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Random Schrodinger operators, Anderson localization, spectrum, mobility edge,  
Green function, constructive criteria, exponential decay. 
---------------9910181845198
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\begin{document}

\title{\vspace*{-.35in}
Finite-volume Criteria for Anderson Localization
% Single-Scale Method for Localization
                 }

\author{Michael Aizenman $^{a,b}$ \ \ \ \  Jeffrey H. Schenker $^b$ \\
  \vspace*{-0.15truein} \\
  Roland M. Friedrich $^c$ \ \ \ \  Dirk Hundertmark $^{a}$ \\
  \vspace*{-0.05truein} \\
\normalsize \it  Departments of Physics$^{(a)}$ and Mathematics$^{(b)}$,
Princeton University \\
\normalsize \it Princeton, NJ 08544, USA. \\
  \vspace*{-0.2truein}  \\
\normalsize \it ${}^{(c)}$ Student, Theoretische Physik,
ETH-Z\"urich, CH--8093, Switzerland.
      }
\maketitle
\thispagestyle{empty}        %removes # on p.1

\begin{abstract}
For random Schr\"odinger operators, and a more general class of 
operators with random potentials of `regular' probability  distributions, 
we present a  family of constructive criteria for the localization 
regime. A technically convenient characterization of localization is 
rapid decay of the Green function's fractional moments. 
In addition to explicit bounds, the constructive criteria 
indicate that the exponential decay of the expectation values of such  
functions may indeed characterize the entire regime of localization. 
This has qualitative consequences --  since the fractional moment 
condition is known to have other significant implications, such as 
dynamical, as well as spectral, localization, and the exponential 
decay of the expectation values of the spectral projection kernels.  
In the converse direction,  
the criteria also rule out fast power-law decay of the Green 
function at mobility edges.  
\end{abstract}

%\vfill
 \noindent {{\bf AMS subject Classification:}  82B44 (Primary), 47B60, 
 60H25.}  \\
%\noindent {\bf Key words:}
\vskip .25truecm
\newpage

  \vskip .25truecm         % Table of contents
    \begin{minipage}[t]{\textwidth}
    \tableofcontents
    \end{minipage}
   \vskip .5truecm
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%********* NEW SECTION *********************************************

\masect{Introduction}
\label{sect:intro}
\vskip-0.6cm

\masubsect{Overview}

Operators with extensive disorder are known to have spectral
regimes (energy ranges) where the spectrum consists of a dense
collection of eigenvalues corresponding to exponentially localized
eigenfunctions. This phenomenon is of relevance  in different
contexts; e.g., it plays a role in the
conductive properties of metals \cite{An,MT,MS},
in  the quantization of
Hall conductance~\cite{Halp82,NTW,AS2,BES,AG}, and in the
emerging subject of optical crystals~\cite{FiKl-SIAM}.


In this work we discuss ``discrete'' random operators, acting in
 $\ell^2(\Z^d)$.
A guiding example is the discrete Schr\"odinger operator
\begin{equation}
H_{\omega} \ = \ T + \lambda V_{\omega} \; ,
\label{eq:proto}
\end{equation}
with $T$ denoting the off-diagonal part, whose matrix elements are
referred to as the {\em hopping terms}, and $V_{\omega}$ a random
multiplication operator -- referred to as the {\em potential}.
The symbol  $\omega$ represents a particular
realization of the disorder, in this case the potential
variables $\{V_{\omega}(x)\}$, and
$\lambda$ serves as the disorder strength parameter.

For the discrete Schr\"odinger operator
\begin{equation}
T_{u,v} \ = \ \begin{cases}
    1 & \mbox{ if }|u-v| = 1 \; ,\\
    0 & \mbox{ if }|u-v| \neq 1 \; ,
\end{cases}
\label{eq:lap}
\end{equation}
and the random potential is given by a collection of independent
identically distributed random variables, $\{V_{\omega}(x)\}_{x\in
\Z^d}$.  However, we shall also consider a more general class of
operators, allowing the incorporation of magnetic fields, periodic
terms, and off-diagonal disorder (see Section~\ref{sect:gen}). We
focus on the case of extensive disorder, where the distribution of
the random operator $H_{\omega}$  is either translation invariant,
or at least gauge equivalent to shifts by multiples of basic
periods (i.e. invariant under periodic magnetic shifts).

Our main goal is to present a sequence of
finite-volume criteria for localization, which permit to conclude
that the following fractional-moment condition is satisfied
in some energy interval $[a,b]\in \R$:
\begin{equation}
\E(|<x| {1 \over H_{\omega}-E - i \eta } | y>|^s) \ \le \ A(s) \
e^{-\mu(s)
|x-y|}  \ , \label{eq:fm}
\end{equation}
for all $E\in [a,b]$, $\eta \in \R$,  and suitable $s\in (0,1)$.
$\E(\cdot )$ represents here the average over the disorder, {\it
i.e.} the random potential.

Needless to say, the bound (\ref{eq:fm}) is of interest mainly in
situations  where the energy $E$ is within the spectrum, {\it
i.e.} $[H_{\omega}-E]^{-1}$ is an unbounded operator
  and the  exponential
 decay occurs only due to the localization of the eigenvalues with
 energies within the interval $[a,b]$.
The fractional moment condition
and some of the techniques used here, were introduced in
a work of Aizenman and Molchanov~\cite{AM},  whose approach we
now extend to new regimes.
As explained there, fractional powers are used in
order to avoid infinity, however the value of $ 0< s < 1$
at which \eq{eq:fm} is derived is of almost no importance
(if \eq{eq:fm} holds for a particular value of $s$,
then it will hold for all $s < \tau$, where $\tau < 1$ is
a number which depends only on the regularity of the probability
distribution of $V_\omega(x)$, see Appendix -- Lemma~\ref{lem:allforone}).
For the systems considered here, \eq{eq:fm} is known to imply
various other properties which are commonly
associated with localization:
\begin{itemize}
\item[i.]  {\it Spectral localization (\cite{AM} - using \cite{SiWo}):}
The spectrum of $H_{\omega}$
within the interval $(a,b)$ is almost-surely of the pure-point type,
and the corresponding eigenfunctions are exponentially localized.
\item[ii.] {\it  Dynamical localization (\cite{Ai94,AM}):}
wave packets with energies in the
specified range do not spread (and in particular the
{\em SULE} condition of \cite{SULE} is met) --
\begin{equation}
\E\left( \sup_{t\in \R} |<x| \ e^{-itH} P_{H\in [a,b]}\ |y>|
\right ) \ \le \tilde A e^{-\tilde \mu |x-y|} \label{eq:dyn}
\end{equation}
\item[iii.] {\it Exponential decay of the projection kernel
(\cite{AG})}; the condition  expressed in a bound similar to
\eq{eq:dyn} for $\E( |<x|\ P_{H\le E}\ |y>|)$ with  $E\in [a,b]$.
This condition plays an important role in the quantization of Hall
conductance, in the ground state of the two dimensional electron
gas with Fermi level $E_{F}\in [a,b]$ \cite{BES,AS2,AG}.
\item[iv.] {\it Absence of level repulsion (\cite{Min}).}
Minami  has shown that \eq{eq:fm} implies, for operators
of the type considered here, that in the
range $[a,b]$ the energy gaps have Poisson-type statistics.
\end{itemize}
The fractional moment condition has already been established for
certain regimes: extreme energies, as well as all energies at high
enough disorder \cite{AM}, and also for weak disorder but far
enough from the unperturbed spectrum \cite{Ai94}. The results
presented here permit to extend it to  band edges where the
required criteria can be shown to be satisfied through `Lifshitz
tail  estimates' on the density of states 
(ref.~\cite{FigPast,BCH,KSS,Stollmann,Klopp99}).  Furthermore, it seems
reasonable to expect that the constructive criteria presented here
may in principle cover the entire region of Anderson localization,
though the calculation needed to apply the test may in some
situations be rather non-practical.


The criteria presented here require input which is somewhat
similar to that employed by the {\em multiscale analysis}, which
since its introduction by Fr\"ohlich and Spencer \cite{FS} has
been an invaluable tool for the analysis of localization in more
than one dimension. There are also similarities, and indeed
relations, between the results. The two methods share the basic
feature that the analysis requires an initial condition which one
may expect to be met in a finite system provided  its linear size
is  of the order of the localization length, or larger. However,
for the method presented here if a suitable input is received on
some scale, then the analysis can proceed using steps, or blocks,
of only that size.  An important difference in the results is that
we obtain bounds with exponential decay for the
\underline{expectation values}, which  are important for some of
the conclusions listed above. (The multiscale analysis bounds on
the error terms decay as $\exp[- (\log L / \log L_o)^{\alpha}]$,
which is faster than any power of $L$, but not fast enough for an
exponential bound for the mean.)

Among the implications of the criteria presented here
is the statement (see Theorem~\ref{thm:tails}) that if, at some
energy $E\in \R$
\begin{equation}
\liminf_{L \rightarrow \infty} L^\xi \P \left [ \dist \left (
\sigma(H_{\Lambda_{L}; \omega}), E \right) \le L^{-\beta} \right ]
\ = \ 0 \; , \label{eq:xicondition1}
\end{equation}
for  some $\xi >  3 (d - 1) $ and  $\beta \in (0,1)$,
then the fractional moment condition (\ref{eq:fm}),
and thus also the consequences listed above,
hold in some open interval containing $E$.
One may note that  the condition (\ref{eq:xicondition1})
is satisfied throughout the regime in which the multiscale
analysis applies.  Thus,  it seems that
for discrete random operators of the type  considered here,
the present work extends
the properties listed above to the entire regime (in terms of
$(E,\lambda)$)  for which
localization can be proven by any of the known methods.

On the other hand, the extension of the present method to
operators in the continuum, for which 
a number of basic localization results have been established using 
the multiscale analysis 
\cite{CH,FiKl,KSS}, still presents some challenges.
Also not covered by the above
summary are discrete operators with potential assuming
discrete values (e.g., $V_{\omega}(x)= \pm 1$ \cite{CKM}).


%%%%%%%%%%%%%%%% NEW SECTION

\masubsect{The finite-volume criteria}
\label{sect:main}

Our main results admit a number of variations.  In this section we
present a formulation which is natural for the prototypical
example of the discrete random Schr\"odinger operators, {\it i.e.}
Hamiltonians of the form (\ref{eq:proto}) with $T$ the discrete
Laplacian (given by (\ref{eq:lap})). In Section~\ref{sect:gen} we
formulate various extensions of the results, including to
operators incorporating magnetic fields and to operators with
hopping terms of unbounded range.

The results are derived under some mild regularity assumptions on
the probability distribution of the variables $\{
V_{\omega}(x)\}_{x\in \Z^d}$ which form the random potential. For
simplicity we address ourselves here to the {\em IID} case: the
potential variables are independent with a common probability
distribution $\rho(dV)$.   The assumption is then that $\rho(dV)$
satisfies the regularity conditions listed below, $R_1(s)$ or
$R_2(s)$. However, the independence is not essential.  What
matters is that  the stated regularity condition be satisfied,
with a uniform constant,  by the  conditional distribution of each
of the potential variables, conditioned on arbitrary values of the
other potentials.

The two regularity conditions mentioned here are:
\begin{description}
  \item[$R_1(s)$:] A probability distribution $\rho(dV)$, on $\R$, is said
  to be {\em $s$-regular}, or to satisfy the condition $R_1(s)$ at
  some $0<s\le 1$,  if there exists $C < \infty$ such that
  \begin{equation}
  \label{eq:rregular}
  \rho(a - \epsilon, a + \epsilon) \leq C \epsilon^s.
  \end{equation}
%  In this case we let $\kappa_s(\rho)$ denote the smallest $C$ such
%  that \eq{eq:rregular} holds.
  \item[$R_2(s)$:] The probability distribution $\rho(dV)$ is said
  to have the {\em decoupling property}  $R_2(s)$, with
  some $0<s\le 1$,  if there exists $C < \infty$ such that
  for any pair of functions  $f$ and $g$ of the form
  \begin{equation}
  f(V) =  {1 \over V - a}, \qquad
  g(V)  =  {V - b \over V - c} \; ,
  \end{equation}
  with $a,b,c \in {\mathbb C}$,
  the expectation of the product can be dominated as follows:
  \begin{equation}
  \E \left (|f(V)|^s |g(V)|^s \right ) \ \leq \ C \ \E \left ( |f(V)|^s
  \right ) \ \E \left ( |g(V)|^s \right ),
  \label{eq:decoupling}
  \end{equation}
  The smallest $C$ such that \eq{eq:decoupling} holds for all
  $a,b,c \in {\mathbb C}$ is called
  here the {\em decoupling constant} for $\rho$, and is denoted by
   $D_s(\rho)$.
\end{description}
A sufficient condition for $R_2(s)$ is
  that $\rho$ have bounded support and satisfy
  $R_1(\tau)$ for some $\tau > 4s$ (see Appendix
  \ref{sect:decoupling}; related discussion is found in
  Refs.~\cite{AM,AG}.)

In Appendix \ref{sect:moment} we show that given any
$\tau$-regular measure $\rho$ and any $s < \tau$, there is a
finite constant $C$ such that for any $2 \times 2$ self adjoint
matrix $A_{2\times 2}$
\begin{equation}
\int \int \rho(du) \rho(dv)  \left | \left [ \left ( A_{2 \times
2} +
\begin{pmatrix}
  u & 0 \\
  0 & v
\end{pmatrix}
\right )^{-1} \right ]_{i,j}\right |^s \ \le \ C \; ,
\label{eq:2by2average}
\end{equation}
where $[ \cdot ]_{i,j}$ denotes the $i,j$ matrix element with $i,j
= 1,2\, $.  Throughout this work, we denote by $C_s$ the smallest 
value of $C$ at which 
(\ref{eq:2by2average}) holds.  For $\rho(dV)$ which also satisfy 
 $R_2(s)$ we let:  
$\widetilde C_s = C_s \cdot D_s(\rho)^2$.

For $\Lambda \subset \Z^d$ we denote by 
$H_{\Lambda ; \omega}$ the operator obtained from
$H_{\omega}$ by ``turning off'' the hopping terms outside
$\Lambda$.  Thus, if $P_\Lambda$ is the orthogonal projection of
$\ell^2(\Z^d)$ onto $\ell^2 (\Lambda)$ (considered as a subspace
of $\ell^2(\Z^d)$), then
\begin{equation}
H_{\Lambda ; \omega} = P_\Lambda T P_\Lambda + \lambda V_\omega.
\end{equation}
Restricted to $\ell^2(\Lambda)$, $H_{\Lambda ; \omega}$
is nothing but
$H_\omega$ with the Dirichlet boundary conditions on the boundary
of $\Lambda$.

We also denote  by $\Gamma(\Lambda)$ the set of the
nearest-neighbor bonds reaching out of $\Lambda$ ({\it i.e.} pairs
with one site in $\Lambda$ and the other outside), by
$\Lambda^{+}$ the collection of sites within distance $1$ from
$\Lambda$, and by $|\Gamma(\Lambda^+)|$ the number of bonds
reaching out of that set.  These notions will be generalized in
Section~2.a.

Following are  our main results for operators of the form
(\ref{eq:proto}).

\begin{thm}
\label{thm:1} Let $H_{\omega}$ be a random Schr\"odinger operator
with the probability distribution of the potential $V(x)$
satisfying the regularity condition $R_1(\tau)$ and fix $s <
\tau$.
If for some $z\in \C$ (possibly real) and some finite region
 $ \Lambda\subset \Z^d $
which contains the origin $O$:
\begin{equation}
b(\Lambda, z) \ := \ \sup_{W\subset \Lambda }
 \ \left\{ |\Gamma(\Lambda^+)| \ {C_s \over \lambda^s} \ \sum_{<u,u'> \in
\Gamma(\Lambda) } \E\left(|<O| {1 \over H_{W ; \omega} - z} |u>|^s
\right) \right\}  \  < \ 1 \; , \label{eq:cond1}
\end{equation}
then there are some $\mu(s) > 0$ and
$A(s) < \infty $ --- which depend on
the energy $z$ only through the bound $b(\Lambda,z)$ --- such
that for any region $\Omega \subset \Z^d$
\begin{equation}
\label{eq:thm1}
 \E_{\pm i 0}\left( |<x| {1 \over  H_{\Omega;\omega} - z }
|y> |^s \right)\ \le \ A(s) \  e^{-\mu(s) \, |x-y| } \; .
\end{equation}
\end{thm}

The subscript of  $\E_{\pm i 0} $, in (\ref{eq:thm1}) is to 
be interpreted as saying  that the bound is
valid for either of the two limiting expressions:
\begin{equation}
\lim_{\eta \searrow 0} \E\left( |<x| {1 \over H_{\Omega;\omega}
- E -\!(+)\, i \eta } |y> |^s \right) \; .
\end{equation}
The ``cutoff'' $\pm i\eta$ is needed for an
unambiguous interpretation in case $z$ is a real energy
($ E $)  within the spectrum of $H$. 
For the random operators considered here it is well understood
that: i) the expectation may be exchanged with the limit $\eta
\searrow 0$, ii) it suffices to verify the uniform bounds
(\ref{eq:thm1}) for finite regions, and iii) the finite volume
expectations are continuous in $\eta$.   
In the proofs we shall be dealing with finite systems; 
the subscript will, therefore,  be omitted there.   

Let us note that already the special case $\Lambda = \{ O\}$ is of
interest.  It provides the following variant of the single-site
criterion of ref.~\cite{AM} (which is, in fact, a bit simpler
since it does not invoke the {\em decoupling lemma}).

\noindent{\bf Corollary} {\it For the random Schr\"odinger
operator a sufficient condition for localization (\ref{eq:fm}) is
that for all $E\in [a,b]$
\begin{equation}
2d (2d -1) \ {C_s \over \lambda^s}  \ \int {1 \over |\lambda V -
E|^s} \ \rho(dV) \ < \ 1 \; .
\label{eq:singlesite} 
\end{equation}
 }  

Just as the main result of ref.~\cite{AM}, the above criterion
permits to easily conclude localization for the cases of high
disorder or extreme energies.  However, we may now  move beyond
that. By testing the hypothesis of Theorem~\ref{thm:1} in the
increasing sequence of volumes $\Lambda = [-L,L]^d$, one may
extend the conclusion to increasing regimes in the `energy
$\times$ disorder plane'.  In fact, it is easy to see that for
each energy at which the strong localization condition
(\ref{eq:thm1}) is satisfied, the hypothesis (\ref{eq:cond1}) will
be met at all sufficiently large $L$.  (This may, however,  be far
from a practical test, as the necessary computation may be rather
difficult for large $L$).

Observant readers may note that the conclusion of
Theorem~\ref{thm:1} provides not only the localization condition
\eq{eq:fm}, but it also rules out {\em extended boundary states}.
The flip side of this observation is that if such states are
present in some geometry, {\it e.g.} the half space, then the
hypothesis of Theorem~\ref{thm:1} will fail to be satisfied even
if the operator exhibits localization in the bulk.   Therefore, we
present also the following result which  permits to establish bulk
localization regardless of the possible presence of extended
boundary states.

\begin{thm}
\label{thm:2} Let $H_{\omega}$ be a random Schr\"odinger operator
with the probability distribution of the potential $V(x)$
satisfying $R_1(\tau)$ and $R_2(s)$, for some $s < \tau$. 
If for some $z \in \C$ and some finite 
region $O \in \Lambda \subset \Z^d$
\begin{equation}
 \left ( 1 + {\widetilde C_s \over \lambda^s} \ |\Gamma(\Lambda)| \right )^2
   \sum_{<u,u'> \in \Gamma(\Lambda)}
            \E \left (
                |<O| {1 \over
                    H_{\Lambda;\omega} - z} |u>|^s
            \right ) \ < \ 1 ,
\label{eq:cond2}
\end{equation}
then $H_{\omega}$ satisfies the
fractional-moment condition (\ref{eq:fm}), 
and there exist $\mu(s) > 0, A(s) < \infty $ so that for any
region $\Omega \subset
\Z^d$,
\begin{equation}
\E_{\pm i 0}\left( |<x| {1 \over  H_{\Omega;\omega}-z} |y> |^s \right)\ \le
\ A(s)\  e^{-\mu(s)\,  \dist_{\Omega}(x,y)} \; ,
\label{eq:thm2}
\end{equation}
with  
\begin{equation}  
\dist_{\Omega}(x,y)= \min\{|x-y|, [\dist(x,\partial \Omega)
+ \dist(y,\partial \Omega)] \} \; . 
\label{eq:dist} 
\end{equation} 
\end{thm}

Let us add that,
as in Theorem~\ref{thm:1}, $A(s)$ and $\mu(s)$ of (\ref{eq:thm2})
depend on $z$ only  through the value of the LHS in \eq{eq:cond2}.

The modified metric, $\dist_{\Omega}(x,y)$,  is a distance
function relative to which the entire boundary of $\Omega$ is
regarded as one point. It permits us to state that there is
exponential decay in the bulk without ruling out non-exponential
decay along the boundary. We supplement the last result by the
following observation.

\begin{thm}
Let $H_{\omega}$ be a random operator given by \eq{eq:proto}, with
the probability distribution of the potential $V(x)$ satisfying
$R_1(\tau)$ and $R_2(s)$, for some $s < \tau$. If at some energy
$E$ (or $z\in \C$) the localization condition  (\ref{eq:fm}) is
satisfied, with
some $A< \infty$ and $\mu> 0$, then for all  large enough 
(but finite) $L $
the condition (\ref{eq:cond2}) is met for $\Lambda = [-L,L]^d$.
\label{thm:3}
\end{thm}

The statement is a bit less immediate than the analogous
claim for Theorem~\ref{thm:1}.  We shall therefore include the
proof below.

%%%%%%%%%%%%%%%% NEW SECTION

\masect{Proofs of the main results} \label{sect:proof}

\masubsect{Some useful notation}

The proofs of the above statements will be presented in terms
which permit a direct extension to operators with more general
hopping terms.  We start by generalizing the notation; in
particular, the sets $\Lambda^{+}$ and $ \Gamma(\Lambda)$ will
be made to depend implicitly on the operator $T$.

%
%  We
% shall make explicit references  to the matrix elements $T_{x,y}$,
% even though in the case of  the discrete Laplacian they take just
% the values $0$ and $1$, and . The purpose is to allow a
% formulation which is valid also for
% operators with more general ``hopping'' terms.

In the study of  $H_{\Omega;\omega}$ we shall often consider
`depleted' Hamiltonians, $ H_{\Omega;\omega}^{(\Gamma)} $, obtained by setting
to zero the operator's non-diagonal matrix elements ({\em hopping terms})
along some collection of ordered pairs of sites (referred to here
as {\em bonds})  $\Gamma \subset \Z^d \times \Z^d$.
The difference is the operator $T^{(\Gamma)}$, with
\begin{equation}
T^{(\Gamma)}_{x,y} = \begin{cases}
        T_{x,y} \ & \mbox{ if } <x,y> \in \Gamma \mbox{ or } <y,x> \in
        \Gamma
        \\
        0 \ & \mbox{ if } <x,y> \not \in \Gamma \mbox{ and } <y,x> \not
        \in \Gamma   \; ,
        \end{cases}
\end{equation}
so that
\begin{equation}
H_{\Omega;\omega} \ = \ H_{\Omega;\omega}^{(\Gamma)} + T^{(\Gamma)}
\; .
\end{equation}

Typically,  $\Gamma$ will be a collection of bonds which forms
the `cut set' of some $W\subset \Z^d$, i.e.,
the set of bonds with $T_{x,y}\neq 0$ connecting sites in $W$ with
sites in its complement.  Thus we denote
\begin{equation}
\Gamma(W) \ = \ \left\{ <u,u'> | \ u \in
          W, u'\in \Z^d
          \backslash W, \mbox{ and }
          T_{u,u'}\neq 0    \right\} \; ,
\end{equation}
and also
\begin{equation}
      W^+ \ = \ W \cup \left \{ u' \in \Z^d |
        \ T_{u,u'} \neq 0
        \mbox{ for some } u \in W \right \} \; .
\end{equation}
The number of elements ({\it i.e.} bonds) in $\Gamma$ is denoted
$|\Gamma|$.

In addition, we use the ``Green function'' notation:
\begin{equation}
G_{\Omega;\omega}(x,y;z) = <x| {1 \over H_{\Omega;\omega} - z} |y>
\; ,
\end{equation}
with $G_{\Omega;\omega}^{(\Gamma)}(x,y;z)$ defined
correspondingly. Often, where it is obvious from context that an
operator is a random variable, we shall suppress the subscript
$\omega$.

In broad terms, the strategy for the proof is to derive a bound on
the average Green function, of the form
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \ \le  \ \sum_{< u ,u' >
\in \Gamma(\Lambda(x))} \gamma_{\Lambda(x)}(< u ,u' >)|T_{u,u'}|^s \
\E \left ( |G_{\Omega}^{(\Gamma(\Lambda(x))}(u',y;z)|^s \right ) \; ,
\label{eq:protobound}
\end{equation}
for all $y \in \Z^d \backslash \Lambda(x) $, where:
$\Lambda(x)= \{ x+y : y\in \Lambda \}$ is a finite neighborhood of $x$,
 translate of some fixed region $\Lambda \ni O$,
and $\gamma_{\Lambda(x)}$ is a quantity which is small when the
typical values of the finite volume Green function between $x$ and
the boundary of $\Lambda(x)$ are small (in a suitable sense).


An inequality of the form (\ref{eq:protobound}) is particularly
useful when
\begin{equation}
\sum_{<u,u'> \in \Gamma(\Lambda(x))} \gamma_{\Lambda(x)}(<u,u'>)
\ |T_{u,u'}|^s
 \ < \ 1 \; ,
\end{equation}
since in that case \eq{eq:protobound} is akin to the statement
that $\E \left ( |G_\Omega(x,y;z)|^s \right )$ is a strictly
subharmonic function of $x$, as long as $|x-y| > {\rm diam} |\Lambda|$,
and thus --- if it is also  uniformly
bounded (which it is) --- it decays exponentially.

The first step towards a bound of the form (\ref{eq:protobound})
is, naturally, the resolvent identity:
\begin{eqnarray}
\nonumber G_{\Omega,\omega} \ & = & \
 G_{\Omega,\omega}^{(\Gamma)} -
   G_{\Omega,\omega}^{(\Gamma)} \cdot T^{(\Gamma)} \cdot
      G_{\Omega,\omega}
        \\
       & & \nonumber \\
       & = & \ G_{\Omega,\omega}^{(\Gamma)} -
  G_{\Omega,\omega} \cdot  T^{(\Gamma)} \cdot
   G_{\Omega,\omega}^{(\Gamma)}
\label{eq:firstorder}
\end{eqnarray}
(written here in the operator form). However, one then reaches an
obstacle, since the quantity whose mean needs to be estimated is a
product of two Green functions which are not independent. In fact,
for some time now this co-dependence has been the main obstacle on
the road to an argument along the lines outlined above, since
otherwise the general strategy applied here is well familiar from
its various successful applications in the context of the
statistical mechanics of homogeneous systems
(\cite{Ham,DoSh,Simon,Lieb,AiNew}), and the other auxiliary tools specific to
the present context have in essence been available since
ref.~\cite{AM}. We solve here this co-dependence problem through a
second application of the resolvent identity (followed by a
decoupling argument of a familiar type).


The two applications of the resolvent identity,
for which the depletion sets $\Gamma_1$ and $\Gamma_2$
need not coincide,  may be combined by
starting our argument from the identity:
\begin{equation}
G_{\Omega} \ = \ 
 G_{\Omega}^{(\Gamma_1)} -
   G_{\Omega }^{(\Gamma_1)} \cdot T^{(\Gamma_1)} \cdot
      G_{\Omega}^{(\Gamma_2)}
      + G_{\Omega }^{(\Gamma_1)} \cdot T^{(\Gamma_1)} \cdot
      G_{\Omega } \cdot T^{(\Gamma_2)} \cdot
      G_{\Omega}^{(\Gamma_2)}  \; \;   .
\label{eq:secondorder}
\end{equation}
Readers familiar with the current techniques may note that once
the middle term $G_\Omega$ is replaced by a uniform bound, the
remaining expression can be made free from co-dependence by an
appropriate choice of $\Gamma_1$ and $\Gamma_2$. The rest are
technicalities, to which we turn next.

\masubsect{Key Lemmas} \label{sect:lemmas}

We shall now present three Lemmas which will be used in the
proofs of our main results.
The first is a known estimate which provides
the afore-mentioned uniform upper bound.

\begin{lem} \label{lem:1}
Let $V(x)$ be a random potential satisfying the regularity
condition $R_1(\tau)$. Then for each $s< \tau$,
%there exists a positive constant $C_s$ such that for
any region $\Omega$,  and any random operator of the form
(\ref{eq:proto})
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right )\  \leq \ {C_s \over
\lambda^s} \; , \label{eq:apriori}
\end{equation}
for all $z \in \C$.
\end{lem}
The statement is an immediate consequence of a version of the Wegner
estimate which we present in the appendix. (See
lemma~\ref{lem:fracmom}; also \eq{eq:conditional} below.)

Next is our new bound.

\begin{lem}
Let $H_{\omega}$ be a random operator given by \eq{eq:proto} with
the probability distribution of the potential $V(x)$ satisfying
the regularity condition $R_1(\tau)$, and let $W$ be a subset of
$\Omega$.  Then, denoting $\widetilde \Gamma = \Gamma(W^{+})$ and
$\Gamma = \Gamma(W)$, % (see Figure~\ref{fig:1}),
 for all $z \in \C$:
\begin{enumerate}
\item
The following `depleted-resolvent bound' holds for any pair of
sites $x \in W$,  $y \in \Omega \backslash W^+$,
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \
    \leq \  \gamma(W) \sum_ {<v,v'> \in \widetilde \Gamma} |T_{v,v'}|^s \
         \E \left (| G_{\Omega \backslash W^+}(v',y;z)|^s \right ) \; ,
\label{eq:d-r}
\end{equation}
with
\begin{equation}
\gamma(W) \ = \  {C_s \over \lambda^s}
         \sum_{<u,u'> \in \Gamma} |T_{u,u'}|^s \
         \E \left ( | G_{W}(x,u;z)|^s \right
         ) \; .
\end{equation}
%where $C_s$ is the constant provided by Lemma~\ref{lem:1}.
\item If, furthermore, the probability distribution of the
potential satisfies also $R_2(s)$ then
%there exists a constant
%$\widetilde C_s$, which depends only on the distribution of $V$,
%such that
the following bound holds for any pair of sites $x \in
W$, $y \in \Omega \backslash W$,
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \
    \leq \   \sum_ {<v,v'> \in \Gamma} \gamma_x(<v,v'>) \ |T_{v,v'}|^s \
         \E \left (| G_{\Omega \backslash W}(v',y;z)|^s \right ) \; ,
\label{eq:d-r2}
\end{equation}
with
\begin{multline}
\gamma_x(<v',v>) \ = \
    \E \left ( | G_W(x,v';z)|^s \right)
    \\ + \ {\widetilde C_s \over \lambda^s} \sum_{<u,u'> \in \Gamma }
    |T_{u,u'}|^s
    \E \left ( | G_W(x,u;z)|^s \right)    \; .
\label{eq:gamma}
\end{multline}
\end{enumerate}
\label{lem:diagram}
\end{lem}

\begin{figure}[htb]
    \begin{center}
    \leavevmode
        \epsfxsize=4in
  \epsfbox{Gbound.eps}
\caption{\footnotesize
\underline{Diagramatic depiction of the bound 
(\ref{eq:diagramatic}) on $G(x,y; z)$, for $x,y \in \Z^d$ and $z\in \C$.}
The long solid lines are `depleted Green functions', the two short 
segments correspond to the hoping terms ($T$) and the double line 
is a full Green function.  Once the latter is replaced by a uniform 
upper bound, the expectation value of the product of the remaining terms 
factorizes. }
\label{fig:beads}
\end{center}
\end{figure}

\begin{proof}
Both results follow from the second-order resolvent identity
\eq{eq:secondorder}, which yields:
\begin{multline}
G_{\Omega}(x,y;z)
 \ =  \ G_{\Omega}^{(\Gamma_1)}(x,y;z) \ - \
 <x|  G_{\Omega}^{(\Gamma_1)} \ T_{\Omega}^{(\Gamma_1)} \
 G_{\Omega}^{(\Gamma_2)} |y> \\ + <x|
G_{\Omega}^{(\Gamma_1)} \ T_{\Omega}^{(\Gamma_1)} \ G_{\Omega} \
T_{\Omega}^{(\Gamma_2)} \  G_{\Omega}^{(\Gamma_2)} |y> \; .
\label{eq:secondordermatelts}
\end{multline}

For the proof of the first claims, we take $\Gamma_1 = \Gamma =
\Gamma(W)$ and $\Gamma_2 = \widetilde \Gamma = \Gamma(W^+)$. Then,
the first term of \eq{eq:secondordermatelts} is zero because
$\Gamma(W)$ decouples $x$ and $y$ and the second term is zero
because $\Gamma(W^+)$ decouples $W^+$ and $y$. Thus
\begin{equation}
G_{\Omega}(x,y;z)
        \ = \ \sum_{ \substack{
        <u,u'> \in \Gamma \\
        <v,v'> \in \widetilde \Gamma}}
    T_{u,u'} \ T_{v,v'} \
    G_{\Omega}^{(\Gamma)}(x,u;z)
    G_\Omega(u',v;z)
    G_{\Omega}^{(\widetilde \Gamma)}(v',y;z) \; \; .
\label{eq:diagramatic} 
\end{equation}
It follows that for any $ s \in (0,1) $
\begin{multline}
\E \left ( |G_{\Omega}(x,y;z)|^s
        \right )  \\ \leq
    \sum_{ \substack{
        <u,u'> \in \Gamma \\
        <v,v'> \in \widetilde \Gamma}}
        |T_{u,u'}|^s |T_{v,v'}|^s \
    \E \left (
        |  G_{\Omega}^{(\Gamma)}(x,u;z)
    G_\Omega(u',v;z)
      G_{\Omega}^{(\widetilde \Gamma)}(v',y;z)|^s
     \right ) \; .
\label{eq:halffirstdiagram}
\end{multline}
(note that for $0 < s < 1$: $|a+b| \leq |a|^s + |b|^s$.)

In estimating the terms on the right hand side of
\eq{eq:halffirstdiagram} let us consider first the conditional
expectation of the central factors, $G_\Omega(u',v;z)$.  Only
these factors depend on the values of the potential at $u'$ and
$v$,
and therefore they can
be replaced by their conditional expectation
$         \E \left ( \left .
           |G_\Omega(u',v;z)|^s
            \right | \{V(q)\}_{q \in \Omega \backslash \{u',v\}}
    \right ) $.
As will be proven in the appendix, under the regularity condition
$R_1(\tau)$ these are uniformly bounded (Lemma~\ref{lem:fracmom}):
\begin{equation}
 \E \left ( \left .
        |G_\Omega(u',v;z)|^s \right |
        \{V(q)\}_{q \in \Omega \backslash \{u',v\}} \right )
    \ \leq  \ {C_s \over \lambda^s} \; .
\label{eq:conditional}
\end{equation}
(The proof involves a reduction to a two-dimensional problem via
the Krein formula, and a two-dimensional Wegner-type estimate.)

Once the central factor in each expectation on the right hand side
of \eq{eq:halffirstdiagram} is replaced by the above bound, what
remains there are  two independent random variables which are ${|
G_{\Omega}^{(\Gamma)}(x,u;z)|^s = |G_W(x,u;z)|^s }$ and ${|
G_{\Omega}^{(\widetilde \Gamma)}(v',y;z)|^s = | G_{\Omega
\backslash W^+ }(v',y;z)|^s}$ . The expectation now factorizes, and
the resulting expression yields the first claim of the Lemma.

For the second claim, we take $\Gamma_1 = \Gamma_2 = \Gamma =
\Gamma(W)$.  Once again the first term of
\eq{eq:secondordermatelts} is zero because $\Gamma(W)$ decouples
$x$ and $y$.  However, the second term is non-zero, and we obtain
\begin{multline}
\E \left ( |G_{\Omega}(x,y;z)|^s
        \right )  \\
\begin{split}
        \leq \ & \sum_{ <v,v'> \in \Gamma}
        |T_{v',v}|^s \E \left (
        | G_{\Omega}^{(\Gamma)}(x,v;z)
         G_{\Omega}^{(\Gamma)}(v',y;z)|^s
        \right ) \\ &  \quad + \
    \sum_{ \substack{
        <u,u'> \in \Gamma \\
        <v,v'> \in \Gamma}}
        |T_{u,u'}|^s |T_{v,v'}|^s \
    \E \left (
        | G_{\Omega}^{(\Gamma)}(x,u;z)
    G_\Omega(u',v;z)
     G_{\Omega}^{(\Gamma)}(v',y;z)|^s
     \right ) \; .
\end{split}
\end{multline}
At this point we may not use the previous argument, since in the
last expectation $V(v)$ affects each of the first two factors and
$V(u')$ affects each of the last two factors.  However, the
dependence of each of these factors on the potentials is of a
particularly simple form: they are ratios of two functions
(determinants) which are separately linear in each potential
variable. Using the decoupling hypotheses, {\it i.e.} the
regularity conditions $R_1(\tau)$  and $R_2(s)$, the expectation
may be bounded by the product of expectations. Specifically, we
prove  in Lemma~\ref{lem:decouplinginequalities} that:
\begin{multline}
\E \left (
        | G_{\Omega}^{(\Gamma)}(x,u;z)
    G_\Omega(u',v;z)
     G_{\Omega}^{(\Gamma)}(v',y;z)|^s
     \right )
    \\ \le \ {\widetilde C_s \over \lambda^s} \E \left (
        | G_{\Omega}^{(\Gamma)}(x,u;z)
         G_{\Omega}^{(\Gamma)}(v',y;z)|^s
        \right ) \; .
\end{multline}
%where $\widetilde C_s$ depends only on the regularity properties
%of the distribution of $V$.

Once again, we are left with a product of two independent random
variables, $| G_{\Omega}^{(\Gamma)}(x,u;z)|^s = | G_W(x,u;z)|^s$
and $| G_{\Omega}^{(\Gamma)}(v',y;z)|^s = | G_{\Omega \backslash W
}(v',y;z)|^s$.  The factorization of the remaining expectation
yields the second claim of the Lemma, \eq{eq:d-r2}.
\end{proof}

The above Lemma provides a bound for the Green function in terms
of its depleted versions.  This suffices for the derivation of the
first of our two main Theorems (Thm~\ref{thm:1}). However, this
does not suffice for the second Theorem, Thm~\ref{thm:2}, for
which we shall use an inequality that is linear in the original
function.  That ``closure'' will be attained with the help of the
following bound on the depleted resolvent in terms of the full
one.

\begin{lem}
Let $H_{\Omega,\omega}$ be a random operator in $\ell^2(\Omega)$, 
 $\Omega \stackrel{\subset}{=} Z^d$, given by \eq{eq:proto}, with
the probability distribution of the potential $V(x)$ satisfying
the regularity conditions $R_1(\tau)$ and $R_2(s)$ for some $s <
\tau$.    Let $W$ be a subset of $\Omega$. Then, the following
holds for any pair of sites 
$u,y \in \Omega \backslash  W$, and every $z \in \C$
\begin{multline}
 \E \left ( | G_{\Omega \backslash W}(u,y;z)|^s \right ) \
        \le \ \E \left ( |G_{\Omega}(u,y;z)|^s \right )
             + \ {\widetilde C_s \over \lambda^s}
            \sum_{<v,v'> \in \Gamma} |T_{v',v}|^s
            \E \left ( |G_{\Omega} (v, y;z) |^s
            \right ) \; ,
 \label{eq:fullbound}
 \end{multline}
with $\Gamma = \Gamma(W)$ the `cut-set' of $W$.
 \label{lem:fullbound}
\end{lem}

\begin{proof}
Starting from the first order
resolvent identity, \eq{eq:firstorder}, and taking expectation
values of its matrix elements, we find:
\begin{multline}
\E \left ( | G^{(\Gamma)}_{\Omega} (u,y;z)|^s \right ) \ \le \
    \E \left ( |G_{\Omega}(u,y;z)|^s \right ) \\
    + \sum_{<v,v'> \in \Gamma(W)} |T_{v',v}|^s
            \E \left ( | G^{(\Gamma)}_{\Omega}(u,v';z)|^s
            |G_{\Omega} (v, y;z) |^s
            \right ) \; ,
\end{multline}
where $\Gamma = \Gamma(W)$, and $G^{(\Gamma)}= G_{\Omega \backslash W}$.
It suffices, therefore, to show that in the last term the factor
 $| G^{(\Gamma)}_{\Omega}(u,v';z)|^s$ may be replaced (for an upper
 bound) by the constant ${\widetilde C_s \over \lambda^s}$.
This follows through a decoupling argument which we present in
the Appendix --- see
Lemma~\ref{lem:decouplinginequalities}.
\end{proof}

\begin{remark}
In the applications we shall use Lemmas~\ref{lem:diagram} and
\ref{lem:fullbound} both in the stated form and in the conjugated
form, with the arguments of the Green functions reversed.
One form of course implies the other (at conjugate energy).
\end{remark}

\masubsect{Proofs of the main results}

We are now ready to derive the results stated in the Introduction.
For simplicity these were stated in the context of the
Schr\"odinger operators, for which $T$ is the discrete Laplacian.
The proofs given in this section will be restricted to this case.
A more generally applicable treatment is presented in the next section.
%
% In the next section, where we discuss more general formulations of
% the results, we shall present an alternative argument for the
% ``sub-harmonic function'' component of the proof, which is better
% suited to operators whose hopping terms $T_{x,y}$  are not of
% finite range.



\begin{proof_of}{Theorem \ref{thm:1}}
%Let $C_s$ be the constant given by lemma \ref{lem:1}.
Assume that for some $z \in \C$ and a finite region $\Lambda$ the
smallness condition (\ref{eq:cond1}) holds. By
Lemma~\ref{lem:diagram} and translation invariance, we learn that
for any region $\Omega$ and any $x,y \in \Omega$ with $y \in \Z^d
\backslash \Lambda^{+}(x)$:
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s
        \right )  \ \leq \ b \cdot {1 \over |\Gamma(\Lambda^+)|}
         \sum_{<v,v'> \in \Gamma(\Lambda^+(x))}
            \E \left (
                |G_{\Omega \backslash \Lambda^+(x)}(v',y;z)|^s
            \right ) \; ,
\label{eq:thm1iterate}
\end{equation}
where $b=b(\Lambda,z)$ of \eq{eq:cond1}, and $\Lambda(x)$ is the
translate of $\Lambda$ by $x$.

By Lemma~\ref{lem:1}, each of the terms in the sum is bounded by
$C_s/\lambda^s$.  Since the sum is normalized by the prefactor $
1/ |\Gamma(\Lambda^+)|$, the inequality (\ref{eq:thm1iterate})
permits to improve that bound for $\E (|G_\Omega(x,y;z)|^s)$  by
the factor $b \ (<1)$. Furthermore, the inequality may be iterated
a number of times, each iteration resulting in an additional
factor of $b$.

One should take note of the fact that the iterations bring in
Green functions corresponding to modified domains. 
%(see Figure~\ref{fig:iterations}).  
It is for this reason that the
initial input assumption was required to hold for modified
geometries, {\it i.e.} not just for $\Lambda$ but also for all its
subsets.

Inequality~(\ref{eq:thm1iterate}) can be iterated
as long as the resulting sequences
($x, v', \ldots, v^{(n)}$) do not get closer to $y$
than the distance
$L= \sup\{ |u| \ | \ u \in \Lambda^+ \}$.
Thus:
\begin{equation}
\E \left ( |G_\Omega(x,y;z) |^s \right ) \ \le \ {C_s \over
\lambda^s} \cdot b^{\lfloor |x-y|/L \rfloor} \ \le \
{C_s \over \lambda^s b } \ e^{- \mu |x-y|}
\; , \label{eq:thm1optimalbound}
\end{equation}
with $\mu = |\ln b| / L$.
\end{proof_of}

Next, let us turn to the proof of the second theorem
(Thm~\ref{thm:2}). The main change is that we now proceed under
the assumption that the smallness condition holds for some region
$\Lambda$ without requiring it to hold also in all subsets.  As
explained in the introduction, the difference may be meaningful if
$H_{\omega}$ has extended boundary states in some geometry.

\begin{proof_of}{Theorem \ref{thm:2}}   Our first goal is to show
that under the assumption (\ref{eq:cond2})
%, with $\widetilde C_s$ the constant given by Lemma~\ref{lem:diagram},
there is $b< 1$
such that for all pairs $\{x,y\}$ with $\Lambda(x) \subset \Omega$
and $y \in \Omega \backslash \Lambda(x)$,
\begin{equation}
\label{eq:thm2iterate} \E
\left ( |G_\Omega(x,y;z)|^s \right ) \ \le \ b \
    \sum_{u \in \Lambda^+(x)  } P^l_x(u) \
         \E \left ( |G_\Omega(u,y;z)|^s \right ) \; ,
\end{equation}
with non-negative weights satisfying:
\begin{equation}
\sum_{u \in \Lambda^+(x)  } P^l_x(u) \ = \ 1 \; .
\label{eq:norm}
\end{equation}
We shall use this inequality along with its conjugate:
\begin{equation}
 \label{eq:reversedthm2iterate} \E
\left ( |G_\Omega(x,y;z)|^s \right ) \ \le \ b
    \sum_{v \in \Lambda^+(y)} P^r_y(v) \
         \E \left ( |G_\Omega(x,v;z)|^s \right ) \; ,
\end{equation}
where $ P^r_y(v)$ satisfy the suitable analog of the
normalization condition (\ref{eq:norm}).

It is important that -- unlike in the inequality
(\ref{eq:thm1iterate}), the functions which appear
on the right hand side of (\ref{eq:thm2iterate}) and
(\ref{eq:reversedthm2iterate}) are computed in the same
domain as those on the left hand side.

The first step is by Lemma~\ref{lem:diagram}, which yields
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \ \leq \
         \sum_ {<u,u'> \in \Gamma(\Lambda(x))}
         \gamma_x(<u,u'>)  \ \E \left (
         | G_{\Omega \backslash \Lambda(x)}(u',y;z)|^s
         \right ) \; ,
\end{equation}
whenever $\Lambda(x) \subset \Omega$ and $y \in \Z^d \backslash \Lambda(x)$,
with $\gamma_x(<u,u'>) $ specified in \eq{eq:gamma}.

Next, we apply Lemma~\ref{lem:fullbound}, \eq{eq:fullbound}, to bound
$\E \left (| G_{\Omega \backslash \Lambda(x)}(u',y;z)|^s
\right ) $ in terms of a sum of quantities of the form
$\E \left ( |G_{\Omega}(v,y;z)|^s \right )$
with  $v\in \Lambda^{+}(x)$.   The result is
initially expressed as a sum over bonds:
\begin{multline}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \ \leq
    \sum_{<u,u'> \in \Gamma(\Lambda(x))}\gamma_x(<u,u'>) \
          \E \left (|G_\Omega(u',y;z)|^s \right )
          \\
    + \ {\widetilde C_s \over \lambda^s} \ \Theta
    \sum_{<u,u'> \in \Gamma(\Lambda(x))}
         \E \left (|G_\Omega(u,y;z)|^s \right ) \; ,
\label{eq:thm2fullbound}
\end{multline}
where, using translation invariance,
\begin{equation}
\nonumber \Theta \ := \ \sum_{<u,u'> \in \Gamma(\Lambda)}
\gamma_O(<u,u'>) \; .
\end{equation}

Collecting terms, and pulling out normalizing factors,
one may cast the inequality (\ref{eq:thm2fullbound}) in
the form (\ref{eq:thm2iterate}) with
\begin{eqnarray}
b  & := & \sum_{<u,u'> \in \Gamma(\Lambda(x))}
        \left( \gamma_x(<u,u'>)  \
        +  {\widetilde C_s \over \lambda^s} \Theta
         \right ) \ = \
       \left ( 1 + {\widetilde C_s \over \lambda^s}
        \ |\Gamma(\Lambda)| \right ) \ \Theta \\
    & = &  \left ( 1 + {\widetilde C_s \over \lambda^s}
        \ |\Gamma(\Lambda)| \right )^2
   \sum_{<u,u'> \in \Gamma(\Lambda)}
            \E \left ( | G_{\Lambda}(O,u;z)|^s
            \right )  \; .
\end{eqnarray}
The smallness condition (\ref{eq:cond2}) is nothing other than  the
assumption that $b  < 1$.

The above argument proves \eq{eq:thm2iterate}. By the
transposition, or time-reflection, symmetry of $H$ ($H^{T}=H$)
also \eq{eq:reversedthm2iterate} holds.
(Such symmetry of $H$ is not essential for our
analysis: it suffices to assume that the smallness condition
\eq{eq:cond2}  holds along with its transpose.)


We proceed in the proof by iterating the inequalities
(\ref{eq:thm2iterate}) and (\ref{eq:reversedthm2iterate}).  However
an adaptation is needed in the argument which was used in the proof of
Theorem~\ref{thm:1} since the iteration can be carried out only
as long as the two points (the arguments of the resolvent) stay at
distance $L= \sup \{ |u| : u \in \Lambda^{+}\}$ not only from
each other but also from the boundary  $\partial \Omega$.
The relevant observation is that for every pair of sites
$x,y\in \Omega$ there is a pair of integers $\{ n, m\} $ such that:
\begin{enumerate}
\item $n+m = \dist_{\Omega}(x,y)$ ,
\item
the ball of radius $n$ centered at $x$ and the ball of radius
$m$ centered at $y$ form a pair of disjoint subsets of $\Omega$.
\end{enumerate}
For the desired bound on $\E \left ( |G_\Omega(x,y;z) |^s \right
)$, we shall iterate  \eq{eq:thm2iterate}
$\lfloor n/L \rfloor$ times from the left,
and (\ref{eq:reversedthm2iterate}) $\lfloor m/L \rfloor$ times
from the right. Similar to
\eq{eq:thm1optimalbound}, we obtain:
\begin{equation}
\E \left ( |G_\Omega(x,y;z) |^s \right ) \ \le  \
{C_s \over \lambda^s b^2 } \ e^{- \mu \ \dist_{\Omega}(x,y)}
\; , \label{eq:thm2optimalbound}
\end{equation}
with $\mu = |\ln b| / L$.
\end{proof_of}

The third Theorem stated in the introduction  (Thm~\ref{thm:3}) is
the claim that the condition which is shown above to be sufficient
for exponential localization, in the sense of \eq{eq:fm}, is also a
necessary one. We shall now prove this to be the case.

\begin{proof_of}{Theorem~\ref{thm:3}}
Suppose that \eq{eq:fm} holds with some $A < \infty$ and $\mu >
0$. We need to show that also in finite systems the Green function
is sufficiently small between an interior point and the boundary.
To bound the finite volume function in terms of the infinite
volume one, we may use lemma~\ref{lem:fullbound}, by which
\begin{multline}
\sum_{<u,u'> \in \Gamma(\Lambda)} \E \left (|G_\Lambda (O,u;z)|^s
    \right ) \ \le \ \sum_{<u,u'> \in \Gamma(\Lambda)}\E \left (
    |G(O,u;z)|^s \right ) \\
+ \ {\widetilde C_s \over \lambda^s} \ |\Gamma(\Lambda)|
            \sum_{<v,v'> \in \Gamma(\Lambda)} |T_{v,v'}|^s \
            \E \left ( |G(O, v'; z) |^s
            \right ) \; ,
\label{eq:noname}
\end{multline}
for any finite region $\Lambda$ containing the origin.
We need to show that for  $\Lambda = [-L,L]^d$ with  $L$ large enough
\begin{equation}
 \left ( 1 + {\widetilde C_s \over
\lambda^s} |\Gamma(\Lambda)| \right )^2
    \sum_{<u,u'> \in \Gamma(\Lambda)}
            \E \left ( |G_\Lambda(o,u;z)|^s
            \right ) \ < \ 1 \; .
\label{eq:local}
\end{equation}
After applying \eq{eq:noname} to the terms on the left side of
 \eq{eq:local} we find that
the number of summands involved and their
prefactors grow only polynomially
in $L$, whereas under our assumption the relevant factors
$ \E \left ( |G(O,u;z)|^s \right )$ are  exponentially small
in $L$.  Hence the condition (\ref{eq:local}) is satisfied
for $L$ large enough.
\end{proof_of}


%%%%%%%%%%%%%%%% NEW SECTION
\newpage

\masect{Generalizations} \label{sect:gen}

\masubsect{Formulation of the general results}

We shall now turn to some generalizations of the theorems which
were presented in Section~\ref{sect:main} for the random
Schr\"odinger operator. The setup may be extended in a number of
ways.

\begin{enumerate}
\item[1.] {\em Addition of magnetic fields.}
The hopping terms $\{T_{x,y}\}$ need not be real.
In particular, the present analysis remains valid when one
includes
in $H_{\omega}$ a constant magnetic field, or a random one with a
translation invariant distribution.
\end{enumerate}


A magnetic field  is incorporated in $T_{x,y}$ through  a factor
$exp( -i  A_{x,y})$, with $A_{x,y}$ an anti-symmetric function of
the bonds.  (It represents the integral of the `vector potential'
$\times (-e/\hbar) $ along the bond $<x,y>$.) Except for the
trivial case, with such a factor $T$ is no longer
 shift invariant.
However, in the case of a constant magnetic field, $T$  will still
be invariant under appropriate``magnetic shifts'', which consist
of ordinary shifts followed by gauge transformations.

Translation-invariance plays a role in our discussion.  However,
since  gauge transformations do not affect the absolute values of
the resolvent, it suffices for us to assume that  $H_{\omega}$ is
{\em stochastically invariant under magnetic shifts} -- in the
sense of Definition~\ref{def:1}.

\begin{enumerate}
\item[2.] {\em Extended hopping terms.}
The discrete Laplacian may be replaced by an operator
with hopping terms of unlimited range.  For exponential
localization we shall however require $\{T_{x,y}\}$ to decay
exponentially in $|x-y|$.

\item[3.] {\em Off-diagonal disorder.}   $\{T_{x,y}\}$ may also
be made random.  It is convenient however to assume exponentially
decaying uniform bounds.  The regularity conditions on the
potential will now be assumed for the conditional distribution of
$V(x)$ at specified off-diagonal disorder.

\item[4.] {\em Periodicity.}  $H_{\omega}$ may also include a
periodic potential, i.e., \eq{eq:proto} may be modified to:
\begin{equation}
H_{\omega} \ = \  T_{x,y; \omega} + U_{per}(x) +
 \lambda V_{\omega}(x)
\end{equation}
This may be further generalized by requiring periodicity only of
the probability distribution of $H$.

\item[5.] {\em More general lattices.}
\end{enumerate}

In the previous discussion, the underlying sets $\Z^d$ may be
replaced by other graphs, with suitable symmetry groups.
The graph structure is relevant if the hopping terms
are limited to graph edges.
However, since we consider also operators
with hoping terms of unlimited range, let us formulate the
result for operators on $\ell^2(\T)$ where the underlying set
is of the form $\T = \G \times S $, with $\G$  a countable
group and $S$ a finite set.
We let $\dist(x,y)$ denote a metric on $\T$ which is
invariant under the natural action of  $\G$ on that set.

For example, this setup allows for $\T$ to be a Bethe lattice, or
a more general Cayley lattice. (Instructive discussion of some
statistical mechanical models in such settings may be found in
refs.\cite{BLPS}).  The set $S$ is included here in order to
leave room for periodic structures. We denote by   $\Cell$ the
``periodicity cell'', which is $\{\imath \}\times S$ where $\imath
$ is the identity in $\G$.

Some of the relevant concepts are summarized in the following
definition.

\begin{df}
\label{def:1} With  $\T = \G \times S$ as above, let  $H_{\omega}$
be a random operator on  $\ell^2(\T)$ (i.e., one with some
specified probability distribution), whose off-diagonal part is
denoted by $T_{\omega}$ and the diagonal part is referred to as
the potential (for consistency, we denote it as $\lambda
V_{\omega}$).
\begin{enumerate}
\item
We say that $H_{\omega}$ is
\underline{stochastically invariant under magnetic shifts}
if for each
$\kappa \in \G$ and almost every $\omega$ there is
a unitary map of the form
\begin{equation}
\left( U_{\kappa, \omega} \psi \right)(x) \ = \ e^{i \phi_{\kappa,
\omega}(x)  } \psi(\kappa x) \; ,
\end{equation}
(with some function $\phi_{\kappa, \omega}(\cdot)$ )
under which
\begin{equation}
U^{*}_{\kappa, \omega} \ H_{\omega} \ U_{\kappa, \omega} \
\stackrel{\mathcal D}{=}  \ H_{\omega} \; ,
\end{equation}
where $\stackrel{\mathcal D}{=} $ means equality of the probability
distributions.
\item
The operator is said to have
\underline{tempered off-diagonal matrix elements}, at a specified
value of $s<1$,
if  there is a kernel $\tau_{x,y}$, and some $m>0$, such that
$T_{x,y;\omega}\ \le \ \tau_{x,y}$, almost surely, and
\begin{equation}
\sup_{x\in \T} \
\sum_{y\in \T} \tau_{x,y}^s \ e^{+\, m \, \dist(x,y)} \ < \
\infty \; . 
\label{eq:deftempered}
\end{equation}
\item We say that the potential  has
an  \underline{$s$-regular distribution} if for some $\tau > s$
the conditional distributions of $\{V_{\omega}(x)\}$, at specified
values of the hopping terms variables $\{T_{u,v;\omega}\}$, are
independent and satisfy the regularity conditions $R_1(\tau)$ and
$R_2(s)$ with uniform constants.
\end{enumerate}
\end{df}

Following is the generalization of Theorem~\ref{thm:1}.


\begin{thm}
Let $H_{\omega}$ be a random operator on $\ell^2(\T)$
($\T=\G\times S$, as above) with an $s$-regular distribution for
the potential $V_{\omega}(\cdot )$, and with tempered off-diagonal
matrix elements ($T_{x,y;\omega}$), which is stochastically
invariant under magnetic shifts. Assume that for some $z\in \C$
and a finite region $\Lambda \subset \T $, which contains the
periodicity cell $\Cell$,  the following is satisfied for all
subsets $W\subset \Lambda$
\begin{equation}
    \left (
        1 + {\widetilde C_s \over \lambda^s} \ \Xi_s(\Lambda)
    \right )
     \sup_{x\in \Cell}
     \sum_{<u,u'> \in \Lambda \times (\T  \backslash \Lambda)}
           \tau(u-u')^s \ \E\left (
                |<x| {1 \over
                    H_{W;\omega} - z  
                    } |u>|^s
            \right ) \ < \  1 \; ,
\label{eq:thmgeneralcond2}
\end{equation}
where
\begin{equation}
\tau(v) = \sup_{u\in \T} \mbox{\rm ess sup}_{\omega} 
     |T_{u,u+v;\omega}|
\; , \qquad
\Xi_s(\Lambda) \ = \  \sum_{<u,u'> \in \Lambda \times (\T
\backslash \Lambda)} \tau(u-u')^s \; .
\label{eq:tau} 
\end{equation}
Then
there exist $\mu > 0$, $A< \infty$, such that for all
$\Omega \subset \T$, and all
$y\in \Omega$,
\begin{equation}
\sum_{x \in \Omega} \E_{\pm i \eta} 
\left( |<x| {1 \over  H_{\Omega; \omega}-z }
|y> |^s \right) \ e^{+  \, \mu \, \dist(x,y)} \ \le \ A
\label{eq:summabledecay2}
\end{equation}
\label{thm:generalized1}
\end{thm}


\noindent{\bf Remarks:}

\noindent {\bf 1.} For graphs which grow at an exponential rate,
such as the Bethe lattice, exponentially decaying functions need
not be summable. The conclusion, \eq{eq:summabledecay2}, was
therefore formulated in the stronger form, which implies both
exponential decay, and almost sure summability. In particular, it
is useful to recall that for $s/2 < 1$:
\begin{equation}
\E\left( \left[ \sum_{y} |G(x,y)|^2 \right ]^{s/2} \right)  \ \le
\ \E\left(  \sum_{y} |G(x,y)|^{s} \right) \; .
\end{equation}

\noindent {\bf 2.} One may note  that in the more general theorem
we do make use of the ``decoupling Lemma'', which was not used in
Theorem~\ref{thm:1}.

\noindent {\bf 3.} Translation invariance played a limited role here: 
the analysis extends readily to random operators with
non-translation invariant distributions, provided only that the 
required bounds are satisfied uniformly for all translates of  
$\Lambda$, and the distribution of the potential is 
uniformly $s$-regular.  To demonstrate the required change 
we cast the next statement in that form.  


As we discussed in the preceding sections, condition
(\ref{eq:thmgeneralcond2}) may fail due to the existence of
extended states at some surfaces.  The following generalization of
Theorem~\ref{thm:2} provides criteria for localization in the bulk
which are less  affected by such surface states.


\begin{thm} Let $H_{\omega}$ be a random operator 
 on $\ell^2(\T)$
($\T=\G\times S$, as above) with an $s$-regular distribution for
the potential $V_{\omega}(\cdot )$, and with tempered off-diagonal
matrix elements ($\{ T_{x,y;\omega} \} $).
Assume that for some $z\in \C$ and a
finite region $\Lambda$, $\Cell \subset \Lambda \subset \T$,
 \begin{equation}
    \left (
        1 + {\widetilde C_s \over \lambda^s} \ \Xi_s(\Lambda)
    \right )^2
    \sup_{x\in \Cell}
     \sum_{\substack {u \in \Lambda(x)  \\
    	            u' \in  \T  \backslash \Lambda(x)} }
           \tau_{u,u'}^s \ \E \left (
                |<x| {1 \over
                    H_{\Lambda;\omega} - z [\bar z] } |u>|^s
            \right ) \ < \  1 \; ,
\label{eq:thmgeneralcond}
\end{equation}
where $\Lambda(x)$ is the unique translate of $\Lambda$, by an 
element of $\G$, which contains $x$, and 
$z [\bar z]$ means that the bound is satisfied for
both $z$ and $\bar z$.  
Then the condition  (\ref{eq:summabledecay2})
holds for the full operator $H_{\omega}$ (i.e., with  $\Omega = \T$),
 and there  exist $B < \infty, \ \tilde \mu > 0$
with which for arbitrary  $\Omega \subset \T$:
 \begin{equation}
\E_{\pm i \eta} 
\left( |<x| {1 \over  H_{\Omega;\omega}-z } |y> |^s \right) \
\le \ B \  e^{- \, \tilde \mu \,  \dist_{\Omega}(x,y)} \; .
\label{eq:Omegadecay}
\end{equation}
\label{thm:generalized2}
\end{thm}

The modified distance $\dist_{\Omega}(x,y)$
is defined by the natural extension of  \eq{eq:dist}.

\masubsect{Derivation of the general results}

The derivation of Theorems \ref{thm:generalized1} and
\ref{thm:generalized2} follows very closely the proofs of
Section~\ref{sect:proof}.  The main difference is in the second
portion of the argument where we encounter a more general
``sub-harmonicity'' relation.

The
first part of the proof rests on  Lemmas \ref{lem:diagram} and
\ref{lem:fullbound} which are easily seen to extend to the setup
described in Theorem~\ref{thm:generalized2}.
(The hopping terms $T_{x,y}$ appearing in
section~\ref{sect:lemmas} are replaced with the uniform
upper-bound $\tau_{x,y}$.) We thus obtain the following
extension of the resolvent bounds.

\begin{lem} Let $H_{\omega}$ be a random operator
with the properties listed in Theorem~\ref{thm:generalized2}, and
let $\Lambda $ be a finite subset of $\T$, containing the periodicity cell
$\Cell$, for which the condition
(\ref{eq:thmgeneralcond2}) is   satisfied.
Then the following bound is valid for
any $x\in \Lambda, y\in \T \backslash \Lambda$,
\begin{equation}
\sup_{\Omega \subset \T} \E \left ( |G_\Omega(x,y;z)|^s \right ) \ \le \
 b \ \sum_{u \in \T}  p_\Lambda(x,u) \ \sup_{\Omega
\subset \T} \E \left ( |G_{\Omega} (u,y; z) |^s \right ) \; ,
\label{eq:lemma3.3a}
\end{equation}
with some $b < 1$ and a ``sub-probability kernel''
$p_\Lambda(x,u)$,  satisfying
\begin{equation} \sum_{u} p_\Lambda(x,u) \ \le \ 1 \; , \text{ and }
\sum_x p_\Lambda(x,u) \ \le \ 1 \; ,
\label{eq:sub-probability}
\end{equation}
which is tempered in the sense that for some $m>0$
\begin{equation}
\sup_x \sum_{u} e^{m \, \dist(x,u)} p_\Lambda(x,u) \ < \ \infty \;
, \text{ and } \sup_u \sum_x e^{m \, \dist(x,u)} p_\Lambda(x,u) \
< \ \infty \;.
\label{eq:tempered}
\end{equation}

Furthermore, assuming (\ref{eq:thmgeneralcond}) instead of
(\ref{eq:thmgeneralcond2}),  the following bound is valid for
any $x\in \Lambda, y\in \T \backslash \Lambda$, and $\Omega
\supset \Lambda  $
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \ \le \ \tilde b \
\sum_{u \in
\T} \widetilde p_\Lambda(x,u) \  \E \left ( |G_{\Omega } (u,y; z) |^s \right
) \; ,
\label{eq:lemma3.3b}
\end{equation}
with some $ \widetilde b < 1$ and $\widetilde p_\Lambda(x,u)$
which satisfies the same conditions as $p_\Lambda(x,u)$.
\label{lem:generalized2}
\end{lem}

The bounds presented in the above lemma may be read as  stating
that the resolvent $\E(|G(x,y;z)|^s)$ is sub-harmonic (we
use this term here in the sense of ``sub-mean'')  with respect
to a tempered probability kernel whenever $x,y$ are
sufficiently far apart. Theorems~\ref{thm:generalized2} and
\ref{thm:generalized1} follow from these bounds via a general
principle which applies to such sub-harmonic functions.  We expect
this principle to be well known, but for completeness we include a
proof here.

\begin{prop} Let $(\T, \dist)$ be a countable metric space,
$\Lambda \subset \T$  a finite subset,
and $g: \T \to \R $ a bounded and non-negative function, which
for all $x \in \T \backslash \Lambda$ satisfies:
 \begin{equation}
    g(x) \ \le \ b \sum_{u} p(x,u) g(u)  \;  ,
\label{eq:g}
\end{equation}
with  a kernel on $\T \times \T$ satisfying
\begin{equation}
\sup_x \sum_u   p(x,u) \ \le \ 1  \; ,
\qquad
\sup_u \sum_x   p(x,u) \ \le \  1 \; ,
\label{eq:kernel-prob}
\end{equation}
which is tempered in the sense of \eq{eq:tempered}.
Then $g(x)$ is exponentially  summable, i.e., for some $\mu > 0$:
\begin{equation}
\sum_y e^{\mu \, \dist(y,\Lambda) } \ g(y) \ < \ \infty \; .
\label{eq:summabledecay}
\end{equation}
\label{prop:subharmonic}
\end{prop}

\begin{proof}  One may read the claim as saying that
the function $g(\cdot)$ lies in the space $\ell^{1;\, \mu}(\T)$
of functions for which the following norm is finite:
\begin{equation}
\|f\|_{1, \mu} \ := \ \sum_{x \in \T} e^{\mu \, \dist (x, \Lambda)}
|f(x)| \; .
\end{equation}
We shall deduce this claim after arriving first at a bound
formulated within the larger  space of bounded functions
$\ell^{\infty}(\T)$.

Let  $P$ be the linear operator with the kernel $p(x,y)$.
Within $\ell^{\infty}(\T)$ the
operator acts as a contraction, since its norm there is
\begin{equation}
\| P \|_{\infty, \infty} \ = \ \sup_{x} \sum_{u} p(x,u) \ \le \ 1
\end{equation}
(using (\ref{eq:kernel-prob}) ).  It is convenient to paraphrase the
assumption on $g(\cdot )$ in the following form, which holds
for all $x\in \T$:
\begin{equation}
g(x) \ \le \ \|g\|_\infty \cdot I_\Lambda(x) \ + \ b \ [P \cdot
g](x) \; ,
\label{eq:globalbound}
\end{equation}
with $I_\Lambda$ the ``indicator function'' of $\Lambda$.
Iterating this relation $N$ times, one obtains a bound in the form of a
finite geometric series with a ``remainder'' which is uniformly
bounded by $ (b \  \| P\|_{\infty, \infty})^N \cdot  \|g\|_\infty $.
As  $N \to \infty$ the reminder vanishes, since
$(b \  \| P\|_{\infty, \infty}) < 1$, and one is left with a bound in
the form of a  convergent series:
\begin{equation}
g(x) \ \le \ \| g \|_\infty \ \sum_{n=0}^{\infty} b^n \
 [P^n \cdot I_\Lambda](x)   \; .
   \label{eq:seriesbound}
\end{equation}

We now note that for a finite region $\Lambda$, the function
$I_\Lambda$ lies in the ``weighted-$\ell^1$ space'' $\ell^{1;\, \mu}$.
The norm of  $P$ as an operator within $\ell^{1;\, \mu}$ is easily seen
to obey:
\begin{equation}
\| P \|_{1,\mu; 1, \mu} \ \le  \ \sup_u \sum_x e^{\mu \, \dist (x,u)}
p(x,u) \; .
\end{equation}
The expression on the right hand side is convex in $\mu$, and
by the temperedness assumption (the analog of \eq{eq:tempered})  it
is finite for small  enough $\mu > 0$.  Since convexity implies
continuity, using (\ref{eq:kernel-prob}) we conclude that there is
some $\mu > 0 $ for which
\begin{equation}
b \ \| P \|_{1,\mu; 1, \mu}\ < \ 1 \; .
\end{equation}
With this choice of $\mu$ we conclude:
\begin{equation}
\sum_{x} e^{\mu \, \dist(x, \Lambda)} \ g(x)  \ \equiv \
\| g \|_{1,\mu} \ \le \
{\| g \|_{\infty} \ | \Lambda | \over 1 - b \, \| P \|_{1,\mu; 1, \mu} }
 \ <  \ \infty  \; .
\end{equation}
\end{proof}

Theorems~\ref{thm:generalized1} and
\ref{thm:generalized2}  now follow by a combination  of the
proposition just shown with Lemma~\ref{lem:generalized2}.

\begin{proof_of}{Theorem \ref{thm:generalized1} }
To establish the claimed bound (\ref{eq:summabledecay2}) fix $y
\in \T$, and let $g(x) = \sup_\Omega \E(|G_\Omega(x,y;z)|^s)$. We
note that for each $x\in \T$ there is a unique element of the
symmetry group, $h_x \in \G$, such that $h_x x \in \Lambda$.
Starting from the kernel $p_\Lambda(h_x x,h_x u)$ which appears in
Lemma~\ref{lem:generalized2}, let us define a shift-invariant
kernel $p(x,y)$ by:
\begin{equation}
p(x,u) \ = \ p_\Lambda(h_x x,h_x u) \; .
\label{eq:pxy}
\end{equation}
Due to the shift invariance of the distribution of $H_{\omega}$,
\eq{eq:lemma3.3a} implies that the function $g(x)$
 is sub-harmonic, in the sense of (\ref{eq:g}), with respect to
the kernel $p(x,u)$, which satisfies (\ref{eq:kernel-prob}) and is
tempered . Thus, a direct application of
Proposition~\ref{prop:subharmonic} yileds now the claimed bound
(\ref{eq:summabledecay2}).
\end{proof_of}

\begin{proof_of}{Theorem \ref{thm:generalized2} }
The situation to be discussed now is different from that 
encountered in the last proof in
that now for each $\Omega$ the basic sub-harmonicity bound
can be assumed only for points which are not too
close to the boundary $\partial \Omega$.
The claim made for the special case $\Omega = \T $ is covered by the
above analysis.
However, the  second claim, i.e., \eq{eq:Omegadecay}, requires a somewhat
different argument.

The argument we shall use shadows the proof of
Proposition~\ref{prop:subharmonic}, replacing there the
weighted-$\ell^{1}$ estimate by its weighted-$\ell^{\infty}$
version. The starting observation is that $\E (
|G_\Omega(x,y;z)|^s )$ has the sub-mean property with respect to
averages over either $x$ or $y$ -- provided the point is at
distance at least  ${\rm diam}(\Lambda)$ from the other and from
the boundary $\partial \Lambda$. (In allowing the averaging
procedure to occur from either side, we rely on the fact that  the
smallness condition holds for both the kernel $G(x,y:z)$ and its
conjugate, or equivalently the fact that the smallness condition
is assumed to hold for both $z$ and $\bar z$ .)

To cast the situation in terms reminiscent of the proof of
Proposition~\ref{prop:subharmonic}, let us  consider the function
$g(<x,y>)= \E ( |G_\Omega(x,y;z)|^s )$ as defined over the space
of pairs, $\Omega \times \Omega$, equipped with the distance
function
\begin{equation}
\dist_\Omega(<x_1,y_1>,<x_2,y_2>) \ = \ \dist_\Omega(x_1,x_2) +
\dist_\Omega(x_2,y_2) \; .
\end{equation}
For $<x,y>$ not in the set
$W :=  \{ <u,v> \ | \ \dist_\Omega(u,v) \le 2 L$,
with $L= {\rm diam}(\Lambda) \} $,
we have the basic sub-mean estimate:
\begin{equation}
g(<x,y>) \ \le \ b \ \sum_{<u,v>} \widetilde p(<x,y>,<u,v>) \ g(<u,v>) \; ,
\end{equation}
with
\begin{equation}
\widetilde p(<x,y>,<u,v>) :=
\left\{
  \begin{array}{ll}
       p(x,u) \, \delta_{y,v} &
          \mbox{\footnotesize if
              \ $\dist_{\Omega}(x,y) >  2 L$ and $\dist(x,\partial
              \Omega) >  L$  } \; , \\
            \delta_{x,u} \, p(y,v)  &
                 \mbox{\footnotesize if
                \   $\dist_{\Omega}(x,y) > 2 L$ and $\dist(x,\partial
                  \Omega) \le L$  }  \; , \\
            \delta_{x,u} \, \delta_{y,v} &
                \mbox{\footnotesize if \  $\dist_{\Omega}(x,y) \le 2
                L$   }       \; ,
          \end{array}
          \right.
   \end{equation}
where $p(x,y)$ is given by \eq{eq:pxy}.

By repeating the arguments seen there we find that
$g(<x,y>)$ obeys the analog of \eq{eq:seriesbound} ---
formulated within the space $\ell^{\infty}( \Omega \times \Omega)$,
 with the set $\Lambda$ replaced by $W$, and the operator $P$ replaced
by $\widetilde P$ defined by the kernel $\widetilde p(<x,y>,<u,v>)$.
Unlike in the previous case, we have no fixed bound on the size of
 the set $W$.  Thus we shall not use here   the weighted-$\ell^{1}$
estimate.  However, we may reuse the argument applying it to
weighted-$\ell^{\infty}$  norm   of $g(\cdot)$, which is defined
as:
\begin{equation}
\| g \|_{\infty; \mu} \ = \ \sup_{<x,y>}  e^{\mu \, \dist(x,y)} \, |g(<x,y>)|
\end{equation}
The conclusion is that there is some $\mu > 0$ at
which $\| g \|_{\infty; \mu} <
\infty$.  Equivalently:
\begin{equation}
\E \left ( |G_\Omega(x,y;z)|^s \right ) \ \le \ \c e^{-\mu \,
\dist_\Omega(x,y)} \; \; ,
\end{equation}
as claimed in Theorem~\ref{thm:generalized2}.
\end{proof_of}


\masect{Some Implications}

We shall now present a number of implications of the finite volume
criteria for localization, focusing on the finite dimensional
lattices $\Z^d$. The  statements will bear some resemblance to
results derived using the multiscale approach, however the
conclusions drawn here go beyond the latter by yielding results on
the exponential decay of the \underline{mean values}.  The 
significance of that was described in the introduction.  

\masubsect{Fast power decay $\Rightarrow$ exponential decay.}

An interesting and useful implication 
(as is seen below) 
is that fast enough power law implies exponential decay. 
In this sense,  random Schr\"odinger operators join 
other statistical mechanical models in which 
such principles have been previously recognized. 
The list includes the general 
Dobrushin-Shlosman results~\cite{DoSh}  and the more specific two-point 
function bounds in: percolation 
(Hammersley\cite{Ham} and Aizenman-Newman~\cite{AiNew}),
Ising ferromagnets 
(Simon~\cite{Simon} and Lieb~\cite{Lieb}),  
certain  $O(N)$ models (Aizenman-Simon~\cite{AiSi}), 
and  time-evolution models (Aizenman-Holley~\cite{AiHo}, 
Maes-Shlosman~\cite{MaSh}.) 

\begin{thm}  Let 
$H_{\omega}$ be a random operator on $\ell^2(\Z^d)$
with an $s$-regular distribution for the potential
($V_{\omega}(x)$)
and  tempered off-diagonal matrix elements ($T_{x,y;\omega}$).
There are $L_o, B_1, B_2 <\infty$, which 
depend only on the temperedness bound (\ref{eq:deftempered}), 
such that if  for some $E\in \R$  and some finite  $L \ge L_o $,
 either
\begin{equation}
           L^{3(d-1)} \ \sup_{\ L / 2 \le \|x-y\| \le L} 
            \E \left(     |<x| {1 \over
                    H_{\Lambda_L(x), \omega} - E} |y>|^s
            \right ) \ \le \ B_1 \; ,  
\label{eq:powerlaw1} 
\end{equation} 
or 
\begin{equation}
     L^{4(d-1)} \ \sup_{\ L / 2 \le \|x-y\| \le L}
      \E \left(
                |<x| {1 \over
                    H_{\omega} - E - i 0} |y>|^s
            \right ) \ \le \ B_2 \; , 
\label{eq:powerlaw2}
\end{equation}
where $\Lambda_L(x)=[-L,L]^d+x$ and $\|x-y\| \equiv \sum_{j} |y_j|$, 
then the \underline{exponential} localization (\ref{eq:fm}) 
holds for all energies in some open interval $(a,b)$ containing $E$. 
\label{thm:power=>exp}
\end{thm}

\begin{proof}
By Theorem~\ref{thm:generalized2}, 
to establish exponential decay at the energy $E$ it suffices to show
 that for each $x\in \Z^d$
\begin{equation}
 \left (
        1 + {\widetilde C_s \over \lambda^s} \ \Xi_s(\Lambda_L)
    \right )^2
    \sum_{\substack {u  \in \Lambda_L(x) \\ u' \in \Z^d \backslash
\Lambda_L(x) } }
               \tau_{u,u'}^s \ \E \left (
                |G_{\Lambda_L(x)}(x,u;E)|^s
            \right ) \ < \  1 \; . 
\label{eq:suff} 
\end{equation}  
Because the off diagonal elements are tempered
we have the following bounds 
\begin{equation}
\tau(u-u')^s \ \le \  \c  \ e^{-m |u-u'|} \; , \qquad
\Xi_s(\Lambda_L) \ \le \  \c  \  L^{d-1} \; ,
\end{equation}
for some $m > 0$, and all $L > 1 $.  
Under the assumption \eq{eq:powerlaw1}: 
\begin{multline} 
  \sum_{\substack {u  \in \Lambda_L(x) \\ u' \in \Z^d \backslash
\Lambda_L(x)}}
               \tau(u-u')^s \ \E \left (
                |G_{\Lambda_L(x)}(x,u;E)|^s
            \right ) \ \le \  \\     
  \le  \   {\widetilde C_s \over \lambda^s}\  \c \ (L/2)^d \   
  e^{-mL}/m  \  + \  \qquad  \  \\ 
     + \ \c \ \sup_{\ L / 2 \le \|x-y\| \le L}
      \E \left(  |<x| {1 \over   H_{\Lambda_L(x), \omega} - E} |y>|^s   \right ) 
      \  e^{-mL}/m  \; .
\label{eq:tiddledoo}
    \end{multline}
For this bound the sum was split according to
 $ \|u-u'\| < ({\rm or} \  \ge)  L/2$, and in the first case 
 we used the uniform upper bound 
$\E( |G(x,u;E)|^s) \le   \widetilde C_s /  \lambda^s$.

It is now easy to see that with an appropriate choice of 
$L_o$ and $B_1$  condition (\ref{eq:powerlaw1}) implies 
the the claimed bound (\ref{eq:suff}) -- for the given 
energy $E$.  The extension to an interval of energies around 
$E$ follows then throught the continuity of the fractional moments of 
\underline{finite volume} Green functions.  

To show the sufficiency of the second condition, we first use 
Lemma~\ref{lem:fullbound} to  bound finite
volume Green functions in terms of the corresponding 
infinite volume funtions
\begin{equation}
 \E \left ( | G_{\Lambda_L(x)}(x,y;E)|^s \right )
  \le   \ \E  \left (| G( x, y;E)|^s \right )
    + \ {\widetilde C_s \over \lambda^s}  \sum_{\substack{
                        u \in \Lambda_L(x) \\ u' \in  \Z^d
                        \backslash \Lambda_L(x) }}
                    \tau_{u',u}^s \ \E  \left
                    ( | G( x, u';E)|^s \right ) \; .
\end{equation} 
Splitting the sum as in \eq{eq:tiddledoo}, we get   
\begin{multline} 
\sup_{\ L / 2 \le \|x-y\| \le L}
      \E \left ( | G_{\Lambda_L(x)}(x,y;E)|^s \right ) \  \le  \\ 
     \le    \left[ {\widetilde C_s \over \lambda^s}\ \right]^2 \c \ (L/2)^d \  
  e^{-mL}/m  \  + \ \qquad  \    \\ 
  + \  \left( 1 + \c \ e^{-mL}/m  \right) \ \times \ L^{d-1} \ 
   \sup_{\ L / 2 \le \|x-y\| \le L} 
   \E \left ( | G(x,y;E)|^s \right )       
\label{eq:tiddledo2} 
\end{multline} 
The combination of \eq {eq:tiddledo2} with (\ref{eq:tiddledoo}), 
yields the claim - for the given energy.  Again, the existence of an open 
interval of energies in which the condition is met 
is implied by the continuity of the finite-volume  expectation 
values.  
\end{proof}

\masubsect{Lower bounds for $G_{\omega}(x,y; E_{{\rm edge} }+i0)$ 
at mobility edges} 

Boundary points of the continuous spectrum  are often referred to as 
a {\em mobility edges}.  (In an ergodic setting 
the location of such points does not depend on the 
realization $\omega$ \cite{KuSu}.)
The proof of the occurance  of 
continuous spectrum for random stochastically  shift-invariant 
operators on $\Z^d$ is still an open problem 
(one may add that we are glossing here over some  
fine distinctions in the dynamical behaviour \cite{dynamics}).  
However it is intersting to note that 
Theorem~\ref{thm:power=>exp} directly yields 
the following pair of lower bounds on the decay rate of the 
Green function at mobility edges, $E_{{\rm edge} }$, for 
stochastically shift invariant random operators with regular 
probability distribution of the potential: 
\begin{equation}
    \ \sup_{\ L / 2 \le \|y\| \le L}
      \E \left(
                |<o| {1 \over
                    H_{[-L,L]^d, \omega} - E_{{\rm edge} } } |y>|^s
            \right ) \ \ge \ B_1 \  L^{-3(d-1)} \; ,  
\label{eq:mobilityedge1} 
\end{equation} 
\begin{equation}
    \ \sup_{\ L / 2 \le \|y\| \le L}
      \E \left(
                |<o| {1 \over
                    H_{\omega} - E_{{\rm edge} } - i 0} |y>|^s
            \right ) \ \ge \ B_2 \  L^{-4(d-1)} \; ,  
\label{eq:mobilityedge2} 
\end{equation} 
with $\|y\| \equiv \sum_{j} |y_j|$.  
We do not expect the power laws provided here to be optimal. 
As mentioned above, vaguely similar bounds are known for the critical 
two-point functions in certain statistical mechanical models 
(percolation, Ising spin systems, and some $O(N)$ spin models).  

\masubsect{Extending off the real axis}

For various applications, such as the decay of the projection
kernel (see  \cite{AG} Sect. 5), it is useful to have bounds
on the resolvent at $z=E+i\eta$ which are uniform in $\eta$.
The following result shows that in order to establish such
uniform bounds it is sufficient to verify our criteria
for real energies in some neighborhood of $E$.

\begin{thm}  Let 
$H_{\omega}$ be a random operator on $\ell^2(\Z^d)$
with an $s$-regular distribution for the potential
($V_{\omega}(x)$)
and  tempered off-diagonal matrix elements ($T_{x,y;\omega}$).
Suppose that for some $E\in \R$, and $\Delta E > 0$, the
following bound holds uniformly for
$\xi \in [E-\Delta E, E+\Delta E ]$:
\begin{equation}
    \E \left(  |<x| {1 \over H_{\omega} - \xi - i 0} |y>|^s
            \right ) \ \le \ A\,  e^{-\mu |x-y|}  \; .
\label{eq:real} 
\end{equation}
Then for all $\eta \in \R$:
\begin{equation}
    \E \left(  |<x| {1 \over H_{\omega} - E - i \eta} |y>|^s
            \right ) \
 \le \ \widetilde A\ \ e^{-\tilde \mu  |x-y|}   \; ,
\label{eq:eq:strip} 
\end{equation}
with some $\widetilde A  < \infty$ and $\tilde \mu > 0$ -- 
which depend on  $\Delta E$ 
and  the bound (\ref{eq:real}). 
\label{thm:strip}
\end{thm}

\noindent{\bf Remark:}
  \noindent  1.   
This result is not needed in situations covered by the 
\underline{single site} version of the criterion provided by 
Theorem~\ref{thm:1}, 
since if \eq{eq:singlesite} 
is satisfied at some $E\in \R$ then
it automatically holds uniformly along the entire line $E + i\R$.
We do not see  
a monotonicity argument for such a deduction in case of
other finite-volumes.  \\ 
\noindent 
2.  One way to derive the statement is by using the fact 
that exponential decay may be tested in finite 
volumes: if a finite volume criterions 
holds for some $E$ then continuity allows one to extend
it to all $E + i \eta$ with $\eta$ sufficiently small.
The Combes-Thomas estimate~\cite{CT} can then be used to cover the
rest of the line $E+i\R$.   
However, by this approach one gets only a weaker decay rate 
for energies off the real axis.   It is tempting to think 
that some contour integration argument could be found to 
significantly improve on that.  The proof given below is a 
step in that direction (though it still leaves one with the feeling 
that a more efficient argument should be possible). 

\begin{proof} 
Assume that the condition (\ref{eq:real}) is satisfied for all 
$\xi \in [E-\Delta E, E+ \Delta E]$.  
We shall show that this implies that for any power $\alpha$
\begin{equation}
    \E \left(  |<x| {1 \over H_{\omega} - \xi - i \eta} |y>|^s
            \right ) \ \le \ 
             { A_{ \alpha} \over |x-y|^{ \alpha} }\;  ,
\label{eq:power2} 
\end{equation}
with the constant $A_{\alpha}< \infty$ uniform in $\eta$.  
The stated conclusion then follows by 
an application of  Theorem~\ref{thm:power=>exp} (and the 
uniform  bounds seen in its proof). 


We shall deal separately 
with  large and small  $|\eta|$, splitting the two regimes at 
$\Delta E \times \pi/ \alpha $.  
 The case $|\eta| \ge \Delta E \times \pi/ \alpha $ 
is covered by the 
general bound of Combes-Thomas~\cite{CT}, which states that: 
\begin{equation}
|G(x,y;E+{\rm i}\eta)| \le (2/\eta){\rm e}^{-m|x-y|}
\label{eq:CT}
\end{equation}
for any $m\ge 0$ such that 
\begin{equation} 
\sum_{x\in \Z^d} \tau(x) \, (e^{m|x|}-1) \ \le \ \eta/2 \; . 
\end{equation} 

To estimate the resolvent for 
 $|\eta| \le \Delta E \times \pi/ \alpha $, we shall use the 
fact that  the  function
\begin{equation}  
f_L(\zeta) \ = \   \E \left( |G_{[-L,L]^d}(x,y; \zeta )|^s \right) 
\end{equation}
is subharmonic in the upper half plane, and continuous 
at the boundary.  The subharmonicity is a general consequence of 
the analyticity of the resolvent in $\zeta$, and the continuity 
is implied through the continuity of the distribution 
of the potential.  $L$ serves as a convenient cutoff, which may be 
removed after the bounds are derived (since 
$H_{[-L,L]^d, \omega} \ \too{L\to \infty} H_{\omega}$ 
in the strong resolvent sense).

Let $D\subset \C$ be the triangular region in the upper half 
in the form of an equilateral triangle based on the real interval 
$[E-\Delta E, E + \Delta E]$ with the side angles equal to 
$ \theta$ -- determined by the condition 
\begin{equation} 
\alpha \ = \ {2 \pi \over \theta } - 1 \; .
\label{eq:theta} 
\end{equation}    
The Poisson-kernel representation of harmonic 
functions yields, for $E+ i \eta \in D$, 
\begin{equation} 
f_L(E + i \eta ) \ \le  \ \int_{\partial D} f_L(\zeta) 
\ P^{D}_{E+ i \eta}(d \zeta) 
\end{equation} 
where $P^{D}_{E+ i \eta}(d \zeta) $ is a certain 
probability measure on $\partial D$. 
We now rely on the fact that this probability measure satisfies 
\begin{equation} 
P^{D}_{E+ i \eta}(d \zeta) \ \le \ \c \  d(\eta^{2 \pi / \theta}) \, /
\Delta E^{2 \pi / \theta} \; .
\end{equation} 
(This is easily understood upon the 
unfolding of $D$ by the map 
$z \mapsto z ^{2 \pi / \theta}$ 
applied from either of the base corners of $D$, 
i.e., from $\zeta = E \pm \Delta E$, and a  comparison with 
the Poisson kernel in the upper half plane.) 
 

For $\zeta \in \partial D \cap \R$ the integrand 
satisfies the exponential  bound (\ref{eq:real}).  
Along the rest of the boundary of $D$ we use the Combes-Thomas bound 
(\ref{eq:CT}).  
Putting it all together we get 
\begin{equation} 
f_L(E + i \eta ) \ \le \  A\,  e^{-\mu |x-y|} \ + \ 
\c  \int_{0}^{\Delta E\,  \theta } 
{2 \over \eta } \, e^{- \c\, |x-y| \, \eta} 
\ d(\eta^{2 \pi / \theta})\ /
\Delta E^{2 \pi / \theta}  \; . 
\end{equation} 
The claimed \eq{eq:power2} follows by simple integration, and the 
relation (\ref{eq:theta}).  
\end{proof} 



\masubsect{Localization in spectral tails.}

The finite volume criteria allow to conclude exponential localization
from suitable bounds on the density of states of the operators
in regions $\Lambda_{L} = [-L,L]^d$.  The following
statement will be useful for such a purpose.

\begin{thm}  Let
$H_{\omega}$ be a random operator on $\ell^2(\Z^d)$ with tempered
off-diagonal matrix elements ($T_{x,y;\omega}$) and a distribution
of the potential which is $s$-regular for all $s$ small enough,
which is stochastically invariant under magnetic shifts. If, at
some energy $E\in \R$:
\begin{equation}
\liminf_{L \rightarrow \infty} L^\xi \P \left [ \dist \left (
\sigma(H_{\Lambda_{L}; \omega}), E \right) \le L^{-\beta} \right ]
\ = \ 0 \; , \label{eq:xicondition}
\end{equation}
for  some $\xi >  3 (d - 1) $, $\beta \in (0,1)$,
then the exponential localization condition (\ref{eq:fm})
holds in some open interval containing $E$.
\label{thm:tails}
\end{thm}

\noindent{\bf Remarks:}
1. As should be clear from the proof, instead of requiring the
$\liminf$ of \eq{eq:xicondition},  to vanish,
it would have sufficed to ask the quantity to reach a small
enough value at  some finite $L$.

2. The input condition (\ref{eq:xicondition}) is similar
to the one used in the multiscale analysis.  In fact, there
it is not important that $\xi > 3(d-1)$, and it suffices
to assume (\ref{eq:xicondition}) with some $\xi >0$.
However, one may
note that if the multiscale analysis applies then its
conclusion allows to deduce the condition as stated here.
Thus, the exponential localization in the stronger sense
discussed in our work applies throughout the regime which
may be reached through the multiscale analysis.

3.  It is of interest to combine the criterion presented above
with Lifshitz tail estimates on the density of states at the
bottom of the spectrum and at band edges. Previous results in this
vein may be found in \cite{FiKl,BCH,KSS,Stollmann}.

\begin{proof} 
We shall demonstrate that under the stated assumption, 
(\ref{eq:xicondition}), 
for all $L$ large enough the 
input condition
(\ref{eq:powerlaw1}) of  Theorem~\ref{thm:power=>exp}: 
\begin{equation}
           L^{3(d-1)} \ \sup_{\ L / 2 \le \|y\| \le L} 
            \E \left(     |<0| {1 \over
                    H_{\Lambda_L, \omega} - E} |y>|^s
            \right ) \ \le \ B_1 \; ,  
\label{eq:indiansummer} 
\end{equation} 
 is satisfied for for all energies
$\widetilde E \in [E-\half L^{-\beta},E + \half L^{-\beta}]$.
Naturally, we shall pick $L$ as the smallest value above 
$L_o$ at which \eq{eq:indiansummer} holds.  Exponential 
localization in the corresponding interval (and strip, 
with $\eta \ne 0$) follows then by 
Theorems~\ref{thm:power=>exp} (and Theorem~\ref{thm:strip}).

We estimate $\E \left (
|G_{\Lambda_L;\omega}(O,u;\widetilde E)|^s \right )$ by
considering separately the contributions from the ``good set'':
\begin{equation}
\Omega_g \ = \ \{ \omega \ | \ \dist \left (
\sigma(H_{\Lambda_{L}; \omega}), E \right) > L^{-\beta} \} \; ,
\end{equation}
and the ``bad set'':
\begin{equation}
\Omega_b \ = \ \Omega_g^c = \ \{ \omega \ | \ \dist \left ( 
\sigma(H_{\Lambda_{L}; \omega}), E \right) \le L^{-\beta} \} \; .
\end{equation}  
For $\omega \in \Omega_g$, $\widetilde E$ is at a small yet 
significant distance from the spectrum ($\Delta E \ge L^{-\beta}$)
of $H_{\Lambda_L;\omega}$ and the Combes-Hislop~\cite{CH}
 argument implies
that
\begin{equation}
|G_{\Lambda_L; \omega}(O,u;\widetilde E)|
    \ \le \ {4 \over  \Delta E} e^{-{1 \over 4} \Delta E
    |u|} \; .
\end{equation}
When $\omega \in \Omega_b$, the resolvent may get to be quite
large.  However, the net contribution to the expectation is small
because $\P(\Omega_b)$ is small. Using the 
H\"older inequality to estimate the latter contribution, we get:
\begin{multline}
\E \left ( |G_{\Lambda_L; \omega}(O,u;\widetilde E)|^s \right ) \\
\begin{split}
 = & \ \E \left (   |G_{\Lambda_L; \omega}(O,u;\widetilde E)|^s
                    \ I[\omega \in \Omega_g] \right )
    \ + \ \E \left ( |G_{\Lambda_L; \omega}(O,u;\widetilde E)|^s
                    \ I[\omega \in \Omega_b] \right ) \\
 \le & \ {4^s L^{s \beta}} e^{- s \, |u|\,  L^{-\beta}\, / 4 } \
    + \ \E \left (|G_{\Lambda_L; \omega}(O,u;\widetilde E)|^t \right  )^{s
\over t}
    \ \E \left ( I[\omega \in \Omega_b]  \right )^{1 -
    {s \over t} } \\
 \le & \ {4^s L^{s \beta}} e^{- s \, |u|\,  L^{-\beta}\, / 4 } \ 
  + \  C_t^{s \over t}/ \lambda^s \, \times  \P \left [ \dist \left (
    \sigma(H_{\Lambda_{L}; \omega}), E \right) \right ]^{1 - {s \over
    t}} \; ,
 \end{split}
 \end{multline}
where  $t$ is any number greater than $s $ for which the distribution 
of the potential is still $t$-regular (i.e., $C_t < \infty$). 

Chosing for the above bound  $s$ small enough so that 
${t \over t-s} 3 (d-1) \le \xi$, we may conclude  that 
\begin{equation} 
   L^{3(d-1)} \ \sup_{\ L / 2 \le \|y\| \le L} 
            \E \left(     |<0| {1 \over
                    H_{\Lambda_L, \omega} - E} |y>|^s
            \right ) \ \too{L \to \infty}  \ 0 \; .
\end{equation} 
This proves the theorem.  \end{proof}  


% \masect{Summary}
To summarize: we have seen here that the localization regime can be 
characterised by finite volume criteria, and that 
this fact carries a range of meaningful implications.  

 
\newpage 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\startappendix

\vspace{1truecm plus 1cm} \noindent {\large\bf Appendix: Auxiliary
Bounds}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%new section
\maappendix{A fractional moment bound}
\label{sect:moment}

The regularity conditions $R_1(\tau)$ and $R_2(s)$ have been used
to give {\em a priori} estimates of certain fractional moments.
Such fractional moment bounds are properties of the general class
of operators with diagonal disorder. Hence, throughout this
appendix, we consider random operators $H_\omega$ on $\ell^2(\T)$
of the form
\begin{equation}
    H_\omega \ = \ T_o + \lambda V_\omega \; , \label{eq:arbitrary}
\end{equation}
where $T_o$ is an arbitrary bounded self adjoint operator and
$V_{\omega}$ is a random potential  for which $V_\omega(x)$ are
independent random variables ($\T$ is any countable set).

\begin{lem}
\label{lem:fracmom} Let $H_{\omega}$ be a random operator given by
\eq{eq:arbitrary} such that for each $x$ the probability
distribution of the potential $V_\omega(x)$ satisfies $R_1(\tau)$
for some fixed $\tau>0$ with constants uniform in $x$. Then there
exists $\kappa_\tau < \infty$ such that for any finite subset
$\Lambda$ of $\T$, any $x, y \in \Lambda$, any $z \in \mathbb C$,
and any $s \in (0, \tau)$
\begin{equation}
\label{eq:fracmom} \E \left ( \left . |<x|
\frac{1}{H_{\Lambda;\omega} - z} |y>|^s \right | \{V(u)\}_{u \in
\Lambda \backslash \{x,y\}} \right) \ \leq \ {\tau \over \tau - s}
\
 { (4 \kappa_{\tau}) \over \lambda^s}^{s/\tau} \; .
\end{equation}
\end{lem}
\begin{proof}
Let us first consider $z=E \in \R$. For such energies
\eq{eq:fracmom} is a consequence of a Wegner type estimate on the
2-dimensional subspace spanned by $|x>, |y>$. The key is to
determine the correct expression for the dependence of $<x|
\frac{1}{H_{\Lambda;\omega} - E} |y>$ on $V_\omega(x)$ and
$V_\omega(y)$. Such an expression is given by the `Krein formula': 
\begin{equation}
 <x| \frac{1}{H_{\Lambda;\omega} - E} |y> \ = \
    <1| \left ( [A]^{-1} +  \lambda \begin{bmatrix}
        V_\omega(x)&0\\0 & V_\omega(y)
        \end{bmatrix}
     \right )^{-1} |2> \; ,
\label{eq:Krein}
\end{equation}
where $[A]$ is a $2\times2$ matrix whose entries do not depend on
$V_\omega(x)$ or $V_\omega(y)$. In fact,
\begin{equation}
    [A] \ = \ \begin{bmatrix}
      <x|\frac{1}{\widehat H_{\Lambda;\omega} - E} |x> &
      <x|\frac{1}{\widehat H_{\Lambda;\omega} - E} |y> \\
      <y|\frac{1}{\widehat H_{\Lambda;\omega} - E} |x> &
      <y|\frac{1}{\widehat H_{\Lambda;\omega} - E} |y>
    \end{bmatrix} \; ,
\end{equation}
where $\widehat H_{\Lambda;\omega}$ denotes the operator obtained
from $H_{\Lambda;\omega}$ by setting $V_\omega(x)$ and
$V_\omega(y)$ equal to zero.

The regularity condition $R_1(\tau)$ implies a Wegner type
estimate: % (see \cite{XXX}):
\begin{equation}
\P \left ( \left \| \left ( [A]^{-1} + \lambda
        \begin{bmatrix}
          V_\omega(x)&0 \\
          0 & V_\omega(y)
        \end{bmatrix}
     \right )^{-1} \right \| > t
        \left . \phantom{a \over b} \right | \; \{V_\omega(u)\}_{u \neq
x,y} \right )
     \ \leq \ {4 \kappa_{\tau} \over (\lambda t)^{\tau}} \; ,
\label{eq:Wegner}
\end{equation}
where $\kappa_\tau$ is any finite number such that for every $v
\in \T$, $a \in \R$, and $\epsilon > 0$
\begin{equation}
 \P \left ( V_\omega(v) \in
(a-\epsilon,a+\epsilon)  \right ) \ \le \ \kappa_\tau
\epsilon^{\tau} \; .
\end{equation}
The desired bound (\ref{eq:fracmom}) follows easily from
\eq{eq:Wegner}. (The factor, $4$, on the right hand side of
(\ref{eq:Wegner}) arises as the square of the ``volume'' of the
region $\{x,y\}$. In the case $x = y$, we could replace this
factor by $1$.)

Although the Krein formula (\ref{eq:Krein}) is true when $E$ is
replaced by any $z \in \C$, the resulting matrix $[A]$ may not be
normal if $z \not \in \R$.  (The resolvent, $1 \over H - z$, {\em
is} normal. However, given an orthogonal projection, $P$, the
operator $P {1 \over H - E} P$ may not be normal!) Yet, the
Wegner-like estimate (\ref{eq:Wegner}) holds only when $[A]$ is a
normal matrix.  At first, this seems to be an obstacle to the
extension of (\ref{eq:fracmom}) to all values of $z$. However,
once the inequality is known for real values of $z$, it follows
for all $z \in \mathbb C$ from analytic properties of the
resolvent. Specifically, the function
\begin{equation}
\phi(z) \ = \ |<x| \frac{1}{H_{\Lambda;\omega} - z} |y>|^s
\end{equation}
is {\em sub-harmonic} in the upper and lower half planes and
decays as $z \rightarrow \infty$.  Hence, $\phi(z)$ is dominated
by the convolution of its boundary values with a Poisson kernel:
\begin{equation}
\phi( E + i \eta) \ \leq \ \int \phi(\widetilde E) {|\eta| \over
(E - \widetilde E)^2 + \eta^2}{d \widetilde E \over \pi} \; .
\end{equation}
By Fubini's theorem and \eq{eq:fracmom} for $\widetilde E \in \R$,
(\ref{eq:fracmom}) is seen to hold for all $z \in \mathbb C$.
\end{proof}

The ``all for one'' principle mentioned previously is actually a
simple consequence of Lemma~\ref{lem:fracmom}.
\begin{lem}
Let $H_{\omega}$ be a random operator as described in
Lemma~\ref{lem:fracmom}, and suppose that there is a distance
function $\dist$ on $\T$ such that for some $s < \tau$ and some $z
\in \C$
\begin{equation}
\E\left( |<x| {1 \over  H_{\omega}-z } |y> |^s \right) \ \ \le \ \
A(s) \  e^{-\mu(s) \, \dist(x,y) } \; , \label{eq:allforone}
\end{equation}
for every $x,y \in \T$. Then, in fact, (\ref{eq:allforone}) holds,
with modified constants $A(r)$ and $\mu(r)$, when $s$ is replaced
by any $r < \tau$. \label{lem:allforone}
\end{lem}
\begin{proof}
Note that given $r,s>0$ with $r < s < \tau$
\begin{multline}
\E \left(|<x| {1 \over  H_{\Lambda;\omega}-E } |y> |^r \right)^{s
\over r} \ \le \ \E \left(
        |<x|{1 \over H_{\Lambda;\omega}-E } |y> |^s
        \right) \\ \le \ \E \left(
        |<x| {1 \over  H_{\Lambda;\omega}-E } |y> |^r
        \right)^{t-s \over t- r}
        \E \left(
        |<x|{1 \over H_{\Lambda;\omega}-E } |y> |^t
        \right)^{s- r \over t - r} \\ \le \
        \left (
        {(4 \kappa_{\tau}) \over \lambda^t}^{t/\tau}
        \right )^{s- r \over t - r}
        \E \left(
        |<x| {1 \over  H_{\Lambda;\omega}-E } |y> |^r
        \right)^{t- s \over t- r} \; ,
\end{multline}
where $t$ is any number with $s < t < \tau$.
\end{proof}

%new section
\maappendix{Decoupling inequalities} \label{sect:decoupling}

\masubappendix{Decoupling inequalities for Green Functions}

The condition $R_2(s)$ plays a crucial role in several of the
arguments presented in this paper.  It has been used to bound
expectations of products of Green functions in terms of products
of expectations.  In this section we demonstrate the validity of
the necessary bounds. The main result is the following:

\begin{lem}
Let $H_{\omega}$ be a random operator given by \eq{eq:arbitrary},
with an $s$ regular distribution of the potential $V_\omega(x)$.
Then %there exists $\widetilde C_s > 0$ such that
\begin{enumerate}
\item For any $\Omega_1, \Omega_2 \subset \T$, any $x,y \in
\Omega_1$, and any $u,v \in \Omega_2$,
\begin{equation}
\E \left ( |G_{\Omega_1}(x,y;z)|^s |G_{\Omega_2}(u,v;z)|^s \right
) \ \le \ {\widetilde C_s \over \lambda^s} \ \E \left (
|G_{\Omega_1}(x,y;z)|^s \right ) \; .
\end{equation}
\item  For any $\Omega_1 \cap \Omega_2 = \emptyset$,
$x,u \in \Omega_1$, $v,y \in \Omega_2$, and $\Omega_3 \subset
\Gamma$,
\begin{multline}
\E \left ( |G_{\Omega_1}(x,u;z)|^s |G_{\Omega_3}(u,v;z)|^s
|G_{\Omega_2}(v,y;z)|^s \right ) \\ \le \ {\widetilde C_s \over
\lambda^s} \ \E \left ( |G_{\Omega_1}(x,u;z)|^s \right ) \E \left
( |G_{\Omega_2}(v,y;z)|^s \right ) \; .
\end{multline}
\end{enumerate}
\label{lem:decouplinginequalities}
\end{lem}

Lemma \ref{lem:decouplinginequalities} is a consequence of the
conditional expectation bound (\ref{eq:fracmom}), the Krein
formula (\ref{eq:Krein}), and the following:
\begin{lem}
Let $V_1,V_2$ be independent real valued random variables which
satisfy $R_2(s)$ for some $s > 0$. Then there exists $D^{(2)}_s>
0$ such that
\begin{equation}
\E \left ( |F(V_1,V_2)|^s |F(V_1,V_2)|^s \right ) \ \le \
D^{(2)}_s \ \E \left ( |F(V_1, V_2)|^s \right ) \ \E \left (
|G(V_1, V_2)|^s \right ) \ ,
\end{equation}
where $F$ and $G$ are arbitrary functions of the form
\begin{eqnarray}
F(V_1,V_2) &=& {1 \over L_1(V_1,V_2)} \\ G(V_1,V_2) &=&
{L_2(V_1,V_2) \over  L_3(V_1,V_2) } \; ,
\end{eqnarray}
with $\{L_i\}$ functions which are linear in each variable
separately.  In fact, we may take $D^{(2)}_s = D_{s;1} D_{s;2}$ ,
where, for $j=1,2$, $D_{s;j}$ is the decoupling constant for
$V_j$.
\end{lem}
%\begin{lem}
%Let $\rho$ satisfy $R_2(s)$.  Then there exists $D^{(2)}_s(\rho)
%> 0$ such that
%\begin{equation}
%\E \left ( |F(V_1,V_2)|^s |F(V_1,V_2)|^s \right ) \ \le \
%D^{(2)}_s(\rho) \ \E \left ( |F(V_1, V_2)|^s \right ) \ \E \left (
%|G(V_1, V_2)|^s \right ) \ ,
%\end{equation}
%where $f$ and $g$ are arbitrary functions of the form
%\begin{eqnarray}
%F(V_1,V_2) &=& {1 \over L_1(V_1,V_2)} \\ G(V_1,V_2) &=&
%{L_2(V_1,V_2) \over  L_3(V_1,V_2) } \; ,
%\end{eqnarray}
%with $\{L_i\}$ functions which are linear in each variable
%separately. In fact, $D^{(2)}_s(\rho) \le (D_s(\rho))^2$.
%($D_s(\rho)$ is the decoupling constant for $\rho$.)
%\end{lem}

\begin{proof}
Let $f(V)$ and $g(V)$ be two functions of the appropriate form for
the decoupling lemma. Then, with $j=1,2$
\begin{equation}
\E \left ( |f(V_j)|^s |g(V_j)|^s \right ) \ \le \ D_{s;1} \ \E
\left ( |f(\widetilde V_j) |^s |g(V_j)|^s \right ) \; ,
\end{equation}
where $\widetilde V_j$ indicates an independent variable
distributed identically to $V_j$.

Now, if $F$ and $G$ are functions of $2$ variables of the given
form, then at fixed values of $V_2$, they satisfy the $1$ variable
decoupling lemma, so
\begin{equation}
\E \left ( |F(V_1,V_2)|^s |G(V_1,V_2)|^s \right ) \ \le \ D_{s;1}
\ \E \left (|F(\widetilde V_1, V_2)|^s |G(V_1, V_2)|^s \right ) \;
.
\end{equation}
For fixed values of $\widetilde V_1$ and $V_1$, $F(\widetilde V_1,
V_2)$ and $G(V_1, V_2)$ (as functions of $V_2$) are again of the
correct form to apply the $1$ variable decoupling lemma.  Thus,
\begin{multline}
\E \left ( |F(V_1,V_2)|^s |G(V_1,V_2)|^s \right ) \ \le \ D_{s;1}
D_{s;2} \ \E \left (|F(\widetilde V_1, \widetilde V_2)|^s |G(V_1,
V_2)|^s \right ) \\ = \ D_{s;1} D_{s;2} \ \E \left (|F( V_1,
V_2)|^s  \right ) \ \E \left ( |G(V_1, V_2)|^s \right ) \; .
\end{multline}
\end{proof}


\masubappendix{A condition for the validity of $R_2(s)$}

Decoupling lemmas have been discussed already in 
references~\cite{AM,Ai94,AG}. 
Though these  contain results similar to those
required here, they do not provide the exact condition 
used in this work. Hence, we briefly present an
elementary condition under which $R_2(s)$ is satisfied.  
The following discussion is by
no means exhaustive. Rather, we simply wish to show that the
condition $R_2(s)$ is not devoid of meaningful examples.

\begin{lem}
Let $\rho$ be a measure with bounded support which satisfies
$R_1(\tau)$. Then for any $s < {\tau \over 4}$, $\rho$ satisfies
$R_2(s)$.
\end{lem}
\begin{proof}
For each $s> 0$, we define
\begin{eqnarray}
\phi_s(z) &=& \int {1 \over |V - z|^s} \rho(dV) \; , \\
\psi_s(z,w) &=& \int {|V-z|^s \over |V-w|^s} \rho(dV) \; , \\
\gamma_s(z,w,\zeta), &=& \int {|V-z|^s \over |V-w|^s |V -
\zeta|^s} \rho(dV) \; .
\end{eqnarray}
Property $R_2(s)$ amounts to the statement that
\begin{equation}
\sup_{z,w, \zeta \in \mathbb C} { \gamma_s(z,w,\zeta) \over
\phi_s(\zeta) \psi_s(z,w)} \ < \ \infty \; . \label{eq:r2true}
\end{equation}

In fact, if we let
\begin{eqnarray}
F_s(z) \ = \ {\sqrt{\phi_{2s}(z)} \over \phi_s(z)} \; ,
\\ G_s(z,w) \ = \ { \sqrt{
\psi_{2s}(z,w)} \over \psi_s(z,w)} \; ,
\end{eqnarray}
then by the Cauchy-Schwartz inequality, it suffices to show that
$F_s$ and $G_s$ are uniformly bounded.  However this is
elementary, for under the assumed conditions, it $F_s$ and $G_s$
are continuous functions which are easily shown to have finite
limits at infinity.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vskip 1truecm

\noindent {\large \bf Acknowledgments\/} \\ 
This work  was supported in part by the NSF Grant PHY-9971149 
(MA).  Jeff Schenker thanks the NSF for
financial support under a Graduate Research Fellowship, 
and Dirk Hundertmark thanks the Deutsche Forschungsgemeinschaft 
for financial support under grant Hu 773/1-1.  

\newpage 
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\end{document}


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%%Trailer
end
%%EOF

---------------9910181845198--
