# Resonant delocalization for random Schr\"odinger operators on tree graphs

**ABSTRACT** We analyse the spectral phase diagram of Schr\"odinger operators $ T +\lambda

V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a

random potential given by iid random variables. The main result is a criterion

for the emergence of absolutely continuous (ac) spectrum due to

fluctuation-enabled resonances between distant sites. Using it we prove that

for unbounded random potentials ac spectrum appears at arbitrarily weak

disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the

spectrum of $T$. Incorporating considerations of the Green function's large

deviations we obtain an extension of the criterion which indicates that, under

a yet unproven regularity condition of the large deviations' 'free energy

function', the regime of pure ac spectrum is complementary to that of

previously proven localization. For bounded potentials we disprove the

existence at weak disorder of a mobility edge beyond which the spectrum is

localized.

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**ABSTRACT:**We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.Annales Henri Poincare 01/2012; 13(8):1745-1766. · 1.53 Impact Factor - SourceAvailable from: Matthias Keller[Show abstract] [Hide abstract]

**ABSTRACT:**We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.Journal d Analyse Mathématique 10/2012; 118(1):363-396. · 0.84 Impact Factor

Page 1

Resonant delocalization for random Schr¨ odinger

operators on tree graphs

Michael AizenmanSimone Warzel

Version of April 7, 2011

Abstract

We map the spectral phase diagram of Schr¨ odinger operators T + λV with un-

bounded random potentials V over regular tree graphs. The main result is a condition

for the existence of absolutely continuous spectrum which supplements a previously

derived criterion for pure-point spectrum. Using it, we show that under weak disor-

der (λ → 0) the regime of absolutely continuous spectrum spreads discontinuously

beyond the spectrum of the unperturbed operator T into a Lifshitz tail regime of very

low density of states. A relevant mechanism for the formation of extended states

there is the occurrence of rare fluctuation-enabled resonances between distant sites.

Keywords. Anderson localization, absolutely continuous spectrum, mobility edge,

Cayley tree

M. Aizenman: Depts. of Physics and Mathematics, Princeton University, Princeton NJ 08544, USA

S. Warzel: Zentrum Mathematik, TU M¨ unchen, Boltzmannstr. 3, 85747 Garching, Germany; e-mail:

warzel@ma.tum.de (corresponding author)

Mathematics Subject Classification (2010): Primary 82B44; Secondary 47B80.

1

arXiv:1104.0969v1 [math-ph] 5 Apr 2011

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Resonant delocalization2

Contents

1 Introduction

1.1 The article’s topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2Past results and the question settled here . . . . . . . . . . . . . . . . . .

3

3

3

2The main result

2.1The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 A criterion for extended states . . . . . . . . . . . . . . . . . . . . . . .

2.3Comparison with a localization criterion . . . . . . . . . . . . . . . . . .

2.4Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

5

6

8

9

3Basic properties of the Green function on tree graphs

3.1Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2Recursion and factorization . . . . . . . . . . . . . . . . . . . . . . . . .

10

10

11

4 The roadmap – Proof part I

4.1A zero-one law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2The key statement – proof of the main result . . . . . . . . . . . . . . . .

4.3A heuristic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

12

12

14

5The moment generating function

5.1Definition, monotonicity and finite-volume estimates . . . . . . . . . . .

5.2Super- and submultiplicativity estimates . . . . . . . . . . . . . . . . . .

5.3Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 Properties of the Lyapunov exponent . . . . . . . . . . . . . . . . . . . .

15

15

16

19

20

6Green function’s extremal fluctuations – Proof part II

6.1 Parameterization of the large-deviation events . . . . . . . . . . . . . . .

6.2The extreme resonance events. . . . . . . . . . . . . . . . . . . . . . .

6.3 The mean number of boosted resonance events

6.4Establishing the events’ occurrence . . . . . . . . . . . . . . . . . . . . .

6.5Proof of the key statement. . . . . . . . . . . . . . . . . . . . . . . . .

21

21

25

26

30

34

. . . . . . . . . . . . . .

A Fractional-moment bounds

A.1 Weak-L1bounds

A.2 Consequences of Assumption D . . . . . . . . . . . . . . . . . . . . . .

36

36

38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B A large deviation principle for triangular arrays

B.1 A large deviation theorem . . . . . . . . . . . . . . . . . . . . . . . . . .

B.2 Application – Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . .

40

40

43

C Lifshitz tails 44

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Resonant delocalization3

1Introduction

1.1The article’s topic

The subject of this work are the spectral properties of random self-adjoint operators in

the Hilbert space ?2(T ) associated with the vertex set T of a regular rooted tree graph of

branching number K > 1. The operators take the form

Hλ(ω) = T + λV (ω),

(1.1)

with T the adjacency matrix and V (ω) an unbounded random potential, i.e., a multiplica-

tion operator which is specified by a collection of random variables on T . The strength of

the disorder is expressed here through the parameter λ ≥ 0.

It is well known that random Schr¨ odinger operators, of which the above tree version

is a relatively more approachable example, exhibit regimes of spectral and dynamical

localization where the operator’s spectrum consists of a dense collection of eigenvalues

with localized eigenfunctions (cf. [9, 22, 26, 17]). However, it still remains an outstand-

ing mathematical challenge to elucidate the conditions for the occurrence of continuous

spectrum, and in particular absolutely continuous (henceforth called ‘ac’) spectrum, in the

presence of homogeneous disorder. Where such is found, the boundary separating contin-

uous spectrum from the regime of localization is referred to as the ‘mobility edge’ [8].

The result presented here answers a puzzle, which has been open since the earlier

works on the subject [1, 2], concerning the location of the mobility edge, and the nature of

the continuous spectrum below it for such operators on regular tree graphs. The result was

given a physics-oriented summary in [7]. As is recalled there, the answer to the question

was not viewed as unambiguous since the regime in which the ac spectrum is found here

includes regions of extremely low density of states of ‘Lifshitz tail’ asymptotics.

1.2Past results and the question settled here

The ‘phase diagram’ summarizing the spectral properties of the operators considered here

was studied already in the early works of Abou-Chacra, Anderson and Thouless [1, 2]. Ar-

guments and numerical work presented in [2] led the authors to surmise that for (centered)

unbounded random potentials, the mobility edge, which separates the localization regime

from that of continuous spectrum, exists at a location which roughly corresponds to the

outer curve in Figure 1. Curiously, for λ ↓ 0 that line approaches energies |E| = K + 1

which is not the edge of the spectrum of the limiting operator T which is given by:1

σ(T) = [−2√K,2√K].

Rigorous results for the above class of operators have established the existence of a

localization regime and, by different arguments, of regions of ac spectrum, leaving how-

ever a gap in with neither analysis applied. More specifically, the following was proven

(1.2)

1Even though the graph T is of constant degree (K + 1), except at the root, the spectrum of T does

not extend to [−(K + 1),(K + 1)]. This is related to the graph’s exponential growth, more precisely to

the positivity of its Cheeger constant. Nevertheless, the larger set does in fact describe the operator’s ?∞-

spectrum.

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Resonant delocalization4

Figure 1: A sketch of the previously known parts of the phase diagram. The outer region

is of proven localization, the smaller hatched region is of proven delocalization. The new

result extends the latter up to the outer curve, assuming ϕλ(1;E) = −logK holds only

along a line. The intersection of the curve with the energy axis is stated exactly, while in

other details the depiction is only schematic.

for the class of operators described above (under assumptions which are somewhat more

general than the conditions A-D below):

Localization regime [3, 4]: For a regime of energies |E| > γ(λ), with

lim

λ↓0γ(λ) = K + 1

(1.3)

(as depicted in Figure 1), with probability one the random operator exhibits spectral

and dynamical localization, at a finite localization length ξλ(E).

Extended states/continuous spectrum [18, 19, 5, 15]: Forenergies|E| < 2√K,atweak

enough disorder, |λ| <?λ(E) (with?λ(E) ↓ 0 for |E| → 2√K), the operator’s spec-

Spectral localization means that in the specified range of energies the operator has

only pure point spectrum, consisting of a dense set of non-degenerate proper eigenvalues

whose eigenfunctions are exponentially localized. The notion of dynamical localization

is explained in Definition 2.3 below.

Thus, the previous results have covered two regimes whose boundaries, sketched in

Figure 1, do not connect. Particularly puzzling has been the region of weak disorder, and

2√K < |E| < K + 1.

As was pointed out in [20], for λ ↓ 0 at those energies the mean density of states vanishes

to all orders in λ (see Appendix C for a precise statement). Such rapid decay is character-

istic of the so-called Lifshitz tail spectral regime, and in finite dimensions it is known to

lead to localization [22, 17]. On tree graphs however, this implication could not be estab-

lished, and localization at weak disorder was successfully proven [4] only for |E| > K+1

trum is almost surely (purely) ac.

(1.4)

Page 5

Resonant delocalization5

(cf. Figure 1 and Proposition 2.4 below). The conclusion of [20] was that for energies E

in the range (1.4) the nature of the spectrum for weak disorder, |λ| ? 1, presents a puzzle

even at the level of heuristics. The main result presented here answers this question.

The analysis is potentially of added interest as it presents a mechanism which does

not seem to have been discussed mathematically before: the formation of extended states

through disorder-enabled resonances. We expect this to be of relevance for disordered

operators also on other graphs where the volume is of exponential growth.

Let us now turn to a more precise statement of our main result.

2 The main result

2.1 The setup

Our discussion will focus on operators of the form (1.1) in the Hilbert space ?2(T ) of

complex-valued, square-summable functions on T , under the following assumptions:

A: T is the vertex set of a rooted tree graph with a fixed branching number K > 1 (the

root being denoted by 0 ∈ T ).

B: T is the adjacency operator of the graph, i.e., (Tψ)(x) :=?

C: {V (x;ω)|x ∈ T } form independent identically distributed (iid) random variables,

with a probability distribution ?(v)dv of continuous density, which is strictly positive

on the entire line R, and has a finite moment (ς ∈ (0,1)):

?

D: Moreover, ?(v) satisfies, for all v0∈ R:

c

ν

dist(x,y)=1ψ(y) for all

ψ ∈ ?2(T ).

|v|ς?(v)dv < ∞.

(2.1)

sup

|v−v0|≤ν?(v) ≤

?

1ν≤|v−v0|<2ν?(v)dv ,

(2.2)

at some uniform ν ∈ (0,∞) and c ∈ (0,∞),

While condition D could be relaxed, let us note that it is satisfied by all probability dis-

tributions whose densities are bounded functions on R of finitely many humps (see Ap-

pendix A). This class includes finite linear combinations of Gaussian, Cauchy, and the

piecewise constant functions.

For ergodic random potentials, a class which includes the iid case, the spectrum of

Hλ(ω) = T + λV (ω) is almost surely a non-random set [9, 22, 17]. Under the present

assumptions, it changes discontinuously from σ(T) at λ = 0 (see (1.2)), to the entire

real line R for λ ?= 0. Furthermore, ergodicity implies the finer statement that the differ-

ent components in the Lebesgue decomposition of the spectrum of Hλ(ω), that is, pure

point (pp), singular continuous (sc), and absolutely continuous (ac) spectrum, are also

Page 6

Resonant delocalization6

given, for almost every ω, by non random sets, some of which may be empty [9, 22, 17].

However, their determination requires a more delicate analysis which is the main point of

this paper.

Naively,onecouldexpectthatatleastintheregimesofeitherverylargeorverysmallλ

the spectrum of T + λV (ω) would resemble that of the dominant term. That, however, is

not quite the case. As is well known, in one dimension randomness has a non-perturbative

effect: even at weak level (|λ| ? 1) it causes complete localization and, in particular, only

pure point spectrum [16, 9, 22]. Somewhat conversely the result presented here shows that

on trees extended states and ac spectrum emerge, through resonances, in regimes where

at first sight one could expect localization to dominate.

2.2A criterion for extended states

The spectral analysis of random operators such as Hλ(ω) proceeds through the study of

the corresponding Green function

Gλ(x,y;ζ,ω) :=?δx,(Hλ(ω) − ζ)−1δy

where ζ ∈ C+:= {ζ ∈ C| Im ζ > 0} and δx∈ ?2(T ) is the Kronecker function local-

ized at x ∈ T . The information about the spectral measure of Hλ(ω) is encoded most

directly in the limiting value Gλ(x,y;E + i0,ω) := limη↓0Gλ(x,y;E + iη,ω). The

existence of this limit for almost every E ∈ R is implied by the theorem of de la Vall´ ee

Poussin, which requires just the self-adjointness of Hλ(ω). More specifically, the spectral

measure µλ,δx(·;ω) associated with Hλ(ω) and δx∈ ?2(T ) is related to the Green function

by the Stieltjes transformation,

?,

(2.3)

Gλ(x,x;ζ,ω) =

?

µλ,δx(dt;ω)

t − ζ

.

(2.4)

The density of the ac componentof µλ,δx(·;ω) is given byπ−1Im Gλ(x,x;E+i0,ω) ≥ 0.

A significant question for our problem is hence whether Gλ(x,x;E +i0,ω) is real or not.

An essential role in our discussion is played by the Green function’s moment generat-

ing function, which we define for s ∈ [−ς,1) and Lebesgue-almost all E ∈ R by:

log E[|Gλ(0,x;E + i0)|s]

ϕλ(s;E) := lim

|x|→∞

|x|

,

(2.5)

where |x| := dist(x,0) and E[·] denotes the average with respect to the underlying prob-

ability measure. The existence of the limit is proven below in Section 5, where we also

show that the function s ?→ ϕλ(s;E), which is obviously convex, is monotone decreas-

ing in s over [−ς,1). As a consequence, the limit at s = 1 is well-defined for almost

all E ∈ R:

ϕλ(1;E) := lim

s↑1ϕλ(s;E).

(2.6)

Our main result is the following criterion for ac spectrum:

Page 7

Resonant delocalization7

Theorem 2.1. For the random operator (1.1) satisfying Assumptions A–D, for any λ > 0

and Lebesgue-almost all E ∈ R at which

ϕλ(1;E) > −logK ,

the operator’s resolvent satisfies almost surely

(2.7)

Im Gλ(0,0;E + i0,ω) > 0.

(2.8)

The spectral implication of (2.8) was discussed above. As commented in [20, 6], this

condition is also of direct relevance for conduction: (2.8) implies that current fed coher-

ently through a wire can be conducted through the graph to infinity.

The proof of Theorem 2.1 reveals a mechanism for the formation of extended states

through rare fluctuation-enabled resonances between distant sites. A more detailed de-

scription is provided in Section 4 where a conditional proof is presented, subject to a

fluctuation analysis whose details are deferred to Section 6.

A sufficient condition for (2.7) which is particularly useful at weak disorder (and,

separately, also for high values of K) can be stated in terms of the Lyapunov exponent

Lλ(E) := −E(log|Gλ(0,0;E + i0)|),

(2.9)

Thanks to convexity ϕλ(s;E) ≥ −sLλ(E) (cf. Section 5), and hence the condition (2.7)

is implied by:

Lλ(E) < logK .

A simple exact calculation2shows that for λ = 0 one has

(2.10)

L0(E) < logK

if and only if

|E| < K + 1.

(2.11)

It is natural to expect Lλ(E) to be continuous in λ and E, a fact which is easily es-

tablished for the Cauchy random potential, i.e., for ?(v) = π−1(v2+ 1)−1(in which case

Lλ(E) = −log|G0(0,0;E + iλ)|). In such a situation the above two observations carry

the implication that any closed energy interval I in the range |E| < K + 1 is within the

regime of absolutely continuous spectrum at sufficiently weak enough disorder. In the

absence of a general continuity result, the following is of relevance here.

Corollary 2.2. Under the assumption of Theorem 2.1, for every closed interval I ⊂

(−K − 1,K + 1) in sufficiently low disorder, i.e. 0 < λ <?λ(I), the condition (2.7)

spectrum in I).

holds at a set of positive measure of energies (and thus there is absolutely continuous

The proof of Corollary 2.2 which is given below in Section 5 yields also an explicit

lower bound on the fraction of I occupied by ac spectrum.

2The Green function G0(0,0;ζ) of the adjacency operator is given by the unique value of Γ in C+which

satisfies the quadratic equation KΓ2+ ζ Γ + 1 = 0; cf. (3.5) below.

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Resonant delocalization8

2.3Comparison with a localization criterion

The significance of the condition (2.7) for ac spectrum may stand out better if one notes

that the opposite inequality implies localization. This is implied by the previously estab-

lished localization results [3, 4], which however have not been known to yield a sharp

criterion for operators on trees. Following is a definition of the various concepts of local-

ization (which extends to arbitrary metric graphs, not just trees).

Definition 2.3. The operator Hλ(ω) is said to exhibit spectral localization in an interval

I ⊂ R if the spectral measures µλ,δx(·;ω) associated to δx ∈ ?2(T ) are almost surely

all of only pure-point type in I. The operator is said to exhibit exponential dynamical

localization in I if for all x ∈ T and R > 0 sufficiently large:

?

dist(x,y)=R

y∈T :

E

?

sup

t∈R|?δx, PI(Hλ)e−itHλδy?|2

?

≤ Cλe−µλ(I)R,

(2.12)

at some µλ(I) > 0, and Cλ< ∞.

For a particle which is initially placed at x ∈ T the left side of (2.12) provides an

upper bound on the probability to be found a time t later at distance R from x, under

the quantum mechanical time-evolution generated by Hλrestricted to states with energies

in I. Of the two conditions, the dynamical localization is a stronger statement: by known

arguments (i.e., the Wiener and RAGE theorem, cf. [17, 26]) it implies also the spectral

localization.

The known localization results can be recast as follows, cf. Thm 1.2, and Eqs. (2.10),

(2.12) in Ref. [4].

Proposition 2.4. Let the random operator (1.1) satisfy Assumptions A–C. If, at a specified

λ > 0, the following condition holds for Lebesgue almost all E within an interval I ⊂ R,

ϕλ(1;E) < −logK − ε,

at some ε > 0, then the operator exhibits exponential dynamical localization in I, in the

sense of (2.12), with some µλ(I) > 0.

Furthermore, the domain in which (2.13) holds includes for each energy |E| > K +1

an interval with a positive range of λ > 0.

(2.13)

The relation of the condition (2.15), which encodes information about the decay of the

Green function, with the time evolution operator is explained by the following relation:

?

which holds for any s ∈ [0,1) and λ > 0 at some constant Cs,λ < ∞. The inequal-

ity (2.14) is a reformulation of a result of [4] on the eigenfunction correlator which was

extended in [23] so as to apply directly to infinite systems. (This relation holds in the

broader context of operators with random potential on arbitrary graphs.)

E

sup

t∈R|?δx, PI(Hλ)e−itHλδy?|2

?

≤ Cs,λ

?

I

E(|G(x,y;E + i0)|s) dE .

(2.14)

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Resonant delocalization9

One may add that if it is only known that for almost all E ∈ I

ϕλ(1;E) < −logK

then one may still conclude [3] that the operator has only pure point spectrum in I

(namely, by establishing liminfη↓0

(0,1) and all x ∈ T , and then invoking the Simon-Wolff criterion [25] instead of (2.14)).

(2.15)

?

y∈TE[|Gλ(x,y;E + iη)|s] < ∞ for some s ∈

2.4Further comments

1. The main result on ac spectrum, Theorem 2.1 (as well as the localization state-

ment Proposition 2.4) extend to the corresponding operator on the fully regular tree

graph B, where every vertex has exactly K+1 neighbors. The Green function of the

operator on B can be computed from the one on the rooted tree T with the help of

the recursion relation (3.3). In particular, this shows that the regime of ac spectrum

of the operator Hλ(ω) on T coincides with that on B.

2. At first sight the ?1-nature of the condition (2.7) for ac spectrum may be surprising

since – ignoring fluctuations – the loss of square summability seems to correspond

to an ?2-condition. The difference is due to the essential role played by extreme

fluctuations, cf. Section 4. The constructive effect of fluctuations here stands in

curious contrast to the fluctuation-reduction arguments which were employed to

prove stability under weak disorder of the ac spectrum for energies E ∈ σ(T) [18,

5, 15].

3. The conditions (2.7) for ac spectrum and (2.15) for localization are not fully com-

plementary since it was not yet proven that the equality ϕλ(1;E) = −logK holds

only along a curve in the phase diagram (as we expect it to be). To fully justify

this it will be good to see a proof that ϕλ(1;E) is differentiable in (λ,E) with only

isolated critical points.

4. A key observation driving our argument is that rare resonances, whose probabilities

of occurrence decay exponentially in the distance, may actually be found to occur

on all distance scales since the volume is also growing exponentially fast (provided

that rate exceeds the other). This causes the emergence of ac spectrum in energies

outside the spectrum of the adjacency operator, including in regimes of very low

density of states (Lifshitz tails).

5. The above mechanism is not applicable for graphs of finite dimension. However we

expectthatTheorem2.1mayadmitextensionstooperatorswithunboundedrandom

potentials on more general hyperbolic graphs, which may include loops, and also to

the analogous random operators on the Poincar´ e disk. Another setup which it will

be of interest to see analyzed are random operators on hypercubes of increasing

dimension, which form the configuration spaces of a many particle system.

Page 10

Resonant delocalization10

3 Basic properties of the Green function on tree graphs

3.1Notation

Analysis on trees, of this as well as of other problems, is aided by the observation that

upon the removal of any site x the tree graph splits into a collection of disconnected

components, which in case x is the root are isomorphic to the original graph. For different

problems on trees this leads to recursion relations in terms of suitably selected quantities

which we shall discuss in the present section. The following notation will facilitate the

formulation of such relations.

1. For a collection of vertices v1,...vnon a tree graph T we denote by Tv1,...vnthe

disconnected subgraph obtained by deleting this collection from T .

2. We denote by HT?, with T?⊂ T , the restriction of H to ?2(T?). E.g., HTv1,...vnis

the operator obtained by eliminating all the matrix elements of H involving any of

the removed sites.

3. The Green function, GT?(x,y;ζ), for a subgraph T?as above, is the kernel of the

resolvent operator (HT?− ζ)−1, with ζ ∈ C+. This function vanishes if x and y

belong to different connected components of T?, and otherwise it stands for the

Green function corresponding to the component which contains the two.

In particular: GTu(x,y;ζ) and GTu,v(x,y;ζ) are the Green functions for the subtree

which is obtained by removing u or, respectively u and v, and all the vertices which

are past the removed site(s) from the perspective of x and y.

4. Given an oriented simple path in T which passes through u ?= 0, we abbreviate

(assuming the path itself is clear within the context):

Γ(u;ζ) ≡ Γ−(u;ζ) := GTu−(u,u;ζ),

Γ+(u;ζ) = GTu+(u,u;ζ),

(3.1)

where u−and u+are the neighboring sites of u on that path. (The paths we shall

encounter below typically start at the root, of a rooted tree, and are oriented away

from it.) For the root 0, we will also use the convention

Γ(0;ζ) := G(0,0;ζ).

(3.2)

5. Any rooted tree T is partially ordered by the relation x ≺ y (resp. x ? y) which

means that x lies on the unique path from the root to y (possibly coinciding with y).

In order to ease the notation, we will drop the superscript on the Green function of the

full rooted tree, i.e., G(x,y;ζ) = GT(x,y;ζ). Moreover, we also drop the dependence of

various quantities on λ at our convenience.

Page 11

Resonant delocalization11

3.2Recursion and factorization

The following properties are part of the general folklore for spectral analysis on trees.

They can be derived by the resolvent identity, or alternatively through a random walk

representation of the Green function, cf. [1, 18, 5, 15].

Proposition 3.1. Let T be the vertex set of a tree graph (not necessarily a regular and

rooted one). Then, at the complex energy parameter ζ ∈ C+, the Green function of the

operator (1.1) satisfies:

1. For any x ∈ T :

G(x,x;ζ) =

?

λV (x) − ζ −

?

y∈Nx

GTx(y,y;ζ)

?−1

,

(3.3)

where Nx:= {y ∈ T | dist(x,y) = 1} denotes the set of neighbors of x.

2. For any pair of partially ordered sites, 0 ≺ x ≺ y,

?

where the ± subscripts on Γ are defined relative to the root.

We will use the following special cases and implication of the above relations:

G(x,y;ζ) = G(x,x;ζ)

x≺u?y

Γ−(u;ζ) = G(y,y;ζ)

?

x?u≺y

Γ+(u;ζ).

(3.4)

1. Denoting by N+

following recursion relation as a special case of (3.3):

0the set of forward neighbors of the root 0 in T , one obtains the

Γ(0;ζ) =

λV (0) − ζ −

?

y∈N+

0

Γ(y;ζ)

−1

(3.5)

2. Asaspecialcaseof(3.4),weconcludethattheGreenfunctionGλ(0,x;ζ)factorizes

into a product of the above variables, taken along the path from the root to x:

G(0,x;ζ) :=

?

0?u?x

Γ(u;ζ).

(3.6)

Moreover, denoting by x−the site preceding x from the direction of the root, (3.4)

also implies:

G(0,x;ζ) = GTx(0,x−;ζ)G(x,x;ζ).

More generally, for any triplet of sites {x,u,y} ⊂ T such that the removal of u

disconnects the other two:

(3.7)

G(x,y;ζ) = GTu(x,u−;ζ) G(u,u;ζ) GTu(u+,y;ζ)

where u−and u+are the neighboring sites of u, on the x and y sides, correspond-

ingly.

(3.8)

Page 12

Resonant delocalization12

4 The roadmap – Proof part I

Our main result, Theorem 2.1, is concerned with a condition under which for certain

energies: Im Γ(0;E + i0) > 0 almost surely. To better convey its essence we split the

proof into two parts. Part I, presented in this section, is a conditional derivation which

relies on a statement which plays an essential role, but whose proof is somewhat technical.

The more technical statement is then established, independently, in the next sections. We

start with some auxiliary observations.

4.1 A zero-one law

As a preparatory step it may be useful to note the following 0-1 law.

Lemma 4.1. For Lebesgue-almost all E ∈ R, the probability that Im Γ(0;E + i0) = 0

holds true is either 0 or 1.

Proof. Taking the imaginary part of (3.5) one gets:

Im Γ(0;ζ) = |G(0,0;ζ)|2?

η +

?

Im Γ(x;ζ),

x∈N+

0

Im Γ(x;ζ)

?

≥ |G(0,0;ζ)|2?

x∈N+

0

(4.1)

with equality in case ζ = E + i0 for those E for which the boundary values exist, that is

for Lebesgue-almost all E ∈ R. Let now q := P(Im Γ(x;E + i0) = 0) which does not

depend does on x. Since the K different terms, Im Γ(x;E+i0), x ∈ N+

variables of the same distribution as Im Γ(0;E + i0), and the factor |G(0,0;E + i0)| is

almost surely non-zero, we may conclude that q = qKor q [1 − qK−1] = 0, and hence

either q = 0 or q = 1.

0, are independent

Thus, in order to prove that for Lebesgue-almost all E ∈ R

P(Im G(0,0;E + i0) > 0) = 1,

(4.2)

it suffices to rule out the following ’no-ac’ hypothesis.

Definition 4.2. For a specified λ ≥ 0, we say that the no-ac hypothesis at E ∈ R holds if

almost surely Im Gλ(0,0;E + i0) = 0.

4.2The key statement – proof of the main result

Iterating (4.1) we conclude that for any n ∈ N and ζ ∈ C+:

Im Γ(0;ζ) ≥

?

x∈Sn

|G(0,x;ζ)|2?

y∈N+

x

Im Γ(y;ζ)

(4.3)

where Sn:= {x ∈ T |dist{0,x} = n}. This relation suggests that the no-ac hypothesis

is false if with uniformly positive probability there are sites x ∈ Snwhich have a forward

Page 13

Resonant delocalization13

neighbor y at which Im Γ(y;E + iη) is not particularly ‘atypical’ and for which at the

same time |G(0,x;ζ)| ? 1.

To turn the above observation into a proof we first introduce the following quantity.

Definition 4.3. For b > 0 and ζ ∈ C+the restricted upper percentile of the distribution

of Im Γ(0;ζ), which will be denoted ξ+

which

P(Im Γ(0;ζ) ≥ t

This quantity is well-defined for the following set of parameters.

b(α,ζ), is the supremum of the values of t ≥ 0 for

and

|Γ(0;ζ)| ≤ b) ≥ α.

(4.4)

Lemma 4.4. For ζ ∈ C+and any 0 < α < P(|Γ(0;ζ)| ≤ b):

Proof. For ζ ∈ C+one has 0 < Im Γ(0;ζ) < (Im ζ)−1. Hence the claim derives from

the following observations:

i. The collection of strictly positive values of t at which (4.4) holds is not empty, since

otherwise Im Γ(0;ζ) = 0 with probability at least P(|Γ(0;ζ)| ≤ b).

ii. The above collection of values of t does not include any value above (Im ζ)−1.

0 < ξ+

b(α,ζ) < ∞.

Now let Mn≡ Mn(ζ;α,b,δ) be the number of sites x ∈ Snat which the following

two conditions are satisfied:

?

|G(0,x;ζ)| ≥ eδ |x|.

The proof of our main result is based on the following statement, which is proven below

in Section 6.

y∈N+

x

Im Γ(y;ζ) ≥ ξ+

b(α,ζ)

(4.5)

(4.6)

Theorem 4.5. For almost all E ∈ R at which (2.7) and the no-ac hypothesis holds, there

are α,b,δ > 0 with α < infη>0P(|Γ(0;E + iη)| ≤ b), and there exist n ∈ N and η0> 0

(which depend on all the above parameters), at which

P(Mn(E + iη;α,b,δ) ≥ 1

for all η ∈ (0,η0).

The proof of Theorem 4.5 involves technical steps which only the more dedicated

reader may care to follow. Let us therefore first show how it is used for the proof of our

main result, and present a heuristic account of the reason for its validity.

and

|Γ(0;E + iη)| ≤ b) > α.

(4.7)

Proof of Theorem 2.1 – Given Theorem 4.5. If there is a site x ∈ Snat which both (4.5)

and (4.6) hold, then by (4.3)

Im Γ(0;E + iη) ≥ e2δ |x|ξ+

b(α,E + iη).

(4.8)

Thus, assuming the no-ac hypothesis and (2.7), Theorem 4.5 implies that (4.4) is valid for

t = eδnξ+

b(α,E + iη), or equivalently (by the definition of ξ+

b):

ξ+

b(α,E + iη) ≥ e2δnξ+

b(α,E + iη).

(4.9)

Page 14

Resonant delocalization14

This is a contradiction, unless ξ+

escape clauses are ruled out by Lemma 4.4. Therefore the no-ac hypothesis is invalidated

for energies at which (2.7) holds. By the 0-1 law of Lemma 4.1, it then follows that at the

given energy Im Γ(0;E + i0) > 0 almost surely.

b(α,E + iη) is either 0 or ∞. However, for η > 0 both

The following heuristic explanation of Theorem 4.5 provides a roadmap for its proof

in Section 6.

4.3A heuristic perspective

Condition(4.6)isarareevent.Apossiblemechanismforitisthesimultaneousoccurrence

of the following two events, at some common value of γ > 0:

|G(x,x;E + iη)| ≥ e(γ+δ)|x|

??GTx(0,x−;E + iη)??≥ e−γ |x|.

(4.10)

(4.11)

(We recall that, x−stands for the vertex preceding x relatively to the root.) These two

conditions imply (4.6) through the relation (3.7).

The first condition (4.10) represents an extremely rare local resonance condition. It

occurs when the random potential at x falls very close to a value at which Gλ(x,x;E+i0)

diverges. By (3.3), such divergence is possible only if GTx(y,y;E + i0) is real at all

y ∈ Nx. Hence, by (3.3) and the continuity of the probabilities in η, under the no-ac

hypothesistheprobabilityof(4.11)occurringatagivensitex ∈ Snisoftheordere−(γ+δ)n

for η sufficiently small (depending on n).

The second condition (4.11) represents a large deviation event, since typically

log|GTx(0,x−;E + iη)| ≈ −L(E)|x|.

(4.12)

By a standard large deviation estimate (which is fully derived below), the probability of

such an event, at γ < −lims↑1

function I(γ) which is related to ϕ(s) ≡ ϕλ(s;E) through the Legendre transform:

I(γ) = − inf

∂ϕ

∂s(s;E) =: ϕ?

−(1), is of the order e−nI(γ)+o(1)with a rate

s∈[0,1)[sγ + ϕ(s)].

(4.13)

The relevant mechanism for the occurrence of (4.11) is the systematic stretching of the

values of |GTx(0,u;E + iη)| along the path 0 ? u ? x−.

Unlike (4.10) and (4.11), the condition (4.5) and the one on |Γ(0,E +iη)| in (4.7) are

not rare events, and their inclusion does not modify significantly the above estimates.

By the above lines of reasoning, and ignoring excessive correlations (a step which is

justified under auxiliary conditions) we arrive at the mean value estimate:

E[Mn] ≈ Knexp(−n[I(γ) + γ + δ + o(1)]) ,

This value is much greater than one for some δ > 0, if

(4.14)

sup

γ

[logK − [I(γ) + γ)] > 0

(4.15)

Page 15

Resonant delocalization15

That is, although the probabilities of the two above events are exponentially small, given

the exponential growth of |Sn| = Kn, under suitable assumptions the mean number of

sites where the conditions occur is large, and even divergent for n → ∞.

To see what (4.15) entails, let us note that by the inverse of the Legendre trans-

form (4.13) :

ϕλ(s;E) ≡ ϕ(s) = −inf

Thus, (4.15) is the condition: ϕ(1;E) > −logK which is mentioned in Theorem 4.5,

and in Theorem 2.1.

γ[I(γ) + sγ)]

(4.16)

In the above discussion it was assumed that the large deviations of the Green func-

tion are described by a good enough rate function, at least for γ arbitrarily close to the

left derivative of ϕλ(s;E) at s = 1. To justify this picture, in the next section we de-

velop some relevant estimates on the moment generating function which allow to apply a

general large deviation principle. For the completeness of presentation a relevant large de-

viations theorem (in which a stronger statement is asserted than what is mentioned above)

is presented in Appendix B.

Totally omitted in the above sketch of the proof of Theorem 4.5 is an important

point whose proof forms the more technical part of the analysis. It concerns the ques-

tion whether the mean value condition E[Mn] ? 1 is a reliable indicator that Mn≥ 1

does occur at probability which does not vanish as n → ∞. (The high mean could be due

to rare fluctuations only.) This will be shown to be the case in Section 6 by establishing

also the second moment condition, that is, an upper bound on E[M2

applying the Paley-Zygmund inequality.

n]/E[Mn]2, and then

5 The moment generating function

Our goal in this section is to establish the existence, monotonicity and some bounds for

the moment generating function ϕλ(s;E) which was introduced in (2.5). However, as it

turns out and perhaps not surprisingly, it is more convenient to carry the analysis first for

complex values of the energy parameter. Thus, we extend our attention to the function

defined by

1

|x|logE[|Gλ(0,x;ζ)|s] =: ϕλ(s;ζ)

for ζ ∈ C+. At the end of this section, we will also compile some properties of the

associated Lyapunov exponent

lim

|x|→∞

(5.1)

Lλ(ζ) := −E[log|Gλ(0,0;ζ)|] ,

(5.2)

and give a proof of Corollary 2.2.

5.1Definition, monotonicity and finite-volume estimates

Theorem 5.1.

function [−ς,∞) ? s ?→ ϕλ(s;ζ) has the following properties.

1. For all ζ ∈ C+and s ∈ [−ς,∞) the limit in (5.1) exists and the

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