now, letting
$\tilde \Gamma \equiv Out(\Gamma) \cup D_{\Lambda_2}$ this shows that, if
$Out(\Gamma) \cup D_{\Lambda_2}$ is $s$-compatible, then, $\Gamma$ is
$s$-compatible.
To prove the converse, we proceed inductively. Let us consider first $n=0$.
Assume that $\Gamma$ is an $s$-compatible
family of contours containing $D_{\Lambda_2}$. We shall show that
$ Out(\Gamma) \cup D_{\Lambda_2}$
is
$s$-compatible.
First of all, note that $Out(\Gamma) $
can be written as $(\gamma_1,\dots,\gamma_n)$ with $\underline \gamma_{i+1}
\subset Int (\gamma_i)$ and $\Lambda_2 \subset V(\gamma_n)$.
Obviously, since $\Gamma$ is compatible, and since $Out(\Gamma) $ contains all
the contours in $\Gamma$
such that $\Lambda_2 \subset V(\gamma)$, the sign in
$(V (\gamma_{i+1}))^c$ must match the one in the component of $\underline
\gamma^c_{i}$ containing $\underline \gamma_{i+1}$; therefore, $Out(\Gamma) $ is
compatible.
To show that $Out(\Gamma) \cup D_{\Lambda_2}$ is $s$-compatible, we consider the
following cases:
a) $\underline
\gamma_n\cap \Lambda_2 \neq \emptyset $. In that case the signs of
$\sigma(\gamma_n)$ must coincide with the signs of $s_x$ for all $B_x$ adjacent
to $\underline
\gamma_n$ (because $\gamma_n$
belongs to $\Gamma$, which is $s$-compatible). But since the signs of $s_x$ are
constant outside $D_{\Lambda_2}$, this implies that
$Out(\Gamma) \cup D_{\Lambda_2}$ is $s$-compatible.
b) $\underline
\gamma_n\cap \Lambda_2 = \emptyset $, which means that $\underline
\gamma\cap \Lambda_2 = \emptyset $, for all $\gamma \in Out(\Gamma)$
($\gamma_n$ is the innermost contour in $ Out(\Gamma)$). It also means that
$s_x=1$ for some
$x\in \partial \Lambda_2$: indeed, otherwise, $\partial \Lambda_2$ would
entirely belong to $D_{\Lambda_2}$, and $\partial \Lambda_2$
would be a part of a contour $\gamma\in \Gamma$ such that
$\Lambda_2 \subset V(\gamma)$; hence
this $\gamma$ would belong to
$ Out(\Gamma)$, but
obviously $\underline
\gamma\cap \Lambda_2 \neq \emptyset $, contradicting our assumption on
$\gamma_n$.
We shall show that the (constant) sign
given by $\sigma(\gamma_n)$ to $\Lambda_2 $ must be $+ 1$. Assuming this result,
we see
that this sign is compatible with those
$s_x=1$ for
$x\in \partial \Lambda_2$, and again, since the signs of $s_x$ are constant
outside
$D_{\Lambda_2}$, this implies that
$Out(\Gamma) \cup D_{\Lambda_2}$ is $s$-compatible.
To show that this sign must be $+ 1$, assume that it is $-1$. This means that
all $x\in \partial \Lambda_2$ with $s_x=1$ must be in $V(\gamma)$
for some contour
$\gamma \in \Gamma \backslash Out(\Gamma)$, since $\Gamma$ is $s$-compatible.
But since all $x\in \partial \Lambda_2$
with $s_x=-1$ belong to $\Gamma\supset D_{\Lambda_2}$, by definition, and since
$\partial \Lambda_2$ is connected, $\partial \Lambda_2$ must be in $V(\gamma)$.
But then $\Lambda_2 \subset V(\gamma)$, which contradicts the fact that $\gamma
\in \Gamma \backslash Out(\Gamma)$.
Let us now proceed inductively:
if $\Gamma'$ is an $s'$-compatible family of
contours on scale $n+1$, then there is an $s$-compatible family $\Gamma \subset
L\Gamma'$. Moreover, $Out(\Gamma )\cup
(D_{\Lambda_2}\backslash {\cal D}_{\Lambda_2}) \subset L(Out(\Gamma') \cup
D'_{\Lambda_2})$. But since, by assumption,
$Out(\Gamma )\cup
D_{\Lambda_2}$ is $s$-compatible,
$Out(\Gamma') \cup D'_{\Lambda_2}$ is $s'$-compatible because of the way
$s'$ and $\sigma(\Gamma')$ were
inductively defined, at the end of the proof of Proposition 3.
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 2}
\vspace{3mm}
Since the subset $\Gamma_1$
of large contours of an $s$-compatible family $\Gamma$ always
satisfies $Out(\Gamma_1)=Out(\Gamma)$, $C_s(\Gamma_1)\neq\emptyset$ means, by
Lemma 1,
that $\Gamma_1 \cup D_{\Lambda_2}$ is $s$-compatible. Thus, any compatible
family $\Gamma \supset \Gamma_1 \cup D_{\Lambda_2}$ is $s$-compatible, by the
argument given at
the beginning of the previous proof.
So, $C_s(\Gamma_1)$ consists of all the families of contours $\Gamma_2$
so that $\Gamma_1\cup \Gamma_2$ is compatible and contains $D_{\Lambda_2}$,
which is equivalent to the statements in the Lemma.
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 3}
\vspace{3mm}
We write
\be
\exp (W({\Gamma_1}, {\Gamma_2}))=
\prod_Y (e^{\Psi(Y, {\Gamma_1}, {\Gamma_2})}-1+1) = \sum_{{\cal Y}} \tau ({\cal
Y}, {\Gamma_1}, {\Gamma_2})
\label{1}
\en
where the product over $Y$ runs over $Y \cap { \underline \Gamma}_2 \neq
\emptyset$, see (4.\ref{4.6}), and
\be
\tau ({\cal Y}, {\Gamma_1}, {\Gamma_2}) = \prod_{Y \in {\cal Y}} (e^{\Psi (Y,
{\Gamma_1}, {\Gamma_2})}
-1).
\label{2}
\en
{}From (3.\ref{3.26}), we have
\be
|e^{\Psi(Y, {\Gamma_1}, {\Gamma_2})}-1| \leq C e^{-\beta |Y|} \label{3}
\en
and, using (4.\ref{4.3}), (3.\ref{3.24}, 3.\ref{3.25}),(3.\ref{3.17b}) and
(3.\ref{3.18a}) we have:
\be
\overline\rho (\gamma) \leq \exp (2\beta k L^{-3\alpha} | \underline{\gamma}
\cap {\cal D}_{\Lambda_2} |- \beta | \underline{\gamma} \backslash {\cal
D}_{\Lambda_2}|)
\label{4}
\en
because $\gamma$ is small, which implies
$\underline \gamma \cap (D_{\Lambda_2}
\backslash {\cal D}_{\Lambda_2}) = \emptyset.$
On the other hand,
\be
| \underline{\gamma} \backslash {\cal D}_{\Lambda_2}| \geq \frac{1}{2d}
L^{-\alpha}
| \underline{\gamma}|,
\;\;\;\; \mbox{if} \;\;\;\; {\gamma}
\not\subset {\cal D}_{\Lambda_2},
\label{5}
\en
which follows from the fact that $\underline{\gamma}$ is connected while the
connected
components of ${\cal D}_{\Lambda_2}$ contain at most $L^{\alpha}$ points, so
that, for each
connected component of ${\cal
D}_{\Lambda_2}$, there exists at least one site in $\underline\gamma
\backslash {\cal D}_{\Lambda_2}$ which is adjacent to that connected component,
and the same site can be adjacent to a fixed
number of such components (the
worst case is when
$
\underline{\gamma} \backslash {\cal D}_{\Lambda_2}$
contains only one site
separating different components of
${\cal D}_{\Lambda_2}$ of size $L^{\alpha}$). Using the fact that we also have
$|\underline{\gamma} \cap {\cal D}_{\Lambda_2}] \leq |\underline \gamma | \leq
2d L^\alpha | \underline \gamma \backslash {\cal D}_{\Lambda_2} |$, we get, for
$\gamma\notin {\cal D}_{\Lambda_2}$ ,
\be
\overline\rho (\gamma) \leq \exp (-c\beta L^{- \alpha}
| \underline{\gamma}|).
\label{6}
\en
and
\be
\overline\rho (\gamma) =1
\label{60}
\en
for $\gamma \in
{\cal D}_{\Lambda_2}$.
Now inserting (\ref{1}) in (4.\ref{4.7}), we get \be
\sum^{\hspace{7mm}{s}}_{\Gamma_2} \overline \rho ( \Gamma_2) \exp (W (\Gamma_1,
\Gamma_2)) = \sum_{\cal Z} \prod_{Z \in {\cal Z}} A (Z, {\Gamma_1})
\label{6c}
\en
where ${\cal Z}=\underline{\Gamma}_2 \cup {\cal Y}$, and $ Z=(\Gamma_2, Y)$ with
$\underline{\Gamma}_2 \cup Y$ connected;
\be
A(Z, \Gamma_1) = \overline\rho
(\Gamma_2) \tau ({\cal Y}, \Gamma_1,\Gamma_2) \label{7}
\en
with ${ \underline \Gamma}_2 \cup { Y} = Z$.
$A (Z, {\Gamma_1})$
depends on
$\Gamma_1 $ only through
$\Gamma_1 \cap Z$ and is a
function of $\{s_x | x \in L^n \overline Z\cap {\Lambda_2}\}$ for $n \geq 1$ or
of $\{s_x |x\in {\Lambda_2}, d(x,Z) \leq 2b\}$ for $n=0$, because it is a
product of factors
with these properties. Combining
(\ref{3}) and (\ref{6}, \ref{60}), we get
\be
A(Z, \Gamma_1) = 1
\en
if $Z = (D_i, \emptyset)$, with $D_i \in {\cal
D}_{\Lambda_2}$, and
\be
|A(Z, \Gamma_1)| \leq \exp (-c\beta L^{-\alpha} |Z|) \label{8}
\en
otherwise.
We shall now see how to apply the polymer formalism (see e.g. \cite{KP}).
There are constraints on $\Gamma_2$
coming from (1,2,3) in Lemma 2: to deal with constraint (3), define a polymer
as $\tilde Z =((\underline{\Gamma}_2\backslash{\cal D}_{\Lambda_2} )
\cup Y$ with $A(\tilde Z, \Gamma_1)=A(Z, \Gamma_1)$ for the corresponding $Z$,
and define $\tilde Z$ to be ``connected" if $\underline{\Gamma}_2
\cup Y$ is connected. Since
$A(Z, \Gamma_1) = 1$,
if $Z = (D_i, \emptyset)$, with $D_i \in {\cal
D}_{\Lambda_2}$, we have $A(\tilde Z, \Gamma_1) =1$ if $\tilde Z=\emptyset$ and
the bound (\ref{8}) allows us to control
the
sum over ``connected" polymers.
The remaining constraints
on $\tilde Z$ come from
(1,2) in Lemma 2. The constraint (1) gives rise to the usual
hard-core constraint between polymers.
To deal with constraint (2), observe that, since the contours in
$\Gamma_2$ are small, if $\gamma \in \Gamma_2$, and if a finite connected
component $V_i$ of $ \underline \gamma^c$
intersects $[L^{-n}\Lambda_2]$, it must be adjacent to $\underline \gamma \cap
\Lambda_2$,
and the constraint
(2) in Lemma 2 is automatically satisfied for that component of $\underline
\gamma^c$, since the signs $\sigma (\gamma)$
must agree (by definition of a contour)
with those of the internal spins
in the blocks adjacent to$\underline \gamma $. On the other hand, if a connected
component $V_i$ does not intersect $\Lambda_2 $, then the $s$-compatibility
reduces there to compatibility, as for the usual Ising contours and
the constraint can be dealt with as in that case.
The claims of the Lemma follow then from the polymer formalism (\cite{KP})
applied to (\ref{6c}), for $\beta_0$ large; $\varphi^+
(Y,\Gamma_1)$ is a sum of products of $A (Z,\Gamma_1)$, with $Z \subset Y$ so
its dependence on
$\{ s_x\}$ and on $\Gamma_1$ follows from the one of the $A(Z,\Gamma_1)$'s
mentioned above. We put
$L^{-2\alpha}$ in (4.\ref{4.8}), instead of $L^{-\alpha}$, in order to control
constants.
To prove that
$\varphi^+ (Y, {\Gamma}_1)=\varphi^- (Y, {\Gamma}_1)$, if $L^n\overline Y\subset
\Lambda_2$, observe that, if $ \underline \gamma \subset
\Lambda_2$, and
$ \underline \gamma \cap
\partial \Lambda_2=\emptyset$, the values of $ \sigma_i (\gamma)$, $i\notin
\underline \gamma$ are determined by the external spins outside $\underline
\gamma$, since, by definition, the internal spins coincide with the external
ones on the boxes adjacent to $\underline \gamma$. Hence, the value of
$\rho(\gamma)$, see (3.\ref{3.13}), is independent of the boundary
conditions on $\Lambda_1$.
It is then easy to see inductively that, on scale $n$, if
$L^n\overline Y\subset \Lambda_2$, $\varphi^+ (Y, {\Gamma}_1)$ is independent of
the boundary conditions.
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 4}
\vspace{3mm}
First, observe that
$\Phi^n_X $ is indeed a function of
$\{s_x\}_{x\in X}$, since, by
Proposition 3 and Lemma 3, $\rho (D_i)$, for $L^n
{\overline D_i}
= X$, and $\varphi^+ (Y)$, for $L^n \overline Y = X$, are
functions of $\{s_x\}_{x\in X}$. Besides, $\Phi^n_X$ is independent of the
boundary conditions on $\Lambda_1$, by the argument given at the end of
the proof of
Lemma 3.
Let us prove the bound
(3.\ref{3.6}). We consider separately each contribution to (4.\ref{4.11}), and
sum them over $n$. From (3.\ref{3.24}, 3.\ref{3.25}, 3.\ref{3.27}), we have:
\be
| \ln \rho^n (D_i)| \leq k_0\beta_n N^n_{\Lambda_2} (D_i) \leq k_0 c \beta_0
L^{nd} |{ D_i}| \leq C(L) \beta_0 L^{nd} \label{9}
\en
where, in the second inequality, we use Lemma 6, and in the third, we use
$|D_i| \leq L^{\alpha}$ since
$D_i $ belongs to ${\cal D}_{\Lambda_2}$. Besides,
\be
d(X) \leq C(L) L^n
\label{10}
\en
if $X= L^n {\overline D_i }$ and
$D_i $ belongs to ${\cal D}_{\Lambda_2}$.
Let, for $x\in {\cal L}$,
\be
n(x,s) = \max \{ n | [L^{-n} x] \in \overline{D_{\Lambda_2}^n} (s) \}.
\label{11}
\en
If $s \in \overline\Omega$, $n(x,s) <\infty$, because $[L^{-n} x] \in
\overline{D_{\Lambda_2}^n} (s)$ means, by (3.\ref{3.18b}), that there exists a
$ y$, with $d(y,[L^{-n} x])\leq 2$, and $N^n_y \neq 0$. However, for any fixed
$x$, $N^n_y = 0$ for all such $y's$, for $n$ large enough, and $s \in
\overline\Omega$.
So we get the bound
\ba
&&
\sum_n \sum_{X \ni x} \sum_{D \in {\cal D}_{\Lambda_2}^n} | \ln \rho^n
(D)| \chi (L^n \overline D = X)
\exp (d(X)^{1-\eta}) \nonumber\\
&\leq& \sum_n \sum_{D \in {\cal D}_{\Lambda_2}^n} |\ln \rho^n (D)| \chi ([L^{-n}
x] \in \overline D) \exp
(C(L) L^{n(1-\eta)}) \nonumber \\
&\leq&C(L)\beta_0
\sum^{n(x,s)}_{n=0} L^{nd} \exp (C(L)L^{n(1-\eta)}) \equiv C_1 (x,s) < \infty,
\label{12}
\ea
since, for each $n$, $[L^{-n} x] \in \overline D$ for at most one $D \in {\cal
D}_{\Lambda_2}^n$.
On the other hand,
\be
d(X) \leq C(L) L^n d(Y) \leq C(L)L^n |Y| \label{13}
\en
if $X= L^n \overline Y$ and $Y$ is connected.
So, with $\eta=4\alpha$,
\ba
&&
\sum_n \sum_{X \ni x} \sum_Y |\varphi^+ (Y)| \chi (L^n \overline Y = X) \exp
(d(X)^{(1-4\alpha)})\nonumber\\
&\leq& \sum_n \sum_{Y \ni [L^{-n} x]} |\varphi^+ (Y)| \exp
(C(L)L^{n(1-4\alpha)}|Y|)\nonumber\\ &\leq& \sum_n \sum_{Y \ni [L^{-n} x]}
|\varphi^+ (Y)| \exp (\frac{\beta_n L^{-2\alpha}|Y|}{2}),
\label{13a}
\ea
where we used
\be
C(L)L^{n(1-4\alpha)}\leq
\frac{\beta_n L^{-2\alpha} }{2},
\label{14}
\en
which holds by (3.\ref{3.27}) for all $n\geq 0$, provided $\beta_0$ is large
enough. Then, the previous sum is bounded by
\be
\sum_n
\sum^{\hspace{7mm}{c}}_{Y \ni [L^{-n} x]} \exp (-\frac{\beta_n L^{-2\alpha}
|Y|}{2})\leq C
\sum_n \exp (-\beta_n\frac{ L^{-2\alpha}}{2}) \equiv C_2
\label{15}
\en
using the bound (4.\ref{4.8}) and the fact that the sum $\sum^c_Y$ runs over
connected sets, since $\varphi^+ (Y)=0$ unless $Y$ is connected.
Now take $C(x,s) = C_1 (x,s) + C_2$ to get (3.\ref{3.6}).
Turning to the proof of (3.{\ref{3.5}),
$$
\| \Phi^n_{X} \|_\infty \leq c \beta_0 |X|, $$
follows, for the second term in (4.\ref{4.11}),
from (\ref{9}) (where $L^{nd} |D_i|
\leq |L^n \overline{D_i}] =
|X|)$, since
a given $X$ can equal $L^n \overline D_i$, for $D_i\in {\cal D}$ for at most one
$n$.
The last term in (4.\ref{4.11}) is controlled by
(4.\ref{4.8}, \ref{15}).
The existence of the limit
$n\to\infty$ follows from the bounds
(\ref{12}, \ref{13a}, \ref{15}).
Turning to the proof of (3.\ref{3.26a}), observe that the last term in
(4.\ref{4.12a}) is independent of $s_{V}$, as long as $d([L^{-n}V],
[L^{-n}\Lambda_2]^c)\geq L$, since each $\rho(D_i)$ is a function of $\{s_x|
x\in L^n \overline D_i\}$,
$L^n \overline D_i\cap \Lambda^c_2\neq \emptyset$, and $|D_i| \leq L^\alpha$.
The contribution to (3.\ref{3.26a}) of the previous term in (4.\ref{4.12a})
is bounded (on scale $n$), using (4.\ref{4.8}) and the fact that $\varphi (Y)$
is a function of $s_{V}$ only for $\overline Y \cap [L^{-n} V] \neq \emptyset$,
by:
\be
|[L^{-n}V]|\exp(-c\beta_n L^{-2\alpha}d([L^{-n}V], [L^{-n}\Lambda_2]^c))
\en
Now, use (3.\ref{3.27}), which implies
$c\beta_n L^{-2\alpha} d([L^{-n}V],
[L^{-n}\Lambda_2]^c) \geq L^{\eta n}
(d(V, \Lambda_2^c))^{1-2\eta}$ (for $\beta_0$ large) and
$|[L^{-n}V]|\leq cL^{-nd} |V|$. The factor $L^{\eta n}$
controls the sum over $n$, and we get (3.\ref{3.26a}).
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 5}
\vspace{3mm}
Consider first $\Psi' (Y',\Gamma')$. From (4.\ref{4.8a}), (3.\ref{3.26}) and
(4.\ref{4.8}) we have
\be
|\widetilde \Psi (Y,\hat \Gamma')| \leq 3 \exp (-\beta L^{-2\alpha} |Y|).
\label{17}
\en
Since $Y$ is connected and all the terms in (4.\ref{4.18}) have $d(Y) \geq
\frac{L}{4}$, we have
\be
|Y| \geq c L |Y'|.
\label{18}
\en
since $[L^{-1}Y]=Y'$.
The number of terms in the sum in (4.\ref{4.18}) is at most $2^{L^d |Y'|}$,
since $Y \subset LY'$, so, using $cL\beta L^{-2\alpha}-L^d\ln 2 \geq \beta
L^{1-4\alpha}$, for $\beta_0$ large, we get (see (4.\ref{4.25}))
\be
|\Psi' (Y',\Gamma')| \leq \exp (-\beta L^{1-4\alpha} |Y'|) \label{19}
\en
for $\beta_0$ large.
Then observe that, by induction and Lemma 3, $\tilde \Psi (Y,\hat \Gamma')=0$
unless $Y$ is connected and $Y\cap \underline\Gamma\neq \emptyset$
(see(4.\ref{4.8a})). Hence, since $[L^{-1} Y] = Y'$, $\Psi' (Y',\hat \Gamma') =
\Psi' (Y',\Gamma')=0$ unless $Y'$ is connected and
$Y'\cap \underline\Gamma'\neq
\emptyset$.
Also by
induction and Lemma 3, $\Psi (Y,\Gamma)$, $\tilde \varphi^+ (Y,\Gamma)$ depend
on
$\Gamma$ only through
$\Gamma \cap Y$; hence, since
$\underline \Gamma \subset L \hat \Gamma'$, $\tilde \Psi (Y,\hat \Gamma')$ and
$\Psi' (Y',\Gamma')$ depend on
$\Gamma'$ only through $\Gamma' \cap Y'$. Likewise, $\Psi (Y,\Gamma)$, $\tilde
\varphi^+ (Y,\Gamma)$ are functions (for $n \geq 1)$ of $\{ s_x | x \in L^n
\overline Y\cap \Lambda_2\}$ but since
$\overline Y
\subset L \overline Y'$ when $[L^{-1} Y] = Y', \Psi' (Y',\Gamma')$ is a function
of $\{ s_x | x \in L^{n+1} \overline Y'\cap \Lambda_2\}$. The same conclusion
holds for $n=0$, because
$\{x | d (x,Y) \leq 2 b\} \subseteq L \overline Y'$ when $[L^{-1} Y] = Y'$, for
$L$ large enough.
Turning to $\rho' (\gamma')$, a similar argument shows that it is a function
of $\{ s_x | x \in L^{n+1} \underline{\overline \gamma'}\cap \Lambda_2\}$. Let
us prove
(3.\ref{3.24}) inductively: observe that it holds trivially for $n=0$ because of
(3.\ref{3.15}). To proceed inductively, we use (4.\ref{4.22}): \be
\rho' (\gamma') =
\sum^{\hspace{7mm}{\ell}}_{(\Gamma, {\cal Y})} \bar \rho (\Gamma) V ({\cal
Y},\Gamma)
\exp (\sum_{i} E(\hat\gamma'_i,\underline\Gamma\cap \hat\gamma'_i))
\label{19a}
\en
with the constraints (4.\ref{4.23}) and (4.\ref{4.24}). Let us bound the
different parts of this expression; Using (4.\ref{4.15}) and the bounds
(3.\ref{3.26}), (4.\ref{4.8}), we have: \be
\sum_{i}
|E(\hat\gamma'_i,\underline\Gamma\cap
\hat\gamma'_i)| \leq C (L) e^{-\beta
L^{-2\alpha}} \sum_{i }|\hat \gamma'_i|
\leq e^{-\beta
L^{-3\alpha}} |\underline \gamma'|
\label{20}
\en
since the sum here runs over disjoint subsets $\hat\gamma'_i$of
$\underline\gamma'$, and the sum in
(4.\ref{4.15}) runs
over connected sets $Y$ that intersect $L \hat\gamma'$. Next using
(4.\ref{4.3}), the bounds (3.\ref{3.24}) and (3.\ref{3.25}), we have:
\be
\bar \rho (\Gamma) \leq \exp \Bigl( \beta k N_{\Lambda_2} ((D_{\Lambda_2}
\backslash {\cal D}_{\Lambda_2})
\cap
\underline \Gamma) + 2 \beta k N_{\Lambda_2} ({\cal D}_{\Lambda_2} \cap
\underline \Gamma) -
\beta |
\underline \Gamma \backslash D_{\Lambda_2} | \Bigr) \label{21}
\en
Since, by (3.\ref{3.18a}), $N_{\Lambda_2}(\underline \Gamma) = N_{\Lambda_2}
(D_{\Lambda_2}
\cap
\underline \Gamma) = N_{\Lambda_2}\Bigl((D_{\Lambda_2} \backslash {\cal
D}_{\Lambda_2}) \cap \underline \Gamma\Bigr) + N_{\Lambda_2} ({\cal
D}_{\Lambda_2} \cap
\underline
\Gamma).$
Now, we can write:
\be
|\underline \Gamma \backslash D_{\Lambda_2}
| \geq c L^{-\alpha} | \underline \Gamma \backslash (D_{\Lambda_2} \backslash
{\cal D}_{\Lambda_2})|. \label{23}
\en
because $\underline \Gamma \backslash D_{\Lambda_2}$ can be written as
$\underline \Gamma \backslash (D_{\Lambda_2} \backslash {\cal
D}_{\Lambda_2})\backslash {\cal D}_{\Lambda_2}$ and, for each connected
component of ${\cal D}_{\Lambda_2}$,
there is always
at least one site in $\underline \Gamma \backslash (D_{\Lambda_2} \backslash
{\cal D}_{\Lambda_2})$ which is adjacent to that component.
By definition of ${\cal D}_{\Lambda_2}$,
\be
N_{\Lambda_2} ({\cal D}_{\Lambda_2} \cap \underline{\Gamma}) \leq L^{-3\alpha} |
{\cal D}_{\Lambda_2} \cap
\underline
\Gamma | \leq L^{-3\alpha} | \underline \Gamma \backslash (D_{\Lambda_2}
\backslash {\cal D}_{\Lambda_2})|. \label{24}
\en
(since ${\cal D}_{\Lambda_2} \cap
\underline
\Gamma \subset
\underline \Gamma
\backslash (D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2})$ ). So we get,
\be
\overline \rho (\Gamma) \leq \exp
\Bigl(
\beta k N_{\Lambda_2} (
(D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}) \cap \underline \Gamma
) -
c \beta L^{-\alpha} | \underline \Gamma \backslash (D_{\Lambda_2}\backslash
{\cal D}_{\Lambda_2})|\Bigr)
\label{25}
\en
We have, by (4.\ref{4.20}), (4.\ref{4.16}),
(3.\ref{3.26}) and Lemma 3,
\be
\Bigl|V({\cal Y}, \Gamma)\Bigr| \leq \prod_{Y\in{\cal Y}} \Bigl(Ce^{-\beta
L^{-2\alpha}|Y|}\Bigr)
\leq
\exp \Bigl(-\beta L^{-3\alpha} |{\cal Y}|\Bigr) \label{26}
\en
where $|{\cal Y}| =\sum_{Y\in{\cal Y}} |Y|$, and we put $ L^{-3\alpha}$ in order
to eliminate the constant $C$.
Inserting (\ref{20}, \ref{25}, \ref{26}) in (\ref{19a}), we get
\be
\rho' (\gamma') \leq \exp \Bigl(e^{-\beta L^{-3\alpha}} | \underline \gamma' | +
\beta' k N'_{\Lambda_2} (\gamma')\Bigr) \sum_{\Gamma,{\cal Y}} \exp \Bigl(
-\beta L^{-3\alpha}|{\cal Y}|-c\beta L^{-\alpha} | \underline \Gamma \backslash
(D_{\Lambda_2}
\backslash {\cal D}_{\Lambda_2})|\Bigr)
\label{27}
\en
with the same constraints as before and we used, by definition (3.\ref{3.27})
of $\beta'$, and (3.\ref{3.17a}) of
$N'_{\Lambda_2}$,
\be
\beta N_{\Lambda_2} \Bigl((D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}) \cap
\underline
\Gamma \Bigr) =
\beta' N'_{\Lambda_2} (\underline \gamma'). \label{27a}
\en
In order to control the sum over $\underline \Gamma,{\cal Y}$, we decompose
$\underline
\Gamma \backslash (D_{\Lambda_2} \backslash {\cal D}_{\Lambda_2}) = \bigcup_i
A_i$ into connected components and call a
component {\em long}
if, either
$A_i=\underline \gamma$, with $V(\gamma) \supset [L^{-n}\Lambda_2]$, or
$A_i \cap L (\underline
\gamma'
\backslash D'_{\Lambda_2}) \neq
\emptyset$ and we call $A_i$ {\em short} otherwise.
Write ${\cal A}^\ell = \bigcup_\ell A_i$ and ${\cal A}^s = \bigcup_s A_i$, where
$\bigcup_\ell (\bigcup_s)$ is the union over the long (short) components. We
claim that
a long component satisfies $d(A_i) \geq L$; to show that, consider two cases:
either
$A_i=\underline \gamma$, with $V(\gamma) \supset [L^{-n}\Lambda_2]$,
and $d(A_i) \geq L$ because
of our restriction on $n$ in Proposition 3; or $A_i$ intersects
$L (\underline
\gamma'
\backslash D'_{\Lambda_2})$, and, since $\Gamma$ contains only long contours,
$A_i$ must intersects
$\overline{D_{\Lambda_2}
\backslash {\cal D}_{\Lambda_2}}$
(see (4.\ref{4.4b}), in which case
it must cross the
corridor of width $L$ given by $LD'_{\Lambda_2} \backslash L ([L^{-1}
(D_{\Lambda_2}\backslash{\cal D}_{\Lambda_2}])$, which implies $d(A_i) \geq L$;
Moreover,
each box in
$L (\underline \gamma'
\backslash D'_{\Lambda_2})$ is intersected either by a long $A_i$ or by a
${ Y}
\in {\cal Y}$ all of which have a diameter at least $\frac{L}{4}$; thus, we have
\be
\sum_{A_i \in {\cal A}^\ell} |A_i| + |{\cal Y}| > cL |\underline \gamma'
\backslash D'_{\Lambda_2}|.
\label{28}
\en
We write then the sum over $\Gamma, {\cal Y}$ as follows
\be
\sum_{{\cal A}^\ell,{\cal Y}} \exp \Bigl( -\beta (cL^{-\alpha} \sum_{A_i \in
{\cal
A}^\ell} | A_i | + L^{-3\alpha} |{\cal Y}|)\Bigr) \sum_{{\cal A}^s} \exp
(-c\beta L^{-\alpha} \sum_{A_i \in {\cal A}^s} |A_i|) \label{29}
\en
using (\ref{28}) and $\beta'=L^{1-4\alpha}\beta$, this sum can be bounded by
$\exp (-2\beta' |
\underline \gamma' \backslash D'_{\Lambda_2}|) \exp (e^{-\beta L^{-3\alpha}}
C(L)|\underline\gamma'|)$, for $L$ large enough, where the last factor bounds
the sums in (\ref{29}).
So, we get, going back to (\ref{27}),
\be
\rho' (\gamma') \leq \exp \Bigl(-\beta' |\underline \gamma' \backslash
D'_{\Lambda_2} | +
\beta' k N'_{\Lambda_2} (\gamma') + ( C(L)+1) e^{-\beta L^{-3\alpha}} |
D'_{\Lambda_2}\cap \underline \gamma' |
\Bigr)
\label{30}
\en
using $|\underline \gamma'| = |\underline \gamma' \backslash D'_{\Lambda_2} | +
|D'_{\Lambda_2}
\cap
\underline
\gamma' |$ and $( C(L) +1) e^{-\beta L^{-3\alpha}} \leq \beta'$.
By (3.\ref{3.18a}), we have
$N'_{\Lambda_2} (\underline \gamma')=
\sum_{D_i\subset \underline \gamma'} N'(D_i) = N'(D'_{\Lambda_2} \cap
\underline \gamma')$, and, using Lemma 6, we can bound
\be
\beta' k N'_{\Lambda_2} (\underline \gamma') + (C(L)+1) e^{-\beta L^{-3\alpha}}
| D'_{\Lambda_2} \cap
\underline \gamma' | \leq \beta' k' N'_{\Lambda_2} (\underline \gamma')
\label{31}
\en
because, by (3.\ref{3.27}), $k' - k = L^{-n} (1 - L^{-1})$ and $e^{-\beta_n
L^{-3\alpha}} << L^{-n}$. This iterates (3.\ref{3.24}).
To prove (3.\ref{3.25}), observe that it follows for $n=0$ because, see
(2.\ref{2.1}),
\be
|H_D ( \sigma_D | \tilde \sigma_{D^c}) | \leq 4d |D| \label{32}
\en
and we obtain a lower bound, with $\beta_0 k_0= 4d\beta$,
by keeping in
(3.\ref{3.12}) only
the term with $T_x = \bar T$.
Inductively, we use
\be
\rho' (D'_i) \geq \exp ( E (D_i')) \rho
\Bigl((D_{\Lambda_2}\backslash {\cal D}_{\Lambda_2}) \cap LD'_i \Bigr)
\label{33}
\en
because all terms in (4.\ref{4.22}) are positive. Then use (3.\ref{3.25})
inductively, the fact, analogous to (\ref{27a}), that \be
\beta N_{\Lambda_2} \Bigl((D_{\Lambda_2}\backslash {\cal D}_{\Lambda_2})\cap
LD'_i\Bigr) =
\beta' N'_{\Lambda_2} (D'_i)
\label{33a}
\en
(\ref{20}) and (\ref{31}) with $\gamma'=D'_i$.
\vspace*{5mm}
\par\noindent
{\bf Proof of Lemma 6}
\vspace{3mm}
The upper bound follows inductively from (3.\ref{3.17}) (for $n=0$), $$
\beta N \Bigl((D(s)\backslash {\cal D}(s))\cap LD'_i\Bigr) =
\beta' N' (D'_i)$$
and from
$$
\sum_{D_j \in (D(s) \backslash {\cal D}(s)) \cap LD'_i} |D_j| \leq L^d |D'_i|
$$
As for the lower bound, it is obvious
for $n=0$ and $\beta_0$ large. As above, we have
\be
\beta' N' (D'_i) = \sum_j \beta N (D_j)
\label{34}
\en
where $\sum_j$ runs over $D_j \in (D(s)\backslash {\cal D}(s)) \cap LD'_i$ and
we have $D'_i = \overline{[L^{-1} (\cup_j D_j)]}$. Write the sum in (\ref{34})
as $\sum^1 + \sum^2$ where in $\sum^1$, we have $| D_j | > L^\alpha$ and in
$\sum^2, |D_j | \leq L^\alpha$. In this latter case we must have $N(D_j) \geq
L^{-3\alpha} |D_j|$ since $D_j \not \in {\cal D}$ (see (3.\ref{3.17b})). For
$\sum^1$, we use Lemma 6 inductively
\be
\sum^{\hspace{7mm}{1}}_j \beta N (D_j) \geq \sqrt{\beta_0} L^{\alpha n/2}
\sum^{\hspace{7mm}{1}}_j | D_j |
\label{35}
\en
and, since we have $|D_j | \geq L^\alpha$, we get
\be
L^{\alpha n/2} \sum^{\hspace{7mm}{1}}_j |D_j| \geq L^{\alpha (n+1)/2} |
D'_{i,1}|
\label{36}
\en
with $D'_{i,1} = \overline{[L^{-1} (\bigcup^{\; 1}_j D_j)]}$. For $\sum^2$, we
have, using the definition (3.\ref{3.27}) of $\beta_n$ and the fact that each
box $Lx'$ with $x' \in D'_{i,2} = \overline{[L^{-1} (\bigcup^2_j D_j)]}$ must
contain or be adjacent to a box
intersected by some $D_j$ in $\sum^2$:
\ba
\sum^{\hspace{7mm}{2}}_j \beta N (D_j) &\geq& L^{-3\alpha} \beta
\sum^{\hspace{7mm}{2}}_j |D_j|\nonumber\\ &\geq& \beta_0 L^{(1-4\alpha)n}
L^{-3\alpha} c |D'_{i,2}|\nonumber\\ &\geq& \sqrt{\beta_0} L^{\alpha(n+1)/2}
|D'_{i,2}| \label{37}
\ea
for $\alpha$ small enough and $\beta_0$ large enough. Obviously
(\ref{34}-\ref{37}) imply the lower bound in (4.\ref{4.25b}) on scale $n+1$
since
\be
|D'_{i,1} | + |D'_{i,2}|\geq |D'_i |.
\label{38}
\en
The same arguments hold for $D^n_{\Lambda_2}$, $N^n_{\Lambda_2}(D_i)$.
\vs{10mm}
\no{\Large\bf Acknowledgments}
\vs{10mm}
We would like to thank A. van Enter, R. Fernandez, C. Maes, C.-E. Pfister, A.
Sokal,
and K. Vande Velde
for
interesting discussions. This work was supported by NSF grant DMS-9205296 and by
EC grant CHRX-CT93-0411.
\addcontentsline{toc}{section}{\bf References} \begin{thebibliography}{10}
\bibitem{benmaroli}
G.~Benfatto, E.~Marinari and E.~Olivieri. \newblock Some numerical results on
the block spin transformation for the $2d$
{I}sing model at the critical point.
\newblock {\em J. Stat. Phys.}, {\bf 78}:731--757, 1995.
\bibitem{BK}
J.~Bricmont and A.~Kupiainen.
\newblock Phase transition in the 3d random field Ising model. \newblock {\em
Commun. Math. Phys.}, {\bf 116}:539--572, 1988.
\bibitem {ciroli96}
E.~N.~M.~Cirillo and E.~Olivieri.
\newblock Renormalization-group at criticality and complete analyticity of
constrained models: a numerical study.
\newblock to appear in {\em J. Stat. Phys.}, 1997.
\bibitem{Dob}
R.~L.~Dobrushin.
\newblock Gibbs states describing a coexistence of phases for the
three-dimensional
Ising model.
\newblock {\em Th. Prob. and its Appl.} {\bf 17}:582-600,
1972.
\bibitem{Do}
R.~L.~Dobrushin.
\newblock Lecture given at the workshop ``Probability and Physics'', Renkum,
(Holland), 28 August- 1 September, 1995.
\bibitem{domgre}
C.~Domb and M.~S.~Green (Eds.).
\newblock {\em Phase transitions and critical phenomena}, Vol.6. \newblock
Academic
Press, New York, 1976.
\bibitem{FP}
R.~Fern\'andez and C.-Ed.~Pfister.
\newblock Global specifications and
non-quasilocality of projections of {G}ibbs measures.
\newblock EPFL preprint, 1996.
\bibitem{GKK}
K.~Gaw{\c e}dzki, R.~Koteck{\'y} and A.~Kupiainen. \newblock
Coarse graining approach to first order phase transitions.
\newblock {\em J. Stat. Phys.} {\bf 47}:701-724,
1987.
\bibitem{Ge}
H.-O.~Georgii.
\newblock {\em Gibbs Measures and Phase Transitions}. \newblock Walter de
Gruyter (de Gruyter Studies in Mathematics, Vol.\ 9), Berlin--New York, 1988.
\bibitem{gol}
N.~Goldenfeld.
\newblock {\em Lectures on phase transitions and the renormalization group.}
\newblock Addison-Wesley, Frontiers in Physics {\bf 85}, 1992.
\bibitem{Gr}
R.~B.~Griffiths
\newblock Nonanalytic behavior above the critical point in a random Ising
ferromagnet
\newblock {\em Phys. Rev. Lett.}, {\bf 23}:17, 1969.
\bibitem{GP1}
R.~B.~Griffiths and P.~A.~Pearce.
\newblock Position-space renormalization-group transformations: {S}ome proofs
and some problems.
\newblock {\em Phys. Rev. Lett.}, {\bf 41}:917--920, 1978.
\bibitem{GP2}
R.~B.~Griffiths and P.~A.~Pearce.
\newblock Mathematical properties of position-space renormalization-group
transformations.
\newblock {\em J. Stat. Phys.}, {\bf 20}:499--545, 1979.
\bibitem{Ke}
K.~Haller and T.~Kennedy.
\newblock Absence of renormalization group pathologies near the critical
temperature--two examples.
\newblock {\em J. Stat. Phys.}, {\bf 85}:607--637, 1996.
\bibitem{hashas}
A.~Hasenfratz and P.~Hasenfratz.
\newblock Singular renormalization group transformations and first order phase
transitions (I).
\newblock {\em Nucl. Phys. B}, {\bf 295}[FS21]:1-20, 1988.
\bibitem{Is}
R.~B.~Israel.
\newblock Banach algebras and {K}adanoff transformations. \newblock In J.~Fritz,
J.~L.~Lebowitz, and D.~Sz{\'a}sz, editors, {\em Random Fields (Esztergom, 1979),
Vol.\ II}, pages 593--608. North-Holland, Amsterdam, 1981.
\bibitem{ken92}
T.~Kennedy.
\newblock Some rigorous results on majority rule renormalization group
transformations near the critical point. \newblock {\em J. Stat. Phys.}, {\bf
72}:15--37,
1993.
\bibitem{KP}
R.~Kotecky, and D. ~Preiss.
\newblock Cluster expansion for abstract polymer models. \newblock {\em Commun.
Math. Phys.}, {\bf 103}:491--498, 1986.
\bibitem{lor94}
J.~L{\"o}rinczi.
\newblock Some results on the projected two-dimensional {I}sing model. \newblock
In M.~Fannes, C.~Maes, and A.~Verbeure,
editors, {\em Proceedings NATO ASI
Leuven Workshop ``On Three Levels''},
pages 373--380, Plenum Press, 1994.
\bibitem{lorvel94}
J.~L{\"o}rinczi and K.~Vande Velde.
\newblock A note on the projection of {G}ibbs measures. \newblock {\em J.\
Stat.\ Phys.}, {\bf 77}:881--887, 1994.
\bibitem{lorwin92}
J.~L{\"o}rinczi and M.~Winnink.
\newblock Some remarks on almost {G}ibbs states. \newblock In N.~Boccara,
E.~Goles, S.~Martinez, and P.~Picco, editors, {\em Cellular Automata and
Cooperative Systems}, pages 423--432, Kluwer, Dordrecht, 1993.
\bibitem{maevel92}
C.~Maes and K.~Vande Velde.
\newblock Defining relative energies for the projected Ising measure. \newblock
{\em Helv. Phys. Acta}, {\bf 65}:1055--1068, 1992.
\bibitem{MV}
C.~Maes and K.~Vande Velde.
\newblock The (non-)Gibbsian nature of states invariant under stochastic
transformations.
\newblock {\em Physica A}, {\bf 206}:587--603, 1994.
\bibitem{maevel96}
C.~Maes and K.~Vande Velde.
\newblock Relative energies for non-Gibbsian states. \newblock Leuven preprint,
1996.
\bibitem{MO1}
F.~Martinelli and E.~Olivieri.
\newblock Some remarks on pathologies of renormalization-group transformations.
\newblock {\em J. Stat. Phys.}, {\bf 72}:1169--1177, 1993.
\bibitem{MO2}
F.~Martinelli and E.~Olivieri.
\newblock Instability of renormalization-group pathologies under decimation.
\newblock {\em J. Stat. Phys.}, {\bf 79}:25--42, 1995.
\bibitem{sch87}
R.~H.~Schonmann.
\newblock Projections of Gibbs measures may be non-Gibbsian. \newblock {\em
Comm. Math. Phys.}, {\bf 124}:1--7, 1989.
\bibitem{vE96}
A.~C.~D.~van Enter.
\newblock Ill-defined block-spin transformations at arbitrarily high
temperatures. \newblock {\em J. Stat. Phys.}, {\bf 83}:761--765, 1996.
\bibitem{vEFK_JSP}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and R.~Koteck{\'y}. \newblock Pathological
behavior of renormalization group maps at high fields and above the transition
temperature.
\newblock {\em J. Stat. Phys.}, {\bf 79}:969--992, 1995.
\bibitem{VFS1}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and A.~D. Sokal. \newblock Renormalization
transformations in the vicinity of first-order phase transitions: {W}hat can and
cannot go wrong. \newblock {\em Phys. Rev. Lett.}, {\bf 66}:3253--3256, 1991.
\bibitem{VFS}
A.~C.~D. van Enter, R.~Fern{\'a}ndez, and A.~D. Sokal. \newblock Regularity
properties and pathologies of position-space renormalization-group
transformations: Scope and limitations of {G}ibbsian theory. \newblock {\em J.
Stat. Phys.}, {\bf 72}:879--1167, 1993.
\bibitem{loren95}
A.~C.~D.~van Enter and J.~L\"orinczi.
\newblock Robustness of the non-Gibbsian property: some examples. \newblock
University of Groningen preprint, 1995, {\em J. Phys. A, Math. and Gen.}, {\bf
29}: 2465--2473, 1996.
\end{thebibliography}
\end{document}