00footnotetext: Date: 00footnotetext: Keywords: Brown measure, matrix Dyson
equation, non-Hermitian random matrix 00footnotetext: MSC2010 Subject
Classifications: 60B20, 15B52.

# Spectrum occupies pseudospectrum for random matrices with diagonal
deformation and variance profile

Johannes Alt${}^{\text{a,}}$ Funding from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ
2047/1, project-id 390685813 is gratefully acknowledged.
Email<EMAIL_ADDRESS>Torben Krüger${}^{\text{b,}}$ Financial
support from VILLUM FONDEN Young Investigator Award (Grant No. 29369) is
gratefully acknowledged.
Email<EMAIL_ADDRESS>
###### Abstract

We consider $n\times n$ non-Hermitian random matrices with independent entries
and a variance profile, as well as an additive deterministic diagonal
deformation. We show that the support of the asymptotic eigenvalue
distribution in the complex plane exactly coincides with the
$\varepsilon$-pseudospectrum in the consecutive limits $n\to\infty$ and
$\varepsilon\to 0$. Furthermore, we provide a description of this support in
terms of a single real-valued function on the complex plane. As a level set of
this locally real analytic function, the spectral edge is a real analytic
variety of dimension at most one.

${}^{\text{a}}$Institute for Applied Mathematics, University of Bonn
${}^{\text{b}}$Department of Mathematics, FAU Erlangen-Nürnberg

##  1 Introduction

The empirical spectral distribution of a non-Hermitian random matrix typically
converges to a non-random probability distribution $\sigma$, the limiting
spectral measure, on the complex plane as its dimension tends to infinity. The
most prominent instance of this phenomenon is the circular law, stating that
the eigenvalues of am appropriately normalised matrix $X$ with centered i.i.d.
entries converges to the uniform distribution on the complex unit disk [25, 8]
(see [47] for optimal moment conditions and [17] for a review).

When a deterministic matrix $A$ is added to $X$, the associated asymptotic
distribution $\sigma$ and its support, the asymptotic spectrum, depend on $A$
in a complicated manner [33]. This distribution can be realised as the Brown
measure [18, 27], which is a generalisation of the spectral measure to non-
normal operators, of an element in a $W^{*}$-probability space with faithful,
tracial state $\langle\mspace{2.0mu}\cdot\mspace{2.0mu}\rangle$. In fact,
$\sigma=\sigma_{A+\mathfrak{c}}$ is the Brown measure of the sum of an
embedding of $A$ into the $W^{*}$-probability space and a circular element
$\mathfrak{c}$ that is $\ast$-free from $A$. In this case, the asymptotic
spectrum $\operatorname{supp}\sigma_{A+\mathfrak{c}}$ coincides with the
closure of
$\mathbb{S}=\\{\zeta\in\mathbb{C}\,:\,\langle(A-\zeta)^{-1}(A^{*}-\overline{\zeta}\\!\,)^{-1})\rangle>1\\}$.
This observation goes back to [33] in the random matrix setting, has been
proven in the infinite dimensional situation in [16] for normal $A$ and
extended to general $A$ in [15, 49]. Subsequently, the regularity of
$\sigma_{A+\mathfrak{c}}$ has been analysed. The measure is absolutely
continuous with respect to the Lebesgue measure on $\mathbb{C}$ [10] and the
density is strictly positive and real analytic on $\mathbb{S}$ [49]. Moreover,
the density typically possesses a jump discontinuity at the edge of
$\mathbb{S}$ [22].

Instead of adding $A$ to the matrix $X$ with i.i.d. entries we may also
introduce more structure into the randomness $X=(x_{ij})_{i,j=1}^{n}$. When
the entries $x_{ij}$ remain independent but admit differing distributions with
entry dependent variances $s_{ij}:=\mathbb{E}\mspace{1.0mu}\lvert
x_{ij}\rvert^{2}$, the density $\sigma$ is still supported on a disk, radially
symmetric and has a jump at the edge, but is in general not constant anymore
on its support [19, 3]. This remains true when the entries of $X$ are
correlated with a decaying correlation structure [6]. In this work we consider
a case in which a nontrivial structure of the randomness and a deterministic
deformation $A$ are present. Our randomness $X$ has independent entries with
variance profile $S=(s_{ij})_{i,j=1}^{n}$ and $A$ is diagonal. In particular,
the matrix $X+A$ considered in this work belongs to the class of Kronecker
matrices discussed in [5].

Non-normal random matrices and a detailed understanding of their spectra play
an important role in many applications, ranging from the stability analysis of
food webs [2, 35, 29] and quantum chaotic scattering [24] to investigating the
transition to chaos in neuronal networks [44, 38]. A persistent challenge in
the analytic study of such matrices $X$ is their spectral instability, i.e.
the fact that tiny changes in the matrix entries may lead to large deviations
of the eigenvalues. To remedy this issue the $\varepsilon$-pseudospectrum
$\operatorname{Spec}_{\varepsilon}(X)$ is introduced (see e.g. [48] for an
overview), which is stable under perturbations, monotonically increasing in
$\varepsilon>0$ and contains the spectrum, namely
$\bigcap_{\varepsilon>0}\operatorname{Spec}_{\varepsilon}(X)=\operatorname{Spec}(X)$.
Especially for high dimensional $X=X_{n}\in\mathbb{C}^{n\times n}$ the
dependence of $\operatorname{Spec}_{\varepsilon}(X_{n})$ on $\varepsilon$ may
be very unstably dependent on $n$ (see e.g. [36, Section 11.6.3] for the
example of a shift operator). In particular, the eigenvalues may accumulate in
a much smaller area than
$\operatorname{Spec}_{0}^{\infty}:=\lim_{\varepsilon\downarrow
0}\lim_{n\to\infty}\operatorname{Spec}_{\varepsilon}(X_{n})$. In the case of
Töplitz matrices $A$ with a very small added randomness $X$ for example, the
spectrum concentrates on curves given by the image of the unit circle by the
Toeplitz symbol inside $\operatorname{Spec}_{0}^{\infty}$ [26, 39, 9, 42].

In contrast, our main result shows that for matrices with independent entries
and diagonal deformation the set $\operatorname{Spec}_{0}^{\infty}$ coincides
with the support of the limiting spectral measure $\sigma$, i.e., that the
spectrum occupies the entire $\varepsilon$-pseudospectrum in the consecutive
limits $n\to\infty$ and $\varepsilon\downarrow 0$. Here we assume that
$s_{ij}=\frac{1}{n}s(\frac{i}{n},\frac{j}{n})$ and
$A=(a(\frac{i}{n})\delta_{ij})_{i,j=1}^{n}$ both have piecewise continuous
limiting profiles $s:[0,1]^{2}\to\mathbb{R}$ and $a:[0,1]\to\mathbb{C}$ to
ensure the existence of the limit as $n\to\infty$. In particular,
$\operatorname{Spec}_{0}^{\infty}$ stably depends on the expectation profile
$a$ and the variance profile $s$. Furthermore, we show that positivity of the
variance profile implies that $\sigma$ is given by a bounded probability
density on the complex plane, which is real analytic and strictly positive on
an open domain $\mathbb{S}:=\\{\beta<0\\}\subset\mathbb{C}$ with boundary
$\partial\mathbb{S}=\\{\beta=0\\}$, where $\beta:\mathbb{C}\to\mathbb{R}$ is a
continuous function that is real analytic in a neighbourhood of
$\partial\mathbb{S}$. From this we obtain that $\partial\mathbb{S}$ is a real
analytic variety of dimension at most $1$. The density $\sigma$ vanishes
outside the closure of $\mathbb{S}$ and typically has a jump discontinuity at
the spectral edge $\partial\mathbb{S}$, except at critical points of $\beta$,
where it vanishes continuously.

These results about the measure $\sigma$ have been shown in the case of
constant variance profile $s$, e.g. when the entries of $X$ are i.i.d., and
for general $A$ in [49, 10, 22]. For constant $s$ our choice of $\beta$
simplifies to $\beta(\zeta)=\frac{1}{n}\operatorname{Tr}\lvert
A-\zeta\rvert^{-2}$, which coincides with the analogous quantity in [49]. In
this situation, the study of properties of $\sigma$ and its relationship to
$\beta$ rely on solving a $\zeta$-dependent family of two coupled scalar
equations, called Dyson equation, for two positive functions
$v_{1},v_{2}:\mathbb{S}\to\mathbb{R}$ that vanish at the boundary
$\partial\mathbb{S}$. The Dyson equation is a self-consistent equation for the
diagonal resolvent entries of the Hermitization of $X+A$ in the $n\to\infty$
limit. In the random matrix setup, the Hermitization idea goes back to [25].
See e.g. [11] for its use in the analysis of Brown measures. For non-constant
$s$, the Dyson equation is no longer finite-dimensional in the $n\to\infty$
limit. Instead, with profiles $a$ and $s$, it becomes a system of two
$\zeta$-dependent equations of the form

$\frac{1}{v_{1}(\zeta)}=Sv_{2}(\zeta)+\frac{\lvert\zeta-a\rvert^{2}}{S^{*}v_{1}(\zeta)}\,,\qquad\frac{1}{v_{2}(\zeta)}=S^{*}v_{1}(\zeta)+\frac{\lvert\zeta-a\rvert^{2}}{Sv_{2}(\zeta)}\,,$
(1.1)

for two positive functions $v_{1},v_{2}:\mathbb{S}\to L^{\infty}[0,1]$, where
$S,S^{*}:L^{\infty}[0,1]\to L^{\infty}[0,1]$ are defined through
$(Sf)(x):=\int s(x,y)f(y)\mathrm{d}y$ and $(S^{*}f)(x):=\int
s(y,x)f(y)\mathrm{d}y$. From $v_{1}$ the probability density $\sigma$ inside
$\mathbb{S}$ is derived through

$\sigma(\zeta):=-\partial_{\zeta}\bigg{\langle}\frac{v_{1}(\zeta)(a-\zeta)}{\pi\mspace{2.0mu}S^{*}v_{1}(\zeta)}\bigg{\rangle}\,,$
(1.2)

where $\langle u\rangle:=\int u(x)\mathrm{d}x$. Taking the derivative in (1.2)
yields a quadratic form of a non-symmetric operator. The main idea for the
proof of positivity of $\sigma$ in the bulk regime, i.e. on $\mathbb{S}$, is
to transform the formula for $\sigma$ into the quadratic form of a strictly
positive operator. Near the spectral edge $\partial\mathbb{S}$, the behaviour
of $\sigma$ is governed by the quantity $\beta$ from the definition of
$\mathbb{S}$. In fact, $\beta(\zeta)$ coincides locally around the spectral
edge with the isolated eigenvalue of the non-symmetric operator $B_{\zeta}$
that is closest to zero, where $B_{\zeta}:L^{\infty}[0,1]\to L^{\infty}[0,1]$
is defined through $B_{\zeta}f:=\lvert a-\zeta\rvert^{2}\mspace{2.0mu}f-Sf$. A
consequence that we derive from this insight is that the jump height of the
edge discontinuity of $\sigma$ at the spectral edge is proportional to
$\lvert\partial_{\zeta}\beta\rvert^{2}$. This requires a careful singular
expansion of $v_{1},v_{2}$ at the spectral edge, where the Dyson equation
(1.1) is unstable. A signature of this instability is that $B_{\zeta}$ is
singular for $\zeta\in\partial\mathbb{S}$ and the main contributions to
$v_{1}$ and $v_{2}$ near $\partial\mathbb{S}$ point into the singular
eigendirections of $B_{\zeta}$. Owing to the dependence of $v_{1}$, $v_{2}$
and $\beta$ on $s$, treating non-constant $s$ and $a$ is a recurring challenge
for the analysis in both regimes.

##  2 Main results

In this section, we state our assumptions and the main results. In the
following, we take $n\in\mathbb{N}$ and write $\llbracket n\rrbracket$ for the
discrete interval $\llbracket n\rrbracket=\\{1,\ldots,n\\}$.

  1. A1

_Independent, centered entries:_ The entries of $X=(x_{ij})_{i,j\in\llbracket
n\rrbracket}$ are independent and centered, i.e. $\\{x_{ij}\colon
i,j\in\llbracket n\rrbracket\\}$ is a family of independent random variables
and $\mathbb{E}\mspace{1.0mu}x_{ij}=0$.

  2. A2

_Finite moments:_ All moments of the entries of $\sqrt{n}X$ are finite, i.e.
there is a sequence of positive constants $C_{\nu}$ such that

$\begin{split}\mathbb{E}\,\lvert
x_{ij}\rvert^{\nu}\,\leq\,C_{\nu}\mspace{2.0mu}n^{-\nu/2}\,,\end{split}$ (2.1)

for all $i,j\in\llbracket n\rrbracket$ and $\nu\in\mathbb{N}$.

  3. A3

_Anticoncentration of entries:_ There is a constant $b\in(0,1)$ such that, for
all $i,j\in\llbracket n\rrbracket$,

$\mathbb{P}\Big{(}b^{-1}\geq\sqrt{n}\lvert x_{ij}-y_{ij}\rvert\geq
b\Big{)}\geq b,$

where $y_{ij}$ is an independent copy of $x_{ij}$.

  4. A4

_Block-continuous variance profile:_ For some $K\in\mathbb{N}$, let $I_{1}$,
…, $I_{K}\subset[0,1]$ be disjoint intervals of positive length such that
$I_{1}\cup\ldots\cup I_{K}=[0,1]$. Let $s\colon[0,1]^{2}\to(0,\infty)$ and
$a\colon[0,1]\to\mathbb{C}$ be functions such that $s|_{I_{l}\times I_{k}}$
and $a|_{I_{l}}$ have continuous extensions to the closures
$\overline{I_{l}}\times\overline{I_{k}}$ and $\overline{I_{l}}$, respectively,
for all $l$, $k\in\llbracket K\rrbracket$. Moreover, we suppose that there is
a constant $c>0$ such that

$\inf_{x,y\in[0,1]}s(x,y)\geq c.$ (2.2)

The constants in the assumptions A1 – A4 are model parameters and independent
of $n$ and, therefore, the respective estimates are uniform in $n$.

The next proposition shows that the empirical spectral measure of non-
Hermitian random matrices with independent entries, a variance profile and a
diagonal expectation has a deterministic limit as the matrix size tends to
infinity. We state and prove this result here, as we did not find it
explicitly stated in the literature, although the tools leading to it and
closely related results are well-known in the community. The independence of
the limit from the entry distributions was shown in [47, Appendix C] and [32,
Theorem 1.3]. When $X$ is a Ginibre matrix, the convergence of the empirical
spectral measure was proved in [43, Theorem 6] and for an $X$ with i.i.d.
entries, in [47, Theorem 1.17].

######  Proposition 2.1 (Convergence of empirical spectral distribution).

Let the functions $s\colon[0,1]^{2}\to(0,\infty)$ and
$a\colon[0,1]\to\mathbb{C}$ satisfy A4. For any $n\in\mathbb{N}$, we set
$A_{n}:=(a_{ij})_{i,j=1}^{n}\in\mathbb{C}^{n\times n}$ with
$a_{ij}:=a\big{(}\frac{i}{n}\big{)}\delta_{ij}$. Let
$(X_{n})_{n\in\mathbb{N}}$ be a sequence of random matrices such that, for
each $n\in\mathbb{N}$ the random matrix $X_{n}\in\mathbb{C}^{n\times n}$ with
$X_{n}=(x_{ij})_{i,j\in\llbracket n\rrbracket}$ satisfies A1, A2 and A3 as
well as $\mathbb{E}\lvert
x_{ij}\rvert^{2}=\frac{1}{n}s\big{(}\frac{i}{n},\frac{j}{n}\big{)}$ for all
$i,j\in\llbracket n\rrbracket$.

Then there exists a unique probability measure $\sigma$ on $\mathbb{C}$ such
that the empirical spectral distribution
$\frac{1}{n}\sum_{\zeta\in\operatorname{Spec}(X_{n}+A_{n})}\delta_{\zeta}$
converges to $\sigma$ weakly in probability as $n\to\infty$, i.e. for every
bounded, continuous function $f\colon\mathbb{C}\to\mathbb{C}$ and
$\varepsilon>0$, we have

$\lim_{n\to\infty}\mathbb{P}\bigg{(}\bigg{\lvert}\frac{1}{n}\sum_{\zeta\in\operatorname{Spec}(X_{n}+A_{n})}f(\zeta)-\int_{\mathbb{C}}f(\zeta)\sigma(\mathrm{d}\zeta)\bigg{\rvert}>\varepsilon\bigg{)}=0.$

Here the sum $\sum_{\zeta\in\operatorname{Spec}(X_{n}+A_{n})}$ is over all
eigenvalues of $X_{n}+A_{n}$, counted with multiplicity.

The proof of Proposition 2.1 is presented in Section 7.1 below.

######  Remark 2.2.

Under the stronger assumptions of Lipschitz-continuity of $a$ and $s$ in
Theorem 2.4 below, Proposition 2.1 can be strengthened to a local law. That
is, $f$ in Proposition 2.1 can be replaced by
$f_{\zeta_{0}}(\zeta):=n^{2\alpha}f(n^{\alpha}(\zeta-\zeta_{0}))$ for
$\alpha\in(0,1/2)$ and $\zeta_{0}\in\mathbb{S}$, where $\mathbb{S}$ denotes
the spectral bulk introduced in (2.8) below. Moreover, $\varepsilon$ can be
chosen to tend to zero, when $n$ tends to infinity, i.e.
$\varepsilon\equiv\varepsilon_{n}=n^{-1+2\alpha+o(1)}$.

The condition A3 is used to control the smallest singular value of $X+A-\zeta$
for $\zeta\in\mathbb{C}$ through the results from [34, 32], where we omit the
index $n$ from our notation in $X=X_{n}$ and $A=A_{n}$. For all other aspects
of our proofs, the weaker condition $n^{-1}\lesssim\min_{i,j\in\llbracket
n\rrbracket}\mathbb{E}\lvert x_{ij}\rvert^{2}$ is sufficient, which is
explicitly listed above as (2.2).

######  Definition 2.3.

The probability measure $\sigma$ from Proposition 2.1 is called _limiting
spectral measure_ associated with $s$ and $a$.

The next theorem states that the pseudospectrum of the $n\times n$-matrix
$X+A$ is asymptotically given by the support of the measure $\sigma$ from
Proposition 2.1 which coincides with the spectrum of $X+A$ by Proposition 2.1
in the limit $n\to\infty$. We first introduce the pseudospectrum of a matrix.
For any $\varepsilon>0$, the _$\varepsilon$ -pseudospectrum_ of a matrix
$R\in\mathbb{C}^{n\times n}$ is defined as the set

$\begin{split}\operatorname{Spec}_{\varepsilon}(R):=\\{\zeta\in\mathbb{C}\colon\lVert(R-\zeta)^{-1}\rVert\geq\varepsilon^{-1}\\}.\end{split}$
(2.3)

Note that $\operatorname{Spec}_{\varepsilon}(R)$ is monotonically increasing
in $\varepsilon$ and
$\operatorname{Spec}(R)=\cap_{\varepsilon>0}\operatorname{Spec}_{\varepsilon}(R)$.

Furthermore, for a sequence $(\Omega_{n})_{n\in\mathbb{N}}$ of sets we use the
customary definitions

$\liminf_{n\to\infty}\Omega_{n}:=\bigcup_{N\in\mathbb{N}}\bigcap_{n\geq
N}\Omega_{n},\qquad\qquad\limsup_{n\to\infty}\Omega_{n}:=\bigcap_{N\in\mathbb{N}}\bigcup_{n\geq
N}\Omega_{n}.$

######  Theorem 2.4 (Spectrum occupies pseudospectrum).

Let $s\colon[0,1]^{2}\to(0,\infty)$, $a\colon[0,1]\to\mathbb{C}$ and
$(I_{k})_{k\in\llbracket K\rrbracket}$ be as in A4. Suppose that
$s|_{I_{l}\times I_{k}}$ and $a|_{I_{l}}$ are Lipschitz-continuous functions
for all $l$, $k\in\llbracket K\rrbracket$. Let $X_{n}$ and $A_{n}$ be as in
Proposition 2.1. Then there exists a monotonically increasing family
$(\operatorname{Spec}_{\varepsilon}^{\infty}(s,a))_{\varepsilon>0}$ of
deterministic subsets of $\mathbb{C}$ such that, almost surely111We assume
that all $X_{n}$ for $n\in\mathbb{N}$ are realised on the same probability
space.,

$\begin{split}\limsup_{n\to\infty}\operatorname{Spec}_{\varepsilon}(X_{n}+A_{n})\subset\operatorname{Spec}_{\varepsilon}^{\infty}(s,a)\subset{\liminf_{n\to\infty}\operatorname{Spec}_{\varepsilon+\delta}(X_{n}+A_{n})}\end{split}$
(2.4)

hold for all $\varepsilon,\delta>0$. Moreover, this family is right
continuous, i.e.
$\cap_{\delta>0}\operatorname{Spec}_{\varepsilon+\delta}^{\infty}(s,a)=\operatorname{Spec}_{\varepsilon}^{\infty}(s,a)$
and the limiting spectral measure $\sigma$ from Proposition 2.1 satisfies

$\bigcap_{\varepsilon>0}\operatorname{Spec}_{\varepsilon}^{\infty}(s,a)=\operatorname{supp}\sigma.$
(2.5)

The proof of Theorem 2.4 is given in Section 7.2 below. We note that the sets
$(\operatorname{Spec}_{\varepsilon}^{\infty}(s,a))_{\varepsilon>0}$ are
monotonically increasing in $\varepsilon$. In particular, Theorem 2.4 implies
that $\operatorname{Spec}(X_{n}+A_{n})$ is eventually almost surely contained
in a neighbourhood of $\operatorname{supp}\sigma$.

We now state additional properties of the limiting spectral measure $\sigma$
and provide a characterisation of $\operatorname{supp}\sigma$ in terms of $a$
and $s$. The measure $\sigma$ itself will later also be given by a formula
that only depends on $a$ and $s$ (cf. (6.3)). In the following, we write
$L^{\infty}[0,1]$ for the space of essentially bounded functions on $[0,1]$,
up to changes of zero measure, when this interval is equipped with the
Lebesgue measure. We denote by $S$ the integral operator on $L^{\infty}[0,1]$
with kernel $s$, i.e.

$S:L^{\infty}[0,1]\to
L^{\infty}[0,1]\,,\quad(Sf)(x):=\int_{0}^{1}s(x,y)f(y)\mathrm{d}y$

For $u\in L^{\infty}[0,1]$, let $D_{u}:L^{\infty}[0,1]\to L^{\infty}[0,1]$ be
the operator $D_{u}f:=uf$ induced by multiplication with $u$. Using these
definitions, for some bounded and measurable function
$a\colon[0,1]\to\mathbb{C}$ and $\zeta\in\mathbb{C}$, we introduce the
operator $B\equiv B_{\zeta}$ on $L^{\infty}[0,1]$ given by

$B_{\zeta}:=D_{\lvert\zeta-a\rvert^{2}}-S.$ (2.6)

Since $B$ maps real-valued functions to real-valued functions, we obtain a
function $\beta\colon\mathbb{C}\to\mathbb{R}$ defined through

$\beta(\zeta):=\inf_{f>0}\sup_{g>0}\frac{\langle{f}\mspace{2.0mu},{B_{\zeta}g}\rangle}{\langle{f}\mspace{2.0mu},{g}\rangle}$
(2.7)

for $\zeta\in\mathbb{C}$, where the infimum and supremum are taken over
bounded functions $f,g\colon[0,1]\to(0,\infty)$ and

$\langle{f}\mspace{2.0mu},{g}\rangle:=\int_{0}^{1}\overline{f(x)}\\!\,g(x)\mathrm{d}x$

is the scalar product on $L^{2}[0,1]$. The definition of $\beta$ is motivated
by the Birkhoff-Varga formula for the spectral radius of a matrix with
positive entries [14]. In terms of $\beta$ we define the set

$\mathbb{S}:=\\{\zeta\in\mathbb{C}\colon\beta(\zeta)<0\\}\,,$ (2.8)

whose closure coincides with $\operatorname{supp}\sigma$, as stated in the
proposition below. We will see in Proposition 5.15 (i) that $\beta$ is a
continuous function and therefore $\mathbb{S}$ is an open set.

######  Proposition 2.5 (Properties of the limiting spectral measure
$\sigma$).

Let $s\colon[0,1]^{2}\to(0,\infty)$, $a\colon[0,1]\to\mathbb{C}$ and
$(I_{k})_{k\in\llbracket K\rrbracket}$ be as in A4. Suppose that
$s|_{I_{l}\times I_{k}}$ and $a|_{I_{l}}$ are Lipschitz-continuous functions
for all $l$, $k\in\llbracket K\rrbracket$. Then the following holds.

  1. (i)

With respect to the Lebesgue measure $\mathrm{d}^{2}\zeta$, the measure
$\sigma$ from Proposition 2.1 has a bounded density on $\mathbb{C}$, which we
also denote by $\sigma$, i.e.
$\sigma(\mathrm{d}\zeta)=\sigma(\zeta)\mathrm{d}^{2}\zeta$.

  2. (ii)

On $\mathbb{S}$, the density $\zeta\mapsto\sigma(\zeta)$ is strictly positive
and real analytic.

  3. (iii)

$\operatorname{supp}\sigma=\overline{\mathbb{S}}$ and this set is bounded.
Furthermore $\\{a(x):x\in[0,1]\\}\subset\mathbb{S}$.

  4. (iv)

$\partial\mathbb{S}$ is a real analytic variety of real dimension at most $1$.

  5. (v)

There exists a unique continuous extension
$\sigma\colon\overline{\mathbb{S}}\to[0,\infty)$ of the density
$\sigma|_{\mathbb{S}}$ to $\overline{\mathbb{S}}$ such that
$\sigma(\zeta)=g(\zeta)\lvert\partial_{\zeta}\beta(\zeta)\rvert^{2}$ for all
$\zeta\in\partial\mathbb{S}$, where $g:\partial\mathbb{S}\to(0,\infty)$ is a
strictly positive function that can be extended to a real analytic function on
a neighbourhood of $\partial\mathbb{S}$.

Proposition 2.5 is a special case of Proposition 6.1 from Section 6, where we
used that by Remark 4.2 Assumption A4 implies Assumption A6 below.

We now give a brief overview about previous results covering (parts of)
Proposition 2.5 for a subclass of the models we analysed. Throughout the
following, $\mathfrak{c}$ is a circular element and $\mathfrak{a}$ is an
element that is $\ast$-free from $\mathfrak{c}$. For some examplary choice of
$\mathfrak{a}$, the Brown measure of $\mathfrak{a}+\mathfrak{c}$ was computed
in [13, Section 5]. In [16, Theorem 1.4], a formula for the support of the
Brown measure of $\mathfrak{a}+\mathfrak{c}$, when $\mathfrak{a}$ is a normal
operator with a Gaussian spectral density, and its absolute continuity with a
smooth density was shown. For general $\mathfrak{a}$, the formula for the
support was derived in [15, Proposition 1.2] under an additional assumption.
When $\mathfrak{a}=\mathfrak{a}^{*}$, [31, Theorems 3.13 and 3.14] provided an
explicit open set such that the support of the Brown measure of
$\mathfrak{a}+\mathfrak{c}$ coincides with the closure of the open set and the
Brown measure has a strictly positive density on the open set as well as
proved a sharp upper bound on the density. Apart from the sharp upper bound,
these results were obtained in [49, Theorems 4.2 and 4.6] for general
$\mathfrak{a}$, where the absence of atoms of the Brown measure and the real
analyticity of the density were also established. Then [10, Theorem 7.10]
proved for general $\mathfrak{a}$ that the Brown measure is absolutely
continuous with respect to the Lebesgue measure on $\mathbb{C}$, i.e.
excluding a singular continuous part in the Brown measure, as well as the
sharp upper bound on the density. In this setup, the edge behaviour of the
Brown measure density was studied in [22, Theorem 2.9] showing a jump
discontinuity or a quadratic growth for the density. In particular, [49,
Theorems 4.2 and 4.6] and [10, Theorem 7.10] cover Proposition 2.5 (i) – (iii)
when $s$ is constant. In this case, [22, Theorem 2.9 and Remark 2.10] yield
(iv) and (v). We note that some of the works listed above considered
generalisations of circular elements such as elliptic elements in addition.
Similar statements about the Brown measure of related models can for example
be found in [12, 21, 28].

######  Remark 2.6 (Necessity of Lipschitz-continuity).

The Lipschitz-continuity assumption on $a$ and $s$ in Proposition 2.5 is
needed to ensure that the image of $a$ remains in a positive distance from the
boundary $\partial\mathbb{S}$ of the spectrum as stated in Proposition 2.5
(iii). For a counterexample, where the Lipschitz-continuity of $a$ is violated
and $\partial\mathbb{S}\cap\\{a(x):x\in[0,1]\\}\neq\emptyset$ we refer to [22,
Proposition 3.1 (iv)].

######  Remark 2.7 (Special cases for $\mathbb{S}$).

In the case when the entries of the random matrix $X_{n}$ are independent and
identically distributed, i.e. when $s=t$ is a constant and
$\mathbb{E}\mspace{1.0mu}\lvert x_{ij}\rvert^{2}=\frac{t}{n}$, we recover from
(2.8) the well-known formula [33, 47]

$\mathbb{S}=\biggl{\\{}{\zeta\in\mathbb{C}:\int_{0}^{1}\frac{\mathrm{d}x}{\lvert
a(x)-\zeta\rvert^{2}}>\frac{1}{t}}\biggr{\\}}\,.$ (2.9)

If $a=0$ then
$\mathbb{S}=\\{\zeta\in\mathbb{C}\colon\lvert\zeta\rvert^{2}<\varrho(S)\\}$,
where $S$ denotes the spectral radius of $S$. This generalises the
corresponding result from [3, 20] to an infinite dimensional setup.

######  Remark 2.8 (Special behaviours of $\partial\mathbb{S}$).

We note that the boundary $\partial\mathbb{S}$ can have isolated points, i.e.
its real dimension can locally be zero (see e.g. [22, Example 3.1 (d)]).
Moreover, $\beta$ can have critical points on the boundary
$\partial\mathbb{S}$, i.e. there can be $\zeta\in\partial\mathbb{S}$ such that
$\partial_{\zeta}\beta(\zeta)=0$ as shown in Example 3.1 below. Even
infinitely many critical points of $\beta$ can occur in $\partial\mathbb{S}$,
see Example 3.2. In particular, these examples reveal that a rich class of
singularities of $\sigma$ can occur at the spectral edge $\partial\mathbb{S}$.

###  2.1 Notations

We now introduce some notations used throughout. We write $\llbracket
n\rrbracket\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{1,\ldots,n\\}$
for $n\in\mathbb{N}$. For $r>0$, we denote by
$\mathbb{D}_{r}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{z\in\mathbb{C}\colon\lvert
z\rvert<r\\}$ the disk of radius $r$ around the origin in $\mathbb{C}$ and by
$\operatorname{dist}(x,A):=\inf\\{\lvert x-y\rvert\colon y\in A\\}$ the
Euclidean distance of a point $x\in\mathbb{C}$ from a set
$A\subset\mathbb{C}$.

We use the convention that $c$ and $C$ denote generic constants that may
depend on the model parameters, but are otherwise uniform in all other
parameters, e.g. $n$, $\zeta$, etc.. For two real scalars $f$ and $g$ we write
$f\lesssim g$ and $g\gtrsim f$ if $f\leq Cg$ for such a constant $C>0$. In
case $f\lesssim g$ and $f\gtrsim g$ both hold, we write $f\sim g$. If the
constant $C$ depends on a parameter $\delta$ that is not a model parameter, we
write $\lesssim_{\delta}$, $\gtrsim_{\delta}$ and $\sim_{\delta}$,
respectively. The notation for inequality up to constant is also used for
self-adjoint matrices/operators $f$ and $g$, where $f\leq Cg$ is interpreted
in the sense of quadratic forms. For complex $f$ and $g\geq 0$ we write
$f=O(g)$ in case $\lvert f\rvert\lesssim g$. Analogously $f=O_{\delta}(g)$
expresses the fact $\lvert f\rvert\lesssim_{\delta}g$.

##  3 Examples

In this section, we present a couple of examples highlighting certain special
behaviours of $\partial\mathbb{S}$.

######  Example 3.1 (Critical points of $\partial\mathbb{S}$).

Let $s\equiv t:=\frac{2}{3}(20-7\sqrt{7})$ be constant on $[0,1]^{2}$,
$\delta=(-17+7\sqrt{7})/8$ and

$a\colon[0,1]\to\mathbb{C},\qquad x\mapsto\begin{cases}1&\text{ if
}x\in[0,1/(2+\delta)),\\\ -1&\text{ if }x\in[1/(2+\delta),2/(2+\delta)),\\\
\mathrm{i}&\text{ if }x\in[2/(2+\delta),1].\end{cases}$

We recall that if $s\equiv t$ then
$\beta(\zeta)=\frac{1}{t}-\int_{0}^{1}\frac{\mathrm{d}x}{\lvert
a(x)-\zeta\rvert^{2}}$ by (2.9) and

$\mathbb{S}=\bigg{\\{}\zeta\in\mathbb{C}\colon\frac{1}{\lvert
1-\zeta\rvert^{2}}+\frac{1}{\lvert
1+\zeta\rvert^{2}}+\frac{\delta}{\lvert\zeta-\mathrm{i}\rvert^{2}}>\frac{2+\delta}{t}\bigg{\\}}.$

We set $y_{0}=(\sqrt{7}-2)/3$ and note that $\beta(\mathrm{i}y_{0})=0$, i.e.
$\mathrm{i}y_{0}\in\partial\mathbb{S}$ and, moreover,
$\beta(\mathrm{i}y_{0}+x+\mathrm{i}y)=(c_{1}x^{2}+c_{2}y^{3})(1+o(1))$ for
small enough $x$, $y\in\mathbb{R}$. Here, $c_{1}$ and $c_{2}$ are two positive
constants and $o(1)$ is meant for $x\to 0$ and $y\to 0$. The boundary of
$\mathbb{S}$ and sampled eigenvalues of the corresponding (Gaussian) random
matrix model are drawn in Figure 1 (LABEL:subfig:x2_y3).

We refer to [5, Example 2.6 and Figure 1] for more examples in the spirit of
Example 3.1.

(a)

(b)

Figure 1:  The solid lines in subfigures (LABEL:subfig:x2_y3) and
(LABEL:subfig:rot_symm_circle_and_at_zero) show the boundary of $\mathbb{S}$
from Examples 3.1 and 3.2, respectively. In addition, the black dots show the
sampled eigenvalues of $X+A$, where $X$ is an $n\times n$ matrix with i.i.d.
$N(0,1)$ standard real normal distributed entries, $n=10000$ and
$A=(\operatorname{diag}(a(i/n))\delta_{ij})_{i,j=1}^{n}$ is a diagonal matrix
and $a$ is chosen as in Examples 3.1 and 3.2, respectively.

######  Example 3.2 (Infinitely many critical points of $\partial\mathbb{S}$).

Let $s\equiv 1$ be constant on $[0,1]^{2}$ and

$a\colon[0,1]\to\mathbb{C},\qquad
x\mapsto\begin{cases}\sqrt{2}\mathrm{e}^{4\pi\mathrm{i}x}&\text{ if
}x\in[0,1/2],\\\ 0&\text{ if }x\in(1/2,1].\end{cases}$

A short computation starting from (2.9) reveals that

$\mathbb{S}=\big{\\{}\zeta\in\mathbb{C}\colon
0\leq\lvert\zeta\rvert^{2}<1\text{ or
}1<\lvert\zeta\rvert^{2}<{(3+\sqrt{5})/2}\big{\\}}.$

Thus, since $\partial\mathbb{D}_{1}\cap\mathbb{S}=\emptyset$ while points on
both sides of $\partial\mathbb{D}_{1}$ belong to $\mathbb{S}$, we conclude
from the analyticity of $\beta$ that
$\partial_{\zeta}\beta(\zeta)=\partial_{\bar{\zeta}}\beta(\zeta)=0$ for all
$\zeta\in\partial\mathbb{D}_{1}$. Hence, there are infinitely many critical
points of $\beta$ on $\partial\mathbb{S}$. The boundary of $\mathbb{S}$ and
sampled eigenvalues of an approximating random matrix model are shown in
Figure 1 (LABEL:subfig:rot_symm_circle_and_at_zero).

Furthermore, using (4.2) and (6.3), we find that

$\sigma(\zeta)=\frac{1}{\pi}\bigg{(}1-\frac{2}{2+x+\frac{x}{(2x-1)^{3}}}\bigg{(}1+\frac{1}{x^{2}}\bigg{)}\bigg{)}\mathbf{1}(\zeta\in\mathbb{S}),$

where $x\in(0,\infty)$ is the unique positive solution of
$\frac{1}{x}+\frac{1}{\sqrt{1+4x+x^{2}-8\lvert z\rvert^{2}+3}}=2$.

##  4 General setup and Dyson equation

In this section, we generalise the setup from the previous section in order to
study an analogue of $\sigma$ when $a$ and $s$ are defined on a probability
space $\mathfrak{X}$ and on $\mathfrak{X}^{2}$ instead of $[0,1]$ and
$[0,1]^{2}$, respectively. Thus, let $(\mathfrak{X},\mathcal{A},\mu)$ be a
probability space, which represents the labels of the main quantities, e.g.
$a$ etc. We denote by $\mathcal{B}=L^{\infty}(\mathfrak{X},\mathcal{A},\mu)$
the measurable essentially bounded functions on $\mathfrak{X}$ up to measure
zero with respect to $\mu$.

Throughout the paper, we fix two measurable functions
$s\colon\mathfrak{X}\times\mathfrak{X}\to[0,\infty)$ and
$a\colon\mathfrak{X}\to\mathbb{C}$ with $a\in\mathcal{B}$. We also assume that

$\sup_{x\in\mathfrak{X}}\int_{\mathfrak{X}}s(x,y)\mu(\mathrm{d}y)<\infty,\qquad\qquad\sup_{y\in\mathfrak{X}}\int_{\mathfrak{X}}s(x,y)\mu(\mathrm{d}x)<\infty.$

Therefore, the two operators $S\colon\mathcal{B}\to\mathcal{B}$ and
$S^{*}\colon\mathcal{B}\to\mathcal{B}$ defined through

$(Su)(x)=\int_{\mathfrak{X}}s(x,y)u(y)\mu(\mathrm{d}y),\qquad(S^{*}u)(x)=\int_{\mathfrak{X}}s(y,x)u(y)\mu(\mathrm{d}y)$
(4.1)

for all $x\in\mathfrak{X}$ and $u\in\mathcal{B}$ are well-defined bounded
linear operators.

We consider two coupled equations for functions in $v_{1},v_{2}\in\mathcal{B}$
with $v_{1}>0$ and $v_{2}>0$, namely

$\displaystyle\frac{1}{v_{1}}$
$\displaystyle=\eta+Sv_{2}+\frac{\lvert\zeta-a\rvert^{2}}{\eta+S^{*}v_{1}}\,,$
(4.2a) $\displaystyle\frac{1}{v_{2}}$
$\displaystyle=\eta+S^{*}v_{1}+\frac{\lvert\zeta-a\rvert^{2}}{\eta+Sv_{2}}\,,$
(4.2b)

for all $\eta>0$ and $\zeta\in\mathbb{C}$. Here, $\eta$ and $\zeta$ are
interpreted as the constant functions on $\mathfrak{X}$ with the respective
value. The equation (4.2) is called the _(vector) Dyson equation_. First, we
clarify the existence and uniqueness of its solution.

######  Lemma 4.1 (Existence and uniqueness).

For each $\eta>0$ and $\zeta\in\mathbb{C}$, there are unique $v_{1}$,
$v_{2}\in\mathcal{B}$ such that $v_{1}>0$ and $v_{2}>0$ and (4.2) holds.

In Appendix A.1, we present the proof of Lemma 4.1 by inferring it from [30]
through a relation of (4.2) to a matrix-valued version.

###  4.1 Assumptions

Throughout the paper, we will impose some of the following assumptions.

  1. A5

_Flatness of $S$:_ There is a constant $C>0$ such that

$\frac{1}{C}\leq s(x,y)\leq C$

for all $x$, $y\in\mathfrak{X}$.

We define the function $\Gamma_{a,s}\colon(0,\infty)\to(0,\infty)$ through

$\Gamma_{a,s}(\tau):=\operatorname*{ess\,inf}_{x\in\mathfrak{X}}\int_{\mathfrak{X}}\frac{1}{\tau^{-1}+\lvert
a(x)-a(y)\rvert+d_{s}(x,y)}\mu(\mathrm{d}y),$ (4.3)

where $d_{s}(x,y):=\operatorname*{ess\,sup}_{q\in\mathfrak{X}}(\lvert
s(x,q)-s(y,q)\rvert+\lvert s(q,x)-s(q,y)\rvert)$. Note that $\Gamma_{a,s}$ is
strictly monotonically increasing.

  1. A6

_Data regularity:_ The data $a$ and $s$ satisfy the regularity assumption

$\lim_{\tau\to\infty}\Gamma_{a,s}(\tau)=\infty$

######  Remark 4.2.

In the case $\mathfrak{X}=[0,1]$ and $\mu$ the Lebesgue-measure on $[0,1]$,
Assumption A4 implies Assumption A6, since in this case $\lvert
a(x)-a(y)\rvert+d_{s}(x,y)\lesssim\lvert x-y\rvert$ for all $x$, $y\in I_{k}$
and all $k\in\llbracket K\rrbracket$.

As we will see in Lemma 5.3 below, A6 together with A5 and $a\in\mathcal{B}$
implies that $v_{1}$ and $v_{2}$ are uniformly bounded in the
$\mathcal{B}$-norm on $\mathfrak{X}$. In addition to $L^{\infty}$, we
introduce the usual $L^{p}$ spaces on $(\mathfrak{X},\mathcal{A},\mu)$. We
denote them by $L^{p}:=L^{p}(\mathfrak{X},\mathcal{A},\mu)$ and the
corresponding norms by $\lVert\mspace{2.0mu}\cdot\mspace{2.0mu}\rVert_{p}$.
For functions $u\in L^{1}$ and $u_{1}$, $u_{2}\in L^{2}$, we define their
average and scalar product as

$\langle u\rangle:=\int
u(x)\mu(\mathrm{d}x)\,,\qquad\langle{u_{1}}\mspace{2.0mu},{u_{2}}\rangle:=\langle\overline{u}\\!\,_{1}u_{2}\rangle\,,$

respectively. By normalisation of the probability measure $\mu$ on
$\mathfrak{X}$ we have $\langle 1\rangle=1$.

##  5 Solution of the Dyson Equation

In this section, we study various properties of the solution $(v_{1},v_{2})$
of the Dyson equation (4.2). We start with simple relations and bounds and
obtain fine properties and expansions later.

From (4.2), we directly conclude

$v_{2}(\eta+S^{*}v_{1})=v_{1}(\eta+Sv_{2})\,,$ (5.1)

which also implies

$\langle v_{1}\rangle=\langle v_{2}\rangle\,.$ (5.2)

Furthermore, we introduce $y$ defined by

$y:=\frac{v_{1}\mspace{2.0mu}(\bar{a}-\bar{\zeta})}{\eta+S^{*}v_{1}}=\frac{v_{2}\mspace{2.0mu}(\bar{a}-\bar{\zeta})}{\eta+Sv_{2}}\,,$
(5.3)

where the second step is a consequence of (5.1). We conclude from (4.2) and
(5.3) that

$v_{1}\leq\eta^{-1},\qquad v_{2}\leq\eta^{-1},\qquad\lvert y\rvert\leq\lvert
a-\zeta\rvert\eta^{-2}$ (5.4)

for all $\zeta\in\mathbb{C}$ and $\eta>0$. Furthermore, we have the identity

$y=\frac{v_{1}(\bar{a}-\bar{\zeta})}{\eta+S^{*}v_{1}}=\frac{1}{a-\zeta}(1-v_{1}(\eta+Sv_{2}))=\frac{1}{a-\zeta}-\frac{v_{1}v_{2}}{\overline{y}\\!\,}$
(5.5)

for any $\zeta\in\mathbb{C}\setminus\operatorname{Spec}(D_{a})$. Here,
$\operatorname{Spec}(D_{a})$ denotes the spectrum of $D_{a}$ considered as
multiplication operator $\mathcal{B}\to\mathcal{B}$, which coincides with the
essential range of $a$.

Throughout the remainder of this section, we assume that $s$ satisfies A5.
This implies

$S^{*}w\sim Sw\sim\langle w\rangle$ (5.6)

for all $w\in\mathcal{B}$ with $w\geq 0$.

###  5.1 Bounds on the solution

This subsection contains bounds on $v_{1}$ and $v_{2}$ under varying
assumptions on $s$ and $a$.

#### Bound in $L^{2}$-norm:

We start with the following bound with respect to the norm on $L^{2}$.

######  Lemma 5.1.

If $s$ satisfies A5 then

$\langle v_{1}^{2}\rangle+\langle v_{2}^{2}\rangle+\langle\lvert
y\rvert^{2}\rangle\lesssim 1$ (5.7)

uniformly for all $\zeta\in\mathbb{C}$ and $\eta>0$.

###### Proof.

We multiply the first relation in (4.2a) by $v_{1}^{2}$ and estimate
$v_{1}\geq v_{1}^{2}Sv_{2}\gtrsim v_{1}^{2}\langle
v_{2}\rangle=v_{1}^{2}\langle v_{1}\rangle$ due to (5.6) and (5.2). Averaging
this estimate and using $\langle v_{1}\rangle>0$ yields $\langle
v_{1}^{2}\rangle\lesssim 1$. The bound $\langle v_{2}^{2}\rangle\lesssim 1$ is
proved analogously. From (5.3), we conclude

$\lvert y\rvert^{2}=\frac{v_{1}^{2}\lvert
a-\zeta\rvert^{2}}{(\eta+S^{*}v_{1})^{2}}\leq\frac{v_{1}^{2}\lvert
a-\zeta\rvert^{2}}{(\eta+S^{*}v_{1})^{2}}+\frac{v_{1}^{2}(\eta+Sv_{2})}{\eta+S^{*}v_{1}}=\frac{v_{1}}{\eta+S^{*}v_{1}},$
(5.8)

where we used (4.2) in the last step. Hence, (5.6) implies

$\langle\lvert y\rvert^{2}\rangle\lesssim\frac{\langle
v_{1}\rangle}{\eta+\langle v_{1}\rangle}\leq 1.\qed$

######  Corollary 5.2.

Let $a\in\mathcal{B}$ and $s$ satisfy A5. Then

$\frac{\eta+\langle v_{i}\rangle}{1+\eta^{2}+\lvert\zeta\rvert^{2}}\lesssim
v_{i}\lesssim\frac{1}{\eta+\langle v_{i}\rangle}\,,\qquad i=1,2\,.$

###### Proof.

We only consider the case $i=1$. The case $i=2$ follows analogously. For the
lower bound, we start from (4.2) and get

$v_{1}=\frac{\eta+S^{*}v_{1}}{(\eta+S^{*}v_{1})(\eta+Sv_{2})+\lvert\zeta-a\rvert^{2}}\gtrsim\frac{\eta+\langle
v_{1}\rangle}{\eta^{2}+\langle v_{1}\rangle^{2}+\lvert\zeta\rvert^{2}+\lVert
a\rVert_{\infty}^{2}}\gtrsim\frac{\eta+\langle
v_{1}\rangle}{1+\eta^{2}+\lvert\zeta\rvert^{2}}\,,$

using $\langle v_{1}\rangle=\langle v_{2}\rangle$ by (5.2), (5.7) and (5.6).
For the upper bound, (4.2), $\langle v_{1}\rangle=\langle v_{2}\rangle$ and
(5.6) imply

$v_{1}\leq\frac{1}{\eta+Sv_{2}}\sim\frac{1}{\eta+\langle v_{1}\rangle}\,.\qed$

#### Bound in supremum norm:

Under stronger assumptions on $s$ and $a$, we can also get a bound on $v_{i}$
in the $L^{\infty}$-norm. Let $\Gamma_{a,s}$ be as defined in (4.3).

######  Lemma 5.3.

Let $a\in\mathcal{B}$, $s$ satisfy A5 and $r>0$. Then there is a constant
$C>0$ with $C\sim_{r}1$ such that $C<\lim_{\tau\to\infty}\Gamma_{a,s}(\tau)$
implies

$\max\\{\lVert v_{1}\rVert_{\infty},\lVert
v_{2}\rVert_{\infty}\\}\leq\Gamma_{a,s}^{-1}(C)\,$

uniformly for $\zeta\in\mathbb{D}_{r}$ and $\eta>0$.

###### Proof.

From (4.2), we obtain

$\displaystyle\frac{1}{v_{1}(y)}$
$\displaystyle\leq\frac{1}{v_{1}(x)}+\bigg{\lvert}\frac{1}{v_{1}(x)}-\frac{1}{v_{1}(y)}\bigg{\rvert}$
$\displaystyle\leq\frac{1}{v_{1}(x)}+\lvert(Sv_{2})(x)-(Sv_{2})(y)\rvert+\bigg{\lvert}\frac{\lvert\zeta-a(x)\rvert^{2}}{\eta+(S^{*}v_{1})(x)}-\frac{\lvert\zeta-a(y)\rvert^{2}}{\eta+(S^{*}v_{1})(y)}\bigg{\rvert}$
$\displaystyle\lesssim_{r}\frac{1}{v_{1}(x)}+d_{s}(x,y)\langle
v_{1}\rangle+\frac{\lvert a(x)-a(y)\rvert}{\langle
v_{1}\rangle}+\frac{d_{s}(x,y)}{\langle v_{1}\rangle}$
$\displaystyle\lesssim\frac{1}{\langle
v_{1}\rangle}\biggl{(}{\frac{1}{v_{1}(x)}+\lvert
a(x)-a(y)\rvert+d_{s}(x,y)}\biggr{)},$

where in the third inequality we used $\lVert a\rVert_{\infty}\lesssim 1$ as
$a\in\mathcal{B}$ and $\langle v_{1}\rangle+\langle v_{2}\rangle\lesssim 1$ by
Lemma 5.1, as well as (5.6) and (5.2). We take the inverse on both sides and
integrate in $y$ with respect to $\mu$ to get
$\Gamma_{a,s}(v_{1}(x))\lesssim_{r}1$ for almost all $x\in\mathfrak{X}$.
Together with the analogous calculation with the roles of $v_{1}$ and $v_{2}$
interchanged, we obtain $\Gamma_{a,s}(\max\\{\lVert
v_{1}\rVert_{\infty},\lVert v_{2}\rVert_{\infty}\\})\leq C$ for some
$C\sim_{r}1$. Thus, the statement of the lemma follows from the monotonicity
of $\Gamma_{a,s}$. ∎

######  Corollary 5.4.

Let $a\in\mathcal{B}$. If $s$ and $a$ satisfy A5 and A6 then

$\lVert v_{1}\rVert_{\infty}+\lVert v_{2}\rVert_{\infty}\lesssim 1$

uniformly for all $\zeta\in\mathbb{C}$ and $\eta>0$.

###### Proof.

Owing to Lemma 5.3, (5.4) and $a\in\mathcal{B}$, it remains to consider the
case $\eta\leq 1$ and $\lvert\zeta\rvert\geq\lVert a\rVert_{\infty}+1$. In
this case, we conclude from (4.2a) that
$\frac{1}{v_{1}}\geq\frac{\lvert\zeta-a\rvert^{2}}{\eta+S^{*}v_{1}}\gtrsim 1$
as $S^{*}v_{1}\lesssim\langle v_{1}\rangle\lesssim 1$ by (5.6) and Lemma 5.1.
Since $v_{1}>0$, we conclude $\lVert v_{1}\rVert_{\infty}\lesssim 1$. The
analogous argument for $v_{2}$ completes the proof of Corollary 5.4. ∎

######  Remark 5.5.

Let $\mathfrak{X}=[0,1]$ and $\mu$ be the Lebesgue measure on $[0,1]$. Let
$I_{1}$, …, $I_{K}$ be disjoint intervals in $[0,1]$ such that
$I_{1}\cup\ldots\cup I_{K}=[0,1]$. If $s\colon[0,1]\times[0,1]\to[0,\infty)$,
$a\colon[0,1]\to\mathbb{C}$ are such that $s|_{I_{l}\times I_{k}}$ and
$a|_{I_{l}}$ are Lipschitz-continuous for every $l$, $k\in\llbracket
K\rrbracket$ then A6 is satisfied and $a\in\mathcal{B}$. In particular, if $s$
satisfies A5 in addition then $v_{1}$ and $v_{2}$ are bounded in
$\lVert\cdot\rVert_{\infty}$ uniformly on $\mathbb{C}\times(0,\infty)$ by
Corollary 5.4. In this case A5 and A6 all hold.

#### Scaling relations

######  Lemma 5.6.

Let $a\in\mathcal{B}$ and $s$ satisfy A5 and A6. Then

$v_{1}\sim\langle v_{1}\rangle=\langle v_{2}\rangle\sim v_{2},\qquad\lvert
y\rvert\lesssim 1$

uniformly for all $\zeta\in\mathbb{C}$ and $\eta>0$. Moreover, for any
sufficiently small positive constant $c\sim 1$ the inequalities $\eta+\langle
v_{1}(\zeta,\eta)\rangle\leq c$ and $\lvert\zeta\rvert\leq 1/c$ imply
$\lvert\zeta-a\rvert\sim 1$ and $\lvert y\rvert\sim 1$.

Before going into the proof of Lemma 5.6, we remark that if $a\in\mathcal{B}$
and $s$ satisfies A5 then

$\lVert
v_{1}(\zeta,\eta)-(1+\eta)^{-1}\rVert\lesssim(1+\lvert\zeta\rvert)\eta^{-2}$
(5.9)

uniformly for $\eta>0$ and $\zeta\in\mathbb{C}$. Indeed, for $\eta\in(0,1]$,
(5.9) is a trivial consequence of (5.4). For $\eta\geq 1$, (5.9) follows by
inverting (4.2a), subtracting $(1+\eta)^{-1}$ on both sides and estimating the
right-hand side using (5.6) and Lemma 5.1.

###### Proof.

We first prove that $v_{1}\sim\langle v_{1}\rangle=\langle v_{2}\rangle\sim
v_{2}$. As $s$ satisfies A5, equation (4.2), $v_{1}$, $v_{2}>0$ and (5.6)
imply

$v_{1}\sim v_{2}\,.$ (5.10)

Hence, it suffices to show $v_{1}\sim\langle v_{1}\rangle$ due to (5.2).

From (5.9), we conclude that $v_{1}\sim(1+\eta)^{-1}$ and $\langle
v_{1}\rangle\sim(1+\eta)^{-1}$ uniformly for $\eta\gtrsim 1$ and
$\lvert\zeta\rvert\lesssim 1$. This proves Lemma 5.6 in that regime. If, on
the other hand, $\lvert\zeta\rvert\geq\lVert a\rVert_{\infty}+1$ then
$\lvert\zeta-a\rvert\sim\lvert\zeta\rvert$. Hence, for such $\zeta$, we
conclude from (4.2a) and (5.6) that

$\frac{1}{v_{1}}\sim\eta+\langle
v_{2}\rangle+\frac{\lvert\zeta\rvert^{2}}{\eta+\langle v_{1}\rangle}.$

As the right-hand side is a constant function on $\mathfrak{X}$, we obtain
$v_{1}\sim\langle v_{1}\rangle$ if $\lvert\zeta\rvert\geq\lVert
a\rVert_{\infty}+1$.

Hence, it remains to consider $\lvert\zeta\rvert\lesssim 1$ and $\eta\lesssim
1$. In particular, $\lvert\zeta-a\rvert\lesssim 1$ as $a\in\mathcal{B}$. Thus,
(4.2a), (5.6) and (5.2) imply

$\eta+\langle v_{1}\rangle\sim v_{1}((\eta+\langle
v_{1}\rangle)^{2}+\lvert\zeta-a\rvert^{2}).$ (5.11)

Together with Lemma 5.1, this yields $\eta+\langle v_{1}\rangle\lesssim
v_{1}$. We conclude $v_{1}\gtrsim\langle v_{1}\rangle$ and $v_{1}\gtrsim\eta$
as well as $\langle v_{1}\rangle\gtrsim\eta$. If $\lvert\zeta-a\rvert\geq c$
for any $c\sim 1$ then $v_{1}\lesssim\eta+\langle v_{1}\rangle\sim\langle
v_{1}\rangle$ by (5.11). Therefore we conclude $v_{1}\sim\langle v_{1}\rangle$
if $\lvert\zeta-a\rvert\geq c$. What remains is the case
$\lvert\zeta-a\rvert\leq c$ and $\eta\leq c$ for some constant $c\sim 1$. As
$v_{1}\lesssim 1$ by A6 and Lemma 5.3, we conclude from (5.11) that
$1\gtrsim\eta+\langle v_{1}\rangle$ or
$1\gtrsim\frac{\lvert\zeta-a\rvert^{2}}{\eta+\langle v_{1}\rangle}$. In the
second case, (5.11) implies $\langle v_{1}\rangle\lesssim\eta+\langle
v_{1}\rangle\sim v_{1}(\lvert\zeta-a\rvert^{4}+\lvert\zeta-a\rvert^{2})$.
Using $\lvert\zeta-a\rvert\leq c$, choosing $c\sim 1$ sufficiently small and
averaging $\langle v_{1}\rangle\lesssim
v_{1}(\lvert\zeta-a\rvert^{4}+\lvert\zeta-a\rvert^{2})$ yield a contradiction
as $\langle v_{1}\rangle>0$. Hence, $\langle v_{1}\rangle\gtrsim 1$ and, thus,
$\langle v_{1}\rangle\sim 1$ by Lemma 5.1 as well as $1\gtrsim v_{1}$ by
(5.11) as $\eta\leq c$ for some small enough $c\sim 1$. Since $v_{1}\lesssim
1$ by Lemma 5.3, this completes the proof of $v_{1}\sim\langle v_{1}\rangle$
uniformly for $\zeta\in\mathbb{C}$ and $\eta>0$.

From $v_{1}\sim\langle v_{1}\rangle$ and (5.8), we conclude $\lvert
y\rvert\lesssim 1$ uniformly for $\zeta\in\mathbb{C}$ and $\eta>0$. Owing to
(5.11) and $v_{1}\sim\langle v_{1}\rangle$, we have $\eta+\langle
v_{1}\rangle\sim\langle v_{1}\rangle(\eta+\langle v_{1}\rangle)^{2}+\langle
v_{1}\rangle\lvert\zeta-a\rvert^{2}$. As $\eta+\langle v_{1}\rangle\leq c$, by
choosing $c\sim 1$ sufficiently small, we can incorporate $\langle
v_{1}\rangle(\eta+\langle v_{1}\rangle)^{2}$ into the left-hand side and
obtain $\langle v_{1}\rangle\lesssim\lvert\zeta-a\rvert^{2}\langle
v_{1}\rangle$. Hence, $1\lesssim\lvert\zeta-a\rvert$ as $\langle
v_{1}\rangle>0$. The bound $\lvert y\rvert\sim 1$ follows from (5.5). This
proves the additional statement and completes the proof of Lemma 5.6. ∎

###  5.2 Relation to Matrix Dyson equation

Let $(v_{1},v_{2})$ be a solution of (4.2). We now relate $(v_{1},v_{2})$ to a
solution $M\in\mathcal{B}^{2\times 2}$ of a matrix equation. To that end, we
define $y$ as in (5.3) and set

$M:=\begin{pmatrix}\mathrm{i}v_{1}&\overline{y}\\!\,\\\
y&\mathrm{i}v_{2}\end{pmatrix}\in\mathcal{B}^{2\times 2}.$ (5.12)

Then $\mathrm{Im}\,M:=\frac{1}{2\mathrm{i}}(M-M^{*})$ is positive definite and
inverting the $2\times 2$ matrix $M$ explicitly shows that $M$ satisfies the
_Matrix Dyson Equation (MDE)_

$-M^{-1}=\begin{pmatrix}\mathrm{i}\eta&\zeta-a\\\
\overline{\zeta-a}&\mathrm{i}\eta\end{pmatrix}+\Sigma[M].$ (5.13)

Here, $\Sigma\colon\mathcal{B}^{2\times 2}\to\mathcal{B}^{2\times 2}$ is
defined through

$\Sigma\bigg{[}\begin{pmatrix}r_{11}&r_{12}\\\
r_{21}&r_{22}\end{pmatrix}\bigg{]}=\begin{pmatrix}Sr_{22}&0\\\
0&S^{*}r_{11}\end{pmatrix}$ (5.14)

for all $r_{11}$, $r_{12}$, $r_{21}$, $r_{22}\in\mathcal{B}$.

On the other hand, if $M\in\mathcal{B}^{2\times 2}$ with $\mathrm{Im}\,M$
positive definite is a solution of (5.13) then it is easy to see that denoting
the diagonal elements of $M$ by $\mathrm{i}v_{1}$ and $\mathrm{i}v_{2}$ yields
a solution of (4.2).

#### Matrix Dyson equation with general spectral parameter and measure
$\rho_{\zeta}$

More generally, we consider the MDE, where the spectral parameter
$\mathrm{i}\eta$ is replaced by $w\in\mathbb{C}$ with $\mathrm{Im}\,w>0$, i.e.

$-M(\zeta,w)^{-1}=\begin{pmatrix}w&\zeta-a\\\
\overline{\zeta-a}\\!\,&w\end{pmatrix}+\Sigma[M(\zeta,w)]$ (5.15)

for $\zeta\in\mathbb{C}$. Then (5.15) has a unique solution
$M(\zeta,w)\in\mathcal{B}^{2\times 2}$ under the constraint that
$\mathrm{Im}\,M(\zeta,w):=\frac{1}{2\mathrm{i}}(M(\zeta,w)-M(\zeta,w)^{*})$ is
positive definite for $\mathrm{Im}\,w>0$ [30].

By [4, Proposition 2.1 and Definition 2.2], the map $w\mapsto\langle
M(\zeta,w)\rangle$ is the Stieltjes transform of a probability measure on
$\mathbb{R}$, where we introduced the short hand notation

$\langle R\rangle:=\frac{1}{2}(\langle r_{11}\rangle+\langle
r_{22}\rangle)\,,\qquad R=\left(\begin{array}[]{cc}r_{11}&r_{12}\\\
r_{21}&r_{22}\end{array}\right)\in\mathcal{B}^{2\times 2}\,.$

######  Definition 5.7.

We denote by $\rho_{\zeta}$ the unique probability measure on $\mathbb{R}$
whose Stieltjes transform is given by $w\mapsto\langle M(\zeta,w)\rangle$.
Through $\rho_{\zeta}$ we define

$\begin{split}\mathbb{S}_{\varepsilon}:=\\{\zeta\in\mathbb{C}:\operatorname{dist}(0,\operatorname{supp}\rho_{\zeta})\leq\varepsilon\\}\end{split}$
(5.16)

for any $\varepsilon\geq 0$.

In the setup of Theorem 2.4, in particular, $\mu$ is the Lebesgue measure on
$\mathfrak{X}=[0,1]$, the measure $\rho_{\zeta}$ from Definition 5.7 is the
asymptotic symmetrized empirical singular value distribution of
$X_{n}+A_{n}-\zeta$ for any $\zeta\in\mathbb{C}$, see [5, Theorem 2.7]. In
this case the set $\operatorname{Spec}_{\varepsilon}^{\infty}(s,a)$ from
Theorem 2.4 is identified in (7.11) below with $\mathbb{S}_{\varepsilon}$ from
(5.16), which is the $n\to\infty$ limit of the $\varepsilon$-pseudospectrum
(2.3) for $R=X+A$.

######  Remark 5.8.

The sets $\mathbb{S}_{\varepsilon}$ defined in (5.16) are monotonously
nondecreasing in $\varepsilon\geq 0$, i.e.
$\mathbb{S}_{\varepsilon_{1}}\subset\mathbb{S}_{\varepsilon_{2}}$ if
$\varepsilon_{1}\leq\varepsilon_{2}$. Moreover, they are bounded, in fact,
$\mathbb{S}_{\varepsilon}\subset\\{\zeta\in\mathbb{C}:\lvert\zeta\rvert\leq\varepsilon+\lVert
a\rVert_{\infty}+2(\lVert S\rVert_{\infty})^{1/2}\\}$ for all $\varepsilon\geq
0$ as a consequence of [4, Proposition 2.1]. Here, $\lVert S\rVert_{\infty}$
denotes the operator norm of $S$ viewed as an operator from $\mathcal{B}$ to
$\mathcal{B}$.

###  5.3 A relation between the derivatives of $M$

In this subsection, we consider derivatives of $M$, the solution of (5.13),
with respect to $\eta$ and $\zeta$ and establish a useful relation between
these in the next lemma.

######  Lemma 5.9.

Let $a\in\mathcal{B}$ and $s\colon\mathfrak{X}\times\mathfrak{X}\to[0,\infty)$
be bounded measurable functions. Then

$\langle\partial_{\eta}M_{21}(\zeta,\mathrm{i}\eta)\rangle=2\mathrm{i}\langle\partial_{\zeta}M(\zeta,\mathrm{i}\eta)\rangle,\qquad\langle\partial_{\eta}M_{12}(\zeta,\mathrm{i}\eta)\rangle=2\mathrm{i}\langle\partial_{\bar{\zeta}}M(\zeta,\mathrm{i}\eta)\rangle$
(5.17)

for every $\zeta\in\mathbb{C}$ and $\eta>0$, where we decomposed

$M(\zeta,\mathrm{i}\eta)=\begin{pmatrix}M_{11}(\zeta,\mathrm{i}\eta)&M_{12}(\zeta,\mathrm{i}\eta)\\\
M_{21}(\zeta,\mathrm{i}\eta)&M_{22}(\zeta,\mathrm{i}\eta)\end{pmatrix}.$

We note that the condition $\sup_{x,y\in\mathfrak{X}}s(x,y)<\infty$ implies
that the operators $S$ and $S^{*}$ from (4.1) can be extended to operators
$L^{2}\to L^{\infty}$.

Before proving Lemma 5.9, we prove the differentiability of $M$ with respect
to $\eta$, $\zeta$ and $\bar{\zeta}$. Let $\mathcal{L}$ be the stability
operator of (5.13), defined as

$\mathcal{L}\colon\mathcal{B}^{2\times 2}\to\mathcal{B}^{2\times 2},\quad
R\mapsto\mathcal{L}[R]:=M^{-1}RM^{-1}-\Sigma[R].$ (5.18)

This operator is invertible for any $\zeta\in\mathbb{C}$ and $\eta>0$ due to
Lemma A.1 below. Therefore, the implicit function theorem applied to (5.13)
and simple computations show that

$\partial_{\zeta}M=\mathcal{L}^{-1}[E_{12}],\qquad\partial_{\bar{\zeta}}M=\mathcal{L}^{-1}[E_{21}],\qquad\partial_{\eta}M=\mathrm{i}\mathcal{L}^{-1}[E_{+}]$
(5.19)

for all $\eta>0$ and $\zeta\in\mathbb{C}$, where we used the notations
$E_{12}$, $E_{21}$ and $E_{+}$ for the elements of $\mathcal{B}^{2\times 2}$
defined through

$E_{12}:=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\qquad
E_{21}:=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix},\qquad
E_{+}:=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$ (5.20)

###### Proof of Lemma 5.9.

As $\langle\partial_{\eta}M_{21}\rangle=\langle
E_{21}^{*}\partial_{\eta}M\rangle$, we get from (5.19) that

$\langle\partial_{\eta}M_{21}\rangle=2\mathrm{i}\langle
E_{21}^{*}\mathcal{L}^{-1}[E_{+}]\rangle=2\mathrm{i}\langle{E_{21}}\mspace{2.0mu},{\mathcal{L}^{-1}[E_{+}]}\rangle.$
(5.21)

Since $\langle\partial_{\eta}M_{12}\rangle=\langle
E_{12}^{*}\partial_{\eta}M\rangle$, there is an analogous statement for
$\langle\partial_{\eta}M_{12}\rangle$.

We start from the first relation in (5.19), use
$(\mathcal{L}[R])^{*}=\mathcal{L}^{*}[R^{*}]$, the invertibility of
$\mathcal{L}^{*}$ and $E_{12}^{*}=E_{21}$ to obtain

$\langle\partial_{\zeta}M\rangle=\langle{(\mathcal{L}^{-1}[E_{12}])^{*}}\mspace{2.0mu},{E_{+}}\rangle=\langle{(\mathcal{L}^{*})^{-1}[E_{12}^{*}]}\mspace{2.0mu},{E_{+}}\rangle=\langle{E_{21}}\mspace{2.0mu},{\mathcal{L}^{-1}[E_{+}]}\rangle.$
(5.22)

Therefore, combining (5.21) and (5.22) proves the first identity in (5.17).
The second one follows analogously. ∎

###  5.4 Stability of Dyson equation and analyticity of its solution

In this section we show how the solution $v_{1},v_{2}$ of (4.2) can be
extended to $\eta=0$. If we stay away from the deterministic analog of the
$\varepsilon$-pseudospectrum, then the solution is extended to $\eta=0$ by
setting $v_{i}=0$ by the following lemma.

######  Lemma 5.10.

Let $\varepsilon>0$. Let
$\zeta\in(\mathbb{C}\setminus\mathbb{S}_{\varepsilon})\cap\mathbb{D}_{1/\varepsilon}$.
Then $v_{i}(\zeta,\eta)\sim_{\varepsilon}\eta$ for all $\eta\in(0,1]$ and
$i=1,2$. In particular, $v_{i}$ is continuously extended to
$\zeta\in\mathbb{C}\setminus\mathbb{S}_{0}$ and $\eta=0$ by setting
$v_{i}(\zeta,0):=0$.

###### Proof.

From (4.2a), we conclude
$v_{1}(\lvert\zeta-a\rvert^{2}+(\eta+Sv_{2})(\eta+S^{*}v_{1}))=\eta+S^{*}v_{1}\geq\eta$.
Thus, $\lvert\zeta\rvert\leq\varepsilon^{-1}$, $a\in\mathcal{B}$, $\eta\leq
1$, (5.6) and Lemma 5.1 imply $v_{1}\gtrsim_{\varepsilon}\eta$ for all
$\eta\in(0,1]$. Similarly, $v_{2}\gtrsim_{\varepsilon}\eta$ for all
$\eta\in(0,1]$. On the other hand, as
$\zeta\in\mathbb{C}\setminus\mathbb{S}_{\varepsilon}$, the statement (v) of
[4, Lemma D.1] holds for $\tau=0$. Hence, [4, Lemma D.1 (i)] implies
$\max\\{v_{1}(\zeta,\eta),v_{2}(\zeta,\eta)\\}\leq\lVert\mathrm{Im}\,m(\mathrm{i}\eta)\rVert\lesssim\eta$
for all $\eta\in(0,c]$ for some sufficiently small $c\sim_{\varepsilon}1$. If
$\eta\in(c,1]$ then the upper bound in Corollary 5.2 yields
$v_{i}\lesssim\eta^{-1}\sim\eta$ for all $\eta\in(c,1]$. This completes the
proof. ∎

The next proposition states that if $\langle v_{1}\rangle=\langle
v_{2}\rangle$ remains bounded away from zero as $\eta\downarrow 0$, then the
solution has an analytic extension to $\eta=0$.

######  Proposition 5.11 (Analyticity in the bulk).

Let $s$ satisfy A5 and $\zeta\in\mathbb{C}$ with $\limsup_{\eta\downarrow
0}\langle v_{1}(\zeta,\eta)\rangle>0$. Then
$v_{1},v_{2}:\mathbb{C}\times(0,\infty)\to(0,\infty)$ has an extension to a
neighbourhood of $(\zeta,0)$ in $\mathbb{C}\times\mathbb{R}$ which is real
analytic in all variables.

To prove this proposition, we show that the Dyson equation (4.2) is stable
even for $\eta=0$. However, the equation does not have a unique solution on
$\mathcal{B}_{+}^{2}$ for $\eta=0$ without the additional constraint $\langle
v_{1}\rangle=\langle v_{2}\rangle$. Therefore, we have to reformulate the
equation to incorporate this constraint. Proposition 5.11 is proved at the end
of this subsection.

We recall that $\mathcal{B}_{+}:=\\{w\in\mathcal{B}:w>0\\}$ and set

$e_{-}=(1,-1)\in{\mathcal{B}}^{2}:=\mathcal{B}\oplus\mathcal{B}\,,\qquad
e_{-}^{\perp}:=\\{h=(h_{1},h_{2})\in\mathcal{B}^{2}:\langle
h_{1}\rangle=\langle h_{2}\rangle\\}.$

For $\eta>0$ and $\zeta\in\mathbb{C}$, we define $J\equiv J_{\zeta,\eta}\colon
e_{-}^{\perp}\cap\mathcal{B}_{+}^{2}\to
e_{-}^{\perp},(w_{1},w_{2})\mapsto(J_{1}(w_{1},w_{2}),J_{2}(w_{1},w_{2}))$
through

$\begin{split}J_{1}(w_{1},w_{2})&:=(\eta+Sw_{2})\biggl{(}{w_{1}-\frac{\eta+S^{*}w_{1}}{(\eta+S^{*}w_{1})(\eta+Sw_{2})+\lvert
a-\zeta\rvert^{2}}}\biggr{)}\,,\\\
J_{2}(w_{1},w_{2})&:=(\eta+S^{*}w_{1})\biggl{(}{w_{2}-\frac{\eta+Sw_{2}}{(\eta+S^{*}w_{1})(\eta+Sw_{2})+\lvert
a-\zeta\rvert^{2}}}\biggr{)}\,.\end{split}$

Then (4.2) takes the form $J(v)=0$ with
$v=(v_{1},v_{2})\in\mathcal{B}_{+}^{2}$.

On $\mathcal{B}^{2}$, we introduce a average and a scalar product defined
through

$\begin{split}\bigg{\langle}\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}\bigg{\rangle}:=\frac{1}{2}\big{(}\langle
x_{1}\rangle+\langle
x_{2}\rangle\big{)},\qquad\qquad\bigg{\langle}{\begin{pmatrix}x_{1}\\\
x_{2}\end{pmatrix}}\,\mspace{2.0mu},\,{\begin{pmatrix}y_{1}\\\
y_{2}\end{pmatrix}}\bigg{\rangle}:=\frac{1}{2}\bigl{(}{\langle\overline{x}\\!\,_{1}y_{1}\rangle+\langle\overline{x}\\!\,_{2}y_{2}\rangle}\bigr{)}\end{split}$
(5.23)

for $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}\in\mathcal{B}$. For
$x\in\mathcal{B}^{2}$, we write $\lVert
x\rVert_{2}:=\sqrt{\langle{x}\mspace{2.0mu},{x}\rangle}$.

For the rest of this section we will assume that $s$ satisfies A5. Until the
proof of Proposition 5.11, we fix $\zeta\in\mathbb{C}$ such that
$\limsup_{\eta\downarrow 0}\langle v_{1}(\zeta,\eta)\rangle\geq\delta$ for
some $\delta>0$. Under these conditions, $J$ remains well defined on
$e_{-}^{\perp}\cap\mathcal{B}_{+}^{2}$ even for $\eta=0$ and we set
$J_{0}:=J_{\zeta,\eta=0}$. We now pick candidates for $v_{1}(\zeta,0)$ and
$v_{2}(\zeta,0)$ by choosing weakly convergent subsequences in the limit
$\eta\downarrow 0$. By Lemma 5.1, there are $v_{0}\in(L^{2})^{2}:=L^{2}\oplus
L^{2}$ and a monotonically decreasing sequence $\eta_{n}\downarrow 0$ in
$(0,1]$ such that $v_{n}=v(\zeta,\eta_{n})$ is weakly convergent to $v_{0}$ in
$(L^{2})^{2}$, i.e. for any $h\in(L^{2})^{2}$,
$\langle{h}\mspace{2.0mu},{v_{n}-v_{0}}\rangle\to 0$ in the limit
$n\to\infty$. We recall that $L^{2}=L^{2}(\mathfrak{X},\mathcal{A},\mu)$.

######  Lemma 5.12.

Then $v_{0}\in\mathcal{B}_{+}^{2}\cap e_{-}^{\perp}$ and $\delta\lesssim
v_{0}\lesssim\frac{1}{\delta}$. Furthermore, $v_{0}$ satisfies (4.2) for
$\eta=0$, i.e. $J_{0}(v_{0})=0$.

For the following arguments, we introduce the operators $S_{o}$ and $S_{d}$ on
$\mathcal{B}^{2}$ defined through

$S_{o}:=\left(\begin{array}[]{cc}0&S\\\ S^{*}&0\end{array}\right)\,,\quad
S_{d}:=\left(\begin{array}[]{cc}S^{*}&0\\\ 0&S\end{array}\right)\,.$ (5.24)

Owing to the upper bound in A5, $S_{o}$ and $S_{d}$ can be extended naturally
to operators on $(L^{2})^{2}$.

###### Proof.

Since $v_{n}\to v_{0}$ weakly and $\langle e_{-}v_{n}\rangle=0$, we conclude
$v_{0}\perp e_{-}$. Furthermore, for any $h\in\mathcal{B}_{+}^{2}$ we get
$\langle hv_{0}\rangle=\lim_{n\to\infty}\langle
hv_{n}\rangle\gtrsim\delta\langle h\rangle$ because of Corollary 5.2 and
$\limsup_{n\to\infty}\langle v_{n}\rangle\geq\delta$. From this we conclude
$v_{0}\gtrsim\delta$. Similarly, Corollary 5.2 implies
$v_{0}\lesssim\frac{1}{\delta}$ and thus $v_{0}\in\mathcal{B}_{+}^{2}$.

The natural extensions of $S$ and $S^{*}$ to operators on $L^{2}$ are Hilbert-
Schmidt operators because $s\in
L^{2}(\mathfrak{X}\times\mathfrak{X},\mu\otimes\mu)$ due to the upper bound in
A5. In particular, $S$ and $S^{*}$ are compact operators on $L^{2}$ and, thus,
$S_{o}v_{n}\to S_{o}v_{0}$ and $S_{d}v_{n}\to S_{d}v_{0}$ in $(L^{2})^{2}$.
The bounds $\delta\lesssim v_{n}\lesssim\delta^{-1}$ then imply that
$J_{\zeta,\eta_{n}}(v_{n})\to J_{0}(v_{0})$ weakly in $(L^{2})^{2}$.
Consequently, $J_{0}(v_{0})=0$. ∎

For the formulation of the next lemma, we note that $\lVert T\rVert_{\infty}$
denotes the operator norm of an operator
$T\colon\mathcal{B}^{2}\to\mathcal{B}^{2}$ and, analogously, $\lVert
T\rVert_{2}$ is the operator norm if $T\colon(L^{2})^{2}\to(L^{2})^{2}$.

######  Lemma 5.13.

Let $v_{0}$ be a weak limit of a sequence $v_{n}=v(\zeta,\eta_{n})$ as above.
Then

$\lVert(\nabla J_{0}|_{w=v_{0}})^{-1}\rVert_{2}+\lVert(\nabla
J_{0}|_{w=v_{0}})^{-1}\rVert_{\infty}\lesssim_{\delta}1.$

###### Proof of Lemma 5.13.

Within this proof we will make use of some results from [3]. Therefore we
introduce notations that match the ones from [3], namely

$\begin{split}\tau:=(\lvert\zeta-a\rvert^{2},\lvert\zeta-a\rvert^{2})\,\end{split}$
(5.25)

and recall the definitions of $S_{o}$ and $S_{d}$ from (5.24). In [3] the
setup $a=0$ was treated and therefore $\tau=\lvert\zeta\rvert^{2}$ was
constant. Here
$\tau=(\tau_{1},\tau_{2})\in\overline{\mathcal{B}_{+}}\\!\,\oplus\overline{\mathcal{B}_{+}}\\!\,$
satisfies $\tau_{1}=\tau_{2}$, which ensures that the necessary computations
from [3] remain applicable. Using the notations (5.24) and (5.25), we write
$J$ in the form

$J(w)=(\eta+S_{o}w)\biggl{(}{w-\frac{1}{\eta+S_{o}w+\frac{\tau}{\eta+S_{d}w}}}\biggr{)}\,.$

Now we take the directional derivative $\nabla_{h}J$ of $J$ in the direction
$h\in\mathcal{B}^{2}$ with $h\perp e_{-}$, i.e. $\langle he_{-}\rangle=0$, and
evaluate at the solution $v=(v_{1},v_{2})$. Thus, we find

$\nabla_{h}J|_{w=v}=(\eta+S_{o}v)\biggl{(}{h+v^{2}S_{o}h-\frac{v^{2}\tau}{(\eta+S_{d}v)^{2}}S_{d}h}\biggr{)}=(\eta+S_{o}v)Lh\,,$
(5.26)

where we used $J(v)=0$ and introduced the linear operator $L\equiv
L_{\zeta,\eta}(v)\colon\mathcal{B}^{2}\to\mathcal{B}^{2}$ as

$\begin{split}Lh:=h+v^{2}S_{o}h-\frac{v^{2}\tau}{(\eta+S_{d}v)^{2}}S_{d}h\end{split}$
(5.27)

to guarantee the last equality. We now restrict our analysis to $\eta=0$ and
use the following lemma that provides a resolvent estimate for
$L_{0}=L_{\zeta,0}(v_{0})$, the operator evaluated on the weak limit $v_{0}$.

######  Lemma 5.14.

There is $\varepsilon_{\ast}\sim_{\delta}1$ such that for any
$\varepsilon\in(0,\varepsilon_{\ast})$ we have the bound

$\begin{split}\sup\Bigl{\\{}{\lVert(L_{0}-\omega)^{-1}\rVert_{\\#}:\omega\not\in\mathbb{D}_{\varepsilon}\cup((1+\mathbb{D}_{1+\varepsilon})\setminus\mathbb{D}_{2\varepsilon})}\Bigr{\\}}\lesssim_{\delta,\varepsilon}1\end{split}$
(5.28)

for $\\#=2,\infty$. Here, $\mathbb{D}_{\varepsilon}$ contains the single
isolated eigenvalue $0$ of $L_{0}$ with corresponding right and left
eigenvectors $v_{-}:=e_{-}v_{0}$ and $S_{o}v_{-}$, i.e.

$\mathbb{D}_{\varepsilon}\cap\operatorname{Spec}(L_{0})=\\{0\\}\,,\qquad\ker
L_{0}^{2}=\mathrm{Span}(v_{-})\,,\qquad L_{0}v_{-}=0\,,\qquad
L_{0}^{*}S_{o}v_{-}=0\,.$

Here, $L_{0}^{*}$ is the adjoint of $L$ with respect to the $L^{2}$-scalar
product introduced in (5.23).

The proof of Lemma 5.14 follows a strategy similar to the one used to prove
stability of the Dyson equation in [3], where the case $a=0$ was treated. For
completeness we present the proof in Appendix A.2 below. Using Lemma 5.14 we
now show that

$\begin{split}\lVert
L_{0}^{-1}|_{(S_{o}v_{-})^{\perp}}\rVert_{\\#}\lesssim_{\delta}1\,,\qquad\\#=2,\infty\,,\end{split}$
(5.29)

from which the claim of Lemma 5.13 immediately follows due to (5.26), A5 and
$v_{0}\gtrsim\delta$. To see (5.29), we will apply [6, Lemma 4.6] to
$C\mspace{1.0mu}L_{0}$ for some appropriately large positive constant
$C\sim_{\delta}1$. The lemma was formulated for $\mathfrak{X}=\\{1,\dots,d\\}$
with the normalized counting measure, i.e. $\mathcal{B}=\mathbb{C}^{d}$.
However, its proof is uniform in the underlying dimension $d$ and it therefore
translates to the current general setup. We now check the assumptions of the
[6, Lemma 4.6]. Note that $L_{0}$ maps $e_{-}^{\perp}$ to
$(S_{o}v_{-})^{\perp}$. By Lemma 5.14 the right and left eigenvectors of
$L_{0}$ corresponding to the eigenvalue $0$ are $v_{-}$ and $S_{o}v_{-}$,
respectively. Moreover,
$\langle{v_{-}}\mspace{2.0mu},{e_{-}}\rangle\gtrsim_{\delta}1$ as
$v_{0}\gtrsim\delta$,
$\lvert\langle{e_{-}}\mspace{2.0mu},{w}\rangle\rvert\leq\lVert w\rVert_{\\#}$
and that $\lVert L_{0}w\rVert_{\\#}\gtrsim_{\delta}\lVert w\rVert_{\\#}$ for
any $w\perp S_{o}v_{-}$ due to Lemma 5.14. By [6, Lemma 4.6] we get $\lVert
L_{0}w\rVert_{\\#}\gtrsim_{\delta}\lVert w\rVert_{\\#}$ for any $w\perp
e_{-}$. Thus, (5.29) is shown. ∎

Now we use the stability at $\eta=0$ to finish the proof of the main result of
this subsection.

###### Proof of Proposition 5.11.

Let $\zeta_{0}\in\mathbb{C}$ be such that $\limsup_{\eta\downarrow 0}\langle
v_{1}(\zeta_{0},\eta)\rangle>0$. Let $\eta_{n}\downarrow 0$ such that
$v_{n}=v(\zeta_{0},\eta_{n})$ is weakly convergent in $(L^{2})^{2}$. This is
possible, because the family $v(\zeta_{0},\eta)$ with $\eta\in(0,1]$ is
bounded in $(L^{2})^{2}$ due to Lemma 5.1. By Lemma 5.12 the weak limit
$v_{0}=\lim_{n\to\infty}v_{n}$ satisfies the Dyson equation,
$J_{\zeta,0}(v_{0})=0$, and by Lemma 5.13 the Dyson equation is stable at
$v=v_{0}$ and $\eta=0$. By the implicit function theorem we find a real
analytic function $\widetilde{v}$, defined on a neighbourhood $U$ of
$(\zeta_{0},0)$ in $\mathbb{C}\times\mathbb{R}$, such that
$\widetilde{v}(\zeta,\eta)$ solves (4.2) and
$\widetilde{v}(\zeta_{0},0)=v_{0}$. Since $v_{0}\gtrsim\delta$ according to
Lemma 5.12, $\widetilde{v}(\zeta,\eta)>0$ on $U$ if the neighbourhood $U$ is
chosen sufficiently small. By uniqueness of the solution to the Dyson equation
we conclude $\widetilde{v}(\zeta,\eta)=v(\zeta,\eta)$ for all $(\zeta,\eta)\in
U$. ∎

###  5.5 Characterisation of $\mathbb{S}$

Throughout this section we assume that $a\in\mathcal{B}$ and $s$ satisfies A5
and A6. To generalize (2.6), (2.7) and (2.8) to the setup introduced in
Section 4, we define an operator $B_{\zeta}\colon\mathcal{B}\to\mathcal{B}$, a
function $\beta\colon\mathbb{C}\to\mathbb{R}$ and a subset
$\mathbb{S}\subset\mathbb{C}$ through

$\begin{split}\beta(\zeta):=\inf_{x\in\mathcal{B}_{+}}\sup_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{B_{\zeta}\mspace{2.0mu}y}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}\,,\qquad
B_{\zeta}:=D_{\lvert
a-\zeta\rvert^{2}}-S\,,\qquad\mathbb{S}:=\\{\zeta\in\mathbb{C}:\beta(\zeta)<0\\}\,.\end{split}$
(5.30)

We also set

$\rho_{\zeta}(0):=\frac{1}{\pi}\lim_{\eta\downarrow 0}\langle
v_{1}(\zeta,\eta)\rangle\,.$ (5.31)

This limit exists, because either $\limsup_{\eta\to 0}\langle
v_{1}(\zeta,\eta)\rangle>0$, in which case $v_{1}$ can be analytically
extended to $\eta=0$ by Proposition 5.11, or $\limsup_{\eta\to 0}\langle
v_{1}(\zeta,\eta)\rangle=0$ in which case the limit equals zero as well. As
explained after Definition 5.7 we can interpret $\rho_{\zeta}(0)$ as the
asymptotic singular value density of $X+A-\zeta$ at zero in case
$(\mathfrak{X},\mu)=([0,1],\mathrm{d}x)$.

In the following we will denote by $\lambda_{\mathrm{PF}}(T)$ the spectral
radius of a compact and positivity preserving operator $T$, i.e.
$\lambda_{\mathrm{PF}}(T)$ is the Perron-Frobenius eigenvalue of $T$. In
particular the operators $S$ and $S^{*}$ are compact as mentioned in the proof
of Lemma 5.12 and therefore so are $D_{x}SD_{y}$ and $D_{x}S^{*}D_{y}$ for
$x,y\in\mathcal{B}$. We use this fact in the statement of the following
proposition.

######  Proposition 5.15.

The following relations between $\beta$, $\mathbb{S}$,
$\mathbb{S}_{\varepsilon}$ and $\rho_{\zeta}$ apply.

  1. (i)

The function $\mathbb{C}\ni\zeta\mapsto\beta(\zeta)$ is continuous and
satisfies $\lim_{\zeta\to\infty}\beta(\zeta)=+\infty$. In particular,
$\mathbb{S}$ is bounded.

  2. (ii)

The spectrum of $D_{a}$ lies inside $\mathbb{S}$, i.e.

$\begin{split}\operatorname{Spec}(D_{a})\subset{\mathbb{S}}\,.\end{split}$
(5.32)

  3. (iii)

The sign of $\beta$ satisfies

$\begin{split}\mathrm{sign}(\beta(\zeta))=\mathrm{sign}\Big{(}1-\lambda_{\mathrm{PF}}\Big{(}SD_{\lvert
a-\zeta\rvert}^{-2}\Big{)}\Big{)}\,,\qquad\zeta\in\mathbb{C}\,.\end{split}$
(5.33)

  4. (iv)

For any $\zeta\in\mathbb{C}$ with $\beta(\zeta)>0$ the operator $B_{\zeta}$ is
invertible. Furthermore, all such $\zeta$ are characterised by

$\begin{split}\\{\zeta\in\mathbb{C}:\beta(\zeta)>0\\}=\\{\zeta\in\mathbb{C}:\operatorname{dist}(0,\operatorname{supp}\rho_{\zeta})>0\\}=\mathbb{C}\setminus\mathbb{S}_{0}\,.\end{split}$
(5.34)

  5. (v)

The set $\mathbb{S}$ is characterised by having a positive singular value
density at the origin, i.e.

$\begin{split}\mathbb{S}=\\{\zeta\in\mathbb{C}:\rho_{\zeta}(0)>0\\}\,.\end{split}$
(5.35)

###### Proof.

Proof of (i): The continuity of $\zeta\mapsto\beta(\zeta)=\beta$ with
$B=B_{\zeta}$ is a consequence of the bound

$\bigg{\lvert}\inf_{x\in\mathcal{B}_{+}}\sup_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{(B+D_{w})y}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}-\beta\bigg{\rvert}\leq\sup_{x\in\mathcal{B}_{+}}\sup_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{D_{\lvert
w\rvert}y}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}\leq\lVert
w\rVert_{\infty}$

for any real valued $w\in\mathcal{B}$. The statement $\beta(\zeta)\to+\infty$
as $\zeta\to\infty$ is obvious.

Before we start with the proof of other individual statements of the
proposition, we show that $\mathbb{S}$ can be classified in terms of the
Perron-Frobenius eigenvalue of $SD_{\lvert a-\zeta\rvert}^{-2}$ in the sense
that

$\begin{split}\mathbb{S}=\\{\zeta\in\mathbb{C}:\lambda(\zeta)>1\\}\,,\end{split}$
(5.36)

where we introduced $\lambda:\mathbb{C}\to[0,\infty]$ as the limit of a
strictly increasing sequence via

$\lambda(\zeta):=\lim_{\varepsilon\downarrow
0}\lambda_{\varepsilon}(\zeta)\,,\qquad\lambda_{\varepsilon}(\zeta):=\lambda_{\mathrm{PF}}\bigl{(}{S(\varepsilon+D_{\lvert
a-\zeta\rvert^{2}})^{-1}}\bigr{)}.$

To show (5.36) let $\varepsilon\in(0,1)$, $D:=D_{\lvert\zeta-a\rvert^{2}}$,
$\lambda_{\varepsilon}=\lambda_{\varepsilon}(\zeta)$ and $C>0$ such that
$1+\lvert\zeta-a\rvert^{2}\leq C$. For $\zeta\in\mathbb{C}$ with
$\beta(\zeta)\geq 0$ we get

$\begin{split}\beta+\varepsilon\leq
C\inf_{x\in\mathcal{B}_{+}}\sup_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{(\varepsilon+B)y}\rangle}{\langle{x}\mspace{2.0mu},{(\varepsilon+D)y}\rangle}=C\biggl{(}{1-\sup_{x\in\mathcal{B}_{+}}\inf_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{S(\varepsilon+D)^{-1}y}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}}\biggr{)}=C\bigl{(}{1-\lambda_{\varepsilon}}\bigr{)},\end{split}$

where we used $\varepsilon+\lvert\zeta-a\rvert^{2}\leq C$ in the first
inequality, and conclude

$\begin{split}\beta\leq C(1-\lambda)\qquad\text{ if }\beta\geq
0\,.\end{split}$ (5.37)

For $\zeta\in\mathbb{C}$ with $\beta(\zeta)<0$ we use

$-\beta-\varepsilon=\sup_{x\in\mathcal{B}_{+}}\inf_{y\in\mathcal{B}_{+}}\frac{-\langle{x}\mspace{2.0mu},{(\varepsilon+B)y}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}$

for sufficiently small $\varepsilon>0$ and find analogously that

$\begin{split}\beta\geq C\bigl{(}{1-\lambda}\bigr{)}\qquad\text{ if
}\beta<0\,.\end{split}$ (5.38)

From (5.37) and (5.38) we conclude (5.36).

We also show that

$\begin{split}\operatorname{Spec}(D_{a})\subset\\{\zeta\in\mathbb{C}:\beta(\zeta)\leq
0\\}.\end{split}$ (5.39)

We will improve this to (5.32) below. Let
$\zeta\in\operatorname{Spec}(D_{a})$. Then
$\operatorname*{ess\,inf}\lvert\zeta-a\rvert=0$. Thus, for any $\varepsilon>0$
we find $x\in\mathcal{B}\setminus\\{0\\}$ with $x\geq 0$ such that
$\lvert\zeta-a\rvert^{2}x\leq\varepsilon x$. In the definition of $\beta$ from
(5.30) we can take the supremum over all
$x\in\overline{\mathcal{B}}\\!\,_{+}$, Thus, we get

$\beta\leq\varepsilon-\inf_{y\in\mathcal{B}_{+}}\frac{\langle{x}\mspace{2.0mu},{Sy}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}\leq\varepsilon\,.$

Since $\varepsilon>0$ was arbitrarily small, we conclude $\beta\leq 0$.

Proof of (iv): Let $\zeta\in\mathbb{C}$ such that $\beta(\zeta)=\beta>0$. Then
(5.37) implies $\lambda_{\mathrm{PF}}(SD^{-1})<1$ with
$D=D_{\lvert\zeta-a\rvert^{2}}$. Here, $D$ is invertible because
$\operatorname*{ess\,inf}\lvert\zeta-a\rvert>0$ due to (5.39). In particular,
$B=(1-SD^{-1})D$ is invertible.

Now we show $\operatorname{dist}(0,\operatorname{supp}\rho_{\zeta})>0$ to see
one inclusion in the characterisation (5.34). The Dyson equation in the matrix
representation, (5.15) is solved by

$\begin{split}M_{0}:=\left(\begin{array}[]{cc}0&(\overline{a-\zeta})^{-1}\\\
(a-\zeta)^{-1}&0\end{array}\right)\end{split}$ (5.40)

at $w=0$. Furthermore, the associated stability operator (cf. (5.18))

$\begin{split}\mathcal{L}_{0}\colon\mathcal{B}^{2\times
2}\to\mathcal{B}^{2\times 2},\quad R\mapsto M_{0}^{-1}RM_{0}^{-1}-\Sigma
R=\left(\begin{array}[]{cc}\lvert
a-\zeta\rvert^{2}r_{22}-Sr_{22}&(a-\zeta)^{2}r_{21}\\\
(\overline{a-\zeta})^{2}r_{12}&\lvert
a-\zeta\rvert^{2}r_{11}-S^{*}r_{11}\end{array}\right)\end{split}$ (5.41)

is invertible because $B_{\zeta}$ is invertible and
$\operatorname*{ess\,inf}\lvert a-\zeta\rvert>0$. Therefore (5.15) can be
uniquely solved for sufficiently small $w$ as an analytic function $w\mapsto
M(\zeta,w)$ with $M(\zeta,0)=M_{0}$ and we get

$\mathcal{L}_{0}[\partial_{w}M(\zeta,w)|_{w=0}]=1\quad\text{and}\quad\partial_{\eta}M(\zeta,\mathrm{i}\eta)|_{\eta=0}=\mathrm{i}\mspace{1.0mu}\mathcal{L}_{0}^{-1}[1]$

In particular, $\operatorname{Im}M(\zeta,\mathrm{i}\eta)$ is positive definite
for sufficiently small $\eta>0$ because $\mathcal{L}_{0}^{-1}$ is positivity
preserving, as can be seen from a Neumann series expansion using that
$\lambda_{\mathrm{PF}}(SD^{-1})<1$. Therefore $M(\zeta,\mathrm{i}\eta)$ is the
unique solution of (5.15) with
$m_{11}(\zeta,\mathrm{i}\eta)=\mathrm{i}v_{1}(\zeta,\eta)$ and
$m_{22}(\zeta,\mathrm{i}\eta)=\mathrm{i}v_{2}(\zeta,\eta)$. Since
$m_{11}|_{\eta=0}=m_{22}|_{\eta=0}=0$ we conclude that
$\rho_{\zeta}(0)=\langle v_{1}(\zeta,0)\rangle=0$. The invertibility of
$\mathcal{L}_{0}$ also implies analyticity of $M(\zeta,w)$ in $w$ in a small
neighbourhood of zero. Thus, $\rho_{\zeta}([-\varepsilon,\varepsilon])=0$ for
$\varepsilon>0$ sufficiently small and
$\operatorname{dist}(0,\operatorname{supp}\rho_{\zeta})>0$.

To see the other inclusion in (5.34), let $\zeta\in\mathbb{C}$ be such that
$\operatorname{dist}(0,\operatorname{supp}\rho_{\zeta})\geq\delta$ for some
$\delta>0$. From [4, Lemma D.1 (iv)] we know that $M=M(\zeta,\mathrm{i}\eta)$
is locally a real analytic function of $\eta$ with an expansion
$M=M_{0}+\mathrm{i}\mspace{1.0mu}\eta\mspace{2.0mu}M_{1}+O(\eta^{2})$, where
$M_{0}=M_{0}^{*}$. Taking the imaginary part of (5.15) at $w=\mathrm{i}\eta$,
dividing both sides by $\eta$ shows that

$\begin{split}(M^{*})^{-1}K_{\eta}M^{-1}=1+\Sigma[K_{\eta}]\,,\end{split}$
(5.42)

where
$K_{\eta}=\frac{1}{\eta}\operatorname{Im}M=\operatorname{Re}M_{1}+O(\eta)$. In
particular, $K_{0}:=\lim_{\eta\downarrow 0}K_{\eta}$ exists and since
$\operatorname{Im}M(\zeta,\mathrm{i}\eta)\sim_{\delta}\eta$ by Lemma 5.10 we
get $K_{0}\sim_{\delta}1$. Evaluating (5.42) at $\eta=0$ yields

$\begin{split}M_{0}(\Sigma K_{0})M_{0}=K_{0}-1\,.\end{split}$ (5.43)

Taking the scalar product of (5.43) with the left Perron-Frobenius eigenvector
of $R\mapsto M_{0}(\Sigma R)M_{0}$ and using $K_{0}\sim_{\delta}1$ we see that
$\lambda_{\mathrm{PF}}(R\mapsto M_{0}(\Sigma R)M_{0})<1$. This is equivalent
to $\lambda:=\lambda_{\mathrm{PF}}(SD^{-1})<1$ with $D:=D_{\lvert
a-\zeta\rvert^{2}}$. Now let $u\in\mathcal{B}_{+}$ be the Perron-Frobenius
eigenvector of $SD^{-1}$. Since $\varepsilon:=\operatorname*{ess\,inf}\lvert
a-\zeta\rvert>0$ we get with $y_{0}:=D^{-1}u$ that

$(D-S)y_{0}=(1-\lambda)Dy_{0}>0\,.$

Thus,

$\beta=\inf_{x>0}\sup_{y>0}\frac{\langle{x}\mspace{2.0mu},{By}\rangle}{\langle{x}\mspace{2.0mu},{y}\rangle}\geq\inf_{x>0}\frac{\langle{x}\mspace{2.0mu},{By_{0}}\rangle}{\langle{x}\mspace{2.0mu},{y_{0}}\rangle}\geq(1-\lambda)\mspace{1.0mu}\varepsilon^{2}>0\,.$

This finishes the proof of (5.34), i.e. of (iv).

Proof of (iii): We have now collected enough information to improve (5.36) to
(5.33). Indeed, by (5.37) and (5.38) it remains to show that $\lambda<1$
implies $\beta>0$. Due to (5.36) we already know $\beta\geq 0$ in case
$\lambda<1$. Now let $\beta=0$ and $\lambda\leq 1$. Then we show that
$\lambda=1$. Indeed by the characterisation (5.34) we have
$0\in\operatorname{supp}\rho_{\zeta}$. Now we consider the identity

$B_{\zeta}v_{2}=\eta-v_{2}(\eta+S^{*}v_{1})(\eta+Sv_{2})$

which follows from (4.2b). For some $\varepsilon>0$ we add
$\varepsilon\mspace{1.0mu}v_{2}$ to both sides and apply the inverse of
$\varepsilon+D$ with $D=D_{\lvert a-\zeta\rvert^{2}}$. Then we take the scalar
product with the right Perron-Frobenius eigenvector
$x_{\varepsilon}\in\mathcal{B}_{+}$ of $S^{*}(\varepsilon+D)^{-1}$
corresponding to its Perron-Frobenius eigenvalue $\lambda_{\varepsilon}>0$.
Note that the Perron-Frobenius eigenvalues of $S^{*}(\varepsilon+D)^{-1}$,
$(\varepsilon+D)^{-1}S$ and $S(\varepsilon+D)^{-1}$ all coincide. Thus we get

$\begin{split}(1-\lambda_{\varepsilon})\langle
x_{\varepsilon}v_{2}\rangle=\eta\mspace{1.0mu}\langle(\varepsilon+D)^{-1}x_{\varepsilon}\rangle-\langle
x_{\varepsilon}(\varepsilon+D)^{-1}(v_{2}(\eta+S^{*}v_{1})(\eta+Sv_{2})-\varepsilon\mspace{1.0mu}v_{2})\rangle\,.\end{split}$
(5.44)

From [4, Corollary D.2] and $\langle v_{i}\rangle\sim v_{i}$ by Lemma 5.6 we
see that $\eta/\langle v_{i}\rangle\to 0$ for $\eta\downarrow 0$. Thus,
dividing (5.44) by $\langle v_{2}\rangle$, taking the limit $\eta\downarrow 0$
and using (5.6) reveals

$(1-\lambda_{\varepsilon})\langle
x_{\varepsilon}\rangle\sim(1-\lambda_{\varepsilon})\langle
x_{\varepsilon}k\rangle=\varepsilon\langle
x_{\varepsilon}(\varepsilon+D)^{-1}k\rangle\sim\varepsilon\langle
x_{\varepsilon}(\varepsilon+D)^{-1}S1\rangle=\varepsilon\langle
S^{*}(\varepsilon+D)^{-1}x_{\varepsilon}\rangle=\varepsilon\mspace{1.0mu}\lambda_{\varepsilon}\langle
x_{\varepsilon}\rangle\,,$

where $k:=\limsup_{\eta\downarrow 0}\frac{v_{2}}{\langle v_{2}\rangle}\sim 1$.
Letting $\varepsilon\downarrow 0$ shows $\lambda=1$. Thus, (5.33) is proven.

Proof of (v): By (5.34) we know that $\rho_{\zeta}(0)>0$ implies
$\beta(\zeta)\leq 0$. Thus, it suffices to show that for $\zeta\in\mathbb{C}$
with $\beta(\zeta)\leq 0$ we get $\beta(\zeta)=0$ if and only if
$\rho_{\zeta}(0)=0$. Now let $\beta=\beta(\zeta)\leq 0$. As above, we consider
the identity (5.44). First, suppose $\rho_{\zeta}(0)>0$, i.e. we can
analytically extend $v$ to $\eta=0$ by Proposition 5.11 and have
$v|_{\eta=0}>0$. Then in the limit $\eta\downarrow 0$ we find

$(\lambda_{\varepsilon}-1)\langle x_{\varepsilon}v_{2}\rangle=\langle
x_{\varepsilon}(\varepsilon+D)^{-1}(v_{2}(S^{*}v_{1}Sv_{2}-\varepsilon))\rangle$

Using $v_{2}\sim\langle v_{2}\rangle\sim\rho_{\zeta}(0)$ for small enough
$\varepsilon>0$ the right hand side satisfies

$\langle
x_{\varepsilon}(\varepsilon+D)^{-1}(v_{2}(S^{*}v_{1}Sv_{2}-\varepsilon))\rangle\sim\rho_{\zeta}(0)^{3}\langle
x_{\varepsilon}(\varepsilon+D)^{-1}S1\rangle\sim\lambda_{\varepsilon}\rho_{\zeta}(0)^{3}\langle
x_{\varepsilon}\rangle.$

Since $\langle x_{\varepsilon}v_{2}\rangle\sim\rho_{\zeta}(0)\langle
x_{\varepsilon}\rangle$ we infer
$\lambda_{\varepsilon}-1\sim\lambda_{\varepsilon}\mspace{2.0mu}\rho_{\zeta}(0)^{2}$.
Thus, $\lambda>1$ and by (5.33) therefore $\beta<0$.

Conversely, let $v|_{\eta=0}=0$. Then we know from Lemma 5.6 that
$\delta:=\operatorname*{ess\,inf}\lvert a-\zeta\rvert>0$. Since
$\beta(\zeta)\leq 0$ the characterisation (5.34) implies
$0\in\operatorname{supp}\rho_{\zeta}$ and by (5.33) we have $\lambda\geq 1$.
Since $\eta/\langle v\rangle\to 0$ for $\eta\downarrow 0$ by [4, Corollary
D.2] we get, dividing (5.44) by $\langle v_{2}\rangle$ and taking the limit
$\eta\downarrow 0$, the scaling behaviour

$1-\lambda_{\varepsilon}\sim\varepsilon\mspace{1.0mu}\lambda_{\varepsilon}\,.$

This implies $\lambda_{\varepsilon}\leq 1$, thus $\lambda=1$, and completes
the proof of (v).

Proof of (ii): Let $\zeta\in\operatorname{Spec}(D_{a})$. By (5.39) we know
$\beta(\zeta)\leq 0$. Suppose $\beta(\zeta)=0$. Then (5.35) would imply
$\rho_{\zeta}(0)=0$. However, this contradicts
$\zeta\in\operatorname{Spec}(D_{a})$ because of Lemma 5.6. This finishes the
proof of the proposition. ∎

###  5.6 Expansion of $v_{1}$ and $v_{2}$ at the spectral edge

In this section we expand the solution $v_{1},v_{2}$ of (4.2) around any edge
point $\zeta_{0}\in\mathbb{C}$. We will see later in Proposition 6.1 that
points in the boundary of the support of the Brown measure $\sigma$, a
probability measure in the complex plane associated with our data $a$ and $s$
and defined in that proposition, satisfy $\beta(\zeta_{0})=0$. Therefore we
consider in this section a fixed $\zeta_{0}\in\mathbb{C}$ with
$\beta(\zeta_{0})=0$. The expansion of $v_{1},v_{2}$ around $\zeta_{0}$ is
based on analytic perturbation theory for $\beta$. Throughout this section we
will always assume $\lvert\zeta-\zeta_{0}\rvert+\eta\leq c$ for some
sufficiently small positive constant $c\sim 1$, i.e. we assume that
$(\zeta,\eta)$ lies within a small neighbourhood of $(\zeta_{0},0)$. We will
see in Corollary 5.18 below that the function $\zeta\mapsto\beta(\zeta)$, used
to define $\mathbb{S}$ in (5.30), coincides locally around $\zeta_{0}$ with
the isolated non-degenerate eigenvalue of $B_{\zeta}$ closest to zero. We
assume A5 and A6 throughout the remainder of this section.

To shorten notation, we denote $v_{i}=v_{i}(\zeta,\eta)$. The identities

$\begin{split}&B_{\zeta}\mspace{1.0mu}v_{2}=\eta-
v_{2}(\eta+S^{*}v_{1})(\eta+Sv_{2}),\qquad
B_{\zeta}^{*}\mspace{1.0mu}v_{1}=\eta-
v_{1}(\eta+S^{*}v_{1})(\eta+Sv_{2}),\end{split}$ (5.45)

which follow from (4.2a) and (4.2b), respectively, are used to expand $v_{1}$
and $v_{2}$ in a neighbourhood of $\zeta_{0}$. We denote by
$b=b_{\zeta}\in\mathcal{B}_{+}$ and $\ell=\ell_{\zeta}\in\mathcal{B}_{+}$ the
right and left eigenvectors of $B=B_{\zeta}$, corresponding to the eigenvalue
$\beta=\beta(\zeta)$ with normalisation $\langle
b\rangle=\langle\ell\rangle=1$, i.e.

$\begin{split}Bb=\beta\mspace{2.0mu}b,\qquad
B^{*}\ell=\beta\mspace{1.0mu}\ell\,.\end{split}$ (5.46)

The existence and uniqueness of $b$ and $\ell$ is a consequence of analytic
perturbation theory and Lemma 5.17 below. This lemma also implies that
$\zeta\mapsto b_{\zeta}$ and $\zeta\mapsto\ell_{\zeta}$ are real analytic
functions. The main result of this section is the following proposition.

######  Proposition 5.16.

Let $s$ and $a$ satisfy A5 and A6. Furthermore, let $\zeta_{0}\in\mathbb{C}$
such that $\beta(\zeta_{0})=0$. Then there is an open neighbourhood
$U\subset\mathbb{C}\times\mathbb{R}^{2}$ of $(\zeta_{0},0,0)$, an open
neighbourhood $V\subset\mathbb{C}\times\mathbb{R}$ of $(\zeta_{0},0)$ and real
analytic functions $\widetilde{w}_{1},\widetilde{w}_{2}\colon U\to\mathcal{B}$
such that

$v_{1}(\zeta,\eta)=\vartheta(\zeta,\eta)\ell_{\zeta}+\widetilde{w}_{i}(\zeta,\eta,\vartheta(\zeta,\eta))\,,\qquad
v_{2}(\zeta,\eta)=\vartheta(\zeta,\eta)b_{\zeta}+\widetilde{w}_{i}(\zeta,\eta,\vartheta(\zeta,\eta))$

for $(\zeta,\eta)\in V$ and $\eta>0$. Furthermore,
$\vartheta=\vartheta(\zeta,\eta)$ satisfies

$\begin{split}\vartheta^{3}\big{\langle}\ell
b(S^{*}\ell)(Sb)\big{\rangle}+\beta\mspace{1.0mu}\vartheta\mspace{1.0mu}\langle\ell\mspace{1.0mu}b\rangle-\eta=g(\zeta,\eta,\vartheta)\,,\end{split}$
(5.47)

for all $(\zeta,\eta)\in V$ where $\ell=\ell_{\zeta}$,
$b=b_{\zeta},\beta=\beta(\zeta)$ and $g\colon U\to\mathbb{R}$ is a real
analytic function, such that

$g(\zeta,\eta,x)=O(\lvert\eta\mspace{1.0mu}x\rvert^{2}+\lvert
x\rvert^{5})\,,\qquad(\zeta,\eta,x)\in U\,.$

The proof of Proposition 5.16 is the content of the remainder of this section
and will be summarised at its end. We remark that as a solution to the cubic
equation (5.47) the quantity $\vartheta$ and with it $v_{1},v_{2}$ are not
analytic at $\zeta=\zeta_{0}$ and $\eta=0$.

The following lemma collects spectral properties of $B_{\zeta_{0}}$. These
properties yield corresponding properties of $B_{\zeta}$ for sufficiently
small $\lvert\zeta-\zeta_{0}\rvert$, using analytic perturbation theory. We
will use this idea throughout the remainder of this section after the
statement of Lemma 5.17.

######  Lemma 5.17 (Properties of $B$).

Let $\zeta_{0}\in\mathbb{C}$ with $\beta(\zeta_{0})=0$ and
$B_{0}:=B_{\zeta_{0}}$. Then there is a constant $\varepsilon>0$ with
$\varepsilon\sim 1$ such that

$\begin{split}\sup\Bigl{\\{}{\lVert(B_{0}-\omega)^{-1}\rVert_{\\#}+\lVert(B_{0}^{*}-\omega)^{-1}\rVert_{\\#}:\omega\in\mathbb{D}_{2\varepsilon}\setminus\mathbb{D}_{\varepsilon}}\Bigr{\\}}\lesssim
1\end{split}$ (5.48)

for $\\#=2,\infty$. Here $\mathbb{D}_{\varepsilon}$ contains a single isolated
non-degenerate eigenvalue $0$ of $B_{0}$, i.e,

$\begin{split}\mathbb{D}_{\varepsilon}\cap\operatorname{Spec}(B_{0})=\\{0\\}\,,\qquad\dim\mathrm{ker}B_{0}^{2}=1\,.\end{split}$
(5.49)

Moreover, the right and left eigenvectors, $b_{0}\in\mathcal{B_{+}}$ and
$\ell_{0}\in\mathcal{B_{+}}$, corresponding to this eigenvalue with
normalisation $\langle b_{0}\rangle=\langle\ell_{0}\rangle=1$ satisfy the
bounds $\ell_{0}\sim b_{0}\sim 1$. Furthermore, if

$P_{0}:=\frac{\langle\ell_{0}\,\cdot\,\rangle}{\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle}b_{0}\,,\qquad
Q_{0}:=1-P_{0}$

denote the associated spectral projections then

$\begin{split}\lVert
B_{0}^{-1}Q_{0}\rVert_{\\#}+\lVert(B_{0}^{*})^{-1}Q_{0}^{*}\rVert_{\\#}\lesssim
1\,.\end{split}$ (5.50)

###### Proof.

Here we present the proofs of the bounds (5.48) and (5.50) for $B_{0}$. The
corresponding bounds for $B_{0}^{*}$ follow analogously. From Proposition 5.15
(ii) and since $\mathbb{S}$ is bounded we know that $\lvert
a-\zeta_{0}\rvert\sim 1$. Thus, $b_{0}$ is the right eigenvector of $D^{-1}S$
with eigenvalue $1$ and $\ell_{0}$ is the right eigenvector of $D^{-1}S^{*}$
with eigenvalue $1$, where $D:=D_{\lvert a-\zeta_{0}\rvert^{2}}$. In
particular, $b_{0},\ell_{0}\in\mathcal{B}_{+}$ by the Krein-Rutman theorem and
the geometric multiplicity of the eigenvalue $0$ of $B_{0}$ is $1$.
Furthermore, the non-degeneracy of the eigenvalue $0$ is a consequence of
$b_{0},\ell_{0}\in\mathcal{B}_{+}$. Indeed, suppose we had
$\dim\mathrm{ker}B^{2}>1$. Then there would be a generalised eigenvector $x$
with $Bx=b_{0}$ and
$\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle=\langle\ell_{0}\mspace{1.0mu}B_{0}x\rangle=0$
which contradicts $\ell_{0}>0$ and $b_{0}>0$. This proves (5.49), which
together with (5.48) implies (5.50). The relation $b_{0}\sim\ell_{0}\sim 1$ is
a direct consequence of $\lvert a-\zeta_{0}\rvert\sim 1$ and (5.6).

We are left with proving (5.48). Instead of controlling the resolvent of
$B_{0}$, it suffices to bound the inverse of $1-SD^{-1}-\omega D^{-1}$ because

$\begin{split}\frac{1}{B_{0}-\omega}=\frac{1}{D}\biggl{(}{\frac{1}{1-SD^{-1}-\omega\mspace{1.0mu}D^{-1}}\widetilde{Q}_{\omega}+\frac{1}{1-SD^{-1}-\omega\mspace{1.0mu}D^{-1}}\widetilde{P}_{\omega}}\biggr{)}\,,\end{split}$
(5.51)

where $\widetilde{P}_{\omega}$ and
$\widetilde{Q}_{\omega}:=1-\widetilde{P}_{\omega}$ are the analytic spectral
projections associated with $SD^{-1}-\omega D^{-1}$ such that

$\widetilde{P}_{0}=\frac{\langle\ell_{0}\,\cdot\,\rangle}{\langle\ell_{0}Db_{0}\rangle}Db_{0}\,.$

Analytic perturbation theory can be applied to $SD^{-1}$ because of Lemma B.2,
which shows that the resolvent of the operator $SD^{-1}$ is bounded in annulus
around its isolated eigenvalue $1$. Consequently, the first summand in (5.51)
is bounded for sufficiently small $\lvert\omega\rvert$. The second summand
admits the expansion

$\frac{1}{1-SD^{-1}-\omega\mspace{1.0mu}D^{-1}}\widetilde{P}_{\omega}=\frac{1}{\widetilde{\beta}(\omega)}\widetilde{P}_{\omega}\,,\qquad\widetilde{\beta}(\omega)=-\omega\mspace{2.0mu}\frac{\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle}{\langle\ell_{0}\mspace{1.0mu}Db_{0}\rangle}+O(\lvert\omega\rvert^{2})$

by standard analytic perturbation formulas, see e.g. [4, Lemma C.1]. Therefore
the second summand is bounded for
$\omega\in\mathbb{C}\setminus\mathbb{D}_{\varepsilon}$ for sufficiently small
$\varepsilon$. ∎

######  Corollary 5.18.

Let $\zeta_{0}\in\mathbb{C}$ with $\beta(\zeta_{0})=0$. Then
$0\in\operatorname{Spec}(B_{\zeta_{0}})$, $\operatorname*{ess\,inf}\lvert
a-\zeta_{0}\rvert>0$ and $\lambda_{\mathrm{PF}}(SD_{\lvert
a-\zeta_{0}\rvert^{2}}^{-1})=1$. Furthermore, there is $\varepsilon>0$ such
that $\beta(\zeta)$ is an isolated non-degenerate eigenvalue of $B_{\zeta}$
for all $\zeta\in\zeta_{0}+\mathbb{D}_{\varepsilon}$. In particular
$\mathbb{D}_{\varepsilon}\ni\zeta\mapsto\beta(\zeta)$ is real analytic and has
the expansion

$\begin{split}\beta(\zeta)&=-2\mathrm{Re}\biggl{[}{\frac{\langle\ell_{0}\mspace{1.0mu}b_{0}\mspace{1.0mu}(a-\zeta_{0})\rangle}{\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle}\overline{(\zeta-\zeta_{0})}\\!\,}\biggr{]}+\biggl{(}{1-2\operatorname{Re}\biggl{[}{\frac{\langle\ell_{0}\overline{(a-\zeta_{0})}\\!\,B_{0}^{-1}Q_{0}[b_{0}(a-\zeta_{0})]\rangle}{\langle\ell_{0}\mspace{2.0mu}b_{0}\rangle}}\biggr{]}}\biggr{)}\lvert\zeta-\zeta_{0}\rvert^{2}\\\
&\quad\qquad-2\operatorname{Re}\biggl{[}{\frac{\langle\ell_{0}(a-\zeta_{0})B_{0}^{-1}Q_{0}[b_{0}(a-\zeta_{0})]\rangle}{\langle\ell_{0}\mspace{2.0mu}b_{0}\rangle}\overline{(\zeta-\zeta_{0})}\\!\,^{2}}\biggr{]}+O(\lvert\zeta-\zeta_{0}\rvert^{3})\,,\end{split}$
(5.52)

which implies the formulas

$\begin{split}\partial_{\zeta}\beta(\zeta_{0})=-\frac{\langle\ell_{0}\mspace{1.0mu}b_{0}\mspace{1.0mu}\overline{(a-\zeta_{0})}\\!\,\rangle}{\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle}\,,\qquad\partial_{\zeta}\partial_{\overline{\zeta}\\!\,}\beta(\zeta_{0})=1-2\operatorname{Re}\biggl{[}{\frac{\langle\ell_{0}\overline{(a-\zeta_{0})}\\!\,B_{0}^{-1}Q_{0}[b_{0}(a-\zeta_{0})]\rangle}{\langle\ell_{0}\mspace{2.0mu}b_{0}\rangle}}\biggr{]}\end{split}$
(5.53)

for the derivatives of $\beta$ at $\zeta=\zeta_{0}$.

###### Proof.

Let $\zeta_{0}\in\mathbb{C}$ be such that $\beta(\zeta_{0})=0$. By Lemma 5.17
we have $0\in\operatorname{Spec}(B_{\zeta_{0}})$ and by Proposition 5.15 (ii)
we get $\operatorname*{ess\,inf}\lvert a-\zeta_{0}\rvert>0$. The fact that
$\lambda_{\mathrm{PF}}(SD_{\lvert\zeta_{0}-a\rvert^{2}}^{-1})=1$ was shown in
(5.33).

Now we show that $\beta(\zeta)$ is an eigenvalue of $B_{\zeta}$ for
sufficiently small $\lvert\zeta-\zeta_{0}\rvert$. Using analytic perturbation
theory, let $b(\zeta)$ and $\ell(\zeta)$ be the right and left eigenvectors of
$B_{\zeta}$ corresponding to the isolated non-degenerate eigenvalue
$\widetilde{\beta}(\zeta)$ with $\widetilde{\beta}(\zeta_{0})=0$ that depends
real analytically on $\zeta$. As $\widetilde{\beta}(\zeta_{0})$ is a real
isolated eigenvalue and $B_{\zeta_{0}}$ as well as $B_{\zeta}-B_{\zeta_{0}}$
are invariant under complex conjugation, $\widetilde{\beta}(\zeta)$,
$b(\zeta)$ and $\ell(\zeta)$ are also real. Since $\ell(\zeta_{0})\sim
b(\zeta_{0})\sim 1$ we have $b(\zeta)$, $\ell(\zeta)\in\mathcal{B}_{+}$ for
sufficiently small $\lvert\zeta-\zeta_{0}\rvert$. Therefore

$\widetilde{\beta}(\zeta)=\inf_{x>0}\frac{\langle{x}\mspace{2.0mu},{B_{\zeta}b(\zeta)}\rangle}{\langle{x}\mspace{2.0mu},{b(\zeta)}\rangle}\leq\beta(\zeta)\leq\sup_{y>0}\frac{\langle{\ell(\zeta)}\mspace{2.0mu},{B_{\zeta}y}\rangle}{\langle{\ell(\zeta)}\mspace{2.0mu},{y}\rangle}=\widetilde{\beta}(\zeta)\,,$

which proves $\widetilde{\beta}=\beta$.

The expansion (5.52) is now a direct consequence of analytic perturbation
theory, as we see e.g. by using [4, Lemma C.1] with $B=B_{0}+E$ and
$E=D_{\lvert a-\zeta\rvert^{2}}-D_{\lvert
a-\zeta_{0}\rvert^{2}}=D_{\lvert\zeta-\zeta_{0}\rvert^{2}-2\mathrm{Re}\,(\overline{(a-\zeta_{0})}(\zeta-\zeta_{0}))}$.
∎

Due to analytic perturbation theory with $\zeta$ in a small neighbourhood of
$\zeta_{0}$ and by Lemma 5.17 we have $b\sim\ell\sim 1$. We split $v_{1}$ and
$v_{2}$ according to the spectral decompositions of $B^{*}$ and $B$, namely

$\begin{split}v_{1}=\vartheta_{1}\mspace{1.0mu}\ell+\widetilde{v}_{1},\qquad
v_{2}=\vartheta_{2}\mspace{1.0mu}b+\widetilde{v}_{2}\end{split}$ (5.54)

with the contributions $\vartheta_{i}=\vartheta_{i}(\zeta,\eta)$ to the
eigendirections $\ell$ and $b$ of $B^{*}$ and $B$ as well as their complements
$\widetilde{v}_{i}=\widetilde{v}_{i}(\zeta,\eta)$ given as

$\vartheta_{1}:=\frac{\langle
b\mspace{1.0mu}v_{1}\rangle}{\langle\ell\mspace{1.0mu}b\rangle}\,,\qquad\vartheta_{2}:=\frac{\langle\ell\mspace{1.0mu}v_{2}\rangle}{\langle\ell\mspace{1.0mu}b\rangle}\,,\qquad\widetilde{v}_{1}:=Q^{*}v_{1}\,,\qquad\widetilde{v}_{2}:=Qv_{2}\,,\qquad
Q:=1-\frac{\langle\ell\,\cdot\,\rangle}{\langle\ell\mspace{1.0mu}b\rangle}b\,.$

To quantify the error terms we introduce

$\alpha:=\lVert v_{1}\rVert_{\infty}+\lVert v_{2}\rVert_{\infty}\,.$

Projecting the identities (5.45) with $Q$ and $Q^{*}$, respectively, leads to

$\begin{split}B_{\zeta}\mspace{1.0mu}\widetilde{v}_{2}=O(\eta+\alpha^{3}),\qquad
B_{\zeta}^{*}\mspace{1.0mu}\widetilde{v}_{1}=O(\eta+\alpha^{3})\,.\end{split}$
(5.55)

Using $\lVert B^{-1}Q\rVert_{\infty}\lesssim 1$, a consequence of (5.50) and
analytic perturbation theory , we find

$\begin{split}\lVert\widetilde{v}_{1}\rVert_{\infty}+\lVert\widetilde{v}_{2}\rVert_{\infty}=O(\eta+\alpha^{3})\,.\end{split}$
(5.56)

Because of $\langle v_{1}\rangle=\langle v_{2}\rangle$, i.e. by (5.2), (5.54)
and the normalisation $\langle b\rangle=\langle\ell\rangle=1$, (5.56) implies

$\begin{split}\vartheta_{1}=\vartheta_{2}+O(\eta+\alpha^{3})\,.\end{split}$
(5.57)

Inserting the decomposition (5.54) into (5.45) and using (5.56), as well as
(5.57), leads to

$\begin{split}&\beta\mspace{1.0mu}\vartheta_{2}\mspace{1.0mu}b+B\widetilde{v}_{2}=\eta-\vartheta^{3}b(Sb)(S^{*}\ell)+O(\eta\alpha^{2}+\alpha^{5})\,,\\\
&\beta\mspace{1.0mu}\vartheta_{1}\mspace{1.0mu}\ell+B^{*}\widetilde{v}_{1}=\eta-\vartheta^{3}\ell(Sb)(S^{*}\ell)+O(\eta\alpha^{2}+\alpha^{5})\,,\end{split}$

where we set $\vartheta:=\frac{1}{2}(\vartheta_{1}+\vartheta_{2})$. Now we
average the first equation against $\ell$ and the second equation against $b$,
use $\langle b\rangle=\langle\ell\rangle=1$ and then take the arithmetic mean
of the resulting equations to find

$\begin{split}\vartheta^{3}\big{\langle}\ell
b(S^{*}\ell)(Sb)\big{\rangle}+\beta\mspace{1.0mu}\vartheta\mspace{1.0mu}\langle\ell\mspace{1.0mu}b\rangle-\eta=O(\eta\mspace{1.0mu}\alpha^{2}+\alpha^{5})\,.\end{split}$
(5.58)

From this approximate cubic equation we conclude the scaling behaviours

$\alpha\sim\vartheta\sim\sqrt{\max\\{0,-\beta\\}}+\frac{\eta}{\eta^{2/3}+\lvert\beta\rvert}\,,\qquad\lVert\widetilde{v}_{1}\rVert_{\infty}+\lVert\widetilde{v}_{2}\rVert_{\infty}\sim\eta+\max\\{0,-\beta\\}^{3/2},$
(5.59)

in the regime of sufficiently small $\alpha$, where we used $\vartheta\geq 0$
and $\vartheta>0$ for $\eta>0$ to choose the correct branch of the solution.
The corresponding argument is summarised in Lemma B.1 in the appendix.

To apply this lemma we absorb the $O(\alpha^{5})$-term on the right hand side
of (5.58) into the cubic term in $\vartheta$ on the left hand side, i.e. we
write $O(\alpha^{5})=\gamma\mspace{2.0mu}\vartheta^{3}$ for some
$\gamma=O(\alpha^{2})$, which we absorb into the coefficient of the
$\vartheta^{3}$-term. Such rewriting is possible since $\alpha=O(\vartheta)$
in the regime where $\alpha$ is sufficiently small. This holds because
$\vartheta\gtrsim\eta$ and $\alpha=O(\vartheta+\eta+\alpha^{3})$ by (5.54),
(5.57) and (5.56). Now we see that $\alpha$ is indeed small for $(\zeta,\eta)$
in a neighbourhood of $(\zeta_{0},0)$. Due to the characterisation of
$\mathbb{S}$ in (5.35) we have $\lim_{\eta\downarrow
0}\alpha|_{\zeta=\zeta_{0}}=0$. With $\beta(\zeta_{0})=0$ and because $\alpha$
is a continuous function of $\eta$ when $\eta>0$, the scaling (5.59) implies
$\alpha|_{\zeta=\zeta_{0}}\sim\eta^{1/3}$. Since $\zeta\mapsto\beta(\zeta)$ is
continuous by Proposition 5.15 (i) and $\alpha$ is a continuous function of
$\zeta$ for any $\eta>0$ the behaviour (5.59) holds as long as
$\eta+\lvert\zeta-\zeta_{0}\rvert$ is sufficiently small.

We now summarise our insights by finishing the proof of Proposition 5.16.

###### Proof of Proposition 5.16.

Following the computation leading to (5.55) we easily see that the right hand
side of these equations are real analytic functions of $\vartheta$, $\eta$,
$\zeta$ and $\widetilde{v}_{i}$. By the implicit function theorem and the
invertibility of $B$ on the range of $Q$ the $\widetilde{v}_{i}$ are real
analytic functions of $\vartheta,\eta$ and $\zeta$. Similarly, the right hand
side of (5.58) is a real analytic function of $\vartheta,\eta$ and $\zeta$.
Together, we have proved Proposition 5.16 with

$\widetilde{v}_{i}(\zeta,\eta)=\widetilde{w}_{i}(\vartheta(\zeta,\eta),\eta,\zeta)\,.\qed$

##  6 Properties of the Brown measure $\sigma$

We now present our main result about the existence and properties of the
measure $\sigma$ in the general setup introduced in Section 4. Here, we
introduce the measure $\sigma$ as a distributional derivative of the function
$L$ defined through

$L(\zeta):=\int_{0}^{\infty}\bigg{(}\langle
v_{1}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{)}\mathrm{d}\eta$ (6.1)

for each $\zeta\in\mathbb{C}$, where $v_{1}$ is the solution of the Dyson
equation (4.2). The existence of this integral in the Lebesgue sense will be
established in Lemma 6.5 below.

In the proof of Proposition 2.1 in Section 7.1 below, we relate this
definition to the limiting measure of the empirical spectral distribution. In
particular, we refer to (7.2), (7.4) and Proposition 7.1 below.

######  Proposition 6.1 (Properties of $\sigma$, general setup).

Let $a\in\mathcal{B}$ and $s$ satisfy A5 and A6. If
$L\colon\mathbb{C}\to\mathbb{R}$ is defined as in (6.1) then the following
holds.

  1. (i)

There is a unique probability measure on $\mathbb{C}$ such that

$\int_{\mathbb{C}}f(\zeta)\sigma(\mathrm{d}\zeta)=-\frac{1}{2\pi}\int_{\mathbb{C}}\Delta
f(\zeta)L(\zeta)\mathrm{d}^{2}\zeta$ (6.2)

for all $f\in C_{0}^{2}(\mathbb{C})$, where $\mathrm{d}^{2}\zeta$ denotes the
Lebesgue measure on $\mathbb{C}$.

  2. (ii)

With respect to the Lebesgue measure, the measure $\sigma$ from (6.2) has a
bounded density on $\mathbb{C}$, which we also denote by $\sigma$, i.e.
$\sigma(\mathrm{d}\zeta)=\sigma(\zeta)\mathrm{d}^{2}\zeta$.

  3. (iii)

On $\mathbb{S}$, the density $\zeta\mapsto\sigma(\zeta)$ is strictly positive
and real analytic.

  4. (iv)

$\operatorname{supp}\sigma=\overline{\mathbb{S}}$ and this set is bounded.
Furthermore $\operatorname{Spec}(D_{a})\subset\mathbb{S}$.

  5. (v)

$\partial\mathbb{S}$ is a real analytic variety of (real) dimension at most
$1$.

  6. (vi)

There exists a unique continuous extension
$\sigma\colon\overline{\mathbb{S}}\to[0,\infty)$ of the density
$\sigma|_{\mathbb{S}}$ to $\overline{\mathbb{S}}$ such that
$\sigma(\zeta)=g(\zeta)\lvert\partial_{\zeta}\beta(\zeta)\rvert^{2}$ for all
$\zeta\in\partial\mathbb{S}$, where $g:\partial\mathbb{S}\to(0,\infty)$ is a
strictly positive function that can be extended to a real analytic function on
a neighbourhood of $\partial\mathbb{S}$.

The proof of Proposition 6.1 is presented at the end of this section, Section
6. In the next subsection, we first establish the existence of the measure
$\sigma$ via (6.2).

We recall that $\sigma$ was called limiting spectral measure (cf. Definition
2.3) in the random matrix setup of Section 2. In the present general setup, we
rename $\sigma$ according to the next definition. The motivation for the
naming of $\sigma$ originates from the Brown measure of operators in von
Neumann algebras and is explained by Proposition D.1 in Appendix D.

######  Definition 6.2.

We call the probability measure $\sigma$ on $\mathbb{C}$, defined by (6.2),
the Brown measure associated with $a$ and $s$.

###  6.1 Representation of the Brown measure $\sigma$

For the next result, we recall the definition of $\mathbb{S}_{\varepsilon}$
from (5.16).

######  Proposition 6.3.

Let $a\in\mathcal{B}$ and $s$ satisfy A5 and let $L$ be defined as in (6.1).
Then there is a unique probability measure $\sigma$ on $\mathbb{C}$ such that
(6.2) holds for all $f\in C_{0}^{2}(\mathbb{C})$. Moreover,
$\operatorname{supp}\sigma\subset\mathbb{S}_{0}$. The measure $\sigma$
satisfies the identity

$\begin{split}\sigma(\zeta)=-\lim_{\eta\downarrow
0}\frac{1}{\pi}\partial_{\overline{\zeta}\\!\,}\langle
y(\zeta,\eta)\rangle\end{split}$ (6.3)

in the sense of distributions, where $y$ is the $(2,1)$ component of $M$ from
(5.12).

The main idea of the proof of Proposition 6.3 will be to show that $-L$ is
subharmonic and, therefore, the distribution $-\frac{1}{2\pi}\Delta L$ is
induced by a measure. Before we present this proof, we establish a few
necessary ingredients. The next lemma will, in particular, imply that $L$ is
well-defined.

######  Lemma 6.4 (Integrating $\langle v_{1}\rangle$ with respect to $\eta$).

Let $a\in\mathcal{B}$ and $s$ satisfy A5. Then, uniformly for
$\zeta\in\mathbb{C}$ and $\eta>0$, we have

$0\leq\langle v_{1}(\zeta,\eta)\rangle\lesssim\frac{1}{1+\eta}.$ (6.4)

Furthermore, uniformly for any $T>0$ and $\zeta\in\mathbb{C}$, we have

$\int_{0}^{T}\bigg{\lvert}\langle
v_{1}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{\rvert}\mathrm{d}\eta\lesssim\min\Big{\\{}T,\sqrt{1+\lvert\zeta\rvert}\Big{\\}},\qquad\int_{T}^{\infty}\bigg{\lvert}\langle
v_{1}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{\rvert}\mathrm{d}\eta\lesssim\frac{1+\lvert\zeta\rvert}{T}.$
(6.5)

###### Proof.

From (5.4) and Lemma 5.1, we immediately conclude (6.4). The bounds in (6.5)
follow directly from (6.4) and (5.9). ∎

In the next lemma, we truncate the lower integration bound in the definition
of $L$ and, thus, obtain $L_{\varepsilon}$. It is an approximate version of
$L$ which is more regular in $\zeta$ and its derivative with respect to
$\zeta$ is given by $\langle y(\zeta,\varepsilon)\rangle/2$.

######  Lemma 6.5 (Definition and derivatives of $L_{\varepsilon}$).

Let $a\in\mathcal{B}$ and $s$ satisfy A5. Then the following holds.

  1. (i)

For each $\varepsilon\geq 0$, the integral

$L_{\varepsilon}(\zeta):=\int_{\varepsilon}^{\infty}\bigg{(}\langle
v_{1}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{)}\mathrm{d}\eta$ (6.6)

exists in Lebesgue sense for every $\zeta\in\mathbb{C}$ and the map
$\mathbb{C}\to\mathbb{R}$, $\zeta\mapsto L_{\varepsilon}(\zeta)$ is
continuous.

  2. (ii)

When $\varepsilon\downarrow 0$ then $L_{\varepsilon}\to L_{0}$ uniformly on
$\mathbb{C}$.

  3. (iii)

For each $\varepsilon>0$, $L_{\varepsilon}$ is infinitely often continuously
differentiable with respect to $\zeta$ and $\bar{\zeta}$ on $\mathbb{C}$.

  4. (iv)

For each $\varepsilon>0$, $y(\cdot,\varepsilon)$ is infinitely often
continuously differentiable with respect to $\zeta$ and $\bar{\zeta}$ and, for
each $\zeta\in\mathbb{C}$,

$\partial_{\zeta}L_{\varepsilon}(\zeta)=\frac{1}{2}\langle
y(\zeta,\varepsilon)\rangle,\qquad\qquad\partial_{\bar{\zeta}}L_{\varepsilon}(\zeta)=\frac{1}{2}\langle\overline{y(\zeta,\varepsilon)}\rangle.$
(6.7)

###### Proof.

The bounds in (6.5) imply that $L_{\varepsilon}(\zeta)$ is well-defined for
every $\varepsilon\geq 0$ and every $\zeta\in\mathbb{C}$. Moreover,
$\zeta\mapsto L_{\varepsilon}(\zeta)$ is continuous on $\mathbb{C}$ for every
$\varepsilon\geq 0$. The first bound in (6.5) implies that $L_{\varepsilon}\to
L_{0}$ uniformly on $\mathbb{C}$ when $\varepsilon\downarrow 0$. To prove the
differentiability of $L_{\varepsilon}$ and $y(\cdot,\varepsilon)$, we conclude
from (5.19) and the bound (A.2) in Lemma A.1 that for some constant $C\equiv
C_{\zeta}>0$,
$\lVert\partial_{\zeta}v_{1}(\zeta,\mathrm{i}\eta)\rVert+\lVert\partial_{\bar{\zeta}}v_{1}(\zeta,\mathrm{i}\eta)\rVert+\lVert\partial_{\eta}v_{1}(\zeta,\mathrm{i}\eta)\rVert\leq
C\max\\{\eta^{-C},\eta^{-2}\\}$ for every $\eta>0$ and every fixed
$\zeta\in\mathbb{C}$. Therefore, $L_{\varepsilon}$ is differentiable with
respect to $\zeta$ and $\bar{\zeta}$ and we can take $\partial_{\zeta}$ and
$\partial_{\bar{\zeta}}$ derivatives of $L_{\varepsilon}(\zeta)$ by exchanging
the derivative and the integral in the definition of $L_{\varepsilon}(\zeta)$.
Hence, we conclude from (5.17) that
$\partial_{\zeta}L_{\varepsilon}(\zeta)=\langle
M_{21}(\zeta,\mathrm{i}\varepsilon)\rangle/2=\langle
y(\zeta,\mathrm{i}\varepsilon)\rangle/2$ for all $\zeta\in\mathbb{C}$ and
$\varepsilon>0$, which implies (6.7). Furthermore, note that $\mathcal{L}$
defined in (5.18) below is the stability operator of the Matrix Dyson
equation, (5.13). As it is invertible when $\eta>0$ by Lemma A.1, the implicit
function theorem implies that $y(\cdot,\varepsilon)$ is infinitely often
differentiable. Therefore, the same holds of $L_{\varepsilon}$ due to (6.7),
which completes the proof of Lemma 6.5. ∎

Together with (6.7) in Lemma 6.5, the next proposition implies that $-\Delta
L_{\varepsilon}\geq 0$.

######  Proposition 6.6.

Let $a\in\mathcal{B}$ and $s$ satisfy A5. Then $-\partial_{\bar{\zeta}}\langle
y(\zeta,\eta)\rangle>0$ for all $\zeta\in\mathbb{C}$ and $\eta>0$.

The proof of Proposition 6.6 is presented in Section 6.2 below. We now have
all ingredients for the proof of Proposition 6.3.

######  Proof of Proposition 6.3.

First, we show that $-L$ is subharmonic since this implies the existence of a
measure $\sigma$ on $\mathbb{C}$ such that (6.2) holds. We know from Lemma 6.5
(i) that $L=L_{0}$ is a continuous function on $\mathbb{C}$. We now verify
that $-L$ satisfies the circle average inequality. Since $L_{\varepsilon}\to
L_{0}=L$ uniformly on $\mathbb{C}$ for $\varepsilon\downarrow 0$ and
$L_{\varepsilon}$ is twice differentiable by Lemma 6.5, it suffices to show
that $-\Delta L_{\varepsilon}\geq 0$ on $\mathbb{C}$. The latter implies that
$-L_{\varepsilon}$ satisfies the circle average inequality and consequently,
$-L$ satisfies it as well. From (6.7) in Lemma 6.5 and Proposition 6.6, we
conclude
$-\Delta_{\zeta}L_{\varepsilon}=-4\partial_{\bar{\zeta}}\partial_{\zeta}L_{\varepsilon}\geq
0$ on $\mathbb{C}$. Therefore, $-L$ is subharmonic on $\mathbb{C}$ and there
exists a positive measure $\sigma$ on $\mathbb{C}$ such that (6.2) holds.

Next, we prove that $\operatorname{supp}\sigma\subset\mathbb{S}_{0}$. Owing to
[4, Lemma D.1], Section 5.2 and (5.19), derivatives of $v_{1}$ and $v_{2}$
with respect to $\zeta$ and $\bar{\zeta}$ are bounded locally uniformly for
$\zeta\in\mathbb{C}\setminus\mathbb{S}_{0}$ and uniformly for
$\eta\in(0,\infty)$. Therefore, $\mathbb{C}\setminus\mathbb{S}_{0}$ is open
and Proposition 5.15 (ii) and (iv) imply
$\operatorname{Spec}(D_{a})\subset\mathbb{S}_{0}$. Moreover, $L$ is twice
continuously differentiable on $\mathbb{C}\setminus\mathbb{S}_{0}$ and for any
$\zeta\in\mathbb{C}\setminus\mathbb{S}_{0}$, the second identity in (5.5),
(6.7) with $\varepsilon\downarrow 0$ and $\lVert
v_{1}(\zeta,\eta)\rVert+\lVert v_{2}(\zeta,\eta)\rVert\to 0$ for
$\eta\downarrow 0$ and such $\zeta$ imply

$\partial_{\zeta}L(\zeta)=\frac{1}{2}\bigg{\langle}\frac{1}{a-\zeta}\bigg{\rangle},\qquad\qquad\partial_{\bar{\zeta}}L(\zeta)=\frac{1}{2}\bigg{\langle}\frac{1}{\bar{a}-\bar{\zeta}}\bigg{\rangle}$
(6.8)

for all $\zeta\in\mathbb{C}\setminus\mathbb{S}_{0}$. In particular, $\Delta
L=4\partial_{\zeta}\partial_{\bar{\zeta}}L=0$ on
$\mathbb{C}\setminus\mathbb{S}_{0}$ and, therefore,
$\operatorname{supp}\sigma\subset\mathbb{S}_{0}$ by (6.2).

What remains is to show that $\sigma$ is a probability measure. The identities
in (6.8) yield $L(\zeta)=-\langle\log\lvert a-\zeta\rvert\rangle+C$ for
$\zeta\in\mathbb{C}\setminus\mathbb{S}_{0}$, where $C\in\mathbb{R}$ is
independent of $\zeta$. (One can check that $C=0$ with some extra effort by
expanding $v_{1}$ for large $\lvert\zeta\rvert$ using (4.2).) Owing to Remark
5.8, it is possible to choose $\varphi\sim 1$ such that $\mathbb{S}_{0}$ is
contained in $\mathbb{D}_{\varphi}$. Note that $\varphi$ can be chosen such
that it depends only on the upper bounds in A5 and any upper bound on $\lVert
a\rVert_{\infty}$. In particular,
$\operatorname{supp}\sigma\subset\mathbb{D}_{\varphi}$. In order to show that
$\sigma$ is a probability measure on $\mathbb{C}$, we pick a rotationally
symmetric function $f\in C_{0}^{\infty}(\mathbb{C})$ such that
$\mathrm{ran}\,f\subset[0,1]$, $f\equiv 1$ on $\mathbb{D}_{\varphi}$ and
$f\equiv 0$ on $\mathbb{C}\setminus\mathbb{D}_{2\varphi}$. Thus, (6.2) and
$\operatorname{supp}\sigma\subset\mathbb{D}_{\varphi}$ imply

$\sigma(\mathbb{C})=\int_{\mathbb{C}}\sigma(\mathrm{d}\zeta)=-\frac{1}{2\pi}\int_{\mathbb{D}_{3\varphi}\setminus\mathbb{D}_{\varphi}}\Delta
f(\zeta)L(\zeta)\mathrm{d}^{2}\zeta=-\frac{1}{2\pi}\int_{\mathbb{D}_{3\varphi}\setminus\mathbb{D}_{\varphi}}(\nabla
f)\cdot(\nabla\langle\log\lvert a-\cdot\rvert\rangle)\mathrm{d}^{2}\zeta,$

where in the second step we used Green’s first identity and that the boundary
terms vanish as $\nabla f\equiv 0$ on
$\partial\mathbb{D}_{3\varphi}\cup\partial\mathbb{D}_{\varphi}$. We change to
polar coordinates and obtain

$\displaystyle-\frac{1}{2\pi}\int_{\mathbb{D}_{3\varphi}\setminus\mathbb{D}_{\varphi}}(\nabla
f)\cdot(\nabla\langle\log\lvert a-\cdot\rvert\rangle)$
$\displaystyle=-\frac{1}{2\pi}\int_{\varphi}^{3\varphi}\partial_{r}f(r)\partial_{r}\bigg{(}\int_{0}^{2\pi}\langle\log\lvert
a-r\mathrm{e}^{\mathrm{i}\theta}\rvert\rangle\mathrm{d}\theta\bigg{)}r\mathrm{d}r$
$\displaystyle=-\int_{\varphi}^{3\varphi}\partial_{r}f(r)\mathrm{d}r=f(\varphi)=1,$

where the second step follows from $\int_{0}^{2\pi}\log\lvert
w-r\mathrm{e}^{\mathrm{i}\theta}\rvert\mathrm{d}\theta=2\pi\log r$ for all
$w\in\mathbb{D}_{r}$ (see e.g. [40, Example 5.7]). This shows that $\sigma$ is
a probability measure on $\mathbb{C}$ and, thus, completes the proof of
Proposition 6.3. ∎

###  6.2 Strict positivity in the bulk

In this subsection we show that the Brown measure has strictly positive
density in the bulk, i.e. inside $\mathbb{S}$ as defined in (5.30).

######  Proposition 6.7 (Strict positivity of Brown measure on $\mathbb{S}$).

Let $a\in\mathcal{B}$ and $s$ satisfy A5 and A6. Then the Brown measure
$\sigma$ is absolutely continuous with respect to the Lebesgue measure on
$\mathbb{C}$ and on $\mathbb{S}$ its density is strictly positive and real
analytic.

For the proof of Proposition 6.7 we compute the Brown measure through the
formula (6.3), i.e. the distributional identity
$\pi\sigma=-2\lim_{\eta\downarrow
0}\overline{\partial}\\!\,_{\zeta}\langle{E_{21}}\mspace{2.0mu},{M}\rangle$.
First we will see that the right hand side in (6.3) is non-negative and is in
fact positive when evaluated at $\eta>0$, i.e. we prove Proposition 6.6. After
that we will see that under assumption A6 the right hand side can be
continuously extended to $\eta=0$ away from $\partial\mathbb{S}$ and remains a
bounded function of $\zeta$, i.e. $\sigma$ has a density.

######  Proof of Proposition 6.6.

For $\eta>0$ and $\zeta\in\mathbb{C}$, we start from the second identity in
(5.19) and compute

$\begin{split}\partial_{\overline{\zeta}\\!\,}\langle{E_{21}}\mspace{2.0mu},{M}\rangle=\Big{\langle}{E_{21}}\,\mspace{2.0mu},\,{\mathcal{L}^{-1}E_{21}}\Big{\rangle}=\Big{\langle}{\mathcal{C}_{M}^{*}E_{21}}\,\mspace{2.0mu},\,{E_{21}}\Big{\rangle}+\Big{\langle}{\mathcal{C}_{M}^{*}E_{21}}\,\mspace{2.0mu},\,{(1-\Sigma\mathcal{C}_{M})^{-1}\Sigma\mathcal{C}_{M}E_{21}}\Big{\rangle}\,,\end{split}$

where $\mathcal{C}_{M}R:=MRM$. With

$\Sigma\mathcal{C}_{M}E_{21}=\left(\begin{array}[]{cc}\mathrm{i}S(v_{2}\overline{y}\\!\,)&0\\\
0&\mathrm{i}S^{*}(v_{1}\overline{y}\\!\,)\end{array}\right)\,,\qquad\mathcal{C}_{M^{*}}E_{21}=\left(\begin{array}[]{cc}-\mathrm{i}v_{1}\overline{y}\\!&\overline{y}\\!\,^{2}\\\
-v_{1}v_{2}&-\mathrm{i}v_{2}\overline{y}\\!\end{array}\right)\,,$

and the action of $\Sigma\mathcal{C}_{M}$ on diagonal matrices in
$\mathcal{B}^{2\times 2}$ given by

$\Sigma\mathcal{C}_{M}\left(\begin{array}[]{cc}r_{1}&0\\\
0&r_{2}\end{array}\right)=\left(\begin{array}[]{cc}S(\lvert
y\rvert^{2}r_{1}-v_{2}^{2}r_{2})&0\\\ 0&S^{*}(-v_{1}^{2}r_{1}+\lvert
y\rvert^{2}r_{2})\end{array}\right)$

this simplifies to

$\begin{split}-\partial_{\overline{\zeta}\\!\,}\langle{E_{21}}\mspace{2.0mu},{M}\rangle=\frac{1}{2}\langle
v_{1}v_{2}\rangle+\bigg{\langle}{\left(\begin{array}[]{c}v_{1}\overline{y}\\!\\\
v_{2}\overline{y}\\!\end{array}\right)}\,\mspace{2.0mu},\,{(1-Y)^{-1}\left(\begin{array}[]{c}S(v_{2}\overline{y}\\!\,)\\\
S^{*}(v_{1}\overline{y}\\!\,)\end{array}\right)}\bigg{\rangle}\,,\end{split}$
(6.9)

where the scalar product on $\mathcal{B}^{2}$ is the one from (5.23) and
$Y\colon\mathcal{B}^{2}\to\mathcal{B}^{2}$ is defined as

$Y\begin{pmatrix}r_{1}\\\ r_{2}\end{pmatrix}=\begin{pmatrix}S(\lvert
y\rvert^{2}r_{1}-v_{2}^{2}r_{2})\\\ S^{*}(-v_{1}^{2}r_{1}+\lvert
y\rvert^{2}r_{2})\end{pmatrix}=\left(\begin{array}[]{cc}SD_{\lvert
y\rvert}^{2}&-SD_{v_{2}}^{2}\\\ -S^{*}D_{v_{1}}^{2}&S^{*}D_{\lvert
y\rvert}^{2}\end{array}\right)\begin{pmatrix}r_{1}\\\ r_{2}\end{pmatrix}\,.$

Now we introduce a symmetrisation of $Y$. For this purpose we define
$\widehat{v}\in\mathcal{B}$ via

$\begin{split}\widehat{v}:=\sqrt{v_{1}(\eta+Sv_{2})}=\sqrt{v_{2}(\eta+S^{*}v_{1})}\,,\end{split}$
(6.10)

where the second identity is due to (5.1), and $V,F,T\in\mathcal{B}^{2\times
2}$ as

$\begin{split}T:=\left(\begin{array}[]{cc}-\hat{v}^{2}&\lvert
a-\zeta\rvert^{2}\frac{v_{1}v_{2}}{\hat{v}^{2}}\\\ \lvert
a-\zeta\rvert^{2}\frac{v_{1}v_{2}}{\hat{v}^{2}}&-\hat{v}^{2}\end{array}\right)\,,\quad
V:=\left(\begin{array}[]{cc}\frac{\hat{v}}{v_{1}}&0\\\
0&\frac{\hat{v}}{v_{2}}\end{array}\right)\,,\quad
F:=V^{-1}S_{o}V^{-1}\end{split}$ (6.11)

analogous to [3, (3.27)]. Then $VFTV^{-1}=Y$ and represented in terms of $F$
and $T$ the formula (6.9) reads

$\begin{split}-2\partial_{\overline{\zeta}\\!\,}\langle{E_{21}}\mspace{2.0mu},{M}\rangle=\bigg{\langle}{\left(\begin{array}[]{c}\widehat{v}\mspace{2.0mu}\overline{y}\\!\\\
\widehat{v}\mspace{2.0mu}\overline{y}\\!\end{array}\right)}\,\mspace{2.0mu},\,{\biggl{(}{\frac{1}{{X}}+\frac{2}{1-{F}{T}}\mspace{2.0mu}{F}}\biggr{)}\left(\begin{array}[]{c}\widehat{v}\mspace{2.0mu}\overline{y}\\!\\\
\widehat{v}\mspace{2.0mu}\overline{y}\\!\end{array}\right)}\bigg{\rangle}\,,\end{split}$
(6.12)

where we introduced

$\begin{split}{X}:=\left(\begin{array}[]{cc}D\bigl{(}{\frac{\hat{v}^{2}}{v_{1}v_{2}}\lvert
y\rvert^{2}}\bigr{)}&0\\\ 0&D\bigl{(}{\frac{\hat{v}^{2}}{v_{1}v_{2}}\lvert
y\rvert^{2}}\bigr{)}\end{array}\right)\,.\end{split}$ (6.13)

In particular, (6.9) is the quadratic form of a self-adjoint operator,
evaluated on a vector in the subspace of $\mathcal{B}^{2}$ with identical
entries in the first and second component. With the orthogonal projection onto
this subspace represented by

${E}:=\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&1\end{array}\right)\in\mathcal{B}^{2\times 2}\,,$

we have ${T}{E}={E}{T}$, ${X}{E}={E}{X}$ and

$\begin{split}{T}=-1+2{X}{E}\,.\end{split}$ (6.14)

The representation (6.14) of $T$ holds because of

$1=\widehat{v}^{2}+\frac{\widehat{v}^{2}}{v_{1}v_{2}}\lvert y\rvert^{2}\,,$

which follows directly from (4.2), (5.3) and the definition of $\widehat{v}$
in (6.10).

Inserting (6.14) into (6.12), we see that proving positivity of the right hand
side of (6.12) reduces to proving that the operator

$\begin{split}{E}\biggl{(}{\frac{1}{{X}}+\frac{2}{1+{F}-2{F}{X}{E}}\mspace{2.0mu}{F}}\biggr{)}{E}&={E}\frac{1}{\sqrt{{X}}}\biggl{(}{1+\frac{2}{1+\widetilde{{F}}\frac{1}{{X}}-2\widetilde{{F}}{E}}\widetilde{{F}}}\biggr{)}\frac{1}{\sqrt{{X}}}{E}\\\
&={E}\frac{1}{\sqrt{{X}}}\biggl{(}{1-\frac{2}{1+\widetilde{{F}}\frac{1}{{X}}}\widetilde{{F}}{E}}\biggr{)}^{-1}\frac{1}{\sqrt{{X}}}{E}={E}\frac{1}{\sqrt{{X}}}\frac{1}{1-{K}}\frac{1}{\sqrt{{X}}}{E}\end{split}$
(6.15)

on $\mathcal{B}^{2}$ is positive definite on the image of $E$, where we
introduced

$\begin{split}\widetilde{{F}}:=\sqrt{{X}}\mspace{2.0mu}{F}\sqrt{{X}}\qquad\text{
and
}\qquad{K}:={E}\frac{2}{1+\widetilde{{F}}\frac{1}{{X}}}\widetilde{{F}}{E}={E}\sqrt{{X}}\frac{2{F}}{1+{F}}\sqrt{{X}}{E}\end{split}$
(6.16)

in the calculation and used $E^{2}=E$ as well as $EX=XE$. Indeed, this is the
case since

$\begin{split}1-{K}=\sqrt{{X}}\biggl{(}{\frac{1}{{X}}-\frac{2{F}_{+}}{1+{F}_{+}}+\frac{2{F}_{-}}{1-{F}_{-}}}\biggr{)}\sqrt{{X}}\geq\sqrt{{X}}\biggl{(}{1-\frac{2{F}_{+}}{1+{F}_{+}}}\biggr{)}\sqrt{{X}}>0\,,\end{split}$
(6.17)

where we first split the self-adjoint operator ${F}={F}_{+}-{F}_{-}$ into its
positive and negative parts, as well as used $0<{X}\leq 1$ and
$\lVert{F}\rVert<1$ for the first inequality. The final inequality follows
from $\frac{2{F}_{+}}{1+{F}_{+}}<1$ because $0\leq F_{+}\leq\lVert F\rVert<1$.
This completes the proof of Proposition 6.6. ∎

In the proof of Proposition 6.6 we have seen that the Brown measure admits the
representation (cf. (6.3), (6.12), (6.14), (6.13) and (6.15))

$\begin{split}\pi\mspace{1.0mu}\sigma=\lim_{\eta\downarrow
0}\bigg{\langle}{\left(\begin{array}[]{c}\overline{e}\\!\,_{y}\sqrt{v_{1}v_{2}}\\\
\overline{e}\\!\,_{y}\sqrt{v_{1}v_{2}}\end{array}\right)}\,\mspace{2.0mu},\,{\frac{1}{1-{K}}\left(\begin{array}[]{c}\overline{e}\\!\,_{y}\sqrt{v_{1}v_{2}}\\\
\overline{e}\\!\,_{y}\sqrt{v_{1}v_{2}}\end{array}\right)}\bigg{\rangle}\end{split}$
(6.18)

in a distributional sense. Here, $e_{y}:=\frac{y}{\lvert
y\rvert}\in\mathcal{B}$ and $K$ is defined in (6.16).

Under the additional assumption A6 we get strict positivity of the density of
the Brown measure inside $\mathbb{S}$.

###### Proof of Proposition 6.7.

For the proof of analyticity of $\sigma$, we recall the definition of $y$ from
(5.3). We conclude from Proposition 5.15 (v), (5.31) and Proposition 5.11 that
$\mathbb{S}\to\mathbb{C}$, $\zeta\mapsto y(\zeta,0)$ is real analytic.
Therefore, (6.3) implies that $\sigma$ is real analytic on $\mathbb{S}$.

To prove a lower bound on $\sigma$, we use (6.18) and see that $1-K$ remains
bounded on the image of $E$ as $\eta\downarrow 0$. Indeed by the identity in
(6.17) the only contribution to $K$ that may potentially be unbounded is the
one associated with $F_{-}$. However, $EF_{-}E|_{\eta=0}\leq 1-\varepsilon$
for some $\varepsilon>0$ because of the spectral gap of $F$ above $-1$ in
Lemma A.2 and the fact that $(\widehat{v},-\widehat{v})$, the eigenvector
corresponding to eigenvalue $-1$, is mapped to zero by $E$. ∎

###  6.3 Edge behaviour of the Brown measure

Here we show that $\sigma$ can be continuously extended to the boundary of
$\mathbb{S}$ and compute its boundary values. Throughout this subsection we
assume A5 and A6.

######  Proposition 6.8 (Boundary values of $\sigma$).

There is a unique continuous extension of $\sigma|_{\mathbb{S}}$ to
$\overline{\mathbb{S}}\\!\,$. This extension satisfies

$\begin{split}\sigma(\zeta_{0})=\frac{1}{\pi}\frac{\lvert\langle(a-\zeta_{0})\ell_{0}\mspace{1.0mu}b_{0}\rangle\rvert^{2}}{\langle\lvert
a-\zeta_{0}\rvert^{4}\ell_{0}^{2}\mspace{1.0mu}b_{0}^{2}\rangle}=\frac{\langle\ell_{0}\mspace{1.0mu}b_{0}\rangle^{2}\lvert\partial_{{\zeta}}\mspace{2.0mu}\beta(\zeta_{0})\rvert^{2}}{\pi\mspace{1.0mu}\langle\lvert
a-\zeta_{0}\rvert^{4}\ell_{0}^{2}\mspace{1.0mu}b_{0}^{2}\rangle}\end{split}$
(6.19)

for any $\zeta_{0}\in\partial\mathbb{S}$, where
$\ell_{0}:=\ell|_{\zeta=\zeta_{0}}$ and $b_{0}:=b|_{\zeta=\zeta_{0}}$.

###### Proof.

We use the identity (6.3) to compute $\sigma$ at some $\zeta$ in a
neighbourhood of $\zeta_{0}$ in terms of $y$. We expand $y$ in terms of
$v_{1},v_{2}$ with the help of (5.5) and expand $v_{1},v_{2}$ in terms of
$\beta$.

For $\eta=0$ equations (5.59) and (5.58) imply that either $v=0$ for
$\beta\geq 0$ or

$\begin{split}\vartheta^{2}\big{\langle}\ell
b(S^{*}\ell)(Sb)\big{\rangle}+\beta\mspace{1.0mu}\langle\ell\mspace{1.0mu}b\rangle=O(\vartheta^{4})\end{split}$
(6.20)

for $\beta<0$, or equivalently for $\zeta\in\mathbb{S}$. Note that according
to Proposition 5.16 the right hand side of (6.20) is a real analytic function
of $\zeta$ and $\vartheta$. In particular, we can write
$\vartheta=\vartheta(\zeta,0)$ as

$\vartheta=\sqrt{\frac{-\beta\langle\ell\mspace{1.0mu}b\rangle}{\big{\langle}\ell^{2}b^{2}\lvert
a-\zeta\rvert^{4}\big{\rangle}}}\mspace{2.0mu}\big{(}1+\beta
h\big{(}\zeta,\sqrt{-\beta}\big{)}\big{)}\mathbbm{1}(\zeta\in\mathbb{S})$
(6.21)

for some real analytic function $h$, where we used (5.46).

According to (6.21) the leading order behaviour of $\vartheta$ is determined
by $\beta$, whose local expansion is given in (5.52). To express $y$ in terms
of $v_{1}$ and $v_{2}$ we recall (5.5). From (5.5), (5.54), (5.56) and (5.57),
we obtain the expansion

$y=\frac{1}{a-\zeta}-\overline{(a-\zeta)}\\!\,v_{1}v_{2}+O(\alpha^{4})=\frac{1}{a-\zeta}-\vartheta^{2}\overline{(a-\zeta)}\\!\,\ell\mspace{1.0mu}b+O(\alpha^{4}+\eta\mspace{1.0mu}\alpha)\,.$

By Proposition 5.16 and (6.21) it is easy to see that the error term is a real
analytic function of $\vartheta$ and $\zeta$ in the regime
$\zeta\in\mathbb{S}$ and for $\eta=0$, i.e.

$y|_{\eta=0}=\frac{1}{a-\zeta}-\vartheta^{2}\overline{(a-\zeta)}\\!\,\ell\mspace{1.0mu}b\big{(}1+\vartheta^{2}f(\zeta,\vartheta)\big{)}\mathbbm{1}(\zeta\in\mathbb{S})$

with a real analytic $f$. Now we differentiate with respect to
$\overline{\zeta}\\!\,$ and use (6.21), in particular
$\lvert\partial_{\overline{\zeta}\\!\,}\vartheta^{2}\rvert\lesssim 1$, to get

$\begin{split}\partial_{\overline{\zeta}\\!\,}y|_{\eta=0}=-\overline{(a-\zeta)}\\!\,\ell\mspace{1.0mu}b\mspace{2.0mu}\partial_{\overline{\zeta}\\!\,}\mspace{2.0mu}\vartheta^{2}+O(\vartheta^{2})=\frac{\overline{(a-\zeta)}\\!\,\ell\mspace{1.0mu}b\langle\ell\mspace{2.0mu}b\rangle}{\langle\lvert
a-\zeta\rvert^{4}\ell^{2}\mspace{2.0mu}b^{2}\rangle}\partial_{\overline{\zeta}\\!\,}\mspace{2.0mu}\beta+O\big{(}\lvert\beta\rvert^{1/2}\big{)}\end{split}$
(6.22)

for $\zeta\in\mathbb{S}$. The right hand side can be continuously extended to
$\overline{\mathbb{S}}\\!\,$. Indeed, by the definition 5.30 of $\mathbb{S}$
and the continuity of $\beta$ from Proposition 5.15 (i) we have
$\beta(\zeta_{0})=0$ for any $\zeta_{0}\in\partial\mathbb{S}$ and thus (6.22)
holds for $\zeta\in\mathbb{S}$ in a neighbourhood of such $\zeta_{0}$ and the
error term vanishes as $\zeta\to\zeta_{0}$. Inserting the formula (5.53) for
$\partial_{\overline{\zeta}\\!\,}\mspace{2.0mu}\beta|_{\zeta=\zeta_{0}}=\overline{\partial_{{\zeta}}\mspace{2.0mu}\beta}\\!\,|_{\zeta=\zeta_{0}}$
into (6.22) and using (6.3) shows the claim (6.19). ∎

######  Lemma 6.9.

Let $\zeta\in\mathbb{C}$ such that $\beta(\zeta)=0$ and
$\partial_{\zeta}\beta(\zeta)=0$. Then $\Delta\beta(\zeta)<0$. In particular,

$\begin{split}\partial\mathbb{S}=\\{\zeta\in\mathbb{C}:\beta(\zeta)=0\\}.\end{split}$
(6.23)

###### Proof.

Let $\zeta\in\mathbb{C}$ with $\beta(\zeta)=0$ and
$\partial_{\zeta}\beta(\zeta)=0$. From (5.53) we read off

$\begin{split}\langle\ell\mspace{1.0mu}b\mspace{1.0mu}(a-\zeta)\rangle=0\,,\qquad\partial_{\zeta}\partial_{\overline{\zeta}\\!\,}\beta=1-2\operatorname{Re}\biggl{[}{\frac{\langle\ell\overline{(a-\zeta)}\\!\,B^{-1}b(a-\zeta)\rangle}{\langle\ell\mspace{2.0mu}b\rangle}}\biggr{]}\,,\end{split}$
(6.24)

where we evaluated the expressions in (5.53) at $\zeta_{0}=\zeta$ and omitted
the projection $Q_{0}$ in the formula for
$\partial_{\zeta}\partial_{\overline{\zeta}\\!\,}\beta$ since
$\langle\ell\mspace{1.0mu}b(a-\zeta)\rangle=0$ implies
$Q_{0}[b(a-\zeta)]=b(a-\zeta)$.

We write $\partial_{\zeta}\partial_{\overline{\zeta}\\!\,}\beta$ in terms of

$K:=D\biggl{(}{\frac{\sqrt{\ell}}{\lvert
a-\zeta\rvert\sqrt{b}}}\biggr{)}SD\biggl{(}{\frac{\sqrt{b}}{\lvert
a-\zeta\rvert\sqrt{\ell}}}\biggr{)}\,,\qquad
x:=\sqrt{\ell\mspace{1.0mu}b}\frac{a-\zeta}{\lvert a-\zeta\rvert}$

and arrive at

$\partial_{\zeta}\partial_{\overline{\zeta}\\!\,}\beta=1-2\operatorname{Re}\biggl{[}{\frac{\langle\overline{x}\\!\,(1-K)^{-1}x\rangle}{\langle\lvert
x\rvert^{2}\rangle}}\biggr{]}=-\frac{1}{\langle\lvert
x\rvert^{2}\rangle}\bigg{\langle}{\frac{1}{1-K}x}\,\mspace{2.0mu},\,{(1-K^{*}K)\frac{1}{1-K}x}\bigg{\rangle}\,.$

Here we used that $(1-K)^{-1}x$ is well-defined since $x\perp k$ due to
(6.24), where $k:=\lvert a-\zeta\rvert\sqrt{\ell\mspace{1.0mu}b}$ is the right
and left Perron-Frobenius eigenvector of $K$, i.e. $(1-K)k=0=(1-K^{*})k$ due
to (5.46) with $\beta=0$. Furthermore, $(1-K^{*}K)k=0$ implies that $k$ is the
Perron-Frobenius eigenvector of $K^{*}K$ and thus $1-K^{*}K$ is strictly
positive definite on $k^{\perp}$, implying $\Delta\beta<0$. Since $\beta$ is
real analytic in a neighbourhood of $\zeta$ with
$\beta(\zeta)=\partial_{\zeta}\beta(\zeta)=0$ according to Corollary 5.18 and
such $\zeta$ cannot be a local minimum of $\beta$ due to
$\Delta\beta(\zeta)<0$ we infer (6.23). ∎

As a consequence of Lemma 6.9 the definition of $\mathbb{S}$ in (5.30) and
Proposition 5.15 (iv) yields

$\mathbb{S}_{0}=\overline{\mathbb{S}}.$ (6.25)

We now have all ingredients to prove Proposition 6.1.

######  Proof of Proposition 6.1.

Part (i) is Proposition 6.3. Items (ii) and (iii) are proved in Proposition
6.7. For the proof of (iv), we conclude
$\operatorname{supp}\sigma\subset\mathbb{S}_{0}=\overline{\mathbb{S}}$ from
Proposition 6.3 and (6.25). Moreover,
$\mathbb{S}\subset\operatorname{supp}\sigma$ follows from (iii), which
completes the proof of (iv). Note that $\partial\mathbb{S}$ is a real analytic
variety due to (6.23) and Corollary 5.18. The dimension of
$\partial\mathbb{S}$ is at most one as $\Delta\beta(\zeta)\neq 0$ if
$\partial_{\zeta}\beta(\zeta)=0$ by Lemma 6.9. This shows (v). Part (vi)
follows from Proposition 6.8 and the fact that $g$ is real analytic by
Corollary 5.18. ∎

##  7 Proof of main results – Proposition 2.1 and Theorem 2.4

This section is devoted to the proofs of our main results, Proposition 2.1 and
Theorem 2.4. They are derived from the results in the previous sections as
well as some inputs from [5, 34, 32]. The underlying idea for both derivations
is the Hermitization approach going back to Girko [25] which allows to
understand the eigenvalue density of $X+A$ by understanding the spectra of the
Hermitian matrices $(H_{\zeta})_{\zeta\in\mathbb{C}}$ defined through

$H_{\zeta}:=\begin{pmatrix}0&X+A-\zeta\\\ (X+A-\zeta)^{*}&0\end{pmatrix}.$
(7.1)

The usefulness of $H_{\zeta}$ becomes apparent from the following properties.
A complex number $\zeta\in\mathbb{C}$ is an eigenvalue of $X+A$ if and only if
$H_{\zeta}$ has a nontrivial kernel. Furthermore, the spectrum of $H_{\zeta}$
is symmetric around zero and its non-negative eigenvalues coincide with the
singular values of $X+A-\zeta$ (with multiplicities).

###  7.1 Proof of Proposition 2.1

After this general explanation, we now focus on the proof of Proposition 2.1.
To that end, we now explain in detail how the empirical spectral distribution
of $X+A$ is expressed in terms of the family
$(H_{\zeta})_{\zeta\in\mathbb{C}}$.

First, as $\log\lvert\,\cdot\,\rvert$ is the fundamental solution for the
Laplace operator on $\mathbb{C}$, we obtain

$\frac{1}{n}\sum_{\xi\in\operatorname{Spec}(X+A)}f(\xi)=\frac{1}{2\pi
n}\sum_{\xi\in\operatorname{Spec}(X+A)}\int_{\mathbb{C}}\Delta
f(\zeta)\log\lvert\xi-\zeta\rvert\mathrm{d}^{2}\zeta=\frac{1}{4\pi
n}\int_{\mathbb{C}}\Delta f(\zeta)\log\lvert\det
H_{\zeta}\rvert\mathrm{d}^{2}\zeta,$ (7.2)

where the last step follows from

$\sum_{\xi\in\operatorname{Spec}(X+A)}\log\lvert\xi-\zeta\rvert=\log\lvert\det(X+A-\zeta)\rvert=\frac{1}{2}\log\lvert\det
H_{\zeta}\rvert.$ (7.3)

We can now express the log-determinant of $H_{\zeta}$ as an integral of the
normalised trace of the resolvent
$G(\zeta,\mathrm{i}\eta):=(H_{\zeta}-\mathrm{i}\eta)^{-1}$ of $H_{\zeta}$ on
the imaginary axis; this expression reads as

$\log\lvert\det H_{\zeta}\rvert=-2n\int_{0}^{T}\mathrm{Im}\,\langle
G(\zeta,\mathrm{i}\eta)\rangle\mathrm{d}\eta+\log\lvert\det(H_{\zeta}-\mathrm{i}T)\rvert$
(7.4)

for any $T>0$ (see [46] for an application of (7.4) in a similar context).
Here and in the following, for a $K\times K$-matrix $R\in\mathbb{C}^{K\times
K}$, we denote by $\langle R\rangle=\frac{1}{K}\operatorname{Tr}R$ the
normalized trace of $R$.

For the proof of Proposition 2.1, we follow the strategy of [6, proof of
Theorem 2.3], which is presented in [6, Section 3.2].

The next proposition, which follows directly from results in [5], shows that
$\langle G(\zeta,\mathrm{i}\eta)\rangle$ is approximately deterministic. Given
(7.2) and (7.4), this explains the origin of the definition of $\sigma$ via
(6.2) and (6.1).

In the next proposition and throughout this section, we use the following
notion of high probability events. We say that a sequence of events
$(\Omega_{n})_{n\in\mathbb{N}}$ occurs _with very high probability_ if for
each $\nu\in\mathbb{N}$, there is a constant $C_{\nu}>0$ (i.e. $C_{\nu}$ does
not depend on $n$) such that $\mathbb{P}(\Omega_{n})\geq 1-C_{\nu}n^{-\nu}$