Eigenvalue statistics for random polymer models: Localization and delocalization

TL;DR

This paper investigates the local eigenvalue statistics of one-dimensional random polymer models, revealing different statistical behaviors at critical energies versus other energies, and connecting these findings to localization and transport phenomena.

Contribution

It proves that LES at critical energies follows a uniform clock process, while at other energies it is a Poisson process, advancing understanding of spectral statistics in random polymer models.

Findings

LES at critical energies is a uniform clock process

LES at non-critical energies is a Poisson point process

Transition in LES is sharp at critical energies

Abstract

We study the local eigenvalue statistics (LES) associated with one-dimensional lattice models of random polymers. We consider models constructed from two polymers. Each polymer is a finite interval of lattice points with a finite potential. These polymers are distributed along $\mathbb{Z}$ according to a Bernoulli distribution. The deterministic spectrum for these models is dense pure point, and is known to contain finitely-many critical energies. In this paper, we prove that the LES centered at these critical energies is described by a uniform clock process, and that the LES for the unfolded eigenvalues, centered at any other energy in the deterministic spectrum, is a Poisson point process. These results add to our understanding of these models that exhibit dynamical localization in any energy interval avoiding the critical energies [Damanik, Sims, Stolz] [De Bievre, Germinet], and nontrivial transport for wave packets with initial states supported at an integer point [Jitomirskaya, Schultz-Baldes, Stolz]. We show that the projection of these initial states onto spectral subspaces associated with any energy interval that contains all of the critical energies exhibit nontrivial transport, refining the connection between nontrivial transport and the critical energies. Finally, we also prove that the transition in the unfolded LES is sharp at the critical energies.

Equations

$$V_{\omega }(2n)=V_{\omega }(2n+1),n\in \mathbb{Z}.$$

$$M_q(T):=\frac{1}{T}{\int }_0^{\infty }{e}^{-\frac{t}{T}}⟨{\delta }_0,e^{i{H}_{\omega }t}|X{|}^q{e}^{-i{H}_{\omega }t}{\delta }_0⟩dt\ge C_{\alpha }{T}^{q-\frac{1}{2}-\alpha },a.s.,$$

$${\hat{t}}_{\pm }$$

$${\hat{v}}_{\pm }$$

$$t_{\omega }$$

$$v_{\omega }$$

$$(H_{\omega }\psi )(n)$$

$$t(n+1)\psi (n+1)$$

$$=$$

$$\displaystyle\left(\begin{array}[]{c}t(n+1)\psi(n+1)\\ \psi(n)\end{array}\right)$$

$$=$$

$$T_{v,t}$$

$$T_{\pm }^E$$

$$T_{\omega }^E(k,m)=T_{{\omega }_{k-1}}^E\dots T_{{\omega }_m}^E,k>m.$$

$$L(E)$$

$$\mu (B):=\underset{L\to \infty }{\lim }\frac{1}{L}Tr(P_{H_{\omega }^L}(B)),$$

$$N(E)={\int }_{-\infty }^{E}d\mu (s).$$

$${\xi }_L:={\xi }_L^{E_0}:=\sum _{j=1}^L{\delta }_{L(N(E_j(L))-N(E_0))}.$$

$$\mathbb{E}\{e^{-{\xi }^{E_0}(f)}\}={\int }_0^{1}d\mu (\theta )e^{-{\sum }_{n\in \mathbb{Z}}f(n-\theta )},$$

$$\mathbb{E}\{e^{-{\xi }^{E_0}(f)}\}=e^{{\int }_{\mathbb{R}}(e^{-f(x)}-1)d\nu (x)},$$

$$\dots E_{-2}^{\prime }(L)<E_{-1}^{\prime }(L)<E_j^c\le E_0^{\prime }(L)<E_1^{\prime }(L)<\dots .$$

$$N(E_{m+1}^{\prime }(L))-N(E_m^{\prime }(L)).$$

$$N(E_{m+1}^{\prime }(L))-N(E_m^{\prime }(L)):=n(E_j^c)(E_{m+1}^{\prime }(L)-E_m^{\prime }(L))+o(L^{-1}).$$

$$\underset{L\to \infty }{\lim }n(E_k^c)L(E_{j+1}^{\prime }(L)-E_j^{\prime }(L))=1,\text{for any }j\in \mathbb{Z},a.s.$$

$${\xi }_L:=\sum _{j=1}^L{\delta }_{n(E_k^c)L(E_j(L)-E_k^c)}.$$

$$T_{\pm }^E$$

$$T_{\omega }^E(k,m)$$

$$MT_{\pm}^{E_{c}}M^{-1}=\left(\begin{array}[]{cc}\cos\eta_{\pm}&-\sin\eta_{\pm}\\ \sin\eta_{\pm}&\cos\eta_{\pm}\end{array}\right).$$

$$|⟨e^{ik{\eta }_{\pm }}⟩|<1,\text{mbox}forallk\in \mathbb{Z}\backslash \{0\};$$

$$0<N(E_c)<1.$$

$$n(E_c)L\left(E_{j+1}^{\prime }(L)-E_j^{\prime }(L)\right)\to 1,a.s.$$

$$|⟨e^{ik{\eta }_{\pm }}⟩{|}^2$$

Full text

1. 1 Introduction and statement of the problem

2. 1.1 The random dimer model

3. 1.2 Random polymer models

4. 1.3 Transfer matrices for RPM

5. 1.4 Main results on eigenvalue statistics for RPM

6. 1.4.1 Local operators and the density of states

7. 1.5 Outline

8. 1.6 Notation

9. 1.7 Acknowledgement

10. 2 Delocalization: LES is uniform clock at critical energies

11. 2.1 The main result on the LES at critical energies

12. 2.2 Modified Prüfer variables

13. 2.3 The characteristic function of $\xi_{L}$

14. 2.4 First convergence result for $\xi_{L}$

15. 2.5 Strong clock behavior

16. 2.6 Verification of Assumptions

17. 2.7 Sharpness of the LES transition

18. 2.8 Summary: Clock statistics at critical energies

19. 3 Localization: LES is Poisson away from critical energies

20. 3.1 Outline of the proof

21. 3.2 Positivity of the Lyapunov exponent

22. 3.3 Wegner estimate

23. 3.4 Bootstrap MSA and SULE bounds

24. 3.5 Minami’s estimate for AB RPM

25. 3.6 Decomposition into subsystems

26. 3.7 Completion of the proof of Theorem 3.3: Poisson limit theorem

27. A Properties of free and modified Prufer phases from [18]

28. B Proof of the increasing property of $x\mapsto\Psi_{L}(x)$

29. C Bootstrap MSA for the AB RPM

30. D Bourgain’s Lemma

31. E Transport and delocalization

32. F Sharpness of the transition

1. Introduction and statement of the problem

Random Schrödinger operators (RSO) in dimension $d\geq 3$ are expected to exhibit a phase transition with respect to the energy for fixed, nonzero disorder. At low energy, or at band-edge energies, the models are expected to exhibit localization, where at high energy, or near the center of the band, there should be delocalized behavior. This general conjecture remains unproven but several ergodic models have been proven to exhibit both localization and delocalization. In one-dimension, the random dimer model on the lattice and, more generally, random polymer models, have been proven to have both properties [15, 18].

This article presents the proof of a sharp localization-delocalization transition for the unfolded local eigenvalue statistics for the random polymer model. Although these models have dense pure point spectrum almost surely, with exponentially decaying eigenvectors, they also have a finite number of critical energies. These are energies at which the Lyapunov exponent vanishes. We prove that the unfolded LES centered at a critical energy is a uniform clock process, and that the unfolded LES centered at any other energy in the deterministic spectrum $\Sigma$ is a Poisson point process.

1.1. The random dimer model

The one-dimensional random dimer model [15] is a discrete random Schrödinger operator on $\ell^{2}({\mathbb{Z}})$ of the form $H_{\omega}=L+V_{\omega}$, where $L$ is the nearest-neighbor, finite-difference Laplacian. The random potential $V_{\omega}$ is defined by a sequence of random variables $V_{\omega}(n)$, for $n\in{\mathbb{Z}}$, with

[TABLE]

A dimer consists of two neighboring sites, say $2n$ and $2n+1$, and a single Bernoulli random variable, say $V_{\omega}(2n)$, taking values in $\{-V,V\}$, $V\in(0,1]$. The random variables $\{V_{\omega}(2n)\}$ are independent and identically distributed (iid). The Anderson-Bernoulli random dimer model (AB RDM) is one for which the probability measure associated with $V_{\omega}(0)$, and consequently all $V_{\omega}(2n)$, is a Bernoulli measure $d\mu(s)=(p\delta(s-V)+(1-p)\delta(s+V))~ds$, with $p\in(0,1)$, and $V\in(0,1]$ . Another way to describe the model is that it is formed from two dimers, one having value $V$ and the other having value $-V$. They are arranged randomly along ${\mathbb{Z}}$, the occurrence of one or the other controlled by a Bernoulli random variable.

The AB RDM, in contrast to the AB random Schrödinger operator (RSO), is characterized by the existence of a pair of critical energies at which the Lyapunov exponent vanishes. For the AB RDM described above, the two critical energies occur at $E_{c}=\pm V$. The standard AB RSO does not have critical energies. The single polymer (a single site) transfer matrices for two different sites never commute almost surely, whereas for the RDM, the two different polymer transfer matrices (products of the single-site transfer matrix over one polymer) commute at the critical energies.

An important consequence of this commutation is that Lyapunov exponent $L(E)$ vanishes at the critical energies. This lead to the conjecture that there are extended states associated with critical energies. In their important study, De Bièvre and Germinet [15] prove that there is dynamical localization on every energy interval not containing a critical energy $E_{c}$. After this work, Jitomirskaya, Schultz-Baldes, and Stolz [18] proved that these models (and the more general random polymer models (RPM)) also exhibit dynamical delocalization in the form of nontrivial transport. The form of dynamical localization proved in [18] is as follows. For any $\alpha>0$, there is a finite constant $C_{\alpha}>0$ so that:

[TABLE]

where $\delta_{0}$ is the function equal to one at $0\in{\mathbb{Z}}$, and zero everywhere else.

Such anomalous transport associated with critical energies had been predicted by physicists (see [15] for references). We note that this estimate may be localized in energy intervals containing all the finitely-many critical energies since, in the complementary set, dynamical localization holds (see Appendix E). Hence, the AB RDM exhibits both almost sure pure point spectrum throughout its spectrum almost surely, and dynamical delocalization.

In this paper, we study the unfolded local eigenvalue statistics for the more general AB RPM, as defined in [18], and prove that there is an energy-dependent transition from Poisson statistics to uniform clock statistics at the critical energies. We also prove that the rescaled eigenvalue spacing exhibits strong clock behavior [25] in this case. Consequently, the transition between these two phases as a function of energy is sharp. These models are the first nontrivial applications of unfolded eigenvalue statistics whose use is necessitated due to the fact that the integrated density of states is not Lipschitz continuous.

1.2. Random polymer models

In this section, we describe the random polymer model (RPM). Following [18, section 2], a polymer of length $L$ with fixed potential $v$ is a finite real sequence $\{v(j),v(j+1),\ldots,v(j+L-1)\}$. We refer to this as polymer $L$. A deterministic polymer model on ${\mathbb{Z}}$ is constructed from two polymers, with possibly different lengths $L_{\pm}$ and different potentials $v_{\pm}$, laid sequentially along ${\mathbb{Z}}$. By randomizing the choice of the polymer in this construction, we obtain the random polymer model (RPM). The randomization is given by Bernoulli random variables: polymer $L_{+}$ occurs with probability $p_{+}:=p$ and polymer $L_{-}$ occurs with probability $p_{-}:=(1-p)$, with $p\in(0,1)$. We refer to this as the Anderson Bernoulli random polymer model (AB RPM). We also include a random hopping coefficient $t_{\omega}(n)$, as in [18]. These models generalize the RDM described above.

In more detail, we associate with each polymer of length $L_{\pm}\in{\bf N}$, the associated $L_{\pm}$-tuples of hopping terms and potentials. We designate these by

[TABLE]

We consider laying these polymers head-to-tail along the lattice ${\mathbb{Z}}$ in a random order to create a random juxtaposition of the two polymers. Each configuration of polymers is labeled by a two-sided sequence $\omega:=(\omega_{\ell})_{\ell\in{\mathbb{Z}}}$ of $\pm$-signs: $\omega_{\ell}=\pm$. . We define sequences $t_{\omega}$ and $v_{\omega}$ by

[TABLE]

Given these definitions, the discrete Schrödinger operator for a RPM is defined by

[TABLE]

acting on $\ell^{2}({\mathbb{Z}})$. As mentioned above, the random variables $\omega_{\ell}$ are iid and distributed according to a Bernoulli distribution with the probability given by ${{\mathbb{P}}}(\omega_{\ell}=\pm)=p_{\pm}$, where $p_{\pm}$ satisfy $p_{\pm}\in(0,1)$, and $p_{+}+p_{-}=1$.

The complete construction of the probability space $({\mathbb{P}},\Omega)$ requires a randomization of the values at the origin, $\hat{v}_{\omega_{0}}(0)$ and $\hat{t}_{\omega_{0}}(0)$. The details are given in [18, section 4.1] and are not essential for the proofs on unfolded LES, but from which standard results known for ergodic Schrödinger operators are also valid for $H_{\omega}$. In particular, it follows that there is a closed set $\Sigma(\subset{\mathbb{R}})$ such that $\sigma(H_{\omega})=\Sigma$, a.s.

1.3. Transfer matrices for RPM

Transfer matrices allow an alternative to the Schrödinger equation $H\psi=E\psi$ in one-dimension. The Schrödinger equation on $\ell^{2}({\mathbb{Z}})$ has the form.

[TABLE]

This implies an equivalent formulation to (1.8):

[TABLE]

where the transfer matrix $T_{v,t}$ is defined by

[TABLE]

Crucial for our discussion are the two polymer transfer matrices associated with each one of the polymers $L_{\pm}$. We denote these single polymer transfer matrices by $T^{E}_{\pm}$. They are given by:

[TABLE]

Definition 1.1**.**

An energy $E_{c}\in{{\mathbb{R}}}$ is critical for the AB RPM $H_{\omega}$ in (1.7) if the single polymer transfer matrices $T^{E}_{\pm}$, defined in (1.23), satisfy:

1. (1)

$|Tr(T^{E_{c}}_{\pm})|<2$, or $T^{E_{c}}_{\pm}=\pm I$, 2. (2)

$[T^{E_{c}}_{+},T^{E_{c}}_{-}]=0$.

The transfer matrix over several polymers is given by the product of the individual polymer transfer matrices. For a configuration $\omega$, we write

[TABLE]

If $m>k$, we have $T^{E}_{\omega}(k,m)=T^{E}_{\omega}(m,k)^{-1}$, and $T^{E}_{\omega}(k,k)=Id$. The Lyapunov exponent is given by

[TABLE]

An important property of a critical energy $E_{c}$ is that the Lyapunov exponent vanishes at that critical energy: $L(E_{c})=0$. Since the Lyapunov exponent is positive on energies in the localization regime, the vanishing of the Lyapunov exponent at a critical energy might imply nontrivial transport for wave packets with initial conditions containing a critical energy.

1.4. Main results on eigenvalue statistics for RPM

Our main results concern the local eigenvalue statistics (LES) for the discrete Schrödinger operator describing random polymer models on $\ell^{2}({\mathbb{Z}})$.

1.4.1. Local operators and the density of states

In order to describe the LES, we define local Schrödinger operators associated with the interval $[0,L-1]\cap{\mathbb{Z}}$. We let $H_{\omega}^{L}$ is the restriction of $H_{\omega}$ to $\{0,1,\cdots,L-1\}$ with Dirichlet boundary conditions at $n=-1$ and at $n=L$. We label the eigenvalues of $H_{\omega}^{L}$ by $\{E_{j}(L)\}_{j=1}^{L}$, in increasing order $E_{1}(L)<E_{2}(L)<\cdots<E_{L}(L)$.

The density of states measure (DOSm) $\mu$ is defined for any bounded Borel subset $B\subset{\mathbb{R}}$ as the limit

[TABLE]

where $P_{H}(B)$ is the spectral projection of an operator $H$ associated to the Borel set $B$. This limit exists weakly due to the ergodicity of the RSO. The integrated density of states ${N}(E)$ is the cumulative distribution function of $\mu$ and is given by

[TABLE]

We will abuse the notation and write $N(I):=\mu(I)$ for the DOS measure of a measurable subset $I\subset{\mathbb{R}}$.

It is known that ${N}(E)$ is Hölder continuous for the AB model [10]. However, we do not know if the density of states measure (DOSm) is absolutely continuous with respect to Lebesgue measure. In the case that the single-site probability measure is absolutely continuous with respect to Lebesgue measure, the density of states function (DOSf) exists and is positive. The DOSf plays a role in the LES in the localization regime. For this reason, we consider the following point process $\xi_{\omega}^{L}$ on ${{\mathbb{R}}}$ centered at $E_{0}\in\Sigma$ formed from the unfolded eigenvalues of $H_{\omega}^{L}$:

[TABLE]

We refer to this random point process, and its limit $\xi^{E_{0}}:=\lim_{L\to\infty}\xi_{L}^{E_{0}}$ as $L\rightarrow\infty$, if it exists, as the local eigenvalue statistics (LES) of $H_{\omega}$ centered at $E_{0}$. $E_{0}$ is called the center energy or the reference energy. For a point process $\xi$ and any $f\in C_{0}({\mathbb{R}})$, we write $\xi(f):=\int f(s)d\xi(s)$. For the AB RPM, we will consider the LES in two mutually-exclusive cases: 1) when $E_{0}\in\mathcal{C}$, the set of critical energies, and 2) when $E_{0}\in\Sigma\backslash\mathcal{C}$.

We are interested in two random point processes.

Definition 1.2**.**

Let $E_{0}\in{\mathbb{R}}$ be fixed. The LES $\xi^{E_{0}}$ centered at $E_{0}$ is clock if there exists a random variable $\theta$ on the unit interval $[0,1)$ with distribution $\mu$ such that the characteristic function satisfies

[TABLE]

for all $f\in C_{0}({\mathbb{R}})$. We also say $\xi^{E_{0}}$ obeys clock $(\mu)$. We call the process a uniform clock process on the interval $[0,1)$ if $\theta$ is uniformly distributed on $[0,1)$ and we write clock $unif[0,1)$.

Definition 1.3**.**

Let $E_{0}\in{\mathbb{R}}$ be fixed. The LES $\xi^{E_{0}}$ centered at $E_{0}$ is Poisson with Lévy measure $\nu$ if the characteristic function satisfies

[TABLE]

for all $0\leq f\in C_{0}({\mathbb{R}})$.

The main result of this article is to characterize the unfolded LES for each $E_{0}\in\Sigma$. Our results are summarized in the following theorem. Assumptions and precise statements may be found in Theorem 2.1 and Theorem 3.3.

Theorem 1.4**.**

Let $H_{\omega}$ be the random polymer model with Bernoulli single-site probability measure. Let $\mathcal{C}:=\{E^{c}_{j}~|~j=1,\ldots,N\}$ be the finite set of critical energies.

1. (1)

Delocalization:* The unfolded LES centered at a critical energy $E^{c}_{j}$ is a uniform clock process.* 2. (2)

Localization*: The unfolded LES centered at an energy $E_{0}\in\Sigma\backslash\mathcal{C}$ is a Poisson point process with intensity measure given by the Lebesgue measure on ${\mathbb{R}}$.*

There are only few cases in which a Hamiltonian shows more than two completely different LES at different reference energies. The only example known so far is the 1D Schrödinger operator with critically decaying random potential[24, 22, 30], where LES is the Sineβ process where $\beta=\beta(E_{0})$ is a non-constant smooth function of the reference energy $E_{0}\in\Sigma$ which takes all values in $(0,\infty)$ [30]. Theorem 1.4 now implies that AB RPM is another example.

We obtain another characterization of the limited behavior of the eigenvalues near a critical energy $E_{j}^{c}\in\mathcal{C}$. We re-number the eigenvalues of $H_{\omega}^{L}$ as

[TABLE]

We are interested in the rescaled, unfolded eigenvalue spacing around $E_{j}^{c}$:

[TABLE]

This can be written in a simplified manner. It follows from [18, Theorem 2] that the IDS is differentiable at $E_{j}^{c}$, with the derivative $n(E_{j}^{c}):={{N}}^{\prime}(E_{j}^{c})>0$. Consequently, the unfolded eigenvalues in (1.32) may be expressed as

[TABLE]

Theorem 1.5**.**

The local eigenvalue spacing near any critical energy $E_{k}^{c}\in\mathcal{C}$ satisfies

[TABLE]

Remark 1.6**.**

(1) Theorem 1.5 implies that eigenvalue spacings asymptotically show a repulsion, and such behavior is called the clock behavior. There are a couple of examples showing clock behavior : (i) The ergodic Jacobi matrices, if $E_{0}$ lies in absolutely continuous spectrum [4], (ii) 1D Schrödinger operators with supercritical decaying random or more general potentials [22, 30, 12], and (iii) 1D Schrödinger operators with sparse random potentials where $E_{0}$ lies in singular continuous spectrum [9].

*Our result is the first example of clock LES for a random Schrödinger operator with pure point spectrum.

(2) We believe that if the LES is clock$(\mu)$, then $\mu$ is generically uniform for translation invariant systems. Note that it is not the case for 1D Schrödinger operator with supercritical random decaying potential [22].*

The method of proof of part (2) of Theorem 1.4 also applies to the standard Anderson-Bernouilli model. So, we mention a corollary for the one-dimensional discrete AB RSO.

Corollary 1.7**.**

Let $H_{\omega}$ be the random Anderson model with Bernoulli single-site probability measure. Assume that $H_{\omega}$ satisfies Assumption 3.2. Then the unfolded LES centered at any energy $E_{0}\in\Sigma$ is a Poisson point process with intensity measure given by the Lebesgue measure on ${\mathbb{R}}$.

1.5. Outline

The main results of this paper apply to the Anderson-Bernoulli (AB) random polymer model (RPM) as introduced by [18], generalizing the AB random dimer model studied by [15]. In section 2, we prove the first part of Theorem 1.4 on uniform clock statistics at critical energies for AB RPM. We also prove Theorem 1.5 on the strong clock behavior of the eigenvalue spacing. To prove these theorems, we basically follow the strategy in [22]. The Poisson characterization of the unfolded LES for the AB RPM at non-critical energies is presented in section 3, roughly following ideas in [29, 17] on the eigenvalue statistics. Because the single-site probability measure is Bernoulli, we prove a Minami estimate partially using the ideas in [7, 20]. We also use bootstrap multiscale analysis (MSA) of Germinet and Klein [16] in order to obtain localization bounds. Several appendices summarize these results on bootstrap and results of [18] used in the main proofs.

1.6. Notation

We use $C>0$ to denote a finite, nonnegative constant whose value may change from line-to-line, but is independent of the parameters of the problem like $L$ and $E$.

1.7. Acknowledgement

We thank Xiaolin Zeng for discussions on the topics of this paper. PDH is partially supported by Simons Foundation Collaboration Grant for Mathematicians No. 843327. FN is partially supported by JSPS KAKENHI Grant Number 20K03659.

2. Delocalization: LES is uniform clock at critical energies

This section is devoted to the proof that the LES are given by a uniform clock process when the approximating processes are centered at a critical energy, by following the strategy in [22]. We compute the limit as $L\rightarrow\infty$ of the characteristic function ${\mathbb{E}}\{e^{-\xi_{L}(f)}\}$, where the process $\xi_{L}$ is defined by the unfolded eigenvalues relative to a critical energy $E_{k}^{c}\in\mathcal{C}$:

[TABLE]

Our calculations hold for any critical energy in $\mathcal{C}$. Consequently we write $E_{c}$ for any $E_{k}^{c}$. We also prove the strong clock characterization that the limit as $L\to\infty$ of the rescaled eigenvalue spacing is a constant.

We recall that as the DOS function $n(E):={N}^{\prime}(E)$ exists at $E=E_{c}$, so that by (1.33), the limit of $\xi_{L}$ in (2.1) coincides with that of (1.28). We also recall that for any random variable $c=(c_{\pm})$ depending on $\omega_{\pm}$, we define the average $\langle c_{\pm}\rangle:=pc_{+}+(1-p)c_{-}$.

2.1. The main result on the LES at critical energies

We recall that the basic transfer matrices for the RPM are defined in (1.16) - (1.23). In particular, the one-polymer transfer matrices are:

[TABLE]

We will also need transfer matrices that map blocks of polymers to other blocks. The block transfer matrix from the $m^{th}$-block to the $k^{th}$-block is given by:

[TABLE]

We recall the definition of a critical energy in Definition 1.1. The first condition says that $E_{c}$ lies in the interior of the spectrum $\sigma(H_{\pm})$ of the corresponding periodic Schrödinger operators constructed from a single polymer $L_{+}$ or $L_{-}$.

By definition, the $L_{\pm}$-polymer transfer matrices $T^{E_{c}}_{\pm}$ can be diagonalized simultaneously. Since $|tr(T^{E_{c}}_{\pm})|<2$, there exist $\eta_{\pm}\in{\mathbb{R}}$, and an invertible matrix $M$, so that

[TABLE]

In other words, the phases $e^{i\eta_{\pm}}$ and $e^{-i\eta_{\pm}}$ are the eigenvalues of $T_{\pm}^{E_{c}}$, respectively. We prove that the LES $\xi_{L}$ converges to clock and that the unfolded eigenvalues satisfy clock behavior. This is the content of the following theorem based on two assumptions: 1) positivity of the IDS at a critical energy, and 2) an assumption on the behavior of the eigenvalues of $T_{\pm}^{E_{c}}$.

Theorem 2.1**.**

We assume that:

1. (1)

The eigenvalues $e^{i\eta_{\pm}},e^{-i\eta_{\pm}}$ of the polymer transfer matrices $T_{\pm}^{E_{c}}$, at a critical energy $E_{c}$, satisfy the condition:

[TABLE] 2. (2)

The integrated density of states at a critical energy is strictly positive:

[TABLE]

Then, the LES $\xi_{L}$ at $E_{c}$, defined in (2.1), satisfies

1. (1)

$\xi_{L}\stackrel{{\scriptstyle d}}{{\to}}clock(unif([0,1))$*, where $unif(I)$ denotes the uniform distribution on the interval $I\subset{\mathbb{R}}$. * 2. (2)

The rescaled eigenvalues, centered around $E_{c}$, exhibit strong clock behavior : for any $j\in{{\mathbb{Z}}}$, we have

[TABLE]

We make some remarks about the two assumptions of Theorem 2.1.

Remark 2.2**.**

1. (1)

Let $\triangle\eta:=\eta_{+}-\eta_{-}$. To understand condition (2.6), we compute

[TABLE]

Thus, the assumption (2.6) is satisfied if for any $k\in{\mathbb{Z}}\backslash\{0\}$, we have $k\triangle\eta\not\equiv 0\,(\rm{mod}~2\pi)$. For the dimer model, $L_{\pm}=2$, direct calculation shows that this condition is satisfied for certain $0<V<1$. 2. (2)

The positivity assumption on the IDS may be replaced by a much weaker one that the number of eigenvalues in $(-\infty,E_{c})$, $(E_{c},\infty)$ goes to infinity as $L\to\infty$, of which the positivity of IDS is a sufficient condition. But the latter one can explicitly be examined for AB RPM. In fact, by **[18, (14)]**, the IDS at a critical energy satisfies,

[TABLE]

The dimer model, $L_{\pm}=2$, with $V\leq 1$, satisfies $\eta_{+}=\pi$ and $\eta_{-}=f(V)$, with $f(V)$ explicitly computable, so that the condition is satisfied: $0<{N}(E_{c})=\frac{1}{2}<1$.

In the next subsections, we present the main technical tools necessary for the proof of Theorem 2.1.

2.2. Modified Prüfer variables

It will be convenient to use modified Prüfer variables, as in [18], to describe the LES. We consider the solution $u$ to the equation $Hu=Eu$ with initial condition

[TABLE]

The modified Prüfer variables $(R_{n}(E),\theta_{n}(E))$ are defined by

[TABLE]

where $M$ is the matrix transforming the polymer transfer matrices at a critical energy to rotations as in (2.5).

We define functions $r:{\mathbb{R}}\to{\mathbb{R}}^{+}$, and $m:{\mathbb{R}}\to{\mathbb{R}}$, by

[TABLE]

Since $e_{\theta+\pi}=-e_{\theta}$, we can take $m(\theta+\pi)=m(\theta)+\pi$. As $r(\theta+\pi)=\|Me_{\theta+\pi}\|$, it follows that $r(\theta+\pi)=r(\theta)$. Finally, these functions are smooth in $\theta$ since $\theta\mapsto e_{\theta}$ is smooth.

For any $x\in{\mathbb{R}}$, we define the fractional part $(x)_{\pi{{\mathbb{Z}}}}$, and the integer part $[x]_{\pi{{\mathbb{Z}}}}$, of $x$ modulo $\pi{\mathbb{Z}}$, by

[TABLE]

For $E\in{\mathbb{R}}$, let $\phi(E,L):=(\theta_{L}(E))_{\pi{{\mathbb{Z}}}}\in[0,\pi)$, be the fractional part, and $m(E,L)\pi:=[\theta_{L}(E)]_{\pi{{\mathbb{Z}}}}\in\pi{\mathbb{Z}}$, be the integer part, of the Prüfer phase $\theta_{L}(E)$, modulo $\pi{\mathbb{Z}}$, respectively, so that

[TABLE]

Following [22], we also define a relative Prüfer angle, divided by $\pi$, by

[TABLE]

2.3. The characteristic function of $\xi_{L}$

We first obtain a representation of the characteristic function of $\xi_{L}$ in terms of the Prufer phase.

Lemma 2.3**.**

The characteristic function of the process $\xi_{L}$ (2.1), associated with the finite-length RPM Schrödinger operator $H_{L}$, satisfies

[TABLE]

where $\Psi_{L}^{-1}$ is the inverse of the function $x\mapsto\Psi_{L}(x)$ defined in (2.13), and $m(\theta)$ is defined in (2.11).

Proof.

Let $x_{n}(L)$ be the $n^{th}$ atom of $\xi_{L}$ :

[TABLE]

Then, by Sturm oscillation theory, the modified Prüfer angle at an eigenvalue satisfies

[TABLE]

and at a critical energy, the modified Prüfer phase satisfies

[TABLE]

We refer to (A.6) for the relation between the free Prufer angle and the modified Prufer angle. From the definition (2.13), we then have

[TABLE]

Since $x\mapsto\Psi_{L}(x)$ is an increasing function (see Appendix B), we obtain

[TABLE]

As $\xi_{L}(f)=\sum_{n=1}^{L}f(x_{n}(L))$, formula (2.15) establishes the result (2.14). ∎

The following lemma from [21, Lemma 3.1] is necessary in order to consider the convergence of the characteristic function.

Lemma 2.4**.**

Let $\{\Psi_{L}(x)\}_{L}$ be a sequence of non-decreasing functions satisfying $\Psi_{L}(x)\to\Psi(x)$ pointwise, and assume that $\Psi$ is an increasing function. If $x_{L}\to x$, then

[TABLE]

Proof.

If the statement is false, we can find $\delta>0$, a point $x$, and a subsequence $L_{n}$ such that

[TABLE]

Taking a subsequence, if necessary, we can assume that

[TABLE]

Since $\Psi_{L_{n}}$ is non-decreasing,

[TABLE]

Letting $n\to\infty$, we have

[TABLE]

leading to a contradiction. ∎

2.4. First convergence result for $\xi_{L}$

Given the form of the characteristic function $\xi_{L}$ in (2.14), we compute the limit as $L\to\infty$ and obtain a first result on the clock process. We make three assumptions whose validity is discussed in Remarks 2.6.

Assumption 2.5**.**

1. (1)

The function $\Psi_{L}$ satisfies $\lim_{L\to\infty}\Psi_{L}(x)=x$, a.s. 2. (2)

There exists a random variable $\phi_{c}\in[0,\pi)$ such that the fractional part $\phi(E_{c},L)$ of the Prüfer phase $\theta_{L}(E_{c})$ converges in distribution to $\phi_{c}$. 3. (3)

The integer part $m(E_{c},L)$ of the Prüfer phase $\theta_{L}(E_{c})$ at a critical energy satisfies

[TABLE]

and

[TABLE]

almost surely.

Remarks 2.6**.**

1. (1)

Part 1, 2 is proven for RPM in section 2.6, under some conditions on $\eta_{\pm}$. 2. (2)

Part 3 follows if the number of eigenvalues of $H_{\omega}^{L}$ in $(-\infty,E_{c})$ and in $(E_{c},\infty)$ goes to infinity as $L\to\infty$. As discussed in Remark 2.2, if the IDS at the critical energy satisfies $0<N(E_{c})<1$, then part 3 follows. 3. (3)

As can be seen later in the proof, the most important condition in Assumption 2.5 is part (1). In fact, without part (2), we can show that the accumulation point of $\xi_{L}$ is a clock process with some $\mu$ (Corollary 2.8), and wituout part (3), the sum $\sum_{n\in{\bf Z}}$ in the definition of the clock process may be replaced by $\sum_{n\in E}$ for some subset $E(\subset{\mathbb{R}})$.

We first present an intermediate result asserting the convergence of the characteristic function to a clock process with probability measure $\mu$.

Theorem 2.7**.**

(Convergence to a clock process) Under Assumption 2.5, the LES $\xi_{L}$ centered at a critical energy satisfies $\xi_{L}\stackrel{{\scriptstyle d}}{{\to}}clock(\mu_{frac})$, where $\mu_{frac}$ is the distribution of the fractional part of the random variable $\frac{1}{\pi}\left(\phi_{c}-m(\frac{\pi}{2})\right)$.

Proof.

By Skorohod’s Theorem [6], we may assume that $\phi(E_{c},L)\to\phi_{c}$, a.s. We write $\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)=\left[\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)\right]+\left\{\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)\right\}$, where $\left[\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)\right]\in{\bf Z}$ and $\left\{\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)\right\}\in[0,1)$ are the integer and fractional part of $\frac{1}{\pi}\left(\phi_{c}-m\left(\frac{\pi}{2}\right)\right)$ respectively. We denote by $\mu_{frac}$ the distribution of this fractional part. Lemma 2.3 provides a representation of the expected value of $e^{-\xi_{L}(f)}$, for any $f\in C_{0}({\mathbb{R}})$:

[TABLE]

By Assumption 2.5 and Lemma 2.4, we find that the limit $L\rightarrow\infty$ of the right side of (2.16) is

[TABLE]

where we write $\theta:=\frac{1}{\pi}(\phi_{c}-m(\frac{\pi}{2})).$ ∎

The proof of Theorem 2.7 implies the following corollary, stating that any accumulation point of $\xi_{L}$ is always a clock process for some $\mu$.

Corollary 2.8**.**

If we only assume Assumption 2.5, parts 1 and 3, then any accumulation point $\xi$ of $\xi_{L}$ is a clock process with some distribution $\mu$ on $[0,1)$.

Proof.

For $f\in C_{c}({\bf R})$, Lemma 2.3 yields

[TABLE]

for sufficiently large $L$, where we set

[TABLE]

Furthermore, if $f\in C_{c}([0,1))$, since $\theta_{L}\in(0,1]$, this yields

[TABLE]

Let $\xi$ be an accumulation point of $\xi_{L}$. Then, we can find a subsequence $\{L_{k}\}_{k}$ such that $\xi_{L_{k}}\stackrel{{\scriptstyle d}}{{\to}}\xi$. Thus, as $k\to\infty$, the limit of the LHS of (2.19) exists and

[TABLE]

which implies that $\{\theta_{L_{k}}\}$ has a limit in distribution: $\lim_{k\to\infty}\theta_{L_{k}}\stackrel{{\scriptstyle d}}{{=:}}\theta$, with distribution $\mu$ supported in $[0,1]$. Taking $L_{k}\to\infty$ in (2.18) now yields

[TABLE]

Therefore, the limiting process $\xi$ is clock $(\mu)$. ∎

In the next section, we prove part 2 of Assumption 2.5 in Proposition 2.12. This will establish the first part of Theorem 2.7 on the uniform nature of the distribution of the limiting random variable $\phi_{c}$. .

2.5. Strong clock behavior

In the clock process problem, we need to study the fractional part of $\frac{1}{\pi}\theta_{L}(E_{c})$ which might not reflect the true nature of the eigenvalue statistics. To further clarify the nature of the LES, we will consider the spacing between consecutive eigenvalues. We let $\{E^{\prime}_{j}(L)\}$ denote the re-indexed eigenvalues of $H_{L}$ centered at a critical energy $E_{c}$:

[TABLE]

To show the strong clock behavior, we need slightly stronger condition than that for the convergence to the clock process :

Assumption 2.9**.**

1. (1)

$\lim_{L\to\infty}\Psi_{L}(x)=x$, a.s., locally uniformly. 2. (2)

$\lim_{L\to\infty}m(E_{c},L)=\infty$, $\lim_{L\to\infty}\left(L-m(E_{c},L)\right)=\infty$.

Theorem 2.10**.**

(Clock behavior: eigenvalue spacing) Under Assumption 2.9, the local eigenvalue spacing around a critical energy almost surely satisfies

[TABLE]

for any $j\in{\mathbb{Z}}$. This establishes part 2 of Theorem 2.1.

Proof.

By part (2) of Assumption 2.9, for any $j\in{\bf Z}$, $E^{\prime}_{j}(L)$ is well defined for sufficiently large $L$. Set $x^{\prime}_{j}(L):={n(E_{c})L}(E^{\prime}_{j}(L)-E_{c})$. Then

[TABLE]

so it suffices to show that RHS of the equation above converges to $1$. On the other hand, since $E^{\prime}_{j}(L),E^{\prime}_{j+1}(L)$ are consecutive eigenvalues, we can find $n_{j}(L)\in{\bf Z}$ such that

[TABLE]

Therefore

[TABLE]

By Lemma 2.11 below, we can find an interval $I=[a,b]$ such that $x^{\prime}_{j}(L),x^{\prime}_{j+1}(L)\in I$ for sufficiently large $L$. Then

[TABLE]

Since $\Psi_{L}(x)\stackrel{{\scriptstyle L\to\infty}}{{\to}}x$ uniformly on $I$, we are done. ∎

Lemma 2.11**.**

For any fixed $j$, the rescaled eigenvalues $x^{\prime}_{j}(L):={n(E_{c})L}(E^{\prime}_{j}(L)-E_{c})$ are bounded with respect to $L$:

[TABLE]

Proof.

The argument given here is deterministic. We compute the modified Prüfer phase. By Sturm oscillation theory,

[TABLE]

On the other hand, by the definition of $\Psi_{L}$,

[TABLE]

These two equations yield the expression

[TABLE]

Since $\phi(E_{c},L)\in[0,\pi)$, we have

[TABLE]

Suppose that $\lim_{L\to\infty}x^{\prime}_{j}(L)=\infty$. For any $\epsilon>0$, we have

[TABLE]

for sufficietly large $L$. On the other hand, since $\Psi_{L}(x)\to x$ pointwise,

[TABLE]

for sufficietly large $L$. Therefore, by (2.22)-(2.23), and the fact that $x\mapsto\Psi_{L}(x)$ is increasing by Proposition B.3, we have

[TABLE]

which contradicts to (2.22). ∎

2.6. Verification of Assumptions

In this section, we verify Assumptions 2.5 and 2.9, that is, $\lim_{L\to\infty}\Psi_{L}(x)=x$ and the convergence of $\phi(E_{c},L)$ in distribution. For this, we use the notation and approach introduced in [18, section 4.2]. The authors in [18] study the transfer matrices at energies $E_{c}+\epsilon$ near the critical energy $E_{c}$. Recalling that the matrix $M\in SL(2,{\mathbb{R}})$ maps $T_{\pm}^{E_{c}}$ to rotations, we define the related transmission $a^{\epsilon}_{\pm}$ and reflection $b^{\epsilon}_{\pm}$ coefficients by

[TABLE]

In order to study the behavior of the Prüfer variables at energies $E_{c}+\epsilon$, we define a general polymer phase shift $S_{\epsilon,\pm}(\theta)$ and amplitude $\rho_{\pm}^{\epsilon}(\theta)$, by

[TABLE]

where the matrix $M$ is defined in (2.5) and vector $e_{\theta}$ is defined in (2.10). Then, the iterated polymer phase shift satisfies

[TABLE]

By equation (2.5), we have, for any $\theta$, $\rho^{0}_{\pm}(\theta)=1$, $\eta_{\pm}=S_{0,\pm}(\theta)-\theta$ up to a multiple of $2\pi$, which is hereby fixed. Setting $\epsilon=0$ in (2.27), we obtain

[TABLE]

yielding

[TABLE]

where $\eta_{\pm}$ are defined in (2.5). Since $\rho^{0}_{\pm}(\theta)=1$, the Lyapunov exponent vanishes at $E_{c}$: $L(E_{c})=0$. Another important formula in [18] is

[TABLE]

where

[TABLE]

We first verify the part 2 of Assuption 2.5: the normalized fractional part of the Prüfer phase $\phi(E_{c},L)/\pi$ converges in distribution to a random variable with the uniform distribution on $[0,1)$.

Proposition 2.12**.**

Suppose that the eigenvalues $\eta_{\pm}$ of the polymer transfer matrices $T^{E_{c}}_{\pm}$ satisfy

[TABLE]

Then, the normalized fractional part of the Prufer phase at a critical energy satisfies

[TABLE]

Proof.

It suffices to show $\phi(E_{c},L)\stackrel{{\scriptstyle d}}{{\to}}unif[0,\pi)$. This will follow from establishing that

[TABLE]

Case 1: Suppose that $L$ is on a polymer node, that is, $L=\sum_{\ell=0}^{N-1}L_{\omega_{\ell}}$, for some $N$. Then $\theta_{L}(E_{c})=\sum_{\ell=0}^{N-1}\eta_{\omega_{\ell}}$. We note that $e^{2ik\theta_{L}(E_{c})}=e^{2ik\phi(E_{c},L)}$, for $k\in{\mathbb{Z}}$. It follows from the independence of the random variables $\omega_{\ell}$, that for any $k\in{\mathbb{Z}}\backslash\{0\}$,

[TABLE]

by the hypothesis on the transfer matrix eigenvalues (2.30).

Case 2: General case : Let $L^{\prime}$ be the maximum number of polymer nodes less than or equal to $L$. Then,

[TABLE]

Case 1 shows that ${{\mathbb{E}}}[\exp(2ik\theta_{L^{\prime}}(E_{c}))]\to 0$. Furthermore, the random variables $\triangle\theta$ and $\theta_{L^{\prime}}(E_{c})$ are independent, so we conclude that $\theta_{L}(E_{c})\stackrel{{\scriptstyle d}}{{\to}}unif[0,\pi)$ in this case too. ∎

When the system size is equal to $L$, a polymer mode, we let $N=N(L)$ be the number of the polymers in $[0,L-1]$. We need a technical lemma from [18] in order to control the sum

[TABLE]

Lemma 2.13**.**

Suppose that $|\langle e^{2i\eta_{\pm}}\rangle|<1$. Let $c_{\omega}:=(c_{\omega_{\ell}})\in{\mathbb{C}}$ be the coefficient sequence defined in (2.29). For almost every $\omega$, there exists an integer $N_{\omega}$, so that for all $N>N_{\omega}$, the sum $I_{\omega,N}^{1}(\theta,\epsilon)$ satisfies

[TABLE]

almost surely, for any $\alpha>0$, and any $\epsilon$ such that $\epsilon\leq\frac{C}{\sqrt{N}}$, for a universal constant $C>0$. Furthermore, we have

[TABLE]

Proof.

(2.35) is proved in [18, Proposition 2], so it suffices to show (2.34). For any $\alpha>0$ and any $N\in{\mathbb{N}}$, we consider the following event:

[TABLE]

Then by [18, Theorem 6], there exist constants $C_{1},C_{2}>0$, such that for any $\theta$, any $N$, and any $\epsilon>0$, with $N\epsilon^{2}<1$, we have

[TABLE]

By the Borel-Cantelli Lemma, for a.e. $\omega$, $|I_{\omega,k}^{1}|\leq N^{\alpha+1/2}$, for sufficiently large $N$. ∎

Proposition 2.14**.**

(Assumption 2.5, Part 1) Suppose that $|\langle e^{2i\eta_{\pm}}\rangle|<1$. Then the family of functions $\Psi_{L}$ satisfy $\lim_{L\to\infty}\Psi_{L}(x)=x$, a.s.

Proof.

Case 1 : Suppose that $L$ is on a polymer node: $L=\sum_{\ell=0}^{N-1}L_{\omega_{\ell}}$. The Prüfer phase $\theta_{L}$ satisfies

[TABLE]

We recall that from [18, section 4.2],

[TABLE]

where the coefficients $c_{\pm}$ and $d_{\pm}$ are defined in (2.29), and $C$ is a universal constant. Thus, substituting $S^{\ell}_{\epsilon,\omega}(\theta)$ to $\theta$ in (2.39), we obtain

[TABLE]

where the remainder satisfies

[TABLE]

Recalling that at a critical energy

[TABLE]

and substituting $\epsilon=\dfrac{x}{n(E_{c})L}$ into (2.40), we get

[TABLE]

where $\epsilon=\dfrac{x}{n(E_{c})L}$. The sum of the exponential term in (2.43) may be estimated as in Lemma 2.13, together with the fact that $N=\mathcal{O}(L)$. We also have that $R_{\epsilon,\ell}=\mathcal{O}(\epsilon^{2})=\mathcal{O}(L^{-2})$, see (2.41). These facts allow us to write, for any $\alpha>0$,

[TABLE]

By the law of large numbers, we have,

[TABLE]

So therefore we get

[TABLE]

By [18, (47)], we have $n(E_{c})=\dfrac{1}{\pi}\dfrac{\langle d_{\pm}\rangle}{\langle L_{\pm}\rangle}$, which yields $\Psi_{L}(x)\to x$, a.s.

Case 2 : General case. As we did in (2.33), we can write

[TABLE]

As in (2.39), the first term $\triangle\Psi$ may be written as

[TABLE]

for appropriate constants $d^{\prime}$, $c^{\prime}$, and $R_{\epsilon}^{\prime}$. ∎

To show the local uniform convergence in Assumption 2.9, we need a little more analysis.

Proposition 2.15**.**

(Assumption 2.9, Part 1) Suppose that $|\langle e^{2i\eta_{\pm}}\rangle|<1$.

Then $\lim_{L\to\infty}\Psi_{L}(x)=x$ locally uniformly, a.s.

Proof.

For given $L$, let $N(L)$ be the number of polymer nodes in $[0,L]$. Note that $\frac{L}{L_{+}}\leq N(L)\leq\frac{L}{L_{-}}$ so that $N(L)/L,L/N(L)=\mathcal{O}(1)$. Fix $\alpha>0$, and let

[TABLE]

where $L\epsilon^{2}<C$. Then by [18, Theorem 6] and standard large deviation estimate for i.i.d. random variables, we have

[TABLE]

On the other hand, proof of Proposition 2.14 implies that, for $\omega\in\Omega\left(L,\dfrac{x}{\rho(E_{c})L}\right)^{c}$, then

[TABLE]

Take $a,b\in{\bf R}$ s.t. $a<b$, and divide the interval $[a,b]$ into small intervals $\{I_{k}\}_{k=1}^{K}$ with $I_{k}=[x_{k},x_{k+1}]$ of size $L^{-\frac{1}{2}+\alpha}$. Suppose that,

[TABLE]

Then for any $y\in(x_{k},x_{k+1})$, we have

[TABLE]

Hence

[TABLE]

Let

[TABLE]

Then $P(\Omega(L))\leq C_{1}L^{\frac{1}{2}-\alpha}e^{-C_{2}L^{\alpha}}$, and for $\omega\in\Omega(L)^{c}$,

[TABLE]

for any $x\in[a,b]$. Let $\Omega_{0}:=\limsup_{L\to\infty}\Omega(L)$. Then $P(\Omega_{0})=0$, and for $\omega\in\Omega_{0}^{c}$, $\Psi_{L}(x)\stackrel{{\scriptstyle L\to\infty}}{{\to}}x$ uniformly in $[a,b]$. ∎

2.7. Sharpness of the LES transition

In this section, we make an exploration of the transition from Poisson to clock statistics as the reference energy $E_{0}$ transitions from the localization regime to a critical energy $E_{c}$. We consider the case when the LES is centered at a noncritical energy $E_{0}(L)$ which is $L$-dependent and $E_{0}(L)\to E_{c}$ as $L\to\infty$:

[TABLE]

Then, in these cases, we can still show that $\Psi_{L}(x)\to x$, locally uniformly. In fact, we have the following extension of Theorem 2.1.

Theorem 2.16**.**

Assume the following two conditions.

1. (1)

$|\langle e^{2i\eta_{\pm}}\rangle|<1$, 2. (2)

$0<\mathcal{N}(E_{c})<1$.

We consider the LES centered at $E_{0}(L)$, with $E_{0}(L)$ as in (2.49). Then

1. (1)

*Any accumulation point of $\xi_{L}$ is a clock process with some probability measure $\mu$. * 2. (2)

The eigenvalue spacing near $E_{0}(L)$ exhibits clock behavior as in (2.51).

Proof.

It suffices to show part (1) of Assumption 2.5 and part (1) of Assumption 2.9 for a relative Prüfer phase defined by

[TABLE]

The idea of proof is similar to that for Propositions 2.14 and 2.15. ∎

Remark 2.17**.**

Theorem 2.16 supports the sharpness of the transition of the unfolded LES as a function of the energy. In the next section, we show that under certain assumptions on the regularity of the IDS, Assumption 3.2, the limiting point process is Poisson at noncritical energies. In remark 3.4, we point out that although the Hölder continuity of the IDS for the AB RPM is known, we do not know of a proof for the Hölder continuity of the inverse function $N^{-1}(E)$. It is possible that these Hölder exponents, $\rho_{1}$ and $\rho_{2}$ vary as the energy approaches a critical energy. In particular if $\rho_{1}\rho_{2}$ no longer satisfies Assumption 3.2, that is, $\frac{2}{3}<\rho_{1}\rho_{2}<1$, our proof of Poisson statistics breaks down. To investigate this, we prove in Appendix F that in the absence of the lower bound of $\rho_{1}\rho_{2}$, the limit points of the point process at noncritical energies are infinitely divisible. Since the clock process is not infinitely divisible (see Theorem F.3), our results show that the limiting point process is infinitely divisible (Poisson) for all noncritical $E_{0}\in\Sigma$, but not infinitely divisible (clock) for all critical energies $E_{0}\in\Sigma$. In this sense, the transition is sharp.

2.8. Summary: Clock statistics at critical energies

We summarize the main results on clock LES at critical energies.

Theorem 2.18**.**

Under these assumptions on the RPM,

1. (1)

The polymer transfer matrices eigenvalues satisfy:

[TABLE] 2. (2)

The IDS satisfies

[TABLE]

we have the following results on the LES at critical energies:

1. (1)

The LES centered at a critical energy satisfies $\xi_{L}\stackrel{{\scriptstyle d}}{{\to}}clock(unif[0,1))$. 2. (2)

Strong clock behavior: The local eigenvalue spacing around a critical energy almost surely satisfies

[TABLE] 3. (3)

If the LES is centered at $E_{0}=E_{c}+\mathcal{O}(L^{-\delta})$ with $\delta>\frac{1}{2}$, then the LES is clock$(\mu)$* with some distribution $\mu$ on $[0,1)$, and we have the strong clock behavior.*

There are examples of RPM for which the two assumptions hold.

3. Localization: LES is Poisson away from critical energies

The deterministic spectrum $\Sigma$ of the Anderson-Bernoulli random polymer model (AB RPM) is purely pure point. It is a natural question to study the unfolded LES (1.28) centered at energies that are not critical. In order to prove that the unfolded LES for AB RPM is Poisson at energies away from the critical energies $\mathcal{C}$, we encounter several new problems:

1. (1)

There is only a weak Wegner estimate proved in [10]; 2. (2)

The standard proofs of the Minami estimate [11, 26] do not apply to Bernoulli probability measures; 3. (3)

The IDS is only Hölder continuous.

This last item means that we do not know if the density of states measure is absolutely continuous with respect to Lebesgue measure. Consequently, we do not know if the density of states function (DOSf) exists. We recall that for random Schrödinger operators with single-site probability measures $d\mu(s)=\rho(s)~ds$, with probability density $\rho\in L^{\infty}({\mathbb{R}})$, the DOSf $n(E):={N}^{\prime}(E)$ determines the intensity of the Poisson point process centered at energy $E$ [26].

To circumvent this difficulty, we work with the LES constructed using the unfolded eigenvalues as in (1.28), see [17, 27]. As is mentioned there, the unfolded eigenvalues are the random variables constructed by filtering the local eigenvalues $\{E_{j}(\Lambda)\}$ through the IDS ${{N}}(E)$. Instead of dealing with the eigenvalues $\{E_{j}(\Lambda)\}=\sigma(H_{\Lambda})$ directly, we study the unfolded eigenvalues $\{{{N}}(E_{j}(\Lambda))\}$. The unfolded LES centered at ${N}({E}_{0})$ is the limit (if it exists) of the finite random point measures given by

[TABLE]

Germinet and Klopp [17] treated the unfolded LES for RSO. They proved that, assuming both Wegner’s and Minami’s estimates, the unfolded LES for random Schrödinger operators in the localization regime is a Poisson point process with an intensity measure given by the Lebesgue measure. We will prove a similar result for the unfolded LES for the AB random polymer models. We recall that at the critical energy $E_{0}\in\mathcal{C}$, the DOSf was proved to exist in [18, Theorem 3]. As a consequence, the unfolded LES at $E_{c}$ is the same as the usual LES constructed with the eigenvalues $\{E_{j}(\Lambda)\}$, as noted in section 1.4.

To state our main result of this section, that the LES is Poisson at noncritical energies, we need to discuss the density of states (DOS) of AB RPM. It is well-known that the DOS plays an important role in the proof of local eigenvalue statistics. We first recall a well-known regularity result of the integrated density of states (IDS) for AB RPM.

Theorem 3.1**.**

[10]** The IDS ${N}:\Sigma\rightarrow[0,1]$ of the AB RPM is locally uniformly Hölder continuous, that is, for any compact interval $I$, we can find positive constants $\rho_{1}\in(0,1]$ and $C<\infty$, such that

[TABLE]

for any $E,E^{\prime}\in I$.

For our LES result, we also need some regularity on the inverse function ${{N}}^{-1}:[0,1]\rightarrow\Sigma$.

Assumption 3.2**.**

(Hölder continuity of ${N}^{-1}$)

1. (1)

${{N}}^{-1}$* is Hölder continuous with an exponent ${\rho_{2}}\in(0,1]$, that is, for any compact interval $I$ we can find positive constants $\rho_{2}\in(0,1]$ and $C<\infty$ such that for $u,u^{\prime}\in[0,1]$,*

[TABLE]

In other words, for $E={N}^{-1}(u)$, and $E^{\prime}={N}^{-1}(u^{\prime})$, we have

[TABLE] 2. (2)

${\rho_{1}\rho_{2}}>\dfrac{2}{3}.$**

By definition, $\rho_{1}\rho_{2}\leq 1$. Part (2) of Assumption 3.2 states that this product $\rho_{1}\rho_{2}$ is not too small.

Our main theorem is the following result on the unfolded LES (3.1) away from the finite set $\mathcal{C}$ of critical energies.

Theorem 3.3**.**

(Poisson statistics) Let $E_{0}\in\Sigma\backslash\mathcal{C}$. We assume that the IDS of the AB RPM satisfies Assumption 3.2. Then, the unfolded LES in (3.1) converges in distribution to the Poisson point process with intensity measure given by the Lebesgue measure:

[TABLE]

Remark 3.4**.**

**

1. (1)

In fact, to prove that the LES is Poisson at a noncritical reference energy $E_{0}$ we only need Assumption 3.2 near the reference energy $E_{0}$. That is, the following local condition is enough:

[TABLE]

Our proof for Poisson statistics breaks down if we do not assume the condition (3.5) with $\rho_{1}\rho_{2}>\frac{2}{3}$, although we do not know of an example for which this condition is satisfied. However, they seems to be the minimum requirement, namely we do not expect Poisson statistics without this condition. In fact, **[17, (1.45) of Theorem 1.10]** also assumes a lower bound on the Hölder continuity of IDS. On the other hand, Theorem 3.3 is model-independent, and valid for general situations such that the Hamiltonian satisfies reasonable conditions such as the independence at a distance, positivity of the Lyaunov exponent at the reference energy, and applicability of MSA. 2. (2)

Avila **[3]** showed that the IDS of the almost Mathieu model in the subcritical region $0<\lambda<1$ has Hölder exponent $\rho_{1}=\frac{1}{2}$ and $N^{-1}$ has Hölder exponent $\rho_{2}=\frac{2}{3}$. This is the only example we know of for which $\rho_{2}$ has been computed. (except the case where the DOS measure has a density so that the IDS is Lipschitz continuous.) 3. (3)

A consequence of the assumed Hölder continuity of $N^{-1}$ is that for any interval $I\subset{\mathbb{R}}$, independent of $L$, we have

[TABLE]

3.1. Outline of the proof

The strategy for the proof is basically due to Minami [26] and Germinet-Klopp [17]: in the localized regime, the original system can be well approximated by the direct sum of subsystems, and the eigenvalues of each subsystem are independent of the eigenvalues in the other subsystems. Hence, for example, the random variables that the subsystems have at least one eigenvalue in an interval are close to independent Bernoulli variables, which is a typical situation where the proof of the Poisson limit theorem works. But since the distribution of the random potential is singular, we need to re-construct the proof from the beginning.

1. (1)

Positivity of Lyapunov exponent: De Bièvre and Germinet [13] proved the positivity of the Lyapunov exponent for the Dimer model and Damanik, Sims, and Stolz [14] proved it for more general one-dimensional discrete models including the AB RPM. We review these results in section 3.2. 2. (2)

Wegner’s estimate : we use “weak-type” Wegner’s bound proved by [10], and we improve the exponent on the bound. 3. (3)

Bourgain’s lemma : Bourgain [7] proved an estimate on the probability that a local Hamiltonian has two eigenvalues in a small interval $I$. We adjust his proof to our situation, and present the proof in Appendix D for completeness. 4. (4)

Bootstrap MSA and SULE bound : We use the conclusion obtained by the bootstrap MSA in Germinet-Klein [16]. The conclusion of bootstrap MSA gives a SULE bound for finite volume systems with both polynomial and sub-exponential probability [17]. 5. (5)

Minami’s estimate : we give the bound on the probability that a local Hamiltonian has more than $k$ eigenvalues on a small interval, where $k\geq 2$. The essential idea is due to [20, 31], but we need some technical modifications to fit to the AB RPM. For related remarks, see Appendix C. 6. (6)

Decomposition of the system into small subsystems : we construct a correspondence between the pairs of eigenvalues and eigenvectors of the original system and those in small subsystems. Then, an argument using the proof of Poisson limit theorem completes the proof. This basic argument is due to Germinet-Klopp[17], but the mathematical construction of these correspondences was initiated in [29].

The first proof of localization for the Anderson-Bernoulli model on ${\mathbb{Z}}$ is due to Carmona, Klein, and Martinelli [10]. The standard proofs of the Wegner estimate for the Anderson model with singular single-site probability measures do not hold since there is no known spectral averaging result, see [11]. In [10], the authors prove a weaker version of the usual Wegner estimate of the form:

[TABLE]

where $P_{H_{\ell}}(I)$ is the spectral projector, see Theorem 3.6. With these results, Carmona, Klein, and Martinelli [10] were able to control the single-energy multiscale analysis (MSA) and prove exponential localization throughout the deterministic spectrum. The same arguments hold in the polymer case. The dimer case $(L_{+},L_{-})=(2,2)$ was treated by De Bièvre and Germinet [15].

The standard proof of LES for discrete RSO [26] and unfolded LES [17] requires a Minami estimate, which is an upper bound on ${\mathbb{P}}\{{\rm Tr}P_{H_{L}}(I)\geq k\}$, for $k\geq 2$. We present a proof of such an estimate for AB RSO and RPM (see Theorem 3.14) drawing from ideas in [7] and [20]. Two other results are needed for localization:

1. The positivity of the Lyapunov exponent, see [14], and
2. and the local Hölder continuity of the IDS, proved for the AB model in [10, Appendix]. Both results play a crucial role in their proof of the exponential decay of the Green’s function. For a recent proof of localization for Anderson models on $\ell^{2}({\mathbb{Z}})$ with singular single-site probability measures, we refer to [8].

3.2. Positivity of the Lyapunov exponent

We recall the definition of the transfer matrix for a finite string of polymers:

[TABLE]

If $m>k$, we have $T^{E}_{\omega}(k,m)=T^{E}_{\omega}(m,k)^{-1}$, and $T^{E}_{\omega}(k,k)=I_{2}$, where $I_{2}$ is the identity matrix. The Lyapunov exponent is given by

[TABLE]

A key component of the proof of localization for the dimer model $L_{+}=L_{-}=2$ by De Bièvre and Germinet [15] is the verification of the two conditions necessary in order to prove the positivity of the Lyapunov exponent. The Furstenberg theorem (see, for example, [14]) requires two properties:

1. (1)

The subgroup $G_{\pm}$ of $SL(2,{\mathbb{R}})$ generated by the two polymer transfer matrices $\{T_{+}^{E},T_{-}^{E}\}$, for $E\neq E_{c}$, is noncompact. 2. (2)

The subgroup $G_{\pm}$ is strongly irreducible.

These conditions were proved to hold in [15] for the dimer model by explicit computation. The proof for the general AB RPM is given in [14]. We cite the result applicable to the AB RPM.

Theorem 3.5**.**

[14, Theorem 6.2]** For the AB RPM, there is a finite subset $\mathcal{C}\subset{\mathbb{R}}$ so that for all $E\in\Sigma\backslash\mathcal{C}$, the subgroup $G_{\pm}(E)$ of $SL(2,{\mathbb{R}})$ is noncompact and strongly irreducible. Consequently, the Lyapunov exponent $L(E)$ for the AB RPM at energies $E\in\Sigma\backslash\mathcal{C}$ is strictly positive.

3.3. Wegner estimate

We use the version of the Wegner’s estimate established in [10] for RSO with singular single-site probability measures, including Bernoulli measures. The proof extends to AB RPM. Briefly, the proof in [10] depends on two lemmas. The first is [10, Lemma 5.1] on the exponential decay of the expectation of products of one-step transfer matrices. For the RPM, one uses the one-polymer transfer matrices $T^{E}_{\pm}$, with $E\in\Sigma\backslash\mathcal{C}$, as defined in (2.2). The main bound is as follows. For any bounded interval $I\subset\Sigma$, there are finite constants $\alpha_{1},\delta>0$, and a positive integer $n_{0}$, so that for $n\geq n_{0}$ we have:

[TABLE]

The proof uses the positivity of the Lyapunov exponent.

The second lemma, [10, Lemma 5.2], concerns the existence of approximate eigenvectors $\widetilde{\varphi}$ of a local Schrödinger operator $H_{\ell}$, and is based on a result of Halperin. This lemma uses the Hölder continuity of the IDS with Hölder exponent $\rho_{1}>0$. For any $E\in\Sigma\backslash{\mathcal{C}}$, there exists a function $\widetilde{\varphi}$ so that

[TABLE]

Given these preliminaries, the Wegner estimate for the AB RPM, based on [10], is:

Theorem 3.6**.**

For any $0<\beta\leq 1$, and for all $\sigma>0$, there exist finite constants $\ell_{0}=\ell_{0}(I,\beta,\sigma)>0$ and $\alpha=\alpha(I,\beta,\sigma)>0$ and a constant $C_{W}=C_{W}(\beta,I)>0$, such that

[TABLE]

for all $E\in I$ and all $\ell>\ell_{0}$.

Remark 3.7**.**

We note that the result in [10] can be extended to include $\beta=1$, but this fact is not used in this paper.

For any $0<\tau<1$, the constant $\alpha$ in (3.11) is bounded by

[TABLE]

where $\rho_{1}$ is the Hölder exponent of the IDS, defined in (3.2), $\kappa:=\dfrac{\tau\alpha_{1}}{2\delta}$, and

[TABLE]

The constants $\alpha_{1},\delta$ are defined in (3.9).

The constant $\sigma\rho_{1}$ in the definition of $\alpha$ is crucial in our argument. Consequently, taking $\tau>0$ sufficiently small such that $\dfrac{\kappa}{2}-\tau\log M>0$, we set

[TABLE]

Then we obtain a second version of the Wegner estimate that will be useful in the proof of the Minami’s estimate used in this paper.

Corollary 3.8**.**

(Wegner Estimate, second version) For any bounded interval $I\subset{\mathbb{R}}$, and for constants $0<\beta\leq 1$, $\sigma>0$, there exists a length scale $\ell_{0}:=\ell_{0}(I,\beta,\sigma)>0$, and an index $q>0$ as in (3.12), so that

[TABLE]

for all $E\in I$, and $\ell\geq\ell_{0}$.

3.4. Bootstrap MSA and SULE bounds

We use the bootstrap multiscale analysis (MSA) developed by Germinet and Klein [16]. We begin with some notation and definitions. Although the usual definitions of subintervals of ${\mathbb{Z}}$ can be extended to the case $L_{+}=L_{-}>1$, when $L_{+}\neq L_{-}$, the position of each polymer nodes is random which may cause inconveniences in the usual decomposition of $H_{\Lambda}$ into subsystems in the proof of Poisson limit theorem. Thus, we work under alternative definition of finite intervals.

Definition 3.9**.**

We always assume that $L\gg L_{\pm}$. For given $\omega=(\omega_{\ell})_{\ell\in{\mathbb{Z}}}$, we fix the configuration $(t_{\omega},v_{\omega})=(\hat{t}_{\omega_{\ell}},\hat{v}_{\omega_{\ell}})_{\ell\in{\mathbb{Z}}}$ such that the left end of $\omega_{0}$ starts at the origin of ${\mathbb{Z}}$. Let

[TABLE]

be the position of the $n$-th polymer node. That is, the left endpoint of $n$-th polymer of length $L_{\omega_{n}}$. Moreover, let $B_{n}\subset{\mathbb{Z}}$ be the corresponding $n$-th block (polymer):

[TABLE]

Then, for any $\omega$, we have ${\mathbb{Z}}=\bigcup_{\ell\in{\mathbb{Z}}}B_{\omega_{\ell}}$. We define the finite interval $\Lambda_{L}$ of size $L$ to be the union of $L$-blocks starting from the origin :

[TABLE]

Then $\Lambda_{L}$ is a random set, but we have $L_{-}L\leq\#\Lambda_{L}\leq L_{+}L$. Likewise, for a polymer node $x\in\{p_{n}\}_{n\in{\mathbb{Z}}}$, we define $\Lambda_{L}(x)$ to be the union of $\left\lfloor\frac{L}{2}\right\rfloor$-intervals to both sides of $x$, that is, if $x=p_{n}$ for some $n\in{\mathbb{Z}}$, then

[TABLE]

We call $\Lambda_{L}$ an interval of size $L$, and call $\Lambda_{L}(x)$ the interval of size $L$ centered at $x$. We note that $\Lambda_{L}$ is not equal to $\Lambda_{L}(0)$.

For the characteristic function of an interval $\Lambda_{L}(x)$, we write $\chi_{x,L}$. For a general interval $\Lambda\subset{\mathbb{Z}}$, we write $\chi_{\Lambda}$ for its characteristic function. We let $\Gamma_{x,L}$ denote the function localized near the boundary points of $\Lambda_{L}(x)$.

The concept of a regular interval for the Green’s function an energy $E$ is central to MSA. For $x,y\in{\mathbb{Z}}$, and $z\in{\mathbb{C}}\backslash\Sigma$, the Green’s function $G_{H}(x,y;z)$ is the integral kernel of the resolvent $R_{H}(z):=(H-z)^{-1}$ of $H$ evaluated at $x$ and $y$: $G_{H}(x,y;z):=\langle\delta_{x},R_{H}(z)\delta_{y}\rangle$. For any $\Lambda\subset{\mathbb{Z}}$, the Green’s function of the restricted Schrödinger operator $H_{\Lambda}:=\chi_{\Lambda}H_{\omega}\chi_{\Lambda}$ is denoted by $G_{\Lambda}(x,y;z)$, or by $G_{\Lambda}(z)$. If the interval $\Lambda$ is of length $L$, we write $H_{L}$ and $G_{L}(x,y;z)$. For a random Schrödinger operator $H_{\omega}$, we sometimes suppress the configuration index $\omega$.

Definition 3.10**.**

The interval $\Lambda_{L}(x)$ of length $L$ centered at $x$, is called $(m,E)$-regular if

[TABLE]

where the operator norm is on $\ell^{2}(\Lambda_{L}(x))$.

Theorem 3.11**.**

(Bootstrap MSA) Let $I$ be any interval $I\subset\Sigma\backslash\mathcal{C}$. For $x,y\in{\mathbb{Z}}$ and $E\in{\mathbb{R}}$, we define the event:

[TABLE]

Then, for any $\zeta\in(0,1)$, and for all $\alpha\in(1,\zeta^{-1})$, and with a scale $L_{k}:=L_{0}^{\alpha^{k}}$, the following estimates hold for $k\in{\mathbb{N}}$:

[TABLE]

Germinet and Klein [16] proved this theorem for the AB RSO. As mentioned by Germinet and De Bièvre [15], the proof extends to the dimer model. The positivity of the Lyapunov exponent is essential in these proofs. The proof can be extended to the AB RPM without major difficulties, see Appendix C.

Two important results of Theorem 3.11 are 1) the subexponential decay of operators of the form $\chi_{x}f(H_{\omega})P_{H_{\omega}}(I)\chi_{y}$ for intervals $I\subset\Sigma\backslash\mathcal{C}$, and 2) strong Hilbert-Schmidt dynamical localization in such intervals.

Among the main results of the bootstrap MSA given in Theorem 3.11 are the essential localization bounds on eigenvectors. These are explicit exponential bounds on the decay of eigenvectors. To state these, we need the notion of a center of localization (COL) of an eigenvector.

Definition 3.12**.**

A center of localization of an eigenvector $\varphi$ of $H_{\Lambda_{L}}$ is any $m\in\Lambda_{L}$ so that $|\varphi(m)|\geq|\varphi(p)|$, for all $p\in\Lambda_{L}$. We will write COL for any such eigenvector maximizer and denote one by $x_{\varphi}(\Lambda_{L})$.

Then the bootstrap MSA result Theorem 3.11 implies a SULE estimate on the eigenvectors of $H$ restricted to an interval $\Lambda_{L}$.

Lemma 3.13**.**

[17, Lemma 1.1]** Let $H_{\Lambda_{L}}$ be the Schrödinger operator for the AB RPM restricted to $\Lambda_{L}\subset{\mathbb{Z}}$. Let $\varphi_{j}^{L}$ be a normalized eigenvector of $H_{\Lambda_{L}}$ associated with eigenvalue $E_{j}(\Lambda_{L})\in I$, and let $x_{j}(\Lambda_{L})\in\Lambda_{L}$ be a COL of $\varphi_{j}^{L}$. Then we have the following localization result for the eigenvectors.

1. (1)

For any $p>0$ and $\xi\in(0,1)$, and for $L>0$ large enough, there exists a set $\mathcal{U}_{L}\subset\Omega$ of configurations such that ${\mathbb{P}}\{\mathcal{U}_{L}\}\geq 1-L^{-p}$, and for any $\omega\in\mathcal{U}_{L}$, one has

[TABLE] 2. (2)

For any $\nu,\xi\in(0,1)$, with $\nu<\xi$, and for $L>0$ large enough, there exists a set $\mathcal{V}_{L}\subset\Omega$ of configurations such that ${\mathbb{P}}\{\mathcal{V}_{L}\}\geq 1-e^{-L^{\nu}}$, and for any $\omega\in\mathcal{V}_{L}$, one has

[TABLE]

We remark that in the localized regime, the distance between any two COL for an eigenvector $\varphi$ of $H_{\Lambda_{L}}$ is $\mathcal{O}((\log L)^{\frac{1}{\xi}})$, with probability $1-L^{-p}$, for any $p>0$ [29, 17]. Lemma 3.13 is proved in [17, Theorem 6.1].

3.5. Minami’s estimate for AB RPM

In what follows, we will consider to decompose the given interval $\Lambda_{L}$ of lattice sites into disjoint union of smaller intervals, such as $\Lambda_{L}=\bigcup_{p}C_{p}$. We also consider the restriction $H_{C_{p}}=H_{\Lambda_{L}}|_{C_{p}}$ of our Hamiltonian $H_{\Lambda}$ to $C_{p}$. In order that $(H_{C_{p}})$ are independent, we henceforth require that the all division points of our decomposition $\{C_{p}\}$ must at successive be polymer nodes. This may cause the slight change of the size of each interval, but it does not affect our estimates.

We divide the interval $\Lambda_{L}$ of size $L$ into smaller intervals $\{C_{p}\}_{p=1}^{N_{L}}$ of size $\ell_{1}:=L^{\beta}$ ($0<\beta<1$), so that $N_{L}=\mathcal{O}(L^{1-\beta})$. We denote by $H_{C_{p}}$ the restriction of $H_{\Lambda_{L}}$ to $C_{p}$. Let $0<\gamma\leq 1$, and let $J\subset{\bf R}$ with $|J|=c_{2}>0$ be a bounded interval (independent of $L$), and set $I:=E_{0}+\dfrac{J}{L^{\gamma}}$, so that $|I|=c_{2}/L^{\gamma}$.

Theorem 3.14**.**

*(Minami estimate for AB RPM)

For $H_{C_{p}}$ and energy interval $I\subset\Sigma\backslash\mathcal{C}$ defined above, we set*

[TABLE]

(1) Suppose that $\beta\leq\rho_{1}\gamma$. Then for any $\epsilon>0$ we can find a positive constant $C_{\epsilon}$ s.t.

[TABLE]

(2) Suppose that $\beta\leq\rho_{1}\gamma$ and $\rho_{1}\gamma>\dfrac{1+\beta}{2}$. Then

[TABLE]

Proof.

We further divide each $C_{p}$ into smaller intervals $\{D^{(p)}_{j}\}_{j}$ of size $\ell_{2}:=(c_{1}\log L)^{1/\xi}$, where $c_{1}>0$, and $\xi\in(0,1)$ is the constant in the SULE estimate in Lemma 3.13. We note that $L=e^{\frac{\ell_{2}^{\xi}}{c_{1}}}$, and $|I|=\dfrac{c_{2}}{L^{\gamma}}=c_{2}\cdot e^{-\frac{\ell_{2}^{\xi}}{c_{1}}\gamma}$. Here, we work under the configuration such that we have a SULE bound, and estimate the error caused by this replacement. To be precise, we take any interval $C_{p}$, and let $\{E_{i}(C_{p})\}_{i=1}^{N(I,C_{p})}$, $\{\varphi^{(C_{p})}_{i}\}_{i=1}^{N(I,C_{p})}$, and $\{x_{i}(C_{p})\}_{i=1}^{N(I,C_{p})}$ be the eigenvalues, corresponding normalized eigenvectors, and their COLs, respectively, of the Hamiltonian $H_{C_{p}}$ in $I$. The index $N(I,C_{P})$ is defined by $N(I,C_{p}):=\#\sigma(H_{C_{p}})\cap I$. Let

[TABLE]

Then by Lemma 3.13, we have, for any $p^{\prime}>0$, we can find $q=q(p^{\prime},d)>0$, s.t. for $L$ large enough,

[TABLE]

Then the error caused by replacing $\Omega$ by $\Omega\cap L(I,p,q)$ turns out to be negligible. In fact, we compute

[TABLE]

Here we set ${\mathbb{P}}(A\,;\,B):={\mathbb{P}}(A\cap B)$. For given $\beta$, by taking $p^{\prime}$ sufficiently large, this error becomes negligible. Thus it suffices to estimate the first term. Here we consider the following two events. For each $p=1,2,\cdots,N_{L}$, let

[TABLE]

We decompose

[TABLE]

We shall show separately that $\sum_{k\geq 2}I_{p,k}$, $\sum_{k\geq 2}II_{p,k}$ satify the estimate part (1) in Theorem 3.14, and then show further that $\sum_{p}\sum_{k\geq 2}I_{p,k}=o(1)$, and $\sum_{p}\sum_{k\geq 2}II_{p,k}=o(1)$. For each $D_{j}^{(p)}$, let $\widetilde{D}_{j}^{(p)}$ be the box of size $2\ell_{2}=2(c_{1}\log L)^{1/\xi}$ which has the same center as $D_{j}^{(p)}$.

(1) Estimate on $I_{p,k}$ :

For $\omega\in\Omega^{b}_{p}\cap L(I,p,q)$, we can find $D_{j}^{(p)}$ and $\varphi_{\sharp}^{(C_{p})}$, with $\sharp=i,i^{\prime}(=1,2,\cdots,N(I,C_{p}))$ such that the corresponding COL’S satisfy $x_{i}(C_{p}),x_{i^{\prime}}(C_{p})\in D_{j}^{(p)}$. Then

[TABLE]

by taking $c_{1}$ sufficiently large. Then in the box $\widetilde{D}^{(p)}_{j}$, $\varphi_{\sharp}$, $\sharp=i,i^{\prime}$ become the approximate eigenfunctions of $H_{p,j}:=H|_{\widetilde{D}^{(p)}_{j}}$ so that $H_{p,j}$ has two eigenvalues in $I$, at the cost of increasing slightly the constant $c_{2}$. By Lemma D.1, we have

[TABLE]

Here we take $\beta=\xi\in(0,1)$ in Lemma 3.13. Note that the constant $c_{1}$ in Lemma D.1 should be replaced by $\dfrac{2^{\xi}c_{1}}{\gamma}$ and moreover, since we have $(\ref{starminami})$ and since $\dfrac{c_{2}}{L^{\gamma}}\ll\dfrac{1}{10\ell_{2}}$, the corresponding eigenvectos of $H_{p,j}$ satisfy the condition (3) in Lemma D.1, under the any choice of $0<t_{0}<\frac{1}{2}$. Now we take the sum $\sum_{k=2}^{|C_{p}|}$ with respect to $k=2,\cdots,|C_{p}|$, where $|C_{p}|=L^{\beta}(1+o(1))$.

[TABLE]

In the last line, we take $c_{1}$ sufficiently large so that the second term in RHS is negligible. We thus proved part (1) of Theorem 3.14 for $I_{p,k}$.

Next we take the sum $\sum_{p}$ with respect to $p=1,2,\cdots,N_{L}$ :

[TABLE]

Since $\rho_{1}\gamma>\dfrac{1+\beta}{2}$, we can take $\epsilon>0$ small enough such that $\dfrac{2\rho_{1}\gamma}{1+\epsilon}>1+\beta$. Therefore

[TABLE]

(2) Estimate on $II_{p,k}$ : First of all, note that we may suppose each COL’s $\{x_{i}(C_{p})\}$ are away from each other in the distance of $3\ell_{2}$. In fact, let

[TABLE]

Then

[TABLE]

Then the argument in (1) shows that the second term $II_{p,k,2}$ in RHS satisfies the same estimate as $I_{p,k}$ above. In fact, we divide $C_{p}$ into boxes $F_{j}^{(p)}$’s of size $a:=3\ell_{2}$ and also into boxes $G_{j}^{(p)}$’s of the same size but centered on the division points of $F_{j}^{(p)}$’s. Then on the event $\Omega^{g}_{p,2}\cap L(I,p,q)$, there exists $j$ such that we can find two COL’s $x_{i}(C_{p})$, $x_{i^{\prime}}(C_{p})$, on the same box $F_{j}^{(p)}$ (or $G_{j}^{(p)}$). Let $\widetilde{F}_{j}^{(p)}$ (resp. $\widetilde{G}_{j}^{(p)}$) be the box with the same center as $F_{j}^{(p)}$ (resp. $G_{j}^{(p)}$) of size $6\ell_{2}$, Then $\widetilde{H}_{j}:=H|_{\widetilde{F}_{j}}$ or $\widetilde{H^{\prime}}_{j}:=H|_{\widetilde{G}_{j}}$ has two eigenvalues in $I$ so that the argument in (1) gives

[TABLE]

We thus showed the part (1) of Theorem 3.14 for $II_{p,k,2}$. Similarly as in (1), we take $\epsilon>0$ small enough such that $\dfrac{2\rho_{1}\gamma}{1+\epsilon}>1+\beta$ which yields

[TABLE]

Thus it suffices to estimate $II_{p,k,1}$. For $\omega\in\Omega_{p,1}^{g}\cap L(I,p,q)$, We can find $k$ boxes $\widetilde{D}_{j_{1}}^{(p)},\widetilde{D}_{j_{2}}^{(p)},\cdots,\widetilde{D}_{j_{k}}^{(p)}$ on which the corresponding Hamiltonians $\widetilde{H}_{p,j_{1}},\cdots,\widetilde{H}_{p,j_{k}}$ have eigenvalues in $I$. Note that, since COL’s satisfy $|x_{i}(C_{p})-x_{i^{\prime}}(C_{p})|>3\ell_{2}$, for all $i,i^{\prime}$, these boxes are disjoint. On the other hand, since $\ell_{2}=(c_{1}\log L)^{1/\xi}$ implies $\dfrac{c_{2}}{L^{\gamma}}=c_{2}\cdot e^{-\frac{\ell_{2}^{\xi}}{c_{1}}\gamma}=c_{2}\cdot e^{-\frac{(2\ell_{2})^{\xi}}{2^{\xi}\cdot c_{1}}\gamma}$, Wegner’s estimate (Corollary 3.8) yields

[TABLE]

In the last line, we used

[TABLE]

Here we take $c_{1}>0$ large enough so that the second term is negligible, at the cost of increasing slightly the constant $C_{W}$. Now we take the sum with respect to $k=2,\cdots,|C_{p}|$.

[TABLE]

Since $\beta\leq\rho_{1}\gamma$, the last factor in RHS is $\mathcal{O}(1)$. We thus showed the part (1) of Theorem 3.14 for $II_{p,k,1}$. Finally, we take the sum with respect to $p=1,2,\cdots,N_{L}$ :

[TABLE]

Therefore by the same argument as in (1), we have

[TABLE]

By (3.22), (3.23), and (3.27), we arrive at the conclusion. ∎

3.6. Decomposition into subsystems

In this subsection, we consider to decompose $\Lambda_{L}$ into small intervals $\Lambda_{\ell}$ of size $\ell$, and relate the eigenvalues of $H_{\Lambda_{L}}$ to those of the family $\{H_{\Lambda_{\ell}(\gamma_{j})}$. We show that, with good probability, number of COL’s of eigenvectors of $H_{\Lambda_{L}}$ with eigenvalues in a given interval $I_{\Lambda}$ is at most one for each $\Lambda_{\ell}(\gamma_{j})$ and, moreover, a COL of $H_{\Lambda_{L}}$ is in $\Lambda_{\ell}(\gamma_{j})$ if and only if $H_{\Lambda_{\ell}(\gamma_{j})}$ has an eigenvalue in a slightly enlarged interval $\widetilde{I}_{\Lambda_{L}}$. We basically follow the strategy and notation in [17, Theorem 1.1]. As above, we work with length scales $\ell:=L^{\beta}$ and $\ell^{\prime}:=L^{\beta^{\prime}}$, for $0<\beta^{\prime}<\beta<1$, and intervals $\Lambda_{L}$ and $\Lambda_{\ell}(\gamma)$, an interval centered at $\gamma\in{\mathbb{Z}}$.

Theorem 3.15**.**

Let $H_{\Lambda_{L}}$ be the AB RMP on an interval of length $L$ and satisfying Assumption 3.2, and let

[TABLE]

Consider the following decomposition of $\Lambda_{L}$: $\Lambda_{L}=\bigcup_{j}\Lambda_{\ell}(\gamma_{j})\cup\Upsilon$ where $\{\Lambda_{\ell}(\gamma_{j})\}_{j}$ are disjoint and $\Upsilon$ is the set $\Lambda_{L}\setminus\bigcup_{j}\Lambda_{\ell}(\gamma_{j})$ enlarged by a length $\ell^{\prime}$, such that

[TABLE]

Then, for $L$ large enough, we can find constants $\alpha>0$, $0<\beta^{\prime}<\beta<1$, and a set of configurations $\mathcal{Z}_{\Lambda_{L}}\subset\Omega$, such that

1. (1)

${\mathbb{P}}(\mathcal{Z}_{\Lambda_{L}})\geq 1-L^{-\alpha}$, 2. (2)

*For $\omega\in\mathcal{Z}_{\Lambda_{L}}$, the COL’s associated to the eigenvalues of $H_{\omega}(\Lambda_{L})$ in $I_{\Lambda_{L}}$ are contained by $\bigcup_{j}\Lambda_{\ell}(\gamma_{j})$. Each interval $\Lambda_{\ell}(\gamma_{j})$ satisfies *

*(i) $H_{\omega}(\Lambda_{\ell}(\gamma_{j}))$ has at most one eigenvalue $E_{j}(\Lambda_{\ell}(\gamma_{j}))$ in $I_{\Lambda_{L}}$, *

*(ii) $\Lambda_{\ell}(\gamma_{j})$ contains at most one COL $x_{k_{j}}(\Lambda_{L})$ corresponding to an eigenvalue $E_{k_{j}}(\Lambda_{L})$ of $H_{\omega}(\Lambda_{L})$. *

(iii) $\Lambda_{\ell}(\gamma_{j})$ contains a COL $x_{k_{j}}(\Lambda_{L})$ of $H_{\omega}(\Lambda_{L})$ if and only if $\sigma(H_{\omega}(\Lambda_{\ell}(\gamma_{j})))\cap I_{\Lambda_{L}}\neq\emptyset$, in which case, denoting $\sigma(H_{\omega}(\Lambda_{\ell}(\gamma_{j})))\cap I_{\Lambda_{L}}=\{E_{j}(\Lambda_{\ell}(\gamma_{j}))\}$, we have

[TABLE]

Remark 3.16**.**

In Theorem 3.15, $\Lambda_{L}$ is defined in Definition 3.9 so that the right end of $\Lambda_{L}$ is in a polymer node $p_{n}\in\mathcal{P}$. However, by adjusting the definition of $\Upsilon$, one can see that the statement of Theorem 3.15 is also valid for any box of the form : $\Lambda_{L}\cup\{p_{n}+1,\cdots,p_{n}+L_{\max}\}$, where $L_{\max}:=\max\{L_{+},L_{-}\}$.

For proof, we need the following definition.

Definition 3.17**.**

Let $\Lambda,\Lambda^{\prime}$ be finite intervals with $\Lambda^{\prime}\subset\Lambda$, and let $E$ be an eigenvalue of $H_{\Lambda}$. We say that $E$** is centered in $\Lambda^{\prime}$

---

if and only if the COL $x_{\varphi}$ of the corresponding eigenvector $\varphi$ of $E$ satisfies $x_{\varphi}\in\Lambda^{\prime}$.*

Proof.

Let $\mathcal{S}_{\ell,L}$, $\widetilde{\mathcal{S}}_{\ell,L}$ be the set of intervals such that

[TABLE]

where $K$ is replaced by some $K^{\prime}>K$ in (3.28), the definition of $I^{\prime}_{\Lambda_{L}}$. Then, by the localization estimate in part (2) of Lemma 3.13, if $\omega\in\mathcal{U}_{\Lambda_{L}}$, we have $\mathcal{S}_{\ell,L}\subset\widetilde{\mathcal{S}}_{\ell,L}$. Next, we apply Theorem 3.14, where we take $\gamma:=\rho_{2}$, and assume

[TABLE]

Then, for any $\epsilon>0$,

[TABLE]

where $\delta:=-1-\beta+\dfrac{2\rho_{1}\rho_{2}}{(1+\epsilon)}$. We can find $\epsilon>0$ such that $\delta>0$, if and only if

[TABLE]

We recall that $\Upsilon$ is the set $\Lambda_{L}\setminus\bigcup_{j}\Lambda_{\ell}(\gamma_{j})$ enlarged by a length $\ell^{\prime}$. We decompose $\Upsilon$ into small intervals of size $\widetilde{\ell}:=(c\log L)^{1/\xi}$ where $0<\xi\leq 1$ is the constant appearing in SULE estimate (Lemma 3.13), and for some $c>0$, and write $\Upsilon=\bigcup_{k}\Lambda_{\widetilde{\ell}}(k)$. Then by localization estimate again,

[TABLE]

Note that, by assumption, $|I_{\Lambda_{L}}|\leq C\left(\dfrac{C}{L}\right)^{\rho_{2}}=\mathcal{O}\left(e^{-\frac{\widetilde{\ell}^{\xi}}{c}\cdot\rho_{2}}\right)$. By Wegner’s estimate(Corollary 3.8), we have

[TABLE]

where we take $c>0$ large enough so that the second term $e^{-q\widetilde{\ell}^{\xi}}$ in RHS is negligible. Then

[TABLE]

Here we have used $\#\{\Lambda_{\widetilde{\ell}}(k)\subset\Upsilon\}\leq\dfrac{L^{1-\beta}\cdot L^{\beta^{\prime}}}{\widetilde{\ell}}$. We can find some $\beta^{\prime}$ such that $0<\beta^{\prime}<\beta$ and $\alpha>0$, if and only if

[TABLE]

Since

[TABLE]

we can find $\beta\in(0,1)$ such that all the conditions (3.29), (3.31), and (3.33) are satisfied if and only if $\rho_{1}\cdot\rho_{2}>\dfrac{2}{3}$. Now, let

[TABLE]

Then the above argument and SULE estimate in Lemma 3.13 yield ${\mathbb{P}}(\mathcal{Z}_{\Lambda})\geq 1-L^{-\alpha}$, for some $\alpha>0$. Moreover, if $\omega\in\mathcal{Z}_{\Lambda}$, (3.32) implies that eigenvalues of $H_{\Lambda_{L}}$ in $I_{\Lambda_{L}}$ are centered in $\bigcup_{j}\Lambda_{\ell-\ell^{\prime}}(\gamma_{j})$ and (3.30) implies there are at most one eigenvalue $E_{k_{j}}$ of $H_{\Lambda_{L}}$ centered on each $\Lambda_{\ell-\ell^{\prime}}(\gamma_{j})$’s. And since (3.30) also implies $H_{\Lambda_{\ell}(\gamma_{j})}$ has at most one eigenvalue $E^{\prime}_{j}$, if there is any, it satisfies $|E_{k_{j}}-E^{\prime}_{j}|\leq e^{-(\ell^{\prime})^{\xi}}$. ∎

3.7. Completion of the proof of Theorem 3.3: Poisson limit theorem

In this subsection, we prove the following statement which completes the proof of Theorem 3.3: the convergence of $\xi_{L}$ to a Poisson process for $E_{0}\in\Sigma\setminus\mathcal{C}$.

Theorem 3.18**.**

Let $I_{1},\cdots,I_{p}\subset{\mathbb{R}}$ be disjoint bounded intervals, and let $k_{1},\cdots,k_{p}\in{\mathbb{N}}\cup\{0\}$. Then, under the hypotheses of Theorem 3.3,

[TABLE]

Remark 3.19**.**

By definition, $\xi_{L}$ is composed of the eigenvalues of $H_{\Lambda_{L}}$, where the right end of $\Lambda_{L}$ is a polymer node $p_{n}\in\mathcal{P}$. However, by remark 3.16, one can see that $\Lambda_{L}$ may be replaced by any box of the form $\Lambda_{L}\cup\{p_{n}+1,\cdots,p_{n}+L_{\max}\}$.

As is already mentioned, we prove Theorem 3.18 by approximating $H_{\Lambda_{L}}$ by the direct sum of small subsystems $\oplus_{j}H_{\Lambda_{\ell}(\gamma_{j})}$ and apply the method of proof of the Poisson limit theorem. Since this decomposition has been done in Theorem 3.15, it suffices to estimate the probability of rare events. In order to do that, we begin by setting up the notation as in [17]. For an interval $J\subset{\mathbb{R}}$, we define

[TABLE]

and, letting ${\bf{1}}_{A}$ denote the characteristic function on a set $A$, the Bernoulli random variables

[TABLE]

We always assume that $J$ is contained in the localized regime of the spectrum such that SULE estimate in Lemma 3.13 is valid. Then, by definition of IDS and by the localization estimate, we expect that ${\mathbb{E}}[N(J,\ell,\ell^{\prime})]$ is close to $N(J)|\Lambda_{\ell}|$ for $\ell$ large enough, where $N(J)$ is the DOS measure of $J$. Next, by Minami’s estimate, Theorem 3.14, we expect that ${\mathbb{P}}(X_{\Lambda_{\ell},I}=1)$ is close to ${\mathbb{E}}[N(I_{\Lambda_{L}},\ell,\ell^{\prime})]$. Lemma 3.20 below proves that these observations are valid, and will be important in the proof of Lemma 3.21 to estimate the probability of the rare events, that is, $\{X_{\Lambda_{\ell},I}=1\}$, necessary to prove the Poisson limit theorem.

Lemma 3.20**.**

For any energy interval $J\subset{\mathbb{R}}$, we have

[TABLE]

Proof.

(1) Let $\mathcal{V}_{\Lambda_{\ell}}\subset\Omega$ be the set of configurations in Lemma 3.13, part (2). For $\omega\in\mathcal{V}_{\Lambda_{\ell}}^{c}$, we use the fact that $\#\sigma(H_{\omega}(\Lambda_{\ell}))\leq|\Lambda_{\ell}|$, so that

[TABLE]

For $\omega\in\mathcal{V}_{\Lambda_{\ell}}$, we use the localization estimate so that

[TABLE]

where $H_{\omega}$ is the infinite volume operator, and

[TABLE]

are the enlarged (respectively, compressed) versions of $J$ to absorb the localization error. We take the expectation on both sides of (3.34). Then, the expectation of the RHS of (3.34) is equal to

[TABLE]

We have used the fact that the IDS $N$ is Hölder continuous with exponent $\rho_{1}$. The expectation of the LHS of (3.34) is equal to

[TABLE]

completing the proof of part (1) of Lemma 3.20.

(2) We let $\eta$ denote the number of eigenvalues of $H_{\Lambda_{\ell}}$ in $I_{\Lambda_{L}}$:

[TABLE]

Since $|I_{\Lambda_{L}}|\leq CL^{-\rho_{2}}$, part (1) of Theorem 3.14 yields

[TABLE]

Then, noting that $X_{\Lambda_{\ell},I}={\bf{1}}(N(I_{\Lambda_{L}},\ell,\ell^{\prime})\geq 1)$,

[TABLE]

Now it suffices to apply part (1) of Lemma 3.20(1) with $J=I_{\Lambda_{L}}$. ∎

To prove Theorem 3.18, let $I_{1},\cdots,I_{p}\subset{\bf R}$ be disjoint bounded energy intervals, and let

[TABLE]

their enlarged and compressed counterparts. Then, we have $I_{j}^{-}\subset I_{j}\subset I_{j}^{+}$, and, for $L$ large, $I_{j}^{+}\cap I_{k}^{+}=\emptyset$, for $j\neq k$. Moreover, we denote the rescaled intervals by $\widetilde{J}^{\pm}_{m}$, and their union $\widetilde{J}_{\pm}$, by

[TABLE]

We denote the number of subintervals $\Lambda_{\ell}(\gamma_{j})$ by $\widetilde{N}=\mathcal{O}\left(L^{1-\beta}\right)(1+o(1))$.

To prove Theorem 3.18, we need to estimate the probability of the rare event ${{\mathbb{P}}}\left(X_{\Lambda_{\ell},I_{m}^{\pm}}=1\right)$ which is done in Lemma 3.21 below.

Lemma 3.21**.**

(1) Suppose that $\rho_{1}\rho_{2}>\dfrac{1}{2}$. Then

[TABLE]

(2) Suppose that $\rho_{1}\rho_{2}>\dfrac{1}{2}$. Then we have

[TABLE]

where we take $\beta,\beta^{\prime}>0$ sufficiently small such that $0<\beta^{\prime}<\beta$ and

[TABLE]

is positive.

Proof.

(1) We first estimate

[TABLE]

We define the counting function

[TABLE]

Since $(\sum_{l}Y_{l}^{\pm}\geq 1)=(\sum_{l}X_{\Lambda_{\ell}(\gamma_{1}),I_{l}^{\pm}}\geq 1)$, it follows that

[TABLE]

Noting that

[TABLE]

it follows from (3.37), together with the Minami estimate, that

[TABLE]

Estimating the expectation of the counting function in (3.38) by part (2) of Lemma 3.20, we then have

[TABLE]

By definition of $\widetilde{J}_{\pm}$, the corresponding IDS is given by

[TABLE]

We estimate the error term in (3.35) from (3.7) and (3.40). This error is decomposed as

[TABLE]

We estimate $I,\cdots,IV$ separately to show they are in the order of $\mathcal{O}(L^{-\delta-(1-\beta)})$. The first and second terms $I$, $II$ satisfy

[TABLE]

To have $I,II=\mathcal{O}(L^{\beta-1-\delta})$ for some $\delta>0$, we need $\beta-\beta^{\prime}>0$. The third term $III$ is exponentially small :

[TABLE]

which is negligible. As for the fourth term $IV$,

[TABLE]

We have $IV=\mathcal{O}(L^{\beta-1-\delta})$ for some $\delta>0$ if and only if $2\rho_{1}\rho_{2}-1>0$. Under this condition, by taking

[TABLE]

we have the statement part (1) of Lemma 3.21.

(2) In part (2) of Lemma 3.20, we replace $I_{\Lambda}$ by

[TABLE]

which yields

[TABLE]

Here we use

[TABLE]

Thus

[TABLE]

As in the proof of part (1) of Lemma 3.21, we can show

[TABLE]

with the same constant $\delta>0$ as in part (1). ∎

We are now in a position to prove Theorem 3.18 based on the argument used to prove Poisson limit theorem.

*Proof of Theorem 3.18

*Take $K>0$ s.t. $\bigcup_{m=1}^{p}I_{m}\subset[-K,K]$. We apply Theorem 3.15 to the following interval:

[TABLE]

Let $\mathcal{Z}_{\Lambda_{L}}\subset\Omega$ be the set of configurations in Theorem 3.15. By the localization estimate, we have

[TABLE]

Let us prove the upper bound. For $\omega\in\bigcap_{m=1}^{p}\left\{\omega\,\middle|\,\#\left\{j\,\middle|\,X_{\Lambda_{\ell}(\gamma_{j}),I_{m}^{+}}=1\right\}=k_{m}\right\}\cap\mathcal{Z}_{\Lambda_{L}}$, let

[TABLE]

be the set of indices $j$ such that $H_{\Lambda_{\ell}(j)}$ has an eigenvalue in $I_{m}^{+}$ centered in $\Lambda_{\ell-\ell^{\prime}}(\gamma_{j})$. Then $\#K_{m}=k_{m}$, and since $\omega\in\mathcal{Z}_{\Lambda_{L}}$, $K_{m}\cap K_{m^{\prime}}=\emptyset$ for $m\neq m^{\prime}$.

Then, we have

[TABLE]

In the last inequality, we used the estimate ${\mathbb{P}}(\mathcal{Z}_{\Lambda_{L}})=1+o(1)$. Since $\{K_{m}\}_{m}$ are disjoint, $\{X_{\Lambda_{\ell}(\gamma_{j}),I_{m}^{+}}\}_{j\in K_{m},\,1\leq m\leq p}$ are independent. Thus

[TABLE]

where $k:=\sum_{m=1}^{p}k_{m}.$ By Lemma 3.21, we have

[TABLE]

On the other hand, note that

[TABLE]

By (3.46), (3.50), (3.51), and (3.52), we have

[TABLE]

which completes the proof of upper bound. Lower bound follows similarly.

Appendix A Properties of free and modified Prufer phases from [18]

Following [18], we define the free Prufer variables $(R^{0}_{n}(E),\theta^{0}_{n}(E))$ by

[TABLE]

[TABLE]

and the modified Prufer variables $(R_{n}(E),\theta_{n}(E))$ by

[TABLE]

and where $M$ is the invertible matrix mapping the polymer transfer matrices $T_{\pm}(E_{c})$ to rotations as in (2.5):

[TABLE]

We recall from (2.11) that $M$ has the property that

[TABLE]

with $r(\theta)>0$, and the smooth function $m$ satisfies $m(\theta+\pi)=m(\theta)+\pi$. It follows from these definitions that free Prufer and modified Prufer phases are related by

[TABLE]

Appendix B Proof of the increasing property of $x\mapsto\Psi_{L}(x)$

We recall the function $\Psi_{L}(x)$ defined by

[TABLE]

We prove that $x\in{\mathbb{R}}\to\Psi_{L}(x)$ is an increasing function.

Lemma B.1**.**

The function $m$ defined in (A.5) is strictly increasing if and only if $\det M>0$.

Proof.

Since $M$ is a $2\times 2$ invertible matrix, we write

[TABLE]

From the definition of $m$ in (2.11), we have

[TABLE]

It follows that

[TABLE]

Consequently, we have,

[TABLE]

∎

Lemma B.2**.**

Without loss of generality, we may suppose that $\det M>0$.

Proof.

We set

[TABLE]

Then, noting that $J^{-1}=J$, we have

[TABLE]

Since $MT_{\pm}(E_{c})M^{-1}=R(\eta)$, applying $J$ to both sides of this equation, we obtain

[TABLE]

Thus, if $\det M<0$, then $\det(JM)=\det J\cdot\det M>0$, and by (B.7), we may replace $M$ and $\eta_{\pm}$ by $JM$ and $-\eta_{\pm}$, respectively. ∎

Proposition B.3**.**

The map $x\in{\mathbb{R}}\mapsto\Psi_{L}(x)$ is increasing.

Proof.

we recall from (A.6) that free Prufer and modified Prufer variables are related by :

[TABLE]

By [18, Lemma 2], $E\mapsto\theta^{0}_{n}(E)$ is increasing for any fixed $n$. By lemmas A.1, A.2, $\theta\mapsto m(\theta)$ is increasing. Therefore $E\mapsto\theta_{n}(E)$ is increasing for any fixed $n$. The increasing property of $\Psi_{L}(x)$ then follows from the definition of $\Psi_{L}(x)$ :

[TABLE]

and these properties. ∎

Appendix C Bootstrap MSA for the AB RPM

The paper of von Dreifus and Klein [32] presents a proof of exponential localization for the AB model on $\ell^{2}({\mathbb{Z}})$ using the positivity of the Lyapunov exponent for the initial length scale estimate, and the Wegner estimate of [10]. The technical conclusion of their analysis is

[TABLE]

The bootstrap MSA improves this probability bound to $\geq 1-e^{L_{k}^{\xi}}$ so that the probability of bad events are subexponentially small.

There is an aspect of the general RPM that differs from the AB model with respect to MSA. When the lengths of the two polymers are the same, $L_{-}=L_{+}$, we can choose intervals of length $L$ that are integer multiples of $L_{+}=L_{-}$, and the MSA proceeds in the usual manner as for the AB model. However, for the general RPM for which $L_{-}\neq L_{+}$, the configurations of the two different polymers in an interval of fixed length $L$ vary. That is, if one fixes an interval $\Lambda_{L}$, the number of random polymers that intersect this interval varies with the configuration $\omega$ and, in general, will contain only parts of a polymer possibly at one or both ends. However, the bootstrap MSA allows for some flexibility in choosing independent intervals. Condition 3 in [16], IAD, independence at a distance, requires only that there exist $\delta>0$, so that $\delta$-nonoverlapping intervals are independent. If we choose $\delta=\max\{L_{-},L_{+}\}$, then the local Schrödinger operators associated with $\Lambda_{L}(x)$ and $\Lambda_{L}(y)$ are independent provided $|x-y|>\delta$.

The other basic assumptions of bootstrap MSA are

1. (1)

SLI. The Simon-Lieb inequality is used to iterate resolvents over lengths scales. 2. (2)

EDI. The exponential decay inequality provides an upper bound on the norm of a generalized eigenvector in terms of a localized resolvent. 3. (3)

IAD. The independence at a distance was discussed above. 4. (4)

NE. The number of eigenvalues is an upper bound on the expected number of eigenvalues of $H_{L}$ in an interval in terms of $L$. 5. (5)

W. The Wegner estimate is discussed in detail in section 3.5.

For AB model or the RPM, let $I\subset\Sigma\backslash\mathcal{C}$ be an interval contained in localization regime minus the critical set $\mathcal{C}$, so that the Lyapunov exponent $L(E)$ is strictly positive for $E\in I$. Then, the conclusion of bootstrap MSA on the interval $I$ is :

[TABLE]

To state the theorems below, we set

[TABLE]

We recall the notations and statements in Germinet-Klein Bootstrap paper [16] for $\ell^{2}({\mathbb{Z}})$.

Definition C.1**.**

[TABLE]

Definition C.2**.**

Let $E\in{{\mathbb{R}}}$, $\theta>0$, $x\in{{\mathbb{Z}}}^{d}$, $L\in 6{{\mathbb{N}}}$. A box $\Lambda_{L}(x)$ is $(\theta,E)$**-suitable****

[TABLE]

Definition C.3**.**

Let $E\in{{\mathbb{R}}}$, $m>0$, $x\in{{\mathbb{Z}}}^{d}$, $L\in 6{{\mathbb{N}}}$. A box $\Lambda_{L}(x)$ is $(m,E)$**-regular****

[TABLE]

Germinet and Klein derived derived four versions of MSA : Theorems 5.1, 5.2, 5.6, and 5.7, appearing as Theorem C.4-C.8 below. By using for theorems successively, the starting hypothesis of Theorem 5.1 leads us to the conclusions of Theorem 5.7.

Theorem C.4**.**

[16, Theorem 5.1]** Let $E_{0}\in I_{0}$, $\theta>bd$, $0<p<\theta-bd$. Suppose that

[TABLE]

Then setting $L_{k+1}:=YL_{k}$, $k=0,1,\cdots$, we have

[TABLE]

Theorem C.5**.**

[16, Theorem 5.2]** Let $E_{0}\in I_{0}$, $\theta>bd$, $0<p^{\prime}<p<\theta-bd$, and $1<\alpha<1+\dfrac{p^{\prime}}{p^{\prime}+2d}$. Suppose that

[TABLE]

Then we can find an interval $I(\delta_{1}):=[E_{0}-\delta_{1},E_{0}+\delta_{1}]\cap I_{0}$ and setting $m_{0}:=\frac{2}{L_{0}}\theta\log L_{0}$, $L_{k+1}:=[L_{k}^{\alpha}]$, $k=0,1,\cdots$, we have

[TABLE]

and moreover

[TABLE]

Definition C.6**.**

For any $\zeta\in(0,1)$, $E\in{{\mathbb{R}}}$, $x\in{{\mathbb{Z}}}^{d}$, $L\in 6{{\mathbb{N}}}$, the interval $\Lambda_{L}(x)$ is $(\zeta,E)$-subexponentially suitable* if*

[TABLE]

Theorem C.7**.**

[16, Theorem 5.6]** $\zeta\in(0,1)$, $E\in{{\mathbb{R}}}$, $x\in{{\mathbb{Z}}}^{d}$, $L\in 6{{\mathbb{N}}}$. $\Lambda_{L}(x)$ is $(\zeta,E)$**-subexponentially suitable****

[TABLE]

Theorem C.8**.**

[16, Theorem 5.7]** Let $E_{0}\in I_{0}$, $\zeta_{0}\in(0,1)$, $Y\gg 1$. For $0<\forall\zeta_{1}<\zeta_{0}$, suppose that

[TABLE]

Then setting $L_{k+1}:=YL_{k}$, we have

[TABLE]

For the AB or random dimer model, the statements in Theorems 5.2, 5.6, and 5.7 are valid with slight modification in the proof. Moreover, the assumption in Theorem 5.7 has been given for AB model by the paper by von Dreifus and Klein [32]. Therefore, by following Theorem 5.7, we have the conclusion in Theorem 5.7 for AB model. We remark that the exponent $\xi$ appearing in the subexponential decay bounds and the eigenvector bounds can be set equal to one for models amenable to the fractional moment method [1]. However, this method has not been extended to models with singular probability measures such as the AB model.

Appendix D Bourgain’s Lemma

As a preliminary to proving Theorem 3.14, we give a proof of a result on the the probability of two eigenvalues of $H_{\ell}$ being in an interval $I\subset{\mathbb{R}}$. This is originally given in [7, Lemma 3] but modified to fit to our model.

Lemma D.1**.**

Let $I:=(E_{0}-\delta,E_{0}+\delta)$, $\ell>0$, $0<\beta\leq 1$, and define length scales as follows: $\delta:=c_{2}e^{-\frac{\ell^{\beta}}{c_{1}}}$ and parameters $0<t_{0}<\frac{1}{2}$ and $0<\epsilon<1$. Let us consider the following event $\Omega_{\delta}$ : There exist two orthonormal vectors $\xi,\xi^{\prime}\in{\mathbb{R}}^{\ell}$ so that:

1. (1)

They are approximate eigenvectors for $H_{\ell}$ and $E_{0}$:

[TABLE]

; 2. (2)

The eigenvector mass is less distributed near the boundary : If $\xi=(\xi_{j})$, then

[TABLE]

We then have the bound for sufficiently large $\ell$ :

[TABLE]

where $\rho_{1}$ is the Hölder exponent for the IDS, and $q$ is defined in (3.12).

Remark D.2**.**

1. (1)

We do not use localization estimate on the approximate eigenvectors $\xi,\xi^{\prime}$, but conditions (1), (2) usually follow from it. 2. (2)

This result was proved by Bourgain for the AB model, that is, rank one perturbations at each point of ${\mathbb{Z}}$ with independent, Bernoulli distributed coefficients. In particular, in the proof, it is important that the local Schrödinger operators associated with two disjoint subintervals of $[0,\ldots,\ell]$ of the from $[0,j_{1}]$ and $[j_{1}+1,\ell]$, are independent. For the AB model this is immediate. For the RPM, however, this is not necessarily true. We will show that we may choose the endpoints $j_{1}$ and $j_{1}+1$ to be polymer nodes, thereby insuring the independence of the restrictions of the Schrödinger operator to these two subintervals.

Proof.

1. Since $\|\xi\|=1$, and condition (2) holds, for $t_{0}\in(0,\frac{1}{2})$, there exists an index $v$, with

[TABLE]

Moreover, since $\xi\perp\xi^{\prime}$, we have

[TABLE]

Hence, we obtain : There exists an index $v_{1}$, with $t_{0}\ell<v_{1}<(1-t_{0})\ell$, so that

[TABLE]

In fact, if this is not so, let $w:=\xi_{v}\xi^{\prime}-\xi^{\prime}_{v}\xi$ . Then any component $w_{k}$ satisfies $|w_{k}|<1/\ell$, for all $k\in[0,\ell]$. This implies

[TABLE]

which is a contradiction to (D.3). Without loss of generality, we can assume $v<v_{1}$. We define a phase $\theta_{j}$ by

[TABLE]

Then by (D.3), we have

[TABLE]

In the last inequality, we used the fact that $\|\xi\|=\|\xi^{\prime}\|=1$ implies $|\xi_{v}|\leq 1$, $|\xi^{\prime}_{v}|\leq 1$ for all indices $v$. This implies:

[TABLE]

2. We begin the construction of approximate eigenvectors for the restriction of $H_{\ell}$ to subintervals of $[0,\ldots,\ell]$. The goal is to find two subintervals on which the restricted Schrödinger operators are independent. For this reason, we consider inner polymer nodes. The endpoints [math] and $\ell$ do not have to be polymer nodes.

**Case 1 ** Suppose there is a polymer node $j_{1}$ so that $v<j_{1}<v_{1}$, and the phases satisfy

[TABLE]

We define another vector $\eta$, a linear combination of $\xi$ and $\xi^{\prime}$, by

[TABLE]

so that $\|\eta\|=1$ and $\eta_{j_{1}}=0$. This vector $\eta$ has entries satisfying

[TABLE]

Here we used (D.3) and (D):

[TABLE]

Furthermore, $\eta$ is an approximate eigenvector for $H_{\ell}$:

[TABLE]

We define restrictions of $H_{\ell}$ to small intervals determined by the index $j_{1}$ by letting

[TABLE]

and corresponding vectors

[TABLE]

By (F.1)-(F.2), these vectors have norms bounded above and below by

[TABLE]

We let $\tilde{\eta}^{(k)}:=\eta{(k)}\|\eta^{(k)}\|^{-1}$ denote the corresponding unit vectors.

These vectors $\tilde{\eta}^{(1)}$ and $\tilde{\eta}^{(1)}$ are approximate eigenvectors for $H^{(1)}$ and $H^{(2)}$, respectively:

[TABLE]

and

[TABLE]

We take $\ell\gg 1$ s.t.

[TABLE]

We take $0<t<1$ so that the size of the box where $H^{(1)}$ lives is equal to $j_{1}=t\cdot\ell$ and that where $H^{(2)}$ lives is equal to $\ell-j_{1}=(1-t)\cdot\ell$. By construction, the two Schrödinger operators $H{(k)}$, for $k=1,2$, are independent so that the second version of the Wegner estimate in Corollary 3.8 can be used to the joint probability that each Schrödinger operator has an eigenvalue in a small interval:

[TABLE]

In the last inequality, we used $xy\leq\frac{x^{2}+y^{2}}{2}$, and $t_{0}<t<1-t_{0}$, $t_{0}<1-t$, and $\beta\leq 1$. $q$ is defined in (3.12). This established the bound in Case 1.

**Case 2 ** Suppose that, for any $v\leq j\leq v_{1}$, either of following two situations happens :

[TABLE]

We denote by $j_{1}$ the smallest polymer node index $v<j_{1}<v_{1}$ so that $|\sin(\theta_{v_{1}}-\theta_{j_{1}})|\leq\frac{1}{10\ell}$. There are two subcases to consider: i) the index $j_{1}$ is a left polymer node, and ii) the index $j_{1}$ is a right polymer node.

**Case 2i ** For this case, the index $j_{1}$ is a left polymer node. We then have that $j_{1}-1$ is a right polymer node index. It follows from the definition of $j_{1}$ that

[TABLE]

We define two vectors

[TABLE]

and

[TABLE]

Note that $\eta^{(1)}_{j_{1}}=0$ and $\eta^{(2)}_{j_{1}-1}=0$. Furthermore, we have the lower bounds on the norms of these vectors:

[TABLE]

for finite constants $C_{k}>0$, $k=1,2$, independent of $\ell$. Here we have used:

[TABLE]

We define the associated local Schrödinger operators as restriction to each subinterval:

[TABLE]

As above, we let $\tilde{\eta}^{(k)}$ denote the corresponding normalized vectors. As a result, we find that $\eta^{(k)}$ are approximate eigenvectors for $H^{(k)}$, respectively:

[TABLE]

The rest of the argument is similar to that in Case 1.

Case 2ii For this case, the index $j_{1}$ is a right polymer node. We then have that $j_{1}-1$ is a left polymer node index and it follows from the definition that

[TABLE]

We define two vectors

[TABLE]

and

[TABLE]

Note that $\eta^{(1)}_{j_{1}-1}=0$ and $\eta^{(2)}_{j_{1}}=0$. Furthermore, we have the lower bounds

[TABLE]

Consequently, the vectors $\eta^{(k)}$ are approximately eigenvectors for $H^{(k)}$, $k=1,2$, and the proof continues as in Case 1. ∎

Appendix E Transport and delocalization

We note that if an initial state is concentrated in an energy interval $I$ containing the set $\mathcal{C}$ of critical energies, then there is dynamical delocalization. That is, the wave packet $e^{-iHt}P(I)\delta_{0}$ exhibits nontrivial transport in the sense of (1.2). Let $\{\delta_{j}\}_{j\in Z}$ denote the canonical orthonormal basis of $\ell^{2}({\mathbb{Z}})$. The authors of [18] proved dynamical delocalization in the sense of (1.2) for the AB RPM and initial state $\delta_{0}$. In this appendix, we show that the initial state $\delta_{0}$ may be replaced by $P_{H}(I)\delta_{j}$, for any $j\in{\mathbb{Z}}$, and for any interval $I\subset\Sigma$ containing all the critical energies. This simple result relies on the complementary fact that if $J\subset\Sigma$ is an interval disjoint from the finite set of critical energies $\mathcal{C}\cap J=\emptyset$, then we have dynamical localization for the initial state $P_{H}(J)\delta_{j}$ as proven for the AB RPM, and other models, in [14, Theorem 2.3].

Proposition E.1**.**

Let $I\subset\Sigma$ be any union of intervals satisfying $\mathcal{C}\subset I$. Then, for any $\alpha>0$, there is a constant $C_{\alpha}(I)>0$ so that for any $j\in{\mathbb{Z}}$,

[TABLE]

Proof.

We first consider $j=0$. Since $P(I)+P(I^{c})={\bf 1}$, we have

[TABLE]

Dynamical localization in $I^{c}$ implies there is a constant $C_{III}>0$ so that

[TABLE]

To bound $II$, the Cauchy-Schwarz inequality gives

[TABLE]

as follows from dynamical localization. From (E.3), (E.4), the lower bound estimate (1.2) of [18], and the equality (E.2), we obtain

[TABLE]

for a possibly different constant. The result follows for arbitrary $j\in{\mathbb{Z}}$ from the unitarity of the translation group, the relation $\delta_{j}=U_{j}\delta_{0}$, and the covariance relation of $H_{\omega}$ with respect to this group. ∎

Appendix F Sharpness of the transition

In this appendix, we provide the details of the results mentioned in Remark 2.17. These remarks concern the transition in the unfolded LES as the centering energy $E_{0}$ passes from a noncritical energy to a critical energy. In the absence of Assumption 3.2, we prove that the unfolded LES changes discontinuously from an infinitely divisible point process at noncritical energies to a non-infinitely divisible point process (a clock process) at a critical energy $E_{0}$. We prove this by showing that certain limiting random variables, constructed from the unfolded LES at noncritical energies, are infinitely divisible whereas the clock process is not infinitely divisible.

Definition F.1**.**

A random variable $X$ is infinitely divisible if and only if for any $J\in{{\mathbb{N}}}$, there exist iid random variables $\{Y_{N,j}\}_{j=1}^{J}$ so that

[TABLE]

We characterize the random variables associated with a clock process. The following theorem characterizes bounded infinitely divisible random variables.

Theorem F.2**.**

[5, Corollary 3]** A bounded random variable is infinitely divisible if and only if it is constant almost surely.

The proof of Theorem F.2 is based on a careful analysis of the Lévy-Khintchine formula [2, Theorem 1.2.1] that provides a representation of the characteristic function of an infinitely divisible random variable.

We first show that, any limit point of LES at critical energies are not infinitely divisible, under a mild condition.

Theorem F.3**.**

Suppose that a critical energy $E_{c}\in\mathcal{C}$ satisfies

*(i) $|\langle e^{2i\eta_{\pm}}\rangle|<1$, and (ii) $0<N(E_{c})<1$.

Let $\xi$ be an accumulation point of $\{\xi_{L}\}$ centered at $E_{c}$. Then $\xi$ is not infinitely divisible.*

Proof.

The condition (i) and Proposition 2.14 yield $\Psi_{L}(x)\to x$ a.s., so together with the condition (ii), Corollary 2.8 implies that an accumulation point $\xi$ of $\xi_{L}$ is a clock process with a probability measure $\mu$ on $[0,1)$. By the condition (i), $\mu$ is not a delta measure at a single point. Thus we can find an interval $I\subset[0,1)$ such that $\xi(I)$ is a non-constant bounded random variable. Therefore, by Theorem F.2, $\xi(I)$, and hence $\xi$ itself, is not infinitely divisible. ∎

In order to show there is a transition, we need to characterize the limit points of the sequence $\{\xi_{L}(I)\}$ for non-critical energies. We still need a condition on the regularity of IDS, but this condition seems to be reasonable.

Assumption F.4**.**

IDS $N$ and its inverse $N^{-1}$ satisfy the Hölder continuity in the following sense near the reference energy $E_{0}$ for some $\rho_{1},\rho_{2}>0$ :

[TABLE]

Theorem F.5**.**

Assume Assumption 2. Let $\xi$ be an accumulation point of the LES $(\xi_{L})_{L}$ centered at $E_{0}\in\Sigma\setminus\mathcal{C}$. Then for each bounded interval $I(\subset{\bf R})$, $\xi(I)$ is infinitely divisible.

Therefore, Theorems F.3 and F.5 show the sharpness of the transition. In what follows, we give proof of Theorem F.5 for completeness, although the idea of which is similar to that for Theorem 3.18. We recall the notations.

$\Lambda_{L}$ : box of size $L$,

$I_{\Lambda_{L}}:=N^{-1}\left(N(E_{0})+\frac{I}{|\Lambda_{L}|}\right)$, where $I$ : bounded interval,

$\ell:=L^{\beta}$, $\ell^{\prime}:=L^{\beta^{\prime}}$, $0<\beta^{\prime}<\beta<1$,

$\Lambda_{\ell}(\gamma)$ : a box of size $\ell$ centered at $\gamma$. Then we decompose $\Lambda_{L}=\bigcup_{j}\Lambda_{\ell}(\gamma_{j})\cup\Upsilon$ where we define $\Upsilon$ to be the set $\Lambda_{L}\setminus\bigcup_{j}\Lambda_{\ell}(\gamma_{j})$ enlarged by a length $\ell^{\prime}$. Let

[TABLE]

The following lemma is an analogue to the decomposition theorem (Theorem 3.15), but under much less requirements.

Lemma F.6**.**

Let

[TABLE]

Then we can choose constants $\beta,\beta^{\prime}>0$ such that

[TABLE]

for some $\alpha>0$.

Proof.

Proof of lemma F.6 is similar to that of Theorem 3.15, but since we do not use Minami’s estimate, we do not need the condition that $\rho_{1}\cdot\rho_{2}>\frac{2}{3}$. In fact, by the argument in the proof of Theorem 3.15, we have

[TABLE]

To have $\alpha>0$, we need $\rho_{1}\cdot\rho_{2}>1-\beta$. We can choose $\beta>0$ sufficiently close to $1$ if necessary such that this condition is satisfied. Hence

[TABLE]

Since ${\mathbb{P}}(\mathcal{U}_{\Lambda_{L}}^{c})\leq L^{-p^{\prime}}$, and since $p^{\prime}$ can be taken arbitrary large, we have

[TABLE]

∎

In the following lemma, we compare the large system $H_{\Lambda_{L}}$ with direct sum of subsystems $\bigoplus_{j}H_{\Lambda_{\ell}(\gamma_{j})}$.

Lemma F.7**.**

Let $I(\subset{\mathbb{R}})$ be a interval. Then there is a set of configurations $\Omega_{\Lambda_{L}}$ with ${\mathbb{P}}(\Omega_{\Lambda_{L}}^{c})\leq L^{-\alpha}$ such that

[TABLE]

Proof.

In the event $\mathcal{U}_{\Lambda_{L}}\cap\mathcal{Y}_{\Lambda_{L}}$, let $\varphi$ be a normalized eigenvector of $H_{\Lambda_{L}}$ in $I_{\Lambda_{L}}$ and let $x_{\varphi}$ be its COL. Since $\omega\in\mathcal{Y}_{\Lambda}$, $x_{\varphi}\in\Lambda_{\ell-\ell^{\prime}}(\gamma_{j})$ for some $\gamma_{j}$. Then

[TABLE]

Then $\varphi|_{\Lambda_{\ell}(\gamma_{j})}$ becomes an approximate eigenvector of $H_{\Lambda_{\ell}(\gamma_{j})}$ so that $H_{\Lambda_{\ell}(\gamma_{j})}$ has an eigenvalue in

[TABLE]

By Wegner’s estimate(Corollary 3.8),

[TABLE]

for some $0<c^{\prime}<c$. We have used the fact that, since $0<\xi<1$, 2nd term in RHS is negligible. Therefore

[TABLE]

satisfies ${\mathbb{P}}(\mathcal{W}_{\Lambda_{L}}^{c}\cap\mathcal{U}_{\Lambda_{L}}\cap\mathcal{Y}_{\Lambda_{L}})\leq e^{-c^{\prime}\rho_{1}L^{\beta^{\prime}\xi}}.$ On the other hand, we have shown that ${\mathbb{P}}(\mathcal{U}_{\Lambda_{L}}^{c})\leq CL^{-p^{\prime}}$, and ${\mathbb{P}}(\mathcal{Y}_{\Lambda_{L}}^{c})\leq CL^{-\alpha}.$ Therefore

[TABLE]

Now, on the event $\mathcal{W}_{\Lambda_{L}}\cap\mathcal{U}_{\Lambda_{L}}\cap\mathcal{Y}_{\Lambda_{L}}$, for each eigenvalue $E_{n}(\Lambda_{L})$ of $H_{\Lambda_{L}}$ in $I_{\Lambda_{L}}$, we can find $j=j(n)$ and corresponding eigenvalue $E_{m(n)}(\Lambda_{\ell}(\gamma_{j}))$ of $H_{\Lambda_{\ell}(\gamma_{j})}$ in $I_{\Lambda_{L}}$. Since eigenfunctions of $H_{\Lambda_{L}}$ are orthogonal each other, this correspondence is one to one [29, Lemma A.4]. Therefore,

[TABLE]

which gives the upper bound. For the lower bound, we set

[TABLE]

Then an analogous argument as in upper bound yields

[TABLE]

By (F.1), (F.2), we have

[TABLE]

with ${\mathbb{P}}\left(\Omega_{\Lambda_{L}}^{c}\right)\leq CL^{-\alpha}.$

∎

*Proof of Theorem F.5

*Let $I(\subset{\mathbb{R}})$ be an interval, and let $K\in{\mathbb{N}}$. It suffices to show the following fact : There exist $K$ i.i.d. random varibles $(\zeta_{k}(I))_{k=1}^{K}$ such that

[TABLE]

We decompose $\Lambda_{L}$ into $K$ disjoint boxes $J_{1},J_{2},\cdots,J_{K}$ : $\Lambda_{L}=\bigcup_{k=1}^{K}J_{k}$. Here we adjust the value of $\beta=\beta_{L}$ and the definition of $\Upsilon$ slightly if necessary such that each $J_{k}$’s contains equal number of $\Lambda_{\ell}(\gamma_{j})$’s. And we also decompose the sum of $\eta_{j,L}$’s into sum of $K$ independent ones.

[TABLE]

Let $H_{k}:=H|_{J_{k}}$, and let $\zeta_{k}^{(L)}$ be the corresponding point process.

[TABLE]

where $\{E_{j}(J_{k})\}_{j}$ are the eigenvalues of $H_{k}$. By Lemma F.7, we have

[TABLE]

with ${\mathbb{P}}\left(\Omega_{L}^{c}\cup\bigcup_{k=1}^{K}\Omega_{J_{k}}^{c}\right)\leq CL^{-\alpha}.$ Therefore, we have

[TABLE]

Since $\xi$ is an accumulation point of $\xi_{L}$, there is a subsequence $\{L(i)\}_{i}$ such that $\xi_{L(i)}\stackrel{{\scriptstyle d}}{{\to}}\xi$. Since $\{\zeta_{k}^{(L(i))}(I)\}_{i=1}^{\infty}$ is tight for any $k=1,\cdots,K$, they are jointly tight. Thus we can further find a subsequence $\{L(i_{\ell})\}_{\ell}$ of $\{L(i)\}_{i}$ and random variables $\zeta_{k}(I)$, $k=1,2,\cdots,K$, such that we jointly have

[TABLE]

Here we use the following fact on the sequence of random variables : $X_{n}\stackrel{{\scriptstyle d}}{{\to}}X$, and $Y_{n}\stackrel{{\scriptstyle P}}{{\to}}0$ yield $X_{n}+Y_{n}\stackrel{{\scriptstyle d}}{{\to}}X$. Therefore, (F.3), (F.4) yield

[TABLE]