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
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