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Ising model, percolation, phase transition, random cluster method
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\begin{document}

\title{{\bf Percolation and number of phases\\ in the 2D Ising 
model}}
\author{ 
Hans-Otto Georgii\\ {\small\sl Mathematisches Institut der 
Universit\"at M\"unchen}\\ {\small\sl Theresienstr.\ 39, D-80333 
M\"unchen, Germany.} 
\and Yasunari Higuchi\\ {\small\sl Department of Mathematics, Faculty of 
Science}\\ {\small\sl Kobe University, Rokko, Kobe 657-8501, Japan.}
}

\date{}
\maketitle

\begin{quote}
We reconsider the percolation approach of Russo, Aizenman and Higuchi 
for showing that there exist only two phases in the Ising model on the 
square lattice. 
We give a fairly short alternative proof which is only based on FKG 
monotonicity and avoids the use of GKS-type inequalities 
originally needed for some background results. Our proof extends to 
the Ising model on other planar lattices such as the triangular 
and honeycomb lattice.
We can also treat the Ising antiferromagnet in an external field and 
the hard-core lattice gas model on $\Z^2$.
\end{quote}

\section{Introduction}

\medskip\noindent
One of the fundamental results on the two-dimensional ferromagnetic 
Ising model is the following theorem obtained independently 
in the late 1970s by Aizenman \cite{Aiz} and Higuchi \cite{Hig}
on the basis of the seminal work of Russo \cite{Rus}.

\medskip\noindent
{\bf Theorem}.
{\em For the ferromagnetic Ising model on $\Z^2$ with no external 
field and inverse temperature $\b>\b_c$, 
there exist precisely two 
extremal Gibbs measures $\mu^+$ and $\mu^-$.}

\medskip\noindent
The basic technique initiated by Russo  
consists of an interplay of three features of the Ising model: 
\bit
\item[--] the strong Markov property for random sets defined by 
geometric conditions involving clusters of constant spin, 

\item[--] the symmetry of the interaction under spin-flip and lattice 
automorphisms, and

\item[--] the ferromagnetic character of the interaction which 
manifests itself in FKG order and positive correlations. 
\eit
These ingredients led to a detailed understanding of the geometric 
features of typical configurations as described by the concepts of 
percolation theory.
In addition to these tools, the authors of \cite{Aiz,Hig,Rus} also 
needed the result that the
limiting Gibbs measure with $\pm$ boundary condition is a mixture of 
the two
pure phases. This result was obtained by quite different means, 
namely some symmetry
correlation inequalities of GKS and Lebowitz type \cite{MM}. While 
these symmetry inequalities are a beautiful and powerful tool, they 
are quite different in character from the FKG inequality and have 
their own restrictions. It is therefore natural to ask whether 
Russo's random cluster method is flexible enough to prove the theorem 
without recourse to symmetry inequalities. On the one hand, this 
would allow to extend the theorem to models with less symmetries, 
while on the other hand one might gain a deeper understanding of 
possible geometric features of typical configurations.

In this paper we propose such a purely geometric reasoning which is 
only based on the three features above and avoids the use of the 
symmetry inequalities of Messager and Miracle-Sole \cite{MM}.
Despite this reduction of tools we could simplify the proof by 
an efficient combination of known geometric arguments. These include 
\bit 
\item [--] the Burton-Keane uniqueness theorem for infinite clusters 
\cite{BK},
\item [--] a version of Zhang's argument for the impossibility of 
simultaneous plus- and minus-percolation in $\Z^2$ (cf.\ Theorem 5.18 of 
\cite{GHM}),
\item [--] Russo's symmetry trick for simultaneous flipping of spins and 
reflection of the lattice \cite{Rus}, and 
\item [--] Aizenman's idea of looking at contour intersections in a 
duplicated system \cite{Aiz}.
\eit
We have tried to keep the paper reasonably 
self-contained, so that the reader will find a complete proof of the 
theorem.
As a payoff of the method we also obtain 
some generalizations. On the one hand, the arguments 
carry over to the Ising model on other planar lattices such as the 
triangular or the hexagonal lattice. On the other hand, in the case 
of the square lattice they cover 
also the antiferromagnetic Ising model in an external field as well 
as the hard-core lattice gas model.

\section{Set-up and basic facts}
\label{facts}

Although we assume that the reader is familiar with the 
definition of the Ising model, let us start recalling a number of 
fundamental facts and introducing some notations. We assume 
throughout that the inverse temperature $\b$ exceeds the Onsager 
threshold $\b_{c}$, and that there is no external field, $h=0$.
The main ingredients we need are the following:
\smallskip

$\bullet$ 
the {\em configuration space\/} $\O=\{-1,1\}^{\Z^2}$, which is 
equipped 
with the Borel $\s$-algebra $\F$ and the local $\s$-algebras $\F_{\L}$
of events depending only on the spins in $\L\subset\Z^2$.

$\bullet$ 
the {\em Gibbs distributions} $\mu_{\L}^\o$ 
in finite regions $\L\subset\Z^2$ with boundary condition $\o\in\O$;
these enjoy the {\em Markov property} which says that 
$\mu_{\L}^\o(A)$ for
$A\in\F_{\L}$ depends only on the restriction of $\o$ 
to the boundary $\partial\L=\{x\not\in\L:|x-y|=1 
\mbox{ for some }y\in\L\}$ of $\L$, and the {\em finite-energy 
property}, which states that $\mu_{\L}^\o(A)>0$.

$\bullet$ 
the {\em Gibbs measures} $\mu$ on $(\O,\F)$ which, by 
definition,
satisfy $\mu(\cdot\,|\F_{\L^c})(\o) =\mu_{\L}^\o$ for $\mu$-almost 
all $\o$ and any finite $\L$;  we write $\G$ for the set 
of all Gibbs measures and $\Gex$
for the set of all extremal Gibbs measures.

$\bullet$ 
the {\em strong Markov property} of Gibbs measures,
stating that 
$\mu(\cdot\,|\F_{\Gamma^c})(\o) =\mu_{\Gamma(\o)}^\o$ for 
$\mu$-almost 
all $\o$ whenever $\Gamma$ is
a {\em random\/} finite subset of $\Z^2$ satisfying
$\{\Gamma=\L\}\in\F_{\L^c}$ for all finite $\L$;
$\F_{\Gamma^c}$ is the set of all events $A$ satisfying 
$A\cap \{\Gamma=\L\}\in\F_{\L^c}$ for all finite $\L$.

$\bullet$ 
the {\em stochastic monotonicity} (or FKG order) of Gibbs 
distributions; writing $\mu\preceq\nu$ when 
$\mu(f)\leq\nu(f)$ for all increasing local (or, equivalently, 
all increasing bounded measurable) real functions $f$ on $\O$,
we have $\mu_{\L}^\o\preceq\mu_{\L}^{\o'}$ when $\o\leq\o'$, and  
$\mu_{\L}^\o\preceq\mu_{\D}^\o$ when $\D\subset\L$ and $\o\equiv 
+1$ on $\L\setminus\D$ (the opposite relation holds when $\o\equiv 
-1$ on $\L\setminus\D$).

$\bullet$ 
the {\em pure phases} 
$\mu^+,\,\mu^-\in\G$ obtained as limits for $\L\uparrow\Z^2$ of 
$\mu_{\L}^\o$ with $\o\equiv+1$ resp.\ $-1$, their invariance 
under all graph automorphisms of $\Z^2$,  
the sandwich relation 
$\mu^-\preceq\mu\preceq\mu^+$ for any other $\mu\in\G$, 
and the resulting extremality of $\mu^+$ and $\mu^-$.

$\bullet$ 
the characterization of extremal Gibbs measures by their 
{\em triviality on the tail $\s$-algebra} 
$\T=\bigcap\{\F_{\L^c}:\L\subset\Z^2\mbox{ finite}\}$; 
the fact that extremal Gibbs measures have {\em positive correlations}; 
and the {\em extremal 
decomposition} representing any Gibbs measure as the barycenter of a 
mass distribution on $\Gex$.

\smallskip\noindent
A general account of Gibbs measures can be found in \cite{Gii}, and 
\cite{GHM} contains an exposition of the 
Ising model and its properties related to stochastic monotonicity.

We will also use a class of transformations of $\O$ which preserve 
the Ising Hamiltonian, and thereby the class $\G$ of Gibbs measures. 
These transformations are
\smallskip

$\bullet$
the {\em spin-flip transformation} 
$T:\o=(\o(x))_{x\in\Z^2}\to (-\o(x))_{x\in\Z^2}\,;$

$\bullet$ the {\em translations} $\th_{x}$, $x\in\Z^2$, which are 
defined by $\th_{x}\o(y)=\o(y-x)$ for $y\in \Z^2$, and in particular 
the horizontal and vertical shifts 
$\th_{\h}=\th_{(1,0)}$ resp.\  $\th_{\v}=\th_{(0,1)}$; and

$\bullet$ the {\em reflections\/} in lines $\ell$ through lattice 
sites: for any $k\in\Z$ we write
\[
R_{k,\h}: \Z^2\ni x=(x_1,x_2) \to (x_1,2k-x_2)\in\Z^2
\]
for the reflection in the horizontal line $\{x_2=k\}$,
and similarly $R_{k,\v}$ for the reflection in the vertical line
$\{x_1=k\}$. For $k=0$ we simply write $R_\h=R_{0,\h}$ and 
$R_\v=R_{0,\v}$. All these reflections act canonically on $\O$.
\medskip

We will investigate the geometric behavior of typical configurations 
in {\em half-planes\/} of $\Z^2$. These are sets of the form 
\[
\pi=\{ x=(x_1, x_2) \in \Z^2 :\ x_i \ge k\}
\]
with $k\in\Z$, $i\in\{1,2\}$, or with `$\ge$' replaced by `$\leq$'.
The line $\ell =\{ x\in \Z^2: \ x_i= k \}$ is called the associated 
{\em boundary line}. In particular, we will consider 
\bit
\item the upper half-plane 
$ \pi_\up = \{ x=(x_1, x_2) \in \Z^2: \ x_2 \geq 0 \} $,
\item the downwards half-plane
$ \pi_\lo = \{ x = (x_1, x_2) \in \Z^2: \ x_2 \leq 0 \} $,
\eit
and the analogously defined right half-plane $ \pi_\ri$
and left half-plane $\pi_\le$. We will also work with
\bit
\item the left horizontal semiaxis
$\ell_\le = \{x=(x_1,x_2) \in \Z^2: \ x_1 \leq 0,\, x_2=0\}$, and
\item the right semiaxis
$ \ell_\ri = \{x = (x_1, x_2) \in \Z^2: \ x_1 \geq 0,\, x_2 = 0 \}$.
\eit
\medskip

In the rest of this section we state three fundamental results on 
percolation in the Ising model. By the symmetry between 
the spin values $+1$ and $-1$, 
these results also hold when `$-$' and `$+$' are 
interchanged. Similarly, all notations introduced with one 
sign will be used accordingly for the opposite sign. 

We assume that the reader is familiar with the basic 
concepts of percolation theory such as
paths and $*$paths, $+$paths and $+*$paths (consisting of plus 
spins), circuits and $*$circuits, semicircuits in half-planes,
clusters, $+$ and $+*$clusters, and so on. These can be found in the 
paper of Russo \cite{Rus}.

The starting point is the following result of \cite{CNPR,Rus}. 
Let $E^+$ denote the 
event that there exists an infinite $+$cluster in 
$\Z^2$, and define $E^-$, $E^{+*}$, $E^{-*}$ analogously.
(Throughout this paper we will use the letter $E$ to denote events 
concerning existence of infinite clusters.) 
%
\begin{lem}{\bf (Existence of infinite clusters)}\label{lem:CNPR}
If $\mu\in\G$ is different from $\mu^-$, there exists with 
positive probability an infinite $+$cluster.
That is, $\mu(E^+)>0$ when $\mu\ne\mu^-$.
\end{lem}
%
{\sl Sketch Proof}. 
Suppose that $\mu(E^+)=0$. Then any given square 
$\D$ is almost surely surrounded by a $-*$circuit, and with 
probability close to $1$ such a circuit can already be found within a 
square $\L\supset\D$ provided $\L$ is large enough. Define a 
random set $\Gamma$ as the largest subset of $\L$ which is the 
interior of such a $-*$circuit. (A largest such set exists because 
the union of such sets is again the interior of a $-*$circuit.) By 
maximality, $\Gamma$ satisfies the conditions of the strong Markov 
property. This together with the stochastic monotonicity 
$\mu^-_\Gamma\preceq \mu^-$ implies (in the limit $\L\uparrow\Z^2$) 
that $\mu\preceq\mu^-$ on $\F_\D$.
Since $\D$ was arbitrary and $\mu^-$ is minimal we find that 
$\mu=\mu^-$, and the lemma is proved. $\Box$

\medskip\noindent
The next lemma is a variant of another result of Russo \cite{Rus}. 
%
\begin{lem}{\bf (Flip-reflection domination)}\label{TR}
Let $\mu \in \G$ and $R$ any reflection, and suppose that for
$\mu$-almost all $\o$ each finite $\D\subset \Z^2$ is surrounded by 
an $R$-invariant 
$*$circuit $c$ such that $\o \ge R\circ T(\o)$ on $c$.
Then $\mu \succeq \mu \circ R \circ T$.
\end{lem}
%
{\sl Proof}. Another way of stating the assumption 
is that for
any finite $R$-invariant $\D$ and $\mu$-almost all $\o$ 
there exists a finite $R$-invariant random set $\Ga(\o)\supset\D$ 
such that $\o \ge R\circ T(\o) \mbox{ on }\partial\Ga(\o)$.
Given any $\e>0$, we can thus find an $R$-invariant $\L$ so large that
with probability at least $1-\e$ such 
an $R$-invariant $\Ga(\o)$ exists within $\L$.
Since the union of any two such $\Ga(\o)$'s enjoys the same 
properties,
we can assume that $\Ga(\o)$ is chosen maximal in $\L$; in the case 
when no 
such $\Ga(\o)$ exists we set $\Ga(\o)=\emptyset$. 
The maximality of $\Ga$ implies that the events $\{\Ga = G\}$ are 
measurable with respect to $\F_{\L\setminus G}$.
For any increasing ${\cal F}_{\D}$-measurable  function $f\geq 0$
we thus get from the strong Markov property
$$
\mu(f) \geq \mu\bigg( \mu_\Ga^\cdot( f)\;;\;\Ga \ne\emptyset\bigg)\;.
$$
However, if $\Ga(\o)\ne\emptyset$ then
$\o \ge R\circ T(\o) \mbox{ on }\partial \Ga(\o)$. By stochastic 
monotonicity, for such $\o$ we have 
$$
\mu_{\Ga(\o)}^\o( f )\ge 
\mu_{\Ga(\o)}^{R\circ T(\o)}( f )=\mu_{\Ga(\o)}^\o( f\circ R\circ T 
)\;,
$$
where the identity follows from the 
$R$-invariance of $\Ga$ and the $R\circ T$ -invariance of the 
interaction. Hence
$$
\mu(f)\geq \mu\bigg( f\circ R\circ T \,;\, \Ga \ne \emptyset\bigg)
\geq \mu ( f\circ R\circ T ) - \e\,\| f \|_\infty \; .
$$
The lemma thus follows by letting $ \e \to 0 $ and $\D\uparrow\Z^2$.
 $\Box$

\medskip\noindent
A third useful result of Russo \cite{Rus} is the following. To state 
it we need to introduce two notations. First, 
let 
\[
\theta=\mu^+(0 \in I^{+*})
\]
be the $\mu^+$-probability that the origin belongs to 
an infinite $+*$cluster. Lemma \ref{lem:CNPR} implies that $\theta>0$.
Secondly, for a half-plane $\pi$ with boundary line $\ell$ and a 
$*$semicircuit $\s$ in $\pi$ we write $\mbox{Int\,}\s$ for the unique 
subset of $\Z^2$ which is invariant under the reflection $R$ in 
$\ell$ 
and satisfies $\pi\cap\partial(\mbox{Int\,}\s)=\s$; we call 
$\mbox{Int\,}\s$ the interior of $\s$. 
%
\begin{lem} {\bf(Point-to-semicircuit lemma)}
\label{point-to-semicircuit}
Let $\pi$ be some half-plane with boundary line $\ell$, $x\in\ell$, 
and $\s$ a $*$semicircuit in $\pi$ with interior 
$\L=\mbox{\rm Int\,}\s\ni x$. Let $\o\in\O$ be such that
$\o\equiv +1$ on $\s$. Then 
\[
\mu_\L^\o\bigg( x \mbox{ \rm is in $\L$ $+*$connected to $\s$} \bigg) \geq 
\theta/2\;.
\]
\end{lem}
%
{\sl Proof}. By hypothesis we have $\o\geq R\circ T(\o)$ on $\partial\L$, 
and therefore
$\mu_\L^\o\succeq\mu_\L^\o\circ R\circ T$. To exploit this relation
we let $B_{x,\s}$ be the event that there exists a $+*$paths in $\L$ 
from $x$ to $\s$, $C_{x,\s}$ the event that $x$ is surrounded by a 
$+*$circuit in $\L$ which is $+*$connected to $\s$, and 
$D_{x,\s}=B_{x,\s}\cup C_{x,\s}$. 
Then $\mu_\L^\o(D_{x,\s}\cup R\circ T(D_{x,\s}))=1$, and 
therefore $\mu_\L^\o(D_{x,\s})\geq1/2$.
Hence 
$$\mu_\L^\o(B_{x,\s}) 
\ge\mu_\L^\o(B_{x,\s}|C_{x,\s})\,\mu(D_{x,\s})
\ge \mu_\L^\o(B_{x,\s}|C_{x,\s})/2\,.
$$
But if $C_{x,\s}$ occurs then there exists a largest random set 
$\Ga\subset\L$ containing $x$ such that $\partial\Ga$ forms a 
$+*$circuit and is $+*$connected to $\s$. Writing $B_{x,\partial\Ga}$ 
for the event that $x$ is $+*$connected to $\partial\Ga$ and
using the strong Markov property we thus find that
$$
\mu_\L^\o(B_{x,\s}|C_{x,\s})= 
\mu_\L^\o\,(\,\mu^+_{\Gamma}(B_{x,\partial\Ga})\,|\,C_{x,\s}\,)
\geq\theta
$$
because $\mu^+_{\Gamma}(B_{x,\partial\Ga})\geq\theta$ by 
stochastic monotonicity. Together with the previous inequality this 
gives the result. $\Box$ 


\section{Percolation in half-planes} 

In this section we will prove that there exist plenty of infinite 
clusters of constant spin in the half-planes of $\Z^2$. 
In particular, this 
will show that all translation invariant $\mu\in{\G}$ 
are mixtures of $\mu^+$ and $\mu^-$.
We will use two pearls of percolation theory, 
the Burton-Keane uniqueness theorem \cite{BK}, and Zhang's argument 
for 
the non-existence of two infinite clusters of opposite sign in $\Z^2$.

For a given half-plane $\pi$ we let $E^+_\pi$ denote the event that 
there exists an infinite $+$cluster in $\pi$. When this occurs, we 
will write $I^+_\pi$ for such an infinite $+$cluster in $\pi$. (As we 
will see, such clusters are unique, so that this notation does not 
lead into conflicts.) In case of the standard half-planes,
we will only keep the 
directional index and omit the $\pi$; for example, we write
$E^+_{\up}$ for $E^+_{\pi_{\mbox{\tiny up}}}$.
Similar notations will be used for $+*$clusters 
and for the sign $-$ instead of $+$.

Let us say that $(\pi,\,\pi')$ is a pair of {\em conjugate 
half-planes\/} if $\pi$ and $\pi'$ share only a common boundary line. 
An 
associated
pair $(I^+_{\pi},I^+_{\pi'})$ or $(I^-_{\pi},I^-_{\pi'})$ of infinite 
clusters of the same sign in $\pi$ and $\pi'$ will be called an
{\em infinite butterfly}. (This name alludes to the assumption that 
the two infinite `wings' have the same `color', but is not meant to 
suggest that they are symmetric and connected to each other, 
although the latter will turn out to be true.)
%
\begin{lem}{\bf(Butterfly lemma)}\label{butterfly}
$\G$-almost surely there exists at least one infinite butterfly.
\end{lem}
%
{\bf Proof: } Suppose the contrary. By the extremal decomposition 
theorem
and the fact that the existence of infinite butterflies is a tail 
measurable event, there is then some $\mu\in{\Gex}$ for which there
exists no infinite butterfly $\mu$-almost surely.
We will show that this is impossible.

{\em Step 1. }First we observe that $\mu$ is $R\circ T$-invariant 
for all reflections $R=R_{k,\h}$ or $R_{k,\v}$, 
and in particular is periodic under translations.
Indeed, let $(\pi,\,\pi')$ be conjugate half-planes with common 
boundary line $\ell$ and $R$ the 
reflection in $\ell$ mapping $\pi$ onto $\pi'$. By the absence of 
infinite butterflies, at least
one of the half-planes $\pi$ and $\pi'$ contains no infinite
$-$cluster, and this or the other half-plane contains no infinite
$+$cluster. In view of the tail triviality of $\mu$, we can assume 
that $\mu(E^-_{\pi})=0$. This means that for $\mu$-almost all $\o$ 
every finite $\D\subset\pi$ is surrounded 
by some $+*$semicircuit $\g$ in $\pi$. For such a $\g$, $c=\g\cup 
R(\g)$
is an $R$-invariant $*$circuit that surrounds $\D\cup R(\D)$
and satisfies $\o \ge R\circ T(\o)$ on $c$. By Lemma \ref{TR}, 
this gives the flip-reflection domination
$\mu \succeq \mu \circ R \circ T$. Since also $\mu(E^+_{\pi})=0$ or
$\mu(E^+_{\pi'})=0$, we conclude in the same way that
$\mu \preceq \mu \circ R \circ T$, so that $\mu=\mu\circ R\circ T$. 
Since both $\th_\h^2$ and $\th_\v^2$ are compositions of two 
reflections, the invariance under the translation group 
$(\th_x)_{x\in 2\,\Z^2}$ follows.

{\em Step 2. }We now take advantage of
the Burton--Keane uniqueness theorem \cite{BK}, stating that for 
every periodic $\mu$ with finite energy there exists at most
one infinite $+$ (resp.\ $-$) cluster, and Zhang's symmetry argument
(cf.\ \cite{GHM}, Theorem 5.18) 
deducing from this uniqueness the absence of simultaneous $+$ and 
$-$percolation.

We start noting that, by the flip-reflection symmetry of $\mu$, 
$\mu$ is different
from $\mu^+$ and $\mu^-$, so that by Lemma \ref{lem:CNPR} and
the Burton--Keane uniqueness theorem there exist both a unique 
infinite 
$+$cluster $I^+$ and a unique infinite $-$cluster $I^-$ 
in the whole plane $\Z^2$ $\mu$-almost surely. 
We now choose a square $\L=[-n,n]^2\cap\Z^2$ so large that
$\mu(\L\cap I^+\ne\emptyset)>1-2^{-12}$.
Let $\partial_k\L$ be the intersection of
$\partial\L$ with the $k$'th quadrant, and let $A^+_{k}$ 
be the increasing event that there exists an 
infinite $+$path in $\L^c$ starting from some site 
in $\partial_k\L$. Define $A^-_{k}$ analogously. Since 
\[
\{\L\cap I^+\ne\emptyset\}\subset \bigcup_{k=1}^4 A^+_{k}
\]
and $\mu$ (as an extremal Gibbs measure) has positive correlations,
it follows that
\[
\prod_{k=1}^4 \mu(\O\setminus A^+_{k})\leq \mu(\bigcap_{k=1}^4
\O\setminus{A^+_{k}})
\leq\mu(\L\cap I^+=\emptyset) < 2^{-12}\;,
\]
whence there exists some $k\in\{1,\ldots,4\}$ such that 
$\mu(\O\setminus{A^+_{k}})<2^{-3}$. For notational convenience 
we assume that $k=1$. 
By the flip-reflection symmetry shown above, we find that
\[
\mu(A^+_{1}\cap A^-_{2} \cap A^+_{3} \cap A^-_{4})>1-4\cdot 
2^{-3}=1/2\;,
\]
which is impossible because on this intersection the infinite clusters
$I^+$ and $I^-$ cannot be both unique. This contradiction concludes 
the proof of the lemma. $\Box$

\medskip\noindent
The butterfly lemma leads immediately to the following result first 
obtained by Messager and Miracle-Sole \cite{MM} by means of 
correlation inequalities of symmetry type; the following proof 
appeared first in \cite{GHM}.
%
\begin{cor}\label{cor:MM}
{\bf (Periodic Gibbs measures) }
Any periodic $\mu\in\G$ is a mixture of $\mu^+$ and $\mu^-$.
\end{cor}
%
{\sl Proof}. Suppose $\mu\in\G$ is invariant under 
$(\th_x)_{x\in p\Z^2}$ for some period $p\geq 1$. Conditioning $\mu$
on any periodic tail event $E$ we obtain again a periodic Gibbs measure.
It is therefore sufficient to show that $\mu(E^+\cap E^-)=0$. Indeed,
the butterfly lemma then shows that $\mu(E^+) +\mu(E^-)=1$, and Lemma 
\ref{lem:CNPR} implies that $\mu(\,\cdot\,|E^+)=\mu^+$ and 
$\mu(\,\cdot\,|E^-)=\mu^-$ whenever these conditional probabilities 
are defined. Hence $\mu=\mu(E^+)\,\mu^+ +\mu(E^-)\,\mu^-$.

Suppose by contraposition that $\mu(E^+\cap E^-)>0$. Since 
$E^+\cap E^-$ is invariant and tail measurable, we can in fact 
assume that $\mu(E^+\cap E^-)=1$; otherwise we replace $\mu$ by 
$\mu(\,\cdot\,|E^+\cap E^-)$. By the butterfly lemma,
there exists a pair $(\pi,\pi')$ of conjugate halfplanes, say 
$\pi_\up$ and $\pi_\lo$, and a sign, say $+$, such that both 
half-planes contain an infinite clusters of this sign with 
positive probability. Since $\mu(E^-)=1$ by assumption, 
we can find a large square $\D$ such that with 
positive probability $\D$ meets infinite $+$clusters in $\pi_\up$ and 
 $\pi_\lo$ and also an infinite $-$cluster. This $-$cluster leaves 
$\D$ either on the left or on the right between the two infinite 
$+$clusters. We can assume that the latter occurs with positive 
probability. By the
finite energy property, it then follows 
that also $\mu(A_0)>0$, where for $k\in p\Z$ we write
$A_k$ for the event that the point $(k,0)$ belongs to a two-sided 
infinite $+$path with its two halves staying in $\pi_\up$ resp.\ 
$\pi_\lo$, and $(k+1,0)$ belongs to an infinite $-$cluster.

Let $A$ be the event that $A_k$ occurs for infinitely many $k<0$ and
infinitely many $k>0$. The horizontal periodicity and Poincar\'e's
recurrence theorem (cf.\ Lemma (18.15) of \cite{Gii}) then show that
$\mu(A_0\setminus A)=0$, and therefore $\mu(A)>0$. But on $A$ there 
exist infinitely many $-$clusters which are separated from each other 
by the infinitely many `vertical' $+$paths. This contradicts the 
Burton--Keane theorem.
$\Box$

\medskip\noindent
The preceding argument actually shows that $\mu(E^{-*}\cap 
E^{+*})=0$ whenever $\mu\in\G$ is periodic. Since $\mu^+(E^+)=1$ by 
Lemma \ref{lem:CNPR} and tail triviality, this shows that in the 
$+$phase the $+$spins 
form an infinite sea with only finite islands.
%
\begin{cor}\label{ocean}
{\bf (Plus-sea in the plus-phase)}
$\mu^+(E^{-*})=0$. Hence, 
 $\mu^+$-almost surely there exists a unique infinite 
$+$cluster $I^+$ in $\Z^2$ which surrounds each finite set. 
\end{cor}
%
We note that in contrast to Zhang's argument 
(cf.\ Theorem 5.18 of \cite{GHM})
our proof of the preceding corollary does not rely on 
the reflection invariance of $\mu^+$ but only on its periodicity, and 
thus can be extended to the setting of Section \ref{extensions} below.

We conclude this section with the observation that percolation in 
half-planes is not affected by spatial shifts.
%
\begin{lem}\label{shift-lemma}
{\bf (Shift lemma)}
Let $\pi$ and $\tilde\pi$ be two half-planes such that 
$\pi\supset\tilde\pi$,
i.e., $\pi$ and $\tilde\pi$ are translates of each other. Then
$E^+_{\pi}= E^+_{\tilde\pi}$ $\G$-almost surely, and similarly with 
$-$ instead of $+$.
\end{lem}
%
{\sl Proof}. Since trivially $E^+_{\pi}\supset E^+_{\tilde\pi}$,
we only need to show that $E^+_{\pi}\subset E^+_{\tilde\pi}$ 
$\G$-almost surely. For definiteness we consider the case when 
$\pi=\pi_\up=\{x_2\ge0 \}$ and 
$\tilde\pi =\{x_2\ge 1\}$. Take any $\mu\in\Gex$, and 
suppose that $\mu( E^+_{\tilde\pi})=0$. Then for almost all $\o$ and 
any $n\geq1$ 
there exists a smallest $-*$semicircuit $\s_n(\o)$ in $\tilde\pi$ 
containing $\D_n\cup\s_{n-1}(\o)$ in its interior; 
here $\D_n= [-n,n]\times[1,n]$ and $\s_0=\emptyset$. 
Let $x_n(\o)\in\ell_\le$ and $y_n(\o)\in\ell_\ri$ be the two points
facing the two endpoints 
of $\s_n(\o)$; these are $\F_{\tilde\pi}$-measurable functions of 
$\o$, and the random sets $\{x_n,y_n\}$ are pairwise disjoint. Let 
$A_n$ 
be the event that the spins at  $x_n$ and $y_n$ take value $-1$. 

We claim that $A_n$ occurs for infinitely many $n$ with probability 
1. Indeed, for each $N\ge1$ and any $x\in\ell_\le$, $y\in\ell_\ri$ 
we have
\bea
&&\mu\bigg(A_{N}\,\bigg|\,x_N=x,y_N=y,\bigcap_{n>N}A_n^c\bigg)\\
&&=\mu\bigg(\mu_{\{x,y\}}^\cdot(\o(x)=\o(y)=-1)\,\bigg|\,
x_N=x,y_N=y,\bigcap_{n>N}A_n^c\bigg)\\
&&\ge \d^2>0
\eea
because the event in the condition is measurable with respect to
$\{x,y\}^c$
and the one-point conditional probabilities of $\mu$
are bounded from below by $\d=[ 1 + e^{8\beta } ]^{-1}$.
Hence $\mu(A_{N}^c|\bigcap_{n>N}A_n^c)\leq 1-\d^2$, and by iteration 
we get
$\mu(\bigcap_{n\ge N}A_n^c)=0$, proving the claim.

We thus conclude that 
with probability $1$ each box $[-n,n]\times[0,n]$ is surrounded
by a $-*$semicircuit in $\pi_\up$, which means 
that $\mu( E^+_{\up})=0$. As $\mu(E^+_{\tilde\pi})$ is either 0 or 1, 
the lemma follows. $\Box$


\section{Uniqueness of semi-infinite clusters}

Our next subject is the uniqueness of infinite 
clusters in half-planes, together with the stronger property that 
such 
clusters touch the boundary line infinitely often.
This result has already been obtained by Russo \cite{Rus}
on the basis of the subsequent Lemma \ref{semi-unique} which we 
derive here differently from the preceding Corollary 
\ref{cor:MM}.
%
\begin{lem} {\bf (Line touching lemma)}\label{line_touching}
For any half-plane $\pi$, there exists $\G$-almost surely at most 
one infinite $+$ (resp.\ $+*$)
cluster $I^+_\pi$  (resp.\ $I^{+*}_\pi$) in $\pi$.
When it exists, this infinite cluster $\G$-almost surely
intersects the boundary line $\ell$ of $\pi$ infinitely often, in the 
sense that outside any finite $\D$ one can find an infinite path in 
this cluster starting from $\ell$.
\end{lem}
%
{\sl Proof}. For definiteness we assume that 
$\pi=\pi_\up$; other half-planes merely correspond to a change of 
coordinates. We consider only infinite 
$+$clusters in $\pi_\up$; the case of $+*$clusters  
is similar. It is also clear that any result proved for the 
$+$sign is also valid with the $-$sign.

{\em Uniqueness: }The uniqueness of infinite $+$clusters in 
$\pi_\up$ is a consequence of the second statement, the line-touching 
property for infinite $-*$clusters. Indeed,
suppose there exists no infinite $-*$cluster in $\pi_\up$; then each 
finite set in $\pi_\up$ is surrounded by a $+$semicircuit, so that 
any 
two infinite $+$paths are necessarily $+$connected to each other. In 
the alternative case when an infinite $-*$cluster $I^{-*}_\up$ in 
$\pi_\up$ exists, 
this $I^{-*}_\up$ meets $\ell_\le$ or $\ell_\ri$ infinitely often, so 
that each infinite $+$cluster must meet the other half-line 
infinitely often. Hence, two such $+$clusters must cross each 
other, and are thus identical. 

{\em Line touching: } Let $\mu\in\Gex$ and $x\in\pi_\up$ and 
consider the event $A^+_x$ that $x$ 
 belongs to an infinite $+$cluster in $\pi_\up$ which does not touch 
the 
horizontal axis $\ell_\h$. We will show that $\mu(A^+_x)=0$. Once 
this is 
established, we can take the union over all $x$ and use the finite 
energy property to see that for each finite $\D$ the event 
``an infinite $+$cluster in $\pi_\up$ is not connected to $\ell_\h$ 
outside 
$\D$'' 
also has probability zero, which means that almost surely any 
infinite 
$+$cluster in $\pi_\up$ must meet $\ell_\h$ infinitely often.

Intuitively, if $A^+_x$ occurs then the infinite $+$cluster 
containing $x$ is separated from $\ell_\h$ by an infinite $-*$path; 
but 
the spins `above' this path feel only the $-$boundary condition and 
thus believe to be in the $-$phase $\mu^-$, so that they will not 
form an 
infinite $+$cluster.

To make this intuition precise we fix some integer $k\geq1$ and 
consider the event $A^+_{x,k}$ that $x$ belongs to a 
$+$cluster of size at least $k$ which does not meet $\ell_\h$. Take a 
large box $\D\subset\pi_\up$ containing $x$.
For $\o\in A^+_{x,k}$ we consider the largest set $\Ga(\o)\subset 
\D$ containing $x$ such that $\o=-1$ on 
$\partial\Ga(\o)\setminus\partial_\up\D$, where 
$\partial_\up\D=\partial\D\cap \pi_\up$. By 
the strong Markov property and the stochastic monotonicity of Gibbs 
distributions, we find that
\[
\mu(A^+_{x,k})\leq \mu\bigg(\mu^\cdot_\Ga(E^+_{x,k}) \bigg)
\leq \mu_\D^{\pm}(E^+_{x,k})\;,
\]
where $E^+_{x,k}$ is the event that $x$ belongs to a $+$cluster in 
$\pi_\up$ of size 
at least $k$ and $\pm$ stands for the configuration which is $+1$ 
on $\pi_\up$ and $-1$ on $\pi_\up^c$.
Now, again by stochastic monotonicity the semi-infinite limit
$\mu^\pm_{\up}=\lim_{\D\uparrow\pi_\up}\mu_\D^{\pm}$ exists. 
Letting first $\D\uparrow\Z^2$ and then $k\to\infty$ we thus find 
that $\mu(A^+_{x})\leq \mu^\pm_{\up}(E^+_\up)$. The lemma thus follows 
from the subsequent lemma. $\Box$

\medskip\noindent
By the argument of Lemma \ref{lem:CNPR}, the following result 
implies the uniqueness of the Gibbs measure on $\pi_\up$ 
with $-$boundary condition in $\pi_\lo$.
%
\begin{lem}\label{semi-unique}
{\bf (No percolation on a bordered half-plane)}
$\mu^\pm_{\up}(E^{+*}_\up)=0$.
\end{lem}
%
{\sl Proof}. To begin we note that $\mu^\pm_{\up}$ is invariant 
under horizontal translations and stochastically maximal 
in the set of all Gibbs measures on $\pi_\up$ with $-$boundary 
condition in $\pi_\lo$. This follows just as in the case of 
the plus-phase $\mu^+$ on the whole lattice. In particular,
$\mu^\pm_{\up}$ is trivial on the $\pi_\up$-tail $\T_\up=
\bigcap \{\F_{\pi_\up\setminus\L}:
{\L\subset\pi_\up\mbox{  finite}}\}$.
 We think of $\mu^\pm_{\up}$ as
a probability measure on $\O$ for which almost all configurations are 
identically equal to $-1$ on $\pi_\up^c$.

Next we consider the downwards translates 
$\mu^+_{n,-}=\mu^\pm_{\up}\circ \th_\v^{-n}$, $n\geq 0$.
Evidently, $\mu^+_{n,-}$ is obtained by an analogous infinite-volume 
limit in the half-plane $\{x_2\geq-n\}$. This shows that 
$\mu^+_{n,-}\preceq \mu^+_{n+1,-}$ by stochastic monotonicity,
so that the stochastically increasing limit 
$\mu^+_{-}=\lim_{n\to\infty}\mu^+_{n,-}$ exists.
Clearly $\mu^+_{-}\in\G$. Also, $\mu^+_{-}$ 
inherits the horizontal invariance of the $\mu^+_{n,-}$ and is in 
addition vertically invariant. Corollary \ref{cor:MM} therefore 
implies that $\mu^+_{-}=a\,\mu^- +(1-a)\mu^+$ for some coefficient 
$a\in[0,1]$.

We claim that $a>0$. For $n\ge1$ let $B_n$ denote the event that the 
origin is $-*$connected to the horizontal line $\{x_2=-n\}$. By the 
horizontal ergodicity of $\mu^+_{n,-}$, there exist for 
$\mu^+_{n,-}$-almost all $\o$ some random integers 
$m_\le(\o)<0<m_\ri(\o)$ 
such that $\o\equiv-1$ on 
$$
\s(\o)=\bigg\{x\in\Z^2: x_1\in\{m_\le(\o),m_\ri(\o)\},\;
-n\leq x_2\leq 0\bigg\}\;.
$$
Together with a segment of the line $\{x_2=-n-1\}$ on which $\o=-1$ 
$\mu^+_{n,-}$-almost surely, $\s(\o)$ forms a $-$semicircuit in 
$\pi_\lo$ surrounding the origin. An immediate application of the 
strong Markov property and the point-to-semicircuit lemma thus 
implies that $\mu^+_{n,-}(B_n)\geq 
\theta/2$. Therefore, writing $E^{-*}_{0,m}$ for the event that the 
origin belongs to some $-*$ cluster of size at least $m$ we find
$\mu^+_{n,-}(E^{-*}_{0,m})\geq \theta/2$ when $n\geq m$. Letting 
first 
$n\to\infty$ and then $m\to\infty$ we see that $\mu^+_{-}(E^{-*}) 
\geq \theta/2$.
Since $\mu^+(E^{-*})=0$ by Corollary \ref{ocean}, it follows that 
$a\geq\theta/2$, and the claim is proved.

To conclude the proof we observe that
\[
\mu^\pm_{\up}(E^{+*}_\up) \leq \mu^+_{-}(E^{+*}) = 1-a <1\;,
\]
again by Corollary \ref{ocean}.
Since $\mu^\pm_{\up}$ is trivial on $\T_\up$, the lemma follows.
$\Box$

\medskip\noindent
The butterfly lemma and shift lemma together still leave the 
possibility that all infinite butterflies have the same orientation, 
either horizontal or vertical. As a consequence of the line touching 
lemma, we can now show that both orientations must occur. 
%
\begin{lem}\label{orthogonal-butterflies} 
{\bf(Orthogonal butterflies)}
$\G$-almost surely there exist both a horizontal infinite butterfly 
in $\pi_\up$ and $\pi_\lo$ as well as a vertical infinite butterfly 
in $\pi_\le$ and $\pi_\ri$.
\end{lem}
% 
{\sl Proof}. Suppose there exists some $\mu \in \Gex$ having almost 
surely no vertical infinite butterfly.
By the first step in the proof of the butterfly lemma, it then follows 
that $\mu=\mu\circ R_{k,\v}\circ T$ for all $k\in\Z$, and thus
$\mu=\mu\circ\th_{\h}^{-2}$.
By the tail triviality, $\mu$ is in fact ergodic under $\th_{\h}^2$.
By the butterfly lemma, horizontal infinite
butterflies do exist, say of color $+$.

We now use an argument similar to that in Corollary \ref{cor:MM},  
with the line touching lemma in place of the Burton-Keane theorem.
Fix any $n\ge1$. For $k\in\Z$ let $A_k$	denote the event that the 
point $(k,0)$ is connected by  
straight $+$paths to infinite $+$clusters in both 
$\pi_{n,\up}=\{x\in\Z^2:x_2\ge n\}$ and  $\pi_{n,\lo}=\{x_2\leq -n\}$.
Let $A$ be the event that $A_k$ occurs for infinitely many $k<0$ 
and infinitely many $k>0$. The finite energy property then shows 
that $\mu(A_0)>0$, and the horizontal ergodicity and Poincar\'e's
recurrence theorem (or the ergodic theorem) imply that
$\mu(A)=1$. But the line 
touching lemma guarantees that the infinitely many doubly-infinite 
`vertical' $+$paths passing through the horizontal axis are connected 
to each other in $\pi_{n,\up}$ and $\pi_{n,\lo}$. As	
$n$ was	arbitrary, it follows that 
almost surely each finite set is surrounded by a  
$+$circuit, and an infinite $-$cluster cannot exist. 
In view of Lemma \ref{lem:CNPR}, this implies that
$\mu=\mu^+$. But $\mu^+$ is not invariant under $R_\v\circ T$, 
in contradiction to what we derived for $\mu$. $\Box$

\medskip\noindent
The preceding argument can be used to derive the result of
Russo \cite{Rus} that $\mu^+$ and $\mu^-$ are the only phases which 
are periodic in one direction. We will not need this intermediate 
result.


\section{Non-coexistence of phases}
\label{inv}

In this section we will prove the following proposition.
%
\begin{prop} \label{invariance}
{\bf (Absence of non-periodic phases) }
Any Gibbs measure $\mu\in\G$ is invariant under translations, i.e., 
$\mu=\mu\circ\th_\h^{-1}$ and $\mu=\mu\circ\th_\v^{-1}$.
\end{prop}
%
Together with Corollary \ref{cor:MM} this will immediately imply the 
main theorem that each Gibbs measure is a mixture of the two phases 
$\mu^+$ and $\mu^-$. Our starting point is the following lemma 
estimating the probability that a semi-infinite cluster can be pinned 
at a prescribed point.
%
\begin{lem}
{\bf (Pinning lemma) } \label{pinning} 
Let $\mu\in\G$, and suppose that there exists an infinite $+*$cluster 
$I^{+*}_\up$ in $\pi_\up$ which meets the right semiaxis $\ell_\ri$ 
infinitely often. Then for each finite
square $\D=[-n,n]^2$ and $x\in\ell\ri$ we have
\[
\mu\bigg(\mbox{\rm $x$ is $+*$connected in 
$(\D\cup\ell_\le)^c$ to $I^{+*}_\up$}\bigg)
\ge \theta/4
\]
provided $x$ lies sufficiently far to the right. The same 
holds when `left' and `right' or `up' and `down' are interchanged.
\end{lem}
%
{\sl Proof}. By hypothesis, the infinite component of 
$I^{+*}_\up\setminus\D$ almost surely contains infinitely many points 
of $\ell_\ri$.
Thus, if $x\in\ell_\ri$ is located far enough to the right 
then, with probability exceeding $1/2$, at least one such 
point can be found left from $x$, and another such point can be found 
right from $x$. This means that $x$ is surrounded by a 
$+*$semicircuit $\s$ in $\pi_\up$
which belongs to $I^{+*}_\up$ and satisfies 
$\D\cap\mbox{Int\,}\s=\emptyset$.

Let $\L$ be a large square box containing $x$. If $\L$ is large 
enough, a semicircuit $\s$ as above can be found within $\L$ with 
probability still larger than $1/2$. We then can assume that $\s$ 
has the largest interior among all such semicircuits in $\L$. Using 
the strong Markov property and the point-to-semicircuit lemma we get 
the result.
$\Box$

\medskip\noindent
Our main task in the following is to analyze the situation when a 
half-plane contains both an infinite $+$cluster and an infinite 
$-$cluster.
(The line-touching lemma still allows this possibility.) 
In this situation it is useful to consider contours.
%
\begin{rem}\label{contours}
{\bf (Contours in half-planes) }
{\rm As is usually done in the Ising model, we draw lines of unit 
length 
between adjacent spins of 
opposite sign. We then obtain a system of polygonal curves running 
through the sites of 
the dual lattice $\Z^2+(\frac12,\frac12)$. A {\em contour\/} in a 
half-plane $\pi$ is a part of 
these polygonal curves which separates a $-$cluster in $\pi$ from a 
$+*$cluster in $\pi$.
This corresponds to the convention that at crossing points the 
contours 
are supposed to bend around the $-$spins. (The artificial asymmetry 
between $+$ and $-$ does not matter, and we could clearly make the 
opposite convention.) 

Suppose now that $\pi$ contains both an infinite $+*$cluster $I^{+*}_\pi$ 
and an infinite $-$cluster $I^-_\pi$. Since these are unique by the 
line touching lemma, it follows that $\pi$ contains a unique 
semi-infinite contour $\g_\pi$ which starts between two points of the 
boundary line $\ell$. On its two sides, $\g_\pi$ is accompanied by an 
infinite $+*$path $f^{+}_\pi$ 
and an infinite $-$path $f^{-}_\pi$. We call $f^{+}_\pi$ and 
$f^-_\pi$ the $+$ resp.\ $-$face of $\g_\pi$. 

By the shift lemma, $\g_\pi$ intersects each line $\ell'\subset\pi$ 
parallel to $\ell$ only finitely often. Indeed, the shifted 
half-plane $\pi'$ with boundary line $\ell'$ also contains a unique
semi-infinite contour $\g_{\pi'}$. Since the faces of $\g_{\pi'}$ are 
connected to $\ell$ by the line touching lemma, $\g_{\pi'}$
necessarily coincides with $\g_\pi$ up to finitely many steps.
}
\end{rem}
%
From now on we consider a fixed extremal Gibbs measure $\mu\in\Gex$.
We want to prove that $\mu$ is horizontally invariant. (The proof of 
vertical invariance is similar.) To this end we consider its 
horizontal translate
$\hat\mu=\mu\circ\th_\h^{-1}$, as well as the product measure
$\hat\nu=\mu\times\hat\mu$ on $\O\times\O$. It is convenient two 
think of the latter as a duplicated system consisting of two 
independent layers.
The following lemma  is a slight variation of a result of Aizenman 
\cite{Aiz} in his 
proof of the main theorem.
%
\begin{lem}\label{intersections}
{\bf (Fluctuations of the infinite contour) }
Suppose $\pi_\up$ contains a semi-infinite contour $\g_\up$ 
$\mu$-almost surely. Then
for $\hat\nu$-almost all 
$(\o,\hat\o)\in\O^2$, $\g_\up(\o)$ and $\g_\up(\hat\o)$ intersect 
each 
other infinitely often.
\end{lem}
%
{\sl Proof}. By the line touching lemma and tail triviality, we can 
assume that $I^-_\up$ intersects $\ell_\ri$ infinitely often and 
$I^{+*}_\up$ intersects $\ell_\le$ infinitely often. (The alternative 
case is analogous.) For  
$n\geq 1$ let $\pi_{n,\up}=\th_\v^n\pi_\up$ be the half-plane 
above the horizontal line through $(0,n)$, $I^{-}_{n,\up}$ 
the infinite $-$cluster in this half-plane and
\[
a_n = \min\{k\in\Z: (k,n)\in I^{-}_{n,\up}\}
\]
the abscissa of the point at which $\g_\up$ enters definitely into 
$\pi_{n,\up}$. 
Since $\g_\up$ is a continuous curve, it is sufficient to show 
that for $\hat\nu$-almost all $(\o,\hat\o)$ we have 
$a_n(\o)\geq a_n(\hat\o)$ and $a_{n+1}(\o)< 
a_{n+1}(\hat\o)$ infinitely often. Using the definition 
$\hat\mu=\mu\circ\th_\h^{-1}$ and setting $\nu=\mu\times\mu$ and 
$d_n(\o,\o')=a_n(\o)-a_n(\o')$ for $\o,\o'\in\O$, we need to 
show that $d_n\geq 1>d_{n+1}$ infinitely often $\nu$-almost 
surely. 

We observe first that $d_n=0$ infinitely often $\nu$-almost surely.
Indeed, the complementary event consists of the two parts 
$A=\{d_n\geq1 \mbox{ eventually}\}$ and 
$B=\{d_n\leq-1 \mbox{ eventually}\}$. Symmetry implies that 
$\nu(A)=\nu(B)$. On the other hand, the tail-triviality of $\mu$ 
implies that $\nu(A)=0$ or $1$. This follows from Fubini's theorem 
because $A$ is measurable with respect 
to the `product-tail' $\T^{(2)}=\bigcap\{\F_{\L^c}^2:\L\subset\Z^2
\mbox{ finite}\}$ in $\O^2$. 
(One should not be mistaken to 
believe that $A$ was measurable with respect to the smaller 
`tail-product' $\T^2$. It is only the case that the 
$\o$-section $A_\o$ of $A$ belongs to $\T$ for any $\o$, and the 
function $\o\to\mu(A_\o)$ is $\T$-measurable.) The disjointness of 
$A$ and $B$ thus implies that $\nu(A)=\nu(B)=0$.

Next we claim that there exists some constant $\d>0$ such that
$$
\nu(d_{n-1}\geq1|d_{n},d_{n+1},\ldots)\geq \d\quad\mbox{ on 
}\,\{d_n=0\}\;.
$$
To see this let $A_{n,k}=\{(\o,\o'):a_n(\o)=a_n(\o')=k \}$,
$\D_{n,k}$ the two-point set consisting of the points $(k,n-1)$ and 
$(k+1,n-1)$, and $B_{n,k}$ the event that $\o=(-1,-1)$ on $\D_{n,k}$ 
and $\o'=(-1,+1)$ on $\D_{n,k}$. Then
\[
\nu(B_{n,k}|\F_{\D_{n,k}^c}^2)(\o,\o')=
\mu^\o_{\D_{n,k}}\times \mu^{\o'}_{\D_{n,k}}(B_{n,k})\geq 
[1+e^{8\b}]^{-4}\equiv\d
\]
and thus
\[
\nu(d_{n-1}\geq1|\F_{n,\up}^2)
\geq \nu(B_{n,k}|\F_{n,\up}^2)
\geq\d \quad \mbox{ on $A_{n,k}\,$ $\nu$-almost surely}
\]
because $A_{n,k}\cap B_{n,k}\subset\{d_{n-1}\geq1\}$ and 
$\pi_{n,\up}\subset \D_{n,k}^c$. Summing over $k$ and 
conditioning on $\s(d_{n},d_{n+1},\ldots)$ we get the claim.

Now, given any integers $1\leq N<L$ let $\t=\t_{N,L}$ be the largest 
$n\in\{N,\ldots,L\}$ such that $d_n=0$, provided such an $n$ 
exists,
and $\t=0$ otherwise. Then 
$\{\t=n\}\in\s(d_{n},d_{n+1},\ldots)$ for 
all $n$, and therefore
\bea
&&\nu\bigg(\exists\, n\geq N: d_{n-1}\geq 1\ >\ d_n\bigg)\geq
\sum_{n=N}^L \nu(d_{n-1}\geq 1,\,\t=n)\\
&&=\sum_{n=N}^L \nu\bigg(\nu(d_{n-1}\geq 
1|d_{n},d_{n+1},\ldots)\,1_{\{\t=n\}}\bigg)
\ \geq\ \d\;\nu\bigg(\exists\, N\leq n\leq L: d_n=0\bigg)\;.
\eea
Letting first $L\to\infty$ and then $N\to\infty$ we see that 
$d_{n-1}\geq 1> d_n$ 
infinitely often with probability at least $\d$. This gives the 
result because $\nu$ is trivial on the product-tail $\T^{(2)}$.
$\Box$

\medskip\noindent 
Our key observation is the following percolation result for the 
duplicated system with distribution $\hat\nu$.
We will say that a path in $\Z^2$ is a $\ls$path 
for a pair $(\o,\hat\o)\in\O^2$ if $\o(x)\leq\hat\o(x)$ for all its 
sites $x$. In the same way we define $\lss$paths, and we can speak 
of $\lss$circuits and $\lss$clusters. 
%
\begin{lem}\label{lss-circuits}
{\bf(No $(+,-)$percolation in the duplicated system) } 
$\hat\nu$-almost surely each finite square $\D=[-n,n]^2$ is 
surrounded by 
a $\lss$circuit in $\Z^2$. 
\end{lem}
%
{\sl Proof}. Consider any two points $x\in\ell_\le$ and 
$y\in\ell_\ri$.
We claim that with $\hat\nu$-probability at least $(\theta/4)^2$ 
there exists a $\lss$path from $x$ to $y$ `above' $\D$, provided $x$ 
and $y$ are located sufficiently far to the left resp.\ to the right. 
We distinguish three cases. 

{\em Case 1: $\mu(E^+_\up)=0$}. By Lemma 
\ref{orthogonal-butterflies}, $\pi_\up$ then almost surely contains an 
infinite $-$cluster $I^-_\up$, and each finite subset of $\pi_\up$ is 
surrounded by a $-*$semicircuit in $\pi_\up$. In other words, an 
infinite $-*$cluster $I^{-*}_\up$ in $\pi_\up$ exists and touches 
both $\ell_\le$ and $\ell_\ri$ infinitely often. By the pinning lemma 
and the positive correlations of $\mu$,
with $\mu$-probability at least $(\theta/4)^2$ both $x$ and $y$ are 
$-*$connected to $I^{-*}_\up$ outside $\D$, and therefore also 
$-*$connected to each other by a $-*$path $p$ above $\D$. However, 
this $-*$path $p$ on the first layer is certainly also a $\lss$path 
for the duplicated system, and the claim follows.

{\em Case 2: $\mu(E^-_\up)=0$}. In this case we also have
$\hat\mu(E^-_\up)=0$. Interchanging $+$ and $-$ and replacing $\mu$ 
by $\hat\mu$ in Case 1, we find that with $\hat\mu$-probability at 
least $(\theta/4)^2$, there exists a $+*$path $\hat p$ in the second 
layer above $\D$ from $x$ to $y$. Since $\hat p$ 
is again a $\lss$path 
for the duplicated system, the claim follows as in the first case.

{\em Case 3: $\mu(E^+_\up)=\mu(E^-_\up)=1$ }. 
Then $\mu$-almost surely there exists a unique semi-infinite contour 
$\g_\up$, and by tail triviality we can assume (for definiteness) 
that $\g_\up$ has its $+$face on the left-hand side $\mu$-almost 
surely, and thus also $\hat\mu$-almost surely. By the pinning lemma 
and the independence of the two layers, the following event has 
$\hat\nu$-probability at least $(\theta/4)^2$: 
\bit
\item[--]in the first layer, $y$ is $-*$connected off $\D$ to 
$I^{-}_\up(\o)$, and thus to the $-$face $f^-_\up(\o)$ of 
$\g_\up(\o)$; that is, there exists an infinite $-*$path $p_y^-(\o)$ 
from $y$ outside $\D$ eventually running along $\g_\up(\o)$;
\item[--]in the second layer, $x$ is $+*$connected off $\D$ to 
$I^{+*}_\up(\hat\o)$, and thus to the $+$face $f^+_\up(\hat\o)$ of 
$\g_\up(\hat\o)$; that is, there exists an infinite $+*$path 
$p_x^+(\hat\o)$ from $x$ outside $\D$ eventually running along 
$\g_\up(\hat\o)$.
\eit
Since $\g_\up(\o)$ and $\g_\up(\hat\o)$ intersect each other 
infinitely often by Lemma \ref{intersections}, the union of 
$p_y^-(\o)$ and $p_x^+(\hat\o)$ contains a 
$*$path from $x$ to $y$ which by construction is a $\lss$path for the 
duplicated system. This proves the claim in the final case. 

To conclude the proof of the lemma, we let $A_{x,y}$ denote the event 
that there exist a $\lss$path from $x$ to $y$ above $\D$, and 
$B_{x,y}$ the event that such a path exists below $\D$. The indicator 
functions of these events can be written as increasing functions $f$ 
resp.\ $g$ of the difference configuration $\hat\o-\o$. Using the 
positive correlations of $\mu$ and $\hat\mu$ we thus obtain
\bea
\hat\nu(A_{x,y}\cap B_{x,y}) &=& \int\mu(d\o)\int\hat\mu(d\hat\o)\; 
f(\hat\o-\o)\, g(\hat\o-\o)\\
&\geq& \int\mu(d\o)\; \hat\mu(f(\cdot-\o))\,\hat\mu(g(\cdot-\o))\\
&\geq& \hat\nu(A_{x,y})\,\hat\nu(B_{x,y})
\ \geq\ (\theta/4)^4\;.
\eea
The last inequality follows from the claim and its analogue for the 
lower half-plane. However, if $A_{x,y}\cap B_{x,y}$ occurs then $\D$ 
is surrounded by a $\lss$circuit for the duplicated system. Letting 
$\D\uparrow\Z^2$ we see that with probability at least $(\theta/4)^4$ 
each finite set is surrounded by a $\lss$circuit. Since this event is 
measurable with respect to the product-tail $\T^{(2)}$ on which 
$\hat\nu$ is trivial, the lemma follows.
$\Box$


\medskip\noindent
It is now easy to complete the proof of Proposition \ref{invariance}.

\medskip\noindent
{\sl Proof of Proposition \ref{invariance}}. Consider any square 
$\D=[-n,n]^2$, and let $\e>0$. By Lemma \ref{lss-circuits}, $\D$ is 
$\hat\nu$-almost surely surrounded by a $\lss$circuit, and with 
probability at least $1-\e$ such a $\lss$circuit can be 
found in a sufficiently large square $\L$. Let $\Ga$ be the interior 
of the largest such $\lss$circuit; if no such $\lss$circuit exists 
let $\Ga=\emptyset$. Then we find for any increasing 
$\F_\D$-measurable function $0\leq f\leq1$, using the strong 
Markov property of $\hat\nu$ and 
the fact that $\mu_\Ga^\o\preceq \mu_\Ga^{\hat\o}$ when 
$\Ga(\o,\o')\ne\emptyset$, 
\bea
\mu(f)&=&\hat\nu(f\otimes 1)\ \leq\ 
\int_{\{\Ga\ne\emptyset\}} d\hat\nu(\o,\o')\;
\mu^{\o}_{\Ga(\o,\o')}(f) +\e\\
&\leq&\int d\hat\nu(\o,\o')\;\mu^{\o'}_{\Ga(\o,\o')}(f) +\e
\ =\ \hat\nu(1\otimes f)+\e\ = \ \hat\mu(f)+\e\;.
\eea
Letting $\e\to0$ and $\D\uparrow\Z^2$ we find that 
$\mu\preceq\hat\mu$. Interchanging $\mu$ and $\hat\mu$ 
(i.e., the roles of the layers) we get the reverse relation. 
Hence $\mu=\hat\mu$, so that $\mu$ is horizontally invariant.
The vertical invariance follows similarly by an interchange of 
coordinates. $\Box$


\section{Extensions}
\label{extensions}

Which properties of the square lattice $\Z^2$ entered into 
the preceding arguments? The only essential feature was its 
invariance 
under the reflections in all horizontal and vertical lines with 
integer coordinates. 
We claim that the theorem remains true for the Ising model 
on any graph $\LL$ with these properties. (The Ising model on 
triangular and honeycomb lattices has already been treated in 
\cite{Fuk}.)

To be more precise, let $\R=\{R_{k,\h},R_{k,\v}:k\in\Z\}$ denote the 
set of all reflections of the Euclidean 
plane ${\bf R}^2$ in horizontal or vertical lines with integer 
coordinates, and suppose $\LL$ is a countable subset of ${\bf R}^2$ 
which (after suitable scaling and rotation) is $R$-invariant
for all $R\in\R$.
Such an $\LL$ is uniquely determined by its finite intersection 
 with the unit cube $[0,1]^2$, and it is periodic with period $2$. 
Suppose further that $\LL$ is equipped with a symmetric 
neighbor relation `$\sim$' satisfying
\bit
\item[(L1)] each $x\in\LL$ has only finitely many `neighbors' 
$y\in\LL$ 
satisfying $x\sim y$;
\item[(L2)] $x\sim y$ if and only if $Rx\sim Ry$ for all $R\in\R$.
\eit
If $x\sim y$ we say that $x$ and $y$ are connected by an edge, which 
is 
visualized by a straight line segment between $x$ and $y$.
The preceding assumptions simply mean that $(\LL,\sim)$ is a locally 
finite graph admitting the reflections $R\in\R$ (and thereby the 
translations $\th_x$, $x\in 2\Z^2$) as graph automorphisms. 
The fundamental further assumption is 
\bit
\item[(L3)] $(\LL,\sim)$ is planar, i.e., 
the edges between different pairs of neighboring points do not cross 
each other except possibly at some lattice points.
\eit
The complement (in ${\bf R}^2$) of 
the union of all edges then splits into connected components called 
the faces of $(\LL,\sim)$.

One basic consequence of planarity is that $(\LL,\sim)$ admits a 
conjugate matching graph $(\LL,\stackrel{*}{\sim})$.  As indicated by 
the 
notation, this conjugate graph has the same set of vertices, but the 
relation $x\stackrel{*}{\sim} y$ holds if either $x\sim y$ or $x$ and 
$y$ are distinct points (on the border) of the same face of $(\LL,\sim)$. 
(Note that this matching dual is in general not planar.) 
The edges of $(\LL,\stackrel{*}{\sim})$ are then used to define the 
concept of 
$*$connectedness. The construction implies that
every path in $(\LL,\sim)$ is also a $*$path (i.e., a path in 
$(\LL,\stackrel{*}{\sim})$), and that the outer boundary of any 
cluster 
is a $*$path, and vice versa.
(The latter property holds for arbitrary matching pairs of graphs
as defined in Kesten \cite{Kes}, e.g. However, we also
used repeatedly the former property which does not extend to general 
matching pairs. In particular, this means that our results do not 
apply to the Ising model on the matching conjugate of $\Z^2$ having
nearest-neighbor interactions {\em and\/} diagonal interactions.) 

Another consequence of planarity is that we can draw contours
separating clusters from $*$clusters. Such contours can either be 
visualized by broken lines passing through the edges of $(\LL,\sim)$,
or simply as a pair consisting of a path and an adjacent $*$path, 
namely the two faces of the contour.
 
In order to see how to work with a graph $(\LL,\sim)$ as above we 
will discuss the proper definition of half-planes and their boundary 
lines. A half-plane $\pi$ is 
still the intersection of $\LL$ with a set of the form $\{x\in {\bf 
R}^2:x_i\geq k\}$, $k\in\Z$, $i\in\{1,2\}$, or with $\leq$ 
instead of $\geq$. However, the `boundary line' $\ell$ is now in 
general not a straight line but rather the set $\ell=\{x\in\pi:x\sim 
y 
\mbox{ for some } y\notin\pi\}$. In particular, $\ell$ is  
not necessarily a line of fixed points for the reflection $R\in\R$ 
mapping $\pi$ onto its conjugate halfplane $\pi'$. However, 
we can simply replace a point $x\in\ell$ by a pair $(x,x')$ 
consisting 
of $x$ and its $R$-image $x'$ (which 
either coincides with $x$ or is a neighbor of $x$).
Similarly, the semiaxes $\ell_\up$ etc.\ should be considered as sets
of such pairs. With these 
modifications, all geometric arguments still work in the obvious way.

 So, as a consequence of the preceding discussion we see that the 
square lattice $\Z^2$ can be replaced, for example, by
\medskip

$\bullet$ the {\em triangular lattice\/} {\bf T}. This is the 
$\R$-invariant lattice satisfying 
${\bf T}\cap[0,1]^2=\{(1,0),(0,1)\}$  and 
$(-1,0)\sim(1,0)\sim(0,1)\sim(2,1)$; the 
remaining edges result from (L2). Since all
faces are triangles, {\bf T} is self-matching. While $\pi_\up$ and
$\pi_\lo$ have a common straight boundary line, the boundaries of 
$\pi_\ri$ and $\pi_\le$ are not straight; besides a common part on 
the vertical axis they also contain the adjacent points $(1,k)$ 
resp.\ $(-1,k)$, $k\in 2\Z$.

$\bullet$ the hexagonal or {\em honeycomb lattice\/} {\bf H}. Here,
for example, ${\bf H}\cap[0,1]^2=\{(\frac13,1),(\frac23,0) \}$
and $(-\frac13,1)\sim(\frac13,1)\sim(\frac23,0)\sim
(\frac43,0)$; all other edges are 
again determined by (L2). As in the triangular 
lattice, $\pi_\up$ and
$\pi_\lo$ have a common straight boundary line, but $\pi_\ri$ and 
$\pi_\le$ 
have no common points.

$\bullet$ the {\em diced lattice}. This is obtained from the 
honeycomb lattice by placing points in the centers of the hexagonal 
faces and connecting them to the three points in the west, north-east 
and south-east of these faces; to obtain reflection symmetry an 
additional shift by $(-\frac13,0)$ is necessary.
See p.\ 16 of \cite{Kes} for more details.

$\bullet$ the covering lattice of the honeycomb lattice, the 
{\em Kagom\'e lattice}, cf.\ p.\ 37 of \cite{Kes}.

\bigskip\noindent
As for the interaction, 
it is neither necessary that the interaction along all bonds is the 
same,
nor that it is invariant under the spin flip. Except for attractivity,
we need only the invariance under simultaneous flip-reflections 
(which in particular 
implies periodicity with period 2).  For example, we did not need 
that 
$\mu^+$ and $\mu^-$ are invariant under all $R\in\R$ and related to 
each other by the spin flip $T$ (cf.\ the comments after Corollary 
\ref{ocean}). We rather needed that $\mu^+=\mu^-\circ 
R\circ T$ for all $R\in\R$ (implying that $\mu^+$ and $\mu^-$ are 
periodic, and that any flip-reflection invariant $\mu$ is different 
from these phases; the latter was used in Lemmas \ref{butterfly} and 
\ref{orthogonal-butterflies}). 
This, however, holds whenever the interaction is invariant
under flip-reflections. The same invariance is sufficient for 
spin-reflection domination and the point-to-semicircuit lemma. So, 
the only thing to observe is that whenever we work with translations 
(as in the proof of Lemma \ref{semi-unique} and in Section \ref{inv} 
\ref{intersections} and below) we have to confine ourselves to {\em 
even\/} translations, which does not raise any problems.
As a result, 
we can consider any system of spins $\o(x)=\pm1$ with formal 
Hamiltonian 
of the form
\be{Ham}
H(\o)=\sum_{x\sim y} U_{x,y}(\o(x),\o(y)) +\sum_{x\in\LL} 
V_x(\o(x))\;,
\ee
where for all $a,b\in\{-1,1\}$ we have $U_{x,y}(a,b)=U_{y,x}(b,a)$ and
\bit
\item[(H1)] $U_{x,y}(1,\cdot)-U_{x,y}(-1,\cdot)$ is decreasing on 
$\{-1,1\}$;
\item[(H2)] $U_{x,y}(a,b)=U_{Rx,Ry}(-a,-b)$ and $V_x(a)=V_{Rx}(-a)$
for all $R\in\R$.
\eit
Assumption (H1) implies that the FKG inequality is applicable.
We thus obtain the following general result.
%
\begin{thm}\label{general}
Consider a planar graph $(\LL,\sim)$ as above and an 
interaction of the form \rf{Ham} satisfying (H1) and (H2). 
Then there exist no more than two extremal Gibbs measures.
\end{thm}
%
The standard case, of course, is the ferromagnetic Ising model 
without external field; this corresponds to the choice 
$U_{x,y}(a,b)=-\b ab$ and $V_x\equiv 0$. But there is also 
another case of particular interest. Consider 
$\LL=\Z^2+(\frac12,\frac12)$, the shifted square lattice with its 
usual 
graph structure. $\LL$ is bipartite, in the sense that $\LL$ splits 
into to disjoint sublattices, $\LL_{even}$ and $\LL_{odd}$, such that 
all 
edges run from one sublattice to the other. If we set 
$U_{x,y}(a,b)=-\b ab$ 
and define a staggered external field
\[
V_x(a)=\left\{\ba{rl}-h a&\mbox{if }x\in \LL_{even}\\
h a& \mbox{if }x\in \LL_{odd}\ea \right.
\]
with $h\in{\bf R}$ then the conditions (H1) and (H2) hold; 
here we take advantage of the fact that the 
reflections $R\in\R$ map 
$\LL_{even}$ into $\LL_{odd}$ and vice versa. 
But it is well-known that this model is isomorphic to the 
{\em anti\/}ferromagnetic Ising model on $\Z^2$ with external field 
$h$; the 
isomorphism consists in flipping all spins in $\LL_{odd}$. This 
gives us the following result.
%
\begin{cor}
For the Ising antiferromagnet on $\Z^2$ for any inverse temperature 
and 
arbitrary external field there exist at most two extremal 
Gibbs measures.
\end{cor}
%
This corollary does not extend to non-bipartite lattices such as the 
triangular lattice. In fact, for the Ising antiferromagnet on {\bf T} 
one expects the existence of three different phases for suitable $h$.

However, there is a similar repulsive model to which our theorem 
applies, namely the hard-core lattice gas on $\Z^2$. In this model, 
the values 
$-1$ and $1$ are interpreted as the absence resp.\ presence of a 
particle, and no particles are allowed to sit on adjacent places.
Interchanging the values $\pm1$ on $\LL_{odd}$ we obtain an 
isomorphic 
model which is defined by setting 
$U_{x,y}(a,b)=\infty$ if $x\in\LL_{even}$ and $a=-b=1$,
 or $x\in\LL_{odd}$ and $a=-b=-1$, and 
$U_{x,y}(a,b)=0$ otherwise;
and
$V_x(a)=-\log\lambda$ if $x\in\LL_{even}$ and $a=1$, 
or $x\in\LL_{odd}$ and $a=-1$, and 
$V_x(a)=0$ otherwise.
The parameter $\lambda>0$ is called the activity.
This model satisfies all conditions of Theorem \ref{general}, except 
that the interaction takes the value $+\infty$. This implies that the 
finite energy condition does not hold as it stands. However, it is 
easily seen that there are still enough admissible configurations to 
satisfy all needs of the Burton-Keane theorem and our other 
applications of 
the finite energy property. This leads us to the following corollary.
%
\begin{cor}
For the hard-core lattice gas on $\Z^2$ at any activity $\lambda>0$ 
there exist at most two extremal Gibbs measures.
\end{cor}
%
\small
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\end{document}
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