Exponential Dynamical Localization for Random Word Models

Abstract

We show that one-dimensional Schrödinger operators whose potentials arise by randomly concatenating words from an underlying set exhibit exponential dynamical localization (EDL) on any compact set which trivially intersects a finite set of critical energies. We do so by first giving a new proof of spectral localization for such operators and then showing that once one has the existence of a complete orthonormal basis of eigenfunctions (with probability one), the same estimates used to prove it naturally lead to a proof of the aforementioned EDL result. The EDL statements provide new localization results for several classes of random Schrödinger operators including random polymer models and generalized Anderson models.

1 Introduction

In this paper, we consider random word models on $\ell ^2(\mathbb {Z})$ given by

$$\begin{aligned} H_{\omega }\psi (n) = \psi (n+1) + \psi (n-1) +V_{\omega }(n)\psi (n). \end{aligned}$$

The potential is a family of random variables defined on a probability space $\Omega$. To construct the potential V above, we fix an $m\in \mathbb {N}$ (a maximum word length) and then consider words $\ldots , \omega _{-1}, \omega _0, \omega _1, \ldots$ which are vectors in $\mathbb {R}^{n}$ with $1 \le n \le m$, so that $V_{\omega }(0)=\omega _{0}(t)$ (where $\omega _{0}(t)$ is entry t in $\omega _{0}$). A precise construction of the probability space $\Omega$ and the random variables $V_{\omega }(n)$ is carried out in Sect. 2.1, and the precise definition of $H_{\omega }$ is given in Eq. (3).

These models are of particular interest not only because they cover a wide class of generalizations of the Anderson model such as the random dimer model, random polymer models, and generalized Anderson models [4], but also because they provide natural examples of the subtleties involved in the various forms of localization: spectral, dynamical, and exponential dynamical (in expectation).

Spectral localization occurs when for almost every $\omega$, $H_{\omega }$ has an orthonormal basis of eigenfunctions $\{\psi _{\omega ,n}\}$ and for each such eigenfunction $\psi _{\omega ,n}$, there are positive constants C, c such that $|\psi _{\omega ,n}(m)| \le Ce^{-c|m|}$ for all $m \in \mathbb {Z}$. In other words, the spectrum is almost surely pure point and all of $H_{\omega }$’s eigenfunctions decay exponentially. A stronger form of localization, dynamical localization, is more closely related to physical manifestations of localization. In particular, dynamical localization implies an absence of transport in the random medium and that the wave packet is suitably confined. We say that $H_{\omega }$ exhibits dynamical localization on the interval I if for a.e. $\omega$, and any $\psi \in \ell ^{2}(\mathbb {Z})$ for which there are $C,c>0$ such that $|\psi (m)|\le Ce^{-c|m|}$ for all $m \in \mathbb {Z}$,

$$\begin{aligned} \sup _{t}\langle P_I(H_\omega )e^{-iH_\omega t}\psi ,|X|^qP_I(H_\omega )e^{-iH_\omega t}\psi \rangle < \infty , \end{aligned}$$

where $P_I(H_\omega )$ is the spectral projection of $H_{\omega }$ onto the set I and $q > 0$.

While it was well known that dynamical localization implies spectral localization, that dynamical localization was a strictly stronger notion was not understood until the authors in [6] constructed an artificial model which was spectrally but not dynamically localized. Indeed, the example in [6] showed that pure point spectrum with exponentially decaying eigenfunctions (spectral localization) could coexist with $\limsup _{t \rightarrow \infty } \frac{||xe^{-itH\delta _{0}}||^{2}}{t^{\alpha }} = \infty$ for all $\alpha < 2$. In short, spectral localization is not a sufficient condition to ensure an absence of transport, while the stronger notion of dynamical localization does imply this absence.

A well-known physically relevant model which sheds more light on these phenomena is the random dimer model. This model was first introduced in [8], and in the random word context introduced above, $\omega _{i}$ takes values $(\lambda ,\lambda )$ or $(-\lambda ,-\lambda )$ with Bernoulli probability. It is known that the spectrum of the operator $H_{\omega }$ is almost surely pure point with exponentially decaying eigenfunctions. On the other hand, when $0 < \lambda \le 1$ (with $\lambda \ne \frac{1}{\sqrt{2}}$), there are critical energies at $E= \pm \lambda$ where the Lyapunov exponent vanishes [5]. These so-called critical energies are precisely what prevent the absence of transport and lead to localization–delocalization phenomena.

In particular, the vanishing Lyapunov exponent at these energies can be exploited to prove lower bounds on quantum transport resulting in almost sure over-diffusive behavior [17]. The authors in [17] show that for almost every $\omega$ and for every $\alpha > 0$, there is a positive constant $C_{\alpha }$ such that

$$\begin{aligned} \frac{1}{T} \int _0^T \langle \delta _0,e^{iH_{\omega }t}|X|^qe^{-iH_{\omega }t}\delta _0 \rangle dt \ge C_\alpha T^{q-\frac{1}{2}-\alpha }. \end{aligned}$$

This was later extended to a sharp estimate in [19].

In light of these remarks, the over-diffusive behavior above contrasts 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 $\pm \lambda$ [5].

We strengthen this last result by showing that there is exponential dynamical localization in expectation (EDL) on any compact set I with $\pm \lambda \notin I$. We say the family of operators $H_{\omega }$ displays EDL on the interval I if there are $C, \alpha > 0$ such that for any $p,q \in \mathbb {Z}$,

$$\begin{aligned} \mathbb {E} \left[ \sup _{t \in \mathbb {R}} |\langle \delta _p, P_{I}(H_{\omega })e^{-itH_{\omega }}\delta _{q} \rangle |\right] \le Ce^{-\alpha |p-q|}. \end{aligned}$$

EDL has several interesting physical consequences including exponential decay of the two point function in the ground state [1], and there is an interest in proving such results in physically relevant contexts such as the dimer and random polymer cases. Our results, however, when taken in conjunction with the over-diffusive behavior above illustrate that the strength of localization does not necessarily impact transport when the localization regime excludes critical energies.

One of the central challenges in dealing with random word models is the lack of regularity of the single-site distribution. The absence of regularity is exactly what allows random word models to encompass singular Anderson models, random dimer models, and more generally, random polymer models. The issues presented by singularity were previously overcome using multi-scale analysis in various stages: first in the Anderson setting [3], then in the dimer case [5, 14], and finally for random word models themselves in [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., [2, 10]). In particular, EDL always follows in the framework of the fractional moment method and the methodology in [7, 22], 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 [20] uses positivity and large deviations 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 complexity 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 [20] and were made explicit in [13]. The authors in [13] 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 developed for the Anderson model in [13, 20] to the random word case. Firstly, in the Anderson setting, a uniform large deviation estimate is immediately available through the results in [24]. Since random word models exhibit local correlations, there are additional steps that need to be taken in order to obtain suitable analogs of large deviation estimates used in [13, 20]. Secondly, random word models may have a finite set of energies where the Lyapunov exponent vanishes and this phenomenon demands care in obtaining estimates on the Green’s functions analogous to those in [13, 20]. Dealing with these issues does, however, produce an unexpected benefit. Since we must consider Green’s functions for non-symmetric intervals, we are able to obtain exponential decay of the Green’s function centered around even and odd points simultaneously, while the arguments in [13, 20] require separate considerations.

Theorem 1

With $H_\omega$ defined in Eq. (3) (and satisfying Eq. (1)), for a.e. $\omega$, $H_{\omega }$ has an orthonormal basis of eigenfunctions $\{\psi _{\omega ,n}\}$ and for each such eigenfunction $\psi _{\omega ,n}$, there are positive constants C, c such that $|\psi _{\omega ,n}(m)| \le Ce^{-c|m|}$ for all $m \in \mathbb {Z}$.

Our main result is:

Theorem 2

For $H_\omega$ defined in Eq. (3) (and satisfying Eq. (1)), there is a finite $D \subset \mathbb {R}$ such that if I is a compact set and $D \cap I = \emptyset$, then $H_{\omega }$ exhibits exponential dynamical localization in expectation on I.

The remainder of the paper is organized as follows: Sect. 2 contains preliminaries needed to discuss the large deviation estimates found in Section . 3. Section 4 contains the precursors needed for localization, Sect. 5 contains the proof of Theorem 1, and Sect. 6 contains the proof of Theorem 2.

2 Preliminaries

2.1 Model Set-up

We begin by providing details on the construction of $\Omega$ and $V_{\omega }(n)$ by following [4].

Fix $m \in \mathbb {N}$ (the maximum word length) and $M > 0$. Set $\mathcal {W} = \bigcup _{j=1}^m \mathcal {W}_{j}$ where $\mathcal W_{j} = [-M,M]^{j}$ and $\nu _{j}$ are finite Borel measures on $\mathcal {W}_{j}$ so that $\sum _{j=1}^{m} \nu _{j} (\mathcal {W}_{j}) = 1$. Let $\tilde{\nu }$ denote the direct sum of the measures $\nu _j$, a probability measure on $\mathcal {W}$.

Additionally, we assume that $(\mathcal {W},\tilde{\nu })$ has two words which do not commute. That is,

$$\begin{aligned} \left\{ \begin{array}{l} \text {For } i = 0,1 \text { there exist }w_{i} \in \mathcal {W}_{j_{i}} \in supp(\tilde{\nu })\text { such that}\\ (w_{0}(1),w_{0}(2),\ldots ,w_{0}(j_{0}),w_{1}(1),w_{1}(2),\ldots ,w_{1}(j_{1}))\\ \text {and }(w_{1}(1),w_{1}(2),\ldots ,w_{1}(j_{1}),w_{0}(1),w_{0}(2),\ldots , w_{0}(j_{0}))\text { are distinct.}\\ \end{array} \right. \end{aligned}$$

(1)

Here, $supp(\tilde{\nu })$ refers to the support of the measure $\tilde{\nu }$.

Set $\Omega _{0} = \mathcal {W}^{\mathbb {Z}}$ and $\mathbb {P}_{0} = \otimes _{\mathbb {Z}} \tilde{\nu }$ on the $\sigma$-algebra generated by the cylinder sets in $\Omega _{0}$.

The average length of a word is defined by $\langle L \rangle = \sum _{j=1}^{m} j \tilde{\nu }(\mathcal {W}_{j})$, and if $w \in \mathcal {W} \cap \mathcal {W}_{j}$, we say w has length j and write $|w| = j$.

We define $\Omega = \bigcup _{j=1}^{m} \Omega _{j} \subset \Omega _{0} \times \{1,...,m\}$ where $\Omega _{j} = \{\omega \in \Omega _{0} : |\omega _{0}| = j\} \times \{1,...,j\}$. We define the probability measure $\mathbb {P}$ on the $\sigma$-algebra generated by the sets $A \times \{k\}$ where $A \subset \Omega _{0}$ such that for all $\omega \in A$, $|\omega _{0}|=j$ and $1 \le k \le j$.

For such sets, we set

$$\begin{aligned} \mathbb {P}[A \times \{k\}] = \frac{\mathbb {P}_{0}(A)}{\langle L \rangle }. \end{aligned}$$

(2)

Remark

The above construction implies that every event $A \subset \Omega _{0}$ gives rise to an event $\tilde{A} \subset \Omega$ with the same probability (up to multiplication by $\langle L \rangle$).

The shifts $T_{0}$ and T on $\Omega _{0}$ and $\Omega$ (respectively) are given by:

$(T_{0}\omega )_n = \omega _{n+1}$ and

$$\begin{aligned} T(\omega ,t)= \left\{ \begin{array}{ll} (\omega , t+1) &{}\quad \text {if}\quad t< |\omega _{0}| \\ (T_0(\omega ),1) &{}\quad \text {if}\quad t = |\omega _{0}|. \\ \end{array} \right. \end{aligned}$$

With this setup, the shift T is ergodic with respect to the probability measure $\mathbb {P}$ and the potential $V_{(\omega ,t)}$ is obtained through $...,\omega _{-1},\omega _{0},\omega _{1},...$ so that $V_{(\omega ,t)}(0) = \omega _{0}(t)$.

That is,

$$\begin{aligned} (H_{(\omega ,t)}u)(n) = u(n+1) + u(n-1) + V_{(\omega ,t)}(n)u(n) \end{aligned}$$

(3)

for all $u \in \ell ^{2}(\mathbb {Z})$.

From the ergodicity of the shift T, the results from [23] imply that the spectrum of $H_{(\omega ,t)}$ is almost surely a non-random set.

Remark 1

For notational convenience, we will often drop the t from the subscript on $H_{(\omega ,t)}$.

2.2 Basic Definitions and Notations

Definition

We call $\psi _{\omega ,E}$ a generalized eigenfunction with generalized eigenvalue E if $\psi _{\omega ,E} \ne 0$, $H_{\omega }\psi _{\omega ,E}=E\psi _{\omega ,E}$, and there is a $C > 0$ such that $|\psi _{\omega ,E}(n)| \le C(1+|n|)$ for all $n \in \mathbb {Z}$.

We denote the restriction of $H_{\omega }$ to the interval $[a,b] \cap \mathbb {Z}$ where $a,b \in \mathbb {Z}$ by $H_{\omega ,[a,b]}$ and for $E\notin \sigma (H_{\omega ,[a,b]})$ the corresponding Green’s function by

$$\begin{aligned} G_{[a,b],E,\omega }=(H_{\omega ,[a,b]}-E)^{-1}. \end{aligned}$$

Additionally, we let

$$\begin{aligned} P_{[a,b],E,\omega }=\det (H_{\omega ,[a,b]}-E) \end{aligned}$$

and

$$\begin{aligned} \tilde{P}_{[a,b],E,\omega ,}=\det (E-H_{\omega ,[a,b]}). \end{aligned}$$

We also let $E_{j,[a,b],\omega }$ denote the jth eigenvalue of the operator $H_{\omega ,[a,b]}$, and note that there are $b-a+1$ many of them (counting multiplicity).

Definition

$x\in \mathbb {Z}$ is called $(c,n_1,n_2,E,\omega )$-regular if:

• (1) $|G_{[x-n_{1},x+n_{2}],E,\omega }(x,x-n_{1})|\le e^{-cn_{1}}$ and

• (2) $|G_{[x-n_{1},x+n_{2}],E,\omega }(x,x+n_{2})|\le e^{-cn_{2}}$.

By a standard decoupling argument, we have for any generalized eigenfunction $\psi _{\omega ,E}$ and any $x \in [a,b]$,

$$\begin{aligned} \psi _{\omega ,E}(x)=-G_{[a,b],E,\omega }(x,a)\psi _{\omega ,E}(a-1)-G_{[a,b],E,\omega }(x,b)\psi _{\omega ,E}(b+1), \end{aligned}$$

(4)

and by Cramer’s rule

$$\begin{aligned} |G_{[a,b],E,\omega }(x,y)|=\frac{|P_{[a,x-1],E,\omega }P_{[y+1,b],E,\omega }|}{|P_{[a,b],E,\omega }|}. \end{aligned}$$

(5)

Remark

The above formula holds with the convention $P_{[a,b],E,\omega } = 1$ whenever $b < a$.

2.3 Transfer Matrices and the Lyapunov Exponent

For $w \in \mathcal {W}$, with $w = (w_{1},...,w_{j})$, we define word transfer matrices by $T_{w,E} = T_{w_{j},E} \cdots T_{w_{1},E}$ where $T_{v,E} = \begin{pmatrix} E-v &{}\quad -1\\ 1 &{}\quad 0 \end{pmatrix}$.

The transfer matrices over several words are given by

$$\begin{aligned} T_{\omega ,E}(k,l) = \left\{ \begin{array}{ll} T_{\omega _{k},E} \cdots T_{\omega _{l},E} &{}\quad \text {if}\quad k > l, \\ \mathbb {I} &{} \quad \text {if}\quad k = l,\\ {T_{\omega ,E}}^{-1}(l,k) &{} \quad \text {if}\quad k < l, \\ \end{array} \right. \end{aligned}$$

(6)

and $T_{[a,b],E,\omega }$ denotes the product of the transfer matrices so that for any generalized eigenfunction $\psi$ with generalized eigenvalue E:

$$\begin{aligned} T_{[a,b],E,\omega }\begin{pmatrix} \psi (a)\\ \psi (a-1)\\ \end{pmatrix}=\begin{pmatrix} \psi (b+1)\\ \psi (b)\\ \end{pmatrix}. \end{aligned}$$

An inductive argument shows

$$\begin{aligned} T_{[a,b],E,\omega }=\begin{pmatrix} \tilde{P}_{[a,b],E,\omega } &{}\quad -\tilde{P}_{[a+1,b],E,\omega }\\ \tilde{P}_{[a,b-1],E,\omega } &{}\quad -\tilde{P}_{[a+1,b-1],E,\omega }\\ \end{pmatrix}. \end{aligned}$$

(7)

We now define two Lyapunov exponents, one via matrix products obtained from $\Omega _{0}$ and the other from $\Omega$. We will see that the two quantities are essentially the same (up to multiplication by a positive constant) and hence provide the same information. It is worth noting, however, that the former is obtained through products of i.i.d. matrices, while the latter is not. In fact, we will need to utilize independence to obtain various estimates on the matrix products, and it is therefore important to verify the relationship between the two Lyapunov exponents.

Since both $T_{0}$ and T are ergodic, Kingman’s subadditive ergodic theorem [21] allows us to define the Lyapunov exponent. Recalling that $\langle L \rangle$ denotes the average length of a word, by the arguments in [4], we have in $\Omega _{0}$,

$$\begin{aligned} \gamma _{0}(E) := \lim _{n \rightarrow \infty } \frac{1}{n} \log || T_{\omega ,E}(n,1)||, \end{aligned}$$

and in $\Omega$,

$$\begin{aligned} \gamma (E) := \lim _{n \rightarrow \infty } \frac{1}{n} \log \left| \left| \begin{pmatrix} E-V_{\omega }(n) &{}\quad -1\\ 1 &{}\quad 0 \end{pmatrix} \cdots \begin{pmatrix} E-V_{\omega }(1) &{}\quad -1\\ 1 &{}\quad 0 \end{pmatrix}\right| \right| . \end{aligned}$$

In both cases, the limit exists for fixed E on a full measure set.

Remark

We note that the limit in $\Omega$ is defined via one-step transfer matrices while the limit in $\Omega _{0}$ is defined via word transfer matrices.

In [4], the authors prove the relationship between the two Lyapunov exponents described in the following theorem.

Theorem 3

$\frac{\gamma _{0}(E)}{\langle L \rangle } = \gamma (E)$.

2.4 Continuity and Uniform Positivity of the Lyapunov Exponent

Let $\mu _E$ denote the smallest closed subgroup of $SL(2,\mathbb {R})$ generated by the ‘word’-step transfer matrices. It is shown in [4] that $\mu _E$ is strongly irreducible and contracting for all E outside of a finite set $D \subset \mathbb {R}$ and these conditions allow for application of Fürstenberg’s theorem [12] resulting in positivity of the Lyapunov exponent whenever $E \notin D$. The irreducibility and contracting conditions in addition to the ones below allow for application of Theorem B by Fürstenberg and Kifer [11] which yields continuity of the Lyapunov exponent and hence uniform positivity outside of D. We begin by defining the maps needed to state the conditions alluded to above.

Suppose $E_k \in \mathbb {R}$ and $E \in \mathbb {R}$ with $E_{k} \rightarrow E$. Let $X_{k}: \Omega _{0} \rightarrow SL(2,\mathbb {R})$ be defined by $X_{k}(\omega )=T_{\omega _{0},E_{k}}$ and $X: \Omega _{0} \rightarrow SL(2,\mathbb {R})$ be defined by $X(\omega )=T_{\omega _{0},E}$. By Theorem B in [11], to prove $\gamma (E_{k}) \rightarrow \gamma (E)$ as $k \rightarrow \infty$, it suffices to show:

• (1) For any $h:SL(2,\mathbb {R}) \rightarrow \mathbb {C}$ with h continuous and of compact support, $\mathbb {E}[h(X_{k})] \rightarrow \mathbb {E}[h(X)]$ as $k \rightarrow \infty$

• (2) $\mathbb {E}[\log ^{+}(||X_{k}||\chi _{\{||X_{k}|| \ge n\}})] \rightarrow 0$ as $n \rightarrow \infty$ uniformly in k

• (3) $\mathbb {E}[\log ^{+}(||X^{-1}||\chi _{\{||X^{-1}|| \ge n\}})] \rightarrow 0$ as $n \rightarrow \infty$.

Remark

Condition (1) is known as weak convergence and conditions (2) and (3) together are known as bounded convergence.

Theorem 4

$\gamma (E)$ is continuous.

Proof

(1) follows by dominated convergence since $X_{k} \rightarrow X$ for a.e. $\omega$ and h is continuous and of compact support. Since the word transfer matrices are uniformly bounded, we may choose $C \in \mathbb {R}$ so that $||X_{k}|| \le C$ for all k. It follows that $\{\log ^{+}||X_{k}||\}$ is a uniformly integrable family, so (2) holds. Finally, (3) follows since $\mathbb {E}[\log ^{+}||X^{-1}||] < \infty$. This completes the proof. $\square$

Theorem 5

If I is a compact interval, $D \cap I = \emptyset$, and $\nu = \inf \{\gamma (E): E \in I\}$, then $\nu >0$.

Proof

Since $\mu _{E}$ is strongly irreducible and contracting for all $E \notin D$, we have $\gamma (E) > 0$ for all $E \in I$. The result now follows by Theorem 4 and compactness of I.

$\square$

Motivated by Eq. (7) and large deviation theorems, we define

$$\begin{aligned} {B^{+}}_{[a,b],\varepsilon }=\left\{ (E,\omega ): E\in I,{|P_{[a,b],E,\omega }|}\ge e^{(\gamma (E)+\varepsilon )(b-a+1)}\right\} , \end{aligned}$$

and

$$\begin{aligned} {B^{-}}_{[a,b],\varepsilon }=\left\{ (E,\omega ): E \in I,{|P_{[a,b],E,\omega }|}\le e^{(\gamma (E)-\varepsilon )(b-a+1)}\right\} , \end{aligned}$$

and the corresponding sections:

$$\begin{aligned} {B^{\pm }}_{[a,b],\varepsilon ,\omega }=\left\{ E: (E,\omega ) \in B^{\pm }_ {[a,b],\varepsilon }\right\} , \end{aligned}$$

and

$$\begin{aligned} {B^{\pm }}_{[a,b],\varepsilon ,E}=\left\{ \omega : (E,\omega ) \in B^{\pm }_ {[a,b],\varepsilon }\right\} . \end{aligned}$$

3 Large Deviation Theorems

The goal of this section is to obtain a uniform large deviation estimate for $P_{[a,b],E}$. In the Anderson model, a direct application of Tsay’s theorem for matrix elements of products of i.i.d. matrices results in both an upper and lower bound for the above determinants. In the general random word case, there are two issues. Firstly, the one-step transfer matrices are not independent. This issue is naturally resolved by considering $\omega _{k}$-step transfer matrices and treating products over each word as a single step. However, in this case, both the randomness in the length of the chain and products involving partial words need to be accounted for. Since matrix elements are majorized by the norm of the matrix and all matrices in question are uniformly bounded, we can obtain an upper bound identical to the one obtained in the Anderson case. Lower bounds on the matrix elements are more delicate and require the introduction of random scales. For the reader’s convenience, we first recall Tsay’s theorem and then give the precise statements and proofs of the results alluded to above.

As remarked above, $\mu _E$ is strongly irreducible and contracting for $E \in I$. In addition, the ‘word’-step transfer matrices (defined in Eq. (6)) are bounded, independent, and identically distributed. These conditions in conjunction with the weak and bounded convergence criteria on the previous page are sufficient for an application of Tsay’s theorem.

Theorem 6

[24] Suppose I is a compact interval and for each $E \in I$, $Z_{1}^{E},..., Z_{n}^{E},...$ are bounded i.i.d random matrices such that the smallest closed subgroup of $SL(2,\mathbb {R})$ generated by the matrices is strongly irreducible, contracting, and the matrices satisfy the weak and bounded convergence criteria (i.e., conditions (1)–(3) in the discussion following Theorem 3). Then for any $\varepsilon > 0$, there is an $\eta > 0$ and an $N\in \mathbb {N}$ such that for any $E \in I$, for any unit vectors u, v and $n > N$,

$$\begin{aligned} \mathbb {P}\left[ e^{(\gamma (E) - \varepsilon )n} \le | \langle Z_{n}^{E} \cdots Z_{1}^{E}u,v \rangle | \le e^{(\gamma (E) + \varepsilon )n}\right] \ge 1 - e^{-\eta n}. \end{aligned}$$

Lemma 1

If I is compact and $I \cap D = \emptyset$, then for any $\varepsilon > 0$ there is an $\eta > 0$ and an N such that if $b \in \mathbb {Z}$ with $b-a+1 > N$ and $E \in I$, then

$$\begin{aligned} \mathbb {P}[B^{+}_{[a,b],\varepsilon ,E}] \le e^{-\eta (b-a+1)}. \end{aligned}$$

Proof

Let $Y_i = |\omega _{i}|$, so $Y_i$ is the length of the ith word and let $S_n = Y_1 +\cdots + Y_n$.

Let $u = \begin{pmatrix} \ 1 \\ \ 0 \end{pmatrix}$ and let $P_{(\omega _{1},\omega _{n}),E} = \det (H_{(\omega _{1},\omega _{n})}-E)$ where $H_{(\omega _{1},\omega _{n})}$ denotes $H_{\omega }$ restricted to the interval where V takes values determined by $\omega _{1}$ through $\omega _{n}$. By Eq. (7) from the previous section, $|P_{(\omega _{1},\omega _{n}),E}| = |\langle T_{\omega ,E}(n,1) u,u \rangle |$.

Letting $\varepsilon > 0$ and applying Theorem 6 to the random products $T_{\omega ,E}(n,1)$, we obtain an $\eta _{1} > 0$ and an $N_1$ such that for $n > N_{1}$, $\mathbb {P}_{0}[\{\omega \in \Omega _{0}: |P_{(\omega _{1},\omega _{n}),E}| \le e^{(\gamma (E) + \varepsilon ) n\langle L \rangle }\}] \ge 1- e^{-\eta _{1}n}$.

Now let $0< \varepsilon _{1} < 1$ so that $\varepsilon _{1}\sup \{\gamma (E): E \in I\} < \varepsilon$. We apply large deviation estimates (e.g., [9]) to the real, bounded, i.i.d. random variables $Y_i$ to obtain an $N_{2}$ and an $\eta _{2} > 0$ such that for $n > N_{2}$, $\mathbb {P}_{0} [S_n - n\varepsilon _{1}< n\langle L \rangle < S_n + n\varepsilon _{1}] \ge 1 - e^{-\eta _{2}n}$.

Denoting the intersection of the above events by $A_n$, we have, on $A_{n}$,

$$\begin{aligned} \begin{aligned} |P_{(\omega _{1},\omega _{n}),E}|&\le e^{(\gamma (E) + \varepsilon )n \langle L \rangle } \\&\le e^{(\gamma (E) + \varepsilon )(S_n + n\varepsilon _{1})} \\&= e^{(\gamma (E) + \varepsilon )S_n +\gamma (E)\varepsilon _{1}n + n\varepsilon \varepsilon _{1}} \\&\le e^{(\gamma (E) + \varepsilon )S_n +\gamma (E)\varepsilon _{1}S_n + S_n\varepsilon \varepsilon _{1}} \\&\le e^{(\gamma (E) + 3 \varepsilon )S_n}. \end{aligned} \end{aligned}$$

(8)

Note that n represents the number of words while $S_{n}$ counts the number of single sites (or letters). Thus, Eq. (8) allows us to obtain an estimate in between two words.

That is, for any $1 \le k \le S_{n+1} - S_{n}$, let $P_{(\omega _{1},\omega _{n}+k),E} = \det (H_{(\omega _{1},\omega _{n} + k)}-E)$ where $H_{(\omega _{1},\omega _{n} + k)}$ denotes $H_{\omega }$ restricted to the interval where $V_{\omega }$ takes values determined by $\omega _{1}$ through the kth letter of $\omega _{n+1}$.

Since the one-step transfer matrices are uniformly bounded, Eqs. (7) and (8) imply for any $1 \le k \le S_{n+1} - S_{n}$, $|P_{[1,S_{n}+k],E,\omega }| \le Ce^{(\gamma (E) + 3 \varepsilon )S_n} \le e^{(\gamma (E) + 4 \varepsilon )S_n}$ on $A_n$ (when n is sufficiently large). Let $\eta _{3} = \min \{\eta _{1},\eta _{2}\}$, and choose $0<\eta <\frac{\eta _{3}}{2m}$.

Having obtained an estimate that counts the number of single sites rather than the number of words (i.e., $e^{\gamma S_{n}}$ instead of $e^{\gamma n}$ in Eq. (8)), we can now use the fact that every event in $\Omega _{0}$ gives rise to an event in $\Omega$ (e.g., Eq. (2)). Using Eq. (8), the estimate above on $P_{[1,S_{n}+k],E,\omega }$, and the remark following Eq. (2), we can apply the shift T in $\Omega$ to conclude that for any sufficiently large n,

$$\begin{aligned} \mathbb {P}[\{|P_{[1,n],E,\omega }| \le e^{(\gamma (E) + 4 \varepsilon )n}\}] \ge 1 - e^{-\eta n}. \end{aligned}$$

The result now follows for any interval [a, b] (with $b-a +1$ sufficiently large) since T preserves the probability of events.

$\square$

We finish the section with a lemma that relies crucially on independence. As above, we will work in $\Omega _{0}$ and ‘lift’ our results to $\Omega$.

The lemma below holds for any fixed $K>1$ and this K will be chosen in the next section.

Lemma 2

There are real-valued random variables $R_{n}$, $R^{'}_{n}$, $Q_{n}$, $Q^{'}_{n}$, and $\tilde{Q}_{n}$ such that:

if I is compact with $I \cap D = \emptyset$ and $\varepsilon > 0$ with

$$\begin{aligned} F_{l,n,\varepsilon }^{3}= & {} \bigcap _{j} \left\{ \omega : E_{j, [l+Q_{n}+k,l+Q^{'}_{n}],\omega } \notin B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon ,\omega } \;\forall k, \quad 0 \le k \le 2m \right\} ,\\ F_{l,n,\varepsilon }^{2+}= & {} \bigcap _{j} \left\{ \omega : E_{j,[l+Q_{n}+k,l+Q^{'}_{n}],\omega } \notin B^{+}_{[y,l+R^{'}_{n}],\varepsilon ,\omega } \right. \\&\left. \forall y \in \left[ l+R_{n}, \lfloor l+R^{'}_{n} - \frac{n}{K} \rfloor \right] , 0 \le k \le 2m\right\} ,\\ F_{l,n,\varepsilon }^{2-}= & {} \bigcap _{j} \left\{ \omega : E_{j,[l+Q_{n}+k,l+Q^{'}_{n}],\omega } \notin B^{+}_{[l+R_{n},y],\varepsilon ,\omega } \right. \\&\left. \forall y \in \left[ \lceil l+R_{n} +\frac{n}{K} \rceil , l+R^{'}_{n}\right] , 0 \le k \le 2m \right\} , \end{aligned}$$

and

$$\begin{aligned} F_{l,n,\varepsilon }^{2} = F_{l,n,\varepsilon }^{2+} \cap F_{l,n,\varepsilon }^{2-}, \end{aligned}$$

then there is N and an $\eta ' > 0$ such that if $n>N$, $l \in \mathbb {Z}$, and $E \in I$:

$$\begin{aligned}&\mathbb {P}[B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon ,E}] \le e^{-\eta '(2n+1)}, \\&\mathbb {P}[B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon ,E}] \le e^{-\eta '(2n+1)}, \\&\mathbb {P}[F_{l,n,\varepsilon }^{3}] \ge 1 - 2m^{2}(2n+3)^{2}e^{-\eta '(2n+1)}, \end{aligned}$$

and

$$\begin{aligned}\mathbb {P}[F_{l,n,\varepsilon }^{2}] \ge 1 - 2m^{4}(2n+3)^{3}e^{-\eta '(\frac{n}{K})}. \end{aligned}$$

Proof

For $\omega \in \Omega _0$, and $a,b \in \mathbb {Z}$, let $H_{(\omega _{a},\omega _{b})}$ denote the restriction of H to the interval where the potential is given by the words $\omega _{a}$ through $\omega _{b}$. Take $u = \begin{pmatrix} \ 1 \\ \ 0 \end{pmatrix}$, and let $P_{(\omega _{a},\omega _{b}),E} = \det (H_{(\omega _{a},\omega _{b})}-E)$ where $H_{(\omega _{a},\omega _{b})}$ denotes $H_{\omega }$ restricted to the interval in which V takes values determined by $\omega _{a}$ through $\omega _{b}$. By Eq. (7) from the previous section, $|P_{(\omega _{-n},\omega _{n}),E}| = |\langle T_{\omega ,E}(n,-n) u,u \rangle |$. Finally, let $S_{(a,b)} = |\omega _{a}|+ \cdots + |\omega _{b}|$. Letting $\varepsilon > 0$ and applying Theorem 6 to the random products $T_{\omega ,E}(n,-n)$, we obtain an $\eta _{1} > 0$ and an $N_1$ such that for $n > N_{1}$ and any $E \in I$,

$$\begin{aligned} \mathbb {P}_{0}[\{\omega \in \Omega _{0} : |P_{(\omega _{-n},\omega _{n}),E}| \ge e^{(\gamma (E) - \varepsilon ) (2n+1)\langle L \rangle }\}] \ge 1- e^{-\eta _{1}(2n+1)}. \end{aligned}$$

(9)

and

$$\begin{aligned} \mathbb {P}_{0}[\{\omega \in \Omega _{0} : |P_{(\omega _{n+1},\omega _{3n+1}),E}| \ge e^{(\gamma (E) - \varepsilon ) (2n+1)\langle L \rangle }\}] \ge 1- e^{-\eta _{1}(2n+1)}.\nonumber \\ \end{aligned}$$

(10)

If $E_{j}$ denotes an eigenvalue corresponding to $H_{(\omega _{n+1},\omega _{3n+1})}$, then by independence

$$\begin{aligned} \mathbb {P}_{0}[\{\omega \in \Omega _{0} : |P_{(\omega _{-n},\omega _{n}),E_{j}}| \ge e^{(\gamma (E_{j}) - \varepsilon )(2n+1)\langle L \rangle }\}] \ge 1- e^{-\eta _{1}(2n+1)} \end{aligned}$$

(11)

whenever $n > N$.

Note that the same argument holds for any eigenvalue of $H_{(\omega _{a},\omega _{b})}$ provided $[a,b] \cap [-n,n] = \emptyset$.

We can also apply large deviation theorems (e.g., [9]) to the (real) random products $S_{(a,b)}$ where $0<\varepsilon _{1} < \min \{1,\varepsilon ,\varepsilon \sup \{\gamma (E): E \in I\}\}$ to obtain an N and an $\eta _{2} > 0$ such that whenever $b-a +1 > N$,

$$\begin{aligned} (b-a+1)(\langle L \rangle - \varepsilon _{1}) \le S_{(a,b)} \le (b-a+1)(\langle L \rangle + \varepsilon _{1}) \end{aligned}$$

(12)

with probability greater than $1-e^{-\eta _{2}(b-a+1)}$.

Using Eq. (11) and applying Eq. (12) with $a=-n$ and $b = n$, we have an event A where:

$$\begin{aligned} \begin{aligned} |P_{(\omega _{-n},\omega _{n}),E_j}|&\ge e^{(\gamma (E) - \varepsilon )(2n+1) \langle L \rangle } \\&\ge e^{(\gamma (E) - \varepsilon )(S_{(-n,n)} - (2n+1)\varepsilon _{1})} \\&= e^{(\gamma (E) - \varepsilon )S_{(-n,n)} -(\gamma (E)\varepsilon _{1})(2n+1) + (2n+1)\varepsilon \varepsilon _{1}}\\&\ge e^{(\gamma (E) - \varepsilon )S_{(-n,n)} -(\gamma (E)\varepsilon _{1})S_{(-n,n)} + (2n+1)\varepsilon \varepsilon _{1}}\\&\ge e^{(\gamma (E) - 2\varepsilon )S_n}. \end{aligned} \end{aligned}$$

(13)

By using a similar argument to deal with the upper bound, we have

$|P_{(\omega _{-n},\omega _{n}),E_j}|\le e^{(\gamma (E) + 3\varepsilon )S_{(-n,n)}}$.

In particular, whenever $\frac{n}{K} > N$, and $y \in [-n,\lfloor n-\frac{n}{K} \rfloor ]$, we have

$$\begin{aligned} |P_{(\omega _{y},\omega _{n}),E_{j}}|\le e^{(\gamma (E) + 3\varepsilon )S_{(y,n)}} \end{aligned}$$

(14)

with probability greater than $1 - e^{\eta _{1}\frac{n}{K}}$ (with a similar estimate holding whenever $y \in [\lceil -n + \frac{n}{K} \rceil ,n]$).

Since the single-step transfer matrices are uniformly bounded and there are at most m single-step transfer matrices in a word transfer matrix, we have a $C>0$ such that:

$$\begin{aligned} \mathbb {P}_{0}[\{ \omega \in \Omega _{0} : |P_{(\omega _{y}+k,\omega _{n}),E_j}| \le Ce^{(\gamma (E_{j}) + 3\varepsilon )S_{(y,n)}}\}] \ge 1 - e^{-\eta _{1}\frac{n}{K}}, \end{aligned}$$

(15)

where $P_{(\omega _{y}+k,\omega _{n})}$ denotes the determinant obtained by restricting H from the kth letter of $\omega _{y}$ to $\omega _{n}$.

Note that the products $S_{(a,b)}$ are also well defined on $\Omega$ and we now are able to define the random variables from the statement of the lemma.

For $(\omega ,t) \in \Omega$ we put,

$$\begin{aligned} \begin{aligned} R^{'}_{n}&= S_{(0,n)} - t, \\ R_{n}&= -S_{(-n,-1)} - t+1, \\ Q_{n}&= R^{'}_{n} + 1, \\ Q^{'}_{n}&= R^{'}_{n} + S_{(n+1,3n+1)}, \text { and } \\ \tilde{Q}_{n}&= R^{'}_{n} + S_{(n+1,2n+1)}. \\ \end{aligned} \end{aligned}$$

(16)

We choose $0< \tilde{\eta _{2}} < \eta _{2}$ and note that $\varepsilon _{1} < \varepsilon$. Thus, applying Eq. (12), we obtain an event with probability greater than $1 - e^{-\tilde{\eta _{2}}n}$ where we have the following estimates on the random variables defined above:

$$\begin{aligned} \begin{aligned} (n+1)(\langle L \rangle - \varepsilon )&\le S_{(0,n)} \le (n+1)(\langle L \rangle + \varepsilon ), \\ n(\langle L \rangle - \varepsilon )&\le S_{(-n,-1)} \le n(\langle L \rangle + \varepsilon ), \\ (2n+1)(\langle L \rangle - \varepsilon )&\le S_{(n+1,3n+1)} \le (2n+1)(\langle L \rangle + \varepsilon ), \text { and }\\ (n+1)(\langle L \rangle - \varepsilon )&\le S_{(n+1,2n+1)} \le (n+1)(\langle L \rangle + \varepsilon ).\\ \end{aligned} \end{aligned}$$

(17)

We choose $\eta '> 0$ to be smaller than $\eta _{1}$ and $\tilde{\eta _{2}}$ so that by the remark after Eq. (2), the definitions of the random variables above, Eqs. (9) and (10), we have for any $E \in I$:

$$\begin{aligned} \mathbb {P}[B^{-}_{[l+R_{n},l+R^{'}_{n}],E}] \le e^{-\eta '(2n+1)} \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}[B^{-}_{[l+Q_{n},l+Q^{'}_{n}],E}] \le e^{-\eta '(2n+1)}. \end{aligned}$$

Moreover, by the same reasoning, after taking the union over all of the possible eigenvalues $E_{j}$ and the ordered pairs (y, n) (and $(-n,y)$), Eq. (13) provides the desired estimate on

$$\begin{aligned} F_{0,n,\varepsilon }^{3} = \bigcap _{j} \left\{ \omega : E_{j,[Q_{n}+k,Q^{'}_{n}],\omega } \notin B^{-}_{[R_{n},R^{'}_{n}],\varepsilon ,\omega } \text{ for } \text{ all } k \text{ with } 0 \le k \le 2m \right\} \end{aligned}$$

while Eqs. (14) and (15) provide the desired estimate on $F_{0,n,\varepsilon }^{2}$.

The result now follows by applying the (measure-preserving) shift T so that the intervals are centered around l rather than 0.

$\square$

Remark

Each of the lemmas above furnish a positive constant: $\eta$, $\eta ^{'}$. For a fixed $\varepsilon$, we call the minimum of these constants the ‘large deviation parameter’ associated with $\varepsilon$ and denote it by $\eta _{\varepsilon }$.

4 Lemmas

We prove localization results on a compact interval $I = [\lambda _{1}, \lambda _{2}]$ where $D \cap I = \emptyset$. In order to do so, we fix a larger interval $\tilde{I} = [\tilde{\lambda _{1}},\tilde{\lambda _{2}}]$ with $\tilde{\lambda _{1}}< \lambda _{1}< \lambda _{2} < \tilde{\lambda _{2}}$ and $D \cap \tilde{I} = \emptyset$, then apply the large deviation theorems from the previous section to $\tilde{I}$.

The following lemmas involve parameters $\varepsilon _0, \varepsilon , \eta _0, \delta _0, \eta _\varepsilon$,K, and the intervals $I,\tilde{I}$. The lemmas hold for any values satisfying the constraints below:

• (1) Let $\nu =\inf \{\gamma (E): E \in \tilde{I}\}$, take $0<\varepsilon _0< \nu /8$ ($\varepsilon _{0} \ll 1)$ and let $\eta _0$ denote the large deviation parameter corresponding to $\varepsilon _0$. Choose any $0<\delta _0<\eta _0$ and let $0<\varepsilon <\min \{(\eta _0-\delta _0)/3m,\varepsilon _{0}/4\}$. Choose $\tilde{M}>0$ so that $|P_{[a,b],E,\omega }|\le \tilde{M}^{b-a+1}$ for all intervals [a, b], $E\in \tilde{I}$, and $\omega \in \Omega .$ Lastly, choose K so that $\tilde{M}^{1/K}<e^{\nu /2}$ and let $\eta _\varepsilon$, $\eta _{\frac{\varepsilon }{4}}$ denote the large deviation parameters corresponding to $\varepsilon$ and $\frac{\varepsilon }{4}$, respectively.

• (2) Any N’s and constants furnished by the lemmas below depend only on the parameters above (i.e., they are independent of $l \in \mathbb {Z}$ and $\omega$).

• (3) We use C to denote various constants which change from proof to proof (and sometimes even within the same proof).

Thus, for the remainder of the work, $\varepsilon _0, \varepsilon , \eta _0, \delta _0, \eta _\varepsilon ,$ and K will be treated as fixed parameters chosen in the manner outlined above.

Remark

Note that the sets $B^{\pm }_{[a,b],\varepsilon }$ are hereafter defined in terms of $\tilde{I}$ rather than I.

Following [13] and [20], we define subsets of $\Omega$ below on which we have regularity of the Green’s functions. This is the key to the proof of all the localization results. As mentioned in the Introduction, the proofs of spectral and dynamical localization given in [20] show that an event formed by the complement of the sets below has exponentially small probability. These estimates were exploited in [13] to provide a proof of exponential dynamical localization for the one-dimensional Anderson model. We follow the example set in these two papers with the appropriate modifications needed to handle the presence of critical energies and the varying length of words.

Let $m_{L}$ denote Lebesgue measure on $\mathbb {R}$.

Lemma 3

If $n\ge 2$ and x is $(\gamma (E)-8\varepsilon _0,n_{1},n_{2},E,\omega )$-singular, then

$$\begin{aligned} (E,\omega )\in {B^-}_{[x-n_{1},x+n_{2}],\varepsilon _0}\cup {B^+}_{[x-n_{1},x-1],\varepsilon _0}\cup {B^+}_{[x+1,x+n_{2}],\varepsilon _0}. \end{aligned}$$

Proof

The result follows by Eq. (5) and the definition of singularity.

$\square$

Let $R_n, R^{'}_n, Q_n, Q^{'}_n,$ and $\tilde{Q}_n$ be the random variables from Lemma 2 and for $l \in \mathbb {Z}$ set

$$\begin{aligned} F_{l,n,\varepsilon _0}^{1}= & {} {} \{\omega : \max \{m_{L}(B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon _0,\omega }),m_{L}(B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon _0,\omega })\} \nonumber \\ {}\le & {} {} e^{-(\eta _0-\delta _0)(2n+1)} \}. \end{aligned}$$

Lemma 4

There is an N such that for $n> N$ and any $l \in \mathbb {Z}$,

$$\begin{aligned} \mathbb {P}[F_{l,n,\varepsilon _{0}}^{1}] \ge 1 - 2m_{L}(\tilde{I})e^{-\delta _{0}(2n+1)}. \end{aligned}$$

Proof

With $0<\varepsilon _0 < 8\nu$ as above, choose N such that the conclusion of Lemma 2 holds. Then for $n > N$,

$$\begin{aligned} m_{L}\times \mathbb {P}(B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon _0})&=\mathbb {E}(m_{L}(B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon _0,\omega }))\\&=\int _{\mathbb {R}}\mathbb {P}(B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon _0,E})\; dm_{L}(E)\\&\le m_{L}(\tilde{I})e^{-\eta _0(2n+1)}.\\ \end{aligned}$$

Applying the same reasoning to $B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon _0}$, by the estimate above and Chebyshev’s inequality,

$$\begin{aligned} e^{-(\eta _0-\delta _0)(2n+1)}\mathbb {P}[(F^{1}_{l,n,\varepsilon _0})^{c}]\le 2m_{L}(\tilde{I})e ^{-\eta _0(2n+1)}. \end{aligned}$$

The result follows by multiplying both sides of the last inequality by $e^{(\eta _{0} - \delta _{0})(2n+1)}$. $\square$

Remark

Lemma 5 is proved in [20] and used there to give a uniform (and quantitative) Craig–Simon estimate similar to the one in [13].

Lemma 5

Let Q(x) be a polynomial of degree $\hat{n}-1$. Let $x_i=\cos \frac{2\pi (i+\theta )}{\hat{n}}$, for $0< \theta <\frac{1}{2}, i=1,2,\ldots , \hat{n}$. If $Q(x_i)\le a^{\hat{n}}$, for all i, then $Q(x)\le C\hat{n}a^{\hat{n}}$, for all $x\in [-1,1]$, where $C=C(\theta )$ is a constant.

Set

$$\begin{aligned} F_{[a,b],\varepsilon }=\{\omega : |P_{[a,b],E,\omega }| \le e^{(\gamma (E)+\varepsilon )(b-a+1)} \text { for all } E \in \tilde{I} \}. \end{aligned}$$

Lemma 6

There are $C >0$ and N such that for $b-a+1 > N$,

$$\begin{aligned} \mathbb {P}[F_{[a,b],\varepsilon }] \ge 1 - C(b-a+2)e^{-\eta _{\frac{\varepsilon }{4}}(b-a+1)}. \end{aligned}$$

Proof

By continuity of $\gamma$ (Theorem 4) and compactness of $\tilde{I}$, if $\varepsilon > 0$, there is $\delta >0$ such that if E,$E^{'} \in \tilde{I}$ with $|E-E^{'}| < \delta$, then

$$\begin{aligned} |\gamma (E) - \gamma (E^{'})| < \frac{1}{4}\varepsilon . \end{aligned}$$

(18)

Note that we will aim to apply Lemma 6 to $P_{[a,b],E,\omega }$, a polynomial of degree $b-a+1$. Thus, we set $\hat{n} = b-a + 2$.

Divide $\tilde{I}$ into sub-intervals of size $\delta$, denoted by $J_{k}=[E_{k}^{\hat{n}},E_{k+1}^{\hat{n}}]$ where $k=1,2,...C$. Additionally, with the $x_{i}$’s given in Lemma 5, let $E_{k,i}^{\hat{n}} = E_{k}^{\hat{n}} + (x_{i} + 1)\frac{\delta }{2}$.

By Lemma 1, there is an N such that for $b-a+1>N$,

$$\begin{aligned} \mathbb {P}[\{\omega : |P_{[a,b],E_{k,i}^{\hat{n}},\omega }| \le e^{(\gamma (E_{k,i}^{\hat{n}}) + \frac{1}{4}\varepsilon )(b-a+1)}\}] \ge 1 - e^{-\eta _{\frac{\varepsilon }{4}}(b-a+1)}. \end{aligned}$$

(19)

Put $F_{[a,b],k,\varepsilon } = \bigcap _{i=1}^{\hat{n}} \{\omega : |P_{[a,b],E_{k,i}^{\hat{n}},\omega }| \le e^{(\gamma (E_{k,i}^{\hat{n}}) + \frac{1}{4}\varepsilon )(b-a+1)}\}$ and $\gamma _{k} = \inf _{E \in J_k} \gamma (E)$. For $\omega \in F_{[a,b],k,\varepsilon }$, by Eq. (18), $|P_{[a,b],E_{k,i}^{\hat{n}},\omega }| \le e^{(\gamma _{k}+ \frac{1}{2}\varepsilon )(b-a+1)}$. Thus, with $Q(x) = P_{[a,b],\omega }(E^{\hat{n}}_{k} + (x+1)\frac{\delta }{2})$, an application of Lemma 5 yields, for any $\omega \in F_{[a,b],k,\varepsilon }$ and $E \in J_{k}$,

$$\begin{aligned} \begin{aligned} |P_{[a,b],E,\omega }|&\le C(b-a+2)e^{(\gamma _{k} + \frac{\varepsilon }{2})(b-a+1)} \\&\le C(b-a+2)e^{(\gamma (E) + \frac{\varepsilon }{2})(b-a+1)} \\&\le e^{(\gamma (E) + \frac{3}{4}\varepsilon )(b-a+1)}.\\ \end{aligned} \end{aligned}$$

(20)

Note that the last inequality follows since we may increase N so that for $(b-a+1)> N$, $C(b-a+2) \le e^{\frac{\varepsilon }{4}(b-a+1)}$.

By Eq. (19), we have

$$\begin{aligned} \mathbb {P}\left[ \bigcap _{k=1}^{C} F_{[a,b],k,\varepsilon }\right] \ge 1 - C(b-a+2)e^{-\eta _{\frac{\varepsilon }{4}}(b-a+2)}. \end{aligned}$$

Thus, since

$$\begin{aligned} \bigcap _{k=1}^{C} F_{[a,b],k,\varepsilon } \subset F_{[a,b],\varepsilon }, \end{aligned}$$

the result follows. $\square$

Let $n_{r,\omega }$ denote the center of localization (if it exists) for $\psi _{\omega ,E}$ (i.e., $n_{r,\omega } \in \mathbb {Z}$ such that $|\psi _{\omega ,E}(n)| \le |\psi _{\omega ,E}(n_{r,\omega })|$ for all $n \in \mathbb {Z}$). Note that by the results in [15], $n_{r,\omega }$ can be chosen as a measurable function of $\omega$.

Put $F^{4}_{l,n,\varepsilon }{=}\left( \!(F_{[l+R_{n},l-1],\varepsilon } \cap F_{[l+1,l+R^{'}_{n}],\varepsilon }\!\right) {\cap }\Big (\!\bigcup _{0 \le k \le 2m} (F_{[l+\tilde{Q}_{n}{+}k+1,l+Q^{'}_{n} ],\varepsilon } \cap F_{[l+Q_{n},l+\tilde{Q}_{n} + k - 1],\varepsilon }\Big )$ and

$$\begin{aligned} J_{l,n,\varepsilon } = F_{l,n,\varepsilon _0}^{1} \cap F_{l,n,\varepsilon }^{2} \cap F_{l,n,\varepsilon }^{3} \cap F^{4}_{l,n,\varepsilon }. \end{aligned}$$

Lemma 7

There is N such that if $n>N$, $\omega \in J_{l,n,\varepsilon }$, with a generalized eigenfunction $\psi _{\omega ,E}$ satisfying either

• (1) $n_{r,\omega }=l$, or

• (2) $|\psi _{\omega }(l)| \ge \frac{1}{2},$

then if $l+\tilde{Q}_{n}+k$ is $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-singular (with $0 \le k \le 2m$), there exist

$$\begin{aligned} l+R_{n}\le y_1 \le y_2 \le l+R^{'}_{n} \end{aligned}$$

and $E_{j}=E_{j,[l+Q_{n} + k,l+Q^{'}_{n}],\omega }$ such that

$$\begin{aligned}&|P_{[l+R_{n},y_1],E_j,\tilde{\omega }}P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\nonumber \\&\quad \ge {} \frac{1}{2m_{L}(\tilde{I})\sqrt{m(2n+1)}}e^{(\gamma (E_j)-\varepsilon )(R^{'}_{n}-R_{n}+1)+(\eta _0-\delta _0)(2n+1)}. \end{aligned}$$

Remark

Note that $y_1$ and $y_2$ depend on $\omega$ and l, but we do not include this subscript for notational convenience. In particular, this is done when the other terms in expressions involving $y_1$ or $y_2$ have the correct subscript and indicate the appropriate dependence.

Proof

Firstly, if $|\psi _{\omega }(l)| \ge \frac{1}{2}$, we may choose $N_1$ such that l is $(\gamma (E)-8\varepsilon _{0}, -R_{n},R^{'}_{n}, E, \omega )$-singular for $n>N_1$. In the case that $n_{r,\omega }=l$, then there is an $N_2$ such that l is naturally, $(\nu -8\varepsilon _{0},-R_{n},R^{'}_{n}, E, \omega )$-singular for all $n>N_2$. Choose $N_3$ so that $e^{\frac{-\nu }{2}n} < \text {dist}(I,\tilde{I})$ for $n>N_3$ and finally choose N to be larger than $N_1$, $N_2$, $N_3$ and the N’s from Lemmas 4, 2 and 6.

Suppose that for some $n>N$, $l+\tilde{Q}_{n}+k$ is $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-singular. By Lemmas 3 and 6, $E\in B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon _0,\omega }$. Note that all eigenvalues of $H_{[l+Q_{n},l+Q^{'}_{n}],\omega }$ belong to $B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon _0,\omega }$. Since $P_{[l+Q_{n},l+Q^{'}_{n}],\tilde{E},\omega }$ is a polynomial in $\tilde{E}$, it follows that $B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\omega ,\varepsilon _{0}}$ is contained in the union of sufficiently small intervals centered at the eigenvalues of $H_{\omega ,[l+Q_{n},l+Q^{'}_{n}]}$. Moreover, Lemma 4 gives

$$\begin{aligned} m(B^{-}_{[l+Q_{n},l+Q^{'}_{n}],\varepsilon _{0},\omega })\le 2m_{L}(\tilde{I})e^{-(\eta _0-\delta _0)(2n+1)}, \end{aligned}$$

so we have the existence of $E_j=E_{j,[l+Q_{n},l+Q^{'}_{n}],\omega }$ so that $|E-E_j|\le 2m_{L}(\tilde{I}) e^{-(\eta _0-\delta _0)(2n+1)}$.

Applying the above argument with l in place of $l+\tilde{Q}_{n} + k$ yields an eigenvalue $E_i=E_{i,\omega ,[l+R_{n},l+R^{'}_{n}]}$ such that $E_i\in B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon _0,\omega }$ and $|E-E_i|\le 2m_{L}(\tilde{I})e^{-(\eta _0-\delta _0)(2n+1)}$. Hence, $|E_i-E_j|\le 4m_{L}(\tilde{I})e^{-(\eta _0-\delta _0)(2n+1)}$. By the previous line and the fact that $E_j\notin B^{-}_{[l+R_{n},l+R^{'}_{n}],\varepsilon ,\omega }$, we see that $||G_{[l+R_{n},l+R^{'}_{n}],E_j,\omega }||\ge \frac{1}{4m_{L}(\tilde{I})}e^{(\eta _0-\delta _0)(2n+1)}$ so that for some $y_1,y_2$ with $l+R_{n}\le y_1 \le y_2 \le l+R^{'}_{n}$,

$$\begin{aligned} |G_{[l+R_{n},l+R^{'}_{n}],E_j,\omega }(y_1,y_2)|\ge \frac{1}{4m_{L}(\tilde{I})\sqrt{m(2n+1)}} e^{(\eta _0-\delta _0)(2n+1)}. \end{aligned}$$

Additionally, another application of Lemma 2 yields $|P_{[l+R_{n},l+R^{'}_{n}],E_j,\omega }|\ge e^{(\gamma (E_j)-\varepsilon )(R^{'}_{n}-R_{n}+1)}$.

Thus, by Eq. (5) we obtain

$$\begin{aligned}&|P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\nonumber \\&\quad \ge {} \frac{1}{4m_{L}(\tilde{I})\sqrt{m(2n+1)}}e^{(\gamma (E_j)-\varepsilon )(R^{'}_{n}-R_{n}+1)+(\eta _0-\delta _0)(2n+1)}. \end{aligned}$$

$\square$

Lemma 8

There is a $\tilde{\eta }>0$ and N such that $n>N$ implies $\mathbb {P}(J_{l,n,\varepsilon })\ge 1-e^{-\tilde{\eta }n}.$

Proof

Let $\mathcal {A}_{1} = [l+2n+1 - m,l+(2n+3)m]$, $\mathcal {A}_{2} = [l+3n+1 - m,l+(3n+3)m]$, and $\mathcal {A}_{3} = [l+n+1 - m,l+(n+3)m]$.

Since there are at most m single-step transfer matrices in word transfer matrix,

$$\begin{aligned} \left( \bigcap _{j_{1} \in \mathcal {A}_{1},j_{2} \in \mathcal {A}_{2}, j_2-j_1\ge n/2}F_{[j_{1},j_{2}],\varepsilon }\right) \cap \left( \bigcap _{j_{3} \in \mathcal {A}_{3},j_{4} \in \mathcal {A}_{1}, j_4-j_3\ge n/2}F_{[j_{3},j_{4}],\varepsilon }\right) \end{aligned}$$

is contained in

$$\begin{aligned} \left( \bigcup _{0 \le k \le 2m} (F_{[l+\tilde{Q}_{n}+k+1,l+Q^{'}_{n} ],\varepsilon } \cap F_{[l+Q_{n},l+\tilde{Q}_{n} + k - 1],\varepsilon })\right) . \end{aligned}$$

By Lemma 6, we also have

$$\begin{aligned}&\mathbb {P}\left[ \left( \bigcup _{j_{1} \in \mathcal {A}_{1},j_{2} \in \mathcal {A}_{2}, j_2-j_1\ge n/2}F^{c}_{[j_{1},j_{2}],\varepsilon }\right) \cup \left( \bigcup _{j_{3} \in \mathcal {A}_{3},j_{4} \in \mathcal {A}_{1}, j_4-j_3\ge n/2}F^{c}_{[j_{3},j_{4}],\varepsilon }\right) \right] \\&\le Cn^{2}e^{-\eta \frac{n}{2}} \end{aligned}$$

for sufficiently large n.

Note that the same reasoning provides a similar estimate on $\mathbb {P}[F_{[l+R_{n},l-1],\varepsilon } \cap F_{[l+1,l+R^{'}_{n}],\varepsilon }]$.

Choose N as in Lemma 7, and note that by Lemmas 2, 4, 6 and the estimate above, for $n>N$,

$$\begin{aligned} \mathbb {P}[J_{l,n,\varepsilon }]\ge & {} 1 - 2m_{L}(\tilde{I})e^{-\delta _{0}(2n+1)} - 2m^{4}(2n+3)^{3}e^{-\eta _{\varepsilon }(\frac{n}{K})}\\&- 2m^{2}(2n+3)^{2}e^{-\eta _{\varepsilon }(2n+1)} - 2Cn^{2}e^{-\eta _{\frac{\varepsilon }{4}}\frac{n}{2}}. \end{aligned}$$

We may choose $\tilde{\eta }$ sufficiently close to 0 and increase N such that for $n>N$ , we have

$$\begin{aligned}&2m_{L}(\tilde{I})e^{-\delta _{0}(2n+1)} + 2m^{4}(2n+3)^{3}e^{-\eta _{\varepsilon }(\frac{n}{K})}\\&\quad + 2m^{2}(2n+3)^{2}e^{-\eta _{\varepsilon }(2n+1)} + 2Cn^{2}e^{-\eta _{\frac{\varepsilon }{4}}\frac{n}{2}} \le e^{-\tilde{\eta }n}, \end{aligned}$$

and the result follows. $\square$

Lemma 9

There is N such that for $n>N$, any $\omega \in J_{l,n,\varepsilon }$, any $y_1$, $y_2$ with $l+R_{n} \le y_1 \le y_2 \le l+R^{'}_{n}$ and any $E_j = E_{j,[l+k + Q_{n},l+Q^{'}_{n}],\omega }$ (with $0 \le k \le 2m$),

$$\begin{aligned} |P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\le e^{(\gamma (E_j)+\varepsilon )(R^{'}_{n}-R_{n}+1)}. \end{aligned}$$

Proof

By choosing N so that Lemma 2 holds for $n>N$, we are led to consider three cases:

• (1) $l+R_{n} +\frac{n}{K} \le y_1 \le y_2 \le l+R^{'}_{n} - \frac{n}{K}$,

• (2) $l+R_{n} +\frac{n}{K} \le y_1 \le l+R_{n}$, while $l+R^{'}_{n} - \frac{n}{K} \le y_2 \le l+R^{'}_{n}$, and finally,

• (3) $l+R_{n} \le y_1 \le l+R_{n} +\frac{n}{K}$ and $l+R_{n} + \frac{n}{K} \le y_2 \le l+R^{'}_{n}$.

In the first case, Lemma 2 immediately yields:

$$\begin{aligned} |P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\le e^{(\gamma (E_j)+\varepsilon )(R^{'}_{n}-R_{n}+1)}. \end{aligned}$$

In the second case, we have $|P_{[y_2,l+R^{'}_{n}],E_j,\omega }| \le \tilde{M}^{\frac{n}{K}}$, while Lemma 2 gives

$$\begin{aligned} |P_{[l+R_{n},y_1],E_j,\omega }| \le e^{(\gamma (E_j)+\varepsilon )(\frac{n}{K})}. \end{aligned}$$

By our choice of K, $\tilde{M}^{\frac{1}{K}} \le e^{\frac{\nu }{2}} \le e^{(\gamma (E_j)+\varepsilon )}$, so we again obtain the desired result.

Finally, in the third case, $|P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\le \tilde{M}^{\frac{2n}{K}} \le e^{(\gamma (E_j)+\varepsilon )(R^{'}_{n}-R_{n}+1)}$ (again by our choice of K). $\square$

5 Spectral Localization

Theorem 7

There is N such that if $n>N$, $0 \le k \le 2m$, and $\omega \in J_{l,n,\varepsilon }$, with a generalized eigenfunction $\psi _{\omega ,E}$ satisfying either

• (1) $n_{r,\omega }=l$, or

• (2) $|\psi _{\omega }(l)| \ge \frac{1}{2}$,

then $l+\tilde{Q}_{n} + k$ is $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-regular.

Proof

Choose N so that Lemmas 7 and 9 hold and

$$\begin{aligned} \frac{1}{4m_{L}(\tilde{I})\sqrt{m(2n+1)}}e^{(\gamma (E_j) - \varepsilon )(R^{'}_{n}-R_{n}+1)+ (\eta _{0} - \delta _{0})(2n+1)} > e^{(\gamma (E_j)+\varepsilon )(R^{'}_{n}-R_{n}+1)} \end{aligned}$$

for $n>N$. This can be done since $\varepsilon < \frac{\eta _{0} - \delta _{0}}{3m}$.

For $n>N$, we obtain the conclusion of the theorem. For if $l+Q_{n}+k$ was not $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-regular, then by Lemma 7

$$\begin{aligned}&|P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\\&\quad \ge {} \frac{1}{4m_{L}(\tilde{I})\sqrt{m(2n+1)}}e^{(\gamma (E_j)-\varepsilon )(R^{'}_{n}-R_{n}+1)+(\eta _0-\delta _0)(2n+1)}. \end{aligned}$$

On the other hand, by Lemma 9, we have

$$\begin{aligned} |P_{[l+R_{n},y_1],E_j,\omega }P_{[y_2,l+R^{'}_{n}],E_j,\omega }|\le e^{(\gamma (E_j)+\varepsilon )(R^{'}_{n}-R_{n}+1)}. \end{aligned}$$

Our choice of N in the first line of the proof yields a contradiction and completes the argument. $\square$

We are now ready to give the proof of Theorem 1. Again, $R_n$, $R^{'}_{n}, Q_n, Q^{'}_{n},$ and $\tilde{Q}_n$ are the scales from Lemma 2.

Proof

(Theorem 1) Following the convention set in Remark 1, for each $k \in \mathbb {Z}$, put

$$\begin{aligned} \Omega _{k} = \{\omega \in \Omega : \text { there exists } N(\omega ) \text { such that } \omega \in J_{k,n,\varepsilon } \text { whenever } n > N(\omega )\}. \end{aligned}$$

By Lemma 8, $\mathbb {P}[\Omega _{k}]=1$ for all k. Thus, if $\tilde{\Omega } = \bigcap _{k} \Omega _{k}$, then $\mathbb {P}[\tilde{\Omega }]=1$.

Since the spectral measures are supported by the set of generalized eigenvalues (e.g., [16]), it suffices to show for all $\omega \in \tilde{\Omega }$, every generalized eigenfunction with generalized eigenvalue $E \in I$ is in fact an $\ell ^{2}(\mathbb {Z})$ eigenfunction which decays exponentially.

Fix an $\omega$ in $\tilde{\Omega }$ and let $\psi = \psi _{\omega ,E}$ be a generalized eigenfunction for $H_{\omega }$ with generalized eigenvalue E. By Eq. (4), and the bounds established on $R_n, R^{'}_n, Q_n, Q^{'}_n,$ ,and $\tilde{Q}_n$ from Lemma 2 (i.e., Eq. (17) in the proof of Lemma 2), it suffices to show that there is $N(\omega )$ such that for $n>N{(\omega )}$, if $0 \le k < 2m$, then $\tilde{Q}_{n}+k$ is $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-regular. We may assume $\psi (0) \ne 0$, and moreover, by rescaling $\psi$, $|\psi (0)| \ge \frac{1}{2}$. Choose N so that for $n > N$, the conclusions of Theorem 7 hold. Additionally, we may choose $N(\omega )$ such that for $n> N(\omega )$, $\omega \in J_{0,n,\varepsilon }$. For $n > \max \{N,N(\omega )\}$, the hypotheses of Theorem 7 are met, and hence $\tilde{Q}_{n}+k$ is $(\gamma (E)-8\varepsilon _0,\tilde{Q}_{n}+k - Q_{n},Q^{'}_{n}-\tilde{Q}_{n}-k,E, \omega )$-regular. $\square$

6 Exponential Dynamical Localization

The strategy used in this section follows [13] with the appropriate modifications needed to deal with the fact that single-step transfer matrices were not used in the large deviation estimates. In particular, the randomness in the conclusion of Theorem 7 will need to be accounted for.

The following lemma was shown in [18], and we state a version below suitable for obtaining EDL on the interval I.

Let $u_{k,\omega }$ denote an orthonormal basis of eigenvectors for $Ran(P_{I}(H_\omega ))$, the range of the spectral projection of $H_{\omega }$ onto the interval I.

Lemma 10

[18] Suppose there is $\tilde{C}>0$ and $\tilde{\gamma } > 0$ such that for any s, $l \in \mathbb {Z}$,

$$\begin{aligned} \mathbb {E}\left[ \sum _{n_{r,\omega }=l}|u_{k,\omega }(s)|^{2}\right] \le \tilde{C}e^{-\tilde{\gamma }|s-l|}. \end{aligned}$$

Then, there are $C>0$ and $\gamma >0$ such that for any $p,q \in \mathbb {Z}$,

$$\begin{aligned} \mathbb {E}[\sup _{t \in \mathbb {R}}|\langle \delta _{p},P_{I}(H_\omega )e^{itH_{\omega }}\delta _{q} \rangle |] \le C(|p-q| + 1)e^{-\gamma |p-q|}. \end{aligned}$$

By Lemma 10, Theorem 2 follows from Theorem 8.

Remark

For almost every $\omega$, we have the existence of an orthonormal basis of eigenfunctions for $H_{\omega }$ by Theorem 1. We denote these functions and the corresponding eigenvalues by $u_{k,\omega }$ and $E_{k,\omega }$, respectively.

Theorem 8

There is $\tilde{C}>0$ and $\tilde{\gamma } > 0$ such that for any s,$l \in \mathbb {Z}$,

$$\begin{aligned} \mathbb {E}\left[ \sum _{n_{r,\omega }=l}|u_{k,\omega }(s)|^{2}\right] \le \tilde{C}e^{-\tilde{\gamma }|s-l|}. \end{aligned}$$

Proof

We choose N so that Theorem 7 and Lemma 8 hold and for $0 \le k \le 2m$, set $\zeta _{j,k}=\min \{\tilde{Q}_j+k-Q_j,Q^{'}_j-\tilde{Q}_j-k\}$.

Choose $0< c < \frac{1}{8}(\nu - 8\varepsilon _{0})$ and increase N so that $(\langle L \rangle + \varepsilon )j > 2m$ for $j > N$. Now since $\frac{\langle L \rangle - \varepsilon }{\langle L \rangle + \varepsilon } > \frac{1}{2}$ and $\frac{j}{3j+1} > \frac{1}{4}$ for $j > 1$, Eq. (17) yields

$$\begin{aligned} \begin{aligned} c(\tilde{Q}_{j} + k)&\le c[(\langle L \rangle + \varepsilon )(2j+1)+2m] \\&<c(\langle L \rangle + \varepsilon )(3j+1) \\&< (\nu -8\varepsilon _{0})(\langle L \rangle - \varepsilon )j \\&\le 2\zeta _{j,k}. \end{aligned} \end{aligned}$$

(21)

We now choose $0< \tilde{\eta }_{1} < \frac{\tilde{\eta }}{(\langle L \rangle + \varepsilon )3}$ and again increase N such that if $j > N$,

$$\begin{aligned} \left( \frac{\tilde{\eta }}{(\langle L \rangle + \varepsilon )3} - \tilde{\eta _{1}}\right) j > \ln (j). \end{aligned}$$

(22)

Now consider s and l in $\mathbb {Z}$.

There are two cases to consider:

• (1) $s-l > 3m(N+1)$,

• (2) $s-l \le 3m(N+1)$.

Now suppose $n_{r,\omega } = l$, $l+\tilde{Q}_{j} \le s < l+\tilde{Q}_{j+1}$, and $\omega \in J_{l,j,\varepsilon }$.

In the first case, using Eq. (17), we have

$$\begin{aligned} \begin{aligned} |s-l|&\le (\langle L \rangle + \varepsilon )(2j+3) \\&< (\langle L \rangle + \varepsilon )3j. \\ \end{aligned} \end{aligned}$$

Thus, $j > \frac{|s-l|}{3(\langle L \rangle + \varepsilon )}$ and in particular, $j > N$.

Since $j > N$, using Theorem 7 and Eq. (4),

$$\begin{aligned} \begin{aligned} |u_{r,\omega }(s)|&\le 2|u_{r,\omega }(l)|e^{-(\gamma (E_{r,\omega })-8\varepsilon _{0}) \zeta _{j,k}} \\&\le 2|u_{r,\omega }(l)|e^{-(\nu -8\varepsilon _{0}) \zeta _{j,k}}. \end{aligned} \end{aligned}$$

By orthonormality and Hölder’s inequality,

$$\begin{aligned} \begin{aligned} \sum _{n_{r,\omega } = l} |u_{r,\omega }(s)|^{2}&\le 4\sum _{n_{r,\omega } = l} |u_{r,\omega }(l)|^{2}e^{-(\nu -8\varepsilon _{0}) 2\zeta _{j,k}}\\&\le 4\sum _{n_{r,\omega } = l} e^{-(\nu -8\varepsilon _{0}) 2\zeta _{j,k}}. \end{aligned} \end{aligned}$$

We need to replace the randomness in the exponent above with an estimate that depends only on the point s.

Writing $s = l + \tilde{Q}_{j} + k$ with $0 \le k \le 2m$, by Eq. (21),

$$\begin{aligned} \sum _{n_{r,\omega } = l} e^{-(\nu -8\varepsilon _{0}) 2\zeta _{j,k}}&\le \sum _{n_{r,\omega } = l} e^{-c |s-l|}. \end{aligned}$$

(23)

Now let $\mathcal {A} = \{j \in \mathbb {Z} : l+\tilde{Q}_{j} \le s < l+\tilde{Q}_{j+1} \text {and} \omega \in J_{l,j,\varepsilon }\}$.

By again using Eq. (17):

$$\begin{aligned} \begin{aligned} (\langle L \rangle - \varepsilon )(2j+1)&\le (\tilde{Q}_{j} + k) \\&\le (\langle L \rangle + \varepsilon )(2j+3) \\&\le (\langle L \rangle + \varepsilon )3j. \end{aligned} \end{aligned}$$

Thus, if $j \in \mathcal {A}$, then

$$\begin{aligned} \frac{|s-l|}{3(\langle L \rangle + \varepsilon )}< j < \frac{|s-l|}{2(\langle L \rangle - \varepsilon )}. \end{aligned}$$

(24)

From Eq. (24), we obtain $|\mathcal {A}| < |s-l|$.

Moreover, if $J = \bigcup _{ j \in \mathcal {A}} J_{l,j,\varepsilon }$, using the estimate provided by Lemma 8 on $\mathbb {P}[J_{l,j,\varepsilon }]$, Eqs. (22) and (24):

$$\begin{aligned} \begin{aligned} \mathbb {P}[J^{c}]&\le |\mathcal {A}|e^{-\frac{\tilde{\eta }|s-l|}{(\langle L \rangle + \varepsilon )3}} \\&\le |s-l|e^{-\frac{\tilde{\eta }|s-l|}{(\langle L \rangle + \varepsilon )3}} \\&\le e^{-\tilde{\eta _{1}}|s-l|}. \end{aligned} \end{aligned}$$

(25)

Thus, by Eqs. (23), (25), and orthonormality, we have

$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ \sum _{n_{r,\omega }=l} |u_{k,\omega }|^{2}\right]&= \mathbb {E}\left[ \sum _{n_{r,\omega }=l} |u_{r,\omega }|^{2} \chi _{J} + \sum _{n_{r,\omega }=l} |u_{r,\omega }|^{2} \chi _{J^{c}}\right] \\&\le C(e^{-c|s-l|} + e^{-\tilde{\eta _{1}}|s-l|}). \end{aligned} \end{aligned}$$

(26)

In the second case (when $s-l \le 3m(N+1)$), again by orthonormality,

$$\begin{aligned} \displaystyle \mathbb {E}\left[ \sum _{n_{r,\omega }=l} |u_{r,\omega }(s)|^{2}\right] \le 1. \end{aligned}$$

(27)

By letting $\tilde{\gamma } = \min \{c,\tilde{\eta _{1}}\}$ and noting that in the second case $e^{-\tilde{\gamma }|s-l|} \ge e^{-\tilde{\gamma }(3m(N+1))}$, Eqs. (26) and (27) imply that for a sufficiently large $\tilde{C}> 0$:

$$\begin{aligned} \mathbb {E}\left[ \sum _{n_{r,\omega }=l}|u_{r,\omega }(s)|^{2}\right] \le \tilde{C}e^{-\tilde{\gamma }|s-l|}. \end{aligned}$$

$\square$