One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

Abstract

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type n^−α for α > 0. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region $\alpha >\frac 12$; a transition from pure point to singular continuous spectrum in the critical region $\alpha =\frac 12$; and pure point spectrum in the sub-critical region $\alpha <\frac 12$. From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.

Appendix A: Some Technical Estimates

1.1 A.1. Unimodular Matrices

The following lemmas correspond to [48, Lemma 2.2 and 8.7]. The first one allows us to establish the upper bound in Lemma 3.2. The proof of Proposition 6.2 is given after the second one. At the end of the section, we state [53, Theorem 8.3] which is used to prove pure point spectrum in the sub-critical regime.

Lemma A.1

Let A be an unimodular matrix and let $\hat \theta = ({\cos \limits } \theta , {\sin \limits } \theta )$. Then, for all pair of angles $|\theta _{1}-\theta _{2}|\leqslant \frac {\pi }{2}$,

$$ \begin{array}{@{}rcl@{}} \| A \| \leqslant \sin \left( \tfrac{|\theta_{1}-\theta_{2}|}{2}\right)^{-1} \max \{ \| A \hat \theta_{1}\|, \| A\hat \theta_{2}\| \}. \end{array} $$

Proof

See [48, Lemma 2.2]. □

The following lemma is used to find eigenfunctions with the proper decay and is the key to Proposition 6.2.

Lemma A.2

For a unimodular matrix with ∥A∥ > 1, define 𝜗 = 𝜗(A) as the unique angle $\vartheta \in (-\frac {\pi }{2},\frac {\pi }{2}]$ such that $\| A \hat \vartheta \| = \| A \|^{-1}$. We also define $r(A)=\left \lVert A\left (\begin {array}{l}1\\0 \end {array}\right ) \right \rVert .\left \lVert A\left (\begin {array}{l}0\\1 \end {array}\right ) \right \rVert ^{-1}$.

Let (A_n)_n be a sequence of unimodular matrices with ∥A_n∥ > 1 and write 𝜗_n = 𝜗(A_n) and r_n = r(A_n). Assume that

• (i) $\underset {n\to \infty }{\lim } \| A_{n}\| = \infty$,

• (ii) $\underset {n\to \infty }{\lim } \frac {\| A_{n+1}A_{n}^{-1}\|}{\| A_{n} \| \| A_{n+1}\|}=0$.

Then,

• (1) (𝜗_n)_n has a limit $\vartheta _{\infty }\in (-\pi /2,\pi /2)$ if and only if (r_n)_n has a limit $r_{\infty }\in [0,\infty )$. If 𝜗_n →±π/2, then $r_{n}\to \infty$ but, if $r_{n}\to \infty$, we can only conclude that |𝜗_n|→ π/2.

• (2) Suppose (𝜗_n)_n has a limit $\vartheta _{\infty }\neq 0, \frac {\pi }{2}$. Then,

  $$ \underset{n\to\infty}{\lim} \frac{\log \| A_{n} \hat\vartheta_{\infty}\|}{\log \| A_{n} \|} = -1 \quad \text{if and only if} \quad \limsup_{n} \frac{\log |r_{n} - r_{\infty}|}{\log \| A_{n} \|} \leqslant -2. $$

  (A.1)

Proof

See [48, Lemma 8.7]. □

We apply this with A_n = T_ω,n in order to prove Proposition 6.2. For positive sequences (b_n)_n and (c_n)_n, we write $b_{n} \simeq c_{n}$ if $\underset {n\to \infty }{\lim } \frac {b_{n}}{c_{n}}=1$ and denote $b_{n} \lesssim c_{n}$ if there exists a constant K > 0 such that $b_{n} \leqslant K c_{n}$ for n large enough, and b_n ≍ c_n if $b_{n} \lesssim c_{n} \lesssim b_{n}$.

Proof of Proposition 6.2

Define

$$ {\Phi}^{(1)}_{n} = \left( \begin{array}{l} \phi^{(1)}_{+,n} \\ \phi^{(1)}_{-,n} \end{array}\right) = \textbf{T}_{\omega,n-1} \left( \begin{array}{l} 1 \\ 0 \end{array}\right), \quad \quad {\Phi}^{(2)}_{n} = \left( \begin{array}{l} \phi^{(2)}_{+,n} \\ \phi^{(2)}_{-,n} \end{array}\right) = \textbf{T}_{\omega,n-1} \left( \begin{array}{l} 0 \\ 1 \end{array}\right), $$

and let $R^{(i)}_{n}, \theta ^{(i)}_{n}, n\geq 1$, i = 1, 2 be the corresponding Prüfer radii and phases. We let $r_{n}=\frac {R^{(1)}_{n}}{R^{(2)}_{n}}$ and 𝜗_n be as in Lemma A.2. Recall the relation ${\Phi }_{n}={\mathcal {P}}_{n} {\Psi }_{n}$. In particular,

$$ {\Phi}_{n}^{(i)} = \left( \begin{array}{l} \phi^{(i)}_{+,n} \\ \phi^{(i)}_{-,n} \end{array}\right) = (-1)^{n-1} R_{n} \left( \begin{array}{l} -\sqrt{p_{2}}\cos(\bar{\theta}_{n}^{(i)}) \\ \sqrt{-p_{1}}\cos(\bar{\theta}_{n}^{(i)}+k) \end{array}\right). $$

Thus it follows from some elementary trigonometry that

$$ \phi^{(1)}_{+,n}\phi^{(2)}_{-,n}-\phi^{(1)}_{-,n}\phi^{(2)}_{+,n} = R^{(1)}_{n} R^{(2)}_{n} \sin(2k) \sin(\theta^{(1)}_{n}-\theta^{(2)}_{n}), $$

where $\bar {\theta }^{(i)}_{n} = \theta ^{(i)}_{n}-(2n-1)k$. On the other hand,

$$ \phi^{(1)}_{+,n}\phi^{(2)}_{-,n}-\phi^{(1)}_{-,n}\phi^{(2)}_{+,n} = \det\left( \textbf{T}_{\omega,n} \left( \begin{array}{ll} 1 0 \\ 0 1 \end{array}\right) \right) =1. $$

This, together with the convergence

$$ \lim_{n\to\infty} \frac{R^{(i)}_{n}}{\log n} = \beta,\quad i=1,2, $$

gives

$$ \lim_{n\to\infty}\frac{\log |\sin(\theta^{(2)}_{n}-\theta^{(1)}_{n})|}{\log n} =\lim_{n\to\infty}\frac{\log |\sin(\bar\theta^{(2)}_{n}-\bar\theta^{(1)}_{n})|}{\log n} = -2\beta. $$

(A.2)

Remember the decomposition (4.2) that we summarize as

$$ \begin{array}{@{}rcl@{}} (R^{(i)}_{n+1})^{2} &=& \left( 1 -\frac{p_{2}}{\sin(2k)}\sin(2\bar\theta_{n}^{(i)})V_{\omega,1}(n)\right.\\ && \left.+\frac{p_{1}}{\sin(2k)}\sin(2(\bar\theta^{(i)}_{n}-k))V_{\omega,2}(n+1) +E_{j}^{(i)} \right) (R^{(i)}_{n})^{2}. \end{array} $$

We have to estimate the difference of the expansions for $\log R^{(1)}_{n}$ and $\log R^{(2)}_{n}$. By (A.2), one has $|\sin \limits (\bar \theta ^{(2)}_{n}-\bar \theta ^{(1)}_{n})| \lesssim n^{-\beta +\epsilon }$, for any 𝜖 > 0. Hence, there exist random sequences $(m_{n})_{n}\subset \mathbb {N}^{*}$ and $({\Delta }_{n})_{n}\subset \mathbb {R}$ such that $\bar \theta ^{(1)}_{n}-\bar \theta ^{(2)}_{n}=m_{n}\pi + {\Delta }_{n}$ and $|{\Delta }_{n}|\lesssim n^{-\beta +\epsilon }$. Therefore,

$$ \sin(2\bar\theta^{(2)}_{n}) = \sin(2\bar\theta^{(1)}_{n}+2{\Delta}_{n}) \simeq \sin(2\bar\theta^{(1)}_{n})+2\cos(2\bar\theta^{(1)}_{n}){\Delta}_{n}. $$

This shows that

$$ \left|V_{\omega,1}(j) \left( \sin(2\bar{\theta}^{(1)}_{j}) - \sin(2\bar{\theta}^{(2)}_{j})\right)\right| \lesssim j^{-\frac{1}{2}-2\beta + \epsilon}. $$

By means of similar arguments, one can show that

$$ \left|V_{\omega,2}(j+1) \left( \sin(2(\bar{\theta}^{(1)}_{j}-k)) - \sin(2(\bar{\theta}^{(2)}_{j}-k))\right)\right| \lesssim j^{-\frac{1}{2}-2\beta + \epsilon}. $$

and

$$ |E_{j}^{(1)}-E_{j}^{(2)}| \lesssim j^{-1-2\beta + \epsilon}. $$

Hence,

$$ \begin{array}{@{}rcl@{}} \log r_{n} &=& -\frac{p_{2}}{\sin(2k)}\sum\limits^{n}_{j=1} \frac{V_{\omega,1}(j)}{\sin k} \left( \sin(2\bar{\theta}^{(1)}_{j}) - \sin(2\bar{\theta}^{(2)}_{j}) \right) \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} && + \frac{p_{1}}{\sin(2k)}\sum\limits^{n}_{j=1} \frac{V_{\omega,2}(j+1)}{\sin k} \left( \sin(2(\bar{\theta}^{(1)}_{j}-k)) - \sin(2(\bar{\theta}^{(2)}_{j}-k)) \right) + \sum\limits^{n}_{j=1} A_{j} \end{array} $$

(A.4)

where the first two sums are convergent martingales by Lemma A.4 with $\gamma =\frac 12 + 2\beta -\epsilon$ and the last one is absolutely convergent as A_j = O(j^{− 1 − 2β+𝜖}). This shows that $r_{n}\to r_{\infty }\in (0,\infty )$ almost surely which implies that 𝜗_n has a limit $\vartheta _{\infty }\neq 0, \frac {\pi }{2}$ by the first part of Lemma A.2.

The equivalence (A.1) in our context corresponds to

$$ \underset{n\to\infty}{\lim}\frac{\log R_{n}(\vartheta_{\infty})}{\log n}=-\beta \quad \text{if and only if} \quad \underset{n}{\limsup} \frac{\log |r_{n}-r_{\infty}|}{\log n} \leqslant -2\beta. $$

Let us denote by $\log r_{n} = M_{n} + S_{n}$ the decomposition (A.4) and $\log r_{\infty }=M_{\infty }+S_{\infty }$, where M_n is the martingale part and $M_{\infty }$ and $S_{\infty }$ are the almost sure limits of M_n and S_n respectively. Then,

$$ \begin{array}{@{}rcl@{}} |r_{\infty}-r_{n}| &=& \mathrm{e}^{M_{\infty}+S_{\infty}}\left| 1-\mathrm{e}^{M_{n}-M_{\infty}+S_{n}-S_{\infty}}\right| \simeq \mathrm{e}^{M_{\infty}+S_{\infty}}\left| M_{n}-M_{\infty}+S_{n}-S_{\infty}\right|\\ &\lesssim& \mathrm{e}^{M_{\infty}+S_{\infty}} n^{-2\beta+2\epsilon}, \end{array} $$

by the last statement of Lemma A.4 with $\gamma = \frac 12 + 2\beta - \epsilon$ and A_j = O(j^{− 1 − 2β+𝜖}). This finishes the proof of Proposition 6.2. □

We state [53, Theorem 8.3] which allowed us to prove pure point spectrum in the sub-critical regime:

Theorem A.3

Let (A_n)_n≥ 1 be 2 × 2 real unimodular matrices and let A_n = A_n⋯A₁ such that

$$ \underset{n\geq 1}{\sum} \frac{\| A_{n+1}\|}{\| \textbf{A}_{n} \|}<\infty. $$

Suppose there exists a monotone increasing function $g:\mathbb {N}^{*} \to (0,\infty )$ such that

$$ \underset{n\to\infty}{\lim} \frac{\log \| A_{n} \|}{g(n)}=0 \quad \text{and} \quad \underset{n\to\infty}{\lim} \frac{\log \| \textbf{A}_{n} \|}{g(n)}=1, $$

and such that

$$ \underset{n\geq 1}{\sum} \mathrm{e}^{-\epsilon g(n)}<\infty, $$

for all 𝜖 > 0. Then, there exists an angle 𝜗₀ such that

$$ \underset{n\to\infty}{\lim} \frac{\log \| \textbf{A}_{n} \widehat{\vartheta}_{0} \|}{g(n)}=-1. $$

1.2 A.2. A Martingale Inequality

The following corresponds to [48, Lemma 8.4]. We formulate it in full generality but provide a short proof under the assumption that V_ω,i(n) = λn^−αω_n,i for uniformly bounded random variables ω_n,i.

Lemma A.4

Let (Z_j)_j be i.i.d. random variables with $\mathbb {E}[Z_{n}]=0$ and $\mathbb {E}[|Z_{n}|^{2}]\leqslant n^{-2\gamma }$ for some γ > 0. Let $\mathcal {G}_{n}=\sigma (Z_{1},\cdots , Z_{n})$ and let $Y_{n} \in \mathcal {G}_{n-1}$ for n ≥ 1 such that $|Y_{n}|\leqslant 1$. Define

$$ M_{n} = \sum\limits^{n}_{j=1} Y_{j}Z_{j} \quad \text{and} \quad s_{n} = \sum\limits^{n}_{j=1} \frac{1}{j^{2\gamma}}. $$

Then, (M_n)_n is a $\mathcal {G}_{n}$-martingale and

• (i) For $\gamma \leqslant \frac 12$ and all ε > 0,

  $$ \underset{n\to\infty}{\lim} s_{n}^{-\frac{1+\varepsilon}{2}}M_{n}= 0,\quad \quad \mathbb{P}-a.s. $$

• (ii) For $\gamma >\frac 12$, (M_n)_n converges $\mathbb {P}$-almost surely to a finite (random) limit $M_{\infty }$ and, for all $\kappa <\gamma - \frac 12$, we have

  $$ \underset{n\to\infty}{\lim} n^{\kappa} \left( M_{\infty}-M_{n}\right) = 0,\quad \mathbb{P}-a.s. $$

Proof

The reader can consult the book [34] for the general properties of martingales used below. The sequence (M_n)_n is a martingale thanks to our hypothesis on (Y_n)_n and (Z_n)_n: indeed, since $M_{n}, Y_{n+1} \in \mathcal {G}_{n}$, Y_n+ 1 is bounded and Z_n+ 1 is independent of $\mathcal {G}_{n}$ and centered, we have, $\mathbb {P}$-almost surely,

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[M_{n+1} | \mathcal{G}_{n}] &=& \mathbb{E}\left[Y_{n+1}Z_{n+1} + M_{n} \Big{|}\mathcal{G}_{n}\right]\\ &=& Y_{n+1}\mathbb{E}[Z_{n+1}] + M_{n} = M_{n}. \end{array} $$

As stated above, we assume Z_n = n^−γX_n with $|X_{n}|\leqslant 1$ and $\mathbb {E}[X_{n}]=0$ to simplify the argument. Let $\gamma \leqslant \frac 12$. We use Azuma’s inequality [5]: let (M_n)_n be a martingale such that $|M_{n}-M_{n-1}|\leqslant c_{n}$ for all n ≥ 1. Then,

$$ \mathbb{P}\left[ |M_{n}-M_{0}| \geq t\right] \leqslant 2 \exp\left\{ - \frac{t^{2}}{2 {\sum}^{n}_{j=1} {c_{j}^{2}}}\right\}. $$

In our case, M₀ = 0, c_j = 2j^−γ, and taking $t=s_{n}^{\frac {1+\varepsilon ^{\prime }}{2}}$ for $0<\varepsilon ^{\prime }<\varepsilon$, we obtain

$$ \mathbb{P}\left[ |M_{n}| \geq s_{n}^{\frac{1+\varepsilon^{\prime}}{2}}\right] \leqslant 2\ \mathrm{e}^{- C n^{\varepsilon^{\prime}}}, $$

for some C > 0. The claim (i) then follows from Borel-Cantelli’s lemma. Now, let $\gamma >\frac 12$. Noticing that, for i < l,

$$ \mathbb{E}[X_{i}Y_{i} X_{l} Y_{l}] = \mathbb{E}[X_{l}]\mathbb{E}[X_{i}Y_{i} Y_{l}] =0, $$

we have

$$ \sup_{n}{\mathbb{E}[M_{n}^{2}}] = \sup_{n} \sum\limits^{n}_{j=1} \frac{{\mathbb{E}[X_{j}^{2}} {Y_{j}^{2}}]}{j^{2\gamma}} \leqslant \underset{j\geq 1}{\sum} \frac{1}{j^{2\gamma}}<\infty. $$

Hence, (M_n)_n is bounded in L² and, as a consequence, converges almost surely, i.e., there exists a random variable $M_{\infty }$ such that $\lim _{n\to \infty }M_{n}=M_{\infty }$, $\mathbb {P}$-a.s.. Finally, applying Azuma’s inequality to the martingale (M_n+i − M_n)_i≥ 0, we obtain

$$ \mathbb{P}\left[ n^{\kappa}\left| M_{n+i}-M_{n}\right| \geq 1\right]\leqslant 2 \exp\left\{ - C n^{2(\gamma-\frac12 -\kappa)}\right\}, $$

for all i ≥ 0. Choosing $\kappa <\gamma -\frac 12$, the last claim follows from Fatou’s lemma, the convergence of (M_n)_n and Borel-Cantelli. □

Remark 1

The result above is proved in [48, Lemma 8.4] under the second moment assumption replacing our use of Azuma’s inequality by Doob’s inequality. For a short proof assuming bounded exponential moments, see [21, Lemma A.1].

1.3 A.3. Control of the Phases

The next lemma provides the control of the Prüfer phases needed to complete the proof of Proposition 4.1. The strategy is taken from [48]. Recalling the definitions of Q_n,1 and Q_n,2 from (4.5),

Lemma A.5

Assume (A3a) and (A4). Let $0<\alpha \leqslant \frac 12$. For each fixed energy corresponding to a value of $k \in (-\pi , -\tfrac {\pi }{2})$ different from $-\frac {5\pi }{8},-\frac {3\pi }{4}$ and $-\frac {7\pi }{8}$,

$$ \underset{n\to\infty}{\lim} \frac{Q_{n,i}}{{\sum}^{n}_{j=1} j^{-2\alpha}} = 0, $$

for i = 1, 2. Moreover, for each compact energy interval I ⊂Σ̈, the convergence is uniform over all initial values 𝜃₀ ∈ [0, 2π) and E ∈ I corresponding to values of k different from $-\frac {5\pi }{8},-\frac {3\pi }{4}$ and $-\frac {7\pi }{8}$.

Proof

We will show that

$$ \limsup_{n\to\infty} \frac{{\sum}^{n}_{j=1}\mathbb{E}[ V_{\omega,j}^{2}]\cos 4 \bar \theta_{j}}{{\sum}^{n}_{j=1} j^{-2\alpha}} = 0, \qquad \mathbb{P}-\text{a.s.,} $$

the other terms being handled similarly. Note that the prefactor accompanying this term in the definition of Q_n,1 is uniformly bounded over compact energy intervals. The computations below are uniform in the initial condition 𝜃₀ and only assume $k\notin \tfrac {\pi }{8}\mathbb {Z}$.

We begin with a simple observation: from (3.17), for any compact interval I ⊂Σ̈, there exists a constant $C=C(I)\in (0,\infty )$ such that

$$ \begin{array}{@{}rcl@{}} && \left| \mathrm{e}^{i(\theta_{n+1}-\theta_{n})}-1\right| = \left| \frac{\zeta_{n+1}}{\zeta_{n}}-1\right| \\ && \leqslant C \left( |V_{\omega,1}(n)|+|V_{\omega,2}(n+1)|+|V_{\omega,1}(n)V_{\omega,2}(n+1)| \right) \leqslant 1, \end{array} $$

for n ≥ n^∗(ω) for some $n^{*}(\omega )=n^{*}(\omega ,I)<\infty$ thanks to (A4). Hence, for n ≥ n^∗(ω), $|\theta _{n+1}-\theta _{n}|< \frac {\pi }{2}$ and, recalling (A4) once more, we have

$$ |\theta_{n+1}-\theta_{n}| \leqslant \frac{\pi}{2} |\sin(\theta_{n+1}-\theta_{n})| \leqslant \frac{\pi}{2} \left|\mathrm{e}^{i(\theta_{n+1}-\theta_{n})}-1\right| \leqslant c_{0}(\omega)\ n^{-\frac{2\alpha}{3}}, $$

for some $c_{0}(\omega )=c_{0}(\omega ,I)\in (0,\infty )$. This can be written in the equivalent form

$$ |\bar{\theta}_{n+1}-\bar{\theta}_{n}+2k|\leqslant c_{0}(\omega) \ n^{-\frac{2\alpha}{3}}, $$

(A.5)

which will be more suitable for our purposes. By possibly increasing the value of c₀(ω), we can assume that (A.5) holds for all n ≥ n^∗(ω) with $c_{0}(\omega )<\infty$$\mathbb {P}$-amost surely. For p ≥ 1, define

$$ \mathcal{E}_{p} = \left\{ \omega: c_{0}(\omega) \leqslant p\right\}, $$

and observe that ${\Omega }=\underset {p\geq 1}{\bigcup } \mathcal {E}_{p}$. The key proof is [48, Lemma 8.5] which states the following: suppose that $y\in \mathbb {R}$ is not in $\pi \mathbb {Z}$. Then, there exists a sequence of integers $q_{l} \to \infty$ such that

$$ \left|\sum\limits^{q_{l}}_{j=1} \cos \theta_{j} \right| \leqslant 1 + \sum\limits^{q_{l}}_{j=1} \left| \theta_{j}-\theta_{0} - jy \right|, $$

(A.6)

for all $(\theta _{j})_{j\geq 0}\subset \mathbb {R}$. We take y = − 8k. Let p ≥ 1 and $\omega \in \mathcal {E}_{p}$. Let n be large enough so that it can be written as n = n₀ + Kq_l with $n_{0}\ge {q_{l}^{2}}$ and $4c_{0}(\omega ) n_{0}^{-\alpha }\le q_{l}^{-2}$. Then,

$$ \begin{array}{@{}rcl@{}} \left|\sum\limits_{j=n_{0}+1}^{n} j^{-2\alpha}\cos(4\bar\theta_{j})\right| &=& \left|\sum\limits_{m=0}^{K} \sum\limits_{r=1}^{q_{l}}(n_{0}+mq_{l}+r)^{-2\alpha}\cos(4\bar\theta(n_{0}+mq_{l}+r))\right| \\ &\leqslant& \sum\limits_{m=0}^{K}(n_{0}+mq_{l})^{-2\alpha} \left|\sum\limits_{r=1}^{q_{l}}\cos(4\bar\theta(n_{0}+mq_{l}+r))\right| \\ && + \sum\limits_{m=0}^{K} \sum\limits_{r=1}^{q_{l}} \left|(n_{0}+mq_{l}+r)^{-2\alpha}-(n_{0}+mq_{l})^{-2\alpha}\right| \\ &=:& A + B. \end{array} $$

We first estimae the term A using (A.6) to get

$$ A \leqslant \sum\limits_{m=0}^{K}(n_{0}+mq_{l})^{-2\alpha}\left( 1+4{\sum}_{r=1}^{q_{l}}|\bar\theta(n_{0}+mq_{l}+r)-\bar\theta(n_{0}+mq_{l})+2kr|\right). $$

Now, by (A.5) it follows that

$$ \begin{array}{@{}rcl@{}} && 4\sum\limits_{r=1}^{q_{l}}|\bar\theta(n_{0}+mq_{l}+r)-\bar\theta(n_{0}+mq_{l})+2kr| \leqslant c_{0} \sum\limits_{r=1}^{q_{l}}\sum\limits_{s=1}^{r}(n_{0}+mq_{l}+r)^{-\frac{2\alpha}{3}} \\ && \leqslant c_{0}(\omega)(n_{0}+mq_{l})^{-\frac{2\alpha}{3}}\sum\limits_{r=1}^{q_{l}} r \leqslant c_{0}(\omega){q_{l}^{2}} n_{0}^{-\frac{2\alpha}{3}} \leqslant 1. \end{array} $$

Thus,

$$ A \leqslant 2 \sum\limits_{m=0}^{K}(n_{0}+mq_{l})^{-2\alpha} \leqslant 2 q_{l}^{-2\alpha} \sum\limits_{m=0}^{K}(n_{0}q_{l}^{-1}+m)^{-2\alpha} \leqslant c_{1} q_{l}^{-2\alpha} \sum\limits_{j=1}^{K}j^{-2\alpha}, $$

for some finite c₁ > 0. To estimate B, we use that

$$ \left| (n_{0}+mq_{l}+r)^{-2\alpha}-(n_{0}+mq_{l})^{-2\alpha}\right| \leqslant c_{2} (n_{0}+mq_{l})^{-2\alpha-1} r, $$

for some finite c₂ > 0 which allows us to write

$$ \begin{array}{@{}rcl@{}} B &\leqslant& c_{2} \sum\limits_{m=0}^{K} \sum\limits_{r=1}^{q_{l}} (n_{0}+mq_{l})^{-2\alpha-1} r \leqslant c_{2} {q_{l}^{2}} n_{0}^{-1} \sum\limits_{m=0}^{K} (1+n_{0}^{-1}mq_{l})^{-1}(n_{0}+mq_{l})^{-2\alpha} \\ &\leqslant& c_{2} \sum\limits_{m=0}^{K} (n_{0}+mq_{l})^{-2\alpha}, \end{array} $$

where we used ${q_{l}^{2}} n_{0}^{-1}\leqslant 1$. This last sum can be estimated as above. Combining, we obtain

$$ \left|\sum\limits^{n}_{j=1} j^{-2\alpha} \cos 4 \bar{\theta}_{j} \right| \leqslant \sum\limits^{n_{0}}_{j=1}j^{-2\alpha} + c_{3} q_{l}^{-2\alpha} \sum\limits_{j=1}^{K}j^{-2\alpha}, $$

for some finite c₃ > 0 and all $\omega \in \mathcal {E}_{p}$. Hence,

$$ \underset{n\to\infty}{\limsup}\frac{\left|{\sum}^{n}_{j=1} j^{-2\alpha} \cos 4 \bar{\theta}_{j} \right|}{{\sum}^{n}_{j=1}j^{-2\alpha}} \leqslant c_{3} q_{l}^{-2\alpha}, $$

for all $\omega \in \mathcal {E}_{p}$. We can then let $l\to \infty$. As the events $\mathcal {E}_{p}$ exhaust Ω, this finishes the proof. □

The next lemma provides the control of the phases needed to complete the proof of Proposition 4.2.

Lemma A.6

Let $0<\alpha \leqslant \frac 12$. Assume (A1)-(A3a) and (A5). For each fixed energy corresponding to a value of $k \in (-\pi , -\tfrac {\pi }{2})$ different from $-\frac {5\pi }{8},-\frac {3\pi }{4}$ and $-\frac {7\pi }{8}$,

$$ \underset{n\to\infty}{\lim} \frac{\mathbb{E}[Q_{n,i}]}{{\sum}^{n}_{j=1} j^{-2\alpha}} = 0, $$

for i = 1, 2. Moreover, for each compact energy interval I ⊂Σ̈, the convergence is uniform over all initial values 𝜃₀ ∈ [0, 2π) and E ∈ I corresponding to values of k different from $-\frac {5\pi }{8},-\frac {3\pi }{4}$ and $-\frac {7\pi }{8}$.

Proof

By Borel-Cantelli ,

$$ |V_{\omega,i}(n)| \leqslant n^{-\frac{2\alpha}{3}-\frac{\varepsilon}{2}}, $$

for all n ≥ τ, i = 1, 2 for some $\mathbb {P}$-almost surely finite τ = τ(ω). Thanks to the uniform control of the previous lemma, we have

$$ \underset{n\to\infty}{\lim} \frac{\mathbb{E}\left[\left|{\sum}^{n}_{j=\tau(\omega)} j^{-2\alpha} \cos 4 \bar{\theta}_{j} \right|\right]}{{\sum}^{n}_{j=1}j^{-2\alpha}} = 0. $$

It is then enough to show that

$$ \mathbb{E}\left[ \sum\limits^{\tau(\omega)}_{j=1} j^{-2\alpha}\right] < \infty. $$

If $\alpha \neq \frac 12$, we have

$$ \begin{array}{@{}rcl@{}} \mathbb{E}\left[ \sum\limits^{\tau(\omega)}_{j=1} j^{-2\alpha}\right] &=& \underset{k\geq 1}{\sum} \left( \sum\limits^{k}_{j=1} j^{-2\alpha} \right) \mathbb{P}[\tau(\omega)=k] \\ &\leqslant& C_{1} \underset{k\geq 1}{\sum} k^{1-2\alpha} \mathbb{P}[\tau(\omega)=k], \end{array} $$

for some finite C₁ > 0. We can estimate the probability inside the sum:

$$ \begin{array}{@{}rcl@{}} \mathbb{P}[\tau(\omega)=k] &\leqslant& \mathbb{P}\left[|V_{j}| > j^{-\frac{2\alpha}{3}-\frac{\varepsilon}{2}}, \forall j<k\right] \\ &=& \prod\limits^{k-1}_{j=1} \mathbb{P}\left[|V_{j}| > j^{-\frac{2\alpha}{3}-\frac{\varepsilon}{2}}\right] \leqslant \prod\limits^{k-1}_{j=1} j^{(\frac{2\alpha}{3}+\frac{\varepsilon}{2})p} \mathbb{E}\left[|V_{j}|^{p}\right] \\ &\leqslant& {C_{2}^{k}} \prod\limits^{k-1}_{j=1} j^{-\varepsilon p} \leqslant {C_{2}^{k}} \mathrm{e}^{-c \varepsilon p k \log k}, \end{array} $$

for some finite C₁ > 0 and c > 0. Hence,

$$ \mathbb{E}\left[ \sum\limits^{\tau(\omega)}_{j=1} j^{-2\alpha}\right] \leqslant C_{1} \underset{k\geq 1}{\sum} k^{1-2\alpha} {C_{2}^{k}} \mathrm{e}^{-c \varepsilon p k \log k} < \infty. $$

The case $\alpha =\frac 12$ is similar. All the above estimates hold uniformly in E ∈ I corresponding to values of k different from $-\frac {5\pi }{8},-\frac {3\pi }{4}$ and $-\frac {7\pi }{8}$. □