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\begin{document}

\title{Non-existence of random gradient Gibbs measures \\ 
%in disordered anharmonic 
%massless 
 in  continuous interface models in $d=2$.} 
%\thanks{Work partially
%%supported by   ?? }}
\author{
Aernout C.D. van Enter%\thanks{Research supported by XXXXXXX}
\footnote{
University of Groningen, 
Department of Mathematics and Computing Sciences, 
Blauwborgje 3,   
9747 AC Groningen, 
The Netherlands
\texttt{aenter@phys.rug.nl}, 
%\texttt{a.c.d. sdf},
%\texttt{http://www..... }}
}\, and
Christof K\"ulske
%\thanks{Research supported by Deutsche Forschungsgemeinschaft}
\footnote{
University of Groningen, 
Department of Mathematics and Computing Sciences, 
Blauwborgje 3,   
9747 AC Groningen, 
The Netherlands
%EURANDOM, LG 1.34,
\texttt{kuelske@math.rug.nl}, 
\texttt{ http://www.math.rug.nl/$\sim$kuelske/ }}
}


\maketitle

\begin{abstract}
We consider statistical mechanics models of 
continuous spins in a disordered environment. 
These models have a natural interpretation as effective interface models. 
%a  disordered continuous spin model  (resp. effective interface model). 
It is well-known that without disorder there are no interface Gibbs measures in 
infinite volume in dimension $d=2$, while there are 
``gradient  Gibbs measures'' describing an infinite-volume distribution for the 
increments of the field, as was shown by  Funaki and Spohn. 

%Funaki and Spohn showed that for convex potentials V there are 
%flat (and tilted) 
%gradient states  
In the present paper we show that adding a disorder term prohibits the existence of 
such gradient Gibbs measures for general interaction potentials in $d=2$. This non-existence 
result generalizes the simple case of Gaussian fields where it follows from  an explicit computation. 

In $d=3$ where random gradient Gibbs measures 
are expected to exist, our method provides a lower bound of the order 
of the inverse of the distance on the decay 
of correlations of Gibbs expectations w.r.t. the distribution of the random environment.  


 \end{abstract}

\smallskip
\noindent {\bf AMS 2000 subject classification:} 60K57, 82B24,82B44.



\section{Introduction} \label{sect:intro}

\subsection{The setup} 
 
Our model is given in terms of the formal 
infinite-volume Hamiltonian
\begin{equation}\begin{split}\label{eins}
&H[\eta]\left(\phi\right)
=\frac{1}{2}\sum_{i,j}p(i-j)V(\phi_i-\phi_j) - \sum_i\eta_i \phi_i\cr
\end{split}
\end{equation}
Here the fields (unbounded continuous spins) $\phi_i\in \R$ represent
height variables of a random surface at the site $i\in \Z^d$. 
Such a model is motivated as an effective model for  
the study of phase boundaries at a mesoscopic level in statistical mechanics.  
The disorder configuration $\eta=(\eta_i)_{i\in \R^d}$ denotes an arbitrary fixed configuration of external fields, modelling a "quenched" (or frozen) random environment.   

For background and various earlier results about both 
continuous and discrete interface models without disorder 
see  \cite{Vel06,BrMeFr86,Fun, Sheff, BiKo06} and references therein, 
for results about discrete  interface models in the presence 
of disorder see \cite{BoKu94, BoKu96}.


\subsubsection*{Assumptions}

The pair potential $V(t)$ is assumed to be even,  $V(t)=V(-t)$, and 
continuously differentiable.  
We require  that  $V$ grows faster than linearly to infinity, i.e. 
$\lim_{|t|\uparrow \infty}\frac{V(t)}{|t|^{1+\varepsilon}}=\infty$ for some positive $\varepsilon$. 
$p(\cdot )$ is the transition kernel of  a simple random walk on $\Z^d$, assumed 
to be symmetric and of finite range.  
We further demand for simplicity that the random fields $\eta_i$ be i.i.d. under the distribution $\P$, 
symmetric, and with finite  nonzero  second moment. We denote the expectation 
w.r.t $\P$ by the symbol $\E$. 




\subsubsection*{Vector Fields, Fields,  and Gradient Fields}

We call the set of sites  $\Z^d$ with oriented edges between 
$i,j$ whenever $p(i-j)>0$, the {\it graph of
the random walk}. 
We call a {\it vector field } (or vector field configuration)
a map from the set of oriented edges $ij$ of this graph to $\R$ such that 
$V_{i,j}=- V_{j,i}$. 
Every field configuration $U=(U_i)_{i\in \Z^d}$ gives rise to 
a vector field configuration $U'=U'(U)$ by $U'_{i,j}=U(i)-U(j)$. 
In this case we call $U'$ {\it the gradient field of $U$}. 
There does not need to exist such a function for a general 
vector field $V$. Its existence is equivalent to the {\it loop condition}  $V_{i,j}+V_{j,k}+V_{k,l}+V_{l,i}=0$ along  
plaquettes 
$i,j,k,l$. We denote the set of all field configurations in infinite volume 
by $\O$ and the set of all gradient field configurations in infinite volume 
by $\O'$. 


\subsubsection*{Gibbs measures and gradient Gibbs measures}

The quenched finite-volume Gibbs measures (or local specification) 
corresponding to the Hamiltonian (\ref{eins})
 in a finite volume $\L\sb \Z^d$, 
a boundary condition $\hat \phi$ and a fixed disorder configuration in $\L$, 
are given by the standard expression  
\begin{equation}\begin{split}\label{localspecification}
&\int \mu^{\hat \phi}_{\L}[\eta](\phi)(F(\phi))\cr
&:=\frac{\int d\phi_{\L}F(\phi_{\L},\hat \phi_{\L^c})
e^{ - \frac{1}{2}\sum_{i,j\in \L}p(i-j)V(\phi_i-\phi_j) 
- \sum_{i\in\L,j\in \L^c}p(i-j)V(\phi_i-\hat\phi_j) 
+  \sum_{i\in \L}\eta_i \phi_i}}{Z_{\L}^{\hat \phi}[\eta]}\cr
\end{split}
\end{equation}
where $Z_{\L}^{\hat \phi}[\eta]$ denotes the normalization constant
that turns the last expression into a probability measure. 
It is a simple matter to see that the growth condition on $V$ guarantees 
the finiteness of the integrals appearing in (\ref{localspecification})
for all arbitrarily fixed choices of $\eta$. 



We note that the Hamiltonian $H[\eta]$ 
changes only by a configuration-independent constant 
under the joint shift $\phi_x\mapsto \phi_x + c$ of all height variables with the same 
$c$. This holds true for any fixed configuration $\eta$.  
Hence finite-volume Gibbs measures tranform under a shift 
of the boundary condition by a shift of the integration variables. 
Using this invariance under height shifts we can lift the finite-volume measures to measures 
on gradient configurations,  defining  the {\it gradient finite-volume Gibbs measures} (gradient local specification). 
These are the probability kernels from $\O'$ to $\O'$ given   by
\begin{equation}
\begin{split}
&\int(\mu')^{\bar \phi'}_{\L}  [\eta](d\phi')G(\phi'):=
\int\mu^{\bar \phi}_{\L}  [\eta](d\phi)G(\phi'(\phi))
\cr
\end{split}
\end{equation}
where $\bar \phi$ is any field configuration whose 
gradient field is $\bar \phi'$. % It is determined up to a constant. 
 
Finally we call $\mu'$ a {\it gradient Gibbs measure} 
if it satisfies the DLR equation, that is 
\begin{equation}
\begin{split}
&\int\mu'(d\bar \phi')\int (\mu')^{\bar \phi'}_{\L}(d\phi')f(\phi')=
\int\mu'(d\phi')f(\phi')
\cr
\end{split}
\end{equation}
Here we dropped the $\eta$ from our notation. 

Gradient Gibbs measures have the advantage that they may exist, even in 
situations where a proper Gibbs measure does not. 
When this happens, 
it means that the interface is locally smooth, although at large scales it 
has too large fluctuations to stay around a given height. 
For background about these 
objects we refer to \cite{Fun,FunSp,Sheff}. 
 


\subsubsection*{Translation-covariant (gradient) Gibbs measures}

Denote by $\t_r \phi=(\phi_{i-r})_{i\in \Z^d}$ the shift of field configurations 
on the lattice by a vector $r\in \Z^d$.  A measurable map $\eta\mapsto \mu[\eta]$ is called 
a {\it translation-covariant random Gibbs measure} if $\mu[\eta]$ is a Gibbs measure 
for almost any $\eta$, and if it behaves appropriately under lattice shifts, i.e. 
$\int\mu[\t_r \eta](d\phi)F(\phi)=\int\mu[\eta](d\phi)F(\t_r\phi)$ for all 
translation vectors $r$. 
This means that the functional dependence of the Gibbs measure on the underlying 
disordered environment is the same at every point in space. 
This notion goes back to \cite{AW} and generalizes the notion 
of a translation-invariant Gibbs measure to the set-up of disordered 
systems. 

Naturally, a measurable map $\eta\mapsto \mu'[\eta]$ is called 
a {\it translation-covariant random gradient Gibbs measure} if 
$\int\mu'[\t_r \eta](d\phi')F(\phi')=\int\mu'[\eta](d\phi')F(\t_r\phi')$ for all $r$. 




%a previous paper 




\subsection{Main results}
A main question to be asked in interface models is 
whether the fluctuations of an interface that is restricted to a finite 
volume  will remain bounded when the volume tends to infinity, 
so that there is an infinite-volume Gibbs measure (or  gradient Gibbs measure)  
describing a localized interface.  
This question is well-understood in translation-invariant 
continuous-height models, and it is the purpose of this note to discuss 
such models in a random environment. 


Let us start by discussing the non-random model in the physically 
interesting dimension $d=2$.  
We start with the Gaussian model where $V(t)=\frac{t^2}{2}$. Gaussian models 
are simple because explicit computations can be done. 
These  show that the Gibbs measure $\mu(d\phi)$ 
does not exist in infinite volume, but the gradient field (gradient Gibbs measure)
does exist in  infinite volume. Both statements may easily be derived by looking 
at the properties of (differences  of)  the matrix elements of 
$(I-P_\L)^{-1}$ which appears here as a covariance matrix, 
where $P_\L$ is the transition operator, restricted to $\L$,  given in terms 
of the random walk kernel $p$.  
Equivalently one may say that the infinite-volume measure exists 
conditioned on the fact that one of the variables $\phi_{i}$ is pinned at the value zero. 
Funaki and Spohn showed more generally that for convex potentials $V$ there are tilted 
gradient Gibbs measures and conversely a  
gradient Gibbs measure is uniquely determined by the tilt \cite{Sheff,Fun,FunSp}.

For (very) non-convex $V$ new phenomena are appearing: 
There may be a first-order phase transition in 
the temperature where the structure of the interface (at zero tilt) changes, 
as shown by Biskup and Kotecky (2006), \cite{BiKo06}.
This phenomenon is related to the phase transition 
seen in rotator models with a very nonlinear potentials exhibited in \cite{ES1, ES2}
The basic mechanism is an energy-entropy transition 
such as was first proven for the Potts model 
for  a sufficiently large number of spin values \cite{KS}. 
 

What can we say for the random model? 
In \cite{KuOr06} the authors showed a quenched deterministic lower bound 
on the fluctuations in the anharmonic model in a finite box 
of the order square root of the sidelength, uniformly in the 
disorder. In particular this implies that there won't be any disordered 
infinite-volume Gibbs measures in $d=2$. This latter statement is not surprising 
since there is already non-existence of the unpinned interface without disorder. 
But what will happen to the gradient Gibbs measure that is known 
to exist without disorder, once  we allow for a disordered environment? 


\subsubsection*{Gaussian results and predictions}

Let us first look in some detail 
at the special case of a Gaussian gradient measure where $V(t)=\frac{t^2}{2}$, 
specializing to nearest neighbor interactions. 
Then, for any fixed configuration $\eta_{\L}$, the finite-volume gradient Gibbs measure 
with zero boundary condition is a Gaussian measure  with expected value 
\begin{equation}\begin{split}\label{randommean}
X_{ij}^\L[\eta]:=\int\mu'_{\L}[\eta](d\phi')(\phi'_{ij})
&= \sum_{y\in \L} T^{\L}_{ij,y} \eta_y \qquad \text{ where }\cr
T^{\L}_{ij,y}&=(-\D_{\L})^{-1}_{i,y}-
 (-\D_{\L})^{-1}_{j,y} 
\end{split}
\end{equation}
The  matrix  $\text{cov}_{\L}(\phi_{i j} ;\phi_{l m} )
= (-\D_{\L})^{-1}_{i,l}-(-\D_{\L})^{-1}_{i,m}- (-\D_{\L})^{-1}_{j,l}+(-\D_{\L})^{-1}_{j,m}$
is the same as in the model without disorder; 
its infinite-volume limit exists in any dimension $d\geq 2$, by a simple computation.  

What about the  infinite-volume 
limit of the mean value (\ref{randommean}), as a function of  the disorder? 
We note that $X^\L[\eta]:=(X^\L_{ij}[\eta])_{{ij \in \L\times\L}\atop{i\sim j}}$  is itself a random vector field 
with covariance 
\begin{equation}\begin{split}\label{mcov}
&C_{\L}(ij,lm)=\E(\eta^2_0)
\sum_{y\in \L} T^{\L}_{ij,y}T^{\L}_{lm,y}\cr
 \end{split}
\end{equation}

Moreover, it is also a gradient vector field, since the loop condition carries over 
from $\phi'$, by linearity of the Gibbs expectation.

For its variance we have in particular 
\begin{equation}\begin{split} 
&C_{\L}(ij,ij)=\E(\eta^2_0)\sum_{y\in \L}(T^{\L}_{ij,y})^2  \cr
 \end{split}
\end{equation}
In two dimensions the infinite-volume limit 
of this expression does not exist since 
 $\int^N r (\frac{d}{dr}\log r)^2 d r \sim \log N$, 
 when the sidelength $N$ of the box diverges to infinity. 
 In dimension $d>2$, we have $\int^N r^{d-1} (\frac{d}{dr} r^{-{(d-2)}})^2 d r \sim 
\int^N r^{-(d-1)} d r$, so the fluctuations stay bounded. 

In particular this explicit computation shows that 
in the Gaussian model there can not be 
infinite-volume random gradient Gibbs measures in $d=2$.
Indeed,  already the local -- 
short-distance -- fluctuations are roughening up the interface.  




It is the main result of this paper to show that this result persists also 
for an anharmonic potential where explicit computations 
are not possible. 








 







\begin{thm} {\bf (Non-existence in $d=2$)}

Suppose $d=2$. 
Then there does not exist a translation-covariant 
random gradient Gibbs measure $\mu'[\eta](d\phi')$ 
that  satisfies the integrability condition 
\begin{equation}
\label{eq:finitenabla}\begin{split}
&\E\Bigl|\int \mu'[\eta](d\phi') V'(\phi'_{ij})\Bigr|<\infty
\cr
\end{split}
\end{equation}
for sites $i\sim j$.
\end{thm}


%It is interesting to contrast this result with the following observation:

%{\bf Remark:} Suppose $d$ is arbitrary. Take $V(t)=\frac{1}{1-t^2}$. 
%Then, at any fixed $\eta$ there does exist  
%(at least one) gradient Gibbs measure $\mu'[\eta](d\phi')$. 
%This follows simply because the  state-space of allowed gradient 
%configurations becomes compact by the choice of the potential. 
%Hence there are gradient Gibbs measures 
%at any fixed choice of $\eta$. Our previous theorem however shows that 
%there can be no nice translation-covariant map that is built from these 
%measures. 





Let us consider  dimension $d=3$ where translation-covariant infinite-volume 
gradient measures are believed to exist. The next result shows that, 
if they do, they must have slow decay of correlations w.r.t. the random environment.  


\begin{thm}{\bf (Slow decay of correlations in 
$d=3$) }

Suppose that $d=3$ and that $\mu'[\eta](d\phi')$ is a random gradient Gibbs measure.  
Put 
\begin{equation}
\label{eq:fin2}\begin{split}
&C(ij,kl):=\E\Bigl( \int \mu'[\eta](d\phi') V'(\phi'_{ij}) 
\int \mu[\eta](d\phi') V'(\phi'_{kl}) 
\Bigr ) 
\cr
\end{split}
\end{equation}
for sites $i\sim j$ and $k\sim l$. 

Then
\begin{equation}
\label{eq:fin3}\begin{split}
&\lim_{r\uparrow\infty}\sup_{  i,j,k,l:|i-k|\geq r  } \frac{|C(ij,kl)|}{ 
r^{-(1 + \e)}}=\infty
\cr
\end{split}
\end{equation}
for all $\e>0$. 
\end{thm}

{\bf Remark: }
Note that in $d=3$  for the case of the quadratic nearest neighbor potential $V(t)=\frac{t^2}{2}$ 
a translation-covariant random gradient Gibbs measure  exists, by explicit computation. 
It is the Gaussian field whose  covariance and mean are given by the infinite-volume limits 
of (\ref{mcov}) and (\ref{randommean}). 
An explicit computation with this Gibbs measure also shows 
that the power of the bound given in the last theorem is optimal, see Subsection 2.3. 

\bigskip 




The method of proof of both theorems relies on a surface-volume comparison.  
It is inspired by the Aizenman-Wehr method (devised in \cite{AW} and used on 
discrete interfaces in \cite{BoKu96}), but different in several aspects.  
As opposed to the Aizenman-Wehr Ansatz, where free energies are considered, 
we derive a discrete divergence equation $\eta=\nabla X$ 
where the external random field (the disorder) acts as a source and 
the vector field $X$ is provided by the expectation of $V'(\phi_i-\phi_j)$ 
with respect to the hypothesized gradient Gibbs measure. 
Exploiting a discrete version of Stokes' theorem and a volume versus surface 
comparison between terms we arrive at a contradiction. 

%
%\vfill\eject

%
%{\bf Let us comment on the dimensionality. 
%Aizenman-Wehr prove the absence 
%of ferromagnetic order in the random field Ising model in dimensions $d\leq 2$ (or more 
%generally bounded discrete spin models.) 
%A variation of it was used to show the absence of translation-covariant localized 
%interfaces in dimension $d=2$. Ref Bovier-Kuelske??
%For bounded continuous spins (rotator models) of random field type  
%Aizenman and Wehr prove absence of ferromagnetic order in $d\leq 4$. 
%This dimensionality arises by a competition 
%of a random volume term with a surface term which is (only) of the order $L^{d-2}$. 

%So a naive 
%guess might be that $d=4$ is the critical dimensionality also 
%in the present model.  
%This is not the case, as the above Gaussian 
%computation shows, and the critical dimensionality $d=2$, as given 
%by our theorem, is the correct one. 
%}

%

%

%

%
%\vfill\eject 




\section{Proof of the Theorems} 
%random fields}



%\subsection{Preliminaries}

%

%

%
%\subsection{Non-existence of non-harmonic random gradient field in $d\leq \frac{\a}{\a-1}$ 
%for $\a$-stable  disorder}

%
%\begin{thm}
%%Be $\P$ 
%Let $\P$ be 
%an i.i.d. $\a$-stable symmetric distribution of random fields, 
%for $\a\in (1,2)$.  
%Suppose $d\leq d(\a):=\frac{\a}{\a-1}$.

%Then there does not exist a translation-covariant 
%random gradient Gibbs  $\mu'[\eta](d\phi')$ 
%that  satisfies the integrability condition 
%\begin{equation}
%\label{eq:finitenabla}\begin{split}
%&\E\Bigl|\int \mu'[\eta](d\phi') V'(\phi'_{ij})\Bigr|<\infty
%\cr
%\end{split}
%\end{equation}
%for sites $i\sim j$.
%\end{thm}

%

%We start with the Proof of Theorem 1 which 
%%explaines 
%explains
%the notion 
%of a vector field and the volume versus surface competition we explore 
%and use to derive a contradiction.  
%\bigskip 

\subsection{Proof of Theorem 1.1} 



The method of proof is to argue from the existence 
of a translation-covariant gradient Gibbs measure to a contradiction in $d=2$.  
To do this we start with the following definition. 


\begin{defn}
%Be 
Let $\mu'$ be an 
%infinite volume 
 infinite-volume 
gradient Gibbs measure, in a random or non-random model. 
Then we call the vector field $X$ 
on the graph of the random walk  given by
\begin{equation}
\label{eq:2.2generalfeldmarschall}\begin{split}
&X_{ij}:=\int\mu'(d\phi')(V' (\phi'_{ij}))\cr
\cr
\end{split}
\end{equation}
the associated vector field. 
\end{defn}
This indeed defines a vector field because, by symmetry, $V'(x)=-V'(-x)$ and hence
 $X_{ij}=-X_{ji}$. Of course the same definition can be made 
 in finite volume, but for our proof we will work immediately in infinite volume. 

We note that a gradient Gibbs measure in the Gaussian model provides even a 
gradient vector field, since $V'(x)=c x$ is a linear function, and 
so the loop condition carries over from $\phi'_{ij}$ 
to $X_{ij}$. For general non-quadratic 
potentials $V$ the vector fields $X_{ij}$ won't  be gradient fields. 
This explains why, in the anharmonic model, we must work 
with vector fields (functions on the edges). 

\begin{prop}
Let $\mu'[\eta](d\phi')$ be a random 
%infinite volume 
infinite-volume 
gradient Gibbs measure 
and $X_{ij}[\eta]:=\int\mu'[\eta](d\phi')(V' (\phi'_{ij}))$ the associated 
vector field on the random walk graph for $i\sim j$.
Then 
\begin{equation}
\label{eq:divergence}\begin{split}
\eta_i= \sum_{j}p(j-i)X_{i j}[\eta]
\cr
\end{split}
\end{equation}
for all $i\in \Z^d$. This equation will be called ``divergence equation''. 
\end{prop}

{\bf Remark: }
This equation is a discrete version of the equation $\eta=\nabla  \cdot X$ 
which holds in continuous space $\R^d$. 

Comparing with Maxwell's equations, 
$X$ plays the role of the electric field, and $\eta$ plays the role of 
the electric charge. 
In the Gaussian case, we know that $X$ is a gradient-field, i.e. curl-free. 
In the case of general potentials $V$ it won't be.  
Note that  the kernel of this equation, that is the vector fields $X$ which are the 
solutions of the equation $0=\nabla  \cdot X$,  consists of all divergence-free 
vector fields,  which are plenty. In particular the divergence equation 
does not  allow to determine $X$ uniquely in terms of $\eta$.                                            
 

\bigskip 

{\bf Proof of the proposition: } We look at the one-site local specification at the site $i$.
Take a single-site integral over 
$\phi_{i}$ appearing in the partition 
function and use partial integration to write 
\begin{equation}
\begin{split}
&
\int d\phi_{i} 
\exp\Bigl(\sum_{j\sim i
}p(j-i)V(\phi_{i} -\phi_{j} )
\Bigr)\Bigl(\eta_i\exp\Bigl(\eta_i\phi_i\Bigr)\Bigr) \cr
&=\int d\phi_{i} \Bigl(-\frac{\partial }{\partial\phi_{i} } \exp\Bigl(-\sum_{j\sim i
}p(j-i)V(\phi_{i} -\phi_{j} )\Bigr)
\Bigr)\exp\Bigl(\eta_i\phi_i\Bigr)\cr
&=\int d\phi_{i}
\sum_{j\sim i
}p(j-i)V'(\phi_{i} -\phi_{j} )
 \exp\Bigl(-\sum_{j\sim i
}p(j-i)V(\phi_{i} -\phi_{j} )+\eta_i\phi_i 
\Bigr)\cr
\cr
\end{split}
\end{equation}
Now, integrate over the remaining $\phi_k$ for $k$ in a finite 
volume $\L$ to see that 
\begin{equation}
\label{eq:divergence1}\begin{split}
\eta_i&= \sum_{j}p(j-i)
\int \mu^{\bar \phi}_{\L}  [\eta](d\phi_{\L})V'(\phi_{i} -\phi_{j} )
\cr
&= \sum_{j}p(j-i)
\int \mu^{\bar \phi'}_{\L}  [\eta](d\phi'_{\L})V'(\phi'_{ij} )
\cr
\end{split}
\end{equation}
Integrating this equation over the boundary condition $\bar \phi'$ w.r.t. 
$\mu'[\eta]$ and using the DLR equation for the gradient 
%field 
measure
implies the proposition. $\Cox$

Summing the divergence equation over $i$ in a finite volume $\L$ 
we note that the contributions of the edges that are contained 
in $\L$ vanish, due to the property of X being a vector field. 
Hence we arrive at the following corollary. 

\begin{cor} (Integral form of divergence equation)
\begin{equation}
\label{eq:obenx}\begin{split}
\sum_{i\in \L}\eta_i 
= \sum_{{i\sim j}\atop{ i\in \L,j\in \L^c}} p(j-i)X_{ij}[\eta]
\cr
\end{split}
\end{equation}
\end{cor}

Note that the sum of boundary terms on the r.h.s. plays the role of 
a surface integral in the Stokes equation.

Let us now specialize to $d=2$ and prove Theorem 1. 
We choose $\L=\{-L,-L+1,\dots, L\}^2$ and normalize by $\frac{1}{L}$. 
 
We remark that the sum over boundary bonds $i j$ decomposes into the 
$4$ sides of a square, whose bonds will be denoted by  
$B_L(\nu)$, $\nu=1,2,3,4$. 
 
So we have 
 \begin{equation}
\label{eq:untenx}\begin{split}
\frac{1}{L}\sum_{i\in \L}\eta_i
=  \frac{1}{L}\sum_{i\in \L,j\in \L^c}p(i-j)X_{ij}[\eta]\cr
=\sum_{\nu=1,2,3,4}\frac{1}{L}\sum_{\langle i,j\rangle \in B_L(\nu)}p(i-j)X_{ij}[\eta]\cr
\end{split}
\end{equation}
By the ergodic theorem, each of the four sums converges to its expected value 
\begin{equation}
\label{eq:untenx1}\begin{split}
\lim_{L\uparrow\infty} \frac{1}{L}
\sum_{\langle i,j\rangle \in B_L(\nu)}p(i-j)X_{ij}[\eta]=
2\sum_{j: j\cdot e_\nu>0}p(j)\E (X_{0 j}[\eta]) \cr
\cr
\end{split}
\end{equation}
in whatever sense (e.g. almost surely or in $L^2$) where $e_1,e_2,e_3=-e_1,e_4=-e_2$ 
are unit vectors on $\Z^d$. 

By the CLT the l.h.s. of equation (\ref{eq:untenx})
does not  converge almost surely to a constant; indeed $\frac{1}{L}\sum_{i\in \L}\eta_i$ converges only in distribution  
to a Gauss distribution with a non-zero variance. This is a contradiction to convergence to 
a delta-distribution, which proves 
Theorem 1.1. $\Cox$
\bigskip
\bigskip


\subsection{Proof of Theorem 1.2}


%\bigskip For an $\a$-stable disorder distribution , in general dimension $d$ we can redo the same. 
%We get 
% \begin{equation}
%\label{eq:untenxx}\begin{split}
%\frac{1}{L^{d-1}}\sum_{i\in V}\eta_i
%=  \frac{1}{L^{d-1}}\sum_{i\in V,j\in V^c}p(i-j)X_{ij}[\eta]\cr
%=\sum_{\nu=1,\dots, 2d } \frac{1}{L^{d-1}}\sum_{\langle i,j\rangle \in B_L(\nu)}p(i-j)X_{ij}[\eta]\cr
%\cr
%\end{split}
%\end{equation}
%We note that 
%$E\exp\Bigl(\frac{i t}{L^{d-1}}\sum_{i\in V}\eta_i \Bigr)
%= e^{- C L^{d - \a(d-1)}|t|}= 
%e^{- C L^{- d (\a-1)+ \a}|t|}$. 

%So, the distribution blows up  
%for $- d (\a-1)+ \a\geq 0$ (or stays the same in the case of equality). 
%On the other hand, 
%assuming integrability, the r.h.s. converges to the delta distribution at the mean, 
%by the ergodic theorem (SLLN). This is a contradiction, which proves the statement. 
%$\Cox$

%

%\vfill\eject



%Let us now turn to dimensions $d=3$. 

%
%Look at the Gaussian model 
%first. When we allow possibly non-Gaussian i.i.d. 
%distributions for the $\eta_i$'s, we see that the random mean of 
%the Gaussian gradient measure with zero boundary condition 
%converges to the the random variable $\sum_{y\in \Z^d} T^{\infty}_{ij,y} \eta_y$
%where $T^{\infty}_{ij,y}$ is the difference of Green's functions taken in infinite volume. 
%(Here we use that the a.s. convergence of the last series is implied 
%by $\sum_{y\in \Z^d}(T^{\infty}_{ij,y})^2 <\infty$, by the Kolmogorov-Khintchin theorem 
%(Theorem IV.2.1 \cite{Shi84})). CHECK!

%Since we have a.s. convergence to a limiting gradient measure 
%we do not need a metastate description here. 

%\bigskip 
%\bigskip 

%{\bf Proof of Theorem 3: }

Take the square of the integral form of the divergence equation 
and take its expectation w.r.t. the measure $\P$. 
\begin{equation}
\label{eq:squareddiv}\begin{split}
\E(\eta^2_0) |\L| 
= \sum_{{i\sim j}\atop{ i\in \L,j\in \L^c}} \sum_{{k\sim l}\atop{ k\in \L,l\in \L^c}} p(j-i)p(k-l)C_{ij,kl} 
\cr
\end{split}
\end{equation}

Assume the bound  
\begin{equation}
\label{contrabound}\begin{split}
&C_{ij,kl} \leq \text{Const} (1+ |i-k |^2 )^{-q}\cr
\end{split}
\end{equation}
Let us take for $\L$ a ball w.r.t. the Euclidean metric, of radius $L$. Then, 
for large $L$ the l.h.s. behaves like a constant times $L^3$.  

The large-$L$ asymptotics of the r.h.s. of (\ref{eq:squareddiv}) is provided by 
the large-$L$ behavior of the double integral 
\begin{equation}
\label{eq:1001}\begin{split}
\int_{L S^2}d\l(x) \int_{L S^2}d\l(y) (1+ |x-y |^2 )^{-q}
\cr
\end{split}
\end{equation}
where $ L S^2$ denotes the two-dimensional sphere with 
radius $L$ and $\l$ is the Lebesgue measure on the sphere. 
By rotation-invariance, this equals 
\begin{equation}
\label{eq:10012}\begin{split}
&c  L^2\int_{L S^2}d\l(y) (1+ |y-L e |^2 )^{-q}\cr
&=c L^4 \int_{S^2}d\l(z) (1+ L^2 |z-e |^2 )^{-q}\cr
\cr
\end{split}
\end{equation}
where $e$ is the North Pole of $S^2$. 
Using polar coordinates $\cos \theta=s$ for the point $z$, 
where $s=1$ is corresponding to the point $e$, we have
$|z-e|^2=2(1-s)$. This gives  
\begin{equation}
\label{eq:10013}\begin{split}
& \int_{S^2}d\l(z) (1+ L^2 |z-e |^2 )^{-q}\cr
&=2 \pi \int_{-1}^1 ds (1+ L^2 2(1-s) )^{-q}=\frac{2 \pi 2^{1-q}}{1-q} (1+ 2L^2)^{-q}
\cr
\end{split}
\end{equation}
By (\ref{eq:squareddiv}) we thus have under the assumption of 
(\ref{contrabound}) that $L^3\leq c L^{4-2 q}$, for large $L$, 
which can only be true for $q\leq \frac{1}{2}$.
This implies that $C_{ij,kl}$ can not decay faster 
than the inverse distance between $i$ and $k$. $\Cox$ 



\subsection{Sharpness of polynomial decay in $d=3$}


\begin{prop}
Suppose $d=3$. 
Let  $\mu'[\eta](d\phi')$ be a random gradient Gibbs measure 
for the Gaussian model $V(x)=\frac{x^2}{2}$  with i.i.d. $\eta_i$ with finite second moment. 
 
Then
\begin{equation}
\label{eq:fin4}\begin{split}
&\lim_{r\uparrow\infty}\sup_{  i,j,k,l:|i-k|\geq r  } \frac{|C(ij,kl)|}{ 
r^{-1 }}<\infty
\cr
\end{split}
\end{equation}
\end{prop}

{\bf Remark: } 
This shows that the fluctuation lower bound is sharp in $d=3$. 




{\bf Proof: }
Let us choose $i=- Re$ and $k=R e$ where $e$ is a unit coordinate vector 
so  $2 R$ is the distance between $i$ and $k$. 
To estimate the large-$R$ asymptotics of the decay of the covariance 
in infinite volume which is given by 

\begin{equation}\begin{split}
&C_{\Z^3}(ij,lm)=E(\eta^2_0)
\sum_{y\in \L} T^{\Z^3}_{ij,y}T^{\Z^3}_{lm,y}\cr
 \end{split}
\end{equation}
we are led to consider the large-$R$ asymptotics 
of the following  integral 
\begin{equation}
\label{eq:fin5}\begin{split}
&I(R):=\int d^3 y \frac{1}{1+|y- R e|^2}\frac{1}{1+|y+ R e|^2}\cr
\end{split}
\end{equation}
Now the proposition follows by an explicit computation. 

Indeed, using cylindrical coordinates this integral can be rewritten 
as 
\begin{equation}
\label{eq:fin6}\begin{split}
&I(R)=2\pi \int_0^\infty dz \int_{0}^\infty d\bar r \bar r \frac{1}{1+(z-R)^2 + \bar r^2} \frac{1}{1+(z+R)^2 + \bar r^2}\cr
&=\frac{\pi}{4 R } \int_0^\infty \frac{dz}{z} \lim_{S\uparrow\infty} \int_{0}^S ds 
\Bigl(\frac{1}{1+(z-R)^2 + s} - \frac{1}{1+(z+R)^2 + s} \Bigr) \cr
&=\frac{\pi}{4 R }J(R)\cr
\end{split}
\end{equation}
Here we have used the substitution $s=r^2$ and 
\begin{equation}
\label{eq:fin7}\begin{split}
&J(R):=\int_0^\infty  \frac{dz}{z} \log \frac{1+(z+R)^2}{1+(z-R)^2}
=\int_0^\infty  \frac{du}{u} \log \frac{R^{-2}+(u+1)^2}{R^{-2}+(u-1)^2}\cr
\end{split}
\end{equation}
Finally, the function $J(R)$ is increasing in $R$ and has the finite limit 
\begin{equation}
\label{eq:fin8}\begin{split}
&\lim_{R\uparrow\infty}J(R)=\int_0^\infty  \frac{du}{u} \log \frac{(u+1)^2}{(u-1)^2}=
8 \int_0^1  \frac{\log x}{x^2-1} dx = \pi^2\cr
\end{split}
\end{equation}
(See an integral table 
%and
or 
check with Mathematica). 
This shows the proposition. $\Cox$



%{\bf Acknowledgements: } 

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\end{thebibliography}

\end{document}
