Oriented Percolation in One--Dimensional $\beta /|x-y|^{2}$\\ Random Cluster Model at $\beta >1$
D. H. U. Marchetti\thanks{% Instituto de F\'{\i}sica
Universidade de S\~{a}o Paulo
Caixa Postal 66318
05315 S\~{a}o Paulo
SP
\; V.{\ }Sidoravicius\thanks{% Instituto de Matem\'{a}tica Pura e Aplicada (IMPA)
Estrada Dona Castorina 110
Jardim Bot\^{a}nico
22460-320 Rio de Janeiro
RJ
Rua Dr. Xavier Sigaud 150
22290-180 Rio de Janeiro
We consider the Fortuin-Kasteleyn random cluster model on $\mathbb{Z}$, with edge occupation probabilities \begin{equation*} p_{\{x,y\} }=\left\{ \begin{array}{ll} p & \mathrm{if}\;\left\vert x-y\right\vert =1, \\ 1-\exp \left\{ -\beta /\left\vert x-y\right\vert ^{2}\right\} & \mathrm{% otherwise,}% \end{array}% \right. \end{equation*}% and weighting factor $\kappa \geq 1$. We prove that oriented percolation occurs when $\beta >1$ and $\kappa \geq 1$ provided $p$ is chosen sufficiently close to $1$. In the particular case of the product measure ($% \kappa =1$), this answers a question posed in \cite{NS}. The proof is based on multi-scale analysis.
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\begin{document}

\author{D. H. U. Marchetti\thanks{%
Instituto de F\'{\i}sica, Universidade de S\~{a}o Paulo, Caixa Postal 66318,
05315 S\~{a}o Paulo, SP, Brasil. E-mail:\textit{\ marchett@if.usp.br} } , \;
V.{\ }Sidoravicius\thanks{%
Instituto de Matem\'{a}tica Pura e Aplicada (IMPA), Estrada Dona Castorina
110, Jardim Bot\^{a}nico, 22460-320 Rio de Janeiro, RJ, Brasil. E-mail:%
\textit{\ vladas@impa.br}} \; and \; M. E. Vares\thanks{%
Centro Brasileiro de Pesquisas F\'{\i}sicas (CBPF), Rua Dr. Xavier Sigaud
150, 22290-180 Rio de Janeiro, RJ, Brasil. E-mail: \textit{eulalia@cbpf.br}}
}
\title{Oriented Percolation in One--Dimensional $\beta /|x-y|^{2}$\\
Random Cluster Model at $\beta >1$}
\date{}
\maketitle

\begin{abstract}
We consider the Fortuin-Kasteleyn random cluster model on $\mathbb{Z}$, with
edge occupation probabilities
\begin{equation*}
p_{\{x,y\} }=\left\{
\begin{array}{ll}
p & \mathrm{if}\;\left\vert x-y\right\vert =1, \\
1-\exp \left\{ -\beta /\left\vert x-y\right\vert ^{2}\right\} & \mathrm{%
otherwise,}%
\end{array}%
\right.
\end{equation*}%
and weighting factor $\kappa \geq 1$. We prove that oriented percolation
occurs when $\beta >1$ and $\kappa \geq 1$ provided $p$ is chosen
sufficiently close to $1$. In the particular case of the product measure ($%
\kappa =1$), this answers a question posed in \cite{NS}. The proof is based
on multi-scale analysis.
\end{abstract}

\section{Introduction \label{0}}

\setcounter{equation}{0} \setcounter{theorem}{0}

It is well known that $1/r^{2}$ gives the \textquotedblleft critical"
falloff for percolation in one-dimensional long range independent edge
percolation models. Moreover, for the one dimensional Fortuin--Kasteleyn
(FK) random cluster model with weighting factor $\kappa \geq 1$ and edge
occupation probabilities of the form $p_{\{x,y\}}=f(\left\vert
x-y\right\vert )$, with $\beta :=\lim\limits_{x\rightarrow \infty
}x^{2}\,f(x)>0$, it is known that for fixed $f(j)<1,j\geq 2$ and varying $%
p=f(1)$, the value $\beta ^{\ast }=1$ is critical in the sense that for $%
\beta \leq 1$ percolation cannot occur unless $p=1$ (see \cite{AN}), while
for $\beta >1$ there is percolation provided $p$ is sufficiently close to
one (see \cite{IN} and \cite{M}). These results are important in the
description of the phase transition diagram for the one-dimensional long
range Ising models studied earlier by Fr\"{o}hlich and Spencer in \cite{FS1}
and for the corresponding Potts models (\cite{ACCN},\cite{IN},\cite{M}), as
these spin systems can be constructed by a random coloring of the clusters
in the FK model with $\kappa =2$, or $\kappa >2$ integer, respectively. For
the particular case of independent edge percolation models ($\kappa =1$)
earlier results were obtained in \cite{NS}, where it was proven that $\beta
^{\ast }\leq 1$ in this case, and that \textit{oriented percolation} occurs
when $\lim\limits_{x\rightarrow \infty }x^{s}\,f(x)>0$ for some $1<s<2$. The
question whether oriented percolation occurs in the boundary case $s=2$
remained unanswered. Theorem \ref{percolation} below gives an affirmative
answer applicable to the FK model: the result is stated for the particular
example of edge probabilities in (\ref{occup}) below, and oriented
percolation is shown when $\beta >1$ and $p<1$ is sufficiently close to one.
The proofs can be easily adapted to include any $f(\cdot )$ satisfying $%
\lim\limits_{x\rightarrow \infty }x^{2}\,f(x)>1$. In this sense $\beta
^{\ast }=1$ remains critical also for oriented percolation.

\noindent \textbf{Preliminaries.} Consider the infinite complete graph with
set of vertices $\mathcal{V}=\mathbb{Z}$ and set of edges $\mathbb{E}%
=\{\{x,y\},\;x\neq y,\;x,y\in \mathbb{Z}\}$, and let $\Omega =\{0,1\}^{%
\mathbb{E}}$. One-dimensional long-range FK random cluster models with
weighting parameter $\kappa >0$ are probability measures on $\sigma (\Omega
) $, the product $\sigma $--algebra on $\Omega $. To define them, let us
first fix $\nu $ the Bernoulli product measure on $\Omega $, with $\nu
(\omega _{\{x,y\}}=1)=p_{\{x,y\}}$ given by

\begin{equation}
p_{\{x,y\} }=\left\{
\begin{array}{ll}
p & \mathrm{if}\;\left\vert x-y\right\vert =1, \\
1-\exp \left\{ -\dfrac{\beta }{\left\vert x-y\right\vert ^{2}}\right\} &
\mathrm{otherwise,}%
\end{array}%
\right.  \label{occup}
\end{equation}
where $0<p<1$ and $\beta >0$ are fixed parameters.

\noindent \textbf{Notation.} We write $q_{\{x,y\} }=1-p_{\{ x,y\} }$; for $%
e=\{x,y\} $ we will write $p_{e}$ instead of $p_{\{x,y\} }$, and say that $e$
\textquotedblleft is open\textquotedblright\ if $\omega _{e}=1$. The length
of an edge $e=\{x,y\}$ is $|x-y|$.

\noindent \textit{Finite volume FK measure.} Given $I\subset \mathbb{Z}$,
consider $\mathbb{E}(I)=\{\{x,y\}\in \mathbb{E}\colon x,y\in I\}$, $\Omega
_{I}=\left\{ 0,1\right\} ^{\mathbb{E}(I)}$, and $\bar{\Omega}_{I}=\left\{
0,1\right\} ^{\mathbb{E}\setminus \mathbb{E}(I^{c})}$, where $I^{c}=\mathbb{Z%
}\backslash I$. Assume that $|I|<\infty $, and fix an \textquotedblleft
external\textquotedblright\ configuration $\tilde{\omega}\in \left\{
0,1\right\} ^{\mathbb{E}(I^{c})}$. The finite volume FK-measure with
external conditions $\tilde{\omega}$ is the probability measure $\mu
_{\kappa ,I}^{\tilde{\omega}}$ on $\bar{\Omega}_{I}$, given by
\begin{equation}
\mu _{\kappa ,I}^{\tilde{\omega}}(A)=\frac{\displaystyle\int_{A}\kappa ^{%
\mathcal{C}_{I}(\omega \circ \tilde{\omega})}~\bar{\nu}_{I}\left( d\omega
\right) }{\displaystyle\int_{\bar{\Omega}_{I}}\kappa ^{\mathcal{C}%
_{I}(\omega \circ \tilde{\omega})}~\bar{\nu}_{I}\left( d\omega \right) }\ ,
\label{mukappa}
\end{equation}%
where $\bar{\nu}_{I}$ is the projection of $\nu $ on $\bar{\Omega}_{I}$, the
configuration $\omega \circ \tilde{\omega}$ is given by $(\omega \circ
\tilde{\omega})_{e}=\omega _{e}$ if $e\in \mathbb{E}\setminus {\ \mathbb{E}%
(I^{c})}$ and $(\omega \circ \tilde{\omega})_{e}=\tilde{\omega}_{e}$ if $%
e\in {\ \mathbb{E}(I^{c})}$, and for any configuration $\omega \in \Omega $,
$\mathcal{C}_{I}(\omega )$ is the number of disjoint connected components
intersecting $I$ in the graph determined by this configuration (i.e. the
graph whose edges coincide with those $e$ such that $\omega _{e}=1$).

Of course we may think of $\mu_{\kappa ,I}^{\tilde{\omega}}$ as a
probability measure on $\Omega _{I}^{ \tilde{\omega}}=\left\{ \omega \in
\Omega : \omega _{e}=\tilde{\omega}_{e},\text{if } e \in {\mathbb{E}(I^c)}%
\right \}$.

Several delicate issues are present in the passage to the infinite volume
limit, specially when $0<\kappa <1$. (See \cite{G1}, \cite{G2}.)

\noindent {\textbf{Throughout the paper we will assume that $\kappa \geq 1$.}%
} Two external configurations $\tilde{\omega}$ play a particular role when
considering the infinite volume limit:

$\tilde{\omega}_{e}=0$ for all $e\in \mathbb{E}(I^{c})$, and the
corresponding measure $\mu _{\kappa ,I}^{\tilde{\omega}}\equiv \mu _{\kappa
,I}^{f}$ is called the \textquotedblleft free\textquotedblright\ measure;

\smallskip

\noindent and \smallskip

$\tilde{\omega}_{e}=1$ for all $e\in \mathbb{E}(I^{c})$, and the
corresponding measure $\mu _{\kappa ,I}^{\tilde{\omega}}\equiv \mu _{\kappa
,I}^{w}$ is called the \textquotedblleft wired\textquotedblright\ measure.

\medskip

\noindent \textit{The infinite volume limit.} On $\Omega $ we consider the
usual partial order: $\omega \leq \omega ^{\prime }$ if $\omega _{e}\leq
\omega _{e}^{\prime }$ for each $e\in \mathbb{E}$. By the FKG inequality
(see \cite{F}, \cite{ACCN}), one has
\begin{equation*}
\mu _{\kappa ,I}^{f}(g)\leq \mu _{\kappa ,I^{\prime }}^{f}(g)\leq \mu
_{\kappa ,I^{\prime }}^{\tilde{\omega}}(g)\leq \mu _{\kappa ,I^{\prime
}}^{w}(g)\leq \mu _{\kappa ,I}^{w}(g)
\end{equation*}%
for any finite intervals $I\subset I^{\prime }\subset\mathbb{Z}$, any
external configuration $\tilde{\omega}$ and any non-decreasing continuous
function $g:\Omega \rightarrow \mathbb{R}$. Thus, as $I\nearrow \mathbb{Z}$
the limit measures $\mu _{\kappa }^{f}$ and $\mu _{\kappa }^{w}$ exist.
Moreover, $\mu _{\kappa }^{f}\le \mu _{\kappa }^{w}$ in FKG sense\footnote{%
That is $\mu \le \mu^\prime$ if $\mu(g) \le \mu^\prime(g)$ for any $g$
continuous and increasing.}, and $\mu _{\kappa }^{f}=\mu _{\kappa }^{w}$ if
and only if there is a unique accumulation point of the set $\mu _{\kappa
,I}^{\tilde{\omega}}$ as $I\nearrow \mathbb{Z}$. If $\kappa =1$, trivially $%
\mu _{\kappa }^{f}=\mu _{\kappa }^{w}=\nu $. Since $p_{\{x,y\} }\equiv
f(|x-y|)$ the measures $\mu _{\kappa }^{f}$ and $\mu _{\kappa }^{w}$ are
translation invariant; both these measures are ergodic. For a more general
discussion on the construction of random cluster measures see e.g. \cite{G1}.

\noindent Fix $\omega \in \Omega $. An alternating sequence of vertices and
edges $x=x_{1},e_{1},x_{2},\dots ,x_{n-1},e_{n-1},x_{n}=y,\;n\geq 1,$ is
called a path connecting $x$ to $y$, and we say that the path is open if $%
\omega _{e_{i}}\equiv \omega _{\{ x_{i},x_{i+1}\} }=1$, $1\leq i\leq n-1$.
We say that $C\subset \mathbb{Z}$ is connected if for any two distinct
vertices $x,y$ in $C$ there exists an open path $\pi $ connecting them. A
maximal connected set is called an \textit{open cluster}, and $C_{x}(\omega
) $ denotes the open cluster containing $x\in \mathbb{Z}$ (we write $%
C_{x}(\omega )=\{x\}$ if $\omega _{\{x,y\} }=0$, for all $y\in \mathbb{Z}%
\setminus \{x\}$). A path $\pi =\left( x_{1},\dots ,x_{n}\right)$ connecting
$x$ to $y,\;x<y$, is called \textit{oriented} if $x_{1}=x<x_{2}<\dots
<x_{n-1}<x_{n}=y$, and we write $x\rightsquigarrow y$ when there is an open
oriented path connecting $x$ to $y$. Analogously we define $%
C_{x}^{+}=\{y\colon x\rightsquigarrow y\}$, and the event
\begin{equation*}
\lbrack x\rightsquigarrow \infty ]=[|C_{x}^{+}|=\infty ].
\end{equation*}

For $L\geq 1$ integer, and use $\mu _{\kappa ,L}^{\tilde{\omega}}$ as a
shorthand for $\mu _{\kappa ,[-L,L]}^{\tilde{\omega}}$.

We are ready to state our main result.

\begin{theorem}
\label{percolation} For any $\beta >1$,$\ \kappa \geq 1$, there exist $%
0<p_{0}<1$ such that, if $p>p_{0}$, then
\begin{equation}
\mu _{\kappa}^{f}(0\not\rightsquigarrow \infty )\leq \epsilon ,  \label{perc}
\end{equation}%
with $\epsilon =\epsilon (p)\searrow 0$ as $p\nearrow 1$.
\end{theorem}

\noindent {\textbf{Remarks}}. \textbf{1}. Since $\mu _{\kappa
}^{f}\leq \mu _{\kappa }^{w}$, the previous statement holds as well
for $\mu _{\kappa }^{w} $.

\noindent \textbf{2}. The proof of this theorem can be adapted to
establish an analogue of Theorem $1.2$ of \cite{M} on the power law
decay of truncated connectivity function,  $\tau ^{\prime }(x):=$
$\mu _{\kappa }^{w}(0\rightsquigarrow x,0\not\rightsquigarrow \infty
)$ (see \cite{IN} and Remarks after Theorem $1.2$ in [M] for the
significance of this quantity in the non-oriented case).  To get the
right decay is still an open problem in the oriented case.


\noindent \textbf{Some related problems.} The type of questions
treated here has various sources of interest and we mention only a
couple of them, which have to do with our own motivations. Consider
the following physical problem: take the one-dimensional Ising model
with pair interactions, the couplings decaying as the square of the
distance between vertices, at inverse temperature $\beta >1$; this
is the model studied by Fr\"{o}hlich and Spencer (\cite{FS1}), for
which a phase transition was established. Take now the finite box
$[-L,L]$ and assume the Dobrushin boundary conditions, i.e. all
spins in $(-\infty ,-L]$ will be taken as $+1$, and all spins in
$[L,+\infty )$ will be taken as $-1$. What can we say about the
behaviour of this model when $L\rightarrow \infty $? Is there any
sort of well defined \textit{interface}? This can be rephrased in a
more general form in the language of the FK representation as a
question about the behavior of connected components of each boundary
in the corresponding dependent percolation model, conditioned not to
touch each other. (Recall that by a random coloring of the clusters,
the FK model gives origin to a spin system which interpolates the
independent percolation model ($\kappa $ $=1$), the Ising model
($\kappa =2$) and the $q$--states Potts model ($\kappa =q>2$,
integer) at inverse temperature $\beta $ and interaction
$J_{\{x,y\}}=\beta ^{-1}\log (\frac{1}{1-p_{\{x,y\}}})$. For details
see \cite{FK,ACCN}). Though we still do not fully understand this
problem, our results shed some light on it and hint that typically
the connected cluster of one of the sides $(-\infty ,-L]$ or
$[L,+\infty )$, (w.p. close to 1/2, each) would take over most of
the \textquotedblleft middle\textquotedblright\ space. This is an
unsolved problem. In \cite{CMR}, the authors obtain a more precise
description for very low temperatures, using cluster expansion
techniques.

An interesting corollary of Theorem \ref{percolation} is as follows.
Consider the Ising model (with $\pm $--valued spins) with interaction $%
J_{\{x,y\}}=|x-y|^{-2}$ if $|x-y|\geq 2$ and $J_{\{x,x+1\}}=J$ at inverse
temperature $\beta $. Let $m_{L}^{0,+}(\beta )$ denote the average spin at
the origin, with \textquotedblleft one-sided\textquotedblright\ $(+)$
boundary conditions in $[L,\infty)$. By the above mentioned FK
representation, we have

\begin{corollary}
Under the conditions of Theorem \ref{percolation},
\begin{equation*}
m_{L}^{0,+}(\beta )\geq \mu _{2,L}^{w}(0\rightsquigarrow \infty )\geq
1-\epsilon
\end{equation*}%
holds for $\epsilon =\epsilon (J)\searrow 0$ as $J\nearrow \infty $
uniformly in $L$. Consequently, there exists a phase transition when the
thermodynamical limit is taken with free boundary conditions on the left
side of the interval $\left[ -L,L\right] $, and $+$ boundary conditions on
the right side.
\end{corollary}

It is also interesting to compare the result on oriented percolation and the
previous corollary with the somehow similar question on the multiplicity of
Gibbs states for Markov chains with infinite connections, where orientation
appears naturally through the time direction. Recently Johansson and {\"{O}}%
berg \cite{JO} showed that if $g$ is a regular specification and
\begin{equation*}
\text{var}_{k}(g)=\sup\{\|g(\sigma)-g(\sigma^\prime)\|_1 \colon
\sigma_i=\sigma^\prime_i, i=1,\dots,k\},
\end{equation*}
then $g$ admits a unique Gibbs measure whenever the sequence $\{{\text{var}}%
_{k}(g)\}_{k=1}^{+\infty }$ is in $\ell ^{2} $. This tells, in particular,
that there are no multiple limiting measures for chains with connections
decaying as $r^{-2}$, as in Example 1 in \cite{JO}. This contrasts with the
two-sided Ising models and, as our Theorem says, with percolation models.
The understanding of Markov chains with infinite connections in the
non-uniqueness regime is still very poor, and it is known as a notoriously
difficult problem. We have strong evidence (see \cite{BHS}) that multi-scale
analysis techniques analogous to those developed in this work could be
turned into a robust tool to study this question.

\medskip

\noindent \textbf{Heuristics of the proof.} The proof relies on Fr\"{o}%
hlich-Spencer multi-scale analysis ideas (\cite{FS}, \cite{FS1}), and we use
the version developed in \cite{KMP} and \cite{M}. In the next few paragraphs
we outline the scheme of the proof, and comment on some key ideas, avoiding
most of consuming technical points. Our goal here is only to give a very
schematic and approximate picture, postponing precise formulations (which
tend to be quite involved) to later in the text.

\medskip

\noindent \textbf{The goal.} We look for an event of positive probability,
whose occurrence implies not only the existence of an infinite open
component, but also guarantees the presence of an oriented infinite open
path. Essentially, we will construct such an event, and show that it has
positive probability. Our key estimate will be: if $\beta >1$, we can find $%
\delta >0$, $\delta^\prime <1$ and $p$ is sufficiently close to $1$, so that
\begin{equation*}
\nu(\exists \,\text{open path } \pi=(x_1,e_1,\dots,x_n)\colon x_1 \le
-L+L^{\delta^\prime},\, x_n \ge L-L^{\delta^\prime}, 0< x_i-x_{i-1} \le
L^{\delta^\prime},\, \forall i\,)\ge 1-L^{-\delta}
\end{equation*}%
for $L=l_k$ as defined below ($k \ge 1$), $l_1$ being sufficiently large,
and where $\nu$ stands for the product measure defined before. Further
control for the dependent model leading to the result for $\mu^{f}_\kappa$
will come along the proof. We will have little control on how close to one $%
p $ has to be (or, on how large $l_1$ we need). \medskip

\noindent \textbf{Scales.} We choose super-exponentially fast growing
scales. Given $1<\alpha <2$, $l_{0}=1$ and $l_{1}$ an integer sufficiently
large, let
\begin{equation}
l_{k}=\lfloor l_{k-1}^{\alpha -1}\rfloor l_{k-1},\quad \;k=2,3,\ldots ,
\label{lk}
\end{equation}%
where as usual $\lfloor z\rfloor =\max\{n\in \mathbb{N}\colon n \le z\}$. We
will use the so--called dynamical blocking argument, in that the size and
location of blocks\footnote{%
Successive blocks share an end-vertex.} will be defined along the procedure
and will depend on the configuration. Still, the length of each block $%
I^{(k)}$ of the $k$--th level (called $k$--block) will be of order $l_{k}$.
More precisely, we shall see that $l_{k}-2l_{k}^{\alpha ^{\prime }/\alpha
}-6l_{k-1}\leq |I^{(k)}|\leq 3l_{k}+6l_{k-1}$, for suitable $1<\alpha
^{\prime }<\alpha $. (In particular, $l_{k-1}\ll l_{k}^{\alpha ^{\prime
}/\alpha }\ll l_{k}$, if $k\geq 2$, and $l_{1}$ is large.)

\noindent \textbf{Defected and good blocks.} Further we will use the
following recursive definition of a ``defected'' block. Fix $1 <
\alpha^\prime < \alpha$ to be specified later.

\textbf{1)} We say that the $0$--block $[i, i+1]$ is \textit{defected} if
the corresponding nearest neighbor edge $\{ i, i+1 \}$ is closed; otherwise
the $0$--block is said to be \textit{good} and the open nearest neighbor
path from $i$ to $i+1$ is called a $0$--\textit{pedestal};

\textbf{2)} For $k\ge 1$, a $k$--block $I^{(k)}=[s, s^\prime]$ is \textit{%
defected} if either it contains two or more defected $(k-1)$--blocks, or it
contains only one defected $(k-1)$--block $[i, i^\prime]$ but there is no
open edge $\{ a, a^\prime \}$ of length at most $l_k^{\alpha^\prime /
\alpha} $ , with $a\le i, \; i^\prime \le a^\prime, a\in \Upsilon,\;
a^\prime \in \Upsilon^\prime$, for some $(k-1)$--pedestals $\Upsilon, \;
\Upsilon^\prime$ contained in $I^{(k)}$. Otherwise $I^{(k)}$ is called
\textit{good}.

Thus, if a $k$--block $[s, s^\prime]$ is good, then it contains an oriented
open path going from $s$ to $s^\prime$: in the case it has no defected $%
(k-1) $--blocks, this path can be obtained by concatenating $(k-1)$%
--pedestals of the good $(k-1)$--blocks which constitute the given $k$%
--block; if it has a (single) defected $(k-1)$--block, and using the above
notation, a similar concatenation yields an oriented open path going from $s$
to $a$, which is followed by an open edge $\{ a, a^\prime \} $, and then
followed by another concatenation of $(k-1)$--pedestals of good $(k-1)$%
--blocks, from $a^\prime$ to $s^\prime$. In both cases, such path from $s$
to $s^\prime $ will be called $k$--\textit{pedestal}, and denoted by $%
\Upsilon$. The part of the cluster between $a$ and $a^\prime$ is again
disregarded in the future construction since we have little control on
oriented connectivity in this segment. The condition $a^\prime -a\le
l_k^{\alpha^\prime / \alpha}$ will be crucial to guarantee that pedestals
are quite dense sets (within the corresponding good blocks), used to push
the construction to higher levels. Some care is needed when treating defects
close to the boundary, which we have disregarded here.

\noindent \textbf{Strategy.} Being ``defected'' doesn't necessarily imply
that there is no oriented open path connecting the endpoints of the block.
Nevertheless, in order to avoid substantial technical difficulties, we will
follow two rules that simplify our construction:

a) once a block is defected, we will assume the worst possible situation,
namely it will be considered as if all edges within this block are closed.

b) once we have at least two defected $(k-1)$--blocks within a $k$--block $I$%
, we will not try to find connections within the $k$--block to fix its
connectivity, but rather will ``push the problem to the next level'', and
try to ``jump over'' this troubled block $I$ by a longer edge of length at
most $l_{k+1}^{\alpha^\prime/ \alpha}$, which starts at the pedestal of some
good $k$--block to the left of $I$, and ends similarly on
%some good block to
the right of $I$.

\noindent \textbf{Estimates.} The scale $l_{1}$ will be taken large enough,
as to be determined later. Assume that $\kappa \geq 1$ is fixed. We begin by
choosing $p$\thinspace:
\begin{equation}
p\geq 1-\kappa ^{-2}\frac{(\ln 2)^{5}}{128}\,l_{1}^{-\delta -1}
\label{choice00}
\end{equation}%
for some $\delta >0$ to be chosen later (see (\ref{del}) below). This
particular dependence on $\kappa $ will allow us to extend the estimate from
the product measure $\nu $ to the random cluster measure $\mu _{\kappa }^{f}$%
.

For $k\geq 2$, let $I^{(k)}$ be a $k$--block of length\footnote{%
This is not exact in general, but holds approximately.} $l_{k}$, which
consists of $N_{k}=l_{k}/l_{k-1}=\lfloor l_{k-1}^{\alpha -1}\rfloor $ blocks
of level $(k-1)$, of length $l_{k-1}$, and written as $\{I_{j}^{(k-1)}
\}_{j=1}^{N_{k}}$. Assume that we have the following estimate
\begin{equation*}
\nu (I_{j}^{(k-1)}\;{\text{is defected}})\leq l_{k-1}^{-\delta },\quad 1\leq
j\leq N_{k}.
\end{equation*}
Under the above assumptions, and if $\delta $ is chosen to satisfy (\ref{del}%
), we see that
\begin{equation*}
\nu \left( \exists \;1\leq i<j\leq N_{k}\colon I_{i}^{(k-1)},I_{j}^{(k-1)}
\text{ are both defected}\right) \leq l_{k}^{-\delta }.
\end{equation*}
When the defected $I_{i}^{(k-1)}$ is unique, we assume for the moment that
it stays at distance larger than $l_{k}^{\alpha ^{\prime }/\alpha }$ from
the boundary of $I^{(k)}$. (Otherwise a sequence of local adjustments of
blocks will be needed, as we shall see in Sect. \ref{I}. The left- and
right-most extremal blocks in our volume are treated differently.) In this
case let $a $ and $a^{\prime }$ be the end-vertices of the unique defected
block $I_{i}^{(k-1)}$. By our construction, there exists an oriented path
starting from the left boundary of $I^{(k)}$ and ending at the vertex $a$
and another open oriented path starting from vertex $a^{\prime }$ and going
to the right boundary of $I^{(k)}$. Both these paths are obtained by
concatenating pedestals of all good $(k-1)$--blocks on the left side of the
defected block $I_{i}^{(k-1)}$ and, respectively, on the right side. We
denote these new left and right pedestals by $\Upsilon $ and $\Upsilon
^{\prime }$, respectively. Given that $I^{(k)}$ has a unique defected $%
I_{i}^{(k-1)}=[a,a^\prime]$, and the pedestals $\Upsilon$ and $%
\Upsilon^\prime$, one has the following upper bound for the conditional
probability of not finding an open edge $\{ x,y\} $ with $x\leq a,a^{\prime
}\leq y,\;x\in \Upsilon ,\,y\in \Upsilon ^{\prime }$ and $y-x\leq
l_{k}^{\alpha ^{\prime }/\alpha }$ :
\begin{equation}  \label{conserto}
\prod_{\substack{ x,y:x\leq a\,<a^{\prime }\leq y,  \\ y-x<l_{k}^{\alpha
^{\prime }/\alpha }  \\ x\in \Upsilon ,\,y\in \Upsilon ^{\prime }}} q_{\{
x,y\} }\,=\exp \Big\{-\sum_{\substack{ x,y:x\leq a\,<a^{\prime }\leq y,  \\ %
y-x<l_{k}^{\alpha ^{\prime }/\alpha }  \\ x\in \Upsilon ,\,y\in \Upsilon
^{\prime }}}\frac{\beta }{\left\vert x-y\right\vert ^{2}}\Big\}\leq
l_{k-1}^{-\beta (1-\eta )(\alpha ^{\prime }-1)},
\end{equation}
where $\eta =\eta (\alpha ,\alpha ^{\prime },l_{1})>0$, and can be taken
arbitrarily small if $l_{1}\rightarrow \infty $.

The precise statement and proof of the above estimate will be given in Lemma %
\ref{est33}. It requires some work, and in order to obtain it for suitable $%
\eta =\eta (\alpha ,\alpha ^{\prime },l_{1})>0$ which can be taken
arbitrarily small if $l_{1}\rightarrow \infty $ we will need to use certain
geometric properties of pedestals $\Upsilon $ and $\Upsilon ^{\prime }$,
which propagate inductively from each level into the next one. Namely, the
pedestals are relatively dense sets (see (\ref{bagunca}) in Sect. \ref{I})
as the construction will show. Using the above estimate, writing
\begin{eqnarray*}
&&\text{$\big\{I^{(k)}$ has a unique defected $(k-1)$--block $[a,a^{\prime
}] $ but remains defected $\big\}=$} \\
&&\text{$\big\{I^{(k)}$ has unique defected $(k-1)$--block $[a,a^{\prime }]%
\big\}\cap $} \\
&&{\text{$\big\{$ there is no open edge $\{ x,y\} $ with $x\leq a,a^{\prime
}\leq y,\;x\in \Upsilon ,\,y\in \Upsilon ^{\prime }$ and $y-x\leq
l_{k}^{\alpha ^{\prime }/\alpha }$ $\big\}$ }},
\end{eqnarray*}%
and since these events depend on disjoint sets of edges, we easily get:
\begin{equation*}
\nu \text{$\big(I^{(k)}$ has a unique defected $(k-1)$--block but remains
defected}\big)\leq l_{k-1}^{\alpha -1-\delta }l_{k-1}^{-\beta (1-\eta
)(\alpha ^{\prime }-1)}\leq l_{k}^{-\delta },
\end{equation*}%
provided
\begin{equation}
\beta (1-\eta )(\alpha ^{\prime }-1)>(\delta +1)(\alpha -1).  \label{delta-2}
\end{equation}%
Since $\beta >1$ and $\eta =\eta (\alpha ,\alpha ^{\prime },l_{1})$ can be
taken very small provided $l_{1}$ is large, it will suffice to suitably fix
the parameters $\alpha $ and $\alpha ^{\prime }$ ($\alpha ^{\prime }$ close
enough to $\alpha $). This is done at the end of Sect. \ref{I}.

\medskip

\noindent \textbf{Difficulties.} To carry on this scheme we have to go
through several \textquotedblleft unpleasant\textquotedblright\ and rather
involved points. The use of a dynamical blocking argument, with the blocks
of a given level depending not only on the size and location of lower level
blocks, but also on their \textquotedblleft status\textquotedblright\
(defected or good), requires a rather tight bookkeeping. This is expressed
through what we call \textquotedblleft itineraries\textquotedblright .

The second difficulty is in the representation of the initial measure, which
should be sensitive to the above mentioned dynamic renormalization. We opt
to write it as a convex combination of a family of probability measures
associated with the above mentioned itineraries.

Once this is achieved, all necessary estimates follow along the scheme of
\cite{FS1} and \cite{KMP}.

The plan of the paper goes as follows: in Section 2 we define the blocks and
describe the dynamic renormalization procedure. In Section 3 we see how the
product measure can be re-written to take advantage of the previously
described blocks in order to recursively estimate the probability of having
a good $k$--block. Finally, in Section 4 we see how to apply the previous
estimates to conclude Theorem \ref{percolation}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Section \ref{mscale}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent \textbf{Acknowledgments} The authors thank C. M. Newman for
suggesting this problem and A.-S. Sznitman for many useful discussions. We
thank E. Presutti for pointing out the \textquotedblleft interface
question\textquotedblright\ and M. Cassandro and I. Merola for discussions
on this topic. We also thank the referees for their careful reading and for
their suggestions which helped us to improve the paper. This work was
supported by Faperj grant E-26/151.905/2000, Fapesp grant 03/01366-7, CNPq -
477259/01-4, Fundacion Andes, and individual CNPq grants. We also thank
various institutions CBPF, IMPA, IF-USP, ETH, UC-Berkeley and MSRI for
financial support and hospitality during periods when this work was carried
out.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Spatial blocks (Dynamic Renormalization) \label{I}}

\setcounter{equation}{0} \setcounter{theorem}{0}

\noindent \textbf{Notation.} For $L\in {\mathbb{N}}$, the construction will
involve the configuration $\omega$ restricted to the set of edges with both
end-vertices in $[-L,L]$. This allows us to prove the result for the measure
$\mu_{\kappa}^f$. We write $\Omega_L$ as a shorthand for $\Omega_{[-L,L]}$.
Scales $\{l_k \}_{k \in {\mathbb{N}}}$ are defined in the following way: $%
l_0 =1$, $l_1$ is an integer large enough to be chosen later (see (\ref{aal}%
) at the end of this Section), and for $1 < \alpha < 2$, we let

\begin{equation}
l_{k}=\lfloor l_{k-1}^{\alpha -1}\rfloor l_{k-1},\quad k\geq 2.
\label{escalas}
\end{equation}
Further we denote $x_{j}^{(k)}=jl_{k},\;j\in {\mathbb{Z}}$.

\noindent For the proof of Theorem \ref{percolation} we may assume that $L =
l_M$, for some $M \in {\mathbb{N}}$.

\medskip

\noindent Throughout the text ${\mathbb{I}}_A$ stands for the indicator
function of an event $A$, i.e. ${\mathbb{I}}_A(\omega)=1$ or 0 according to $%
\omega \in A$ or not.

\bigskip

\noindent \textbf{Decomposition of events. Level 0.} We set $I_i^{(0)} = [i,
i+1]$. They are called $0$--blocks, and for $i$ such that $I_i^{(0)} \subset
[-L,L] $ we define the events:
\begin{equation*}
G(I_i^{(0)} )= \{ \omega : \omega_{\{i,i+1\}}=1\}, \qquad B(I_i^{(0)} )= \{
\omega : \omega_{\{i,i+1\}}=0\}.
\end{equation*}
$I_{i}^{(0)}$ is said to be defected when $B(I_{i}^{(0)})$ occurs; otherwise
it is said to be a good $0$--block. \medskip

\noindent \textbf{Level 1.} Consider the intervals $\widetilde{I}
_{j}^{(1)}=[jl_{1},(j+1)l_{1}]$ and for each $j$ such that $\widetilde{I}
_{j}^{(1)}\subset [-L,L]$ we define the following partition of $\Omega _{L}$%
:
\begin{eqnarray}
G(\widetilde{I}_{j}^{(1)})
&=&\bigcap_{i=jl_{1}}^{(j+1)l_{1}-1}G(I_{i}^{(0)}),  \notag \\
H_{i}(\widetilde{I}_{j}^{(1)})&=& B(I_{i}^{(0)})\cap \bigcap\limits
_{\substack{ {s=jl_{1}}  \\ {s\neq i}}}^{(j+1)l_{1}-1}G(I_{s}^{(0)})\quad {%
\text{ for $\; i \in [jl_{1}, (j+1)l_{1}-1]$}},  \notag \\
H(\widetilde{I}_{j}^{(1)}) &=&\bigcup_{i=jl_{1}}^{(j+1)l_{1}-1}H_{i}(%
\widetilde{I}_{j}^{(1)}),  \notag \\
B(\widetilde{I}_{j}^{(1)}) &=&\left( G(\widetilde{I}_{j}^{(1)})\cup H(%
\widetilde{I}_{j}^{(1)})\right) ^{c},  \label{eventos111}
\end{eqnarray}
where $G$ stands for \textit{good}, $H$ for \textit{hopeful} and $B$ for
\textit{bad}, and accordingly, $\widetilde{I}_{j}^{(1)}$ is said to be good
(for given $\omega$) if it contains no defected $0$--blocks, ``hopeful" if
it contains only one defected $0$--block, and is said to be ``bad"
otherwise. When $H_{i}(\widetilde{I}_{j}^{(1)})$ occurs, $I_{i}^{(0)}$ is
called the defected 0-block in $\widetilde{I}_{j}^{(1)}$.

\noindent \textbf{Adjustment.} Given $\omega $, we first consider the set of
all $j$'s such that $\omega \in H_{i_j}(\widetilde{I}_{j}^{(1)})\subset H(%
\widetilde{I}_{j}^{(1)})$ and such that the index $i_j$ of the (unique)
defected block $I_{i_{j}}^{(0)}\subset \widetilde{I}_{j}^{(1)}$ verifies $%
jl_{1}\leq i_{j}\leq jl_{1}+\lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor
-1$ (resp.\ $(j+1)l_{1}-\lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor
\leq i_{j}\leq (j+1)l_{1}-1$).

If this set is empty in both cases, we set $I^{(1)}_j = \widetilde I^{(1)}_j
$ for all $j$'s, and say that $G(I_{j}^{(1)})$, $H(I_{j}^{(1)})$ or $%
B(I_{j}^{(1)})$ occurs if $G(\widetilde{I}_{j}^{(1)})$, $H(\widetilde{I}%
_{j}^{(1)})$ or $B(\widetilde{I}_{j}^{(1)})$ do occur.

If this set is not empty, we take arbitrarily one of such indices $j$; if $%
\widetilde I^{(1)}_j $ is not the interval which contains $-L$ (resp. $L$),
to be treated in case 3) below, we check if $\widetilde{I}_{j-1}^{(1)}$
(resp.\ $\widetilde{I}_{j+1}^{(1)}$) has a defected $0$--block in the
sub-interval $[jl_{1}-2\lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor
,jl_{1}]$ (resp.\ $[(j+1)l_{1},(j+1)l_{1}+2\lfloor l_{1}^{\alpha ^{\prime
}/\alpha }\rfloor -1]$).

1) If yes, then we consider a new interval $I_{j-1}^{(1)} = \widetilde
I_{j-1}^{(1)}\cup \widetilde I_j^{(1)}$ (resp.\ $I_{j}^{(1)} = \widetilde
I_{j}^{(1)}\cup \widetilde I_{j+1}^{(1)}$) and say that the event $%
B(I_{j-1}^{(1)} )$ (resp.\ $B(I_{j}^{(1)} )$) occurs. (This is motivated by
the fact that for the chosen $\omega$ the new interval will contain at least
two defected $0$--blocks.)

\medskip

2) If not, then we consider two new intervals $%
I_{j-1}^{(1)}=[(j-1)l_{1},jl_{1}-\lfloor l_{1}^{\alpha ^{\prime }/\alpha
}\rfloor ]$ and $I_{j}^{(1)}=[jl_{1}-\lfloor l_{1}^{\alpha ^{\prime }/\alpha
}\rfloor ,(j+1)l_{1}]$ (resp.\ $I_{j}^{(1)}=[jl_{1},(j+1)l_{1}+\lfloor
l_{1}^{\alpha ^{\prime }/\alpha }\rfloor ]$ and $I_{j+1}^{(1)}=[(j+1)l_{1}+%
\lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor ,(j+2)l_{1}]$) and say that
$G(I_{j-1}^{(1)})$, $H(I_{j-1}^{(1)})$ or $B(I_{j-1}^{(1)})$ occurs if the
event $H(I_{j}^{(1)})$ occurs and, resp.\ $G(\widetilde{I}_{j-1}^{(1)})$, $H(%
\widetilde{I}_{j-1}^{(1)})$ or $B(\widetilde{I}_{j-1}^{(1)})$, do occur
(resp.\ $G(I_{j+1}^{(1)})$, $H(I_{j+1}^{(1)})$ or $B(I_{j+1}^{(1)})$ occurs
if the event $H(I_{j}^{(1)})$ occurs and, resp.\ $G(\widetilde{I}%
_{j+1}^{(1)})$, $H(\widetilde{I}_{j+1}^{(1)})$ or $B(\widetilde{I}%
_{j+1}^{(1)})$ do occur). In this case the adjustment moves the boundary
\textquotedblleft away\textquotedblright\ from the unique defected block in $%
I_{j}^{(1)}$, but doesn't change the number of the defected $0$--blocks in
the adjusted intervals.

\medskip

3) If the interval $\widetilde I_{j}^{(1)}$ under consideration is the
leftmost (resp. the rightmost) interval in $[-L,L]$, we set $I_{j}^{(1)} =
\widetilde I_{j}^{(1)}$ and say that $G(I_{j}^{(1)})$ occurs.

\medskip

4) We set $I^{(1)}_j = \widetilde I^{(1)}_j $ if $\widetilde I^{(1)}_j $ was
not involved in the previous adjustment, and say that $G(I_{j}^{(1)})$, $%
H(I_{j}^{(1)})$ or $B(I_{j}^{(1)})$ occurs if accordingly \ $G(\widetilde{I}%
_{j}^{(1)})$, $H(\widetilde{I}_{j}^{(1)})$, or $B(\widetilde{I}_{j}^{(1)})$
occur.

If after this step we are still left with intervals $I_j^{(1)}$ for which $%
H( I_j^{(1)} )$ occurs and its defected $0$--block stays within distance $%
\lfloor l_1^{\alpha^{\prime}/ \alpha}\rfloor$ from the boundary of $%
I_j^{(1)} $, we repeat the above procedure, now applied to the intervals
already adjusted in the previous step. After finitely many steps of such
adjustment procedure there are left no intervals $I_j^{(1)}$ for which the
event $H(I_j^{(1)})$ occurs and its defected $0$--block stays within
distance $\lfloor l_1^{\alpha^{\prime}/ \alpha}\rfloor $ from the boundary,
and the adjustment procedure is then stopped. \medskip

\noindent \textbf{Remark.} Notice that the adjustment procedure is well
defined, i.e. the final partition does not depend on the order in which we
do adjustments and in which order we pick the intervals that still need to
be adjusted (in case if we have more than one). It also has \textit{a
locality} property, i.e. the final modification of each initial interval $%
\widetilde I^{(1)}_j$ depends on the values of the configuration in the
nearest neighbor and, at most, in the next nearest neighbor intervals only.

\medskip

Once the adjustment is completed, the obtained intervals are called $1$%
-blocks, and re-numerated from left to right as $I^{(1)}_j,\, j=1,\dots $.
Notice that $l_1 - 2\lfloor l_1^{\alpha^{\prime}/ \alpha}\rfloor \leq |
I^{(1)}_j | \leq 3l_1$, and $\cup_j I^{(1)}_j=[-L,L]$. \medskip

In other words, values of $\omega $ on the nearest neighbor edges of $[-L,L]$
determine, through the above procedure, a division $I^{(1)}(\omega)\equiv
\{I_{j}^{(1)}(\omega )\}_{j}$ of the interval $[-L,L]$ into $1$--blocks,
with the property that any two adjacent blocks are sharing their end-vertex.
This is the final state of the ``adjustment" procedure. Values of $\omega $
on the nearest neighbor edges in $[-L,L]$ also determine where the defected
0-blocks are located within each 1-block, and we denote by $%
D_{j}^{(1)}(\omega)$ the set of indices of the defected $0$--blocks within $%
I_{j}^{(1)}(\omega)$, and $D^{(1)}(\omega)\equiv \{D_{j}^{(1)}(\omega
)\}_{j} $ .

A collection of intervals $\overline{I}^{(1)}\equiv \{\overline{I}%
_{j}^{(1)}\}_{j}$ and sets of indices $\overline{D}^{(1)}\equiv \{\overline{D%
}_{j}^{(1)}\}_{j}$ is called \textbf{admissible} if there exists an $\omega $
such that $I_{j}^{(1)}(\omega )=\overline{I}_{j}^{(1)}$ and $%
D_{j}^{(1)}(\omega )=\overline{D}_{j}^{(1)}$ for all $j$'s. A pair $%
J_{L}^{(1)}:=\{I_{j}^{(1)},D_{j}^{(1)}\}$ is called an itinerary at level 1,
or an 1-itinerary, and we denote by
\begin{equation*}
\mathcal{I}_{L}^{(1)}\equiv \{J_{L}^{(1)}\}
\end{equation*}%
the (finite) set of all admissible level 1-itineraries. For fixed $\overline{%
J}_{L}^{(1)}:=\{\overline{I}_{j}^{(1)},\overline{D}_{j}^{(1)}\}$ we will
denote
\begin{equation*}
\mathcal{E}(\overline{J}_{L}^{(1)})=\{\omega :\;I^{(1)}(\omega )=\overline{I}%
^{(1)},\;D^{(1)}(\omega )=\overline{D}^{(1)}\}.
\end{equation*}

\medskip

\noindent \textbf{1-Pedestals.} Fix an admissible pair $J^{(1)}_L =\{
I^{(1)}, \, D^{(1)}\}$ and consider all 1-blocks $I_{j}^{(1)}$ for which $%
|D_{j}^{(1)}| = 1 $. (It implies that on $\mathcal{E}(J^{(1)}_L)$ some event
$H_{i_j}(I_{j}^{(1)})$ occurs.) For such 1-blocks, the set of vertices $x\in
I_{j}^{(1)}$ to the left (resp. right) of the defected $0$--block in $%
I_{j}^{(1)}$ will be called \textit{left 1-pedestal of $I_{j}^{(1)}$} (resp.
\textit{right 1-pedestal}) and denoted by $\Upsilon _{\mathcal{L}%
}(I_{j}^{(1)})$ (resp. $\Upsilon _{\mathcal{R}}(I_{j}^{(1)})$). The vertices
in each of these 1-pedestals are connected by open nearest neighbor edges.
Define
\begin{eqnarray}
G^{+}(I_{j}^{(1)}, D_{j}^{(1)}) &=&\mathcal{E}(J^{(1)}_L) \cap \lbrack
\omega :\exists x\in \Upsilon _{\mathcal{L}}(I_{j}^{(1)}),y\in \Upsilon _{%
\mathcal{R}}(I_{j}^{(1)}),\;y-x\leq \lfloor l_{1}^{\alpha ^{\prime }/\alpha
}\rfloor :\omega _{\{x,y\}}=1],  \notag \\
B^{+}(I_{j}^{(1)}, D_{j}^{(1)}) &=&\mathcal{E}(J^{(1)}_L) \setminus
G^{+}(I_{j}^{(1)}).  \label{2.8888}
\end{eqnarray}

If $G^{+}(I_{j}^{(1)}, D_{j}^{(1)})$ occurs, and the pair $(x,y)$ such that $%
x\in \Upsilon _{\mathcal{L}}(I_{j}^{(1)}),y\in \Upsilon _{\mathcal{R}%
}(I_{j}^{(1)}),\;y-x\leq \lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor$, $%
\omega _{\{x,y\}}=1$ is not unique, we choose one in arbitrary way, and,
once the pair $(x,y)$ is chosen, the interval $[x+1,y-1]$ will be called
defected part of $I_{j}^{(1)}$, and denoted by $\mathcal{D}(I_{j}^{(1)})$.
In this case we define $\Upsilon (I_{j}^{(1)})=\left( \Upsilon _{\mathcal{L}%
}(I_{j}^{(1)})\cup \Upsilon _{\mathcal{R}}(I_{j}^{(1)})\right) \setminus
\mathcal{D}(I_{j}^{(1)})$. If we are in the case 3) as described above, for
the leftmost interval we take $\Upsilon (I_{j}^{(1)})= \Upsilon _{\mathcal{R}
}(I_{j}^{(1)})$ (resp. $\Upsilon (I_{j}^{(1)})= \Upsilon _{\mathcal{L}
}(I_{j}^{(1)})$).

On the other hand, for those $j$ such that $D_{j}^{(1)} = \emptyset$, we get
that on $\mathcal{E}(J^{(1)}_L)$ the event $G(I_{j}^{(1)})$ occurs, which
means that there are no closed nearest neighbor edges in $I_{j}^{(1)}$, and
we define the pedestal $\Upsilon (I_{j}^{(1)})=I_{j}^{(1)}$.

This completes the first step, associating with each $\omega$ an itinerary $%
J^{(1)}_L (\omega)=\{ I^{(1)}(\omega), \, D^{(1)}(\omega)\}$, and the
``state" $G,\, G^{+}, \, B$ or $B^{+}$ to each $1$--block of the itinerary $%
J^{(1)}_L$. For the next step of the procedure it is irrelevant which
particular sub-type $G$ or $G^{+}$ (resp. $B$ or $B^{+}$) occurs, without
creating ambiguities, we will simply denote their occurrence by the letter $%
G $ (resp. $B$).

In particular, a $1$--pedestal $\Upsilon (I_{j}^{(1)})$ is given by the
vertices of open oriented path with all edges, except possibly one, being
nearest neigbhour, and this larger edge has length at most $\lfloor
l_1^{\alpha^\prime/\alpha}\rfloor$. For each $1$--block, except possibly the
two which contain the extremes $-L$ or $L$, the pedestal connects left and
right endpoints of the interval.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbp]
\centering
\includegraphics[scale=0.35]{Bmsv11.eps}
\caption{Adjustments: part $a)$ shows the deterministic $1$--blocks $%
\widetilde{I}_{j}^{(1)}$, bold-face segments show location of the
defects. Part $b)$ shows how these blocks were adjusted.
$\widetilde{I}_{5}^{(1)}$ and $\widetilde{I}_{6}^{(1)}$, merge into
a single $1$--block $I_{5}^{(1)}$. } \label{fig1}
\end{figure}

%%KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK
%%

\noindent \textbf{Level {\textit{k} }.} Let $2\le k\le M$, and assume that
we have completed the step $(k-1)$ of the recursion. This means that for
each $\omega \in \Omega_L$ and any $r=1,\dots ,k-1$ the following objects
are defined:

\begin{itemize}
\item the collection of $r$--blocks $I^{(r)}(\omega)=\{I_{j}^{(r)}(\omega)
\}_{j} $, such that $\cup_{j}I_{j}^{(r)}(\omega)=[-L,L]$, and any two
adjacent intervals share exactly an endpoint. Moreover, the uniform bound
holds:
\begin{eqnarray}
l_{r}-(2\lfloor l_{r}^{\alpha ^{\prime }/\alpha }\rfloor
+6l_{r-1})<|I_{j}^{(r)}(\omega)|\leq 3l_{r}+6l_{r-1},
\end{eqnarray}

\item each of the $I_{j}^{(r)}(\omega)$ can be in two possible states $G$ or
$B$: \newline
If $I_{j}^{(r)}(\omega)$ is in the state $B$ (or $B(I_{j}^{(r)}(\omega))$
occurs), we say that $I_{j}^{(r)}(\omega)$ is \textit{defected}. \newline
If $I_{j}^{(r)}(\omega)$ is in the state $G$ (or $G(I_{j}^{(r)}(\omega))$
occurs) and it is not the leftmost or the rightmost interval of the
partition, then $\omega$ has an $r$--pedestal $\Upsilon (I_{j}^{(r)})$ given
by vertices of an open oriented path from the left to the right boundary of $%
I_{j}^{(r)}(\omega)$. If $I_{j}^{(r)}(\omega)$ is the leftmost (resp. the
rightmost) interval, an $r$--pedestal $\Upsilon (I_{j}^{(r)})$ is given by
vertices of an open oriented path which starts from some vertex $x \in [-L,
-L + \lfloor l_r^{\alpha^{\prime}/ \alpha}\rfloor]$ and ends at the right
boundary of $I_{j}^{(r)}(\omega)$ (resp. starts from the left boundary of $%
I_{j}^{(r)}(\omega)$ and ends at some vertex $x \in [L- \lfloor
l_r^{\alpha^{\prime}/ \alpha}\rfloor, L]$.

\item the collection $D^{(r)}(\omega)=\{D_{j}^{(r)}(\omega) \}_{j} $, where $%
D_{j}^{(r)}(\omega)$ is the set of labels of the defected $(r-1)$--blocks
which are contained in $I_{j}^{(r)}(\omega)$.
\end{itemize}

For $\omega $ fixed, the sequence of pairs
\begin{equation*}
J_{L}^{(k-1)}(\omega )=\{(I^{(1)}(\omega ),D^{(1)}(\omega )),\dots
,(I^{(k-1)}(\omega ),D^{(k-1)}(\omega ))\},
\end{equation*}%
is called $(k-1)$--\textit{itinerary}, and each element $(I^{(r)},D^{(r)}),%
\;1\leq r\leq k-1$ is called $r$--th step of the itinerary. For fixed $%
\overline{J}_{L}^{(k-1)}=\{(\overline{I}^{(1)},\overline{D}^{(1)}),\dots ,(%
\overline{I}^{(k-1)},\overline{D}^{(k-1)})\}$ a possible realization of a $%
(k-1)$--itinerary, we denote
\begin{equation*}
\mathcal{E}(\overline{J}_{L}^{(k-1)})=\{\omega :J_{L}^{(k-1)}(\omega )=%
\overline{J}_{L}^{(k-1)}\},
\end{equation*}%
and we shall now see how to define the $k$--blocks and the continuation to a
$k$--itinerary. When $k=M$ we will end up with only two intervals.

\noindent \textbf{Construction of} $k$ - \textbf{blocks.} Fix $%
J^{(k-1)}_L=\{(I^{(1)},D^{(1)}),\dots ,(I^{(k-1)}, D^{(k-1)}) \}$ a possible
realization of a $(k-1)$--itinerary. For each $z\in [-L,L]$ we set $%
j_{z}^{k}=\min \{j:z\in I_{j}^{(k-1)}\}$, $\hat{\jmath}%
_{i}^{k}=j_{x_{i}^{(k)}}^{k}$, cf. notation after (\ref{escalas}), $%
i=-l_M/l_k,\dots, l_M/l_{k}-1$, and define the intervals:
\begin{equation*}
\widetilde{I}_{i}^{(k)}=\bigcup_{s=\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath}%
_{i+1}^{k}}I_{s}^{(k-1)}=:[a^{(k)}_i,a^{(k)}_{i+1}]
\end{equation*}
as well as the following partition of $\mathcal{E}(J^{(k-1)}_L)$:

\begin{eqnarray}
G(\widetilde{I}_{i}^{(k)}) &=&\mathcal{E}(J^{(k-1)}_L)\cap\displaystyle %
\bigcap\limits_{s=\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath}
_{i+1}^{k}}G(I_{s}^{(k-1)}),  \notag \\
H_s(\widetilde{I}_{i}^{(k)}) &=&\mathcal{E}(J^{(k-1)}_L)\cap\displaystyle %
\left[ B(I_{s}^{(k-1)})\cap \displaystyle\bigcap\limits_{u=\hat{\jmath}%
_i^k+1,u\neq s}^{\hat{\jmath}_{i+1}^{k}}G(I_{u}^{(k-1)})\right] ,  \notag \\
H(\widetilde{I}_{i}^{(k)}) &=&\displaystyle \bigcup\limits_{s=\hat{\jmath}%
_{i}^{k}+1}^{\hat{\jmath}_{i+1}^{k}} H_s(\widetilde{I}_{i}^{(k)}) ,  \notag
\\
B(\widetilde{I}_{i}^{(k)}) &=&\mathcal{E}(J^{(k-1)}_L)\setminus\left( G(
\widetilde{I}_{i}^{(k)})\cup H(\widetilde{I}_{i}^{(k)})\right).
\label{eventosk}
\end{eqnarray}

\noindent \textbf{Adjustment.} Given $\omega \in \mathcal{E}(J^{(k-1)}_L)$
consider all $i$ for which $H_s( \widetilde{I}_{i}^{(k)})$ occurs for $s$
such that the distance of the defected $(k-1)$--block $I_{s}^{(k-1)}\subset
\widetilde{I}_{i}^{(k)}$ to the left endpoint $a^{(k)}_i$ (right endpoint $%
a^{(k)}_{i+1}$, resp.) is less than $\lfloor l_{k}^{\alpha^{\prime} /
\alpha}\rfloor$. If this set is non-empty take arbitrarily any such $%
\widetilde{I}_{i}^{(k)}$.

If the selected $\widetilde{I}_{i}^{(k)}$ is the leftmost (resp. the
rightmost) interval in $[-L,L]$, we set ${I}_{i}^{(k)} = \widetilde{I}%
_{i}^{(k)}$, and say that $G({I}_{i}^{(k)})$ occurs (or that ${I}_{i}^{(k)}$
is in $G$ state for this $\omega$).

If the selected $\widetilde{I}_{i}^{(k)}$ is not one of these two extremal
intervals, we then check if $\widetilde{I} _{i-1}^{(k)}$ (resp.\ $\widetilde{%
I}_{i+1}^{(k)}$) has a defected block $I_{r}^{(k-1)}$ at distance at most $%
3\lfloor l_{k-1}^{\alpha ^{\prime }/\alpha }\rfloor$ from $a^{(k)}_i$ (resp.
from $a^{(k)}_{i+1}$), and

1) If yes, then we consider a new interval $I_{i-1}^{(k)} = \widetilde
I_{i-1}^{(k)}\cup \widetilde I_i^{(k)}$ (respectively $I_{i}^{(k)}
=\widetilde I_{i}^{(k)}\cup \widetilde I_{i+1}^{(k)}$) and say that $%
B(I_{i-1}^{(k)} )$ (resp. $B(I_{i}^{(k)} )$) occurs, or that the
corresponding interval is in state $B$;

\smallskip

2) If not, then we consider two new intervals:
\begin{eqnarray}
I_{i-1}^{(k)}=\bigcup_{s=\hat{\jmath}_{i-1}^{k}+1}^{j_{a_{i}^{(k)}-l_{k}^{%
\alpha ^{\prime }/\alpha }}^{k}-1}I_{s}^{(k-1)}, &\qquad \quad
&I_{i}^{(k)}=\bigcup_{s=j_{a_{i}^{(k)}-l_{k}^{\alpha ^{\prime }/\alpha
}}^{k}}^{\hat{\jmath}_{i+1}^{k}}I_{s}^{(k-1)}  \label{carnaval4} \\
\Big(\text{respectively,}\qquad I_{i}^{(k)}=\bigcup_{s=\hat{\jmath}%
_{i}^{k}+1}^{j_{a_{i+1}^{(k)}+l_{k}^{\alpha ^{\prime }/\alpha
}}^{k}}I_{s}^{(k-1)} &\qquad \quad
&I_{i+1}^{(k)}=\bigcup_{s=j_{a_{i+1}^{(k)}+l_{k}^{\alpha ^{\prime }/\alpha
}}^{k}+1}^{\hat{\jmath}_{i+2}^{k}}I_{s}^{(k-1)}\Big ),  \label{carnaval5}
\end{eqnarray}%
and say that $H(I_{i}^{(k)})$ and $G(I_{i-1}^{(k)})$, $H(I_{i-1}^{(k)})$ or $%
B(I_{i-1}^{(k)})$ occurs if respectively $G(\widetilde{I}_{i-1}^{(k)})$, $H(%
\widetilde{I}_{i-1}^{(k)})$ or $B(\widetilde{I}_{i-1}^{(k)})$ do occur
(respectively $H(I_{i-1}^{(k)})$ and $G(I_{i+1}^{(k)})$, $H(I_{i+1}^{(k)})$
or $B(I_{i+1}^{(k)})$ occurs if respectively $G(\widetilde{I}_{i+1}^{(k)})$,
$H(\widetilde{I}_{i+1}^{(k)})$ or $B(\widetilde{I}_{i+1}^{(k)})$ do occur).

Finally we set $I_{i}^{(k)}=\widetilde{I}_{i}^{(k)} $ if $\widetilde{I}%
_{i}^{(k)}$ was not involved in the adjustment and say $G(I_{i}^{(k)})$, $%
H(I_{i}^{(k)})$ or $B(I_{i}^{(k)})$ occurs if respectively $G(\widetilde{I}%
_{i}^{(k)})$, $H( \widetilde{I}_{i}^{(k)})$ or $B(\widetilde{I}_{i}^{(k)})$
do occur.

If after this step we are still left with intervals $I_{i}^{(k)}$ for which $%
H(I_{i}^{(k)})$ occurs and its defected interval $I_{s}^{(k-1)}$ stays
within distance $\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor $ from one
of the endpoints of $I_{i}^{(k)}$, then we repeat the above procedure. After
finitely many steps of this adjustment procedure all $I_i^{(k)}$ for which $%
H(I_i^{(k)})$ occurs have their defected $(k-1)$--block at distance larger
than $\lfloor l_k^{\alpha^{\prime} / \alpha}\rfloor$ from the boundary of $%
I_i^{(k)}$.

Once the adjustments are completed, the final intervals are called $k$%
-blocks, and re-numerated from left to right as $I_{j}^{(k)},\;j=1,\dots $.
We then consider the collection $D^{(k)}=\{D_j^{(k)}\}_{j}$ where $D_j^{(k)}$
gives the labels of the defected $(k-1)$--blocks contained in $I_{j}^{(k)}$.
A sequence $(I^{(k)},D^{(k)})$ which can be obtained in this way (for some $%
\omega \in \Omega_L$) is said to be \textbf{admissible} for $J^{(k-1)}_L$,
in which case $J^{(k)}_L :=\{ (I^{(s)}, D^{(s)}),\, 1\le s\le k\}$ is a $k$%
--itinerary.

By $\mathcal{I}^{(k)}(J^{(s)}_L)$ we denote the collection of all $k$%
--itineraries that are continuation of $J^{(k-1)}_L$.

We can always write $I_{j}^{(k)}=\displaystyle
\bigcup\nolimits_{s_{0}(j)}^{s_{1}(j)}I_{s}^{(k-1)}$, and simple recursive
estimate holds:
\begin{equation}
l_{k}-(2\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor
+6l_{k-1})<|I_{j}^{(k)}|\leq 3l_{k}+6l_{k-1}.  \label{vno}
\end{equation}

\medskip

\noindent $k$\textbf{-Pedestals.} For $\omega$ fixed, let $%
I_{j}^{(k)}(\omega)$ be a $k$--block for which after adjustment $%
G(I_{j}^{(k)})$ or $H(I_{j}^{(k)})$ occurs. For such $k$--blocks we will
define $k$--pedestals.

\begin{itemize}
\item if $G(I_{j}^{(k)})$ occurs and if $I_{j}^{(k)}(\omega)$ is not
leftmost (resp. rightmost) interval in $[-L,L]$, then all its sub-blocks $%
I_{s}^{(k-1)}$ are in $G$ state, and we define $\Upsilon (I_{j}^{(k)})= %
\displaystyle\bigcup\nolimits_{s_{0}(j)}^{s_{1}(j)}\Upsilon (I_{s}^{(k-1)})$.%
\newline

\item if $G(I_{j}^{(k)})$ occurs and $I_{j}^{(k)}(\omega)$ is the leftmost
(resp. rightmost) interval in $[-L,L]$ which contains unique defected $(k-1)$%
--block $I_{r}^{(k-1)}$ within distance $\lfloor
l_k^{\alpha^{\prime}/\alpha} \rfloor$ from its left (resp. right) boundary
then $\Upsilon (I_{j}^{(k)})= \displaystyle\bigcup\nolimits_{r+1}^{s_{1}(j)}%
\Upsilon (I_{s}^{(k-1)})$ (resp. $\Upsilon (I_{j}^{(k)})= \displaystyle%
\bigcup\nolimits_{s_{0}(j)}^{r-1}\Upsilon (I_{s}^{(k-1)})$). \newline

\item if $H_r (I_{j}^{(k)})$ occurs\footnote{%
i.e. $I_{r}^{(k-1)}$ is the only defected block within $I_{j}^{(k)}$}, we
define $\Upsilon _{\mathcal{L}}(I_{j}^{(k)})=\displaystyle\bigcup
\nolimits_{s_{0}(j)}^{r-1}\Upsilon (I_{s}^{(k-1)})$ and $\Upsilon _{\mathcal{%
R}}(I_{j}^{(k)})=\displaystyle\bigcup\nolimits_{r+1}^{s_{1}(j)} \Upsilon
(I_{s}^{(k-1)}), $ called left and right pedestals\footnote{%
From the occurrence of $G(I_{s}^{(k-1)})$ for all other $(k-1)$--blocks
within $I_{j}^{(k)}$, we know there exists an open oriented path connecting
the left boundary of $I_{j}^{(k)}$ to the right boundary of $I_{r-1}^{(k-1)}$
and an open oriented path connecting the left boundary of $I_{r+1}^{(k-1)}$
to the right boundary of $I_{j}^{(k)}$. These paths are obtained by
concatenation of the corresponding $\Upsilon (I_{s}^{(k-1)}),\;{s_{0}(j)}%
\leq s\leq {r-1}$ and $r+1\leq s\leq s_{1}(j)$, respectively.} of $%
I_{j}^{(k)}$, and check if there exists $x\in \Upsilon _{\mathcal{L}%
}(I_{j}^{(k)})$ and $y\in \Upsilon _{\mathcal{R}}(I_{j}^{(k)})$ with $%
y-x\leq \lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor $ such that $\omega
_{\{x,y\}}=1$:

-- If yes, we say that $I_{j}^{(k)}$ is in $G$ state, and if the pair $(x,y)$
with $x\in \Upsilon _{ \mathcal{L}}(I_{j}^{(k)}),y\in \Upsilon _{\mathcal{R}%
}(I_{j}^{(k)}),\;y-x \leq \lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor $%
, and $\omega _{\{x,y\}}=1$ is not unique, we choose one in an arbitrary
way, and, once $(x,y)$ is chosen, denote $\mathcal{D} (I_{j}^{(k)}) =
[x+1,y-1]$, and define
\begin{equation*}
\Upsilon (I_{j}^{(k)})=\left( \Upsilon _{\mathcal{L}}(I_{j}^{(k)})\cup
\Upsilon _{\mathcal{R}}(I_{j}^{(k)})\right) \setminus \mathcal{D}%
(I_{j}^{(k)}).
\end{equation*}
-- If such an open edge $\{x,y\}$ does not exist we say that $I_{j}^{(k)}$
is in $B$ state.
\end{itemize}

In other words, we consider the event
\begin{eqnarray*}
G^{+}(I_{j}^{(k)}) &=& H(I_{j}^{(k)})\cap \lbrack \omega :\exists x\in
\Upsilon _{\mathcal{L}}(I_{j}^{(k)}),y\in \Upsilon _{\mathcal{R}%
}(I_{j}^{(k)}),\;y-x\leq \lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor
:\omega _{\{x,y\}}=1],
\end{eqnarray*}
whose occurrence implies the $k$--block to be in $G$ state, i.e. when
dealing at larger scales we do not distinguish if $I_{j}^{(k)}$ is in $G$
state due to the occurrence of $G(I_{j}^{(k)})$ or $G^{+}(I_{j}^{(k)})$.

This completes the $k$--th step, associating with each itinerary $J^{(k-1)}$
its continuation with a sequence of $k$--blocks $I^{(k)}= \{I^{(k)}_j \}_{j}$%
, re-numerated from left to right. Moreover, with each $k$--block we
associate one of the states $G$ or $B$. \medskip

\noindent \textbf{Structure of pedestals.} First we state a simple geometric
property of pedestals, which will be used in estimating the conditional
probability of the event $G^+$ defined above, given $H$. Our goal is to show
that if a $k$--block, $k\ge 1$, $I^{(k)}=[s,s^{\prime }]$ contains only one
defected $(k-1)$--block, here denoted by $[a,a^{\prime }]$, and there are
corresponding left and right pedestals $\Upsilon_{\mathcal{L}} $ and $%
\Upsilon_{\mathcal{R}}$, spanning from $s$ to $a$ and from $a^{\prime }$ to $%
s^{\prime }$, respectively, then there exists a constant $C \equiv C(\alpha,
\alpha^{\prime}) $ such that
\begin{equation}
\Big|\Upsilon_{\mathcal{L}} \cap \lbrack a-\lfloor l_{k}^{\alpha ^{\prime
}/\alpha }\rfloor ,\,a]\Big|\ge Cl_{k}^{\alpha ^{\prime }/\alpha } \quad
\text{and}\quad \Big|\Upsilon_{\mathcal{R}}\cap \lbrack a^{\prime
},\,a^{\prime }+\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor ]\Big|\ge
Cl_{k}^{\alpha ^{\prime }/\alpha }.  \label{bagunca}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbp]
\centering
\includegraphics[scale=1.2]{Bmsv12.eps}
\caption{Pedestals and defects: part $a)$ shows the deterministic
$1$--blocks $I^{(1)}_i, \; 1 \le i \le 10$, located in the
$2$--block $I_j^{(2)}$; bold-face segments show location of the
$0$--level defects. Part $b)$ shows construction of $1$--level
pedestals, marked by light-gray strips. The block $ I_6^{(1)}$ is a
defected $1$--block. The segments $(x_i,y_i)$ are ``enlarged''
defects in $I_i^{(1)}$. Part $c)$ shows creation of $2$--level
pedestals, marked by dark-gray strips, concatenated by long range
edges. The segment $(x_j, y_j)$ is enlarged defect for $I_j^{(2)}$.}
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Inequality (\ref{bagunca}) follows trivially from the following recursive
relation: if we have a $k$--block $I^{(k)}= \cup_{s_0}^{s_1} I^{(k-1)}_s$
which is in $G$ state, then
\begin{equation*}
\big| \Upsilon(I^{(k)}) \big| \ge \displaystyle \sum_{s\colon
[G(I^{(k-1)}_s) \, \text{occurs}]} \big| \Upsilon( I^{(k-1)}_s) \big| -
l_{k}^{\alpha ^{\prime }/\alpha }.
\end{equation*}
\smallskip We will end this section with the estimate which will be
important for the recursive step in the next section, and after that we also
fix the parameters which will determine the choice of $p$ close to one, as
in (\ref{choice00}). Recall that by $H_z(I_j^{(k)})\subset H(I_j^{(k)})$ we
denote the event, for which the unique defected $(k-1)$--block within $%
I_j^{(k)}$ has index $z$, and by construction stays at distance larger than $%
\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor$ from the boundaries of $%
I_j^{(k)}$.

\begin{lemma}
\label{est33} There exists $\eta \equiv \eta (\alpha ,\alpha ^{\prime
},l_{1})$ with $\eta \searrow 0$ as $l_{1}\nearrow +\infty $ and such that
the following estimate for the conditional probability with respect to the
product measure (defined right above (\ref{occup}))
\begin{equation}
\nu \big\lbrack \exists x\in \Upsilon _{\mathcal{L}}(I_{j}^{(k)}),y\in
\Upsilon _{\mathcal{R}}(I_{j}^{(k)}),\;y-x\leq \lfloor l_{k}^{\alpha
^{\prime }/\alpha }\rfloor :\omega _{\{x,y\}}=1\big|\,H(I_{j}^{(k)})%
\big\rbrack\geq 1-l_{k-1}^{-\beta (1-\eta )(\alpha ^{\prime }-1)}
\label{lax}
\end{equation}
holds for $k \ge 2$. For $k=1$ the r.h.s in (\ref{lax}) is replaced by $%
1-l_{1}^{-\beta (1-\eta )(\alpha ^{\prime }-1)/\alpha}$ .
\end{lemma}

\smallskip \noindent \textbf{Proof.} We show the above estimate by
conditioning on $H_{z}(I_{j}^{(k)})$, uniformly in $z$, and we make repeated
use of the following upper and lower bounds: if $I$ and $I^{\prime }$ are
two intervals, and $3\le d=\text{dist}(I,I^{\prime })$, then

\begin{equation}
C^{-}\ J(I,I^{\prime })\leq \sum_{\substack{ x\in I\cap \mathbb{Z}  \\ y\in
I^{\prime }\cap \mathbb{Z}}}\frac{1}{|x-y|^{2}}\leq C^{+}\ J(I,I^{\prime }),
\label{Bsumm}
\end{equation}%
holds with $C^{\pm }=\left( 1\pm 2/d\right) ^{2}$ and
\begin{equation}
\ J(I,I^{\prime })=\int_{I\times I^{\prime }}dx\,dy\frac{1}{|x-y|^{2}}=\ln
\frac{(|I|+d)(|I^{\prime }|+d)}{d\,(|I|+|I^{\prime }|+d)}.  \label{BJ}
\end{equation}%
Notice that we have $C^{-}\left( \left\vert x-y\right\vert -2\right)
^{-2}\leq \left\vert x-y\right\vert ^{-2}\leq C^{+}\left( \left\vert
x-y\right\vert +2\right) ^{-2}$ for $\left\vert x-y\right\vert \ge d$. We
shall need also the inequality%
\begin{equation}
J(I,I^{\prime })\leq 4\frac{\left\vert I^{\prime }\right\vert }{\left\vert
I^{\prime \prime }\right\vert }J\left( I,I^{\prime \prime }\right)
\label{inequal}
\end{equation}%
which holds for every $I$, $I^{\prime }$ and $I^{\prime \prime }$ such that $%
I^{\prime }\subset I^{\prime \prime }$ and $d^{\prime }=\text{dist}%
(I,I^{\prime \prime })\geq \left\vert I^{\prime \prime}\right\vert $.
Indeed, setting $f(x)=\int_{I}dy~|x-y|^{-2}$, for $x \in I^{\prime\prime}$,
straightforward calculations give that under the above conditions:
\begin{equation*}
f(x^\prime)\le 4f(x^{\prime\prime}) \quad {\text{for each }}\quad x^\prime
\in I^{\prime },x^{\prime\prime} \in I^{\prime\prime}
\end{equation*}%
from where the inequality (\ref{inequal}) follows upon integration.

If $k \ge 2$ and $H_{z}(I_{j}^{(k)})$ occurs, we have the left $k$--pedestal
$\Upsilon _{\mathcal{L}}(I_{j}^{(k)})$ spanning from the left endpoint of $%
I_{j}^{(k)}$ to the left endpoint of $I_{z}^{(k-1)}$, and the right $k$%
-pedestal $\Upsilon _{\mathcal{R}}(I_{j}^{(k)})$, spanning from the right
endpoint of $I_{z}^{(k-1)}$ to the right endpoint of $I_{j}^{(k)}$. Take two
segments $S_{z}^{\mathcal{L}}$ and $S_{z}^{\mathcal{R}}$, such that $|S_{z}^{%
\mathcal{L}}|=|S_{z}^{\mathcal{R}}|=\lfloor l_{k}^{\alpha ^{\prime }/\alpha
}\rfloor/3$, lying immediately to the left and, respectively, to the right
of $I_{z}^{(k-1)}$. Denote
\begin{eqnarray*}
\widehat{\Upsilon }_{\mathcal{L}}(I_{j}^{(k)}) &=&\Upsilon _{\mathcal{L}
}(I_{j}^{(k)})\cap S_{z}^{\mathcal{L}}, \\
\widehat{\Upsilon }_{\mathcal{R}}(I_{j}^{(k)}) &=&\Upsilon _{\mathcal{R}
}(I_{j}^{(k)})\cap S_{z}^{\mathcal{R}}.
\end{eqnarray*}
Then
\begin{equation*}
\nu[{\text{all edges $\{ x,y\} ,\;x\in \widehat{\Upsilon }_{\mathcal{L}%
}(I_{j}^{(k)}),\,y\in \widehat{\Upsilon }_{\mathcal{R}}(I_{j}^{(k)})$ are
closed}}]
\end{equation*}%
\begin{equation*}
\le{\prod_{\substack{ x\in S_{z}^{\mathcal{L}}  \\ y\in S_{z}^{\mathcal{R}}}}%
q_{\{ x,y\} }}{\prod_{\substack{ x\in S_{z}^{\mathcal{L}}\setminus \widehat{%
\Upsilon }_{\mathcal{L}}(I_{j}^{(k)})  \\ y\in S_{z}^{\mathcal{R}}}}q_{\{
x,y\} }^{-1}}{\prod _{\substack{ x\in S_{z}^{\mathcal{L}}  \\ y\in S_{z}^{%
\mathcal{R}}\setminus \widehat{\Upsilon }_{\mathcal{R}}(I_{j}^{(k)})}}q_{\{
x,y\} }^{-1}}.
\end{equation*}

\noindent Applying (\ref{Bsumm}) to $S_{z}^{\mathcal{L}}$ and $S_{z}^{%
\mathcal{R}}$ we immediately get the following bound:
\begin{equation}
{\prod_{\substack{ x\in S_{z}^{\mathcal{L}}  \\ y\in S_{z}^{\mathcal{R}}}}%
q_{\{ x,y\} }}=\exp \big\{-{\sum_{\substack{ x\in S_{z}^{\mathcal{L}}  \\ %
y\in S_{z}^{\mathcal{R}}}}\frac{\beta }{|x-y|^{2}}}\big\}\leq
l_{k-1}^{-\beta (\alpha ^{\prime }-1)(1-b)},  \label{gru}
\end{equation}%
where $b\equiv b(\alpha ^{\prime },l_{1})$ and $b\searrow 0$ when $%
l_{1}\nearrow +\infty $. The same computation gives that if a 1-block $I$
has a unique closed edge $\{ a, a+1\}$ with both $a, a+1$ at distance larger
than $l_1^{\alpha^\prime/\alpha}$ from the endpoints of $I$, then the
probability that there is an open edge $\{ x,y\}$ with $x<a<y$, $y-x \le
l_1^{\alpha^\prime/\alpha}$ is larger than or equal of $1-l_1^{-\beta(1-%
\eta)(\alpha^\prime -1)/\alpha}$.

On the other hand denoting by $\mathcal{D}_{n}(S_{z}^{\mathcal{L}}\setminus
\widehat{\Upsilon }_{\mathcal{L}}(I_{j}^{(k)})),\;0\leq n\leq k-2$ (resp. $%
\mathcal{D}_{n}(S_{z}^{\mathcal{R}}\setminus \widehat{\Upsilon }_{\mathcal{R}%
}(I_{j}^{(k)})))$ the set of vertices which belong to all $n$--level defects
contained in the segment $S_{z}^{\mathcal{L}}$ (resp. $S_{z}^{\mathcal{R}}$%
), we get
\begin{equation*}
{\prod_{\substack{ x\in S_{z}^{\mathcal{L}}\setminus \widehat{\Upsilon }_{%
\mathcal{L}}(I_{j}^{(k)})  \\ y\in S_{z}^{\mathcal{R}}}}q_{\{ x,y\} }}%
=\prod_{n=0}^{k-2}{\prod_{\substack{ x\in \mathcal{D}_{n}(S_{z}^{\mathcal{L}%
}\setminus \widehat{\Upsilon }_{\mathcal{L}}(I_{j}^{(k)}))  \\ y\in S_{z}^{%
\mathcal{R}}}}q_{\{ x,y\} }}=\exp \Big\{-\sum_{n=0}^{k-2}\sum_{\substack{ %
x\in \mathcal{D}_{n}(S_{z}^{\mathcal{L}}\setminus \widehat{\Upsilon }_{%
\mathcal{L}}(I_{j}^{(k)}))  \\ y\in S_{z}^{\mathcal{R}}}}\frac{\beta }{%
|x-y|^{2}}\Big\}.
\end{equation*}%
Once again, applying (\ref{Bsumm}) for each $0\leq n\leq k-2$ and taking
into account the structure of $n$--level pedestals together with (\ref%
{inequal}), we have%
\begin{eqnarray*}
\sum_{n=0}^{k-2}\sum_{\substack{ x\in \mathcal{D}_{n}(S_{z}^{\mathcal{L}%
}\setminus \widehat{\Upsilon }_{\mathcal{L}}(I_{j}^{(k)}))  \\ y\in S_{z}^{%
\mathcal{R}}}}\dfrac{\beta }{|x-y|^{2}} &\leq
&C^{+}\sum_{n=0}^{k-2}\sum_{\nu }J\left( I_{\nu }^{\prime },I^{R}\right) \\
&\leq &C\sum_{n=0}^{k-2}\frac{l_{n+1}^{\alpha ^{\prime }/\alpha }}{l_{n+1}}%
\sum_{\nu }J\left( I_{\nu }^{\prime \prime },I^{R}\right) \\
&\leq &C^{\prime }l_{1}^{\alpha ^{\prime }/\alpha -1}J\left( I^{\mathcal{L}%
},I^{\mathcal{R}}\right)
\end{eqnarray*}%
where $I^{\mathcal{L}(\mathcal{R})}$, $I_{\nu }^{\prime }$ and $I_{\nu
}^{\prime \prime }$ are intervals in $\mathbb{R}$ such that $I^{\mathcal{L}(%
\mathcal{R})}\cap \mathbb{Z}=S_{z}^{\mathcal{L}(\mathcal{R})}$, $%
\bigcup_{\nu }\left( I_{\nu }^{\prime }\cap \mathbb{Z}\right) =\mathcal{D}%
_{n}(S_{z}^{L}\setminus \widehat{\Upsilon }_{\mathcal{L}}(I_{j}^{(k)}))$ and
the sum $\sum_{\nu }$ is taken over all indices $\nu $ of $(n+1)$--blocks $%
I_{\nu }^{(n+1)}=I_{\nu }^{\prime \prime }\cap \mathbb{Z}$, where the
defected $n$--blocks are located. The condition to apply (\ref{inequal}) in
the first inequality above follows from $3l_{k-1}+6l_{k-2}\le
l_k^{\alpha^\prime/\alpha}$ which is true for any $k\ge 2$, provided $l_1$
has geen taken large enough. From this we can easily get that
\begin{equation}
{\prod_{\substack{ x\in S_{z}^{\mathcal{L}}\setminus \widehat{\Upsilon }_{%
\mathcal{L}}(I_{j}^{(k)})  \\ y\in S_{z}^{\mathcal{R}}}}q_{\{ x,y\} }}\geq
l_{k-1}^{-\beta (\alpha ^{\prime }-1)b^{\prime }},  \label{hto}
\end{equation}%
where $b^{\prime }\equiv b^{\prime }(\alpha ,\alpha ^{\prime },l_{1})$ and $%
b^{\prime }\searrow 0$ when $l_{1}\nearrow +\infty $. Analogous bound holds
for the third term. Finally, from the upper bound for the length of a $(k-1)$%
-block, we have
\begin{eqnarray}
&&[\omega :\exists x\in \Upsilon _{\mathcal{L}}(I_{j}^{(k)}),y\in \Upsilon _{%
\mathcal{R}}(I_{j}^{(k)}),\;y-x\leq \lfloor l_{k}^{\alpha ^{\prime }/\alpha
}\rfloor :\omega _{\{x,y\}}=1]^c  \notag \\
&\subseteq &[{\text{all edges $\{ x,y\} ,\;x\in \widehat{\Upsilon }_{%
\mathcal{L}}(I_{j}^{(k)}),\,y\in \widehat{\Upsilon }_{\mathcal{R}%
}(I_{j}^{(k)})$ are closed}}],~  \label{vno1}
\end{eqnarray}%
the statement of the Lemma follows from (\ref{gru}) and (\ref{hto}). $%
\square $

\medskip

\noindent \textbf{Fixing the parameters.} For fixed $\beta >1$, which is the
first main parameter of the model we choose the pair $\alpha ,\;\alpha
^{\prime }$ with $1<\alpha ^{\prime }<\alpha <2$ such that
\begin{equation}
\beta (\alpha ^{\prime }-1)-\frac{2(\alpha -1)^{2}}{2-\alpha }>\alpha -1,
\label{aa}
\end{equation}%
i.e. $\beta (\alpha ^{\prime }-1)>\alpha \left( \alpha -1\right) /(2-\alpha
) $. Now we fix $l_{1}>1$ so large that the parameter $\eta =\eta (\alpha
,\alpha ^{\prime },l_{1})$ in (\ref{lax}) becomes so close to zero, that
\begin{equation}
\beta (1-\eta )(\alpha ^{\prime }-1)-\frac{2(\alpha -1)^{2}}{2-\alpha }%
>\alpha -1.  \label{aal}
\end{equation}%
Finally we fix
\begin{equation}
\delta >\frac{2(\alpha -1)}{2-\alpha }  \label{del}
\end{equation}%
such that
\begin{equation}
\beta (1-\eta )(\alpha ^{\prime }-1)-\delta (\alpha -1)>\alpha -1.
\label{aald}
\end{equation}

\noindent Inequalities (\ref{lax}), (\ref{del}) and (\ref{aald}) will be
crucial for the inductive estimates.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Decomposition of the product measure. Basic estimates \label{II}}

\setcounter{equation}{0} \setcounter{theorem}{0} \noindent Let $\nu _{L}$ be
the product measure on $\Omega _{L}$, the space of configurations over edges
with both end-vertices in $[-L,L]$. Looking at successive projections of
this measure, we need a notation that emphasize the set of edges which have
been integrated out. This explains the somehow heavy notation below.

\begin{proposition}
\label{decomposition}For $L=l_{M}$ and $k=0,\dots,M$\!, the measure $\nu
_{L} $ can be written as
\begin{equation}
\nu _{L}=\sum_{J_{L}^{(k)}\in \mathcal{I}_{L}^{(k)}}c({J}_{L}^{(k)})%
\prod_{I_{i}^{(k)}\in {\ I}^{(k)}}\Big(\gamma _{_{I_{i}^{(k)}}}+\Delta
_{_{I_{i}^{(k)}}}\Big)\nu _{J_{L}^{(k)}}^{o},  \label{indk0}
\end{equation}
\noindent where

\noindent $\bullet $ the sum $\sum_{J_{L}^{(k)}\in \mathcal{I}^{(k)}}$ is
taken over all possible itineraries $J_{L}^{(k)}=\{\{I^{(i)},D^{(i)}%
\}_{i=0}^{k}\}$ of length $k$;

\noindent \noindent the product $\prod_{I_{i}^{(k)}\in I^{(k)}}$ is taken
over all $k$--blocks $I_{i}^{(k)}$ which form partition of $L$ at the last
step $I^{(k)}$of the itinerary $J_{L}^{(k)}$.

\noindent $\bullet $ The measure $\nu _{{J}_{L}^{(k)}}^{o}$ is the marginal
of $\nu _{L}$, after integrating out all variables corresponding to the
edges $e\in \cup _{i}\ \mathbb{E}(I_{i}^{(k)}),\;I_{i}^{(k)}\in I^{(k)}$.

\noindent $\bullet $ $c({J}_{L}^{(k)})\geq 0,\quad \sum_{J_{L}^{(k)}\in
\mathcal{I}_{L}^{(k)}}c({J}_{L}^{(k)})=1$; \smallskip

\noindent \medskip \noindent $\bullet $ $\gamma _{_{I_{i}^{(k)}}}$ and $%
\Delta _{_{I_{i}^{(k)}}}$ are non-negative functions satisfying the
following properties:

\noindent 1)
\begin{equation}
\gamma_{_{I_i^{(k)}}}(\omega) \Delta_{_{I_i^{(k)}}} (\omega)=0.
\label{estm00123k}
\end{equation}

\noindent 2) If $\Delta_{_{I_i^{(k)}}}(\omega )>0$, in which case we say
that $I_{i}^{(k)}$ is defected for $\omega $, then there is no $k$--pedestal
connecting the end-vertices of $I_{i}^{(k)}$. Moreover,
\begin{eqnarray}
\sup_\omega\,\kappa^{\mathcal{C}_{I_i^{(k)}}(\omega)}
\Delta_{_{I_i^{(k)}}}(\omega) \,\le l_{k}^{-\delta}.  \label{estm11k}
\end{eqnarray}

\noindent 3) If $\gamma_{_{I_{i}^{(k)}}}(\omega )>0$, the interval $%
I_{i}^{(k)}$ contains a $k$--pedestal which, unless $I_{i}^{(k)}$ is the
rightmost or the leftmost for $I^{(k)}$, connects its end-vertices;
otherwise, the $k$--pedestal is as explained in Sect. \ref{I}. Moreover, for
all configurations $\omega $ with at least one closed nearest neighbour edge
in $I_{i}^{(k)}$, one has
\begin{equation}
\kappa ^{\mathcal{C}_{I_{i}^{(k)}}(\omega )}\gamma _{_{I_{i}^{(k)}}}(\omega
)\leq 1.  \label{estm123k}
\end{equation}%
When all nearest neighbour edges in $I_{i}^{(k)}$ are open we only have the
trivial bound $\gamma _{_{I_{i}^{(k)}}}(\omega )\leq 1$.
\end{proposition}

\noindent \textbf{Proof.} Through this section we will repeatedly use the
following straightforward identity:

\noindent Let $N>0$, $c_{N}=2^{1/N}-1$, and $\{\gamma _{i}\}_{1}^{N}$ and $%
\{\Delta _{i}\}_{1}^{N}$ any two sets of random variables. Then
\begin{equation}
\prod_{j=1}^{N}(\gamma _{i}+\Delta _{i})=\sum_{\substack{ J\subset \left\{
1,\ldots ,N\right\} :  \\ J\neq \emptyset }}c_{N}^{|J|}\left(
\prod_{j=1}^{N}\gamma _{i}+c_{N}^{-|J|}\prod_{k\in J^{c}}\gamma
_{k}\prod_{j\in J}\Delta _{j}\right) ,  \label{ident}
\end{equation}%
\begin{equation}
\sum_{\substack{ J\subset \left\{ 1,\ldots ,N\right\} :  \\ J\neq \emptyset
}}c_{N}^{|J|}=1, \qquad\qquad c_{N}^{-1}\leq N/\ln 2.  \label{ident22}
\end{equation}

\noindent {\textbf{Step 1. {\textit{k }= 0}}}. In this case $\mathcal{I}%
_{L}^{(0)}$ has only one element $J_{L}^{(0)}=\{I^{(0)},D^{(0)}\}$ , and $%
I^{(0)}=\{I_{i}^{(0)}\}_{i=-L}^{L-1}$, $D^{(0)}=\{D_{i}^{(0)}
\}_{i=-L}^{L-1} $ with $I_{i}^{(0)}=[i,i+1]$, and $D_{i}^{(0)}=\emptyset $
for all $i$. Thus we trivially have:%
\begin{equation}
\nu _{L}=\prod_{I_{i}^{(0)}\in I^{(0)}}\left( p\,{\mathbb{I}}%
_{G(I_{i}^{(0)})}+(1-p){\mathbb{I}}_{B(I_{i}^{(0)})}\right) \nu
_{J_{L}^{(0)}}^{o}=\prod_{I_{i}^{(0)}\in I^{(0)}}\left( \gamma
_{_{I_{i}^{(0)}}}+\Delta _{_{I_{i}^{(0)}}}\right) \nu _{J_{L}^{(0)}}^{o},
\label{ind0}
\end{equation}%
where $\gamma _{_{I_{i}^{(0)}}}=p\,{\mathbb{I}}_{G(I_{i}^{(0)})}$ and $%
\Delta _{_{I_{i}^{(0)}}}=(1-p){\mathbb{I}}_{B(I_{i}^{(0)})}$. For $\omega $
such that ${\mathbb{I}}_{B(I_{i}^{(0)})}(\omega )=1$ we have $\mathcal{C}%
_{I_{i}^{(0)}}(\omega )=2$, since in this case in $I_{i}^{(0)}$ the nearest
neighbor edge is closed and we have two end-vertices as two disjoint
connected components. Thus, by our choice of $p$ in (\ref{choice00}), we
have
\begin{equation}
\sup_{\omega }\,\kappa ^{2}\Delta _{_{I_{i}^{(0)}}}(\omega )\,\leq \frac{%
(\ln 2)^{5}}{128}l_{1}^{-\delta -1}~.  \label{step0}
\end{equation}%
Item 3) is trivially checked when $k=0$.

\noindent \textbf{Proof of the recursive step. } Though simpler, the
procedure involved to get the estimates for $k=1$ has minor differences with
respect to the case $k\ge 2$. We point them out in the proof.

Let $k \ge 1$, and assume that we have the decomposition at step $(k-1)$,
satisfying the above described properties:
\begin{equation}
\nu_{L} = \sum_{J_L^{(k-1)}\in \mathcal{I}_L^{(k-1)}}c(J_L^{(k-1)})
\prod_{I_{u}^{(k-1)}\in {\ I}^{(k-1)}}\Big(\gamma_{_{I_u^{(k-1)}}}
+\Delta_{_{I_u^{(k-1)}}}\Big)\nu^o_{J_L^{(k-1)}}.  \label{indk}
\end{equation}

\noindent To obtain the decomposition and the estimates for the $k$ - th
step we proceed as follows: first, for each fixed itinerary $J^{(k-1)}\in
\mathcal{I}^{(k-1)}$ we rewrite
\begin{equation}
\prod_{I_{u}^{(k-1)}\in {\ I}^{(k-1)}}\Big(\gamma_{_{I_u^{(k-1)}}}
+\Delta_{_{I_u^{(k-1)}}}\Big)  \label{indk2}
\end{equation}
as a convex sum of product-type expressions, where the sum will be taken
over all possible choices of ${I}^{(k)}$ admissible from ${I}^{(k-1)}$ as
the $k$--th step of the itinerary which starts from the last step of $%
J_L^{(k-1)}$. The products will be taken over all such $k$--blocks ${I}%
^{(k)}_i \in {I}^{(k)}$.

\medskip

\noindent Recalling the construction of $k$--blocks in Sect. \ref{I}, and
the notation $\hat{\jmath }^k_i=\min\{s\colon x^{(k)}_i \in I^{(k-1)}_s\}$,
we write
\begin{equation}
\prod_{I_{u}^{(k-1)}\in I^{(k-1)}}\Big(\gamma_{_{I_u^{(k-1)}}}+%
\Delta_{_{I_u^{(k-1)}}}\Big) = \prod_{i:\ x^{(k)}_i \in [-L,L]}\ \prod_{s=%
\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath} _{i+1}^{k}}\Big(%
\gamma_{_{I_s^{(k-1)}}}+\Delta_{_{I_s^{(k-1)}}}\Big).  \label{carnaval3}
\end{equation}
For each $i$ in the product, take $D_i^{(k)}\subseteq \{ \hat{\jmath}%
_{i}^{k}+1,\dots, \hat{\jmath}_{i+1}^{k}\}, \; D_i^{(k)} \neq \emptyset$,
and using the identity (\ref{ident}) we rewrite r.h.s. of (\ref{carnaval3})
as
\begin{equation}
\sum_{\{D_{i}^{(k)}\}_{i}}c(\{ D_{i}^{(k)}\} _{i}) \prod_{i}\Big( \prod_{s=%
\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath}_{i+1}^{k}}\gamma_{_{I_s^{(k-1)}}}+ c_{%
\hat{\jmath}_{i+1}^{k}-\hat{\jmath}_{i}^{k}-1}^{-|D_{i}^{(k)}|}
\prod_{s^{\prime }\in\lbrack\hat{\jmath}_{i}^{k}+1,\hat{\jmath}
_{i+1}^{k}]\setminus
D_{i}^{(k)}}\gamma_{_{I_{s^\prime}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i}^{(k)}}\Delta_{_{I_{s^{\prime \prime}}^{(k-1)}}}\Big),  \label{zyze}
\end{equation}
\smallskip

\noindent where the sum $\sum_{\{D_{i}^{(k)}\}_{i}}$ is taken over all
possible collections $\{D_{i}^{(k)}\}_{i}$, $i \in \{-l_M/l_k,\dots,
l_M/l_k-1\}$, and the product $\prod_{i}$ is taken over all
(not-yet-modified) intervals $\tilde I^{(k)}_i=\cup_{s=\hat{\jmath}%
_{i}^{k}+1}^{\hat{\jmath} _{i+1}^{k}}I_{s}^{(k-1)} \equiv[%
a^{(k)}_i,a^{(k)}_{i+1}]$. Moreover
\begin{equation*}
c(\{ D_{i}^{(k)}\} _{i})\ge 0, \qquad \sum_{\{D_{i}^{(k)}\}_{i}}c( \{
D_{i}^{(k)}\} _{i}) =1,
\end{equation*}
and $D_{i}^{(k)}$ is the set of indices of the defected $(k-1)$--blocks $%
I_{s}^{(k-1)}$ in $\tilde I^{(k)}_i$.

\noindent \textbf{Rearrangement of factors at step} $k$\textbf{.} Recall
that at this stage we are working with the fixed itinerary $J_{L}^{(k-1)}$,
and now we will analyze each term in (\ref{zyze}). Thus $\{D_{i}^{(k)}\}_{i}$
is fixed, and we rearrange each product
\begin{equation}
\prod_{i}\Big(\prod_{s=\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath}%
_{i+1}^{k}}\gamma _{_{I_{s}^{(k-1)}}}+c_{\hat{\jmath}_{i+1}^{k}-\hat{\jmath}%
_{i}^{k}-1}^{-|D_{i}^{(k)}|}\prod_{s^{\prime }\in \lbrack \hat{\jmath}%
_{i}^{k}+1,\hat{\jmath}_{i+1}^{k}]\setminus D_{i}^{(k)}}\gamma
_{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s^{\prime \prime }\in D_{i}^{(k)}}\Delta
_{_{I_{s^{\prime \prime }}^{(k-1)}}}\Big),  \label{zyzefru}
\end{equation}%
according to locations of the defected $(k-1)$--blocks $I_{s}^{(k-1)},\;s\in
D_{i}^{(k)}$ in $\tilde{I}_{i}^{(k)}=[a_{i}^{(k)},a_{i+1}^{(k)}]$. \medskip

\noindent \textbf{i) Boundary shift case.} We check, as explained in Section %
\ref{I}, if there is an $i$ so that $|D_{i}^{(k)}|=1 $, say $%
D_{i}^{(k)}=\{s\}$, and the corresponding defected $I_{s}^{(k-1)}$ stays at
distance not more than $l_{k}^{\alpha ^{\prime }/\alpha }$ from an
end-vertex of $\tilde{I}_{i}^{(k)}$, say $a_{i}^{(k)}$. For each such $i$,
we check if $D_{i-1}^{(k)}$ contains an index $v$ so that the corresponding $%
I_{v}^{(k-1)}$ stays at distance not more than $3l_{k}^{\alpha ^{\prime
}/\alpha }$ from $a_{i}^{(k)}$. If it does not, then we shift the boundary
between the blocks $\tilde{I}_{i-1}^{(k)}$ and $\tilde{I}_{i}^{(k)}$, as in (%
{\ref{carnaval4}). In agreement, we rewrite the product of the two
corresponding factors in (\ref{zyze})
\begin{eqnarray}
&&\Big(\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{\hat{\jmath}_{i}^{k}}%
\gamma_{_{I_s^{(k-1)}}} +c_{\hat{\jmath}_{i}^{k}-\hat{\jmath}
_{i-1}^{k}-1}^{-|D_{i-1}^{(k)}|}\prod_{s^{\prime }\in \lbrack \hat{\jmath}
_{i-1}^{k}+1,j_{i}^{k}]\setminus
D_{i-1}^{(k)}}\gamma_{_{I_{s^\prime}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i-1}^{(k)}}\Delta_{_{I_{s^{\prime \prime}}^{(k-1)}}}\Big)  \notag \\
&\times &\Big(\prod_{s=\hat{\jmath}_{i}^{k}+1}^{\hat{\jmath}
_{i+1}^{k}}\gamma_{_{I_s^{(k-1)}}}+c_{\hat{\jmath}_{i+1}^{k}-\hat{\jmath}
_{i}^{k}-1}^{-|D_{i}^{(k)}|}\prod_{s^{\prime }\in \lbrack \hat{\jmath}%
_{i}^{k}+1, \hat{\jmath}_{i+1}^{k}]\setminus
D_{i}^{(k)}}\gamma_{_{I_{s^\prime}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i}^{(k)}}\Delta_{_{I_{s^{\prime \prime}}^{(k-1)}}}\Big)  \label{shiftlk}
\end{eqnarray}
as (our condition implies, in particular that $D_{i-1}^{(k)}\cap \lbrack
j_{a_{i}^{(k)}-\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor }^{k}+1,\hat{%
\jmath}_{i}^{k}]=\emptyset $):
\begin{eqnarray}
&\;&\Big(\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{j_{a_{i}^{(k)}-\lfloor
l_{k}^{\alpha ^{\prime }/\alpha }\rfloor
}^{k}}\gamma_{_{I_s^{(k-1)}}}+c_{j_{i}^{k}-j_{i-1}^{k}-1}^{-|D_{i-1}^{(k)}|}%
\prod_{s^{\prime }\in \lbrack j_{i-1}^{k}+1,j_{a_{i}^{(k)}-\lfloor
l_{k}^{\alpha ^{\prime }/\alpha }\rfloor }^{k}]\setminus
D_{i-1}^{(k)}}\gamma_{_{I_{s^\prime}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i-1}^{(k)}}\Delta_{_{I_{s^{\prime \prime}}^{(k-1)}}}\Big)  \notag \\
&\times &\Big(\prod_{s=j_{a_{i}^{(k)}-\lfloor l_{k}^{\alpha ^{\prime
}/\alpha }\rfloor }^{k}+1}^{\hat{\jmath}_{i+1}^{k}}\gamma_{_{I_s^{(k-1)}}}
+c_{j_{i+1}^{k}-j_{i}^{k}-1}^{-|D_{i}^{(k)}|}\prod_{s^{\prime }\in \lbrack
j_{a_{i}^{(k)}-\lfloor l_{k}^{\alpha ^{\prime }/\alpha }\rfloor }^{k}+1,\hat{%
\jmath}_{i+1}^{k}]\setminus
D_{i}^{(k)}}\gamma_{_{I_{s^\prime}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i}^{(k)}}\Delta_{_{I_{s^{\prime \prime}}^{(k-1)}}}\Big).  \label{shiftlkaz}
\end{eqnarray}%
}

\noindent Analogous rewriting is performed in the case the defect stays
close to the right boundary of $\tilde I^{(k)}_i$ and $\tilde I^{(k)}_{i+1}$
has no defect within distance $3l_k^{\alpha^\prime/\alpha}$ from its left
boundary.

\noindent \textbf{ii) Two blocks merge case.} Consider only the indices $i$
for which $|D_{i}^{(k)}|=1$ and the corresponding defected $(k-1)$--block
stays at distance not more than $l_{k}^{\alpha ^{\prime }/\alpha }$ from an
end-vertex of $\tilde{I}_{i}^{(k)}$, say $a_{i}^{(k)}$. Assume now that
differently from the previous case i) the block $\tilde{I}_{i-1}^{(k)}$ has
a defected $(k-1)$--block within distance $3l_{k}^{\alpha ^{\prime }/\alpha
} $ from $a_{i}^{(k)}$. As explained in Section 2, in this situation we can
not shift the boundary between the two blocks, and instead we will consider
them together, as one defected block at level $k$. In this case the product
of two terms, as in (\ref{shiftlk}), but now corresponding to two merging
blocks, is rewritten using (\ref{ident}) for a two factor product:%
\begin{eqnarray}
&&c_{2}\Big(\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{\hat{\jmath}%
_{i+1}^{k}}\gamma _{_{I_{s}^{(k-1)}}}+c_{2}^{-1}c_{\hat{\jmath}_{i}^{k}-\hat{%
\jmath}_{i-1}^{k}-1}^{-|D_{i-1}^{(k)}|}\prod_{s^{\prime }\in \lbrack \hat{%
\jmath}_{i}^{k}+1,\hat{\jmath}_{i+1}^{k}]\setminus D_{i}^{(k)}}\gamma
_{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s^{\prime \prime }\in D_{i}^{(k)}}\Delta
_{_{I_{s^{\prime \prime }}^{(k-1)}}}\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{\hat{%
\jmath}_{i+1}^{k}}\gamma _{_{I_{s}^{(k-1)}}}\Big)  \notag \\
&+&c_{2}\Big(\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{\hat{\jmath}%
_{i+1}^{k}}\gamma _{_{I_{s}^{(k-1)}}}+c_{2}^{-1}c_{\hat{\jmath}_{i+1}^{k}-%
\hat{\jmath}_{i}^{k}-1}^{-|D_{i}^{(k)}|}\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{%
\hat{\jmath}_{i}^{k}}\gamma _{_{I_{s}^{(k-1)}}}\prod_{s^{\prime }\in \lbrack
\hat{\jmath}_{i-1}^{k}+1,\hat{\jmath}_{i}^{k}]\setminus D_{i-1}^{(k)}}\gamma
_{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s^{\prime \prime }\in
D_{i-1}^{(k)}}\Delta _{_{I_{s^{\prime \prime }}^{(k-1)}}}\Big)  \notag \\
&+&c_{2}^{2}\Big(\prod_{s=\hat{\jmath}_{i-1}^{k}+1}^{\hat{\jmath}%
_{i+1}^{k}}\gamma _{_{I_{s}^{(k-1)}}}+c_{2}^{-2}c_{\hat{\jmath}_{i+1}^{k}-%
\hat{\jmath}_{i}^{k}-1}^{-|D_{i}^{(k)}|}c_{\hat{\jmath}_{i}^{k}-\hat{\jmath}%
_{i-1}^{k}-1}^{-|D_{i-1}^{(k)}|}\prod_{s^{\prime }\in \lbrack \hat{\jmath}%
_{i}^{k}+1,\hat{\jmath}_{i+1}^{k}]\setminus D_{i}^{(k)}}\gamma
_{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s^{\prime \prime }\in D_{i}^{(k)}}\Delta
_{_{I_{s^{\prime \prime }}^{(k-1)}}}  \notag \\
&\times &\prod_{s^{\prime }\in \lbrack \hat{\jmath}_{i-1}^{k}+1,\hat{\jmath}%
_{i}^{k}]\setminus D_{i-1}^{(k)}}\gamma _{_{I_{s^{\prime
}}^{(k-1)}}}\prod_{s^{\prime \prime }\in D_{i-1}^{(k)}}\Delta
_{_{I_{s^{\prime \prime }}^{(k-1)}}}\Big) .  \label{mergk}
\end{eqnarray}

\noindent Analogous rewriting is performed in the case the defect stays
close to the right boundary of $\tilde I^{(k)}_i$ and $\tilde I^{(k)}_{i+1}$
has a defected $(k-1)$--block within distance $3l_k^{\alpha^\prime/\alpha}$
from its left boundary.

\medskip

\noindent \textbf{iii) No modification case.} If during the adjustment
procedure the segment $\tilde{I}^{(k)}_i$ remains unchanged, then the
corresponding term in (\ref{zyze}) is not modified.

\smallskip

\noindent As long as the adjustment procedure calls on the rearrangement of
blocks we continue rewriting the factors as described above, taking into
account all modifications which have been performed up to this current step.
After a finite number of steps the procedure is stopped and the $k$--blocks
are renamed from left to right as $I^{(k)}_r$.

\medskip

When all modifications are completed we can write the r.h.s. of the last
equality in (\ref{zyze}) as follows:
\begin{equation}
\sum_{(I^{(k)},\{D_{r}^{(k)}\}_{r})}\prod_{r}c({I_{r}^{(k)}})\prod_{r}\Big( %
\prod_{s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}\gamma_{_{I_s^{(k-1)}}}+\widehat{
\zeta }(I_{r}^{(k)})\,\prod_{\substack{ {s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}
\\ {s\not\in D_{r}^{(k)}}}}\gamma_{_{I_s^{(k-1)}}}\prod_{s^{\prime}\in
D_{r}^{(k)}}\Delta_{_{I_{s^\prime}^{(k-1)}}} \Big) , \label{zyze2}
\end{equation}
where, for the fixed sequence $I^{(k-1)}$ (see just before (\ref{zyze})), we
have:

\noindent $\bullet$ the sum $\sum_{(I^{k},\{D_{r}^{(k)}\}_r)}$ is taken over
all possible choices of pairs $(I^{(k)},\{D_{r}^{(k)}\}_r)$ such that

\noindent 1) $I^{(k)}\equiv I^{(k)}(\{D_{r}^{(k)} \}_i) = \{I^{(k)}_{r}\}_r$
is admissible from $I^{(k-1)}$. It depends on the location of the defected $%
(k-1)$--blocks. The set of indices of defected $(k-1)$--blocks in $I^{(k-1)}$
equals to $\cup_r D_{r}^{(k)} $;

\noindent 2) $D_{r}^{(k)}$ is the set of indices of the defected $(k-1)$%
--blocks within $I^{(k)}_{r}$, for each $r$ (we use the same notation for
the $\{D_{r}^{(k)}\} $ before and after the adjustment procedure, but some
of these sets might have merged; of course $\cup_i D_{i}^{(k)} $ does not
change during the adjustment procedure;

\noindent $\bullet $ the constants $c({I_{r}^{(k)}})\ge 0$ depend on the all
itinerary $J^{(k)}$, and
\begin{equation*}
\sum_{(I^{k},\{D_{r}^{(k)}\}_{r})}\prod_{r}c({I_{r}^{(k)}})=1;
\end{equation*}

\noindent $\bullet $ If $k \ge 2$, $\widehat{\zeta}(I_{r}^{(k)})$ satisfies
the following bound:
\begin{equation}
\widehat{\zeta }(I_{r}^{(k)})\leq c_{2}^{-4}c_{8{\lfloor l_{k-1}^{\alpha
-1}\rfloor }}^{-|D_{r}^{(k)}|}\leq \frac{16}{(\ln 2)^{4}}\left( \frac{8{%
\lfloor l_{k-1}^{\alpha -1}\rfloor }}{\ln 2}\right) ^{|D_{r}|},
\label{topdog}
\end{equation}%
while for $k=1$ we should replace $\lfloor l_{k-1}^{\alpha -1}\rfloor$ by $%
l_1$ on the r.h.s. of (\ref{topdog}). To show (\ref{topdog}) we consider the
case when three adjacent intervals merge in the adjustment procedure. The
other cases are similar and are left to the reader. Assuming that three
blocks $\tilde{I}_{i}^{(k)},\tilde{I}_{i+1}^{(k)}$ and $\tilde{I}%
_{i+2}^{(k)} $ merge, and applying repeatedly (\ref{mergk}) first to the
terms corresponding to $\tilde{I}_{i}^{(k)}$ and $\tilde{I}_{i+1}^{(k)}$,
and then to $\tilde{I}_{i+2}^{(k)}$, we get that the largest corresponding
constant is

\begin{equation}
c_2^{-4} c_{j^k_{i+1} - \hat{\jmath}^k_{i}-1}^{-|D_{i}^{(k)}|}c_{\hat{\jmath}%
^k_{i+2} - \hat{\jmath}^k_{i+1}-1}^{-|D_{i+1}^{(k)}|}c_{\hat{\jmath}^k_{i+3}
- \hat{\jmath}^k_{i+2}-1}^{-|D_{i+3}^{(k)}|}.  \label{topdog1}
\end{equation}

\noindent On the other hand, the estimate (\ref{vno}) gives the uniform
bound:
\begin{equation}
\hat{\jmath}_{i+1}^{k}-\hat{\jmath}_{i}^{k}-1\leq \frac{3l_{k}+6l_{k-1}}{%
l_{k-1}-(2\lfloor l_{k-1}^{\alpha ^{\prime }/\alpha }\rfloor +6l_{k-2})}\leq
8\lfloor l_{k-1}^{\alpha -1}\rfloor ,  \label{topdog2}
\end{equation}%
for $k \ge 2$, provided $l_{1}$ is large. For $k=1$ we have $\hat{\jmath}%
_{i+1}^{1}-\hat{\jmath}_{i}^{1}-1\leq 3l_{1}+6\le 8l_1$. Recalling that $%
c_{N}^{-1}\leq N/\ln 2$, (\ref{topdog2}) and (\ref{topdog1}) imply (\ref%
{topdog}). \medskip

\noindent For $I_{i}^{(k)}$ such that $|D_{i}^{(k)}|=1$, say $%
D_{i}^{(k)}=\{z\}$, we define

\begin{equation*}
g_{_{I_{i}^{(k)}}}=\prod_{e\in \mathbb{E}^{^{\ast
}}(I_{i}^{(k)})}p_{e}^{\omega _{e}}q_{e}^{1-\omega _{e}}\mathbb{I}_{\omega
\in G_{z}^{+}(I_{i}^{(k)})},\ \qquad d_{_{I_{i}^{(k)}}}=\prod_{e\in \mathbb{E%
}^{^{\ast }}(I_{i}^{(k)})} p_{e}^{\omega _{e}}q_{e}^{1-\omega _{e}}\mathbb{I}
_{\omega \in B_{z}^{+}(I_{i}^{(k)}),}
\end{equation*}%
where $\mathbb{E}^{^{\ast }}(I_{i}^{(k)})=\mathbb{E}(I_{i}^{(k)})\setminus
\cup _{s}\mathbb{E}(I_{s}^{(k-1)})$. When $|D_{i}^{(k)}|>1$ we define $%
g_{_{I_{i}^{(k)}}}\equiv 0,\,d_{_{I_{i}^{(k)}}}=\prod\limits_{_{e\in \mathbb{%
E}^{^{\ast }}(I_{i}^{(k)})}}p_{e}^{\omega _{e}}q_{e}^{1-\omega _{e}}$.

Thus, for $I^{(k-1)}$ fixed and $I^{(k)}=\{I_{r}^{(k)}\}_{r}$, \ obtained
from $I^{(k-1)}$, with the collection of sets of defected indices $D\equiv
D^{(k)}=\{D_{r}^{(k)}\}_{r}$, we write
\begin{eqnarray}
&\;&\prod_{_{I_{r}^{(k)}}}\Big(\prod_{s:I_{s}^{(k-1)}\subset
I_{r}^{(k)}}\gamma _{_{I_{s}^{(k-1)}}}+\widehat{\zeta }(I^{(k)}_r)\,%
\displaystyle\prod_{\substack{ s:I_{s}^{(k-1)}\subset I_{r}^{(k)}  \\ %
s\notin D_{r}^{(k)}}}\gamma _{_{I_{s}^{(k-1)}}}\prod_{s^{\prime }\in
D_{r}^{(k)}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}\Big)\nu _{J_{L}^{(k-1)}}^{o}
\notag \\
&\;&\prod_{_{I_{r}^{(k)}}}\big(\prod_{s:I_{s}^{(k-1)}\subset
I_{r}^{(k)}}\gamma _{_{I_{s}^{(k-1)}}}+\widehat{\zeta }(I^{(k)}_r)\, %
\displaystyle\prod_{\substack{ s:I_{s}^{(k-1)}\subset I_{r}^{(k)}  \\ %
s\notin D_{r}^{(k)}}}\gamma _{_{I_{s}^{(k-1)}}}\prod_{s^{\prime }\in
D_r^{(k)}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}\big)%
(g_{_{I_{r}^{(k)}}}+d_{_{I_{r}^{(k)}}})\,\nu _{J_{L}^{(k)}}^{o}  \notag \\
&=&\prod_{_{I_{r}^{(k)}}}\Big(\gamma _{_{I_{r}^{(k)}}}+\Delta
_{_{I_{r}^{(k)}}}\Big)\nu _{J_{L}^{(k)}}^{o},  \label{boutk}
\end{eqnarray}%
where (products over $s$ carry the condition $I_{s}^{(k-1)}\subset
I_{r}^{(k)}$)
\begin{eqnarray}
\gamma _{_{I_{r}^{(k)}}}
&=&(g_{_{I_{r}^{(k)}}}+d_{_{I_{r}^{(k)}}})\prod_{s}\gamma
_{_{I_{s}^{(k-1)}}}+\widehat{\zeta} (I_{r}^{(k)})\,g_{_{I_{r}^{(k)}}}%
\prod_{s\notin D_r^{(k)}}\gamma _{_{I_{s}^{(k-1)}}}\prod_{s^{\prime }\in
D_r^{(k)}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}  \label{monday} \\
\Delta _{_{I_{r}^{(k)}}} &=&\widehat{\zeta} (I_{r}^{(k)})%
\,d_{_{I_{r}^{(k)}}}\prod_{s\notin D_r^{(k)}}\gamma
_{_{I_{s}^{(k-1)}}}\prod_{s^{\prime }\in D_r^{(k)}}\Delta _{_{I_{s^{\prime
}}^{(k-1)}}}.  \label{mondaybis}
\end{eqnarray}%
and $\nu _{J_{L}^{(k)}}^{o}$ is the marginal of $\nu _{J_{L}^{(k-1)}}^{o}$
(or of $\nu _{L}$) with all variables corresponding to the edges $e\in \cup
_{i}\mathbb{E}(I_{i}^{(k)})$ integrated out.

\medskip

\noindent Thus, taking (\ref{boutk}) into account, (\ref{indk}) can be
continued as:
\begin{eqnarray}
\nu_{L} &=&\sum_{{I}^{(k-1)}\in \mathcal{I}^{(k-1)}}c({I} ^{k-1})\sum_{%
\{D_{r}\}_{r}}\prod_{I_{r}^{(k)}}c({I_{r}^{(k)}})\prod_{I_{r}^{(k)}}\Big(%
\gamma_{_{I_{r}^{(k)}}}+\Delta_{_{I_{r}^{(k)}}}\Big)\nu _{J_L^{(k)}}  \notag
\\
&=&\sum_{{I}^{(k-1)}\in \mathcal{I}^{(k)}}c({I}^{k})\prod_{I_{r}^{(k)}\in
\mathcal{I}^{(k)}}\Big(\gamma_{_{I_{r}^{(k)}}}+\Delta_{_{I_{r}^{(k)}}}\Big)%
\nu^o _{J_L^{(k)}} . \label{deck}
\end{eqnarray}%
\medskip

\medskip

\noindent \textbf{Estimates.} First of all we observe that using (\ref%
{topdog}) and (\ref{mondaybis}) we get:

\begin{equation}
\Delta_{_{I_{r}^{(k)}}}(\omega )\leq \frac{16}{(\ln 2)^{4}}\left( \frac{8{%
\lfloor l_{k-1}^{\alpha -1}\rfloor }}{\ln 2}\right)
^{|D_{r}^{(k)}|}d_{_{I_{r}^{(k)}}}(\omega )\prod_{s\not\in
D_{r}^{(k)}}\gamma_{_{I_{s}^{(k-1)}}}(\omega )\prod_{s^{\prime }\in
D_{r}^{(k)}}\Delta_{_{I_{s^{ \prime }}^{(k-1)}}}(\omega ),  \label{e44}
\end{equation}
for $k\ge2$, while in the case $k=1$, $\lfloor l_{k-1}^{\alpha -1}\rfloor $
is replaced by $l_1$. \medskip

\noindent Given an itinerary $J$ we will define the following sets:
\begin{eqnarray*}
{\widetilde{D}}_{r,0}^{(k)}(J) &\equiv &{\widetilde{D}}_{r,0}^{(k)}=\{s%
\colon I_{s}^{(k-1)}\subset I_{r}^{(k)},{\text{all nearest neighbour edges
in $I_{s}^{(k-1)}$ are open}}\}, \\
{\widetilde{D}}_{r,1}^{(k)}(J) &\equiv &{\widetilde{D}}_{r,1}^{(k)}=\{s%
\notin D_{r}^{(k)}\colon I_{s}^{(k-1)}\subset I_{r}^{(k)},{\text{some
nearest neighbour edges in $I_s^{(k-1)}$ are closed}}\}.
\end{eqnarray*}%
We will use the following inequality:%
\begin{equation}  \label{e47}
\kappa ^{\mathcal{C}_{I_{r}^{(k)}}} \le \prod_{\substack{ {\
s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s \in \widetilde D_{r,1}^{(k)}}}}%
\kappa ^{\mathcal{C } _{I_{s}^{(k-1)}}}\prod_{\substack{ {\
s^{\prime}:I_{s^{\prime}}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s^{\prime}\in
D_r^{(k)}}}}\kappa ^{{\mathcal{C}}_{I_{s^{\prime }}^{(k-1)}}},
\end{equation}
where it is taken into account that presence of blocks $I_{s}^{(k-1)}\subset
I_{r}^{(k)},\;s\in {\widetilde{D}}_{r,0}^{(k)}(J)$ does not in\-crea\-se the
total number of disjoint connected clusters.

Using (\ref{lax}), (\ref{estm11k}), (\ref{estm123k}) and (\ref{e44}), we
get, uniformly in $\omega$:
\begin{eqnarray}
&\!&\kappa ^{\mathcal{C}_{I_{r}^{(k)}}}\Delta _{_{I_{r}^{(k)}}}  \notag \\
&\leq& \kappa ^{\mathcal{C}_{I_{r}^{(k)}}}\frac{16}{(\ln 2)^{4}}\left( \frac{%
8{\ \lfloor l_{k-1}^{\alpha -1}\rfloor }}{\ln 2}%
\right)^{|D_{r}^{(k)}|}d_{_{I_{r}^{(k)}}}\prod_{\substack{ {%
s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s\not\in D_r^{(k)}}}}\gamma
_{_{I_{s}^{(k-1)}}}\prod_{\substack{ {s^{\prime }:I_{s^{\prime
}}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s^{\prime }\in D_r^{(k)}}}}\Delta
_{_{I_{s^{\prime }}^{(k-1)}}}  \notag \\
&\leq &\kappa ^{\mathcal{C}_{I_{r}^{(k)}}}\frac{16}{(\ln 2)^{4}}\left( \frac{%
8{\ \lfloor l_{k-1}^{\alpha -1}\rfloor }}{\ln 2}\right)
^{|D_{r}^{(k)}|}d_{_{I_{r}^{(k)}}}\prod_{s\in {\widetilde{D}}%
_{r,0}^{(k)}(J)}\gamma _{_{I_{s^{\prime \prime }}^{(k-1)}}}\prod_{s\in {%
\widetilde{D}}_{r,1}^{(k)}(J)}\gamma _{_{I_{s}^{(k-1)}}}\prod_{s^{\prime
}\in D_{r}^{(k)}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}  \notag \\
&\leq & \frac{16}{(\ln 2)^{4}}\left( \frac{8{\ \lfloor l_{k-1}^{\alpha
-1}\rfloor }}{\ln 2}\right) ^{|D_{r}^{(k)}|}d_{_{I_{r}^{(k)}}}\prod
_{\substack{ {s: I_{s}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s\in {\widetilde{D}}
_{r,1}^{(k)}(J)}}}\kappa ^{\mathcal{C}_{I_{s}^{(k-1)}}}\gamma
_{_{I_{s}^{(k-1)}}}\prod_{\substack{ {s^{\prime }:I_{s^{\prime
}}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s^{\prime }\in D_{r}^{(k)}}}}\kappa ^{%
\mathcal{C }_{I_{s^{\prime }}^{(k-1)}}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}.
\label{e45}
\end{eqnarray}%
Recall that, by construction, the variables $d_{_{I_{r}^{(k)}}}$, $\gamma
_{_{I_{s}^{(k-1)}}}$, $\Delta _{_{I_{s^{\prime }}^{(k-1)}}}$, with $s\neq
s^{\prime }$, all depend on disjoint sets of edges. For the last inequality
of (\ref{e45}) we used ({\ref{e47})} and the trivial bound $\prod_{s\in {%
\widetilde{D}}_{r,0}^{(k-1)}(J)}\gamma _{_{I_{s^{\prime \prime
}}^{(k-1)}}}(\omega )\leq 1$. Having (\ref{e45}), we consider two cases $%
|D_r^{(k)}|=1$ and $|D_r^{(k)}|>1$. If $|D_r^{(k)}|>1$ we use the trivial
bound $d_{_{I_{r}^{(k)}}}\le 1$ and properties stated in 1)-3) at scale $%
(k-1)$: $\kappa ^{\mathcal{C} _{I_{s}^{(k-1)}}}\gamma
_{_{I_{s}^{(k-1)}}}\leq 1$ for $s\in \widetilde{D}_{r,1}^{(k)}(J)$, and $%
\kappa ^{\mathcal{C}_{I_{s^{\prime }}^{(k-1)}}}\Delta _{_{I_{s^{\prime
}}^{(k-1)}}}\leq l_{k-1}^{-\delta }$ for $s^\prime\in D_{r}^{(k)}$, and we
get that the r.h.s. of last inequality of (\ref{e45}) is bounded by
\begin{equation*}
\frac{128}{(\ln 2)^{4}}l_{k-1}^{(\alpha -1-\delta )|D_r^{(k)}|}\leq \frac{128%
}{(\ln 2)^{4}}l_{k-1}^{2(\alpha -1-\delta )}\leq l_{k}^{-\delta }.
\end{equation*}%
If $|D_r^{(k)}|=1$ we again use $\kappa ^{\mathcal{C}_{I_{s}^{(k-1)}}}\gamma
_{_{I_{s}^{(k-1)}}}\leq 1$ and $\kappa ^{\mathcal{C}_{I_{s^{\prime
}}^{(k-1)}}}\Delta_{_{I_{s^{\prime }}^{(k-1)}}}\leq l_{k-1}^{-\delta }$ as
previously, but now we use Lemma \ref{est33} to estimate $d_{_{I_{r}^{(k)}}}$%
. In this case the r.h.s. of last inequality of (\ref{e45}) is bounded by
\begin{equation*}
\frac{128}{(\ln 2)^{5}}l_{k-1}^{-\delta +\alpha -1}l_{k-1}^{-\beta (1-\eta
)(\alpha ^{\prime }-1)}\leq l_{k}^{-\delta },
\end{equation*}%
and we get the estimate (\ref{estm11k}).

Now we estimate $\kappa ^{\mathcal{C}_{I_{r}^{(k)}}(\omega )}\gamma
_{_{I_{r}^{(k)}}}(\omega )$. First observe that for each $\omega $ only one
of two summands in (\ref{monday})

\begin{eqnarray}
(g_{_{I_{r}^{(k)}}} + d_{_{I_{r}^{(k)}}})\prod_{s :I_{s}^{(k-1)}\subset
I_{r}^{(k)}} \gamma_{_{I_{s}^{(k-1)}}} \quad \text{or }\quad \widehat{\zeta}
(I_{r}^{(k)})\, g_{_{I_{r}^{(k)}}} \Delta_{_{I_{s^{\prime }}^{(k-1)}}} \prod
_{\substack{ {s :I_{s}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s \notin D_{r}}}}
\gamma_{_{I_{s}^{(k-1)}}}
\end{eqnarray}
is non-zero, corresponding to the cases $D_{r}^{(k)} = \emptyset$ or $%
D_{r}^{(k)} = \{s^{\prime} \}$. The first term can be estimated (recall that
we are in the case ${\widetilde{D} }_{r,1}^{(k)}(J) \neq \emptyset $):

\begin{eqnarray}
\kappa ^{\mathcal{C}_{I_{r}^{(k)}}}(g_{_{I_{r}^{(k)}}}+d_{_{I_{r}^{(k)}}})%
\prod_{s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}\gamma _{_{I_{s}^{(k-1)}}}
&=&\kappa ^{\mathcal{C}%
_{I_{r}^{(k)}}}(g_{_{I_{r}^{(k)}}}+d_{_{I_{r}^{(k)}}})\prod_{s^{\prime
\prime }\in {\widetilde{D}}_{r,0}^{(k)}(J)}\gamma _{_{I_{s^{\prime \prime
}}^{(k-1)}}}\prod_{s\in {\widetilde{D}}_{r,1}^{(k)}(J)}\gamma
_{_{I_{s}^{(k-1)}}}  \notag \\
&\leq &\prod_{s\in {\widetilde{D}}_{r,1}^{(k)}(J)}\kappa ^{\mathcal{C}%
_{I_{s}^{(k-1)}}}\gamma _{_{I_{s}^{(k-1)}}} \leq 1,  \label{prdsr2}
\end{eqnarray}%
where we used trivial bound $\gamma _{_{I_{s^{\prime \prime
}}^{(k-1)}}}(\omega )\leq p_{1}^{|I_{s^{\prime \prime }}^{(k-1)}|}<1$ for $%
s^{\prime \prime }\in {\widetilde{D}}_{r,0}^{(k)}$. For the second term we
have:
\begin{eqnarray}
&\!&\kappa ^{\mathcal{C}_{I_{r}^{(k)}}}\widehat{\zeta} (I_{r}^{(k)})%
\,g_{_{I_{r}^{(k)}}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}\prod_{\substack{ {%
s:I_{s}^{(k-1)}\subset I_{r}^{(k)}}  \\ {s\notin D_r^{(k)}}}}\gamma
_{_{I_{s}^{(k-1)}}}  \notag \\
&=&\kappa ^{\mathcal{C}_{I_{r}^{(k)}}}\widehat{\zeta} (I_{r}^{(k)})%
\,g_{_{I_{r}^{(k)}}}\Delta _{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s\in {%
\widetilde{D}}_{r,0}^{(k)}(J)}\gamma _{_{I_{s^{\prime \prime
}}^{(k-1)}}}\prod_{s\in {\widetilde{D}}_{r,1}^{(k)}(J)}\gamma
_{_{I_{s}^{(k-1)}}}  \notag \\
&\leq &\kappa ^{\mathcal{C}_{I_{s^{\prime }}^{(k-1)}}}\Delta
_{_{I_{s^{\prime }}^{(k-1)}}}\prod_{s\in {\widetilde{D}}_{r,1}^{(k)}(J)}%
\kappa ^{\mathcal{C}_{I_{s}^{(k-1)}}}\gamma _{_{I_{s}^{(k-1)}}} \leq 1.
\label{dsr2}
\end{eqnarray}%
From (\ref{prdsr2}) and (\ref{dsr2}) we get (\ref{estm123k}). To conclude
the proof when $k=1$, we recall the modification of (\ref{e44}) in this
case, which, together with (\ref{e45}) for $k=1$, and the estimate (\ref%
{step0}), gives
\begin{equation*}
\kappa ^{\mathcal{C}_{I_{r}^{(1)}}}\Delta _{_{I_{r}^{(1)}}}(\omega )\leq
\frac{16}{(\ln 2)^{4}}\left( \frac{8(\ln 2)^{4}l_{1}^{-\delta }}{128}\right)
^{|D_{r}^{(1)}|}\leq l_{1}^{-\delta }
\end{equation*}%
for every $|D_{r}^{(1)}|\geq 1$. The other estimates (\ref{prdsr2}) and (\ref%
{dsr2}) follow exactly in the same way. This concludes the proof of
Proposition \ref{decomposition}.$\Box $

\medskip

\noindent Since $L=l_{M}$, at the $M$--th step of any itinerary, there will
be only two intervals $I_{1}^{(M)},I_{2}^{(M)}$. From (\ref{estm11k}) with $%
k=M$ , we easily see that
\begin{equation}
\mu _{\kappa ,L}^{f}\Big(x\rightsquigarrow y,\;\text{for some}\,x\in \big[%
-L,-L+\lfloor L^{\alpha ^{\prime }/\alpha }\rfloor \big]\text{ and some}%
\,y\in \big[L-\lfloor L^{\alpha ^{\prime }/\alpha }\rfloor ,L\big]\Big)\geq
1-2L^{-\delta }.  \label{terminamos!}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Cluster of the origin \label{III}}

\setcounter{equation}{0} \setcounter{theorem}{0}

The occurrence of the event in the l.h.s. of (\ref{terminamos!}) does not
imply that the event
\begin{equation*}
0 \rightsquigarrow y, \; {\text{for some}} \; y \in \big[L - \lfloor l_{{M}%
}^{\alpha ^{\prime }/\alpha }\rfloor , L\big]
\end{equation*}
also occurs. Next we give the uniform lower bound for the measure of this
last event.

\bigskip

\noindent Recalling that $j_{z}^{k}=\min \{j:z\in I_{j}^{(k-1)}\}$, for any $%
1\leq k\leq {M}$ we define the following events:

\begin{equation}
\psi ^{(k)}=\bigcap_{i=j_{0}^{k}-\lfloor \lfloor l_{k}^{\alpha ^{\prime
}/\alpha }\rfloor /l_{k-1}\rfloor }^{j_{0}^{k}+\lfloor \lfloor l_{k}^{\alpha
^{\prime }/\alpha }\rfloor /l_{k-1}\rfloor }G(I_{i}^{(k-1)})
\label{eventos1}
\end{equation}
and consider
\begin{equation}
\Psi _{L}=\bigcap_{j=1}^{{M}}\psi ^{(j)}.  \label{eventos2}
\end{equation}
Notice that we have at most two $M$--blocks, denoted by $I^{(M)}_i$, where $%
1\le i \le s$ and $s(\omega)\in\{1,2\}$.

The occurrence of $\bigcap_{k=1}^{n}\psi ^{(k)}$, $1\leq n\leq M$ implies
that the origin $0$ is the right (resp. left) end--vertex of a $(n-1)$%
--block $I_{j_{0}^{n}}^{(n-1)}$ (resp. $I_{j_{0}^{n}+1}^{(n-1)}$) for each $n
$, since no adjustments are performed in this case, and necessarily it
belongs to the pedestals $\Upsilon (I_{j_{0}^{n}}^{(n-1)})$ and $\Upsilon
(I_{j_{0}^{n}+1}^{(n-1)})$ for any $\;1\leq n\leq {M}$. In particular, for $%
\omega \in \Psi _{L}$ we have $s(\omega )=2$. Moreover, in the event $\Psi
_{L}\cap G(I_{1}^{(M)})\cap G(I_{2}^{(M)})$, the origin 0 belongs to the
open cluster connecting vertices $x$ and $y$ as described above.

\noindent Finally, using (\ref{estm11k}) and FKG, and under our choice of
parameters $1<\alpha ^{\prime }<\alpha <2$, $p,\,l_{1}$ we get
\begin{equation*}
\mu _{\kappa ,L}^{w}(\Psi _{L}\cap G(I_{1}^{(M)})\cap G(I_{2}^{(M)}))\geq
p^{\lfloor l_{1}^{\alpha ^{\prime }/\alpha }\rfloor }\prod_{k=2}^{{M}%
}(1-l_{k-1}^{-\delta })^{\lfloor \lfloor l_{k}^{\alpha ^{\prime }/\alpha
}\rfloor /l_{k-1}\rfloor }>c(\alpha ,\alpha ^{\prime },p,l_{1})>0
\end{equation*}%
since, by (\ref{del}) we have
\begin{equation*}
\sum_{k=2}^{\infty }l_{k-1}^{-\delta }\lfloor \lfloor l_{k}^{\alpha ^{\prime
}/\alpha }\rfloor /l_{k-1}\rfloor \leq \sum_{k=2}^{\infty }l_{k-1}^{-\delta
+\alpha ^{\prime }-1}<\infty ~.
\end{equation*}%
\noindent This completes the proof of the Theorem \ref{percolation}. $\Box $

\medskip

%The proof of Theorem \ref{conn} goes through the same lines, and uses
%arguments very similar to the arguments of the present Section, and
%therefore we omit it. EXPLICAR?? NAO MUITO CLARO PARA MIM...

\begin{thebibliography}{ACCN}
\bibitem[AN]{AN} M. Aizenman and C. M. Newman. Discontinuity of the
percolation density in one-dimensional $1/|x-y|^{2}$ models. Commun. Math.
Phys. \textbf{107}, 611 (1986).

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

\bibitem[BHS]{BHS} N. Berger, C. Hoffman, V. Sidoravicius, Non-uniqueness
for specifications in $\ell^{2+\epsilon}$. Preprint PR/0312344 v4 (2005)

\bibitem[CMR]{CMR} M. Cassandro, I. Merola, U. Rozikov. Phase cohexistence
in one dimensional Ising models with long range interactions. (In
preparation)

\bibitem[F]{F} C. M. Fortuin. On the random-cluster model: II. The
percolation model. Physica \textbf{58}, 393 (1972).

\bibitem[FK]{FK} C. M. Fortuin and P. W. Kasteleyn. On the random-cluster
model: I. Introduction and relation to other models. Physica \textbf{57},
536 (1972).

\bibitem[FS]{FS} J. Fr\"{o}hlich and T. Spencer. The Kosterlitz - Thouless
transition in two-dimensional Abelian spin system and the Coulomb gas.
Commun. Math. Phys. \textbf{81}, 527 (1981).

\bibitem[FS1]{FS1} J. Fr\"{o}hlich and T. Spencer. The phase transition $%
1/r^{2}$. Commun. Math. Phys. \textbf{84}, 87 (1982).

\bibitem[G1]{G1} G. Grimmett. The stochastic random-cluster process and the
uniqueness of random-cluster measures. Ann. Prob. \textbf{23}, 1461 (1995).

\bibitem[G2]{G2} G. Grimmett. {\textit{The Random-Cluster Model.}} Springer.
(2006)

\bibitem[IN]{IN} J. Z. Imbrie and C. M. Newman. An Intermediate Phase with
Slow Decay of Correlations in One Dimension $1/\left\vert x-y\right\vert
^{2} $ Percolation, Ising and Potts Models. Commum. Math. Phys. \textbf{118}%
, 303 (1988).

\bibitem[JO]{JO} A. Johansson and A. {\"{O}}berg. Square summability of
variations of $g$--functions and uniqueness in $g$--measures. Math. Res.
Lett. \textbf{10}, No 5-6, 587-601 (2003).

\bibitem[M]{M} Domingos H. U. Marchetti. Upper bound on the truncated
connectivity in one-dimensional $\beta /\left| x-y\right| ^{2}$ percolation
models at $\beta >1$. Rev. Math. Phys. \textbf{7}, 723-742 (1995).

\bibitem[KMP]{KMP} A. Klein, D. H. U. Marchetti and J. F. Perez. Power-law
fall off in the Kosterlitz--Thouless phase of a two-dimensional lattice
Coulomb gas. Journ. Stat. Phys. \textbf{60}, 137-166 (1990).

\bibitem[NS]{NS} C. M. Newman and L. S. Schulman. One dimensional $%
1/|x-y|^{s}$ percolation models: the existence of a transition for $s\leq 2$%
, Commun. Math. Phys. \textbf{104}, 547 (1986).

%\bibitem[NSp]{NSp} C. M. Newman and H. Spohn. Private communication. (1998).
\end{thebibliography}

\end{document}

math0506404.extracted_bib
\begin{thebibliography}{ACCN}
\bibitem[AN]{AN} M. Aizenman and C. M. Newman. Discontinuity of the
percolation density in one-dimensional $1/|x-y|^{2}$ models. Commun. Math.
Phys. \textbf{107}, 611 (1986).

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

\bibitem[BHS]{BHS} N. Berger, C. Hoffman, V. Sidoravicius, Non-uniqueness
for specifications in $\ell^{2+\epsilon}$. Preprint PR/0312344 v4 (2005)

\bibitem[CMR]{CMR} M. Cassandro, I. Merola, U. Rozikov. Phase cohexistence
in one dimensional Ising models with long range interactions. (In
preparation)

\bibitem[F]{F} C. M. Fortuin. On the random-cluster model: II. The
percolation model. Physica \textbf{58}, 393 (1972).

\bibitem[FK]{FK} C. M. Fortuin and P. W. Kasteleyn. On the random-cluster
model: I. Introduction and relation to other models. Physica \textbf{57},
536 (1972).

\bibitem[FS]{FS} J. Fr\"{o}hlich and T. Spencer. The Kosterlitz - Thouless
transition in two-dimensional Abelian spin system and the Coulomb gas.
Commun. Math. Phys. \textbf{81}, 527 (1981).

\bibitem[FS1]{FS1} J. Fr\"{o}hlich and T. Spencer. The phase transition $%
1/r^{2}$. Commun. Math. Phys. \textbf{84}, 87 (1982).

\bibitem[G1]{G1} G. Grimmett. The stochastic random-cluster process and the
uniqueness of random-cluster measures. Ann. Prob. \textbf{23}, 1461 (1995).

\bibitem[G2]{G2} G. Grimmett. {\textit{The Random-Cluster Model.}} Springer.
(2006)

\bibitem[IN]{IN} J. Z. Imbrie and C. M. Newman. An Intermediate Phase with
Slow Decay of Correlations in One Dimension $1/\left\vert x-y\right\vert
^{2} $ Percolation, Ising and Potts Models. Commum. Math. Phys. \textbf{118}%
, 303 (1988).

\bibitem[JO]{JO} A. Johansson and A. {\"{O}}berg. Square summability of
variations of $g$--functions and uniqueness in $g$--measures. Math. Res.
Lett. \textbf{10}, No 5-6, 587-601 (2003).

\bibitem[M]{M} Domingos H. U. Marchetti. Upper bound on the truncated
connectivity in one-dimensional $\beta /\left| x-y\right| ^{2}$ percolation
models at $\beta >1$. Rev. Math. Phys. \textbf{7}, 723-742 (1995).

\bibitem[KMP]{KMP} A. Klein, D. H. U. Marchetti and J. F. Perez. Power-law
fall off in the Kosterlitz--Thouless phase of a two-dimensional lattice
Coulomb gas. Journ. Stat. Phys. \textbf{60}, 137-166 (1990).

\bibitem[NS]{NS} C. M. Newman and L. S. Schulman. One dimensional $%
1/|x-y|^{s}$ percolation models: the existence of a transition for $s\leq 2$%
, Commun. Math. Phys. \textbf{104}, 547 (1986).

%\bibitem[NSp]{NSp} C. M. Newman and H. Spohn. Private communication. (1998).
\end{thebibliography}