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first-order phase transition, entropy-energy conflict,
staggered phase, Widom-Rowlinson lattice gas, plane-rotor model,
ferrofluid, percolation, chessboard estimate, reflection positivity
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%%%%% greek letters
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\def\d{\delta}
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\def\inv{^{-1}}
\begin{document}
\title{{\bf Entropy-driven phase transitions\\ in multitype lattice
gas models}}
\author{%
Hans-Otto Georgii\\ {\small\sl Mathematisches Institut der
Universit\"at M\"unchen}\\ {\small\sl Theresienstr.\ 39, D-80333
M\"unchen, Germany}
\and Valentin Zagrebnov\\{\small\sl Universit\'e de la M\'editerran\'ee
and Centre de Physique Th\'eorique,}\\
{\small\sl CNRS-Luminy-Case 907, 13288 Marseille Cedex 9, France}}
\date{}
\maketitle
\renewcommand{\baselinestretch}{1.0}
\small
\begin{quote}
\date{February 23, 2000} \\[-0.6ex]
\hspace*{0ex}\hrulefill\hspace*{0ex}
In multitype lattice gas models with hard-core interaction of
Widom--Rowlinson type, there is a competition between the entropy due
to the large number of types, and the positional energy and geometry
resulting from the exclusion rule and the activity of particles. We
investigate this phenomenon in four different models on the square
lattice: the multitype Widom--Rowlinson model with diamond-shaped
resp.\ square-shaped exclusion between unlike particles, a
Widom--Rowlinson model with additional molecular exclusion, and a
continuous-spin Widom--Rowlinson model. In each case we show that
this competition leads to a first-order phase transition at some
critical value of the activity, but the number and character of phases
depend on the geometry of the model. Our technique is based on
reflection positivity and the chessboard estimate.
\\[-0.6ex]
\hspace*{0ex}\hrulefill\hspace*{0ex}
KEY WORDS: first-order phase transition, entropy-energy conflict,
staggered phase, Widom--Rowlinson lattice gas, plane-rotor model,
ferrofluid, percolation, chessboard estimate, reflection positivity.
\end{quote}
\renewcommand{\baselinestretch}{1.1}\normalsize
\section{Introduction}
Although the most familiar examples of phase transitions in lattice
models originate from a degeneracy of ground states and therefore
occur at low temperatures, this is not the only situation in which
phase transitions can occur. Another possible source of criticality
is a conflict of energy and entropy. This was noticed first by
Dobrushin and Shlosman \cite{DS} in the case of an asymmetric
double-well potential with two (sharp resp.\ mild) local minima
separated by a barrier. They found that for some specific temperature
energy and entropy attain a balance leading to the coexistence of
high- and low-temperature phases corresponding to the two wells; cf.\
also Section 19.3.1 of \cite{Gii}. Later on, Koteck\'y and Shlosman
\cite{KS} observed that such a first-order phase transition can occur
even in the absence of an energy barrier, provided there is an
``explosion'' of entropy. They demonstrated this in particular on the
prototypical case of the $q$-state Potts model on $\Z^d$ for large
$q$, showing that for a critical temperature there exist $q$
distinct ordered low-temperature phases as well as one disordered
high-temperature phase; see also Section 19.3.2 of \cite{Gii}.
This paper has the objective of studying entropy-driven first-order phase
transitions of similar kind in multitype lattice gas models with
type-dependent hard core interaction. In such models the crucial
parameter is the activity instead of temperature, and the
entropy-energy conflict turns into a competition between the
entropy of particle types and the positional energy and geometry
resulting from the exclusion rule and the activity of particles. One
is asking for a critical activity with coexistence of low-density and
high-density phases.
The basic example of this kind is the multicomponent Widom--Rowlinson
lattice gas model investigated first by Runnels and Lebowitz \cite{RL}
in 1974 and studied later (theoretically and numerically) by Lebowitz
et al.\ \cite{LMNS}, cf.\ also \cite{NL}. If the number $q$ of types
is large enough (the numerical estimates give $q\ge 7$), there exist
three different regimes: besides the low-density uniqueness regime and
a high-density regime with $q$ ``demixed'' phases for $z>z_c(q)$,
there exists an intermediate domain of activities $z_0(q)0$
which governs the overall particle density; we assume that the activity
does {\em not\/} depend on the particle color. Accordingly, the Gibbs
distribution in a finite region $\L\subset\Z^2$ with boundary
condition $\eta$ in $\L^c=\Z^2\setminus\L$ is given by
\be{Gibbs}
\mu_{\L,\eta}^{z,q}(\s)= 1_{\{\s\equiv\eta \mbox{ \scriptsize off }\L\}}\;
(Z_{\L,\eta}^{z,q})^{-1}\; z^{N_\L(\s)} \exp\Big[-
\sum_{\langle ij\rangle\cap\L\ne\emptyset} U(\s_i,\s_j)\Big]\;,
\ee
where $N_\L(\s)=|\{i\in\L:\s_i\ne 0\}|$ is the number of particles
in $\L$, and $Z_{\L,\eta}^{z,q}$ is a normalizing constant.
Alternatively, we may think of $\mu_{\L,\eta}^{z,q}$ as obtained by
conditioning a Bernoulli measure on the set of admissible
configurations. Let
\[
\O=\{ \s\in E^{\Z^2}: \forall \langle ij\rangle\
\s_i\s_j=0\mbox{ or }\s_i=\s_j\}
\]
be the set of all admissible configurations on $\Z^2$. Given any
such configuration $\s\in\O$ and any subset $\L$ of $\Z^2$, we write
$\s_\L$ for the restriction of $\s$ to $\L$. We also write
$\O_{\L,\eta}$ for the set of all
admissible configurations $\s\in E^{\L}$ in $\L$ which are compatible
with some $\eta\in\O$, in the sense that the composed
configuration $\s\eta_{\L^c}$ belongs to $\O$. In particular, we write
$\O_\L=\O_{\L,0}$ for the set of all admissible configurations in
$\L$, which are compatible with the empty configuration $0$ outside $\L$.
It is then easy to see that
\[
\mu_{\L,\eta}^{z,q} = \pi_{\L}^{z,q}(\dot|\O_{\L,\eta})\;,
\]
where $\pi_{\L}^{z,q}=\bigotimes_{i\in\L} \pi_i^{z,q}$ is the $\L$-product
of the measures
$\pi_i^{z,q}=(\frac1{1+qz},\frac{z}{1+qz},\ldots,\frac{z}{1+qz})$ on
$E$.
Given the Gibbs distributions $\mu_{\L,\eta}^{z,q}$, we define the
associated class $\G(z,q)$ of (infinite volume) Gibbs measures on $\O$ in the
usual way \cite{Gii}. Our main result below shows that for large $q$
there exist two different activity regimes in which $\G(z,q)$
contains several phases of quite different behavior. These regimes
meet at a critical activity $z_c(q)$ and produce a first-order phase
transition.
The different phases admit a geometric description in percolation
terms. Let $\Z^2$ be equipped with the usual graph structure
(obtained by drawing edges between sites of Euclidean distance 1).
Given any $\s\in\O$, a subset $S$ of $\Z^2$ will be called an
\emph{occupied cluster} if $S$ is a maximal connected subset of
$\{i\in\Z^2: \s_i\ne 0\}$, and an \emph{occupied sea} if, in addition,
each finite subset $\D$ of $\Z^2$ is surrounded by a circuit (i.e.,
closed lattice path) in $S$. In other words, an occupied sea is an infinite
occupied cluster with interspersed finite `islands'. If in fact
$\s_i=a$ for all $i\in S$ we say $S$ is an \emph{occupied sea of color
$a$}. We consider also the dual graph structure of $\Z^2$ with
so-called $*$edges between sites of distance 1 or $\sqrt 2$, and the
associated concept of $*$connectedness. An \emph{even occupied
$*$sea} is a maximal $*$connected subset of $\{i=(i_1,i_2)\in\Z^2:
i_1+i_2 \mbox{ even, } \s_i\ne 0\}$ containing $*$circuits around
arbitrary finite sets $\D$. Likewise, an \emph{odd empty $*$sea} is a
maximal $*$connected subset of $\{i=(i_1,i_2)\in\Z^2: i_1+i_2 \mbox{
odd, } \s_i=0\}$ surrounding any finite $\D$.
%
\begin{thm}\label{th:WR}
If the number $q$ of colors exceeds some $q_0$, there exists an
activity threshold $z_c(q)\in \;]\,q/5,5q\,[$ and numbers
$0<\e(q)<1/3$ with $\e(q)\to 0$ as $q\ti$ such that the following
hold:
{\rm (i) }For $z>z_c(q)$, there exist $q$ distinct translation
invariant `colored' phases $\mu_a\in\G(z,q)$, $a\in\{1,\ldots,q\}$.
Relative to $\mu_a$, there exists almost surely an occupied sea of
color $a$ containing any given site with probability at least
$1-\e(q)$.
{\rm (ii) }For $q_0/q\le zz_c(q)$, there exist $q$ distinct translation
invariant `colored' phases $\mu_a\in\G(z,q)$, $a\in\{1,\ldots,q\}$.
Relative to $\mu_a$, there exists almost surely an occupied sea of
color $a$ containing any given site with probability at least
$1-\e(q)$.
{\rm (ii) }For $q_0/q\le zz_c(q)$ there exist $2q$ distinct colored and
staggered phases
$\mu_{a,\even},\, \mu_{a,\odd}\in\G(z,q)$,
$a\in\{1,\ldots,q\}$, which are invariant under even translations.
Relative to $\mu_{a,\even}$ there exist almost surely both an even
occupied $*$sea of color $a$ and an odd empty $*$sea, and any
two adjacent sites belong to these $*$seas with probability at least
$1-\e(q)$. $\mu_{a,\odd}$ is obtained from
$\mu_{a,\even}$ by a one-step translation.
{\rm (ii) }For $q_0^{1/7}/q\le z0$ are defined by their densities
with respect to the product measure $\nu^\L$, which are again given by
the right-hand side of equation \rf{Gibbs}. We write $\G(z,\a)$ for
the associated set of Gibbs measures. Since $U$ preserves the
$O(2)$-symmetry of particle orientations, the
Mermin--Wagner--Dobrushin--Shlosman theorem (cf.\ Theorem (9.20) of
\cite{Gii}) implies that each such Gibbs measure is invariant under
simultaneous rotations of particle orientations.
%
\begin{thm}\label{th:contspin}
If $\a$ is less than some sufficiently small $\a_0$,
there exist a critical activity $z_c(\a)\in\;]\,\a^{-2}/18,5\,\a^{-2}\,[$
and numbers $0<\e(\a)<1/3$ with $\e(\a)\to 0$ as $\a\to 0$ such that
the following hold:
{\rm (i) }For $z>z_c(\a)$ there exists a dense `ordered' phase
$\mu_\ord\in\G(z,\a)$ exhibiting the translation invariance and
$O(2)$-symmetry of the model. Relative to $\mu_\ord$, there exists
almost surely an occupied sea containing any fixed site with
probability at least $1-\e(\a)$ (and on which the orientations of
adjacent particles differ only by the angle $2\pi\a$).
{\rm (ii) }For $\a_0\inv\le zz_c(\a)$, and there is a
second-order transition from the staggered regime to the low-activity
uniqueness regime. We thus have the following phase diagram.
%
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(7.6,1)
\put(0.14,0){\line(1,0){1}}
\put(1.4,-0.024){$\ldots$}
%\put(1.2,0){\dashbox{0.05}(1,0)}
\put(2.2,0){\vector(1,0){6.4}}
\put(8.7,-0.1){$z$}
\put(0,-0.31){$\mbox{$|$}\atop\mbox{0}$}
\put(5,-0.32){$\mbox{$|$}\atop\mbox{$z_c(q)$}$}
\put(-1.4,0.2){\small phases:}
\put(0.5,0.2){\small 1}
\put(2.6,0.2){\small 2 staggered}
\put(6.5,0.2){\small ordered}
\put(5.3,0.5){$3$}
\end{picture}
\end{center}
\end{figure}
\medskip\noindent
{\bf Remark \thethm } \ The model above is a continuous-spin
counterpart of the standard Widom--Rowlinson model considered in
Section \ref{sec:WR}. It is rather straightforward to modify our
techniques for investigating
analogous continuous-spin variants of the square-shaped
Widom--Rowlinson model and of the model with diagonal molecular hard
core. In the first case, we obtain a phase diagram of the form
\begin{figure}[ht!]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(7.6,1)
\put(0.14,0){\line(1,0){1}}
\put(1.4,-0.024){$\ldots$}
\put(2.2,0){\vector(1,0){6.4}}
\put(8.7,-0.1){$z$}
\put(0,-0.31){$\mbox{$|$}\atop\mbox{0}$}
\put(5,-0.32){$\mbox{$|$}\atop\mbox{$z_c(q)$}$}
\put(-1.4,0.2){\small phases:}
\put(0.5,0.2){\small 1}
\put(2.6,0.2){\small disordered}
\put(6.5,0.2){\small ordered}
\put(5.3,0.5){$2$}
\end{picture}
\end{center}
\end{figure}
%
\noindent
and in the second case we find
\begin{figure}[h!]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(7.6,1)
\put(0.14,0){\line(1,0){1}}
\put(1.4,-0.024){$\ldots$}
\put(2.2,0){\vector(1,0){6.4}}
\put(8.7,-0.1){$z$}
\put(0,-0.31){$\mbox{$|$}\atop\mbox{0}$}
\put(4.1,-0.32){$\mbox{$|$}\atop\mbox{$z_c(q)$}$}
\put(-1.4,0.2){\small phases:}
\put(0.5,0.2){\small 1}
\put(2.4,0.2){\small disordered}
\put(5.2,0.2){\small 2 staggered ordered}
\put(4.5,0.5){$3$}
\end{picture}
\end{center}
\end{figure}
%
\noindent
The details are left to the reader.
\section{Proof of Theorem \ref{th:WR}}
\label{sec:pf_WR}
The proof of all four theorems follows the general scheme described in
Chapters 18 and 19 of Georgii \cite{Gii}, which is similar in spirit
to that of Dobrushin and Shlosman \cite{DS} and Koteck\'y and Shlosman
\cite{KS}. This scheme consists of two parts: a model-specific
contour estimate implying percolation of ``good plaquettes'', and a
general part deducing from this percolation the first-order transition
and the properties of phases. We describe the general part first and
defer the contour estimate to a second subsection. Many of the
details presented here for the Widom--Rowlinson model carry over to
the other models, so that for the proofs of Theorems \ref{th:sqWR} to
\ref{th:contspin} we only need to indicate the necessary changes. We
note that our arguments can easily be extended to the higher
dimensional lattices $\Z^d$ using either the ideas of Chapter 18 of
\cite{Gii} or those of \cite{Fuku}.
\subsection{Competition of staggered and ordered plaquettes}
\label{sec:compete}
We consider the standard plaquette $C=\{0,1\}^2$ in $\Z^2$ as well as
its translates $C+i$, $i\in\Z^2$. Two plaquettes $C+i$ and $C+j$ will
be called adjacent if $|i-j|=1$, i.e., if $C+i$ and $C+j$ share a
side. We are interested in plaquettes with a specified configuration
pattern. Each such pattern will be specified by a subset $F$ of
$\O_C$, the set of admissible configurations in $C$. For any such $F$
we define a random set $V(F)$ as follows. Let $r_1$ and $r_2$ be the
reflections of $C$ in the vertical resp.\ horizontal line in the
middle of $C$, and $r^i= r_1^{i_1} r_2^{i_2}$ the reflection
associated to $i=(i_1,i_2) \in\Z^2$. We then let
\be{V}
V(F): \s \to \{i\in\Z^2: r^i\s_{C+i}\in F\}
\ee
be the mapping associating with each $\s\in\O$ the set of
plaquettes on which $\s$ shows the pattern specified by $F$. (The
reflections $r^i$ need to be introduced for reasons of consistency:
they guarantee that two adjacent plaquettes may both belong to $V(F)$
even when $F$ is not reflection invariant, as e.g.\ the sets $G_\even
$ and $G_\odd$ below.)
We are interested in the case when $F$ is one of the following sets
of `good' configurations on $C$. These sets are distinguished
according to their occupation pattern. Describing a configuration on
$C$ by a $2\times 2$ matrix in the obvious way, we define
\bit
\item $G_\st = G_\even \cup G_\odd \equiv
\{{0\;b \choose a\;0}: 1\le a,b\le q\}\cup
\{{a\;0 \choose 0\;b}: 1\le a,b\le q\}$, the set of all
{\em staggered\/} configurations with `diagonal occupations'.
\item $G_\ord=\bigcup_{1\le a\le q} G_a\equiv
\bigcup_{1\le a\le q}\{{a\;a \choose a\;a}\}$, the set of all
fully {\em ordered\/} configurations with four particles
of the same color.
\item $G=G_\st\cup G_\ord$, the set of all good configurations.
\eit
Our first objective is to establish percolation of good plaquettes,
i.e., of plaquettes in which the configuration is good; the other
plaquettes will be called bad. We want to establish this kind of
percolation for suitable Gibbs measures \emph{uniformly in the
activity $z$} (provided $z$ is not too small). A suitable class of
Gibbs measures is that obtained by infinite-volume limits with
periodic boundary conditions.
For any integer $L\ge1$ we consider the rectangular box
\be{LaL}
\L_L=\{-12\,L+1,\ldots,12\,L\}\times \{-14\,L+1,\ldots,14\,L\}
\ee
in $\Z^2$ of size $v(L)=24\,L\times 28\, L$. (The reason for this
particular choice will become clear in the proofs of Lemmas \ref{B3}
and \ref{B2}.) We write $\mu_{L,\per}^{z,q}$ for the Gibbs
distribution in $\L_L$ with parameters $z,q$ and periodic boundary
condition, and $\G_\per(z,q)$ for the set of all limiting measures of
$\mu_{L,\per}^{z,q}$ as $L\ti$ (relative to the weak topology of
measures). The basic result is the following {\em contour estimate\/}
which shows that bad plaquettes have only a small chance to occur.
%
%\subsubsection{contour-est}
\begin{prop}\label{contour-est} For any $\d>0$ there exists a
number $q_0\in\N$ such that
\be{cont-est}
\mu(\D \cap V(G)=\emptyset) \le \d^{|\D |}
\ee
whenever $q\ge q_0$, $zq\ge q_0$, $\mu\in \G_\per(z,q)$, and $\D \subset
\Z^2$ is finite.
\end{prop}
%
In the above, $\{\D \cap V(G)=\emptyset\}$ is a short-hand for the event
consisting of all $\s$ for which all plaquettes $C+i$, $i\in \D $, are bad;
similar abbreviations will also be used below.
The proof of the proposition takes advantage of reflection positivity and
the chessboard estimate, cf.\ Corollary (17.17) of \cite{Gii}, and is
deferred to the next section. We mention here only that $q_0$ is
chosen so large that
\be{q_0}
\d(q)\equiv q^{-1/56}+q^{-1/12}+q^{-1/4}+q^{-1/2}\le\d
\ee
when $q\ge q_0$. It will be essential in the following that the
contour estimate is uniform for $z\ge q_0/q$.
As an immediate consequence of the contour estimate we obtain the
existence of a \emph{sea of good plaquettes}. We will say that a set of
plaquettes forms a sea if the set of their left lower corners is
connected and surrounds each finite set. It is then evident that the
existence of a sea of completely occupied plaquettes implies the
existence of an occupied sea; likewise, the existence of a sea of
plaquettes which are occupied on their even points implies the
existence of an even occupied $*$sea. In this way, the concept of a
sea of plaquettes is general enough to include all concepts of seas
introduced in Section \ref{results}.
Specifically, for any $F\subset \O_C$ we
define $\sea(F)$ as the largest sea in $V(F)$ whenever $V(F)$ contains
a sea, and let $\sea(F)=\emptyset$ otherwise.
For $z\ge q_0/q$ we
write $\bar\G_\per(z,q)$ for the set of all accumulation points (in
the weak topology) of measures $\mu_n\in\G_\per(z_n,q)$ with $z_n\to
z$, $z_n\ge q_0/q$. The graph of the correspondence $z\to
\bar\G_\per(z,q)$ is closed; this will be needed in the proof of
property (A2) below.
%
%\subsubsection{sea}
\begin{prop}\label{sea} For any $\e>0$ there exists a
number $q_0\in\N$ such that
\[
\mu(0\in \sea(G))\ge 1-\e
\]
whenever $\mu\in\bar\G_\per(z,q)$, $q\ge q_0$ and $z\ge q_0/q$.
\end{prop}
%
\proof Note first that the contour estimate \rf{cont-est} involves
only local events and therefore extends immediately to all $\mu\in
\bar\G_\per(z,q)$. The statement then follows directly from
Proposition \ref{contour-est} together with Lemmas (18.14) and (18.16)
of \cite{Gii}. The number $\d$ has to be chosen so small that
$4\d(1-5\d)^{-2}\le\e\;$. $\Box$
\medskip\noindent
What is the advantage of having a sea of good plaquettes?
The key property is that the sets $G_\st$ and $G_\ord$ have disjoint
side-projections. That is, writing $b=\{(0,0),(1,0)\}$ for the two points
on the bottom side of $C$ we have
\[
\s\in G_\st, \s'\in G_\ord \Rightarrow \s_b\ne\s_b'\;,
\]
and similarly for the other sides of $C$. As a consequence, if two
adjacent plaquettes are good then they are both of the same type,
either staggered or ordered. Therefore each sea of good plaquettes is
either a sea of staggered plaquettes, or a sea of ordered plaquettes.
Hence
\[
\{\sea(G)\ne\emptyset\}=\{\sea(G_\st)\ne\emptyset\}\cup
\{\sea(G_\ord)\ne\emptyset\}\;,
\]
and the two sets on the right-hand side are disjoint.
Moreover, the sets $G_\even$ and $G_\odd$ also have disjoint
side-projections, and so do the sets $G_a$, $1\le a\le q$.
Therefore, the event $\{\sea(G_\st)\ne\emptyset\}$ splits into the
two disjoint subevents $\{\sea(G_\even)\ne\emptyset\}$ and
$\{\sea(G_\odd)\ne\emptyset\}$, and $\{\sea(G_\ord)\ne\emptyset\}$
splits off into the disjoint subevents $\{\sea(G_a)\ne\emptyset\}$,
$1\le a\le q$. In other words, each sea of good plaquettes has a
characteristic occupation pattern or color corresponding to a
particular phase, and we only need to identify the activity regimes
for which the different phases do occur.
To this end we fix any $\e>0$. We will need later that $\e<1/6$. As
in the proof of Proposition \ref{sea}, we choose some $0<\d<1/25$ such
that $4\d(1-5\d)^{-2}\le\e\;$, and we let $q_0$ be so large that
condition \rf{q_0} holds for all $q\ge q_0$. For such $q_0$ and $q$
we consider the two activity domains
\[
A_\st = \Big\{z\ge q_0/q: \mu(0\in V(G_\st))\ge \mu(0\in V(G_\ord))
\mbox{ for some }\mu\in \bar\G_\per(z,q)\Big\}
\]
and
\[
A_\ord = \Big\{z\ge q_0/q: \mu(0\in V(G_\ord))\ge \mu(0\in V(G_\st))
\mbox{ for some }\mu\in \bar\G_\per(z,q)\Big\}\;.
\]
Our next result shows that these sets describe the regimes in which
staggered resp.\ ordered phases exist. The \emph{mean particle
density} $\r(\mu)$ of a measure $\mu$ which is periodic under
translations with period 2 is defined by
\be{rho}
\r(\mu)= \mu(N_C)/|C|\;;
\ee
recall that $N_C$ is the number of particles in $C$.
%
%\subsubsection{phases}
\begin{prop}\label{phases}
{\rm(a) } For each $z\in A_\st$ there exist two `staggered' Gibbs
measures $\mu_\even,\mu_\odd\in\G(z,q)$ invariant under even
translations of $\Z^2$ and permutations of particle colors.
$\mu_\even$-almost surely we have $\sea(G_\even)\ne\emptyset$, and all
occupied clusters are finite and have independently distributed random
colors. In addition, $\mu_\even(0\in\sea(G_\even))\ge1-2\e$, and in
particular $\r(\mu_\even)\le \frac12+\e$. $\mu_\odd$ has the
analogous properties.
{\rm(b) } For each $z\in A_\ord$ there exist $q$ `colored' translation
invariant Gibbs measures $\mu_a\in\G(z,q)$, $a\in\{1,\ldots,q\}$ .
Each $\mu_a$ satisfies $\mu_a(\sea(G_a)\ne\emptyset)=1$,
$\mu_a(0\in\sea(G_a))\ge1-2\e$, and in particular
has mean particle density $\r(\mu_a)\ge 1-2\e$.
\end{prop}
%
\proof (a) Let $z\in A_\st$ be given and $\mu\in\bar\G_\per(z,q)$
be such that $\mu(0\in V(G_\st))\ge \mu(0\in V(G_\ord))$. Then
$\mu(0\in V(G_\ord))\le 1/2$ and therefore
\bea
\mu\Big(0\in\sea(G_\st)\Big)&=&
\mu\Big(0\in\sea(G),\, 0 \not\in V(G_\ord)\Big)\\
&\ge&1-\e -\frac12= \frac12 -\e>0\;.
\eea
But $G_\st$ splits into
the two parts $G_\even$ and $G_\odd$ which are related to each other
by the reflection in the line $\{x_1=1/2\}$, and $\mu$ is invariant
under this reflection. Hence
\[
p\equiv \mu\Big(0\in\sea(G_\even)\Big)= \mu\Big(0\in\sea(G_\odd)\Big)\ge
\frac12\left(\frac12-\e\right)>0\;.
\]
We can therefore define the conditional probabilities
$\mu_\even=\mu(\dot|\sea(G_\even)\ne\emptyset)$ and
$\mu_\odd=\mu(\dot|\sea(G_\odd)\ne\emptyset)$. Since the events in
the conditions are tail measurable, these measures belong to $\G(z,q)$.
It is clear that these conditional probabilities inherit all common
invariance properties of $\mu$ and the conditioning events.
Moreover, we find
\bea
\mu\Big(\sea(G_\even)\ne\emptyset\Big)&=&\frac12\,
\mu\Big(\sea(G_\st)\ne\emptyset\Big)\\
&\le&
\frac12\,\mu\Big(0\in\sea(G_\st)\Big)+\frac12\,\mu\Big(0\not\in\sea(G)\Big)\\
&\le& p +\frac{\e}{2}\;,
\eea
and therefore
$$\mu_\even\Big(0\in\sea(G_\even)\Big)\ge\frac{p}{p+\e/2}\ge 1-2\e\;.$$
In particular, it follows that
\[
\r(\mu_\even) \le \frac12\; \mu_\even\Big(0\in V(G_\even)\Big) +
\mu_\even\Big(0\not\in V(G_\even)\Big)\le \frac12 + \e\;.
\]
Finally, we show that $\mu_\even$-almost surely all occupied clusters
are finite, and their colors are conditionally independent and
uniformly distributed when all particle positions are fixed. Indeed,
since $\mu_\even(\sea(G_\even)\ne\emptyset)=1$ there exists
$\mu_\even$-almost surely an odd empty $*$sea. This means that any
box $\D$ is almost surely surrounded by an empty $*$circuit. On the
one hand, this shows that all occupied clusters must be finite almost
surely. On the other hand, for any $\eta>0$ we can find a box
$\D'\supset\D$ containing an empty $*$circuit around $\D$ with
probability at least $1-\eta$. Let $\Gamma$ be the largest set with
$\D\subset\Gamma\subset\D'$ such that there are no particles on its
outer boundary $\partial\Gamma$; if no such set exists we set
$\Gamma=\emptyset$. The events $\{\Gamma=\L\}$ then depend only on
the configuration in $\Z^2\setminus \L$. By the strong Markov
property of $\mu_\even$, we conclude that on
$\{\Gamma\ne\emptyset\}$ the distribution of colors of the occupied
clusters meeting $\D$ is governed by the Gibbs distribution in
$\Gamma$ with empty boundary condition. The symmetry properties of
the latter thus imply that these colors are conditionally independent
and uniformly distributed. Letting $\eta\to 0$ and $\D\uparrow\Z^2$ we
find that this statement holds in fact for all occupied clusters.
By construction, $\mu_\odd$ is obtained from $\mu_\even$ by a
one-step translation, and thus has the analogous properties.
(b) The proof of this part is quite similar. Pick any
$z\in A_\ord$ and $\mu\in\bar\G_\per(z,q)$ such that
$\mu(0\in V(G_\ord)|0\in V(G))\ge 1/2$. Since $\mu$ is invariant under
permutations of colors it then follows in the same way that
\[
p\equiv \mu\Big(0\in\sea(G_a)\Big)\ge
\frac1q\left(\frac12-\e\right)>0\;,
\]
so that we can define the conditional probabilities $\mu_a=
\mu(\dot|\sea(G_a)\ne\emptyset)\in\G(z,q)$, $a\in\{1,\ldots,q\}$.
Also,
\[
\mu\Big(\sea(G_a)\ne\emptyset\Big)
\le
\frac1q\,\mu\Big(0\in\sea(G_\ord)\Big)+\frac1q\,\mu\Big(0\not\in\sea(G)\Big)
\le p +\frac{\e}{q}\;,
\]
whence $\mu_a(0\in\sea(G_a))\ge{p}/({p+\e/q})\ge 1-2\e$ and
$
\r(\mu_a)\ge \mu_a(0\in V(G_a))\ge 1-2\e
$.
$\Box$
\medskip%\noindent
According to the preceding proposition, Theorem \ref{th:WR} will be
proved once we have shown that there exists a critical activity
$z_c(q)\in \;]\,q/5,5q\,[$ such that $A_\st =[q_0/q,z_c(q)]$ and
$A_\ord=[z_c(q),\infty[$. To this end we will establish the following
items:
\newcounter{num}
\begin{list}{(A\arabic{num}) }{\usecounter{num}
\setlength{\leftmargin}{9ex} \setlength{\labelsep}{2ex}}
\item[(A1)] $A_\st\cup A_\ord= [q_0/q,\infty[\;$.
\item[(A2)] $A_\st$ and $A_\ord$ are closed.
\item[(A3)] $A_\ord\cap [q_0/q,{q/5}] \;=\emptyset\;$.
\item[(A4)] $A_\st\cap [5 q,\infty[ \;=\emptyset\;$.
\item[(A5)] $|A_\st\cap A_\ord|\le 1\;$.
\end{list}
Statement (A1) follows trivially from the definitions of $A_\st$ and $A_\ord$.
Assertion (A2) is also obvious because these definitions involve only
local events, and the graph of the correspondence
$z\to\bar\G_\per(z,q)$ is closed by definition.
Property (A3) corresponds to the discovery of Runnels and Lebowitz
\cite{RL} that staggered phases do exist in a nontrivial activity regime,
and follows directly from the next result.
%
%\subsubsection{stag}
\begin{lem}\label{stag}
For $z\le q/5$ and $\mu\in \bar\G_\per(z,q)$ we have
\[
\mu\Big(0\in V(G_\ord)\Big|0\in V(G)\Big) <1/2\;.
\]
\end{lem}
%
\proof Consider the Gibbs distribution $\mu_{L,\per}^{z,q}$ in
the box $\L_L$ with periodic boundary condition, and let
\be{Gord}
G_{\ord,L}=\Big\{\s\in \O_{L,\per}: \s_{C(i)}\in G_\ord
\mbox{ for all }i\in\L_L\Big\}\,;
\ee
here we write $\O_{L,\per}$ for the set of admissible configurations
in the torus $\L_L$ (including nearest-neighbor bonds between the left
and the right sides as well as between the top and bottom
sides of $\L_L$),
and $C(i)$ for the image $C+i \mbox{ mod }\L_L$ of $C$ under the
periodic shift of $\L_L$ by $i$.
(As $G_\ord$ is reflection-symmetric, we can omit the
reflections $r^i$ which appear in \rf{V}.)
The chessboard estimate (cf.\ Corollary (17.17) of \cite{Gii})
then implies that
\[
\mu_{L,\per}^{z,q}\Big(0\in V(G_\ord)\Big) \le
\mu_{L,\per}^{z,q}(G_{\ord,L})^{1/v(L)}\;.
\]
We compare the latter probability with that of the event
\be{Geven}
G_{\even,L}=\Big\{\s\in \O_{L,\per}: r^i\s_{C(i)}\in G_\even \mbox{ for
all }i\in\L_L\Big\}\;.
\ee
This gives
\[
\mu_{L,\per}^{z,q}(G_{\ord,L})\le
\mu_{L,\per}^{z,q}(G_{\ord,L})\Big/\mu_{L,\per}^{z,q}(G_{\even,L})
= z^{v(L)}q \;z^{-v(L)/2}q^{-v(L)/2}
\]
because $G_{\ord,L}$ contains only the $q$ distinct close packed
monochromatic configurations, while for $\s\in G_{\even,L}$ the
$v(L)/2$ particles can have independent colors. Taking the $v(L)$'th
root and letting $L\ti$ we find for $\mu\in\bar\G_\per(z,q)$
\[
\mu\Big(0\in V(G_\ord)\Big)\le (z/q)^{1/2}\le 5^{-1/2}<
(1-\d)/2\;.
\]
The last inequality comes from the choice of $\d$.
Since $\mu(0\in V(G))\ge 1-\d$ by Proposition \ref{contour-est},
the lemma follows.
$\Box$
\medskip%\noindent
Assertion (A4) corresponds to the well-known fact that $q$ ordered
phases exist when the activity is large. For $q=2$ this was already
shown by Lebowitz and Gallavotti \cite{LG}, and for arbitrary $q$
by Runnels and Lebowitz
\cite{RL}. This is again a simple application of the chessboard
estimate.
%
%\subsubsection{order}
\begin{lem}\label{order}
For $z\ge 5q$ and $\mu\in \bar\G_\per(z,q)$ we have
\[
\mu\Big(0\in V(G_\st)\Big|0\in V(G)\Big) <1/2\;.
\]
\end{lem}
%
\proof Let $G_{\ord,L}$ be as in \rf{Gord}, and define $G_{\st,L}$
analogously. By the chessboard estimate we find
\bea
\mu_{L,\per}^{z,q}\Big(0\in V(G_\st)\Big) &\le&
\mu_{L,\per}^{z,q}(G_{\st,L})^{1/v(L)}\\
&\le& \Big(
\mu_{L,\per}^{z,q}(G_{\st,L})\Big/\mu_{L,\per}^{z,q}(G_{\ord,L})\Big)^{1/v(L)}\\
&\le& 2^{1/v(L)}z^{1/2}q^{1/2} \;z^{-1}q^{-1/v(L)}
\eea
because $G_{\st,L}=G_{\even,L}\cup G_{\odd,L}$ contains
$2\,q^{v(L)/2}$ distinct
configurations of particle density $1/2$. We can now complete the
argument as in the preceding proof.
$\Box$
\medskip%\noindent
For the proof of (A5) we will use a thermodynamic argument, namely the
convexity of the pressure as a function of $\log z$.
For any translation invariant probability measure $\mu$ on $\O$ we
consider the {\em entropy per volume}
\[
s(\mu)=\lim_{|\L|\to\infty} |\L|^{-1}\, S(\mu_\L)\;.
\]
Here we write $\mu_\L$ for the restriction of $\mu$ to
$\O_\L$,
\[
S(\mu_\L)=-\sum_{\s\in\O_\L}\mu_\L(\s) \,\log \mu_\L(\s)
\]
is the entropy of $\mu_\L$, and the notation $|\L|\to\infty$ means that
$\L$ runs through a
specified increasing sequence of square boxes;
for the existence of $s(\mu)$ we refer to \cite{Gii,Rue}.
We define the thermodynamic \emph{pressure} by
\be{p}
P(\log z)=\max_\mu \Big[\r(\mu)\,\log z +s(\mu)\Big]\;;
\ee
the maximum extends over all
translation invariant probability measures $\mu$ on $\O$,
and $\r(\mu)=\mu(\s_0\ne 0)$ is the associated mean particle density,
cf. \rf{rho}.
(Since $\O$ is defined as the set of all admissible
configurations, the hard-core intercolor repulsion is taken into
account automatically.)
By definition, $P$ is a convex function of $\log z$, and the
variational principle (see Theorems 4.2 and 3.12 of \cite{Rue})
asserts that the maximum in \rf{p} is attained precisely on $\G_\Th(z,q)$,
the set of all translation invariaion invariant elements of $\G(z,q)$.
By standard arguments (cf.\ Remark (16.6) and Corollary (16.15) of
\cite{Gii}) it follows that $P$ is strictly convex, and
\be{tangent}
P_-'(\log z) \le \r(\mu) \le P_+'(\log z)\mbox{ for all
}\mu\in\G_\Th(z,q)\;;
\ee
here we write $P_-'$ and $P_+'$ for the left-hand resp.\
right-hand derivative of $P$.
By strict convexity, $P_-'$ and $P_+'$ are strictly
increasing and almost everywhere identical. Assertion (A5) thus
follows from the lemma below.
%
%\subsubsection{z_c}
\begin{lem}\label{z_c}
For each $z\in A_\st\cap A_\ord$ we have
$P_-'(\log z) \le 2/3 \le P_+'(\log z)$.
\end{lem}
%
\proof This has already been shown essentially in
Proposition \ref{phases}. Pick any $z\in A_\st\cap A_\ord$, and
let $\mu\in\bar\G_\per(z,q)$ be as in the
proof of Proposition \ref{phases}(a).
Consider the conditional probability
$\mu_\st=\mu(\dot|\sea(G_\st)\ne\emptyset)=\frac12\,\mu_\even+
\frac12\,\mu_\odd$. By the arguments there, $\mu_\st$ is
well-defined, belongs to $\G_\Th(z,q)$, and satisfies
$\r(\mu_\st)\le 1/2 +\e < 2/3$. On the other hand, the measures
$\mu_a$ constructed in Proposition \ref{phases}(b) also belong to
$\G_\Th(z,q)$ and satisfy $\r(\mu_a)\ge 1-2\e>2/3$. The lemma thus
follows from \rf{tangent}.
$\Box$
\medskip%\noindent
We can now complete the proof of Theorem \ref{th:WR}.
Properties (A1) to (A4) together imply that $A_\st\cap A_\ord
\ne\emptyset$. This is because the interval $[q_0/q,\infty[$ is
connected and therefore cannot be the union of two disjoint non-empty
closed sets. Combining this with (A5) we find
that $A_\st\cap A_\ord$ consists of a unique value
$z_c(q)$. In particular, $A_\ord$ cannot contain any value
$zz_c(q)$. Hence $A_\st =[q_0/q,z_c(q)]$ and
$A_\ord=[z_c(q),\infty[$, and Theorem \ref{th:WR} follows from
Proposition \ref{phases}.
\subsection{Contour estimates}
\label{sec:cont_est}
In this subsection we will prove Proposition \ref{contour-est}.
Consider the set $\O_C$ of all
admissible configurations in $C$, and the set $B=\O_C\setminus G$ of all
bad configurations in $C$. We split $B$ into the following subsets
which are distinguished by their occupation pattern:
\bit
\item $B_0 =\{{0\;0 \choose 0\;0}\}$,
the singleton consisting of the empty configuration in $C$.
\item $B_1 =
\{{0\;0 \choose a\;0}, {0\;0 \choose 0\;a}, {0\;a \choose 0\;0},
{a\;0 \choose 0\;0}: 1\le a\le q\}$,
the set of all configurations with a single particle in $C$.
\item $B_{2}=\{{a\;a \choose 0\;0},{a\;0 \choose a\;0}, {0\;0 \choose
a\;a},{0\;a \choose 0\;a} : 1\le a\le q\}$, the set of admissible
configurations for which one side of $C$ is occupied, and the other
side is empty.
\item $B_3
=\{{a\;a \choose 0\;a}, {a\;a \choose a\;0}\}, {a\;0 \choose a\;a},
{0\;a \choose a\;a}: 1\le a\le q\}$, the set of all
admissible configurations with three particles in $C$.
\eit
We then clearly have
$B = \bigcup_{k=0}^3 B_k$.
The four different kinds of ``badness'' of a plaquette
will be treated separately in the three lemmas below. We start with the most
interesting case of plaquettes with three particles.
For any $L\ge1$ and $k\in \{0,\ldots,3\}$ let
\[
B_{k,L}=\{\s\in \O_{L,\per}: \s_{C(i)}\in B_k \mbox{ for
all }i\in\L_L\}\;,
\]
where $C(i)$ is as in \rf{Gord}. Consider the quantities
$p_{k,L}^{z,q}= \mu_{L,\per}^{z,q}(B_{k,L})^{1/v(L)}$ and
$p_k^{z,q}=\limsup_{L\ti}p_{k,L}^{z,q}$.
%
%\subsubsection{B3}
\begin{lem}\label{B3}
$p_3^{z,q}\le q^{-1/56}$ for all $z>0$ and $q\in\N$.
\end{lem}
%
\proof
Fix any integer $L\ge 1$ and consider the set $B_{3,L}$ of
configurations $\s$ in $\L_L$ having a single empty site in each
plaquette. We claim that $|B_{3,L}|< q\;2^{14L+2}$. First of all, for
each $\s\in B_{3,L}$
the occupied sites in $\L_L$ form a connected set, so that all
particles have the same color. Thus there are only $q$ possible colorings,
and we only need to count the possible occupation patterns for $\s\in B_{3,L}$.
It is easy to see that the plaquettes $C(i)$ with $\s(i)=0$ form a
partition of $\L_L$. For each such partition, the
plaquettes are either arranged in rows or in
columns. In the first case, each row is determined by its parity (even or odd),
namely the parity of $i_1$ for each $C(i)$ in this row;
likewise, in the second case each column is determined by its parity.
We can therefore count all such partitions as follows. There are 4
possibilities of choosing the plaquette containing the origin. If
this plaquette is fixed, there are no more than
$2^{14L-1}$ possibilities of arranging all plaquettes in rows and choosing
the parity of each row. Similarly, there are at
most $2^{12L-1}$ possibilities of arranging the plaquettes in
columns. The number of such partitions is therefore no larger than
$4(2^{14L-1}+2^{12L-1})$, and the claim follows.
To estimate $\mu_{L,\per}^{z,q}(B_{3,L})$ we will rearrange the positions of
all particles so that many different colors become possible. More
precisely, we divide $\L_L$ into $(3L)(4L)$ rectangular cells $\D(j)$ of
size $8\times7$. Let $\D_0(j)$ be the
rectangular cell of size $7\times6$ situated in the left lower corner
of $\D(j)$, and consider the set
\[
F_{3,L}=\{\s\in\O_{L,\per}: \s\ne 0\mbox{ on }\D_0(j), \;
\s\equiv 0\mbox{ on }\D(j)\setminus\D_0(j)\mbox{ for all }j\}\;.
\]
Since $|\D(j)\setminus\D_0(j)|=8\cdot 7- 7\cdot 6= |\D(j)|/4$ for
all $j$, each $\s\in F_{3,L}$ has particle number $3\,v(L)/4$, just as
the configurations in $B_{3,L}$. As the colors of the particles in
the blocks $\D_0(j)$ can be chosen independently, we have
$|F_{3,L}|=q^{12L^2}=q^{v(L)/56}$. (The above construction, together
with a similar construction in the proof of the next lemma, explains
our choice of the rectangle $\L_L$.)
Now we can write
\[
\mu_{L,\per}^{z,q}(B_{3,L}) \le
\frac{\mu_{L,\per}^{z,q}(B_{3,L})}{\mu_{L,\per}^{z,q}(F_{3,L})}
= \frac{|B_{3,L}|}{|F_{3,L}|} \le 2^{14L+2}\; q^{1-v(L)/56}\;.
\]
The proof is completed by taking the $v(L)$'th root and letting $L\ti$.
$\Box$
\medskip%\noindent
Next we estimate the probability of plaquettes with two adjacent
particles at one side of $C$.
%
%\subsubsection{B2}
\begin{lem}\label{B2}
$p_2^{z,q}\le q^{-1/12}$ for all $z>0$ and $q\in\N$.
\end{lem}
%
\proof Fix any $L\ge1$, and let $\s\in B_{2,L}$. Then the particles
are either arranged in alternating occupied and empty rows, or in
alternating occupied and empty columns. The colors in all rows resp.\
columns can be chosen independently of each other. Hence
$|B_{2,L}|= 2(q^{14L}+q^{12L})\le 4\,q^{14L}$. Moreover, each $\s\in
B_{2,L}$ has particle number $v(L)/2$. As in the last proof, we
construct a set $F_{2,L}$ of configurations with the same particle number but
larger color entropy as follows.
We partition $\L_L$ into $(8L)(7L)$ rectangular cells $\D(j)$ of
size $3\times4$, and let $\D_0(j)$ be the
rectangular cell of size $2\times3$ in the left lower corner
of $\D(j)$. We then define
\[
F_{2,L}=\{\s\in\O_{L,\per}: \s\ne 0\mbox{ on }\D_0(j), \;
\s\equiv 0\mbox{ on }\D(j)\setminus\D_0(j)\mbox{ for all }j\}\;.
\]
Since $|\D(j)\setminus\D_0(j)|=3\cdot 4- 2\cdot 3= |\D(j)|/2$ for
all $j$, each $\s\in F_{2,L}$ has particle number $v(L)/2$.
As the particle colors in
the blocks $\D_0(j)$ can be chosen independently, we have
$|F_{3,L}|=q^{56\,L^2}=q^{v(L)/12}$. As in the last proof, we thus
find
\[
\mu_{L,\per}^{z,q}(B_{2,L}) \le
\frac{\mu_{L,\per}^{z,q}(B_{2,L})}{\mu_{L,\per}^{z,q}(F_{2,L})}
= \frac{|B_{2,L}|}{|F_{2,L}|} \le 4\; q^{14L-v(L)/12}\;.
\]
Taking the $v(L)$'th root and letting $L\ti$ we obtain the result.
$\Box$
\medskip%\noindent
Finally we consider the probability of `diluted' plaquettes with a single
or no particle.
%
%\subsubsection{B10}
\begin{lem}\label{B10}
$p_0^{z,q}\le (zq)^{-1/2}$ and $p_1^{z,q}\le (zq)^{-1/4}$ for all
$z>0$, $q\in\N$.
\end{lem}
%
\proof We consider first the case of no particle.
For each $L\ge1$ we can write
\[
\mu_{L,\per}^{z,q}(B_{0,L}) \le
\frac{\mu_{L,\per}^{z,q}(B_{0,L})}{\mu_{L,\per}^{z,q}(G_{\even,L})}
= \frac{1}{z^{v(L)/2}q^{v(L)/2}}\;,
\]
where $G_{\even,L}$ is defined by \rf{Geven}.
The identity follows from the facts that
$B_{0,L}$ contains only the empty configuration, whereas
each configuration in $G_{\even,L}$ consists of $v(L)/2$
particles with arbitrary colors. The first result is thus obvious.
Turning to the case of a single particle per plaquette, we note that
each $\s\in B_{1,L}$ consists of $v(L)/4$
particles with arbitrary colors, and
there are no more than $2^{14L+2}$ distinct occupation patterns for
these particles; the latter follows as in the proof of Lemma \ref{B3}
(by interchanging empty and occupied sites). Hence
\[
\mu_{L,\per}^{z,q}(B_{1,L}) \le
\frac{\mu_{L,\per}^{z,q}(B_{1,L})}{\mu_{L,\per}^{z,q}(G_{\even,L})}
\le \frac{2^{14L+2} z^{v(L)/4}q^{v(L)/4}}{z^{v(L)/2}q^{v(L)/2}}\;,
\]
and the second result follows by taking the $v(L)$'th root and letting
$L\ti$.
$\Box$
\medskip\noindent
{\em Proof of Proposition \ref{contour-est}. }Let $\mu\in
\G_\per(z,q)$ and a finite $\D \subset\Z^2$ be given. Then we can write
\bea
\mu(\D \cap V(G)=\emptyset) &=& \sum_{\g:\D \to\{0,\ldots,3\}}
\mu(\s: \s_{C+i}\in
B_{\g(i)} \mbox{ for all }i\in \D )\\
&\le& \sum_{\g:\D \to\{0,\ldots,3\}} \limsup_{L\ti}
\mu_{L,\per}^{z,q}(\s: \s_{C(i)}\in
B_{\g(i)} \mbox{ for all }i\in \D )\\
&\le&\sum_{\g:\D \to\{0,\ldots,3\}} \limsup_{L\ti} \prod_{i\in \D }
p_{\g(i),L}^{z,q}\\
&\le& \bigg( \sum_{k=0}^3 p_{k}^{z,q} \bigg)^{|\D |}
\;.
\eea
In the third step we have used the chessboard estimate, see Corollary (17.17)
of \cite{Gii}. Inserting the estimates of Lemmas \ref{B3}, \ref{B2}
and \ref{B10} and choosing $q_0$ as in \rf{q_0} we get the result.
$\Box$
\section{Proof of Theorem \ref{th:sqWR}}
\label{pf:sqWR}
Here we indicate how the proof of Theorem \ref{th:WR} can be adapted
to obtain Theorem \ref{th:sqWR}. First of all, the different geometry
of the present model leads to a new
classification of good and bad plaquettes: the ordered configurations
in $G_\ord$ are still good, but the (former good) configurations in
$G_\st$ are now bad and will be denoted by $B_\st$, while the
configurations in $B_1$ are now good, and we set $G_\dis=B_1$.
We first need an analog of the contour estimate,
Proposition \ref{contour-est}. Remarkably, the estimates of Lemmas
\ref{B3} and \ref{B2} carry over without any change. To deal with $B_\st$
we can proceed exactly as in Lemma \ref{B2}, noting that each
configuration in $B_{\st,L}$ is monochromatic, so that
$|B_{\st,L}|=2\,q$.
This shows that also $p_\st^{z,q}\le q^{-1/12}$. Finally, for $B_0$
we compare the set $B_{0,L}$ with
\be{F1L}
F_{1,L}=\{\s\in\O_{L,\per}: \s_i\ne 0\mbox{ iff }i\in 2\,\Z^2\}\;;
\ee
this gives $p_0^{z,q}\le
(zq)^{-1/4}$. The counterpart of Proposition \ref{contour-est} thus holds
as soon as $q_0$ is so large that $q_0^{-1/56}+
2\,q_0^{-1/12}+q_0^{-1/4}\le\d$.
With the contour estimate in hand we can then proceed as in Section
\ref{sec:compete}. Proposition \ref{phases} carries over verbatim;
the only difference is that $G_\st$ is replaced by $G_\dis$ (which is
not divided into two parts with disjoint side-projections), and
$\r(\mu_\dis)\le \frac 14(1-2\e)+2\e=\frac 14 +\frac{3\e}{2}$.
By the latter estimate, the
assumption $\e<1/6$ is slightly stronger than necessary
for adapting Lemma \ref{z_c} to the present case,
but we stick to it for simplicity.
The counterparts of Lemmas \ref{stag} and \ref{order} are obtained as
follows. On the one hand, we have the estimate
\[
\mu_{L,\per}^{z,q}(G_{\ord,L})\le
\mu_{L,\per}^{z,q}(G_{\ord,L})\Big/\mu_{L,\per}^{z,q}(F_{1,L})
\le z^{v(L)}q \;z^{-v(L)/4}q^{-v(L)/4}\;,
\]
showing that
\[
\mu\Big(0\in V(G_\ord)\Big)\le (z^3/q)^{1/4}
\le 3^{-3/4}< (1-\d)/2
\]
when $\mu\in\bar\G_\per(z,q)$ and $z\le q^{1/3}/3$. On the other hand,
as in Lemma \ref{B10} we find
\[
\mu_{L,\per}^{z,q}(G_{\dis,L})\le
\mu_{L,\per}^{z,q}(G_{\dis,L})\Big/\mu_{L,\per}^{z,q}(G_{\ord,L})
\le \frac{2^{14L+2} z^{v(L)/4}q^{v(L)/4}}{z^{v(L)}q}
\]
and therefore
\[
\mu\Big(0\in V(G_\dis)\Big)\le (q/z^3)^{1/4}
\le 3^{-3/4}< (1-\d)/2
\]
when $\mu\in\bar\G_\per(z,q)$ and $z\ge 3\,q^{1/3}$. With these
ingredients it is now straightforward to complete the proof of
Theorem \ref{th:sqWR} along the lines of Section
\ref{sec:compete} .
\section{Proof of Theorem \ref{th:WRhc}}
\label{sec:pf_hc}
Here we consider the Widom--Rowlinson model with molecular hard-core
exclusion. We look again at good configurations in plaquettes. The
set $\O_C$ of admissible configurations in $C$ splits into the
good sets
\[
G_\ord= G_\even\cup G_\odd =\bigcup_{1\le a\le q} G_{a,\even} \cup
G_{a,\odd} \equiv \bigcup_{1\le a\le q}\{\textstyle{0\;a \choose a\;0}\}
\cup \{\textstyle{a\;0 \choose 0\;a}\}
\]
of \emph{ordered staggered} configurations, the good set
\[
G_\dis =B_1=\{{\textstyle{0\;0 \choose a\;0}, {0\;0 \choose 0\;a},
{0\;a \choose 0\;0},{a\;0 \choose 0\;0}}: 1\le a\le q\}
\]
of \emph{disordered} configurations, and the only bad set $B_0$
consisting of the empty configuration. The main technical problem
which is new in the present model is that the sets $G_\ord$ and
$G_\dis$ \emph{fail} to have disjoint side-projections (although this
is the case for the sets $G_{a,\even}$ and $G_{a,\odd}$). We
therefore cannot simply consider sets of good plaquettes, but need to
consider the sets of ``good plaquettes with neighbors in the same phase''.
Accordingly, we introduce the random sets
\bea
\hat V(G_\ord)&=& \{i\in V(G_\ord):i+(1,0),i+(0,1)\in V(G_\ord)\}\;,\\
\hat V(G_\dis)&=& \{i\in V(G_\dis):i+(1,0),i+(0,1)\in V(G_\dis)\}\;,
\eea
and $\hat V(G)=\hat V(G_\ord)\cup\hat V(G_\dis)$. By definition, a
sea in $\hat V(G)$ then contains either a sea in $\hat V(G_\ord)$ or a
sea in $\hat V(G_\dis)$. To establish the existence of such a sea we
use the following contour estimate.
%
%\subsection{contour-est-hc}
\begin{prop}\label{contour-est-hc} For any $\d>0$ there exists a
number $q_0\in\N$ such that
\[
\mu(\D \cap \hat V(G)=\emptyset) \le \d^{|\D |}
\]
whenever $q\ge q_0$, $zq\ge q_0^{1/7}$, $\mu\in \G_\per(z,q)$, and
$\D \subset \Z^2$ is finite.
\end{prop}
%
\proof Let us start by introducing some notations. We consider the
sublattices
\[
\LL_{1,\even}=\{i=(i_1,i_2)\in \Z^2: i_1\mbox{ is even}\}\;,\quad
\LL_{1,\odd}= \Z^2\setminus \LL_{1,\even}\;,
\]
and their rotation images $\LL_{2,\even}$ and $\LL_{2,\odd}$ which
are similarly defined. We also introduce the horizontal
double-plaquette
\[
D_1=C\cup (C+(1,0)) =\{0,1,2\}\times\{0,1\}
\]
and the event
\[
E_1 =\Big\{ \s\in E^{D_1}: \s_C\in G_\ord,\, \s_{C+(1,0)}\in G_\dis,
\mbox{ or vice versa}\Big\}
\]
that the two sub-plaquettes of $D_1$ are good but of different type.
$E_1$ thus consists of the configurations of the form ${a\;0\;0
\choose 0\;a\;0}$ with $1\le a\le q$, and their reflection images.
In the same way, we define the vertical double-plaquette $D_2=C\cup
(C+(0,1))$ and the associated event $E_2$. With these notations we
have
\[
\Z^2\setminus \hat V(G) \subset \bigcup_{k=1}^7 W_k\;,
\]
where the random subsets $W_k$ of $\Z^2$ are given by
\bea
W_1 &=& V(B_0)\;,\quad W_2\ =\ V(B_0)-(1,0)\,,\quad W_3\ =\ V(B_0)-(0,1)\,,\\
W_4 &=& \{i\in \LL_{1,\even}: \s_{D_1+i}\in E_1 \}\;,\quad
W_5\ =\ \{i\in \LL_{1,\odd} : \s_{D_1+i}\in E_1 \}\;,\\
W_6 &=& \{i\in \LL_{2,\even}: \s_{D_2+i}\in E_2 \}\;,\quad
W_7\ =\ \{i\in \LL_{2,\odd} : \s_{D_2+i}\in E_2 \}\;.
\eea
(The sets $W_k$ are not necessarily disjoint.) So, for each
$\mu\in\G_\per(z,q)$ we can write
\be{cases}
\mu(\D\cap\hat V(G)=\emptyset) \le \sum_{\D_1\cup\ldots\cup
\D_7=\D }
\min_{1\le k\le 7}\mu(\D_k\subset W_k)\;,
\ee
where the sum extends over all disjoint partitions of $\D$.
We estimate now each term.
Consider first the case $k=1$. Just as in Lemma \ref{B10} we obtain
from the chessboard estimate
\[
\mu(\D_1\subset W_1)^{1/|\D_1|}\le \limsup_{L\ti}
\frac{\mu_{L,\per}^{z,q}(B_{0,L})^{1/v(L)}}{\mu_{L,\per}^{z,q}
(F_{1,L})^{1/v(L)}}= (zq)^{-1/4}\;,
\]
where $F_{1,L}$ is given by \rf{F1L}. The same estimate holds in the
cases $k=2,3$ because these merely correspond to a translation.
Next we turn to the case $k=4$. Let $L$ be so large that $\L_L\supset
\D_4$. Using reflection positivity in the lines through the sites
of $\LL_{1,\even}$, we conclude from the chessboard estimate that
\[
\mu_{L,\per}^{z,q}(\D_4\subset W_4)^{1/|\D_4|}\le
\mu_{L,\per}^{z,q}(E_{1,L})^{2/v(L)}
\]
for the event
\[
E_{1,L} =\Big\{ \s\in \O_{L,\per}: \s_{D_1(i)}\in E_1
\mbox{ for all }i\in\L_L\cap\LL_{1,\even}\Big\}\;.
\]
In the above, $D_1(i)$ stands for the image $D_1+i \mbox{ mod }\L_L$ of
$D_1$ under the periodic shift by $i$ of the torus $\L_L$.
Each $\s\in E_{1,L}$ has the following structure: every fourth
vertical line (with horizontal coordinate either 0 or 2
modulo 4) is empty, and on each group of three vertical lines
between these empty lines every second site is occupied, with the
coordinates of occupied sites being either even-odd-even in these
three lines, or odd-even-odd; see the figure below.
\scriptsize
\[
\ba{*{9}{c}}
\c & \b & \c & \b & \c & \c & \b & \c & \c \\
\c & \c & \b & \c & \c & \b & \c & \b & \c \\
\c & \b & \c & \b & \c & \c & \b & \c & \c \\
\c & \c & \b & \c & \c & \b & \c & \b & \c \\
\c & \b & \c & \b & \c & \c & \b & \c & \c \\
\c & \c & \b & \c & \c & \b & \c & \b & \c \\
\c & \b & \c & \b & \c & \c & \b & \c & \c
\ea
\]\normalsize
Of course, the interaction implies that
the color of particles is constant in each of these groups of three
vertical lines. Consequently, each such $\s$ has particle number
$3v(L)/8$, and $|E_{1,L}| = 2 (2q)^{6L}$; recall the definition
\rf{LaL} of $\L_L$.
We now make a construction similar to that in Lemma \ref{B3}. We
divide $\L_L$ into $12L^2$ rectangular cells $\D(j)$ of size
$8\times7$. Let $\D_0(j)$ be the rectangular cell of size $7\times6$
situated in the left lower corner of $\D(j)$, and consider the set
\[
F_L=\{\s\in\O_{L,\per}: \s_i\ne 0\mbox{ iff }i_1+i_2
\mbox{ is even and }i\in\D_0(j)\mbox{ for some }j\}\;.
\]
Since $|\D(j)\setminus\D_0(j)|=|\D(j)|/4$ for all $j$, each $\s\in
F_L$ has particle number $3\,v(L)/8$, just as the configurations in
$E_{1,L}$. As the colors of the particles in the blocks $\D_0(j)$ can
be chosen independently, we have $|F_L|=q^{12L^2}=q^{v(L)/56}$. Hence
\[
\mu_{L,\per}^{z,q}(E_{1,L}) \le
\frac{\mu_{L,\per}^{z,q}(E_{1,L})}{\mu_{L,\per}^{z,q}(F_L)}
= \frac{|E_{1,L}|}{|F_L|} \le 2 (2q)^{6L}\; q^{-v(L)/56}
\]
and therefore, by taking the $2/v(L)$'th power and letting $L\ti$,
we obtain
\[
\mu(\D_4\subset W_4)^{1/|\D_4|}\le q^{-1/28}\;.
\]
The same estimate holds in the cases $k=5,6,7$, as these are obtained
by a translation or interchange of coordinates.
We now combine all previous estimates as follows. Let $q_0$ be so
large that $7\, (q_0^{-1/28})^{1/7}<\d$, and suppose that $q\ge q_0$
and $zq\ge q_0^{1/7}$. Then $\mu(\D_k\subset W_k)\le
q_0^{-|\D_k|/28}$ for all $k$ and thus, in view of \rf{cases} and
since $|\D_k|\ge |\D|/7$ for at least one $k$,
\[
\mu(\D\cap\hat V(G)=\emptyset) \le
\sum_{\D_1\cup\ldots\cup \D_7=\D } (q_0^{-1/28})^{|\D|/7}
<\d^{|\D|}.
\]
The proof of the contour estimate is therefore complete.
$\Box$
\medskip
To prove Theorem \ref{th:WRhc} we can now proceed as in Section
\ref{sec:compete}. Let $\hat\sea(G)$ be the largest sea in $\hat V(G)$
if the latter contains a sea, and $\hat\sea(G)=\emptyset$ otherwise.
It is then immediate that a counterpart of Proposition \ref{sea}
holds, and the definition of $\hat V(G)$ implies that
\[
\{\hat\sea(G)\ne\emptyset\}=\{\hat\sea(G_\dis)\ne\emptyset\}\cup
\{\hat\sea(G_\ord)\ne\emptyset\}\;,
\]
where the two sets on the right-hand side are disjoint. Moreover,
\[
\{\hat\sea(G_\ord)\ne\emptyset\}\subset
\bigcup_{a=1}^q\{\sea(G_{a,\even})\ne\emptyset\}
\cup \{\sea(G_{a,\odd})\ne\emptyset\}\;.
\]
By the argument of Proposition \ref{phases} we thus obtain the
existence of $2q$ ordered phases (as described in Theorem
\ref{th:WRhc}(i)) whenever $z$ is such that
$\mu(0\in V(G_\ord))\ge \mu(0\in V(G_\dis))$
for some $\mu\in \bar\G_\per(z,q)$, and the existence of a disordered
phase $\mu_\dis$ whenever the reverse inequality holds for such a $\mu$.
We have $\r(\mu_\dis)\le \frac 14 + \frac{3\e}{2}$ and
$\r(\mu_{a,\even})= \r(\mu_{a,\odd})\ge \frac 12 -\e$.
The topological argument of Section \ref{sec:compete} together with
obvious counterparts of Lemmas \ref{stag} and \ref{order} then show
that both cases must occur simultaneously for some $z=z_c(q)$, and
this $z$ is unique by the convexity argument of Lemma \ref{z_c}.
(For the latter we need to assume that $\e<1/10$.)
We conclude this section with a comment on the model with
nearest-particle color repulsion and a molecular hard-core exclusion
between next-nearest neighbors.
\medskip\noindent
{\bf Comment on Remark \ref{th:WRhc} (3).}
If the r\^oles of $\Phi$ and $U$ are interchanged, the good ordered
configurations in
$C$ are those with two particles of the same color on one side
of $C$, and no particle on the opposite side; we call this set again
$G_\ord$. For large $z$, one can
easily establish a contour estimate implying the existence of a sea
$S(G_\ord)$, and thus by symmetry also the existence of the
four phases mentioned in Remark \ref{th:WRhc} (3). The disordered good
plaquettes are again described by the set $G_\dis$.
As in the case of the Hamiltonian \rf{Ham_hc}, the sets $G_\ord$ and
$G_\dis$ have no disjoint side-projections. However, whereas in
that case we were able to show an entropic disadvantage
in having adjacent $G_\ord$- and
$G_\dis$-plaquettes, this is not true in the present case. The
configurations resulting from iterated reflections of a double
plaquette of type ${\c\,\c\,\c\choose\b\,\b\,\c}$ have the maximal
entropy possible for this particle number. Therefore the system
can freely combine ordered and disordered
plaquettes, and our argument for a first-order transition breaks down.
So it seems likely that the transition from the ordered to the
disordered phase is of second order.
\section{Proof of Theorem \ref{th:contspin}}
\label{sec:pf_cs}
The analysis of the plane-rotor Widom--Rowlinson model is very similar
to that of the standard Widom--Rowlinson model; only a few
modifications are necessary. We define again the set $\O_C$ of
admissible configurations in the plaquette $C$ in the obvious way,
introduce the sets $G_\st=G_\even\cup G_\odd$ as in Section
\ref{sec:compete} (replacing $\{1,\ldots,q\}$ by $S^1$), and set
$G_\ord=\{\s\in \O_C: \s_i\in S^1 \mbox{ for all }i\in C\}$ and
$G=G_\ord\cup G_\st$. The main task is to obtain a counterpart of the
contour estimate, Proposition \ref{contour-est}. To this end we
consider the same classes $B_k$, $k\in\{0,\ldots,3\}$ of bad
configurations as in Section \ref{sec:cont_est} (with the obvious
modifications), and the sets $B_{k,L}$ and the associated quantities
$p_k^{z,\a}$.
To deal with the case $k=3$ we proceed as in Lemma \ref{B3}, arriving
at the inequality
\[
\mu_{L,\per}^{z,\a}(B_{3,L}) \le
\frac{\mu_{L,\per}^{z,\a}(B_{3,L})}{\mu_{L,\per}^{z,\a}(F_{3,L})}
= \frac{\nu^\L(B_{3,L})}{\nu^\L(F_{3,L})}\;.
\]
Now, $\nu^\L(B_{3,L})\le 2^{14L+2} (2\a)^{3\,v(L)/4-1}$;
the first factor estimates the number of possible occupation
patterns, and the second term bounds the probability that the
configuration is admissible (by keeping only the bonds in a tree
spanning all occupied positions). On the other hand,
$\nu^\L(F_{3,L})\ge (\a^{7\cdot 6-1})^{v(L)/56}$, as can be seen
by letting the spins in each block $\D_0(k)$ follow a ``leader spin'' up
to the angle $2\pi\a/2$. Hence
\[
\frac{\nu^\L(B_{3,L})}{\nu^\L(F_{3,L})}\le
2^{14L+2}\;2^{3\,v(L)/4}\;\a^{v(L)/56-1}
\]
and therefore $p_3^{z,\a}\le 2^{3/4}\,\a^{1/56}$.
In the case $k=2$ we proceed as in the proof of Lemma \ref{B2}. On the one
hand,
\[
\nu^\L(B_{2,L})\le
2((2\a)^{24\,L-1})^{14\,L}+2((2\a)^{28\,L-1})^{12\,L}
\le 4 (2\a)^{v(L)/2-14L}
\]
since the spins are ordered in separate rows or columns, and $2\a<1$. On the
other hand, $\nu^\L(F_{2,L})\ge (\a^{2\cdot 3-1})^{v(L)/12}$ by the
same argument as above. Hence
\[
\mu_{L,\per}^{z,\a}(B_{2,L}) \le
\frac{\nu^\L(B_{2,L})}{\nu^\L(F_{2,L})}
\le 4\cdot 2^{v(L)/2}\; \a^{v(L)/12}\; (2\a)^{-14L}
\]
and therefore $p_2^{z,\a}\le 2^{1/2}\,\a^{1/12}$.
Finally, for $k=0$ we obtain %as in Lemma \ref{B10}
\[
\mu_{L,\per}^{z,\a}(B_{0,L}) \le
\frac{\mu_{L,\per}^{z,\a}(B_{0,L})}{\mu_{L,\per}^{z,\a}(G_{\even,L})}
= \frac{1}{z^{v(L)/2}}
\]
and thus $p_0^{z,\a}\le z^{-1/2}$. Likewise, in the case $k=1$ we get
as in Lemma \ref{B10}
\[
\mu_{L,\per}^{z,\a}(B_{1,L}) \le
\frac{\mu_{L,\per}^{z,\a}(B_{1,L})}{\mu_{L,\per}^{z,\a}(G_{\even,L})}
\le \frac{2^{14L+2}z^{v(L)/4}}{z^{v(L)/2}}
\]
and thereby $p_1^{z,\a}\le z^{-1/4}$. Combining these estimates as in the
proof of Proposition \ref{contour-est} we arrive at the counterpart
of \rf{cont-est} as soon as $\a_0$ is so small that
\[
2^{3/4}\,\a_0^{1/56}+2^{1/2}\,\a_0^{1/12}+\a_0^{1/4}+\a_0^{1/2}\le\d
\]
and $\a\le\a_0$, $z\ge 1/\a_0$.
To complete the proof of Theorem \ref{th:contspin} as in Section
\ref{sec:compete} we still need to adapt Lemmas \ref{stag} and
\ref{order}. Writing
\[
\mu_{L,\per}^{z,\a}(G_{\ord,L})\le
\frac{\mu_{L,\per}^{z,\a}(G_{\ord,L})}{\mu_{L,\per}^{z,\a}(G_{\even,L})}
\le \frac{z^{v(L)}(2\a)^{v(L)-1} }{z^{v(L)/2}}
\]
we find that for $z\le \a^{-2}/18$ and $\mu\in\bar\G_\per(z,\a)$
\[
\mu\Big(0\in V(G_\ord)\Big)\le z^{1/2}2\a\le 2/\sqrt{18} <
(1-\d)/2\;.
\]
Likewise, since
\[
\mu_{L,\per}^{z,\a}(G_{\st,L})\le
\frac{\mu_{L,\per}^{z,\a}(G_{\st,L})}{\mu_{L,\per}^{z,\a}(G_{\ord,L})}
\le \frac{2\;z^{v(L)/2} }{z^{v(L)}\,\a^{v(L)-1}}\;,
\]
we see that for $z\ge 5\,\a^{-2}$ and $\mu\in\bar\G_\per(z,\a)$
\[
\mu\Big(0\in V(G_\st)\Big)\le z^{-1/2}\a\inv <
(1-\d)/2\;.
\]
The remaining arguments of Section \ref{sec:compete} can be taken over
with no change to prove Theorem \ref{th:contspin}.
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\end{document}
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