for all $n\in\mathbb{N}$.

######  Proposition 7.1 (Deterministic approximation of resolvent of
$H_{\zeta}$, averaged version).

Let $X$ and $A=D(a)$ for some $a=(a_{i})_{i=1}^{n}\in\mathbb{C}^{n}$ satisfy
A1, A2 and $\lVert a\rVert_{\infty}=\max_{i=1}^{n}\lvert a_{i}\rvert\lesssim
1$. Let $(v_{1}^{(n)},v_{2}^{(n)})$ be the solution of (4.2) with
$\mathfrak{X}=\llbracket n\rrbracket$, $\mu$ the normalized counting measure
on $\llbracket n\rrbracket$ as well as $a$ and $s$ with
$s(i,j):=n\mathbb{E}\lvert x_{ij}\rvert^{2}$ for $i,j\in\llbracket
n\rrbracket$ interpreted as functions on $\llbracket n\rrbracket$ and
$\llbracket n\rrbracket^{2}$, respectively. Let $\varphi>0$ be fixed. Then
there are universal constants $\delta>0$ and $P\in\mathbb{N}$ such that

$\lvert\langle G(\zeta,\mathrm{i}\eta)\rangle-\mathrm{i}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle\rvert\leq\frac{n^{P\delta}}{(1+\eta^{2})n}$

with very high probability uniformly for all $n\in\mathbb{N}$,
$\eta\in[n^{-\delta},\infty)$ and $\zeta\in\mathbb{D}_{\varphi}$.

######  Proof.

The matrix $X+A$ is a Kronecker matrix according to [5, Definition 2.1] with
the choices $L=1$, $\ell=1$, $\widetilde{\alpha}_{1}=1$, $X_{1}=X$,
$\beta_{1}=0$, $Y_{1}=0$ and $\widetilde{a}_{i}=a_{i}$ for all $i\in\llbracket
n\rrbracket$. In particular, the Hermitization $H_{\zeta}$ defined in (7.1) is
also a Kronecker matrix. Moreover, $H_{\zeta}$ satisfies the assumptions of
[5, Lemma B.1 (ii)] due to A1, A2 and $\lVert a\rVert_{\infty}\lesssim 1$.
Since the Hermitized matrix Dyson equation from [5, eq.s (2.2) – (2.6)]
coincides with the matrix Dyson equation, (5.13), associated with (4.2) for
$(v_{1}^{(n)},v_{2}^{(n)})$ (see Section 5.2 for more explanations), [5, eq.
(B.5)] and [5, eq. (4.46)] imply Proposition 7.1. ∎

To control the integral in (7.4), we need to ensure that the smallest singular
value $\operatorname{s_{\min}}(X+A-\zeta)$ of $X+A-\zeta$, i.e. the smallest,
in modulus, eigenvalue of $H_{\zeta}$, is not too small. This is ensured by
the next proposition. It follows easily from the main results of [34] in the
real case, and [32] in the complex case.

######  Proposition 7.2 (Smallest singular value of $X+A-\zeta$).

Let $X\in\mathbb{C}^{n\times n}$ satisfy A1, A2 for $\nu=2$ and all
$i,j\in\llbracket n\rrbracket$ and A3. Then the following holds.

  1. (i)

For any constant $K>0$, there are constants $C>0$ and $c>0$ such that

$\mathbb{P}\big{(}\operatorname{s_{\min}}(X+A-\zeta)\leq\varepsilon
n^{-1}\big{)}\leq C\varepsilon+2\mathrm{e}^{-cn}$

for all $\varepsilon\geq 0$, $\zeta\in\mathbb{D}_{K}$ and
$A\in\mathbb{C}^{n\times n}$ satisfying $\lVert A\rVert_{\mathrm{hs}}^{2}\leq
Kn$.

  2. (ii)

Let $K$, $A$ and $\zeta$ be as in (i). For any $\delta>0$, the event
$\\{\operatorname{s_{\min}}(X+A-\zeta)\geq\mathrm{e}^{-n^{\delta}}\\}$ holds
with very high probability.

###### Proof.

We check the conditions of [32, Theorem 1.1] for $A_{n}=\sqrt{n}(X+A-\zeta)$.
The assumptions of Proposition 7.2 on $A$ and $\zeta$ as well as A2 with
$\nu=2$ imply $\mathbb{E}\lVert X+A-\zeta\rVert_{\mathrm{hs}}^{2}\lesssim n$.
The second condition of [32, Theorem 1.1] is identical to A3. Therefore,
Proposition 7.2 follows from [32, Theorem 1.1]. ∎

In addition to the control on the smallest singular value of $X+A-\zeta$, we
also need to bound the number of its small singular values. This is the
content of the next lemma, which follows simply from Proposition 7.1 and an
upper bound on $\lvert\langle M(\zeta,\mathrm{i}\eta)\rangle\rvert$.

######  Lemma 7.3 (Number of small singular values of $X+A-\zeta$).

Let $X$ satisfy A1 and A2 and let $A=D(a)$ for some $a\in\mathbb{C}^{n}$ with
$\lVert a\rVert_{\infty}\lesssim 1$. Let $\varphi>0$ be fixed. Then there is a
universal constant $\delta>0$ such that

$\lvert\operatorname{Spec}(H_{\zeta})\cap[-\eta,\eta]\rvert\lesssim n\eta$

with very high probability uniformly for all $\eta\in[n^{-\delta},\infty)$ and
$\zeta\in\mathbb{D}_{\varphi}$.

###### Proof.

Proposition 7.1 and (6.4) imply that the trace of $G(\zeta,\mathrm{i}\eta)$ is
bounded by $n$ with very high probability. More precisely,
$\lvert\operatorname{Tr}G(\zeta,\mathrm{i}\eta)\rvert\lesssim n$ with very
high probability uniformly for all $\eta\in[n^{-\delta},\infty)$ and
$\zeta\in\mathbb{D}_{\varphi}$. Hence, we set
$\Sigma_{\eta}:=\operatorname{Spec}(H_{\zeta})\cap[-\eta,\eta]$ and estimate

$\frac{\lvert\Sigma_{\eta}\rvert}{2\eta}\leq\sum_{\lambda\in\Sigma_{\eta}}\frac{\eta}{\lambda^{2}+\eta^{2}}\leq\mathrm{Im}\,\operatorname{Tr}G(\zeta,\mathrm{i}\eta)\lesssim
n.\qed$

We apply the previous results, i.e. Proposition 7.1, Proposition 7.2 and Lemma
7.3, to the right-hand side of (7.7) by discretizing the integral in $\zeta$
through the next lemma.

######  Lemma 7.4 (Monte Carlo Sampling).

Let $\Omega\subset\mathbb{C}$ be bounded and of positive Lebesgue measure. Let
$\mu$ be the normalised Lebesgue measure on $\Omega$ and
$F\colon\Omega\to\mathbb{C}$ square-integrable with respect to $\mu$. Let
$m\in\mathbb{N}$ and $\xi_{1}$, …, $\xi_{m}$ be independent random variables
distributed according to $\mu$. Then, for any $\delta>0$, we have

$\mathbb{P}\bigg{(}\bigg{\lvert}\frac{1}{m}\sum_{i=1}^{m}F(\xi_{i})-\int_{\Omega}F\mathrm{d}\mu\bigg{\rvert}\leq\frac{1}{\sqrt{m\delta}}\Big{(}\int_{\Omega}\Big{\lvert}F-\int_{\Omega}F\mathrm{d}\mu\Big{\rvert}^{2}\Big{)}^{1/2}\bigg{)}\geq
1-\delta.$

Lemma 7.4 is a special case of [46, Lemma 36]. For the convenience of the
reader, we present the very short proof here.

###### Proof.

Each of the i.i.d. random variables $F(\xi_{1})$, …, $F(\xi_{m})$ has
expectation $\int_{\Omega}F\mathrm{d}\mu$ and variance $\int_{\Omega}\lvert
F-\int_{\Omega}F\mathrm{d}\mu\rvert^{2}\mathrm{d}\mu$. Hence, Chebysheff’s
inequality yields Lemma 7.4. ∎

The final ingredient for the proof of Proposition 2.1 is the following remark
which asserts that all eigenvalues of $X+A$ are contained in
$\mathbb{S}_{\varepsilon}$ defined in (5.16) with very high probability.

######  Remark 7.5 (No outlier eigenvalues of $X+A$).

If $X$ satisfies A1 and A2 and $A=D(a)$ for some $a\in\mathbb{C}^{n}$ with
$\lVert a\rVert_{\infty}\lesssim 1$ then, for every $\varepsilon>0$ and
$\delta\in(0,\varepsilon)$, all eigenvalues of $X+A$ are contained in
$\mathbb{S}_{\varepsilon}$ with very high probability, i.e. for each $\nu>0$,
there is a constant $C\equiv C_{\varepsilon,\delta,\nu}>0$ such that

$\mathbb{P}\big{(}\operatorname{Spec}(X+A)\subset\operatorname{Spec}_{\varepsilon-\delta}(X+A)\subset\mathbb{S}_{\varepsilon}\big{)}\geq
1-Cn^{-\nu}$

for all $n\in\mathbb{N}$. This follows directly from [5, Lemma 6.1]. Here, we
used that $X+A$ is a Kronecker matrix according to [5, Definition 2.1] and
that the Dyson equation (5.13) and [5, eq. (2.6)] coincide as explained in the
proof of Proposition 7.1.

We have now collected all ingredients for the proof of Proposition 2.1, which
we present next.

######  Proof of Proposition 2.1.

Let $(v_{1}^{(n)},v_{2}^{(n)})$ and $\sigma^{(n)}$ be defined222 We note that
$(v_{1}^{(n)},v_{2}^{(n)})$ from Corollary C.2 can be naturally identified
with $(v_{1}^{(n)},v_{2}^{(n)})$ from Proposition 7.1. In particular, $\langle
v_{1}^{(n)}\rangle$ yields the same result for either definition and this is
the only quantity derived from $(v_{1}^{(n)},v_{2}^{(n)})$ that plays a role
in the following. as in Corollary C.2. Because of Corollary C.2, it suffices
to show that for each $f\in C_{b}(\mathbb{C})$,

$\lim_{n\to\infty}\mathbb{P}\bigg{(}\bigg{\lvert}\frac{1}{n}\sum_{\xi\in\operatorname{Spec}(X+A)}f(\xi)-\int_{\mathbb{C}}f\mathrm{d}\sigma^{(n)}\bigg{\rvert}>\varepsilon\bigg{)}=0.$
(7.5)

The main part of the proof will be to show the existence of a constant
$\delta>0$ such that

$\bigg{\lvert}\frac{1}{n}\sum_{\zeta\in\operatorname{Spec}X}f(\zeta)-\int_{\mathbb{C}}f(\zeta)\sigma^{(n)}(\mathrm{d}\zeta)\bigg{\rvert}\leq
n^{-\delta}\lVert\Delta f\rVert_{\mathrm{L}^{1}}+n^{-10}\lVert\Delta
f\rVert_{\mathrm{L}^{3}}$ (7.6)

with very high probability uniformly for all $f\in C_{0}^{2}(\mathbb{C})$
satisfying $\operatorname{supp}f\subset\mathbb{D}_{\varphi}$ for any fixed
constant $\varphi>0$.

We now explain how (7.6) implies (7.5), i.e. Proposition 2.1. If $f\in
C_{0}^{2}(\mathbb{C})$ then this is obvious. Let $f\in
C_{b}(\mathbb{C})\setminus C_{0}^{2}(\mathbb{C})$. Owing to Remark 7.5, we
know that $\operatorname{Spec}(X+A)\subset\mathbb{S}_{1}$ with very high
probability. We note that $\mathbb{S}_{1}\subset\mathbb{D}_{\varphi}$ for some
$\varphi\sim 1$ by Remark 5.8. By possibly increasing $\varphi\sim 1$, we also
have $\operatorname{supp}\sigma^{(n)}\subset\mathbb{D}_{\varphi}$ due to
Corollary C.2 (ii). Therefore, it suffices to consider $f\in
C_{b}(\mathbb{C})$ with $\operatorname{supp}f\subset\mathbb{D}_{\varphi+1}$.
Then we find $f_{\varepsilon}\in C_{0}^{2}(\mathbb{C})$ such that $\lVert
f-f_{\varepsilon}\rVert_{\mathrm{L}^{\infty}}\leq\varepsilon/2$,
$\operatorname{supp}f_{\varepsilon}\subset\mathbb{D}_{\varphi+1}$ and
$\lVert\Delta f_{\varepsilon}\rVert_{\mathrm{L}^{1}}+\lVert\Delta
f_{\varepsilon}\rVert_{\mathrm{L}^{3}}\lesssim_{\varepsilon}1$. Hence,
approximating $f$ by $f_{\varepsilon}$ in (7.5) and using (7.6) for
$f_{\varepsilon}$ shows that (7.6) implies (7.5).

It remains to show (7.6). We fix a constant $\varphi>0$ and set
$\Omega=\mathbb{D}_{\varphi}$. For any $T>0$, we conclude from (7.2), (7.4),
Corollary C.2 (ii) and the second bound in (6.5) that

$\frac{1}{n}\sum_{\xi\in\operatorname{Spec}(X+A)}f(\xi)-\int_{\mathbb{C}}f(\zeta)\sigma^{(n)}(\mathrm{d}\zeta)=\int_{\Omega}F(\zeta)\frac{\mathrm{d}^{2}\zeta}{\lvert\Omega\rvert}+O\big{(}T^{-1}\lVert\Delta
f\rVert_{\mathrm{L}^{1}}),$ (7.7)

where

$F(\zeta):=\frac{\lvert\Omega\rvert}{\pi}(\Delta f(\zeta))h(\zeta),\qquad
h(\zeta):=\frac{1}{n}\sum_{\xi\in\operatorname{Spec}(X+A)}\log\lvert\xi-\zeta\rvert+\int_{0}^{T}\bigg{(}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{)}\mathrm{d}\eta.$

Note that $h$ and, thus, $F$ depend on the choice of $T$.

We now estimate
$\int_{\Omega}F(\zeta)\frac{\mathrm{d}^{2}\zeta}{\lvert\Omega\rvert}$ by
applying Lemma 7.4 to it. Since $\zeta\mapsto\log\lvert\xi-\zeta\rvert$ lies
in $\mathrm{L}^{p}(\Omega)$ for every $p\in[1,\infty)$, the first bound in
(6.5) implies that, for every $p\in[1,\infty)$, $\lVert
h\rVert_{\mathrm{L}^{p}(\Omega)}\lesssim_{p}1$ uniformly for $T>0$. Therefore,
$F\in L^{2}(\Omega)$ and Lemma 7.4 with $\delta=n^{-\nu}$ and $m=n^{\nu-130}$
yields

$\bigg{\lvert}\int_{\Omega}F(\zeta)\frac{\mathrm{d}^{2}\zeta}{\lvert\Omega\rvert}-\frac{1}{m}\sum_{i=1}^{m}F(\xi_{i})\bigg{\rvert}\leq
n^{-10}\lVert F\rVert_{\mathrm{L}^{2}}\lesssim n^{-11}\lVert\Delta
f\rVert_{\mathrm{L}^{3}}$ (7.8)

with very high probability, where $\xi_{1},\ldots,\xi_{m}$ are independent
random variables distributed according to the normalized Lebesgue measure on
$\Omega$.

What remains for the proof of (7.6) is to bound
$\frac{1}{m}\sum_{i=1}^{m}F(\xi_{i})$. To that end, we set $T=n^{100}$ and
show in the following that, for all small enough $\delta>0$,

$\lvert F(\zeta)\rvert\leq n^{-\delta}\lvert\Delta f(\zeta)\rvert$ (7.9)

with very high probability uniformly for all $\zeta\in\Omega$. We set
$\eta_{*}:=n^{-\delta}$ and introduce

$\displaystyle h_{1}(\zeta)$ $\displaystyle:=\int_{\eta_{*}}^{T}\big{(}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle-\mathrm{Im}\,\langle
G(\zeta,\mathrm{i}\eta)\rangle\big{)}\mathrm{d}\eta,$ $\displaystyle
h_{2}(\zeta)$ $\displaystyle:=-\int_{0}^{\eta_{*}}\mathrm{Im}\,\langle
G(\zeta,\mathrm{i}\eta)\rangle\mathrm{d}\eta$ $\displaystyle h_{3}(\zeta)$
$\displaystyle:=\frac{1}{4n}\sum_{\lambda\in\operatorname{Spec}(H_{\zeta})}\log\bigg{(}1+\frac{\lambda^{2}}{T^{2}}\bigg{)}-\log\bigg{(}1+\frac{1}{T}\bigg{)},$
$\displaystyle h_{4}(\zeta)$
$\displaystyle:=\phantom{-}\int_{0}^{\eta_{*}}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle\mathrm{d}\eta.$

Hence, owing to (7.3), (7.4) and
$\int_{0}^{T}(1+\eta)^{-1}\mathrm{d}\eta=\log(1+T)$, we obtain the
decomposition $h(\zeta)=h_{1}(\zeta)+h_{2}(\zeta)+h_{3}(\zeta)+h_{4}(\zeta)$.

Next, we estimate the terms $h_{1},\ldots,h_{4}$ individually. For $h_{1}$,
Proposition 7.1, a union bound and a continuity argument in $\eta$ imply
$\lvert h_{1}(\zeta)\rvert\leq n^{-1+P\delta+\delta}$ with very high
probabilityfootnote 2. A simple computation shows that

$-h_{2}(\zeta)=\frac{1}{4n}\sum_{\lambda\in\operatorname{Spec}(H_{\zeta})}\log\bigg{(}1+\frac{\eta_{*}^{2}}{\lambda^{2}}\bigg{)}\leq\frac{1}{4n}\sum_{\lambda\in\operatorname{Spec}(H_{\zeta})\cap[-\eta_{*}^{1/2},\eta_{*}^{1/2}]}\log\bigg{(}1+\frac{\eta_{*}^{2}}{\lambda^{2}}\bigg{)}+\eta_{*},$

where in the last step we used that
$\log(1+\eta_{*}^{2}\lambda^{-2})\leq\log(1+\eta_{*})\leq\eta_{*}$ if
$\lvert\lambda\rvert>\eta_{*}^{1/2}$. To estimate the remaining sum, we
conclude from Proposition 7.2 and Lemma 7.3 that

$\frac{1}{4n}\sum_{\lambda\in\operatorname{Spec}(H_{\zeta})\cap[-\eta_{*}^{1/2},\eta_{*}^{1/2}]}\log\bigg{(}1+\frac{\eta_{*}^{2}}{\lambda^{2}}\bigg{)}\lesssim\frac{\log\eta_{*}+\lvert\log\min_{\lambda\in\operatorname{Spec}(H_{\zeta})}\lvert\lambda\rvert\rvert}{n}\lvert\operatorname{Spec}(H_{\zeta})\cap[-\eta_{*}^{1/2},\eta_{*}^{1/2}]\rvert\lesssim
n^{\varepsilon}\eta_{*}^{1/2}$

with very high probability for any $\varepsilon>0$. Therefore, $\lvert
h_{2}(\zeta)\rvert\lesssim n^{-\delta/2+\varepsilon}$, which yields $\lvert
h_{2}(\zeta)\rvert\leq n^{-\delta}$ by shrinking $\delta$. To estimate
$h_{3}$, we use $\log(1+x)\leq x$ and obtain

$\lvert
h_{3}(\zeta)\rvert\leq\frac{1}{4nT^{2}}\operatorname{Tr}(H_{\zeta})^{2}+T^{-1}=\frac{1}{2nT^{2}}\sum_{i,j=1}^{n}(\overline{x_{ji}}+(\bar{a}_{i}-\bar{\zeta})\delta_{ji})(x_{ij}+(a_{i}-\zeta)\delta_{ij})+T^{-1}\lesssim
T^{-1}$

since $\lvert x_{ij}\rvert\leq n^{-1/2+\varepsilon}$ with very high
probability due to Assumption A2 and $\lvert
a_{i}\rvert+\lvert\zeta\rvert\lesssim 1$ as $\lVert a\rVert_{\infty}\lesssim
1$ and $\zeta\in\mathbb{D}_{\varphi}$. Since $a^{(n)}$ and $s^{(n)}$ from
(C.1) satisfy $\lVert a^{(n)}\rVert_{\infty}\lesssim 1$ and A5 with the same
constants as $a$ and $s$, Lemma 5.1 implies $\lvert
h_{4}(\zeta)\rvert\lesssim\eta_{*}$ uniformly for all $n\in\mathbb{N}$. This
completes the proof of (7.9).

The bound (7.9) implies

$\frac{1}{m}\sum_{i=1}^{m}\lvert
F(\xi_{i})\rvert\leq\frac{n^{-\delta}}{m}\sum_{i=1}^{m}\lvert\Delta
f(\xi_{i})\rvert\leq n^{-\delta}\lVert\Delta
f\rVert_{\mathrm{L}^{1}}+n^{-11}\lVert\Delta f\rVert_{\mathrm{L}^{2}}$ (7.10)

with very high probability, where the second step follows from Lemma 7.4 with
$F=\Delta f$.

Finally, we combine (7.7), (7.8) and (7.10) and, thus, obtain (7.6), which
completes the proof of Proposition 2.1. ∎

###  7.2 Proof of Theorem 2.4

We recall the definition of $\mathbb{S}_{\varepsilon}$ from (5.16) and
specialise to the case $\mathfrak{X}=[0,1]$ with $\mu$ the Lebesgue-measure on
$[0,1]$. Then we set

$\operatorname{Spec}_{\varepsilon}^{\infty}(s,a):=\mathbb{S}_{\varepsilon}.$
(7.11)

With this definition, (2.5) follows from Proposition 6.1 (iv) and
$\cap_{\varepsilon>0}\operatorname{Spec}_{\varepsilon}^{\infty}(s,a)=\mathbb{S}_{0}=\overline{\mathbb{S}}$
due to Remark 5.8 and (6.25).

Now we verify (2.4). First we see that for any $\varepsilon,\delta>0$ the
inclusion

$\limsup_{n\to\infty}\operatorname{Spec}_{\varepsilon}(X_{n}+A_{n})\subset\operatorname{Spec}_{\varepsilon+\delta}^{\infty}(s,a)=\mathbb{S}_{\varepsilon+\delta}$

holds almost surely by Remark 7.5 and the Borel-Cantelli lemma. Since
$\cap_{\delta>0}\mathbb{S}_{\varepsilon+\delta}=\mathbb{S}_{\varepsilon}$ by
definition this shows the first inclusion in (2.4).

The second inclusion in (2.4) follows from

$\begin{split}\mathbb{S}_{\varepsilon}\subset\operatorname{Spec}_{\varepsilon+\delta}(X_{n}+A_{n})=\\{\zeta\in\mathbb{C}:\operatorname{dist}(0,\operatorname{Spec}(H_{\zeta}))\leq\varepsilon+\delta\\}\end{split}$
(7.12)

eventually almost surely for any $\varepsilon,\delta>0$. Here $H_{\zeta}$ is
the Hermitisation of $X_{n}+A_{n}$ from (7.1). To prove (7.12) we see that the
global law from [5, Theorem 2.7] holds almost surely when all random matrices
in the statement are realised on the same probability space. This can be seen
easily from its proof. Indeed, the global law is an immediate consequence of
[5, (B.5)], which holds with very high probability. Thus, the Borel-Cantelli
lemma ensures almost sure convergence in

$\frac{1}{2n}\operatorname{Tr}f(H_{\zeta})\to\int_{\mathbb{R}}f(\tau)\rho_{\zeta}(\mathrm{d}\tau)$

for every compactly supported continuous function $f$. ∎

##  Appendix A Existence and uniqueness of solution to Dyson equation

###  A.1 Proof of Lemma 4.1

Owing to the identification of solutions to (4.2) and (5.13) in Section 5.2,
we can now infer the existence and uniqueness of the solution to (4.2) to the
existence and uniqueness of the solution to (5.13). Indeed, the latter is a
very simple case of the general existence and uniqueness result [30, Theorem
2.1]. This proves Lemma 4.1. ∎

###  A.2 Stability operator of Matrix Dyson equation

Let $M$ be the solution of (5.13). The stability operator of (5.13) is given
by (5.18).

######  Lemma A.1.

If $a$ and $s$ satisfy the assumptions of Lemma 5.9, then the following holds.

  1. (i)

For all $\eta>0$ and $\zeta\in\mathbb{C}$, we have

$\lVert M\rVert\leq\eta^{-1}.$ (A.1)

  2. (ii)

For all $\eta>0$ and $\zeta\in\mathbb{C}$, the stability operator
$\mathcal{L}$ and its adjoint $\mathcal{L}^{*}$ are invertible. Moreover, for
any constant $K>0$, there is a constant $C\geq 2$ such that

$\lVert\mathcal{L}^{-1}\rVert_{2}+\lVert\mathcal{L}^{-1}\rVert_{\infty}\leq
C\max\\{\eta^{-2},\eta^{-C}\\}$ (A.2)

for all $\eta>0$, $a\in\mathcal{B}$ and $\zeta\in\mathbb{C}$ satisfying
$\lvert\zeta\rvert+\lVert a\rVert_{\infty}\leq K$.

###### Proof.

For a proof of (A.1), we refer to [1, eq. (4.1)], where the proof was carried
out in the finite dimensional setting. In our setting the proof follows the
same argument. The invertibility of $\mathcal{L}$ and its adjoint
$\mathcal{L}^{*}$ as well as the bound (A.2) are obtained by translating the
proofs of [5, Lemmas 3.4 and 3.7] to the present setup using (A.1). ∎

######  Proof of Lemma 5.14.

We recall that $\zeta\in\mathbb{C}$ is fixed such that
$\limsup_{\eta\downarrow 0}\langle v_{1}(\zeta,\eta)\rangle\geq\delta$ for
some $\delta>0$ and that $v_{n}=v(\zeta,\eta_{n})\to v_{0}$ weakly in
$(L^{2})^{2}$, where $v_{0}\sim_{\delta}1$. First we use the identities

$Lv_{-}=-\eta\frac{\tau\mspace{2.0mu}v^{2}}{(\eta+S_{d}v)^{2}}\,,\qquad
L^{*}(e_{-}(\eta+S_{o}v))=\eta\mspace{1.0mu}e_{-}$

with $v_{-}=ve_{-}$ and $v=v_{n}$. In the limit $\eta\downarrow 0$ we see
$L_{0}v_{-}=0$ and $L_{0}^{*}S_{o}v_{-}=0$. Here we used that $v$ satisfies
the Dyson equation. For the rest of this proof we drop the $0$-index from our
notation. We introduce $T$, $V$, $F:\mathcal{B}^{2}\to\mathcal{B}^{2}$ as in
(6.11), evaluated at $v=v_{0}$ and $\eta=0$. In terms of $T,F$ and $V$ we
obtain

$L=V^{-1}(1-TF)V\,.$ (A.3)

We consider the natural extensions $F\colon(L^{2})^{2}\to(L^{2})^{2}$ and
$T\colon(L^{2})^{2}\to(L^{2})^{2}$. These operators are self-adjoint. We
import results about their spectral properties from [3, Lemma 3.4, Lemma 3.6,
(3.44) and (3.45)]. The proofs of these properties translate immediately to
the current setup, although originally formulated for
$\mathcal{B}=\mathbb{C}^{d}$, since they are uniform in the dimension $d$.
Furthermore, the proofs from [3] are not affected by the changed definition of
$\tau=(\tau_{1},\tau_{2})$ from (5.25) since we still have
$\tau_{1}=\tau_{2}$.

######  Lemma A.2 (Spectral properties of $F$ and $T$).

The Hermitian operator $F\colon(L^{2})^{2}\to(L^{2})^{2}$ satisfies the
following properties:

  1. (i)

$F$ has non-degenerate isolated eigenvalues at $\pm 1$ and a spectral gap
$\varepsilon\sim_{\delta}1$, i.e.

$\operatorname{Spec}(F)\subset\\{-1\\}\cup[-1+\varepsilon,1-\varepsilon]\cup\\{1\\}\,.$

  2. (ii)

The eigenvectors corresponding to the eigenvalues $\pm 1$ are

$FVv=Vv\,,\qquad FVv_{-}=-Vv_{-}\,,$

where $v=(v_{1},v_{2})$ and $v_{-}=(v_{1},-v_{2})$.

The Hermitian operator $T\colon(L^{2})^{2}\to(L^{2})^{2}$ satisfies the
following properties:

  1. (iii)

The spectrum is bounded away from $1$ by a gap of size
$\varepsilon\sim_{\delta}1$, i.e.

$\operatorname{Spec}(T)\subset[-1,1-\varepsilon]\,.$

  2. (iv)

For $x\in\\{(y,-y):y\in\mathcal{B}\\}$ we have $Tx=-x$.

We now prove Lemma 5.14 by tacitly using the properties of $F$ and $T$ from
Lemma A.2. Since $f_{-}:=Vv_{-}\in\\{(y,-y):y\in\mathcal{B}\\}$ we see that
$F$ and $T$ both leave the subspace $(f_{-})^{\perp}$ invariant. Now, using
(A.3), we rewrite the resolvent of $L$ as

$\frac{1}{L-\omega}=V^{-1}\frac{1}{1-\omega-TF}V\,.$

The operator $V$ and its inverse satisfy the bound $\lVert
V\rVert_{\\#}+\lVert V^{-1}\rVert_{\\#}\lesssim_{\delta}1$ for $\\#=2,\infty$,
where we used $\delta\lesssim v\lesssim\frac{1}{\delta}$ from Lemma 5.12.
Thus, it suffices to show (5.28) with $L$ replaced by $1-TF$. Furthermore, we
can restrict to the case $\\#=2$, since [6, Lemma 4.5] is applicable because

$\lVert TF\rVert_{\infty}+\lVert TF\rVert_{\infty\to 2}\lVert
TF\rVert_{2\to\infty}\lesssim_{\delta}1\,.$

Here, $\lVert TF\rVert_{\infty\to 2}$ and $\lVert TF\rVert_{2\to\infty}$
denote the operator norms of $TF$ viewed as operator
$\mathcal{B}^{2}\to(L^{2})^{2}$ and $(L^{2})^{2}\to\mathcal{B}^{2}$,
respectively.

Since $\lVert TF\rVert_{2}\leq 1$ we have the bound
$\lVert(1-TF-\omega)^{-1}\rVert_{2}\lesssim_{\varepsilon,\delta}1$ for any
$\omega\not\in 1+\mathbb{D}_{1+\varepsilon}$ and any $\varepsilon>0$. Now we
can decompose

$\begin{split}\frac{1}{1-TF-\omega}=(1-P)\frac{1}{1-TF-\omega}(1-P)-\frac{1}{\omega}P\,,\end{split}$
(A.4)

where $P$ is the orthogonal projection onto the span of $f_{-}$. Provided
$\varepsilon>0$ is chosen sufficiently small, the first summand in (A.4) is
bounded for $\omega\in\mathbb{D}_{2\varepsilon}$. Indeed $1-TF$ has a bounded
inverse on $f_{-}^{\perp}$ by applying a generalisation of [3, Lemma 3.7] from
the finite dimensional case ${\mathcal{B}}=\mathbb{C}^{d}$ to the general
setup here. The proof of [3, Lemma 3.7] is not affected by this
generalisation. Finally, the second summand in (A.4) is bounded on
$\mathbb{C}\setminus\mathbb{D}_{\varepsilon}$. Altogether the bound (5.28) is
proven. The non-degeneracy of the eigenvector $v_{-}$ of $L$ also follows from
the decomposition (A.4). ∎

##  Appendix B Auxiliary results

######  Lemma B.1.

Let $y>0$ be the unique solution to the equation
$y^{3}+\beta\mspace{1.0mu}y=x$ for $x>0$ and $\beta\in\mathbb{R}$. Then

$y\sim\sqrt{\max\\{0,-\beta\\}}+\frac{x}{x^{2/3}+\lvert\beta\rvert}\,.$

###### Proof.

First we consider the case $\beta\geq 0$. Then clearly

$y\sim\frac{x}{x^{2/3}+\lvert\beta\rvert}\,.$

Now let $\beta<0$, then we must have $y=\sqrt{-\beta}(1+\varepsilon)$ for some
$\varepsilon>0$ since $y^{2}+\beta\mspace{1.0mu}y>0$. For this $\varepsilon$
we get the equation
$\varepsilon^{3}+3\varepsilon^{2}+2\varepsilon=x\mspace{1.0mu}\lvert\beta\rvert^{-3/2}\,.$
Thus, we have the scaling

$\varepsilon\sim\min\biggl{\\{}{\frac{x}{\lvert\beta\rvert^{3/2}},\frac{x^{1/3}}{\lvert\beta\rvert^{1/2}}}\biggr{\\}}\,.$

Therefore we conclude

$y=\sqrt{\lvert\beta\rvert}+\min\biggl{\\{}{\frac{x}{\lvert\beta\rvert},x^{1/3}}\biggr{\\}}\sim\sqrt{\lvert\beta\rvert}+\frac{x}{x^{2/3}+\lvert\beta\rvert}\,,$

which is the claim of the lemma. ∎

######  Lemma B.2.

Let $S\colon\mathcal{B}\to\mathcal{B}$ be an integral operator as in (4.1)
with a kernel $s:\mathfrak{X}^{2}\to(0,\infty)$ that satisfies the bounds
$\varepsilon\leq s(x,y)\leq\frac{1}{\varepsilon}$ for all $x$,
$y\in\mathfrak{X}$ and some constant $\varepsilon>0$ and such that the
spectral radius of $S$ is normalised to $\varrho(S)=1$. Then there are
constants $\delta,C>0$, depending only on $\varepsilon$, such that

$\sup\Big{\\{}\lVert(S-z)^{-1}\rVert_{2}\colon z\not\in D_{1-\delta}(0)\cup
D_{\delta}(1)\Big{\\}}\leq C$

and $\operatorname{Spec}(S)\cap D_{\delta}(1)=\\{1\\}$ is non-degenerate.

###### Proof.

We follow line by line the arguments from the proof of [23, Lemma A.1], where
the finite dimensional case with $\lvert\mathfrak{X}\rvert<\infty$ and $\mu$
the counting measure is carried out. ∎

##  Appendix C Discretizing the Dyson equation

In order to formulate the next lemma under the weakest assumptions, we first
relax the condition A4 by avoiding the lower bound on $s$ in the following
assumption.

  1. A7

_Piecewise continuity of $s$ and $a$: _ Let $I_{1}$, …, $I_{K}\subset[0,1]$ be
disjoint intervals with some $K\in\mathbb{N}$ 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
$\overline{I_{l}}\times\overline{I_{k}}$ and $\overline{I_{l}}$, respectively,
for all $l,k\in\llbracket K\rrbracket$.

Throughout this section, we write
$\mathbb{C}_{+}:=\\{w\in\mathbb{C}\colon\mathrm{Im}\,w>0\\}$. Let
$(v_{1},v_{2})$ be the solution of (4.2) on $\mathfrak{X}=[0,1]$ and with
$\mu$ the Lebesgue measure on $[0,1]$, $M$ the unique solution of (5.15) and
$\rho_{\zeta}$ the associated probability measure from Definition 5.7.

######  Lemma C.1.

Let $s$ and $a$ satisfy A7. For $n\in\mathbb{N}$, define the functions
$a^{(n)}$ on $[0,1]$ and $s^{(n)}$ on $[0,1]^{2}$ through

$a^{(n)}:=\sum_{i=1}^{n}a\bigg{(}\frac{i}{n}\bigg{)}\mathbf{1}_{[(i-1)/n,i/n)},\qquad
s^{(n)}:=\sum_{i,j=1}^{n}\frac{1}{n}s\bigg{(}\frac{i}{n},\frac{j}{n}\bigg{)}\mathbf{1}_{[(i-1)/n,i/n)\times[(j-1)/n,j/n)},$
(C.1)

where $\mathbf{1}_{\Omega}$ denotes the indicator function of the set
$\Omega$. Let $\Sigma^{(n)}$ be defined analogously to (5.14) with $s$
replaced by $s^{(n)}$. If $M^{(n)}$ is the unique solution of (5.15) with
$a^{(n)}$ and $\Sigma^{(n)}$ instead of $a$ and $\Sigma$, $\delta>0$ is
constant and $\zeta\in\mathbb{C}$ is fixed, then

$\lim_{n\to\infty}\lVert M^{(n)}(\zeta,w)-M(\zeta,w)\rVert_{2}=0$

uniformly for all $w\in\mathbb{C}_{+}$ satisfying
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$. Here,
$\lVert R\rVert_{2}:=\lVert\operatorname{Tr}(R^{*}R)\rVert_{1}^{1/2}/\sqrt{2}$
for any $R\in\mathcal{B}^{2\times 2}$, where $\operatorname{Tr}(R^{*}R)$ is
considered as a function on $[0,1]$, and $\lVert f\rVert_{p}$ is the
$L^{p}([0,1],\mu)$-norm for $f\colon[0,1]\to\mathbb{C}$.

If $(v_{1}^{(n)},v_{2}^{(n)})$ is the solution of (4.2) on $[0,1]$ with the
Lebesgue measure $\mu$ and $a$ and $s$ replaced by $a^{(n)}$ and $s^{(n)}$
from (C.1), then for any (fixed) $\zeta\in\mathbb{C}$ and $\eta>0$, we have

$\lim_{n\to\infty}\,\max\big{\\{}\lVert
v_{1}^{(n)}(\zeta,\eta)-v_{1}(\zeta,\eta)\rVert_{2},\,\lVert
v_{2}^{(n)}(\zeta,\eta)-v_{2}(\zeta,\eta)\rVert_{2}\big{\\}}=0.$

Throughout the remainder of this section, a operators appear that map
$\mathcal{B}^{2\times 2}$ to $\mathcal{B}^{2\times 2}$. We write
$\lVert\,\cdot\,\rVert_{\ast\to\\#}$ with $\ast$, $\\#\in\\{2,\infty\\}$ for
the operator norm if the definition space is equipped with the norm
$\lVert\,\cdot\,\rVert_{\ast}$ and the target space with
$\lVert\,\cdot\,\rVert_{\\#}$. If $\ast=\\#$ then we simply write
$\lVert\,\cdot\,\rVert_{\ast}$ for the corresponding operator norm.

###### Proof.

Owing to the explanations in Section 5.2, especially, (5.12), it suffices to
show that $\lVert M^{(n)}(\zeta,w)-M(\zeta,w)\rVert_{2}\to 0$ if $n\to\infty$.

Fix $\delta>0$ and $\zeta\in\mathbb{C}$. We introduce the matrices
$A\in\mathcal{B}^{2\times 2}$ and $A^{(n)}\in\mathcal{B}^{2\times 2}$ through

$A:=\begin{pmatrix}0&a\\\ \overline{a}&0\end{pmatrix},\qquad
A^{(n)}:=\begin{pmatrix}0&a^{(n)}\\\ \overline{a^{(n)}}&0\end{pmatrix}.$

For $w\in\mathbb{C}_{+}$ satisfying
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$ and $t\geq
0$, we set $M^{(n)}=M^{(n)}(\zeta,w+\mathrm{i}t)$, $M=M(\zeta,w+\mathrm{i}t)$
and $L[R]:=R-M\Sigma[R]M$ for all $R\in\mathcal{B}^{2\times 2}$. With
$\Delta:=M^{(n)}-M$, a short computation starting from (5.15) and the
analogous relation with $M^{(n)}$, $a^{(n)}$ and $\Sigma^{(n)}$ yields

$\begin{split}L[\Delta]=M\Sigma[\Delta]\Delta+M(\Sigma^{(n)}-\Sigma)[M^{(n)}]M^{(n)}+M(A-A^{(n)})M^{(n)}.\end{split}$
(C.2)

We now invert $L$ and estimate the resulting relation in
$\lVert\,\cdot\,\rVert_{2}$. We collect a few auxiliary bounds. From [5, eq.s
(3.22), (3.11a), (3.11c)], we conclude the existence of a constant $C_{1}>0$,
depending only on $\delta$ but independent of $w$ and $t$, such that $\lVert
L^{-1}\rVert_{2}\leq C_{1}$ for all $w\in\mathbb{C}_{+}$ with
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$ and $t\geq
0$. As $\lVert
M(\zeta,w)\rVert\leq(\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta}))^{-1}$
by [5, eq. (3.11a)]333The proof in [5] is given in the finite dimensional
setup; the proof in the setup of this article is identical., we have $\lVert
M\rVert\leq(\max\\{\delta,t\\})^{-1}$ and $\lVert
M^{(n)}\rVert\leq(\max\\{\delta,t\\})^{-1}$ for all $t\geq 0$. Owing to [4,
Lemma B.2(i)], the upper bound on $s$ following from its piecewise continuity
implies that there is a constant $C_{2}\geq 1$ such that
$\lVert\Sigma\rVert_{2\to\infty}\leq C_{2}$. Therefore, there is a constant
$C>0$ depending only on $\delta$ but not on $w$ or $t$ such that

$\lVert\Delta\rVert_{2}\leq
C(\lVert\Delta\rVert_{2}^{2}+\Psi_{n}),\qquad\Psi_{n}:=\lVert\Sigma^{(n)}-\Sigma\rVert_{2}+\lVert
A-A^{(n)}\rVert_{2}$ (C.3)

for all $w\in\mathbb{C}_{+}$ with
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$ and all
$t\geq 0$. Here, $\Delta\equiv\Delta(\zeta,w+\mathrm{i}t)$.

Owing to A7, we get $\Psi_{n}\to 0$ as $n\to\infty$. Thus, we find
$n_{0}\in\mathbb{N}$ such that $2\Psi_{n}C^{2}\leq 1/4$ for all $n\geq n_{0}$.
Fix $w\in\mathbb{C}_{+}$ with
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$. We set
$t_{*}:=\sup\\{t\geq 0\colon\lVert\Delta(\zeta,w+\mathrm{i}t)\rVert_{2}\geq
2C\Psi_{n}\\}$. Since $\lVert M^{(n)}\rVert+\lVert M\rVert\to 0$ for
$t\to\infty$, we obtain $t_{*}<\infty$. Next, we conclude $t_{*}=0$. Suppose
$t_{*}>0$. Hence, $\lVert\Delta(\zeta,w+\mathrm{i}t_{*})\rVert_{2}=2C\Psi_{n}$
by continuity. As $2\Psi_{n}C^{2}\leq 1/4$, we conclude from (C.3) that
$\lVert\Delta(\zeta,w+\mathrm{i}t_{*})\rVert_{2}\leq
3C\Psi_{n}/2<2C\Psi_{n}=\lVert\Delta(\zeta,w+\mathrm{i}t_{*})\rVert_{2}$. This
contradiction implies $t_{*}=0$. Note that this holds for any
$w\in\mathbb{C}_{+}$ as long as
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$ and $n\geq
n_{0}$. Thus, for $n\geq n_{0}$, we obtain $\lVert
M^{(n)}(\zeta,w)-M(\zeta,w)\rVert_{2}=\lVert\Delta(\zeta,w)\rVert_{2}\leq
2C\Psi_{n}$ for all $w\in\mathbb{C}_{+}$ with
$\operatorname{dist}(w,\operatorname{supp}\rho_{\zeta})\geq\delta$, which
concludes the proof of Lemma C.1 as $\Psi_{n}\to 0$ with $n\to\infty$. ∎

######  Corollary C.2.

Let $s$ and $a$ satisfy A4 and $(v_{1}^{(n)},v_{2}^{(n)})$ be as in Lemma C.1.
Then the following holds.

  1. (i)

For each $n\in\mathbb{N}$, replacing $v_{1}$ by $v_{1}^{(n)}$ in (6.1) yields
a well-defined continuous function $L^{(n)}\colon\mathbb{C}\to\mathbb{R}$.

  2. (ii)

For each $n\in\mathbb{N}$, there exists a probability measure $\sigma^{(n)}$
on $\mathbb{C}$ such that (6.2) holds with $\sigma$ and $L$ replaced by
$\sigma^{(n)}$ and $L^{(n)}$, respectively. Furthermore, there is $\varphi\sim
1$ such that $\operatorname{supp}\sigma^{(n)}\subset\mathbb{D}_{\varphi}$ for
all $n\in\mathbb{N}$.

  3. (iii)

Moreover, $\sigma^{(n)}$ converges to $\sigma$ weakly as $n$ tends to
infinity.

###### Proof.

Clearly, $a^{(n)}$ and $s^{(n)}$ from (C.1) satisfy $\lVert
a^{(n)}\rVert_{\infty}\lesssim 1$ and A5 (with the same constants as $a$ and
$s$). Hence, Lemma 6.5, Proposition 6.3 and Remark 5.8 imply the well-
definedness of $L^{(n)}$ and the existence of probability measures
$\sigma^{(n)}$ satisfying (6.2) for $L^{(n)}$ as well as
$\operatorname{supp}\sigma^{(n)}\subset\mathbb{D}_{\varphi}$, respectively.

Since $\sigma^{(n)}$ for all $n\in\mathbb{N}$ and $\sigma$ are probability
measures on $\mathbb{C}$, for the weak convergence it suffices to show
$\int_{\mathbb{C}}f\mathrm{d}\sigma^{(n)}\to\int_{\mathbb{C}}f\mathrm{d}\sigma$
as $n\to\infty$ for all $f\in C_{0}^{2}(\mathbb{C})$. Fix $f\in
C_{0}^{2}(\mathbb{C})$. As $a^{(n)}$ and $s^{(n)}$ from (C.1) satisfy $\lVert
a^{(n)}\rVert_{\infty}\lesssim 1$ and A5 with the same constants as $a$ and
$s$, we conclude from Lemma 5.1 and (5.9) that $\big{\lvert}\Delta
f(\zeta)\big{(}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\big{)}\big{\rvert}\lesssim\frac{\lvert\Delta
f(\zeta)\rvert}{1+\eta^{2}}$ uniformly $\eta>0$, $\zeta\in\mathbb{C}$ and
$n\in\mathbb{N}$. That is the implicit constant hidden by $\lesssim$ does not
depend on $\eta$, $\zeta$ and $n$. Owing to the integrability of the right-
hand side with respect to $\zeta$ and $\eta$ over
$\mathbb{C}\times(0,\infty)$, we obtain from (6.2), Fubini and dominated
convergence that

$\int_{\mathbb{C}}f\mathrm{d}\sigma^{(n)}=\int_{\mathbb{C}}\int_{0}^{\infty}\Delta
f(\zeta)\bigg{(}\langle
v_{1}^{(n)}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{)}\mathrm{d}\eta\,\mathrm{d}^{2}\zeta\\\
\longrightarrow\int_{\mathbb{C}}\int_{0}^{\infty}\Delta
f(\zeta)\bigg{(}\langle
v_{1}(\zeta,\eta)\rangle-\frac{1}{1+\eta}\bigg{)}\mathrm{d}\eta\,\mathrm{d}^{2}\zeta=\int_{\mathbb{C}}f\mathrm{d}\sigma$

as $n\to\infty$. This completes the proof of Corollary C.2. ∎

######  Corollary C.3.

Let $s$ and $a$ satisfy A7 and let $M^{(n)}$ be as in Lemma C.1. Fix
$\zeta\in\mathbb{C}$. Let $\rho_{\zeta}^{(n)}$ be the self-consistent density
of states associated with the Hermitization of $A_{n}+X_{n}$, i.e.

$\begin{split}\int_{\mathbb{R}}\frac{\rho_{\zeta}^{(n)}(\mathrm{d}\tau)}{\tau-w}=\frac{1}{2}\operatorname{Tr}\langle
M^{(n)}(\zeta,w)\rangle\end{split}$ (C.4)

for all $w\in\mathbb{C}_{+}$. Here, $\operatorname{Tr}$ denotes the trace of
$2\times 2$ matrices in $\mathbb{C}^{2\times 2}$ and the average
$\langle\mspace{2.0mu}\cdot\mspace{2.0mu}\rangle$ is taken entrywise. Then

  1. (i)

$\rho_{\zeta}^{(n)}\to\rho_{\zeta}$ weakly as $n\to\infty$.

  2. (ii)

$\limsup_{n\to\infty}\operatorname{supp}\rho_{\zeta}^{(n)}\subset\operatorname{supp}\rho_{\zeta}$.

###### Proof.

Item (i) follows directly from the convergence of the Stieltjes transforms,
i.e. for all $w\in\mathbb{C}_{+}$,
$\langle\operatorname{Tr}M^{(n)}(\zeta,w)\rangle\to\langle\operatorname{Tr}M(\zeta,w)\rangle$
as $n\to\infty$, due to Lemma C.1.

For the proof of (ii), we fix $\delta>0$ and show that
$\operatorname{supp}\rho_{\zeta}^{(n)}\subset\operatorname{supp}\rho_{\zeta}+(-\delta,\delta)$
for all sufficiently large $n$. If $\tau\in\mathbb{R}$ satisfies
$\operatorname{dist}(\tau,\operatorname{supp}\rho_{\zeta})\geq\delta$ then, by
[4, Lemma D.1], $M=M(\zeta,\tau)=\lim_{\eta\downarrow
0}M(\zeta,\tau+\mathrm{i}\eta)$ exits and is self-adjoint. Moreover, $\lVert
L^{-1}\rVert_{2}+\lVert M\rVert\lesssim_{\delta}1$ uniformly for $\eta\geq 0$
and $\tau\in\mathbb{R}$ with
$\operatorname{dist}(\tau,\operatorname{supp}\rho_{\zeta})\geq\delta$. We
recall the definition $L[R]=R-M\Sigma[R]M$ for $R\in\mathcal{B}^{2\times 2}$
from the proof of Lemma C.1. As $\lVert M\rVert\lesssim_{\delta}1$, Lemma C.1
implies $\lVert
M^{(n)}(\zeta,\tau+\mathrm{i}\eta)\rVert_{2}\lesssim_{\delta}1$ uniformly for
all $\eta>0$, $\tau\in\mathbb{R}$ with
$\operatorname{dist}(\tau,\operatorname{supp}\rho_{\zeta})\geq\delta$ and all
sufficiently large $n$. Arguing similarly as in the proof of Lemma 5.3, we
conclude $\lVert M^{(n)}(\zeta,\tau+\mathrm{i}\eta)\rVert\lesssim_{\delta}1$
uniformly for $\eta$, $\tau$ and $n$ as before. We set
$M^{(n)}:=M^{(n)}(\zeta,\tau+\mathrm{i}\eta)$ and
$L^{(n)}[R]=R-M^{(n)}\Sigma^{(n)}[R]M^{(n)}$ for $R\in\mathcal{B}^{2\times
2}$. For such $\eta$, $\tau$ and $n$, we obtain
$\lVert(L^{(n)})^{-1}\rVert_{2}\lesssim_{\delta}1$ by perturbation theory from
$\lVert M\rVert+\lVert M^{(n)}\rVert+\lVert
L^{-1}\rVert_{2}\lesssim_{\delta}1$,
$\lVert\Sigma^{(n)}\rVert_{2\to\infty}+\lVert\Sigma\rVert_{2\to\infty}\lesssim
1$ and $\lVert M^{(n)}-M\rVert_{2}\to 0$ for $n\to\infty$. Hence, by the
implicit function theorem, for all sufficiently large $n$, the function
$\eta\mapsto M^{(n)}(\zeta,\tau+\mathrm{i}\eta)$ is continuous on
$[\eta_{0}-\varepsilon,\eta_{0}+\varepsilon]$ for some $\varepsilon>0$
independent of $\eta_{0}>0$. In particular, we can extend $M^{(n)}$
continuously to $\eta=0$ in a unique way.

Let $\tau\in\mathbb{R}$ with
$\operatorname{dist}(\tau,\operatorname{supp}\rho_{\zeta})\geq\delta$. For
$M=M(\zeta,\tau)$, we now consider the relation

$L[\Delta]=\frac{1}{2}\big{(}K_{n}(\Delta,\widetilde{\Sigma},\widetilde{A})+K_{n}(\Delta^{*},\widetilde{\Sigma},\widetilde{A})^{*}\big{)}\,,\quad
K_{n}(\Delta):=M\Sigma[\Delta]\Delta+M\widetilde{\Sigma}[M+\Delta](M+\Delta)+M\widetilde{A}(M+\Delta)\,,$

with variables $\Delta\in\mathcal{B}^{2\times 2}$,
$\widetilde{A}=\widetilde{A}^{*}\in\mathcal{B}^{2\times 2}$,
$\widetilde{\Sigma}\colon\mathcal{B}^{2\times 2}\to\mathcal{B}^{2\times 2}$
such that $\widetilde{\Sigma}[R]^{*}=\widetilde{\Sigma}[R^{*}]$ for all
$R\in\mathcal{B}^{2\times 2}$. Since $\lVert
L^{-1}\rVert_{2}\lesssim_{\delta}1$, by the implicit function theorem, this
relation has a unique solution $\Delta$ as long as
$\lVert\widetilde{\Sigma}\rVert_{2}$ and $\lVert\widetilde{A}\rVert_{2}$ are
sufficiently small, as $L[0]=0$ and $K_{n}(0,0,0)=0$. Moreover, this solution
satisfies $\Delta=\Delta^{*}$ as $L[R]^{*}=L[R^{*}]$ for all
$R\in\mathcal{B}^{2\times 2}$ due to $M^{*}=M$. Owing to (C.2) and $M=M^{*}$,
we have
$L[M^{(n)}-M]=(K_{n}(M^{(n)}-M,\Sigma^{(n)}-\Sigma,A-A^{(n)})+K_{n}((M^{(n)}-M)^{*},\Sigma^{(n)}-\Sigma,A-A^{(n)}))/2$
with $M^{(n)}=M^{(n)}(\zeta,\tau)$. Hence, as
$\lVert\Sigma^{(n)}-\Sigma\rVert_{2}+\lVert A-A^{(n)}\rVert_{2}\to 0$ for
$n\to\infty$ by the proof of Lemma C.1, we get $\Delta=M^{(n)}-M$ for all
sufficiently large $n$ and, therefore, $M^{(n)}=(M^{(n)})^{*}$ for such $n$.
Since this holds for any $\delta>0$ we conclude that
$\operatorname{Im}M^{(n)}(\zeta,\tau+\omega)=0$ for sufficiently small
$\lvert\omega\rvert$ with $\omega\in\mathbb{R}$. Because of (C.4) this implies
that $\tau\not\in\operatorname{supp}\rho_{\zeta}^{(n)}$. ∎

##  Appendix D Representation of $\sigma$ as Brown measure

In Proposition D.1 of this appendix, we represent $\sigma$ from Proposition
6.1 as the Brown measure of an operator in a von Neumann algebra. This is the
motivation behind Definition 6.2.

The definition of a Brown measure is given after the next proposition, which
is formulated in the language of operator-valued free probability444For the
necessary definitions in free probability, we refer to the recent monograph
[36]..

######  Proposition D.1 (Representation of $\sigma$ as Brown measure).

Let $a\in\mathcal{B}$ and $s$ satisfy A5. Then there is an operator-valued
probability space $(\mathcal{A},E,\mathcal{B})$ and an operator
$\mathfrak{c}\in\mathcal{A}$ such that

  * •

$E\colon\mathcal{A}\to\mathcal{B}$ is a positive conditional expectation.

  * •

$(\mathcal{A},\langle E[\,\cdot\,]\rangle)$ is a tracial $W^{*}$-probability
space.

  * •

the Brown measure of $a+\mathfrak{c}$ on $(\mathcal{A},\langle
E(\cdot)\rangle)$ coincides with $\sigma$.

  * •

$\begin{pmatrix}0&\mathfrak{c}\\\ \mathfrak{c}^{*}&0\end{pmatrix}$ is an
operator-valued semi-circular element in the operator-valued probability space
$(\mathcal{A}^{2\times 2},\mathrm{id}\otimes E,\mathcal{B}^{2\times 2})$,
where we identified $\mathcal{A}^{2\times 2}=\mathbb{C}^{2\times
2}\otimes\mathcal{A}$, whose covariance is given by $\Sigma$ from (5.14). In
particular,

$M(\zeta,\mathrm{i}\eta)=\mathrm{id}\otimes
E\bigg{[}\begin{pmatrix}-\mathrm{i}\eta&\mathfrak{c}+a-\zeta\\\
(\mathfrak{c}+a-\zeta)^{*}&-\mathrm{i}\eta\end{pmatrix}^{-1}\bigg{]}$ (D.1)

satisfies the Matrix-Dyson equation, (5.13).

  * •

$E[\mathfrak{c}b\mathfrak{c}]=0$ for all $b\in\mathcal{B}$.

Before we show Proposition D.1, we introduce the notion of a Brown measure.
Let $(\mathcal{A},\tau)$ be a tracial $W^{*}$-probability space. The Brown
measure is a generalisation of the spectral distribution of normal operators
to non-normal ones. Let $\mathfrak{a}\in\mathcal{A}$. The _Brown measure_
$\mu_{\mathfrak{a}}$ of $\mathfrak{a}$ is the unique compactly supported
probability measure on $\mathbb{C}$ with

$\int_{\mathbb{C}}\log\lvert\zeta-\xi\rvert\mu_{\mathfrak{a}}(\mathrm{d}\xi)=\log
D(\mathfrak{a}-\zeta)$ (D.2)

for all $\zeta\in\mathbb{C}$. Here, $D(\mathfrak{a}-\zeta)$ denotes the
_Fuglede-Kadison determinant_ of $\mathfrak{a}-\zeta$. The Fuglede-Kadison
determinant of an arbitrary $\mathfrak{b}\in\mathcal{A}$ is defined by

$D(\mathfrak{b}):=\lim_{\varepsilon\downarrow
0}\exp(\tau(\log(\mathfrak{b}^{*}\mathfrak{b}+\varepsilon)^{1/2}))\in[0,\infty).$

Originally introduced in [18], the Brown measure was revived in [27]. An
introduction to the Brown measure and the Fuglede-Kadison determinant can be
found in [36, Chapter 11].

######  Proof of Proposition D.1.

As $\mathcal{B}$ is a commutative C∗-algebra, it is a standard result that $S$
and $S^{*}$ are completely positive maps (see e.g. [37, Theorem 3.9 or Theorem
3.11]). Hence, $\Sigma\colon\mathcal{B}^{2\times 2}\to\mathcal{B}^{2\times 2}$
from (5.14) is also a completely positive map. From the constructions in [45,
Sections 4.3 and 4.6], we obtain a von Neumann algebra $\widehat{\mathcal{A}}$
such that $\mathcal{B}^{2\times 2}\subset\widehat{\mathcal{A}}$ is a sub-von
Neumann algebra with the same unit as well as a positive conditional
expectation $\widehat{E}\colon\widehat{\mathcal{A}}\to\mathcal{B}^{2\times 2}$
and an operator-valued semicircular element
$\mathfrak{H}=\mathfrak{H}^{*}\in\widehat{\mathcal{A}}$ such that

$\widehat{E}[\mathfrak{H}B\mathfrak{H}]=\Sigma[B]$ (D.3)

for all $B\in\mathcal{B}^{2\times 2}$. A concise summary of this construction
is given in [41, Section 3.5]555We also refer to Lemma 8.2 and its proof in
the first arXiv-version of [4], which can be found at arXiv:1804.07752v1..

We recall the definitions of $E_{12}$ and $E_{21}\in\mathcal{B}^{2\times 2}$
from (5.20) and define $E_{11}$ and $E_{22}\in\mathcal{B}^{2\times 2}$
analogously. Let

$\mathcal{A}:=\mathrm{vN}(E_{1i}\mathfrak{A}E_{j1}\colon
i,j=1,2\,,\;\mathfrak{A}\in\widehat{\mathcal{A}})$

be the sub-von Neumann algebra of $\widehat{\mathcal{A}}$ generated by
$E_{1i}\mathfrak{A}E_{j1}$ for $i$, $j=1$, $2$ and
$\mathfrak{A}\in\widehat{\mathcal{A}}$. Note that $E_{11}\in\mathcal{A}$ is
the unit of $\mathcal{A}$. We introduce the map

$\Phi\colon\mathcal{A}^{2\times
2}\to\widehat{\mathcal{A}}\,,\qquad\left(\begin{array}[]{cc}\mathfrak{a}&\mathfrak{b}\\\
\mathfrak{c}&\mathfrak{d}\end{array}\right)\mapsto
E_{11}\mathfrak{a}E_{11}+E_{11}\mathfrak{b}E_{12}+E_{21}\mathfrak{c}E_{11}+E_{21}\mathfrak{d}E_{12},$

which is clearly an injective ∗-algebra homomorphism. Surjectivity follows
from

$\Phi\left(\begin{array}[]{cc}E_{11}\mathfrak{A}E_{11}&E_{11}\mathfrak{A}E_{21}\\\
E_{12}\mathfrak{A}E_{11}&E_{12}\mathfrak{A}E_{21}\end{array}\right)=E_{11}\mathfrak{A}E_{11}+E_{11}\mathfrak{A}E_{22}+E_{22}\mathfrak{A}E_{11}+E_{22}\mathfrak{A}E_{22}=\mathfrak{A}\,$

as $E_{11}+E_{22}$ is the unit of $\mathcal{B}^{2\times
2}\subset\widehat{\mathcal{A}}$. Moreover, $\Phi$ maps the unit of
$\mathcal{A}^{2\times 2}$ to the unit $\widehat{\mathcal{A}}$. Hence, $\Phi$
is bijective, unital ∗-algebra homomorphism between unital C∗-algebras, which
is consequently also isometric. We define $E\colon\mathcal{A}\to\mathcal{B}$
through setting $E[\mathfrak{a}]$ as the $(1,1)$-entry of
$\widehat{E}[\mathfrak{a}]\in\mathcal{B}^{2\times 2}$ for all
$\mathfrak{a}\in\mathcal{A}$. Then $E\colon\mathcal{A}\to\mathcal{B}$ is a
positive conditional expectation and, hence, $(\mathcal{A},E,\mathcal{B})$ is
an operator-valued probability space. It follows that $(\mathcal{A},\langle
E[\,\cdot\,]\rangle)$ is a tracial $W^{*}$-probability space666This is also
detailed in the proof of Lemma 8.2 in the first arXiv-version of [4],
available at arXiv:1804.07752v1..

Using $\Phi$ implicitly, we now identify $\mathfrak{H}$ with an element in
$\mathcal{A}^{2\times 2}$, i.e. we find $\mathfrak{a}$, $\mathfrak{b}$,
$\mathfrak{c}\in\mathcal{A}$ satisfying $\mathfrak{a}=\mathfrak{a}^{*}$ and
$\mathfrak{b}=\mathfrak{b}^{*}$ such that

$\mathfrak{H}=\begin{pmatrix}\mathfrak{a}&\mathfrak{c}\\\
\mathfrak{c}^{*}&\mathfrak{b}\end{pmatrix}.$

Moreover, by identifying $\mathcal{A}^{2\times 2}$ with $\mathbb{C}^{2\times
2}\otimes\mathcal{A}$, we obtain

$\mathrm{id}\otimes E\bigg{[}\begin{pmatrix}\mathfrak{a}&\mathfrak{c}\\\
\mathfrak{c}^{*}&\mathfrak{b}\end{pmatrix}\begin{pmatrix}r_{11}&r_{12}\\\
r_{21}&r_{22}\end{pmatrix}\begin{pmatrix}\mathfrak{a}&\mathfrak{c}\\\
\mathfrak{c}^{*}&\mathfrak{b}\end{pmatrix}\bigg{]}=\widehat{E}\bigg{[}\mathfrak{H}\begin{pmatrix}r_{11}&r_{12}\\\
r_{21}&r_{22}\end{pmatrix}\mathfrak{H}\bigg{]}=\begin{pmatrix}S[r_{22}]&0\\\
0&S^{*}[r_{11}]\end{pmatrix}$

for all $r_{11}$, $r_{12}$, $r_{21}$, $r_{22}\in\mathcal{B}$. This implies
$E[\mathfrak{a}\mathfrak{a}]=0$ and $E[\mathfrak{b}\mathfrak{b}]=0$, hence,
$\mathfrak{a}=\mathfrak{b}=0$ as $\mathfrak{a}=\mathfrak{a}^{*}$ and
$\mathfrak{b}=\mathfrak{b}^{*}$. Moreover,
$E[\mathfrak{c}r_{21}\mathfrak{c}]=0$ for all $r_{21}\in\mathcal{B}$.

Owing to the standard relation between the covariance of an operator-valued
semicircular element and its $R$-transform, see e.g. [36, Theorem 11 of
Chapter 9], we conclude that $M(\zeta,\mathrm{i}\eta)$ as defined in (D.1)
satisfies (5.13).

Given the above construction, it remains to show that the Brown measure of
$a+\mathfrak{c}$ coincides with $\sigma$. This proof proceeds analogously to
[6, proof of Proposition 2.9]. We explain the necessary replacements. First,
analogously to [6, proof of (5.28)], we obtain

$-L(\zeta)=\log D(a+\mathfrak{c}-\zeta)$

for all $\zeta\in\mathbb{C}$.

Proposition 6.3 and standard results from potential theory (cf. [7, Chapter
4.3]) imply

$\int_{\mathbb{C}}\log\lvert\zeta-\xi\rvert\sigma(\mathrm{d}\xi)=-L(\zeta)+h(\zeta)$

for all $\zeta\in\mathbb{C}$ and some harmonic function
$h\colon\mathbb{C}\to\mathbb{C}$. In the proof of Proposition 6.3, we showed
that $L(\zeta)=-\langle\log\lvert a-\zeta\rvert\rangle+C$ for all sufficiently
large $\zeta\in\mathbb{C}$ with some constant $C\in\mathbb{R}$, independent of
$\zeta$. An expansion of $v_{1}$ starting from (4.2) for large
$\lvert\zeta\rvert$ reveals that $C=0$. Hence, $h(\zeta)\to 0$ for
$\zeta\to\infty$ and, therefore, $h\equiv 0$ as it is harmonic. ∎

## References

  * [1] O. H. Ajanki, L. Erdős, and T. Krüger, _Stability of the matrix Dyson equation and random matrices with correlations_ , Probab. Theory Related Fields 173 (2019), no. 1-2, 293–373.
  * [2] S. Allesina and S. Tang, _The stability–complexity relationship at age 40: a random matrix perspective_ , Population Ecology 57 (2015), no. 1, 63–75.
  * [3] J. Alt, L. Erdős, and T. Krüger, _Local inhomogeneous circular law_ , Ann. Appl. Probab. 28 (2018), no. 1, 148–203.
  * [4] J. Alt, L. Erdős, and T. Krüger, _The Dyson equation with linear self-energy: spectral bands, edges and cusps_ , Doc. Math. 25 (2020), 1421–1539. MR 4164728
  * [5] J. Alt, L. Erdős, T. Krüger, and Yu. Nemish, _Location of the spectrum of Kronecker random matrices_ , Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019), no. 2, 661–696. MR 3949949
  * [6] J. Alt and T. Krüger, _Inhomogeneous circular law for correlated matrices_ , J. Funct. Anal. 281 (2021), no. 7, Paper No. 109120, 73\. MR 4271784
  * [7] D. H. Armitage and S. J. Gardiner, _Classical potential theory_ , Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001.
  * [8] Z. D. Bai, _Circular law_ , Ann. Probab. 25 (1997), no. 1, 494–529.
  * [9] A. Basak, E. Paquette, and O. Zeitouni, _Regularization of non-normal matrices by Gaussian noise—the banded Toeplitz and twisted Toeplitz cases_ , Forum Math. Sigma 7 (2019), Paper No. e3, 72. MR 3912376
  * [10] S. Belinschi, Z. Yin, and P. Zhong, _The Brown measure of a sum of two free random variables, one of which is triangular elliptic_ , Adv. Math. 441 (2024), Paper No. 109562. MR 4710866
  * [11] S. T. Belinschi, P. Śniady, and R. Speicher, _Eigenvalues of non-Hermitian random matrices and Brown measure of non-normal operators: Hermitian reduction and linearization method_ , Linear Algebra Appl. 537 (2018), 48–83. MR 3716236
  * [12] H. Bercovici and P. Zhong, _The Brown measure of a sum of two free random variables, one of which is R-diagonal_ , preprint (2022), arXiv:2209.12379.
  * [13] P. Biane and F. Lehner, _Computation of some examples of Brown’s spectral measure in free probability_ , Colloq. Math. 90 (2001), no. 2, 181–211. MR 1876844
  * [14] G. Birkhoff and R. S. Varga, _Reactor criticality and nonnegative matrices_ , J. Soc. Indust. Appl. Math. 6 (1958), 354–377. MR 100984
  * [15] C. Bordenave and M. Capitaine, _Outlier eigenvalues for deformed i.i.d. random matrices_ , Comm. Pure Appl. Math. 69 (2016), no. 11, 2131–2194. MR 3552011
  * [16] C. Bordenave, P. Caputo, and D. Chafa\̈mathrm{i}, _Spectrum of Markov generators on sparse random graphs_ , Comm. Pure Appl. Math. 67 (2014), no. 4, 621–669. MR 3168123
  * [17] C. Bordenave and D. Chafa\̈mathrm{i}, _Around the circular law_ , Probab. Surv. 9 (2012), 1–89. MR 2908617
  * [18] L. G. Brown, _Lidskiui’s theorem in the type II case_ , Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., Harlow, 1986, pp. 1–35.
  * [19] N. Cook, W. Hachem, J. Najim, and D. Renfrew, _Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs_ , Electron. J. Probab. 23 (2018), Paper No. 110, 61.
  * [20] N. Cook, W. Hachem, J. Najim, and D. Renfrew, _Non-Hermitian random matrices with a variance profile (II): Properties and examples_ , J. Theoret. Probab. 35 (2022), no. 4, 2343–2382. MR 4509071
  * [21] B. K. Driver, B. Hall, and T. Kemp, _The Brown measure of the free multiplicative Brownian motion_ , Probab. Theory Related Fields 184 (2022), no. 1-2, 209–273. MR 4498510
  * [22] L. Erdős and H. C. Ji, _Density of Brown measure of free circular Brownian motion_ , preprint (2023), arXiv:2307.08626.
  * [23] L. Erdős, T. Krüger, and D. Renfrew, _Power law decay for systems of randomly coupled differential equations_ , SIAM J. Math. Anal. 50 (2018), no. 3, 3271–3290. MR 3816180
  * [24] Y. V. Fyodorov and H.-J. Sommers, _Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time-reversal invariance_ , vol. 38, 1997, Quantum problems in condensed matter physics, pp. 1918–1981. MR 1450906
  * [25] V. L. Girko, _The circular law_ , Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 669–679. MR 773436
  * [26] A. Guionnet, P.M. Wood, and O. Zeitouni, _Convergence of the spectral measure of non-normal matrices_ , Proc. Amer. Math. Soc. 142 (2014), no. 2, 667–679. MR 3134007
  * [27] U. Haagerup and F. Larsen, _Brown’s spectral distribution measure for $R$-diagonal elements in finite von Neumann algebras_, J. Funct. Anal. 176 (2000), no. 2, 331–367.
  * [28] B. C. Hall and C.-W. Ho, _The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element_ , Lett. Math. Phys. 112 (2022), no. 2, Paper No. 19, 61. MR 4392479
  * [29] H. M. Hastings, F. Juhasz, and M. A. Schreiber, _Stability of structured random matrices_ , Proceedings: Biological Sciences 249 (1992), no. 1326, 223–225.
  * [30] J. W. Helton, R. Rashidi Far, and R. Speicher, _Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints_ , Int. Math. Res. Not. IMRN (2007), no. 22, Art. ID rnm086, 15. MR 2376207
  * [31] C.-W. Ho and P. Zhong, _Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions_ , J. Eur. Math. Soc. (JEMS) 25 (2023), no. 6, 2163–2227. MR 4592867
  * [32] V. Jain and S. Silwal, _A note on the universality of ESDs of inhomogeneous random matrices_ , ALEA Lat. Am. J. Probab. Math. Stat. 18 (2021), no. 2, 1047–1059. MR 4282181
  * [33] B. Khoruzhenko, _Large- $N$ eigenvalue distribution of randomly perturbed asymmetric matrices_, J. Phys. A 29 (1996), no. 7, L165–L169. MR 1395506
  * [34] G. V. Livshyts, K. Tikhomirov, and R. Vershynin, _The smallest singular value of inhomogeneous square random matrices_ , Ann. Probab. 49 (2021), no. 3, 1286–1309. MR 4255145
  * [35] R. M. May, _Will a large complex system be stable?_ , Nature 238 (1972), 413–414.
  * [36] J. A. Mingo and R. Speicher, _Free probability and random matrices_ , Fields Institute Monographs, vol. 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017. MR 3585560
  * [37] V. Paulsen, _Completely bounded maps and operator algebras_ , Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
  * [38] K. Rajan and L. F. Abbott, _Eigenvalue spectra of random matrices for neural networks_ , Phys. Rev. Lett. 97 (2006), 188104.
  * [39] L. Reichel and L.N. Trefethen, _Eigenvalues and pseudo-eigenvalues of Toeplitz matrices_ , Linear Algebra Appl. 162/164 (1992), 153–185, Directions in matrix theory (Auburn, AL, 1990). MR 1148398
  * [40] E. B. Saff and V. Totik, _Logarithmic potentials with external fields_ , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom. MR 1485778
  * [41] D. Shlyakhtenko, _Random Gaussian band matrices and freeness with amalgamation_ , Internat. Math. Res. Notices (1996), no. 20, 1013–1025. MR 1422374
  * [42] J. Sjöstrand and M. Vogel, _General Toeplitz matrices subject to Gaussian perturbations_ , Ann. Henri Poincaré 22 (2021), no. 1, 49–81. MR 4201590
  * [43] P. Śniady, _Random regularization of Brown spectral measure_ , J. Funct. Anal. 193 (2002), no. 2, 291–313. MR 1929504
  * [44] H. Sompolinsky, A. Crisanti, and H.-J. Sommers, _Chaos in random neural networks_ , Phys. Rev. Lett. 61 (1988), no. 3, 259–262.
  * [45] R. Speicher, _Combinatorial theory of the free product with amalgamation and operator-valued free probability theory_ , Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898
  * [46] T. Tao and V. Vu, _Random matrices: universality of local spectral statistics of non-Hermitian matrices_ , Ann. Probab. 43 (2015), no. 2, 782–874. MR 3306005
  * [47] T. Tao, V. Vu, and M. Krishnapur, _Random matrices: Universality of ESDs and the circular law_ , Ann. Probab. 38 (2010), no. 5, 2023–2065.
  * [48] L. N. Trefethen and M. Embree, _Spectra and pseudospectra_ , Princeton University Press, Princeton, NJ, 2005, The behavior of nonnormal matrices and operators. MR 2155029
  * [49] P. Zhong, _Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra_ , preprint (2021), arXiv:2108.09844.