# Avila’s acceleration via zeros of determinants, and applications to
Schrödinger cocycles

Rui Han  Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA<EMAIL_ADDRESS>and  Wilhelm Schlag  Department of
Mathematics
Yale University
New Haven, CT 06511, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper we give a characterization of Avila’s quantized acceleration of
the Lyapunov exponent via the number of zeros of the Dirichlet determinants in
finite volume. As applications, we prove $\beta$-Hölder continuity of the
integrated density of states for supercritical quasi-periodic Schrödinger
operators restricted to the $\ell$-th stratum, for any
$\beta<(2(\ell-1))^{-1}$ and $\ell\geq 2$. We establish Anderson localization
for all Diophantine frequencies for the operator with even analytic potential
function on the first supercritical stratum, which has positive measure if it
is nonempty.

R. Han is partially supported by NSF DMS-2143369. W. Schlag is partially
supported by NSF grant DMS-1902691. We would like to thank Michael Goldstein
for his interest and encouragement, and for suggesting Remark 1.9

##  1\. Introduction

This paper studies the one-dimensional quasi-periodic Schrödinger operator:

(1.1)
$\displaystyle(H_{\alpha,\theta}\phi)_{n}=\phi_{n+1}+\phi_{n-1}+f(\theta+n\alpha)\phi_{n},$

with real-valued analytic potential $f(\theta)\in
C_{\eta}^{\omega}(\mathbb{T})$, $\eta>0$. Here $\alpha$ is called the
frequency and $\theta$ is called the phase, and
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$ is the one-dimensional torus, and we shall
use $\|\cdot\|_{\mathbb{T}}$ to denote the torus norm.

We study the operator in the positive Lyapunov exponent regime. The goal of
this paper is to build a bridge between the large deviation estimates and
avalanche principle developed in a series of papers by Bourgain-Goldstein
[BG], Goldstein-Schlag [GS1, GS2] on the one hand, with Avila’s quantized
acceleration of the Lyapunov exponent [Av3] on the other hand. In particular,
we obtain a sharp characterization of the acceleration in terms of the number
of zeros of the Dirichlet determinants, see Theorem 1.2. As applications, we
improve the Hölder exponent of the integrated density of states in [GS2] (see
Theorem 1.3), and we prove Anderson localization of the operator on the first
supercritical stratum, on which the acceleration equals $1$, and for all
Diophantine frequencies (see Theorem 1.6).

To be more specific, let

$\displaystyle D_{n}(\theta,E):=\det(H_{\alpha,\theta}-E)|_{[0,n-1]}.$

Since $f$ is analytic, we can write $D_{n}(\theta,E)$ as a function of
$e^{2\pi i\theta}$. Replacing $e^{2\pi i\theta}$ with a complex variable $z$,
we shall also denote $D_{n}(\theta,E)$ by $D_{n}(z,E)$. We assume throughout
that $\alpha$ satisfies the Diophantine condition

(1.2)
$\displaystyle\mathrm{DC}_{c,a}:=\Big{\\{}\alpha\in\mathbb{T}:\,\|n\alpha\|_{\mathbb{T}}\geq\frac{c}{n(\log
n)^{a}},\text{ for all }|n|\geq 1,\Big{\\}}$

where $c>0$ and $a>1$. It is well-known that for any $a>1$,
$\cup_{c>0}\mathrm{DC}_{c,a}$ is a full measure set.

For an energy $E$, let the Lyapunov exponent $L(E,\varepsilon)$ of the
complexified operator $H_{\alpha,\theta+i\varepsilon}$ be defined as usual,
see (2.15) below. As a corner stone of his global cocycle theory [Av3], Avila
exhibited quantization of the slope of the Lyapunov exponent as a convex
function of $\varepsilon$: the right derivative $\kappa(E,\varepsilon)$ of
$L(E,\varepsilon)/(2\pi)$ in $\varepsilon$ is always an integer, see Section
2.2. Using the quantized acceleration, Avila stratified the spectrum
$\sigma(H_{\alpha,\theta})$ into

$\displaystyle\sigma(H_{\alpha,\theta})=\bigcup_{\ell\geq
1}\mathcal{S}_{\ell},$

where

$\displaystyle\mathcal{S}_{\ell}:=\sigma(H_{\alpha,\theta})\cap\\{E:\,\kappa(E,0)=\ell-1\\}.$

The set $\mathcal{S}_{\ell}$ is called the $\ell$-th stratum of the spectrum.

For $E\in\mathcal{S}_{1}$, it is necessary that $L(E,0)=0$ since otherwise the
corresponding Schrödinger cocycle is uniformly hyperbolic which leads to a
contradiction with $E\in\sigma(H_{\alpha,\theta})$. In fact, $\mathcal{S}_{1}$
is refereed to as the subcritical regime. This stratum is usually studied by
KAM and reducibility methods, see e.g. [DS, E1, P, Av1, AJ, Av2].

For each $\ell\geq 2$, we further divide

$\displaystyle\mathcal{S}_{\ell}=\mathcal{S}_{\ell}^{+}\cup\mathcal{S}_{\ell}^{0},$

where $\mathcal{S}_{\ell}^{+}:=\mathcal{S}_{\ell}\cap\\{E:\,L(E,0)>0\\}$ and
$\mathcal{S}_{\ell}^{0}:=\mathcal{S}_{\ell}\cap\\{E:\,L(E,0)=0\\}$. The
energies in $\cup_{\ell\geq 2}\mathcal{S}_{\ell}^{0}$ are referred to as the
critical energies, and those in $\cup_{\ell\geq 2}\mathcal{S}_{\ell}^{+}$ are
supercritical.

One of the central examples of quasi-periodic Schrödinger operator is the
almost Mathieu operator:

(1.3)
$\displaystyle(H^{\mathrm{AMO}}_{\lambda,\alpha,\theta}\phi)_{n}=\phi_{n+1}+\phi_{n-1}+2\lambda\cos
2\pi(\theta+n\alpha)\phi_{n}.$

The supercritical, critical and subcritical regime corresponds to the regions
where the coupling constant satisfies $|\lambda|>1$, $|\lambda|=1$, and
$0<|\lambda|<1$, respectively. A special feature for the almost Mathieu
operator is that for $|\lambda|>1$,
$\sigma(H^{\mathrm{AMO}}_{\lambda,\alpha,\theta})=\mathcal{S}_{2}^{+}$, hence
the entire spectrum is contained in the first supercritical stratum.

Our analysis focuses on general analytic potentials in the supercritical
regime. We begin with an immediate corollary of [Av3] and [DGSV] on the
Lebesgue measure of strata.

######  Theorem 1.1.

Let $\alpha\in\mathrm{DC}_{c,a}$ for some $c>0$ and $a>1$. For any $k\geq 2$,
if

(1.4) $\bigcup_{j=2}^{k}\mathcal{S}_{j}^{+}\neq\emptyset,\text{\ \ then
}\Big{|}\bigcup_{j=2}^{k}\mathcal{S}_{j}^{+}\Big{|}>0$

In particular, if $\mathcal{S}_{2}^{+}\neq\emptyset$, then
$|\mathcal{S}_{2}^{+}|>0$.

Indeed, let $E_{0}$ be in the set in (1.4). Then by the upper semicontinuity
and quantization of the acceleration [Av3, Theorem 5], as well as the
continuity of the Lyapunov exponent

$I_{0}\subset\Big{\\{}E\in\mathbb{R}\>:\>\kappa(E,0)<k+\frac{1}{2},\;L(E,0)>0\Big{\\}}$

for some interval $I_{0}=(E_{0}-\delta,E_{0}+\delta)$. By [DGSV, Theorem H],
we see that $|\sigma(H_{\alpha,\theta})\cap I_{0}|>\frac{1}{2}\delta$ if
$\delta$ is small enough. By [Av3, Theorem 6] one has $\kappa(E,0)\geq 1$ on
$\sigma(H_{\alpha,\theta})\cap I_{0}$, hence $\sigma(H_{\alpha,\theta})\cap
I_{0}=\cup_{j=2}^{k}\mathcal{S}_{j}^{+}$, whence the theorem.

We now turn to the main theorem of our work, which characterizes Avila’s
acceleration via the number of zeros. Throughout we denote annuli by
$A_{r}:=\\{z\in\mathbb{C}:\,1/r<|z|<r\\}$ and

$N_{n}(E,\varepsilon):=\\#\\{z\in\overline{A_{e^{2\pi\varepsilon}}}:\,D_{n}(z,E)=0\\}$

is the number of zeros relative to $z$ of a determinant in finite volume.

######  Theorem 1.2.

Let $L(E)\geq\tau>0$ and let $\varepsilon\in(0,\eta)$ be such that

$L(E,\varepsilon)=L(E,0)+2\pi\kappa(E,0)\varepsilon.$

Then for $n>N(\tau,E,f,\eta,\alpha,\gamma)$,

(1.5)
$\displaystyle\left|\frac{1}{2n}N_{n}(E,\varepsilon/2)-\kappa(E,0)\right|\leq\varepsilon^{-1}n^{-\gamma}.$

This theorem gives a characterization of the acceleration $\kappa(E,0)$ in
terms of the number of zeros of $D_{n}(z,E)$ in a neighborhood of the unit
circle. Theorem 1.2 is part of the more general Theorem 4.1 (see also Theorem
4.4), where the characterization of
$\kappa(E,\varepsilon)-\kappa(E,\varepsilon^{-})$, for non-zero $\varepsilon$,
is given in terms of zeros of $D_{n}(z,E)$ near the circle with radius
$e^{2\pi\varepsilon}$. An analogous result for the Riesz mass of
$\log\|M_{n}(z,E)\|$, where $M_{n}$ is the transfer matrix (2.11), is given in
Theorem 5.1.

A key new ingredient in the proof is a Riesz representation of subharmonic
function on the annulus, compared to the one on disks that were used to cover
the annulus in [GS1, GS2]. It turns out the annulus version of the
representation is more naturally connected to the Lyapunov exponent, hence to
the acceleration. As one of the applications of Theorem 1.2, we improve on the
known regularity of the integrated density of states (IDS) as a function of
the energy.

######  Theorem 1.3.

Assume the potential function $f$ is analytic and fix
$\alpha\in\mathrm{DC}_{c,a}$ for some $c>0$ and $a>1$. Then for any $\ell\geq
2$, on the open set $\bigcup_{j=1}^{\ell}\mathcal{S}_{j}^{+}$ the IDS of
$H_{\alpha,\theta}$ is $\beta$-Hölder continuous with any
$0<\beta<\frac{1}{2(\ell-1)}$.

This result proves the conjecture by You in [Y] that the Hölder exponent for
IDS is at least $1/(2\kappa(E,0))$ (but we cannot decide equality of the
exponent to this ratio). Note that $\bigcup_{j=1}^{\ell}\mathcal{S}_{j}^{+}$
includes gaps in the spectrum on which the IDS is constant. But by Theorem
1.1, if this set of energies is nonempty, then it contains a positive measure
subset of the spectrum. The proof of Theorem 1.3 is not self-contained. In
fact, Theorem 1.3 follows by combining the proof of [GS2, Theorem 1.1] with
the characterization of the acceleration (and thus the stratum) in Theorem
1.2. Also, recall from [GS2] that the IDS is Lipschitz off a zero-measure set
of energies.

######  Remark 1.4.

In [GS2, Theorem 1.1], it was proved that for
$f(\theta)=\sum_{n=-k_{0}}^{k_{0}}c_{n}e^{2\pi in\theta}$, a trigonometric
polynomial of degree $k_{0}\geq 1$ and assuming $\omega\in{DC}_{c,a}$ and
$L(E,0)>0$ the IDS is $\beta$-Hölder continuous for any
$\beta<\frac{1}{2k_{0}}$. Since for trigonometric $f$ one has $\kappa(E,0)\leq
k_{0}$, Theorem 1.3 is a refinement of [GS2, Theorem 1.1].

######  Remark 1.5.

For the almost Mathieu operator and $\alpha\in\mathrm{DC}$ (see (1.8)), the
IDS was proved to be $1/2$-Hölder if $|\lambda|\neq 0,1$ [AJ]. The
$1/2$-Hölder exponent was also proved for Diophantine $\alpha$ and
$f=\lambda\,g$ with $g$ being analytic and the coupling constant $|\lambda|$
being small, first in the perturbative regime (smallness depends on
$\alpha,g$) in [Am], and then in the non-perturbative regime [AJ] (with
dependence on $\alpha$ removed). For quasi-periodic long-range operators with
large trigonometric polynomial potentials and Diophantine frequencies, the
Hölder exponent in [GS2] was improved recently in the perturbative regime in
[GYZ]. These results are proved using the reducibility method.

Our next application concerns Anderson localization (pure point spectrum with
exponentially decaying eigenfunctions). Anderson localization for quasi-
periodic operators was first studied perturbatively [FSW, Sin, E2] where
$f=\lambda\,g$ with large $\lambda$ whose largeness depends on both $\alpha$
and $g$. The first non-perturbative Anderson localization was obtained by
Jitomirskaya for the almost Mathieu operator [J1, J2] for any $|\lambda|>1$
and all Diophantine frequencies. In [BG], Bourgain and Goldstein proved
Anderson localization for the general analytic potential in the supercritical
regime for an implicit full measure set of $\alpha$. In this paper we show
that quasi-periodic operators with even potentials in the first supercritical
stratum $\mathcal{S}_{2}^{+}$ is always Anderson localized for all Diophantine
frequencies. This result thus includes that of the supercricitical almost
Mathieu operator as a special case. Let

(1.6) $\displaystyle\Theta_{c^{\prime},b}:=\bigcap_{k\geq 1}\bigcup_{|n|\geq
k}\Big{\\{}\theta\in\mathbb{T}:\,\|2\theta+n\alpha\|_{\mathbb{T}}<\frac{c^{\prime}}{|n|^{b}}\Big{\\}},$

where $c^{\prime}>0$ and $b>1$. Clearly the complement
$(\Theta_{c^{\prime},b})^{c}$ is a full measure set.

######  Theorem 1.6.

Assume the potential function $f$ is even. For $\alpha\in\mathrm{DC}_{c,a}$
for some $c>0$ and $a>1$, and $\theta\in(\Theta_{c^{\prime},b})^{c}$ for some
$c^{\prime}>0$ and $b>1$, $H_{\alpha,\theta}$ has pure point spectrum with
exponentially decaying eigenfunctions in $\mathcal{S}_{2}^{+}$.

Recall that by Theorem 1.1 if $\mathcal{S}_{2}^{+}$ is non-empty, then this
set has to be of positive measure. As a corollary of Theorem 1.1, we can prove
Anderson localization for analytic perturbations of the supercritical almost
Mathieu operator, which has recently been studied by Ge-Jitomirskaya-Zhao in
[GJZ].

######  Corollary 1.7.

Let the perturbed almost Mathieu operator be defined as

(1.7)
$\displaystyle(H_{\lambda,\alpha,\theta,g,\varepsilon}\phi)_{n}=\phi_{n+1}+\phi_{n-1}+(2\lambda\cos
2\pi(\theta+n\alpha)+\varepsilon g(\theta+n\alpha))\phi_{n},$

where $g\in C^{\omega}_{\mathbb{T}_{\eta}}$ is even and real-valued. For
$|\lambda|>1$, there exists
$\varepsilon_{1}=\varepsilon_{1}(\lambda,\|g\|_{\mathbb{T}_{\eta}})>0$ such
that if $|\varepsilon|<\varepsilon_{1}$, and $\alpha\in\mathrm{DC}_{c,a}$ for
some $c>0$ and $a>1$, and $\theta\in(\Theta_{c^{\prime},b})^{c}$ for some
$c^{\prime}>0$ and $b>1$, $H_{\lambda,\alpha,\theta,g,\varepsilon}$ has pure
point spectrum with exponentially decaying eigenfunctions.

This corollary follows from combining Theorem 1.6 with the following lemma of
Avila.

######  Lemma 1.8 (Lemma 25, [Av3]).

For $|\lambda|>1$ and $\alpha\in\mathbb{R}\setminus\mathbb{Q}$, there exists
$\varepsilon_{1}=\varepsilon_{1}(\lambda,\|g\|_{\mathbb{T}_{\eta}})>0$ such
that for $|\varepsilon|<\varepsilon_{1}$,
$\sigma(H_{\lambda,\alpha,\theta,g,\varepsilon})=\mathcal{S}_{2}^{+}$.

######  Remark 1.9.

It is possible to prove Theorem 1.6 and Corollary 1.7 for $\alpha$ satisfying
a weaker Diophantine condition:

(1.8)
$\displaystyle\mathrm{DC}:=\bigcup_{c>0}\bigcup_{\rho>1}\left\\{\alpha\in\mathbb{T}:\,\|n\alpha\|_{\mathbb{T}}\geq\frac{c}{|n|^{\rho}},\text{
for any }n\neq 0\right\\}.$

Replacing our $\mathrm{DC}_{c,a}$ with the weaker condition will lead to less
sharp large deviation estimates in Section 2.3, but these will still suffice
for the proof of Anderson localization.

Next let us comment on the proof of Theorem 1.6. The key to prove Anderson
localization is to eliminate “double resonance”, which roughly speaking occurs
when both $D_{n}(\theta,E)$ and $D_{n}(\theta+k\alpha,E)$ are close to $0$ for
$k\simeq n^{C}$. Jitomirskaya’s [J1, J2] argument for the almost Mathieu
operator uses crucially that the potential is cosine. She proved
$D_{n}(\theta-\frac{n-1}{2}\alpha,E)$ is a polynomial of $\cos(2\pi\theta)$ of
degree at most $n$, and used the Lagrange interpolation formula to eliminate
double resonances. Bourgain and Goldstein [BG] developed a large deviation
estimate for the set on which the norm of the transfer matrix is small, and
then used semi-algebraic tools to study the complexity of this set. In order
to eliminate double resonances, they then devised a “steep line” argument,
which requires deleting an implicit zero measure of $\alpha$’s. In this paper,
on the one hand we appeal to the large deviation estimates in [GS2] for the
sub-exponentially small measure of the deviation set $\mathcal{B}_{n}$ (see
(6.79)), on the other hand we combine the $\kappa=1$ case of Theorem 1.2 with
a Cartan set argument to show that $\mathcal{B}_{n}$ consists of at most
$2n+o(n)$ many intervals. Using that the potential is even, the $2n+o(n)$
intervals form $n+o(n)$ pairs under reflection $\theta\to-\theta$. The
arithmetic conditions on $\alpha,\theta$ prevent that two points in the
trajectory $\\{\theta+j\alpha\\}$ fall into the same pair (at the appropriate
scales). This eliminates double resonances. It is worth mentioning that
although the $\kappa=1$ case of Theorem 1.2 ensures that $D_{n}(z,E)$ has at
most $2n+o(n)$ many zeros, $D_{n}(\theta-\frac{n-1}{2}\alpha,E)$ is not a
polynomial of degree at most $2n+o(n)$ in $\cos(2\pi\theta)$, so the Lagrange
interpolation argument as in [J1, J2] does not apply.

The evenness assumption on the potential is crucial for our argument. It is
not clear if one can improve on [BG] without making this symmetry assumption
in the sense that there is an explicit condition on the phases $\theta$ which
guarantees localization. Recently, Forman and VandenBoom [FV] proved Anderson
localization perturbatively for large cosine-like potentials without evenness
assumption, thus removing the symmetry requirement from the classical work by
Fröhlich, Spencer, and Wittwer [FSW]. However, while [FSW] eliminate resonant
phases via an arithmetic condition, [FV] do so by an implicit procedure.

Finally, let us mention that it is an interesting question as to whether the
results in Theorems 1.3 and 1.6 hold for Liouville frequencies. Related
questions have been addressed for the almost Mathieu operator in e.g. [AD,
Av1, AYZ, JL]. For general analytic potentials, large deviation estimates have
been developed for some Liouville $\alpha$’s in [YZ, HZ], leading to the proof
of Hölder continuity of the IDS, albeit with non-sharp exponent. It would be
interesting to see if one can combine the techniques in [YZ, HZ] with ours.

This paper is organized as follows. In Section 2 we collect some preliminary
results; in Section 3 we develop the Riesz representation of subharmonic
functions on an annulus; in Sections 4 and 5 we apply the Riesz representation
to the functions $u_{n}$ and $v_{n}$ and establish the zero count of Theorem
1.2. Finally, Anderson localization as in Theorem 1.6 is obtained in Section
6, and Theorem 1.3 on Hölder continuity of the IDS is proved in Section 7.

##  2\. Preliminaries

Notations: Let $A_{R}:=\\{z\in\mathbb{C}:\,1/R<|z|<R\\}$ be the annulus and
$\mathcal{C}_{R}:=\\{z\in\mathbb{C}:\,|z|=R\\}$ be the circle. For a set
$U\subset\mathbb{R}$, let $|U|$ be its Lebesgue measure.

###  2.1. Determinant and transfer matrix

Since $f\in C_{\eta}^{\omega}(\mathbb{T})$, the determinant $D_{n}(z,E)$ is
holomorphic in the annulus $A_{e^{2\pi\eta}}$. Moreover, as $f$ is real-valued
for $\theta\in\mathbb{R}$, one has
$D_{n}(z,E)=\overline{D_{n}(1/\overline{z},E)}$ for $z\in\mathcal{C}_{1}$,
hence

(2.9) $\displaystyle D_{n}(z,E)=\overline{D_{n}(1/\overline{z},E)},\text{ for
}z\in A_{e^{2\pi\eta}}.$

In particular, we note the following fact that will be used many time
throughout the paper:

######  Fact 2.1.

If $w\notin\mathcal{C}_{1}$ is a zero of $D_{n}(z,E)$, then $1/\overline{w}$
is also a zero.

Let

(2.10) $\displaystyle u_{n}(z,E):=\frac{1}{n}\log|D_{n}(z,E)|.$

and as usual,

(2.11) $\displaystyle M(\theta,E)=\left(\begin{matrix}E-f(\theta)\ &-1\\\
1&0\end{matrix}\right)$

denotes the transfer matrix, and

$\displaystyle M_{n}(\theta,E):=M(\theta+(n-1)\alpha,E)\cdots M(\theta,E)$

be the n-step transfer matrix. Let

(2.12) $\displaystyle v_{n}(z,E):=\frac{1}{n}\log\|M_{n}(z,E)\|.$

Both $u_{n}$ and $v_{n}$ are subharmonic functions in the annulus $A_{R}$. The
starting point of our study is an effective Riesz representation of
subharmonic functions in $A_{R}$, which we develop in Section 3.

Recall the well-known connection between $D_{n}$ and $M_{n}$, viz.

(2.13) $\displaystyle
M_{n}(z,E)=\left(\begin{matrix}D_{n}(z,E)\,&-D_{n-1}(ze^{2\pi i\alpha},E)\\\
D_{n-1}(z,E)&-D_{n-2}(ze^{2\pi i\alpha},E)\end{matrix}\right).$

This implies that

(2.14) $\displaystyle u_{n}(z,E)\leq v_{n}(z,E),$

which will be used multiple times in this paper.

###  2.2. Lyapunov exponent and acceleration

Let

(2.15) $\displaystyle L_{n}(E,\varepsilon):=\int_{0}^{1}v_{n}(e^{2\pi
i(\theta+i\varepsilon)},E)\,\mathrm{d}\theta,$

and

(2.16) $\displaystyle
L(E,\varepsilon):=\lim_{n\to\infty}L_{n}(E,\varepsilon).$

The Lyapunov exponent is even in $\varepsilon$ due to $f$ being real-valued,
and is a convex function in $\varepsilon$ by Hadamard’s three circle theorem.
We define its derivative from the right as

(2.17) $\displaystyle\kappa(E,\varepsilon):=\lim_{\delta\to
0^{+}}\frac{L(E,\varepsilon+\delta)-L(E,\varepsilon)}{2\pi\delta}.$

The function $\kappa(E,\varepsilon)$ is referred to as the acceleration of the
Lyapunov exponent.

In [Av3], Avila proved the following:

######  Theorem 2.2.

[Av3, Theorem 5] For any irrational $\alpha$ and any $E,\varepsilon$, the
acceleration $\kappa(E,\varepsilon)$ is always an integer.

Using the quantized acceleration, Avila further introduced the stratification
of the spectrum: $X_{1}:=\sigma(H_{\alpha,\theta})$, and $X_{j}$ with $j\geq
2$ is defined as

$\displaystyle X_{j}:=\\{E\in X_{1}:\,\kappa(E,0)\geq j-1\\}.$

The set

(2.18) $\displaystyle\mathcal{S}_{j}:=X_{j}\setminus
X_{j+1}=\\{E\in\sigma(H_{\alpha,\theta}):\,\kappa(E,0)=j-1\\}$

is called the $j$-th stratum of the stratification. It is clear that
$\\{\mathcal{S}_{j}\\}_{j=1}^{\infty}$ are pairwise disjoint and

$\displaystyle\bigcup_{j\geq
1}\mathcal{S}_{j}=X_{1}=\sigma(H_{\alpha,\theta}).$

###  2.3. Large deviation estimates

Our analysis depends crucially on the large deviation estimates (and their
consequences) for $u_{n}$ and $v_{n}$ developed in a series of papers [BG,
GS1, GS2], in combination with the avalanche principle. We list the results
that are crucial to our analysis below. Recall that the potential $f\in
C^{\omega}(\mathbb{T}_{\eta})$. The first one is a uniform upper bound for
$v_{n}$, the logarithm of the norm of the transfer matrix.

######  Lemma 2.3.

[GS2, Proposition 4.3] Assume $L(E,0)>\tau>0$. Then for all $n\geq 1$, and
$0\leq|\varepsilon|<\eta/2$,

$\displaystyle\sup_{\theta\in\mathbb{T}}v_{n}(e^{2\pi
i(\theta+i\varepsilon)},E)\leq L_{n}(E,\varepsilon)+C\frac{(\log
n)^{C_{0}}}{n}.$

for some constants $C=C(\|f\|_{\eta},\eta,c,a,\tau,E)$ and $C_{0}=C_{0}(a)$,
with $c$, $a$ as in (1.2).

The next is a quantitative convergence rate of $L_{n}(E,\varepsilon)$ to
$L(E,\varepsilon)$.

######  Theorem 2.4.

[GS1, Theorem 5.1] If $L(E,0)>\tau>0$, then for $n\geq 1$, one has that for
$|\varepsilon|<\eta/2$,

$\displaystyle 0\leq L_{n}(E,\varepsilon)-L(E,\varepsilon)\leq\frac{C}{n},$

where $C=C(\tau,E,f,\eta,\alpha)$.

The next is an average lower bound for $u_{n}$.

######  Lemma 2.5.

[GS2, Lemma 2.10] If $L(E)>\tau>0$, there exists a positive constant
$\gamma_{1}>0$ such that for $|\varepsilon|<\eta/2$, and all
$n>N(\tau,E,f,\eta,\alpha)$,

$\displaystyle\int_{0}^{1}u_{n}(e^{2\pi
i(\theta+i\varepsilon)},E)\,\mathrm{d}\theta>L_{n}(E,\varepsilon)-n^{-\gamma_{1}}.$

The final tool is a large deviation estimate for $u_{n}$.

######  Lemma 2.6.

[GS2, Proposition 2.11] If $L(E)>\tau>0$, there exists a positive constant
$\gamma_{2}>0$ such that for $n>N(\tau,E,f,\eta,\alpha)$,

$\displaystyle\left|\\{\theta\in\mathbb{T}:\,u_{n}(e^{2\pi
i\theta},E)<L_{n}(E)-n^{-\gamma_{2}}\\}\right|<e^{-n^{\gamma_{2}}}.$

###  2.4. Eigenfunction expansion

Let $\phi$ be a solution to $H_{\alpha,\theta}\phi=E\phi$, where
$H_{\alpha,\theta}$ is the Schrödinger operator in (1.1). Then the following
eigenfunction expansion holds: let $y\in[\ell_{1},\ell_{2}]$,

(2.19)
$\displaystyle\phi_{y}=-\frac{D_{\ell_{2}-y}(\theta+(y+1)\alpha,E)}{D_{\ell_{2}-\ell_{1}+1}(\theta+\ell_{1}\alpha,E)}\phi_{\ell_{1}-1}-\frac{D_{y-\ell_{1}}(\theta+\ell_{1}\alpha,E)}{D_{\ell_{2}-\ell_{1}+1}(\theta+\ell_{1}\alpha,E)}\phi_{\ell_{2}+1}.$

This identity is a consequence of Cramer’s rule.

##  3\. Effective Riesz representation for the annulus

This section develops some basic potential theory on annuli. We begin with the
Green’s function on an annulus, which is standard.

###  3.1. Green’s function for the annulus.

It is worth pointing out that we never use the explicit expression of the
Green’s function, rather only the fact that it can be decomposed as in (3.22)
below. For the sake of completeness, we state the precise Green’s kernel,
which can be derived by the method of images.

######  Lemma 3.1.

The Green’s function on the annulus $A_{R}$ is given by

(3.20) $\displaystyle G_{R}(z,w)=\frac{\log(|z|/R)\log(|w|/R)}{4\pi\log
R}+K_{R}(z,w),$

where

(3.21) $\displaystyle
K_{R}(z,w)=\frac{1}{2\pi}\log\left(|z/R-w/R|\cdot\frac{\prod_{k=1}^{\infty}|1-\frac{1}{R^{4k}}\frac{z}{w}|\cdot|1-\frac{1}{R^{4k}}\frac{w}{z}|}{\prod_{k=1}^{\infty}|1-\frac{1}{R^{4k-2}}w\overline{z}|\cdot|1-\frac{1}{R^{4k-2}}\frac{1}{\overline{z}w}|}\right).$

The Green’s function is symmetric and invariant under rotations:
$G_{R}(z,w)=G_{R}(w,z)$ and $G_{R}(z,w)=G_{R}(e^{i\phi}z,e^{i\phi}w)$.

###### Proof.

The symmetry properties are evident from the formula. Second, by inspection

(3.22) $\displaystyle G_{R}(z,w)$
$\displaystyle=\frac{1}{2\pi}\log|z-w|+H_{R}(z,w),\quad z\in
A_{R},w\in\overline{A_{R}}$ $\displaystyle\Delta_{z}H_{R}(z,w)$
$\displaystyle=0$

whence $\Delta_{z}G(z,w)=\delta_{0}(z-w)$. Third, if $w=R$, then

$G_{R}(z,w)=K_{R}(z,w)=\frac{1}{2\pi}\log\left(\frac{|z/R-1|}{|1-\frac{1}{R}\overline{z}|}\right)=0$

This holds for all $|w|=R$ by rotational invariance of $G_{R}$. If
$|w|=R^{-1}$, then

$G_{R}(z,1/R)=-\frac{1}{2\pi}\log(|z|/R)+\frac{1}{2\pi}\log\left(\frac{|z/R-1/R^{2}|}{|1-1/(R\bar{z})|}\right)=0$

and this again holds for all $|w|=R^{-1}$ by rotational invariance. The lemma
follows from the uniqueness of the Green’s function. ∎

We evaluate the integral of the Green’s function along the circle
$z\in\mathcal{C}_{r}$ with $1/R\leq r\leq R$. These integrals will be used
later in Section 4 to estimate the number of zeros of $D_{n}(z,E)$, and the
Riesz mass of $v_{n}(z,E)$ in Section 5.

######  Lemma 3.2.

For $1/R\leq r\leq R$ and $w\in A_{R}$, we have

(3.23) $\displaystyle I(\log r,\log R,w)$
$\displaystyle:=2\pi\int_{0}^{1}G_{R}(re^{2\pi i\theta},w)\,\mathrm{d}\theta$
(3.24) $\displaystyle=(2\log R)^{-1}\begin{cases}\log(rR)\log|w/R|,\text{ if
}|w|\geq r\\\ \\\ \log(r/R)\log|wR|,\text{ if }|w|<r.\end{cases}$

###### Proof.

(3.22) yields

(3.25) $\displaystyle 2\pi\int_{0}^{1}G_{R}(re^{2\pi
i\theta},w)\,\mathrm{d}\theta$
$\displaystyle=\int_{0}^{2\pi}\frac{1}{2\pi}\log|re^{i\theta}-w|\,d\theta+\int_{0}^{2\pi}H_{R}(re^{i\theta},w)\,\mathrm{d}\theta$
$\displaystyle=:J_{1}(w)+J_{2}(w)$

where $J_{1}(w)=\log|w|$ if $|w|\geq r$ and $J_{1}(w)=\log r$ if $|w|\leq r$.
$J_{2}(w)$ is harmonic in $w\in A_{R}$, radial, and continuous on
$\overline{A_{R}}$. Thus, $J_{2}(w)=a\log|w|+b$. Setting $|w|=R$, respectively
$|w|=R^{-1}$ shows that

$J_{2}(w)=\frac{\log(r/R)}{2\log R}\log(|w|/R)-\log R$

which implies (3.23). ∎

###  3.2. Effective Riesz representation

We now turn to the basic Riesz representation of subharmonic functions. In
contrast to [GS1, GS2], which analyzed supercricial cocycles by means of
potential theory on small disks, here we conduct this analysis globally on
annuli.

######  Lemma 3.3.

Let $v$ be a subharmonic function in a neighborhood of $\overline{A_{R}}$, and
assume $v|_{\partial A_{R}}$ is a continuous function. Let $G_{R}$ be the
Green’s function for $A_{R}$, as in (3.20). There exists a positive finite
measure $\mu$ on $A_{R}$, and a harmonic function $h_{R}$ on $A_{R}$, such
that

$\displaystyle v(w)=\int_{A_{R}}2\pi G_{R}(z,w)\,\mu(\mathrm{d}z)+h_{R}(w),$

where

(3.26) $\displaystyle h_{R}(w)=\int_{\partial
A_{R}}v(z)\,\nu(w,A_{R})(\mathrm{d}z),$

where $\nu(w,A_{R})$ is the harmonic measure of $A_{R}$ with pole at $w$. In
particular,

(3.27) $\displaystyle h_{R}(z)=v(z),\text{ for }z\in\partial A_{R}.$

###### Proof.

Without loss of generality, we may assume that $v$ is smooth. If this is not
the case, we convolve $v$ with a radial nonnegative mollifier. The submean
property then guarantees monotone convergence. We skip these technical
details.

By Green’s second identify, with the Green’s function $G_{R}$ defined in
(3.20),

$\displaystyle v(w)-\int_{A_{R}}G_{R}(z,w)\Delta
v(z)\,m(\mathrm{d}z)=\int_{\partial A_{R}}v(z)\frac{\partial G_{R}}{\partial
n_{z}}(z,w)\,\sigma(\mathrm{d}z),$

where $m$ is Lebesgue measure and $\sigma$ is the (unnormalized) arclength
measure on $\partial A_{R}$. Since $v$ is smooth and subharmonic, $\Delta v$
is a non-negative, continuous function, and defines a positive measure
$2\pi\mu=\Delta v\,m$. Therefore

(3.28) $\displaystyle v(w)=\int_{A_{R}}2\pi
G_{R}(z,w)\,\mu(\mathrm{d}z)+h_{R}(w),$

where

(3.29) $\displaystyle h_{R}(w):=$ $\displaystyle\int_{\partial
A_{R}}v(z)\frac{\partial G_{R}}{\partial n_{z}}(z,w)\,\sigma(\mathrm{d}z)$
(3.30) $\displaystyle=$ $\displaystyle\int_{\partial
A_{R}}v(z)\,\nu(w,A_{R})(\mathrm{d}z)$

is the harmonic part. ∎

######  Remark 3.4.

By the maximum principle, we have

(3.31) $\displaystyle\sup_{w\in A_{R}}h_{R}(w)\leq\max_{z\in\partial
A_{R}}v(z).$

The harmonic measure of the annulus $A_{R}$ can be computed explicitly from
$G_{R}$ above, but we have no need for that. We only require a basic well-
known bound on the density of the harmonic measure, namely that it is
controlled by the inverse distance to $\partial A_{R}$.

##  4\. Riesz representation for $u_{n}$

Recall that $D_{n}(z,E)$ is a holomorphic function in $A_{e^{2\pi\eta}}$. For
$0\leq\varepsilon<\eta$, let

$N_{n}(E,\varepsilon):=\\#\\{z\in\overline{A_{e^{2\pi\varepsilon}}}:\,D_{n}(z,E)=0\\},$

in particular

$N_{n}(E,0)=\\#\\{z\in\mathcal{C}_{1}:\,D_{n}(z,E)=0\\}.$

###  4.1. Avila’s acceleration via a zero count

We establish the following more general form of Theorem 1.2.

######  Theorem 4.1.

For some energy $E\in\mathbb{R}$ assume that $L(E,0)\geq\tau>0$. Assume
further that $0<\varepsilon_{0}<\eta$ is such that there exists
$\varepsilon_{1}>0$,
$[\varepsilon_{0}-\varepsilon_{1},\varepsilon_{0}+\varepsilon_{1}]\subset[0,\eta)$,
$D_{n}(z,E)$ is zero-free on $\partial A_{e^{2\pi
i(\varepsilon_{0}+\varepsilon_{1})}}$, and

(4.32) $\displaystyle\kappa(E,\varepsilon_{0}+\varepsilon)\equiv\kappa,$

for $|\varepsilon|<\varepsilon_{1}$. Then for the constant $\gamma_{1}>0$ in
Lemma 2.5, for $n>N(\tau,E,f,\eta,\alpha,\gamma_{1})$ and some absolute
constant $C_{1}>0$,

$\displaystyle\left|\frac{1}{2n}N_{n}(E,\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})-\kappa\right|\leq
C_{1}\varepsilon_{1}^{-2}n^{-\gamma_{1}}.$

######  Remark 4.2.

Suppose that for $\varepsilon_{0}\in(0,\eta)$ there exists $\varepsilon_{1}>0$
such that
$(\varepsilon_{0}-\varepsilon_{1},\varepsilon_{0}+\varepsilon_{1})\subset[0,\eta)$
and for $0<\varepsilon\leq\varepsilon_{1}$,

$\displaystyle\kappa(E,\varepsilon_{0}-\varepsilon)\equiv\kappa_{1}<\kappa_{2}\equiv\kappa(E,\varepsilon_{0}+\varepsilon).$

Then Theorem 4.1 yields for $n>N(\tau,E,f,\eta,\alpha,\gamma_{1})$,

$\displaystyle\left|\frac{1}{2n}\left(N_{n}(E,\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})-N_{n}(E,\varepsilon_{0}-\frac{1}{3}\varepsilon_{1})\right)-(\kappa_{2}-\kappa_{1})\right|\leq
C_{1}\varepsilon_{1}^{-2}n^{-\gamma_{1}}.$

Hence one can also characterize the change of slopes of the piece-wise linear
function $L(E,\varepsilon)$ in terms of zero counts of $D_{n}(z,E)$.

###### Proof.

In the proof we shall omit the dependence of various parameters on $E$ for
simplicity. By the assumption (4.32) and Theorem 2.4, one has for
$0\leq|\varepsilon|\leq\varepsilon_{1}$,

(4.33) $\displaystyle L(\varepsilon_{0})+2\pi\kappa\varepsilon\leq
L(\varepsilon_{0}+\varepsilon)\leq L_{n}(\varepsilon_{0}+\varepsilon)\leq
L(\varepsilon_{0}+\varepsilon)+\frac{C}{n}=L(\varepsilon_{0})+2\pi\kappa\varepsilon+\frac{C}{n}.$

Let $\tilde{R}:=e^{2\pi(\varepsilon_{0}+\varepsilon_{1})}$ and
$\tilde{N}:=N_{n}(\varepsilon_{0}+\varepsilon_{1})$. Denote by
$w_{1},w_{2},...,w_{\tilde{N}}$ the zeros of $D_{n}(z)$ in $A_{\tilde{R}}$.
Define

$G_{\tilde{R},n}(z):=\frac{1}{n}\sum_{k=1}^{\tilde{N}}G_{\tilde{R}}(z,w_{k}),$

where $G_{R}(\cdot,\cdot)$ is the Green’s function in (3.20). Lemma 3.3, with
$R=\tilde{R}$, applied to $u_{n}$ yields

(4.34) $\displaystyle u_{n}(z)=2\pi G_{\tilde{R},n}(z)+h_{\tilde{R},n}(z),$

First, we estimate the harmonic part.

######  Lemma 4.3.

With the constant $\gamma_{1}$ in Lemma 2.5 and the constant $C_{0}$ in Lemma
2.3, for $z\in A_{e^{2\pi\varepsilon}}$,
$0\leq\varepsilon<\varepsilon_{0}+\varepsilon_{1}$, and
$n>N(\tau,E,f,\eta,\alpha)$

(4.35) $\displaystyle
L_{n}(E,\varepsilon_{0}+\varepsilon_{1})-\frac{C}{\tilde{R}-e^{2\pi\varepsilon}}n^{-\gamma_{1}}\leq
h_{\tilde{R},n}(z,E)\leq L_{n}(E,\varepsilon_{0}+\varepsilon_{1})+C\frac{(\log
n)^{C_{0}}}{n},$

###### Proof.

In the proof we shall omit the dependence on $E$ for simplicity. Note that the
harmonic part satisfies $h_{\tilde{R},n}=u_{n}$ on $\partial A_{\tilde{R}}$,
due to (LABEL:eq:un_sym). By Lemma 2.3 and (2.14), one has that for
$r=\tilde{R}$ or $1/\tilde{R}$ and $n$ large, uniformly in $\theta$,

$\displaystyle h_{\tilde{R},n}(re^{2\pi i\theta})=u_{n}(re^{2\pi i\theta})\leq
v_{n}(re^{2\pi i\theta})\leq
L_{n}(\varepsilon_{0}+\varepsilon_{1})+C\frac{(\log n)^{C_{0}}}{n}.$

Hence by the maximum principle (3.31),

(4.36) $\displaystyle h_{\tilde{R},n}(z)\leq
L_{n}(\varepsilon_{0}+\varepsilon_{1})+C\frac{(\log n)^{C_{0}}}{n},\text{ for
}z\in\overline{A_{\tilde{R}}}.$

We also have by Lemma 2.5 that for $n$ large enough,

(4.37) $\displaystyle\int_{0}^{1}h_{\tilde{R},n}(\tilde{R}e^{2\pi
i\theta})\,\mathrm{d}\theta=\int_{0}^{1}u_{n}(\tilde{R}e^{2\pi
i\theta})\,\mathrm{d}\theta\geq
L_{n}(\varepsilon_{0}+\varepsilon_{1})-\frac{1}{n^{\gamma_{1}}}.$

Let

(4.38)
$\displaystyle\tilde{h}_{\tilde{R},n}(z):=L_{n}(\varepsilon_{0}+\varepsilon_{1})+C\frac{(\log
n)^{C_{0}}}{n}-h_{\tilde{R},n}(z)\geq 0,$

where we invoked (4.36). In view of (4.37), for $n$ large,

(4.39) $\displaystyle\int_{0}^{1}\tilde{h}_{\tilde{R},n}(\tilde{R}e^{2\pi
i\theta})\,\mathrm{d}\theta\leq\frac{2}{n^{\gamma_{1}}}.$

By (3.26) and (4.39), and the well-known estimate on the harmonic measure

$0\leq\frac{\mathrm{d}\nu(w,A_{R})(z)}{\mathrm{d}\sigma(z)}\leq
C(\mathrm{dist}(w,\partial A_{R}))^{-1},$

with arclength measure $\sigma$, one has that for $z\in
A_{e^{2\pi\varepsilon}}$, with
$0\leq\varepsilon<\varepsilon_{0}+\varepsilon_{1}$,

$\displaystyle
0\leq\tilde{h}_{\tilde{R},n}(z)\leq\frac{C}{\tilde{R}-e^{2\pi\varepsilon}}\int_{0}^{1}\tilde{h}_{\tilde{R},n}(\tilde{R}e^{2\pi
i\theta})\,\mathrm{d}\theta\leq\frac{C}{\tilde{R}-e^{2\pi\varepsilon}}n^{-\gamma_{1}}.$

This combined with (4.36) yields the claimed result. ∎

Next, we evaluate the integrals of $G_{\tilde{R},n}(z)$ along circles. For
$1\leq r\leq\tilde{R}$,

$\displaystyle I_{n}(\log r,\log\tilde{R}):=$ $\displaystyle\int_{0}^{1}2\pi
G_{\tilde{R},n}(re^{2\pi i\theta})\,d\theta$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{k=1}^{\tilde{N}}I(\log r,\log\tilde{R},w_{k}),$

where $I(\log r,\log\tilde{R},w)$ is defined as in (3.23). By (3.23),

$\displaystyle I_{n}(\log r,\log\tilde{R})=$
$\displaystyle\frac{1}{n}\sum_{|w_{k}|>r}\frac{\log
r+\log\tilde{R}}{2\log\tilde{R}}\log\frac{|w_{k}|}{\tilde{R}}$
$\displaystyle+\frac{1}{n}\sum_{|w_{k}|<1/r}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|+\frac{1}{n}\sum_{1/r\leq|w_{k}|\leq
r}\frac{\log r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|.$

In view of Fact 2.1,

$\displaystyle\sum_{|w_{k}|>r}\frac{\log
r+\log\tilde{R}}{2\log\tilde{R}}\log\frac{|w_{k}|}{\tilde{R}}+\sum_{|w_{k}|<1/r}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|$ $\displaystyle=$
$\displaystyle\sum_{|w_{k}|>r}\left(\frac{\log
r+\log\tilde{R}}{2\log\tilde{R}}\log\frac{|w_{k}|}{\tilde{R}}+\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log\frac{\tilde{R}}{|w_{k}|}\right)$
$\displaystyle=$
$\displaystyle\sum_{|w_{k}|>r}\log\frac{|w_{k}|}{\tilde{R}}=-\sum_{k}\int_{r}^{\tilde{R}}\chi_{[|w_{k}|,\tilde{R})}(x)\,\frac{\mathrm{d}x}{x}$
(4.40) $\displaystyle=$
$\displaystyle-\int_{r}^{\tilde{R}}\\#\\{w_{k}:\,r<|w_{k}|\leq
x\\}\frac{\mathrm{d}x}{x}$

where $\chi$ is the characteristic function. By Fact 2.1,

(4.41) $\displaystyle\\#\\{w_{k}:\,r<|w_{k}|\leq
x\\}=\frac{1}{2}\big{[}N_{n}(\frac{\log x}{2\pi})-N_{n}(\frac{\log
r}{2\pi})\big{]}.$

Plugging this into (4.1) yields

$\displaystyle\sum_{|w_{k}|>r}\frac{\log
r+\log\tilde{R}}{2\log\tilde{R}}\log\frac{|w_{k}|}{\tilde{R}}+\sum_{|w_{k}|<1/r}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{r}^{\tilde{R}}(N_{n}(\frac{\log
x}{2\pi})-N_{n}(\frac{\log r}{2\pi}))\,\frac{\mathrm{d}x}{x}$ (4.42)
$\displaystyle=$ $\displaystyle-\pi\int_{\frac{\log
r}{2\pi}}^{\frac{\log\tilde{R}}{2\pi}}(N_{n}(\varepsilon)-N_{n}(\frac{\log
r}{2\pi}))\,\mathrm{d}\varepsilon.$

Similarly,

$\displaystyle\sum_{1/r\leq|w_{k}|\leq r}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|$ $\displaystyle=$
$\displaystyle\sum_{1<|w_{k}|\leq r}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|+\sum_{1/r\leq|w_{k}|<1}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log|w_{k}\tilde{R}|$
$\displaystyle+\sum_{|w_{k}|=1}\frac{\log
r-\log\tilde{R}}{2\log\tilde{R}}\log\tilde{R}$ $\displaystyle=$
$\displaystyle\sum_{1<|w_{k}|\leq r}(\log
r-\log\tilde{R})+\frac{N_{n}(0)}{2}(\log r-\log\tilde{R})$ (4.43)
$\displaystyle=$ $\displaystyle\frac{1}{2}(\log
r-\log\tilde{R})N_{n}(\frac{\log r}{2\pi}).$

Therefore, combing (4.1) and (4.1) yields

(4.44) $\displaystyle I_{n}(\log
r,\log\tilde{R})=-\frac{\pi}{n}\int_{\frac{\log
r}{2\pi}}^{\frac{\log\tilde{R}}{2\pi}}N_{n}(\varepsilon)\,\mathrm{d}\varepsilon.$

Integrating (4.34) along $z\in\mathcal{C}_{r_{j}}$, $1\leq
r_{1}<r_{2}\leq\tilde{R}$, and combining with (4.44), one obtains

$\displaystyle\int_{0}^{1}u_{n}(r_{j}e^{2\pi
i\theta})\,\mathrm{d}\theta=-\frac{\pi}{n}\int_{\frac{\log
r_{j}}{2\pi}}^{\frac{\log\tilde{R}}{2\pi}}N_{n}(\varepsilon)\,\mathrm{d}\varepsilon+\int_{0}^{1}h_{\tilde{R},n}(r_{j}e^{2\pi
i\theta})\,\mathrm{d}\theta.$

Taking the difference of the equations above between $r_{1}$ and $r_{2}$, we
arrive at

$\displaystyle\int_{0}^{1}u_{n}(r_{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}u_{n}(r_{1}e^{2\pi
i\theta})\,\mathrm{d}\theta=$ $\displaystyle\frac{\pi}{n}\int_{\frac{\log
r_{1}}{2\pi}}^{\frac{\log
r_{2}}{2\pi}}N_{n}(\varepsilon)\,\mathrm{d}\varepsilon$ (4.45)
$\displaystyle+\int_{0}^{1}h_{\tilde{R},n}(r_{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}h_{\tilde{R},n}(r_{1}e^{2\pi
i\theta})\,\mathrm{d}\theta.$

By Lemma 2.5, we have for $n$ large,

(4.46) $\displaystyle\int_{0}^{1}u_{n}(r_{j}e^{2\pi i\theta})\,d\theta\geq
L_{n}(\frac{\log r_{j}}{2\pi})-\frac{1}{n^{\gamma_{1}}}.$

while it follows from Lemma 2.3 and (2.14) that for $C=C(\tau,E,f,\eta,c,a)>0$
and $C_{0}=C_{0}(a)$,

(4.47) $\displaystyle u_{n}(r_{j}e^{2\pi i\theta})\leq v_{n}(r_{j}e^{2\pi
i\theta})\leq L_{n}(\frac{\log r_{j}}{2\pi})+C\frac{(\log n)^{C_{0}}}{n}.$

By Lemma 4.3, one has for $n$ large,

(4.48) $\displaystyle\left|\int_{0}^{1}h_{\tilde{R},n}(r_{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}h_{\tilde{R},n}(r_{1}e^{2\pi
i\theta})\,\mathrm{d}\theta\right|\leq\frac{C}{\tilde{R}-r_{2}}n^{-\gamma_{1}}.$

Hence plugging the estimates (4.46) and (4.47) into (4.1), one concludes that
for $n$ large,

$\displaystyle\frac{1}{2n}(\log r_{2}-\log r_{1})N_{n}(\frac{\log
r_{1}}{2\pi})\leq$ $\displaystyle\frac{\pi}{n}\int_{\frac{\log
r_{1}}{2\pi}}^{\frac{\log
r_{2}}{2\pi}}N_{n}(\varepsilon)\,\mathrm{d}\varepsilon$ (4.49)
$\displaystyle\leq$ $\displaystyle L_{n}(\frac{\log
r_{2}}{2\pi})-L_{n}(\frac{\log
r_{1}}{2\pi})+\frac{C}{\tilde{R}-r_{2}}n^{-\gamma_{1}},$

and

$\displaystyle\frac{1}{2n}(\log r_{2}-\log r_{1})N_{n}(\frac{\log
r_{2}}{2\pi})\geq$ $\displaystyle\frac{\pi}{n}\int_{\frac{\log
r_{1}}{2\pi}}^{\frac{\log
r_{2}}{2\pi}}N_{n}(\varepsilon)\,\mathrm{d}\varepsilon$ (4.50)
$\displaystyle\geq$ $\displaystyle L_{n}(\frac{\log
r_{2}}{2\pi})-L_{n}(\frac{\log
r_{1}}{2\pi})-\frac{C}{\tilde{R}-r_{2}}n^{-\gamma_{1}}.$

Taking $r_{1}=\exp(2\pi(\varepsilon_{0}+\frac{1}{3}\varepsilon_{1}))$ and
$r_{2}=\exp(2\pi(\varepsilon_{0}+\frac{2\varepsilon_{1}}{3}))$ in (4.1) yields

(4.51)
$\displaystyle\frac{\pi\varepsilon_{1}}{3n}N_{n}(\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})\leq
L_{n}(\varepsilon_{0}+\frac{2}{3}\varepsilon_{1})-L_{n}(\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})+C\varepsilon_{1}^{-1}n^{-\gamma_{1}}.$

Setting $r_{1}=\exp(2\pi\varepsilon_{0})$ and
$r_{2}=\exp(2\pi(\varepsilon_{0}+\frac{\varepsilon_{1}}{3}))$ in (4.1) yields

(4.52)
$\displaystyle\frac{\pi\varepsilon_{1}}{3n}N_{n}(\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})\geq
L_{n}(\varepsilon_{0}+\frac{\varepsilon_{1}}{3})-L_{n}(\varepsilon_{0})-C\varepsilon_{1}^{-1}n^{-\gamma_{1}}.$

Combining (4.51), (4.52) with (4.33), we infer that for some absolute constant
$C_{1}>0$ and $n>N(\tau,E,f,\eta,\alpha,\gamma_{1})$,

(4.53)
$\displaystyle\left|\frac{1}{2n}N_{n}(\varepsilon_{0}+\frac{1}{3}\varepsilon_{1})-\kappa\right|\leq
C_{1}\varepsilon_{1}^{-2}n^{-\gamma_{1}}.$

This proves the claimed result. ∎

###  4.2. Riesz representation for $u_{n}$ via zeros

We can now draw the following conclusions about the structure of the
logarithms of $|D_{n}|$ where $D_{n}$ are the determinants in finite volume.

######  Theorem 4.4.

Let $E\in\mathbb{R}$ be such that $L(E,0)\geq\tau>0$. Suppose
$\varepsilon_{2}\in(0,\eta)$ satisfies

(4.54) $\displaystyle L(E,\varepsilon)=L(E,0)+2\pi\kappa(E,0)\varepsilon,$

for all $0\leq\varepsilon\leq\varepsilon_{2}$, and $D_{n}(z.E)$ is zero-free
on $\partial A_{e^{2\pi\varepsilon_{2}}}$. Let $R=e^{2\pi\varepsilon_{2}}$ and
$w_{1},...,w_{N_{n}(E,\varepsilon_{2})}$ be the zeros of $D_{n}(z,E)$ in
$A_{R}$ and define

$\displaystyle
G_{R,n}(z,E)=\frac{1}{n}\sum_{k=1}^{N_{n}(E,\varepsilon_{2})}G_{R}(z,w_{k}),$

where $G_{R}$ is the Green’s function in (3.20). Then

$\displaystyle u_{n}(z,E)=2\pi G_{R,n}(z,E)+h_{R,n}(z,E),$

where the harmonic part satisfies $h_{R,n}=u_{n}$ on $\partial A_{R}$.
Furthermore,

  * •

with the constant $\gamma_{1}$ in Lemma 2.5 and the constant $C_{0}$ in Lemma
2.3, for any $z\in A_{r}$, $1\leq r<R$, and $n>N(\tau,E,f,\eta,\alpha)$ the
harmonic part satisfies

(4.55) $\displaystyle
L_{n}(E,\varepsilon_{2})-\frac{C}{R-r}\frac{1}{n^{\gamma_{1}}}\leq
h_{R,n}(z,E)\leq L_{n}(E,\varepsilon_{2})+C\frac{(\log n)^{C_{0}}}{n},$

for some constant $C=C(\tau,E,f,\eta,\alpha)>0$.

  * •

with the constant $\gamma_{1}$ in Lemma 2.5 and some absolute constant $C>0$,
for all $0\leq\varepsilon\leq\frac{2}{3}\varepsilon_{2}$, and
$n>N(\tau,E,f,\eta,\alpha,\gamma_{1})$

(4.56) $\displaystyle|\frac{1}{2n}N_{n}(E,\varepsilon)-\kappa(E,0)|\leq
C\varepsilon^{-2}n^{-\gamma_{1}}.$

In particular, when the acceleration $\kappa(E,0)=1$,

(4.57) $\displaystyle|\frac{1}{2n}N_{n}(E,\varepsilon)-1|\leq
C\varepsilon^{-2}n^{-\gamma_{1}}.$

###### Proof.

Taking
$\varepsilon_{0}=\varepsilon_{1}=\frac{3}{4}\varepsilon\leq\frac{1}{2}\varepsilon_{2}$,
it is clear that (4.56) follows from Theorem 4.1. Taking
$\varepsilon_{0}=\varepsilon_{1}=\frac{1}{2}\varepsilon_{2}$, (4.35) follows
from Lemma 4.3. ∎

##  5\. Riesz representation for $v_{n}$

It is natural to ask if the acceleration can be characterized by the Riesz
mass of the function $v_{n}$ rather than through the number of zeros of
$D_{n}$, or equivalently, the Riesz mass of $u_{n}$. For future reference, we
show here that this is indeed the case.

######  Theorem 5.1.

Let $E\in\mathbb{R}$ be such that $L(E,0)\geq\tau>0$. Let
$\varepsilon_{2}\in(0,\eta)$ be such that

(5.58) $\displaystyle L(E,\varepsilon)=L(E,0)+2\pi\kappa(E,0)\varepsilon,$

for $0\leq\varepsilon\leq\varepsilon_{2}$ and set $R=e^{2\pi\varepsilon_{2}}$.
Then

(5.59) $\displaystyle v_{n}(w,E)=\int_{A_{R}}2\pi
G_{R}(z,w)\,\mu_{n,E}(\mathrm{d}z)+h_{R,n}(w,E),$

where the harmonic part satisfies $h_{R,n}=v_{n}$ on $\partial A_{R}$.
Furthermore, for any $1<r<R^{1/3}$, for some constant $C>0$, and $n$ large,

(5.60)
$\displaystyle|\mu_{n,E}(A_{r})-2\kappa(E,0)|\leq\frac{C\varepsilon_{2}^{-1}}{\log
r}\cdot\frac{(\log n)^{C_{0}}}{n}.$

With the constant $C_{0}$ in Lemma 2.3 and some constant $C>0$, for any $1\leq
r<R$ and $n$ large enough

(5.61) $\displaystyle L_{n}(E,\varepsilon_{2})-\frac{C}{R-r}\frac{(\log
n)^{C_{0}}}{n}\leq h_{R,n}(w,E)\leq L_{n}(E,\varepsilon_{2})+C\frac{(\log
n)^{C_{0}}}{n}$

for all $w\in A_{r}$.

###### Proof.

We omit the dependence on $E$ in the proof for simplicity. The analysis of the
harmonic part (5.61) is similar to that of (4.35), and we leave the details to
the reader. Integrating (5.59) along $w\in\mathcal{C}_{r}$ and $C_{r^{2}}$,
$1<r\leq R^{1/3}$, and subtracting one from the other yields in analogy to
(4.1) that

(5.62) $\displaystyle L_{n}(\frac{2\log r}{2\pi})-L_{n}(\frac{\log
r}{2\pi})=\int_{0}^{1}v_{n}(r^{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}v_{n}(re^{2\pi
i\theta})\,\mathrm{d}\theta$ (5.63) $\displaystyle=$
$\displaystyle\int_{A_{R}}\left(\int_{0}^{1}2\pi G_{R}(z,r^{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}2\pi
G_{R}(z,re^{i\theta})\,\mathrm{d}\theta\right)\,\mu_{n,E}(\mathrm{d}z)$ (5.64)
$\displaystyle+\int_{0}^{1}h_{R,n}(r^{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}h_{R,n}(re^{2\pi
i\theta})\,\mathrm{d}\theta.$

By Theorem 2.4 and (5.58), for large $n$

(5.65) $\displaystyle|L_{n}(\frac{2\log r}{2\pi})-L_{n}(\frac{\log
r}{2\pi})-\kappa(0)\log r|\leq\frac{C}{n}.$

By (5.61),

(5.66) $\displaystyle\left|\int_{0}^{1}h_{R,n}(r^{2}e^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}h_{R,n}(re^{2\pi
i\theta})\,\mathrm{d}\theta\right|\leq\frac{C}{R-r^{2}}\frac{(\log
n)^{C_{0}}}{n}.$

By Lemma 3.2 and the symmetry $G_{R}(z,w)=G_{R}(w,z)$,

(5.67) $\displaystyle\int_{0}^{1}2\pi G_{R}(z,re^{2\pi
i\theta})\,\mathrm{d}\theta-\int_{0}^{1}2\pi G_{R}(z,e^{2\pi
i\theta})\,\mathrm{d}\theta$ (5.68) $\displaystyle=$
$\displaystyle\begin{cases}\frac{\log r}{2\log R}\log\frac{|z|}{R}\,\text{ if
}|z|\geq r^{2}\\\ \\\ \frac{\log r-2\log R}{2\log R}\log|z|+\frac{3}{2}\log
r,\,\text{ if }r\leq|z|<r^{2}\\\ \\\ \frac{\log r}{2\log R}\log(|z|R),\text{
if }|z|<r\end{cases}$

Since $v_{n}(z)=v_{n}(1/\overline{z})$, and $\Delta v_{n}=\mu_{n}$, the
measure $\mu_{n}$ exhibits reflection symmetry

$\mu_{n}(\mathrm{d}z)=\mu_{n}(\mathrm{d}(1/\overline{z})).$

In combination with (5.67) we conclude that

(5.69) $\displaystyle\int_{|z|\geq r^{2}}\frac{\log r}{2\log
R}\log\frac{|z|}{R}\,\mu_{n}(\mathrm{d}z)+\int_{|z|\leq r^{-2}}\frac{\log
r}{2\log R}\log(|z|R)\,\mu_{n}(\mathrm{d}z)=0,$

as well as

$\displaystyle\int_{r\leq|z|<r^{2}}\left(\frac{\log r-2\log R}{2\log
R}\log|z|+\frac{3}{2}\log r\right)\,\mu_{n}(\mathrm{d}z)$
$\displaystyle+\int_{r^{-2}<|z|\leq r^{-1}}\frac{\log r}{2\log
R}\log(|z|R)\,\mu_{n}(\mathrm{d}z)$ $\displaystyle=$
$\displaystyle\int_{r\leq|z|<r^{2}}\left(\frac{\log r-2\log R}{2\log
R}\log|z|+\frac{3}{2}\log r+\frac{\log r}{2\log
R}\log\frac{R}{|z|}\right)\,\mu_{n}(\mathrm{d}z)$ (5.70) $\displaystyle=$
$\displaystyle\int_{r\leq|z|<r^{2}}\log\frac{r^{2}}{|z|}\,\mu_{n}(\mathrm{d}z),$

and

$\displaystyle\int_{1<|z|<r}\frac{\log r}{2\log
R}\log(|z|R)\,\mu_{n}(\mathrm{d}z)$
$\displaystyle+\int_{r^{-1}<|z|<1}\frac{\log r}{2\log
R}\log(|z|R)\,\mu_{n}(\mathrm{d}z)+\int_{|z|=1}\frac{\log
r}{2}\,\mu_{n}(\mathrm{d}z)$ (5.71) $\displaystyle=$ $\displaystyle\frac{\log
r}{2}\cdot\mu_{n}(A_{r}).$

Combining (5.65), (5.66), (5.69), (5) and (5) with (5.62), one obtains

(5.72) $\displaystyle\kappa(0)\log r-\frac{C}{R-r^{2}}\frac{(\log
n)^{C_{0}}}{n}\leq$ $\displaystyle\frac{\log
r}{2}\cdot\mu_{n}(A_{r})+\int_{r\leq|z|<r^{2}}\log\frac{r^{2}}{|z|}\,\mu_{n}(\mathrm{d}z)$
(5.73) $\displaystyle\leq$ $\displaystyle\frac{\log
r}{2}\cdot\mu_{n}(A_{r})+\log r\cdot\mu_{n}(r\leq|z|<r^{2})$ (5.74)
$\displaystyle\leq$ $\displaystyle\frac{\log r}{2}\cdot\mu_{n}(A_{r^{2}}),$

as well as

$\displaystyle\kappa(0)\log r+\frac{C}{R-r^{2}}\frac{(\log n)^{C_{0}}}{n}\geq$
$\displaystyle\frac{\log
r}{2}\cdot\mu_{n}(A_{r})+\int_{r\leq|z|<r^{2}}\log\frac{r^{2}}{|z|}\,\mu_{n}(\mathrm{d}z)$
(5.75) $\displaystyle\geq$ $\displaystyle\frac{\log r}{2}\mu_{n}(A_{r}).$

In view of (5.72) (replacing $r^{2}$ with $r$) and (5) one has

$\displaystyle|\mu_{n}(A_{r})-2\kappa(0)|\leq\frac{C\varepsilon_{2}^{-1}}{\log
r}\cdot\frac{(\log n)^{C_{0}}}{n}.$

This proves the claimed result. ∎

##  6\. Anderson localization

In this section, we prove Theorem 1.6. Let $E\in\mathcal{S}_{2}^{+}$. Then
$L(E,0)=\tau>0$ and there exists $\varepsilon_{2}\in(0,\eta)$ such that

(6.76) $\displaystyle L(E,\varepsilon)=L(E,0)+2\pi\varepsilon,$

for $0\leq\varepsilon\leq\varepsilon_{2}$ and $D_{n}(z,E)$ is zero-free on
$\partial A_{e^{2\pi\varepsilon_{2}}}$. Let $R=e^{2\pi\varepsilon_{2}}$. Note
that since the potential $f$ is assumed to be even, $f(\theta)=f(-\theta)$,
and thus
$D_{n}(\theta-\frac{n-1}{2}\alpha,E)=D_{n}(-\theta-\frac{n-1}{2}\alpha,E)$ as
well as

(6.77) $\displaystyle D_{n}(z\cdot e^{-\pi i(n-1)\alpha},E)=D_{n}(e^{-\pi
i(n-1)\alpha}/z,E),\text{ for }z\in A_{R}.$

We have by (6.77) the following

######  Fact 6.1.

If $w\in A_{R}$ is a zero of $D_{n}(z,E)$, then $e^{-2\pi i(n-1)\alpha}/w$ is
also a zero.

In the following, we shall fix an energy $E$ and omit the dependence on $E$
for simplicity. We shall also write $L_{n}(E,0)=L_{n}$ and $L(E,0)=L$.

###  6.1. Geometric structure of the large deviation set

Let $\varepsilon$ be a small constant such that
$0<\varepsilon<\min(L(E,0)/20,\varepsilon_{2})$, and we set
$R^{\prime}:=e^{2\pi\varepsilon}$ and $N^{\prime}:=N_{n}(\varepsilon)$ for
simplicity. Let $w_{1},...,w_{N^{\prime}}$ be the zeros of $D_{n}(z)$ in
$A_{R^{\prime}}$. By the Riesz representation theorem, see Theorem 4.4,
applied to $u_{n}$ with $R^{\prime}$ instead of $R$,

(6.78) $\displaystyle u_{n}(z)=G_{R^{\prime},n}(z)+h_{R^{\prime},n}(z).$

We need to control the complexity of the large deviation set

(6.79) $\displaystyle\mathcal{B}_{n}:=\\{\theta\in\mathbb{T}:\,u_{n}(e^{2\pi
i\theta})<L_{n}-n^{-\gamma_{2}}\\},$

where $\gamma_{2}>0$ is the constant in Lemma 2.6. We are dropping $E$ from
some of the notation for simplicity.

######  Lemma 6.2.

For $n$ large enough, there exists an integer
$N^{\prime\prime}\leq\frac{N^{\prime}}{2}+1$, and a collection
$\mathcal{F}_{n}=\\{U_{j}\\}_{j=1}^{N^{\prime\prime}}$ intervals in
$\mathbb{T}$ such that

$\displaystyle\mathcal{B}_{n}\subset\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup(-(n-1)\alpha-
U_{j})),$

where the notation $x-U:=\\{x-\theta:\,\theta\in U\\}$ for $x\in\mathbb{T}$
and $U\subseteq\mathbb{T}$. Furthermore, for each $1\leq j\leq
N^{\prime\prime}$,

(6.80) $\displaystyle|U_{j}|\leq e^{-n^{\gamma_{2}/2}},$

where $\gamma_{2}>0$ is the constant in Lemma 2.6.

###### Proof.

This essentially follows from combining Theorem 1.2 and [GS2, Lemma 2.17]. We
first recall Cartan’s estimate [L, Theorem 4, Page 79] as it appears in [GS2,
Lemma 2.15]

######  Definition (Cartan set).

For an arbitrary subset
$\mathcal{P}\subset\mathcal{D}(z_{0},1)\subset\mathbb{C}$, where
$\mathcal{D}(z_{0},1)$ is the disk, we say that
$\mathcal{P}\in\mathrm{Car}(H,K)$ if
$\mathcal{P}\subset\cup_{k=1}^{k_{0}}\mathcal{D}(z_{k},r_{k})$ with $k_{0}\leq
K$, and

(6.81) $\displaystyle\sum_{j}r_{j}<e^{-H}.$

By Wiener’s covering lemma we can assume that $\mathcal{D}(z_{k},r_{k})$ are
pairwise disjoint, at the expense of a factor of $3$ in (6.81).

######  Lemma 6.3.

Let $\varphi$ be an analytic function defined in a disk
$\mathcal{D}:=\mathcal{D}(z_{0},1)$. Let
$M\geq\sup_{z\in\mathcal{D}}\log|\varphi(z)|$, $m\leq\log|\varphi(z_{0})|$.
Given $H\gg 1$, there exists a set $\mathcal{P}\subset\mathcal{D}$,
$\mathcal{P}\in\mathrm{Car}(H,K)$, $K=CH(M-m)$ for some absolute constant
$C>0$, such that

(6.82) $\displaystyle\log|\varphi(z)|>M-CH(M-m),$

for any $z\in\mathcal{D}(z_{0},1/6)\setminus\mathcal{P}$.

###### Proof.

See [GS2, Lemma 2.15]. ∎

By Lemma 2.6, for $n$ large enough, we have

$\displaystyle|\mathcal{B}_{n}|\leq e^{-n^{\gamma_{2}}}.$

Hence we can find $\\{\theta_{j}\\}_{j=1}^{j_{n}}$ such that for any $j$,
$\theta_{j}\notin\mathcal{B}_{n}$, and also $j_{n}\leq 20n$,
$\mathbb{T}=\bigcup_{j}(\theta_{j}-\frac{1}{12n},\theta_{j}+\frac{1}{12n})$,
and thus

(6.83) $\displaystyle\mathcal{C}_{1}\subset\bigcup_{j}\mathcal{D}(e^{2\pi
i\theta_{j}},\frac{\pi}{6n}).$

Let $z_{0,j}:=e^{2\pi i\theta_{j}}$. Consider
$\varphi_{j}(z):=D_{n}(\frac{2\pi}{n}(z-z_{0,j})+z_{0,j},E)$ as an analytic
function on $\mathcal{D}_{j}:=\mathcal{D}(z_{0,j},1)$. Since
$\theta_{j}\notin\mathcal{B}_{n}$, we have

(6.84) $\displaystyle\log|\varphi_{j}(z_{0,j})|=n\cdot u_{n}(e^{2\pi
i\theta_{j}})\geq nL_{n}-n^{1-\gamma_{2}}.$

Also since for $z\in\mathcal{D}_{j}$, we have
$\left|\frac{2\pi}{n}(z-z_{0,j})+z_{0,j}\right|\in[1-\frac{2\pi}{n},1+\frac{2\pi}{n}]$,
hence

(6.85) $\displaystyle\frac{2\pi}{n}(z-z_{0,j})+z_{0,j}\in A_{e^{4\pi/n}}.$

Hence by (2.14), and Lemma 2.3, we have for $z\in\mathcal{D}_{j}$,

(6.86) $\displaystyle\log|\varphi_{j}(z)|\leq\sup_{w\in A_{e^{4\pi/n}}}n\cdot
v_{n}(w)\leq\sup_{|\varepsilon|\leq 2/n}nL_{n}(\varepsilon)+C(\log
n)^{C_{0}}.$

By Theorem 2.4 and (6.76), we have

(6.87) $\displaystyle\sup_{|\varepsilon|\leq 2/n}L_{n}(\varepsilon)\leq
L_{n}+\frac{C}{n}.$

Hence for $z\in\mathcal{D}_{j}$, by (6.86) and (6.87), we have

(6.88) $\displaystyle\log|\varphi_{j}(z)|\leq nL_{n}+C(\log n)^{C_{0}}.$

By Lemma 6.3 with $H=n^{\gamma_{2}/2}$ and the estimates in (6.84) and (6.88),
there exists $\mathcal{P}_{j}\subset\mathcal{D}_{j}$,
$\mathcal{P}_{j}\in\mathrm{Car}(H,K)$, $K=CH(M-m)$ such that

(6.89) $\displaystyle\log|\varphi_{j}(z)|>nL_{n}+C(\log n)^{C_{0}}-CH(C(\log
n)^{C_{0}}+n^{1-\gamma_{2}})>nL_{n}-C_{1}n^{1-\frac{1}{2}\gamma_{2}},$

for some constant $C_{1}$ and any
$z\in\mathcal{D}(z_{0,j},1/6)\setminus\mathcal{P}_{j}$. Next we show

######  Lemma 6.4.

For any $\theta_{0}$, such that

(6.90) $\displaystyle u_{n}(e^{2\pi
i\theta_{0}})<L_{n}-C_{1}n^{-\frac{1}{2}\gamma_{2}},$

there exists a zero $w_{\ell}$ of $D_{n}(z)$, such that

(6.91) $\displaystyle|e^{2\pi
i\theta_{0}}-w_{\ell}|\leq\frac{4\pi}{n}e^{-n^{\gamma_{2}/2}}.$

###### Proof.

By (6.83), there exists $\theta_{j_{*}}$, $1\leq j_{*}\leq j_{n}$, such that

$|e^{2\pi i\theta_{0}}-e^{2\pi i\theta_{j_{*}}}|\leq\frac{\pi}{6n},$

which implies

(6.92) $\displaystyle z_{0,j_{*}}+\frac{n}{2\pi}(e^{2\pi
i\theta_{0}}-z_{0,j_{*}})\in\mathcal{D}_{j_{*}}(z_{0,j_{*}},\frac{1}{12}).$

Then by (6.90) and that

$\log|\varphi_{j_{*}}(z_{0,j_{*}}+\frac{n}{2\pi}(e^{2\pi
i\theta_{0}}-z_{0,j_{*}}))|=n\cdot u_{n}(e^{2\pi i\theta_{0}}),$

whence

(6.93) $\displaystyle\log|\varphi_{j_{*}}(z_{0,j_{*}}+\frac{n}{2\pi}(e^{2\pi
i\theta_{0}}-z_{0,j_{*}}))|<nL_{n}-C_{1}n^{1-\frac{1}{2}\gamma_{2}}.$

By (6.89), it is necessary that

$z_{0,j_{*}}+\frac{n}{2\pi}(e^{2\pi
i\theta_{0}}-z_{0,j_{*}})\in\mathcal{P}_{j_{*}}\subset\bigcup_{k=1}^{k_{*}}\mathcal{D}(z_{k,j_{*}},r_{k,j_{*}}).$

Let $k_{0}$ be such that

(6.94) $\displaystyle z_{0,j_{*}}+\frac{n}{2\pi}(e^{2\pi
i\theta_{0}}-z_{0,j_{*}})\in\mathcal{D}_{j_{*}}(z_{0,j_{*}},\frac{1}{12})\cap\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}}).$

Then since $r_{k_{0},j_{*}}\ll 1/6$,

(6.95)
$\displaystyle\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}})\subset\mathcal{D}_{j_{*}}(z_{0,j_{*}},\frac{1}{6}).$

Hence by (6.89), we have

(6.96) $\displaystyle\log|\varphi_{j}(z)|\geq
nL_{n}-C_{1}n^{1-\frac{1}{2}\gamma_{2}},\text{ for
}z\in\partial\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}}).$

Assume $\phi_{j}(z)\neq 0$ for
$z\in\mathcal{D}_{j_{*}}(z_{0,j_{*}},\frac{1}{6})$. Then, by the maximal
principle for harmonic functions, it follows from (6.96) that

(6.97) $\displaystyle\log|\varphi_{j}(z)|\geq
nL_{n}-C_{1}n^{1-\frac{1}{2}\gamma_{2}},\text{ for
}z\in\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}})$

But this leads to a contradiction with (6.92) and (6.93). Hence

$\varphi_{j}(\tilde{z})=D_{n}\big{(}\frac{2\pi}{n}(\tilde{z}-z_{0,j_{*}})+z_{0,j_{*}},E\big{)}=0$

for some $\tilde{z}\in\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}})$. This
implies

(6.98)
$\displaystyle\mathcal{D}(z_{k_{0},j_{*}},r_{k_{0},j_{*}})\ni\tilde{z}=z_{0,j*}+\frac{n}{2\pi}(w_{\ell}-z_{0,j_{*}}),\text{
for some }\ell.$

Combining (6.94) with (6.98), we have

$\displaystyle\frac{n}{2\pi}\left|w_{\ell}-e^{2\pi i\theta_{0}}\right|\leq
2r_{k_{0},j_{*}}\leq 2e^{-H}=2e^{-n^{\gamma_{2}/2}}.$

This proves (6.91). ∎

By Lemma 6.4 and Theorem 1.2, we have

(6.99)
$\displaystyle\mathcal{B}_{n}\subset\bigcup_{\ell=1}^{N^{\prime}}\\{\theta\in\mathbb{T}:\,|e^{2\pi
i\theta}-w_{\ell}|\leq e^{-n^{\gamma_{2}/2}}\\}.$

By Fact 6.1,

(6.100)
$\displaystyle\bigcup_{\ell=1}^{N^{\prime}}\\{\theta\in\mathbb{T}:\,|e^{2\pi
i\theta}-w_{\ell}|\leq
e^{-n^{\gamma_{2}/2}}\\}=\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup(-(n-1)\alpha-
U_{j})),$

for some $N^{\prime\prime}\leq\frac{N^{\prime}}{2}+1$, as claimed by Lemma
6.2. ∎

###  6.2. Proof of Anderson localization

To prove Anderson localization, by Shnol’s theorem [B, Sim, Sch], it suffices
to show that any generalized eigenfunction $\phi$ with the property that

(6.101) $\displaystyle\max(|\phi_{0}|,|\phi_{-1}|)=1,\text{ and
}|\phi_{y}|\leq C|y|,$

decays exponentially. In the following, let $\phi$ be a solution of
$H_{\alpha,\theta}\phi=E\phi$, satisfying (6.101). Combining Theorem 4.4 (with
$\kappa=1$) with Lemma 6.2 yields

$\displaystyle\mathcal{B}_{n}\subset\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup(-(n-1)\alpha-
U_{j})),$

with $N^{\prime\prime}\leq n+C\varepsilon_{2}^{-1}n^{1-\gamma_{1}}$.
Furthermore, for each $j$, one has the measure estimate

(6.102) $\displaystyle|U_{j}|\leq e^{-n^{\gamma_{2}}}.$

For $x\in\mathbb{R}$, let $[x]$ be the integer part of $x$.

######  Lemma 6.5.

For any $n$ large enough, and any $y\in\mathbb{Z}$ such that111The proof for
negative $y$ is analogous by symmetry. $n<y<10n$, let

(6.103) $\displaystyle I_{1}:=$
$\displaystyle[-[\frac{7}{8}n],-[\frac{1}{8}n]]$ (6.104) $\displaystyle
I_{2}:=$ $\displaystyle[y-[\frac{7}{8}n],y-[\frac{1}{8}n]]$

There exists $\ell\in I_{1}\cup I_{2}$ such that

$\displaystyle\theta+\ell\alpha\notin\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup((n-1)\alpha-
U_{j})).$

###### Proof.

Note that the cardinality

(6.105) $\displaystyle\\#I_{1}+\\#I_{2}\geq\frac{3}{2}n-2\geq
N^{\prime\prime},$

for $n$ large enough. It suffices to prove each pair $U_{j}\cup(-(n-1)\alpha-
U_{j})$ consists of at most one point in $\\{\theta+\ell\alpha\\}_{\ell\in
I_{1}\cup I_{2}}$. We argue by contradiction, suppose there exist
$\ell_{1},\ell_{2}$ such that

$\displaystyle\theta+\ell_{1}\alpha\in U_{j},\text{ and
}\theta+\ell_{2}\alpha\in U_{j}.$

Then, since $\alpha\in\mathrm{DC}_{c,a}$ and that $|\ell_{1}-\ell_{2}|<11n$,

$\displaystyle|U_{j}|\geq\|\theta+\ell_{1}\alpha-(\theta+\ell_{2}\alpha)\|_{\mathbb{T}}=\|(\ell_{1}-\ell_{2})\alpha\|_{\mathbb{T}}\geq\frac{c^{\prime\prime}}{n(\log
n)^{a}},$

for some constant $c^{\prime\prime}$ depending on $c$. But this contradicts
with (6.102). The case when

$\displaystyle\theta+\ell_{1}\alpha\in(-(n-1)\alpha-U_{j}),\text{ and
}\theta+\ell_{2}\alpha\in(-(n-1)\alpha-U_{j}).$

is similar. Suppose there exist $\ell_{1},\ell_{2}$ such that

$\displaystyle\theta+\ell_{1}\alpha\in U_{j},\text{ and
}\theta+\ell_{2}\alpha\in(-(n-1)\alpha-U_{j}).$

Since $\theta\in(\Theta_{c^{\prime},b})^{c}$, for $k$ large enough, one has

$\displaystyle\|2\theta+k\alpha\|_{\mathbb{T}}\geq\frac{c^{\prime}}{|k|^{b}}.$

Using that $n/4\leq\ell_{1}+\ell_{2}+n\leq 11n$, we infer that

$\displaystyle|U_{j}|\geq\|\theta+\ell_{1}\alpha-(-\theta-\ell_{2}\alpha-(n-1)\alpha)\|_{\mathbb{T}}=\|2\theta+(\ell_{1}+\ell_{2}+n-1)\alpha\|_{\mathbb{T}}\geq\frac{\tilde{c}}{n^{b}},$

for some constant $\tilde{c}$ depending on $c^{\prime}$. This contradicts with
(6.102) again. Thus the claimed results hold. ∎

Next, we show the following.

######  Lemma 6.6.

For any $\ell\in I_{1}$, one has
$\theta+\ell\alpha\in\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup(-(n-1)\alpha-
U_{j}))$.

###### Proof.

Argue by contradiction. Suppose there exists $\ell_{1}\in I_{1}$ such that

$\theta+\ell_{1}\alpha\notin\bigcup_{j=1}^{N^{\prime\prime}}(U_{j}\cup(-(n-1)\alpha-
U_{j})).$

By Lemma 6.2, it is necessary that
$\theta+\ell_{1}\alpha\notin\mathcal{B}_{n}$, which implies

(6.106)
$\displaystyle\frac{1}{n}\log|D_{n}(\theta+\ell_{1}\alpha)|=u_{n}(e^{2\pi
i(\theta+\ell_{1}\alpha)})\geq L_{n}-n^{-\gamma_{2}/2}\geq L-\varepsilon,$

where we used that $L_{n}\geq L$ due to Theorem 2.4. Let
$\ell_{2}:=\ell_{1}+n-1$. By Lemma 2.3, Theorem 2.4 and (2.14), we see that
for $k>k(\varepsilon)$ large enough

(6.107) $\displaystyle\frac{1}{k}\log|D_{k}(\theta)|=u_{k}(e^{i\theta})\leq
v_{k}(e^{i\theta})\leq L_{k}+C\frac{(\log k)^{C_{0}}}{k}\leq L+\varepsilon.$

Combining (6.106), (6.107) with (2.19),

$\displaystyle|\phi_{0}|\leq$
$\displaystyle\frac{|D_{\ell_{2}}(\theta+\alpha)|}{|D_{n}(\theta+\ell_{1}\alpha)|}|\phi_{\ell_{1}-1}|+\frac{|D_{-\ell_{1}}(\theta+\ell_{1}\alpha)|}{|D_{n}(\theta+\ell_{1}\alpha)}|\phi_{\ell_{2}+1}|$
$\displaystyle\leq$ $\displaystyle
e^{\ell_{1}(L-20\varepsilon)}|\phi_{\ell_{1}-1}|+e^{-\ell_{2}(L-20\varepsilon)}|\phi_{\ell_{2}+1}|$
(6.108) $\displaystyle\leq$ $\displaystyle
Ce^{\ell_{1}(L-20\varepsilon)}|\ell_{1}|+Ce^{-\ell_{2}(L-20\varepsilon)}|\ell_{2}|<\frac{1}{2},$

invoking (6.101) and $\min(|\ell_{1}|,|\ell_{2}|)\geq[n/8]$. Similarly, one
shows that $|\phi_{-1}|<1/2$. Hence we arrive at a contradiction with the
assumption that $\max(|\phi_{0}|,|\phi_{-1}|)=1$. ∎

Combining Lemmas 6.5, 6.6 with Lemma 6.2 yields

######  Corollary 6.7.

There exists $\ell_{3}\in I_{2}$ such that
$\theta+\ell_{3}\alpha\notin\mathcal{B}_{n}$.

The proof of Anderson localization then follows from a similar argument as in
the proof of Lemma 6.6. Indeed, let $\ell_{4}:=\ell_{3}+n-1$. Similar to
(6.2), we have

(6.109) $\displaystyle|\phi_{y}|\leq
Ce^{-(y-\ell_{3})(L-20\varepsilon)}\ell_{3}+Ce^{-(\ell_{4}-y)(L-20\varepsilon)}\ell_{4}.$

By construction of $I_{2}$,

$\displaystyle\min(y-\ell_{3},\ell_{4}-y)\geq\Big{[}\frac{1}{8}n\Big{]}\geq\frac{1}{90}y,\text{
and }\max(\ell_{3},\ell_{4})\leq y+\frac{7}{8}n<2y.$

Plugging the above estimates into (6.109) yields

$\displaystyle|\phi_{y}|\leq e^{-\frac{1}{100}(L-20\varepsilon)y}.$

This proves the claimed result.

######  Remark 6.8.

It is possible to modify the proof to show the following asymptotics:

$\displaystyle\lim_{|y|\to\infty}-\frac{\ln(|\phi_{y}|^{2}+|\phi_{y-1}|^{2})}{2|y|}=-L.$

##  7\. Hölder regularity of the IDS

We now indicate the modifications needed of [GS2, Theorem 1.1] to prove
Theorem 1.3. The main change occurs on page 848, in terms of the zero count of
the Determinants $D_{N}$ in the annulus $A_{R}$. In fact, the $2k_{0}N$
estimate of the total number of zeros in terms of the $k_{0}=\deg V$ (where
$V$ is the potential function), is now replaced by the sharper Theorem 1.2.
Thus, [GS2, Corollary 14.14] can be improved to read

$k(\widehat{A}_{j_{0}},\zeta_{j_{0}},r^{(2)})\leq 2\kappa(E,0)\leq 2(\ell-1)$

for all energies $E\in\bigcup_{j=1}^{\ell}\mathcal{S}_{j}^{+}$. By the proof
of Theorem 1.4 on page 849 of [GS2] we conclude that that theorem holds with
$k_{0}\leq 2\kappa(E,0)$ for all $E\in\cup_{j=1}^{\ell}\mathcal{S}_{j}^{+}$.
This in turn then improves on [GS2, Lemma 17.5], with the same $k_{0}$.
Finally, this improvement of Lemma 17.5 allows for the Holder exponents stated
in Theorem 1.3, see [GS2, Section 18].

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