\documentstyle{article}
\begin{document}
\title{A NOTE ABOUT A BURTON KEANE'S THEOREM}
\author{BY\\
H. Guiol\thanks{Partially supported by FAPESP grant 96/04859-9}\\
{\sl Instituto de Matem\'atica e Estat\'\i stica}\\{\sl Universidade de
S\~ao
Paulo}\\
{\sl BP 66281, 05389-970 S\~ao Paulo SP, Brasil}\\
{\sl e-mail: herve@ime.usp.br}}
\date{march 1997}
\maketitle
\newcommand{\carn}{\hfill\rule{0.25cm}{0.25cm}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\begin{abstract}
In BURTON-KEANE'S
structural paper \cite{bk1} a situation has not been lightened,
such that the proof of their result about the topological structure of
ribbons in the two-dimensional site percolation has a gap.
We introduce the confluent zone notion and prove,
using Burton-Keane's arguments, the almost surely
absence of these zones (This completes the proof of
the topological structure of ribbons announced by
the authors). In the end we give a alternative proof of the
corollary of Theorem 2 in \cite{bk1}.
\end{abstract}
\section {Definitions and recalls :}
We begin by following line by line what BURTON and KEANE have stated
in \cite{bk1}.
\subsection{Basic Clustering}
{\renewcommand{\baselinestretch}{3}
We will use the classic Euclidean distance $d$ between
two points $x$ and $y$ of ${\bf Z}^2$ given by the quadratic
norm $\Vert .\Vert_2$.
}
We say that two points of ${\bf Z}^2$ are {\bf neighbors} if
they are at distance 1 from each other.
We call {\bf path} a sequence (finite or not) $(x_1,...,x_n)$
such that $d(x_i,x_{i+1})=1$ for all $i=1,...,n-1$ and
$x_i\not=x_j$ if $i\not=j$ {\it i.e.} to us a path is a
chain of neighboring points with no loops.
\begin{definition}
Let $S$ be a subset of ${\bf Z}^2$, and define the relation
${\cal R}$ by
\[
x\,{\cal R}\,y\quad\Leftrightarrow
\mbox{ There exists a path from $x$ to $y$ in $S$.}
\]
${\cal R}$ is of course an equivalent relation over $S$.
We call {\bf components} of $S$ the equivalent
classes of ${\cal R}$.
\end{definition}
We are now going to distinguish finite and infinite
components.
\begin{definition}
We set
$int(S):=\bigcup\{\,K\,:\,K\mbox{ finite component
of }{\bf Z}^2\setminus S\,\}$ the {\bf interior} of $S$;
$ext(S):=\bigcup\{\,K\,:\,K\hbox{ infinite
component of }{\bf Z}^2\setminus S\}$ the {\bf exterior} of $S$;
$cl(S):=\,S\,\cup\,int(S)$ the {\bf closure} of $S$.
\end{definition}
Now for a given realization of site percolation in ${\bf Z}^2$
{\it i.e.} for any $x$ in $\{0,1\}^{{\bf Z}^2}$ and
for any $i$ in $\{0,1\}$, we set
\[
G_i:=\{z\in{\bf Z}^2\,:\,x(z)=i\}.
\]
\begin{definition}
The components of $G_i$ are called $i${\bf -clusters}.
\end{definition}
\begin{remark}
-A cluster in general will be either a 0-cluster or
an 1-cluster.
-The collection of clusters for a given
$x\in \{0,1\}^{{\bf Z}^2}$ partitions ${\bf Z}^2$.
\end{remark}
\begin{definition}
Let $C$ and $C'$ be two clusters, and suppose
$C\subseteq cl(C')$. Then, we write $C\prec C'$ and
\item{}say that $C'$ {\bf encloses} $C$, or $C$ is enclosed
by $C'$.
\end{definition}
Notice that the enclosure relation defines a partial
ordering on the set of clusters.
\begin{definition}
Maximal elements of this ordering are called {\bf essentials clusters},
and the others are called inessentials clusters.
\end{definition}
\begin{remark}
Any inessential cluster is finite.
\end{remark}
We can now distinguish two sorts of realizations :
\begin{definition}
A realization $x\in\{0,1\}^{{\bf Z}^2}$ is called
$(i)$ {\bf essential} if each cluster is enclosed by an
(unique) essential cluster.
$(ii)${\bf infinite cascade} if each cluster is
inessential (and finite).
\end{definition}
The following proposition, proved in \cite{bk1}, shows that there
exists only two possibilities :
\begin{proposition}
A realization is either an essential or an infinite cascade.
\end{proposition}
\begin{remark}
-In \cite{bk1} it is proved, with an ergodic assumption,
that these events have probability zero or one.
-We will focus on essentials realizations which is
in general the most interesting and the most frequent case.
\end{remark}
When a cluster is essential it can be finite or not.
Therefore, if we are interested in finite essential
clusters, it is natural to characterize their positions from
each other. In order to do this, R. Burton and M. Keane changed
the previous definition of neighborhood and introduced
the following $*$-neighborhood notion.
Two points in ${\bf Z}^2$ are said to be
$*${\bf -neighbors} if their Euclidean distance is either 1
or $\sqrt 2$.
As before a finite or infinite sequence
$(x_1,...,x_n,...)$ is said to be a $*${\bf -path} if
$d(x_i,x_{i+1})\leq\sqrt 2$ and $x_i\not=x_j$ for all indices
$i,j$, $i\not=j$.
For a given (essential) realization $x$ we set
\[
H:=\bigcup\{cl(C)\,:\,C\mbox{ is an essential finite
cluster (of }x)\}.
\]
We also introduce the equivalence relation $\cal R^*$ on
$H$ by
\[
x\,{\cal R}^*\,y\Leftrightarrow\mbox{ there exists a $*$-path
in $H$ from $x$ to }y.
\]
\begin{definition}
We call
{\bf rocks} equivalence classes of
$\cal R^*$ which are of finite cardinality;
{\bf $*$-ribbons} the infinite ones;
On $H^c$ we define the
{\bf 0-ribbon} : is the union of all clusters
enclosed in a fix infinite (essential) 0-cluster.
{\bf 1-ribbon} : is the union of all clusters
enclosed in a fix infinite (essential) 1-cluster.
\end{definition}
We have now the following proposition (\cite{bk1}) :
\begin{proposition}
If $x$ is an essential realization, then ${\bf Z}^2$ is partitioned
into 0-ribbons, 1-ribbons, $*$-ribbons and rocks such that :
(i) Ribbons of the same type are never neighbors;
(ii) $*$-ribbons and rocks are never $*$-neighbors;
(iii) Ribbons are infinite, and rocks are finite;
(iv) The closure of any cluster is contained in a
ribbon or a rock.
\end{proposition}
\begin{remark}
R. Burton and M. Keane noted that a similar procedure to
the above one should be possible for realizations of site
percolation in higher dimensions in ${\bf Z}^d,\,d\geq 3$.
\end{remark}
\subsection{Stationary Clustering}
We are going to introduce a dynamic on $\{0,1\}^{{\bf Z}^2}$
and this will provide us more things about the "topological
structure" of rocks and ribbons.
We denote $X:=\{0,1\}^{{\bf Z}^2}$ and we consider the probability
space $(X,{\cal B}_X,\mu)$.
We suppose that the measure $\mu$ is stationary and
ergodic.
\begin{remark}
>From now on it is clear that the event $\{x\in X$ ; $x$ is
infinite cascade $\}$ is ${\bf Z}^2$-invariant and so by
ergodicity have probability 0 or 1. Of course, it is the
same for its complementary event $\{x\in X$ ; $x$ is
essential $\}$.
\end{remark}
\begin{definition}
We say that $\mu$ is {\bf essential} if and only if
$\mu$-almost all realization is essential. Otherwise
$\mu$ is said to be an infinite cascade.
\end{definition}
\section{Topological structure of ribbons.}
Up to now we assume that $\mu$ is essential.
\subsection{Clustering in an Essential Process}
R. Burton and M. Keane have given two theorems. In the first one, they
show that rocks could be eliminated (cf page 6-7 of \cite{bk1}).
The second one says
\begin{theorem}\label{vois}
Let $Q$ be the event "$x$ contains a ribbon
whose complement in ${\bf Z}^2$ has at least 3 components".
Then $\mu(Q)=0$.
\end{theorem}
But this theorem is not sufficient to conclude that ribbons
are topologically strips as is suggested by the following
counter example :
\begin{definition}
Let $x\in \{0,1\}^{{\bf Z}^2}$. We call
{\bf confluent zone} for the realization $x$
a square of four points $z_1=(u,v), z_2=(u+1,v), z_3=(u,v+1),
z_4=(u+1,v+1)$
in ${\bf Z}^2$ such that $x(z_1)=x(z_4)\not=x(z_2)=x(z_3)$ and
such that this zone has at least three (distinct)
ribbons as neighbors.
\end{definition}
If such a zone exists then it is very easy to see that some ribbon may
have
two different neighbors on the same side. So that the main result of
\cite{bk1}
should fail.
Fortunately for a stationary ergodic measure this
counter-example has measure zero.
\begin{theorem}\label{zone}
Let $T:=\{x\in \{0,1\}^{{\bf Z}^2}\,:\,x$ has a confluent
zone $\}$. Then $\mu(T)=0$.
\end{theorem}
{\bf Proof :}
Clearly, $T$ is ${\bf Z}^2$-invariant. Recall that $\mu$
is ergodic, and assume that $\mu(T)=1$.
So there exists an
$\alpha>0$ such that $\mu\{\,x\in T\,,\,0$ {\sl is the (lexicographic)
smallest
point in a confluent
zone of} $x\}=\alpha$.
Scheme of the proof is similar to the one given in \cite{bk1}
for theorem 1.
(Let $A_z:=\{x\in T\,,\,z\mbox{ \sl is the smallest point in a confluent
zone
of }x\}$, we have $T=\cup_{z\in{\bf Z}^2}A_z$. Therefore there
exists
$z_0$ such that $\mu(A_{z_0})>0$ and by stationarity
$\mu(A_0)>0)$.
Now, apply the Tempel'man's ergodic theorem to the event
\[
A:=\{\,0\hbox{ \sl is the smallest point in a confluent zone }\}
\]
and deduce that for some $n$ (large enough), one can find
$\varepsilon n^2$ confluent zones in the box
$K_n=[0,n]\times [0,n]$.
>From three
neighbors of each of these zones we construct
three (infinite) distinct paths in three (distinct) ribbons.
For each confluent zones their three paths intersect
with the boundary of the box $K_n$.
If we now start from $(0,0)$ on the boundary of $K_n$ and
go throw the boundary in a clockwise way, we call
the second of these three points we meet the {\bf central}
point. An elementary application of the Jordan's curve
theorem shows that the central points of different
confluent zones must be distinct. Hence, the boundary of
$K_n$ contains at least $\varepsilon n^2$ points which is absurd
for large $n$. $\carn$
\begin{remark}
$\mu(Q)=0$ and $\mu(T)=0$ mean that for
$\mu$-almost every realization $x\in \{0,1\}^{{\bf Z}^2}$,
a ribbon of $x$ is topologically a "strip",
{\it i.e.} it has either
-one neighbor on each "side" (a real strip);
-only one neighbor (a half plane);
-no neighbor (the whole plane).
\end{remark}
\section{Finite energy measure}
\begin{definition}
For $z\in{\bf Z}^2$ and $i\in\{0,1\}$ define
$\tau\,:\,X\rightarrow X$ by
\[
\tau(x)(z'):=\left\{
\begin{array}{l}
x(z') \mbox{ if } z'\not=z\\
i \mbox{ if } z'=z.\\
\end{array}
\right.
\]
The measure $\mu$ is said to be of {\bf finite energy}
if for any event $A$ with $\mu(A)>0$, any $z\in{\bf Z}^2$ and
any $i\in\{0,1\}$ one has
\[
\mu(\tau(A))>0.
\]
\end{definition}
\begin{corollary}\label{ef} (Burton-Keane \cite{bk1})
If $\mu$ has finite energy, then almost every
$x\in X=\{0,1\}^{{\bf Z}^2}$ contains at most two ribbons.
\end{corollary}
The proof in \cite{bk1} does not seem to be clear. Here we
give another proof using a result in \cite{bk2} :
\begin{theorem}\label{unic} (Burton-Keane \cite{bk2})
Let $\mu$ be of finite energy and choose
$i\in\{0,1\}$
then $\mu$-almost every $x\in X$ has at most
one infinite $i$-cluster.
\end{theorem}
{\bf Proof of corollary \ref{ef}}
If it is not $\mu\{\,x$ {\sl has a ribbon with two
neighbors} $\}>0$, and thus we have only the three
following possibilities :
(Recall that, from theorem \ref{vois} and
theorem \ref{zone}, ribbons are topologically strips.)
First case :
We have three different type of ribbons, then changing
the realization on large enough finite box we obtain a ribbon
whose complement in ${\bf Z}^2$ has 3 components. So, by theorem
\ref{vois}, this event has measure zero.
Second case :
Two ribbons are of the same type but are not $*$-ribbons,
then this event has measure zero, by theorem \ref{unic} on the
uniqueness of $i$-clusters.
Third case :
We have two $*$-ribbons, then changing the realization
on a large enough finite box we go back to the preceding
case. $\carn$
\appendix{\bf Acknowledgments}
The author wants to thank M. Decking, P. Liardet and E. Andjel to
introduce
him in this percolation problematic. Special thanks also to R. Burton,
E. Saada,
C. Bezuidenhout, G. Grimmett, V. Belitsky, P. Ferrari, L.R. Fontes, J.
Neves,
S. Martinez and A. Nogueira for interesting discussions.
\begin{thebibliography}{9}
\bibitem{bk1} Burton, R.M., Keane, M., Topological and metric properties
of infinite clustering in stationary two-dimensional site
percolation. {\sl Isr. J. Math.} {\bf 76}, No 3, 299-316 (1991) .
\bibitem{bk2} Burton, R.M., Keane, M., Density and uniqueness in
percolation. {\sl Commun. Math. Phys.} {\bf 121}, 501-505 (1989).
\end{thebibliography}
\end{document}