$\displaystyle\widehat{\mathcal{Q}}\,\Big{[}S\otimes T\Big{]}$
$\displaystyle=\sum_{i,j,\alpha,\beta=1}^{n}s_{ij}t_{\alpha\beta}\,\Phi\,\big{(}\mathcal{Q}\otimes\mathrm{Id}_{n}\big{)}\,\Phi\,\Big{[}E_{ij}\otimes
E_{\alpha\beta}\Big{]}$
$\displaystyle=\sum_{i,j,\alpha,\beta=1}^{n}s_{ij}t_{\alpha\beta}\,\Phi\,\big{(}\mathcal{Q}\otimes\mathrm{Id}_{n}\big{)}\,\Big{[}E_{i\beta}\otimes
E_{\alpha j}\Big{]}$
$\displaystyle=\frac{1}{\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,\alpha,\beta,m=1}^{n}s_{ij}t_{\alpha\beta}\,\langle\operatorname{\mathrm{Im}}M_{0},E_{i\beta}\rangle\,\Phi\,\Big{(}\ell_{m}\ell_{m}^{t}\otimes
E_{\alpha j}\Big{)}$
$\displaystyle=\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,\alpha,\beta,m}s_{ij}t_{\alpha\beta}\,\Big{(}\ell_{m}e_{j}^{t}\otimes
e_{\alpha}e_{\beta}^{t}\operatorname{\mathrm{Im}}M_{0}\,e_{i}\,\ell_{m}^{t}\Big{)}$
$\displaystyle=\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,m}s_{ij}\,\Big{(}\ell_{m}e_{j}^{t}\otimes
T\operatorname{\mathrm{Im}}M_{0}\,e_{i}\,\ell_{m}^{t}\Big{)}.$

After evaluating the above operator at $\varphi(B)$ and taking $\varphi^{-1}$,
we get

$\displaystyle\varphi^{-1}\bigg{(}\widehat{\mathcal{Q}}\,\Big{[}S\otimes
T\Big{]}\varphi\big{(}B\big{)}\bigg{)}$
$\displaystyle=\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,m=1}^{n}s_{ij}\,\varphi^{-1}\bigg{(}\,\Big{(}\ell_{m}e_{j}^{t}\otimes
T\operatorname{\mathrm{Im}}M_{0}e_{i}\,\ell_{m}^{t}\Big{)}\varphi\big{(}B\big{)}\bigg{)}$
$\displaystyle=\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,m=1}^{n}s_{ij}\,\ell_{m}e_{j}^{t}B\ell_{m}e_{i}^{t}\operatorname{\mathrm{Im}}M_{0}T^{t}$
$\displaystyle=\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\sum_{i,j,m=1}^{n}s_{ij}\,\ell_{m}\ell_{m}^{t}B^{t}e_{j}e_{i}^{t}\operatorname{\mathrm{Im}}M_{0}T^{t}$
$\displaystyle=\frac{\operatorname{\mathrm{Im}}M_{0}\,B^{t}\,S^{t}\,\operatorname{\mathrm{Im}}M_{0}\,T^{t}}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}},$

where in the second step we used the definition of $\varphi$ in (4.6).

Now we apply the above formula with $S=G_{lk}(z)$ and $T=G_{kl}(\zeta)$, and
use that $(G_{kl})^{t}=G_{lk}$ together with the linearity of $\varphi$ and
$\widehat{\mathcal{Q}}$, to rewrite (4.17) as

$\displaystyle\varphi^{-1}\bigg{(}\widehat{\mathcal{Q}}\,\Big{[}\frac{1}{N}\sum_{k,l=1}^{N}G_{lk}(z)\otimes
G_{kl}(\zeta)\Big{]}\varphi\big{(}B\big{)}\bigg{)}$
$\displaystyle=\frac{\operatorname{\mathrm{Im}}M_{0}}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\,B^{t}\frac{1}{N}\sum_{k,l=1}^{N}G_{kl}(z)\operatorname{\mathrm{Im}}M_{0}G_{lk}(\zeta)$
$\displaystyle=\frac{\operatorname{\mathrm{Im}}M_{0}}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\,B^{t}\,G^{\operatorname{\mathrm{Im}}M_{0}}(z,\zeta).$

Lemma 3.8 implies that

$\frac{1}{n\|\operatorname{\mathrm{Im}}M_{0}\|_{\textrm{HS}}^{2}}\operatorname{\mathrm{Im}}M_{0}\,B^{t}\,G^{\operatorname{\mathrm{Im}}M_{0}}(z,\zeta)=\vartheta\,\frac{2\mathrm{i}\,}{z-\zeta}\frac{\operatorname{\mathrm{Im}}M_{0}B^{\,t}\operatorname{\mathrm{Im}}M_{0}}{n\langle\operatorname{\mathrm{Im}}M_{0}\rangle}+\mathcal{E}^{B}(z,\zeta),$

where
$\mathbb{E}[\|\mathcal{E}^{B}(z,\zeta)\|]=O_{\prec}\Big{(}1+N^{-1}\widehat{\eta}^{\,-2}+\|B\|\big{(}1+N^{-1/2}\widehat{\eta}^{\,-3/2}\big{)}\Big{)}$,
and $\vartheta=1$ for $(z,\zeta)\in\Omega^{+}\times\Omega^{-}$ and
$\vartheta=-1$ for $(z,\zeta)\in\Omega^{-}\times\Omega^{+}$. This, together
with the estimate of the last term in (4.15), completes the proof of the
lemma. ∎

Now we state and prove the analog of Lemma 3.4 for $\beta=1$.

######  Lemma 4.2.

Let $\gamma\in(0,1)$ and
$\tau\in\big{(}0,\min\\{\gamma,(1-\gamma)\\}/7\big{)}$. Then

$\frac{d}{dt}\mathbb{E}[\mathfrak{e}(t)]=-\frac{t}{\pi^{2}}\int\displaylimits_{\Omega\times\Omega}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,(S_{ij}+\widetilde{S}_{ij})\Big{)}\mathfrak{e}(t)\Big{]}d^{2}\\!\zeta
d^{2}\\!z+\mathcal{E}$ (4.18)

where $|\mathcal{E}|\prec N^{-\tau}$, and

$\displaystyle
S_{ij}(z,\zeta):=\frac{1}{N}\sum_{k,l=1}^{N}G_{lk}(z)\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\,\Gamma[E_{ij}]G_{kl}(\zeta),$
(4.19)
$\displaystyle\widetilde{S}_{ij}(z,\zeta):=\frac{1}{N}\sum_{k,l=1}^{N}G_{lk}(z)\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\,\widetilde{\Gamma}[E_{ij}]G_{lk}(\zeta).$
(4.20)

###### Proof.

Recall that the operator $\mathscr{S}^{(1)}$ was defined in (4.1). We keep the
notation
$\mathscr{S}[\bm{R}]=\Gamma\Big{[}\frac{1}{N}\sum_{j=1}^{N}R_{jj}\Big{]}\otimes\bm{I}_{N}$
as in Section 3, and denote
$\widetilde{\mathscr{S}}[\bm{R}]:=\frac{1}{N}\sum_{i,j=1}^{N}\widetilde{\Gamma}[R_{ji}]\otimes\bm{E}_{ij}$,
so that

$\mathscr{S}^{(1)}=\mathscr{S}+\widetilde{\mathscr{S}}.$ (4.21)

The lines (3.25)-(3.30) are repeated exactly as in the proof of Lemma 3.4 with
$\mathscr{S}$ replaced by $\mathscr{S}^{(1)}$. Then we apply the decomposition
(4.21) on the right-hand side of (3.32) and treat the part containing
$\widetilde{\mathscr{S}}$ as an additional error term. We end up with the
equation

$\displaystyle-\mathbb{E}\Big{[}\mathfrak{e}(t)\big{(}1-\mathbb{E}\big{)}\big{[}\mathscr{S}^{(1)}[\bm{G}(z)]\bm{G}(z)\big{]}\Big{]}=\mathbb{E}\Big{[}\mathfrak{e}(t)\Big{(}\frac{1}{\bm{M}(z)}+z\bm{I}_{nN}-\bm{K}_{0}\Big{)}\big{(}1-\mathbb{E}\big{)}\bm{G}(z)\Big{]}$
$\displaystyle-\mathbb{E}\Big{[}\mathfrak{e}(t)\big{(}1-\mathbb{E}\big{)}\big{[}\mathscr{S}[\bm{G}(z)]\bm{M}(z)\big{]}\Big{]}+\mathcal{E}_{3}(z,\zeta)+\widetilde{\mathcal{E}}_{3}(z,\zeta),$

where $\mathcal{E}_{3}$ is defined in (3.33) and the second error term above
is given by

$\widetilde{\mathcal{E}}_{3}(z,\zeta):=-\mathbb{E}\Big{[}\mathfrak{e}(t)\big{(}1-\mathbb{E}\big{)}\big{[}\widetilde{\mathscr{S}}[\bm{G}(z)]\bm{G}(z)\big{]}\Big{]}.$

After using (3.29) (with $\mathscr{S}^{(1)}$ instead of $\mathscr{S}$) and
(3.26) we obtain the same equation as in (3.34) with an additional error term

$\mathscr{B}_{z}\,\mathbb{E}\Big{[}\mathfrak{e}(t)\big{(}1-\mathbb{E}\big{)}\big{[}\bm{G}\big{]}\Big{]}=-\mathbb{E}\Big{[}\widetilde{\bm{W}}\bm{G}\,\nabla_{\widetilde{\bm{W}}}\big{(}\mathfrak{e}(t)\big{)}\Big{]}\frac{1}{\bm{M}}-\Big{(}\mathcal{E}_{2}+\mathcal{E}_{3}+\widetilde{\mathcal{E}}_{3}\Big{)}\frac{1}{\bm{M}}.$
(4.22)

We see that the operator $\widetilde{\mathscr{S}}$ does not explicitly appear
in leading expressions of (4.22) anymore and the terms have exactly the same
form as in (3.34). Therefore, we repeat (3.34)-(3.41) in the proof of Lemma
3.4 line by line with the error term $\mathcal{E}_{3}$ replaced by
$\mathcal{E}_{3}+\widetilde{\mathcal{E}}_{3}$. The operator
$\mathscr{S}^{(1)}$ reappears again when we repeat the computations in
(3.42)-(3.44). After taking the partial expectation with respect to
$\widetilde{\bm{W}}$, the operator $\mathscr{S}$ in (3.44) is replaced by
$\mathscr{S}^{(1)}=\mathscr{S}+\widetilde{\mathscr{S}}$. More precisely, we
have

$\displaystyle\mathbb{E}\Big{[}\operatorname{{\mathrm{Tr}}}\Big{(}\mathscr{B}_{z}^{-1}\Big{[}\widetilde{\bm{W}}\bm{G}(z)\frac{1}{\bm{M}(z)}\Big{]}\Big{)}\operatorname{{\mathrm{Tr}}}\Big{(}\widetilde{\bm{W}}\bm{G}(\zeta)\Big{)}\mathfrak{e}(t)\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\sum_{p,q=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}(E_{ji}\otimes\bm{E}_{qp})\bm{G}(z)\frac{1}{\bm{M}(z)}\big{(}(\mathscr{B}_{z}^{*})^{-1}[\bm{I}_{nN}]\big{)}^{*}\mathscr{S}^{(1)}[E_{ij}\otimes\bm{E}_{pq}]\,\bm{G}(\zeta)\Big{)}\mathfrak{e}(t)\Big{]}.$

By applying the decomposition (4.21) and using (3.47)) we rewrite the term
containing $\mathscr{S}$ as

$\displaystyle\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\sum_{p,q=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}(E_{ji}\otimes\bm{E}_{qp})\bm{G}(z)\frac{1}{\bm{M}(z)}\big{(}(\mathscr{B}_{z}^{*})^{-1}[\bm{I}_{nN}]\big{)}^{*}\mathscr{S}[E_{ij}\otimes\bm{E}_{pq}]\,\bm{G}(\zeta)\Big{)}\mathfrak{e}(t)\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad=\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\sum_{k,l=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}G_{lk}(z)\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\Gamma[E_{ij}]G_{kl}(\zeta)\Big{)}\mathfrak{e}(t)\frac{1}{N}\Big{]},$

which gives $S_{ij}(z,\zeta)$ in (4.18). In order to obtain the term in (4.18)
that contains $\widetilde{S}_{ij}(z,\zeta)$, we notice that

$\widetilde{\mathscr{S}}\big{[}E_{ij}\otimes\bm{E}_{pq}\big{]}=\frac{1}{N}\widetilde{\Gamma}[E_{ij}]\otimes\bm{E}_{qp}$

for any $E_{ij}\otimes\bm{E}_{pq}$. Therefore,

$\displaystyle\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\sum_{p,q=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}(E_{ji}\otimes\bm{E}_{qp})\bm{G}(z)\frac{1}{\bm{M}(z)}\big{(}(\mathscr{B}_{z}^{*})^{-1}[\bm{I}_{nN}]\big{)}^{*}\widetilde{\mathscr{S}}[E_{ij}\otimes\bm{E}_{pq}]\,\bm{G}(\zeta)\Big{)}\mathfrak{e}(t)\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\frac{\mathfrak{e}(t)}{N}\sum_{i,j=1}^{n}\sum_{p,q=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}(E_{ji}\otimes\bm{E}_{qp})\bm{G}(z)\frac{1}{\bm{M}(z)}\big{(}(\mathscr{B}_{z}^{*})^{-1}[\bm{I}_{nN}]\big{)}^{*}\big{(}\,\widetilde{\Gamma}[E_{ij}]\otimes\bm{E}_{qp}\big{)}\,\bm{G}(\zeta)\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\frac{\mathfrak{e}(t)}{N}\sum_{i,j=1}^{n}\sum_{k,l=1}^{N}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}G_{lk}(z)\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\widetilde{\Gamma}[E_{ij}]G_{lk}(\zeta)\Big{)}\Big{]},$

which gives rise to the summand containing $\widetilde{S}_{ij}(z,\zeta)$ from
(4.20) in (4.18).

It remains to estimate the error term
$\mathcal{E}=\mathcal{E}_{1}+\widetilde{\mathcal{E}}_{1}$, where
$\widetilde{\mathcal{E}}_{1}$ is defined through (3.40) and (3.36) with
$\mathcal{E}_{3}$ replaced by $\widetilde{\mathcal{E}}_{3}$. The error term
$\mathcal{E}_{1}$ satisfies the same bound as in Lemma 3.4, namely
$|\mathcal{E}_{1}|\prec N^{-\tau}$. For $\widetilde{\mathcal{E}}_{1}$ we see
that

$\displaystyle(\mathrm{Id}_{n}\otimes\operatorname{{\mathrm{Tr}}}_{N})\Big{[}\widetilde{\mathscr{S}}[\bm{G}(z)]\bm{G}(z)\Big{]}=\frac{1}{N}\sum_{k,l=1}^{N}\widetilde{\Gamma}[G_{lk}(z)]G_{lk}(z)$
$\displaystyle\qquad\qquad\qquad\qquad=\sum_{\alpha=1}^{d}\Big{(}L_{\alpha}\frac{1}{N}\sum_{k,l=1}^{N}G_{lk}(z)L_{\alpha}G_{lk}(z)+L_{\alpha}^{\,t}\frac{1}{N}\sum_{k,l=1}^{N}G_{lk}(z)L_{\alpha}^{\,t}G_{lk}(z)\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad=\sum_{\alpha=1}^{d}\Big{(}L_{\alpha}\,\widetilde{G}^{L_{\alpha}}(z,z)+L_{\alpha}^{\,t}\,\widetilde{G}^{L_{\alpha}^{t}}(z,z)\Big{)}.$

From the first statement in Lemma 4.1 applied to
$\widetilde{G}^{L_{\alpha}}(z,z)$ and $\widetilde{G}^{L^{t}_{\alpha}}(z,z)$ we
have

$\mathbb{E}\bigg{[}\Big{\|}(\mathrm{Id}_{n}\otimes\operatorname{{\mathrm{Tr}}}_{N})\Big{[}\widetilde{\mathscr{S}}[\bm{G}(z)]\bm{G}(z)\Big{]}\Big{\|}\bigg{]}=O_{\prec}\Big{(}1+\frac{1}{N^{1/2}|\operatorname{\mathrm{Im}}z|^{\,3/2}}\Big{)},$

and thus (see (3.36))

$\Big{\|}\mathcal{B}_{z}^{-1}(\mathrm{Id}_{n}\otimes\operatorname{{\mathrm{Tr}}}_{N})\Big{[}\widetilde{\mathcal{E}}_{3}\frac{1}{\bm{M}(z)}\Big{]}\Big{\|}=O_{\prec}\Big{(}1+\frac{1}{N^{1/2}|\operatorname{\mathrm{Im}}z|^{3/2}}\Big{)}$

uniformly on $\Omega\times\Omega$. Finally, integrating the trace of the above
expression as in (3.52) yields the estimate

$|\widetilde{\mathcal{E}}_{1}|\prec
N^{3\tau/2}\big{(}\|g\|_{1}+\|g^{\prime}\|_{1}\big{)}\frac{1}{(N\eta_{0})^{1/2}}.$

Choosing $\tau$ such that $1-\gamma>7\tau$ ensures that both
$|\mathcal{E}_{1}|$ and $|\widetilde{\mathcal{E}}_{1}|$ are bounded by
$N^{-\tau}$. ∎

###  4.3 Proof of Theorem 2.1 for $\beta=1$

With Lemma 4.2 established, we proceed with the proof of Theorem 2.1 for
$\beta=1$. Consider the integral

$\mathcal{V}=\frac{1}{\pi^{2}}\int_{\Omega\times\Omega}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,S_{ij}(z,\zeta)+E_{ji}\widetilde{S}_{ij}(z,\zeta)\Big{)}\mathfrak{e}(t)\Big{]}d^{2}\\!\zeta
d^{2}\\!z$ (4.23)

appearing in (4.18) in Lemma 4.2 with $S_{ij}(z,\zeta)$ and
$\widetilde{S}_{ij}(z,\zeta)$ defined in (4.19) and (4.20). Using the notation
introduced in (4.2) we can write $S_{ij}(z,\zeta)$ and
$\widetilde{S}_{ij}(z,\zeta)$ in (4.23) as

$S_{ij}(z,\zeta)=G^{B_{ij}(z)}(z,\zeta),\quad\widetilde{S}_{ij}(z,\zeta)=\widetilde{G}^{\widetilde{B}_{ij}(z)}(z,\zeta)$
(4.24)

with

$B_{ij}(z):=\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\,\Gamma[E_{ij}],\quad\widetilde{B}_{ij}(z):=\frac{1}{M(z)}\mathcal{B}_{z}^{-1}[I_{n}]\,\widetilde{\Gamma}[E_{ij}].$

Similarly as in (3.109), we have

$B_{ij}(z)=B_{ij}(E_{0})+O(|\operatorname{\mathrm{Im}}z|),\qquad\widetilde{B}_{ij}(z)=\widetilde{B}_{ij}(E_{0})+O(|\operatorname{\mathrm{Im}}z|).$
(4.25)

It follows from the argument in (3.113)-(3.116) that if for some
$\dagger,\star\in\\{-,+\\}$ the function $h$ is analytic on
$\Omega^{\dagger}\times\Omega^{\star}$ and satisfies on this set the bound
$|h(z,\zeta)|\prec|\operatorname{\mathrm{Im}}z|^{-\alpha}|\operatorname{\mathrm{Im}}\zeta|^{-\beta}$,
then

$\bigg{|}\frac{1}{\pi^{2}}\int_{\Omega^{\dagger}\times\Omega^{\star}}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}h(z,\zeta)d^{2}\\!\zeta
d^{2}\\!z\,\bigg{|}\prec\frac{N^{(\alpha+\beta+1)\tau}}{\eta_{0}^{\alpha+\beta-1}}.$
(4.26)

Now, following (3.110)-(3.111), we split the integral over
$\Omega\times\Omega$ into four parts

$\mathcal{V}=\mathcal{V}^{\,(+,+)}+\mathcal{V}^{\,(+,-)}+\mathcal{V}^{\,(-,+)}+\mathcal{V}^{\,(-,-)}$

determined by the signs of $\operatorname{\mathrm{Im}}z$ and
$\operatorname{\mathrm{Im}}\zeta$. For $\mathfrak{e}=\mathfrak{e}(t)$, denote

$h_{1}(z,\zeta):=\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}G^{B_{ij}(z)}(z,\zeta)\Big{)}\mathfrak{e}\Big{]},\,\,\,\tilde{h}_{1}(z,\zeta):=\mathbb{E}\Big{[}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,\widetilde{G}^{\widetilde{B}_{ij}(z)}(z,\zeta)\Big{)}\mathfrak{e}\Big{]}.$

Using the representation (4.24) together with the bounds (3.98), (4.3) and
(4.25) we obtain that

$|h_{1}(z,\zeta)|+|h_{2}(z,\zeta)|\prec
1+\frac{1}{N^{1/2}\widehat{\eta}^{\,3/2}}$

uniformly on
$(z,\zeta)\in\big{(}\Omega^{+}\times\Omega^{+}\big{)}\cup\big{(}\Omega^{-}\times\Omega^{-}\big{)}$.
Then (4.26) implies that

$\big{|}\mathcal{V}^{\,(+,+)}\big{|}+\big{|}\mathcal{V}^{\,(-,-)}\big{|}\prec
N^{\tau}\eta_{0}+\frac{N^{5/2\tau}}{(N\eta_{0})^{1/2}}.$

Taking $\gamma\in(0,1)$ and $\tau\in(0,\min\\{(1-\gamma)/7,\gamma/2\\})$
ensures that
$\big{|}\mathcal{V}^{\,(+,+)}\big{|}+\big{|}\mathcal{V}^{\,(-,-)}\big{|}\prec
N^{-\tau}$.

We compute $\mathcal{V}^{\,(+,-)}$ and $\mathcal{V}^{\,(-,+)}$ in three steps.
First, we split the integrands in (4.23) with the identities (4.24) into
leading and error terms using the approximations (3.102)-(3.103) and
(4.4)-(4.5). The integrals of the _error_ terms that arise from these
decompositions are treated analogously to $\mathcal{V}^{\,(+,+)}$ and
$\mathcal{V}^{\,(-,-)}$. Indeed, if we replace $h(z,\zeta)$ in (4.26) with the
error terms from (3.102)-(3.103) and (4.4)-(4.5), and take $\gamma\in(0,1)$
and $\tau\in(0,\min\\{(1-\gamma)/7,\gamma/2\\})$, then the resulting integrals
are of order $O_{\prec}(N^{-\tau})$. Therefore, we can replace
$S_{ij}(z,\zeta)$ and $\widetilde{S}_{ij}(z,\zeta)$ in (4.23) by their
deterministic approximations defined in (3.102) and (4.4).

In the second step, exactly as in (3.120)-(3.121), we use (4.25) to replace
$B_{ij}(z)$ and $\widetilde{B}_{ij}(z)$ in the deterministic approximations of
$S_{ij}(z,\zeta)$ and $\widetilde{S}_{ij}(z,\zeta)$ with $B_{ij}(E_{0})$ and
$\widetilde{B}_{ij}(E_{0})$ correspondingly. After integrating, under the
assumption that $\gamma\in(0,1)$ and
$\tau\in(0,\min\\{(1-\gamma)/7,\gamma/2\\})$, this replacement gives a term of
size $O_{\prec}(N^{-\tau})$.

After applying the simplifications from the first two steps, we arrive at

$\displaystyle\mathcal{V}^{\,(+,-)}$
$\displaystyle=\frac{\mathbb{E}\big{[}\mathfrak{e}(t)\big{]}}{\pi^{2}}\int\displaylimits_{\Omega^{+}\times\Omega^{-}}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\frac{1}{z-\zeta}\Big{(}\phi^{\,(+,-)}+\tilde{\phi}^{\,(+,-)}\Big{)}\,d^{2}\\!\zeta
d^{2}\\!z+O_{\prec}(N^{-\tau}),$ $\displaystyle\mathcal{V}^{\,(-,+)}$
$\displaystyle=\frac{\mathbb{E}\big{[}\mathfrak{e}(t)\big{]}}{\pi^{2}}\int\displaylimits_{\Omega^{-}\times\Omega^{+}}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\frac{1}{z-\zeta}\,\Big{(}\phi^{\,(-,+)}+\tilde{\phi}^{\,(-,+)}\Big{)}d^{2}\\!\zeta
d^{2}\\!z+O_{\prec}(N^{-\tau}),$

where we introduce the $(z,\zeta)$-independent constants

$\displaystyle\phi^{\,(+,-)}$
$\displaystyle=\frac{2\mathrm{i}\,}{\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,\Big{\langle}\operatorname{\mathrm{Im}}M_{0},\,\frac{1}{M_{0}}M_{0}^{\prime}\,\Gamma[E_{ij}]\,\Big{\rangle}\operatorname{\mathrm{Im}}M_{0}\Big{)},$
(4.27) $\displaystyle\tilde{\phi}^{\,(+,-)}$
$\displaystyle=\frac{2\mathrm{i}\,}{n\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\operatorname{\mathrm{Im}}M_{0}\,\Big{(}\frac{1}{M_{0}}M_{0}^{\prime}\,\widetilde{\Gamma}[E_{ij}]\Big{)}^{\,t}\operatorname{\mathrm{Im}}M_{0}\Big{)},$
(4.28) $\displaystyle\phi^{\,(-,+)}$
$\displaystyle=\frac{-2\mathrm{i}\,}{\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,\Big{\langle}\operatorname{\mathrm{Im}}M_{0},\,\frac{1}{M_{0}^{*}}(M_{0}^{\prime})^{*}\,\Gamma[E_{ij}]\,\Big{\rangle}\operatorname{\mathrm{Im}}M_{0}\Big{)},$
(4.29) $\displaystyle\tilde{\phi}^{\,(-,+)}$
$\displaystyle=\frac{-2\mathrm{i}\,}{n\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\operatorname{\mathrm{Im}}M_{0}\Big{(}\frac{1}{M_{0}^{*}}(M_{0}^{\prime})^{*}\,\widetilde{\Gamma}[E_{ij}]\Big{)}^{\,t}\operatorname{\mathrm{Im}}M_{0}\Big{)}.$
(4.30)

For $\dagger,\star\in\\{+,-\\},\dagger\neq\star$, the scalar quantities
$\phi^{\,(\dagger,\star)}$ arise from the deterministic approximations (3.102)
of $G^{B_{ij}(E_{0})}(z,\zeta)$ for
$(z,\zeta)\in\Omega^{\dagger}\times\Omega^{\star}$, and
$\tilde{\phi}^{\,(\dagger,\star)}$ from the approximations (4.4) of
$\widetilde{G}^{\widetilde{B}_{ij}(E_{0})}(z,\zeta)$ for
$(z,\zeta)\in\Omega^{\dagger}\times\Omega^{\star}$. When expressing
$\phi^{\,(\dagger,\star)}$ and $\tilde{\phi}^{\,(\dagger,\star)}$ in
(4.27)-(4.30) we used that $\vartheta=1$ for
$(z,\zeta)\in\Omega^{+}\times\Omega^{-}$ and $\vartheta=-1$ for
$(z,\zeta)\in\Omega^{-}\times\Omega^{+}$. Moreover, we applied the identity
(3.124), introduced again the shorthand notations $M_{0}:=\lim_{y\downarrow
0}M(E_{0}+\mathrm{i}\,y)$ and $M_{0}^{\prime}:=\lim_{y\downarrow
0}M^{\prime}(E_{0}+\mathrm{i}\,y)$, and used $\lim_{\Omega^{-}\ni z\to
E_{0}}M(z)=M_{0}^{*}$ and $\lim_{\Omega^{-}\ni z\to
E_{0}}M^{\prime}(z)=(M_{0}^{\prime})^{*}$. It has been shown in
(3.122)-(3.125) that

$\phi^{\,(+,-)}=\frac{2\mathrm{i}\,}{\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\Big{\langle}\Gamma\Big{[}\operatorname{\mathrm{Im}}M_{0}\,\frac{1}{M_{0}}M^{\prime}_{0}\Big{]}\,\operatorname{\mathrm{Im}}M_{0}\Big{\rangle}=-1.$
(4.31)

Since $M(z)$ and $M^{\prime}(z)$ are symmetric and
$\big{(}\widetilde{\Gamma}[R]\big{)}^{t}=\widetilde{\Gamma}[R^{t}]$ for any
$R\in\mathbb{C}^{n\times n}$, we have

$\displaystyle\frac{1}{n}\sum_{i,j=1}^{n}\operatorname{{\mathrm{Tr}}}\Big{(}E_{ji}\,\operatorname{\mathrm{Im}}M_{0}\Big{(}\frac{1}{M_{0}}M^{\prime}_{0}\,\widetilde{\Gamma}[E_{ij}]\Big{)}^{t}\operatorname{\mathrm{Im}}M_{0}\Big{)}$
$\displaystyle=\sum_{i,j=1}^{n}\Big{\langle}E_{ji}\operatorname{\mathrm{Im}}M_{0}\widetilde{\Gamma}[E_{ji}]\Big{(}\frac{1}{M_{0}}M_{0}^{\prime}\Big{)}^{t}\operatorname{\mathrm{Im}}M_{0}\Big{\rangle}$
$\displaystyle=\sum_{i,j=1}^{n}\sum_{\alpha=1}^{d}\Big{\langle}E_{ji}\operatorname{\mathrm{Im}}M_{0}\Big{(}L_{\alpha}E_{ji}\Big{(}\operatorname{\mathrm{Im}}M_{0}\frac{1}{M_{0}}M_{0}^{\prime}L_{\alpha}^{t}\Big{)}^{t}+L_{\alpha}^{t}E_{ji}\Big{(}\operatorname{\mathrm{Im}}M_{0}\frac{1}{M_{0}}M_{0}^{\prime}L_{\alpha}\Big{)}^{t}\,\Big{)}\Big{\rangle}$
$\displaystyle=\sum_{i,j=1}^{n}\sum_{\alpha=1}^{d}\Big{\langle}E_{ii}\operatorname{\mathrm{Im}}M_{0}L_{\alpha}E_{jj}\operatorname{\mathrm{Im}}M_{0}\frac{1}{M_{0}}M_{0}^{\prime}L_{\alpha}^{t}\Big{\rangle}+\Big{\langle}E_{ii}\operatorname{\mathrm{Im}}M_{0}L_{\alpha}^{t}E_{jj}\operatorname{\mathrm{Im}}M_{0}\frac{1}{M_{0}}M_{0}^{\prime}L_{\alpha}\Big{\rangle}$
$\displaystyle=\Big{\langle}\operatorname{\mathrm{Im}}M_{0}\,\Gamma\Big{[}\operatorname{\mathrm{Im}}M_{0}\frac{1}{M_{0}}M_{0}^{\prime}\Big{]}\Big{\rangle}.$

Plugging this into (4.28) and using (4.31) yields

$\tilde{\phi}^{\,(+,-)}=\frac{2\mathrm{i}\,}{\langle\operatorname{\mathrm{Im}}M_{0}\rangle}\Big{\langle}\Gamma\Big{[}\operatorname{\mathrm{Im}}M_{0}\,\frac{1}{M_{0}}M^{\prime}_{0}\Big{]}\,\operatorname{\mathrm{Im}}M_{0}\Big{\rangle}=-1.$

By applying the same method we show that
$\phi^{\,(-,+)}=\tilde{\phi}^{\,(-,+)}=-1$. We conclude that

$\mathcal{V}=-\frac{2}{\pi^{2}}\mathbb{E}\big{[}\mathfrak{e}(t)\big{]}\int\displaylimits_{(\Omega^{+}\times\Omega^{-})\cup(\Omega^{-}\times\Omega^{+})}\frac{\partial\tilde{f}(z)}{\partial\overline{z}}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\frac{1}{z-\zeta}\,d^{2}\\!\zeta
d^{2}\\!z+O\big{(}N^{-\tau}\big{)}.$

Finally, using (3.127), (3.130) and the same argument as at the end of Section
3.5 we obtain that in the real symmetric case
$\lim_{N\to\infty}\mathbb{E}[e(t)]=e^{-\frac{t^{2}}{2}V[g]}$, where

$V[g]=\frac{1}{2\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{(g(x)-g(y))^{2}}{(x-y)^{2}}dxdy.$

This finishes the proof of Theorem 2.1.

##  Appendix A Kronecker random matrix models with $L$-flat self-energy

In this section we state several important properties of the solution to the
matrix Dyson equation (2.6) under the assumption that the self-energy operator
$\Gamma$ satisfies the $L$-flatness property (A). Most of these results are
established by following the proofs of the corresponding results in
Propositions 2.2, 4.2 and 4.7 of [2], where the matrix Dyson equation (2.6)
has been studied under the stronger assumption of $1$-flat self-energy (2.10).
A statement similar to that of Proposition A.1 below for the general $L$-flat
self-energy can be found in an early arXiv version (version v3) of [2]. For
the reader’s convenience, we present the full proof, adjusted to our current
setup, and show how the modifications of the arguments from [2] yield
Proposition A.1.

Consider the Kronecker random matrix model $\bm{H}^{(\beta)}$,
$\beta\in\\{1,2\\}$, defined in (2.1). Let $M(z)$ be the solution to the
corresponding matrix Dyson equation (2.6). Recall that by [42, Theorem 2.1],
for any $z\in\mathbb{C}_{+}$, the solution to (2.6) satisfying
$\operatorname{\mathrm{Im}}M(z)>0$ exists and is unique. The solution also
admits the Stieltjes transform representation (2.8) (see, e.g., [4,
Proposition 2.1]). Denote

$\rho(z):=\frac{1}{\pi}\big{\langle}\operatorname{\mathrm{Im}}M(z)\big{\rangle}$

for $z\in\mathbb{C}_{+}$, and recall that for $T\in\mathbb{C}^{n\times n}$ we
denote by $\mathcal{C}_{T}$ an operator on $\mathbb{C}^{n\times n}$ defined in
(3.19) by $\mathcal{C}_{T}[R]=TRT$ for all $R\in\mathbb{C}^{n\times n}$.

######  Proposition A.1 (Properties of the solution to the MDE).

Suppose that the self-energy operator $\Gamma$ satisfies the $L$-flatness
property (A) for some $L\in\mathbb{N}$. Then the following holds.

  * (i)

$M(z)$ and $M^{-1}(z)$ are uniformly bounded: The solution $M(z)$ satisfies
the bounds

$\big{\|}M(z)\big{\|}\lesssim 1,\qquad\big{\|}M^{-1}(z)\big{\|}\lesssim 1+|z|$
(A.1)

uniformly for $z\in\mathbb{C}_{+}$.

  * (ii)

$\operatorname{\mathrm{Im}}M(z)$ and $\rho(z)I_{n}$ are comparable: The
relation

$\operatorname{\mathrm{Im}}M(z)\sim\rho(z)I_{n}$

holds uniformly on $\mathbb{C}_{+}$.

  * (iii)

Linear stability: The bound

$\Big{\|}\big{(}\mathrm{Id}_{n}-\mathcal{C}_{M(z)}\Gamma\big{)}^{-1}\Big{\|}\lesssim
1+\frac{1}{(\rho(z)+\mathrm{dist}(z,\mathrm{supp}\,\rho))^{2}}$ (A.2)

holds uniformly for all $z\in\mathbb{C}_{+}$.

  * (iv)

Regularity of $M(z)$ and the density of states: The continuous extension of
$M(z)$ to $\mathbb{C}_{+}\cup\mathbb{R}$ exists and satisfies
$\|M(z_{1})-M(z_{2})\|\lesssim|z_{1}-z_{2}|^{1/3}$ uniformly for all
$z_{1},z_{2}\in\mathbb{C}_{+}\cup\mathbb{R}$. In particular, the density of
states $\rho(x)$ defined in (2.13) is $1/3$-Hölder continuous on $\mathbb{R}$,
i.e.,

$|\rho(x_{1})-\rho(x_{2})|\lesssim|x_{1}-x_{2}|^{1/3}$ (A.3)

uniformly for all $x_{1},x_{2}\in\mathbb{R}$. Moreover, the density $\rho(x)$
is real analytic on the open set $\\{x\in\mathbb{R}\,:\,\rho(x)>0\\}$.

  * (v)

Analytic continuation of $M(z)$: For any $x_{0}\in\mathbb{R}$ satisfying
$\rho(x_{0})>0$ the solution $M(z)$ can be analytically extended to a
neighborhood of $x_{0}$ in $\mathbb{C}$.

The hidden constants in $\lesssim$ and $\sim$ depend on $L,d\in\mathbb{N}$ and
the structure matrices $K_{0},L_{1},\ldots,L_{d}\in\mathbb{C}^{n\times n}$.

###### Proof.

We split the proof of Proposition A.1 into four steps corresponding to the
statements (i), (ii), (iii) and (iv) - (v). To make the presentation lighter,
we will often suppress the $z$-dependence in the notation.

Proof of (i). By taking the imaginary part of the Dyson equation (2.6) and
multiplying it by $M^{*}$ from the left and $M$ from the right we get

$\operatorname{\mathrm{Im}}M=\operatorname{\mathrm{Im}}zM^{*}M+M^{*}\Gamma[\operatorname{\mathrm{Im}}M]M\geq\Gamma[\operatorname{\mathrm{Im}}M]$
(A.4)

for all $z\in\mathbb{C}_{+}$. Since $\operatorname{\mathrm{Im}}M$ is positive
definite, the $L$-flatness (2.9) gives

$\operatorname{\mathrm{Im}}M\gtrsim\sum_{k,l=1}^{n}z_{kl}\big{(}\operatorname{\mathrm{Im}}M\big{)}_{kk}M^{*}E_{ll}M\gtrsim\sum_{k=1}^{n}\big{(}\operatorname{\mathrm{Im}}M\big{)}_{kk}M^{*}E_{kk}M,$
(A.5)

where in the last step we used that $z_{kk}=1$ for $1\leq k\leq n$. If in
(A.4) we multiply by $M$ from the left and $M^{*}$ from the right, then
instead of (A.5) we obtain

$\operatorname{\mathrm{Im}}M\gtrsim\sum_{k,l=1}^{n}z_{kl}\big{(}\operatorname{\mathrm{Im}}M\big{)}_{kk}ME_{ll}M^{*}\gtrsim\sum_{k=1}^{n}\big{(}\operatorname{\mathrm{Im}}M\big{)}_{kk}ME_{kk}M^{*}.$
(A.6)

By looking at the diagonal entries of (A.5) and (A.6) we get

$w_{j}\gtrsim\sum_{k,l=1}^{n}z_{kl}w_{k}t_{lj}\gtrsim\sum_{k=1}^{n}w_{k}t_{kj}$
(A.7)

for all $1\leq j\leq n$, where we denoted
$w_{j}:=\big{(}\operatorname{\mathrm{Im}}M\big{)}_{jj}$ and
$t_{kj}:=|M_{kj}|^{2}+|M_{jk}|^{2}$. The matrix
$T:=(t_{kj})_{k,j=1}^{n}\in\mathbb{R}^{n\times n}$ has nonnegative entries,
and the vector $\mathrm{w}=(w_{j})_{j=1}^{n}\in\mathbb{R}^{n}$ has strictly
positive entries. For convenience, we rewrite (A.7) as

$\mathrm{w}^{t}\gtrsim\mathrm{w}^{t}\,Z\,T\geq\mathrm{w}^{t}\,T$ (A.8)

with vector inequalities understood in the entrywise sense. If we denote by
$\mathrm{v}\in\mathbb{R}^{n}$ the right Perron-Frobenius eigenvector of $T$,
then after comparing $\mathrm{w}^{t}\mathrm{v}$ and
$\mathrm{w}^{t}T\mathrm{v}=\|T\|\mathrm{w}^{t}\mathrm{v}$, and using (A.8) we
find that the spectral norm of $T$ satisfies the bound $\|T\|\lesssim 1$
uniformly on $z\in\mathbb{C}_{+}$. Thus, we conclude that (A.1) holds
uniformly on $z\in\mathbb{C}_{+}$.

Moreover, if we take the norm on both sides of the matrix Dyson equation
(2.6), then (A.1) yields the bound

$\big{\|}M^{-1}(z)\big{\|}\lesssim 1+|z|$ (A.9)

uniformly on $z\in\mathbb{C}_{+}$, and therefore, for any $C>0$ we have

$\|M\|\sim\|T\|\sim 1$ (A.10)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$.

Proof of (ii). Since the measure $V(dx)$ is compactly supported, it is easy to
see from the Stieltjes transform representation (2.8) and the normalization
$V(\mathbb{R})=I_{n}$ that for a sufficiently large $C>0$ the equivalences

$\operatorname{\mathrm{Im}}M(z)\sim\frac{\operatorname{\mathrm{Im}}z}{|z|^{2}}I_{n}\sim\rho(z)I_{n}$
(A.11)

hold uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\geq C\\}$.

Suppose that $|z|\leq C$. Then applying the first inequality in (A.8) $2L$
times gives

$\mathrm{w}^{t}\gtrsim\mathrm{w}^{t}\,(Z\,T\,Z\,T)^{L}\gtrsim\mathrm{w}^{t}\,(Z\,T^{\,2})^{L}.$
(A.12)

where in the second inequality we again used that $z_{kk}=1$ for $1\leq k\leq
n$. The bounds (A.9) and (A.10) imply that $M^{*}M\sim I_{n}$, from which we
have

$\sum_{j=1}^{n}|M_{jk}|^{2}\sim 1$ (A.13)

for all $1\leq k\leq n$. Now, using (A.13), we estimate the diagonal entries
of $T^{\,2}$ from below

$\big{(}T^{\,2}\big{)}_{kk}=\sum_{j=1}^{n}t_{jk}^{2}\gtrsim\Big{(}\sum_{j=1}^{n}t_{jk}\Big{)}^{2}\geq\Big{(}\sum_{j=1}^{n}|M_{jk}|^{2}\Big{)}^{2}\gtrsim
1$

for $1\leq k\leq n$, which yields

$\mathrm{w}^{t}\,(Z\,T^{2})^{L}\gtrsim\mathrm{w}^{t}\,Z^{L}.$ (A.14)

Since $Z^{L}$ has all entries greater than or equal to $1$, we conclude from
(A.12) and (A.14) that

$w_{j}\gtrsim\sum_{k=1}^{n}w_{k}\sim\langle\operatorname{\mathrm{Im}}M(z)\rangle$

for $1\leq j\leq n$. Now
$\big{(}\operatorname{\mathrm{Im}}M(z)\big{)}_{jj}\sim\rho(z)$ for all $1\leq
j\leq n$, and from (A.5) and (A.9) we get

$\operatorname{\mathrm{Im}}M(z)\gtrsim\rho(z)M^{*}(z)M(z)\geq\rho(z)\big{\|}M^{-1}(z)\big{\|}^{2}I_{n}\gtrsim\rho(z)I_{n}$
(A.15)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$. To estimate
$\operatorname{\mathrm{Im}}M(z)$ from above, we notice that since the
dimension of the equation (2.6) is fixed, we trivially have
$\Gamma[\operatorname{\mathrm{Im}}M(z)]\lesssim\rho(z)I_{n}$. Therefore, by
taking again the imaginary part of the MDE (2.6) we get

$\operatorname{\mathrm{Im}}M=\operatorname{\mathrm{Im}}z\,M^{*}M(z)+M^{*}\Gamma[\operatorname{\mathrm{Im}}M]M\lesssim(\operatorname{\mathrm{Im}}z+\rho(z))M^{*}M.$
(A.16)

On the other hand, from the imaginary part of the Stieltjes transform
representation (2.8) we see that
$\operatorname{\mathrm{Im}}M(z)\gtrsim\operatorname{\mathrm{Im}}zI_{n}$ for
$|z|\leq C$, which together with (A.16) and (A.10) gives

$\operatorname{\mathrm{Im}}M(z)\lesssim\rho(z)\|M(z)\|^{2}I_{n}\lesssim\rho(z)I_{n}$
(A.17)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$. Combining (A.11),
(A.15) and (A.17) we obtain

$\operatorname{\mathrm{Im}}M(z)\sim\rho(z)I_{n}$ (A.18)

uniformly on $z\in\mathbb{C}_{+}$.

Proof of (iii). We now establish (A.2). Following the proof of the
invertibility of the stability operator from [2, Section 4.2], we define the
_saturated_ self-energy operator
$\mathcal{F}=\mathcal{F}(z):\mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n}$
given by

$\mathcal{F}=\mathcal{C}^{*}\,\Gamma\,\mathcal{C},$ (A.19)

where
$\mathcal{C}:=\mathcal{C}_{\sqrt{\operatorname{\mathrm{Im}}M}}\,\mathcal{C}_{W}$
and

$W:=\Big{(}I_{n}+\big{(}\mathcal{C}_{\sqrt{\operatorname{\mathrm{Im}}M(z)}}^{-1}[\operatorname{\mathrm{Re}}M(z)]\big{)}^{2}\Big{)}^{1/4}$

is a positive definite matrix. Then (A.2) can be obtained by first
establishing the existence of the uniform spectral gap for $\mathcal{F}$, and
then applying the Rotation-Inversion Lemma (see Lemmas 4.7, 4.9 and the proof
of Proposition 4.4 in [2] for the details). The crucial ingredient in this
approach is the study of the spectrum of $\mathcal{F}$ in [2, Lemma 4.7]. We
gather the spectral properties of $\mathcal{F}$ is the following lemma, that
will be proven at the end of this section.

######  Lemma A.2 (Spectrum of $\mathcal{F}$).

Let $\mathcal{F}=\mathcal{F}(z)$ be the operator defined in (A.19), and
suppose that $\Gamma$ satisfies the $L$-flatness property (A). Then the
following holds.

  * (i)

For any $x\in\mathbb{R}$ the operator $\mathcal{F}(x)$ possesses a simple
Perron-Frobenius eigenvalue $\|\mathcal{F}(x)\|_{2}=1$, so that

$\mathcal{F}(x)[F]=F,$ (A.20)

where $F\in\mathbb{C}^{n\times n}$ is the corresponding Perron-Frobenius
eigenvector satisfying $\|F\|_{\mathrm{HS}}=1$ and
$\operatorname{\mathrm{Im}}F>0$, and $\|\mathcal{F}\|_{2}$ denotes the
operator norm of $\mathcal{F}$ induced by the Hilbert-Schmidt norm on
$\mathbb{C}^{n\times n}$.

  * (ii)

There exists $C>0$, sufficiently large, such that

$\mathcal{F}^{L}[R]\sim\langle R\rangle I_{n}$ (A.21)

for all positive definite $R\in\mathbb{C}^{n\times n}$ uniformly on
$\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$ .

  * (iii)

There exists $\kappa>0$, sufficiently small, such that

$\mathrm{Spec}(\mathcal{F})\subset[-1+\kappa,1-\kappa]\cup\\{1\\}$ (A.22)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$.

Given the existence of the spectral gap for $\mathcal{F}$ in (A.22), the
linear stability bound (A.2) follows by repeating the argument presented in
[2, Section 4.2]. From (A.10) we also find the term
$(\rho(z)+\mathrm{dist}(z,\mathrm{supp}\,\rho))^{-2}$ on the right-hand side
of the (A.2) with the explicit exponent $2$. Indeed, using (A.10) and the fact
that the dimension of the matrix Dyson equation is $N$-independent, we get
from Eq. (4.41) in [2] that

$\big{\|}\big{(}\mathrm{Id}-\mathcal{C}_{M(z)}\Gamma\big{)}^{-1}\big{\|}\lesssim\big{\|}\big{(}\mathcal{C}_{U}-\mathcal{F}\big{)}^{-1}\big{\|}$
(A.23)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$, where $U$ is a unitary
matrix and $\mathcal{F}$ was defined in (A.19). Then the Rotation-Inversion
Lemma [2, Lemma 4.9] implies that

$\big{\|}\big{(}\mathcal{C}_{U}-\mathcal{F}\big{)}^{-1}\big{\|}\lesssim\frac{1}{\max\\{1-\|\mathcal{F}\|,|1-\langle
F,\mathcal{C}_{U}[F]\rangle|\\}}$ (A.24)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$, where $F$ is the
normalized Perron-Frobenius eigenvector of $\mathcal{F}$. With (A.10), the
lower bound in Eq. (4.45) in [2] becomes
$1-\|\mathcal{F}\|\gtrsim\mathrm{dist}(z,\mathrm{supp}\,\rho)^{2}$, and the
last inequality in the proof of Lemma 4.4 in [2] gives $|1-\langle
F,\mathcal{C}_{U}[F]\rangle|\gtrsim\rho^{2}$, which together with (A.23) and
(A.24) establish (A.2).

Proof of (iv) and (v). With the linear stability (A.2) established, the
regularity of $M(z)$ and the density of states (A.3), as well as the real-
analyticity of $\rho(x)$, is obtained by following exactly the lines of the
proof of [2, Proposition 2.2]. ∎

###### Proof of Lemma A.2.

Part (i) is obtained using exactly the same argument as in part (i) of Lemma
4.7 in [2].

In order to prove part (ii), we split (A.21) into the upper and lower bounds.
The upper bound in (A.21) is obtain in the same way as the upper bound in Eq.
(4.33) of [2], which can then be iterated $L$ times. For the lower bound in
(A.21), using the proof by induction, we show that

$\mathcal{F}^{l}[R]\gtrsim\sum_{k,j=1}^{n}\big{(}Z^{l}\big{)}_{kj}\big{\langle}\mathcal{C}^{*}[E_{kk}]R\big{\rangle}\,\mathcal{C}^{*}[E_{jj}]$
(A.25)

for all $1\leq l\leq n$. The case $l=1$ follows directly from (2.9) and the
definition of $\mathcal{F}$ in (A.19). Suppose that (A.25) holds for $l<L$.
Then after applying $\mathcal{F}$ on both sides of (A.25) and using (2.9) we
get

$\mathcal{F}^{l+1}[R]\gtrsim\sum_{k,j=1}^{n}\big{(}Z^{l}\big{)}_{kj}\big{\langle}\mathcal{C}^{*}[E_{kk}]R\big{\rangle}\,\sum_{k^{\prime},j^{\prime}=1}^{n}z_{k^{\prime}j^{\prime}}\big{\langle}\mathcal{C}^{*}[E_{k^{\prime}k^{\prime}}]\mathcal{C}^{*}[E_{jj}]\big{\rangle}\mathcal{C}^{*}[E_{j^{\prime}j^{\prime}}].$

Since
$\big{\langle}\mathcal{C}^{*}[E_{k^{\prime}k^{\prime}}]\mathcal{C}^{*}[E_{jj}]\big{\rangle}\geq
0$ for all $1\leq k^{\prime},j\leq n$, we can further estimate

$\mathcal{F}^{l+1}[R]\gtrsim\sum_{k,j,j^{\prime}=1}^{n}\big{(}Z^{l}\big{)}_{kj}\big{\langle}\mathcal{C}^{*}[E_{kk}]R\big{\rangle}\,z_{jj^{\prime}}\Big{\langle}\big{(}\mathcal{C}^{*}[E_{jj}]\big{)}^{2}\Big{\rangle}\,\mathcal{C}^{*}[E_{j^{\prime}j^{\prime}}].$
(A.26)

In order to estimate the right-hand side of (A.26) from below, we notice that
exactly as in [2, Lemma 4.6] we have that

$\rho^{1/2}(z)W\sim I_{n}$ (A.27)

uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$ for any sufficiently
large $C>0$. Therefore, from (A.18), (A.27) and the definition of
$\mathcal{C}$ we get
$\big{\langle}\big{(}\mathcal{C}^{*}[E_{jj}]\big{)}^{2}\big{\rangle}\gtrsim
1$, which concludes the proof of the induction step. Now (A.25) with $l=L$
yields

$\mathcal{F}^{L}[R]\gtrsim\sum_{k,j=1}^{n}\big{(}Z^{L}\big{)}_{kj}\big{\langle}\mathcal{C}^{*}[E_{kk}]R\big{\rangle}\,\mathcal{C}^{*}[E_{jj}]\gtrsim\big{\langle}\mathcal{C}^{*}[I_{n}]R\big{\rangle}\,\mathcal{C}^{*}[I_{n}],$

which after applying (A.18) and (A.27) to
$\mathcal{C}^{*}[I_{n}]=W\operatorname{\mathrm{Im}}M(z)\,W$ gives
$\mathcal{F}^{L}[R]\gtrsim\langle R\rangle I_{n}$. Combined with the upper
bound, this proves (A.21).

With (A.21) established, part (iii) follows from [2, Lemma 4.8]. Indeed, by
applying [2, Lemma 4.8] to the operator $\mathcal{F}^{L}$ we find that
$\mathcal{F}^{L}$ possesses a spectral gap uniformly on
$\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$. From this and the symmetry of
$\mathcal{F}$ we conclude that $\mathcal{F}$ also has a spectral gap, and thus
(A.22) holds uniformly on $\\{z\in\mathbb{C}_{+}\,:\,|z|\leq C\\}$. ∎

##  Appendix B Proof of Lemma 3.3

The crucial ingredient of the proof of Lemma 3.3 is the following cumulant
expansion formula. This formula is the complex analog of [31, Proposition 3.2]
and follows immediately from the real case presented there.

######  Lemma B.1.

Let $\bm{\xi}:=(\xi_{1},\dots,\xi_{K})\in\mathbb{C}^{K}$ be a complex random
vector with finite moments up to order $R+1$ for $R\in\mathbb{N}$, and let
$\varphi:\mathbb{C}^{K}\to\mathbb{C}$ be $R+1$ times differentiable with
bounded partial derivatives. Then

$\mathbb{E}\Big{[}\xi_{1}\varphi(\bm{\xi})\Big{]}=\sum_{m=0}^{R-1}\frac{1}{m!}\mathbb{E}\Big{[}\big{(}\kappa^{\nabla}_{m+1}[\xi_{1},\bm{\xi}]\varphi\big{)}(\bm{\xi})\Big{]}+\frac{1}{R!}\mathbb{E}\bigg{[}\int_{0}^{1}\big{(}K^{\nabla}_{R+1,s}[\xi_{1},\bm{\xi}]\varphi\big{)}(s\bm{\xi})ds\bigg{]},$
(B.1)

where for a complex random variable $\theta$ we set
$\kappa^{\nabla}_{1}[\theta,\bm{\xi}]\varphi:=\mathbb{E}[\theta]\varphi$,
$K_{m+1,s}^{\nabla}$ is a random differential operator defined for $m\geq 1$
through

$K_{m+1,s}^{\nabla}[\theta,\bm{\xi}]:=m!\int_{[0,1]^{m-1}}\nabla_{\bm{\xi}}(\mathbbm{1}_{s\leq
t_{m-1}}-\mathbb{E})\nabla_{\bm{\xi}}(\mathbbm{1}_{t_{m-1}\leq
t_{m-2}}-\mathbb{E})\cdots\nabla_{\bm{\xi}}(\mathbbm{1}_{t_{1}\leq
1}-\mathbb{E})\theta d\bm{t}$

with $d\bm{t}:=dt_{1}\dots dt_{m-1}$,
$\nabla_{\bm{\xi}}:=\sum_{i=1}^{K}(\xi_{i}\partial_{i}+\overline{\xi}_{i}\overline{\partial}_{i})$,
and where

$\kappa^{\nabla}_{m+1}[\theta,\bm{\xi}]:=\mathbb{E}\bigg{[}\int_{0}^{1}K^{\nabla}_{m+1,s}[\theta,\bm{\xi}]ds\bigg{]}$

is a non-random differential operator.

The derivation of the above lemma is identical to the real case treated in
[31, Proposition 3.2], thus omitted. We now proceed to establishing Lemma 3.3.

###### Proof of Lemma 3.3.

For $\star\in\\{1,2\\}$ we have

$\mathbb{E}[\bm{W}\mathscr{F}_{\star}]=\sum_{i,j=1}^{N}\sum_{l=1}^{N}\mathbb{E}\Big{[}W_{il}F_{\star,lj}\Big{]}\otimes\bm{E}_{ij},$
(B.2)

where $F_{1,lj}:=G_{lj}$ and $F_{2,lj}:=G_{lj}\mathfrak{e}(t)$. We apply Lemma
B.1 to two cases, namely
$\varphi=\varphi(W_{il},W_{li}):=F_{1,lj}(W_{il},W_{li})$ and
$\varphi=\varphi(W_{il},W_{li}):=F_{2,lj}(W_{il},W_{li})$, where we keep the
entries of $W_{ab}$ with
$(a,b)\in\\{1,\dots,N\\}^{2}\setminus\\{(i,l),(l,i)\\}$ fixed, i.e., we take
the partial expectation $\mathbb{E}_{il}$ with respect to $W_{il}$ first. For
$i=l$ we interpret this as $\varphi=\varphi(W_{ii})$.

In order to accommodate the matrix setting of (B.2), where both $W_{il}$ and
$F_{\star,lj}(W_{il},W_{li})$ are $n\times n$ matrices, we denote by
$\nabla^{ij}_{V}:=\sum_{\alpha,\beta=1}^{n}v_{\alpha\beta}\partial_{w_{ij}^{\alpha\beta}}$
the directional derivative with respect to the $(i,j)$-th block
$W_{ij}=(w_{ij}^{\alpha\beta})_{\alpha,\beta=1}^{n}$ in the direction
$V=(v_{\alpha\beta})_{\alpha,\beta=1}^{n}\in\mathbb{C}^{n\times n}$. The
analyticity of $F_{\star,lj}(W_{il},W_{li})$ for $\star\in\\{1,2\\}$ implies
that $\overline{\partial}_{w_{ik}^{\alpha\beta}}F_{\star,lj}=0$ for all $1\leq
i,k\leq N$ and $1\leq\alpha,\beta\leq n$.

With this notation, the cumulant expansion (B.1) for $R=3$ reads

$\mathbb{E}\Big{[}W_{il}F_{\star,lj}\Big{]}=\mathbb{E}\bigg{[}\int_{0}^{1}\big{(}\nabla^{il}_{\widetilde{W}_{il}}+\nabla^{li}_{\widetilde{W}_{li}}\big{)}(\mathbbm{1}_{s\leq
1}-\mathbb{E}_{\widetilde{\bm{W}}})\widetilde{W}_{il}ds\,F_{\star,lj}(W_{il},W_{li})\\\
+\sum_{\sigma_{1},\sigma_{2}\in\\{il,li\\}}\int_{[0,1]^{2}}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}(\mathbbm{1}_{s\leq
t_{1}}-\mathbb{E}_{\widetilde{\bm{W}}})\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}(\mathbbm{1}_{t_{1}\leq
1}-\mathbb{E}_{\widetilde{\bm{W}}})\widetilde{W}_{il}dt_{1}ds\,F_{\star,lj}(W_{il},W_{li})\\\
+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}\in\\{il,li\\}}\int_{0}^{1}K^{\nabla,(\sigma_{1},\sigma_{2},\sigma_{3})}_{4,s}[W_{il},(W_{\sigma_{1}},W_{\sigma_{2}},W_{\sigma_{3}})]F_{\star,lj}(sW_{il},sW_{li})ds\bigg{]},$
(B.3)

where we introduced a random differential operator

$K^{\nabla,\bm{\sigma}}_{4,s}[V_{0},\bm{V}]:=\int_{0}^{1}\int_{0}^{1}\nabla_{V_{1}}^{\sigma_{1}}(\mathbbm{1}_{s\leq
t_{2}}-\mathbb{E})\nabla_{V_{2}}^{\sigma_{2}}(\mathbbm{1}_{t_{2}\leq
t_{1}}-\mathbb{E})\nabla_{V_{3}}^{\sigma_{3}}(\mathbbm{1}_{t_{1}\leq
1}-\mathbb{E})V_{0}dt_{1}dt_{2}$

with random matrices $V_{0}$, $(V_{1},V_{2},V_{3})=:\bm{V}$ and indices
$\bm{\sigma}:=(\sigma_{1},\sigma_{2},\sigma_{3})$. In the above formulas
$\widetilde{\bm{W}}$ is an independent copy of $\bm{W}$, and
$\mathbb{E}_{\widetilde{\bm{W}}}$ denotes the partial expectation with respect
to $\widetilde{\bm{W}}$. In (B.3) we also used that $W_{il}$ is centered,
which implies that the term corresponding to $m=0$ in (B.1) vanishes.

We now treat each term on the right-hand side of (B.3) individually. From the
direct computation of the first term we see that for
$\sigma_{1}\in\\{il,li\\}$

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{1}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}(\mathbbm{1}_{s\leq
1}-\mathbb{E}_{\widetilde{\bm{W}}})\widetilde{W}_{il}dsF_{\star,lj}(W_{il},W_{li})\bigg{]}=\mathbb{E}\bigg{[}\widetilde{W}_{il}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}F_{\star,lj}(W_{il},W_{li})\bigg{]}.$

Taking the sum of the above expression for $\sigma_{1}\in\\{il,li\\}$ and
$l\in\\{1,\ldots,N\\}$, and using the independence of $\widetilde{W}_{il}$ and
$\widetilde{W}_{ab}$ for $ab\notin\\{il,li\\}$ gives
$\mathbb{E}\big{[}\big{(}\widetilde{\bm{W}}\nabla_{\widetilde{\bm{W}}}\mathscr{F}_{\star}(\bm{W})\big{)}_{ij}\big{]}$.
Together with (B.2) we recover the first term in (3.14).

Similar computations for the second term give

$\displaystyle\mathbb{E}\bigg{[}\sum_{l=1}^{N}\sum_{\sigma_{1},\sigma_{2}\in\\{il,li\\}}\int_{[0,1]^{2}}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}(\mathbbm{1}_{s\leq
t_{1}}-\mathbb{E}_{\widetilde{\bm{W}}})\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}(\mathbbm{1}_{t_{1}\leq
1}-\mathbb{E}_{\widetilde{\bm{W}}})\widetilde{W}_{il}dt_{1}ds\,F_{\star,lj}(W_{il},W_{li})\bigg{]}$
$\displaystyle=\frac{1}{2}\mathbb{E}\bigg{[}\sum_{l=1}^{N}\sum_{\sigma_{1},\sigma_{2}\in\\{il,li\\}}\widetilde{W}_{il}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}F_{\star,lj}(\bm{W})\bigg{]}=\frac{1}{2}\mathbb{E}\Big{[}\Big{(}\widetilde{\bm{W}}\big{(}\nabla_{\widetilde{\bm{W}}}\big{)}^{2}\mathscr{F}_{\star}(\bm{W})\Big{)}_{ij}\Big{]}.$
(B.4)

We now estimate the above expression when $F_{1,lj}(\bm{W})=G_{lj}(z)$, i.e.,
$\mathscr{F}_{1}(\bm{W})=\bm{G}(z)$. From the properties of the resolvent
(3.27) we find that

$\nabla^{\sigma_{1}}_{V_{1}}\dots\nabla_{V_{k}}^{\sigma_{k}}G_{lj}=(-1)^{k}\sum_{\tau\in
S_{k}}G_{li_{\tau(1)}}V_{\tau(1)}G_{j_{\tau(2)}i_{\tau(2)}}\cdots
G_{j_{\tau(k-1)}i_{\tau(k)}}V_{\tau(k)}G_{j_{\tau(k)}j}$ (B.5)

for any $V_{1},\ldots,V_{k}\in\mathbb{C}^{n\times n}$ and double indices
$\sigma_{p}=i_{p}j_{p}$. Using the local law (3.7) together with (3.57), (B.5)
and moment bounds (2.4) we obtain for $\sigma_{1},\sigma_{2}\in\\{il,li\\}$
the estimates

$\widetilde{W}_{il}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}G_{lj}(\bm{W})=\left\\{\begin{array}[]{ll}O_{\prec}\Big{(}\frac{1}{N^{3/2}}\Big{)}\,,&\quad
l=j\,,\\\
O_{\prec}\bigg{(}\frac{1}{\sqrt{N|\operatorname{\mathrm{Im}}z|}}\,\frac{1}{N^{3/2}}\bigg{)}\,,&\quad
l\neq j\,.\end{array}\right.$ (B.6)

uniformly on $z\in\Omega$. When $z\in\Omega$ the functions $G_{ij}(z)$ are
deterministically bounded by $N$, thus the stochastic domination bounds (B.6)
also hold in expectation. Therefore, if in (B.4) we take the sum with respect
to $l\in\\{1,\ldots,N\\}$ and use the estimate (B.6) in expectation, we obtain
the bound

$\bigg{\|}\mathbb{E}\Big{[}\frac{1}{2}\widetilde{\bm{W}}(\nabla_{\widetilde{\bm{W}}})^{2}\bm{G}\Big{]}\bigg{\|}_{\max}=O_{\prec}\Big{(}\frac{1}{N|\operatorname{\mathrm{Im}}z|^{1/2}}\Big{)}=O_{\prec}\Big{(}\frac{N^{\tau/2}}{N\sqrt{\eta_{0}}}\Big{)}.$
(B.7)

To analyze the case $\mathscr{F}_{2}(\bm{W})=\bm{G}\mathfrak{e}(t)$,
$F_{2,lj}(\bm{W})=G_{lj}(z)\mathfrak{e}(t)$, we derive convenient formulas for
the repeated directional derivatives of $\mathfrak{e}(t)$, similar to (B.5).
For this, we define

$\begin{split}\mathcal{I}([\sigma,V])&:=\sum_{\tau\in
S_{k}}(-1)^{k}\frac{\mathrm{i}\,t}{\pi}\int\displaylimits_{\Omega}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\operatorname{{\mathrm{Tr}}}\sum_{l=1}^{N}G_{li_{\tau(1)}}(\zeta)V_{\tau(1)}G_{j_{\tau(1)}i_{\tau(2)}}(\zeta)\cdots
V_{\tau(k)}G_{j_{\tau(k)}l}(\zeta)d^{2}\\!\zeta\\\
&=\frac{(-1)^{k+1}}{k}\frac{\mathrm{i}\,t}{\pi}\sum_{\tau\in
S_{k}}\int\displaylimits_{\Omega}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\operatorname{{\mathrm{Tr}}}G_{j_{\tau(k)}i_{\tau(1)}}(\zeta)V_{\tau(1)}\cdots
G_{j_{\tau(k-1)}i_{\tau(k)}}(\zeta)V_{\tau(k)}d^{2}\\!\zeta\end{split}$

for a multiset $[\sigma,V]:=[(\sigma_{1},V_{1}),\dots,(\sigma_{k},V_{k})]$,
where $\sigma_{p}=i_{p}j_{p}$ are double indices, and
$V_{1},\ldots,V_{k}\in\mathbb{C}^{n\times n}$. In particular, we have

$\nabla^{\sigma_{0}}_{V_{0}}\mathcal{I}([{\sigma},{V}])=\mathcal{I}\big{(}\big{[}(\sigma_{0},V_{0}),(\sigma_{1},V_{1}),\ldots,(\sigma_{k},V_{k})\big{]}\big{)}\,.$

Using this and
$\nabla^{\sigma_{1}}_{V_{1}}\mathfrak{e}(t)=\mathfrak{e}(t)\mathcal{I}\big{(}[(\sigma_{1},V_{1})]\big{)}$
we see by induction that

$\nabla^{\sigma_{1}}_{V_{1}}\dots\nabla^{\sigma_{k}}_{V_{k}}\mathfrak{e}(t)=\mathfrak{e}(t)\sum_{\pi\in\mathcal{P}([\sigma,V])}\mathcal{I}(\pi)\,,$
(B.8)

where
$\mathcal{P}([\sigma,V])=\mathcal{P}([(\sigma_{1},V_{1}),\dots,(\sigma_{k},V_{k})])$
denotes all partitions of $[(\sigma_{1},V_{1}),\dots,(\sigma_{k},V_{k})]$, and
for $\pi\in\mathcal{P}([\sigma,V])$ we set

$\mathcal{I}(\pi)=\prod_{\lambda\in\pi}\mathcal{I}(\lambda)\,.$ (B.9)

From (B.6) and $|\mathfrak{e}(t)|\leq 1$ we have that for all
$i,j,l\in\\{1,\ldots,N\\}$ and $\sigma_{1},\sigma_{2}\in\\{il,li\\}$ the
estimate

$\mathfrak{e}(t)\widetilde{W}_{il}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}G_{lj}=\left\\{\begin{array}[]{ll}O_{\prec}\Big{(}\frac{1}{N^{3/2}}\Big{)}\,,&\quad
l=j\,,\\\
O_{\prec}\bigg{(}\frac{N^{\tau/2}}{\sqrt{N\eta_{0}}}\,\frac{1}{N^{3/2}}\bigg{)}\,,&\quad
l\neq j\,,\end{array}\right.$ (B.10)

holds uniformly on $z\in\Omega$. For the mixed derivatives (B.5) and (B.8)
yield

$\widetilde{W}_{il}\Big{(}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}G_{lj}\Big{)}\Big{(}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}\mathfrak{e}(t)\Big{)}=-\widetilde{W}_{il}G_{li_{1}}\widetilde{W}_{i_{1}j_{1}}G_{j_{1}j}\,\mathfrak{e}(t)\,\mathcal{I}\big{(}[(\sigma_{2},\widetilde{W}_{\sigma_{2}})]\big{)}.$
(B.11)

The local law (3.7) implies that for any $i,l\in\\{1,\ldots,N\\}$,
$\sigma_{2}=i_{2}j_{2}\in\\{il,li\\}$, and $A\in\mathbb{C}^{n\times n}$

$\mathcal{I}\big{(}[(\sigma_{2},A)]\big{)}=\frac{\mathrm{i}\,t}{\pi}\int_{\Omega}\frac{\partial\tilde{f}(\zeta)}{\partial\overline{\zeta}}\frac{\partial}{\partial\zeta}\operatorname{{\mathrm{Tr}}}\Big{[}G_{j_{2}i_{2}}(\zeta)A\Big{]}d^{2}\\!\zeta=\left\\{\begin{array}[]{ll}O_{\prec}\Big{(}N^{\tau}|t|\|A\|\Big{)},&\,l=i,\\\
O_{\prec}\bigg{(}\frac{N^{3\tau/2}|t|\|A\|}{\sqrt{N\eta_{0}}}\bigg{)},&\,l\neq
i,\end{array}\right.$ (B.12)

where we applied Stokes’ theorem to rewrite the above expression using a
contour integral as in (3.53). Combining (B.11), (B.12), the local law bounds
(3.7) for $G$ and the bounds for $\widetilde{W}$ in (2.2)-(2.4) and (3.57) we
get

$\widetilde{W}_{il}\Big{(}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}G_{lj}\Big{)}\Big{(}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}\mathfrak{e}(t)\Big{)}=\left\\{\begin{array}[]{ll}O_{\prec}\bigg{(}\frac{N^{\tau}|t|}{N^{3/2}}\bigg{)}\,,&\quad
l=i,\\\
O_{\prec}\bigg{(}\frac{1}{N^{3/2}}\,\frac{N^{3\tau/2}|t|}{\sqrt{N\eta_{0}}}\bigg{)}\,,&\quad
l\neq i\,.\end{array}\right.$ (B.13)

Finally, using (B.8), the structure of $\mathcal{I}[\sigma,V]$, and applying
Stoke’s theorem again we find that for all
$\sigma_{1},\sigma_{2}\in\\{il,li\\}$

$\mathcal{I}\Big{(}\big{[}(\sigma_{1},\widetilde{W}_{\sigma_{1}}),(\sigma_{2},\widetilde{W}_{\sigma_{2}})\big{]}\Big{)}=O_{\prec}\bigg{(}\frac{N^{\tau}|t|}{N}\bigg{)}.$

Together with (B.12), (B.8) and (B.9) this gives the bound for the second
order derivatives

$\displaystyle\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}\mathfrak{e}(t)=O_{\prec}\bigg{(}\frac{N^{2\tau}|t|^{2}}{N}+\frac{N^{3\tau}|t|^{2}}{N^{2}\eta_{0}}+\frac{N^{\tau}|t|}{N}\bigg{)}=O_{\prec}\bigg{(}\frac{N^{2\tau}(|t|+|t|^{2})}{N}\bigg{)}\,,$
(B.14)

where we also used that $\tau<1-\gamma$ to absorb all three terms in (B.14)
into one. Now (B.14), (3.7), (2.2)-(2.4) and (3.57) imply

$\widetilde{W}_{il}G_{lj}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}\mathfrak{e}(t)=\left\\{\begin{array}[]{ll}O_{\prec}\bigg{(}\frac{N^{2\tau}(|t|+|t|^{2})}{N^{3/2}}\bigg{)}\,,&\quad
l=j\,,\\\
O_{\prec}\bigg{(}\frac{N^{2\tau}(|t|+|t|^{2})}{N^{3/2}}\,\frac{N^{\tau/2}}{\sqrt{N\eta_{0}}}\bigg{)}\,,&\quad
l\neq j\,.\end{array}\right.$ (B.15)

From the Leibniz rule, (B.10), (B.13) and (B.15) we establish the bound of the
term in (B.4) for $F_{2,lj}=G_{lj}\mathfrak{e}(t)$, namely

$\displaystyle\sum_{l=1}^{N}\sum_{\sigma_{1},\sigma_{2}\in\\{il,li\\}}\widetilde{W}_{il}\nabla^{\sigma_{1}}_{\widetilde{W}_{\sigma_{1}}}\nabla^{\sigma_{2}}_{\widetilde{W}_{\sigma_{2}}}\Big{(}G_{lj}\mathfrak{e}(t)\Big{)}$
$\displaystyle=O_{\prec}\bigg{(}\frac{N^{2\tau}(1+|t|^{2})}{N^{3/2}}+\frac{N^{5\tau/2}(1+|t|^{2})}{N\sqrt{\eta_{0}}}\bigg{)}$
$\displaystyle=O_{\prec}\bigg{(}\frac{N^{5\tau/2}(1+|t|^{2})}{N\sqrt{\eta_{0}}}\bigg{)}$

holding uniformly for $z\in\Omega$ and $i,j\in\\{1,\ldots,N\\}$. Using the
same argument as in the case $\star=1$ to extend the above bound to its
expectation we conclude that

$\bigg{\|}\mathbb{E}\Big{[}\frac{1}{2}\widetilde{\bm{W}}(\nabla_{\widetilde{\bm{W}}})^{2}\big{(}\bm{G}\mathfrak{e}(t)\big{)}\Big{]}\bigg{\|}_{\max}=O_{\prec}\bigg{(}\frac{N^{5\tau/2}(1+|t|^{2})}{N\sqrt{\eta_{0}}}\bigg{)}$
(B.16)

uniformly for $z\in\Omega$.

It remains to estimate the last term in (B.3). The local law bound
$\|\bm{G}\|_{\max}\prec 1$ implies

$|\nabla^{\sigma_{1}}_{V_{1}}\dots\nabla^{\sigma_{k}}_{V_{k}}\mathfrak{e}(t)|\prec(1+|t|^{k})N^{k\tau}\prod_{i=1}^{k}\|V_{i}\|\,,\qquad\|\nabla^{\sigma_{1}}_{V_{1}}\dots\nabla_{V_{k}}^{\sigma_{k}}G_{ab}\|\prec\prod_{i=1}^{k}\|V_{i}\|.$

If we now use $\|W_{ab}\|\prec N^{-1/2}$ following from (2.2)-(2.4), we see
that the term in the remainder of (B.3) satisfies the bound

$\|K^{\nabla,(\sigma_{1},\sigma_{2},\sigma_{3})}_{4,s}[W_{il},(W_{\sigma_{1}},W_{\sigma_{2}},W_{\sigma_{3}})]F_{\star,lj}(sW_{il},sW_{li})\|=O_{\prec}\Big{(}\frac{(1+|t|^{3})N^{3\tau}}{N^{2}}\Big{)}$

uniformly for $i,l,j\in\\{1,\ldots,N\\}$,
$\sigma_{1},\sigma_{2},\sigma_{3}\in\\{il,li\\}$, $s\in(0,1)$ and
$z\in\Omega$. After plugging this bound into (B.3), taking the sum for
$l\in\\{1,\ldots,N\\}$ in (B.2) and using (B.4) we get the following expansion
formula

$\displaystyle\mathbb{E}\Big{[}\bm{W}\mathscr{F}_{\star}(\bm{W})\Big{]}=\mathbb{E}\Big{[}\widetilde{\bm{W}}\,\nabla_{\widetilde{\bm{W}}}\mathscr{F}_{\star}(\bm{W})\Big{]}+\frac{1}{2}\mathbb{E}\Big{[}\widetilde{\bm{W}}(\nabla_{\widetilde{\bm{W}}})^{2}\mathscr{F}_{\star}(\bm{W})\Big{]}+\mathcal{E}_{\star}(z),$

with $\|\mathcal{E}_{\star}(z)\|_{\rm max}\lesssim(1+|t|^{3})N^{3\tau}N^{-1}$.
We finish the proof by denoting

$\mathcal{D}_{\star}(z):=\frac{1}{2}\mathbb{E}\Big{[}\widetilde{\bm{W}}(\nabla_{\widetilde{\bm{W}}})^{2}\mathscr{F}_{\star}(\bm{W})\Big{]}+\mathcal{E}_{\star}(z)$

and using the bounds (B.7) and (B.16). ∎

## References

  * [1] O. H. Ajanki, L. Erdős, and T. Krüger.  Universality for general Wigner-type matrices.  Probab. Theory Related Fields, 169(3-4):667–727, 2017.
  * [2] O. H. Ajanki, L. Erdős, and T. Krüger.  Stability of the matrix Dyson equation and random matrices with correlations.  Probab. Theory Related Fields, 173(1-2):293–373, 2019.
  * [3] J. Aljadeff, D. Renfrew, and M. Stern.  Eigenvalues of block structured asymmetric random matrices.  J. Math. Phys., 56(10):103502, 14, 2015.
  * [4] J. Alt, L. Erdős, and T. Krüger.  The Dyson equation with linear self-energy: spectral bands, edges and cusps.  Doc. Math., 25:1421–1539, 2020.
  * [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(2):661–696, 2019\.
  * [6] G. W. Anderson.  Convergence of the largest singular value of a polynomial in independent Wigner matrices.  Ann. Probab., 41(3B):2103–2181, 2013.
  * [7] G. W. Anderson.  A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices.  Ann. Inst. Henri Poincaré Probab. Stat., 51(3):809–841, 2015\.
  * [8] Z. Bai and J. Yao.  On the convergence of the spectral empirical process of Wigner matrices.  Bernoulli, 11(6):1059 – 1092, 2005.
  * [9] Z. D. Bai and J. W. Silverstein.  CLT for linear spectral statistics of large-dimensional sample covariance matrices.  Ann. Probab., 32(1A):553–605, 2004.
  * [10] M. Banna and G. Cébron.  Operator-valued matrices with free or exchangeable entries.  Ann. Inst. Henri Poincaré Probab. Stat., 59(1):503–537, 2023\.
  * [11] Z. Bao, K. Schnelli, and Y. Xu.  Central limit theorem for mesoscopic eigenvalue statistics of the free sum of matrices.  Int. Math. Res. Not. IMRN, (7):5320–5382, 2022.
  * [12] R. Bauerschmidt, P. Bourgade, M. Nikula, and H.-T. Yau.  The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem.  Adv. Theor. Math. Phys., 23(4):841–1002, 2019.
  * [13] F. Bekerman and A. Lodhia.  Mesoscopic central limit theorem for general $\beta$-ensembles.  Ann. Inst. Henri Poincaré Probab. Stat., 54(4):1917–1938, 2018\.
  * [14] S. Belinschi, M. Capitaine, S. Dallaporta, and M. Fevrier.  Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices.  ArXiv:2107.10031, 2021.
  * [15] J. Berstel and C. Reutenauer.  Noncommutative rational series with applications, volume 137 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 2011.
  * [16] P. Bourgade, L. Erdős, and H.-T. Yau.  Universality of general $\beta$-ensembles.  Duke Math. J., 163(6):1127–1190, 2014.
  * [17] A. Boutet de Monvel and A. Khorunzhy.  Asymptotic distribution of smoothed eigenvalue density. I. Gaussian random matrices.  Random Oper. Stochastic Equations, 7(1):1–22, 1999.
  * [18] A. Boutet de Monvel and A. Khorunzhy.  Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices.  Random Oper. Stochastic Equations, 7(2):149–168, 1999.
  * [19] G. Cipolloni, L. Erdős, J. Henheik, and O. Kolupaiev.  Gaussian fluctuations in the equipartition principle for wigner matrices.  ArXiv:2301.05181, 2023.
  * [20] G. Cipolloni, L. Erdős, J. Henheik, and D. Schröder.  Optimal lower bound on eigenvector overlaps for non-hermitian random matrices.  ArXiv:2301.03549, 2023.
  * [21] G. Cipolloni, L. Erdős, and D. Schröder.  Eigenstate thermalization hypothesis for Wigner matrices.  Comm. Math. Phys., 388(2):1005–1048, 2021.
  * [22] G. Cipolloni, L. Erdős, and D. Schröder.  Mesoscopic central limit theorem for non-hermitian random matrices.  ArXiv:2210.12060, 2022.
  * [23] G. Cipolloni, L. Erdős, and D. Schröder.  Optimal multi-resolvent local laws for Wigner matrices.  Electron. J. Probab., 27:Paper No. 117, 38, 2022.
  * [24] G. Cipolloni, L. Erdős, and D. Schröder.  Functional central limit theorems for Wigner matrices.  Ann. Appl. Probab., 33(1):447–489, 2023.
  * [25] B. Collins, J. A. Mingo, P. Śniady, and R. Speicher.  Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants.  Doc. Math., 12:1–70, 2007.
  * [26] M. Duits and K. Johansson.  On mesoscopic equilibrium for linear statistics in Dyson’s Brownian motion.  Mem. Amer. Math. Soc., 255(1222):v+118, 2018.
  * [27] L. Erdős and A. Knowles.  The Altshuler-Shklovskii formulas for random band matrices II: The general case.  Ann. Henri Poincaré, 16(3):709–799, 2015.
  * [28] L. Erdős, T. Krüger, and Yu. Nemish.  Local laws for polynomials of Wigner matrices.  J. Funct. Anal., 278(12):108507, 59, 2020.
  * [29] L. Erdős, T. Krüger, and Yu. Nemish.  Scattering in quantum dots via noncommutative rational functions.  Ann. Henri Poincaré, 22(12):4205–4269, 2021.
  * [30] L. Erdős, T. Krüger, and D. Renfrew.  Randomly coupled differential equations with elliptic correlations.  ArXiv:1908.05178, 2019.
  * [31] L. Erdős, T. Krüger, and D. Schröder.  Random matrices with slow correlation decay.  Forum Math. Sigma, 7:e8, 89, 2019.
  * [32] L. Erdős, B. Schlein, and H.-T. Yau.  Local semicircle law and complete delocalization for Wigner random matrices.  Comm. Math. Phys., 287(2):641–655, 2009.
  * [33] Y. V. Fyodorov, B. A. Khoruzhenko, and N. J. Simm.  Fractional Brownian motion with Hurst index $H=0$ and the Gaussian unitary ensemble.  Ann. Probab., 44(4):2980–3031, 2016.
  * [34] A. Garg, L. Gurvits, R. Oliveira, and A. Wigderson.  Operator scaling: theory and applications.  Found. Comput. Math., 20(2):223–290, 2020.
  * [35] F. Gesztesy and E. Tsekanovskii.  On matrix-valued Herglotz functions.  Math. Nachr., 218:61–138, 2000.
  * [36] U. Haagerup and S. Thorbjørnsen.  Random matrices and $K$-theory for exact $C^{\ast}$-algebras.  Doc. Math., 4:341–450, 1999.
  * [37] U. Haagerup and S. Thorbjørnsen.  A new application of random matrices: ${\rm Ext}(C^{*}_{\rm red}(F_{2}))$ is not a group.  Ann. of Math. (2), 162(2):711–775, 2005.
  * [38] H. M. Hastings, F. Juhasz, and M. A. Schreiber.  Stability of structured random matrices.  Proceedings: Biological Sciences, 249(1326):223–225, 1992.
  * [39] Y. He and A. Knowles.  Mesoscopic eigenvalue statistics of Wigner matrices.  Ann. Appl. Probab., 27(3):1510–1550, 2017.
  * [40] Y. He, A. Knowles, and R. Rosenthal.  Isotropic self-consistent equations for mean-field random matrices.  Probab. Theory Related Fields, 171(1-2):203–249, 2018.
  * [41] J. W. Helton, T. Mai, and R. Speicher.  Applications of realizations (aka linearizations) to free probability.  J. Funct. Anal., 274(1):1–79, 2018.
  * [42] 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, (22):Art. ID rnm086, 15, 2007.
  * [43] J. Huang and B. Landon.  Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $\beta$ and potentials.  Probab. Theory Related Fields, 175(1-2):209–253, 2019.
  * [44] H. C. Ji and J. O. Lee.  Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices.  Random Matrices Theory Appl., 9(3):2050011, 79, 2020.
  * [45] K. Johansson.  On fluctuations of eigenvalues of random Hermitian matrices.  Duke Mathematical Journal, 91(1):151 – 204, 1998.
  * [46] A. M. Khorunzhy, B. A. Khoruzhenko, and L. A. Pastur.  Asymptotic properties of large random matrices with independent entries.  J. Math. Phys., 37(10):5033–5060, 1996.
  * [47] I. Klep, J. E. Pascoe, and J. Volčič.  Regular and positive noncommutative rational functions.  J. Lond. Math. Soc. (2), 95(2):613–632, 2017.
  * [48] T. Krüger and Yu. Nemish.  Mesoscopic spectral CLT for quadratic polynomials in random matrices.  In preparation, 2023.
  * [49] G. Lambert.  Mesoscopic fluctuations for unitary invariant ensembles.  Electron. J. Probab., 23:Paper No. 7, 33, 2018.
  * [50] B. Landon, P. Lopatto, and P. Sosoe.  Single eigenvalue fluctuations of general Wigner-type matrices.  Probability Theory and Related Fields, pages 1–62, 1 2023.
  * [51] B. Landon and P. Sosoe.  Applications of mesoscopic CLTs in random matrix theory.  Ann. Appl. Probab., 30(6):2769–2795, 2020.
  * [52] J. O. Lee, K. Schnelli, B. Stetler, and H.-T. Yau.  Bulk universality for deformed Wigner matrices.  The Annals of Probability, 44(3):2349 – 2425, 2016.
  * [53] Y. Li, K. Schnelli, and Y. Xu.  Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices.  Ann. Inst. Henri Poincaré Probab. Stat., 57(1):506–546, 2021\.
  * [54] Y. Li and Y. Xu.  On fluctuations of global and mesoscopic linear statistics of generalized Wigner matrices.  Bernoulli, 27(2):1057–1076, 2021.
  * [55] A. Lodhia and N. J. Simm.  Mesoscopic linear statistics of Wigner matrices.  ArXiv:1503.03533, 2015.
  * [56] T. Mai, R. Speicher, and S. Yin.  The free field: realization via unbounded operators and Atiyah property.  ArXiv:1905.08187, 2019.
  * [57] K. Rajan and L. F. Abbott.  Eigenvalue spectra of random matrices for neural networks.  Phys. Rev. Lett., 97:188104, Nov 2006.
  * [58] V. Riabov.  Mesoscopic eigenvalue statistics for wigner-type matrices.  ArXiv:2301.01712, 2023.
  * [59] R. E. Skelton, T. Iwasaki, and K. M. Grigoriadis.  A unified algebraic approach to linear control design.  The Taylor & Francis Systems and Control Book Series. Taylor & Francis Group, London, 1998.
  * [60] T. P. Speed.  Cumulants and partition lattices.  Austral. J. Statist., 25(2):378–388, 1983.
  * [61] S. Thorbjørnsen.  Mixed moments of Voiculescu’s Gaussian random matrices.  J. Funct. Anal., 176(2):213–246, 2000.
  * [62] H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors.  Handbook of semidefinite programming, volume 27 of International Series in Operations Research & Management Science.  Kluwer Academic Publishers, Boston, MA, 2000.  Theory, algorithms, and applications.