\newcount\mgnf\newcount\tipi\newcount\tipoformule\newcount\greco

\tipi=2          %uso caratteri: 2=cmcompleti, 1=cmparziali, 0=amparziali
\tipoformule=0   %=0 da numeroparagrafo.numeroformula; se no numero
                 %assoluto
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INCIPIT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI
%%%
%%% per assegnare un nome simbolico ad una equazione basta
%%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o,
%%% nelle appendici, \Eqa(...) o \eqa(...):
%%% dentro le parentesi e al posto dei ...
%%% si puo' scrivere qualsiasi commento;
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%%% All' inizio di ogni paragrafo si devono definire il
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%%% corrispondente file .aux; basta includere all'inizio l'istruzione
%%%           \include{nomefile}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\write16{ EQ #1 ==> \equ(#1)  }}
\def\etichettaa(#1){(A\veraappendice.\veraformula)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi
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\openout15=\jobname.aux%\write15
\openout13=\jobname.bib
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\let\eqas=\eqa
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%  Numerazione pagine

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\def\data{\number\day/\ifcase\month\or january \or february \or march \or
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%\footline={\rlap{\hbox{\copy200}}\hss\tenrm\folio\hss}
\footline={\hss\tenrm\folio\hss}

%%%%%%%%
%% am
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\font\setterm=amr7 %\font\settei=ammi7
%\font\settesy=amsy7 \font\settebf=ambx7 %\font\setteit=amit7
%%%%% cambiamenti di formato %%%
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\textfont0=\setterm   %\textfont1=\settei
%\textfont2=\settesy   %\textfont3=\setteit
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%\textfont\bffam=\settebf  \def\bf{\fam\bffam\settebf}
\normalbaselineskip=9pt\normalbaselines\rm
}\let\nota=\settepunti}
%%%%%%%

%%cm completo
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\newfam\msbfam   %per uso in \TIPITOT
\newfam\truecmr  %per uso in \TIPITOT
\newfam\truecmsy %per uso in \TIPITOT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Scelta dei caratteri
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\newskip\ttglue
\ifnum\tipi=0\TIPIO \else\ifnum\tipi=1 \TIPI\else \TIPITOT\fi\fi



\global\newcount\numpunt

\magnification=\magstephalf
\baselineskip=16pt
\parskip=8pt



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\def\b{\beta}
\def\d{\delta}
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\def\f{\phi}
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\def\s{\sigma}
\def\t{\tau}
\def\th{\theta}
\def\vp{\varphi}
\def\z{\zeta}
\def\o{\omega}
\def\D{\Delta}
\def\L{\Lambda}
\def\G{\Gamma}
\def\O{\Omega}
\def\S{\Sigma}
\def\Th{\Theta}
\def\del #1{\frac{\partial^{#1}}{\partial\l^{#1}}}

\def\h{\eta}

\def\E{{I\kern-.25em{E}}}
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%\def\f1{{1\kern-.25em\hbox{\vsm I}}}

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\def\P{{I\kern-.25em{P}}}


%\def\P{\hskip.2em\hbox{\rm P\kern-0.8em{I}\hskip.7em}}


\def\der{\frac{d}{dx}}
\def\del{\partial}
\def\tr{\hbox{tr}}
% Spezielle Definitionen
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\def\AA{{\cal A}}
\def\BB{{\cal B}}
\def\CC{{\cal C}}
\def\DD{{\cal D}}
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\def\JJ{{\cal J}}
\def\KK{{\cal K}}
\def\SS{{\cal S}}
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\def\Pp#1{\Psi_{\b,N,\d}(#1)}
\def\pp#1{\phi_{\b,\d}(#1)}
\def\adb{a_{\d,\b}}
\def\aa{a}
\def\ee{\frac{\xi^t\xi}{N}}
\def\Rr#1{r(M,#1)}
\def\rr#1{r^*(M ,#1)}
\def\pat#1{\xi^{#1}}
\def\ov#1{\overline{#1}}
\def\jg{J_\g}

%   Non-character macros

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\def\lemma #1{\noindent{\thbf Lemma #1: }}
\def\definition #1{\noindent{\thbf Definition #1: }}
\def\corollary #1{\noindent{\thbf Corollary #1: }}
\def\proof{{\noindent\pr Proof: }}
\def\proofof #1{{\noindent\pr Proof of #1: }}
\def\endproof{$\diamondsuit$}
\def\remark{\noindent{\bf Remark: }}
\def\thanks{\noindent{\bf Aknowledgements: }}
\font\pr=cmbxsl10
%\font\thbf=cmcsc10 scaled\magstep1
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% Font-Definitions

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%\numsec=1\numfor=1




\font\tit=cmbx12
\font\aut=cmbx12
\font\aff=cmsl12
\def\s{\char'31}
\nopagenumbers
{$  $}
\vskip2truecm
\centerline{\tit THERE ARE NO NICE INTERFACES IN 2+1 DIMENSIONAL}
\vskip.2truecm
\centerline{\tit SOS-MODELS IN RANDOM MEDIA\footnote{${}^1$}{\ftn Work
partially supported by the Commission of the European Communities
under contract No. CHRX-CT93-0411 and CHBGT-CT93-0244}}
%\vskip.2truecm
%\centerline{\tit }
\vskip2truecm
%\centerline{\tit }
%\vskip2.5cm
\centerline{\aut Anton Bovier
\footnote{${}^2$}{\ftn e-mail:
bovier@iaas-berlin.d400.de}}
\vskip.1truecm
\centerline{\aff Weierstra\s{}-Institut }
\centerline{\aff f\"ur Angewandte Analysis und Stochastik}
\centerline{\aff Mohrenstrasse 39, D-10117 Berlin, Germany}
\vskip.5truecm
\centerline{\aut  Christof K\"ulske\footnote{${}^3$}{\ftn
e-mail: kuelske@levy.univ-rennes1.fr}}
\vskip.1truecm
\centerline{\aff Institut de Recherche Math\'ematique de Rennes}
\centerline{\aff Universit\'e de Rennes 1}
\centerline{\aff Campus de Beaulieu}
\centerline{\aff F-35042 Rennes Cedex, France}
\vskip1.5truecm\rm
\noindent {\bf Abstract:} We prove that in dimension $d\leq 2$
translation covariant  Gibbs states describing
 rigid interfaces in  a disordered solid-on-solid (SOS) 
cannot exist for any value of the temperature, in contrast to 
the situation in $d\geq 3$. The prove relies on an adaptation 
of a theorem of Aizenman and Wehr.
  
\noindent {\bf Key Words: } Disordered systems, interfaces,
SOS-model


\vfill
     ${}$

\eject
\count0=1

\footline={\hss\tenrm\folio\hss}


\overfullrule=0pt
\chap{1. Introduction}1

In this short note we want to conclude our analysis on the
properties of interfaces in random environments by complementing
our proof [BK] of the existence of Gibbs measures describing
rigid interfaces in the SOS model with random surface tension
(at low temperatures and weak disorder) in dimension $d\geq 3$
by showing that on the contrary, in dimension $d\leq 2$, such
Gibbs states cannot exist at any temperature as soon as there is any disorder
present. In contrast to the technically rather involved existence proof,
the proof of the converse statement is simple; in fact it is a fairly
straightforward application of a beautiful theorem of Aizenman and Wehr
[AW] which they used to prove the uniqueness of the Gibbs state
in the two-dimensional random field Ising model.
For a extensive discussion of the history of the problem we refer to
the introduction of our previous paper [BK].

The model we consider is defined as follows. A surface is described by
$\Z$-valued variables $h_x\in\Z$, $x\in \Z^d$. The Hamiltonian
is given (formally) by
$$
{\cal H}(h)=\sum_{<x,y>}|h_x-h_y|+\e\sum_{x, k}\eta_x(k)\1_{h_x=k}
\Eq(1.1)
$$
where $\{\eta_x(k)\}_{x\in\Z^d,k\in Z}$ is a family of independent identically
distributed random variables on some abstract probability space
$(\O,\FF,\P)$,  with non-degenerate distribution $\P$. We assume that
$\E[\eta_x(k)]=0$, $\E[\eta_x(k)^2]=1$, where $\E$ denotes the 
expectation w.r.t. the distribution $\P$.
As a matter of fact, our result will apply to a far more general
class of Hamiltonians but we stick to the specific example for
clarity. In [BK] we have proven that under suitable conditions on the
temperature and on the distribution $\P$, for $d\geq 3$
infinite volume Gibbs states $\mu_H$ for this model can be constructed
as weak limits of finite volume Gibbs measures where the heights on the
boundary were set to a fixed and constant value $H$. This reflected the
fact that ground states of the Hamiltonian with such boundary conditions
tend to be mostly flat interfaces with only rare and localized
fluctuations provoked by some large deviations of the random fields.
In lower dimensions this is not expected to be the case; rather,on the
basis of the Imry-Ma argument [IM], fluctuations
are expected to grow without bounds as the volumes increase, resulting in the
fact that in the limit as the volume tends to infinity, the probability
to observe the interface near the center of the volume at any given height
should tend to zero meaning that an infinite volume Gibbs state does not
exist. We want to prove a result that reflects this expectation.

To this end
we define, following Aizenman and Wehr [AW], the random equivalent
of translation invariant Gibbs states, namely {\it translation covariant
} Gibbs states.
Let us first note that in the context of random systems, the corresponding
random  Gibbs measures
are most naturally viewed a Gibbs-measure valued random variables on the space
$(\O,\FF,\P)$, i.e.  a   measurable map from $(\O,\FF)$ into the space of
Gibbs measures on the measure space of the dynamical variables, in our
case
$(\Z^{\Z^d},\BB)$, where $\Z^{\Z^d}$ is equipped with the product topology
of the discrete topology on $\Z$ and $\BB$ is the corresponding finitely
generated sigma-algebra (a recent exposition on some formal aspects
of random Gibbs measures is given in  [S]).

\definition{1}[AW]  {\it
A random Gibbs  state
$\mu(\eta)$
is called translation covariant
iff it satisfies, almost surely,

\item{(i)}
$$
\eqalign{
&\mu
\left(\left(\eta_\L(h)+\D\eta_\L(h),
\eta_{\L^c}(h)\right)_{h\in\Z}\right)
\left(\quad\cdot\quad\right)\cr
&=\frac{\mu
\left(\left(\eta_\L(h),\eta_{\L^c}(h)\right)_{h\in\Z}\right)
\left(\quad\cdot\quad
\exp(-\b \e\sum_{x\in\L}\D\eta_x(h_x))
\right)}
{\mu
\left(\left(\eta_\L(h),\eta_{\L^c}(h)\right)_{h\in\Z}\right)
\left(
\exp(-\b \e\sum_{x\in\L}\D\eta_x(h_x))
\right)}
\cr
}
\Eq(2.2)
$$
for any finite volume perturbation $\D\eta_\L(h)$ of the random fields, and
\item{(ii)}
$$
\eqalign{
&\mu
\left(\left(\eta_{x+y}(h)\right)_{x\in\Z^d,h\in\Z}\right)
\left(f\left(h_x\right)_{x\in\Z^d}\right)\cr
&=\mu
\left(\left(\eta_{x}(h)\right)_{x\in\Z^d,h\in\Z}\right)
\left(f\left(h_{x-y}\right)_{x\in\Z^d}\right)\cr
\cr
}
\Eq(2.3)
$$
for all $y\in\Z^d$.

%\item{(iii)}

%for all $H\in\Z$.
}

Let us note that if one translation covariant Gibbs state, say $\mu_0$,
exists, than there exists an infinite family of them, $\mu_H$, for all
$H\in \Z$, where
$$
\eqalign{
&\mu_{H}
\left(\left(\eta_{x}(h)\right)_{x\in\Z^d,h\in\Z}\right)
\left(f\left(h_x\right)_{x\in\Z^d}\right)\cr
&=\mu_{0}
\left(\left(\eta_{x}(h+H)\right)_{x\in\Z^d,h\in\Z}\right)
\left(f\left(h_{x}+H\right)_{x\in\Z^d}\right)\cr
\cr
}
\Eq(2.4)
$$

We will prove the following Theorem:

\theo{1}{\it Suppose that the distribution $\P$ of $\eta_x(h)$  either
\item{(i)}  has no isolated atoms or
\item{(ii)} has compact support,

then, if $d\leq 2$, $\e\neq 0$, for all $\b<\infty$,
%for almost all realizations of the random fields $\eta$,
the SOS model defined through \eqv(1.1) does not permit
translation covariant random Gibbs states.
}


\remark Translation covariant Gibbs states are the nice things one expects to
get as weak limits with simple boundary conditions which in particular
should not be too knowledgeable of the disorder. In particular, property
\eqv(2.2) can only be violated if $\mu_H$ was constructed as a weak limit
with boundary conditions that depended on the random fields in the finite set
$\L$.  It is quite conceivable that rather artificial Gibbs states violating
the conditions \eqv(2.2) and \eqv(2.3) can be constructed in this model. E.g.
it might be possible to  choose a sequence of volumes $\L_n\uparrow\Z^d$ and a
 sequence of random boundary conditions  carefully in such a way as to
ensure
that the corresponding ground states have height $h_0=0$ at the origin. It is
conceivable that such a sequence of measures could converge, but clearly
they are `physically' irrelevant. 

\remark To prove Theorem 1 we will show that the assumption of
translation covariant Gibbs states leads to a contradiction. 
 One might hope that a more direct approach 
based e.g. on the renormalization group method could also work and give 
more precise information on finite volume quantities. Such an approach, however,
appears to be exceedingly difficult. In [K] a result on the absence of
stable interfaces based on that idea was proven, but only in a specific 
mean-field type limit of a  hierarchical model. The reader may find 
it instructive to study that paper, since it hints at the complexities 
occurring in the problem.

\vskip1cm

\chap{2. Proof of the theorem}2



We will show that the assumption that there exist
translation covariant states in $d\leq 2$  leads to a contradiction.
Having realized what it is that we want to prove, the adaptation of the
arguments of Aizenman and Wehr to our situation is almost trivial.
To do so, we define the `order parameters'
$$
\eqalign{
&M(h,h')\equiv\E\left[
\mu_{0}\left(h_x=h\right) \right]
-\E\left[
\mu_{0}\left(h_x=h'\right) \right]
}
\Eq(2.5)
$$
The point here is that {\it if} these quantities vanish, than
the we have the following contradiction
$$
1=\E \sum_{h\in \Z}\mu_{0}\left(h_x=h\right) =\sum_{h\in \Z}
\E \mu_{0}\left(h_x=h\right) =\sum_{h\in \Z}\E\mu_{0}\left(h_x=h'\right)
\Eq(2.5bis)
$$
for any $h'$. In fact, if
$h^*$ denotes any value for which $\E \mu_{0}\left(h_x=h^*\right)>0$,
to arrive at the same
contradiction it is enough to show that
there exists an infinite number of values $h$ such
that $M(h^*,h)=0$.
%$\mu_0(h_x=h)=0$ for all $h$, almost surely,  contradicting the assumption that
%$\mu_0$ is a probability measure.
%Namely, $M(h,h')\equiv 0$ implies that for
%all $h\in \Z$, $\E \mu_0(h_x=h)=q$ for some constant $q\geq 0$. Moreover,
%since for all $N\in \N$, $1\geq \E\sum_{h=-N}^N\mu_0 (h_x=h)=(2N+1)q$, $q=0$
%is the only possibility. But $\E \mu_0(h_x=h)=0$ implies that
%$\mu_o(h_x=h) =0 $ with probability one.

Thus to prove the theorem, we only have to show  that this is the case.
Let us define, for fixed $h$, $H$, $\b$, and
finite volume $\L$ the generating functions
$$
\eqalign{
&\G_\L(h,H)\equiv \frac{1}{\b}
\E\left[
\ln\mu_{0}\left(
\exp(\b\e \sum_{x\in\L}\eta_x(h_x))
\right)
-\ln\mu_{H}\left(
\exp(\b \e\sum_{x\in\L}\eta_x(h_x))
\right)
\Bigl|
%\left(\eta_x(h)\right)_{x\in\L}
\FF_{\L,h}
\right]
}
\Eq(2.6)
$$
where $\FF_{\L,h}$ denotes the sigma-algebra that is generated by the
random variables $\{\eta_x(h)\}_{x\in\L}$.
Define further the random variable
$$
\eqalign{
&\t_x(h,H)
=\E\left[
\mu_{0}\left(h_x=h\right)
-\mu_{H}\left(h_x=h\right)
\Bigl|\FF_{\Z^d,h}
%\left(\eta_x(h)\right)_{x\in\Z^d}
\right]
}
\Eq(2.7)
$$
Then we have the following

\lemma {1} {\it The functions $\G_\L(h,H)$ and $\t_x(h,H)$ have the following 
properties:
\item{(0)}
$$
\eqalign{
&\t_z(h,H)\left(\left(\eta_{x+y}(h)\right)_{x\in\Z^d}\right)
=\t_{z-y}(h,H)\left(\left(\eta_{x}(h)\right)_{x\in\Z^d}\right)
}
\Eq(2.8)
$$

\item{(i)}
For all $x\in\L$
$$
\eqalign{
&\frac{\del}{\del \eta_x(h)}\G_\L(h,H)
=\e\E\left[\t_x(h,H)\Bigl|
\FF_{\L,h}
%\left(\eta_x(h)\right)_{x\in\L}
\right]
}
\Eq(2.9)
$$

\item{(ii)}
$$
\eqalign{
&\E\left[\t_x(h,H)\right]=M(h,h-H)
}
\Eq(2.10)
$$

\item{(iii)}
For all positive  $\b,\e$,
$$
\eqalign{
&|\t_x(h,H)|\leq 1
}
\Eq(2.11)
$$
and
$$
\eqalign{
&\left|\frac{\del}{\del \eta_x(h)}\t_x(h,H)\right|\leq \frac{\e\b}{4}
}
\Eq(2.12)
$$

\item{(iv)}
$$
\eqalign{
&\E[\G_\L(h,H)]=0
}
\Eq(2.13)
$$
}

\proof
\eqv(2.8) follows from \eqv(2.3).
\eqv(2.9) follows from  \eqv(2.2).
\eqv(2.10) is a consequence of the
`covariance w.r.t. height shift' expressed by \eqv(2.4).
The bound \eqv(2.11) is obvious. To prove \eqv(2.12), just note that
$$
\eqalign{
&\frac{\del}{\del \eta_x(h)}\t_x(h,H)
=\e\b\E\left[\mu_{0}\left(h_x=h\right)-\mu_{0}\left(h_x=h\right)^2
-\mu_{H}\left(h_x=h\right)+\mu_{H}\left(h_x=h\right)^2
\Biggl|
\FF_{\Z^d,h}
\right]
}
\Eq(2.14)
$$
\eqv(2.13) follows again from \eqv(2.4).

\endproof

Lemma 1 ensures that we are in the situation
of [AW] Prop.6.1. which allows us to bound the
fluctuations of $\G_\L(h,H)$ from below.
In particular we have from Prop.6.1.
$$
\eqalign{
&\liminf_{\L=[-L,L]^d,L\uparrow\infty}
\E\left[\exp\left(t\G_\L(h,H)/\sqrt{|\L|}\right)\right]
\geq\exp\left(\frac{t^2 b^2}{2} \right)
}
\Eq(2.13bis)
$$
where  (see [AW], Eq.(6.24))
$$
b^2\geq \E \left[\E\left[\G_\L(h,H)|\FF_{0,h}\right]^2\right]
\Eq(2.131)
$$
We distinguish the cases (i) and (ii) in the hypothesis of our theorem.
In case (i), Eq. A.3.2 and Proposition A.3.2, case ii. of [AW]
immediately gives that
$$
b\geq \e\theta_{\P}(M(h,H),1/(\e\b))>0
\Eq(2.132)
$$
if $M(h,H)\neq 0$.

Under the assumption of case (ii), we show the following

\lemma{2}{\it  Let $h^*$ be such that $\E\mu_0(h_x=h^*)>0$. Then there 
exists $H_0<\infty$ such that for all $H\geq H_0$
$$
b\geq \e\g_{\P}(M(h^*,H),1/(\e\b))>0
\Eq(2.133)
$$
if $M(h^*,H)\neq 0$.}

\proof 
>From Proposition A.3.2, case iii. of [AW] \eqv(2.133) follows if 
 the function $\eta_0(h^*)\mapsto g\left(\eta_0(h^*)\right)
\equiv\E\left[\G(h,H)|\FF_{0,h^*}\right]\left(\eta_0(h^*)\right)$
is monotone for all $\eta_0(h^*)\in[A,B]$,
where $[A,B]$ is the convex hull of the support of
the one field distribution $\P$.

To prove the monotonicity, we proceed as follows. From
\eqv(2.9)
we get
$$
\eqalign{
\frac{\del  g}{\del \eta_0(h^*)}
&=\e\E\left[\t_x(h^*,H)\Bigl|\FF_{0,h^*}\right]\cr
&=\e\E\left[\mu_{0}\left(h_x=h^*\right)\Bigl|\FF_{0,h^*}\right]
-\e\E\left[\mu_{H}\left(h_x=h^*\right)\Bigl|\FF_{0,h^*}\right]
}
\Eq(2.91)
$$
It is easy to see that \eqv(2.3) implies the deterministic bounds
$$
\eqalign{
\d_H\exp\left(-2\b\e(B-A)   \right)
\leq\E\left[\mu_{H}\left(h_x=h^*\right)\Bigl|\FF_{0,h^*}\right]\left(
\eta_0(h^*)\right)
\leq \d_H\exp\left(2\b\e(B-A)   \right)
}
\Eq(2.92)
$$
for all $\eta_0(h^*)\in[A,B]$
where $\d_H\equiv \E\left[\mu_{H}\left(h_x=h^*\right)\right]$.
Since
$$
\sum_{H\in\Z}\d_H
=\sum_{H\in\Z}\E\left[\mu_{0}\left(h_x=h^*-H\right)\right]=1
\Eq(2.92bis)
$$
there exists  $H_0$ such that
$\d_H\leq
\d_0\exp\left(-4\b\e(B-A)   \right)$ for all $H\geq H_0$.
This implies by \eqv(2.91) the  desired monotonicity for all  $H\geq H_0$.
\endproof


To conclude the proof of the theorem, we thus only have to to show that
\eqv(2.12) with $b^2>0$ leads to a contradiction.
This relies on the following lemma:

\lemma{3}{\it
$$
\eqalign{
&|\G_\L(h,H)|\leq |H||\del \L|
}
\Eq(2.14bis)
$$
where $|\del \L|=|\{(x,y)|x\in\L,y\in\L^c,|x-y|_2=1\}|$.
}


In fact, \eqv(2.14bis) implies
$$
\eqalign{
&\E\left[\exp\left(t\G_\L(h,H)/\sqrt{|\L|}\right)\right]
\leq\exp\left(|t| |H|\frac {|\del\L|}{\sqrt{|\L|}}\right)
}
\Eq(2.15)
$$
which is the desired contradiction if $\L$ is chosen as e.g. a $d$-dimensional
cube and $d\leq 2$. This  concludes the proof of the
theorem, if we assume Lemma 3. \endproof

To conclude we prove  Lemma 3.

\proof (of Lemma 3)
We focus on one summand in (2.2) and write
$$
\eqalign{
&\E\left[\ln\mu_{H}
\left(
\left(\eta_\L(h),\eta_{\L^c}(h)\right)_{h\in\Z}
\right)
\left(
\exp(\b \e\sum_{x\in\L}\eta_x(h_x))
\right)\Biggl|
\FF_{\L,h}
\right]\cr
&=-\E\left[\ln\mu_{H}
\left(
\left(0_\L,\eta_{\L^c}(h)\right)_{h\in\Z}
\right)
\left(
\exp(-\b \e\sum_{x\in\L}\eta_x(h_x))
\right)\Biggl|
\FF_{\L,h}
\right]\cr
&=-\E\left[\ln\mu_{H}
\left(
\left(0_\L,\eta_{\L^c}(h-H)\right)_{h\in\Z}
\right)
\left(
\exp(-\b \e\sum_{x\in\L}\eta_x(h_x))
\right)\Biggl|
\FF_{\L,h}
\right]\cr
&=-\E\left[\ln\mu_{0}
\left(
\left(0_\L,\eta_{\L^c}(h)\right)_{h\in\Z}
\right)
\left(
\exp(-\b \e\sum_{x\in\L}\eta_x(h_x+H))
\right)\Biggl|
\FF_{\L,h}
\right]\cr
}
\Eq(2.16)
$$
where the first equality is due to the transformation law (1.2)
w.r.t. local perturbations,
the second to the stationarity of the distribution of the random fields
under the shift
$h_x\mapsto h_x+H$ for $x\in\L^c$, the third to (1.4).

Let us now employ the DLR-equations (see [Ge]) to write
$$
\eqalign{
&\mu_{0}
\left(
\left(0_\L,\eta_{\L^c}(h)\right)_{h\in\Z}
\right)
\left(
\exp(-\b \e\sum_{x\in\L}\eta_x(h_x+H))
\right)
=\int\mu_{0}
\left(
\left(0_\L,\eta_{\L^c}(h)\right)_{h\in\Z}
\right)(d\bar h_{\L^c})\times\cr
&\times\frac{\sum_{h_\L\in\Z^\L}
\exp\left(-\b\sum_{{<x,y>}\atop{x,y\in\L}} |h_x-h_y|
-\b\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x-\bar h_y|
-\b \e\sum_{x\in\L}\eta_x(h_x+H))
\right)}
{\sum_{h_\L\in\Z^\L}
\exp\left(-\b\sum_{{<x,y>}\atop{x,y\in\L}} |h_x-h_y|
-\b\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x-\bar h_y|
\right)}
}
\Eq(2.17)
$$
Note that only the numerator is $H$-dependent. Therefor
we introduce $h'_x=h_x+H$ for $x\in\L$
estimate the boundary term in the `surface-energy'
in the exponential in the numerator uniformly by
$$
\eqalign{
&\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x-\bar h_y|
=\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x'-\bar h_y-H|\cr
&\leq \sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x'-\bar h_y|
+|H||\del\L|
}
\Eq(2.18)
$$
and
$$
\eqalign{
&\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x-\bar h_y|
\geq \sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x'-\bar h_y|
-|H||\del\L|
}
\Eq(2.19)
$$
>From this we have
$$
\eqalign{
&\sum_{h_\L\in\Z^\L}
\exp\left(-\b\sum_{{<x,y>}\atop{x,y\in\L}} |h_x-h_y|
-\b\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h_x-\bar h_y|
-\b \e\sum_{x\in\L}\eta_x(h_x+H))
\right)\cr
&\leq \exp\left(\b|H||\del\L|\right)
\sum_{h'_\L\in\Z^\L}
\exp\left(-\b\sum_{{<x,y>}\atop{x,y\in\L}} |h'_x-h'_y|
-\b\sum_{{<x,y>}\atop{x\in\L,y\in\L^c}} |h'_x-\bar h_y|
-\b \e\sum_{x\in\L}\eta_x(h'_x))
\right)
}
\Eq(2.20)
$$
and a similar lower bound. Substituting these bounds in (2.12)
and comparing the $H=0$-term gives (2.10) directly.
\endproof

To summarize the gist of the proof, Lemma 3 roughly the fact that when we deform
a interface aver a local region $\L$ by shifting it up by a distance $H$, then
this `costs' no more than to build a boundary wall, i.e $H|\del L|$. 
On the other hand, the Aizenman-Wehr theorem says that there are always
regions around where such a price is compensated by a corresponding 
gain in random energy. In that sense, the proof really builds along the 
Imry-Ma argument. On the other hand, we see that to make this 
argument rigorous, one has to proceed quite carefully in order to avoid 
possible pathologies that could be produced by very ``exotic' constructions 
of Gibbs states. This somewhat restricts the generality of our 
statement (namely that we only exclude translation covariant Gibbs states
rather then `any' Gibbs states) but such a restriction does not appear
physically  unreasonable.

\vskip 0.5cm
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\end