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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Past results and the question settled here . . . . . . . . . . . . . . . . . .
3
3
3
2 The main result
2.1The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A criterion for extended states . . . . . . . . . . . . . . . . . . . . . . .
2.3 Comparison with a localization criterion . . . . . . . . . . . . . . . . . .
2.4Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
8
9
3 Basic properties of the Green function on tree graphs
3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Recursion and factorization . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
11
4 The roadmap – Proof part I
4.1A zero-one law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The key statement – proof of the main result . . . . . . . . . . . . . . . .
4.3 A heuristic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
12
14
5 The moment generating function
5.1Definition, monotonicity and finite-volume estimates . . . . . . . . . . .
5.2 Super- and submultiplicativity estimates . . . . . . . . . . . . . . . . . .
5.3Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4Properties of the Lyapunov exponent . . . . . . . . . . . . . . . . . . . .
15
15
16
19
20
6 Green function’s extremal fluctuations – Proof part II
6.1 Parameterization of the large-deviation events . . . . . . . . . . . . . . .
6.2 The 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 tails44
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Resonant delocalization3
1 Introduction
1.1 The 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)
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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
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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:
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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 ∈
2.4 Further 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.
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Resonant delocalization10
3Basic properties of the Green function on tree graphs
3.1 Notation
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.
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Resonant delocalization 11
3.2 Recursion 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)
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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 delocalization 13
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.
4.3 A 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
Page 16
Resonant delocalization16
i) ϕλ(s;ζ) is convex and non-increasing in s ∈ [−ς,∞).
ii) For s ∈ [0,2]:
− sLλ(ζ) ≤ ϕλ(s;ζ) ≤ −s log
√K .
(5.3)
iii) There are constants C+,C−∈ (0,∞), which in case s ∈ [−ς,1) are uniformly
bounded in Im ζ ∈ (0,1], such that for all x ∈ T :
(C+C−)−2e|x|ϕλ(s;ζ)≤ E[|Gλ(0,x;ζ)|s] ≤ (C+C−)2e|x|ϕλ(s;ζ)
(5.4)
2. For Lebesgue-almost all E ∈ R and all s ∈ [−ς,1) the limit in (2.5) exists and is
finite. The function [−ς,1) ? s ?→ ϕλ(s;E) coincides with the limiting value of ϕλ,
i.e., for all s ∈ [−ς,1) and all E ∈ R:
ϕλ(s;E) = lim
η↓0ϕλ(s;E + iη)
= lim
|x|→∞
η↓0
1
|x|logE[|Gλ(0,x;E + iη)|s] .
(5.5)
In particular, ϕλ(s;E) shares the properties listed in i), ii) and iii), within the re-
duced range: s ∈ [−ς,1).
Remark 5.2. Recall that ς ∈ (0,1) is the potential’s moment which is assumed to be
finite, i.e., E[|V (0)|ς] < ∞. The relation (5.5) in particular asserts that for s ∈ [−ς,1) the
limits η ↓ 0 and |x| → ∞ commute. This does not generally extend to s ≥ 1, in which
case the limit η ↓ 0 may produce a divergence if taken first (for E in the localization
regime of pure-point spectrum), while the quantity on the right is finite and non-increasing
in s for all s ≥ −ς.
For the proof of Theorem 5.1, which is presented in Section 5.3, we first derive some
auxiliary bounds, in particular super and submultiplicativity estimates which are related
to the Green function’s factorization properties. In this context, we recall that a supermul-
tiplicative positive sequence is one satisfying: αm+n≥ B αmαn> 0. By Fekete’s lemma
[14] for such sequences the following limit exists:
lim
n→∞
1
nlogαn=: Ψ,
(5.6)
and αm ≤ B−1emΨfor every m ∈ N. For submultiplicative sequences the reversed
inequalities hold.
5.2Super- and submultiplicativity estimates
Our proof of Theorem 5.1 is based on the super- and submultiplicativity of moments of
the Green function. Following is therefore an essential observation.
Page 17
Resonant delocalization17
Lemma 5.3. For any two vertices 0 ≺ u ≺ x and in case s ∈ [−ς,∞) and ζ ∈ C+or
s ∈ [−ς,1) and ζ = E + i0:
1
C−
with u±and x−as defined in (3.8), and constants 0 < C+,C−< ∞, which only depend
on s,λ,ζ, and which are uniformly bounded in Im ζ ∈ (0,1] in case s ∈ [−ς,1).
Proof. Using the factorization representation (3.8), and the statistical independence of the
two factors which are in the denominator of (5.7) we may write:
E?|GTx(0,x−;ζ)|s?
where Av(s)
E?|GTu(0,u−;ζ)|s|GTu,x(u+,x−;ζ)|s× Q?
To estimate this quantity we note that by (3.3):
≤
E?|GTx(0,x−;ζ)|s?
E(|GTu(0,u−;ζ)|s) E(|GTu,x(u+,x−;ζ)|s)
≤ C+
(5.7)
E(|GTu(0,u−;ζ)|s) E(|GTu,x(u+,x−;ζ)|s)
u (·) represents the weighted probability average:
= Avu
?|GTx(u,u;ζ)|s?
(5.8)
Av(s)
u(Q) =
E(|GTu(0,u−;ζ)|s) E(|GTu,x(u+,x−;ζ)|s)
(5.9)
GTx(u,u;ζ) =
1
v∈NuGTu,x(v,v;ζ) λV (u) − ζ −?
(5.10)
1. The upper bound: In case s ≥ 1, the operator-theoretic bound |GTx(u,u;ζ)|
(Im ζ)−1yields the upper bound in (5.7) with C+:= (Im ζ)−1.
In case s ∈ [0,1), the expression (5.10) and (A.5) readily imply that:
?|GTx(u,u;ζ)|s?
In case s ∈ [−ς,0), the expression (5.10) together with the inequality (|a| + |b|)σ≤
|a|σ+ |b|σfor σ ∈ [0,1] also implies:
Av(s)
u
≤
Av(s)
u
≤
???s
(1 − s)λs
∞
(=: C+) .
(5.11)
?|GTx(u,u;ζ)|s?
≤ λ−sE?|V (u)|−s?+|ζ|−s+
?
v∈Nu
Av(s)
u
?|GTu,x(v,v;ζ)|−s?.
(5.12)
To bound the terms v ?∈ {u−,u+}, we use (5.11) to conclude that
?|GTu,x(v,v;ζ)|−s?≤
In the remaining cases v ∈ {u−,u+}, we use the factorization property (3.7), Jensen’s
inequality and (5.11) to conclude:
?|GTu(u−,u−;ζ)|−s?
≤ Av(s)
Av(s)
u
λs
(1 + s)???s
∞
.
(5.13)
Av(s)
u
=
?Av(s)
≤
u−
?|GTu(u−,u−;ζ)|s??−1
(1 + s)???s
u−
?|GTu(u−,u−;ζ)|−s?
λs
∞
(=: C+) ,
(5.14)
Page 18
Resonant delocalization18
and similarly for u+. (Note that in case u−= 0, the definition of Av(s)
u−extends naturally.)
2. The lower bound: First assume that s > 0. The expression (5.10) implies for any t > 0:
?
?
[λεE(|V (0)|ε) + (K + 1)εtε]s/ε
where the last inequality holds for any ε ∈ (0,min{ς,s}]. It derives from that fact that
the random variables appearing in the numerator and V (u) are independent (even with
respect to Av(s)
E[|Q|ε]−s/ε. We now choose t large enough, so that Av(s)
1/2. In case v ?∈ {u−,u+} this is quantified in the estimate (A.6), and in case v ∈
{u−,u+} in (A.21).
If s ∈ [−ς,0], we use the Jensen inequality together with (5.11) to conclude that
?|GTx(u,u;ζ)|s?≥
which completes the proof of (5.7).
Av(s)
u
?|GTx(u,u;ζ)|s?
v∈N(u)Av(s)
≥ Av(s)
u
1?For all v ∈ Nu:
|GTu,x(v,v;ζ)| ≤ t?
[λ|V (u)| + |ζ| + (K + 1)t]s
?
≥
u
?1?|GTu,x(v,v;ζ)| ≤ t??
(=: C−) ,
(5.15)
u (·)), and Jensen’s inequality, which yields E[|Q|−s] ≥ E[|Q|−ε]s/ε≥
u
?1?|GTu,x(v,v;ζ)| ≤ t??
≥
Av(s)
u
1
Av(s)
u (|GTx(u,u;ζ)|−s)
≥
(1 + s)λs
???s
∞
(=: C−) ,
(5.16)
The above lemma addresses the Green function restricted to subgraphs. Arguments
used in the proof also imply that the full Green function may in fact be compared with its
restricted versions. Moreover, the effect of peeling off one vertex is bounded:
Lemma 5.4. Under the assumptions of Lemma 5.3, let x−−stand for the neighbor of x−
towards the root:
E?|GTx(0,x−;ζ)|s?
1
C+C−
where x−−is the neighbor of x−towards the root.
Proof. For the proof of (5.17) we use the factorization of the Green function:
1
C−
≤
E?|GTx−(0,x−−;ζ)|s? ≤ C+,
E(|GTx(0,x−;ζ)|s)
(5.17)
≤
E(|G(0,x−;ζ)|s)
≤ C+C−,
(5.18)
GTx(0,x−;ζ) = GTx−(0,x−−;ζ)GTx(x−,x−;ζ).
Since the last factor is of the form (5.10), the same strategy as in the proof of Lemma 5.3
yields (5.17).
For a proof of (5.18) we employ the factorization:
(5.19)
G(0,x;ζ) = GTx(0,x−;ζ) G(x,x;ζ).
(5.20)
Thus,byargumentsasintheproofofLemma5.3,thequantityE(|G(0,x;ζ)|s)isbounded
from above and below in terms of E?|GTx(0,x−;ζ)|s?. Since the latter lacks x, we ap-
ply (5.17) to append this vertex.
Page 19
Resonant delocalization19
5.3Proof of Theorem 5.1
Proof of Theorem 5.1. In the following we pick a simple path in T to infinity, and label
its vertices by 0 =: x0,x1,x2,.... We first show that
αn(ζ) := E???GTxn+1(x0,xn;ζ)??s?
2. s ∈ [−ς,1) and ζ = E + i0. In both cases, the factorization property (3.8), Lemma 5.3
and (5.17) imply for all n,m ∈ N:
αn+m+1(ζ) ≥ C−1
By Fekete’s lemma [14], the limit Ψ(ζ) := limn→∞n−1logαn(ζ) hence exists.
Analogous reasoning as above using Lemma 5.3 and (5.17) also show submultiplica-
tivity, i.e., for all n,m ∈ N:
αn+m+1(ζ) ≤ C+αn(ζ)αm(ζ) ≤ C+C−αn+1(ζ)αm(ζ).
By super- and submultiplicativity, the limit Ψ(ζ) hence serves as an upper and lower
bound on αm(ζ) for any m ∈ N:
(C+C−)−1emΨ(ζ)≤ αm(ζ) ≤ C+C−emΨ(ζ).
To establish the existence of the limits (5.1) and (2.5), we use (5.24) and (5.18) which
reads
(C+C−)−1αn(ζ) ≤ E[|G(x0,xn;ζ)|s] ≤ C+C−αn(ζ).
Hence the limits (5.1) and (2.5) agree with Ψ(ζ) = ϕλ(s;ζ) in both cases: 1. s ∈ [−ς,∞)
and ζ ∈ C+and 2. s ∈ [−ς,1) and ζ = E + i0.
Since the constants C+,C−are uniformly bounded in case s ∈ [−ς,1), the conver-
gence (5.1) is in fact uniform with respect to Im ζ ∈ (0,1]. This proves (5.5), namely that
the limits η ↓ 0 and |x| → ∞ can be taken in any order.
The finite-volume bounds (5.4) now follow from (5.24) and (5.25).
It remains to establish the properties listed in i) and ii). Since the prelimits are convex
functions of s, the limit is convex. Since for any ? ≥ 0
E?|G(0,x;ζ)|s+??≤ (Im ζ)−?E[|G(0,x;ζ)|s] ,
the limit (5.1) is non-increasing.
The first inequality in (5.3) is a consequence of convexity and the factorization prop-
erty (3.6) of the Green function,
(5.21)
is supermultiplicative in the two cases of interest: 1. s ∈ [−ς,∞) and ζ ∈ C+and
−αn(ζ)αm(ζ) ≥ (C+C−)−1αn+1(ζ)αm(ζ).
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
logE[|G(0,x;ζ)|s] ≥ sE[log|G(0,x;ζ)|] = −s|x|Lλ(ζ).
The second inequality in (5.3) relies on the following bound on the sums of squares of
Green functions
?
(5.27)
|x|=n
|Gλ(0,x;ζ)|2≤
?
x∈T
|Gλ(0,x;ζ)|2=Im Gλ(0,0;ζ)
Im ζ
≤
1
(Im ζ)2.
(5.28)
Page 20
Resonant delocalization20
From the finite-volume bounds (5.4), we conclude that for any n = dist(x,0) ∈ N:
Knenϕλ(2;ζ)≤ (C+C−)2KnE?|Gλ(0,x;ζ)|2?
= (C+C−)2E
??
|x|=n
|Gλ(0,x;ζ)|2?
≤(C+C−)2
(Im ζ)2.
(5.29)
Since the right side is independent of n, we thus have ϕλ(2;ζ)+logK ≤ 0. By convexity,
this implies ϕλ(s;ζ) ≤ −s log√K for all s ∈ [0,2].
5.4 Properties of the Lyapunov exponent
We now compile some properties of the Lyapunov exponent (5.2) (see also (2.9)), which
lead us to a proof of Corollary 2.2. We first note that Lλ(ζ) is a harmonic function, since it
is the negative real part of the Herglotz function Wλ(ζ) =: E[logGλ(0,0;ζ)]. As a con-
sequence, the limit Lλ(E) = −limη↓0Re Wλ(E + iη) exists for almost all E ∈ R. The
following continuity property is hence a straightforward consequences of the continuity
of the harmonic measure associated with Lλ.
Lemma 5.5. For any bounded interval I ⊂ R the function [0,∞) ? λ ?→?
lim
λ↓0
I
Proof. Since Im Wλ∈ (0,π), the harmonic conjugate of Lλ= −Re Wλhas a definite
sign and locally integrable boundary values. This implies that the harmonic measure σλ
associated with Lλ(ζ) = π−1?Im(E − ζ)−1σλ(dE) is purely ac [12, Thm. 3.1, Cor. 1],
The asserted continuity thus follows from the vague continuity of the measure σλ,
which in turn follows from the resolvent convergence Gλ(0,0;ζ,ω) → Gλ0(0,0;ζ,ω) as
λ → λ0for all ζ ∈ C+and all ω.
In particular, Lemma 5.5 ensures that the mean value of the Lyapunov exponent over
any bounded, non-empty interval I,
1
|I|
is continuous in λ ≥ 0.
Poof of Corollary 2.2. Since Lλ(E) ≥ log√K by (5.3), we may employ the Chebychev
inequality to control the Lebesgue measure of that subset of I on which (2.10) is violated:
?
ILλ(E)dE is
continuous, and, in particular:
?
Lλ(E)dE =
?
I
L0(E)dE .
(5.30)
and one has σλ(I) =?
ILλ(E)dE.
Mλ(I) :=
?
I
Lλ(E)dE ,
(5.31)
|{E ∈ I |Lλ(E) ≥ logK}| ≤
I
Lλ(E) − log√K
log√K
dE = |I|Mλ(I) − log√K
log√K
.
(5.32)
Theassertionthusfollowsfromthecontinuity(5.30)andthefactthatlog√K ≤ M0(I) <
logK for all closed intervals I ∈ (−K − 1,K + 1) by a computation.
Note that M0(I) = log√K for all I ⊂ (−2√K,2√K). Hence, in this case the
measure in (5.32) tends to |I| as λ ↓ 0.
Page 21
Resonant delocalization21
6Green function’s extremal fluctuations – Proof part II
Our aim in this section is to prove the key estimate, Theorem 4.5. To to so, we follow the
outline given in Section 4 and construct events which serve as amplifiers of the of the
imaginary part of the Green function. Under the no-ac hypothesis, such events will occur
at a positive probability.
6.1Parameterization of the large-deviation events
For the remainder of this section, we fix the disorder parameter λ > 0 and an energy
E ∈ R such that (2.7) holds, i.e.,
∆ ≡ ∆λ(E) := logK + ϕλ(1;E) ∈?0,1
In particular, it will be assumed throughout that at the given energy
2logK?.
(6.1)
a) ϕλ(t;E) = limη↓0ϕλ(t;E + iη) exists for all t ∈ [−ς,1),
b) theboundaryvaluesoftheGreenfunctionsGT(u,v)(x,y;E+i0,ω)existsimultaneously
for all x,y ∈ T , (u,v) ∈ T , and P-almost all ω.
The assumptions (a) and (b) are valid, regardless of (6.1), at almost every energy. Since
E ∈ R and λ > 0 are fixed throughout the section they will be omitted from the notation
at our convenience.
Let us note that due to the convexity of ϕ(s) and (5.3), under the assumption (6.1) the
left derivative of ϕ satisfies (see Figure 2):
0 < −ϕ?
−(1) ≤ ∆.
(6.2)
We proceed by associating to the given λ and E certain parameters (namely γ, β, κ,
?, and τ) which will also be kept fix for the remainder of this section. These parameters
feature in the definition of the fluctuation events which will be associated with vertices on
the sphere Sn=?x ∈ T
Sκ
associated with the parameter κ which we pick at the range:
?
where β > 0 is fixed satisfying the constraint (6.17) below. The thinned sphere Sκ
radius shall be larger that 4?κ−1?, is characterized by the length scales
nκ:= 2?κn
The first one is only a fraction of the second length scale, i.e.
??dist(x,0) = n?of a given radius n ∈ N. Since we later need
n⊂ Sn
to control the correlations among such events, we consider vertices on the thinned sphere
(6.3)
κ ∈
0, min
?
∆
16β,1
4
??
,
(6.4)
n, whose
2? ∈ 2N,Nκ:= n − nκ.
(6.5)
1
2κn ≤ nκ≤ κn,nκ≤
κ
1−κNκ≤4
3κNκ.
(6.6)
Page 22
Resonant delocalization 22
Figure 2: Sketch of the moment-generating function in case ϕλ(1;E) > −logK. Regard-
less of this assumption the curve does not enter the shaded region. The parameter γ is the
negative slop of the tangent at s and the value of the rate function I(γ) = −ϕ(s)−sγ can
be read off as the negative value at the intersection of that tangent with the vertical axis.
Then Sκ
them, cf. Figure 3.
nis uniquely determined by having KNκvertices with 2nκ+1 vertices separating
We now pick a value s ≡ sλ(E) ∈ (0,1) at which the moment-generating generating
function t ?→ ϕλ(t;E) ≡ ϕ(t) is differentiable, and such that
a) the derivative at s, satisfies
γ := −ϕ?(s) ≥ ∆ > 0,
(6.7)
b) the following condition holds
I(γ) + γ = −[ϕ(s) + (1 − s)ϕ?(s)] ≤ logK −7
8∆,
(6.8)
c) and in addition (1 − s) < 1/16 and ϕ(s) < −1
In view of (6.1) and (6.2), and the convexity of ϕ, the above conditions are satisfied at a
dense collection of values of s approaching 1 from below (see Figure 2). (Condition (c) is
only imposed to simplify some of the estimates.)
The parameter γ ≡ γ(s) will be used as a target-value for the decay of the Green
function in the large deviation events Lxdefined below.
For any site x ∈ Snwe label the vertices of the unique path from the root to x as
x0= 0,x1,...,xn= x, and we denote as
2logK.
?Tx:= Txnκ−1,x
(6.9)
Page 23
Resonant delocalization 23
Figure 3: The geometry of the resonance-boosted large-deviation event.
the tree truncated beyond the segment of length Nκwhose end points are {xnκ−1,x}
(cf. Figure 3). Associated with this segment there are the two collections of variables
{Γ+(j;η)}Nκ
Γ+(j;η) := GTxn−j−1,x(xn−j,xn−j;E + iη),
Γ−(j;η) := GTxnκ−1,xnκ+j(xnκ−1+j,xnκ−1+j;E + iη).
By Eq. (3.4):
j=1and {Γ−(j;η)}Nκ
j=1:
(6.10)
G?Tx(xnκ,xN−1;E + iη) =
Nκ
?
j=1
Γ+(j;η) =
Nκ
?
j=1
Γ−(j;η).
(6.11)
Definition 6.1. We refer to the following as the large-deviation events associated with
sites x ∈ Snand η, ? > 0
Nκ
?
2nκ
Lx:= L(bc)
x
∩
k=1
?L(k,+)
x
∩ L(k,−)
x
?
[ ≡ Lx(η;?) ] ,
(6.12)
where for any k ∈ {1,...,Nκ}:
L(k,±)
x
:=
?
?|Γ+(Nκ;η)| ≤b
k?
j=1
|Γ±(j;η)| ∈ e−γk?e−?k, e?k??
2
?≡ L(k,±)
2
x
(η;?)?,
L(bc)
x
:=
?∩?|Γ−(Nκ;η)| ≤b
?.
When not explicitly needed, we will suppress the dependence on η and ? (whose value
is fixed below).
Page 24
Resonant delocalization24
The boundary events L(bc)
the probability of Rxgiven below in Lemma 6.8 and ii) the estimate (6.35) on the size of
the self-energy at x are derived only under the condition L(bc)
a value large enough so that,
?
andthisassuresthattheprobabilityoftheeventL(bc)
(the numbers being largely arbitrary) uniformly in η > 0, cf. (A.6) and (A.21).
To fix the parameter ?, we invoke the following implication of the large-deviation
theory which is presented in the Appendix B:
x
play a role in the following context: i) the lower bound on
x
. The parameter b is fixed at
b
2
≥ max2,16???∞
λ
, (νλ)−1min
?
1,
?
15
21+sc
(1 − s)
?
1
1−s??
,
(6.13)
x
isboundedfrombelowby1−2
16=7
8
Theorem 6.2. For any ? > 0 there is η0> 0 and n0> 0 such that for all η ∈ (0,η0) and
all n = dist(x,0) ≥ k ≥ n0:
P(Lx(η;?)) ≥ e−Nκ(I(γ)+2?),
P?L(k,±)
The proof is presented in Appendix B, based on the general Theorem B.1, which is
also proven there.
We now fix ? at a value at which:
2? ∈?0, min?∆
This parameter will be used in controlling the probabilities of various large deviation
events.
(6.14)
(6.15)
x
(η;?)?
≤ e−(I(γ)−2?)k.
24,κ∆
4
??.
(6.16)
Before turning to the main definitions, we introduce yet another event which refers to
the behavior of the Green function between x0and xnκ−1, for which we require a certain
(largely arbitrary) minimal decay rate
β > ς−1ϕλ(−ς;E + i0)(> 0),
(6.17)
combined with two conditions at the end points.
Definition 6.3. We refer to the following as the regular events associated with sites x ∈
Snand η > 0:
Rx:= R(bc)
x
∩?|GTx(0,xnκ−1;E + iη)| ∈?e−nκβ,1??
:=?||GTx(0,0;E + iη)| ≤b
This event is regular in the sense that it occurs with a probability of order one, which
is independent of n, cf. Lemma 6.8 below. The reason for its inclusion in the paper is
mainly of technical origin: in the subsequent proof of a second moment bound, Theo-
rem 6.10 below, we cannot allow the large deviation event Lxto extend down to the root,
but we nevertheless need some control on the Green function on this segment.
[≡ Rx(η)]
(6.18)
where
R(bc)
x
2
?∩?||GTx(xnκ−1,xnκ−1;E + iη)| ≤b
2
?.
Page 25
Resonant delocalization 25
6.2 The extreme resonance events
A final parameter τ will set the scale of exponential blow-up of the Green function at x in
the definition of the resonance-boosted large-deviation event below:
??
The choice of this parameter is taylored to: i) compensate the decay of the Green function
on the segment preceeding x, cf. (6.24) below, and ii) ensure that for n large enough and
η small enough:
τ−1KNκP(Lx) ≥ exp?Nκ
16
τ := expγ +3
4∆
?
Nκ
?
.
(6.19)
?logK − (γ + I(γ)) − 2? −3
4∆??
≥ exp?Nκ
∆
?,
(6.20)
by (6.14), (6.8) and (6.16). The fact that this term can be made arbitrarily large as n → ∞
will be essential in the subsequent argument.
Having fixed the basic parameters, we now turn to the precise definition of the events.
Definition 6.4. For each x ∈ Snand η > 0 we define
1. the resonance-boosted large-deviation event,
Dx:= Ex∩ Lx∩ Rx
[ ≡ Dx(η) ]
(6.21)
which consists of the following three events:
a) extreme deviation event:
b) large deviation event:
c) regular event:
Ex := {|G(x,x;E + iη)| ∈ τ [1,2]}.
(cf. Definition 6.1)
(cf. Definition 6.3)
Lx
Rx
2. the regular current event
Ix :=
?
[ ≡ Ix(η,α) ] .
y∈N+
x
?Im Γ(y;E + iη) ≥ ξb
+(E + iη,α) and |Γ(y;E + iη)| ≤ b?
(6.22)
which is parametrized by α ∈ (0,1).
The joint event Dx∩ Ixwill be referred as a boosted resonance event at x.
Several remarks are in order:
1. The resonance-boosted large-deviation event are taylored so that in the event Dx≡
Dx(η) the Green function associated with the root and x exhibits an exponential
blow-up. Namely, by the factorization property of the Green function,
G(0,x;ζ) = GTx(0,xn−1;ζ)G(x,x;ζ)
= GTx(0,xnκ−1;ζ)GTLx(xnκ,xn−1;ζ)G(x,x;ζ).
(6.23)
Page 26
Resonant delocalization26
For ζ = E + iη, the first term is controlled by Rx. The large deviation event Lx
controls the second factor and the extreme fluctuation event Excompensates for the
decay of the first two terms. Using (6.6), (6.4), and (6.16), we hence arrive at the
estimate:
|G(0,x;E + iη)| ≥ e−nκβe−(γ+?)Nκτ
≥ exp?Nκ
?3
4∆ − ? −4
2∆Nκ
3κβ??
≥ exp?1
?
≥ exp?3
4∆n?.
(6.24)
2. We recall from Definition 4.3 that the value ξb
+(E + iη,α) is taylored such that
P(Ix) ≥ αK ≥ α.
(6.25)
6.3 The mean number of boosted resonance events
The aim of this subsection is to show that on average the number
Nn(η,α) :=
?
x∈Sκ
n
1Dx(η)∩Ix(η,α)
(6.26)
of amplified current events on the thinned sphere is bounded below. In order to present a
reasonably explicit estimate, we introduce the constant
?b≡ ?b(E) := inf
I⊆2(K+1)[−b,b]
P(λV (x) ∈ I + E)
|I|
> 0.
(6.27)
which is a lower bound on the probability density of our basic random variables and
strictly positive by assumption. Our result is valid under the no-ac hypothesis, cf. Defini-
tion 4.2.
Theorem 6.5. Under the no-ac hypothesis, for all α ∈ (0,1) and all n sufficiently large
there exists η0≡ η0(α,n) such that for all η ∈ (0,η0):
E[Nn(η,α)] ≥
Moreover, for any α ∈ (0,1) the right side can be made arbitrarily large by choosing n
sufficiently large.
1
16?bτ−1KNκP(Lx) P(Ix) .
(6.28)
Proof. The bound (6.28) is a straightforward consequence of Lemma 6.7 below. The sec-
ond claim follows from the exponential estimate (6.20) on τ−1KNκP(Lx) which over-
whelms P(Ix) ≥ α > 0 if n is chosen large enough.
Before diving into the details of the estimates behind the above proof, let us explain the
idea behind this result. The essence for the validity of (6.28) is to show that the probability
of the occurrence of the extreme fluctuation Exis of order τ−1. Rewriting this event,
Ex=?|λV (x) − E − iη − Σ(x;E + iη)| ∈ τ−1?1
2,1??
(6.29)
Page 27
Resonant delocalization 27
thereby exposing the dependence of G(x,x;ζ) on the potential at x and the associated
self-energy,
Σ(x;ζ) :=
?
y∈Nx
GTx(y,y;ζ),
(6.30)
one realizes that if the latter has a non-zero imaginary part, the Green function stays
bounded and no resonance mechanism kicks in. On the other hand, if η ≤ (8τ)−1and in
the event Sx∩ Tx, where
Sx:= {|Σ(x,E + iη)| ≤ (K + 1)b}
Tx:=?Im Σ(x;E + iη) ≤ (8τ)−1?,
the imaginary part of the term in the right side of (6.29) is bounded by (4τ)−1and the real
part of the self-energy is bounded by 2(K + 1)b. As a consequence, we may estimate
the conditional probability of Exconditioned on the sigma algebra Axgenerated by the
random variables V (y), y ?= x:
P?Ex
≥ 1Sx∩Tx
=
(6.31)
??Ax
?
≥ 1Sx∩TxP?|λV (x) − E − Re Σ(x;E + iη)| ∈
inf
1
4τ[2,3]??Ax
?
|σ|≤2(K+1)bP?|λV (x) − E − σ| ∈
4?bτ−11Sx∩Tx.
1
4τ[2,3]??Ax
?
1
(6.32)
where ?bwas defined in (6.27). Now, Sxis a regular event, i.e., it occurs with positive
probability which is independent of n. Under the no-ac hypothesis, the subsequent lemma
shows that the probability of the event Txis also (arbitrarily) close to one.
Lemma 6.6. Under the no-ac hypothesis, Im Σ(x;E + i0,ω) = 0 for P-almost all ω
and all x ∈ T .
Proof. Recall that the self-energy coincides with the sum (6.30) of Green functions asso-
ciated with the neighbors of x. The Green function associated with the forward neighbors,
y ?= x−, are identically distributed to Γ(0;E + i0) and hence Im GTx(y,y;E + i0,ω) =
0 for Lebesgue × P-almost all (E,ω). The Green function associated with the back-
ward neighbor x−differs by a finite-rank perturbation from a variable which is identi-
cally distributed to Γ(0;E + i0) (i.e., the surgery which renders the rooted to into a full
tree). Since finite-rank perturbations do not change the ac spectrum, we also conclude
Im GTx(x−,x−;E + i0,ω) = 0 for Lebesgue × P-almost all (E,ω).
The bound (6.32) quantifies the essence of the resonance mechanism and leads to the
following
Lemma 6.7. Under the no-ac hypothesis, for every n and α ∈ (0,1) there exists η0≡
η0(α,n) > 0 such that for all η ∈ (0,η0) and all x ∈ Sn:
P(Dx∩ Ix) ≥
1
8?bτ−1P(Rx∩ Lx) P(Ix) ≥
1
16?bτ−1P(Lx) P(Ix) .
(6.33)
Page 28
Resonant delocalization28
Proof. In order to estimate the probability of the joint occurrence of the events Dxand Ix,
we first condition on the sigma algebra Axand use (6.32) to obtain:
P(Dx∩ Ix) = E?1Rx∩Lx∩IxP?Ex
≥
≥
≥
=
??Ax
??
1
4?bτ−1P(Rx∩ Lx∩ Ix∩ Sx∩ Tx)
1
4?bτ−1P(Rx∩ Lx∩ Ix∩ Tx)
1
4?bτ−1[P(Rx∩ Lx∩ Ix) − (1 − P(Tx))]
1
4?bτ−1[P(Rx∩ Lx) P(Ix) + P(Tx) − 1] .
(6.34)
Here the second inequality used Rx∩ Lx∩ Ix⊂ Sxwhich derives from the following
facts:
i) in the event Ix, each of the K terms in (6.30) corresponding to a forward neighbor
of x is bounded in modulus by b.
ii) second order perturbation theory ensures that in the event Rx∩ Lxthe term corre-
sponding to the backward neighbor x−of x is bounded according to
|GTx(x−,x−;E + iη)| ≤ |G?Tx(x−,x−;E + iη)|
+ |GTx(xnκ−1,xnκ−1;E + iη)| |G?Tx(xnκ,x−;E + iη)|2
≤
To proceed with our estimate on the right side in (6.34) we first condition on the sigma
algebra A generated by a the random variables V (y), xnκ? y, and use Lemma 6.8 below
to conclude that for some η0> 0 and some n0∈ N and all η ∈ (0,η0) and n ≥ n0:
P(Rx∩ Lx) = E?1LxP?Rx
We now use Lemma 6.6 which implies that under the no-ac hypothesisand for any x ∈ T
and any ε > 0:
lim
b
2+b
2= b .
(6.35)
??A??
≥
1
2P(Lx) .
(6.36)
η↓0P(Im Σ(x;E + iη) > ε) = 0.
(6.37)
Since P(Ix) ≥ α > 0, and
P(Rx∩ Lx) ≥
1
2P(Lx) ≥
1
2
inf
η∈(0,1]P(Lx(η)) > 0,
(6.38)
is strictly positive by (6.14), we conclude that there is some η0(n,α) ∈ (0,1] such that for
all η < η0(n,α):
1 − P(Tx) ≤1
This concludes the proof of (6.33).
2P(Rx∩ Lx) P(Ix) .
(6.39)
It remains to proof the following lemma.
Lemma 6.8. Let A be the sigma-algebra generated by the random variables V (y) with
xnκ? y. Then there is η0 > 0 and n0 > 0 such that for all η ∈ (0,η0) and all n =
dist(x,0) ≥ n0:
P?Rx(η)??A?
≥
1
21L(bc)
x
.
(6.40)
Page 29
Resonant delocalization 29
Proof. The idea is to control the conditional probability of the complement of the four
events hidden in the definition of Rxfrom above provided that L(bc)
the proof appearing constants C will be independent of n,η.
Asapreparation,weexposetheinfluencetheconditioningonA hasonGTx(0,xnκ−1;E+
iη) using the factorization property of the Green function into the two factors:
x
occurs. Throughout
G(η) := GTx(xnκ−1,xnκ−1;E + iη)
?G(η) := GTxnκ−1(0,xnκ−2;E + iη) = GTx(0,xnκ−1;E + iη)?G(η).
with Axnκthe sigma-algebra generated by V (y), y ?= xnκ, we thus have
?
≤ C eϕ(s;E+iη)nκ.
Since supη∈(0,η0]ϕ(s;E + iη) < 0 for some η0> 0, the right side is arbitrarily small if n
is chosen large enough. Furthermore, for any B > 0 we have
?
(6.41)
?≤ C
By Chebychev’s inequality and the uniform boundedness, supη>0E?|G(η)|s??Axnκ
P
≤ E
|?G(η)G(η)| > 1??A
??
|?G(η)G(η)|s??A
?
≤ C E
?
|?G(η)|s?
(6.42)
P
|?G(η)G(η)| < e−nκβ??A
The event in the second term takes the form
???λV (xnκ−1) − E − iη −
In the event L(bc)
x
, there is B > 0 (which in independent of n and η) such that for all
η ∈ (0,1]:
P?|G(η)| < B−1??A?1L(bc)
and the finite-volume estimates (5.4) and Lemma 5.4:
?
≤ C B exp(nκ(ϕ(−ς;E + iη) − βς)) .
By choosing η sufficiently small, the exponent in the right is strictly negative thanks to the
choice of β. Hence the term can be made arbitrarily small by choosing n large enough.
The proof of (6.40) concludes with the observation that by the choice of b
?
≤ P
?
|?G(η)| < B e−nκβ?
+ P?|G(η)| < B−1??A?.
(6.43)
?
y∈Nxnκ−1
G?Tx(y,y;E + iη)
??? > B .
x
≤
1
161L(bc)
x
.
(6.44)
The first term on the right in (6.43) is estimated with the help of Chebychev’s inequality
P
|?G(η)| < B e−nκβ?
≤ B e−nκβςE
?
|?G(η)|−ς?
(6.45)
1 − P?R(bc)
x
(η)??A?≤
2
16
=
1
8
(6.46)
cf. (6.13).
Page 30
Resonant delocalization30
6.4Establishing the events’ occurrence
The mere fact that the mean number of events diverges, for n → ∞, does not yet imply
thatsucheventsdooccurwithuniformlypositiveprobability.Thealternativeisthatthedi-
vergence reflects an increasingly rare but also increasingly correlated occurrence of these
events. To prove that boosted resonances do occur regularly, on sufficiently large spheres
Sn, we use the second-moment criterion which is provided by the Paley and Zygmund
inequality [21]:
Proposition 6.9. Let X be a real random variable with a finite, non-zero second moment.
Then for any θ ∈ (0,1):
P(X > θE[X]) ≥ (1 − θ)2E[X]2
E[X2].
(6.47)
Thus,ouraiminthissubsectionistoprovideauniformupperboundonE[N2]/E[N]2,
for N = Nn=?
Theorem 6.10. Under the no-ac hypothesis, there exists some constant C < ∞ such
that for all α ∈ (0,1), all n sufficiently large there is η0 ≡ η0(α,n) such that for all
η ∈ (0,η0):
E[Nn(η,α)2]
E[Nn(η,α)]2≤ C < ∞,
and, as a consequence:
x∈Sκ
n1Dx∩Ix, which counts the number of boosted resonance events on
the thinned sphere.
(6.48)
P?Nn(η,α) ≥1
2E[Nn(η,α)]?
≥
1
4C
> 0.
(6.49)
Proof. Throughout the proof we will suppress the dependence on η,α at our convenience.
Appearing constants c, C will be independent of n, η and α.
Since E[Nn] ≥ 1 for all n sufficiently large by Theorem 6.5, it is enough to bound
from above the following average in terms of E[Nn]2:
?
x?=y
E[Nn(Nn− 1)] =
x,y∈Sκ
n
P(Dx∩ Dy∩ Ix∩ Iy) = |Sκ
n|
?
y∈Sκ
n\{x}
P(Dx∩ Dy∩ Ix∩ Iy) .
(6.50)
The last equality holds for arbitrary x ∈ Sκ
metry, the joint probability P(Dx∩ Dy∩ Ix∩ Iy) depends only on the distance of the last
common ancestor x ∧ y to the root. It is therefore useful to introduce the ratio
P(Dx∩ Dy∩ Ix∩ Iy)
P(Dx∩ Ix) P(Dy∩ Iy)
The sum in (6.50) may then be organized in terms of the last common ancestor x ∧ y
on the path P0,x= {x0,...,xn} connecting the root with x. In fact, since Sκ
x ∧ y belongs to the shortened path Pκ
nwhich we will fix in the following. By sym-
:= r(j) δdist(x∧y,0),j .
(6.51)
nis thinned,
?. Moreover, for
0,x:=?u ∈ P0,x
??dist(u,0) < Nκ
Page 31
Resonant delocalization31
a given x ∧ y ∈ Pκ
common ancestor, is |Sκ
0,x, the number of vertices y ∈ Sκ
n|K−dist(x∧y,0)such that
n, which for fixed x have the same
E[Nn(Nn− 1)]
E[Nn]2
=
Nκ−1
?
j=0
r(j)
Kj.
(6.52)
In order to estimate the sum in the right side of (6.52), we always drop the condition Rx
in the definition of Dx:
r(j) ≤P(Lx∩ Ly∩ Ex∩ Ey∩ Ix∩ Iy)
P(Dx∩ Ix) P(Dy∩ Iy)
For an estimate on the numerator in the right side, we first focus on the extreme fluctu-
ation events and aim to integrate out the random variable associated with x and y using
Theorem A.2 in the Appendix. In general, what stands in the way of this procedure is the
dependence of Lxon V (y) and Lyon V (x), respectively. We therefore relax the condi-
tions in the large deviation events and pick suitable
δdist(x∧y,0),j.
(6.53)
?Lx,j⊃ Lx,
(and hence
?Ly,j⊃ Ly)
(6.54)
such that?Lx,jand?Ly,jare independent of both V (x) and V (y). Postponing the details of
using Theorem A.2 in the Appendix:
?
≤ C
these choices which will depend on j, we bound the numerator on the right side in (6.53)
P(Lx∩ Ly∩ Ex∩ Ey∩ Ix∩ Iy) ≤ E
?
1?Lx,j∩?Ly,j∩Ix∩IyP(Ex∩ Ey|Ax,y)
?
τ−2P
??Lx,j∩?Ly,j
?
+ C τ−1E
?
1?Lx,j∩?Ly,jmin????Gx,y
??,1???
P(Ix)P(Iy)
(6.55)
where we have abbreviated by Ax,ythe sigma algebra generated by the variables V (ξ),
ξ ?∈ {x,y} and
?Gx,y:= GTx,y(xn−1,yn−1;E + iη).
Under the assumptions of Lemma 6.7 and Lemma 6.8, the denominator in the right
side of (6.53) is bounded from below by cτ−2P(Lx)P(Ly)P(Ix)P(Iy) provided n is
sufficiently large and η is sufficiently small. The terms on the right side in (6.55) hence
give rise to two terms, r(j) ≤ r1(j)+r2(j), which for fixed j = dist(x∧y,0) are defined
as:
r1(j) := CP??Lx,j∩?Ly,j
r2(j) :=
P(Lx) P(Ly)E
(6.56)
This quantity measures the strength of the interaction of the events Exand Ey.
?
P(Lx) P(Ly)
C τ
(6.57)
?
1?Lx,j∩?Ly,jmin?|?Gx,y|,1??
(6.58)
For the precise definition of the events?Lx,jand?Ly,jwe distinguish three cases:
Page 32
Resonant delocalization32
Case 0 ≤ j < nκ: The events Lxand Lyare already independent of the potential at x
and y. Therefore we choose
?Lx,j= Lx.
bounded in n and η:
nκ−1
?
For an estimate on r2(j), we drop the indicator function in the right side of (6.58)
and use the fact that min{|x|,1} ≤ |x|σfor any σ ∈ [0,1); in particular, for σ = s:
C τ
P(Lx) P(Ly)E?|?Gx,y|s?
Here the second inequality derives from the finite-volume estimates (5.4). Since
ϕ(s) < −1
inequalities is dominated by its last term:
(6.59)
As a consequence, the corresponding sum involving r1(j) is seen to be uniformly
j=0
r1(j)
Kj
≤ C
∞
?
j=0
1
Kj.
(6.60)
r2(j) ≤
≤
C τ
P(Lx) P(Ly)e2(n−j)ϕ(s).
(6.61)
2logK by assumption on s, the geometric sum in the following chain of
nκ−1
?
j=0
r2(j)
Kj
≤
C τ
P(Lx) P(Ly)
nκ−1
?
e2Nκϕ(s)
Knκ
j=0
e2(n−j)ϕ(s)
Kj
≤
C τ
P(Lx) P(Ly)
.
(6.62)
Using the large deviation result, Theorem 6.2, and the fact that −ϕ(s) = I(γ)+γ s,
we estimate
τ
P(Lx) P(Ly)e2Nκϕ(s)≤ e4Nκ?τ e−2Nκγs≤ eNκ
≤ eNκ(15
since 2s > 15/8.
??7
4−2s)
?
∆+4?
?
8−2s)∆≤ C ,
(6.63)
Case nκ≤ j ≤3
2nκ: We choose
?Lx,j= L
y
(Nκ−1
x
2nκ−1,+)
,
(6.64)
which is independent of?Ly,j= L
P??Lx
Here the first inequality follows from the large deviation result, Theorem 6.2, and
holds for n large enough and η sufficiently small. In this situation, the third in-
equality also applies since I(γ) ≤ logK −15
(Nκ−1
2nκ−1,+)
. An estimate on r1(j) hence requires
to bound the ratio:
?
P(Lx)
≤ Ce−(n−3
2nκ−2)(I(γ)−2?)
e−Nκ(I(γ)+2?)
≤ C e4Nκ?e
nκ
2I(γ)≤ C Knκ/2.
(6.65)
8∆ by (6.8) and (6.7), and 4Nκ? ≤
Page 33
Resonant delocalization33
∆κNκ/4 ≤ ∆nκ/2. As a consequence, the sum corresponding to r1(j) is bounded
uniformly in n:
3
2nκ
?
j=nκ
r1(j)
Kj
≤ C Knκ
∞
?
j=nκ
1
Kj≤ C
∞
?
j=0
1
Kj.
(6.66)
For an estimate on the sum corresponding to r2(j) we use (6.61) again which yields
3
2nκ
?
j=nκ
r2(j)
Kj
≤
C τ
P(Lx) P(Ly)
e(2Nκ−nκ)ϕ(s)
3
2nκ
K
≤
C τ
P(Lx) P(Ly)
e2Nκϕ(s)
Knκ/2≤ C (6.67)
by (6.63).
Case3
2nκ< j < Nκ: In this main case, we pick
?Lx,j = L(j−nκ−1,−)
x
∩ L(Nκ+nκ−j−1,+)
and L(Nκ+nκ−j−1,+)
x
x
,
(6.68)
Note that L(j−nκ−1,−)
independent. We may hence estimate the numerator in the definition of r1(j) using
the large deviation result, Theorem 6.2 to conclude that for all n sufficiently large
and η sufficiently small:
P??Lx,j∩?Ly,j
≤ C P(Lx)P(Ly) e8Nκ?e−I(γ)(nκ−j).
Since I(γ) < logK, the corresponding sum is hence uniformly bounded in n:
x
= L(j−nκ−1,−)
y
and L(Nκ+nκ−j−1,+)
y
are
?≤ P?L(j−nκ−1,−)
≤ C e−(I(γ)−2?)(2n−j−nκ)
x
?P?L(Nκ+nκ−j−1,+)
x
?P?L(Nκ+nκ−j−1,+)
y
?
(6.69)
Nκ−1
?
2nκ+1
j=3
r1(j)
Kj
≤ C e8Nκ?
Nκ
?
2nκ
nκ
2I(γ)
j=3
e−I(γ)(nκ−j)
Kj
≤ C e8Nκ?e
K
3
2nκ
≤ Ce8Nκ?
Knκ
≤ C ,
(6.70)
cf. (6.65).
For an estimate on r2(j) we drop conditions in the indicator function and use
min{|x|,1} ≤ |x|sagain:
E?1L(nκ,j−1)
The Green function in the numerator is a product of three terms,?Gx,y= Gj?Gx?Gy
Gj:= GTx,y(xj,yj)
?Gx:= GTxj,x(xj+1,xn−1)
r2(j) ≤ C τ
x
|?Gx,y|s?
τ P(Lx) P(Ly)
(6.71)
with
(6.72)
?Gy:= GTyj,y(yj+1,yn−1)
Page 34
Resonant delocalization34
of which only the first one depends on V (j). Since L(nκ,j−1)
we may hence condition on the potential elsewhere and use the uniform bound
E[|Gj|s|V (y)y ?= xj] ≤ C to estimate the numerator in (6.71):
E?1L(nκ,j−1)
x
is independent of V (j)
x
|?Gx,y|s?≤ C E?1L(nκ,j−1)
≤ C e−(j−nκ)(I(γ)−2?)e2(n−j)ϕ(s).
x
|?Gx?Gy|s?
= C P?L(nκ,j−1)
x
?E
?
|?Gx|s?
E
?
|?Gy|s?
(6.73)
Summing over j with a weight K−jwe again obtain a geometric sum which is in
this case bounded by the number of terms times the maximum of its first and last
term. Therefore we conclude that
?
2nκ+1
?e−(Nκ−nκ)(I(γ)−2?)e2nκϕ(s)
In the first case, we use ϕ(s) < −I(γ) and Corollary 6.2 to conclude that the term
is uniformly bounded in n:
Nκ−1
j=3
r2(j)
Kj
≤
Nκ−1
?
j=nκ
r2(j)
Kj
≤ Nκ
C τ
P(Lx) P(Ly)
(6.74)
× max
KNκ
,e2Nκϕ(s)
Knκ
?
.
Nκ
C τ
P(Lx) P(Ly)
e−Nκ(I(γ)−2?)
KNκ
≤ Nκ
≤ C Nκe−Nκ(1
C eNκ(I(γ)+γ+3
4∆+6?)
KNκ
8∆−6?)≤ C ,
(6.75)
since ? < ∆/48.
In the second case, we use (6.63) to conclude that the term is uniformly bounded
in n:
C τ
P(Lx) P(Ly)Knκ
since 2s >15
This concludes the proof of (6.48). Hence, (6.49) is a consequence of the Paley-Zygmund
inequality (with θ =1
Nκ
e2Nκϕ(s)
≤ C NκeNκ(15
8−2s)
≤ C ,
(6.76)
8.
2).
6.5Proof of the key statement
We are now ready to give a proof of our key statement.
Proof of Theorem 4.5. Consider (Lebesgue-almost all) energies E satisfying (6.1) and fix
b as in (6.13), which in particular guarantees that P(|Γ(0;E + iη)| ≤ b) ≥ 1 −
uniformly in η > 0. Moreover, we pick
α = min?1
where C is the constant from Theorem 6.10 and δ =3
we now argue that Nn(η;α) ≥ 1 implies that Mn(E + iη,α,b,δ) ≥ 1 and |Γ(0;E +
iη)| ≤ b. Namely, if Nn(η;α) ≥ 1, then there is some x ∈ Snsuch that:
1
16=15
16,
4C,1
4
?,
(6.77)
8∆. With this choice of parameters,
Page 35
Resonant delocalization35
i) the condition (4.6) is satisfied with δ =3
8∆ by (6.24).
ii) the condition (4.5) is satisfied by definition of the event Ix(η;α).
iii) by second-order perturbation theory, we have
|Γ(0;E + iη)| ≤
??GTx(0,0;E + iη)??
≤b
≤b
+??GTx(0,x−;E + iη)??2|G(x,x;E + iη)|
2+ 2 ≤ b.
2+ 2 exp?−(γ − 2? −3
4∆)Nκ
?
(6.78)
Here the second estimate is based on the factorization of the Green function (6.23)
and follow from the fact that Rx(η) ∩ Lx(η;?) occurred.
Under the no-ac hypothesis, by Theorem 6.10, for all n sufficiently large there is η0 ≡
η0(α,n) such that for all η ∈ (0,η0):
P(Mn(E + iη,α,b,δ) ≥ 1 and |Γ(0;E + iη)| ≤ b)
≥ P(Nn(η;α) ≥ 1) ≥
1
4C≥ α.
(6.79)
This completes the proof of Theorem 4.5.
Page 36
Resonant delocalization 36
AFractional-moment bounds
The aim of this appendix is to present some basic weak-L1bounds on Green functions of
random operators, and related fractional moment estimates. Theorem A.2, which presents
suchboundsforpairsofGreenfunctions,isanewresultwhichisneededhereinSection6,
and which may also be of independent interest. In the last subsection we discuss the
related implications of the regularity Assumption D.
The discussion in this appendix is carried within the somewhat broader context of
operators of the form:
Hλ(ω) = H0+ λV (ω),
acting the Hilbert space ?2(G), with λ ≥ 0 the disorder-strength parameter and:
I G the vertex set of some metric graph,
II H0a self-adjoint operator in ?2(G), and
III V (ω) a random potential such that the random variables {V (x)|x ∈ G} are iid with
a probability distribution whose density is (essentially) bounded, ? ∈ L∞(R).
(A.1)
A.1Weak-L1bounds
We recall that according to the Krein formula, the Green function of Hλ(ω) restricted to
the sites x,y is in its dependence on V (x) and V (y) of the form
?Gλ(x,x;ζ) Gλ(x,y;ζ)
where Aλ(ζ) is given by the inverse of the left side for V (x) = V (y) = 0. In particular,
Gλ(x,x;ζ) = (λV (x) − a)−1with some a ∈ C which is independent of V (x).
The assumed boundedness of the density ? of the distribution of V (x) trivially implies
bounds on probabilities of weak-L1-type:
?
Since the dependence of the Green function Gλ(x,x;ζ) on V (x) is of the above form,
this implies that the following well-known weak-L1bound, and hence the boundedness
of fractional moments (cf. [3]).
Gλ(y,x;ζ) Gλ(y,y;ζ)
?
=
??λV (x)0
0 λV (y)
?
+ Aλ(ζ)
?−1
,
(A.2)
sup
a∈C
1|v−a|<1
t?(v)dv ≤
???∞
t
.
(A.3)
Proposition A.1. For a random operator Hλ(ω) = H0+ λV (ω) on ?2(G) satisfying
assumptions I–III, at any complex energy parameter ζ ∈ C+and for any t > 0 and
s ∈ (0,1), the Green function satisfies:
P?|Gλ(x,x;ζ)| > t??Ax
?
≤
???∞
λt
???s
(1 − s)λs,
,
(A.4)
E?|Gλ(x,x;ζ)|s??Ax
?
≤
∞
(A.5)
where Axdenotes the sigma-algebra generated by V (y) , y ?= x.
Page 37
Resonant delocalization37
One trivial, but useful consequence of (A.4) is that for any p ∈ (0,1) and t ≥
P?|Gλ(x,x;ζ)| ≤ t??Ax
cerns the joint conditional probability of events as in (A.4) associated with two (distinct)
sites
???∞
λ(1−p):
(A.6)
?
≥ p.
Our new result, which was vital in our second-moment analysis in Theorem 6.10, con-
Theorem A.2. In the situation of Proposition A.1, consider two sites x ?= y in a graph.
Then for any t > 0 and ζ ∈ C+:
?
≤
λ2t
P
|Gλ(x,x;ζ)| > t and |Gλ(y,y;ζ)| > t??Axy
min
?
2???∞
?
4???∞
????Aλ(x,y;ζ)????Aλ(y,x;ζ)??+ t−1
?
, 1
?
, (A.7)
where Aλ(x,y;ζ) are the off-diagonal matrix elements of Aλ(ζ) in (A.2), and Axyis the
the sigma-algebra generated by V (ξ), ξ ?∈ {x,y}.
In case of a tree graph, G = T , the off-diagonal matrix elements of Aλ(ζ) simplify:
Gλ(x,y;ζ)
Gλ(x,x;ζ)Gλ(y,y;ζ) − Gλ(x,y;ζ)Gλ(y,x;ζ)= GTx,y
This is most easily proven by noting that the ratio does not depend on V (x) and V (y) so
that we may take them to infinity. In this limit the ratio
Aλ(x,y;ζ) =
λ
(x−,y−;ζ).
(A.8)
Gλ(x,y;ζ)/[Gλ(x,x;ζ)Gλ(y,y;ζ)]
tends to GTx,y
λ
(x−,y−;ζ) and its numerator vanishes.
Proof of Theorem A.2. Let Aλ(x,y;ζ) denote the matrix elements of Aλ(ζ) in the rank-
two Krein formula (A.2) and abbreviate
u := λV (x) + Aλ(x,x;ζ)
v := λV (y) + Aλ(y,y;ζ) ,
andα := Aλ(x,y;ζ),β := Aλ(y,x;ζ).Thelowerboundson|Gλ(x,x;ζ)and|Gλ(y,y;ζ)|
translate to:
????u −αβ
u
v
????≤
1
t
1
t.
(A.9)
????v −αβ
????≤
(A.10)
The claim will be proven on the basis of the following two observations:
1. For any set of specified values of {α,β,A(x,x;ζ),A(y,y,;ζ)}, and of v, the set
of Re u for which (A.9) holds is an interval of length at most 2/t, and a similar
statement holds for v and u interchanged and Eq. (A.9) replaced by (A.10).
Page 38
Resonant delocalization 38
2. For any solution of (A.9) and (A.10):
min{|u|,|v|} ≤ |α| + t−1.
(A.11)
The first statement is fairly obvious once one focuses on the condition on the real part in
(A.9). To prove the second assertion, let
?
Assuming (A.9) and (A.10) we have:
w :=
|u| · |v| ≥ min{|u|,|v|}
(A.12)
|u||v| − |α||β| ≤ |uv − αβ| ≤min{|u|,|v|}
t
≤
?|u||v|
t
(A.13)
where the first relation is by the triangle inequality, and the second by (A.9) and (A.10).
Hence, under the assumed condition, the real quantity w := |u||v| satisfies:
w2− |α||β| ≤
w
t.
(A.14)
Solving the quadratic equation we find:
w ≤1
2t+
?
1
(2t)2+ |α||β| ≤
1
2t+
?1
2t+
?
|α||β|
?
,
(A.15)
which implies (A.11).
To bound the probability in (A.7), let us consider the set of values of V (x) and V (y)
for which the event occurs, at specified values of the 2 × 2 matrix Aλ(ζ). Let S ⊂ R2be
the corresponding range of values of {Re u,Re v}. Then by 2., S is contained within the
union of two strips, one parallel to the Re v axis and the other parallel to the Re u axis. To
bound the measure of its intersection with the first one, we note that the relevant values of
Re u are contained in an interval of length at most 2
of u the range of values of Re v is of Lebesgue measure not exceeding 2/t (by 2.). Hence
the measure of the intersection of S with this strip is at most4
similar bound applies to the intersection of S with the second one. Adding the two, one
gets the bound claimed in (A.7).
?
1
t+?|α||β|
?
1
, and for each value
t
?
t+?|α||β|
?
, and a
A.2Consequences of Assumption D
The class of probability densities satisfying Assumption D (see Eq. (2.2)) includes those
? which have a single hump. More precisely, suppose there is some m ∈ R such that ?
is monotone increasing for v < m and monotone decreasing for v > m. If one picks
ν > 0 such that ?(m)/max{?(m − 2ν), ?(m + 2ν)} =: c < ∞, then (2.2) is satisfied
for all v0∈ R at that value of ν and c. Examples of single-hump probability densities are
Gaussian and the Cauchy densities.
Page 39
Resonant delocalization 39
Similarly as above one sees that any finite linear combination of single-hump func-
tions also lead to probability densities which satisfy (2.2).
Our next goal is to illuminate some of the consequences of (2.2). Clearly, if ? satis-
fies (2.2), then ? ∈ L∞(R) and (A.3) applies. In fact, the assumption is taylored to provide
the following extension of (A.3).
Lemma A.3. If ? ≥ 0 satisfies (2.2) (with constants ν,c > 0), then for any s ∈ (0,1),
a ∈ C and tν ≥ 1:
?
Proof. We start by estimating the left side
?
2
(1 − s)t1−s
since ν ≥ t−1. Using (2.2) we then conclude that the last factor is bounded from above by
c
ν
c 2s
ν1−s
The above two estimates imply the assertion.
1|v−a|<1
t
?(v)dv
|v − a|s≤
?
1 +(1 − s) (tν)1−s
21+sc
?−1?
?(v)dv
|v − a|s.
(A.16)
1|v−a|<1
t
?(v)dv
|v − a|s≤
sup
|v−a|<1
t
?(v)
?
2
|v−a|<1
t
dv
|v − a|s
=
sup
|v−a|<1
t
?(v) ≤
(1 − s)t1−s
sup
|v−a|<ν?(v),
(A.17)
?
1ν≤|v−a|≤2ν?(v)dv ≤
c (2ν)s
ν
?
1 1
1ν≤|v−a|≤2ν
?(v)dv
|v − a|s
≤
?
t≤|v−a|
?(v)dv
|v − a|s.
(A.18)
In view of (A.2) this lemma bears the following consequences for weighted averages
of the following type:
E[|Gλ(x,y;ζ)|sQ]
E[|Gλ(x,y;ζ)|s]
where x,y ∈ G, ζ ∈ C+and s ∈ (0,1). We denote by P(x,y)
measure.
Proposition A.4. In the situation of Proposition A.1, assume additionally that ? satis-
fies (2.2) (with constants ν,c > 0). Then, at any complex energy parameter ζ ∈ C+and
for any s ∈ (0,1) and t ≥ (λν)−1, the Green function satisfies:
?
where Axdenotes the sigma-algebra generated by V (y) , y ?= x.
Analogously to (A.6), we conclude from (A.20) that for any p ∈ (0,1) and all t ≥
(νλ)−1max?1,?21+sc
P(x,y)
s
uniformly in y ∈ G, the choice of the graph G and ζ ∈ C+.
E(x,y)
s
[Q] :=,
(A.19)
s
the corresponding probability
P(x,y)
s
(|Gλ(x,x;ζ)| > t|Au) ≤
1 +(1 − s) (tνλ)1−s
21+sc
?−1
,
(A.20)
1−s
p
1−p
?1/(1−s)?:
?|Gλ(x,x;ζ)| ≤ t??Ax
?
≥ p,
(A.21)
Page 40
Resonant delocalization40
BA large deviation principle for triangular arrays
In our analysis of the Green function’s large deviations we make use of a large deviation
principle. The statement and its proof are similar to large deviation theorems which are
familiar in statistical mechanics and probability theory [10, 11, 13]. However since a close
enough reference could not be located we enclose the proof here.
B.1A large deviation theorem
The following theorem should be regarded as a stand-alone statement. It is intended to be
read disregarding fact that the symbols which appear there (Γ and η ) were assigned a spe-
cific meaning elsewhere in the paper. The similarity does however indicate the application
of this theory to the main discussion of this work.
Theorem B.1. Let {Γ(N)
random variables indexed by η ≥ 0, satisfying the following two conditions, at some
r,ς ∈ (0,1) and Cr< ∞:
a. The functions
1
NlogE
j
(η)}N
j=1with N ∈ N, be a family of a triangular arrays of
ΨN(t;η) :=
?N
j=1
?
|Γ(N)
j
(η)|t
?
(B.1)
converge pointwise in (−ς,1):
Ψ(t) := lim
N→∞
η↓0
ΨN(t;η).
(B.2)
b. For all 1 ≤ k < N, and t1,t2∈ [0,r]
?
i=1
E
k?
|Γ(N)
i
(η)|t1
N
?
j=k+1
|Γ(N)
j
(η)|t2
?
≤ Cre(N−k)[ΨN(t1,η)−ΨN(t2,η)]E
?N
?
i=1
|Γ(N)
i
(η)|t2
?
. (B.3)
Then for every γ which coincides with −Ψ?(s) at a point s ≡ s(γ) ∈ (0,r) where
the function Ψ(s) is differentiable, and for any ε > 0, there are? N ≡? N(ε,γ) < ∞ and
1. Given the rate function I(γ) := −inft∈(0,r)[Ψ(t) + tγ] one has:
?N
j=1
ˆ η ≡ ˆ η(ε,γ) > 0 such that for all N ≥? N and 0 < η < ˆ η the following estimates hold:
P
?
|Γ(N)
j
(η)| ≥ e−(γ+ε)N
?
≤ e−I(γ)Neε(r+1)N
(B.4)
Page 41
Resonant delocalization 41
2. With respect to the s-tilted probability average defined by
Ps(Q) =
E
?
E
IQ×?N
j=1|Γ(N)
j=1|Γ(N)
j
(η)|s?
(η)|s?
??N
j
,
(B.5)
for any ? ∈ {0,...,N}:
Ps
?
?
??
??
j=1
|Γ(N)
j
(η)| ≥ e−(γ−ε)?
?
?
≤ Cre−κ(ε,γ)?/3
(B.6)
Ps
j=1
|Γ(N)
j
(η)| ≤ e−(γ+ε)?
≤ Cre−κ(ε,γ)?/3
(B.7)
where κ(ε,γ) := min{κ−(ε,γ), κ+(ε,γ)} > 0 and
κ±(ε,γ) :=
sgn∆=±
0≤s+∆≤r
sup[Ψ(s) + (Ψ?(s) ± ε)|∆| − Ψ(s + ∆)] .
(B.8)
3. For any event Q:
P(Q) ≥ e−I(γ)Ne−ε(r+1)N?Ps(Q) − Cre−κ(ε,γ)N/3?
Several remarks apply:
(B.9)
1. The function Ψ is convex, assuming the limit (B.2) exists, and therefore the above
value of I(γ) can also be presented as
I(γ) = −[Ψ(s) + γs] .
(B.10)
The error margins κ±(ε,γ) defined in (B.8) are strictly positive for any ε > due to
convexity of Ψ.
2. The proof of Theorem B.1 follows a standard procedure for such bounds: what
is a large deviation for the value of
N
probability measure becomes a regular occurrence once the measure is suitably
tilted, i.e. modified by the factor?N
to this standard procedure the observation that under the condition (B.3) the global
tilt of the measure shifts the typical values of the sample mean of logΓjfor all the
partial sums, to values in the vicinity of (−γ).
In the proof we make use of the following fact on convergence of convex functions.
1
?N
j
j=1logΓ(N)
j
with respect the the initial
j=1|Γ(N)
|sat suitable s. The statement is then
derived by relating the original and the tilted probabilities. In Theorem B.1 we add
Lemma B.2. Under the condition (B.2), one has the uniform convergence
lim
N→∞
η↓0
sup
s∈[0,r]
|ΨN(s;η) − Ψ(s)| = 0.
(B.11)
Page 42
Resonant delocalization42
Proof. This follows from the fact that if a family of convex functions converges pointwise
over an open interval, then its convergence is uniform on compact subsets, cf. [24].
Proof of Theorem B.1. Since the superscript of Γ(N)
occasionally omitted (it takes a common value for all terms within each statement).
We will choose? N ≡? N(ε,γ) < ∞ and ˆ η ≡ ˆ η(ε,γ) > 0 using Lemma B.2 such that
RN(η) := sup
s∈[0,r]
The proof of (B.4) relies on an elementary Chebychev estimate with s ∈ [0,r]:
?N
j=1
= eεsNe−NI(γ)eN[ΨN(s;η)−Ψ(s)]≤ eε(r+1)Ne−NI(γ)
for any N ≥? N and 0 < η < ˆ η by (B.12).
for any ∆ ∈ (0,r − s]:
?
j=1
≤ Cre[ΨN(s+∆;η)−ΨN(s;η)]?e∆(γ−ε)?
Infimizing over ∆ ∈ (0,r − s], we hence conclude that the left side in (B.14) is bounded
by
Cre−κ+(ε,γ)?e2?RN(η)≤ Cre−κ+(ε,γ)?/3
for any N ≥? N and 0 < η < ˆ η by (B.12).
Ps
j=1
≤ Cre[ΨN(s−∆;η)−ΨN(s;η)]?e−∆(γ+ε)?
for any ∆ ∈ (0,s]. Infimizing over this parameter, we hence conclude that the left side
in (B.16) is bounded by Cre−κ−(ε,γ)?e2?RN(η)≤ Cre−κ−(ε,γ)?/3by (B.12).
j
is somewhat redundant it will be
for all N ≥? N(ε,γ) and 0 < η < ˆ η(ε,γ):
|ΨN(s;η) − Ψ(s)| < min?ε,1
3κ(ε,γ)?,
(B.12)
P
?
|Γj(η)| ≥ e−(γ+ε)N
?
≤ eN[s(γ+ε)+ΨN(s;η)]
(B.13)
For a proof of (B.6) we again employ the Chebychev inequality and (B.3) to conclude
Ps
??
|Γj(η)| ≥ e−(γ−ε)?
?
≤ Es
?
??
j=1
|Γj(η)|∆?
e∆(γ−ε)?
(B.14)
(B.15)
The proof of (B.7) proceeds similarly. It starts from the observation that
?
??
|Γj(η)| ≤ e−(γ+ε)?
?
≤ Es
?
N
?
j=?+1
|Γj(η)|−∆?
e−∆(γ+ε)?
(B.16)
For a proof of (B.9) we estimate the regular probability of in terms of the one defined
via the tilted measure:
?
?
P(Q) ≥ eNΨN(s;η)es(γ−ε)NPs
Q and
N
?
j=1
|Γj(η)| ≤ e−(γ−ε)N
?N
j=1
?
≥ eNΨN(s;η)es(γ−ε)N
Ps(Q) − Ps
?
|Γj(η)| ≥ e−(γ−ε)N
??
.
(B.17)
Page 43
Resonant delocalization43
The first terms are estimated from below similarly as in (B.13) by e−I(γ)e−(r+1)εN. The
second term in the bracket is bounded by Cre−κ(ε,γ)N/3for any N ≥? N and 0 < η < ˆ η
B.2 Application – Proof of Theorem 6.2
according to (B.6).
The aim of this subsection is to establish a proof of the main application of the large-
deviation bounds in this paper.
Proof of Theorem 6.2. We first check the applicability of Theorem B.1. By construction,
{Γ±(j;η)}Nκ
satisfy the consistency condition (6.11). As a consequence, the quantity defined in (B.1)
agrees for both cases:
????G?Tx(xnκ,xN−1;E + iη)
j=1, which were defined in (6.10), are two families of triangular arrays. They
ΨNκ(s;η) =
1
Nκlog E
???
s?
.
(B.18)
Lemma 5.4 and Theorem 5.1 imply that for any t ∈ [−ς,1):
ϕ(t;E) ≡ ϕ(t) =lim
Nκ→∞
η↓0
ΨNκ(t;η).
(B.19)
Moreover, these bound also ensure the validity of (B.3) with arbitrary r ∈ (0,1). For a
proof of this assertion, one integrates out the random variable associated with the first
vertex on which t2occurs, cf. (5.7).
The upper bound (6.15) is hence a consequence of (B.4). For a proof of the lower
bound (6.14) we employ (B.9). We first note that the choice of b is taylored to ensure
?L(bc)
cf. (6.13) and the subsequent remark. Furthermore, using (B.6) and (B.7) we conclude
that there are? N ≡? N(?,γ) and ? η ≡ ? η(?,γ) such that for all Nκ≥? N and η ∈ (0, ? η):
1 − Ps
k=1
2nκ
Nκ
?
2nκ
Nκ
?
2nκ
Ps
x
?≥7
8,
(B.20)
?
Nκ
?
?
L(k,±)
x
(η;?)
?
≤
k=1
Ps
?
k?
j=1
|Γ±(j;η)| ≥ e−(γ−ε)??
+ Ps
?
k?
j=1
|Γ±(j;η)| ≤ e−(γ+ε)???
≤ 2Cr
k=1
e−κ(ε,γ)k/3≤
6Cr
κ(ε,γ)e−κ(ε,γ)nκ/6.
(B.21)
By choosing nκsufficiently large, this term can be made arbitrarily small since κ(ε,γ) >
0. As a consequence, we conclude that there is some n0and η0such that for all |x| ≥ n0
and η ∈ (0,η0):
Ps(Lx(η;?)) ≥1
2.
(B.22)
Page 44
Resonant delocalization44
Using this estimate in (B.9) concludes the proof of (6.15), since the second term in (B.9)
is seen to be arbitrarily small for n large enough and any factor may be absorbed for
sufficiently large Nκby decreasing the prefactor e−Nk(I(γ)+(1+r)?)in (B.9).
C Lifshitz tails
The integrated density of states of the random operator (1.1) on the tree graph T is given
by
Nλ(E) := E??δ0, P(−∞,E)(Hλ)δ0
cf. [20]. Below the spectrum of the unperturbed operator, it is expected to decay rapidly
for λ ↓ 0. Such a behavior is referred to as Lifshitz tailing. The following result is based
on a standard estimate used to establish such this decay and to our knowledge goes back
to L. Pastur, cf. [22, Thm. 9.1].
??,
(C.1)
Proposition C.1. Consider a random operator Hλ(ω) = T + V (ω), satisfying assump-
tions A–C, and assume that the random variable V (0) has exponential moments. Then
for any E ≤ inf σ(H0) and any λ ≥ 0:
Nλ(E) ≤ inf
t≥0
One may further estimate the right side in (C.2) to simplify the bound:
?e−t(∆E)E?e−λtV (0)??,
where (∆E) := −(E + 2√K) ≥ 0. In case of centered Gaussian variables, one has
E?e−λtV (0)?= eλ2t2/2which implies Nλ(E) ≤ exp(−(∆E)2/[2λ2]) (in accordance with
supσ(T).
??δ0,e−t(H0−E)δ0? E?e−λtV (0)??.
(C.2)
Nλ(E) ≤ inf
t≥0
(C.3)
the results in [20]). This bound naturally extends by symmetry also to energies E ≥
Proof of Proposition C.1. We pick t ≥ 0 and use the spectral representation (2.4):
??E
The semigroup generated by Hλ(ω) in ?2(T ) may be represented in terms of a Feynman-
Kac formula which takes the form
?
whereνtissome(non-negative)measureonajumpprocess{xt}onT ,cf.[9,Prop.II.3.4].
The result now follows from Jensen’s inequality, e−λ?t
together with Fubini’s theorem, which allows to perform the probabilistic expectation E[·]
first.
Nλ(E) = E
−∞
µλ,δ0(dξ)
?
≤ etEE
??
e−tξµλ,δ0(dξ)
?
= etEE??δ0, e−tHλδ0
??.
(C.4)
?δ0, e−tHλ(ω)δ0
?
= exp
?
−λ
?t
0
V (xs;ω)ds
?
νt(dx)
(C.5)
0V (xs;ω)ds≤ t−1?t
0e−λtV (xs;ω)ds,
Page 45
Resonant delocalization45
Acknowledgments
We thank the Departments of Physics and Mathematics at the Weizmann Institute of Sci-
ence for hospitality at visits during which some of the work was done. This research was
supported in part by NSF grants DMS-0602360 (MA) and DMS-0701181 (SW), and a
Sloan Fellowship (SW).
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