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\begin{document}
\newcommand{\mB}{\mathcal{B}}
\newcommand{\omB}{\overline{\mathcal{B}}}
\newcommand{\delz}{\delta_{[\sigma_{x} = 0]}}
\newcommand{\omr}{\omega_{r}}
\newcommand{\omg}{\omega_{g}}
\newcommand{\oL}{\overline{\Lambda}}
\newcommand{\omy}{\omega_{y}}
\newcommand{\omw}{\omega_{w}}
\newcommand{\omb}{\omega_{b}}
\title[Asymmetric Random Cluster Model]
{The Asymmetric Random Cluster Model and Comparison of Ising and Potts
Models}
\author{Kenneth S. Alexander}
\address{Department of Mathematics DRB 155\\
University of Southern California\\
Los Angeles, CA 90089-1113}
\email{alexandr@math.usc.edu}
\thanks{Research supported by NSF grants DMS-9504462 and DMS-9802368.}
\keywords{random cluster model, Potts model, Ising model, Potts lattice
gas, FKG property, dilution, critical point}
\subjclass{Primary: 60K35; Secondary: 82B20,82B43}
\date{\today}
\begin{abstract}
We introduce the asymmetric random cluster (or ARC) model, which is a
graphical representation of the Potts lattice gas, and establish its basic
properties. The ARC model allows a rich variety of comparisons (in the FKG
sense) between models with different parameter values; we give, for example,
values $(\beta, h)$ for which the 0's configuration in the Potts lattice gas
is dominated by the ``+'' configuration of the $(\beta,h)$ Ising model.
The Potts model, with possibly an external field applied to one of the spins,
is a special case of the Potts lattice gas, which allows our comparisons
to yield rigorous bounds on the critical temperatures of Potts models. For
example, we obtain $.571 \leq 1 - \exp(-\beta_{c}) \leq .600$ for the 9-state
Potts model on the hexagonal lattice. Another comparison bounds the movement
of the critical line when a small Potts interaction is added to a lattice gas
which otherwise has only interparticle attraction. ARC models can also be
compared to related models such as the partial FK model, obtained by
deleting a fraction of the nonsingleton clusters from a realization of
the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds
on the effects of small annealed site dilution on the critical temperature of
the Potts model.
\end{abstract}
\maketitle
\section{Introduction} \label{S:intro}
Random cluster models, or graphical representations, have become an
increasingly important tool in the study of lattice models. Most prominently,
the Fortuin-Kasteleyn random cluster model (or simply, the \emph{FK model}),
introduced in \cite{FK}, \cite{Fo1} and \cite{Fo2}, has been used to analyze
aspects of the Potts and Ising models, including critical behavior
\cite{LMR}, long-range versions \cite{ACCN}, mean-field behavior in high
dimensions \cite{KS}, covariance structure \cite{BC}, mixing properties
\cite{Al98} and efficient simulation \cite{SW}.
Wiseman and Domany \cite{WD} and Pfister and Velenik \cite{PV}
considered graphical representations of the
Ashkin-Teller model, and graphical representations for large classes of models
have been considered in the contexts of efficient simulation
(\cite{CM1},\cite{CM2}) and conditions for Gibbs uniqueness \cite{AC}.
A principal advantage of random cluster models is that the configuration
space, typically $\{0,1\}^{\mathcal{B}}$ for some set $\mathcal{B}$
of bonds, is partially ordered in a natural way, meaning that it makes
sense to speak of one configuration being ``larger than'' another,
or of one measure on configurations dominating another, in the FKG sense.
The standard comparison theorem of \cite{FK} (see also \cite{ACCN})
says (in standard notation---see (\ref{E:FKweight})) that if
\[
1 \leq q \leq q^{\prime} \quad \text{and} \quad
\frac{p}{q(1-p)} \leq \frac{p^{\prime}}{q^{\prime}(1 - p^{\prime})}
\]
then the FK model with parameters $(p^{\prime},q^{\prime})$ dominates
the model with parameters $(p,q)$ in the FKG sense (that is, increasing
events have larger probabilities at $(p^{\prime},q^{\prime})$.) This
yields information about the smoothness of the critical line in the
$(p,q)$-parameter space, among other things; see \cite{ACCN}. Another
comparison inequality for the FK model appears in \cite {Gr95}.
A principal disadvantage of the standard comparison theorem is that it is not
very sharp. Rephrased, the theorem says that as one moves in $(p,q)$-space
up any line $p/q(1-p) = c$, the configurations of the FK model get larger.
In a sharper result, the corresponding lines would approximately parallel the
presumed critical line, given by $p^{2}/q(1-p)^{2} = 1$ for the
two-dimensional integer lattice, which clearly not so for the lines in the
standard comparison theorem; the situation in higher dimensions is even
worse. A second disadvantage is that an external field in the Potts model
cannot be incorporated into the FK model in a very natural way.
In this paper we introduce a new model, the asymmetric random cluster model,
(or simply, the \emph{ARC model}), which is a random cluster representation
of the Potts lattice gas (that is, the annealed site-diluted Potts model.)
As is well known, the $q$-state Potts lattice gas includes the $(q+1)$-state
Potts model as a special case. We will show that the ARC model allows quite
sharp comparison theorems between different parameter values of the
Potts lattice gas. This leads to a variety of consequences. We obtain
rigorous bounds $\beta_{1} \leq \beta_{c} \leq \beta_{2}$ on the critical
inverse temperature of the Potts model on various lattices (within about
5\%, in many cases, and sometimes much less, of numerical or other
nonrigorous estimates in the literature), and establish standard
properties of the high-temperature regime, such as exponential decay
of correlations and weak mixing, up to the lower bound $\beta_{1}$.
by contrast, existing methods for establishing such properties are generally
perturbative, working only for very small values of the inverse temperature.
(For exceptions, generally involving $q = 2$ or $q$ large, see e.g.
\cite{Al98}, \cite {KS}, \cite{LMR}, \cite{SS}, \cite{vEFSS}.) We also obtain
bounds on critical line, or critical surface, locations in the parameter space
of the Potts lattice gas. We bound the change in the critical temperature, or
in the critical line, when certain models are perturbed by adding a small term
to the Hamiltonian. One such perturbation is small annealed site dilution
added to a standard Potts model. In another perturbation, we begin with a
lattice gas with only one species of particle, or essentailly equivalently, a
Potts latttice with $q$ species but with no additional energy associated with
adjacent particles of mismatched species; we then add a small Potts interaction
between the different species.
Numerous aspects of the phase diagram of the Potts lattice gas (though not the
ones we consider here) have been studied in \cite{CKS}.
To enrich the set of possible set of comparisons which can be made using the
ARC model, we also introduce and analyze what we call the \emph{partial FK
model}, which is obtained from the usual FK model by deleting a fraction of
the nonsingleton clusters. Comparisons between ARC models and partial FK
models are used in our analysis of site dilution.
Like the FK model, the ARC model is useful in constructing couplings
between measures under different boundry conditions; we will demonstrate an
application of such a construction. Further, we will use the ARC model
to examine the question of when the distribution of the set of empty sites of
a Potts lattice gas---that is, the distribution of $\{\delta_{[\sigma_{x}
= 0]}: x \in \Lambda\}$, where $\Lambda$ is a subset of the lattice---has
the FKG property. This includes, as a special case, the FKG property for the
distribution of the set of sites of any one species in the Potts model,
which was recently established by L. Chayes \cite{Ch}.
Of course the advantages of the ARC model over the FK model--principally
sharper comparison theorems and more natural incorporation of external
fields--do not come without a price. For example, there is phase coexistence
in the Potts model precisely when there is percolation (under wired boundary
conditions) in the corresponding FK model. In the ARC model, by contrast,
there are two bond configurations, corresponding to the two pair
interactions (Potts interaction and interparticle attraction), and the
relation between percolation and phase transition is more complex. Further,
correlations in the Potts model are given (under free boundary conditions)
by connectivities in the corresponding FK model; the ARC model has no such
property. So the ARC model supplements, but does not replace, the FK model.
\section{Preliminaries and Description of the Models} \label{S:prelim}
By a \emph{lattice} we mean a periodic graph embedded in Euclidean space. The
\emph{degree} of a site (that is, vertex) of a graph is the number of bonds
emanating from that site. When the degree is the same for every site of a
lattice, this degree is called the \emph{coordination number} of the lattice.
The $q$-\emph{state Potts lattice gas} on a finite subset $\Lambda$ of a
lattice $\mathbb{L}$ is described by variables $\sigma_{x} \in \{0,1,..,q\}$
at each site $x \in \Lambda$; 0 denotes an empty site, and $1,..,q$ are
possible spins, or species, for a particle at $x$. Let $n_{x} =
\delta_{[\sigma_{x} \in \{1,..,q\}]}$ be the indicator of the presence of a
particle at $x$. We write the Hamiltonian as
\begin{equation} \label{E:Hamilt}
H(\sigma) = - J \sum_{\langle xy \rangle} n_{x}n_{y}\delta_{[\sigma_{x} =
\sigma_{y}]} - \kappa \sum_{\langle xy \rangle} n_{x}n_{y} -
\sum_{x} \mu_{x}n_{x},
\end{equation}
where the first two sums are over adjacent unordered pairs (that is, bonds)
$\langle xy \rangle$ with $x, y \in \Lambda$; when there is a boundary
condition we include also ajacent pairs with only one of $x, y$ in
$\Lambda$. We call $J$ the \emph{interaction strength}, $\kappa$ the
\emph{interparticle attraction}, and $\mu_{x}$ the \emph{chemical
potential} at $x$. Note that when $\kappa = 0$, adjacent mismatched particles
are energentically equivalent to adjacent empty sites, whereas adjacent matched
particles have a lower energy. When $\kappa = 0$ we call the Potts lattice
gas \emph{neutral}. Let $\partial \Lambda$ denote the set of sites in
$\Lambda^{c}$ which are adjacent to $\Lambda$; by a \emph{boundary condition}
for the Potts lattice gas we mean a configuration $\eta \in
\{0,1,..,q\}^{\partial \Lambda}$. The corresponding Hamiltonian is denoted
$H_{\Lambda,\eta}$, and the partition function for the Potts lattice
gas at $(\beta,J,\kappa,\{\mu_{x}\})$ is
\[
Z(\Lambda,\eta,\beta,J,\kappa,\{\mu_{x}\}) = \sum_{\sigma}
e^{-\beta H_{\Lambda,\eta}(\sigma)}.
\]
When $\mu_{x} = \mu$ for all $x$, the corresponding measure on
$\{0,..,q\}^{\Lambda}$ is denoted
$P^{PLG}_{\Lambda,\eta,q,\beta,J,\kappa,\mu}$. There are really only three
free numerical parameters (or sets of parameters, if $\mu$ depends on $x$)
in the partition function, so the inverse temperature $\beta$ is a
redundant parameter, though at times convenient; we will generally take
$\beta$ to be 1.
A configuration $\sigma$ together with a boundary condition $\eta$ on
$\partial \Lambda$ (or on $\Lambda^{c}$) yields a combined configuration on
$\overline{\Lambda} = \Lambda \cup \partial \Lambda$ (or on $\mathbb{L}$)
which we denote $(\sigma \eta)$.
A graph $G$ is designated by a pair $(\Lambda,\mathcal{B})$, where $\Lambda$
is a set of sites and $\mathcal{B}$ is a set of bonds. The set of sites of
$G$ is also denoted $S(G)$, and the set of bonds is also denoted $B(G)$.
Let $\Lambda$ be a finite set of sites of a lattice and let
$\mathcal{B}(\Lambda) = \{\langle xy \rangle: x, y \in \Lambda\}$ and
$\omB(\Lambda) = \{\langle xy \rangle: x \in \Lambda \text{ or } y \in
\Lambda\}$. Given a subgraph, either $G = (\overline{\Lambda},\mB)$ or
$G = (\Lambda,\mB)$, of $(\overline{\Lambda},\omB(\Lambda))$ and given a
boundary condition $\eta$ and a configuration $\sigma$ on $\Lambda$, we define
variables $N_{**} = N_{**}(G,(\sigma \eta))$ by
\begin{align} \label{E:Ndefs}
N_{00} &= |\{\langle xy \rangle \in \mB: (\sigma \eta)_{x} = (\sigma
\eta)_{y} = 0\}|, \\
N_{ss} &= |\{\langle xy \rangle \in \mB: (\sigma \eta)_{x} = (\sigma
\eta)_{y} \in \{1,..,q\}\}|, \notag \\
N_{ss^{\prime}} &= |\{\langle xy \rangle \in \mB: (\sigma \eta)_{x}, (\sigma
\eta)_{y} \in \{1,..,q\}, (\sigma \eta)_{x} \neq
(\sigma \eta)_{y}\}|, \notag \\
N_{0s} &= |\{\langle xy \rangle \in \mB: (\sigma \eta)_{x} \in \{1,..,q\},
(\sigma \eta)_{y} = 0\}|, \notag \\
N_{s} &= |\{x \in \Lambda: \sigma_{x} \in \{1,..,q\}\}|, \notag \\
N_{0} &= |\{x \in \Lambda: \sigma_{x} = 0\}|, \notag
\end{align}
so that
\begin{equation} \label{E:Hequiv}
H(\sigma) = -(\kappa + J)N_{ss} - \kappa N_{ss^{\prime}}
- \sum_{x} \mu_{x}n_{x}
\end{equation}
and
\begin{equation} \label{E:Nsum}
|\mB| = N_{00} + N_{ss} + N_{ss^{\prime}} + N_{0s}.
\end{equation}
For $x \in \Lambda$ let
\[
m_{x} = |\{y: \langle xy \rangle \in \mB\}|.
\]
In the case of $\mathbb{L}$ with coordination number $m$ and $\mB =
\omB(\Lambda)$ we have $m_{x} = m$ for all $x$. In general,
\[
\sum_{x} m_{x}\delta_{[\sigma_{x} = 0]} = 2N_{00} + N_{0s}.
\]
Subtracting this from (\ref{E:Nsum}), multiplying by $\kappa$ and adding the
result to (\ref{E:Ndefs}) gives
\begin{equation} \label{E:newH}
H(\sigma) = -J N_{ss} - \kappa N_{00} + \sum_{x} (\mu_{x} +
\kappa m_{x}) \delz + c(G)
\end{equation}
where $c(G)$ is a nonrandom constant. If $\kappa = J$ this becomes
\[
H(\sigma) = -\mathcal{I}(N_{ss} + N_{00} + \sum_{x} h_{x}\delz) + c(G),
\]
where $h_{x}$ is given by
\[
J h_{x} = - (\mu_{x} + Jm_{x}).
\]
The Hamiltonian for the $(q+1)$-state Potts model with external field
$h_{x}$ applied to spin 0 at each $x$ is
\begin{equation} \label{E:PottsHam}
H(\sigma) = -N_{ss} - N_{00} - \sum_{x} h_{x}\delz,
\end{equation}
so the $q$-state Potts lattice gas at $(1,J,J,\{\mu_{x}\})$ is the same as the
$(q+1)$-state Potts model with inverse temperature $\beta$ and external fields
$\{h_{x}\}$ given by
\begin{equation} \label{E:hdef}
\beta = J \quad \text{and} \quad \beta h_{x} = -(\mu_{x} + Jm_{x}).
\end{equation}
Note that in the case of fixed coordination number and chemical potential, say
$m_{x} = m, \mu_{x} = \mu$ for all $x \in \Lambda$, we have that $h = h_{x}$
does not depend on $x$, and
\[
\sum_{x} h_{x}\delz = hN_{0}.
\]
In the further special case of the Ising model ($q + 1 = 2$ states), it is
natural (since the external field is applied to spin 0) to relabel 0 as
``+''and 1 as ``-'', and of course $N_{ss^{\prime}} = 0$. The computation
yielding (\ref{E:Nsum}), done in reverse, is then just the standard
lattice-gas transformation of the Ising model:
\begin{align} \label{E:lgtransf}
H(\sigma) &= -(N_{--} + N_{++} + hN_{+}) \\
&= -2N_{--} + (m + h)N_{-} + c(G). \notag
\end{align}
Note that in some formulations in the literature, this would be the Hamiltonian
corresponding to an external field of $h/2$. From (\ref{E:Nsum}) and
(\ref{E:hdef}) we obtain the standard fact that when $q = 1$ and $J = 0$, the
Potts lattice gas (which is then called a \emph{binary lattice gas}) is
equivalent, under the same relabeling, to an Ising model with parameters
$(\beta,h)$ given by
\begin{equation} \label{E:bhdefs}
\beta = \frac{\kappa}{2},
\quad \beta h = -\left(\mu + \frac{\kappa m}{2}\right).
\end{equation}
To construct the ARC model, we begin by rewriting the partition function of
the Potts lattice gas, as was done in \cite{FK} for the Potts model. Let
$G = (\Lambda,\mB)$ be a finite subgraph of a lattice $\mathbb{L}$. For
simplicity we first consider free boundary conditions, with $\mu_{x} = \mu$
for all $x \in \Lambda$. Let $\Omega = \{0,1\}^{\mB}$. A \emph{bond
configuration} is an element $\omega \in \Omega$; when convenient we
alternatively view $\omega$ as a subset of $\mB$ or as a subgraph of
$(\Lambda,\mB)$. Bonds $e$ with $\omega_{e} = 1$ are \emph{open} in $\omega$;
those with $\omega_{e} = 0$ are \emph{closed}. Let $C(\omega)$ denote the
number of open clusters in $\omega$, let $\omega \vee \omega^{\prime}$ and
$\omega \wedge \omega^{\prime}$ denote the coordinatewise maximum and minimum,
respectively, and define
\begin{align}
|\omega| &= \{e \in \mB: e \text{ is open}\}, \notag \\
\mathcal{I}(\omega) &= \{x \in S(G): x \text{ is an isolated
site of the graph } \omega\}, \notag \\
I(\omega) &= |\mathcal{I}(\omega)|. \notag
\end{align}
Here an isolated site means a singleton cluster. The partition function
corrresponding to the Hamiltonian (\ref{E:Ndefs}), with $\beta = 1$, is
\begin{align} \label{E:partn1}
Z(&\Lambda,J,\kappa,\mu) \\
&= \sum_{\sigma} \exp\left((\kappa + J)N_{ss} + \kappa N_{ss^{\prime}}
+ \mu N_{s} \right) \notag \\
&= e^{\mu |\Lambda|} \sum_{\sigma} e^{-\mu N_{0}} \prod_{\langle xy \rangle}
(1 + (e^{\kappa} - 1)n_{x}n_{y}) \prod_{\langle xy \rangle}
(1 + (e^{J} - 1)\delta_{[\sigma_{x} = \sigma_{y} \neq 0]}). \notag
\end{align}
Expanding out the products over bonds yields
\begin{align} \label{E:partn2}
Z(&\Lambda,J,\kappa,\mu) \\
&= e^{\mu |\Lambda|} \sum_{\omega_{g} \in \Omega} \sum_{\omega_{r} \in
\Omega} \sum_{A \subset \mathcal{I}(\omega_{g} \vee \omega_{r},\Lambda)}
e^{-\mu |A|} (e^{\kappa} - 1)^{|\omega_{g}|}
(e^{J} - 1)^{|\omega_{r}|} q^{C(\omega_{r}) - |A|} \notag \\
&= e^{\mu |\Lambda|} \sum_{\omega_{g} \in \Omega} \sum_{\omega_{r} \in
\Omega} \left( 1 + \frac{e^{-\mu}}{q}\right)^{I(\omega_{g} \vee
\omega_{r})} e^{-\mu |A|} (e^{\kappa} - 1)^{|\omega_{g}|}
(e^{J} - 1)^{|\omega_{r}|} q^{C(\omega_{r})}. \notag
\end{align}
Note that the sets $A$ in the expansion (\ref{E:partn2}) correspond to
sets $\{x: \sigma_{x} = 0\}$ in (\ref{E:partn1}), the spin values
$\sigma_{x}$ in the terms of (\ref{E:partn1}) are constant on the clusters of
$\omega_{r}$ in the corresponding terms in (\ref{E:partn2}), and the values
$n_{x}$ are all 1 on each nonsingleton cluster of $\omega_{g}$. The
expression (\ref{E:partn2}) motivates us to define the \emph{ARC model on}
$(\Lambda,\mB)$ \emph{with parameters} $(p_{r},p_{g},q,Q)$ \emph{and
free boundary conditions} to be the measure on $\Omega \times \Omega$ given
by the weights
\begin{equation} \label{E:ARCweight}
W(\omr,\omg) = p_{g}^{|\omg|}(1 - p_{g})^{|\mB| - |\omg|}p_{r}^{|\omr|}
(1 - p_{r})^{|\mB| - |\omr|}q^{C(\omr)}Q^{I(\omr \vee \omg)}.
\end{equation}
Here $p_{r}, p_{g} \in [0,1], q > 0$ and $Q \geq 1$.
Edwards and Sokal \cite{ES} observed that the Potts and FK model could be
constructed on a common probability space. The analog of their result is
valid here as well, provided $J, \kappa \geq 0$. Specifically, we relate
parameters of the $q$-state Potts lattice gas and the ARC model by
\begin{equation} \label{E:ARCparams}
p_{r} = 1 - e^{-J}, \quad p_{g} = 1 - e^{-\kappa}, \quad
Q = 1 + \frac{e^{-\mu}}{q};
\end{equation}
the parameter $q$ takes the same value in both models. We call an ARC model
and a Potts lattice gas \emph{corresponding} when their parameters are related
by (\ref{E:ARCparams}). We view $\Omega \times \Omega$ as a
set of configurations on a lattice in which
there are two bonds---one green and one red---between each adjacent pair of
sites of $(\Lambda,\mB)$. Given a site configuration $\sigma$, we obtain
a green-bond configuration $\omg$ from independent bond percolation at density
$p_{g}$ on $\{\langle xy \rangle: n_{x} = n_{y} = 1\}$
and a red-bond configuration $\omr$ from independent bond percolation at
density $p_{r}$ on $\{\langle xy \rangle: \sigma_{x} = \sigma_{y} \in
\{1,..,q\}\}$. Conversely, given the bond configurations $\omr$ and $\omg$, we
obtain a site configuration $\sigma$ by choosing a spin from $\{1,..,q\}$
independently and uniformly for each cluster of $\omr$ which is not an
isolated site of $\omr \vee \omg$; for each isolated site of $\omr \vee \omg$,
we choose a spin independently from $\{0,1,..,q\}$ with probability
proportional to $e^{-\mu}$ for 0 and proportional to 1 for each of $1,..,q$.
Thus
\begin{align} \label{E:labelprobs}
P(\sigma_{x} = 0 \mid x \in \mathcal{I}(\omr \vee \omg))
&= \frac{e^{-\mu}}{q+e^{-\mu}}
= \frac{Q - 1}{Q}, \\
P(\sigma_{x} = i \mid x \in \mathcal{I}(\omr \vee \omg))
&= \frac{1}{q+e^{-\mu}}
= \frac{1}{qQ} \quad \text{for each } i = 1,..,q. \notag
\end{align}
For either construction, the result is a joint distribution of site and bond
configurations for which the marginal distribution of the sites in the Potts
lattice gas and the marginal of the bonds is the ARC model.
To allow the chemical potential $\mu_{x}$ to vary with $x$, we can modify the
ARC model to allow $Q = Q_{x}$ to depend on $x$; we merely replace the term
$Q^{I(\omr \vee \omg)}$ in (\ref{E:ARCweight}) with
\[
\prod_{x \in \mathcal{I}(\omr \vee \omg)} Q_{x}.
\]
Via similar constructions, we can obtain the ARC model with either a site
boundary condition or a bond boundary condition, defined as follows. A
\emph{bond boundary condition on} $\mB^{c}$ is a configuration $\rho =
(\rho_{r},\rho_{g})$ in $\{0,1\}^{\mB^{c}} \times \{0,1\}^{\mB^{c}}$.
Under a bond boundary condition, the \emph{ARC model on} $(\Lambda,\mB)$
\emph{with parameters} $(p_{r},p_{g},q,Q)$ \emph{and boundary condition}
$\rho$ is again given by the weights $W(\omr,\omg)$ of (\ref{E:ARCweight}),
except that now $C(\omr)$ (also written $C(\omr \mid \rho_{r})$)
is defined to be the number of clusters of
$(\omr\rho_{r})$ which intersect $\Lambda$. We denote this model by
$P^{ARC}_{\Lambda,\rho,p_{r},p_{g},q,Q}$. When $\rho_{r}$ is all 1's, the
configuration $\rho_{g}$ is irrelevant (that is, it does not affect the
weights $W(\omr,\omg)$), and we say the resulting ARC model has
\emph{red-wired boundary condition}. When also $\mB = \omB(\Lambda)$ we
denote the corresponding measure on bond configurations by
$P_{\Lambda,rw,p_{r},p_{g},q,Q}^{ARC}$.
For integer $q$, a \emph{site boundary condition} is given by a boundary
condition for the corresponding Potts lattice gas, that is, an element
$\eta \in \{0,1,..,q\}^{\partial \Lambda}$. Site boundary conditions are
defined only when $G$
has form $(\overline{\Lambda},\omB(\Lambda))$. Define the events
\begin{align} \label{E:Devents}
D_{r}(\Lambda,\eta) &= \{\omr \in \{0,1\}^{\omB(\Lambda)}: \eta_{x}
= \eta_{y} \text{ for every } x,y \in \partial \Lambda \text{ for} \notag
\\
&\qquad \qquad \text{which }
x \leftrightarrow y \text{ in } \omr, \text{ and } \{x \in \partial
\Lambda: \eta_{x} = 0\} \subset \mathcal{I}(\omr)\}, \\
D_{g}(\Lambda,\eta) &= \{\omg \in \{0,1\}^{\omB(\Lambda)}: \{x \in \partial
\Lambda: \eta_{x} = 0\} \subset \mathcal{I}(\omg)\}, \notag \\
D(\Lambda,\eta) &= \{(\omr,\omg): \omr \in D_{r}(\Lambda,\eta), \omg \in
D_{g}(\Lambda,\eta)\}. \notag
\end{align}
Here $x \leftrightarrow y$ means there is a path of open bonds connecting $x$
to $y$. Once again, the \emph{ARC model on} $(\oL,\omB(\Lambda))$ \emph{with
parameters} $(p_{r},p_{g},q,Q)$ \emph{and site boundary condition} $\eta$,
denoted $P^{ARC}_{\Lambda,\eta,p_{r},p_{g},q,Q}$,
is given by the weights in (\ref{E:ARCweight}), with $C(\omega_{r})$ now
defined to be the number of clusters of $\omr$ which do not intersect
$\partial \Lambda$, and with $I(\omr \vee \omg)$ now defined to be the number
of isolated sites of $\omr \vee \omg$ in $\Lambda$ (instead of $\oL$), except
that weight 0 is assigned to configurations not in $D(\Lambda,\eta)$. This is
equivalent to the red-wired ARC model conditioned on the event
$D(\Lambda,\eta)$. More generally, one can allow the boundary spins
$\eta_{x}$ to take values in an arbitrary finite set $V$ containing 0, in
place of $\{0,1,..,q\}$, since the definition of $D(\Lambda,\eta)$ caries over
to such situations; this will be useful when $q$ is not an integer. We call
such a boundary condition a \emph{generalized site boundary condition}.
Since the definitions of $C(\omega), \mathcal{I}(\omega)$ and
$I(\omega)$ depend on the
boundary condition, when ambiguity is possible we will use the notation
$C(\omega \mid \rho)$ for the number of clusters of $\omega$ when the
bond boundary condition is $\rho,\ C(\omega,\Lambda)$ for the number of
clusters having all sites in $\Lambda$, and $\mathcal{I}(\omega,\Lambda)$ and
$I(\omega,\Lambda)$ respectively for the set and the number of isolated sites
in the set $\Lambda$.
When $\eta$ is all 0's, $D(\Lambda,\eta)$ is the event that every site of
$\partial \Lambda$ is isolated; we therefore call site boundary condition
$\eta$ the \emph{isolated boundary condition} and denote the corresponding
measure $P_{\Lambda,iso,p_{r},p_{g},q,Q}^{ARC}$. The ARC model on
$(\oL,\omB(\Lambda))$ with isolated boundary condition is equivalent to the
ARC model on
$(\Lambda,\mB(\Lambda))$ with free boundary condition. (But see Remark
\ref{R:freeiso} below.)
In the ARC model, a green bond and a red bond connect each adjacent pair of
sites. It is convenient to add a third bond, colored black, which we define
to be open precisely when either the red or the green bond is open. Thus the
corresponding configuration of black bonds is $\omb = \omr \vee
\omb$. It is easy to see that only the black and red bonds (not the
green) are needed when one constructs a Potts lattice gas configuration by
labeling the clusters of an ARC model configuration. To each ARC model there
thus corresponds what we call a \emph{red/black ARC model with parameters}
$(p_{b},p_{rb},q,Q)$ given by the weights
\begin{align} \label{E:RBARCweight}
W(\omb,\omr) &= p_{b}^{|\omb|}(1 - p_{b})^{|\mB| - |\omb|}
p_{rb}^{|\omr|}(1 - p_{rb})^{|\omb| - |\omr|} q^{C(\omr)}
Q^{I(\omb)} \\
&\qquad \qquad \text{for all } (\omb,\omr) \text{ with } \omr \subset
\omb, \notag
\end{align}
where $p_{b}$ and $p_{rb}$ are given by
\begin{equation} \label{E:RBARCparams}
1 - p_{b} = (1 - p_{r})(1 - p_{g}),\quad p_{rb} = p_{r}/p_{b}.
\end{equation}
The weights (\ref{E:RBARCweight}) are obtained by first rewriting the
``independent bonds'' weight:
\begin{align} \label{E:indepbond}
p_{g}^{|\omg|}&(1 - p_{g})^{|\mB| - |\omg|} p_{r}^{|\omr|}(1 - p_{r})^{|\mB|
- |\omr|} \\
&= p_{b}^{|\omb|}(1 - p_{b})^{|\mB| - |\omb|}
p_{rb}^{|\omr|}(1 - p_{rb})^{|\omb| - |\omr|}p_{g}^{|\omg \wedge
\omr|}(1 - p_{g})^{|\omr| - |\omg \wedge \omr|}, \notag
\end{align}
then summing over all choices of $\omg \wedge \omr$ for a given
$(\omb,\omr)$. Equality (\ref{E:indepbond}) reflects the fact that
one can choose red and green configurations by first choosing a black
configuration, then a red configuration which is a subset of the black one,
then a green configuration which is a subset of the red one, then adding
this green configuration an open green bond wherever there is an open black
bond but a closed red bond.
There are three important special cases of the ARC model. The first is the
\emph{neutral ARC model}, in which $p_{g} = 0$. From (\ref{E:ARCparams})
a neutral ARC model corresponds precisely to a neutral Potts lattice gas, that
is, one with $\kappa = 0$. There are no green bonds so the weights
(\ref{E:ARCweight}) become
\[
W(\omr) = p_{r}^{|\omr|}(1 - p_{r})^{|\mB| - |\omr|}q^{C(\omr)}Q^{I(\omr)}.
\]
The second special case is the \emph{Potts ARC model}, in which $p_{g} =
p_{r}$. Recall that the $q$-state Potts lattice gas at $(1,J,J,\{\mu_{x}\})$
is the same as the $(q+1)$-state Potts model at $(\beta,\{h_{x}\})$ with
$\beta$ and $h_{x}$ given by (\ref{E:hdef}). From (\ref{E:ARCparams}), the
condition $\kappa = J$ is equivalent to $p_{g} = p_{r}$, so for integer $q$ a
Potts ARC model corresponds to a $(q+1)$-state Potts model. More precisely,
the $(q+1)$-state Potts model at $(\beta,\{h_{x}\})$ corresponds to a Potts ARC
model with parameters $(p,p,q,\{Q_{x}\})$ where
\begin{equation} \label{E:PARCparams}
p = 1 - e^{-\beta} \quad \text{and} \quad Q_{x} = 1 +
\frac{e^{\beta(m_{x}+h_{x})}}{q}.
\end{equation}
In the absense of an extenal field, the Potts model is of course symmetric in
the spin variables $0,1,..,q$, except for boundary conditions. By contrast,
in constructing the Potts ARC model from the $(q+1)$-state Potts model by
independent percolation (open red bonds with probability $p_{r}$
on matching pairs with spins $1,..,q$;
open green bonds with probability $p_{g}$
on general pairs with spins $1,..,q$), the spin values are
clearly treated asymmetrically, with 0 given special treatment. This
asymmetric treatment of a symmetric model is a key part of what makes the
Potts ARC model a useful tool.
\begin{remark} \label{R:freeiso}
It was mentioned above that the ARC model on $(\oL,\omB(\Lambda))$ with
isolated boundary condition is equivalent to the ARC model on
$(\Lambda,\mB(\Lambda))$ with free boundary condition. In the case of the
Potts ARC model, it should be noted that the values of $m_{x}$ are different
for these two graphs, which affects the translation between the external
fields $h_{x}$ in the Potts model and the parameters $Q_{x}$ in the Potts ARC
model. Consider for example a Potts model on a finite subset $\Lambda$
of a lattice of coordination number
$m$, with constant external field $h$. A free boundary condition on this
model corresponds to a Potts ARC model with parameters
$\{Q_{x}\}$ that are different for sites $x$ adjacent to $\partial
\Lambda$. The constant-$Q$ Potts ARC model corresponds instead to the Potts
model with 0's boundary condition.
\end{remark}
The third special case is the \emph{Ising ARC model}, in which $q = 1$. This
is really a special case of the black-bond configuration in the red/black ARC
model (\ref{E:RBARCweight}). The red bonds are removed, or summed out,
because they are irrelevant when $q = 1$; the open red bonds are just obtained
from independent percolation on the open black bonds. The \emph{Ising ARC
model with parameters} $(p,Q)$ is given by the weights
\[
W(\omega) = p^{|\omega|}(1 - p)^{|\mB| - |\omega|}Q^{I(\omega)}.
\]
Values $Q_{x}$ depending on $x$ are allowed as before. For a lattice with
coordination number $m$, the Ising model on $\Lambda$ at $(\beta,h)$ with
boundary condition $\eta$ corresponds to an Ising ARC model model on
$(\oL,\omB(\Lambda))$ with site boundary condition $\eta$ and parameters
\begin{equation} \label{E:IARCparams}
p = 1 - e^{-2\beta}, \quad Q = 1 + e^{\beta(m+h)},
\end{equation}
provided we relabel 0 as ``+''and 1 as ``-''. An Ising ARC model
configuration can be obtained by independent percolation at density $p$ on the
``- -'' bonds of an Ising configuration. Conversely an Ising configuration
can be obtained by labeling each isolated site independently, according to the
following analog of (\ref{E:labelprobs}):
\begin{align} \label{E:Ilabelprobs}
P(\sigma_{x} = + \mid x \in \mathcal{I}(\omr \vee \omg))
&= \frac{Q-1}{Q}, \\
P(\sigma_{x} = 1 \mid x \in \mathcal{I}(\omr \vee \omg))
&= \frac{1}{Q}. \notag
\end{align}
Note that $1/Q$ is precisely the probability that a site is ``-'' given that
all its neighbors are ``+''.
By contrast, the FK model (with $q = 2$) is obtained from independent
percolation on both ``++'' and ``- -'' bonds of an ising configuration, at the
lower density $p = 1 - e^{-\beta}$. The density is higher for the Ising ARC
model because the Ising ARC model configuration is essentially the union of the
red and green configurations, each of which is obtained by independent
percolation at density $1 - e^{-\beta}$ on ``- -'' bonds.
Of course, one could equally well construct a joint Ising/Ising ARC model
configuration using independent percolation on the ``++'' bonds of an Ising
configuration, though as we have defined things, the Ising ARC model would then
have the opposite site boundary condition from the Ising model. We will refer
to this as the \emph{reversed polarity} construction of the Ising ARC model.
\begin{remark} \label{R:noninteger}
Even when $q > 0$ is not an integer, one can still construct a joint site-bond
configuration with site variables $n_{x} \in \{0,1\}$, by using the first half
of (\ref{E:labelprobs}) to label the isolated sites of an ARC model
configuration; all sites not labeled 0 are labeled 1. If the ARC model has
parameters $(p_{r},p_{g},q,\{Q_{x}\})$ with $q$ not an integer, and $J,
\kappa, \{\mu_{x}\}$ are given by (\ref{E:ARCparams}), we call the resulting
site-bond model the $q$-\emph{state Potts lattice gas with parameters}
$(1,J,\kappa,\{\mu_{x}\})$, thereby extending the definition to noninteger
$q$. A bond or generalized site boundary condition can be applied in the
natural way. If the ARC model is a Potts ARC model, and $\beta, \{h_{x}\}$ are
given by (\ref{E:PARCparams}), we similarly call the site-bond model the
$(q+1)$-\emph{state Potts model with parameters} $(\beta,\{h_{x}\})$. To
distinguish things when necessary, we will refer to the standard
site-variables-only Potts model or Potts lattice gas with integer $q$ as the
\emph{usual} model, and refer to the joint site-bond model just defined for
general $q$ as the \emph{particle/bond} model. We call the random variable
$\{\delta_{[n_{x} = 0]}: x \in \Lambda\}$ (or its distribution, in a harmless
abuse of terminology) the \emph{0's configuration} of the (usual or
particle/bond) Potts lattice gas or (usual or particle/bond) Potts model.
When appropriate, the \emph{j's configuration} is defined similarly for $j
\neq 0$.
\end{remark}
Recently and independently, L. Chayes and J. Machta
(\cite{CM1}, \cite{CM2}) introduced
particle/bond random cluster models for a wide class of lattice gases, in the
context of efficient simulation. Our particle/bond Potts lattice gas is one
example of this class.
We turn next to our other new models, the partial and bicolored FK models. The
FK model on $(\Lambda,\mB)$ with parameters $(p,q)$ assigns weights
\begin{equation} \label{E:FKweight}
W(\omega) = p^{|\omega|}(1 - p)^{|\mB| - |\omega|}q^{C(\omega)}
\end{equation}
to bond configurations. As shown in \cite{ES}, for $\beta$ given by $p = 1 -
e^{-\beta}$, a configuration of the usual $q$-state Potts model at inverse
temperature $\beta$ can be obtained from a configuration $\omega$ of the FK
model at $(p,q)$, by choosing a label for each cluster of $\omega$
independently and uniformly from $\{0,1,..,q-1\}$; this construction yields a
joint site-bond configuration for which the sites are a Potts model and the
bonds are an FK model. Fix an integer $0 < t < q-1$ and suppose that we color
yellow all open bonds in such a joint configuration with (necessarily
matching) endpoints labeled $0,..,t-1$, and color white all open bonds with
endpoints labeled $t,..,q-1$. The weight of a given yellow/white bond
configuration is then
\begin{align} \label{E:BFKweight}
W(\omy,\omw) &= p^{|\omy| + |\omw|} (1 - p)^{|\mB| - |\omy| - |\omw|}
q^{C(\omy \vee \omw)} \left(\frac{t}{q}\right)^{C(\omy)-I(\omy)} \notag \\
&\qquad \qquad \cdot \left(1 - \frac{t}{q}\right)^{C(\omw) - I(\omw)}
\delta_{E}\bigl((\omy,\omw)\bigr) \notag \\
&= p^{|\omy|} (1 - p)^{|\mB| - |\omy|}
t^{C(\omy)} t^{-I(\omy)} \left(\frac{p}{1-p}\right)^{|\omw|} (q -
t)^{C(\omw) +
I(\omy \vee \omw) - I(\omw)} \notag \\
&\qquad \qquad \cdot \left(\frac{q}{q-t}\right)^{I(\omy \vee \omw)}
\delta_{E}\bigl((\omy,\omw)\bigr)
\end{align}
where
\begin{align}
E = E(\Lambda,\mB) = \{(\omy,\omw): &\text{ no site is an endpoint of both an
open yellow bond} \notag \\
&\qquad \text{and an open white bond}\}. \notag
\end{align}
One can obtain such a
yellow/white configuration directly from an FK configuration, without the
intermediate step of the joint Potts/FK configuration, by independently
coloring each FK cluster (including singletons) yellow with probability $t/q$,
and white with probability $1 - t/q$. Thus we neeed not restrict $t$ or $q$
to be an integer; any $0 < t < q$ will do. We call the distribution of the
yellow/white site-bond configuration the \emph{bicolored FK model on}
$(\Lambda,\mB)$ \emph{with parameters}
$(p,q,t)$ (and free boundary.) When $(\Lambda,\mB)$ is a subgraph of a
lattice, a bond boundary condition for the
bicolored FK model can be imposed by specifying a bond boundary condition
$\rho$ for the uncolored FK model, then specifying a color for each cluster
of $\rho$. Alternately, as a special case of generalized site boundary
conditions one can specify a color, yellow or white, for each site of
$\partial
\Lambda$. Under such a bicolored site boundary condition $\eta$, the event
$E$ in (\ref{E:BFKweight}) should be replaced by
\begin{align}
A(\Lambda,\eta) = E(\oL,\omB(\Lambda)) \ \cap \ &\{(\omy,\omw): \eta_{x} =
\eta_{y} \text{ for} \notag \\
&\qquad \text{every } x, y \in \partial \Lambda \text{ for which }
x \leftrightarrow y \text{ in } \omy \vee \omw\}. \notag
\end{align}
We use the notation $C(\omega,\Lambda)$, and $C(\omega
\mid \rho)$ for (bicolored) bond boundary
conditions $\rho$, as we do for the ARC model.
Summing (\ref{E:BFKweight}) over $\omw$ for a given $\omy$ yields the weight
of the yellow configuration $\omy$, under bicolored site boundary condition
$\eta \in
\{yellow,white\}^{\partial \Lambda}$:
\begin{equation} \label{E:PFKweight}
W(\omy) = p^{|\omy|} (1 - p)^{|\mB| - |\omy|} t^{C(\omy,\Lambda)}
t^{-I(\omy,\Lambda)} F(\omy),
\end{equation}
where
\begin{align} \label{E:Fdef}
F(\omy) = \sum_{\omw \in \{0,1\}^{\omB(\mathcal{I}(\omy,\Lambda))}}
&\left(\frac{p}{1-p}\right)^{|\omw|}
(q-t)^{C(\omw,\Lambda)+I(\omy \vee \omw,\Lambda)-I(\omw,\Lambda)} \\
&\qquad \cdot \left(\frac{q}{q-t}\right)^{I(\omy \vee \omw,\Lambda)}
\delta_{A(\Lambda,\eta)}\bigl((\omy,\omw)\bigr). \notag
\end{align}
We call the model given by the weights (\ref{E:PFKweight}) the \emph{partial
FK model on} $(\oL,\omB(\Lambda))$ \emph{with parameters} $(p,q,t)$ \emph{and
bicolored site boundary condition} $\eta$. Note that the exponent
$C(\omw,\Lambda) + I(\omy \vee
\omw,\Lambda) - I(\omw,\Lambda)$ in (\ref{E:Fdef}) is the number of clusters
of $\omw$ not intersecting $\partial \Lambda$ which have (all) sites in
$I(\omy,\Lambda)$; from this observation we see that $F(\omy)$ is precisely the
partition function of the neutral ARC model on
$(\mathcal{I}(\omy,\Lambda),\omB(\mathcal{I}(\omy,\Lambda))$ with parameters
$(p,0,q-t,\tfrac{q}{q-t})$, boundary condition $\eta$ on white sites in
$\partial \Lambda$, and 0's (or free) boundary condition on $\Lambda
\backslash \mathcal{I}(\omy,\Lambda)$ and on yellow sites in $\partial
\Lambda$. This, together with (\ref{E:labelprobs}), proves the following
result.
\begin{proposition} \label{P:PFKcondl}
Let $p \in [0,1], q \geq 1$ and $0 < t < q$. Let $\Lambda$ be a finite
subset of the sites of a lattice $\mathbb{L}$ and let $\eta$ be a bicolored
generalized site boundary condition for the bicolored FK model on
$(\oL,\omB(\Lambda))$. Conditionally on the yellow-bond configuration $\omy$
of the bicolored FK model with parameters $(p,q,t)$,
\begin{enumerate}
\item[(i)] the the white bonds form a neutral ARC model on
$(\mathcal{I}(\omy,\Lambda),\omB(\mathcal{I}(\omy,\Lambda)))$ with
parameters
$(p.0,q-t,\tfrac{q}{q-t})$
and boundary condition $\eta$ on the white sites in $\partial \Lambda$, and
0's (or free) boundary condition on both $\Lambda \backslash
\mathcal{I}(\omy,\Lambda)$ and
the yellow sites in $\partial \Lambda$;
\item[(ii)] the yellow sites in $\mathcal{I}(\omy,\Lambda)$ have the
distribution of the 0's
configuration of a $(q-t)$-state neutral Potts lattice gas on
$\mathcal{I}(\omy)$
with the same boundary condition as in (i), with parameters $(1,J,0,\mu)$
given by
\[
p = 1 - e^{-J}, \quad t = e^{-\mu}.
\]
\end{enumerate}
\end{proposition}
We call the neutral Potts lattice gas of Proposition \ref{P:PFKcondl}(ii) the
\emph{conditional neutral Potts lattice gas} of the bicolored FK model.
\begin{remark} \label{R:qminus1PFK}
The case $t = q-1$ in Proposition \ref{P:PFKcondl} is of particular interest,
for $F$ is then the partition function of an Ising ARC model. In this case,
for integer $q$, part (ii) says that for the joint Potts/FK configuration,
conditionally on the bonds with (matching) endpoints in $\{1,..,q-1\}$, the 0
and non-0 sites left isolated by these bonds form an Ising model. Further,
for $\beta$ given by $p = 1 - e^{-\beta}$, let $Q = 1 + e^{\beta m}/(q-1)$
(corresponding to a Potts model with no external field---cf.
(\ref{E:PARCparams}).) Then the yellow bonds of the bicolored FK model with
parameters $(p,q,q-1)$ have the same distribution as the red bonds of the
Potts ARC model with parameters $(p,p,q-1,Q)$, as both configurations are
obtained from a $q$-state Potts model configuration by independent percolation
at density $p$ on $\{\langle xy \rangle \in \mB: \sigma_{x} = \sigma_{y} \neq
0\}$.
\end{remark}
One type of bicolored generalized site boundary condition specifies only a
color for each site; equivalently, all white boundary sites are 0's and all
yellow boundary sites have a second spin, say 1. Thus we may, for
example, have a bicolored FK model with all-white or all-yellow site boundary
condition.
\section{Statement of Main Results} \label{S:main}
In this section we describe our main results; proofs appear in later sections.
Let us use ``$\leq$'' to denote the natural 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 $A$. This is equivalent to the statement
that there exists a coupling $\tilde{P}$ of $\{0,1\}^{\mB} \times
\{0,1\}^{\mB}$ with marginals $P_{1}$ and $P_{2}$ for which
$\tilde{P}(\{(\omega,\omega^{\prime}): \omega \geq \omega^{\prime}\}) = 1$.
As is well-known, if $P_{1}$ and $P_{2}$ are determined by weights $W_{1}$ and
$W_{2}$ respectively, $P_{1}$ or $P_{2}$ has the FKG property, and
$W_{1}/W_{2}$ is an increasing function on $\{0,1\}^{\mB}$, then $P_{1}$
dominates $P_{2}$.
Let $\eta^{i}$ denote the all-$i$'s boundary condition; if $i \neq 0$ we call
$\eta^{i}$ a \emph{constant-species boundary condition}. More generally, we
say that a generalized site boundary condition $\eta$ has a \emph{single
particle species} if there exists $i \neq 0$ such that $\eta_{x} = 0$ or $i$
for all $x \in \partial \Lambda$. Note that for the Ising ARC model ($q = 1$),
every (nongeneralized) site boundary condition has just a single particle
species.
L. Chayes \cite{Ch} proved that for the usual Potts model without external
field, the 0's configuration satisfies the FKG lattice condition and thus has
the FKG property. Of course, 0 can be replaced by any other spin. In two
dimensions, Chayes obtained as a consequence that Gibbs nonuniqueness is
characterized by the percolation of spin $i$ under boundary condition
$\eta^{i}$. Chayes' proof of the FKG property allows for boundary conditions
specified in terms of the variables $\delz, x \in \partial \Lambda$, but does
not cover general site boundary conditions. There is good reason for this as
the following example shows.
\begin{example} \label{EX:noFKG}
Consider a 3-state Potts model with spin space $\{0,1,2\}$ at inverse
temperature $\beta$ without external field, on $\Lambda = \{x,y,z\} \subset
\mathbb{Z}^{3}$, where $x = (1,0,0), y = (2,0,0), z = (3,0,0)$. The boundary
condition is as follows: of the five boundary sites adjacent to $x$, three
have spin 1 and two have spin 2; of the four oundary sites adjacent to $y$, two
have spin 1 and two have spin 2; and of the five boundary sites adjacent to
$z$, two have spin 1 and three have spin 2. We write $ijk$ for the
configuration $\sigma$ with $\sigma_{x} = i, \sigma_{y} = j, \sigma_{z} = k$.
For large $\beta$, most of the probability is concentrated on the
energy-minimizing configurations 111, 112, 122, 222 so the partition function
$Z$ corresponding to the Hamiltonian (\ref{E:newH}) satisfies $Z \sim
4e^{9\beta}$ as $\beta \to \infty$. Hence as $\beta \to \infty$ we have
\[
P(\sigma_{x} = \sigma_{z} = 0) = \frac{3e^{2\beta}}{Z},
\]
\[
P(\sigma_{z} = 0) = P(\sigma_{x} = 0) \sim P(022) = \frac{e^{6\beta}}{Z}
\sim \frac{e^{-3\beta}}{4}
\]
and hence
\[
P(\sigma_{z} = 0 \mid \sigma_{x} = 0) \sim 3e^{-4\beta}.
\]
Thus for large $\beta$, the events $[\sigma_{z} = 0]$ and $[\sigma_{x} = 0]$
are negatively correlated, so the FKG property fails for the 0's configuration.
\end{example}
This brings up the more general question of just when the FKG property holds
for the 0's configuration of a Potts lattice gas. A sufficient condition is
given by the following result; Example \ref{EX:noFKG} demonstrates the need to
restrict to single-particle-species boundary conditions.
\begin{theorem} \label{T:0sFKG}
The 0's configuration of any particle/bond Potts lattice gas on a finite
subset $\Lambda$, of the sites of a lattice, with $J, \kappa \geq 0$, under
free boundary conditions or under
any site boundary condition $\eta$ which has a single particle species,
satisfies the FKG lattice condition.
\end{theorem}
We say that \emph{percolation of spin} $i$ occurs under a measure $P$ if with
probability 1, there exists an infinite self-avoiding lattice path on which
all sites have spin $i$. Using Theorem \ref{T:0sFKG} we will establish the
following.
\begin{corollary} \label{C:percunique}
Consider a $q$-state Potts lattice gas ($q$ and integer) with parameters
$(\beta,J,\kappa,$ $\mu)$, with $J, \kappa \geq 0$, on a planar lattice
$\mathbb{L}$. If percolation of 0's occurs under boundary condition
$\eta^{i}$ for some $i \in\{1,..,q\}$, then there is a unique Gibbs
distribution at $(\beta,J,\kappa,\mu)$.
\end{corollary}
We turn next to conditions for Gibbs uniqueness and weak mixing, and to the
bounds on critical points that can be obtained by establishing such properties
throughout most of the high-temperature regime. A bond or site model with
specified parameters (but unspecified boundary condition) is said to have the
\emph{weak mixing property} if there exist $C,\lambda$ as follows. Given
finite sets
$\Delta \subset \Lambda$ and any two boundary conditions (bond or generalized
site) $\eta_{1}$ and $\eta_{2}$ for the model on $(\oL,\omB(\Lambda))$, the
corresponding distributions $P_{1}$ and $P_{2}$ of the configuration on
$(\Delta, \mB(\Delta))$ satisfy
\[
\Var(P_{1},P_{2}) \leq C \sum_{x \in \Delta,y \notin \Lambda}
e^{-\lambda |y - x|},
\]
where $\Var(\cdot,\cdot)$ denotes total variation distance.
Loosely this says that the maximum influence, on a
fixed region, of the boundary condition decays exponentially to 0 as the
boundary recedes to infinity. Turning to the FK model, fix $p,q$ and for each
finite subset $\Lambda$ of the sites of a lattice $\mathbb{L}$, let
$P_{\Lambda,w}^{FK}$ denote the model at $(p,q)$ on $(\oL,\omB(\Lambda))$ with
wired boundary condition. The infinite-volume limit, denoted $P_{w}^{FK}$, is
said to have \emph{exponential decay of local wired-boundary connectivities} if
there exist $C, \lambda > 0$ such that for every finite $\Lambda \ni 0$,
\[
P_{\Lambda,w}^{FK}(0 \leftrightarrow \partial \Lambda \text{ by a path of
open bonds}) \leq Ce^{-\lambda r(\Lambda)},
\]
where $r(\lambda) = \min\{|x|: \in \partial \Lambda\}$. Note this is
stronger than the usual notion of exponential decay of connectivities (for the
infinite-volume limit), as studied e.g. in \cite{GP},
though for the FK model on planar lattices the two
notions have been proven equivalent \cite{Al98}. It is not hard to show (see
\cite{Al98}) that if the FK model at $(p,q)$ has exponential decay of local
wired-boundary connectivities, then it has the weak mixing property, as does
the corresponding Potts model if $q$ is an integer. Weak mixing for the Potts
model has a variety of useful consequences, particularly in two dimensions;
see \cite{MOS}.
A planar lattice $\mathbb{L}$ divides the plane into polygonal faces. The
\emph{dual lattice} $\mathbb{L}^{*}$ is constructed by placing a \emph{dual
site} at the center of each such face, and then a \emph{dual bond} between each
pair of dual sites for which the corresponding faces have a bond (that is, an
edge) in common. For example, the dual of the triangular lattice is the
hexagonal lattice, and vice versa. When necessary for clarity, bonds of
$\mathbb{L}$ are called \emph{regular bonds}. To each regular bond $e$ there
is associated a unique dual bond $e^{*}$ connecting the centers of the two
faces of which $e$ is an edge. The dual bond $e^{*}$ is defined to be open
precisely when $e$ is closed, so that for each bond configuration $\omega$ on
$\mathbb{L}$, there is unique dual configuration $\omega^{*}$ on
$\mathbb{L}^{*}$. For each $q > 0$, for $p \in [0,1]$ the value $p^{*}$
\emph{dual to} $p$ \emph{at level} $q$ is given by
\[
\frac{p}{q(1-p)} = \frac{1 - p^{*}}{p^{*}}.
\]
If the regular bonds are distributed as the infinite-volume FK model at
$(p,q)$ on $\mathbb{L}$ with wired boundary condition, then the dual bonds
form the infinite-volume FK model at $(p^{*},q)$ on $\mathbb{L}^{*}$ with free
boundary condition (see \cite{Gr96}.) If $p$ is the \emph{self-dual point}
\[
p_{sd}(q) = \frac{\sqrt{q}}{1 + \sqrt{q}},
\]
then $p = p^{*}$. For the FK model on the square lattice it is conjectured
that the percolation critical point is the self-dual point for all $q$; this
is known for $q \geq 25.72$, from \cite{LMR}.
For the Potts model, the value $\beta^{*}$ dual to a given $\beta$ can be
obtained using FK duality and the correspondence $p = 1 - e^{-\beta}$.
Let $p_{c}^{FK}(q,\mathbb{L})$ denote the percolation critical point of the FK
model on a lattice $\mathbb{L}$, and let $\beta_{c}^{Potts}(q,\mathbb{L})$
denote the critical inverse temperature of the $q$-state Potts model on
$\mathbb{L}$, so that
\[
p_{c}^{FK}(q,\mathbb{L}) = 1 - e^{-\beta_{c}^{Potts}(q,\mathbb{L})}.
\]
If $q$ is not an integer, we take this as the definition of
$\beta_{c}^{Potts}(q,\mathbb{L})$. For $q = 2$ we alternately write
$\beta_{c}^{Ising}(\mathbb{L})$.
It is believed that weak mixing should hold for the $q$-state Potts model for
all $q \geq 1$ and all subcritical $\beta$, but for the most part it has only
been established perturbatively, for $\beta$ near 0. Exceptions include the
Ising model \cite{Hi1}, and $q \geq 25.72$ on the square lattice \cite{LMR}.
Here we will establish weak mixing for the Potts model on planar lattices
throughout most of the subcritical region, that is, nearly up to
$\beta_{c}^{Potts}$. As a byproduct we obtain rigorous bounds on the Potts
and FK critical points on such lattices. The specifics are as follows.
\begin{theorem} \label{T:planarbounds}
Let $\mathbb{L}$ be a planar lattice of coordination number $m$, and suppose
the dual lattice $\mathbb{L}^{*}$ has coordination number $m^{*}$. Let $q >
1$ and define $\beta_{1} = \beta_{1}(q+1,m)$ and $p_{1} = p_{1}(q+1,m)$ by
\begin{equation} \label{E:betadefs}
e^{\beta_{1}} = \frac{q-1}{q^{(m-2)/m} - 1}, \quad
p_{1} = 1 - e^{-\beta_{1}} = \frac{1 - q^{-2/m}}{1 - q^{-1}}.
\end{equation}
\begin{enumerate}
\item[(i)] The FK model on $\mathbb{L}$ at $(p,q+1)$ has exponential decay
of local wired-boundary connectivities, and has the weak mixing propety, for
all $p < p_{1}(q+1,m)$. Its critical point $p_{c} =
p_{c}^{FK}(q+1,\mathbb{L})$ satisfies $p_{1} \leq p_{c} \leq p_{2}$, where
$p_{2} = p_{1}(q+1,m^{*})^{*}$ is the value dual to $p_{1}(q+1,m^{*})$ at
level $q+1$.
\item[(ii)] If $q$ is an integer, the $(q+1)$-state Potts model at
$(\beta,0)$ on $\mathbb{L}$ has the weak mixing property for all $\beta <
\beta_{1}(q+1,m)$. Its critical point $\beta_{c} =
\beta_{c}^{Potts}(q+1,\mathbb{L})$ satisfies $\beta_{1} \leq \beta_{c} \leq
\beta_{2}$, where $\beta_{2} = \beta_{1}(q+1,m^{*})^{*}$ is the value dual
to $\beta_{1}(q+1,m^{*})$ at level $q+1$.
\item[(iii)] The lower bounds $p_{1}$ and $\beta_{1}$ remain valid even if
there is no $m^{*}$ (that is, not all sites of $\mathbb{L}^{*}$ have the same
degree.)
\end{enumerate}
\end{theorem}
We now apply Theorem \ref{T:planarbounds} to some examples.
\begin{example} \label{EX:sqlatt}
For the square lattice with certain values of $q$, it is known \cite{LMR} that
the FK critical point is the self-dual point:
\[
p_{c}^{FK}(q,\mathbb{Z}^{2}) = \frac{\sqrt{q}}{1 + \sqrt{q}} \text{ for }
q = 1,2 \text{ and all } q \geq 25.72,
\]
and this is believed to hold for all $q \geq 1$; it is known (see \cite{Gr96})
that $p_{c}^{FK}(q,\mathbb{Z}^{2}) \geq p_{sd}(q)$ for all $q \geq 1$. By
contrast, the lower bound given by Theorem \ref{T:planarbounds} is
\[
p_{1}(q,4) =
\frac{\sqrt{q-1}}{1 + \sqrt{q-1}} = p_{sd}(q-1) =
p_{sd}(q) - \frac{1}{2q^{3/2}} -
O\left(\frac{1}{q^{2}}\right) \quad \text{as } q \to \infty,
\]
and the upper bound is
\[
\frac{\sqrt{q}}{\sqrt{\tfrac{q-1}{q}} + \sqrt{q}} = p_{sd}(q) +
\frac{1}{2q^{3/2}} +
O\left(\frac{1}{q^{2}}\right).
\]
For $q = 10$, for example, we obtain $.760 \leq p_{c} \leq .769$, compared to
the conjectured value $p_{sd} = .760$, and we establish exponential decay for
all $p < .750$.
\end{example}
\begin{example} \label{EX:duallatts}
For the triangular and hexagonal lattices, there are computations of the Potts
critical point in the physics literature using the star-triangle transformation
and variants thereof (\cite{HKW},\cite{Wu1}), but it is not clear whether these
can be made rigorous. Since the triangular and hexagonal lattices are dual
to each
other, the upper bounds on each of these lattices come from lower bounds on
the other lattice. For example, the values .513 and .740 are dual at level $q
= 3$, as are .413 and .810.
For many other planar lattices, there are
only estimates obtained by series expansion methods, renormalization group
methods and/or simulation; see
\cite{Wu2} for a summary and references. For the Kagom\'e lattice lower bounds
are computed in
\cite{KW}, but again, the level of rigor is unclear.
Our rigorous bounds, and
corresponding nonrigorous values from the physics literature, are summarized in
Table \ref{TB:crit}. The nonrigorous values for the triangular and hexagonal
lattices are from the presumably exact general formula in
\cite{HKW}. The nonrigorous values for the Kagom\'e lattice are from
\cite{KW} for $q = 4$ and from the conjectured general formula in
\cite{Wu1} for $q = 9,30$; it should be noted
that this general formula was found in
\cite{EW} to be incorrect. The lower bound from \cite{KW} for the Kagom\'e
lattice with $q = 4$ is .672, better than our rigorous bound .634.
The accuracy of the bounds becomes quite high for
larger values of
$q$. Nonetheless, this table should perhaps be seen less as a source of new
information about critical points and more as a numerical quantification of the
sharpness of the comparison result, Proposition \ref{P:comptoplus}.
\end{example}
\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}[h]{| l | c | c | c | c |}
\hline
\textbf{Lattice} & \textbf{m} & \textbf{q} & \textbf{Rigorous
Bounds on } $p_{c}^{FK}(q,\mathbb{L})$
& \textbf{Nonrigorous Value} \\ \hline
Triangular & 6 & 3 & $.413 \leq p_{c} \leq .513$
& .468 \\ \hline
Triangular & 6 & 4 & $.460 \leq p_{c} \leq .532$
& .500 \\ \hline
Triangular & 6 & 9 & $.571 \leq p_{c} \leq .600$
& .588 \\ \hline
Triangular & 6 & 30 & $.699 \leq p_{c} \leq .706$
& .703 \\ \hline
Hexagonal & 3 & 3 & $.740 \leq p_{c} \leq .810$
& .773 \\ \hline
Hexagonal & 3 & 4 & $.779 \leq p_{c} \leq .824$
& .800 \\ \hline
Hexagonal & 3 & 9 & $.857 \leq p_{c} \leq .871$
& .863 \\ \hline
Hexagonal & 3 & 30 & $.926 \leq p_{c} \leq .928$
& .927 \\ \hline
Kagom\'e & 4 & 4 & $.634 \leq p_{c}$
& .686 \\ \hline
Kagom\'e & 4 & 9 & $.739 \leq p_{c}$
& .761 \\ \hline
Kagom\'e & 4 & 30 & $.843 \leq p_{c}$
& .851 \\ \hline
\end{tabular}
\\[12pt]
\caption{Nonrigorous values and rigorous bounds for the critical point of
the FK model on planar lattices.}\label{TB:crit}
\end{center}
\end{table}
The idea behind the proof of Theorem \ref{T:planarbounds} is as follows. The
main step is to prove that there is
exponential decay of local wired-boundary
connectivities for $p < p_{1}$; this establishes weak mixing which shows
that $p_{1} \leq p_{c}^{FK}$, and then, using duality, that $p_{c} \leq
p_{2}$. The bounds on $\beta_{c}^{Potts}$ follow from the Potts/FK
correspondence using (\ref{E:PARCparams}). The wired-boundary FK model on
a finite $\Lambda$ can be obtained from percolation
at density $p$ on the matching bonds in the Potts model with a constant,
say all-0, boundary condition. To obtain the exponential decay of local
wired-boundary connectivities, then, we need only consider percolation
on ``00'' bonds. It is enough to show that the probability of a path
in the Potts model from the origin to $\partial \Lambda$ on which all spins
are 0's decays exponentially in $r(\Lambda)$. For this, it is enough to
show that the 0's configuration of the Potts model is dominated by the
configuration of some other species in some other model where this
exponential decay property is known to hold. The latter role is played by
the ``+'' configuration of an Ising model with negative external field.
The ARC model, and in particular the fact that the 0's configuration is
obtained by independent site percolation on the isolated sites of the ARC
configuration, is used to facilitate the comparison of the Potts 0's to the
Ising ``+'' configuration.
We will prove a somewhat weaker analog of Theorem \ref{T:planarbounds}
for higher dimensions. It requires an assumption on the Ising model to
which the Potts 0's configuration is compared, as follows. Given an inverse
temperature $\beta$, a value of $q$ and a lattice with coordination number $m$,
define $\beta^{\prime\prime}$ by
\begin{equation} \label{E:betapp}
e^{\beta^{\prime\prime}} = \frac{q - 1 + e^{\beta}}{q^{(m-1)/m}}
\end{equation}
and let $\beta_{1}$ be as in \ref{E:betadefs}. We will need the following
condition:
\begin{align} \label{E:nonperc}
&\text{independent percolation at density } 1 - e^{-\beta} \text{ on the ``++''
bonds of the} \\
&\text{minus phase of the Ising model at } (\beta^{\prime\prime},0) \text{
produces no infinite cluster, a.s.} \notag
\end{align}
Unfortunately we have no way to verify this for $d > 2$, except when we are in
the Peierls regime $e^{\beta^{\prime\prime}} > 3$, where it is known that
``+'' spins do not percolate in the minus phase. From (\ref{E:betapp}),
a sufficient condition for $e^{\beta^{\prime\prime}} > 3$ is that $q > 3^{2d}$.
This leads to the following two results.
\begin{theorem} \label{T:highdbounds}
For $q > 1$, consider the FK model at $(p,q+1)$ on $\mathbb{Z}^{d}$, and,
for integer $q$, the corresponding $(q+1)$-state Potts model at $(\beta,0)$,
with $p = 1 - e^{-\beta}$. If (\ref{E:nonperc}) holds, with
$\beta^{\prime\prime}$ given by (\ref{E:betapp}), and
\begin{equation} \label{E:pcond}
p < \frac{1 - q^{-1/d}}{1 - q^{-1}}
\end{equation}
or equivalently
\begin{equation} \label{E:betacond}
e^{\beta} < \frac{q - 1}{q^{(d-1)/d} - 1},
\end{equation}
then there is no percolation in the FK model, and (for integer $q$) the
Potts model has a unique Gibbs distribution. If (\ref{E:nonperc}) holds for
all $\beta$ in a neighborhood of $\beta_{1}(q+1,2d)$, then
\[
p_{c}^{FK}(q+1,\mathbb{Z}^{d}) < \frac{1 - q^{-1/d}}{1 - q^{-1}}.
\]
\end{theorem}
\begin{corollary} \label{C:bigq}
For all $d \geq 2$ and $q > 3^{2d}$, the FK model on $\mathbb{Z}^{d}$ with
parameters $(p,q+1)$ has no percolation provided (\ref{E:pcond}) holds.
If $q > 3^{2d}$ is an integer, the $(q+1)$-state Potts model on
$\mathbb{Z}^{d}$ at inverse temperature $\beta$ has a unique Gibbs
distribution provided (\ref{E:betacond}) holds.
\end{corollary}
\begin{remark} \label{R:sharpcond}
The equivalent conditions (\ref{E:pcond}) and (\ref{E:betacond}) are apparently
quite sharp for $\mathbb{Z}^{d}$, even for small $q$. For example, for the
4-state Potts model ($q = 3$) in dimension 3, we have
\[
\frac{1 - q^{-1/d}}{1 - q^{-1}} = .460
\]
while nonrigorous estimates of $p_{c}$ in the physics literature range from
.468 to .477 (see \cite{Wu2}.) It is not hard to show that
$\beta/2 \leq \beta^{\prime\prime} \leq \beta$ for all $\beta \leq \beta_{1}$.
Since the FK model has at most one infinite
cluster a.s., we know that independent percolation at density
$1 - e^{-\beta^{\prime\prime}}$ on the ``++'' bonds of the minus phase of the
Ising model at $(\beta^{\prime\prime},0)$ produces no infinite cluster a.s.;
in (\ref{E:nonperc}) we replace $1 - e^{-\beta^{\prime\prime}}$ with
the larger percolation density $1 - e^{-\beta}$. Percolation at the
still-larger density $1 - e^{-2\beta^{\prime\prime}}$
on the ``++'' bonds of the minus phase of the
Ising model at $(\beta^{\prime\prime},0)$ produces the Ising ARC model (with
reversed polarity), so to establish (\ref{E:nonperc}) it is enough to show that
\begin{align} \label{E:nonperc2}
&\text{there is no percolation in the Ising ARC model, corresponding to an
Ising} \\
&\text{model at } (\beta^{\prime\prime},0), \text{ with isolated boundary
condition.} \notag
\end{align}
\end{remark}
\begin{remark} \label{R:meanfield}
We consider (\ref{E:nonperc}) and (\ref{E:nonperc2}) in the mean field limit,
$m = 2d \to \infty$, for the integer lattice. Let
$\beta_{1}^{\prime\prime}(q+1,m)$ be the value of $\beta^{\prime\prime}$ when
$\beta = \beta_{1}(q+1,m)$, that is,
\begin{equation} \label{E:beta1pp}
e^{\beta^{\prime\prime}_{1}(q+1,m)} = \frac{q - 1}{q^{1/m}(q^{(m-2)/m} - 1)}
= \frac{1}{q^{1/m}}e^{\beta_{1}(q+1,m)}
\end{equation}
(cf. (\ref{E:betapp}).) We have from (\ref{E:betadefs}) that
\[
\lim_{m \to \infty} m\beta_{1}(q+1,m) = \frac{2q \log q}{q - 1},
\]
which is the ``right'' mean field limit, in that it is believed that
\begin{equation} \label{E:meanflim}
\lim_{d \to \infty} 2d\beta_{c}(q+1,\mathbb{Z}^{d}) = \frac{2q \log q}{q-1}
\end{equation}
and it is proved in \cite{KS} that
\[
\limsup_{d \to \infty} 2d\beta_{c}(q+1,\mathbb{Z}^{d}) \leq
\frac{2q \log q}{q-1}.
\]
The limit (\ref{E:meanflim}) is known for the Ising model, $q + 1 = 2$, with
the RHS interpreted as 2. Thus if for all sufficiently large $d$ one could
establish (\ref{E:nonperc}) for all $\beta$ in a neighborhood of $\beta_{1}$,
it would prove (\ref{E:meanflim}). From (\ref{E:beta1pp}),
\[
\lim_{m \to \infty} m\beta^{\prime\prime}_{1}(q+1,m) =
\frac{(q+1) \log q}{q-1},
\]
and the latter is an increasing function of $q > 1$, so for all $q > 1$,
\[
\beta^{\prime\prime}_{1}(q+1,2d) > \beta_{c}^{Potts}(2,\mathbb{Z}^{d})
\quad \text{for all sufficiently large } d.
\]
Thus for (\ref{E:meanflim}) it would be enough to prove (\ref{E:nonperc}) or
(\ref{E:nonperc2}) when $\beta^{\prime\prime}$ is above the Ising critical
point $\beta_{c}^{Potts}(2,\mathbb{Z}^{d})$. Define the percolation threshhold
\begin{align}
p_{c}^{Ising}(\beta^{\prime\prime},\mathbb{L}) = \inf \{p &\in [0,1]:
\text{ independent percolation at density } p \text{ on the ``++''}
\notag \\
&\text{bonds of the minus phase of the Ising model at }
(\beta^{\prime\prime},0)
\text{ on } \mathbb{L} \notag \\
&\text{produces an infinite cluster a.s.}\}. \notag
\end{align}
From Remark \ref{R:sharpcond} we know
$p_{c}^{Ising}(\beta^{\prime\prime},\mathbb{L}) \geq 1 -
e^{-\beta^{\prime\prime}}$ and for (\ref{E:nonperc}) it suffices that
\begin{align} \label{E:nbdcond}
p_{c}^{Ising}(\beta^{\prime\prime},\mathbb{L}) &> 1 - e^{-\beta}, \quad
\text{for all } \beta \text{ in a neighborhood of form }
(\beta_{1} - \epsilon,\beta_{1}) \\
&\text{and for } \beta^{\prime\prime} \text{ as in (\ref{E:betapp})},
\notag
\end{align}
where $\epsilon > 0$.
We are unable to verify (\ref{E:nbdcond}) for the integer lattice but we can
verify the analog for the Cayley tree $\mathbb{T}_{m}$ with large coordination
number $m$ as follows. It is easily checked that
\begin{equation} \label{E:pcasymp}
p_{c}^{Ising}(\beta^{\prime\prime},\mathbb{T}_{m}) \sim
(mP^{Ising}_{\mathbb{T}_{m},-,\beta_{m}^{\prime\prime},0}
(\sigma_{x} = +))^{-1} \quad \text{as } m \to \infty,
\end{equation}
where $x$ is arbitrary and
$P^{Ising}_{\mathbb{T}_{m},-,\beta_{m}^{\prime\prime},0}$ denotes
the infinite-volume Ising model on $\mathbb{T}_{m}$ at
$(\beta_{m}^{\prime\prime},0)$ with minus boundary condition.
This is just the branching-process
approximation which says that the critical percolation density is such that
the mean number of sites
$x$ adjacent to 0 for which both $\sigma_{x} = +$ and $\langle 0x \rangle$ is
open is approximately 1. Suppose we have a sequence $\{\beta_{m}\}$ with
$m\beta_{m} \to c$ for some $c$ ``sufficiently close'' to $2q(\log q)/(q-1)$;
then from (\ref{E:betapp}), $m\beta_{m}^{\prime\prime} \to a(c) = \tfrac{c}{q}
+ \log q$, while from the mean field limit for the magnetization on a tree
(which is straightforward),
\begin{equation} \label{E:Ppluslimit}
P^{Ising}_{\mathbb{T}_{m},-,\beta_{m}^{\prime\prime},0}
(\sigma_{x} = +) \to \frac{1 - M_{a(c)}}{2}
\end{equation}
where $M_{a}$ is the positive solution of $M = \tanh(aM/2)$. We claim that
\begin{equation} \label{E:magineq}
\frac{2}{1 - M_{a(c)}} > c \quad \text{for all } c \text{ in a neighborhood
of } \frac{2q \log q}{q - 1},
\end{equation}which with (\ref{E:pcasymp}) and (\ref{E:Ppluslimit}) establishes
(\ref{E:nbdcond}) for $\mathbb{T}_{m}$ with $m$ large. Thus it seems
plausible that (\ref{E:nbdcond}) may hold for the integer lattice when $q >
1$, at least in high dimensions. It is sufficient to prove (\ref{E:magineq})
for $c = 2q(\log q)/(q-1)$, so $c > 2$ and $a(c) = (q+1)(\log q)/(q-1)$. Now
(\ref{E:magineq}) for this $c$ is equivalent to $M_{a(c)} > 1 - \tfrac{2}{c}$
or
\[
\tanh\left(\frac{a(c)}{2}\left(1 - \frac{2}{c}\right)\right) > 1 -
\frac{2}{c}
\]
or
\begin{equation} \label{E:tanh}
\tanh\left(\frac{q+1}{2} \left(\frac{\log q}{q-1} - \frac{1}{q}\right)\right)
> 1 - \frac{q-1}{q \log q},
\end{equation}
and (\ref{E:tanh}) can be verified for all $q > 1$ by a tedious but
straightforward calculation, using $e^{x} > 1 + x + x^{2}/2$ for $x > 0$.
Thus (\ref{E:magineq}) and hence (\ref{E:nbdcond}) hold for $m$ large, so for
each $q > 1$, for $\mathbb{T}_{m}$ with $m$ large, (\ref{E:nonperc}) holds
for all $\beta$ in a neighborhood of $\beta_{1}$, as required.
\end{remark}
The obvious problem with Theorem \ref{T:highdbounds} is the difficulty
of verifying (\ref{E:nonperc}). We give next an alternate theorem which
has more readily verifiable hypotheses (at least in certain limits.) The
price paid for this is that the resulting bound on $p_{c}^{FK}$ or
$\beta_{c}^{Potts}$ is weaker, particularly for small $q$.
The \emph{magnetization} $M(\beta,h) = M(\beta,h,\mathbb{L})$ of the
Ising model at $(\beta,h)$ on the lattice $\mathbb{L}$ is the mean value
of $\sigma_{0}$ in the infinite-volume plus phase. For $h \geq 0$, the
\emph{susceptibility} at $(\beta,h)$ is the quantity
\[
\chi(\beta,h) = \frac{1}{\beta}\frac{\partial}{\partial h} M(\beta,h).
\]
\begin{theorem} \label{T:chibeta}
For $q > 1$ consider the FK model at $(p,q+1)$ on $\mathbb{Z}^{d}$,
and, for integer $q$, the corresponding $(q+1)$-state Potts model
at $(\beta,0)$, with $p = 1 - e^{-\beta}$. Suppose
\begin{equation} \label{E:chibeta}
2\chi\left(\frac{\beta}{2},0\right) <
1 + M\left(\frac{\beta}{2},0\right)
\end{equation}
and suppose
\begin{equation} \label{E:pbound}
p < 1 - \frac{1}{q^{1/d}}
\end{equation}
or equivalently
\[
e^{\beta} < q^{1/d}.
\]
Then there is no percolation in the FK model, and (for integer $q$)
the Potts model has a unique Gibbs distribution at inverse
temperature $\beta$. If (\ref{E:chibeta}) holds for all $\beta$ in
a neighborhood of $(\log q)/d$, then
\[
p_{c}(q+1,\mathbb{Z}^{d}) \geq 1 - \frac{1}{q^{1/d}}.
\]
\end{theorem}
Condition (\ref{E:pbound}) is clearly less sharp than (\ref{E:pcond}),
especially for small $q$; in particular one could not hope to obtain
the mean-field limit (\ref{E:meanflim}) using (\ref{E:pbound}).
For example, for $q + 1 = 4$ and $d = 3$, (\ref{E:pbound})
allows $p < .306$ while (\ref{E:pcond}) allows $p < .460$
(see Remark \ref{R:sharpcond}.) For $q + 1 = 30$ and $d = 10$,
there is less difference, as (\ref{E:pbound}) allows $p < .286$
while (\ref{E:pcond}) allows $p < .296$.
We can verify (\ref{E:chibeta}) in certain limits, which leads
to the following.
\begin{corollary} \label{C:larged}
Suppose $q + 1 \geq 10.56$. For all sufficiently large $d$, the FK
model satisfies
\[
p_{c}^{FK}(q+1,\mathbb{Z}^{d}) \geq 1 - \frac{1}{q^{1/d}}.
\]
\end{corollary}
\begin{remark}\label{R:KScompare}
For the Potts model, Kesten and Schonmann \cite{KS} established
a lower bound for $q + 1 \geq 2$ of the form
\[
\beta_{c}^{Potts}(q+1,\mathbb{Z}^{d}) \geq \frac{c(q)}{d}
\quad \text{for all } d.
\]
Their $c(q)$ is very close to the mean-field value $(q \log q)/(q - 1)$
for small $q$, which is better than what Corollary \ref{C:larged}
gives, but for large $q$ their $c(q)$ is only about half the
mean-field value, so Corollary \ref{C:larged} is better for large $q$.
\end{remark}
\begin{remark} \label{R:largebeta}
Since $\chi(\beta) \to 0$ as $\beta \to \infty$, (\ref{E:chibeta}) holds
for all sufficiently large $\beta$. Therefore for fixed $d$, Theorem
\ref{T:chibeta} shows $p_{c}^{FK}(q+1,\mathbb{Z}^{d}) \geq
1 - q^{-1/d}$ for all sufficiently large $q$. But this is of less
interest that large $d$, because a series expansion for
$p_{c}^{FK}(q+1,\mathbb{Z}^{d})$ for large $q$ is known
\cite{LMMRS}.
\end{remark}
We turn next to the analysis of certain perturbations of the (zero-field)
Potts and Ising models. The main question of interest to us is how
these perturbations affect the critical inverse temperature of the
model. We consider first annealed site dilution, that is, a Potts
lattice gas in which 0's are rare. It should be pointed out that the
coresponding problem for annealed bond dilution is much simpler,
because bond dilution is essentially just a change of temperature
(see \cite{St}.)
We define the \emph{dilution parameter} $\theta$ of a $q$-state
Potts lattice gas at $(1,J,\kappa,\mu)$ on a lattice with
coordination number $m$ by
\[
\theta = e^{-(\mu + \kappa m)}.
\]
For the corresponding ARC model, this becomes
\begin{equation} \label{E:ARCtheta}
\theta = q(Q - 1)(1 - p_{g})^{m}.
\end{equation}
Heuristically, and to an extent quantifiably (see Lemma
\ref{L:rare0}), $\theta$ gives the order of magnitude of the typical
fraction of 0's in the system. Define the ARC model critical point
for red bonds by
\begin{align}
p_{c}^{ARC}(p_{g},q,Q,\mathbb{L}) = \inf \{p_{r}:
&\text{ percolation of red bonds occurs in the} \notag \\
&\text{in the red-wired ARC model at } (p_{r},p_{g},q,Q)
\text{ on } \mathbb{L} \} \notag
\end{align}
and the corresponding Potts lattice gas critical point by
\begin{align}
J_{c}^{PLG}(q,\kappa,\mu,\mathbb{L}) = \inf\{J:
&\text{ there is symmetry breaking in the} \notag \\
&q-\text{state Potts lattice gas at } (1,J,\kappa,\mu)
\text{ on } \mathbb{L} \} \notag
\end{align}
As is easy to show (see Proposition \ref{P:percunique2}), if $(p_{g},Q)$ and
$(\kappa,\mu)$ are related as in (\ref{E:ARCparams}), we have
\begin{equation} \label{E:pbetacrit}
p_{c}^{ARC}(p_{g},q,Q,\mathbb{L}) = 1 -
e^{-J_{c}^{PLG}(q,\kappa,\mu,\mathbb{L})}.
\end{equation}
In the next theorem, the underlying heuristic is that when
small annealed site dilution is added to the Potts model, the
change in critical temperature, the fraction of empty sites,
and the dilution parameter $\theta$ are all of the same order
of magnitude. What we actually prove are one-sided bounds
consistent with this picture. These bounds essentially compare
the effect of the dilution to the effect of change much
simpler to analyze: replacing the parameter $q$ with $q + \theta$
in the FK model.
\begin{theorem} \label{T:dilution}
Let $\mathbb{L}$ be a lattice of coordination number $m$.
\begin{enumerate}
\item[(i)] If $q,Q \geq 1, p_{g} \in [0,1], (Q - 1)(1 - p_{g})^{m/2}
\leq 1$ and the dilution parameter $\theta$ is sufficiently small,
then
\[
p_{c}^{FK}(q,\mathbb{L}) \leq
p_{c}^{ARC}(p_{g},q,Q,\mathbb{L}) \leq p_{c}^{FK}(q
+ \theta,\mathbb{L}).
\]
\item[(ii)] If $\kappa \geq 0, \mu \in \mathbb{R}, q \geq 1,
\mu + \tfrac{1}{2} \kappa m + \log q \geq 0$ and the dilution
parameter $\theta = e^{-(\mu + \kappa m)}$ is sufficiently small,
then
\[
\beta_{c}^{Potts}(q,\mathbb{L}) \leq
J_{c}^{PLG}(q,\kappa,\mu,\mathbb{L})
\leq \beta_{c}^{Potts}(q+\theta,\mathbb{L}).
\]
\end{enumerate}
\end{theorem}
\begin{remark} \label{R:compare}
Let us compare Theorem \ref{T:dilution} to what can be obtained
by much simpler techniques, similar to the standard comparison
theorem discussed in the introduction. As we will show (see Lemma
\ref{L:compare}), since $Q^{2C(\omega) - I(\omega)}$ is a decreasing function
of $\omega$, these
simpler techniques show that the ARC model, call it $P$, at
$(p_{r},p_{g},q,Q)$ dominates the ARC model, call it $P^{\prime}$, at
$(p_{r},p_{g},qQ^{2},1)$. Hence the red bond configuration under
$P^{\prime}$, which forms the FK model at $(p_{r},qQ^{2})$, is dominated by
the red bond configuration under $P$. This shows that
$p_{c}^{ARC}(p_{g},q,Q,
\mathbb{L}) \leq p_{c}^{FK}(qQ^{2},\mathbb{L})$. But
$qQ^{2} \geq q + 2q(Q - 1)$, so this result is worse than
Theorem \ref{T:dilution} by a factor of at least 2 in the correction
$\theta$, and by a much larger factor if $p_{g}$ is near 1, or $Q$
is large.
\end{remark}
For the square lattice, and $q \geq 25.72$, the FK critical point is
known exactly \cite{LMR}. This tells us the exact change in
$p_{c}^{FK}$ when $q$ is replaced by $q + \theta$, so we can
get more detailed information
from Theorem \ref{T:dilution}
about the change in critical point induced
by the dilution. We summarize this in the following corollary.
\begin{corollary} \label{C:Z2case}
Suppose $\kappa \geq 0, \mu \in \mathbb{R}, q \geq 25.72$, and
$\mu + \tfrac{1}{2}\kappa m + \log q \geq 0$. Then for the square
lattice, for $\theta = e^{-(\mu + \kappa m)}$, as $\theta \to 0$,
\begin{equation} \label{E:Jcexpansion}
\beta_{c}^{Potts}(q,\mathbb{Z}^{2}) \leq
J_{c}^{PLG}(q,\kappa,\mu,\mathbb{Z}^{2}) \leq
\beta_{c}^{Potts}(q,\mathbb{Z}^{2}) + \frac{1}{2}
\sqrt{q}(1 + \sqrt{q})\theta + o(\theta).
\end{equation}
\end{corollary}
We do not expect that the factor $\tfrac{1}{2}\sqrt{q}(1 + \sqrt{q})$
multiplying $\theta$ in (\ref{E:Jcexpansion}) is sharp, but we do
expect, though we cannot prove, that the true correction term is of order
$\theta$.
A special case of ``dilution'' of the $q$-state Potts model is the
$(q+1)$-state Potts model with a large negative external field
applied to one of the spins. This is the subject of our next result.
\begin{theorem} \label{T:hifield}
Let $\mathbb{L}$ be a lattice and $q \geq 1$, and for $h \geq 0$, let
\[
\theta(h) = \exp\bigl(-\beta_{c}^{Potts}(q,\mathbb{L})h\bigr).
\]
Then
\begin{equation} \label{E:hifield}
\beta_{c}^{Potts}(q,\mathbb{L}) \leq
\beta_{c}^{Potts}(q + 1,-h,\mathbb{L}) \leq
\beta_{c}^{Potts}(q + \theta(h),\mathbb{L}).
\end{equation}
\end{theorem}
Note that (\ref{E:hifield}) is an equality both for $h = 0$ and in the limit
$h \to \infty$.
We turn now to a different perturbation: the Potts lattice gas with small $J$.
When $J = 0$ the particles form a binary lattice gas with the Gibbs weight
multiplied by an entropy factor $q^{N_{s}}$, where $N_{s}$ is the number of
particles; this factor just adds $\log q$ to the chemical potential. Thus
small $J$ may be considered a perturbation of a binary lattice gas, or
equivalently, of an Ising model. More precisely, presuming we relabel
particles as ``-''and empty sites as ``+'', the $J = 0$
Potts lattice gas becomes an Ising model with parameters $(\beta_{0},h_{0})$
given by (cf. (\ref{E:bhdefs}))
\begin{equation} \label{E:beta0h0}
\beta_{0} = \frac{\kappa}{2}, \quad \beta_{0}h_{0} =
-(\mu + \frac{\kappa m}{2} + \log q).
\end{equation}
We call $h_{0}$ the \emph{effective external field} of the $J = 0$ Potts
lattice gas. Thus for $J = 0$, the phase diagram in $(\kappa,\mu)$-space is
known---there is a critical line $\mu = -\tfrac{1}{2}\kappa m - \log q,\
\kappa > 2\beta_{c}^{Ising}(\mathbb{L})$ where there is phase coexistence, and
Gibbs uniqueness holds everywhere outside the closure of this critical line.
For fixed $\kappa > 2\beta_{c}^{Ising}(\mathbb{L})$, as $\mu$
increases, there is a first-order aggregation transition at $\mu =
-\tfrac{1}{2}\kappa m - \log q$ from an empty-dominated regime to a
particle-dominated regime. If the lattice is planar these regimes are
characterized by the percolation of empty sites and of particles,
respectively. For small $J$ one expects this phase diagram to be perturbed
only slightly. It is outside the scope of this work to make this phrase
``only slightly'' into a rigorous statement---this nontrivial problem would
involve showing that for small $J$ the transition remains sharp, meaning there
is no interval of $\mu$ values in which there is an intermediate phase, and
showing that the minimum value of $\kappa$ for which the transition is
first-order remains near
$2\beta_{c}^{Ising}(\mathbb{L})$ for small $J$. (See \cite{CKS} for more on
the phase diagram for positive $J$.) Instead we will establish a
one-sided bound---adding a positive $J$ reduces the critical $\mu$ by at
least a certain function of $J$. Define
\begin{align}
\mu_{c}^{PLG}(q,J,\kappa,\mathbb{L}) = \sup\{ \mu &\in \mathbb{R}:
\text{ there is percolation of 0's in the infinite-volume }
\notag \\
&q-\text{state Potts lattice gas at } (1,J,\kappa,\mu) \text{ on }
\mathbb{L} \text{ with 0's} \notag \\
&\text{boundary condition} \} \notag
\end{align}
so that for planar $\mathbb{L}$, from the preceding discussion,
\[
\mu_{c}^{PLG}(q,0,\kappa,\mathbb{L}) = - \frac{\kappa m}{2} - \log q
\quad \text{for all } \kappa > 2\beta_{c}^{Ising}(\mathbb{L}).
\]
\begin{theorem} \label{T:smallJ}
Let $\mathbb{L}$ be a planar lattice of coordination number $m$. For $ q
\geq 1, J \geq 0$ and $\kappa > 2\beta_{c}^{Ising}(\mathbb{L})$,
\begin{align} \label{E:mubound}
\mu_{c}^{PLG}(q,J,\kappa,\mathbb{L}) &\leq
\mu_{c}^{PLG}(q,0,\kappa,\mathbb{L}) - \frac{m}{2}
\log \left(\frac{1}{q}e^{J} + \frac{q-1}{q} \right) \\
&= \mu_{c}^{PLG}(q,0,\kappa,\mathbb{L}) - \frac{m}{2q}J
- \frac{m(q-1)}{4q^{2}}J^{2}
+ O(J^{3}) \quad \text{as } J \to 0. \notag
\end{align}
\end{theorem}
We expect, but do not prove, that the order-$J$ term in the RHS of
(\ref{E:mubound}) is the true first-order correction, and that the
order-$J^{2}$ terms is correct as well for certain lattices, including the
square and hexagonal lattices, which have the property that for a bond
$\langle xy \rangle$, the length of the shortest path from $x$ to $y$ outside
$\langle xy \rangle$ is more than 2. To see why, note first that the Potts
lattice gas with small $J$ has particles of species which are approximately
independent and uniform in $\{1,..,q\}$, so that for any bond, the endpoints
match species with probability approximately $1/q$. Conditionally on the
particle locations, there is approximate pairwise independence, but not mutual
independence, among the variables $\delta_{[\sigma_{x} = \sigma_{y}]}$ as
$\langle xy \rangle$ varies over bonds with particles at both ends. If these
variables $\delta_{[\sigma_{x} = \sigma_{y}]}$ \emph{were} mutually
independent, then each bond with particles at both ends would make a
contribution to the Gibbs weight $e^{-H_{\Lambda,\eta}(\sigma)}$ of
$e^{\kappa}$ from the interparticle attraction and of
\[
\frac{1}{q}e^{J} + \frac{q-1}{q}
\]
from the Potts interaction. Defining $\kappa_{0}$ by
\[
e^{\kappa_{0}} = e^{\kappa}\left( \frac{1}{q}e^{J} + \frac{q-1}{q} \right)
\]
we see that the effect of positive $J$
under mutual independence would be merely to change the
interparticle attraction from $\kappa$ to $\kappa_{0}$; we would still have
effectively just a binary lattice gas. What actually happens is a slight
variation of this, as follows. The contribution to the Gibbs weight from a
bond $\langle xy \rangle$ with particles at both ends is
\[
e^{\kappa}\bigl( \lambda e^{J} + (1 - \lambda) \bigr),
\]
where the random value $\lambda = \lambda(x,y,\{n_{z}\})$
is the probability that $\sigma_{x} =
\sigma_{y}$ for a Potts model at inverse temperature $J$ on the particles, but
with the interaction between $x$ and $y$ ``turned off,'' and
$\lambda e^{J} + (1 - \lambda)$ is the ratio of two partition functions for
the Potts model on the particles, one with the $\langle xy \rangle$ Potts
interaction ``turned on'' and one with it ``turned off.'' Assuming ``nice''
boundary conditions, e.g. free or having a single particle species,
we have $\lambda \geq 1/q$. We therefore call
$\kappa_{0}$ (which corresponds to $\lambda = 1/q$) the
\emph{minimum effective
interparticle atttraction} of the
$q$-state Potts lattice gas at $(1,J,\kappa,\mu)$. The heuristic content of
Theorem \ref{T:smallJ} is that since $\mu_{c}^{PLG}(q,0,\kappa,\mathbb{L})$
is a decreasing function of $\kappa$, and the small-$J$ system is roughly like
a $J = 0$ system with interparticle attraction $\kappa_{0}$ or greater, we
should expect $\mu_{c}^{PLG}(q,J,\kappa,\mathbb{L}) \leq
\mu_{c}^{PLG}(q,0,\kappa_{0},\mathbb{L})$. This is fairly sharp so long as
$\lambda$ is typically close to $1/q$. The deviation $\lambda - 1/q$ should
be of order of the probability of a path of open bonds in the FK model from
$x$ to $y$ outside $\langle xy \rangle$, which is of order $J^{n}$ for
small $J$ when the shortest possible path has length $n$. Thus the inequality
in (\ref{E:mubound}) should be accurate to within order $J^{n}$.
The $J = 0$ Potts lattice gas with interparticle attraction $\kappa_{0}$,
which we are effectively using as a bound for the positive-$J$ system, is
equivalent (when we ignore particle species) to an Ising model with parameters
$(\beta^{\prime},h^{\prime})$ given by
\begin{equation} \label{E:effectives}
\beta^{\prime} = \frac{\kappa_{0}}{2}, \quad \beta^{\prime}h^{\prime} =
-\left( \mu + \frac{1}{2}\kappa_{0}m + \log q \right)
\end{equation}
or equivalently
\begin{equation} \label{E:effectives2}
e^{2\beta^{\prime}} = e^{\kappa}(\frac{1}{q}e^{J} + \frac{q-1}{q}),
\quad e^{\beta^{\prime}(m + h^{\prime})} = \frac{e^{-\mu}}{q}.
\end{equation}
We call $h^{\prime}$ the \emph{maximum effective external field}, and
$\beta^{\prime}$ the \emph{minimum effective inverse temperature}, of the
$q$-state Potts lattice gas at $(1,J,\kappa,\mu)$.
Our final topic is couplings. Couplings have been a useful tool
in studying how the boundary condition influences probabilities
under a Gibbs distribution (see \cite{Ne},\cite{AC}.) For a Potts
lattice gas configuration $\sigma$ on a finite set $\Lambda$ with
boundary condition $\eta$ on $\partial \Lambda$, we define the
\emph{boundary particle cluster}
\begin{align}
C(\partial \Lambda, \sigma) = \{x \in \Lambda: x &\text{ is connected
to } \partial \Lambda \text{ by a path in which} \notag \\
&\text{all sites } x \in \Lambda
\text{ have } n_{x} = 1\}. \notag
\end{align}
We say that the Potts lattice gas at $(1,J,\kappa,\mu)$ on a lattice
$\mathbb{L}$ has the \emph{boundary coupling property with respect
to particles} if for every finite $\Lambda$ and every boundary condition
$\eta$ on $\partial \Lambda$, there exists a coupling $\tilde{P}$ of
the measures with boundary conditions $\eta$ and $\eta^{1}$ satisfying
\[
\tilde{P}(\{ (\sigma,\sigma^{\prime}): \sigma \text{ and }
\sigma^{\prime} \text{ agree on } \Lambda \backslash C(\partial
\Lambda,\sigma^{\prime}) \}) = 1.
\]
This property is instrumental in \cite{Al98} in establishing the
following property for the Potts model on a planar lattice with
nonnegative external field applied to spin 0: exponential decay of
the (infinite-volume) probability of connecting two sites by a
path with no 0 spins implies weak mixing.
\begin{theorem} \label{T:coupling}
For every lattice $\mathbb{L}$ and all $J, \kappa \geq 0, q \geq 1$
and $\mu \in \mathbb{R}$, the $q$-state Potts lattice gas at $(1,J,
\kappa,\mu)$ on $\mathbb{L}$ has the boundary coupling property
with respect to particles.
\end{theorem}
\section{Basic Properties of the ARC Model and the Partial FK Model.}
We begin with the FKG property for the ARC model.
Note that an ARC model is a probability distribution on pairs of
configurations. Hence if $\omega = (\omr,\omg)$ and $\omega^{\prime} =
(\omr^{\prime},\omg^{\prime})$, then $\omega \geq
\omega^{\prime}$ means precisely that both $\omr \geq \omr^{\prime}$ and $\omg
\geq \omg^{\prime}$. In particular, there is never a comparison of a red bond
to a green one, and one should not view red and green as two possible states
of a single bond $\langle xy \rangle$, but rather as two parallel bonds
between $x$ and $y$.
\begin{lemma} \label{L:FKG}
Let $P$ be an ARC model on a graph $(\Lambda,\mB)$, possibly with a boundary
condition,
given by the weights (\ref{E:ARCweight}) for
all $\omega = (\omega_{r},\omega_{g}) \in \{0,1\}^{\mB} \times
\{0,1\}^{\mB}$, with $q,Q \geq 1$. Then $P$ satisfies the FKG lattice
condition (\ref{E:FKGlatt}). Consequently, $P$ has the FKG
property.
\end{lemma}
\begin{proof}
As in the analogous proof for the FK model (see \cite{Gr97}), we have
\[
C(\omega_{r} \vee \omega_{r}^{\prime}) - C(\omega_{r}) \geq
C(\omega_{r}^{\prime}) - C(\omega_{r} \wedge \omega_{r}^{\prime}).
\]
Similarly, setting $K(\omega) = I(\omega_{r} \vee \omega_{g})$,
\[
K(\omega \vee \omega^{\prime}) - K(\omega) \geq
K(\omega^{\prime}) - K(\omega \wedge \omega^{\prime}),
\]
and (\ref{E:FKGlatt}) follows easily. That (\ref{E:FKGlatt})
implies the FKG property is a result of \cite{FKG}.
\end{proof}
\begin{remark} \label{R:FKG}
Lemma \ref{L:FKG} applies to the ARC model under any bond boundary
condition, but does not apply to the ARC model with site boundary
conditions in general, because the weights (\ref{E:ARCweight})
then only apply to a restricted set of configurations $\omega$.
Constant-species boundary conditions are covered by Lemma
\ref{L:FKG} since the event $D(\Lambda,\eta)^{c}$ of
(\ref{E:Devents}) is empty in such cases. More generally,
suppose we have a generalized site boundary condition $\eta$ which
has a single particle species. Let $F^{\prime} = \{ x \in \partial
\Lambda: \eta_{x} = i\}$ and $E^{\prime} = \{x \in \partial \Lambda:
\eta_{x} = 0\}$, and let $(\Lambda^{\prime},\mB^{\prime})$ be the
graph obtained by deleting $E^{\prime}$ and all bonds
with an endpoint in $E^{\prime}$ from $(\oL,\omB(\Lambda))$.
Then the ARC model on $(\oL,\omB(\Lambda))$ with boundary condition
$\eta$ on $\partial \Lambda$ is equivalent to the ARC model on
$(\Lambda^{\prime},\mB^{\prime})$ with all-$i$'s boundary condition
on $F^{\prime}$. Thus Lemma \ref{L:FKG} applies under
boundary conditions having a single particle species as well. Here we are
using the fact that when the graph if $(\oL,\omB(\Lambda))$, the proof of
Lemma \ref{L:FKG} is not changed if $I(\cdot)$ means $I(\cdot,\Lambda)$ and
not $I(\cdot,\oL)$.
\end{remark}
Let us call an ARC model on a graph $(\Lambda,\mB)$
\emph{unconditioned} if the weights (\ref{E:ARCweight}) apply to
all confiturations in $\{0,1\}^{\mB} \times \{0,1\}^{\mB}$. Thus
ARC models with free or bond boundary conditions are unconditioned,
and an ARC model with site boundary condition having a single
particle species is equivalent to an unconditioned ARC model on an
appropriate subgraph, as in Remark \ref{R:FKG}.
Exactly as for the FK model (see \cite{Gr96},Theorem 3.1), we
obtain the following using Lemma \ref{L:FKG}.
\begin{corollary} \label{C:infvol}
For fixed $p_{r} \in [0,1], p_{g} \in [0,1], q \geq 1$ and $Q \geq
1$, the measures on $\{0,1\}^{\mB(\mathbb{L})} \times
\{0,1\}^{\mB(\mathbb{L})}$ given by the weak limits
\[
P_{rw,p_{r},p_{g},q,Q}^{ARC} = \lim_{\Lambda \nearrow S(\mathbb{L})}
P_{\Lambda,rw,p_{r},p_{g},q,Q}^{ARC}
\quad \text{and} \quad P_{iso,p_{r},p_{g},q,Q}^{ARC} =
\lim_{\Lambda \nearrow S(\mathbb{L})} P_{\Lambda,iso,p_{r},p_{g},q,Q}^{ARC}
\]
exist, and are translation-invariant and ergodic.
\end{corollary}
We continue with an analog for the ARC model of the standard
comparison theorem of \cite{FK} (see also \cite{ACCN}) for the
FK model.
\begin{lemma} \label{L:compare}
Consider an unconditioned ARC model on a finite graph (possibly
with a boundary condition). Let $p_{r},p_{g},p_{r}^{\prime},p_{g}^{\prime}
\in [0,1]$
and let $q,Q,Q^{\prime} \geq 1$
and $q^{\prime} > 0$.
The model at $(p_{r}^{\prime},p_{g}^{\prime},q^{\prime},Q^{\prime})$
dominates the model at $(p_{r},p_{g},q,Q)$ under any of the
following conditions:
\begin{enumerate}
\item[(i)] $p_{r} \leq p_{r}^{\prime}, p_{g} \leq p_{g}^{\prime},
q = q^{\prime}$ and $Q = Q^{\prime}$;
\item[(ii)] $p_{r} = p_{r}^{\prime}, p_{g} = p_{g}^{\prime},
q \geq q^{\prime}$ and $Q \geq Q^{\prime}$;
\item[(iii)] $p_{r} \leq p_{r}^{\prime}, p_{g} = p_{g}^{\prime},
Q = Q^{\prime}$ and
\[
\frac{p_{r}}{q(1 - p_{r})} \leq \frac{p_{r}^{\prime}}{q^{\prime}(1 -
p_{r}^{\prime})};
\]
\item[(iv)] $p_{r} \leq p_{r}^{\prime}, p_{g} \leq p_{g}^{\prime},
q = q^{\prime}$ and
\[
\frac{p_{r}}{Q^{2}(1 - p_{r})} \leq
\frac{p_{r}^{\prime}}{(Q^{\prime})^{2}(1 - p_{r}^{\prime})}.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
Let $W^{\prime}(\omr,\omg)$ and $W(\omr,\omg)$ be the weight functions
as in (\ref{E:ARCweight}), for the two parameter choices. It is easy
to see that $|\omr|, |\omg|, |\omr| + C(\omr), 2|\omr| + I(\omr \vee \omg)$
and $2|\omg| + I(\omr \vee \omg)$ are increasing functions
of $(\omr,\omg)$, while
$C(\omr)$ and $I(\omr \vee \omg)$ are decreasing functions.
It follows easily that in all four cases, $W^{\prime}/W$ is an
increasing function.
\end{proof}
Suppose we have two models, $P_{A}$ and $P_{B}$, for configurations
on a finite $\Lambda$, each with boundary conditions, and we have
species $i$
appearing under $P_{A}$ and $j$ appearing under $P_{B}$. As a
shorthand terminology, we say that the $i$'s configuration under
$P_{A}$ dominates the $j$'s configuration under $P_{B}$ if
$P_{A}(\{ \delta_{[\sigma_{x} = i]}: x \in \Lambda \} \in \cdot )$
dominates $P_{B}(\{ \delta_{[\sigma_{x} = j]}: x \in \Lambda \}
\in \cdot )$. Note these are measures on $\{0,1\}^{\Lambda}$.
For a Potts lattice gas on a finite set $\Lambda$, define
\[
X_{0} = X_{0}(\Lambda,\sigma) = \{x \in \Lambda: \sigma_{x} = 0\}.
\]
\begin{lemma} \label{L:zerocompare}
Consider a $q$-state Potts lattice gas and a $q^{\prime}$-state
Potts lattice gas on a finite set $\Lambda$ under respective site
boundary conditions $\eta$ and $\eta^{\prime}$, with parameter
values $(1,J,\kappa,\mu)$ and $(1,J^{\prime},\kappa^{\prime},
\mu^{\prime})$ respectively, satisfying $J, \kappa, J^{\prime},
\kappa^{\prime} \geq 0$. Let $(p_{r},p_{g},q,Q)$ and
$(p_{r}^{\prime},p_{g}^{\prime},q^{\prime},Q^{\prime})$ be the
parameters of the corresponding ARC models. Suppose $Q \geq
Q^{\prime}$ and the black configuration $\omb = \omr \vee \omg$
of the ARC model under
$P^{ARC}_{\Lambda,\eta^{\prime},p_{r}^{\prime},p_{g}^{\prime},
q^{\prime},Q^{\prime}}$ dominates the black configuration under
$P^{ARC}_{\Lambda,\eta,p_{r},p_{g},q,Q}$. Then the 0's configuration of
the Potts lattice gas under $P^{PLG}_{\Lambda,\eta,q,
1,J,\kappa,\mu}$ dominates the 0's configuration under
$P^{PLG}_{\Lambda,\eta^{\prime},q^{\prime},1,J^{\prime},\kappa^{\prime},
\mu^{\prime}}$.
\end{lemma}
\begin{proof}
Let $P$ and $P^{\prime}$ denote the ARC models at $(p_{r},p_{g},q,Q)$
and at $(p_{r}^{\prime},p_{g}^{\prime},q^{\prime},Q^{\prime})$,
respectively. There exists a coupling $\tilde{P}$ of $P$ and $P^{\prime}$ for
which $\tilde{P}(\{(\omb,\omb^{\prime}): \omb \leq \omb^{\prime}\}) = 1$,
and hence $\mathcal{I}(\omb,\Lambda) \supset \mathcal{I}(\omb^{\prime},
\Lambda)$ a.s. From (\ref{E:labelprobs}), since $Q \geq Q^{\prime}$,
the ARC configurations $\omb$ and
$\omb^{\prime}$ can therefore be labeled to produce lattice-gas configurations
$\sigma$ and $\sigma^{\prime}$ satisfying $X_{0}(\Lambda,\sigma) \supset
X_{0}(\Lambda,\sigma^{\prime})$.
\end{proof}
Applying Lemmas \ref{L:compare} and \ref{L:zerocompare} to the Ising model and
Ising ARC model yields the following result, obtained by Schonmann and
Shlosman (\cite{SS}, Lemma 1) using different methods.
\begin{lemma} \label{L:pluscompare}
(\cite{SS})
Consider the Ising model on a finite subset
$\Lambda$ of a lattice with coordination
number $m$, with boundary condition $\eta$. Suppose that
\begin{equation} \label{E:betah3}
\beta^{\prime}(m - h^{\prime}) \geq \beta(m - h);
\end{equation}
\begin{equation} \label{E:betah4}
\beta^{\prime}(m + h^{\prime}) \leq \beta(m + h).
\end{equation}
Then the ``+'' configuration
on $\Lambda$ at $(\beta,h)$ dominates the ``+'' configuration
at $(\beta^{\prime},h^{\prime})$.
\end{lemma}
Note that if $-m \leq h^{\prime} \leq m$, then (\ref{E:betah3}) and
(\ref{E:betah4}) imply $h \geq h^{\prime}$.
\begin{proof}[Proof of Lemma \ref{L:pluscompare}]
The comparison is made by way of the model with a third set of
parameters, $(\beta^{\prime\prime},h^{\prime\prime})$. Define these by
\[
\beta^{\prime}(m + h^{\prime}) = \beta^{\prime\prime}(m + h^{\prime\prime}),
\quad \beta(m - h) = \beta^{\prime\prime}(m - h^{\prime\prime}).
\]
It is easy to check that $\beta^{\prime\prime} \leq
\min(\beta,\beta^{\prime})$. From Lemma \ref{L:zerocompare} and Lemma
\ref{L:compare}(i) we have the following two conclusions:
\begin{enumerate}
\item[(i)] the ``+'' configuration at
$(\beta^{\prime\prime},h^{\prime\prime})$ dominates the ``+''
configuration at $(\beta^{\prime},h^{\prime})$;
\item[(ii)] the ``+'' configuration at
$(\beta^{\prime\prime},-h^{\prime\prime})$ dominates the ``+''
configuration at $(\beta,-h)$.
\end{enumerate}
We can restate (ii) as:
\begin{enumerate}
\item[(iii)] the ``-'' configuration at $(\beta,-h)$
dominates the ``-''
configuration at $(\beta^{\prime\prime},-h^{\prime\prime})$.
\end{enumerate}
Since (iii) is valid under arbitrary boundary condition, we can interchange
the roles of ``+''and ``-'' in (iii) to obtain:
\begin{enumerate}
\item[(iv)] the ``+'' configuration at $(\beta,h)$ dominates the ``+''
configuration at $(\beta^{\prime\prime},h^{\prime\prime})$.
\end{enumerate}
Now (i) and (iv) prove the lemma.
\end{proof}
For the lattice $\mathbb{Z}^{d}$, a \emph{plaquette} is a face of a
unit hypercube centered at a lattice site. Each plaquette is
the perpendicular
bisector of a unique bond. A \emph{dual surface}
(consisting of plaquettes) is
the outer boundary of a connected set which is the union of a finite
collection of such hypercubes.
\begin{proposition} \label{P:percunique2}
Consider the red-wired ARC model on a lattice $\mathbb{L}$ with
parameters $(p_{r},p_{g},$ $q,Q)$, with $q \in \mathbb{Z}$, and the
(usual) $q$-state Potts lattice gas with corresponding parameters
$(1,J,\kappa,\mu)$, given by (\ref{E:ARCparams}).
\begin{enumerate}
\item[(i)] If the ARC model with red-wired boundary condition
has no percolation in the black
configuration $\omb = \omr \vee \omg$, then the Potts lattice gas
has a unique Gibbs distribution.
\item[(ii)] The red bonds of the ARC model with red-wired boundary condition
percolate if and only if
the Potts lattice gas exhibits symmetry breaking, that is, there is
a Gibbs distribution not symmetric in $\{1,..,q\}$.
\end{enumerate}
\end{proposition}
\begin{proof}
We give the proof for the integer lattice only; for other lattices
one need only extend the notion of an ``outermost dual surface'' in the
appropriate way. By Lemma \ref{L:FKG}, on a finite
$(\oL,\omB(\Lambda))$ the ARC model with red-wired boundary dominates the
ARC model with any other generalized site boundary condition $\eta$.
If $\Delta \subset \Lambda$ and in some configuration $\omb$ there is
no path of open bonds from $\partial \Lambda$ to $\Delta$, then
there is a unique outermost dual surface $\Gamma = \Gamma(\omb)$
surrounding $\Delta$ which is crossed by no open black bond. Let
$P_{1}$ and $P_{2}$ denote the red-wired measure
(that is, the measure under boundary condition $\eta^{1}$)
and the measure under
$\eta$, respectively, for the ARC model on $(\oL,\omB(\Lambda))$.
As is well-known in the context of the FK model (see e.g. \cite{Ne}),
the coupling $\tilde{P}$ of $P_{1}$ and $P_{2}$ can be chosen so that
\[
\tilde{P}\bigl(\{(\omb,\omb^{\prime}): \omb \text{ and }
\omb^{\prime}
\text{ agree inside } \Gamma(\omb)\} \mid \partial \Lambda
\not\leftrightarrow \Delta \text{ in } \omb \bigr) = 1.
\]
When $\omb$ and $\omb^{\prime}$ agree inside $\Gamma(\omb)$, clusters
of $\omb$ and $\omb^{\prime}$ can be labeled identically to create
Potts configurations, under boundary conditions $\eta^{1}$ and $\eta$,
which also agree inside $\Gamma(\omb)$. Letting $\Lambda \nearrow
S(\mathbb{L})$ we have $\tilde{P}(\partial \Lambda \not\leftrightarrow
\Delta \text{ in } \omb) \to 1$ and (i) follows. The proof of (ii)
is similar to the the proof for the FK model (see \cite{ACCN}.)
\end{proof}
The next result is an analog of Proposition \ref{P:PFKcondl}.
\begin{lemma} \label{L:ARCcondl}
Let $\Lambda$ be a finite subset of a lattice $\mathbb{L}$ with
coordination number $m$. Consider a $q$-state particle/bond Potts
lattice gas on $(\oL,\omB(\Lambda))$ with a generalized site boundary
condition $\eta$, with parameters $(1,J,\kappa,\mu)$ satisfying
$\kappa, J \geq 0$. Then conditionally on $\omega_{r}$, the 0's
of the Potts lattice gas form the ``+'' configuration of an Ising
model on $\mathcal{I}(\omr,\Lambda)$ with parameters $(\tilde{\beta},
\tilde{h})$ given by
\[
\tilde{\beta} = \frac{\kappa}{2}, \quad
e^{\tilde{\beta}(m + \tilde{h})} = \frac{e^{-\mu}}{q}
\]
and with boundary condition as follows: ``-'' on
$\Lambda \backslash \mathcal{I}(\omega_{r},\Lambda)$ and on $\{x \in \partial
\Lambda:
\eta_{x} \neq 0 \}$, ``+'' on $\{ x \in \partial \Lambda: \eta_{x}
= 0 \}$. In particular, for the $(q+1)$-state particle/bond
Potts model at $(\beta,0)$, we have
\begin{equation} \label{E:condlparams}
\tilde{\beta} = \frac{\beta}{2}, \quad e^{\beta \tilde{h}/2}
= \frac{e^{\beta m/2}}{q}.
\end{equation}
\end{lemma}
\begin{proof}
It is immediate from (\ref{E:ARCweight}) that conditionally on
$\omr$, the green bonds of the ARC model in $\omB(\Gamma)$
form an Ising ARC model on
$(\overline{\Gamma},\omB(\Gamma))$, where $\Gamma = \mathcal{I}(\omr,
\Lambda)$, with ``-'', or equivalently wired, boundary condition.
Applying (\ref{E:PARCparams}) yields the result.
\end{proof}
We call the Ising model of Lemma \ref{L:ARCcondl} the \emph{conditional
Ising model} of the (particle/bond) Potts lattice gas or of the equivalent
ARC model. Note that conditionally on $\omr$, in addition to the Ising ARC
model formed by the green bonds in $\omB(\Gamma)$, the remaining green bonds of
the ARC model---those with neither endpoint isolated in
$\omr$---are independently open with probability $p_{g}$.
For $\mathbb{L}= \mathbb{Z}^{d}$ we have from (\ref{E:condlparams})
that $\tilde{h} < 0$ if and only if $e^{\beta} < q^{1/d}$. This
is very close to the condition that $\beta$ is subcritical, at least
for large $q$ \cite{LMR}. Thus, except perhaps near the critical point,
0's are favored relative to particles on $\mathcal{I}(\omr,\Lambda)$ when
$\beta$ is supercritical, and particles are favored when $\beta$ is
subcritical.
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.) The ratio
$U(\omega)/V(\omega)$ of two functions is an increasing function
if and only if
\begin{equation} \label{E:incrfunc}
\frac{U(\omega \vee e)}{U(\omega)} \geq
\frac{V(\omega \vee e)}{V(\omega)} \quad \text{for all } \omega
\text{ and } e.
\end{equation}
In some situations of interest, the ratios appearing in
(\ref{E:incrfunc}) can be interpreted as probabilities, as the next
three lemmas show. Let $P^{Ising}_{\Lambda,\eta,\beta,h}$ denote the
distribution of the Ising model with parameters $(\beta,h)$ on
a finite set $\Lambda$ with boundary condition $\eta$ on $\partial
\Lambda$, and let $P^{Ising}_{+,\beta,h}$ and $P^{Ising}_{-,\beta,h}$ denote
the infinite volume limits under ``+'' and ``-'' boundary conditions,
respectively, on the full lattice
$\mathbb{L}$.
\begin{lemma} \label{L:Gratio}
Let $p \in (0,1)$ and $Q \geq 1$, and let $\Lambda$ be a finite set
of sites of a lattice $\mathbb{L}$ with coordination number $m$.
For $\omega \in \{0,1\}^{\omB(\Lambda)}$ define
\[
G(\omega) = \sum_{\omr \in
\{0,1\}^{\mB(\mathcal{I}(\omega,\Lambda))}} \
\left(\frac{p}{1-p}\right)^{|\omr|}
Q^{I(\omr \vee \omega,\Lambda)}.
\]
Then for $e = \langle xy \rangle \in \omB(\Lambda)$,
\begin{equation} \label{E:Gratio}
\frac{G(\omega \vee e)}{G(\omega)} = (Q - 1)^{I(\omega \vee e,
\Lambda) - I(\omega,\Lambda)} \ P^{Ising}_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -),
\end{equation}
where $\beta$ and $h$ are given by
\begin{equation} \label{E:betah}
p = 1 - e^{-\beta}, \quad Q = 1 + e^{\beta(m - h)/2}.
\end{equation}
\end{lemma}
If $x \notin \Lambda$ then $P^{Ising}_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -)$ should be interpreted as
$P^{Ising}_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h}(\sigma_{y}$ $ = -)$. Similarly if both $x, y \notin \Lambda$
then $P^{Ising}_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -)$ should be interpreted
as 1. Also, in the event that $\mB(\mathcal{I}(\omega,\Lambda))$ is empty,
we define $G(\omega)$ to be $Q^{I(\omega,\Lambda)}$.
Given an ARC-model red-bond configuration $\omega$, there is a
conditional Ising model on $\mathcal{I}(\omega,\Lambda)$, and a corresponding
reversed-polarity Ising ARC model. $G(\omega)$ is a version of the
partition function for this reversed-polarity Ising ARC model. In the
case that the original ARC model is a Potts ARC model, we may view $\omega$
as the FK portion of a joint FK/Potts configuration, with the ``00'' bonds
deleted. The configurations $\omr$ summed to obtain $G(\omega)$ are
just the possible choices for the set of ``00'' bonds to make $\omega \vee
\omr$ a full FK configuration. This is made precise in Lemma
\ref{L:completetoFK}.
\begin{proof}[Proof of Lemma \ref{L:Gratio}]
Fix $\omega$ and $e = \langle xy \rangle$.
Let $\Gamma = \mathcal{I}(\omega,\Lambda)$ and
$\Delta = I(\omega,\Lambda) - I(\omega \vee e,\Lambda) = |\{x,y\} \cap
\Gamma|$. Consider an Ising ARC model, which we denote $P_{1}$,
on $(\overline{\Gamma},\omB(\Gamma))$ with parameters $(p,Q)$ and ``+'',
or equivalently isolated, boundary conditions. This is equivalent (see
Remark \ref{R:freeiso}) to an Ising ARC model $P_{2}$ on $(\Gamma,
\mB(\Gamma))$ with the same parameters $(p,Q)$ but with free boundary.
The weights for $P_{2}$ are given by the terms of the sum $G(\omega)$:
\[
W_{2}(\omr) = \left(\frac{p}{1-p}\right)^{|\omr|}Q^{I(\omr,\Gamma)},
\quad \omr \in \{0,1\}^{\mB(\Gamma)}.
\]
Let $P_{3}$ be the Ising ARC model on $(\Gamma \backslash \{x,y\},
\mB(\Gamma \backslash \{x,y\}))$ with parameters $(p,Q)$ and free
boundary, and let $\mathcal{A} = \mB(\Gamma) \backslash \mB(\Gamma
\backslash \{x,y\})$. Note $e \in \mathcal{A}$. Each $\omr$ for which all
bonds of $\mathcal{A}$ are closed (that is, $\{x,y\} \cap \Gamma
\subset \mathcal{I}(\omr,\Gamma))$
corresponds to a unique configuration
$\alpha_{r}$ which is
the restriction of $\omr$ to $\mB(\Gamma \backslash \{x,y\})$, and
conversely.
The corresponding weight is
\begin{equation} \label{E:W3def1}
W_{3}(\alpha_{r}) = \left( \frac{p}{1-p} \right)^{|\alpha_{r}|}
Q^{I(\alpha_{r},\Gamma \backslash \{x,y\})} =
W_{2}(\omr)Q^{-\Delta}.
\end{equation}
Summing $W_{3}(\alpha_{r})$ over all $\alpha_{r}$ yields $G(\omega \vee e)$,
so summing (\ref{E:W3def1}) and dividing by $G(\omega)$ yields
\begin{equation} \label{E:Gratio2}
\frac{G(\omega \vee e)}{G(\omega)} = P_{2}\bigl( \{x,y\} \cap \Gamma
\subset \mathcal{I}(\omr,\Gamma)) \bigr) Q^{-\Delta} = P_{1}\bigl( \{x,y\}
\cap \Gamma \subset \mathcal{I}(\omr,\Gamma) \bigr) Q^{-\Delta}.
\end{equation}
But from (\ref{E:labelprobs}),
\begin{align}
P^{Ising}_{\Gamma,-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -)
&= P^{Ising}_{\Gamma,+,\beta/2,-h}(\sigma_{x} = \sigma_{y} = +)
\notag \\
&= P_{1}\bigl( \{x,y\} \cap \Gamma \subset \mathcal{I}(\omr,\Gamma) \bigr)
\left( \frac{Q-1}{Q} \right)^{\Delta}, \notag
\end{align}
and (\ref{E:Gratio}) follows.
\end{proof}
We have viewed $G(\omega)$ as the partition function of a reversed-polarity
Ising ARC model. We could do the same without reversing the polarity. This
yields a different partition function $T(\omega)$, expressed below as a sum
over configurations $\omg$. These configurations $\omg$ are precisely those
of the Ising ARC model which appeared in the proof of Lemma \ref{L:ARCcondl}.
\begin{lemma} \label{L:Tratio}
Let $p \in (0,1)$ and $Q \geq 1$, and let $\Lambda$ be a finite subset
of a lattice $\mathbb{L}$. For $\omega \in \{0,1\}^{\omB(\Lambda)}$
define $b(\omega) = |\omB(\mathcal{I}(\omega,\Lambda))|$ and
\[
T(\omega) = \sum_{\omg \in \{0,1\}^{\omB(\mathcal{I}(\omega,\Lambda))}}
p^{|\omg|} (1 - p)^{b(\omega) - |\omg|} Q^{I(\omega \vee \omg,
\Lambda)}.
\]
Then for $e = \langle xy \rangle \in \omB(\Lambda)$,
\begin{equation} \label{E:Tratio}
\frac{T(\omega \vee e)}{T(\omega)} = P_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -),
\end{equation}
where $\beta$ and $h$ are given by
\begin{equation} \label{E:betah2}
p = 1 - e^{-\beta}, \quad Q = 1 + e^{\beta(m + h)/2}.
\end{equation}
\end{lemma}
Note that the configurations $\omr$ in Lemma \ref{L:Gratio} are on
$\mB(\mathcal{I}(\omega,\Lambda))$, while the configurations $\omg$ in
Lemma \ref{L:Tratio} are on $\omB(\mathcal{I}(\omega,\Lambda))$.
\begin{proof}[Proof of Lemma \ref{L:Tratio}]
As in the proof of Lemma \ref{L:Gratio}, fix $\omega$ and
$e = \langle xy \rangle$ and let
$\Gamma = \mathcal{I}(\omega,\Lambda), \Delta = I(\omega,\Lambda) -
I(\omega \vee e,\Lambda) = |\{x,y\} \cap \Lambda|$ and
$b(\omega) = |\omB(\Gamma)|$.
This time consider an
Ising ARC model $P_{1}$ on $(\overline{\Gamma},\omB(\Gamma))$ with
parameters $(p,Q)$ and with ``-'', or equivalently wired, boundary condition.
The weights for $P_{1}$ are given by the terms of the sum $T(\omega)$:
\[
W_{1}(\omg) = p^{|\omg|} (1 - p)^{b(\omega) - |\omg|}
Q^{I(\omega \vee \omg,\Lambda)}, \quad \omg \in \{0,1\}^{\omB(\Gamma)}.
\]
Let $P_{3}$ be the Ising ARC model on $(\Gamma \backslash \{x,y\},
\omB(\Gamma \backslash \{x,y\}))$ with parameters $(p,Q)$ and ``-''
boundary condition, and let $\mathcal{C} = \omB(\Gamma) \backslash \omB(\Gamma
\backslash \{x,y\})$. Note $e \in \mathcal{C}$. Each $\omg$ for which all
bonds of $\mathcal{C}$ are open corresponds to a unique configuration
$\zeta_{g}$ which is
the restriction of $\omg$ to $\omB(\Gamma \backslash \{x,y\})$, and
conversely.
The corresponding weight is
\begin{equation} \label{E:W3def}
W_{3}(\zeta_{g}) = p^{|\zeta_{g}|} (1 - p)^{b(\omega) - |\mathcal{C}|
- |\zeta_{g}|} Q^{I(\zeta_{g},\Gamma \backslash \{x,y\})} =
W_{1}(\omg)p^{-|\mathcal{C}|}.
\end{equation}
Summing $W_{3}(\zeta_{g})$ over all $\zeta_{g}$ yields $T(\omega \vee e)$, so
summing (\ref{E:W3def}) and dividing by $T(\omega)$ yields
\[
\frac{T(\omega \vee e)}{T(\omega)} = P_{1}(\text{all bonds of }
\mathcal{C} \text{ are open}) p^{-|\mathcal{C}|}.
\]
But from (\ref{E:labelprobs}),
\[
P_{1}(\text{all bonds of } \mathcal{C} \text{ are open})
= P^{Ising}_{\Gamma,-,\beta/2,h}(\sigma_{x} = \sigma_{y} = -)
p^{|\mathcal{C}|}
\]
and (\ref{E:Tratio}) follows.
\end{proof}
Here is a related result for the FK model. Write $\omega \backslash \{e\}$
for the configuration obtained by closing the bond $e \in \omega$.
\begin{lemma} \label{L:Rratio}
Let $p \in (0,1)$ and $q \geq 1$, let $\Lambda$ be a finite set of sites in
a lattice $\mathbb{L}$, and let $\eta$ be a generalized site boundary
condition. For $\omega \in \{0,1\}^{\omB(\Lambda)} \cap
D_{g}(\Lambda,\eta)$ define
\[
R(\omega) = \sum_{\omr \subset \omega} p^{|\omr|} (1 - p)^{|\omega| -
|\omr|} q^{C(\omr,\Lambda)} \delta_{D_{r}(\Lambda,\eta)}(\omr).
\]
Then for $e \in \omega$,
\begin{equation} \label{E:Rratio}
\frac{R(\omega \backslash \{e\})}{R(\omega)} = \frac{1}{1-p}
P^{FK}_{\omega}(e \text{ is closed}),
\end{equation}
where $P^{FK}_{\omega}$ denotes probability for the FK model on the graph
$(\oL,\omega)$ with parameters $(p,q)$ and site boundary condition $\eta$.
Further,
\begin{equation} \label{E:Rincr}
\frac{R(\omega)}{(\tfrac{p}{q} + 1 - p)^{|\omega|}}
\text{ is an increasing function.}
\end{equation}
\end{lemma}
\begin{proof}
The proof of (\ref{E:Rratio}) is similar to those of Lemmas \ref{L:Gratio}
and \ref{L:Tratio} so we omit it. From \cite{ACCN} we have
\[
P^{FK}_{\omega}(e \text{ is closed}) \leq \frac{q(1-p)}{p + q(1-p)}
\]
so (\ref{E:Rincr}) follows from (\ref{E:Rratio}) and the criterion
(\ref{E:incrfunc}).
\end{proof}
We have used red/green coloring for the ARC model and its cousin the
particle/bond Potts model, and yellow/white coloring
for the bicolored FK model, to help avoid confusion between the models.
In the next lemma we want to add bonds to the particle/bond Potts model red
configuration to obtain a bicolored FK model. To maintain our color scheme,
this requires thinking of these red bonds as instead being yellow. Thus we
refer to the particle/bond Potts model ``with the red bonds recolored yellow.''
\begin{lemma} \label{L:completetoFK}
Let $\Lambda$ be a finite subset of a lattice $\mathbb{L}$, and consider
a $(q+1)$-state particle/bond Potts model on $(\oL,\omB(\Lambda))$ at
$(\beta,0)$ with 0's boundary condition, with the red bonds recolored
yellow. The yellow bonds of this model, supplemented by independent
percolation of white bonds at density $p = 1 - e^{-\beta}$ on the ``00''
bonds,
form a bicolored FK model on $(\oL,\omB(\Lambda))$ with parameters
$(p,q+1,q)$ and all-white boundary condition.
\end{lemma}
\begin{proof}
From Remark \ref{R:qminus1PFK}, the yellow bonds of the particle/bond Potts
model form a partial FK model on $(\oL,\omB(\Lambda))$ with parameters
$(p,q+1,q)$ and all-white boundary condition. From Lemma \ref{L:ARCcondl},
conditionally on $\omega_{y}$ (or equivalently $\omr$), the 0's (relabeled
``-'') and the particles (relabeled ``+'') form an Ising model on
$\mathcal{I}(\omega_{y},\Lambda)$ at $(\tilde{\beta},\tilde{h})$, where
$\tilde{\beta}$ and $\tilde{h}$ are given by (\ref{E:condlparams}),
with boundary condition ``+'' on $\Lambda \backslash \mathcal{I}(\omega_{y}
,\Lambda)$
and ``-'' on $\partial \Lambda$. Hence, still conditionally on $\omega_{y}$,
the white bonds from the independent percolation form an Ising ARC model at
$(p,Q)$ with this same boundary condition, where $Q = 1 + e^{\tilde{beta}(m +
\tilde{h})} = q + 1$. However, by Proposition \ref{P:PFKcondl}(i), for the
bicolored FK model with parameters $(p,q+1,q)$, the white bonds have exactly
this same Ising ARC model as their conditional distribution given
$\omega_{y}$. The result follows.
\end{proof}
The next proposition is the key to the proof of Theorem \ref{T:planarbounds}.
\begin{proposition} \label{P:comptoplus}
Let $\Lambda$ be a finite set of sites of a lattice $\mathbb{L}$ with
coordination number $m$, let $J, \kappa \geq 0$ and $q \geq 1$ and
consider a $q$-state Potts lattice gas on $\Lambda$ at $(1,J,\kappa,\mu)$
with site boundary condition $\eta$. Let $\beta^{\prime}$ be the minimum
effective inverse temperature, and $h^{\prime}$ the maximum effective
external field, of this Potts lattice gas, as given in (\ref{E:effectives2}),
and define an Ising-model boundary condition $\eta^{\prime}$ by
$\eta^{\prime}_{x} = +$ if $\eta_{x} = 0$, $\eta^{\prime}_{x} = -$
if $\eta_{x} \neq 0$. Then the following hold:
\begin{enumerate}
\item[(i)] The 0's configuration of this Potts lattice gas is dominated by
the ``+'' configuration of an Ising model on $\Lambda$ at $(\beta^{\prime},
h^{\prime})$ with boundary condition $\eta^{\prime}$.
\item[(ii)] If $h^{\prime} < 0$, this ``+'' configuration is further
dominated by the ``+'' configuration of an Ising model on $\Lambda$ at
$(\beta^{\prime\prime},0)$ with boundary condition $\eta^{\prime}$, where
\[
\beta^{\prime\prime} = \beta^{\prime}\frac{m - h^{\prime}}{m},
\]
or equivalently,
\[
e^{\beta^{\prime\prime}} = (qe^{\mu})^{1/m}e^{\kappa}(\frac{1}{q}e^{J} +
\frac{q-1}{q}).
\]
\end{enumerate}
In particular:
\begin{enumerate}
\item[(iii)] The 0's configuration of a $(q+1)$-state Potts model at
$(\beta,h)$ with boundary condition $\eta$ is dominated by the ``+''
configuration of an Ising model on $\Lambda$ at $(\beta^{\prime},
h^{\prime})$ (and at $(\beta^{\prime\prime},0)$, if $h^{\prime} < 0$)
with boundary condition $\eta^{\prime}$, where $(\beta^{\prime},
h^{\prime})$ is given by
\begin{equation} \label{E:betahprime2}
e^{2\beta^{\prime}} = e^{\beta}(\frac{1}{q}e^{\beta} +
\frac{q-1}{q}), \quad e^{\beta^{\prime}(m + h^{\prime})} =
\frac{e^{\beta(m + h)}}{q}
\end{equation}
and $\beta^{\prime\prime} = \beta^{\prime}(m - h^{\prime})/m$,
or equivalently
\begin{equation} \label{E:betapp2}
e^{\beta^{\prime\prime}} = \frac{q - 1 + e^{\beta}}{q^{(m-1)/m}}
e^{\beta h/m}.
\end{equation}
\end{enumerate}
\end{proposition}
\begin{proof}
Corresponding to the Potts lattice gas there is a red/black ARC model
on $(\oL,\omB(\Lambda))$ with site boundary condition $\eta$ and parameters
given by (\ref{E:ARCparams}) and (\ref{E:RBARCparams}):
\[
p_{b} = 1 - e^{-(\kappa + J)}, \quad p_{rb} = \frac{1 - e^{-J}}{1 -
e^{-(\kappa + J)}}, \quad Q = 1 + \frac{e^{-\mu}}{q}.
\]
Summing (\ref{E:RBARCweight}) over $\omr$, we see that the black configuration
has weights given by
\[
W(\omb) = p_{b}^{|\omb|} (1 - p_{b})^{|\omB(\Lambda)| - |\omb|}
Q^{I(\omb,\Lambda)} R(\omb) \delta_{D_{g}(\Lambda,\eta)}(\omb)
\]
where
\[
R(\omb) = \sum_{\omr \subset \omb} p_{rb}^{|\omr|} (1 - p_{rb})^{|\omb| -
|\omr|} q^{C(\omr,\Lambda)} \delta_{D_{r}(\Lambda,\eta)}(\omr).
\]
Since the Ising ARC model with arbitrary (nongeneralized) site boundary
condition has the FKG property, by (\ref{E:Rincr}) in
Lemma \ref{L:Rratio} this black
configuration dominates an Ising ARC model with parameters $(p^{\prime},Q)$
and site boundary condition $\eta^{\prime}$, where $p^{\prime}$ is given by
\[
\frac{p^{\prime}}{1 - p^{\prime}} = \left( \frac{p_{rb}}{q} + 1 - p_{rb}
\right) \frac{p_{b}}{1 - p_{b}} = 1 + e^{\kappa}\left( \frac{1}{q}e^{J} +
\frac{q-1}{q} \right).
\]
But then $p^{\prime} = 1 - e^{-2\beta^{\prime}}$, so from (\ref{E:IARCparams})
this Ising ARC model corresponds to an Ising model at
$(\beta^{\prime},h^{\prime})$ with boundary condition $\eta^{\prime}$. Now
(i) follows from Lemma
\ref{L:zerocompare}, and then (ii) from Lemma \ref{L:pluscompare}; (iii) is a
special case of (i) and (ii).
\end{proof}
\begin{remark} \label{R:hprimeneg}
Proposition \ref{P:comptoplus} is particularly useful when $h^{\prime} < 0$,
for then the dominating Ising ``+'' configuration is a minority spin.
Particularly for the $(q+1)$-state Potts model in two dimensions, we will see
that the comparison can be used to transfer known properties of the Ising
model to the Potts model. This is useful because a number of properties are
easier to prove for the Ising model, where one has tools such as symmetry
inequalities which are not available for the Potts model in general. From
(\ref{E:effectives2}) we have
\[
h^{\prime} < 0 \quad \text{if and only if} \quad \frac{e^{-\mu}}{q} <
e^{\beta^{\prime}m},
\]
or equivalently
\begin{equation} \label{E:hpnegcond}
h^{\prime} < 0 \quad \text{if and only if} \quad \frac{e^{-(\mu +
\tfrac{1}{2}\kappa m)}}{q} < \left( \frac{1}{q} e^{J} + \frac{q-1}{q}
\right)^{m/2}.
\end{equation}
\end{remark}
\begin{lemma} \label{L:hpos}
In any infinite-volume limit of the Potts ARC model corresponding to a Potts
model at $(\beta,h)$ with $h > 0$, red bonds a.s. do not percolate.
\end{lemma}
\begin{proof}
As is well-known (see \cite{Al98}), by use of a ``ghost site'' one can
construct an FK model corresponding to a (usual) Potts model with a positive
external field applied to species 0. This model has the finite energy property
(see \cite{NS} or \cite{BK} for the definition) so a configuration a.s. has at
most one infinite cluster \cite{BK}. It follows easily that in the joint
Potts/FK configuration, there is no percolation of open bonds whose endpoints
$x,y$ have species $\sigma_{x} = \sigma_{y} \neq 0$. These are precisely the
red bonds of the ARC model.
\end{proof}
The next result will be used in the proof of Theorem \ref{T:dilution}, when we
compare the ARC model corresponding to
a $q$-state Potts lattice gas with dilution parameter $\theta$ to a partial FK
model at $(p,q+\theta,q)$. The red bonds of this ``$q$-state'' ARC model may
be viewed loosely as an FK model with the same $q$, diluted by the addition of
some 0 sites, just as the original Potts lattice gas is a diluted Potts
model. Similarly, the bicolored FK model at $(p,q+\theta,q)$ is, again
loosely, an FK model at $(p,q)$, with bonds colored yellow, diluted by the
addition of some white bonds. It is thus reasonable to try to compare these
two types of models, particularly when the dilution is small. The main
question to be answered is, given an ARC model, what is the comparable value
of $\theta$ in the partial FK model? An answer, or more precisely a one-sided
bound on an answer, comes from the following.
\begin{proposition} \label{P:comparedilut}
Let $\Lambda$ be a finite set of sites of a lattice $\mathbb{L}$ with
coordination number $m$ and consider an ARC model $P$ on
$(\oL,\omB(\Lambda))$ at $(p_{r},p_{g},q,Q)$, with $q > 1$, and with
single-species site boundary condition $\eta$. Let $\theta = q(Q-1)(1 -
p_{g})^{m}$ be the dilution parameter.
\begin{enumerate}
\item[(i)] If
\begin{equation} \label{E:paramcompare}
p_{g} \leq p_{r} \quad \text{and} \quad \frac{p_{g}}{1 - p_{g}}
\leq \frac{p_{r}}{\theta(1 - p_{r})},
\end{equation}
then the red-bond configuration under $P$ dominates the partial
FK model at $(p_{r},q+\theta,q)$ on $(\oL,\omB(\Lambda))$
with site boundary condition $\eta^{\prime}$
given by $\eta_{x}^{\prime} = $ ``white'' if $\eta_{x} = 0, \eta_{x}^{\prime}
= $ ``yellow'' if $\eta_{x} \neq 0$.
\item[(ii)] Let $\beta/2$ and $h$ be the parameters of the conditional Ising
model of the ARC model P, given by
\begin{equation} \label{E:betah5}
p_{g} = 1 - e^{-\beta}, \quad \frac{q}{\theta} = e^{\beta(m - h)/2},
\end{equation}
and suppose that, in the infinite volume limit, for some $0 < \delta < 1/2$,
\begin{equation} \label{E:plusprob}
P^{Ising}_{-,\beta/2,h}(\sigma_{x} = + \text{ for some } x \text{
adjacent to } 0 \mid \sigma_{0} = +) < \delta.
\end{equation}
Suppose also that $\eta$ is a constant-species (equivalently, red-wired)
boundary condition. Then the red-bond configuration under $P$
dominates the partial FK model at $(p_{r},q + \theta^{\prime},q)$ on
$(\oL,\omB(\Lambda))$ with
all-yellow boundary condition, where
\[
\theta^{\prime} = \frac{1 - \delta}{1 - 2\delta} \theta.
\]
\end{enumerate}
\end{proposition}
Since the Ising model in (\ref{E:plusprob}) is (except for boundary condition)
the conditional Ising model of
the ARC model $P$, the ``+''spins in (\ref{E:plusprob}) correspond
approximately to the 0's configuration of the ARC model $P$. Loosely,
(\ref{E:plusprob}) holds when 0's are so rare that most 0's are isolated from
any other 0's, and this will be true when $\theta$ is small. Thus
(\ref{E:plusprob}) is a substitute for (\ref{E:paramcompare}) when the
dilution is very small. The values of greatest interest in Proposition
\ref{P:comparedilut} are small $\theta$ and $p_{r}$ near the FK critical
point $p_{c}^{FK}(q,\mathbb{L})$. Since $\theta^{\prime} > \theta$, the
conclusion in (ii) is weaker than (i), but for small $\delta$ the difference
is small, and for the aformentioned values of greatest interest, we only
expect our $\theta$ to be sharp up to a constant depending on $q$ anyway; see
Remark \ref{R:thetasharp}.
\begin{proof}[Proof of Proposition \ref{P:comparedilut}]The basic technique is
roughly to compare the conditional Ising model of the ARC model $P$ to the
conditional neutral Potts lattice gas of the partial FK model. For clarity of
exposition we give the proof of (i) only when $\eta$ is the all-1's, or
equivalently red-wired, boundary condition (so that $\eta^{\prime}$ is
all-yellow); the general case is quite
similar. The weights for the partial FK model are given by
(\ref{E:PFKweight}) and (\ref{E:Fdef}):
\[
W^{PFK}(\omy) = p_{r}^{|\omy|} (1 - p_{r})^{|\omB(\Lambda)| - |\omy|}
q^{C(\omy,\Lambda)} q^{-I(\omy,\Lambda)} F(\omy), \quad \omy \in
\{0,1\}^{\omB(\Lambda)},
\]
where
\[
F(\omy) = \sum_{\omw \in \{0,1\}^{\mB(\mathcal{I}(\omy,\Lambda))}}
\left( \frac{p_{r}}{1 - p_{r}} \right)^{|\omw|}
\theta^{C(\omw,\mathcal{I}(\omy,\Lambda))}
\left( \frac{q + \theta}{\theta} \right)^{I(\omy \vee \omw,\Lambda)}.
\]
Here we use the fact that under the all-yellow boundary condition, we have
$(\omy,\omw) \in A(\Lambda,\eta)$ if and only if $(\omw)_{e} = 0$ for all
$e \notin \mB(\mathcal{I}(\omy,\Lambda))$, meaning that effectively
$\omw \in
\{0,1\}^{\mB(\mathcal{I}(\omy,\Lambda))}$. The weights for the ARC-model red
bonds are given by summing (\ref{E:ARCweight}):
\begin{equation} \label{E:ARCredwgt}
W^{ARC}(\omr) = p_{r}^{|\omr|} (1 - p_{r})^{|\omB(\Lambda)| - |\omr|}
q^{C(\omr,\Lambda)} T(\omr),
\quad \omr \in \{0,1\}^{\omB(\Lambda)},
\end{equation}
where
\begin{equation} \label{E:Tdef}
T(\omr) = \sum_{\omg \in \{0,1\}^{\omB(\mathcal{I}(\omr,\Lambda))}}
p_{g}^{|\omg|} (1 - p_{g})^{b(\omr) - |\omg|} Q^{I(\omr \vee \omg,\Lambda)}
\end{equation}
with $b(\omr) = |\omB(\mathcal{I}(\omr,\Lambda))|$. Note that green bonds not
in $\omB(\mathcal{I}(\omr,\Lambda))$ have been summed out; this is possible
because the states of such bonds (open or closed) do not affect the factor
$Q^{I(\omr \vee \omg,\Lambda)}$. Define
\[
Q^{\prime} = 1 + \frac{1}{(Q - 1)(1 - p_{g})^{m}} = \frac{q +
\theta}{\theta},
\]
so that the values $\beta, h$ obtained from $p_{g}$ and $Q$ via
(\ref{E:betah}) are the same as the values $\beta,h$ obtained from $p_{g}$ and
$Q^{\prime}$ via (\ref{E:betah2}). Define
\[
G(\omega) = \sum_{\omw \in \{0,1\}^{\mB(\mathcal{I}(\omega,\Lambda))}}
\left( \frac{p_{g}}{1 - p_{g}} \right)^{|\omw|} (Q^{\prime})^{I(\omw \vee
\omega,\Lambda)}.
\]
By Lemmas \ref{L:Gratio} and \ref{L:Tratio},
\begin{equation} \label{E:Tratio2}
\frac{T(\omega \vee e)}{T(\omega)} = (Q^{\prime} - 1)^{I(\omega,\Lambda) -
I(\omega \vee e,\Lambda)} \frac{G(\omega \vee e)}{G(\omega)} \quad
\text{for all } \omega \text{ and } e.
\end{equation}
Let $P^{IA}_{\mathcal{I}(\omega,\Lambda)}$ be the Ising ARC model on
$(\mathcal{I}(\omega,\Lambda),\mB(\mathcal{I}(\omega,\Lambda)))$ at
$(p_{g},Q^{\prime})$ with free boundary, and let
$P^{NA}_{\mathcal{I}(\omega,\Lambda)}$ be the neutral ARC model on
$(\mathcal{I}(\omega,\Lambda),\mB(\mathcal{I}(\omega,\Lambda)))$ at
$(p_{r},0,\theta,\tfrac{q+\theta}{\theta})$ with free boundary. Note that
$P^{IA}_{\mathcal{I}(\omega,\Lambda)}$ is also the ARC model at
$(p_{g},0,1,Q^{\prime})$. By (\ref{E:Gratio2}) in the proof of Lemma
\ref{L:Gratio},
\begin{equation} \label{E:Gratio3}
\frac{G(\omega \vee e)}{G(\omega)} = P^{IA}_{\mathcal{I}(\omega,\Lambda)}
\bigl( \{ \omw: \{x,y\} \cap \mathcal{I}(\omega,\Lambda) \subset
\mathcal{I}(\omw,\Lambda) \} \bigr) \left( \frac{1}{Q^{\prime}}
\right)^{I(\omega,\Lambda) - I(\omega \vee e,\Lambda)}.
\end{equation}
By an argument similar to the proof of (\ref{E:Gratio2}) we have
\begin{equation} \label{E:Fratio}
\frac{F(\omega \vee e)}{F(\omega)} = P^{NA}_{\mathcal{I}(\omega,\Lambda)}
\bigl( \{ \omw: \{x,y\} \cap \mathcal{I}(\omega,\Lambda) \subset
\mathcal{I}(\omw,\Lambda) \} \bigr) \left( \frac{1}{q + \theta}
\right)^{I(\omega,\Lambda) - I(\omega \vee e,\Lambda)}.
\end{equation}
Under (\ref{E:paramcompare}), by Lemma \ref{L:compare},
$P^{NA}_{\mathcal{I}(\omega,\Lambda)}$ dominates
$P^{IA}_{\mathcal{I}(\omega,\Lambda)}$. Hence (\ref{E:Fratio}),
(\ref{E:Tratio2}) and (\ref{E:Gratio3}) show that
\begin{align} \label{E:Fbound}
\frac{F(\omega \vee e)}{F(\omega)} &\leq \left( \frac{Q^{\prime}}{q + \theta}
\right)^{I(\omega,\Lambda) -
I(\omega \vee e,\Lambda)} \frac{G(\omega \vee e)}{G(\omega)} \\
&= \left( \frac{1}{q} \right)^{I(\omega,\Lambda) -
I(\omega \vee e,\Lambda)} \frac{T(\omega \vee e)}{T(\omega)}. \notag
\end{align}
Using (\ref{E:incrfunc}) this shows that $W^{ARC}/W^{PFK}$ is an increasing
function. Since from Lemma \ref{L:FKG} the ARC model (with
boundary condition $\eta$) has the FKG property, so does the red-bond
configuration alone, and (i) follows.
Now suppose (\ref{E:plusprob}) holds. The value of $I(\omega,\Lambda) -
I(\omega \vee e,\Lambda) = |\{x,y\} \cap \mathcal{I}(\omega,\Lambda)|$ is
either 0, 1 or 2; the case of 0 is trivial because then (\ref{E:Tratio2}),
(\ref{E:Gratio3}) and (\ref{E:Fratio}) are all equal to 1. Let us assume
$I(\omega,\Lambda) - I(\omega \vee e,\Lambda) = 2$; the case of 1 is similar.
Let $U$ denote the event that $\sigma_{z} = +$ for some $z$ adjacent to 0. We
have
\begin{align}
P^{IA}_{\mathcal{I}(\omega,\Lambda)}&\bigl( \{ \omw: x \notin
\mathcal{I}(\omega \vee \omw,\Lambda) \} \bigr) \notag \\
&\leq P^{Ising}_{\mathcal{I}(\omega,\Lambda),-,\beta/2,h}(\sigma_{x} = +,
\sigma_{z} = + \text{ for some } z \text{ adjacent to }x) \notag \\
&\leq P^{Ising}_{-,\beta/2,h}([\sigma_{0} = +] \cap U) \notag \\
&< \frac{P^{Ising}_{-,\beta/2,h}([\sigma_{0} = +] \cap
U)}{P^{Ising}_{-,\beta/2,h}(U^{c})} \notag \\
&= \frac{P^{Ising}_{-,\beta/2,h}(\sigma_{0} = + \mid U^{c})
P^{Ising}_{-,\beta/2,h}(U \mid \sigma_{0} = +)}{P^{Ising}_{-,\beta/2,h}
(U^{c} \mid \sigma_{0} = +)} \notag \\
&\leq \frac{\delta}{Q^{\prime}(1 - \delta)} \notag
\end{align}
and similarly for $y$, so, using the FKG property,
\begin{equation} \label{E:problower}
P^{IA}_{\mathcal{I}(\omega,\Lambda)} \bigl( \{ \omw: \{x,y\} \subset
\mathcal{I}(\omega \vee \omw,\Lambda) \} \bigr) \geq \left(1 -
\frac{\delta}{Q^{\prime}(1 - \delta)} \right)^{I(\omega,\Lambda) -
I(\omega \vee e,\Lambda)}.
\end{equation}
Let
\[
\tau = 1 - \frac{\delta}{Q^{\prime}(1 - \delta)}.
\]
Combining (\ref{E:problower}) with (\ref{E:Tratio2}) and (\ref{E:Gratio3})
yields
\begin{align}
\frac{T(\omega \vee e)}{T(\omega)} &= (Q^{\prime} - 1)^{I(\omega,\Lambda) -
I(\omega \vee e,\Lambda)} \frac{G(\omega \vee e)}{G(\omega)} \notag \\
&\geq \left( \frac{(Q^{\prime} - 1)\tau}{Q^{\prime}}
\right)^{I(\omega,\Lambda) - I(\omega \vee e,\Lambda)}. \notag
\end{align}
It follows using the criterion (\ref{E:incrfunc})
that the red-bond configuration of the ARC model $P$ dominates the
red-bond configuration of the neutral ARC model at
$(p_{r},0,q,Q^{\prime\prime})$ on $(\oL,\omB(\Lambda))$ with boundary
condition $\eta$, where
\[
Q^{\prime\prime} = \frac{Q^{\prime}}{(Q^{\prime} - 1)\tau}
= \frac{q + \theta}{q\tau}.
\]
But this neutral ARC model satisfies the hypotheses of (i), so its red-bond
configuration dominates the partial FK model at
$(p_{r},q+\theta^{\prime\prime},q)$ on $(\oL,\omB(\Lambda))$ with all-yellow
(equivalently wired) boundary condition, where
\[
\theta^{\prime\prime} = q(Q^{\prime\prime} - 1) =
\frac{q + \theta}{\tau} - q.
\]
A short calculation shows $\theta^{\prime\prime} < \theta^{\prime}$, and (ii)
follows.
\end{proof}
\begin{remark} \label{R:thetasharp}
It is apparent from the proof of Proposition \ref{P:comparedilut} that if we
could find a smaller value of $\theta$ (so that $Q^{\prime} < 1 + q/\theta$)
for which the inequality between the first and last terms of (\ref{E:Fbound})
were reversed, the domination in (i) and (ii) would then also be reversed.
Now from (\ref{E:labelprobs}), if we extend the neutral ARC model to its
particle/bond form, we have
\begin{align}
P^{NA}_{\mathcal{I}(\omega,\Lambda)} &\bigl( \{ \omw: \{x,y\} \subset
\mathcal{I}(\omega \vee \omw,\Lambda) \} \bigr) \left( \frac{q}{q + \theta}
\right)^{I(\omega,\Lambda) - I(\omega \vee e,\Lambda)} \notag \\
&= P^{NA}_{\mathcal{I}(\omega,\Lambda)}(n_{x} = n_{y} = 0); \notag
\end{align}
note that this particle/bond neutral ARC model has 0's boundary condition on
$\partial \mathcal{I}(\omega,\Lambda)$. thus to establish the reverse of
(\ref{E:Fbound}), it is enough to choose $\theta$ so that for the
particle/bond neutral ARC model at $(p_{r},0,\theta,\tfrac{q +
\theta}{\theta})$ we have (in the notation of the last proof)
\begin{equation} \label{E:P0ineq}
P^{NA}_{\mathcal{I}(\omega,\Lambda)}(n_{x} = n_{y} = 0) \geq
\left( \frac{Q^{\prime} - 1}{Q^{\prime}}
\right)^{I(\omega,\Lambda) - I(\omega \vee e,\Lambda)}.
\end{equation}
The bonds of this neutral ARC model are precisely the white bonds of the
bicolored FK model from which it was obtained, and the 0's of the neutral ARC
model are precisely the isolated yellow sites of this bicolored FK model. Thus
we might expect that for fixed $\omega$, when $x \in
\mathcal{I}(\omega,\Lambda)$,
\begin{align} \label{E:particleprob}
P^{NA}_{\mathcal{I}(\omega,\Lambda)}&(n_{x} = 1) \\
&= P^{BFK}(x \text{ is white} \mid \omy = \omega) \notag \\
&\approx P^{BFK}(x \text{ is white } \mid x \in \mathcal{I}(\omy,\Lambda))
\notag \\
&= \frac{P^{BFK}(x \text{ is white})}{P^{BFK}(x \text{ is white or isolated
yellow})} \notag
\end{align}
where $P^{BFK}$ denotes the bicolored FK model at $(p_{r},q + \theta,q)$ on
$(\oL,\omB(\Lambda))$ with boundary condition $\eta^{\prime}$. (It is
only the approximation that is nonrigorous here.) Since
\[
P^{BFK}(x \text{ is white}) \leq \frac{\theta}{q + \theta} \quad \text{and}
\quad P^{BFK}(x \text{ is yellow } \mid x \text{ is isolated}) =
\frac{q}{q+\theta},
\]
the right side of (\ref{E:particleprob}) is at most
\begin{equation} \label{E:RHSbound}
\frac{\theta}{\theta + qP^{FK}(x \text{ is isolated})},
\end{equation}
where $P^{FK}$ denotes the FK model at $(p_{r},q+ \theta)$ on
$(\oL,\omB(\Lambda))$ with generalized site boundary condition
$\eta^{\prime}$. We might expect (\ref{E:P0ineq}) to hold when
(\ref{E:RHSbound}) is approximately $1/Q^{\prime}$, that is,
\[
\theta \approx \frac{qP^{FK}(x \text{ is isolated})}{Q^{\prime}},
\]
which for small $\theta$ is within approximately a factor of $P^{FK}(x$ is
isolated) of the value $\theta = q/(Q^{\prime} - 1)$ of Proposition
\ref{P:comparedilut}. Thus we expect that the value of $\theta$ in the
Proposition is sharp ``up to a constant,'' but we are unable to prove this;
the main obstacle is the absense of the FKG property for the neutral ARC model
at $(p_{r},0,\theta,\tfrac{q + \theta}{\theta})$ when $\theta < 1$.
\end{remark}
Our next lemma shows that 0's are rare in the Potts lattice gas roughly when
the dilution parameter $\theta$ is small and the effective external field (on
empty sites--see (\ref{E:beta0h0})) is negative for the binary lattice gas
obtained by replacing parameters $J,\kappa$ with $0, J + \kappa$.
\begin{lemma} \label{L:rare0}
Suppose $\kappa_{k} \geq 0$ and $\mu_{k} \in \mathbb{R}$ for each $k \geq
1$, and $q \geq 1, J \geq 0$. Let $\mathbb{L}$ be a lattice of
coordination number $m$, and let $P_{k}$ be the infinite-volume $q$-state
Potts lattice gas at $(1,J,\kappa_{k},\mu_{k})$ with all-1's boundary
condition. If
\begin{equation} \label{E:toinfty}
\mu_{k} + \kappa_{k}m \to \infty
\end{equation}
and
\begin{equation} \label{E:poseffec}
\mu_{k} + \frac{1}{2}\kappa_{k}m + \log q \geq 0 \quad \text{for all
sufficiently large } k,
\end{equation}
then $P_{k}(n_{0} = 0) \to 0$ as $k \to \infty$. Conversely if $P_{k}(n_{0}
= 0) \to 0$ then (\ref{E:toinfty}) holds and
\begin{equation} \label{E:liminf}
\liminf_{k \to \infty}\frac{\mu_{k} + \tfrac{1}{2}\kappa_{k}m + \log
q}{\kappa_{k}} \geq 0.
\end{equation}
(The quantity $(\mu_{k} + \tfrac{1}{2}\kappa_{k}m + \log q)/\kappa_{k}$
should be interpreted as $+\infty$ if $\kappa_{k} = 0$.)
\end{lemma}
\begin{proof}
We may assume our Potts lattice gas is in particle/bond form.
Let $(p_{r},(p_{g})_{k},q,Q_{k})$ be the parameters of the ARC model
corresponding to $P_{k}$ (see (\ref{E:ARCparams}). The conditional Ising
model of $P_{k}$ has parameters $(\beta_{k}/2,h_{k})$ given by Lemma
\ref{L:ARCcondl}:
\[
\beta_{k} = \kappa_{k}, \quad -\frac{1}{2}\beta_{k}h_{k} = \mu_{k} +
\frac{1}{2}\kappa_{k}m + \log q,
\]
so
\[
\frac{1}{2}\beta_{k}(m - h_{k}) = \mu_{k} + \kappa_{k}m + \log q.
\]
Thus (\ref{E:toinfty}) and (\ref{E:poseffec}) are equivalent to
\begin{equation} \label{E:Isingequiv1}
\beta_{k}(m - h_{k}) \to \infty \quad \text{and} \quad h_{k} \leq 0,
\end{equation}
while (\ref{E:toinfty}) and (\ref{E:liminf}) are equivalent to
\begin{equation} \label{E:Isingequiv2}
\beta_{k}(m - h_{k}) \to \infty, \quad \text{and} \quad
\limsup_{k \to \infty} h_{k} \leq 0.
\end{equation}
Now the ARC-model red-bond configuration on the full lattice under $P_{k}$ is
dominated by the FK model at $(p_{r},q)$ so $P_{k}(\Lambda \subset
\mathcal{I}(\omr,S(\mathbb{L})))$ stays bounded away from 0 as $k \to \infty$
for each finite set of sites $\Lambda$. Hence in particular
\begin{equation} \label{E:small0prob}
P_{k}(n_{0} = 0) \to 0 \quad \text{if and only if} \quad P_{k}\bigl(n_{0} = 0
\mid 0 \in \mathcal{I}(\omr,S(\mathbb{L}))\bigr) \to 0.
\end{equation}
But this latter probability is just the probability of a ``+'' at 0 for the
conditional Ising model on $\mathcal{I}(\omr,S(\mathbb{L}))$ which has ``-''
boundary condition, and this probability is bounded above by the same
probability for the infinite-volume minus phase of the Ising model on
$\mathbb{L}$. That is,
\begin{equation} \label{E:0vsplus}
P_{k}\bigl(n_{0} = 0 \mid 0 \in \mathcal{I}(\omr,S(\mathbb{L}))\bigr)
\leq P^{Ising}_{k}(\sigma_{0} = +),
\end{equation}
where $P^{Ising}_{k}$ denotes the infinite-volume minus phase of the Ising
model at $(\beta_{k}/2,h_{k})$.
If (\ref{E:toinfty}) and (\ref{E:poseffec}) hold, or equivalently
(\ref{E:Isingequiv1}) holds, then for large $k$ either $-h_{k}$ is large or
both $\beta_{k}$ is large and $h_{k} \leq 0$; either way we obtain
$P^{Ising}_{k}(\sigma_{0} = +) \to 0$ and hence from (\ref{E:small0prob}) and
(\ref{E:0vsplus}), $P_{k}(n_{0} = 0) \to 0$.
Conversely suppose $P_{k}(n_{0} = 0) \to 0$. Analogously to (\ref{E:0vsplus})
we have
\begin{align}
P_{k}\bigl(n_{0} = 0 \mid 0 \in \mathcal{I}(\omr,S(\mathbb{L}))\bigr)
&\geq P^{Ising}_{k}(\sigma_{0} = + \mid \sigma_{x} = - \text{ for all } x
\text{ adjacent to } 0) \notag \\
&= \frac{1}{1 + e^{\beta(m - h_{k})/2}}, \notag
\end{align}
so the first half of (\ref{E:Isingequiv2}) follows from (\ref{E:small0prob}).
To prove the second half of (\ref{E:Isingequiv2}), suppose the first half
holds but $\limsup h_{k} > \epsilon$ for some $\epsilon > 0$; we may assume
$h_{k} \geq \epsilon$ for all $k$, so $\beta_{k} \to \infty$. If $\Lambda$ is
sufficiently large then for all large $k$,
\[
P_{k}\bigl(n_{0} = 0 \mid \Lambda \subset
\mathcal{I}(\omr,S(\mathbb{L}))\bigr) \geq P^{Ising}_{k}(\sigma_{0} = +
\mid \sigma_{x} = - \text{ for all } x \in \partial \Lambda) > \frac{1}{2}.
\]
Since $P_{k}(\Lambda \subset \mathcal{I}(\omr,S(\mathbb{L})))$ is bounded away
from 0, it follows that $P_{k}(n_{0} = 0)$ is bounded away from 0.
\end{proof}
In terms of ARC model parameters, (\ref{E:poseffec}) can be restated as
\begin{equation} \label{E:poseffecARC}
(Q_{k} - 1)(1 - (p_{g})_{k})^{m/2} \leq 1.
\end{equation}
\begin{remark} \label{R:0probto0}
The proof of Lemma \ref{L:rare0} contains the fact that (\ref{E:Isingequiv1})
implies $P^{Ising}_{k}(\sigma_{0} = +) \to 0$. One can actually obtain the
stronger conculsion from (\ref{E:Isingequiv1}) that
\[
P^{Ising}_{k}(\sigma_{x} = + \text{ for some } x \text{ adjacent to } 0 \mid
\sigma_{0} = +) \to 0.
\]
Indeed, (\ref{E:Isingequiv1}) implies that for large $k$, either $-h_{k}$ is
very large or both $h_{k} \leq 0$ and $\beta_{k}$ is large enough that a
Peierls-type argument shows that a cluster of two or more ``+'' spins is much
less likely than a single isolated ``+'' spin.
\end{remark}
\section{Proofs of the Main Theorems}
\begin{proof}[Proof of Theorem \ref{T:0sFKG}]
The Hamiltonian for the $q$-state Potts lattice gas is as given by
(\ref{E:Hamilt}); for simplicity we assume $\mu_{x} = \mu$ for all $x$ and
assume there is a site boundary condition which has a single particle species
$i$. Let $F^{\prime} = \{ x \in \partial \Lambda:\eta_{x} = i \}$. As in
Remark \ref{R:FKG}, the corresponding ARC model on a subgraph
$(\Lambda^{\prime},\mB^{\prime})$ of $(\oL,\omB(\Lambda))$ is unconditioned.
We use $P^{PLG}$ to denote the Potts lattice gas and $P^{ARC}$ to denote this
ARC model on $(\Lambda^{\prime},\mB^{\prime})$, which has weights
$W(\omr,\omg)$ given by (\ref{E:ARCweight}), parameters given by
(\ref{E:ARCparams}) and site boundary condition $\eta$ equivalent to a
red-wired boundary condition on $F^{\prime}$. Let $W^{FK}(\omr)$ and $P^{FK}$
denote the weights and probabilities, respectively, for an FK model on
$(\Lambda^{\prime},\mB^{\prime})$ at $(p_{r},q)$
with wired boundary condition on
$F^{\prime}$. Let $P^{ind}$ denote probability corresponding to independent
percolation at density $p_{g}$ on $\mB^{\prime}$. For $A, B \subset \Lambda$
we have using (\ref{E:labelprobs}):
\begin{align} \label{E:Aprob}
P^{PLG}(X_{0} = A) &= \sum_{K \supset A} P^{ARC}(\mathcal{I}(\omb,\Lambda) =
K) \left( \frac{Q-1}{Q} \right)^{|A|} \left( \frac{1}{Q}
\right)^{|K| - |A|} \\
&= (Q - 1)^{|A|} Z_{ARC}^{-1} \sum_{(\omr,\omg):\mathcal{I}(\omr \vee \omg,
\Lambda) \supset A} \frac{W(\omr,\omg)}{Q^{I(\omr \vee \omg)}} \notag \\
&= (Q - 1)^{|A|} Z_{ARC}^{-1} \sum_{\omr:\mathcal{I}(\omr,
\Lambda) \supset A} W^{FK}(\omr) \sum_{\omg: \mathcal{I}(\omg,\Lambda)
\supset A} P^{ind}(\omg) \notag \\
&= (Q - 1)^{|A|} Z_{ARC}^{-1} Z_{FK} P^{FK}(\omB(A) \text{ all closed})
P^{ind}(\omB(A) \text{ all closed}), \notag
\end{align}
where $Z_{ARC}$ and $Z_{FK}$ are the partition functions of the ARC and FK
models, respectively. The FKG property of the measure $P^{FK}(\cdot \mid
\omB(A \cap B \text{ all closed})$ yields that
\[
\frac{P^{FK}(\omB(A \cup B) \text{ all closed})}{P^{FK}(\omB(A) \text{ all
closed})} \geq \frac{P^{FK}(\omB(B) \text{ all closed})}{P^{FK}(\omB(A
\cap B) \text{ all closed})},
\]
and similarly for $P^{ind}$. This and (\ref{E:Aprob}) readily yield
\[
P^{PLG}(X_{0} = A \cup B)P^{PLG}(X_{0} = A \cap B) \geq
P^{PLG}(X_{0} = A)P^{PLG}(X_{0} = B),
\]
which is the FKG lattice condition. The proof under free boundary condition,
or for nonconstant $\mu_{x}$, is essentially similar.
\end{proof}
\begin{proof}[Proof of Corollary \ref{C:percunique}]
For a joint ARC/Potts lattice gas configuration on $\mathbb{L}$, black bonds
cannot percolate if each site is surrounded by a circuit on which every site
$x$ has $\sigma_{x} = 0$. As in \cite{Ch}, with this observation the
corollary is a direct consequence of Proposition \ref{P:percunique2},
Theorem \ref{T:0sFKG} and the main result of \cite{GKR}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:planarbounds}]
Consider first the FK model with $p < p_{1}$; define $\beta$ by $p = 1 -
e^{-\beta}$. Let $\beta^{\prime}$ be the minimum
effective inverse temperature, and $h^{\prime}$ the maximum effective
external field, of the corresponding $(q+1)$-state Potts model at $(\beta,0)$,
as given by (\ref{E:betahprime2}). It is easily checked that
\[
h^{\prime} < 0 \quad \text{if and only if} \quad \beta < \beta_{1}.
\]
Let $\beta^{\prime\prime}$ be as in (\ref{E:betapp2}) and let $\Lambda$ be a
finite subset of the sites of $\mathbb{L}$, with $0 \in \Lambda$. By
Proposition \ref{P:comptoplus}(iii), the 0's configuration of the $q$-state
Potts model at $(\beta,0)$ with 0's boundary condition on $\partial \Lambda$
is dominated by the ``+'' configuration of the Ising model at
$(\beta^{\prime},h^{\prime})$ with ``+'' boundary condition on $\partial
\Lambda$. Therefore by Lemma \ref{L:completetoFK}, the white-bond
configuration of the bicolored FK model at $(p,q+1,q)$ on
$(\oL,\omB(\Lambda))$ with all-white boundary condition is dominated by
independent percolation at density $p$ on the ``++''bonds of this Ising
model. Since $h^{\prime} < 0$, results from \cite{Hi1} and \cite{SS} say that
the Ising model has the weak mixing property, and results from \cite{Hi2} and
\cite{CCS} say that it has exponential decay of ``+'' connectivity. (These
proofs are only written for the square lattice, but everything works for
general planar lattices; the key fact used in \cite{CCS} is the result of
\cite{ABF} that the Ising model has exponential decay of correlations for all
$\beta < \beta_{c}^{Ising}(\mathbb{L})$, and this proof works on general
periodic lattices with minor modifications \cite{Ai}.) Therefore there exist
constants
$C, \lambda > 0$, not depending on $\Lambda$, such that
\begin{align} \label{E:plusdecay}
P^{Ising}_{\Lambda,+,\beta^{\prime},h^{\prime}}&(0 \leftrightarrow \partial
\Lambda \text{ by a lattice path on which all sites } x \text{ have }
\sigma_{x} = +) \\
&\leq Ce^{-\lambda r(\Lambda)}. \notag
\end{align}
Define probability measures as follows:
$P^{Ising,++}_{\Lambda,+,\beta^{\prime},h^{\prime},p}$ for the distribution of
the bond-site configuration produced by independent percolation at density
$p$ on ``++'' bonds, in $(\oL,\omB(\Lambda))$, of the Ising model at
$(\beta^{\prime},h^{\prime})$ on $\Lambda$ with ``+'' boundary condition;
$P^{BFK}_{\Lambda,wh}$ for the bicolored FK model on $(\oL,\omB(\Lambda))$
with all-white boundary condition; and $P^{FK}_{\Lambda,w}$ for the FK model
on $(\oL,\omB(\Lambda))$ with wired boundary condition. From
(\ref{E:plusdecay}),
\begin{equation} \label{E:opendecay}
P^{Ising,++}_{\Lambda,+,\beta^{\prime},h^{\prime},p}(0 \leftrightarrow
\partial \Lambda \text{ by a path of open bonds}) \leq Ce^{-\lambda
r(\Lambda)}.
\end{equation}
Hence by the domination,
\[
P^{BFK}_{\Lambda,wh}(0 \leftrightarrow
\partial \Lambda \text{ by a path of open white bonds}) \leq Ce^{-\lambda
r(\Lambda)},
\]
which is equivalent to
\begin{equation} \label{E:opendecay2}
P^{FK}_{\Lambda,w}(0 \leftrightarrow
\partial \Lambda \text{ by a path of open bonds}) \leq Ce^{-\lambda
r(\Lambda)};
\end{equation}
this proves exponential decay of local wired-boundary connectivities, and thus
also weak mixing, for the FK model. This proves (iii). Applying this result
to the dual lattice $\mathbb{L}^{*}$, we see that when $p > p_{2}$, that is,
$p^{*} < p_{1}(q+1,m^{*})$, the dual configuration has exponential decay of
local wired-boundary connectivities, and thus has the weak mixing property.
But weak mixing for the dual configuration is equivalent to weak mixing for
the regular configuration, and (i) follows. (ii) is an immediate consequence
of (i).
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:highdbounds}]
For simplicity we restrict attention to the integer lattice. Consider the FK
model in $d$ dimensions at $(p,q+1)$ with $p < p_{1}(q+1,2d)$. Let
$\beta^{\prime},h^{\prime},\beta^{\prime\prime}$ be as in the proof of Theorem
\ref{T:planarbounds}. In order to show that there is no percolation in the
wired-boundary infinite-volume limit, one must establish the following analog
of (\ref{E:opendecay2}):
\[
\lim_{\Lambda_{0} \nearrow \mathbb{Z}^{d}}
\lim_{\Lambda \nearrow \mathbb{Z}^{d}} P^{FK}_{\Lambda,w}(0 \leftrightarrow
\partial \Lambda_{0} \text{ by a path of open bonds}) = 0.
\]
For this it is enough to establish the following analog of
(\ref{E:opendecay}):
\begin{equation} \label{E:opendecay3}
\lim_{\Lambda_{0} \nearrow \mathbb{Z}^{d}}
\lim_{\Lambda \nearrow \mathbb{Z}^{d}}
P^{Ising,++}_{\Lambda,+,\beta^{\prime},h^{\prime},p}(0 \leftrightarrow
\partial \Lambda_{0} \text{ by a path of open bonds}) = 0.
\end{equation}
Since
$h^{\prime} < 0$, the Ising model at $(\beta^{\prime},h^{\prime})$ has a
unique Gibbs distribution, so one can replace the ``+'' boundary condition
with ``-'' in (\ref{E:opendecay3}). From Proposition
\ref{P:comptoplus}(ii), one can then also replace
$(\beta^{\prime},h^{\prime})$ with $(\beta^{\prime\prime},0)$ in
(\ref{E:opendecay3}). Further, since the FK model is monotone in $p$, one need
only consider $\beta$ close to $\beta_{1}$. Thus the proof of Gibbs
uniqueness in Theorem \ref{T:planarbounds} goes through since we have assumed
(\ref{E:nonperc}) for appropriate $\beta$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:chibeta}]
The idea is to show that the corresponding Potts ARC model is dominated by
another Potts ARC model corresponding to a Potts model with a positive
external field on 0's; then Lemma \ref{L:hpos} can be applied.
We claim that for some $\epsilon > 0$, we have
\begin{align} \label{E:derivbound}
\frac{2}{\beta} \frac{\partial}{\partial h}
P^{Ising}_{\Lambda,+,\beta/2,h}&(\sigma_{x} = +) \leq \left( \frac{1}{2} -
\epsilon \right) P^{Ising}_{\Lambda,+,\beta/2,h}(\sigma_{x} = +) \\
&\text{for all } h > 0 \text{ and all finite } \Lambda \text{ and } x \in
\Lambda. \notag
\end{align}
Indeed, as is standard, from symmetry inequalities the left side of
(\ref{E:derivbound}) is a decreasing function of $h$ and an increasing
function of $\Lambda$, while from the FKG property the right side is a
decreasing function of $\Lambda$. So it is enough to verify
(\ref{E:derivbound}) in the limit as $h \searrow 0$ and $\Lambda \nearrow
\mathbb{Z}^{d}$, but this is exactly (\ref{E:chibeta}). Thus
(\ref{E:derivbound}) is proved. By symmetry, (\ref{E:derivbound}) is
equivalent to
\begin{align} \label{E:derivbound2}
- \frac{2}{\beta} \frac{\partial}{\partial h}
P^{Ising}_{\Lambda,-,\beta/2,h}&(\sigma_{x} = -) \leq \left( \frac{1}{2} -
\epsilon \right) P^{Ising}_{\Lambda,-,\beta/2,h}(\sigma_{x} = -) \\
&\text{for all } h < 0 \text{ and all finite } \Lambda \text{ and } x \in
\Lambda. \notag
\end{align}
Let $Q = 1 + e^{2d\beta}/q$, so the Potts ARC model corresponding to the Potts
model at $(\beta,0)$ has parameters $(p,p,q,Q)$. It is sufficient to show
that this Potts ARC model, with red-wired boundary condition, has no
percolation of red bonds in the infinite-volume limit. By Lemma \ref{L:hpos},
for this it is enough to find $q^{\prime}$ and $h^{\prime} > 0$ such that,
letting $(p,p,q^{\prime},Q^{\prime})$ be the parameters of the Potts ARC model
corresponding to a $q^{\prime}$-state Potts model at $(\beta,h^{\prime})$,
this Potts ARC model dominates the Potts ARC model at $(p,p,q,Q)$, both models
having red-wired boundary condition on some $(\oL,\omB(\Lambda))$.
The external field $h_{I}$ of the
conditional Ising model of the latter Potts ARC model is given by
\[
Q = 1 + e^{\beta(2d + h_{I})/2},
\]
so $e^{\beta h_{I}/2} = e^{d\beta}/q < 1$ and thus $h_{I} < 0$. Hence we can
choose $0 > h_{I}^{\prime} > h_{I}$, then choose $q^{\prime} < q$ and $0 <
h^{\prime} \leq \epsilon(h_{I}^{\prime} - h_{I})$ satisfying
\[
e^{\beta(2d + h^{\prime}_{I})/2} = \frac{e^{\beta(2d +
h^{\prime})}}{q^{\prime}},
\]
and set
\[
Q^{\prime} = 1 + \frac{e^{\beta(2d + h^{\prime})}}{q^{\prime}}.
\]
We now show that under red-wired boundary condition, the red-bond configuration
of the Potts ARC model at $(p,p,q^{\prime},Q^{\prime})$ dominates the red-bond
configuration of the Potts ARC model at $(p,p,q,Q)$. The weights for the
ARC model red-bond configuration at $(p,p,q,Q)$ on $(\oL,\omB(\Lambda))$ with
red-wired boundary condition,
and the definition of $T(\omr)$, are given by (\ref{E:ARCredwgt}) and
(\ref{E:Tdef}). We let $W^{\prime}(\omr)$ and $T^{\prime}(\omr)$ denote the
corresponding quantities for the model at $(p,p,q^{\prime},Q^{\prime})$. To
establish the desired domination, by (\ref{E:incrfunc}) it is sufficient to
show that
\begin{equation} \label{E:Tcompare}
\left( \frac{q}{q^{\prime}} \right)^{C(\omega,\Lambda) - C(\omega \vee e,
\Lambda)} \frac{T^{\prime}(\omega \vee e)}{T^{\prime}(\omega)}
\geq \frac{T(\omega \vee e)}{T(\omega)} \quad \text{for all } \omega
\text{ and all } e = \langle xy \rangle.
\end{equation}
If neither $x$ nor $y$ is in $\mathcal{I}(\omega,\Lambda)$, then both sides of
(\ref{E:Tcompare}) are 1. Let us assume both $x$ and $y$ are in
$\mathcal{I}(\omega,\Lambda)$, so $C(\omega,\Lambda) - C(\omega \vee e,
\Lambda) = 1$; the case in which exactly one is in
$\mathcal{I}(\omega,\Lambda)$ is similar. By Lemma \ref{L:Tratio},
\begin{align} \label{E:Tratio3}
\frac{T(\omega \vee e)}{T(\omega)} &= P^{Ising}_{\mathcal{I}(\omega,\Lambda),
-,\beta/2,h_{I}}(\sigma_{x} = \sigma_{y} = -) \\
&= P^{Ising}_{\mathcal{I}(\omega,\Lambda),-,\beta/2,h_{I}}(\sigma_{x} = -)
P^{Ising}_{\mathcal{I}(\omega,\Lambda) \cup
\{x\},-,\beta/2,h_{I}}(\sigma_{y} = -), \notag
\end{align}
and similarly for $T^{\prime}$, with $h_{I}$ replaced by $h_{I}^{\prime}$.
Also,
\begin{equation} \label{E:qratio}
\frac{q}{q^{\prime}} = e^{-\beta(h_{I} - h_{I}^{\prime})/2} e^{-\beta
h^{\prime}} \geq e^{(\tfrac{1}{2} - \epsilon)\beta(h_{I}^{\prime} -
h_{I})}.
\end{equation}
Integrating (\ref{E:derivbound2}) gives
\[
\log P^{Ising}_{\mathcal{I}(\omega,\Lambda),-,\beta/2,h_{I}^{\prime}}
(\sigma_{x} = -) - \log P^{Ising}_{\mathcal{I}(\omega,\Lambda),-,
\beta/2,h_{I}}(\sigma_{x} = -) \geq -\left( \frac{1}{2} - \epsilon \right)
\frac{\beta (h_{I}^{\prime} - h_{I})}{2}
\]
and
\begin{align}
\log P^{Ising}_{\mathcal{I}(\omega,\Lambda) \cup
\{x\},-,\beta/2,h_{I}^{\prime}}
&(\sigma_{y} = -) - \log P^{Ising}_{\mathcal{I}(\omega,\Lambda) \cup
\{x\},-,\beta/2,h_{I}}(\sigma_{y} = -) \notag \\
&\geq -\left( \frac{1}{2} - \epsilon \right)
\frac{\beta (h_{I}^{\prime} - h_{I})}{2}, \notag
\end{align}
which with (\ref{E:Tratio3}) and (\ref{E:qratio}) proves (\ref{E:Tcompare}).
\end{proof}
\begin{proof}[Proof of Corollary \ref{C:larged}]
We will show that for large $d$, (\ref{E:chibeta}) holds for all $\beta$ in a
neighborhood of $(\log q)/d$. Consider the mean-field magnetization
$M_{0}(\beta_{0},h_{0})$, which for $\beta_{0} > 2$ and $h_{0} \geq 0$ is the
positive solution of
\begin{equation} \label{E:tanheq}
M_{0} = \tanh\left( \frac{\beta_{0}(M_{0} + h_{0})}{2} \right).
\end{equation}
We claim that for $\beta_{0} > 2.257$ we have
\begin{equation} \label{E:derivbound3}
\frac{1}{\beta_{0}} \frac{\partial M_{0}}{\partial h_{0}} (\beta_{0},0)
< 1 + M_{0}(\beta_{0},0).
\end{equation}
In fact, differentiating (\ref{E:tanheq}) yields
\[
\frac{2}{\beta_{0}} \frac{\partial M_{0}}{\partial h_{0}} =
\frac{1 - M_{0}^{2}}{1 - \tfrac{1}{2}\beta_{0}(1 - M_{0}^{2})}
\]
so (\ref{E:derivbound3}) is equivalent to
\[
\frac{1 - M_{0}}{2} < 1 - \frac{1}{2}\beta_{0}(1 - M_{0}^{2})
\]
at $(\beta_{0},0)$. For this it suffices that
\[
M_{0} > 1 - \frac{1}{\beta_{0}},
\]
which is equivalent to
\begin{equation} \label{E:tanhineq}
\tanh \left( \frac{\beta_{0} - 1}{2} \right) > 1 - \frac{1}{\beta_{0}}.
\end{equation}
A routine calculation verifies that (\ref{E:tanhineq}) holds for $\beta_{0} >
2.257$.
Now suppose $\{ \beta_{d} \}$ is a sequence with $d\beta_{d} \to \log q$.
Since $q + 1 > 10.56$ we have $\log q > 2.257$. Also, from convergence to
mean-field limits \cite{KS},
\[
M\left( \frac{\beta_{d}}{2},0 \right) \to M_{0}(\log q,0)
\]
and
\[
\chi \left( \frac{\beta_{d}}{2},0 \right) \to \frac{1}{\log q}
\frac{\partial M_{0}}{\partial h_{0}}(\log q,0).
\]
With (\ref{E:derivbound3}) this proves (\ref{E:chibeta}) for all $\beta$ in a
neighborhood of $(\log q)/d$, for large $d$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:dilution}]
Fix $\delta > 0$ and let $\theta^{\prime} = \theta(1 + \delta)/(1 +
2\delta)$. Fix $p_{r} > p_{c}^{FK}(q + \theta^{\prime},\mathbb{L})$. From
Lemma \ref{L:rare0}, Remark \ref{R:0probto0} and (\ref{E:ARCtheta}), if
$\theta$ is sufficiently small then (\ref{E:plusprob}) holds for $\beta,h$ as
in (\ref{E:betah5}). Hence by Proposition \ref{P:comparedilut}(ii) the red
bonds percolate in the infinite-volume
ARC model on $\mathbb{L}$ at $(p_{r},p_{g},q,Q)$ with
red-wired boundary condition. Thus $p_{c}^{ARC}(p_{g},q,Q,\mathbb{L}) \leq
p_{c}^{FK}(q + \theta^{\prime},\mathbb{L})$. Since $\delta$ is arbitrary, (i)
follows. Part (ii) is just a restatement of (i) in the context of the Potts
lattice gas, using (\ref{E:pbetacrit}).
\end{proof}
\begin{proof}[Proof of Corollary \ref{C:Z2case}]
This is immediate from Theorem \ref{T:dilution}(ii) and the result from
\cite{LMR} that $\exp(\beta_{c}^{Potts}(q,\mathbb{Z}^{2})) = 1 + \sqrt{q}$ for
all $q \geq 25.72$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:hifield}]
Let $\beta > \beta_{c}^{Potts}(q + \theta(h),\mathbb{L})$. Since
$\beta_{c}^{Potts}(q+t,\mathbb{L})$ is an increasing function of $t$, we
have $\beta > \beta_{c}^{Potts}(q + e^{-\beta h},\mathbb{L})$. Let
$(p_{r},p_{g},q,Q)$ be the parameters of the Potts ARC model corresponding to
the Potts model at $(\beta,-h)$. The dilution parameter of this Potts ARC
model is $e^{-\beta h} \leq 1$, and $p_{r} = p_{g} = 1 - e^{-\beta}$, so
Proposition \ref{P:comparedilut} applies and shows that the red bond
configuration of this ARC model, with red-wired boundary condition, dominates
the partial FK model at $(p_{r},q + e^{-\beta h},q)$. Since
$\beta > \beta_{c}^{Potts}(q + e^{-\beta h},\mathbb{L})$, there is percolation
in the infinite-volume wired-boundary FK model at $(p_{r},q + e^{-\beta h})$,
so there is also percolation a.s. in the yellow-boundary partial FK model at
$(p_{r},q + e^{-\beta h},q)$. Therefore the red bonds of our Potts ARC model
percolate, meaning $\beta \geq \beta_{c}^{Potts}(q+1,-h,\mathbb{L})$, and the
theorem follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:smallJ}]
Since the Ising model has the FKG property, on a planar lattice there is no
percolation of ``+'' spins when the external field is negative (see
\cite{Hi1}, \cite{Hi2}.) Hence the theorem is an immediate consequence of
Proposition \ref{P:comptoplus} and (\ref{E:hpnegcond}).
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:coupling}]
Fix a finite $\Lambda$ and a boundary condition $\eta$. The corresponding ARC
model on $(\oL,\omB(\Lambda))$ with boundary condition $\eta^{1}$
(equivalently, red-wired) dominates the ARC model with boundary condition
$\eta$, so as in the proof of Proposition \ref{P:percunique2}, the Potts
lattice gases under the two boundary conditions can be coupled, creating a
pair of configurations $((\omr,\omg),(\omr^{\prime},\omg^{\prime}))$ which
agree everywhere inside each dual surface which is crossed by no open bond
(red or green) of the red-wired ARC model configuration $(\omr,\omg)$. But
every site not inside such a dual surface is necessarily in the boundary
particle cluster, and the result follows.
\end{proof}
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