Lifshitz tails for random band matrices

Lifshitz tails for random band matrices

Benedikt Staffler

Master Thesis

Theoretical and Mathematical Physics

LMU Munich

Advisor: Prof. Laszlo Erdos, Ph.D.

September 18, 2013

Declaration of Authorship

I declare that this thesis was composed by myself and that the work contained therein is myown, except where explicitly stated otherwise in the text.

Benedikt StafflerSeptember 18, 2013

iv

Contents

1 Introduction 1

1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Random Schrodinger operators 5

2.1 The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Neumann-Dirichlet bracketing . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Lifshitz tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Random band matrices 25

3.1 Wigner band matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Random banded covariance matrices . . . . . . . . . . . . . . . . . . . . . . . 263.3 The Anderson band matrix model . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Lifshitz tails for the Anderson band matrix model 35

4.1 Bounds on the density of states . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Lifshitz tails for random banded covariance matrices . . . . . . . . . . . . . . 404.3 Lifshitz tails for Wigner band matrices . . . . . . . . . . . . . . . . . . . . . . 42

vi CONTENTS

Chapter 1

Introduction

1.1 Preface

In this master thesis we investigate the occurrence of Lifshitz tails for the density of statesfor two ensembles of random Schrodinger operators. The density of states measure is an im-portant quantity in solid-state and condensed matter physics. Roughly speaking it measuresthe number of states in a given energy interval per unit volume. Given a random Hamiltonoperator HL on ℓ2(ΛL), where ΛL ⊂ Z

d is finite, we can define the empirical density of statesas the measure νL = 1

|ΛL|∑

λ∈σ(HL)δλ, where σ(HL) is the spectrum of HL and δλ is the

Dirac-measure at the point λ. We can extend this approach to operators H on the wholed-dimensional lattice Zd by taking finite-volume versions HL of H and consider the thermody-namical limit ν = limL→∞

1|ΛL|

∑

λ∈σ(HL)δλ = limL→∞ νL, where the convergence is usually

in the vague sense for measures. The expectation value E[ν] given by E[ν](A) = E[ν(A)] iscalled the density of states and its distribution function N(E) := E[ν]((−∞, E)) is called theintegrated density of states.

When we introduce disorder in a quantum mechanical system the behavior of the inte-grated density of states near the bottom of the spectrum changes drastically. This was firstobserved by the physicist I. Lifshitz. More precisely, Lifshitz found that for an ordered system(e.g. a periodic potential) the integrated density of states has a power-law decay near thebottom of the spectrum, i.e. N(E) ∼ C(E − E0)

d/2 for E ց E0, where E0 is the infimum ofthe spectrum. However, if we consider a disordered system, the integrated density of statesdecays exponentially, i.e. N(E) ∼ C1e

−C(E−E0)−d/2for E ց E0. This behavior is called

Lifshitz tails.

The intuitive reason is the following: The behavior of N(E) is determined by the proba-bility that there is an eigenvalue of magnitude E. Consider the associated eigenvector. Forthe kinetic energy to be small the eigenvector needs to be spread out - this is a standarduncertainty principle. In that case the eigenvector is exposed to a considerable amount ofpotential energy. But a typical random potential does not offer a large space free of potentialbarriers - typically the probability of a large area having very low potential decays exponen-tially. Such estimates are usually referred to as large deviation theorems. We will see bothof these principles emerge in the proofs of Lifshitz tails for the different random Schrodingeroperator ensembles.

In Chapter 2 we introduce the Anderson model. It will serve as our main example torigorously introduce the terms mentioned above and as a prototype for the ideas developed

2 1. Introduction

later on. The content of this chapter is based on [1] and [8] and covers the important stepson the way to Lifshitz tails for the Anderson model.

In Chapter 3 we are going to generalize the Anderson model by allowing off-diagonaldisorder. Therefore, we will define two ensembles of random band matrices, namely theWigner band matrices in Section 3.1 and the random banded covariance matrices in Section3.2. While the non-vanishing entries of a Wigner band matrix are independent and identicallydistributed random variables (apart from the necessary dependence due to the Hermitianstructure) a random banded covariance matrix is of the form B∗B for some matrix B where thenon-vanishing entries are independent and identically distributed random variables. Hence,the entries of a banded covariance matrix are not independent and it is automatically apositive operator. Due to these differences we will also distinguish between the two bandmatrix ensembles in Chapter 4. In Section 3.3 we will define a random Schrodinger operatorwhich we call the Anderson band matrix model. In addition to the usual Laplacian weintroduce random jump rates that are expressed by a random band matrix and we will treatit similarly as the on-site potential is treated in the random Schrodinger case. For short wewill call it potential.

1.2 Main results

In Chapter 4 we prove Lifshitz tails for the Anderson band matrix model where the potentialis a Wigner band matrix. We also present our approach for the covariance matrix ensemble.This work focuses on band matrices with fixed bandwidth and bounded matrix entries.

For Wigner band matrices the main result is contained in Theorem 4.5 in Section 4.3. Itproves Lifshitz tails in form

limEցE0

ln | lnN(E)|ln(E − E0)

= −d2. (1.1)

The proof is consisting of an upper and a lower bound on the integrated density of states.Using a variational argument we will first establish a finite-volume bound on the integrateddensity of states in Lemma 4.1 in Section 4.1 of the form

1

|ΛL|E[N(HL, E)] ≥ 1

|Λl|E[N(Hl, E)], for all L > l > 0, (1.2)

and further estimate it to yield the lower bound in (1.1). For the upper bound we reduce theproblem to a diagonal potential consisting of the entries of the corresponding Wigner bandmatrix. This allows us to apply the result on Lifshitz tails for the Anderson model and derivethe upper bound in (1.1).

For the banded covariance matrices we established a lower bound on the integrated densityof states in Section 4.2 in the well-known form

limEցE0


≥ −d2. (1.3)

It is just a straightforward application of 4.1. We further derive a finite-volume bound on theintegrated density of states. It consists of a localization formula for random band matrices inLemma 4.2 which we use to derive the estimate

N(E) ≤ 1

|Λl|E[N(−∆N

l + χlB∗Bχl, E +R(E))], (1.4)

1.2 Main results 3

in Lemma 4.3, where B∗B is a banded covariance matrix, R(E) ≥ 0 and R(E) → 0 as E → 0and χl is a localization function to a box Λl. For details and proofs we refer to Section 4.1.

Acknowledgements

First and foremost, I would like to thank my supervisor, Prof. Laszlo Erdos, Ph.D., forthe valuable guidance and advice. In addition, I wish to express my sincere gratitude toAlessandro Michelangeli, Ph.D. for his counsel not only in matters of mathematics. Besides,I would also like to thank Nikolai Leopold und Claudio Llosa Isenrich for joining me for”Mensa” almost every day. Last but not least, I would like to thank Jenny Pfeiffer for hernever-ending patience and support.

4 1. Introduction

Chapter 2

Random Schrodinger operators

The goal of this chapter is to give a complete proof of Lifshitz tails for the Anderson model.In Section 2.1 we start with defining the setup and fixing some basic notions which will beused throughout the thesis. We will also briefly introduce the framework of ’ergodic stochasticprocesses’, which allows us to answer some basic questions in a very convenient way. In Section2.2 we make a short excursion about boundary conditions for the discrete Laplacian. We willintroduce Neumann and Dirichlet boundary conditions on graphs and derive the discreteanalogue for the Dirichlet-Neumann-bracketing. Based on that we will define the density ofstates and its finite-volume approximations in Section 2.3. Using all this ingredients we willbe able to prove the occurrence of Lifshitz tails for the Anderson model in Section 2.4. Thischapter is mainly based on [1] and the references therein.

2.1 The Anderson model

The (discrete) Anderson model is a random Schrodinger operator on the Hilbert space ofsquare-summable functions over the d-dimensional cubic lattice Z

d

ℓ2(Zd) =

ψ : Zd → C

∣

∣

∣

∣

∣

∑

x∈Zd

|ψ(x)|2 <∞

(2.1)

with its standard norm

‖ψ‖2 =

∑

x∈Zd

|ψ(x)|2

1/2

. (2.2)

The kinetic energy is given by the discrete or graph Laplacian

∆ : ℓ2(Zd) → ℓ2(Zd)

(∆ψ)(x) :=∑

y∈Zd

‖x−y‖1=1

(ψ(y)− ψ(x)), (2.3)

6 2. Random Schrodinger operators

where ‖ · ‖1 is the usual 1-norm on Zd. Its quadratic form is given by

〈φ,∆ψ〉 = −1

2

∑

x∈Zd

∑

y∈Zd

‖x−y‖1=1

(φ(x)− φ(y)) (ψ(x) − ψ(y)) . (2.4)

In contrast to the continuous case this operator is bounded by

‖∆ψ‖2 =

∑

x∈Zd

∣

∣

∣

∣

∣

∑

‖j‖1=1

(ψ(x+ j)− ψ(x))

∣

∣

∣

∣

∣

2

1/2

≤∑

‖j‖1=1

∑

x∈Zd

∣

∣

∣ψ(x+ j)

∣

∣

∣

2

1/2

+

∑

x∈Zd

∣

∣

∣ψ(x)

∣

∣

∣

2

1/2

≤ 4d‖ψ‖2. (2.5)

Furthermore, it is easy to see that it is symmetric and hence a self-adjoint operator. Tocalculate its spectrum we define the Fourier transform

F : ℓ2(Zd) → L2([0, 2π]d)

(Fψ)(k) :=1

(2π)d/2

∑

x∈Zd

e−ik·xψ(x), (2.6)

which is a unitary operator and transforms the discrete Laplacian to the multiplication oper-ator

(F∆F−1φ)(k) = 2

d∑

ν=1

(cos(kν)− 1)φ(k) =: h(k)φ(k), (2.7)

for all φ ∈ L2([0, 2π]d), where k = (k1, . . . , kd) ∈ [0, 2π]d. Hence, we see that the spectrum ofthe discrete Laplacian σ(∆) equals the spectrum of the multiplication operator σ(F∆F−1) =[−4d, 0] and therefore

σ(−∆) = σac(−∆) = [0, 4d], (2.8)

which also implies that ‖∆‖ = 4d.

The potential V is given by a multiplication operator with a real function V (x), x ∈ Zd,

i.e.

(V ψ)(x) := V (x)ψ(x). (2.9)

The simplest way to make it random is to define V (x) as random variables for all x ∈ Zd.

Definition 2.1. Let I be some index set and (Ω,A,P) be a probability space. A family(X(i))i∈I of random variables X(i) : Ω → R is called stochastic process with index set I.

2.1 The Anderson model 7

We consider a stochastic process (Vω(x))x∈Zd , where the random variables Vω(x) are in-dependent and identically distributed with common distribution P0, i.e. for any Borel set Aand any x ∈ Z

d

P0(A) = P(V (x) ∈ A). (2.10)

Let supp P0 be the support of P0, i.e.

supp P0 = x ∈ R | ∀ε > 0 : P0((x− ε, x+ ε)) > 0. (2.11)

If supp P0 is compact, i.e. supp P0 ⊂ [−C,C] for some C > 0, then

supx∈Zd

|Vω(x)| ≤ C (2.12)

with probability one. In this case the operator

Hω = −∆+ Vω (2.13)

is bounded and hence self-adjoint on ℓ2(Zd), where Vω acts on ψ ∈ ℓ2(Zd) as

(Vωψ)(x) = Vω(x)ψ(x). (2.14)

If supp P0 is not compact, then we still can define the multiplication operator Vω on D(Vω) =ψ ∈ ℓ2(Zd)|Vωψ ∈ ℓ2. Since Vω(x) is real-valued for all x ∈ Z

d, the operator Vω is self-adjoint on D(Vω). Using the Kato-Rellich theorem, we further get that Hω = −∆ + Vω isessentially self-adjoint on the functions of compact support, i.e. on

ℓ20(Zd) = ψ ∈ ℓ2(Zd) | ψ(x) = 0 for all but finitely many x ∈ Z

d. (2.15)

Putting things together, we have the Hamiltonian

Hω = −∆+ Vω (2.16)

acting in ℓ2(Zd), where ∆ and Vω are defined as above. We call this operator the Andersonmodel.

In the following we will briefly introduce the more general notion of ’ergodic processes’and ’ergodic operators’ which allows us to make some conclusions about the properties of theAnderson model. Therefore, let (Ω,A,P) be a probability space and T : Ω → Ω a measurablemapping. T is called a measure preserving transformation if

P(T−1(A)) = P(A) (2.17)

for all A ∈ A. For a family (Ti)i∈I , I any index set, of measure preserving transformationswe call a set A ∈ A invariant under (Ti)i∈I , if

T−1i A = A (2.18)

for all i ∈ I.

Definition 2.2. A family (Ti)i∈Zd of measure preserving transformations on a probabilityspace (Ω,A,P) is called ergodic if any invariant A ∈ A has probability zero or one. A stochas-tic process (X(i))i∈Zd is called ergodic if there exists an ergodic family of measure preservingtransformations (Ti)i∈Zd such that

X(i, Tjω) = X(i− j, ω). (2.19)


The particular choice of Ω is irrelevant. For the Anderson model it is reasonable toconsider the canonical probability space Ω = R

Zdwith the σ−algebra A = B(RZd

). Thelatter is generated by the cylinder sets

Z(Ix1 , . . . , Ixn) := ω ∈ Ω | ∀k = 1, . . . , n : ωxk ∈ Ixk , (2.20)

with Ixk ⊂ R Borel.On (Ω,A) a family of random variables (X(i))i∈Zd can be realized as X(i, ω) = ωi. A

random potential Vω(x) consisting of independent and identically distributed random variablesis given by X(x, ω) = Vω(x) for x ∈ Z

d. Due to the independence of the random variablesthe probability measure P on Ω is just the product measure of the probability measure P0 onR given by P0(A) = P(Vω(0) ∈ A), where P0 is the distribution of Vω(0).

We define the family of shifts Tx : Ω → Ω with x ∈ Zd as

(Txω)y = ωx−y (2.21)

for y ∈ Zd. They leave the probability of any cylinder set Z(Ix1 , . . . , Ixn) invariant:

P (Z(Ix1 , . . . , Ixn)) =

n∏

k=1

P0(Ixk) = P(

T−1y Z(Ix1 , . . . , Ixn)

)

. (2.22)

This property extends to the whole σ-algebra and, hence, the shifts are a family of measurepreserving transformations. They are also ergodic. This can be seen by showing that theyare mixing, i.e. for A,B ∈ A

P(

T−1x A ∩B

)

→ P(A)P(B) (2.23)

as ‖x‖∞ → ∞. For A and B cylinder sets this is obvious. Moreover, the system of sets A,B for which (2.23) holds is a σ-algebra. Thus (2.23) is true on A. Now let M ∈ A be aninvariant set. From (2.23) we get

P(M) = P(M ∩M) = P(T−1x M ∩M) → P(M)2 (2.24)

proving that M has either probability zero or one.One of the most important theorems in ergodic theory is Birkhoff’s theorem. It generalizes

the strong law of large numbers to ergodic processes. We will need this result later to definethe density of states of a random Schrodinger operator.

Theorem 2.3 (Birkhoff). Let (X(i))i∈Zd be an ergodic stochastic process and suppose thatE[|X(0)|] <∞. Then for ΛL := x ∈ Z

d | ‖x‖∞ ≤ L ⊂ Zd and |ΛL| := (2L+1)d the volume

of ΛL

limL→∞

1

|ΛL|∑

i∈ΛL

X(i) = E[X(0)], (2.25)

for P-almost all ω.

For a proof of this result see for example [9].The next step is to consider operator-valued function on the probability space.

Definition 2.4. Let (Ω,A,P) be a probability space. A family of self-adjoint operatorsH(ω), ω ∈ Ω, on a separable Hilbert space is called weakly measurable, if the functionsω 7→ 〈φ, f(H(ω))ψ〉 are measurable for all f ∈ L∞(R) and all φ,ψ ∈ H.

2.1 The Anderson model 9

We will not care too much about measurability here and refer to [2] for details.

Definition 2.5. Let (Ω,A,P) be a probability space with a family (Ti)i∈Zd of ergodic trans-formations. A family of measurable self-adjoint operator H(ω), ω ∈ Ω, is called ergodic ifH(Tiω) is unitarily equivalent to H(ω).

On ℓ2(Zd) we define the translation operators

(Uyψ)(x) := ψ(x− y), (2.26)

which are clearly unitary. Given any ergodic process (V (x))x∈Zd , the corresponding multipli-cation operator Vω on ℓ2(Zd) transforms as UxVωU

−1x = VTxω. Since −∆ is invariant under

conjugation by Ux, each operator of the form

Hω = −∆+ Vω (2.27)

is ergodic.The Anderson model has another nice property. Let δx ∈ ℓ2(Zd) be the Kronecker delta,

i.e. for x, y ∈ Zd

δx(y) =

1, if x = y

0, if x 6= y.(2.28)

Then, for Ty and Uy as above and f ∈ C0(R), the set of continuous functions with compactsupport,

〈δx, f(HTyω)δx〉 = 〈δx, Uyf(Hω)U∗y δx〉

= 〈U∗y δx, f(Hω)U

∗y δx〉

= 〈δx−y, f(Hω)δx−y〉. (2.29)

Definition 2.6. A family of ergodic operators H(ω) in ℓ2(Zd) is said to satisfy the condition(C1), if it satisfies

〈δx, f(HTyω)δx〉 = 〈δx−y, f(Hω)δx−y〉 (2.30)

for the shifts Tx defined in (2.21).

A first benefit of the theory of ergodic operators is the following theorem by Pastur aboutthe spectrum of ergodic operators, which was originally proven in [3].

Theorem 2.7. (Pastur) The spectrum of a family H(ω) of ergodic operators is P-almostsurely non-random, i.e. there is Σ ⊂ R such that

P(σ(H(ω)) = Σ) = 1. (2.31)

Moreover, the same applies to any subset in the Lebesgue-decomposition of the spectrum, i.e.there are sets Σac, Σsc, Σpp ⊂ R such that

P(σ#(H(ω)) = Σ#) = 1, # = ac, sc, pp, (2.32)

where ac is the absolutely continuous part of the spectrum, sc is the singular continuous partand pp is the pure point part.


The next theorem tells us explicitly what the spectrum of the Anderson model is. Themore general result for a special class of ergodic operators is due to [5]. For the Andersonmodel the statement is the following.

Theorem 2.8. Let Hω be the Anderson model. Then for P-almost all ω we have

σ(Hω) = [0, 4d] + supp P0. (2.33)

Proof. ( [1], [8]) The spectrum of a deterministic multiplication operator (V (x))x∈Zd is givenby the closure of its essential range, i.e.

σ(V ) = V (x) | x ∈ Zd. (2.34)

Hence, σ(Vω) = supp P0 P-almost surely. Now we are using the fact that the distance of thespectra of two operators A,B : D(A) → H in a Hilbert space H with common domain D(A),whose difference is a bounded operator, is bounded by

dist (σ(A), σ(B)) ≤ ‖A−B‖. (2.35)

Set A = Hω and B = Vω + 2d. Then ‖A−B‖ ≤ 2d and since σ(B) = supp P0 + 2d we get

σ(Hω) ⊂ supp P0 + [0, 4d]. (2.36)

For the converse inclusion we are going to use the Weyl criterion (see e.g. [10]) whichstates

λ ∈ σ(Hω) ⇔ ∃φn ∈ D0, ‖φn‖ = 1 : ‖(Hω − λ)φn‖ → 0 as n→ ∞, (2.37)

where D0 is any vector space such that Hω is essentially self-adjoint on D0. The sequence φnis called a Weyl sequence for Hω and λ.

Let λ ∈ [0, 4d] + supp P0. We write λ = λ0 + λ1, where λ0 ∈ σ(−∆) = [0, 4d] andλ1 ∈ supp P0. Take a Weyl sequence φn for −∆ and λ0, i.e. ‖(−∆ + λ0)φn‖ → 0, asn → ∞ and ‖φn‖ = 1. Since −∆ is essentially self-adjoint on D0 = ℓ20(Z

d) we may supposeφn ∈ ℓ20(Z

d). For k ∈ N we define the events

Ωk :=

ω ∈ Ω

∣

∣

∣

∣

∃j ∈ Zd : sup

x∈supp φk|Vω(x+ j)− λ1| <

1

k

. (2.38)

For λ1 ∈ supp P0 and we see that

P(Ωk) ≥ P

(

∀x ∈ supp φk : |V (x)− λ1| <1

k

)

≥ P

(

|V (0)− λ1| <1

k

)|supp φk|

≥ P0((λ1 −1

k, λ1 +

1

k))|supp φk|

> 0, (2.39)

where we used independence of the random variables V (x) in the second line. Hence theevents Ωk have a non-zero probability. Furthermore, it is easy to see that the events Ωk are

2.2 Neumann-Dirichlet bracketing 11

invariant under the shifts (Txω)y = ωx−y defined in equation (2.21). Thus, by ergodicity,P(Ωk) = 1 for each k ∈ N. Therefore, their intersection

Ω0 =⋂

k∈NΩk (2.40)

is almost certain too, i.e. P(Ω0) = 1.

By construction, for each ω ∈ Ω0, there is a sequence (jn)n∈N ⊂ Zd such that

supx∈supp φn

|Vω(x+ jn)− λ1| → 0, as n→ ∞. (2.41)

Hence, the sequence defined by

ψn := φn(· − jn), (2.42)

is a Weyl sequence for Hω and λ, since

‖(Hω − λ)ψn‖ ≤ ‖(−∆− λ0)ψn‖+ ‖(Vω − λ1)ψn‖≤ ‖(−∆− λ0)φn‖+ sup

x∈supp φn|Vω(x+ jn)− λ1|‖φn‖, (2.43)

which goes to zero as n→ ∞.

Remark 2.9. The proof of Theorem 2.8 can easily be generalized to an ergodic operator ofthe form Hω = −∆+Vω on ℓ2(Zd) which satisfies condition (C1), where Vω is a multiplicationoperator but we do not assume that the Vω(x) are independent and identically distributed.Associated to Vω, we define the quantities

supp1(P) := λ ∈ R | ∀ε > 0 : P (|Vω(x)− λ| < ε) > 0 (2.44)

and

supp2(P) :=

λ ∈ R

∣

∣

∣∀ε > 0,Λ ⊂ Z

d finite : P

(

supx∈Λ

|Vω(x)− λ| < ε

)

> 0

. (2.45)

The following result is due to Kunz and Souillard [5] and was generalized in [12].

Theorem 2.10. For the ergodic operator Hω = −∆+ Vω on ℓ2(Zd) which satisfies condition(C1) we have

[0, 4d] + supp2(P) ⊂ σ(Hω) ⊂ [0, 4d] + supp1(P), (2.46)

for P-almost all ω.

In the case that (Vω(x))x∈Zd are independent and identically distributed random variables,supp1(P) = supp2(P) and we recover Theorem 2.8.

2.2 Neumann-Dirichlet bracketing

Before we define the density of states for a random Schrodinger operator, we need to introduceboundary conditions for the Laplacian. These will be needed to restrict our operator Hω tosome finite subset Λ ⊂ Z

d. This short excursion on boundary conditions is based on [1], [4]


and [8]. In the following we will develop the notion of Dirichlet and Neumann Laplacians ona graph with vertex set G and edge set EG. The degree degG(x) of a vertex x ∈ G is given by

degG(x) := |y ∈ G | (x, y) ∈ EG|. (2.47)

The graph Laplacian is defined by

(∆Gψ)(x) :=∑

(x,y)∈EG

(ψ(y)− ψ(x)) . (2.48)

If the degree of the vertices are uniformly bounded, then ∆G is a bounded operator with‖∆G‖ ≤ 2 supx∈G degG(x). Note, that for G = Z

d and EZd = (x, y) ∈ Zd × Z

d | ‖x− y‖1 =1 = 2d, the notion of the graph Laplacian defined in the previous section coincides with thisdefinition. Now let (G′, EG′) be a subgraph of (G, EG). On this subgraph we will introducetwo different types of Laplacians: The Neumann Laplacian on ℓ2(G′) coincides with the graphLaplacian on the subgraph. It is given by

(∆NG′ψ)(x) :=

∑

(x,y)∈EG′

(ψ(y)− ψ(x)) . (2.49)

The Dirichlet Laplacian on ℓ2(G′) is given by

(∆DG′ψ)(x) := (∆N

G′ψ)(x) + 2(degG′(x)− degG(x))ψ(x). (2.50)

While the Neumann Laplacian is so to say not aware of the bigger graph G, the DirichletLaplacian increases the diagonal entry by one for each adjacent edge in G \G′. It is easy tosee that

0 ≤ −∆NG′ ≤ −∆D

G′. (2.51)

The benefit of this choice becomes apparent is the discrete analogue of the Neumann-Dirichlet bracketing.

Proposition 2.11. (Neumann-Dirichlet bracketing) Let (G, EG) be a graph with uniformlybounded degree and let (G1, EG1) and (G2, EG2) be disjoint subgraphs. Then

−∆NG1

⊕∆NG2

≤ −∆NG1∪G2

≤ −∆DG1∪G2

≤ −∆DG1

⊕∆DG2

(2.52)

on ℓ2(G1 ∪G2) = ℓ2(G1)⊕ ℓ2(G2).

Proof. The quadratic form of −∆NG1∪G2

on (G1 ∪G2, EG1∪G2) is given by

〈ψ,−∆NG1∪G2

ψ〉 =∑

(x,y)∈EG1∪G2

|ψ(x)− ψ(y)|2. (2.53)

Omitting all terms of the sum, where neither (x, y) ∈ EG1 nor (x, y) ∈ EG2 , yields the firstinequality. The second inequality is (2.51). The third inequality follows from

〈ψ, −∆DG1

⊕∆DG2ψ〉 − 〈ψ,−∆D

G1∪G2ψ〉

=∑

(x,y)∈EG1∪G2(x,y)/∈EG1

∪EG2

|ψ(x) + ψ(y)|2 ≥ 0. (2.54)

We will use this proposition in the next section to derive some useful finite-volume boundson the density of states.

2.3 The density of states 13

2.3 The density of states

In this section we will introduce the density of states, a quantity of great importance instatistical and condensed matter physics. The density of states roughly speaking measuresthe number of states per unit volume in a given energy interval. This is fairly easy if we workin a finite dimensional Hilbert space: For an N × N -matrix H we can define the empiricaldensity of states measure as a sum over the N eigenvalues En of the matrix (counted withmultiplicity)

ν =1

N

N∑

n=0

δEn , (2.55)

where δx is the Dirac-measure at the point x. This measure is also called the empiricaleigenvalue distribution. The situation is different in infinite dimensions. Since the spectrumof our random Schrodinger operator Hω is not discrete, we can not simply count eigenvaluesor, equivalently, take the dimension of the corresponding spectral projections, since those areeither zero or infinite. To overcome this problem, we will first restrict our operator Hω to afinite box ΛL ⊂ Z

d given by

ΛL := x ∈ Zd | ‖x‖∞ ≤ L (2.56)

with appropriate boundary conditions. We arrived at the situation of a finite dimensionalHilbert space, where we can apply the definition above. The density of states for the initialsystem will then be given by taking the limit L→ ∞.

For the rigorous treatment we will follow the argument in [1] and [8]. To prove the existenceof this limit we will again use the language of ergodic operators using a thermodynamic limit.Let Hω = −∆+ Vω be an ergodic operator in ℓ2(Zd) which satisfies the condition (C1) fromDefinition 2.6. For f ∈ C0(R), the set of continuous functions with compact support, we candefine the function νL : C0(R) → R by

νL(f) :=1

|ΛL|tr(χΛL

f(Hω)), (2.57)

where χΛLis the characteristic function of the box ΛL. Since νL is a bounded linear functional

on the continuous functions with compact support, by the Riesz representation theorem, thereis a measure, which we will also denote by νL, such that

νL(f) =

∫

R

f(u)dνL(u). (2.58)

The following theorem tells us, that νL converges vaguely to a non-randommeasure as L→ ∞.

Theorem 2.12. Let Hω = −∆+Vω be an ergodic operator which satisfies the condition (C1).Then there is a set Ω0 of probability one, such that

limL→∞

1

|ΛL|tr(χΛL

f(Hω)) = E[〈δ0, f(Hω)δ0〉], (2.59)

for all f ∈ C0(R) and all ω ∈ Ω0.

From Theorem 2.12, we see that the measure ν defined by∫

Z

f(u)dν(u) := E[〈δ0, f(Hω)δ0〉], (2.60)

is a probability measure. We use this to define the density of states:


Definition 2.13. The probability measure ν defined by

ν(A) = E[〈δ0, χA(Hω)δ0〉], (2.61)

where A is a Borel set, is called the density of states measure. The distribution function Nof ν, defined by

N(E) = ν((−∞, E]), (2.62)

is called the integrated density of states.

Proof of Theorem 2.12. Let f ∈ C0(R), then

1

|ΛL|tr(χΛL

f(Hω)) =1

|ΛL|∑

x∈ΛL

〈δx, f(Hω)δx〉

=1

|ΛL|∑

x∈ΛL

〈δ0, f(HTxω)δ0〉. (2.63)

Since |〈δ0, f(HTxω)δ0〉| ≤ ‖f‖∞, we can apply Birkhoff’s ergodic theorem (Theorem 2.3) whichyields that for every f ∈ C0(R) there is a set Ωf of probability one such that

limL→∞

1

|ΛL|tr(χΛL

f(Hω)) = E[〈δ0, f(Hω)δ0〉] (2.64)

for all ω ∈ Ωf . Now let D0 be a countable dense set in C0(R) in the uniform topology. Thenthe set

Ω0 =⋂

f∈D0

Ωf , (2.65)

also has probability one, where Ωf is the set of probability one such that (2.64) holds. Forany f ∈ C0(R), there is fε ∈ D0 such that ‖f − fε‖∞ < ε. Hence, we can estimate

|∫

fdν −∫

fdνL|

≤ |∫

fdν −∫

fεdν|+ |∫

fεdν −∫

fεdνL|+ |∫

fεdνL −∫

fdνL|

≤ 2‖f − fε‖∞ + |∫

fεdν −∫

fεdνL|. (2.66)

The first term can be made arbitrarily small by choosing ε small while the third term will besmall if we choose L large.

Equation (2.64) in the proof of Theorem 2.12 is still valid if we assume f to be onlymeasurable and bounded. In this case we have the following corollary.

Corollary 2.14. Let f : R → R be measurable and bounded. Then there is a set Ωf ofprobability one such that

limL→∞

1

|ΛL|tr(χΛL

f(Hω)) = E[〈δ0, f(Hω)δ0〉] (2.67)

for all ω ∈ Ωf .


Our next aim is to replace the infinite volume operator Hω by a finite-dimensional matrixwith appropriate boundary conditions. This is the purpose of the next theorem. For betterreadability we will drop the subscript ω from the operators.

Theorem 2.15. Let H be an ergodic operator which satisfies the condition (C1) and let(HL)L∈N be a sequence of operators such that for P-almost all ω ∈ Ω:

1. χLHL = HLχL

2. χLHL − χLH is trace class satisfying

1

|ΛL|tr |χLHL − χLH| → 0, as L→ ∞. (2.68)

Then there is a set Ω0 of probability one such that

limL→∞

1

|ΛL|trχLf(HL) =

∫

R

f(u)dν(u) (2.69)

for all f ∈ C0(R) and ω ∈ Ω0, where ν is the density of states measure.

Proof. We define the measure µL by

∫

f(u)dµL(u) =1

|ΛL|trχLf(HL). (2.70)

The statement of the theorem is that µL vaguely converges to ν. Therefore, it is enough toshow that

∫

f(u)dµL(u) →∫

f(u)dν(u) (2.71)

for all functions of the form f(u) = rz(u) = (u − z)−1 with z ∈ C \ R because linearcombinations of these functions are dense in C0(R) by the Stone-Weierstrass theorem. Usingthat

∫

rz(u)dµL(u) =1

|ΛL|trχL(rz(HL)) (2.72)

and∫

rz(u)dνL(u) =1

|ΛL|trχL(rz(H)) (2.73)

we can estimate∣

∣

∣

∣

1

|ΛL|trχL(rz(HL))−

1

|ΛL|trχL(rz(H))

∣

∣

∣

∣

=1

|ΛL||trχL[rz(HL)− rz(H)]|

=1

|ΛL||trχL[rz(HL)(H −HL)rz(H)]|

≤ 1

|ΛL||ℑz|−2 tr |χL(H −HL)| −→ 0, as L→ ∞, (2.74)


where we used the resolvent formula in the third line, the estimate ‖Rz(H)‖ ≤ 1ℑz in the

fourth line and our second assumption to conclude convergence. Hence, we have shown that∣

∣

∣

∣

∫

rz(u)dµL(u)−∫

rz(u)dνL(u)

∣

∣

∣

∣

→ 0, as L→ ∞. (2.75)

But from Theorem 2.12 we already know that∫

rz(u)dνL(u) →∫

rz(u)dν(u). (2.76)

Therefore, we can conclude that also µL vaguely converges to ν.

We will use Theorem 2.15 to approximate the infinite-volume operator H by finite-volumeversions HL of itself. Before we do so we will derive some general properties which will alsobe valid for the Anderson model. For an operator HL that is acting on the finite-dimensionalHilbert space ℓ2(ΛL) we define the normalized eigenvalue counting function by

1

|ΛL|N(HL, E) :=

1

|ΛL||n|En(HL) < E| , (2.77)

which counts the number of eigenvalues below E. By definition, it is the distribution functionof µL, i.e.

1

|ΛL|N(HL, E) =

∫

χ(−∞,E](u)dµL(u). (2.78)

An immediate consequence of Corollary 2.14 is that for a fixed E the distribution functionof νL converges to N almost surely on an E-dependent set. The next corollary shows that weeven have convergence on a E-independent set for all E ∈ R.

Corollary 2.16. In the situations of Theorem 2.12, or respectively Theorem 2.15, we have

N(E) = limL→∞

νL((−∞, E]), (2.79)

or respectively

N(E) = limL→∞

µL((−∞, E]) = limL→∞

1

|ΛL|N(HL, E) (2.80)

for P-almost all ω and all E ∈ R.

Proof. We will do the proof for (2.79). The proof for (2.80) is analogous. Since N is monotoneincreasing, the set of discontinuity points of N is at most countable (see Lemma 2.17 below).Consequently, there is a countable set S of continuity points of N which is dense in R. ByCorollary 2.14 there is a set of full probability such that

∫

χ(−∞,E](u)dνL(u) → N(E), as L→ ∞ (2.81)

for all E ∈ S. Let ε > 0 be arbitrary and E ∈ S. Then, there are E+, E− ∈ S withE− ≤ E ≤ E+ such that N(E+)−N(E−) <

ε2 . Using that N is monotone we can estimate

N(E)−∫

χ(−∞,E](u)dνL(u)

≤ N(E+)−∫

χ(−∞,E−](u)dνL(u)

≤ N(E+)−N(E−) +

∣

∣

∣

∣

N(E−)−∫

χ(−∞,E−](u)dνL(u)

∣

∣

∣

∣

≤ ε, (2.82)


for L large enough and analogous

N(E) −∫


≥ N(E−)−N(E+) +

∣

∣

∣

∣

N(E+)−∫

χ(−∞,E+](u)dνL(u)

∣

∣

∣

∣

≥ −ε. (2.83)

Hence, we get∣

∣

∣

∣

N(E) −∫


∣

∣

∣

∣

→ 0, as L→ ∞. (2.84)

This proves (2.79) for the continuity points of N . But since there are at most countably manydiscontinuity points of N , again by Corollary 2.14, there are at most countably many sets offull probability such that

N(E) = limL→∞

νL((−∞, E]), (2.85)

where E is a discontinuity point of N . But since the intersection of countably many sets offull probability is still a set of full probability, we conclude that (2.79) holds for all E.

Lemma 2.17. Let f : R → R be a monotone increasing function. Then f has at mostcountably many points of discontinuity.

Proof. Since f is monotone, the limits f(t−) = limsրt f(x) and f(t+) = limsցt f(s) exist. Iff is discontinuous at t ∈ R then f(t+)− f(t−) > 0. We define

Dn =

t ∈ R

∣

∣

∣

∣

f(t+)− f(t−) >1

n

. (2.86)

The set D of discontinuity points of f is then given by⋃

n∈NDn. Assume now that D isuncountable. Then, since the countable union of countable sets is countable, at least one ofthe Dn must be uncountable. Since f is monotone and defined the whole real line R it mustbe bounded on any bounded interval. Thus Dn ∩ [−M,M ] is finite for any M > 0. Hence,Dn =

⋃

M∈N(Dn∩ [−M,M ]) is countable. This is a contradiction to the conclusion above.

Now let us return to the Anderson model Hω = −∆ + V . In Theorem 2.8 we saw thatthe spectrum of Hω is given by the non-random set [0, 4d] + supp P0. Since the density ofstates measure ν counts the number of states, we would expect it to be supported only onthe spectrum of Hω This is the statement of the next proposition.

Proposition 2.18. For the Anderson model Hω = −∆ + V , the corresponding density ofstates measure satisfies

supp (ν) = σ(Hω) = [0, 4d] + supp P0. (2.87)

Proof. Assume first that λ /∈ σ(Hω). Then there is an ε > 0 such that χ(λ−ε,λ+ε)(Hω) = 0P-almost surely, and hence

ν((λ− ε, λ+ ε)) = E[

〈δ0, χ(λ−ε,λ+ε)(Hω)δ0〉]

= 0, (2.88)

i.e. λ /∈ supp (ν).


Assume now that λ ∈ σ(Hω). Then χ(λ−ε,λ+ε)(Hω) 6= 0 P-almost surely for any ε >

0. Furthermore, since χ(λ−ε,λ+ε)(Hω) is a projection, there is some x ∈ Zd such that

E[

〈δx, χ(λ−ε,λ+ε)(Hω)δx〉]

6= 0. Hence, we get

0 6= E[

〈δx, χ(λ−ε,λ+ε)(Hω)δx〉]

= E[

〈δ0, χ(λ−ε,λ+ε)(Hω)δ0〉]

= ν((λ− ε, λ+ ε)), (2.89)

where we used in the second line that the Anderson model satisfies condition (C1) in the form

〈δx, f(Hω)δx〉 = 〈δ0, f(HTxω)δ0〉 (2.90)

and that the shifts Tx are measure preserving, i.e.

E[

〈δ0, χ(λ−ε,λ+ε)(Hω)δ0〉]

= E[

〈δ0, χ(λ−ε,λ+ε)(HTxω)δ0〉]

. (2.91)

This shows that λ ∈ supp (ν). We conclude that supp (ν) = σ(Hω).

Our next aim is to use the boundary conditions defined in Section 2.2 to approximate theinfinite-volume operator H by finite-volume versions of itself. We define

H#L := −∆# + V, (2.92)

where # specifies the boundary conditions Dirichlet (# = D) or Neumann (# = N). SinceV is diagonal, an immediate consequence of Proposition 2.11 is the following:

Corollary 2.19. Let ΛL ⊂ Zd be the disjoint union of smaller boxes Λk, k = 1, . . . K. Then⊕

k

HNΛk

≤ HNL ≤ HD

L ≤⊕

k

HDΛk. (2.93)

It is easy to prove that the operators H#L satisfy the assumptions of Theorem 2.15. We

can combine this with Corollary 2.16 to get an estimate of the integrated density of statesinvolving only finite-volume operators.

Lemma 2.20. For the Anderson model H = −∆+V one has for all l > 0 and for all E ∈ R

1

|Λl|E[N(HD

l , E)] ≤ N(E) ≤ 1

|Λl|E[N(HN

l , E)]. (2.94)

Proof. For L > l > 0, consider the box ΛL and assume that ΛL is the disjoint union of smallerboxes Λl,k of volume |Λl,k| = |Λl| = (2l + 1)d. Denote by H#

L the Anderson model with

boundary conditions # on ΛL and respectively by H#l,k the Anderson model with boundary

conditions # on Λl,k. Applying Corollary 2.19 and the min-max-principle yields∑

k

N(HDl,k, E) ≤ N(HD

L , E) ≤ N(HNL , E) ≤

∑

k

N(HNl,k, E). (2.95)

Taking expectation yields

|ΛL||Λl|

E[N(HDl , E)] ≤ E[N(HD

L , E)] ≤ E[N(HNL , E)] ≤ |ΛL|

|Λl|E[N(HN

l , E)], (2.96)

since there are |ΛL||Λl| terms in each of the two sums which are independent and equally dis-

tributed. Dividing by |ΛL| and letting L→ ∞, the two terms in the middle converge to N(E)by Corollary 2.16, which yields the desired result.

2.4 Lifshitz tails 19

A useful corollary of this is the following, which will be our starting point in the nextsection.

Corollary 2.21. In the situation of Lemma 2.20 we have

1

|Λl|P(E0(H

Dl ) < E) ≤ N(E) ≤ P(E0(H

Nl ) < E). (2.97)

Proof. By the definition of N(H#l , E), where # denotes the boundary condition, we get

E[N(H#l , E)] = E

|Λl|−1∑

i=0

χ(−∞,E](Ei(H#l ))

=

|Λl|−1∑

i=0

P(Ei(H#l ) < E). (2.98)

Since P(Ei(H#l ) < E) ≥ 0 we have

E[N(H#l , E)] ≥ P(E0(H

#l ) < E). (2.99)

On the other hand P(Ei(H#l ) < E) ≤ P(E0(H

#l ) < E). Hence

E[N(H#l , E)] ≤ |Λl|P(E0(H

#l ) < E). (2.100)

Together with Lemma 2.20 this yields the claim.

2.4 Lifshitz tails

The physicist Lifshitz observed that for low energies the integrated density of states changesdrastically if one introduces disorder in a system. In the ordered case he found that theintegrated density of states behaves as

N(E) ∼ (E − E0)d/2 for E ց E0, (2.101)

where E0 is the infimum of the spectrum. However, if we consider a disordered system theintegrated density of states decays exponentially, i.e.

N(E) ∼ C1e−C(E−E0)−d/2

for E ց E0. (2.102)

This behavior is called Lifshitz tails. In this section we will prove Lifshitz tails for theAnderson model.

Let Hω = −∆ + Vω be a random operator on ℓ2(Zd). We will assume that the randomvariables (V (x))x∈Zd are independent and identically distributed with common distributionP0. Let a0 := inf supp P0 > −∞. We will further assume that P0 is non trivial, i.e. notconcentrated on a single point and that for some C, κ > 0 and all ε ∈ (0, 1]:

P0([a0, a0 + ε]) ≥ Cεκ. (2.103)

In this case E0 = inf σ(Hω) = a0 P-almost surely by Theorem 2.8 and Proposition 2.18.We will prove a weak version of (2.102).


Theorem 2.22. (Lifshitz tails) Under the assumptions above we have that

limEցE0


= −d2. (2.104)

Proof. We will follow the proof given in [1]. Further we will assume that a0 = 0. In particularthis implies that Vω(x) ≥ 0 P-almost surely for all x ∈ Z

d. This can always be achieved byadding a positive constant to the potential. From Corollary 2.21 we have

1

|Λl|P(E0(H

Dl ) < E) ≤ N(E) ≤ P(E0(H

Nl ) < E). (2.105)

We will now further estimate the upper and lower bound in this equality to obtain thedesired result.

We start with the lower bound:

Proof of Lifshitz tails - the lower bound

To proceed we need an upper bound on E0(HDl ). Therefore, we use the min-max principle

E0(HDl ) = inf

ψ∈ℓ2(Λl)‖ψ‖=1

〈ψ,HDl ψ〉

≤ infψ∈ℓ2(Λl)‖ψ‖=1

〈ψ,−∆Dl ψ〉 +max

x∈Λl

Vω(x). (2.106)

Now we try to find an appropriate ψ which minimizes the right hand side in (2.106). Wechoose the ”tent function”

ψ(x) = l − ‖x‖∞, for all x ∈ Λl (2.107)

and

ψ(x) =1

‖ψ‖ψ(x). (2.108)

For this function we calculate

∑

x∈Λl

|ψ(x)|2 ≥∑

x∈Λl/2

|ψ(x)|2 ≥ |Λl/2|(

l

2

)2

≥ c1ld+2 (2.109)

and

〈ψ,−∆Dl ψ〉 =

∑

(x,y)∈Λl×Λl

‖x−y‖1=1

|ψ(y)− ψ(x)|2

≤ |(x, y) ∈ Λl × Λl | ‖x− y‖1 = 1|≤ c2l

d. (2.110)

In total we get

E0(HDl ) ≤ c0

l2+max

x∈Λl

Vω(x). (2.111)


Now we choose l =√

2c0E . Then we can further estimate

N(E) ≥ 1

|Λl|P (∀x ∈ Λl : V (x) < E/2)

=1

|Λl|(P0([0, E/2)))

|Λl|

≥ 1

|Λl|e|Λl| ln(CEκ)

≥ c4Ed/2ec3 ln(E)E−d/2

(2.112)

which yields the lower bound in (2.104).

Proof of Lifshitz tails - the upper bound

The proof of the upper bound is more involved, since we need to get a lower bound on E0. Itwill be provided by Temple’s inequality. For its proof we refer to [13].

Lemma 2.23. (Temple’s inequality) Let A be a self-adjoint operator with an isolated nondegenerate eigenvalue E0 = inf σ(H) and let E1 = inf(σ(H) \E0). If ψ ∈ D(A) with ‖ψ‖ = 1satisfies

〈ψ,Aψ〉 < E1, (2.113)

then

E0 ≥ 〈ψ,Aψ〉 − 〈ψ,A2ψ〉 − 〈ψ,Aψ〉2E1 − 〈ψ,Aψ〉 . (2.114)

To apply Temple’s inequality for the operator HNl we need a good test function. A

convenient guess is the ground state of −∆Nl , namely

ψ0(x) =1

|Λl|1/2for all x ∈ Λl. (2.115)

One easily sees that −∆Nl ψ0 = 0 and, furthermore,

〈ψ,HNl ψ〉 = 〈ψ, V ψ〉 = 1

|Λl|∑

x∈Λl

Vω(x). (2.116)

This is an arithmetic mean of independent and identically distributed random variableswhich is typically close to E[V (0)] > 0 for large l. To apply Temple’s inequality we wouldneed that

1

|Λl|∑

x∈Λl

Vω(x) < E1(HNl ), (2.117)

which is certainly wrong for large l since E1(HNl ) → 0. But we can further estimate

E1(HNl ) ≥ E1(−∆N

l ) ≥ cl−2, (2.118)

where the second inequality can be obtained by direct calculation for some fixed constant c.We now define the cut-off potential


V lω(x) = min

Vω(x),c

3l2

. (2.119)

The random variables (V lω(x))x∈Zd are still independent and identically distributed random

variables, but their distribution depends on l. For the corresponding Schrodinger operator

HNl := −∆N

l + V lω, (2.120)

we get E0(HNl ) ≥ E0(H

Nl ) by the min-max principle. Moreover,

〈ψ0, HNl ψ0〉 =

1

|Λl|∑

x∈Λl

V lω(x) ≤

c

3l2, (2.121)

by the definition of V lω and, consequently,

〈ψ0, HNl ψ0〉 ≤

c

3l2< E1(−∆N

l ) ≤ E1(HNl ). (2.122)

Thus, by Temple’s inequality applied to HNl and ψ0:

E0(HNl ) ≥ E0(H

Nl )

≥ 〈ψ0, HNl ψ0〉 −

〈ψ0, (HNl )2ψ0〉

cl−2 − 〈ψ0, HNl ψ0〉

≥ 1

|Λl|∑

x∈Λl

V lω(x)−

1|Λl|∑

x∈Λl(V lω(x))

2

(c− c3 )l

−2

≥ 1

|Λl|∑

x∈Λl

V lω(x)−

1

|Λl|∑

x∈Λl

V lω(x)

c3 l

−2

2c3 l

−2

≥ 1

2|Λl|∑

x∈Λl

V lω(x). (2.123)

Collecting these estimates we arrive at

N(E) ≤ P(

E0(HNl ) < E

)

≤ P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

. (2.124)

Now we choose an appropriate l. Since V lω ≤ c

3l2, the right hand side of (2.124) will be

one if l is too big. We certainly need that c3l2 >

E2 . We set

l := ⌊βE− 12 ⌋, (2.125)

for some β small enough and ⌊·⌋ the floor function. In this setting one can show the followinglarge deviation estimate:

Lemma 2.24. For l = ⌊βE− 12 ⌋ with β small and l large enough,

P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

≤ e−γ|Λl| (2.126)

with some γ > 0.


Given this lemma we proceed

N(E) ≤ P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

≤ e−γ|Λl|

= e−γ⌊βE− 1

2 ⌋d+1

≤ e−γ′E−d

2 , (2.127)

which gives the desired upper bound in (2.104).

To complete the proof we have to prove Lemma 2.24. It is basically a large deviationestimate but for the fact that the random variables V l

ω depend on l, which in turn implicitlydepends on E.

Proof of Lemma 2.24 ([1]).

By the choice of l we can estimate

P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

≤ P

1

|Λl|∑

x∈Λl

V lω(x) < 2β2l−2

≤ P

(

#

x | V lω(x) <

c

3l−2

≥(

1− 6β2

c

)

|Λl|)

, (2.128)

where c is the constant from (2.119). The second inequality follows from the fact that if less

than (1 − 6β2

c )|Λl| of the V lω(x) are below c

3 l−2, then more than 6β2

c |Λl| of them are at leastc3 l

−2. In this case

1

|Λl|∑

x

V lω(x) ≥

1

|Λl|6β2

c|Λl|

c

3l−2 = 2β2l−2. (2.129)

Since P(V lω(x) > 0) > 0 by (2.103), there is γ > 0 such that q := P(V l

ω(x) < γ) < 1. Therefore,we can define the new random variables

ξx =

1, if V lω(x) < γ

0, otherwise.(2.130)

The ξx are independent and identically distributed with E[ξx] = q. Let r = 1− 6β2

c and chooseβ small enough such that q < r < 1. For l sufficiently large, we can continue our estimate:

P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

≤ P

(

#

x | V lω(x) <

c

3l−2

≥ r|Λl|)

≤ P

(

#

x | V lω(x) < γ

≥ r|Λl|)

≤ P

1

|Λl|∑

x∈Zd

ξx ≥ r

. (2.131)


We arrived at a standard large deviation problem. Using the inequality

P(X > a) ≤ e−taE[etX ] for t ≥ 0, (2.132)

we obtain

P

1

|Λl|∑

x∈Λl

V lω(x) < 2E

≤ e−|Λl|rtE

∏

x∈Λl

etξx

= e−|Λl|(rt−lnE[etξ0 ]). (2.133)

Finally, we have to show that we can choose t such that f(t) := rt − lnE[etξ0 ] > 0. Wecompute

f ′(t) = r − E[ξ0etξ0 ]

E[etξ0 ]. (2.134)

Since f ′(0) = r− q > 0 and f(0) = 0, there is a t > 0 with f(t) > 0. This completes the proofof the lemma.

Chapter 3

Random band matrices

In this chapter we are going to generalize the Anderson model by allowing off-diagonal dis-order. We introduce random jump rates that are expressed by a random band matrix andwe will treat it similarly as the on-site potential is treated in the random Schrodinger case.For short we will call it potential. Therefore, we will consider two kinds of random bandmatrix models, namely Wigner band matrices and random banded covariance matrices. Wewill define these matrix ensembles in Section 3.1 and 3.2 in a form which is convenient for thefurther treatment. In Section 3.3 we will define the Anderson band matrix model, a randomSchrodinger operator where the potential is a random band matrix making it an intermediatemodel between random Schrodinger theory and random matrix theory. Afterwards we are go-ing to use the general theory developed in Chapter 2 to derive some immediate consequencesof these definitions. The material covered here is taken from [6] and [7].

To be consistent with the notation we will denote the matrix entries of an N ×N matrixM by M(x, y) for x, y = 1, . . . , N . More generally, for an operator M : Zd → Z

d we define itsmatrix entries as M(x, y) = 〈δx,Mδy〉 for x, y ∈ Z

d.

3.1 Wigner band matrices

One of the most important examples of random matrices are the Wigner matrix ensembles.LetM = (M(x, y))Nx,y=1 be an N×N Hermitian matrix where the matrix elements M(x, y) =

M(y, x), x ≤ y, are independent random variables with zero mean, i.e.

EM(x, y) = 0 for x, y = 1, 2, . . . , N. (3.1)

Let the variancesσ(x, y)2 = E|M(x, y)|2 = σ(x, y)2. (3.2)

satisfy the following normalization condition for any fixed y

∑

x

σ(x, y)2 = 1. (3.3)

Definition 3.1. An N × N Hermitian random matrix M = (M(x, y))Nx,y=1 is called gener-alized Wigner matrix (ensemble) if the matrix elements are centered (3.1), their variancesσ(x, y)2 = E|M(x, y)|2 satisfy (3.3) and M(x, y) | x ≤ y are independent. In the caseσ(x, y)2 = 1

N , we call the ensemble Wigner matrices.

26 3. Random band matrices

The celebrated Wigner semicircle law is essentially the non-commutative analogue to thecentral limit theorem.

Theorem 3.2. Let MN be an N ×N Wigner matrix. Then the empirical eigenvalue distri-bution νN = 1

N

∑

i δλi , where λi’s are the eigenvalues of MN , converges weakly, in probabilityto the semicircle distribution with density

ρsc(x) =1

2π

√

4− x2χ|x|≤2, (3.4)

i.e. for all ε > 0 and for all f ∈ Cb(Z), the space of bounded and continuous functions on thereal line, we have that

limN→∞

P

(∣

∣

∣

∣

∫

fdνN −∫

fdρsc

∣

∣

∣

∣

> ε

)

= 0. (3.5)

In comparison to the Anderson model, the Wigner matrices model a mean-field systemwith random transition rates rather then the short range transitions with one-site random-ness of the Anderson model. An intermediate model is given by the random band matriceswhich we are going to consider. As above, let B = (B(x, y))Nx,y=1 be an N × N Hermitian

matrix where the matrix elements B(x, y) = B(y, x), x ≤ y, are independent and identicallydistributed random variables with zero mean. For W =W (N) ∈ N let the variances satisfy

σ(x, y)2 =

1W , if |x− y| ≤W

0, if |x− y| > W.(3.6)

We call W the bandwidth of the band matrix B. As a consequence of this definition thematrix entries B(x, y) will be zero for |x− y| > W .

The generalization to Λ ⊂ Zd for some d ≥ 1 is straightforward. We simply replace | · |

by the distance ‖ · ‖1 on the graph and normalize the variances correctly. I.e. we call alinear Hermitian operator B : Λ → Λ a Wigner band matrix, when the entries B(x, y) areindependent (apart from the necessary dependence due to the Hermitian structure), centeredrandom variables whose variances

E|B(x, y)|2 = σ(x, y)2 = σ(y, x)2 (3.7)

satisfy

σ(x, y)2 =

1W d , if ‖x− y‖1 ≤W

0, if ‖x− y‖1 > W.(3.8)

3.2 Random banded covariance matrices

The other important class of random matrices which we are going to consider are the randomcovariance matrices.

Definition 3.3. A random covariance matrix M is a matrix of the form

M = B∗B, (3.9)

where B is an N × N matrix with centered independent and identically distributed entrieswith variances

E|B(x, y)|2 = N−1. (3.10)

3.3 The Anderson band matrix model 27

In this case the entries of M are not independent but they are generated by the entries ofB in a straightforward way. Furthermore, the eigenvalues of M are the singular values of B.The counterpart of the semicircle law is the Marchenko-Pastur law:

Theorem 3.4. The empirical eigenvalue distribution of a random covariance matrix con-verges weakly, in probability to the distribution with density

ρmp(x) =1

2π

√

4− x

xχ[0,4](x). (3.11)

As before, we are interested in a band matrix model of this random matrix ensemble.Therefore, we will relax the condition (3.10), namely we will assume that B is a band matrixof bandwidth W/2, i.e. such that the entries B(x, y) of B vanish for |x− y| > W/2 for someeven W > 0 and that

E|B(x, y)|2 =W−1, (3.12)

for |x − y| ≤ W . As a result, the covariance matrix M will also be a band matrix withbandwidth W , where W is even. The generalization to arbitrary dimensions d ≥ 1 can bedone as for the Wigner band matrices. We will call this band matrix ensemble the bandedcovariance matrices.

3.3 The Anderson band matrix model

We will now define our working model, which we will be concerned with for the rest of thethesis.

Definition 3.5. (Anderson band matrix model) Let L > 0 and ΛL ⊂ Zd. We define two

different models of random Schrodinger operators HL on ℓ2(ΛL), namely

1. H(1)L = −∆L +B, where B is a Wigner band matrix,

2. H(2)L = −∆L +M , where M is a banded covariance matrix.

In both cases ∆L is the graph Laplacian with Neumann boundary conditions. We will callthese models Anderson band matrix models. Furthermore, we say that the bandwidth W isfixed, if W is independent of L (in particular W = O(1) as L→ ∞).

Since M is a covariance matrix, there is a matrix B such that M = B∗B which contains

all informations about M . Therefore we will write H(2)L = −∆+B∗B. We will also drop the

indices (1) and (2) and write HL if it is obvious which operator we talk about.

To be able to work with this new matrix model we are going to determine some of itsproperties. We will focus on the case that the bandwidth is fixed. In this situation, we candirectly define an operator H on ℓ2(Zd) to which HL will converge as L→ ∞, namely

(Hψ)(x) =∑

‖x−y‖≤WH(x, y)ψ(y) =

∑

‖k‖≤WH(x, x+ k)ψ(x + k). (3.13)

The following Proposition tells us that the theory we developed in Chapter 2 can also beapplied to the Anderson band matrix model.


Proposition 3.6. If H is a Anderson band matrix model with fixed bandwidth then it is anergodic operator. In particular, the spectrum is almost surely deterministic and the density ofstates is well-defined for this operator.

Proof. As for the Anderson model we will consider the canonical realization of the independentand identically distributed matrix entries of the potential. We will show that the Andersonband matrix model with fixed bandwidth is an ergodic operator. In this case, the conclusionthat the spectrum is almost surely deterministic is just an application of Theorem 2.7. Toshow ergodicity, let B be the operator defined by

(Bωψ)(x) =∑

‖k‖≤WBω(x, x+ k)ψ(x+ k), (3.14)

where the matrix entries are realized on the canonical probability space Ω = RZd × R

Zdby

B(x, y) : RZd × RZd → R

ω 7→ Bω(x, y) := ω(x,y). (3.15)

On Ω we define the shifts for x, y, z ∈ Zd

Tz : Ω → Ω, (Tzω)(x,y) = ω(x−z,y−z). (3.16)

Analogous to the Anderson model one can show that these shifts are an ergodic family. Alsothe family of random variable B(x, y) is ergodic since for x, y, z ∈ Z

d and ω ∈ Ω

BTzω(x, y) = (Tzω)(x,y) = ω(x−z,y−z) = Bω(x− z, y − z). (3.17)

Now we define the unitary operators Ux : ℓ2(Zd) → ℓ2(Zd) for x ∈ Zd by

(Uxψ)(y) = ψ(x− y), (3.18)

for y ∈ Zd. We calculate the behavior of the potential under Ux

(UyBωU∗yψ)(x) = (BωU

∗yψ)(x − y)

=∑

‖k‖≤WBω(x− y, x− y + k)ψ(x+ k)

=∑

‖k‖≤WBTyω(x, x+ k)ψ(x + k)

= (BTyωψ)(x). (3.19)

This shows that B is ergodic. Since the Laplacian is also invariant under Ux we conclude thatalso H is ergodic.

In the case of the Wigner band matrix we will assume for simplicity that d = 1, i.e. weare considering one-dimensional lattice. Further we will assume that the matrix entries arebounded, i.e. there is C > 0 such that

B(x, y) ∈[

− C√W,C√W

]

, (3.20)


for all x, y ∈ ΛL. Then, we can estimate

‖Bψ‖2ℓ2(Λl)≤

∑

x

(

∑

y

|B(x, y)||ψ(y)|)2

≤ C2

W(2W + 1)

∑

x

∑

y|x−y|≤W

|ψ(y)|2

≤ C2

W(2W + 1)2‖ψ‖2, (3.21)

where we used in the second line that the matrix entries are bounded by (3.20) and the factthat the matrix is banded with bandwidth W , i.e. there are (2W + 1) non-vanishing entriesin each row.

This calculation shows that σ(−∆L+B) ⊂ [− (2W+1)C√W

,− (2W+1)C√W

+4], where the additional

summand of 4 is due the the Laplacian. If we assume that the matrix entries B(x, y) can bearbitrarily close to the lower boundary with a positive probability, then we will also have

inf σ(−∆L +B) → −(2W + 1)C√W

, (3.22)

for L→ ∞. This is the statement of the next theorem.

Theorem 3.7. Let d = 1 and let B be a Wigner band matrix with fixed bandwidth W and

with bounded entries, i.e. B(x, y) ∈[

− C√W, C√

W

]

for some C > 0. Assume that for all δ > 0

and for all x, y ∈ ΛL

P

(

B(x, y) < − C√W

+ δ

)

> 0. (3.23)

Then

inf σ(−∆L +B) → −(2W + 1)C√W

(3.24)

as L→ ∞.

Proof. As in the proof of Proposition 3.6 we are working on the canonical probability spaceand assume that the operators are defined on the whole space ℓ2(Z). We will show that wecan make the kinetic energy arbitrarily small while the potential energy is almost sure veryclose to the lower boundary of the spectrum specified in (3.24).

Let ε > 0 and let l, a ∈ N. Define the function ψ ∈ ℓ2(Z) given by ψ(x) = 1 for all x ∈ Λl,

ψ(x) = l+a−‖x‖∞a for all x ∈ Λl+a and ψ(x) = 0 otherwise. We can estimate its norm from

below by

‖ψ‖2 =∑

x∈Λl

|ψ(x)|2 +∑

x∈Λl+a\Λl

|ψ(x)|2

≥ |Λl|+∑

x∈Λl+a/2\Λl

|ψ(x)|2

≥ |Λl|+1

4|Λl+a/2 \ Λl|, (3.25)


where we used in the third line that ψ(x) ≥ 12 on the set Λl+a/2\Λl. Furthermore, we calculate

〈ψ,−∆ψ〉 =1

2

∑

x∈Λl+a+1\Λl−1

∑

y:‖x−y‖1=1

|ψ(x) − ψ(y)|2

=1

a2|Λl+a+1 \ Λl−1| , (3.26)

where we used in the second line that |ψ(x)−ψ(y)| = 1a by construction of ψ on the specified

set and that every point has 2 neighbors. Combining these two estimates we get

〈ψ,−∆ψ〉‖ψ‖2 ≤ 1

a2|Λl+a+1 \ Λl−1|

|Λl|+ 14 |Λl+a/2 \ Λl|

. (3.27)

Using that |Λl| = 2l + 1 this can be simplified to

〈ψ,−∆ψ〉‖ψ‖2 ≤ 1

a28a

8l + a+ 4. (3.28)

The fraction on the right hand side is monotone decreasing in l. We choose l > a. Then8a

8l+a+4 ≤ 1 and we arrive at

〈ψ,−∆ψ〉‖ψ‖2 ≤ 1

a2. (3.29)

We choose a big enough such that

〈ψ,−∆ψ〉‖ψ‖2 ≤ ε

2, (3.30)

but at least a > W .

Now we estimate the potential. We are going to look for large areas where the potentialis small, i.e. we are considering the event

Ωδ :=

ω ∈ Ω

∣

∣

∣

∣

∃j ∈ Z : supx,y∈supp ψ+ΛW

|B(x+ j, y + j) +C√W

| < δ

. (3.31)

It has a positive probability since

P(Ωδ) = P

(

∀x, y ∈ supp ψ +ΛW : B(x, y) < − C√W

+ δ

)

≥ P

(

B(0, 0) < − C√W

+ δ

)|supp ψ+ΛW |

> 0, (3.32)

where we used the independence of the matrix entries in the second line and the assumption(3.23) in the third line. Moreover, Ωδ is invariant under the shifts on the canonical probabilityspace defined in (3.16). By ergodicity we conclude that P(Ωδ) = 1.

By construction, for every ω ∈ Ωδ we can find j ∈ Z such that each matrix entry B(x+j, y+ j) < − C√

W+ δ =: bδ for x, y ∈ supp ψ+ΛW . We define a shifted version of the function


ψ by ψ(j)(x) = ψ(x− j). Then we can estimate the potential by

〈ψ(j), Bψ(j)〉 =∑

x,y

ψ(j)(x)B(x, y)ψ(j)(y)

< bδ

∑

x∈Λl−W

∑

y∈Z‖y−x‖1≤W

ψ(j)(x)ψ(j)(y) +∑

x∈Λl+a\Λl

∑


ψ(j)(x)ψ(j)(y)

= bδ

|Λl−W |(2W + 1) +∑

x∈Λl+a\Λl

∑


ψ(j)(x)ψ(j)(y)

, (3.33)

where used that B(x, y) < bδ on the specified set in the second line and the definition of ψ(j)

in the third line. We further estimate the second term∑

x∈Λl+a\Λl

∑


ψ(j)(x)ψ(j)(y)

≥∑

x∈Λl+a−W \Λl

ψ(j)(x)(2W + 1)

(

ψ(j)(x)− W

a

)

≥ 1

4(2W + 1)|Λl+a/2 \ Λl| − (2W + 1)

W

a

1

2|Λl+a/2 \ Λl|

=1

2(2W + 1)|Λl+a/2 \ Λl|

(

1

2+W

a

)

, (3.34)

where we used that ψ(j)(y) ≥ ψ(j)(x)− Wa on the specified set in the second line and the same

estimate as in (3.25) for the sums in the third line. In total we get

〈ψ(j), Bψ(j)〉‖ψ(j)‖2 ≤ bδ(2W + 1)

|Λl−W |+ 12(2W + 1)|Λl+a/2 \ Λl|

(

12 + W

a

)

|Λl|+ 12 |Λl+a \ Λl|

= bδ(2W + 1)

[

l −W + 12

l + 12 + a

4

+a4

(

12 +

Wa

)

l + 12 +

a4

]

. (3.35)

For l → ∞, the terml−W+ 1

2

l+ 12+ a

4

converges to 1 from below, whilea4(

12+W

a )l+ 1

2+ a

4

converges to 0 from

above. I.e. for any ε > 0 we can choose l such that

l −W + 12

l + 12 + a

4

+a4

(

12 +

Wa

)

l + 12 +

a4

> 1− ε. (3.36)

Then

〈ψ(j), Bψ(j)〉 ≤ −(2W + 1)C√W

+(2W + 1)C√

Wε+ δ(1 − ε). (3.37)

We choose ε and δ small such that

(2W + 1)C√W

ε+ δ(1 − ε) <ε

2. (3.38)


This is possible because choosing ε small amounts to choosing l large and δ > 0 was arbitrary.Consequently

〈ψ(j), Bψ(j)〉 ≤ −(2W + 1)C√W

+ε

2(3.39)

and we finally arrive at the estimate

〈ψ(j), (−∆+B)ψ(j)〉 < −(2W + 1)C√W

+ ε (3.40)

for every ε > 0, where we used that the Laplacian is invariant under shifts, i.e.

−∆ψ = −∆ψ(j). (3.41)

Since

inf σ(−∆+B) = infψ∈ℓ2(Z)

〈ψ, (−∆+B)ψ〉 ≤ 〈ψ(j), (−∆+B)ψ(j)〉, (3.42)

and σ(−∆L +B) ⊂ [− (2W+1)C√W

,− (2W+1)C√W

+ 4] the claim follows.

In the case of random banded covariance matrices, we have for ψ ∈ ℓ2(ΛL)

〈ψ,B∗Bψ〉 = ‖Bψ‖2 ≥ 0. (3.43)

Hence, σ(−∆L + B∗B) ⊂ [0,∞). On the other hand, if the entries of B can be arbitrarilyclose to 0 with a positive probability, then inf σ(−∆L + B∗B) → 0 as L → ∞ which is theresult of the following theorem.

Theorem 3.8. Let B∗B be a random banded covariance matrix with fixed bandwidth W .Assume that for all δ > 0

P (|B(x, y)| < δ) > 0, (3.44)

for all x, y ∈ ΛL. Then

inf σ(−∆L +B∗B) → 0, (3.45)

as L→ ∞.

Proof. The proof is analogous to the proof of Theorem 2.8. Again we are working on thecanonical probability space and assume that the operators are defined on ℓ2(Zd). Let ε > 0.Since 0 ∈ σ(−∆) there is ψε with ‖ψε‖ = 1 such that

‖ −∆ψε‖ <ε

2(3.46)

by the Weyl criterion. We may assume that ψε ∈ ℓ20(Zd). We set ψ

(j)ε (x) = ψε(x − j). Now

define the event

Ωδ :=

ω ∈ Ω∣

∣

∣∃j ∈ Z

d : ‖B∗B1supp ψ

(j)ε‖ < δ

. (3.47)

This event has a positive probability. Indeed, if all |B(x, y)| < δ on supp ψ(j)ε , then

|B∗B(x, y)| ≤∑

z

|B(z, x)||B(z, y)| ≤ (2W + 1)δ2. (3.48)


Hence, by (3.21), ‖B∗B1supp ψ

(j)ε‖ ≤ (2W + 1)2δ2. If we choose δ small enough we get

‖B∗B1supp ψ

(j)ε‖ < δ. But the probability that all |B(x, y)| < δ on supp ψ

(j)ε is positive which

follows directly from the assumption (3.44).Hence, P(Ωδ) > 0 and Ωδ is clearly invariant under the shifts on the canonical probability

space defined in (3.16). By ergodicity, we conclude that P (Ωδ) = 1 for all δ > 0.Let ω ∈ Ωδ. By construction there is j ∈ Z

d such that ‖B∗B1supp ψ

(j)ε‖ < δ. Hence,

choosing δ = ε2 we can estimate

‖(−∆+B∗B)ψ(j)ε ‖ ≤ ‖ −∆ψ(j)

ε ‖+ ‖B∗Bψ(j)ε ‖

≤ ε

2+ ‖B∗B1

supp ψ(j)ε‖‖ψ(j)

ε ‖≤ ε. (3.49)

Setting ε = 1n for n ∈ N we see that ψ

(j)1n

is a Weyl sequence for −∆+ B∗B and the value 0.

On the other hand −∆+B∗B ≥ 0 and hence inf σ(−∆+B∗B) = 0.


Chapter 4

Lifshitz tails for the Anderson band

matrix model

In this chapter we study the behavior of the integrated density of states near the bottom ofthe spectrum of the Anderson band matrix model defined in the previous chapter with fixedbandwidth. Our goal is to prove Lifshitz tails for the Anderson band matrix model wherethe potential is a Wigner band matrix. We will also present a localization formula for bandmatrices which we use to derive a finite-volume bound on the integrated density of states forthe Anderson band matrix model where the potential is a banded covariance matrix.

4.1 Bounds on the density of states

The starting point in the proof of Lifshitz tails for the Anderson model were the finite-volume bounds on the integrated density of states in Lemma 2.20. The proof solely relied onthe Neumann-Dirichlet bracketing and this was feasible since the potential in the Andersonmodel is local - it is a diagonal matrix. Since this is not the case anymore we would liketo have a substitute for this result. As usual, the lower bound can be established easily byapplying a variational argument, which is valid for both band matrix ensembles.

Lemma 4.1. (Lower bound on the DOS) Let L > 0 and HL = −∆L + B be a randomSchrodinger operator, where B is either a Wigner band matrix or a random banded covariancematrix. Then, we have for all L > l > 0

1

|ΛL|E[N(HL, E)] ≥ c

|Λl|E[N(HD

l , E)], (4.1)

where c > 0 is a constant only depending on W and HDl is the operator HD

l = −∆Dl +B on

Λl.

Proof. For a lower bound on the DOS we need an upper bound on the operator HL. Asdepicted in Figure 4.1 we divide the box ΛL into smaller boxes Λl,j of size l plus a securitydistance such that for i 6= j

(Λl,j + ΛW ) ∩ Λl,i = ∅, (4.2)

where ΛW is the box at 0 with side length W . This assures that the boxes Λl,j are separatedby a distance of W and we define Λ(s) := ΛL \⋃j Λl,j. We will refer to the separation zone

36 4. Lifshitz tails for the Anderson band matrix model

Figure 4.1: Dividing ΛL into smaller boxes plus security distance

Λ(s) as the security distance of the boxes Λl,j. It ensures, that the operator HL on the boxΛl,j is independent of all other boxes Λl,i for i 6= j. Define the operator HL on

D(HL) =

ψ ∈ ℓ2(ΛL) | ψ(x) = 0 for x ∈ Λ(s)

(4.3)

by

HL = −⊕

j

∆Dl,j +B. (4.4)

We want to compare the eigenvalues of HL and HL. Therefore, let φ1, . . . , φn ∈ ℓ2(ΛL),where n < dim(D(HL)). Then

infψ∈span(φ1,...,φn)

〈ψ,HLψ〉 ≤ infψ∈span(φ1,...,φn)

ψ∈D(HL)

〈ψ, HLψ〉. (4.5)

Taking supφ1,...,φn on both sides we get by the min-max-principle

En(HL) ≤ En(HL). (4.6)

Now we extend HL to the security zone in such a way such that HL is bigger than HL inoperator sense. We will denote the extension also by HL. Since ‖B‖ = O(1) there exists aC > 0 such that ‖HL‖ < C. We define

HL(x, x) = C for all x ∈ Λ(s). (4.7)

and for x, y ∈ ΛL with x 6= yHL(x, y) = 0, (4.8)

if either x or y or both are in Λ(s). Then by the min-max principle

HL ≤ HL, (4.9)

4.1 Bounds on the density of states 37

and hence, for all E ∈ R,N(HL, E) ≥ N(HL, E) (4.10)

and thereforeE[N(HL, E)] ≥ E[N(HL, E)]. (4.11)

Therefore, by definition of HL we get for E < C

E[N(HL, E)] = E[N(⊕jHil , E)]

= E

∑

j

N(Hjl , E)

= c|ΛL||Λl|

E[N(Hl, E)], (4.12)

where Hjl is the operator Hl on the j-th box Λl,j. Letting C → ∞ we get the conclusion for

all E ∈ R.

In the case of random banded covariance matrices we will also derive an upper bound onthe density of states. Therefore, we will prove a suitable localization formula to restrict thepotential to independent boxes with an affordable error. The key lemma is the following.

Lemma 4.2. Let M be a band matrix with bandwidth W and χk : ΛL → [0, 1], k = 1, . . . , N ,functions such that

∑

k

χk(x)2 = 1 for all x ∈ ΛL. (4.13)

We think of the χk as localization functions to subsets of ΛL and define the correspondingmultiplication operators Xk by

Xk(x, y) := δxyχk(x). (4.14)

If the localization functions satisfy

|χk(x)− χk(y)| ≤W

l(4.15)

for some l > 0, then

M ≥∑

k

XkMXk −(

W

l

)2

supx

∑

y

|M(x, y)|. (4.16)

Proof. We can estimate M in the following way:

〈ψ,Mψ〉 =∑

x,y

ψ(x)M(x, y)ψ(y)

=∑

x,y

∑

k

1

2

(

χ2k(x) + χ2

k(y))

ψ(x)M(x, y)ψ(y)

=∑

x,y

ψ(x)

(

∑

k

χk(x)M(x, y)χk(y)

)

ψ(y)

−1

2

∑

x,y

ψ(x)∑

k

χk(x)M(x, y) (χk(y)− χk(x))ψ(y)

−1

2

∑

x,y

ψ(x)∑

k

(χk(x)− χk(y))M(x, y)χk(y)ψ(y). (4.17)


The first sum at the right hand side of (4.17) is already the one appearing in (4.16). Theother two sum form the error term which we can write as

−1

2

∑

x,y

ψ(x)∑

k

χk(x)M(x, y) (χk(y)− χk(x))ψ(y)

−1

2

∑

x,y

ψ(x)∑

k

(χk(x)− χk(y))M(x, y)χk(y)ψ(y)

= −1

2

∑

x,y

ψ(x)M(x, y)ψ(y)∑

k

[

χk(x) (χk(y)− χk(x)) + (χk(x)− χk(y))χk(y)]

=1

2

∑

x,y


k

(χk(y)− χk(x))2 . (4.18)

We need to estimate this quantity from below. Using |χk(x) − χk(y)| ≤ Wl we can further

estimate

1

2

∑

x,y


k

(χk(y)− χk(x))2

≥ −(

W

l

)2∑

x,y

|ψ(x)||M(x, y)||ψ(y)|

≥ −2

(

W

l

)2∑

x,y

|ψ(x)|2|M(x, y)|

≥ −(

W

l

)2

supx

∑

y

|M(x, y)|, (4.19)

which yields the desired result.

We use this localization formula now to get an upper bound on the integrated density ofstates for random banded covariance matrices. The precise statement is the following.

Lemma 4.3. Let HL = −∆L +B∗B be the Anderson band matrix model on ΛL ⊂ Z, wherethe potential is a random covariance matrix with bandwidth W . For simplicity we will alsoassume that the matrix entries of B are bounded, i.e. there is some C > 0 such that

|B(x, y)| < C

W, (4.20)

for all x, y ∈ ΛL. Furthermore, denote by χl a localization function to a box Λl ⊂ ΛL, i.e.supp χl ⊂ Λl and χl = 1 on Λl−W ⊂ Λl. Then for all l > 0 and E > 0

N(E) ≤ 1

|Λl|E[N(−∆N

l + χlB∗Bχl, E +R(E))], (4.21)

where R(E) ≥ 0 and R(E) → 0 as E → 0 for l ≫W 1√E.

4.1 Bounds on the density of states 39

Figure 4.2: Localization functions

Proof. First, divide ΛL into boxes Λl,k of size 0 < l < L plus a security distance Λ(s), suchthat the boxes Λl,k have a distance of at least W from each other as in the proof of Lemma4.1. Now choose localization functions χk which satisfy the assumptions of Lemma 4.2 suchthat each χk localizes to a different box Λl,k as illustrated in Figure 4.2, i.e. supp χk ⊂ Λl,kand χk = 1 on the biggest possible subset of Λl,k without violating the assumptions of Lemma4.2. We can choose the χk in such a way that they are only shifted versions of one localizationfunction χl. The localization to the security zone Λ(s) will be denoted by χ(s). An applicationof Lemma 4.2 to M = B∗B with the localization functions defined above yields

B∗B ≥∑

k

χkB∗Bχk + χ(s)B∗Bχ(s) −

(

W

l

)2

supx

∑

y

|B∗B(x, y)|. (4.22)

Since χ(s)B∗Bχ(s) ≥ 0 we can drop this term and we are left with

B∗B ≥∑

k

χkB∗Bχk −

(

W

l

)2

supx

∑

y

|B∗B(x, y)|. (4.23)

The benefit of the security distance is that the matrices (χkB∗Bχk)k are independent of each

other.To estimate the error term we use that, by assumption, there is C > 0 such that for all

x, y ∈ ΛL

|B(x, y)| < C

W. (4.24)

Then for all x, y ∈ ΛL∑

y

|B∗B(x, y)| = O(1) (4.25)

as L→ ∞ and hence(

W

l

)2

supx

∑

y

|B∗B(x, y)| ∼(

W

l

)2

O(1). (4.26)


Since infλ|λ ∈ σ(HL) = 0, choosing l ≫ W 1√E, the error term will be of order o(E) for

E → 0. Hence, the term R :=(

Wl

)2supx

∑

y |B∗B(x, y)| satisfies the condition given in thelemma.

To estimate the Laplacian we use the Neumann-Dirichlet bracketing from Proposition 2.11on the boxes Λl,k and Λ(s) defined above, which yields

−∆L ≥⊕

k

(

−∆Nk

)

⊕ (−∆Ns ), (4.27)

where ∆Nk is the Neumann-Laplacian on the box Λl,k respectively ∆N

s on the set Λ(s). Butsince ∆s ≥ 0 we can again estimate this by

−∆L ≥⊕

k

(

−∆Nk

)

, (4.28)

where the right hand side is to be understood to vanish on Λ(s).

Putting things together we arrive at

E[N(HL, E)] ≤ EN

(

⊕

k

(

−∆Nk

)

+∑

k

χkB∗Bχk, E −R(E)

)

= E

∑

k

N(

−∆Nk + χkB

∗Bχk, E −R(E))

=|ΛL||Λl|

EN(

−∆Nk χlB

∗Bχl, E −R(E))

, (4.29)

where we used in the second line that the boxes Λl,k do not overlap and in the third line thatthe matrices on different boxes Λl,k are independent and that the χk are only shifted versionsof a common χl. Dividing both side by |ΛL| and letting L→ ∞ yields the desired result.

The approach for the upper bound to Lifshitz tails for Wigner band matrices is differentand does not require an upper bound on the integrated density of states of this kind.

4.2 Lifshitz tails for random banded covariance matrices

With the auxiliary statements from the previous section we are able to establish a lower boundon the integrated density of states for banded covariance matrices with fixed bandwidth. Theprecise statement is the following.

Theorem 4.4. Let H = −∆ + B∗B be the Anderson band matrix model, where B∗B is arandom banded covariance matrix with fixed bandwidth W . Assume that the matrix entriesB(x, y) of B have a common distribution P0 which satisfies for all ε ∈ (0, 1]

P0((−ε, ε)) ≥ cεk, (4.30)

for some c, k > 0. Then

limEց0

ln | lnN(E)|ln(E)

≥ −d2. (4.31)

4.2 Lifshitz tails for random banded covariance matrices 41

Proof. Using inequality (4.1), we get for any l > 0

N(E) ≥ c

|Λl|E[N(HD

l , E)]

≥ c

|Λl|P(E0(H

Dl ) < E). (4.32)

To get an upper bound on E0(HDl ) we use the min-max-principle applied to the tent

function ψ from equation (2.108), i.e.

ψ(x) =1

‖ψ‖ψ(x), (4.33)

whereψ(x) = l − ‖x‖∞, for all x ∈ Λl. (4.34)

Then, by (2.110) and (2.109)

〈ψ,−∆Dl ψ〉 ≤

c0l2, (4.35)

for some constant c0 and hence

E0(Hl) ≤ c0l2

+ 〈ψ0, B∗Bψ0〉

=c0l2

+ ‖Bψ0‖2

≤ c0l2

+ ‖B‖2. (4.36)

Choosing l =√

2c0E we get

N(E) ≥ c

|Λl|P(

‖B‖2 ≤ E/2)

=c

|Λl|P

(

‖B‖ ≤√

E/2)

≥ c

|Λl|P

(

∀x, y ∈ Λl : |B(x, y)| ≤ c1√E)

, (4.37)

where we used in the last inequality that, since W is fixed, there is c1 > 0 such that

P (‖B‖ < E/2) ≥ P (∀x, y ∈ Λl : |B(x, y)| ≤ c1E) . (4.38)

This follows from equation (3.21).Now we can proceed with the estimate of the integrated density of states. Since B(x, y)

are independent, we have for E small enough

N(E) ≥ c

|Λl|P

(

|B(x, y)| < c1√E)c2|Λl|

≥ c

|Λl|P0

(

(−c1√E, c1

√E))c2|Λl|

≥ c

|Λl|(

c(c1√E)k

)c2|Λl|

=c

|Λl|e|Λl|c3 ln(c4Ek/2). (4.39)


By our choice l ∼ E−1/2 we finally arrive at

N(E) ≥ c5Ed/2eE

−d/2c3 ln(c4Ek/2), (4.40)

which yields (4.31).

We expect that the reverse inequality in (4.31) can be established too.

4.3 Lifshitz tails for Wigner band matrices

In this section we prove Lifshitz tails for Wigner band matrices. As usual the proof is splitinto two parts - one for the lower and one for the upper bound. For the lower bound weuse Lemma 4.1 derived at the beginning of this chapter. To establish the upper bound wereduce the problem to a diagonal potential and apply Theorem 2.22 on Lifshitz tails for theAnderson model. The statement is the following.

Theorem 4.5. Let HL = −∆+B be the Anderson band matrix model, where B is a Wignerband matrix with fixed bandwidth W . For simplicity we assume that B is symmetric and thatd = 1. Assume that there is 0 ≤ b ≤ 2

√W such that

B(x, y) ∈[

− b√W,b√W

]

. (4.41)

Furthermore, assume that

P

(

B(x, y) ≤ − b√W

+ ε

)

≥ Dεk. (4.42)

for some D, k > 0 and all ε ∈ (0, 1], x, y ∈ Z. Then

limEցE0

ln | lnN(E − E0)|ln(E − E0)

= −d2, (4.43)

where E0 = − (2W+1)b√W

by equation (3.22).

Proof. We start again with the lower bound.

The lower bound

Applying Lemma 4.1, we have for all l > 0

N(E) ≥ c

|Λl|P(E0(H

Dl ) < E). (4.44)

As in the proof of Theorem 4.4 we choose the tent function ψ from equation (2.108) as testfunction. Then, we have 〈ψ,−∆D

l ψ〉 ≤ c0l2 for some constant c0 and, hence,

〈ψ, (−∆Dl +B)ψ〉 ≤ c0

l2+ 〈ψ,Bψ〉. (4.45)

4.3 Lifshitz tails for Wigner band matrices 43

Since we want to consider the asymptotics E → inf σ(H) = − b(2W+1)√W

, we rewrite E =

E0 + ε = − b(2W+1)√W

+ ε for some small ε > 0. Now we choose l =√

2c0ε . Then, we can further

estimate

N(E) ≥ c

|Λl|P

(

〈ψ,Bψ〉 < E0 + ε/2)

≥ c

|Λl|P

(

∀x, y ∈ Λl : B(x, y) < − b√W

+ε

2(2W + 1)

)

. (4.46)

The second inequality follows from the fact that if B(x, y) < − b√W

+ ε2(2W+1) for all x, y ∈ Λl,

then

∑

x,y

ψ(x)B(x, y)ψ(y) <

(

− b√W

+ε

2(2W + 1)

)

∑

x,y

ψ(x)ψ(y)

≤(

− b√W

+ε

2(2W + 1)

)

(2W + 1)∑

x

ψ(x)2

≤ E0 + ε/2. (4.47)

We continue the estimate

N(E) ≥(

∀x, y ∈ Λl : B(x, y) < − b√W

+ε

2(2W + 1)

)

≥ c

|Λl|P

(

B(0, 0) < − b√W

+ε

2(2W + 1)

)|Λl|

≥ c

|Λl|

(

D

(

ε

2(2W + 1)

)k)|Λl|

≥ c

|Λl|ec1|Λl| ln εk , (4.48)

with some constant c1 depending on W . By our choice l−2 ∼ ε we get that

N(E) ≥ cεd/2eε−d/2c2 ln ε = c(E − E0)

d/2e(E−E0)−d/2c2 ln(E−E0), (4.49)

from which the desired lower bound in (4.43) follows.

The upper bound

We sacrifice part of the kinetic energy to reduce the problem to a diagonal potential. Fix asmall 1 ≥ β ≥ 0. For ψ ∈ ℓ2(ΛL) we calculate

〈ψ, (−β∆L +B)ψ〉 = β∑

x,y

|ψ(x) − ψ(y)|2 +∑

x,y

ψ(x)B(x, y)ψ(y)

=∑

x

(

∑

y

B(x, y)

)

|ψ(x)|2

+∑

x,y

(

β − 1

2B(x, y)

)

|ψ(x) − ψ(y)|2. (4.50)


Choosing β ≥ b2√W

we get

〈ψ, (−β∆L +B)ψ〉 ≥∑

x

(

∑

y

B(x, y)

)

|ψ(x)|2. (4.51)

We define the diagonal potential V as

V (x) :=∑

y

B(x, y). (4.52)

Note, that the entries of V are not independent. We can further estimate V in the followingway

V (x) =∑

y

B(x, y) ≥∑

y≥xB(x, y)−W

b√W

=: V (x)− b√W, (4.53)

where V (x) defines a new diagonal potential V , whose entries are independent now, since byDefinition 3.1 the matrix entries B(x, y) | x ≤ y are independent. In total we get

−∆L +B ≥ (1− β)(−∆L) + V − b√W. (4.54)

We reduced the problem to a diagonal potential consisting of the entries of the band matrix.By Corollary 2.21 we get the following bound on the density of states for every l > 0

N(E) ≤ P

(

E0

(

(1− β)(−∆Nl ) + V − b

√W ))

< E)

. (4.55)

We are interested in the asymptotics E → −2W+1√W

b (which is, as it should be, also the infimum

of the spectrum of the ”new” diagonal operator V − b√W ). Rewriting E = − b(2W+1)√

W+ ε as

in the proof of the lower bound we get

N(E) ≤ P

(

E0

(

(1− β)(−∆Nl ) + V

)

< ε− bW + 1√W

)

≤ P

(

E0

(

(1− β)(−∆Nl ) + V + b

W + 1√W

)

< ε

)

. (4.56)

Since

V (x) + bW + 1√W

≥ 0 (4.57)

for any x ∈ Λl, we have V + bW−1√W

≥ 0. Now we are in the situation of the upper bound in

Theorem 2.22 for the operator (1− β)(−∆N ) + V + bW+1√W

. Using (2.127) we get

N(E) ≤ e−γ′ε−d/2

= e−γ′

(

E+b (2W+1)√W

)−d/2

(4.58)

for some γ′ > 0 and l−2 ∼ ε. This yields the desired exponential decay and thereby the upperbound in (4.43).

Bibliography

[1] W. Kirsch, An Invitation to Random Schrodinger operators, arXiv:0709.3707 [math-ph](2007)

[2] W. Kirsch, F. Martinelli, On the ergodic properties of the spectrum of general randomoperators, J. Reine Angew. Math. 334, 141-156 (1982)

[3] L.A. Pastur, Spectral properties of disordered systems in the one-body approximation, Com-mun. Math. Phys. 75, 179-196 (1980)

[4] M. Reed, B. Simon, Methods of modern mathematical physics. IV: Analysis of operators,New York - San Francisco - London: Academic Press (1978)

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Lifshitz tails for random band matrices

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