# Symmetric matrices with banded heavy tail noise: local law and eigenvector
delocalization

Yi HAN  Department of Pure Mathematics and Mathematical Statistics, University
of Cambridge<EMAIL_ADDRESS>
###### Abstract.

In this work we consider deterministic, symmetric matrices with heavy-tailed
noise imposed on entries within a fixed distance $K$ to the diagonal. The most
important example is discrete 1d random Schrödinger operator defined on
$0,1,\cdots,N$ where the potentials imposed on the diagonal have heavy-tailed
distributions and in particular may not have a finite variance. We assume the
noise is of the form $N^{-\frac{1}{\alpha}}\xi$ where $\xi$ are some i.i.d.
random potentials. We investigate the local spectral statistics under various
assumptions on $\xi$: when it has all moments but the moment explodes as $N$
gets large; when it has finite $\alpha+\delta$-moment for some $\delta>0$; and
when it is the $\alpha$-stable law. We prove in the first two cases that a
local law for each element of Green function holds at the almost optimal scale
with high probability. As a bi-product we derive Wegner estimate, eigenvalue
rigidity and eigenvector de-localization in the infinity norm. For the case of
$\alpha$-stable potentials imposed on discrete 1d Laplacian, we prove that (i)
Green function entries are bounded with probability tending to one, implying
eigenvectors are de-localized in the infinity norm; (ii) with positive
probability some entries of the Green function do not converge to that of the
deterministic matrix; and (iii) the trace of Green function converges to the
Stieltjes transform of arcsine law with probability tending to one. These
findings are in contrast to properties of Levy matrices recently uncovered. We
extend our results to other scaling in front of the noise and derive local
laws on the corresponding intermediate scales, and further extend to Wigner
matrices perturbed by finite band heavy-tail noise.

Supported by EPSRC grant EP/W524141/1.

##  1\. Introduction

In this paper we investigate deterministic, symmetric matrices perturbed by
noise acting on its diagonal, or only acting on elements sufficiently close to
the diagonal. A primary example is the 1d random Schrödinger operator defined
on $\\{0,1,\cdots,N\\}$ with zero boundary condition

$H_{n}\varphi(k)=\varphi(k-1)+\varphi(k+1)+\mathfrak{a}(k)\varphi(k),\quad\varphi(0)=\varphi(N)=0.$
(1.1)

where $\mathfrak{a}(k)$ are random potentials that have variance
$\frac{1}{N}$, so that the effect of noise induces fluctuations on microscopic
scales but the global density of states is same as the discrete Laplacian.
Another motivating example is the tridiagonal matrix models for beta ensembles
[17] and their generalizations, see Example 1.34 for details.

A typical assumption in the literature for these matrices is that the random
potentials only act on the diagonal and sub-diagonal, and the randomness has
sub-Gaussian tails, or at least they should have a finite second moment. Under
both assumptions, the method based on transfer matrix recursions and Prüfer
coordinates are applicable and lead to a detailed description of the bulk and
edge behavior, see for example [33], [36], [28], [7], [15].

Motivated by recent progress in Levy matrices [3], [2] (symmetric matrices
with i.i.d. entries having $\alpha$-stable distribution), in this paper we
investigate the statistical properties of these matrices assuming that the
random potentials no longer have a finite second moment. Moreover, we are not
restricted to tridiagonal or Jacobi matrices, and the noise can act on all
matrix elements having distance at most $K$ to the diagonal. Meanwhile, the
method of moments, which is also applicable to tridiagonal matrices with all
moments [37], [32], is far from effective when we are in the heavy-tailed
setting.

Our investigation of these matrices proceed through a direct computation of
its Green function. We will assume the knowledge of a local law for the matrix
when all random potentials are set to be zero. Then we carry out a resolvent
expansion to investigate to what extent the local law continues to hold for
the noisy matrix. We uncover a complex picture: when we impose $\alpha$-stable
noise, then the trace of Green function should converge to the correct limit
(the Stieltjes transform of arcsine law, for example) with high probability,
but with high probability some individual entries of Green function will not
converge to the deterministic limit. Nonetheless, with high probability Green
function entries are bounded, so eigenvectors are de-localized in the
$L^{\infty}$ norm. The results we propose depend strongly on the tails of the
random variable $\xi$, so we state the theorems separately under the
corresponding tail assumptions. In several theorems we prove, we only prove
some property holds with probability tending to 1, but counterexamples imply
that they should not hold with overwhelming probability (i.e. with probability
at least $1-N^{-D}$ for any $D>0$). These properties are all specific to the
finite band structure of the noise that we impose, and are known to be false
for Levy matrices [35], [3] and most other random matrix models.

###  1.1. Models and assumptions

In this paper we will mostly investigate matrices of the form

$H_{N}=H_{N}^{\infty}+A_{N}$ (1.2)

where $H_{N}^{\infty}$ is a (deterministic) symmetric matrix and $A_{N}$ is
the noisy matrix with a finite bandwidth. The notation $H_{N}^{\infty}$
suggests that we are looking at a zero temperature model where the noise is
frozen.

The primary example of deterministic symmetric matrices falling under the
assumption of this paper is the discrete Laplacian in 1d, that is,

$H_{N}^{\infty}=\begin{pmatrix}0&1&0&\cdots&0\\\ 1&0&1&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ 0&\cdots&1&0&1\\\
0&\cdots&0&1&0\end{pmatrix}.$ (1.3)

We first introduce some general notations. Define the resolvent (or Green’s
function) for a matrix $H_{N}$ and its trace $m_{N}$ as

$G(z):=\frac{1}{H_{N}-z},\quad
m_{N}(z):=\frac{1}{N}\operatorname{Tr}G(z),\quad z\in\mathbb{C}^{+},$ (1.4)

where $\mathbb{C}_{+}\equiv\\{E+i\eta\in\mathbb{C}:\eta>0\\}$.

We denote by $G^{\infty}(z)$ the resolvent matrix of $H_{N}^{\infty}$. The
following estimate is the main assumption on $H_{N}^{\infty}$ that we will
need throughout this paper.

######  Proposition 1.1.

Given any small $\epsilon>0$ and $\kappa>0$, define

$\mathcal{S}\equiv\mathcal{S}(\epsilon,\kappa):=\\{z=E+i\eta:|E|\leq
2-\kappa,N^{-1+\epsilon}\leq\eta\leq 1\\}.$ (1.5)

Then the resolvent matrix $G^{\infty}$ of $H_{N}^{\infty}$ defined in (1.3)
satisfies the following (deterministic) upper bound: there exists a constant
$C$ depending only on $\epsilon,\kappa$ such that

$\sup_{z\in\mathcal{S}}\max_{1\leq i,j\leq
N}\left|G^{\infty}_{ij}(z)\right|\leq C.$ (1.6)

The method to prove Proposition 1.1 is to compute explicitly the resolvent
matrix $G^{\infty}$ and check its elements are bounded on $\mathcal{S}$.

We will be working more generally with other symmetric matrices
$H_{N}^{\infty}$ satisfying the same estimate.

######  Definition 1.2.

( $H_{N}^{\infty}$: deterministic, symmetric matrix with bounded resolvent)

$H_{N}^{\infty}$ is an $N\times N$ symmetric square matrix with elements
$h_{ij}=h_{ji}$ that satisfy: for any $\kappa>0$ and $\epsilon>0$, there
exists a constant $C$ depending only on $\epsilon,\kappa$ that gives

$\sup_{z\in\mathcal{S}(\epsilon,\kappa)}\max_{1\leq i,j\leq
N}\left|G^{\infty}_{ij}(z)\right|\leq C.$ (1.7)

where $\mathcal{S}(\epsilon,\kappa)$ is defined in (1.5).

######  Remark 1.3.

This assumption and the forthcoming proofs can be modified, after multiplying
by a scalar, to cover symmetric matrices that satisfy the Green function
estimate (1.6) with $E=\Re(z)$ satisfying $|E|\leq D-k$ rather than
$2-\kappa$, for any $D>0$.

Now we introduce our assumptions on the noisy matrix $A_{N}$.

In the classical context of Anderson localization [5], [31],[26], the standard
assumption is
$A_{N}=\operatorname{diag}(\mathfrak{a}_{1},\cdots,\mathfrak{a}_{N})$, where
$(\mathfrak{a}_{1},\cdots,\mathfrak{a}_{N})$ is a class of i.i.d. random
potentials with $N$-independent variance. In such cases however, the
eigenvectors are localized and decay exponentially fast [4] [22].

To restore de-localization, one needs to reduce the variance of random
potentials. Typical assumptions on $A_{N}$ in the literature [27] [28] can be
summarized as follows:

######  Assumption 1.4.

(Standard assumption on noise, tridiagonal)

$A_{ij}=A_{ji},\quad A_{ij}=0\text{ for all }|i-j|>1,\quad
A_{ij}=\frac{1}{\sqrt{N}}\xi_{ij},$ (1.8)

where $\xi_{ij},$ $1\leq i\leq j\leq n,|i-j|\leq 1$, are a set of independent
random variables (possibly depending on $N$) that satisfy, for some
$\delta>0$,

$\mathbb{E}[\xi_{ij}]=0,\quad\mathbb{E}[|\xi_{ij}|^{2+\delta}]\leq C<\infty.$
(1.9)

In this paper we make the following new assumptions on the noisy matrix
$A_{N}$, which cover all $\alpha$-stable laws that have infinite variance.
Also, the matrix $A_{N}$ need not be diagonal or tridiagonal: it has a finite
but arbitrary bandwidth $K$. As illustrated in the proof, we can let $K$ be
slowly growing, comparable to $\log N$.

In the heavy-tailed case, we take the scaling $\frac{1}{N^{\frac{1}{\alpha}}}$
in the definition of $A_{N}$. This scaling is consistent with the Levy matrix
literature, and (as we will show) this scaling is the typical one that leads
to random fluctuations on microscopic scales but guarantees some form of local
laws on all mesoscopic scales $[N^{-1+\epsilon},1]$.

######  Definition 1.5.

(Finite band noisy matrix $A_{N}$) Given any $\alpha\in(0,2)$, let
$A_{N}=(A_{ij})$ be an $N\times N$ matrix that satisfy, for a fixed integer
(bandwidth) $K>0$:

$A_{ij}=A_{ji},\quad A_{ij}=0\text{ for all }|i-j|>K,\quad
A_{ij}=\frac{1}{N^{\frac{1}{\alpha}}}\xi_{ij},\text{ for all }|i-j|\leq K.$
(1.10)

where $\xi_{ij},1\leq i\leq j\leq n,$ $|i-j|\leq K$, are independent random
variables that satisfy one of the following three assumptions: (for $i>j$ we
set $\xi_{ij}=\xi_{ji}$)

######  Assumption 1.6.

(Finite but exploding moments) Let $A_{N}$ be the square matrix defined in
Definition 1.5 for some $\alpha\in(0,2)$. We assume that for some $\omega>0$,
setting $q=N^{\frac{\omega}{10\alpha}}$, we have the following moment
estimates: for some $\delta>0$,

$|\mathbb{E}[A_{ij}]|\leq\frac{C}{N^{1+\delta}},\quad\mathbb{E}[|A_{ij}|^{p}]\leq\frac{C}{Nq^{p-\alpha}}\text{
for any }p\geq 2.$ (1.11)

Typical cases covered are
$\xi_{ij}=\widetilde{\xi}_{ij}1_{|\widetilde{\xi}_{ij}|\leq
N^{\frac{1}{\alpha}}q^{-1}}$, where $\widetilde{\xi}$ has a finite $\alpha$-th
moment and $\widetilde{\xi}$ has a symmetric law. To check this it suffices to
note that $|A_{ij}|^{p}\leq(q^{-1})^{p-\alpha}|A_{ij}|^{\alpha}$.

######  Assumption 1.7.

(Finite $\alpha+\delta$-moment) Let $A_{N}$ be the square matrix defined in
Definition 1.5 for some $\alpha\in(0,2),$ and we assume that the random
variables $\xi_{ij}$ satisfy, for some $\delta>0,$

$\xi_{ij}\overset{\operatorname{Law}}{=}-\xi_{ij},\quad\mathbb{E}[|\xi_{ij}|^{\alpha+\delta}]<C<\infty,\quad\text{
for all }j-K\leq i\leq j.$ (1.12)

######  Assumption 1.8.

($\alpha$-stable noise) Let $A_{N}$ be the square matrix defined in Definition
1.5 for some $\alpha\in(0,2),$ and we assume that the random variables
$\xi_{ij}$ are i.i.d. modulo symmetric restriction, have a symmetric law
$\xi_{ij}\overset{\operatorname{Law}}{=}-\xi_{ij},$ and for a bounded function
$L(x)$

$G(x):=\mathbb{P}(|\xi_{ij}|\geq x)=L(x)x^{-\alpha},\quad x\geq 1.$ (1.13)

######  Remark 1.9.

After some modifications our result generalizes to the case where $L(x)$ is a
slow varying function. The boundedness assumption in (1.13) is only meant to
simplify some computations. All the following results can be easily
generalized to the case where $\xi_{ij}$ is replaced by $\xi_{ij}+c_{ij}$
where the constant $c_{ij}$ satisfies
$\frac{1}{N^{\frac{1}{\alpha}}}|c_{ij}|\leq\frac{C}{N^{1+\delta}}$ for some
$\delta>0$.

######  Remark 1.10.

The assumption that $\xi_{ij}$ has symmetric law is used to guarantee that
truncated versions of $\xi_{ij}$ have mean zero. This assumption can possibly
be removed with some additional effort, and is not used in other places of
proof.

###  1.2. Finite but exploding moments: local laws and corollaries

The main theorem of local laws under Assumption 1.6 are presented as follows:

######  Theorem 1.11.

(Local law assuming all moments) Assume that the deterministic matrix
$H_{N}^{\infty}$ satisfies Assumption 1.2 and the noisy matrix satisfies
Assumption 1.6. Let $G(z)$ be the green function of
$H_{N}:=H_{N}^{\infty}+A_{N}$, and $G^{\infty}(z)$ be the Green function of
$H_{N}^{\infty}$. Then for any $\epsilon>0$ and $\kappa>0$, we can find
constant $C>0$ and $\nu>0$ depending only on $\epsilon$, $\kappa$,
$(\xi_{ij})_{i,j}$ and the upper bound in (1.7) such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S(\epsilon,\kappa)}}\max_{1\leq i,j\leq
N}|G-G^{\infty}|_{ij}(z)\geq C\log N^{\log\log
N}\left(\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N\eta}}\right)\right)\leq
e^{-\nu\log N^{\log\log N}}.$ (1.14)

######  Example 1.12.

(No entry-wise local law with overwhelming probability) We show by a
counterexample that Assumption 1.6 is almost optimal for such a local law, and
the claimed local law may fail under Assumption 1.7 or 1.8. To see this, we
choose $H_{N}^{\infty}$ to be the matrix (1.3), and let $A_{N}$ be a diagonal
matrix with only its $(i,i)$ entry nonzero, with
$A_{ii}=\frac{1}{N^{\alpha}}\xi_{ii}$ and $\xi_{ii}$ is the $\alpha$-stable
law. From resolvent expansion one must have
$G_{ii}(z)-G^{\infty}_{ii}(z)=-G_{ii}(z)A_{ii}G^{\infty}_{ii}(z)$. Properties
of the arcsine law imply that $G^{\infty}_{ii}(z)$ is bounded away from zero.
If (1.14) were true in this setting then $G_{ii}$ would be close to
$G^{\infty}_{ii}$ with overwhelming probability (i.e., with probability at
least $1-N^{-D}$ for any $D>0$), but with probability $N^{-1}$ we have
$|A_{ii}|>1$, leading to a contradiction. For more discussion, see Theorem
1.21 (3).

The local law we just proved has the following corollaries:

######  Corollary 1.13.

(Wegner’s estimate) Assume that $H_{N}^{\infty}+A_{N}$ satisfies all the
assumptions in Theorem 1.11. For any interval $I\subset\mathbb{R}$ let
$\mathcal{N}_{I}$ denote the number of eigenvalues of $H_{N}^{\infty}+A_{N}$
in $I$. Then for any $\epsilon>0$ and $\kappa>0$, one can find constants $C>0$
and $\nu>0$ depending on $\epsilon,\kappa,\alpha$ such that

$\mathbb{P}\left(\sup_{I\subset I_{\kappa,\epsilon}}\mathcal{N}_{I}\geq
C|I|N\right)\leq e^{-\nu\log N^{\log\log N}}$ (1.15)

where $I_{\kappa,\epsilon}$ is the set of all intervals
$I\subset[-2+\kappa,2-\kappa]$ with length $N^{-1+\epsilon}<|I|<1$.

The local law we just proved helps us to obtain convergence rates of the
density of states to the arcsine law, and to obtain rigidity estimates of
eigenvalues around their classical locations.

######  Corollary 1.14.

(Local arcsine law and eigenvalue rigidity) Assume the matrix $H_{N}^{\infty}$
is the 1d Laplacian (1.3) and $A_{N}$ satisfies Assumption 1.6. Denote by
$\rho^{as}$ the arcsine law, i.e. the probability law on $\mathbb{R}$ with
density

$\rho^{as}=\frac{1}{2\pi\sqrt{1-E^{2}/4}}1_{|E|<2}.$ (1.16)

Denote by $\mu_{N}$ the empirical measure of eigenvalues of
$H_{N}^{\infty}+A_{N}$, i.e. the the measure
$\frac{1}{N}\sum_{i}\delta_{\lambda_{i}}$ where $\lambda_{i}$ are eigenvalues
of $H_{N}^{\infty}+A_{N}$.

Then given any $\kappa>0$ we can find $\nu>0$ and (sufficiently small)
$c_{*}>0$ depending on $\kappa$ and $\alpha$, such that

$\mathbb{P}\left(\sup_{I\subset[-2+\kappa,2-\kappa]}|\mu_{N}(I)-\rho(I)|\geq
N^{-c_{*}}\right)\leq e^{-\nu\log N^{\log\log N}}$ (1.17)

where the supremum ranges over all intervals $I\subset[-2+\kappa,2-\kappa]$.

Without loss of generality we consider eigenvalues in $[0,2-\kappa]$. For
$i=1,\cdots,\frac{N}{2}$ define the classical location of the arcsine law
$\rho$ as follows: the classical location of the $i$-th positive eigenvalue is
the constant $\gamma_{i}$ that satisfies

$N\int_{0}^{\gamma_{i}}\rho(dx)=i-\frac{1}{2},\quad i=1,\cdots,\frac{N}{2}.$
(1.18)

Let $0\leq\lambda_{1}^{\prime}\leq\lambda_{2}^{\prime}\leq\cdots$ be the
ordered non-negative eigenvalues of the $N\times N$ random matrix
$H_{N}^{\infty}+A_{N}$. Then for any sufficiently small $\kappa>0$ we can find
positive constants $C$, $\nu$ and (sufficiently small) $c_{*}$ depending on
$\kappa$ and $\alpha$ such that

$\mathbb{P}\left(\sup_{i\geq 1:\lambda_{i}^{\prime}\leq
2-\kappa}|\lambda_{i}^{\prime}-\gamma_{i}|\geq CN^{-c_{*}}\right)\leq
e^{-\nu\log N^{\log\log N}}.$ (1.19)

The proof of this corollary is in Section 4.2. In these estimates we have
excluded the spectral edge $\pm 2$ because the arcsine law has unbounded
density there. To obtain nontrivial scaling limit at the edge one should take
a different scaling of potentials (scale them by $N^{-\frac{3}{2}}$ rather
than $N^{-\frac{1}{2}}$ when they have finite second moments, see for example
[23]).

######  Remark 1.15.

In the case of tridiagonal matrix (taking $K=1$) and when the random variables
$\xi_{ij}$ have sub-Gaussian tails, the results of bulk rigidity and
eigenvector delocalization are not exactly new as they can be derived from the
SDE characterization of microscopic bulk scaling limits in [36], see [8] for
details. These results however seem to be new when we have larger bandwidth
$K>1$, when we have heavy tails, and when we consider the matrices
$H_{N}^{\sigma}$ (1.24) in which random potentials $\xi_{ij}$ do not have the
diffusive scaling.

######  Remark 1.16.

We could allow the bandwidth $K$ in definition (1.8) to be $N$-dependent, in
such a way that $K=K(N)=o(N^{\sigma})$ as $N\to\infty$, for any $\sigma>0$. As
an example one can take $K=\log N$. In this case, the results of Theorem 1.11
and Corollary 1.13 remain true, and the proof is essentially unchanged.

A limitation of Corollary 1.14 is that the convergence rate is very slow and
we do not expect this to be optimal. The reason is that in the estimate
(1.14), no matter the value of $\eta$, the convergence rate is no faster than
$N^{-\frac{\epsilon}{10\alpha}}$, which is much slower than the
$N^{-1+\epsilon}$ rate that can be easily derived when all moments are finite.
This is a typical feature for random matrices with sparsity or heavy tails,
and we are not quite able to improve these estimates.

###  1.3. Local laws assuming more moment

######  Theorem 1.17.

(Entry-wise local law given additional moments) Assume that $H_{N}^{\infty}$
satisfies Definition 1.2. Assume that the noisy matrix $A=(A_{ij})$ satisfies
Assumption 1.7, that is, we assume each $\xi_{ij}$ has finite
$\alpha+\delta$-th moment. Then for any $\epsilon>0$ and $\kappa>0$, we can
find constant $C>0$ and some (small) $c_{*}>0$ depending only on
$\epsilon,\kappa$ and $\xi_{ij}$ such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S(\epsilon,\kappa)}}\max_{1\leq i,j\leq
N}\left|G_{ij}(z)-G_{ij}^{\infty}(z)\right|\geq
C\left(N^{-\frac{\epsilon}{20\alpha}}+N^{\frac{\epsilon}{40}}\sqrt{\frac{1}{N\eta}}\right)\right)\leq
N^{-c_{*}\epsilon}.$ (1.20)

The proof is in Section 3.3.

######  Corollary 1.18.

(Eigenvector de-localization) Under the same assumption as Theorem 1.17, for
any $\epsilon>0$ and $\kappa>0$, we can find a sufficiently small constant
$c_{*}>0$ and a large constant $C>0$ depending on $\epsilon,\kappa,\alpha$
such that the following is true: the possibility that $H_{N}^{\infty}+A_{N}$
has an eigenvalue $\lambda\in[-2+\kappa,2-\kappa]$ whose corresponding
normalized eigenvector $\mathbf{v}_{\lambda}$ satisfies
$\|\mathbf{v}_{\lambda}\|_{L^{2}}=1$ and
$\|\mathbf{v}_{\lambda}\|_{L^{\infty}}\geq CN^{-\frac{1}{2}+\epsilon}$ is at
most $N^{-c_{*}\epsilon}.$ If instead we assume that $A_{N}$ satisfies
Assumption 1.6, then the said event has probability at most $e^{-\nu\log
N^{\log\log N}}$.

The proof of this corollary is standard and omitted (see also the proof of
Corollary 1.21). See for example [10], Theorem 2.10. To prove the first part
we use the local law in Theorem 1.17 and to prove the second part we use the
local law in Theorem 1.11.

###  1.4. Local laws and de-localization for alpha- stable noise

When elements $A_{ij}$ of $A_{N}$ have $\alpha$\- stable laws with the
$\frac{1}{N^{\frac{1}{\alpha}}}$ scaling in front of it, the proof of
eigenvector de-localization is significantly more complicated and will be the
task of this section.

Before the proof, we first show that eigenvector de-localization should not be
expected to take place with overwhelming probability (probability larger than
$1-N^{-D}$ for any $D$).

######  Example 1.19.

(Localized approximate eigenfunctions) Assume that $H_{N}^{\infty}$ is the
matrix (1.3), and let $A_{N}$ be a diagonal matrix with all diagonal elements
$A_{ii}=\frac{1}{N}\xi_{ii}$, where $\xi_{ii}$ are i.i.d. variables with
Cauchy distribution. We can construct an approximate eigenfunction of $H_{N}$
as follows: there is an an eigenvector $\mathbf{v}$ of $H_{N}$ that has
eigenvalue $\lambda$. Assume without of generality that we can find three
consecutive induces $i-1,i,i+1$ such that three coordinates
$\mathbf{v}_{i-1},\mathbf{v}_{i},\mathbf{v}_{i+1}$ are non-zero, and that
$|\mathbf{v}_{i}|=\|\mathbf{v}\|_{\infty}$. In step 1, we re-sample the value
of $A_{ii}$, and assume that its new value $A_{ii}^{\prime}$ satisfies
$|\lambda-A_{ii}^{\prime}|\leq N^{-0.5}|\lambda-A_{ii}|$. Then for the
relation $\mathbf{v}_{i-1}+\mathbf{v}_{i+1}=(\lambda-
A_{ii}^{\prime})\mathbf{v}_{i}^{\prime}$ to be satisfied with
$\mathbf{v}_{i-1},\mathbf{v}_{i+1}$ fixed, one needs to have
$|\mathbf{v}_{i}^{\prime}|\geq N^{0.5}|\mathbf{v}_{i}|\geq 1$. In step 2, we
resample the value of $A_{i-1,i-1}$ and $A_{i+1,i+1}$, and assume that with
the new value, we have both equations
$|\mathbf{v}_{i-2}+\mathbf{v}_{i}^{\prime}-(\lambda-
A_{i-1,i-1}^{\prime})\mathbf{v}_{i-1}|<0.1N^{-0.5}$ and
$|\mathbf{v}_{i}^{\prime}+\mathbf{v}_{i+2}-(\lambda-
A_{i+1,i+1}^{\prime})\mathbf{v}_{i+1}|<0.1N^{-0.5}$ hold. Then we define a new
vector
$\mathbf{v}^{\prime}=(\mathbf{v}_{1},\cdots,\mathbf{v}_{i-1},\mathbf{v}_{i}^{\prime},\mathbf{v}_{i+1},\cdots,\mathbf{v}_{n})$.
Re-normalize $\mathbf{v}^{\prime}$ so that it has $L^{2}$ norm one, we have by
construction that $\mathbf{v}^{\prime}$ is completely localized in the
$L^{\infty}$ sense: $\|\mathbf{v}^{\prime}\|_{\infty}\sim 1$. Moreover,
$\mathbf{v}^{\prime}$ is an approximate eigenvector of $H_{N}^{\prime}$ with
$A_{i-1,i-1},A_{ii},A_{i+1,i+1},$ re-sampled, in the sense that
$|H_{N}^{\prime}\mathbf{v}^{\prime}|\leq N^{-0.5}$, and the probability that
these three random variables can be sampled to satisfy the aforementioned
condition is at least $N^{-3}$, which is much larger than the possibility of
an overwhelmingly small event.

This hints that the obstruction to eigenvector de-localization is that some of
the atypical indices of $A_{ii}$ (the indices $i$ such that $|A_{ii}|\sim 1$)
are too close to each other. With this idea, we prove the following de-
localization result that holds with probability tending to one. We first
introduce some definitions:

######  Definition 1.20.

(Removal set) For any non-negative integers $K$ and $p$, define $\Delta_{K}$
as the set of $E\in(-2,2)$ such that $\sin(\ell\arccos(\frac{1}{2}E))=0$ for
some $\ell=1,2,\cdots,K$. Then define
$\Delta_{K}^{p}:=\Delta_{K}+[-10^{-p},10^{-p}]$, which is a neighborhood of
$\Delta_{K}$ consisting of intervals of length $2*10^{-p}$, centered at
elements in $\Delta_{K}$. Define a parameter region

$\mathcal{S}(\epsilon,\kappa,K,p):=\\{z=E+i\eta:E\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p},\quad
N^{-1+\epsilon}\leq\eta\leq 1\\}.$ (1.21)

Note that if $K=0$ (diagonal matrix) or $K=1$ (tridiagonal matrix), then the
set $\Delta_{K}$ is empty, so that $\Delta_{K}^{p}$ is empty as well, and
$\mathcal{S}(\epsilon,\kappa,K,p)$ is simply $\mathcal{S}(\epsilon,\kappa)$.

######  Theorem 1.21.

(Entry-wise Green function bounds in $\alpha$-stable case) Assume that
$H_{N}^{\infty}$ is the matrix (1.3) of 1d Laplacian, and $A_{N}$ satisfies
Assumption 1.8, Then, with $G$ denoting the Green function of
$H_{N}^{\infty}+A_{N}$,

  1. (1)

Green function is bounded: we can find a (large) $C>0$ and a (small) $c_{*}>0$
depending on $\epsilon,\kappa,\alpha,K,p$ such that for $N$ large,

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa,K,p)}\max_{1\leq
i,j\leq N}\left|G_{ij}(z)\right|\leq C\right)\geq 1-N^{-c_{*}\epsilon}.$
(1.22)

  2. (2)

Entry-wise local law fails with positive probability: we can find some
constant $C_{0}>0$ and some $P_{0}\in(0,1)$ depending on
$\epsilon,\kappa,\alpha,K,p$ such that for $N$ large,

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa,K,p)}\max_{1\leq
i,j\leq N}|G_{ij}(z)-G_{ij}^{\infty}(z)|\geq C_{0}\right)\geq P_{0}.$ (1.23)

In particular, if $K=0$ (diagonal) or $K=1$ (tridiagonal), then the constants
$C$ and $C_{0}$ do not depend on $K$ and $p$, and the estimates hold for all
$z\in\mathcal{S}(\epsilon,\kappa)$.

The proof of this theorem is given in Section 5. It is interesting to compare
this result with Theorem 1.17: Theorem 1.17 deals with the case where
$\xi_{ij}$ has $\alpha+\delta$ moments, so that the effect of noise is much
smaller, and $G_{ij}$ converges to $G_{ij}^{\infty}$ with probability going to
one; whereas this theorem (1.21) deals with the critical case where the noise
has compelling effect with the Laplacian: the Green function is still bounded
with high probability but most likely $G_{ij}$ does not converge to
$G_{ij}^{\infty}$ for some entries $(i,j)$.

######  Remark 1.22.

The estimates in Theorem 1.21 can be made uniform over
$z\in\mathcal{S}(\epsilon,\kappa)$ when $K=0$ or $K=1$, i.e. when we consider
tridiagonal or Jacobi matrices. This is also the case covered by most existing
literature on random Schrödinger operators. When $K\geq 2$, our estimates hold
outside a removal set defined in 1.20, and it seems hard to make the estimates
uniform over $\mathcal{S}(\epsilon,\kappa)$. We are not sure if this is merely
a technical restriction or is truly a barrier that the Green function will be
unbounded near those removed sets. Nonetheless, by choosing $p$ large enough,
we have derived Green function estimates for Lebesgue a.e.
$E\in(-2+\kappa,2-\kappa)$ despite the bound being not uniform.

The proof of Theorem 1.21 uses more properties on the matrix $H_{N}^{\infty}$
than those stated in Definition 1.2: every entry of $G^{\infty}_{ij}(z)$ is
bounded. We use the fact that the off-diagonal elements of
$G^{\infty}_{ij}(z)$ decay exponentially fast, see Proposition (2.3); and that
the imaginary parts of $G^{\infty}_{ij}(z)$ have a certain pattern when
$(i,j)$ is close to but not on the diagonal, see Proposition 2.4. For this
reason we have not stated the result for general matrices $H_{N}^{\infty}$
satisfying Definition 1.2.

######  Corollary 1.23.

(Eigenvector de-localization) Under the same assumption as in Theorem 1.21,
for any $\epsilon>0$ and $\kappa>0$, we can find a sufficiently small constant
$c_{*}>0$ and a large constant $C>0$ depending on $\epsilon,\kappa,\alpha,K,p$
such that the following is true: the possibility that $H_{N}^{\infty}+A_{N}$
has an eigenvalue $\lambda\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$ such
that its corresponding normalized eigenvector $\mathbf{v}_{\lambda}$ satisfies
$\|\mathbf{v}_{\lambda}\|_{L^{2}}=1$ and
$\|\mathbf{v}_{\lambda}\|_{L^{\infty}}\geq CN^{-\frac{1}{2}+\epsilon}$ is at
most $N^{-c_{*}\epsilon}.$ In particular if $K=0$ or $1$, then the claim holds
for all eigenvalues in $[-2+\kappa,2-\kappa]$.

The proof is given in Section 5.2.1.

###  1.5. Different scaling of noise

We now investigate the case where, instead of the
$\frac{1}{N^{\frac{1}{\alpha}}}$ leading coefficient, we have a different
scaling in front of $\xi_{ij}$. The techniques in this paper still apply, and
we prove local laws at certain intermediate scales.

######  Theorem 1.24.

(Different scaling: when all moments exist) For any
$\sigma\in(0,\frac{1}{\alpha}),$ consider the matrix

$H_{N}^{\sigma}:=H_{N}^{\infty}+N^{\sigma}A_{N},$ (1.24)

where $H_{N}^{\infty}$ satisfies Definition 1.2, $A_{N}$ satisfies the
following variant of Assumption 1.6: for any $\omega>\frac{\epsilon}{8}>0$ and
$\delta>0$,

$|\mathbb{E}[A_{ij}]|\leq\frac{C}{N^{1+\sigma+\delta}},\quad\mathbb{E}[|A_{ij}|^{p}]\leq\frac{C}{N(N^{\sigma+\omega})^{p-\alpha}}\text{
for any }p\geq 2.$ (1.25)

Denote the Green function of $H_{N}^{\sigma}$ by $G^{\sigma}(z)$. Then
$G^{\sigma}$ satisfies the following local law: we can find constants $C$ and
$\nu$ depending on $\epsilon,\omega,\kappa$ such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa,\sigma)}\max_{1\leq
i,j\leq N}\left|G^{\sigma}_{ij}(z)-G^{\infty}_{ij}(z)\right|\geq
CN^{-\min(\frac{\omega}{2},\frac{\epsilon}{4})}\right)\leq e^{-\nu\log
N^{\log\log N}},$ (1.26)

where

$\mathcal{S}(\epsilon,\kappa,\sigma):=\\{z=E+i\eta:|E|\leq
2-\kappa,N^{-1+\epsilon+\sigma\alpha}\leq\eta\leq 1\\}.$

As a corollary, Wegner’s estimate (1.15) holds for $H_{N}^{\sigma}$ for all
intervals $I\subset[-2+\kappa,2-\kappa]$ with length
$N^{-1+\epsilon+\sigma\alpha}<|I|<1$. Moreover, eigenvector de-localization
holds in the following form: with probability at least $1-e^{-\nu\log
N^{\log\log N}}$, for any eigenvalue $\lambda\in[-2+\kappa,2-\kappa]$ its
$L^{2}$-normalized eigenvector $\mathbf{v}_{\lambda}$ satisfies
$\|\mathbf{v}_{\lambda}||_{\infty}\leq
CN^{-\frac{1-\epsilon-\sigma\alpha}{2}}$.

The proof of this theorem is contained in Section 3.2.1. We have essentially
covered all the scales where there is a polynomially decaying factor in front
of $\xi_{ij}$, as we only impose $\sigma\in(0,\frac{1}{\alpha})$. Remark that
compared to Assumption 1.6, we have enlarged the value of $q$, thus imposing a
much stronger growth condition on the large values $A_{ij}$ can take. Concrete
examples where (1.25) is satisfied is when $\xi_{ij}$ is defined from a
symmetric random variable $\xi$ having finite $\alpha$-moment, conditioning
that $|\xi|\leq N^{\frac{1}{\alpha}-\sigma-\omega}$.

######  Theorem 1.25.

(Different scaling: $\alpha$-stable case) In this case assume $H_{N}^{\infty}$
is the matrix of 1d Laplacian (1.3) and assume $A_{N}$ satisfies Assumption
1.8. For any $\sigma\in(0,\frac{1}{2\alpha})$, denote by $G^{\sigma}$ the
Green function of $H_{N}^{\sigma}$ as in (1.24). Then all the three claims of
Theorem 1.21 continue to hold for $G^{\sigma}$, after replacing every
$\mathcal{S}(\epsilon,\kappa,K,p)$ by
$\mathcal{S}(\epsilon,\kappa,2\sigma,K,P),$ where

$\mathcal{S}(\epsilon,\kappa,2\sigma,K,p):=\\{z=E+i\eta:E\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p},\quad
N^{-1+2\sigma\alpha+\epsilon}\leq\eta\leq 1\\}.$ (1.27)

In particular, eigenfunction de-localization in Corollary 1.23 continues to
hold in the following form: we can find a small constant $c_{*}>0$ and a large
constant $C>0$ depending on $\epsilon,\kappa,\alpha,K,p$ such that, with
probability at least $1-N^{-c_{*}\epsilon}$, given any eigenvalue
$\lambda\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$, its
$L^{2}$-normalized eigenvector $\mathbf{v}_{\lambda}$ must satisfy
$|\mathbf{v}_{\lambda}|_{L^{\infty}}\leq
CN^{-\frac{1-\epsilon-2\sigma\alpha}{2}}$.

The proof of this theorem is given in Section 5.3.

######  Remark 1.26.

Despite its effectiveness, our method cannot detect local statistics at
intermediate scale $[N^{-1},N^{-1+\sigma\alpha}]$. Some other methods are
needed for a finer analysis. For example: are eigenvectors de-localized or
localized at this intermediate scale? Can we describe a scaling limit of
eigenvalues?

######  Remark 1.27.

In the $\alpha$-stable case, Theorem 1.25 imposes a condition
$\sigma\in(0,\frac{1}{2\alpha})$. These conditions define in some sense a sub-
critical regime, in which the locations $(i\leq j)$ where $N^{\sigma}A_{ij}$
takes atypically large values are sparse with high probability. On the other
hand, if $\sigma\in(\frac{1}{2\alpha},\frac{1}{\alpha})$, then the global
density of states still converges to the arcsine law [25], but the local
statistics cannot be investigated via techniques in this paper. The issue is
that the locations $(i\leq j)$ where $N^{\sigma}A_{ij}$ may take atypically
large values are no longer sparse with high probability, significantly
complicating the analysis.

###  1.6. A simple criterion for local laws of Stieltjes transform

To complete the investigation, we show that even if entry-wise local law fails
for the matrix, one can still prove the trace of Green function converges to
the correct limit with probability tending to one. The reason behind it is
that only a small fraction of Green function entries will deviate from the
expected limit, and the trace takes a law of large numbers.

######  Theorem 1.28.

(Local law for Stieltjes transform: 1d Laplacian with $\alpha$-stable noise)
Assume that $H_{N}^{\infty}$ is the matrix given in (1.3) and $A_{N}$
satisfies Assumption 1.8. Recall that $m$ and $m^{\infty}$ denote the trace of
Green function of $H_{N}^{\infty}+A_{N}$ and $H_{N}^{\infty}$ respectively.
Then for any $\epsilon>0$ and $\kappa>0$ we can find constants $C$ and $c_{*}$
depending on $\epsilon,\kappa,\alpha$ such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa)}|m(z)-m^{\infty}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq N^{-c_{*}\epsilon}$ (1.28)

and the same holds if we replace $m^{\infty}(z)$ by
$m^{as}(z)=\frac{1}{\sqrt{z^{2}-4}}$ for $z\in\mathbb{C}_{+},$ which is the
Stieltjes transform of the arcsine law.

More generally, for any $\sigma\in(0,\frac{1}{2\alpha})$, consider
$H_{N}^{\infty}+N^{\sigma}A_{N}$, where the two matrices satisfy the
assumptions stated above. Denote the trace of the Green function of
$H_{N}^{\infty}+N^{\sigma}A_{N}$ by $m^{\sigma}(z)$. Then for any $\epsilon>0$
and $\kappa>0$ we can find positive constants $C$ and $c_{*}$ such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa,2\sigma)}|m^{\sigma}(z)-m^{\infty}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq N^{-c_{*}\epsilon},$ (1.29)

where

$\mathcal{S}(\epsilon,\kappa,\sigma):=\\{z=E+i\eta:|E|\leq
2-\kappa,N^{-1+\epsilon+\sigma\alpha}\leq\eta\leq 1\\}.$

The proof is given in Section 6.

######  Remark 1.29.

We again impose a technical restriction $\sigma\in(0,\frac{1}{2\alpha})$ and
do not work to the whole decaying regime $\sigma\in(0,\frac{1}{\alpha})$. The
reason is that we need to compute the Green function of $H_{N}^{\infty}$ when
a certain number of rows and columns with the same index are removed, and this
restrictive hypothesis $\sigma\in(0,\frac{1}{2\alpha})$ guarantees with high
probability that such a Green function bound is available via Proposition 1.1.
It would be possible to get rid of this technical assumption if one can adopt
a different estimate on the Green function of $H_{N}^{\infty}$ after this type
of row and column removal.

######  Corollary 1.30 (Corollary of local law).

Let $H_{N}^{\infty}$ and $A_{N}$ satisfy the same assumptions as in Theorem
1.28 .Then with probability at least $1-N^{-c_{*}\epsilon}$ the following
statements are true: (i) the Wegner’s estimate in Corollary 1.13 also holds
for $H_{N}^{\infty}+A_{N}$. (ii) The Wegner’s estimate also holds for
$H_{N}^{\infty}+N^{\sigma}A_{N}$ provided that the interval $I$ in the
statement of Corollary 1.13 has length $|I|>N^{-1+2\sigma\alpha+\epsilon}$.
(iii)The rigidity and convergence rate estimates of eigenvalues stated in
Corollary 1.14 hold for both $H_{N}^{\infty}+A_{N}$ and
$H_{N}^{\infty}+N^{\sigma}A_{N}$.

The proof of Corollary 1.30 is essentially the same as Corollary 1.13 and 1.14
: in the proof of these two results we only used the local law for the trace
of Green function, and entry-wise local law is not used.

###  1.7. Comparison to Wigner matrices and Levy matrices

In this paper we introduced a class of matrices that are heavy tail
perturbations of square symmetric matrices, where the heavy tail random
variables are sufficiently close to the diagonal. We choose a scaling factor
$\frac{1}{N^{\frac{1}{\alpha}}}$ in front of the noise, and some more general
scaling factors are considered as well, so that the density of states of
$H_{N}$ is asymptotically close to the noiseless part $H_{N}^{\infty}$.

The effect of noise takes place on a local scale, where we prove that at the
almost optimal scale $N^{-1+\epsilon}$, the Stieltjes transform converges with
high probability to the correct limit but individual entries of Green function
may not converge with a non-vanishing probability. We prove Wegner estimates
and eigenvalue rigidity, and with probability converging to 1 the
eigenfunctions are de-localized in the $L^{\infty}$ norm sense. However,
localized approximate eigenfunctions may arise with a polynomially small
probability. In the forthcoming Theorem 1.32, we will prove that we have
similar phenomenon if we replace the deterministic matrix $H_{N}^{\infty}$ by
an independent Wigner matrix $W_{N}$.

The statistical properties of these matrices are very different from those
revealed in the recent investigation of Levy matrices with $\alpha$-stable
distribution [16], [6], [35]. [13], [12], [9]. When $\alpha\in(1,2)$, [3]
proves that entr-ywise local law holds with overwhelming probability, and
eigenfunctions are de-localized with overwhelming probability. When
$\alpha\in(0,1)$, [2] proves that there is a mobility edge: above this energy
the eigenvectors are localized and below this the eigenvectors are de-
localized with probability tending to one.

The primary difference of our model to Levy matrices is possibly that our
noise is essentially one dimensional, so that individual peaks will have a
huge effect. We cannot use analytical tools like ergodicity of Dyson Brownian
motion to smooth the peaks out.

###  1.8. Other generalizations

Finally we discuss other generalizations of the method in this paper.

######  Example 1.31.

(Wigner matrix with heavy tail perturbations) Our method of proof can be
easily extended to cover Wigner matrices perturbed by heavy tailed noise. That
is, we consider $W_{N}+A_{N}$ where $W_{N}$ and $A_{N}$ are independent,
$A_{N}$ satisfies the same assumption as before but $W_{N}$ is an $N\times N$
Wigner matrix whose entries have sub-exponential decay.

For the Wigner matrix $W_{N}$, entry-wise local semicircle law is satisfied
with overwhelming probability [20], which trivially implies that Green
function entries are bounded, (1.7). Now we can condition on $W_{N}$ and apply
the same proof procedure in this paper. Thus results in Theorem 1.11 and 1.17
can be extended without much difficulty to the matrix $W_{N}+A_{N}$.

We can also extend Theorem 1.28 to $W_{N}+A_{N}$. This time we can cover all
$\sigma\in(0,\frac{1}{\alpha})$.

######  Theorem 1.32.

(Local law for Stieltjes transform: Wigner matrix with banded $\alpha$-stable
perturbation) Let $W_{N}=(W_{ij})$ be an $N\times N$ real symmetric matrix
with independent entries modulo symmetry restriction, and satisfy
$\mathbb{E}[W_{ij}]=0$, $\mathbb{E}[|W_{ij}|^{2}]=\frac{1}{N}$ and $W_{ij}$
has sub-exponential decay: for some $\vartheta>0$, $x>1$ and all
$i,j\in[1,N]:$

$\mathbb{P}(|W_{ij}|>N^{-\frac{1}{2}}x)\leq\vartheta^{-1}\exp(-x^{\vartheta}).$

Let $A_{N}$ be the matrix satisfying Assumption 1.8 and assume that $W_{N}$
and $A_{N}$ are independent. Let $m(z)$ denote the trace of Green function of
$W_{N}+A_{N}$ and $m^{sc}(z)$ the Stieltjes transform of the semicircle law:
for all $z\in\mathbb{C}$ with $\Im z>0$,
$m^{sc}(z)=\frac{-z+\sqrt{z^{2}-4}}{2}$. Then for any $\epsilon>0$ we can find
constants $C$ and $\nu$ depending on $\epsilon,\alpha$ such that

$\mathbb{P}\left(\sup_{z\in\hat{\mathcal{S}}(\epsilon)}|m(z)-m^{sc}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}},$ (1.30)

where

$\hat{\mathcal{S}}(\epsilon):=\\{z=E+i\eta:|E|\leq
5,N^{-1+\epsilon}\leq\eta\leq 1\\}.$

More generally, for any $\sigma\in(0,\frac{1}{\alpha})$, consider
$W_{N}+N^{\sigma}A_{N}$, where the two matrices satisfy the assumptions stated
above. Denote the trace of the Green function of
$H_{N}^{\infty}+N^{\sigma}A_{N}$ by $m^{\sigma}(z)$. Then for any $\epsilon>0$
we can find constants $C$ and $\nu$ such that

$\mathbb{P}\left(\sup_{z\in\hat{\mathcal{S}}(\epsilon,\sigma)}|m^{\sigma}(z)-m^{sc}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}},$ (1.31)

where

$\hat{\mathcal{S}}(\epsilon,\sigma):=\\{z=E+i\eta:|E|\leq
5,N^{-1+\epsilon+\sigma\alpha}\leq\eta\leq 1\\}.$

The proof is given in Section 6.1.

######  Remark 1.33.

In this theorem we can cover the whole decaying regime
$\sigma\in(0,\frac{1}{\alpha})$ and the local law holds with overwhelming
probability. This is because removing the rows and columns of a Wigner matrix
indexed by a certain index set, the submatrix is again a Wigner matrix and its
local law estimates are available. Moreover, in this local law we can cover
the spectral edge as the estimate holds throughout $E\in[-5,5]$. Finally, we
can check that the counterexamples in 1.12 continue to hold for $W_{M}+A_{N}$,
i.e. entry-wise local law does not hold with overwhelming probability.

######  Example 1.34.

(Matrix models for beta-ensembles) Another important example where our method
can possibly be applied concerns the matrix models for beta ensembles. In
Dumitriu,-Edelman [17], they showed that a form of random Jacobi matrices
$H_{\beta}$ given below, where $N(0,\cdot)$ is the normal distribution and
$\chi_{\cdot}$ is the chi-distribution with specified parameters, are matrix
models for the Gaussian $\beta$-ensemble,

$H_{\beta}=\frac{1}{\beta}\begin{pmatrix}N(0,2)&\chi_{(N-1)\beta}&0&\cdots&\cdots&0\\\
\chi_{(N-1)\beta}&N(0,2)&\chi_{(N-2)\beta}&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\
0&\cdots&\chi_{3\beta}&N(0,2)&\chi_{2\beta}&0\\\
0&\cdots&0&\chi_{2\beta}&N(0,2)&\chi_{\beta}\\\
0&\cdots&\cdots&0&\chi_{\beta}&N(0,2)\end{pmatrix}.$ (1.32)

One can replace these Gaussian and chi distributions by general random
variables that have the same mean and variance and study the resulting random
Jacobi matrix. This is usually conducted via an approach based on transfer
matrix recursions or Prüfer coordinates (see for example [15],[7], [29],
[30],) assuming the random variables have sub-Gaussian tails.

We can instead regard such matrices as noisy perturbations of the following
deterministic matrix

$H_{\infty}=2\begin{pmatrix}0&\sqrt{(N-1)/N}&0&&\\\
\sqrt{(N-1)/N}&0&\sqrt{(N-2)/N}\\\ &\ddots&\ddots&&\\\ &&0&\sqrt{2/N}&0\\\
&&\sqrt{2/N}&0&\sqrt{1/N}\\\ &&0&\sqrt{1/N}&0\end{pmatrix}.$ (1.33)

and apply the results established in this paper, so that we can cover the case
when the random variables have heavy tails. A technical challenge is that the
Green function estimates of $H_{\infty}$ can not be obtained very easily. A
proof of local law of the trace of the Green function $H_{\infty}$ can be
found in [34] via Hermite polynomials and the steepest descent method.
However, one still needs an entry-wise local law so that the results of this
paper can be applicable.

##  2\. Arcsine laws and Green function for the symmetric matrix

In this section we cover some properties of the arcsine law (1.16) and the
matrix model of 1d Laplacian (1.3). These results can be derived from simple
computations, but they seem to be new in the literature and can possibly be
used in other contexts.

###  2.1. Green’s function for the 1d Laplacian

In this section we prove Proposition 1.1, i.e. the Green function $G^{\infty}$
of $H_{N}^{\infty}$ defined in (1.3) is uniformly bounded from above when
$\eta>>N^{-1}$.

###### Proof of Proposition 1.1.

The following argument is taken from [24]: for $D\in(-2,2)$, we can explicitly
compute $(H_{N}^{\infty}+DI_{N})^{-1}$ as follows:

Let $M_{k}$ denote the determinant of $H_{k}^{\infty}+DI_{k}$, then we have
the recursive formula

$M_{k+1}=DM_{k}-M_{k-1},\quad M_{0}=1,M_{1}=D.$ (2.1)

Assume that $D=-2\cos\lambda$, then the solution to this recursion is

$M_{k}=(-1)^{k}\sin((k+1)\lambda)/\sin\lambda.$ (2.2)

Then thanks to the tridiagonal structure of $M_{k}$, if we denote by $R_{ij}$
the $(i,j)$-th element of the inverse matrix, then

$R_{ij}=(-1)^{i+j}M_{i-1}M_{N-j}/M_{N},\quad i\leq j.$ (2.3)

that is,

$R_{ij}=\frac{\sin(i\lambda)\sin((N-j+1)\lambda)}{\sin\lambda\sin((N+1)\lambda)}=\frac{\cos((N+1-|j-i|)\lambda)-\cos((N+1-j-i)\lambda)}{2\sin\lambda\sin((N+1)\lambda)}.$
(2.4)

This expression admits an analytic extension to the complex plane when
$\lambda$ is complex valued. By definition of the domain $\mathcal{S}$ and the
trigonometric expansion

$\cos(x-iy)=\cos x\cosh y-i\sin x\sinh y,$ (2.5) $\sin(x-iy)=\sin x\cosh
y-i\cos x\sinh y,$ (2.6)

one sees that for any $D\in\mathcal{S}$ we can find a unique $\lambda$ with
$D=-2\cos\lambda$, such that
$\Re\lambda\in[\kappa^{\prime}\pi,\pi-\kappa^{\prime}\pi]$ for some
sufficiently small $\kappa^{\prime}>0$ depending on $\kappa$. Moreover, from
the expression $\frac{\partial}{\partial
x}\arccos(x+iy)=-\frac{1}{1-(x+iy)^{2}}$ and $\frac{\partial}{\partial
y}\arccos(x+iy)=-i\frac{1}{1-(x+iy)^{2}}$, we can see that for
$D\in\mathcal{S}$ and $\lambda=\arccos\frac{D}{2}$, one must have
$|\Im\lambda|\geq C^{\prime}N^{-1+\epsilon}$ for some $C^{\prime}$ depending
only on $\kappa$.

Now we turn back to the inverse formula (2.4) and show this expression is
uniformly upper bounded. Concerning the term $\sin\lambda$, one can show that
$|\sin\lambda|$ is uniformly bounded away from 0 since $\Re\lambda$ is away
from $0$ and $\pi$, and $\Im\lambda$ is not too large. Concerning the term
$\sin((N+1)\lambda)),$ thanks to the fact that $|\Im\lambda|\geq
C^{\prime}N^{-1+\epsilon}$, one sees from trigonometric expansion that
$|\sin((N+1)\lambda))|\sim e^{N|\Im\lambda}|$ and that this factor dominates
both terms of the numerator of the right hand side of (2.4). Therefore,
$|R_{ij}|$ is uniformly bounded from above for all indices $i,j\in[1,N]$. This
completes the proof of Proposition 1.1. ∎

###  2.2. Properties of the arcsine law

######  Proposition 2.1.

The Stieltjes transform $m^{as}(z)$ of the arcsine law (1.16) is given by

$m^{as}(z)=\frac{1}{2\sqrt{z^{2}/4-1}},\quad z\in\mathcal{S}.$ (2.7)

###### Proof.

This is not a new result but we sketch its proof. A standard result from
residue calculus shows that for any complex constant $a$ outside $[-1,1]$, we
have

$\int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}(x-a)}=-\frac{\pi}{\sqrt{a^{2}-1}}.$
(2.8)

After rearranging the constants, for any complex $z$ outside $[-2,2]$, we have

$m^{as}(z)=-\int_{-2}^{2}\frac{dx}{2\pi\sqrt{1-x^{2}/4}(x-z)}=\frac{1}{2\sqrt{z^{2}/4-1}}.$
(2.9)

∎

######  Lemma 2.2.

Let $m^{\infty}(z)$ denote the Stieltjes transform of $H_{N}^{\infty}$ and
$m^{as}(z)$ denote the Stieltjes transform of the arcsine law, for any
$z\in\mathcal{S}(\epsilon,\kappa)$. Then we can find a constant $C>0$
depending on $\epsilon$ and $\kappa$ such that

$|m^{\infty}(z)-m^{as}(z)|\leq
C(N^{-0.5\epsilon}+e^{-N^{1-0.5\epsilon}\eta}).$ (2.10)

###### Proof.

Recall from Section 2.1 that for $\lambda=\arccos\frac{1}{2}z$, the $(j,j)$-th
diagonal term of the resolvent matrix $G^{\infty}:=(H_{N}^{\infty}-z)^{-1}$ is
given by (we avoid using subscript $i$ to avoid confusion with the imaginary
unit)

$G^{\infty}_{jj}=\frac{\cos((N+1)\lambda)-\cos((N+1-2j)\lambda)}{2\sin\lambda\sin((N+1)\lambda))}.$
(2.11)

We also argued in Section 2.1 that for $z\in\mathcal{S}(\epsilon,\kappa)$,
necessarily we have $|\Im\lambda|>C\eta>N^{-1+\epsilon}$ for some $C>0$, and
$\Im\lambda<0$.

From the trigonometric expansion (2.5) of $\sin(x)$ and $\cos(x)$, one sees
that whenever $j\in[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}]$, necessarily
$|\cos((N+1-2j)\lambda)|\leq e^{-N^{1-0.5\epsilon}\eta}|\cos((N+1)\lambda|$.
Thanks to the fact that $G_{jj}^{\infty}$ is uniformly bounded (proved in
Section 2.1), we conclude that

$\left|G_{jj}^{\infty}-\frac{1}{2\sin\lambda\tan((N+1)\lambda)}\right|\leq
Ce^{-N^{1-0.5\epsilon}\eta},\quad
j\in[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}].$ (2.12)

In the next step, from the trigonometric expansion

$\tan(x+iy)=\frac{\sin 2x+i\sinh 2y}{\cos 2x+\cosh 2y},$

and from the assumption $\Im\lambda<-C\eta<-CN^{-1+\epsilon}$, we see that

$\left|\frac{1}{\tan((N+1)\lambda)}-i\right|\leq e^{-N\eta}.$ (2.13)

Since $|\frac{1}{\sin\lambda}|$ is bounded from above on $\mathcal{S}$, we
conclude that

$\left|G_{jj}^{\infty}-\frac{i}{2\sin\lambda}\right|\leq
Ce^{-N^{1-0.5\epsilon}\eta},\quad
j\in[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}].$ (2.14)

For indices $i$ that are outside $[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}]$,
the above estimate does not apply but these terms are all uniformly upper
bounded. There are altogether $2N^{1-0.5\epsilon}$ such terms. Finally, note
the following trigonometric identity

$\sin(\arccos(\frac{z}{2}))=\sqrt{1-z^{2}/4}=i\sqrt{z^{2}/4-1}.$

Summing everything up, we deduce that for any
$z\in\mathcal{S}(\epsilon,\kappa)$,

$|m^{\infty}(z)-m^{as}(z)|\leq
2\frac{N^{1-0.5\epsilon}}{N}+Ce^{-N^{1-0.5\epsilon}\eta}.$ (2.15)

This completes the proof. ∎

######  Proposition 2.3.

(Off-diagonal decay of Green function). The Green function
$G_{ij}^{\infty}(z)$ satisfies the following decay estimate: for any
$\epsilon>0,\kappa>0$ we can find a constant $C$ such that, setting
$L=N^{1-0.5\epsilon}$, then for any $\eta\in[N^{-1+\epsilon},1]$ and
$E\in[-2+\kappa,2-\kappa]$ we must have

$|G_{ij}^{\infty}(z)|\leq e^{-N^{0.5\epsilon}},\quad|i-j|>L.$ (2.16)

More generally, setting $L=N^{1-\sigma\alpha-0.5\epsilon}$, then for any
$\eta\in[N^{-1+\sigma\alpha+\epsilon},1]$ we must have

$|G_{ij}^{\infty}(z)|\leq e^{-N^{0.5\epsilon}},\quad|i-j|>L.$ (2.17)

###### Proof.

We only prove the first claim, as the second one is exactly analogous. Recall
from Section 2.1 that for $\lambda=\arccos\frac{1}{2}z$, the $(j,k)$-th term
of $G^{\infty}:=(H_{N}^{\infty}-z)^{-1}$ is given by

$G^{\infty}_{jk}=\frac{\cos((N+1-|j-k|)\lambda)-\cos((N+1-j-k)\lambda)}{2\sin\lambda\sin((N+1)\lambda))}.$
(2.18)

Then we have $N+1-|j-k|\in[1,N-L]$ and $N+1-j-k\in[-N+L,N-L]$. Then thanks to
the fact that $|\Im\lambda|>N^{-1+\epsilon}$ and the trigonometric identity
(2.5), this immediately implies that

$\left|\frac{\cos((N+1-|j-k|)\lambda)}{\sin((N+1)\lambda))}\right|\leq
e^{-N^{0.5\epsilon}},\quad\left|\frac{\cos((N+1-j-k)\lambda)}{\sin((N+1)\lambda))}\right|\leq
e^{-N^{0.5\epsilon}},$

which completes the proof. ∎

We will also make crucial use of an estimate on the imaginary part of Green
function entries. As the computation shows, we have to take a special care in
the general case $K\geq 2$.

######  Proposition 2.4.

(Imaginary part of Green function) For any $z=E+i\eta$ with
$E\in[-2+\kappa,2-\kappa]$ and $\eta\in[N^{-1+\epsilon},1]$, we derive the
following estimates:

  1. (1)

Imaginary part of Stieltjes transform: we can find some constant
$C_{\kappa}>0$ such that

$\Im m^{as}(z)>C_{\kappa}.$ (2.19)

The real part is small: for any $\kappa>0$ we can find sufficiently small
$d_{\kappa}>0$ such that for all $E\in[-2+\kappa,2-\kappa]$ and
$\eta\in[N^{-1+\epsilon},d_{\kappa}]$, we have

$|\Re m^{as}(z)|\leq 10^{-3}C_{\kappa}.$ (2.20)

  2. (2)

In the case $K=1$, given any $\kappa>0$, we can find $C_{\kappa,1}\in[0,1)$
and a sufficiently small $d_{\kappa}>0$ such that

for all $i,j\in[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}]$ and any
$\eta=\Im(z)\in[N^{-1+\epsilon},d_{\kappa}]$, we have

$|\Im G^{\infty}_{ij}(z)|\leq C_{\kappa,1}\Im G^{\infty}_{ii}(z),\quad\text{
for any }|i-j|\leq 1.$

More generally, given any $\sigma\in(0,\frac{1}{\alpha})$ the estimate also
holds if
$i,j\in[N^{1-\sigma\alpha-0.5\epsilon},N-N^{1-\sigma\alpha-0.5\epsilon}]$ and
$\eta=\Im(z)\in[N^{-1+\sigma\alpha+\epsilon},d_{\kappa}]$.

  3. (3)

For general positive integers $K$, given any $p\in\mathbb{N}_{+}$ and
$\kappa>0$, we may find $C_{\kappa,K,p}\in[0,1)$ and small $d_{\kappa}>0$ such
that

for all $i,j\in[N^{1-0.5\epsilon},N-N^{1-0.5\epsilon}]$, any
$\eta=\Im(z)\in[N^{-1+\epsilon},d_{\kappa}]$ and any
$E\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$, we have

$|\Im G^{\infty}_{ij}(z)|\leq C_{\kappa,K,p}\Im G^{\infty}_{ii}(z),\quad\text{
for any }|i-j|\leq K,$

where $\Delta_{K}^{p}:=\Delta_{K}+[-10^{-p},10^{-p}]$ is a neighborhood of
$\Delta_{K}$ and $\Delta_{K}$ is the set of all $E\in[-2,2]$ such that
$\sin(\ell\arccos(\frac{1}{2}E))=0$ for some $\ell=2,\cdots,K$. More
generally, given any $\sigma\in(0,\frac{1}{\alpha})$ the estimate also holds
if $i,j\in[N^{1-\sigma\alpha-0.5\epsilon},N-N^{1-\sigma\alpha-0.5\epsilon}]$
and $\eta=\Im(z)\in[N^{-1+\sigma\alpha+\epsilon},d_{\kappa}]$.

###### Proof.

The first claim follows by direct computation:

$m^{as}(E+i\eta)=\frac{1}{\sqrt{E^{2}-\eta^{2}-4+2Ei\eta}}=\frac{i\sqrt{\eta^{2}+4-E^{2}+2Ei\eta}}{\sqrt{(E^{2}-\eta^{2}-4)^{2}+4E^{2}\eta^{2}}}.$
(2.21)

Then noting that we assume $E\in[-2+\kappa,2-\kappa]$ and $\eta\in[0,1]$, the
claimed lower bound follows from elementary complex analysis, and can be made
to depend only on $\kappa$.

Also, from (2.21) we see that assuming $\eta\in[N^{-1+\epsilon},d_{\kappa}]$
for some very small positive constant $d_{\kappa}$, we have

$|\Re m^{as}(E+i\eta)|\leq 10^{-3}C_{\kappa}.$ (2.22)

This completes the proof of part (1).

For the second and third claim, first observe that by our assumptions on
indices $i$ and $j$, and the assumption on $\eta$, one must have, in both case
$\sigma=0$ and $\sigma>0$, that

$|\cos((N+1-j-k)\lambda)|\leq
Ce^{-N^{0.5\epsilon}}|\cos((N+1-|j-k|)\lambda)|,$

so we only need to consider the first term in the numerator of (2.18). If
$i=j$, then following the steps in the proof of Lemma 2.2, one can show that
$G_{ii}^{\infty}(z)$ is close to $m^{ac}(z)$ with an exponentially small
error. Thus the desired conclusion follows from the first claim.

The more general case $0<|i-j|\leq K$ is somewhat more elaborate. We assume
for simplicity that $|i-j|=K$. Then by trigonometric identities

$\frac{\cos((N+1-K)\lambda)}{\cos((N+1)\lambda)}=\cos(K\lambda)+\cot((N+1)\lambda)\sin(K\lambda).$
(2.23)

The ratio (2.23) is of interest because, up to a vanishing error,

$\frac{G_{ij}^{\infty}(z)}{G_{ii}^{\infty}(z)}=\eqref{cosdifference}(1+O(e^{-N^{0.5\epsilon}})).$
(2.24)

Thanks to the fact that $G_{ii}^{\infty}(z)$ is very close to $m^{as}(z)$ (see
proof of Lemma 2.2) and that by part (1) of this Proposition, the real part of
$m^{as}(z)$ is very small if $\eta\in[0,d_{\kappa}]$, we may approximately
regard $G_{ii}^{\infty}(z)$ as a purely imaginary number and we hope for the
case that (2.23) is not close to a real number, so that $\Im
G_{ij}^{\infty}(z)$ is strictly smaller than $\Im G_{ii}^{\infty}(z)$.

Now we prove that the said claim holds. First, thanks to (2.13), we have that

$|\cot((N+1)\lambda)-i|\leq e^{-N^{0.5\epsilon}}.$

Write $\lambda=a+ib$. Recall that $\lambda=\arccos(z/2)$, and $\Im z$ is very
small, by properties of the arcsine function, we have the estimate $|b|\leq
d_{\kappa}^{\prime}d_{\kappa}$ and $\min(a,\pi-a)\geq e_{\kappa}^{\prime}$ for
some $d_{\kappa}^{\prime}$, $e_{\kappa}^{\prime}$ that only depend on
$\kappa$.

For $z\in\mathcal{S}(\epsilon,\kappa)$, assuming in addition that $d_{\kappa}$
is chosen sufficiently small so that $d_{\kappa}<10^{-3}e_{\kappa}^{\prime}$,
then $\cos(K\lambda)=\cos(Ka)(1+O(10^{-3}))$ and
$\sin(K\lambda)=\sin(Ka)(1+O(10^{-3}))$, so that they are both real numbers up
to a very small error.

We have to distinguish the case $K=1$ (tridiagonal) and $K>1$ (general
banded). In the case $K=1$, since $\sin(a)$ is nowhere vanishing and
$\cosh(b)$ is at least 1, it is easy to verify that, combining (2.24) and
(2.23) and the fact that $G_{ii}^{\infty}(z)$ is an imaginary number up to a
very small error, we have

$|\Im G_{ij}^{\infty}(z)|\leq C_{\kappa,1}\Im G_{ii}^{\infty}(z),$ (2.25)

where $C_{\kappa,1}\in[0,1)$ depends only on $\kappa$ and is strictly smaller
than 1.

For the general case $K\geq 2$, we would like to remove the sets of
$z=2\cos(a+bi)$ such that $\sin(Ka)=0$, as this would lead to
$|\Re\eqref{cosdifference}|$ being arbitrarily close to one and
$|\Im\eqref{cosdifference}|$ being arbitrarily close to zero. As we have
required that $\eta=\Im(z)<d_{\kappa}$ for a sufficiently small $d_{\kappa}$,
we only have to remove a very small neighborhood of each
$E\in[-2+\kappa,2-\kappa]$ such that $\sin(\ell\arccos(\frac{1}{2}E))=0$ for
some $\ell=1,2,\cdots,K$. Let $\Delta_{K}$ denote the set of all such $E$, and
write $\Delta_{K}^{p}:=\Delta_{K}+[-10^{-p},10^{-p}]$ the union of short
intervals with length $2\times 10^{-p}$ centered at one of the points in
$\Delta_{K}$, for any $p\in\mathbb{N}_{+}$.

Then whenever $E\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$ and
$\eta\in[N^{-1+\epsilon},d_{\kappa}]$, we still have

$|\Im G_{ij}^{\infty}(z)|\leq C_{\kappa,K,p}\Im G_{ii}^{\infty}(z),$ (2.26)

where $C_{\kappa,K,p}\in[0,1)$ depends on $\kappa,K,p$ and is strictly less
than one. Then the proof of part (3) is finished.

∎

##  3\. Entry-wise local law

###  3.1. Green’s function identities and concentration estimates

For given matrices $B$ and $C$ of the same dimension, we have the identity

$B^{-1}-C^{-1}=B^{-1}(C-B)C^{-1}.$ (3.1)

Now we take $B=H_{N}^{\infty}+A_{N}-zI$ and $C=H_{N}^{\infty}-zI$, then
$B^{-1}=G$ and $C^{-1}=G^{\infty}$, and we obtain

$G-G^{\infty}=-GA_{N}G^{\infty}.$ (3.2)

We will use frequently the Ward’s identity: for any $H$ an $N\times N$ matrix,
and $G=(H-z)^{-1}$ its Green function for any given $z\in\mathbb{C}_{+}$ with
$\eta=\Im z>0$, we have

$\sum_{j=1}^{N}|G_{jk}|^{2}=\frac{\Im G_{kk}}{\eta}.$ (3.3)

The following comparison result of Green’s function will also be used:

######  Proposition 3.1 ([1],Appendix B).

Given $E\in\mathbb{R}$, $\eta,\eta^{\prime}\in\mathbb{R}_{>0}$, and any
$N\times N$ matrix $H$. Denote by $z=E+i\eta$,
$z^{\prime}=E+i(\eta+\eta^{\prime})$, $G(z)=(H-z)^{-1}=\\{G_{jk}\\}$ and
$G^{\prime}(z)=(H-z^{\prime})^{-1}=\\{G^{\prime}_{jk}\\},$ then for any
$j,k\in[1,N]$ we have

$|G_{jk}-G_{jk}^{\prime}|\leq\frac{\eta^{\prime}}{2\eta}(|\Im
G_{jj}^{\prime}|+|\Im G_{kk}|).$ (3.4)

In particular we have for each $j\in[1,N]$,

$\frac{\min(|G_{jj}^{\prime}|,|G_{jj}|)}{\max(|G_{jj}^{\prime}|,|G_{jj}|)}>1-\frac{\eta^{\prime}}{\eta}.$
(3.5)

We will need the following large deviations estimate for sums of independent
random variables with growing moments:

######  Proposition 3.2 ([21],Lemma 3.8).

Given $(a_{i})$ a family of centered independent random variables that satisfy

$\mathbb{E}|a_{i}|^{p}\leq\frac{C^{p}}{N^{\gamma}q^{ap+b}}$ (3.6)

for any $2\leq p\leq(\log N)^{\log\log N}$, for some given $a\geq 0$ and
$b,\gamma\in\mathbb{R}$. Then there exists some $\nu>0$ such that for all
$2\leq\xi\leq\log\log N$, we have

$\mathbb{P}\left(\left|\sum_{i}\Psi_{i}a_{i}\right|\geq(\log
N)^{\xi}\left[\frac{\sup_{i}|\Psi_{i}|}{q^{a}}+(\frac{1}{N^{\gamma}q^{b+2a}}\sum_{i}|\Psi_{i}|^{2})^{1/2}\right]\right)\leq
e^{-\nu(\log N)^{\xi}}.$ (3.7)

The following generalization is easy to obtain: assume $(a_{i})$ are not
necessarily centered but still satisfies the moment condition (3.6) and
satisfy, for some $\delta>0$ and constant $C$,

$|\mathbb{E}[a_{i}]|\leq\frac{C}{N^{1+\delta}},$ (3.8)

then we have, outside a set of probability at most $e^{-\nu(\log N)^{\xi}}$,
the following estimate

$\left|\sum_{i}\Psi_{i}a_{i}\right|\leq(\log
N)^{\xi}\left[(\frac{1}{q^{a}}+\frac{1}{N^{\delta}})\sup_{i}|\Psi_{i}|+(\frac{1}{N^{\gamma}q^{b+2a}}\sum_{i}|\Psi_{i}|^{2})^{1/2}\right].$
(3.9)

###  3.2. Green’s function estimate in the heavy-tailed case

The following Theorem is a generalization of Theorem 1.11, and constitutes the
essential building block for all other proofs in this paper. Similar proof
techniques will be used extensively in the sequel.

######  Theorem 3.3.

Let $\overline{H}_{N}^{\infty}$ be a real symmetric matrix as in Definition
1.2. Let $\overline{A}_{N}$ be an $N\times N$ real symmetric random matrix
with independent elements $\overline{A}_{ij}$ for all $|i-j|\leq K$ and zero
otherwise. Assume that we can find some $\delta>0$ and $\gamma>0$ such that

$\mathbb{E}[|A_{ij}|]\leq\frac{C}{N^{1+\delta}},\quad\mathbb{E}[|A_{ij}|^{p}]\leq\frac{C}{N^{\gamma}q^{p-\alpha}},\text{
for each }p\geq 2,$

where $q$ is $N$-dependent and $q>>\log N^{\log\log N}$.

Then for any $\kappa>0$, $\epsilon>0$ we can find positive constants $C$ and
$\nu$ depending only on $\kappa,\epsilon$ such that

$\mathbb{P}\left(\max_{\begin{subarray}{c}z\in\mathcal{S}\\\ \eta\geq
N^{-\gamma+0.5\epsilon}\end{subarray}}\max_{i,j}|\overline{G}_{ij}-\overline{G}^{\infty}_{ij}|\geq
C\log N^{\log\log
N}\left(\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N^{\gamma}\eta}}\right)\right)\leq
e^{-\nu\log N^{\log\log N}},$ (3.10)

where $\overline{G}$ is the Green function of
$\overline{H}_{N}^{\infty}+\overline{A_{N}}$ and $\overline{G}^{\infty}$ is
the Green function of $\overline{H}_{N}^{\infty}$.

###### Proof.

We start with the resolvent identity (3.2) which gives that

$[\overline{G}-\overline{G}^{\infty}]_{ij}=-\sum_{p,l\in[1,N]}\overline{G}_{i,p}\overline{A}_{p,l}\overline{G}^{\infty}_{lj}=-\sum_{t=-K}^{K}J_{t},$
(3.11)

where each $J_{t}$ contains the terms in the summation with a term
$\overline{A}_{kl}$ that satisfies $k-l=t$:

$J_{t}=\sum_{p\in[1,N]}\overline{G}_{i,p}\overline{A}_{p,p-t}\overline{G}^{\infty}_{p-t,j},$
(3.12)

since the matrix $\overline{A}_{N}$ has bandwidth $K$. Here we adopt the
convention that $\overline{A}_{p,p-t}=0$ if $p-t<0$ or $p-t>N$.

Then we apply concentration inequalities to each term $J_{t}$. The resolvent
matrix $\overline{G}^{\infty}$ is deterministic but $\overline{G}$ is not
independent of $\overline{A}$. We will carry out an inductive argument such
that at each step, the matrix elements of $\overline{G}$ are uniformly bounded
with overwhelming probability.

First fix a very large constant $C$, which is larger than the upper bound in
(1.7). Fix some $E\in(-2+\kappa,2-\kappa)$ and for each $m\in[1,N]$ define an
event $P_{m}$ as follows: (The event $P_{m}$ depends on $N$ but we suppress
the $N$-dependence to simplify notation)

$P_{m}\overset{\operatorname{def}}{=}\left\\{\sup_{i,j\in[1,N]}\left|\overline{G}_{i,j}\left(z_{m}\right)\right|<C\right\\},$
(3.13)

where we abbreviate

$z_{m}=E+i\frac{N+1-m}{N}.$ (3.14)

Obviously, $\mathbb{P}(P_{1})=1$ as the constant $C$ is chosen sufficiently
large. We will show via an inductive argument that for some $\nu$ to be
determined later,

$\mathbb{P}(P_{m}^{c})\leq\mathbb{P}(P_{m-1}^{c})+e^{-\nu\log N^{\log\log N}}$
(3.15)

for any $m\leq N-N^{1-\gamma+0.5\epsilon}$.

Assume that we have verified (3.15) for each event $P_{1},\cdots,P_{m}$
((3.15) is an empty statement if $m=1$).

Thanks to (3.5) and the inductive assumption, we deduce that

$|\overline{G}_{ii}(z_{m+1})|\leq 2C,\quad\omega\in P_{m}$

for each $i\in[1,N]$. Then apply (3.4) and the inductive hypothesis, we deduce
that

$|\overline{G}_{ij}(z_{m+1})|\leq 2C,\quad\omega\in P_{m}$

for each $i,j\in[1,N]\times[1,N].$

We now wish to apply the concentration inequality in Proposition 3.2 to
estimate each summation $J_{t}$. Two complications arise at this point: first,
$\overline{G}_{i,p}$ are not independent from $\overline{A}_{p.l}$, and
second, entries of $\overline{G}$ are bounded by $2C$ with high probability
but not almost surely. To solve the second issue we work on the event $P_{m}$,
and to solve the first issue we note that the proof of Proposition 3.2 only
involves computing very high powers of the random variables involved, so if
$\overline{G}_{ij}$’s are bounded a.s. by some constant $C$, then we can drag
the constant out (see Appendix A) and consequently the dependence of
$\overline{A}$ and $\overline{G}$ does not ruin the analysis.

More concretely, we define

$\widetilde{\overline{G}}_{i,p}=\begin{cases}\overline{G}_{i,p},\quad\omega\in
P_{m},\\\ 2C,\quad\omega\in P_{m}^{c}\end{cases}.$ (3.16)

and consider

$\widetilde{J}_{t}=\sum_{p\in[1,N]}\widetilde{\overline{G}}_{i,p}\overline{A}_{p,p-t}\overline{G}^{\infty}_{p-t,j}.$
(3.17)

Then $J_{t}=\widetilde{J}_{t}$ when $\omega\in P_{m}$, and by definition,
$\widetilde{\overline{G}}_{i,p}$ is almost surely bounded by $2C$ for all
indices $(i,p)$. Now we apply Proposition (3.2) (indeed, we apply its
generalization (3.9)) to each $\widetilde{J}_{t}$ (to eliminate the dependence
between $\overline{A}$ and $\widetilde{\overline{G}}$, we use the almost sure
upper bound on $\widetilde{\overline{G}}$ and drag it out of the expectation
when computing high moments, see Appendix A for details) and deduce: for each
$2\leq\xi\leq N$ and for each $t=-K,-K+1,\cdots,K-1,K$,

$\mathbb{P}(|\widetilde{J}_{t}|\geq(\log
N)^{\xi}[\frac{2C}{q}+\frac{2C}{N^{\delta}}+(\frac{1}{N^{\gamma}}\sum_{k\in[1,N]}|G_{k-t,j}^{\infty}|^{2})^{1/2}])\leq
e^{-\nu(\log N)^{\xi}}.$ (3.18)

By Ward’s identity (3.3), note that we have set $G_{p-t,j}^{\infty}$ to zero
when $p-t\notin[1,N]$:

$\sum_{p\in[1,N]}|G_{p-t,j}^{\infty}|^{2}\leq\sum_{p\in[1,N]}|G_{p,j}^{\infty}|^{2}=\frac{\Im
G_{j,j}^{\infty}}{\eta}\leq\frac{C}{\eta}.$ (3.19)

Since $q>1$, we can ignore the $q$ factor and just write, for
$2\leq\xi\leq\log\log N$:

$\mathbb{P}\left(|\widetilde{J}_{t}|\geq C(\log
N)^{\xi}\left[\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N^{\gamma}\eta}}\right]\right)\leq
e^{-\nu(\log N)^{\xi}}.$ (3.20)

We take a union bound over $t=-K,\cdots,K$: upon slightly changing $\nu$,

$\mathbb{P}\left(\sup_{t=-K,\cdots,K}|\widetilde{J}_{t}|\geq C(\log
N)^{\xi}\left[\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N^{\gamma}\eta}}\right]\right)\leq
e^{-\nu(\log N)^{\xi}}.$ (3.21)

Since $\widetilde{J}_{t}=J_{t}$ on $P_{m}$ for each $t=-K,\cdots,K$ , we
deduce that

$\mathbb{P}\left(\sup_{t=-K,\cdots,K}|{J}_{t}|\geq C(\log
N)^{\xi}\left[\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N^{\gamma}\eta}}\right]\right)\leq
e^{-\nu(\log N)^{\xi}}+\mathbb{P}(P_{m}^{c}).$ (3.22)

We plug in these estimates into the resolvent expansion in (3.11). Then one
sees that for any $m<N-N^{1-\gamma+0.5\epsilon}$,

$\mathbb{P}\left(\left|[\overline{G}-\overline{G}^{\infty}]_{ij}(z_{m})\right|\geq
C(\log N)^{\log\log
N}\left[\frac{1}{q}+\frac{1}{N^{\delta}}+\sqrt{\frac{1}{N^{\gamma}\eta}}\right]\right)\leq\mathbb{P}[P_{m}^{c}]+e^{-\nu(\log
N)^{\log\log N}}$ (3.23)

for each $i,j\in[1,N]^{2}$. In particular, thanks to the boundedness of
$\overline{G}^{\infty}$ from Definition 1.2 and our assumption
$\eta>N^{-\gamma+0.5\epsilon}$, we have shown $|\overline{G}_{ij}(z_{m+1})|<C$
for each $i,j$ on an event of probability at least
$1-\mathbb{P}[P_{m}^{c}]-e^{-\nu(\log N)^{\log\log N}}$. Thus we have proved
that (3.15) holds for $m+1$ in place of $m$, after slightly changing the value
of $\nu>0$,

We can now run this inductive procedure from $m=1$ up to
$m=N-N^{1-\gamma+0.5\epsilon}$, and after slightly modifying the value of
$\nu$, we have proved that (3.23) holds for any $z_{m}$ outside a set of
probability at most $e^{-\nu(\log N)^{\log\log N}}$. Combined with a standard
continuity estimate (that is, find a set of $N^{6}$ mesh points in
$\mathcal{S}$ with distance $N^{-3}$ between two neighboring ones, and note
that the Green function $\overline{G}$ is $N^{2}$ -Lipschitz continuous in
$z$), we can upgrade the above estimate to be uniform over $z\in\mathcal{S}$.
This completes the proof of Proposition 3.3.

∎

####  3.2.1. Local law under different scaling

In this section we prove Theorem 1.24. This will be essentially a corollary of
Theorem 3.3.

###### Proof of Theorem 1.24.

We first compute the moments as follows:

$|\mathbb{E}[N^{\sigma}A_{ij}]|\leq\frac{C}{N^{1+\delta}}\quad\mathbb{E}[|N^{\sigma}A_{ij}|^{p}]\leq\frac{C^{p}}{N^{1-\sigma\alpha}N^{\omega(p-\alpha)}},\quad\text{
for each }p\geq 2.$

Then we apply Theorem 3.3 with $\gamma=1-\sigma\alpha$ and $q=N^{\omega}$.
Note also that we made the assumption $\eta\geq N^{-1+\epsilon+\sigma\alpha}$.
This completes the proof, and see the next section for the statements
concerning Wegner’s estimate and eigenvector de-localization. ∎

###  3.3. Local law given additional moments

In this section we prove Theorem 1.17.

###### Proof of Theorem 1.17.

We set $q=N^{\frac{\epsilon}{10\alpha}}$ for some sufficiently small
$\epsilon$. Since $\xi_{ij}$ are independent and have a finite
$\alpha+\delta$-th moment, using a union bound and choosing the $\epsilon>0$
sufficiently small, we can find some $c_{*}$ such that on an event $P_{N}$
with probability at least $1-N^{-c_{*}\epsilon}$, all the elements $A_{ij}$
have absolute value less than $q^{-1}$. Let $\widehat{A}_{ij}$ denote the
random variables $A_{ij}$ conditioned on $P_{N}$. Then since $\xi_{ij}$ has a
symmetric law we must have $\mathbb{E}[\widehat{A}_{ij}]=0$. Moreover, for any
$p\geq 2$ we have $\mathbb{E}[|\widehat{A}_{ij}|^{p}]\leq
q^{-(p-\alpha)}\mathbb{E}[|\widehat{A}_{ij}|^{\alpha}]\leq\frac{C}{Nq^{p-\alpha}}$
as our conditioning only decreases the variance and
$A_{ij}=\frac{1}{N^{\frac{1}{\alpha}}}\xi_{ij}$ with $\xi_{ij}$ having finite
$\alpha$-moment. Then the claimed local law is a direct corollary of Theorem
3.3. ∎

##  4\. Proof of Corollaries

###  4.1. Wegner estimate

We first prove the Wegner estimate in Corollary 1.13. The Wegner estimates in
Theorem 1.24 and Corollary 1.30 can be derived in exactly the same way.

###### Proof of Corollary 1.13.

The Wegner estimate is a simple consequence of the local law established in
Theorem 1.11 and the following identity from [18], Proposition 2.1 : for any
probability distribution $F(x)$ whose Stieltjes transform is given by $m(z)$,
given $z=E+i\eta$ and consider $I=[E-\eta/2,E+\eta/2]$, then

$\mathcal{N}_{I}=N\int_{I}dF(x)\leq\frac{5}{4}N\eta\int_{E-\eta/2}^{E+\eta/2}\frac{\eta
dF(x)}{(x-E)^{2}+\eta^{2}}\leq\frac{5}{4}N\eta\Im m(z).$ (4.1)

For Theorem 1.24 and Corollary 1.30, one only needs to use the local laws
derived in the respective settings. ∎

###  4.2. Proof of local arcsine law and eigenvalue rigidity

The proof of Corollary 1.14 on the local arcsine law follows standard
procedure but has some slight differences.

###### Proof of Corollary 1.14, local arcsine law.

The proof is essentially an adaptation of standard methods, based on an
application of the Helffer-Sjöstrand formula. See for example [10], Section 8.
However we are dealing with the arcsine law rather than semicircle law, and in
our case the convergence rate is very slow, so we give a sketch of proof and
illustrate the differences. Let $\eta=N^{-1+\epsilon}$, consider interval
$I\subset[-2+\kappa,2-\kappa]$ and define $f\equiv
f_{I,\eta}\in\mathcal{C}_{c}^{\infty}(\mathbb{R},[0,1])$ such that $f(x)=1$
when $x\in I,$ $f(x)=0$ when $\operatorname{dist}(x,I)>\epsilon$,
$\|f\|_{\infty}\leq C\eta^{-1}$ and $\|f^{\prime\prime}\|_{\infty}\leq
C\eta^{-2}$. Consider smooth even function
$\chi\in\mathcal{C}_{c}^{\infty}(\mathbb{R},[0,1])$ with $\chi(y)=1$ for
$|y|\leq 1$, $\chi(y)=0$ for $|y|>2$, and $\|\chi^{\prime}\|_{\infty}\leq C$.
Denote by

$\hat{\mu}:=\mu-\rho,\quad\hat{m}(z)=m(z)-m^{as}(z).$ (4.2)

Then by the Helffer-Sjöstrand formula,

$\displaystyle\int f(\lambda)\hat{\mu}(d\lambda)=$
$\displaystyle-\frac{1}{2\pi}\int
dx\int_{|y|\leq\eta}dyf^{\prime\prime}(x)\chi(y)\Im\hat{m}(x+iy)$ (4.3)
$\displaystyle-\frac{1}{2\pi}\int
dx\int_{|y|>\eta}dyf^{\prime\prime}(x)\chi(y)y\Im\hat{m}(x+iy)$ (4.4)
$\displaystyle+\frac{1}{2\pi}\int dx\int
dy(f(x)+iyf^{\prime}(x))\chi^{\prime}(y)\hat{m}(x+iy)).$ (4.5)

From the local law in Theorem 1.11, we can find some $c_{*}>0$ such that with
probability at least $1-e^{-\nu\log N^{\log\log N}}$, we have

$|\hat{m}(x+iy)|\leq N^{-c_{*}}$

for all $x+iy\in\mathcal{S}(\epsilon,\kappa)$. We note that in the local law
Theorem 1.11, the convergence rate is very slow even at macroscopic
scales–this is a barrier that cannot be overcome here when the $\xi_{ij}$’s do
not have sub-Gaussian tails.

Then we can bound (4.5) thanks to the fact that
$\operatorname{Supp}\chi^{\prime}\in[-2,2]\setminus[-1,1]$:

$|\eqref{3.23}|\leq CN^{-c_{*}}.$

From the estimate

$y\Im\hat{m}(x+iy)\leq y\Im m(x+iy)\leq\eta\Im m(x+i\eta)\leq C\eta$

where the last step follows from the monotonicity of $y\to y\Im m(x+iy)$ for
the Stieltjes transform of any probability measure, we can bound (4.4) as

$|\eqref{3.22}|\leq\eta^{-1}\int_{|y|\leq\eta}dyC\eta=C\eta.$

To bound (4.3), we integrate by parts in $x$ and then in $y$:

$\displaystyle|\eqref{3.21}|\leq\int dx\int_{\eta}^{\infty}dy$
$\displaystyle|f^{\prime}(x)\chi(y)\hat{m}(x+iy)|$ (4.6) $\displaystyle+\int
dx\int_{\eta}^{\infty}dy|f^{\prime}(x)y\chi^{\prime}(y)\hat{m}(x+iy)|+\int
dx|f^{\prime}(x)\eta\hat{m}(x+i\eta)|.$

Then on the given event of high probability, the second line is bounded by
$CN^{-c_{*}}$. The first line is bounded by

$C\int_{\eta}^{2}dy\frac{N^{-c_{*}}}{y}\leq CN^{-c_{*}}\log\eta\leq
CN^{-c_{*}/2}.$

Combining everything, we see that

$\left|\int f(\lambda)\hat{\mu}d\lambda\right|\leq CN^{-c_{*}/2}.$

One may choose $2c_{*}$ in place of $c_{*}$. The rest of the proof follows
from standard arguments, see for example [10], Section 8.

∎

The statements concerning local arcsine law in Corollary 1.30 can be proved in
exactly the same way, even if the local law is only derived on the scale
$\eta\in[N^{-1+\sigma\alpha+\epsilon},1]$. This is because the convergence
rate we derive is arbitrarily slow, so that any mesoscopic values of $\eta$
would be sufficient for deriving such a slow convergence rate.

###### Proof of Corollary 1.30, local arcsine law.

We follow exactly the same step as in the previous proof, and the only
difference is we take $\eta=N^{-1+\sigma\alpha+\epsilon}$ in place of
$N^{-1+\epsilon}$. ∎

Then we prove the second part: eigenvalue rigidity estimates.

###### Proof of Corollary 1.14, eigenvalue rigidity.

We again follow the standard proof techniques, see for example [10], Section
9.

For $E\in[0,2-\kappa]$ define $f(E)=m^{ac}([0,E]).$ Recall that we only
consider eigenvalues in $[0,2-\kappa]$. Then by the first part of Corollary
1.14, recalling $\mu_{N}$ is the empirical measure of the eigenvalues of
$H_{N}^{\infty}+A_{N}$, we have that with probability at least $1-e^{-\nu\log
N^{\log\log N}}$:

$\frac{i}{N}=f(\gamma_{i})+\frac{1}{2N}=\mu_{N}([0,\lambda_{i}^{\prime}])=f(\lambda_{i}^{\prime})+N^{-c_{*}},$

for all $i$ such that $[0,\lambda_{i}^{\prime}]\subset[0,2-\kappa]$.

Note that $f^{\prime}(\lambda)=\frac{1}{2\pi\sqrt{1-\lambda^{2}/4}}$ is
bounded from above and below on $[0,2-\kappa]$, so we approximately get
$f^{\prime}(\gamma_{i})\approx f^{\prime}(\lambda_{i}^{\prime})$ and then by
the mean value theorem,

$|\gamma_{i}-\lambda_{i}^{\prime}|\leq\frac{f(\gamma_{i})-f(\lambda_{i}^{\prime})}{f^{\prime}(\gamma_{i})}\leq
CN^{-c_{*}}.$

This holds jointly for all $i\geq 0$ such that $\lambda_{i}^{\prime}\leq
2-\kappa$, with probability at least $1-e^{-\nu\log N^{\log\log N}}$. This
completes the proof. ∎

The proof of eigenvalue rigidity statements in Corollary 1.30 is exactly the
same and hence omitted.

##  5\. Eigenvector delocalization for alpha-stable laws

In this section we assume $A_{N}$ satisfies Assumption 1.8. The proof under
Assumption 1.8 is much harder than the previous cases.

###  5.1. A slightly weaker version

To illustrate the main ideas, we first prove the following slightly weaker
version of Theorem 1.21. The proof of Theorem 1.21 follows similar lines. We
stress that in this Proposition, we can take all
$z\in\mathcal{S}(\epsilon,\kappa)$ and do not need to restrict to
$\mathcal{S}(\epsilon,\kappa,K,p)$.

######  Proposition 5.1.

Under the same assumption as Theorem 1.21, assume for simplicity that $A_{N}$
is a diagonal matrix (i.e. $K=0$). Then we can find some $P_{*}\in(0,1)$ such
that for $N$ sufficiently large, we have the following bound (where $C$ is the
constant given in (1.7))

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa)}\max_{1\leq i,j\leq
N}\left|G_{ij}(z)\right|\leq 4C\right)\geq P_{*}.$ (5.1)

###### Proof.

In the proof we assume that $\eta\in[N^{-1+\epsilon},1]$. We fix a length
scale $L=N^{1-0.5\epsilon}$ and a cutoff value
$q=N^{\frac{0.01\epsilon}{\alpha}}$. Then for each $i$,
$\mathbb{P}[|A_{ii}|>q^{-1}]\leq N^{-1+0.01\epsilon}$.

We first show that it is very unlikely to have a pair $(i,j)\in[1,N]^{2}$ such
that $|A_{ii}|>q^{-1}$ and $|A_{jj}|>q^{-1}$ for $|i-j|\leq 2L$: denote this
event by $D_{N}^{1}$, then

$\mathbb{P}(D_{N}^{1})\leq
NN^{-1+0.01\epsilon}2LN^{-1+0.01\epsilon}=N^{-0.48\epsilon},$ (5.2)

where the $N$ factor is the choice of the location for the first such $A_{ii}$
and the $2L$ factor is the choice of the location for $A_{jj}$ which has
distance at most $L$ to $A_{ii}$. Moreover, it is also unlikely to have some
$i\in[0,L]\cup[N-L,N]$ with $|A_{ii}|\geq q^{-1}$: denote this event by
$D_{N}^{2}$,

$\mathbb{P}(D_{N}^{2})\leq 2LN^{-1+0.01\epsilon}=N^{-0.49\epsilon}.$ (5.3)

Denote $D_{N}=D_{N}^{1}\cup D_{N}^{2}$. Then

$\mathbb{P}(D_{N})\leq 2N^{-0.48\epsilon}.$ (5.4)

It is more convenient to re-sample the entries of $A_{N}$ via the following
procedure. We first sample the set of labels $(T,F)$ independently on each
label $(i,i)$, where a label $T$ is assigned with probability
$\mathbb{P}[|A_{ii}|<q^{-1}]=1-N^{-1+0.01\epsilon}$ and a label $F$ is
assigned with probability $\mathbb{P}[|A_{ii}|>q^{-1}]=N^{-1+0.01\epsilon}$.
Then with probability at least $1-2N^{-0.48\epsilon}$ the event $D_{N}$ does
not happen. Having sampled the labels $(T,F)$, we then sample
$\xi_{ii}:=N^{\frac{1}{\alpha}}A_{ii}$ for $(i,i)$ with label $T$ from the
probability law (1.13) conditioned to take values in
$\xi_{ii}\in[-N^{\frac{1-0.01\epsilon}{\alpha}},N^{\frac{1-0.01\epsilon}{\alpha}}]$,
and we sample $\xi_{ii}$ for $(i,i)$ with label $F$ from the probability law
(1.13) conditioned to take values in
$\xi_{ii}\in(-\infty,-N^{\frac{1-0.01\epsilon}{\alpha}}]\cup[N^{\frac{1-0.01\epsilon}{\alpha}},\infty)$.
It is not hard to check that the matrix $A_{N}$ sampled from the said
procedure, with 0 filled in the remaining elements, has the same distribution
as $A_{N}$ in Assumption 1.8, conditioned on $D_{N}^{c}$.

Now we make an additional assumption that none of $A_{ii}$ is large: define

$E_{N}:=\left\\{|A_{ii}|\geq\frac{1}{4\sup_{z\in\mathcal{S},i,j\in[1,N]^{2}}|G^{\infty}_{ij}(z)|}\text{
for some }i\in[1,N]\right\\}.$

Thanks to the derivation in Section 2.1, the supremum in the denominator is
non-zero and we denote the supremum by $C$. By definition of the probability
law (1.13),

$\mathbb{P}(E_{N}^{c})\geq(1-\frac{(4C)^{\alpha}}{N})^{N}\geq C_{*}>0$ (5.5)

for some constant $C_{*}$ independent of $N$.

We now prove that the entries $G_{ij}$ are bounded with positive probability,
via induction on $\eta$ from $\eta=1$ to $\eta=N^{-1+\epsilon}$, where
$z=E+\eta i$. Recall that we abbreviate

$z_{m}=E+i\frac{N+1-m}{N},\quad m=1,\cdots,N,$

and now we consider the event

$Q_{m}\overset{\operatorname{def}}{=}\left\\{\sup_{i,j\in[1,N]^{2}}\sup_{E\in[-2+\kappa,2-\kappa]}\left|{G}_{ij}\left(z_{m}\right)\right|<4C\mid
E_{N}^{c}\cap D_{N}^{c}\right\\},$ (5.6)

The goal is to show by induction that for some $\nu>0$,

$\mathbb{P}(Q_{m}^{c})\leq\mathbb{P}(Q_{m-1}^{c})+e^{-\nu\log N^{\log\log
N}}.$ (5.7)

Before we proceed, we mention that conditioning on $E_{N}^{c}$ does not break
the independence of the $A_{ii}$’s, and conditioning on $D_{N}^{c}$ does not
break the independence of the $A_{ii}$’s such that $(i,i)$ has label $T$, i.e.
$|A_{ii}|<q^{-1}$. Further, the conditioned versions of $A_{ii}$’s are still
centered thanks to the fact that $\xi_{ii}$ is assumed to have a symmetric
law. Finally, for any label $(i,i)$ with label $T$, our conditioning procedure
reduces the variance and high moments of these $A_{ii}$’s. In the following we
still use the notation $A_{ii}$ to denote $A_{ii}$ conditioned on
$E_{N}^{c}\cap D_{N}^{c}$.

For the initial state $m=1$, $Q_{m}$ holds almost surely thanks to the fact
that $\Im z_{1}=1$. Now we condition on $Q_{m-1}$ and work towards the
estimate for $Q_{m}$. Thanks to 3.5, we must have

$\sup_{i,j\in[1,N]^{2}}\sup_{E\in[-2+\kappa,2-\kappa]}|G_{ij}(z_{m})|<8C,\quad\omega\in
Q_{m-1}.$

Consider an arbitrary index pair $(i,j)\in[1,N]^{2}$. Then from resolvent
expansion,

$G_{ij}-G_{ij}^{\infty}=-\sum_{k}G_{ik}A_{kk}G_{kj}^{\infty}=T_{ij}^{1}+T_{ij}^{2}+T_{ij}^{3},$
(5.8)

where we decompose the summation into three terms, with

$T_{ij}^{1}=-\sum_{|k-j|\leq L,|A_{kk}|<q^{-1}}G_{ik}A_{kk}G_{kj}^{\infty},$
$T_{ij}^{2}=-\sum_{|k-j|\leq L,|A_{kk}|>q^{-1}}G_{ik}A_{kk}G_{kj}^{\infty},$

and

$T_{ij}^{3}=-\sum_{|k-j|\geq L}G_{ik}A_{kk}G_{kj}^{\infty}.$

The summation $T_{ij}^{2}$ has at most one term as we condition on the event
$D_{N}^{c}$. By definition of $E_{N}^{c}$, it is immediate that
$|T_{ij}^{2}|\leq\frac{1}{4}\cdot 8C=2C$ on the event $Q_{m-1}$.

The term $T_{ij}^{3}$ can be bounded directly: (i) by the exponential decay
property in Proposition 2.3, we have $|G_{kj}^{\infty}|\leq
e^{-N^{0.5\epsilon}}$; (ii) there are at most $N$ terms in the summation;
(iii) $|G_{ik}|$ is bounded by $8C$ and (iv) $|A_{kk}|$ is bounded by
$\frac{1}{4C}$. Combining these facts we have $|T_{ij}^{3}|\leq
2Ce^{-N^{0.25\epsilon}}$ and is thus very small.

We bound the term $T_{ij}^{1}$ via concentration inequality. Just as in
equations (3.16) and (3.17), we introduce an (almost surely bounded) random
variable $\widetilde{G}_{ik}(z_{m})$ which equals $G_{ik}(z_{m})$ on $Q_{m-1}$
and equals $8C$ on $Q_{m-1}^{c}$. Then define $\widetilde{T}_{ij}^{1}$ as
$T_{ij}^{1}$ replacing $G_{ik}$ by $\widetilde{G}_{ik}$. Thus by definition,
we see $T_{ij}^{1}=\widetilde{T}_{ij}^{1}$ on $Q_{m-1}$. We will use an
estimate derived as in (6.4), which further implies, thanks to our
conditioning procedure, that for any index $(i,i)$ with label $T$, we have

$\mathbb{E}[|A_{ii}|^{p}]\leq\frac{C}{N^{1-10^{-3}\epsilon}q^{p-\alpha}},\quad
q=N^{\frac{\epsilon}{100\alpha}}.$

Now we apply Proposition 3.2, which implies that with probability at least
$1-e^{-\nu\log N^{\log\log N}}$ we have

$|\widetilde{T}_{ij}^{1}|\leq
N^{0.001\epsilon}\left[\frac{8C^{2}}{N^{\frac{0.001\epsilon}{\alpha}}}+C^{2}\sqrt{\frac{1}{N^{1-0.001\epsilon}\eta}}\right]\leq
C^{2}N^{-0.004\epsilon},$

where we used Ward’s identity and the assumption $\eta\geq N^{-1+\epsilon}$.
This bound can be upgraded to hold for all indices $(i,j)\in[1,N]^{2}$ and any
$E\in[\kappa-2,2-\kappa]$ simultaneously with probability at least
$1-e^{-\nu\log N^{\log\log N}}$ for some $\nu>0$ thanks to the fact that each
$A_{kk}$ is bounded by 1, each $G_{ik}$ is $N^{2}$-Lipschitz in $z$ and that
$e^{-\log N^{\log\log N}}$ decays faster than any polynomial rate $N^{-D}$. We
finally conclude with

$\mathbb{P}(|T_{ij}^{1}|\geq C^{2}N^{-0.004\epsilon})\leq e^{-\nu\log
N^{\log\log N}}+\mathbb{P}(Q_{m-1}^{c}).$ (5.9)

Now turn back to (5.8). With probability at least $1-(e^{-\nu\log N^{\log\log
N}}+\mathbb{P}(Q_{m-1}^{c})),$ we must have

$|G_{ij}|\leq|G_{ij}^{\infty}|+|T_{ij}^{1}|+|T_{ij}^{2}|+|T_{ij}^{3}|\leq
C+2C+C^{2}N^{-0.005\epsilon}\leq 4C,$

for $N$ large, so one necessarily deduces that $|G_{ij}|\leq 4C$ for all
$(i,j)$. This establishes (5.7) for subscript $m$. Running the induction
procedure down to $\eta=N^{-1+\epsilon}$, this proves the boundedness of all
$G_{ij}$.

This finishes the proof of Proposition 5.1 thanks to the estimate (5.7), (5.4)
and (5.5).

∎

###  5.2. Proof of main result

Now we are ready to prove Theorem 1.21. We will use crucially the property
that $G_{ij}^{\infty}$ has an non-vanishing imaginary part when $i$ and $j$
are close, see Proposition 2.4.

###### Proof of Theorem 1.21, (1).

We again choose $\eta\in[N^{-1+\epsilon},1]$, $L=N^{1-0.5\epsilon}$ and
$q=N^{\frac{0.01\epsilon}{\alpha}}$. We work under the general assumption that
$A_{N}$ has bandwidth $K$, i.e. $A_{ij}=0$ whenever $|i-j|>K$.

Define as in the previous proof the event that atypical $A_{ij}$ locations are
not distant:

$D_{N}^{1}:=\\{\text{there exists }i\leq j,i^{\prime}\leq
j^{\prime}\in[1,N]^{4}:|A_{ij}|>q^{-1},|A_{i^{\prime}j^{\prime}}|>q^{-1},|i-i^{\prime}|\leq
2L\\}.$ (5.10) $D_{N}^{2}:=\\{\text{there exists }i\leq
j\in[1,N]^{2}:|A_{ij}|>q^{-1},i\in[0,L]\cup[N-L,N]\\},$ (5.11)

and $D_{N}:=D_{N}^{1}\cup D_{N}^{2}$. Since $L$ is rapidly growing in $N$ and
$K$ is fixed, $|i-i^{\prime}|\leq 2L$ implies $|j-j^{\prime}|\leq 2L+2K\sim
2L$, and $i\leq L$ implies $j\leq L+K\sim L$. We show as in the previous proof
that $\mathbb{P}(D_{N})$ is very small: for some fixed constant $c_{*}$:

$\mathbb{P}(D_{N})\leq
K^{2}NLN^{2(-1+0.01\epsilon)}+2KLN^{-1+0.01\epsilon}\leq N^{-c_{*}\epsilon}.$

As in the previous proof, we assign labels $T$ and $F$ to entries $(i,j)$
independently, where a label $T$ is assigned with probability
$P:=\mathbb{P}(|\xi_{ij}|\leq N^{\frac{1}{\alpha}}q^{-1})$ ans a label $F$ is
assigned with probability $1-P$. We then resample the matrix $A_{N}$ via first
re-sampling the labels $T$ and $F$, then sample $A_{ij}$ based on their
labels.

For some very large $C$ depending on the constant $C_{\kappa,K,p}$ in
Proposition 2.4 (the value of $C$ will be later specified), We consider the
event

$R_{m}\overset{\operatorname{def}}{=}\left\\{\sup_{i,j\in[1,N]}\sup_{E\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}}\left|{G}_{i,j}\left(z_{m}\right)\right|<4C,\mid
D_{N}^{c}\right\\},$ (5.12)

and we aim to show by induction that for some $\nu>0$,

$\mathbb{P}(R_{m}^{c})\leq\mathbb{P}(R_{m-1}^{c})+e^{-\nu\log N^{\log\log
N}}.$ (5.13)

Instead of starting from $m=1$, we now start from a fixed number $m_{0}$ such
that $\frac{N+1-m_{0}}{N}<d_{\kappa}$, where $d_{\kappa}$ is given in
Proposition 2.4. Upon choosing $C:=\hat{C}$ sufficiently large we can assume
(5.13) is already satisfied for $m=1,\cdots,m_{0}$ with the bound $\hat{C}$.

Assume that we have verified (5.13) up to $m-1\geq m_{0}$. We now verify it
holds for $m$. Again thanks to 3.5, we must have

$\sup_{i,j\in[1,N]^{2}}\sup_{E\in[-2+\kappa,2-\kappa]}|G_{ij}(z_{m})|<8C,\quad\omega\in
R_{m-1}.$

In the following we say a site $(i,j)$ is atypical if $|A_{ij}|\geq q^{-1}$.
We now divide the possible index pairs $(i,j)$ into three cases:

  * •

In case (A), j is at distance at least $L$ to any index $k$ such that
$|A_{j^{\prime}k}|>q^{-1}$ for some $j^{\prime}$. Informally, this means for
any $k$ within distance $L$ to $j$, there is no atypical sites of $A_{*k}$
entering the sum (5.14). Such index pairs are safe to analyze.

  * •

In case (B), within distance $0<|j-k|<L$ there exists an index $k$ and another
index $j^{\prime}\notin\\{i,j\\}$ such that $|A_{j^{\prime}k}|>q^{-1}$ (on
$D_{N}^{c}$ there is at most one such index pair $(j^{\prime},k)$ modulo
symmetry constraint). Informally, this means that there must be an atypical
$A_{j^{\prime}k}$ in the summation such that $k$ is within distance $L$ to the
endpoint $j$, but $A_{j^{\prime}k}$ and $A_{kj^{\prime}}$ are not adjacent to
$G_{ij}$. These index pairs are the most dangerous to analyze.

  * •

In case (C), one can find an index $k$ with $|A_{jk}|>q^{-1}$. Informally,
this means that $j$ is the first label of an atypical value $A_{jk}$ which
enters into the sum. The analysis of case (C) will be the prerequisite for
analyzing case (B).

In any of these cases, by resolvent expansion, we have

$G_{ij}-G_{ij}^{\infty}=-\sum_{k,l}G_{ik}A_{kl}G_{Lj}^{\infty}=T_{ij}^{1}+T_{ij}^{2}+T_{ij}^{3},$
(5.14)

where we decompose the sum into three terms, with all the summations over
$k,l\in[1,N]^{2}:$

$T_{ij}^{1}=-\sum_{|l-j|\leq L,|A_{kl}|<q^{-1}}G_{ik}A_{kl}G_{lj}^{\infty},$

$T_{ij}^{2}=-\sum_{|l-j|\leq L,|A_{kl}|>q^{-1}}G_{ik}A_{kl}G_{lj}^{\infty},$

and

$T_{ij}^{3}=-\sum_{|l-j|\geq L}G_{ik}A_{kl}G_{lj}^{\infty}.$

First consider case (A), where the $T_{ij}^{2}$ term does not present. Then we
decompose $T_{ij}^{1}$ into $2K+1$ sub-sums, consisting of the appearance of
$A_{kl}$ with $k-l=-K,\cdots,K$. For each term we use concentration inequality
(Proposition 3.2, and defining the truncated version $\widetilde{G}_{ik}$ and
$\widetilde{T}_{ij}^{1}$ as in (3.16) and (3.17)), we deduce that with
probability at least $1-e^{-\nu\log N^{\log\log N}}-\mathbb{P}(R_{m-1}^{c})$,
we have

$|T_{ij}^{1}|\leq
N^{0.001\epsilon}\left[\frac{8C^{2}N^{0.002\epsilon}}{N^{\frac{0.01\epsilon}{\alpha}}}+C^{2}\sqrt{\frac{1}{N^{1-10^{-3}\epsilon}\eta}}\right]\leq
C^{2}N^{-0.001\epsilon}.$ (5.15)

For the term $T_{ij}^{3}$, we use the exponential decay of $G_{lj}^{\infty}$
for large $|l-j|$ (Proposition 2.3) to bound almost surely $|T_{ij}^{3}|\leq
e^{-N^{c_{*}\epsilon}}$ for some $c_{*}>0$. Taking these terms back to the
resolvent expansion we necessarily have $|G_{ij}(z_{m})|\leq 4C$. We are done
for case (A).

Then we consider case (C). The bounds for $T_{ij}^{1}$ and $T_{ij}^{3}$ are
exactly the same as in case (A) and they can be upper bounded by a vanishing
quantity $N^{-0.001\epsilon}$. As we are in case (C), $T_{ij}^{2}$ contains
only one or two terms, depending on whether the atypical value $A_{jl}$ is on
the diagonal or not, which are

$T_{ij}^{2}=\begin{cases}-G_{ij}A_{jl}G_{lj}^{\infty},\quad j=l,\\\
-G_{ij}A_{jl}G_{lj}^{\infty}-G_{il}A_{lj}G_{jj}^{\infty},\quad j\neq
l.\end{cases}$

(there is one by definition of case (C), and there is only one modulo symmetry
restriction as we work on $D_{N}^{c}$). To fix ideas, we first work in the
case $j=l$, so we only have one term in $T_{ij}^{2}$. Then we can rewrite the
resolvent expansion as

$G_{ij}(1+A_{jl}G_{lj}^{\infty})=G_{ij}^{\infty}+T_{ij}^{1}+T_{ij}^{3}.$
(5.16)

This is the most critical point of the proof. We argue that whatever the value
of $A_{jl}$, we can always invert $1+A_{jl}G_{lj}^{\infty}$ and the norm of
the inverse is uniformly bounded, so that $|G_{ij}|$ has an upper bound that
is independent of $N$ and $m$ ! Indeed, if $|A_{jl}|<\frac{1}{4C_{*}}$, where
$C_{*}$ is the supremum of the norms of $G_{ij}^{\infty}$ on $\mathcal{S}$,
then the real part of $1+A_{jl}G_{lj}^{\infty}$ is at least $\frac{3}{4}$. If
otherwise, then $|A_{jl}|>\frac{1}{4C_{*}}$, then since $j=l$, $\Im
G_{lj}^{\infty}$ is bounded strictly away from 0 uniformly for $E=\Re
z\in[-2+\kappa,2-\kappa]$ by Proposition 2.4 (1), so the imaginary part of
$1+A_{jl}G_{lj}^{\infty}$ is also bounded strictly away from zero. Combining
both cases, we can derive the following very useful estimate: there exists
another constant $C_{1}$ depending only on $C_{*}$ and the constant
$C_{\kappa}$ in Proposition 2.4 (1) such that for any pair $(i,j)$ in case
(C),

$|G_{ij}(z_{m})|\leq C_{1}\min(1,\frac{1}{|A_{jl}|})$ (5.17)

with probability at least $1-e^{-\nu\log N^{\log\log
n}}-\mathbb{P}(R_{m-1}^{c})$.

Now we consider the non-diagonal case $j\neq l$, so that there are two terms
in $T_{ij}^{2}$. Then the resolvent expansion for $G_{ij}$ and $G_{il}$ are:

$\begin{cases}G_{ij}(1+A_{jl}G_{lj}^{\infty})+G_{il}A_{lj}G_{jj}^{\infty}=G_{ij}^{\infty}+T_{ij}^{1}+T_{ij}^{3},&\\\
G_{il}(1+A_{lj}G_{jl}^{\infty})+G_{ij}A_{jl}G_{ll}^{\infty}=G_{il}^{\infty}+T_{il}^{1}+T_{il}^{3}.\end{cases}$
(5.18)

To solve $G_{ij}$ and $G_{il}$, we have to invert the matrix

$\Lambda_{jl}:=\begin{pmatrix}1+A_{jl}G_{lj}^{\infty}&A_{lj}G_{jj}^{\infty}\\\
A_{jl}G_{ll}^{\infty}&1+A_{lj}G_{jl}^{\infty}\end{pmatrix}.$ (5.19)

We compute its determinant:

$\displaystyle\det\Lambda_{jl}$
$\displaystyle=(1+A_{jl}G_{lj}^{\infty})^{2}-A_{jl}^{2}G_{ll}^{\infty}G_{jj}^{\infty}$
(5.20) $\displaystyle=(1+A_{jl}\Re G_{lj}^{\infty})^{2}-A_{jl}^{2}|\Im
G_{lj}^{\infty}|^{2}+A_{jl}^{2}|\Im G_{ll}^{\infty}|\Im
G_{jj}^{\infty}|(1+o(1))$ $\displaystyle+2i(1+A_{jl}\Re
G_{lj}^{\infty})A_{jl}\Im G_{lj}^{\infty},.$

where in the second equality we used the fact that $G_{ii}^{\infty}$ is very
close to a purely imaginary number, thanks to Proposition 2.4, (1). Now thanks
to Proposition 2.4, (2) and (3), and the fact that $\Im G_{ll}^{\infty}(z)$ is
sufficiently close to $m^{as}(z),$ we can find constants $D_{\kappa,1}>0$ and
$D_{\kappa,K,p}>0$ such that (if $K=0,1$ use the constant $D_{\kappa,1}$,
otherwise use the constant $D_{\kappa,K,p}$.)

$|\det\Lambda_{jl}|\geq\Re\det\Lambda_{jl}\geq
D_{\kappa,K,P}|A_{jl}|^{2}|m^{as}(z)|^{2}.$ (5.21)

This bound will be used when $|A_{jl}|$ is large. When $|A_{jl}|$ is small,
say $|A_{jl}|\leq\frac{1}{10C_{*}}$ where $C_{*}$ is the supremum of
$|G_{ij}^{\infty}|$ on $\mathcal{S}$, then a simple computation shows that

$|\det\Lambda_{jl}|\geq\Re\det\Lambda_{jl}\geq 0.81.$ (5.22)

Now we can invert the matrix and solve the linear system (5.18). We do the
computations separately in the two cases ($|A_{jl}|$ small and $|A_{jl}|$
large) and use the corresponding estimates on $\det\Lambda_{jl}$. With high
probability $T_{ij}^{1},T_{ij}^{3},T_{il}^{1},T_{il}^{3}$ are all negligible,
so that we can find $C_{1}$ depending only on $C_{*}$ and $D_{\kappa,K,p}$
such that

$|G_{ij}(z_{m})|\leq C_{1}\min(1,\frac{1}{|A_{jl}|}),\quad|G_{il}(z_{m})|\leq
C_{1}\min(1,\frac{1}{|A_{jl}|})$ (5.23)

with probability at least $1-e^{-\nu\log N^{\log\log
n}}-\mathbb{P}(R_{m-1}^{c})$. This finishes the inductive proof for case (C),

Now we are ready for case (B). The terms involved in $T_{ij}^{1}$ and
$T_{ij}^{3}$ can be bounded just as the previous two cases and are with
probability $1-e^{-\nu\log N^{\log\log N}}-\mathbb{P}(R_{m-1}^{c})$ smaller
than $N^{-0.001\epsilon}$. The term $T_{ij}^{2}$ equals
$G_{ik}A_{kl}G_{lj}^{\infty}$ if $k=l$, or
$G_{ik}A_{kl}G_{lj}^{\infty}+G_{il}A_{lk}G_{kj}^{\infty}$ if $k\neq l$. We
have proved in the previous paragraph that any such pair $(i,k)$ (that
corresponds to a unique pair $(k\leq l)$ with $|A_{kl}|>q^{-1}$) must satisfy
$|G_{ik}|<C_{1}\min(1,\frac{1}{|A_{kl}|})$. Then we easily derive from the
resolvent identity that with probability at least $1-e^{-\nu\log N^{log\log
N}}-\mathbb{P}(R_{m-1}^{c})$, for $(i,j)$ in case $(B),$

$|G_{ij}(z_{m})|\leq 2C_{1}C_{*}.$

It suffices to take $C=\max(4C_{1}C_{*},\hat{C})$ in the definition (5.12) of
$R_{m}$. This finishes the inductive step from $m-1$ to $m$, and hence
finishes the proof of the whole theorem.

∎

Finally we prove the second part of Theorem 1.21, that is, we show with
positive probability there exists some entry $(i,j)$ such that $G_{ij}(z)$
does not converge to $G_{ij}^{\infty}(z)$.

###### Proof of Theorem 1.21, (2).

The proof follows from analyzing (5.16). We have proved that with overwhelming
probability $T_{ij}^{1}$ and $T_{ij}^{3}$ are smaller than
$N^{-0.001\epsilon}$. In the case $l=j$, $|G_{lj}^{\infty}|$ is uniformly
bounded away from zero thanks to Proposition 2.4 (1), so that once
$A_{jl}\in[\frac{1}{2},\infty)$ we must be able to find a constant $C_{0}$
with $|G_{ij}-G_{ij}^{\infty}|\geq C_{0}$. With positive probability there
exists some $j=l$ with $A_{jl}\in[\frac{1}{2},\infty)$. This completes the
proof. ∎

####  5.2.1. Eigenvector de-localization

In the end we justify Corollary 1.23.

###### Proof of Corollary 1.23.

Let $u_{i}(k),$ $i,k=1,\cdots,N$ denote the $k$-th coordinate of the
normalized eigenfunction that corresponds to the $i$-th smallest eigenvalue
$\lambda_{i}$ of $H_{N}^{\infty}+A_{N}$. We restrict to
$\lambda_{i}\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$. In the following
we choose a random spectral parameter $z:=\lambda_{i}+i\eta$ where
$\eta=N^{-1+\epsilon}$. Then from Theorem 1.21, (1), we can find $c_{*}>0$
sufficiently small, such that with probability at least
$1-N^{-c_{*}\epsilon}$, the following holds for all $k\in[1,N]$ and all
$\lambda_{i}\in[-2+\kappa,2-\kappa]\setminus\Delta_{K}^{p}$:

$C>\Im
G_{kk}(z)=\sum_{j=1}^{N}\frac{\eta}{(\lambda_{j}-\lambda_{i})^{2}+\eta^{2}}|u_{j}(k)|^{2}\geq\frac{1}{\eta}|u_{i}(k)|^{2}=N^{1-\epsilon}|u_{i}(k)|^{2}.$
(5.24)

This completes the proof. ∎

###  5.3. Green function estimates under different scaling

In this section we prove Theorem 1.25. This will be a careful adaptation of
the proof of Theorem 1.21.

###### Proof of Theorem 1.25.

In the proof we take $\eta\in[N^{-1+2\sigma\alpha+\epsilon},1]$. We fix a
length scale $L=N^{1-2\sigma\alpha-0.5\epsilon}$ and a cutoff value
$q=N^{\frac{0.01\epsilon}{\alpha}}$. We only give a sketch by mostly following
the proof of Theorem 1.21, and only highlight where some changes are needed.

Define as in the previous proof the event that atypical $A_{ij}$ locations are
not distant:

$\displaystyle D_{N}:=$ $\displaystyle\\{\text{there exists }i\leq
j,i^{\prime}\leq
j^{\prime}\in[1,N]^{4}:N^{\sigma}|A_{ij}|>q^{-1},N^{\sigma}|A_{i^{\prime}j^{\prime}}|>q^{-1},|i-i^{\prime}|\leq
2L\\}$ (5.25) $\displaystyle\cup\\{\text{ there exists }i\leq
j\in[1,N]^{2}:N^{\sigma}|A_{ij}|>q^{-1},i\in[1,L]\cup[N-L,N].\\}$

Recall the assumption
$A_{ij}=N^{\sigma}\frac{1}{N^{\frac{1}{\alpha}}}\xi_{ij}$, so we have

$\mathbb{P}(D_{N})\leq KN\cdot(2K-1)2L\cdot
N^{2(-1+\sigma\alpha+0.01\epsilon)}+2KLN^{-1+\sigma\alpha+0.01\epsilon}\leq
N^{-0.48\epsilon},$ (5.26)

where we use that
$\mathbb{P}(N^{\sigma}|A_{ij}|>q^{-1})=\mathbb{P}(|\xi_{ij}|\geq
N^{\frac{1-0.01\epsilon}{\alpha}-\sigma})$.

We adopt the same resolvent expansion as in (5.14). This time we have the
three terms

$T_{ij}^{1}=-\sum_{|l-j|\leq
L,N^{\sigma}|A_{kl}|<q^{-1}}N^{\sigma}G_{ik}A_{kl}G_{lj}^{\infty},$

$T_{ij}^{2}=-\sum_{|l-j|\leq
L,N^{\sigma}|A_{kl}|>q^{-1}}N^{\sigma}G_{ik}A_{kl}G_{lj}^{\infty},$

and

$T_{ij}^{3}=-\sum_{|l-j|\geq L}N^{\sigma}G_{ik}A_{kl}G_{lj}^{\infty}.$

We first estimate $T_{ij}^{1}$. To derive a sharp estimate we will use an
estimate (6.14) derived in later chapters. Granted with (6.14), we apply
concentration inequality just as in (5.15) to derive that, with probability at
least $1-e^{-\nu\log N^{\log\log N}}$,

$|T_{ij}^{1}|\leq
N^{0.001\epsilon}\left[\frac{8C^{2}}{N^{\frac{0.01\epsilon}{\alpha}}}+C^{2}\sqrt{\frac{1}{N^{1-\sigma\alpha-10^{-3}\epsilon}\eta}}\right]\leq
C^{2}N^{-0.001\epsilon},$

thanks to the assumption that $\eta>N^{-1+\sigma\alpha+\epsilon}$. (We have
omitted the truncation procedure and the induction procedure on the length
scale of $\eta$: they are exactly the same as in the previous proofs.)

To estimate $T_{ij}^{3}$, note that with probability tending to 1, we may
assume that $|N^{\sigma}A_{ij}|\leq N^{1.1\sigma}$ for all pairs $(i,j)$, and
we condition on this event. Thanks to the decay property (Proposition 2.3) of
$G^{\infty}$: $|G^{\infty}_{ij}(z)|\leq Ce^{-0.5\epsilon}$ for all $|i-j|>L$,
and that $\eta>N^{-1+2\sigma\alpha+\epsilon}$, we may bound $|T_{ij}^{3}|\leq
CN^{-c_{*}\epsilon}$ just as in the previous proof, assuming that we have an
a-priori bound $|G_{ij}|\leq C$ for all $i,j$.

Granted with these estimates, we can categorize indices $(i,j)$ into the three
different classes (A),(B) and (C) as in the previous proof, but now we say
$(i,j)$ is atypical if $N^{\sigma}|A_{ij}|>q^{-1}$. More precisely, each
$(i,j)$ lies in one of the following three cases: in case (A), j is at
distance at least $L$ to any index $k$ such that
$N^{\sigma}|A_{j^{\prime}k}|>q^{-1}$ for some $j^{\prime}$. In case (B),
within distance $0<|j-k|<L$ there exists an index $k$ and another index
$j^{\prime}\neq j$ such that $N^{\sigma}|A_{j^{\prime}k}|>q^{-1}$ (on
$D_{N}^{c}$ there is at most one such index pair $(j^{\prime},k)$). In case
(C), one can find an index $k$ with $N^{\sigma}|A_{jk}|>q^{-1}$. Then we
follow the induction argument (5.12) and (5.13) up until
$\eta=N^{-1+2\sigma\alpha+\epsilon}$.

For $(i,j)$ in case (A), we follow exactly the same argument as in the
previous proof, and the desired estimate on $|G_{ij}|$ readily follows. The
induction step from $m$ to $m+1$ is verified for these $A_{ij}$’s.

For $(i,j)$ in case (C), we again use Proposition 2.4 to solve the linear
system (5.18) and deduce that for all such pairs $(i,j)$ that correspond to a
unique $(j,l)$ with $N^{\sigma}|A_{jl}|>q^{-1}$, we can find $C_{1}>0$
depending only on $C_{*}$ and $C_{\kappa,K,p}$ that

$|G_{ij}(z_{m})|\leq
C_{1}\min(1,\frac{1}{N^{\sigma}|A_{jl}|}),\quad|G_{il}(z_{m})|\leq
C_{1}\min(1,\frac{1}{N^{\sigma}|A_{lj}|}).$ (5.27)

Finally for $(i,j)$ in class (B), as in the previous proof, the $T_{ij}^{1}$
and $T_{ij}^{3}$ terms are vanishing with high probability. The $T_{ij}^{2}$
term $-G_{ik}\cdot N^{\sigma}A_{kl}\cdot G_{lj}^{\infty}$ or $-G_{ik}\cdot
N^{\sigma}A_{kl}\cdot G_{lj}^{\infty}-G_{il}\cdot N^{\sigma}A_{lk}\cdot
G_{kj}^{\infty}$ is bounded by $4C_{1}C_{*}$ thanks to (5.27). Thus we have
justified the inductive hypothesis at $m$. For the remaining parts, we exactly
follow the proof of Theorem 1.21.

∎

##  6\. Local law for the Stieltjes transform

In this section we prove both Theorem 1.28 and Theorem 1.32. We begin with the
first part, that is, $H_{N}=H_{N}^{\infty}+A_{N}$.

We first show that with overwhelming probability, most of the random variables
in $A_{N}$ have a small upper bound. Without loss of generality assume the
function $L(x)$ in (1.13) is bounded by 1 for all $x\in[0,\infty)$. The
general case proceeds after some minor change.

######  Lemma 6.1.

Assuming that Assumption 1.8 is satisfied. Let $\Delta_{\epsilon}$ denote the
event that there are at least $2(K+1)N^{\frac{\epsilon}{4}}$ elements of the
matrix $A_{N}$ that have absolute value larger than
$(K+1)^{\frac{1}{\alpha}}N^{-\frac{\epsilon}{10\alpha}}$. Then
$\mathbb{P}[\Delta_{\epsilon}]\leq e^{-CN^{\epsilon}\log N}$ for some
universal constant $C>0$.

###### Proof.

Thanks to the fact that Assumption 1.8 is satisfied for $L(x)$ bounded by 1,

$\displaystyle\mathbb{P}[\Delta_{\epsilon}]$
$\displaystyle\leq\sum_{\ell=(K+1)N^{\frac{\epsilon}{4}}}^{(K+1)N}\binom{(K+1)N}{\ell}\mathbb{P}[|\xi_{ij}|\geq(K+1)^{\frac{1}{\alpha}}N^{\frac{1}{\alpha}-\frac{\epsilon}{10\alpha}}]^{\ell}$
(6.1)
$\displaystyle\leq\sum_{\ell=(K+1)N^{\frac{\epsilon}{4}}}^{(K+1)N}\binom{(K+1)N}{\ell}((K+1)N)^{-\ell}N^{\frac{\epsilon}{10}\ell}.$

By a simple combinatorial estimate,

$\mathbb{P}[\Delta_{\epsilon}]\leq\sum_{\ell=((K+1)N^{\frac{\epsilon}{4}}}^{(K+1)N}\frac{1}{\ell!}N^{\frac{\epsilon}{10}\ell}.$

From the elementary inequality
$\ell!\geq(\lfloor\frac{\ell}{2}\rfloor)!N^{\frac{\epsilon}{10}(K+1)\ell}$
given $\ell\geq(K+1)N^{\frac{\epsilon}{4}}$, we deduce that

$\mathbb{P}[\Delta_{\epsilon}]\leq\sum_{\ell=(K+1)N^{\epsilon/4}/2}^{\infty}\frac{1}{\ell!}\leq\frac{e}{((K+1)N^{\epsilon/4}/2)!}\leq
e^{-CN^{\epsilon}\ln N}$

where the last step follows from Stirling’s formula. ∎

Then we outline the following sampling procedure that will be used in both
Theorem 1.28 and Theorem 1.32. We however use different notions of admissible
labels.

######  Sampling 6.2.

We outline a procedure to sample and truncate the noisy matrix $A_{N}$.

First, we introduce a label $L=(L_{ij})$, $i-K\leq j\leq i\leq N$, that has
independent entries. For each site $(i,j)$, we assume that $L_{ij}=T$ with
probability
$P:=\mathbb{P}(|\xi_{ij}|<(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}})$
and $L_{ij}=F$ with probability $1-P$.

Then thanks to Lemma 6.1, with probability at least $1-e^{-cN^{\epsilon}\ln
N}$, $L$ has no more than $(K+1)N^{\frac{\epsilon}{4}}$ elements with label
$F$. We call such a label $L$ admissible.

Moreover, we say a label $L$ is separably admissible if $L_{ij}=F$, $i\leq j$
implies that the label
$i\notin[1,N^{1-0.5\epsilon}]\cup[N-N^{1-0.5\epsilon},N]$, and that for any
other $(k\leq l)$ such that $L_{kl}=F$ one must have
$|i-k|>N^{1-0.5\epsilon}$. Then the computation in (5.26) shows that

$\mathbb{P}(L\text{ is {separably admissible}})\geq 1-N^{-0.48\epsilon}.$

Let $\xi$ be a random variable with symmetric law satisfying (1.13). Let
$\xi^{T}$ be the probability distribution of $\xi$ conditioned to take value
in
$[-(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}},(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}}]$,
and let $\xi^{F}$ be the probability distribution of $\xi$ conditioned to take
value in
$\mathbb{R}\setminus[-(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}},(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}}]$.

For any given label $L$, we sample the $N\times N$ matrix $A_{N}(L)$ as
follows. For any $i-K\leq j\leq i$, if $L_{ij}=T$ we sample $A_{ij}$ from the
law $\xi^{T}$, and if $L_{ij}=F$ we sample $A_{ij}$ from the law $\xi^{F}$.
All these $A_{ij}$s are sampled independently. For $i-K>j$ we set $A_{ij}=0$.
Then extend by symmetry via $A_{ij}=A_{ji}$ for all $i<j$ to define the whole
matrix $A_{N}$. It is not hard to check that the matrix obtained by first
sampling $L$, then sampling $A_{N}$ from $L$ has the same law as the original
matrix $A_{N}$ from Assumption 1.8. To stress the dependence, we use the
symbol $A_{N}(L)$ to mean a matrix $A_{N}$ sampled from $L$ via this
procedure.

Then we define the truncation procedure. For any label $L$, let
$\mathbf{T}^{L}$ be the subset of $[1,N]$ consisting of indices $i$ such that
$L_{ik}=F$ or $L_{ki}=F$ for some $k\in[1,N]$. Denote by
$A_{N}^{(\mathbf{T}^{L})}$ the matrix obtained by removing the rows and
columns of $A_{N}(L)$ with indices in $\mathbf{T}^{L}$, and denote by
$H_{N}^{\infty,(\mathbf{T}^{L})}$ the matrix obtained by removing from
$H_{N}^{\infty}$ the rows and columns with indices in $\mathbf{T}^{L}$. Note
that $A_{N}^{(\mathbf{T}^{L})}$ is independent of $\mathbf{T}^{L}$.

In the following we will first prove a local law for
$H_{N}^{\infty,(\mathbf{T}^{L})}+A_{N}^{(\mathbf{T}^{L})}$ for any (separably)
admissible label $L$. Then we show how this implies a local law for
$H_{N}^{\infty}+A_{N}$.

######  Lemma 6.3.

for any separably admissible label $L$, denote by $m^{(\mathbf{T}^{L})}$ the
trace of the Green function of
$H_{N}^{\infty,(\mathbf{T}^{L})}+A_{N}^{(\mathbf{T}^{L})}$, and denote by
$m^{\infty,(\mathbf{T}^{L})}$ that of $H_{N}^{\infty,(\mathbf{T}^{L})}$. Then
we can find positive constants $C$, $\nu$ depending on $\kappa,\epsilon$ such
that

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon,\kappa)}|m^{(\mathbf{T}^{L})}(z)-m^{\infty,(\mathbf{T}^{L})}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}}$ (6.2)

and the constants can be chosen uniformly over all admissible labels $L$.

###### Proof.

We verify that the assumptions in Theorem 3.3 are satisfied by both
$H_{N}^{\infty,(\mathbf{T}^{L})}$ and $A_{N}^{(\mathbf{T}^{L})}$. First, since
$L$ is separably admissible, denoting the elements of $\textbf{T}^{L}$ as
$\\{k_{1}<k_{2}<\cdots<k_{s}\\}$, then we must have $k_{1}>N^{1-0.5\epsilon}$,
and $k_{r}-k_{r-1}>N^{1-0.5\epsilon}$ for each $r=2,\cdots,s$. Writing out the
matrix form of 1-d Laplacian, one can see that
$H_{N}^{\infty,(\mathbf{T}^{L})}$ is a block diagonal matrix with block size
$k_{1}-1,k_{2}-k_{1}-1,\cdots,k_{s}-k_{s-1}-1$ and each block has the form
$H_{k_{r}-k_{r-1}-1}^{\infty}$. Thanks to this block-diagonal form, when
computing the Green function of $H_{N}^{\infty,(\mathbf{T}^{L})}$ we need only
compute the Green function of each block, which is already done in Section
2.1. Now each block has size at least $N^{1-0.5\epsilon}$ and
$\eta>N^{-1+\epsilon}$, so that Proposition 1.1 is applicable and leads to

$\sup_{z\in\mathcal{S}}\max_{i,j}\left|G^{\infty,(\mathbf{T}^{L})}_{ij}(z)\right|\leq
C,$ (6.3)

where $C$ is uniform over any separably admissible label $L$.

We have verified that $H_{N}^{\infty,(\mathbf{T}^{L})}$ satisfies the
assumption of Theorem 3.3. For $A_{ij},$ since $\xi_{ij}$ is assumed to have a
symmetric law, we have that $\mathbb{E}[\xi^{T}]=0$. To compute higher moments
we use Lemma B.1, which implies that for some slow varying function $L_{0}$,

$\mathbb{E}[|N^{-\frac{1}{\alpha}}\xi^{T}|^{p}]\leq
L_{0}(N^{\frac{1-\epsilon/10}{\alpha}})C_{\alpha}\frac{C}{Nq^{p-\alpha}},\quad
p\geq 2,$

where $q=N^{\frac{\epsilon}{10\alpha}}$. Since $L_{0}(x)$ is slow-varying, we
may assume $L_{0}(N^{\frac{1-\epsilon/10}{\alpha}})\leq CN^{10^{-3}\epsilon}$,
then

$\mathbb{E}[|N^{-\frac{1}{\alpha}}\xi^{T}|^{p}]\leq\frac{C}{N^{1-10^{-3}\epsilon}q^{p-\alpha}}.$
(6.4)

Then the result follows immediately from Theorem 3.3, noting that the square
matrix $H_{N}^{\infty,(\mathbf{T}^{L})}+A_{N}^{(\mathbf{T}^{L})}$has size at
least $N-(2K+1)N^{\frac{\epsilon}{2}}\sim N$ for any separably admissible $L$.
∎

Finally we show that $m^{(\mathbf{T}^{L})}$ is very close to $m$ for any
admissible $L$, i.e. the error induced by truncation is negligible.
Inequalities concerning the trace of row and column removed Green function can
be found for example in [19], Lemma 4.2 or [14]. The formula is as follows:
given any $N\times N$ matrix $H$, for any index st
$\mathbf{T}=\\{k_{1},\cdots,k_{s}\\}\subset[1,N],$ let $H^{(\mathbf{T})}$
denote the matrix $T$ with the rows and columns labelled $k_{1},\cdots,k_{s}$
removed, and let $G^{(\mathbf{T})}$ denote the resolvent matrix of
$H^{(\mathbf{T})}$. Then for any distinct indices $i,j,k$ that are not in
$\mathbf{T}$, we have

$G_{ij}^{(\mathbf{T})}-G_{ij}^{(k\mathbf{T})}=G_{ik}^{(\mathbf{T})}G_{kj}^{(\mathbf{T})}(G_{kk}^{(\mathbf{T})})^{-1}$
(6.5)

where $k\mathbf{T}$ means adjoining $k$ to $\mathbf{T}$. Denote by
$m^{(\mathbf{T})}$ the trace of $G^{(\mathbf{T})}$, then

$m^{(\mathbf{T})}-m^{(\mathbf{kT})}=\frac{1}{N}\sum_{i}\frac{G_{ik}^{(\mathbf{T})}G_{ki}^{(\mathbf{T})}}{G_{kk}^{(\mathbf{T})}}=\frac{1}{N}\frac{[(G^{\mathbf{(T)}})^{2}]_{kk}}{G_{kk}^{(\mathbf{T})}}$
(6.6)

Thanks to the elementary inequality
$|(\lambda-\omega)^{-2}|=\eta^{-1}\Im[(\lambda-z)^{-1}]$,
$\lambda\in\mathbb{R}$, we deduce that $|[G^{\mathbf{T}^{2}}_{ii}|=\frac{\Im
G^{\mathbf{T}}_{ii}}{\eta}$ and we conclude with

$|m^{(\mathbf{T})}-m^{(\mathbf{kT})}|\leq\frac{1}{N\eta},$ (6.7)

and more generally

$|m-m^{(\mathbf{T})}|\leq\frac{|\mathbf{T}|}{N\eta}.$ (6.8)

Now we are ready to give a proof of Theorem 1.28.

###### Proof of Theorem 1.28: the first part.

We first sample the label $L$ following the rule in Sampling 6.2. Then

$\mathbb{P}(L\text{ is separably admissible })\geq 1-N^{-c_{*}\epsilon}.$
(6.9)

By Lemma 6.3, for any separably admissible $L$,

$\mathbb{P}\left(|m^{(\mathbf{T}^{L})}-m^{\infty,(\mathbf{T}^{L})}|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}},$ (6.10)

where the constants $C$ and $\nu$ are independent of the separably admissible
label $L$.

Combined with (6.8) applied to both $m$ and $m^{\infty}$, setting
$\mathbf{T}:=\mathbf{T}^{L}$, we deduce after taking expectation over all
separably admissible $L$, that

$\mathbb{P}\left(|m-m^{\infty}|\geq
CN^{-\frac{\epsilon}{40}}+\frac{2(2K+1)N^{\frac{\epsilon}{2}}}{N\eta}\mid
L\text{ separably admissible}\right)\leq e^{-\nu\log N^{\log\log N}},$ (6.11)

for some constant $C$ depending only on $\alpha$, $\kappa$ and $\epsilon$,
where we use the fact that for any separably admissible $L$,
$|\mathbf{T}^{L}|\leq(2K+1)N^{\frac{\epsilon}{2}}$. Combining with (6.9) and
the fact that $\eta>N^{-1+\epsilon}$, this completes the proof of (1.28) in
the first part of Theorem 1.28.

Finally replacing $m^{\infty}(z)$ by $m^{as}(z)$ leads to a vanishing error
thanks to Lemma 2.2. ∎

###  6.1. Wigner matrix with banded perturbation

In this section we prove Theorem 1.32. Before that, we need an extension of
Theorem 3.3:

######  Theorem 6.4.

Let $\overline{W}_{N}$ be a Wigner matrix satisfying the definitions in
Theorem 1.32. Let $\overline{A}_{N}$ be an $N\times N$ real symmetric random
matrix with independent elements $\overline{A}_{ij}$ for all $|i-j|\leq K$ and
zero otherwise. Assume that we can find some $\delta>1$ and $\gamma>0$ such
that

$\mathbb{E}[|A_{ij}|]\leq\frac{C}{N^{1+\delta}},\quad\mathbb{E}[|A_{ij}|^{p}]\leq\frac{C}{N^{\gamma}q^{p-\alpha}},\text{
for each }p\geq 2,$

where $q$ is $N$-dependent and $q>>\log N^{\log\log N}$. Assume that
$\overline{W}_{N}$ and $\overline{A}_{N}$ are independent.

Then for any $\epsilon>0$ we can find positive constants $C$ and $\nu$
depending on $\epsilon$ such that

$\mathbb{P}\left(\max_{\begin{subarray}{c}z\in\mathcal{S}(\epsilon)\\\
\eta\geq
N^{-\gamma+0.5\epsilon}\end{subarray}}\max_{i,j}|\overline{G}_{ij}-m^{sc}\delta_{ij}|(z)\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}},$ (6.12)

where $\overline{G}$ is the Green function of
$\overline{W}_{N}+\overline{A_{N}}$ and $\delta_{ij}$ satisfies
$\delta_{ij}=1$ if $i=j$.

###### Proof.

Let $G^{\overline{W}}$ denote the Green function of $\overline{W}_{N}$. Then
$\overline{W}_{N}$ satisfies the following local law [20], where $m^{sc}(z)$
is the Stieltjes transform of semicircle law:

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\epsilon)}\sup_{1\leq i,j\leq
N}|G^{\overline{W}}_{ij}-\delta_{ij}m^{sc}(z)|\geq
CN^{-\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}}.$ (6.13)

Since $\overline{W}_{N}$ and $\overline{A}_{N}$ are independent, we may
condition on this event with probability $1-e^{-\nu\log N^{\log\log N}}$ (so
that $\overline{W}_{N}$ is now a deterministic matrix and satisfies the entry-
wise Green function upper bound (1.7)) and use Theorem 3.3 to complete the
proof.

∎

We now give the proof of the first part of Theorem 1.32.

###### Proof of Theorem 1.32, the first part..

We follow the sampling procedure in Sampling 6.2. With probability at least
$1-e^{-N^{\epsilon}\log N}$, the sampled label $L$ is admissible. Let
$W_{N}^{(\mathbf{T}^{L})}$ denote the matrix $W_{N}$ with rows and columns
indexed by indices in $\mathbf{T}^{L}$ removed. Since $W_{N}$ is a Wigner
matrix, $W_{N}^{(\mathbf{T}^{L})}$ is again a Wigner matrix with size
$N-|\mathbf{T}^{L}|$. Therefore we can apply Theorem 6.4 to derive a local law
for $W_{N}^{(\mathbf{T}^{L})}+A^{(\mathbf{T}^{L})}$, with the constants
uniform over all choices of admissible label $L$. Then we use (6.8) to
complete the proof of the first part of Theorem 1.32. See next section for the
proof of the second part.

∎

###  6.2. Trace of Green function under different scaling

Now we prove the second part of Theorem 1.28 and Theorem 1.32, i.e. when we
have an $N^{\sigma}$ scaling in front of $A_{ij}$.

We begin with some moment computations and use again Lemma B.1. We choose
$q=N^{\frac{\epsilon}{10\alpha}}$ and choose
$x=N^{\frac{1}{\alpha}}N^{-\sigma}q^{-1}$, then for any $p\geq 2:$

$\mathbb{E}[|\xi_{ij}|^{p}1_{|\xi_{ij}|\leq x}]\leq
C_{\alpha}L_{0}(x)N^{\frac{p}{\alpha}}\frac{1}{Nq^{p-\alpha}N^{\sigma(p-\alpha)}}.$

where $C_{\alpha}=\frac{2}{2-\alpha}.$ Then

$\mathbb{E}[|N^{-\frac{1}{\alpha}}N^{\sigma}\xi_{ij}|^{p}1_{|\xi_{ij}|\leq
x}]\leq C_{\alpha}L_{0}(x)\frac{1}{N^{1-\sigma\alpha}q^{p-\alpha}}.$

Since $L_{0}$ is a slow-varying function, we can assume that $L_{0}(x)\leq
C_{\alpha}N^{10^{-3}\epsilon}$. Recalling the definition of $A_{N}$, we can
rewrite the last expression as

$\mathbb{E}[|N^{\sigma}A_{ij}|^{p}1_{N^{\sigma}|A_{ij}|\leq q^{-1}}]\leq
C_{\alpha}\frac{1}{N^{1-\sigma\alpha-10^{-3}\epsilon}q^{p-\alpha}}.$ (6.14)

We also need to prove the following analogue of Lemma 6.1:

######  Lemma 6.5.

Assuming that Assumption 1.8 is satisfied. Let $\Delta^{\sigma}_{\epsilon}$
denote the event that there are at least
$2(K+1)N^{\sigma\alpha+\frac{\epsilon}{4}}$ elements of the matrix
$N^{\sigma}A_{N}$ that have absolute value larger than
$(K+1)^{\frac{1}{\alpha}}N^{-\frac{\epsilon}{10\alpha}}$. Then
$\mathbb{P}[\Delta^{\sigma}_{\epsilon}]\leq e^{-CN^{\epsilon}\ln N}$ for some
universal constant $C>0$.

###### Proof.

We work as in Lemma 6.1:

$\displaystyle\mathbb{P}[\Delta^{\sigma}_{\epsilon}]$
$\displaystyle\leq\sum_{\ell=(K+1)N^{\frac{\epsilon}{4}+\sigma\alpha}}^{(K+1)N}\binom{(K+1)N}{\ell}\mathbb{P}[|\xi_{ij}|\geq(K+1)^{\frac{1}{\alpha}}N^{\frac{1}{\alpha}-\sigma-\frac{\epsilon}{10\alpha}}]^{\ell}$
(6.15)
$\displaystyle\leq\sum_{\ell=(K+1)N^{\frac{\epsilon}{4}+\sigma\alpha}}^{(K+1)N}\binom{(K+1)N}{\ell}((K+1)N)^{-\ell}N^{(\frac{\epsilon}{10}+\sigma\alpha)\ell}.$

By a simple combinatorial estimate,

$\mathbb{P}[\Delta_{\epsilon}]\leq(K+1)\sum_{\ell=((K+1)N^{\frac{\epsilon}{4}+\sigma\alpha}}^{(K+1)N}\frac{1}{\ell!}N^{\frac{\epsilon}{10}\ell+\sigma\alpha\ell}.$

At this point we expand the factorial $\ell!$ and see that, arranging the
$\ell$ terms in the product from large to small, for all the terms that are
larger than $N^{\frac{\epsilon}{10}+\sigma\alpha}$, their product is at least

$N^{(\frac{\epsilon}{6}+\sigma\alpha)(\ell-N^{\frac{\epsilon}{10}+\sigma\alpha})},$

and, observing that $N^{\frac{\epsilon}{10}+\sigma\alpha}\leq
N^{-\frac{3\epsilon}{20}}\ell$, when $N$ large we have

$\frac{N^{(\frac{\epsilon}{10}+\sigma\alpha)\ell}}{N^{(\frac{\epsilon}{6}+\sigma\alpha)(\ell-N^{\frac{\epsilon}{10}+\sigma\alpha})}}\leq\frac{N^{N^{-\frac{3\epsilon}{20}}(\frac{\epsilon}{6}+\sigma\alpha)\ell}}{N^{\frac{\epsilon}{15}\ell}}\leq
1.$

Thus we conclude with

$\mathbb{P}[\Delta^{\sigma}_{\epsilon}]\leq\sum_{\ell=(K+1)N^{\frac{\epsilon}{10}}}^{\infty}\frac{1}{\ell!}\leq\frac{e}{((K+1)N^{\frac{\epsilon}{10}})!}\leq
e^{-CN^{\epsilon}\ln N}$

where the last step follows from Stirling’s formula.

∎

Now we are ready to prove the second part of Theorem 1.28.

###### Proofof Theorem 1.28, the second part..

We define as in Sampling 6.2 a sampling procedure as follows: first sample a
label $L$, independently on each entry $(i,j)$, via the following rule:
$L_{ij}=T$ with probability
$P^{\sigma}:=\mathbb{P}(|\xi_{ij}|<(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}-\sigma})$,
and $L_{ij}=F$ with probability $1-P^{\sigma}$. Then thanks to Lemma 6.5, with
probability at least $1-e^{-CN^{\epsilon}\ln N}$, $L$ has at most
$(K+1)N^{\frac{\epsilon}{4}+\sigma\alpha}$ elements with label $F$, and we
call such label $L$ an admissible label. Meanwhile, by the computation (5.26),
we see that with probability at least $1-N^{-0.48\epsilon}$, any label $(i\leq
j)$ such that $L_{ij}=F$ must satisfy
$i\notin[1,N^{1-2\sigma\alpha-0.5\epsilon,}]\cup[N-N^{1-2\sigma\alpha-0.5\epsilon,}N]$
and that for any other $(k\leq l)$ with $L_{kl}=F$ necessarily we have
$|i-k|\geq N^{1-2\sigma\alpha-0.5\epsilon}$, and we call such label $L$
separably admissible. Let $\xi$ be a probability distribution with a symmetric
law that satisfies (1.13). Let $\xi^{T}_{\sigma}$ be the probability
distribution of $\xi$ conditioned to take value in
$[-(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}-\sigma},(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}-\sigma}]$,
and let $\xi^{F}_{\sigma}$ be the probability distribution of $\xi$
conditioned to take value in
$\mathbb{R}\setminus[-(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}-\sigma},(K+1)^{\frac{1}{\alpha}}N^{\frac{1-\epsilon/10}{\alpha}-\sigma}]$.
For any site $(i,j)$ with $L_{ij}=T$, we sample $A_{ij}$ from the law
$\xi^{T}_{\sigma}$; and for any site with $L_{ij}=F$ we sample $A_{ij}$ from
the law $\xi^{F}_{\sigma}$. These samplings are independent modulo the
symmetry constraint.

In the first step we prove, for any separably admissible label $L$, a local
law for the Stieltjes transform of
$H_{N}^{\infty,(\mathbf{T}^{L})}+N^{\sigma}A_{N}^{(\mathbf{T}^{L})}$.

Since $L$ is separably admissible, we must have that
$H_{N}^{\infty,(\mathbf{T}^{L})}$ is block diagonal with each block of size
larger than $N^{1-2\sigma\alpha-0.5\epsilon}$. Then using
$\eta>N^{-1+2\sigma\alpha+\epsilon}$, we can verify after a slight
modification of Proposition 1.1 that

$\sup_{z\in\mathcal{S}(\epsilon,\kappa,2\sigma)}\max_{i,j}\left|G^{\infty,(\mathbf{T}^{L})}_{ij}(z)\right|\leq
C,$ (6.16)

where $C$ can be chosen independent of the separably admissible label. Then
thanks to the moment estimate (6.14), we can apply Proposition 3.3 to deduce
that there are positive constants $C$, $\nu$ depending on $\kappa,\epsilon$
such that

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\kappa,\epsilon)}|m^{(\mathbf{T}^{L})}(z)-m^{\infty,(\mathbf{T}^{L})}(z)|\geq
CN^{\frac{\epsilon}{800}}(\frac{1}{q}+\sqrt{\frac{1}{N^{1-\sigma\alpha-10^{-3}\epsilon}\eta}})\right)\leq
e^{-\nu\log N^{\log\log N}}$ (6.17)

and the constants can be chosen uniformly over all admissible labels $L$, and
$q=N^{\frac{\epsilon}{10\alpha}}$.

Assuming that $\eta>N^{-1+\sigma\alpha+\epsilon}$, we deduce

$\mathbb{P}\left(\sup_{z\in\mathcal{S}(\kappa,\epsilon),\eta\geq
N^{-1+\sigma\alpha+\epsilon}}|m^{(\mathbf{T}^{L})}(z)-m^{\infty,(\mathbf{T}^{L})}(z)|\geq
CN^{\frac{\epsilon}{40}}\right)\leq e^{-\nu\log N^{\log\log N}}.$ (6.18)

In the second step, we use (6.8) to deduce that, similarly as in the previous
proof,

$\mathbb{P}\left(|m-m^{\infty}|\geq
CN^{-\frac{\epsilon}{40}}+\frac{2(2K+1)N^{\sigma\alpha+\frac{\epsilon}{2}}}{N\eta}\mid
L\text{ separably admissible}\right)\leq e^{-\nu\log N^{\log\log N}},$ (6.19)

Then the proof is finished thanks to our assumption on $\eta$.

∎

###### Proof of Theorem 1.32, the second part..

We follow the sampling procedure discussed in the previous paragraph (which is
different from the sampling procedure in 6.2). Thanks to Lemma 6.5, with
probability at least $1-e^{-N^{\epsilon}\log N}$, the sampled label $L$ is
admissible. Let $W_{N}^{(\mathbf{T}^{L})}$ denote the matrix $W_{N}$ with rows
and columns indexed by indices in $\mathbf{T}^{L}$ removed. Since $W_{N}$ is a
Wigner matrix, $W_{N}^{(\mathbf{T}^{L})}$ is again a Wigner matrix with size
$N-|\mathbf{T}^{L}|$. Therefore we can apply Theorem 6.4 to derive a local law
for $W_{N}^{(\mathbf{T}^{L})}+A^{(\mathbf{T}^{L})}$ that holds for all
$\eta=\Im(z)\in[N^{-1+\sigma\alpha+\epsilon},1]$, and the constants are chosen
uniform over all admissible label $L$. Then we use (6.8) to complete the proof
of the second part of Theorem 1.32.

∎

##  Appendix A Concentration inequality via high moments

In this appendix we briefly recall the proof of Proposition 3.2 from [21], to
show that Proposition 3.2 remains true when $\Psi_{i}$ and $a_{i}$ may be
dependent, but that $\Psi_{i}=C_{i}D_{i}$ where $D_{i}$ is deterministic and
$C_{i}$ is almost surely bounded by some constant.

###### Proof sketch.

We compute high moments of the summation: for any positive integer $r$,

$\mathbb{E}|\sum_{i}\Psi_{i}a_{i}|^{2r}=\mathbb{E}\sum_{i_{1},\cdots,i_{2r}}\bar{\Psi}_{i_{1}}\bar{\Psi}_{i_{2}}\cdots\bar{\Psi}_{i_{r}}\Psi_{i_{r+1}}\cdots\Psi_{i_{2r}}\bar{a}_{i_{1}},\cdots\bar{a}_{i_{r}}a_{i_{r+1}}\cdots
a_{i_{2r}}$

Each factor in the summation defines a partition $(i_{1},\cdots,i_{2r})$ of
indices $1,2,\cdots,r$ by requiring that $j$ and $k$ belong to the same
equivalent class if $i_{j}=i_{k}$. Then we first prescribe a partition
$\Gamma$ of the indices, then sum over labels yielding configuration $\Gamma$,
then sum over all partitions. Since $a_{i}$ are centered, each equivalence
class has cardinality at least 2. Denote by $r_{s}$ the size of equivalent
class $s$, then $r_{1}+\cdots+r_{l}=2r$ where $l$ is the number of equivalence
classes of $\Gamma$. The contribution from a partition $\Gamma$ in the sum is
bounded in absolute value by

$\sum_{i_{1},\cdots,i_{l}}\prod_{s=1}^{l}\mathbb{E}\left[|\Psi_{i_{s}}|^{r_{s}}|a_{i_{s}}|^{r_{s}}\right].$

Now assuming that for each $i$, we can write $\Psi_{i}=C_{i}D_{i}$ where
$D_{i}$ are deterministic, and $|C_{i}|\leq C_{0}$ almost surely, then the
above sum is bounded by

$(C_{0})^{2r}\sum_{i_{1},\cdots,i_{l}}\prod_{s=1}^{l}|D_{i_{s}}|^{r_{s}}\mathbb{E}[|a_{i_{s}}|^{r_{s}}].$

We have taken the $C_{0}$ factor out, and we are now reduced to the case where
$D_{i}$ is deterministic, which is the case covered by [21], Lemma 3.8. The
proof completes after rearranging the summation as in [21] and applying a high
moment Markov inequality. ∎

##  Appendix B Moments of alpha-stable laws

We quote the following very useful result:

######  Lemma B.1.

([11], Lemma 5.8) Given a non-negative random variable $A$ such that for any
$x\geq 0$,

$\mathbb{P}(A\geq x)=\ell(x)x^{-\alpha},$ (B.1)

for a slowly varying function $\ell$ and some $\alpha>0$. Then we can find a
slowly varying function $L_{0}$ such that for any $k\in\mathbb{N}_{+}$ and any
$x\geq 0$,

$\mathbb{E}[A^{k}1_{|A|\leq x}]\leq\begin{cases}L_{0}(x),\quad\text{ if
}k\leq\alpha,\\\ L_{0}(x)\frac{k}{k-\alpha}x^{k-\alpha},\quad\text{ if
}k>\alpha.\end{cases}$ (B.2)

## References

  * [1] Amol Aggarwal  “Bulk universality for generalized Wigner matrices with few moments”  In _Probability Theory and Related Fields_ 173 Springer, 2019, pp. 375–432
  * [2] Amol Aggarwal, Charles Bordenave and Patrick Lopatto  “Mobility Edge of L$\backslash$’evy Matrices”  In _arXiv preprint arXiv:2210.09458_ , 2022
  * [3] Amol Aggarwal, Patrick Lopatto and Horng-Tzer Yau  “GOE statistics for Lévy matrices”  In _arXiv preprint arXiv:1806.07363_ , 2018
  * [4] Michael Aizenman and Stanislav Molchanov  “Localization at large disorder and at extreme energies: An elementary derivations”  In _Communications in Mathematical Physics_ 157 Springer, 1993, pp. 245–278
  * [5] Philip W Anderson  “Absence of diffusion in certain random lattices”  In _Physical review_ 109.5 APS, 1958, pp. 1492
  * [6] Gérard Ben Arous and Alice Guionnet  “The spectrum of heavy tailed random matrices”  In _Communications in Mathematical Physics_ 278.3 Springer, 2008, pp. 715–751
  * [7] Fanny Augeri, Raphael Butez and Ofer Zeitouni  “A CLT for the characteristic polynomial of random Jacobi matrices, and the G $\beta$ E”  In _Probability theory and related fields_ 186.1-2 Springer, 2023, pp. 1–89
  * [8] Zhigang Bao and Zhonggen Su  “Local Semicircle law and Gaussian fluctuation for Hermite beta ensemble”  In _arXiv preprint arXiv:1104.3431_ , 2011
  * [9] Serban Belinschi, Amir Dembo and Alice Guionnet  “Spectral measure of heavy tailed band and covariance random matrices”  In _Communications in Mathematical Physics_ 289.3 Springer, 2009, pp. 1023–1055
  * [10] Florent Benaych-Georges and Antti Knowles  “Lectures on the local semicircle law for Wigner matrices”  In _arXiv preprint arXiv:1601.04055_ , 2016
  * [11] Florent Benaych-Georges and Sandrine Péché  “Localization and delocalization for heavy tailed band matrices”  In _Annales de l’IHP Probabilités et statistiques_ 50.4, 2014, pp. 1385–1403
  * [12] Charles Bordenave and Alice Guionnet  “Delocalization at small energy for heavy-tailed random matrices”  In _Communications in Mathematical Physics_ 354 Springer, 2017, pp. 115–159
  * [13] Charles Bordenave and Alice Guionnet  “Localization and delocalization of eigenvectors for heavy-tailed random matrices”  In _Probability Theory and Related Fields_ 157.3-4 Springer, 2013, pp. 885–953
  * [14] Paul Bourgade, Horng-Tzer Yau and Jun Yin  “Local circular law for random matrices”  In _Probability Theory and Related Fields_ 159.3-4 Springer, 2014, pp. 545–595
  * [15] Reda Chhaibi, Joseph Najnudel and Thomas Madaule  “On the maximum of the C$\beta$E field”  In _Duke Mathematical Journal_ 167, 2016
  * [16] Pierre Cizeau and Jean-Philippe Bouchaud  “Theory of Lévy matrices”  In _Physical Review E_ 50.3 APS, 1994, pp. 1810
  * [17] Ioana Dumitriu and Alan Edelman  “Matrix models for beta ensembles”  In _Journal of Mathematical Physics_ 43.11 American Institute of Physics, 2002, pp. 5830–5847
  * [18] László Erdős, Benjamin Schlein and Horng-Tzer Yau  “Wegner estimate and level repulsion for Wigner random matrices”  In _International Mathematics Research Notices_ 2010.3 OUP, 2010, pp. 436–479
  * [19] László Erdős, Horng-Tzer Yau and Jun Yin  “Bulk universality for generalized Wigner matrices”  In _Probability Theory and Related Fields_ 154.1-2 Springer, 2012, pp. 341–407
  * [20] László Erdős, Horng-Tzer Yau and Jun Yin  “Rigidity of eigenvalues of generalized Wigner matrices”  In _Advances in Mathematics_ 229.3 Elsevier, 2012, pp. 1435–1515