# Scaling limit of the conductivity of random resistor networks on simple
point processes

Alessandra Faggionato  Alessandra Faggionato. Dipartimento di Matematica,
Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy
<EMAIL_ADDRESS>
###### Abstract.

We consider random resistor networks with nodes given by a simple point
process on $\mathbb{R}^{d}$ and with random conductances. The length range of
the electrical filaments can be unbounded. We assume that the randomness is
stationary and ergodic w.r.t. the action of the group $\mathbb{G}$, given by
$\mathbb{R}^{d}$ or $\mathbb{Z}^{d}$. This action is covariant w.r.t.
translations on the Euclidean space. Under minimal assumptions we prove that
a.s. the suitably rescaled directional conductivity of the resistor network
along the principal directions of the effective homogenized matrix $D$
converges to the corresponding eigenvalue of $D$ times the intensity of the
simple point process. We also prove the a.s. scaling limit of the conductivity
along any direction perpendicular to the kernel of $D$ by suitably modifying
the shape of the region of physical observation. Our results cover plenty of
models including e.g. the standard conductance model on $\mathbb{Z}^{d}$, the
Miller-Abrahams resistor network for conduction in amorphous solids (to which
we can now extend the bounds in agreement with Mott’s law previously obtained
in [9, 20, 21] for Mott’s random walk), resistor networks on the supercritical
cluster in lattice and continuum percolations, resistor networks on crystal
lattices and on Delaunay triangulations.

Keywords: simple point process, resistor network, Miller-Abrahams random
resistor network, random conductance model, supercritical percolation,
homogenization, 2-scale convergence.

AMS 2010 Subject Classification: 60G55, 74Q05, 82D30

This work has been partially supported by the ERC Starting Grant 680275 MALIG
and by the Grant PRIN 20155PAWZB ”Large Scale Random Structures”

##  1\. Introduction

Random resistor networks in ${\mathbb{R}}^{d}$ are an effective tool to
investigate transport in disordered media and have been much investigated both
in Physics and Probability (cf. e.g. [5, 27, 28, 37] and references therein).
Randomness can affect both the conductances of the electric filaments and the
location of the nodes. It describes micro-inhomogeneities which can be of
different physical nature. For example, one can consider mixtures of
conducting and non conducting materials (cf. [28, Section II]), thus
motivating the study of resistor networks on percolation clusters. One can
also consider amorphous solids as doped semiconductors in the regime of strong
Anderson localization. In this case the doping impurities have random
positions $x_{i}$ (described mathematically by a simple point process) and the
Mott’s variable range hopping (v.r.h.) of the conducting electrons can be
modeled by the Miller-Abrahams (MA) random resistor network. In this network
nodes are given by the impurities and, between any pair of impurities located
at $x_{i}$ and $x_{j}$, there is an electrical filament of conductance

$c_{x_{i},x_{j}}(\omega):=\exp\Big{\\{}-\frac{2}{\gamma}|x_{i}-x_{j}|-\frac{\beta}{2}(|E_{i}|+|E_{j}|+|E_{i}-E_{j}|)\Big{\\}}\,,$
(1)

where $\gamma$ is the localization length, $\beta=1/kT$ is the inverse
temperature and the $E_{i}$’s are random energy marks associated to the
impurities (cf. [4, 33, 36, 38] and Section 3 below). Typically, the energy
marks are taken as i.i.d. random variables with distribution
$p(E)\propto|E|^{\alpha}dE$ on an interval containing the origin, for some
$\alpha\geq 0$. Mott’s v.r.h. was introduced by Mott to explain the anomalous
low-temperature conductivity decay, which (for isotropic materials) behaves as

$\sigma(\beta)\asymp
A\exp\left\\{-c\beta^{\frac{\alpha+1}{\alpha+1+d}}\right\\}\,,$ (2)

where $A$ has a negligible $\beta$-dependence, while $c>0$ is
$\beta$-independent (Mott considered the case $\alpha=0$, while Efros and
Shklovskii introduced $\alpha$ to model a possible Coulomb pseudogap in the
density of states). Eq. (2) is usually named Mott’s law. We refer to Mott’s
Nobel Lecture [32] and the monographs [36, 38] for more details.

As in Mott’s law, a fundamental physical quantity is given by the conductivity
of the resistor network along a given direction. One considers a box centered
at the origin of ${\mathbb{R}}^{d}$ with two opposite faces orthogonal to the
fixed direction, where the electric potential takes value $0$ and $1$,
respectively. Then the conductivity is the total electric current flowing
across any section orthogonal to the given direction and equals the total
dissipated energy (cf. [5, Section 1.3], [27, Section 11] and Section 2
below).

In this work we consider generic random resistor networks on
${\mathbb{R}}^{d}$ with random conductances and nodes at random positions,
hence described by a simple point process. The electric filaments can be
arbitrarily long. We assume that the randomness is stationary and ergodic
w.r.t. the action of the group $\mathbb{G}$, given by $\mathbb{R}^{d}$ or
$\mathbb{Z}^{d}$. This action is covariant w.r.t. the action of ${\mathbb{G}}$
by translations on the Euclidean space. Under minimal assumptions we prove
that, as the box size diverges, a.s. the suitably rescaled directional
conductivity of the resistor network along the principal directions of the
effective homogenized matrix $D$ converges to the corresponding eigenvalue of
$D$ times the intensity of the simple point process (cf. Theorems 1 and 2).
$D$ admits a variational characterization which can be used to get upper and
lower bounds (cf. e.g. [12, 20, 21]). The conductivity scaling limit along non
principal directions is treated in our Theorem 3. Information on the limiting
behavior of the electrical potential is provided in Proposition 2.8. We point
out that our finite-moment conditions (A7) in Section 2 are the optimal ones,
as they are necessary to define the effective homogenized matrix
(integrability of $\lambda_{2}$ in (A7)) and the space of square integrable
forms (integrability of $\lambda_{0}$ in (A7)).

Our target has been to achieve a universal qualitative result, hence our
theorems apply to a large variety of geometric structures. In particular we
obtain the scaling limit of the conductivity for the resistor network on the
supercritical percolation cluster on ${\mathbb{Z}}^{d}$, providing a solution
to the the open Problem 1.18 in [5] which goes far beyond Bernoulli bond
percolation (cf. also [1, 10, 24, 27] and references therein). As a byproduct,
we get also a proof of the strictly positivity of $D$ for this model
alternative to the original one in [12]. Our results cover also the MA
resistor network allowing to extend to its asymptotic directional
conductivities the bounds in agreement with Mott’s law (2) previously obtained
in [9, 20, 21] for Mott’s random walk as detailed in Corollary 3.1 in Section
3 (Theorem 1 will be used also to fully prove Mott’s law for several
environments in [17]). For the standard conductance model on
${\mathbb{Z}}^{d}$ our results improve the existing ones (see [30] and the
discussion below). The above examples are discussed in Section 3. We also
mention some other examples, as resistor networks on crystal lattices, on
Delaunay triangulations, on supercritical clusters in continuum percolation.
We point out that there isn’t a prototype random resistor network to deal with
as a leading example. For example, the underlying graph of the MA resistor
network is the complete graph on an infinite simple point process, which is
completely different from the supercritical percolation cluster. This
geometric heterogeneity requires a geometric abstract setting to get the
desired universality.

Our proof is based on stochastic homogenization via 2-scale convergence (cf.
[16, 40, 41] and references therein) and the theory of simple point process.
In [41] Piatnitski and Zhikov have proved homogenization for the massive
Poisson equation $\lambda u+{\mathbb{L}}u=f$ by 2-scale convergence on bounded
domains also with mixed Dirichlet-Neumann b.c., ${\mathbb{L}}$ being the
generator of a diffusion in random environments. In [41, Section 7] the above
result has been applied to get that the magnitude of the effective homogenized
matrix $D$ equals the limiting rescaled “directional conductivity” for a
diffusion on the skeleton of the supercritical percolation cluster in
Bernoulli bond percolation. The proof relies on the a priori check that $D>0$,
based on previous results on left–right crossings valid in the Bernoulli case.
We have developed here a direct proof of the scaling limit of the direction
conductivity, which avoids the constraint $D>0$ (whose check usually requires
further assumptions) and previous investigations of the massive Poisson
equation (which would require the cut-off procedures developed in [16,
Sections 14,16] in order to deal with arbitrarily long conductances). In
general, we avoid any assumption on the left-right crossings of the resistor
network (usually of difficult investigation if the FKG inequality is
violated). We stress that the existing proofs for random diffusions also with
different b.c. (cf. [26, 41]) do not adapt well to our general discrete
setting as the presence of arbitrarily long conductances in an amorphous
setting forces to deal with amorphous local gradients, which keep trace of the
function variation along any filament exiting from any given point and which
are very irregular objects.

Concerning previous results, we point out that the case of i.i.d. random
conductances between nearest-neighbor sites of ${\mathbb{Z}}^{d}$ with value
in a fixed interval $(\delta_{0},1-\delta_{0})$, $\delta_{0}>0$, has been
considered by Kozlov in [30], but - as discussed in Section 3 below - in [30]
some Neumann b.c. for the electric potential are imposed at the microscopic
level, while in general they are not satisfied in the presence of micro-
inhomogeneities. As stated for example in [7], in the case of stationary
ergodic random conductances between nearest-neighbor sites of
${\mathbb{Z}}^{d}$ having value in $(\delta_{0},1-\delta_{0})$
($\delta_{0}>0$) and with potential at the boundary of the box given by a
fixed linear function, one can prove the scaling limit of the dissipated
energy by adapting the methods developed for the continuous case (cf. [26, 29,
35] and the technical results collected in [31]). We also point out that
previously, in [39, p. 26], Zhikov obtained the scaling limit of the
“directional conductivity” of the standard diffusion with partial Dirichlet
b.c. in a perforated domain built by fattening the supercritical percolation
cluster. We point out that the b.c. in [39, Eq. (1.22)] does not correspond to
the effective one for the resistor network on the supercritical percolation
cluster as the Neumann part is missing.

As a further step of investigation we plan to derive quantitative results on
the scaling of the directional conductivity at cost of additional technical
assumptions (cf. e.g. [2] for some quantitative stochastic homogenization
results on the supercritical percolation cluster on ${\mathbb{Z}}^{d}$).
Finally, we point out that the present work is an extension and improvement of
our unpublished notes [15].

Outline of the paper. In Section 2 we present our models and main results
(i.e. Theorems 1, 2, 3 and Proposition 2.8). In Section 3 we discuss relevant
examples. The rest of the paper is devoted to proofs.

##  2\. Models and main results

We start with a probability space $(\Omega,\mathcal{F},\mathcal{P})$ encoding
all the randomness of the system. Elements $\omega$ of $\Omega$ are called
_environments_. We denote by ${\mathbb{E}}[\cdot]$ the expectation associated
to $\mathcal{P}$.

We denote by $\mathcal{N}$ the space of locally finite subsets
$\\{x_{i}\\}\subset{\mathbb{R}}^{d}$, $d\geq 1$. As common, we will identify
the set $\\{x_{i}\\}$ with the counting measure $\sum_{i}\delta_{x_{i}}$. In
particular, if $\xi=\\{x_{i}\\}\in\mathcal{N}$, then $\int
d\xi(x)f(x)=\sum_{i}f(x_{i})$ and $\xi(A)=\sharp(\\{x_{i}\\}\cap A)$ for
$f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ and $A\subset{\mathbb{R}}^{d}$. On
$\mathcal{N}$ one defines a special metric $d$ (cf. [11, App. A2.6]) such that
a sequence $(\xi_{n})$ converges to $\xi$ in $\mathcal{N}$ if
$\lim_{n\to\infty}\int d\xi_{n}(x)f(x)=\int d\xi(x)f(x)$ for any bounded
continuous function $f:{\mathbb{R}}^{d}\to{\mathbb{R}}$ vanishing outside a
bounded set. Then the $\sigma$–algebra of Borel sets of $(\mathcal{N},d)$ is
generated by the sets $\\{\xi\in\mathcal{N}\,:\,\xi(A)=k\\}$ with $A$ and $k$
varying respectively among the Borel sets of ${\mathbb{R}}^{d}$ and in
${\mathbb{N}}$. In what follows, we think of $\mathcal{N}$ as measure space
endowed with the $\sigma$–algebra of Borel sets.

We consider a _simple point process_ on ${\mathbb{R}}^{d}$ defined on
$(\Omega,\mathcal{F},\mathcal{P})$, i.e. a measurable map
$\Omega\ni\omega\mapsto\hat{\omega}\in\mathcal{N}$. We also consider the
_group_ ${\mathbb{G}}$ given by ${\mathbb{R}}^{d}$ or ${\mathbb{Z}}^{d}$
acting on the Euclidean space ${\mathbb{R}}^{d}$ by the translations
$\tau_{g}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$, where $\tau_{g}x=x+g$.

######  Warning 2.1.

To simplify here the presentation, when ${\mathbb{G}}={\mathbb{Z}}^{d}$ we
assume that $\hat{\omega}\subset{\mathbb{Z}}^{d}$ for all $\omega\in\Omega$
(in Section 2.1 we will remove this assumption).

We assume that ${\mathbb{G}}$ acts also on $\Omega$ and, with a slight abuse
of notation made non ambiguous by the context, we denote by
$(\tau_{g})_{g\in{\mathbb{G}}}$ also the action of ${\mathbb{G}}$ on $\Omega$.
In particular, this action is given by a family of ${\mathbb{G}}$–parametrized
maps $\tau_{g}:\Omega\to\Omega$ such that $\tau_{0}={\mathbb{I}}$,
$\tau_{g}\circ\tau_{g^{\prime}}=\tau_{g+g^{\prime}}$ for all
$g,g^{\prime}\in{\mathbb{G}}$,
${\mathbb{G}}\times\Omega\ni(g,\omega)\mapsto\tau_{g}\omega$ is measurable
(${\mathbb{R}}^{d}$, ${\mathbb{Z}}^{d}$ are endowed with the Euclidean metric
and the discrete topology, respectively). As common, a subset $A\subset\Omega$
is called _translation invariant_ if $\tau_{g}A=A$ for all $g\in{\mathbb{G}}$.
The name comes from the fact that the action of ${\mathbb{G}}$ on $\Omega$
describes how the environment changes when applying translations on the
Euclidean space (cf. Assumption (A4) below). We will assume that $\mathcal{P}$
is stationary and ergodic w.r.t. the action of ${\mathbb{G}}$ on $\Omega$. We
recall that stationarity means that $\mathcal{P}(\tau_{g}A)=A$ for any
$A\in\mathcal{F}$ and $g\in{\mathbb{G}}$, while ergodicity means that
$\mathcal{P}(A)\in\\{0,1\\}$ for any translation invariant set
$A\in\mathcal{F}$.

Due to our assumptions stated below, the simple point process has finite
positive _intensity_ $m$, where

$m:=\begin{cases}{\mathbb{E}}\bigl{[}\hat{\omega}([0,1]^{d})\bigr{]}&\text{ if
}{\mathbb{G}}={\mathbb{R}}^{d}\,,\\\ \mathcal{P}(0\in\hat{\omega})&\text{ if
}{\mathbb{G}}={\mathbb{Z}}^{d}\,.\end{cases}$ (3)

As a consequence, the _Palm distribution_ $\mathcal{P}_{0}$ associated to the
simple point process is well defined (cf. [16, Section 2] and references
therein). We recall that $\mathcal{P}_{0}$ is the probability measure on
$(\Omega,\mathcal{F})$ with support in

$\Omega_{0}:=\\{\omega\in\Omega\,:\,0\in\hat{\omega}\\}\,,$ (4)

such that, for any $A\in\mathcal{F}$,

$\mathcal{P}_{0}(A):=\begin{cases}\frac{1}{m}\int_{\Omega}d\mathcal{P}(\omega)\int_{[0,1]^{d}}d\hat{\omega}(x)\mathds{1}_{A}(\tau_{x}\omega)&\text{
if }{\mathbb{G}}={\mathbb{R}}^{d}\,,\\\ \mathcal{P}(A|\Omega_{0})&\text{ if
}{\mathbb{G}}={\mathbb{Z}}^{d}\,.\end{cases}$ (5)

In the rest, we will denote by ${\mathbb{E}}_{0}[\cdot]$ the expectation
w.r.t. $\mathcal{P}_{0}$.

We fix a measurable function (describing the _random conductance field_)

${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\times\Omega\ni(x,y,\omega)\mapsto
c_{x,y}(\omega)\in[0,+\infty)$

such that $c_{x,x}(\omega)=0$ for all $x\in{\mathbb{R}}^{d}$. The value of
$c_{x,y}(\omega)$ will be relevant only for $x\not=y$ in $\hat{\omega}$. For
later use we define the function $\lambda_{k}:\Omega_{0}\to[0,+\infty]$ as

$\lambda_{k}(\omega):=\int_{{\mathbb{R}}^{d}}d\hat{\omega}(x)c_{0,x}(\omega)|x|^{k}\,,$
(6)

where $|x|$ denotes the Euclidean norm of $x\in{\mathbb{R}}^{d}$.

Recall that temporary assumption in Warning 2.1

Assumptions. We make the following assumptions:

  * (A1)

$\mathcal{P}$ is stationary and ergodic w.r.t. the action of ${\mathbb{G}}$ on
$\Omega$;

  * (A2)

the intensity $m$ given in (3) is finite and positive;

  * (A3)

$\mathcal{P}(\omega\in\Omega:\tau_{g}\omega\not=\tau_{g^{\prime}}\omega\;\forall
g\not=g^{\prime}\text{ in }{\mathbb{G}})=1$;

  * (A4)

for all $\omega\in\Omega$, $g\in{\mathbb{G}}$ and
$x,y\in\widehat{\tau_{g}\omega}$, it holds

$\displaystyle\widehat{\tau_{g}\omega}=\tau_{-g}(\hat{\omega})\,,$ (7)
$\displaystyle c_{x,y}(\tau_{g}\omega)=c_{\tau_{g}x,\tau_{g}y}(\omega)\,;$ (8)

  * (A5)

for all $\omega\in\Omega$ the weights $c_{x,y}(\omega)$ are symmetric, i.e.
$c_{x,y}(\omega)=c_{y,x}(\omega)$ $\forall x,y\in\hat{\omega}$;

  * (A6)

for $\mathcal{P}$–a.a. $\omega$ the graph with vertex set $\hat{\omega}$ and
edges given by $\\{x,y\\}$ with $x\not=y$ in $\hat{\omega}$ and
$c_{x,y}(\omega)>0$ is connected;

  * (A7)

$\lambda_{0},\lambda_{2}\in L^{1}(\mathcal{P}_{0})$;

  * (A8)

$L^{2}(\mathcal{P}_{0})$ is separable.

We point out that the above assumptions are the same presented in [16, Section
2.4] when the rates $r_{x,y}(\omega)$ there are symmetric (hence coinciding
with our $c_{x,y}(\omega)$). We now comment the above assumptions (recalling
also some remarks from [16]). Due to [11, Proposition 10.1.IV], (A1) and (A2),
for $\mathcal{P}$–a.a. $\omega$ the set $\hat{\omega}$ is infinite. (A3)
implicitly includes that the event under consideration is measurable as common
in the applications (measurability is automatic if
${\mathbb{G}}={\mathbb{Z}}^{d}$). (A3) is usually a rather fictitious
assumption. Indeed, usually by free one can add some randomness enlarging
$\Omega$ to assure (A3) (similarly to [12, Remark 4.2-(i)]). For example, if
$(\Omega,\mathcal{F},\mathcal{P})$ is simply the space $\mathcal{N}$ endowed
with a probability measure and if it describes a simple point process on
${\mathbb{R}}^{d}$ obtained by spatially periodizing a simple point process on
$[0,1)^{d}$, then to gain (A3) it would be enough to mark points by i.i.d.
random variables with non-degenerate distribution. (A4) describes how the
Euclidean translations influence the randomness. (A5) is natural due to the
interpretation of conductance of $c_{x,y}(\omega)$ discussed below. (A6) is a
technical assumption assuring that a measurable function $u$ on $\Omega_{0}$
such that, $\mathcal{P}_{0}$–a.s., $u(\tau_{x}\omega)=u(\omega)$ for all
$x\in\hat{\omega}$ with $c_{0,x}(\omega)>0$ is constant $\mathcal{P}_{0}$–a.s.
(cf. [16][Lemma 8.5]). In Section 3.2 we will discuss a relevant example where
(A6) does not hold but anyway the application of our Theorem 1 allows to
derive the scaling limit of the directional conductivity. By [8, Theorem 4.13]
(A8) is fulfilled if $(\Omega_{0},\mathcal{F}_{0},\mathcal{P}_{0})$ is a
separable measure space where
$\mathcal{F}_{0}:=\\{A\cap\Omega_{0}\,:\,A\in\mathcal{F}\\}$ (i.e. there is a
countable family $\mathcal{G}\subset\mathcal{F}_{0}$ such that the
$\sigma$–algebra $\mathcal{F}_{0}$ is generated by $\mathcal{G}$). For
example, if $\Omega_{0}$ is a separable metric space and
$\mathcal{F}_{0}=\mathcal{B}(\Omega_{0})$ (which is valid if $\Omega$ is a
separable metric space and $\mathcal{F}=\mathcal{B}(\Omega)$) then (cf. [8, p.
98]) $(\Omega_{0},\mathcal{F}_{0},\mathcal{P}_{0})$ is a separable measure
space and (A8) is valid. Note that we are not assuming that $\Omega$ is a
compact metric space as in e.g. [41], hence (A8) becomes relevant to have
countable families of test functions for the 2-scale convergence (we refer to
[16] for further comments on this issue).

######  Definition 2.1.

We define the effective homogenized matrix $D$ as the $d\times d$ nonnegative
symmetric matrix such that

$a\cdot Da=\inf_{f\in L^{\infty}(\mathcal{P}_{0})}\frac{1}{2}\int
d\mathcal{P}_{0}(\omega)\int d\hat{\omega}(x)c_{0,x}(\omega)\left(a\cdot
x-\nabla f(\omega,x)\right)^{2}\,,$ (9)

where $\nabla f(\omega,x):=f(\tau_{x}\omega)-f(\omega)$.

By (A7) the above definition is well posed. In general, $a\cdot b$ denotes the
Euclidean scalar product of the vectors $a$ and $b$.

Given $\ell>0$ we consider the box, stripe and half-stripes111The term
_stripe_ is appropriate for $d=2$. We keep the same terminology for all
dimensions $d$.

$\begin{cases}\Lambda_{\ell}:=(-\ell/2,\ell/2)^{d}\,,&S_{\ell}:={\mathbb{R}}\times(-\ell/2,\ell/2)^{d-1}\\\
S_{\ell}^{-}:=\\{x\in
S_{\ell}\,:\,x_{1}\leq-\ell/2\\}\,,&S_{\ell}^{+}:=\\{x\in
S_{\ell}\,:\,x_{1}\geq\ell/2\\}\,.\end{cases}$ (10)

######  Warning 2.2.

We denote by $e_{1},\dots,e_{d}$ the canonical basis of ${\mathbb{R}}^{d}$. In
what follows we focus on the direction determined by $e_{1}$. In the general
case, when considering the direction determined by a unit vector $e$, one has
just to refer our results to the regions $O(\Lambda_{\ell})$, $O(S_{\ell})$,
$O(S_{\ell}^{-})$ and $O(S_{\ell}^{+})$, where $O$ is a fixed orthogonal
linear map such that $O(e_{1})=e$.

We define $\Omega_{1}$ as the set of $\omega\in\Omega$ satisfying the
connectivity property in (A6) and the bounds (cf. (6))

$\lambda_{0}(\tau_{x}\omega)=\sum_{y\in\hat{\omega}}c_{x,y}(\omega)<+\infty\;\;\;\forall
x\in\hat{\omega}\,.$ (11)

Note that $\Omega_{1}$ is a translation invariant measurable set with
$\mathcal{P}(\Omega_{1})=1$ (use (A7) and Lemma 5.1 below).

######  Definition 2.2 (Resistor network ${\rm(RN)}^{\omega}_{\ell}$).

Given $\omega\in\Omega_{1}$ we consider the $\ell$–parametrized resistor
network ${\rm(RN)}^{\omega}_{\ell}$ on $S_{\ell}$ with node set
$\hat{\omega}\cap S_{\ell}$. To each unordered pair of nodes $\\{x,y\\}$, such
that $\\{x,y\\}\cap\Lambda_{\ell}\not=\emptyset$ and $c_{x,y}(\omega)>0$, we
associate an electrical filament of conductance $c_{x,y}(\omega)$ (see Figure
1-(left)).

We can think of ${\rm(RN)}^{\omega}_{\ell}$ as a weighted non-oriented graph
with vertex set $\hat{\omega}\cap S_{\ell}$, edge set

${\mathbb{B}}^{\omega}_{\ell}:=\bigl{\\{}\\{x,y\\}\subset(\hat{\omega}\cap
S_{\ell})\,:\,\\{x,y\\}\cap\Lambda_{\ell}\not=\emptyset\,,\;c_{x,y}(\omega)>0\bigr{\\}}$
(12)

and weight of the edge $\\{x,y\\}$ given by the conductance $c_{x,y}(\omega)$.

Figure 1. (Left) The resistor network ${\rm(RN)}^{\omega}_{\ell}$,
conductances are omitted. The box and the stripe correspond to
$\Lambda_{\ell}$ and $S_{\ell}$, respectively. (Right) Functions in
$H^{1,\varepsilon}_{0,\omega}$ take any value on the nodes surrounded by an
annulus, and have value zero on all other nodes (the box corresponds here to
$\Lambda$ and has side length $1$, the underlying graph is
$\mathcal{G}^{\varepsilon}_{\omega}$).

######  Definition 2.3 (Electric potential).

Given $\omega\in\Omega_{1}$ we denote by $V_{\ell}^{\omega}$ the _electric
potential_ of the resistor network (RN)${}^{\omega}_{\ell}$ with values $0$
and $1$ on $S_{\ell}^{-}$ and $S_{\ell}^{+}$, respectively, taken by
convention equal to zero on the connected components of
(RN)${}^{\omega}_{\ell}$ included in $\Lambda_{\ell}$. In particular,
$V_{\ell}^{\omega}$ is the unique function $V_{\ell}^{\omega}:\hat{\omega}\cap
S_{\ell}\to{\mathbb{R}}$ such that

${\sum}_{y\in\hat{\omega}\cap
S_{\ell}}c_{x,y}(\omega)\left(V^{\omega}_{\ell}(y)-V^{\omega}_{\ell}(x)\right)=0\qquad\forall
x\in\hat{\omega}\cap\Lambda_{\ell}\,,$ (13)

and satisfying

$\begin{cases}V^{\omega}_{\ell}(x)=0&\text{ if
}x\in\hat{\omega}\cap\Lambda_{\ell},\text{ $x$ is not connected to
$\hat{\omega}\setminus\Lambda_{\ell}$ in {\rm(RN)${}^{\omega}_{\ell}$}}\,,\\\
V^{\omega}_{\ell}(x)=0&\text{ if }x\in\hat{\omega}\cap S^{-}_{\ell}\,,\\\
V^{\omega}_{\ell}(x)=1&\text{ if }x\in\hat{\omega}\cap
S_{\ell}^{+}\,.\end{cases}$ (14)

Note that (13) corresponds to Kirchhoff’s law. As discussed in Section 6, the
above electric potential exists and is unique and has values in $[0,1]$. We
recall that, given $(x,y)$ with $\\{x,y\\}\in{\mathbb{B}}^{\omega}_{\ell}$
(cf. (12)),

$i_{x,y}(\omega):=c_{x,y}(\omega)\bigl{(}V^{\omega}_{\ell}(y)-V^{\omega}_{\ell}(x)\bigr{)}$
(15)

is the electric current flowing from $x$ to $y$ under the electric potential
$V^{\omega}_{\ell}$, due to Ohm’s law. For simplicity, $\ell$ is understood in
the notation $i_{x,y}(\omega)$.

######  Definition 2.4 (Directional effective conductivity).

Given $\omega\in\Omega_{1}$ we call $\sigma_{\ell}(\omega)$ the _effective
conductivity_ of the resistor network $({\rm RN})^{\omega}_{\ell}$ along the
first direction under the electric potential $V^{\omega}_{\ell}$. More
precisely, $\sigma_{\ell}(\omega)$ is given by

$\begin{split}\sigma_{\ell}(\omega)&:=\sum_{x\in\hat{\omega}\cap
S_{\ell}^{-}}\;\sum_{y\in\hat{\omega}\cap\Lambda_{\ell}}i_{x,y}(\omega)=\sum_{x\in\hat{\omega}\cap
S_{\ell}^{-}}\sum_{y\in\hat{\omega}\cap\Lambda_{\ell}}c_{x,y}(\omega)\bigl{(}V_{\ell}^{\omega}(y)-V_{\ell}^{\omega}(x)\bigr{)}\,.\end{split}$

It is simple to check that, for any $\gamma\in[-\ell/2,\ell/2)$,
$\sigma_{\ell}(\omega)$ equals the current flowing through the hyperplane
$\\{x\in{\mathbb{R}}^{d}\,:\,x_{1}=\gamma\\}$:

$\sigma_{\ell}(\omega)=\sum_{\begin{subarray}{c}x\in\hat{\omega}\cap
S_{\ell}:\\\
x_{1}\leq\gamma\end{subarray}}\;\sum_{\begin{subarray}{c}y\in\hat{\omega}\cap
S_{\ell}:\\\
\\{x,y\\}\in{\mathbb{B}}^{\omega}_{\ell},\,y_{1}>\gamma\end{subarray}}i_{x,y}(\omega)\,.$
(16)

$\sigma_{\ell}(\omega)$ also equals the total dissipated energy:

$\sigma_{\ell}(\omega)=\sum_{\\{x,y\\}\in{\mathbb{B}}^{\omega}_{\ell}}c_{x,y}(\omega)\bigl{(}V^{\omega}_{\ell}(x)-V^{\omega}_{\ell}(y)\bigr{)}^{2}\,.$
(17)

Indeed, by collapsing all nodes in $\hat{\omega}\cap S_{\ell}^{-}$ into a
single node and similarly for $\hat{\omega}\cap S_{\ell}^{+}$, one reduces to
the same setting of [13, Section 1.3] where (17) is proved.

We can now state our first main result concerning the infinite volume
asymptotics of $\sigma_{\ell}(\omega)$ (the proof is given in Sections 7 and
14):

######  Theorem 1.

Suppose that $e_{1}$ is an eigenvector of $D$. Then there exists a translation
invariant measurable set $\Omega_{\rm typ}\subset\Omega_{1}$ with
$\mathcal{P}(\Omega_{\rm typ})=1$ such for all $\omega\in\Omega_{\rm typ}$ it
holds $\lim_{\ell\to+\infty}\ell^{2-d}\sigma_{\ell}(\omega)=mD_{1,1}$.

The notation $\Omega_{\rm typ}$ refers to the fact that elements of
$\Omega_{\rm typ}$ are typical environments, as $\mathcal{P}(\Omega_{\rm
typ})=1$. From

######  Remark 2.5.

From the description of $\Omega_{\rm typ}$ in Sections 7 and 10 and from the
proof, it is simple to check that the same set $\Omega_{\rm typ}$ leads to the
conclusion of Theorem 1 (and similarly of Proposition 2.8 and Theorem 2 below)
for any eigenvector of $D$. Similarly, the set $\Omega_{\rm typ}$ leads to the
conclusion of Theorem 3 below for any vector in ${\rm Ker}(D)^{\perp}$.

To clarify the link with homogenization and state our further results, it is
convenient to rescale space in order to deal with fixed stripe and box. More
precisely, we set $\varepsilon:=1/\ell$. Then $\varepsilon>0$ is our scaling
parameter. We set

$\begin{cases}\Lambda:=(-1/2,1/2)^{d}\,,&S:={\mathbb{R}}\times(-1/2,1/2)^{d-1}\,,\\\
S^{-}:=\\{x\in S:x_{1}\leq-1/2\\}\,,&S^{+}:=\\{x\in S:x_{1}\geq
1/2\\}\,.\end{cases}$ (18)

Note that $\Lambda_{\ell}=\ell\Lambda$, $S_{\ell}=\ell S$,
$S_{\ell}^{\pm}=\ell S^{\pm}$. Here and below, $\omega\in\Omega_{1}$. We write
$V_{\varepsilon}:\varepsilon\hat{\omega}\cap S\to[0,1]$ for the function given
by $V_{\varepsilon}(\varepsilon x):=V_{\ell}^{\omega}(x)$ (note that the
dependence on $\omega$ in $V_{\varepsilon}$ is understood, as for other
objects below).

We introduce the atomic measures

$\mu^{\varepsilon}_{\omega,\Lambda}:=\varepsilon^{d}\sum_{x\in\varepsilon\hat{\omega}\cap\Lambda}\delta_{x}\,,\qquad\nu^{\varepsilon}_{\omega,\Lambda}:=\varepsilon^{d}\sum_{(x,y)\in\mathcal{E}_{\varepsilon}(\omega)}c_{x/\varepsilon,y/\varepsilon}(\omega)\delta_{(x,(y-x)/\varepsilon)}\,,$
(19)

where

$\mathcal{E}_{\varepsilon}(\omega):=\\{(x,y)\,:\,x,y\in\varepsilon\hat{\omega}\cap
S,\,c_{x,y}(\omega)>0\text{ and }\\{x,y\\}\cap\Lambda\not=\emptyset\\}\,.$
(20)

Note that $\mu^{\varepsilon}_{\omega,\Lambda}$ and
$\nu^{\varepsilon}_{\omega,\Lambda}$ have finite total mass (for the latter
use that $\omega\in\Omega_{1}$).

Given a function $f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$, we define
the _amorphous gradient_ $\nabla_{\varepsilon}f$ on pairs $(x,z)$ with
$x\in\varepsilon\hat{\omega}\cap S$ and $x+\varepsilon
z\in\varepsilon\hat{\omega}\cap S$ as

$\nabla_{\varepsilon}f(x,z)=\frac{f(x+\varepsilon z)-f(x)}{\varepsilon}\,.$
(21)

Moreover, we define the operator

${\mathbb{L}}^{\varepsilon}_{\omega}f(x):=\varepsilon^{-2}\sum_{y\in\varepsilon\hat{\omega}\cap
S}c_{x/\varepsilon,y/\varepsilon}\left[f(y)-f(x)\right]\,,\qquad
x\in\varepsilon\hat{\omega}\cap\Lambda\,,$ (22)

whenever the series in the r.h.s. is absolutely convergent. Let
$f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$ be a bounded function. Then
${\mathbb{L}}^{\varepsilon}_{\omega}f(x)$ is well defined for all
$x\in\varepsilon\hat{\omega}\cap\Lambda$ as $\omega\in\Omega_{1}$. As the
amorphous gradient $\nabla_{\varepsilon}f$ is bounded too, we have that
$\nabla_{\varepsilon}f\in L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})$.
Moreover, if in addition $f$ is zero outside $\Lambda$, it holds (cf. Lemma
6.1)

$\langle
f,-{\mathbb{L}}^{\varepsilon}_{\omega}f\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}=\frac{1}{2}\langle\nabla_{\varepsilon}f,\nabla_{\varepsilon}f\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,.$
(23)

######  Definition 2.6 (Graph $\mathcal{G}_{\omega}^{\varepsilon}$ and sets
$\mathcal{C}_{\omega,\Lambda}^{\varepsilon}$,
$\mathcal{C}_{\omega}^{\varepsilon}$).

Given $\varepsilon>0$ and $\omega\in\Omega_{1}$, we consider the non-oriented
graph $\mathcal{G}_{\omega}^{\varepsilon}$ with vertex set
$\varepsilon\hat{\omega}\cap S$ and edges given by the unordered pairs
$\\{x,y\\}$ such that $c_{x,y}(\omega)>0$ and $\\{x,y\\}$ intersects
$\Lambda$. We write $\mathcal{C}_{\omega,\Lambda}^{\varepsilon}$ and
$\mathcal{C}_{\omega}^{\varepsilon}$ for the union of the connected components
in $\mathcal{G}_{\omega}^{\varepsilon}$ included in $\Lambda$ and,
respectively, intersecting $S^{-}\cup S^{+}$.

$\mathcal{G}_{\omega}^{\varepsilon}$ equals the $\varepsilon$-rescaling of the
graph obtained by disregarding the weights in the weighted graph
(RN)${}^{\omega}_{\ell}$, where $\ell=1/\varepsilon$. Note that
$\mathcal{G}_{\omega}^{\varepsilon}$ has no edge between
$\varepsilon\hat{\omega}\cap S^{-}$ and $\varepsilon\hat{\omega}\cap S^{+}$
and that $\varepsilon\hat{\omega}\cap
S=\mathcal{C}_{\omega,\Lambda}^{\varepsilon}\cup\mathcal{C}_{\omega}^{\varepsilon}$.
The edges of $\mathcal{G}_{\omega}^{\varepsilon}$ coincide with the edges
obtained from $\mathcal{E}_{\varepsilon}(\omega)$ when disregarding the
orientation.

######  Definition 2.7 (Functional spaces $H^{1,\varepsilon}_{0,\omega}$,
$K^{\varepsilon}_{\omega}$).

Given $\omega\in\Omega_{1}$ we define the Hilbert space

$H^{1,\varepsilon}_{0,\omega}:=\Big{\\{}f:\varepsilon\hat{\omega}\cap
S\to{\mathbb{R}}\text{ such that }f(x)=0\;\;\forall
x\in\mathcal{C}^{\varepsilon}_{\omega,\Lambda}\cup(\mathcal{C}^{\varepsilon}_{\omega}\setminus\Lambda)\Big{\\}}$

endowed with the squared norm
$\|f\|^{2}_{H^{1,\varepsilon}_{0,\omega}}:=\|f\|^{2}_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}+\|\nabla_{\varepsilon}f\|^{2}_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$.
In addition, we define $K^{\varepsilon}_{\omega}$ as the set of functions
$f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$ such that $f(x)=0$ for all
$x\in(\varepsilon\hat{\omega}\cap
S^{-})\cup\mathcal{C}_{\omega,\Lambda}^{\varepsilon}$ and $f(x)=1$ for all
$x\in\varepsilon\hat{\omega}\cap S^{+}$.

We refer to Figure 1-(right) for a graphical comment on the graph
$\mathcal{G}^{\varepsilon}_{\omega}$ and functions in
$H^{1,\varepsilon}_{0,\omega}$. Given $\omega\in\Omega_{1}$, in Section 6 we
will derive that, due to (13) and (14), $V_{\varepsilon}$ is the unique
function in $K^{\varepsilon}_{\omega}$ such that
${\mathbb{L}}^{\varepsilon}_{\omega}V_{\varepsilon}(x)=0$ for all
$x\in\varepsilon\hat{\omega}\cap\Lambda$ (cf. Lemma 6.2). We point out that,
by (17), the rescaled conductivity $\ell^{2-d}\sigma_{\ell}(\omega)$ equals
the flow energy associated to $V_{\varepsilon}$:

$\ell^{2-d}\sigma_{\ell}(\omega)=\frac{1}{2}\langle\nabla_{\varepsilon}V_{\varepsilon},\nabla_{\varepsilon}V_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\,.$
(24)

The limit in Theorem 1 can therefore be restated as

$\lim_{\varepsilon\downarrow
0}\frac{1}{2}\langle\nabla_{\varepsilon}V_{\varepsilon},\nabla_{\varepsilon}V_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}=mD_{1,1}\,,\;\;\forall\omega\in\Omega_{\rm
typ}\,.$ (25)

To prove Theorem 1 we will distinguish the cases $D_{1,1}=0$ and $D_{1,1}>0$.
The proof for $D_{1,1}=0$ (which is simpler) is given in Section 7, while the
proof for $D_{1,1}>0$ will take the rest of our investigation and will be
concluded in Section 14. In the case $D_{1,1}>0$ we can say more on the
behavior of $V_{\varepsilon}$. To this aim we introduce the function
$\psi:S\to[0,1]$ as

$\psi(x):=x_{1}+\frac{1}{2}\text{ if }x\in\Lambda\,,\;\;\;\psi(x):=0\text{ if
}x\in S^{-}\,,\;\;\;\psi(x):=1\text{ if }x\in S^{+}\,.$ (26)

######  Proposition 2.8.

Suppose that $e_{1}$ is an eigenvector of $D$ with eigenvalue $D_{1,1}>0$.
Then, for all $\omega\in\Omega_{\rm typ}$, $V_{\varepsilon}\in
L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})$ converges weakly and 2-scale
converges weakly to $\psi\in L^{2}(\Lambda,dx)$.

The definition of the above convergences is recalled in Section 11.

As discussed in Section 8, $\psi$ is the unique weak solution on $\Lambda$ of
the so–called effective equation given by $\nabla_{*}\cdot(D\nabla_{*}\psi)=0$
with suitable mixed Dirichlet-Neumann boundary conditions, where $\nabla_{*}$
denotes the projection of the standard gradient $\nabla$ on the subspace of
${\mathbb{R}}^{d}$ orthogonal to the kernel of $D$ (cf. Definition 8.7). Due
to Proposition 2.8, $\nabla_{*}\cdot(D\nabla_{*}\psi)=0$ represents the
effective macroscopic equation of the electric potential $V_{\varepsilon}$ in
the limit $\varepsilon\downarrow 0$, when $D_{1,1}>0$.

###  2.1. Extensions

We detail here the extended setting where our Theorem 1 and Proposition 2.8
still hold, with suitable modifications.

First we point out that the action of the group ${\mathbb{G}}$ on
${\mathbb{R}}^{d}$ can be a generic proper action, i.e. given by translations
$\tau_{g}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ of the form
$\tau_{g}x:=x+g_{1}v_{1}+\cdots+g_{d}v_{d}$ for a fixed basis
$v_{1},v_{2},\dots,v_{d}$. Indeed, one can always reduce to the previous case
$\tau_{g}x=x+g$ at cost of a linear isomorphism of the Euclidean space (we
refer to [16, Section 2] for the form of $D$ when applying a generic proper
action).

Let us keep in what follows $\tau_{g}x:=x+g$ for $x\in{\mathbb{R}}^{d}$ and
$g\in{\mathbb{G}}$. Before (see Warning 2.1), to simplify the presentation,
for ${\mathbb{G}}={\mathbb{Z}}^{d}$ we have restricted to the case
$\hat{\omega}\subset{\mathbb{Z}}^{d}$ for all $\omega\in\Omega$. This is a
real restriction, since it does not allow to cover interesting examples given
e.g. by crystal lattices (consider for example a random conductance model on
the hexagonal lattice, see [16, Section 5.6]). For the general case we have
the following result:

######  Theorem 2.

When ${\mathbb{G}}={\mathbb{Z}}^{d}$ and not necessarily
$\hat{\omega}\subset{\mathbb{Z}}^{d}$, the content of Theorem 1 and
Proposition 2.8 remain valid under Assumptions (A1),..,(A8) with $m$,
$\mathcal{P}_{0}$, $\lambda_{k}$ and $D$ defined respectively as in [16, Eq.
(10)],[16, Eq. (11)], [16, Eq. (15)] and [16, Def. 3.6].

For the reader’s convenience, we point out that in the above formulas from
[16] one has to replace $\theta_{g}$ by $\tau_{g}$, $r_{x,y}(\omega)$ by
$c_{x,y}(\omega)$, $\mu_{\omega}$ by $\hat{\omega}$, $\Delta$ by $[0,1)^{d}$.

###  2.2. Beyond principal directions

We explain here how to modify the definition of directional conductivity to
get a quenched scaling limit along directions determined by unit vectors in
${\rm Ker}(D)^{\perp}$, the subspace perpendicular to the kernel of $D$ ($D$
is thought of as linear map on ${\mathbb{R}}^{d}$). Obstructions when the unit
vector does not belong to ${\rm Ker}(D)^{\perp}$ are discussed in Section 8.1.
Trivially, ${\rm Ker}(D)^{\perp}$ is left invariant by the linear map
$x\mapsto Dx$. On ${\rm Ker}(D)^{\perp}$ this map is an isomorphism and the
quadratic form $x\mapsto x\cdot Dx$ is strictly positive definite.

Suppose that $e_{1}\in{\rm Ker}(D)^{\perp}$. We fix an orthonormal basis
$\mathfrak{e}_{1}=e_{1},\mathfrak{e}_{2},\dots,\mathfrak{e}_{d_{*}}$ of ${\rm
Ker}(D)^{\perp}$ and an orthonormal basis
$\mathfrak{e}_{d_{*}+1},\dots,\mathfrak{e}_{d}$ of ${\rm Ker}(D)$, where
$d_{*}:={\rm dim}\bigl{(}{\rm Ker}(D)^{\perp}\bigr{)}$ (this notation will be
in agreement with Definition 8.1 later).

We fix a unit vector ${\bf w}\in{\rm Ker}(D)^{\perp}$ orthogonal to
$D\mathfrak{e}_{2},\dots,D\mathfrak{e}_{d_{*}}$. Then ${\bf w}\cdot
e_{1}\not=0$ (by contradiction, if
$w=\alpha_{2}\mathfrak{e}_{2}+\dots\alpha_{d_{*}}\mathfrak{e}_{d_{*}}$, we
would have ${\bf w}\cdot D{\bf w}=\alpha_{i}\sum_{i=1}^{d_{*}}{\bf w}\cdot
D\mathfrak{e}_{i}=0$, thus implying that ${\bf w}\in{\rm Ker}(D)$). Hence,
without loss, we take ${\bf w}\cdot e_{1}>0$. Note that, by this constraint,
${\bf w}$ is unique as $D\mathfrak{e}_{2},\dots,D\mathfrak{e}_{d_{*}}$ are
$d_{*}-1$ independent vectors of the $d_{*}$–dimensional space ${\rm
Ker}(D)^{\perp}$.

We define $\Lambda$ as the open parallelepiped

$\Lambda:=\\{x\in{\mathbb{R}}^{d}\,:\,|{\bf w}\cdot x|<c{\bf w}\cdot
e_{1}\text{ and }|x\cdot\mathfrak{e}_{k}|<1/2\text{ for }2\leq k\leq d\\}\,,$

where $c$ is such that the volume of $\Lambda$ equals $1$. Note that for each
$x\in\Lambda$ the set $x+{\mathbb{R}}e_{1}$ is a segment between $F_{-}$ and
$F_{+}$, where $F_{\pm}:=\\{x\in\bar{\Lambda}\,:\,{\bf w}\cdot x=\pm c{\bf
w}\cdot e_{1}\\}$. We define
$S:=\\{x\in{\mathbb{R}}^{d}\,:\,|x\cdot\mathfrak{e}_{k}|<1/2\text{ for }2\leq
k\leq d\\}$. We set $S^{-}:=\\{x\in S\,:\,{\bf w}\cdot x\leq-c{\bf w}\cdot
e_{1}\\}$, $S^{+}:=\\{x\in S\,:\,{\bf w}\cdot x\geq c{\bf w}\cdot e_{1}\\}$,
$\Lambda_{\ell}:=\ell\Lambda$, $S_{\ell}:=\ell S$, $S_{\ell}^{\pm}:=\ell
S_{\pm}$. Keep Definitions 2.2, 2.3 and 2.4. We define $\psi:S\to{\mathbb{R}}$
by setting $\psi(x):=0$ for $x\in S^{-}$, $\psi(x):=1$ for $x\in S^{+}$ and
$\psi(x):={\bf a}\cdot(x+ce_{1})$ otherwise, where ${\bf a}:=(2c{\bf w}\cdot
e_{1})^{-1}{\bf w}$.

Comments motivating the above new setting are given in Section 8.1.

######  Theorem 3.

Suppose that $e_{1}\in{\rm Ker}(D)^{\perp}$. Then the conclusion of Theorem 1
and Proposition 2.8 holds, with the exception that in Theorem 1 the limiting
rescaled conductivity is now $m\,{\bf a}\cdot D{\bf a}$ instead of
$mD_{1,1}=m\,e_{1}\cdot De_{1}$. Moreover, the analogous of Theorem 2 remains
valid.

######  Warning 2.3.

Since the geometrical objects involved in Theorem 3 are slightly different, in
order to avoid some heavy notation, up to Section 14 we will deal with the
setting of Theorems 1 and 2 and Proposition 2.8. In Appendix A we will comment
how to modify our arguments to prove Theorem 3.

##  3\. Examples

Our results can be applied to plenty of random resistor networks. Our
assumptions (A1),…,(A8) (with the modifications indicated in Theorem 2 for the
general case with ${\mathbb{G}}={\mathbb{Z}}^{d}$) equal the ones in [16] when
$n_{x}(\omega)\equiv 1$ there (corresponding to the fact that
$r_{x,y}(\omega)$ in [16] is symmetric and equals our $c_{x,y}(\omega)$).
Hence, to all the examples discussed in [16, Section 5] with symmetric rates
one can apply the present Theorems 1, 2, 3 and Proposition 2.8. Below we focus
on three of these examples particularly relevant for transport in disordered
media, adding some comments. As some other examples discussed in [16] we just
mention resistor networks with random long conductances on ${\mathbb{Z}}^{d}$
and on crystal lattices. By the same techniques one can consider also e.g.
resistor networks on supercritical percolation clusters in continuum
percolation and on Delaunay triangulations [22].

###  3.1. Nearest-neighbor random conductance model on ${\mathbb{Z}}^{d}$

We take $\Omega:=(0,+\infty)^{{\mathbb{E}}_{d}}$, where ${\mathbb{E}}_{d}$ is
the set of non-oriented edges of the standard lattice ${\mathbb{Z}}^{d}$. We
take $\hat{\omega}:={\mathbb{Z}}^{d}$ for all $\omega\in\Omega$ and set
$c_{x,y}(\omega):=\omega_{\\{x,y\\}}$ if $\\{x,y\\}\in{\mathbb{E}}_{d}$ and
$c_{x,y}(\omega)=0$ otherwise. The group ${\mathbb{G}}={\mathbb{Z}}^{d}$ acts
in a natural way on ${\mathbb{R}}^{d}$ by translations and on $\Omega$ by the
maps $\tau_{g}\omega=(\omega_{x-g,y-g})_{\\{x,y\\}\in{\mathbb{E}}_{d}}$ if
$\omega=(\omega_{\\{x,y\\}})_{\\{x,y\\}\in{\mathbb{E}}_{d}}$. We recall that
Assumption (A3) is fictitious. Then all the other Assumptions (A1),(A2) and
(A4),..,(A8) are satisfied whenever $\mathcal{P}$ is stationary and ergodic
w.r.t. ${\mathbb{Z}}^{d}$–translations and ${\mathbb{E}}[c_{0,e}]<+\infty$ for
all $e$ in the canonical basis of ${\mathbb{Z}}^{d}$. A previous _almost_
result for the directional conductivity in the random conductance model on
${\mathbb{Z}}^{d}$ was provided by Kozlov in [30, Section 3] for i.i.d.
conductances with value in a strictly positive interval
$(\delta_{0},1-\delta_{0})$, $\delta_{0}>0$. Indeed, as mentioned in the
Introduction, in [30] the author introduced some Neumann boundary conditions
at the microscopic level (cf. [30, Eq. (3.3)] which in general are not
fulfilled by the electric potential (compare for example with [13] and [27,
Section 11.3]). To see why they can be violated, consider the resistor network
in Figure 2. When $\varepsilon\ll 1$, the resistor network is well
approximated by a linear series of $5$ electrical filaments with conductance
$1$. Hence, for $\varepsilon\ll 1$, the electrical potential is around $1/5$
at node $A$ and around $2/5$ at node $B$, hence it cannot coincide at $A$ and
$B$ as in [30, Eq. (3.3)] (note that in [30] vertical and horizontal faces are
exchanged w.r.t. our setting). Finally we point out that, as $D$ in [30,
Section 3] is a multiple of the identity, the special role played by the
principal directions of $D$ does not emerge there.

Figure 2. Square resistor network with conductances equal to $1$ on bold edges
and equal to $\varepsilon$ on thin edges. The electric potential is $0$ ($1$)
on the left (right) vertical side.

###  3.2. Resistor network on infinite percolation clusters

We take $d\geq 2$ and $\Omega=\\{0,1\\}^{{\mathbb{E}}_{d}}$.
${\mathbb{E}}_{d}$ and the action of ${\mathbb{G}}:={\mathbb{Z}}^{d}$ are as
in the previous example. Let $\mathcal{P}$ be a probability measure on
$\Omega$ stationary, ergodic and fulfilling (A3). We assume that for
$\mathcal{P}$–a.a. $\omega$ there exists a unique infinite connected component
$\mathcal{C}(\omega)\subset{\mathbb{Z}}^{d}$ in the graph given by the edges
$\\{x,y\\}$ in ${\mathbb{E}}_{d}$ with $\omega_{x,y}=1$. Given
$\omega\in\Omega$ we set $\hat{\omega}:=\mathcal{C}(\omega)$ when
$\mathcal{C}(\omega)$ exists and $\hat{\omega}=\emptyset$ otherwise. We set
$c_{x,y}(\omega):=1$ for all $x,y\in\hat{\omega}$ such that
$\\{x,y\\}\in{\mathbb{E}}_{d}$ and $\omega_{\\{x,y\\}}=1$, and
$c_{x,y}(\omega):=0$ otherwise. Then all assumptions (A1),…,(A8) are
satisfied.

To simplify the notation, we suppose that $\mathcal{P}$ is invariant by
coordinate permutations. Then by symmetry $D=c\mathbb{I}$ and Theorem 1
implies that, along any direction of ${\mathbb{R}}^{d}$, $\mathcal{P}$–a.s.
$\lim_{\ell\to+\infty}\ell^{2-d}\sigma_{\ell}(\omega)=c\,m$, where $m$ is the
probability that the origin belongs to the infinite cluster, i.e.
$m=\mathcal{P}(0\in\hat{\omega})$. Since
$\mathcal{P}_{0}:=\mathcal{P}(\cdot|0\in\hat{\omega})$, we can rewrite the
variational characterization of limiting value $c\,m$ as

$c\,m=\inf_{f}\frac{1}{2}\int
d\mathcal{P}(\omega)\sum_{\begin{subarray}{c}e:|e|=1\,,\\\
\omega_{\\{0,e\\}}=1\end{subarray}}\mathds{1}_{0\in\hat{\omega}}\left(e\cdot
e_{1}-\nabla f(\omega,e)\right)^{2}\,,$ (27)

where $f$ varies among all bounded Borel functions on
$\Omega_{0}=\\{0\in\hat{\omega}\\}$ and $\nabla
f(\omega,e):=f(\tau_{e}\omega)-f(\omega)$. This result solves the open problem
stated in [5, Problem 1.18], even without restricting to Bernoulli bond
percolation.

From now on, here and in the next paragraph, we focus on the case that
$\mathcal{P}$ is a Bernoulli bond percolation on ${\mathbb{Z}}^{d}$ with
supercritical parameter $p$ (which fulfills all the above properties). It is
known that $c>0$ (cf. [6, 12, 34]). The proof originally provided in [12,
Section 4.2] can now be simplified due to our result and due to [25]. Indeed,
calling $N_{\ell}(\omega)$ the maximal number of vertex-disjoint left-right
crossings of $\Lambda_{\ell}$ and calling $s_{j}$ the number of bonds in the
$j$–th path, as pointed out in [10, Eq. (3.3),(3.4)], it holds

$\sigma_{\ell}(\omega)\geq\sum_{j=1}^{N_{\ell}(\omega)}\frac{1}{s_{j}}\geq
N_{\ell}(\omega)^{2}\bigl{(}\sum_{j=1}^{N_{\ell}(\omega)}s_{j}\bigr{)}^{-1}\,.$
(28)

In fact, the first bound follows by computing exactly the conductivity of
$N_{\ell}(\omega)$ linear resistor chains, while the second bound follows from
Jensen’s inequality. Since, for some positive constant $C$, $\mathcal{P}$–a.s.
$N_{\ell}(\omega)\geq C\,\ell^{d-1}$ for $\ell$ large (combine [25, Remark (d)
in Section 5] with Borel-Cantelli lemma) and since
$\sum_{j=1}^{N_{\ell}(\omega)}s_{j}$ is upper bounded by the total number of
bonds in $\Lambda_{\ell}$, we conclude that
$c=m^{-1}\lim_{\ell\to+\infty}\ell^{2-d}\sigma_{\ell}(\omega)>0$.

Differently from [5, Problem 1.18], where the resistor network in
$\Lambda_{\ell}$ is made only by the bonds in the supercritical percolation
cluster, in e.g. [10] the resistor network in $\Lambda_{\ell}$ is made by all
bonds $\\{x,y\\}\in{\mathbb{E}}_{d}$ with $\omega_{\\{x,y\\}}=1$. On the other
hand, we claim that $\mathcal{P}$–a.s. the two resistor networks have the same
conductivity along a given direction for $\ell$ large enough (hence the
scaling limit is the same). Indeed, by combining Borel-Cantelli lemma with
Theorems (8.18) and (8.21) in [23], we get that the following holds for some
$C>0$ $\mathcal{P}$–a.s. eventually in $\ell$: for all
$x\in\Lambda_{\ell}\cap{\mathbb{Z}}^{d}$, if the cluster in the bond
percolation $\omega$ containing $x$ is finite, then it has radius at most
$C\ln\ell$. As a consequence of this result, fixed a given direction,
$\mathcal{P}$–a.s. the bonds not belonging to the infinite percolation cluster
do not contribute to the directional conductivity if the box side length
$\ell$ is large enough. As known, by similar arguments (apply [23, Theorem
(5.4)]) one gets that $\mathcal{P}$–a.s. the directional conductivity
$\sigma_{\ell}(\omega)$ is zero for $\ell$ large when considering the
subcritical bond percolation and the resistor network on $\Lambda_{\ell}$
given by all bonds $\\{x,y\\}\in{\mathbb{E}}_{d}$ with $\omega_{\\{x,y\\}}=1$.

Results similar to the above ones can be derived for site percolation or for
positive random conductances on the supercritical cluster as in [16, Section
5].

###  3.3. Miller-Abrahams (MA) random resistor network

The MA random resistor network plays a fundamental role in the study of
electron transport in amorphous media as doped semiconductors in the regime of
strong Anderson localization (cf. [4, 33, 36, 38]). With our notation, it is
obtained by taking as $\Omega$ the space of configurations of marked simple
point processes [11]. In particular, an element of $\Omega$ is a countable set
$\omega=\\{(x_{i},E_{i})\\}$ with $\\{x_{i}\\}\in\mathcal{N}$, such that if
$x_{i}=x_{j}$ then $E_{i}=E_{j}$. $E_{i}\in{\mathbb{R}}$ is a real number,
called _energy mark_ of $x_{i}$. The simple point process
$\Omega\ni\omega\mapsto\hat{\omega}\in\mathcal{N}$ is given by
$\hat{\omega}:=\\{x_{i}\\}$ if $\omega=\\{(x_{i},E_{i})\\}$. The group
${\mathbb{G}}:={\mathbb{R}}^{d}$ acts on $\Omega$ as
$\tau_{x}\omega:=\\{(x_{i}-x,E_{i})\\}$ if $\omega=\\{(x_{i},E_{i})\\}$.
Moreover, between any pair of nodes $x_{i}$ and $x_{j}$, there is a filament
with conductance (1). In the physical applications (cf. [21, 38] and
references therein), $\mathcal{P}$ is obtained by starting with a simple point
process $\\{x_{i}\\}$ with law $\hat{}\mathcal{P}$ and marking points with
i.i.d. random variables $\\{E_{i}\\}$ (independently from the rest) with
common distribution of the form $\nu(dE)=c\,|E|^{\alpha}dE$ with finite
support $[-A,A]$, for some exponent $\alpha\geq 0$ and some $A>0$. Then (A1)
and (A2) are satisfied if $\hat{}\mathcal{P}$ is stationary and ergodic (see
[11]) with positive finite intensity. (A3), (A4), (A5), (A6) are automatically
satisfied. As discussed in [16, Section 5.4] (A7) is satisfied when
${\mathbb{E}}[\hat{\omega}([0,1]^{d})^{2}]<+\infty$, while (A8) is always
satisfied.

As by Theorem 1 the asymptotic rescaled directional conductivities can be
expressed in terms of $D=D(\beta)$, which coincides with the diffusion matrix
of Mott’s random walk, we get the following:

######  Corollary 3.1.

The asymptotic rescaled directional conductivities of the MA resistor network,
along the principal directions of the effective homogenized $D(\beta)$,
satisfy for $\beta\uparrow+\infty$ the bounds in agreement with Mott’s law
obtained for the diffusion matrix of Mott’s random walk stated in [20, Thm. 1]
and [21, Thm. 1] for $d\geq 2$ and in [9, Thm. 1.2] for $d=1$.

For other rigorous results on the MA resistor network we refer to [18, 19].

##  4\. Reduction to the case ${\mathbb{G}}={\mathbb{R}}^{d}$

In this section we explain how to prove Theorem 2 (and therefore also Theorem
1 for ${\mathbb{G}}={\mathbb{Z}}^{d}$) once proved Theorem 1 and Proposition
2.8 for ${\mathbb{G}}={\mathbb{R}}^{d}$. To this aim we point out that there
exists a canonical procedure to pass from the model with group
${\mathbb{Z}}^{d}$ to a related model with group ${\mathbb{R}}^{d}$ that we
now briefly recall, referring to [16, Section 6] for a detailed discussion.

Let us consider the context of Theorem 2 with ${\mathbb{G}}={\mathbb{Z}}^{d}$.
Again, without loss of generality, we take $\tau_{g}x=x+g$ for all
$x\in{\mathbb{R}}^{d}$ and $g\in{\mathbb{Z}}^{d}$. Similarly, we have the
translation action of the group ${\mathbb{R}}^{d}$ given by
$\bar{\tau}_{g}x:=x+g$ for all $x,g\in{\mathbb{R}}^{d}$.

We consider the probability space $\bar{\Omega}:=\Omega\times[0,1)^{d}$ with
probability $\bar{}\mathcal{P}:=\mathcal{P}\times dx$ and $\sigma$–field
$\bar{\mathcal{F}}:=\mathcal{F}\otimes\mathcal{B}([0,1)^{d})$,
$\mathcal{B}([0,1)^{d})$ being the family of Borel sets of $[0,1)^{d}$. Given
$y\in{\mathbb{R}}^{d}$, we define the integer part $[y]$ of $y$ as the unique
element $z\in{\mathbb{Z}}^{d}$ such that $y\in z+[0,1)^{d}$. We then set
$\beta(y):=y-[y]\in[0,1)^{d}$. The group ${\mathbb{R}}^{d}$ acts on
$\bar{\Omega}$ by means of the maps $(\bar{\tau}_{g})_{g\in{\mathbb{R}}^{d}}$
with $\bar{\tau}_{g}(\omega,a):=(\tau_{[g+a]}\omega,\beta(g+a))$.

Finally, to $(\omega,a)\in\bar{\Omega}$ we associate the locally finite set
$\widehat{(\omega,a)}:=\hat{\omega}-a$ and consider the conductances
$\bar{c}_{x-a,y-a}(\omega,a):=c_{x,y}(\omega)$ for $x,y\in\hat{\omega}$.

Then, under the assumptions of Theorem 2, by the observations collected in
[16, Section 6] the setting given by
$(\bar{\Omega},\bar{}\mathcal{F},\bar{\mathcal{P}})$, the group
$\bar{\mathbb{G}}:={\mathbb{R}}^{d}$ with the above two actions, the simple
point process $\widehat{\bar{\omega}}$ and the conductance field
$\bar{c}_{x,y}(\bar{\omega})$ satisfies assumptions (A1),…,(A8). Hence, once
proven Theorem 1 and Proposition 2.8 for ${\mathbb{G}}={\mathbb{R}}^{d}$, one
gets the following (taking $e_{1}$ as principal direction of the effective
homogenized matrix $\bar{D}$ for the new setting). There exists a translation
invariant measurable set $\bar{\Omega}_{\rm typ}$ (for the
${\mathbb{R}}^{d}$–action) with $\bar{}\mathcal{P}({\bar{\Omega}}_{\rm
typ})=1$ such that for all $(\omega,a)\in{\bar{\Omega}}_{\rm typ}$ it holds
$\lim_{\ell\to+\infty}\ell^{2-d}\sigma_{\ell}(\omega,a)=\bar{m}\bar{D}_{1,1}$.
As discussed in [16, Section 6], $\bar{m}=m$ and $\bar{D}=D$. On the other
hand, since ${\bar{\Omega}}_{\rm typ}$ is translation invariant, for
$(\omega,a)\in\bar{\Omega}_{\rm typ}$ we have
$(\omega,0)=\bar{\tau}_{-a}(\omega,a)\in{\bar{\Omega}}_{\rm typ}$. Note also
that $\sigma_{\ell}(\omega,0)=\sigma_{\ell}(\omega)$ as
$\widehat{(\omega,0)}=\hat{\omega}$ and
$\bar{c}_{x,y}(\omega,0)=c_{x,y}(\omega)$ for $x,y\in\hat{\omega}$. Hence, if
$(\omega,a)\in{\bar{\Omega}}_{\rm typ}$, then
$\lim_{\ell\to+\infty}\ell^{2-d}\sigma_{\ell}(\omega)=mD_{1,1}$. One can then
check that Theorem 1 is fulfilled with $\Omega_{\rm
typ}:=\\{\omega\in\Omega\,:(\omega,a)\in\bar{\Omega}_{\rm typ}\text{ for some
}a\in[0,1)^{d}\\}$ ($\Omega_{\rm typ}$ is translation invariant as
$(\tau_{g}\omega,a)=\bar{\tau}_{g}(\omega,a)$ for all $g\in{\mathbb{Z}}^{d}$,
while the issue concerning $\Omega_{1}$ can be settled using that
$\bar{\lambda}_{k}(\omega,a)=\lambda_{k}(\omega)$ as observed in [16, Section
6]). By similar arguments one obtains Proposition 2.8.

######  Warning 4.1.

Due to the above result, in the rest we just take
${\mathbb{G}}={\mathbb{R}}^{d}$ and focus on Theorem 1 and Proposition 2.8. We
treat Theorem 3 in Appendix A.

##  5\. Preliminary facts on the Palm distribution $\mathcal{P}_{0}$

In this short section we recall some basic facts on the Palm distribution
$\mathcal{P}_{0}$ used in what follows. Recall (4).

######  Lemma 5.1.

[16, Lemma 7.1] Given a measurable subset $A\subset\Omega_{0}$, the following
facts are equivalent: (i) $\mathcal{P}_{0}(A)=1$; (ii)
$\mathcal{P}\left(\omega\in\Omega\,:\,\tau_{x}\omega\in A\;\forall
x\in\hat{\omega}\right)=1$; (iii)
$\mathcal{P}_{0}\left(\omega\in\Omega_{0}\,:\,\tau_{x}\omega\in A\;\forall
x\in\hat{\omega}\right)=1$.

By ergodicity (cf. (A1)), given a measurable function
$g:\Omega_{0}\to[0,\infty)$ it holds

$\lim_{s\to\infty}\frac{1}{(2s)^{d}}\int_{(-s,s)^{d}}d\hat{\omega}(x)\,g(\tau_{x}\omega)=m\,{\mathbb{E}}_{0}[g]\qquad\mathcal{P}\text{--a.s.}$
(29)

One can indeed refine the above result. To this aim we define
$\mu^{\varepsilon}_{\omega}$ as the atomic measure on ${\mathbb{R}}^{d}$ given
by
$\mu^{\varepsilon}_{\omega}:=\varepsilon^{d}\sum_{x\in\hat{\omega}}\delta_{\varepsilon
x}$. Then it holds:

######  Proposition 5.2.

[16, Prop. 3.1] Let $g:\Omega_{0}\to{\mathbb{R}}$ be a measurable function
with $\|g\|_{L^{1}(\mathcal{P}_{0})}<+\infty$. Then there exists a translation
invariant measurable subset $\mathcal{A}[g]\subset\Omega$ such that
$\mathcal{P}(\mathcal{A}[g])=1$ and such that, for any
$\omega\in\mathcal{A}[g]$ and any $\varphi\in C_{c}({\mathbb{R}}^{d})$, it
holds

$\lim_{\varepsilon\downarrow 0}\int
d\mu_{\omega}^{\varepsilon}(x)\varphi(x)g(\tau_{x/\varepsilon}\omega)=\int
dx\,m\varphi(x){\mathbb{E}}_{0}[g]\,.$ (30)

The above proposition (which is the analogous e.g. of [41, Theorem 1.1]) is at
the core of 2-scale convergence. It follows again from ergodicity. The
variable $x$ appears in the l.h.s. of (30) at the macroscopic scale in
$\varphi(x)$ and at the microscopic scale in $g(\tau_{x/\varepsilon}\omega)$.

######  Definition 5.3.

Given a function $g:\Omega_{0}\to[0,+\infty]$ such that
${\mathbb{E}}_{0}[g]<+\infty$, we define $\mathcal{A}[g]$ as
$\mathcal{A}[g_{0}]$ (cf. Proposition 5.2), where
$g_{0}:\Omega_{0}\to{\mathbb{R}}$ is defined as $g$ on $\\{g<+\infty\\}$ and
as $0$ on $\\{g=+\infty\\}$.

######  Remark 5.4.

By playing with suitable test functions $\varphi$, one derives from
Proposition 5.2 that the limit (29) holds indeed for all $\omega$ varying in a
suitable translation invariant measurable subset of $\Omega$ with
$\mathcal{P}$–probability $1$.

##  6\. The Hilbert space $H^{1,\varepsilon}_{0,\omega}$ and the amorphous
gradient $\nabla_{\varepsilon}f$

Let $\omega\in\Omega_{1}$. Recall Definition 2.6 of
$\mathcal{G}_{\omega}^{\varepsilon}$,
$\mathcal{C}_{\omega,\Lambda}^{\varepsilon}$ and
$\mathcal{C}_{\omega}^{\varepsilon}$ and Definition 2.7 of
$H^{1,\varepsilon}_{0,\omega}$ and $K^{\varepsilon}_{\omega}$. For later use,
we point out that
$K^{\varepsilon}_{\omega}=H^{1,\varepsilon}_{0,\omega}+\psi_{\varepsilon}$,
where $\psi_{\varepsilon}:\varepsilon\hat{\omega}\cap S\to[0,1]$ is the
function (depending also on $\omega$)

$\psi_{\varepsilon}(x):=\begin{cases}x_{1}+\frac{1}{2}&\text{ if
}x\in\mathcal{C}_{\omega}^{\varepsilon}\cap\Lambda\,,\\\ 0&\text{ if
}x\in(\varepsilon\hat{\omega}\cap
S^{-})\cup\mathcal{C}_{\omega,\Lambda}^{\varepsilon}\,,\\\ 1&\text{ if
}x\in\varepsilon\hat{\omega}\cap S^{+}\,.\end{cases}$ (31)

As discussed in Section 2, if $f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$
is bounded, then $f\in L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})$,
$\nabla_{\varepsilon}f\in L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})$ and
${\mathbb{L}}^{\varepsilon}_{\omega}f\in
L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})$. By definition of
$\nu^{\varepsilon}_{\omega,\Lambda}$, given bounded functions
$f,g:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$, we have

$\langle\nabla_{\varepsilon}f,\nabla_{\varepsilon}g\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}=\varepsilon^{d-2}\sum_{(x,y)\in\mathcal{E}_{\varepsilon}(\omega)}c_{x/\varepsilon,y/\varepsilon}(\omega)\bigl{(}f(y)-f(x)\bigr{)}\bigl{(}g(y)-g(x)\bigr{)}\,.$
(32)

######  Lemma 6.1.

Let $\omega\in\Omega_{1}$. Given $f,g:\varepsilon\hat{\omega}\cap
S\to{\mathbb{R}}$ with $f\equiv 0$ on
$\varepsilon\hat{\omega}\setminus\Lambda$ and $g$ bounded, it holds

$\langle
f,-{\mathbb{L}}^{\varepsilon}_{\omega}g\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}=\frac{1}{2}\langle\nabla_{\varepsilon}f,\nabla_{\varepsilon}g\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\,.$
(33)

###### Proof.

Since $f\equiv 0$ outside $\Lambda$ we have

$\begin{split}\langle
f,-{\mathbb{L}}^{\varepsilon}_{\omega}g\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}=-\sum_{x\in\varepsilon\hat{\omega}\cap
S}\varepsilon^{d-2}f(x)\sum_{y\in\varepsilon\hat{\omega}\cap
S}c_{x/\varepsilon,y/\varepsilon}(\omega)\bigl{(}g(y)-g(x)\bigr{)}\,.\end{split}$
(34)

The r.h.s. is an absolutely convergent series as $\omega\in\Omega_{1}$, $f$
and $g$ are bounded and $f\equiv 0$ on $S\setminus\Lambda$, hence we can
freely permute the addenda. Due to the symmetry of the conductances, the
r.h.s. of (34) equals

$-\sum_{y\in\varepsilon\hat{\omega}\cap
S}\varepsilon^{d-2}f(y)\sum_{x\in\varepsilon\hat{\omega}\cap
S}c_{x/\varepsilon,y/\varepsilon}(\omega)\bigl{(}g(x)-g(y)\bigr{)}\,.$

By summing the above expression with the r.h.s. of (34), we get

$\langle
f,-{\mathbb{L}}^{\varepsilon}_{\omega}g\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}=\frac{\varepsilon^{d-2}}{2}\sum_{x\in\varepsilon\hat{\omega}\cap
S}\sum_{y\in\varepsilon\hat{\omega}\cap
S}c_{x/\varepsilon,y/\varepsilon}(\omega)\bigl{(}f(y)-f(x)\bigr{)}\bigl{(}g(y)-g(x)\bigr{)}\,.$

As the generic addendum in the r.h.s. is zero if
$(x,y)\not\in\mathcal{E}_{\varepsilon}(\omega)$ since $f\equiv 0$ on
$S\setminus\Lambda$, by (32) we get (33). ∎

######  Lemma 6.2.

Given $\omega\in\Omega_{1}$, the following holds:

  * (i)

$V_{\varepsilon}$ is the unique function $v\in K_{\omega}^{\varepsilon}$ such
that ${\mathbb{L}}^{\varepsilon}_{\omega}v(x)=0$ for all
$x\in\varepsilon\hat{\omega}\cap\Lambda$;

  * (ii)

$V_{\varepsilon}$ is the unique function $v\in K^{\varepsilon}_{\omega}$ such
that
$\langle\nabla_{\varepsilon}u,\nabla_{\varepsilon}v\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}=0$
for all $u\in H^{1,\varepsilon}_{0,\omega}$;

  * (iii)

$V_{\varepsilon}$ is the unique minimizer of the following variational
problem:

$\inf\bigl{\\{}\langle\nabla_{\varepsilon}v,\nabla_{\varepsilon}v\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\,:\,v\in
K^{\varepsilon}_{\omega}\bigr{\\}}\,.$ (35)

###### Proof.

On the finite dimensional Hilbert space $H^{1,\varepsilon}_{0,\omega}$ we
consider the bilinear form
$a(f,g):=\frac{1}{2}\langle\nabla_{\varepsilon}f,\nabla_{\varepsilon}g\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$.
Trivially, $a(\cdot,\cdot)$ is a continuous symmetric bilinear form. If
$a(f,f)=0$, then $f$ is constant on the connected components of the graph
$\mathcal{G}_{\omega}^{\varepsilon}$. This fact and the definition of
$H^{1,\varepsilon}_{0,\omega}$ imply that $f\equiv 0$. As
$H^{1,\varepsilon}_{0,\omega}$ is finite dimensional, we conclude that
$a(\cdot,\cdot)$ is also coercive. By writing
$v=f_{\varepsilon}+\psi_{\varepsilon}$, the function $v\in
K^{\varepsilon}_{\omega}$ in Item (i) is the one such that $f_{\varepsilon}\in
H^{1,\varepsilon}_{0,\omega}$ and

${\mathbb{L}}^{\varepsilon}_{\omega}f_{\varepsilon}(x)=-{\mathbb{L}}^{\varepsilon}_{\omega}\psi_{\varepsilon}(x)\qquad\forall
x\in\varepsilon\hat{\omega}\cap\Lambda\,.$ (36)

As $f_{\varepsilon}\equiv 0\equiv\psi_{\varepsilon}$ on
$\mathcal{C}^{\varepsilon}_{\omega,\Lambda}$, which is a union of connected
components of $\mathcal{G}_{\omega}^{\varepsilon}$, (36) is automatically
satisfied for $x\in\mathcal{C}^{\varepsilon}_{\omega,\Lambda}$. Hence in (36)
we can restrict to $x\in\mathcal{C}^{\varepsilon}_{\omega}\cap\Lambda$. On the
other hand, functions in $H^{1,\varepsilon}_{0,\omega}$ are free on
$\mathcal{C}^{\varepsilon}_{\omega}\cap\Lambda$, and zero elsewhere. Due to
the above observations, (36) is equivalent to requiring that $\langle
u,-{\mathbb{L}}^{\varepsilon}_{\omega}f_{\varepsilon}\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}=\langle
u,{\mathbb{L}}^{\varepsilon}_{\omega}\psi_{\varepsilon}\rangle_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}$
for any $u\in H^{1,\varepsilon}_{0,\omega}$. Hence, by Lemma 6.1,
$f_{\varepsilon}\in H^{1,\varepsilon}_{0,\omega}$ satisfying (36) can be
characterized also as the solution in $H^{1,\varepsilon}_{0,\omega}$ of the
problem

$a(f_{\varepsilon},u)=-\frac{1}{2}\langle\nabla_{\varepsilon}\psi_{\varepsilon},\nabla_{\varepsilon}u\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\qquad\forall
u\in H^{1,\varepsilon}_{0,\omega}\,.$ (37)

By the Lax–Milgram theorem we conclude that there exists a unique function
$f_{\varepsilon}$ satisfying (37), thus implying Item (i). Since
$a(f_{\varepsilon},u)=\frac{1}{2}\langle\nabla_{\varepsilon}f_{\varepsilon},\nabla_{\varepsilon}u\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$,
the uniqueness of the solution $f_{\varepsilon}$ of (37) leads to Item (ii).
Moreover, always by the Lax–Milgram theorem, $f_{\varepsilon}$ is the unique
minimizer of the functional $H^{1,\varepsilon}_{0,\omega}\ni
v\mapsto\frac{1}{2}a(v,v)+\frac{1}{2}\langle\nabla_{\varepsilon}\psi_{\varepsilon},\nabla_{\varepsilon}v\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$,
and therefore of the functional $H^{1,\varepsilon}_{0,\omega}\ni
v\mapsto\frac{1}{4}\langle\nabla_{\varepsilon}(v+\psi_{\varepsilon}),\nabla_{\varepsilon}(v+\psi_{\varepsilon})\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$.
This proves Item (iii). ∎

######  Remark 6.3.

As $V_{\varepsilon}$ is harmonic on $\varepsilon\hat{\omega}\cap\Lambda$ (cf.
Lemma 6.2–(i)), $V_{\varepsilon}$ has values in $[0,1]$.

######  Remark 6.4.

Given $(x,y)\in\mathcal{E}_{\varepsilon}(\omega)$, $\\{x,y\\}$ is included
either in $\mathcal{C}^{\varepsilon}_{\omega,\Lambda}$ or in
$\mathcal{C}^{\varepsilon}_{\omega}$. In the first case we have
$\nabla_{\varepsilon}V_{\varepsilon}(x,(y-x)/\varepsilon)=0$. This observation
and Lemma 6.2–(ii) imply that
$\langle\nabla_{\varepsilon}f,\nabla_{\varepsilon}V_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}=0$
for all $f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$ such that $f(x)=0$
$\forall
x\in\varepsilon\hat{\omega}\cap(S\setminus\Lambda)=\mathcal{C}_{\omega}^{\varepsilon}\setminus\Lambda$.

Fo the next result, recall (26) and (31).

######  Lemma 6.5.

There exists a translation invariant measurable subset
$\Omega_{2}\subset\Omega_{1}$ such that $\mathcal{P}(\Omega_{2})=1$ and, for
all $\omega\in\Omega_{2}$,

$\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|\psi\|_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,,\,\qquad\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}\psi\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,,$
(38) $\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|\psi_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,,\qquad\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}\psi_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,,$
(39) $\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|V_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,,\qquad\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}V_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty\,.$
(40)

###### Proof.

By Proposition 5.2 and Remark 5.4 there exists a translation invariant
measurable set $\Omega_{2}\subset\Omega_{1}$ such that, for any
$\omega\in\Omega_{2}$, $\lim_{\varepsilon\downarrow
0}\mu^{\varepsilon}_{\omega}(\Lambda)=m$ and $\lim_{\varepsilon\downarrow
0}\int_{\Lambda}d\mu^{\varepsilon}_{\omega}(x)\lambda_{2}(\tau_{x/\varepsilon}\omega)=m{\mathbb{E}}_{0}[\lambda_{2}]$.

Let us take $\omega\in\Omega_{2}$. Since
$\psi,\psi_{\varepsilon},V_{\varepsilon}$ have value in $[0,1]$ and
$\mu^{\varepsilon}_{\omega,\Lambda}$ has mass
$\mu^{\varepsilon}_{\omega}(\Lambda)\to m$, we get the first bounds in (38),
(39) and (40).

Let us prove that $\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}\psi\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty$.
We have (recall (32))

$\begin{split}\|\nabla_{\varepsilon}\psi\|^{2}_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}&=\varepsilon^{d-2}\sum_{(x,y)\in\mathcal{E}_{\varepsilon}(\omega)}c_{x/\varepsilon,y/\varepsilon}(\omega)\left(\psi(y)-\psi(x)\right)^{2}\\\
&\leq\varepsilon^{d-2}\sum_{(x,y)\in\mathcal{E}_{\varepsilon}(\omega)}c_{x/\varepsilon,y/\varepsilon}(\omega)\left(y_{1}-x_{1}\right)^{2}\\\
&\leq
2\varepsilon^{d-2}\sum_{x\in\varepsilon\hat{\omega}\cap\Lambda}\sum_{y\in\varepsilon\hat{\omega}\cap
S}c_{x/\varepsilon,y/\varepsilon}(\omega)\left(y_{1}-x_{1}\right)^{2}\,.\end{split}$
(41)

We can rewrite the last expression as
$2\varepsilon^{d}\sum_{x\in\hat{\omega}\cap(\varepsilon^{-1}\Lambda)}\sum_{y\in\hat{\omega}\cap(\varepsilon^{-1}S)}c_{x,y}(\omega)(y_{1}-x_{1})^{2}$,
which is bounded by
$2\varepsilon^{d}\sum_{x\in\hat{\omega}\cap(\varepsilon^{-1}\Lambda)}\lambda_{2}(\tau_{x}\omega)=2\int_{\Lambda}d\mu^{\varepsilon}_{\omega}(x)\lambda_{2}(\tau_{x/\varepsilon}\omega)$.
The last integral converges to $2m{\mathbb{E}}_{0}[\lambda_{2}]<+\infty$ as
$\varepsilon\downarrow 0$ since $\omega\in\Omega_{2}$. This concludes the
proof that $\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}\psi\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty$.

To prove that $\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}\psi_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty$
it is enough to use the above result and note that
$c_{x,y}(\omega)|\psi_{\varepsilon}(y)-\psi_{\varepsilon}(x)|\leq
c_{x,y}(\omega)|\psi(y)-\psi(x)|$ $\forall x,y\in\hat{\omega}$.

Since $V_{\varepsilon}$ minimizes (35) and $\psi_{\varepsilon}\in
K^{\varepsilon}_{\omega}$, we have
$\|\nabla_{\varepsilon}V_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\leq\|\nabla_{\varepsilon}\psi_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}$.
Hence $\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}V_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}<+\infty$
by the second bound in (39). ∎

###  6.1. Some properties of the amorphous gradient $\nabla_{\varepsilon}$

In Section 2 we have defined $\nabla_{\varepsilon}f$ for functions
$f:\varepsilon\hat{\omega}\cap S\to{\mathbb{R}}$. Definition (21) can by
extended by replacing $S$ with any set $A\subset{\mathbb{R}}^{d}$ by requiring
that $x,x+\varepsilon z\in\varepsilon\hat{\omega}\cap A$. Given
$f,g:\varepsilon\hat{\omega}\to{\mathbb{R}}$, it is simple to check the
following Leibniz rule:

$\nabla_{\varepsilon}(fg)(x,z)=\nabla_{\varepsilon}f(x,z)g(x)+f(x+\varepsilon
z)\nabla_{\varepsilon}g(x,z)\,.$ (42)

Given $\varphi\in C_{c}({\mathbb{R}}^{d})$, take $\ell$ such that
$\varphi(x)=0$ if $|x|\geq\ell$ and fix $\phi\in C_{c}({\mathbb{R}}^{d})$ with
values in $[0,1]$, such that $\phi(x)=1$ for $|x|\leq\ell$ and $\phi(x)=0$ for
$|x|\geq\ell+1$. Then, by applying respectively the mean value theorem and
Taylor expansion, we get for some positive constant $C(\varphi)$ that

$\displaystyle\bigl{|}\nabla_{\varepsilon}\varphi(x,z)\bigr{|}\leq\|\nabla\varphi\|_{\infty}|z|\bigl{(}\phi(x)+\phi(x+\varepsilon
z)\bigr{)}\text{ if }\varphi\in C^{1}_{c}({\mathbb{R}}^{d})\,,$ (43)
$\displaystyle\bigl{|}\nabla_{\varepsilon}\varphi(x,z)-\nabla\varphi(x)\cdot
z\bigr{|}\leq\varepsilon C(\varphi)|z|^{2}\bigl{(}\phi(x)+\phi(x+\varepsilon
z)\bigr{)}\text{ if }\varphi\in C^{2}_{c}({\mathbb{R}}^{d})\,.$ (44)

##  7\. Proof of Theorem 1 when $D_{1,1}=0$

Due to (25) we need to prove that $\lim_{\varepsilon\downarrow
0}\langle\nabla_{\varepsilon}V_{\varepsilon},\nabla_{\varepsilon}V_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}=0$
for all $\omega$ varying in a good set, where _good set_ means a translation
invariant measurable subset of $\Omega_{1}$ with $\mathcal{P}$-probability
equal to $1$. Trivially it is enough to prove the following claim: given
$\delta>0$, $\varlimsup_{\varepsilon\downarrow
0}\frac{1}{2}\langle\nabla_{\varepsilon}V_{\varepsilon},\nabla_{\varepsilon}V_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\leq\delta$
for all $\omega$ in a good set. Let us prove our claim.

As $D_{1,1}=0$ and by (9), given $\delta>0$ we can fix $f\in
L^{\infty}(\mathcal{P}_{0})$ such that

$m{\mathbb{E}}_{0}\Big{[}\int
d\hat{\omega}(x)c_{0,x}(\omega)\left(x_{1}-\nabla
f(\omega,x)\right)^{2}\Big{]}\leq\delta\,.$ (45)

Given $\varepsilon>0$ we define the functions
$v_{\varepsilon},\tilde{v}_{\varepsilon}:\varepsilon\hat{\omega}\cap
S\to{\mathbb{R}}$ as $v_{\varepsilon}(x):=\psi(x)-\varepsilon
f(\tau_{x/\varepsilon}\omega)$ if $x\in\Lambda$; $v_{\varepsilon}(x):=0$ if
$x\in S^{-}$; $v_{\varepsilon}(x):=1$ if $x\in S^{+}$ and
$\tilde{v}_{\varepsilon}(x):=v_{\varepsilon}(x)\mathds{1}_{S\setminus\mathcal{C}^{\varepsilon}_{\omega,\Lambda}}(x)$
for all $x$. As $\tilde{v}_{\varepsilon}(x)\in K^{\varepsilon}_{\omega}$, by
Lemma 6.2-(iii) it is enough to prove that $\varlimsup_{\varepsilon\downarrow
0}\langle\nabla_{\varepsilon}\tilde{v}_{\varepsilon},\nabla_{\varepsilon}\tilde{v}_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\leq\delta$
for all $\omega$ in a good set. To this aim we bound

$\frac{1}{2}\langle\nabla_{\varepsilon}\tilde{v}_{\varepsilon},\nabla_{\varepsilon}\tilde{v}_{\varepsilon}\rangle_{L^{2}(\nu^{\varepsilon}_{\omega,\Lambda})}\leq\varepsilon^{d-2}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda}\sum_{y\in\hat{\omega}\cap\varepsilon^{-1}S}c_{x,y}(\omega)\bigl{(}v_{\varepsilon}(\varepsilon
y)-v_{\varepsilon}(\varepsilon x)\bigr{)}^{2}\,.$

We split the sum in the r.h.s. into three contributions $C(\varepsilon)$,
$C_{-}(\varepsilon)$ and $C_{+}(\varepsilon)$, corresponding respectively to
the cases $y\in\hat{\omega}\cap\varepsilon^{-1}\Lambda$,
$y\in\hat{\omega}\cap\varepsilon^{-1}S^{-}$ and
$y\in\hat{\omega}\cap\varepsilon^{-1}S^{+}$, while in all the above
contributions $x$ varies among $\hat{\omega}\cap\varepsilon^{-1}\Lambda$.

If $x,y\in\hat{\omega}\cap\varepsilon^{-1}\Lambda$, then
$v_{\varepsilon}(\varepsilon y)-v_{\varepsilon}(\varepsilon
x)=\varepsilon(y_{1}-x_{1}-\nabla f(\tau_{x}\omega,y-x))$. Hence, we can bound

$C(\varepsilon)\leq\varepsilon^{d}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda}\;\sum_{y\in\hat{\omega}}c_{x,y}(\omega)(y_{1}-x_{1}-\nabla
f(\tau_{x}\omega,y-x))^{2}\,.$ (46)

By ergodicity (cf. (29) and Remark 5.4) for all $\omega$ varying in a suitable
good set the r.h.s. of (46) converges to the l.h.s of (45), and therefore it
is bounded by $\delta$. Hence, $\varlimsup_{\varepsilon\downarrow
0}C(\varepsilon)\leq\delta$.

We now consider $C_{-}(\varepsilon)$ and prove that
$\lim_{\varepsilon\downarrow 0}C_{-}(\varepsilon)=0$. If
$x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda$ and
$y\in\hat{\omega}\cap\varepsilon^{-1}S^{-}$, then
$(v_{\varepsilon}(\varepsilon x)-v_{\varepsilon}(\varepsilon
y))^{2}=\varepsilon^{2}(x_{1}-f(\tau_{x}\omega))^{2}\leq
2\varepsilon^{2}x_{1}^{2}+2\varepsilon^{2}\|f\|_{\infty}^{2}\leq
2\varepsilon^{2}(x_{1}-y_{1})^{2}+2\varepsilon^{2}\|f\|_{\infty}^{2}$. Hence
it remains to show that

$\varepsilon^{d}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda}\;\sum_{y\in\hat{\omega}\cap\varepsilon^{-1}S^{-}}c_{x,y}(\omega)[(x_{1}-y_{1})^{2}+1]$
(47)

goes to zero as $\varepsilon\downarrow 0$. Given $\rho\in(0,1)$ we set
$\Lambda_{\rho}:=\rho\Lambda$. We denote by $A_{1}(\rho,\varepsilon)$ the sum
in (47) restricted to $x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda_{\rho}$ and
$y\in\hat{\omega}\cap\varepsilon^{-1}S^{-}$. We denote by
$A_{2}(\rho,\varepsilon)$ the sum coming from the remaining addenda. We
observe that
$A_{1}(\rho,\varepsilon)\leq\varepsilon^{d}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda_{\rho}}g_{(1-\rho)/2\varepsilon}(\tau_{x}\omega)$,
where
$g_{\ell}(\omega):=\sum_{z\in\hat{\omega}}c_{0,z}(\omega)[z_{1}^{2}+1]\mathds{1}(|z|_{\infty}\geq\ell)$.
By (29) and Remark 5.4, for all $\omega$ varying in a good set
$\Omega^{\prime}$ it holds $\lim_{\varepsilon\downarrow
0}\varepsilon^{d}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}\Lambda_{\rho}}g_{\ell}(\tau_{x}\omega)=m\rho^{d}{\mathbb{E}}_{0}[g_{\ell}]$
for all $\ell\in{\mathbb{N}}$. As $g_{(1-\rho)/2\varepsilon}\leq g_{\ell}$ for
$\varepsilon$ small and by (A7) and dominated convergence, we obtain for all
$\omega\in\Omega^{\prime}$ that $\varlimsup_{\varepsilon\downarrow
0}A_{1}(\rho,\varepsilon)\leq\varlimsup_{\ell\uparrow\infty}m\rho^{d}{\mathbb{E}}_{0}[g_{\ell}]=0$.
We move to $A_{2}(\rho,\varepsilon)$. We can bound $A_{2}(\rho,\varepsilon)$
by

$\varepsilon^{d}\sum_{x\in\hat{\omega}\cap\varepsilon^{-1}(\Lambda\setminus\Lambda_{\rho})}\sum_{y\in\hat{\omega}}c_{x,y}(\omega)[(x_{1}-y_{1})^{2}+1]\,.$
(48)

By Proposition 5.2 with suitable test functions, for all $\omega$ varying in a
suitable good set we get that (48) converges as $\varepsilon\downarrow 0$ to
$m{\mathbb{E}}_{0}[\lambda_{2}+\lambda_{0}]\ell(\Lambda\setminus\Lambda_{\rho})$,
where here $\ell(\cdot)$ denotes the Lebesgue measure. To conclude the proof
that $\lim_{\varepsilon\downarrow 0}C_{-}(\varepsilon)=0$, it is therefore
enough to take the limit $\rho\uparrow 1$.

By the same arguments used for $C_{-}(\varepsilon)$, one proves that
$\lim_{\varepsilon\downarrow 0}C_{+}(\varepsilon)=0$.

##  8\. Effective equation with mixed boundary conditions

In this section we assume that $d_{*}:={\rm dim}({\rm Ker}(D)^{\perp})\geq 1$
and, given a domain $A\subset{\mathbb{R}}^{d}$, $L^{2}(A)$ and $H^{1}(A)$
refer to the Lebesgue measure $dx$, which will be omitted from the notation.
Here, and in the other sections, given a unit vector $v$ we write
$\partial_{v}f$ for the weak derivative of $f$ along the direction $v$ (if
$v=e_{i}$, then $\partial_{v}f$ is simply the standard weak derivative
$\partial_{i}f$).

We are interested in elliptic operators with mixed (Dirichlet and Neumann)
boundary conditions. By denoting $\bar{\Lambda}$ the closure of $\Lambda$, we
set

$F_{-}:=\\{x\in\bar{\Lambda}\,:\,x_{1}=-1/2\\}\,,\;\,F_{+}:=\\{x\in\bar{\Lambda}\,:\,x_{1}=1/2\\}\,,\,\;F:=F_{-}\cup
F_{+}\,.$

######  Definition 8.1.

We fix an orthonormal basis
$\mathfrak{e}_{1},\mathfrak{e}_{2},\dots,\mathfrak{e}_{d_{*}}$ of ${\rm
Ker}(D)^{\perp}$ and an orthonormal basis
$\mathfrak{e}_{d_{*}+1},\dots,\mathfrak{e}_{d}$ of ${\rm Ker}(D)$, where
$d_{*}:={\rm dim}\bigl{(}{\rm Ker}(D)^{\perp}\bigr{)}$ (when $D$ is non
degenerate, one can simply take $\mathfrak{e}_{k}:=e_{k}$).

######  Definition 8.2.

We introduce the following three functional spaces:

  * •

We define $H^{1}(\Lambda,d_{*})$ as the Hilbert space given by functions $f\in
L^{2}(\Lambda)$ with weak derivative $\partial_{\mathfrak{e}_{i}}f$ in
$L^{2}(\Lambda)$ for any $i=1,\dots,d_{*}$, endowed with the squared norm
$\|f\|^{2}_{1,*}:=\|f\|^{2}_{L^{2}(\Lambda)}+\sum_{i=1}^{d_{*}}\|\partial_{\mathfrak{e}_{i}}f\|^{2}_{L^{2}(\Lambda)}$.
Moreover, given $f\in H^{1}(\Lambda,d_{*})$, we define

$\nabla_{*}f:=\sum_{i=1}^{d_{*}}\left(\partial_{\mathfrak{e}_{i}}f\right)\mathfrak{e}_{i}\,.$
(49)

  * •

We define $H^{1}_{0}(\Lambda,F,d_{*})$ as the closure in
$H^{1}(\Lambda,d_{*})$ of

$\Big{\\{}\varphi_{|\Lambda}\,:\,\varphi\in
C^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)\Big{\\}}\,.$

  * •

We define the functional set $K$ as (cf. (26))

$K:=\\{\psi_{|\Lambda}+f\,:\,f\in H^{1}_{0}(\Lambda,F,d_{*})\\}\,.$ (50)

We stress that, if $\mathfrak{e}_{i}=e_{i}$ for all $i=1,\dots,d_{*}$ (as in
many applications), then

$\nabla_{*}f:=(\partial_{1}f,\dots,\partial_{d_{*}}f,0,\dots,0)\in
L^{2}(\Lambda)^{d}\,.$ (51)

The definition of $H^{1}(\Lambda,d_{*})$ is indeed intrinsic, i.e. it does not
depend on the particular choice of
$\mathfrak{e}_{1}$,$\mathfrak{e}_{2}$,…,$\mathfrak{e}_{d_{*}}$ as orthonormal
basis of ${\rm Ker}(D)^{\perp}$: by linearity, $f\in H^{1}(\Lambda,d_{*})$ if
and only if $f\in L^{2}(\Lambda)$ and $\partial_{v}f\in L^{2}(\Lambda)$ for
any unit vector $v\in{\rm Ker}(D)^{\perp}$ ($\|f\|^{2}_{1,*}$ is also basis-
independent). Moreover, when $f$ is smooth, $\nabla_{*}f$ is simply the
orthogonal projection of $\nabla f$ on ${\rm Ker}(D)^{\perp}$.

######  Remark 8.3.

Suppose that $\mathfrak{e}_{i}=e_{i}$ for all $1\leq i\leq d_{*}$. Let $f\in
H^{1}(\Lambda,d_{*})$. Given $1\leq i\leq d_{*}$, by integrating
$\partial_{i}f$ times
$\varphi(x_{1},\dots,x_{d_{*}})\phi(x_{d*+1},\dots,x_{d})$ with $\varphi\in
C_{c}^{\infty}({\mathbb{R}}^{d_{*}})$ (varying in a suitable countable subset)
and $\phi\in C_{c}^{\infty}({\mathbb{R}}^{d-d_{*}})$, one gets that the
function $f(\cdot,y_{1},\dots,y_{n-d_{*}})$ belongs to
$H^{1}\left((-1/2,1/2)^{d_{*}}\right)$ for a.e.
$(y_{1},\dots,y_{n-d_{*}})\in(-1/2,1/2)^{n-d_{*}}$.

Being a closed subspace of the Hilbert space $H^{1}(\Lambda,d_{*})$,
$H^{1}_{0}(\Lambda,F,d_{*})$ is a Hilbert space. We also point out that in the
definition of $K$ one could replace $\psi_{|\Lambda}$ by any other function
$\phi\in H^{1}(\Lambda,d_{*})\cap C(\bar{\Lambda})$ such that $\phi\equiv 0$
on $F_{-}$ and $\phi\equiv 1$ on $F_{+}$ as follows from the next lemma:

######  Lemma 8.4.

Let $u\in H^{1}(\Lambda,d_{*})\cap C(\bar{\Lambda})$ satisfy $u\equiv 0$ on
$F$. Then $u\in H^{1}_{0}(\Lambda,F,d_{*})$.

###### Proof.

For simplicity of notation we suppose that $\mathfrak{e}_{i}=e_{i}$ for all
$1\leq i\leq d_{*}$ (in the general case, replace $\partial_{i}$ by
$\partial_{\mathfrak{e}_{i}}$ below). We use some idea from the proof of [8,
Theorem 9.17]. We set $u_{n}(x):=G(nu(x))/n$, where $G\in C^{1}({\mathbb{R}})$
satisfies: $|G(t)|\leq|t|$ for all $t\geq 0$, $G(t)=0$ for $|t|\leq 1$ and
$G(t)=t$ for $|t|\geq 2$. Note that
$\partial_{i}u_{n}(x)=G^{\prime}(nu(x))\partial_{i}u(x)$ for $1\leq i\leq
d_{*}$ (cf. [8, Prop. 9.5]). Hence, $u_{n}\to u$ a.e. and
$\partial_{i}u_{n}\to\partial_{i}u$ a.e. In the last identity, we have used
that $\partial_{i}u=0$ a.e. on $\\{u=0\\}$ which follows as a byproduct of
Remark 8.3 and Stampacchia’s theorem (see Thereom 3 and Remark (ii) to Theorem
4 in [14, Section 6.1.3]). By dominated convergence one obtains that $u_{n}\to
u$ in $H^{1}(\Lambda,d_{*})$. Since $H^{1}_{0}(\Lambda,F,d_{*})$ is a closed
subspace of $H^{1}(\Lambda,d_{*})$, it is enough to prove that $u_{n}\in
H^{1}_{0}(\Lambda,F,d_{*})$. Due to our hypothesis on $u$ and the definition
of $G$, $u_{n}\equiv 0$ in a neighborhood of $F$ inside $\bar{\Lambda}$. Hence
the conclusion follows by applying the implication (iii)$\Rightarrow$ (i) in
Proposition 8.5 below. Equivalently, it is enough to observe that, by adapting
[8, Cor. 9.8] or [14, Thm. 1, Sec. 4.4], there exists a sequence of functions
$\varphi_{k}\in C_{c}^{\infty}({\mathbb{R}}^{d})$ such that
${\varphi_{k}}_{|\Lambda}\to u_{n}$ in $H^{1}(\Lambda,d_{*})$. Since
$u_{n}\equiv 0$ in a neighborhood of $F$, it is easy to find $\phi\in
C_{c}^{\infty}({\mathbb{R}}^{d}\setminus F)$ such that
${(\phi\varphi_{k})}_{|\Lambda}\to u_{n}$ in $H^{1}(\Lambda,d_{*})$. Hence
$u_{n}\in H^{1}_{0}(\Lambda,F,d_{*})$. ∎

One can adapt the proof of [8, Prop. 9.18] to get the following criterion
assuring that a function belongs to $H^{1}_{0}(\Lambda,F,d_{*})$:

######  Proposition 8.5.

Given a function $u\in L^{2}(\Lambda)$, the following properties are
equivalent:

  * (i)

$u\in H^{1}_{0}(\Lambda,F,d_{*})$;

  * (ii)

there exists $C>0$ such that

$\Big{|}\int_{\Lambda}u\,\partial_{\mathfrak{e}_{i}}\varphi dx\Big{|}\leq
C\|\varphi\|_{L^{2}(\Lambda)}\quad\forall\varphi\in
C_{c}^{\infty}(S)\,,\;\forall i:1\leq i\leq d_{*}\,;$ (52)

  * (iii)

the function

$\bar{u}(x):=\begin{cases}u(x)&\text{ if }x\in\Lambda\,,\\\ 0&\text{ if }x\in
S\setminus\Lambda\,,\end{cases}$ (53)

belongs to $H^{1}(S,d_{*})$ (which is defined similarly to
$H^{1}(\Lambda,d_{*})$ by replacing $\Lambda$ with $S$). Moreover, in this
case it holds
$\partial_{\mathfrak{e}_{i}}\bar{u}=\overline{\partial_{\mathfrak{e}_{i}}u}$
for $1\leq i\leq d_{*}$, where $\overline{\partial_{\mathfrak{e}_{i}}u}$ is
defined by extending $\partial_{\mathfrak{e}_{i}}u$ as zero on
$S\setminus\Lambda$.

######  Lemma 8.6 (Poincaré inequality).

It holds $\|f\|_{L^{2}(\Lambda)}\leq\|\partial_{1}f\|_{L^{2}(\Lambda)}$ for
any $f\in H^{1}_{0}(\Lambda,F,d_{*})$.

###### Proof.

Given $f\in\mathcal{C}^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)$, by Schwarz
inequality, for any $(x_{1},x^{\prime})\in\Lambda$ with
$x_{1}\in(-\frac{1}{2},\frac{1}{2})$ we have
$f(x_{1},x^{\prime})^{2}=\bigl{(}\int_{-1/2}^{x_{1}}\partial_{1}f(s,x^{\prime})ds\bigr{)}^{2}\leq\int_{-1/2}^{1/2}\partial_{1}f(s,x^{\prime})^{2}ds$.
By integrating over $\Lambda$ we get the desired estimate for
$f\in\mathcal{C}^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)$. Since
$\mathcal{C}^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)$ is dense in
$H^{1}_{0}(\Lambda,F,d_{*})$, we get the thesis. ∎

######  Definition 8.7.

We say that $v$ is a weak solution of the equation

$\nabla_{*}\cdot(D\nabla_{*}v)=0$ (54)

on $\Lambda$ with boundary conditions

$\begin{cases}v(x)=0&\text{ if }x\in F_{-}\,,\\\ v(x)=1&\text{ if }x\in
F_{+}\,,\\\ D\nabla_{*}v(x)\cdot\mathbf{n}(x)=0&\text{ if
}x\in\partial\Lambda\setminus F\,,\end{cases}$ (55)

if $v\in K$ (cf. (50)) and if $\int_{\Lambda}\nabla_{*}u\cdot
D\nabla_{*}v\,dx=0$ for all $u\in H^{1}_{0}(\Lambda,F,d_{*})$.

Above $\mathbf{n}$ denotes the outward unit normal vector to the boundary in
$\partial\Lambda$.

######  Remark 8.8.

In the above definition it would be enough to require that
$\int_{\Lambda}\nabla_{*}u\cdot D\nabla_{*}vdx=0$ for all $u\in
C^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)$ since the functional
$H^{1}_{0}(\Lambda,F,d_{*})\ni u\mapsto\int_{\Lambda}\nabla_{*}u\cdot
D\nabla_{*}vdx\in{\mathbb{R}}$ is continuous.

We shortly motivate the above Definition 8.7. Just to simplify the notation we
suppose that $\mathfrak{e}_{i}=e_{i}$ for $1\leq i\leq d_{*}$. By Green’s
formula we have

$\int_{\Lambda}(\partial_{i}f)g\,dx=-\int_{\Lambda}f(\partial_{i}g)\,dx+\int_{\partial\Lambda}fg(\mathbf{n}\cdot
e_{i})dS\,,\qquad\forall f,g\in C^{1}(\bar{\Lambda})\,,$ (56)

where $dS$ is the surface measure on $\partial\Lambda$. By taking $1\leq
i,j\leq d_{*}$, $f=\partial_{j}v$ and $g=u$ in (56), we get

$\int_{\Lambda}u\nabla_{*}\cdot(D\nabla_{*}v)dx=-\int_{\Lambda}\nabla_{*}u\cdot(D\nabla_{*}v)dx+\int_{\partial\Lambda}u(\nabla_{*}v\cdot(D\mathbf{n}))dS\,,$
(57)

for all $v\in C^{2}(\bar{\Lambda})$ and $u\in C^{1}(\bar{\Lambda})$. Hence,
$v\in C^{2}(\bar{\Lambda})$ satisfies $\nabla_{*}\cdot(D\nabla_{*}v)=0$ on
$\Lambda$ and $\nabla_{*}v\cdot(D\mathbf{n})\equiv 0$ on
$\partial\Lambda\setminus F$ if and only if
$\int_{\Lambda}\nabla_{*}u\cdot(D\nabla_{*}v)dx=0$ for any $u\in
C^{1}(\bar{\Lambda})$ with $u\equiv 0$ on $F$. Such a set $\mathcal{C}$ of
functions $u$ is dense in $H^{1}_{0}(\Lambda,F,d)$. Indeed $\mathcal{C}\subset
H^{1}_{0}(\Lambda,F,d)$ by Lemma 8.4, while
$C^{\infty}_{c}({\mathbb{R}}^{d}\setminus F)\subset\mathcal{C}$. Hence, we
conclude that $v\in C^{2}(\bar{\Lambda})$ satisfies
$\nabla_{*}\cdot(D\nabla_{*}v)=0$ on $\Lambda$ and
$\nabla_{*}v\cdot(D\mathbf{n})\equiv 0$ on $\partial\Lambda\setminus F$ if and
only if $\int_{\Lambda}\nabla_{*}u\cdot(D\nabla_{*}v)dx=0$ for any $u\in
H^{1}_{0}(\Lambda,F,d)$. We have therefore proved that $v\in
C^{2}(\bar{\Lambda})$ is a classical solution of (54) and (55) if and only if
it is a weak solution in the sense of Definition 8.7.

######  Lemma 8.9.

Suppose that $e_{1}\in{\rm Ker(D)}^{\perp}$. Then there exists a unique weak
solution $v\in K$ of the equation $\nabla_{*}\cdot(D\nabla_{*}v)=0$ with
boundary conditions (55). Furthermore, $v$ is the unique minimizer of
$\inf_{h\in K}\int\nabla_{*}h\cdot D\nabla_{*}h\,dx$.

###### Proof.

To simplify the notation, in what follows we write $\psi$ instead of
$\psi_{|\Lambda}$. We define the bilinear form
$a(f,g):=\int_{\Lambda}\nabla_{*}f\cdot D\nabla_{*}gdx$ on the Hilbert space
$H^{1}_{0}(\Lambda,F,d_{*})$. The bilinear form $a(\cdot,\cdot)$ is symmetric
and continuous. Due to the Poincaré inequality (cf. Lemma 8.6) and since
$e_{1}\in{\rm Ker}(D)^{\perp}$, $a(\cdot,\cdot)$ is also coercive (use that
$\int|\partial_{1}f|^{2}dx=\int|\nabla_{*}f\cdot
e_{1}|dx\leq\int|\nabla_{*}f|^{2}dx\leq a(f,f)/C$, where $C$ is the minimal
positive eigenvalue of $D$).

By definition we have that $v\in K$ is a weak solution of equation
$\nabla_{*}\cdot(D\nabla_{*}v)=0$ with b.c. (55) if and only if, setting
$f:=v-\psi$, $f\in H^{1}_{0}(\Lambda,F,d_{*})$ and $f$ satisfies

$\int\nabla_{*}f\cdot D\nabla_{*}udx=-\int\nabla_{*}\psi\cdot
D\nabla_{*}u\,dx\qquad\forall u\in H^{1}_{0}(\Lambda,F,d_{*})\,.$ (58)

Note that the r.h.s. is a continuous functional in $u\in
H^{1}_{0}(\Lambda,F,d_{*})$. Due to the above observations and by Lax–Milgram
theorem, we conclude that there exists a unique such function $f$, hence there
is a unique weak solution $v$ of equation $\nabla_{*}\cdot(D\nabla_{*}v)=0$
with b.c. (55). Moreover $f$ satisfies

$\frac{1}{2}a(f,f)+\int\nabla_{*}\psi\cdot D\nabla_{*}f\,dx=\inf_{g\in
H^{1}_{0}(\Lambda,F,d_{*})}\Big{\\{}\frac{1}{2}a(g,g)+\int\nabla_{*}\psi\cdot
D\nabla_{*}g\,dx\Big{\\}}\,.$

By adding to both sides $\frac{1}{2}\int\nabla_{*}\psi\cdot
D\nabla_{*}\psi\,dx$, we get that $\frac{1}{2}\int\nabla_{*}v\cdot
D\nabla_{*}v=\inf_{h\in K}\frac{1}{2}\int\nabla_{*}h\cdot D\nabla_{*}h\,dx$. ∎

As $\nabla_{*}\psi_{|\Lambda}(x)=e_{1}$ and $De_{1}$ is proportional to
$e_{1}$ if $e_{1}$ is a $D$–eigenvector, from the above lemma we immediately
get:

######  Corollary 8.10.

If $e_{1}$ is an eigenvector of $D$ with eigenvalue $D_{1,1}>0$, then the
function $\psi_{|\Lambda}$ is the unique weak solution $v\in K$ of the
equation $\nabla_{*}\cdot(D\nabla_{*}v)=0$ with boundary conditions (55).

###  8.1. Comments on the effective equation (specific for the case
$\Lambda=(-1/2,1/2)^{d}$)

When $e_{1}\cdot De_{1}=0$, automatically $e_{1}\in{\rm Ker}(D)$ and Theorems
1 and 2 hold. On the other hand, if $e_{1}\cdot De_{1}>0$ but
$e_{1}\not\in{\rm Ker}(D)^{\perp}$, then the conclusion of Lemma 8.9 can fail.
Take for example $d=2$, $d_{*}=1$. Then, if $|\mathfrak{e}_{1}\cdot e_{1}|\ll
1$, functions of the form $f(x):=\varphi(x\cdot\mathfrak{e}_{2})$, with
$\varphi\in C^{1}({\mathbb{R}})$, $\varphi(u)\equiv 0$ for $u<-\varepsilon$,
$\varphi(u)\equiv 1$ for $u>\varepsilon$ and $\varepsilon>0$ small, satisfy
$f\in K$ and $\nabla_{*}f\equiv 0$. Hence Lemma 8.9 fails.

From now on let us restrict to $e_{1}\in{\rm Ker}(D)^{\perp}$. Then Lemma 8.9
holds. The problem is now that the unique weak solution $v$ of (54) with b.c.
(55) is not necessarily affine. To be affine one should have $v(x)=e_{1}\cdot
x+b$ for some $b\in{\mathbb{R}}$ (because of the values on $F_{-}$ and
$F_{+}$), but then we would need $De_{1}$ to be orthogonal to
$e_{2},\dots,e_{d}$ due to (55), and therefore we would need again that
$e_{1}$ is an eigenvector. To overcome this obstruction one has to change the
shape of $\Lambda$ as discussed in Section 2.2 for Theorem 3.

##  9\. Square integrable forms and effective homogenized matrix

As common in homogenization theory, the variational formula (9) defining the
effective homogenized matrix $D$ admits a geometrical interpretation in the
Hilbert space of square integrable forms. For later use we recall here this
interpretation. We also collect some facts taken from [16].

We define $\nu$ as the Radon measure on $\Omega\times{\mathbb{R}}^{d}$ such
that

$\int d\nu(\omega,z)g(\omega,z)=\int d\mathcal{P}_{0}(\omega)\int
d\hat{\omega}(z)c_{0,z}(\omega)g(\omega,z)$ (59)

for any nonnegative measurable function $g(\omega,z)$. We point out that $\nu$
has finite total mass since
$\nu(\Omega\times{\mathbb{R}}^{d})={\mathbb{E}}_{0}[\lambda_{0}]<+\infty$.
Elements of $L^{2}(\nu)$ are called _square integrable forms_.

Given a function $u:\Omega_{0}\to{\mathbb{R}}$, its gradient $\nabla
u:\Omega\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ is defined as

$\nabla u(\omega,z):=u(\tau_{z}\omega)-u(\omega)\,.$ (60)

If $u$ is defined $\mathcal{P}_{0}$–a.s., then $\nabla u$ is well defined
$\nu$–a.s. by Lemma 5.1. If $u$ is bounded and measurable, then $\nabla u\in
L^{2}(\nu)$. The subspace of _potential forms_ $L^{2}_{\rm pot}(\nu)$ is
defined as the following closure in $L^{2}(\nu)$:

$L^{2}_{\rm pot}(\nu):=\overline{\\{\nabla u\,:\,u\text{ is bounded and
measurable}\\}}\,.$

The subspace of _solenoidal forms_ $L^{2}_{\rm sol}(\nu)$ is defined as the
orthogonal complement of $L^{2}_{\rm pot}(\nu)$ in $L^{2}(\nu)$.

######  Definition 9.1.

Given a square integrable form $v\in L^{2}(\nu)$ its divergence ${\rm
div}\,v\in L^{1}(\mathcal{P}_{0})$ is defined as ${\rm div}\,v(\omega)=\int
d\hat{\omega}(z)c_{0,z}(\omega)(v(\omega,z)-v(\tau_{z}\omega,-z))$.

The r.h.s. in the above identity is well defined since it corresponds
$\mathcal{P}_{0}$–a.s. to an absolutely convergent series (cf. [16, Section 8]
where (A3) is used). For any $v\in L^{2}(\nu)$ and any bounded and measurable
function $u:\Omega_{0}\to{\mathbb{R}}$, it holds (cf. [16, Section 8]) $\int
d\mathcal{P}_{0}(\omega){\rm div}\,v(\omega)u(\omega)=-\int
d\nu(\omega,z)v(\omega,z)\nabla u(\omega,z)$. As a consequence we have that,
given $v\in L^{2}(\nu)$, $v\in L^{2}_{\rm sol}(\nu)$ if and only if ${\rm
div}\,v=0$ $\mathcal{P}_{0}$–a.s. By using (A6) one also gets (cf. [16,
Section 8]):

######  Lemma 9.2.

The functions $g\in L^{2}(\mathcal{P}_{0})$ of the form $g={\rm div}\,v$ with
$v\in L^{2}(\nu)$ are dense in $\\{w\in
L^{2}(\mathcal{P}_{0})\,:\,{\mathbb{E}}_{0}[w]=0\\}$.

As $\lambda_{2}\in L^{1}(\mathcal{P}_{0})$, given $a\in{\mathbb{R}}^{d}$ the
form $u_{a}(\omega,z):=a\cdot z$ belongs to $L^{2}(\nu)$. We note that the
effective homogenized matrix $D$ defined in (9) satisfies, for any
$a\in{\mathbb{R}}^{d}$,

$\begin{split}q(a):=a\cdot Da&=\inf_{v\in L^{2}_{\rm
pot}(\nu)}\frac{1}{2}\|u_{a}+v\|^{2}_{L^{2}(\nu)}=\frac{1}{2}\|u_{a}+v^{a}\|^{2}_{L^{2}(\nu)}\,,\end{split}$
(61)

where $v^{a}=-\Pi u_{a}$ and $\Pi:L^{2}(\nu)\to L^{2}_{\rm pot}(\nu)$ denotes
the orthogonal projection of $L^{2}(\nu)$ on $L^{2}_{\rm pot}(\nu)$. Hence
$v^{a}$ is characterized by the properties

$v^{a}\in L^{2}_{\rm pot}(\nu)\,,\qquad v^{a}+u_{a}\in L^{2}_{\rm
sol}(\nu)\,.$ (62)

Moreover it holds (cf. [16, Section 9]):

$Da=\frac{1}{2}\int d\nu(\omega,z)z\bigl{(}a\cdot
z+v^{a}(\omega,z)\bigr{)}\qquad\forall a\in{\mathbb{R}}^{d}\,.$ (63)

By (61) the kernel ${\rm Ker}(q)$ of the quadratic form $q$ is given by ${\rm
Ker}(q):=\\{a\in{\mathbb{R}}^{d}\,:\,q(a)=0\\}=\\{a\in{\mathbb{R}}^{d}\,:\,u_{a}\in
L^{2}_{\rm pot}(\nu)\\}$. Note that ${\rm Ker}(q)={\rm Ker}(D)$. The following
result is the analogous of [41, Lemma 5.1] (cf. [16, Section 9]):

######  Lemma 9.3.

${\rm Ker}(q)^{\perp}=\Big{\\{}\int d\nu(\omega,z)b(\omega,z)z\,:\,b\in
L^{2}_{\rm sol}(\nu)\Big{\\}}$.

It is simple to check that Definition 8.1 and Lemma 9.3 imply the following:

######  Corollary 9.4.

${\rm
Span}\\{\mathfrak{e}_{1},\mathfrak{e}_{2},\dots,\mathfrak{e}_{d_{*}}\\}=\Big{\\{}\int
d\nu(\omega,z)b(\omega,z)z\,:\,b\in L^{2}_{\rm sol}(\nu)\Big{\\}}$.

######  Definition 9.5.

Let $b(\omega,z):\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a
measurable function with $\|b\|_{L^{1}(\nu)}<+\infty$. We define the
measurable function $c_{b}:\Omega_{0}\to[0,+\infty]$ as

$c_{b}(\omega):=\int d\hat{\omega}(z)c_{0,z}(\omega)|b(\omega,z)|\,,$ (64)

the measurable function $\hat{b}:\Omega_{0}\to{\mathbb{R}}$ as

$\hat{b}(\omega):=\begin{cases}\int
d\hat{\omega}(z)c_{0,z}(\omega)b(\omega,z)&\text{ if
}c_{b}(\omega)<+\infty\,,\\\ 0&\text{ if }c_{b}(\omega)=+\infty\,,\end{cases}$
(65)

and the measurable set
$\mathcal{A}_{1}[b]:=\\{\omega\in\Omega\,:\,c_{b}(\tau_{z}\omega)<+\infty\;\forall
z\in\hat{\omega}\\}$.

$\mathcal{A}_{1}[b]$ is a measurable translation invariant set and
$\mathcal{P}(\mathcal{A}_{1}[b])=\mathcal{P}_{0}(\mathcal{A}_{1}[b])=1$ if
$\|b\|_{L^{1}(\nu)}<+\infty$ (cf. [16, Section 10]). Trivially
$\|\hat{b}\|_{L^{1}(\nu)}\leq\|b\|_{L^{1}(\nu)}$.

######  Definition 9.6.

Given a measurable function
$b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ we define
$\tilde{b}:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ as

$\tilde{b}(\omega,z):=\begin{cases}b(\tau_{z}\omega,-z)&\text{ if
}z\in\hat{\omega}\,,\\\ 0&\text{ otherwise}\,.\end{cases}$ (66)

######  Definition 9.7.

Let $b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a measurable
function with $\|b\|_{L^{1}(\nu)}<+\infty$. If
$\omega\in\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]\cap\Omega_{0}$, we
set ${\rm
div}_{*}b(\omega):=\hat{b}(\omega)-\hat{\tilde{b}}(\omega)\in{\mathbb{R}}$.

One can check (cf. [16, Section 11]) that given a measurable function
$b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ with
$\|b\|_{L^{2}(\nu)}<+\infty$, then
$\|b\|_{L^{2}(\nu)}=\|\tilde{b}\|_{L^{2}(\nu)}$,
$\mathcal{P}_{0}(\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}])=1$, ${\rm
div}_{*}b={\rm div}\,b=-{\rm div}\,\tilde{b}$ in $L^{1}(\mathcal{P}_{0})$. In
particular, for $\|b\|_{L^{2}(\nu)}<+\infty$, $b\in L^{2}_{\rm sol}(\nu)$ if
and only if $\tilde{b}\in L^{2}_{\rm sol}(\nu)$.

######  Definition 9.8.

Let $b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a measurable
function with $\|b\|_{L^{2}(\nu)}<+\infty$ and such that its class of
equivalence in $L^{2}(\nu)$ belongs to $L^{2}_{\rm sol}(\nu)$. Then we set

$\mathcal{A}_{d}[b]:=\\{\omega\in\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]\,:\,{\rm
div}_{*}b(\tau_{z}\omega)=0\;\forall z\in\hat{\omega}\\}\,.$ (67)

$\mathcal{A}_{d}[b]$ is a translation invariant measurable set and
$\mathcal{P}(\mathcal{A}_{d}[b])=1$ (cf. [16, Section 11]).

Recall the set $\mathcal{A}[g]$ introduced in Proposition 5.2 and Definition
5.3. The following lemma corresponds to [16, Lemma 18.2]. Here we have
isolated the properties on $\omega$ used in the proof there.

######  Lemma 9.9.

Suppose that $\omega$ belongs to the sets $\mathcal{A}_{1}[1]$,
$\mathcal{A}[\lambda_{0}]$, $\mathcal{A}_{1}[|z|^{2}]$ and $\mathcal{A}[\int
d\hat{\omega}(z)c_{0,z}(\omega)|z|^{2}\mathds{1}(|z|\geq\ell)]$ for all
$\ell\in{\mathbb{N}}$. Then $\forall\varphi\in C_{c}^{2}({\mathbb{R}}^{d})$ we
have

$\lim_{\varepsilon\downarrow 0}\int
d\nu^{\varepsilon}_{\omega}(x,z)\bigl{[}\nabla_{\varepsilon}\varphi(x,z)-\nabla\varphi(x)\cdot
z\bigr{]}^{2}=0\,.$ (68)

##  10\. The set $\Omega_{\rm typ}$ of typical environments

In this section we describe the set $\Omega_{\rm typ}$ of typical environments
for which Theorem 1 and Proposition 2.8 hold when $D_{1,1}>0$, and for which
Theorem 3 holds when ${\bf a}\cdot D{\bf a}>0$. To define $\Omega_{\rm typ}$
and also 2-scale convergence in the next section, we need to isolate suitable
countable dense sets of $L^{2}(\mathcal{P}_{0})$ and $L^{2}(\nu)$.

Recall that $L^{2}(\mathcal{P}_{0})$ is separable due to (A8). Then one can
prove that also $L^{2}(\nu)$ is separable (cf. [16, Section 12]). The
separability of $L^{2}(\mathcal{P}_{0})$ and $L^{2}(\nu)$ will be used below
to construct suitable countable dense subsets. The functional sets
$\mathcal{G}_{1},\mathcal{H}_{1},\mathcal{G}_{2},\mathcal{H}_{2},\mathcal{W}$
presented below are as in [16, Section 12] (we recall their definition for the
reader’s convenience).

$\bullet$ The functional sets $\mathcal{G}_{1},\mathcal{H}_{1}$. We fix a
countable set $\mathcal{H}_{1}$ of measurable functions
$b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that
$\|b\|_{L^{2}(\nu)}<+\infty$ for any $b\in\mathcal{H}_{1}$ and such that
$\\{{\rm div}\,b\,:\,b\in\mathcal{H}_{1}\\}$ is a dense subset of $\\{w\in
L^{2}(\mathcal{P}_{0})\,:\,{\mathbb{E}}_{0}[w]=0\\}$ when thought of as set of
$L^{2}$–functions (recall Lemma 9.2). For each $b\in\mathcal{H}_{1}$ we define
the measurable function $g_{b}:\Omega_{0}\to{\mathbb{R}}$ as (cf. Definition
9.7)

$g_{b}(\omega):=\begin{cases}{\rm div}_{*}b(\omega)&\text{ if
}\omega\in\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]\,,\\\ 0&\text{
otherwise}\,.\end{cases}$ (69)

Note that $g_{b}={\rm div}\,b$, $\mathcal{P}_{0}$–a.s. (see Section 9). We set
$\mathcal{G}_{1}:=\\{g_{b}\,:\,b\in\mathcal{H}_{1}\\}$.

$\bullet$ The functional sets $\mathcal{G}_{2},\mathcal{H}_{2}$. We fix a
countable set $\mathcal{G}_{2}$ of bounded measurable functions
$g:\Omega_{0}\to{\mathbb{R}}$ such that the set $\\{\nabla
g\,:\,g\in\mathcal{G}_{2}\\}$, thought in $L^{2}(\nu)$, is dense in
$L^{2}_{\rm pot}(\nu)$. We define $\mathcal{H}_{2}$ as the set of measurable
functions $h:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that
$h=\nabla g$ for some $g\in\mathcal{G}_{2}$.

$\bullet$ The functional set $\mathcal{W}$. We fix a countable set
$\mathcal{W}$ of measurable functions
$b:\Omega_{0}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that, thought of as
subset of $L^{2}(\nu)$, $\mathcal{W}$ is dense in $L^{2}_{\rm sol}(\nu)$.
Since $\tilde{b}\in L^{2}_{\rm sol}(\nu)$ for any $b\in L^{2}_{\rm sol}(\nu)$,
at cost to enlarge $\mathcal{W}$ we assume that $\tilde{b}\in\mathcal{W}$ for
any $b\in\mathcal{W}$.

We introduce the following functions on $\Omega_{0}$ (note that they are
$\mathcal{P}_{0}$–integrable), where $n,i,b$ vary respectively in
${\mathbb{N}}$, $\\{1,\dots,d\\}$, $\mathcal{W}$:

$\displaystyle f_{n}(\omega):=\int
d\hat{\omega}(z)c_{0,z}(\omega)|z|^{2}\mathds{1}(|z|\geq n)\,,$ (70)
$\displaystyle u_{b,i}(\omega):=\int
d\hat{\omega}(z)c_{0,z}(\omega)z_{i}{b}(\omega,z)\,,$ (71) $\displaystyle
u_{b,i,n}(\omega):=|u_{b,i}(\omega)|\mathds{1}\bigl{(}|u_{b,i}(\omega)|\geq
n\bigr{)}\,.$ (72)

######  Definition 10.1 (Functional set $\mathcal{G}$).

We define $\mathcal{G}$ as the union of the following countable sets of
measurable functions on $\Omega_{0}$, which are $\mathcal{P}_{0}$–square
integrable: $\\{1\\}$, $\mathcal{G}_{1}$, $\mathcal{G}_{2}$ and
$\\{u_{b,i}\mathds{1}(|u_{b,i}|\leq n)\\}$ with $b\in\mathcal{W}$,
$i\in\\{1,\dots,d\\}$, $n\in{\mathbb{N}}$.

######  Definition 10.2 (Functional set $\mathcal{H}$).

We define $\mathcal{H}$ as the union of the following countable sets of
measurable functions on $\Omega_{0}\times{\mathbb{R}}^{d}$, which are
$\nu$–square integrable: $\mathcal{H}_{1}$, $\mathcal{H}_{2}$, $\mathcal{W}$,
$\\{(\omega,z)\mapsto z_{i}\,:\,1\leq i\leq d\\}$, $\\{b\mathds{1}(|b|\leq
n)\,:\,b\in\mathcal{W},\;n\in{\mathbb{N}}\\}$.

######  Definition 10.3 (Set $\Omega_{\rm typ}$ of typical environments).

The set $\Omega_{\rm typ}\subset\Omega$ of typical environments is the
intersection of the following sets:

  * •

$\mathcal{A}[gg^{\prime}]$ for all $g,g^{\prime}\in\mathcal{G}$ (recall that
$1\in\mathcal{G}$);

  * •

$\mathcal{A}_{1}[bb^{\prime}]\cap\mathcal{A}[\widehat{bb^{\prime}}]$ as
$b,b^{\prime}\in\mathcal{H}$;

  * •

$\Omega_{2}$ (cf. Lemma 6.5);

  * •

$\mathcal{A}_{1}[|z|^{k}]\cap\mathcal{A}[\lambda_{k}]$ for $k=0,2$;

  * •

$\mathcal{A}[f_{n}]$ for all $n\in{\mathbb{N}}$;

  * •

$\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]\cap\mathcal{A}_{1}[b^{2}]\cap\mathcal{A}_{1}[\tilde{b}^{2}]$
for all $b\in\mathcal{H}$;

  * •

$\mathcal{A}[\widehat{\;b^{2}\;}]\cap\mathcal{A}[\widehat{\;\tilde{b}^{2}\;}]\cap\mathcal{A}[\widehat{\,|b|\,}]\cap\mathcal{A}[\widehat{\,|\tilde{b}|\,}]$
for all $b\in\mathcal{H}$;

  * •

$\mathcal{A}_{1}[\tilde{b}(\omega,z)z_{i}]$ for $1\leq i\leq d$ for all
$b\in\mathcal{W}$;

  * •

$\mathcal{A}[u_{b,i,n}]$ for all $b\in\mathcal{W}$, $1\leq i\leq d$ and
$n\in{\mathbb{N}}$;

  * •

$\mathcal{A}_{d}[b]$ for all $b\in\mathcal{W}$ (recall (67)).

As $\lambda_{0},\lambda_{2}\in L^{1}(\mathcal{P}_{0})$, due to Proposition 5.2
and the discussion in Section 9, and since we are dealing with a countable
family of constraints, $\Omega_{\rm typ}$ is a measurable subset of
$\Omega_{2}\subset\Omega_{1}$ with $\mathcal{P}(\Omega_{\rm typ})=1$. Since
moreover $\Omega_{2}$, $\mathcal{A}[\cdot]$, $\mathcal{A}_{1}[\cdot]$ and
$\mathcal{A}_{d}[\cdot]$ are translation invariant sets as already pointed
out, we conclude that $\Omega_{\rm typ}$ is also translation invariant.

Above we have listed the properties characterizing $\Omega_{\rm typ}$ as they
will be used along the proof (anyway, we will point out the specific property
used in each step under consideration).

##  11\. Weak convergence and 2-scale convergence

Recall $\mu^{\varepsilon}_{\omega,\Lambda}$ and
$\nu^{\varepsilon}_{\omega,\Lambda}$ given in (19). We also set (recalling the
definition of $\mu^{\varepsilon}_{\omega}$ given before Proposition 5.2)

$\displaystyle\mu^{\varepsilon}_{\omega}:=\varepsilon^{d}\sum_{x\in\varepsilon\hat{\omega}}\delta_{x}\,,\qquad\qquad\nu^{\varepsilon}_{\omega}:=\varepsilon^{d}\sum_{x\in\varepsilon\hat{\omega}}\sum_{y\in\varepsilon\hat{\omega}}c_{\frac{x}{\varepsilon},\frac{y}{\varepsilon}}(\omega)\delta_{(x,\frac{y-x}{\varepsilon})}\,,$
(73)
$\displaystyle\mu^{\varepsilon}_{\omega,S}:=\varepsilon^{d}\sum_{x\in\varepsilon\hat{\omega}\cap
S}\delta_{x}\,,\qquad\nu^{\varepsilon}_{\omega,S}:=\varepsilon^{d}\sum_{x\in\varepsilon\hat{\omega}\cap
S}\sum_{y\in\varepsilon\hat{\omega}\cap
S}c_{\frac{x}{\varepsilon},\frac{y}{\varepsilon}}(\omega)\delta_{(x,\frac{y-x}{\varepsilon})}\,.$
(74)

In this section $\Delta$ equals $S$ or $\Lambda$.

######  Definition 11.1.

Fix $\omega\in\Omega$ and a family of $\varepsilon$–parametrized functions
$v_{\varepsilon}\in L^{2}(\mu^{\varepsilon}_{\omega,\Delta})$. We say that the
family $\\{v_{\varepsilon}\\}$ _converges weakly_ to the function $v\in
L^{2}(\Delta,mdx)$, and write $v_{\varepsilon}\rightharpoonup v$, if the
family $\\{v_{\varepsilon}\\}$ is bounded (in the sense:
$\varlimsup_{\varepsilon\downarrow
0}\|v_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\omega,\Delta})}<+\infty$) and
$\lim_{\varepsilon\downarrow 0}\int
d\mu^{\varepsilon}_{\omega,\Delta}(x)v_{\varepsilon}(x)\varphi(x)=\int_{\Delta}dx\,mv(x)\varphi(x)$
for all $\varphi\in C_{c}(\Delta)$.

######  Definition 11.2.

Fix $\tilde{\omega}\in\Omega_{\rm typ}$, an $\varepsilon$–parametrized family
of functions $v_{\varepsilon}\in
L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})$ and a function $v\in
L^{2}\bigl{(}\Delta\times\Omega,mdx\times\mathcal{P}_{0}\bigr{)}$. We say that
_$v_{\varepsilon}$ is weakly 2-scale convergent to $v$_, and write
$v_{\varepsilon}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}v$, if the
family $\\{v_{\varepsilon}\\}$ is bounded, i.e.
$\varlimsup_{\varepsilon\downarrow
0}\|v_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})}<+\infty$,
and

$\lim_{\varepsilon\downarrow 0}\int
d\mu_{\tilde{\omega},\Delta}^{\varepsilon}(x)v_{\varepsilon}(x)\varphi(x)g(\tau_{x/\varepsilon}\tilde{\omega})=\int
d\mathcal{P}_{0}(\omega)\int_{\Delta}dx\,mv(x,\omega)\varphi(x)g(\omega)\,,$
(75)

for any $\varphi\in C_{c}(\Delta)$ and any $g\in\mathcal{G}$.

As $\tilde{\omega}\in\Omega_{\rm typ}\subset\mathcal{A}[g]$ for all
$g\in\mathcal{G}$, by Proposition 5.2 one gets that
$v_{\varepsilon}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}v$ where
$v_{\varepsilon}:=\varphi\in L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})$
and $v:=\varphi\in L^{2}(\Delta,mdx)$ for any $\varphi\in C_{c}(\Delta)$.

One can prove the following fact by using the first item in Definition 10.3
and by adapting the proof of [16, Lemma 13.5]:

######  Lemma 11.3.

Let $\tilde{\omega}\in\Omega_{\rm typ}$. Then, given a bounded family of
functions $v_{\varepsilon}\in
L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})$, there exists a subsequence
$\\{v_{\varepsilon_{k}}\\}$ such that
$v_{\varepsilon_{k}}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}v$ for some
$v\in L^{2}(\Delta\times\Omega,mdx\times\mathcal{P}_{0})$ with
$\|v\|_{L^{2}(\Delta\times\Omega,mdx\times\mathcal{P}_{0})}\leq\varlimsup_{\varepsilon\downarrow
0}\|v_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})}$.

Recall the definition of the measure $\nu$ given in (59).

######  Definition 11.4.

Given $\tilde{\omega}\in\Omega_{\rm typ}$, an $\varepsilon$–parametrized
family of functions $w_{\varepsilon}\in
L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})$ and a function $w\in
L^{2}\bigl{(}\Delta\times\Omega\times{\mathbb{R}}^{d}\,,mdx\times
d\nu\bigr{)}$, we say that _$w_{\varepsilon}$ is weakly 2-scale convergent to
$w$_, and write $w_{\varepsilon}\stackrel{{\scriptstyle
2}}{{\rightharpoonup}}w$, if $\\{w_{\varepsilon}\\}$ is bounded in
$L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})$, i.e.
$\varlimsup_{\varepsilon\downarrow
0}\|w_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})}<+\infty$,
and

$\lim_{\varepsilon\downarrow 0}\int
d\nu_{\tilde{\omega},\Delta}^{\varepsilon}(x,z)w_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)\\\
=\int_{\Delta}dx\,m\int d\nu(\omega,z)w(x,\omega,z)\varphi(x)b(\omega,z)\,,$
(76)

for any $\varphi\in C_{c}(\Delta)$ and any $b\in\mathcal{H}$.

One can prove the following fact by using the second item in Definition 10.3
and by adapting the proof of [16, Lemma 13.7]:

######  Lemma 11.5.

Let $\tilde{\omega}\in\Omega_{\rm typ}$. Then, given a bounded family of
functions $w_{\varepsilon}\in
L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})$, there exists a subsequence
$\\{w_{\varepsilon_{k}}\\}$ such that
$w_{\varepsilon_{k}}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}w$ for some
$w\in L^{2}(\Delta\times\Omega\times{\mathbb{R}}^{d}\,,\,mdx\times\nu)$ with
$\|w\|_{L^{2}(\Delta\times\Omega\times{\mathbb{R}}^{d}\,,\,mdx\times\nu)}\leq\varlimsup_{\varepsilon\downarrow
0}\|w_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})}$.

##  12\. $2$-scale limits of uniformly bounded functions

We fix $\tilde{\omega}\in\Omega_{\rm typ}$ and we assume that $d_{*}={\rm
dim}({\rm Ker}(D)^{\perp})\geq 1$. The domain $\Delta$ below can be
$\Lambda,S$. We consider a family of functions $\\{f_{\varepsilon}\\}$ with
$f_{\varepsilon}:\varepsilon\widehat{\tilde{\omega}}\cap S\to{\mathbb{R}}$
such that

$\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|f_{\varepsilon}\|_{\infty}<+\infty\,,$ (77)
$\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|f_{\varepsilon}\|_{L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})}<+\infty\,,$
(78) $\displaystyle\varlimsup_{\varepsilon\downarrow
0}\|\nabla_{\varepsilon}f_{\varepsilon}\|_{L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})}<+\infty\,.$
(79)

Due to Lemmas 11.3 and 11.5, along a subsequence $\\{\varepsilon_{k}\\}$ we
have

$\displaystyle L^{2}(\mu^{\varepsilon}_{\tilde{\omega},\Delta})\ni
f_{\varepsilon}\stackrel{{\scriptstyle 2}}{{\rightharpoonup}}v\in
L^{2}(\Delta\times\Omega,mdx\times\mathcal{P}_{0})\,,$ (80) $\displaystyle
L^{2}(\nu^{\varepsilon}_{\tilde{\omega},\Delta})\ni\nabla_{\varepsilon}{f}_{\varepsilon}\stackrel{{\scriptstyle
2}}{{\rightharpoonup}}w\in
L^{2}(\Delta\times\Omega\times{\mathbb{R}}^{d}\,,\,mdx\times\nu)\,,$ (81)

for suitable functions $v,w$.

######  Warning 12.1.

In this section (with exception of Lemma 12.1 and Claim 12.4), when taking the
limit $\varepsilon\downarrow 0$, we understand that $\varepsilon$ varies along
the subsequence $\\{\varepsilon_{k}\\}$ satisfying (80) and (81). Moreover, we
set

$\bar{f}_{\varepsilon}(x):=0\text{ for
}x\in\varepsilon\widehat{\tilde{\omega}}\setminus S\,.$ (82)

The structural results presented below (cf. Propositions 12.2 and 12.3)
correspond to a general strategy in homogenization by 2-scale convergence (see
Propositions 15.1 and 17.1 in [16], Lemmas 5.3 and 5.4 in [41], Theorems 4.1
and 4.2 in [40]). Condition (77) would not be strictly necessary, but it
allows important technical simplifications, and in particular it allows to
avoid the cut-off procedures developed in [16, Sections 14,16] in order to
deal with the long jumps in the Markov generator (22). Differently from [16]
here we have to control also several boundary contributions. We will apply
Propositions 12.2 and 12.3 only to the following cases: $\Delta=\Lambda$ and
$f_{\varepsilon}={V_{\varepsilon}}$; $\Delta=S$ and
$f_{\varepsilon}=V_{\varepsilon}-\psi$.

In what follows we will use the following control on long filaments:

######  Lemma 12.1.

Given $\tilde{\omega}\in\Omega_{\rm typ}$, $\ell>0$ and $\varphi\in
C_{c}({\mathbb{R}}^{d})$, it holds

$\lim_{\varepsilon\downarrow 0}\varepsilon^{-2}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\,|\varphi(x)|\,\mathds{1}(|z|\geq\ell/\varepsilon)=0\,.$
(83)

###### Proof.

Recall the definition (70) of $f_{n}$. Given $n\in{\mathbb{N}}$ we take
$\varepsilon$ small so that $\ell/\varepsilon>n$. Then we can bound

$\begin{split}&\varepsilon^{-2}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\,|\varphi(x)|\,\mathds{1}(|z|\geq\ell/\varepsilon)\leq\ell^{-2}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|\,\mathds{1}(|z|\geq\ell/\varepsilon)|z|^{2}\\\
&\leq\ell^{-2}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|\,\mathds{1}(|z|\geq
n)|z|^{2}=\ell^{-2}\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)|\varphi(x)|f_{n}(\tau_{x/\varepsilon}\omega)\,.\end{split}$

As $\omega\in\Omega_{\rm
typ}\subset\mathcal{A}[f_{n}]\cap\mathcal{A}_{1}[|z|^{2}]$, we have (cf. (A7))

$\lim_{\varepsilon\downarrow 0}\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)|\varphi(x)|f_{n}(\tau_{x/\varepsilon}\omega)=\int
dxm|\varphi(x)|{\mathbb{E}}_{0}[f_{n}]<+\infty\,.$

The last expression goes to zero as $n\to+\infty$ by dominated convergence. ∎

######  Proposition 12.2.

For $dx$–a.e. $x\in\Delta$, the map $v(x,\cdot)$ given in (80) is constant
$\mathcal{P}_{0}$–a.s.

###### Proof.

Recall the definition of the functional sets $\mathcal{G}_{1},\mathcal{H}_{1}$
given in Section 10. We claim that $\forall\varphi\in C^{1}_{c}(\Delta)$ and
$\forall\psi\in\mathcal{G}_{1}$ it holds

$\int_{\Delta}dx\,m\int
d\mathcal{P}_{0}(\omega)v(x,\omega)\varphi(x)\psi(\omega)=0\,.$ (84)

Having (84) it is standard to conclude. Indeed, (84) implies that, $dx$–a.e.
on $\Delta$, $\int d\mathcal{P}_{0}(\omega)v(x,\omega)\psi(\omega)=0$ for any
$\psi\in\mathcal{G}_{1}$. We conclude that, $dx$–a.e. on $\Delta$,
$v(x,\cdot)$ is orthogonal in $L^{2}(\mathcal{P}_{0})$ to $\\{w\in
L^{2}(\mathcal{P}_{0})\,:\,{\mathbb{E}}_{0}[w]=0\\}$. Hence
$v(x,\omega)={\mathbb{E}}_{0}[v(x,\cdot)]$ for $\mathcal{P}_{0}$–a.a.
$\omega$.

It now remains to prove (84). We first note that, by (75), (80) and since
$\tilde{\omega}\in\Omega_{\rm typ}$ and
$\psi\in\mathcal{G}_{1}\subset\mathcal{G}$,

$\text{l.h.s. of }\eqref{chiavetta}=\lim_{\varepsilon\downarrow 0}\int
d\mu^{\varepsilon}_{\tilde{\omega},\Delta}(x)f_{\varepsilon}(x)\varphi(x)\psi(\tau_{x/\varepsilon}\tilde{\omega})\,.$

Let us take $\psi=g_{b}$ with $b\in\mathcal{H}_{1}$ as in (69). By [16, Lemma
11.7] and since $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]$, we have

$\begin{split}\int
d\mu^{\varepsilon}_{\tilde{\omega},\Delta}(x)f_{\varepsilon}(x)\varphi(x)\psi(\tau_{x/\varepsilon}\tilde{\omega})&=\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\bar{f}_{\varepsilon}(x)\varphi(x)\psi(\tau_{x/\varepsilon}\tilde{\omega})\\\
&=-\varepsilon\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}(\bar{f}_{\varepsilon}\varphi)(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,.\end{split}$

By (42) we have

$\displaystyle-\varepsilon\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}(\bar{f}_{\varepsilon}\varphi)(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)=-\varepsilon
C_{1}(\varepsilon)+\varepsilon C_{2}(\varepsilon)\,,$ $\displaystyle
C_{1}(\varepsilon):=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,,$
$\displaystyle C_{2}(\varepsilon):=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x+\varepsilon
z)\nabla_{\varepsilon}\varphi(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,.$

To get (84) we only need to show that $\lim_{\varepsilon\downarrow
0}\varepsilon C_{1}(\varepsilon)=0$, $\lim_{\varepsilon\downarrow
0}\varepsilon C_{2}(\varepsilon)=0$.

We start with $C_{1}(\varepsilon)$. By Schwarz inequality and since
$\tilde{\omega}\in\Omega_{\rm typ}\subset\mathcal{A}_{1}[b^{2}]$

$|C_{1}(\varepsilon)|\leq\Big{[}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}\Big{]}^{1/2}\Big{[}\int\mu^{\varepsilon}_{\tilde{\omega}}(x)|\varphi(x)|\widehat{\,b^{2}\,}(\tau_{x/\varepsilon}\tilde{\omega})\Big{]}^{1/2}\,.$

Since $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b^{2}]\cap\mathcal{A}[\widehat{\,b^{2}\,}]$, the
last integral in the r.h.s. converges to a finite constant as
$\varepsilon\downarrow 0$. It remains to prove that $\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}$
remains bounded from above as $\varepsilon\downarrow 0$. We call $\ell$ the
distance between the support of $\varphi$ (which is contained in $\Delta$ as
$\varphi\in C^{1}_{c}(\Delta)$) and $\partial\Delta$. Then, between the pairs
$(x,z)$ with $x+\varepsilon z\not\in S$ contributing to the above integral,
only the pairs $(x,z)$ such that $x\in\Delta$ and $|z|\geq\ell/\varepsilon$
can give a nonzero contribution. In both cases $\Delta=\Lambda$ and $\Delta=S$
we can estimate

$\begin{split}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|&\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}\leq\int
d\nu^{\varepsilon}_{\tilde{\omega},\Delta}(x,z)|\varphi(x)|\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}\\\
&+\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}\mathds{1}(|z|\geq\ell/\varepsilon)\,.\end{split}$
(85)

The first addendum in the r.h.s. of (85) is bounded due to (79). The second
addendum goes to zero due to (77) (implying that
$|\nabla_{\varepsilon}\bar{f}|\leq C/\varepsilon$ for small $\varepsilon$) and
Lemma 12.1. Hence the l.h.s. of (85) remains bounded as $\varepsilon\downarrow
0$. This completes the proof that $\lim_{\varepsilon\downarrow 0}\varepsilon
C_{1}(\varepsilon)=0$.

We move to $C_{2}(\varepsilon)$. Let $\phi$ be as in (43). Using (77), (43)
and afterwards [16, Lemma 11.3–(i)], for some $\varepsilon$–independent
constants $C$’s (which can change from line to line), for $\varepsilon$ small
we can bound

$\begin{split}&|C_{2}(\varepsilon)|\leq C\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bigl{|}\nabla_{\varepsilon}\varphi(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)\bigr{|}\\\
&\leq C\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|z|\,|b(\tau_{x/\varepsilon}\tilde{\omega},z)|\bigl{(}\phi(x)+\phi(x+\varepsilon
z)\bigr{)}\\\ &\leq C\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)|z|(|b|+|\tilde{b}|)(\tau_{x/\varepsilon}\tilde{\omega},z)\\\
&\leq C\Big{[}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)|z|^{2}\Big{]}^{1/2}\Big{[}2\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)(b^{2}+\tilde{b}^{2})(\tau_{x/\varepsilon}\tilde{\omega},z)\Big{]}^{1/2}\,.\end{split}$
(86)

The first integral in the last line of (86) equals $\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\phi(x)\lambda_{2}(\tau_{x/\varepsilon}\tilde{\omega})$.
Since $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[|z|^{2}]\cap\mathcal{A}[\lambda_{2}]$, this
integral converges to a finite constant as $\varepsilon\downarrow 0$. The
second integral in the last line of (86) equals $\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\phi(x)(\widehat{\,b^{2}\,}+\widehat{\,\tilde{b}^{2}\,})(\tau_{x/\varepsilon}\tilde{\omega})$
as $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b^{2}]\cap\mathcal{A}_{1}[\tilde{b}^{2}]$. Since
also $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}[\widehat{\,b^{2}\,}]\cap\mathcal{A}[\widehat{\,\tilde{b}^{2}\,}]$,
the last integral converges to a finite constant. This implies that
$\lim_{\varepsilon\downarrow 0}\varepsilon C_{2}(\varepsilon)=0$. ∎

Due to Proposition 12.2 we can write $v(x)$ instead of $v(x,\omega)$, where
$v$ is given by (80). Recall Definition 8.1. We extend the notation (49) for
$\nabla_{*}$ also to functions with domain different from $\Lambda$.

######  Proposition 12.3.

Let $v$ and $w$ be as in (80) and (81). Then it holds:

  * (i)

$v$ has weak derivatives $\partial_{\mathfrak{e_{j}}}v\in L^{2}(\Delta,dx)$
for $j:1\leq j\leq d_{*}$;

  * (ii)

$w(x,\omega,z)=\nabla_{*}v(x)\cdot z+v_{1}(x,\omega,z)$, where $v_{1}\in
L^{2}\bigl{(}\Delta,dx;L^{2}_{\rm pot}(\nu)\bigr{)}$.

Above, $L^{2}\bigl{(}\Delta,dx;L^{2}_{\rm pot}(\nu)\bigr{)}$ denotes the space
of square integrable maps $f:\Delta\to L^{2}_{\rm pot}(\nu)$, where $\Delta$
is endowed with the Lebesgue measure.

###### Proof.

Given a square integrable form $b$, we define $\eta_{b}:=\int
d\nu(\omega,z)zb(\omega,z)$. Note that $\eta_{b}$ is well defined as $b\in
L^{2}(\nu)$ and by (A7). We observe that $\eta_{b}=-\eta_{\,\tilde{b}}$ as
derived in [16, Proof of Prop. 17.1] by using (A3). We claim that for each
solenoidal form $b\in L^{2}_{\rm sol}(\nu)$ and each function $\varphi\in
C^{2}_{c}(\Delta)$, it holds

$\int_{\Delta}dx\,m\varphi(x)\int
d\nu(\omega,z)w(x,\omega,z)b(\omega,z)=-\int_{\Delta}dx\,mv(x)\nabla\varphi(x)\cdot\eta_{b}\,.$
(87)

Having (87) one can conclude the proof of Proposition 12.3 by rather standard
arguments. Indeed, having Corollary 9.4, it is enough to apply the same
arguments presented in [16] to derive [16, Prop. 17.1] from [16, Eq. (131)].
The notation is similar, one has just to restrict to $x\in\Delta$.

Let us prove (87). Since both sides of (87) are continuous as functions of
$b\in L^{2}_{\rm sol}(\nu)$, it is enough to prove it for $b\in\mathcal{W}$.
Since $\tilde{\omega}\in\Omega_{\rm typ}$, along $\\{\varepsilon_{k}\\}$ it
holds $\nabla_{\varepsilon}f_{\varepsilon}\stackrel{{\scriptstyle
2}}{{\rightharpoonup}}w$ as in (81) and since
$b\in\mathcal{W}\subset\mathcal{H}$ (cf. (76)) we can write

$\begin{split}\text{l.h.s. of }\eqref{kokeshi}&=\lim_{\varepsilon\downarrow
0}\int
d\nu^{\varepsilon}_{\tilde{\omega},\Delta}(x,z)\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,.\end{split}$
(88)

Since $b\in\mathcal{W}\subset L^{2}_{\rm sol}(\nu)$ and
$\tilde{\omega}\in\Omega_{\rm typ}\subset\mathcal{A}_{d}[b]$, from [16, Lemma
11.7] we get

$\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}(\bar{f}_{\varepsilon}\varphi)(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)=0\,.$

Using the above identity and (42), we get

$\begin{split}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)&\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)\\\
&=-\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x+\varepsilon
z)\nabla_{\varepsilon}\varphi(x,z)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,.\end{split}$

As $\omega\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]$, by applying now
[16, Lemma 11.3-(ii)] to the above r.h.s., we get

$\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x)\nabla_{\varepsilon}\varphi(x,z)\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)\,.$

By combining (88) with the above identity, we obtain

$\text{l.h.s. of }\eqref{kokeshi}=\lim_{\varepsilon\downarrow
0}\left(-R_{1}(\varepsilon)+R_{2}(\varepsilon)\right)\,,$ (89)

where

$\displaystyle R_{1}(\varepsilon):=\int
d\left[\nu^{\varepsilon}_{\tilde{\omega}}-\nu^{\varepsilon}_{\tilde{\omega},\Delta}\right](x,z)\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)\,,$
$\displaystyle R_{2}(\varepsilon):=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x)\nabla_{\varepsilon}\varphi(x,z)\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)\,.$

We claim that $\lim_{\varepsilon\downarrow 0}R_{1}(\varepsilon)=0$. We call
$\ell$ the distance between the support $\Delta_{\varphi}\subset\Delta$ of
$\varphi$ and $\partial\Delta$. Then in $R_{1}(\varepsilon)$ the contribution
comes only from pairs $(x,z)$ such that $x\in\Delta_{\varphi}$ and
$x+\varepsilon z\not\in S$ and therefore from pairs $(x,z)$ such that
$x\in\Delta$ and $|z|\geq\ell/\varepsilon$:

$|R_{1}(\varepsilon)|\leq\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)\varphi(x)b(\tau_{x/\varepsilon}\tilde{\omega},z)|\mathds{1}(x\in\Delta\,,\;|z|\geq\ell/\varepsilon)\,.$

By Schwarz inequality we have therefore that $R_{1}(\varepsilon)^{2}\leq
I_{1}(\varepsilon)I_{2}(\varepsilon)$, where

$\displaystyle I_{1}(\varepsilon):=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\nabla_{\varepsilon}\bar{f}_{\varepsilon}(x,z)^{2}|\varphi(x)|\mathds{1}(|z|\geq\ell/\varepsilon)\,,$
$\displaystyle I_{2}(\varepsilon):=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)|\varphi(x)|b(\tau_{x/\varepsilon}\tilde{\omega},z)^{2}=\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)|\varphi(x)|\widehat{\,b^{2}\,}(\tau_{x/\varepsilon}\tilde{\omega})\,.$

The last identity concerning $I_{2}(\varepsilon)$ uses that
$\tilde{\omega}\in\Omega_{\rm typ}\subset\mathcal{A}_{1}[b^{2}]$. It holds
$\lim_{\varepsilon\downarrow 0}I_{1}(\varepsilon)=0$ due to Lemma 12.1 and
(77), while $\varlimsup_{\varepsilon\downarrow 0}I_{2}(\varepsilon)<+\infty$
since $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}[\widehat{\,b^{2}\,}]$. This proves that
$R_{1}(\varepsilon)\to 0$.

We now move to $R_{2}(\varepsilon)$. To treat this term we will use the
following fact:

######  Claim 12.4.

We have

$\lim_{\varepsilon\downarrow 0}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\Big{|}\bar{f}_{\varepsilon}(x)\bigl{[}\nabla_{\varepsilon}\varphi(x,z)-\nabla\varphi(x)\cdot
z\bigr{]}\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)\Big{|}=0\,.$ (90)

###### Proof of Claim 12.4.

We fix $k\in{\mathbb{N}}$ and $\phi\in C_{c}({\mathbb{R}}^{d})$ with values in
$[0,1]$ such that $\varphi(x)=0$ if $|x|\geq k$, $\phi(x)=1$ for $|x|\leq k$
and $\phi(x)=0$ for $|x|\geq k+1$. Given $\ell\in{\mathbb{N}}$ we write the
integral in (90) as $A_{\ell}(\varepsilon)+B_{\ell}(\varepsilon)$, where
$A_{\ell}(\varepsilon)$ is the contribution coming from $z$ with $|z|\leq\ell$
and $B_{\ell}(\varepsilon)$ is the contribution coming from $z$ with
$|z|>\ell$. Due to (44),(77), [16, Lemma 11.3-(i)] we have

$\begin{split}A_{\ell}(\varepsilon)&\leq C\ell^{2}\varepsilon\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bigl{(}\phi(x)+\phi(x+\varepsilon
z)\bigr{)}|\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)|\\\ &\leq
C\ell^{2}\varepsilon\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)(|b|+|\tilde{b}|)(\tau_{x/\varepsilon}\tilde{\omega},z)\,.\end{split}$

Since $\omega\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b]\cap\mathcal{A}_{1}[\tilde{b}]$, the last
expression above can be written as $C\ell^{2}\varepsilon\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\phi(x)\bigl{[}\widehat{\,|b|\,}+\widehat{\,|\tilde{b}|\,}\bigr{]}(\tau_{x/\varepsilon}\tilde{\omega})$.
Since $\omega\in\Omega_{\rm
typ}\subset\mathcal{A}[\widehat{\,|b|\,}]\cap\mathcal{A}[\widehat{\,|\tilde{b}|\,}]$,
the above integral converges to a finite constant as $\varepsilon\downarrow
0$. This allows to conclude that $\lim_{\varepsilon\downarrow
0}A_{\ell}(\varepsilon)=0$.

It remains to prove that
$\lim_{\ell\uparrow\infty}\varlimsup_{\varepsilon\downarrow
0}B_{\ell}(\varepsilon)=0$. We argue as above but now we apply (43). Due to
(43), (77) and [16, Lemma 11.3-(i)], we can bound

$B_{\ell}(\varepsilon)\leq C\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)(|b|+|\tilde{b}|)(\tau_{x/\varepsilon}\tilde{\omega},z)|z|\mathds{1}(|z|\geq\ell)\,.$

Recall (70). Then, by Schwarz inequality $B_{\ell}(\varepsilon)\leq
C\,C_{\ell}(\varepsilon)^{1/2}D_{\ell}(\varepsilon)^{1/2}$, where (as
$\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}_{1}[b^{2}]\cap\mathcal{A}_{1}[\tilde{b}^{2}]$)

$\displaystyle C_{\ell}(\varepsilon)$ $\displaystyle:=2\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)(b^{2}+\tilde{b}^{2})(\tau_{x/\varepsilon}\tilde{\omega},z)=2\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\phi(x)\bigl{(}\widehat{b^{2}}+\widehat{\tilde{b}^{2}}\bigr{)}(\tau_{x/\varepsilon}\tilde{\omega})$
$\displaystyle D_{\ell}(\varepsilon)$ $\displaystyle:=\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\phi(x)|z|^{2}\mathds{1}(|z|\geq\ell)=\int
d\mu^{\varepsilon}_{\tilde{\omega}}(x)\phi(x)f_{\ell}(\tau_{x/\varepsilon}\tilde{\omega})\,.$

As $\tilde{\omega}\in\Omega_{\rm
typ}\subset\mathcal{A}[\widehat{b^{2}}]\cap\mathcal{A}[\widehat{\tilde{b}^{2}}]\cap\mathcal{A}_{1}[|z|^{2}]\cap\mathcal{A}[f_{\ell}]$,
we conclude that $\varlimsup_{\varepsilon\downarrow
0}B_{\ell}(\varepsilon)\leq
C^{\prime}{\mathbb{E}}_{0}[\widehat{b^{2}}+\widehat{\tilde{b}^{2}}]^{1/2}{\mathbb{E}}_{0}[f_{\ell}]^{1/2}$,
and the r.h.s. goes to zero as $\ell\to\infty$ by dominated convergence and
(A7). ∎

We come back to (87). By combining (89), (90) and the limit
$R_{1}(\varepsilon)\to 0$, we conclude that

$\text{l.h.s. of }\eqref{kokeshi}=\lim_{\varepsilon\downarrow 0}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x)\nabla\varphi(x)\cdot
z\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)\,.$ (91)

Due to (91) and since $\eta_{\tilde{b}}=-\eta_{b}$ as already observed, to
prove (87) we only need to show that

$\lim_{\varepsilon\downarrow 0}\int
d\nu^{\varepsilon}_{\tilde{\omega}}(x,z)\bar{f}_{\varepsilon}(x)\nabla\varphi(x)\cdot
z\tilde{b}(\tau_{x/\varepsilon}\tilde{\omega},z)=\int
dx\,mv(x)\nabla\varphi(x)\cdot\eta_{\tilde{b}}\,.$ (92)

To this aim we observe that (recall (71))

$\int