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homogenization, random environment.
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\title[Homogenization in Asymmetric Environment]{Homogenization of Random Walk 
in Asymmetric Random Environment}


\author{Joseph G. Conlon }

\address{University of Michigan\\ Department of Mathematics\\ Ann Arbor,
  MI 48109-1109}
\email{conlon@math.lsa.umich.edu}


\begin{document}

\maketitle

\begin{abstract}
In this paper, the author investigates the scaling limit of a partial difference
  equation on the d dimensional integer lattice $\Z^d$, corresponding to a
  translation invariant random walk perturbed by a random vector field. In the
  case when the translation invariant  walk scales to a Cauchy process he
  proves convergence to an effective equation on $\R^d$. The effective equation
  corresponds to a Cauchy process perturbed by a constant vector field. In the
  case when the translation invariant  walk scales to  Brownian motion he
  proves that the scaling limit, if it exists, depends on dimension. For
  $d=1,2$  the scaling limit cannot be diffusion. 
\end{abstract}


\section{Introduction.}
In this paper we shall be concerned with homogenization of a nondivergence form 
elliptic equation.  Let $(\Om, {\cF}, \mu)$ be a probability space and 
${\bb} : \Om \ra \R^d$ be a bounded measurable function 
${\bb}(\om) = (b_1(\om), ..., b_d(\om))$ with 
\begin{equation} \label{L1}
|{\bf b}(\om)|^2=\sum_{i=1}^d|b_i(\om)|^2 \le 1 \ \ , \om\in\Om \   .
\end{equation}  
We assume that $\Z^d$ acts on $\Om$ by translation operators 
$\tau_x : \Om \ra \Om$, $x\in \Z^d$, which are measure preserving ergodic and 
satisfy the properties $\tau_x \tau_y = \tau_{x+y}$, \\ $\tau_0 $ = identity, $x,y \in \Z^d$.  Using these operators we can define a measurable function ${\bb} : \Z^d \times \Om \ra \R^d$ by ${\bb}(x, \om) = {\bb}(\tau_x \om) , \ x \in \Z^d, \ \om \in \Om$.

For $\ve > 0$ let $\Z^d_\ve = \ve \Z^d$ be the $\ve$ scaled integer lattice.  We define the integral of a function $g : \Z^d_\ve \ra \R$ by
\[	\int_{\Z^d_\ve} g(x)dx \ \  \DEF  \ \ \ve^d \sum_{x\in \Z^d_\ve} g(x) \ .		\]
The space $L^2(\Z^d_\ve)$ is then the space of square integrable functions 
$u : \Z^d_\ve \ra \C$ with norm,	$\| u \|_\ve$ satisfying
\[	\|u\|^2_\ve = \int_{\Z^d_\ve} |u(x)|^2 dx .	\]
For $i=1,...,d$, let ${\bf e}_i \in \Z^d$ be the element with entry 1 in the ith
position and 0 in the other positions.  Suppose $\ga \in \C$ and $f : \R^d \ra \R$ is a $C^\infty$ function with compact support.  Let $p : \Z^d \ra \R$ be a probability density, whence
  \[	p(x) \ge 0, \ x \in \Z^d, \ \ \sum_{x\in \Z^d} p(x) = 1 \ .    \]
With any such $p$ we can associate a translation invariant operator $A_p$ on functions $u_\ve : \Z^d_\ve \ra \R$ by
\[	A_p u_\ve (x) = u_\ve(x) - \sum_{y\in \Z^d} p(y) u_\ve(x + \ve y) .  \]
For some probability densities $p$ we can define the degree of the symbol for 
$A_p$.  We denote this degree by $|A_p|$.  If $p$ is given by
\begin{equation} \label{A1}
p(\pm {\bf e}_i) = 1/2 d, \ \ i = 1, ..., d \ ,
\end{equation}
then $|A_p| = 2$. We can also define a $p$ with $|A_p| = 1$.  To do this 
consider the standard nearest neighbor random walk on $\Z^{d+1}$ started at the 
point $(0, -1)$, \\ $ 0 \in \Z^d$.  Then
\begin{multline} \label{B1}
p(x) = {\rm probability \ the \ walk \ first \ hits \ the \ hyperplane} \\

 \{ (y,0) : y \in \Z^d\} \ {\rm in \ the \ point} \ (x,0), \ x \in \Z^d.    
\end{multline}
We shall be interested in solutions to the equation,
\begin{eqnarray} \label{C1}
A_p u_\ve(x, \om) &-& \ga \sum^d_{i=1} b_i \left( \frac x \ve , \om \right) \left[u_\ve(x+\ve{\bf e}_i, \om) - u_\ve(x-\ve{\bf e}_i, \om) \right] \\
+ \ \ve^{|A_p|} u_\ve(x,\om) &=& \ve^{|A_p|} f(x), \ x \in \Z^d_\ve \ , \ \ \om \in \Om, \nn
\end{eqnarray}
when $p$ is given by either (\ref{A1}) or (\ref{B1}).  For a function 
$g : {\bf X} \ra \C$ where ${\bf X}=\R^d$ or ${\bf X}=\Z_\ve^d$, let $\hat g$ 
be its Fourier transform,
\[	\hat g(\xi) = \int_{\bf X} g(x) e^{ix\cd \xi} dx, \ \ \xi \in \R^d.	\]
Note that when 
${\bf X}=\Z_{\ve}^d$, then $\hat g(\xi)$ is periodic in $\xi$ 
whence we
can restrict $\xi$ to the cube $[-\pi/\ve, \ \pi/\ve]^d$.
Suppose now ${\bb} \equiv 0$ in (\ref{C1}) and $p$ is given by (\ref{A1}).  Then it is easy to see that (\ref{C1}) has a unique solution $u_\ve(x)$ in $L^2(\Z^d_\ve)$.  Further, if $u(x), \ x \in \R^d$, is the function which satisfies
\[    [1 + |\xi|^2 / 2d ] \hat u(\xi) = \hat f (\xi), \ \ \xi \in \R^d,	\]
then $\lim_{\ve \ra 0} \| u_\ve - u\|_\ve = 0$.  Similarly if ${\bb} \equiv 0$ 
and $p$ is given by (\ref{B1}) then (\ref{C1}) has a unique solution $u_\ve(x)$ 
in $L^2(\Z^d_\ve)$.  We show in $\S 2$ that if $u(x) , x \in \R^d$, is the 
function which satisfies
\begin{equation} \label{D1}
[1 + |\xi|] \hat u(\xi) = \hat f(\xi), \ \ \xi \in \R^d \ ,
\end{equation}
then
\begin{equation} \label{E1}
\lim_{\ve \ra 0} \| u_\ve - u\|_\ve = 0 \ .
\end{equation}
We can also prove a corresponding theorem when ${\bb} \not\equiv 0$ in the case 
$p$ is given by (\ref{B1}). In the following the expectation value of a random
variable $X$ on $\Om$ is denoted by $<X>$.
\begin{theorem}  For $0 < \ve \le 1$,  there exists a constant $\ga_d>0$,
depending only on $d$, such that if $\ga\in\C$ satisfies $|\ga| < \ga_d$ 
then  (\ref{C1}) has a unique solution $u_\ve(\cd,\om) \in L^2(\Z^d_\ve), \ 
\om \in \Om$.  There exists a vector $q_{\ga} \in \C^d$, 
$ |q_{\ga}|<1$, analytic in $\ga$, 
such that if $u(x), \ x \in \R^d$, is the function which satisfies 
\begin{equation} \label{F1}
[ 1 + |\xi| + iq_{\ga} \cd \xi] \hat u(\xi) = \hat f (\xi), \ \ \xi \in \R^d,	
\end{equation}
then there is the limit
\begin{equation} \label{G1}
\lim_{\ve \ra 0} \left< \| u_\ve( \ , \cd) - u\|^2_\ve \right> = 0.
\end{equation}
\end{theorem}
Theorem 1.1 is a homogenization result for $p$ given by (\ref{B1}).  It says 
that the scaling limit of the random walk, with transition probability 
(\ref{B1}), in a random environment described by ${\bb}(x,\om)$, is a Cauchy 
process with a constant drift $q_{\ga}$. Corresponding results for uniformly 
elliptic partial differential equations
in divergence form were first proved by Kozlov \cite{KOZLOV} and later by Papanicolaou
and Varadhan \cite{VARADHAN}. In this case the scaling limit is Brownian
motion. One can also prove a Brownian motion scaling limit for parabolic
equations in divergence form when the coefficients are random in time as well
as space \cite{LOY}. It is possible to relax somewhat the uniform ellipticity 
assumption and still prove the scaling limit \cite{FANJIANG}. For a
comprehensive survey of homogenization results in partial differential
equations see the book of Oleinik at al \cite{ZKO}. Here we obtain the scaling
limit of an asymmetric partial difference equation. The scaling limit
of a divergence form partial difference equation has been obtained by
K\"unnemann \cite{KUNNEMANN} and of a symmetric partial difference equation in
non-divergence form by Lawler \cite{LAWLER}.

Evidently $q_{\ga}$ is in general nonzero even in the case $<{\bb}(\cd)> = 0$. 
Nevertheless it is possible to impose some general conditions on ${\bb}(\cd)$ 
to ensure $q_\ga = 0$.  We say that the vector field ${\bb}(\cd)$ is reflection invariant if it has the property that 
\begin{multline} \label{H1}
\left< \prod^n_{i=1} b_{k_i} (\tau_{x_i} \cd) \right> = (-1)^n \left< 
\prod^n_{i=1} b_{k_i}(\tau_{-x_i} \cd) \right>, \\
x_i \in \Z^d, \ 1 \le k_i \le d, \  \ i=1,...n, \ \ n \ge  1 . 
\end{multline}
Clearly (\ref{H1}) implies that $<\bb(\cd)> = 0$. We show at the end of $\S2$
that if $\bb(\cdot)$ satisfies (\ref{H1}) then $q_\gamma=0$ in (\ref{F1}).

Suppose now (\ref{H1}) holds. Let $u_\ve(x)=<u_\ve(x,\cdot)>, \ x\in\Z^d_\ve$,
where $u_\ve(x,\om)$ is the unique solution to (\ref{C1}) given by Theorem 1.1.
Suppose $\hat p : [-\pi, \pi]^d \ra \R$ is the Fourier 
transform of the function $p$ of (\ref{B1}) and $\hat f_\ve$ the Fourier 
transform of the function $f$ restricted to the lattice $\Z^d_\ve$.
If $\hat{u}_\ve(\xi)$ is the Fourier transform of $u_\ve(x)$, then one can see
that there is a $d\times d$ matrix $q_{\ga,\ve}(\zeta), \ \zeta\in [-\pi, \
\pi]^d$, continuous in $\zeta$ and analytic in $\ga$, such that
\begin{equation} \label{I1}
\hat{u}_\ve(\xi)[1+\ve^{-1}\{1-\hat{p}(\ve\xi)\}-\ve^{-1}
e(\ve\xi)q_{\ga,\ve}(\ve\xi)e(-\ve\xi)]=\hat{f}_\ve(\xi) \ , \ 
\xi\in[-\pi/\ve,\ \pi/\ve] \ ,
\end{equation}
where $e(\zeta), \ \zeta\in\R^d$, is the vector
$e(\zeta)=(e_1(\zeta),\cdot\cdot\cdot,e_d(\zeta))$, with $e_k(\zeta)=1-
e^{i{\bf e}_k\cdot\zeta}$.
\begin{theorem}  Suppose the variables 
${\bb}(\tau_x \cd), \ x\in \Z^d$, are independent with \\ 
$<{\bb}(\cd)> = 0$ and $q_{\ga,\ve}(\zeta)$ is defined by (\ref{I1}) for 
$|\ga| < \ga_d$, where $\ga_d$ is the constant of Theorem 1.1.  
Then there exists a positive constant $\ga'_d < \ga_d$, depending only on $d$, 
such that if 
$|\ga| < \ga'_d$ the matrix $q_{\ga,\ve}(\zeta)$ converges uniformly in $\zeta$
as $\ve\ra 0$  to a matrix $q_{\ga}(\zeta)$, $\zeta\in [-\pi, \ \pi]^d$.
\end{theorem}
Theorem 1.2 is a considerably deeper theorem than Theorem 1.1.  One has to use 
the Calderon-Zygmund theorem \cite{GILBARG,STEIN} that the Hilbert transform 
is bounded on $L^p$ spaces for $p \not= 2$.  In contrast, Theorem 1.1 can be 
proved by just using $L^2$ theory.

Next we consider the situation when $p$ is given by (\ref{A1}) and 
${\bf b}\not\equiv 0$. We then have the following theorem:
\begin{theorem}  For $0 < \ve \le 1$ and   $\ga\in\C$ satisfying 
$|\ga| < \ve/\sqrt{2d}$, 
equation (\ref{C1}) has a unique solution $u_\ve(\cd,\om) \in L^2(\Z^d_\ve), \ 
\om \in \Om$.  If (\ref{H1}) holds then there is a   
$d\times d$ matrix $q_{\ga,\ve}(\zeta), \ \zeta\in [-\pi, \
\pi]^d$, continuous in $\zeta$ and analytic in $\ga$, such that the Fourier
transform $\hat{u}_\ve(\xi)$ of $<u_\ve(x,\cdot)>$ satisfies,
\begin{equation} \label{M1}
\hat{u}_\ve(\xi)[1+\ve^{-2}\{1-\hat{p}(\ve\xi)\}-\ve^{-2}
e(\ve\xi)q_{\ga,\ve}(\ve\xi)e(-\ve\xi)]=\hat{f}_\ve(\xi) \ , \ 
\xi\in[-\pi/\ve,\ \pi/\ve] \ .
\end{equation}
\end{theorem}
Note that (\ref{M1}) is simply the analogue for $p$ given by (\ref{A1}) of
(\ref{I1}) for $p$ given by (\ref{B1}). Since $q_{\ga,\ve}(\zeta)$ is analytic
in $\ga$, we can write it in a series expansion,
\begin{equation} \label{N1}
q_{\ga,\ve}(\zeta)=\sum_{m=2}^{\infty} \ga^m q_{m,\ve}(\zeta) \ , 
\end{equation}
which converges for $|\ga| < \ve/\sqrt{2d}$. In $\S 5$ we investigate the limiting
behavior of $q_{m,\ve}$ as $\ve\ra 0$. We restrict ourselves to the situation
when the ${\bf b}(\tau_x\cdot), \ x\in\Z^d$, are given by independent Bernoulli
variables. Thus we assume that $b_j(\cdot)\equiv 0, \ j>1$, and
$b_1(\tau_x\cdot)=Y_x, \ x\in\Z^d$, where the variables $Y_x, \ x\in\Z^d$, are
assumed to be i.i.d. Bernoulli, $Y_x=\pm 1$ with equal probability. In that
case we have the following:
\begin{theorem} For $m=2,3,\cdot\cdot\cdot$ the $d\times d$ matrix 
$q_{m,\ve}(\zeta)$ has the properties: \\
(a) $q_{m,\ve}(\zeta)\equiv 0$ if $m$ is odd or $m=2$. \\
(b) The entries of the matrix $q_{m,\ve}(\zeta)=[q_{m,\ve,k,k'}(\zeta)], \ 1\le
k,k'\le d$, are zero if $k,k'> 1$. 
\\
(c) The entries of the matrix $q_{m,\ve}(0)=[q_{m,\ve,k,k'}(0)]$ are 
zero if $k+k'> 2$. \\ 
(d) Assume $d=1$ and $\mathcal{K}\subset \R$ is a compact subset. Then  
$\ve^{m/2}q_{m,\ve}(\ve\xi)$ converges uniformly for
$\xi\in \mathcal{K}$ to a function $q_{m}(\xi)$,  as $\ve\ra 0$. The function 
$q_{4}(\xi)$ is given by the formula, 
\begin{equation} \label{O1}
q_{4}(\xi)=\frac{32}{18+\xi^2} \ .
\end{equation}
(e) If $d=2$ then  $q_{4,\ve,1,1}(0)$ converges to $+\infty$ as $\ve\ra 0$. \\
(f) If $d\ge 3$  then $q_{4,\ve}(\zeta)$ converges uniformly for
$\zeta\in[-\pi,\pi]^d$ to a matrix $q_{4}(\zeta)=[q_{4,k,k'}(\zeta)]$  
satisfying $q_{4,1,1}(0)>0$, as $\ve\ra 0$. \\
\end{theorem}
Unlike in the previous theorems, the results of Theorem 1.4 depend on the
dimension $d$. For $d=1$ the theorem suggests that if we set
$\ga=\ga'\sqrt{\ve}$  in (\ref{C1}), fix $\ga'$and let $\ve\ra 0$, then we
should obtain a scaling limit. Note however that one expects the resulting
series obtained from (\ref{N1}) by letting $\ve\ra 0$ to be at most asymptotic 
in $\ga'$, not analytic. The limit here is related to the limits obtained by
Sinai\cite{SINAI} and Kesten\cite{KESTEN} for one dimensional random walk in
random environment. For $d=2$ the theorem suggests that if we fix $\ga$ 
in (\ref{C1}) and let $\ve\ra 0$ no scaling limit exists. This appears to
contradict a conjecture of Fisher\cite{FISHER} that two dimensional  random walk 
in random environment  has a diffusive scaling limit. In contrast Derrida and
Luck\cite{DERRIDA} conjecture a nondiffusive scaling limit in two dimensions.
A numerical study of the two dimensional equation (\ref{C1}) was made in
\cite{CV} using a multigrid algorithm. It was observed that the algorithm gave
considerable acceleration over pure iterative methods such as Gauss-Seidel.
This suggests that there is some stability in the  Fourier space behavior of the
solution to (\ref{C1}) as $\ve\ra 0$.
For $d\ge 3$ the theorem suggests a diffusive scaling limit for $d$ dimensional
random walk in random environment and that the effective diffusion constant is
smaller than for the zero noise case. 

There has been some recent work \cite{PISZTORA,SZNITMAN1,SZNITMAN2} on (\ref{C1}) with $p$ given by (\ref{A1}) and
under the assumption $<{\bf b}(\cdot)>\ne 0$. This situation is very different
to the situation studied in Theorem 1.4 since one expects now the drift to
dominate diffusion. The methods used in \cite{SZNITMAN1,SZNITMAN2} are related
to methods used to prove Anderson localisation for the random Schrodinger
equation.
\section{Proof of Theorem 1.1}
We first prove (\ref{E1}).  Thus we consider (\ref{C1}) with ${\bb} \equiv 0$ and $p$ given by (\ref{B1}).  To obtain an expression for $p$, let us denote a point in $\Z^{d+1}$ by $(x,y)$ with $x \in \Z^d, \ y \in \Z$.  For $n=0,1,2,...$ let $G(x,y,n)$ be the probability density of the standard random walk in $\Z^{d+1}$, started at  the origin, after $n$ steps.  Thus
\begin{equation} \label{A2}
\sum_{x\in\Z^d} \sum_{y\in\Z} G(x,y,n) e^{i[x\cd\xi + y \zeta]} = \left\{ \frac 1 {(d+1)} \left[ \sum^d_{i=1} \cos ({\bf e}_i \cd \xi) + \cos \zeta \right] \right \}^n .
\end{equation}
By the reflection principle, the probability that the $d+1$ dimensional walk, started at $(0, -1)$ first hits the hyperplane $\{ (y,0) : y \in \Z^d \}$ after $n+1$ steps at the point $(x,0)$ is given by
\[	\frac 1 {2(d+1)} \left[ G(x,0,n) - G(x,2,n) \right] .	\]
Hence we have
\begin{equation} \label{H2}
p(x) = \frac 1 {2(d+1)} \sum^\infty_{n=0} \left[ G(x,0,n) - G(x,2,n) \right] 
\end{equation}
We can compute the Fourier transform of $p$ by using (\ref{A2}).  Thus
\begin{equation} \label{B2}
\hat p(\xi) = \sum_{x \in \Z^d} p(x)e^{ix\cd \xi} = \frac 1 {4\pi(d+1)} 
\int^\pi_{-\pi} d\zeta \frac{1-\cos 2\zeta}{1 - \left[ \sum^d_{i=1} \cos ({\bf e}_i \cd \xi) + \cos \zeta \right] /(d+1)}.
\end{equation}

\begin{lem}  The limit (\ref{E1}) holds.
\end{lem}
\begin{proof}  We rewrite the formula of (\ref{B2}) for $\hat p(\xi)$ as
\begin{equation} \label{C2}
\hat p(\xi) = 1 - \frac 1{2\pi} \int^\pi_{-\pi} d\zeta 
\frac{(1 + \cos \zeta) \sum^d_{i=1} [1-\cos ({\bf e}_i \cd \xi)]} 
{\left[ (1 - \cos \zeta) + \sum^d_{i=1} [1-\cos ({\bf e}_i \cd \xi)] \right]} 
\ ,
\end{equation}
whence, on performing the integration with respect to $\zeta$, we obtain the
formula,
\begin{equation} \label{AO2}
1- \hat p(\xi)=\left\{\left[1+\frac{2}{\sum^d_{i=1} [1-\cos ({\bf e}_i \cd
\xi)]}\right]^{1/2}-1\right\}\sum^d_{i=1} [1-\cos ({\bf e}_i \cd \xi)] \ .
\end{equation}
It follows from (\ref{B2}), (\ref{C2}) that $0 < \hat p(\xi) \le 1$.  We can 
also easily see from (\ref{AO2}) that
\begin{equation} \label{F2}
\lim_{\ve \ra 0} \ve^{-1} [1 - \hat p(\ve\xi)] = |\xi|, \ \ \ \ \xi \in \R^d .
\end{equation}
Consider now (\ref{C1}) with ${\bb} \equiv 0$.  Taking
the Fourier transform of (\ref{C1}), 
we see that (\ref{C1}) is equivalent to the equation,
\begin{equation} \label{D2}
[1 + \ve - \hat p(\ve \xi)] \hat u_\ve(\xi) = \ve \hat f_\ve(\xi) , \ \ \xi \in \left[\frac{-\pi}{\ve}, \frac \pi \ve \right]^d ,
\end{equation}
where $ \hat f_\ve$ is the Fourier transform of $f$ regarded as a function on $\Z^d_\ve$.  Since $f$ is $C^\infty$ of compact support it is easy to see that (\ref{D2}) is uniquely solvable in $L^2(\left[\frac{-\pi}{\ve}, \frac \pi \ve \right]^d )$ and
\begin{equation} \label{E2}
\hat u_\ve(\xi) =  \hat f_\ve(\xi) \big/ [ 1 + \ve^{-1} \left\{1-\hat p(\ve \xi)
\} \right] , \ \ \  \xi \in \left[\frac{-\pi}{\ve}, \  \frac \pi \ve \right]^d .
\end{equation}
Let $\hat u(\xi)$ be the function given in (\ref{D1}).  We decompose its Fourier inverse $u(x), \ \ x \in \R^d$, as a sum, $u(x) = v_\ve(x) + w_\ve(x)$, where 
\[
v_\ve(x) = \frac 1{(2\pi)^d} \int_{ [\frac{-\pi}{\ve}, \frac \pi \ve ]^d }
\hat u (\xi)e^{-ix\cd\xi} d\xi . 
\]
Since $\hat f(\xi)$ is rapidly decreasing in $\xi$ it follows that $\lim_{\ve \ra 0} \| w_\ve\|_\ve = 0$.  We can see this from the identity
\[
\|w_\ve\|^2_\ve = \frac 1{(2\pi)^d} \int_{[\frac{-\pi}{\ve}, \frac \pi \ve ]^d }
\left| \sum_{n\in\Z^d\backslash \{0\} } \hat u (\xi + \frac{2\pi n}{\ve})\right|^2 d\xi . 
\]
Thus we have
\begin{eqnarray*}
\lim_{\ve \ra 0} \| u_\ve - u\|_\ve &=& \lim_{\ve \ra 0} \| u_\ve - v_\ve\|_\ve, \\
\|u_\ve - v_\ve\|^2_\ve &=& \frac 1{(2\pi)^d} \int_{[\frac{-\pi}{\ve}, \frac \pi \ve ]^d }
|\hat u_\ve(\xi) - \hat u(\xi)|^2 d\xi,
\end{eqnarray*}
where $\hat u_\ve(\xi)$ is given by (\ref{E2}) and $\hat u(\xi)$ by (\ref{D1}). 
The result follows from (\ref{F2}) and the dominated convergence theorem. \ \ \ 
\ \ \end{proof}

Next we wish to prove existence of a solution to (\ref{C1}) provided $\ga$ is sufficiently small, depending only on $d$.  To do this we need to define the Green's function $G_{p,\eta}(x), \ x \in \Z^d$, associated with $p$.  Thus for $\eta > 0$, $G_{p,\eta}(x)$ is the solution to the equation 
\begin{equation} \label{G2}
A_p \; G_{p,\eta}(x) + \eta \; G_{p,\eta}(x) = \de(x), \ \ \ x \in \Z^d ,
\end{equation}
where $\de$ is the Kronecker $\de$ function, $\de(0) = 1, \ \de(x) = 0, \ x \not= 0$.  The Fourier transform $\hat G_{p,\eta}(\xi)$ of $G_{p,\eta}(x)$ is given by the formula
\begin{equation} \label{M2}
\hat G_{p,\eta}(\xi) = \left[ 1 + \eta - \hat p(\xi) \right]^{-1}, \ \ \ \xi \in [-\pi, \pi]^d .
\end{equation}
It is easy to see from (\ref{AO2}) that
\[	0 \le \hat G_{p,\eta}(\xi) \le C / [\eta + |\xi| ]\ ,	\]
for some universal constant $C$.  We can also obtain a corresponding decay rate on the function $G_{p,\eta}(x)$.  

\begin{lem}  Let $G_{p,\eta}(x), \ x \in \Z^d$, be the solution to (\ref{G2}), and $0 < \eta \le 1$.  Then there is a constant $C_d$, depending only on $d$ such that if $d \ge 2$, 
\begin{equation} \label{J2}
0 \le G_{p,\eta}(x) \le C_d \big/ \left[ 1 + \eta^2 |x|^2 \right] \left[ 1 + |x|^{d-1} \right], \ \ \ x \in \Z^d.
\end{equation}
\end{lem}
\begin{proof}  Consider the random walk on $\Z^d$ with transition probability $p$ given by (\ref{H2}).  Let $p_n(x), n = 0,1,2,...$ be the probability that the walk, started at the origin hits $x$ after $n$ steps.  Thus $p_0(x) = \de(x)$ and $p_1(x) = p(x), \ \ x \in \Z^d$.  It is easy to see that the solution of (\ref{G2}) is given by the formula,
\begin{equation} \label{I2}
G_{p,\eta}(x) = \sum^\infty_{m=0} (1 + \eta)^{-m-1} p_m(x), \ \ \ \ x \in \Z^d.
\end{equation}
We can derive a formula for $p_m(x), \ \ m \ge 2$, similar to the formula (\ref{H2}) for $p(x)$.  In fact we have
\begin{equation} \label{L2}
p_m(x) = \frac 1 {2(d+1)} \sum^\infty_{n=0} [G(x, m-1, n) - G(x, m+1, n)]. 
\end{equation}
It is easy to see from this and (\ref{I2}) that 
\begin{equation} \label{K2}
G_{p,0}(x) = \de(x) + \frac 1{2(d+1)} \sum^\infty_{n=0} [G(x,0,n) + G(x,1,n)].
\end{equation}
If we use now the bound
\[
0 \le G(x,y,n) \le \frac{C_d} {1+n^{(d+1)/2}} \exp \left[-\min\{|x| + |y|, (|x|^2 + |y|^2)/(n+1)\}/C_d \right],
\]
for the standard random walk on $\Z^{d+1}$, then the inequality (\ref{J2}) for 
$\eta = 0$ follows from (\ref{K2}).  To obtain (\ref{J2}) for $\eta > 0$ we need
to use the inequality,
\begin{multline*}
0 \le G(x, y-1, n) - G(x, y+1, n) \le	\\

\frac{C_d y} {1+n^{(d+3)/2}} \exp \left[ -\min \{ |x| + |y|, \ (|x|^2 + |y|^2)
 / (n+1) \} / C_d \right], \\

	x \in \Z^d, \ y \in \Z, \ \ y \ge 0.
\end{multline*}	
In follows from this and (\ref{L2}) that
\[
0 \le p_m(x) \le \frac{C_d m}{1 + [|x|^2 + m^2]^{(d+1)/2} } \ , \ \ \ \ m \ge 1.
\]
The inequality (\ref{J2}) follows now from this last inequality by estimating 
the sum (\ref{I2}). \ \ \ \ \ \end{proof}

\begin{lem}
Suppose $d \ge 2$ and $f : \R^d \ra \R$ is a $C^\infty$ function with compact 
support.  Then there exists a constant $\ga_d$, depending only on $d$ such that 
if $|\ga| < \ga_d$ and $0 < \ve \le 1$, then (\ref{C1}) has a unique solution 
$u_\ve (x, \om)$ in $L^2(\Z^d_\ve)$.  Further, there is a constant $C_d$, 
depending only on $d$, such that $\|u_\ve(\cd, \om)\|_\ve \le C_d \|f\|_\ve$ .
\end{lem}
\begin{proof} Suppose $u_\ve (x, \om)$ is in $L^2(\Z^d_\ve)$ and satisfies (\ref{C1}).  Let $v_\ve(x, \om)$ be defined by 
\[	v_\ve(x, \om) = \ve^{-1} [A_p u_\ve(x,\om) + \ve u_\ve(x,\om)]. \] 
It is easy to see that $v_\ve(x, \om) \in L^2(\Z^d_\ve)$.  In view of (\ref{G2}) it follows that 
\begin{equation} \label{P2}
u_\ve(x, \om) = \ve^{1-d} \int_{\Z^d_\ve} dy \ G_{p,\ve} \left( \frac{x-y} \ve 
\right)  v_\ve(y, \om).
\end{equation}
Now for $i=1,...,d$ let us define operators $T_i$ on $L^2(\Z^d_\ve)$ by
\[    T_i w(x) = \ve^{-d} \int_{\Z^d_\ve} dy \left[ G_{p,\ve} \left( \frac{x-y} 
\ve + {\bf e} _i \right) - G_{p,\ve} \left( \frac{x-y} \ve - {\bf e} _i \right) 
\right] w(y)  \  .
\]
It is easy to see that $T_i$ is a bounded operator on $L^2(\Z^d_\ve)$.  It follows from (\ref{M2}) that the norm of $T_i$ is bounded by 
\begin{equation} \label{N2}
\| T_i \| \le \sup_{\xi \in [-\pi,\pi]^d} \left[ \frac{2|\sin({\bf e}_i \cd \xi)
|}{1+\ve - \hat p(\xi)}
\right].
\end{equation}
We can see from (\ref{AO2}) that there is a constant $c_d ,\  0<c_d<1$, depending only on $d$ such that
\begin{equation} \label{AG2}
1 - \hat p(\xi) \ge c_d |\xi |, \ \ \ \xi \in [-\pi, \pi]^d.
\end{equation}
It follows from this and (\ref{N2}) that $\|T_i\| \le 2/c_d$.  Observe now that (\ref{C1}) implies that $v_\ve(x,\om) \in L^2(\Z^d_\ve)$ satisfies the equation
\begin{equation} \label{O2}
v_\ve(x,\om) - \ga \sum^d_{i=1} b_i(x/\ve,\om) T_i v_\ve(x,\om) = f(x).
\end{equation}
In view of (\ref{L1}), if we take $\ga_d = c_d/2$ it follows that for $|\ga| < \ga_d$ the 
function $v_\ve(x,\om)$ is uniquely determined by (\ref{O2}).  In view of 
(\ref{P2}) it follows that $u_\ve(x,\om) \in L^2(\Z^d_\ve)$ is the unique 
solution to (\ref{C1}).  To prove existence note that (\ref{O2}) has a 
solution $v_\ve(x,\om) \in L^2(\Z^d_\ve)$ provided $|\ga| < \ga_d$.  If we then 
define $u_\ve(x,\om)$ by (\ref{P2}) one can easily see that $u_\ve(x,\om)$ 
satisfies (\ref{C1}). \ \ \ \ \end{proof}

We have proven existence of a unique solution $u_\ve(x,\om)$ to (\ref{C1}) 
provided $|\ga| < \ga_d$. Next we follow a development similar to the one in 
\cite{CN}.  Thus we put $v_\ve(x,\om) = u_\ve(x, \tau_{-x/\ve} \om)$, $x \in \Z^d_\ve, \om \in \Om$, whence $u_\ve(x,\om) = v_\ve(x, \tau_{x/\ve} \om)$.    Rewriting (\ref{C1}) in terms of $v_\ve$, we obtain the equation,
\begin{eqnarray*}
v_\ve(x, \tau_{x/\ve} \om) &-& \sum_{y \in \Z^d} p(y) v_\ve(x + \ve y, \tau_y \tau_{x/\ve} \om) \\
&-& \ga \sum^d_{i=1} b_i (\tau_{x/\ve} \om) [v_\ve(x + \ve{\bf e}_i, \tau_{{\bf e}_i}\tau_{x/\ve} \om) \\
&-& v_\ve(x - \ve{\bf e}_i, \tau_{-{\bf e}_i}\tau_{x/\ve} \om) ] + \ve v_\ve(x, \tau_{x/\ve} \om) = \ve f(x), \ \ \ x \in \Z^d_\ve, \ \ \ \om \in \Om.
\end{eqnarray*}
This last equation is evidently the same as
\begin{eqnarray} \label{Q2} 
v_\ve(x, \om) &-& \sum_{y \in \Z^d} p(y) v_\ve(x + \ve y, \tau_y \om) \\
&-& \ga \sum^d_{i=1} b_i  (\om) [v_\ve(x + \ve{\bf e}_i, \tau_{{\bf e}_i} \om) -
 v_\ve(x - \ve{\bf e}_i, \tau_{-{\bf e}_i} \om) ] \nn \\
&+& \ve v_\ve(x,  \om) = \ve f(x), \ \ \ x \in \Z^d_\ve, \ \ \ \om \in \Om. \nn
\end{eqnarray}
We write now  $v_\ve(x, \om) = u_\ve(x) + \psi_\ve(x,\om)$, $\ x \in \Z^d_\ve, \ \ \om \in \Om$, where $< \psi_\ve(x, \cd) > = 0$, $\ x \in \Z^d_\ve$.  It follows that
\[
u_\ve(x) = \Big< v_\ve(x, \cd) \Big> = \Big< u_\ve(x, \cd) \Big> , \ \ x \in \Z^d_\ve \ .
\]
Substituting into (\ref{Q2}) we obtain the equation
\begin{eqnarray} \label{R2}
u_\ve(x) &-& \sum_{y \in \Z^d} p(y) u_\ve(x + \ve y) \\
&-& \ga \sum^d_{i=1} b_i  (\om) [u_\ve(x + \ve{\bf e}_i) - u_\ve(x - \ve{\bf e}_i)] + \ve u_\ve(x) \nn \\
&+& \psi_\ve(x,\om) - \sum_{y\in\Z^d} p(y) \psi_\ve(x + \ve y, \tau_y \om) \nn \\
&-& \ga \sum^d_{i=1} b_i(\om)  \left[\psi_\ve(x +\ve{\bf e}_i, \tau_{{\bf e}_i} \om)  -
\psi_\ve(x - \ve{\bf e}_i, \tau_{-{\bf e}_i} \om)\right]  \nn \\
&+& \ve \psi_\ve(x,  \om) = \ve f(x), \ \ \ x \in \Z^d_\ve, \ \ \ \om \in \Om. \nn
\end{eqnarray}
Taking the expected value of this last equation, we obtain the equation,
\begin{eqnarray} \label{S2}
A_p u_\ve(x) &-& \ga \sum^d_{i=1} <b_i (\cd)> [u_\ve(x + \ve{\bf e}_i) - u_\ve(x - \ve{\bf e}_i)] + \ve u_\ve(x)  \\
&-& \ga \left< \sum^d_{i=1} b_i(\cd) \left[\psi_\ve(x +\ve{\bf e}_i, \tau_{{\bf e}_i} \cd)  -
\psi_\ve(x - \ve{\bf e}_i, \tau_{-{\bf e}_i} \cd)\right] \right> \nn \\
& =&  \ve f(x), \ \ x \in \Z^d_\ve \ \nn . 
\end{eqnarray}
If we subtract now (\ref{S2}) from (\ref{R2}) we obtain
\begin{multline} \label{T2}
\psi_\ve(x,\om) - \sum_{y\in\Z^d} p(y) \psi_\ve(x + \ve y, \tau_y \om)  \\
- \ga P  \sum^d_{i+1} b_i(\om)  [\psi_\ve(x +\ve{\bf e}_i, \tau_{{\bf e}_i} \om)  -
\psi_\ve(x - \ve{\bf e}_i, \tau_{-{\bf e}_i} \om)]   \\
+ \ve \psi_\ve(x, \om) = \ga \sum^d_{i=1}[b_i(\om) -<b_i (\cd)>] [u_\ve(x + 
\ve{\bf e}_i) - u_\ve(x - \ve{\bf e}_i) ],  \ x \in \Z^d_\ve,  \ \om \in 
\Om,   
\end{multline}
where $P : L^2(\Om) \ra L^2(\Om)$ is the projection orthogonal to the constant 
function.  Thus a solution $u_\ve(x,\om)$ of (\ref{C1}) which is in 
$L^2(\Z^d_\ve \times \Om)$ yields functions $u_\ve(x)$ in $L^2(\Z^d_\ve)$ and 
$\psi_\ve(x, \om)$ in $L^2(\Z^d_\ve \times \Om)$ which satisfy 
$< \psi_\ve(x, \cd) > = 0, \ \ x \in \Z^d_\ve$, and the equations 
(\ref{S2}), (\ref{T2}).  Conversely, if we can find functions $u_\ve(x)$ in 
$L^2(\Z^d_\ve)$ and $\psi_\ve(x, \om)$ in $L^2(\Z^d_\ve \times \Om)$ which have 
the property that $< \psi_\ve(x, \cd) > = 0, \ \ x \in \Z^d_\ve$, and satisfy 
(\ref{S2}), (\ref{T2}) then we can construct from them the solution 
$u_\ve(x,\om)$ of (\ref{C1}) and $u_\ve(x,\om)$ is in 
$L^2(\Z^d_\ve \times \Om)$.

We concentrate now on finding solutions to the system (\ref{S2}), (\ref{T2}) of equations.  To do this we Fourier transform the equations.  The Fourier transform of (\ref{S2}) is given by
\begin{multline} \label{X2}
\hat u_\ve(\xi) \left\{ 1 + \ve^{-1} \left[1 - \hat p(\ve \xi)\right] + 
2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin({\bf e}_j \cd \ve \xi)}{\ve} \right\}
\\     + \ga \ve^{-1} \left< \sum^d_{j=1} b_j(\cd) \left[ e^{i\ve {\bf e}_j \cd 
\xi} \hat \psi_\ve(\xi, \tau_{-{\bf e}_j} \cd) -e^{-i\ve {\bf e}_j \cd \xi} 
\hat \psi_\ve(\xi, \tau_{{\bf e}_j} \cd) \right] \right> = \hat f_\ve(\xi),
 \ \ \xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d.
\end{multline}
The Fourier transform of (\ref{T2}) is given by
\begin{multline} \label{W2}
\hat \psi_\ve(\xi, \om) - \sum_{y\in \Z^d} p(y) e^{-i\ve y \cd \xi} \hat \psi_\ve(\xi, \tau_y \om)

\\
 + \ga P \sum^d_{j=1} b_j(\om) \left[ e^{i\ve {\bf e}_j \cd  \xi} \hat \psi_\ve(\xi, \tau_{-{\bf e}_j} \om) -e^{-i\ve {\bf e}_j \cd  \xi} \hat \psi_\ve(\xi, \tau_{{\bf e}_j} \om) \right] 

+ \ve \hat \psi_\ve(\xi, \om) \\= \ga \sum^d_{j=1} \left[ b_j(\om) - \left< 
b_j(\cd) \right>\right] \left[ e^{-i\ve {\bf e}_j \cd  \xi} - 
e^{i\ve {\bf e}_j \cd  \xi}\right] \hat u_\ve(\xi),  \ 
\xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d, \ \ \om \in \Om.
\end{multline}
Suppose now that $\Psi^k_\ve(\zeta, \om), \ 1 \le k \le d, \ \zeta \in [-\pi, \pi]^d, \ \om \in \Om$, are functions in $L^2([-\pi, \pi]^d \times \Om)$ which satisfy the equation,
\begin{multline} \label{V2}
\Psi^k_\ve(\zeta, \om) - \sum_{y\in \Z^d} p(y) e^{-iy \cd \zeta} \Psi^k_\ve(\zeta, \tau_y  \om)
\\
 + \ga P \sum^d_{j=1} b_j(\om) \left[ e^{i{\bf e}_j \cd  \zeta}  \Psi^k_\ve(\zeta, 
\tau_{-{\bf e}_j} \om) -e^{-i {\bf e}_j \cd  \zeta} \Psi^k_\ve(\zeta, \tau_{{\bf e}_j} \om) \right] 
\\
+ \ve  \Psi^k_\ve(\zeta, \om) = b_k(\om)  - < b_k(\cd) >, \ 
\zeta \in [ -\pi, \pi]^d, \ \ \om \in \Om, \ 1 \le k \le d.
\end{multline}
Then it is clear that the function 
\begin{equation} \label{U2}
\hat \psi_\ve(\xi, \om) = -2 i \ga \hat u_\ve(\xi) \sum^d_{k=1} \sin (\ve {\bf e}_k\cd\xi)\Psi^k_\ve(\ve \xi, \om)
\end{equation}
is in $L^2 (\left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d \times \Om)$ provided $\hat u_\ve(\xi)$ is bounded.  Furthermore $\hat \psi_\ve(\xi, \om)$ given by (\ref{V2}), (\ref{U2}) satisfies (\ref{W2}).  If we substitute for $\hat \psi_\ve$ from (\ref{U2}) into (\ref{X2}) then we obtain the equation, 
\begin{multline} \label{AC2}
\hat u_\ve(\xi) \bigg\{ 1 + \ve^{-1} \left[1 - \hat p(\ve \xi)\right] + 2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin(\ve {\bf e}_j \cd \xi)}{\ve} 
\\
 - 2i \ga^2  \bigg< \sum^d_{j,k=1} b_j(\cd) \frac{\sin(\ve {\bf e}_k \cd \xi)}{\ve}
\bigg[ e^{i\ve {\bf e}_j \cd  \xi} \Psi^k_\ve(\ve \xi, \tau_{-{\bf e}_j} \cd) 
\\
-e^{-i\ve {\bf e}_j \cd \xi} \Psi^k_\ve(\ve \xi, \tau_{{\bf e}_j} \cd) \bigg] \bigg> \bigg\} = \hat f_\ve(\xi), \ \ \xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d.
\end{multline}
\begin{lem}  Suppose $d \ge 2$ and $\ga_d=c_d/2$, where $c_d$ satisfies
(\ref{AG2}). Then if $|\ga| < \ga_d$ and $0 < \ve \le 1$, the equation 
(\ref{V2}) 
has a unique solution $\Psi^k_\ve(\zeta, \om) \in L^2(\Om)$.  Furthermore, 
$\Psi^k_\ve(\zeta, \om)$ viewed as a function from $[-\pi, \pi]^d$ to $L^2(\Om)$
is continuous and satisfies an inequality,
\begin{equation} \label{AB2} 
\ve^2 \sup_{\zeta \in [-\pi, \pi]^d} \left< |\Psi^k_\ve(\zeta, \cd)|^2 
\right> \le C_d,  
\end{equation}
where $C_d$ depends only on $d$.
\end{lem}
\begin{proof}  We follow an argument similar to that in Lemma 2.3.  Suppose 
$\Psi^k_\ve(\zeta, \om) \in L^2(\Om)$ and satisfies (\ref{V2}).  Let $\Phi^k_\ve(\zeta, \om)$ be defined by
\begin{equation} \label{Y2}
\Phi^k_\ve(\zeta, \om) = \Psi^k_\ve(\zeta, \om) - \sum_{y\in\Z^d} p(y)e^{-iy\cd \zeta} \Psi^k_\ve(\zeta, \tau_y \om) + \ve \Psi^k_\ve(\zeta, \om).
\end{equation}
It is easy to see that $\Phi^k_\ve(\zeta, \om) \in L^2(\Om)$.  Further, it follows from (\ref{G2}) that
\begin{equation} \label{Z2}
\Psi^k_\ve(\zeta, \om) =  \sum_{x\in\Z^d} G_{p,\ve}(x)e^{-ix\cd \zeta} 
\Phi^k_\ve(\zeta, \tau_x \om).
\end{equation}
Now for $\eta > 0$, $\zeta \in [-\pi,\pi]^d, \ j=1, ..., d$ we define operators $T_{j,\eta,\zeta}$ on $L^2(\Om)$ by
\begin{equation} \label{AD2}
T_{j,\eta,\zeta} \vp(\om) = \sum_{x\in \Z^d} \left[ G_{p,\eta}(x-{\bf e}_j) - 
G_{p,\eta}(x+{\bf e}_j)\right] e^{-ix\cd\zeta} \varphi(\tau_x \om).
\end{equation}
We can see from Bochner's theorem \cite{SIMON} that $T_{j,\eta,\zeta}$ is bounded on $L^2(\Om)$ and
\begin{equation} \label{AE2}
\|T_{j,\eta,\zeta}\| \le \sup_{\zeta' \in [-\pi,\pi]^d} \frac{2|\sin({\bf e}_j 
\cd \zeta')|}{1+\eta - \hat p(\zeta ')} \le \frac{2}{c_d} \ ,
\end{equation}
where the constant $c_d$ is from (\ref{AG2}).  Further, (\ref{V2}), (\ref{Y2}), (\ref{Z2}) imply that $\Phi^k_\ve(\zeta, \omega)$ satisfies the equation,
\begin{equation} \label{AA2} 
\Phi^k_\ve(\zeta, \omega) - \ga P \sum^d_{j=1} b_j(\om) T_{j,\ve,\zeta}     \Phi^k_\ve(\zeta, \omega) = b_k(\om) - < b_k(\cd) >.
\end{equation}
Hence if we take $\ga_d = c_d/2$, it follows that for $|\ga| < \ga _d$ the function 
$\Phi^k_\ve(\zeta, \omega)$ is uniquely determined by the equation (\ref{AA2}). 
In view of (\ref{Z2}) it follows that $\Psi^k_\ve(\zeta, \om)$ is the unique 
solution of (\ref{V2}) in  $L^2(\Om)$.  We can similarly prove existence for 
$|\ga| < \ga_d$.  The inequality of (\ref{AB2}) follows from (\ref{Z2}) on using 
Bochner's theorem and the fact that $\Phi^k_\ve(\zeta, \cd) \in L^2(\Om)$ is 
bounded independently of $\zeta, \ve$.  \end{proof}

It is clear now that we can construct the solutions to (\ref{S2}), (\ref{T2}),  by using Lemma 2.4 and the equation (\ref{U2}), (\ref{AC2}). Next we wish to show that the limit of the coefficient of $\hat u_\ve(\xi)$ in (\ref{AC2}) as $\ve \ra 0$ exists.  Evidently
\begin{multline*}
\lim_{\ve \ra 0} \left\{ 1 + \ve^{-1} [1 - \hat p(\ve \xi)] + 2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin(\ve{\bf e}_j \cd \xi)}{\ve} \right\}
\\
= 1 + |\xi| + 2\ga i \sum^d_{j=1} < b_j(\cd) > ({\bf e}_j  \cd \xi) \ ,
\end{multline*}
so we are left to show existence of the term involving the function $\Psi^k_\ve$.  To prove this first observe from (\ref{Z2}), (\ref{AD2}) that
\[
e^{-i {\bf e}_j \cd \zeta} \Psi^k_\ve(\zeta, \tau_{{\bf e}_j} \om) 
- e^{i {\bf e}_j \cd \zeta} \Psi^k_\ve(\zeta, \tau_{-{\bf e}_j} \om) = T_{j,\ve,\zeta} \Phi^k_\ve(\zeta, \om).
\]
Next we prove a version of the von Neumann ergodic theorem \cite{SIMON}. 
\begin{lem} Let $\varphi \in L^2(\Om)$ and $\xi \in \R^d$.  Then if $< \varphi > = 0$ the function $T_{j,\ve,\ve\xi} \varphi$ converges in $L^2(\Om)$ to the function $T_{j,0,0} \varphi$.
\end{lem}
\begin{proof}  First note that we can define the operator 
$T_{j,0,\zeta}, \ \zeta \in [-\pi, \pi]^d$, by (\ref{AD2}) and from (\ref{AE2}) it is a bounded operator on $L^2(\Om)$.  To be more precise, observe that for $\eta >0, \ T_{j,\eta,\zeta} \varphi = 0$ if $\varphi \in L^2(\Om)$ satisfies $\partial_{j,\zeta} \varphi = 0$, where $\partial_{j,\zeta}$ is the operator on $L^2(\Om)$ given by
\[
\partial_{j,\zeta} \varphi(\om) = e^{-i{\bf e}_j \cd \zeta} \varphi(\tau_{{\bf e}_j} \om) 
-e^{i{\bf e}_j \cd \zeta} \varphi(\tau_{-{\bf e}_j} \om).
\]
Observe that the adjoint $\pa^*_{j,\zeta}$ of $\pa_{j,\zeta}$ is given by $\pa^*_{j,\zeta} =   - \pa_{j,\zeta}$.  Suppose now $\varphi = \pa_{j,\zeta} \psi$ where $\psi \in L^2(\Om)$.  Then from (\ref{AD2}) we have that
\begin{equation} \label{AF2}
T_{j,\eta,\zeta} \varphi(\om) = \sum_{x\in\Z^d} \left[ G_{p,\eta}(x+2{\bf e}_j) 
+ G_{p,\eta}(x - 2{\bf e}_j) -2 G_{p,\eta}(x)\right] e^{-ix \cd \zeta} \psi(\tau_x \om).
\end{equation}
We can readily see, using the argument of Lemma 2.2 that there is a constant $C_d$, depending only on $d$ such that
\begin{eqnarray*}
|G_{p,\eta}(x+2{\bf e}_j) &+& G_{p,\eta}(x-2{\bf e}_j) - 2G_{p,\eta}(x) | \\
&\le& C_d / [1+\eta^2|x|^2][1 + |x|^{d+1}], \ \ x \in \Z^d.
\end{eqnarray*}
Hence if $\varphi = \pa_{j,\zeta} \psi, \ \psi \in L^2(\Om)$, we can define $T_{j,0,\zeta}\varphi$ by setting $\eta = 0$ in the formula (\ref{AF2}).   If $\pa_{j,\zeta} \varphi = 0$ we set $T_{j,0,\zeta}\varphi = 0$.  In view of (\ref{AE2}) this defines $T_{j,0,\zeta}$ as a bounded operator on $L^2(\Om)$ which has the property that, for any $\varphi \in L^2(\Om)$, $T_{j,\eta,\zeta}\; \varphi$ converges in $L^2(\Om)$ to $T_{j,0,\zeta}\varphi$ as $\eta \ra 0$.

So far we have not used the ergodicity of the translation operators $\tau_x, \ x \in \Z^d$.  Suppose now $\varphi \in L^2(\Om)$ and $< \phi > = 0$.  Then ergodicity of the translation operators implies that for any $\de > 0$ there exists $\psi_\de \in L^2(\Om)$ such that $\| \varphi - \pa_{j,0} \psi_\de \| < \delta$.  Now using the fact that 
\[  
\pa_{j,0} \psi_\de = \pa_{j,\ve \xi} \psi_\de + \psi'_{\de,\ve}
\]
where $\psi'_{\de,\ve} \in L^2(\Om)$ satisfies 
$\| \psi'_{\de,\ve}\| \le 2\ve |{\bf e}_j \cd \xi | \|\psi_\de\|,$ it follows 
from (\ref {AE2}), (\ref{AF2}) that $T_{j,\ve,\ve \xi}[\pa_{j,0} \psi_\de]$ 
converges in $L^2(\Om)$ as $\ve \ra 0$ to the function 
$T_{j,0, 0 }[\pa_{j,0} \psi_\de]$.  The result then follows from 
(\ref{AE2}) on letting $\de \ra 0$.  \end{proof}

Lemma 2.5 enables us to define the vector $q_\ga \in \R^d$ of (\ref{F1}).  
\begin{lem} There exists a constant $\ga_d>0$, depending only on $d$, such that
if $|\ga| < \ga_d$ and $0 < \ve \le 1$, the equation (\ref{AC2}) has a unique 
solution $\hat u_\ve \in L^2 ([\frac {-\pi}\ve, \frac \pi \ve ]^d)$.  There is 
a vector $q_\ga \in \C^d$, analytic in $\ga$ and satisfying $|q_\ga|<1$, such that if $u(x), \ x 
\in \R^d$, has Fourier transform satisfying (\ref{F1}) and $u_\ve(x), \ x \in \Z^d_\ve$, is the Fourier inverse of $\hat u_\ve$, then 
\begin{equation} \label{AI2}
\lim_{\ve \ra 0} \| u_\ve - u \|_\ve = 0.
\end{equation}
\end{lem}
\begin{proof}  First observe that we can rewrite (\ref{AC2}) as
\begin{multline} \label{AH2}
\hat u_\ve(\xi) \bigg\{ 1 + \ve^{-1} \left[1 - \hat p(\ve \xi)\right] + 2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin(\ve {\bf e}_j \cd \xi)}{\ve} 
\\
  + 2i \ga^2  \bigg< \sum^d_{j,k=1} b_j(\cd) \frac{\sin(\ve {\bf e}_k \cd \xi)}{\ve}
T_{j,\ve,\ve\xi} \Phi^k_\ve(\ve\xi, \cd) \bigg> \bigg\}
= \hat f_\ve(\xi), \ \ \xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d,
\end{multline}
where $\Phi^k_\ve(\zeta, \om)$ is the solution of (\ref{AA2}).  Note that from 
(\ref{AH2}),(\ref{AG2}) and (\ref{AE2})  one has  that 
$\hat u_\ve \in L^2 ([\frac {-\pi}\ve, \frac \pi \ve ]^d)$ provided 
$0 < \ve \le 1$, $|\ga| < \ga_d$ and $\ga_d$ is sufficiently small, 
depending only on $d$.  It is also clear from Lemma 2.5 and (\ref{AE2}) that for any $\varphi \in L^2(\Om)$ with $<\varphi> = 0$ and nonnegative integer $m$, 
\[
\lim_{\ve \ra 0} T^m_{j,\ve,\ve \xi} \vp  = T^m_{j,0,0} \vp \ ,
\]
where the convergence is in the $L^2$ norm.  It follows then from (\ref{AA2}) that
\[
\lim_{\ve \ra 0} T_{j,\ve,\ve \xi} \Phi^k_\ve(\ve \xi, \cd) = T_{j,0,0} \Phi^k_0(0, \cd).
\]
Hence from (\ref{AH2}) we define the vector $q_\ga = (q_1,..., q_d)$ by
\begin{equation} \label{AL2}
q_k = 2\ga <b_k(\cd)> + 2\ga^2 \sum^d_{j=1} \left< b_j(\cd) T_{j,0,0} \Phi^k_0(0,\cd)\right>.
\end{equation}
Evidently we may choose $\ga_d$ sufficiently small, depending only on $d$, such
that $|q_\ga|<1$.
The limit (\ref{AI2}) now follows from the argument of Lemma 2.1.   \end{proof}

\begin{lem}
Let $\Psi^k_\ve(\zeta, \om)$ be the function defined in Lemma 2.4 for $|\ga| < \ga_d, \ 0 < \ve \le 1$.  Then for any $\xi \in \R^d$, there is the limit, 
\begin{equation} \label{AJ2}
\lim_{\ve \ra 0} \ \ve^2 \left< |\Psi^k_\ve(\ve \xi, \cd)|^2 \right> = 0.
\end{equation}
\end{lem}
\begin{proof}  We first consider the zero{\it th} order contribution in $\ga$ to $\Psi^k_\ve$.  Thus in (\ref{AA2}) we make the approximation $\Phi^k_\ve(\zeta, \om) \simeq b_k(\om) - <b_k(\cd)>$.  In this approximation we have from (\ref{Z2}) and Bochner's theorem the identity,
\begin{equation} \label{AK2}
\ve^2 \left< |\Psi^k_\ve(\ve \xi, \cd)|^2 \right> = \int_{[-\pi,\pi]^d} d\mu_\varphi(\zeta) \left[ \frac \ve {1+\ve - \hat p(\zeta - \ve \xi)} \right]^2,
\end{equation}
where $d\mu_\varphi$ is the finite positive Borel measure on $[-\pi,\pi]^d$, satisfying 
\[
\left< \varphi(\tau_x \cd) \overline{\varphi(\cd)} \right > =  
\int_{[-\pi,\pi]^d} e^{ix \cd \zeta} d\mu_\varphi(\zeta), \ \ \ \ 
x \in \Z^d \ ,
\]
and $\varphi(\om) = b_k(\om) - <b_k(\cd) >$.  Observe now that
\begin{eqnarray}
 \left[ \frac \ve {1 + \ve - \hat p(\zeta - \ve \xi)} \right]^2 &\le& 1, \ \ \ \ \zeta \in [-\pi, \pi]^d, \\
\lim_{\ve \ra 0} \left[ \frac \ve {1 + \ve - \hat p(\zeta - \ve \xi)} \right]^2 &=& 0, \ \ \ \ \zeta \in [-\pi, \pi]^d \backslash \{0 \}.
\end{eqnarray}
Since $< \varphi > = 0$, one has by ergodicity of the translation operators $\tau_x, \ \ x \in \Z^d$, that $\mu_\varphi(\{ 0 \}) = 0$.  It follows now by dominated convergence that (\ref{AJ2}) holds in this approximation.

To deal with the general case note that (\ref{AK2}) holds with $\varphi(\om) = \varphi_\ve(\om) = \Phi^k_\ve(\ve \xi, \om)$.  The result follows then from the above argument if we use the fact that $d\mu_{\phi_\ve} \le 2d\mu_{\phi_0} + 2 d\mu_{\phi_\ve-\phi_0}$ and the fact that  $\lim_{\ve \ra 0} \mu_{\phi_\ve-\phi_0} ([-\pi, \pi]^d)$ 
$= \|\phi_\ve - \phi_0 \|^2 = 0$.  \end{proof} 
\textit{Proof of Theorem 1.1}:  We just need to prove (\ref{G1}).  In view of Lemma 2.6 and (\ref{U2}) it will be sufficient to show that
\[
\lim_{\ve \ra 0} \int_{[\frac{-\pi}{\ve} , \frac \pi \ve ]^d} |\hat 
f_\ve(\xi)|^2 |\xi|^2 \ve^2
\left< |\Psi^k_\ve(\ve \xi, \cd)|^2  \right> d\xi = 0.
\]
This follows by dominated convergence from (\ref{AB2}) and Lemma 2.7.   
\qquad \endproof
Finally we show that if (\ref{H1}) holds then the vector $q_\ga$ defined by (\ref{AL2}) is zero.
\begin{lem}
Suppose (\ref{H1}) holds and the conditions of Theorem 1.1.  Then $q_\ga=0$ in (\ref{F1}).
\end{lem}
\begin{proof}  In view of (\ref{AL2}) we need to show that
\begin{equation} \label{AM2}
\left< b_j(\cd) T_{j,0,0} \Phi^k_0 (0, \cd) \right> = 0, \ \ 1 \le j,k \le d,
\end{equation}
where $\Phi^k_0$ satisfies (\ref{AA2}).  To do this let $L^2_R(\Om)$ be the subset of functions $\Phi \in L^2(\Om)$ which have the property that
\begin{multline*}
\left< \Phi(\tau_x \cd) \prod^n_{i=1} b_{k_i}(\tau_{x_i} \cd) \right> = (-1)^{n+1} 
\left< \Phi(\tau_{-x} \cd) \prod^n_{i=1} b_{k_i}(\tau_{-x_i} \cd) \right>,
\\
  x, x_i \in \Z^d, \ \ \ 1 \le k_i \le d, \ \ \  i = 1,...,n, \ \ \ n \ge 0 \ .
\end{multline*}
Evidently $\Phi \in L^2_R(\Om)$ implies $<\Phi> = 0$.  In view of (\ref{H1}) $b_i(\cd) \in L^2_R(\Om), \ 1 \le i \le d$.  Observe now that for any $\eta > 0$ the operator $b_j(\om)T_{j,\eta,0}$ takes $L^2_R(\Om)$ into $L^2_R(\Om)$ .  This follows from the fact that $G_{p,\eta}(x) = G_{p,\eta}(-x), \ x \in \Z^d$.  Now let $\Phi^k_\eta(\om)$ be the solution in $L^2_R(\Om)$ to the equation,
\begin{equation} \label{AN2}
\Phi^k_\eta(\om) - \ga \sum^d_{j=1} b_j(\om) T_{j,\eta,0} \Phi^k_\eta(\om) = b_k(\om).
\end{equation}
Clearly a unique solution $\Phi^k_\eta$ exists if $|\ga| < \ga_d$.  Comparing 
(\ref{AN2}) and (\ref{AA2}) one sees that 
$\Phi^k_\ve(0,\om) = \Phi^k_\ve(\om) \in L^2_R(\Om)$.  Taking the limit as 
$\ve \ra 0$ we conclude that $\Phi^k_0(0,\om) \in L^2_R(\Om)$, whence 
(\ref{AM2}) follows.   \ \ \ \ \end{proof}

\section{Proof of Theorem 1.2}

Since $<{\bb}(\cd) > = 0$ we have from (\ref{AA2}), (\ref{AH2}) that
\begin{equation} \nonumber
e(\ve\xi)q_{\ga,\ve}(\ve\xi)e(-\ve\xi) = - 2i \sum^d_{k=1} 
\sin(\ve {\bf e}_k \cd \xi) \sum^\infty_{n=1} \ga^{n+1} 
\left< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,\ve\xi} P \right]^n b_k(\cd) \right>.
\end{equation}
It is easy to see from the argument of Lemma 2.8 that if (\ref{H1}) holds, then
\begin{equation} \label{K3}
\left< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,0} P \right]^n b_k(\cd) \right> = 0, \ \ 
1 \le k \le d, \ \ n \ge 1.
\end{equation}
Hence we may write
\begin{multline} \label{A3}
e(\ve\xi)q_{\ga,\ve}(\ve\xi)e(-\ve\xi) = - 2i \sum^d_{k=1} 
\sin(\ve {\bf e}_k \cd \xi) \sum^\infty_{n=0} \ga^{n+2} \sum^n_{m=0}
\\
\bigg< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,0} P \right]^m 
 \left[ \sum^d_{j=1} b_j(\cd) \left\{T_{j,\ve,\ve \xi} - T_{j,\ve,0} \right\} P \right]
\\
 \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,\ve \xi} P \right]^{n-m} b_k(\cd) \bigg>  \ .
\end{multline}
If $x\in\Z^d$ with $x\cdot{\bf e}_1>0$, we denote by $L_1(x)$ all the points 
of $\Z^d$ which lie on the
line segment joining the point $x-(x\cdot{\bf e}_1){\bf e}_1$ and $x$, excluding 
the point $x$. If $x\cdot{\bf e}_1=0$, then $L_1(x)$ is the empty set. 
If $x\cdot{\bf e}_1<0$, then $L_1(x)$ is all the points 
of $\Z^d$ which lie on the
line segment joining the point $x-(x\cdot{\bf e}_1){\bf e}_1$ and $x$,
excluding the point $x-(x\cdot{\bf e}_1){\bf e}_1$. Similarly if 
$x\cdot{\bf e}_2>0$ we denote by 
$L_2(x)$ all the points of $\Z^d$ which lie on the
line segment joining the point $x-(x\cdot{\bf e}_1){\bf e}_1-(x\cdot{\bf e}_2)
{\bf e}_2$ and 
$x-(x\cdot{\bf e}_1){\bf e}_1$, excluding the point $x-(x\cdot{\bf e}_1)
{\bf e}_1$ etc. For 
$\eta>0$, $\zeta\in[-\pi,\pi]^d$, $k=1,\cdot\cdot\cdot,d$, we define
operators $S_{k,\eta,\zeta}$ on $L^2(\Om)$ by 
\begin{equation} \label{L3}
S_{k,\eta,\zeta}\varphi(\om)=
\sum^d_{j=1} b_j(\om)\sum_{x\in \Z^d} \left[ G_{p,\eta}(x-{\bf e}_j) - 
G_{p,\eta}(x+{\bf e}_j)\right] 
\mathrm{sgn}(x\cdot{\bf e}_k)\sum_{y\in L_k(x)}e^{-iy\cd\zeta} 
\varphi(\tau_x \om) \ .
\end{equation}
We can see now from (\ref{A3}) and (\ref{AD2}) that the matrix 
$q_{\ga,\ve}(\zeta)=[q_{\ga,\ve,k,k'}(\zeta)], \ 1\le k,k'\le d$, is given by 
the formula
\begin{multline} \label{J3}
q_{\ga,\ve,k,k'}(\zeta)=-\sum^\infty_{n=0} \ga^{n+2} \sum^n_{m=0}

\bigg< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,0} P \right]^m 
S_{k',\ve,\zeta}P 
\left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve, \zeta} P \right]^{n-m} b_k(\cd) \bigg>
\\  
-
\sum^\infty_{n=0} \ga^{n+2} \sum^n_{m=0}

\bigg< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,0} P \right]^m 
e^{-i{\bf e}_k\cdot\zeta}S_{k,\ve,\zeta}P 
\left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve,\zeta} P \right]^{n-m} b_{k'}(\cd) \bigg>
\ .
\end{multline}

We proceed as in \cite{CN} to consider the case when the 
${\bb}(\tau_x \cd), \ x \in \Z^d$, are given by independent Bernoulli variables.
Thus we assume that $b_j({\cd}) \equiv 0, \ j > 1$, and 
$b_1(\tau_x \cd) = Y_x \ , x \in \Z^d$, where the variables $Y_x, \ x \in \Z^d$,
are assumed to be  i.i.d. Bernoulli, $Y_x = \pm 1$ with equal probability.  
Using the notation of $\S4$ of \cite{CN} we introduce the Fock spaces 
${\cF}^r(\Z^d), \ 1 < r < \infty$, with ${\cF}^2 (\Z^d)$  isomorphic to 
$L^2(\Om)$.  Now we have seen from (\ref{AE2}) that $T_{1,\ve,\zeta}$ is a bounded 
operator on 
$L^2(\Om)$ with norm $\|T_{1,\ve,\zeta}\|_2 \le 2/c_d$.  We have also seen in 
Lemma 2.5 that
\begin{equation} \label{M3}
\lim_{\ve \ra 0} \ \| [T_{1,\ve,\zeta}  - T_{1,0,\zeta} ]\varphi\|_2 = 0, \ \  \varphi \in L^2(\Om).
\end{equation}
We note now as in \cite{CN} that the Calderon-Zygmund theorem implies that a 
similar result holds for all Fock spaces ${\cF}^r(\Z^d), \ 1 < r < \infty$. We
shall show that the limit in (\ref{M3}) is uniform for $\zeta\in[-\pi,\pi]^d$,
provided one projects orthogonal to the constant function.
\begin{lem}  Suppose $1 < r < \infty$ and regard $T_{1,\ve,\zeta}$ as an 
operator on ${\cF}^r(\Z^d)$.  Then there is a constant $C_{r,d}$ depending only 
on $r,d$ such that $\| T_{1,\ve,\zeta}\|_r \le C_{r,d}$.  There is also the 
limit,
\begin{equation} \label{B3}
\lim_{\ve \ra 0} 
\left\{\sup_{\zeta\in[-\pi,\pi]^d} \| [T_{1,\ve,\zeta}P  - 
T_{1,0,\zeta} P]\varphi \|_r\right\} = 0, \ \ \ 
\varphi \in {\cF}^r(\Z^d).
\end{equation}
\end{lem}
\begin{proof} Let $T_{\ve,\zeta}$ be the integral operator on $L^r(\Z^d)$
defined by
$$ T_{\ve,\zeta}f(x)=\sum_{y\in\Z^d} K_\ve(x-y)e^{-i(x-y)\cdot\zeta}f(y), \
x\in\Z^d \ ,
$$
where $K_\ve(x)=G_{p,\ve}(x-{\bf e}_1)-G_{p,\ve}(x+{\bf e}_1)$. It follows from
(\ref{AG2}) that 
\begin{equation} \label{O3}
|\hat{K}_\ve(\xi)|\le 2/c_d, \ \ \  \xi\in[-\pi,\pi]^d \ .
\end{equation}
We may also apply the argument of Lemma 2.2 to see that there is a constant 
$C_d >0$, depending only on $d$, such that
\begin{eqnarray} \label{C3}
|K_\ve(x)| &\le& C_d / [1+\ve^2 |x|^2] [1 + |x|^d], \ \ \ x \in \Z^d \ , \\
|K_\ve(x + {\bf e}_i) - K_\ve(x) | &\le& C_d / [1+\ve^2 |x|^2] [1 + |x|^{d+1}], 
 \ \ \ x \in \Z^d \ .\nonumber 
\end{eqnarray}
It follows now from (\ref{O3}), (\ref{C3}) that $T_{\ve,\zeta}$ is a bounded 
operator on $L^r(\Z^d)$ and there is a constant $C_{r,d}$ depending only on 
$r, d$ such that $\|T_{\ve,\zeta}\|_r \le C_{r,d}$.  This is a discrete version of 
the Calderon-Zygmund Theorem \cite{STEIN}. Now, arguing as in \cite{CN}, we see
that this also implies 
$\| T_{1,\ve,\zeta}\|_r \le C_{r,d}$.

Next we show that 
\begin{equation} \label{P3}
\lim_{\ve \ra 0} 
\left\{\sup_{\zeta\in[-\pi,\pi]^d} \| [T_{\ve,\zeta}  - 
T_{0,\zeta} ]f \|_r\right\} = 0, \ \ \ 
f \in L^r(\Z^d).
\end{equation}
We first show that (\ref{P3}) holds for $r=2$. We can do this by going to the
Fourier representation of $T_{\ve,\zeta}$, which we denote 
$\hat{T}_{\ve,\zeta}$. Thus,
\begin{equation} \label{R3}
\hat{T}_{\ve,\zeta}\hat{f}(\xi)=\hat{K}_\ve(\xi-\zeta)\hat{f}(\xi), \ 
\xi\in[-\pi,\pi]^d \ . 
\end{equation}
Observe now that since $\hat{f}\in L^2([-\pi,\pi]^d)$, for any $\delta>0$ there
exists $\nu>0$ such that
\begin{equation} \label{Q3}
\int_{|\xi-\zeta|<\nu}|\hat{f}(\xi)|^2d\xi  \ < \ \delta, \ \ \ 
\zeta\in [-\pi,\pi]^d \ .
\end{equation}
It is easy to see that for any $\nu>0$, $\hat{K}_\ve(\xi)$ converges uniformly 
to  $\hat{K}_0(\xi)$ in the region $\xi\in[-\pi,\pi]^d, \ |\xi|>\nu$. It follows
from this and (\ref{O3}), (\ref{R3}), (\ref{Q3}) that (\ref{P3}) holds for $r=2$.

To prove (\ref{P3}) for general $r, \ 1<r<\infty$, we define for
$n=1,2\cdot\cdot\cdot, \ \zeta\in[-\pi,\pi]^d$ an operator $A_{n,\zeta}$ on
functions $f:\Z^d\ra\C$ by
$$
A_{n,\zeta}f(x)=\sum_{z\in\Z^d} \varphi_n(z)e^{iz\cdot\zeta} f(x+z), \ \
x\in\Z^d \ .
$$
Here $\varphi_n$ is the probability density for the standard random walk in
$\Z^d$ after $n$ steps of the walk. Thus $\varphi_n(z)\ge 0, \ z\in\Z^d$, and
$$ \sum_{z\in\Z^d} \varphi_n(z)=1 \ . $$
It follows that $A_{n,\zeta}$ is a bounded operator on $L^r(\Z^d)$ with
$\|A_{n,\zeta}\|_r\le 1$. Observe now that
$$ T_{\ve,\zeta}[f-A_{n,\zeta}f](x)= 
\sum_{y\in\Z^d} K_{\ve,n}(x-y)e^{-i(x-y)\cdot\zeta}f(y), \ \ x\in\Z^d,
$$
where the function $K_{\ve,n}$ is given by
$$K_{\ve,n}(x)=\sum_{z\in\Z^d}[K_\ve(x)-K_\ve(x+z)]\varphi_n(z) \ . $$
It is easy to see from (\ref{C3}) that $K_{\ve,n}\in L^1(\Z^d)$ with norm 
$\|K_{\ve,n}\|_1$ satisfying $\|K_{\ve,n}\|_1\le C_{d,n}$, where the constant 
$C_{d,n}$ depends only on $d$ and $n$. Further, for any $x\in\Z^d$,
$K_{\ve,n}(x)$ converges to $K_{0,n}(x)$. We conclude that for any function
$f\in L^r(\Z^d)$ one has
$$
\lim_{\ve \ra 0} 
\left\{\sup_{\zeta\in[-\pi,\pi]^d} \| [T_{\ve,\zeta}  - 
T_{0,\zeta} ][f-A_{n,\zeta}f] \|_r\right\} = 0,
$$
for any $n=1,2,\cdot\cdot\cdot$. The result (\ref{P3}) follows then if we can
show that for any $f\in L^r(\Z^d)$, $1<r<\infty$, and $\delta>0$, there exists
a positive integer $N$ such that
$$ \|A_{n,\zeta}f\|_r< \delta, \ \ n\ge N, \ \ \zeta\in[-\pi,\pi]^d \ . $$
This last inequality follows by approximating $f\in L^r(\Z^d)$ with a function
of finite support. A similar argument implies (\ref{B3}).
\end{proof}
\begin{lem} Suppose $d\ge 2, \ 1 < r < d$ and $q<\infty$ satisfies $1/r>1/d+1/q$. 
Then the operator $S_{k,\eta,\zeta}$ of (\ref{L3}) is a bounded operator from 
the space ${\cF}^r(\Z^d)$ to ${\cF}^q(\Z^d)$. There is a constant $C_{r,q,d}$, 
depending only on $r,q,d$, such that the norm $\|S_{k,\eta,\zeta} \|_{r,q}$ of
the operator $S_{k,\eta,\zeta}$  satisfies the inequality  
$\|S_{k,\eta,\zeta} \|_{r,q}\le C_{r,q,d}$.  There is also the 
limit,
\begin{equation} \label{E3}
\lim_{\ve \ra 0} 
\left\{\sup_{\zeta\in[-\pi,\pi]^d} \| [S_{k,\ve,\zeta}P  - 
S_{k,0,\zeta} P]\varphi \|_{r,q}\right\} = 0, \ \ \ 
\varphi \in {\cF}^r(\Z^d).
\end{equation}
\end{lem}
\begin{proof} We proceed as in Lemma 3.1. Let $S_{\ve,\zeta}$ be the operator
on functions $f\in L^r(\Z^d)$ defined by 
$$ S_{\ve,\zeta}f(x)=\sum_{y\in\Z^d} K_{\ve,\zeta}(x-y) f(y), \
x\in\Z^d \ ,
$$
where 
$$
K_{\ve,\zeta}(x)=[G_{p,\ve}(x-{\bf e}_1)-G_{p,\ve}(x+{\bf e}_1)]
\mathrm{sgn}(x\cdot{\bf e}_k)\sum_{z\in L_1(x)}e^{-iz\cd\zeta} \ .
$$ 
It follows from (\ref{C3}) that there is a constant 
$C_d >0$, depending only on $d$, such that
$$
|K_{\ve,\zeta}(x)| \le C_d / [1 + |x|^{d-1}], \ \ \ x \in \Z^d \ . 
$$
Hence $K_{\ve,\zeta}\in L^s(\Z^d)$ for any $s>d/(d-1)$. It follows now by
Young's inequality that $S_{\ve,\zeta}f\in L^q(\Z^d)$ for any $q>0$ satisfying
$1/r>1/d+1/q$. Further, there is a constant $C_{r,q,d}$, depending only on
$r,q,d$, such that the norm $\|S_{\ve,\zeta}\|_{r,q}$ of $S_{\ve,\zeta}$ as an
operator from $L^r(\Z^d)$ to $L^q(\Z^d)$ satisfies  
$\|S_{\ve,\zeta}\|_{r,q}\le C_{r,q,d}$. It is easy to see that for any $x\in\Z^d$,
$$
\lim_{\ve\ra 0}\left\{\sup_{\zeta\in[-\pi,\pi]^d} 
|K_{\ve,\zeta}(x)-K_{0,\zeta}(x)|\right\}=0 \ .
$$
Hence, if  $S_{0,\zeta}$ is the operator with kernel $K_{0,\zeta}$, one has
$$
\lim_{\ve \ra 0} 
\left\{\sup_{\zeta\in[-\pi,\pi]^d} \| [S_{\ve,\zeta}  - 
S_{0,\zeta} ]f \|_q\right\} = 0, \ \ \ 
f \in L^r(\Z^d) \ ,
$$
provided $q<\infty$ satisfies $1/r>1/d+1/q$. Now, arguing as in \cite{CN} we
obtain the result. 
\end{proof}
\begin{corollary} Suppose $b_j(\cd) \equiv 0, \ j > 1$ and 
$b_1(\tau_x \cd) = Y_x$, 
$x \in \Z^d$, where the $Y_x, \ x \in \Z^d$, are independent Bernoulli variables 
with zero mean.  Then Theorem 1.2 holds.
\end{corollary}
\begin{proof} Observe that $b_k(\cdot)\in \cF^r(\Z^d)$ for all $r\ge 1$. The
result follows now from Lemmas 3.1, 3.2 and the representation 
(\ref{J3}) for $q_{\ga,\ve}(\zeta)$.
\end{proof}
Next we wish to generalize our method to all ${\bb}(\cd)$ such that 
${\bb}(\tau_x\cd), \ x \in \Z^d$, are independent random variables.  To do this 
we pursue the method developed in $\S 5$ of \cite{CN}.  Thus we define a spin space $S$ as all $s=(m_1,...,m_d)$ where the $m_i$ are nonnegative integers satisfying $|s| = m_1 + ... + m_d \ge 1.$  For $s \in S$ we define a random variable $b_s$ by
\[	b_s(\cd) = \prod^d_{i=1} \ b_i(\cd)^{m_i} \ , \ \ \ s = (m_1, ..., m_d),  \]
and a random variable $Y_{0,s} = b_s - <b_s>$.  Then for 
$s \in S, \ \ n \in \Z^d$ we define the variable $Y_{n,s}$ to be the translate 
of $Y_{0,s} $ i.e. $Y_{n,s}(\cd) = Y_{0,s}(\tau_n \cd)$.  For $1 < r < \infty$ 
we may define Fock spaces ${\cF}^r_S(\Z^d)$ of many particle wave functions 
where the particles move in $\Z^d$ and have spin in $S$.  We can also, for any
$\delta>0$, define a 
mapping $U_{\delta}$ from ${\cF}^2_S(\Z^d)$ to functions on $\Om$ by
\begin{equation} \label{H3}
U_{\delta} \psi = \psi_0 + \sum^\infty_{N=1} 
\begin{array}[t]{c}
{\displaystyle\sum_{\{n_1,...,n_N\}\in \Z^{d,N}}} \\
{\scriptstyle s_i \in S, \ 1 \le i \le N}
\end{array} \psi_N(n_1,s_1,n_2,s_2,...,n_N,s_N)
\end{equation}
\[ \times \delta^{|s_1|+...+|s_N|} \ Y_{n_1,s_1} Y_{n_2,s_2} ... Y_{n_N,s_N} \ .\]
Following the argument of Lemma 5.1 of \cite{CN} we have
\begin{lem}
There exists $\delta>0$, depending only on $d$, such that the mapping 
$U_{\delta}$ of (\ref{H3}) is a bounded operator from ${\cF}^2_S(\Z^d)$ to 
$L^2(\Om)$ satisfying $\| U_{\delta}\| \le 1$.
\end{lem}
We shall use the operator $U_{\delta}=U$ of Lemma 3.3 to transfer the equation of (\ref{AA2}) from 
$L^2(\Om)$ to ${\cF}^2_S(\Z^d)$.  Thus we shall need to construct operators 
$P_{\cF}, \ B_{j,{\cF}}$, $T_{j,\ve,\zeta,{\cF}}, \ 1 \le j \le d$, with the 
property that they are bounded operators on ${\cF}^2_S(\Z^d)$ and satisfy
\begin{equation} \label{T3}
UP_{\cF} = PU, \ \ U B_{j,{\cF}} = b_j(\cd)U, \  UT_{j,\ve,\zeta,{\cF}} = 
T_{j,\ve,\zeta} U \ .
\end{equation}
The simplest of these operators to construct is $P_{\cF}$.  Thus if 
$\psi = \{\psi_N : N=0,1,2,...\} \in {\cF}^2_S (\Z^d)$ then 
$P_{\cF}\psi = \{ (P_{\cF}\psi)_N : N=0,1,2,...\}$, where 
$(P_{\cF}\psi)_0 = 0$, $(P_{\cF}\psi)_N = \psi_N, \ N \ge 1$.  Similarly we 
have $T_{j,\ve,\zeta,{\cF}} \psi = $ 
$\{ (T_{j,\ve,\zeta,{\cF}} \psi)_N : N=0,1,2,... \}$, where
\[	(T_{j,\ve,\zeta,{\cF}} \psi)_0 = \left[ e^{-i\zeta\cd {\bf e}_j} - e^{i\zeta\cd {\bf e}_j}\right] \hat G_{p,\ve}(-\zeta) \psi_0, \]
\[	(T_{j,\ve,\zeta,{\cF}} \psi)_N (n_1, s_1, ..., n_N, s_N) = \]
\[ \sum_{x\in \Z^d} \left[ G_{p,\ve}(x - {\bf e}_j) - G_{p,\ve}(x + {\bf e}_j) \right] e^{-ix\cd \zeta} \psi_N(n_1-x, s_1, ..., n_N - x, s_N), \ \ N \ge 1. \]
The most complicated of the operators to define is $B_{j,\cF}$.  We have
\[	(B_{j,{\cF}} \psi)_N (n_1, s_1, ..., n_N, s_N)  = < b_j(\cd) > \psi_N(n_1, s_1, ..., n_N, s_N) \]
\[   +\sum_{s\in S} \left[ <b_j b_s > - <b_j> <b_s> \right] 
\delta^{|s|} \psi_{N+1} (0, s, n_1, s_1, ..., n_N, s_N) , \]
\[  {\rm if} \ \ \ \ n_k \not= 0, \ \ \ 1 \le k \le N, \ \ \ N \ge 0;	\]
\begin{eqnarray*}
(B_{j,{\cF}} \psi)_N (0, s_1, n_2, s_2, ..., n_N ,s_N)  &=& 0,  \\
{\rm if} \ \ n_k \not= 0, \ \ 2 \le k \le N, \ \ s_1 &=&  (m_1, ..., m_d), \ m_j = 0; \\
(B_{j,{\cF}} \psi)_N (0, s_1, n_2, s_2, ..., n_N ,s_N)  &=& \delta^{-1} \psi_N
(0, s'_1, n_2, s_2, ..., n_N ,s_N) , \\
{\rm if} \ \  n_k \not= 0, \ \ 2 \le k \le N, \ \ s_1 &=&  (m_1, ..., m_d), \ 
m_j > 0, \ |s_1|>1, \\
				       s'_1 &=& (m_1, ...,m_{j-1}, m_j-1, m_{j+1},..., m_d); \\ 
(B_{j,{\cF}} \psi)_N (0, s_1, n_2, s_2, ..., n_N ,s_N)  &=& \delta^{-1} \psi_{N-1} (n_2, s_2, ..., n_N ,s_N)
\end{eqnarray*}
\[  -\sum_{s\in S}   <b_s>  \delta^{|s|-1} \psi_N (0, s, n_2, s_2, ..., n_N, s_N) , \]
\[  {\rm if} \ \ \ \ n_k \not= 0, \ \ \ 2 \le k \le N \ \ \ {\rm and} \ \ \ 
s_1=(0,0,..,1,0, ..., 0),  \]
with  1  in the ith position.
\begin{lem}  For $1 \le k \le d$ let $\vp_k \in {\cF}^2_S(\Z^d)$ be the wave 
function $\vp_k = \{ \vp_{k,N} : N=0,1,2,... \}$ where $\vp_{k,N} \equiv 0$ if 
$N \not= 1$ and $\vp_{k,1}$ is defined by
\[	\vp_{k,1}(n, s) = 1 \ \ \ {\rm if} \ \ \ n=0, \ \ 
s = (0,..,0,1,0,...,0),	\]
with  1  in the  kth position,
$\vp_{k,1}(n,s) = 0$, otherwise.
Then there exists a constant $\ga_d$ depending only on $d$ such that if 
$|\ga| < \ga_d$, 
the equation
\begin{equation} \label{I3}
\Phi^k_{\ve,{\cF}} (\zeta)  -\ga  P_{{\cF}} \sum^d_{j=1} B_{j,{\cF}}  
T_{j,\ve,\zeta,{\cF}} \Phi^k_{\ve,{\cF}}(\zeta) = \vp_k \ ,
\end{equation}
has a unique solution $\Phi^k_{\ve,{\cF}} (\zeta) \in {\cF}^2_S(\Z^d)$.  
Further the function $\Phi^k_\ve(\zeta, \om) = U \Phi^k_{\ve,{\cF}} (\zeta)  
\in L^2(\Om)$ is the solution to (\ref{AA2}).
\end{lem}
\begin{proof}  Evidently $P_{\cF}$ is a bounded operator on ${\cF}^2_S(\Z^d)$ 
with norm 1.  We can also easily see that $T_{j,\ve,\zeta, {\cF}}$ is a bounded 
operator on   ${\cF}^2_S(\Z^d)$ with norm bounded by the RHS of (\ref{AE2}).  Also 
$B_{j, {\cF}}$ is bounded with norm depending only on $d$.  Hence we may choose $\ga_d$ 
small enough,    depending only on $d$, such that (\ref{I3}) is uniquely 
solvable.  The fact that $U  \Phi^k_{\ve,{\cF}} (\zeta)$ solves (\ref{AA2}) 
comes from (\ref{T3}) and $U \vp_k = b_k(\cd) - <b_k>$.
\end{proof}
\textit{Proof of Theorem 1.2}: Observe from (\ref{AH2}) that $q_{\ve,\ga}(\zeta)$
is defined in terms of 
$\Phi^k_\ve(\zeta, \om)$. In view of Lemma 3.4 we get a
representation of $q_{\ve,\ga}(\zeta)$ similar to (\ref{J3}) but with the 
operators $T_{j,\ve,\zeta},  \ S_{k,\ve,\zeta},\ b_j(\cdot)$ on $L^2(\Om)$ 
replaced by the corresponding operators 
$T_{j,\ve,\zeta,\cF},  \ S_{k,\ve,\zeta,\cF},\ B_{j,{\cF}}$ on Fock space. Here
the operator $S_{k,\ve,\zeta,\cF}$ satisfies 
$US_{k,\ve,\zeta,\cF}=S_{k,\ve,\zeta}U$. We now argue exactly as in Corollary
3.1.
\qquad \endproof

\section{Proof of Theorem 1.3}
We assume in this section that the function $p$ is given by (\ref{A1}). Let
$G_{p,\eta}(x)$ be the function defined by (\ref{G2}). Then
$\hat{G}_{p,\eta}(\xi)$ satisfies the inequality,
$$
0\le \hat{G}_{p,\eta}(\xi)\le C/[\eta+|\xi|^2] \ ,
$$
for some universal constant $C$. One can also see that if $d\ge 2$ and 
$0<\eta<1$, there is a constant $C_d$, depending only on $d$, such that
\begin{equation} \label{I4}
|G_{p,\eta}(x-{\bf e}_k)-G_{p,\eta}(x+{\bf e}_k)|\le
C_d\exp\left[-\sqrt{\eta}|x|/C_d\right]/[1+|x|^{d-1}] \ .
\end{equation}
Next we prove the analogue of Lemma 2.3.
\begin{lem}
Suppose $d \ge 2$ and $f : \R^d \ra \R$ is a $C^\infty$ function with compact 
support.  Then if $\ga\in\C$ satisfies $|\ga| < \ve/\sqrt{2d}$ and $0 < \ve \le 1$, 
the equation (\ref{C1}) has a unique solution 
$u_\ve (x, \om)$ in $L^2(\Z^d_\ve)$.  Further, there is a constant $C_d$, 
depending only on $d$, such that $\|u_\ve(\cd, \om)\|_\ve \le C_d \|f\|_\ve$ .
\end{lem}
\begin{proof} Suppose $u_\ve (x, \om)$ is in $L^2(\Z^d_\ve)$ and satisfies (\ref{C1}).  Let $v_\ve(x, \om)$ be defined by 
\[	v_\ve(x, \om) = \ve^{-2} [A_p u_\ve(x,\om) + \ve^2 u_\ve(x,\om)]. \] 
It is easy to see that $v_\ve(x, \om) \in L^2(\Z^d_\ve)$.  In view of (\ref{G2}) it follows that 
\begin{equation} \label{A4}
u_\ve(x, \om) = \ve^{2-d} \int_{\Z^d_\ve} dy \ G_{p,\ve^2} \left( \frac{x-y} \ve 
\right)  v_\ve(y, \om).
\end{equation}
Now for $i=1,...,d$ let us define operators $T_i$ on $L^2(\Z^d_\ve)$ by
\[    T_i w(x) = \ve^{1-d} \int_{\Z^d_\ve} dy \left[ G_{p,\ve^2} \left( \frac{x-y} 
\ve + {\bf e} _i \right) - G_{p,\ve^2} \left( \frac{x-y} \ve - {\bf e} _i \right) 
\right] w(y)  \  .
\]
It is easy to see that $T_i$ is a bounded operator on $L^2(\Z^d_\ve)$.  
It follows from (\ref{M2}) that the norm of $T_i$ is bounded by 
\begin{equation} \label{B4}
\| T_i \| \le \sup_{\xi \in [-\pi,\pi]^d} \left[ \frac{2\ve|\sin({\bf e}_i \cd \xi)
|}{1+\ve^2 - \hat p(\xi)}
\right].
\end{equation}
Now (\ref{B4}) and (\ref{L1}) imply that the operator
$\sum^d_{i=1} b_i(\cdot,\om) T_i$ is bounded on 
$L^2(\Z^d_\ve)$ and has norm
satisfying the inequality,
\begin{eqnarray*}
\|\sum^d_{i=1} b_i(\cdot,\om) T_i\|&\le &
\sup_{\xi \in [-\pi,\pi]^d} \left[ \frac{2\ve
\left\{\sum_{i=1}^d\sin^2({\bf e}_i \cd \xi)\right\}^{1/2}}
{1+\ve^2 - \hat p(\xi)}
\right] \\
 &\le &4\sup_{z>0}\left[\frac{\ve z}{\ve^2+2z^2/d}\right] \ = \ \sqrt{2d} \ .
\end{eqnarray*}
Observe now that (\ref{C1}) implies that $v_\ve(x,\om) \in L^2(\Z^d_\ve)$ satisfies the equation
\begin{equation} \label{C4}
v_\ve(x,\om) - \ga\ve^{-1} \sum^d_{i=1} b_i(x/\ve,\om) T_i v_\ve(x,\om) = f(x).
\end{equation}
Hence if $|\ga| < \ve/\sqrt{2d}$, the 
function $v_\ve(x,\om)$ is uniquely determined by (\ref{C4}).  In view of 
(\ref{A4}) it follows that $u_\ve(x,\om) \in L^2(\Z^d_\ve)$ is the unique 
solution to (\ref{C1}).  To prove existence note that (\ref{C4}) has a 
solution $v_\ve(x,\om) \in L^2(\Z^d_\ve)$ provided $|\ga| < \ve/\sqrt{2d}$.  
If we then 
define $u_\ve(x,\om)$ by (\ref{A4}) one can easily see that $u_\ve(x,\om)$ 
satisfies (\ref{C1}). \ \ \ \ \end{proof}
In order to obtain the equation (\ref{M1}) we follow a development similar to the 
one following Lemma 2.3. Thus we put 
$v_\ve(x,\om) = u_\ve(x, \tau_{-x/\ve} \om)$, $x \in \Z^d_\ve, \om \in \Om$, 
and write  $v_\ve(x, \om) = u_\ve(x) + \psi_\ve(x,\om)$, 
$\ x \in \Z^d_\ve, \ \ \om \in \Om$, where $< \psi_\ve(x, \cd) > = 0$, 
$\ x \in \Z^d_\ve$.  It follows that
\[
u_\ve(x) = \Big< v_\ve(x, \cd) \Big> = \Big< u_\ve(x, \cd) \Big> , \ \ x 
\in \Z^d_\ve \ .
\]
Furthermore, the Fourier transform $\hat{u}_\ve(\xi)$ of 
$u_\ve(x)$ satisfies the equation,
\begin{multline} \label{D4}
\hat u_\ve(\xi) \left\{ 1 + \ve^{-2} \left[1 - \hat p(\ve \xi)\right] + 
2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin({\bf e}_j \cd \ve \xi)}{\ve^2} \right\}
\\     + \ga \ve^{-2} \left< \sum^d_{j=1} b_j(\cd) \left[ e^{i\ve {\bf e}_j \cd 
\xi} \hat \psi_\ve(\xi, \tau_{-{\bf e}_j} \cd) -e^{-i\ve {\bf e}_j \cd \xi} 
\hat \psi_\ve(\xi, \tau_{{\bf e}_j} \cd) \right] \right> = \hat f_\ve(\xi),
 \ \ \xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d.
\end{multline}
The Fourier transform of $\psi_\ve(x,\om)$ is given in terms of 
$\hat u_\ve(\xi)$ by
\begin{equation} \label{E4}
\hat \psi_\ve(\xi, \om) = -2 i \ga \hat u_\ve(\xi) \sum^d_{k=1} \sin (\ve {\bf e}_k\cd\xi)\Psi^k_\ve(\ve \xi, \om)
\end{equation}
Here $\Psi^k_\ve(\zeta, \om), \ 1 \le k \le d, \ \zeta \in [-\pi, \pi]^d, \ \om \in \Om$, are functions in $L^2([-\pi, \pi]^d \times \Om)$ which satisfy the equation,
\begin{multline} \label{F4}
\Psi^k_\ve(\zeta, \om) - \sum_{y\in \Z^d} p(y) e^{-iy \cd \zeta} \Psi^k_\ve(\zeta, \tau_y  \om)
\\
 + \ga P \sum^d_{j=1} b_j(\om) \left[ e^{i{\bf e}_j \cd  \zeta}  \Psi^k_\ve(\zeta, 
\tau_{-{\bf e}_j} \om) -e^{-i {\bf e}_j \cd  \zeta} \Psi^k_\ve(\zeta, \tau_{{\bf e}_j} \om) \right] 
\\
+ \ve^2  \Psi^k_\ve(\zeta, \om) = b_k(\om)  - < b_k(\cd) >, \ 
\zeta \in [ -\pi, \pi]^d, \ \ \om \in \Om, \ 1 \le k \le d.
\end{multline}
If we substitute for $\hat \psi_\ve$ from (\ref{E4}) into 
(\ref{D4}) then we obtain the equation, 
\begin{multline} \label{G4}
\hat u_\ve(\xi) \bigg\{ 1 + \ve^{-2} \left[1 - \hat p(\ve \xi)\right] + 
2\ga i \sum^d_{j=1} <b_j(\cd)> \frac{\sin(\ve {\bf e}_j \cd \xi)}{\ve^2} 
\\
 - 2i \ga^2  \bigg< \sum^d_{j,k=1} b_j(\cd) \frac{\sin(\ve {\bf e}_k \cd \xi)}
{\ve^2}
\bigg[ e^{i\ve {\bf e}_j \cd  \xi} \Psi^k_\ve(\ve \xi, \tau_{-{\bf e}_j} \cd) 
\\
-e^{-i\ve {\bf e}_j \cd \xi} \Psi^k_\ve(\ve \xi, \tau_{{\bf e}_j} \cd) \bigg] \bigg> \bigg\} = \hat f_\ve(\xi), \ \ \xi \in \left[ \frac {-\pi}\ve, \frac \pi \ve \right]^d.
\end{multline}

Let us assume now that (\ref{H1}) holds. Then, following the development at 
the beginning of $\S 3$, we have that the matrix 
$q_{\ga,\ve}(\zeta)=[q_{\ga,\ve,k,k'}(\zeta)], \ 1\le k,k'\le d$ of (\ref{M1}) 
is given by the formula
\begin{multline} \label{H4}
q_{\ga,\ve,k,k'}(\zeta)=-\sum^\infty_{n=0} \ga^{n+2} \sum^n_{m=0}

\bigg< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve^2,0} P \right]^m 
S_{k',\ve^2,\zeta}P 
\left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve^2, \zeta} P \right]^{n-m} b_k(\cd) \bigg>
\\  
-
\sum^\infty_{n=0} \ga^{n+2} \sum^n_{m=0}

\bigg< \left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve^2,0} P \right]^m 
e^{-i{\bf e}_k\cdot\zeta}S_{k,\ve^2,\zeta}P 
\left[ \sum^d_{j=1} b_j(\cd) T_{j,\ve^2,\zeta} P \right]^{n-m} b_{k'}(\cd) 
\bigg>
\ ,
\end{multline}
where the operator  $T_{j,\eta, \zeta}$ is defined by (\ref{AD2}) and 
$S_{k,\eta,\zeta}$ by (\ref{L3}), with $p$ given by (\ref{A1}). One can see
from the proof of Lemma 4.1 that $q_{\ga,\ve}(\zeta)$ is continuous in
$\zeta$ and analytic in $\ga$ provided $|\ga|<\ve/\sqrt{2d}$.

\section{Proof of Theorem 1.4}
The matrix $q_{m,\ve}(\zeta)$ is given by the coefficient of $\ga^m$ in the
expansion (\ref{H4}). Since the ${\bf b}(\tau_x\cdot)$ are given by Bernoulli
variables, (a) and (b)  of Theorem 1.4 follow easily from (\ref{H4}). Next we obtain a
formula for $q_{4,\ve}(\zeta)$. Let $K_\ve(x)$ be defined by
\begin{equation} \label{D5}
K_\ve(x)= G_{p,\ve^2}(x-{\bf e}_1)- G_{p,\ve^2}(x+{\bf e}_1), \ x\in\Z^d \ .
\end{equation}
For $x\in\Z^d$, let $x^R$ be the reflection of $x$ in the plane through the
origin with normal ${\bf e}_1$. It is clear that
\begin{equation} \label{A5}
K_\ve(x^R) \ =  \ -K_\ve(x), \ x\in\Z^d \ .
\end{equation}
For $1\le k\le d, \ \zeta\in[-\pi,\pi]^d$, define $a_k(\zeta), \ b_k(\zeta)$ by
\begin{eqnarray}  \label{F5}
a_{k,\ve}(\zeta)&=& \sum_{x\in\Z^d} [K_\ve(x)]^3 
\mathrm{sgn}(x\cdot{\bf e}_k)\sum_{y\in L_k(x)}e^{-iy\cd\zeta} \ , \\
b_{k,\ve}(\zeta)&=& \sum_{x\in\Z^d} [K_\ve(x)]^3 e^{ix\cd\zeta}
\mathrm{sgn}(x\cdot{\bf e}_k)\sum_{y\in L_k(x)}e^{-iy\cd\zeta} \ . \nonumber 
\end{eqnarray}
Note that
\begin{equation} \label{H5} 
a_{k,\ve}(0)=b_{k,\ve}(0)=
\sum_{x\in\Z^d} [K_\ve(x)]^3(x\cdot{\bf e}_k) \ .
\end{equation}
Then one can see from (\ref{H4}) that
\begin{equation} \label{B5}
q_{4,\ve,1,1}(\zeta)= [1+e^{-i{\bf e}_1\cd\zeta}][2a_{1,\ve}(\zeta)-
b_{1,\ve}(\zeta)] \ .
\end{equation}
One also has for $1<k\le d$,
\begin{eqnarray} \label{C5}
q_{4,\ve,1,k}(\zeta)&=& 2a_{k,\ve}(\zeta)-b_{k,\ve}(\zeta) \ , \\
q_{4,\ve,1,k}(\zeta)&=& e^{-i{\bf e}_k\cd\zeta}[2a_{k,\ve}(\zeta)-
b_{k,\ve}(\zeta)] \ .
\nonumber
\end{eqnarray}
Since $a_{k,\ve}(0)=b_{k,\ve}(0)$ it follows that 
$q_{4,\ve,1,1}(0)=2a_{1,\ve}(0)$. From (\ref{A5}) and (\ref{H5}) we see that 
$a_{k,\ve}(0)=0$ if
$k>1$. Hence $q_{4,\ve,k,k'}(0)=0$ for $k+k'>2$.
\begin{lem}(a) Let $d=1$ and $\mathcal{K}\subset \R$ be a compact set. Then 
$\ve^2a_{1,\ve}(\ve\xi)$ converges uniformly for
$\xi\in \mathcal{K}$ to a function $a_1(\xi)$, as $\ve\ra 0$. Similarly
$\ve^2b_{1,\ve}(\ve\xi)$ converges to a function $b_1(\xi)$. \\
(b) If $d\ge 3$ and $1\le k\le d$, then $a_{k,\ve}(\zeta)$ converges uniformly for
$\zeta\in[-\pi,\pi]$ to a function $a_k(\zeta)$, as $\ve\ra 0$. Similarly
$b_{k,\ve}(\zeta)$ converges to a function $b_k(\zeta)$.
\end{lem}
\begin{proof} (a)  It is easy to see that for $d=1, \ \eta>0$, one has
$$
G_{p,\eta}(x)=\frac{\left[1+\eta-\sqrt{2\eta+\eta^2}\right]^{|x|}}
{\sqrt{2\eta+\eta^2}} \ , \ \ \ x\in\Z \ .
$$
Hence if for $\ve>0$, we define $\nu(\ve)$ by
\begin{equation} \label{E5}
\nu(\ve)= 1+\ve^2-\ve\sqrt{2+\ve^2} \ ,
\end{equation}
then the function $K_\ve(x)$ of (\ref{D5}) is given by 
\begin{equation} \label{G5}
K_\ve(x)=2\nu(\ve)^{|x|}, \ x\in\Z, \ x> 0 \ . 
\end{equation}
Substituting  from (\ref{G5}) into (\ref{F5}) we obtain the identity,
$$
a_{1,\ve}(\zeta)= \frac{8\nu(\ve)^3}{1-\nu(\ve)^3}
\left[ \frac{1}{1-\nu(\ve)^3e^{-i\zeta}}+
\frac{e^{i\zeta}}{1-\nu(\ve)^3e^{i\zeta}}\right] \ .
$$
One can see that
$$
\lim_{\ve\ra 0} \frac{\ve}{1-\nu(\ve)^3e^{-i\ve\xi}}=
\frac{1}{3\sqrt{2}+i\xi} \ .
$$
We conclude that
$$
\lim_{\ve\ra 0} \ve^2a_{1,\ve}(\ve\xi)= \frac{16}{18+\xi^2} \ .
$$
Since for $d=1$ one has $a_{1,\ve}(\zeta)=b_{1,\ve}(-\zeta)$, 
the formula (\ref{O1}) follows now from the last equation and (\ref{B5}). \\ 
(b) This follows from the fact that for $d\ge 3$, the limit $\lim_{\ve\ra 0}
K_\ve(x)=K_0(x)$ exists for all $x\in\Z^d$, and the bound (\ref{I4}).
\end{proof}
\begin{lem} (a) For $d\ge 3$, one has $a_1(0)>0$. \\
(b) For $d=2$ one has $\lim_{\ve\ra 0} a_{1,\ve}(0)=+\infty$.
\end{lem}
\begin{proof} (a) In view of (\ref{A5}) and (\ref{H5}) it is sufficient to show
that $K_0(x)>0$ if $x\in\Z^d$ satisfies $x\cdot{\bf e}_1>0$. To see this
observe from (\ref{G2}) and (\ref{A5}) that $K_\ve(x)$ satisfies the equation
$$
A_p K_\ve(x)+\ve^2 K_\ve(x)= g(x), \ \ x\in\Z^d, \ x\cdot{\bf e}_1>0  \ ,
$$
with Dirichlet boundary condition $K_\ve(x)=0, \ x\cdot{\bf e}_1=0$. Here $g$
is the function, $g(x)=1, \ x={\bf e}_1$, $g(x)=0$ otherwise. The positivity of 
$K_0(x)$ follows now from the maximum principle. \\
(b) We shall show that there exists $\alpha>0$ and a positive constant 
$c_\alpha$, depending only on  $\alpha$ such that for $d=2$,
\begin{equation} \label{I5}
\lim_{\ve\ra 0}K_\ve(n{\bf e}_1+m{\bf e}_2)\ge \frac{c_\alpha}{n}, \ \ \ \ \ 
|m|<\alpha n, \ n=1,2,\cdot\cdot\cdot \ .
\end{equation}
To prove (\ref{I5}) we use the fact that for $d=1$ one has
$$
K_\ve(n)= 2\nu(\ve)^n=\frac{2}{\pi}\int_0^\pi\frac{\sin(n\zeta)\sin(\zeta)}
{2\sin^2(\zeta/2)+\ve^2}  d\zeta \  \ \ \ \   n\ge 1 \ .
$$
Since one also has for $d=2$ the representation,
$$
K_\ve(n{\bf e}_1+m{\bf e}_2)=\frac{2}{\pi^2}\int_0^\pi\int_0^\pi
\frac{\cos(m\xi)\sin(n\zeta)\sin(\zeta)}
{\sin^2(\zeta/2)+\sin^2(\xi/2)+\ve^2}  d\zeta d\xi \ ,
$$
we obtain for $d=2$ the formula,
$$
\lim_{\ve\ra 0} K_\ve(n{\bf e}_1+m{\bf e}_2)= \frac{4}{\pi}\int_0^\pi
\nu(\sqrt{2}\sin(\xi/2))^n \cos(m\xi)d\xi \ ,  \  \ \ m,n\in\Z, \ n\ge 1.
$$
Now there exist universal positive constants $c,C$ such that,
$$
\exp[-Cn\xi]\le\nu(\sqrt{2}\sin(\xi/2))^n\le \exp[-cn\xi], \ \  \ 
n\ge1, \ 0\le\xi\le \pi.
$$
We conclude that the LHS of (\ref{I5}) is bounded below by
$$
\frac{4}{\pi}\int_0^\pi\exp[-Cn\xi] \ d\xi=\frac{4}{C\pi n}\left\{1-
\exp[-C\pi n]\right\} \ ,
$$
if $m=0$, and by
\begin{multline*}
\frac{2\sqrt{2}}{\pi}\int_0^{\pi/4|m|}\exp[-Cn\xi] \ d\xi -
\frac{4}{\pi}\int_{\pi/4|m|}^\infty\exp[-cn\xi] \ d\xi  \\
=\frac{2\sqrt{2}}{C\pi n}\left\{1-
\exp\left[-\frac{C\pi n}{4|m|}\right]\right\} \ - \ 
\frac{4}{c\pi n}\exp\left[-\frac{c\pi n}{4|m|}\right] \ ,
\end{multline*}
if $m\ne 0$. The inequality (\ref{I5}) clearly follows from the last two
identities. The result follows now from (\ref{H5}) and (\ref{I5}) since the
argument of part (a) implies $K_\ve(n{\bf e}_1+m{\bf e}_2)>0, \ m,n\in\Z, \
n>0$.   
\end{proof}
We are left now to prove  (c) and (d) of Theorem 1.4 for $m>4$. In order to do
this we need to obtain a suitable representation of $q_{m,\ve}(\zeta)$ for
general $m$. We first consider the situation $\zeta=0$. Recall that we are
assuming the   $b_j(\cdot)\equiv 0, \ j>1$, and
$b_1(\tau_x\cdot)=Y_x, \ x\in\Z^d$, where the variables $Y_x, \ x\in\Z^d$, are
assumed to be i.i.d. Bernoulli, $Y_x=\pm 1$ with equal probability.
\begin{lem} Suppose $m\ge 4$ is even. Then $q_{m,\ve,k,k'}(0)=0$ if $k+k'>2$
and $q_{m,\ve,1,1}(0)$ has the representation
\begin{multline}  \label{J5}
q_{m,\ve,1,1}(0)=-2\sum_{x_1,\cdot\cdot\cdot,x_{m-1}\in\Z^d}
\left(\sum_{\alpha=1}^{m-1}x_\alpha\cdot{\bf e}_1\right)
K_\ve(x_1)\cdot\cdot\cdot K_\ve(x_{m-1}) \\
\left< Y_0Y_{x_1}Y_{x_1+x_2}\cdot\cdot\cdot
Y_{x_1+\cdot\cdot\cdot+x_{m-1}}\right> \ .  
\end{multline}
\end{lem}
\begin{proof} First note that by the argument of Lemma 2.8  the
representation (\ref{H4}) for $q_{\ga,\ve}(0)$ continues to hold when we delete 
the projection operator $P$ from the RHS of the equation. It is easy to see
from this that (\ref{J5}) holds. If $k>1$ then one has a similar representation
to (\ref{J5}) for $q_{m,\ve,1,k}(0)$ and  $q_{m,\ve,k,1}(0)$ but with ${\bf
e}_1$ in (\ref{J5}) replaced by ${\bf e}_k$. In that case (\ref{A5}) implies
$q_{m,\ve,k,1}(0)=q_{m,\ve,1,k}(0)=0$. 
\end{proof}
In order to prove (d) of Theorem 1.4 for general $m$ we use (\ref{J5}) to 
obtain a representation of $q_{m,\ve,1,1}(0)$ as a sum indexed by certain types
of graph. We shall use the terminology of  \cite{BOLLOBAS}. For 
$q=2,3,\cdot\cdot\cdot$ let $\cF_q$ be the set of  unlabeled, connected, directed 
multigraphs on $q$ vertices  with the  properties: 

\noindent (A) The graph has no loops, \\
(B)  Each vertex has equal indegree and outdegree, \\
(C) The degree of each vertex is a multiple of 4, \\
(D) The number of edges in the graph is $4q$.

For a graph $G\in \cF_q$ let $V[G]$ denote the vertex set of $G$ and $E[G]$ be 
the set of directed edges of $G$. Each 
directed edge $e$ has two vertices $e_+$ and $e_-$, with the direction of the
edge being from $e_-$  to  $e_+$. We associate with $G$ and a directed edge 
$e\in E[G]$, a number $K_\ve(G,e)$ defined by,
\begin{equation} \label{K5}
K_\ve(G,e)= \sum_{\{y_v\in\Z^d: v\in V[G]\}} 
(y_{e_-}\cdot{\bf e}_1)\delta(y_{e_+})
\prod_{e'\in E[G], \ e'\ne e} K_\ve(y_{e'_+}- y_{e'_-}) \ .
\end{equation}
In (\ref{K5}) $\delta(x), \ x\in\Z^d$ is the Kronecker delta, 
$\delta(0)=1, \ \delta(x)=0, \ x\ne 0$.
\begin{lem} Let $m\ge 4$ be an even integer and $q=m/2$. Then 
\begin{equation} \label{L5}
q_{m,\ve,1,1}(0)=\sum_{G\in\cF_q, \ e\in E[G]} c(G,e)K_\ve(G,e) \ ,
\end{equation}
for some integers $c(G,e)$ depending only on the graph $G$ and the directed 
edge $e\in E[G]$.
\end{lem}
\begin{proof} Observe that 
$$
K_\ve(x_1)\cdot\cdot\cdot K_\ve(x_{m-1}) 
\left< Y_0Y_{x_1}Y_{x_1+x_2}\cdot\cdot\cdot
Y_{x_1+\cdot\cdot\cdot+x_{m-1}}\right> \ne 0,
$$
only if the sequence $0, x_1,x_1+x_2, \cdot\cdot\cdot,
x_1+\cdot\cdot\cdot+x_{m-1}$ of points in $\Z^d$ has the property that it
consists of $q$ pairs of  points with the property that
no two adjacent points of the sequence are the same. The $q$ pairs of points
donot have to be distinct. Hence we may write,
\begin{equation} \label{M5}
q_{m,\ve,1,1}(0)=-2\sum_{i=1}^q a_i \ ,
\end{equation}
where $a_i$ corresponds to the term on the RHS of (\ref{J5}) where $i$ of the 
$q$ pairs are distinct. For a graph $G\in\cF_q$ let $K'_\ve(G,e)$ be defined in
the same way as $K_\ve(G,e)$ except that the summation on the RHS of (\ref{K5})
is over all $y_v\in\Z^d: v\in V[G]$ such that the $q$ points 
$y_v\in\Z^d$ are distinct.  Define $\cF'_q$ as the subset of 
$\cF_q$ consisting of
regular graphs of degree 4 i.e. the degree of each vertex is exactly 4. Now by
(B) of the definition of $\cF_q$ and Euler's theorem \cite{BOLLOBAS} a graph 
$G\in \cF_q$ has an Eulerian path. Let $c(G)$ be the number of Eulerian paths.
For a graph $G\in \cF_q$, one can group directed edges into equivalence classes
where two directed edges $e,e'$ are equivalent if there is a graph isomorphism  
of $G$ taking $e$ to $e'$. Let $E'[G]$ be  this set of equivalence classes.
Then we may define $K'_\ve(G,\hat{e})$ for $\hat{e}\in E'[G]$ by 
$K'_\ve(G,\hat{e})= K'_\ve(G,e)$ for any $e\in\hat{e}$. Then one can see that 
$a_q$ of (\ref{M5}) is given by,
$$
a_q=\sum_{G\in\cF'_q, \ \hat{e}\in E'[G]} c(G)K'_\ve(G,\hat{e}) \ .
$$
Since we can obtain a similar representation for the $a_i, \ 1\le i<q$, we have
the formula (\ref{L5}) up to replacing $K'_\ve(G,e)$ by $K_\ve(G,e)$. The result
therefore holds if we can show that for any graph $G\in \cF_q$ and $e\in E[G]$
there exist integers $c_{G,e}(G',e'), \ G'\in \cF_q, \ e'\in E[G']$ such that
\begin{equation} \label{O5}
K'_\ve(G,e)=\sum_{G'\in\cF_q, \ e'\in E[G']} c_{G,e}(G',e')K_\ve(G',e') \ .
\end{equation}
The previous identity follows from Mayer's trick to obtain an expansion for
an non-ideal gas \cite{FORD}. That is for a function $\varphi:\Z^d\ra \R$, one
writes
\begin{equation} \label{N5}
\exp\left[-\sum_{1\le i<j\le N} \varphi(y_i-y_j)\right]=
\prod_{1\le i<j\le N}[1+f_{i,j}] \ ,
\end{equation}
where $f_{i,j}= \exp[-\varphi(y_i-y_j)]-1$, and expands the RHS of (\ref{N5})
out. Now (\ref{O5}) follows from (\ref{N5}) on taking $\varphi$ to be the
function, $\varphi(y)=0, \ y\ne 0, \ \varphi(0)=+\infty$.
\end{proof}
\begin{lem} Suppose $d=1, \ q\ge 2$ and $G\in \cF_q, \ e\in G$. If $|G|$ is the
number of vertices of $G$, then the limit,
\begin{equation} \label{P5}
\lim_{\ve\ra 0} \ve^{|G|}K_\ve(G,e) = K(G,e)
\end{equation}
exists.
\end{lem}
\begin{proof} Evidently it is sufficient to show that (\ref{P5}) holds with 
$K_\ve(G,e)$ replaced by $K'_\ve(G,e)$. Suppose $|G|=k\le q$ and label the 
vertices of $G$ as $1,2,\cdot\cdot\cdot, k$ with $e_+=1, \ e_-=k$. Then
$$
K'_\ve(G,e)=\sum_{\pi\in \mathcal{S}_k} K_{\ve,\pi}(G,e) \ ,
$$
where $\mathcal{S}_k$ is the group of permutations on $1,2,\cdot\cdot\cdot , k$ 
and
$$
K_{\ve,\pi}(G,e)=  \sum_{y_{\pi 1}<y_{\pi 2}< \cdot\cdot\cdot
<y_{\pi k}} 
\delta(y_1)y_k 
\prod_{e'\in E[G], \ e'\ne e} K_\ve(y_{e'_+}- y_{e'_-}) \ . 
$$
In view of (\ref{G5}) there are positive integers $N_j, \ 1\le j\le k-1$ such
that
$$
K_{\ve,\pi}(G,e)  =  \pm \sum_{y_{\pi 1}<y_{\pi 2}< \cdot\cdot\cdot
<y_{\pi k}} 
\delta(y_1)y_k 
\prod_{j=1}^{k-1} 2^{N_j}\nu(\ve)^{N_j[y_{\pi(j+1)}-y_{\pi(j)}]} \  .
$$
Suppose now that $\pi k_1=1, \ \pi k_2=k$ and $k_1<k_2$. Then
\begin{equation} \label{Q5}
K_{\ve,\pi}(G,e)=\sum_{i=k_1}^{k_2-1} K_{\ve,\pi,i}(G,e) \ ,
\end{equation}
where
$$
K_{\ve,\pi,i}(G,e)  =  \pm \sum_{y_{\pi 1}<y_{\pi 2}< \cdot\cdot\cdot
<y_{\pi k}} 
\delta(y_1)[y_{\pi(i+1)}- y_{\pi(i)}]
\prod_{j=1}^{k-1} 2^{N_j}\nu(\ve)^{N_j[y_{\pi(j+1)}-y_{\pi(j)}]} \  .
$$
It is easy to see now just as in Lemma 5.1 that the limit
$$
\lim_{\ve\ra 0} \ve^k K_{\ve,\pi,i}(G,e)
$$
exists. Hence from (\ref{Q5}) the limit,
\begin{equation} \label{S5}
\lim_{\ve\ra 0} \ve^k K_{\ve,\pi}(G,e)
\end{equation} 
exists if $k_1<k_2$. A similar argument shows that the limit (\ref{S5}) also
exists if $k_1>k_2$. 
It follows then from the previous identities that the limit (\ref{P5}) exists.
\end{proof}
\begin{remark} Lemma 5.5 proves that (d) of Theorem 1.4 holds with $\xi=0$.
Observe that the number of graphs in $\cF_q$ is of order $q!$. One can see
this from an asymptotic formula for the number $A_q$ of labeled 4-regular simple
graphs on $q$ vertices \cite{READ},
$$
A_q\sim [(4q)!e^{-15/4}]/[(2q)!96^q] \ .
$$
\end{remark}

Next we wish to obtain the analogue of Lemma 5.4 for $q_{m,\ve,k,k'}(\zeta)$
with $\zeta\in[-\pi,\pi]^d$ and at least one of $k,k'$ being $1$. To do this we
define for any $G\in\cF_q, \ e\in E[G]$, a function $K_\ve(G,e,\zeta)$ by
\begin{equation} \label{R5}
K_\ve(G,e,\zeta)=
\sum_{\{y_v\in\Z^d: v\in V[G]\}} 
\exp[-iy_{e_-}\cdot\zeta]\delta(y_{e_+})
\prod_{e'\in E[G], \ e'\ne e} K_\ve(y_{e'_+}- y_{e'_-}) \ .
\end{equation}
Note that $K_\ve(G,e,\zeta)$ is defined similarly to $K_\ve(G,e)$ of (\ref{K5})
and has the property $K_\ve(G,e,0)=0$. For $1\le k\le d$ we define 
$K_{\ve,k}(G,e,\zeta)$ by
\begin{multline} \label{T5}
K_{\ve,k}(G,e,\zeta)=
\sum_{\{y_v\in\Z^d: v\in V[G]\}}
\delta(y_{e_+})\left[\prod_{e'\in E[G], \ e'\ne e} K_\ve(y_{e'_+}- y_{e'_-}) 
\right] \\
\left[\sum_{e'\in E[G], \ e'\ne e}
\exp[i(y_{e'_+}- y_{e_-})\cdot\zeta] \  
{\rm sgn}[(y_{e'_+}- y_{e'_-})\cdot{\bf e}_k]
\sum_{y\in L_k(y_{e'_+}- y_{e'_-})} e^{-iy\cdot\zeta}  \right]  \ .
\end{multline}
Note that $K_\ve(G,e)$ of (\ref{K5}) is given by 
$K_\ve(G,e)=K_{\ve,1}(G,e,0)$. In the following lemma we need to define $\cF_q$ 
for $q=0,1$. The set $\cF_1$ is the single point vertex graph $G$ with 
$K_\ve(G,e,\zeta)=0$. The set $\cF_0$ is also the single point vertex graph $G$ 
with $K_\ve(G,e,\zeta)=1$. 
\begin{lem} Let $m\ge 4$ be an even integer and $q=m/2$. Define  
$A_{m,\ve,k}(\zeta)$ by
\begin{multline} \label{U5}
A_{m,\ve,k}(\zeta)=  \sum_{q'=2}^q
 \  \ \sum_{G\in\cF_{q'}, \ G'\in\cF_{q-q'}} \\ 
\sum_{ e\in E[G], \ e'\in E[G']} c(G,e,G',e')
K_{\ve,k}(G,e,\zeta)K_\ve(G',e',\zeta) \ ,
\end{multline}
for suitable integers $c(G,e,G',e')$ depending only on the graphs $G,G'$ and 
the directed edges $e\in E[G], \ e'\in E[G']$. Then,
$$
\begin{array}{cccr}  
q_{m,\ve,1,1}(\zeta)&=& [1+e^{-i{\bf e}_1\cdot\zeta}] \ A_{m,\ve,1}(\zeta) \ ,
& \\
q_{m,\ve,1,k}(\zeta)&=& A_{m,\ve,k}(\zeta) \ ,   & 1<k\le d, \\
q_{m,\ve,k,1}(\zeta)&=& e^{-i{\bf e}_k\cdot\zeta}A_{m,\ve,k}(\zeta) \ ,
 & 1<k\le d.
\end{array}
$$
\end{lem}
\begin{proof} We define  $A_{m,\ve,k'}(\zeta)$ as the coefficient of $\ga^m$ in
the first term on the RHS of (\ref{H4}). Evidently the formulas for 
$q_{m,\ve,k,k'}(\zeta)$ in the statement of the lemma follow from this. We need
then to establish (\ref{U5}). This follows by the same argument as in Lemma 5.4.
Note now that, unlike in Lemmas 5.3 and 5.4,  the projection operators $P$ in
(\ref{H4}) make a contribution. Thus we obtain the factorisations 
$K_{\ve,k}(G,e,\zeta)K_\ve(G',e',\zeta)$ in (\ref{U5}) with $G\in\cF_{q'}, \ 
G'\in\cF_{q''}$, where $q'+q''=q$.
\end{proof}
\begin{lem} Suppose $d=1$ and $\mathcal{K}\subset\R$ is a compact set. Let 
$q\ge 2$ and $G\in \cF_q, \ e\in G$. If $|G|$ is the
number of vertices of $G$, then the limits,
\begin{eqnarray} \label{V5}
\lim_{\ve\ra 0} \ve^{|G|-1}K_\ve(G,e,\ve\xi) &=& K(G,e,\xi)  \ , \\
\lim_{\ve\ra 0} \ve^{|G|}K_{\ve,1}(G,e,\ve\xi) &=& K_1(G,e,\xi) \ , \label{W5}
\end{eqnarray}
exist, uniformly for $\xi\in \mathcal{K}$.
\end{lem}
\begin{proof} The proof of (\ref{V5}) follows exactly the proof of Lemma 5.5.
To prove (\ref{W5}) we proceed similarly. Thus we focus on one term in the
summation over $e'\in E[G], \ e'\ne e$ in (\ref{T5}). As in Lemma 5.5 we fix a
permutation $\pi$ and consider the sum over 
$y_{\pi 1}<y_{\pi 2}< \cdot\cdot\cdot<y_{\pi k}$. Suppose that $\pi k_1=e'_-,
\ \pi k_2= e'_+$ and $k_1<k_2$. Then
\begin{multline*}
{\rm sgn}(y_{e'_+}- y_{e'_-})
\sum_{y\in L_1(y_{e'_+}- y_{e'_-})} e^{-iy\zeta} = \\
\sum_{i=k_1}^{k_2-1} \left(\exp[- i(y_{\pi(i)}-y_{\pi(k_1)})\zeta]
\sum_{z=0}^{y_{\pi(i+1)}- y_{\pi(i)}-1}e^{-iz\zeta}\right) \ .
\end{multline*}
If we consider a fixed $i$ on the RHS of the previous equation and sum over 
$y_{\pi 1}<y_{\pi 2}< \cdot\cdot\cdot<y_{\pi k}$, then we see as in Lemma 5.5
that the contribution of this summation to (\ref{W5}) converges uniformly for
$\xi\in\mathcal{K}$ as $\ve\ra 0$. Since we can similarly argue for all the
other contributions we conclude that the uniform limit (\ref{W5}) exists. 
\end{proof}
It is clear that (d) of Theorem 1.4 follows from the previous two Lemmas.
\begin{thebibliography}{9}
\bibitem{BOLLOBAS} Bollobas, B.:
\newblock Modern Graph Theory,
\newblock Springer, New York 1998, MR 99h:05001.  
\bibitem{CN}
Conlon, J. and Naddaf, A.:
\newblock On homogenisation of elliptic equations with random
coefficients, 
\newblock \textit{Electronic Journal of Probability} \textbf{5}, paper 9, 1-58
(2000), MR 1768843.
\bibitem{CV}
Conlon, J. and vonDohlen, B.: 
\newblock Numerical simulations of random walk in random environment,
\newblock \textit{J. Statist. Phys.} \textbf{92}, 571-586 (1998), 
MR 99j:82039.
\bibitem{DERRIDA}
\newblock Derrida, B. and Luck, J.:
\newblock Diffusion on a random lattice: Weak-disorder expansion in arbitrary
dimension,
\newblock \textit{ Phys. Rev. B} \textbf{28}, 7183-7190 (1983).
\bibitem{FANJIANG} Fanjiang, A. and Papanicolaou, G.: 
\newblock Diffusion in Turbulence , 
\newblock \textit{ Prob. Theory Relat. Fields} \textbf{105}, 279-334 (1996),
MR 98d:60156.
\bibitem{FISHER} Fisher, D.:
\newblock Random walks in random environment,
\newblock \textit{Phys, Rev. A} \textbf{30}, 960-964 (1984),
MR 85h:82028.
\bibitem{FORD} Ford, G. and Uhlenbeck, G.:
\newblock Lectures in Statistical Mechanics,
\newblock American Math Society, Providence 1963, MR 27\#1241.
\bibitem{GILBARG} Gilbarg, D. and Trudinger, A.:
\newblock Elliptic Partial Differential
Equations of Second Order, 
\newblock 2nd edition, Springer, New York 1983, MR 86c:35035.
\bibitem{KESTEN} Kesten, H.:
\newblock The limit distribution of Sinai's random walk in random environment,
\newblock \textit{Phys. A.} \textbf{138}, 299-309 (1986),
MR 88b:60165.
\bibitem{KOZLOV} Kozlov, S.:
\newblock Averaging of random structures,
\newblock \textit{Dokl. Akad. Nauk. SSSR} \textbf{241}, 1016-1019 (1978),
MR 80e:60078.
\bibitem{KUNNEMANN}
K\"unnemann, R.:
\newblock The diffusion limit for reversible jump processes on $\Z$ with
  ergodic random bond conductivities,
\newblock \textit{Commun. Math. Phys.} \textbf{90}, 27-68 (1983), 
MR 84m:60102.
\bibitem{LAWLER}
Lawler, G.:
\newblock Weak convergence of a random walk in a random environment,
\newblock \textit{Commun. Math. Phys.} \textbf{87}, 81-87 (1982),
MR 84b:60093.
\bibitem{LOY}
Landim, C., Olla, S. and Yau, H.-T.:
\newblock Convection-diffusion equation with space-time ergodic random 
flow,
\newblock  \textit{Probab. Theory Relat. Fields} \textbf{112}, 203-220 (1998), 
MR 99j:35084.
\bibitem{VARADHAN} Papanicolaou, G. and Varadhan, S.: 
\newblock Boundary value problems with rapidly oscillating random coefficients, 
\newblock Volume 2 of \textit{
Coll. Math. Soc. Janos Bolya}, \textbf{27},  Random fields, Amsterdam,
North Holland Publ. Co. 1981, pp. 835-873, MR 84k:58233.
\bibitem{PISZTORA} Pisztora, A., Povel, T. and Zeitouni, O.:
\newblock Large deviation estimates for a one-dimensional walk in a random
environment,
\textit{Probab. Theory Related Fields} \textbf{113}, 191-219 (1999),
MR 99m:60048. 
\bibitem{READ} Read, R. and Wormald, N.:
\newblock Number of Labeled 4-Regular Graphs,
\newblock \textit{J. Graph Theory} \textbf{4}, 203-212 (1980),
MR 81e:05085.
\bibitem{SIMON} Reed, M. and Simon, B.: 
\newblock Methods of Mathematical Physics I -Functional Analysis, 
\newblock Academic Press, 1972, MR 85e:46002.
\bibitem{SINAI} Sinai, Y.:
\newblock Limiting behavior of a one-dimensional random walk in a random
medium,
\newblock \textit{Theory Prob. Appl.} \textbf{27}, 256-268 (1982),
MR 83k:60078.
\bibitem{STEIN} Stein, E.: 
\newblock Singular integrals and differentiability properties of functions, 
\newblock Princeton University Press, 1970, MR 58\#2467.
\bibitem{SZNITMAN1} Sznitman, A.:
\newblock Slowdown and neutral pockets for a random walk in random environment,
\newblock \textit{Probab. Theory Related Fields} \textbf{115}, 287-323 (1999),
MR 2001a:60035.
\bibitem{SZNITMAN2} Sznitman, A.:
\newblock  Slowdown estimates and central limit theorem for random walks in
random environment,
\newblock \textit{J. Eur. Math. Soc.} \textbf{2}, 93-143 (2000),
MR 1763302.
\bibitem{ZKO}
Zhikov, V., Kozlov, S. and Oleinik, O.:
\newblock Homogenization of Differential Operators and Integral Functionals,
\newblock Springer Verlag, Berlin, 1994, MR 96h:35003b.
\end{thebibliography}

\end{document}
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