Document [original]

The Random Conductance Model: Local times large

deviations, law of large numbers and effective conductance

vorgelegt von

Master of Science Michele Salvi

Rom

Von der Fakult¨at II - Mathematik und Naturwissenschaften

der Technischen Universit¨at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr.rer.nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Etienne Emmrich

Berichter/Gutachter: Prof. Dr. Wolfgang K¨onig

Berichter/Gutachter: Prof. Dr. Noam Berger

Tag der wissenschaftlichen Aussprache: 22. April 2013

Berlin 2013

D 83

May, 2013

Acknowledgements

As an immigrant, my first thank has to go to Germany: This country recived me,

welcomed me, fed me without asking much in exchange and finally accepted me. I

thank Germany and forgive her for the food and the weather. Secondly, my deepest

thanks go to germans. It took a while to understand each other, but after almost three

years I can say that I feel a little german, and I hope the germans I dealt with feel a

little bit italian, too.

The first person to whom I would like to express my gratitude is perhaps the most

german of them all, Wolfgang. He introduced me to real mathematical research in the

smoothest way possible and accompanied me to the first result of my career, which was

one of the greatest satisfactions of my entire life. I thank him for his patience, precision,

support and for valuing my ideas. I also had the honor to work with three other great

scientists: Noam, with his wisdom, Marek, sharing his life along with his knowledge

(and a pint of beer), and of course Tilman, the honeybadger of maths.

TU was a perfect environment for my Ph.D., where I found good friends besides

good mathematics. I thank all of the people who alternated in the long corridor and in

particular Sebastian, for sharing three years of contractions and for introducing me to

pasta Miracoli, Stefano, for the zigolo giallo and for not resigning himself to the idea

that it is too late for everything, and my two bosses, the Ur-one who broke up with

mathematics too early and the tiny one who is still struggling with it.

Thanks to all of the Pohls for being my family (with a special grandmother) when

the original one was too far away and to the Piccolos (with a special Santa).

A special thanks go to Giulia, for the snail, woodpecker, salamander, owl, chinchilla,

bumble-bee and sugar glider.

Many people start with jumping and end up flying away. I due to the Blue Straw-

berries if I’m not flown away from Italy, yet: thanks to Miles, even if over 200Kg, to

Turi, even if turned Milanese, to Mongo, even if down, to the Great, even if he has no

’even if’, to Bomba, even if to the other side of the ocean, to Sabbath, even if he does

not like the ’amatriciana’, and to Suatoni, even if he follows Previti’s path.

Being italian I left the family at last: thanks to my mum, for calling the government

when I’m sick, to my dad, for the sausages in Configno, and to Siriana, for the dinners

we had and the dinners to come. And very last, thanks to my cats, who don’t even

know what a dedication is.

to Schr¨odinger, so that those who don’t know him don’t understand this.

iii

Entre braves, messieurs les Officiers, doit-on pas toujours finir par sentendre ?

Vive la France alors, nom de Dieu ! Vive la France !

Louis-Ferdinand Cline, Voyage au bout de la nuit

Abstract

Reversible random walks in random environment are called random walks among

random conductances (RWRC) and they naturally arise in many branches of science as

models for physical phenomena. In this thesis we first introduce RWRC, highlighting

the connections with electrical networks, and give a substantial background on previous

literature. Then, we present a series of original results.

The first one is the proof of an annealed large deviation principle (LDP) for the

local times of a RWRC forced to stay in a finite domain. We give an explicit expression

for the rate function and obtain as a byproduct of the LDP asymptotic formulas for the

non-exit probabilities from the given domain. This result has relevant applications in

the parabolic Anderson model and in the study of random Schr¨odinger operators.

The second result deals with the law of large numbers for the endpoint of a RWRC.

We show that whenever the α-log moments of the conductances are finite for some

α > 1, the limiting speed is zero almost surely. On the other hand, finite log moments

for α < 1 do not imply zero speed: we construct ad hoc counterexamples based on

geometrical constructions of random trees.

Finally we analyze the fluctuations of the minimum of the Dirichlet energy in the

random conductance model. This quantity, known as effective conductance, describes

the total electric current flowing through an electric network and has a central role in

homogenization theory. We establish a central limit theorem for the effective conduc-

tance under the assumptions of Dirichlet boundary conditions and conductances with

small ellipticity contrast.

vii

viii

Zusammenfassung

Reversible Irrfahrten in zuf¨alliger Umgebung (random walks in random environment,

RWRE) werden Irrfahrten unter zuf¨alligen Leitf¨ahigkeiten (random walks among ran-

dom conductances, RWRC) genannt und treten in vielen Teilbereichen des Wissenschaft

als naturliche Modelle f¨ur physikalische Ph¨anomene auf. In dieser Arbeit f¨uhren wir zu

erst RWRC ein und heben dabei die Verbindungen zu elektronischen Netzwerken hervor.

Gleichzeitig geben wir eine ¨

Ubersicht ¨uber die existierende Literatur. Danach stellen

wir unsere Ergebnisse vor.

Zuerst geben wir einen Beweis f¨ur ein annealed Prinzip der großen Abweichungen

(Large Deviation Principle, LDP) f¨ur die Lokalzeiten einer RWRC, die sich innerhalb

einer Region mit festem Durchmesser befindet. Wir geben eine explizite Darstellung

der Ratenfunktion an und erhalten durch das LDP asymptotische Formeln f¨ur die

Wahrscheinlichkeiten der RWRC in der gegebenen Region zu bleiben. Dieses Ergebnis

findet wichtige Anwendungen bei der Betrachtung von zuf¨alligen Schr¨odinger-Operatoren

und des parabolischen Anderson-Modells (Parabolic Anderson Model, PAM).

Das zweite Resultat behandelt das Gesetz der grossen Zahlen f¨ur den Endpunkt eines

RWRC. Wir zeigen, dass die asymptotische Geschwindigkeit fast sicher gleich Null ist,

sobald die α-log Momente der Leitf¨ahigkeiten f¨ur ein α > 1 endlich sind. Auf der

anderen Seite implizieren endliche log Momente nicht Null-Geschwindigkeit: Wir geben

ad hoc Gegenbeispiele mit Hilfe von geometrischen Konstruktionen zuf¨alliger B¨aume.

Schließlich analysieren wir die Fluktuationen vom Minimum der Dirichlet Energie

im zuf¨alligen Leitf¨ahigkeits Modell. Diese Quantit¨at, bekannt als effektive Leitf¨ahigkeit

(effective conductance), repr¨asentiert den totalen elektrischen Strom, der in einem elek-

trischen Netzwerk fließt, und spielt eine zentrale Rolle in der Homogenisierungstheo-

rie. Wir beweisen einen zentralen Grenzwertsatz unter den Annahmen von Dirichlet-

Randbedingungen und zuf¨allige Leitf¨ahigkeiten mit kleinem elliptischen Kontrast.

Contents

Acknowledgements i

Dedication iii

Abstract vii

Introduction 1

Two reasons for studying random walks in random environment . . . . . . . . 1

Structureofthechapters ............................. 2

1 Model and results 4

1.1 The random conductance model and the related walks . . . . . . . . . . 4

1.1.1 Themodel .............................. 4

1.1.2 RWRC: discrete vs continuous time . . . . . . . . . . . . . . . . 5

1.1.3 Electrical networks . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Previous and new results . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Large deviations and the local times . . . . . . . . . . . . . . . . 12

1.2.2 Local times large deviations for an RWRC . . . . . . . . . . . . . 13

1.2.3 Law of large numbers and the point of view of the particle . . . . 16

1.2.4 Moments conditions for non-zero speed of RWRC’s . . . . . . . . 18

1.2.5 Effective conductance and homogenization theory . . . . . . . . . 19

1.2.6 A central limit theorem for the effective conductance . . . . . . . 21

2 Large deviations for the occupation measure 24

2.1 Heuristicderivation.............................. 24

2.2 ProofofTheorem1.3............................. 26

2.2.1 Proof of the lower bound . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Proof of the upper bound . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Proof of Corollary 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Outlook: growing domains . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 The speed of the RWRC 37

3.1 Moment conditions for speed zero . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Trees...................................... 39

3.2.1 TheBZZtree............................. 40

3.2.2 The Diagonal tree . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Theenvironment............................... 49

3.4 ProofofTheorem1.8............................. 57

4 A Central Limit Theorem for the effective conductance 59

4.1 Keyingredients................................ 59

4.1.1 Martingale approximation . . . . . . . . . . . . . . . . . . . . . . 59

4.1.2 Stationary edge ordering . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.3 An explicit form of martingale increment . . . . . . . . . . . . . 61

4.1.4 Input from homogenization theory . . . . . . . . . . . . . . . . . 63

4.1.5 Perturbed corrector and variance formula . . . . . . . . . . . . . 65

4.1.6 Organization ............................. 68

4.2 ProofoftheCLT............................... 68

4.3 TheMeyersestimate............................. 73

4.3.1 L2 bounds and convergence . . . . . . . . . . . . . . . . . . . . . 73

4.3.2 The Meyers estimate in finite volume . . . . . . . . . . . . . . . . 76

4.3.3 Interpolation ............................. 79

4.3.4 Weak type-(1,1) estimate . . . . . . . . . . . . . . . . . . . . . . 81

4.3.5 Triple gradient of finite-volume Green’s function . . . . . . . . . 85

4.4 Perturbed harmonic coordinate . . . . . . . . . . . . . . . . . . . . . . . 89

References 92

xii

Introduction

Two reasons for studying random walks in random environ-

ment

Random walks in random environment were first introduced in the sixties as problems

coming from biology. The first track we could find in the literature is the article by Cher-

nov [Che62] in 1962, where the random dynamics on a structure that is also random

was introduced as a toy-model for the replication of DNA-chains. In 1972 an analo-

gous mathematical model arose in the context of crystallography in a paper by Temkin

(see [Tem72]). The study of these models had then a huge impact in the field of disor-

dered physical media and energy conduction for irregular materials, see e.g. Kirkpatrick

[Kir72]. This list could go on for long, but here is a first reason for being interested

in random walks in random media: their applications and the request for theoretical

results pop out from many different scientific areas, also very distant from each other,

including biology, social sciences, theory of communications and, of course, physics.

The second aspect is genuinely mathematical: random walks in random environment

proved to be an endless source of beautiful mathematical problems, where beautiful in-

cludes (at least) the meanings of interesting, challenging and deep. The tools and the

applications to other mathematical subjects testify this fact: Methods from functional

analysis, graph theory, theoretical physics, homogenization theory, geometry are indis-

pensable for solving problems that may appear purely probabilistic at a first glance.

In this thesis we will analyze a particular kind of motion in random media, namely

the walk associated with the random conductance model. The reason for this choice

has the name of reversibility: when the walk is starting from the stationary measure,

one cannot distinguish whether the time is flowing forward or backwards. At a tech-

nical level, this feature allows one to use results from classical harmonic analysis and

carry out calculations much more explicitly than in the general case. The random con-

ductance model covers itself a huge amount of different scenarios (the random walk on

the percolation cluster is a remarkable example) and is deeply connected with physical

electrical resistor networks and with stochastic homogenization theory.

We will deal in particular with three aspects of this model. The first one is the

study of the local times of the walk, roughly speaking the amount of time the walk

spends in each site. We will prove that, when we take the average on all possible

environments, the graphic of the local times approximates a deterministic shape as the

time becomes larger and larger. The second one is a law of large numbers. In all the

classical examples of random walks among random conductances, the limiting speed of

the walk, that is the displacement of the walker over the number of performed steps, is

always zero. Is it possible to have a different behaviour? The answer is yes if we allow

”very big” conductances: We will construct ingenious examples where the limiting speed

is strictly positive almost surely. Finally, we will deal with the longstanding problem

of the description of the effective conductance, representing the total electric current

flowing through an electric network when the boundary vertices are kept at a given

voltage. Under some restriction on the law of the conductances, we will prove the

gaussian nature of the fluctuations of the effective conductance around its mean.

The community has spent a lot of efforts for understanding the random conductance

model and thousands of pages have been written, with a particular rebirth of the interest

and a peak in the production in the last five years. Nevertheless the field is still fertile,

and many misteries ask to be understood.

Structure of the thesis

In order to get no one lost, here is the organization of the thesis.

Chapter 1 is divided in two main sections. The first one introduces the random

conductance model (Section 1.1.1), the different types of random walks that can be

defined on it (Sec. 1.1.2) and its connections with electrical networks (Sec. 1.1.3). The

second part (Sec. 1.2) collects previous results in the field and introduces the original

ones presented in the thesis. After a general overview of known results, the attention

is focused on large deviations for random walks in random environment (Sec. 1.2.1)

and a large deviation principle for the local times of a random walk among random

conductances (RWRC) is stated (Sec. 1.2.2). The second topic is the law of large

numbers for the end-point of an RWRC (Sec. 1.2.3) and Section 1.2.4 provides moment

conditions on the conductances for having non-zero limiting speed of the walk. The

final part of Section 1.2 deals with homogenization theory and the study of the effective

conductance (Sec. 1.2.5). A central limit theorem for the latter is established in Section

1.2.6.

In Chapter 2 we present the complete proof of the large deviation principle for the

local times, following the lines of the article [KSW12]. We also add to the article a final

section (Sec. 2.4) on recent developments and possible future research on this subject.

Chapter 3 deals with the proof of the results of Section 1.2.4 as in [BS12]: Section

3.1 gives a sufficient condition for the RWRC to have zero speed, while Sections 3.2 and

3.3 give the construction of the counterexamples when such conditions are not fulfilled.

The proof of the central limit theorem for the effective conductance is the main

object of Chapter 4, reporting the results obtained in the paper [BSW12].

Chapter 1

Model and results

1.1 The random conductance model and the related walks

1.1.1 The model

Take a graph G= (V, E), where Vis the set of its vertices and Ethe set of its edges.

The graphs we consider will not contain double edges or loops and the edges will be

undirected. In fact, the support for our conductance model will always be Zdor a subset

of Zd, unless explicitly specified otherwise.

Let (Ω,F) be the couple of the product space Ω := [0,∞)Eof all possible config-

urations of non-negative weights assigned to the bonds of the graph and the relative

Borel sigma-algebra F. Each element ω∈Ω is a collection of numbers {ωxy}x∼y, called

conductances, where x, y are two elements of Vand the symbol ”∼” means that there

exists a bond b= (x, y)∈Econnecting xand y. The name ’conductance’ has to do with

the strict relation between this model and electrical networks, which we will analyze in

detail in Section 1.1.3. Depending on the situations, we will use for convenience also

the notations ωx,y,ω(x, y), ωbor ω(x, x ±ei) in the lattice case, where eiis an element

of the canonical base of Zd, for i= 1, ..., d. By definition, these weights are symmetric,

that is ωxy =ωyx for all (x, y)∈E. As a convention, ωxy = 0 when (x, y)6∈ E.

Let Pbe a probability measure on Ω. We say that Pis elliptic if for all (x, y)∈E

one has P(ωxy >0) = 1. We call Pstrongly elliptic if the support of each conductance

is bounded away from zero and infinity, that is, there exists λ > 0 such that

P(λ≤ωxy ≤1

λ) = 1.(1.1)

In the case of discrete lattice model, we call the shift by a vector z∈Zdof a

configuration ω∈Ω the map τz: Ω →Ω such that for all x∼y∈Zdwe have

(τzω)x,y =ωx+z,y+z. We say that Pis shift-invariant if for any event A∈ F and z∈Zd

we have

P(A) = P(τzA),

where of course (τzA) := {ω∈Ω : τ−zω∈A}. Recall also that Pis said shift-ergodic

if, whenever P(τzA) = P(A) for every z∈Zdand some event A, then P(A)∈ {0,1}.

We indicate with Ethe expectation with respect to P(in Chapter 2 we will make

also use of the notation h·i for the same object).

1.1.2 RWRC: discrete vs continuous time

From now on we will restrict, unless explicitly stated, to the d-dimensional euclidean

square lattice Zd, where the conductances are present only on edges connecting nearest

neighbours, that is, x∼yif and only if kx−yk1= 1, where k·k1is the usual `1distance.

Given a realization ω∈Ω of conductances we can introduce many kinds of random

walks exhibiting different behaviours. We illustrate here the three most studied walks:

1) the discrete-time RWRC;

2) the variable-speed random walk (VSRW);

3) the constant-speed random walk (CSRW).

1) The discrete-time random walk among random conductances performes one step

at each interval of time and chooses its next position proportionally to the weight of the

bond that brings it there. More precisely, Pω

zis the law of the random walk starting in

z∈Zdand with transition probabilities given by

Pω

z(Xn+1 =y|Xn=x) = ωxy

πω(x),(1.2)

for y∼x,n∈N0and where

πω(x) = X

y0∈Zd:y0∼x

ωxy0.(1.3)

If πω(x) = 0 than the random walk stands still forever when present at x.

2) The VSRW is the continuous-time process (Xt)t≥0generated by the modified

discrete Laplace operator ∆ω. This is given by

∆ωf(x) = X

y∈Zd, y∼x

ωxy(f(y)−f(x)) f:Zd→R, x ∈Zd.(1.4)

Note that because of the symmetry of the conductances, ∆ωis a symmetric operator.

Described in words: When at point x∈V, the VSRW waits an exponential time with

parameter πω(x) = Pw∼xωxw (i.e., with mean 1/π(x)) and then jumps to the next

point according to (1.2).

3) The CSRW behaves exactly like the VSRW, but the waiting times are exponential

random variables with parameter 1 at each point. The generator is then

∆ωf(x) = X

y∈Zd, y∼x

ωxy

πω(x)(f(y)−f(x)) f:Zd→R, x ∈Zd.(1.5)

In all the three models, we call the measure on paths for a fixed environment ω∈Ω

the quenched measure. We will be also interested in taking the expectation with respect

to the conductances of such a measure. For an event A∈ F, an event Bon the space

of trajectories of the random walk and a starting vertex x∈Zd, we define the annealed

measure as

Px(A×B) = ZA

Pω

x(B)dP(ω),(1.6)

with the convention Px(A) = RΩPω

x(A)dP(ω).

Averaging over the conductances has the advantage to ’regularize’ the environment,

in the sense that the main contribution to the annealed measure is given by typical

configurations of conductances. Furthermore, if Pis translation invariant, then P·is

also translation invariant. On the other hand under this measure the process loses the

Markov property: information from the past can indeed specify characteristic of the

discovered environment and influence the probabilities of the future steps.

One of the characteristics that causes anomalous behaviours of RWRC’s is the pres-

ence of traps. The different nature of traps in the discrete, constant-speed and variable-

speed cases is one of the main differences between the three models. For example, an

edge with a huge conductance can trap both the discrete time walk and the CSRW:

both will go back and forth on that edge for a long time with high probability. This

effect becomes particularly strong if the conductances are not bounded from above. On

the other hand, the VSRW will jump many times over the edge (in average as many

times as the other two processes), but it will do it so fast that the total effect will be

negligible from the point of view of the time spent there.

The property that makes the RWRC (in discrete or continuous time) so impor-

tant among the huge family of Random Walks in Random Environment (RWRE) is

reversibility. A Markov chain is said reversible with respect to a measure µif, choosing

the starting point according to µ, the distribution of (X0, X1, ..., Xn−1, Xn) is equal to

that of (Xn, Xn−1, ..., X1, X0). That is, it is not possible to recognize whether the chain

is running forward or backwards in time.

Calling πωthe (not necessarily finite) measure on Vdescribed in (1.3) one can easily

check that:

πω(x)Pω(x, y) = πω(x)ωxy

πω(x)=ωxy =ωyx =πω(y)Pω(y, x).(1.7)

This is known as detailed balance equation and iterating gives reversibility in the discrete-

time setting.

Note also that every chain on a graph that is reversible with respect to some measure

µand with transition probabilities (p(x, y))x,y∈Vcan be represented as a random walk

among random conductances, setting ωxy =µ(x)p(x, y).

In continuous time the notion of reversibility translates into

Pω

x(Xt=y) = Pω

y(Xt=x),∀t≥0,for the VSRW,

and Pω

x(Xt=y)

πω(y)=Pω

y(Xt=x)

πω(x),∀t≥0,for the CSRW.

1.1.3 Electrical networks

In [DS84] Doyle and Snell present in a very readable way the deep connections between

our model and real-world electrical networks, justifying our use of terms borrowed from

physics literature. A more rigorous, though simple, introduction to this relation is given

by [LP12], Chapters 2 and 9.

For this section we go back considering a general finite connected graph G= (V, E)

and look at it as an electrical network in the physical sense, where edges are made

of conducting wires. Two distinct sets of vertices, A, Z ⊂V(it is easier to think of

singletons), are attached to a battery that keeps a constant difference of, say, a unit

voltage between the two sets (that is, v(a) = 1 for all a∈Aand v(z) = 0 for z∈Z,

where vis the voltage at a given point). Every edge (x, y)∈Ehas a resistance rxy

and therefore a conductance cxy =1

rxy . We call ixy =v(x)−v(y)

rxy the current flowing from

xto y(Ohm’s Law). Summing this quantity over all the neighbours of xmust give

0, since Kirkhoff’s Laws from classical Physics state that the current flowing into any

point x∈V,x6∈ A∪Z, must be the same as the current flowing out of it. A little

algebra shows therefore that the voltage vat xis the weighted mean of the voltage of

the neighbours of x, i.e. is harmonic in the sense that

v(x) = X

y∼x

cxy

π(x)v(y).

For the random walk among conductances ωxy =cxy, the corresponding quatity is

p(x) = Pω

x(the random walk reaches Abefore Z).

This is in fact also an harmonic function that assumes values 1 in Aand 0 in Z, and by

unicity v(x) = p(x) for all x∈G.

The function i:E→Ris a flow between Aand Z, that is, a function fon the

directed pairs of neighbours in Gsatisfying f(x, y) = f(y, x), and Py∼xf(x, y) = 0 for

x6∈ A∪Z. Consider now the case A={a}. Let pesc be the probability, starting in a, of

returning to abefore ”escaping”, that is, reaching a point in Z. Then the probabilistic

interpretation of the current is the following: ixy is proportional to the expected number

of times that a walker, starting at aand wondering around until reaching Z, will jump

over the edge from xto yminus the times that he jumps from yto x. The constant of

proportionality is exactly the effective resistance Reff of the network, given by

R−1

eff =pesc X

y∼a

ωay =1

Py∼aiay

.(1.8)

We will call the quantity in (1.8) also effective conductance,Ceff . The name comes

from the fact that we could substitute the entire circuit with a unique wire of conduc-

tance Ceff between aand Z. Note that Ceff does not depend on the difference of voltage

between aand Z, it is an intrinsic property of the network.

We can now regulate the voltage at aso that the total current coming out of a(i.e.,

Py∼aiay) is 1. Among all the flows with this property (called unit flows), the current

is the one that minimizes the dissipation of energy, defined as

Edis(θ) = 1

x,y

θ(x, y)2rxy.(1.9)

This is known as Thomson’s principle ([TT79]).

Another probabilistic interpretation of the voltage can be given via the Green func-

tion. Let GZ(x, y) be the number of expected visits to y∈Vbefore touching set Zfor

the random walk started in x∈V. Then, setting the voltage to have a unit current

flow from ato Z, the voltage at x∈Vis equal to

v(x) = GZ(a, x)

Py∼xωxy

The previous notions can be extended to infinite connected graphs, though losing

a bit of their appealing realism. It is important to underline that the parallel beteen

real electrical network and the random conductance model is not purely aesthetic. The

techniques inherited from the physics literature can lead to very important theoretical

results. One remarkable example is the theorem that says that a random walk on an

infinite connected graph Gwith conductances {ωxy}is transient if and only if there

exists a unit flow of finite Edis energy (in this setting we have called a unit flow from a

point a∈Va function fsatisfying the previous properties with A={a}, but Z=∅,

and with Py∼af(a, y) = 1). Another example, closer to the content of this thesis, is

[Ros12], where an ”energy dissipation approach” is used for proving a result close to

that of 1.2.6.

1.2 Previous and new results

The random conductance model has gained growing attention from the community

in the last decade. One of the reasons is its relation to other important models

in Statistical Physics, such as the gradient fields (e.g., [BS11]), reinforced random

walk ([MR09, ACK12, ST12] among the others) and percolation theory (e.g., [GKZ93],

[BB07], [MP07], [BDCKY11]).

RWRC offers a wide range of problems to work on, but in the following chapters we

will mainly deal with three topics: large deviations for the local times of the RWRC,

the law of large numbers (LLN) for the endpoint of the walk and the central limit

theorem (CLT) for the effective conductance. In the next subsections we present in

greater detail the previous works on these particular subjects, but now a little detour

for a more general overview is incumbent. Refer to [Bis11] for a quite complete picture

of the known results and of the open problems on the RCM and its random walk.

The questions that are traditionally most studied for the RWRC are those of re-

currence and transience,heat kernel estimates and functional central limit theorems

(FCLT). References about the first subject include [GKZ93] for the random walk on

the infinite percolation cluster (see below for a precise definition of it), [Ber02] for the

random walk on the infinite cluster of long-range percolation in dimensions d= 1,2,

[ACK12] and [ST12] for recent results on the closely related model of edge reinforced

random walk. Heat kernel estimates have been addressed, e.g., in [Bar04] and [MR04]

for the percolation cluster, in [Del99] for the uniformly elliptic case, in [BD10] for

i.i.d. conductances bounded from below but not from above and in [BBHK08] for gen-

eral distributions of conductances between 0 and 1.

Much efforts have been put in the study of FCLTs, that is, the convergence in some

sense, after a space time rescaling, of the random walk to some process in continu-

ous space (often a Brownian motion). The case of uniformly elliptic conductances is

somewhat the easiest to study. In [SS04] the authors give the first complete proof of a

quenched functional central limit theorem (QFCLT) for any dimension when the con-

ductances are i.i.d.. An extension to stationary symmetric ergodic environments can

be found as a particular case in Section 6 of [BD10]. In [BD10] the QFCLT is derived

for i.i.d. conductances bounded away from 0 but not from infinity, i.e. they can reach

arbitrarily high values. When the expectation of the conductances is infite, the CSRW

rescaled in the proper way converges in law to a fractional kinetic motion ([Bˇ

C11]).

Of great interest is the case of conductances that can assume value zero. This

means that the random walk is not allowed to cross certain bonds and the labyrinth-

type geometry of the environment makes the analysis of the model more complicated.

In order to make things meaningful in this framework, one has to assume in dimension

dthat P(ωxy = 0) > pc(d), where pc(d) is the critical probability for bond percolation

in Zd. This guarantees the existence of an infinite cluster and one has only to condition

on the event that the starting point of the random walk lies indeed in this infinite

component. In the case of the simple random walk on the supercritical percolation

cluster (i.e. the conductances can assume only values 0 or 1) a QFCLT has been proven

first in [SS04] for d≥4 and then in [BB07] and [MP07] at the same time for the

remaining dimensions. The invariance principle can be also proven when the support

of the (i.i.d.) conductances is more generally contained between 0 and 1, see [BP07]

and [Mat08]. In [BBHK08] (for d≥5) and [BB12] (for d= 4) the authors study the

probabilities of return to the origin after 2nsteps of the walk and prove that, quite

surprisingly, the usual Gaussian upper estimates for the heat kernel do not hold. As a

consequence, we find ourselves in the unusual case where a CLT holds but a local CLT

does not. Finally, [ABDH10] deals basically with all the previous setting at once: The

quenched invariance principle is here proven for the CSRW and VSRW with general

i.i.d. conductances with values in [0,∞).

A weaker or annealed version of the previous results goes back to the seminal work

[DMFGW89], where the authors assume conductances which are translation invariant,

ergodic and have finite mean.

In the last year a couple of articles appeared showing a uniform quenched CLT

([GP12a]) and a conditioned (the random walk is forced to stay positive) quenched

CLT ([GP12b]) in the case of a RWRC whose jumps are unbounded (with polynomial

bounds on the tails of the jumps) in one dimension. These results have interesting

applications to Knudsen billiards (see [CP12]), modeling problems of transport and

diffusion in nanotubes.

Another recent topic that is gaining growing attention is that of random walks in

dynamic random environment. For dynamic random conductances, i.e. the weights

on the bonds may vary in time, an invariant principle has been proven in [And12] for

stationary ergodic conductances uniformly bounded from above and below, polynomially

mixing in space and time.

1.2.1 Large deviations and the local times

Large deviations is the study of unlikely events, the probability of which decreases

exponentially fast as time passes. [DZ10] is a prominent reference for the general theory,

while [dH00] offers a smoother introduction to the topic for non-experts. We recall the

general definition of a Large Deviation Principle (LDP) in a formulation similar to that

of Varadhan [Var66].

Definition 1.1. Let (E, d)a metric space and BEthe relative Borel-σ-algebra. Let

{γn}n∈Na sequence of positive numbers with γn→ ∞ and I:E→[0,∞]a function

such that I6≡ ∞. We say that a sequence of probability measures {µn}n∈Nsatisfies a

Large Deviation Principle (LDP) with rate function Iand speed γnif

(i) lim inf

n→∞

γn

log µ(O)≥ − inf

x∈OI(x)for all open sets O⊂E

(ii) lim sup

n→∞

γn

log µ(C)≤ − inf

x∈CI(x)for all closed sets C⊂E

(iii)The level sets Φ(s) := {x∈E:I(x)≤s}are compact.

There is a large amount of literature dedicated to large deviations statements for

RWRE’s, but the results are mostly dedicated to the limiting speed of the random walks

(see [GdH94], [CGZ00], [Var03] and many many others). LDP’s for other features of

the walks, which usually are well understood in the non-random-environment case, are

much less addressed.

One significant example is that of the local times, or occupation times, of the process.

Define

`t(z) := Zt

δz(Xs) ds z ∈Zd, t > 0,(1.10)

to be the time spent by a random walk (Xs)s≥0in the point z∈Zdup to time t. Here

δ·(·) denotes the usual Dirac delta, assuming value 1 when its two arguments are the

same and 0 otherwise.

Consider now (Xs)s≥0to be a random walk among fixed conductances {ωxy}x,y∈Zd.

Take a box BL= [−L, L]d∩Zd. An interesting question is: What is the probability

that the walker starting at the origin will not leave BLup to time T0, that is,

Pz∈BL`T(z) = T? But one can be much more precise and ask: What is the probability

that the normalized local time is close in some sense to a given function gwith support

in BL, i.e., 1

T`T≈g? The answer can be found, going back to the eighties, in a series

of seminal papers by Donsker and Varadhan [DV75b, DV75a, DV76, DV83] on the west

side of the world and G¨artner [G¨

77] on the east side, which built the basis for the

theory of large deviations for the occupation time measures of various types of Markov

processes.

Theorem 1.2. The sequence Pω

0(1

t`t∈ ·|supp(`t)⊂BL)satisfies a large deviation

principle on the space M1(BL)of the probability measures on BLwith speed tand rate

function ILgiven by

IL(µ) = −∆ω√µ, √µ−CL=X

x,y∈Zd:x∼y

ωxy√µ(y)−√µ(x)2−CL,(1.11)

where

CL= inf

µ∈M1(BL)X

x,y∈BL:x∼y

ωxy√µ(y)−√µ(x)2.(1.12)

Here we have trivially extended µto the whole Zdand therefore we can include also in

the sum in (1.12) all x, y ∈Zd, with x∼y.

A proof of the theorem similarly stated can be found in [K¨

06].

It was not possible for us to find references addressing the same problem when the

conductances are also random.

1.2.2 Local times large deviations for an RWRC

As pointed out in Section 1.2.1, no LDP for the local times `t(z) = Rt

0δXs(z) dshad been

proven before in the case of random weights on the bonds. In [KSW12] we derive the

annealed analogon of Theorem 1.2 in the random environment setting. As a byproduct

we obtain the asymptotics of the non-exit probability from a finite set B⊂Zd(not

necessarily a box) and the lower tails of the principal (i.e., smallest) eigenvalue λω(B)

of −∆ωin Bwith zero boundary condition, which can be seen as a Schr¨odinger operator.

We concentrate on the interesting case where the conductances are positive, but can

assume arbitrarily small values. Here the annealed behaviour comes from a combined

strategy of the conductances and the walk, and the description of their interplay is the

focus of our study. Loosely speaking, the optimal joint strategy of the conductances

and the walk to meet the non-exit condition X[0,t]⊂Bfor large tis that the conduc-

tances assume extremely small t-dependent values and the walker realizes very large

t-dependent holding times and/or trajectories that do not leave B. We will informally

describe this picture in greater detail.

Our main assumption on the i.i.d. field ωof conductances is that, for any {x, y} ∈ E,

ωxy ∈(0,∞) and essinf (ωxy)=0.(1.13)

More specifically, we require some regularity of the lower tails, namely the existence of

two parameters η, D ∈(0,∞) such that

log Pr(ωxy ≤ε)∼ −Dε−η, ε ↓0.(1.14)

That is, the edge weights can attain arbitrarily small values with prescribed probabili-

ties.

Our main theorem is the following large deviation principle for the normalised local

times before exiting B. That is, we restrict to the event {X[0,t]⊂B}={supp(`t)⊂B}.

EB:= {{x, y}:x∈B, y ∈Zd, y ∼x}(1.15)

we denote the set of edges connecting the sites of Bwith their neighbours both in B

and outside.

Theorem 1.3 (Annealed LDP for 1

t`t).Assume that ωsatisfies (1.13) and (1.14). Fix

a finite connected set B⊂Zdcontaining the origin. Then the process of normalized local

times, (1

t`t)t>0, under the annealed sub-probability law hPω

0(· ∩{X[0,t]⊂B})isatisfies

an LDP on M1(B), the space of probability measures on B, with speed tη

η+1 and rate

function Jgiven by

J(g2) := Kη,D X

{x,y}∈EB

|g(y)−g(x)|2η

η+1 , g ∈`2(Zd),supp(g)⊂B, kgk2= 1,(1.16)

where Kη,D =1 + 1

η(Dη)1

η+1 .

The proof of Theorem 1.3 is given in Chapter 2. More explicitly, it says

lim inf

t→∞ t−η

η+1 log DPω

01

t`t∈O, X[0,t]⊂BE≥ − inf

g2∈OJ(g2) for O⊂ M1(B) open,

(1.17)

lim sup

t→∞

t−η

η+1 log DPω

01

t`t∈C, X[0,t]⊂BE≤ − inf

g2∈CJ(g2) for C⊂ M1(B) closed,

(1.18)

and that the rate function Jhas compact level sets. Our convention is to extend

any probability measure on Btrivially to a probability measure on Zd; note the zero

boundary condition in Bthat is induced in this way.

Interestingly, we can see how the boundary case η=∞formally reconstructs the

result of Theorem 1.2.

Remark 1.4. As can be seen from its proof, Theorem 1.3 holds literally true if Zdis

replaced by an (infinite or finite) graph and Bby some finite subgraph.

A heuristic explanation of the speed and rate function is given in Section 2.1. It

turns out there that the conductances that give the most contribution to the LDP are

of order t−1/(1+η)and assume a certain deterministic shape.

With the special choice O=C=M1(B), we obtain the following corollary.

Corollary 1.5 (Non-exit probability from B).Under the assumptions of Theorem 1.3,

lim

t→∞t−η

η+1 log DPω

0X[0,t]⊂BE=−Kη,DLη(B),(1.19)

where

Lη(B) = inf

g2∈M1(B)X

{x,y}∈EB

|g(y)−g(x)|2η

η+1 .(1.20)

From Theorem 1.3, we also derive the precise logarithmic lower tails of the principal

(i.e., smallest) eigenvalue λω(B) of −∆ωin Bwith zero boundary condition.

Corollary 1.6 (Lower tails for the bottom of the spectrum of ∆ω).Under the assump-

tions of Theorem 1.3,

lim

ε↓0εηlog Pr(λω(B)≤ε) = −DLη(B)η+1.

The proof of this Corollary is postoponed to Section 2.3.

1.2.3 Law of large numbers and the point of view of the particle

While the question of the convergence of RWRC’s to some continuous diffusion is pretty

delicate to handle, the law of large numbers (LLN) is much more well understood.

If (Xn)n∈Nis our RWRC, call

lim

n→∞

the limiting speed, or just speed, of the walk. We ask: What are the conditions on the

distribution Pof the conductances in order to guarantee that such limit exists and is

the same for P-almost every environment ωand Pω

0-almost every trajectory?

Let us consider the Zdgrid. The following result is well known, see [Bis11], Theorem

2.4, for a more general case including the possibility of finite-mean jumps of the walk.

Theorem 1.7. Let {ωxy}x,y∈Zdbe nearest neighbor conductances sampled from a shift-

ergodic elliptic distribution Pwith E[ωxy]<∞. Then

Pω

0lim

n→∞

n= 0= 1 for P-almost every ω∈Ω.

The proof of this fact involves a quite standard tool for RWRE’s, namely the so called

motion from the point of view of the particle (see [KV86], [Koz85], [PV81] and [PV82] for

some earlier results). The technique consists in looking at the Markov chain (τXnω)n∈N

defined on the space of the environments Ω with transition probabilities

K(ω, ˜ω) = X

x∼0

Pω

0(X1=x)δτxω(˜ω).

This chain has, in the setting of the previous theorem,

Q(dω) := π0(ω)

E[π0(ω)]P(dω)

as a stationary measure, which is (as the formula shows) absolutely continuous with re-

spect to the original measure (this in general doesn’t happen for non-reversible RWRE’s).

The price paid for moving to a much bigger state space is rewarded with the stationar-

ity of the increments of the chain, which the original dynamics did not have. It can be

shown that the chain (τXnω)n∈Nis ergodic and the result of Theorem 1.7 follows then

quite easily from the usual ergodic theorem and representing (Xn)n∈Nas an additive

functional of (τXnω)n∈N.

A similar result can be easily proven for the simple random walk on the percolation

cluster. A much more interesting model is that of the biased random walk on the

percolation cluster (in which the walker ’prefers’ to go in one direction), where atipical

and unexpected behaviors of the speed occur: It may happen that the model switches

from a ballistic to a subballistic regime as the drift in the preferred direction increases.

The literature for physical motivations and background includes [Dha84] and [DS98],

while a mathematical coverage can be found in [BGP03], [Szn03], the more recent [Fri10]

and [FH11], and others.

Note that the conditions of Theorem 1.7 do not require any restriction on the mixing

properties of the environment, i.e. on how much far-away conductances are correlated.

A complementary well known result deals with the strongest mixing condition possible:

the i.i.d. case.

Theorem 1.8. Let the environment ωbe sampled with an i.i.d. law. Then

Plim

n→∞

n= 0= 1.

It is important to underline that no assumptions on the distribution of a single conduc-

tance were made.

We could not find any reference of such result in the literature. In Section 3.4 we will

give a sketch of the proof in two dimensions (for which we thank Prof. Noam Berger).

Both Theorem 1.7 and Theorem 1.8 give sufficient conditions for the RWRC to have

almost sure zero speed. To our knowledge there were no examples of walks among

ergodic random environments in Zdexhibiting non-zero speed in the literature before

[BS12].

Finally we would like to mention that a lot of research has been carried out for

RWRC’s when the underlying graph is other than the square lattice. One of the most

remarkable examples are random walks on Galton-Watson trees. For this model the

speed (defined as the limit of the distance from the root divided by the number of steps

performed) shows somewhat unexpected features: It is not monotonous in the strength

of the bias when the tree is allowed to have leaves (see [LPP96] and [A¨

11]), while its

behavious is still not fully understood in the no-leaves case ([BAFS11]).

1.2.4 Moments conditions for non-zero speed of RWRC’s

For this result, we let Pbe a measure on Ω which satisfies the following two conditions:

(i) Pis invariant and ergodic w.r.t. the group of spatial moves in Z2.

(ii) The marginal distribution of ωeis the same for all choices of the edge e, i.e. vertical

and horizontal edges have the same distribution.

Note that this is weaker than invariance w.r.t. rotations. Condition (ii) can be weakened

significantly, but for simplicity we keep it as is.

The two Theorems 1.7 and 1.8 give as result an almost sure zero-speed. There seems

to be two types of criteria involved: The first is moment conditions that control the size

of the conductances, and the second is mixing conditions saying that if the environment

mixes fast enough then the speed is zero. In [BS12] we only consider the first type,

and show that the sharp condition is that the logarithm of the conductances has high

enough moments.

Our main result is as follows.

Theorem 1.9. Let ebe an edge in E2.

1. If there exists α > 1such that

E[logαωe]<∞,(1.21)

then

Plim

n→∞

n= 0= 1.

2. For every α < 1there exists a distribution Pon environments such that E[logαωe]<

∞, but

Plim

n→∞

n= 0= 0.

Furthermore, in this case it is possible to choose Pso that either

P



lim

n→∞

n



∞>0= 1

Plim

n→∞

ndoes not exist= 1.

Remark 1.10. Our proofs will deal with conductances bounded away from zero, but

would work in the same way including the possibility of zero conductances. Note also

that the choice of dimension 2has been made in order to have easier and more intu-

itive proofs. We are confident that the same results can be proven with the very same

techniques in higher dimensions, with critical αequal to d−1.

1.2.5 Effective conductance and homogenization theory

As is well known, most materials, regardless how pure they may seem at the macro-

scopic level, have a rather complicated microscopic structure. It may then come as a

surprise that physical phenomena such as heat or electric conduction are described so

well using differential equations with smooth, sometimes even constant, coefficients. An

explanation has been offered by homogenization theory (see the monograph by [JKO94]

for an overview on the subject and its history): rapid oscillations at the microscopic

level average out, or homogenize, at the macroscopic scale. However, this does not mean

that the microscopic structure is simply washed out. Indeed, while it disappears from

the structure of the resulting equations, it remains embedded in the values of effective

material constants, e.g., the coefficients.

An illustrative example of a homogenization problem is that of effective conductance

for the RCM. For any Λ ⊂Zd, let B(Λ) be the edges with at least one endpoint in Λ.

Given an f:Zd→Rand a finite Λ ⊂Zd, let

QΛ(f) := X

hx,yi∈B(Λ)

ωxyf(y)−f(x)2,(1.22)

where each pair (x, y) is counted only once. This is the electrostatic Dirichlet energy for

the potential fwith Dirichlet boundary condition on the boundary vertices of Λ (note

the analogy with (1.12), where zero boundary conditions were considered instead).

For simplicity we will consider the square box ΛL:= [0, L)d∩Zd. A quantity of prime

interest for us is the effective conductance, which we already defined in the context of

electrical networks (compare with 1.8),

Ceff

L(t) := infQΛL(f): f(x) = t·x, ∀x∈∂ΛL,(1.23)

where t∈Rdand where ∂Λ are those vertices outside Λ that have an edge into Λ.

By Kirchhoff’s and Ohm’s laws (see, e.g., [DS84]), Ceff

L(t) is the total electric current

flowing through the network when the boundary vertices are kept at voltage t·x.

For homogeneous resistor networks, i.e., when ωxy := afor all hx, yi, the infimum

(1.23) is achieved by f(x) := t·xand so Ceff

L(t) = a|t|2Ld(1 + o(1)). A question of

(reasonably) practical interest is then what happens when the conductances {ωxy}are

no longer constant, but remain close to a constant.

A comparison of QΛwith these ωxy’s and the homogeneous case shows that Ceff

L(t)

is still of the order of |t|2Ld. Moreover, thanks to the choice of the linear boundary

condition, by subadditivity arguments the limit

ceff(t) := lim

L→∞

LdCeff

L(t) (1.24)

exists almost surely for any ergodic distribution of the conductances. The problem left

to resolve is thus a computation of the limit value.

Although ceff(t) can be computed only in a handful of (periodic) cases, it can be

characterized in large generality: Suppose that ωis a sample from a shift-ergodic law P

on the product space Ω := [λ, 1

λ]B(Zd)indexed by edges of Zd, for some λ > 0. As is well

known,

ceff(t) = inf

g∈L∞(P)EX

x=ˆe1,...,ˆed

a0,x(ω)t·x+∇xg(ω)2.(1.25)

Here ˆe1,...,ˆedare the unit coordinate vectors in Rdand ∇xg(ω) := g◦τx(ω)−g(ω)

is the gradient of gin direction of x∈Zd. The expression in (1.25) can be inter-

preted as the Dirichlet energy density with the spatial average naturally replaced by

the ensemble average. This object has not been introduced in the form of the limit

1.24 (see [PV82], [Koz86], [K¨un83] and the book [JKO94]), and proving the equivalence

with the original formulation requires a bit of work (it can be deduced, e.g., through

Proposition 4.3 of Chapter 4).

Once the (deterministic) leading-order of Ceff

L(t) has been identified, the next natural

question is that of fluctuations. It is obvious e.g., by checking the explicitly computable

d= 1 case — that no universal limit law can be expected for general conductance

distributions, but progress could perhaps be made for the (physically most appealing)

case of i.i.d. conductances. However, even here establishing just the order of magnitude

of the fluctuations turned out to be an arduous task. Indeed, more than a decade ago

Wehr [Weh97] showed that Var(Ceff

L)≥cLdfor some c > 0 but a corresponding upper

bound has been furnished only recently by Gloria and Otto [GO11]. Both of these results

contain important technical caveats: Wehr requires continuously distributed ωxy’s while

Gloria and Otto express their results under a “massive” cutoff.

Gloria and Otto drew important ideas from an earlier unpublished note by Naddaf

and Spencer [NS98] where (optimal) upper bounds on the variance are derived for

certain correlated conductance laws. The main tool of [NS98] is the Meyers estimate

(cf Meyers [Mey63]), to be used heavily in the proof of Theorem 1.11 as well. Other

noteworthy earlier derivations of (suboptimal) variance upper bounds include an old

paper by Yurinskii [Yur86] and a more recent paper by Benjamini and Rossignol [BR08].

Closely related to these estimates are recent derivations of quantitative central limit

theorems for random walk among random conductances and approximations of the

limiting diffusivity matrix, e.g., Caputo and Ioffe [CI03], Bourgeat and Piatnitski [BP04],

Boivin [Boi09], Mourrat [Mou12], etc. Incidentally, the Meyers estimate is also the key

tool in [CI03].

1.2.6 A central limit theorem for the effective conductance

In the article [BSW12] we prove that, for i.i.d. conductances which are (deterministi-

cally) not too far from a constant, the asymptotic law of Ceff

L(t) (defined in (1.23)) is

in fact Gaussian. Let N(µ, σ2) denote the normal random variable with mean µand

variance σ2. Then we have:

Theorem 1.11. Suppose the conductances ωxy are i.i.d. For each d≥1, there is

λ=λ(d)∈(0,1) such that the following holds: If (1.1) is satisfied P-a.s. with this λ,

then for each t∈Rdthere is σ2

t∈[0,∞)such that

Ceff

L(t)−ECeff

L(t)

|ΛL|1/2

law

−→

L→∞ N(0, σ2

t).(1.26)

Whenever the conductance law is non-degenerate we have σ2

t>0for all t6= 0.

The proof also immediately yields:

Corollary 1.12. Under the conditions of Theorem 1.11,

|ΛL|VarCeff

L(t)−→

L→∞ σ2

t,(1.27)

where σ2

tis as in (1.26).

A few remarks are in order:

Remarks 1.13. (1) Notice that (1.26) does not give us much information on the “order

expansion” of Ceff

L(t). Indeed, we know that ECeff

L(t)is to the leading order equal to

ceff(t)|ΛL|but when this order is subtracted, the next-order term is (presumably) of

boundary size. In d≥3, this is still larger than the typical size of the fluctuations.

Notwithstanding, what (1.26) does tell us is the character of the leading order random

term.

(2) There is in fact a formula for σ2

t, see Theorem 4.7 below, which also shows that

t7→ σ2

tis a bi-quadratic (and thus smooth) function of t. However, the formula involves

complicated conditioning and does not seem very useful for practical computations.

(3) There is no restriction on the single-conductance law other than (1.1). In par-

ticular, this law can have a non-absolutely continuous part including atoms. Certain

technical problems do arise at this level of generality; see Section 4.1.5 which, we be-

lieve, is of independent interest.

We prove Theorem 1.11 by reducing it to the Martingale Central Limit Theorem.

There are two main technical ingredients: homogenization theory (which enables a sta-

tionary martingale approximation of Ceff

L(t)) and analytical estimates for finite-volume

harmonic coordinates (by which we control the errors in the martingale approximation).

The restrictions to rectangular boxes, linear boundary conditions and small ellipticity

contrasts permit us to encapsulate the analytical input into a single step, the Meyers

estimate, cf Proposition 4.4 and Theorem 4.15. These restrictions can be relaxed but

not without additional arguments not all of which have been handled satisfactorily at

the time [BSW12] was uploaded on the arXiv. These are deferred to a follow-up paper.

We remark that two recent preprints have been brought to our attention at the time

this work was first announced in conference talks. First, Nolen [Nol11] has established

a normal approximation to the effective conductance defined over a periodic environ-

ment, in the limit when the period tends to infinity. Second, in a preprint that was

posted at the time of writing the present note, Rossignol [Ros12] formulates and proves

a central limit law for the effective resistance for the corresponding problem on a torus.

Nolen’s defines the problem over continuum, albeit with a rather strong assumption on

an underlying Gaussian i.i.d. structure. Rossignol’s setting is based on minimizing the

electrostatic energy over currents (rather than potentials) subject to a restriction on the

total current flowing around the torus. By a well known reciprocity relation between

effective conductance and resistance, these papers appear to address similar problems

(see Section 1.1.3).

Our work differs from both Nolen [Nol11] and Rossignol [Ros12] primarily in its

emphasis on fixed (Dirichlet), as opposed to periodic, boundary conditions. Indeed, a

majority of our technical work is aimed at controlling the resulting boundary effects.

Also the way a Gaussian limit law is established is quite different: Nolen appeals to

Stein’s method, Rossignol uses concentration of measure while we invoke the Martingale

Central Limit Theorem. A deficiency of our result compared to [Nol11] and [Ros12] is

the limitation on ellipticity contrast. Nolen overcomes this by an appeal to Gloria

and Otto [GO11], although this ultimately precludes the most interesting conclusion

in d= 2. Rossignol’s approach appears to work seamlessly for all elliptic product laws.

While the Gloria-Otto method can be adapted to our situation as well, just as for

Nolen [Nol11] it fails to deliver the desired conclusion in d= 2 (the issue is that the

method yields bounds on the moments of the corrector, which diverge in d= 2, while

we need only moments of the gradients of the corrector). The moment bounds thus

seem to be a separate technical matter, so for our first paper we decided to sacrifice

on generality of the distribution and derived the CLT only in the simplest, albeit still

physically interesting, case.

Chapter 2

Large deviations for the

occupation measure

In Section 2.1 we present a heuristic derivation of Theorem 1.3. Section 2.2 is dedicated

to the rigorous proof of the main result: Subsection 2.2.1 covers the lower bound (1.17)

while Subsection 2.2.2 takes care of the upper bound (1.18). Finally we prove Corollary

1.6 in Section 2.3. Section 2.4 offers a brief outlook of possible future research on the

subject.

2.1 Heuristic derivation

We now give a formal derivation of the LDP in Theorem 1.3. Given a fixed realisation

ϕ={ϕxy :{x, y} ∈ EB} ∈ (0,∞)EBof the conductances, the probability that the

normalised local time resembles some realisation g2∈ M1(B) is roughly

Pϕ

01

t`t≈g2≈exp −tIϕ(g2),(2.1)

where the corresponding Donsker-Varadhan rate function is given by

Iϕ(g2) = −∆ϕg, g=X

{x,y}∈EB

ϕxy|g(x)−g(y)|2.(2.2)

This is a formal application of the LDP for the normalized occupation times of a Markov

process with symmetric generator ∆ϕas in [DV75b] and [G¨

77] (see Theorem 1.2); by

(·,·) we denote the standard inner product on `2(Zd). Note that the event {X[0,t]⊂B}

is contained in {1

t`t≈g2}, therefore we drop it from the notation.

Taking random conductances into account, we expect an LDP on a slower scale than

t, as small t-dependent values of the conductances lead to a slower decay of the annealed

probability of the event {1

t`t≈g2}. Therefore, we rescale ωby a factor trwith some

r > 0 to be determined later, and approximate

Pr trω≈ϕ= Pr ∀{x, y} ∈ EB:ωxy ≈t−rϕxy=Y

{x,y}∈EB

Pr ωxy ≈t−rϕxy

≈exp −trηH(ϕ),(2.3)

where the rate function for the conductances is given by

H(ϕ) := DX

{x,y}∈EB

ϕ−η

xy .(2.4)

Here we made use of the tail assumptions in (1.14). Hence, combining (2.1) and (2.3),

DPω

01

t`t≈g21l{trω≈ϕ}E≈Pt−rϕ

01

t`t≈g2Pr ω≈t−rϕ

≈exp n−tIt−rϕ(g2)−trηH(ϕ)o

≈exp n−X

{x,y}∈EBt1−rϕxyg(x)−g(y)2+trηDϕ−η

xy o.

(2.5)

We obtain the slowest decay by choosing rsuch that t1−r=trη, which means r=

(1 + η)−1. Then the right-hand side has scale tη

η+1 , which is the scale of the desired

LDP. In order to find the rate function, we optimize over ϕand obtain that the choice

ϕ=ϕ(g)with

ϕ(g)

xy = (Dη)1

η+1 |g(y)−g(x)|−2

η+1 ,{x, y} ∈ EB,(2.6)

contributes most to the joint probability. Therefore, we have the result

DPω

01

t`t≈g2E≈exp n−tη

η+1 J(g2)o,

where the rate function is identified as

J(g2) = inf

ϕIϕ(g2) + H(ϕ)=Iϕ(g)(g2) + H(ϕ(g)) = Kη,D X

{x,y}∈EB

|g(y)−g(x)|2η

η+1 .

(2.7)

The tail assumptions we have made on the environment distribution lead to a fairly

remarkable interaction between the random influences of the environment on the one

hand and the random walk on the other. Under more general assumptions, e.g.,

log Pr(ωxy ≤ε)∼ −α(ε), ε →0

for some sufficiently regular nonincreasing function α:R+→R+, we would expect an

analogous result to hold. However, if α(ε) is not a polynomial in ε, the scale and rate

function of a corresponding LDP certainly would not have such an explicit form.

2.2 Proof of Theorem 1.3

In this section, we prove Theorem 1.3. This amounts to showing the two inequalities in

(1.17) and (1.18), since the compactness of the level sets follows immediately from the

continuity of Jand compactness of the space M1(B). The two inequalities are proven

in the next two sections.

2.2.1 Proof of the lower bound

In order to prove (1.17), we need to control the transition from one realization of the

environment to another. To this end, we first identify the density of this transition on

process level. We feel that this should be generally known, but could not find a suitable

reference. For ϕ:E→(0,∞) we abbreviate ¯ϕ(x) := Py∼xϕ(x, y). We also write ϕxy

instead of ϕ(x, y).

Lemma 2.1. Assume that ϕ, ψ:E→(0,∞)are bounded both from above and away

from zero. Denote by S(t)the number of jumps the process X= (Xs)s∈[0,t]makes up

to time tand by 0< τ1< . . . < τS(t)the corresponding jump times. Fix some starting

point x∈Zdand put τ0= 0. Then, for all t∈[0,∞),

Φt(X) :=

S(t)

i=1 ϕ(Xτi−1, Xτi)

ψ(Xτi−1, Xτi)e−(τi−τi−1)[¯ϕ(Xτi−1)−¯

ψ(Xτi−1)]e−(t−τS(t))[¯ϕ(Xt)−¯

ψ(Xt)]

is the Radon-Nikodym density of Pϕ

xwith respect to Pψ

xwith time horizon t.

Proof. We will write Φtinstead of Φt(X). Obviously, Φt>0 almost surely. We

start showing that, for all t≥0, the expectation of Φtunder Pψ

xis one. Then, we

use Kolmogorov’s extension theorem to show the existence of a measure Pxsuch that

Px(A) = Eψ

x(Φt1lA) for all A∈ Ft, where (Ft)t∈[0,∞)is the natural filtration generated

by X. It remains to show that the process Xunder Pxis a Markov process and that it

is generated by ∆ϕ, which implies Px=Pϕ

Let us start by showing that the expectation of Φtunder Pψ

xis one. Consider the

discrete-time process

Zn:=

i=1 ϕ(Xτi−1, Xτi)

ψ(Xτi−1, Xτi)e−(τi−τi−1)[¯ϕ(Xτi−1)−¯

ψ(Xτi−1)].

We have, for x∈Zd,

Eψ

x[Z1] = X

y∼x

ψxy

ψ(x)

ϕxy

ψxy Z∞

ψ(x)e−¯

ψ(x)s−( ¯ϕ(x)−¯

ψ(x))sds=X

y∼x

ϕxy

¯ϕ(x)= 1.

Combining this equation with the strong Markov property, we see that (Zn)nis a mar-

tingale with respect to the filtration (Fτn)n∈Ngenerated by the jumping times and that

Eψ

x"ϕ(Xt, XτS(t)+1 )

ψ(Xt, XτS(t)+1 )e−(τS(t)+1−t)[¯ϕ(Xt)−¯

ψ(Xt)]Ft#=Eψ

Xt[Z1] = 1 (2.8)

Pψ

x-almost surely for all x∈Zd. Then, we obtain

Eψ

x[Φt] = Eψ

x[ZS(t)+1], x ∈Zd,

by inserting the first term of (2.8) under the expectation and using that Φtis Ft-

measurable. Consequently, it remains to show that Eψ

x[ZS(t)+1] = 1. As S(t) + 1 is

an unbounded, but almost surely finite stopping time with respect to the filtration

(Fτn)n∈N, the optional sampling theorem yields that Eψ

x[ZS(t)+1]≤1. On the other

hand, for all integers k > 0,

Eψ

x[ZS(t)+1]≥Eψ

x[ZS(t)+11lS(t)+1≤k] = Eψ

x[ZS(t)+1∧k]−Eψ

x[Zk1lS(t)≥k]=1−Eψ

x[Zk1lS(t)≥k].

(2.9)

To show that the last term is arbitrarily close to one for large k, we recall that on

{S(t)≥k}

Zk≤maxx∈Zd, y∼xϕxy

minx∈Zd, y∼xψxy k

etmax{|ϕxy−ψxy|:{x,y}∈E}=: αk,

so Eψ

x[Zk1lS(t)≥k] is bounded from above by αkPψ

x(S(t)≥k). As all jumping times are

exponentially distributed with a parameter smaller than γ:= maxx∈Zd¯

ψ(x), we may

estimate

Pψ

x(S(t)≥k)≤eγt ∞

n=k

(γt)n

n!.

The tail of an exponential series is super-exponentially small, which means αkPψ

x(S(t)≥

k)→0 for k→ ∞. Since (2.9) was true for all k, we see that Eψ

x[ZS(t)+1] = 1.

For arbitrary k∈Nand t1, . . . , tk≥0 define ˆ

t= maxi∈{1,...,k}tiand a measure

Qt1,...,tkon (Zd)kby

Qt1,...,tk(x1, . . . , xk) = Eψ

x[Φˆ

t1l{Xt1=x1,...,Xtk=xk}], x1, . . . , xk∈Zd.

We verify without much effort that Eψ

x[Φt+s1lA] = Eψ

x[Φt1lA] for all A∈ Ftand t, s >

0, which implies consistency of the family of measures above. Thus, by Kolmogorov’s

extension theorem, there exists a measure Pxwith finite-dimensional distributions as

above, and we have Px(A) = Eψ

x[Φt1lA] for all t > 0 and A∈ Ft. We show that the

process Xunder Pxsatisfies the Markov property, i.e.,

Ex[1l{Xt+s=y}|Ft] = PXt(Xs=y)Px-a.s. for all y∈Zd, s, t > 0 (2.10)

where Exdenotes expectation with regard to Px. Note that PXtis defined as we have

considered an arbitrary starting point xin what we have shown so far. Indeed, for all

A∈ Ft

ExEx[1l{Xt+s=y}|Ft]1lA=Ex[1l{Xt+s=y}1lA] = Eψ

x[Φt+s1l{Xt+s=y}1lA]

=Eψ

xEψ

x[Φt+s1l{Xt+s=y}|Ft]1lA

(∗)

=Eψ

xΦtEψ

Xt[Φs1l{Xs=y}]1lA

=ExEXt[1l{Xs=y}]1lA,

where equation (∗) is due to the fact that Xsatisfies the Markov property under Pψ

and Φt+sΦ−1

t1l{Xt+s=y}depends only on X[t,t+s]. Consequently, we have shown (2.10)

and Xis a Markov process under Pxwith a unique infinitesimal generator. Elementary

calculations show that

tEψ

x[f(Xt)Φt]−f(x)t→0

−−→ ∆ϕf(x)

for arbitrary x∈Zdand f:Zd→R. This implies Px=Pϕ

xand the proof is complete.

Now we use Lemma 2.1 to compare probabilities for two environments that are close

to each other.

Corollary 2.2. Let ϕ, ψ :E→(0,∞)with 0< ψxy −ε≤ϕxy ≤ψxy +εfor some

ε > 0and all {x, y} ∈ E. Moreover, let Fbe some event that depends on the process

(Xs)s∈[0,t]up to time tonly. Then

Pϕ

0F≥e−4dεtPψ−ε

0F.

Proof. Let Φtdenote the Radon-Nikodym density of Pϕ

0with respect to Pψ−ε

0up to time

t. Employing the representation given in Lemma 2.1, we have

Φt≥

S(t)

i=1 e−(τi−τi−1)[¯ϕ(Xτi−1)−¯

ψ(Xτi−1)+2dε]e−(t−τS(t))[¯ϕ(Xt)−¯

ψ(Xt)+2dε]

≥

S(t)

i=1 e−(τi−τi−1)4dεe−(t−τS(t))4dε ≥e−4dεt.

The desired inequality follows immediately.

Remark 2.3. If the event Ais contained in {supp(`t)⊂B}, it suffices to require

0< ψxy −ε≤ϕxy ≤ψxy +εfor some ε > 0and all {x, y} ∈ EB.

Let us now show (1.17). Fix an open set O⊂ M1(B). As the event {X[0,t]⊂B}is

contained in {1

t`t∈O}, we omit it in the notation. Observe that the distributions of

t`tunder Pω

0and 1

t1−r`t1−runder Ptrω

0coincide for all 0 < r < 1. Hence

lim inf

t→∞

tη

η+1

log DPω

01

t`t∈OE= lim inf

t→∞

tlog DPt1

ηω

01

t`t∈OE,

which will simplify the application of a classical Donsker-Varadhan LDP for random

walks in fixed environment later. Choose an element g2∈Oarbitrarily. For M > 0

define ϕ(g)

M:EB→(0,∞) by

ϕ(g)

M(x, y) = 





(Dη)1

η+1 |g(y)−g(x)|−2

η+1 if |g(y)−g(x)|>0,

Motherwise.

Next, we introduce the set

A=ϕ:EB→(0,∞)ϕ(g)

M−ε≤ϕ≤ϕ(g)

M,(2.11)

where ε > 0 is picked smaller than 1

2minEBϕ(g)

M. By dint of Corollary 2.2,

DPt1

ηω

01

t`t∈OE≥DPt1

ηω

01

t`t∈O1lt1

ηω∈AE

≥inf

ϕ∈APϕ

01

t`t∈OPr t1

ηω∈A)

≥e−4dεtPϕ(g)

M−ε

01

t`t∈OPr t1

ηω∈A).(2.12)

Using the tail assumption in (1.14), we see that

lim

t→∞

tlog Pr t1

ηω∈A) = −H(ϕ(g)

M),

where His given in (2.4). Furthermore, we apply the lower bound of the classical

Donsker-Varadhan LDP (see [DV75b] or [G¨

77]) to get

lim inf

t→∞

tlog Pϕ(g)

M−ε

01

t`t∈O≥ −inf

OIϕ(g)

M−ε,

where Iϕis given in (2.2). Hence, from (2.12) we obtain

lim inf

t→∞

tlog DPt1

ηω

01

t`t∈OE≥ −4dε −inf

OIϕ(g)

M−ε−H(ϕ(g)

≥ −4dε −inf

OIϕ(g)

M−H(ϕ(g)

≥ −4dε −Iϕ(g)

(g2)−H(ϕ(g)

M),

since Iϕ(g)

M−ε≤Iϕ(g)

and g2∈O. Now we send εto zero and Mto ∞, to obtain

lim inf

t→∞

tlog DPt1

ηω

01

t`t∈OE≥ −Iϕ(g)(g2)−H(ϕ(g)) = −J(g2),

where ϕ(g)= limM→∞ ϕ(g)

Mis given in (2.6), and we used (2.7). The desired lower bound

follows by passing to the infimum over all g2∈O.

2.2.2 Proof of the upper bound

In this section we prove (1.18). Let us first fix some configuration ϕ∈(0,∞)Eand

start with an estimate for the probability Pϕ

0(1

t`t∈ ·). This approach has actually

been used by other authors before, but we provide an independent proof for the sake of

completeness.

Lemma 2.4. Fix an arbitrary set A⊂ M1(B). Then

Pϕ

01

t`t∈A≤f(0)

minBfexp ntsup

h2∈AX

x∈B

∆ϕf(x)

f(x)h2(x)o(2.13)

for arbitrary f:Zd→[0,∞)with supp(f) = Band t > 0.

Proof. We consider the Cauchy problem







∂tu(x, t)=∆ϕu(x, t) + V(x)u(x, t), x ∈Zd, t > 0,

u(x, 0) = f(x), x ∈Zd,

(2.14)

with

V=−∆ϕf

f1lB.

Obviously, u(·, t)≡f(·) solves (2.14). On the other hand, by the Feynman-Kac formula,

any nonnegative solution usatisfies

u(x, t) = Eϕ

xheRt

0V(Xs)dsu(Xt, t)i, x ∈Zd, t ≥0.(2.15)

Therefore, we may estimate

f(0) = Eϕ

0he−Rt

∆ϕf(Xs)

f(Xs)dsf(Xt)i

≥Eϕ

0he−Px∈B

∆ϕf(x)

f(x)`t(x)f(Xt)1l{1

t`t∈A}i

≥min

Bfexp n−tsup

h2∈AX

x∈B

∆ϕf(x)

f(x)h2(x)oPϕ

01

t`t∈A,

which is a rearrangement of the assertion.

Now fix some closed set C⊂ M1(B). As a closed subset of a finite-dimensional

space, Cis compact with respect to the Euclidean topology. We are going to apply a

standard compactness argument, which is in the spirit of the proof of the upper bound

in Varadhan’s lemma [DZ98, Thm. 4.3.1]. The idea is to cover Cwith certain open

balls, where ‘open’ refers to the Euclidean topology.

Fix δ > 0. For g2∈Cdefine

dg= min |g(y)−g(x)|:{x, y} ∈ E, g(x)6=g(y)∈(0,∞),

where we recall that g2is defined on the entire Zdand is zero outside B. Consider the

open ball in M1(B) of radius δg:= min{d4

g, δ}centered at g2. Fixing a configuration

ϕ∈(0,∞)E, we can apply Lemma 2.4 with f(·) := g(·) + pδg1lBand obtain

Pϕ

01

t`t∈Bδg(g2)≤1 + pδg

pδg

exp ntsup

h2∈Bδg(g2)X

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

h2(x)o.(2.16)

In what follows, we show

sup

h2∈Bδg(g2)X

x∈B

∆ϕ(g+√δg1lB)(x)

g(x) + √δg

h2(x)≤ −Iϕ(g2)(1 −7δ1

4),(2.17)

where we recall from (2.2) that Iϕ(g2) = P{x,y}∈Eϕxy|g(x)−g(y)|2=−(∆ϕg, g). To

that end, we replace h2by (g+pδg1lB)2and control the error terms.

sup

h2∈Bδg(g2)X

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

h2(x)

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

(g(x) + pδg)2

+ sup

h2∈Bδg(g2)X

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg(h2(x)−g2(x)) −2pδgg(x)−δg.

(2.18)

The first sum is easily estimated against the standard Donsker-Varadan rate func-

tion:

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

(g(x) + pδg)2=∆ϕ(g+pδg1lB), g +pδg1lB

≤∆ϕg, g=−Iϕ(g2),

where we have used the symmetry of the operator ∆ϕand that g= 0 outside B. In

order to estimate the last term in (2.18), we treat the contribution of every summand

within the square brackets separately. We begin with the first part and observe that

|h2(x)−g2(x)|=|h(x)−g(x)||h(x) + g(x)| ≤ 2δgfor all h2∈Bδg(g2) and x∈B. Thus

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

(h2(x)−g2(x))

{x,y}∈E:

x,y∈B

ϕxy

g(y)−g(x)

g(x) + pδg

(h2(x)−g2(x)) −X

{x,y}∈E:

x∈B,y6∈B

ϕxy(h2(x)−g2(x))

≤X

{x,y}∈E

x,y∈B

ϕxy |g(x)−g(y)|

pδg

2δg+X

{x,y}∈E:

x∈B,y6∈B

ϕxy2δg

≤4δ1

4Iϕ(g2).

The last step is due to the fact that δ

g≤g(x)−g(y) whenever g(x)−g(y)>0. Secondly,

x∈B

∆ϕ(g+√δg1lB)(x)

g(x) + pδg

(−2pδgg(x))

≤X

{x,y}∈E:

x,y∈B

ϕxy|g(x)−g(y)|2pδgg(x)

g(x) + pδg−2pδgg(y)

g(y) + pδg+X

{x,y}∈E:

x∈B,y6∈B

ϕxy2pδgg(x)

≤X

{x,y}∈E:

x,y∈B

ϕxy|g(x)−g(y)|22δg

pδgdg

{x,y}∈E:

x∈B,y6∈B

ϕxy2pδg|g(x)−g(y)|

≤2δ1

4Iϕ(g2).

Here, we have used δ

g≤dg. The only part left is

x∈B

∆ϕ(g+pδg1lB)(x)

g(x) + pδg

(−δg)

≤X

{x,y}∈E:

x,y∈B

ϕxy|g(x)−g(y)|1

g(x) + pδg−1

g(y) + pδgδg+X

{x,y}∈E:

x∈B,y6∈B

ϕxyδg

≤X

{x,y}∈E:

x,y∈B

ϕxy|g(x)−g(y)|21

pδgdg

δg+X

{x,y}∈E:

x∈B,y6∈B

ϕxyδg

≤δ1

4Iϕ(g2).

Combining (2.18) with the last three estimates, we obtain (2.17) and in particular

Pϕ

01

t`t∈Bδ(g2)≤1 + pδg

pδgY

{x,y}∈E

exp −t ϕxy|g(x)−g(y)|2(1 −7δ1

4).(2.19)

The balls Bδg(g2) with g2∈Ccover Cand since this set is compact, we may extract a

finite subcovering of C. Denote by (g2

i)i=1,...,N the centers of the balls in this subcovering.

Then, applying (2.19) for ϕ=t1

ηω, we obtain

lim sup

t→∞

tlog DPt1

ηω

01

t`t∈CE

≤max

i=1,...,N lim sup

t→∞

tlog DPt1

ηω

01

t`t∈Bδgi(g2

i)E

≤max

i=1,...,N X

{x,y}∈EB

lim sup

t→∞

tlog Dexp −t1+η

ηωxy|gi(y)−gi(x)|2(1 −7δ1

4)E.

According to de Bruijn’s exponential Tauberian theorem [BGT89, Theorem 4.12.9],

the tail assumption (1.14) is equivalent to the condition that, for any M > 0 and

{x, y} ∈ E,

lim

t→∞

tlog Dexp −t1+η

ηωxyME=−Kη,DMη

1+η,(2.20)

where we recall Kη,D =1 + 1

η(Dη)1

η+1 from Theorem 1.3. Thus, with δso small that

1−7δ1

4>0, we obtain

lim sup

t→∞

tlog DPt1

ηω1

t`t∈CE≤max

i=1,...,N X

{x,y}∈EB

−Kη,D|gi(y)−gi(x)|2η

1+η(1 −7δ1

4)η

1+η

≤ −(1 −7δ1

4)η

1+ηinf

g2∈CJ(g2)

with Jas in (2.7). Since we may choose δarbitrarily small, the proof of (1.18) is

complete.

2.3 Proof of Corollary 1.6

Proof. A Fourier expansion shows that, Pr -almost surely,

Pω

0(X[0,t]⊂B) = |B|

i=1

e−tλω

ivω

i(0)(vω

i,1l) ≤|B|

i=1

e−tλω

i|B| ≤ |B|2e−tλω(B),

where 0 < λω(B) = λω

1≤ ··· ≤ λω

|B|are the eigenvalues of ∆ωwith zero boundary

condition in Band (vω

i)i=1,...,|B|a corresponding orthonormal base of eigenvectors. We

also have, Pr -almost surely,

e−tλω(B)≤|B|

i=1

e−tλω

i(vω

i,1l)2≤X

z∈B

Pω

z(X[0,t]⊂B).

Applying Theorem 1.3 to B−zand using the shift-invariance of ω, we see that the

expectation of the right-hand side has the same logarithmic asymptotics as hPω

0(X[0,t]⊂

B)i. Therefore, the two above inequalities show that

log De−tλω(B)E∼log DPω

0(X[0,t]⊂B)E, t → ∞.(2.21)

Now de Bruijn’s exponential Tauberian theorem [BGT89, Theorem 4.12.9], together

with (1.19) yields the desired asymptotics.

2.4 Outlook: growing domains

The LDP in Theorem 1.3 is formulated for the simplest domain possible, that is a finite

set of points. What happens if we let the domain grow with time?

Imagine for simplicity to have a box Band to blow it up by a factor αt. We require

of course that αtgrows in time, but also that αtt1/2in order to give to the random

walk the time to fill the whole blown up box αtB. The problem emerging from this new

scenario is the meaning of an LDP on a sequence of different spaces (the box changes

its size as the time passes). In order to make sense out of it, we must rescale the boxes

mapping them to a common space, say the initial box B. This brings to highly non-

trivial analytical problems, as we pass from a discrete to a contionuous setting, wishing

to replace discrete gradients with a derivatives.

An interesting choice for the rate of growth could be αt=t1

d+2 . In fact, this

guarantees that the exponential rate of decay of the probability for the random walk

to stay in the box αtB(of the order t

α2

t) matches the exponential rate of decay for the

probability of ”controlling” the value of the d·αd

tconductances in the box.

K¨onig and Wolff have already investigated the case when the i.i.d. conductances have

a double stretched exponential tail near zero as in Theorem 1.3 and found a variety of

very interesting possible behaviours.

Theorem 2.5. Assume that ωsatisfies (1.13) and (1.14) with η > d

2(η > 1if d= 1),

and in addition that ωxy1l{ωxy<ε}has for some ε > 0a continuous increasing density.

Suppose 1αtt1

d+2 is increasing and fix a set G⊂Rdthat is open, connected,

bounded, with sufficiently regular boundary and containing the origin. Let F:= {f2:

f∈L2(G),||f||2= 1}equipped with the weak topology of integrals against bounded

continuous functions. Then the process of (properly rescaled and normalized) local times

Lt(x) := αd

t`t(bαtxc)satisfies

lim inf

t→∞

γt

log Pω

0(Lt∈O|supp`t⊂αtG≥ − inf

f2∈OJ0(f2)for all O⊂ F open,

where the speed is γt=t

1+ηα

d−2η

1+η

t,J0=J−infg∈F Jand

J(f2) := 





Kη,D Pd

i=1 RG|∂if(y)|2η

1+ηdyif f∈H1

0(G)

∞otherwise

.(2.22)

Kη,D is the same constant as in Theorem 1.3. Furthermore, J0has compact level sets

and for the non-exit probabilities also the corresponding upper bound holds:

lim inf

t→∞

γt

log Pω

0(supp`t⊂αtG≤ − inf

f2∈F J(f2).(2.23)

The proof of this theorem can be found in the Ph.D. thesis of Tilman Wolff [Wol13].

In fact, it is expected that a full LDP with the rate function described in (2.22) holds.

For the non-exit probability it is also shown therein that the exponent η=1

2is critical:

below this threshold the speed of the LDP is no longer the same. This is due to the

fact that, thanks to the fat tails, it is possible for the conductances to assume extremely

small values in bounded regions, trapping the random walk.

While in the finite-box case the assumption of conductances that can attain arbi-

trarily small values was fundamental in order to have interesting results, this is not

true anymore for the growing-box case. The limiting shape of the local times is hard to

predict even in the simplest settings, for example when the conductances can assume

only two values, say 1 and 2. Homogenization Theory may be the key ingredient (see

e.g. the harmonic coordinate technique in Chapter 4) to be combined with the tools

provided in the previous sections.

It is also worth noticing that the growing-domain case brings as a natural question

the quenched behaviour of the local times, which was trivial in the finite-domain setting.

The simplest case, i.e. strongly elliptic conductances, has also been treated in [Wol13].

Chapter 3

The speed of the RWRC

In Section 3.1 we show Part 1 of Theorem 1.9, which ends up being a simple application

of the Varopoulos-Carne Theorem. In Sections 3.2 and 3.3 we show Part 2 of Theorem

1.9. The construction builds upon the example constructed by Bramson, Zeitouni and

Zerner in [BZZ06].

Finally, in Section 3.4 we give a proof of Theorem 1.8.

3.1 Moment conditions for speed zero

In this section we prove Part 1 of Theorem 1.9.

In order to prove it, we will use the well known Varopoulos-Carne bound. For proof

see, e.g., [Car85].

Lemma 3.1 (Varopoulos-Carne).Let Lbe an irreducible Markov transition kernel with

reversible measure π. For states xand y, denote d(x, y) = min{n:Ln(x, y)>0}. Then

for every x,yand n,

Ln(x, y)≤2qπ(y)

π(x)·e−d(x,y)2

2n.(3.1)

Proof of Part 1 of Theorem 1.9. The measure πon Z2, defined by π(x) = Py∼xω{x,y}

is a reversible measure for our random walk. As in (1.21), let

D=E[logαωe]<∞.

For n∈N, consider the points x∈Z2such that ||x||∞=n, and call Enthe set of

edges having at least one end in these points. Note that |En|= 24n.

Then by Markov’s inequality, for every n∈Nand K > 0 we get

P∃e∈Ens.t. ωe>K

4≤24nD

logα(K

4).

In particular, if K=enβwith 1/α < β < 1, then

P∃e∈Ens.t. ωe>K

4≤Cn1−αβ,

for some constant C > 0.

Observe that 1 −αβ < 0. Therefore, by Borel-Cantelli lemma, for an integer κ >

(αβ −1)−1, a.s. for all nlarge enough and every edge e∈Enκ, we have

ωe≤1

4enκβ .

Therefore, for every xs.t. kxk∞=nκ, we have that π(x)≤enκβ .

Now fix M∈Nand assume that Mis large. For every nlarge enough,

PωkXMnκk∞> nκ≤Pω∃k≤Mnκ:kXMnκk∞=nκ

≤

Mnκ

k=1 X

x:kxk∞=nκ

Pω(Xk=x)

≤

Mnκ

k=1 X

x:kxk∞=nκ

2qπ(x)

π(0) e−n2κ

≤C0π(0)−1/2exp nκβ

2−nκ

2M,

for some constant C0>0.

Therefore, again by Borel-Cantelli, almost surely for all nlarge enough,

kXMnκk∞≤nκ.

From here we immediately get that almost surely

lim sup

n→∞ kXnk∞

n≤2

and in fact, since Mis arbitrary,

Plim

n→∞

n= 0= 1.

3.2 Trees

In this section and in the next one we prove Part 2 of Theorem 1.9. The section is divided

into two different subsections. In Subsection 3.2.1 we create the structure for the random

environment where, with probability one, the sequence Xn

ndoes not converge, and in

Subsection 3.2.2 we create another example where with probability one the sequence

Xn

nconverges to a speed which is not zero. In both cases E[logαωe]<∞for arbitrary

α < 1. The example in Subsection 3.2.1 is a direct application of the tree construction

of Bramson, Zeitouni and Zerner [BZZ06]. For the construction in Subsection 3.2.2, we

need to modify the tree of [BZZ06]. The construction is inspired by the construction in

[BZZ06], but we need to change quite a few details in order for the speed to converge.

In both cases, we adapt trees into environments for the random walk in the exact

same fashion. This is done in Section 3.3. Now, we give a short introduction with the

necessary terms from [BZZ06], and then, in Subsection 3.2.1 and 3.2.2, we create the

actual trees.

An ancestral function is a (in our case random) function a:x∈Z2→a(x)∈Z2

with the following properties:

•xand a(x) are nearest neighbours;

•a(a(x)) 6=x;

•the set of edges Fa:= {{x, a(x)}:x∈Z2}is a forest (i.e. the graph (Z2, Fa)

contains no cycles).

Every connected component of Fais an infinite tree. a(x) can be seen as the parent

of xand we denote by an(x) the n-th generation ancestor of x, for n≥0 (with the

convention a0(x) = x).

We also say that an ancestral function is directed if for some i, j ∈ {+1,−1}and for

every x∈Z2,a(x)−x∈ {(0, i),(j, 0)}.

The length of the longest branch starting in x(or the distance from xof its farthest

descendant, if one prefers the genealogical metaphore) is

h(x) := sup{n≥0 : ∃y∈Z2such that an(y) = x}.(3.2)

We are interested in the distribution of h(0) in the case of a random translation

invariant ancestral function.

Theorem 1 in [BZZ06] says that for any stationary ancestral function there exists a

constant c≥0 such that

lim inf

n→∞

nP(h(0) ≥n)≥c. (3.3)

In the same article the authors show that this is in fact the best lower bound achiev-

able. We give the 2-dimensional version of Theorem 2 in that paper:

Theorem 3.2 ([BZZ06], Theorem 2).There exists a stationary directed ancestral func-

tion (a(x))x∈Z2that is polinomially mixing of order 1 and for which

lim sup

n→∞ nP(h(0) ≥n)<∞.(3.4)

We now describe the BZZ tree, as appearing in [BZZ06].

3.2.1 The BZZ tree

We provide now the construction of the ancestral function used in [BZZ06], restricted

to the 2 dimensional case. We will make use of the same notations as [BZZ06] with an

additional tilde.

Let {e1, e2}be the canonical basis of Z2, with e1parallel to the x-axis. Fix two

constants ˜

θand ˜n0∈Nsuch that 2√2≤˜

θ≤˜n2

0. For every x∈Z2let ˜

L(x) be i.i.d.

random variables with atomless distribution and satisfying

P(˜

L(x)> t) = ˜

t2for t≥˜n0.(3.5)

We define an umbrella of intesity tto be

Ut=[

i=1,2

Ui,t (3.6)

where

Ui,t =y= (y1, y2)∈Z2:yi= 0, yj∈(0, t], j 6=i(3.7)

are the sides of the umbrella. The strength of the umbrella is also defined to be

equal to its intensity.

For every x∈Z2we will open the umbrella x+˜

U˜

L(x). Informally, one can think of

the ancestral function as a drop of rain trying to fall towards the up-right direction of the

plane and sliding on the sides of the umbrellas. Whenever two or more umbrellas overlap,

the water will consider only the strongest of them and penetrate the perpendicular ones.

Formally, one defines for every x∈Z2the strongest umbrella passing through that

point perpendicular to direction ei, for i∈ {1,2}, as

λi(x) = sup

y∈Z2:x∈y+˜

Ui,˜

L(y)

L(y).(3.8)

Note that the sup is taken over a non-empty set and it is easy to show that ˜

λi(x) is

also a.s. finite.

Since the distributions of the ˜

L(x)’s are atomless, the direction I(x)∈ {1,2}such

that

λI(x)(x) = min{˜

λi(x), i = 1,2}

is well defined. The ancestral function we are looking for is

˜a(x) = x+eI(x).(3.9)

The set of edges {x, ˜a(x)}, x ∈Z2through which the drops of rain have flown forms

a random forest (which can be shown to be in fact a random tree spanning the whole

Z2). This is the ancestral function used to prove Theorem 3.2, and we will call the

graph obtained with it the BZZ tree.

3.2.2 The Diagonal tree

We will now slightly modify the example seen in the previous subsection. Our aim is to

build a new tree for which the behaviour of h(0) is essentially the same as in the BZZ

tree, but with a different shape of the graph. Roughly speaking, it will not allow to have

long strips that are ”too horizontal” or ”too vertical”. This feature and its importance

will become more clear when we will describe the dynamics on these trees.

Fix suitable constants θand n0∈Nsuch that 10 ≤θ≤n2

0and so that following

equation (3.10) makes sense. For every x∈Z2consider i.i.d. random variables L(x)>1

with atomless distributions fulfilling

P(L(0) > t) = θlog t

t2for all t≥n0.(3.10)

Figure 3.1: Both in the straight umbrellas case a) and in the narrow umbrellas case

b) the drop of water follows the side of the biggest umbrella met. Note that in b) the

longest umbrellas are also the ones that are the narrowest.

The new umbrellas we want to open are a bit different from the tilde-umbrellas of

the previous section.

Define an umbrella of intensity tas

Ut=[

i=1,2

Ui,t (3.11)

where U2,t is the best Z2-grid lower approximation of the open segment of length tthat

makes an angle of π

4−1

log twith the x-axis, living in the first quadrant and starting

in the origin. U1,t is the reflection of U2,t with respect to the bisecting line of the first

quadrant. U1,t and U2,t are the sides of the umbrella. Note that this time the intensity

gives us the strength, the length but also the width of the umbrella. In particular, the

longer the umbrella, the more narrow it is.

We can think once more that drops of rain pouring from every point of the lattice

try to fall towards the up-right direction and that every time they reach a new vertex,

they are deflected by the strongest umbrella that passes through that vertex (see Figure

3.1).

In analogy with the straight-umbrellas case we define the strongest umbrella through

xperpendicular to direction ei, for i, j ∈ {1,2}and i6=j, as

λi(x) = sup

y∈Z2: [x,x+ej]∈y+UL(y)

L(y).(3.12)

Note that since L(0) >1 and since we are taking the lower (for the first component)

and upper (for the second) approximations of the segments described above, [x, x+e1]∈

U2,L(x−e1)and [x, x +e2]∈U1,L(x−e2), so that the sup on the right hand side of (3.12)

is taken over a non-empty set. It requires slightly more work compared to the straight-

umbrellas case to prove that it is also a.s. finite and therefore well defined.

We need some more notations. Similarly to [BZZ06], for m, n ∈Zcall Sn

mthe slab

m=x= (x1, x2)∈Z2:m≤x1+x2≤n.

The protecting area G(see Figure 3.2) is defined as

G:= nx= (x1, x2)∈ −N2∃n∈N:

x∈S−n

−nand −x1∈hyn·cos π

4−αn, yn·cos π

4+αnio,(3.13)

where αn= arctan √2

3 log nand yn=n

3√2 log np2 + 9 log2n. These values guarantee that

every segment S−m

−m∩Gis 2m

3 log mlong, and therefore contains √2m

3 log mpoints of Z2(up to

one unit, at most).

Note that every umbrella x+Uswith x∈G,−(x1+x2) = nand s∈[n, n2],

”protects” the origin O, meaning that Olies inside the ”Z2-triangle” generated by the

sides x+U1,s and x+U2,s.

Lemma 3.3. There is a constant csuch that for i= 1,2and t>n0,

P(λi(0) > t)≤clog t

t.(3.14)

Proof. This is a straightforward calculation.

P(λi(0) > t)≤CZ∞

[s]log s

s3ds

≤C∞

k=0 Z2k+1t

2kt

slog s

s3ds ≤C∞

k=0

log 2kt

2kt

=C1

∞

k=0

2k[log t+ log 2k]≤clog t

Figure 3.2: The protecting area Gis the region of the plane from which we can have um-

brellas that protect the origin. In particular, having a suitably strong umbrella starting

in the part of Gdelimited by the slab S−n

−n0will ensure h(0) < n with high probability.

Also in this case, the fact that the distributions of the L(x)’s are atomless guarantees

the uniqueness of a direction I(x)∈ {1,2}such that

λI(x)(x) = min{λi(x), i = 1,2}.

For example, if I(x) = 1, it means that the strongest vertical umbrella through xis

weaker than the strongest horizontal one. I(x) is the direction which the drop of water

will follow.

We can therefore define the new ancestral function

a(x) = x+eI(x).(3.15)

By its construction, it follows automatically that a:Z2→Z2is stationary and directed.

Theorem 3.4. The random ancestral function described in (3.15) is such that

lim sup

n→∞

log2nP(h(0) > n)<∞.(3.16)

3.2.3 Proof of Theorem 3.4

We closely follow the proof of Theorem 2 in [BZZ06].

We say that an umbrella Upenetrates a weaker umbrella Vin point x∈Z2if one

side of Uintersects one side of Vand xis the upper-right point of their intersection.

The following lemma bounds the probability that an umbrella of intensity tstarting in

the origin gets penetrated by another stronger umbrella in a given point z.

Lemma 3.5. Fix any t>n0. Let z∈Z2such that [z, z +ei]∈Uj,t, for some i, j ∈

{1,2}. Then there exists a constant c > 0independent of tsuch that

PI(z)6=i|L(0) = t≤clog t

t.(3.17)

Proof. For convenience, we shift the umbrella so that zis translated to the origin. We

look first at the event Ekthat the umbrella gets penetrated in the origin by an umbrella

of intensity s∈[k, k +1], for k+1 > t. Note that such a penetrating umbrella can come

only from S−1

−k−1. Furthermore, on every Sm

m,m∈ {−k−1, ..., −1}, there are almost

surely at most four points that can generate it, since for all the others the slope of the

sides would prevent them from penetrating the original umbrella in the origin. Hence

P(Ek)≤

k+1

m=1

4θlog k

k2−θlog(k+ 1)

(k+ 1)2≤c0log k

k2,

for some constant c0.

It is now easy to see that

PI(0) 6=i|L(−z) = t≤∞

k=btc

P(Ek)≤clog t

For n≥n0, define now the random variables Mn∈ {n0−1, ..., n}as following:

Mn:= max m∈ {n0, ..., n}:∃x∈S−m

−m∩Gwith m<L(x)< m2,(3.18)

with the convention Mn=n0−1 whenever the set on the right hand side is empty.

Proving that, for some constant c,

P(h(0) > m, Mn=m)≤clog2n

n2∀m=n0, ..., n, (3.19)

would imply

P(h(0) > n)≤

m=n0−1

P(h(0) > m, Mn=m)≤clog2n

n,(3.20)

that is the statement of the theorem.

We first prove (3.19) in the easy case m=n0−1.

P(h(0) > n0−1, Mn=n0−1) ≤P(Mn=n0−1)

=Pfor all m=n0, ..., n and x∈S−m

−m∩G, L(x)6∈ (m, m2)

m=n01−P(L(0) ≥m) + P(L(0) > m2)#(S−m

−m∩G)

m=n01−θlog m

m2+θlog m2

m4√2

log m

≤

m=n01−θ1−2

0log m

m2√2

log m

≤e−θ1−2

0√2

3Pn

m=n0

m≤c n−2(3.21)

by the choice of θ, for some c > 0.

For the more complex cases m=n0, ..., n we faithfully follow [BZZ06] once again.

For i, j, r ∈Z,i≤jand x∈Z2, define the events

i(x, r) = L(y)6∈ −y·~

1 + r, (−y·~

1 + r)2for all y∈Sj

i∩(x+G).(3.22)

Firstly note that

Ph(0) > m, Mn=m≤X

x∈S−m

−m∩G

Ph(0) > m, L(x)∈(m, m2), A−m−1

−n(0,0)

x∈S−m

−m∩G

Ph(−x)> m, L(0) ∈(m, m2), A−1

m−n(−x, m)

x∈Sm

m∩−G

Ph(x)> m, L(0) ∈(m, m2), A−1

m−n(x, m),

where we have used stationarity to obtain the second line and we write −G=x=

(x1, x2) : (−x1,−x2)∈G.

Consider now the segment joning the points in Sm

m∩ −G, divide it in eight parts

of the same length (approximately 1

3 log mlong) and call them I1, ..., I8. For every

j∈ {1, ..., 8}, consider ˆxjand ˇxj, the points with respectively the highest and the

lowest y-coordinate on Ij. Draw the infinite cones ˆ

Cjand ˇ

Cjwith amplitude β=

arctan 2

3 log mwhose bisector makes an angle of 5

4πwith the x-axis and with vertices

ˆxjand ˇxjrespectively. Observe that the points in the area ˆ

Cj∩ˇ

Cj∩S−1

−n+mare contained

in S−1

−n+m∩(x+G) for every x∈Ij. Therefore the event

Ej(m, n) := L(y)6∈ −(y1+y2) + m, (−(y1+y2) + m)2

for all y= (y1, y2)∈ˆ

Cj∩ˇ

Cj∩S−1

−n+m

is contained in the event A−1

m−n(x, m) for all x∈Ij. Hence

x∈Sm

m∩−G

Ph(x)> m, L(0) ∈(m, m2), A−1

m−n(x, m)

≤

j=1 X

x∈Ij

P(h(x)> m, L(0) ∈(m, m2), A−1

m−n(x, m))

≤

j=1 X

x∈Ij

P(h(x)> m, L(0) ∈(m, m2), Ej(m, n))

j=1

Eh#{x∈Ij:h(x)> m};L(0) ∈(m, m2); Ej(m, n)i.(3.23)

The interval (m, m2) can be divided in a finite number of disjoint subintervals such

that the Z2approximation of every umbrella with intensity in a given subinterval looks

the same at least up to the first medges. More precisely, there exists M∈Nand

there exist {m1=m<m2< ... < mM=m2}such that, for any k∈ {1,2, ..., M},

∀h, l ∈(mk, mk+1), one has Uh|m=Ul|m, where Uh|mis the umbrella of intensity h

whose sides are restricted to the first medges (going from bottom-left towards up-right).

Therefore, we can rewrite (3.23) as

j=1

l=1

Eh#{x∈Ij:h(x)> m};L(0) ∈(ml, ml+1); Ej(m, n)i.(3.24)

For any point x∈Sm

m∩−Gto have h(x)> m, there must be a branch coming out of

xthat perforates the protecting umbrella generated by the origin (since L(0) ∈(m, m2)).

That is, at least one point zon UL(0)|mmust be penetrated by another umbrella. On

the other hand, every penetrated zcan give rise to at most one of such x’s. Hence, for

any l= 1, ..., M, given L(0) ∈(ml, ml+1),

#{x∈Sm

m∩−G:h(x)> m} ≤ X

i=1,2X

[z,z+ei]∈UL(0)|m

1l{I(z)6=i}.(3.25)

Plugging this in (3.24) gives

Ph(0) > m, Mn=m

≤

j=1

l=1 X

i=1,2X

[z,z+ei]∈UL(0)|m

PI(z)6=i, L(0) ∈(ml, ml+1), Ej(m, n).

The intersection of the first two events inside the last probability is not independent

of Ej(m, n), but there is a negative correlation between them. We obtain therefore the

upper bound

Ph(0) > m, Mn=m

≤

j=1

l=1 X

i=1,2X

[z,z+ei]∈UL(0)|m

PI(z)6=i, L(0) ∈(ml, ml+1)PEj(m, n).

We can now directly compute the right hand side of last expression. For [z, z +ei]∈

UL(0)|mwe have, by Lemma 3.5,

PI(z) = i;L(0) ∈(ml, ml+1)=Zml+1

PI(z) = i|L(0) = td

dtP(L(0) ≤t)dt

≤Zml+1

clog t

t3(2 log t−1)dt

≤Klog2m

m4(ml+1 −ml),(3.26)

for some constant K.

Summing over the directions i= 1,2 and over all the z∈Z2such that [z, z +ei]∈

UL(0)|mand then summing over l= 1, ..., M, one is left with a factor of order log2m

m2.

In order to evaluate the probability of any Ej(m, n), note that, for k≥m, every

Cj∩ˇ

Cj∩S−k+m

−k+mcontains more than 1

log kpoints of the lattice. In fact, each cone ˆ

Cjand

Cjintersects S(k), the hyperplane containing S−k+m

−k+m, on the segments ˆ

kand ˇ

k, each

of length bigger than k

√2·2

3 log m(they are, in fact, the double of the cathetus of a right

triangle, whose opposite angle measures β

2radians and with the other cathetus k

√2long).

Since ˆxjand ˇxjare roughly 1

3 log mfar apart, the intersection of ˆ

kand ˇ

kis longer

than √2k

3 log m−1

3 log m≥1

log k. Being the distance between close points on ˆ

Cj∩ˇ

Cj∩S−k

−k

equivalent to √2, the total number of points is bigger than 1

√2

log k≥1

log k. By the

independence of the (L(x))x∈Z2

PEj(m, n)≤

k=m+1 1−θlog k

k21

log k

≤exp n−θ

k=m+1

≤exp n−θ

5Zn

m+1

sdso

=n

m+ 1−θ

5.(3.27)

Putting all together and reminding that θ≥10, we finally obtain, for some constant

Ph(0) > m, Mn=m≤clog2m

m2n

m+ 1−θ

≤c(m+ 1)θ

5−2n−θ

5log2m

≤c n−2log2n. (3.28)

3.3 The environment

The two random trees constructed in the previous sections will provide, in some sense,

the support for our random environments. In both cases, the ω’s are constructed in the

following way.

Sample a realization of the tree as described above. For every z∈Z2, the edge

{z, a(z)}will have a conductance value of ω{z,a(z)}= e(h(z)+1)A, where a:Z2→Z2is

the ancestral function used for constructing the sampled tree and A > 1 is a constant.

We set all the other conductances to be equal to one.

For both the BZZ and the Diagonal tree, the conductances have infinite α-logmoments

for any α > 1. On the other hand, choosing appropriately the constant A > 1, we can

obtain conductances with finite α-logmoments for αarbitrarily close to 1 from below.

Proposition 3.6. Take ¯α < 1. Then, the conductances of the random environments

described above with 1<A< 1

¯αare such that

Elogαωe≤ ∞ ∀α≤¯α(3.29)

and

Elogαωe=∞ ∀α≥1.(3.30)

Proof. We first prove it for the random environment built on the BZZ-tree support.

Elogαωe=Z∞

P(logαωe> t)dt

=∞

k=0 Z(k+1)αA

kαA

P(logαωe> t)dt

≤∞

k=0

P(logαωe> kαA)((k+ 1)αA −kαA)

=∞

k=0

P(h(0) > k)((k+ 1)αA −kαA) (3.31)

By equations (3.3) and (3.4) we know that for all sufficiently large k∈N, say k≥K,

k≤P(h(0) > k)≤c0

Therefore, on the one hand, taking α≤¯α,

Elogαωe≤C+∞

k=K

kαA(k+ 1)αA−1<∞,

where C > 0 is the finite contribution of the first K−1 terms of the sum, while, on the

other hand, when α≥1 we obtain, with a minor modification of (3.31),

Elogαωe>∞

k=K

kαAkαA−1=∞.

Note that the very same proof is valid for the random environment built over the

Diagonal tree structure, since the log2-correction in Theorem 3.4 doesn’t change the

behaviour of the series (3.31).

Proposition 3.7. For almost every environment ωsampled from the constructions of

the previous section, the random walk among the conductances ωwill eventually follow

the tree. This means that almost surely there exists ¯n < ∞such that for all n≥¯n, if

Xn=xthen Xn+1 =a(x), where a:Z2→Z2is the ancestral function used to construct

the tree underlying the environment.

Proof. The probability that, starting in a point x∈Z2, the random walk will follow the

tree forever is, by the independece of the jumps, bigger than

∞

k=1

ekA

2e(k−1)A+ ekA+ 1.(3.32)

It is easy, in fact, to get convinced that this is a very pessimistic estimate. It represents

the case in which we start from a leaf of the tree (that is, a vertex that is ancestor of

no other vertices) and where every time ω{Xn,a(Xn)}is of order k(that is, equal to ekA),

then the two edges under and at the left of Xnare of order k−1.

Call T1, T2, ... the times in which the random walk doesn’t go in the direction of

the ancestral function. After each of these times, a new attempt to follow the tree is

performed. Therefore if we show that the product (3.32) is a constant strictly bigger

than zero, than the sum of the probabilities of succeeding in following the tree in one of

the attempts is infinite. By Borel-Cantelli lemma, this means that almost surely there

will be a finite time from which we will always follow the tree.

We are left to show that (3.32) is bigger than zero, or, equivalently, that its log is

bigger than −∞:

log ∞

k=1

ekA

2e(k−1)A+ ekA=−∞

k=1

log 1+2e(k−1)A

ekA

>−2∞

k=1

e(k−1)A−kA

>−2A∞

k=1

e−(k−1)A−1>−∞,(3.33)

where we have used the mean value theorem for the bound kA−(k−1)A≥A(k−1)A−1.

Proposition 3.8. The random walk among random conductances with environment

built on the BZZ tree, as described above, has almost surely no limiting speed.

Proposition 3.9. The random walk among random conductances with environment

built on the diagonal tree, as described above, has almost surely a limiting speed which

is not zero.

Proof of Proposition 3.8. From Proposition 3.7 we know that with probability 1 there

exists a finite time from which the random walk will use only edges pointing the right or

up direction with respect to its current position. Without loss of generality we can think

this time to be time 0. In order to study the limiting speed of the process, we have to

go back to the underlying structure of the tree on which we have built the environment.

Note that every time the random walk makes a step in the direction of the ancestral

function, it finds several new umbrellas perpendicular to its previous step and a new

parallel one. If the strongest perpendicular umbrella is stronger than any other umbrella

on the direction of the previous step, the branch of tree changes orientation and the

next step of the random walk will follow it; otherwise, it will perform another step in

the same direction as before.

The distribution of the length ¯

Lof the new perpendicular umbrellas met at each

step is easy to calculate:

P(¯

L(0) > t) = P∃j∈Nsuch that ˜

L((0,−j)) >max{t, j}

= 1 −btc

j=1

P(˜

L((0,−j)) ≤t)∞

j=btc+1

P(˜

L((0,−j)) ≤j)

= 1 −1−˜

t2btc∞

j=btc+1 1−˜

j2(3.34)

so that, by straightforward calculations,

t≤P(¯

L(0) > t)≤c00

t,(3.35)

for some c0, c00 >0.

Now, being on a branch of the tree, what is the probability of passing from the

umbrella that has generated that part of the branch to a stronger one before the umbrella

itself ends? Suppose that that the random walk is on a branch of the tree generated

by an umbrella of length k > 2n2

0. Then the probability of not meeting a stronger

perpendicular umbrella or a stronger umbrella on the same direction of the current one

before leaving the present umbrella is bigger than

1−c00

kk1−˜

k2k>e−2c00−1,

that is a constant strictly smaller than 1 and independent of k.

Considering only the strongest umbrellas through each point, call rush a sequence

of intersecting umbrellas each bigger of the previous one that determines a part of the

final tree.

Starting on any rush, the probability of leaving it (that is, of travelling the whole

length of one of the umbrellas without meeting a stronger one) after having visited

N∈Ndifferent umbrellas is

P(Leave the rush after more than Numbrellas) <1−ce−2c00N,(3.36)

for some c > 0.

This means, by Borel-Cantelli lemma, that with probability 1 the random walk will

leave any rush in finite time. Given a realization of the walk, call τ(1) ∈Nthe time in

which the random walk leaves the first rush, τ(2) the time in which it leaves the second

one and so on. τ(1) < τ(2) < ... is a sequence of (almost surely finite) integer stopping

times that goes to infinity.

Fix T > 1 and define the times T1=T,τ1= mini=1,2,...{τ(i) : τ(i)> T1}and

recursively

Tk=τk−1+τk−1Tk∀k > 1,

τk= min

i=1,2,...{τ(i) : τ(i)> Tk−1} ∀k > 1.(3.37)

Our aim is now to show that in the intervals of the form (Tk−1, Tk), the longest

umbrella met is of length of the order τk−1Tk. We don’t want the longest umbrella to

be much longer than this, otherwise it could ”interfere” with the next intervals: consider

the event

Ek={In the interval (Tk−1, Tk) the longest umbrella met

is stronger than Tk+1τk}.

Its probability can be bounded from above by

P(Ek)<1−1−˜

τkTk+1 Tk

<1−e−2Tk

τkTk

Tk+1 ,

for some constant c > 0, since Tk≤τk. By Borel-Cantelli lemma, P(Eki.o.) = 0.

On the other hand, we don’t want the longest umbrella to be shorter than that. This

is because we want it to be long a positive fraction of the entire time interval (Tk−1, Tk).

In fact, the interval (Tk−1, Tk) is long about τk−1Tk. Furthermore, we want the random

walk to follow this umbrella for a positive fraction (say an ε > 0 fraction) of its length

before leaving the time interval. This two events guarantee a relevant contribution to

the speed up to time Tk. Therefore take, for a fixed ε > 0 small,

Fk=In the interval (Tk−1, Tk(1 −ε)) the longest umbrella met is stronger

than τk−1Tkand is bigger than the biggest umbrella in (Tk(1 −ε), Tk).

By the independence of the new umbrellas discovered at each step, we have, for all

k∈N,

P(Fk)>(1 −ε)P(one of the Tk−Tk−1umbrellas is longer than τk−1Tk)

= (1 −ε)1−1−c0

τk−1TkTk−Tk−11−˜

τ2

k−1T2kTk−Tk−1

>(1 −ε)1−1−c0

τk−1TkTk−Tk−1

>(1 −ε)1−e−c0

τk−1Tk(τk−1+τk−1Tk)

=C(3.38)

where C > 0 is a constant not depending on k. By the second Borel-Cantelli lemma,

there are almost surely infinitely many intervals (Tk−1, Tk) for which Fkhappens.

Hence, almost surely there exists a ¯

k∈N(depending eventually on the realization

of the environment and of the random walk) such that Ekdoes not happen for every

k > ¯

kwhile Fkholds infinitely many times. Take now the strongest umbrella met up

to time T¯

k. Its length L > 0 is almost surely finite, so that κ:= min{k:Tk> T¯

k+L}

is well defined. Note that ∀k > κ + 1, in the interval (Tk, Tk+1) there is no umbrella

longer than Tk+1τkmet in the past.

Take the infinite subsequence κ<k1< k2< ... such that Fkiholds true for every

i∈Nand such that the longest umbrella met in the ki’th interval (Tki−1, Tki) is followed

by the random walk at least for a positive fraction 0 < η <  of its length. Note that

since there is no longer umbrella coming from a previous interval, once the random walk

meets this umbrella it follows it until its end or at least until the end of the interval

itself, and the probability of meeting the umbrella before the last ηfraction of its length

is strictly positive. This implies that we have such a sequence (ki)i=1,2,... almost surely.

Suppose now that a limiting speed v= (v[1], v[2]) existed. We want to show that in

each of those intervals there is at least one time tat which the ratio Pt

j=1 Xj/t is far

from v, bringing to a contradiction. Call ti∈Nthe time at which the longest umbrella of

the interval (Tki−1, Tki) is met and ti={”Time of the last point of the umbrella”∧Tki}.

By definition, this umbrella is longer than τki−1Tki, it is met before time Tki(1 −ε) and

before the last η-fraction of its length. Call

j=1

Xj= (vi[1], vi[2]) =: vi

and

j=1

Xj= (vi[1], vi[2]) =: vi

the partial speeds up to time tiand tirespectively. Without loss of generality suppose

that we met the longest umbrella on its horizontal side. Note that

vi[1] = 1

tivi[1]ti−ti+ti

and that ti

>1 + ητki−1Tki

(1 −ε)τki−1(Tki+ 1) >1 + η

2(1 + ε)=: β > 0.

Further suppose v[1], v[2] 6∈ {0,1}. Then if v[1] > vi[1]

|v[1] −vi[1]|=v[1] −vi[1]ti

ti+ti

ti−1>(β−1)(1 −v[1]) >0,(3.39)

while if v[1] ≤vi[1]

max |v[1] −vi[1]|,|v[1] −vi[1]|≥1

2(vi[1] −vi[1])

2vi[1] −vi[1]ti

ti+ti

ti−1

2(β−1)(1 −v[1]).(3.40)

In both cases the distance from the limiting speed is bigger than a constant that is

independent of kiand strictly bigger than zero.

The cases v[1] = 1 and v[1] = 0 have probability 0. In fact, the probability of

meeting in any interval (Tki−1,(1 −ε)Tki) a vertical (respectively, horizontal) umbrella

of order τki−1Tkithat is stronger of any other horizontal (vertical) umbrellas met before

(and of following it for a time of O(t)) is strictly positive, for the reasons mentioned

above.

Proof of Proposition 3.9. Let v= (0.5,0.5). We claim that, almost surely,

lim

n→∞

n=v.

As in the previous proof, let N0be such that for every n>N0, we have Xn+1 =a(Xn),

where ais the ancestral function. By a minor modification of Proposition 3.7, N0is

almost surely finite. We need to prove that for every ε > 0 there exists a (random)

finite Msuch that for every n>M, we have kXn/n −vk< ε, where we write k·kfor,

e.g., the usual `1-norm. To this end, we need to understand the various umbrellas that

the random walk traverses. By the construction of the diagonal tree, there exists K > 0

such that for every umbrella which is stronger than K, for every two points xand yon

the umbrella whose distance is larger than some U=U(ε), we have



y−x

ky−xk−v



< ε.

For umbrellas which are not stronger than K, their distribution is symmetric w.r.t. the

diagonal, and their directions are i.i.d. and therefore they give an average of v.

Therefore,

lim

n→∞

n=v.

3.4 Proof of Theorem 1.8

Here we give a sketch of the proof of Theorem 1.8 for the two dimensional case.

Fix a threshold M > 0 such that P(ωxy ≤M) is bigger than the critical probability

for bond percolation. For any configuration of conductances ω, an edge is good if its

conductance is smaller or equal to Mand bad otherwise. Standard percolation estimates

tell us that all the connected components of bad edges are finite and that for any bad

edge eone has P(|Ue|> n)≤e−cn for some constant c > 0, where |Ue|is the number

of bad edges in the component of e. Now we define a new set of conductances ω0such

that

ω0

e=





2M(|Ue|+|∂Ue|) if eis bad or a boundary edge

ωeotherwise

where a good edge is a boundary edge if it shares an endpoint with a bad edge (i.e. it

is its neighbour) and ∂Ueis the set of all (good) neighbours of the bad edges in the

component of e.

Note that the new environment ω0is ergodic and that

E[ω0

xy]≤∞

i=1

2Mn2P(|Ue|+|∂Ue|=n) + M < ∞.

Hence, the well known Nash-Williams criterion (see, e.g., [LP12]) implies easily the

recurrence of the random walk on ω0. This is equivalent to say (compare Section 1.1.3)

that for every flow θthe dissipation energy Eω0

dis(θ) in 1.9 relative to ω0must be infinite.

If we show that for every flow θwe have Eω

dis(θ)≥Eω0

dis(θ), then we are done.

For ν=ω, ω0we have

Eν

dis(θ) = X

egood not boundary

θ2(e)

νe

ebad or boundary

θ2(e)

νe

egood not boundary

θ2(e)

νe

Ubad connected component X

e∈U∪∂U

θ2(e)

νe

where one should be careful not to count more than once boundary edges in order to

have exact equality (in any case, counting them more than once does not change the

finiteness of Eν

dis).

We just need to show that for every bad component Uone has

e∈U∪∂U

θ2(e)

ωe

=: Eω,U

dis ≥Eω0,U

dis .

Note that for every e∈U∪∂U one has |θ(e)| ≤ Pe0∈∂U |θ(e0)|by the definition of

dissipation energy and therefore θ2(e)≤ |∂U|Pe0∈∂U θ2(e0). One finally obtains

Eω0,U

dis =X

e∈U∪∂U

θ2(e)

ω0

≤Pe∈U∪∂U θ2(e)

2M(|U|+|∂U|)2

≤|U∪∂U||∂U|Pe∈∂U θ2(e)

2M(|U|+|∂U|)2

≤1

e∈∂U

θ2(e)

≤X

e∈U∪∂U

θ2(e)

ωe

=Eω,U

dis .(3.41)

Chapter 4

A Central Limit Theorem for the

effective conductance

In Section 4.1 we discuss the strategy of the proof of Theorem 1.11 and state its principal

ingredients in the form of suitable propositions. In Subsection 4.1.6 we describe the

organization of the rest of the chapter.

Note that in this chapter we will make use of the sign axy for the conductances rather

than ωxy, being the first notation more used in the Homogenization Theory lterature.

4.1 Key ingredients

4.1.1 Martingale approximation

A standard way to control fluctuations of a function of i.i.d. random variables is by way

of a martingale approximation. Let us order the random variables {axy :hx, yi ∈ B(ΛL)}

in any (for now) convenient way and let Fkto be the σ-algebra generated by the first k

of them. (Since we only aim at a distributional convergence, the σ-algebras may depend

on L.) Then

Ceff

L(t)−ECeff

L(t) = |B(ΛL)|

k=1

Zk,(4.1)

where

Zk:= ECeff

L(t)Fk−ECeff

L(t)Fk−1.(4.2)

Obviously, the quantity Zkis a martingale increment. In order to show distributional

convergence to N(0, σ2), it suffices to verify the (Lindenberg-Feller-type of) conditions

of the Martingale Central Limit Theorem due to Brown [Bro71]:

(1) There exists σ2∈[0,∞) such that

|ΛL|

|B(ΛL)|

k=1

E(Z2

k|Fk−1)−→

L→∞ σ2(4.3)

in probability, and

(2) for each  > 0,

|ΛL|

|B(ΛL)|

k=1

EZ2

k1l{|Zk|>|ΛL|1/2}Fk−1−→

L→∞ 0 (4.4)

in probability.

The sums on the left suggest invoking the Spatial Ergodic Theorem, but for that we

would need to ensure that the individual terms in the sum are (at least approximated

by) functions that are stationary with respect to shifts of Zd. This necessitates the

following additional input:

(i) a specific choice of the ordering of the edges, and

(ii) a more explicit representation for Zk.

We will now discuss various aspects of these in more detail.

4.1.2 Stationary edge ordering

Recall that B(Zd) denotes the set of all (unordered) edges in Zd. We will order B(Zd)

as follows: Let denote the lexicographic ordering of the vertices of Zd. Explicitly, for

x= (x1, . . . , xd) and y= (y1, . . . , yd) we have xyif either x=yor x6=yand there

exists i∈ {1, . . . , d}such that xj=yjfor all j < i and xi< yi. We will write x≺yif

x6=yand xy.

For the purpose of defining a stationary ordering of the edges, and also easier nota-

tion in some calculations that are to follow, we now identify B(Zd) with the set of pairs

(x, i), where x∈Zdand i∈ {1, . . . , d}, so that (x, i) corresponds to the edge between

the vertices xand x+ ˆei. We will then write

(x, i)(y, j) if 





either x≺y

or x=yand i≤j.

(4.5)

Again, (x, i)≺(y, j) if (x, i)(y, j) but (x, i)6= (y, j). It is easy to check that is a

complete order on B(Zd). A key fact about this ordering is its stationarity with respect

to shifts:

Lemma 4.1. If (x, i)(y, j)then also (x+z, i)(y+z, j)for all z∈Zd.

Proof. This is a trivial consequence of the definition.

Now we proceed to identify the sigma algebras {Fk}in the martingale representation

above. Recall that Ω := [λ, 1/λ]B(Zd)denotes the set of conductance configurations

satisfying (1.1). Writing ωfor elements of Ω we use axy =axy(ω), for hx, yi ∈ B(Zd),

to denote the coordinate projection corresponding to edge hx, yi. Given L≥1, set

N:= |B(ΛL)|and let b1, . . . , bNbe the enumeration of B(ΛL) induced by the ordering

of edges defined above. Then we set

Fk:= σ(ωb:bbk), k = 1, . . . , N, (4.6)

with

F0:= σ(ωb:b≺b1).(4.7)

By definition F0is independent of the edges in B(ΛL) while FNdetermines the entire

configuration in B(ΛL). Note also that Fkincludes information about edges that are

not in B(ΛL). This will be of importance once we replace Zkby a random variable that

depends on all of ω.

4.1.3 An explicit form of martingale increment

Having addressed the ordering of the edges, and thus the definition of the σ-algebras

Fk, we now proceed to derive a more explicit form of the quantity Zkfrom (4.2). Given

ω∈Ω, define the operator Lωon (Ror Rd-valued) functions on the lattice via

(Lωf)(x) := X

y:hx,yi∈B(Zd)

axy(ω)f(y)−f(x).(4.8)

This is an elliptic finite-difference operator — a random Laplacian — that arises as the

generator of the random walk among random conductances {axy(ω)}(see, e.g., Biskup [Bis11]

for a review of these connections). The existence/uniqueness for the associated Dirichlet

problem implies that for any finite Λ ⊂Zdthere is a unique ΨΛ: Ω ×(Λ ∪∂Λ) →Rd

such that x7→ ΨΛ(ω, x) obeys





LωΨΛ(ω, x) = 0, x ∈Λ,

ΨΛ(ω, x) = x, x ∈∂Λ.

(4.9)

It is then easily checked that f(x) := t·ΨΛ(ω, x) is the unique minimizer of f7→ QΛ(f)

over all functions fwith the boundary values f(x) = t·xfor x∈∂Λ. In particular, we

have

Ceff

L(t) = QΛLt·ΨΛL(4.10)

for all t∈Rd. The function x7→ ΨΛ(ω, x) will sometimes be referred to as a finite-

volume harmonic coordinate. (The first line in (4.9) justifies this term.)

The minimum value QΛ(t·ΨΛ) is a non-decreasing, continuous and concave function

of {axy :hx, yi ∈ B(Λ)}. Thanks to the uniqueness of the solution to (4.9), QΛ(t·ΨΛ)

is also continuously differentiable in axy’s with

∂

∂axy

QΛ(t·ΨΛ) = t·ΨΛ(ω, y)−t·ΨΛ(ω, x)2,hx, yi ∈ B(Λ).(4.11)

This relation is of fundamental importance for what is to come.

Abusing the notation slightly, let ω1, . . . , ωN, with N:= |B(Λ)|, denote the compo-

nents of the configuration ωover B(Λ) labeled in the order induced by defined above.

Let

q(ω1, . . . , ωN) := QΛ(t·ΨΛ) (4.12)

mark explicitly the dependence of the right-hand side on these variables. The product

structure of the underlying probability measure then allows us to give a more explicit

expression for the increment Zk=Zk(ω1, . . . , ωk):

Zk=ZP(dω0

k). . . P(dω0

N)q(ω1, . . . , ωk, ω0

k+1, . . . , ω0

−q(ω1, . . . , ωk−1, ω0

k, . . . , ω0

N)

=ZP(dω0

k). . . P(dω0

N)Zωk

ω0

d˜ωk

∂

∂˜ωk

q(ω1, . . . , ωk−1,˜ωk, ω0

k+1, . . . , ω0

N),

(4.13)

with the inner integral in Riemann sense. A key point is that the last partial derivative

is (modulo notational changes) given by (4.11), i.e., Zkis the modulus-squared of the

gradient of t·ΨΛover the k-th edge in B(Λ) integrated over part of the variables. To see

that Zkis a martingale increment note that the Riemann integral changes sign when

its limits are interchanged.

4.1.4 Input from homogenization theory

In order to apply the Spatial Ergodic Theorem to the sums on the left of (4.3) (4.4), we

will substitute for Zka quantity that is stationary with respect to the shifts of Zd. This

will be achieved by replacing the discrete gradient of ΨΛ— which by (4.11) enters as

the partial derivative of qin the formula for Zk— by the gradient of its infinite-volume

counterpart, to be denoted by ψ. The existence and properties of the latter object are

standard:

Proposition 4.2 (Infinite-volume harmonic coordinate).Suppose the law of the con-

ductances is ergodic with respect to the shifts of Zdand assume (1.1) for some λ∈(0,1).

Then there is a function ψ: Ω ×Zd→Rdsuch that

(1) (ψis Lω-harmonic) Lωψ(ω, x) = 0 for all xand P-a.e. ω.

(2) (ψis shift covariant) For P-a.e. ωwe have ψ(ω, 0) := 0 and

ψ(ω, y)−ψ(ω, x) = ψ(τxω, y −x), x, y ∈Zd.(4.14)

(3) (ψis square integrable)

EX

x=ˆe1,...,ˆed

a0,x(ω)ψ(ω, x)2<∞.(4.15)

(4) (ψis approximately linear) The corrector χ(ω, x) := ψ(ω, x)−xsatisfies

lim

|x|→∞

E|χ(ω, x)|2

|x|2= 0.(4.16)

Proof. Properties (1-3) are standard and follow directly from the construction of ψ

(which is done, essentially, by showing that a minimizing sequence in (1.25) converges

in a suitable L2-sense; see, e.g., Biskup [Bis11, Section 3.2] for a recent account of this).

As to (4), a moment’s thought reveals that it suffices to show this for xof the form nˆei,

where n→ ±∞. This follows from the Mean Ergodic Theorem, similarly as in [Bis11,

Lemma 4.8].

The replacement of (the gradients of) ΨΛby ψnecessitates developing means to

quantify the resulting error. For this we introduce an Lp-norm on functions f: Ω×(Λ∪

∂Λ) →Rdby the usual formula

k∇fkΛ,p := 1

|Λ|X

hx,yi∈B(Λ)

Ef(ω, y)−f(ω, x)p1/p

.(4.17)

Analogously, we also introduce a norm on functions ϕ: Ω ×Zd→Rdby

k∇ϕkp:= X

x=ˆe1,...,ˆed

Eϕ(ω, x)−ϕ(ω, 0)p1/p.(4.18)

Here we introduced the symbol ∇ffor an Rd-valued function whose i-th component

at xis given by ∇if(x) := f(x+ ˆei)−f(x) — abusing our earlier use of this notation.

It is reasonably well known, albeit perhaps not written down explicitly anywhere, that

the gradients of ΨΛand ψare close in k·kΛ,2-norm (see, however, Proposition 3.1 of

Caputo and Ioffe [CI03] for a torus version of this statement).

Proposition 4.3. Suppose the law Pon conductances {axy}is ergodic with respect to

shifts of Zdand obeys (1.1) for some λ∈(0,1). Then



∇(ΨΛL−ψ)

ΛL,2−→

L→∞ 0.(4.19)

As we will elaborate on later (see Remark 4.14), this is exactly what is needed to

establish the representation (1.25) for the limit value ceff(t) of the sequence L−dCeff

L(t).

However, in order to validate the conditions (4.3–4.4) of the Martingale Central Limit

Theorem, more than just square integrability is required. For this we state and prove:

Proposition 4.4 (Meyers’ estimate).Suppose Pis ergodic with respect to shifts. For

each d≥1, there is λ=λ(d)∈(0,1) such that if (1.1) holds P-a.s. with this λ, then

for some p > 4,

k∇ψkp<∞(4.20)

and

sup

L≥1

∇(ΨΛL−ψ)

ΛL,p <∞.(4.21)

Proposition 4.4 is the sole reason for our restriction on ellipticity contrast. We

believe that, on the basis of the technology put forward in Gloria and Otto [GO11], no

such restriction should be needed. To attest this we note that versions of the above

bounds actually hold pointwise for a.e. ω∈Ω satisfying (1.1); i.e., for norms without

the expectation E. In addition, from [GO11, Proposition 2.1] we in fact know (4.20) for

all p∈(1,∞) when d≥3. A torus version of Proposition 4.4 appeared in Theorem 4.1

of Caputo and Ioffe [CI03].

4.1.5 Perturbed corrector and variance formula

Unfortunately, a direct attempt at the substitution of (the gradients of) ΨΛby ψin

(4.13) reveals another technical obstacle: As (4.13) relies on the Fundamental Theorem

of Calculus, the replacement of ΨΛby ψrequires the latter function to be defined for ω

that may lie outside of the support of P. This is a problem because ψis generally

determined by conditions (1-4) in Proposition 4.2 only on a set of full P-measure. Im-

posing additional assumptions on P— namely, that the single-conductance distribution

is supported on an interval with a bounded and non-vanishing density — would allow

us to replace the Lebesgue integral in (4.13) by an integral with respect to P(d˜ωk) and

thus eliminate this problem. Notwithstanding, we can do much better by invoking a

rank-one perturbation argument which we describe next.

Fix an index i∈ {1, . . . , d}and recall the notation ∇if(x) := f(x+ ˆei)−f(x). For

a vertex x∈Zdand a finite set Λ ⊂Zdsatisfying x∈Λ or x+ ˆei∈Λ, let g(i)

Λ(ω, x) be

defined by

g(i)

Λ(ω, x)−1:= infQΛ(f): f(x+ ˆei)−f(x)=1, f∂Λ= 0,(4.22)

where 0−1:= ∞. (In Section 4.4 we will see that g(i)

Λis also a double gradient of the

Green function for operator Lω.) Note that (4.13) and (4.11) ask us to understand how

∇iΨΛ(ω, x) changes when the coordinate of ωover hx, x + ˆeiiis perturbed. Somewhat

surprisingly, this change takes a multiplicative form:

Proposition 4.5 (Rank-one perturbation).Let Λ⊂Zdbe finite and x, y ∈Λbe nearest

neighbors; y=x+ˆeifor some i∈ {1, . . . , d}. For any ω, ω0that agree everywhere except

at edge b:= hx, yi,

∇iΨΛ(ω0, x) = 1−(ω0

b−ωb)g(i)

Λ(ω0, x)∇iΨΛ(ω, x).(4.23)

For the prefactor we alternatively get

1−(ω0

b−ωb)g(i)

Λ(ω0, x) = expn−Zω0

ωb

d˜ωbg(i)

Λ(˜ω, x)o,(4.24)

where ˜ωcoincides with ωexcept at b, where it equals ˜ωb. In particular, 1−(ω0

b−

ωb)g(i)

Λ(ω0, x)is bounded away from 0 and ∞uniformly in ω∈Ωand Λ⊂Zd.

It is worthy a note that (4.23) is a special case of a more general rank-one perturba-

tion formula; cf Lemma 4.22, which may be of independent interest. Incidentally, such

formulas have proved extremely useful in the analysis of random Schr¨odinger operators.

The Λ ↑Zd-limit of the right-hand side can now be controlled uniformly in ω∈Ω:

Proposition 4.6. Suppose (1.1) holds for some λ∈(0,1). Then Λ7→ g(i)

Λ(ω, x)is non-

decreasing and bounded away from zero and infinity uniformly in Λ⊂Zdand ω∈Ω.

In particular, for all ω∈Ωand all x∈Zdthe limit

g(i)(ω, x) := lim

Λ↑Zdg(i)

Λ(ω, x) (4.25)

exists and satisfies

g(i)(ω, x)−1= infQZd(f): f(x+ ˆei)−f(x) = 1,|supp(f)|<∞,(4.26)

where supp(f) := {x∈Zd:f(x)6= 0}. In particular, (ω, x)7→ g(i)(ω, x)is stationary

in the sense that g(i)(τzω, x +z) = g(i)(ω, x)holds for all ω∈Ωand all x, z ∈Zd.

Before we wrap up the outline of the proof of Theorem 1.11, let us formulate a

representation for the limiting variance σ2

tfrom Theorem 1.11: For x∈Zdand i∈

{1, . . . , d}, let bdenote the edge corresponding to the pair (x, i) and let

h(ω, x, i) := ZP(dω0

b)Zωb

ω0

d˜ωb1−(˜ωb−ωb)g(i)(˜ω, x)2,(4.27)

where ˜ωis the configuration equal to ωexcept at b, where it equals ˜ωb. Define the

matrix ˆ

Z(x, i) := {ˆ

Zjk(x, i)}j,k=1,...,d by the quadratic form

t, ˆ

Z(x, i)t:= Eh(·, x, i)∇i(t·ψ)(·, x)2σωb0:b0(x, i),(4.28)

where (x, i) represents the edge hx, x + ˆeiiand t∈Rd. Then we have:

Theorem 4.7 (Limiting variance).Under the assumptions of Theorem 1.11, the matrix

elements of ˆ

Z(x, i)are square integrable. In particular, σ2

tfrom Theorem 1.11 is given

σ2

i=1

Et, ˆ

Z(0, i)t2, t ∈Rd.(4.29)

As an inspection of (4.28) reveals, the limiting variance is thus a bi-quadratic form in

t. Although concisely written, the expression is not very useful from the practical point

of view; particularly, due to the unwieldy conditioning in (4.28). The representation

using the h-function also adds to this; it is no longer obvious, albeit still true, that

E(t, ˆ

Z(x, i)t)σωb0:b0≺(x, i)= 0,(4.30)

i.e., that (t, ˆ

Z(x, i)t) is a martingale increment. A question of interest is whether an

expression can be found for σ2

tthat is more amenable to computations.

Remark 4.8. Since t7→ Ceff

L(t)is quadratic in t, the above actually implies that, as

L→ ∞, the joint law of the random variables

Ceff

L(t)−ECeff

L(t)

|ΛL|1/2:t∈Rd(4.31)

tends to the law of multivariate Gaussian {Gt:t∈Rd}with

E(Gt)=0 and EGtGs) =

i=1

Et, ˆ

Z(0, i)ts, ˆ

Z(0, i)s,(4.32)

where ˆ

Z(0, i)is as in (4.28). Naturally, t7→ Gtis a quadratic form as well.

4.1.6 Organization

The proofs (and the rest of the paper) are organized as follows. In Section 4.2 we

assemble the ingredients — following the steps outlined in the present section — into

the proofs of Theorems 1.11 and 4.7 and Corollary 1.12. In Section 4.3 we then show

that the finite-volume harmonic coordinate approximates its full lattice counterpart

in an L2-sense as stated in Proposition 4.3 and establish the Meyers estimate from

Proposition 4.4. A key technical tool is the Calder´on-Zygmund regularity theory and a

uniform bound on the triple gradient of the Green function of the simple random walk

in finite boxes. Finally, in Section 4.4, we prove Propositions 4.5 and 4.6 dealing with

the harmonic coordinate over environments perturbed at a single edge.

4.2 Proof of the CLT

In this section we verify the conditions (4.3) (4.4) of the Martingale Central Limit

Theorem and thus prove Theorems 1.11 and 4.7. All derivations are conditional on

Propositions 4.3–4.6 the proofs of which are postponed to later sections. Throughout

we will make use of the following simple but useful consequence of H¨older’s inequality:

Lemma 4.9. For any p0>p>2,α:= 2

p0−p

p0−2and β:= p0

p−2

p0−2,



∇(ΨΛL−ψ)

ΛL,p ≤

∇(ΨΛL−ψ)

α

ΛL,2

∇(ΨΛL−ψ)

β

ΛL,p0.

Proof. Apply H¨older’s inequality to the function f:= |∇(ΨΛL−ψ)|.

Assume now the setting developed in Section 4.1; in particular, the ordering of edges

and sigma-algebras Fkfrom Section 4.1.2 and the martingale increment Zkfrom (4.2)

and its representation (4.13) from Section 4.1.3. In analogy with equation (4.27), we

also define

hΛ(ω, x, i) := ZP(dω0

b)Zωb

ω0

d˜ωb1−(˜ωb−ωb)g(i)

Λ(˜ω, x)2,(4.33)

where b:= hx, x + ˆeiiand ˜ωis the configuration equal to ωexcept at b, where it equals

˜ωb. By Proposition 4.5, we may write the martingale increment Zkas

Zk=EhΛL(·, xk, ik)∇ik(t·ΨΛL)(·, xk)2Fk,(4.34)

where xkand ikare the vertex and the edge direction corresponding to bk, i.e., bk=

hxk, xk+ ˆeiki. Recall the notation for ˆ

Z(x, i) from (4.28) and note that this is well

defined and finite P-a.s. thanks to the estimates (4.20–4.21) as well as boundedness

of h. Note the dependence of Zkon L.

Proposition 4.10 (Martingale CLT — first condition).Assume that the premises (and

thus conclusions) of Propositions 4.3–4.6 hold. Then Zk∈L2(P)for all kand

|ΛL|

|B(ΛL)|

k=1

E(Z2

k|Fk−1)−→

L→∞

i=1

Et, ˆ

Z(0, i)t2(4.35)

in P-probability and L1(P).

Proof. Fix t∈Rd. Thanks to Lemma 4.1 and Proposition 4.2(2), for each i∈ {1, . . . , d},

the collection of conditional expectations

Et, ˆ

Z(x, i)t2σωb:b≺(x, i):x∈Zd(4.36)

is stationary with respect to the shifts on Zdand, by Proposition 4.4, uniformly bounded

in L1(P). Labeling the edges in B(ΛL) according to the order , the Spatial Ergodic

Theorem yields

|ΛL|

|B(ΛL)|

k=1

E(t, ˆ

Z(xk, ik)t)2Fk−1−→

L→∞

i=1

Et, ˆ

Z(0, i)t2(4.37)

with the limit P-a.s. and in L1(P). To see how this relates to our claim, abbreviate

Ak:= hΛL(·, xk, ik)∇ik(t·ΨΛL)(·, xk)2,(4.38)

Bk:= h(·, xk, ik)∇ik(t·ψ)(·, xk)2,(4.39)

and denote

RL,k := EhEAkFk2−EBkFk2Fk−1i.(4.40)

By (4.34) we have Zk=E(AkFk), while (4.28) reads (t, ˆ

Z(xk, ik)t) = E(BkFk).

Hence, as soon as we show that

|ΛL|

|B(ΛL)|

k=1

E|RL,k|−→

L→∞ 0,(4.41)

the claim (4.35) will follow.

The proof of (4.41) will proceed by estimating E|RL,k|which will involve applications

of the Cauchy-Schwarz inequality (in order to separate terms) and Jensen’s inequality

(in order to eliminate conditional expectations). First we note

E|RL,k| ≤ E(Ak−Bk)21/2E(Ak+Bk)21/2.(4.42)

Writing Ak=Bk+ (Ak−Bk) and noting (a+b)2≤2a2+ 2b2tells us

E(Ak+Bk)2≤2E(Ak−Bk)2+ 8EB2

k.(4.43)

Summing over kand applying Cauchy-Schwarz, we find that

|ΛL|

|B(ΛL)|

k=1

E|RL,k|≤qα2α+ 8β,(4.44)

where

α:= 1

|ΛL|

|B(ΛL)|

k=1

E(Ak−Bk)2and β:= 1

|ΛL|

|B(ΛL)|

k=1

E(B2

k).(4.45)

By inspection of (4.44) we now observe that it suffices to show that βstays bounded

while αtends to zero in the limit L→ ∞.

The boundedness of βfollows from (4.20) and the fact that h(·, x, i) is bounded;

indeed, these yield E(|Bk|2)≤ khk2

∞|t|4k∇ψk4

4uniformly in kand L. Concerning the

terms constituting α, using (a+b)2≤2a2+ 2b2we first separate terms as

E(Ak−Bk)2≤2EhΛL(·, xk, ik)2|∇ik(t·ΨΛL)(·, xk)|2−|∇ik(t·ψ)(·, xk))|22

+ 2EhΛL(·, xk, ik)−h(·, xk, ik)2∇ik(t·ψ)(·, xk)4.(4.46)

Since hΛis uniformly bounded, the average over kof the first term is bounded by a

constant times the product of (k∇ΨΛLkΛL,4+k∇ψkΛL,4)2and k∇(ΨΛL−ψ)k2

ΛL,4. The

latter tends to zero as L→ ∞ by Proposition 4.4, Proposition 4.3 and Lemma 4.9 (with

the choices p:= 4 and p0>4 but sufficiently close to 4).

For the second term in (4.46) we pick p > 4 and use H¨older’s inequality to get

|ΛL|

|B(ΛL)|

k=1

EhΛL(·, xk, ik)−h(·, xk, ik)2∇ik(t·ψ)(·, xk)4

≤ |t|4k∇ψk4

ΛL,p1

|ΛL|

|B(ΛL)|

k=1

EhΛL(·, xk, ik)−h(·, xk, ik)2q1/q

,(4.47)

where qsatisfies 4/p+1/q= 1. The norm of k∇ψkΛL,p is again bounded by Proposition 4.4

as long as pis sufficiently close to 4; to apply (4.20), we need to invoke the stationarity

of ∇ψto realize k∇ψkΛL,p =k∇ψkp.

For the second term in (4.47) we first need to show that for each  > 0 there is

N≥1 so that for all ω∈Ω,

dist`1(Zd)(x, Λc

L)≥N⇒hΛL(ω, x, i)−h(ω, x, i)< . (4.48)

For this we use that, thanks to (4.27), (4.33) and (1.1),

hΛ(ω, x, i)−h(ω, x, i)≤CZ1/λ

d˜ωbg(i)

Λ(˜ω, x)−g(i)(˜ω, x)(4.49)

for some constant C=C(λ)<∞. To estimate the right-hand side, by the monotonicity

of Λ 7→ g(i)

Λ(˜ω, x) and its stationarity with respect to shifts, we have

g(i)

Λ(ω, x)−g(i)(ω, x)≤g(i)

ΛN(τxω, 0) −g(i)(τxω, 0), ω ∈Ω,(4.50)

as soon as x+ ΛN⊂Λ. Then (4.48) follows from (4.49) and the fact that the difference

on the right-hand side of (4.50) converges to zero uniformly in ω∈Ω.

We now bound the last term in (4.47) as follows. The terms for which xkis at least N

steps away from ΛLare bounded by thanks to (4.49); the sum over the remaining terms

is of order NLd−1thanks to the uniform boundedness of hΛ−h. Hence, in the limit

L→ ∞, the second term in (4.47) is of order 1/q; taking ↓0 shows that αtends to zero

as L→ ∞. Invoking (4.44), this finishes the proof of (4.41) and the whole claim.

Proposition 4.11 (Martingale CLT — second condition).Assume that the premises

(and thus conclusions) of Propositions 4.3–4.6 hold. Then for each  > 0,

|ΛL|

|B(ΛL)|

k=1

EZ2

k1l{|Zk|>|ΛL|1/2}Fk−1−→

L→∞ 0,(4.51)

in P-probability.

Proof. This could be proved by strengthening a bit the statement of Proposition 4.10

(from squares of the Z’s to a slightly higher power), but a direct argument is actually

easier.

First we note that it suffices to show convergence in expectation. Let p > 4 be such

that the statements in Proposition 4.4 hold. By H¨older’s and Chebyshev’s inequalities

we have

EZ2

k1l{|Zk|>|ΛL|1/2}≤1

|ΛL|1/2p−4

2E|Zk|p/2.(4.52)

Since hΛLis bounded, Jensen’s inequality yields

E|Zk|p/2≤CEhE∇ik(t·ΨΛ)(·, xk)2Fkip/2≤CE∇ik(t·ΨΛ)(·, xk)p.

(4.53)

It follows that

|ΛL|

|B(ΛL)|

k=1

E|Zk|p/2≤C|t|pk∇ΨΛLkp

ΛL,p.(4.54)

The right-hand side is bounded uniformly in L. Using this in (4.52), the claim follows.

We can now finish the proof of our main results:

Proof of Theorems 1.11 and 4.7 from Propositions 4.3–4.6. The distributional conver-

gence in (1.26) is a direct consequence of the Martingale Central Limit Theorem whose

conditions (4.3–4.4) are established in Propositions 4.10 and 4.11. The limiting variance

σ2

tis given by the right-hand side of (4.35), in agreement with (4.29). It remains to

prove that σ2

t>0 whenever t6= 0 and the law Pis non-degenerate.

Suppose on the contrary that σ2

t= 0. Then for each iwe would have E((t, ˆ

Z(0, i)t)2) =

0 and thus (t, ˆ

Z(0, i)t) = 0 P-a.s. Denoting b:= h0,ˆeii, (4.27–4.28) imply that, for P-

a.e. ωb,

ZP(dω0

b)Zωb

ω0

d˜ωbE1−(˜ωb−ωb)g(i)

Λ(˜ω, 0)∇i(t·ψ)(ω, 0)2F(0,i)= 0,(4.55)

where F(0,i):= σ(ωb). Let Ω1⊂[λ, 1/λ] be the set of ωbwhere this holds. Then P(Ω1) = 1

and, since Pis non-degenerate, Ω1contains at least two points. The expectation in (4.55)

is independent of ω0

b; subtracting the expression for two (generic) choices of ωbin Ω1then

shows that the inner integral must vanish for all ωb, ω0

b∈Ω1. But (4.24) tells us that

the prefactor in square brackets, and thus the conditional expectation, is non-negative

(in fact, it is bounded away from zero). Hence, this can only happen when

∇i(t·ψ)(·,0) = 0,P-a.s. for all i= 1, . . . , d. (4.56)

But then ceff(t) = 0, which cannot hold for t6= 0 when (1.1) is in force.

Proof of Corollary 1.12 from Propositions 4.3–4.6. Thanks to (4.1–4.2) and Proposi-

tion 4.10, Ceff

L(t) is a martingale whose increments, Zkare square integrable. Therefore,

VarCeff

L(t)=|B(ΛL)|

k=1

E(Z2

k).(4.57)

But the right-hand side is the expectation of the quantity on the left of (4.35). Since

the convergence in (4.35) occurs in L1(P), the claim follows.

4.3 The Meyers estimate

The goal of this section is to give proofs of Propositions 4.3 and 4.4. The former is

a simple consequence of the Hilbert-space structure underlying the definition of a har-

monic coordinate; the latter (to which this section owes its name) is a far less immediate

consequence of the Calder´on-Zygmund regularity theory for singular integral operators.

4.3.1 L2 bounds and convergence

Recall our notation Lωfor the operator in (4.8). We begin by noting an explicit rep-

resentation of the minimum of f7→ Qλ(f) as a function of the (Dirichlet) boundary

condition:

Lemma 4.12. Let Λ⊂Zdbe finite and fix an ω∈Ω. Then there is K:∂Λ×∂Λ→

[0,∞), depending on Λand ω, such that for any hthat obeys Lωh(x) = 0 for x∈Λ,

QΛ(h) = 1

x,y∈∂Λ

K(x, y)h(y)−h(x)2.(4.58)

Moreover, K(x, y) = K(y, x)for all x, y ∈∂Λand

y∈∂Λ

K(x, y) = X

z∈Λ

hx,zi∈B(Λ)

axz (4.59)

for all x∈∂Λ.

Proof. “Integrating” by parts we obtain

QΛ(h) = −X

y∈Λ

h(y)(Lωh)(y) + X

y∈∂Λ, x∈Λ

hx,yi∈B(Λ)

axy h(y)−h(x)h(y).(4.60)

Employing the fact that his Lω-harmonic, the first sum drops out. For the second sum

we recall that h(x) = Pz∈∂ΛpΛ(x, z)h(z), where pΛ(x, z) is the discrete Poisson kernel

which can be defined by pΛ(x, z) := Px

ω(Xτ∂Λ=z) for τ∂Λdenoting the first exit time

from Λ of the random walk in conductances ω. Now set

K(y, z) := X

x∈Λ

hx,yi∈B(Λ)

axypΛ(x, z) (4.61)

and note that Pz∈∂ΛK(y, z) = Px∈Λ,hx,yi∈B(Λ) axy. It follows that

y∈∂Λ, x∈Λ

hx,yi∈B(Λ)

axy h(y)−h(x)h(y) = X

y,z∈∂Λ

K(y, z)h(y)−h(z)h(y).(4.62)

The representation using the random walk and its reversiblity now imply that Kis

symmetric. Symmetrizing the last sum then yields the result.

Remark 4.13. We note that Lemma 4.12 holds even for vector valued functions; just

replace [h(y)−h(x)]2by the norm squared of h(y)−h(x). This applies to several

derivations that are to follow; a point that we will leave without further comment.

We can now prove Proposition 4.3 dealing with the convergence of ∇ΨΛto ∇ψin

k·kΛ,2-norm, as Λ := ΛLfills up all of Zd.

Proof of Proposition 4.3. Abbreviate h(x) := ψ(ω, x)−ΨΛL(ω, x). The bound (1.1)

implies



∇(ΨΛL−ψ)

2

ΛL,2≤1

|ΛL|EX

hx,yi∈B(ΛL)

axyh(y)−h(x)2.(4.63)

Let f: Λ ∪∂Λ→Rdbe the minimizer of

infX

hx,yi∈B(ΛL)f(y)−f(x)2, f(z) = χ(z) for all z∈∂ΛL.(4.64)

Since his the minimizer of the corresponding Dirichlet energy with conductances {axy}

and boundary condition χ, we get using (1.1)

hx,yi∈B(ΛL)

axyh(y)−h(x)2≤X

hx,yi∈B(ΛL)

axyf(y)−f(x)2

≤1

λX

hx,yi∈B(ΛL)f(y)−f(x)2.

(4.65)

Writing the last sum coordinate-wise and applying Lemma 4.12, we thus get

hx,yi∈B(ΛL)

axyh(y)−h(x)2≤1

2λX

x,y∈∂ΛL

K(x, y)χ(ω, y)−χ(ω, x)2,(4.66)

where the kernel K(x, y) pertains to the homogeneous problem, i.e., the simple random

walk. Note that these bounds hold for all configurations satisfying (1.1).

By shift covariance and sublinearity of the corrector (cf Proposition 4.2(2,4)), for

each ε > 0 there is A=A(ε) such that

Eχ(·, x)−χ(·, y)2≤A+ε|x−y|2.(4.67)

Using this and (4.66) in (4.63) yields



∇(ΨΛL−ψ)

2

ΛL,2≤1

2λ2

|ΛL|X

x,y∈∂ΛL

K(x, y)A+ε|x−y|2.(4.68)

But Py∈∂ΛLK(x, y)≤1 for each x∈∂ΛLwhile Px,y∈∂ΛLK(x, y)|x−y|2is, by

Lemma 4.12, the Dirichlet energy of the function x7→ xfor conductances all equal

to 1. Hence, the last sum in (4.68) is bounded by A|∂ΛL|+ε|B(ΛL)|. Taking L→ ∞

and ε↓0 finishes the proof.

Remark 4.14. As alluded to in the introduction, the L2-convergence ∇ΨΛL→ ∇ψ

permits us to prove the formula (1.25) for ceff(t). The argument is similar to (albeit

much easier than) what we used in the proof of Proposition 4.10. Indeed, we trivially

decompose

Ceff

L(t) = QΛLt·ΨΛL=QΛL(t·ψ) + QΛLt·ΨΛL−QΛL(t·ψ).(4.69)

The stationarity of the gradients of ψand the Spatial Ergodic Theorem imply that for

any ergodic law Pon conductances, P-a.s. and in L1(P),

|ΛL|QΛL(t·ψ)−→

L→∞ EX

x=ˆe1,...,ˆed

a0,x(ω)t·ψ(ω, x)2.(4.70)

It follows from the construction of the harmonic coordinate that expression on the right

coincides with the infimum in (1.25). (There is no gradient on the right-hand side of

(4.70) because ψ(ω, 0) := 0.) It remains to control the difference on the extreme right

of (4.69).

Using the quadratic nature of QΛ, the ellipticity assumption (1.1) and Cauchy-

Schwarz,

EQΛt·ΨΛ−QΛ(t·ψ)

|Λ|

≤1

λ|t|2

∇(ΨΛ−ψ)

2

Λ,2+2

λ|t|2k∇ψk2

∇(ΨΛ−ψ)

Λ,2.(4.71)

By Proposition 4.3 — which holds for any shift-ergodic (elliptic) law on conductances

— the right-hand side tends to zero as Λ := ΛLincreases to Zd. Since we know that

|ΛL|−1Ceff

L(t)is bounded and converges almost surely (e.g., by the Subadditive Ergodic

Theorem), it converges also in L1(P). We conclude that the limit value ceff(t)is given

by (1.25).

4.3.2 The Meyers estimate in finite volume

Key to the proof of Proposition 4.4 is the Meyers estimate. The term owes its name to

Norman G. Meyers [Mey63] who discovered a bound on Lp-continuity (in the right-hand

side) of the solutions of Poisson equation with second-order elliptic differential operators

in divergence from, provided the associated coefficients are close to a constant. The

technical ingredient underpinning this observation is the Calder´on-Zygmund regularity

theory for certain singular integral operators in Rd. (Incidentally, as noted in [Mey63],

Meyers’ argument is a generalization of earlier work of Boyarskii, cf [Mey63, ref. 2 and 3]

for systems of first-order PDEs and a version of his result was also derived, though not

published, by Calder´on himself; cf [Mey63, page 190]).

To ease the notation, in addition to (4.18), we will use the notation kfkpalso for

the canonical norm in `p(Λ),

kfkp:= X

x∈Λf(x)p1/p,(4.72)

throughout the rest of this section.

Let us review the gist of Meyers’ argument for functions on Zd. Our notation is

inspired by that used in Naddaf and Spencer [NS98] and Gloria and Otto [GO11]. A

general form of the second order difference operator Lin divergence form is

L:= ∇?·A·∇,(4.73)

where A={Aij(x): i, j = 1, . . . , d, x ∈Zd}are x-dependent matrix coefficients, ∇f(x)

is a vector whose i-th component is ∇if(x) := f(x+ ˆei)−f(x) and ∇?is its conjugate

acting as ∇?

if(x) := f(x)−f(x−ˆei). The above Lis explicitly given by

(Lf)(x) =

i,j=1Ai,j(x)f(x+ˆei)−f(x)−Ai,j(x−ˆej)f(x+ˆei−ˆej)−f(x−ˆej).(4.74)

Now, if Ais close to the identity matrix, it makes sense to write

L= ∆ + ∇?·(A−id) ·∇,(4.75)

where we noted that the standard lattice Laplacian ∆ corresponds to ∇?·id ·∇. This

formula can be used as a starting point of perturbative arguments.

Consider a finite set Λ ⊂Zdand let g: Λ ∪∂Λ→Rd. Let fbe a solution to the

Poisson equation

−Lf=∇?·g, in Λ,(4.76)

with f:= 0 on ∂Λ. Employing (4.75), we can rewrite this as

−∆f=∇?·g+ (A−id) ·∇f.(4.77)

The function on the right has vanishing total sum over Λ and hence it lies in the domain

of the inverse (∆)−1

Λof ∆ with zero boundary conditions. Taking this inverse followed

by one more gradient, and denoting

KΛ:= ∇(−∆)−1

Λ∇?,(4.78)

this equation translates to

∇f=KΛ·g+ (A−id)∇f.(4.79)

A first noteworthy point is that this is now an autonomous equation for ∇f. A second

point is that, if kKΛkpis the norm of KΛas a map (on vector valued functions) `p(Λ) →

`p(Λ) and kA−idk∞is the least a.s. upper bound on the coefficients of A(x)−id,

uniform in x, we get

k∇fkp≤ kKΛkpkA−idk∞k∇fkp+kKΛkpkgkp.(4.80)

Assuming kKΛkpkA−idk∞<1 this yields

k∇fkp≤kKΛkpkgkp

1−kKΛkpkA−idk∞

.(4.81)

Furthermore, the condition kKΛkpkA−idk∞<1 ensures the very existence of a unique

solution ∇fto (4.79) via a contraction argument; (4.81) then implies the continuity of

g7→ ∇fin `p(Λ).

The aforementioned general facts are relevant for us because Lωis of the form (4.73).

Indeed, set Aij(x) := δijax,x+ˆeiand note that (4.74) reduces to (4.8). The finite-volume

corrector

χΛ(ω, x) := ΨΛ(ω, x)−x(4.82)

then solves the Poisson equation

−LωχΛ=∇?·g, where g(x) := (ax,x+ˆe1, . . . , ax,x+ˆed).(4.83)

Thanks to (1.1), this gis bounded uniformly so, in order to have (4.81) for all finite

boxes, our main concern is the following claim:

Theorem 4.15. For each p∈(1,∞), the operator KΛLis bounded in `p(ΛL), uniformly

in L≥1.

Proof of Proposition 4.4 from Theorem 4.15. Let p∗>4. Since (in our setting) kA−

idk∞≤λ−1−1, we may choose λ∈(0,1) close enough to one so that supL≥1kKΛLkp∗kA−

idk∞<1. From the above derivation it follows

sup

L≥1k∇χΛLkΛL,p∗<∞.(4.84)

We claim that this implies

k∇χkp<∞, p < p∗.(4.85)

Indeed, pick α > 0 and note that, for any ∈(0, α),

x∈ΛL

1l{|∇χ(·,x)|>α}≤X

x∈ΛL

1l{|∇χΛL(·,x)|>α−}+X

x∈ΛL

1l{|∇χΛL(·,x)−∇χ(·,x)|>}.(4.86)

Taking expectations and dividing by |ΛL|, the left hand side becomes P(|∇χ(·,0)|> α),

while the second sum on the right can be bounded by −2k∇χΛL−∇χk2

ΛL,2, which tends

to zero as L→ ∞ by Proposition 4.3. Applying Chebyshev’s inequality to the first sum

on the right and taking L→ ∞ followed by ↓0 yields

P|∇χ(·,0)|> α≤1

αp∗sup

L≥1k∇χΛLkp∗

ΛL,p∗.(4.87)

Multiplying by αp−1and integrating over α > 0 then proves (4.85).

Returning to the claims in Proposition 4.4, inequality (4.85) is a restatement of

(4.20). Since (4.84–4.85) imply the uniform boundedness of k∇(χΛL−χ)kΛL,p, for each

p<p∗, Lemma 4.9 then shows k∇(χΛL−χ)kΛL,p →0, as L→ ∞ for all p<p∗. This

proves (4.21) as well.

4.3.3 Interpolation

In the proof of Theorem 4.15 we will follow the classical argument — by and large due

to Marcinkiewicz — that is spelled out in Chapter 2 (specifically, proof of Theorem 1

in Section 2.2) of Stein’s book [Ste70]. The reasoning requires only straightforward

adaptations due to discrete setting and finite volume, but we still prefer to give a full

argument to keep the present paper self-contained. A key idea is the use of interpola-

tion between the strong `2-type estimate (Lemma 4.16) and the weak `1-type estimate

for KΛL(Lemma 4.17). Both of these of course need to hold uniformly in L≥1.

Lemma 4.16. For any finite Λ⊂Zd, the `2(Λ)-norm of KΛsatisfies kKΛk2≤1.

Proof. Let Hbe a Hilbert space and Ta positive self-adjoint, bounded and invertible

operator. Then for all h∈ H,

h, T−1h= sup

g∈H2(g, h)−(g, Tg).(4.88)

We will apply this to Hgiven by the space (of R-valued functions) `2(Λ), T:= −∆

and h:= ∇?·ffor some f: Λ →Rdwith zero boundary conditions outside Λ. Then

∇?·f, (ε−∆)−1∇?·f= sup

g∈`2(Λ)2(g, ∇?·f)−ε(g, g)+(g, ∆g)

= sup

g∈`2(Λ)2(∇g, f)−ε(g, g)−(∇g, ∇g)−(f, f)+ (f, f)

= sup

g∈`2(Λ)−(∇g−f, ∇g−f)+ (f, f)

≤(f, f),

(4.89)

where we used that ∇?is the adjoint of ∇in the space of Rd-valued functions `2(Λ) and

where the various inner products have to be interpreted either for R-valued or Rd-valued

functions accordingly. Taking ↓0, the left-hand side becomes (f, KΛ·f). The claim

follows.

The second ingredient turns out to be technically more involved.

Lemma 4.17. KΛLis of weak-type (1-1), uniformly in L > 1. That is, there exists b

such that, for all L > 1,f∈`1(ΛL)and α > 0,

{z∈ΛL:|KΛLf(z)|> α}≤b

K1kfk1

α.(4.90)

Deferring the proof of this lemma to the next subsection, we now show how this

enters into the proof of Theorem 4.15.

Proof of Theorem 4.15 from Lemma 4.17. We follow the proof in Stein [Ste70, Theo-

rem 5, page 21]. We begin with the case 1 <p<2. Let f∈`p(ΛL) and pick α > 0.

Let f1:= f1l{|f|>α}and f2:= f1l{|f|≤α}. Then

{z∈ΛL:|KΛLf(z)|>2α}≤{z∈ΛL:|KΛLf1|> α}

+{z∈ΛL:|KΛLf2|> α}.(4.91)

Lemmas 4.16 and 4.17 then yield

{z∈ΛL:|KΛLf(z)|> α}≤b

K1kf1k1

α+b

K2kf2k2

α2,(4.92)

with b

K1and b

K2independent of L. Multiplying by αp−1and integrating, we infer

kKΛLfkp

p=pZ∞

αp−1{z∈ΛL:|KΛLf(z)|> α}dα

≤pX

zZ∞

0b

K1αp−2|f(z)|1l{|f|>α}+b

K2αp−3|f(z)|21l{|f|≤α}dα

=pb

K1X

z|f(z)|Z|f(z)|

αp−2dα+pb

K2X

z|f(z)|2Z∞

|f(z)|

αp−3dα

=pb

p−1X

z|f(z)|p+pb

2−pX

z|f(z)|p,

(4.93)

proving the assertion in the case 1 <p<2.

For p∈(2,∞), the fact that KΛis obviously symmetric implies that kKΛkp=kKΛkq,

where qis the index dual to p. Hence supL≥1kKΛLkp<∞for all p∈(1,∞).

4.3.4 Weak type-(1,1) estimate

It remains to prove Lemma 4.17. The strategy is to represent the operator using a

singular kernel that has a “nearly `1-integrable” decay. Let GΛ(x, y) be the Green

function (i.e., inverse) of the Laplacian ∆ on Λ with zero boundary condition on ∂Λ.

Lemma 4.18. The operator KΛadmits the representation

ˆei·KΛ·f(x)=X

y∈Λ

j=1∇(1)

i∇(2)

jGΛ(x, y)fj(y),(4.94)

where the superscripts on the ∇’s indicate which of the two variables the operator is

acting on.

Proof. Since both GΛand fvanish outside Λ, we have

ˆei·KΛ·f(x)=∇iX

y∈Λ

GΛ(·, y)∇?·f(y)(x)

y∈ZdGΛ(x+ ˆei, y)−GΛ(x, y)d

j=1

[fj(y−ˆek)−fj(y)]

j=1 X

y∈ZdGΛ(x+ ˆei, y + ˆej)−GΛ(x, y + ˆej)fj(y)

−

j=1 X

y∈ZdGΛ(x+ ˆei, y)−GΛ(x, y)fj(y).

(4.95)

This is exactly the claimed expression.

Crucial for the proof of the weak-type (1,1)-estimate in Lemma 4.17 is an integrable

decay estimate on the gradient of the kernel of the operator KΛ:

Proposition 4.19. There exists C > 0independent of Lsuch that

∇(2)

i∇(1)

j∇(2)

kGΛL(x, y)≤C

|x−y|d+1 (4.96)

for all x, y ∈ΛLand i, j, k ∈ {1, . . . , d}.

Although (4.96) is certainly not unexpected, and perhaps even well-known, we could

not find an exact reference and therefore provide an independent proof in Section 4.3.5.

With this estimate at hand, we can now turn to the proof of Lemma 4.17.

Proof of Lemma 4.17 from Proposition 4.19. To ease the notation, we will write Λ :=

ΛL(note that all bounds will be uniform in L) and, resorting to components, write KΛ

for the scalar-to-scalar operator with kernel K(i,j)

Λ(x, y) := ∇(1)

i∇(2)

jGΛ(x, y) for some

fixed i, j ∈ {1, . . . , d}. For the most part, we adapt the arguments in Stein [Ste70,

pages 30-33].

Take some function f: Λ →R, extended to vanish outside Λ, and pick α > 0. Con-

sider a partition of Zdinto cubes of side 3r, where ris chosen so large that 3−rdkfk1≤α.

Naturally, each of the cubes in the partition further divides into 3dequal-sized sub-cubes

of side 3r−1, which subdivide further into sub-cubes of side 3r−2, etc. We will now des-

ignate these to be either good cubes or bad cubes according to the following recipe. All

cubes of side 3rare ex definitio good. With Qbeing one of these sub-cubes of side 3r−1,

we call Qgood if 1

|Q|X

z∈Qf(z)≤α, (4.97)

and bad otherwise. For each good cube, we repeat the process of partitioning it into 3d

equal-size sub-cubes and designating each of them to be either good or bad depending

on whether (4.97) holds or not, respectively. The bad cubes are not subdivided further.

Iterating this process, we obtain a finite set Bof bad cubes which covers the

(bounded) region B:= SQ∈B Q. We define G:= Zd\B, the good region, and note that

f(z)≤α, z ∈G, (4.98)

and

α < 1

|Q|X

z∈Qf(z)≤3dα, Q ∈ B,(4.99)

where the last inequality is due to the fact that the parent cube of a bad cube is good.

Next we define the “good” function

g(z) := 





f(z), z ∈G

|Q|Pz∈Qf(z), z ∈Q∈ B.

(4.100)

The “bad” function, defined by b:= f−g, then satisfies

b(z) = 0, z ∈G,

z∈Q

b(z) = 0, Q ∈ B.(4.101)

Since KΛf=KΛg+KΛb, as soon as

{z:|KΛg(z)|>α/2}≤b

K1kfk1

2αAND {z:|KΛb(z)|>α/2}≤b

K1kfk1

2α,(4.102)

the desired bound (4.90) will hold. We will now show these bounds in separate argu-

ments.

Considering gfirst, we note that kgk2

2is bounded by a constant times αkfk1. Indeed,

for z∈Blet Qzdenote the bad cube containing z. Then

z∈Zd

g(z)2=X

z∈G

f(z)2+X

z∈B

g(z)2

≤αX

z∈Gf(z)+X

z∈B1

|Qz|X

y∈Qz

f(z)2

≤αkfk1+ 3dαX

z∈B

|Qz|X

y∈Qzf(z)

≤(3d+ 1)αkfk1

(4.103)

by using (4.98) on Gand (4.99) on B. By Chebychev’s inequality and Lemma 4.16,

{z:|KΛg(z)|> α}≤kKΛgk2

α2≤(3d+ 1)kKΛk2

2kfk1

α.(4.104)

Note that this yields an estimate that is uniform in Λ := ΛLbecause kKΛk2≤1 by

Lemma 4.16.

Let us turn to the estimate in (4.102) concerning b. Let {Qk:k= 1,...,|B|} be

an enumeration of the bad cubes and let bk:= b1lQkbe the restriction of bonto Qk.

Abusing the notation to the point where we write KΛ(x, y) for the kernel governing KΛ,

from (4.101) we then have

KΛbk(z) = X

y∈QkKΛ(z, y)−KΛ(z, yk)b(y),(4.105)

where ykis the center of Qk(remember that all cubes are odd-sized). Let ˜

Qkdenote the

cube centered at ykbut of three-times the size — i.e., ˜

Qkis the union of Qkwith the

adjacent 3d−1 cubes of the same side. The bound now proceeds depending on whether

z∈˜

Qkor not.

For z6∈ ˜

Qk, the distance between zand any y∈Qkis proportional to the distance

between zand yk. Proposition 4.19 thus implies

KΛ(z, y)−KΛ(z, yk)≤Cdiam(Qk)

|z−yk|d+1 , z 6∈ ˜

Qk.(4.106)

Moreover, thanks to (4.100),

y∈Qk|b(y)| ≤ X

y∈Qk|f(y)|+|g(y)|≤2X

y∈Qk|f(y)|.(4.107)

Using these in (4.105) yields

|KΛbk(z)| ≤ Cdiam(Qk)

|z−yk|d+1 X

y∈Qk|f(y)|.(4.108)

Summing over all z6∈ ˜

Qkand taking into account that |z−yk| ≥ diam(Qk) for z∈˜

Qk,

we conclude

z∈Λ\˜

|KΛbk(z)| ≤ Cdiam(Qk)X

y∈Qk|f(y)|X

z:|z−yk|≥diam(Qk)

|z−yk|d+1

≤˜

y∈Qk|f(y)|

(4.109)

for some constant ˜

C. Setting ˜

B:= Sk˜

Qkand summing over k, we obtain

z∈Λ\˜

B|KΛb(z)| ≤ ˜

y∈B|f(y)| ≤ ˜

Ckfk1,(4.110)

which by an application of Chebychev’s inequality yields

{z∈Λ\˜

B:|KΛb(z)| ≥ α}≤˜

Ckfk1

α.(4.111)

i.e., a bound of the desired form.

To finish the proof, we still need to take care of z∈˜

B. Here we get (and this is the

only step where we are forced to settle on weak-type estimates),

{z∈˜

B:|KΛb(z)| ≥ α}≤ |˜

B| ≤ 3dX

k|Qk|

≤3dX

αX

z∈Qkf(z)≤3dkfk1

α.

(4.112)

The bound (4.90) then follows by combining (4.104), (4.111) and (4.112).

4.3.5 Triple gradient of finite-volume Green’s function

In order to finish the proof of Theorem 4.15, we still need to establish the decay estimate

in Proposition 4.19. This will be done by invoking a corresponding bound in the full

lattice and reducing it onto a box by reflection arguments. (This is the sole reason

why we restrict to rectangular boxes; more general domains require considerably more

sophisticated methods.)

For ε > 0, let Gεdenote the Green function associated with the discrete Laplacian

∆ on Zdwith killing rate ε > 0, i.e., Gε(·,·) is the kernel of the bounded operator

(ε−∆)−1on `2(Zd). This function admits the probabilistic representation

Gε(x, y) = ∞

k=0

PxXk=y

(1 + ε)k+1 ,(4.113)

where Xis the simple random walk and Pxis the law of Xstarted at x. This func-

tion depends only on the difference of its arguments, so we will interchangeably write

Gε(x, y) = Gε(x−y). We now claim:

Lemma 4.20. There exists b

C > 0such that, for all ε > 0, all i, j, k ∈ {1,...d}and all

x6= 0,

∇i∇j∇kGε(x)≤b

|x|d+1 .(4.114)

Sketch of proof. This is a mere extension (by adding one more gradient) of the estimates

from in Lawler [Law91, Theorem 1.5.5]. (Strictly speaking, this theorem is only for the

transient dimensions but, thanks to ε > 0, the same proofs would apply here.) The

main idea is to use translation invariance of the simple random walk to write G(x) as

a Fourier integral and then control the gradients thereof under the integral sign. We

leave the details as an exercise to the reader.

We now state and prove a stronger form of Proposition 4.19.

Lemma 4.21. There exists C > 0such that, for all L > 1,ε > 0and arbitrary

i, j, k ∈ {1,...d},

|∇(2)

i∇(1)

j∇(2)

kGε

Λ(x, y)| ≤ C

|x−y|d+1 (4.115)

for all x, y ∈Λand all i, j, k ∈ {1, . . . , d}. Here, the superscripts on the operators

indicate the variable the operator is acting on.

Proof. Throughout, we fix L∈Nand denote Λ := ΛL. The proof is based on the

Reflection Principle for the simple random walk on Zd. To start, denote

Λ0:= ΛL={0, . . . , L}d,

Λi:= Zi×{0, . . . , L}d−i, i = 1, . . . , d −1,

Λd:= Zd,

(4.116)

(abusing our earlier notation), write X(i)for the i-th component of Xand let

τi

0:= inf{k≥0: X(i)

k= 0}, τi

L:= inf{k≥0: X(i)

k=L}.

For y∈Λiwith components y= (y1, . . . , yd), and integer-valued indices n∈Z, put

2n(y) := (y1,...,2nL +yi, . . . , yd)

2n+1(y) := (y1,...,2(n+ 1)L−yi, . . . , yd).

Our first claim is that, for i∈ {1, . . . , d},

PxXk=y, τi

0> k, τi

L> k=X

n∈Z

(−1)nPxXk=ri

n(y).(4.117)

In order to show (4.117), fix i∈ {1, . . . , d}and x, y ∈Λiand let Ak

mfor k, m ∈Ndenote

the set of paths of length kstarting in xand ending in ri

n(y) (for some n∈Z) that visit

the set {xi=LZ}exactly mtimes. Moreover, for a path p, let s(p) := 0 if the path p

ends in an even vertex (that is, ri

2n(y) for some n) and s(p) := 1 if it ends in an odd

vertex. We note that, for m > 0,

p∈Ak

(−1)s(p)= 0.(4.118)

To see this, we consider the mapping from Ak

monto itself defined by taking a path and

reflecting the segment between the last visit to LZand the endpoint around the point

where it last visited LZ. This is obviously a bijection from Ak

monto itself which changes

the sign of (−1)s(p). It follows that the sum must vanish. As all paths in Ak

mhave the

same probability, we may in each summand in (4.118) multiply the probability of each

respective path and obtain

0 = X

p∈Ak

(−1)s(p)Px(X0,...,k =p) = X

p∈Ak

m, n∈Z

(−1)s(p)Px(X0,...,k =p, Xk=ri

n(y))

p∈Ak

m, n∈Z

(−1)nPx(X0,...,k =p, Xk=ri

n(y))

n∈Z

(−1)nPx(X0,...,k ∈Ak

m, Xk=ri

n(y)) for all m≥0

(4.119)

with X0,...,k denoting the path of the random walk up to time k. We now verify (4.117)

PxXk=y, τi

0> k, τi

L> k=PxX0,...,k ∈Ak

0

n∈Z

(−1)nPxX0,...,k ∈Ak

0, Xk=ri

n(y)

(4.119)

m≥0, n∈Z

(−1)nPxX0,...,k ∈Ak

m, Xk=ri

n(y)

n∈Z

(−1)nPxXk=ri

n(y).

This obviously holds regardless of any restriction of the other components of the walk,

which means that we have in particular

PxXk=y, τj

0> k, τj

L> k ∀j > i=X

n∈Z

(−1)nPxXk=ri+1

n(y), τj

0> k, τj

L> k ∀j > i+1

(4.120)

for each i∈ {0, . . . , d−1}. Let us now establish the desired representation for the Green

function. For any i∈ {0, . . . , d}, the Green function Gε

Λion Λiwith zero boundary

condition is given by

Gε

Λi(x, y) = ∞

k=0

(1 + )−k−1PxXk=y, τj

0> k, τj

L> k ∀j > i.(4.121)

Applying (4.120) to every probability term, we obtain for each i∈ {0, . . . , d −1}

Gε

Λi(x, y) = X

n∈Z

(−1)nGε

Λi+1 (x, ri+1

n(y)).(4.122)

Consecutive application of this equality gives

Gε

Λ(x, y) = X

z∈Zd

(−1)z1+...+zdGε

Zd(x, rz(y)) (4.123)

for all x, y ∈Λ, where we abbreviate rz=r1

z1◦··· ◦ rd

zd. From Lemma 4.20, we thus

obtain

∇(2)

i∇(1)

j∇(2)

kGε

Λ(x, y)≤X

z∈Zd∇?

i∇j∇?

kGε(x−rz(y))≤X

z∈Zdb

|x−rz(y)|d+1 (4.124)

for all x, y ∈Λ. Now we are ready to conclude the argument. Let x, y ∈Λ and

abbreviate

zmax =d

max

i=1 |zi|.

Whenever zmax ≤1, we have |x−rz(y)| ≥ |x−y|as reflection always increases the dis-

tance between points in Λ. If zmax >1, we may even estimate |x−rz(y)| ≥ d−1/2L|z| ≥

d−1|x−y||z|.The latter is verified quickly using d1/2zmax ≥ |z| ≥ zmax and the fact

that zmax is at least 2 in this case. Therefore, we obtain

∇(2)

i∇(1)

j∇(2)

kGε

Λ(x, y)≤X

z:zmax≤1b

|x−y|d+1 +X

z:zmax>1

dd+1 b

|x−y|d+1|z|d+1

≤b

|x−y|d+1 3d+dd+1 X

z6=0

|z|d+1 ,

(4.125)

which is the desired estimate.

We are now ready to complete the proof of Theorem 4.15:

Proof of Proposition 4.19. Although the ε↓0 limit of Gexists only in d≥3, for gradi-

ents we have ∇G(x, y) = limε↓0∇G(x, y) in all d≥1. Since the bound in Lemma 4.21

holds uniformly in ε > 0, we get the claim in all d≥1.

4.4 Perturbed harmonic coordinate

In this section we will prove Propositions 4.5 and 4.6. Abandoning our earlier notation,

let

GΛ(x, y;ω)=(−Lω)−1(x, y) (4.126)

denote the Green function in Λ with Dirichlet boundary condition for conductance

configuration ω. (Thus, the simple-random walk Green function from Section 4.3 cor-

responds to ω:= 1.) The Green function is the fundamental solution to the Poisson

equation, i.e., 









−LωGΛ(x, z, ω) = δx(z) if z∈Λ,

GΛ(x, z, ω)=0,if z∈∂Λ,

(4.127)

where δx(z) is the Kronecker delta. Note that GΛis defined for all ω∈Ω. The solution

to (4.127) is naturally symmetric,

GΛ(x, y;ω) = GΛ(y, x;ω), x, y ∈Λ,(4.128)

and so we can extend it to a function on Λ ∪∂Λ by setting GΛ(x, ·;ω) = 0 whenever

x∈∂Λ. Here is a generalized form of the representation (4.23):

Lemma 4.22 (Rank-one perturbation).For a finite Λ⊂Zdlet x, y ∈Λbe nearest

neighbors. For any ω, ω0such that ω0

b=ωbexcept at b:= hx, yi, and any z∈Λ∪∂Λ,

ΨΛ(ω0, z)−ΨΛ(ω, z)

=−(ω0

xy −ωxy)GΛ(z, y;ω0)−GΛ(z, x;ω0)ΨΛ(ω, y)−ΨΛ(ω, x).(4.129)

Proof. Suppose ω, ω0∈Ω are such that ω0equals ωexcept at the edge b:= hx, yi, where

ω0

b:= ωb+. Define the function ΦΛ: Λ ∪∂Λ→Rdby

ΦΛ(z) := ΨΛ(ω, z)−GΛ(z, y;ω0)−GΛ(z, x;ω0)ΨΛ(ω, y)−ΨΛ(ω, x).(4.130)

We claim that

Lω0ΦΛ= 0 in Λ.(4.131)

Since for z∈∂Λ we have ΦΛ(z) = ΨΛ(ω, z) = z, this will imply ΦΛ(·) = ΨΛ(ω0,·)

thanks to the uniqueness of the solution of the Dirichlet problem.

In order to show (4.131), we first use (4.127–4.128) to get

Lω0ΦΛ(z) = Lω0ΨΛ(ω, z)−δy(z)−δx(z)ΨΛ(ω, y)−ΨΛ(ω, x).(4.132)

To deal with the term Lω0ΨΛ(ω, z), we think of of Lω0as a matrix of dimension |Λ|. For

its coefficients Lω(z, z0) := hδz,Lωδz0i`2(Λ) we obtain

Lω0(z, z0) = Lω(z, z0) + δy(z)−δx(z)δy(z0)−δx(z0).(4.133)

Using that LωΨΛ(ω, z) = 0 for z∈Λ, we now readily confirm (4.131).

Proof of Proposition 4.5. Set y:= x+ ˆeiand denote ∇if(z) := f(z+ ˆei)−f(z).

Lemma 4.22 shows

∇iΨΛ(ω0, x) = h1−(ω0

b−ωb)∇(1)

i∇(2)

iGΛ(x, x, ω0)i∇iΨΛ(ω, x),(4.134)

where the superindices on ∇indicate which variable is the operator acting on. To prove

the claim we need to show

∇(1)

i∇(2)

iGΛ(x, x, ω)−1= infQΛ(f): f(y)−f(x)=1, f∂Λ= 0,(4.135)

where the conductances in QΛcorrespond to ω. For this, let fbe the minimizer of the

right-hand side. The method of Largrange multipliers shows

−Lωf(z) = αδy(z)−δx(z).(4.136)

Thanks to (4.127), this is solved by

f(z) = αGΛ(y, z;ω)−GΛ(x, z;ω)=α∇(1)

iGΛ(x, z;ω) (4.137)

which in light of the constraint f(y)−f(x) = 1 gives α= [∇(1)

i∇(2)

iGΛ(x, x, ω)]−1.

Since also QΛ(f) = hf, −Lωfi`2(Λ), (4.136) gives QΛ(f) = αand so (4.135) holds. The

correspondence (4.23) then follows from (4.134–4.135); the identity (4.24) results by

differentiation of the left-hand side with respect to ω0

Finally, it remains to establish the limit (4.25), including all of its stated properties:

Proof of Proposition 4.6. Thanks to ellipticity restriction (1.1), we have a bound on this

quantity in terms of the lattice Laplacian. This shows that, for some c=c(λ)∈(0,1),

c < ∇(1)

i∇(2)

iGΛ(x, x, ω0)<1/c (4.138)

uniformly in Λ. Moreover, Λ 7→ ∇(1)

i∇(2)

iGΛ(x, x, ω0) is obviously non-decreasing in Λ

and so the limit exists. The formula (4.26) and the claimed stationarity then follow as

well.

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