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\title[Decay in Finite Volumes]
{Mixing Properties and Exponential Decay\\
for Lattice Systems in Finite Volumes}
\author{Kenneth S. Alexander}
\address{Department of Mathematics DRB 155\\
University of Southern California\\
Los Angeles, CA 90089-1113 USA}
\email{alexandr@math.usc.edu}
\thanks{Research supported by NSF grant DMS-9802368.}
\keywords{exponential decay of correlations, exponential decay of
connectivities, FK model, Potts model, weak mixing, strong mixing}
\subjclass{Primary: 60K35; Secondary: 82B20}
\date{\today}
\begin{abstract}
An infinite-volume mixing or exponential-decay property in a spin system
or percolation model reflects the inability of the influence of the
configuration in one region to propagate to distant regions, but in some
circumstances where such properties hold, propagation can nonetheless occur in
finite volumes endowed with boundary conditions. We establish the absense of
such propagation, particularly in two dimensions in finite volumes which are
simply connected, under a variety of conditions, mainly for the Potts model and
the Fortuin-Kasteleyn (FK) random cluster model, allowing external fields. For
example, for the FK model in two dimensions we show that exponential decay of
connectivity in infinite volume implies exponential decay in simply connected
finite volumes, uniformly over all such volumes and all boundary conditions, and
implies a strong mixing property for such volumes with certain types of boundary
conditions. For the Potts model in two
dimensions we show that exponential decay
of correlations in infinite volume implies a strong mixing property in simply
connected finite volumes, which includes exponential decay of correlations in
simply connected finite volumes, uniformly over all such volumes and all
boundary conditions.
\end{abstract}
\maketitle
\section{Introduction and Preliminaries} \label{S:intro}
Many models encountered in statistical mechanics
exhibit exponential decay of the
two-point function at sufficiently high temperatures. Typical spin systems
exhibit exponential decay of correlations, and many standard percolation models
are known or believed to have exponential decay of connectivities for those
non-critical parameter values at which there is no percolation. In particular
this can be said for many random cluster models (graphical representations)
corresponding to spin systems. Such exponential decay is by its nature an
infinite-volume property, but it does have a finite-volume analog which has
apparently been little studied. Let $R$ be a bounded subset of $\RRn$.
For a spin system on the lattice $\ZZn$, we
consider the truncated correlation (covariance)
$\langle \delta_{[\sigma_{x} = i]}; \delta_{[\sigma_{y} = j]}
\rangle_{\Lambda,\eta}^{\beta}$ for the system on $\Lambda = R \cap
\ZZn$ at inverse temperature $\beta$ under boundary condition
$\eta$. For a percolation model we consider the
probability $P_{\mathcal{B},\rho}(x
\lra y)$ of the event $x \lra y$ that there exists a path of open bonds from $x$
to $y$, for the model on the set $\mathcal{B}$ of all bonds contained in $R$,
under boundary condition $\rho$. We may ask, do there exists constants $C$ and
$\lambda$,
\emph{not} depending on the region $R$
or on the boundary condition, such that
for all $x, y \in \Lambda$ this finite-volume correlation or connectivity is
bounded above by
$C \exp(-\lambda d(x,y))$? There are two natural choices for the metric $d$
here: the Euclidean metric $d_{2}$ and the \emph{restricted-path metric}
\[
d_{R}(x,y) = \min \{ n \geq 0: \text{ there exists a lattice path of
length } n \text{ in } R \text{ from } x \text{ to } y \}.
\]
When such $C, \lambda$ exist we say there is
\emph{uniform exponential decay of finite-volume correlations} or
\emph{connectivities}. When the uniformity is only over some limited class
$\mkC$ of regions, or regions with boundary
conditions, we refer to the uniform exponential decay as being \emph{for the
class} $\mkC$. If the metric is not clear from the context, we refer
to the decay as being \emph{in the Euclidean metric} or \emph{in the
restricted-path metrics}. We will establish here sufficient conditions for such
uniform exponential decay in two dimensions, in both metrics.
What makes the study of correlations, and analogously connectivities, under a
boundary condition difficult is that the influence of one
spin might in principle
propagate (in part) along the boundary to affect distant spins much more
than would occur in infinite volume. Such propagation
was studied by Martinelli,
Olivieri and Schonmann \cite{MOS} in showing that in two dimensions, weak mixing
implies a form of strong mixing. But these authors restricted attention to very
regular regions $\Lambda$, specifically large
squares or unions of large squares;
we will consider more general regions here. We will not, however, consider spin
systems as general as those in \cite{MOS}.
The region $\Lambda$ cannot be completely arbitrary. The examples in (\cite{MO}
pp. 458--459), one due to R. H. Schonmann, show that when $\Lambda$ is so
irregular that the boundary
$\pL$ permeates through the bulk of $\Lambda$,
the influence of a single spin can
propagate much more than is the case in infinite volume. The hypothesis we will
make on $\Lambda$ is a lattice version of
simple connectedness, which we will see
is enough to avoid this problem.
The main percolation model of interest to us is the Fortuin-Kasteleyn random
cluster model (briefly, the \emph{FK model}) of \cite{Fo1}, \cite{Fo2},
\cite{FK}, including the version with external fields. The FK model is a
graphical representation of the Potts model. When possible, however, we will
state results for more general models.
In two dimensions one can also consider
decay of dual connectivities in finite volumes, and here the effect of boundary
conditions can be dramatic. For example, the FK model corresponding to an Ising
model at subcritical temperature exhibits exponential decay of dual connectivity
in infinite volume, but for a large square with Dobrushin boundary conditions
(plus on the top half, minus on the bottom half), a long dual connection is
forced to exist, meaning the exponential decay is destroyed.
For the Ising model in finite volume with
boundary condition in which all boundary
spins are plus or 0 (free), as observed by C. Pfister \cite{Pf}, the boundary
condition may be viewed as an infinite external field applied to the boundary
plus spins, together with the ``turning off''
of the couplings for all bonds with
an endpoint at a boundary 0 spin. Standard symmetry inequalities then show that
the truncated correlation in finite volume is
bounded above by the infinite-volume
truncated correlation, so exponential decay of correlations implies at
least a weaker form of uniform exponential decay of finite-volume
connectivities, in which we restrict the allowed boundary conditions. But
symmetry inequalities tell us nothing here under Ising-model boundary conditions
which mix plus and minus spins. And for other models of interest, such as the
Potts model, symmetry inequalities aren't even available.
Instead, our techniques
involve first establishing uniform exponential decay of finite-volume
connectivities in percolation models, then transferring
these results to some spin
systems using random cluster representations.
Beyond exponential decay of the two-point function, we consider mixing
properties. Consider finite regions $\Delta \subset \Lambda$ with boundary
condition on $\Lambda^{c}$. Roughly, weak mixing is
the property that the maximum
influence (measured additively) of the boundary condition on the probability of
any event occuring in any fixed $\Delta$ decays to 0 exponentially in
$d_{2}(\Delta,\Lambda)$ as
$\pL$ recedes to infinity, and ratio weak mixing is a similar but stronger
property with influence measured multiplicatively.
In strong mixing, the maximum
(additive) influence of a change in the boundary condition on
the probability of any event occurring in any subset
$\Theta$ of $\Lambda^{c}$ decays exponentially in $d_{2}(\Delta,\Theta)$. Thus
weak mixing allows the influence of a region $\Theta \subset \Lambda^{c}$ to
propagate along the boundary, but strong mixing does not allow this. In
\cite{Al98mix} the following facts are proved. For bond percolation models in
two dimensions, under mild hypotheses (satisfied, for example, by the FK model),
exponential decay of connectivities implies weak mixing.
In any dimension, under
additional hypotheses satisfied by the FK model, weak mixing and exponential
decay of connectivity imply ratio weak mixing. For the Potts model without
external fields above the critical temperature in two dimensions, exponential
decay of correlations implies weak mixing, and in arbitrary dimension, weak
mixing implies ratio weak mixing. Further, in
\cite{MOS} it is shown that in two dimensions, weak mixing is equivalent to a
restricted version of strong mixing in which $\Lambda$ is required to be a union
of large squares. We consider here uniform versions of these results in finite
volumes.
The condition in \cite{MOS} that $\Lambda$ be a union of large squares is more
restrictive than it might first appear, for the following reason. It is often
of interest to take two subsets $\Delta, \Sigma \subset \Lambda$ and, under a
boundary condition specified on $\partial \Lambda$, consider the influence of an
event or configuration on $\Sigma$ on the probabilities of events occuring on
$\Delta$. One can hope to apply the results of \cite{MOS} in this situation by
treating $\Sigma$ as part of the boundary, but this requires that
$\Lambda \bs \Sigma$ be a union of large squares, which is
not generally natural.
Turning to formalities,
a \emph{bond}, denoted $\langle xy \rangle$, is an unordered pair of
nearest neighbor sites of $\ZZn$. When convenient
we view bonds as being closed line segments in the plane; this should be
clear from the context. In particular for $R \subset
\RRn, \ \mathcal{B}(R)$
denotes the set of all bonds for which the corresponding closed line segments
are contained in $R$, and when we refer to distances between sets of
bonds, we mean distances between the corresponding sets of line
segments. The exception is for $\Lambda \subset
\ZZn$, for which we set
$\mathcal{B}(\Lambda) = \{\langle xy \rangle: x, y \in \Lambda\}$.
(Again, this should be clear from the context.)
For a set $\mD$ of bonds we let
$V(\mD)$ denote the set of all endpoints of bonds in
$\mD$, and
\[
\partial \mD = \{\langle xy \rangle: x \in V(\mD),
y \notin V(\mD)\}, \quad \overline{\mD} =
\mD \cup \partial \mD.
\]
We write $\omB(\Lambda)$ for
$\overline{\mathcal{B}(\Lambda)}$.
A \emph{bond configuration} is an element $\omega \in
\{0,1\}^{\mathcal{B}(\ZZn)}$.
For $n=2$, the \emph{dual lattice} is the translation of the integer lattice
by (1/2,1/2); we write $x^{*}$ for $x + (1/2,1/2)$.
To each (regular)
bond $e$ of the lattice there corresponds a \emph{dual bond}
$e^{*}$ which is its perpendicular bisector; the dual bond is
defined to be open in a configuration $\omega$ precisely
when the regular bond is closed, and the corresponding configuration
of dual bonds is denoted $\omega^{*}$. We write
$(\ZZ)^{*}$ for $\{x^{*}: x \in \ZZ\}$.
A \emph{cluster} in a given configuration is a connected
component of the graph with site set $\ZZn$ and all open
bonds; for $n=2$, \emph{dual clusters} are defined analogously for open dual
bonds. (In contexts where there is a boundary condition consisting of
a configuration on the complement
$\mathcal{B}^{c}$ for some set $\mathcal{B}$ of bonds,
a cluster may include bonds in $\mathcal{B}^{c}$.) For a configuration on
$\omB(\Lambda)$ for some finite $\Lambda$, a
\emph{boundary cluster} is a cluster
which intersects $\pL$ and a \emph{non-boundary cluster} is one which does not.
Given a set $\mD$ of bonds, we write $\mD^{*}$ for
$\{e^{*}: e \in \mD\}$. The set of all endpoints of bonds in
$\mD^{*}$ is denoted
$V^{*}(\mD)$ or $V^{*}(\mD^{*})$.
For $\Lambda \subset \ZZn$ or $\Lambda \subset (\ZZ)^{*}$ we define
\[
\partial \Lambda = \{x \notin \Lambda: x \text{ adjacent to }
\Lambda\}, \quad \partial_{in}\Lambda =
\{x \in \Lambda: x \text{ adjacent to }
\Lambda^{c}\}
\]
where adjacency is in the appropriate lattice $\ZZn$ or $(\ZZ)^{*}$.
A \emph{(dual) path} is a sequence $\gamma =
(x_{0},\langle x_{0}x_{1} \rangle, x_{1},\ldots
x_{n-1},\langle x_{n-1}x_{n} \rangle, x_{n})$ of alternating (dual)
sites and bonds. $\gamma$ is \emph{self-avoiding} if all sites are
distinct. We write $x \lra y$ ($x \lrad y$) in $\omega$
if there is a path of open (dual) bonds from $x$ to $y$ in $\omega$.
By a \emph{bond percolation model} we mean a probability
measure $P$ on
$\{0,1\}^{\mathcal{B}(\ZZn)}$.
The finite-volume
distribution for the model $P$ under boundary condition $\rho \in
\{0,1\}^{\mathcal{B}^{c}}$ is
\[
P_{\mathcal{B},\rho} = P(\cdot \mid \omega_{e}
= \rho_{e} \text{ for all } e \in \mathcal{B}^{c}),
\]
where $\mathcal{B} \subset \mathcal{B}(\ZZn)$. We write $\rho^{i}$ for the
all-$i$ boundary condition. We say a bond percolation
model $P$ has \emph{bounded energy} if there exists $p_{0} > 0$ such that
\begin{equation} \label{E:boundener}
1 - p_{0} > P(\omega_{e} = 1 \mid \omega_{b}, b \neq e) > p_{0} \quad
\text{for all } \{\omega_{b}, b \neq e\}.
\end{equation}
Write $\omega_{\mD}$ for $\{\omega_{e}: e \in \mD\}$
and let $\mG_{\mD}$ denote the
$\sigma$-algebra generated by $\omega_{\mD}$.
$P$ has the \emph{weak mixing property} if for some $C, \lambda > 0$, for all
finite sets $\mD,\mE$ with $\mD \subset \mE$,
\begin{align}
\sup \{\Var(&P_{\mE,\rho}(\omega_{\mD} \in \cdot),
P_{\mE,\rho^{\prime}}(\omega_{\mD} \in \cdot)):
\rho, \rho^{\prime} \in \{0,1\}^{\mE^{c}}\} \notag \\
&\leq C \sum_{x \in V(\mD),y \in V(\mE^{c})}
e^{-\lambda |x - y|}, \notag
\end{align}
where $\Var(\cdot,\cdot)$ denotes total variation distance between measures.
Roughly, the influence of the boundary condition on a finite region decays
exponentially with distance from that region. Equivalently, for some
$C, \lambda > 0$, for all sets $\mE, \mF \subset
\mathcal{B}(\ZZ)$,
\begin{align} \label{E:weakmix}
\sup \{|&P(E \mid F) - P(E)|: E \in \mG_{\mE}, F \in
\mG_{\mF}, P(F) > 0\} \\
&\leq C \sum_{x \in V(\mE),y \in V(\mF)}
e^{-\lambda |x - y|}. \notag
\end{align}
$P$ has the \emph{ratio weak mixing property} if for some $C, \lambda > 0$,
for all sets $\mE, \mF \subset
\mathcal{B}(\ZZ)$,
\begin{align} \label{E:rweakmix}
\sup &\left\{ \left| \frac{P(E \cap F)}{P(E)P(F)} - 1 \right| : E \in
\mG_{\mE}, F \in
\mG_{\mF}, P(E)P(F) > 0 \right\} \\
&\leq C \sum_{x \in V(\mE),y \in V(\mF)}
e^{-\lambda |x - y|}, \notag
\end{align}
whenever the right side of (\ref{E:rweakmix}) is less than 1. Note that in weak
mixing the influence of the event $F$ on the probability of $E$ is measured
additively, but in ratio weak mixing it is measured multiplicatively---$F$ can
alter the probability of $E$ by at most a factor near 1.
A multiplicative result
is much stronger when dealing with events $E$ of probability much smaller than
the additive constant, i.e. the right side of (\ref{E:weakmix}).
The finite-volume analog of weak mixing is strong mixing, as studied for spin
systems in
\cite{MOS} and elsewhere, sometimes under other names. We say $P$ has the
\emph{strong mixing property} (for a class $\mkC$
of regions and boundary conditions, in the metric
$d$) if for some
$C, \lambda > 0$, for all $(\mB,\rho) \in \mkC$
and all $\mE,\mF \subset \mathcal{B}$,
\begin{align} \label{E:strongmix}
\sup \{|&P_{\mB,\rho}(E \mid F) - P_{\mB,\rho}(E)|: E \in
\mG_{\mE}, F \in
\mG_{\mF}, P_{\mB,\rho}(F) > 0\} \\
&\leq C \sum_{x \in V(\mE),y \in V(\mF)}
e^{-\lambda d(x,y)}. \notag
\end{align}
The \emph{ratio strong mixing property} is defined
similarly with (\ref{E:strongmix}) replaced by
\begin{align} \label{E:ratiosm}
\sup &\left\{ \left| \frac{P_{\mB,\rho}(E \cap
F)}{P_{\mB,\rho}(E)P_{\mB,\rho}(F)} - 1 \right| : E \in
\mG_{\mE}, F \in
\mG_{\mF}, P_{\mB,\rho}(E)P_{\mB,\rho}(F) > 0 \right\} \\
&\leq C \sum_{x \in V(\mE),y \in V(\mF)}
e^{-\lambda d(x,y)}. \notag
\end{align}
Here $d$ is either $d_{2}$ or the restricted-path metric $d_{\mB}$. As we will
see (Remark \ref{R:tunnel}), in contrast to the situation for spin systems,
for bond percolation models these properties are not always quite the right ones
to consider, as influence may be transmitted, in effect, through the boundary
configuration $\rho$. Hence later, for the FK model we will be restricting
$\mE,
\mF$ in (\ref{E:strongmix}) and (\ref{E:ratiosm}) in a manner depending
on $\rho$.
Given
$\rho \in \{0,1\}^{\omB(\Lambda)^{c}}$ we define $(\omega, \rho)$ to be the bond
configuration on the full lattice which coincides with $\omega$
on $\omB(\Lambda)$
and with $\rho$ on $\omB(\Lambda)^{c}$.
Let us use ``$\leq$'' to denote the coordinatewise partial ordering on
$\{0,1\}^{\mB}$. An event $A$ is called \emph{increasing} if $\omega \in A,
\omega \leq \omega^{\prime}$ imply $\omega^{\prime} \in A$, and
\emph{decreasing} if its complement is increasing.
A probability measure $P$ on $\{0,1\}^{\mB}$
is said to have the \emph{FKG property} if
\[
P(A \cap B) \geq P(A)P(B) \quad \text{for all increasing events } A, B.
\]
$P$ is said to satisfy the \emph{FKG lattice condition}
if
\begin{equation} \label{E:FKGlatt}
P(\omega \vee \omega^{\prime}) P(\omega \wedge \omega^{\prime})
\geq P(\omega) P(\omega^{\prime}) \quad \text{for all } \omega,
\omega^{\prime}.
\end{equation}
As proved in \cite{FKG}, the FKG lattice condition implies the FKG property.
For $P_{1}$ and $P_{2}$ probability measures on $\{0,1\}^{\mB}$, we say
$P_{1}$ \emph{dominates} $P_{2}$ (in the FKG sense) if $P_{1}(A) \geq
P_{2}(A)$ for all increasing events $A$.
We say that an (infinite-volume) bond percolation model $P$
has \emph{exponential
decay of connectivities} if there exist $C, \lambda > 0$ such that for all $x$
and $y$,
\[
P(x \lra y) \leq C e^{-\lambda |y - x|}.
\]
If $P$ has the FKG property, then
$-\log P(0^{*} \leftrightarrow x^{*})$ is a subadditive function
of $x$, and therefore the limit
\begin{equation} \label{E:surftens}
\tau(x) =
\lim_{k \to \infty} -\frac{1}{k}\log
P(0 \leftrightarrow kx),
\end{equation}
exists for $x \in \mathbb{Q}^{2}$, provided we take the limit
through values of $k$ for which $kx \in \ZZn$. This definition
extends to $\RRn$ by continuity (see \cite{Al97pwr}); the
resulting $\tau$ is a norm on $\RRn$.
By standard subadditivity results, the limit in (\ref{E:surftens}) is approached
from above, so that, for $\theta = x/|x|$,
\begin{equation} \label{E:conupr}
P(0 \leftrightarrow x) \leq e^{-\tau(x)} = e^{-\tau(\theta)|x|} \quad
\text{for all} \ x.
\end{equation}
For $\Lambda
\subset \ZZn$ finite, $\rho \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}$, and
$\Gamma \subset \Lambda^{c}$ finite, we call $\mathcal{B}(\Gamma)$ a
\emph{controlling region} for $\overline{\mathcal{B}}(\Lambda)$ and $\rho$
if for every $\rho^{\prime} \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}$ such that
$\rho = \rho^{\prime}$ on $\mathcal{B}(\Gamma)$, we have
$P_{\Lambda,\rho} = P_{\Lambda,\rho^{\prime}}$. We say $P$ has
\emph{exponentially bounded controlling regions} if there exist constants
$C, \lambda > 0$ such that for every choice of disjoint finite sets $\Lambda$
and $\Gamma$,
\begin{align}
P(&\{\rho \in \{0,1\}^{\mathcal{B}(\Lambda^{c})}: \mathcal{B}(\Gamma)
\ \text{is not a controlling region for} \
\overline{\mathcal{B}}(\Lambda) \
\text{and} \ \rho \}) \notag \\
&\leq C \sum_{x \in \Lambda,y \in \Lambda^{c} \backslash \Gamma}
e^{-\lambda |x - y|}. \notag
\end{align}
Note that when $P(E)$ is much smaller than the right side of (\ref{E:weakmix}),
the weak mixing condition (\ref{E:weakmix}) allows $P(E \mid F)$ to be many
times larger than $P(E)$, but the ratio weak mixing condition (\ref{E:rweakmix})
does not allow this. Nonetheless, it is proved in \cite{Al98mix} that if $P$ is
translation invariant and has exponentially bounded controlling regions and the
weak mixing property, then
$P$ has the ratio weak mixing property. (The hypothesis of translation
invariance should have been included in the statement of this result in
\cite{Al98mix}.)
We call $\mD \subset \mB(\ZZ)$ \emph{simply lattice-connected} if $\mD$ and
$\mB(\ZZ) \backslash \mD$ are connected. We call $\Lambda \subset \ZZ$ simply
lattice-connected if $\omB(\Lambda)$ has that property.
We can now state our first main result for percolation models, essentially that
when open paths do not propagate far in infinite volume, neither can they
propagate in finite volumes, along the boundary or through the bulk, provided
the model has the weak mixing property in infinite volume. The statement
is given for the square lattice for ease of exposition, but the result is valid
for any planar lattice, as all cited results used in the proof similarly so
extend. We defer all proofs to Section \ref{S:proofs}.
\begin{theorem} \label{T:percmain}
Let $P$ be a translation-invariant bond percolation model on $\mB(\ZZ)$
having exponential decay of connectivities and the weak mixing property.
Then $P$ has uniform exponential decay of finite-volume connectivities for the
class of all simply lattice-connected subsets of $\mB(\ZZ)$ with arbitrary bond
boundary conditions, for both the Euclidean and restricted-path metrics.
\end{theorem}
This result for the restricted-path metric clearly implies the result for the
Euclidean metric, so the last phrase is really just for emphasis. The same
applies to the other results of this paper.
We do not know whether the rate of exponential decay in infinite volume and the
uniform rate of exponential decay in finite volumes are the same in Theorem
\ref{T:percmain}.
For $p \in [0,1], q > 0$ and $\mB \subset \mB(\ZZn)$, the \emph{FK model}
$P_{\Lambda,w}^{p,q}$ on the graph $(V(\mB),\mB)$
with parameters $(p,q)$ and free boundary condition is defined by the weights
\begin{equation} \label{E:FKweight}
W(\omega) = p^{|\omega|}(1 - p)^{|\mB| - |\omega|}q^{K(\omega)}
\end{equation}
Here $|\omega|$ means the number of open bonds in $\omega$
and $K(\omega)$ denotes the
number of open clusters in $\omega$.
More generally, let $K(\omega \mid \rho)$ be the number
of open clusters of $(\omega, \rho)$ which intersect $V(\mB)$. The FK model
$P_{\Lambda,\rho}^{p,q}$ with
bond boundary condition $\rho$ is given by the
weights in (\ref{E:FKweight}) with
$K(\omega)$ replaced by $K(\omega \mid \rho)$. The boundary condition
$\rho^{1}$ is also called the \emph{wired boundary condition} and we write
$K_{w}(\omega)$ for $K(\omega \mid \rho^{1})$; the corresponding weights are
\begin{equation} \label{E:FKweight2}
W_{w}(\omega) = p^{|\omega|}(1 - p)^{|\mB| - |\omega|}q^{K_{w}(\omega)}
\end{equation}
Alternately, we consider site boundary conditions. For notational
convenience we allow an additional spin value 0 at boundary sites, i.e.
$\eta \in \{0,1,..,q\}^{\Lambda}$; taking $\eta_{x} = 0$ makes the boundary
condition free at $x$.
Specifically,
suppose $\mB = \omB(\Lambda)$ for some finite $\Lambda \subset \ZZn$;
given $\eta \in \{0,1,..,q\}^{\partial \Lambda}$ define
\begin{align} \label{E:Devent}
J(\Lambda,\eta) = \bigl\{\omega \in \{0,1\}^{\omB(\Lambda)}: \eta_{x}
&= \eta_{y} \text{ for every } x,y \in \partial \Lambda \text{ for which }
x \lra y \text{ in } \omega, \\
&\omega_{e} = 0 \text{ for all } e \in \{\langle xy \rangle:
x \in \Lambda, y \in \partial\Lambda, \eta_{y} = 0\} \bigr\}. \notag
\end{align}
Here $x \lra y$ means there is a path of open bonds connecting $x$
to $y$. The FK model $P^{p,q}_{\omB(\Lambda),\eta}$ with
site boundary condition $\eta$ is given by the weights in (\ref{E:FKweight2}),
multiplied by $\delta_{J(\Lambda,\eta)}(\omega)$. We write $\eta^i$ for the
all-$i$ site boundary condition.
Taking $\eta = \eta^0$ gives the FK model with
free boundary condition; we denote
it $P^{p,q}_{\mB(\Lambda),f}$. Let
\[
\mB^+(\Lambda,\eta) =
\mB(\Lambda) \cup \{ \langle xy \rangle: x \in \Lambda, y \in \pL,
\eta_y \neq 0 \}.
\]
Generally $\eta$ is understood from the context and we suppress it in the
notation, writing $\mB^+(\Lambda)$. The the model on $\omB(\Lambda)$ with
boundary condition $\eta$ is equivalent to the model on $\mB^+(\Lambda)$ with
site boundary condition defined only on $\{ y \in \pL: \eta_y \neq 0 \}$.
We call $\eta$ a \emph{single-species
boundary condition} if for some $i$,
$\eta_{x} \in \{0,i\}$ for all $x$.
For a summary of basic
properties of the FK model, see \cite{Gr95}. In particular, since we are in two
dimensions, for $p \neq \sqrt{q}/(1 + \sqrt{q})$ there is a unique
infinite-volume FK measure on $\mB(\ZZ)$, which can be
obtained as the limit of $P^{p,q}_{\omB(\Lambda),w}$ as $\Lambda \nearrow \ZZ$;
we denote this measure $P^{p,q}$ and we say that
\emph{random-cluster uniqueness}
holds for the FK model at $(p,q)$. For $q \geq 1$, the
FK model satisfies the FKG
lattice condition, under any bond or single-species site boundary
condition.
For the FK model with external fields
$h_{i}$, $i = 1,\ldots,q$ and free boundary,
the factor $q^{K(\omega)}$ in the weight $W(\omega)$ is
replaced by
\begin{equation} \label{E:FKexternal}
\prod_{C \in \mC(\omega)} \left((1-p)^{h_{1}s(C)} + (1-p)^{h_{2}s(C)} +
\ldots + (1-p)^{h_{q}s(C)}\right),
\end{equation}
where $\mC(\omega)$ is the set of clusters in $(\oL,\omB(\Lambda))$ in the
configuration $\omega$ and
$s(C)$ denotes the number of sites in the cluster $C$. The parameters are then
$(p,q,\{h_{i}\})$; $q$ must be an integer, and
we may omit $\{h_{i}\}$ when all external fields are 0. We
need only consider
$0 = h_{1} \geq h_{2} \geq \ldots \geq h_{q}$, so we henceforth assume this in
our notation. Species $i$ is called \emph{stable} if
$h_{i}$ is maximal, i.e. $h_{i} = h_{1} = 0$. For bond
boundary conditions $\rho$ we replace (\ref{E:FKexternal}) with
\begin{equation} \label{E:FKextbond}
\prod_{C \in \mC(\omega \mid \rho)} \left((1-p)^{h_{1}s(C)} +
(1-p)^{h_{2}s(C)} +
\ldots + (1-p)^{h_{q}s(C)}\right),
\end{equation}
where $\mC(\omega \mid \rho)$ is the set of open
clusters of $(\omega\rho)$ which
intersect $V(\mB)$. If $\mB(\ZZn) \bs \mB$ is
connected then for stable $i$ the wired boundary
condition is equivalent to the site boundary condition $\eta^i$; we therefore
refer to $\eta^i$ as the \emph{i-wired} boundary condition. For general
site boundary conditions $\eta$ for the model on $\omB(\Lambda)$ the factor
(\ref{E:FKexternal}) is multiplied by
\begin{align} \label{E:FKextsite}
\prod_{C \in \mC_{int}(\omega)} &\left((1-p)^{h_{1}s(C)} +
(1-p)^{h_{2}s(C)} +
\ldots + (1-p)^{h_{q}s(C)}\right) \\
&\times \quad
\prod_{C \in \mC_{\partial}(\omega)} (1-p)^{h_{i(C)}s(C)} \quad \times \quad
\delta_{J(\Lambda,\eta)}(\omega), \notag
\end{align}
where $\mC_{\partial}(\omega)$ (respectively $\mC_{int}(\omega)$) is the set of
clusters in the configuration
$\omega$ which do (respectively don't) intersect
$\partial \Lambda$ and $i(C)$ is
the species for which $\eta_{x} = i$ for all $i \in \partial \Lambda \cap C$.
(The existence of such an $i$ is forced by the event $J(\Lambda,\eta)$.)
We call $\eta$ a \emph{single-stable-species boundary
condition} if for some stable $i$,
$\eta_{x} \in \{0,i\}$ for all $x$.
For $q \geq 1$, the FK model with external fields satisfies the FKG lattice
condition, under any bond or single-stable-species site boundary
condition. It should be noted that a single-stable-species site boundary
condition $\eta$ is equivalent to a bond boundary condition in which all sites
$x \in \pL$ where $\eta_x \neq 0$ are part of a single infinite cluster in the
boundary $\omB(\Lambda)^c$.
Let $p_{c}(q,n,\{h_i\})$ denote the percolation critical point of the FK
model on $\ZZn$; we omit the $\{h_i\}$ when there are no external fields. The
following facts are known for $n=2$. For $q = 1, q = 2,$ and
$q \geq 25.72$, we have $p_{c}(q,2) =
\tfrac{\sqrt{q}}{1 + \sqrt{q}}$ \cite{LMR}, and the connectivity decays
exponentially for all $p < p_{c}(q,2)$ \cite{Gr97}. This is believed to be true
for all $q$; for $2 < q < 25.72$ the connectivity is known to
decay exponentially at least for all $p < \tfrac{\sqrt{q-1}}
{1 + \sqrt{q-1}}$, and analogous results hold for other planar lattices
\cite{AlARC}. For general $q \geq 1$, if the
connectivity decays exponentially then the model has the ratio weak mixing
property \cite{Al98mix}. (This result is actually given
assuming a nonnegative external field applied to
at most one species, but the proof carries over without change to arbitrary
external fields; the necessary FKG property is
proved in \cite{BBCK}.) From this
and Theorem \ref{T:percmain} we immediately get the following.
\begin{theorem} \label{T:FKcase}
Let $P = P^{p,q,\{h_{i}\}}$ be an FK model on $\mB(\ZZ)$
with $p < p_c(q,2,\{h_i\})$, and suppose
$P$ has exponential decay of connectivities. Then
$P$ has uniform exponential decay of finite-volume connectivities
for the class of all simply lattice-connected subsets of $\mB(\ZZ)$
with arbitrary
(site or bond) boundary condition, for both the Euclidean and restricted-path
metrics.
\end{theorem}
Theorem \ref{T:FKcase} is implicitly stated here, and later proved, for $q$ an
integer, but in the absense of external fields it is valid for noninteger $q$ as
well. The proof requires only minor modifications. Similar considerations hold
for all our other results.
When long paths of open bonds exist for subcritical $p$, the standard heuristic
picture is that the path has a ``string of beads'' structure, meaning that
``blobs'' of open bonds, containing many double connections between pairs of
sites and having a linear scale on the order of the correlation length, are
connected by short single paths of open bonds. Such a picture has been made
rigorous in \cite{CCC} and \cite{CI} for Bernoulli percolation. The next
corollary establishes a small part of such a picture for the FK model in two
dimensions, by showing that long double connections have an exponential cost
over and above the cost of a single connection.
\begin{corollary} \label{C:twopath}
Let $P = P^{p,q,2,\{h_{i}\}}$ be an FK model on
$\mB(\ZZ)$ with $p < p_c(q,2,\{h_i\})$,
and suppose $P$ has exponential decay of connectivities. Then
there exists $\epsilon > 0$ such that for all $x \in \ZZ$, for $\theta = x /
|x|$,
\[
P(\text{there exist two bond-disjoint open paths } 0 \lra x)
\leq e^{-(\tau(\theta) + \epsilon) |x|}.
\]
\end{corollary}
\begin{remark} \label{R:tunnel}
For a bond percolation model on a finite set
$\mB$ of bonds under bond boundary condition $\rho$, influence can
propagate not only along the boundary inside $\mB$
but also through the exterior via $\rho$. For example,
for the FK model, suppose
$\langle uv \rangle, \langle xy \rangle \in \mB$ and in $\rho$ there are open
paths from $u$ to $x$ and from $v$ to $y$, and these two paths are in different
clusters of the configuration $\rho$. That is, the two paths form a tunnel from
$\langle uv \rangle$ to $\langle xy \rangle$. It is then
straightforward to show
that the events that $\langle uv \rangle$ and $\langle xy \rangle$ are open can
have a correlation bounded away from 0 uniformly in the length of the tunnel.
Tunnels of dual bonds can cause similar problems and effectively can exist
even under site boundary conditions (see Example \ref{E:nodecay} below.)
Nonetheless, restricting some of our results to site boundary conditions makes
tunneling a manageable problem. As we have noted,
single-stable-species site boundary conditions are equivalent to bond boundary
conditions, and are thus effectively a natural class of bond boundary conditions
which do not allow the tunneling phenomenon to be a problem. The results we
state under such boundary conditions can be generalized to some other bond
boundary conditions for which tunneling does not occur in such a way as to be a
problem.
\end{remark}
For $x \in \RRn$ and $l > 0$ let $Q_{l}(x)$ denote the closed cube of side $2l$
centered at $x$. Write $Q(x)$ for $Q_{1/2}(x)$, and for $\Lambda \subset \ZZn$
let $Q(\Lambda) = \cup_{x \in \Lambda} Q(x)$.
\begin{example} \label{E:nodecay}
Define bonds of the square lattice
\[
b_n^+ = \langle (n,0),(n+1,0) \rangle, \qquad
b_n^- = \langle (-n-1,0),(-n,0) \rangle
\]
and consider the FK model on $\omB(Q_n(0))$, without external fields, at $(1 -
\epsilon,q)$ where $\epsilon > 0$ and $q > 1$, with site boundary condition
$\eta_x = 1$ if $x = (n+1,0)$ or $x = (-n-1,0)$, $\eta_x = 0$ otherwise. It is
straightforward to verify the following statements:
\begin{align}
&P((-n,0) \lra (n,0)) = 1 - O(\epsilon^3); \notag \\
&P(b_n^+ \text{ is closed} \mid (-n,0) \lra (n,0)) = P(b_n^- \text{ is closed}
\mid (-n,0) \lra (n,0))
= \epsilon + O(\epsilon^2); \notag \\
&P(b_n^+ \text{ and } b_n^- \text{ are closed} \mid (-n,0) \lra (n,0)) =
q\epsilon^2 + O(\epsilon^3). \notag
\end{align}
In all cases the $O(\cdot)$ is uniform in $n$.
The first statement implies that the last two are true without
the conditioning on
$[(-n,0) \lra (n,0)]$. It follows that when $\epsilon$ is
sufficiently small, the
correlation between the events $[b_n^+ \text{ is closed}]$ and $[b_n^- \text{ is
closed}]$ does not decay to 0 as $n \to \infty$.
The boundary condition $\eta$ is equivalent to a bond boundary
condition in which
an open path connects $(-n-1,0)$ to $(n+1,0)$ outside $\omB(Q_n(0))$.
Thus strong mixing fails for bond and
for single-species-site boundary conditions, although exponential decay of dual
connectivity and weak mixing both hold in infinite volume, and uniform
exponential decay of finite-volume dual connectivities holds, by Theorem
\ref{T:FKcase}.
This same phenomenon persists in the presence of external fields, provided there
is more than one stable species. In Theorem \ref{T:siteratiosm}
we will see that
when there is a unique stable species, we do obtain strong mixing.
Replacing regular bonds with dual bonds throughout this example (in
the case of no
external fields), including in the bond form of the boundary
condition, we obtain
an example in which there is exponential decay of
connectivity and weak mixing in
infinite volume, and uniform exponential decay of finite-volume connectivities,
but strong mixing for general bond boundary conditions fails.
This is an example
of the tunneling phenomenon of Remark \ref{R:tunnel}.
\end{example}
Given a metric $d$ we call a class $\mkC$ of subsets of $\mB(\ZZn)$
\emph{inheriting} with respect to $d$ if for all $\mB \in
\mkC, x \in V(\mB)$ and $r > 0$, the connected component of
$x$ in $\{ e \in \mB:
d(e,x) \leq r \}$ is in $\mkC$. The class of all simply lattice-connected
subsets of $\mB(\ZZ)$ is clearly inheriting, with respect to the Euclidean and
restricted-path metrics.
We define a \emph{closure subset} of $\mB(\ZZn)$ to be a subset of form
$\omB(\Lambda)$ for some finite $\Lambda$.
These are the subsets which can take site boundary conditions. If $\mE =
\omB(\Lambda)$ is a simply lattice-connected closure subset of $\mB(\ZZn)$, then
$\mE_{x,r} = \mE \cap\{ e \in
\mB(\ZZn): d(e,x) \leq r \}$ is simply lattice-connected but is not typically a
closure subset, so strictly speaking the class of all simply lattice-connected
subsets of $\mB(\ZZn)$ is not inheriting. However we can add a few bonds to
$\mE_{x,r}$---specifically, those in $\partial \mE_{x,r} \bs \partial
\mB(\Lambda)$---to create a closure subset. Classes which are ``almost''
inheriting in this sense work perfectly well in our proofs, so we tacitly treat
them as if they were inheriting in what follows.
The next result, for the FK model in general dimension, says in effect that when
paths cannot propagate in any of a class of finite regions, then neither can
influence (of one event on another distant one) so propagate,
under site boundary
conditions, even when this influence is measured multiplicatively. This result
is not restricted to 2 dimensions. The underlying idea for establishing mixing
from uniform exponential decay occurs in \cite{AC}
and \cite{Ne}, though without
the ``ratio'' aspect, which appears in \cite{Al98mix}.
\begin{theorem} \label{T:siteratiosm}
Let $P = P^{p,q,\{h_{i}\}}$ be the FK model at $(p,q,\{h_{i}\})$ on
$\mB(\ZZn)$ and suppose random-cluster uniqueness holds. Let $d$ be either the
Euclidean or restricted-path metric, and let $\mkC$ be a class
of closure subsets of $\mB(\ZZn)$ which is inheriting with respect to $d$.
Suppose $P$ has uniform exponential decay of
finite-volume connectivities for the class
$\mkC$ with wired boundary conditions, for the metric
$d$. Then $P$ has the ratio strong mixing property for the
class $\mkC$ and arbitrary site boundary conditions, for the metric $d$.
\end{theorem}
For dimension $n=2$, as in Theorem \ref{T:FKcase} we need not assume uniformity
of the exponential decay to draw a similar conclusion. Specifically, we will
prove the following.
\begin{theorem} \label{T:siteratiosm2}
Let $P = P^{p,q,\{h_{i}\}}$ be the FK model at $(p,q,\{h_{i}\})$ on
$\ZZ$, and let $\mkC$ be the class of all
simply lattice-connected closure subsets of $\mB(\ZZ)$. Suppose random-cluster
uniqueness holds at $(p,q,\{h_{i}\})$. Let $d$ be either the Euclidean or the
restricted-path metric.
(i) Suppose
$P$ has exponential decay of connectivities. Then
$P$ has the ratio strong mixing property for the
class $\mkC$ with arbitrary site
boundary conditions.
(ii) Suppose $P$ has
exponential decay of dual connectivities. Then $P$ has the ratio strong mixing
property for the class $\mkC$ with free and wired boundary conditions.
(iii) Suppose there is a unique stable species and $P$ has
exponential decay of dual connectivities. Then $P$ has the ratio strong mixing
property for the class $\mkC$ with arbitrary site boundary
conditions.
\end{theorem}
From Theorem
\ref{T:siteratiosm2}(i) we see that for the FK model on $\mB(\ZZ)$, exponential
decay of connectivity in infinite volume is enough to ensure that in finite
regions, influence cannot propagate along the boundary or
through the bulk, under
site boundary conditions.
Example
\ref{E:nodecay} shows that significantly more general boundary conditions cannot
be allowed in Theorem \ref{T:siteratiosm2}(ii), in contrast to Theorem
\ref{T:siteratiosm2}(iii).
From \cite{BBCK}, the assumption of random-cluster uniqueness in Theorem
\ref{T:siteratiosm2} is satisfied except possibly at the (unique) percolation
critical point, which we denote $p_c(q,n,\{h_{i}\})$.
We turn our attention now to spin systems. We restrict
attention mainly to the $q$-\emph{state Potts model} (with possible
external fields), which is the spin system with single-spin space $S =
\{1,..,q\}$ and Hamiltonian
\[
H_{\Lambda,\eta}(\sigma_{\Lambda}) = -\sum_{\langle xy \rangle \in
\omB(\Lambda)} \delta_{[(\sigma\eta)_{\Lambda}(x)
= (\sigma\eta)_{\Lambda}(y)]} - \sum_{i = 1}^{q} \sum_{x \in \Lambda}
h_{i} \delta_{[\sigma_{x} = i]}
\]
for the model on $\Lambda$ with boundary condition $\eta$; the corresponding
finite-volume Gibbs distribution at inverse temperature $\beta$ is given by
\[
\mu_{\Lambda,\eta}^{\beta,\{h_i\}}(\sigma_{\Lambda}) =
\frac{1}{Z_{\Lambda,\eta}^{\beta,\{h_i\}}} e^{-\beta
H_{\Lambda,\eta}(\sigma_{\Lambda})}, \quad \sigma_{\Lambda} \in S^{\Lambda}
\]
where $Z_{\Lambda,\eta}^{\beta,\{h_i\}}$ is the partition function.
We denote the critical inverse temperature of
the model on $\ZZn$, without external fields, by $\beta_{c}(q,d)$.
As shown by Edwards and Sokal in \cite{ES}, for $\beta$ given by $p = 1 -
e^{-\beta}$, a configuration of the $q$-state Potts model (without external
fields) on
$\Lambda$ with boundary condition $\eta$ at inverse temperature $\beta$ can be
obtained from a configuration
$\omega$ of the FK model at $(p,q)$ with site boundary condition $\eta$, by
choosing a label for each non-boundary cluster of
$\omega$ independently and uniformly from $\{1,...,q\}$. For a cluster
intersecting $\{ x \in \pL: \sigma_{x} = i \}$ (which can happen for at most one
$i$), all sites are labeled $i$. This
\emph{cluster-labeling construction} yields a joint site-bond configuration,
which we call an \emph{Edwards-Sokal configuration}, in
which the sites are a Potts model and the bonds are an FK model. When
the parameters are related in this way, we call the Potts and FK models
\emph{corresponding}. Alternatively, if one selects a Potts configuration
$\sigma_{\Lambda}$ and does independent percolation at density $p$ on the set of
bonds
\[
\{\langle xy \rangle \in \omB(\Lambda): (\sigma\eta)_{x} =
(\sigma\eta)_{y} \},
\]
the resulting bond configuration is a realization of the corresponding FK model.
We call this the \emph{percolation construction} of the FK model.
For the $q$-state Potts model (without external
fields) at inverse temperature $\beta < \beta_{c}(q,n)$ ,
for $p = 1 - e^{-\beta}$ and the FK model at $(p,q)$, the covariance in
the Potts model and the connectivity in the FK model are related by
\begin{equation} \label{E:covarconn}
q^{2} \cov(\delta_{[\sigma_{0} = i]},\delta_{[\sigma_{x} = i]})
= (q - 1)P(0 \lra x), \quad i = 1,..,q;
\end{equation}
see \cite{ACCN} or \cite{Gr95}. Thus exponential decay of connectivities
in the FK model is equivalent to exponential decay of correlations in
the corresponding Potts model. Further, we have
\begin{equation} \label{E:critpts}
p_{c}(q,d) = 1 - e^{-\beta_{c}(q,n)};
\end{equation}
again see \cite{ACCN} or \cite{Gr95}. When external fields are present, we can
take (\ref{E:critpts}) as the definition of an inverse temperature
$\beta_c(q,d,\{ h_i \})$, that is,
\begin{equation} \label{E:critpts2}
p_c(q,n,\{ h_i \}) = 1 - e^{-\beta_{c}(q,n,\{ h_i \})}.
\end{equation}
This $\beta_{c}(q,n,\{ h_i \})$ is not necessarily a true critical point,
however, if there is a unique stable species.
Under external fields $\{h_{i}\}$, the cluster-labeling construction is modified
as follows. Let $s(C)$ denote the number of sites in a cluster $C$. For
non-boundary clusters $C$, the spin $i$ is chosen for the cluster
$C$ with probability proportional to $e^{\beta h_{i} s(C)}$.
The (ratio) weak mixing and (ratio) strong mixing properties extend
straightforwardly to spin systems. We do not write out a separate definition
explicitly; see \cite{AlARC}.
The following is the Potts-model analog of Theorem \ref{T:siteratiosm2}.
\begin{theorem} \label{T:spinmain}
Let $\mu^{\beta,q,\{ h_i \}}$ be the $q$-state Potts model at $(\beta,\{ h_i
\})$ on $\ZZ$. Let $\mkC$ be the class of all
finite simply lattice-connected subsets of $\ZZ$ with arbitrary boundary
condition, and let $d$ be either the Euclidean or the restricted-path metric.
(i) If there are multiple stable species, $\beta < \beta_c(q,2,\{ h_i \})$ and
$\mu^{\beta,q,\{ h_i \}}$ has exponential decay of correlations, then it has
the ratio strong mixing property for the class $\mkC$.
(ii) If $\beta < \beta_c(q,2,\{ h_i \})$ and
the corresponding FK model has exponential decay of connectivities, then
$\mu^{\beta,q,\{ h_i \}}$ has the ratio strong mixing property for the
class $\mkC$.
(iii) If there is a unique stable species, $\beta > \beta_c(q,2,\{ h_i \})$ and
the corresponding FK model has exponential decay of dual connectivities, then
$\mu^{\beta,q,\{ h_i \}}$ has the ratio strong mixing property for the
class $\mkC$.
\end{theorem}
Note that in Theorem \ref{T:spinmain} we are implicitly viewing $\mkC$ as a
class of pairs consisting of a set and a boundary condition on that set, rather
than just as a class of sets. We will use each meaning of $\mkC$ at various
times; which one we are using should be clear from the context.
From the relation (\ref{E:critpts}) and the remarks preceding Theorem
\ref{T:FKcase}, we see that the hypothesis in Theorem \ref{T:spinmain}(i) of
exponential decay of correlations is known to be satisfied whenever $e^{\beta} <
e^{\beta_{c}} = 1 + \sqrt{q}$ if $q = 2$ or $q
\geq 26$, and whenever $e^{\beta} < 1 + \sqrt{q-1}$ if $3 \leq q \leq 25$, in
the absense of external fields.
The Potts-model analog of Theorem \ref{T:siteratiosm} is contained
in the following theorem.
\begin{theorem} \label{T:spinZdconn}
Consider the $q$-state Potts model $\mu^{\beta,q,\{ h_{i} \}}$ on $\ZZn$ at
inverse temperature $\beta$ with external fields $h_{i}$. Let $d$ be either
the Euclidean or the restricted path metric and let $\mkD$ be a class of subsets
of $\ZZn$ such that the class $\mkC = \{
\omB(\Lambda): \Lambda \in \mkD \}$ is inheriting with respect to $d$. Suppose
that the corresponding FK model has uniform exponential decay of finite-volume
connectivities for the class
$\mkC$ and wired boundary condition, for the metric $d$. Then
$\mu^{\beta,q,\{ h_{i} \}}$ has the ratio strong mixing property for the class
$\mkD$ and arbitrary boundary conditions, for the same metric.
\end{theorem}
\begin{remark} \label{R:nonPotts}
The notion of a ``corresponding random cluster model'' makes sense
in the context
of any random cluster representation of any spin system. The analog of Theorem
\ref{T:spinZdconn} will be valid for a nearest-neighbor spin system provided the
following conditions are satisfied. (1) There exists a bond percolation model
$P$ on
$\ZZn$ such that for every finite $\Lambda$, $P_{\omB(\Lambda),w}$ FKG-dominates
the random cluster representation on $\omB(\Lambda)$
with arbitrary spin boundary
condition, and $P$ has uniform exponential decay of finite-volume
connectivities. (In Theorem \ref{T:spinZdconn}, $P$ is the FK model itself.)
(2) The joint site-bond configuration obtained in the cluster-labeling
construction has the following Markov property for
open dual surfaces: for every
$\Delta \subset \Lambda$, every boundary condition $\eta$ on
$\partial \Lambda$, and every configuration $\sigma_{\Delta}$ on $\Delta$, given
$\sigma_{\Delta}$ and given that all bonds in $\omB(\Lambda) \cap \partial
\mB(\Delta)$ are closed, the configuration $\sigma_{\Lambda \bs \Delta}$ has
distribution corresponding to the boundary condition on $\partial (\Lambda
\bs \Delta)$ which is given by
$\eta_{x}$ for $x \in \partial (\Lambda \bs \Delta) \cap \pL$, free for $x \in
\partial (\Lambda \bs \Delta) \cap \Lambda$. Condition (2) is valid provided
that there is no interaction between clusters in the random cluster
representation, that is, the weight assigned to a
bond configuration is a product
of weights assigned to clusters. This is true for the FK model (with or without
external fields) and for the random cluster representations of the models given
studied in \cite{AC}, which are given as follows: the
single-spin space $S$ is a
compact group, $\mE: S \to
\mathbb{R}$ is a function, and the Hamiltonian is
\begin{equation} \label{E:groupHam}
H_{\Lambda,\eta}(\sigma_{\Lambda}) = -\half \sum_{\langle xy \rangle \in
\omB(\Lambda)} \mE(\sigma_{x}^{-1} \sigma_{y}).
\end{equation}
This includes the Potts model without external fields.
\end{remark}
\section{Proofs} \label{S:proofs}
Throughout the paper $c_{1}, c_{2},...$ and $\epsilon_{1}, \epsilon_{2},...$
represent constants which depend only on the infinite-volume model under
discussion, with $\epsilon_{i}$ used for ``sufficiently small'' constants.
$d_{p}$ denotes the $l^{p}$ metric.
We begin with a result needed for the proof of Theorem \ref{T:percmain}. For
$\mB \subset \mB(\ZZn)$ and $r > 0$, let
\[
\mB_{r} = \{ e \in \mB: d(e,\mB(\ZZn \bs \mB)) \geq r \}.
\]
By a \emph{chain of l-cubes} (from $x_{1}$ to $x_{k}$) we
mean a finite sequence
$Q_{l}(x_{1}),..,Q_{l}(x_{k})$ of cubes with disjoint interiors,
with $x_{i} \in l\ZZn$ for all $i \leq k$ and
$d_{\infty}(x_{i},x_{i+1}) = 2l$ for all $i \leq k - 1$.
\begin{proposition} \label{P:avoidbdry}
Let $P$ be a translation-invariant bond percolation model on $\ZZn$ having
exponential decay of connectivities and the weak mixing property. Then for
each sufficiently large $r$ there exist $C_{r}, \lambda_{r} > 0$ such that for
all $\mB \subset \mB(\ZZn)$, all $x, y \in
V(\mB_{r})$, and every boundary condition $\rho \in \{ 0,1\}^{\mB_{r}^{c}}$,
\[
P_{\mB,\rho}(x \lra y \text{ via a path in } \mB_{r} ) \leq
C_{r}e^{-\lambda_{r}d_{2}(x,y)}.
\]
\end{proposition}
\begin{proof}
From the weak mixing property, there
exist $c_{1}, c_{2}, \epsilon_{1}, \epsilon_{2} > 0$ such
that if $z \in \ZZn$ and $r
\in \mathbb{Z}$ is sufficiently large, then for every boundary
condition $\theta$ on $\bigl( \mB \cap \mB(Q_{2r}(z)) \bigr)^{c}$,
\begin{align} \label{E:badcube}
P_{\mB \cap \mB(Q_{2r}(z)),\theta} &\bigl( \text{there is an open path in }
\mB_{r} \cap \mB(Q_{r}(z)) \text{ from } u \text{ to } v \\
&\qquad \qquad \text{ for some } u, v \text{ with } d_{2}(u,v) \geq r/2 \bigr)
\notag \\
&\leq P \bigl( \text{there is an open path in }
\mB_{r} \cap \mB(Q_{r}(z)) \text{ from } u \text{ to } v \notag \\
&\qquad \qquad \text{ for some } u, v \text{ with } d_{2}(u,v) \geq r/2 \bigr)
+ c_{1}e^{-\epsilon_1 r} \notag \\
&\leq c_{2} e^{-\epsilon_{2}r}. \notag
\end{align}
For fixed $r$ we call a cube $Q_{2r}(z)$ \emph{bad} if the event on the
left side of (\ref{E:badcube}) occurs, and \emph{good} otherwise. We define a
site percolation process on $\ZZn$ by setting $\xi_{q} = 1$ if
$Q_{2r}(2rq)$ is bad, and $\xi_{q} = 0$ otherwise. By (\ref{E:badcube}), for
every set of cubes $Q_{2r}(2rq_{1}),..,Q_{2r}(2rq_{k})$ having
disjoint interiors,
with
$q_{i} \in \ZZn$, the configuration $\{ \xi_{q_{i}}: i \leq k \}$ is
FKG-dominated by Bernoulli site percolation at density $c_{2}
e^{-\epsilon_{2}r}$.
In particular, the probability for all $2r$-cubes in a given chain of length $k$
to be bad is at most $( c_{2} e^{-\epsilon_{2}r} )^{k}$.
If $d_{2}(x,y) \geq r/2$ and $x \lra y$ via a path in $\mB_{r}$,
it is easy to see that there exist $z, w \in 2r\ZZn$ such that $x
\in Q_{2r}(w), y \in Q_{2r}(z)$, and there is a
chain of bad $2r$-cubes from $z$
to $w$. The number of chains of $2r$-cubes from $z$ to $w$ of length $k$ is at
most $5^{dk}$, so the proposition now follows from a routine Peierls-type
argument.
\end{proof}
The next lemma establishes a weaker analog of Theorem \ref{T:percmain}, which we
will later bootstrap using renormalization.
\begin{lemma} \label{L:almostexp}
Let $P$ be a translation-invariant bond percolation model on $\mB(\ZZ)$
having bounded energy, exponential decay of connectivities and the weak mixing
property. Let $\mkC$ be the class of all finite simply lattice-connected
subsets of $\mB(\ZZ)$. There exist constants
$c_{i}, \epsilon_{3}$ such that for
all $\mB \in
\mkC$, all boundary conditions $\rho$ and all $x,y \in V(\mB)$ with
$d_{2}(x,y) \geq c_{3}$,
\[
P_{\mB,\rho}(x \lra y) \leq c_{4}
e^{-\epsilon_{3}d_{2}(x,y) / \log d_{2}(x,y)}.
\]
\end{lemma}
\begin{proof}
Let $r, C_{r}, \lambda_{r}$ be as in Proposition \ref{P:avoidbdry}, and fix $x,
y, \mB$ with $d_{2}(x,y) \geq 30r$ and $x, y \in V(\mB)$. Let $\gamma$ be the
lattice dual circuit which is the outer boundary of $Q(V(\mB))$. Let $x^{\pr},
y^{\pr}$ be the dual sites in $\gamma$ closest to $x$ and $y$, respectively
(breaking ties arbitrarily), and let $\gamma_{R}$ and $\gamma_{L}$ denote the
segments of $\gamma$ from $x^{\pr}$ to $y^{\pr}$ (counterclockwise) and from
$y^{\pr}$ to $x^{\pr}$, respectively. Let $\epsilon > 0$.
\emph{Case 1}: Either $d_{2}(x,\gamma) \geq \frac{1}{10} d_{2}(x,y)$ or
$d_{2}(y,\gamma) \geq \frac{1}{10} d_{2}(x,y)$; we may assume the former. Then
by Proposition \ref{P:avoidbdry}, if
$d_{2}(x,y)$ is sufficiently large (depending on $r$), then for every boundary
condition $\rho$,
\begin{align} \label{E:bdryfar}
P_{\mB,\rho}&(x \lra y) \\
&\leq P_{\mB,\rho} \bigl( x \lra z \text{ via a path in } \mB_{r} \text{ for
some } z \text{ with } d_{2}(x,z) \geq \frac{1}{20}d_{2}(x,y)
\bigr) \notag \\
&\leq c_{5}d_{2}(x,y)C_{r} e^{-\lambda_{r}d_{2}(x,y)/20}. \notag
\end{align}
\emph{Case 2}:
\[
\max\bigl( d_{2}(x,\gamma),d_{2}(y,\gamma) \bigr) < \frac{1}{10}d_{2}(x,y).
\]
Let $\mS$ denote the set of bonds in $\mB^{c}$ which are surrounded by
$\gamma$. Since $\mB$ is simply lattice-connected, every dual bond in $\mS^{*}$
must be connected to exactly one of $\gamma_{L}$,
$\gamma_{R}$, by a path in $\mS^{*}$. This partitions $\mS^{*}$
into two subsets, call them $\mS_{L}^{*}$ and $\mS_{R}^{*}$, and we define
$\tilde{\gamma}_{L} = \gamma_{L} \cup \mS^{*}_{L}, \tilde{\gamma}_{R} =
\gamma_{R} \cup \mS^{*}_{R}$.
For $k \geq 1$ let
\[
\Omega_{k} = \{ z \in V(\mB): (k-1)c_{6} \log d_{2}(x,y) < d_{2}(z,x^{\pr})
\leq k c_{6} \log d_{2}(x,y) \},
\]
where $c_{6} = c_{6}(r)$ is a (large) constant to be specified, and let
\[
\mD_{k} = \{ \langle uv \rangle \in \mB: u \in \Omega_{k}, v \in \Omega_{k+1}
\}.
\]
We may assume $x,y$ satisfy
\[
\log d_{2}(x,y) > \frac{4r}{c_{6}}.
\]
Let $k_{\min} - 1$ and
$k_{\max}$ be the smallest and largest integers $k$, respectively, for which
$d_{2}(x^{\pr},x) < k c_{6} \log d_{2}(x,y) < d_{2}(x^{\pr},y)$. Then
\begin{equation} \label{E:kminmax}
k_{\max} - k_{\min} \geq \frac{d_{2}(x,y)}{2c_{6} \log d_{2}(x,y)}.
\end{equation}
For each $k_{\min} \leq k
\leq k_{\max}$, every path from $x$ to $y$ must pass through $\mB(\Omega_{k})$,
and $\omB(\Omega_{k})^{*}$ must include bonds of both $\tilde{\gamma_{L}}$ and
$\tilde{\gamma_{R}}$. Among the connected components of $\mD_{k}^{*}$ there is
at least one which has one endpoint in $\tilde{\gamma_{L}}$ and the other in
$\tilde{\gamma_{R}}$; let us call such a component $k$-\emph{crossing}. Among
the $k$-crossing components there is one,
which we denote $\mE_{k}^{*}$, with the
property that for every path $\varphi$ from
$x$ to $y$, $\mE_{k}^{*}$ is the last $k$-crossing crossed by $\varphi$.
Now $\mE_{k-1}^{*}$ and $\mE_{k}^{*}$ divide $\mB$ into 3
pieces, and we call the middle one $\mC_{k}$; more precisely, $\mC_{k}$ is the
connected component of $\mB \bs (\mE_{k-1} \cup \mE_{k})$ for which both
$\mE_{k-1} \subset \partial \mC_{k}$ and $\mE_{k} \subset \partial \mC_{k}$.
For $t > 0$ let
\[
\Xi_{k}^{L,t} = \{ z \in V(\mC_{k}): d_{1}(z,\tilde{\gamma_{L}}) \leq t \},
\quad
\Xi_{k}^{R,t} = \{ z \in V(\mC_{k}): d_{1}(z,\tilde{\gamma_{R}}) \leq t \},
\]
\[
\Delta_{k}^{+} = V(\mC_{k}) \cap V(\mE_{k}), \quad
\Delta_{k}^{-} = V(\mC_{k}) \cap V(\mE_{k-1}),
\]
and
\[
\hat{\mC}_{k} = \mC_{k-1} \cup \mD_{k-1} \cup \mC_{k} \cup \mD_{k} \cup
\mC_{k+1}.
\]
\emph{Case 2a}. $\Xi_{k}^{L,3r} \cap \Xi_{k}^{R,3r} \neq \phi$. Then there is a
dual path of length at most $6r$ from $\tilde{\gamma}_{L}$ to
$\tilde{\gamma}_{R}$ in
$\hat{\mC}_{k}^{*}$. If this dual path is open, then there can be no open path
in $\hat{\mC_{k}}$ from
$\Delta_{k-1}^{-}$ to $\Delta_{k+1}^{+}$. For $p_{0}$ as in
(\ref{E:boundener}) we therefore have for all boundary conditions $\rho$ and
all bond configurations $\theta_{\mB \bs \hat{\mC}_{k}}$:
\begin{equation} \label{E:closesides}
P_{\mB,\rho} \bigr( \Delta_{k-1}^{-} \lra \Delta_{k+1}^{+}
\text{ via a path in }
\hat{\mC_{k}} \mid \omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr) \leq 1 - p_{0}^{6r}.
\end{equation}
\emph{Case 2b}.
$\Xi_{k}^{L,3r} \cap \Xi_{k}^{R,3r} = \phi$. Let $\mA_k = \mC_{k}
\bs \omB(\Xi_{k}^{L,2r} \cup
\Xi_{k}^{R,2r})$. Then there exist paths from $x$ to $y$ in $\mB$ outside
$\omB(\Xi_{k}^{L,2r} \cup
\Xi_{k}^{R,2r})$, and there is a unique connected component $\mF_{k}^{*}$ of
$\mE_{k-1}^* \cap \partial Q(V(\mA_k))$ such that all such paths cross
$\mF_{k}^{*}$. Necessarily $\mF_{k}^{*}$ has one endpoint in $\partial
Q(\Xi_{k}^{L})$ and the other in $\partial Q(\Xi_{k}^{R})$.
From duality, there are two
possibilities: either there is an open path from $\Delta_{k}^{-}$ to
$\Delta_{k}^{+}$ in $\mA_k$, or there is an open dual path from $\partial
Q(\Xi_{k}^{L})$ to $\partial Q(\Xi_{k}^{R})$ in $\mA_{k}^{*}$. Since
$\mA_{k} \subset \mB_{r}$, it follows from Proposition \ref{P:avoidbdry} that,
provided $c_{6}$ is large, for all boundary conditions $\rho$ and
all bond configurations $\theta_{\mB \bs \hat{\mC}_{k}}$,
\begin{align} \label{E:middlepath}
P_{\mB,\rho} &\bigl( \partial Q(\Xi_{k}^{L,2r}) \lrad \partial
Q(\Xi_{k}^{R,2r}) \text{ via a dual path in } \mA_{k}^{*} \mid
\omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr) \\
&\geq 1 - P_{\mB,\rho} \bigl( \Delta_k^- \lra \Delta_k^+
\text{ via a path in }
\mA_k \mid \omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr) \notag \\
&\geq 1 - |\Delta_{k}^{-}| |\Delta_{k}^{+}| C_{r} e^{-\lambda_{r} c_{6} (\log
d_{2}(x,y))/2} \notag \\
&\geq 1 - (c_{4} d_{2}(x,y))^{2} C_{r} e^{-\lambda_{r} c_{6} (\log
d_{2}(x,y))/2} \notag \\
&\geq \half. \notag
\end{align}
When there is an open dual path in $\mA_{k}^{*}$ from $\partial Q(\Xi_{k}^{L})$
to $\partial Q(\Xi_{k}^{R})$, there is a unique such path, which we denote
$\alpha_{k}$, which is ``closest to $\mF_{k}^{*}$'' (analogous to the ``lowest
occupied crossing'' of e.g. \cite{Ke}.) Let
$Y_{k}^{L}$ and $Y_{k}^{R}$ denote the (random) initial and final sites,
respectively, of $\alpha_{k}$.
For every dual site $u$ in $\partial Q(\Xi_{k}^{L,2r}) \cap V(\mA_k^*)$ there is
a dual path of length at most $2r + 2$ from $u$ to $\tilde{\gamma}_{L}$ outside
$\mA_{k}^{*}$; we denote this path by
$\beta_{u}^{L}$, making an arbitrary choice of $\beta_{u}^{L}$
if more than one is
possible. We define $\beta_{u}^{R}$ analogously for $u$ in $\partial
Q(\Xi_{k}^{R}) \cap V(\mA_k^*)$. Conditionally on the connection event on the
left side of (\ref{E:middlepath}), the probability that $\beta_{Y_{k}^{L}}^{L}$
and
$\beta_{Y_{k}^{R}}^{R}$ are both open dual paths is at least $p_{0}^{4r+4}$.
Further, when these two dual paths are both open, there is an open dual path in
$\hat{\mC}_{k}^*$ from $\tilde{\gamma}_{L}$ to $\tilde{\gamma}_{R}$. From this
and (\ref{E:middlepath}) we conclude that for all boundary conditions $\rho$ and
all bond configurations $\theta_{\mB \bs \hat{\mC}_{k}}$,
\begin{align}
P_{\mB,\rho} &\bigl( \tilde{\gamma}_{L} \lrad \tilde{\gamma}_{R}
\text{ via a dual path in } \hat{\mC}_{k}^* \mid
\omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}}\bigr) \notag \\
&\geq p_{0}^{4r+4} P_{\mB,\rho} \bigl( \partial Q(\Xi_{k}^{L})
\lrad \partial Q(\Xi_{k}^{R})
\text{ via a dual path in } \mA_{k}^{*} \mid
\omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr) \notag \\
&\geq \half p_{0}^{4r+4}. \notag
\end{align}
Using duality again, this shows that, again for all boundary
conditions $\rho$ and
all bond configurations $\theta_{\mB \bs \hat{\mC}_{k}}$,
\begin{equation} \label{E:Gjbound}
P_{\mB,\rho} \bigl( \Delta_{k}^{-} \lra \Delta_{k}^{+} \text{ via a path in }
\hat{\mC}_{k} \mid \omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr)
\leq 1 - \half p_{0}^{4r+4}.
\end{equation}
We have shown that provided $r$ is large, in both Cases 2a and 2b,
\begin{equation} \label{E:ringbound}
P_{\mB,\rho} \bigr( \Delta_{k-1}^{-} \lra \Delta_{k+1}^{+}
\text{ via a path in }
\hat{\mC_{k}} \mid \omega_{\mB \bs \hat{\mC_{k}}} = \theta_{\mB \bs
\hat{\mC_{k}}} \bigr) \leq 1 - p_{0}^{6r}.
\end{equation}
Let $G_{j}$ denote the event $[\Delta_{3j-1}^{-} \lra \Delta_{3j+1}^{+}
\text{ via a path in } \hat{\mC}_{3j}]$ and let $j_{\min}$ and $j_{\max}$ be the
smallest and largest integers, respectively, in $\{ j \in \mathbb{Z}: k_{\min}
< 3j < k_{\max} \}$. From (\ref{E:ringbound}) we have
\[
P_{\mB,\rho} (G_{j} \mid G_{j_{\min}} \cap .. \cap G_{j-1}) \leq
1 - \half p_{0}^{4r+4} \quad \text{for all } j_{\min} \leq j \leq j_{\max},
\]
and hence
\[
P_{\mB,\rho} (x \lra y) \leq P_{\mB,\rho} (\cap_{j =
j_{\min}}^{j_{\max}} G_{j})
\leq (1 - \half p_{0}^{4r+4})^{j_{\max} - j_{\min} + 1}.
\]
From (\ref{E:kminmax}), provided $d_{2}(x,y)$ is sufficiently large,
\[
j_{\max} - j_{\min} + 1 \geq \frac{1}{3}\left( k_{\max} - k_{\min} \right) - 2
\geq \frac{d_{2}(x,y)}{7c_{6}\log d_{2}(x,y)},
\]
and the lemma follows from this and (\ref{E:bdryfar}).
\end{proof}
We call a subset $\Lambda$ of $\ZZn$ $l^{\infty}$-\emph{connected}
if for all $x,
y \in \Lambda$ there is a sequence of sites $x = x_{1}, \ldots, x_{n} = y$ in
$\Lambda$ with $d_{\infty}(x_{i},x_{i+1}) \leq 1$ for all $i$.
\begin{proof}[Proof of Theorem \ref{T:percmain}]
From Lemma \ref{L:almostexp} we see that given $\epsilon > 0$ there exists $l$
such that for all $\mB \in \mkC$, all boundary conditions $\rho$, and all
$z \in \ZZ$,
\begin{equation} \label{E:badprob}
P_{\mB \cap \mB(Q_{l}(z) \cap \ZZ),\rho} \bigl( u \lra v
\text{ for some } u,v \in Q_{l}(z) \text{ with } d_{2}(u,v) \geq
\frac{l}{4} \bigr) < \epsilon.
\end{equation}
Fix $\mB \in \mkC$.
For $z \in l\ZZ$ we call $Q_{l}(z)$ \emph{bad} if the event on the left
side of (\ref{E:badprob}) occurs, and \emph{good} otherwise. As in the proof of
Proposition \ref{P:avoidbdry}, for disjoint cubes the configuration of bad cubes
is FKG-dominated by Bernoulli site percolation at density $\epsilon$. Further,
if $x \lra y$ for some $x,y$ with $d_{2}(x,y) \geq 8l$, then there is an
chain of bad $l$-cubes from $w$ to $z$, of length at
least $d_{2}(x,y)/4l$, where
$w,z$ are sites in $l\ZZ$ with $x \in Q_{l}(w), y \in Q_{l}(z)$. Provided
$\epsilon$ is sufficiently small, a routine Peierls
argument therefore gives that
for all boundary conditions
$\theta$ and all $x,y$ with $d_{2}(x,y) \geq 8l$,
\[
P_{\mB,\theta}(x \lra y) \leq c_{7} e^{-d_{2}(x,y)/4l}.
\]
This proves the theorem for the metric $d_{2}$.
Next we consider $d_{\mB}$. Suppose $x \lra y$ for some $x,y$ with
$d_{\mB}(x,y) \geq 15l^{2}$. Fix $z \in l\ZZ$ with $x \in
Q_{l/2}(z)$ and let
$A$ be the connected component of $Q_{l}(z)$ in $\cup \{ Q_{l}(w): w \in l\ZZ,
Q_{l}(w) \text{ is bad} \}$. Let
\[
\Theta = \{ w \in \ZZ: Q_{l}(lw) \subset A, Q_{l}(lw) \text{ is bad} \}.
\]
Note that since any one bond is contained in at most 4 sets $Q_{l}(lw)$ with $w
\in \ZZ$, there must be a subset $f(\Theta) \subset \Theta$ with
$|f(\Theta)|
\geq |\Theta|/4$ and $\{ Q_{l}(lw), w \in f(\Theta) \}$ mutually disjoint.
Let $C_{x}$ be the open cluster of $x$ and $A^{t} = \{ u \in \RR:
d_{2}(u,A) \leq t \}$; then
\[
C_{x} \subset A^{l/4}, \qquad |\{ \text{bonds in } C_{x} \}| \
geq d_{\mB}(x,y).
\]
There are at most $9l^{2}$ bonds in a cube $Q_{l}(w)$, and hence at most
$9|\Theta|l^{2}$ in $\mB(A)$ and at most $15|\Theta|l^{2}$ in $\mB(A^{l/4})$.
Therefore $|\Theta| \geq d_{\mB}(x,y)/15l^{2}$. Now $\Theta$ must be an
$l^{\infty}$-connected subset of $\ZZ$ with $z \in \Theta$. The number of
possible such lattice animals
$\Theta$ with $|\Theta| = n$ is at most $c_{8}^{n}$ for all $n$, by
the argument of (\cite{Ke2}, p. 85.) Thus provided $\epsilon$ is sufficiently
small we have, again by a routine Peierls-type argument,
\begin{equation}
P_{\mB,\rho} ( x \lra y )
\leq P_{\mB,\rho} \left( |\Theta| \geq
\frac{d_{\mB}(x,y)}{15l^{2}} \right)
\leq P_{\mB,\rho} \left( |f(\Theta)| \geq
\frac{d_{\mB}(x,y)}{60l^{2}} \right)
\leq c_{9} e^{-\epsilon_{4} d_{\mB}(x,y)/60l^{2}}, \notag
\end{equation}
which proves the theorem for the metric $d_{\mB}$.
\end{proof}
Let $B_{\tau}(x,r)$ denote the $\tau$-ball of radius $r$ centered at $x$.
Consider distinct points $x,y \in \mathbb{R}^{2}$ with $|y - x| \geq
4\sqrt{2}$. We let
$S(x,y)$ denote the closed slab between the tangent line to
$\partial B_{\tau}(x,\tau(y-x))$ at $y$ and the parallel line through $x$; we
call $S(x,y)$ the \emph{natural slab} of $x$ and $y$. (Note the tangent
line is not necessarily unique; if it is not we make some arbitrary choice.)
Due to the 8-fold symmetry of the lattice, $S(x,y)$ makes an angle of
not more than $45^{\circ}$ with $y-x$. Note that if $u,v$ are on opposite sides
of $S(x,y)$, then
\begin{equation} \label{E:natslab}
\tau(v-u) \geq \tau(y-x).
\end{equation}
\begin{proof}[Proof of Corollary \ref{C:twopath}]
Fix $x$ and let $R$ be the closed parallelogram of which two of the sides are
segments of $\partial S(0,x)$ of length $8|x|$ centered at 0 and $x$
respectively. Note the other two sides are parallel to $x$.
We denote the sides containing 0 and $x$ by $l_0$
and $l_x$, respectively. We view $l_0$ as the left side of the parallelogram,
which specifies which short side is the bottom. Let $\Gamma_R$ denote the
lowest open crossing of $R$ from $l_0$ to $l_x$, when an open crossing exists.
Given a path $\gamma$ from $l_0$ to $l_x$, we let $U_{\gamma}$ denote the
portion of
$R$ which is strictly above $\gamma$, and let $V_{\gamma,0}$ (respectively
$V_{\gamma,x}$) denote the set of sites in $V(\mB(U_{\gamma}))$ which are
endpoints of bonds intersecting
$l_0$ (respectively $l_x$.) By Theorem \ref{T:FKcase} there exists $\epsilon
> 0$ such that for each such $\gamma$ we have
\[
P_{\mB(U_{\gamma}),w}(V_{\gamma,0} \lra V_{\gamma,x}) \leq e^{-2\epsilon |x|}.
\]
Therefore provided $x$ is sufficiently large,
\begin{align} \label{E:twopath1}
P&(\text{there exist two bond-disjoint open paths } 0 \lra x) \\
&\leq P(0 \lra y \text{ for some $y$ with } |y| \geq 2|x|) \notag \\
&\quad + P(\text{there exist two bond-disjoint open paths } l_0 \lra l_x
\text{ in } R) \notag \\
&\leq c_{10}e^{-\frac{3}{2}\tau(\theta)|x|}+ \sum_{\gamma} P(\Gamma_R =
\gamma; l_0 \lra l_x \text{ via an open
path in } U_{\gamma}) \notag \\
&\leq c_{10}e^{-\frac{3}{2}\tau(\theta)|x|}+ \sum_{\gamma} P(\Gamma_R
= \gamma) P_{\mB(U_{\gamma}),w}(V_{\gamma,0}
\lra V_{\gamma,x}) \notag \\
&\leq c_{10}e^{-\frac{3}{2}\tau(\theta)|x|} + e^{-2\epsilon |x|} P(l_0 \lra
l_x) \notag \\
&\leq c_{10}e^{-\frac{3}{2}\tau(\theta)|x|}
+ e^{-(\tau(\theta) +\epsilon) |x|} \notag
\end{align}
where the third inequality uses the FKG property and the fifth uses
(\ref{E:natslab}), and the sums are over all self-avoiding paths $\gamma$ from
$l_0$ to $l_x$. This completes the proof.
\end{proof}
We turn next to the proof of Theorem \ref{T:siteratiosm}. We first establish
strong mixing, and later obtain ratio strong mixing as a consequence.
We need to
use certain variants of the Markov property. A \emph{dual plaquette} is a face
of a hypercube $Q(x)$ for some $x \in \ZZn$. As with dual bonds, a dual
plaquette is defined to be open precisely when the bond which it perpendicularly
bisects is closed. For a set $A$ of bonds (or dual plaquettes), let $\Open(A)$
denote the event that all bonds (or dual plaquettes) in $A$ are open. By
a \emph{dual surface} we mean a set of dual plaquettes
which is the outer boundary
of $Q(\Delta)$ for some finite $\Delta \subset
\ZZn$ for which the interior of $Q(\Delta)$ is connected.
We say a bond percolation model
$P$ has the \emph{Markov property for open dual surfaces} if for every dual
surface $S$, the bond configurations inside and outside $S$ are
independent given
the event $\Open(S)$.
The FK model with arbitrary external fields has this property. In two
dimensions, the
\emph{Markov property for open circuits} (of regular bonds) is defined
analogously; the FK model has this property if and only if there are no external
fields (see \cite{Al00}.) However, we can come close to the
Markov property even
under external fields: if
$P$ is the infinite-volume
$k$-wired FK model on $\mB(\ZZ)$ for some stable $k$, then
letting $\omega_{int}$
and $\omega_{ext}$ denote the bond configurations inside and outside
$\gamma$, respectively, we have (see \cite{Al00}) for some $C, a > 0$
\begin{align} \label{E:Markov}
\sup &\left\{ \left|\frac{P(\omega_{int} \in A \mid
\Open(\gamma), \omega_{ext}
\in B)}{P(\omega_{int} \in A \mid \Open(\gamma))} - 1\right|:
A \in \mathcal{G}_{\mathcal{B}(\Int(\gamma))},
B \in \mathcal{G}_{\mathcal{B}(\Ext(\gamma))}\right\} \\
&\qquad \leq Ce^{-a|\gamma|}
\qquad \text{for all } \gamma \notag
\end{align}
where $\Int$ and $\Ext$ denote the interior and exterior, respectively.
When (\ref{E:Markov}) holds we say $P$ has the \emph{near-Markov
property for open circuits}.
A \emph{coupling} of two measures $P_1$ and $P_2$ on $\{ 0,1 \}^{\mB}$ is a
measure $\mathbb{P}$ on $\{ 0,1 \}^{\mB} \times \{ 0,1 \}^{\mB}$ with marginals
$P_1$ and $P_2$ (in order.) A standard way of constructing couplings
(see \cite{Al98mix}, \cite{AC}, \cite{Ne}) is via what
we call a \emph{construction algorithm}, which is a rule specifying for each
subset $\mE$ of $\mB$ and each pair $(\omega^1_{\mE},\omega^2_{\mE})$ of
configurations on $\mE$ a choice of a bond $b =
b(\mE,\omega^1_{\mE},\omega^2_{\mE})$ and a choice of a ``single-bond'' coupling
of $P_1(\omega_b = \cdot \mid \omega^1_{\mE})$ and $P_2(\omega_b = \cdot \mid
\omega^2_{\mE})$ on $\{ 0,1 \}^2$. In particular there is an initial bond
$b_1 = b(\phi)$ and an initial single-bond coupling on $b_1$. We construct a
coupled pair of configurations by first choosing
$(\omega^1_{b_1},\omega^2_{b_1})$ under the initial single-bond coupling,
then applying the rule to determine both the second bond $b_2 = b(\{b_1
\},\omega^1_{b_1},\omega^2_{b_1})$ and the single-bond coupling on $b_2$,
then choosing $(\omega^1_{b_2},\omega^2_{b_2})$ using this
single-bond coupling on
$b_2$, and iterating in this manner until the entire configuration is
constructed. We let $\mE_n$ denote the (random) set consisting of the first $n$
bonds chosen. We also consider \emph{stopped construction algorithms} in which
the construction stops after a random number $\tau$ of steps, with $\tau$ a
stopping time relative to $\{ \mathfrak{S}_n \}$, where $\mathfrak{S}_n$ is the
$\sigma$-field generated by the first $n$ steps of the construction. It is easy
to see that, given a set $A$ of bonds or sites, $\tau$ may be chosen so that
$S_{\tau}$ is the closure of the cluster of $A$; one merely ``builds the cluster
outwards from $A$.''
We will use two particular types of construction algorithms.
The first type are
\emph{independent (stopped) construction algorithms}, in which
$\omega^1_b$ and
$\omega^2_b$ are chosen independently of each other in each iteration, i.e. the
single-bond coupling is product measure. The second type are
\emph{FKG (stopped)
construction algorithms} \cite{Ne} which exist if $P_1$ or $P_2$ satisfies the
FKG lattice condition and $P_1(\omega) / P_2(\omega)$ is an increasing function
of $\omega$. This ensures that $P_1(\omega_{\mE^c} \in \cdot \mid \omega_{\mE} =
\rho_{\mE} )$ dominates $P_2(\omega_{\mE^c} \in \cdot \mid \omega_{\mE} =
\rho_{\mE} )$ for all
$\mE$ and $\rho_{\mE}$. In turn this means that at each step of the
construction, the single-bond coupling can be chosen so $\omega^1_b \geq
\omega^2_b$, and thus at the end, $\omega^1_{\mE_{\tau}} \geq
\omega^2_{\mE_{\tau}}$.
\begin{proposition} \label{P:sitesm}
Let $P = P^{p,q,\{ h_i \}}$ be the FK model at $(p,q,\{ h_i \})$ on $\ZZn$, let
$d$ be either the Euclidean or the restricted path metric and let $\mkC$ be a
class of subsets of $\ZZn$ which is inheriting with respect to $d$.
Suppose $P$ has uniform
exponential decay of finite-volume connectivities for
the class $\mkC$ with wired
boundary conditions. Then $P$ has
the strong mixing property for the
class $\mkC$ and arbitrary site boundary conditions, for the metric $d$.
\end{proposition}
\begin{proof}
Fix a finite
$\mB = \omB(\Lambda) \in \mkC$, a site boundry condition $\eta$ and
$\mE, \mF \subset \mB$.
For a bond configuration $\omega_{\mB \bs \mF}$ let
\[
C_{\mF}(\omega_{\mB \bs \mF}) = \cup_{x \in
V(\mB \bs \mF) \cap V(\mF)} \ C_{x}(\omega_{\mB \bs \mF}),
\]
where
$C_{x}(\omega_{\mB \bs \mF})$ is the cluster of $x$ in $\omega_{\mB \bs \mF}$.
Note that all bonds in $\partial C_{\mF}(\omega_{\mB \bs \mF})
\cap (\mB \bs \mF)$
are closed. Also,
\begin{align}\label{E:Markovextn}
&\text{conditionally on the event $[C_{\mF} = \mA]$, the bond
configuration on
$\mB \bs (\mA \cup \mF)$} \\
&\text{is the FK model with site boundary
condition given by} \notag \\
&\text{$\eta$ on $V(\mB \bs (\mA \cup \mF)) \cap \pL$, 0 on $V(\mB \bs (\mA
\cup \mF)) \cap \Lambda$}. \notag
\end{align}
(This is a straightforward extension of the Markov property for open dual
surfaces.) Further, since $P_{\mB, w}$ has the strong FKG property,
the measure $P_{\mB \bs \mF, w}$ FKG-dominates
$P_{\mB,\eta}(\omega_{\mB \bs \mF}
\in \cdot \mid \omega_{\mF} = \theta_{\mF})$ for all $\theta_{\mF}$.
From these observations, as in \cite{Ne} and \cite {AC}, we see
that for each pair of configurations $\theta_{\mF}, \theta_{\mF}^{\pr}$ it is
possible, using an FKG construction algorithm, to construct a coupling measure
$\mathbb{P}$ on $(\{ 0,1 \}^{\mB \bs \mF})^{3}$ such that
\begin{enumerate}
\item[(i)] the marginals of $\mathbb{P}$ are (in order)
$P_{\mB,\eta}(\omega_{\mB
\bs \mF}
\in \cdot \mid \omega_{\mF} = \theta_{\mF}), P_{\mB,\eta}(\omega_{\mB \bs \mF}
\in \cdot \mid \omega_{\mF} = \theta_{\mF}^{\pr})$, and $P_{\mB \bs \mF, w}$;
\item[(ii)] with $\mathbb{P}$-probability 1, the configuration $(\omega,
\omega^{\pr},\omega^{\pr\pr})$ satisfies
\[
\omega \leq \omega^{\pr\pr}, \quad \omega^{\pr} \leq \omega^{\pr\pr}, \quad
\omega_{\mB \bs (\mF \cup \mC_{\mF}(\omega_{\mB \bs \mF}^{\pr\pr}))} =
\omega_{\mB \bs (\mF \cup \mC_{\mF}(\omega_{\mB \bs \mF}^{\pr\pr}))}^{\pr}.
\]
\end{enumerate}
That is, the first 2 configurations agree outside the cluster of $\mF$ in
the largest configuration $\omega^{\pr\pr}$. Then for
$E \in \mG_{\mE}$,
\begin{align} \label{E:couple}
|P_{\mB,\eta}&(E \mid \omega_{\mF} = \theta_{\mF}) - P_{\mB,\eta}(E \mid
\omega_{\mF} = \theta_{\mF}^{\pr})| \\
&\leq \mathbb{P}(\omega \neq \omega^{\pr}) \notag \\
&\leq P_{\mB \bs \mF, w}(C_{\mF}(\omega_{\mB \bs \mF}) \cap \mE \neq \phi)
\notag \\
&= P_{\mB \bs \mF, w}(x \lra y \text{ for some } x \in V(\mF), y \in V(\mE)).
\notag
\end{align}
We claim that for some $c_{11}, \epsilon_{5}$
(not depending on $\mB, \mE, \mF$),
the right side of (\ref{E:couple}) is bounded above by
\begin{equation} \label{E:claim}
\sum_{x \in V(\mE), y \in V(\mF)} c_{11} e^{-\epsilon_{5} d(x,y)}.
\end{equation}
(This is not immediate from the uniform exponential decay assumption because we
do not assume $\mB \bs \mF \in \mkC$.) Let $\mF^{+} = \{ x \in \ZZn:
d(x,\mF) \leq d(x,\mE) \}$. For $z \in \partial \mF^{+} \cap \Lambda$ let
$U_{z}$ be the connected component of $z$ in the ``ball'' $W_{z} = \{ e \in \mB:
d(z,e) \leq
\half d(z,\mF) \}$. Then from the uniform exponential decay of connectivities,
the FKG property and the inheriting property of $\mkC$, for some $c_{i}$ and
$\epsilon_{6}$,
\begin{align} \label{E:connect}
P_{\mB \bs \mF, w}&(x \lra y \text{ for some } x \in V(\mE), y \in V(\mF)) \\
&\leq P_{\mB \bs \mF, w}(z \lra V(\mB \bs W_{z}) \text{ for some } z \in
\partial \mF^{+} ) \notag \\
&\leq \sum_{z \in \partial \mF^{+}} P_{U_{z}, w}(z \lra V(\mB \bs W_{z}))
\notag \\
&\leq \sum_{z \in \partial \mF^{+}} c_{12} d(z,\mF)^{d-1} e^{-\epsilon_{6}
d(z,\mF)} \notag \\
&\leq \sum_{z \in \partial \mF^{+}} c_{13} e^{-\epsilon_{6} d(z,\mF)/2}.
\notag
\end{align}
For each $z \in \partial \mF^{+}$ let $f(z)$ be the site in $V(\mF)$ which is
closest to $z$ (in the metric $d$, breaking ties arbitrarily.) Then for $x \in
V(\mF)$ and $z \in f^{-1}(x)$, from simple geometry
\[
d(x,\mE) \geq d(z,\mE) - d(z,x) \geq d(z,x) - 2
\]
and hence
\[
|f^{-1}(x)| \leq c_{14}(d(x,\mE) + 2)^{d}.
\]
Therefore
\begin{align} \label{E:connectsum}
\sum_{z \in \partial \mF^{+}} &e^{-\epsilon_{6} d(z,\mF)/2}
\\
&\leq \sum_{x \in V(\mF)} \sum_{z \in f^{-1}(x)} e^{-\epsilon_{6}
d(z,x)/2} \notag \\
&\leq \sum_{x \in V(\mF)} c_{15} (d(x,\mE) + 2)^{d}
e^{-\epsilon_{6}d(x,\mE)/2}
\notag \\
&\leq \sum_{x \in V(\mF)} c_{16} e^{-\epsilon_{6}d(x,\mE)/4} \notag \\
&\leq \sum_{x \in V(\mF), y \in V(\mE)} c_{16} e^{-\epsilon_{6}d(x,y)/4}.
\notag
\end{align}
This and (\ref{E:connect}) prove the claim of the bound (\ref{E:claim}).
Combining the claim with (\ref{E:couple}), and
averaging over $\theta_{\mF}$ and then over $\theta_{\mF}^{\pr}$ shows that
\[
|P_{\mB,\eta}(E \mid F) - P_{\mB,\eta}(E)| \leq
\sum_{x \in V(\mF), y \in V(\mE)} c_{17} e^{-\epsilon_{6}d(x,y)/4},
\]
which proves the proposition.
\end{proof}
The next two results form the analog of the last proposition for dual
connectivity. The proof does not carry over, due to the fact that, under
external fields, the FK model has only the near-Markov property for open
circuits, not the full Markov property for open circuits, and the extension
(\ref{E:Markovextn}) of the Markov property breaks down completely. We evade
this difficulty by using the ARC model of \cite{AlARC}. For $q > 0, Q \geq 1$
and $p_r, p_g \in [0,1]$, the ARC model on $\graph$ with parameters
$(p_r,p_g,q,Q)$ is given by the weights
\begin{equation} \label{E:ARCweight}
W(\omega_r,\omega_g) = p_r^{|\omega_r|} (1-p_r)^{|\omB(\Lambda)| - |\omega_r|}
p_g^{|\omega_g|} (1-p_g)^{|\omB(\Lambda)| - |\omega_g|} q^{K(\omega_r)}
Q^{I(\omega_r \vee \omega_g)},
\end{equation}
assigned to configurations $(\omega_r,\omega_g) \in \{0,1\}^{\omB(\Lambda)}
\times \{0,1\}^{\omB(\Lambda)}$. Here $I(\omega)$ denotes
the number of isolated
sites (singleton open clusters) in $\Lambda$ in the configuration $\omega$, and
$\omega_r \vee \omega_g$ denotes the coordinatewise maximum.
Bonds in $\omega_r$ are called \emph{red
bonds}, and bonds in $\omega_g$ are called \emph{green bonds}. The \emph{black
configuration} is defined to be $\omega_b = \omega_r \vee \omega_g$, the
bondwise maximum. The ARC model is a
graphical representation of the $q$-state Potts lattice gas, and hence, as a
special case, of the $(q+1)$-state Potts model on $\Lambda \subset \ZZn$ at
inverse temperature
$\beta$ with an external field applied to one species only, say $h_1$ applied to
species 1, when we take
\begin{equation} \label{E:ARCparams}
p_r = p_g = 1 - e^{-\beta}, \qquad Q = 1 + \frac{e^{\beta(2d + h_1)}}{q}.
\end{equation}
Under site boundary conditions $\eta$ with the property that, for some $i
\in \{2,..,q+1\}, \eta_x \in \{0,1,i\}$ for all $x \in \pL$, the ARC model
satisfies the FKG lattice condition \cite{AlARC}.
It should be noted
here that in \cite{AlARC}, the species for the $(q+1)$-state Potts model were
given as $0,1,..,q$ instead of $1,2,..,q+1$, and the
external field was applied to
species 0. Here we instead retain the meaning of 0 as a free boundary
condition.
As with the FK model, for integer $q$ it is straightforward in the ARC model to
allow external fields applied to all species, by replacing the factor
$q^{K(\omega_r)}$ in the ARC weight (\ref{E:ARCweight}) with the weight
\[
\prod_{C \in \mC(\omega_r)} \left((1-p)^{h_{2}s(C)} + (1-p)^{h_{3}s(C)} +
\ldots + (1-p)^{h_{q+1}s(C)}\right)
\]
as in (\ref{E:FKexternal}) under free boundary conditions, with modifications
analogous to (\ref{E:FKextbond}), (\ref{E:FKextsite}) for bond and site boundary
conditions. It is readily checked that the FKG lattice condition still
holds under all bond boundary conditions and under site boundary conditions with
$\eta_x \in \{0,1,i\}$ for all $x$, where
$i \in \{2,..,q\}$. The parameters are $(p_r,p_g,q,Q,\{ h_i \})$.
In the ARC model with external fields we need not require
that $h_1 \geq h_2 \geq ..
\geq h_{q+1}$ but rather only that $h_2 \geq .. \geq h_{q+1}$; the position of
$h_1$ relative to the other $h_i$'s is arbitrary. To make the model a graphical
representation of the Potts model with external fields, one should
replace $q$ in
(\ref{E:ARCparams}) with
\[
e^{\beta h_2} + \ldots + e^{\beta h_{q+1}}.
\]
For parameters as in (\ref{E:ARCparams}) (modified as above for external fields,
if necessary), the cluster-labeling construction of the Potts model from the ARC
model works as follows \cite{AlARC}. Each non-boundary cluster
$C$ of $\omega_r$
which is not a single isolated site in $\omega_b$ is independently given label
$i$ $(2
\leq i
\leq q+1)$ with probability proportional to $e^{\beta h_i s(C)}$;
no such cluster
is ever given label $i = 1$. In each boundary cluster of
$\omega_r$ intersecting $\{ x \in \pL: \sigma_x = i \}$, all sites are labeled
$i$. Each site in $\Lambda$ which is isolated in $\omega_b$ is independently
given label
$i$ with probability proportional to $e^{\beta h_i}$ for $2 \leq i \leq q+1$ and
proportional to $e^{\beta(2d + h_i)}$ for $i = 1$. (Note the resulting
probability for label 1 is $(Q-1)/Q$.) Thus in the Edwards-Sokal-type joint
construction, sites with species 1 are always isolated
in the black configuration
$\omega_b$.
In certain contexts it is useful to consider the model
obtained from the ARC model
when only a portion of the cluster labeling is done. Specifically, given an
ARC-model configuration with parameters as in (\ref{E:ARCparams}) we can
independently label each isolated site as being of species 1 with probability
$(Q - 1) / Q$, leaving all other sites unlabeled. As in \cite{AlARC} we call
the resulting measure on site/bond configurations the
\emph{particle-bond form} of the ARC model. Given a realization of the
particle-bond ARC model on some set $\omB(\Lambda)$,
with red-bond configuration
$\omega_r$, we can enlarge the set of open red bonds by doing independent
percolation, with red bonds, at density $p_r$ on the set of bonds
$\{ \langle xy
\rangle: \sigma_x = \sigma_y = 1 \}$ (noting that such
$x$ and $y$ are necessarily
isolated in $\omega_r$). The resulting enlarged red configuration is a
realization of the FK model on $\omB(\Lambda)$ at
$(p_r,q,\{h_i\})$ \cite{AlARC}.
We call this the
\emph{ARC-based percolation construction} of the FK model.
We say that a $q$ -state Potts model $\mu$ (in infinite volume) has
\emph{exponential decay of non-1 connectivities} if
there exists $C, \lambda > 0$
such that for all $x,y$,
\[
\mu(\text{there exists a lattice path from $x$ to $y$ in which every site $z$
has } \sigma_z \neq 1) \leq Ce^{-\lambda |y-x|}.
\]
We say that \emph{Gibbs uniqueness holds} at $(\beta,\{h_i \})$ if there is a
unique infinite-volume Gibbs distribution for the Potts model with those
parameters. If Gibbs uniqueness holds and $\beta > \beta_{c}(q,n,\{ h_i \})$,
then there must be a unique stable species. Conversely, if there is a unique
stable species and the dimension $n = 2$, then random-cluster uniqueness holds
for the corresponding FK model except possibly at $p = p_{c}(q,2,\{ h_i \})$
\cite{BBCK}, and it follows readily from the cluster-labeling construction that
Gibbs uniqueness holds for the Potts model except possibly at $\beta =
\beta_{c}(q,2,\{ h_i
\})$. This exception is a real one, at least for large $q$ and small
fields---the existence of a unique stable species need not imply Gibbs
uniqueness at $\beta_{c}(q,2,\{ h_i \})$
\cite{Ch}.
\begin{proposition} \label{P:dualsitesm}
Let $\mu^{\beta,q,\{ h_i \}}$ be the $q$-state Potts model at
$(\beta,\{ h_i \})$
on $\ZZ$. Suppose Gibbs uniqueness holds. Let
$d$ be either the Euclidean or the restricted path metric.
Suppose one of the following holds:
\begin{enumerate}
\item[(i)] the corresponding FK model has
exponential decay of connectivities;
\item[(ii)] the Potts model $\mu^{\beta,q,\{ h_i \}}$ has exponential decay of
non-1 connectivities;
\item[(iii)] the corresponding FK model has
exponential decay of dual connectivities;
\item[(iv)] the corresponding ARC model black configuration has exponential
decay of connectivities.
\end{enumerate}
Then $\mu^{\beta,q,\{ h_i \}}$ (respectively,
the corresponding FK and ARC models)
has the strong mixing property for the
class of all simply lattice-connected subsets of $\ZZ$ (respectively, of
$\mB(\ZZ)$) and arbitrary site boundary conditions, for the metric $d$.
\end{proposition}
By the remarks preceding Proposition \ref{P:dualsitesm}, if (ii) or (iii)
(and Gibbs uniqueness) hold, then there must be a unique stable species, but
(i) and (iv) allow multiple stable species, if the temperature is supercritical.
Also, (i) can only be valid at high temperatures ($\beta <
\beta_{c}(q,2,\{ h_i \})$) and (iii) can only be valid at low temperatures
($\beta >
\beta_{c}(q,2,\{ h_i \})$) but (ii) and (iv) may be valid at arbitrary
temperatures.
Gibbs uniqueness in the
Potts model implies random-cluster uniqueness in the corresponding FK and ARC
models, as is apparent from the percolation construction,
so (i), (iii) and (iv)
are not ambiguous.
\begin{proof}[Proof of Proposition \ref{P:dualsitesm}]
Suppose (iii) holds. For the corresponding Potts model define
\[
\Sigma_1 = \{ x: \sigma_x = 1 \}, \qquad \Sigma_u = \{ x: \sigma_x \in
\{2,..,q\} \}.
\]
Consider a joint Potts-FK-ARC configuration.
In infinite volume, every
connected component of $\mB(\Sigma_u)$ in the Potts
configuration must be finite,
and must be surrounded by an open dual circuit in the FK configuration. It
follows that (ii) holds. Since all $x$ with $\sigma_x = 1$
are isolated sites in
the ARC model, (ii) implies (iv). Thus, except under (i), we may assume (iv).
The ARC model then has the weak mixing property, by Theorem 3.1 of
\cite{Al98mix}. (Strictly speaking, that theorem is stated for a single
configuration whereas the ARC model has separate red and green configurations,
but the extension is trivial.) Therefore by Theorem \ref{T:percmain}, the ARC
model has uniform exponential decay for the class of
all simply lattice-connected
subsets of $\mB(\ZZ)$, with arbitrary site or bond boundary conditions. The
proof of Proposition \ref{P:sitesm} then goes through (with 0 replaced by 1 in
(\ref{E:Markovextn})) to show that the ARC model has the strong mixing property
for the class of all simply lattice-connected subsets of $\mB(\ZZ)$ with
arbitrary site boundary conditions.
In fact, the proof of Proposition \ref{P:sitesm} shows more. Consider the
joint Potts-ARC configuration under a site boundary condition $\eta$
on $\pL$, and let
$\omega = (\omega_r,\omega_g)$ denote a generic ARC model configuration. Let
$\Delta, \Gamma \subset
\Lambda$ and let $\xi_{\Gamma}, \xi_{\Gamma}^{\prime}$ be Potts-model
configurations on $\Gamma$. Define the ARC-model cluster of $\Gamma$ by
\[
C_{\Gamma}(\omega_{\omB(\Lambda) \bs \mB(\Gamma)}) =
\cup_{x \in \Gamma} \ C_x(\omega_{\omB(\Lambda) \bs \mB(\Gamma)}),
\]
where $C_x(\omega_{\omB(\Lambda) \bs \mB(\Gamma)})$
is the cluster of $x$ in the
ARC model black configuration $(\omega_{\omB(\Lambda) \bs \mB(\Gamma)})_b$.
Analogously to (\ref{E:Markovextn}), conditionally on the event $[C_{\Gamma} =
\mA]$, the site configuration on $\Lambda \bs V(\mA)$ is the Potts model
with site boundary condition given by $\eta$ on
$\partial(\Lambda \bs V(\mA)) \cap
\pL$, 0 on
$\partial(\Lambda \bs V(\mA)) \cap \Gamma$. Observe that the ARC model measures
on $\omB(\Lambda) \bs \mB(\Gamma)$ with site boundary conditions $\eta$ on $\pL$
and $\xi_{\Gamma}$ or $\xi_{\Gamma}^{\prime}$ on $\Gamma$ are both
FKG-dominated by the wired measure on $\omB(\Lambda) \bs \mB(\Gamma)$, and the
latter has the FKG property. Therefore these three ARC-model measures can be
coupled so that the configurations agree outside the cluster of
$\Gamma$ in the largest (wired) configuration. Then via the cluster-labeling
construction, the corresponding three Potts
configurations can be coupled so that
they agree outside this same cluster of $\Gamma$. As in (\ref{E:couple}) --
(\ref{E:connectsum}) this shows that for any event $E \in
\mG_{\Delta}$,
\begin{align}
|\mu_{\Lambda,\eta}^{\beta,q,\{ h_i \}}&(E \mid
\sigma_{\Gamma} = \xi_{\Gamma})
- \mu_{\Lambda,\eta}^{\beta,q,\{ h_i \}}(E \mid \sigma_{\Gamma} =
\xi_{\Gamma}^{\prime})| \notag \\
&\leq P_{\omB(\Lambda),w}(x \lra y \text{ for some } x \in \Gamma, y \in
\Delta) \notag \\
&\leq \sum_{x \in \Gamma, y \in \Lambda} c_{18}e^{-\epsilon_7 d(x,y)}, \notag
\end{align}
where $P_{\omB(\Lambda),w}$ is the wired ARC model. This proves that
$\mu^{\beta,q,\{ h_i \}}$ has the desired strong mixing property.
Under (i), essestially the same proof works, with the FK model substituted for
the ARC model, to yield that
$\mu^{\beta,q,\{ h_i \}}$ has the desired strong mixing property.
Given $\Gamma \subset \Lambda$ and a joint Potts-FK configuration
$(\xi_{\Gamma},\rho_{\mB(\Gamma)})$ on
$(\Gamma,\mB(\Gamma))$, it is easy to see that conditioning on both
$\xi_{\Gamma}$ and $\rho_{\mB(\Gamma)}$ is the same as conditioning only on
$\xi_{\Gamma}$.
As in the proof of (\cite{Al98mix}, Theorem 6.1(i)), this observation and the
percolation and cluster-labeling constructions of the FK model
can be used to show that whenever the Potts model has the stated strong mixing
property, so does the corresponding FK model.
\end{proof}
Proposition \ref{P:dualsitesm} does not cover the case in which
there are multiple stable species and the FK model has exponential decay of dual
connectivities. Of course, in that case the Potts model itself will not satisfy
Gibbs uniqueness so will not have any reasonable sort of mixing property, but
the corresponding FK model still may. The next proposition, like Theorem
\ref{T:siteratiosm}(iii), removes the assumption, in Proposition
\ref{P:dualsitesm}(iii), of Gibbs uniquesness in the corresponding Potts model,
allowing multiple stable species, at the expense of great restriction on the
boundary condition. The proof, which will follow some preliminary lemmas, is
also significantly more complex.
\begin{proposition} \label{P:dualnonunsm}
Let $P=P^{p,q,\{h_i\}}$ be the FK model at $(p,q,\{h_i\})$ on $\ZZ$. Suppose
random-cluster uniqueness holds and $P$ has
exponential decay of dual connectivities. Let $d$ be either the Euclidean or
restricted path metric. Then $P^{p,q,\{h_i\}}$ has the strong mixing property
for the class of all simply lattice-connected closure subsets of $\mB(\ZZ)$
with free and wired boundary conditions, for the metric $d$.
\end{proposition}
Since, under the assumptions of Proposition \ref{P:dualnonunsm}, there is not
necesssarily Gibbs uniqueness in the corresponding Potts and ARC models, (iii)
and especially (iv) in Proposition \ref{P:dualsitesm} become potentially
ambiguous statements and thus the proof of Proposition \ref{P:dualsitesm} does
not carry over. Instead we will make use of the following. Consider a
$q$-state Potts model at $(\beta, \{ h_i \})$ on a finite
$\Lambda$ with stable species $1,..,k$ and
unstable species $k+1,..,q$. Starting
from the corresponding Edwards-Sokal joint Potts/FK model, with $p = 1 -
e^{-\beta}$, as in \cite{AlARC} we color each site
$x$ yellow if $\sigma_x$ is stable, white if $\sigma_x$
is unstable, and designate
each open bond to have the color of its (necessarily matching) endpoints.
We call the corresponding configuration of colored bonds and sites (without the
values $\sigma_x$) the
\emph{bicolored FK model}. When necessary for clarity, we call the usual FK
model the
\emph{uncolored FK model}. Using the Edwards/Sokal joint Potts/FK model, we see
that in either the bicolored or uncolored FK model, one can have bond boundary
conditions $\rho \in \{0,1 \}^{\omB(\Lambda)^c}$ for which each cluster is
designated to be one of 3 types: yellow, white, or uncolored (see Remark
\ref{R:genbdry} below.) Note that the yellow-wired boundary condition is the
same as the uncolored wired boundary condition, since infinite clusters can
only be yellow. The configuration of open yellow bonds
$\{ \delta_{[b
\text{ open and yellow}]}: b \in \omB(\Lambda) \}$ (or its distribution, in a
harmless abuse of terminology) is called the \emph{stable partial FK model}, or
briefly the
\emph{SPFK model}, with parameters $(p,q,k,\{ h_i \})$, and its distribution is
denoted $P^{(p,q,k,\{ h_i\})}_{SPFK,\omB(\Lambda),\rho}$. For the SPFK model we
can think of all open bonds, including those in the boundary condition, as being
yellow. Define
\[
\mI(\omega,\Lambda) = \{ x \in \Lambda: x \text{ is an isolated site of }
\omega \},\qquad I(\omega,\Lambda) = |\mI(\omega,\Lambda)|,
\]
\[
r_n = r_n(p,h_{k+1},..,h_q) = \sum_{i=k+1}^{q} (1-p)^{-nh_i}
\]
It is easily calculated (see \cite{AlARC}) that the weight associated to
an SPFK configuration $\omega$ under a bond boundary condition $\rho$ is
\[
W_{SPFK}(\omega) = p^{|\omega|} (1-p)^{|\omB(\Lambda)| - |\omega|}
k^{K(\omega \mid \rho)} G(\omega)
\]
where
\[
G(\omega) = k^{-I((\omega,\rho),\oL)}
\sum_{\omega_w \in \{ 0,1 \}^{\mB(\mI((\omega, \rho),\oL)) \cap
\omB(\Lambda)}}
\left( \frac{p}{1-p} \right)^{|\omega_w|} \left( \prod_{C \in \mC(\omega_w)}
r_{|C|} \right) \left( \frac{k+r_1}{r_1} \right)^{I((\omega
\vee \omega_w, \rho), \oL)}.
\]
Here the configurations $\omega_w$ correspond to white configurations in the
bicolored FK model. Given a bond configuration $\omega$ and a bond $e$, we let
$\omega \vee e$ denote the configuation obtained by adding the bond
$e$ to $\omega$ (that is, by declaring $e$ to be open.)
\begin{remark} \label{R:interest}
The reason for interest in the SPFK model is as follows. Consider a cicuit
$\gamma$ and let $\Lambda$ be the set of sites surrounded by $\gamma$, with
boundary condition in which all bonds of $\gamma$ are open. The FK model with
external fields lacks the Markov property for open dual circuits because the
weight attached to the boundary cluster in $\omB(\Lambda)$ depends on the number
of sites outside $\gamma$ in that cluster, hence is affected by the bond
configuration outside $\gamma$. In the SPFK model, the same boundary cluster
gets weight $k$ in all configurations, so it is easy to see that the Markov
property for open dual circuits does hold.
\end{remark}
\begin{remark} \label{R:genbdry}
A boundary condition $\rho$ for the SPFK model can be viewed as a boundary
condition for the full FK model, with all non-singleton clusters of $\rho$
conditioned to be be labeled with stable species in the cluster-labeling
construction, and with the label types (stable/unstable) for singleton clusters
not specified. This idea can be extended to allow boundary conditions $\rho$
for the SPFK model which are bond configurations for the full FK model, with
stable/unstable labels specified for some clusters, with both singleton and
nonsingleton clusters of $\rho$ allowed to have specified or unspecified
labels. We call such a $\rho$ a \emph{partly labeled bond boundary condition}.
Consider an unlabeled nonsingleton boundary cluster $C$ in a partly labeled bond
boundary condition $\rho$, with $C \cap \pL \neq \phi$. The effect of the
presence of $C$ is to identify the sites $C \cap \pL$ into a single ``condensed
site'' which should be treated as one site in determining the set $
\mI((\cdot,\rho),\oL)$ of isolated sites in the formula for $G(\omega)$.
However, if the condensed site is isolated the weight associated to it in the
FK model is $k + r_{s(C)}$, not $k + r_1$, so the factor
$(k + r_1) / r_1)$ associated to such an isolated condensed site should be
replaced by $(k + r_{s(C)}) / r_{s(C)}$. similarly, if the condensed site $C$
consistts of $j$ original sites and is not isolated, all $j$ must be counted in
calculating the size $s(\cdot)$ of the cluster containing $C$.
The proofs of Lemmas \ref{L:SPFK-FKG} and \ref{L:SPFKweakmix} are both valid for
partly labeled bond boundary conditions, with no significant modifications.
\end{remark}
\begin{lemma} \label{L:SPFK-FKG}
For every finite $\Lambda$, every $(p,q,k,\{ h_i \})$ and every bond boundary
condition $\rho$, the SPFK model $P^{p,q,k,\{
h_i\}}_{SPFK,\Lambda,\rho}$satisfies the FKG lattice condition.
\end{lemma}
\begin{proof}
As is standard, it is sufficient to show that, for every bond $e$,
$W_{SPFK}(\omega
\vee e)/W_{SPFK}(\omega)$ is an increasing function of $\omega$. Thus fix $e
= \langle xy \rangle \in \omB(\Lambda)$. Since
$K(\omega \vee e \mid \rho) - K(\omega \mid \rho)$ is increasing, it is
sufficient to show that $G(\omega \vee e)/G(\omega)$ is increasing. We
proceed as in the proof of (\cite{AlARC}, Proposition 4.16). Let $F(\omega) =
k^{I((\omega,\rho),\oL)} G(\omega)$ denote the sum in $G(\omega)$ and let
$\mD(\omega) = \mB(\mI((\omega, \rho),\oL)) \cap \omB(\Lambda)$ denote the
set of bonds on which the configurations $\omega_w$ exist.
Note $F(\omega)$ is the partition function of the ARC model on $\mD(\omega)$ at
$(p,0,q-k,\frac{k + r_1}{r_1},\{ h_{k+1},..,h_q \})$ with free boundary; we
denote this model by $P^{ARC}_{\omega}$. Let
\[
\Delta(\omega) = I((\omega,\rho),\oL) - I((\omega \vee e,\rho),\oL)
= |\{ x,y \} \cap \mI((\omega,\rho),\oL)|.
\]
Terms in the sum
$F(\omega \vee e)$, each multiplied by
$(k + r_1)^{\Delta(\omega)}$, correspond precisely to those configurations
$\omega_w$ of this ARC model in which $\{ x,y \} \cap \mI((\omega,\rho),\oL)
\subset \mI((\omega \vee \omega_w,\rho),\oL)$, or equivalently, in which all
bonds in $\mD(\omega) \cap \overline{ \{ e \} }$ are closed. It follows that
\[
\frac{F(\omega \vee e)}{F(\omega)} = (k + r_1)^{-\Delta(\omega)}
P^{ARC}_{\omega} \bigl( \bigl\{ \omega_w: \text{ all bonds
in $\mD(\omega) \cap \overline{ \{ e \} }$ are closed in } \omega_w
\bigr\} \bigr)
\]
and then that
\begin{equation} \label{E:FKGcond}
\frac{G(\omega \vee e)}{G(\omega)} = \left( \frac{k}{k+r_1}
\right)^{\Delta(\omega)} P^{ARC}_{\omega} \bigl( \bigl\{ \omega_w:
\text{ all bonds in $\mD(\omega) \cap \overline{ \{ e \} }$ are closed in }
\omega_w \bigr\}\bigr).
\end{equation}
It is easy to see that $\left( \frac{k}{k+r_1}
\right)^{\Delta(\omega)}$ is an increasing function of $\omega$.
It is proved in
\cite{AlARC} that the ARC model without external fields
satisfies the FKG lattice
condition. With external fields the proof is similar except that one must
establish the straightforward fact that $r_{n+m}/r_n r_m$ is an
increasing function of $m$ and $n$. Thus
$P^{ARC}_{\omega}$ satisfies the FKG lattice condition. Let $b = \langle uv
\rangle$ be a bond which is closed in $\omega \vee e$. Then
\[
P^{ARC}_{\omega \vee b}(\cdot) =
P^{ARC}_{\omega} \bigl( (\omega_w)_{\mD(\omega \vee b)} \in
\cdot \mid \text{ all bonds
in $\mD(\omega) \cap
\overline{ \{ b \} }$ are closed in } \omega_w \bigr)
\]
and the conditioning here is on a decreasing event. Therefore
\begin{align} \label{E:FKGcomp}
P^{ARC}_{\omega \vee b} &\bigl( \bigl\{ \omega_w:
\text{ all bonds in $\mD(\omega \vee b) \cap \
overline{ \{ e \} }$ are closed
in } \omega_w \bigr\}\bigr) \\
&= P^{ARC}_{\omega} \bigl( \text{ all bonds in
$\mD(\omega \vee b) \cap \overline{ \{ e \} }$ are closed
in } \omega_w \notag \\
&\qquad \qquad \qquad \qquad \mid \text{ all bonds
in $\mD(\omega) \cap
\overline{ \{ b \} }$ are closed in } \omega_w \bigr) \notag \\
&\geq P^{ARC}_{\omega} \bigl( \text{ all bonds in
$\mD(\omega) \cap \overline{ \{ e \} }$ are closed
in } \omega_w \bigr), \notag
\end{align}
so $G(\omega \vee e)/G(\omega)$ is increasing, as desired.
\end{proof}
It follows from Lemma \ref{L:SPFK-FKG} that the free- and wired-boundary SPFK
models have infinite-volume limits, and there is a unique infinite-volume limit
if and only if the free and wired models are equal.
\begin{lemma} \label{L:SPFKweakmix}
Let $P=P^{p,q,\{h_i\}}$ be the FK model at $(p,q,\{h_i\})$ on $\ZZ$. Suppose
random-cluster uniqueness holds and $P$ has
exponential decay of dual connectivities. Then the
corresponding SPFK model has
the weak mixing property and has uniform
exponential decay of finite-volume dual connectivities
for the class of all simply lattice-connected subsets of $\mB(\ZZ)$ with
arbitrary bond boundary condition, for both the Euclidean and
restricted-path metrics.
\end{lemma}
\begin{proof}
It is clear that the SPFK model also satisfies random-cluster uniqueness.
Further, in the infinite-volume FK model there is with probability 1 a unique
infinite cluster with finite ``holes'' defined by exterior open dual circuits,
with an exponentially decreasing tail for the distribution
of hole sizes. In the
Edwards-Sokal joint construction the infinite cluster is stable with probability
1, so the same infinite cluster with the same holes is present in the SPFK
configuration. It follows that the SPFK model has exponential decay of dual
connectivities.
Let $\gamma_n$ be the dual circuit forming the boundary of
$[-n-\half,n+\half]^2$
and let $\Lambda_n = [-n,n]^2 \cap \ZZ$. Let $k$ be the number of stable
species. Given a bond boundary condition
$\rho$ on $\omB(\Lambda_n)^c$, by Lemma \ref{L:SPFK-FKG} and the Markov property
for open circuits (see Remark \ref{R:interest}), there exists a coupling of the
measures $P^{p,q,k,\{h_i\}}_{SPFK,\omB(\Lambda_n),\rho}$ and
$P^{p,q,k,\{h_i\}}_{SPFK,\omB(\Lambda_n),f}$ with the property that the
respective configurations $\omega, \omega^f$ on $\omB(\Lambda)$ agree outside
the set
\[
\bigr\{ b \in \omB(\Lambda_n): b^* \lrad \gamma_n \text{ in } (\omega^f)^*
\bigr\}.
\]
(Such couplings can be made using an FKG construction algorithm---see
\cite{Al98mix}, \cite{AC} or
\cite{Ne}.) Therefore by (\cite{Al98mix}, Theorem 3.1) the SPFK model has the
weak mixing property. The uniform exponential decay now follows from Theorem
\ref{T:percmain}.
\end{proof}
The next lemma controls the size of the portion of the FK-model boundary cluster
which is attached to boundary sites occupied by unstable species.
\begin{lemma} \label{L:unstsite}
Let $P = P^{p,q,\{h_i\}}$ be an FK model on $\mB(\ZZ)$, let $d$ be the
Euclidean
or restricted-path metric. There exist constants $C, \lambda$ such that for
every finite $\Gamma \subset \ZZ$, every site boundary condition $\eta$,
every $x \in \Gamma$ and
every unstable species $j$, for $(\partial \Gamma)_j = \{ y
\in \partial \Gamma:
\sigma_y = j \}$,
\[
P_{\Gamma,\eta}(x \lra (\partial \Gamma)_j) \leq Ce^{-\lambda d(x,
(\partial \Gamma)_j)}.
\]
\end{lemma}
\begin{proof}
We will do the proof for the Euclidean metric; the modifications for the
restricted-path metric are minor. Fix $\Gamma, \eta$ and
$x \in \Gamma$. Let $k$
be the number of stable species. Recall that
$Q_r(x)$ denotes the closed square of side $2r$ centered at $x$ and let $\mD_m$
denote the set of all bonds which cross $\partial Q_{m + \half}(x)$. Let
\[
m_{\min} = \lfloor \frac{1}{3} d(x,(\partial \Gamma)_j) \rfloor, \qquad
m_{\max} = \lfloor \frac{2}{3} d(x,(\partial \Gamma)_j) \rfloor,
\]
where $\lfloor \cdot \rfloor$ denotes the integer part.
Let $C_x$ denote the open cluster of
$x$, fix $m_{\min} \leq m \leq m_{\max}$ and define events
\[
B_{\Theta} = [ x \lra (\partial \Gamma)_j, C_x \cap \mD_m = \Theta ]
\]
for $\Theta \subset \mD_m$. We fix such a
$\Theta$ and write bond configurations
$\omega$ as $(\omega_{\omB(\Gamma) \bs \Theta},
\omega_{\Theta})$. Thus we define
\[
\tilde{B}_{\Theta} = \{ \omega_{\omB(\Gamma) \bs \Theta}:
(\omega_{\omB(\Gamma) \bs \Theta}, \rho_{\Theta}^1) \in B_{\Theta} \},
\]
where, as will be recalled, $\rho_{\Theta}^i$ denotes the all-$i$ configuration
on $\Theta$.
Given a configuration $\omega \in B_{\Theta}$, closing all bonds in $\Theta$
breaks the cluster $C_x$ into one or more clusters in $Q_m(x)$, and one or more
clusters outside the interior of $Q_{m+1}(x)$. More precisely, the first
group is
\[
\hat{\mC}_x(\omega_{\omB(\Gamma) \bs \Theta}) = \{ \text{all clusters of }
(\omega_{\omB(\Gamma) \bs \Theta}, \rho_{\Theta}^0 ) \text{ contained in }
C_x((\omega_{\omB(\Gamma) \bs \Theta},\rho_{\Theta}^1)) \cap Q_m(x) \},
\]
and the FK weight assigned to each cluster $C$ in this group is $k + r_{|C|}
\geq 1$. Let $N(\omega_{\omB(\Gamma) \bs \Theta})$ denote the total number of
sites in the clusters in $\hat{\mC}_x(\omega_{\omB(\Gamma)
\bs \Theta})$, and let
$W(\omega)$ denote the FK weight of $\omega$. We then have, for
$\omega_{\omB(\Gamma) \bs \Theta} \in \hat{B}_{\Theta}$, $N(\omega_{\omB(\Gamma)
\bs \Theta}) \geq m$ and therefore
\begin{align} \label{E:joinbound}
P\bigl( (\omega_{\omB(\Gamma) \bs \Theta},\rho_{\Theta}^1) \mid
\omega_{\omB(\Gamma) \bs \Theta} \bigr) &\leq
\frac{W((\omega_{\omB(\Gamma) \bs \Theta},\rho_{\Theta}^1))}
{W((\omega_{\omB(\Gamma) \bs \Theta},\rho_{\Theta}^0))} \\
&\leq \left( \frac{p}{1-p} \right)^{|\Theta|} (1-p)^{-h_{k+1}
N(\omega_{\omB(\Gamma) \bs \Theta})} \notag \\
&\leq \left( \frac{p}{1-p} \vee 1 \right)^{|\Theta|} (1-p)^{-h_{k+1} m}.
\notag
\end{align}
Therefore
\[
P(B_{\Theta}) \leq \max \left\{ P\bigl( (\omega_{\omB(\Gamma) \bs \Theta},
\rho_{\Theta}^1) \mid \omega_{\omB(\Gamma) \bs \Theta} \bigr):
\omega_{\omB(\Gamma) \bs \Theta} \in \tilde{B}_{\theta} \right\}
\leq \left( \frac{p}{1-p} \vee 1 \right)^{|\Theta|} (1-p)^{-h_{k+1} m}.
\]
Fix $M > 0$ to be specified. We have
\begin{align} \label{E:smallint}
P(x &\lra (\partial \Gamma)_j, |C_x \cap \mD_m| \leq M) \\
&\leq \sum_{\Theta \subset \mD_m: |\Theta| \leq M} P(B_{\Theta}) \notag \\
&\leq c_{19} |\mD_m|^M \left( \frac{p}{1-p} \vee 1 \right)^M
(1-p)^{-h_{k+1} m} \notag
\end{align}
so that, provided $d(x,(\partial \Gamma)_j)$ is sufficiently large (depending on
$M$),
\begin{align} \label{E:smallint2}
P(x &\lra (\partial \Gamma)_j, |C_x \cap \mD_m| \leq M \text{ for some }
m_{\min} \leq m \leq m_{\max}) \\
&\leq c_{19} d(x,(\partial \Gamma)_j) |\mD_m|^M \left( \frac{p}{1-p} \vee 1
\right)^M (1-p)^{-h_{k+1} m_{\min}} \notag \\
&\leq c_{20} (1-p)^{-h_{k+1} d(x,(\partial \Gamma)_j)/6}. \notag
\end{align}
Next we consider configurations satisfying
\[
x \lra (\partial \Gamma)_j, \quad |C_x \cap \mD_m| > M \text{ for all }
m_{\min} \leq m \leq m_{\max}.
\]
Taking $m = m_{\max}$ and $\Theta$ as above,
in (\ref{E:joinbound}) we have for such configurations $N(\omega_{\omB(\Gamma)
\bs\Theta}) \geq M d(x,(\partial \Gamma)_j)/3$. Therefore
\begin{align} \label{E:largeint}
P(x &\lra (\partial \Gamma)_j, |C_x \cap \mD_m| > M \text{ for all }
m_{\min} \leq m \leq m_{\max}) \\
&\leq \sum_{\Theta \subset \mD_{m_{\max}}} P\left( B_{\Theta} \cap
\left[ N(\omega_{\omB(\Gamma)
\bs\Theta}) \geq \frac{1}{3} M d(x,(\partial \Gamma)_j)) \right] \right)
\notag \\
&\leq 2^{|\mD_{m_{\max}}|} \left( \frac{p}{1-p} \vee 1
\right)^{|\mD_{m_{\max}}|} (1-p)^{-h_{k+1} M d(x,(\partial \Gamma)_j)/3}
\notag \\
&\leq c_{21} (1-p)^{-h_{k+1} M d(x,(\partial \Gamma)_j)/6} \notag
\end{align}
provided $M$ is sufficiently large. With (\ref{E:smallint2})
this completes the
proof.
\end{proof}
\begin{proof}[Proof of Proposition \ref{P:dualnonunsm}]
Let $\Lambda$ be a finite subset of $\ZZ$ and $\mE, \mF \subset \omB(\Lambda)$,
let $\eta$ be the free or wired boundary condition on $\pL$ and let $\rho_{\mE},
\rho^{\prime}_{\mE}$ be bond configurations on $\mE$. We wish to show roughly
that
$P_{\omB(\Lambda),\eta}(\omega_{\mF} \in \cdot \mid \omega_{\mE} = \rho_{\mE})$
and $P_{\omB(\Lambda),\eta}(\omega_{\mF} \in \cdot \mid \omega_{\mE} =
\rho^{\prime}_{\mE})$ differ by an exponentially small amount. Let $\mathbb{P}$
denote the distribution of the Edwards-Sokal joint Potts/FK configuration
$(\sigma,\omega)$ on $\omB(\Lambda)$. Define
\[
U(\sigma) = \{x \in V(\mE): \sigma_x \text{ is stable} \}
\]
and consider the measures
\[
\mathbb{P}( \cdot \mid \omega_{\mE} = \rho_{\mE}, U(\sigma)
= U_0), \quad \mathbb{P}( \cdot \mid \omega_{\mE} =
\rho_{\mE}^1, U(\sigma) = \phi)
\]
where $U_0 \subset V(\mE)$ is \emph{compatible with} $\rho_{\mE}$,
that is, $U_0$
contains either all or none of each cluster of $\rho_{\mE}$. We call the first
of these measures the \emph{lower measure} and the second the \emph{upper
measure}. It is sufficient to show that for arbitrary compatible
$U_0$, these two measures can be coupled so that the corresponding bond
configurations agree on
$\mF$, except with an exponentially small probability. Note that if $\eta$ is
the wired boundary condition, the Potts model under
$\mathbb{P}$ is conditioned on the event that all boundary spins are the same
stable species. Also, we may assume $\mE = \mB(E) \cap \omB(\Lambda)$ for some
$E \subset \overline{\Lambda}$. Then
$\omB(\Lambda) \bs \mE = \omB(\Lambda \bs E)$.
The coupling is constructed as follows. The lower and upper bond
configurations are denoted $\omega_{\omB(\Lambda) \bs \mE}$ and
$\omega_{\omB(\Lambda) \bs \mE}^{\prime}$,
respectively. First we use a stopped construction algorithm to create
$\overline{C}_{U_0}(\omega_{\omB(\Lambda) \bs \mE})$,
the closure of the cluster of
$U_0$, in the lower FK configuration $\omega_{\omB(\Lambda) \bs \mE}$.
Suppose the resulting
value of the cluster closure is $\overline{C}_{U_0}(\omega_{\omB(\Lambda)
\bs \mE}) = \omB(G)$ for some
$G \subset \Lambda \bs E$. (Note the cluster closure always has such a form for
some $G$.) We then choose the stable part of the upper
bond configuration on this
same set
$\omB(G)$, using the SPFK model, making this choice of
upper SPFK configuration independently of the lower FK configuration we
selected on $\omB(G)$. Let $\mE_+ = \mE \cup \omB(G)$. Note that in the lower
configuration, all bonds in
$\partial\mB(G) \cap \omB(\Lambda) \bs \mE$ are closed. We now treat the
upper and lower configurations on
$\omB(G)$ as parts of extended boundary conditions for the remaining upper
and lower configurations on $\omB(\Lambda) \bs \mE_+$.
(Here ``extended'' refers
to the fact that these boundary conditions are configurations on
$\omB(\Lambda)^c \cup \mE_+$, not just on $\omB(\Lambda)^c$.) Since all
of the open bonds relevant to these extended boundary conditions have stable
species at their endpoints, we can treat them as
boundary conditions for the SPFK
model, rather than for the full FK model. (Those with unstable species at their
endpoints are irrelevant because they cannot have
endpoints in $\omB(\Lambda) \bs
\mE_+$.) Let $\tilde{\omega}_{\omB(\Lambda) \bs
\mE_+}$ and $\tilde{\omega}_{\omB(\Lambda) \bs \mE_+}^{\prime}$ respectively
denote the lower and upper SPFK configurations. Note that the extended boundary
condition for the upper configuration is always larger, in the usual ordering,
than the extended boundary condition for the lower configuration. Therefore by
Lemma
\ref{L:SPFK-FKG}, the upper SPFK configuration on $\omB(\Lambda) \bs \mE_+$
FKG-dominates the lower one.
Note in part of what follows, we will use the domain $\mB^+(\Lambda)$ instead of
$\omB(\Lambda)$, because for $\eta$ free we do not want to allow the use of the
boundary dual bonds $(\partial \mB(\Lambda))^*$, which are always open,
in forming our dual paths.
We next use another stopped construction algorithm to create
$\overline{C^*_{\mE_+^*}} (\tilde{\omega}_{\mB^+(\Lambda) \bs \mE_+})$,
the closure of the dual cluster of
$\mE_+^*$ in the lower SPFK configuration $\tilde{\omega}_{\mB^+(\Lambda) \bs
\mE_+}$, while simultaneously creating the upper SPFK configuration on
the same (random) set of dual bonds. By the FKG domination, this can be done
with the upper SPFK configuration larger than the lower one. Suppose the
resulting values of this dual-cluster closure and the dual-cluster boundary (in
$\mB^+(\Lambda) \bs \mE_+$) are $\overline{C_{\mE_+^*}^*}
(\tilde{\omega}_{\mB^+(\Lambda) \bs \mE_+}) = \mH^*$ and
$\partial C_{\mE_+^*}^* (\tilde{\omega}_{\mB^+(\Lambda) \bs \mE_+}) =
\mD^*$ for some $\mD, \mH \subset \mB^+(\Lambda) \bs \mE_+$. Then all (regular)
bonds in $\mD$ are open, in both the upper and lower SPFK configurations. Let
$\mE_{++} = \mE_+ \cup \mH$. As before with $\mE_+$, the upper and lower SPFK
configurations on $\mE_{++}$ can each be treated as further-extended boundary
conditions for the corresponding configurations on $\omB(\Lambda) \bs \mE_{++}$.
Because all bonds in $\mD$ are open, these further-extended boundary conditions
are both the same---wired on each component of $\mD$ if
$\eta$ is free, and fully
wired if $\eta$ is wired. Here we are using an extension, analogous to
(\ref{E:Markovextn}), of the Markov property for open
surfaces valid for the SPFK
model---see Remark \ref{R:interest}---and for $\eta$ free we are using simple
lattice-connectedness of $\omB(\Lambda)$ to ensure that no two components of
$\mD$ are connected to each other both by an open path in $\mE_{++}$ and by a
path in $\omB(\Lambda)$ outside $\mE_{++}$. Even when the SPFK
configurations we
have constructed on portions of
$\mE_{++}$ are extended to full FK configurations on the same portions of
$\mE_{++}$, the boundary condition for configurations on
$\omB(\Lambda) \bs \mE_{++}$ is not affected. (Here ``portions of $\mE_{++}$''
means $\mH$ in the lower configuration and all of $\mE_{++}$ in the upper
configuration.) Thus in completing the upper and
lower FK configurations we can choose $\omega_{\omB(\Lambda) \bs \mE_{++}}$ and
$\omega_{\omB(\Lambda) \bs \mE_{++}}^{\prime}$ to be equal. The set $\mE_{++} =
\mE_{++}(\omega_{\omB(\Lambda) \bs \mE})$ is of course random, and we have
\begin{align} \label{E:agree}
|\mathbb{P}&(F \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0) -
\mathbb{P}(F \mid \omega_{\mE} = \rho_{\mE}^1, U(\sigma) = \phi)| \\
&\leq \mathbb{P}(\mE_{++}(\omega_{\omB(\Lambda) \bs \mE}) \cap \mF \neq
\phi \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0),
\quad F \in \mG_{\mF}.
\notag
\end{align}
For $0 < \alpha < 1$ let
\[
\Gamma(\mE,\alpha) = \left\{ x \in \Lambda: d(x,\mE) \leq \frac{\alpha}{1 -
\alpha} d(x,\mF) \right\},
\]
\[
\Gamma^*(\mE,\alpha) = \left\{ x \in V^*(\omB(\Lambda)): d(x,\mE) \leq
\frac{\alpha}{1 - \alpha} d(x,\mF) \right\}.
\]
These are the points which are, roughly speaking, at most $\alpha$ fraction
of the way from $\mE$ to $\mF$. Then
\begin{align} \label{E:twoways}
\mathbb{P}&(\mE_{++}(\omega_{\omB(\Lambda) \bs \mE}) \cap \mF \neq
\phi \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0) \\
&\leq
\mathbb{P}\left( \mE_+(\omega_{\omB(\Lambda) \bs \mE}) \cap
\Gamma\biggl( \mE,\frac{1}{3} \biggr)^c
\neq \phi \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0 \right) \notag \\
&\qquad + \mathbb{P}\left( \mE_+(\omega_{\omB(\Lambda) \bs \mE}) \cap
\Gamma\biggl( \mE,\frac{1}{3} \biggr)^c
= \phi, \mE_{++}(\omega_{\omB(\Lambda) \bs \mE}) \cap \mF \neq \phi
\mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0 \right) \notag
\end{align}
Since conditioning on $\omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0$ is just an
average of certain site boundary conditions, Lemma \ref{L:unstsite} yields that
for some $c_{22},\epsilon_{8}$,
\begin{equation} \label{E:longunstbl}
\mathbb{P}\left( \mE_+(\omega_{\omB(\Lambda) \bs \mE}) \cap
\Gamma\biggl( \mE,\frac{1}{3} \biggr)^c
\neq \phi \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0 \right)
\leq c_{22}\sum_{x \in \mE, y \in
\Gamma(\mE,\frac{1}{3})^c} e^{-\epsilon_{8}
d(x,y)}
\end{equation}
for some constants $c_i$. As in the proof of (\cite{Al98mix}, Theorem 3.3) we
have for some $c_{23},\epsilon_{9}$,
\begin{equation} \label{E:sums}
\sum_{x \in \mE, y \in \Gamma(\mE,\frac{1}{3})^c} e^{-\epsilon_{8} d(x,y)}
\leq c_{23} \sum_{x \in \mE, z \in \mF} e^{-\epsilon_{9} d(x,z)}.
\end{equation}
Define for $x \notin \mE$
\[
B_x = \{y \in \RR: d(x,y) \leq \frac{1}{4}d(x,\mE) \}, \quad
B_x^+ = \{y \in \RR: d(x,y) \leq \frac{1}{4}d(x,\mE) + 2 \}
\]
By Lemma \ref{L:SPFKweakmix}, the SPFK model has
uniform exponential decay of finite-volume dual connectivities for the class of
all simply lattice-connected subsets of $\mB(\ZZ)$ with arbitrary bond boundary
conditions. Letting $k$ be the number of stable species, it follows readily
that
\begin{align} \label{E:longdual}
\mathbb{P}&\left( \mE_+(\omega_{\omB(\Lambda) \bs \mE}) \cap
\Gamma\left( \mE,\frac{1}{3} \right)^c
= \phi, \mE_{++}(\omega_{\omB(\Lambda) \bs \mE}) \cap \mF \neq \phi
\mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0 \right) \\
&\leq \sup_{\rho} P^{p,q,k,\{h_i\}}_{SPFK,\omB(\Lambda) \bs
\omB(\Gamma(\mE,\frac{1}{3})),\rho}
(\text{there exists an open dual path which starts within }
\notag \\
&\qquad \qquad \qquad \qquad \qquad \text{distance 1 of }
\Gamma(\mE,\tfrac{1}{3}) \text{ and ends
within distance 1 of } \mF) \notag \\
&\leq \sup_{\rho} \sum_{x \in \partial \Gamma^*(\mE,\tfrac{2}{3})}
P^{p,q,k,\{h_i\}}_{SPFK,\omB(\Lambda) \bs
\omB(\Gamma(\mE,\frac{1}{3})),\rho} (x \lrad B_x^c)
\notag \\
&\leq \sup_{\rho} \sum_{x \in \partial \Gamma^*(\mE,\tfrac{2}{3})}
P^{p,q,k,\{h_i\}}_{SPFK,\omB(\Lambda) \cap \mB(B_x^+),\rho} (x \lrad B_x^c).
\notag \\
&\leq \sum_{x \in \partial \Gamma^*(\mE,\tfrac{2}{3})} c_{24}
e^{-\epsilon_{10}
d(x,\mE)/4} \notag
\end{align}
where the sup is over all bond boundary conditions, and
where for the last inequality we use Theorem \ref{T:percmain} and the fact that
$\omB(\Lambda) \cap
\mB(B_x^+)$ is simply lattice-connected. Similarly to (\ref{E:sums}), we have
\begin{equation} \label{E:sums2}
\sum_{x \in \partial \Gamma^*(\mE,\tfrac{2}{3})} c_{24} e^{-\epsilon_{10}
d(x,\mE)/4} \leq \sum_{y \in \mE,z \in \mF} c_{25} e^{-\epsilon_{11}d(y,z)}.
\end{equation}
Now (\ref{E:agree})---(\ref{E:sums2}) prove that
\begin{align} \label{E:disagree}
|\mathbb{P}&(F \mid \omega_{\mE} = \rho_{\mE}, U(\sigma) = U_0) -
\mathbb{P}(F \mid \omega_{\mE} = \rho_{\mE}^1, U(\sigma) = \phi)| \\
&\leq c_{26} \sum_{x \in \mE, y \in \mF} e^{-\epsilon_{12} d(x,y)}. \notag
\end{align}
Since $U_0$ is arbitrary, this shows that
\begin{equation} \label{E:disagree2}
|P_{\omB(\Lambda),\eta} (F \mid \omega_{\mE} = \rho_{\mE}) -
P_{\omB(\Lambda),\eta}(F \mid \omega_{\mE}
= \rho^{\prime}_{\mE})|
\leq c_{26} \sum_{x \in \mE, y \in \mF} e^{-\epsilon_{12} d(x,y)},
\end{equation}
which completes the proof.
\end{proof}
To move from strong mixing to ratio strong mixing in our varied
contexts, we need
to extend some definitions and results in \cite{Al98mix}. A \emph{component
partition} of a set
$\mE \subset \mB(\ZZn)$ is a partition
$\mE = \mE_1 \cup .. \cup \mE_m$ such that
each component of $\mE$ is contained in some $\mE_i$.
For a bond percolation model $P$ on
$\mB(\ZZn)$, for $\Lambda \subset \ZZn$ finite, $\eta$ a site boundary condition
on $\pL$, $\mB \subset \omB(\Lambda)$, $\omB(\Lambda) \bs \mB = \mE \cup
\mF$ a component partition, $\epsilon > 0$, and $\rho \in
\{0,1\}^{\omB(\Lambda)}$ we say
$\mB$ is an
$\epsilon$-\emph{near blocking region for} $\mE, \mF$ \emph{in} $\rho$ if for
every configuration $\rho^{\prime}$ such that
$\rho = \rho^{\prime}$ on $\mB$ and
every event $A \in
\mG_{\mE}$ we have
\[
(1 - \epsilon)P_{\Lambda,\eta}(A \mid \omega_{\mB \cup \mF}
= \rho_{\mB \cup \mF}^{\prime}) \leq
P_{\Lambda,\eta}(A \mid \omega_{\mB \cup \mF} = \rho_{\mB \cup \mF})
\leq (1 + \epsilon)
P_{\Lambda,\eta}(A \mid \omega_{\mB \cup \mF} =
\rho_{\mB \cup \mF}^{\prime}).
\]
In other words, the configuration on $\mB$ blocks the
configuration on $\mF$ from
influencing probabilities for events on $\mE$ by more than a factor of
$1 \pm \epsilon$. A 0-blocking region is also called a \emph{fully blocking
region}. Blocking regions are the finite-volume analogs of contolling regions.
Let $\mkC$ be a class of subsets of $\mB(\ZZn)$.
We say that $P$ has \emph{exponentially bounded blocking regions} for
the class $\mkC$ and metric $d$ if there exist $C, \lambda$ such that for every
$\Lambda, \eta$ and $\mB \subset
\omB(\Lambda)$ with $(\omB(\Lambda),\eta) \in\mkC$ and every component partition
$\omB(\Lambda) \bs \mB =
\mE \cup \mF$, for $\epsilon = C \sum_{x \in V(\mE),y \in V(\mF)} e^{-\lambda
d(x,y)}$, we have
\begin{equation} \label{E:blocking}
P_{\omB(\Lambda), \eta}(\mB \text{ is not an $\epsilon$-near blocking region
for }
\mE, \mF) < \epsilon.
\end{equation}
We say that $P$ has \emph{exponentially bounded fully blocking regions} for
the metric $d$ if in place of (\ref{E:blocking}) we
have
\[
P_{\omB(\Lambda),\eta}(\mB \text{ is not a fully blocking region for }
\mE, \mF) < \epsilon.
\]
Essentially the same definition applies under bond boundary conditions.
\begin{lemma} \label{L:nearblock}
Let $P=P^{p,q,\{h_i\}}$ be an FK model on $\mB(\ZZn)$,
let $d$ be the Euclidean
or restricted-path metric and let
$\mkC$ be a class of closure subsets of $\ZZn$ which is
inheriting with respect
to $d$.
(i) If $P$ has uniform
exponential decay of finite-volume connectivities for the class $\mkC$ with
arbitrary site boundary conditions,
then $P^{p,q,\{h_i\}}$ has
exponentially bounded fully blocking regions for the class $\mkC$ with
arbitrary site boundary conditions, in the metric $d$.
(ii) If $n=2$, $\mkC$ is the class of all simply lattice-connected
closure subsets and
$P$ has
exponential decay of connectivities, then $P^{p,q,\{h_i\}}$ has
exponentially bounded fully blocking regions for the class $\mkC$ with
arbitrary site boundary conditions, in the metric $d$.
(iii) If $n=2$ and
$P$ has a unique stable species, then
$P^{p,q,\{h_i\}}$ has
exponentially bounded blocking regions for the class $\mkC$ with arbitrary
single-stable-species site
boundary conditions, in the metric $d$. If $d$ is the Euclidean metric, then
arbitrary bond boundary conditions can be included as well.
(iv) If $n = 2$, $\mkC$ is the class of all simply
lattice-connected subsets and
$P^{p,q,\{h_i\}}$ has
exponential decay of dual connectivities, then $P^{p,q,\{h_i\}}$ has
exponentially bounded blocking regions for the class
$\mkC$ with free and wired
boundary conditions, in the metric $d$.
\end{lemma}
\begin{proof}
Fix $\Lambda \subset \ZZn$ finite, $\eta$ a site boundary condition
on $\pL$, $\mB \subset \omB(\Lambda)$, and $\omB(\Lambda) \bs \mB = \mE \cup
\mF$ a component partition.
If in a configuration $\omega \in \{ 0,1
\}^{\omB(\Lambda)}$ there is no open path from $V(\mE)$ to $V(\mF)$, then there
exists a collection of dual surfaces which together separate $\mE$ from $\mF$,
such that none of these dual surfaces is crossed by an open bond in
$\omB(\Lambda)$. (These dual surfaces may include
plaquettes dual to bonds outside $\omB(\Lambda)$.) It therefore follows from a
straightforward minor extension of the Markov property for open dual circuits
(cf. (\ref{E:Markovextn})) that
\begin{align} \label{E:blockcond}
P_{\omB(\Lambda),\eta}&(\mB \text{ is not a fully blocking region for }
\mE, \mF) \\
&\leq P_{\omB(\Lambda),\eta}(x \lra y \text{ for some } x \in V(\mE), y
\in V(\mF) ), \notag
\end{align}
and (i) follows.
Under the assumptions of (ii), it follows from Theorem \ref{T:FKcase} that
the assumptions of (i) are satisfied.
Under the assumptions of (iii), with single-stable-species site boundary
condition $\eta$, let
$\rho, \rho^{\prime}$ be configurations such that $\rho = \rho^{\prime}$ on
$\mB$, and let $\Theta = \{ x \in \pL: \eta_x \neq 0 \}$ be the set of boundary
sites where the stable species resides. The difference between the
measures
$P_{\omB(\Lambda),\eta}(\omega_{\mE}
\in \cdot \mid \omega_{\mB \cup \mF} = \rho_{\mB \cup \mF})$ and
$P_{\omB(\Lambda),\eta}(\omega_{\mE} \in \cdot \mid
\omega_{\mB \cup \mF} = \rho_{\mB \cup \mF}^{\prime})$ appears
only in the weight
assigned to clusters of $(\omega_{\mE},\rho_{\mB \cup \mF})$ which intersect
$V(\mE)$ and $V(\mF)$ but not
$\Theta$. More precisely, considering the first measure, let
$\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup
\mF})$ denote the set of all such clusters. For $C \in
\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})$ let $\hat{s}(C) = s(C)$ if $C
\cap \Theta = \phi$ and $\hat{s}(C) = \infty$ if $C \cap \Theta \neq \phi$.
Then the combined weight
assigned to all clusters in $\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})$
under the measure
$P_{\omB(\Lambda),\eta}(\omega_{\mE}
\in \cdot \mid \omega_{\mB \cup \mF} = \rho_{\mB \cup \mF})$ is (cf.
(\ref{E:FKexternal})---(\ref{E:FKextsite}))
\begin{align} \label{E:factor}
\prod_{C \in \mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})} &\left(
1 + (1-p)^{- h_2 \hat{s}(C)} + \ldots + (1-p)^{- h_{q+1} \hat{s}(C)} \right)
\\
&\leq \prod_{C \in \mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})} \left(
1 + q(1 - p)^{-h_2 s(C)} \right) \notag \\
&\leq \prod_{x \in V(\mE)} \left( 1 + c_{28}e^{-\epsilon_{13} d(x,V(\mF))}
\right) \notag
\end{align}
since $h_2 < 0$; here we interpret $(1-p)^{\infty}$ as 0. Let
\[
\delta = \sum_{x \in V(\mE)} 2c_{28}e^{-\epsilon_{13} d(x,V(\mF))}
\]
and suppose $\delta \leq 1$. Then by (\ref{E:factor}), the left side of
(\ref{E:factor}) is between 1 and $1 + \delta$, and the same holds with
$\rho_{\mB \cup \mF}$ replaced by $\rho_{\mB \cup \mF}^{\prime}$. It follows
that, in every configuration $\omega_{\omB(\Lambda)}$, $\mB$ is a $\delta$-near
blocking region for $\mE,\mF$. On the other hand, if $\delta > 1$, then
certainly $P_{\omB(\Lambda),\eta}(\mB \text{ is not a
$\delta$-near blocking region
for } \mE, \mF) < \delta$. This proves (iii) for single-stable-species site
boundary conditions.
Next, consider (iii) with $d$ Euclidean and a bond boundary condition
$\tilde{\rho}$. We proceed similarly to the site boundary condition case but in
place of
$\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})$ we use the class
$\mC_{\mE,\mF}^1(\omega_{\mE},\rho_{\mB \cup \mF})$ consisting of all clusters
of $(\omega_{\mE},\rho_{\mB \cup \mF},\tilde{\rho})$ which intersect both
$V(\mE)$ and $V(\mF)$.
(Such clusters now need not pass through $\mB$ and may connect $V(\mE)$
to $V(\mF)$ via the boundary configuration $\tilde{\rho}$.) As before, the
difference between the measures given $\rho_{\mB\cup
\mF} $ and given $\rho_{\mB \cup \mF}^{\prime}$ lies only in the combined weight
assigned to such clusters. Since $d$ is the Euclidean distance, for $C \in
\mC_{\mE,\mF}^1(\omega_{\mE},\rho_{\mB \cup \mF})$ and $x \in C \cap V(\mE)$ we
have $s(C) \geq d(x,V(\mF))$. Therefore proceeding as in (\ref{E:factor}) we
obtain (iii).
Finally, under the assumptions of (iv), we again consider the class
$\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF})$. Let $k$ be the number of
stable species. Suppose there is no open dual path from $V^*(\mE^*)$ to
$V^*(\mF^*)$ in
$\mB^+(\Lambda)^* \cap \mB^*$. If $\eta$ is wired, this means every cluster
intersecting both $V(\mE)$ and $V(\mF)$ must also intersect $\pL$ in $\mB$.
This in turn means $\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup \mF}) = \phi$
and hence $P_{\omB(\Lambda),\eta}(\omega_{\mE}
\in \cdot \mid \omega_{\mB \cup \mF} = \rho_{\mB \cup
\mF}) = P_{\omB(\Lambda),\eta}(\omega_{\mE} \in \cdot \mid
\omega_{\mB \cup \mF} = \rho_{\mB \cup \mF}^{\prime})$, so $\mB$ is a fully
blocking region. Thus
\begin{align}
P_{\omB(\Lambda),\eta}&(\mB \text{ is not a fully blocking region
for } \mE, \mF) \notag \\
&\leq P_{\omB(\Lambda),\eta}(V^*(\mE^*) \lrad V^*(\mF^*)) \notag \\
&\leq \sum_{x \in V(\mE), y \in V(\mF)} c_{29}e^{-\epsilon_{14} d(x,y)},
\notag
\end{align}
completing the proof for $\eta$ wired.
If instead $\eta$ is free, the absense of an open dual path from
$V^*(\mE^*)$ to $V^*(\mF^*)$ in
$\mB^+(\Lambda)^* \cap \mB^*$ means that for
each connected component
$\mE_i$ of $\mE$ and each connected component $\mF_j$ of $\mF$, there is at most
one cluster in
$\mC_{\mE,\mF}^0(\omega_{\mE},\rho_{\mB \cup
\mF})$ which intersects both $V(\mE_i)$ and $V(\mF_j)$. If this cluster, call
it $C_{ij}$, exists, the weight assigned to it is
\[
k + (1-p)^{- h_{k+1} s(C_{ij})} + \ldots + (1-p)^{- h_{q+1} s(C_{ij})}
\]
which is between $k$ and $k + qe^{-\epsilon_{103} d(\mE_i,\mF_j)}$, uniformly in
$\omega_{\omB(\Lambda)}$. This means that $\mB$ is an $\epsilon$-near blocking
region for $\mE, \mF$, for
\[
\epsilon = \sum_{i,j} \epsilon_{12}e^{-\epsilon_{104} d(\mE_i,\mF_j)}
\leq \sum_{x \in V(\mE), y \in V(\mF)}
\epsilon_{12}e^{-\epsilon_{104} d(x,y)}.
\]
Thus we have
\begin{align}
P_{\omB(\Lambda),\eta}&(\mB \text{ is not an $\epsilon$-near blocking region
for } \mE, \mF) \notag \\
&\leq P_{\omB(\Lambda),\eta}(V^*(\mE^*) \lrad V^*(\mF^*)) \notag \\
&\leq \sum_{x \in V(\mE), y \in V(\mF)} c_{29}e^{-\epsilon_{14} d(x,y)},
\notag
\end{align}
completing the proof for $\eta$ free.
\end{proof}
The weak-mixing analog of the following theorem is (\cite{Al98mix}, Theorem
3.3). The proof for strong mixing is essentially the same so we
do not include it
here.
\begin{theorem} \label{T:ratiostrong}
Let $P$ be a bond percolation model on $\mB(\ZZn)$, let $\mkC$ be a class
of subsets of $\mB(\ZZn)$ together with (site or bond) boundary conditions
and let $d$ be the Euclidean or restricted-path metric.
Suppose $P$ has the strong mixing property for the class $\mkC$ in the metric
$d$, and suppose $P$ has exponentially bounded blocking regions. Then $P$ has
the ratio strong mixing property for the class $\mkC$ in the metric $d$.
\end{theorem}
\begin{remark} \label{R:spinanalog}
The obvious spin-system analog of Theorem \ref{T:ratiostrong} is also valid. In
particular, nearest-neighbor systems such as the Potts model always have
exponentially bounded blocking regions, so that for such systems, strong mixing
implies ratio strong mixing, for any class $\mkC$.
\end{remark}
Write $x \lraf y$ for
the event $[x \lra y, x \not\lra \infty]$ (if the context is infinite volume)
or the event $[x \lra y, x \not\lra \pL]$ (if the context is a finite volume
$\Lambda$.) Let $e_i$ denote the $i$th unit coordinate vector.
The following lemma was proved in
\cite{BC96} in the absense of external fields. The proof there also works under
external fields, but it is complex and interwoven with other proofs, so we
present a short direct proof here.
\begin{lemma} \label{L:finiteconn}
Consider an FK model $P=P^{p,q,\{ h_i \}}$ on $\mB(\ZZ)$ with $p >
p_c(q,2,\{ h_i
\})$. If $P(0 \lraf x)$ decays exponentially in $|x|$, then
$P$ has exponential decay of dual connectivities.
\end{lemma}
\begin{proof}
Suppose the dual connectivity does not decay exponentially, that is, $\tau(x) =
0$ for all $x$. We claim first that the dual connectivity also does not decay
exponentially in halfspaces. Let $\epsilon > 0$ and let $p_{\infty} = P(0 \to
\infty)$,
$H_t = \{(x_1,x_2): x_2 \leq t \}$ and $\tilde{H}_t = \{(x_1,x_2): x_1 \geq
t\}$. Let $k$ be large enough so $P(0^*
\lrad (ke_1)^*) \geq e^{-\epsilon k}$. When there is an open dual path from
$0^*$ to
$(ke_1)^*$ we choose an uppermost point $Y$ of the path,
making an arbitrary choice of path and/or of $Y$ if more than one choice is
possible. We may assume $y_1 \geq k/2$; the other case is symmetric. There
is also a leftmost point of the path, which we call $X$. Let
\[
c_{30} = \left( \sum_{x \in \ZZ} \frac{1}{(1 + |x|)^3} \right)^{-1}.
\]
There must exist $x,y \in (\ZZ)^*$ such that
\begin{align}
P&(0^* \lrad (ke_1)^*, X = x, Y = y) \notag \\
&\geq \frac{c_{30}}{(1 + |x|)^3}\frac{c_{30}}{(1 + |y-x|)^3}
P(0^* \lrad (ke_1)^*) \notag \\
&\geq \frac{c_{30}}{(1 + |x|)^3}\frac{c_{30}}{(1 + |y-x|)^3} e^{-\epsilon k}.
\end{align}
Consider first the case of $|x| \leq |y-x|$. We have two
subcases, according to
which of
$|y_1 - x_1|, |y_2 - x_2|$ is larger; we may assume $|y_1 - x_1| \geq |y_2 -
x_2|$, as the other subcase is similar after a $90^{\circ}$ rotation. We have
\[
P(x \lrad y \text{ via an open dual path in } H_{y_2}) \geq P(0^* \lrad
(ke_1)^*, X = x, Y = y)
\geq \frac{c_{31}}{|y-x|^6} e^{-\epsilon k}.
\]
Let $z$ be the reflection of $y$ across the vertical line through $x$, and $m =
y_1 - x_1$. Then since $m \geq k/2$,
\begin{align} \label{E:noexp}
P&(0^* \lrad (2me_1)^* \text{ via an open dual path in } H_{1/2}) \\
&= P(z \lrad y \text{ via an open dual path in } H_{y_2}) \notag \\
&\geq P(x \lrad y \text{ via an open dual path in } H_{y_2})^2 \notag \\
&\geq \frac{c_{32}^2}{m^{12}} e^{-4\epsilon m}. \notag
\end{align}
Since $\epsilon$ is arbitrary, it follows that the left side of (\ref{E:noexp})
does not decay exponentially in $m$, proving the claim.
The other case, $|y-x| \leq |x|$, is similar, using $0^* \lrad x$ in place of $x
\lrad y$.
Since $p \neq p_c(q,2,\{h_i\})$, random cluster uniqueness holds \cite{BBCK}.
Fix
$l \geq 1$; we have by the FKG property
\[
P(0 \lra le_1) \geq P(|C_0| = \infty, |C_{le_1}| = \infty) \geq p_{\infty}^2.
\]
Let $\Theta_{j,r} = [-r,j+r] \times [-r,r]$. By random cluster uniqueness, if
$r$ is sufficiently large (depending on $l$),
\[
\inf_{\rho} P_{\Theta_{l,r},\rho}(0 \lra le_1 \text{ via an open path in }
\Theta_{l,r}) \geq \half p_{\infty}^2,
\]
where the inf is over all boundary conditions $\rho$ on $\Theta_{l,r}$. From
this we obtain, using the FKG property again and choosing $r$ of form $s + 1/2$
for some integer $s$, that for $\delta > 0$ and $r,n$ sufficiently large,
\begin{align} \label{E:noexp2}
P&(0 \lraf nle_1) \notag \\
&\geq P \Bigl(0 \lra nle_1 \text{via an open path in }
\Theta_{nl,r}, \notag \\
&\qquad \qquad (-r,-r) \lrad
(-r,r) \lrad (nl + r,r) \lrad (nl + r,-r) \lrad (-r,-r) \notag \\
&\qquad \qquad \text{ via open dual
paths outside } \Theta_{nl,r} \Bigr) \notag \\
&\geq \left( \half p_{\infty}^2 \right)^n e^{-\delta (8r + 2nl)}. \notag
\end{align}
Since $\delta$ and $l$ are arbitrary, it follows that $P(0 \lraf ne_1)$ does
not decay exponentially in $n$.
\end{proof}
Let $\mu^{\beta,\{ h_i \}}_{\ZZ,i}$ denote the infinite-volume Potts model on
$\ZZ$ at
$(\beta,\{ h_i\})$ with species-$i$ boundary
condition, for stable $i$. Part (i)
of the next lemma is well-known in the absense of external fields.
\begin{lemma} \label{L:corrconn}
Consider the $q$-state Potts model $\mu^{\beta,q,\{ h_i \}}$ on $\ZZ$.
Suppose there are multiple stable species.
(i) Suppose Gibbs uniqueness holds and $\mu^{\beta,q,\{ h_i \}}$ has
exponential decay of
correlations. Then the corresponding FK model has exponential decay of
connectivities.
(ii) Suppose $\beta > \beta_c(q,2,\{ h_i \})$ and $\mu^{\beta,q,\{ h_i
\}}_{\ZZ,1}$ has exponential decay of
correlations. Then the corresponding FK model has exponential decay of
dual connectivities.
\end{lemma}
\begin{proof}
Suppose there are $k$ stable species, and let $\mathbb{P}$ denote the
distribution of the corresponding Edwards-Sokal joint
Potts-FK configuration in a
finite volume $\Lambda$ with all-1 boundary condition. Under $\mathbb{P}$ we
have conditional covariance, given the bond configuration, given by
\begin{align}
\cov(\delta_{[\sigma_x = 1]},\delta_{[\sigma_y = 1]} \mid \omega)
&= \begin{cases}
\frac{1}{k+r_n}\left(1 - \frac{1}{k+r_n}\right), &\text{if } x \lraf y
\text{ and } s(C_x) = s(C_y) = n\\
0, &\text{otherwise}
\end{cases} \notag \\
&\geq \frac{1}{k+r_1}\left( 1 - \frac{1}{k+r_1} \right) \delta_{[x \lraf y]}
\notag
\end{align}
and conditional expectation
\[
M_x = \mathbb{E}(\delta_{[\sigma_x = 1]} \mid \omega)
= 1 - \left(1 - \frac{1}{k+r_{s(C_x)}} \right)\delta_{[x \not\lra \pL]}.
\]
Since this conditional expectation
is an increasing function of $\omega$, by the FKG property of the FK model we
have $\cov(M_x,M_y) \geq 0$ for all $x,y$. Therefore
\[
\cov(\delta_{[\sigma_x = 1]},\delta_{[\sigma_y = 1]}) \geq
\frac{1}{k+r_1}\left( 1 - \frac{1}{k+r_1} \right) P(x \lraf y),
\]
where $P$ is the finite-volume FK measure.
This same inequality then holds in infinite volume. (Note the infinite-volume
FK measure is necessarily unique under both
(i) and (ii)---see the remarks preceding Proposition \ref{P:dualsitesm}.)
It follows that
$P(x \lraf y)$ decays exponentially, This
proves (i). Under (ii) there is percolation in the FK model, so (ii) follows
from Lemma \ref{L:finiteconn}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:siteratiosm}]
The theorem follows from Proposition \ref{P:sitesm}, Lemma \ref{L:nearblock}(i),
and Theorem \ref{T:ratiostrong}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:siteratiosm2}]
(i) is an immediate consequence of Theorems \ref{T:FKcase} and
\ref{T:siteratiosm}. (ii) follows from Proposition \ref{P:dualnonunsm}, Lemma
\ref{L:nearblock}(iv), and Theorem \ref{T:ratiostrong}. (iii) follows from
Proposition \ref{P:dualsitesm}(iii), Lemma \ref{L:nearblock}(iii) and Theorem
\ref{T:ratiostrong}.
\end{proof}
Note that we don't need Proposition \ref{P:dualnonunsm} in the proof of Theorem
\ref{T:siteratiosm2} if there are no external fields, as (ii) then follows from
(i) and duality.
\begin{proof}[Proof of Theorem \ref{T:spinmain}]
Let $P$ be the corresponding FK model.
Under the hypotheses of (i), by Lemma \ref{L:corrconn}(i) $P$
has exponential decay of connectivities. Hence under (i) or (ii), by
Proposition \ref{P:dualsitesm}(i), $\mu^{\beta,q,\{h_i\}}$ has the strong
mixing property for the class $\mkC$ with arbitrary boundary conditions. By
Remark \ref{R:spinanalog}, the desired ratio strong mixing property holds.
Under (iii), Gibbs uniqueness holds, so by Proposition \ref{P:dualsitesm}(iii),
$\mu^{\beta,q,\{h_i\}}$ has the strong
mixing property for the class $\mkC$ with arbitrary boundary conditions. Again
by Remark \ref{R:spinanalog}, the desired ratio strong mixing property holds.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:spinZdconn}]
By the method of proof of Proposition \ref{P:dualsitesm}(i) (noting that,
due to our stronger assumption, we do not need Theorem \ref{T:FKcase}), using
the FK model, $\mu^{\beta,q,\{h_i\}}$ has the strong
mixing property for the class $\mkC$ with arbitrary boundary conditions. By
Remark \ref{R:spinanalog}, the desired ratio strong mixing property holds.
\end{proof}
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\end{document}