# Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner
Matrices

Jacob Fronk  Department of Mathematical Sciences, University of Copenhagen
Torben Krüger  Department of Mathematical Sciences, University of Copenhagen
Department of Mathematics, FAU Erlangen-Nürnberg  Yuriy Nemish  Department of
Mathematics, University of California, San Diego

###### Abstract

We study Hermitian non-commutative quadratic polynomials of multiple
independent Wigner matrices. We prove that, with the exception of some
specific reducible cases, the limiting spectral density of the polynomials
always has a square root growth at its edges and prove an optimal local law
around these edges. Combining these two results, we establish that, as the
dimension $N$ of the matrices grows to infinity, the operator norm of such
polynomials $q$ converges to a deterministic limit with a rate of convergence
of $N^{-2/3+o(1)}$. Here, the exponent in the rate of convergence is optimal.
For the specific reducible cases, we also provide a classification of all
possible edge behaviours.

11footnotetext:  Partially supported by VILLUM FONDEN research grant no. 29369
E-mail addresses<EMAIL_ADDRESS>(J. Fronk<EMAIL_ADDRESS>(T. Krüger),
<EMAIL_ADDRESS>(Yu. Nemish)

_Keywords: polynomials of random matrices, local laws, Dyson equation, extreme
eigenvalues_
AMS Subject Classification: 60B20, 15B52

##  1 Introduction

The empirical spectral distribution of a random matrix is typically well
approximated by a deterministic measure as its dimension grows to infinity. A
clear contender for the most famous example of such a convergence is the
celebrated semi-circle law. It states that the spectral measure of a Wigner
matrix, a Hermitian $N\times N$-matrix $\mathbf{X}$ with centered i.i.d.
entries $x_{ij}$ above the diagonal and $\mathbb{E}|x_{ij}|^{2}=N^{-1}$,
converges to the semi-circle distribution, supported on the interval $[-2,2]$
[51]. In particular, the largest and smallest eigenvalues of $\mathbf{X}$
converge to the respective edges of the support, implying the convergence
$\|\mathbf{X}\|\to 2$ of the operator norm, provided the fourth moments of the
entries of $\sqrt{N}{\mathbf{X}}$ are finite [7, 8].

For non-commutative Hermitian polynomials
$\mathbf{Q}=q(\mathbf{X}_{1},\dots,\mathbf{X}_{l})$ in several independent
Wigner matrices $\mathbf{X}_{i}$ an analogous statement holds. In this setup
the limit of the eigenvalue distribution equals the distribution associated to
the polynomial $\mathfrak{q}=q(\mathfrak{s}_{1},\dots,\mathfrak{s}_{l})$,
where the matrices are replaced by free semicircular random variables within a
non-commutative probability space, i.e. $\mathfrak{s}_{i}$ can be interpreted
as operators acting on an infinite dimensional Hilbert space. This result was
first established for Gaussian random matrices in [50] and extended to Wigner
matrices in [18]. Similarly, the convergence of the norms
$\|\mathbf{Q}\|\to\|\mathfrak{q}\|$ was first shown by Haagerup-Thorbjørnsen
[31] in the Gaussian case and the Wigner case was proven by Anderson in [2].
Such results have also been shown when some $\mathbf{X}_{i}$ are replaced by
non-random matrices [12, 39]. For non-Hermitian polynomials convergence of the
spectral measure to the limiting Brown measure, predicted by free probability
theory, is known only for very specific cases, e.g. for products of random
matrices [29, 41] and for quadratic polynomials [17].

Determining the limiting spectral measure $\rho$ of Hermitian polynomials
$\mathbf{Q}$, or equivalently the distribution of $\mathfrak{q}$, becomes a
nontrivial task beyond particular computable cases. Several works have been
devoted to the analysis of the regularity properties of $\rho$. It has been
shown that $\rho$ has a single interval support [30] and does not contain any
atoms [38, 44]. The cumulative distribution function of $\rho$ is Hölder-
continuous [9], and if $q$ is a monomial or a homogeneous quadratic polynomial
then $\rho$ is even absolutely continuous [15, 21].

In the present paper we consider the case when $q$ is a general polynomial of
degree two, i.e. we study self-adjoint polynomials in $l$ independent Wigner
matrices $\mathbf{X}_{1},\ldots,\mathbf{X}_{l}$ of the form

$q(\mathbf{X}_{1},\ldots,\mathbf{X}_{l})=\sum_{i,j=1}^{l}\mathbf{X}_{i}A_{ij}\mathbf{X}_{j}\mathbf{+}\sum_{i=1}^{l}b_{i}\mathbf{X}_{i}+c,$
(1.1)

where $0\neq A=(A_{ij})\in\mathbb{C}^{l\times l}$ is a Hermitian matrix,
$b=(b_{i})\in\mathbb{R}^{l}$ and $c\in\mathbb{R}$. For these polynomials we
classify the edge behaviour of $\rho$ and show (see Proposition 2.7 below)
that at both edges the limiting spectral measure is absolutely continuous and
apart from specific reducible cases its density exhibits a square root growth.
The reducible cases are, up to a shift and change in sign, of the form
$\mathbf{Y}^{*}\mathbf{Y}$, where $\mathbf{Y}$ is an affine combination of the
underlying Wigner matrices. Such polynomials still have a square root edge at
the rightmost point of the spectrum, but have a density blow-up at the
leftmost point if it is equal to zero. All these cases are classified in
Proposition 2.8 below.

The square root growth of the limiting spectral distribution is a well-known
phenomenon, that is already present in the semicircle law for Wigner matrices
$\mathbf{X}$. In this setup the rate of convergence for the norm of
$\mathbf{X}$ is $\|\mathbf{X}\|=2+\mathcal{O}(N^{-2/3+o(1)})$ with very high
probability [20] and several tail estimates have been established [5, 6, 23].
In fact, for Wigner matrices the distribution of the largest eigenvalue is
known to be universal and given by the Tracy-Widom law [25, 45, 46]. This
distribution was first identified by Tracy and Widom for the Gaussian
ensembles [48, 49] and necessary and sufficient conditions for its
universality in the context of Wigner matrices identified in [37].

Edge universality has been extended to many other Hermitian random matrix
models, including invariant ensembles [14], covariance matrices [26, 43],
deformed Wigner matrices [36], deterministic matrices with Gaussian
perturbations [35] and models with correlations [1, 42]. Such universality
results often rely on control of the eigenvalue location on mesoscopic scales
between $O(1)$ and $O(N^{-1})$, i.e. on a local law.

Local laws arose in the context of Wigner matrices [20, 47] and have
subsequently been extended to the more complex models listed above. For non-
commuting polynomials of several random matrices local laws are known only in
specific cases, starting with Anderson’s work on the anti-commutator
$\mathbf{X}_{1}\mathbf{X}_{2}+\mathbf{X}_{2}\mathbf{X}_{1}$ of Wigner matrices
[3] that controls the deviation of bulk eigenvalues from their expected
position on the scale $\mathcal{O}(N^{-1/2})$. For general polynomials, that
satisfy certain checkable conditions, an optimal local law in the bulk regime
was proved in [21]. Related results are [10] and [11], where it was shown that
for two random matrices satisfying a local law and whose eigenvectors are in
generic direction to each other also their sum satisfies a local law at the
edge and in the bulk. This result covers e.g.
$\mathbf{X}_{1}^{2}+\mathbf{X}_{2}^{3}$ if one of these matrices is a Gaussian
unitary ensemble. For non-Hermitian polynomials the results [40] and [28]
cover products of independent matrices with i.i.d. entries. and [17] quadratic
polynomials.

Currently the best estimate on the convergence rate of the norm for general
polynomials of GUE matrices is
$-N^{-\varepsilon}\leq\|\mathbf{Q}\|-\|\mathfrak{q}\|\leq CN^{-1/4}$, for some
$\varepsilon<\frac{1}{3}$ and $C>0$ established in [16]. Our work improves
this bound for polynomials
$\mathbf{Q}=q(\mathbf{X}_{1},\ldots,\mathbf{X}_{l})$ of the form (1.1) to the
optimal rate of $N^{-2/3+o(1)}$ with square root growth at the edges of the
spectral density and extends the result to Wigner matrices. The main novelty
here is a detailed analysis of the Dyson equation, describing a generalized
resolvent of the linearization matrix associated with the polynomial in the
limit $N\to\infty$ and, consequently, the resolvent of $\mathbf{Q}$ itself.
The idea of linearising polynomials of random matrices in this way stems from
[30, 31] and has been used in many works since, in particular in [2, 4, 13,
32, 33]. In particular, we perform a comprehensive stability analysis of the
Dyson equation that allows us (i) to prove a square root growth of the
limiting spectral density $\rho$ and (ii) to establish an optimal bound on the
difference between the solution to the Dyson equation and the generalized
resolvent by using a modification of the bound on the random error matrix in
the Dyson equation from [22]. The main insight is that the matrix Dyson
equation for the linearization, which has a linear self-energy term, can be
reduced to a scalar equation for a function $m=m(z)$ of the form

$-\frac{1}{m}=z+\gamma(m),$ (1.2)

where the self-energy term $\gamma(m)$ is now a non-linear function of $m$.
This representation allows us to identify the values of the spectral edges in
terms of the coefficients of $q$ and study the quadratic singularity at these
edge points.

### Notations

In this section, we introduce some definitions commonly used throughout the
paper. The standard scalar product of vectors $v,w\in\mathbb{C}^{n}$ will be
denoted by $\langle v,w\rangle$ and the standard euclidean norm by
$\|v\|=\sqrt{\langle v,v\rangle}$. A vector $v$ is called normalized if
$\|v\|=1$.

Matrices $R\in\mathbb{C}^{k\times n}$ for (fixed) $k,n\in\mathbb{N}$ are
usually denoted by non-boldfaced roman letters and matrices
$\mathbf{R}\in\mathbb{C}^{kN\times nN}$ are usually denoted by boldfaced roman
letters. In particular, we denote the identity matrix on $\mathbb{C}^{k\times
k}$ by $I_{k}$ and the identity matrix on $\mathbb{C}^{kN\times kN}$ by
$\mathbf{I}_{kN}$. For any $k,n\in\mathbb{N}$ we embed $\mathbb{C}^{k\times
n}$ in $\mathbb{C}^{k\times n}\otimes\mathbb{C}^{N\times N}$ by identifying
$R\in\mathbb{C}^{k\times n}$ with
$R\otimes\mathbf{I}_{N}\in\mathbb{C}^{k\times n}\otimes\mathbb{C}^{N\times
N}\cong\mathbb{C}^{kN\times nN}$ and write compactly

$R=R\otimes\mathbf{I}_{N}\in\mathbb{C}^{(l+1)N\times(l+1)N}.$ (1.3)

Matrices $R\in\mathbb{C}^{k\times n}$, which are embedded into
$\mathbb{C}^{kN\times nN}$, get still denoted by non-boldfaced letters.

For $R,T\in\mathbb{C}^{n\times n}$ we denote the normalized trace by $\langle
R\rangle=\frac{1}{n}\operatorname{Tr}R$ and define a scalar product by
$\langle R,T\rangle:=\langle R^{*}T\rangle$. We use the standard operator norm
and the Hilbert-Schmidt norm, which are given by

$\|R\|=\sup_{\|x\|\leq
1}\|Rx\|\quad\text{and}\quad\|R\|_{\mathrm{hs}}=\sqrt{\langle R^{*}R\rangle}.$
(1.4)

for vectors of matrices $V=(V_{i})_{i\in\llbracket
l\rrbracket}\in(\mathbb{C}^{n\times n})^{l}$ we denote by $\|V\|$ the maximum
of the operator norms of the entries, i.e.

$\|V\|=\max_{i\in\llbracket l\rrbracket}\|V_{i}\|.$ (1.5)

For random matrices $\mathbf{R}\in\mathbb{C}^{kN\times kN}$ the isotropic and
averaged p-norms are defined by

$\|\mathbf{R}\|_{p}:=\sup_{\|\mathbf{x}\|,\|\mathbf{y}\|\leq
1}\left(\mathbb{E}|\langle\mathbf{x},\mathbf{R}\mathbf{y}\rangle|^{p}\right)^{\frac{1}{p}}\quad\mathrm{and}\quad\|\mathbf{R}\|^{\mathrm{av}}_{p}:=\sup_{\|\mathbf{B}\|\leq
1}\left(\mathbb{E}|\langle\mathbf{B}\mathbf{R}\rangle|^{p}\right)^{\frac{1}{p}}.$
(1.6)

For a block matrix $\mathbf{R}\in\mathbb{C}^{kN\times nN}$ with blocks
$\mathbf{R}_{ij}\in\mathbb{C}^{N\times N}$, $i\in\llbracket k\rrbracket$ and
$j\in\llbracket n\rrbracket$, we define the blockwise (averaged) trace
$\underline{\mathbf{R}}\in\mathbb{C}^{k\times n}$ by

$\underline{\mathbf{R}}_{ij}=\langle\mathbf{R}_{ij}\rangle.$ (1.7)

A matrix $R$ is said to be positive definite if $\langle v,Rv\rangle>0$ for
all $v\in\mathbb{C}^{n\times n}\setminus\\{0\\}$ and we write $R>0.$ It is
called positive semi-definite if $\langle v,Rv\rangle\geq 0$ for all
$v\in\mathbb{C}^{n\times n}\setminus\\{0\\}$ and we write $R\geq 0$. $R$ is
called negative (semi-)definite, denoted by $R<0$ ($R\leq 0$), if $-R$ is
positive (semi)-definite. For $S\in\mathbb{C}^{n\times n}$ Hermitian we write
$S\geq R$ if $S-R\geq 0$ and $S>R$ if $S-R>0$.

For linear operators acting on matrix spaces, we denote by
$\|\cdot\|_{\mathrm{sp}}$ the norm induced by the Hilbert-Schmidt norm,
$\|\cdot\|_{\mathrm{hs}}.$ The identity map between matrix spaces is denoted
by $\mathbb{1}$, i.e. $\mathbb{1}[R]=R$ for all $R\in\mathbb{C}^{k\times n}$.

The upper half-plane will be denoted by $\mathbb{H}$, i.e.

$\mathbb{H}=\\{z\in\mathbb{C}:\operatorname{\mathrm{Im}}z>0\\}$ (1.8)

and $\llbracket n\rrbracket:=\\{1,2,\ldots,n\\}$ is used for the natural
numbers up to $n$. If $X$ and $Y$ are positive quantities, we use the notation
$X\lesssim Y$ if there is some constant $c>0$ such that $X\leq cY$. The
constant will in general depend on the coefficients of $q$. If it also depends
on some other parameters $\alpha$, we write $X\lesssim_{\alpha}Y$. In
particular, the constant will never depend on the dimension of our random
matrices, $N$. If both $X\lesssim Y$ and $Y\lesssim X$ hold true , we write
$X\sim Y$.

### Acknowledgements

We thank Peter Henning Thomsen Krarup for his valuable contributions in the
early stages of the project.

##  2 Main results

######  Assumption.

Let $\zeta_{0}$ be a real-valued and $\zeta_{1}$ be a complex-valued random
variable and let $\zeta_{0}$ and $\zeta_{1}$ be independent. For $i=0,1$, they
are to satisfy

$\mathbb{E}[\zeta_{i}]=0,\quad\mathbb{E}[|\zeta_{i}|^{2}]=1\quad\text{and}\quad\mathbb{E}[|\zeta_{i}|^{p}]\leq
C_{p}$ (2.1)

for all $p\in\mathbb{N}$ and some constants $C_{p}>0$, depending on $p$. Let
$\mathbf{W}\in\mathbb{C}^{N\times N}$ be a Hermitian random matrix
characterized by its entry distribution:

  1. 1.

The diagonal entries $\\{w_{ii}:\,i\in\llbracket N\rrbracket\\}$ and off-
diagonal entries $\\{(w_{ij},w_{ji}):\,i,j\in\llbracket N\rrbracket,\,i<j\\}$
are independent;

  2. 2.

$\\{w_{ii}:\,i\in\llbracket N\rrbracket\\}$ consists of independent copies of
$\frac{1}{\sqrt{N}}\zeta_{0}$,

  3. 3.

$\\{(w_{ij},w_{ji}):\,i,j\in\llbracket N\rrbracket,\,i<j\\}$ consists of
independent copies of $\frac{1}{\sqrt{N}}(\zeta_{1},\bar{\zeta}_{1})$.

For a fixed $l\in\mathbb{N}$ we define
$\mathbf{X}=(\mathbf{X}_{i})_{i\in\llbracket
l\rrbracket}\in(\mathbb{C}^{N\times N})^{l}$, a vector of random matrices,
where each $\mathbf{X}_{i}$, $i\in\llbracket l\rrbracket$, is an independent
copy of $\mathbf{W}$.

To present our results we first need to distinguish between shifted reducible
and non-reducible quadratic polynomials.

######  Definition 2.1 (Shifted reducible and non-reducible second degree
polynomial).

We call any non-commutative quadratic polynomial of the matrices
$\mathbf{X}=(\mathbf{X}_{i})_{i\in\llbracket l\rrbracket}$ which is of the
form

$q_{r}(\mathbf{X})=\alpha(v^{*}\mathbf{X}-\xi)(v^{*}\mathbf{X}-\xi)^{*}-\beta$
(2.2)

for some $\alpha,\beta,\xi\in\mathbb{R}$, $\alpha\neq 0$, $\xi\geq 0$ and
$v\in\mathbb{C}^{l}$ with $\|v\|=1$ a shifted reducible quadratic polynomial.
Any polynomial not of this form is called a non-reducible quadratic
polynomial.

######  Remark.

Shifted reducible quadratic polynomials are exactly the polynomials of the
form (1.1) with coefficients $A=\alpha vv^{*}$, $b=-\alpha\xi(v+\bar{v})$ and
$c=\alpha|\xi|^{2}-\beta$ for some $\alpha,\beta,\xi\in\mathbb{R}$ with
$\alpha\neq 0$, $\xi\geq 0$ and normalised $v\in\mathbb{C}^{l}$. Note that our
definition of a shifted reducible polynomial also allows for polynomials of
the form (2.2) with $\xi\in\mathbb{C}$ since
$(v^{*}\mathbf{X}-\xi)(v^{*}\mathbf{X}-\xi)^{*}=(e^{\mathrm{i}\varphi}v^{*}\mathbf{X}-e^{\mathrm{i}\varphi}\xi)(e^{\mathrm{i}\varphi}v^{*}\mathbf{X}-e^{\mathrm{i}\varphi}\xi)^{*}$
for all $\varphi\in\mathbb{R}$ and we used this invariance to restrict to
$\xi\in\mathbb{R}_{\geq 0}$ w.l.o.g..

Shifted reducible polynomials are those where the edge characteristics reduce
to understanding the singular value statistics of some (not necessarily
Hermitian) first order polynomial in $\mathbf{X}$, whereas non-reducible
polynomials are those where such a simplification is not possible. The main
focus of this work are non-reducible polynomials, but we also characterize the
limiting spectral measure of the shifted reducible polynomials.

For non-reducible polynomials, we prove the convergence of the norm of
$q(\mathbf{X})$ to a deterministic $\tau_{*}$ in the following sense:

######  Theorem 2.2 (Convergence of the matrix norm).

Let $q$ be a non-reducible quadratic polynomial of the form (1.1). There is a
deterministic $\tau_{*}>0$, only depending on the coefficients $A,b,c$ of $q$,
such that for all $\varepsilon,D>0$ the operator norm of $q(\mathbf{X})$
satisfies the estimate

$\mathbb{P}\left(|\|q(\mathbf{X})\|-\tau_{*}|\geq
N^{-\frac{2}{3}+\varepsilon}\right)\leq C_{\varepsilon,D}N^{-D}.$

######  Remark.

The deterministic value $\tau_{*}$ in Theorem 2.2 is the value of the norm as
predicted by the limiting spectral measure

$\rho:=\lim_{N\to\infty}\frac{1}{N}\sum_{\mu\in\operatorname{Spec}(q(\mathbf{X}))}\delta_{\mu},$
(2.3)

where $\delta_{\mu}$ denotes the Dirac measure at point $\mu$ and the sum runs
over all eigenvalues of $q(\mathbf{X})$ accounting also for multiplicity. That
is, we have $\tau_{*}=\max\\{|\tau_{+}|,|\tau_{-}|\\}$, where
$\operatorname{supp}(\rho)=[\tau_{-},\tau_{+}]$ (see Definition 2.8 below).
The points $\tau_{+}$ and $\tau_{-}$ can be obtained by solving an explicit
polynomial equation, for details see Lemma 3.5 below.

To obtain the main theorem we need several intermediate results. They
establish that the eigenvalue density of any non-reducible polynomial
$q(\mathbf{X})$ approximately shows a square root behaviour around its edge.
For reducible polynomials, we classify the different edge behaviours.

The central object of our interest is the Stieltjes transform of the limiting
spectral measure $\rho$, which we denote by $m$. The function $m$ is uniquely
defined by the following proposition.

######  Proposition 2.3 (Existence and uniqueness of the Stieltjes transform).

Let $A\in\mathbb{C}^{l\times l}$, $A=A^{*}$, $A\neq 0$, $b\in\mathbb{R}^{l}$
and $c\in\mathbb{R}$. There is a unique function $m:\mathbb{H}\to\mathbb{H}$
such that

  1. 1.

$m$ is complex analytic on all of $\mathbb{H}$;

  2. 2.

$\lim_{z\to\infty}zm(z)=-1$;

  3. 3.

For all $z\in\mathbb{H}$ the equation

$-m^{-1}=z+\gamma(m),$ (2.4)

with

$\gamma(m):=-\operatorname{Tr}A(I_{l}+Am)^{-1}+mb^{t}\left((I_{l}+2m\widehat{A})^{-2}(I_{l}+m\widehat{A})\right)b-c,$
(2.5)

is satisfied for $m=m(z)$. Here, the notation
$\widehat{A}=\frac{1}{2}(A+A^{t})$ was used.

The proof of Proposition 2.3 is deferred to Appendix A.3.

######  Definition 2.4 (Self-consistent density of states).

By Conditions 1 and 2 the function $m$ in Proposition 2.3 is the Stieltjes
transform of a unique probability measure $\rho$ on the real line, i.e.

$m(z)=\int_{\mathbb{R}}\frac{\rho(\mathrm{d}x)}{x-z}$ (2.6)

for all $z\in\mathbb{H}$. We call $\rho$ the self-consistent density of states
corresponding to $q(\mathbf{X})$.

######  Remark.

By the global law, [21, Proposition 2.17], the self-consistent density of
states is indeed the limiting spectral measure.

Note that since $A$ is Hermitian, the matrix
$\widehat{A}=\frac{1}{2}(A+A^{t})$ denotes the entrywise real part of $A$.
This should not be confused with the algebraic definition of the real part of
a matrix, $\operatorname{\mathrm{Re}}R=\frac{1}{2}(R+R^{*})$. Let
$I\subset\mathbb{R}$ be an interval and $E\in I$. Due to the Stieltjes
inversion formula, the limiting spectral measure and its Stieltjes transform
are related by the equation

$\rho(E)=\lim_{\eta\searrow
0}\frac{1}{\pi}\operatorname{\mathrm{Im}}m(E+\mathrm{i}\eta),$ (2.7)

whenever that limit exists for all $E\in I$ (see e.g. [27, Equation (1.4)]).
It was shown in [44, Theorem 1.1 (3)] that $\operatorname{supp}(\rho)$ is a
single compact interval on the real line. In particular, this means that
$\rho$ has no internal edges. We use the following definition.

######  Definition 2.5 (Edges of the limiting spectral measure).

Let $\tau_{+}$ denote the position of the right edge of $\rho$ and let
$\tau_{-}$ denote the position of the left edge of $\rho,$ i.e., we have

$\operatorname{supp}(\rho)=[\tau_{-},\tau_{+}]$ (2.8)

We also introduce the notion of a regular edge.

######  Definition 2.6.

Let $\tau_{0}\in\\{\tau_{-},\tau_{+}\\}$. The limiting spectral measure $\rho$
is said to have a regular edge at $\tau_{0}$ if
$\rho(\mathrm{d}E)=\rho(E)\mathrm{d}E$ has a Lebesgue density (also denoted by
$\rho$) in a neighborhood of $\tau_{0}$ and

$\lim_{\begin{subarray}{c}E\in\mathrm{supp}(\rho)\\\
E\to\tau_{0}\end{subarray}}\frac{\rho(E)}{\sqrt{|\tau_{0}-E|}}$ (2.9)

exists and does not equal 0.

In other words, regular edges are those that show a square root decay of the
density $\rho$. Our next two results concern the edge characterization of
reducible and non-reducible polynomials.

######  Proposition 2.7 (Edges of non-reducible polynomials).

Let $q$ be a non-reducible quadratic polynomial and $\rho$ its associated
limiting spectral measure defined in (2.3). Then the measure $\rho$ has
regular edges both at $\tau_{-}$ and at $\tau_{+}$.

Next, we consider shifted reducible polynomials of the form

$q_{r}(\mathbf{X})=(v^{*}\mathbf{X}-\xi)(v^{*}\mathbf{X}-\xi)^{*}$ (2.10)

for some $\xi\in\mathbb{R}_{\geq 0}$ and normalized $v\in\mathbb{C}^{l}$.

######  Remark.

Compared to the general case defined in (2.2) we restrict here to $\alpha=1$
and $\beta=0$. As these constants only constitute a scaling and a shift
respectively, the following result, Proposition 2.8, generalizes in a
straightforward manner to all $\alpha\neq 0$ and $\beta\in\mathbb{R}.$ For
$\alpha<0$ the roles of the left and the right edge are reversed.

We introduce the quantities $\sigma=\|\operatorname{\mathrm{Re}}v\|,$
$\mu=\langle\operatorname{\mathrm{Re}}v,\operatorname{\mathrm{Im}}v\rangle$
and for $\sigma\in(0,1)$ we define $s>0$ with

$s^{2}:=\begin{cases}\left((\sigma^{2}+a^{2}_{+}(1-\sigma^{2})+2a_{+}\mu)(\sigma^{2}+a_{+}\mu)\right)^{-1}\\\
\qquad+\left((\sigma^{2}+a^{2}_{-}(1-\sigma^{2})+2a_{-}\mu)(\sigma^{2}+a_{-}\mu)\right)^{-1}&\text{if
}\mu\neq 0,\\\ \sigma^{-4}&\text{if }\mu=0,\\\ \end{cases}$ (2.11)

where for $\mu\neq 0$ we also set the constant $a_{\pm}\in\mathbb{R}$ to

$a_{\pm}=\frac{1-2\sigma^{2}}{2\mu}\pm\sqrt{\left(\frac{1-2\sigma^{2}}{2\mu}\right)^{2}+1}.$
(2.12)

The edges of the shifted reducible polynomial are then characterized as
follows.

######  Proposition 2.8 (Edges of shifted reducible polynomials).

Let $q_{r}$ be a shifted reducible polynomial as in (2.10) for some
$\xi\in\mathbb{R}_{\geq 0}$ and normalized $v\in\mathbb{C}^{l}$. Then the
limiting spectral measure $\rho$ has a regular edge at
$\tau_{+}=\max\operatorname{supp}(\rho)$. Furthermore there is a $\kappa>0$
such that the behaviour of $\rho$ on $(\tau_{-},\tau_{-}+\kappa)$ with
$\tau_{-}=\min\operatorname{supp}(\rho)$ is given by

$\rho(E)\sim\begin{cases}(E-\tau_{-})^{-\frac{1}{2}}\quad\text{if }\xi=0\\\
(E-\tau_{-})^{-\frac{1}{2}}\quad\text{if }v\in\mathbb{R}^{l}\text{ and
}\xi<2\\\ (E-\tau_{-})^{-\frac{1}{4}}\quad\text{if }v\in\mathbb{R}^{l}\text{
and }\xi=2\\\ (E-\tau_{-})^{-\frac{1}{2}}\quad\text{if
}v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all
}\varphi\in(-\pi,\pi]\text{ and }s\xi<2\\\
(E-\tau_{-})^{-\frac{1}{3}}\quad\text{if }v\in\mathbb{C}^{l}\setminus
e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all }\varphi\in(-\pi,\pi]\text{
and }s\xi=2.\end{cases}$ (2.13)

In all other cases the left edge is also regular, i.e.
$\rho(E)\sim(E-\tau_{-})^{1/2}.$

######  Remark.

The reason behind this result is that the edge behaviour near the left edge
follows from the distribution of small singular values of
$v^{*}\mathbf{X}-\xi$. It follows that $\rho$ has singularities precisely if
$\xi$ is in the asymptotic spectrum of $v^{*}\mathbf{X}$, with stronger
singularities being observed if $\xi$ is in the bulk of the spectrum and
weaker ones if $\xi$ is on the edge of the spectrum.

Secondly we prove that $m\mathbf{I}_{N}$, where $\mathbf{I}_{N}$ is the
identity matrix on $\mathbb{C}^{N\times N}$, well approximates the resolvent
$\mathbf{g}=(q(\mathbf{X})-z\mathbf{I}_{N})^{-1}\in\mathbb{C}^{N\times N}$ of
$q(\mathbf{X})$ around any regular edge and away from the spectrum. More
precisely we prove uniform convergence of the quadratic form of $\mathbf{g}$
on the set

$\mathbb{D}_{\gamma}^{\kappa_{0}}:=\\{E+\mathrm{i}\eta\in\mathbb{H}:\,|E-\tau_{0}|\leq\kappa_{0},\,N^{-1+\gamma}\leq\eta\leq
1\\}$ (2.14)

for some $\kappa_{0}\sim 1$ and for all $\gamma>0$, as well as on the set

$\mathbb{G}_{\gamma}^{C,\eta_{0}}:=\\{E+\mathrm{i}\eta\in\mathbb{H}:\,C^{-1}\leq\operatorname{dist}(E,\operatorname{supp}(\rho))\leq
C,\,N^{-1+\gamma}\leq\eta\leq\eta_{0}\\}$ (2.15)

for all $C,\gamma>0$ and some $\eta_{0}$ depending on $C$.

######  Theorem 2.9 (Local law for regular edges of polynomials).

Let $q$ be a polynomial of the form (1.1) and let $\rho$ have a regular edge
at $\tau_{0}\in\\{\tau_{-},\tau_{+}\\}$. There is a $\kappa_{0}>0$, depending
only on the coefficients of $q$, such that for all $\varepsilon,\gamma,D>0$
and $z=E+\mathrm{i}\eta\in\mathbb{D}_{\gamma}^{\kappa_{0}}$ the isotropic
local law

$\mathbb{P}\left(|\langle\mathbf{x},(\mathbf{g}-m)\mathbf{y}\rangle|>\|\mathbf{x}\|\|\mathbf{y}\|N^{\varepsilon}\left(\frac{\operatorname{\mathrm{Im}}m}{\sqrt{N\eta}}+\frac{1}{N\eta}\right)\right)\lesssim_{\varepsilon,\gamma,D}N^{-D}$
(2.16)

holds for all deterministic $\mathbf{x},\mathbf{y}\in\mathbb{C}^{N}$.
Moreover, the averaged local law

$\mathbb{P}\left(\left|\frac{1}{N}\operatorname{Tr}(\mathbf{B}(\mathbf{g}-m))\right|>\|\mathbf{B}\|\frac{N^{\varepsilon}}{N\eta}\right)\lesssim_{\varepsilon,\gamma,D}N^{-D}$
(2.17)

holds for all deterministic $\mathbf{B}\in\mathbb{C}^{N\times N}$. If
additionally $E\notin\operatorname{supp}(\rho),$ an improved averaged local
law of the form

$\mathbb{P}\left(\left|\frac{1}{N}\operatorname{Tr}(\mathbf{B}(\mathbf{g}-m))\right|>\|\mathbf{B}\|N^{\varepsilon}\left(\frac{1}{N|z-\tau_{0}|}+\frac{1}{(N\eta)^{2}\sqrt{|z-\tau_{0}|}}\right)\right)\lesssim_{\varepsilon,\gamma,D}N^{-D}$
(2.18)

holds.

Away from the spectrum, we will also prove the following form of an averaged
local law

######  Proposition 2.10 (Local law away from the spectrum).

Let $q$ be a polynomial of the form (1.1). For all $C>0$ there is an
$\eta_{0}>0$, depending only on the coefficients of $q$, such that an averaged
local law of the form

$\mathbb{P}\left(\left|\frac{1}{N}\operatorname{Tr}(\mathbf{B}(\mathbf{g}-m))\right|>\|\mathbf{B}\|N^{\varepsilon}\left(\frac{1}{N}+\frac{1}{(N\eta)^{2}}\right)\right)\lesssim_{\varepsilon,\gamma,D,C}N^{-D}$
(2.19)

holds true for all deterministic $\mathbf{B}\in\mathbb{C}^{N\times N}$,
$\gamma,\varepsilon,D>0$ and $z\in\mathbb{G}_{\gamma}^{C,\eta_{0}}$.

The local laws, Theorem 2.9 and Proposition 2.10, are proved in Section 5.
They give us very precise control over the spectral properties of
$q(\mathbf{X})$ close to any regular edge as can be seen from the following
corollaries,

######  Corollary 2.11 (Edge eigenvector delocalization).

Let the assumptions of Theorem 2.9 be satisfied and $\mathbf{v}$ be a
normalized eigenvector of $q(\mathbf{X})$ with eigenvalue $\lambda$. Then
there is a $\kappa_{0}>0$, only depending on the coefficients of $q$, such
that if $|\lambda-\tau_{0}|\leq\kappa_{0}$ then we have for all
$\varepsilon,D>0$ that

$\sup_{\mathbf{x}\in\mathbb{C}^{N\times
N},\|\mathbf{x}\|=1}\mathbb{P}\left(|\langle\mathbf{x},\mathbf{v}\rangle|\geq
N^{-\frac{1}{2}+\varepsilon}\right)\lesssim_{D,\varepsilon}N^{-D}$ (2.20)

######  Corollary 2.12 (Eigenvalue rigidity).

Denote the classical index of the eigenvalue close to energy
$E\in\mathrm{supp}(\rho)$ by

$k(E):=\left\lceil
N\int_{-\infty}^{E}\rho(E^{\prime})\mathrm{d}E^{\prime}\right\rceil,$ (2.21)

with $\lceil\cdot\rceil$ being the ceiling function. Let $\rho$ have a regular
edge at $\tau_{0}$. All eigenvalues around $\tau_{0}$ are close to their
classical position in the following sense. There is a $\kappa_{0}>0$, only
depending on the coefficients of $q$, such that

$\mathbb{P}\left(\mathrm{sup}_{E}|\lambda_{k(E)}-E|\geq\min\left\\{\frac{N^{\varepsilon}}{N|E-\tau_{0}|},\frac{N^{\varepsilon}}{N^{\frac{2}{3}}}\right\\}\right)\lesssim_{D,\varepsilon}N^{-D}.$
(2.22)

holds for all $\varepsilon,D>0$ and $E\in\mathrm{supp}(\rho)$ with
$|E-\tau_{0}|\leq\kappa_{0}$.

###### Proof of Theorem 2.2.

As the norm of any Hermitian matrix $H$ with non-decreasing eigenvalues
$\lambda_{i}$, $i\in\llbracket n\rrbracket$, is given by
$\|H\|=\max\\{|\lambda_{1}|,|\lambda_{n}|\\}$, Theorem 2.2 follows as a
special case of Corollary 2.12 in conjunction with Proposition 2.7, which
states that all edges of non-reducible polynomials are regular. ∎

##  3 Properties of the spectral density

We first state two propositions which describe the behaviour of the Stieltjes
transform in the upper half-plane close to the edges of the spectrum. The
first proposition concerns non-reducible polynomials, whereas the second one
covers shifted reducible polynomials. Subsequently, we conclude Propositions
2.7 and 2.8 from Propositions 3.1 and 3.2 and state Corollary 3.3. It
summarizes important properties of the Stieltjes transform close to any
regular edge and is used to prove the local law, Theorem 2.9. Afterwards, we
prove Propositions 3.1 and 3.2 in the remainder of the section.

######  Proposition 3.1 (Stieltjes transform for non-reducible polynomials).

Let $q$ be a non-reducible quadratic polynomial. Then we have the following
behaviour of $m$ close to the edges.

  1. 1.

There are $m_{+}<0$, $c_{+}>0$ and $u>0$ such that for all $z\in\mathbb{H}$
with $|z-\tau_{+}|\leq u$ the Stieltjes transform of the limiting spectral
density $m=m(z)$ satisfies

$m-m_{+}=c_{+}\sqrt{z-\tau_{+}}+\mathcal{O}(|z-\tau_{+}|).$ (3.1)

  2. 2.

There are $m_{-}>0$, $c_{-}>0$ and $u>0$ such that for all $z\in\mathbb{H}$
with $|z-\tau_{+}|\leq u$ the Stieltjes transform of the limiting spectral
density $m=m(z)$ satisfies

$m-m_{-}=-c_{-}\sqrt{\tau_{-}-z}+\mathcal{O}(|\tau_{-}-z|).$ (3.2)

Here, $\sqrt{\cdot}$ denotes the square root function that maps the positive
real axis to itself and with a branch cut along the negative real axis.

######  Proposition 3.2 (Stieltjes transform for shifted reducible
polynomials).

Let $q$ be a shifted reducible polynomial of the form (2.10). Then we have the
following behaviour of $m$ close to the edges.

  1. 1.

There are $m_{+}<0$, $c_{+}>0$ and $u>0$ such that for all $z\in\mathbb{H}$
with $|z-\tau_{+}|\leq u$ the function $m=m(z)$ satisfies

$m-m_{+}=c_{+}\sqrt{z-\tau_{+}}+\mathcal{O}(|z-\tau_{+}|).$ (3.3)

  2. 2.

There are $c_{-}>0$ and $u>0$ such that for all $z\in\mathbb{H}$ with
$|z-\tau_{-}|\leq u$ the function $m=m(z)$ satisfies

$m=\begin{cases}c_{-}(\tau_{-}-z)^{-\frac{1}{2}}+\mathcal{O}(1)&\text{if
}\xi=0\\\ c_{-}(\tau_{-}-z)^{-\frac{1}{2}}+\mathcal{O}(1)&\text{if
}v\in\mathbb{R}^{l}\text{ and }\xi<2\\\
c_{-}(\tau_{-}-z)^{-\frac{1}{4}}+\mathcal{O}(1)&\text{if
}v\in\mathbb{R}^{l}\text{ and }\xi=2\\\
c_{-}(\tau_{-}-z)^{-\frac{1}{2}}+\mathcal{O}(1)&\text{if
}v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all
}\varphi\in(-\pi,\pi]\text{ and }s\xi<2\\\
c_{-}(\tau_{-}-z)^{-\frac{1}{3}}+\mathcal{O}(1)&\text{if
}v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all
}\varphi\in(-\pi,\pi]\text{ and }s\xi=2,\end{cases}$ (3.4)

where $s$ was defined in (2.11). If none of the above conditions is satisfied,
there is additionally an $m_{-}>0$ such that for all $z\in\mathbb{H}$ with
$|z-\tau_{-}|\leq u$ we have

$m-m_{-}=-c_{-}\sqrt{\tau_{-}-z}+\mathcal{O}(|\tau_{-}-z|).$ (3.5)

The function $\zeta\mapsto\zeta^{p}$ is chosen such that the positive real
axis is mapped to itself and with a branch cut along the negative real axis
for all $p\in\mathbb{R}\setminus\mathbb{Z}.$

######  Corollary 3.3.

Let $q(\mathbf{X})$ have a regular edge at $\tau_{0}$ and $m_{0}=m(\tau_{0})$.
There is a $u>0$ such that the function $m=m(z)$ satisfies

$|z-\tau_{0}|\sim|m-m_{0}|^{2}$ (3.6)

and

$\operatorname{\mathrm{Im}}m\sim\begin{cases}\sqrt{|E-\tau_{0}|+\eta}&\text{if
}E\in\operatorname{supp}(\rho)\\\
\frac{\eta}{\sqrt{|E-\tau_{0}|+\eta}}&\text{if
}E\notin\operatorname{supp}(\rho)\end{cases}$ (3.7)

for all $z=E+\mathrm{i}\eta\in\mathbb{H}$ with $|z-\tau_{0}|\leq u$.

###### Proof of Proposition 2.7 and Proposition 2.8.

By (2.7) we have $\rho(E)=\lim_{\eta\searrow 0}\pi^{-1}m(E+\mathrm{i}\eta)$ on
any interval $I\subset\mathbb{R}$ on which the limit exists for all $E\in I$.
For the cases in Proposition 3.1 and Proposition 3.2 where $m-m_{0}$ shows a
square root behaviour around an edge $\tau_{0}$, we take the limit on
$(\tau_{0}-u,\tau_{0}+u)$ to prove that the edge is regular. For the cases in
Proposition 3.2 where $m$ diverges at $\tau_{-}$, we take the limit on
$(\tau_{-}-u,\tau_{-}+u)\setminus\\{\tau_{-}\\}$ to obtain the respective
asymptotic behaviour. ∎

Let $\mu_{i}\in\mathbb{R}$ denote the eigenvalues of $A$,
$\widehat{\mu_{i}}\in\mathbb{R}$ the eigenvalues of $\widehat{A}$ and let
$w_{i}\in\mathbb{R}^{l}$ be an orthonormal set of eigenvectors of
$\widehat{A}$ corresponding to the $\widehat{\mu_{i}}$. By (2.5) the function
$\gamma$ is defined in terms of these quantities as

$\gamma(m)=-\sum_{i=1}^{l}\frac{\mu_{i}}{1+m\mu_{i}}+\sum_{i=1}^{l}\frac{|\langle
w_{i},b\rangle|^{2}(1+m\widehat{\mu}_{i})}{(1+2m\widehat{\mu}_{i})^{2}}m-c.$
(3.8)

From now on we consider $\gamma$ to be defined on its maximal domain,
$\mathbb{C}\setminus\mathscr{S}(\gamma)$, where

$\mathscr{S}(\gamma):=\\{-\mu_{i}^{-1}\in\mathbb{R}:\,\mu_{i}\neq
0\\}\cup\\{-(2\widehat{\mu}_{i})^{-1}\in\mathbb{R}:\,\widehat{\mu}_{i}\neq
0\text{ and }\langle w_{i},b\rangle\neq 0\\}$ (3.9)

denotes the poles of $\gamma$. We also require the function $h$, which we
define as follows.

######  Definition 3.4.

Let $h:\mathbb{C}\setminus\mathscr{S}(h)$ be given by

$h(m):=\frac{1}{m^{2}}-\gamma^{\prime}(m)=\frac{1}{m^{2}}-\sum_{i=1}^{l}\frac{\mu_{i}^{2}}{(1+m\mu_{i})^{2}}-\sum_{i=1}^{l}\frac{|\langle
w_{i},b\rangle|^{2}}{(1+2m\widehat{\mu}_{i})^{3}}.$ (3.10)

Here, $\mathscr{S}(h):=\mathscr{S}(\gamma)\cup\\{0\\}$ denotes the set of
poles of $h$ and $\gamma^{\prime}(m)$ is the derivative of $\gamma$ with
respect to $m$.

Proposition 2.3 asserts that $m$ is a Stieltjes transform of the measure
$\rho,$ which has real and compact support. Thus $m$ is analytic and has
positive imaginary part on $\mathbb{H}$. We take the derivative of (2.4) with
respect to $z$ on $\mathbb{H}$ to find

$m^{\prime}=\frac{1}{{h}(m)}.$ (3.11)

As $m$ is a Stieltjes transform of $\rho$, its analyticity extends to
$\mathbb{C}\setminus\operatorname{supp}(\rho)$ and the above equation holds on
$\overline{\mathbb{H}}\setminus\operatorname{supp}(\rho),$ where
$\overline{\mathbb{H}}=\mathbb{H}\cup\mathbb{R}$. Next, we define

$\displaystyle m_{-}^{*}$
$\displaystyle=\begin{cases}\min((0,\infty)\cap\mathscr{S}(h))&\text{if
}(0,\infty)\cap\mathscr{S}(h)\neq\emptyset\\\
\infty&\text{otherwise},\end{cases}$ (3.12) $\displaystyle m_{+}^{*}$
$\displaystyle=\begin{cases}\max((-\infty,0)\cap\mathscr{S}(h))&\text{if
}(-\infty,0)\cap\mathscr{S}(h)\neq\emptyset\\\
-\infty&\text{otherwise.}\end{cases}$ (3.13)

In other words, $(m_{+}^{*},0)$ and $(0,m_{-}^{*})$ are the maximal intervals
to the left and the right of the origin, where $h$ is continuous. The
following lemmata describe the existence and characterization of roots of $h$
both for the shifted reducible and the non-reducible case on $(m_{+}^{*},0)$
and $(0,m_{-}^{*})$.

######  Lemma 3.5.

Let $q$ be a non-reducible quadratic polynomial. Then we have the following.

  1. 1.

The function $h$ has a unique root $m_{-}$ in $(0,m_{-}^{*})$ and $h$ has a
unique root $m_{+}$ in $(m_{+}^{*},0)$. Both of them are of first order. More
precisely, they satisfy

$\pm h^{\prime}(m_{\pm})>0.$ (3.14)

  2. 2.

The positions of the edges $\tau_{\pm}$, defined in Definition 2.8, are given
in terms of $m_{\pm}$ by

$\tau_{\pm}=-m_{\pm}^{-1}-\gamma(m_{\pm}).$ (3.15)

######  Lemma 3.6.

Let $h$ be the function introduced in Definition 3.4 for a reducible
polynomial $q=q_{r}$ of the form (2.10) with normalized $v\in\mathbb{C}^{l}$
and $\xi\geq 0$. Then the following holds true.

  1. 1.

The function $h$ has a unique root $m_{+}$ in $(m_{+}^{*},0)$, which is of
first order and satisfies

$h^{\prime}(m_{+})>0.$ (3.16)

  2. 2.

The function $h$ is continuous on $(0,\infty)$. It has no root in $(0,\infty)$
if and only if one of the following hold true:

    1. (a)

$\xi=0$;

    2. (b)

or $v\in\mathbb{R}^{l}$ and $\xi\leq 2$;

    3. (c)

or $v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}$ for all
$\varphi\in(-\pi,\pi]$ and $s\xi\leq 2$.

If none of the above conditions are satisfied, $h$ does have a unique root
$m_{-}\in(0,\infty)$, which is of first order and satisfies

$h^{\prime}(m_{-})<0.$ (3.17)

If $h$ does not have any roots in $(0,\infty)$, the following asymptotic
behaviour is observed for $m\to\infty$:

$h(m)\sim\begin{cases}m^{-3}\quad\text{if }\xi=0\\\ m^{-3}\quad\text{if
}v\in\mathbb{R}^{l}\text{ and }\xi<2\\\ m^{-5}\quad\text{if
}v\in\mathbb{R}^{l}\text{ and }\xi=2\\\ m^{-3}\quad\text{if
}v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all
}\varphi\in(-\pi,\pi]\text{ and }s\xi<2\\\ m^{-4}\quad\text{if
}v\in\mathbb{C}^{l}\setminus e^{\mathrm{i}\varphi}\mathbb{R}^{l}\text{ for all
}\varphi\in(-\pi,\pi]\text{ and }s\xi=2.\end{cases}$ (3.18)

Here $s$ was defined in (2.11).

  3. 3.

The positions of the edges $\tau_{\pm}$, defined in Definition 2.8, are given
by

$\tau_{+}=-m_{+}^{-1}-\gamma(m_{+})$ (3.19)

and

$\tau_{-}=\begin{cases}-m_{-}^{-1}-\gamma(m_{-})&\text{if }m_{-}>0\text{ with
}h(m_{-})=0\text{ exists;}\\\ 0&\text{otherwise.}\end{cases}$ (3.20)

The proof of the lemmata is deferred until after the proof of Propositions 3.1
and 3.2.

###### Proof of Propositions 3.1 and 3.2.

In both the non-reducible and the shifted reducible cases the function $h$ has
a unique root $m_{+}$ in $(m_{+}^{*},0)$. For all $m\in(m_{+},0)$ the
derivative of the inverse $z=z(m)$ of $m=m(z)$ with respect to $m$ is given by

$\frac{\mathrm{d}}{\mathrm{d}m}z=h(m).$ (3.21)

Then, by Lemma 3.5 and Lemma 3.6, there is an $r\sim 1$ and a $c_{+}>0$, only
depending on the coefficients of $q$, such that for all $m\in(m_{+},m_{+}+r)$
we have

$\frac{\mathrm{d}}{\mathrm{d}m}z=h(m)=2c_{+}^{2}(m-m_{+})(1+\mathcal{O}(m-m_{+}))$
(3.22)

and thus

$z-\tau_{+}=c_{+}^{2}(m-m_{+})^{2}(1+\mathcal{O}(m-m_{+})).$ (3.23)

Taking the square root of the above equation, we obtain

$m-m_{+}=c_{+}\sqrt{z-\tau_{+}}+\mathcal{O}(|z-\tau_{+}|)$ (3.24)

for all $z\in(\tau_{+},\tau_{+}+u)$ for some $u>0.$ As $m$ is holomorphic on
$\mathbb{H}\cup(\tau_{+},\infty)$ and has positive imaginary part on
$\mathbb{H}$, the relation (3.24) also holds for all $z\in\mathbb{H}$ with
$|z-\tau_{+}|\leq u$ for some $u>0$.

In all cases, where $h$ has a root $m_{-}$ in $(0,m_{-}^{*})$, we find by the
same argument that

$m-m_{-}=-c_{-}\sqrt{\tau_{-}-z}+\mathcal{O}(|\tau_{-}-z|)$ (3.25)

for all $z\in\mathbb{H}$ with $|z-\tau_{+}|\leq u$ for some $u,c_{-}>0.$

Finally, in the cases where $h$ does not have a root in $(0,\infty)$, it is
continuous on the whole interval and we have that for all $m\in(0,\infty)$ the
derivative of $z$ with respect to $m$ is given by

$\frac{\mathrm{d}}{\mathrm{d}m}z=h(m).$ (3.26)

Then, by Lemma 3.6, there is an $R\sim 1$ and a $c_{0}>0$, only depending on
the coefficients of $q$, such that for all $m\in(R,\infty)$ we have

$\frac{\mathrm{d}}{\mathrm{d}m}z=h(m)=c_{0}m^{-p}(1+\mathcal{O}(m^{-1}))\quad\text{and
thus}\quad z=-\frac{c_{0}}{p-1}m^{-(p-1)}(1+\mathcal{O}(m^{-1})),$ (3.27)

with $p\in\\{3,4,5\\}.$ Inverting the relation, we obtain

$m=c_{-}(-z)^{-\frac{1}{p-1}}+\mathcal{O}(1)$ (3.28)

for all $z\in(-u,0)$ and some $u,c_{-}>0.$ Again, as $m$ is holomorphic on
$\mathbb{H}\cup(-\infty,0)$ and has positive imaginary part on $\mathbb{H}$,
the relation (3.28) also holds for all $z\in\mathbb{H}$ with $|z|\leq u$ for
some $u>0$.

∎

###### Proof of Lemma 3.5 and Lemma 3.6.

We prove both lemmata simultaneously and remark, where it becomes necessary to
distinguish the different polynomials $q.$ We first investigate the existence
and order of roots of $h$. Let $h_{q}$ denote the rational function (3.10)
with the dependence on $q$ being made explicit. We have $h_{q}(-m)=h_{-q}(m)$.
Therefore any root of $h_{q}$ on $(0,\infty)$ corresponds to a root of
$h_{-q}$ on $(-\infty,0).$ For simplicity, the proofs are thus only formulated
for $m\in(-\infty,0)$. To obtain the corresponding result for $m\in(0,\infty)$
for the function $h_{q}$, we consider the function $h_{-q}$ on $(-\infty,0)$
instead.

We now prove for all polynomials $q$ as in (1.1) that $h^{\prime}(m)>0$ for
all $m\in(m_{+}^{*},0)$ with $h(m)\geq 0$ and thus $h$ can only have first-
order roots on the interval.

Note that on $(m_{+}^{*},0)$ the inequality $h\geq 0$ is equivalent to

$\sum_{i=1}^{l}\frac{(m\mu_{i})^{2}}{(1+m\mu_{i})^{2}}+\sum_{\begin{subarray}{c}i=1,\\\
\langle w_{i},b\rangle\neq 0\end{subarray}}^{l}(m|\langle
w_{i},b\rangle|)^{2}\frac{1}{(1+2m\widehat{\mu}_{i})^{3}}\leq 1$ (3.29)

and that on $(m_{+}^{*},0)$ the inequality $h^{\prime}>0$ is equivalent to

$\sum_{i=1}^{l}\frac{(m\mu_{i})^{3}}{(1+m\mu_{i})^{3}}+\sum_{\begin{subarray}{c}i=1,\\\
\langle w_{i},b\rangle\neq 0\end{subarray}}^{l}(m|\langle
w_{i},b\rangle|)^{2}\frac{3(m\widehat{\mu}_{i})}{(1+2m\widehat{\mu}_{i})^{4}}<1.$
(3.30)

For $m\in(m_{+}^{*},0)$ we have also that $m\mu_{i}>-1$ for all
$i\in\llbracket l\rrbracket$ and $m\widehat{\mu}_{i}>-\frac{1}{2}$ for all
$i\in\llbracket l\rrbracket$ such that $\langle w_{i},b\rangle\neq 0$ by the
definition of $m_{+}^{*}$ in (3.13). Furthermore, Lemma A.1 ensures that
$\max_{i\in\llbracket l\rrbracket}\\{m\mu_{i}\\}\geq\max_{i\in\llbracket
l\rrbracket}\\{m\widehat{\mu}_{i}\\}$. Thus the desired implication, $h\geq 0$
implies $h^{\prime}>0$, follows directly from the following lemma by
identifying (3.31) and (3.32) right below with (3.29) and (3.30) respectively.

######  Lemma 3.7.

Let $n,k\in\mathbb{N}$ with $n\geq k$. Let $y_{i}$, $i\in\llbracket
n\rrbracket$ and let $\widehat{y}_{j}$, $j\in\llbracket k\rrbracket$ be
collections of real numbers, both of them sorted in non-increasing order,
$y_{1}\geq y_{2}\geq\ldots\geq y_{n}$ and
$\widehat{y}_{1}\geq\widehat{y}_{2}\geq\ldots\geq\widehat{y}_{k}$. Suppose
they satisfy

  1. 1.

$y_{1}\geq\widehat{y}_{1}$;

  2. 2.

$y_{n}>-1$ and $\widehat{y}_{k}>-\frac{1}{2}$.

Furthermore let $c_{j}>0$, $j\in\llbracket k\rrbracket$. Then the inequality

$\sum_{i=1}^{n}\frac{y_{i}^{2}}{(y_{i}+1)^{2}}+\sum_{j=1}^{k}c_{j}^{2}\frac{1}{(2\widehat{y}_{j}+1)^{3}}\leq
1$ (3.31)

implies

$\sum_{i=1}^{n}\frac{y_{i}^{3}}{(y_{i}+1)^{3}}+3\sum_{j=1}^{k}c_{j}^{2}\frac{\widehat{y}_{j}}{(2\widehat{y}_{j}+1)^{4}}\leq
1-\nu$ (3.32)

for some $\nu>0$, depending only on $y_{1}.$

The proof of the lemma is deferred to the end of the section.

To characterize the existence of roots of $h$ we introduce the following
notions. If $A$ is a rank one matrix with real entries, then $\widehat{A}=A$
and $\widehat{A}$ is a rank one matrix as well. We will denote its non-zero
eigenvalue by $\mu$ and the corresponding normalized eigenvector by $w$. If,
on the other hand, $A$ is a rank one matrix with $A\in\mathbb{C}^{l\times
l}\setminus\mathbb{R}^{l\times l}$, then $\widehat{A}$ is a rank two matrix by
Lemma A.1 in the appendix. We denote its non-zero eigenvalues by $\mu_{\pm}$
and the corresponding normalized eigenvectors by $w_{\pm}.$

######  Lemma 3.8.

Let $m_{+}^{*}$ be defined as in (3.13). The function $h$ has no root in
$(m_{+}^{*},0)$ if and only if $A$ is a negative semi-definite rank one
matrix, $b\in\operatorname{Image}(\widehat{A})$ and either

  1. 1.

$A\in\mathbb{R}^{l\times l}$ and $\|b\|\leq 4\|A\|$;

  2. 2.

or $A\in\mathbb{C}^{l\times l}\setminus\mathbb{R}^{l\times l}$ and

$\frac{|\langle b,w_{+}\rangle|^{2}}{r_{+}^{3}}+\frac{|\langle
b,w_{-}\rangle|^{2}}{r_{-}^{3}}\leq(4\|A\|)^{2},$ (3.33)

with $r_{\pm}:=-\|A\|^{-1}\mu_{\pm}.$

If $h$ has no root in $(m_{+}^{*},0)$, then we have $m_{+}^{*}=-\infty$ and
the asymptotic behaviour of $h$ for $m\to-\infty$ is given by

$h(m)\sim\begin{cases}\frac{1}{(-m)^{3}}&A\in\mathbb{R}^{l\times l}\text{and
}\|b\|<4\|A\|\\\ \frac{1}{(-m)^{5}}&A\in\mathbb{R}^{l\times l}\text{and
}\|b\|=4\|A\|\\\ \frac{1}{(-m)^{3}}&A\in\mathbb{C}^{l\times
l}\setminus\mathbb{R}^{l\times l}\text{ and }\frac{|\langle
b,w_{+}\rangle|^{2}}{|r_{+}|^{3}}+\frac{|\langle
b,w_{-}\rangle|^{2}}{|r_{-}|^{3}}<(4\|A\|)^{2}\\\
\frac{1}{(-m)^{4}}&A\in\mathbb{C}^{l\times l}\setminus\mathbb{R}^{l\times
l}\text{ and }\frac{|\langle b,w_{+}\rangle|^{2}}{|r_{+}|^{3}}+\frac{|\langle
b,w_{-}\rangle|^{2}}{|r_{-}|^{3}}=(4\|A\|)^{2}.\end{cases}$ (3.34)

The proof of the lemma is also deferred to the end of the section.

Note that $h$ can have at most one root on $(m_{+}^{*},0)$. To see this,
recall that $h(m)\geq 0$ implies $h^{\prime}(m)>0$. Therefore, if $m_{+}$ is a
root of $h$ on $(m_{+}^{*},0)$, we know that $h$ is strictly monotonously
increasing on $(m_{+},0)$. Thus, any root of $h$ on $(m_{+}^{*},0)$ is also
the largest one on the interval; therefore it must be the unique root.

Now let $q$ be a non-reducible polynomial. If $\operatorname{rank}A\geq 2$,
then $h$ has a root in $(-\infty,0)$ by Lemma 3.34. If
$\operatorname{rank}A=1$, we can express $A$ as $A=\alpha ww^{*}$ for some
$\alpha\in\mathbb{R}\setminus\\{0\\}$ and normalized $w\in\mathbb{C}^{l}$. The
vector $b$ cannot be of the form
$b=\alpha_{1}\operatorname{\mathrm{Re}}w+\alpha_{2}\operatorname{\mathrm{Im}}w$
for any $\alpha_{1},\alpha_{2}\in\mathbb{R}$ as $q$ would be reducible
otherwise. Thus $b\notin\operatorname{Image}A$ and again $h$ has a root in
$(-\infty,0)$ by Lemma 3.34, completing the proof of Statement 1 of Lemma 3.5.

Next, consider a shifted reducible polynomial $q=q_{r}$ for $q_{r}$ as in
(2.10). If we write $q$ in terms of (1.1) we have $A=vv^{*}\geq 0$. Applying
Lemma 3.5 we obtain that $h$ has a root in $(-\infty,0)$ and thereby prove
Statement 1 of Lemma 3.6.

Finally, consider $q=-q_{r}$ for $q_{r}$ as in (2.10). Then the coefficients
$A$ and $b$ associated with $q$ are given by $A=-vv^{*}$ and
$b=2\xi\operatorname{\mathrm{Re}}v.$ The matrix $A$ is a negative semi-
definite rank one matrix. For
$\mu=\langle\operatorname{\mathrm{Re}}v,\operatorname{\mathrm{Im}}v\rangle\neq
0$ the non-zero eigenvalues of
$\widehat{A}=-\frac{1}{2}(v^{*}v+vv^{*})=-(\operatorname{\mathrm{Re}}v)^{*}\operatorname{\mathrm{Re}}v-(\operatorname{\mathrm{Im}}v)^{*}\operatorname{\mathrm{Im}}v$
are given by

$\mu_{\pm}=\sigma^{2}+\mu a_{\pm},$ (3.35)

where $\sigma=\|\operatorname{\mathrm{Re}}v\|$ and $a_{\pm}$ was defined in
(2.12). The corresponding normalized eigenvectors are

$w_{\pm}=\frac{1}{\sigma^{2}+a_{\pm}(1-\sigma^{2})+2a_{\pm}\mu}\left(\operatorname{\mathrm{Re}}v+a_{\pm}\operatorname{\mathrm{Im}}v\right).$
(3.36)

For $\mu=0$ the eigenvector-eigenvalue pairs are given by

$(w_{+},\mu_{+})=\left(\frac{\operatorname{\mathrm{Im}}v}{1-\sigma^{2}},1-\sigma^{2}\right),\quad(w_{-},\mu_{-})=\left(\frac{\operatorname{\mathrm{Re}}v}{\sigma^{2}},\sigma^{2}\right).$
(3.37)

Now recall that $h_{q}(m)=h_{-q}(-m)=h_{q_{r}}(-m)$ and Statement 2 of Lemma
3.6 now follows from applying Lemma 3.34 to $h_{q}$.

It remains to prove the relations between $m_{\pm}$ and $\tau_{\pm}$ stated in
(3.15), (3.19) as well as (3.20). Let $q$ be either a non-reducible polynomial
or a reducible polynomial of the form (2.10). For
$\widetilde{m}\in(m_{+}^{*},0)$ define

$z(\widetilde{m}):=-\frac{1}{\widetilde{m}}-\gamma(\widetilde{m}).$ (3.38)

The function $z$ is real analytic on $(m_{+}^{*},0)$ with derivative
$z^{\prime}(\widetilde{m})=h(\widetilde{m})$. By Lemma 3.5, Part 1 and Lemma
3.6, Part 1, the function $h$ has a unique root $m_{+}$ in $(m_{+}^{*},0)$ and
$h(\widetilde{m})>0$ for all $\widetilde{m}\in(m_{+},0)$. Therefore,the
function $z$ is invertible on $(m_{+},0)$ and its inverse function
$\widetilde{m}:(z_{+},\infty)\to(m_{+},0)$, with $z_{+}:=z(m_{+})$, is also
real analytic and monotonically increasing. Since
$z^{\prime}(m_{+})=h(m_{+})=0$, the analyticity of $\widetilde{m}$ cannot be
extended onto any neighbourhood of $z_{+}$.

By (3.38), the function $\widetilde{m}$ satisfies (2.4) and has the asymptotic
behaviour $\widetilde{m}=-z^{-1}+\mathcal{O}(z^{2})$ for $z\to\infty$. At the
same time, $m$, the function uniquely defined by Proposition 2.3, is a
Stieltjes transform of a probability measure with support
$[\tau_{-},\tau_{+}]$. As such, it can be analytically extended to
$\mathbb{C}\setminus[\tau_{-},\tau_{+}]$ but not to a neighbourhood of
$\tau_{+}$ and the extension is real-valued on
$\mathbb{R}\setminus[\tau_{-},\tau_{+}]$. As an extension it also satisfies
(2.4) on $\mathbb{C}\setminus[\tau_{-},\tau_{+}]$ and has the asymptotic
behaviour $m=-z^{-1}+\mathcal{O}(z^{2})$. In particular, the restriction of
the extension to $\mathbb{R}\setminus[\tau_{-},\tau_{+}]$, called
$m_{\mathbb{R}}$, is a real analytic function that also satisfies (2.4), has
the asymptotic behaviour $m_{\mathbb{R}}=-z^{-1}+\mathcal{O}(z^{2})$ and
cannot be analytically extended to a neighbourhood of $\tau_{+}$. As
satisfying (2.4) and having the asymptotic behaviour
$m=-z^{-1}+\mathcal{O}(z^{2})$ uniquely define an analytic function on a
neighbourhood of $\infty$, we have $\widetilde{m}=m_{\mathbb{R}}$ on
$(C,\infty)$ for some $C>0$. If we assume that $\tau_{+}>z_{+},$ then
$\widetilde{m}$ would be an analytical continuation of $m_{\mathbb{R}}$ to
some neighbourhood of $\tau_{+}$, which is a contradiction and vice versa.
Therefore we must have $z_{+}=\tau_{+}.$ In particular, we have

$\tau_{+}=z(m_{+})=-\frac{1}{m_{+}}-\gamma(m_{+}).$ (3.39)

By an analogous argument to the left of the spectrum we find

$\tau_{-}=z(m_{-})=-\frac{1}{m_{-}}-\gamma(m_{-})$ (3.40)

if $m_{-}$ exists. Now, let $q=q_{r}$ be a shifted reducible polynomial of the
form (2.10) such that $h$ has no root on $(0,\infty)$. Then the function
$z(\widetilde{m})$ defined on $(0,\infty)$ by (3.38) is a monotonously
increasing analytic function on the entirety of its domain. It is therefore
invertible and its inverse function, $\widetilde{m}$, is analytic on
$(-\infty,z_{\infty})$ with $z_{\infty}:=\lim_{x\to\infty}z(x)$ but cannot be
analytically extended to a neighbourhood of $z_{\infty}$. Along the lines of
the above argument, $m_{\mathbb{R}}$ would be an analytical continuation of
$\widetilde{m}$ in a neighbourhood of $z_{\infty}$ if $\tau_{-}>z_{\infty}$
and $\widetilde{m}$ would be an analytical continuation of $m_{\mathbb{R}}$ in
a neighbourhood of $\tau_{-}$ if $z_{\infty}>\tau_{-}$. Thus we have
$\tau_{-}=z_{\infty}$ and in particular

$\tau_{-}=\lim_{x\to\infty}-\frac{1}{x}-\gamma(x).$ (3.41)

For reducible $q_{r}$ as in (2.10), we have $A=vv^{*}$, $b=-\xi(v+\bar{v})$
and $c=\xi^{2}$. Taking the limit (3.41) with these parameters we obtain
$\tau_{-}=0$. ∎

###### Proof of Lemma 3.7.

Let $y\in(-1,\infty)$ and $\widehat{y}\in(-\frac{1}{2},\infty)$. We define

$h_{1}(y)=\frac{y}{y+1}\quad\mathrm{and}\quad
h_{2}(\widehat{y})=\frac{1}{2\widehat{y}+1}.$ (3.42)

Using this notation (3.31) can be expressed as

$g(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}):=\sum_{i=1}^{n}h_{1}(y_{i})^{2}+\sum_{j=1}^{k}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})\leq
1$ (3.43)

and (3.32) as

$f(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}):=\sum_{i=1}^{n}h_{1}(y_{i})^{3}+3\sum_{j=1}^{k}c_{j}^{2}\widehat{y}_{j}h^{4}_{2}(\widehat{y}_{j})\leq
1-\nu$ (3.44)

To prove the lemma it is then sufficient to show that $g\leq 1$ implies $f\leq
1-\nu$.

First assume $g=0$. Since all summands of $g$ are non-negative, it follows
that $y_{i}=0$ for all $i\in\llbracket n\rrbracket$ and $c_{j}=0$ for all
$j\in\llbracket k\rrbracket$ and so $f$ vanishes as well. Thus we assume
$g\neq 0$ from now on and we will prove that $g\leq 1$ implies $f<g$.

We will prove the cases $\widehat{y}_{1}\leq\frac{1}{2}$,
$\frac{1}{2}<\widehat{y}_{1}<1$ and $\widehat{y}_{1}\geq 1$ separately and
start with $\widehat{y}_{1}\leq\frac{1}{2}$. Then
$\widehat{y}_{j}\leq\frac{1}{2}$ for all $j$. Note that
$\widehat{y}_{j}h_{2}^{4}(\widehat{y}_{j})\leq\widehat{y}_{1}h_{2}(\widehat{y}_{1})h_{2}^{3}(\widehat{y}_{j})$
and $h_{1}^{3}(y_{i})\leq h_{1}(y_{1})h_{1}^{2}(y_{i})$ hold for all $i$ and
$j$ since $\widehat{y}\mapsto\widehat{y}h_{2}(\widehat{y})$ and $h_{1}$ are
monotonously increasing and the $\widehat{y}_{j}$ and $y_{i}$ are sorted in
non-increasing order. Thus

$\begin{split}f(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})&\leq
h_{1}(y_{1})\sum_{i=1}^{n}h_{1}^{2}(y_{i})+3\widehat{y}_{1}h_{2}(\widehat{y}_{1})\sum_{j=1}^{k}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})\\\
&\leq
h_{1}(y_{1})\sum_{i=1}^{n}h_{1}^{2}(y_{i})+\frac{3}{4}\sum_{j=1}^{k}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})\leq\max\left\\{\frac{y_{1}}{1+y_{1}},\frac{3}{4}\right\\}g(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}).\end{split}$
(3.45)

The second inequality holds since
$3\widehat{y}_{1}h_{2}(\widehat{y}_{1})\leq\frac{3}{4}$ due to
$\widehat{y}_{1}\leq\frac{1}{2}$ and the third one because all summands in
both sums are non-negative. The relation $f\leq 1-\nu$ for $g\leq 1$ then
follows in this regime.

For $\frac{1}{2}<\widehat{y}_{1}<1$ we estimate every summand in $f$ but the
$y_{1}$ term by the corresponding term in $g$ and we find

$f(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})\leq
g(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})-h_{1}(y_{1})^{2}+h_{1}(y_{1})^{3}.$
(3.46)

This upper bound is valid since $\widehat{y}_{j}\leq\widehat{y}_{1}<1$ for all
$j.$ Thus for $g\leq 1$ we find

$f(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})\leq
g(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})-\left(\frac{y_{1}}{y_{1}+1}\right)^{2}\frac{1}{1+y_{1}}<1-\frac{2}{27},$
(3.47)

where in the last step we used $y_{1}\geq\widehat{y}_{1}>\frac{1}{2}.$

Now let $\widehat{y}_{1}\geq 1$. Let $k_{0}$ be the largest integer such that
$\widehat{y}_{k_{0}}\geq 1.$ Then

$f_{0}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}):=\sum_{i=2}^{n}h_{1}^{3}(y_{i})+\sum_{j=k_{0}+1}^{k}3c_{j}^{2}\widehat{y}_{j}h^{4}_{2}(\widehat{y}_{j})\leq\sum_{i=2}^{n}h_{1}^{2}(y_{i})+\sum_{j=k_{0}+1}^{k}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})=:g_{0}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})$
(3.48)

is concluded by again estimating each term individually (either of the sums
might be empty). Note that $g_{0}\geq 0$. Therefore we only need to prove that

$g_{1}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}):=(g-g_{0})(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})=h_{1}^{2}(y_{1})+\sum_{j=1}^{k_{0}}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})\leq
1$ (3.49)

implies

$f_{1}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c}):=(f-f_{0})(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})=h_{1}^{3}(y_{1})+\sum_{j=1}^{k_{0}}3c_{j}^{2}\widehat{y}_{j}h^{4}_{2}(\widehat{y}_{j})\leq
g_{1}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})-\nu$ (3.50)

for some $\nu>0$, only depending on $y_{1}$, to conclude the proof of the
lemma.

For $\widehat{y}\in[1,\infty)$ the inequality $h^{3}_{2}(\widehat{y})\leq
3\widehat{y}h^{4}_{2}(\widehat{y})$ is satisfied and consequently

$\sum_{j=1}^{k_{0}}h^{3}_{2}(\widehat{y}_{j})\leq\sum_{j=1}^{k_{0}}3\widehat{y}_{j}h^{4}_{2}(\widehat{y}_{j})$
(3.51)

holds. Therefore for any fixed $(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})$
and $r\in\mathbb{R}_{+}$ the difference
$(g_{1}-f_{1})(\mathbf{y},\widehat{\mathbf{y}},r\mathbf{c})$ decreases
monotonically in $r$. Hence we only need to prove
$(g_{1}-f_{1})(\mathbf{y},\widehat{\mathbf{y}},r^{*}\mathbf{c})\geq\nu$ for

$r^{*}:=\sup\\{r\geq
0|\,g_{1}(\mathbf{y},\widehat{\mathbf{y}},r\mathbf{c})\leq 1\\}.$ (3.52)

On the other hand $g_{1}(\mathbf{y},\widehat{\mathbf{y}},r\mathbf{c})$
increases monotonically in $r$ and therefore
$g_{1}(\mathbf{y},\widehat{\mathbf{y}},r^{*}\mathbf{c})=1$. It is thus
sufficient to prove that $g_{1}=1$ implies $f_{1}\leq 1-\nu$. Hence we assume
$g_{1}=1$. Then

$\sum_{j=1}^{k_{0}}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})=1-h_{1}^{2}(y_{1})$
(3.53)

and

$\begin{split}f_{1}(\mathbf{y},\widehat{\mathbf{y}},\mathbf{c})&=h_{1}^{3}(y_{1})+\sum_{j=1}^{k_{0}}3c_{j}^{2}\widehat{y}_{j}h^{4}_{2}(\widehat{y}_{j})\leq
h_{1}^{3}(y_{1})+\max_{j\in\llbracket
k_{0}\rrbracket}\\{3\widehat{y}_{j}h_{2}(\widehat{y}_{j})\\}\sum_{j=1}^{k_{0}}c_{j}^{2}h^{3}_{2}(\widehat{y}_{j})\\\
&\leq
h_{1}^{3}(y_{1})+3\widehat{y}_{1}h_{2}(\widehat{y}_{1})(1-h_{1}^{2}(y_{1}))\leq
h_{1}^{3}(y_{1})+3y_{1}h_{2}(y_{1})(1-h_{1}^{2}(y_{1}))\\\
&=\frac{1}{(1+y_{1})^{3}}(y_{1}^{3}+3y_{1}^{2}+3y_{1})=1-\frac{1}{(1+y_{1})^{3}}.\end{split}$
(3.54)

The inequalities holds because $yh_{2}(y)$ increases monotonically in $y$ and
$y_{1}\geq\widehat{y}_{1}\geq\widehat{y}_{j}$ for all $j$. Combining this with
(3.45) and (3.47) concludes the proof of the lemma. ∎

###### Proof of Lemma 3.34.

Note that $h(m)>0$ for $m$ close to 0, since the $m^{-2}$ term is dominating
$h$ from (3.10) in this regime. First, we consider the situation, where $h$
has a pole in $(-\infty,0)$. In other words, we have $m_{+}^{*}>-\infty$,
where $m_{+}^{*}$ was defined in (3.13). All poles of $h$ on $(-\infty,0)$ are
either of order two or of order three so the pole at $m_{+}^{*}$ will be as
well. If the pole is of order two, then the behaviour around $m^{*}$ is given
by $\lim_{m\to m_{+}^{*}}h(m)=-\infty$ and if it is of order three $h$ behaves
like $\lim_{m\searrow m_{+}^{*}}h(m)=-\infty$ and $\lim_{m\nearrow
m_{+}^{*}}h(m)=\infty$. Since $h$ is continuous outside of its poles, the
existence of a pole thus implies the existence of a root in $(m_{+}^{*},0)$.
Now let $h$ have no poles in $(-\infty,0),$ i.e. $m_{+}^{*}=-\infty.$ Then $h$
is analytic on $(-\infty,0)$ and $h$ has a root in $(-\infty,0)$ if and only
if $h(m)$ is negative for some $m\in(-\infty,0).$

We separate multiple cases.

######  Case 1 ($\operatorname{rank}(A)\geq 2$).

Either $h$ has a pole in $(-\infty,0)$ or $b^{t}(1+2m\widehat{A})^{-3}b\geq 0$
for all $m\in(-\infty,0),$ thus

$h(m)\leq\frac{1}{m^{2}}-\operatorname{Tr}A^{2}(1+mA)^{-2}=-\frac{\operatorname{rank}(A)-1}{m^{2}}+\mathcal{O}(m^{-3})<0$
(3.55)

for sufficiently large $-m$ and $h$ has a root either way.

######  Case 2 ($\operatorname{rank}(A)=1$ and $A\geq 0$).

The matrices $A$ and $\widehat{A}$ have a positive eigenvalue, therefore $h$
will have a pole in $(-\infty,0)$ and thus also a root.

######  Case 3 ($\operatorname{rank}(A)=1$, $A\leq 0$ and
$b\notin\operatorname{Image}\widehat{A}$).

In this case, $h$ has no pole on $(-\infty,0)$ and there is a vector $w_{0}$
in the kernel of $\widehat{A}$ such that $\langle w_{0},b\rangle\neq 0.$ From
(3.10) we have that $h$ is estimated

$h(m)\leq\frac{1}{m^{2}}-|\langle w_{0},b\rangle|^{2}<0,$ (3.56)

where the last inequality holds for $-m$ sufficiently large and so $h$ has a
root as well.

######  Case 4 ($\operatorname{rank}(A)=1$, $A\leq 0$,
$b\in\operatorname{Image}\widehat{A}$ and $A\in\mathbb{R}^{l\times l}$).

Since $A$ is real-symmetric, we have $A=\widehat{A}$ and $\widehat{A}$ is also
a rank one matrix and $h$ has no poles in $(-\infty,0)$. Either $b=0$ or $b$
is an eigenvector of $\widehat{A}$ with eigenvalue $\mu=-\|A\|$ since
$A=\widehat{A}$, $\operatorname{rank}A=1$ and $A\leq 0.$ In both cases $h$
becomes

$h(m)=\frac{1}{m^{2}}\left(1-\frac{(\mu m)^{2}}{(1+\mu
m)^{2}}\right)-\|b\|^{2}\frac{1}{(1+2\mu m)^{3}}$ (3.57)

To find the asymptotics for $m\to-\infty$ consider $0<\zeta<1$, with
$\zeta=(\mu m)^{-1}$. We find

$\begin{split}h(m)&=\|A\|^{2}\zeta^{2}\left(1-\frac{1}{(\zeta+1)^{2}}\right)-\frac{\|b\|^{2}}{8}\frac{\zeta^{3}}{(\frac{1}{2}\zeta+1)^{3}}\\\
&=\zeta^{3}\left(\|A\|^{2}\sum_{k=0}^{\infty}(k+2)(-\zeta)^{k}-\frac{\|b\|^{2}}{8}\sum_{k=0}^{\infty}\frac{(k+1)(k+2)}{2}(-\frac{1}{2}\zeta)^{k}\right)\\\
&=\left(2\|A\|^{2}-\frac{\|b\|^{2}}{8}\right)\zeta^{3}+3\left(\frac{\|b\|^{2}}{16}-\|A\|^{2}\right)\zeta^{4}+\left(4\|A\|^{2}-\frac{3\|b\|^{2}}{8}\right)\zeta^{5}+\mathcal{O}(\zeta^{6}).\end{split}$
(3.58)

Therefore $h$ has a root if $\|b\|>4\|A\|$. If $\|b\|\leq 4\|A\|$ then the
leading order coefficient of the expansion is positive. Since $h(m)\geq 0$
implies $h^{\prime}(m)>0$ this implies $h(m)>0$ for all $m\in(-\infty,0)$.
Therefore there is no root and the first two cases in (3.34) follow.

######  Case 5 ($\operatorname{rank}(A)=1$, $A\leq 0$,
$b\in\operatorname{Image}\widehat{A}$ and $A\in\mathbb{C}^{l\times
l}\setminus\mathbb{R}^{l\times l}$).

Since $A$ is not real-symmetric, we have $\operatorname{rank}\widehat{A}=2$ by
Lemma A.1 and $h$ has no poles in $(-\infty,0)$. Since
$b\in\operatorname{Image}\widehat{A}$, the function $h$ can be expressed as

$h(m)=\frac{1}{m^{2}}\left(1-\frac{(\mu m)^{2}}{(1+\mu m)^{2}}\right)-|\langle
b,w_{+}\rangle|^{2}\frac{1}{(1+2\mu_{+}m)^{3}}-|\langle
b,w_{-}\rangle|^{2}\frac{1}{(1+2\mu_{-}m)^{3}}$ (3.59)

To find the asymptotics for $m\to-\infty$ consider $0<\zeta<1$, with
$\zeta=(-\|A\|m)^{-1}$. We find

$\begin{split}h(m)=&\|A\|^{2}\zeta^{2}\left(1-\frac{1}{(\zeta+1)^{2}}\right)-\frac{|\langle
b,w_{+}\rangle|^{2}}{8r_{+}^{3}}\frac{\zeta^{3}}{(\frac{1}{2r_{+}}\zeta+1)^{3}}-\frac{|\langle
b,w_{-}\rangle|^{2}}{8r_{-}^{3}}\frac{\zeta^{3}}{(\frac{1}{2r_{+}}\zeta+1)^{3}}\\\
=&\left(2\|A\|^{2}-\frac{|\langle
b,w_{+}\rangle|^{2}}{8r_{+}^{3}}-\frac{|\langle
b,w_{-}\rangle|^{2}}{8r_{-}^{3}}\right)\zeta^{3}\\\
&\qquad\qquad\qquad\qquad+3\left(\frac{|\langle
b,w_{+}\rangle|^{2}}{16r_{+}^{4}}-\frac{|\langle
b,w_{-}\rangle|^{2}}{16r_{-}^{4}}-\|A\|^{2}\right)\zeta^{4}+\mathcal{O}(\zeta^{5}).\end{split}$
(3.60)

By an analogous consideration to the above case, $h$ has a root if and only if

$\frac{|\langle b,w_{+}\rangle|^{2}}{r_{+}^{3}}+\frac{|\langle
b,w_{-}\rangle|^{2}}{r_{-}^{3}}>(4\|A\|)^{2}$ (3.61)

and the remaining two cases in (3.34) follow.

∎

##  4 Linearization

Throughout the remainder of the paper, we suppress the index of the identity
matrix if it is in dimension $l+1$ or in $(l+1)N$, i.e.
$I:=I_{l+1}\in\mathbb{C}^{(l+1)\times(l+1)}$ and
$\mathbf{I}:=\mathbf{I}_{(l+1)N}\in\mathbb{C}^{(l+1)N\times(l+1)N}$. In all
other cases, we still write out the dimension in the index.

To prove that $\mathbf{g}$ converges towards $m$ in the sense laid out in
Theorem 2.9, we use the linearization method, which we briefly introduce here.
For more details on the construction of linearizations in the context of
random matrices, see e.g. [21, 30, 31]. First, assume that $A$ is invertible.
We define the linearization $\mathbf{L}$ of $q$ as

$\mathbf{L}=K_{0}+\sum_{j=1}^{l}K_{j}\otimes\mathbf{X}_{j}\in\mathbb{C}^{(l+1)N\times(l+1)N},$
(4.1)

where

$K_{0}=\begin{pmatrix}c&0\\\
0&-A^{-1}\end{pmatrix}\in\mathbb{C}^{(l+1)\times(l+1)}\quad\text{and}\quad
K_{j}=\begin{pmatrix}b_{j}&e_{j}^{t}\\\
e_{j}&0\end{pmatrix}\in\mathbb{C}^{(l+1)\times(l+1)}$ (4.2)

for $j\in\llbracket l\rrbracket$ and $e_{j}$ being the $j^{\mathrm{th}}$
standard Euclidean base vector. Here we made use of our convention to identify
any matrix $R\in\mathbb{C}^{k\times n}$, $k,n\in\mathbb{N}$ with
$R\otimes\mathbf{I}_{N}\in\mathbb{C}^{kN\times nN}$ introduced in Section 1.
Let $J\in\mathbb{C}^{(l+1)\times(l+1)}$ be the orthogonal projection onto the
first entry. For $\delta\in[0,1]$ and $z=E+\mathrm{i}\eta\in\mathbb{H}$ we
define

$\mathbf{G}_{\delta}=(\mathbf{L}-zJ-\mathrm{i}\eta\delta(\mathbf{I}-J))^{-1}\in\mathbb{C}^{(l+1)N\times(l+1)N}$
(4.3)

$\mathbf{G}_{\delta}$ is a generalized resolvent and using the Schur
complement formula we obtain

$\mathbf{G}_{\delta}=\begin{pmatrix}\mathbf{g}_{\delta}&\mathbf{g}_{\delta}\mathbf{X}^{t}A_{\delta}\\\
A_{\delta}\mathbf{X}\mathbf{g}_{\delta}&-A_{\delta}+A_{\delta}\mathbf{X}\mathbf{g}_{\delta}\mathbf{X}^{t}A_{\delta}\end{pmatrix},$
(4.4)

where

$A_{\delta}=A(I_{l}+\mathrm{i}\delta\eta A)^{-1}\in\mathbb{C}^{l\times
l}\quad\text{and}\quad\mathbf{g}_{\delta}=\left(\mathbf{X}^{t}A(I_{l}+\mathrm{i}\delta\eta
A)^{-1}\mathbf{X}+b^{t}\mathbf{X}+c-z\right)^{-1}\in\mathbb{C}^{N\times N}.$
(4.5)

Note that $I_{l}+\mathrm{i}\delta\eta A$ is invertible for all $\eta>0$ and
$\delta\in[0,1]$ as $A$ is Hermitian. In particular we find

$(\mathbf{G}_{0})_{11}=\mathbf{g}\in\mathbb{C}^{N\times N}.$ (4.6)

This justifies, why $\mathbf{L}$ is called the linearization of $q$. The
$\delta\neq 0$ case adds an additional regularization, which we use to prove
the local law, Proposition 4.3, for $\delta=0$. $\mathbf{G}_{\delta}$
satisfies the equation

$\mathbf{I}+(zJ+\mathrm{i}\eta\delta(\mathbf{I}-J)-K_{0}+\mathcal{S}[\mathbf{G}_{\delta}])\mathbf{G}_{\delta}=\mathbf{D},$
(4.7)

where

$\mathcal{S}[\mathbf{R}]=\mathbb{E}[(\mathbf{L}-\mathbb{E}[\mathbf{L}])\mathbf{R}(\mathbf{L}-\mathbb{E}[\mathbf{L}])]$
(4.8)

is called the self-energy operator and $\mathbf{D}$ is the error term. It is
defined by

$\mathbf{D}:=\mathcal{S}[\mathbf{G}_{\delta}]\mathbf{G}_{\delta}+(\mathbf{L}-K_{0})\mathbf{G}_{\delta}.$
(4.9)

We split the self energy term into $\widetilde{\Gamma}$ and
$\mathcal{S}_{\mathrm{o}}$, i.e.
$\mathcal{S}=\widetilde{\Gamma}+\mathcal{S}_{\mathrm{o}}$. The first part,
$\widetilde{\Gamma}$, depends only on the averaged trace of its argument, i.e.
for all $R\in\mathbb{C}^{(l+1)N\times(l+1)N}$ we have

$\widetilde{\Gamma}[\mathbf{R}]=\Gamma[\underline{\mathbf{R}}].$ (4.10)

The blockwise trace, $\underline{\mathbf{R}}$, was introduced in (1.7) and
$\Gamma:\mathbb{C}^{(l+1)\times(l+1)}\to\mathbb{C}^{(l+1)\times(l+1)}$ is
given by

$\Gamma\left[\begin{pmatrix}\omega&v^{t}\\\
w&T\end{pmatrix}\right]=\begin{pmatrix}\omega\|b\|^{2}+b^{t}(v+w)+\operatorname{Tr}T&\omega
b^{t}+w^{t}\\\ \omega b+v&\omega I_{l}\end{pmatrix},$ (4.11)

where $\omega\in\mathbb{C}$, $v,w\in\mathbb{C}^{l}$ and
$T\in\mathbb{C}^{l\times l}$. In (4.10) we have made use of our convention to
identify $\Gamma[\underline{\mathbf{R}}]$ with
$\Gamma[\underline{\mathbf{R}}]\otimes\mathbf{I}_{N}$

The second term
$\mathcal{S}_{\mathrm{o}}:\mathbb{C}^{(l+1)N\times(l+1)N}\to\mathbb{C}^{(l+1)N\times(l+1)N}$
is given by

$\mathcal{S}_{\mathrm{o}}\left[\begin{pmatrix}\bm{\omega}&\mathbf{V}^{t}\\\
\mathbf{W}&\mathbf{T}\end{pmatrix}\right]=\frac{1}{N}\mathbb{E}\zeta_{1}^{2}\begin{pmatrix}\|b\|^{2}\bm{\omega}^{(o)}+b^{t}(\mathbf{V}^{(o)}+\mathbf{W}^{(o)})+\operatorname{Tr}_{\mathrm{b}}\mathbf{T}^{(o)}&\bm{\omega}^{(o)}b^{t}+\mathbf{W}^{(o)}\\\
\bm{\omega}^{(o)}b+(\mathbf{V}^{t})^{(o)}&I_{l}\otimes\bm{\omega}^{(o)}\end{pmatrix},$
(4.12)

where $\bm{\omega}\in\mathbb{C}^{N\times N}$,
$\mathbf{V}=(\mathbf{V}_{i})_{i\in\llbracket
l\rrbracket}\in(\mathbb{C}^{N\times N})^{l}$,
$\mathbf{W}=(\mathbf{W}_{i})_{i\in\llbracket
l\rrbracket}\in(\mathbb{C}^{N\times N})^{l}$ and
$\mathbf{T}\in\mathbb{C}^{lN\times lN}$. For any
$\mathbf{R}=(\mathbf{R}_{ij})_{i\in\llbracket k\rrbracket,j\in\llbracket
n\rrbracket}\in\mathbb{C}^{kN\times nN}$, $n,k\in\mathbb{N}$, we define
$\mathbf{R}^{(o)}$ by
$(\mathbf{R}^{(o)})_{ij}:=\mathbf{R}_{ji}^{t}-\operatorname{diag}(\mathbf{R}_{ji})\in\mathbb{C}^{N\times
N}$. For $\mathbf{R}=(\mathbf{R}_{ij})_{i,j\in\llbracket
l\rrbracket}\in\mathbb{C}^{lN\times lN}$ the block-trace
$\operatorname{Tr}_{\mathrm{b}}$ is defined by

$\operatorname{Tr}_{\mathrm{b}}\mathbf{R}=\sum_{i=1}^{l}\mathbf{R}_{ii}\in\mathbb{C}^{N\times
N}.$ (4.13)

Equation (4.7) without the error term and $\mathcal{S}_{\mathrm{o}}$ is called
the Dyson equation (DE) and its solution is denoted by
$M_{\delta}=M_{\delta}(z)\in\mathbb{C}^{(l+1)\times(l+1)}$, i.e.

$I+(zJ+\mathrm{i}\eta\delta(I-J)-K_{0}+\Gamma[M_{\delta}])M_{\delta}=0\in\mathbb{C}^{(l+1)\times(l+1)}.$
(4.14)

In the next subsection, we will lay out in what sense $\mathbf{G}_{\delta}$ is
close to $M_{\delta}$. [21, Lemma 2.6] asserts the existence of a unique
analytic solution to (4.14) for $\delta=0$ and [34, Theorem 2.1] guarantees a
unique solution for $\delta\in(0,1]$. For $m_{\delta}\in\mathbb{C}$,
$v_{\delta},w_{\delta}\in\mathbb{C}^{l}$ and
$\widehat{M}_{\delta}\in\mathbb{C}^{l\times l}$ we partition $M_{\delta}$ as

$M_{\delta}:=\begin{pmatrix}m_{\delta}&v_{\delta}^{t}\\\
w_{\delta}&\widehat{M}_{\delta}\end{pmatrix}.$ (4.15)

Since $M_{\delta}$ solves (4.14) it is invertible and by the Schur complement
formula its inverse is given by

$M_{\delta}^{-1}=\begin{pmatrix}m^{-1}+m^{-2}v^{t}_{\delta}\left(\widehat{M}_{\delta}-w_{\delta}m_{\delta}^{-1}v_{\delta}^{t}\right)^{-1}w_{\delta}&-m^{-1}v_{\delta}^{t}\left(\widehat{M}_{\delta}-w_{\delta}m_{\delta}^{-1}v_{\delta}^{t}\right)^{-1}\\\
-m^{-1}\left(\widehat{M}_{\delta}-w_{\delta}m_{\delta}^{-1}v_{\delta}^{t}\right)^{-1}w_{\delta}&\left(\widehat{M}_{\delta}-w_{\delta}m_{\delta}^{-1}v_{\delta}^{t}\right)^{-1}\end{pmatrix}.$
(4.16)

At the same time by (4.14) the inverse of $M_{\delta}$ can be expressed as

$M_{\delta}^{-1}=\begin{pmatrix}-z-m_{\delta}b^{t}b-b^{t}(v_{\delta}+w_{\delta})-\operatorname{Tr}\widehat{M}_{\delta}&-m_{\delta}b^{t}-w_{\delta}^{t}\\\
-m_{\delta}b-v&-A^{-1}-(m_{\delta}+\mathrm{i}\eta\delta)I_{l}\end{pmatrix}.$
(4.17)

Comparing the two expressions for $M_{\delta}^{-1}$, we obtain a set of
equations for $m_{\delta},v_{\delta},w_{\delta}$ and $\widehat{M}_{\delta}$.
Solving them we find explicit expressions for $v_{\delta},w_{\delta}$ and
$\widehat{M}_{\delta}$ in terms of $m_{\delta}$. In other words,
$M_{\delta}=M_{\delta}[m_{\delta}]$ can be expressed purely in terms of its
(1,1) entry with $M_{\delta}[x]$ given by

$M_{\delta}[x]:=\begin{pmatrix}x&-xb^{t}V_{\delta}(x)A_{\delta}\\\
-xA_{\delta}V_{\delta}(x)b&-A_{\delta}(I_{l}+xA_{\delta})^{-1}+xA_{\delta}V_{\delta}(x)bb^{t}V_{\delta}(x)A_{\delta}\end{pmatrix}$
(4.18)

with

$V_{\delta}(x):=x(I_{l}+2x\widehat{A}_{\delta})^{-1}.$ (4.19)

Note that we use square brackets to denote $M_{\delta}$ as a function of its
(1,1) entry. This is done to avoid confusion with $M_{\delta}(z)$, which
denotes $M_{\delta}$ as a function of the spectral parameter $z.$ The two
functions are related by $M_{\delta}[m_{\delta}(z)]=M_{\delta}(z)$. The entry
$m_{\delta}$ satisfies the equation

$-m_{\delta}^{-1}=z+\gamma_{\delta}(m_{\delta}),$ (4.20)

where

$\gamma_{\delta}(x):=-\operatorname{Tr}A_{\delta}(I_{l}+A_{\delta}x)^{-1}+b^{t}(V_{\delta}(x)-V_{\delta}(x)\widehat{A}_{\delta}V_{\delta}(x))b-c.$
(4.21)

and $\widehat{A}_{\delta}=\frac{1}{2}(A_{\delta}+A_{\delta}^{t})$. For
$\delta=0$, Equation (4.20) corresponds precisely to (2.4). From here on we no
longer assume that $A$ is invertible and instead, we use the Equations
$\eqref{resolvent delta}$, (4.9) and (4.18) as the definition for
$\mathbf{G}_{\delta}$, $\mathbf{D}$ and $M_{\delta}=M_{\delta}[m_{\delta}]$
respectively. Note that this is also well defined for (4.9) as $K_{0}$ and
$\mathbf{L}$, defined in (4.2) and (4.1) respectively, depend on $A^{-1}$, but
$\mathbf{L}-K_{0}$ does not. The function $m_{\delta}$ is uniquely defined by
the following lemma.

######  Lemma 4.1.

Let $\rho$ have a regular edge at $\tau_{\pm}$ and let $\delta\in(0,1]$. The
function $m_{\delta}$ is uniquely defined by the following criteria around the
edge and away from the spectrum.

  1. 1.

There is a $u>0$ only depending on the coefficients of $q$ such that for all
$z\in\mathbb{H}$ with $|z-\tau_{\pm}|<u$ there is a unique function
$m_{\delta}=m_{\delta}(z)$ that solves (4.20) and satisfies

$\operatorname{\mathrm{Im}}m_{\delta}\sim\begin{cases}\sqrt{\kappa+\eta}&\text{if
}E\in\operatorname{supp}(\rho)\\\ \frac{\eta}{\sqrt{\kappa+\eta}}&\text{if
}E\notin\operatorname{supp}(\rho)\end{cases}\quad\text{as well
as}\quad|m_{\delta}-m_{\pm}|^{2}\sim|z-\tau_{\pm}|,$ (4.22)

with $\kappa=|E-\tau_{\pm}|$ and $m_{\pm}=\lim_{\eta\searrow
0}m(\tau_{\pm}+\mathrm{i}\eta)$.

  2. 2.

For all $C>0$ there is an $\eta_{0}$ such that for all
$z=E+\mathrm{i}\eta\in\mathbb{H}$ with
$C^{-1}<\operatorname{dist}(E,\operatorname{supp}(\rho))<C$ and
$\eta<\eta_{0}$ there is a unique function $m_{\delta}=m_{\delta}(z)$ that
solves (4.20) and satisfies

$\operatorname{\mathrm{Im}}m_{\delta}\sim_{C}\eta\quad\text{as well
as}\quad|m_{\delta}(z)-m(E)|\sim\eta,$ (4.23)

with $m(E)=\lim_{\eta\searrow 0}m(E+\mathrm{i}\eta)$.

In each case $M_{\delta}=M_{\delta}[m_{\delta}]$ is also the unique solution
to (4.14) with $\operatorname{\mathrm{Im}}M_{\delta}>0$ if the coefficient
matrix $A$ is invertible.

The lemma is proven in Appendix A.3.

###  4.1 Local law for the linearization

######  Definition 4.2 (Shifted square of a Wigner matrix).

A polynomial $q$ as in (1.1) with $A\in\mathbb{R}^{l\times l},$
$\operatorname{rank}A=1$ and $b=0$ is called a shifted square of a Wigner
matrix.

######  Remark.

If a polynomial $q$ satisfies the above definition, then there is some
$v\in\mathbb{R}^{l}$ with $v\neq 0$ and

$q(\mathbf{X})=\pm(v^{t}\mathbf{X})^{2}+c.$ (4.24)

Here, $\mathbf{W}:=\frac{1}{\|v\|}v^{t}\mathbf{X}$ is a Wigner matrix,
normalised such that $\mathbb{E}|w_{ij}|^{2}=\frac{1}{N}$ and we have

$q(\mathbf{X})=\pm\|v\|^{2}\mathbf{W}^{2}+c.$ (4.25)

This justifies the terminology in the above definition.

In case $q$ is not a shifted square of a Wigner matrix, we will prove that
$M_{0}$ and $\mathbf{G}_{0}$ are close to each other around any regular edge.
As $m$ and $\mathbf{g}$ are sub-matrices of $M_{0}$ and $\mathbf{G}_{0}$
respectively, this also implies closeness of $\mathbf{g}$ and $m$. This
result, a local law for the linearization, is presented below in Propositions
4.3 and 4.4. For $q$ being a shifted square of a Wigner matrix, we will prove
the local law, Theorem 2.9 and Proposition 2.10, directly without the use of a
linearization. The reason why we prove this case separately is that the
stability operator $\mathscr{L}$, defined below in (4.33), does have an
additional unstable direction.

######  Proposition 4.3 (Edge local law for the linearization).

Let $q$ be a polynomial of the form (1.1) that is not a shifted square of a
Wigner matrix and let the corresponding $\rho$ have a regular edge at
$\tau_{0}$. There is a $\kappa_{0}>0$ depending only on the parameters of $q$
such that for all $\varepsilon,\gamma,D>0$, $\delta\in\\{0,1\\}$ and
$z=E+\mathrm{i}\eta\in\mathbb{D}_{\gamma}^{\kappa_{0}}$ the isotropic local
law

$\mathbb{P}\left(|\langle\mathbf{x},(\mathbf{G}_{\delta}-M_{\delta})\mathbf{y}\rangle|>\|\mathbf{x}\|\|\mathbf{y}\|N^{\varepsilon}\left(\frac{\operatorname{\mathrm{Im}}m}{\sqrt{N\eta}}+\frac{1}{N\eta}\right)\right)\lesssim_{\varepsilon,\gamma,D}N^{-D}$
(4.26)

holds for all deterministic $\mathbf{x},\mathbf{y}\in\mathbb{C}^{(l+1)N}$.
Moreover, the averaged local law

$\mathbb{P}\left(|\langle\mathbf{B}(\mathbf{G}_{\delta}-M_{\delta})\rangle|>\|\mathbf{B}\|\frac{N^{\varepsilon}}{N\eta}\right)\lesssim_{\varepsilon,\gamma,D}N^{-D}$
(4.27)

holds for all deterministic $\mathbf{B}\in\mathbb{C}^{(l+1)N\times(l+1)N}$.
For $E\notin\operatorname{supp}(\rho)$, an improved averaged local law of the
form

$\mathbb{P}\left(|\langle\mathbf{B}(\mathbf{G}_{\delta}-M_{\delta})\rangle|>\|\mathbf{B}\|N^{\varepsilon}\left(\frac{1}{N(\kappa+\eta)}+\frac{1}{(N\eta)^{2}\sqrt{\kappa+\eta}}\right)\right)\lesssim_{\varepsilon,\gamma,D}N^{-D},$
(4.28)

with $\kappa=|E-\tau_{0}|$, is obtained.

######  Proposition 4.4 (Local law for the linearization away from the
spectrum).

Let $q$ be a polynomial of the form (1.1) that is not a shifted square of a
Wigner matrix. For all $C>0$ there is an $\eta_{0}>0$, depending only on the
coefficients of $q$, such that an averaged local law of the form

$\mathbb{P}\left(|\langle\mathbf{B}(\mathbf{G}_{\delta}-M_{\delta})\rangle|>\|\mathbf{B}\|N^{\varepsilon}\left(\frac{1}{N}+\frac{1}{(N\eta)^{2}}\right)\right)\lesssim_{\varepsilon,\gamma,D,C}N^{-D}$
(4.29)

holds true for all deterministic
$\mathbf{B}\in\mathbb{C}^{(l+1)N\times(l+1)N}$, $\varepsilon,\gamma,D>0$ and
$z\in\mathbb{G}_{\gamma}^{C,\eta_{0}}$.

To obtain Theorem 2.9, we only require the $\delta=0$ case, but to prove it we
will require the $\delta=1$ case, thus we state both cases together in the
proposition. The proof of Proposition 4.3 has two major ingredients. For one
we show that the error term $\mathbf{D}$ in (4.7) is indeed small, this is
done in Proposition 4.5, which we import from [22, Theorem 4.1] and adjust to
our setting. Additionally, we need to prove that (4.14) is stable under small
perturbations. This is done in Proposition 4.6.

######  Proposition 4.5.

Let $\varepsilon>0$, $p\in\mathbb{N}$ and $\delta\in[0,1]$. Then there is a
$c>0$ such that

$\|\mathbf{D}\|_{p}\lesssim_{p,\varepsilon}N^{\varepsilon}\sqrt{\frac{\|\mathbf{G}_{\delta}\mathbf{G}_{\delta}^{*}\|_{p_{0}}}{N}}\left(1+\|\mathbf{G}_{\delta}\|_{p_{0}}\right)^{c}\left(1+N^{-\frac{1}{4}}\|\mathbf{G}_{\delta}\|_{p_{0}}\right)^{cp}$
(4.30)

and

$\|\mathbf{D}\|_{p}^{\mathrm{av}}\lesssim_{p,\varepsilon}N^{\varepsilon}\frac{\|\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}\|_{p_{0}}}{N}\left(1+\|\mathbf{G}_{\delta}\|_{p_{0}}\right)^{c}\left(1+N^{-\frac{1}{4}}\|\mathbf{G}_{\delta}\|_{p_{0}}\right)^{cp}$
(4.31)

where we defined $p_{0}=c\frac{p^{4}}{\varepsilon}$.

###### Proof.

First, consider the case of $A$ being invertible. Then
$\mathbf{G}_{\delta}=(\mathbf{L}-zJ-\mathrm{i}\eta\delta(\mathbf{I}-J))^{-1}$
and our proof follows the proof of [22, Theorem 4.1] line by line with the
exception that the Ward identities, (51a) and (51b), do not apply for
$\delta=0$. Thus, we cannot replace the
$\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}$ terms by
$\eta^{-1}\operatorname{\mathrm{Im}}\mathbf{G}_{\delta}$ and we instead are
left with the upper bounds (4.30) and (4.31).

Now consider $A$ being non-invertible. Then $A+\varepsilon$ is invertible for
all $\varepsilon\in(0,u)$ for some $u\sim 1$. We denote the
$\mathbf{G}_{\delta}$ associated with $A+\varepsilon$ by
$\mathbf{G}_{\delta}^{\varepsilon}$ and we have $\lim_{\varepsilon\to
0}\mathbf{G}_{\delta}^{\varepsilon}=\mathbf{G}_{\delta}$. We thus only need to
prove that the constants in (4.30) and (4.31) are uniform in $\varepsilon$ to
obtain the proposition in this case. This is non-trivial as [22, Theorem 4.1]
states as a condition that $\mathbb{E}[\mathbf{L}]$ is bounded and this is
clearly not the case in the $\varepsilon\to 0$ limit. The assumption is only
used, however, to ensure that $\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}$
satisfies the lower bound
$\|\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}\|_{p_{0}}\gtrsim 1$. In our
case, this follows instead from
$\|\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}\|_{p_{0}}\geq\|\mathbf{g}_{\delta}^{*}\mathbf{g}_{\delta}\|_{p_{0}}\gtrsim_{p_{0}}\|\mathbf{g}^{*}\mathbf{g}\|_{p_{0}}$.
Since $\mathbf{g}$ is the resolvent of $q(\mathbf{X})$ we have
$\|\mathbf{g}^{*}\mathbf{g}\|_{q}\geq\mathbb{E}[\|q(\mathbf{X})\|^{-1}]\gtrsim
1$ uniformly in $z$ for bounded $z$. Here, the last inequality holds true
since $q(\mathbf{X})$ satisfies the inequality

$\|q(\mathbf{X})\|\leq\sum_{i,j=1}^{l}|A_{ij}|\|\mathbf{X}_{i}\|\|\mathbf{X}_{j}\|+\sum_{i=1}^{l}|b_{i}|\|\mathbf{X}_{i}\|+|c|$
(4.32)

and $\mathbb{E}[\|\mathbf{X}_{i}\|]\lesssim 1$ for all $i\in\llbracket
l\rrbracket$ since the $\mathbf{X}_{i}$ are Wigner matrices. Therefore the
proposition also holds for non-invertible $A$. ∎

The following result concerns the stability of the Dyson equation. The
stability operator
$\mathscr{L}:\mathbb{C}^{(l+1)N\times(l+1)N}\to\mathbb{C}^{(l+1)N\times(l+1)N}$
is given by

$\mathscr{L}[\mathbf{R}]=\mathbf{R}-M_{\delta}\mathcal{S}[\mathbf{R}]M_{\delta}$
(4.33)

and we will prove

######  Proposition 4.6 (Control of $\mathscr{L}$).

Let $q$ be a polynomial of the form (1.1) that is not a shifted square of a
Wigner matrix and let the corresponding $\rho$ have a regular edge at
$\tau_{0}$. There exists a $u\sim 1$ such that for all
$z=E+\mathrm{i}\eta\in\mathbb{H}$ with $|z-\tau_{0}|<u$ and $\delta\in[0,1]$
there exists an eigenvalue $\beta$ with corresponding left and right
eigenvectors $L,B\in\mathbb{C}^{(l+1)\times(l+1)}$ of $\mathscr{L}$ such that

$\begin{split}\|\mathscr{L}^{-1}\|_{\mathrm{sp}}\sim(\kappa+\eta)^{-\frac{1}{2}},&\quad\|(\mathbb{1}-\mathscr{P})\mathscr{L}^{-1}\|_{\mathrm{sp}}\lesssim
1,\quad|\beta|\sim(\kappa+\eta)^{\frac{1}{2}},\\\ |\langle L,B\rangle|\sim
1,&\quad\|L\|+\|B\|\sim 1,\quad|\langle
L,M_{\delta}\mathcal{S}[B]B\rangle|\sim 1,\end{split}$ (4.34)

with $\mathbb{1}$ being the identity operator, $\mathscr{P}$ being the
spectral projection onto $B$, i.e.

$\mathscr{P}=(\langle
L\otimes\mathbf{I}_{N},B\otimes\mathbf{I}_{N}\rangle)^{-1}\langle
L\otimes\mathbf{I}_{N},\cdot\rangle(B\otimes\mathbf{I}_{N})$ (4.35)

and $\kappa=|E-\tau_{0}|$.

Furthermore, for any $C>0$ and $E=\operatorname{\mathrm{Re}}z$ with
$C^{-1}<\operatorname{dist}(E,\operatorname{supp}(\rho))<C$ there is an
$\eta_{0}>0$ such that we have

$\|\mathscr{L}^{-1}\|_{\mathrm{sp}}\sim_{C}1$ (4.36)

uniformly for all $\eta\leq\eta_{0}.$

The proof will be given in Section 4.2.

######  Remark.

$B$ and $L$ being right and left eigenvectors of $\mathscr{L}$ with eigenvalue
$\beta$ is understood in the sense of

$\mathscr{L}[B]=\beta B\quad\text{and}\quad\mathscr{L}^{*}[L]=\bar{\beta}L.$
(4.37)

Here, we used the notation
$R=R\otimes\mathbf{I}_{N}\in\mathbb{C}^{(l+1)N\times(l+1)N}$ introduced in
(1.3). The adjoint is defined with respect to the scalar product
$\langle\mathbf{R},\mathbf{T}\rangle=\langle\mathbf{R}^{*}\mathbf{T}\rangle$.

Corollary 3.3, as well as Propositions 4.5 and 4.6 are the main ingredients to
Proposition 4.3 and are in fact sufficient for $\delta=1.$ For $\delta=0$
however, extra care is needed as the Ward identity for resolvents
$\mathbf{G}$,

$\mathbf{G}\mathbf{G}^{*}=\frac{\operatorname{\mathrm{Im}}\mathbf{G}}{\eta},$
(4.38)

does not translate to generalized resolvents. Instead, we will estimate
$\mathbf{G}_{0}\mathbf{G}_{0}^{*}$ by $\mathbf{G}_{1}\mathbf{G}_{1}^{*}$,
which allows us to obtain the local law for $\delta=0$ from the $\delta=1$
case. The proof of Proposition 4.3 will be given in Section 5 and follows the
general strategy from [1], modified to accommodate for the lack of a Ward
identity.

###  4.2 Proof of Proposition 4.6

###### Proof of Proposition 4.6.

We split the stability operator into
$\mathscr{L}=\mathscr{L}^{(0)}+\mathscr{L}^{(1)}$ with

$\mathscr{L}^{(0)}[\mathbf{R}]:=\mathscr{L}[\underline{\mathbf{R}}]\quad\text{and}\quad\mathscr{L}^{(1)}[\mathbf{R}]:=\mathscr{L}[\mathbf{R}-\underline{\mathbf{R}}].$
(4.39)

By (4.11) and (4.12), we have
$\tilde{\Gamma}[\mathbf{R}-\underline{\mathbf{R}}]=\mathcal{S}_{\mathrm{o}}[\underline{\mathbf{R}}]=0.$
Thus we have

$\mathscr{L}^{(0)}[\mathbf{R}]=\mathcal{L}[\underline{\mathbf{R}}]\otimes\mathbf{I}_{N}\quad\text{and}\quad\mathscr{L}^{(1)}[\mathbf{R}]=\mathbf{R}-\underline{\mathbf{R}}-M_{\delta}\mathcal{S}_{\mathrm{o}}[\mathbf{R}]M_{\delta}$
(4.40)

with
$\mathcal{L}:\mathbb{C}^{(l+1)\times(l+1)}\to\mathbb{C}^{(l+1)\times(l+1)}$
defined as

$\mathcal{L}[R]:=R-M_{\delta}\Gamma[R]M_{\delta}.$ (4.41)

The image of $\mathscr{L}^{(0)}$ is given by

$\\{\mathbf{R}\in\mathbb{C}^{(l+1)N\times(l+1)N}:\,\underline{\mathbf{R}}=\mathbf{R}\\}=:\mathcal{U}$
(4.42)

and its kernel is given by $\mathcal{U}^{\perp}$, the orthogonal complement of
$\mathcal{U}.$ At the same time the image of $\mathscr{L}^{(1)}$ is contained
in $\mathcal{U}^{\perp}$ and $\mathcal{U}$ is contained in the kernel of
$\mathscr{L}^{(1)}$. That is, $\mathscr{L}$ decomposes into
$\mathscr{L}^{(0)}$ acting on $\mathcal{U}$ and $\mathscr{L}^{(1)}$ acting on
its orthogonal complement.

The behaviour of $\mathcal{L}$ is summarized in the following lemma.

######  Lemma 4.7.

Let $q$ be a polynomial of the form (1.1) that is not a shifted square of a
Wigner matrix and let the corresponding $\rho$ have a regular edge at
$\tau_{0}$. There exists an $u\sim 1$ such that for all
$z=E+\mathrm{i}\eta\in\mathbb{H}$ with $|z-\tau_{0}|<u$ and $\delta\in[0,1]$
there exists an eigenvalue $\beta$ with corresponding normalized left and
right eigenvectors $L$ and $B$ of $\mathcal{L}$ such that

$\begin{split}\|\mathcal{L}^{-1}\|_{\mathrm{sp}}\sim(\kappa+\eta)^{-\frac{1}{2}},&\quad\|(\mathbb{1}-\mathcal{P})\mathcal{L}^{-1}\|_{\mathrm{sp}}\lesssim
1,\quad|\beta|\sim(\kappa+\eta)^{\frac{1}{2}},\\\ |\langle L,B\rangle|\sim
1,&\quad|\langle L,M_{\delta}\Gamma[B]B\rangle|\sim 1,\end{split}$ (4.43)

with $\mathcal{P}$ being the spectral projection onto $B$, i.e.
$\mathcal{P}=(\langle L,B\rangle)^{-1}\langle L,\cdot\rangle B$ and
$\kappa=|E-\tau_{+}|$.

Furthermore, for any $C>0$ and $E=\operatorname{\mathrm{Re}}z$ with
$C^{-1}\leq\operatorname{dist}(E,\operatorname{supp}(\rho))\leq C$ there is an
$\eta_{0}>0$ such that we have

$\|\mathcal{L}^{-1}\|_{\mathrm{sp}}\sim_{C}1$ (4.44)

uniformly for all $\eta\leq\eta_{0}.$

The proof of Lemma 4.7 is deferred to the end of the section. For
$\mathcal{S}_{\mathrm{o}}$ we find

$\|{\mathcal{S}_{\mathrm{o}}}[\mathbf{R}]\|_{\mathrm{hs}}\lesssim\frac{1}{N}\|\mathbf{R}\|_{\mathrm{hs}}$
(4.45)

for all $\mathbf{R}\in\mathbb{C}^{(l+1)N\times(l+1)N}$. Thus
$\mathcal{S}_{\mathrm{o}}$ is bounded by

$\|\mathcal{S}_{\mathrm{o}}\|_{\mathrm{sp}}\lesssim\frac{1}{N}.$ (4.46)

By Corollary 3.3, Lemma 4.1 and (4.18) we have $\|M_{\delta}\|\lesssim 1$ for
all $z$ such that $|z-\tau_{0}|\leq u$ and some $u>0$. Combined with (4.46) it
follows that there is a $C>0$ such that

$\|\mathscr{L}^{(1)}\|_{\mathrm{sp}}\leq
1+CN^{-1}\quad\text{and}\quad\operatorname{Spec}\left(\mathscr{L}^{(1)}|_{\mathcal{U}^{\perp}}\right)\subset
B_{CN^{-1}}(1),$ (4.47)

where $B_{\varepsilon}(x)$ denotes the $\varepsilon$ neighborhood of $x$. Thus
for sufficiently large $N$ the smallest eigenvalue of $\mathscr{L}$ equals
that of $\mathcal{L}$ and the corresponding left and right eigenvectors of
$\mathscr{L}$ are given by $L\otimes\mathbf{I}_{N}$ and
$B\otimes\mathbf{I}_{N}$. The norm of the inverse of $\mathscr{L}$ is bounded
by

$\|\mathscr{L}^{-1}\|_{\mathrm{sp}}\leq\max\\{1+CN^{-1},\|\mathcal{L}^{-1}\|_{\mathrm{sp}}\\}\quad\text{and}\quad\|(1-\mathcal{P})\mathscr{L}^{-1}\|_{\mathrm{sp}}\leq\max\\{1+CN^{-1},\|\mathcal{L}^{-1}\|_{\mathrm{sp}}\\},$
(4.48)

completing the proof of Proposition 4.6 ∎

###### Proof of Lemma 4.7.

First, we prove that $\mathcal{L}$ has exactly one vanishing eigenvalue at
$\tau_{0}$, and from there on we conclude the proof with the help of a
perturbative argument. We define

$\mathcal{C}_{J}[R]=JRJ,$ (4.49)

i.e. $\mathcal{C}_{J}$ is the projection onto the $(1,1)$ entry and in
particular $\mathcal{C}_{J}[M_{0}]=mJ$. We also set
$\mathcal{C}_{J}^{\perp}R:=R-\mathcal{C}_{J}R$ and
$\widetilde{\mathbb{C}}^{(l+1)\times(l+1)}:=\operatorname{Image}\mathcal{C}_{J}^{\perp}$.
For any $R\in\mathbb{C}^{(l+1)\times(l+1)}$ we denote $r:=R_{11}$ and
$\widetilde{R}:=\mathcal{C}_{J}^{\perp}[R]$, i.e. $R=rJ+\widetilde{R}$. Let
$T_{z}$ be the matrix

$T_{z}=\begin{pmatrix}1&0\\\
0&zA\end{pmatrix}\in\mathbb{C}^{(l+1)\times(l+1)}$ (4.50)

as well as

$F:\begin{cases}\mathbb{C}\times\widetilde{\mathbb{C}}^{(l+1)\times(l+1)}\times\mathbb{H}&\to\mathbb{C}^{(l+1)\times(l+1)}\\\
(r,\widetilde{R},z)&\mapsto
rJ+\widetilde{R}+T_{z}(z+\Gamma[rJ+\widetilde{R}]T_{z})^{-1}.\end{cases}$
(4.51)

Then $F(m,\widetilde{M},z)=0$ for $M=M_{0}[m]$ defined in (4.18). For
$\delta=0$, the stability operator $\mathcal{L}$ is the derivative of $F$ in
the sense that

$\begin{split}\mathcal{L}[R]&=D_{R}F(m,\widetilde{M},z),\end{split}$ (4.52)

where $D_{R}F=\frac{\mathrm{d}}{\mathrm{d}\varepsilon}F(m+\varepsilon
r,\widetilde{M}+\varepsilon\widetilde{R},z)|_{\varepsilon=0}$ is the
directional derivative of $F$ in the direction $R$. We first consider the case
when $A$ is invertible. The case of non-invertible $A$ will be treated
afterwards. We define $\mathcal{B}:=\mathcal{C}_{M^{-1}}\mathcal{L}$ on
$z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$, where $m(z)$ is defined as
the unique analytical continuation to
$\mathbb{C}\setminus\operatorname{supp}(\rho)$. Since $F(m,\widetilde{M},z)=0$
we have

$\begin{split}\mathcal{B}[R]&=D_{R}[\mathcal{C}_{M^{-1}}F]=D_{rJ}\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F]+D_{\widetilde{R}}\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F]+D_{rJ}\mathcal{C}_{J}^{\perp}[\mathcal{C}_{M^{-1}}F]+D_{\widetilde{R}}\mathcal{C}_{J}^{\perp}[\mathcal{C}_{M^{-1}}F],\end{split}$
(4.53)

where we used the linearity of the derivatives as well as the linearity of
$\mathcal{B}$ in the second equality and we omitted the arguments of $F$. The
above equation decomposes $\mathcal{B}$ into a two-by-two block operator with
diagonal blocks
$\mathcal{B}_{11}[rJ]:=D_{rJ}\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F]$,
$\mathcal{B}_{22}[\widetilde{R}]:=D_{\widetilde{R}}\mathcal{C}_{J}^{\perp}[\mathcal{C}_{M^{-1}}F]$
and off-diagonal blocks
$\mathcal{B}_{12}[\widetilde{R}]:=D_{\widetilde{R}}\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F]$,
$\mathcal{B}_{21}[rJ]:=D_{rJ}\mathcal{C}^{\perp}_{J}[\mathcal{C}_{M^{-1}}F]$.

A tedious but straightforward calculation shows that $\mathcal{B}_{22}$ is
invertible on the image of $\mathcal{C}_{J}^{\perp}$ and its inverse is given
by

$(\mathcal{B}_{22})^{-1}\left[\begin{pmatrix}0&r_{12}^{t}\\\
r_{21}&\widehat{R}\end{pmatrix}\right]=\begin{pmatrix}0&h_{12}^{t}\\\
h_{21}&\widehat{H}\end{pmatrix}$ (4.54)

with

$\begin{split}h_{12}=&-A^{t}V_{0}(mA(r_{21}-m^{-1}\widehat{R}w_{0})+(1+mA)(r_{12}-m^{-1}\widehat{R}^{t}v_{0}))\\\
h_{21}=&-AV_{0}(mA^{t}(r_{12}-m^{-1}\widehat{R}^{t}v_{0})+(1+mA^{t})(r_{21}-m^{-1}\widehat{R}w_{0}))\\\
\widehat{H}=&\frac{A}{1+mA}\widehat{R}\frac{A}{1+mA}-AV_{0}(mA^{t}(r_{12}-m^{-1}\widehat{R}^{t}v_{0})+(1+mA^{t})(r_{21}-m^{-1}\widehat{R}w_{0}))m^{-1}v^{t}\\\
&\qquad-
wm^{-1}((r_{21}^{t}-m^{-1}w_{0}^{t}\widehat{R})mA^{t}+(r_{12}^{t}-m^{-1}v_{0}^{t}\widehat{R})(1+mA^{t}))V_{0}A,\end{split}$
(4.55)

where $V_{0}=V_{0}(m)$ was introduced in (4.19) and $v_{0}$ and $w_{0}$ where
defined in (4.15). Their explicit form in terms of $m=m_{0}$ is given in
(4.18). From $F=0$, we also have
$\mathcal{C}_{J}^{\perp}\mathcal{C}_{M^{-1}}F=0$ and both
$\widetilde{M}=\widetilde{M}_{0}$ and $z$ are uniquely defined by $m$ (see
(4.18) and (2.4)). Therefore the total derivative of
$\mathcal{C}_{J}^{\perp}\mathcal{C}_{M^{-1}}F$ with respect to $m$ is well
defined and vanishes as well, i.e.

$0=\frac{\mathrm{d}}{\mathrm{d}m}\mathcal{C}_{J}^{\perp}\mathcal{C}_{M^{-1}}F=\mathcal{C}_{J}^{\perp}D_{J}\mathcal{C}_{M^{-1}}F+\mathcal{C}_{J}^{\perp}D_{\widetilde{M}^{\prime}}\mathcal{C}_{M^{-1}}F+z^{\prime}(m)\frac{\partial}{\partial
z}\mathcal{C}_{J}^{\perp}\mathcal{C}_{M^{-1}}F=\mathcal{B}_{21}[J]+\mathcal{B}_{22}[\widetilde{M}^{\prime}]$
(4.56)

where $\widetilde{M}^{\prime}:=\frac{\partial\widetilde{M}}{\partial m}$ and
$z^{\prime}$ denote the derivative of $\mathcal{C}_{J}^{\perp}[M[m]]$ and $z$
with respect to $m$. In the last step we also used $\frac{\partial}{\partial
z}\mathcal{C}_{J}^{\perp}\mathcal{C}_{M^{-1}}F=0$, which follows from (4.14).
Since $\mathcal{B}_{22}$ is invertible on its image, (4.56) is equivalent to

$(\mathcal{B}_{22})^{-1}\mathcal{B}_{21}[J]=-\widetilde{M}^{\prime}.$ (4.57)

Therefore the Schur complement of $\mathcal{B}_{22}$ is given by

$(\mathcal{B}_{11}-\mathcal{B}_{12}(\mathcal{B}_{22})^{-1}\mathcal{B}_{21})[J]=\frac{\partial}{\partial
m}C_{J}[\mathcal{C}_{M^{-1}}F]+D_{\widetilde{M}^{\prime}}\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F].$
(4.58)

In other words, the Schur complement of $\mathcal{B}_{22}$ is the total
derivative of $\mathcal{C}_{J}[\mathcal{C}_{M^{-1}}F]$ with respect to $m$ for
fixed $z$. Calculating it, we find

$(\mathcal{B}_{11}-\mathcal{B}_{12}(\mathcal{B}_{22})^{-1}\mathcal{B}_{21})[J]=\left(\frac{1}{m^{2}}-\gamma^{\prime}(m)\right)J=h(m)J,$
(4.59)

where $h$ was introduced in Definition 3.4. By Lemma 3.5, Lemma 3.6 and Lemma
A.3 we have $h(m)\neq 0$ for all
$z\in\mathbb{R}\setminus\operatorname{supp}(\rho).$ Therefore, the Schur
complement of $\mathcal{B}_{22}$ is invertible on the image of
$\mathcal{C}_{J}$ for all $z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$.
Since both $\mathcal{B}_{22}$ and its Schur complement are invertible on their
respective images for all $z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$,
the operator $\mathcal{B}$ for all
$z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$ is also invertible with its
inverse given by the Schur complement formula

$\begin{split}\mathcal{B}^{-1}[R]&=\left(\mathbb{1}-(\mathcal{B}_{22})^{-1}\mathcal{B}_{21}\right)\left(\mathcal{B}_{11}-\mathcal{B}_{12}(\mathcal{B}_{22})^{-1}\mathcal{B}_{21}\right)^{-1}\left[rJ-\mathcal{B}_{12}(\mathcal{B}_{22})^{-1}[\widetilde{R}]\right]+(\mathcal{B}_{22})^{-1}[\widetilde{R}]\\\
&=h(m)^{-1}\left(r-\left\langle
J,\mathcal{B}_{12}(\mathcal{B}_{22})^{-1}[\widetilde{R}]\right\rangle\right)M^{\prime}+(\mathcal{B}_{22})^{-1}[\widetilde{R}],\end{split}$
(4.60)

where $M^{\prime}=J+\widetilde{M}^{\prime}$ is the derivative of $M$ with
respect to $m$ and the inverse of the Schur complement is acting on
$\operatorname{span}(J)$. Therefore, $\mathcal{L}$ is also invertible with
inverse $\mathcal{L}^{-1}=\mathcal{B}^{-1}\mathcal{C}_{M^{-1}}$ for
$z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$. Now let $A$ be non-
invertible. Then, $A+\varepsilon$ is invertible for all $\varepsilon\in(0,u]$
for some $u>0$. We use an $\varepsilon$ subscript to denote quantities with
$A$ being replaced by $A^{\varepsilon}:=A+\varepsilon$, e.g.
$\mathcal{L}^{\varepsilon}$, $(\mathcal{B}^{-1})^{\varepsilon}$ etc. We have
$\mathcal{L}^{\varepsilon}(\mathcal{B}^{-1}\mathcal{C}_{M^{-1}})^{\varepsilon}=\mathbb{1}$
for all $\varepsilon\in(0,u]$ and both the limits $\lim_{\varepsilon\searrow
0}\mathcal{L}^{\varepsilon}$ and $\lim_{\varepsilon\searrow
0}(\mathcal{B}^{-1}\mathcal{C}_{M^{-1}})^{\varepsilon}$ exist. Indeed,
$\mathcal{L}^{\varepsilon}$ is the derivative of $F^{\varepsilon}$, which is
smooth at $\varepsilon=0$, proving the existence of $\lim_{\varepsilon\searrow
0}\mathcal{L}^{\varepsilon}$. Furthermore we explicitly calculate the
expression for $(\mathcal{B}^{-1}\mathcal{C}_{M^{-1}})^{\varepsilon}$ in terms
of $A^{\varepsilon}$ and $m^{\varepsilon}$ by using (4.17), (4.54), (4.55) and
(4.60). In the resulting expression, $(A^{\varepsilon})^{-1}$, appearing in
(4.17), cancels and therefore the limit $\lim_{\varepsilon\searrow
0}(\mathcal{B}^{-1}\mathcal{C}_{M^{-1}})^{\varepsilon}$ exists. We leave the
details to the reader. Therefore, $\mathcal{L}$ is also invertible for
$z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$ and we have
$\mathcal{L}^{-1}=\lim_{\varepsilon\searrow
0}(\mathcal{B}^{-1}\mathcal{C}_{M^{-1}})^{\varepsilon}$.

For any $C>0$ the norm of the operator $\mathcal{L}^{-1}$ is bounded on the
compact subset $C^{-1}\leq\operatorname{dist}(z,\operatorname{supp}(\rho))\leq
C$ of $\mathbb{R}\setminus\operatorname{supp}(\rho)$, i.e.
$\|\mathcal{L}^{-1}\|_{\mathrm{sp}}\sim_{C}1$. By continuity, there is some
$\eta_{0}>0,$ depending on $C$, such that
$\|\mathscr{L}^{-1}\|_{\mathrm{sp}}\sim_{C}1$ still holds for
$z=E+\mathrm{i}\eta\in\mathbb{H}$ with
$C^{-1}\leq\operatorname{dist}(E,\operatorname{supp}(\rho))\leq C$ and
$\eta\leq\eta_{0}$ and we have therefore shown (4.44).

Let $\mathcal{Q}_{M^{\prime}}$ be the projection onto the orthogonal
complement of $M^{\prime}$. By (4.54), (4.55) and (4.60) we have

$\|\mathcal{Q}_{M^{\prime}}\mathscr{L}^{-1}\|_{\mathrm{sp}}\lesssim\left\|\frac{1}{1+Am}\right\|+\|V_{0}\|\lesssim\operatorname{dist}\left(-1,\operatorname{Spec}(mA)\right)+\operatorname{dist}\left(-\frac{1}{2},\operatorname{Spec}(m\widehat{A})\right)$
(4.61)

uniformly in $z\in\mathbb{R}\setminus\operatorname{supp}(\rho)$. At
$z=\tau_{0}$ we have $h(m)=0$ and thus

$\left\|\frac{(mA)^{2}}{(1+mA)^{2}}\right\|\leq\operatorname{Tr}\left(\frac{(mA)^{2}}{(1+mA)^{2}}\right)+b^{t}\left(\frac{V_{0}}{m}\right)^{3}b=1.$
(4.62)

In the above equation, we have equality if and only if
$\operatorname{rank}A=1$ and $b=0$. Therefore, we have
$\operatorname{Spec}(mA)\subset[-\frac{1}{2},\infty)$ and
$-\frac{1}{2}\in\operatorname{Spec}(A)$ if and only if
$\operatorname{rank}A=1$ and $b=0$. By Lemma A.1, we have
$\min\operatorname{Spec}(m\widehat{A})=\min\operatorname{Spec}(mA)$ if and
only if $mA\in\mathbb{R}^{l\times l}$. Therefore we have
$-\frac{1}{2}\in\operatorname{Spec}(m\widehat{A})$ if and only if
$\operatorname{rank}A=1$, $A\in\mathbb{R}^{l\times l}$ and $b=0$. That is,
$-\frac{1}{2}\in\operatorname{Spec}(m\widehat{A})$ at $\tau_{0}$ if and only
if $A$ is a shifted square of a Wigner matrix. If $A$ is not a shifted square
of a Wigner matrix, then (4.61) is uniformly bounded in some neighbourhood of
$\tau_{0}.$ Thus $\mathcal{L}$ can at most have one vanishing eigenvalue at
$\tau_{0}$. Indeed let

$H(m,z):=F(m,\widetilde{M}[m],z)=M[m]+T_{z}(z+\Gamma[M[m]]T_{z})^{-1},$

where $M[m]$ is given by the right hand side of (4.18). The spectral parameter
$z$ is also uniquely defined by $m$ (see (2.4)), i.e. $H(m_{,}z(m))=0$ with

$z(m):=-\frac{1}{m}-\gamma(m).$

Taking the derivative with respect to $m$ we find

$0=\frac{\mathrm{d}}{\mathrm{d}m}H(m,z(m))=\partial_{m}H+\frac{\mathrm{d}z}{\mathrm{d}m}\partial_{z}H.$
(4.63)

By Corollary 3.3 the derivative $\frac{\mathrm{d}z}{\mathrm{d}m}$ vanishes at
the edge. Therefore at $z=\tau_{0}$ the above equation simplifies to

$\partial_{m}H=0.$ (4.64)

Calculating this we obtain

$0=M^{\prime}-M\Gamma[M^{\prime}]M=\mathcal{L}[M^{\prime}],$ (4.65)

with $M^{\prime}=M^{\prime}[m]$ the derivative $M$ with respect to $m$ and we
used $H=0$. Therefore $B=M^{\prime}[m]$ is the critical right eigenvector at
$\tau_{0}$. Next, we obtain the critical left eigenvector $L$. The adjoint of
the stability operator $\mathcal{L}$ is

$\mathcal{L}^{*}[R]=R-\Gamma[M^{*}RM^{*}].$ (4.66)

At any regular edge, we have $\operatorname{\mathrm{Im}}M=0$, i.e. $M=M^{*}.$
Therefore

$0=\Gamma[\mathcal{L}[B]]=\Gamma[B]-\Gamma[\mathcal{C}_{M}[\Gamma[B]]]=\mathcal{L}^{*}[\Gamma[B]]$
(4.67)

at $z=\tau_{0}$. Thus the critical left eigenvector is given by

$L=\Gamma[B].$ (4.68)

Note that $L\neq 0$ since $L_{ii}=B_{11}=1$ for all $1<i\leq l+1$ by the
expression for $\Gamma$ in (4.11). As $L$ and $B$ belong to the same non-
degenerate eigenvalue they cannot be orthogonal and since $\mathcal{L}$ does
not depend on $N$ they satisfy

$|\langle L,B\rangle|\sim 1$ (4.69)

at $\tau_{0}$. Equation (4.69) is also satisfied in a $u$ neighbourhood of
$\tau_{0}$, with $u\sim 1$, since $L$ and $B$ vary continuously in $z$.

To obtain $|\langle L,M\Gamma[B]B\rangle|$ we calculate the second total
derivative

$\begin{split}0&=\frac{\mathrm{d}^{2}}{\mathrm{d}{m}^{2}}H(m,z(m))\\\
&=\partial_{m}^{2}H(m,z(m))+\frac{\mathrm{d}^{2}z}{\mathrm{d}m^{2}}\partial_{z}H(m,z(m))\\\
&\quad+\frac{\mathrm{d}z}{\mathrm{d}m}\left(\frac{\mathrm{d}z}{\mathrm{d}m}\partial_{z}^{2}H(m,z(m))+2\partial_{m}\partial_{z}H(m,z(m))\right).\end{split}$
(4.70)

At the edge, $\frac{\mathrm{d}z}{\mathrm{d}m}$ once again vanishes, whereas
$\frac{\mathrm{d}^{2}z}{\mathrm{d}m^{2}}=c_{0}\neq 0$ does not by Corollary
3.3 and Lemma 4.1. Thus calculating the derivatives we arrive at

$\begin{split}0&=-c_{0}MJM+M^{\prime\prime}-M\Gamma[M^{\prime\prime}]M-2M\Gamma[M^{\prime}]M\Gamma[M^{\prime}]M\\\
&=-c_{0}MJM+\mathcal{L}[M^{\prime\prime}]-2M\Gamma[M^{\prime}]M^{\prime},\end{split}$
(4.71)

where we used $H=0$ multiple times and (4.65) in the last step. Next, we solve
for the last term, use $M^{\prime}=B$ and take an inner product with $L$:

$\begin{split}\langle L,M\Gamma[B]B\rangle&=\frac{1}{2}\left(\langle
L,\mathcal{L}[M^{\prime\prime}]\rangle-c_{0}\langle L,MJM\rangle\right)\\\
&=\frac{1}{2}\left(\langle\mathcal{L}^{*}[L],M^{\prime\prime}\rangle-
c_{0}\langle MLM,J\rangle\right)=-\frac{c_{0}}{2}B_{11}\neq 0.\end{split}$
(4.72)

In the last equality, $MLM=M\Gamma[B]M=B$ was used. By continuity the relation
$|\langle L,M\Gamma[B]B\rangle|\sim 1$ also holds in a $u$ neighbourhood of
$\tau_{0}$ with $u\sim 1$. At $\tau_{0}$, the operator $\mathcal{L}$ is
independent of $\delta$ and therefore its vanishing eigenvectors $L$ and $B$
are as well. Consequently, (4.69) and (4.72) also hold for all
$\delta\in[0,1]$ and as both expressions are also continuous in $z$ for all
$\delta$, they also hold in some order one neighborhood of $\tau_{0}.$

Next we study how $\mathcal{L}$ varies in $z$ around $\tau_{0}$ for arbitrary
$\delta\in[0,1]$ and we make the dependence explicit by writing
$\mathcal{L}=\mathcal{L}^{z}$, $M_{\delta}=M^{z}_{\delta}$, etc. and define
$\mathcal{E}^{z}:=\mathcal{L}^{z}-\mathcal{L}^{\tau_{0}}.$ As the eigenvalues
of $\mathcal{L}^{z}$ depend continuously on $z$, there is a $u>0$ such that
$\mathcal{L}^{z}$ has an isolated small eigenvalue $\beta^{z}$ for all
$|z-\tau_{0}|<u.$ By perturbation theory $\beta_{z}$ is given by

$\beta^{z}=(\langle L^{\tau_{0}},B^{\tau_{0}}\rangle)^{-1}\langle
L^{\tau_{0}},\mathscr{E}^{z}(B^{\tau_{0}})\rangle+\mathcal{O}(\|\mathscr{E}^{z}\|^{2}_{\mathrm{sp}}).$
(4.73)

To estimate the right hand side, we first evaluate
$\mathcal{E}^{z}[B^{\tau_{0}}]$ and find

$\mathcal{E}^{z}[B^{\tau_{0}}]=\mathcal{L}^{z}[B^{\tau_{0}}]-\mathcal{L}^{\tau_{0}}[B^{\tau_{0}}]=\mathcal{C}_{M^{z}_{\delta}}[L^{\tau_{0}}]-C_{M^{\tau_{0}}_{\delta}}[L^{\tau_{0}}]=M^{\tau_{0}}_{\delta}L^{\tau_{0}}(M^{\tau_{0}}_{\delta}-M^{z}_{\delta})+(M^{\tau_{0}}_{\delta}-M^{z}_{\delta})L^{\tau_{0}}M^{z}_{\delta}.$
(4.74)

We take the scalar product with $L^{\tau_{0}}$ and use the cyclic invariance
of the trace to obtain

$\begin{split}|\langle
L^{\tau_{0}},\mathcal{E}^{z}[B^{\tau_{0}}]\rangle|&=|\langle
L^{\tau_{0}}(M^{\tau_{0}}_{\delta}+M^{z}_{\delta})L^{\tau_{0}}(M^{\tau_{0}}_{\delta}-M^{z}_{\delta})\rangle|\\\
&=2|\langle
L^{\tau_{0}}M^{\tau_{0}}_{\delta}L^{\tau_{0}}(M^{\tau_{0}}_{\delta}-M^{z}_{\delta})\rangle|+\mathcal{O}(|m^{z}_{\delta}-m_{\delta}^{\tau_{0}}|^{2})\\\
&=2|\langle
L^{\tau_{0}}M^{\tau_{0}}_{\delta}L^{\tau_{0}}(M^{\tau_{0}}_{0})^{\prime}\rangle(m_{\delta}^{\tau_{0}}-m_{\delta}^{z})|+\mathcal{O}(|m_{\delta}^{z}-m_{\delta}^{\tau_{0}}|^{2}+\eta\delta)\\\
&=2|\langle
L^{\tau_{0}},M^{\tau_{0}}_{\delta}\Gamma[B^{\tau_{0}}]B^{\tau_{0}}\rangle||m_{\delta}^{\tau_{0}}-m_{\delta}^{z}|+\mathcal{O}(|m_{\delta}^{z}-m_{\delta}^{\tau_{0}}|^{2}+\eta\delta)\\\
&\sim|m_{\delta}^{\tau_{0}}-m_{\delta}^{z}|\sim\sqrt{\kappa+\eta}.\end{split}$
(4.75)

Here, the third equality follows from the fact that
$M_{\delta}[m_{\delta}]=M_{0}(m_{\delta})+\mathcal{O}(\eta\delta)$ and $M_{0}$
is analytic in $m_{\delta}$ at $m_{\delta}^{\tau_{0}}=m^{\tau_{0}}$. In the
fourth line we used (4.68), $M^{\prime}=B$ and in the fifth line that
$|\langle
L^{\tau_{0}},M^{\tau_{0}}_{\delta}\Gamma[B^{\tau_{0}}]B^{\tau_{0}}\rangle|$ is
non-vanishing. In the last relation, we used Corollary 3.3 and Lemma 4.1.
Taking the absolute value (4.73), the asymptotic behaviour of $|\beta^{z}|$
follows from (4.75) and (4.69), i.e.

$|\beta^{z}|\sim\sqrt{\kappa+\eta}+\mathcal{O}(\|\mathscr{E}^{z}\|^{2}_{\mathrm{sp}})\sim\sqrt{\kappa+\eta}.$
(4.76)

In the last step, we used that

$\|\mathscr{E}^{z}\|_{\mathrm{sp}}=\mathcal{O}(|m_{\delta}^{z}-m_{\delta}^{\tau_{0}}|).$
(4.77)

∎

##  5 Proof of the local law

In this section, we prove Theorem 2.9 and Proposition 2.10. For shifted
squares of Wigner matrices (see Definition 4.2), we provide a direct proof
below in Lemma 5.1. The Stieltjes transform $m$ and the resolvent $\mathbf{g}$
are submatrices of $M_{0}$ and $\mathbf{G}_{0}$, introduced in (4.18) and
(4.4), respectively. For polynomials that are not shifted squares of Wigner
matrices, the local laws are therefore a direct consequence of Proposition 4.3
and Proposition 4.4, the local laws for the linearization around regular edges
and away from the spectrum, and we will spend most of the section proving
them. The main steps of our proof for the edge local law, Proposition 4.3,
follow the general strategy of [1, Proposition 3.3]. Here, we briefly describe
the main ideas of the proof. Throughout this section we will use the notation
$\mathbf{\Delta}_{\delta}:=\mathbf{G}_{\delta}-M_{\delta}$. First, we
establish a global law away from the spectrum, stated below in Proposition
5.7. Then, we use the global law as a starting point for a bootstrapping
process. The bootstrapping proposition, Proposition 5.8, establishes a local
law iteratively on scales ever closer to the optimal scale, $\eta\sim
N^{-1+\varepsilon}$. Lemmata 5.4, 5.5 and 5.6 are auxiliary results used in
the bootstrapping process. Lemma 5.5 establishes a bound for
$\mathbf{\Delta}_{\delta}$ in terms of the error $\mathbf{D}$ and
$\Theta_{\delta}$, the projection of $\mathbf{\Delta}_{\delta}$ onto its
unstable direction, as well as an approximate quadratic equation for
$\Theta_{\delta}$. Lemma 5.6 transforms the quadratic bound on
$\Theta_{\delta}$ into a linear bound. The most crucial difference between our
proof and that of [1, Proposition 3.3] is addressed in Lemma 5.4. It provides
a naive upper bound for $\mathbf{G}_{0}^{*}\mathbf{G}_{0}$ and allows us to
estimate $\mathbf{G}_{0}^{*}\mathbf{G}_{0}$ in terms of
$\mathbf{G}_{1}^{*}\mathbf{G}_{1}$. Unlike $\mathbf{G}_{0}$, the matrix
$\mathbf{G}_{1}$ is a resolvent and thus satisfies
$\mathbf{G}_{1}^{*}\mathbf{G}_{1}=\eta^{-1}\operatorname{\mathrm{Im}}\mathbf{G}_{1}$.
As $\mathbf{D}$ is in turn bounded by
$\mathbf{G}_{\delta}^{*}\mathbf{G}_{\delta}$ this step is necessary to obtain
the correct upper bound for $\mathbf{G}_{0}-M_{0}$. The proof of the local law
away from the spectrum, Proposition 4.4, makes use of a similar but more
simple strategy since $\mathscr{L}$ does not have an unstable direction away
from $\operatorname{supp}\rho$.

######  Lemma 5.1.

Let $q$ be a shifted square of a Wigner matrix as introduced in Definition 4.2
and let $\rho$ have a regular edge at $\tau_{0}$.

  1. 1.

There is a $\kappa_{0}>0$, depending only on the coefficients of $q$, such
that for all $\varepsilon,\gamma,D>0$ and
$z\in\mathbb{D}_{\gamma}^{\kappa_{0}}$ the isotropic local law (2.16) holds
for all deterministic $\mathbf{x},\mathbf{y}\in\mathbb{C}^{N}$ and the
averaged local law (2.17) holds for all deterministic
$\mathbf{B}\in\mathbb{C}^{N\times N}$. If additionally
$E\notin\operatorname{supp}(\rho),$ we also have the improved local law
(2.18).

  2. 2.

For all $C>0$ there is an $\eta_{0}>0$, depending only on the coefficients of
$q$, such that the averaged local law (2.19) holds true for all deterministic
$\mathbf{B}\in\mathbb{C}^{N\times N}$, $\gamma,\varepsilon,D>0$ and
$z\in\mathbb{G}_{\gamma}^{C,\eta_{0}}$.

###### Proof.

By assumption, $q$ is a shifted square of a Wigner matrix. Thus it is of the
form

$q(\mathbf{X})=q_{a,c}(\mathbf{W}):=a\mathbf{W}^{2}+c,$ (5.1)

with $a,c\in\mathbb{R}$, $a\neq 0$ and $\mathbf{W}$ being a Wigner matrix. By
$\mathbf{g}_{a,c}$ we denote the resolvent of (5.1) and by $m_{a,c}$ the
solution of (2.4) for (5.1) with explicitly stated dependence on $a,c$. We
have

$\mathbf{g}_{a,c}(z)=\frac{1}{a\mathbf{W}^{2}+c-z}=\frac{1}{a}\mathbf{g}_{1,0}\left(\frac{z-c}{a}\right)$
(5.2)

and

$m_{a,c}(z)=\frac{1}{a}m_{1,0}\left(\frac{z-c}{a}\right).$ (5.3)

Therefore $q_{a,c}$ has a regular edge at $a\tau_{0}+c$ if and only if
$q_{1,0}$ has a regular edge at $\tau_{0}$ and Theorem 2.9 for general $a,c$
follows from $a=1$, $c=0$. Thus let w.l.o.g. $a=1$, $c=0$ and we drop the
subscripts again from $m$ and $\mathbf{g}$.

For $\zeta\in\mathbb{H}$ let
$\mathbf{g}_{\mathbf{W}}:=\frac{1}{\mathbf{W}-\zeta}$ be the resolvent of
$\mathbf{W}$ at spectral parameter $\zeta$. We have

$\mathbf{g}(z)=\frac{1}{2\sqrt{z}}\left(\mathbf{g}_{\mathbf{W}}(\sqrt{z})+\mathbf{g}_{-\mathbf{W}}(\sqrt{z})\right),$
(5.4)

where again the square root function is chosen such that the positive real
axis is mapped to itself and with a branch cut along the negative real axis.
Let $\rho_{\mathrm{sc}}(x):=\frac{1}{2\pi}\sqrt{(4-x^{2})_{+}}$ with
$(y)_{+}:=\max\\{y,0\\}$ be the semi-circle density and $m_{\mathrm{sc}}$ its
Stieltjes transform. By explicitly solving (2.4) for $q=\mathbf{W}^{2}$ and
$m_{\mathrm{sc}}$ we find

$m(z)=\frac{1}{\sqrt{z}}\left(-\sqrt{z}+\sqrt{z-4}\right)=\frac{1}{\sqrt{z}}m_{\mathrm{sc}}(\sqrt{z}).$
(5.5)

for all $z\in\mathbb{H}$. The semi-circle density $\rho_{\mathrm{sc}}$ has
regular edges at $\pm 2$, therefore $\rho$ has exactly one regular edge at
$\tau_{0}=4$. Its left edge is a hard edge and has a singularity at zero.
Combining (5.4) and (5.5) we find

$\mathbf{g}(z)-m(z)=\frac{1}{2\sqrt{z}}\left((\mathbf{g}_{\mathbf{W}}(\sqrt{z})-m_{\mathrm{sc}}(\sqrt{z}))+(\mathbf{g}_{-\mathbf{W}}(\sqrt{z})-m_{\mathrm{sc}}(\sqrt{z}))\right).$
(5.6)

Note that $-\mathbf{W}$ is also a Wigner matrix. Therefore, Statement 1 of
Lemma 5.1 around $\tau_{0}=4a+c$ follows from [1, Theorem 2.6] and Statement 2
of Lemma 5.1 Proposition 2.10 from [22, Theorem 2.1]. ∎

From now on we assume that $q$ is not a shifted square of a Wigner matrix. To
prove Proposition 4.3 and Proposition 4.4, we will require several
intermediate results, which we state below. Throughout the remainder of the
section, we assume w.l.o.g. that there is some $\alpha>0$ such that

$\|\mathbf{X}_{i}\|\lesssim N^{\alpha}.$ (5.7)

The assumption can be removed by a standard argument using Chebyshev’s
inequality and our moment assumption (2.1).

We also introduce stochastic domination, a commonly used notation used to
state high probability bounds in a way well adapted to our needs. It has first
been introduced in a slightly different form in [19], for the form stated here
see e.g. [20].

######  Definition 5.2 (Stochastic domination).

Let $X=(X^{(N)})_{N\in\mathbb{N}}$ and $Y=(Y^{(N)})_{N\in\mathbb{N}}$ be
families of non-negative random variables. We say $X$ is stochastically
dominated by $Y$ if for all (small) $\varepsilon>0$ and (large) $D>0$

$\mathbb{P}\left(X^{(N)}\geq
N^{\varepsilon}Y^{(N)}\right)\lesssim_{D,\varepsilon}N^{-D}.$ (5.8)

We denote this relation by $X\prec Y$. If the constant in the definition
depends on any other parameters $\alpha$, we write $X\prec_{\alpha}Y$.

Furthermore, we introduce the norm
$\|\cdot\|_{*}:=\|\cdot\|_{*}^{K,\mathbf{x},\mathbf{y}}$ for deterministic
$\mathbf{x},\mathbf{y}\in\mathbb{C}^{kN}$ and $k,K\in\mathbb{N}$. Our
definition follows that of [1] and it goes back to [22] and we refer to those
two works for details. First, we define the set

$I_{0}:=\\{\mathbf{x},\mathbf{y}\\}\cup\\{\mathbf{e}_{a},L^{*}_{a\cdot}:\,a\in\llbracket
kN\rrbracket\\},$ (5.9)

where $\mathbf{e}_{a}$ is the $a^{\mathrm{th}}$ standard base vector.
Replacing a scalar index by a dot denotes the vector that runs over the entire
range of the index, e.g.
$\mathbf{R}_{a\cdot}:=(\mathbf{R}_{ab})_{b\in\llbracket kN\rrbracket}$ is the
$a^{\mathrm{th}}$ row vector of a matrix $\mathbf{R}$. For $j\in\mathbb{N}$ we
define the set $I_{j}$ recursively by

$I_{j+1}:=I_{j}\cup\\{M_{\delta}\mathbf{u}:\,\mathbf{u}\in
I_{j}\\}\cup\\{\kappa_{c}((M_{\delta}\mathbf{u})a,b\cdot),\kappa_{d}((M_{\delta}\mathbf{u})a,\cdot
b):\,\mathbf{u}\in I_{j},a,b\in\llbracket kN\rrbracket\\},$ (5.10)

where $\kappa(ab,cd)$ denotes the cumulant of the $(a,b)$ entry and the
$(c,d)$ entry of $\sum_{j=1}^{l}K_{j}\otimes\mathbf{X}_{j}$ and $\kappa_{c}$
and $\kappa_{d}$ denote the decomposition of $\kappa$ into its direct and its
cross contribution according to the Hermitian symmetry. In (5.10) we also use
the shorthand notation $\kappa(\mathbf{x}b,cd):=\sum_{a}x_{a}\kappa(ab,cd)$.
Then $\|\cdot\|_{*}$ is defined as

$\|\mathbf{R}\|_{*}:=\sum_{0\leq j\leq
K}N^{-\frac{j}{2K}}\|\mathbf{R}\|_{I_{j}}+N^{-\frac{1}{2}}\max_{\mathbf{u}\in
I_{K}}\frac{\|\mathbf{R}\mathbf{u}\|}{\|\mathbf{u}\|},\quad\|\mathbf{R}\|_{I_{j}}:=\max_{\mathbf{u},\mathbf{v}\in
I_{j}}\frac{|\langle\mathbf{v},\mathbf{R}\mathbf{u}\rangle|}{\|\mathbf{u}\|\|\mathbf{v}\|}$
(5.11)

The notion of stochastic domination is closely related to bounds in p-norms as
can be seen from the following lemma.

######  Lemma 5.3 ([22, Lemma 5.4]).

Let $\mathbf{R}\in C^{kN\times kN}$, $k\in\mathbb{N}$, be a random matrix and
$\Phi$ be a stochastic control parameter. Then the following holds true

  1. 1.

If $\Phi\gtrsim N^{-C}$ and $\|\mathbf{R}\|\lesssim N^{C}$ for some $C>0$ and
$|\langle\mathbf{x},\mathbf{R}\mathbf{y}\rangle|\prec\Phi\|\mathbf{x}\|\|\mathbf{y}\|$
for all $\mathbf{x},\mathbf{y}$, then
$\|\mathbf{R}\|_{p}\lesssim_{\varepsilon,p}N^{\varepsilon}\Phi$ for all
$\varepsilon>0$, $p\in\mathbb{N}$.

  2. 2.

If $\|\mathbf{R}\|_{p}\lesssim_{\varepsilon,p}N^{\varepsilon}\Phi$ for all
$\varepsilon>0$, $p\in\mathbb{N}$ then
$\|\mathbf{R}\|_{*}^{K,\mathbf{x},\mathbf{y}}\prec\Phi$ for any fixed
$K\in\mathbb{N}$ and $\mathbf{x},\mathbf{y}\in\mathbb{C}^{N}$.

######  Lemma 5.4 (A priori bound).

$\mathbf{G}_{0}$ satisfies the a priori bound

$\mathbf{G}_{0}^{*}\mathbf{G}_{0}\lesssim\frac{1}{\eta^{2}}(1+\|\mathbf{X}\|^{4})$
(5.12)

uniformly in $z\in\mathbb{H}$ with $\eta\lesssim 1$.

###### Proof.

We recall that $\mathbf{G}_{0}$ is given by

$\mathbf{G}_{0}=\begin{pmatrix}\mathbf{g}&\mathbf{g}\mathbf{X}^{t}A\\\
A\mathbf{X}\mathbf{g}&-A+A\mathbf{X}\mathbf{g}\mathbf{X}^{t}A\end{pmatrix}.$
(5.13)

We estimate $\mathbf{G}_{0}$ blockwise to obtain

$\mathbf{G}_{0}\mathbf{G}_{0}^{*}\leq\|\mathbf{G}_{0}\|^{2}\lesssim\|\mathbf{g}\|^{2}(1+\|\mathbf{X}\|^{4})\leq\frac{1}{\eta^{2}}(1+\|\mathbf{X}\|^{4}).$
(5.14)

The last inequality holds because $\mathbf{g}$ is a resolvent. ∎

The lemma is used in the last step of the following estimate on
$\mathbf{G}_{0}^{*}\mathbf{G}_{0}$,

$\begin{split}\mathbf{G}_{0}^{*}\mathbf{G}_{0}&\leq
2(\mathbf{G}_{1}^{*}\mathbf{G}_{1}+(\mathbf{G}_{0}-\mathbf{G}_{1})^{*}(\mathbf{G}_{0}-\mathbf{G}_{1}))\\\
&=2(\mathbf{G}_{1}^{*}\mathbf{G}_{1}+\eta^{2}\mathbf{G}_{1}^{*}(I_{l+1}-J)\mathbf{G}_{0}^{*}\mathbf{G}_{0}(I_{l+1}-J)\mathbf{G}_{1})\lesssim(1+\|\mathbf{X}\|^{4})\mathbf{G}_{1}^{*}\mathbf{G}_{1}.\end{split}$
(5.15)

In the first inequality, we applied Lemma A.2 from the appendix with
$R=\mathbf{G}_{1}$ and $T=\mathbf{G}_{0}-\mathbf{G}_{1}$. In the second step,
we used

$\mathbf{G}_{0}-\mathbf{G}_{1}=\mathbf{G}_{0}(\mathbf{G}_{1}^{-1}-\mathbf{G}_{0}^{-1})\mathbf{G}_{1}=-\mathrm{i}\eta\mathbf{G}_{0}(I_{l+1}-J)\mathbf{G}_{1}.$
(5.16)

Taking the p-norm on both sides of (5.15) and applying Hölder’s inequality we
obtain

$\|\mathbf{G}_{0}^{*}\mathbf{G}_{0}\|_{p}\lesssim_{p}(1+\|\|\mathbf{X}\|^{4}\|_{2p})\|\mathbf{G}_{1}^{*}\mathbf{G}_{1}\|_{2p}\lesssim_{p}\|\mathbf{G}_{1}^{*}\mathbf{G}_{1}\|_{2p}=\frac{\|\operatorname{\mathrm{Im}}\mathbf{G}_{1}\|_{2p}}{\eta}$
(5.17)

for all $p\in\mathbb{N}$. Here we use $\|Y\|_{p}:=(\mathbb{E}[|Y|^{p}])^{1/p}$
for scalar random variables $Y$. In the second to last step, we used
$\|\mathbf{X}\|_{p}\lesssim_{p}1$, which follows from Lemma 5.3 since
$\|\mathbf{X}\|\prec 1$ and $\|\mathbf{X}\|\leq N^{C}$ by assumption (5.7).

We define the sets

$\mathbb{D}^{\kappa_{0}}:=\\{z\in\mathbb{H}:\,|E-\tau_{0}|\leq\kappa_{0}\\},\quad\mathbb{G}^{C,\eta_{0}}:=\\{z\in\mathbb{H}:\,C^{-1}<\operatorname{dist}(E,\operatorname{supp}(\rho))<C,\eta\leq\eta_{0}\\}$
(5.18)

and the random variable

$\Theta_{\delta}:=\frac{\langle
L\otimes\mathbf{I}_{N},\mathbf{\Delta}_{\delta}\rangle}{\langle
L\otimes\mathbf{I}_{N},B\otimes\mathbf{I}_{N}\rangle}.$ (5.19)

With $\Theta_{\delta}$ we can write the projection of
$\mathbf{\Delta}_{\delta}$ onto the critical direction (see (4.35)) as
$\mathscr{P}[\mathbf{\Delta}_{\delta}]=\Theta_{\delta}B$. Furthermore, let
$\chi(A)$ denote the indicator function on event $A$. We import following
lemmata from [1, Proposition 3.3] and [1, Lemma 3.9].

######  Lemma 5.5 ([1, Proposition 3.3]).

Let $\delta\in\\{0,1\\}$ and
$\|\cdot\|_{*}=\|\cdot\|_{*}^{K,\mathbf{x},\mathbf{y}}$. There is a
$\kappa_{0}\sim 1$ and deterministic matrices $\mathbf{R}_{i}$ with
$\|\mathbf{R}_{i}\|\lesssim 1$ for $i=1,2$ such that

$\mathbf{\Delta}_{\delta}\chi(\|\mathbf{\Delta}_{\delta}\|_{*}\leq
N^{-\frac{3}{K}})=(\Theta_{\delta}B-\mathscr{L}^{-1}[(\mathbb{1}-\mathscr{P})[M_{\delta}\mathbf{D}]]+\mathcal{E})\chi(\|\mathbf{\Delta}_{\delta}\|_{*}\leq
N^{-\frac{3}{K}}),$ (5.20)

with an error function $\mathcal{E}$ of size

$\|\mathcal{E}\|_{*}=\mathcal{O}\left(N^{\frac{2}{K}}(|\Theta_{\delta}|^{2}+\|\mathbf{D}\|_{*}^{2})\right)$
(5.21)

and $\Theta_{\delta}$ satisfying the approximate quadratic equation

$\left(\xi_{1}\Theta_{\delta}+\xi_{2}\Theta_{\delta}^{2}\right)\chi(\|\mathbf{\Delta}_{\delta}\|_{*}\leq
N^{-\frac{3}{K}})=\mathcal{O}\left(N^{\frac{2}{K}}\|\mathbf{D}\|_{*}^{2}+|\langle\mathbf{R}_{1},\mathbf{D}\rangle|+|\langle\mathbf{R}_{2},\mathbf{D}\rangle|\right)$
(5.22)

with $\xi_{1}\sim\sqrt{\kappa+\eta}$, $\xi_{2}\sim 1$ uniformly in
$\mathbf{x},\mathbf{y}\in\mathbb{C}^{(l+1)N}$ and
$z\in\mathbb{D}^{\kappa_{0}}$.

###### Proof.

The proof of [1, Proposition 3.3] uses [1, Proposition 3.1] as an input. After
replacing [1, Proposition 3.1] by our analogous Proposition 4.6 the proof
follows theirs line by line. ∎

######  Lemma 5.6 ([1, Lemma 3.9]).

Let $d=d(\eta)$ be a monotonically decreasing function in $\eta\geq N^{-1}$
and assume $0\leq d\lesssim N^{-\varepsilon}$ for some $\varepsilon>0$.
Suppose there are $\kappa_{0},\gamma>0$ such that

$|\xi_{1}\Theta_{\delta}+\xi_{2}\Theta_{\delta}^{2}|\lesssim d\text{ for all
}z\in\mathbb{D}_{\gamma}^{\kappa_{0}}\quad\text{and}\quad|\Theta_{\delta}|\lesssim\min\left\\{\frac{d}{\sqrt{\kappa+\eta}},\sqrt{d}\right\\}\text{for
some }z_{0}\in\mathbb{D}_{\gamma}^{\kappa_{0}}.$ (5.23)

Then also $|\Theta_{\delta}|\lesssim\min\\{d/\sqrt{\kappa+\eta},\sqrt{d}\\}$
for all $z^{\prime}\in\mathbb{D}_{\gamma}^{\kappa_{0}}$ with
$\operatorname{\mathrm{Re}}z^{\prime}=\operatorname{\mathrm{Re}}z_{0}$ and
$\operatorname{\mathrm{Im}}z^{\prime}\leq\operatorname{\mathrm{Im}}z_{0}$.

######  Proposition 5.7 (Global Law).

For all $C>0$ and some $\kappa_{0}>0$ with $\kappa_{0}\sim 1$ there is an
$\eta_{0}>0$ such that for all $\delta\in[0,1]$,
$z\in\mathbb{D}^{\kappa_{0}}\cup\mathbb{G}^{C,\eta_{0}}$ and $\varepsilon,D>0$
the isotropic global law,

$\mathbb{P}\left(|\langle\mathbf{x},\mathbf{\Delta}_{\delta}\mathbf{y}\rangle|>\|\mathbf{x}\|\|\mathbf{y}\|\frac{N^{\varepsilon}}{\sqrt{N}}\right)\lesssim_{\varepsilon,D,\eta,C}N^{-D}$
(5.24)

holds for all deterministic $\mathbf{x},\mathbf{y}\in\mathbb{C}^{(l+1)N}$.
Additionally, we have a global averaged law,

$\mathbb{P}\left(|\langle\mathbf{B}\mathbf{\Delta}_{\delta}\rangle|>\|\mathbf{B}\|\frac{N^{\varepsilon}}{N}\right)\lesssim_{\varepsilon,D,\eta,C}N^{-D},$
(5.25)

for all deterministic $\mathbf{B}\in\mathbb{C}^{(l+1)N\times(l+1)N}$.

###### Proof.

First, consider $\delta=1$ and $A$ being invertible. Then for any
$z\in\mathbb{H}$ the matrix $\mathbf{G}_{1}$ is a resolvent of the Hermitian
random matrix $\mathbf{L}-EJ$ with spectral parameter $\mathrm{i}\eta$. In
this regime the global law is covered by [22, Theorem 2.1].

Now consider $A$ to be non-invertible. Then there is a $u>0$ s.t.
$A+\varepsilon$ is invertible for all $0<\varepsilon<u$ and a global law for
$A+\varepsilon$ also follows from [22, Theorem 2.1]. For $C>0$ fix a
$z\in\mathbb{D}^{\kappa_{0}}\cup\mathbb{G}^{C,\eta_{0}}$ with
$\kappa_{0},\eta_{0}$ sufficiently small for Proposition 4.6 to be applicable.
$\mathscr{L}$ is continuous in $\varepsilon\in[0,u]$ for sufficiently small
$u\sim 1$ and Proposition 4.6 holds true uniformly for sufficiently small
$\varepsilon$. As $M_{1}$ and $\mathbf{G}_{1}$ are also continuous in
$\varepsilon\in[0,u]$ for some $u\sim 1$, the proposition follows from a
stochastic continuity argument.

For $\delta\in[0,1)$ and arbitrary $A$ the global law follows from another
stochastic continuity argument as $M_{\delta}$, $\mathbf{G}_{\delta}$ and
$\mathscr{L}$ are also continuous in $\delta$. ∎

Before starting our bootstrapping argument, we introduce the following notions
for isotropic and averaged stochastic dominance for random matrices
$\mathbf{R}$ and deterministic control parameters $\Lambda$,

$\begin{split}|\mathbf{R}|\prec\Lambda\text{ in
}\mathbb{D}\Leftrightarrow\|\mathbf{R}\|_{*}^{K,\mathbf{x},\mathbf{y}}\prec\Lambda\text{
uniform in }\mathbf{x},\mathbf{y}\text{ and }z\in\mathbb{D}\\\
|\mathbf{R}|_{\mathrm{av}}\prec\Lambda\text{ in
}\mathbb{D}\Leftrightarrow\frac{|\langle\mathbf{B}\mathbf{R}\rangle|}{\|\mathbf{B}\|}\prec\Lambda\text{
uniform in }\mathbf{B}\neq 0\text{ and }z\in\mathbb{D}.\end{split}$ (5.26)

######  Proposition 5.8 (Bootstrapping).

Assume the following:
1\. The isotropic and averaged local laws,

$|\bm{\Delta}_{\delta}|\prec
N^{\frac{2}{K}}\left(\sqrt{\frac{\operatorname{\mathrm{Im}}m}{N\eta}}+\frac{N^{\frac{2}{K}}}{N\eta}\right),\quad|\bm{\Delta}_{\delta}|_{\mathrm{av}}\prec\begin{cases}\displaystyle{\frac{N^{\frac{2}{K}}}{N\eta}},&E\in\mathrm{supp}{\rho},\\\\[10.0pt]
\displaystyle{\frac{N^{\frac{2}{K}}}{N(\eta+\kappa)}+\frac{N^{\frac{4}{K}}}{N^{2}\eta^{2}\sqrt{\eta+\kappa}}},&E\notin\mathrm{supp}{\rho},\end{cases}$
(5.27)

hold on $z=E+\mathrm{i}\eta\in\mathbb{D}_{\gamma_{0}}^{\kappa_{0}}$ for some
$\gamma_{0},\kappa_{0},K$ and $\delta\in\\{0,1\\}$.
2\. The isotropic and averaged local laws,

$|\bm{\Delta}_{\delta}|\prec_{C}N^{\frac{2}{K}}\left(\frac{1}{\sqrt{N}}+\frac{N^{\frac{2}{K}}}{N\eta}\right),\quad|\bm{\Delta}_{\delta}|_{\mathrm{av}}\prec_{C}\left(\frac{1}{N}+\frac{N^{\frac{4}{K}}}{(N\eta)^{2}}\right)$
(5.28)

hold on $z=E+\mathrm{i}\eta\in\mathbb{G}_{\gamma_{0}}^{C,\eta_{0}}$ for some
$\gamma_{0},C,\eta_{0},K$ and $\delta\in\\{0,1\\}$.

Then, for all $\gamma>0$ with $\gamma\geq\frac{100}{K}$ there is a
$\gamma_{s}>0$ independent of $\gamma_{0}$ such that the local laws (5.27)
also hold on $\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}$ with
$\gamma_{1}=\max\\{\gamma,\gamma_{0}-\gamma_{s}\\}$ and the local laws (5.28)
also hold on $\mathbb{G}_{\gamma_{1}}^{C,\eta_{0}}$ with
$\gamma_{1}=\max\\{\gamma,\gamma_{0}-\gamma_{s}\\}$.

###### Proof.

We first prove the local law (5.27) for
$z\in\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}$ and then comment on the
modifications necessary to prove (5.28) for
$z\in\mathbb{G}_{\gamma_{1}}^{C,\eta_{0}}$. We begin by proving that
$\eta\mapsto\eta\|\mathbf{G}_{\delta}\|_{p}$ is monotonically non-decreasing
in $\eta$. For $\delta=1$ the proof is given in the proof of [22, Proposition
5.5]. For $\delta=0$ the proof modifies as follows. Let $\varepsilon>0$. For
fixed $E$ we define $f(\eta):=\eta\|\mathbf{G}_{0}(E+\mathrm{i}\eta)\|_{p}$.
It satisfies

$\begin{split}\liminf_{\varepsilon\to
0}\frac{f(\eta+\varepsilon)-f(\eta)}{\varepsilon}&=\liminf_{\varepsilon\to
0}\|\mathbf{G}_{0}(E+\mathrm{i}(\eta+\varepsilon))\|_{p}+\frac{\eta(\|\mathbf{G}_{0}(E+\mathrm{i}(\eta+\varepsilon))\|_{p}-\|\mathbf{G}_{0}(E+\mathrm{i}\eta)\|_{p})}{\varepsilon}\\\
&\geq\|\mathbf{G}_{0}(E+\mathrm{i}\eta)\|_{p}-\lim_{\varepsilon\to
0}\eta\left\|\frac{\mathbf{G}_{0}(E+\mathrm{i}(\eta+\varepsilon))-\mathbf{G}_{0}(E+\mathrm{i}\eta)}{\varepsilon}\right\|_{p}\\\
&=\|\mathbf{G}_{0}(E+\mathrm{i}\eta)\|_{p}-\eta\|\mathbf{G}_{0}(E+\mathrm{i}\eta)J\mathbf{G}_{0}(E+\mathrm{i}\eta)\|_{p}\end{split}$
(5.29)

To obtain a bound for the last term we estimate

$\eta|\langle\mathbf{x},\mathbf{G}_{0}J\mathbf{G}_{0}\mathbf{y}\rangle|\leq\frac{\eta}{2}(\langle\mathbf{x},\mathbf{G}_{0}J\mathbf{G}_{0}^{*}\mathbf{x}\rangle+\langle\mathbf{y},\mathbf{G}_{0}^{*}J\mathbf{G}_{0}\mathbf{y}\rangle)=\frac{1}{2}(\langle\mathbf{x},\operatorname{\mathrm{Im}}\mathbf{G}_{0}\mathbf{x}\rangle+\langle\mathbf{y},\operatorname{\mathrm{Im}}\mathbf{G}_{0}\mathbf{y}\rangle)$
(5.30)

for $\mathbf{x},\mathbf{y}\in\mathbb{C}^{(l+1)N}$. In the last step, we used
the Ward identity for generalized resolvents,

$\mathbf{G}_{0}J\mathbf{G}_{0}=\frac{\operatorname{\mathrm{Im}}\mathbf{G}_{0}}{\eta}.$
(5.31)

Thus we find

$\eta\|\mathbf{G}_{0}J\mathbf{G}_{0}\|_{p}\leq\sup_{\|\mathbf{x}\|,\|\mathbf{y}\|=1}\left(\mathbb{E}\left(\frac{1}{2}(\langle\mathbf{x},\operatorname{\mathrm{Im}}\mathbf{G}_{0}\mathbf{x}\rangle+\langle\mathbf{y},\operatorname{\mathrm{Im}}\mathbf{G}_{0}\mathbf{y}\rangle)\right)^{p}\right)^{\frac{1}{p}}\leq\|\mathbf{G}_{0}\|_{p},$
(5.32)

where we used

$|\langle\mathbf{x},\operatorname{\mathrm{Im}}\mathbf{R}\mathbf{x}\rangle|\leq|\langle\mathbf{x},\mathbf{R}\mathbf{x}\rangle|$
(5.33)

in the last step. Therefore

$\liminf_{\varepsilon\to 0}\frac{f(\eta+\varepsilon)-f(\eta)}{\varepsilon}\geq
0$ (5.34)

and the claim follows.

For $\delta=0,1$, assume the local law (5.27) on
$\mathbb{D}_{\gamma_{0}}^{\kappa_{0}}$. Then
$\|\mathbf{G}_{\delta}\|_{p}\sim_{p}1$ on
$\mathbb{D}_{\gamma_{0}}^{\kappa_{0}}$ and by monotonicity of
$\eta\|\mathbf{G}_{\delta}\|_{p}$ we find

$\|\mathbf{G}_{\delta}\|_{p}\lesssim_{p}N^{\gamma_{s}}.$ (5.35)

on $\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}$. We choose $\gamma_{s}<\frac{1}{4}$.
Then Proposition 4.5 yields

$\|\mathbf{D}\|_{p}\lesssim_{p,\varepsilon}N^{\varepsilon+c\gamma_{s}}\sqrt{\frac{\|\mathbf{G}_{\delta}\mathbf{G}_{\delta}^{*}\|_{q}}{N}}\quad\text{and}\quad\|\mathbf{D}\|_{p}^{\mathrm{av}}\lesssim_{p,\varepsilon}N^{\varepsilon+c\gamma_{s}}\frac{\|\mathbf{G}_{\delta}\mathbf{G}_{\delta}^{*}\|_{q}}{N}.$
(5.36)

For $\delta=0$ estimate the quadratic term by making use of (5.17) and then
use the Ward identity on $\mathbf{G}_{1}\mathbf{G}_{1}^{*}$ for both
$\delta=0,1$ to obtain

$\|\mathbf{D}\|_{p}\lesssim_{p,\varepsilon}N^{\varepsilon+c\gamma_{s}}\sqrt{\frac{\|\operatorname{\mathrm{Im}}\mathbf{G}_{1}\|_{2q}}{\eta
N}}\lesssim_{p}\frac{N^{\frac{c^{\prime}}{2}\gamma_{s}}}{\sqrt{\eta
N}}\quad\text{and}\quad\|\mathbf{D}\|_{p}^{\mathrm{av}}\lesssim_{p,\varepsilon}N^{\varepsilon+c\gamma_{s}}\frac{\|\operatorname{\mathrm{Im}}\mathbf{G}_{1}\|_{2q}}{\eta
N}\lesssim_{p}\frac{N^{c^{\prime}\gamma_{s}}}{\eta N}.$ (5.37)

on $\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}$ for sufficiently small
$\varepsilon>0$ and some modified $c^{\prime}>0$. Note that we are allowed to
exchange the $q$ norm in the bound for $\delta=1$ by a $2q$ norm as the norm
is increasing in $q$. After using Lemma 5.3 to turn the p-norm bound in the
first equation of (5.37) into a bound on the *-norm bound, we get from Lemma
5.5

$|\xi_{1}\Theta_{\delta}+\xi_{2}\Theta_{\delta}^{2}|\chi(\|\bm{\Delta}_{\delta}\|_{*}\leq
N^{-\frac{3}{K}})\prec\frac{N^{\frac{2}{K}+c^{\prime}\gamma_{s}}}{\eta N}$
(5.38)

on $\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}.$ The left hand side is also
Lipschitz continuous with Lipschitz constant $\prec\eta^{-2}\leq N^{2}$. Thus
the inequality

$|\xi_{1}\Theta_{\delta}+\xi_{2}\Theta_{\delta}^{2}|\chi(\|\bm{\Delta}_{\delta}\|_{*}\leq
N^{-\frac{3}{K}})\leq N^{-\frac{10}{K}}$ (5.39)

holds with very high probability on all of
$\mathbb{D}_{\gamma_{1}}^{\kappa_{0}}$ for sufficiently large $K$ and
sufficiently small $\gamma_{s}$. By our assumption the local law, (5.27),
holds on $\mathbb{D}_{\gamma_{0}}^{\kappa_{0}}$. In conjunction with another
stochastic continuity argument, we also get the bound

$|\Theta_{\delta}|\leq\min\left\\{\frac{N^{-\frac{10}{K}}}{\sqrt{\kappa+\eta}},N^{-\frac{5}{K}}\right\\}$
(5.40)

on all of $\mathbb{D}_{\gamma_{0}}^{\kappa_{0}}$ with very high probability.
Therefore Lemma 5.6 can be applied with $d=N^{-10/K}$ and we obtain

$|\Theta_{\delta}|\chi(\|\bm{\Delta}_{\delta}\|_{*}\leq N^{-\frac{3}{K}})\prec
N^{-\frac{5}{K}}.$ (5.41)

Next, we turn this into a bound on $\bm{\Delta}_{\delta}.$ To do so we again
use the *-norm bound obtained from applying Lemma 5.3 to (5.37) to find
$\|\mathbf{D}\|_{*}\prec N^{-\frac{7}{K}}$ for sufficiently small $\gamma_{s}$
and sufficiently large $K$. Using both bounds on (5.20) we get