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\topmatter
\leftheadtext\nofrills
{\headerfont L. Chayes, D. McKellar, B. Winn}
\rightheadtext\nofrills
{\headerfont Percolation and the 2-d Ashkin--Teller Model}
\title Percolation and Gibbs States Multiplicity\\ for Ferromagnetic
Ashkin--Teller Models on $\Bbb Z^2$ \endtitle
\author
\hbox{\hsize=5in
\vtop{\centerline{L. Chayes, D. McKellar, B. Winn}
\centerline{{\it Department of
Mathematics}}
\centerline{{\it UCLA}}}}
\endauthor
\address
L. Chayes, D. McKellar, B. Winn
\hfill\newline
Department of Mathematics
\hfill\newline
University of California
\hfill\newline
Los Angeles, California 90095--1555
\endaddress
\email
lchayes\@math.ucla.edu, a540dmm\@pic.ucla.edu, bwinn\@ucla.edu
\endemail
\keywords Percolation, Potts Models, Ashkin--Teller Models, Random Cluster
Models, FKG inequalities, Gibbs states \endkeywords
\abstract
\baselineskip = 18pt
For a region of the nearest neighbor ferromagnetic Ashkin--Teller
spin systems on $\Bbb Z^2$, we characterize the existence of
multiple Gibbs states via percolation. In particular, there are multiple
Gibbs states if and only if there exists percolation of any
of the spin types. (I.e., the magnetized states are characterized
by percolation of the dominant species.) This result was previously known only
for the Potts models on $\Bbb Z^2$.
\endabstract
\thanks
L\. Chayes was supported by the NSA under Grant \# MDA904-98-1-0518.
B. Winn and D. McKellar would like to extend a special thanks to
Irving \& Jean Stone, and to the Naumbergs for their Honors Research Awards
given through the UCLA Honors Undergraduate Summer Research
Program.
\endthanks
\endtopmatter
\TagsOnRight
\define\<{\left<}
\define\>{\right>}
\define\K{\Cal K_{\*}}
\define\Kt{\Cal K_{\<\widetilde{i,j}\>}}
\redefine\kt{\Cal k_{\<\widetilde{i,j}\>}}
\define\ijt{\<\widetilde{i,j}\>}
\define\Kuv{\Cal K_{\*__}}
\define\Kiu{\Cal K_{\__*}}
\define\Kiv{\Cal K_{\**}}
\define\kuv{k_{\*__}}
\define\k{k_{\__*}}
\define\w{\omega}
\define\bb{\text{\underbar{$b$}}}
\redefine\aa{\text{\underbar{$a$}}}
\define\ssigma{\text{\underbar{$\sigma$}}}
\define\ttau{\text{\underbar{$\tau$}}}
\define\ijg{\langle i,j \rangle \in \Cal G}
\define\hdge{\ge}
\define\hdle{\le}
\define\Hemp{H_{\emptyset}}
\def\E#1{\Bbb E_{#1}}
\define\Lk{\Lambda_{\text{k}}}
\define\Lkk{\Lambda_{\text{k}+1}}
\define\Xk{\Xi_{\text{k}}}
\define\Ll{\Lambda_{\text{L}}}
\define\nug{\nu_{\Cal G}}
\define\rcl{\rho^{C_x}_{\Lk}}
\define\rwl{\rho^{{\W}}_{\Lk}}
\define\rc{\rho^{C_x}}
\define\rw{\rho^{{\W}}}
\define\G{\Cal G}
\define\Gt{\widehat{\Cal G}}
\define\Gtil{\tilde{\Cal G}}
\define\W{\Cal B}
\define\lk{\Bbb L, \Bbb K}
\define\Rk{R_{\G}^{\Bbb L,\Bbb K,\Cal B}}
\define\Rkk{R_{\G}^{\Bbb L,\Bbb K',\Cal B}}
\define\Rkt{R_{\Gt}^{\widehat{\Bbb L},\widehat{\Bbb K}}}
\define\rkt{\rho_{\Gt}^{\widehat{\Bbb L},\widehat{\Bbb K}}}
\define\rk{\rho^{\Bbb L,\Bbb K,\Cal B}}
\define\rkk{\rho^{\Bbb L,\Bbb K',\Cal B}}
\define\lkb{\Bbb L,\Bbb K,\Cal B}
\define\lkkb{\Bbb L,\Bbb K',\Cal B}
\define\dg{\partial \G}
\define\wt{\widehat{\w}}
\define\kp{\Bbb K}
\define\kph{\widehat{\Bbb K}}
\newpage
\heading Introduction
\endheading
An issue that sometimes arises in statistical mechanics concerns the connection
between percolation and phase transitions.
For the Potts models on $\Bbb Z^2$ there are characterization theorems
relating the uniqueness of the Gibbs states and the
absence of spin-system percolation \cite{CNPR$_{1\&2}$},
\cite {C$_1$}. Explicitly, for the Ising magnet, the region of non-uniqueness
is {\it characterized} by percolation of $+$ spins in the $+$ state. The
analogous result holds for the Potts model and a number of similar results,
for various systems,
were established in \cite{GLM}.
In this paper, we will establish such a result for a region of
the Ashkin--Teller models. Specifically, there are multiple limiting Gibbs
states precisely at those temperatures which foster
percolation of one of the spin types.
We begin with a description of a general Ashkin-Teller model on an
arbitrary graph with
spins at each vertex.
There are four possible spin types, labelled:
blue, red$+$, yellow, and red$-$. The spins may be regarded as
lying equidistant on the unit circle,
occurring clockwise in the order just named, with blue at 12 o'clock.
There is complete symmetry around the circle, so that
interactions receive energy assignments based solely on the relative
positions of the spin colors on the circle.
Here the model is ``completely'' ferromagnetic: colors opposite to each other
receive the
highest energy assignments; the like-like interactions the lowest, and the
adjacent colors receive an
intermediate energy. Without loss of generality, we may set
this intermediate energy level $= 0$. For positive $\K$, $\k$, we set the
like-like interaction between sites $i$ and $j$ along the edge $\**= \k-\K$,
and the interaction for spin pairs with colors opposite to each other to $\k$.
Although the $\Bbb Z^2$ Ashkin-Teller model in our theorem has uniform
couplings (and
at most one edge between any two sites),
some of our proofs will use the flexibility
of multiple edges between sites and non-uniform coupling constants.
In this paper, we will confine attention to
the parameter region $\k \le \K /2$ for all $\**$.
Let $\vec s_i$ denote
the Cartesian coordinates of the site $i$'s color on the
unit circle.
Then the explicit energy value $\Cal E(\vec s_i, \vec s_j)$ between sites
$i$ and $j$ is:
$$
\Cal E(\vec s_i, \vec s_j) = \Gamma\vec s_i \cdot \vec s_j +
\gamma(\vec s_i \cdot \vec s_j)^2 \tag{1}
$$
where $-\Gamma = \frac{\K}{2}$ and $-\gamma = \frac{\K-2\k}{2}$.
For any finite graph $\Cal G$ the Hamiltonian is given by
$H=\sum_{\ijg}
{\Cal E(\vec s_i,\vec s_j)}$, and the Boltzmann weight of any spin configuration
is $e^{-\beta H}$ where $1/\beta\propto$ temperature.
The phase diagram of the Ashkin-Teller model is depicted in figure 1; this
is the upper right hand quadrant of Baxter's diagram 12.12 \cite{B},
slightly tilted.
In the notation of Baxter's book, the change of variables (for the uniform case)
is as follows:
$\epsilon_3=k$, $\epsilon_0=k-K$, and $\epsilon_1=\epsilon_2=0$.
The present work focuses on the region $0\in\w}{\delta_{\sigma_i\sigma_j}} \tag{7}
$$
where $\w\subset\Theta_{\Cal G}$ is an Ising FK bond configuration, and
$B^{\kp}_p(\w)$ is the Bernoulli weight for $\w$ with
probability $p_{\**}=1-e^{-\beta \K b_ib_j}$ of the bond $\**$ being
occupied. Specifically,
$$
B^{\kp}_p(\w)=\prod_{\**\in\w}p_{\**}\prod_{\**\notin\w}(1-p_{\<
i,j\>}).
$$
Let $C(\w)=$ the number of connected components of the configuration
$\w$
(where sites not touching bonds are considered to be individual components.)
Summing over $\ssigma$ and $\bb$, we arrive at the marginal distribution:
$$
\mu_{\G}(\w)\propto\Cal Z_{\aa}^{I,\Bbb K}e^{\beta\psi^{\Bbb K}(\aa)}
\sum_{\bb}e^{\beta\psi^{\Bbb K}(\bb)} B^{\kp}_p(\w)2^{C(\w)}.\tag{8}
$$
Notice that $p_{\**}$ is nonzero only if
$b_i$ and $b_j$ are one; it is observed that $\w$ bonds
represent full spin alignment so that each connected cluster must be
monochrome -- either of the blue or yellow type.
So far, we have only considered free boundary conditions on the graph $\Cal G$.
Also of interest are {\it blue} boundary conditions. Let $\dg$ denote a set
of ``boundary''
sites in $\Cal G$. Consider the analogous developments under the boundary conditions
that all sites of $\dg$ are fixed at blue; we denote the corresponding
measures by
$\nug^{\Cal B}(-)$, $\phi_{\Cal G}^{\Cal B}(-)$ and $\mu_{\Cal G}^{\Cal
B}(-)$ respectively.
Of course, $\nu^{\Cal B}_{\Cal G}(-)$ is just the marginal distribution of
a Canonical Gibbs
measure. In equation (7), the terms $\delta_{\sigma_i \sigma_j}$ must be
modified if $i$
and/or $j$ is a boundary site, and in addition the partition function $\Cal
Z_{\aa}^{I,\Bbb K}$
has to be recomputed. Finally, in the counting of clusters one arrives at
$2^{C_{\text w}(\w)-1}$
where $C_{\text w}(\w)$ is the number of components counted as though all
sites of $\dg$ are
identified as a single site. Hence, in the $\mu_{\Cal G}^{\Cal B}(-)$
measure, the connected
component of the boundary represents sites that are all blue.
We are now ready for the first direction of our characterization proof.
\proclaim{Theorem 1a}
In the region $0\le\k\le \K/2$ of the above described Ashkin-Teller model
on $\Bbb Z^2$,
the presence of multiple Gibbs states implies that there
is percolation of blue spins in the ``blue'' state: the state obtained as
the limit of finite volume conditional measures with all
boundary spins set to blue.
\endproclaim
\demo{Proof}
Let $\vec s_0$ be the spin at the origin, and let $\hat e_y$ be the unit
vector in the blue direction. The superscript ``$\W$'' will
denote
blue boundary conditions on $\G$. By use of yet another (bi-layer)
graphical representation, Theorem III.7 in \cite{CM$_1$} demonstrates that
non-uniqueness of Gibbs states in the region $\k\le\K/2$ of the
Ashkin--Teller model is equivalent to positive spontaneous magnetization. So
for this direction of the argument,
it suffices to assume that
we have this positive magnetization.
Let $\<-\>^{\W}_{\G,w}$ be the expectation with respect to a measure $w$ under
blue boundary conditions on $\G$.
From positivity of the magnetization, we have:
$$
\_{\G,\nu}^{\Cal B}=
\<\vec s_0\cdot \hat e_y\>^{\W}_{\G,\nu} \ge \epsilon >0 \tag{9}
$$
for some $\epsilon >0$, for all finite $\G\subset\Bbb Z^2$.
Let $E$ be the event that
the origin is connected to the boundary of $\G$ through $\w$ bonds.
Recalling the measure described in (7), we see that:
$$
\split
\^{\Cal B}_{\G,\nu}
&=\^{\Cal B}_{\G,\phi}\\
&=\^{\Cal B}_{\G,\phi}\phi^{\Cal
B}_{\G}(E)+\^{\Cal B}_{\G,\phi}\phi^{\Cal
B}_{\G}(E^c).\endsplit
\tag{10}
$$
Given $E$, $b_0\sigma_0=1$; it is easy to see that the second term vanishes.
Thus,
$$
\^{\Cal B}_{\G,\nu} =\phi^{\Cal B}_{\G}(E) = \mu^{\Cal
B}_{\G}(E).
\tag{11}
$$
Hence, (9) implies
$\mu^{\Cal B}_{\G}(E)\ge\epsilon>0\quad\forall$ finite $\G\subset\Bbb Z^2$.
So we have percolation of $\w$ bonds. The blue boundary condition now
forces the percolating
cluster to, in fact, be blue. In the thermodynamic limit, this gives us
percolation of blue spins. \qed
\enddemo
For the second direction of the argument we will make use of a
result by Gandolfi, Keane,
and Russo \cite{GKR}. Their result requires a measure on $\Bbb Z^2$ that
$\bullet$ is invariant under translations and axis reflections
$\bullet$ is ergodic under vertical and horizontal translations
$\bullet$ satisfies the FKG condition: positive events are positively
correlated.
\noindent Under these three conditions, if there is percolation, then an
infinite cluster
is unique with probability one. Furthermore, all other spin-types lie in finite
star-connected clusters.
(The definition of star-connectedness is as follows: two sites
are said to be star-connected if they are nearest neighbors or
next nearest neighbors; i\.e\. if neither their $x$ nor their
$y$ coordinates differ, in modulus, by more than one.)
Let $\rw(\bb) \overset{\text {def}}\to= \underset{\G\nearrow{\Bbb
Z^2}}\to{\text{lim}}
{\rw_{\G}(\bb)}$, where ${\rw_{\G}(\bb)}$ is the $b-$marginal distribution of
$\nu_{\G}^{\Cal B}(\bb,\ssigma).$
We will demonstrate in the Appendix that this measure satisfies the above conditions.
\proclaim{Theorem 1b}
In the region $k\le K/2$ of the Ashkin--Teller model on $\Bbb Z^2$,
percolation of blues
implies the
existence of multiple Gibbs states.
\endproclaim
\demo {Proof}
We remind the reader that $b=1$ for the blue and yellow spins, whereas $b=0$
for the red spins.
The FKG property of $\rw(\bb)$ (see Appendix), then, actually establishes
the FKG property
for the ordering blue, yellow $\ge$ reds.
Suppose that we have percolation of blues. By Theorem 1a, if there were no
percolation of blues in the blue state, then
we would not see percolation in any purported state, all states being
equivalent.
Thus, blues are percolating in the blue state.
Then certainly the blue-yellow spin combination percolates under these
conditions.
Since our blue measure satisfies the
conditions of
the GKR theorem, the blue-yellow infinite cluster is unique WP1, and all red
clusters lie in finite star-connected clusters.
Now we may produce at least 2 distinct Gibbs states: one corresponding to
the blue-yellow percolation -- a ``green'' state,
and one for red percolation -- a ``red'' state. We have just learned that
these are mutually exclusive situations:
Consider the event that the origin is
part of an infinite cluster, given that the origin is blue or yellow; this
event has positive probability in the green state, but
has {\sl zero} probability in the red state. Hence, these states are
distinct, and we have non-uniqueness of Gibbs states. $\qed$
\enddemo
Together, Theorems 1a \& 1b give us our desired characterization.
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\heading Appendix
\endheading
To facilitate the
oncoming
handling of boundary conditions, we will consider
$\Cal G$ to be a general finite graph (not necessarily a subset of $\Bbb Z^2$)
and we will use the more general Hamiltonian
$H=H_{\aa,\ttau}^{\Bbb L}+H_{\bb,\ssigma}^{\Bbb K}$ where $\Bbb L$,
analogous to $\Bbb K$,
represents a configuration of $\Cal L_{\**}$,\ $ l_{\**}$ on
$\Theta_{\G}$ for the $a$
$\&$ $\tau$ variables.
At this point, we take interest in the marginal distribution
$\rho_{\G}^{\lk}(\bb)=\sum_{\ssigma}{\nug^{\lk}(\bb,\ssigma)}$,
with weights
denoted by $\Cal R_{\G}^{\lk}(\bb)$:
$$\Cal R_{\G}^{\lk}(\text{\underbar{$b$}}) = \Cal Z_{\aa}^{I,\Bbb L} \Cal
Z_{\bb}^{I,\Bbb K}e^{\beta\psi^{\Bbb K}(\bb)+\beta\psi^{\Bbb L}(\aa)}
\tag{12}$$
where $\psi^{\Bbb K}(\bb)={ \sum_{\ijg}(\K-\k)b_i b_j}$, and
$\psi^{\Bbb L}(\aa)={\sum_{\ijg}(\Cal L_{\**}-l_{\**})a_ia_j}$.
\proclaim{Lemma 1}
The measure $\rho_{\G}^{\lk}(\bb)$ is strong FKG.
\endproclaim
\demo {Proof}
This is almost identical to a result found in \cite{C$_2$}, with slightly
different measures, so here it is:
In this case the strong FKG property (which does imply the usual ``weaker''
FKG condition) is equivalent to the lattice condition. For this measure,
the lattice condition
states that for any two configurations $\eta_1$ and $\eta_2$ of
$b$-variables on our
graph, $\rho(\eta_1\wedge\eta_2)\rho(\eta_1\vee\eta_2) \ge
\rho(\eta_1)\rho(\eta_2)$.
(Here all subscripts and superscripts have been dropped.) The object
$\eta_1\vee\eta_2$ is
a new configuration for which each site $i$ chooses the higher value
between $b_i$ from
$\eta_1$ and $b_i$ from $\eta_2$. Similarly, $\eta_1\wedge\eta_2$ chooses
the lower value at
each site.
Since $b_i$ can only take the values
$0$ and $1$, it is necessary and sufficient to check that:
$$\Cal R_{\G}^{\lk}(\bb^*)|_{b_u,b_v=0} \Cal
R_{\G}^{\lk}(\bb^*)|_{b_u,b_v=1} \ge \Cal R_{\G}^{\lk}(\bb^*)|_{b_u=0,b_v=1}
\Cal R_{\G}^{\lk}(\bb^*)|_{b_u=1,b_v=0}\tag{13}$$
for arbitrary sites $u$, $v$, and for $\bb^*=$ a {\sl fixed} configuration
of spins on all sites of $\G$, excluding $u$ \& $v$.
Since $a_i=1-b_i$, and because $H_{\aa,\ttau}^{\Bbb L}$ and
$H_{\bb,\ssigma}^{\Bbb K}$ are identical in form,
it is sufficient to check this lattice condition for
$H=H_{\bb,\ssigma}^{\Bbb K}$.
Our desired inequality is as follows:
$$
\multline
\left.\left(\Cal Z_{\bb}^{I,\Bbb K} e^{\beta \psi^{\Bbb
K}(\bb)}\right)\right|_{b_u,b_v=0}
\left.\left(\Cal Z_{\bb}^{I,\Bbb K} e^{\beta \psi^{\Bbb
K}(\bb)}\right)\right|_{b_u,b_v=1} \\ \hdge
\left.\left(\Cal Z_{\bb}^{I,\Bbb K} e^{\beta \psi^{\Bbb
K}(\bb)}\right)\right|_{b_u=0,b_v=1}
\left.\left(\Cal Z_{\bb}^{I,\Bbb K} e^{\beta \psi^{\Bbb
K}(\bb)}\right)\right|_{b_u=1,b_v=0}.
\endmultline \tag{14}
$$
For ease of notation, we define:
$$-H_{\emptyset}\overset\text{def}\to= \sum_{\langle i,j \rangle:i,j
\notin \{u,v\}}\K b_i b_j(\delta_{\sigma_i \sigma_j} -1)$$
$$-H_u \overset\text{def}\to= \sum_{\langle i,u \rangle: i\not= v}\Kiu
b_i(\delta_{\sigma_i \sigma_j} -1)$$ and $-H_v$ is
defined accordingly. With this notation, after canceling the factor
$e^{2\beta \sum_{\langle i,j \rangle \not= \langle u,v \rangle}{(\K-\k)b_i
b_j}}_{b_u,b_v=0}$ from both sides, equation (14) reduces to:
$$
\multline
e^{\beta(\Kuv-\kuv)}
\text{Tr}[e^{-\beta(\Hemp+H_u+H_v)}e^{\beta \Kuv(\delta_{\sigma_u
\sigma_v}-1)}]
\text{Tr}[e^{-\beta \Hemp}]\\ \hdge
\text{Tr}[e^{-\beta(\Hemp +H_u)}]\text{Tr}[e^{-\beta(\Hemp+H_v)}]
\endmultline \tag{15}
$$
where the trace is understood to be taken over $\ssigma$.
Now we will
simplify things further by proving the following inequality:
$$
e^{\beta(\Kuv-\kuv)}\text{Tr}[e^{-\beta(\Hemp+H_u+H_v)}e^{\beta
\Kuv(\delta_{\sigma_u \sigma_v}-1)}] \ge
\text{Tr}[e^{-\beta(\Hemp+H_u+H_v)}]. \tag{16}
$$
Dividing by the RHS, and letting $\E{H}(-)$ be the Ising expectation with
respect to the Hamiltonian $H$, this inequality is equivalent to
$$
e^{\beta (\Kuv-\kuv)}\E{\Hemp+H_u+H_v}[(1-e^{-\beta \Kuv})\delta_{\sigma_u
\sigma_v}+e^{-\beta \Kuv}] \ge 1.\tag{17}
$$
Using the fact that $\E{\Hemp+H_u+H_v}(\delta_{\sigma_u \sigma_v}) \ge 1/2$,
the left hand side of the equation is bounded below by
$e^{\beta(\frac{\Kuv}{2}-\kuv)}\cosh{(\frac{\beta \Kuv}{2})}$, which is
always $\ge 1$, since we are in the region where
$\frac{\K}{2} \ge \k$.
Having shown (16), the Lemma is implied by the following ``alteration'' of
(15):
$$
\text{Tr}[e^{-\beta(\Hemp+H_u+H_v)}]\text{Tr}[e^{-\beta \Hemp}] \hdge
\text{Tr}[e^{-\beta(\Hemp+H_u)}]\text{Tr}[e^{-\beta(\Hemp+H_v)}], \tag{18a}
$$
which is tantamount to: $\E{\Hemp+H_u}(e^{-\beta H_v}) \hdge
\E{\Hemp}(e^{-\beta H_v})$.
Let $N(v)= \{\text{nearest neighbors}$ $\text{of $v$, excluding $u$}\}$,
$\{T\}=$ the set of all subsets of $N(v)$, and
$$
\Upsilon(T)=\prod_{i \in T}(1-e^{-\beta b_i\Kiv})\prod_{i \notin T}e^{-\beta b_i
\Kiv}.\tag{19}
$$
Now expanding FK style, (18a) continues to transform
into:
$$
\sum_{T}\left[\Upsilon(T)\E{\Hemp+H_u}\left(\prod_{i\in
T}\delta_{\sigma_i\sigma_v}\right)\right]\hdge
\sum_{T}\left[\Upsilon(T)\E{\Hemp}\left(\prod_{i\in
T}\delta_{\sigma_i\sigma_v}\right)\right]. \tag{18b}
$$
This will certainly be true if for arbitrary $T$, the individual
expectations in (18b)
obey the inequality. As in \cite{C$_1$}, these expectations can be
expressed with the Ising FK representation, in terms of probabilities of
cluster events. Let $\Bbb P_H^{\text{RC}}(-)$,
$\<-\>^{\text{RC}}_H$ be the Random Cluster (RC) probability and expectation
corresponding to the Ising Hamiltonian $H$, and let $|T|$ be the
number of sites in $T$. Given a bond configuration $\w$ from this
representation,
let $C_T(\w)$ be the number of connected
components containing sites in $T$. Converting from spin system expectations:
$$
\align
\E{H}\left(\prod_{i\in T}\delta_{\sigma_i
\sigma_v}\right)&=\sum_{n=1}^{|T|}(\frac{1}{2})^{n-1}\Bbb
P_H^{\text{RC}}\{C_T=n\} \\
\ &=\<\frac{1}{2^{C_T-1}} \>^{\text{RC}}_H.
\tag{20}
\endalign $$
So we want:
$$
\<\frac{1}{2^{C_T-1}} \>_{\Hemp+H_u}^{\text{RC}} \hdge
\<\frac{1}{2^{C_T-1}} \>_{\Hemp}^{\text{RC}}\tag{21}
$$
which is true because $(\frac{1}{2})^{C_T-1}$ is an increasing function of
bond configurations (added bonds $\rightarrow$ smaller $C_T$),
and $\Bbb P_{\Hemp+H_u}^{\text{RC}}(-)$ FKG dominates $\Bbb
P_{\Hemp}^{\text{RC}}(-)$.
By definition, one measure FKG dominates a second measure if it assigns
higher probabilities
to all positive
events than does the second measure.
In this case, positive events will receive higher probabilities with $H_u$ added to the Hamiltonian $\Hemp$.
$\qed$
\enddemo
Let us now consider $\rk_{\G}(-)$, which is defined as the marginal distribution
obtained from the blue measure $\nu_{\G}^{\lkb}(-)$, and which
has weights:
$$
\Rk(\bb)=e^{\beta\psi^{\Bbb K}(\bb)+\beta\psi^{\Bbb L}(\aa)}\Cal Z^{I,\Bbb
L}_{\aa}\sum_{\ssigma}{e^{\beta I^{\Bbb K}_{\bb,\ssigma}}}
\chi_{(\ssigma_{\dg}=1)}\chi_{(\bb_{\dg}=1)}\tag{22}
$$
where $\chi$ is the indicator function. Expanding into $\w$ bonds with the
constraint that all Ising spins on the boundary $\dg$
are fixed at $\sigma_i=1$,
$$
\Rk(\bb)=e^{\beta\psi^{\Bbb K}(\bb)+\beta\psi^{\Bbb L}(\aa)}\Cal Z^{I,\Bbb
L}_{\aa}\sum_{\w}{B^{\kp}_p(\w)2^{C_{\text w}(\w)-1}}\chi_{(\bb_{\dg}=1)}.
\tag{23}
$$
\proclaim{Corollary}
$\rk_{\G}(-)$ is strong FKG.
\endproclaim
\demo{Proof} Without loss of generality, we may assume
that $u,v\notin\dg$ and that $b_i=1\quad\forall i\in\dg$. If either of these
conditions are violated, the lattice condition holds
trivially. Let us define $\Gt=\G\cup\Gtil$, where $\Gtil=\{\ijt\}$, a set of new
edges connecting each boundary site in $\G$ to every other
boundary site in $\G$. For each added edge $\ijt$, set the values $\Cal
L_{\ijt}=l_{\ijt}=k_{\ijt}=0$ and $\Kt\gg1$.
Notice that for all
$\wt$ on $\Gt$ such that all $\ijt$ bonds are occupied, $C(\wt)=C_{\text
w}(\w)$. Considering the limit
as all $\Kt\rightarrow\infty$, we find that $p_{\ijt}=1-e^{-\beta\Kt
b_ib_j}\rightarrow1$; consequently, for all $\wt$ on $\Gt$ having
a vacant $\ijt$ bond, $B^{\kph}_p(\wt)\rightarrow 0$. It follows that
$\sum_{\wt {\text{ on
}}\Gt}B^{\kph}_p(\wt)2^{C(\wt)}\underset{\{\Kt\}\rightarrow\infty}\to
\longrightarrow
\sum_{\w {\text{ on }}\G}B^{\kp}_p(\w)2^{C_{\text w}(\w)}$.
The key is that $\rk(\bb)=\lim_{\{\Kt\}\to\infty}\rkt(\bb)$. Recalling
that $b_ib_j=1$ for all $\ijt\in\Gt$, we find that
the ratio of the respective weights $=$
$$e^{\beta\sum_{\Gtil}\Kt(b_ib_j-1)}
\frac{\sum_{\wt}B^{\kph}_p(\wt)2^{C(\wt)}}
{\sum_{\w}B^{\kp}_p(\w)2^{C_{\text w}(\w)}}
\rightarrow{\text {const.}}\tag{24}$$
And our result is clear.
$\qed$
\enddemo
Define the partial ordering
$\Bbb K'\succ\Bbb K$ if for each $\ijg$,
$\K'\ge\K, \k'=\k$. To establish the unique existence of an infinite volume
limiting measure $\rk(-)$, we need the
following proposition.
\proclaim{Proposition}
For $\Bbb K'\succ\Bbb K$, $$\rkk_{\G}(-)\underset{\text {FKG}}\to\ge
\rk_{\G}(-)$$
\endproclaim
\noindent where the subscript on the inequality denotes FKG domination of
measures, as discussed
after equation (21).
\demo{Proof}
As in \cite{C$_2$}, we will express the ratio of pertinent weights as the
Ising Random Cluster expectation of an increasing function.
It is sufficient to consider the case where $\Bbb K'$ and $\Bbb K$ differ
only in that $\Kuv'-\Kuv=\Delta_{uv}\ge0$. Assume that we have
$\bb$ such that $b_i=1\quad\forall i\in\dg$.
If $u$ and $v$ are both in $\dg$, the ratio of weights is simply
$e^{\beta\Delta_{uv}b_ub_v}$, a decidedly increasing function of $\bb$.
For the case where $u$ and $v$ are not both on the boundary, a bit of
manipulation gives:
$$
\frac{\Rkk(\bb)}{\Rk(\bb)}=1+(e^{\beta\Delta_{uv}b_ub_v})
\frac{\sum_{\w}B^{\Bbb
K'}_p(\w)\sum_{\ssigma}\prod_{\**\in\w}\delta_{\sigma_i\sigma_j}
\chi_{(\ssigma_{\dg}=1)}\delta_{\sigma_u\sigma_v}}
{\sum_{\w}B^{\kp}_p(\w)2^{C_{\text w}(\w)-1}}.\tag{25}
$$
Defining $T_{uv}$ to be the event that $u$ and $v$ are connected by $\w$
bonds, and splitting the summation in the numerator according
to this criterion, the ratio (25) becomes:
$$
1+(e^{\beta\Delta_{uv}b_ub_v})\sinh(\frac{1}{2}\beta\Delta_{uv}b_ub_v)[1+<\c
hi_{(T_{uv})}>^{\text {RC}}];
\tag {26}
$$
indeed, another increasing function of $\bb$.
$\qed$
\enddemo
Let $\rk_{\Lk}(-)$ be the measure on a finite graph $\Lk$, and
let $\{\Lk\}$ be a nested sequence of finite graphs such that $\Lkk \supset
\Lk$, and
$\Lk \nearrow \Bbb Z^2$ as ${\text{k}}\rightarrow\infty$. By the previous
proposition, $\rkk_{\Lkk}(-)\left.\right|_{\Lk}
\underset{\text {FKG}}\to\ge \rk_{\Lkk}(-)\left.\right|_{\Lk}$
for $\Bbb K'\succ\Bbb K$.
Let $Q_{\text k}=\{\**\in\Theta_{\Lkk}\backslash\Theta_{\Lk} \}$. Notice
that for the measure $\rkk_{\Lkk}(-)\left.\right|_{\Lk}$,
letting $\K\rightarrow\infty\quad\forall \**\in Q_{\text k}$ essentially
wires the boundary of $\Lk$
to the boundary of $\Lkk$ with $\w$ bonds. This forces the limiting measure
to have blue boundary conditions on $\Lk$, so that:
$$
\lim\Sb{\K\to\infty}\\{\forall \**\in Q_{\text k}}\endSb
\rkk_{\Lkk}(-)\left.\right|_{\Lk} = \rk_{\Lk}(-).\tag{27}
$$
It follows that
$$
\rk_{\Lk}(-) \underset{\text{FKG}}\to \ge
{\rk_{\Lkk}(-)\left.\right|_{\Lk}}.\tag{28}
$$
Hence the limiting measure $\rk(-) \overset{\text {def}}\to= \lim_{{\text
k}\rightarrow\infty}{\rk_{\Lk}(-)}$ exists, and
by a squeezing argument is independent of the particular nesting sequence: it is
unique.
Notice that $\rk(-)$ must also satisfy the lattice condition, and thus
retains the
strong FKG property.
We now restore $\Bbb L=\Bbb K$, with uniform couplings on $\Cal G\subset
\Bbb Z^2$.
\proclaim{Lemma 2}
$\rho^{{\W}}(-)$ is ergodic under translations, and invariant under
translations and axis re${f\!\!\!\ l}$ections.
\endproclaim
\demo{Proof}
Without loss of generality, consider positive local cylinder events $A$ and
$B$. Let $x\in \Bbb Z^2$, and let
$T_x$ be the linear translation operator which moves a spin at the origin to
the site $x$. Translation invariance follows by
translating any sequence $\{\Lk\}$ that defines $\rho^{\Cal B}(A)$
and using this to obtain $\rho^{\Cal B}(T_xA)$.
Axis re$\text{{f\!\!\ l}}$ection invariance may be seen by simply choosing
a nested
sequence of $\Lk$'s symmetric about the axis in question.
For ergodicity, we use
our measure's FKG property and translation invariance
to get $\rho^{{\W}}(AT_xB)\ge\rho^{{\W}}(A)\rho^{{\W}}(T_xB)=
\rho^{{\W}}(A)\rho^{{\W}}(B)$
$\forall x$. Conditioning on the event that the supports of $A$ and $T_xB$
are each
surrounded by disjoint circuits of blue, the reverse
inequality is demonstrated in the limit as $|x|\rightarrow\infty$ by
applying FKG dominance of measures.
$\qed$
\enddemo
\bigskip
\bigskip
\bigskip
\newpage
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\bigskip
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\bigskip
\bigskip
\endRefs
\enddocument
\end
*