\documentclass[11pt]{article}
\input{epsf.sty}
\usepackage{hyperref}
%\usepackage{showlabels}
\usepackage{amsmath, amssymb}
\textwidth 146 mm
\textheight 230 mm
\oddsidemargin 7mm \evensidemargin -1mm \topmargin -4mm
\newtheorem {thm}{Theorem}[section]
\newtheorem {prop}[thm]{Proposition}
\newtheorem {lem}[thm]{Lemma}
\newtheorem {cor}[thm]{Corollary}
\newtheorem {defn}[thm]{Definition}
\newtheorem {conj}[thm]{Conjecture}
\newtheorem {rem}[thm]{Remark}
\newtheorem {cond}[thm]{Condition}
\def\Cox{\hfill \Box}
\def\del{\partial}
\def\N{{\Bbb N}}
\def\Z{{\Bbb Z}}
\def\R{{\Bbb R}}
\def\P{{\Bbb P}}
\def\Q{{\Bbb Q}}
\def\E{{\Bbb E}}

%\def\Z{{\bf Z}}
%\def\R{{\bf R}}
%\def\P{{\bf P}}
%\def\Q{{\bf Q}}
\def\0{{\bf 0}}
\def\cW{{\cal W}}
\def\eps{\varepsilon}
\def\dist{{\rm dist}}
\def\parent{{\rm parent}}
\def\leqd{\stackrel{\cal D}{\preceq}}
\def\ba{{\backslash}}
\def\sb{{\subset}}
\def\ov{\overline}

\def\a{\alpha}

\def\b{\beta}

\def\d{\delta}

\def\e{\varepsilon}

\def\f{\varphi}
\def\phi{\varphi}

\def\g{\gamma}

\def\l{\lambda}

\def\k{\kappa}

\def\r{\rho}

\def\s{\sigma}

\def\t{\tau}

\def\th{\theta}

\def\x{\xi}

\def\z{\zeta}

\def\o{\omega}

\def\D{\Delta}

\def\L{\Lambda}

\def\G{\Gamma}

\def\O{\Omega}

\def\S{\Sigma}

\def\T{\T}

\def\Th{\Theta}

\def\X{\Xi}

\def\AA{{\cal A}}

\def\PP{{\cal P}}
\def\NN{{\cal N}}
\def\Norm{\text{Norm}\,}
\def\Const{\text{Const}\,}
\def\const{\text{const}\,}





\begin{document}

\title{Continuous interfaces with disorder:\\
Even strong pinning is too weak in $2$ dimensions } 
 %\thanks{Work partially
% supported by   MURST (2004-06),  Cofin:  Prin 2004028108.}
%\thanks{Work partially
%%supported by   ?? }}
\author{
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/ }}\, and
Enza Orlandi%\thanks{Research supported by XXXXXXX}
\footnote{
Dipartimento di Matematica, 
Universit\'a  degli Studi  Roma Tre, 
Largo San Leonardo Murialdo, 1, 
00146 Roma,  ITALY, 
\texttt{orlandi@mat.uniroma3.it },
\texttt{http://www.mat.uniroma3.it/users/orlandi/}}
%\footnote{
%University of Groningen, 
%???????????? 
%Centre for Theoretical Physics, 
%Nijenborgh 4,
%9747AG Groningen
%The Netherlands 
%\texttt{aenter@phys.rug.nl}, 
%\texttt{a.c.d. sdf},
%\texttt{http://www..... }}
}


\maketitle

 
\begin{abstract}
We consider statistical mechanics models of continuous height effective interfaces in the presence 
of a delta-pinning of strength $\e$ at height zero.  
There is a detailed mathematical understanding of the depinning transition in $2$ dimensions without 
disorder.    
Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for $\e>0$ 
and diverges like $|\log \e|$ for  $\e\downarrow 0$.
How does the presence of a quenched disorder term in the Hamiltonian 
modify this transition?  
We show that an arbitrarily weak random field term is enough to beat 
an arbitrarily strong delta-pinning in $2$ dimensions and will cause 
delocalization. The proof is based on a rigorous lower bound for the  
overlap between local magnetizations and random fields in finite volume. 
In $2$ dimensions it  implies growth faster than the volume which is a contradiction to localization.    
We also derive a simple complementary inequality which shows that in higher dimensions 
the fraction of pinned sites converges to one with $\e\uparrow \infty$. 


\end{abstract}

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



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

\subsection{The setup} 
 
The study of lattice effective interface models, continuos and discrete, 
has a long tradition in statistical 
mechanics  \cite{Vel06,BrMeFr86, FoLiNi86, Fun, Sheff, BiKo06,BoKu94, BoKu96}.  
The model we study is given in terms of  variables  $\phi_i\in \R$ which, physically speaking,  
are thought to represent
height variables of a random surface at the sites $i\in \Z^d$. 
Mathematically,   they are just continuous unbounded (spin) variables. 
The  model is defined in terms of: a  pair potential $V$, 
a quenched random term, and a pinning 
term at interface height zero.   
More precisely, we are interested 
in the behavior of the  quenched finite-volume Gibbs measures  
 in a finite volume $\L\sb \Z^d$ with  
fixed boundary condition at height zero, given by
\begin{equation}\begin{split}\label{localspecification}
&\mu_{\e,\L}[\eta](d\phi_{\L})\cr
&=\frac{
e^{ - \frac{1}{4 d }\sum_{\langle i,j \rangle \in \L}V(\phi_i-\phi_j) 
- \frac{1}{4 d} \sum_{  i\in\L,j\in \L^c, |i-j|=1 }V(\phi_i) 
+  \sum_{i\in \L}\eta_i \phi_i}      \prod_{i\in \L}(d\phi_i +\e \d_0(d\phi_i))    }
{Z_{\e,\L}[\eta]},
\cr
\end{split}
\end{equation}
where the partition function $Z_{\e,\L}[\eta]$ denotes the normalization constant
that turns the last expression into a probability measure.  
%Such a model is motivated as an effective model for  
%the study of phase boundaries at a mesoscopic level in statistical mechanics.  
The Dirac-measures at the interface height zero are multiplied with the parameter 
$\e$, having the meaning of a coupling strength. 
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.    
What do we expect for such a model? 
Recall that the variance of a free massless interface in a finite box diverges like the logarithm 
of the sidelength when there are no random fields. 
 The corresponding interface is said to be delocalized, while it remains finite for $d \ge 3$. See \cite  {Vel06}  for a review on the subject.   Adding an arbitrarily small pinning $\e$ (without disorder) always localizes 
the interface uniformly in the volume, with the variance of the field behaving 
on the scale $|\log \e|$ when $\e$ tends to zero. 
Indeed, there is a beautiful and complete mathematical understanding of the model without disorder, in the case 
of both Gaussian and uniformly elliptic potentials (see \cite{BoVe01, DeuVel00}) with precise asymptotics 
as the pinning force tends to zero. These results follow from the analysis 
of the distribution of pinned sites and the random walk (arising 
from the random walk representation of the covariance of the $\phi_i$'s) 
with killing at these sites.
In this sense there is already a random system that needs to be analyzed 
even without disorder in the original model. 

What do we expect if we turn on randomness in the model and add the $\eta_i$'s ? 
Let us review first what we know about the same model without a pinning force. 
In $d=2$  we recently proved the deterministic lower bound 
$\mu_{\L_N}[\eta](|\phi_0| \geq t \sqrt {\log L} )  \geq c \exp(- c t^2)$ uniformly  over all  
fixed disorder configuration $\eta$, for general potentials $V$ (assuming 
not too slow growth at infinity)  \cite{KuOr06}.  So, it is not possible to stabilize 
an interface by cleverly choosing a random field configuration (one could 
think e.g. that this might be possible with a staggered field).  As this result holds 
at any arbitrary fixed configuration here we don't need any assumptions 
on the distribution of random fields. This result clearly excludes the existence of an infinite-volume Gibbs measure describing a two dimensional 
interface in infinite volume in the presence of random fields. 
In another paper  \cite{EnKu06} the question of existence of {\it gradient Gibbs measures}  (Gibbs 
distributions of the increments of the interface) in infinite volume  was raised. 
Note that while interface 
states may not exist in the infinite volume such gradient states may very well exist, 
as the example of the two-dimensional Gaussian free field shows, by computation.  
(For existence beyond the Gaussian case which is far less trivial, see  \cite{Fun,FunSp}.)  
It was proved in \cite{EnKu06} that there are no such gradient Gibbs measures in the random model in dimension $d=2$.

Now, turn to the full model in $d=2$.   
In view of the localization taking place at any positive pinning force $\e$ without disorder, a natural 
guess might be that with disorder at least at very large $\e$ there would be pinning. 
However, we show as  a result of the present paper that  
this is not the case, somewhat to our own surprise, 
and an arbitrarily strong pinning does not suffice to keep the interface bounded. 
 

\subsection{Main results}



\subsubsection*{Delocalization in $d=2$ - superextensivity of the overlap }


Denote by $\L_L\subset \Z^2$ the square of sidelength $2 L +1$ centered 
at the origin.  Denote by $ \E(\cdot)$ the mean with respect to the disorder  $\eta$.  

In this subsection we consider the 
disorder average of the overlap between local magnetizations and random fields in $\L_L$  showing 
that it grows faster than the volume. This in particular implies that in two dimensions there 
is never pinning, for arbitrarily weak random field and arbitrarily large 
pinning forces $\e$. Here is the result. 

\begin{thm}\label{overlapthm2d}  Assume that $V(t)=V(-t)$ is a twice differentiable 
potential, $\sup_{t}V''(t)\leq 1$, $\liminf_{|t|\uparrow\infty} \frac{\log V(t)}{\log |t|}>1$,   and let 
  $\eta_i$ be symmetrically 
   distributed, i.i.d. 
 with finite second moment. 
Let $d=2$.  Then there is a constant $a>0$, independent of the distribution of the random fields  
and the pinning strength $\e\geq 0$, such that 
\begin{equation}\begin{split}\label{main1}
& \liminf_{L\uparrow \infty } \frac{1}{ L^2 \log L } \sum_{i\in \L_L}
\E \Bigl( \eta_i\mu_{\e,\L_L}[\eta](\phi_i)\Bigr)\geq a \, \E(\eta_0^2).  \cr 
\end{split}
\end{equation}
\end{thm}
\bigskip 


Note that the growth condition on $V$ includes the quadratic case and ensures  
the finiteness of the integrals appearing in (\ref{localspecification})
for all arbitrarily fixed choices of $\eta$, even at $\e=0$.  

Generalizations to interactions that are non-nearest neighbor 
are obvious; all results go through e.g. for 
finite range and we skip them in this presentation 
for the sake of simplicity. 
We like to exhibit the case of Gaussian random fields (and not necessarily Gaussian potential $V$) 
since the bound acquires a form that looks even more striking because it becomes 
independent of the size of the variance of the $\eta_i$'s (as long as this is strictly positive). 

\begin{cor}
Let us assume that the random fields $\eta_i$ have an i.i.d.  Gaussian distribution with mean zero and 
strictly positive variance of arbitrary size. 

Then, with the same constant  $a$ as above, we have the bound   
\begin{equation}\begin{split}\label{cor1}
& \liminf_{L\uparrow \infty } \frac{1}{ L^2 \log L } \sum_{i\in \L_L}
 \E \Bigl( \mu_{\e,\L_L}[\eta](\phi^2_i)
-  \mu_{\e,\L_L}[\eta](\phi_i)^2 \Bigr) 
\geq a >0 \cr 
\end{split}
\end{equation}
for any $0\leq \e<\infty$. 
\end{cor}

The equation (\ref{cor1}) follows from (\ref{main1}) by partial integration w.r.t. the Gaussian disorder 
average (transforming the overlap into the variance of the $\phi_i$'s).  

Note that, even in the unpinned case of $\e=0$,  Theorem \ref{overlapthm2d} 
is not entirely trivial in the case of general potentials $V$. Here it provides 
an alternative simple way to see the delocalization in the presence of 
random fields (while the explicit lower bound on the tails of \cite{KuOr06} provides
more information). 



\bigskip 

\subsubsection*{Lower bound on overlap in $d\geq 3$ }


The analogue of Theorem \ref{overlapthm2d}  for higher dimensions
is the following. 


\begin{thm}\label{overlapthmdgeq3} Let $d\geq 3$ and let $\e\geq 0$ be arbitrary 
and assume the same conditions on $V$ and $\eta_i$ as in Theorem \ref{overlapthm2d}.  

There are positive constants $B_1, B_2 <\infty $,  
independent of the distribution of the random fields  
and the pinning strength $\e\geq 0$, such that 
\begin{equation}\begin{split}\label{main3}
& \liminf_{L\uparrow \infty } \frac{1}{ L^d } \sum_{i\in \L_L}
\E \Bigl( \eta_i\mu_{\e,\L_L}[\eta](\phi_i)\Bigr)\geq  \frac{\E(\eta_0^2) \, (-\D^{-1})_{0,0}}{2}
- \log (B_1 + B_2 \e),  \cr 
\end{split}
\end{equation}
where the positive constant 
$(-\D^{-1})_{0,0}$ is the diagonal element of the inverse of the infinite-volume lattice Laplace 
operator whose existence is guaranteed in  $d\geq 3$. 

\end{thm}

We recall that the infinite-volume lattice Laplace 
operator $\D=(\D_{i,j})_{i,j\in \Z^d }$ is the matrix with entries $\D_{i,j}=- 1 $ 
if $i=j$, $\D_{i,j}=\frac{1}{2d} $ if $i$ and $j$ are nearest neighbors and zero otherwise. 
The existence of its inverse in $d\geq 3$ follows e.g. from the random walk expansion \cite{BrFrSp82,Law}.

\subsubsection*{Lower bound on the pinned volume in $d\geq 3$ }

We complement the previous lower bounds on the overlaps 
which are depinning-type of results by a pinning-type result.  
It is a lower bound on the disorder average 
of the quenched Gibbs-expectation of the 
fraction of pinned sites.  Contrary to before, when we needed an upper bound 
on the interaction potential $V$, now we assume  a lower bound on 
$V$.  
%While we needed an upper bound 
%on the interaction potential $V$ before we are assuming now a lower bound on 
%$V$.  


\begin{thm}\label{345}  Let $d\geq 3$. Assume that $\inf_{t}V''(t)= c_-  >0$ 
 and let   $\eta_i$ be 
 symmetrically  distributed, i.i.d. 
 with finite second moment. 

Then there exist dimension-dependent constants $C_1,C_2>0$, independent 
of the distribution of the disorder,  such that, for all $\e>1$ and for all volumes $\L$, 
the disorder average of the fraction of pinned sites obeys the estimate 
\begin{equation}\begin{split}\label{allyouneed}
&\frac{1}{|\L|}\sum_{i\in \L}\E\Bigl( \, \mu_{\e,\L}[\eta](\phi_i=0) \Bigr) 
\geq 1- \frac{C_1 + C_2 \E (\eta_0^2) }{\log \e}. 
\end{split}
\end{equation}

\end{thm}

This shows pinning for the large $\e$ regime in the "thermodynamic sense" that the 
fraction of pinned sites can be made arbitrarily close to one, uniformly 
in the volume. As usual this result does not allow to make statement 
about the Gibbs measure itself. 


\bigskip

The proofs follows from "thermodynamic reasoning". 
The first "depinning-type" result follows from taking the log of the partition function 
and differentiating and integrating back w.r.t. the coupling strength of the random fields. 
Exploiting the linear form of the random fields, convexity, comparison 
of non-Gaussian with the Gaussian partition functions, and asymptotics of Green's functions
the results follow, see Chapter 2.  




\bigskip
\bigskip
\bigskip 



\section{Proof of Depinning-type results}

The estimates in formulas (\ref{main1}), (\ref{cor1}), and (\ref{main3})
are immediate consequences of the next  fixed-disorder estimate  stated in Proposition \ref {e1}. 
To formulate it,  we denote by $\D_{\L}=(\D_{\L;i,j})_{i,j\in \L }$ the Dirichlet Laplacian in the volume $\L\sb \Z^d $,  
that is the matrix with entries $\D_{\L; i,j}=- 1 $ 
if $i=j$, $\D_{\L; i,j}=\frac{1}{2d} $ if $i$ and $j$ are nearest neighbors in $\L$ and zero otherwise. 
$\L$ may be finite or infinite here. 

For convenience of the reader we recall that 
the matrix inverse has a well-known probabilistic interpretation, see for instance  \cite{BrFrSp82, Law}.   
 Indeed the matrix 
$(-\D_{\L})^{-1}_{i,j}, i, j \in \Lambda$  is  the Green's function of the  simple  random walk   $X$     killed as it exits  $ \Lambda$:
  $$(-\D_{\L})^{-1}_{i,j}:= \sum_{n \ge 0} \P_i[ X_n=j, \tau_\Lambda>n ]\quad i \in \Lambda, $$
 with
  $ \tau_{\Lambda}:= \min \{ n: X_n \notin \Lambda \} $. So,  the matrix element 
  $(-\D_{\L})^{-1}_{i,j}$ is the mean expected number of visits to $j\in \Lambda$ starting from  $i\in \Lambda$ up to the killing time $ \tau_{\Lambda}$.  It is well known that if $ \Lambda \neq \Z^d$ then $(-\D_{\L})^{-1}_{i,j}$ is finite in all dimensions.    When $ \Lambda = \Z^d$,  $(-\D_{\Z^d})^{-1}_{i,j}$ is finite for $d \ge 3$, see \cite {Law} for  further details.  %and all the  properties used in the following. 
  We  denote   $(-\D)^{-1}_{i,j}:=(-\D_{\Z^d})^{-1}_{i,j}$. 
 With this in mind we have the following proposition. 
      
\begin{prop} \label {e1} 
Assume  that $\sup_{t}V''(t)\leq 1$, $\liminf_{|t|\uparrow\infty} \frac{\log V(t)}{\log |t|}>1$,   and let 
  $\eta_i$ be symmetrically 
   distributed, i.i.d. 
 with finite second moment. 

For any dimension 
$d$,  there are  constants $C_{\text{nG}, d}<\infty $ and $c_{\text{G},d}>0$
such that, for 
all fixed configurations of local fields $\eta$, we have  
\begin{equation}\begin{split}\label{localspecification1}
& \frac{1}{2}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j - |\L| \log\frac{ C_{\text{nG}, d} + \e}{c_{\text{G},d}}
\leq \sum_{i\in \L} \eta_i\mu_{\e,\L}[\eta](\phi_i). 
\end{split}
\end{equation}
\end{prop}
\noindent {\bf Proof:} 
Let us see  what comes out when we differentiate and integrate 
back the free energy in finite volume w.r.t. strength of the random fields, 
\begin{equation}\begin{split}\label{q1}
& \frac{d}{dh}\log Z_{\e,\L}[h\eta]
=\sum_{i\in \L}\eta_i\mu_{\e,\L}[h\eta](\phi_i).
\end{split}
\end{equation}
At every fixed $\eta$, this quantity is a monotone function 
of $h$. This is seen by another differentiation w.r.t. $h$ 
which produces the variance of  the overlap $q$ 
between local magnetizations and random fields
w.r.t. to the Gibbs measures, that is 
\begin{equation}\begin{split}\label{q1}
\Bigl( \frac{d}{dh}\Bigr)^2\log Z_{\e,\L}[h\eta]
&=\mu_{\e,\L}[h\eta](q^2 ) -  \bigl( \mu_{\e,\L}[h\eta](q)\bigr)^2\geq 0 \text{  where }\cr 
&q=\sum_{i\in \L}\eta_i \phi_i. \cr 
\end{split}
\end{equation}
As a consequence we have 
 \begin{equation}\begin{split}\label{q3}
& \log \frac{Z_{\e,\L}[\eta]}{Z_{\e,\L}[0]}
=\sum_{i\in \L}\int_0^1 dh\,\, \eta_i\mu_{\e,\L}[h \eta](\phi_i)
\leq \sum_{i\in \L}  \eta_i \, \mu_{\e,\L}[\eta](\phi_i).\cr
\end{split}
\end{equation}
 Dropping the pinning term and retaining only the terms in the partition function 
 corresponding to the Lebesgue measure we have the lower bound 
 \begin{equation} \label{q4a}  Z_{\e,\L}[\eta] 
\geq Z_{\e=0,\L}[\eta]. 
 \end{equation} 
We may assume that  $V(0)=0$ since a constant drops out 
of the Gibbs-expectation, without changing the hypothesis over $V$. 
Then,  from the upper bound $ \sup_{t}V''(t)\leq 1$, follows $V(t)\leq \frac{t^2}{2}$. 
Applying this estimate to the exponent of the partition function  we immediately obtain 
 \begin{equation} \label{q4b}
Z_{\e=0,\L}[\eta] 
 \geq Z_{\e=0,\L}^{\text{Gauss}}[\eta], \end{equation}
 where  we have denoted by $ Z_{\e=0,\L}^{\text{Gauss}}[\eta] $ 
the Gaussian partition function with potential $V(t)=\frac{t^2}{2}$. 
 This Gaussian partition function can be easily computed: 
 Using the linearity of the random external field $\eta$,  
 the Gaussian integral produces the quadratic form of the covariance matrix $(-\D_{\L})^{-1}$ and 
the partition function without disorder:  
 \begin{equation} \label{q4}
Z_{\e=0,\L}^{\text{Gauss}}[\eta]=\exp\Bigl( \frac{1}{2}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j 
\Bigr)Z_{\e=0,\L}^{\text{Gauss}}[0].
\end{equation}
The Gaussian partition function in zero field is given by  a determinant, 
$Z_{\e=0,\L}^{\text{Gauss}}[0] = (2 \pi)^\frac{|\L|}{2}\sqrt{\det (-\D_{\L})^{-1}}
=(2 \pi)^\frac{|\L|}{2}\exp -\frac{1}{2} \text{Tr}\log (-\D_{\L})$ which, by expansion of the logarithm, 
 again has a standard random walk representation 
\begin{equation} \begin{split}
\label{q4a}\text{Tr}\log (-\D_{\L})&= - \sum_{i\in \L}\sum_{n \ge 0} \frac{1}{n}\P_i[ X_n=i, \tau_\Lambda>n ]\cr 
&\geq -|\L|\sum_{n \ge 0} \frac{1}{n}\P_0[ X_n=0].
\end{split}
\end{equation}
The equality is well known: it is a probabilistic rewriting of formula (6.18) of \cite{BrFrSo83}.  
The inequality is obtained by dropping the killing outside the volume, and provides 
for us the uniformity in the volume $\L$. 
Indeed, the quantity on the r.h.s. is finite even in all dimensions $d\geq 1$ (see \cite{Law}) 
which proves that there is a constant such that 
$Z_{\e=0,\L}^{\text{Gauss}}[0] \geq c_{\text{G},d}^{|\L|}$.  
Using this, along with  \eqref {q4} we  have arrived at
$$
 Z_{\e,\L}[\eta] 
  \geq \exp\Bigl( \frac{1}{2}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j 
\Bigr)c_{\text{G},d}^{|\L|}. 
$$
This is the first ingredient to bound the l.h.s. of (\ref{q3}) from below.

Let us now explain how to get an upper bound on $Z_{\e,\L}[0]$.  
  We expand the integration measure in $ Z_{\e,\L}[0]$ as a sum 
  $\prod_{i \in \Lambda} ( d\phi_i+ \e \delta_0(d\phi_i))=\sum_{A\subset \L} \e^{|A|} 
  \prod_{i\in A}\delta_0(d\phi_i))
   \prod_{j\in \L\backslash A} d\phi_j $. 
   This has been a standard trick 
  in all treatments of models with delta-pinning, see e.g. \cite{BoVe01}.  
 For our purposes  we use that the lower bound 
on $V(t)$   implies that   $\int d\phi_i \exp\Bigl( - \frac{V(\phi_i) }{4 d}\Bigr)=:C_{\text{nG},d}$
is finite.  Let us choose a spanning tree for $\L\sb \Z^d$. Let us call the 
edge set $E$. Let us single out one vertex at the boundary of $\L$ and call it $i_0$. 
Then, dropping all the terms in the exponent appearing in the partition function that 
are not corresponding to the tree edges $(i,j)\in E$, we may bound 
\begin{equation}\begin{split}\label{upperbound}
Z_{\e=0,\L}[0]& =\int e^{ - \frac{1}{4 d }\sum_{\langle i,j \rangle \in \L}V(\phi_i-\phi_j) 
- \frac{1}{4 d} \sum_{  i\in \L,j\in \L^c, |i-j|=1 }V(\phi_i) }  \prod_{i\in E} d\phi_i  \cr
&\leq \int e^{ - \frac{1}{4 d }\sum_{( i,j )\in E}V(\phi_i-\phi_j) 
- \frac{1}{4 d} V(\phi_{i_0}) }  \prod_{i\in \L} d\phi_i  \cr
&= C_{\text{nG},d}^{|\L|}.
\cr
\end{split}
\end{equation}
This is the desired upper bound. 
These two  ingredients give 
\begin{equation}\begin{split}\label{q5}
&Z_{\e,\L}[0] 
= \sum_{A\subset \L} \e^{|A|} Z_{\e=0,\L\backslash A}[0] \cr
&\leq   \sum_{A\subset \L} \e^{|A|} C_{\text{nG},d}^{|\L\backslash A|}
=(C_{\text{nG},d}+\e )^{|\L|}. 
\end{split}
\end{equation}
So the desired estimate on the overlap  \eqref   {localspecification1} follows from (\ref{q3}),(\ref{q4}) and (\ref{q5}).
$\Cox$

\bigskip 
\bigskip 
 Theorems
\ref{overlapthm2d} and \ref{overlapthmdgeq3}  
can be easily derived from  Proposition \ref {e1}. Indeed, 
taking a disorder average  we have 
\begin{equation}\begin{split}\label{localspecification2}
& \frac{\E (\eta_0^2)} {2}\sum_{i\in \L}(-\D_{\L})^{-1}_{i,i} -  |\L|\log\frac{ C_{\text{nG}, d} + \e}{c_{\text{G},d}}
\leq \sum_{i\in \L} \E \Bigl( \eta_i\mu_{\e,\L}[\eta](\phi_i)\Bigr). 
\end{split}
\end{equation}
 {\bf Proof of  Theorem  \ref {overlapthm2d} }   
  It is known, see \cite {Law}, page 40,  that in $d=2$,      the asymptotics of the Green's-function 
in a square,  at fixed $i$  is  $(-\D_{\L_L})^{-1}_{i,i}\sim \log L$. This and \eqref {localspecification2} imply immediately \eqref {main1}. $\Cox$

 
\bigskip 

Finally let us note in passing that a constant magnetic field 
is always winning against an arbitrarily strong pinning, and even more strongly 
than a random field.  
Indeed, let $d\geq 2$, let $\eta_i=h\geq 0$ for all sites $i$ 
and let $\e\geq 0$ be arbitrary.   
Then, there is a constant $c_d >0$, independent of $h$ and $\e$, such that 
\begin{equation}\begin{split}\label{localspecification3}
& \liminf_{L\uparrow \infty } \frac{1}{ L^{d+2} } \sum_{i\in \L_L}
\mu_{\e,\L_L}[h](\phi_i)\geq  c_d h.
 \cr 
\end{split}
\end{equation}
This again follows from    Proposition \ref {e1}, using $\sum_{i,j\in \L}(-\D_{\L_L})^{-1}_{i,j} \sim L^{d+2}$. 
 



\bigskip 
\bigskip 


\section{Proof of Pinning-type results } 

To prove the lower bound on the fraction of pinning sites 
in dimension $d\geq 3$  given in Theorem \ref{345} we will in fact prove the following 
fixed-disorder lower bound:  

\begin{prop} \label {prope} Under the assumptions given in Theorem \ref{345},
for all finite volumes $\L$ and 
for all realizations $\eta$ we have, for any $\e_0$, $\e > \e_0>0$,  
 \begin{equation}\begin{split}\label{laballballon}
&\frac{1}{|\L|}
\sum_{i\in \L}\mu_{\e,\L}[\eta](\phi_i=0)\cr
&\geq \frac{1}{\log \frac{\e}{\e_0}}
\Biggl(\log \frac{\e  c_-^\frac{d}{2} }{ \Bigl(1+  \frac{\e_0 c_-^\frac{d}{2}}{(2\pi)^\frac{d}{2}} \Bigr) C_{\text{G},d}}
- \frac{1}{2 c_- |\L|}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j  \Biggr) 
\end{split}
\end{equation}
with a constant $C_{\text{G},d}$ defined in (\ref{18}). 
\end {prop}

 
 
 
\noindent {\bf Proof of Theorem \ref{345}}: The proof     follows from the existence of the infinite-volume Green's-function 
in $d\geq 3$.     Taking a disorder-expectation  of 
\eqref {laballballon} and setting $\e_0=1$ we get the thesis. 
$\Cox$ 


\bigskip 

\noindent {\bf Proof of  Proposition \ref{prope}: } 
The proof is based on the trick to differentiate and integrate back the log of the partition 
function, now w.r.t. $\e$: Differentiation gives 
 \begin{equation}\begin{split}\label{localspecification4}
&\e \frac{d}{d\e}\log Z_{\e,\L}[\eta]
= \sum_{i\in \L}\mu_{\e,\L}[\eta](\phi_i=0).
\end{split}
\end{equation}
We integrate this relation back, and it will be important 
for us to do it starting from a positive $\e_0>0$. So we get     
 \begin{equation}\begin{split}\label{12}
&\log \frac{Z_{\e,\L}[\eta]}{ 
Z_{\e_0,\L}[\eta]}
= \int_{\e_0}^\e \frac{d \tilde \e}{\tilde \e}\sum_{i\in \L}\mu_{\tilde \e,\L}[\eta](\phi_i=0)
\leq \log \frac{\e}{\e_0}\cdot \sum_{i\in \L}\mu_{\e,\L}[\eta](\phi_i=0),
\end{split}
\end{equation}
where we have used that $\sum_{i\in \L}\mu_{\tilde \e,\L}[\eta](\phi_i=0)$ is a monotone 
function of $\tilde \e$. 
Note that  the integrand itself is not a monotone function. 
(Compare \cite{CaVel00} for a related non-random pinning scenario, with back-integration from zero.)   
Expanding as before, see  \eqref {q5}, and
keeping 
only the contribution in the expansion where all  sites are pinned
we have 
   \begin{equation}\label{13}
 Z_{\e,\L}[\eta]  
= \sum_{A\subset \L} \e^{|A|} Z_{\e=0,\L\backslash A}[\eta]\geq \e^{|\L|}. \end{equation}
For the upper bound on the partition function 
of the full model (at $\e_0$) we first
use the lower bound on the potential $V(t)\geq \frac{c_{-} t^2}{2}$ 
giving us a comparison with a Gaussian partition function with curvature $c_-$: 
  \begin{equation}\begin{split}\label{14}
&Z_{\e_0,\L}[\eta]
\leq Z_{\e_0,\L}^{\text{Gauss},c_-  }[\eta].\cr
\end{split}
\end{equation}
It is a simple matter to rescale the Gaussian curvature away
 \begin{equation}\begin{split}\label{15}
& Z_{\e_0,\L}^{\text{Gauss},c_-  }[\eta]
={c_-}^{-\frac{d |\L|}{2}} Z_{\e_0 {c_-}^{\frac{d}{2}} ,\L}^{\text{Gauss}}[{c_-}^{-\frac{1}{2}} \eta],
\cr
\end{split}
\end{equation}
where the partition function on the r.h.s. is taken with unity curvature potential. 
For the Gaussian partition function we claim the upper bound (writing 
again in the original parameters) of the form 
\begin{equation}\begin{split}\label{16}
&Z_{\e ,\L}^{\text{Gauss}}[\eta] \leq \Bigl(1+  \frac{\e}{(2\pi)^\frac{d}{2}} \Bigr) ^{|\L|} 
Z_{\e=0,\L}^{\text{Gauss}}[\eta].  \cr
\end{split}
\end{equation}
Here is an elementary proof: 
We will replace successively the single-site integrations 
involving the Dirac measure by integrations only over the Lebesgue measure with 
the appropriately adjusted prefactor. Indeed, 
consider one site $i$ and compute the contribution 
to the partition function while 
fixing the values of $\phi_j$ for $j$ not equal to $i$. 
Then use that  
\begin{equation}\begin{split}\label{17}
&\int\Bigl(  d\phi_i + \e \d_0(d\phi_i)
\Bigr)\exp\Bigl( -\frac{\phi_i^2}{2}+ (\sum_{j\sim i}\phi_j + \eta_i )\phi_i 
\Bigr) \cr
&=(2\pi)^\frac{d}{2}\exp\Bigl( \frac{(\sum_{j\sim i}\phi_j + \eta_i )^2}{2}
\Bigr) + \e 
 \cr
 &\leq \Bigl(1+  \frac{\e}{(2\pi)^\frac{d}{2}} \Bigr) 
 (2\pi)^\frac{d}{2}\exp\Bigl( \frac{(\sum_{j\sim i}\phi_j + \eta_i )^2}{2}\Bigr)\cr
 &=  \Bigl(1+  \frac{\e}{(2\pi)^\frac{d}{2}} \Bigr) 
  \int d\phi_i  
\exp\Bigl( -\frac{\phi_i^2}{2}+ (\sum_{j\sim i}\phi_j + \eta_i )\phi_i 
\Bigr) \cr
\end{split}
\end{equation}
and iterate over the sites. 
For the Gaussian unpinned partition function use, see \eqref {q4},  
\begin{equation}\begin{split}\label{18}
&Z_{\e=0,\L}^{\text{Gauss}}[\eta]  
=\exp\Bigl( \frac{1}{2}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j 
\Bigr)Z_{\e=0,\L}^{\text{Gauss}}[0] \cr
&\leq \exp\Bigl( \frac{1}{2}\sum_{i,j\in \L}(-\D_{\L})^{-1}_{i,j}\eta_i\eta_j 
\Bigr)C_{\text{G},d}^{|\L|}\cr
\end{split}
\end{equation} 
with a suitable constant. 
From here (\ref{allyouneed}) follows from (\ref{12},\ref{13},\ref{14},\ref{15},\ref{16},\ref{18})
$\Cox$ 

\bigskip 
\bigskip 


{\bf Acknowledgements: } 
The authors thank Pietro Caputo for an interesting discussion, Aernout van Enter 
for comments on a previous draft of the manuscript.  C.K. thanks 
the University Roma Tre for hospitality. 




\begin{thebibliography}{19}

\small


%\bibitem{AW} M. Aizenman and J. Wehr, {\em Rounding effects on quenched randomness 
%on first-order phase transitions.} Commun. Math. Phys. 130, 489--528,  1990. 

\bibitem{BoVe01} E. Bolthausen, Y. Velenik, {\em 
Critical behavior of the massless free field at the depinning transition.} 
Comm. Math. Phys. 223, 161�-203, 2001.




\bibitem{BiKo06}
M. Biskup and R. Koteck\'y, 
{\em  
Phase coexistence of gradient Gibbs states.}  Probab. Theory Rel. Fields 139, 1--39, 2007. 


%\bibitem{Bi06}
%M. Biskup, {\em  
%Reflection positivity and phase transitions in lattice spin models}, 
%lecture notes from Prague Summer School on Mathematical Statistical Mechanics, 
%available at  http://www.math.ucla.edu/ $\sim$ biskup/

\bibitem{BoKu94}
A. Bovier and C. K\"ulske, {\em A rigorous renormalization group method 
for interfaces in random media.}  Rev. Math. Phys. 6, 413--496, 1994.  

\bibitem{BoKu96}
A. Bovier and C. K\"ulske, {\em There are no nice interfaces in $(2 + 1)$-dimensional 
SOS models in random media},  J. Statist. Phys., 83:  751--759, 1996. 

%\bibitem{BL76}
% H. J. Brascamp and E. H. Lieb, {\em 
% On extensions of the Brunn-Minkowski and 
%Prekopa-Leindler theorems, including inequalities for log concave functions, 
%and with an application to the diffusion equation.} J. Functional Analysis 
%22,  366--389, 1976. 


\bibitem{BrMeFr86}
 J. Bricmont, A. El Mellouki, and J. Fr\"ohlich, 
 {\em Random surfaces in statistical mechanics: roughening, rounding, wetting, $\dots$} 
  J. Statist. Phys.  42, 743--798, 1986. 

%\bibitem{ES1}
%A.C.D. van Enter and S.B. Shlosman,
%{\em First-order transitions for n-vector models in two and more 
%dimensions: Rigorous proof.}
%Phys.Rev. Lett. 89, 285702, 2002.
%
%\bibitem{ES2}
%A.C.D. van Enter and S.B. Shlosman,
%{\em Provable first-order transitions for nonlinear vector and 
%gauge models.}
%Comm. Math. Phys. 255, 21--32, 2005.

\bibitem{BrFrSp82}  D. Brydges, J. Fr\"ohlich, Th. Spencer, 
{\em The Random Walk Representation of Classical Spin Systems and Correlation Inequalitities,}
Comm Math. Phys 83, 123--150, 1982.  

\bibitem{BrFrSo83}  D. Brydges, J. Fr\"ohlich, A. Sokal, 
{\em The Random-Walk Representation of Classical 
Spin Systems and Correlation Inequalities,}
Commun. Math. Phys. 91, 117--139, 1983. 

\bibitem{CaVel00}  P. Caputo, Y.  Velenik, 
{\em A note on wetting transition for gradient fields. }�
Stochastic Process. Appl. 87, 107--113, 2000.  


\bibitem{DeuVel00} J.-D. Deuschel, Y. Velenik, {\em 
Non-Gaussian surface pinned by a weak potential.} 
Probab. Theory Related Fields 116, 359�-377, 2000. 



\bibitem{EnKu06} A. C. D. van Enter,  C. K\"ulske, 
{\em Non-existence of random gradient Gibbs measures 
in continuous interface models in $d=2$.}, math.PR/0611140, 
to be published in Annals of Applied Probability 


\bibitem{FoLiNi86}
G. Forgacs, R. Lipowski and Th.M. Nieuwenhuizen, {\em The Behaviour of Interfaces in Ordered and Disordered Systems}, in Phase Transitions and Critical Phenomena, vol. 14, edited by C. Domb and J.L. Lebowitz, Academic Press, 1986.

\bibitem{Fun}
T. Funaki,
{\em Stochastic Interface models.}
2003 Saint Flour lectures, Springer Lecture Notes in 
Mathematics, 1869, 103--294, 2005.

\bibitem{FunSp}
T. Funaki and H. Spohn,
{\em Motion by mean curvature from the Ginzburg-Landau 
$\nabla \phi$ interface model.}
Comm. Math. Phys. 185, 1--36, 1997.

% \bibitem{G} G. Giacomin,
% {\em 
% Aspect of Statistical Mechanics of random surfaces. Notes of lectures given at IHP, fall %2001????}
 

%\bibitem{IoShloVel02}
%D. Ioffe, S. B. Shlosman, and Y. Velenik, {\em 2D models of statistical physics with continuous 
%symmetry: the case of singular interactions},  
%Comm. Math. Phys. 226 no. 2, 433--454,  2002.  

%\bibitem{KS} R. Koteck\'y and S. B. Shlosman.
%{\em First-order transitions in large entropy lattice models.}
% Comm. Math. Phys. 83, 493--515, 1982.

%\bibitem{K99a} C. K\"ulske,   {\em The continuous-spin random field model:
%Ferromagnetic ordering in  $d\geq 3$}, 
% Rev.Math.Phys. 11, %No. 10, 
%1269--1314, 1999.

 \bibitem{Law}
 G. F. Lawler, 
 {\em  Intersections of random walks}, 
 Basel-Boston Birkhauser, 1991. 


\bibitem{KuOr06} C. K\"ulske, E. Orlandi, 
{\em A simple fluctuation lower bound for a disordered 
massless random continuous spin model in $d=2$.} 
Electronic Comm. Probab.  {\bf 11} 200-205, 2006.

\bibitem{Sheff}
S. Sheffield, 
{\em Random surfaces, large deviations principles and gradient Gibbs measure
classifications.}
arXiv math.PR{/}0304049, 
Asterisque 304, 2005.

\bibitem{Vel06} Y. Velenik, {\em 
Localization and delocalization of random interfaces.} 
Probability Surveys  3, 112-169, 2006.



\end{thebibliography}

\end{document}
