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\topmatter
%
%
\leftheadtext\nofrills {\draftfont M.~Biskup, L.~Chayes,
R.~Koteck\'y } \rightheadtext\nofrills {\draftfont Continuity in
2d Ferromagnets}
%
%
\title
On the Continuity of the Magnetization \\
and the Energy Density for Potts Models \\ on Two-dimensional Graphs
\endtitle
\author
M.~Biskup\footnotemark"${}^{\dag}$",
L.~Chayes\footnotemark"${}^{\ddag}$"\
and
R.~Koteck\'y\footnotemark"${}^{\flat}{}^{\natural}$"
\endauthor
\footnotetext"${}^\dag$"{Microsoft Research, Redmond}
\footnotetext"${}^\ddag$"{Department of Mathematics, UCLA,
Los Angeles.}%; partly supported by the NSA under grant MDA904-98-1-0518}
\footnotetext"${}^\flat$"{Center for Theoretical Study, Charles
University, Prague.} \footnotetext"${}^\natural$"{Department of
Theoretical
Physics, Charles University, Prague.}%;
\address
Marek Biskup \hfill
\newline
Microsoft Research\hfill\newline One Microsoft Way, \hfill\newline
Redmond WA 98052
\endaddress \email biskup\@microsoft.com \endemail
\address Lincoln Chayes \hfill\newline
Department of Mathematics, UCLA, \hfill\newline Los Angeles,
California 90095-1555
\endaddress
\email lchayes\@math.ucla.edu \endemail
\address
Roman Koteck\'y \hfill\newline Center for Theoretical Study,
Charles University,\hfill\newline Jilsk\'a 1, 110 00 Praha 1,
Czech Republic \hfill\newline \phantom{18.}and \hfill\newline
Department of Theoretical Physics, Charles
University,\hfill\newline V~Hole\v sovi\v ck\'ach~2,
180~00~Praha~8, Czech Republic \endaddress
\email kotecky\@cucc.ruk.cuni.cz \endemail
\abstract
We consider the $q$-state Potts model on two-dimensional
planar graphs. Our only assumptions concerning the graph and
interaction are that the associated graphical representations
satisfy the conclusion of the theorem of Gandolfi, Keane and Russo
\cite{GKR}. In addition to $\Bbb Z^2$, the class of graphs we
consider contains, for example, the triangular, honeycomb, and
Kagom\'e lattices. Under these conditions
we show that the only possible point of discontinuity of the
magnetization and the energy density is at the onset of the
magnetic ordering transition (i.e., at the threshold for bond
percolation in the random-cluster model). The result generalizes
to any model with a natural dual, appropriate FKG monotonicity
properties and a percolation characterization of the Gibbs
uniqueness.
\endabstract
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\document
\baselineskip = 14pt \head 1. Introduction
\endhead
In a recent paper \cite{BC} coauthored by one of us, the
continuity of the energy density in the Potts
(and generalized Ashkin-Teller)
ferromagnets on $\Bbb Z^2$ was investigated.
As is well known, the Potts model on the square lattice is
self-dual. Using this property
it was established that if
there is any discontinuity in the energy density, this must take place at
the point where the model and its dual ostensibly coincide (i.e.,
at the self-dual point). Whenever such a discontinuity actually
happens, the self-dual point was proved to coincide with the
magnetic ordering transition. Thus, away from the transition point
the energy is continuous. (The opposite case, where the energy
density is presumed
to be continuous even at the self-dual point
was investigated in \cite{CS}, where partial information about the
{\it critical\/} nature of the transition was obtained.)
Here we investigate the problem in a general two-dimensional
context. We consider Potts models on two-dimensional planar graphs
that
satisfy certain hypotheses to be detailed below. For these
systems, there is always a well defined ordering temperature,
whose inverse we denote by $\beta_t$. The quantity $\beta_t$ is
characterized by ``onset of spontaneous magnetization'' and/or
``onset of percolation'' in the graphical representation. We show
that for all $\beta$ except possibly $\beta_t$ the energy density
is continuous. As a corollary, we obtain that for all
$\beta>\beta_t$, there are exactly $q$ pure phases (for
$\beta<\beta_t$ there is a unique Gibbs state \cite{ACCN}). Then,
as our final result, we show that for all temperatures below
$\beta_t^{-1}$ the magnetization is also continuous in $\beta$.
Our setting is genuinely two-dimensional, no self-duality is
required. Thus,
above and beyond the square lattice,
a large class of two-dimensional graphs can be treated.
In order to ensure that both the magnetization and the energy density
are well defined, we restrict ourselves to planar
graphs that arise from Bravais lattices.
Given an {\it elementary cell\/} $\Bbb V$
(i.e., a finite connected graph in $\Bbb R^2$),
the latter arise as orbits of $\Bbb V$ under the action of an infinite
group isomorphic to $\Bbb Z^2$.
The only other (key)
ingredient is the assumption is that
the conclusion of Gandolfi-Keane-Russo's theorem \cite{GKR} holds.
%Thus
Letting
$\PP$ denote the set of graphs we consider,
$\PP$ includes all graphs $\GG$ such that
\roster
\item"$\bullet$"
$\GG$ is a {\it planar Bravais lattice\/} with a basis (for a definition see e.g\. \cite{AM}, page 75).
\endroster
Furthermore, the interaction (Hamiltonian) must be such that
\roster
\item"$\bullet$"
The associated
%%
% Potts
% Redundant.
%%
random cluster model on $\GG$ excludes the simultaneous
percolation of occupied bonds and dual percolation of vacant
bonds.
\endroster
As recently remarked by Georgii and Higuchi \cite{GH},
Zhang's argument streamlining the proof
of the GKR-theorem (as described in \cite{GHM} based on its
reproduction in the forthcoming book \cite{G1})
can be extended to rather general planar graphs.
Thus, in particular, $\PP$ includes
any planar Bravais lattice that is invariant under reflections through
the horizontal and vertical axes; i.e., the triangular, honeycomb,
Kagom\'e, dice and octagonal lattices, to name just a few.
Moreover, models with asymmetric couplings can also be considered,
provided the reflection symmetry is preserved.
The remaining sections of this paper are organized as follows: In the
next section we collect the necessary facts about the Potts model and
the associated random cluster models. In the third section we state
and prove the theorem (Theorem A) on the continuity of the energy density on
any graph in $\PP$ as well as some
corollaries on
%%
% concerning
% Either this goes or the "on" below has to go.
%%
the number of phases below the ordering transition and on
the location of the transition line in the asymmetric model
on $\Bbb Z^2$.
The last section is devoted to the proof of continuity of the
magnetization (Theorem B).
\head 2. The Potts models
\endhead
Consider a two-dimensional graph $\GG=(\Bbb S,\Bbb B)$, where
$\Bbb S$ denotes the vertex and $\Bbb B$ the edge sets,
respectively. The Potts Hamiltonian is given by
$$
\HH=-\sum_{\langle i,j\rangle\in\Bbb B} J_{ij}
\delta_{\sigma_i,\sigma_j},
\tag 2.1
$$
with $J_{ij}>0$ and $\sigma_i\in\{1,\dots,q\}$ the usual Potts
variables.
Let $\LL\subset\GG$ be a subgraph consisting of a collection of
vertices $\Bbb S_{\LL}$ together with the set $\Bbb B_{\LL}$ of
all edges connecting pairs of vertices in $\Bbb S_{\LL}$. We use
the notation ${\LL}\Subset\GG$ to indicate that $\LL\subset\GG$ is
finite and connected.
We define $\partial \Bbb S_{\LL}$ to be the sites in $\Bbb
S_{\LL}^c$ with at least one neighbor in $\Bbb S_{\LL}$:
$$
\partial\Bbb S_{\LL}=\{j\in\Bbb S_{\LL}^c|
\exists\langle i,j\rangle\in\Bbb B\text{ for some }
i\in\Bbb S_{\LL}\}.
\tag 2.2
$$
The Gibbs measure on $\LL\Subset\GG$ with boundary conditions
$\boldsymbol\sigma_{\partial\Bbb
S_{\LL}}=\{\sigma_i|i\in\partial\Bbb S_{\LL}\}$ is defined by the
usual Boltzmann weights in accord with the Hamiltonian $\HH$. If
$\#$ is a boundary condition, or a convex combination thereof, we
denote the corresponding finite volume Gibbs measure by
$\langle-\rangle^\#_{\beta \HH;\LL}$. Of additional interest are
the free boundary conditions (denoted by $\#=f$) which are
obtained by setting the $J_{ij},$ $i\in\Bbb S_{\LL}$, $j\in\Bbb
S_{\LL}^c$, to zero.
Let $R_{ij}=e^{\beta J_{ij}}-1$ and let $\LL\Subset\GG$. For a
given
boundary condition $\#$ that is provided by a {\it
fixed\/} spin configuration at $\partial\Bbb S_{\LL}$, the
associated random cluster measure in $\LL$ is defined by the
relation
$$
\mu_{\beta \HH;\LL}^\#(\boldsymbol\omega)\varpropto
\biggl(\prod_{\langle i,j\rangle\in\boldsymbol\omega}
R_{ij}\biggr)
q^{\CC^\#(\boldsymbol\omega)}\chi^\#(\boldsymbol\omega). \tag 2.3
$$
Here $\boldsymbol\omega$ is a subset of the edges of $\LL$
together with the boundary edges (namely,
$\boldsymbol\omega\subset\{\langle i,j\rangle|i\in\Bbb S_{\LL},
j\in\Bbb S\}$, while $\CC^\#$ and $\chi^\#$ are defined as
follows: Let $\partial\Bbb S_{\LL}^{(1)},\dots,\partial\Bbb
S_{\LL}^{(q)}$ denote the set of sites of $\partial\Bbb S_{\LL}$
where the spins take on the values $1,\dots, q$, respectively.
Then $\chi^\#(\boldsymbol\omega)$ is zero if there is a connection
between any of these components and it is one otherwise. Given
that $\chi^\#(\boldsymbol\omega)=1$, $\CC^\#(\boldsymbol\omega)$
is the number of connected components of $\boldsymbol\omega$
(including the isolated sites in $\Bbb S_{\LL}$), counting each
connected component of each $\partial\Bbb
S_{\LL}^{(1)},\dots,\partial\Bbb S_{\LL}^{(q)}$ as a single
component. In general, we may also consider any convex combination
of such measures.
Every (infinite volume) spin Gibbs measure $\langle-\rangle_{\beta
\HH}^\#$ has a natural random cluster counterpart. Namely, this
can be directly seen from the construction $\rho^\#_{\beta
\HH|\LL}(-)= \bigl\langle \mu_{\beta
\HH;\LL}^\circleddash(-)\bigr\rangle_{\beta \HH}^\#$, where
$\circleddash$ denotes the {\it spin\/} boundary condition of the
above type that is subject to the expectation under the state
$\langle-\rangle_{\beta \HH}^\#$. The random cluster measures
$(\rho^\#_{\beta \HH|\LL})_{\LL\Subset\GG}$ form a consistent
family (meaning that the projection of $\rho^\#_{\beta \HH|\LL}$
onto any $\LL^\prime\subset\LL$ is precisely $\rho^\#_{\beta
\HH|\LL^\prime}$), hence they are finite-volume projections of a
unique infinite-volume random cluster measure $\rho^\#_{\beta
\HH}$. The relation between $\langle-\rangle_{\beta \HH}^\#$ and
$\rho^\#_{\beta \HH}$ can be characterized as follows: Every event
of the type $\{\omega_b=1|b=b_1,\dots,b_n\}$ yields the same
expectation under $\langle-\rangle_{\beta\HH}^\#$ as does the
product of functions $\frac{R_b}{1+R_b}\delta_{\sigma_i\sigma_j}$,
$b=\langle i,j\rangle=b_1,\dots,b_n$, under $\rho^\#_{\beta \HH}$.
An alternative route to this relationship is by using the
Edwards-Sokal Gibbs measures as worked out in \cite{BBCK}.
Among all possible random cluster measures, of a particular
interest are the ones generated by the free and wired boundary
conditions ($\#=f,w$). The latter are defined by setting
$\chi^f(\boldsymbol\omega)=\chi^w(\boldsymbol\omega)\equiv1$, and
interpreting the quantity $\CC^f(\boldsymbol\omega)$ as the usual
number of components while $\CC^w(\boldsymbol\omega)$ is counting
all clusters attached to the boundary as a single component. The
free measure corresponds to the free boundary condition in the
spin-system, whereas the wired measure corresponds to all boundary
spins set to the same spin state $r$, i.e., $\partial\Bbb
S_{\LL}^{(s)}=\emptyset$ for $s\not=r$ and $\partial\Bbb
S_{\LL}^{(r)}=\partial\Bbb S_{\LL}$.
We state without proof the following results that will be needed
in subsequent developments. The proofs of these results can be found in (or easily extended from)
\cite{ACCN, G2} and the various other references stated.
\roster
\widestnumber\item{\!\!\!}%\!\!\!}
\item"{(i)}" In the partial order defined by putting occupied bond above vacant one, the free and wired
measures are (strong) FKG with
$$
\mu_{\beta \HH;\LL}^w\underset{\FKG}\to\ge \mu_{\beta \HH;\LL}^f.
\tag 2.4
$$
Further, for any boundary condition $\#$ as described above,
$$
\mu_{\beta \HH;\LL}^w\underset{\FKG}\to\ge \mu_{\beta \HH;\LL}^\#,
\tag 2.5
$$
even if the latter measure is not FKG.
\item"{(ii)}"
For the free and wired cases, infinite volume (weak) limits exist:
if $({\LL}_k)$ is a sequence of volumes with
${\LL}_k\subset{\LL}_{k+1}\Subset \GG$ which eventually exhaust
the entire graph $\GG$, limiting measures emerge independent of
the details of $({\LL}_k)$. The limiting objects will be denoted
by $\mu_{\beta \HH}^w$ and $\mu_{\beta \HH}^f$.
In the cases at hand, where the lattice can be described as a Bravis lattice
with a basis and the couplings are invariant under translations by the
primitive (generating) vectors, then the measures
$\mu_{\beta \HH}^w$ and $\mu_{\beta \HH}^f$ are also invariant under these
translations.
\item"{(iii)\!\!}"
Percolation in these models is defined in the strongest possible
sense. Supposing that a site $x$ is contained in $\Bbb
S_{{\LL}_k}$, let $P^\beta_{\LL_k}(x)=\mu_{\beta
\HH;{\LL}_k}^w(x\leftrightarrow\partial\Bbb S_{\LL_k})$ denote the
probability that the origin is connected to the boundary in the
wired state. Then the limit
$$
P^\beta_{\infty}(x)=\lim_{k\to\infty} P^\beta_{\LL_k}(x) \tag 2.6
$$
exists and is independent of the sequence $({\LL}_k)$. Moreover,
even though $P^\beta_{\infty}(x)$ in principle depends on $x$, the
positivity of this quantity does not: Either
$P^\beta_{\infty}(x)>0$ for all $x\in\Bbb S$ or
$P^\beta_{\infty}(x)=0$ for all $x\in\Bbb S$. The necessary and
sufficient condition for unicity of the Gibbs state is that
$P_\infty^\beta(0)=0$ (see \cite{ACCN}). Under the condition that
$P_\infty^\beta(0)>0$, the spontaneous magnetization defined as
$$
\goth m(\beta)=\frac\partial{\partial\goth h}\Bigl(
\lim_{k\to\infty}\frac1{|\Bbb S_{{\LL}_k}|}
\log\sum_{\boldsymbol\sigma}
e^{-\beta\HH_{{\LL}_k}(\boldsymbol\sigma)+\goth h
{\textstyle\sum_{i\in\Bbb S_{{\LL}_k}}}
\delta_{\sigma_i,1}}\Bigr)\Big|_{\goth h=0^+}
\tag 2.7
$$
is also strictly positive. (Here $\HH_{{\LL}_k}$ is the
Hamiltonian restricted to $\LL_k$ and $\boldsymbol\sigma$ denotes
spin configurations in $\Bbb S_{\LL_k}$.) In fact, $\goth
m(\beta)=P_\infty^\beta(0)$ for the homogeneous cases (i.e.,
$J_{ij}=J$ independent of $i,j$ and $\GG$ a homogeneous graph). In
general, if $\GG\in\PP$, then
$$
\goth m(\beta)=\frac1{|\Bbb V|}\sum_{x\in \Bbb
V}P_\infty^\beta(x),
\tag 2.8
$$
where $\Bbb V$ is the elementary lattice cell.
\item"{(iv)\kern-3pt}"
For any $\beta_1<\beta_2$, we have
$$
\mu_{\beta_1 \HH}^w\le \mu_{\beta_2 \HH}^f.
\tag 2.9
$$
The proof is based on a ``free energy'' argument and can be found
in \cite{G2,BCK,BBCK}. In conjunction with the FKG domination
stated in (i), this FKG bound implies that $\beta\mapsto
P_\infty^\beta(0)$ and, consequently, $\beta\mapsto\goth m(\beta)$
are monotone increasing (in fact $\beta\mapsto P_\infty^\beta(0)$
is right continuous). Hence $\beta_t=\inf\{\beta|\goth
m(\beta)>0\}$ is a well-defined percolation threshold. In
particular, we have that $\mu_{\beta_2
\HH}^f(x\leftrightarrow\infty)\ge \mu_{\beta_1
\HH}^w(x\leftrightarrow\infty)$, so the free measure at
$\beta>\beta_t$ exhibits percolation almost surely.
\item"{(v)}"
Using Strassen's theorem (see \cite{St} or \cite{L}, page 75),
the FKG domination bound from (i) implies that $\mu_{\beta
\HH}^f=\mu_{\beta \HH}^w$ whenever $\mu_{\beta
\HH}^f(\omega_b=1)=\mu_{\beta \HH}^w(\omega_b=1)$ for all bonds
$b$. The same argument applies when $\mu_{\beta \HH}^f$ is
replaced by any (subsequential-)limiting state~$\mu_{\beta
\HH}^\#$.
\item"{(vi)\!\kern-1pt}"
Let $\langle i,j\rangle$ be a nearest-neighbor bond. There is a
one-to-one correspondence between the energy density $\goth
e^\#_{ij}(\beta)$ in the state $\langle-\rangle_{\beta \HH}^\#$
and the bond density $\goth b^\#_{ij}(\beta)$ in the corresponding
random-cluster measure $\mu_{\beta \HH}^\#$, with these quantities
defined as
$$
\aligned \goth
e^\#_{ij}(\beta)&=-J_{ij}\bigl\langle\delta_{\sigma_i,\sigma_j}\bigr\rangle_{\beta
\HH}^\#\\ \goth b^\#_{ij}(\beta)&=\mu_{\beta
\HH}^\#(\omega_{ij}=1).
\endaligned
\tag 2.11
$$
The relation reads
$$
\goth b^\#_{ij}(\beta)=-\frac 1{J_{ij}}\frac
{R_{ij}}{1+R_{ij}}\goth e^\#_{ij}(\beta) \tag 2.12
$$
(for a derivation see \cite{MCLSC}).
It follows by convexity of the free energy that both
$\beta\mapsto \goth b^\#_{ij}(\beta)$ and $\beta\mapsto \goth
e^\#_{ij}(\beta)$ are continuous except for countably many values
of $\beta$. The overall energy density $\goth e(\beta)$ is defined
by averaging the values of $\goth e^\#_{ij}(\beta)$ over the bonds
in the elementary lattice cell $\Bbb V$ and taking the supremum
over all boundary conditions $\#$. By convexity arguments, $\goth
e(\beta)$ can alternatively be defined as the right-derivative of
the free energy with respect to $\beta$. At the points of
continuity of the latter, $\goth e(\beta)$ is the energy density
for all states.
\item"{(vii)\!\kern-2pt}"
Let $\Cal G^*$ denote the dual graph of $\Cal G$ (note that $\Cal G^*$ exist
because $\Cal G$ is planar). We denote by
$\boldsymbol\omega^*$ the complementary configuration of
$\boldsymbol\omega$, where $\boldsymbol\omega^*$ is occupied at
the dual bond whenever the direct bond is vacant in
$\boldsymbol\omega$ and vice versa. Then, in the case of free and
wired boundary conditions, the dual of a random cluster measure is
again a random cluster measure with parameters
$$
R^*_{ij}=\frac q{R_{ij}} \tag 2.13
$$
and with free and wired boundary conditions interchanged.
\endroster
\head 3. Continuity of the energy density
\endhead
We begin by extending a theorem established in \cite{BC} for the usual Potts model on $\Bbb Z^2$ that was proved
using the {\it
self\/}-duality of the lattice. It turns
out that the reference to self-duality is not essential; it only
matters that the model has a natural dual that can readily be
analyzed.
\proclaim{Theorem A} For the two-dimensional Potts model on any
graph $\Cal G\in\PP$, the overall energy density $\goth e(\beta)$
is continuous at every $\beta \neq \beta_t$.
\endproclaim
\demo{\sserif Proof} First note that, whenever a discontinuity in
the energy density occurs, there are (at least) two coexisting
pure spin states, one with the higher and the other with the lower
value of the energy density, as follows by a limiting argument for
which we refer e.g\. to \cite{BC} or any standard textbook on
statistical mechanics.
If $\beta<\beta_t$, which means no percolation, the
claim in (iii) implies uniqueness of the spin Gibbs state. This
rules out discontinuity in this region.
For $\beta>\beta_t$, let us consider the dual system. This is a
similar Potts spin system, however, now at dual values of the
parameters, which correspond to high temperatures. Indeed, for any
$\beta>\beta_t$, there is percolation in the free measure (as was
observed in statement (iv) above). Invoking the \cite{GKR} result,
which asserts that percolation implies finiteness of all
connected components of ``dual-to-vacant'' bonds, we have that there is no
percolation in the wired state of the dual model. The latter rules
out percolation in all states, hence, the dual model has a unique
Gibbs state and thus a continuous energy/bond density. Since the
free energies of the model and its dual are equal up to an analytic factor,
the desired result is established.
\qed
\enddemo
\proclaim{Corollary I} For the two-dimensional Potts model on any
$\Cal G\in\PP$, at every $\beta>\beta_t$ there are exactly $q$
pure (i.e., translation invariant extremal) states.
\endproclaim
\remark{\sserif Remark} The result that the continuity of the
energy at $\beta>\beta_t$ implies the existence of exactly $q$
pure phases in the Potts model was established in \cite{Pf}.
Additional results along these lines for other spin systems were
also established there, however, the arguments %of \cite{Pf}
were restricted to systems on $\Bbb Z^d$. Presumably, this is not
essential but, in any case, certain modifications would have to be
implemented.
With the help of Pirogov-Sinai theory, even more can be proven in
the case of very large values of $q$. Namely, the class of all
translation invariant states is exhausted by $q$ phases for
$\beta>\beta_t$, a single phase at $\beta<\beta_t$, and $q+1$
phases at $\beta=\beta_t$ \cite{M}. Again, even though these
results are explicitly proven only for $\Bbb Z^d$ they are directly
extendable to other
periodic lattices, provided $q$ stays large.
Here we use an alternative method, which is perhaps less
generalizable (e.g., in the direction of non-Potts type systems
and for $d>2$), but is more in accord with the percolation spirit
of the present work.
\endremark
\demo{\sserif Proof of Corollary I}
Let $\langle-\rangle^\odot_{\beta\HH}$ be
a translation-invariant Gibbs measure. As argued previously, there
is a unique (translation-invariant) random cluster measure
$\rho^\odot_{\beta\HH}$ associated with this measure. In
particular, the corresponding bond and energy densities $\goth
b_{ij}^\odot(\beta)$ and $\goth e_{ij}^\odot(\beta)$ satisfy the
same relationship as stated in (vi). Since at $\beta>\beta_t$ the
energy density is equal for all states, we conclude that
$\rho^\odot_{\beta\HH}$ has the same bond density as the wired
state $\mu_{\beta \HH}^w$ (namely, a jump in any $\goth b_{ij}$,
with $\langle i,j\rangle $
%%
% $b$
% ThatŐs what makes the link with "_{ij}" at \goth b.
%%
being a bond in $\Bbb V$, implies a jump in the overall
bond and hence also energy density). Moreover, by the FKG
domination
$$
\mu_{\beta \HH}^w\underset{\FKG}\to\ge \rho^\odot_{\beta\HH},
\tag 3.1
$$
so Strassen's theorem implies that $\mu_{\beta \HH}^w=
\rho^\odot_{\beta\HH}$. We conclude, in particular, that
$\rho^\odot_{\beta\HH}$ has a unique infinite cluster and only
finite components of the dual bonds to vacant bonds.
Given $\epsilon>0$ and $\ell$ an integer, with probability at
least $1-\epsilon$ under $\rho^\odot_{\beta\HH}$ there is a
circuit of occupied bonds enclosing a box $\Lambda_\ell$ of size
$\ell$ such that it is contained in a large-enough box $\Lambda$,
both centered at the origin (otherwise there is a dual connection
between the boundaries of the boxes). However, this means that
with probability $\ge1-\epsilon$ under $\langle-\rangle^\odot_{\beta\HH}$,
there is a circuit of sites in the ring $\Lambda\setminus\Lambda_\ell$
whereupon the spin is constant. Hence, $\langle-\rangle^\odot_{\beta\HH}$ can be
approximated by a convex combination of finite-volume measures
with constant boundary conditions: If $\lambda_{\Gamma;k}$ is the probability
that the circuit $\Gamma$ occurs and the spin thereupon is equal to $k$,
then
$$
\sum_{\Gamma,k}\lambda_{\Gamma;k}
\langle \Cal O\rangle^{[k]}_{\beta\HH;\LL(\Gamma)}-\epsilon
\le \langle \Cal O\rangle^\odot_{\beta\HH}\le
\epsilon+\sum_{\Gamma,k}\lambda_{\Gamma;k}
\langle \Cal O\rangle^{[k]}_{\beta\HH;\LL(\Gamma)},
\tag 3.2
$$
for any observable $\Cal O$ in $\Lambda_\ell$ with
$\Vert \Cal O\Vert_\infty\le 1$.
Here $\langle-\rangle^{[k]}_{\beta\HH;\LL(\Gamma)}$ denotes the spin
state in $\Int\Gamma$ with boundary condition $\sigma_i=k$ at
$i\in\Gamma$.
As $\LL(\Gamma)\nearrow\GG$,
each such state has a unique thermodynamic limit
\cite{C, BBCK}, which altogether
give rise to $q$ distinct states for
$\beta>\beta_t$. By taking a subsequential limit of the
coefficients $\lambda_k=\sum_\Gamma\lambda_{\Gamma,k}$ (and noting
that $\sum_{k}\lambda_{k}$ tends to one as $\ell\to\infty$) we see that
$\langle-\rangle^\odot_{\beta\HH}$ is indeed a mixture of these
$q$ states.
\qed
\enddemo
\proclaim{Corollary II} Consider the asymmetric Potts models on
$\Bbb Z^2$ with couplings $J_{ij}=K$ in the vertical direction and
$J_{ij}=L$ in the horizontal direction (with $0\le K,L\le\infty$).
If this model has a discontinuous transition (which is the case
for large $q$), then it occurs exactly at $\beta_t(K,L)$
determined by the equation $(e^{\beta_tK}-1)(e^{\beta_tL}-1)=q$.
\endproclaim
\demo{\sserif Proof} Abbreviating $\kk=e^{\beta K}-1$ and
$\ll=e^{\beta L}-1$, let us define $\kk^*$ and $\ll^*$ by the
formula
$$
\aligned \kk \,\ll^*\!&=q\\ \kk^*\ll &=q.
\endaligned
\tag 3.3
$$
Since the original model is on $\Bbb Z^2$, it is easy to verify
that its dual is the same model with $(\kk,\ll)$ replaced by
$(\kk^*,\ll^*)$ (and the wired and free boundary measures
interchanged). The $(\kk,\ll)$-parameter space
$\varSigma=\{(\kk,\ll)|\kk\ge0,\,\ll\ge 0\}$ splits into three
disjoint parts: $\varSigma_0$ and $\varSigma_\infty$, with the
former containing the point $(0,0)$ and the latter containing
$(\infty,\infty)$, and the self-dual line $\CcC=\{(\kk,\ll)|\kk\ll=q\}$.
On the other hand, the free and wired random cluster measures are
both increasing in $\kk$ and~$\ll$, hence we can define a unique
transition line $\widetilde{\CcC}=\{(\kk_\alpha,\ll_\alpha)|0<\alpha<\infty\}$,
where
$\ll_\alpha=\alpha\kk_\alpha$ and where $\kk_\alpha$ is the
infimum of all $\kk$ such that there is percolation in the model
with parameters $(\kk,\alpha\kk)$. Similarly, $\widetilde{\CcC}$
induces a trichotomy: $\varSigma$ splits into $\widetilde{\CcC}$,
the high-temperature part $\widetilde{\varSigma}_0$, and the
low-temperature part $\widetilde{\varSigma}_\infty$. It is of
importance that $\widetilde{\CcC}$ is parametrizable as a
function either of $\kk$ or of $\ll$ as follows from monotonicity
analogous to claim (iv) above. As a consequence,
$\widetilde{\varSigma}_\infty$ is mapped into
$\widetilde{\varSigma}_0$ under the duality map, and thus
$\widetilde{\CcC}\subset \CcC\cup
\widetilde{\varSigma}_\infty$.
Consider now the
%%
% a
% According to my understanding of the use of the articles.
%%
curve $\gamma$ that $(\kk,\ll)$ sweeps out as
$\beta$ increases from $0$ to $\infty$ and suppose that there is a
discontinuity in the energy density at some purported
$(\kk^\dag,\ll^\dag)\in\gamma$. As before, a limiting argument
ensures the existence of two Gibbs measures at
$(\kk^\dag,\ll^\dag)$ exhibiting the two values of $\goth e$.
However, if $(\kk^\dag,\ll^\dag)\in\widetilde{\varSigma}_0$, then
this contradicts the no-percolation uniqueness theorem whereas if
$(\kk^\dag,\ll^\dag)\in\varSigma_\infty$ (note the absence of
``tilde'') the same is applies to the dual model. Consequently,
$(\kk^\dag,\ll^\dag)\in\varSigma\setminus(\widetilde{\varSigma}_0
\cup\varSigma_\infty)\subset \CcC$. \qed
\enddemo
\head
4. Continuity of the magnetization
\endhead
Here we state and prove the following claim:
\proclaim{Theorem B} For the Potts model on any graph $\GG\in\PP$,
the spontaneous magnetization $\goth m(\beta)$ is continuous for
all $\beta>\beta_t$.
\endproclaim
\remark{\sserif Remark} We need only a slight variant of the
argument in the proof of Theorem A. Namely, such a discontinuity
implies, it would seem, two different translation invariant states
at the purported point of discontinuity (distinguished by the
value of the magnetization), which in turn implies different
energy densities. However, this time the argument is not quite so
straightforward because there is no guarantee that the lower state
will be pure---conceivably, it can be a particular mixture of the
various upper states craftily tuned to reduce the magnetization.
Indeed this phenomenon presumably occurs in the one-dimensional
$1/r^2$ models: There is a discontinuity in the magnetization at
the critical point (the Thouless effect) and yet at the critical
point, there is no extremal state with zero magnetization. Thus we must
proceed with caution.
\endremark
\demo{\sserif Proof of Theorem B} Suppose the magnetization is
discontinuous at a $\beta=\beta^\circledast>\beta_t$. In the
inhomogeneous case, at least one sublattice magnetization (defined
by restricting the effect of the field $\goth h$ in (2.7) to the
sublattice)---say the one of the vertex-type of the origin---necessary
undergoes a jump at $\beta=\beta^\circledast$,
because the magnetization of {\it each\/} sublattice increases.
The whole problem is thus converted to
the sublattice containing the origin to which we now restrict our attention
(and to which we will not make any further explicit reference).
Obviously, the wired state at $\beta=\beta^\circledast$ gives rise
to the upper value of the magnetization. Let us use $\goth m_+$ to
denote this value and let $\goth m_-$ be the lower value, i.e.,
$\goth m_-=\lim_{\beta\uparrow\beta^\circledast}\goth m(\beta)$.
Define $\Delta=(\goth m_+)^2-(\goth m_-)^2$. Then, for any $x$ and
$y$,
$$
\mu_{\beta^\circledast \HH}^w(x\leftrightarrow y)={\textstyle\frac
q{q-1}}\bigl\langle \delta_{\sigma_x\sigma_y}-{\textstyle\frac
1q}\bigr\rangle_{\beta^\circledast \HH}^{[k]}\ge (\goth m_+)^2,
\tag 4.1
$$
where $[k]$, $k=1,\dots,q$, denotes any of the $q$ ordered states
obtained as the limit of finite volume states with all the
boundary spins set to the $k$-th spin state.
Let $\epsilon>0$ be a small number, in particular, we demand
$\epsilon<\Delta/6$. We shall show that for $L$ large enough there
is a Gibbs state $\langle-\rangle^\#_{\beta^\circledast \HH}$ such
that
$$
{\textstyle\frac q{q-1}} \bigl\langle
\delta_{\sigma_0\sigma_{2L}}-{\textstyle
\frac1q}\bigr\rangle^\#_{\beta^\circledast \HH} \le (\goth m_+)^2
-\Delta+6\epsilon,
\tag 4.2
$$
where $\sigma_{2L}$ is a shorthand for the spin at the site
$(2L,0)$.
To show this, notice first that at any inverse temperature
$\beta$, the quantity $\frac q{q-1}\langle
\delta_{\sigma_x\sigma_y}-{\textstyle \frac1q}\rangle^f_{\beta
\HH}=\mu_{\beta \HH}^f(x\leftrightarrow y)$ is decomposed into two
events. Namely, the event $\{[x\leftrightarrow y]_F\}$ that $x$
and $y$ are in the same finite cluster and
%%
% or
% ThatŐs what we want to say.
%%
the event
$\Pi_\infty(x)\cap\Pi_\infty(y)$ that both are in an infinite
cluster. Let $\beta_0\in (\beta_t,\beta^\circledast)$ and let
$L\gg\ell\gg 1$ be two length scales that satisfy the following
two conditions for the measure $\mu_{\beta_0 \HH}^f$: \roster
\item"(a)"
With probability larger than $1-\epsilon$, a site at the boundary
of the square vessel $\Lambda_\ell$ centered at the origin is
connected to infinity.
\item"(b)"
With probability larger than $1-\epsilon$, there is a circuit of
occupied bonds in the region
$\Lambda_{L}\setminus(\Lambda_\ell\cup\partial\Lambda_\ell)$.
\endroster
We denote the events described in these conditions by $\Cal A$,
$\Cal B$, respectively. Clearly, (a) can be achieved because
there is percolation whereas (b) holds because there is no dual
percolation.
Let $\Cal C$ be the event that there is no dual circuit
surrounding the origin containing any site a distance greater than
$2L$ away. Since $\Cal C\supset\Cal A\cap\Cal B$, it occurs with
probability that is, assuming (a) and (b), larger than
$1-2\epsilon$. Moreover, it is obvious that all
three events are increasing. Hence, if $\mu_{\beta_0 \HH}^f(\Cal
D)>1-\epsilon$, then $\mu_{\beta \HH}^f(\Cal D)>1-\epsilon$ for
all $\beta>\beta_0$ and $\Cal D=\Cal A,\Cal B$, while for
$\Cal D=\Cal C$ the same holds with $\epsilon$ replaced by $2\epsilon$.
For the part $\{[0\leftrightarrow 2L]_F\}$ of the event
$0\leftrightarrow 2L$, let us first note that
$\{[0\leftrightarrow 2L]_F\}\subset\Cal C^c$.
We thus have, for $\beta\ge\beta_0$,
the bound
$$
\mu_{\beta \HH}^f\bigl([0\leftrightarrow 2L]_F\bigr)\le \mu_{\beta
\HH}^f(\Cal C^c)<2\epsilon.
\tag 4.3
$$
Concerning the event $\Pi_\infty(x)\cap\Pi_\infty(y)$, define
$\Lambda_L$ to be a box of size $2L-1$ centered at $0$ and let
$P_L^\beta(0)$ be the probability that the origin is connected to
$\partial\Lambda_L$ in the finite volume system with wired
boundary conditions. Clearly,
$$
\mu_{\beta \HH}^f\bigl(\Pi_\infty(0)\cap\Pi_\infty(2L)\bigr)\le
\bigl[ P_L^\beta(0)\bigr]^2,
\tag 4.4
$$
which means that we just need to bound the quantity $P_L^\beta(0)$
in terms of $P_\infty^\beta(0)=\goth m(\beta)$.
Let
%%
% now
% Redundant.
%%
$\Cal F$ denote the event that there is a circuit of
occupied bonds in the region
$\Lambda_L\setminus(\Lambda_\ell\cup\partial\Lambda_\ell)$ that is
connected to infinity. Clearly $\Cal A\cap \Cal B\subset\Cal F$.
Under the condition $\Cal F$, it is not hard to see that the
probability of $\Pi_\infty(0)$ exceeds $P_L^\beta(0)$. Indeed,
conditioning further on the outermost circuit in $\Lambda_L$ that
satisfies the requirements for $\Cal F$, all that is required is a
connection from the origin to this circuit. Let $\Bbbone_\Gamma$
indicate that the outermost such circuit is precisely $\Gamma$.
Then we have
$$
\aligned
\mu_{\beta \HH}^w\bigl(\Pi_\infty(0)\cap\Cal
F\bigr)&=\sum_\Gamma \mu_{\beta \HH}^w \bigl(\,\Bbbone_\Gamma\,
\mu_{\beta
\HH;\operatorname{Int}\Gamma}^w(0\leftrightarrow\Gamma)\bigr)\\
&\ge \mu_{\beta
\HH;\Lambda_L}^w(0\leftrightarrow\partial\Lambda_L)\mu_{\beta
\HH}^w(\Cal F),
\endaligned
\tag 4.5
$$
where we used that conditioning on $\Gamma$ yields the wired measure
for the interior of $\Gamma$ and that
$\Lambda\mapsto\mu_{\beta\HH,\Lambda}^w(0\leftrightarrow\partial\Lambda)$
is monotone decreasing in $\Lambda$. Finally, we summed over
$\Gamma$ to get $\mu_{\beta \HH}^w(\Cal F)$ back.
Invoking the bound $\mu_{\beta \HH}^w(\Cal
F)>1-2\epsilon$, we have
$$
P_\infty^\beta(0)\ge \mu_{\beta \HH}^w(\Cal F) P_L^\beta(0)>
P_L^\beta(0)(1-2\epsilon).
\tag 4.6
$$
whenever $\beta>\beta_0$. Now let $(\beta_k)_{k\ge0}$ denote a
sequence of inverse temperatures increasing to $\beta^\circledast$
and let $\#$ denote the Gibbs state defined as a
%%
% the
% we donŐt know that the limit exists.
%%
$k\to\infty$
limit of the free Gibbs states at $\beta_k$.
By putting (4.3), (4.4) and (4.6) together, we have for all $k$ that
$$
{\textstyle\frac q{q-1}} \bigl\langle
\delta_{\sigma_0\sigma_{2L}}-{\textstyle\frac1q}\bigl\rangle_{\beta_k
\HH}^f < \goth m(\beta_k)^2+6\epsilon,
\tag 4.7
$$
where we remind the reader that $\epsilon$ is uniform in $k$.
Hence
$$
{\textstyle\frac q{q-1}} \bigl\langle \delta_{\sigma_0
\sigma_{2L}}-{\textstyle\frac1q}\bigl\rangle_{\beta^\circledast
\HH}^\# \le (\goth m_-)^2+6\epsilon=(\goth
m_+)^2-\Delta+6\epsilon<(\goth m_+)^2.
\tag 4.8
$$
A comparison with (4.1) reveals that the state $\langle-\rangle_{\beta^\circledast
\HH}^\# $ as well as the random cluster measure $\mu^\#_{\beta^\circledast \HH}$
associated with this state are evidently different from the wired
state. However, by the last display, $\mu^\#_{\beta^\circledast \HH}$
is lower than the wired state, which implies that the bond density is
strictly below the value in the wired state. Thence, the existence of a
discontinuity of the magnetization at $\beta^\circledast>\beta_t$
implies a discontinuity in the energy density, which contradicts
the result of Theorem A.
\qed
\enddemo
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\enddocument