SPECTRAL AND DYNAMICAL PROPERTIES OF CERTAIN RANDOM JACOBI MATRICES WITH GROWING PARAMETERS

Transcription

1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 6, June 00, Pages S Article electronically published on January 0, 00 SPECTRAL AND DYNAMICAL PROPERTIES OF CERTAIN RANDOM JACOBI MATRICES WITH GROWING PARAMETERS JONATHAN BREUER Abstract In this paper, a family of rom Jacobi matrices with off-diagonal terms that exhibit power-law growth is studied Since the growth of the romness is slower than that of these terms, it is possible to use methods applied in the study of Schrödinger operators with rom decaying potentials A particular result of the analysis is the existence of operators with arbitrarily fast transport whose spectral measure is zero dimensional The results are applied to the infinite Dumitriu-Edelman model 00, its spectral properties are analyzed Introduction For a sequence of positive numbers, {an} n, a sequence, {bn} n,ofreal numbers, let J {an} n, {bn} n denote the Jacobi matrix with off-diagonal elements given by {an} n diagonal elements given by {bn} n on the diagonal That is, b a 0 0 a b a 0 J {an} n, {bn} n 0 a b3 a3 For η 0, λ > 0, let J λ,η be the Jacobi matrix whose parameters are a λ,η n λ n η b λ,η n 0 Since η <, J η is a self-adjoint operator on l N [] For any such operator we may define the spectral measure, µ, asthe unique measure satisfying δ, J z dµx δ z C \ R, R x z where, denotes the inner product in l It follows from the work of Janas Naboko [8] that J λ,η has absolutely continuous spectrum covering the whole real line This paper deals with rom perturbations of J λ,η that are weak in the sense that the variance of the perturbing rom parameters grows like n η with η <η Received by the editors November 3, 007, in revised form, June 6, Mathematics Subject Classification Primary 47B36; Secondary 60H5 36 c 00 American Mathematical Society Reverts to public domain 8 years from publication

2 36 J BREUER The case an, η < 0 η 0 no perturbation off the diagonal is the extensively studied family of one-dimensional discrete Schrödinger operators with a rom decaying potential [3, 4, 5, 3, 4, 9] For these Schrödinger operators it has been established that when η <, the absolutely continuous spectrum of the Laplacian is as preserved When η >, however, the disorder wins over, the spectrum is pure point with eigenfunctions that decay at a super-polynomial rate At the critical point η the spectral behavior exhibits a sensitive dependence on the coupling constant: The generalized eigenfunctions decay polynomially, the spectral measure is pure point or singular continuous according to whether these generalized eigenfunctions are in l or not for a comprehensive treatment of discrete Schrödinger operators with rom decaying potentials, see Section 8 of [3] From this perspective, the extension presented in this paper is in allowing growth of the off-diagonal terms Intuitively, these terms are responsible for transport, thus, their growth should have an effect on the spectrum similar to that of the decay of the potential diagonal terms Indeed, a particular case of our analysis is that of η 0, namely, the diagonal terms are iid rom variables We show that the critical point here is η Below this value, the spectrum is as pure point, whereas above it the spectral measure is one-dimensional More generally, let {X ω n} n be a sequence of iid rom variables Let {Y ω n} n be another such sequence the distributions of the X s the Y s need not be the same Assume the following is satisfied: i For all n X ω n Y ω n 0 where fω fωdp Ω, F,dp is the underlying probability space Ω ii For any k N 3 iii 4 Yω n k <, Xω n, Xω n k < Yω n 4 iv The common distribution of X ω n is absolutely continuous with respect to Lebesgue measure Given the quadruple Υ η,η,λ,λ with0<η <, η <η λ,λ > 0, define the two rom sequences b Υ,ω n b λ,η ;ωn λ n η X ω n, α λ,η ;ωn λ n η Y ω n Let define a Υ,ω n a λ,η n+α λ,η ;ωn 5 J Υ,ω J {a Υ,ω n} n, {b Υ,ωn} n The assumptions on the parameters defining J Υ,ω do not exclude the possibility that some of the off-diagonal terms will vanish However, with probability one, this

3 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 363 may happen only a finite number of times, so that J Υ,ω has an infinite part with strictly positive off-diagonal entries In the following, when we refer to J Υ,ω,we refer to this part We shall prove Theorem For the model above, let γ η η let Λ λ /λ The following holds with probability one: If γ> the spectrum of J Υ,ω is R µ Υ,ω, the spectral measure of J Υ,ω,is one-dimensional, meaning that it does not give weight to sets of Hausdorff dimension less than In the case γ the spectrum of J Υ,ω is R, we have the following two possibilities: a If Λ > η,thenµ Υ,ω is pure point with eigenfunctions decaying like ψ E ω n n Λ+η b If Λ η, then µ Υ,ω is purely singular continuous with exact Hausdorff dimension equal to Λ η 3 If γ<, then the spectrum is pure point with eigenfunctions decaying like ψ E ω n e Λn γ In this case, if η > η, then the spectrum fills R Remark We say that a measure, µ, has exact Hausdorff dimension, ϱ, if it is supported on a set of Hausdorff dimension ϱ does not give weight to sets of Hausdorff dimension less than ϱ For more information concerning the decomposition of general measures with respect to their Hausdorff-dimensional properties, consult [5] the references therein Remark Similar to the Schrödinger case, one would expect to have an absolutely continuous spectrum for γ>/ Unfortunately, though we believe this is true, one-dimensional spectral measure is all we could get Remark The requirement that {X ω n} n {Y ωn} n be identically distributed sequences is not really necessary is noted here only to simplify the discussion In resemblance of the Schrödinger case, the proof of this theorem follows by analyzing the asymptotics of solutions to the formal eigenfunction equation Jψ Eψ Namely, we shall analyze the solutions to the difference equation 6 anψn ++bnψn+an ψn Eψn, n > By a theorem of Kiselev Last [, Theorem ], the results obtained have implications for the quantum dynamics associated with J; namely the behavior of a given vector ψ under the operation of the one parameter unitary group e itj More precisely, let ˆX be the position operator defined by ˆXψ n nψn

4 364 J BREUER Theorem of [], which can be seen to hold in our setting, says that Theorem Proposition 7 below lead to Theorem Assume γ Λ η namely, case b of Theorem Then for any ε>0, m N, T>0 ψ l, T 7 ˆXm e itj Υ,ω ψ, e itj Υ,ω ψ dt Cω, ψ, m, εt m/ η ε T with probability one 0 Note that η may be chosen arbitrarily close to, while the spectral measure may have any dimension in [0, Thus, by tuning the parameters we obtain operators with any local spectral dimensions having arbitrarily fast transport As an application of our general analysis, we study the Gaussian β ensembles arising naturally in the context of Rom Matrix Theory: The eigenvalue distribution functions for the three classical Gaussian ensembles are given by 8 f β,n E,,E N exp N E j E j E k β G βn j<k N with β, 4 for the Gaussian Orthogonal Ensemble, Gaussian Unitary Ensemble Gaussian Symplectic Ensemble, respectively A family of rom matrix ensembles, indexed by β, having f β,n as their eigenvalue distribution function, for any positive value of β, was recently constructed by Dumitriu Edelman [6] The matrices in these ensembles are finite rom Jacobi matrices with the distribution of the off diagonal terms depending on β: Definition 3 Fix β > 0 The rom family of Jacobi matrices J β,ω J {a β,ω n} n, {b β,ωn} n is defined by: The rom variables {a β,ω n} n, {b β,ωn} n are all independent b β,ω n are all stard Gaussian variables that is, with zero mean variance, irrespective of β n 3 The probability distribution function of a β,ω n isgivenby 9 P {ω a β,ω n <C} Γ βn C 0 x βn e x dx In [6], Dumitriu Edelman showed that the eigenvalue distribution function of the finite matrix, obtained as the restriction of J β,ω to the N N upper left corner, is f β,n for any β>0 From property 3 above, it follows that 0 a β,ω n βn+ βn βn Γ a β,ω ndω Ω Γ a β,ω n a β,ω n 4 + O n + O 4βn Thus we see that the family J β,ω corresponds to the case η /, η 0of the general matrices introduced above Technically, the following theorem is not a n 3,

5 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 365 corollary of Theorem because of the O n / term in 0 the O n term in The proof of Theorem, however, is robust with respect to such a change, we have Theorem 4 For any β, the spectrum of J β,ω is R with probability one If β<, then, with probability one, the spectral measure, µ β,ω, corresponding to J β,ω δ, is pure point with eigenfunctions decaying as ψ ω n n + β If β, then with probability one, for any ε>0, µ β,ω has exact dimension β with probability one Furthermore, for β, we have that, almost surely, for any ψ, ε>0 m T ˆXm e itj β,ω ψ, e itj β,ω ψ dt Cω, ψ, m, εt m ε T 0 This result, without the dynamical part, was announced in [] We note that the analogous Circular β Ensembles can be realized as eigenvalues of truncated CMV matrices This was shown by Killip Nenciu [0] later used by Killip Stoiciu in their analysis of level statistics for ensembles of rom CMV matrices [] The bulk spectral properties of the appropriate matrices were analyzed by Simon [, Section 7] The proof of Theorem is given in the next section Since the proof of the spectral part of Theorem 4 is precisely the same, it is not given separately As noted earlier, the dynamical part of our analysis Theorem the corresponding statement in Theorem 4 follows immediately from Theorem Proposition 7, by Theorem of [] The method we use is a variation on the one used by Kiselev-Last-Simon [3, Section 8] in their analysis of the Schrödinger case described above A notable difference is the fact that, due to the growth of the an, the effective energy E parameter, an, vanishes in the limit This, in addition to requiring a modification in the technique of proof see Lemma 4 Proposition 5 below, leads to the fact that the asymptotics of the generalized eigenfunctions are constant over R At the critical point η η, this implies uniformity of the local Hausdorff dimensions of the spectral measure A modified Combes-Thomas estimate, for operators with unbounded off-diagonal terms, enters our analysis in the identification of the spectrum of J Υ,ω Such an estimate may be of independent interest, thus is presented in the Appendix Proof of Theorem We begin with a simple lemma that shows that, in a certain sense, J Υ,ω is a rom relatively decaying perturbation of J λ,η Lemma For any ε>0 there exists, with probability one, a constant C Cω, ε for which n η X ω n n η C n γ ε

6 366 J BREUER n η Y ω n n η C n γ ε, where γ η η Proof By 3 Chebyshev s inequality we have for any k N { P n P ω n η X ω n n η } n γ ε Ck n kε By choosing k >ε we see that P n < n now follows from the Borel-Cantelli Lemma The proof of is the same As stated in the Introduction, we follow the strategy of [3] In particular, we will deduce the spectral properties of J Υ,ω from the asymptotics of the solutions to the corresponding eigenfunction equation In order to fix notation, for a given Jacobi matrix J {an} n, {bn} n afixede R, denotebyψ E a solution to the equation 3 anψ E n ++bnψ E n+an ψ E n Eψ E n, n > It is customary to extend this equation to n by defining a0 Clearly, the space of sequences {ψ E n} n0 solving 3 is a two-dimensional vector space, any such sequence is completely determined by its values at 0 We let ψφ En st for the solution of 3 satisfying 4 ψφ E 0 sinφ, ψφ E cosφ We note that, formally, ψ0 E n satisfies Jψ0 E Eψ0 E Let Then, for any φ, so S E n E bn an an an 0 T E n S E n S E n S E ψ E φ n + ψ E φ n T E n ψ T E E n φ ψφ E0 ψ E 0 n + ψ E π ψ0 E n ψ E π, n + n We call the matrices S E n defined above one-step transfer matrices, for the matrices T E n weusethenamen-step transfer matrices Our main technical result is Theorem Let J Υ,ω be the family of rom Jacobi matrices described in the Introduction Then, for any E R, the following holds with probability one

7 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 367 If γ> log Tω E n 5 lim η n logn If γ log Tω E n 6 lim Λ η n logn 3 If γ< log Tω E n 7 lim n n γ Λ The EFGP transform see [3] is a useful tool for studying the asymptotic behavior of T E n in the Schrödinger case an For an, certain modifications are needed We proceed to present a version that is suitable for our purposes Let J {a ω n} n{b ω n} n be a Jacobi matrix whose entries are all independent rom variables Let ãn a ω n α ω a ω n ãn, assume that 8 lim ãn n that 9 lim n α ω n ãn 0 with probability one These properties clearly hold for J Υ,ω see Lemma In the analysis that follows we keep E R fixed, so we omit it from the notation Define Then, K ω n 0 0 a ω n S ω n K ω nsnk ω n E bω n a ω n a ω n a ω n 0 T ω n S ω n S ω n S ω K ω nt ω n For any φ, define the sequences {u ω,φ n} n {v ω,φn} n by uω,φ n v ω,φ n T cosφ ω n, sinφ so that, from the definition of ψ ω,φ ψ φ for the rom Jacobi parameters, we see that 0 u ω,φ n ψ ω,φ n +, v ω,φ n a ω nψ ω,φ n By 8 we see that for any E R sufficiently large n, we may define k n 0,πby cosk n E ãn Clearly, k n π as n

8 368 J BREUER Now, define R ω,φ n θ ω,φ n through R ω,φ n sinθ ω,φ n v ω,φ n sink n 3 R ω,φ ncosθ ω,φ n ãnu ω,φ n v ω,φ ncosk n, so that using 0 R ω,φ n v ω,φ n +ãn u ω,φ n ãnu ω,φ nv ω,φ ncosk n a ω n ψ ω,φ n +ãn ψ ω,φ n + a ω neψ ω,φ n +ψ ω,φ n, which leads to 4 R ω,φ n ãn ψ ω,φ n + ψ ω,φ n + + α ω nãnψ ω,φ n ãn ψ ω,φ n + ψ ω,φ n + α ω n ψ ω,φ n + ãn ψ ω,φ n + ψ ω,φ n + Ea ωnψ ω,φ nψ ω,φ n + ãn ψ ω,φ n + ψ ω,φ n + By 9, it follows that the right h side converges to one with probability one, uniformly in φ, so that almost surely, for sufficiently large n, there are constants C,C > 0 such that C R ω,φ n ãn ψ ω,φ n + ψ ω,φ n + C R ω,φ n Now, by a straightforward adaptation of Lemma of [3], it follows that for any two angles φ φ, there are constants C 3,C 4 > 0 such that 5 C 3 maxr ω,φ n,r ω,φ n ãn T ω n C 4 maxr ω,φ n,r ω,φ n Thus, we are led to examine the asymptotic properties of log R ω,φ n Let us formulate a recursion relation for R ω,φ n : 3 mean Rω,φ n sinθ ω,φ n R ω,φ ncosθ ω,φ n 0 sinkn uω,φ n ãn cosk n v ω,φ n 0 sinkn 0 ψω,φ n + ãn cosk n 0 a ω n ψ ω,φ n We also know that ψω,φ n + ψ ω,φ n + ψω,φ n + S ω n + ψ ω,φ n,

9 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 369 so Rω,φ n + sinθ ω,φ n + 6 R ω,φ n +cosθ ω,φ n + 0 sink n+ 0 ãn + cosk n+ 0 a ω n sinkn 0 a ω n ãn cosk n 0 sink n+ S ãn + cosk n+ ω n + 0 sinkn Rω,φ n sinθ ω,φ n ãn cosk n R ω,φ ncosθ ω,φ n S ω n + Rω,φ n sinθ ω,φ n R ω,φ ncosθ ω,φ n Now, write S ω n + E bω n+ a ω n+ a ω n+ a ω n + 0 E ãn + ãn+ ãn+ a ω n + ãn b ωn+ ãn+ 0 a ω n+ ãn+ ãn+ 0 We define Z ω n + 0 sink n+ ãn + cosk n+ 0 sinkn ãn cosk n E ãn+ ãn+ ãn + 0 sinθω,φ n cosθ ω,φ n W ω n + 0 sink n+ ãn + cosk n+ 0 sinkn ãn cosk n b ωn+ ãn+ 0 a ω n+ ãn+ sinθω,φ n cosθ ω,φ n ãn+ 0 we ignore the dependence on φ since we keep it fixed Then, from 6 we see that 7 R ω,φ n + ãn + R ω,φ n a ω n + Z ωn ++W ω n +

10 370 J BREUER θ ω,φ n satisfies a recurrence relation as well: From 0 we have that 8 v ω,φ n +a ω n +u ω,φ n 9 a ω n +u ω,φ n ++b ω n +u ω,φ n+a ω nu ω,φ n Eu ω,φ n Write, using -3, that cotθ ω,φ n + ãn +u ω,φn + cosk n+ v ω,φ n + sink n+ v ω,φ n + 0 ãn +u ω,φn + a ω n +cosk n+ u ω,φ n sink n+ a ω n +u ω,φ n Furthermore, observing that R ω,φ n sinθ ω,φ n+k n ãn sink n u ω,φ n R ω,φ ncosθ ω,φ n+k n ãncosk n u ω,φ n v ω,φ n ãncosk n u ω,φ n a ω nu ω,φ n, we may write cotθ ω,φ n+k n ãncosk nu ω,φ n a ω nu ω,φ n ãn sink n u ω,φ n Substituting u ω,φ n + from 9 into 0 then u ω,φ n from into the resulting equation, we get ãn +ãn sink n cotθ ω,φ n + a ω n + sink n+ cotθ ω,φn+k n ãn + +cotk n+ a ω n + ãn + sink n a ω n + b ωn + κ ω n +cot θ ω,φ n + ζ ω n +, where θ ω,φ n θ ω,φ n+k n By 5 picking two angles φ φ, Theorem follows from Proposition 3 Let J Υ,ω be the family of rom Jacobi matrices described in the Introduction Then, for any E R for any φ, the following holds with probability one: If γ> log Rω,φ E 3 lim n η n logn If γ log Rω,φ E 4 lim n Λ+η n logn

11 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 37 3 If γ< log Rω,φ E 5 lim n n n γ Λ Proof As in [3] we shall prove the statement by using the recursion relation for R ω n equation 7 Namely, we shall prove that 6 F γ n log Z ω j+w ω j aω j log ãj converges to the appropriate limit, where F γ n logn forγ F γn n γ otherwise From this point on, a ω n a Υ,ω n b ω n b Υ,ω n We shall need some estimate on the behavior of θ ω n We start with Lemma 4 For any ε>0, there exists, with probability one, a constant C Cω, ε, such that 7 θ ω,φ n + θ ω,φ n C maxn γ+ε,n Proof of Lemma 4 We start by proving a similar statement for κ ω n + + ζ ω n + : Recall that k n π Thus, for large enough n, sink n > Moreover, cosk n n η It follows that sink n sink n+ cosk n cosk n+ sink n +sink n+ C n +η Thus, κ ω n + + ζ ω n + sink n ãn +ãn sink n+ a ω n + + sink n sink n+ sink n+ + ãn + cotk n+ a ω n + + ãn + sink n a ω n + b ωn + I + I + I 3 + I 4 Now, choosing ε<η η, by Lemma I ãn +ãn ãn + a ω n + +4 α ω n +ãn + a ω n + + α ω n + a ω n + C ω, εmaxn γ+ε,n, I 3 cosk n+ α ω n +ãn ++α ω n + a ω n + C 3ω, ε n γ ε,

12 37 J BREUER I 4 ãn +b ω n + a ω n + C 4ω, ε n γ ε almostsurelywehaveshownabovethat I C n +η, so we see that there exists, with probability one, a constant C 5 ω, ε for which 8 κ ω n + + ζ ω n + C 5 max n γ+ε,n Now use e ix +icot x to see that, if κ ω n + + ζ ω n + < which indeed happens almost surely, for large enough n, then e iθω,φn+ e i θ ω,φ n 4 κ ω n + + ζ ω n +, which implies by e ix x π that θ ω,φ n + θ ω,φ n π κ ω n + + ζ ω n + This, together with 8, implies 7 concludes the proof of the lemma The direct consequence of this is Proposition 5 Assume fn is a function that satisfies fn o n r fn + fn o n r, with γ if γ / η > 0, γ if γ</ η > 0, 9 r η if γ / η 0, η γ if γ</ η 0 Then, for any φ, 30 lim fjcosθ ω,φ j 0 n F γ n almost surely The same statement holds with θ replaced by θ Proof of Proposition 5 We shall prove the statement for θ By summation by parts, fjcosθ ω,φ j 3 n fn j cosθ ω,φ j cosθ ω,φ l fj + fj l

13 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 373 Thus we are led to examine j l cosθ ω,φl Assume that j m is even Then j cosθ ω,φ l m cosθ ω,φ l cosθ ω,φ l + π l l m θ ω,φ l θ ω,φ l π 3 l m θ ω,φ l θ ω,φ l + k l π l C ω maxj γ+ε,j η almost surely, by Lemma 4 by k n π cosk n, which holds for sufficiently large n Thus, for any j, weget j cosθ ω,φ l C ω maxj γ+ε,j η + 33 l C ω +maxj γ+ε,j η A simple calculation finishes the proof for θ The proof for θ follows the same argument, with an additional k l k l term in 3 We abbreviate 34 A ω j a ωj ãj ãj α ωj ãj + α ωj ãj λ Y ω j λ Y ω j λ j γ + j γ λ 35 B ω j b ωj ãj λ X ω j λ j γ By a straightforward calculation we get W ω j sin θ ω j ãj sin k j ãj A ω j + B ω j +cosk j B ω ja ω j, Z ω j +sin θ ω j ãj ãj ãj + ãj sin θ ω j ãj sin k j sin k j sin k j

14 374 J BREUER 38 Z ω j, W ω j ãj ãj A ω j sin θ ω j sin k j ãj ãj sin θ ω j cosk j sinθ ω j B ωjsin θ ω j sink j Recall θ ω j θ ω,φ j θ ω,φ j+k j θ ω j+k j Let ε< γ 8 By Lemma, the fact that cosk j ãj the identity sin x sin y cos x cos y, it follows that W ω j O ω j γ+ε, Z ω j, W ω j O ω j γ+ε Zω j Oj with probability one, where the notation O ω indicates that the implicit constant depends on ω Thus, we can use log + x x x + Ox3, together with the observation that almost surely Z ω j +Z ω j, W ω j+ W ω j 4Z ω j, W ω j + O ω j 3γ ε, to see that, with probability one, for large enough j we have that log + Z ω j +Z ω j, W ω j+ W ω j 39 Z ω j + Z ω j, W ω j+ W ω j Z ω j, W ω j + O ω j 3γ ε + j +γ ε Therefore, since the addition of a finite number of terms is inconsequential, it follows that 40 lim n F γ n log Z ω j+w ω j lim n F γ n Z ω j, W ω j Z ω j + Z ω j, W ω j+ W ω j with probability one, in the sense that both limits exist together are equal if they do

15 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 375 Similarly, 4 lim n F γ n aω j log ãj lim n F γ n lim n F γ n log + A ω j A ω j A ωj Thus, our problem is reduced to computing the limits: ξ Z lim n F γ n Zω j, ξ W lim n F γ n ξ ZW lim n F γ n ξ ZW lim n F γ n ξ A lim n F γ n W ω j, Z ω j, W ω j, Z ω j, W ω j, A ω j A ωj By 37, Z ω j sin ãj θ ω j ãj ãj + ãj sin θ ω j sin k j sin k j ãj sin k j The last term on the right is absolutely summable On η, so we only need to look at sin θ ω j ãj ãj ãj But,bysin α cosα by Proposition 5, we see that recall ãj λ j η 4 ξ Z lim n F γ n lim n F γ n ãj ãj η j ãj { η if γ /, 0 otherwise

16 376 J BREUER For the other four limits, we shall use extensively Lemma 84 of [3] in order to replace A ω B ω by their means For example, write 43 W ω j sin θ ω j ãj A sin k j ãj ω j A ω j + sin θ ω j ãj B sin k j ãj ω j B ω j + sin θ ω j ãj sin k j ãj cosk j B ω ja ω j + sin θ ω j ãj sin k j ãj Aω j + B ω j Then the first three terms have mean zero, so, by Lemma 84 of [3], we get that ξ W lim n F γ n sin θ ω j ãj sin k j ãj Aω j + B ω j Exping A ω j, throwing out terms that are oj γ, applying Proposition 5 writing, again, sin α cosα, we get 44 γ W lim n F γ n j η j η sin k j Λj γ 4Y ω j + X ω j { lim Λj γ 0 if γ>/, n F γ n Λ otherwise recall Y ω j 4 Applying the same procedure to ξ ZW ξ A we get ξ ZW lim n F γ n lim n F γ n ξ A lim n F γ n lim n F γ n j η sin θ ω j Λ Y ωj j η sin k j j γ { Λ 0 if γ>/, j γ Λ otherwise Λ Y ωj j γ { Λ 0 if γ>/, j γ Λ otherwise

17 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 377 The computation of ξ ZW involves sin 4 θ ω 3 8 cos θ ω + cos4 θ ω 8, for which Proposition 5 is useless Luckily, the cos4 θ ω j cancels out As before, ξ ZW Λ lim n F γ n j γ 4Yω j sin 4 θ ω j j η sin k j j η Xω j sin θ ω j + 4 Now write 4Yω j sin 4 θ ω j j η sin k j j η + Xω j sin θ ω j 4 sin 4 θ ω j j η sin k j j η + sin θ ω j 4 3j η 8j η sin k j + 8 j η + j η sin k j cos θ ω j j η + cos4 θ ω j j η sin k j 3j η 8j η sin k j + 8 j η + j η sin k j cos θ ω j + cos4 θ ω j On +On η to see that { 0 if γ>/, 47 ξ ZW Λ otherwise, where the cos θ ω term vanishes by Proposition 5 limits, the proposition is proved Summing up the various Proof of Theorem By 5, the theorem follows from Proposition 3 Theorem follows almost immediately from Theorem Proposition 3 As in the Schrödinger situation, the case γ requires some subtle reasoning

18 378 J BREUER We have established that, in this case, with probability one, equation 3 has a solution, ψ, with 48 ψn n Λ η In order to use subordinacy theory, we need the existence of another solution with faster decay at infinity The following is Lemma 87 of [3], formulated for general regular matrices: Lemma 6 Let u φ cosφ, sin φ R For any matrix A GL R with deta d>0, letφa be the unique φ π, π ] with d Au φ d A Define ρa Au 0 Au π/ Let A n be a sequence of matrices in GL R with deta n d n > 0 that satisfy A dn A n d n+ A n n A n A n+ 0 as n Let ρ n ρa n φ n φa n Then: φ n has a limit φ if only if lim n ρ n ρ exists ρ is allowed, but then we only have φ n π Suppose φ n has a limit φ 0, π equivalently, ρ 0, Then log A n u log d n lim n log A n log d n if only if log ρ n ρ lim sup n log A n log d n Proposition 7 Let J Υ,ω be the family of rom Jacobi matrices described in Theorem, withγ Then, for any E R, there exists, with probability one, cosφω an initial condition Ψ φω such that sinφω log Tω E nψ φω 49 lim n log n Λ + η Proof We imitate the proof of [3, Lemma 88] By Proposition 3 log R ω,0 n lim n log n Λ + η log R ω,π/ n lim n log n Λ + η for almost every ω By -3, R ω,0 nr ω,π/ n sinθ ω,π/ n θ ω,0 n ãnsink n so that, for almost every ω, log θ ω,0 n θ ω,π/ n 50 lim Λ n log n recall ãn λ n η

19 RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 379 Let ρ ω n R ω,0n R ω,π/ n Then 5 L ω n log ρ ω n + log ρ ω n log+x ω,0 n log + X ω,π/ n, where 5 X ω,φ n Z ω,φ n ++W ω,φ n + Since X ω,φ j O j almost surely, we may apply a finite Taylor expansion to the above using 36, to see that, with probability one, for large enough n, L ω n ωn sin θ ω,0 n sin θ ω,π/ n + ωn sin θ ω,0 n sin θ ω,π/ n + 3 ωn sin θ ω,0 n sinθ ω,0 n sin θ ω,π/ n sinθ ω,π/ n + O n +Λ ε, where ω n ωn 3 ωn 0 i ω n C j n, i,, 3 The O n +Λ ε for the remainder follows from Lemma 50 A stard application of Kolmogorov s inequality the Borel Cantelli Lemma shows that with probability one, for large enough k, foreachi,, 3, m 53 sup i ωj m k +,, k j k + k also k 54 sup i ωj m k +,, k m k Combining this with the fact that, with probability one, for large enough n, sin θ ω,0 n sin θ ω,π/ n O n Λ+ε, sin θω,0 n sin θ ω,π/ n O n Λ+ε sin θ ω,0 n sinθ ω,0 n sin θ ω,π/ n sinθ ω,π/ n O n Λ+ε, it follows that L ω n n

20 380 J BREUER exists L ω n C ωn Λ+ε nn almost surely Thus, with proability one, lim n R ω,0 n R ω,π/ n lim n ρ ω n ρ ω exists log ρ ω n ρ ω lim sup Λ n log n Lemma 6 completes the proof note that d n det Tω E n a ω n We are now ready to complete the Proof of Theorem By Theorem, Fubini s Theorem, the fact that the distribution of X ω n is absolutely continuous with respect to Lebesgue measure, the theory of rank-one perturbations [0], it follows that, with probability one, the spectral measure is supported on the set of energies where, for any ε>0 sufficiently large n, Tω E n n η+ε From Corollary 44 of [9] it now follows that the spectral measure is continuous with respect to ε-dimensional Hausdorff measure, for any ε > 0 Thus the spectral measure is one-dimensional Since ψn δ, J z δ n solves the eigenvalue equation for z away from n 0, Theorem A Wronskian conservation imply that if z R were to be outside of the spectrum, the transfer matrices would have to exhibit exponential growth Since this is not the case, it follows that the spectrum is R In this case again, the fact that the spectrum is R follows from the polynomial bound on the transfer matrices in Theorem Theorem A below As for the properties of the spectral measure, these follow from Theorem in [9] using 48, Proposition 7 the theory of rank one perturbations 3 The existence, with probability one, of an exponentially decaying eigenfunction, for every E R, follows from Theorem Theorem 83 of [6] Fubini the theory of rank one perturbations imply that the spectral measure is supported, with probability one, on the set where these eigenfunctions exist Comparing powers of n in the exponent, Theorem A implies that, as long as η > η, the spectrum fills R Appendix A A Combes-Thomas estimate for Jacobi matrices with unbounded parameters This section presents a Combes-Thomas estimate, suitable for application to Jacobi matrices with unbounded off-diagonal terms Also see [8] for a related result Theorem A Let J J {an} n, {bn} n be a self-adjoint Jacobi matrix such that 0 <an fn for a nondecreasing function fn Let z C be such that distz, SpecJ σ where SpecJ is the spectrum of J Then A δ, J z e δ N σ e α N N,

21 where α N min A RANDOM JACOBI MATRICES WITH GROWING PARAMETERS 38, σ 4efN Inparticular, δ, J Υ,ω z e η δ N e Cσ,ω N σ almost surely Remark The monotonicity of f is not essential One may instead replace fn in the formula for α N by maxf,,fn Proof Let R N be the diagonal matrix defined by { e α N A3 R N n, n n, n N, e α N N, n > N Then e α n N δ, J z δ N δ,r N J z R N δ N, so it suffices to bound R N J z R N Noting that R N J z R N R N J z R N CN z, we may apply the resolvent identity to get C N z J z + C N z J z R N J z R N J z A simple computation shows that J z R N J z R N efnα N if α N Thus, by J z σ,weseethat C N z σ + C N z efn α N, σ so, by α N σ efn,weseethat C N z σ, which finishes the proof Acknowledgments We are grateful to Peter Forrester Uzy Smilansky for presenting us with the problem that led to this paper We also thank Yoram Last Uzy Smilansky for many useful discussions We thank the referee for helpful remarks This research was supported in part by THE ISRAEL SCIENCE FOUNDA- TION grant no 69/06 by grant no from the United States-Israel Binational Science Foundation BSF, Jerusalem, Israel References [] J Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, TranslMathMonographs, Vol 7, Amer Math Soc, Providence, RI, 968 MR078 36:5768 [] J Breuer, P Forrester U Smilansky, Rom discrete Schrödinger operators from rom matrix theory, JPhysA:MathTheor40 007, F F8 MR c:8073 [3] F Delyon, Appearance of a purely singular continuous spectrum in a class of rom Schrödinger operators, JStatPhys40 985, MR j:8039 [4] F Delyon, B Simon B Souillard, From power-localized to extended states in a class of one-dimensional disordered systems, PhysRevLett5 984, [5] F Delyon, B Simon B Souillard, From power pure point to continuous spectrum in disordered systems, Ann Inst H Poincaré Phys Théor 4 985, MR d:35098 [6] I Dumitriu A Edelman, Matrix models for beta ensembles, JMathPhys43 00, MR g:8044

22 38 J BREUER [7] D J Gilbert D B Pearson, On subordinacy analysis of the spectrum of onedimensional Schrödinger operators, J Math Anal Appl 8 987, MR a:34033 [8] J Janas S Naboko, Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries, J Funct Anal 9 00, MR d:4704 [9] S Jitomirskaya Y Last, Power-law subordinacy singular spectra, I Half-line operators, ActaMath83 999, 7 89 MR a:47033 [0] R Killip I Nenciu, Matrix models for circular ensembles, Int Math Res Not, , MR h:8003 [] R Killip M Stoiciu, Eigenvalue statistics for CMV matrices: From Poisson to clock via CβE, preprint math-ph/ [] A Kiselev Y Last, Solutions, spectrum dynamics for Schrödinger operators on infinite domains, Duke Math J 0, 000, 5 50 MR i:354 [3] A Kiselev, Y Last, B Simon, Modified Prüfer EFGP transforms the spectral analysis of one-dimensional Schrödinger operators, Commun Math Phys , 45 MR g:3467 [4] S Kotani N Ushiroya, One-dimensional Schrödinger operators with rom decaying potentials, Commun Math Phys 5 988, MR f:60069 [5] Y Last, Quantum dynamics decompositions of singular continuous spectra, JFunct Anal 4 996, MR k:8044 [6] Y Last B Simon, Eigenfunctions, transfer matrices, absolutely continuous spectrum of one-dimensional Schrödinger operators, InventMath35 999, MR f:47060 [7] M Reed B Simon, Methods of Modern Mathematical Physics, I FunctionalAnalysis, Academic Press, New York, 97 MR :49a [8] J Sahbani, Spectral properties of Jacobi matrices of certain birth death processes, J Operator Theory , MR e:4708 [9] B Simon, Some Jacobi matrices with decaying potential dense point spectrum, Commun Math Phys 87 98, MR d:47033 [0] B Simon Spectral analysis of rank one perturbations applications In Proc Mathematical Quantum Theory, II Schrödinger Operators, CRM Proceedings Lecture Notes 8, edited by J Feldman, R Froese L Rosen, American Mathematical Society, Providence, RI 995 MR c:47008 [] B Simon, Orthogonal Polynomials on the Unit Circle, vol American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, Rhode Isl, 005 MR a:400b Department of Mathematics 53-37, California Institute of Technology, Pasadena, California 95 address: jbreuer@caltechedu Current address: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 9904, Israel