Resonant delocalization for random Schrödinger operators on tree graphs
RResonant Delocalizationfor Random Schr ¨odinger Operatorson Tree Graphs
Michael Aizenman Simone Warzel
December 14, 2011
Abstract
We analyse the spectral phase diagram of Schr¨odinger operators T + λV on regular treegraphs, with T the graph adjacency operator and V a random potential given by iid randomvariables. The main result is a criterion for the emergence of absolutely continuous ( ac ) spec-trum due to fluctuation-enabled resonances between distant sites. Using it we prove that forunbounded random potentials ac spectrum appears at arbitrarily weak disorder ( λ (cid:28) inan energy regime which extends beyond the spectrum of T . Incorporating considerations ofthe Green function’s large deviations we obtain an extension of the criterion which indicatesthat, under a yet unproven regularity condition of the large deviations’ ’free energy function’,the regime of pure ac spectrum is complementary to that of previously proven localization.For bounded potentials we disprove the existence at weak disorder of a mobility edge beyondwhich the spectrum is localized. Keywords.
Anderson localization, absolutely continuous spectrum, mobility edge, Cayleytree
Dedicated to Hajo Leschke on the occasion of his 66th birthday.
M. Aizenman: Depts. of Physics and Mathematics, Princeton University, Princeton NJ 08544, USAS. Warzel: Zentrum Mathematik, TU M¨unchen, Boltzmannstr. 3, 85747 Garching, Germany; e-mail:[email protected] (corresponding author)
Mathematics Subject Classification (2010):
Primary 82B44; Secondary 47B80. a r X i v : . [ m a t h - ph ] D ec esonant delocalization Contents
Im Γ(0; E + iη ) . . . . . . . . . . . . . . . . 194.2 A conditional proof of the criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Heuristics of the resonance mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Resonances based on the Lyapunov behavior . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Lower bound on the mean number of resonant sites . . . . . . . . . . . . . . . . . . . . . 244.6 The enabling second moment upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A Fractional-moment bounds 44
A.1 Weak- L bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.2 The regularity assumption D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B A large deviation principle for triangular arrays 48
B.1 A general large deviation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.2 Applications to Green function’s large deviations . . . . . . . . . . . . . . . . . . . . . . 51 esonant delocalization The subject of this work are the spectral properties of random self-adjoint operators in the Hilbertspace (cid:96) ( T ) associated with the vertex set T of a regular rooted tree graph of a fixed branchingnumber K > . The operators take the form H λ ( ω ) = T + λ V ( ω ) , (1.1)with T the adjacency matrix and V ( ω ) a random potential, i.e., a multiplication operator which isspecified by a collection of random variables indexed by T . For simplicity we focus on the caseof independent identically distributed ( iid ) random variables of absolutely continuous distribution, (cid:37) ( v ) dv . The strength of the disorder is expressed through the parameter λ ≥ . Some of the resultspresented below will be formulated for unbounded random potentials, in which case the support ofthe distribution of V ( x ) is assumed to be the full line. For other results we assume that the rangeof values of V ( x ) is the interval [ − , .It is well known that random Schr¨odinger operators, of which the above tree version is a rela-tively more approachable example, exhibit regimes of spectral and dynamical localization wherethe operator’s spectrum consists of a dense collection of eigenvalues with localized eigenfunctions(cf. [14, 32, 36, 26]). However, it still remains an outstanding mathematical challenge to elucidatethe conditions for the occurrence of continuous spectrum, and in particular absolutely continuous (henceforth called ‘ ac ’) spectrum, in the presence of homogeneous disorder. The significance ofthe ac spectrum from the scattering perspective, or a schematic conduction experiment, is illus-trated in Figure 1. In the operator’s ( E, λ ) phase diagram , the boundary separating the regime oflocalization from the regime of continuous spectrum , assuming such is found, is referred to as the mobility edge [10].The results presented here focus on a new resonance-driven mechanism by which ac spectrumoccurs for operators such as H λ ( ω ) in the setup described above. Following is a summary of themain points.1. A new sufficiency criterion is derived for ac spectrum on tree graphs in terms of a relatedLyapunov exponent.The guiding observation for . is that localized modes join into extended states when their energydifferences are smaller that the corresponding tunneling amplitudes. The latter decay exponen-tially in the distance at the rate whose typical values is given by the Lyapunov exponent. Hencethe probability of a mixing resonance between localized modes at specified location is exponen-tially small. However, when the volume of the relevant configuration space increases exponentiallyresonances will be found, and delocalization prevails. This criterion is particularly applicable atweak and moderate disorder. It is applied here for two results, which apply separately for boundedand for unbounded random potentials:2. For unbounded potentials we show that ac spectrum appears ’discontinuously’ at arbitrarilyweak disorder in regimes with very low density of states (of Lifshits tail asymptotic falloff).This 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 spectrumbelow it.3. For bounded random potentials it is shown that at weak disorder there is no mobility edgebeyond which the states are localized. This has the surprising implication that for this casethe standard picture of the phase diagram needs to be corrected. esonant delocalization Ψ( ξ ) = e ikξ + R e − ikξ Figure 1: A model setup for quantum conduction through the graph (after [30]): particles aresent at energy E = k + U wire down a wire which is attached to the graph at x = 0 . In thestationary state the particles’ wave function is described along the wire by the combination ofplane waves e ik E ξ + R E e − ik E ξ , and along the graph it is given by a decaying solution of theSchr¨odinger equation. The natural matching conditions relate the reflection coefficient R E to theGreen function, and it is found that | R E | < exactly if Im (cid:104) δ , ( H λ − E − i − δ (cid:105) (cid:54) = 0 , whichis also the condition for E to be in the support of the ac spectrum of H λ .In essence, . and . show that while in one dimension arbitrary weak level of disorder yieldslocalization, on trees the ac spectrum is quite robust.4. Extending the analysis which yields the criterion . through considerations of the Greenfunction’s large-deviations, we obtain an improved sufficiency criterion for ac spectrumwhich appears to be complimentary to the previously derived criterion for localization. Toreduce technicalities, the derivation of the extended criterion is limited to unbounded poten-tials with support in R .The last point is an indication that the mechanism which is discussed here is in essence the relevantone, in the tree setup.A physics-oriented summary of the results . and . was given in [8] and, correspondingly, [9].Our purpose here is to provide the detailed derivation of the above statements. In the proof we donot present the direct construction of extended states, but instead focus on properties of the Greenfunction which in essence convey the same information. By a simple calculation, cf. (3.6), σ ( T ) = [ − √ K, √ K ] . (1.2)For ergodic random potentials, a class which includes the iid case, the spectrum of H λ ( ω ) = T + λV ( ω ) is almost surely given by a non-random set, which under the present assumptionsis [14, 32, 26]: σ ( H λ ) = σ ( T ) + λ supp ρ . (1.3) Even 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 Cheegerconstant. Nevertheless, this larger set does describe the operator’s (cid:96) ∞ -spectrum. esonant delocalization λ = 0 upward:1. In the unbounded case, of potentials with supp (cid:37) = R , the spectrum of H λ ( ω ) changesdiscontinuously from an interval to the full line.2. In the bounded case the spectrum changes continuously, spreading at a linear rate whichequals if supp (cid:37) = [ − , .The determination of the nature of the spectral measures whose support spans σ ( H λ ) requireshowever a more detailed consideration. The spectral analysis proceeds through the study of thecorresponding Green function G λ ( x, y ; ζ, ω ) := (cid:68) δ x , ( H λ ( ω ) − ζ ) − δ y (cid:69) , (1.4)where ζ ∈ C + := { ζ ∈ C | Im ζ > } and δ x ∈ (cid:96) ( T ) is the Kronecker function localizedat x ∈ T . In particular, the spectral measure µ λ,δ x ( · ; ω ) associated with H λ ( ω ) and δ x ∈ (cid:96) ( T ) isrelated to the Green function through the Stieltjes transform: G λ ( x, x ; ζ, ω ) = (cid:90) µ λ,δ x ( du ; ω ) u − ζ . (1.5)Of particular interest is the limiting value G λ ( x, x ; E + i , ω ) := lim η ↓ G λ ( x, x ; E + iη, ω ) ,which exists for almost every E ∈ R (by the general theory of the Stieltjes transform [17, 14, 32]).The different spectra of H λ ( ω ) are associated with the Lebesgue decomposition of the mea-sures µ λ,δ x ( · ; ω ) into their different components: pure point ( pp ), singular continuous ( sc ), andabsolutely continuous ( ac ), not all of which need be present. Ergodicity, combined with the proofof equivalence of the local measures [24, 25], implies that the supports of the different componentsof µ λ,δ x ( du ; ω ) are also almost surely non-random [14, 32, 26], and coincide for all x ∈ T .The spectral characteristics are related to the dynamical properties of the unitary time evolutiongenerated by H λ ( ω ) (cf. the RAGE theorem in [36, 26]) and to questions of conduction.The absolutely continuous component of µ λ,δ x ( · ; ω ) is given by µ ( ac ) λ,δ x ( du ; ω ) = π − Im G λ ( x, x ; u + i , ω ) du , (1.6)which is not zero provided the non-negative function satisfies Im G λ ( x, x ; E + i , ω ) (cid:54) = 0 ona positive measure set of energies. As noted in [30, 7], this condition is equivalent also to thestatement that current which is injected coherently at energy E down a wire attached at a site x will be conducted through the graph to infinity, see Figure 1.Another possible behavior is localization : Definition 1.1.
The operator H λ ( ω ) associated with a metric graph (not necessarily a tree) is saidto exhibit:i. spectral localization in an interval I ⊂ R if the spectral measures µ λ,δ x ( · ; ω ) associated to δ x ∈ (cid:96) ( T ) are almost surely all of only pure-point type in I .ii. exponential dynamical localization in I if for all x ∈ T and R > sufficiently large: (cid:88) y ∈T :dist( x,y )= R E (cid:18) sup t ∈ R |(cid:104) δ x , P I ( H λ ) e − itH λ δ y (cid:105)| (cid:19) ≤ C λ e − µ λ ( I ) R , (1.7)at some µ λ ( I ) > , and C λ < ∞ , with E [ · ] denoting the average with respect to the underly-ing probability measure. esonant delocalization ϕ λ (1; E ) = − log K holds onlyalong a line. The intersection of the curve with the energy axis is stated exactly, while in otherdetails the depiction is only schematic.For a particle which is initially placed at x ∈ T the left side of (1.7) provides an upper boundon the probability to be found a time t later at distance R from x , under the quantum mechanicaltime-evolution generated by H λ restricted to states with energies in I . Dynamical localization isthe stronger of the two statements. By known arguments (i.e., the Wiener and RAGE theorem,cf. [26, 36]) it implies also the spectral localization. The spectral ‘phase diagram’ of the operators considered here was studied already in the earlyworks of Abou-Chacra, Anderson and Thouless [1, 2]. Arguments and numerical work presentedin [2] led the authors to surmise that for (centered) unbounded random potentials, the mobilityedge, which separates the localization regime from that of continuous spectrum, exists at a locationwhich roughly corresponds to the outer curve in Figure 2. Curiously, for λ ↓ that line approachesenergies | E | = K + 1 which is not the edge of the spectrum of the limiting operator T .Rigorous results for the above class of operators have established the existence of a localizationregime and of regions of ac spectrum, leaving however a gap in which neither analysis applied.More specifically, the following was proven for the class of operators described above (underassumptions which are somewhat more general than the conditions A-D below): Localization regime [4, 5]:
For any unbounded random potential with supp ρ = R , whose prob-ability distribution satisfies also a mild regularity condition, there is a regime of energies ofthe form: | E | > γ ( λ ) , with lim λ ↓ γ ( λ ) = K + 1 , (1.8)where with probability one, H λ ( ω ) has only pure point spectrum, and where it also exhibitsdynamical localization. Extended states / continuous spectrum [27, 28, 6, 20]:
For energies | E | < √ K and at weakenough disorder, i.e. | λ | < (cid:98) λ ( E ) (with (cid:98) λ ( E ) ↓ for | E | → √ K ), the operator’s spectrumis almost surely (purely) absolutely continuous. esonant delocalization Localization (p.p. spectrum)Extended states (a.c. spectrum) λ − √ K √ K E λ min ! ≥ ( √ K − " Figure 3: Sketch of the previously expected phase diagram for the Anderson model on the Bethelattice (the solid line) and the correction presened here (dashed line). Our analysis suggests thatat weak disorder there is no localization and the spectrum is purely ac . While the proof of that isincomplete, we prove that for λ ≤ ( √ K − / near the spectral edges the spectrum is purelyabsolutely continuous.Thus, the previous results have covered two regimes whose boundaries, sketched in Figure 2,do not connect. Particularly puzzling has been the region of weak disorder and √ K < | E | < K + 1 . (1.9)At those energies the mean density of states vanishes to all orders in λ , for λ ↓ [30]. Such rapiddecay is characteristic of the so-called Lifshits tail spectral regime. In finite dimensions it is knownto lead to localization [32, 26]. On tree graphs however, this implication could not be established,and localization at weak disorder was successfully proven [5] only for | E | > K + 1 (cf. Figure 2and Proposition 2.6 below). For energies E in the range (1.9) the nature of the spectrum at weakdisorder has been a puzzle even at the level of heuristics [30]. The question is answered by thesecond of the results mentioned above. It has been expected that for bounded random potentials the phase diagram of the random operators(1.1) looks qualitatively as depicted in Fig. 3 (c.f. [2, 12]), the key points being:1. At weak and moderate disorder a mobility edge has been expected to occur, within whichthe spectrum is absolutely continuous and beyond which it is pure point - consisting thereof a dense countable collection of eigenvalues with proper eigenfunctions.2. The extended states disappear at strong enough disorder ( λ > λ sd ( K ) ), where completelocalization prevails.Significant parts of this picture have been supported by rigorous results, in particular completelocalization at strong disorder [4, 5], and the persistence of ac spectrum at weak disorder [27, 6, 20](though some questions remain as to the precise asymptotics of λ sd ( K ) for K → ∞ . However,as stated in 3. above, at weak and moderate disorder, for regular trees this picture needs to bemodified.Let us now turn to a more precise formulation of the statements listed above. esonant delocalization Our discussion will focus on operators of the form (1.1) in the Hilbert space (cid:96) ( 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 > (the root beingdenoted by ∈ T ).B: T is the adjacency operator of the graph, i.e., ( T ψ ) ( x ) := (cid:80) dist( x,y )=1 ψ ( y ) for all ψ ∈ (cid:96) ( T ) .C: { V ( x ; ω ) | x ∈ T } form independent identically distributed ( iid ) random variables, with aprobability distribution (cid:37) ( v ) dv with (cid:37) ∈ L ∞ ( R ) ,which has a finite moment, i.e., for some ς ∈ (0 , : (cid:90) | v | ς (cid:37) ( v ) dv < ∞ . (2.1)D: The probability density (cid:37) is bounded relative to its minimal function , which we define as M ( v ) := inf ν ∈ (0 , (2 ν ) − (cid:82) | x − v |≤ ν (cid:37) ( x ) dx . I.e., for Lebesgue-almost all v ∈ R : (cid:37) ( v ) ≤ c M ( v ) , (2.2)with a finite constant c .In case of unbounded potentials, we will mostly restrict our attention to those which addition-ally satisfy the following assumption:E: For all k < ∞ : inf | v |≤ k (cid:37) ( v ) > .While condition D could be relaxed, let us note that it is satisfied by all probability distributionswhose densities are bounded functions on R of finitely many humps (see Appendix A). This classincludes finite linear combinations of Gaussian, Cauchy, and the piecewise constant functions. For a criterion which is particularly useful at weak disorder (and, separately, also for high valuesof K ) let us introduce the Lyapunov exponent, which we define for the rooted tree (with the rootat x = 0 ) as: L λ ( E ) := − E (log | G λ (0 , E + i | ) . (2.3)Since Lyapunov exponents are usually associated with dynamical systems, let us just commentthat the relevance of such a perspective can be seen from the recursive structure of the rooted tree,and the factorization of the Green function which are discussed in Proposition 3.1 below.The first of the results listed in the introduction is: Theorem 2.1.
For the random operator H λ ( ω ) as in (1.1) , with λ > , satisfying Assumptions A–D: at Lebesgue-almost every E ∈ R at which L λ ( E ) < log K , (2.4) the operator’s Green function satisfies almost surely: Im G λ (0 , E + i > . (2.5) esonant delocalization Proposition 2.2.
Assume that the distribution of V (0; · ) conditioned on the values of the potentialat all other sites is almost surely absolutely continuous. If for some interval I ⊂ R , the condi-tion (2.5) holds for almost every E ∈ I then with probability one within I the spectral measure µ λ,δ ( du ; ω ) is absolutely continuous. If the analogous conditions holds for all sites x , then thespectrum of H λ ( ω ) is almost surely purely absolutely continuous in I . The proof combines the characterization (due to Aronszajn [11]) of the support the singularcomponent of µ λ,δ ( du ; ω ) as the set of energies where condition (2.5) fails, with the spectralaveraging principle which implies that if this set is of zero Lebesgue measure than also the spectralmeasure of this set is zero for almost all realizations of the potential. This argument applies as wellto all other choices for the graph and for the unperturbed operator T . A simple exact calculation (cf. Subsection 3.2) shows that for λ = 0 one has L ( E ) < log K if and only if | E | < K + 1 . (2.6)Curiously, the energy range defined by the above condition is strictly larger that the (cid:96) -spectrumof T (cf. (1.2)).It seems natural to expect L λ ( E ) to be continuous in ( λ, E ) , a fact which is easily estab-lished for the Cauchy random potential, i.e., for (cid:37) ( v ) = π − (cid:0) v + 1 (cid:1) − , in which case L λ ( E ) = − log | G (0 , E + iλ ) | . In such a situation, Theorem 2.1 together with Proposition 2.2 carry theimplication that for any closed energy interval I in the range | E | < K +1 , at weak enough disorderthe random operator H λ ( ω ) has almost surely purely absolutely continuous spectrum in I .While we do not have a general proof of the continuity of L λ ( E ) , one can show that its averagesover intervals are continuous. Using this weaker continuity we arrive at the following conclusion. Corollary 2.3.
For unbounded random potentials with supp (cid:37) = R , under the assumption of The-orem 2.1 in every closed interval I ⊂ ( − K − , K + 1) there is absolutely continuous spectrumat sufficiently low disorder, i.e. the condition (2.11) holds at a set of positive measure of energiesprovided < λ < (cid:98) λ ( I ) at some (cid:98) λ ( I ) > . The proof of Corollary 2.3 which is given below in Section 6.1 yields also an explicit lowerbound on the fraction of I occupied by ac spectrum.For bounded potentials we prove, through other estimates of L λ ( E ) which are provided inSection 6.2: Corollary 2.4.
For bounded random potentials with supp (cid:37) = [ − , , under the assumptionof Theorem 2.1 for λ < [ √ K − / (2.7) with probability one H λ ( ω ) has purely absolutely continuous spectrum at the spectral edges, i.e.within a range of energies of the form | E λ | − δ ( λ ) ≤ | E | ≤ | E λ | . (2.8) at some δ ( λ ) > , with E λ = inf σ ( H λ ) = − √ K − λ . esonant delocalization The criterion provided by Theorem 2.1 can be improved by taking into account large deviationeffects. The pertinent observation here is that while typically log | G λ (0 , x ; E + i | / | x | ≈ − L λ ( E ) , (2.9)with | x | := dist( x, , there typically also are exponentially many sites to which the Green func-tion (which can be viewed as expressing the tunneling amplitude) exhibits a slower decay rate. Anotable feature of the resulting improved criterion is that it appears to be complementary to thepreviously developed criterion for localization.Information about the large deviations can be recovered from a suitable free energy function,which we define for s ∈ [ − ς, by ϕ λ ( s ; E ) := lim | x |→∞ log E [ | G λ (0 , x ; E + i | s ] | x | , (2.10)and for s = 1 by ϕ λ (1; E ) := lim s ↑ ϕ λ ( s ; E ) .The existence of the limit (for Lebesgue-almost all E ∈ R ) is proven below in Section 3.3.We also show there that the function s (cid:55)→ ϕ λ ( s ; E ) , which is obviously convex, is monotonedecreasing in s over [ − ς, , and thus the limit at s = 1 is well-defined for almost all E ∈ R .Following is the improved version of Theorem 2.1. To avoid an additional complication in thederivation, we establish it here for potentials with supp (cid:37) = R only. Theorem 2.5.
Under Assumptions A–E, for any λ > and Lebesgue-almost all E ∈ R at which ϕ λ (1; E ) > − log K , (2.11) the operator’s Green function satisfies almost surely Im G λ (0 , E + i > . (2.12)By convexity arguments ϕ λ ( s ; E ) ≥ − s L λ ( E ) (cf. Section 3.3) and hence the condition (2.4)of Theorem 2.1 is satisfied whenever (2.11) holds.For a better appreciation of the criterion provided by the condition (2.11), let us note that theopposite inequality implies localization. This is implied by the previously established localizationresults [4, 5] which can be recast as follows (cf. Thm 1.2, and Eqs. (2.10), (2.12) in Ref. [5]). Proposition 2.6.
Under Assumptions A–C, if the following condition holds for an interval I anda specified λ > E ∈ I ϕ λ (1; E ) < − log K , (2.13) then the operator H λ ( ω ) exhibits exponential dynamical localization in I , in the sense of (1.7) with some µ λ ( I ) > .Furthermore, the domain in which (2.13) holds includes for each energy | E | > K + 1 aninterval with a positive range of λ > . The relation of the condition (2.13), which encodes information about the decay of the Greenfunction, with the time evolution operator is explained by the following bound: E (cid:18) sup t ∈ R |(cid:104) δ x , P I ( H λ ) e − itH λ δ y (cid:105)| (cid:19) ≤ C s,λ (cid:90) I E ( | G ( x, y ; E + i | s ) dE . (2.14) esonant delocalization s ∈ [0 , and λ > at some constant C s,λ < ∞ . This inequality is areformulation of a result of [5] on the eigenfunction correlator which was extended in [33] so asto apply directly to infinite systems. (This relation holds in the broader context of operators withrandom potential on arbitrary graphs.)One may add that if it is only known that for almost all E ∈ Iϕ λ (1; E ) < − log K (2.15)then one may still conclude [4] that the operator has only pure point spectrum in I , though not nec-essarily of uniform localization length. (The argument proceeds by establishing lim inf η ↓ (cid:80) y ∈T E [ | G λ ( x, y ; E + iη ) | s ] < ∞ for some s ∈ (0 , and all x ∈ T , and then invoking the Simon-Wolff criterion [35] instead of (2.14)).
1. The spectral criteria provided by Theorems 2.1 and Theorems 2.5 for for ac spectrum, andProposition 2.6 for localization extend to the corresponding operator on the fully regulartree graph B , where every vertex has exactly K + 1 neighbors. The Green function of theoperator on B can be computed from the one on the rooted tree T with the help of therecursion relation (3.3) below. In particular, this implies coincidence of the regimes of ac spectra of the operator H λ on T and B .2. At first sight the (cid:96) -nature of the condition (2.11) for ac spectrum may be surprising since– ignoring fluctuations – the loss of square summability seems to correspond to an (cid:96) -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 thefluctuation-reduction arguments which were employed to prove stability under weak dis-order of the ac spectrum for energies E ∈ σ ( T ) [27, 6, 20].3. The conditions (2.11) for ac spectrum and (2.15) for localization are not fully complemen-tary since it was not yet proven that the equality ϕ λ (1; E ) = − log K holds in the phasediagram only along a curve. Hence it will be good to see a proof that ϕ λ (1; E ) is differen-tiable in ( λ, E ) with only isolated critical points, and that it is likewise regular in E for eachgiven λ . This could allow to conclude that the phase diagram of H λ includes only regimes oflocalization and regimes of purely ac spectrum (i.e., no sc spectrum), separated by a curveor curves, which are the mobility edge(s).4. The key observation that rare resonances, whose probabilities of occurrence decay expo-nentially in the distance, may actually be found to occur on all distance scales when thevolume is also growing exponentially fast, is not applicable to graphs of finite dimension.However, it may be of relevance for random operators on other hyperbolic graphs whichmay include loops (examples of which were considered in [21, 22, 29]), and also for theanalogous random operators on the Poincar´e disk. Another setup which it will be of interestto see analyzed are random operators on hypercubes of increasing dimension, which formthe configuration spaces of a many particle system. esonant delocalization Analysis on trees, of this as well as of other problems, is aided by the observation that upon theremoval of any site x the tree graph splits into a collection of disconnected components, which incase x is the root are isomorphic to the original graph. For different problems on trees this leads torecursion relations in terms of suitably selected quantities. The following notation will facilitatethe formulation of such relations in the present context.1. For a collection of vertices v , ...v n on a tree graph T we denote by T v ,...v n the disconnectedsubgraph obtained by deleting this collection from T .2. We denote by H T (cid:48) , with T (cid:48) ⊂ T , the restriction of H to (cid:96) ( T (cid:48) ) . E.g., H T v ,...vn is theoperator obtained by eliminating all the matrix elements of H involving any of the removedsites.3. The Green function, G T (cid:48) ( x, y ; ζ ) , for a subgraph T (cid:48) as above, is the kernel of the resolventoperator ( H T (cid:48) − ζ ) − , with ζ ∈ C + . This function vanishes if x and y belong to differentconnected components of T (cid:48) , and otherwise it stands for the Green function correspondingto the component which contains the two.In particular: G T u ( x, y ; ζ ) and G T u,v ( x, y ; ζ ) are the Green functions for the subtree whichis obtained by removing u or, respectively u and v , and all the vertices which are past theremoved site(s) from the perspective of x and y .4. Given an oriented simple path in T which passes through u (cid:54) = 0 , we abbreviate (assumingthe path itself is clear within the context): Γ( u ; ζ ) ≡ Γ − ( u ; ζ ) := G T u − ( u, u ; ζ ) , (3.1) Γ + ( u ; ζ ) = G T u + ( u, u ; ζ ) , where u − and u + are the neighboring sites of u on that path. (The paths we shall encounterbelow typically start at the root, of a rooted tree, and are oriented away from it.) For the root , we will also use the convention Γ(0; ζ ) := G (0 , ζ ) . (3.2)5. Any rooted tree T is partially ordered by the relation x ≺ y (resp. x (cid:22) 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 rootedregular tree, i.e., G ( x, y ; ζ ) = G T ( x, y ; ζ ) . Moreover, we also drop the dependence of variousquantities on λ at our convenience. 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: esonant delocalization
1. For any x ∈ T : G ( x, x ; ζ ) = (cid:16) λV ( x ) − ζ − (cid:88) y ∈N x G T x ( y, y ; ζ ) (cid:17) − , (3.3) where N x := { y ∈ T | dist( x, y ) = 1 } denotes the set of neighbors of x .2. For any pair of partially ordered sites, ≺ x ≺ y , G ( x, y ; ζ ) = G ( x, x ; ζ ) (cid:89) x ≺ u (cid:22) y Γ − ( u ; ζ ) = G ( y, y ; ζ ) (cid:89) x (cid:22) u ≺ y Γ + ( u ; ζ ) . (3.4) where the ± subscripts on Γ are defined relative to the root. These relations are among the generally used tools for spectral analysis on trees. They canbe derived by the resolvent identity, or alternatively through a random walk representation of theGreen function, cf. [1, 27, 6, 20]. We will use the following implication of the above.1. The relation (3.3) yields the recursion relation : Γ(0; ζ ) = (cid:16) λ V (0) − ζ − (cid:88) y ∈N +0 Γ( y ; ζ ) (cid:17) − , (3.5)where N +0 is the set of forward neighbors of the root in T .In particular: the Green function G (0 , ζ ) of the adjacency operator T is given by theunique value of Γ in C + which satisfies the quadratic equation K Γ + ζ Γ + 1 = 0 . (3.6)From this, one can directly determine that T has the spectrum given by (1.2), and the spectralmeasure µ ,δ ( dE ) is ac with density (cid:112) (4 K − E ) + / (2 πK ) .2. As a special case of (3.4), the Green function G (0 , x ; ζ ) factorizes into a product of theabove variables, taken along the path from the root to x : G (0 , x ; ζ ) := (cid:89) (cid:22) u (cid:22) x Γ( u ; ζ ) . (3.7)Moreover, denoting by x − the site preceding x from the direction of the root, (3.4) alsoimplies: G (0 , x ; ζ ) = G T x (0 , x − ; ζ ) G ( x, x ; ζ ) . (3.8)More generally, for any triplet of sites { x, u, y } ⊂ T such that the removal of u disconnectsthe other two: G ( x, y ; ζ ) = G T u ( x, u − ; ζ ) G ( u, u ; ζ ) G T u ( u + , y ; ζ ) (3.9)where u − and u + are the neighboring sites of u , on the x and y sides, correspondingly. esonant delocalization To conclude qualitative information on the rate at which | G λ (0 , x ; E + i | decays in x , we shallnow establish the existence, monotonicity (in s ), and finite volume bounds for the Green function’sfree energy (2.10). It is more convenient to carry the analysis first for complex values of the energyparameter. Thus, we extend the domain of the function to include also C + = { z ∈ C | Im z > } ,where the function is defined simply as ϕ λ ( s ; ζ ) := lim | x |→∞ | x | log E [ | G λ (0 , x ; ζ ) | s ] , (3.10)for all ζ ∈ C + . For the following statement, we recall that ς ∈ (0 , is a moment for which it isassumed that E [ | V (0) | ς ] < ∞ . Theorem 3.2. At any value of the energy parameter in the upper half-plane, ζ ∈ C + : Forall s ∈ [ − ς, ∞ ) the limit in (3.10) exists and the function [ − ς, ∞ ) (cid:51) s (cid:55)→ ϕ λ ( s ; ζ ) has thefollowing properties:(a) ϕ λ ( s ; ζ ) is convex and non-increasing in s ∈ [ − ς, ∞ ) .(b) For s ∈ [0 , : − s L λ ( ζ ) ≤ ϕ λ ( s ; ζ ) ≤ − s log √ K , (3.11) where L λ ( ζ ) := − E [log | G λ (0 , ζ ) | ] is the Lyapunov exponent.(c) For any s ∈ [ − ς, ∞ ) and x ∈ T : C ± ( s ; ζ ) − e | x | ϕ λ ( s ; ζ ) ≤ E [ | G λ (0 , x ; ζ ) | s ] ≤ C ± ( s ; ζ ) e | x | ϕ λ ( s ; ζ ) (3.12) with C ± ( s ; ζ ) ∈ (0 , ∞ ) , which at any fixed s ∈ [ − ς, are bounded uniformly in ζ ∈ K + i (0 , for any compact K ⊂ R .(d) The derivative at s = 0 is given by the (negative) Lyapunov exponent, i.e. for all ζ ∈ C + : ∂ϕ λ ∂s (0; ζ ) = − L λ ( ζ ) . (3.13) At Lebesgue-almost all real energies, E ∈ R : for all s ∈ [ − ς, the limit in (2.10) existsand is finite. The function [ − ς, (cid:51) s (cid:55)→ ϕ λ ( s ; E ) coincides with the limiting value of ϕ λ ,i.e., for all s ∈ [ − ς, and all E ∈ R : ϕ λ ( s ; E ) = lim η ↓ ϕ λ ( s ; E + iη )= lim | x |→∞ η ↓ | x | log E [ | G λ (0 , x ; E + iη ) | s ] . (3.14) In particular, within the reduced range: s ∈ [ − ς, , the function ϕ λ ( s ; E ) shares the prop-erties listed in (a)-(c), and the Lyapunov exponent relation (3.13) also holds for almost allreal values of ζ (= E ) . The relation (3.14) in particular asserts that for s ∈ [ − ς, the limits η ↓ and | x | → ∞ commute. This does not generally extend to s ≥ , in which case the limit η ↓ may diverge iftaken first (for E in the regime of pure-point spectrum), while the quantity on the left is finite andnon-increasing in s for all s ≥ − ς . However, let us add that under certain conditions the constraint s < could be lifted. As it should be clear from the proof in Section 3.3.2, the relevant condition esonant delocalization s and E = Re ζ thesuper- and sub-multiplicativity bounds of Lemma 3.3 and Lemma 3.4 hold with constants whichare uniform in Im ζ . This condition could be satisfied even at s ≥ if, for instance, the s -momentsof the Green function factors which yield these constants stay finite as η (cid:38) due to a smoothingeffect of the absolutely continuous spectrum. Our proof of Theorem 3.2 is based on super- and sub-multiplicativity in | x | of the Green function’smoments, properties which are related to the Green function’s factorization.Following is the essential statement. Lemma 3.3.
If either s ∈ [ − ς, ∞ ) and ζ ∈ C + , or s ∈ [ − ς, and ζ = E + i , then for any twovertices ≺ u ≺ x (and u ± and x − defined in (3.9) ): C − ( s ; ζ ) − ≤ E (cid:0) | G T x (0 , x − ; ζ ) | s (cid:1) E ( | G T u (0 , u − ; ζ ) | s ) E ( | G T u,x ( u + , x − ; ζ ) | s ) ≤ C + ( s ; ζ ) (3.15) with some < C + ( s ; ζ ) , C − ( s ; ζ ) < ∞ which, at fixed s ∈ [ − ς, are uniformly bounded in ζ ∈ K + i (0 , for any compact K ⊂ R . Furthermore for fixed s and ζ , within the above range, lim s → C − ( s ; ζ ) = lim s → C + ( s ; ζ ) = 1 . (3.16) Proof.
Using the factorization representation (3.9), and the statistical independence of the twofactors which are in the denominator of (3.15) we may write: E (cid:0) | G T x (0 , x − ; ζ ) | s (cid:1) E ( | G T u (0 , u − ; ζ ) | s ) E ( | G T u,x ( u + , x − ; ζ ) | s ) = Av ( s ) u (cid:0) | G T x ( u, u ; ζ ) | s (cid:1) (3.17)where Av ( s ) u ( · ) represents the weighted probability average: Av ( s ) u ( Q ) = E (cid:0) | G T u (0 , u − ; ζ ) | s | G T u,x ( u + , x − ; ζ ) | s × Q (cid:1) E ( | G T u (0 , u − ; ζ ) | s ) E ( | G T u,x ( u + , x − ; ζ ) | s ) (3.18)To estimate this quantity we note that by (3.3): G T x ( u, u ; ζ ) = (cid:16) λV ( u ) − ζ − (cid:88) v ∈N u G T u,x ( v, v ; ζ ) (cid:17) − (3.19)
1. The upper bound:
In case s ≥ , the operator-theoretic bound | G T x ( u, u ; ζ ) | ≤ (Im ζ ) − yields the upper bound in (3.15) with C + := (Im ζ ) − .In case s ∈ [0 , , the expression (3.19) and (A.5) readily imply that: Av ( s ) u (cid:0) | G T x ( u, u ; ζ ) | s (cid:1) ≤ s (cid:107) (cid:37) (cid:107) s ∞ (1 − s ) λ s (=: C + ) . (3.20)In case s ∈ [ − ς, , the expression (3.19) together with the inequality ( | a | + | b | ) σ ≤ | a | σ + | b | σ for σ ∈ [0 , also implies: Av ( s ) u (cid:0) | G T x ( u, u ; ζ ) | s (cid:1) ≤ λ − s E (cid:2) | V ( u ) | − s (cid:3) + | ζ | − s + (cid:88) v ∈N u Av ( s ) u (cid:0) | G T u,x ( v, v ; ζ ) | − s (cid:1) . (3.21) esonant delocalization v (cid:54)∈ { u − , u + } , we use (3.20) to conclude that Av ( s ) u (cid:0) | G T u,x ( v, v ; ζ ) | − s (cid:1) ≤ λ s (1 + s ) 2 s (cid:107) (cid:37) (cid:107) s ∞ . (3.22)In the remaining cases v ∈ { u − , u + } , we use the factorization property (3.8), Jensen’s inequalityand (3.20) to conclude: Av ( s ) u (cid:0) | G T u ( u − , u − ; ζ ) | − s (cid:1) = (cid:104) Av ( s ) u − (cid:0) | G T u ( u − , u − ; ζ ) | s (cid:1)(cid:105) − ≤ Av ( s ) u − (cid:0) | G T u ( u − , u − ; ζ ) | − s (cid:1) ≤ λ s (1 + s ) 2 s (cid:107) (cid:37) (cid:107) s ∞ (=: C + ) , (3.23)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 > . The expression (3.19) implies for any t > and any ε ∈ (0 , min { ς, s } ] : Av ( s ) u (cid:0) | G T x ( u, u ; ζ ) | s (cid:1) ≥ Av ( s ) u (cid:32) (cid:2) For all v ∈ N u : | G T u,x ( v, v ; ζ ) | ≤ t (cid:3) [ λ | V ( u ) | + | ζ | + ( K + 1) t ] s (cid:33) ≥ (cid:81) v ∈N u Av ( s ) u (cid:0) (cid:2) | G T u,x ( v, v ; ζ ) | ≤ t (cid:3)(cid:1) [ λ ε E ( | V (0) | ε ) + | ζ | ε + ( K + 1) ε t ε ] s/ε . (3.24)The last inequality derives from that fact that the random variables appearing in the numeratorand V ( u ) are independent (even with respect to Av ( s ) u ( · ) ), and Jensen’s inequality, which yields E [ | Q | − s ] ≥ E [ | Q | − ε ] s/ε ≥ E [ | Q | ε ] − s/ε . We now choose t ≡ t ( s ) large enough, so that Av ( s ) u (cid:0) (cid:2) | G T u,x ( v, v ; ζ ) | ≤ t (cid:3)(cid:1) ≥ − s . In case v (cid:54)∈ { u − , u + } this is quantified in the esti-mate (A.6), and in case v ∈ { u − , u + } in (A.21).If s ∈ [ − ς, , we use the Jensen inequality together with (3.20) to conclude that Av ( s ) u (cid:0) | G T x ( u, u ; ζ ) | s (cid:1) ≥ Av ( s ) u ( | G T x ( u, u ; ζ ) | − s ) ≥ (1 + s ) λ s s (cid:107) (cid:37) (cid:107) s ∞ (cid:0) =: C − − (cid:1) , (3.25)which completes the proof of (3.15), and by inspection also of (3.16).The above lemma addresses the Green function restricted to subgraphs. Arguments used in theproof 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 3.4.
Under the assumptions of Lemma 3.3, let x −− stand for the neighbor of x − towardsthe root: C − ( s ; ζ ) − ≤ E (cid:0) | G T x (0 , x − ; ζ ) | s (cid:1) E (cid:16) | G T x − (0 , x −− ; ζ ) | s (cid:17) ≤ C + ( s ; ζ ) , (3.26) [ C + ( s ; ζ ) C − ( s ; ζ )] − ≤ E ( | G (0 , x − ; ζ ) | s ) E ( | G T x (0 , x − ; ζ ) | s ) ≤ C + ( s ; ζ ) C − ( s ; ζ ) , (3.27) where x −− is the neighbor of x − towards the root. esonant delocalization Proof.
For the proof of (3.26) we use the factorization of the Green function: G T x (0 , x − ; ζ ) = G T x − (0 , x −− ; ζ ) G T x ( x − , x − ; ζ ) . (3.28)Since the last factor is of the form (3.19), the argument used in the proof of Lemma 3.3 yields (3.26).For a proof of (3.27) we employ the factorization: G (0 , x ; ζ ) = G T x (0 , x − ; ζ ) G ( x, x ; ζ ) . (3.29)Thus, by arguments as in the proof of Lemma 3.3, the quantity E ( | G (0 , x ; ζ ) | s ) is bounded fromabove and below in terms of E (cid:0) | G T x (0 , x − ; ζ ) | s (cid:1) . Since the latter lacks x , we apply (3.26) toappend this vertex. We now turn to the main results on the free energy function. In this context, we recall that asupermultiplicative positive sequence is one satisfying: α m + n ≥ B α m α n > . By Fekete’slemma [19] for such sequences the limit lim n →∞ n − log α n =: Ψ , exists and α m ≤ B − e m Ψ for every m ∈ N . For submultiplicative sequences the reversed inequalities hold. Proof of Theorem 3.2.
In the following we pick a simple path in T to infinity, and label its verticesby x , x , x , . . . . We first show that α n ( ζ ) := E (cid:104)(cid:12)(cid:12)(cid:12) G T xn +1 ( x , x n ; ζ ) (cid:12)(cid:12)(cid:12) s (cid:105) (3.30)is supermultiplicative in the two cases of interest: 1. s ∈ [ − ς, ∞ ) and ζ ∈ C + and 2. s ∈ [ − ς, and ζ = E + i . In both cases, the factorization property (3.9), Lemma 3.3 and (3.26) imply forall n, m ∈ N : α n + m +1 ( ζ ) ≥ C − − α n ( ζ ) α m ( ζ ) ≥ ( C + C − ) − α n +1 ( ζ ) α m ( ζ ) . (3.31)By Fekete’s lemma [19], the limit Ψ( ζ ) := lim n →∞ n − log α n ( ζ ) exists.Analogous reasoning using Lemma 3.3 and (3.26) also show submultiplicativity, i.e., for all n, m ∈ N : α n + m +1 ( ζ ) ≤ C + α n ( ζ ) α m ( ζ ) ≤ C + C − α n +1 ( ζ ) α m ( ζ ) . (3.32)By super- and sub-multiplicativity, the limit Ψ( ζ ) provides both an upper and lower bound on α m ( ζ ) for any m ∈ N : ( C + C − ) − e m Ψ( ζ ) ≤ α m ( ζ ) ≤ C + C − e m Ψ( ζ ) . (3.33)To establish the existence of the limits (3.10) and (2.10), we use (3.33) and (3.27) which reads C − ± α n ( ζ ) ≤ E [ | G ( x , x n ; ζ ) | s ] ≤ C ± α n ( ζ ) . (3.34)with C ± := C + C − . Hence the limits (3.10) and (2.10) agree with Ψ( ζ ) = ϕ λ ( s ; ζ ) in both cases: i. s ∈ [ − ς, ∞ ) and ζ ∈ C + and ii. s ∈ [ − ς, and ζ = E + i .Since for any fixed s ∈ [ − ς, and E ∈ R the constants C + , C − , C ± are bounded uniformlyin Im ζ ∈ (0 , , the convergence (3.10) is also uniform with respect to Im ζ ∈ (0 , , and thelimits η ↓ and | x | → ∞ can be taken in any order. This proves (3.14).The finite-volume bounds (3.12) now follow from (3.33) and (3.34). esonant delocalization (a) , (b) and (d) . Since the prelimits are convexfunctions of s , the limit is convex. Since for any (cid:15) ≥ E (cid:2) | G (0 , x ; ζ ) | s + (cid:15) (cid:3) ≤ (Im ζ ) − (cid:15) E [ | G (0 , x ; ζ ) | s ] , (3.35)the limit (3.10) is non-increasing in s . This concludes the proof of (a) .The first inequality in (3.11) is a consequence of convexity and the factorization property (3.7)of the Green function. In fact, if either 1. s ∈ [ − ς, ∞ ) and ζ ∈ C + or 2. s ∈ [ − ς, and ζ = E + i : log E [ | G (0 , x ; ζ ) | s ] ≥ s E [log | G (0 , x ; ζ ) | ] = − s | x | L ( ζ ) . (3.36)The second inequality in (3.11) relies on the following bound on the sums of squares of Greenfunctions (cid:88) | x | = n | G (0 , x ; ζ ) | ≤ (cid:88) x ∈T | G (0 , x ; ζ ) | = Im G (0 , ζ )Im ζ ≤ ζ ) . (3.37)From the finite-volume bounds (3.12), we conclude that for any n = dist( x, ∈ N : K n e n ϕ (2; ζ ) ≤ C ± K n E (cid:2) | G (0 , x ; ζ ) | (cid:3) = C ± E (cid:104) (cid:88) | x | = n | G (0 , x ; ζ ) | (cid:105) ≤ C ± (Im ζ ) . (3.38)The right side is independent of n , and thus ϕ (2; ζ ) + log K ≤ . Since ϕ (0; ζ ) = 0 , convexityimplies ϕ ( s ; ζ ) ≤ − s log √ K for all s ∈ [0 , . This concludes the proof of (b) .Let us now turn to the differentiability property (d) . If either s ∈ [ − ς, ∞ ) and ζ ∈ C + or s ∈ [ − ς, and ζ = E + i , the factorization property (3.7) of the Green function, (3.11) and thefinite-volume bounds (3.12) imply: ≤ ϕ ( s ; ζ ) + s L ( ζ ) ≤ | x | (log E [ | G (0 , x ; ζ ) | s ] − E [log | G (0 , x ; ζ ) | s ]) + log C ± | x |≤ s | x | E (cid:104) (log | G (0 , x ; ζ ) | ) ( | G (0 , x ; ζ ) | s + 1) (cid:105) + log C ± | x | . (3.39)Here the last inequality derives from the two elementary bounds e α ≤ α + α ( e α + 1) / and β ≤ e β valid for all α, β ∈ R . Using the fractional moment bounds (A.5) and thefactorization property of the Green function, it is easy to check that there is some constant C < ∞ such that for all s ∈ (0 , / and x ∈ T the first factor is bounded by Cs | x | . Furthermore,since log C ± ( s ; ζ ) = o (1) as s → by (3.16), the claim (3.13) follows by choosing | x | = (cid:98) s − (log C ± ) / (cid:99) . The properties established in Theorem 3.2 for the free energy function ϕ λ ( s ; E ) allow one toestablish decay properties of the Green function which are important for the resonance analysiswhich is presented below. The typical behavior is determined by the Lyapunov exponent: esonant delocalization ϕ λ (1; E ) > − log K . Regardless ofthis assumption the curve does not enter theshaded region. The parameter γ is the neg-ative slop of the tangent at s and the valueof the rate function I ( γ ) = − ϕ λ ( s ; E ) − sγ can be read off as the negative value at theintersection of that tangent with the verticalaxis. Theorem 3.5.
For almost all E ∈ R and all (cid:15) > there is some η > such that for all η ∈ (0 , η ) : lim | x |→∞ P (cid:16) | G (0 , x ; E + iη ) | ∈ e − L ( E ) | x | (cid:2) e − (cid:15) | x | , e (cid:15) | x | (cid:3)(cid:17) = 1 . (3.40) The same applies to G T x (0 , x − ; E + iη ) (when substituting G (0 , x ; E + iη ) ). The proof is presented in Appendix B, based on the general and more comprehensive large-deviation Theorem B.1. The latter is established through some standard arguments for which en-abling bounds are provided by Theorem 3.2.Other values of | x | − log | G (0 , x ; E + iη ) | can also be observed, but these represent largedeviations for which the rate function is given by the Legendre transform: I ( γ ) := − inf s ∈ [ − ς, [ ϕ λ ( s ; E ) + sγ ] . (3.41)More explicitly, for any γ which is attainable as γ = − ∂ϕ λ ( s ; E ) /∂s at s ∈ [ − ς, : P (cid:16) | G (0 , x ; E + iη ) | ∈ e − γ | x | (cid:2) e − (cid:15) | x | , e (cid:15) | x | (cid:3)(cid:17) ≈ e − I ( γ )] | x | , (3.42)where ≈ means that the ratio of the two terms is of the order e o ( | x | ) for large | x | . A strongerlarge-deviation principle is presented in Theorem 5.2. Our goal in this section is to prove Theorem 2.1. We start with some useful preparatory observa-tions.
Im Γ(0; E + iη ) Lemma 4.1.
For Lebesgue-almost all E ∈ R , the probability that Im Γ(0; E + i
0) = 0 is either or . esonant delocalization Proof.
Taking the imaginary part of (3.5) one gets:
Im Γ(0; E + iη ) = | G (0 , E + iη ) | (cid:104) η + (cid:88) x ∈N +0 Im Γ( x ; E + iη ) (cid:105) ≥ | G (0 , E + iη ) | (cid:88) x ∈N +0 Im Γ( x ; E + iη ) , (4.1)with equality in case η = 0 for those E for which the boundary values exist, that is for Lebesgue-almost all E ∈ R . Let now q := P (Im Γ(0; E + i
0) = 0) . The factor | G (0 , E + i | is al-most surely non-zero, since, for example, E [ | G (0 , E + i − ς ] < ∞ , using the recursion rela-tion (3.5), Assumption C and the finiteness of fractional moments. Since the K different terms, Im Γ( x ; E + i , x ∈ N +0 , are independent variables of the same distribution as Im Γ(0; E + i ,and | G (0 , E + i | (cid:54) = 0 almost surely, we may conclude that q = q K or q [1 − q K − ] = 0 ,and hence either q = 0 or q = 1 .In order to quantify the way the distribution of Im Γ(0; ζ ) settles on its limit as Im ζ ↓ , weintroduce the following quantity. Definition 4.2.
For ζ ∈ C + and α ∈ (0 , the upper percentile ξ ( α, ζ ) of the distribution of Im Γ(0; ζ ) is the supremum of the values of t ≥ for which P (Im Γ(0; ζ ) ≥ t ) ≥ α . (4.2) Lemma 4.3.
For ζ ∈ C + and any α ∈ (0 , : < ξ ( α, ζ ) < ∞ .Proof. For ζ ∈ C + one has < Im Γ(0; ζ ) ≤ (Im ζ ) − . Hence the claim derives from thefollowing observations: i. The collection of strictly positive values of t at which (4.2) holds is notempty, since otherwise Im Γ(0; ζ ) = 0 with probability one. ii. The above collection of values of t does not include any value above (Im ζ ) − .Iterating (4.1) we conclude that for any n ∈ N and ζ ∈ C + : Im Γ(0; ζ ) ≥ (cid:88) x ∈S n | G (0 , x ; ζ ) | (cid:88) y ∈N + x Im Γ( y ; ζ ) (4.3)where S n := { x ∈ T | dist(0 , x ) = n } . As a first consequence of this important relation, we notethat the distribution of Im Γ(0; ζ ) does not broaden too fast as Im ζ ↓ . As a measure of the(relative) width of the distribution we use the ratios ξ ( α ; ζ ) /ξ ( β ; ζ ) . Lemma 4.4.
For any E ∈ R the distribution of Im Γ(0; E + iη ) remains relatively tight in thelimit η ↓ in the sense that for any pair α, β ∈ (0 , : lim inf η ↓ ξ ( α ; E + iη ) ξ ( β ; E + iη ) > . (4.4) Proof.
We fix α, β ∈ (0 , (by monotonicity it would suffice to consider the case α > β ) and pickan arbitrary < (cid:15) < − β . For a given x ∈ S n , let us consider the event R x := { ( | G (0 , x ; E + iη ) | ≥ e − n(cid:96) } , where (cid:96) > L ( E ) is fixed at an arbitrary value. We now choose n ∈ N large enoughand η > small enough such that for all η ∈ (0 , η ) simultaneously P ( R cx ) ≤ α (cid:32) − (cid:114) β − (cid:15) (cid:33) and K n α (cid:114) − (cid:15)β ≥ β(cid:15) , (4.5) esonant delocalization n = | x | large enough, it follows from Theorem 3.5 that also the first re-quirement can be met. In order to control the sum in (4.3) we also introduce the event I x := (cid:83) y ∈N + x { Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) } . From (4.3) and the Cauchy-Schwarz inequality it thenfollows that P (cid:16) Im Γ(0; ζ ) ≥ e − (cid:96)n ξ ( α ; E + iη ) (cid:17) ≥ P ( N ≥ ≥ E [ N ] E [ N ] , (4.6)where N := (cid:80) x ∈S n R x ∩ I x denotes the number of joint events R x ∩ I x on the sphere S n . Theright side in (4.6) is estimated using the independence of the events I x for all x ∈ S n : E (cid:2) N (cid:3) − E [ N ] = E [ N ( N − ≤ (cid:88) x,y ∈S n x (cid:54) = y P ( I x ) P ( I y ) ≤ K n P ( I x ) . (4.7)Together with the lower bound E [ N ] = K n P ( R x ∩ I x ) ≥ K n ( P ( I x ) − P ( R cx )) ≥ K n ( α − P ( R cx )) ≥ β(cid:15) , (4.8)the inverse of the right side in (4.6) is bounded from above using (4.5): E (cid:2) N (cid:3) E [ N ] ≤ E [ N ] + (cid:18) − P ( R cx ) α (cid:19) − ≤ (cid:15)β + 1 − (cid:15)β = 1 β . (4.9)From the definition of the upper percentile and (4.6) together with (4.9) it hence follows ξ ( β ; E + iη ) ≥ e − (cid:96)n ξ ( α ; E + iη ) . The proof is concluded by noting that the first factor in the right sideis independent of η and strictly positive. We prove Theorems 2.1 and 2.5 by contradicting the following ‘no-ac’ hypothesis.
Definition 4.5.
For a specified λ ≥ , we say that the no-ac hypothesis at E ∈ R holds if almostsurely Im G (0 , E + i
0) = 0 .The relation (4.3) suggests that the no-ac hypothesis is false if with uniformly positive proba-bility there are sites x ∈ S n with | G (0 , x ; ζ ) | (cid:29) , and a forward neighbor y with a not particularly‘atypical’ value of Im Γ( y ; E + iη ) . A key step is: Theorem 4.6.
For almost all E ∈ σ ( H λ ) , if either1. L ( E ) < log K , or (Lyapunov exponent criterion) ϕ (1; E ) > − log K , and Assumption E, (large-deviation criterion) and the no-ac hypothesis holds true, then there are δ, p > and n ≥ such that for all n ≥ n : lim inf η ↓ P (cid:18) max x ∈S n | G (0 , x ; E + iη ) | max y ∈N + x Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) ≥ e δn (cid:19) ≥ p . (4.10)A heuristic argument for the validity of Theorem 4.6 is given in Subsection 4.3 below. Theproof is split: the Lyapunov exponent criterion is established in Subsection 4.4, whereas the proofof the large-deviation criterion, which is a bit more involved, is given separately in Section 5. Firsthowever let us show how Theorem 4.6 is used for the proof of our main results. esonant delocalization Proof of Theorem 2.1 and Theorem 2.5 – given Theorem 4.6.
We will argue by contraction. As-sume the no-ac hypothesis for the given energy E ∈ σ ( H λ ) . From Lemma 4.6 and (4.3) it thenfollows that there are α, δ, η , p > and n ≥ such that for all η ∈ (0 , η ) and all n ≥ n : P (cid:16) Im Γ(0; E + iη ) ≥ e δn ξ ( α ; E + iη ) (cid:17) ≥ P (cid:18) max x ∈S n | G (0 , x ; E + iη ) | max y ∈N + x Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) ≥ e δn (cid:19) ≥ p . (4.11)As a consequence, we conclude ξ ( p ; E + iη ) ≥ e δn ξ ( α ; E + iη ) , and since n can be takenarbitrarily large lim η ↓ ξ ( α ; E + iη ) ξ ( p ; E + iη ) = 0 . (4.12)This however contradicts the relative tightness condition (4.4). A possible mechanism for the rare events featured in (4.10) is the simultaneous occurrence of thefollowing two events, at some common value of γ > : | G ( x, x ; E + iη ) | ≥ e ( γ + δ ) | x | (4.13) (cid:12)(cid:12) G T x (0 , x − ; E + iη ) (cid:12)(cid:12) ≥ e − γ | x | . (4.14)These two conditions imply | G (0 , x ; E + iη ) | ≥ e δ | x | through the relation (3.8).The first, (4.13), represents an extremely rare local resonance condition. It occurs when therandom potential at x falls very close to a value at which G ( x, x ; E + i diverges. By (3.3), suchdivergence is possible if G T x ( y, y ; E + i is real at all y ∈ N x . By (3.3) and the continuity ofthe probabilities in η , under the no-ac hypothesis the probability of (4.13) occurring at a given site x ∈ S n is of the order e − ( γ + δ ) n for η sufficiently small (depending on n ).The second condition, (4.14), representsi) a typical event, in case γ = L ( E ) (cf. Theorem 3.5),ii) a large deviation event, in case γ < L ( E ) (cf. (3.42)).In the first case, the mean number of sites in the sphere S n on which (4.13) and (4.14) occuris E [ N ] ≈ K n e − ( L ( E )+ δ ) n (cid:29) provided < δ < log K − L ( E ) . Unlike (4.13), the condi-tions Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) are not rare events, and their inclusion does not modifysignificantly the above estimate.In the second case, by a standard large deviation estimate as in (3.42), the probability of theevent (4.14) with γ ≈ − lim s ↑ ∂ϕ∂s ( s ; E ) =: ϕ (cid:48)− (1) is of the order e − nI ( γ )+ o (1) with a rate function I ( γ ) which is related to ϕ ( s ) ≡ ϕ λ ( s ; E ) through the Legendre transform. The relevant mecha-nism for the occurrence of (4.14) is the systematic stretching of the values of | G T x (0 , u ; E + iη ) | along the path (cid:22) u (cid:22) x − . By the above lines of reasoning, and ignoring excessive correla-tions (a step which is justified under auxiliary conditions) we arrive at the mean value estimate E [ N ] ≈ K n exp ( − n [ I ( γ ) + γ + δ + o (1)]) . This value is much greater than for some δ > ,provided sup γ [log K − [ I ( γ ) + γ )] > (4.15) esonant delocalization |S n | = K n , under suitable assumptions E [ N ] → ∞ for n → ∞ . To see what(4.15) entails, let us note that by the inverse of the Legendre transform (3.41): ϕ ( s ; E ) ≡ ϕ ( s ) = − inf γ [ I ( γ ) + sγ )] (4.16)Thus, (4.15) is the condition ϕ (1; E ) > − log K which is mentioned in Theorem 4.6, and inTheorem 2.5.The analysis which relates to the first condition i) yields the Lyapunov exponent criterionwhich we shall prove first. The proof of the more complete result, which uses the condition ii) is abit more involved, and is therefore postponed the next section. The aim of this subsection is to prove the first criterion of Theorem 4.6. Thus, we fix the disorderparameter λ > and the energy E ∈ R , assuming that L λ ( E ) < log K . In view of the generalbound L λ ( E ) > log √ K , for which the strict inequality was shown in [6, Thm. 4.1] (the weakinequality is explained by (3.11)), the assumption is equivalent to: δ := log K − L λ ( E ) ∈ (cid:16) , log √ K (cid:17) . (4.17)In accordance with the above heuristics, we consider the following three events. Definition 4.7.
For each x ∈ S n and η > we associate the following events:i. The extreme deviation event , at blow-up parameter τ := e ( L ( E )+2 δ ) n E x := {| G ( x, x ; E + iη ) | ≥ τ } . ii. The regular decay event at decay rate (cid:96) := L ( E ) + δR x := (cid:110) | G T x (0 , x − ; E + iη ) | ≥ e − (cid:96)n (cid:111) . iii. The α -marginality event , at probability α ∈ (0 , I x := (cid:91) y ∈N + x { Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) } . We will suppress the dependence of these events on α, η > . The parameter τ is chosen suchthat i. τ − K n = e δn and ii. in the event E x ∩ R x : | G (0 , x ; E + iη ) | = | G T x (0 , x − ; E + iη ) | | G ( x, x ; E + iη ) | ≥ e δn , (4.18)by the factorization (3.8) of the Green function. The decay rate (cid:96) is chosen so that the event R x occurs asymptotically as n → ∞ with probability one (cf. Theorem 3.5).We will monitor the number of simultaneous occurrences of the three events listed above,which is given by the random number N := (cid:88) x ∈S n E x ∩ R x ∩ I x . (4.19) esonant delocalization n → ∞ , of the expectation value E [ N ] does not on its own implythat the probability of N > has a positive limit. However, such a conclusion can be drawnfrom suitable information on the first two moments, e.g. using the following consequence of theCauchy-Schwarz inequality P ( N ≥ ≥ E [ N ] E [ N ] . (4.20)We shall next derive bounds on the first two moments which will enable the proof that the aboveprobability is bounded below. Our lower bound on E [ N ] is based on a relation of the probability of extreme deviation eventsto the mean (local) density of states D ( E ) associated with fully regular Caley tree B in which every vertex has exactly K + 1 neighbors. This density of states is given, for almost all E ∈ R ,by [30, 3]: D ( E ) := lim η ↓ π E (cid:2) Im G B ( x, x ; E + iη ) (cid:3) . (4.21)Since ζ (cid:55)→ E [ G ( x, x ; ζ )] is a Herglotz function, the limit exists for almost all E ∈ R . Moreover,due to homogeneity it is independent of x ∈ B . The following property is well known, cf. [3, 14],but very important for us. Proposition 4.8.
The support of D coincides with the almost-sure spectrum, i.e., for Lebesgue-almost all E ∈ σ ( H λ ) one has D ( E ) > . Varying the potential at x is a rank-one perturbation of the operator H λ ( ω ) , and the responseof the corresponding Green function’s diagonal element is particularly simple: G B ( x, x ; ζ ) = ( λV ( x ) − σ x ( ζ )) − , σ x ( ζ ) := ζ + (cid:88) y ∈N x G B x ( y, y ; ζ ) , (4.22)(which a special case of (3.3)). This allows us to relate the aforementioned probability of extremedeviation events to the density of states D ( E ) . It is at this point that the regularity Assumption Dplays a helpful role. Lemma 4.9.
For Lebesgue-almost all E ∈ R , under the no-ac hypothesis the following holds forall x ∈ B :1. Im σ x ( E + i
0) = 0 almost surely.2. D ( E ) = E (cid:2) (cid:37) (cid:0) λ − σ x ( E + i (cid:1)(cid:3) λ .3. for any ˆ τ ≥ λ − and any event Z x which is independent of V ( x ) : D ( E ) ≤ c λ ˆ τ P (cid:0)(cid:8) | G B ( x, x ; E + i | ≥ ˆ τ (cid:9) ∩ Z x (cid:1) + (cid:107) (cid:37) (cid:107) ∞ λ P ( Z cx ) , (4.23) where c ∈ (0 , ∞ ) is the constant from Assumption D. esonant delocalization Proof.
The proof of the first assertion is based on the observation that, under the no-ac hypothesis, Im G B x ( y, y ; E + i , ω ) = 0 for P -almost all ω , all x ∈ T and all y ∈ N x . This follows fromthe fact that the Green functions G B x ( y, y ; E + i associated with the neighbors, y ∈ N x , areidentically distributed to Γ(0; E + i and hence Im G B x ( y, y ; E + i , ω ) = 0 for Lebesgue × P -almost all ( E, ω ) .The proof of the representation . is based on (4.22). We first condition on the sigma-algebra A x generated by the random variables V ( y ) , y (cid:54) = x , and write E (cid:2) Im G B ( x, x ; E + iη ) | A x (cid:3) = (cid:90) (cid:37) ( v ) Im ( λv − σ x ( E + iη )) − dv . (4.24)Since lim η ↓ σ x ( E + iη ) = σ x ( E + i for almost all E ∈ R and the distribution of σ x ( E + i iscontinuous, Lebesgue’s differentiation theorem implies that for Lebesgue × P -almost all ( E, ω ) : lim η ↓ π (cid:90) (cid:37) ( v ) Im ( λv − σ x ( E + iη ; ω )) − dv = (cid:37) ( λ − σ x ( E + i ω )) λ . (4.25)This together with the dominated convergence theorem, which is based on the Wegner bound E (cid:2) Im G B ( x, x ; E + iη ) | A x (cid:3) ≤ π (cid:107) (cid:37) (cid:107) ∞ λ , (4.26)concludes the proof of the representation . We may now refine . by first inserting an indicator function of any event Z x which is indepen-dent of V ( x ) and its complement Z cx . The equalities (4.24) and (4.25) together with (4.26) thenimply: D ( E ) ≤ λ − E (cid:2) (cid:37) ( λ − σ x ( E + i ω )) 1 Z x (cid:3) + (cid:107) (cid:37) (cid:107) ∞ λ P ( Z cx ) . (4.27)Using Assumption D, the first term on the right side is now seen to relate to the probability ofextreme deviation events. More precisely, for any ˆ τ ≥ λ − almost surely λ (cid:37) ( λ − σ x ( E + i ω )) ≤ c λ ˆ τ (cid:90) (cid:37) ( v ) 1 | λv − σ x ( E + i ω ) |≤ ˆ τ − dv = 2 c λ ˆ τ P (cid:0) | G B ( x, x ; E + i | ≥ ˆ τ | A x (cid:1) (4.28)This concludes the proof of (4.23).Based on the above estimates, we may now provide a lower bound on E [ N ] . Corollary 4.10.
For Lebesgue-almost every E ∈ σ ( H λ ) under the no-ac hypothesis there are α ∈ (0 , , C, η ∈ (0 , ∞ ) and n ≥ such that for all n ≥ n and η ∈ (0 , η ) : E [ N ] = K n P ( R x ∩ E x ∩ I x ) ≥ K n D ( E ) C τ ≥ D ( E ) C > . (4.29) Proof.
The continuity lim η ↓ P (cid:0)(cid:8) | G B ( x, x ; E + iη ) | ≥ τ (cid:9) ∩ Z x (cid:1) = P (cid:0)(cid:8) | G B ( x, x ; E + i | ≥ τ (cid:9) ∩ Z x (cid:1) (4.30)for almost every E ∈ R , guarantees the validity of (4.23) with c replaced by c and all η smallenough. To extend this estimate to the Green function associated with the regular rooted tree T , we esonant delocalization (cid:96) ( T ) into (cid:96) ( B ) and use perturbation theory, the general recursion relation (3.3)and the multiplicativity (3.4): (cid:12)(cid:12) G B ( x, x ; ζ ) − − G T ( x, x ; ζ ) − (cid:12)(cid:12) ≤ (cid:12)(cid:12) Γ B x ( x − ; ζ ) − Γ T x ( x − ; ζ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) G B x (0 − , x − ; ζ ) (cid:12)(cid:12) (cid:12)(cid:12) G T x (0 , x − ; ζ ) (cid:12)(cid:12) = (cid:12)(cid:12) G B x (0 − , − ; ζ ) (cid:12)(cid:12) (cid:12)(cid:12) G T x (0 , x − ; ζ ) (cid:12)(cid:12) . (4.31)For all E ∈ R such that D ( E ) > there exists t > such that according to (A.6) the event ˆ B x := { (cid:12)(cid:12) G B x (0 − , − ; E + iη ) (cid:12)(cid:12) ≤ t } has for all η > a probability of at least P ( ˆ B x ) ≥ − λD ( E )8 (cid:107) (cid:37) (cid:107) ∞ > . (4.32)Moreover, according to Theorem 3.5 and since e − δn τ − = K − n > e − nL ( E ) , there is n ≥ and η ∈ (0 , ∞ ) such that for all n ≥ n and η ∈ (0 , η ) the event ˆ R x := { (cid:12)(cid:12) G T x (0 , x − ; E + iη ) (cid:12)(cid:12) ≤√ e − δn τ − } has a probability of at least P ( ˆ R x ) ≥ − λD ( E )8 (cid:107) (cid:37) (cid:107) ∞ > . (4.33)Summarizing the above estimates, we conclude that there is n ≥ and η ∈ (0 , ∞ ) such that forall n ≥ n and η ∈ (0 , η ) and any event Z x which is independent of V ( x ) : D ( E ) ≤ c λ τ P (cid:16)(cid:8) | G B ( x, x ; E + iη ) − | ≤ (2 τ ) − (cid:9) ∩ ˆ B x ∩ ˆ R x ∩ Z x (cid:17) + (cid:107) (cid:37) (cid:107) ∞ λ P (cid:16) ˆ B cx ∪ ˆ R cx ∪ Z cx (cid:17) ≤ c λ τ P ( E x ∩ Z x ) + (cid:107) (cid:37) (cid:107) ∞ λ P ( Z cx ) + 14 D ( E ) . (4.34)We apply this bound to Z x = R x ∩ I x . Since P ( R cx ∪ I cx ) ≤ P ( R cx ) + P ( I cx ) ≤ P ( R cx ) + 1 − α .By Theorem 3.5, there is n ≥ n and η ∈ (0 , η ] such that for all n ≥ n and η ∈ (0 , η ) P ( R x ) ≥ − λD ( E )8 (cid:107) (cid:37) (cid:107) ∞ > . (4.35)Choosing α := 1 − λD ( E )8 (cid:107) (cid:37) (cid:107) ∞ completes the proof of (4.29). The mere fact that the mean number of events diverges, for n → ∞ (cf. (4.29)) does not yet implythat such events do occur with uniformly positive probability. The alternative is that the divergencereflects an increasingly rare but also increasingly correlated occurrence of these events. To provethat the resonances do occur regularly, on sufficiently large spheres S n , we use the second-momentmethod which is based on the following estimate. Lemma 4.11.
Assuming L ( E ) < log K , there is C ∈ (0 , ∞ ) such that for all n ≥ , all η > and all α ∈ (0 , : E [ N ( N − ≤ C τ − K n . (4.36) esonant delocalization Proof.
Throughout the proof appearing constants C ∈ (0 , ∞ ) will be independent of n , η and α .We start from the observation that E [ N ( N − (cid:88) x,y ∈S n x (cid:54) = y P ( R x ∩ E x ∩ I x ∩ R y ∩ E y ∩ I y ) ≤ (cid:88) x,y ∈S n x (cid:54) = y P ( E x ∩ E y ) . (4.37)The probability in the right side is estimated using the weak- L bound for pairs of Green functionin Theorem A.2 below. Denoting by A xy the sigma-algebra generated by the random variables V ( u ) , u (cid:54)∈ { x, y } , it yields P ( E x ∩ E y ) = E (cid:2) P (cid:0) E x ∩ E y (cid:12)(cid:12) A xy (cid:1)(cid:3) ≤ Cτ (cid:18) τ + E (cid:2) min (cid:8) , | G T x,y ( x − , y − ; E + iη ) | (cid:9)(cid:3)(cid:19) , (4.38)with some constant C ∈ (0 , ∞ ) . The first term is already of the desired form since the number ofterms in the sum in (4.37) is bounded by K n . To estimate the second term we use min { , | x |} ≤| x | s valid for any s ∈ [0 , . Choosing s := L ( E ) + 2 δ log K ∈ (0 , , (4.39)we estimate the factional-moment with the help of the finite-volume bounds (3.12) and the upperbound in (3.11): E (cid:2) || G T x,y ( x − , y − ; E + iη ) | s (cid:3) ≤ C K − s dist( x,y ) (4.40)with some constant C ∈ (0 , ∞ ) . The corresponding sum contributing to (4.37) is estimated byfixing x ∈ S n and summing over the distance of the least common ancestor of x and y to the root: (cid:88) x,y ∈S n x (cid:54) = y E (cid:2) || G T x,y ( x − , y − ; E + iη ) | s (cid:3) ≤ C K n n − (cid:88) j =0 K n − j K − s ( n − j ) ≤ C K (2 − s ) n = C τ − K n , (4.41)where the last inequality is based on (4.39).We are now ready for the proof of the main result of this section. Proof of Theorem 4.6; the Lyapunov exponent criterion.
By Corollary 4.10 and Lemma 4.11, thereare α ∈ (0 , (which is one of the parameters in the definition of N ), C, η ∈ (0 , ∞ ) and n ≥ such that for all n ≥ n and η ∈ (0 , η ) : E (cid:2) N (cid:3) E [ N ] = 1 E [ N ] + E [ N ( N − E [ N ] ≤ C . (4.42)Hence, second-moment bound (4.20) allows us to conclude that P ( N ≥ ≥ C − uniformly in n > n and η ∈ (0 , η ) .However, whenever N ≥ one may conclude that the quantity which appears in the left sideof (4.10) satisfies max x ∈S n | G (0 , x ; E + iη ) | max y ∈N + x Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) ≥ e δn . (4.43)Taken together, (4.43) and the above probability estimate directly imply the part of Theorem 4.6which relates to the Lyapunov exponent criterion, with p = C − ).As was shown in Section 4.6, the above result implies the Lyapunov exponent criterion whichis stated in Theorem 2.1. esonant delocalization As explained in the introduction, while the Lyapunov exponent criterion is very useful it does notyet cover the full regime of extended states. Our next aim is to establish an extended version of thiscriterion, improved through the incorporation in the argument of the large deviation considerations.The result is stated above as the second part of Theorem 4.6. We now turn to its proof, followingthe outline which is given in Section 4.3. The strategy has much in common with the derivationof the Lyapunov exponent criterion, however the proof involves some additional technicalities.Since the applications which are discussed in the introduction rely on just the Lyapunov exponentcriterion, only the more dedicated reader may wish to follow this Section.
For the remainder of this subsection, we fix the disorder parameter λ > and an energy E ∈ R such that ϕ ( t ) ≡ ϕ ( t ; E ) = lim η ↓ ϕ ( t ; E + iη ) exists for all t ∈ [ − ς, and (2.11) holds, i.e., ∆ := log K + ϕ (1; E ) ∈ (cid:0) , log K (cid:1) . (5.1)Due to the convexity of ϕ ( s ) and (3.11), under the assumption (5.1) the left derivative of ϕ satisfies(see Figure 4): < − ϕ (cid:48)− (1) ≤ ∆ . (5.2)We proceed by associating to the given λ and E certain parameters ( γ , β , κ , (cid:15) , and τ ) whichwill also be kept fixed for the remainder of this section. These parameters feature in the definitionof the resonance events which will be associated with vertices on the sphere S n of radius n ∈ N . Tocontrol the correlations among such events we restrict to vertices on the thinned sphere S κn ⊂ S n associated with the parameter κ which we pick in the range: κ ∈ (cid:0) , min (cid:8) ∆16 (cid:96) , (cid:9)(cid:1) , (5.3)where (cid:96) > L ( E ) is fixed (largely arbitrary). The thinned sphere S κn , whose radius shall be largerthan (cid:100) κ − (cid:101) , is characterized by the length scales n κ := 2 (cid:98) κn (cid:99) ∈ N and N κ := n − n κ . Thefirst one is only a fraction of the second length scale, i.e. κ n ≤ n κ ≤ κ n , n κ ≤ κ − κ N κ ≤ κ N κ . (5.4)Then S κn is uniquely determined by having K N κ vertices with n κ + 1 vertices separating them,cf. Figure 5.We now pick a value s ∈ (0 , at which the free energy function t (cid:55)→ ϕ ( t ) is differentiable,and such thata) the derivative at s , satisfies γ := − ϕ (cid:48) ( s ) ≥ ∆ > , (5.5)b) the following condition holds I ( γ ) + γ = − (cid:2) ϕ ( s ) + (1 − s ) ϕ (cid:48) ( s ) (cid:3) ≤ log K − ∆ , (5.6)c) and in addition (1 − s ) < / and ϕ ( s ) < − log K . esonant delocalization ϕ , the above conditions are satisfied at a densecollection of values of s approaching from below (see Figure 4). (Condition c) is only imposedto simplify some of the estimates.)The parameter γ will be used as a target-value for the decay of the Green function in the largedeviation events L x defined below. For any site x ∈ S n we label the vertices of the unique pathfrom the root to x as x = 0 , x , . . . , x n = x , and we denote as (cid:98) T x := T x nκ − ,x (5.7)the tree truncated beyond the segment of length N κ whose end points are { x n κ − , x } (cf. Fig-ure 5). Associated with this segment there are the two collections of variables { Γ + ( j ; η ) } N κ j =1 and { Γ − ( j ; η ) } N κ j =1 : Γ + ( j ; η ) := G T xn − j − ,x ( x n − j , x n − j ; E + iη ) , Γ − ( j ; η ) := G T xnκ − ,xnκ + j ( x n κ − j , x n κ − j ; E + iη ) , (5.8)such that by (3.4): G (cid:98) T x ( x n κ , x n − ; E + iη ) = N κ (cid:89) j =1 Γ + ( j ; η ) = N κ (cid:89) j =1 Γ − ( j ; η ) . (5.9) Definition 5.1.
We refer to the following as the large-deviation events associated with sites x ∈ S n and η, (cid:15) > L x := L (bc) x ∩ N κ (cid:92) k = 12 n κ (cid:16) L ( k, +) x ∩ L ( k, − ) x (cid:17) , (5.10)where for any k ∈ { , . . . , N κ } : L ( k, ± ) x := (cid:110) k (cid:89) j =1 | Γ ± ( j ; η ) | ∈ e − γk (cid:2) e − (cid:15)k , e (cid:15)k (cid:3)(cid:111) ,and L (bc) x := (cid:8) | Γ + ( N κ ; η ) | ≤ b (cid:9) ∩ (cid:8) | Γ − ( N κ ; η ) | ≤ b (cid:9) . esonant delocalization η and (cid:15) (whose value is fixed below).The boundary events L (bc) x play a role in the following context: i. the lower bound on theprobability of R x given below in Lemma 5.7, and ii. the estimate (5.28) on the size of the self-energy at x are derived only under the condition L (bc) x . The parameter b is fixed at a value largeenough so thata) b ≥ (cid:107) (cid:37) (cid:107) ∞ λ max (cid:8) , (cid:0) − (3 / K (cid:1) − (cid:9) , andb) P s (cid:16) L (bc) x (cid:17) ≥ , cf. (B.5),the latter being possible thanks to (A.21). (The numbers are largely arbitrary.)To fix the parameter (cid:15) , we invoke the following large-deviation statement which is derived inthe Appendix B. Theorem 5.2.
For any (cid:15) > there is η > and n > such that for all η ∈ (0 , η ) and all n = dist( x, ≥ k ≥ n : P ( L x ( η ; (cid:15) )) ≥ e − N κ ( I ( γ )+2 (cid:15) ) , (5.11) P (cid:16) L ( k, ± ) x ( η ; (cid:15) ) (cid:17) ≤ e − ( I ( γ ) − (cid:15) ) k . (5.12)We now fix (cid:15) at a value at which: (cid:15) ∈ (cid:0) , min (cid:8) ∆24 , κ ∆4 (cid:9)(cid:1) . (5.13)This parameter will be used in controlling the probabilities of various large deviation events.Before turning to the main definitions, we introduce yet another event which refers to thebehavior of the Green function between x and x n κ − , for which we require the (largely arbitrary)minimal decay rate (cid:96) > L ( E ) combined with a condition at an end point. Definition 5.3.
We refer to the following as the regular events associated with sites x ∈ S n and η > : R x := R (bc) x ∩ (cid:110) | G T x (0 , x n κ − ; E + iη ) | ∈ (cid:2) e − n κ (cid:96) , (cid:3)(cid:111) (5.14)where R (bc) x := (cid:8) | G T x ( x n κ − , x n κ − ; E + iη ) | ≤ b (cid:9) .This event is regular in the sense that it occurs with a probability of order one, which is in-dependent of n , cf. Theorem 3.5. The reason for its inclusion in the paper is mainly of technicalorigin: in the subsequent proof of a second moment bound, Theorem 5.8 below, we cannot allowthe large deviation event L x to extend down to the root, but we nevertheless need some control onthe Green function on this segment.Having fixed the basic parameters, we now turn to the precise definition of the events. Definition 5.4.
For each x ∈ S n and η > we definei. the resonance-boosted large-deviation event , D x := E x ∩ L x ∩ R x (5.15)which consists of the following three events: esonant delocalization τ := exp (cid:0)(cid:0) γ + ∆ (cid:1) N κ (cid:1) : E x := {| G ( x, x ; E + iη ) | ≥ τ } , b) large deviation event: L x (cf. Definition 5.1)c) regular event: R x (cf. Definition 5.3)ii. the α -marginality event at probability α ∈ (0 , : I x := (cid:91) y ∈N + x { Im Γ( y ; E + iη ) ≥ ξ ( α ; E + iη ) } . The joint event D x ∩ I x will be referred to as a resonance event at x .Several remarks are in order:1. The resonance-boosted large-deviation events are tailored so that in the event D x the Greenfunction associated with the root and x exhibits an exponential blow-up. Namely, by thefactorization property of the Green function, G (0 , x ; ζ ) = G T x (0 , x n − ; ζ ) G ( x, x ; ζ )= G T x (0 , x n κ − ; ζ ) G (cid:98) T x ( x n κ , x n − ; ζ ) G ( x, x ; ζ ) . (5.16)For ζ = E + iη , the first term is controlled by R x . The large deviation event L x controls thesecond factor and the extreme fluctuation event E x compensates for the decay of the firsttwo terms. Using (5.4), (5.3), and (5.13), we hence arrive at the estimate: | G (0 , x ; E + iη ) | ≥ e − n κ (cid:96) e − ( γ + (cid:15) ) N κ τ ≥ exp (cid:0) N κ (cid:0) ∆ − (cid:15) − κ(cid:96) (cid:1)(cid:1) ≥ exp (cid:0) ∆ N κ (cid:1) ≥ exp (cid:0) ∆ n (cid:1) . (5.17)‘2. The choice of the blow-up scale τ is tailored to: i. compensate the decay of the Greenfunction on the segment preceeding x , cf. (5.17), and ii. ensure that for n large enough and η small enough: τ − K N κ P ( L x ) ≥ exp (cid:0) N κ (cid:0) log K − ( γ + I ( γ )) − (cid:15) − ∆ (cid:1)(cid:1) ≥ exp (cid:0) N κ ∆16 (cid:1) , (5.18)by (5.11), (5.6) and (5.13). The fact that this term can be made large as n → ∞ will beessential in the subsequent argument.3. We recall from Definition 4.2 that the value ξ ( α ; E + iη ) ensures that P ( I x ) ≥ α . Postponing the proof of the occurrence of the above resonance events, the proof of our key state-ment, the large-deviations criterion of Theorem 4.6, is along the same lines as in the Lyapunovregime. esonant delocalization Proof of Theorem 4.6 the large-deviation criterion.
We monitor the number N := (cid:88) x ∈S κn D x ∩ I x (5.19)of resonances on the thinned sphere and note that the event N ≥ implies the event the right sideof (4.10) for δ = ∆ > using (5.17).According to Theorems 5.6 and 5.8, there are α ∈ (0 , , C, η ∈ (0 , ∞ ) and n ≥ such thatfor all n ≥ n and η ∈ (0 , η ) : E (cid:2) N (cid:3) E [ N ] = 1 E [ N ] + E [ N ( N − E [ N ] ≤ C . (5.20)Together with (4.20), this concludes the proof.The second-moment method on which the the above proof is based requires a lower bound onthe mean number of events as well as an upper bound on their second moment. These will be thetopics of the remaining subsections.
The main idea behind a lower bound on the average number of resonances is that the probabilityof the occurrence of the extreme fluctuation E x is of order τ − . Rewriting this event, E x = (cid:8) | λV ( x ) − σ x ( E + iη ) | ≤ τ − (cid:9) (5.21)thereby exposing the dependence of G ( x, x ; ζ ) on the potential at x and on σ x ( E + iη ) := E + iη + (cid:88) y ∈N x G T x ( y, y ; ζ ) , (5.22)one realizes that if the latter has a non-zero imaginary part, the Green function stays bounded andno resonance mechanism kicks in. On the other hand, in the event S x ∩ T x , where S x := (cid:92) y ∈N x S x ( y ) , with S x ( y ) := (cid:8)(cid:12)(cid:12) G T x ( y, y ; ζ ) (cid:12)(cid:12) ≤ b (cid:9) T x := (cid:8) Im σ x ( E + iη ) ≤ (2 τ ) − (cid:9) , (5.23)the imaginary part of the term in the right side of (5.21) is bounded by (2 τ ) − and the real partis bounded by ( K + 1) b . As a consequence, we may estimate the conditional probability of E x conditioned on the sigma algebra A x generated by the random variables V ( y ) , y (cid:54) = x : P (cid:0) E x (cid:12)(cid:12) A x (cid:1) ≥ S x ∩ T x P (cid:0) | λV ( x ) − E − Re σ x ( E + iη ) | ≤ τ (cid:12)(cid:12) A x (cid:1) ≥ S x ∩ T x inf | σ |≤ ( K +1) b P (cid:0) | λV ( x ) − E − σ | ≤ τ (cid:12)(cid:12) A x (cid:1) ≥ (cid:37) b τ − S x ∩ T x . (5.24)where the last estimate relied on Assumption D and we introduced (cid:37) b := inf v ∈ ( K +1) [ − b, b ] ( cλ ) − (cid:37) (cid:16) v + Eλ (cid:17) > . (5.25)Now, S x is a regular event, i.e., it occurs with positive probability which is independent of n .Under the no-ac hypothesis the probability of the event T x is (arbitrarily) close to one. esonant delocalization Lemma 5.5.
Under the no-ac hypothesis, Im σ x ( E + i , ω ) = 0 for P -almost all ω and all x ∈ T .Proof. Recall that σ x coincides with the sum (5.22) of Green functions associated with the neigh-bors of x . The Green function associated with the forward neighbors, y (cid:54) = x − , are identicallydistributed to Γ(0; E + i and hence Im G T x ( y, y ; E + i , ω ) = 0 for Lebesgue × P -almostall ( E, ω ) . The Green function associated with the backward neighbor x − differs by a finite-rankperturbation from a variable which is identically distributed to Γ(0; E + i (i.e., the surgery whichrenders the rooted to into a full tree). Since finite-rank perturbations do not change the ac spectrum,we also conclude Im G T x ( x − , x − ; E + i , ω ) = 0 for Lebesgue × P -almost all ( E, ω ) .The bound (5.24) quantifies the essence of the resonance mechanism and leads to the following Theorem 5.6.
Under the no-ac hypothesis, for every n large enough there exists η > such thatfor all η ∈ (0 , η ) , and α ∈ [1 / , and all x ∈ S n : E [ N ] = K N κ P ( D x ∩ I x ) ≥ (cid:37) b τ − K N κ P ( L x ) . (5.26) The right side can be made arbitrarily large by choosing n sufficiently large.Proof. In order to estimate the probability of the joint occurrence of the events D x and I x , we firstcondition on the sigma algebra A x and use (5.24) to obtain: P ( D x ∩ I x ) = E (cid:2) R x ∩ L x ∩ I x P (cid:0) E x (cid:12)(cid:12) A x (cid:1)(cid:3) ≥ (cid:37) b τ − P ( R x ∩ L x ∩ I x ∩ S x ∩ T x ) ≥ (cid:37) b τ − [ P ( R x ∩ L x ∩ I x ∩ S x ) − (1 − P ( T x ))]= (cid:37) b τ − (cid:2) P (cid:0) R x ∩ L x ∩ S − x (cid:1) P (cid:0) I x ∩ S + x (cid:1) + P ( T x ) − (cid:3) , (5.27)where we abbreviated S − x := S x ( x − ) and S + x := (cid:84) y ∈N + x S x ( y ) . The first term simplifies using:i) the inclusion R x ∩ L x ⊂ S − x . This derives from second order perturbation theory. Moreprecisely, in the event R x ∩ L x the term corresponding to the backward neighbor x − of x isbounded according to | G T x ( x − , x − ; E + iη ) | ≤ | G (cid:98) T x ( x − , x − ; E + iη ) | + | G T x ( x n κ − , x n κ − ; E + iη ) | | G (cid:98) T x ( x n κ , x − ; E + iη ) | ≤ b + b = b . (5.28)ii) the estimate P ( I x ∩ S + x ) ≥ P ( I x ) + P ( S + x ) − ≥ α + (1 − (cid:107) (cid:37) (cid:107) ∞ ( λb ) − ) K − ≥ . Herethe last inequality used α ≥ / and the particular choice of b .To proceed with our estimate on the right side in (5.27) we use Lemma 5.7 below which guaranteesthat for some η > and some n ∈ N and all η ∈ (0 , η ) and n ≥ n : P (cid:0) R x ∩ L x ∩ S − x (cid:1) = P ( R x ∩ L x ) ≥ P ( L x ) . (5.29)We now use Lemma 5.5 which implies that under the no-ac hypothesis and for any x ∈ T and any ε > : lim η ↓ P (Im σ x ( E + iη ) > ε ) = 0 . (5.30) esonant delocalization inf η ∈ (0 , P ( L x ( η )) > is strictly positive by (5.11), we conclude that there is some η ( n ) ∈ (0 , η ] such that for all η ∈ (0 , η ( n )) : − P ( T x ) ≤ P ( L x ) . (5.31)This concludes the proof of (5.26). The exponential estimate (5.18) finally shows that the rightside in (5.26) is arbitrarily large if n is chosen large.It remains to prove the following lemma. Lemma 5.7.
There is η > and n > such that for all η ∈ (0 , η ) and all n = dist( x, ≥ n : P ( R x ∩ L x ) ≥ P ( L x ) . (5.32) Proof.
The idea is to control the conditional probability conditioned on the sigma-algebra A generated by the random variables V ( y ) with x n κ (cid:22) y . The assertion follows from the fact thatthere is η > and n > such that for all η ∈ (0 , η ) and all n = dist( x, ≥ n : P (cid:0) R x (cid:12)(cid:12) A (cid:1) L (bc) x ≥ L (bc) x . (5.33)As a preparation, we expose the influence the conditioning on A has on the Green function usingits factorization property: G ( η ) := G T x ( x n κ − , x n κ − ; E + iη ) (cid:98) G ( η ) := G T xnκ − (0 , x n κ − ; E + iη ) = G T x (0 , x n κ − ; E + iη ) (cid:14) G ( η ) . (5.34)By the choice of the parameter b , one has P (cid:0) R (bc) x | A (cid:1) ≥ / and hence P (cid:0) R x (cid:12)(cid:12) A (cid:1) ≥ P (cid:16) | (cid:98) G ( η ) G ( η ) | ∈ (cid:2) e − (cid:96)n κ , (cid:3) (cid:12)(cid:12) A (cid:17) − ≥ P (cid:16) | (cid:98) G ( η ) | ∈ (cid:2) B e − (cid:96)n κ , b − (cid:3)(cid:17) + P (cid:0) | G ( η ) | ∈ (cid:2) B − , b (cid:3) (cid:12)(cid:12) A (cid:1) − , ≥ P (cid:16) | (cid:98) G ( η ) | ∈ (cid:2) B e − (cid:96)n κ , b − (cid:3)(cid:17) + P (cid:0) | G ( η ) | ≥ B − (cid:12)(cid:12) A (cid:1) − , (5.35)where the last inequalities hold for any B ∈ [1 , ∞ ) . By Theorem 3.5 the first term converges toone as n κ → ∞ . The event in the second term takes the form (cid:12)(cid:12)(cid:12) λV ( x n κ − ) − E − iη − (cid:88) y ∈N xnκ − G (cid:98) T x ( y, y ; E + iη ) (cid:12)(cid:12)(cid:12) ≤ B .
In the event L (bc) x , there is B > (which is independent of n and η ) such that for all η ∈ (0 , : P (cid:0) | G ( η ) | < B − (cid:12)(cid:12) A (cid:1) L (bc) x ≤ L (bc) x . (5.36)This completes the proof. esonant delocalization Our aim in this subsection is to provide a uniform upper bound on E (cid:2) N (cid:3) / E [ N ] , for N = (cid:80) x ∈ S κn D x ∩ I x , which counts the number of resonance events on the thinned sphere. Theorem 5.8.
Under the no-ac hypothesis, there exists some constant
C < ∞ such that for all n sufficiently large there is η ≡ η ( n ) such that for all η ∈ (0 , η ) , α ∈ [1 / , : E [ N ( N − E [ N ] ≤ C < ∞ . (5.37) Proof.
Throughout the proof we will suppress the dependence on n , η and α at our convenience.Appearing constants c, C will be independent of n , η and α . We write E [ N ( N − (cid:88) x,y ∈S κn x (cid:54) = y P ( D x ∩ D y ∩ I x ∩ I y ) = |S κn | (cid:88) y ∈S κn \{ x } P ( D x ∩ D y ∩ I x ∩ I y ) . (5.38)The last equality holds for arbitrary x ∈ S κn which we will fix in the following. By symmetry, thejoint probability P ( D x ∩ D y ∩ I x ∩ I y ) depends only on the distance of the last common ancestor x ∧ y to the root. It is therefore useful to introduce the ratio P ( D x ∩ D y ∩ I x ∩ I y ) P ( D x ∩ I x ) P ( D y ∩ I y ) := r ( j ) δ dist( x ∧ y, ,j . (5.39)The sum in (5.38) may then be organized in terms of the last common ancestor x ∧ y on thepath P ,x = { x , . . . , x n } connecting the root with x . In fact, since S κn is thinned, x ∧ y belongsto the shortened path P κ ,x := (cid:8) u ∈ P ,x (cid:12)(cid:12) dist( u, < N κ (cid:9) . Moreover, for a given x ∧ y ∈P κ ,x , the number of vertices y ∈ S κn , which for fixed x have the same common ancestor, is | S κn | K − dist( x ∧ y, such that E [ N ( N − E [ N ] = N κ − (cid:88) j =0 r ( j ) K j . (5.40)In order to estimate the sum in the right side of (5.40), we always drop the condition R x in thedefinition of D x : r ( j ) ≤ P ( L x ∩ L y ∩ E x ∩ E y ∩ I x ∩ I y ) P ( D x ∩ I x ) P ( D y ∩ I y ) δ dist( x ∧ y, ,j . (5.41)For an estimate on the numerator in the right side, we first focus on the extreme fluctuation eventsand aim to integrate out the random variable associated with x and y using Theorem A.2 in theAppendix. In general, what stands in the way of this procedure is the dependence of L x on V ( y ) and L y on V ( x ) , respectively. We therefore relax the conditions in the large deviation events andpick suitable (cid:98) L x,j ⊃ L x , (and hence (cid:98) L y,j ⊃ L y ) (5.42)such that (cid:98) L x,j and (cid:98) L y,j are independent of both V ( x ) and V ( y ) . Postponing the details of thesechoices which will depend on j , we bound the numerator on the right side in (5.41) using Theo-rem A.2 in the Appendix: P ( L x ∩ L y ∩ E x ∩ E y ∩ I x ∩ I y ) ≤ E (cid:104) (cid:98) L x,j ∩ (cid:98) L y,j P ( E x ∩ E y | A x,y ) (cid:105) ≤ C (cid:16) τ − P (cid:16)(cid:98) L x,j ∩ (cid:98) L y,j (cid:17) + τ − E (cid:104) (cid:98) L x,j ∩ (cid:98) L y,j min (cid:8)(cid:12)(cid:12) (cid:98) G x,y (cid:12)(cid:12) , (cid:9)(cid:105)(cid:17) , (5.43) esonant delocalization A x,y the sigma algebra generated by the variables V ( ξ ) , ξ (cid:54)∈ { x, y } and (cid:98) G x,y := G T x,y ( x n − , y n − ; E + iη ) . (5.44)This quantity measures the strength of the interaction of the events E x and E y .Under the assumptions of Theorem 5.6, the denominator in the right side of (5.41) is boundedfrom below by c τ − P ( L x ) P ( L y ) provided n is sufficiently large and η is sufficiently small. Theterms on the right side in (5.43) hence give rise to two terms, r ( j ) ≤ r ( j ) + r ( j ) , which forfixed j = dist( x ∧ y, are defined as: r ( j ) := C P (cid:0)(cid:98) L x,j ∩ (cid:98) L y,j (cid:1) P ( L x ) P ( L y ) (5.45) r ( j ) := C τ P ( L x ) P ( L y ) E (cid:104) (cid:98) L x,j ∩ (cid:98) L y,j min (cid:8) | (cid:98) G x,y | , (cid:9)(cid:105) (5.46)For the precise definition of the events (cid:98) L x,j and (cid:98) L y,j we distinguish three cases: Case ≤ j < n κ : The events L x and L y are already independent of the potential at x and y .Therefore we choose (cid:98) L x,j = L x . (5.47)As a consequence, the corresponding sum involving r ( j ) is seen to be uniformly boundedin n and η : n κ − (cid:88) j =0 r ( j ) K j ≤ C ∞ (cid:88) j =0 K j . (5.48)For an estimate on r ( j ) , we drop the indicator function in the right side of (5.46) and usethe fact that min {| x | , } ≤ | x | σ for any σ ∈ [0 , ; in particular, for σ = s : r ( j ) ≤ C τ P ( L x ) P ( L y ) E (cid:2) | (cid:98) G x,y | s (cid:3) ≤ C τ P ( L x ) P ( L y ) e n − j ) ϕ ( s ) . (5.49)Here the second inequality derives from the finite-volume estimates (3.12). Since ϕ ( s ) < − log K by assumption on s , the geometric sum in the following chain of inequalities isdominated by its last term: n κ − (cid:88) j =0 r ( j ) K j ≤ C τ P ( L x ) P ( L y ) n κ − (cid:88) j =0 e n − j ) ϕ ( s ) K j ≤ C τ P ( L x ) P ( L y ) e N κ ϕ ( s ) K n κ . (5.50)Using the large deviation result, Theorem 5.2, and the fact that − ϕ ( s ) = I ( γ ) + γ s , weestimate τ P ( L x ) P ( L y ) e N κ ϕ ( s ) ≤ e N κ (cid:15) τ e − N κ γs ≤ e N κ (cid:16)(cid:16) − s ) (cid:17) ∆+4 (cid:15) (cid:17) ≤ e N κ ( 158 − s )∆ ≤ C , (5.51)since s > / . esonant delocalization Case n κ ≤ j ≤ n κ : We choose (cid:98) L x,j = L ( N κ − n κ − , +) x , (5.52)which is independent of (cid:98) L y,j = L ( N κ − n κ − , +) y . An estimate on r ( j ) hence requires tobound the ratio: P (cid:0)(cid:98) L x (cid:1) P ( L x ) ≤ C e − ( n − n κ − I ( γ ) − (cid:15) ) e − N κ ( I ( γ )+2 (cid:15) ) ≤ C e N κ (cid:15) e n κ I ( γ ) ≤ C K n κ / . (5.53)Here the first inequality follows from the large deviation result, Theorem 5.2, and holds for n large enough and η sufficiently small. In this situation, the third inequality also appliessince I ( γ ) ≤ log K − ∆ by (5.6) and (5.5), and N κ (cid:15) ≤ ∆ κN κ / ≤ ∆ n κ / . As aconsequence, the sum corresponding to r ( j ) is bounded uniformly in n : n κ (cid:88) j = n κ r ( j ) K j ≤ C K n κ ∞ (cid:88) j = n κ K j ≤ C ∞ (cid:88) j =0 K j . (5.54)For an estimate on the sum corresponding to r ( j ) we use (5.49) again which yields n κ (cid:88) j = n κ r ( j ) K j ≤ C τ P ( L x ) P ( L y ) e (2 N κ − n κ ) ϕ ( s ) K n κ ≤ C τ P ( L x ) P ( L y ) e N κ ϕ ( s ) K n κ / ≤ C (5.55)by (5.51). Case n κ < j < N κ : In this main case, we pick (cid:98) L x,j = L ( j − n κ − , − ) x ∩ L ( N κ + n κ − j − , +) x , (5.56)Note that L ( j − n κ − , − ) x = L ( j − n κ − , − ) y and L ( N κ + n κ − j − , +) x and L ( N κ + n κ − j − , +) y are in-dependent. We may hence estimate the numerator in the definition of r ( j ) using the largedeviation result, Theorem 5.2 to conclude that for all n sufficiently large and η sufficientlysmall: P (cid:0)(cid:98) L x,j ∩ (cid:98) L y,j (cid:1) ≤ P (cid:16) L ( j − n κ − , − ) x (cid:17) P (cid:16) L ( N κ + n κ − j − , +) x (cid:17) P (cid:16) L ( N κ + n κ − j − , +) y (cid:17) ≤ C e − ( I ( γ ) − (cid:15) )(2 n − j − n κ ) ≤ C P ( L x ) P ( L y ) e N κ (cid:15) e − I ( γ )( n κ − j ) . (5.57)Since I ( γ ) < log K , the corresponding sum is hence uniformly bounded in n : N κ − (cid:88) j = 32 n κ +1 r ( j ) K j ≤ C e N κ (cid:15) N κ (cid:88) j = 32 n κ e − I ( γ )( n κ − j ) K j ≤ C e N κ (cid:15) e n κ I ( γ ) K n κ ≤ C e N κ (cid:15) K n κ ≤ C , (5.58)cf. (5.53). esonant delocalization r ( j ) we drop conditions in the indicator function and use min {| x | , } ≤| x | s again: r ( j ) ≤ C τ E (cid:2) L ( nκ,j − x | (cid:98) G x,y | s (cid:3) P ( L x ) P ( L y ) (5.59)The Green function in the numerator is a product of three terms, (cid:98) G x,y = G j (cid:98) G x (cid:98) G y with G j := G T x,y ( x j , y j ) (5.60) (cid:98) G x := G T xj,x ( x j +1 , x n − ) (cid:98) G y := G T yj,y ( y j +1 , y n − ) of which only the first one depends on V ( x j ) . Since L ( n κ ,j − x is independent of V ( x j j ) wemay hence condition on the potential elsewhere and use the uniform bound E (cid:2) | G j | s | A x j (cid:3) ≤ C to estimate the numerator in (5.59): E (cid:2) L ( nκ,j − x | (cid:98) G x,y | s (cid:3) ≤ C E (cid:2) L ( nκ,j − x | (cid:98) G x (cid:98) G y | s (cid:3) = C P (cid:0) L ( n κ ,j − x (cid:1) E (cid:104) | (cid:98) G x | s (cid:105) E (cid:104) | (cid:98) G y | s (cid:105) ≤ C e − ( j − n κ )( I ( γ ) − (cid:15) ) e n − j ) ϕ ( s ) . (5.61)Summing over j with a weight K − j we again obtain a geometric sum which is in this casebounded by the number of terms times the maximum of its first and last term. Therefore weconclude that N κ − (cid:88) j = 32 n κ +1 r ( j ) K j ≤ N κ − (cid:88) j = n κ r ( j ) K j ≤ N κ C τ P ( L x ) P ( L y ) (5.62) × max (cid:40) e − ( N κ − n κ )( I ( γ ) − (cid:15) ) e n κ ϕ ( s ) K N κ , e N κ ϕ ( s ) K n κ (cid:41) . In the first case, we use ϕ ( s ) < − I ( γ ) and Corollary 5.2 to conclude that the term isuniformly bounded in n : N κ C τ P ( L x ) P ( L y ) e − N κ ( I ( γ ) − (cid:15) ) K N κ ≤ N κ C e N κ ( I ( γ )+ γ + 34 ∆+6 (cid:15) ) K N κ ≤ C N κ e − N κ ( ∆ − (cid:15) ) ≤ C , (5.63)since (cid:15) < ∆ / .In the second case, we use (5.51) to conclude that the term is uniformly bounded in n : N κ C τ P ( L x ) P ( L y ) e N κ ϕ ( s ) K n κ ≤ C N κ e N κ ( − s ) ≤ C , (5.64)since s > .This concludes the proof of (5.37). esonant delocalization As we saw in Section 2.3, the applications of the conditions which are derived here for absolutelycontinuous spectrum still require some additional information on the function ϕ λ (1; E ) , or at leaston the Lyapunov exponent L λ ( E ) . While we do not have useful independent bounds on ϕ λ (1; E ) ,in this section we present some partial continuity results for L λ ( E ) which enable the derivation ofthe main conclusions which were drawn in Corollaries 2.3 and 2.4 on the spectral phase diagram.Let us start with some general observations:1. The Lyapunov exponent is the negative real part of the Herglotz function (cf. [17, 32]) givenby W λ ( ζ ) := E [log Γ λ (0; ζ )] . As such, its boundary values lim η ↓ L λ ( E + iη ) exist forLebesgue-almost all E ∈ R and. The latter coincides with L λ ( E ) defined in (2.3), as isseen using a variant of Vitali’s convergence theorem whose use is based on the fact thatthe fractional moments of Γ λ (0; E + iη ) with positive and negative power are uniformlybounded in η .2. In the absence of disorder, the Lyapunov exponent is easy to compute, L ( ζ ) = − log | Γ ( ζ ) | ,where Γ ( ζ ) is the unique solution of K Γ + ζ Γ + 1 = 0 in C + , and one finds: L ( E ) = log √ K | E | ≤ √ K , ∈ (cid:16) log √ K, log K (cid:17) √ K < | E | < K + 1 , ≥ log K | E | ≥ K + 1 . (6.1)3. In general, L λ ( ζ ) is related to the free energy function ϕ λ ( s ; ζ ) through the relation (3.13)and the inequality (3.11) from which one concludes the bound L λ ( ζ ) ≥ log √ K which issaturated if and only if λ = 0 and | E | ≤ √ K . Thanks to the (weak) continuity of the harmonic measure associated with L λ , energy averages turnout to be continuous in the disorder parameter λ ≥ . Theorem 6.1.
For any bounded interval I ⊂ R the function [0 , ∞ ) (cid:51) λ (cid:55)→ (cid:82) I L λ ( E ) dE iscontinuous, and, in particular: lim λ ↓ (cid:90) I L λ ( E ) dE = (cid:90) I L ( E ) dE . (6.2) Proof.
Since the harmonic measure σ λ ( I ) := (cid:82) I L λ ( E ) dE associated with L λ ( ζ ) = π − (cid:82) Im ( E − ζ ) − σ λ ( dE ) is absolutely continuous, the asserted continuity thus follows from the vague con-tinuity of σ λ , which in turn follows from the (weak) resolvent convergence G λ (0 , ζ, ω ) → G λ (0 , ζ, ω ) as λ → λ for all ζ ∈ C + and all ω .In particular, Theorem 6.1 ensures that the mean value of the Lyapunov exponent over anybounded, non-empty interval I , M λ ( I ) := 1 | I | (cid:90) I L λ ( E ) dE , (6.3)is continuous in λ ≥ . This immediately implies Corollary 2.3, namely that the condition L λ ( E ) < log K holds on a positive fraction of every interval I ⊂ ( − ( K + 1) , K + 1) . esonant delocalization Poof of Corollary 2.3.
Since L λ ( E ) ≥ log √ K , we may employ the Chebychev inequality tocontrol the Lebesgue measure of that subset of I on which (2.4) is violated: |{ E ∈ I | L λ ( E ) ≥ log K }| ≤ (cid:90) I L λ ( E ) − log √ K log √ K dE = | I | M λ ( I ) − log √ K log √ K . (6.4)The assertion thus follows from the continuity (6.2) and the fact that log √ K ≤ M ( I ) < log K for all closed intervals I ⊂ ( − K − , K + 1) by a computation, cf. (6.1).Note that M ( I ) = log √ K for all I ⊂ ( − √ K, √ K ) . Hence, in this case the measurein (6.4) tends to as λ ↓ . Let us now turn to the proof of Corollary 2.4. Accordingly, for the remainder of this section, wewill assume that supp (cid:37) = [ − , such that almost surely σ ( H λ ) = [ −| E λ | , | E λ | ] with E λ = − √ K − λ .The main ideas behind the conditions in Corollary 2.4 are:a) At the (lower) spectral edge the Lyapunov exponent is bounded according to: L λ ( E λ ) ≤ L ( E λ − λ ) . (6.5)(An analogous bound applies to the upper edge). This inequality derives from the operatormonotonicity of the function (0 , ∞ ) (cid:51) x (cid:55)→ x − and the estimate ≤ H λ − E λ ≤ T +2 √ K + 2 λ , which implies Γ λ (0; E λ ) ≥ Γ ( E λ − λ ) .b) Using the explicit formula for the Lyapunov exponent in case λ = 0 (cf. (6.1)), we concludethat the condition L ( E λ − λ ) < log K holds if and only if E λ − λ > − ( K + 1) or equivalentlyif (2.7) holds.The following theorem extends the bound (6.5) to energies near E λ in the spectrum. Analogousarguments yield an upper bound near − E λ . Theorem 6.2.
For a random potential satisfying Assumptions A–D with supp (cid:37) = [ − , , for all λ > : lim sup E ↓ E λ L λ ( E ) ≤ L ( E λ − λ ) . (6.6)Following the arguments above, this theorem in particular implies Corollary 2.4. Proof of Corollary 2.4.
Without loss of generality, we restrict the discussion to the region near thelower edge E λ of σ ( H λ ) . For fixed λ < ( √ K − / we may pick ε ( λ ) := log K − L ( E λ − λ ) > which is strictly positive if and only if (2.7) holds. We hence conclude from Theorem 6.2 thatthere is δ ( λ ) > such that L λ ( E ) < log K for any E ≤ E λ + δ ( λ ) . In the proof of Theorem 6.2, we consider the finite-volume restriction of the operator to the Hilbert-space over B R := { x ∈ T | dist(0 , x ) < R } , i.e., H ( R ) λ := 1 B R H λ B R on (cid:96) ( B R ) . (6.7) esonant delocalization E ∈ R : (cid:12)(cid:12)(cid:12) Γ λ (0; E + i − Γ ( R ) λ (0; E ) (cid:12)(cid:12)(cid:12) ≤ (cid:88) x ∈S R (cid:12)(cid:12) G ( R ) λ (0 , x − ; E ) (cid:12)(cid:12) | G λ (0 , x ; E ) | =: S ( R ) λ ( E ) . (6.8)The proof idea for Theorem 6.2 is to choose R such that:a) The following event has a good probability, Z := (cid:110) E λ + ∆ ≤ inf σ ( H ( R ) λ ) (cid:111) . (6.9)In this event and for any E ∈ [ E λ , E λ + ∆) one can use the operator monotonicity of (0 , ∞ ) (cid:51) x (cid:55)→ x − together with the bound ≤ H ( R ) λ − E ≤ H ( R )0 + λ − E which implies Γ ( R ) λ (0; E ) ≥ Γ ( R )0 (0; E − λ ) ≥ Γ ( R )0 (0; E λ − λ ) ≥ Γ (0; E λ − λ ) − S ( R )0 ( E λ − λ ) ≥ Γ (0; E λ − λ ) (cid:16) − K R e − RL ( E λ − λ ) (cid:17) . (6.10)Here, the second inequality holds for all E λ ≤ E < inf σ ( H ( R ) λ ) , the third is a special case of(6.8), and the last inequality follows from the fact that ≤ Γ ( R )0 ( x ; E ) ≤ Γ ( x ; E ) , (6.11)which, using the factorization property of the Green function, implies S ( R )0 ( E ) ≤ K R e − RL ( E ) Γ (0; E ) for any E ∈ R .b) The error terms on the right side of (6.8) and (6.10) are small compared to Γ (0; E λ − λ ) ≥ in the sense that also the event Z := (cid:110) S ( R ) λ ( E ) ≤ Γ (0; E λ − λ ) K − δR/ (cid:111) (6.12)occurs with a good probability. For reasons will become clear in the next subsection, we willchoose δ := log(1 + λ √ K )64 (cid:107) (cid:37) (cid:107) ∞ K log K (6.13)The probability of failure of the first event Z is bounded with the help of the following lemma.Due to Lifshits tailing, this estimate is far from optimal and one expects the probability in (6.14)to be exponentially small (see [13] and references therein for a precise conjecture). Lemma 6.3.
There is some
C > such that for all R > and all ∆ > P (cid:16) inf σ ( H ( R ) λ ) < E λ + ∆ (cid:17) ≤ C K R ∆ / (6.14) Proof.
By Chebychev’s inequality the left side is bounded from above by E (cid:104) tr 1 ( −∞ ,E ) ( H ( R ) λ ) (cid:105) ≤ tr 1 ( −∞ ,E + λ ) ( H ( R )0 ) ≤ e t ( E + λ ) tr e − tH ( R )0 ≤ e t ( E + λ ) tr 1 B R e − tH B R ≤ C K R e t ∆ t − / , (6.15)where E := E λ + ∆ and the last inequality stems form the explicitly known form of the kernel ofthe (infinite-volume) semigroup. Taking t = ∆ − yields the result. esonant delocalization Z are more involved. Postponing thedetails of this probabilistic estimate, which will be the topic of the next subsection, the proof ofTheorem 6.2 proceeds as follows: Proof of Theorem 6.2.
Abbreviating Z := Z ∩ Z , we write L λ ( E ) = − E [1 Z log | Γ λ (0; E + i | ] − E [1 Z c log | Γ λ (0; E + i | ] (6.16)In the event Z and assuming E ∈ [ E λ , E λ + ∆) , one may use (6.8) and (6.10) to estimate | Γ λ (0; E + i | ≥ Γ ( R )0 ( x ; E ) − S ( R ) λ ( E ) ≥ Γ (0; E λ − λ ) (cid:16) − K R e − RL ( E λ − λ ) − K − δR/ (cid:17) . (6.17)The right side is strictly positive for any λ > provided R is large enough. In this case, the abovebound and the monotonicity of the logarithm yields the following bound on the first term on theright in (6.16): − E [1 Z log | Γ λ (0; E + i | ] ≤ L ( E λ − λ ) − log (cid:16) − K R e − RL ( E λ − λ ) − K − δR/ (cid:17) . (6.18)The second term in (6.16) is estimated using the Cauchy-Schwarz inequality − E [1 Z c log | Γ λ (0; E + i | ] ≤ (cid:112) P ( Z c ) (cid:114) E (cid:104) | log | Γ λ (0; E + i || (cid:105) (6.19)Since | log | x || ≤ | x | / + | x | − / ) the second factor is bounded with the help of fractional-moment estimates and (3.3) by a constant which only depends on λ . The probability of failureof the event Z is estimated using Lemma 6.3 and Lemma 6.4 which prove that under the condi-tion (6.21) below: P ( Z c ) ≤ P ( Z c | Z ) + P ( Z c ) ≤ C ( λ ) K − δ δ R + 2 − R + C K R ∆ / . (6.20)We pick ∆ := ( E − E λ ) /c ( λ ) with c ( λ ) from (6.21) and R := (cid:100) log ∆ − log K (cid:101) . The proof is completedby noting that for any λ > : i. ∆ → as E → E λ and ii. R → ∞ as ∆ → . The remaining task concerns the estimate on the error in (6.8). We will prove
Lemma 6.4.
For every λ > there exists a finite C ( λ ) such that if E ≤ E λ + ∆ (cid:34) − exp (cid:32) − log(1 + λ √ K )64 (cid:107) (cid:37) (cid:107) ∞ K log K (cid:33)(cid:35) [=: E λ + c ( λ ) ∆] . (6.21) then P (cid:0) Z c (cid:12)(cid:12) Z (cid:1) ≤ C ( λ ) K − δ δ R + 2 − R . For a proof of this auxiliary estimate, we need to control the first factor in the right side of (6.8)in case
E < inf σ ( H ( R ) λ ) . This is done with the help of the following lemma, which might be ofindependent interest. esonant delocalization Lemma 6.5.
1. Assume a ≤ b < inf σ ( H ( R ) λ ) , then Γ ( R ) λ ( x ; a ) ≤ Γ ( R ) λ ( x ; b ) ≤ (cid:32) b − a inf σ ( H ( R ) λ ) − b (cid:33) Γ ( R ) λ ( x ; a ) . (6.22)
2. Assume a ≤ − √ K and x ∈ B R , then Γ ( R ) λ ( x ; a − λ ) ≤ Γ ( R )0 ( x ; a ) (cid:32) λ √ K − a (cid:33) − ( V ( x )+1) . (6.23) Proof.
The inequalities (6.22) follow from the spectral representation (cid:82) ( u − ζ ) − µ ( R ) λ,δ x ( du ) =Γ ( R ) λ ( x ; ζ ) and elementary inequalities for the integrand.The second claim is based on the observation that a − λ ≤ inf σ ( H λ ) ≤ inf σ ( H ( R ) λ ) for any R > . We may hence differentiate for any λ ≥ : − d Γ ( R ) λ ( x ; a − λ ) dλ ≥ ( V ( x ) + 1) Γ ( R ) λ ( x ; a − λ ) . (6.24)One of the last factors is estimated by Γ ( R ) λ ( y, y ; a − λ ) − ≤ (cid:10) δ y , ( H ( R ) λ + λ − a ) δ y (cid:11) ≤ √ K +2 λ − a . Integrating the resulting inequality yields (6.23).In the following, we suppose E λ + ∆ := inf σ ( H ( R ) λ ) > E > E λ such that ξ λ ( E ) := E − E λ inf σ ( H ( R ) λ ) − E ∈ (0 , ∞ ) . (6.25)Then Lemma 6.5 and the factorization property (3.7) of the Green function imply for all x ∈ S R : ≤ G ( R ) λ (cid:0) , x − ; E (cid:1) ≤ (1 + ξ λ ( E )) R G ( R ) λ (0 , x − ; E λ ) ≤ (1 + ξ λ ( E )) R K R/ (cid:18) λ √ K (cid:19) − σ ( x ) , (6.26)where σ ( x ) := (cid:80) (cid:22) y ≺ x ( V ( y ) + 1) ≥ . To further estimate the right side, we will consider theevent Z := (cid:40) min x ∈S R σ ( x ) ≥ δ log K + 2 log(1 + ξ λ ( E ))log(1 + λ √ K ) R (cid:41) , (6.27)with δ > from (6.13). This event is tailored such that G ( R ) λ (cid:0) , x − ; E (cid:1) ≤ K − R ( δ + ) and hence E (cid:20)(cid:12)(cid:12) S ( R ) λ ( E ) (cid:12)(cid:12) δ δ (cid:12)(cid:12) Z ∩ Z (cid:21) ≤ K R E (cid:20)(cid:12)(cid:12)(cid:12) G ( R ) λ (cid:0) , x − ; E (cid:1) G λ (0 , x ; E ) (cid:12)(cid:12)(cid:12) δ δ (cid:12)(cid:12) Z ∩ Z (cid:21) ≤ C ± K − δ R , (6.28)where the last inequality is based on (3.12) and the upper bound in (3.11). The constants C + , C − depend (also through δ ) on λ . Chebychev’s inequality hence leads to P (cid:0) Z c (cid:12)(cid:12) Z ∩ Z (cid:1) ≤ C ( λ ) K − δ δ R (6.29)with a finite constant C ( λ ) which only depends on λ . For an estimate on the probability of theevent Z we use the following esonant delocalization Lemma 6.6.
For any < α ≤ (8 (cid:107) (cid:37) (cid:107) ∞ K ) − : P (cid:0) min x ∈S R σ ( x ) < αR (cid:1) ≤ K R (cid:16) (cid:112) (cid:107) (cid:37) (cid:107) ∞ α (cid:17) R . (6.30) Proof.
Since there are K R vertices with dist(0 , x ) = R , it suffices for the proof of (6.30) to fix x and estimate P (cid:0) σ ( x ) < αR (cid:1) ≤ (cid:16) e αt E (cid:104) e − t ( V (0)+1) (cid:105)(cid:17) R , (6.31)for any t > , where we employed the help of a Chebychev inequality and the fact that therandom variables (cid:0) V ( y ) (cid:1) are iid . Inserting indicator functions on the set { V (0) + 1 ≥ α } andits complement, we further bound e αt E (cid:2) e − t ( V (0)+1) (cid:3) ≤ e − tα + 2 α (cid:107) (cid:37) (cid:107) ∞ e tα . Choosing t = − (2 α ) − log(4 α (cid:107) (cid:37) (cid:107) ∞ ) > , yields the result.We may now finally give a Proof of Lemma 6.4.
The choice of δ in (6.13) and the condition (6.21) together with Lemma 6.6imply that P ( Z c ) ≤ − R . We have thus established that P (cid:0) Z c (cid:12)(cid:12) Z (cid:1) ≤ P (cid:0) Z c (cid:12)(cid:12) Z ∩ Z (cid:1) + P ( Z c ) ≤ C ( λ ) K − δ δ R + 2 − R . (6.32) AppendixA Fractional-moment bounds
The aim of this appendix is to present some basic weak- L bounds on Green functions of randomoperators, and related fractional moment estimates. Theorem A.2, which presents such bounds forpairs of Green functions, is a new result which is needed here in the proof of our criteria, and whichmay also be of independent interest. In the last subsection we discuss the related implications ofthe regularity Assumption D.The discussion in this appendix is carried within the somewhat broader context of operators ofthe form: H λ ( ω ) = H + λ V ( ω ) , (A.1)acting in the Hilbert space (cid:96) ( G ) , with λ ≥ the disorder-strength parameter and:I G the vertex set of some metric graph,II H a self-adjoint operator in (cid:96) ( G ) , andIII V ( ω ) a random potential such that the random variables { V ( x ) | x ∈ G} are iid with a prob-ability distribution whose density is (essentially) bounded, (cid:37) ∈ L ∞ ( R ) . esonant delocalization A.1 Weak- L bounds We recall that according to the Krein formula, the Green function of H λ ( ω ) restricted to the sites x, y is in its dependence on V ( x ) and V ( y ) of the form (cid:18) G λ ( x, x ; ζ ) G λ ( x, y ; ζ ) G λ ( y, x ; ζ ) G λ ( y, y ; ζ ) (cid:19) = (cid:20)(cid:18) λ V ( x ) 00 λ V ( y ) (cid:19) + A λ ( ζ ) (cid:21) − , (A.2)where A λ ( ζ ) is given by the inverse of the left side for V ( x ) = V ( y ) = 0 . In particular, G λ ( x, x ; ζ ) = ( λV ( x ) − a ) − with some a ∈ C which is independent of V ( x ) .The assumed boundedness of the density (cid:37) of the distribution of V ( x ) trivially implies boundson probabilities of weak- L -type: sup a ∈ C (cid:90) | v − a | < t (cid:37) ( v ) dv ≤ (cid:107) (cid:37) (cid:107) ∞ t . (A.3)Since the dependence of the Green function G λ ( x, x ; ζ ) on V ( x ) is of the above form, this impliesthat the following well-known weak- L bound, and hence the boundedness of fractional moments(cf. [4]). Proposition A.1.
For a random operator H λ ( ω ) = H + λ V ( ω ) on (cid:96) ( G ) satisfying assump-tions I–III, at any complex energy parameter ζ ∈ C + and for any t > and s ∈ (0 , , the Greenfunction satisfies: P (cid:0) | G λ ( x, x ; ζ ) | > t (cid:12)(cid:12) A x (cid:1) ≤ (cid:107) (cid:37) (cid:107) ∞ λ t , (A.4) E (cid:2) | G λ ( x, x ; ζ ) | s (cid:12)(cid:12) A x (cid:3) ≤ s (cid:107) (cid:37) (cid:107) s ∞ (1 − s ) λ s , (A.5) where A x denotes the sigma-algebra generated by V ( y ) , y (cid:54) = x . One trivial, but useful consequence of (A.4) is that for any p ∈ (0 , and t ≥ (cid:107) (cid:37) (cid:107) ∞ λ (1 − p ) : P (cid:0) | G λ ( x, x ; ζ ) | ≤ t (cid:12)(cid:12) A x (cid:1) ≥ p . (A.6)Our new result, which was vital in our second-moment analysis in Lemma 4.11 and The-orem 5.8, concerns the joint conditional probability of events as in (A.4) associated with two(distinct) sites Theorem A.2.
In the situation of Proposition A.1, consider two sites x (cid:54) = y in a graph. Then forany t > and ζ ∈ C + : P (cid:16) | G λ ( x, x ; ζ ) | > t and | G λ ( y, y ; ζ ) | > t (cid:12)(cid:12) A xy (cid:17) ≤ (cid:107) (cid:37) (cid:107) ∞ λ t min (cid:26) (cid:107) (cid:37) (cid:107) ∞ (cid:18)(cid:113)(cid:12)(cid:12) A λ ( x, y ; ζ ) (cid:12)(cid:12) (cid:12)(cid:12) A λ ( y, x ; ζ ) (cid:12)(cid:12) + t − (cid:19) , (cid:27) , (A.7) where A λ ( x, y ; ζ ) are the off-diagonal matrix elements of A λ ( ζ ) in (A.2) , and A xy is the thesigma-algebra generated by V ( ξ ) , ξ (cid:54)∈ { x, y } . esonant delocalization G = T , the off-diagonal matrix elements of A λ ( ζ ) simplify: A λ ( x, y ; ζ ) = G λ ( x, y ; ζ ) G λ ( x, x ; ζ ) G λ ( y, y ; ζ ) − G λ ( x, y ; ζ ) G λ ( y, x ; ζ ) = G T x,y λ ( x − , y − ; ζ ) . (A.8)This is most easily proven by noting that the ratio does not depend on V ( x ) and V ( y ) so that wemay take them to infinity. In this limit the ratio G λ ( x, y ; ζ ) / [ G λ ( x, x ; ζ ) G λ ( y, y ; ζ )] tends to G T x,y λ ( x − , y − ; ζ ) and its numerator vanishes. Proof of Theorem A.2.
Let A λ ( x, y ; ζ ) denote the matrix elements of A λ ( ζ ) in the rank-two Kreinformula (A.2) and abbreviate u := λV ( x ) + A λ ( x, x ; ζ ) v := λV ( y ) + A λ ( y, y ; ζ ) , and α := A λ ( x, y ; ζ ) , β := A λ ( y, x ; ζ ) . The lower bounds on | G λ ( x, x ; ζ ) and | G λ ( y, y ; ζ ) | translate to: (cid:12)(cid:12)(cid:12)(cid:12) u − αβv (cid:12)(cid:12)(cid:12)(cid:12) ≤ t (A.9) (cid:12)(cid:12)(cid:12)(cid:12) v − αβu (cid:12)(cid:12)(cid:12)(cid:12) ≤ t . (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 /t , and a similar statement holds for v and u interchanged and Eq. (A.9) replaced by (A.10).2. For any solution of (A.9) and (A.10): min {| u | , | v |} ≤ | α | + t − . (A.11)The first statement is fairly obvious once one focuses on the condition on the real part in (A.9). Toprove the second assertion, let w := (cid:112) | u | · | v | ≥ min {| u | , | v |} (A.12)Assuming (A.9) and (A.10) we have: | u | | v | − | α | | β | ≤ | u v − αβ | ≤ min {| u | , | v |} t ≤ (cid:112) | 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: w − | α | | β | ≤ wt . (A.14) esonant delocalization w ≤ t + (cid:115) t ) + | α | | β | ≤ t + (cid:18) t + (cid:112) | α | | β | (cid:19) , (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 whichthe event occurs, at specified values of the × matrix A λ ( ζ ) . Let S ⊂ R be the correspondingrange of values of { Re u, Re v } . Then by , S is contained within the union of two strips, oneparallel to the Re v axis and the other parallel to the Re u axis. To bound the measure of itsintersection with the first one, we note that the relevant values of Re u are contained in an intervalof length at most (cid:16) t + (cid:112) | α | | β | (cid:17) , and for each value of u the range of values of Re v is ofLebesgue measure not exceeding /t (by ). Hence the measure of the intersection of S with thisstrip is at most t (cid:16) t + (cid:112) | α | | β | (cid:17) , and a similar bound applies to the intersection of S with thesecond one. Adding the two, one gets the bound claimed in (A.7). A.2 The regularity assumption D
The class of probability densities satisfying Assumption D (see Eq. (2.2)) includes those (cid:37) whichhave a single hump. More precisely, suppose there is some m ∈ R such that (cid:37) is monotoneincreasing for v < m and monotone decreasing for v > m . If one picks ν > such that (cid:37) ( m ) / min { (cid:37) ( m − ν ) , (cid:37) ( m + ν ) } =: c < ∞ , then (2.2) is satisfied for all v ∈ R and c = 2 max { , c /ν } Examples of single-hump probability densities are Gaussian and the Cauchydensities. Similarly as above one sees that any finite linear combination of single-hump functionsalso lead to probability densities which satisfy (2.2).Our next goal is to illuminate some of the consequences of (2.2). Clearly, if (cid:37) satisfies (2.2),then (cid:37) ∈ L ∞ ( R ) and (A.3) applies. In fact, the assumption is tailored to provide the followingextension of (A.3). Lemma A.3. If (cid:37) ≥ satisfies (2.2) (with constant c > ), then for any s ∈ (0 , , a ∈ C and t ≥ : (cid:90) | v − a | < t (cid:37) ( v ) dv | v − a | s ≤ c (1 − s ) t − s (cid:90) (cid:37) ( v ) dv | v − a | s . (A.16) Proof.
We start by estimating the left side (cid:90) | v − a | < t (cid:37) ( v ) dv | v − a | s ≤ sup | v − a |≤ t (cid:37) ( v ) (cid:90) | v − a | < t dv | v − a | s = 2(1 − s ) t − s sup | v − a | < t (cid:37) ( v ) . (A.17)Using (2.2) we then conclude that the last factor in the right side is bounded from below by (cid:90) (cid:37) ( v ) dv | v − a | s ≥ (cid:90) | v − a |≤ (cid:37) ( v ) dv ≥ c sup | v − a |≤ (cid:37) ( v ) . (A.18)The above two estimates imply the assertion. esonant delocalization E ( x,y ) s [ Q ] := E [ | G λ ( x, y ; ζ ) | s Q ] E [ | G λ ( x, y ; ζ ) | s ] , (A.19)where x, y ∈ G , ζ ∈ C + and s ∈ (0 , . We denote by P ( x,y ) s the corresponding probabilitymeasure. Proposition A.4.
In the situation of Proposition A.1, assume additionally that (cid:37) satisfies (2.2) (with constant c > ). Then, at any complex energy parameter ζ ∈ C + and for any s ∈ (0 , and t ≥ λ − , the Green function satisfies: P ( x,y ) s ( | G λ ( x, x ; ζ ) | > t | A x ) ≤ c (1 − s ) ( λt ) − s , (A.20) where A x denotes the sigma-algebra generated by V ( y ) , y (cid:54) = x . Analogously to (A.6), we conclude from (A.20) that for any p ∈ (0 , and all t ≥ λ − ( c/ [(1 − s )(1 − p )] / (1 − s ) : P ( x,y ) s (cid:0) | G λ ( x, x ; ζ ) | ≤ t (cid:12)(cid:12) A x (cid:1) ≥ p , (A.21)uniformly in y ∈ G , the choice of the graph G and ζ ∈ C + . B A 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 statisticalmechanics and probability theory [15, 16, 18]. However since a close enough reference could notbe located we enclose the proof here.
B.1 A general large deviation theorem
The following theorem should be regarded as a stand-alone statement. It is intended to be readdisregarding fact that the symbols which appear there ( Γ and η ) were assigned a specific meaningelsewhere in the paper. The similarity does however indicate the application of this theory to themain discussion of this work. Theorem B.1.
Let { Γ ( N ) j ( η ) } Nj =1 with N ∈ N , be a family of a triangular arrays of randomvariables indexed by η ≥ , satisfying the following two conditions, at some r < r and C < ∞ :a. The functions Ψ N ( t ; η ) := 1 N log E N (cid:89) j =1 | Γ ( N ) j ( η ) | t (B.1) converge pointwise in [ r , r ] ⊂ ( − , : Ψ ( t ) := lim N →∞ η ↓ Ψ N ( t ; η ) . (B.2) esonant delocalization b. For all ≤ k < N , and t , t ∈ [ r , r ] E k (cid:89) i =1 | Γ ( N ) i ( η ) | t N (cid:89) j = k +1 | Γ ( N ) j ( η ) | t ≤ C e ( N − k )[ Ψ N ( t ,η ) − Ψ N ( t ,η )] E (cid:32) N (cid:89) i =1 | Γ ( N ) i ( η ) | t (cid:33) . (B.3) Then for every γ which coincides with − Ψ (cid:48) ( s ) at a point s ≡ s ( γ ) ∈ ( r , r ) where the function Ψ ( s ) is differentiable, and for any ε > , there are (cid:98) N ≡ (cid:98) N ( ε, γ ) < ∞ and ˆ η ≡ ˆ η ( ε, γ ) > suchthat for all N ≥ (cid:98) N and < η < ˆ η the following estimates hold:1. Given the rate function I ( γ ) := − inf t ∈ [ r ,r ] [ Ψ ( t ) + tγ ] one has: P N (cid:89) j =1 | Γ ( N ) j ( η ) | ≥ e − ( γ + ε ) N ≤ e − I ( γ ) N e εN (B.4)
2. With respect to the s -tilted probability average defined by P s ( Q ) = E (cid:16) I Q × (cid:81) Nj =1 | Γ ( N ) j ( η ) | s (cid:17) E (cid:16)(cid:81) Nj =1 | Γ ( N ) j ( η ) | s (cid:17) , (B.5) for any (cid:96) ∈ { , . . . , N } : P s (cid:96) (cid:89) j =1 | Γ ( N ) j ( η ) | ≥ e − ( γ − ε ) (cid:96) ≤ C e − κ ( ε,γ ) (cid:96)/ (B.6) P s (cid:96) (cid:89) j =1 | Γ ( N ) j ( η ) | ≤ e − ( γ + ε ) (cid:96) ≤ C e − κ ( ε,γ ) (cid:96)/ (B.7) where κ ( ε, γ ) := min { κ − ( ε, γ ) , κ + ( ε, γ ) } > and κ ± ( ε, γ ) := sup sgn ∆= ± r
3. For any event Q : P ( Q ) ≥ e − I ( γ ) N e − εN (cid:104) P s ( Q ) − C e − κ ( ε,γ ) N/ (cid:105) (B.9)Several remarks apply: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 con-vexity of Ψ . esonant delocalization N (cid:80) Nj =1 log Γ ( N ) j with respect the the initial probability measurebecomes a regular occurrence once the measure is suitably tilted, i.e. modified by the factor (cid:81) Nj =1 | Γ ( N ) j | s at suitable s . The statement is then derived by relating the original and thetilted probabilities. In Theorem B.1 we add to this standard procedure the observation thatunder the condition (B.3) the global tilt of the measure shifts the typical values of the samplemean of log Γ j for 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. Lemma B.2.
Under the condition (B.2) , one has the uniform convergence: lim N →∞ η ↓ sup s ∈ [ r ,r ] | Ψ N ( s ; η ) − Ψ ( s ) | = 0 . (B.11) Proof.
This follows from the fact that if a family of convex functions converges pointwise over anopen interval, then its convergence is uniform on compact subsets, cf. [34].
Proof of Theorem B.1.
Since the superscript of Γ ( N ) j is somewhat redundant it will be occasionallyomitted (it takes a common value for all terms within each statement).We will choose (cid:98) N ≡ (cid:98) N ( ε, γ ) < ∞ and ˆ η ≡ ˆ η ( ε, γ ) > using Lemma B.2 such that for all N ≥ (cid:98) N ( ε, γ ) and < η < ˆ η ( ε, γ ) : R N ( η ) := sup s ∈ [ r ,r ] | Ψ N ( s ; η ) − Ψ ( s ) | < min (cid:8) ε , κ ( ε, γ ) (cid:9) , (B.12)The proof of (B.4) relies on an elementary Chebychev estimate with s ∈ ( r , r ) : P N (cid:89) j =1 | Γ j ( η ) | ≥ e − ( γ + ε ) N ≤ e N [ s ( γ + ε )+ Ψ N ( s ; η )] = e εsN e − NI ( γ ) e N [ Ψ N ( s ; η ) − Ψ ( s )] ≤ e εN e − NI ( γ ) (B.13)for any N ≥ (cid:98) N and < η < ˆ η by (B.12).For a proof of (B.6) we again employ the Chebychev inequality and (B.3) to conclude for any ∆ such that s + ∆ ∈ ( r , r ) : P s (cid:96) (cid:89) j =1 | Γ j ( η ) | ≥ e − ( γ − ε ) (cid:96) ≤ E s (cid:104) (cid:96) (cid:89) j =1 | Γ j ( η ) | ∆ (cid:105) e ∆( γ − ε ) (cid:96) ≤ C e [ Ψ N ( s +∆; η ) − Ψ N ( s ; η )] (cid:96) e ∆( γ − ε ) (cid:96) (B.14)Infimizing over ∆ , we hence conclude that the left side in (B.14) is bounded by C e − κ + ( ε,γ ) (cid:96) e (cid:96) R N ( η ) ≤ C e − κ + ( ε,γ ) (cid:96)/ (B.15)for any N ≥ (cid:98) N and < η < ˆ η by (B.12).The proof of (B.7) proceeds similarly. It starts from the observation that P s (cid:96) (cid:89) j =1 | Γ j ( η ) | ≤ e − ( γ + ε ) (cid:96) ≤ E s (cid:104) N (cid:89) j = (cid:96) +1 | Γ j ( η ) | − ∆ (cid:105) e − ∆( γ + ε ) (cid:96) ≤ C e [ Ψ N ( s − ∆; η ) − Ψ N ( s ; η )] (cid:96) e − ∆( γ + ε ) (cid:96) (B.16) esonant delocalization ∆ such that s − ∆ ∈ ( r , r ) . Infimizing over this parameter, we hence conclude that theleft side in (B.16) is bounded by C e − κ − ( ε,γ ) (cid:96) e (cid:96) R N ( η ) ≤ C e − κ − ( ε,γ ) (cid:96)/ by (B.12).For a proof of (B.9) we estimate the regular probability of in terms of the one defined via thetilted measure: P ( Q ) ≥ e NΨ N ( s ; η ) e s ( γ − ε ) N P s Q and N (cid:89) j =1 | Γ j ( η ) | ≤ e − ( γ − ε ) N ≥ e NΨ N ( s ; η ) e s ( γ − ε ) N P s ( Q ) − P s N (cid:89) j =1 | Γ j ( η ) | ≥ e − ( γ − ε ) N . (B.17)The first terms are estimated from below similarly as in (B.13) by e − I ( γ ) e − εN . The second termin the bracket is bounded by Ce − κ ( ε,γ ) N/ for any N ≥ (cid:98) N and < η < ˆ η according to (B.6). B.2 Applications to Green function’s large deviations
The aim of this subsection is to establish the two main large-deviation statements which are usedin this paper, which were asserted in Theorems 3.5 and 5.2. We start with the latter.
Proof of Theorem 5.2.
We first check the applicability of Theorem B.1. By construction, the vari-ables { Γ ± ( j ; η ) } N κ j =1 , which were defined in (5.8), are two families of triangular arrays. Theysatisfy the consistency condition (5.9). As a consequence, the quantity defined in (B.1) agrees forboth cases: Ψ N κ ( s ; η ) = 1 N κ log E (cid:104)(cid:12)(cid:12)(cid:12) G (cid:98) T x ( x n κ , x N − ; E + iη ) (cid:12)(cid:12)(cid:12) s (cid:105) . (B.18)Lemma 3.4 and Theorem 3.2 imply that for any t ∈ ( − ς, : ϕ ( t ; E ) ≡ ϕ ( t ) = lim N κ →∞ η ↓ Ψ N κ ( t ; η ) . (B.19)Moreover, these bound ensure the validity of (B.3) with r = − ς and arbitrary r ∈ (0 , . For aproof of this assertion, one integrates out the random variable associated with the first vertex onwhich t occurs, cf. (3.15).The upper bound (5.12) is hence a consequence of (B.4). For a proof of the lower bound (5.11)we employ (B.9). We first note that the choice of b is tailored to ensure P s (cid:16) L (bc) x (cid:17) ≥ . Further-more, using (B.6) and (B.7) we conclude that there are (cid:98) N ≡ (cid:98) N ( (cid:15), γ ) and (cid:98) η ≡ (cid:98) η ( (cid:15), γ ) such that forall N κ ≥ (cid:98) N and η ∈ (0 , (cid:98) η ) : − P s (cid:16) N κ (cid:92) k = 12 n κ L ( k, ± ) x ( η ; (cid:15) ) (cid:17) ≤ N κ (cid:88) k = 12 n κ P s (cid:16) k (cid:89) j =1 | Γ ± ( j ; η ) | ≥ e − ( γ − ε ) (cid:96) (cid:17) + P s (cid:16) k (cid:89) j =1 | Γ ± ( j ; η ) | ≤ e − ( γ + ε ) (cid:96) (cid:17) ≤ C N κ (cid:88) k = 12 n κ e − κ ( ε,γ ) k/ ≤ Cκ ( ε, γ ) e − κ ( ε,γ ) n κ / . (B.20) esonant delocalization n κ sufficiently large, this term can be made arbitrarily small since κ ( ε, γ ) > . As aconsequence, we conclude that there is some n and η such that for all | x | ≥ n and η ∈ (0 , η ) : P s ( L x ( η ; (cid:15) )) ≥ . (B.21)Using this estimate in (B.9) concludes the proof of (5.12), since the second term in (B.9) is seento be arbitrarily small for n large enough and any factor may be absorbed for sufficiently large N κ by decreasing the prefactor e − N k ( I ( γ )+2 (cid:15) ) in (B.9). Proof of Theorem 3.5.
In a similar way as in the proof of Theorem 5.2, the assertion follows fromTheorem B.1 in the special case of s = 0 . This choice is admissible since, according to (3.13), thefree energy function ϕ ( s ; E ) , which emerges in the limit (B.19), is differentiable at s = 0 withderivative given by the negative Lyapunov exponent. Acknowledgments.
It is a pleasure to thank Mira Shamis for bringing Proposition 2.2 to our attention. We thank theDepartments of Physics and Mathematics at the Weizmann Institute of Science for hospitality at visits during whichsome of the work was done. This research was supported in part by NSF grants PHY-1104596 and DMS-0602360(MA), DMS-0701181 (SW), and a Sloan Fellowship (SW).
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