EXPONENTIAL DYNAMICAL LOCALIZATION FOR RANDOM WORD MODELS 1. Abstract We give a new proof of spectral localization for one-dimensional Schr¨odingeropera- tors whose potentials arise by randomly concatenating words from an underlying set. We then show that once one has the existence of a complete orthonormal basis of eigen- functions (with probability one), the same estimates used to prove it naturally lead to a proof of exponential dynamical localization in expectation (EDL) on any compact set which trivially intersects a finite set of critical energies. 2. Introduction 2 We consider the random word models on l (Z) given by H! (n) = (n + 1) + (n − 1) + V!(n) (n): The potential is a family of random variables defined on a probability space Ω. To construct the potential V above, we consider words :::;!−1;!0;!1;::: which are vectors n in R with 1 ≤ n ≤ m, so that V!(0) corresponds to the kth entry in !0. A precise construction of the probability space Ω and the random variables V!(n) is carried out in the next section. One well-known example of random word models is the random dimer model. In this situation, !i takes on values (λ, λ) or (−λ, −λ) with Bernoulli probability. First introduced in [6], the random dimer model is of interest to both physicists and math- ematicians because it exhibits a delocalization-localization phenomenon. It is known that the spectrum of the operator H! is almost surely pure point with exponentially decaying eigenfunctions. On the other hand, when 0 < λ ≤ 1 (with λ 6= p1 ), there are 2 critical energies at E = ±λ where the Lyapunov exponent vanishes [5]. The vanishing Lyapunov exponent at these energies can be exploited to prove lower bounds on quan- tum transport resulting in almost sure overdiffusive behavior [13]. The authors in [13] show that for almost every ! and for every α > 0 there is a positive constant Cα such that T Z 1 1 iH!t q −iH!t q− −α hδ0; e jXj e δ0idt ≥ CαT 2 : T 0 This was later extended to a sharp estimate in [16]. Again, the over-diffusive behavior above is to be contrasted with the fact that not only does the random dimer model display spectral localization, but also dynamical localization on any compact set I not containing the critical energies ±λ [5]: −iH!t q −iH!t suphPI (H!)e ; jXj PI (H!)e i < 1; (1) t Here PI (H!) is the spectral projection of H! onto the set I. 1 2 EXPONENTIAL DYNAMICAL LOCALIZATION FOR RANDOM WORD MODELS We strengthen this last result by showing that there is exponential dynamical local- ization in expectation (EDL) on any compact set I with ±λ2 = I. That is, there are C; α > 0 such that for any p; q 2 Z, −itH! −αjp−qj E sup jhδp;PI (H!)e δqij ≤ Ce : (2) t2R Another well-known example of random word models is the Anderson model. One of the central challenges in dealing with this model in full generality arises when there is a lack of the regularity of the single-site distribution. The absence of regularity was previously overcome using multi-scale analysis; first, in the Anderson setting [2], then in the dimer case [5, 10], and finally in random word models [4]. The multi-scale approach leads to weaker dynamical localization results than those where sufficient regularity of the single-site distribution allows one to instead appeal to the fractional moment method (e.g. [8], [1]). In particular, EDL always follows in the framework of the fractional moment method, but of course regularity is required. Loosely speaking, the multi-scale analysis shows that the complement of the event where one has exponential decay of the Green's function has small probability. One of the consequences of this method is that while this event does have small probability, it can only be made sub-exponentially small. A recent new proof of spectral and dynamical localization for the one-dimensional Anderson model for arbitrary single-site distributions [14] uses positivity and large de- viations of the Lyapunov exponent to replace parts of the multi-scale analysis. The major improvement in this regard (aside from a shortening of the length and complex- ity of localization proofs in one-dimension) is that the complement of the event where the Green's function decays exponentially can be shown to have exponentially (rather than sub-exponentially) small probability. These estimates were implicit in the proofs of spectral and dynamical localization given in [14] and were made explicit in [9]. The authors in [9] then used these estimates to prove EDL for the Anderson model and we extend those techniques to the random word case. There are, however, several issues one encounters when adapting the techniques de- veloped for the Anderson model in [9, 14] to the random word case. Firstly, in the Anderson setting, a uniform large deviation estimate is immediately available using a theorem in [17]. Since random word models exhibit local correlations, there are addi- tional steps that need to be taken in order to obtain suitable analogs of large deviation estimates used in [9, 14]. Secondly, random word models may have a finite set of ener- gies where the Lyapunov exponent vanishes and this phenomena demands some care in obtaining estimates on the Green's functions analogous to those in [9, 14]. As above, with PI (H!) denoting the spectral projection of H! onto the interval I, we have the following result: Theorem 1. The spectrum of H! is almost surely pure point with exponentially decaying eigenfunctions. Our main result is: EXPONENTIAL DYNAMICAL LOCALIZATION FOR RANDOM WORD MODELS 3 Theorem 2. There is a finite D ⊂ R such that if I is a compact set and D \ I = ; −itH! −αjp−qj then there are C > 0 and α > 0 such that E[sup jhδp;PI (H!)e δqij] ≤ Ce for t2R any p; q 2 Z. The remainder of the paper is organized as follows: section 3 contains preliminaries needed to discuss the large deviation estimates found in section 4 and the lemmas (which are established in section 6) needed for localization found in section 5. Finally, section 7 contains the proof of Theorem 1 and section 8 contains the proof of Theorem 2. 3. Preliminaries We begin by providing details on construction of Ω and V!(n) by following [4]. Sm j Fix m 2 N and M > 0. Set W = j=1 Wj where Wj = [−M; M] and νj are finite Pm Borel measures on Wj so that j=1 νj(Wj) = 1. Additionally, we assume that (W; ν) has two words which do not commute. That is, 8 For i = 0; 1 there exist w 2 W 2 supp(ν) such that < i ji (w0(1); w0(2); :::; w0(j0); w1(1); w1(2); :::; w1(j1)) (3) : and (w1(1); w1(2); :::; w1(j1); w0(1); w0(2); :::; w0(j0)) are distinct. Z Set Ω0 = W and P0 = ⊗Zν on the σ-algebra generated by the cylinder sets in Ω0. Pm The average length of a word is defined by hLi = j=1 jν(Wj) and if w 2 W \ Wj, we say w has length j and write jwj = j. Sm We define Ω = j=1 Ωj ⊂ Ω0 × f1; :::; mg where Ωj = f! 2 Ω0 : j!0j = jg × f1; :::; jg. We define the probability measure P on the σ-algebra generated by the sets A × fkg where A ⊂ Ω0 such that for all ! 2 A, j!0j = j and 1 ≤ k ≤ j. P0(A) For such sets we set P[A × fkg] = hLi . The shifts T0 and T on Ω0 and Ω (respectively) are given by: (T0!)n = !n+1 and (!; k + 1) if k < j! j T (!; k) = 0 (4) (T0(!); 1) if k = j!0j: With this set-up, the shift T is ergodic and the potential V!;k is obtained through :::; !−1;!0;!1; ::: so that V!;k(0) = !0(k). Thus, the results from [3] can be used to show the spectrum of H! is almost surely a non-random set. Definition. We call !;E a generalized eigenfunction with generalized eigenvalue E if H! !;E = E !;E and j !;E(n)j ≤ (1 + jnj). We denote H! restricted to the interval [a; b] by H!;[a;b] and for E2 = σ(H!;[a;b]) the corresponding Green's function by −1 G[a;b];E;! = (H!;[a;b] − E) : Additionally, we let P[a;b];!;E = det(H!;[a;b] − E): We also let Ej;!;[a;b] denote the jth eigenvalue of the operator H!;[a;b] and note that there are b − a + 1 many of them (counting multiplicity). 4 EXPONENTIAL DYNAMICAL LOCALIZATION FOR RANDOM WORD MODELS Definition. x 2 Z is called (c; n; E; !)-regular if there is a c > 0 so that: −cn (1) jG[x−n;x+n];E;!(x; x − n)j ≤ e and −cn (2) jG[x−n;x+n];E;!(x; x + n)j ≤ e . It is well-known that for any generalized eigenfunction !;E and any x 2 [a; b], (x) = −G[a;b];E;!(x; a) (a − 1) − G[a;b];E;!(x; b) (b + 1); (5) and using Cramer's rule on the b − a + 1-dimensional matrix H!;[a;b], one obtains: jP[a;x−1];E;!P[y+1;b];E;!j jG[a;b];E;!(x; y)j = : (6) jP[a;b];E;!j For w 2 W, with w = (w1; :::; wj), we define word transfer matrices by Tw;E = E − v −1 T :::T where T = .