Commun. Math. Phys. 370, 311–324 (2019) Communications in Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-019-03502-8 Mathematical Physics
Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model
Svetlana Jitomirskaya, Xiaowen Zhu Department of Mathematics, University of California, Irvine, USA. E-mail: [email protected]; [email protected]
Received: 28 March 2018 / Accepted: 17 December 2018 Published online: 6 July 2019 – © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: The proof of Anderson localization for the 1D Anderson model with arbi- trary (e.g. Bernoulli) disorder, originally given by Carmona–Klein–Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large devia- tion estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig–Simon’s bound that works in high generality and may be of independent interest.
1. Introduction
Anderson localization for the Anderson model can be proved in several different ways if the common distribution of the i.i.d.r.v’s is absolutely continuous. Without that condition (or at least some Hölder regularity) it remains an open question for d ≥ 2, and the number of approaches that work for d = 1 also drops dramatically. Such is the situation, for example, for the Bernoulli–Anderson model. Anderson localization for arbitrary 1D disorder (with only a moment condition on the distribution) was first proved in [6]. The approach was based on certain regularity of the Lyapunov exponents coming from the (analysis around) the Furstenberg theorem to obtain an analogue of Wegner’s lemma (automatic in the absolutely continuous case). After that the proof was reduced to multi- scale analysis, with initial scale coming again from the positive Lyapunov exponent. Another argument was later presented in [17], where an approach to positivity and regularity of the Lyapunov exponent using replica trick was given, again reducing the proof to multi-scale analysis. Multi-scale analysis is a method that allows to achieve Green’s function decay and ultimately localization from high probability of decay at the initial scale. It works in a variety of settings. Originally developed by Fröhlich and Spencer [11], it was significantly simplified in [20] but remains somewhat involved. 312 S. Jitomirskaya, X. Zhu
It should be noted that in the multidimensional case no shortcuts such as Furstenberg theorem or replica trick are available, and the multi-scale analysis is used to reach conclusions analogous to the positivity of the Lyapunov exponent simultaneously with the proof of localization. Yet in the one-dimensional case positivity of the Lyapunov exponent essentially provides the averaged decay statement, thus a large portion of the conclusion of the multi-scale analysis, making its machinery seem redundant. A method to effectively exploit positive Lyapunov exponent for a localization proof, based on the analysis of the large deviation set for the Lyapunov exponent, was first de- veloped in [15], for the almost Mathieu operator, with a precursor in [14]. This initiated the so-called non-perturbative approach, in contrast with earlier proofs based on some form of multi-scale analysis [12,18]. A robust method based on subharmonic function theory and the theory of semianalytic sets was then developed in [3] and other papers summarized in [2], to conclude localization from positive Lyapunov exponents for an- alytic quasiperiodic and some other deterministic potentials. The fact that those ideas can be applicable also to the Anderson model was mentioned in some talks by one of the authors circa 2000, but the details were never developed. One goal of this paper is to obtain a proof of Anderson localization for the 1D Anderson model in the spirit of [15] but with appropriate simplifications due to randomness. Another proof, also based on large deviations and also avoiding multi-scale analysis was recently developed in [5]. The proof of [5] is based on deterministic ideas close to the ones in [4], which we believe may be somewhat more complicated than needed for the random case. We mention that yet another, purely dynamical, proof of localization for the bounded 1D Anderson model appears in [13]. One ingredient in our simple argument for spectral localization, Theorem 3.3,is Craig–Simon’s upper bound based on subharmonicity of the Lyapunov exponent [7], a statement that holds for any ergodic potential. In order to prove dynamical localization we need a uniform in energy and quantitative version of this statement, that we prove for general ergodic potentials satisfying certain large deviation bounds, a result that could be of independent interest. We note that our proof does not explicitly use subharmonicity. The rest of this paper is organized as follows. Section 2 contains the preliminaries, the statement of the spectral localization result, Theorem 2.1, and its quick reduction to Theorem 2.2. We then prove the preparatory Lemmas 3.1, 3.2, 3.5, and Corollary 3.4 in Sect. 3. Then we complete the proof of Theorem 2.2 in Sect. 4. Our proof effectively establishes a more precise result, Theorem 4.1, which in turn immediately implies the Lyapunov behavior at all eigenvalues, Theorem 4.2. We formulate and prove the general uniform Craig–Simon-type statement in Sect. 5, and use it in Sect. 6 to prove dynamical localization.
2. Preliminaries
The one dimensional Anderson model is given by a discrete Schrödinger operators Hω
(Hω)(n) = (n +1) + (n − 1) + ωn(n), (2.1) where ωn ∈ R are independent identically distributed random variables with common Borel probability distribution μ. We will assume that S ⊂ R, the topological support of μ, is compact , and contains at least two points. We will denote the probability space Z Z [a,b]∩Z [a,b]∩Z = S , with elements {ωn}n∈Z ∈ . Denote μ as P.LetP[a,b] be μ on S . Also let T be the shift T ωi = ωi−1. Finally, we denote Lebesgue measure on R by m.We Large Deviations of the Lyapunov Exponent and Localization 313 say that Hω has spectral localization in I if for a.e. ω, Hω has only pure point spectrum in I and its eigenfunctions (n) decay exponentially in n. Definition 1. We call E a generalized eigenvalue (g.e.), if there exists a nonzero poly- nomially bounded function (n) such that Hω = E. We call (n) a generalized eigenfunction.
Since the set of g.e. supports the spectral measure of Hω (e.g. [8]), we only need to show:
Theorem 2.1. For a.e. ω, for every g.e. E, the corresponding generalized eigenfunction ω,E (n) decays exponentially in n.
For [a, b] an interval, a, b ∈ Z, define H[a,b],ω to be operator Hω resticted to [a, b] with zero boundary conditions outside [a, b]. Note that it can be expressed as a “b−a+1”- dimensional matrix. The Green’s function for Hω restricted to [a, b] with energy E ∈/ σ[a,b],ω is −1 G[a,b],E,ω = (H[a,b],ω − E) . Note that this can also be expressed as a “b − a + 1”-dimensional matrix. Denote its (x, y) entry as G[a,b],E,ω(x, y). It is well known that
(x) =−G[a,b],E,ω(x, a)(a − 1) − G[a,b],E,ω(x, b)(b +1), x ∈[a, b] (2.2) and we have
σ := σ(Hω) =[−2, 2] + Sa.e.ω. (2.3) Definition 2. For c > 0, n ∈ Z,wesayx ∈ Z is (c, n, E,ω)-regular, if −cn G[x−n,x+n],E,ω(x, x − n) ≤ e −cn G[x−n,x+n],E,ω(x, x + n) ≤ e . Otherwise, we call it (c, n, E,ω)-singular. By (2.2) and definition 2, Theorem 2.1 follows from
Theorem 2.2. There exists 0 with P(0) = 1, such that for every ω˜ ∈ 0, for any g.e. ˜ ˜ ˜ EofHω˜ , there exist N = N(E, ω),˜ C = C(E), such that for every n > N, 2n, 2n +1 are (C, n, E˜, ω)˜ -regular. Some other standard basic settings are below. Denote
P[a,b],E,ω = det(H[a,b],E,ω − E), a ≤ b.
If a > b,letP[a,b],E,ω = 1. Then P[ , − ], ,ω P[ , ], ,ω ( , ) = a x 1 E y+1 b E , ≤ . G[a,b],E,ω x y x y (2.4) P[a,b],E,ω
If we denote the transfer matrix T[a,b],E,ω as the matrix such that (b) (a) = T[ , ], ,ω (b − 1) a b E (a − 1) 314 S. Jitomirskaya, X. Zhu where solves Hω = E, then P[a,b],E,ω −P[a+1,b],E,ω T[a,b],E,ω = . P[a,b−1],E,ω −P[a+1,b−1],E,ω The Lyapunov exponent exists by Kingman’s subadditive ergodic theorem and is given by 1 1 1 γ(E) = lim log T[ , ], ,ω dP(ω) = lim log T[ , ], ,ω , a.e.ω. (2.5) →∞ 0 n E →∞ 0 n E n n 0 n n
Let ν = inf E∈σ γ(E). By the Furstenberg’s theorem ν>0. It follows from (2.4) that the desired exponential decay of the Green’s function can be achieved if all the P[a,b] (b−a)γ (E) in (2.4) behave as e , thus leading to the study of deviations of ln P[a,b] from its mean. In fact, the key estimates underlying the analysis of [6] are precisely large deviation bounds for the Lyapunov exponent due to Le Page [16]. Here we will use a corresponding statement for the matrix elements [19]. Lemma 2.3 (“uniform-LDT”). For any >0, there exists η = η() > 0 such that, there exists N0 = N0(), such that for every b − a > N0, and any E in a compact set, 1 −η(b−a+1) P ω : log P[a,b],E,ω −γ(E) ≥ ≤ e . (2.6) b − a +1 It will also be convenient to use the general subharmonicity upper bound due to Craig–Simon [7] Theorem 2.4 (Craig–Simon [7]). For a.e. ω for all E, we have
log T[ , ], ,ω lim 0 n E ≤ γ(E). (2.7) n→∞ n +1
3. Main Lemmas Denote + = ( ,ω):| |≥ (γ (E)+)(b−a+1) B[a,b], E P[a,b],E,ω e (3.1) − = ( ,ω):| |≤ (γ (E)−)(b−a+1) B[a,b], E P[a,b],E,ω e (3.2) ± ={ω : ( ,ω) ∈ ± } ± ={ : ( ,ω) ∈ ± } and denote B[a,b],,E E B[a,b], , B[a,b],,ω E E B[a,b], , = + ∪ − B[a,b],∗ B[a,b],∗ B[a,b],∗. ω| = (ω ,...,ω ) Let E j,(ωa ,...,ωb) be eigenvalues of H[a,b],ω with [a,b] a b . Large deviation theorem gives us the estimate that for all E, a, b, P( ± ) ≤ −η(b−a+1). B[a,b],,E e (3.3) = < 1 ν Assume 0 8 is fixed for now, so we omit it from the notations until Lemma 3.5.Letη0 = η(0) be the corresponding parameter from Lemma 2.3
Lemma 3.1. For n ≥ 2,ifxis(γ (E) − 80, n, E,ω)-singular, then ( ,ω)∈ − ∪ + ∪ + . E B[x−n,x+n] B[x−n,x−1] B[x+1,x+n] Large Deviations of the Lyapunov Exponent and Localization 315
Remark 1. Note that from (3.3), for all E, x, n ≥ 2, − −η ( ) P( ∪ + ∪ + ) ≤ 0 n+1 . B[x−n,x+n],E B[x−n,x−1],E B[x+1,x+n],E 3e Proof. Follows imediately from the definition of singularity and (2.4).
Now we will use the following three lemmas to find the proper 0 for Theorem 2.2.
Lemma 3.2. Let 0 <δ0 <η0. For a.e. ω (we denote this set as 1), there exists N1 = N1(ω), such that for every n > N1, − − −(η −δ )( ) { ( ), ( )}≤ 0 0 2n+1 . max m B[n+1,3n+1],ω m B[−n,n],ω e Proof. By (3.3),
− −η ( ) × P( ) ≤ (σ) 0 2n+1 m B[n+1,3n+1] m e − −η ( ) × P( ) ≤ (σ) 0 2n+1 . m B[−n,n] m e If we denote − −(η0−δ0)(2n+1) δ ,n,+ = ω : m(B[ , ],ω) ≤ e 0 n+1 3n+1 − −(η −δ )( ) = ω : ( ) ≤ 0 0 2n+1 . δ0,n,− m B[−n,n],ω e
We have by Tchebyshev,
−δ ( ) P(c ) ≤ m(σ)e 0 2n+1 . (3.4) δ0,n,± By Borel–Cantelli lemma, we get for a.e.ω,
− − −(η −δ )( ) { ( ), ( )}≤ 0 0 2n+1 , max m B[n+1,3n+1],ω m B[−n,n],ω e for n > N1(ω). Remark 2. Note that we can actually shift the operator and use center point l instead of 0. Then we will get 1(l) instead of 1, N1(l,ω)instead of N1(ω). And if we pick N1(l,ω)in the theorem as the smallest integer satisfying the conclusion, we can estimate 2 when we will have N1(l,ω) ≤ ln |l|, which is very useful in the proof for dynamical localization in Sect. 6. The next results follows from:
Theorem 3.3. For a.e. ω(we denote this set as 2), for all E, we have log T[− , ], ,ω log T[ , ], ,ω max lim n 0 E , lim 0 n E ≤ γ(E) (3.5) n→∞ n→∞ n +1 n +1 log T[ , ], ,ω log T[ , ], ,ω max lim n+1 2n+1 E , lim 2n+1 3n+1 E ≤ γ(E). (3.6) n→∞ n +1 n→∞ n +1 Remark 3. (3.5) is a direct reformulation of the result of [7], Theorem 2.4, while (3.6) follows by exactly the same proof. 316 S. Jitomirskaya, X. Zhu
Corollary 3.4. For every ω ∈ 2, for every E, there exists N2 = N2(ω, E), such that for every n > N2,
(γ (E)+)(n+1) max{ T[−n,0],E,ω , T[0,n],E,ω } < e (γ (E)+)(n+1) max{ T[n+1,2n+1],E,ω , T[2n+1,3n+1],E,ω } < e .
Lemma 3.5. Let >0, K > 1, For a.e. ω(we denote this set as 3 = 3(, K )), = (ω) > there exists N3 N3 , so that for every n N3, for every E j,(ωn+1,...,ω3n+1), for every n n y1, y2 satisfying −n ≤ y1 ≤ y2 ≤ n, |−n − y1|≥ , and |n − y2|≥ , we have ∈/ ∪ ∪ K K E j,(ωn+1,...,ω3n+1) B[−n,y1],,ω B[y2,n],,ω B[−n,n],,ω. Remark 4. Note that and K are not fixed yet, we’re going to determine them later in Sect. 4. ¯ Proof. Let P be the probability that there are some y1, y2, j with
∈ ∪ ∪ . E j,(ωn+1,...,ω3n+1) B[−n,y1],,ω B[y2,n],,ω B[−n,n],,ω
Note that for any fixed ωc,...,ωd , with [c, d]∩[a, b]=∅, by independence,
−η0(b−a+1) P[ , ]c (B[ , ],, ,(ω ,...,ω ) ) = P[ , ](B[ , ],, ,(ω ,...,ω ) ) ≤ e . c d a b E j c d a b a b E j c d
Applying to [a, b]=[−n, y1] or [y2, n], [c, d]=[n +1, 3n +1] and integrating over ω ,...,ω ω ,...,ω −n y1 or y2 n, we get
−η ( n ) 0 K +1 P(B[− , ],, ,(ω ,...,ω ) ∪ B[ , ],, ,(ω ,...,ω ) ) ≤ 2e , n y1 E j n+1 3n+1 y2 n E j n+1 3n+1 and
−η0(2n+1) P(B[−n,n],,ω) ≤ e so
¯ 3 −η ( n +2) P ≤ (2n +1) 2e 0 K . (3.7)
Thus by Borel–Cantelli, we get the result.
Remark 5. Similar to Remark 2, we can get 3(l), N3(l,ω)for an operator shifted by . .ω (ω) instead, and get the result that for a e (we denote this set as N3 ), there exists L3 , 2 such that for any |l| > L3, N3(l,ω)≤ ln |l|. This will be of use in Sect. 6 for proving dynamical localization.
4. Proof of Theorem 2.2
We will only provide a proof that 2n +1is(c, n, E,ω)-regular, the argument for 2n being similar. Large Deviations of the Lyapunov Exponent and Localization 317
Proof. Let be small enough such that
(b−a+1) |P[a,b],E,ω| < M , ∀E ∈ σ, ω. Pick K big enough such that
1 M K < L. Let τ>0 be such that
1 M K ≤ L − τ
˜ −(η0−δ0)(2n+1) |E − E j,[n+1,3n+1],ω˜ |≤e .
By the same argument, there exists Ei,[−n,n],ω˜ , such that
˜ −(η0−δ0)(2n+1) |E − Ei,[−n,n],ω˜ |≤e .
−(η0−δ0)(2n+1) Thus |Ei,[−n,n],ω˜ − E j,[n+1,3n+1],ω˜ |≤2e . However, by Theorem 3.5, one has E j,[n+1,3n+1],ω˜ ∈/ B[−n,n],,ω˜ , while Ei,[−n,n],ω˜ ∈ B[−n,n],,ω˜ This will give us a contradiction below. −(η0−δ0)(2n+1) Since |Ei,[−n,n],ω˜ − E j,[n+1,3n+1],ω˜ |≤2e and Ei,[−n,n],ω˜ is the e.v. of H[−n,n],ω˜ ,
1 (η −δ )(2n+1) G[− , ], ,ω˜ ≥ e 0 0 . n n E j,[n+1,3n+1],ω˜ 2
Thus there exist y1, y2 ∈[−n, n] and such that 1 (η −δ )(2n+1) G[− , ], ,ω˜ (y , y ) ≥ e 0 0 . n n E j,[n+1,3n+1],ω˜ 1 2 2n 318 S. Jitomirskaya, X. Zhu
Let E j = E j,[n+1,3n+1],ω˜ .WehaveE j ∈/ B[−n,n],,ω˜ , thus
(γ ( )−)( ) | |≥ E j 2n+1 P[−n,n],,E j ,ω˜ e so by (2.4),
1 (η −δ )(2n+1) (γ (E )−)(2n+1) P[− , ],, ω˜ P[ , ],, ,ω˜ ≥ e 0 0 e j . (4.3) n y1 E j y2 n E j 2n Then for the left hand side of (4.3), there are three cases: n n (1) both |−n − y1| > and |n − y2| > K |− − | >Kn | − |≤ n (2) one of them is large, say n y1 K while n y2 K (3) both small. For (1),
1 (η −δ γ( )−)( ) (γ ( ) ) e 0 0+ E j 2n+1 ≤ e2n E j + . 2n
Since by our choice (4.1), η0 − δ0 + γ(E j ) − >γ(E j ) + ,forn large enough, we get a contradiction. For (2),
1 (η −δ +γ(E )−)(2n+1) (γ (E )+)(2n+1) n e 0 0 j ≤ e j (M) K 2n is in contradiction with (4.1) and (4.2). For (3), with (4.1) and (4.2)
1 (η −δ +γ(E )−)(2n+1) 2n 2n (η −δ +γ(E )−) 2n e 0 0 j ≤ M K ≤ (L − τ) ≤ (e 0 0 j − τ) , 2n also a contradiction. ˜ ˜ Thus our assumption that 2n + 1 is not (γ (E) − 80, n, E, ω)˜ -regular is false. Theo- rem 2.2 follows. Note that we have established the following more precise version of Theorem 2.2.
Theorem 4.1. There exists 0 with P(0) = 1, such that for every ω˜ ∈ 0, for any g.e. ˜ ˜ EofHω˜ , and >0, there exists N = N(E, ω,˜ ), such that for every n > N,2n, 2n +1 are (γ (E) − , n, E˜, ω)˜ -regular.
It is a standard patching argument (e.g. proof of Theorem 3 in [15]) that this im- −(γ (E)−)n plies |E (n)|≤CE,e for any >0. Combined with Theorem 2.4,this immediately implies that we have Lyapunov behavior at every generalized eigenvalue.
Theorem 4.2. For a.e. ω for all generalized eigenvalues E, we have
log T[ , ], ,ω lim 0 n E = γ(E). (4.4) n→∞ n +1 Large Deviations of the Lyapunov Exponent and Localization 319
5. Uniform and Quantitative Craig–Simon
Craig–Simon Theorem 2.4 implies that for a.e. ω and every E ∈ σ there exists N(ω, E) (n+1)(γ (E)+) such that for n > N, T[0,n],E,ω ≤e . For the proof of dynamical localiza- tion one however needs a statement of this type with N uniform in E. Such a statement is the goal of this section. We will show that it holds for any ergodic dynamical sys- tem satisfying the uniform LDT (Large Deviation Type) condition: Lemma 2.3. Thus this result has more general nature than the rest of the paper and may be of indepen- dent interest. In particular, it is applicable to quasiperiodic dynamics with Diophantine frequencies and analytic sampling functions. We note that uniform LDT condition can also be replaced by a combination of a pointwise LDT condition and continuity of the Lyapunov exponent. We have:
Theorem 5.1. Let the ergodic family Hω satisfy Lemma 2.3.Fix0 > 0. For a.e. ω (we denote this set as 2 = 2(0)), there exists N2(ω), such that for any n > N2(ω), E ∈ σ ,
(γ (E)+0)(n+1) |P[0,n],E,ω|≤e .
An immediate corollary is
Corollary 5.2. Let Hω,0 be as above. Then there exists 2 with P(2) = 1, such that for ω ∈ 2, there exists N2(ω) such that
(γ (E)+30)(n+1) max |P[0,n],E,ω|, |P[−n,0],E,ω|, |P[n+1,2n+1],E,ω|, |P[2n+1,3n+1],E,ω| ≤ e .
Thus we can replace Corollary 3.4 with this uniform version.
Proof. We start with the following
( ) − = 2π(i+θ) <θ< Lemma 5.3. Let Q x be a polynomial of degree n 1. Let xi cos n , 0 n n 1/2,i = 1, 2,...,n. If Q(xi ) ≤ a , for all i, then Q(x) ≤ Cna , for all x ∈[−1, 1], where C = C(θ) is a constant.
Proof. By Lagrange interpolation, we have
n x − x j Q(x) = Q(xi ) . (5.1) xi − x j i=1 j=i
Note that π(i + j +2θ) π(i − j) ln |xi − x j |= ln sin +lnsin +ln2 n n j=i j=i =: A + B + (n − 1) ln 2.
We will use the following lemma without giving a proof. 320 S. Jitomirskaya, X. Zhu
Lemma 5.4 (Lemma9.6in[1]). Let p and q be relatively prime. Let 1 ≤ k0 ≤ qbe such that
| sin 2π(x + k0 p/(2q))|= min | sin 2π(x + kp/(2q))|. 1≤k≤q
Then q ln q +ln(2/π) < ln | sin 2π(x + kp/(2q))| + (q − 1) ln 2 ≤ ln q. (5.2) k=1 k=k0
For B,wetakep = 1, q = n, x =−i/(2n), k = j. Then k0 = i, and we get B ≥ ln n +ln(2/π) − (n − 1) ln 2.
For A,weestimatebyLemma5.4 with p = 1, q = n, x = (i +2θ)/2n, k = j.If = | π(i+ j+2θ)| k0 j0 is the minimum term of ln sin n , then π( θ) π( θ) 2i +2 i + j0 +2 A ≥ ln n +ln(2/π) − (n − 1) ln 2 − ln sin +lnsin . n n For 0 <θ<1/4, we have
π( θ) π( θ) | sin 2i+2 | | sin 2i+2 | n = n ≤ 1 = ( ). π( θ) π· θ π· θ O n | i+ j0+2 | | sin 2 | | sin 2 | sin n n n Thus
ln |xi − x j |≥−(n − 1) ln 2 + ln n + C. j=i
= 2πa Writing x cos n , by Lemma 5.4, we get
ln |x − x j |≤−(n − 1) ln 2 + 2 ln n + C. j=i
Thus x − x j ≤ Cn x − x j=i i j and we have
Q(x) ≤ Cnan.
Now we can finish the proof of Theorem 5.1. We know that σ is compact, so contained in some bounded closed interval. Assume we are dealing with [a, a + A]. Unifrom LDT implies that γ is a continuous function of E [10]. Since γ(E) is uniformly continuous, for any 0, there exists δ0 such that Large Deviations of the Lyapunov Exponent and Localization 321
|γ(Ex ) − γ(Ey)|≤0, if |Ex − Ey|≤δ0. (5.3)
Divide the interval [a, a + A] into length-δ0 sub-intervals. There are K =[A/δ0] + 1 of them (the last one may be shorter). Denote them as Ik,fork = 1,...,K .For Ik =[Ek,n, Ek+1,n],letEk1,n,...,Ekn,n be distributed as in Lemma 5.3. Namely, set Eki,n = Ek,n + (xi +1)δ0/2, where xi areasinLemma5.3,0<θ<1/2. Note that for any Ex , Ey ∈[Ek1,n, Ekn,n], |γ(Ex ) − γ(Ey)|≤0. Since by the uniform-LDT condition (γ ( ) )( ) −η ( ) P ω :∃ = ,..., , . . | |≥ Eki,n + 0 n+1 ≤ 0 n+1 , i 1 n s t P[0,n],Eki,n,ω e ne by Borel–Cantelli, for a.e. ω, (we denote this set as (k)), there exists N(k,ω), such that for all n > N(k,ω), (γ ( ) )( ) | |≤ Eki,n + 0 n+1 , ∀ = ,..., . P[0,n],Eki,n,ω e i 1 n γ = γ( ) If we denote k,n inf E∈[Ek1,n,Ekn,n] E , then by (5.3) (γ ( ) )( ) (γ )( ) | |≤ Eki,n + 0 n+1 ≤ k,n+2 0 n+1 , ∀ = ,..., . P[0,n],Eki,n,ω e e i 1 n γ( ) ( ) Let M be big enough such that, for any n > M, C(n+2)e E +2 0 ≤ e 0 n+1 . Thus by ( ) = | ( )δ Lemma 5.3, applied to Q x P[0,n],E,ω = x+1 0 , a polynomial of degree n +1, E Ek,n+ 2 we have that, if n > max{N(k,ω),M},forE ∈[Ek,n, Ek+1,n]
(γk,n+20)(n+2) (γ (E)+20)(n+2) (γ (E)+30)(n+1) |P[0,n],E,ω|≤C(n +2)e ≤ C(n +2)e ≤ e ˜ ˜ Let 2 = (k), N(ω) = maxk{N(k,ω),M}. Then for any n > N(ω), k
(γ (E)+30)(n+1) |P[0,n],E,ω|≤e , ∀E ∈[a, a + A]. This allows us to also obtain a quantitative version of Theorem 5.1. Assume the N2(ω) in Theorem 5.1 is chosen to be the smallest satisfying the condition. Let l ∈ Z, ( ,ω)= ( l ω) ¯ = l N2 l N2 T .Let 2 l∈Z T 2. ˜ Lemma 5.5. For a.e. ω (we denote this set as 2), there exists L2 = L2(ω), such that 2 2 for all |l| > L2,N2(l,ω)≤ ln |l|. In particular, if n > ln |l|, then
(γ (E)+0)(n+1) |P[l,l+n],E,ω|≤e , forallE ∈ σ. ¯ ¯ Proof. Let ω ∈ 2, l ∈ Z, k ∈ N. By Theorem 5.1, has full measure. We have ∞ ∞ P{ω : ( ,ω)≥ }≤ P{ω : ( ,ω)= }≤ P( + ) N2 l k N2 l n B[l,l+n−1],E n=k n=k ∞ −(γ ( ) ) −(γ ( ) ) ≤ Ce E + 0 n ≤ Ce E + 0 k. n=k Thus 2 2 −(γ (E)+0)(ln |l|) P{ω : N2(l,ω)≥ ln |l|} ≤ Ce . ˜ By Borel–Cantelli lemma, we get the result and the corresponding 2. 322 S. Jitomirskaya, X. Zhu
6. Dynamical Localization
Now we have established the spectral localization for 1-d Anderson model. With some more effort, we can get the dynamical localization. We say that Hω exhibits dynamical localization if for a.e.ω, for any >0, there exists α = α(ω) > 0, C = C(, ω), such that for all x, y ∈ Z:
−itHω |y| −α|x−y| sup |δx , e δy| ≤ Ce e . t
According to [9], we only need to prove that for a.e. ω, Hω has SULE (Semi-Uniformly Localized Eigenfunction). We say H has SULE if H has a complete set {ϕE } of or- thonormal eigenfunctions, such that there is α>0, and for each >0, a C such that for any eigenvalue E, there exists l = lE ∈ Z, such that
|lE | −α|x−lE | |ϕE (x)|≤Ce e , x ∈ Z.
C ln2(1+|l |) −α|x−l | In fact, we will prove that |ϕE (x)|≤Ce E e E ,see(6.3), (6.5). In order to do this, we need to modify Lemmas 3.2 and 3.5 using the same method as in Lemma 5.5. Assume the Ni (ω), i = 1, 3 in Lemmas 3.2 and 3.5 are chosen to be the smallest param- ∈ Z ( ,ω) = ( l ω) ¯ = l eters satisfying the condition. Let l , Ni l Ni T .Let i l∈Z T i , i = 1, 3. ˜ Lemma 6.1. For a.e. ω (we denote this set as 1,3), there are L1(ω), L3(ω) such that for any |l| > max{L1, L3},
2 max{N1(l,ω),N3(l,ω)}≤ln |l|.
¯ Proof. Let ω ∈ 1, l ∈ Z, k ∈ N, then by (3.4)
∞ ∞ −δ0(2n+1) −δ0(2k+1) P{ω : N1(l,ω)>k}≤ P(δ,n,±) ≤ 2m(σ)e ≤ Ce . n=k n=k
Thus
2 2 −δ0(2ln |l|+2) P{ω : N1(l,ω)>ln |l|} ≤ Ce .
By Borel–Cantelli lemma, we can get the result. The same argument works for N3. Then we rebuild the criteria for regularity around a singular point l.
Lemma 6.2. For a.e. ω (we denote this set as ˜ ), for any l, there exists N(l,ω),such that for any n > N(l,ω)and for all E ∈ σ either l or l +2n +1, and either l or l −2n −1 are (γ (E) − 80, n, E,ω)-regular.
Proof. In Sect. 4, we proved that either 0 or 2n +1is(γ (E) − 80, n, E,ω)-singular for all n > N(ω), with N(ω) = max{N1(ω), N2(ω), N3(ω)}. Here we set N(l,ω) = max{N(T l ω), N(T −l ω)}, and modify ˜ accordingly. Large Deviations of the Lyapunov Exponent and Localization 323
˜ ˜ ˜ ˜ ˜ Now, take 4 = 2 ∩ 1,3 ∩ and fix ω ∈ 4.Weomitω from notations from now on. By Lemmas 5.5 and 6.1, there exist L1, L2, L3 such that for all |l| > max{L1, L2, L3},
2 Ni (l) ≤ ln |l|, ∀i = 1, 2, 3 for all E ∈ σ . Let l be a position of the maximum point of ϕ .TakeL with ln2 L ≥[ ln 2 ]+ E E 4 4 γ(E)−80 2 1. For any n ≥ ln L4, and any e.v. E, lE is naturally (μ − 80, n, E)-singular by (2.2). Let L = max{L , L , L , L }, N(l) := max{N (l), N (l), N (l), ln 2 }. Then 1 2 3 4 1 2 3 γ(E)−80 for any |l| > L,
N(l) ≤ ln2 |l|. (6.1)
If |lE | > L, then for any n ≥ N(lE ), lE is (γ (E) − 80, n, E)-singular, so x = lE ± (2n +1) is (γ (E) − 80, n, E)-regular. By (2.2), for any |x − lE |≥N(lE )
−(γ (E)−80)|x−lE | |ϕE (x)|≤2e . (6.2)
Since ϕE is normalized, in fact for all x,
(γ (E)−80)N(lE ) −(γ (E)−80)|x−lE | |ϕE (x)|≤2e e .
By (6.1), for any ,
2 (γ (E)−80) ln (1+|lE |) −(γ (E)−80)|x−lE | |ϕE (x)|≤2e e . (6.3)
If |lE |≤L, for any ,forn ≥ N(lE ), we use the same argument as (6.2) and get
2 −(γ (E)−80)|x−lE | ln (1+|lE |) −(γ (E)−80)|x−lE | |ϕE (x)|≤2e ≤ 2e e . (6.4)
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