\section{Introduction}
Since their origin in the early $20$th century, the study of Hardy inequalities has been a vigorous field of research.
On the one hand, this stems from the great variety of applications Hardy inequalities have in analysis and mathematical physics, for example as uncertainty principles in quantum mechanics, \cite{Fr11}, for the solvability and growth control of differential equations \cite{AGG06} and in spectral graph theory \cite{Nag04}. On the other hand, they provide intriguing examples of functional inequalities where explicit sharp constants and asymptotics of minimizers or ground states can be studied. While most of the literature focuses on  the continuum setting of differential operators, the inequality was originally phrased and proven by Hardy in the discrete setting; see \cite{KMP06} for a review of the history.

Classically, the Hardy weight $ w $ is expressed as the  inverse of the distance function to some power which is for example given by $ w(x)=\frac{(d-2)^2}{4}|x|^{-2} $ for the most familiar setting in $ {\mathbb R}^{d} $, $ d\ge 3 $. This weight is optimal in a precise sense and in particular the constant $ \tfrac{(d-2)^2 }{4}$ is sharp. In 2014, Devyver, Fraas, and Pinchover \cite{DFP} (see also \cite{DP}) presented a method---the so-called supersolution construction---to construct \emph{optimal} Hardy weights for general (not necessarily symmetric) positive second-order elliptic operators   on non-compact Riemannian manifolds. This method was later extended in \cite{KePiPo2} to weighted graphs where it was also shown that for the standard Laplacian on $ {\mathbb Z}^{d} $, $ d\ge 3 $, there is an optimal Hardy weight $ w $ which asymptotically satisfies the continuum asymptotics $ w(x)\sim \frac{(d-2)^2}{4}|x|^{-2}  $, improving the constant previously obtained by Kapitanski and Laptev \cite{KaL}. While the method using super solutions such as the Green's function to obtain Hardy inequalities and the ground state transform  seem  to be folklore, see e.g. \cite{Fi00,FS08,FSW08,Gol14}, the novelity of the approach lies in the use of solutions with specific properties and the proof of optimality. For further recent work on optimal decay of Hardy weights in the continuum, see \cite{BGGP,DPP,KP,PV}.

\subsection{Summary of main results about optimal Hardy weights}
In this paper we establish inverse-square behavior at large distances of optimal Hardy weights for elliptic operators on $ {\mathbb Z}^{d} $ with $d\geq 3$. Specifically, we prove \textit{inverse-square bounds on optimal Hardy weights} in the following settings.
\begin{itemize}
	\item For general coefficients, upper bounds on annular averages and lower bounds on sectorial averages (Theorem~\ref{thm:mainspatial}).
	\item For ergodic coefficients satisfying a logarithmic Sobolev inequality, pointwise almost-sure upper bounds (Theorem~\ref{thm:mainrandom}).
	\item For weakly random i.i.d.\ coefficients, pointwise lower bounds in expectation (Theorem \ref{thm:mainrandompert}).
\end{itemize}

An upper bound on an optimal Hardy weight informs on what is the best possible expected behavior, while a lower bound corresponds to a concrete Hardy inequality that is useful in applications.

We comment on the fact that all of the bounds we prove here involve some kind of averaging or exclude a probability-zero set. One may naively hope for stronger purely deterministic bounds without averaging which are uniform in the underlying coefficients (or rather, only depend on their ellipticity ratio).  However, such bounds are not to be expected because optimal Hardy weights are closely related to Green's function derivatives and from elliptic regularity theory \cite{A1,A2,deGiorgi,GT,LSW63,Moser,Nash} it is well-known that pointwise bounds of the expected scaling cannot hold with constants that are uniform in the coefficients;  see Section \ref{sect:ERT}. Here, we address this problem by proving suitably weakened versions of inverse square behavior.

Our main focus lies on the treatment of the discrete case, since it is more interesting from a technical point of view. Analogous results in the continuum case do not seem to have appeared previously but can be derived in a parallel manner; see e.g.\ Section \ref{sect:continuum}.

\subsection{Green's function estimates and asymptotics}
On a technical level, we leverage and extend techniques developed in the wider context of homogenization theory for controlling Green's functions. This is a wide field; see \cite{AKM,BG,BGO1,BGO2,B,C,CKS,CN,DD,DGL,DGO,GNO,GO,KL,MO1,MO2} and references therein. Specifically, we draw on upper bounds proved by Marahrens-Otto \cite{MO1,MO2} and add some new results on lower bounds and asymptotics of Green's functions and their derivatives. More precisely:

\begin{itemize}
\item[(a)] We provide a general argument for lower bounds on sectorial averages (with the expected $|x|^{1-d}$-scaling) of Green's function derivatives, cf.\ Lemma \ref{lm:onedirection}. This is a simple one-page argument using only the Aronson-type bounds \eqref{eq:aronsondiscrete}, the sectorial geometry, and the Cauchy-Schwarz inequality.
\item[(b)] We provide a deterministic asymptotic expansion of the averaged (or ``annealed'') Green's function for weakly random i.i.d.\ coefficients, i.e., at ``small ellipticity contrast'' (Theorem \ref{thm:GFasymptotic}). To our knowledge, this is the first asymptotic formula of this kind. It draws on a recent novel approach to stochastic homogenization by Bourgain \cite{B}, which gives a precise description of the averaged Green's function, in the nearly sharp rendition from \cite{KL}. We combine these results with the Fourier-space techniques of Uchiyama \cite{Uch98}.
\item[(c)] Theorem \ref{thm:GFasymptotic} implies an analogous asymptotic expansion for Green's function derivatives, up to third order in $d=3$ and up to order $d+1$ for $d\geq 4$ (Corollary~\ref{cor:nablaGFasymptotic}). These bounds may become useful more widely, since especially lower bounds on Green's function derivatives are usually hard to come by. In fact, upper bounds on higher derivatives of order $>2$ were only proved recently \cite[Corollary~1.5]{KL}, also at small ellipticity contrast. For upper bounds on derivatives of the averaged Green's function up to second-order valid in more general contexts, see \cite{CGO,CN,DD}
\item[(d)] We can combine Theorem~\ref{thm:GFasymptotic} with concentration results from \cite{MO1} to obtain the surprisingly universal formula \eqref{eq:randomasymptoticinformal} for the long-distance asymptotic of the random Green's function at small ellipticity contrast (Corollary~\ref{cor:randomGFasymptotic}). Statements of a similar flavor in related settings can be found in \cite{BG,BGO1,BGO2}.
\end{itemize}

\subsection{Organization of the paper}
The paper is organized as follows.
\begin{itemize}

\item In \textbf{Section \ref{sect:prelim}}, we recall the elliptic graph setting, review the definition of optimal Hardy weights and their graph-intrinsic supersolution construction \cite{KePiPo2}.

\item In \textbf{Sections \ref{sect:main1}} and \textbf{\ref{sect:main2}}, we state our main results on optimal Hardy weights. The former section contains the determistic results with annular averages and the latter section contains the results for elliptic operators with ergodic coefficients.

\item In \textbf{Section \ref{sect:GF}}, we present some novel facts about discrete Green's functions. Here, they are mainly used to control optimal Hardy weights but they may turn out to be interesting in their own right.

\item The brief \textbf{Section \ref{sect:general}} contains rather general upper and lower bounds relating optimal Hardy weights and Green's functions on any elliptic graph.

\item In \textbf{Section \ref{sect:pfmainspatial}},  we prove the main results of \textbf{Sections \ref{sect:main1}} with spatial averaging.
\item In \textbf{Section \ref{sect:pfmainrandom}}, we prove the main results of \textbf{Sections \ref{sect:main2}} and \textbf{\ref{sect:GF}} which are probabilistic.
\item In the short \textbf{Section \ref{sect:rellich}}, we apply the results to Rellich inequalities on elliptic graphs. Specifically, we use the techniques of \cite{KePiPo4} to derive the expected $|x|^{-4}$ scaling in a probabilistic sense.

\item The \textbf{Appendix} contains the proof of a non-standard but elementary fact about the free Green's function (it is nowhere locally constant) and a self-contained derivation of the leading term in Theorem \ref{thm:GFasymptotic} based on \cite[Theorem 1.1]{KL} and elementary tools from harmonic analysis.
\end{itemize}

Two open questions are discussed in Remarks \ref{rmk:T} and \ref{rmk:open}.

\section{Preliminaries}\label{sect:prelim}
\subsection{The elliptic graph setting}
Let $ X $ be a discrete countable set and let $b $ be a  \emph{graph} over $ X $. That is $ b:X\times X\to[0,\infty) $ is a symmetric map with zero diagonal. We denote $ x\sim y $
whenever $ b(x,y)>0 $. The graph  $ b $ is assumed  to be \emph{connected} that is if for any to vertices $ x,y\in X $ there is a path $ x=x_{0}\sim \ldots\sim x_{n}=y $.
We denote
by $ \deg $ the \emph{weighted vertex degree}
\begin{align*}
\deg(x)=\sum_{y\in X}b(x,y).
\end{align*}

We assume that the graph satisfies the following ellipticity condition for some $ E>0 $
\begin{align}\label{E}
 b(x,y)\ge E\sum_{z\in X}b(x,z)
\end{align}
for all $ x\sim y $.
It is not hard to see that \eqref{E} implies that the graph has bounded combinatorial degree, i.e.,
\begin{align*}
	 \sup_{x\in X}\#\{y\in X\mid b(x,y)>0\} <\infty.
\end{align*}
 Hence, $ \deg $ takes  finite values and the graph is locally finite.

Let $ {L} $ be the operator acting on $ C(X) $, the real valued functions on $ X $, via
\begin{equation}\label{eq:Ldefn}
{L} f(x)=\sum_{\substack{y\in X}}b(x,y)(f(x)-f(y)).
\end{equation}
Restricting $ {L} $ to the compactly supported functions $ C_{c}(X) $, we obtain a symmetric operator on $ \ell^{2}(X) $ whose Friedrich extension  is with slight abuse of notation also  denoted by $ L $ with domain $ D(L) $.
Note that $ L $ is bounded if and only if $ \deg $ is bounded and in this case $ D(L)=\ell^{2}(X) $, see e.g. \cite[Theorem~9.3]{HKLW}. The associated quadratic form
$$
{Q}(f)=  \sum_{x, y\in X}b(x,y)(f(x)-f(y))^{2}
$$
is the Dirichlet energy of $f\in C(X)$ and for $ f\in D(L) $ we have
\begin{align*}
{Q}(f)=2\langle f,Lf\rangle.
\end{align*}
Denote by $ G: X\times X\to[0, \infty] $ the Green function of $L$ which is given by
\begin{align*}
G(x,y)=\lim_{{\alpha}\searrow 0}(L+{\alpha})^{-1}1_{x}(y)
\end{align*}
where $ 1_{x} $ is the characteristic function of the vertex $ x $.
In case $ G(x,y) $ is finite for some $ x,y\in X $ it is finite for all $ x,y \in X $ due to connectedness of the graph. In this case the graph is called \emph{transient}. We say the Green's function is \emph{proper} if $ G(o,\cdot) $ is a proper function for some vertex $ o\in X $. (We recall that a function is proper if the preimages of compact sets are compact.)

In the following, whenever we consider elliptic operators on $ {\mathbb Z}^{d} $, we will always have that $ G(o,\cdot) $ is proper. Also, we will often denote $$ G(x)=G(o,x) $$
and choose $ o =0$ whenever $ X={\mathbb Z}^{d} $.
\subsection{Optimal Hardy weights}
In this paper, we are interested in the large-distance behavior of optimal Hardy weights.

We say $ w:X\to [0,\infty) $ is a \emph{Hardy weight} for the graph $ b $ if all $ f\in C_{c}(X)$ satisfy the \emph{Hardy inequality} with weight $w$,
\begin{equation}\label{eq:hardy}
\begin{aligned}
{Q}(f)=\sum_{x, y\in X}b(x,y)(f(x)-f(y))^{2}\ge \sum_{x\in X}w(x)f(x)^{2}
\end{aligned}
\end{equation}
This inequality can be extended to the extended Dirichlet space, i.e., the closure with respect to $ {Q}^{1/2} $ which becomes a norm as we assumed that the graph is transient, see e.g. \cite{FOT} for general Dirichlet forms or \cite[Theorem~B.2]{KLSW} in the graph case. In particular, the constant function $ 1 $ is not in the extended Dirichlet space of transient Dirichlet forms. If $ \deg $ is bounded, then $ L $ is bounded and $ \ell^{2}(X) $ is included in the extended Dirichlet space, see \cite{HKLW}.

A \emph{ground state} for an operator $ H $ is a positive non-trivial solution $ u\in C(X) $, $ u\ge 0 $ to $ Hu=0 $ such that for any other super-solution $ v \ge 0$ to $ Hv\ge0 $ outside of a compact set with $ u\ge v\ge 0 $  there exists $ C\ge0 $ such that $ v\ge Cu $. The ground state of ${L}$ always exists in our situation of locally finite graphs by the virtue of the Allegretto-Piepenbrink theorem, \cite[Theorem~3.1]{HK}.

The following definition of optimal Hardy weights was first made by Devyver, Fraas, and Pinchover \cite{DFP}, see also \cite{DP}, for elliptic operators in the continuum and it was later investigated in \cite{KePiPo2} for graphs.

\begin{definition}\label{defn:optimal}
We say that $w$ is an \textit{optimal Hardy weight} if it satisfies the following three conditions.
\begin{itemize}
	\item[(a)]	\textit{Criticality}: For every $\tilde w\geq w$ but $\tilde w\neq w$, the Hardy inequality fails.
	\item[(b)] \textit{Null-criticality:} The formal Schr{\"o}dinger operator ${L}-w$ does not have a ground state in $\ell^2(X)$.
	\item[(c)] \textit{Optimality near infinity:} For any $\lambda>0$ and any compact set $K$, the function $(1+\lambda)w$ fails to be a Hardy weight for compactly supported functions in $X\setminus K$.
	 \end{itemize}
\end{definition}

We note that (a) and (b) imply (c) as can be seen from the proof in \cite{KePiPo2} and it was shown in the continuum in \cite{KP}. Another answer to how large  Hardy weight can be is given in \cite{KP} in the continuum case in terms of integrability in at infinity.

\subsection{Connection to the Green's function}
The results of \cite{KePiPo2} (building on \cite{DFP,DP}) derived optimal Hardy weights in an intrinsic way via superharmonic functions. More precisely, by \cite[Theorem~0.1]{KePiPo2}, the function $ w:X\to[0,\infty) $ defined by
	\begin{align*}
	w(x)=2\frac{{L} u^{\frac{1}{2}}(x)}{u^{\frac{1}{2}}(x)}
	\end{align*}
	is an optimal Hardy weight whenever $ u $ is a proper positive superharmonic function which is harmonic outside of a finite set and satisfies
	the growth condition $ \sup_{x\sim y} u(x)/u(y)<\infty$. Now, $ G(o,\cdot)=G(\cdot) $ is a  positive superharmonic function which  is and harmonic outside of $ \{o\} $. Then, \cite[Theorem~0.2]{KePiPo2} implies that
	\begin{align}\label{eq:wG}
	w_G(x)=2\frac{{L} G(x)^{\frac{1}{2}}}{G(x)^{\frac{1}{2}}}=1_{o}(x)+\frac{1}{G(x)}
	\sum_{\substack{y\in X: \\y\sim x}}b(x,y)(G(x)^{\frac{1}{2}}-G(y)^{\frac{1}{2}})^{2}
	\end{align}
	is an optimal Hardy weight. The condition  $ \sup_{x\sim y} G(x)/G(y)<\infty$ is satisfied by  \eqref{E}, see Lemma~\ref{lm:basic} below, and, therefore,  $ w_{G} $ is optimal in the sense of Definition~\ref{defn:optimal} whenever $ G $ is proper.

We will make use of the facts discussed in this section as follows:

\begin{enumerate}
	\item We can use bounds on the Green's function and its derivatives to obtain upper and lower bounds on $w_G$ via \eqref{eq:wG}.
	\item We can use the optimality of the Hardy weight $w_G$ to conclude asymptotic bounds on all possible Hardy weights $w$.
\end{enumerate}

In previous works \cite{KePiPo2,KePiPo3}, this has been implemented for graphs where the Green's function is, at least asymptotically, exactly computable. Here we show for the first time that the Hardy weight scaling of $|x|^{-2}$ is robust in various senses for a wide class of elliptic graphs defined on ${\mathbb Z}^d$ whose Green's functions are no longer explicitly computable.

\subsection{From upper bounds on $w_G$ to asymptotic bounds on all Hardy weights}
In this short section, we make precise Step (2) mentioned above, i.e., how to go from upper bounds on $w_G$ to asymptotic bounds on all Hardy weights. Note that the Hardy inequality \eqref{eq:hardy} is stronger if the weight $w$ is large, so we are interested in limiting how large a general Hardy weight can be, given a bound on the special optimal weight $w_G$.

Let $ d $ be some metric on $ X $ and $ \|x\|=d(x,o) $ for $ x\in X $ and $ o\in X $ which was used in the definition of $ G(x)=G(o,x) $. Below when we consider $ X={\mathbb Z}^{d} $ we will use the Euclidean norm $ |\cdot| $ for $ \|\cdot\| $
and $ o=0 $.

\begin{proposition}\label{prop:wGw} Assume that $b$ is a connected transient graph over $X$ which satisfies \eqref{E} and $ G $ is proper.
Assume that there exists a constant $C>0$ and a power $p>0$ such that
\begin{equation}\label{eq:wGrate}
w_G(x)\leq \frac{C}{1+\|x\|^{p}},\qquad  x\in X.
\end{equation}
Suppose a function $\tilde w:X\to (0,\infty)$ satisfies
$$
\tilde w(x)\geq (1+{\lambda})\frac{C}{1+\|x\|^p},\qquad  x\in X\setminus K,
$$
for some ${\lambda}>0$ and some compact set $K\subseteq X$. Then $\tilde w$ is not a Hardy weight.
\end{proposition}

In particular, Proposition \ref{prop:wGw} shows that the asymptotic decay rate of $w_G$ controls the asymptotics of all Hardy weights: Assuming we know that \eqref{eq:wGrate} holds, there cannot exist a Hardy weight $\tilde w$ satisfying $\tilde w(x)\gtrsim \|x\|^{{\varepsilon}-p}$ asymptotically (meaning outside of compacts), for any ${\varepsilon}>0$. Here, $\|x\|^{\varepsilon}$ can be replaced by any function that goes to infinity as $\|x\|\to\infty$, say $\log\log\|x\|$.

\begin{proof}[Proof of Proposition \ref{prop:wGw}]
By optimality at infinity of $w_G$, we know that or every ${\lambda}>0$ and every compact set $K$, there exists a function $f$ supported in $X\setminus K$ such that the Hardy inequality with weight $(1+{\lambda})w_G$ fails. By the assumed bound on $w_G$, this implies that the Hardy inequality also fails for any $\tilde w$ such that
$$
\tilde w(x)\geq \frac{C(1+{\lambda})}{\|x\|^{p}}\geq (1+{\lambda})w_G(x),\qquad \textnormal{ on } X\setminus K.
$$
This proves Proposition \ref{prop:wGw}.
\end{proof}

We remark that Proposition \ref{prop:wGw} only uses optimality near infinity of $w_G$, i.e., property (c) of Definition~\ref{defn:optimal}. Properties (a) and (b) also transfer information from optimal Hardy weights to general Hardy weights, but we focus on property (c) because there the mechanism is the most clear.

\section{Bounds on optimal Hardy weights with spatial averaging}\label{sect:main1}
In this section, we discuss our first class of main results, namely upper and lower bounds on the optimal Hardy weight $w_G$ for models on ${\mathbb Z}^d$, $d\geq 3$. These bounds translate to information on the best possible large-distance decay of any Hardy weight via Proposition~\ref{prop:wGw}.

\subsection{The benchmark: the free Laplacian on ${\mathbb Z}^d$}
We are interested in the long-distance behavior of the optimal Hardy weight $w_G$ from \eqref{eq:wG}. To understand this, we require asymptotic control on $G$ and $\nabla G$ (for a definition of $ \nabla G $ see the next subsection) which is made explicit Proposition \ref{prop:wGbounds} below.

The benchmark is the case of the free Laplacian $ \Delta $ on ${\mathbb Z}^d$, $d\geq 3$,
which is given via the standard edge weights $ b(x,y)=1 $ if $ |x-y|=1 $ and zero otherwise. The Green's function $ G_{0} $ of $ \Delta $ is asymptotically explicitly computable and satisfies  $G_0(x)\sim |x|^{2-d}$ and $|\nabla G_0(x)|\sim |x|^{1-d}$; see e.g.\ \cite{Spitzer,Uch98}, where $$  |x|=(|x_1|^{2}+\ldots+|x_d|^{2})^{1/2}  $$ denotes the Euclidian norm. Using asymptotic formulas, the optimal Hardy weight, including the sharp constant, is computed as
\begin{align}\label{eq:wG0}
w_{G_0}(x)={2}\frac{{\Delta}[G_0^{1/2}(x)]}{G_0^{1/2}(x)}=\frac{\big( d-2 \big)^2}{4} \,\frac{1}{|x|^2} + {\mathcal{O}}\Bigg( \frac{1}{|x|^3} \Bigg), \quad\,\, x\in {\mathbb Z}^d,
\end{align}
in \cite[Theorem~8.1]{KePiPo2}.
See \cite{KaL} for a similar estimate with a different constant.

Our goal in this paper is to show that the inverse square decay is robust, in various senses, within the class of elliptic graphs on ${\mathbb Z}^d$, whose Green's functions are no longer known to be explicitly computable.

\subsection{General elliptic operators on ${\mathbb Z}^d$}
\label{sect:ERT}
Let us introduce the setting of elliptic operators on ${\mathbb Z}^d$,  $d\geq 3$.  Let $\mathbb E^d$ denote the set of undirected edges $[x,y]$ with nearest neighbors $x,y\in {\mathbb Z}^d$, i.e. $ |x-y|=1 $. For  ${\lambda}\in (0,1]$, we define the class of uniformly elliptic coefficient fields
\begin{equation}\label{eq:setting1}
\mathcal{A}_{{\lambda}}:=\setof{a:\mathbb E^d\to {\mathbb R}}{{\lambda} \leq a\leq {\lambda}^{-1}}.
\end{equation}

We will study the following class of examples. For $a\in \mathcal{A}_{{\lambda}}$, we set
\begin{equation}\label{eq:setting2}
X={\mathbb Z}^d,
\qquad b(x,x+e_j)=b(x+e_j,x)=a([x,x+e_j])
\end{equation}
for all $ x\in{\mathbb Z}^{d} $ and the standard basis $ e_{j} $, $1\leq j\leq d$. One can easily verify that the graph defined by \eqref{eq:setting2} satisfies the ellipticity condition \eqref{E} for some number $E=E(d,{\lambda})>0$.
We define the associated operator ${L}$ via \eqref{eq:Ldefn} and also denote by $ L $ corresponding self-adjoint operator on $ \ell^{2}({\mathbb Z}^{d}) $. Equivalently, we may express this as $$ {L}=\nabla^* a \nabla $$ for appropriately defined $\nabla,\nabla^*$,  cf. \cite{MO1},
 i.e.,
\begin{align*}
	\nabla :C({\mathbb Z}^{d})\to C({\mathbb Z}^{d})^{d}, \quad
	\nabla f(x )=
	\left(\begin{matrix}
		f(x+e_{1})-f(x)\\
		\vdots\\
		f(x+e_{d})-f(x)
	\end{matrix}\right),
\end{align*}
$ a $ is identified with the operator induced by the  $ d\times d $ diagonal matrix
\begin{align*}
	\left(\begin{matrix}
		a([\cdot,\cdot+e_{1}])&&0\\
	&	\ddots&\\
0	&&a([\cdot,\cdot +e_{d}])
	\end{matrix}\right)
\end{align*}
and $ \nabla ^{*} $ is the formal adjoint of $\nabla  $ which acts as
\begin{align*}
	\nabla^{*} :C({\mathbb Z}^{d})^{d}\to C({\mathbb Z}^{d}), \quad \nabla ^{*}
\left(\begin{matrix}
	f_1(x)\\
	\vdots \\
	f_d(x)
\end{matrix}\right)= \sum_{j=1}^{d}(f_j(x-e_{j} )-f_j(x)).
\end{align*}
With slight abuse of notation we also write for $ f\in C({\mathbb Z}^{d}) $ an edge $ e=[x,x+e_{j}] $
\begin{align*}
	\nabla f(e)=f(x)-f(x+e_{j})
\end{align*}

In order to correctly estimate the optimal Hardy weights of these graphs defined by \eqref{eq:wG}, the ideal situation would be to  have upper and lower bounds on their Green's functions which agree with the behavior $G_0(x)\sim |x|^{2-d}$ and $|\nabla G_0(x)|\sim |x|^{1-d}$ of the free Green's function $G_0$ that we reviewed above. Deriving such bounds on $\nabla G$ and its continuum analog are old problems in potential theory and probability theory with close connections to the elliptic regularity theory developed by de Giorgi, Nash, and Moser in the 1950s \cite{deGiorgi,Moser,Nash}.

On the one hand, the situation for $G$ itself is relatively pleasant. Based on Nash's parabolic comparison estimates to the free Laplacian, Littman, Stampacchia, and Weinberger \cite{LSW63} and Aronson \cite{A1,A2} proved that the $|x|^{2-d}$-decay is preserved uniformly in the class $\curly{A}_{{\lambda}}$, i.e., there exists a constant $C_{d,{\lambda}}>1$ such that
\begin{equation}\label{eq:aronson}
C^{-1}_{d,{\lambda}} |x-y|^{2-d}\leq G(x,y)\leq C_{d,{\lambda}} |x-y|^{2-d},\qquad  x, y\in {\mathbb R}^d, x\neq y.
\end{equation}
We emphasize that $C_{d,{\lambda}}$ depends on the coefficient field $a$ only through the ellipticity parameter ${\lambda}$. To be precise, \eqref{eq:aronson} was proved in the continuum setting. For the extension to the discrete case, see \cite{DD,SC} and references therein. In the discrete case, we may use the fact that the Green's function is bounded on the diagonal to write
\begin{equation}\label{eq:aronsondiscrete}
C^{-1}_{d,{\lambda}} (1+|x-y|)^{2-d}\leq G(x,y)\leq C_{d,{\lambda}} (1+|x-y|)^{2-d},\qquad  x,y\in{\mathbb Z}^d.
\end{equation}
We remark that the bounds in \eqref{eq:aronson} and \eqref{eq:aronsondiscrete} establish that the Green's function is proper. We capture this fact because of its relevance in the following lemma.

\begin{lemma}Whenever, $ a\in \mathcal{A}_{\lambda} $ for $ \lambda \in(0,1]$, the  Green's function $G(y,\cdot)$ is proper for any fixed $y\in {\mathbb Z}^d$.
\end{lemma}

On the other hand, bounds for $\nabla G$ are a more delicate matter. In fact, within the general elliptic framework as defined above, there are counterexamples that show one cannot expect a pointwise upper bound $|\nabla G(x)|\leq C|x|^{1-d}$ to hold with an $a$-uniform constant $C$. However, it turns out that derivative bounds for general elliptic equations (discrete and continuum) are much more subtle. Indeed, the standard elliptic regularity theory \cite{deGiorgi,Nash,Moser} only yields
$
|\nabla G(x)|\leq C(d,{\lambda}) |x|^{2-d-{\alpha}(d,{\lambda})}
$
for some ${\alpha}(d,{\lambda})>0$, cf.\ \cite{MO1,MO2}. Moreover, it is well-known that in the continuum situation, an $a$-uniform bound of the form $|\nabla G(x,y)|\lesssim |x-y|^{1-d}$ cannot hold; explicit counterexamples can be constructed using the theory of quasi-conformal mappings; see p.\ 795 in \cite{GO} and p.\ 299 in \cite{GT}.

Still, there is hope to prove weaker (for example average-case) bounds on $\nabla G$ of the right order and making this precise is one of the main applications of homogenization theory \cite{AKM,B,C,CKS,CN,DD,DGL,DGO,GNO,GO,KL,MO1,MO2}. The main idea of our paper is to build on and in some cases extend the Green's function estimates connected to homogenization theory and to apply them to optimal Hardy weights.

\subsection{Upper and lower bounds for annular averages}
We follow the order laid out in the introduction and begin with the most general results which hold uniformly in the coefficient field $a\in \curly{A}_{{\lambda}}$. This comes at the price of a spatial average, either over a dyadic annulus (upper bound) or a dyadic sector (lower bound).

To define a discrete sector, we introduce the cone $\curly C_{j,{\delta}}\subseteq {\mathbb R}^d$ in direction $j\in \{1,\ldots,d\}$ with opening angle given in terms of ${\alpha}\in (0,1)$, i.e.,
$$
\curly C^{j, {\alpha}}=\setof{x\in {\mathbb R}^d}{\scp{x}{e_j}>(1-{\alpha})|x|}.
$$
We then define the associated discrete sector of radial size $\ell>0$ on scale $R>0$ by
$$
\curly S^{j,{\alpha}}_{R,\ell}=\setof{x\in{\mathbb Z}^d\cap \curly C_{j,{\alpha}}}{R\leq |x|\leq \ell R}.
$$

\begin{theorem}[Inverse-square  bounds on spatial averages]\label{thm:mainspatial}
Let $d\geq 3$ and ${\lambda}\in (0,1]$. There exist constants $c_{d,\lambda}>1$  and  $\ell=\ell_d>1$ such that for every $a\in\curly{A}_{{\lambda}}$ and every radius $R\geq 1$, $j\in \{1,\ldots,d\}$ and ${\alpha}\in (0,1)$,
\begin{align}
\label{eq:mainspatialub}
 R^{-d}\sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}} w_G(x)
 \leq& c_{d,{\lambda}} R^{-2}\\
 \label{eq:mainspatiallb}
R^{-d}\sum_{x\in \curly S^{j,{\alpha}}_{R,\ell}} w_G(x)
\geq& c_{d,{\lambda}}^{-1} R^{-2}.
 \end{align}

\end{theorem}

Note that the number of summands in both sums is of the order $R^d$, so up to a change in the constant $c_{d,{\lambda}}$, \eqref{eq:mainspatialub} and \eqref{eq:mainspatiallb} are indeed upper and lower bounds on normalized spatial averages. The proof is given in Section~\ref{sect:pfmainspatial}.

We can use Theorem \ref{thm:mainspatial} to obtain matching upper and lower bounds on annular averages.
\begin{corollary}[Inverse-square behavior of annular averages]
\label{cor:ublb}
Let $d\geq 3$ and ${\lambda}\in (0,1]$. There exist constants $c_{d,\lambda},\ell_d>1$ so that for all $a\in\curly{A}_{{\lambda}}$ and all $R\geq 1$,
\begin{equation}
 \frac{1}{c_{d,{\lambda}}} R^{-2}
 \leq
  R^{-d}\sum_{\substack{x \in {\mathbb Z}^d:\\ R\leq |x|\leq \ell_d R}} w_G(x)\leq c_{d,{\lambda}} R^{-2}.
\end{equation}
\end{corollary}

Corollary \ref{cor:ublb} says that the inverse square behavior of the optimal Hardy weight holds for completely general elliptic coefficients on ${\mathbb Z}^d$ after annular averaging.

\begin{proof}[Proof of Corollary \ref{cor:ublb}]
The lower bound follows from \eqref{eq:mainspatiallb} and $\curly S^{j,{\alpha}}_{R,\ell}\subseteq \{x:\, R\leq|x|\leq \ell_d R\}$.
The upper bound follows by covering the annulus $\{x:\, R\leq|x|\leq  \ell_d R\}$ with finitely many dyadic annuli and using \eqref{eq:mainspatialub} on each dyadic annulus.
\end{proof}

\subsection{The continuum case}\label{sect:continuum}
While we focus on the discrete case, we briefly mention that our statements have continuum analogs which have not
appeared in the literature. Optimal Hardy weights were originally considered in the continuum in work of Devyver, Fraas, and Pinchover, \cite{DFP,DP}. The notion of optimality is analogous to Definition \ref{defn:optimal}; see Definition 2.1 in \cite{DFP}.

We consider an elliptic, divergence form operator
$$
P=\mathrm{div}(A\nabla)
$$
on ${\mathbb R}^d$ with a scalar function $A$ belonging to
$$
\curly{A}^{cont}_{\lambda}=\setof{\tilde A:{\mathbb R}^d\to{\mathbb R}}{ \textnormal{ H\"older continuous, }{\lambda}\leq \tilde A\leq 1 \textnormal{ a.e.}}
$$
 for some ${\lambda}>0$. Moreover, \cite{DFP} mention that the assumption of H\"older continuity of $\tilde A$ can be relaxed to any assumption guaranteeing standard elliptic regularity theory, which in addition to ${\lambda}\leq \tilde A\leq 1$ means just measurability \cite{deGiorgi,Moser,Nash}

  A Hardy weight is defined as a non-zero function $W:{\mathbb R}^d\setminus\{0\}\to[0,\infty)$ verifying
$$
\scp{\varphi}{P\varphi}\geq \int_{{\mathbb R}^d\setminus\{0\}} W(x) |\varphi(x)|^2 \mathrm{d} x,\qquad  \varphi \in C_0^\infty({\mathbb R}^d\setminus\{0\}).
$$
According to Theorem 2.2 in \cite{DFP}, an optimal Hardy weight can be defined in terms of the minimal positive Green's function with a pole at the origin, $G$, in the following way
\begin{equation}\label{eq:WG}
W_G(x)=\l|\nabla\log \sqrt{G(x)}\r|_A^2=\frac{|\nabla G(x)|_A^2}{4G(x)^2},\qquad \text{where } |\xi|_A^2=\xi\cdot A\xi.
\end{equation}
One can prove analogous bounds on this continuum optimal Hardy weight. For the sake of brevity, we only state one result that establishes the inverse-square behavior with upper and lower bounds on annular averages, namely a continuum analog of Corollary \ref{cor:ublb}. For readers interested in continuum analogs of the later results in the perturbative probabilistic setting of Section \ref{sect:perturbativesetup}, we note that an extension of the results in \cite{KL} to the continuum setting is in preparation in \cite{DLP}.

\begin{proposition}[Inverse square behavior of annular averages --- continuum version]\label{prop:continuum}
Let $d\geq 3$ and ${\lambda}\in(0,1]$. There exists constants $c_{d,{\lambda}},\ell_d>1$ so that for every $A\in \curly{A}^{cont}_{\lambda}$ and every $R\geq 1$, it holds that
\begin{equation}
 \frac{1}{c_{d,{\lambda}}} R^{-2}
 \leq
  R^{-d}\int\limits_{\substack{x\in{\mathbb R}^d\setminus\{0\}:\\
  R\leq |x|\leq \ell_d R}} W_G(x) \mathrm{d} x \leq c_{d,{\lambda}} R^{-2}
\end{equation}
\end{proposition}

The proof is a straightforward continuum analog of the proof of Theorem \ref{thm:mainspatial}; see Section \ref{sect:continuumpf}.

\section{Bounds on optimal Hardy weights for ergodic coefficients}\label{sect:main2}
\subsection{Upper bound for ergodic coefficients}
We can remove the need for spatial averaging and prove pointwise bounds for elliptic operator with ergodic coefficients. Note that due to the random environment, the Green's function is not exactly computable in this setting.
We now describe the precise setup. Let $\mathbb P$ be a probability measure on the measure space $\curly{A}_{{\lambda}}$ endowed with the product topology. We write $\qmexp{\cdot}$ for the associated expectation value.

\begin{assumption}\label{ass:P}
We make the following two assumptions on $\mathbb P$.
\begin{enumerate}
\item $\mathbb P$ is stationary, i.e., for any $z\in{\mathbb Z}^d$, the translated coefficient field $a(\cdot+z)$ is also distributed according to $\mathbb P$.
\item $\mathbb P$ satisfies a logarithmic Sobolev inequality with constant $\rho>0$, i.e., for any random variable $\zeta:\curly{A}_{{\lambda}}\to {\mathbb R}$, it holds that
\begin{equation}\label{eq:LSI}
\qmexp{\zeta^2 \log\frac{\zeta^2}{\qmexp{\zeta^2}}}\leq \frac{1}{2\rho}\qmexp{\sum_{e\in \mathbb E^d} \l(\mathrm{osc}_{e}\zeta\r)^2},
\end{equation}
where the oscillation of $\zeta$ is a new random variable defined by
$$
\l(\mathrm{osc}_{e}\zeta\r)(a)=\sup_{\substack{\tilde a\in \curly{A}_{{\lambda}}:\\ \tilde a=a \mbox{\scriptsize { on }}\mathbb{E}^{d}\setminus\{e\}}} \zeta(\tilde a)-\inf_{\substack{\tilde a\in \curly{A}_{{\lambda}}:\\  \tilde a=a \mbox{\scriptsize { on }}\mathbb{E}^{d}\setminus\{e\}}} \zeta(\tilde a).
$$
\end{enumerate}
\end{assumption}

In a nutshell, Assumption \ref{ass:P} specifies that the probability measure $\mathbb P$ is stationary and ``sufficiently'' ergodic. Indeed, the logarithmic Sobolev inequality implies a spectral gap and thus mixing time estimates for the associated Glauber dynamics \cite{GNO}. The idea that Assumption \ref{ass:P} is useful in the context of Green's function estimates in stochastic homogenization goes back to an unpublished manuscript of Naddaf-Spencer. We quote a well-known result which immediately provides lots of examples satisfying this assumption.

\begin{lemma}[The case of i.i.d.\ coefficients; see e.g.\ Lemma 1 in \cite{MO1}]
\label{lm:iid}
If the coefficients $\l(a(e)\r)_{e\in \mathbb E^d}$ are chosen in an independent and identically distributed (i.i.d.) way with values in $[{\lambda},1]$, then the induced measure $\mathbb P$ on $\curly{A}_{{\lambda}}$ satisfies Assumption~\ref{ass:P} (with $\rho=1/8$).
\end{lemma}

\begin{remark}
The logarithmic Sobolev inequality \eqref{eq:LSI} is a slightly weaker version of the more standard one in which the oscillation is replaced by the partial derivative $\frac{\partial \zeta}{\partial a(e)}$. The version \eqref{eq:LSI}  has the slight advantage that the above lemma holds for the most general choices of i.i.d.\ coefficient fields \cite{MO1}.
\end{remark}

We are now ready to state the main result in the probabilistic setting, which provides a pointwise upper bound on the random function $w_G:\Omega\times{\mathbb Z}^d\to{\mathbb R}_+$ with probability $1$. Here $\Omega$ denotes the underlying probability space. We warn the reader that we will suppress the $\omega$-dependence of $w_G$ on occasion.

\begin{theorem}[Pointwise upper bound]\label{thm:mainrandom}
Let $d\geq 3$, and suppose Assumption \ref{ass:P} holds. For every $p\geq 1$, there exists $C_{d,{\lambda},p}>0$ so that
\begin{equation}\label{eq:wGannealed}
\qmexp{w_G(x)^p}\leq C_{d,{\lambda},p} (1+|x|)^{-2p}, \qquad x\in{\mathbb Z}^d.
\end{equation}
Moreover, for every ${\varepsilon}>0$, there exists  finite sets $K_{\varepsilon}(\omega)\subseteq {\mathbb Z}^{d}$ so that
\begin{equation}\label{eq:wGprob1}
w_G(\omega,x)\leq (1+|x|)^{-2+{\varepsilon}},\qquad  \omega\in \Omega,\, x\in{\mathbb Z}^d\setminus K_{{\varepsilon}}(\omega)
\end{equation}
holds with probability $1$.
\end{theorem}

The first estimate \eqref{eq:wGannealed} is a pointwise estimate for the ``annealed'' Hardy weight $\qmexp{w_G}$. The second part of the statement holds pointwise with probability $1$, at the modest price of an ${\varepsilon}$-loss in the exponent. The proof is given in Section~\ref{sect:pfmainrandom}.\medskip

Since $w_G$ is an optimal Hardy weight, this theorem yields information on the best-possible decay of an arbitrary Hardy weight via Proposition~\ref{prop:wGw}. For instance, we have the following corollary.

\begin{corollary}[Pointwise decay of optimal Hardy weights]\label{cor}
Let $d\geq 3$ and suppose Assumption \ref{ass:P} holds. Let $p>0$ and let $w:\Omega\times {\mathbb Z}^d\to{\mathbb R}_+$ be a function so that for some $C(\omega)>0$,
\begin{equation}\label{eq:cor}
w(\omega,x)\geq C(\omega) (1+|x|)^{-p},\qquad  x\in{\mathbb Z}^d
\end{equation}
holds with positive probability. If $w$ is a Hardy weight, then  $p\geq 2$.
\end{corollary}

\begin{proof}
We prove the contrapositive statement. Suppose that \eqref{eq:cor} holds for some $p<2$ with positive probability. We have
$$
\mathbb P\l(\inf_{x\in{\mathbb Z}^d}w(x)(1+|x|)^p>0\r)>0
$$
Let ${\varepsilon}_0=\frac{2-p}{2}>0$ and apply \eqref{eq:wGprob1} of Theorem \ref{thm:mainrandom} to obtain
$$
\mathbb P\l(\inf_{x\in{\mathbb Z}^d}w_G(x)(1+|x|)^{2-{\varepsilon}_0}<\infty\r)=1
$$
On the positive-probability event where both estimates hold,  one can use $p<2-{\varepsilon}_0$ and apply Proposition \ref{prop:wGw} with $K$ taken to be an appropriate ball to conclude that $w$ is not a Hardy weight. This proves Corollary \ref{cor}.
\end{proof}

\subsection{The perturbative setting of small randomness}\label{sect:perturbativesetup}
While Theorem \ref{thm:mainrandom} gives a pointwise upper bound, the fact that the constant in \eqref{eq:wGprob1} depends on the random realization $\omega$ and that there is no matching lower bound leave something to be desired. In this last main result, we show that both of these restrictions can be remedied in a clean way in the perturbative setting of weak i.i.d.\ randomness (also known as small ellipticity contrast). For this, we build on a recent breakthrough of Bourgain \cite{B}, subsequently refined in \cite{KL}, which gives a precise description of the averaged Green's function; see also Section \ref{sect:GFasymptotics}.

We begin by reviewing the setup and main results of \cite{B,KL}.
We consider a family of  mean-zero i.i.d.\ random variables $\{\omega_x\}_{x\in{\mathbb Z}^d}$ which are taking values in $ [-1,1] $. We let ${\delta}\in (0,1)$ and define an elliptic operator through \eqref{eq:setting2} with
\begin{equation}\label{eq:pertcoefficients}
a([x,x+e_j])=1+{\delta} \omega_x,\qquad  j\in\{1,\ldots,d\}.
\end{equation}
Note that $a\in \curly{A}_{1-{\delta}}$ is elliptic. In  \cite{KL},  which this section is based on, the model is studied also for non-isotropic $ a $.

In \cite{B}, it is shown that there exists $c_d>0$ so that for all ${\delta}\in (0,c_d)$, the averaged Green's function $\qmexp{G}$ arises itself as a Green's function of a translation-invariant operator $\curly{L}$, which is the harmonic mean of the original random operator. This ``parent operator'' has an explicit representation of the form
\begin{equation}\label{eq:harmonicmeanoperator}
\curly{L}={\Delta} +\nabla^* \mathbf{K}^{\delta} \nabla,
\end{equation}
where $ \Delta $ is the free Laplacian and  $\mathbf{K}^{\delta}$ is a convolution operator-valued $d\times d$ matrix whose components satisfy the key decay estimate \cite[Theorem 1.1]{KL}
\begin{equation}\label{eq:Kdeltadecay}
|K^{\delta}_{j,k}(x-y)|\leq C_d {\delta}^2 (1+|x-y|)^{-3d+1/2},\qquad  j,k\in \{1,\ldots,d\}.
\end{equation}
Equivalently, the operator $\curly{L}$ is a Fourier multiplier with the symbol $m: \mathbb T^d\to \mathbb C$, on the torus $\mathbb T^d=({\mathbb R}/2\pi {\mathbb Z})^d$, given by
\begin{equation}\label{eq:mthetadefn}
m(\theta)=2\sum_{j=1}^d (1-\cos\theta_j)+\sum_{1\leq j,k\leq d}
 (e^{-i\theta_j}-1)\widehat{K^{\delta}_{j,k}}(\theta) (e^{i\theta_k}-1).
\end{equation}
The decay bound \eqref{eq:Kdeltadecay} implies that $$ \widehat{K^{\delta}_{j,k}}\in C^{2d-1}(\mathbb T^d) $$ for all $j,k\in \{1,\ldots,d\}$ and this regularity will be useful in the following.

\subsection{Results in the perturbative setting}\label{sect:perturbativeresults}

The following theorem can be considered the matching lower bound to Theorem \ref{thm:mainrandom}.

\begin{theorem}[Pointwise lower bounds]\label{thm:mainrandompert}
\mbox{}
Let $d\geq 3$. There exists $c_d>0$ so that for all ${\delta}\in (0,c_d)$ and all $p\in (\frac{1}{2},\infty)$, there is a constant $C_{d,{\delta},p}>0$ so that the following holds.
\begin{enumerate}
\item[(i)] For $d\in \{3,4\}$, there is a radius $R_0>1$ so that
\begin{equation}\label{eq:mainrandompert1}
\langle w_G(x)^p\rangle \geq C_{d,{\delta},p} (1+|x|)^{-2p}, \qquad  |x|\geq R_0.
\end{equation}
\item[(ii)] For $d\geq 5$,
\begin{equation}\label{eq:mainrandompert2}
\langle w_G(x)^p\rangle \geq C_{d,{\delta},p} (1+|x|)^{-2p}, \qquad  x\in{\mathbb Z}^d.
\end{equation}
\end{enumerate}
\end{theorem}

Theorem \ref{thm:mainrandompert} is proved in Section \ref{sect:mainrandompertpf}. At the heart of the proof are novel Green's function asymptotics which are described in Section \ref{sect:GFasymptotics}.

We may combine Theorems \ref{thm:mainrandom} and \ref{thm:mainrandompert} to conclude that, e.g., for $p\geq 1$ and $d\geq 5$, there exist constants $C_{d,{\delta},p} ,C_{d,{\delta},p}' >0$ so that
\begin{equation}
C_{d,{\delta},p} (1+|x|)^{-2p}\leq \langle w_G(x)^p\rangle \leq C_{d,{\delta},p}' (1+|x|)^{-2p},\qquad  x\in{\mathbb Z}^d.
\end{equation}
These results thus confirm the inverse-square scaling of the optimal Hardy weight $w_G$ in an averaged sense.

\begin{remark}
Note that in dimensions $d\in \{3,4\}$ we have a lower bound at large distances $|x|\geq R_0$ only, while in dimensions $d\geq 5$ we have a stronger, global lower bound, which is positive everywhere on ${\mathbb Z}^d$. Indeed, in general the optimal Hardy weight $w_G:{\mathbb Z}^d\to [0,\infty)$ may have zeros, a possibility we can exclude for $d\geq 5$ in a probabilistic sense with this result.
\end{remark}

Since we now have control over various moments of the pointwise optimal Hardy weight $w_G$, the second moment method can be used to obtain pointwise lower bounds  with controlled positive probability.

\begin{corollary}[Pointwise lower bounds with positive probability]\label{cor:smm}
\mbox{}
Let $d\geq 3$. There exists $c_d>0$ so that for all ${\delta}\in (0,c_d)$, there are constants $c_{d,{\delta}},C_{d,{\delta}}>0$ so that the following holds.
\begin{enumerate}
\item[(i)] For $d\in \{3,4\}$, there is a radius $R_0>1$ so that
\begin{equation}\label{eq:corrandompert1}
\mathbb P\l(w_G(x)>C_{d,{\delta}} (1+|x|)^{-2}\r)\geq c_{d,{\delta}}>0, \qquad  |x|\geq R_0.
\end{equation}
\item[(ii)] Let  $d\geq 5$. Then
\begin{equation}\label{eq:corrandompert2}
\mathbb P\l(w_G(x)>C_{d,{\delta}} (1+|x|)^{-2}\r)\geq c_{d,{\delta}}>0,\qquad  x\in{\mathbb Z}^d.
\end{equation}
\end{enumerate}
\end{corollary}

\begin{proof}[Proof of Corollary \ref{cor:smm}] We use the Paley-Zygmund inequality which says that for all $\theta\in(0,1)$
$$
\mathbb P\big(w_G(x)>\theta \qmexp{w_G(x)}\big)\geq (1-\theta)^2 \frac{\qmexp{w_G(x)}^2}{\qmexp{w_G(x)^2}}.
$$
We apply this with $\theta=\frac{1}{2}$. Then we use Theorem \ref{thm:mainrandompert} with $p=1$ to obtain a lower bound on $\qmexp{w_G(x)}$ and Theorem \ref{thm:mainrandom} with $p=2$ to obtain an upper bound on $\qmexp{w_G(x)^2}$. This proves Corollary \ref{cor:smm}.
\end{proof}

\begin{remark}[Probabilistic interpretation of $\curly{L}$]
\label{rmk:T}
In the setting of \cite{B,KL} described above, one may ask whether the operator $\curly{L}$ from \eqref{eq:harmonicmeanoperator} is again the generator of a random walk on ${\mathbb Z}^d$. More precisely, one can write $m(\theta)=4d\l(1-\hat T(\theta)\r)$ with the function $T:{\mathbb Z}^d\to{\mathbb R}$ defined by
\begin{equation}\label{eq:Tdefn}
\begin{aligned}
T(x)=\frac{1}{2}{\delta}_{x=0}+\frac{1}{4d}{\delta}_{|x|=1}+\frac{1}{4d}\sum_{j,k=1}^d \big(&-K^{\delta}_{j,k}(x)+K^{\delta}_{j,k}(x-e_j)\\
&+K^{\delta}_{j,k}(x-e_k)-K^{\delta}_{j,k}(x-e_j-e_k)\big).
\end{aligned}
\end{equation}
So, a curious question is the following: Is $T(x)\geq 0$ for all $x\in{\mathbb Z}^d$? If this is the case, it can be interpreted as the transition function of a random walk with generator $\mathcal L$ (up to a multiplicative factor $4d$). This would mean that probabilistic averages of solutions are themselves governed by \textit{bona fide} diffusion process which may hold non-trivial information about the non-averaged processes. In our opinion, such a direct physical meaning of the operator $\mathcal L$ is not at all obvious and would be remarkable. We encountered this question when considering the Green's function asymptotics in Theorem \ref{thm:GFasymptotic}, which is a necessary tool for Theorem \ref{thm:mainrandompert} and would follow immediately from work of Uchiyama \cite{Uch98} if $T(x)\geq 0$. According to our initial investigation, identifying the conditions under which $T(x)\geq 0$ is true appears to be connected to subtle questions concerning componentwise positivity of matrix inverses \cite{JS}.
\end{remark}

\section{Green's function estimates}
\label{sect:GF}
Our investigations of optimal Hardy weights make use of existing upper bounds on Green's function gradients, specifically those of Marahrens-Otto \cite{MO1,MO2}, but they also require us to prove some new lower bounds on Green's function gradients. Since such bounds are largely absent from the literature, we believe they may be of independent interest to readers concerned with homogenization theory.

We recall our notation $G(x)=G(x,0)$.

\subsection{Lower bound on sectorial averages of Green's function derivatives}\label{sect:onedirection}
The following lemma gives a general lower bound on sectorial averages of Green's function derivatives for all elliptic coefficient fields. To this end, recall the definition of a sector
$ \curly S^{j,{\alpha}}_{R,\ell}=\setof{x\in{\mathbb Z}^d}{\langle x,e_{j}\rangle>(1-{\alpha})|x|,\;R\leq |x|\leq \ell R} $ for $ j\in\{1,\ldots,d\} $ and $ R,\ell >0 $,  $ {\alpha} \in(0,1)$.

\begin{lemma}[Lower bound on sectorial averages]\label{lm:onedirection}
Let $d\geq 3$. There exist constants $\ell=\ell_d>0$ and $C_{d,\lambda}>1$ such that for every $a\in\curly{A}_{{\lambda}}$, every $1\leq j\leq d$, every $ {\alpha}\in(0,1) $ and every radius $R\geq 1$,
$$
R^{-d}\sum_{x\in \curly S^{j,{\alpha}}_{R,\ell}} |G(x)-G(x+e_j)|^2 \geq  C_{d,{\lambda}} R^{2-2d}.
$$
\end{lemma}

The one-page proof of this lemma relies only on Aronson's pointwise bounds \eqref{eq:aronsondiscrete}, the sectorial geometry, and the Cauchy-Schwarz inequality; see Section~\ref{sect:onedirectionpf} for the details. Lemma \ref{lm:onedirection} is used to prove the lower bound in Theorem \ref{thm:mainspatial}.

\subsection{Green's function asymptotics}\label{sect:GFasymptotics}

In this section, we discuss the second class of main results about asymptotics of Green's functions and of optimal Hardy weights in a perturbative model with weak randomness previously studied in \cite{B,DGL,KL}.

By Taylor expansion of the Fourier multiplier \eqref{eq:mthetadefn} at the origin, we find that the lowest order is quadratic of the form $\scp{\theta}{\mathbf Q \theta}$ with the $d\times d$ Hessian matrix
\begin{equation}\label{eq:Qdefn}
\mathbf Q=\mathrm{Hess}(m)(0)=\mathbf{I}_d+\widehat{\mathbf{K}^\delta}(0).
\end{equation}

The matrix $\mathbf Q$ plays an important role for characterizing the leading-order behavior at large distances. In \cite{KL} a detailed analysis of $  \mathbf{K}^\delta$ is carried which yields that
the $d\times d$ matrix $\widehat{\mathbf{K}^\delta}(0)$ is symmetric\footnote{The symmetry can be seen from the power series representation in equation (1.14) of \cite{KL}. Indeed, the so called Helmholtz projection $\mathbf{K}={\nabla \Delta^{-1}\nabla^*}$ appearing therein has a kernel $\mathbf K(x-y)$ taking values in the real-valued $d\times d$ matrices and satisfying $\mathbf K^T(x)=\mathbf{K}(-x)$. Combining this with the invariance of the underlying probability distribution of $\{\omega_x\}_x$  under reflection  yields the symmetry.} and satisfies \eqref{eq:Kdeltadecay} which in turn implies $$ \|\widehat{\mathbf{K}^\delta}(0)\|\leq C_d{\delta}^2. $$
 This observation readily yields the following proposition which is used for our analysis below.

\begin{proposition}\label{prop:Ksymm}
The $d\times d$ matrix $\mathbf{Q}$ is symmetric and  $\mathbf{Q}\geq 1-C_d {\delta}^2$. In particular, $\mathbf{Q}$ is positive definite for sufficiently small ${\delta}>0$.
\end{proposition}

From now on we assume that $ \delta $ is sufficiently small such that $ \mathbf{Q} $ is positive definite.

We introduce the modified spatial variable
\begin{equation}\label{eq:tildexdefn}
\tilde x = \sigma \mathbf{Q}^{-1/2} x,\qquad \textnormal{with } \sigma=(\det\mathbf{Q})^{1/(2d)},
\end{equation}
and the universal constant
\begin{equation}\label{eq:kappadefn}
\kappa_d=\frac{1}{2}\pi^{-d/2}\Gamma(d/2-1).
\end{equation}

For $d\geq 3$, we denote
\begin{equation}\label{eq:mddefn}
m_d=
\begin{cases}
3,\qquad &\textnormal{if } d=3,\\
d+1,\qquad &\textnormal{if } d\geq 4.
\end{cases}
\end{equation}

\begin{thm}[Averaged Green's function asymptotics in the perturbative setting]\label{thm:GFasymptotic}
For $d\geq 3$, there exists $c_d>0$ so that for all ${\delta}\in (0,c_d)$ the following holds. There exist polynomials $U_1,\ldots,U_{m_d}$ with $U_k$ having degree at most $3k$ so that
\begin{equation}\label{eq:GFasymptotic}
\qmexp{G(x)}=\frac{\kappa_d}{\sigma^2} |\tilde x|^{2-d}
+\sum_{k=1}^{m_d} U_k\l(\frac{\tilde x}{|\tilde x|}\r) |\tilde x|^{2-d-k} +o(|\tilde x|^{2-d-m_d}), \qquad \textnormal{as } |x|\to\infty.
\end{equation}
\end{thm}

Recall that $\tilde x =\sigma\mathbf{Q}^{-1/2} x$. We have
\begin{equation}\label{eqn:xtilde}
C_{d,{\delta}}^{-1}|x|\leq |\tilde x|\leq C_{d,{\delta}}|x|
\end{equation}
 for an appropriate constant $C_{d,{\delta}}>1$ and $ \delta>0 $ small enough. The reason is that $\mathbf{Q}=\mathbf I_d +\widehat{\mathbf{K}^\delta}(0)$ is close to the identity by Proposition \ref{prop:Ksymm}. In particular, we have that $|x|\to\infty$ and $|\tilde x|\to\infty$ are equivalent and $o(|x|^{-k})=o(|\tilde x|^{-k})$.

The proof of Theorem \ref{thm:GFasymptotic} generalizes a delicate Fourier analysis developed by Uchiyama \cite{Uch98} in the probabilistic setting of random walks; see Section \ref{sect:GFasymptoticpf}. The key input is the decay estimate \eqref{eq:Kdeltadecay} which is the main result of \cite{KL}.

The polynomials $U_k$ are Fourier transforms of fractions of the form $\tfrac{P_{2d-2+k}(\xi)}{|\xi|^{2d}}$ where $P_{2d-2+k}$ is a homogeneous polynomial of degree $2d-2+k$. These polynomial can be explicitly computed as moments of the function $T$ defined in \eqref{eq:Tdefn}, cf.\  \cite{Uch98}. For instance, we have
$$
U_1(\omega)= \int_{\mathbb R^d} \frac{P_{2d-1}(\xi)}{|\xi|^{2d}} e^{-i\omega \cdot \xi}\mathrm{d} \xi,
\qquad
P_{2d-1}(\xi)
=-\frac{2i}{3\sigma^4 (2\pi)^d} |\xi|^{2d-4} \sum_{x\in{\mathbb Z}^d} T(x) (\xi\cdot x)^3.
$$
Here the Fourier transform is defined in the sense of tempered distributions on $\mathbb R^d\setminus\{0\}$ and the integral can be explicitly computed via Lemma 2.1 in \cite{Uch98}.

\begin{remark} For readers with a background in homogenization theory, we point out that the $U_k$ which depend on $T$ from \eqref{eq:Tdefn} are therefore computable from the operator $\mathbf{K}^\delta$ alone. This operator has several interesting properties some of which are still under investigation: (a) It fully characterizes the law of the probablity measure \cite[Proposition 1.8]{KL} and (b) its moments can be viewed as higher-order correctors in the context of homogenization theory with small ellipticity contrast \cite{DGL}. The latter this is related to the so-called Bourgain-Spencer conjecture \cite{D,DGL,DLP}.
\end{remark}

\begin{remark}\label{rmk:firstterm}
For readers interested in the underlying Fourier theory and the emergence of the matrix $\mathbf{Q}$, we give a short self-contained derivation of the lowest-order asymptotic order in Theorem \ref{thm:GFasymptotic} in Appendix \ref{sect:direct} based on \cite[Theorem 1.1]{KL}. The idea is to reduce it to the Green's function of the free Laplacian through Taylor expansion around the origin in Fourier space and then one uses basic harmonic analysis to control the error terms.
\end{remark}

\subsection{Consequences of Theorem \ref{thm:GFasymptotic}}

In the discrete setting, pointwise asymptotics yield pointwise asymptotics of derivatives of one order less. Moreover, this procedure can be iterated for higher derivatives, with a loss of one asymptotic order per derivative. Since our expansion \eqref{eq:GFasymptotic} has $m_d+1$ terms, we can describe asymptotics of the derivatives $\qmexp{\nabla^\alpha G}$ with $|\alpha|\leq m_d$ up to order $m_d-|\alpha|$. (Recall that $m_d$ defined in \eqref{eq:mddefn} is $d+1$ for $d\geq 4$.)

We recall the notation of the derivative $\nabla$ along edges given
$$
\nabla f([x,x+e_j]) = f(x+e_j)-f(x).
$$
We also note that by linearity $\qmexp{\nabla f}=\nabla \qmexp{f}$.

\begin{corollary}[Asymptotics of Green's function derivatives]\label{cor:nablaGFasymptotic}
Let $1\leq j\leq d$, $s\in\{ \pm 1\}$ and let $\alpha\in {\mathbb N}_0^d$ be a multi-index with $|{\alpha}|\leq m_d$. We make the assumptions as in Theorem \ref{thm:GFasymptotic}.
Then
\begin{equation}\label{eq:nablaalphaGFasymptotic}
\begin{aligned}
\qmexp{\nabla^\alpha G([x,x+s e_j])}=&\nabla^\alpha\l(\frac{\kappa_d}{\sigma^2} |\tilde x|^{2-d}
+\sum_{k=1}^{m_d-|\alpha|} U_k\l(\frac{\tilde x}{|\tilde x|}\r) |\tilde x|^{2-d-k}\r)\\
&+o(|\tilde x|^{2-d-m_d}), \qquad \textnormal{as } |x|\to\infty.
\end{aligned}
\end{equation}
\end{corollary}

A few comments on this are in order. Recall notation \eqref{eq:tildexdefn} and the fact that each $U_k$ is a polynomial. By the mean value theorem, we can bound the discrete derivative by the corresponding continuum derivative, cf.\ \cite[Lemma, p.~6]{Don}. Together with Proposition \ref{prop:Ksymm}, this implies
\begin{equation}\label{eq:Uorder}
\nabla^\alpha U_k\l(\frac{\tilde x}{|\tilde x|}\r) |\tilde x|^{2-d-k}=O(|\tilde x|^{2-d-k-|\alpha|}).
\end{equation}
In other words, the different summands appearing in \eqref{eq:nablaalphaGFasymptotic} belong to successively smaller powers of $|\tilde x|$. Therefore, as mentioned above, \eqref{eq:nablaalphaGFasymptotic} is indeed an asymptotic expansion comprising $m_d-|\alpha|$ orders. For $|{\alpha}|=m_d$, \eqref{eq:nablaalphaGFasymptotic} reduces to
$$
\qmexp{\nabla^\alpha G([x,x+s e_j])}=\frac{\kappa_d}{\sigma^2} \nabla^\alpha  |\tilde x|^{2-d}+o(|\tilde x|^{2-d-m_d}), \qquad \textnormal{as } |x|\to\infty.
$$
so it just manages to capture the leading order asymptotic of the $m_d$-th derivative.

The leading terms do not involve $U_k$ and are therefore particularly easy to compute. For example, we have the following leading-order gradient asymptotic
\begin{equation}\label{eq:nablaGFasymptotic}
\qmexp{\nabla G([x,x+s e_j])}=s \frac{2-d}{2}\frac{\kappa_d}{\sigma^2} |\tilde x|^{1-d} \frac{\scp{\tilde x_j}{\tilde e_j}}{|\tilde x|}
+{\mathcal{O}}(|\tilde x|^{-d}), \qquad \textnormal{as } |x|\to\infty.
\end{equation}
This particular result will be used to prove Theorem \ref{thm:mainrandompert} about Hardy weights.

As a second and final consequence of Theorem \ref{thm:GFasymptotic}, we use concentration bounds from \cite{MO1} to obtain surprisingly universal asymptotic information on the random Green's function $G$.

\begin{corollary}[Universal asymptotics of the random Green's function]\label{cor:randomGFasymptotic}
With the same assumptions as in Theorem \ref{thm:GFasymptotic}, let ${\varepsilon}>0$. There exist constants $C_{d,{\delta},{\varepsilon}},C_{d,{\delta},{\varepsilon}}'>0$ so that
\begin{align}
\label{eq:randomGFasymptotic}
\sup_{x\in{\mathbb Z}^d} \frac{\l|G(x)-\frac{\kappa_d}{\sigma^2} |\tilde x|^{2-d}\r|}{(1+|x|)^{1-d+{\varepsilon}}} &\leq C_{d,{\delta},{\varepsilon}}
\end{align}
 holds with probability $1$.
\end{corollary}

Stated informally, Corollary \ref{cor:randomGFasymptotic} says that, with probability $1$,
\begin{equation}\label{eq:randomasymptoticinformal}
\begin{aligned}
G(x)=-\frac{\kappa_d}{\sigma^2} |\tilde x|^{2-d}+{\mathcal{O}}(|x|^{1-d+{\varepsilon}}),\qquad \textnormal{as } |x|\to\infty.
\end{aligned}
\end{equation}
To our knowledge, Corollary \ref{cor:randomGFasymptotic} is the first result identifying such universal spatial asymptotic of a random Green's function.

Corollaries \ref{cor:nablaGFasymptotic} and \ref{cor:randomGFasymptotic}, as well as Formula \eqref{eq:nablaGFasymptotic}, are proved in Section \ref{sect:corproofs}.

\begin{remark}[An open question]
\label{rmk:open}
It would be very interesting to generalize Corollary \ref{cor:randomGFasymptotic} to describe the asymptotic of the gradient of the random Green's function $\nabla G([x,x+e_j])$. Such a result would have immediate consequences in the context of Hardy weights (by improving Theorem \ref{thm:mainrandompert} and Corollary \ref{cor:smm} on pointwise lower bounds), but also well beyond, because there are almost no workable lower bounds on random Green's function gradients in the literature as we mentioned in the discussion preceding Lemma~\ref{lm:onedirection}.

Since the averaged Green's function gradient is understood by Corollary \ref{cor:nablaGFasymptotic}, which as we mentioned can be easily refined, the main outstanding technical roadblock to such an extension is the lack of concentration bound \eqref{eq:GFconcentration} which is due to the fact that the ``vertical derivative'' $\mathrm{osc}_e \nabla G([x,x+e_j],0)$ behaves like $\nabla\nabla G([x,x+e_j],e) \nabla G(e,0)$ due to an asymmetry that arises from differentiating the standard resolvent equation; see also formula (54) in \cite{MO1}
\end{remark}

\section{General pointwise estimates}\label{sect:general}
In this short section, we  present some convenient general pointwise estimates that are used in more specific situations later and are collected here for the convenience of the reader. We derive from the formula for $ w_{{G}} $ given by \eqref{eq:wG} upper and lower bounds which directly involve the discrete derivative $|G(x)-G(y)|$ for $x\sim y$ as summarized in the following proposition. These bounds hold for all elliptic graphs and without any averaging.

Recall that $E>0$ is a constant so that \eqref{E} holds.

\begin{proposition}[General pointwise bound]\label{prop:wGbounds}
Assume that $b$ is a connected transient graph over $X$ which satisfies \eqref{E}. Then,
\begin{align}
w_G(x)
\leq& 1_{o}(x)+\frac{1}{G(x)^2}
	\sum_{\substack{y\in X}}b(x,y)(G(x)-G(y))^2,\\
	w_G(x)
\geq& 1_{o}(x)+(1+E^{-1/2})^{-1}\frac{1}{G(x)^2}
	\sum_{\substack{y\in X}}b(x,y)(G(x)-G(y))^2.
\end{align}
\end{proposition}

We mention in passing that these bounds yield rough decay estimates on $w_G$ that do not require any information on $\nabla G$, though we will not use these rough bounds in the following. Indeed, from ${Q}(f)=2\scp{f}{Lf}$ and the defining property of the Green's function we have that
$$
\sum_{x\in X} w_G(x) G(x)^2 \leq C G(o) <\infty.
$$
which for example implies a weak decay estimate on ${\mathbb Z}^3$ via the Aronson bound \eqref{eq:aronsondiscrete}.

The proof of Proposition \ref{prop:wGbounds} uses the following basic comparison property of the Green function of strongly elliptic graphs with bounded combinatorial degree.

\begin{lemma}\label{lm:basic}
Let $ b $ be a connected transient graph over $ X $  which satisfies the ellipticity condition \eqref{E}. Then, for all $ x\sim y $
	\begin{align*}
	G(x)\geq E  G(y)
	\end{align*}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lm:basic}]
Note that the Green's function is superharmonic, more specifically, it satisfies
\begin{align*}
{L} G(o,\cdot)=1_{o}.
\end{align*}
 By the virtue of the ellipticity condition \eqref{E} we have for all $ x\sim y $
\begin{align*}
 G(x) \ge\frac{1}{\sum_{z\in X}b(x,z) } \sum_{z\in X}b(x,z) G(z)\ge E G(o,y)
\end{align*}
which proves Lemma \ref{lm:basic}.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:wGbounds}]
For the upper bound, we note that
$$
|G(x)^{1/2}-G(y)^{1/2}|=\frac{|G(x)-G(y)|}{G(x)^{1/2}+G(y)^{1/2}}
\leq
\frac{|G(x)-G(y)|}{G(x)^{1/2}}.
$$
For the lower bound, we use that by Lemma \ref{lm:basic}, we have $G(y)\leq E^{-1}G(x)$ and so
$$
|G(x)^{1/2}-G(y)^{1/2}|=\frac{|G(x)-G(y)|}{G(x)^{1/2}+G(y)^{1/2}}	\geq (1+E^{-1/2})^{-1}\frac{|G(x)-G(y)|}{G(x)^{1/2}}.
$$
Squaring both sides and applying the resulting estimates to the formula \eqref{eq:wG} for $ w_{G} $  establishes Proposition \ref{prop:wGbounds}.
\end{proof}

\section{Proof of Theorem \ref{thm:mainspatial} on spatial averaging}
\label{sect:pfmainspatial}

\subsection{Proof of the upper bound \eqref{eq:mainspatialub}}

Let $a\in\curly{A}_{{\lambda}}$ and $R\geq 1$ be arbitrary and recall the setting given by \eqref{eq:setting2}. By the upper bound in Proposition \ref{prop:wGbounds}, the lower Aronson-type bound in \eqref{eq:aronsondiscrete} and $a\leq {\lambda}^{-1}$, we have
$$
\begin{aligned}
&\sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}} w_G(x)\\
&\leq \sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}} \frac{1}{G(x)^2}\sum_{\substack{j=1,\ldots,d\\ s=\pm1}}
a([x,x+se_j])\l|G(x)-G(x+se_j)\r|^2\\
& \leq {\lambda}^{-1} C_{d,{\lambda}}^2 R^{2(d-2)}	\sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}}\sum_{\substack{j=1,\ldots,d\\ s=\pm1}} |G(x)-G(x+se_j)|^2.
\end{aligned}
$$

To bound the annular averages of discrete derivatives of $G$, we employ \cite[Lemma~1.4]{MO2}. To verify the conditions, we recall  that the class $\curly{A}_{\lambda}$ is related to the class used in \cite{MO2} (where the upper bound on the coefficient is equal to $1$ instead of ${\lambda}^{-1}$) by simple rescaling which only changes the ${\lambda}$-dependence of the constant below. In conclusion, we obtain the existence of a constant $\tilde C_{d,{\lambda}}>0$ such that for every $a\in\curly{A}_{{\lambda}}$ and every radius $R\geq 1$,
$$
R^{-d}\sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}}\sum_{\substack{j=1,\ldots,d\\ s=\pm1}} |G(x)-G(x+se_j)|^2\leq \tilde C_{d,{\lambda}}R^{2(1-d)}.
$$
This yields
$$
\begin{aligned}
R^{-d}\sum_{\substack{x\in{\mathbb Z}^d:\\ R\leq |x|\leq 2R}} w_G(x)
 \leq c_{d,{\lambda}} R^{-2},
\end{aligned}
$$
which  is the desired upper bound \eqref{eq:mainspatialub}.\medskip

 It now remains to prove the lower bound \eqref{eq:mainspatiallb}.

\subsection{Proof of Lemma~\ref{lm:onedirection}}
\label{sect:onedirectionpf}

The proof of the lower bound in Theorem \ref{thm:mainspatial} uses Lemma~\ref{lm:onedirection} which we prove now.

Let ${\delta}>0$. We consider the case $j=1$ without loss of generality. We let $\ell>0$ to be determined later. The idea of the proof is that by using both parts of the Aronson-type bound \eqref{eq:aronsondiscrete}, we know that $G$ must have decayed somewhat between the interior and the exterior boundary of the sector.

In a preliminary step, we replace the sector
$$
\curly{S}^{1,{\alpha}}_{R,\ell}=\setof{x\in{\mathbb Z}^d}{\scp{x}{e_1}>(1-{\alpha})|x|,\;R\leq |x|\leq \ell R}
$$
by a cuboid. Observe that there are constants $c_{in}, c_{out},c_{orth}>0$ depending only on $d$, so that the cuboid
$$
\mathrm{Cub}=\setof{(x_1,\ldots,x_d)\in {\mathbb Z}^d}{c_{in} R\leq x_1\leq c_{out} \ell R \text{ and } |x_i|\leq c_{orth} {\alpha} R\mbox{ for all } i\ge 2}
$$
is contained in the sector $\curly{S}^{1,{\alpha}}_{R,\ell}$. Without loss of generality, we assume that $c_{in} R$ and $c_{out} \ell R$ are integers.  Note further that for any fixed $\ell$, $\mathrm{Cub}$ has cardinality of order $R^d$ as $R\to\infty$. We conclude that it suffices to prove the claim with $\mathrm{Cub}$ in place of $\curly{S}^{1,{\alpha}}_{R,\ell}$.

We introduce the inner and outer faces of the cuboid
$$
\begin{aligned}
F_{in}=&\setof{x=(c_{in} R,x_2,\ldots,x_d)}{|x_i|\leq c_{orth} {\alpha} R\,\mbox{ for all } i\ge 2},\\
F_{out}=&\setof{x=(c_{out} \ell R,x_2,\ldots,x_d)}{|x_i|\leq c_{orth} {\alpha} R\,\mbox{ for all }i\ge 2}.
\end{aligned}
$$

By the two-sided Aronson-type bounds \eqref{eq:aronsondiscrete}, we have
$$
\begin{aligned}
G(x)\geq& C_{d,\lambda}^{-1} (1+|x|)^{2-d}\geq C'_{d,\lambda} c_{in}^{2-d} R^{2-d},\qquad  x\in F_{in},\\
G(x)\leq& C_{d,\lambda} (1+|x|)^{2-d}\leq C_{d,\lambda} c_{out}^{2-d} (\ell R)^{2-d},\qquad  x\in F_{out}.
\end{aligned}
$$
Given $x\in F_{in}$, note that the point $x+R(c_{out}\ell-c_{in})e_1\in F_{out}$. By choosing $\ell=\ell_d>0$ sufficiently large, depending on $d$ but not on $R$, these bound imply that  the Greens' function differs by order $R^{2-d}$ between these two points. Namely, there exists  $C''_{d,\lambda} >0$ so that
\begin{equation}\label{eq:glb}
G(x)-G(x+Rc e_1)\geq C''_{d,\lambda} R^{2-d}, \qquad   x\in F_{in}.
\end{equation}
where we introduced $c:=c_{out}\ell_d-c_{in}$.

Since $F_{in}$ contains an order of $(c_{orth}{\alpha} R)^{d-1}\sim R^{d-1}$ many sites, this implies
$$
\sum_{x\in F_{in}} \l(G(x)-G(x+Rce_1)\r)\geq C'''_{d,\lambda} R.
$$

By telescoping (recall that $cR\in{\mathbb Z}$) and the Cauchy-Schwarz inequality, we can upper bound the left-hand side as follows
\begin{equation}\label{eq:telescopic}
\begin{aligned}
&\sum_{x\in F_{in}} \l(G(x)-G(x+Rce_1)\r)\\
&=\sum_{x\in F_{in}} \sum_{n=1}^{cR}\l(G(x+(n-1)e_1)-G(x+ ne_1)\r)\\
&\leq C_{d} R^{d/2}\sqrt{\sum_{x\in F_{in}} \sum_{n=1}^{cR} |G(x+(n-1)Re_1)-G(x+ ne_1)|^2}\\
&\leq C_{d} R^{d/2} \sqrt{\sum_{x\in \mathrm{Cub}} |G(x)-G(x+e_1)|^2}.
\end{aligned}
\end{equation}
We combine this with \eqref{eq:glb} to conclude
$$
C_{d,\lambda}'' R\leq  R^{d/2}\sqrt{\sum_{x\in \mathrm{Cub}}  |G(x)-G(x+e_1)|^2},
$$
Since it suffices to prove the claim for $\mathrm{Cub}$, Lemma \ref{lm:onedirection} follows.
\qed

\subsection{Proof of the lower bound \eqref{eq:mainspatiallb}}
Recall the definition of a sector $ \curly{S}^{j,{\alpha}}_{R,\ell}=\setof{x\in{\mathbb Z}^d}{\scp{x}{e_j}>(1-{\alpha})|x|,\;R\leq |x|\leq \ell R} $ for $ j\in\{1,\ldots,d\} $ and ${\alpha}, R,\ell>0 $.

By the lower bound in Proposition \ref{prop:wGbounds}, the upper Aronson-type bound \eqref{eq:aronsondiscrete} and $a\geq {\lambda}$, we have
$$
\begin{aligned}
\sum_{\curly{S}^{j,{\alpha}}_{R,\ell}} w_G(x)
&\geq (1+E^{-1/2})^{-1}\sum_{x\in\curly{S}^{j,{\alpha}}_{R,\ell}} \frac{1}{G(x)^2}\sum_{\substack{i=1,\ldots,d\\ s=\pm1}}  a([x,x+se_j])|G(x)-G(x+se_i)|^2\\
& \geq (1+E^{-1/2})^{-1}C_{d,{\lambda}}^{-2} \lambda (1+(\ell_dR)^{d-2})^{-2}	\sum_{\curly{S}^{j,{\alpha}}_{R,\ell}}\sum_{\substack{i=1,\ldots,d\\ s=\pm1}}  |G(x)-G(x+se_i)|^2\\
& \geq C'_{d,{\lambda}} R^{2(2-d)}	\sum_{x\in\curly{S}^{j,{\alpha}}_{R,\ell}} |G(x)-G(x+e_j)|^2.
\end{aligned}
$$
By Lemma \ref{lm:onedirection}, we conclude that
$$
R^{-d} \sum_{x\in\curly{S}^{j,{\alpha}}_{R,\ell}} w_G(x)
\geq C''_{d,{\lambda}} R^{-2},
$$
which is \eqref{eq:mainspatiallb}. This completes the proof of Theorem \ref{thm:mainspatial}.
\qed

\subsection{Proof of Proposition \ref{prop:continuum} on the continuum case}
\label{sect:continuumpf}
The proof follows along the same lines as Theorems \ref{thm:mainspatial} and Corollary \ref{cor:ublb}. Here we summarize the necessary modifications.

In the continuum, Definition \eqref{eq:WG} is already in the general form achieved in the discrete setting by Proposition \ref{prop:wGbounds}. For the upper bound, we then use Aronson's bound \eqref{eq:aronson} in the original continuum version and note that Lemma 1.4 of \cite{MO2} is also proved in the continuum in Section 2 of \cite{MO2}; see also Lemma 2.1 in \cite{LSW63}. For the lower bound, when proving Lemma \ref{lm:onedirection} in the continuum, we argue analogously and replace the telescopic sum in \eqref{eq:telescopic} by an application of the fundamental theorem of calculus. The details are left to the reader.
\qed

\section{Proofs for random coefficients}
\label{sect:pfmainrandom}
In this section, we first prove Theorem \ref{thm:mainrandom} for general elliptic coefficients subject to Assumption \ref{ass:P}. Afterwards, we turn to the perturbative setting and prove Theorems \ref{thm:GFasymptotic} and \ref{thm:mainrandompert}, in that order.

\subsection{Proof of Theorem \ref{thm:mainrandom} on pointwise upper bounds}
We first prove the annealed estimate \eqref{eq:wGannealed}.  Let $p\geq 1$. The case $ x=0 $ is clear by   Proposition~\ref{prop:wGbounds}. For $x\neq 0$, by  Proposition~\ref{prop:wGbounds}  and the Aronson bound \eqref{eq:aronsondiscrete}, we have
\begin{equation}\label{eq:bigsum}
 \begin{aligned}
 \qmexp{w_G(x)^p}
&\leq
\left\langle\frac{1}{G(x)^{2p}}
\left(\sum_{\substack{i=1,\ldots,d\\ s=\pm1}} a([x,x+se_j])|G(x)-G(x+se_j)|^2\right)^{p}
	\right\rangle\\
		&
	\leq
\left\langle
C_{d,{\lambda}}^{2p} \l(1+|x|\r)^{2p(d-2)}
\left(\sum_{\substack{i=1,\ldots,d\\ s=\pm1}} a([x,x+se_j])|G(x)-G(x+se_j)|^2\right)^{p}
	\right\rangle\\
\end{aligned}
\end{equation}

To estimate the discrete derivative of $G$, we use $|a|\leq 1$ and \cite[Theorem 1]{MO1} which applies under Assumption \ref{ass:P} and gives
$$
\qmexp{|G(x)-G(x\pm e_j)|^{2p}}^{\frac{1}{2p}}\leq C'_{d,{\lambda},\rho,p} (1+|x|)^{1-d},
$$
for every $1\leq j\leq d$. Using this and the elementary inequality $\l(\sum_{j=1}^d b_j\r)^{p}\leq 2^{p-1} \sum_{j=1}^d b_j^p$, we find
$$
\begin{aligned}
 \qmexp{w_G(x)^p}
\leq&
C_{d,{\lambda}}^{2p} 2^{p-1} \l(1+|x|\r)^{2p(d-2)}\sum_{\substack{i=1,\ldots,d\\ s=\pm1}} \qmexp{|G(x)-G(x+e_j)|^{2p}}\\
\leq & C_{d,{\lambda}}^{2p}2^{p}d(C'_{d,{\lambda},\rho,p})^{2p} (1+|x|)^{-2p}.
\end{aligned}
$$
This proves \eqref{eq:wGannealed} when $x\neq 0$.

It remains to prove the almost-sure estimate \eqref{eq:wGprob1}. Let ${\varepsilon}>0$ and set $p_{\varepsilon}=\frac{2d}{{\varepsilon}}$. By Markov's inequality and the estimate \eqref{eq:wGannealed} proven right above, we have
$$
\sum_{x\in {\mathbb Z}^d}\mathbb P\l(w_G(x)\geq (1+|x|)^{-2+{\varepsilon}}\r)
\leq
C_{d,{\varepsilon},{\lambda}}\sum_{x\in {\mathbb Z}^d}(1+|x|)^{-2{\varepsilon} p_{\varepsilon}}<\infty.
$$
Now, the Borel-Cantelli lemma implies that $w_G(x)\geq (1+|x|)^{-2+{\varepsilon}}$ can occur for at most finitely many $x\in {\mathbb Z}^d$.  This proves Theorem \ref{thm:mainrandom}.
\qed

\subsection{Proof of Theorem \ref{thm:GFasymptotic}}\label{sect:GFasymptoticpf}
Recall that the averaged Green's function $\qmexp{G}$ is the Green's function of the operator $ \mathcal{L} $ defined in \eqref{eq:harmonicmeanoperator}. Since the latter is a Fourier multiplier, as discussed in \cite[Appendix~A]{KL}, we have the Fourier-space representation
\begin{equation}\label{eq:GavgFourierrep}
\qmexp{G(x)}=\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{1}{m(\theta)} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
\end{equation}
with
$$
m(\theta)=2\sum_{j=1}^d (1-\cos\theta_j)+\sum_{1\leq j,k\leq d}
 (e^{-i\theta_j}-1)\widehat{K^{\delta}_{j,k}}(\theta) (e^{i\theta_k}-1)
$$
and $\widehat{K^{\delta}_{j,k}}\in C^{2d-1}(\mathbb T^d)$ for all $j,k\in \{1,\ldots,d\}$ by \cite[Theorem 1.1]{KL}.
For $ \delta $ small enough $ m $ vanishes only at the origin $ \theta=0 $ by the  bound on $ \mathbf K^{\delta} $, \eqref{eq:Kdeltadecay}. Thus, by Taylor expansion, $\frac{1}{m(\theta)}$ only has a quadratic singularity at the origin and so the integral is well-defined for $d\geq 3$.

The goal is to perform asymptotic analysis of \eqref{eq:GavgFourierrep} as $|x|\to\infty$. This is a delicate stationary phase argument which has to take special care of the singularity at the origin in Fourier space as this becomes non-integrable after the last derivative is taken. The same problem was treated in detail by Uchiyama \cite{Uch98} in a probabilistic setting and we show below that his argument extends to our case. (This is perhaps not so surprising considering that there are only few Fourier-theoretic arguments that leverage positivity conditions.)
 To make contact with the probabilistic perspective, we denote
\begin{equation}\label{eq:mrewrite}
m(\theta)=4d\l(1-\hat T(\theta)\r)
\end{equation}
with $T:{\mathbb Z}^d\to{\mathbb R}$ given as in Remark \ref{rmk:T}, i.e.,
\begin{equation}\label{eq:Tdefn'}
\begin{aligned}
T(x)=\frac{1}{2}{\delta}_{x=0}+\frac{1}{4d}{\delta}_{|x|=1}+\frac{1}{4d}\sum_{j,k=1}^d \big(&-K^{\delta}_{j,k}(x)+K^{\delta}_{j,k}(x-e_j)\\
&+K^{\delta}_{j,k}(x-e_k)-K^{\delta}_{j,k}(x-e_j-e_k)\big).
\end{aligned}
\end{equation}
Note that we produced the term $\frac{1}{2}{\delta}_{x=0}$ by adding and subtracting a constant in \eqref{eq:mrewrite}. This is a common technical trick in the context of discrete random walks to remove periodicity, cf.\ \eqref{eq:ap} below.\medskip

\textit{Step 1.} As mentioned above, our goal is to extend  \cite[Theorem 2]{Uch98} to our situation. In a first step, we verify the assumptions of that theorem with the exception of $T(x)\geq 0$. The function $T$ satisfies the following properties assumed in \cite{Uch98} for small $ \delta $. For all these properties the decay bound  \eqref{eq:Kdeltadecay}, which is $ |K^{\delta}_{j,k}(x)|\leq C_{d}\delta^{2} (1+|x|^{-3d+1/2}) $,  from \cite[Theorem 1.1]{KL} is of the essence.

\begin{enumerate}[label=(\roman*)]
\item  $T$ has zero mean. Indeed, by \eqref{eq:Kdeltadecay}, we can use Fubini and a change of variables to see
\begin{equation}
\label{eq:Uass1}
\begin{aligned}
\sum_{x\in{\mathbb Z}^d} xT(x)
=&\frac{1}{4d}\sum_{x\in{\mathbb Z}^d} x {\delta}_{|x|=1}\\
&+\frac{1}{4d}\sum_{j,k=1}^d \sum_{x\in{\mathbb Z}^d} K^{\delta}_{j,k}(x) \l(-x+(x+e_j)+(x+e_k)-(x+e_j+e_k)\r)\\
=&0.
\end{aligned}
\end{equation}

\item The smallest subgroup of ${\mathbb Z}^d$ generated by
\begin{equation}\label{eq:ap}
\setof{x\in{\mathbb Z}^d}{T(x)>0}
\end{equation}
is equal to ${\mathbb Z}^d$. This is a kind if aperiodicity property. To see it is true, note that we can use the decay bound \eqref{eq:Kdeltadecay}, to conclude that for all sufficiently small ${\delta}>0$, we have $T(0)>0$ and $T(\pm e_j)>0$ for $j=1,\ldots,d$.

\item The decay bound \eqref{eq:Kdeltadecay} also implies the summability of
\begin{equation}\label{eq:Uass2}
\begin{aligned}
&\sum_{x\in{\mathbb Z}^d}  |T(x)||x|^{2+m_d} <\infty,\qquad &d=3 \textnormal{ or } d\geq 5,\\
&\sum_{x\in{\mathbb Z}^d}  |T(x)||x|^{2+m_d} \ln |x| <\infty,\qquad &d=4,
\end{aligned}
\end{equation}
were $m_d=2d-3$, $ d=3,4 $  and $ m_{d}=d+1 $, $ d\ge 5 $, was defined in \eqref{eq:mddefn}.

\end{enumerate}

Together (i)-(iii) verify the assumptions of Theorem 2 in \cite{Uch98} with $m$ equal to $m_d$ and $T$ called $p$ there, with the exception of non-negativity. \medskip

\textit{Step 2.} We confirm that the fact that $T$ may be negative does not pose any problems in the proof. This step uses Proposition \ref{prop:Ksymm} and the extension is applicable as long as $\mathbf Q$ is strictly positive semidefinite.

The proof of \cite[Theorem 2]{Uch98} is contained in Section 4 of that paper.
The proof makes use of  general estimates on Fourier integrals taken from Sections 2 and 3 of \cite{Uch98} which do not depend on the non-negativity of $ p $. This concerns  Lemma~2.1, Lemma~3.1 and Corollary 3.1 from \cite{Uch98}. These are used in Section~4
together with the absolute summability \eqref{eq:Uass2} to control the error terms. The only step where the loss of non-negativity requires a short argument is the proof of  $c(\theta)^2+s(\theta)^2>0$ which is obtained on page 226 of \cite{Uch98} from positivity and aperiodicity. We now verify this condition to our context.

For $\theta\in\mathbb T^d$, we set
 $$
 c(\theta)=\sum_{x\in{\mathbb Z}^d} T(x)(1-\cos(\theta\cdot x)),\qquad  s(\theta)=\sum_{x\in{\mathbb Z}^d} T(x)\sin(\theta\cdot x).
 $$

 \begin{lemma}\label{lm:cstheta}
For sufficiently small ${\delta}>0$, we have $c(\theta)^2+s(\theta)^2>0$ for $\theta\in\mathbb T^d\setminus\{0\}$.
 \end{lemma}

 \begin{proof}[Proof of Lemma \ref{lm:cstheta}]
 By Taylor expansion around $\theta=0$, we obtain
 $$
 \begin{aligned}
 c(\theta)=&\frac{1}{2}\sum_{x\in{\mathbb Z}^d} T(x) (\theta\cdot x)^2+{\mathcal{O}}(\theta^4),\\
 s(\theta)=&\sum_{x\in{\mathbb Z}^d} T(x) (\theta\cdot x)+{\mathcal{O}}(\theta^5),
 \end{aligned}
 $$
 where the error terms are controlled the decay bound of $ \mathbf{K}^{\delta} $, \eqref{eq:Kdeltadecay} which  enters via the definition of $ T $ given in \eqref{eq:Tdefn}. By the definitions of $ \mathbf Q $,  \eqref{eq:Qdefn}, and $ T $, \eqref{eq:Tdefn}, we have $$  \mathbf{Q}=\mathrm{Hess}(m)(0)=-4d \mathrm{Hess}(\hat T)(0). $$ Using this for $ c $ and the zero mean property  \eqref{eq:Uass1} of $ T $, for $ s $ we obtain $$
 c(\theta)=\frac{1}{2d}\scp{\theta}{\mathbf{Q}\theta}+{\mathcal{O}}(\theta^4),
 \qquad s(\theta)={\mathcal{O}}(\theta^5).
 $$
 Now Proposition \ref{prop:Ksymm} says that for sufficiently small ${\delta}>0$ the matrix $\mathbf Q$ is positive definite, i.e.,
  $$
 \scp{\theta}{\mathbf{Q}\theta}\geq (1-{\delta}^2 C_d)\theta^2
 $$
 and therefore $c(\theta)^2+s(\theta)^2>0$ holds for all $|\theta|<r$ for some small $r>0$.

 It remains to prove a lower bound over the set  $K=\setof{\theta\in\mathbb T^d}{|\theta|\geq r}$. To this end, let $c_0(\theta),s_0(\theta)$ denote the analogs of $c(\theta),s(\theta)$ with ${\delta}=0$. Then we have $c_0(\theta)^2+s_0(\theta)^2>0$ for all $\theta\in K$ by the aperiodicity of the simple random walk. On the one hand, the continuous function $c_0(\theta)^2+s_0(\theta)^2$ takes its minimum on the compact set $K$; call it $\mu>0$. On the other hand, by \eqref{eq:Tdefn'} and \eqref{eq:Kdeltadecay}, we have
 $$
 \sup_{\theta\in K}|c(\theta)^2+s(\theta)^2-\l(c_0(\theta)^2+s_0(\theta)^2\r)|\leq {\delta}^2 C_{d}.
 $$
 Thus, choosing ${\delta}$ small enough that ${\delta}^2 C_{d}\leq \mu/2$, we conclude that $c(\theta)^2+s(\theta)^2>0$ holds on $K$ as well. This proves Lemma \ref{lm:cstheta}.
 \end{proof}

\textit{Step 3.}  We are now ready to complete the proof of Theorem \ref{thm:GFasymptotic}. Thanks to (i)-(iii), Lemma \ref{lm:cstheta} and the paragraph preceding it, the proof of Theorem 2 from \cite{Uch98} extends to our situation and yields an asymptotic expansion similar to \eqref{eq:GFasymptotic}. Namely,  taking account of the rescaling by $4d$ that we introduced in \eqref{eq:mrewrite}, we have the asymptotic expansion
\begin{equation}\label{eq:Uchexpansion}
 \qmexp{G(x)}=\frac{1}{4d}\frac{\kappa_d}{(\sigma')^2} |x'|^{2-d}
+\sum_{k=1}^{m_d} U_k\l(\frac{x'}{|x'|}\r) |x'|^{2-d-k} +o(|x'|^{2-d-m_d}), \quad \textnormal{as } |x|\to\infty,
\end{equation}
where $x'=\sigma'(\mathbf{Q}')^{-1/2} x$ and $\mathbf{Q}'$ is the $d\times d$ matrix generating the second-moment functional
$$
\scp{\theta}{\mathbf Q' \theta} = \sum_{x\in{\mathbb Z}^d} T(x) (x\cdot \theta)^2,\qquad \sigma'=(\det\mathbf Q')^{1/(2d)}.
$$

When we compare this with our claim  \eqref{eq:GFasymptotic}, we see that the latter features $\tilde x =\sigma\mathbf{Q}^{-1/2} x$ instead, with the matrix $\mathbf{Q}$ defined in \eqref{eq:Qdefn}. These are related via
\begin{equation}\label{eq:Q'Qconnect}
\mathbf{Q}'=\frac{1}{4d}\mathbf Q
\end{equation}
To see this, we use the Fourier representation and recall \eqref{eq:mrewrite} and \eqref{eq:mthetadefn} to find
$$
 \begin{aligned}
\mathbf{Q}_{i,j}'
=\sum_{x\in{\mathbb Z}^d} T(x) x_i x_j
=- \l(\frac{\partial^2}{\partial \theta_i\partial \theta_j}\hat T\r)(0)
=\frac{1}{4d}\l(\frac{\partial^2}{\partial \theta_i\partial \theta_j}m\r)(0)
= \frac{1}{4d} \mathbf{Q}_{i,j}
 \end{aligned}
$$
for all $i,j\in\{1,\ldots,d\}$.

Finally, we employ the identity \eqref{eq:Q'Qconnect} and its consequence $\sigma'=(4d)^{-1/2}\sigma$ in \eqref{eq:Uchexpansion}. For the leading term, we note that $|x'|=|x|$ and $\frac{1}{4d(\sigma')^2}=\frac{1}{\sigma^2}$. For the subleading term, we note that $\frac{x'}{|x'|}=\frac{\tilde x}{|\tilde x|}$. Absorbing the factors of $(4d)^{d+k-2}$ into the $U_1,\ldots,U_m$ then yields \eqref{eq:GFasymptotic}. This proves Theorem \ref{thm:GFasymptotic}.
 \qed\\

\subsection{Proofs of Corollaries \ref{cor:nablaGFasymptotic} and \ref{cor:randomGFasymptotic} on Green's function asymptotics}
\label{sect:corproofs}

\begin{proof}[Proof of Corollary \ref{cor:nablaGFasymptotic}]
Let $1\leq j\leq d$, $s\in\{ \pm 1\}$ and let $\alpha\in {\mathbb N}_0^d$ be a multi-index with $|\alpha|\leq m_d$. We apply Theorem \ref{thm:GFasymptotic}. Regarding the $U_k$ terms, \eqref{eq:Uorder} shows that
$$
\nabla^\alpha \sum_{k=m_d-|{\alpha}|+1}^{m_d} U_k\l(\frac{\tilde x}{|\tilde x|}\r) |\tilde x|^{2-d-k}\in O(|\tilde x|^{1-d-m_d})\subseteq o(|\tilde x|^{2-d-m_d}).
$$
Regarding the $o(|\tilde x|^{2-d-m_d})$ error term appearing in Theorem \ref{thm:GFasymptotic}, we note that
for $f\in o(|\tilde x|^{2-d-m_d})$, the triangle inequality implies $\nabla^\alpha f([x,x+se_j])
\in o(|\tilde x|^{2-d-m_d})$, so the error does not get worse under discrete differentiation. This proves Corollary~\ref{cor:nablaGFasymptotic}.
\end{proof}

\begin{proof}[Proof of Formula \eqref{eq:nablaGFasymptotic}]
We recall the notation \eqref{eq:tildexdefn}, i.e., $\tilde x=\sigma \mathbf{Q}^{-1/2} x$. We first use Corollary \ref{cor:nablaGFasymptotic} and the bound \eqref{eq:Uorder} for all $k\geq 1$ to find
\begin{align*}
\qmexp{\nabla G([x,x+s e_j])}=\frac{\kappa_d}{\sigma^2} ( |\tilde x+s\tilde e_j|^{2-d}- |\tilde x|^{2-d}) +{\mathcal{O}}(|\tilde x|^{-d})
\end{align*}
To compute the leading term, we expand $(1+y)^q=1+qy+{\mathcal{O}}(y^2)$ to obtain
$$
\begin{aligned}
 |\tilde x+s\tilde e_j|^{2-d}- |\tilde x|^{2-d}
 &=|\tilde x|^{2-d}\l(\l(1+\scpp{\frac{\tilde x}{|\tilde x|^2}}{s\tilde e_j}+\frac{|\tilde e_j|^2}{|\tilde x|^2}\r)^{\frac{2-d}{2}}-1\r)\\
 &=|\tilde x|^{1-d} \frac{2-d}{2} \scpp{\frac{\tilde x}{|\tilde x|}}{s\tilde e_j}+{\mathcal{O}}(|\tilde x|^{-d}),\qquad \textnormal{as } |x|\to\infty,
\end{aligned}
$$
where we also made use of the equivalence of  the norms $|x|  $ and $ |\tilde x| $, cf.\ \eqref{eqn:xtilde}.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor:randomGFasymptotic}]
By Theorem \ref{thm:GFasymptotic}, the assertion\eqref{eq:randomGFasymptotic} follows once we show that
\begin{equation}\label{eq:randomGFasymptotic'}
\sup_{x\in{\mathbb Z}^d} \frac{\l|G(x)-\qmexp{G(x)}\r|}{(1+|x|)^{1-d+{\varepsilon}}} \leq C_{d,{\delta},{\varepsilon}}
\end{equation}
holds with probability $1$. Let $p\in (1,\infty)$. By Markov's inequality, we have
$$
\mathbb P\l(\l|G(x)-\qmexp{G(x)}\r|>(1+|x|)^{1-d+{\varepsilon}}\r)
\leq \frac{\qmexp{\l|G(x)-\qmexp{G(x)}\r|^{2p}}}{(1+|x|)^{2p(1-d+{\varepsilon})}}
$$
for every $x\in {\mathbb Z}^d$. We bound the numerator through the concentration bound from Corollary 1 of \cite{MO1}, which says that
\begin{equation}\label{eq:GFconcentration}
\qmexp{|G(x)-\qmexp{G(x)}|^{2p}}^{\frac{1}{p}}\leq C_{d,{\delta},p} (1+|x|)^{2-2d},\qquad  x\in {\mathbb Z}^d.
\end{equation}
We note that this is applicable because the choice of coefficients \eqref{eq:pertcoefficients} satisfies the logarithmic Sobolev inequality by Lemma \ref{lm:iid}. We obtain
$$
\mathbb P\l(\l|G(x)-\qmexp{G(x)}\r|>(1+|x|)^{1-d+{\varepsilon}}\r)
\leq C_{d,{\delta},p} (1+|x|)^{-2p{\varepsilon}}.
$$
This is summable over $x\in {\mathbb Z}^d$ for the choice $p=\frac{d}{{\varepsilon}}$. Hence, the Borel-Cantelli lemma yields \eqref{eq:randomGFasymptotic'} and Corollary \ref{cor:randomGFasymptotic} is proven.
\end{proof}

\subsection{Proof of Theorem \ref{thm:mainrandompert} on pointwise lower bounds}
\label{sect:mainrandompertpf}
Let $p\geq \frac{1}{2}$. We set out to show the lower bound \eqref{eq:mainrandompert2} which reads as
\begin{align*}
	\langle w_G(x)^p\rangle \geq C_{d,{\delta},p} (1+|x|)^{-2p}
\end{align*}
for $  x\in{\mathbb Z}^d $.
We first use the lower bound in Proposition \ref{prop:wGbounds}, the lower Aronson-type bound \eqref{eq:aronsondiscrete} and that $b$ is induced by $a\in \curly{A}_{1-{\delta}}$ to find
$$
\begin{aligned}
w_G(x)^p
\geq& (1+E^{-1/2})^{-p}\frac{1}{G(x)^{2p}}
	\l(\sum_{\substack{y\in X}}b(x,y)(G(x)-G(y))^2\r)^p\\
\geq& (1-{\delta}) C_{d,{\delta}} (1+|x|)^{2p(d-2)}\l(\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x}}(G(x)-G(y))^2\r)^p\\
\geq& C_{d,{\delta},p} (1+|x|)^{2p(d-2)}\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x}}(G(x)-G(y))^{2p}.
\end{aligned}
$$
Next, we take the average $\qmexp{\cdot}$ of both sides and use Jensen's inequality with the convex function $z\mapsto z^{2p}$ to obtain
\begin{equation}\label{eq:wGplb}
\begin{aligned}
\qmexp{w_G(x)^p}
\geq& C_{d,{\delta},p} (1+|x|)^{2p(d-2)}\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x}}\qmexp{(G(x)-G(y))^{2p}}\\
\geq& C_{d,{\delta},p} (1+|x|)^{2p(d-2)}  \sum_{j=1}^d \sum_{s=\pm 1} (\qmexp{\nabla G([x,x+s e_j])})^{2p}.
\end{aligned}
\end{equation}
From Corollary \ref{cor:nablaGFasymptotic}, specifically \eqref{eq:nablaGFasymptotic}, we obtain the lower bound
$$
\begin{aligned}
\qmexp{w_G(x)^p}
\geq C_{d,{\delta},p} (1+|x|)^{2p(d-2)} |\tilde x|^{2p(1-d)}, \qquad \textnormal{as } |x|\to\infty
\end{aligned}
$$
By  \eqref{eqn:xtilde}, we have $|\tilde x|\leq C_{d,{\delta}} |x|$ and so
$$
\begin{aligned}
\qmexp{w_G(x)^p}
\geq C_{d,{\delta},p} (1+|x|)^{-2p}, \qquad \textnormal{as } |x|\to\infty.
\end{aligned}
$$
Let now $R_0>2$ be sufficiently large so that
\begin{equation}\label{eq:wGplbfar}
\begin{aligned}
\qmexp{w_G(x)^p}
\geq \frac{C_{d,{\delta},p}}{2} (1+|x|)^{-2p}
\end{aligned}
\end{equation}
for $ |x|\geq R_0 $.
This proves \eqref{eq:mainrandompert1} for $d\in \{3,4\}$. To prove the assertion \eqref{eq:mainrandompert2} for all $ x $ and  $d\geq 5$, we need a lower bound on $\qmexp{w_G(x)^p}$ over the ball $B_{R_0}=\setof{x\in{\mathbb Z}^d}{|x|<R_0}$. In view of \eqref{eq:wGplb}, it suffices to prove that for every $x_0\in B_{R_0}$ there exists $C_{d,{\delta},x_0}>0$ so that
\begin{equation}\label{eq:positivePball}
\mathbb P\l(\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x_0}}|G(x)-G(y)|>C_{d,{\delta},x_0}\r) >0.
\end{equation}

Given $\beta\in [-1,1]$, we write $G_\beta$ for the Green's function of $(1+{\delta}\beta)\Delta$, a simple rescaling of the well-understood Green's function $G_0$ of the free Laplacian.

We will now prove \eqref{eq:positivePball} by using that with positive probability $G$ is uniformly close to some $G_\beta$ on the entire ball $B_{R_0}$.

\begin{lemma}[Green's function comparison lemma]
\label{lm:GFcomparison}
Let $d\geq 5$ and $R_0$ be given as above. For any $\beta\in [-1,1]$ in the support of $ \omega_{0} $ and any ${\varepsilon}>0$, we have
\begin{equation}
\mathbb P\l(\sup_{x_1\in B_{2R_0}}|G(x_1)-G_\beta(x_1)|\leq  {\varepsilon}\r)>0.
\end{equation}
\end{lemma}

We postpone the proof of Lemma \ref{lm:GFcomparison} for now and continue with the proof of \eqref{eq:positivePball} to conclude Theorem \ref{thm:mainrandompert}.

Let $d\geq 5$ and fix $x_0\in B_{R_0}$. According to Lemma \ref{lm:GFcomparison}, for any ${\varepsilon}>0$ there is a positive probability that
\begin{equation}\label{eq:GFreplacement}
\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x_0}}|G(x_0)-G(y)|
\geq\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x_0}}|G_\beta(x_0)-G_\beta(y)|-4d{\varepsilon}.
\end{equation}
Finally, we recall that $G_\beta=(1+\delta\beta)^{-1} G_0$ with $G_0$ the Green's function of the free Laplacian. Lemma \ref{lm:G0} implies that there exists $C_{d,R_0}$ so that
$$
\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x_0}}|G_\beta(x_0)-G_\beta(y)|
>(1+\delta\beta)^{-1} C_{d,R_0}, \qquad  |x|\leq 2R_0.
$$
Taking ${\varepsilon}$ sufficiently small in \eqref{eq:GFreplacement} then yields \eqref{eq:positivePball}. Thus, we have shown \eqref{eq:wGplbfar} for $ d\ge 5 $ and all $ x $ and we have proven Theorem~\ref{thm:mainrandompert}.
\qed\\

It remains to prove Lemma \ref{lm:GFcomparison}.
\begin{proof}[Proof of Lemma \ref{lm:GFcomparison}]
We take $\beta\in [-1,1]$ to be any point in the support of the random variable $\omega_0$. By definition, this means that for all ${\varepsilon}>0$, there exists $p_{\varepsilon}>0$ so that
$$
\mathbb P(\omega_0\in (\beta-{\varepsilon},\beta+{\varepsilon}))\geq p_{\varepsilon}.
$$
Let ${\varepsilon}>0$ and define
$$ R_{\varepsilon}=\max\{4R_0,{\varepsilon}^{-2}\} .$$
Since $\{\omega_x\}_{x\in {\mathbb Z}^d}$ are independent and identically distributed, we can strengthen the above to the statement that there exists $\tilde p_{\varepsilon}>0$ so that
\begin{equation}\label{eq:betasupport}
\mathbb P\l(\omega_x\in (\beta-{\varepsilon},\beta+{\varepsilon}) \mbox{ for all }  x\in B_{R_{\varepsilon}}\r)\geq \tilde p_{\varepsilon}.
\end{equation}
From now on, we fix a realization of $\{\omega_x\}_{x\in {\mathbb Z}^d}$ which we assume satisfies $\omega_x\in (\beta-{\varepsilon},\beta+{\varepsilon})$ for all $x\in B_{R_{\varepsilon}}$ with an ${\varepsilon}$ to be chosen later.

We now replace the variables $\omega_x$ one-by-one by $\beta$ via the resolvent identity. To make this precise, it is convenient to use edge variables following \cite[Section 7.3]{MO1}. Recall the original setup \eqref{eq:setting1} and \eqref{eq:setting2}. Let $F\subset \mathbb E^d$ and let $F'=F\cup \{e'\}$ for some fixed edge $e'\not\in F$. For ${\gamma}\in\{F,F'\}$ let $a^{\gamma}=\{a^{{\gamma}}(e)\}_{e\in \mathbb E^d}$ be two families of coefficients in $\mathcal A_{1-{\delta}}$ satisfying
$$
a^F(e')\neq a^{F'}(e')  \quad \textnormal{ and } \quad a^F(e)= a^{F'}(e),\qquad  e\neq e'.
$$
We write $G^{\gamma}$, ${\gamma}\in\{F,F'\}$, for the corresponding Green's functions. The resolvent identity gives
\begin{equation}\label{eq:resolventidentity}
G^F(x,0)-G^{F'}(x,0)=(a^F(e')- a^{F'}(e)) \nabla G^{F}(x,e') \nabla G^{F'}(e',0)\qquad  x\in {\mathbb Z}^d,
\end{equation}
confer Equation~(53) in \cite{MO1}.
Let $\{e_n\}_{n\geq 0}=\mathbb E^d$ be an enumeration of the edge set. We define
$$
F_N:=\bigcup_{1\leq n\leq N} \{e_n\},\qquad  N\geq 0
$$
with the convention that $F_0=\emptyset$ and $F_\infty=\mathbb E^d$. We recall \eqref{eq:pertcoefficients} and note that every edge $e$ has a unique representation as $e=[x,x+e_j]$ with $x\in{\mathbb Z}^d$ and $j\in \{1,\ldots,d\}$. We use this to define the coefficients
\begin{equation}\label{eq:aedefn}
a^{F_N}(e)=
\begin{cases}
1+{\delta} \omega_x,\qquad &\textnormal{if } e=[x,x+e_j]\in F_N,\\
1+{\delta} \beta,\qquad &\textnormal{if } e\not\in F_N.
\end{cases}
\end{equation}

We can then use telescoping and the resolvent identity \eqref{eq:resolventidentity} to write, for every $x_1\in B_{2R_0}$,
\begin{equation}\label{eq:GminusGbeta}
\begin{aligned}
	&G(x)-G_\beta(x_1)=G^{F_0}(x_1)-G^{F_\infty}(x_1)\\
	&=\sum_{N\geq 0} G^{F_N}(x_1)-G^{F_{N+1}}(x_1)\\
	&=\sum_{N\geq 0} (a^{F_N}(e_{N+1})- a^{F_{N+1}}(e_{N+1})) \nabla G^{F_N}(x_1,e_{N+1}) \nabla G^{F_{N+1}}(e_{N+1},0).
\end{aligned}
\end{equation}
We decompose the last sum in $N$ as follows. Let $\mathcal I\subset \mathbb N_0$ be the finite index set of non-negative integers $N\geq 0$ so that $e_{N+1}=[x,x+e_j]$ with $x\in B_{R_{\varepsilon}}$. On this set, we may apply \eqref{eq:betasupport} and use \eqref{eq:aedefn} to obtain
$$
|a^{F_N}(e_{N+1})- a^{F_{N+1}}(e_{N+1})|\leq {\varepsilon}
$$
We combine this with the a priori estimate
\begin{equation}\label{eq:GFapriori}
|\nabla G^{F_N}(x_1,e_{N+1}) \nabla G^{F_{N+1}}(e_{N+1},0)|
\leq 4 \l(\sup_{N\geq 0} \sup_{x,y\in {\mathbb Z}^d} G^{F_N}(x,y)\r)^2
\leq C_{d,{\delta}},
\end{equation}
where the latter estimate holds by \eqref{eq:aronsondiscrete} and the fact that $a^{F_N}\in \curly{A}_{1-{\delta}}$ for all $N\geq 0$. Employing these estimates for $n\in\mathcal I$ in \eqref{eq:GminusGbeta} gives
\begin{equation}\label{eq:GimplementI}
\begin{aligned}
	&|G(x_1)-G_\beta(x_1)|\\
	&\leq{\varepsilon} | \mathcal I| C_{d,{\delta}}+(1+{\delta})^2\sum_{N\in \mathbb N_0\setminus \mathcal{I}}
	|\nabla G^{F_N}(x_1,e_{N+1})| |\nabla G^{F_{N+1}}(e_{N+1},0)|,
\end{aligned}
\end{equation}
where $| \mathcal I|<\infty$ denotes the cardinality of $\mathcal I$.

To estimate the sum over $\mathbb N_0\setminus \mathcal{I}$, we first reparametrize it using the fact that there is a bijective map $N:B_{R_{\varepsilon}}^c \times \{1,\ldots,d\}\to \mathbb N_0\setminus \mathcal I$ so that $e_{N(x,j)}=[x,x+e_j]$.
$$
\begin{aligned}
	&\sum_{N\in \mathbb N_0\setminus \mathcal{I}} |\nabla G^{F_N}(x_1,e_{N+1})| |\nabla G^{F_{N+1}}(e_{N+1},0)|\\
	&=\sum_{x\in B_{R_{\varepsilon}}^c} \sum_{j=1}^d |\nabla G^{F_{N(x,j)}}(x_1,[x,x+e_j])| |\nabla G^{F_{N(x,j)+1}}([x,x+e_j],0)|\\
\end{aligned}
$$
Next, we refine the a priori estimate \eqref{eq:GFapriori} via \eqref{eq:aronsondiscrete}  to
$$
\begin{aligned}
	|\nabla G^{F_{N(x,j)}}(x_1,[x,x+e_j])| |\nabla G^{F_{N(x,j)+1}}([x,x+e_j],0)|\leq C_{d,{\delta}} |x_1-x|^{2-d} |x|^{2-d}.
\end{aligned}
$$
We observe that $|x-x_1|\geq |x|/2$ for all $ x\in B_{R_{{\varepsilon}}^{c}} $ because $x_1\in B_{2R_0}$ and $R_{\varepsilon}\geq 4R_0$. Hence $|x_1-x|^{2-d}\leq C |x|^{2-d}$
and  since $ \sum_{x\in {\mathbb Z}^{d}} |x|^{-d-1/2}\leq C_{d}<\infty$ we have shown that
$$
\sum_{x\in B_{R_{\varepsilon}}^c} \sum_{j=1}^d |\nabla G^{F_{N(x,j)}}(x_1,[x,x+e_j])| |\nabla G^{F_{N(x,j)+1}}([x,x+e_j],0)|
\leq C_{d,{\delta}} R_{\varepsilon}^{4-d-{1}/{2}}.
$$
Whenever $ d\ge5 $, we have $  R_{\varepsilon}^{4-d-{1}/{2}}\leq {\varepsilon} $ since $R_{\varepsilon}\geq {\varepsilon}^{-2}$.
In view of  \eqref{eq:GimplementI}, this implies
$$
|G(x_1)-G_\beta(x_1)|\leq C_{d,{\delta},R_0} {\varepsilon}.
$$
Note that all estimates are uniform in $x_1\in B_{2R_0}$. Since the required assumption \eqref{eq:betasupport} holds with positive probability, this proves Lemma \ref{lm:GFcomparison}.
\end{proof}

\section{Application to Rellich inequalities}\label{sect:rellich}
In this short section, we explain the implications of the results from Section \ref{sect:perturbativeresults} for Rellich inequalities on graphs. The first Rellich inequality was presented by F.~Rellich at the ICM 1954 in Amsterdam \cite{R}. Very recently, \cite{KePiPo4} provided a general mechanism for generating Rellich inequalities from strictly positive Hardy weights on graphs. Adapting \cite[Corollary 4.1]{KePiPo4} to the case of elliptic operators on ${\mathbb Z}^d$ and subharmonic function $u=G$, we obtain the following result.

\begin{thm}[Rellich inequality on ${\mathbb Z}^d$, \cite{KePiPo4}]\label{thm:rellich}
Let $ a\in \mathcal{A}_{\lambda} $ for $ \lambda\in(0,1] $, $W={\mathrm {supp}\,} w_G\subseteq {\mathbb Z}^{d}$  and  ${\alpha}\in (0,1)$. Then we have the Rellich inequality
\begin{equation}\label{eq:rellich}
\|\mathbf{1}_\varphi \Delta \varphi \|_{\frac{G^{\alpha}}{w_G}} \geq (1-{\gamma}) \|\varphi\|_{G^{\alpha} w_G}
\end{equation}
for all $ {\varphi}\in C_{c}(W) $, where $\mathbf{1}_{{\varphi}} $ is the characteristic function of the support of $ {\varphi} $ and
$$
{\gamma}=\l(\frac{1-(\lambda^{2}/2d)^{{\alpha}/2}}{1-(\lambda^{2}/2d)^{1/2}}\r)^2.
$$
\end{thm}
\begin{proof}
In view of \cite[Remark~3.4]{KePiPo4} the assumption of standard weights can replaced by the ellipiticity condition \eqref{E} which is satisfied in our setting with $ E= \lambda^{2}/2d $. Precisely, in  \cite[Theorem~3.3]{KePiPo4} from which   \cite[Corollary 4.1]{KePiPo4} is deduced one can replaced the constant $ D $ with  $ 1/E $.
\end{proof}

For the free Laplacian on ${\mathbb Z}^d$, $d\geq 5$, and ${\alpha}=\tfrac{2}{d-2}$ the standard Green's function asymptotic shows that $\frac{G_0^{\alpha}}{w_{G_0}} \sim C$ and $G_0^{\alpha} w_{G_0}\sim C'  |x|^{-4}$ \cite[Example 4.5]{KePiPo4}. This matches the $|x|^{-4}$-scaling of the original Rellich weight on ${\mathbb R}^d$ \cite{R}.

We consider this question in the weakly random setting of Section \ref{sect:perturbativesetup}. We can use the lower probabilistic bounds established on $w_G$ to derive the same scaling behavior for Rellich weights of general elliptic operators on ${\mathbb Z}^d$, $d\geq 5$, that was found for the free Laplacian, again in a probabilistic sense.

\begin{proposition}[$|x|^{-4}$-scaling for Rellich weights]
For $d\geq 5$, let ${\alpha}=\tfrac{2}{d-2}$. Then, there exists $c_d>0$ so that for all ${\delta}\in (0,c_d)$ there exists $ C_{d,{\delta}} >0$ such that
$$
\begin{aligned}
\left\langle G(x)^{\alpha} w_G(x)  \right\rangle \geq C_{d,{\delta}} (1+|x|)^{-4}
\end{aligned}
$$
and there exist   $ c_{d,{\delta}}>0 $ such that
$$
\mathbb P\l(\frac{G(x)^{\alpha}}{w_G(x)}<C_{d,{\delta}}\r)\geq c_{d,{\delta}},\qquad  x\in {\mathbb Z}^d.
$$
\end{proposition}

Note that the direction of the bounds are the right ones to be applicable in \eqref{eq:rellich}.

\begin{proof}
The first bound follows from \eqref{eq:aronsondiscrete} and Theorem \ref{thm:mainrandompert}.
The second bound follows  from \eqref{eq:aronsondiscrete} and Corollary \ref{cor:smm}.
\end{proof}

\section*{Acknowledgments} The authors are grateful to Yehuda Pinchover for valuable comments. They would like to thank Peter Bella, Arianna Giunti, and Felix Otto for helpful comments on a draft version of the paper. MK acknowledges the financial support of the German Science Foundation. The authors would like to thank the organizers of the program ``Spectral Methods in Mathematical Physics'' held in 2019 at Institut Mittag-Leffler where this project was initiated.
\begin{appendix}

\section{The free Green's function is never locally constant}
The following lemma is used in the main text to derive lower bounds on Hardy weights near the origin. It is elementary, but does not appear to be completely standard. We learned of the argument through \cite{mathoverflow} and include it here for the convenience of the reader.

We recall that $G_0$ is the Green's function of the free Laplacian on ${\mathbb Z}^d$.

\begin{lemma}\label{lm:G0}
Let $d\geq 1$. Then
\begin{equation}
\sum_{\substack{y\in{\mathbb Z}^d:\\ y\sim x}}|G_0(x)-G_0(y)| >0,\qquad  x\in{\mathbb Z}^d.
\end{equation}
\end{lemma}

\begin{proof}
We write $\mathbb{P}^0$ for the probability measure of the symmetric simple  random walk $ S $ started at the origin. A random-walk representation of the Green's function is given in  \cite{LL} via
$$
G_0(x)=C_d\sum_{n\geq 0} \mathbb{P}^0(S_{n}=x)
$$
Let $ x\in {\mathbb Z}^{d} $ and without loss of generality, we suppose that the first coordinate $x_1\geq 1$. Moreover, we assume that $\sum_{i=1}^d x_i$ is odd; the even case can be argued analogously. Observe that then
$$
\mathbb{P}^0(S_{n}=x+e_1)=\mathbb{P}^0(S_{n}=x-e_1)=0,\qquad  n\geq 0 \textnormal{ odd},
$$
due to parity considerations.

Now we claim that if $\mathbb{P}^0(S_{n}=x+e_1)>0$, then
\begin{equation}\label{eq:G0claim}
\mathbb{P}^0(S_{n}=x+e_1)<\mathbb{P}^0(S_{n}=x-e_1),\qquad  n\geq 0 \textnormal{ even}.
\end{equation}
This implies the assertion of the lemma after summation in $n$.

It remains to prove \eqref{eq:G0claim}. Given an index set $\mathcal I_n\subset \{1,\ldots,n\}$, let $A(\mathcal I_n)$ be the random event that the steps $\mathcal I_n$ occurr in the $\pm e_1$ direction and the steps in $\{1,\ldots,n\}\setminus \mathcal I_n$ do not. By conditioning we have
$$
\mathbb{P}^0(S_{n}=x\pm e_1)=\sum_{\mathcal I_n\subset \{1,\ldots,n\}}  \mathbb{P}^0(S_{n}=x\pm e_1\vert A(\mathcal I_n))\, \mathbb{P}^0(A(\mathcal I_n)
$$
Therefore, it suffices to prove that if $\mathbb{P}^0(S_{n}=x+e_1\vert A(\mathcal I_n))>0$, then
\begin{align*}
	\mathbb{P}^0(S_{n}=x+e_1\vert A(\mathcal I_n))<\mathbb{P}^0(S_{n}=x-e_1\vert A(\mathcal I_n)),\qquad  n\geq 0 \textnormal{ even}.
\end{align*}To see this, we use the fact that conditional on $A(\mathcal I_n)$, we have two independent random walks: one in the $\pm e_1$ direction of $\tilde n=|\mathcal I_n|$ steps and one in the remaining $d-1$ directions of $n-\tilde n$ steps. For any $y\in{\mathbb Z}^d$, we write $y=(y_1,P(y))$ with $P(y)\in {\mathbb Z}^{d-1}$. Note that $P(x+e_1)=P(x-e_1)=P(x)$. Hence, if $\mathbb{P}^0(S_{n}=x+e_1\vert A(\mathcal I_n))>0$, then
$$
\begin{aligned}
\mathbb{P}^0(S_{n}=x+e_1\vert A(\mathcal I_n))
&=\mathbb{P}^0(P(S_{n})=P(x) \vert A(\mathcal I_n)) \binom{\tilde n}{\tfrac{\tilde n+x_1+1}{2}} 2^{-\tilde n}\\
&<\mathbb{P}^0(P(S_{n})=P(x) \vert A(\mathcal I_n)) \binom{\tilde n}{\tfrac{\tilde n+x_1-1}{2}} 2^{-\tilde n}\\
&=\mathbb{P}^0(S_{n}=x-e_1\vert A(\mathcal I_n))
\end{aligned}
$$
where we used the elementary estimate that
$$
\binom{\tilde n}{\tfrac{\tilde n+x_1+1}{2}}<\binom{\tilde n}{\tfrac{\tilde n+x_1-1}{2}}
$$
because $\tilde n\ge x_1\geq 1$. This proves Lemma \ref{lm:G0}.
\end{proof}

\section{Direct argument for the leading order in Theorem \ref{thm:GFasymptotic}}\label{sect:direct}
In this appendix, we give a self-contained proof of the lowest order asymptotic in Theorem \ref{thm:GFasymptotic}, i.e.,
\begin{equation}\label{eq:direct}
\qmexp{G(x)}=\frac{\kappa_d}{\sigma^2}|\tilde x|^{2-d} +{\mathcal{O}}(|x|^{1-d}),\qquad \textnormal{as } |x|\to\infty.
\end{equation}
The approach is to use a Taylor expansion around the origin in Fourier space which is justified since $\widehat{K^{\delta}_{j,k}}\in C^{2d-1}(\mathbb T^d)$ by \cite[Theorem 1.1]{KL}. The leading term is reduced to the free Green's function by an appropriate substitution, whose leading order asymptotic we assume here. The error terms are controlled by adapting the dyadic decomposition argument in Appendix A of \cite{KL}.

\begin{proof}[Proof of \eqref{eq:direct}]
We recall Definition \eqref{eq:mthetadefn} of $m(\theta)$ and the fact that $\widehat{K^{\delta}_{j,k}}\in C^{2d-1}(\mathbb T^d)$ for all $j,k\in \{1,\ldots,d\}$. We first isolate the lowest, quadratic order of $m(\theta)$ by setting
$$
m(\theta)=m_0(\theta)+\tilde m(\theta),
\qquad \textnormal{where }
m_0(\theta)=\scpp{\theta}{\mathbf{Q}\theta}.
$$
Observe that $\tilde m\in C^{2d-1}(\mathbb T^d)$ satisfies
\begin{equation}\label{eq:tildembounds}
|D^{\alpha} \tilde m(\theta)|\leq C_{d}|\theta|^{3-|{\alpha}|},\qquad  {\alpha}\in \mathbb N_0^d \textnormal{ with } |{\alpha}|\leq 2d-1.
\end{equation}
We can decompose
\begin{equation}\label{eq:1/mdecomp}
\frac{1}{m(\theta)} =\frac{1}{m_0(\theta)} -\frac{\tilde m(\theta)}{m_0(\theta)m(\theta)}.
\end{equation}
The next lemma then implies \eqref{eq:direct}.

\begin{lemma}\label{lm:GFsubleading}
For all ${\delta}\geq 0$ sufficiently small, we have
\begin{equation}\label{eq:G_hot}
\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{\tilde m(\theta)}{m_0(\theta)m(\theta)}\frac{\mathrm{d}^d \theta}{(2\pi)^d}={\mathcal{O}}(|x|^{1-d}),\qquad \textnormal{as } |x|\to\infty.
\end{equation}
\begin{equation}\label{eq:G_lt}
\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{1}{m_0(\theta)} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
=\frac{\kappa_d}{\sigma^2} |x'|^{2-d} +{\mathcal{O}}(|x|^{1-d}),\qquad \textnormal{as } |x|\to\infty.
\end{equation}
\end{lemma}

\begin{proof}[Proof of Lemma \ref{lm:GFsubleading}]
We loosely follow Appendix A in \cite{KL} where a similar problem is treated and start with the proof of \eqref{eq:G_hot}. We define the function $ F $ on $ \mathbb{T}^{d} $
$$
F(\theta)=\frac{\tilde m(\theta)}{m_0(\theta)m(\theta)}.
$$
and note that, due to \eqref{eq:tildembounds} and the quadratic vanishing order of $m(\theta)$ and $m_0(\theta)$ at the origin, we have
\begin{equation}\label{eq:DalFestimate}
|F(\theta)|\leq C_d |\theta|^{-1}.
\end{equation}
In essence, we aim to use integration by parts to transfer $d-1$ derivatives onto $F$. This leads to a singular Fourier integral, which we analyze through the following dyadic scale decomposition of the frequencies. Let $\varphi:[0,\infty)\to{\mathbb R}$ be a smooth cutoff function with $\varphi=1$ on $[0,2\pi]$ which is supported on $[0,4\pi]$. Define $\psi(r):=\varphi(r)-\varphi(2r)$ and $\psi_l(r)=\psi(2^l r)$ for all $l\geq 1$ and $ r\ge 0 $. Note that this defines a partition of unity $\sum_{l\geq 0}\psi_l(r)=1$ for all $r\neq 0$. We decompose
\begin{equation}\label{eq:fldecompose}
\int_{\mathbb T^d} e^{ix\cdot \theta} F(\theta)\frac{\mathrm{d}^d \theta}{(2\pi)^d}
=\sum_{l\geq 0} f_l(x).
\end{equation}
where we rescaled and introduced
$$
f_l(x)=2^{-ld}\int_{\mathbb T^d} e^{i2^{-l}x\cdot \theta} F_l(\theta)\frac{\mathrm{d}^d \theta}{(2\pi)^d},\qquad F_l(\theta)=\psi(|\theta|) F(2^{-l}\theta).
$$
Note that \eqref{eq:DalFestimate} implies $|F_l(\theta)|\leq C_d 2^{l} |\theta|^{-1}$, so we can use the triangle inequality to obtain
\begin{equation}\label{eq:flestimate1}
|f_l(x)|\leq C_d 2^{-l(d-1)}.
\end{equation}
This is useful for for $|x|2^{-l}\leq 1$, while for $|x|2^{-l}\geq 1$ it can be improved to
\begin{equation}\label{eq:flestimate2}
|f_l(x)|\leq C_d \frac{2^{-l(d-1)}}{(|x|2^{-l})^{d}}.
\end{equation}
To prove \eqref{eq:flestimate2}, we assume without loss of generality that $|x_1|=\max_{1\leq j\leq d}|x_j|$ and use $d$-fold integration by parts to write
\begin{equation}\label{eq:flibp}
f_l(x) =i^d \frac{2^{-ld}}{(x_1 2^{-l})^d}
\int_{\mathbb T^d} e^{i2^{-l}x\cdot \theta} \partial_{\theta_1}^d F_l(\theta)\frac{\mathrm{d}^d \theta}{(2\pi)^d}.
\end{equation}
Next, we observe that \eqref{eq:tildembounds} and the quadratic behavior of both $m_0$ and $m$ at the origin yield that $\|\partial_{\theta_1}^d F_l\|_\infty\leq C_d 2^{-l(d-1)}$; compare Lemma A.1 in \cite{KL}. Applying this bound to \eqref{eq:flibp} yields \eqref{eq:flestimate2}.

We use \eqref{eq:flestimate1} and \eqref{eq:flestimate2} and bound the resulting geometric series to find
$$
\begin{aligned}
\sum_{l\geq 0} |f_l(x)|
\leq C_d \sum_{l\geq \log_2 |x|} 2^{-l(d-1)}+C_d \sum_{0\leq l\leq \log_2 |x|}\frac{2^{-l(d-1)}}{(|x|2^{-l})^{d}}
\leq C_d |x|^{1-d}.
\end{aligned}
$$
In view of \eqref{eq:fldecompose}, this proves \eqref{eq:G_hot}.

Next, we turn to the proof of \eqref{eq:G_lt}. By Proposition \ref{prop:Ksymm}, the matrix $\mathbf{Q}$ is symmetric and positive definite. Hence, by substitution,
$$
\begin{aligned}
\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{1}{m_0(\theta)} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
=&\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{1}{\scp{\mathbf{Q}^{1/2}\theta}{\mathbf{Q}^{1/2}\theta}} \frac{\mathrm{d}^d \theta}{(2\pi)^d}\\
=&\sigma^{-d}
\int_{\mathbb T^d} e^{i (\mathbf{Q}^{-1/2}x)\cdot \theta} \frac{1}{|\theta|^2} \frac{\mathrm{d}^d \theta}{(2\pi)^d}.
\end{aligned}
$$
For the volume element, we used that $\det(\mathbf{Q}^{1/2})= (\det\mathbf{Q})^{1/2}=\sigma^d$ since $\mathbf{Q}$ is symmetric.

By applying \eqref{eq:G_hot} with ${\delta}=0$, we obtain
$$
\begin{aligned}
\int_{\mathbb T^d} e^{i \mathbf{Q}^{-1/2}x\cdot \theta} \frac{1}{|\theta|^2} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
=\int_{\mathbb T^d} e^{i \mathbf{Q}^{-1/2}x\cdot \theta} \frac{1}{2\sum_{j=1}^d (1-\cos\theta_j)} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
+{\mathcal{O}}(|x|^{1-d}).
\end{aligned}
$$
We recognize the first integral as the Green's function of the free Laplacian on ${\mathbb Z}^d$ evaluated at the point $\mathbf Q^{-1/2}x$. The standard asymptotic formula for the free Laplacian, see e.g.\ \cite{Spitzer,Uch98}, gives
$$
\int_{\mathbb T^d} e^{ix\cdot \theta} \frac{1}{m_0(\theta)} \frac{\mathrm{d}^d \theta}{(2\pi)^d}
=\frac{\kappa_d}{\sigma^d} |\mathbf{Q}^{-1/2}x|^{2-d} +{\mathcal{O}}(|x|^{1-d})
=\frac{\kappa_d}{\sigma^2} |x'|^{2-d} +{\mathcal{O}}(|x|^{1-d}).
$$
This proves \eqref{eq:G_lt}.
\end{proof}
Combining \eqref{eq:1/mdecomp} in the beginning of the proof with \eqref{eq:GavgFourierrep}, \eqref{eq:G_lt} and \eqref{eq:G_hot} yields \eqref{eq:direct} which finishes the proof.
\end{proof}

\end{appendix}

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