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\begin{document}
\def\entier{{\bf Z}}
\def\reel{{\bf R}}
\def\dom{\mathop{\rm dom}}
\title{\Large\bf Interface Models and Boundary Conditions}
\author{\normalsize\sc Salvador Miracle-Sole$^{(1)}$ and Jean Ruiz$^{(2)}$}
\date{\small Centre de Physique Th\'eorique, CNRS,
\\ Luminy, Case 907, F-13288 Marseille Cedex 9, France }
\maketitle
\markright{S. Miracle-Sole and J. Ruiz / Interface Models}
\bibliographystyle{alpha}
\footnotetext[1]{ E-mail address: \textit{\it
miracle@cpt.univ-mrs.fr\/}} \footnotetext[2]{ E-mail address:
\textit{\it ruiz@cpt.univ-mrs.fr\/}} \setcounter{footnote}{2}
\footnotesize
\begin{quote}
{\sc Abstract:} We study certain aspects of the thermodynamic
formalism of interface models. Under appropriated conditions, we
prove a conjecture proposed by Spohn \cite{Sp} few years ago, and
as a consequence the validity of exact results on the equilibrium
shape associated to certain Solid-On-Solid models.
\\[3pt]
{\sc Key words:} SOS models, surface tension, crystal shapes,
Gibbs ensembles, Legendre transform.
\end{quote}
\normalsize \vskip15pt
Interface models have been introduced as simple models for a
microscopic description of the phase separation surface between
coexisting phases. In these models the interface is represented as
the graph of a function defined on a reference plane. At each site
$i$ of a finite square lattice $\Lambda\subset{\cal L}=\entier^2$
an integer variable $\phi(i)$ is assigned which represents the
height of the interface at this site. A statistical mechanical
model is obtained by defining the energy of each configuration
$\phi= \{\phi(i)\}$. The standard examples are of the form
\begin{equation}
H_\Lambda (\phi) =\sum_{\vert
i-j\vert=1}U(\phi(i)-\phi(j)) \label{1}
\end{equation}
with
$U(r)\ge0$. Thus, the case $U(r)=r^2$ corresponds to the discrete
Gaussian model, while for $U(r)=\vert r\vert$ one obtains the
solid-on-solid (SOS) model. Restricted SOS models, in which
$U(r)=+\infty$, except for a finite number of values of $r$, can
also be considered. We assume
\begin{equation}
\sum_{k\in\entier}e^{-\beta U(k)}= K(\beta)<\infty
\label{2}
\end{equation}
if $\beta\ge0$. The weight of a given
configuration, at the inverse temperature $\beta$, is proportional
to the Boltzmann factor \ $\exp \big( -\beta H_\Lambda(\phi)
\big)$.
These models provide an approximate description of the microscopic
interface separating two phases at equilibrium, such as the
positively and negatively magnetized phases of the
three-dimensional Ising model. Actually the SOS model may be
obtained as the limit of the anisotropic Ising model with nearest
neighbour interactions, when we let the interaction parameter, in
the vertical direction, tend to infinity. Here the Ising model is
defined inside a box with boundary conditions which enforce an
interface with a given average slope between the positive and
negative phases. Moreover, if all interaction parameters tend to
infinity or, equivalently, if $\beta$ tends to infinity, the
interface is then described by a restricted SOS model.
The statistical mechanics of interface models is, naturally,
rather different from that of bounded spin systems. Indeed, the
thermodynamic free energy of such models represents an interfacial
free energy per unit projected area. It thus depend on the chosen
boundary condition. Moreover, the interface can be rough, as it is
the case in many situations, implying that the Gibbs states do not
exit. For such models, a detailed analysis related to these two
properties is still lacking, as it is pointed out by Spohn in
ref.\ \cite{Sp} (see appendix B).
Concerning the properties of Gibbs states, he proposed a study of
the equilibrium measure in terms of the heights differences
satisfying a local constraint. He conjectured in particular that
for a given fixed boundary condition, the corresponding Gibbs
state is unique (see also ref.\ \cite{FS}).
Concerning the free energy, one is naturally led to introduce a
conjugate Gibbs ensemble with respect to the interface boundaries.
As it is conjectured in ref.\ \cite{Sp}, the free energy
associated to this conjugate ensemble should coincide with the
Legendre transform of the interfacial free energy. It is the aim
of this article to examine certain aspects of the corresponding
thermodynamic formalism. To this end, we shall first introduce the
appropriated definitions for the free energies associated to the
different Gibbs ensembles. For a class of appropriated
conditions, we shall then prove the validity of this last
mentioned conjecture.
We take $\Lambda=\Lambda(N_1,N_2)$ as a rectangular box of sides
$2N_1$ and $2N_2$ centered at the origin, i. e., as the set of
sites $i=(i_1,i_2)\in{\cal L}$ such that $\vert i_1\vert\le N_1$
and $\vert i_2\vert\le N_2$. Its area is denoted by
$\vert\Lambda\vert = 4N_1N_2$. The boundary $\partial\Lambda$ is
the set of sites $i \in \Lambda$ such that $\vert i_1\vert=N_1$ or
$\vert i_2\vert=N_2$. It is understood in equation (\ref{1}) that
the bond $\{ i,j \}$ belongs to $\Lambda$ and can intersect or be
included in $\partial\Lambda$.
In order to define the free energy of the macroscopic interface
corresponding to a given average slope $p=(p_1,p_2)$, we introduce
the Gibbs ensemble ${\cal E}^{\rm clos}(p,\Lambda)$ which consists
of all configurations, in the box $\Lambda$, satisfying the
(``closed'') boundary conditions
\begin{equation}
\phi(i)={\bar\phi}_p (i)=[p\cdot i], \quad i\in\partial\Lambda
\label{3}
\end{equation}
where $p\cdot i = p_1i_1+p_2i_2$ and $[\,
\cdot \, ]$ represents the integer part. The partition function is
\begin{equation}
Z^{\rm clos} (p,\Lambda) = \sum _{\phi\in {\cal
E}^{\rm clos}(p,\Lambda) } \exp \big( -\beta H_\Lambda(\phi) \big)
\label{4}
\end{equation}
where the sum runs over all
configurations in $\Lambda$ satisfying conditions (\ref{3}). The
associated free energy per unit projected area is defined as
\begin{equation}
f^{\rm clos}(p, \Lambda) = - {{1}\over{ \beta\,
\vert\Lambda\vert}} \ln Z^{\rm clos} (p,\Lambda) \label{5}
\end{equation}
\begin{Th}
\label{T1} The thermodynamic limit of {\rm(\ref{5})}, which
defines the projected surface tension $f(p)$, exists. It is a
convex and Lipschitz continuous function of $p$ in the interior of
the effective domain of $f$.
\end{Th}
The validity of the above statements is known. See, for instance,
ref.\ \cite{MMR}, for a proof of them in a more general setting.
The effective domain of a convex function $f$ is the set $\dom f =
\{p : f(p) < \infty\}$.
%Notice that because of condition (2) the set
%${\cal E}^{\rm c}(\Lambda)$ can be empty
%for certain values of $p$ (which gives
%$Z^{\rm c}=0$).
The surface tension, which represents the interfacial free energy
per unit area of the mean interface, is
$$
\tau (p) = (1+p^2_1+p^2_2)^{-1/2} f(p)
$$
The convexity of $f$ is equivalent to the fact that the surface
tension $\tau$ satisfies a stability condition called the
pyramidal inequality (see refs.\ \cite{MMR}, \cite{DS}).
The boundary conditions considered above can be interpreted as a
``canonical'' constraint. We are going to discuss the conjugate
Gibbs ensemble, which can be viewed as a ``grand canonical''
ensemble with respect to the interface boundaries.
It is useful to introduce the variables $\xi(\ell) = \xi(i,j) =
\phi(i) - \phi(j)$ associated to the oriented bonds of the lattice
$\ell=\{ i,j \}$, $\vert i-j\vert =1$. Note that $-\ell=\{ j,i \}$
and $\xi(\ell)=-\xi(-\ell)$. The admissible configurations $\xi$,
being the gradient of $\phi$, satisfy $\sum_{\ell\in\lambda}
\xi(\ell) =0$ for any closed loop $\lambda$ in $\Lambda$.
Equivalently,
\begin{equation}
\xi(i,j)+\xi(j,k)+\xi(k,\ell)+\xi(\ell,i)=0 \label{6}
\end{equation}
for every plaquette (elementary square loop) $P=\{
i,j,k,\ell \}$ in $\Lambda$.
Periodic boundary conditions in the box $\Lambda$ are defined with
respect to the $\xi$ variables. Namely, for all
$i_1=-N_1,\dots,N_1-1$, $i_2=-N_2,\dots,N_2-1$, it is assumed that
\begin{eqnarray}
\phi(N_1,i_2+1) - \phi(N_1,i_2) &=& \phi(-N_1,i_2+1) -
\phi(-N_1,i_2), \cr \phi(i_1+1,N_2) - \phi(i_1,N_2)
&=&\phi(i_1+1,-N_2) - \phi(i_1,-N_2) \label{11}
\end{eqnarray}
We introduce the boundary terms
\begin{eqnarray}
{\cal S}_1(\phi)
&=& \sum_{\ell\in\ell_1(\Lambda)} \xi(\ell) =\sum_{-N_1\le i_2\le
N_1} \big( \phi(N_1,i_2) - \phi(-N_1,i_2) \big) \cr {\cal
S}_2(\phi) &=& \sum_{\ell\in\ell_2(\Lambda)} \xi(\ell)
=\sum_{-N_2\le i_1\le N_2} \big(\phi(i_1,N_2) -
\phi(i_1,-N_2)\big)
\end{eqnarray}
where $\ell_1(\Lambda)$ and
$\ell_2(\Lambda)$ are the sets of all bonds in $\Lambda$ parallel
to the $i_1$ and to the $i_2$ axis, respectively, oriented
according to increasing coordinates. The grand canonical
prescription, which is convenient to consider, consists in adding
to the energy a term of the form
\begin{equation} x_1 {\cal S}_1(\phi) + x_2 {\cal S}_2(\phi)
\label{8} \end{equation} where $x = (x_1,x_2)\in \reel^2$
represent the slope chemical potentials. The associated partition
function and free energy are
\begin{eqnarray}
{\Xi}^{\rm free}
(x,\Lambda) &=&\sum _{\phi,\phi(0)=0} \exp \big( -\beta
H_\Lambda(\phi) + x_1 {\cal S}_1(\phi) + x_2 {\cal S}_2(\phi)
\big)
\label{9} \\
\varphi^{\rm free} (x,\L) &=& -{{1}\over{\beta \,
\vert\Lambda\vert} } \ln {\Xi}^{\rm free} (x,\Lambda)
\label{10}
\end{eqnarray}
in the case of free boundary conditions. To break the
translation symmetry, we pinned the height $\phi(0)$ at zero.
The periodic partition function is defined as the sum in (\ref{9})
with the constraint (\ref{11}) on the configurations. This
function and the corresponding free energy are denoted
\begin{equation}
{\Xi}^{\rm per}(x,\Lambda),\quad \varphi^{\rm
per} (x)
\label{10a}
\end{equation}
From the definitions it can be proved that the thermodynamic
limit of (\ref{10}) exists. We shall not discuss this point here
and analyze instead the problem of the equivalence of the
conjugate Gibbs ensembles described by $Z$ and $\Xi$. Notice that
(10) and (11) are not defined as the equivalent ensemble of
(\ref{10a}) with closed boundary conditions.
We introduce the set ${\cal E}^{\rm per} (p,\Lambda)$, of
configurations in $\Lambda$ which satisfy the boundary conditions
\begin{equation}
\phi(-N_1,i_2) - \phi(N_1,i_2) = [2N_1p_1],\quad
\phi(i_1,-N_2) - \phi(i_1,N_2) = [2N_2p_2]
\label{12}
\end{equation}
for all $i_1=-N_1+1,\dots,N_1$,
$i_2=-N_2+1,\dots,N_2$ (and also $\phi(0)=0$), contains the set
(\ref{3}) used in the definition of the surface tension, and
describes also a set of interfaces with average slope $p$. Under
periodic boundary conditions, condition (\ref{12}) can
equivalently be written as
\begin{equation} {\cal S}_1(\phi) =
[\vert\Lambda\vert p_1],\quad {\cal S}_2(\phi) =
[\vert\Lambda\vert p_2] \label{13}
\end{equation}
In this case,
(\ref{12}) is satisfied as soon as it is satisfied for some $i_1$
and some $i_2$. This set of configurations, satisfying conditions
(\ref{11}) and (\ref{13}), defines the canonical Gibbs ensemble
which corresponds to the grand canonical ensemble described above.
We denote by
\begin{eqnarray} Z^{\rm per} (p,\Lambda) &=& \sum
_{\phi\in{\cal E}^{\rm per} (p,\Lambda),\phi(0)=0} \exp \big(
-\beta H_\Lambda(\phi)\big)
\label{14}
\\
f^{\rm per}(p,\Lambda) &=& - {{1}\over{\beta | \Lambda| }} \ln
Z^{\rm per}(p,\Lambda)
\label{15}
\end{eqnarray}
the associated
``canonical'' partition function and free energy. Our purpose is
now to prove that the two partition functions $Z^{\rm per}$ and
$Z^{\rm clos}$, with periodic and closed boundary conditions,
define the same free energy.
\begin{Th}
\label{T2} The thermodynamic limit of {\rm(\ref{15})} exists and
coincides with the projected surface tension
\begin{equation}
\lim_{N_1,N_2\to\infty} f^{\rm per}\big( p,\L(N_1, N_2) \big)
=f(p)
\label{26}
\end{equation}
\end{Th}
This theorem is proved in the Appendix.
%{\it Proof.} The proof is given in section 3.
%\medskip
In the next Theorem we study the grand canonical ensemble (with
respect to the interface boundaries), defined in equation
(\ref{9}), with periodic boundary conditions. We introduce the
Legendre transform
\begin{equation} -\varphi (x) = \sup_p \big(
p\cdot x - f(p)\big)
\label{32}
\end{equation}
\begin{Th}
\label{T3}
Let $D=\{x : \varphi (x) > -\infty\}$
and write, respectively, $D^{\rm int}$, $\bar D$, and $\partial
D$, for the interior, the closure, and the boundary of the convex
set $D$. Then,
\begin{eqnarray}
&&{\it for}\ x\in D^{\rm
int},\hphantom{\reel^2\setminus{\bar D}\partial D} \lim_{N_1,
N_2\to\infty}\varphi^{\rm per} (x,\Lambda)=\varphi (x), \hfill\cr
&&{\it for}\ x\in \reel^2\setminus{\bar D},\hphantom{D^{\rm
int}\partial D} \lim_{N_1, N_2\to\infty}\varphi^{\rm per}
(x,\Lambda)= -\infty, \hfill\cr &&{\it for}\ x\in\partial
D,\hphantom{\reel^2\setminus{\bar D}D^{\rm int}} \limsup_{N_1,
N_2\to\infty}\varphi^{\rm per} (x, \Lambda) \le \limsup_{x'\to
x,x'\in D^{\rm int}}\varphi (x').
%\hfill\cr}$$
\end{eqnarray}
\end{Th}
\proof Since, from Theorem \ref{T2}, we know that $f =\lim_{N_1
N_2 \to \infty} f^{\rm per}$, the above statements, together with
the relation (\ref{32}) between the free energies, express the
thermodynamic equivalence of the Gibbs ensembles with partition
functions $Z^{\rm per}(p)$ and ${\Xi}\,^{\rm per}(x)$, and can be
proved in the same way as Theorem 4 in ref.\ \cite{MMR}.
\medskip
These relations imply that the surface $z = \varphi (x_1,x_2)$
gives, according to the Wulff construction, or its equivalent, the
Andreev construction, the equilibrium shape of the crystal
associated to the system (see \cite{An}, \cite{MMR}).
We next examine the particular case of the horizontal interfaces
(slope $p=0$), in which stronger properties can be proved.
\begin{Th}
\label{T4} The following limits exist and coincide
\begin{equation}
\lim_{N_1,N_2\to\infty} \varphi^{\rm
free}(0,\Lambda(N_1,N_2)) = \lim_{N_1,N_2\to\infty} \varphi^{\rm
per}(0,\Lambda(N_1,N_2)) = \varphi(0)
\label{33}
\end{equation}
Moreover,
\begin{equation} \varphi (0) = f(0) \label{34} \end{equation}
\end{Th}
This theorem is proved in the Appendix.
%{\it Proof.} The proof is given in section 3.
\medskip
These results apply, in particular, also to the restricted
solid-on-solid models. Of particular interest are some of these
models which are exactly solvable, in which the height differences
\begin{equation}
\xi(i,j) = \phi(j)-\phi(i)
\end{equation} for nearest neighbour sites,
are restricted to have only two values. One of these models is the
body-centered solid-on-solid model ({\sc bcsos}) of van Beijeren
\cite{vB}, defined on a square lattice with the restriction
$\xi(i,j) = \pm1$. Another is the triangular Ising solid-on-solid
model ({\sc tisos}) of Bl\"ote and Hilhorst \cite{BH}, in which
the height variables are associated to the sites of a triangular
lattice and $\xi(i,j)$ is allowed to take the values $1$ or $-2$.
These two models appear in the description of the ground state
interfaces of the Ising model on a body-centered cubic lattice
with nearest and next-nearest neighbour interactions \cite{KM}.
The latter model describes also the ground state interfaces of the
usual Ising model on a cubic lattice with nearest neighbour
interactions \cite{BH}, \cite{Mi}.
Taking into account the compatibility condition (\ref{6}) between
the difference variables $\xi(i,j)$, it can easily be seen that
there is a one to one correspondence between the set of
configurations of the {\sc bcsos} model and the set of
configurations of the six-vertex model, the compatibility
condition being equivalent to the ice rule. Similarly, the
configurations of the {\sc tisos} model are in one to one
correspondence with the ground state configurations of the Ising
antiferromagnet on a triangular lattice. This explain why the free
energy of the six-vertex model or of the triangular Ising model
depends on the boundary conditions. Theorem \ref{T4} proves then
the equivalence between the free and the periodic boundary
conditions for these models in the case of symmetric interactions
with respect to the axes.
The crystal shape associated to these particular models was
obtained from the (``grand canonical'') partition function
(\ref{10a}). In this way the van Beijeren model is equivalent to a
six-vertex model with polarizations, and the Bl\"ote-Hilhorst
model to a zero-temperature triangular Ising antiferromagnet with
external fields. The equilibrium shape of the corresponding
crystals is directly related to the free energy of these models
and may be exactly computed. See the original work by Jayaprakash
{\it et al.} \cite{JST} and Nienhuis {\it et al.} \cite{NHB} for a
more detailed discussion, including the study of the shape of the
facets of these crystals and their roughening transitions. Theorem
\ref{T3} above proves then the rigorous validity of these exact
results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% regarder pour la redaction ref.
%%% H. van Beijeren and I. Nolden:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\medskip
\centerline{\large\sc Appendix}
\bigskip
\noindent To prove theorems \ref{T2} and \ref{T4} we first
establish the following two lemmas. For concreteness we shall
consider the solid-on-solid models. The same proof applies to
other interface models, as the discrete gaussian model. Its
extension to restricted SOS models is explained in the remark
after the proof of Theorem \ref{2}.
\begin{Lm}
\label{L1} The partition function $Z^{\rm per}$ satisfies the
subadditivity property
\begin{equation}
Z^{\rm per} \big( p,
\Lambda(2N_1,N_2) \big) \ge \Big( Z^{\rm per} \big(
p,\Lambda(N_1,N_2) \big) \Big)^2 K(\beta)^{-2N_2}
\label{16}
\end{equation}
\end{Lm}
\proof Consider the rectangular box $\Lambda(N_1,N_2)$ with the
specified configurations of bonds $\bar\xi = \{\bar\xi(\ell),$
$\ell \in \partial_2\Lambda\}$ on the two sides parallel to the
$i_2$-axis of the rectangle, and let $Z^{\rm per}( p, \Lambda\mid
\bar\xi)$ be the partition function $Z^{\rm per}$ with these
imposed constraints. Then
\begin{equation}
Z^{\rm per}(p,\Lambda)
= \sum_{\bar\xi} Z^{\rm per}(p,\Lambda\mid \bar\xi)
\label{17}
\end{equation}
If we paste two such boxes to form a $(2N_1,N_2)$ rectangle
then
\begin{equation}
Z^{\rm per}\big( p,\Lambda(2N_1,N_2) \mid
\bar\xi \big) \ge \Big( Z^{\rm per}\big( p,\Lambda(N_1,N_2) \mid
\bar\xi \big) \Big)^2
e^{\beta H_\lambda(\bar\xi) }
\label{18}
\end{equation}
since we can always impose the
configuration $\bar\xi$ on each component of the box
$\Lambda(2N_1,N_2)$ and the energy associated to their common side
$\lambda$ is
\begin{equation}
H_\lambda(\bar\xi)= \sum_{\ell\in\lambda}U
\big(\bar\xi(\ell) \big)
\label{19}
\end{equation}
From
Schwartz inequality, we have
\begin{eqnarray}
&&\Big(\sum_{\bar\xi}
Z^{\rm per}\big(p,\Lambda(N_1,N_2) \mid \bar\xi \big)\Big)^2 \cr
&&\qquad\le \Big(\sum_{\bar\xi} \big( Z^{\rm per}\big(
p,\Lambda(N_1,N_2) \mid \bar\xi \big) \big)^2 e^{\beta
H_\lambda(\bar\xi) } \Big) \Big( \sum_{\bar\xi} e^{-\beta
H_\lambda(\bar\xi)} \Big) \label{20}
\end{eqnarray}
and, on the
other hand,
\begin{equation}
\sum_{\bar\xi} e^{-\beta H_\lambda(\bar\xi)} \le \Big(
\sum_{\bar\xi (\ell)} e^{-\beta U( \bar\xi (\ell) )} \Big)^{2N_2}
=\big( K(\beta ) \big)^{2N_2}
\label{21}
\end{equation}
by
considering the $\bar\xi(\ell)$, $\ell\in\lambda$ as independent
variables. The application of inequalities (\ref{18}), (\ref{20})
and (\ref{21}) to equation (\ref{17}) gives the proof of the
Lemma.
\medskip
Now let $\Lambda_0=\Lambda(N_1,N_2)$ be an arbitrary, but
henceforth fixed rectangle and form the standard sequence
$\Lambda_k=\Lambda\big( 2^k N_1,2^k N_2 \big)$ with $k$ integral.
Using the definition (\ref{15}), we have, by arguing as in the
proof of (\ref{21}),
\begin{equation} f^{\rm per}(p,\Lambda_k) \ge
-{{1}\over{\beta}} \ln K(\beta)
\label{22}
\end{equation}
and, as
a consequence of Lemma \ref{L1},
\begin{equation} f^{\rm
per}(p,\Lambda_{k+1}) \le f^{\rm per}(p,\Lambda_k) + 2^{-k} \big(
{1\over{2N_1}}+{1\over{2N_2}}\big) {1\over\beta} \ln K( \beta )
\label{23}
\end{equation}
Hence, the sequence $\{ f^{\rm per}(p,\Lambda_k)\}$,
$k=0,1,\dots$, is essentially a decreasing sequence and since it
is bounded below it has a limit.
Next, it will be convenient to restrict the set of configurations
on $\Lambda$ in such a way that the interface be contained in a
parallelepiped of height $M$. Namely, we impose the condition
\begin{equation}
\vert \phi(i) - p\cdot i \vert \le M
\label{24}
\end{equation}
We denote by $Z^{\rm per}(p,\Lambda,M)$ and $f^{\rm
per}(p,\Lambda,M)$ the associated partition functions and free
energies.
\begin{Lm}
\label{L2} With the above notations, we have
\begin{equation}
f^{\rm per}(p,\Lambda_{k},M) \le f^{\rm per}(p,\Lambda_0,M) + 2
\big( {1\over{2N_1}}+{1\over{2N_2}}\big) {1\over\beta} \ln
K(\beta)
\label{25}
\end{equation}
\end{Lm}
\proof Since Lemma \ref{L1} is still valid for the restricted set
of configurations (\ref{23}), we obtain the Lemma from equation
(\ref{20}), by iteration.
\medskip
\noindent{\it Proof of Theorem \ref{T2}.} We first compute the
partition function (\ref{3}) on a rectangle of sides $2(2^kN_1+1)$
and $2(2^kN_2+1)$, which contains the standard rectangle
$\Lambda_k$ and has the same center. We denote by ${\cal B}$ the
set of bonds which join the boundary of this rectangle to the
boundary of $\Lambda_k$.
%, and by ${\cal B}'$ the set of bonds contained in $\partial\Lambda_k$.
The sum $\sum'$ below is over the set of configurations in
$\Lambda_k$ which satisfy condition (\ref{24}). This set of
configurations is furthermore restricted by conditions (\ref{11}),
(\ref{13}) and $\phi(0)=0$. This gives the first inequality in the
expression below. The second inequality follows from condition
(\ref{21}) which implies $\vert\phi(j) - \phi(i)\vert \le 2M$.
\begin{eqnarray}
&&Z^{\rm clos}\big( p,\Lambda (2^kN_1+1,2^kN_2+1)\big) \cr
&\ge&
\sum_{\phi} {}'%\ {\rm in}\ \Lambda_k}
\exp\Big\{ - \beta \Big( \sum_{ \{i,j\}\in {\cal B}} \vert \phi
(j) - \bar\phi_p (i)\vert - \sum_{ \{i,j\}\in \partial \L }
\vert\phi(j) - \phi(i)\vert - H_{\Lambda_k}({\phi}) \Big) \Big\}
\cr &\ge&\ \exp\big(- \beta 2^{k} (2N_1+2N_2) 4M \big) Z^{\rm
per} (p,\Lambda_k,M)
\label{27}
\end{eqnarray}
Taking the
logarithms and dividing by $-\beta\vert\Lambda_k\vert$,
we get
\begin{equation} \alpha_k f^{\rm clos}(p,\Lambda_k) \le f^{\rm
per}(p,\Lambda_k,M) + 2^{-k} 4M
\bigg({1\over{2N_1}}+{1\over{2N_2}}\bigg)
\label{28}
\end{equation}
where
\begin{equation}
\alpha_k = (2^kN_1+1)(2^kN_2+1)
2^{-2k}(N_1N_2)^{-1} \label{29}
\end{equation} which tends to 1
when $k\to\infty$.
Then, from Lemma \ref{L2},
\begin{eqnarray}
\alpha_k f^{\rm clos}(p,\Lambda_k) &\le& f^{per} (p,\Lambda_0,M_k)
+ 2^{-k} 4M_k \bigg({1\over{2N_1}}+{1\over{2N_2}}\bigg) \cr &+& 2
\bigg( {1\over{2N_1}}+{1\over{2N_2}}\bigg) {1\over\beta} \ln K(
\beta )
\label{30}
\end{eqnarray}
This equation holds for any $M$.
Taking $M=M_k$ in such a way that $M_k\to\infty$ and $2^{-k}M_k\to
0$, when $k\to\infty$, we obtain
\begin{equation} f(p) \le f^{\rm
per}(p,\Lambda_0) + 2 \bigg( {1\over{2N_1}}+{1\over{2N_2}}\bigg)
{1\over\beta} \ln K( \beta)
\label{31}
\end{equation}
Since, on
the other side $f^{\rm per}(p,\Lambda)\le f^{\rm
clos}(p,\Lambda)$, as a direct consequence of their definitions,
the theorem follows from the last inequality.
\medskip
\noindent {\it Remark.} It is easy to see that Lemma \ref{L1} is
still valid for restricted SOS models and hence also Lemma
\ref{L2}, provided that the expression $(1+e^{-\beta})/(1-
e^{-\beta})$, in (\ref{18}) and (\ref{28}), is replaced by the
number $2$. This comes from the fact that equation (\ref{21}) now
reads $H_\lambda(\bar\xi)=0$. The proof of Theorem \ref{T2} has to
be reviewed since an expression analogous to the first inequality
(\ref{18}) cannot be obtained simply by increasing by 2 the length
of the sides of the box. In order that the required configuration
could be admissible in the restricted SOS models we have to
increase it by $L_k=2M_k$. But, since $2^{-k}L_k\to 0$ when
$k\to\infty$, this does not affect the thermodynamic limit and,
hence, the theorem can be proved similarly.
\medskip
\noindent{\it Proof of Theorem \ref{T4}.}
From the corresponding
definitions it follows that
\begin{equation} {\Xi}^{\rm free}(0,
\Lambda) \ge {\Xi}^{\rm per}(0, \Lambda) \ge Z^{\rm clos}
(0,\Lambda) \label{35}
\end{equation}
The appropriate converse
inequalities can be established by arguing as in the proof of
Theorem \ref{T2}, itself a consequence of Lemmas \ref{L1} and
\ref{L2}. The main point is to prove the validity of Lemma
\ref{L1} for the partition function ${\Xi}^{\rm free}$.
Consider the partition function with free boundary conditions in
the rectangular box $\Lambda(2N_1,N_2)$, obtained by pasting two
boxes $\Lambda(N_1,N_2)$ along one of the sides parallel to the
$i_2$-axis. Let $\lambda$ be this common line, inside the box
$\Lambda(2N_1,N_2)$, and let $\bar\xi$ be the configuration $\{
\bar\xi(\ell)\}$ on the bonds $\ell$ belonging to $\lambda$. We
denote by ${\Xi}^{\rm free}\big(0, \Lambda(N_1,N_2)\mid \bar\xi
\big)$ the partition function over all configurations in
$\Lambda(N_1,N_2)$ whose restriction to one of the sides parallel
to the $i_2$-axis coincides with the given configuration
$\bar\xi$, and having free boundary conditions on the other three
sides. Because of the symmetry of the system with respect to the
$\lambda$-axis, we have
\begin{equation}
{\Xi}^{\rm free}\big(0,
\Lambda(2N_1,N_2) \big) \ge \sum_{\bar\xi} \Big( {\Xi}^{\rm
free}\big(0, \Lambda(N_1,N_2)\mid \bar\xi \big) \Big)^2 e^{\beta
H_\lambda(\bar\xi)}
\label{36}
\end{equation}
where $H_\lambda(\bar\xi)$ is given by (\ref{19}). Then,
inequality (\ref{36}) allows us to derive Lemma \ref{1}, the
subadditivity property, for the function ${\Xi}^{\rm free}$. All
the other steps in the proof of Theorem \ref{T2} follow in the
same way as above and lead to the conclusion that the first limit
in (\ref{33}) is equal to $f(0)$. Then inequalities (\ref{35}),
Theorems \ref{T1} and \ref{T3} imply that this limit coincides
with the second limit in (\ref{33}) and with $\varphi (0)$. This
ends the proof of the theorem.
\bigskip
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\end{document}