step bounds, a telescopic argument will yield the estimate (5.4) also for
$H^{(\bm{w})}$. These ”sufficiently good” one-step bounds are essentially
required to accommodate the large number $O(N^{2})$ of necessary replacements
in order to arrive at $H^{(\gamma(N))}$. A key feature of our proof, in
contrast to previous applications of the replacement strategy, is that the
error will not always be $o(N^{-2})$ in each step, but their cumulative size
after summation is still $o(1)$.

The proof of Proposition 3.4 is divided in two main parts: At first, in Part
(a), Section 5.2, we show the isotropic part of (5.4), that is $\Psi_{k}^{\rm
iso}\prec 1$, via a double induction on the number $k\in{\mathbb{N}}$ of
traceless matrices and the moment $p\in{\mathbb{N}}$ taken of $\Psi_{k}^{\rm
iso}$, i.e. ${\mathbb{E}}|\Psi_{k}^{\rm iso}|^{p}$. Thereby, we crucially use
that the $\prec$-bound is (essentially) equivalent to controlling arbitrarily
high moments up to an $N^{\xi}$-error with arbitrarily small $\xi>0$.
Afterwards, in Part (b), Section 5.3, using Part (a) as an input, we will
demonstrate $\Psi_{k}^{\rm av}\prec 1$ (and thus conclude the proof of
Proposition 3.4 for $\mathfrak{I}_{k+1}=\emptyset$ resp.
$\mathfrak{I}_{k}=\emptyset$) for every fixed $k$ via a single induction on
the moment $p$. The main reason for this order of the argument is that the
one-by-one replacement in step $\gamma$ is conducted via resolvent expansion
focusing on the differing matrix entries at positions
$(i,j)=\phi^{-1}(\gamma)$ and $(j,i)$, and thereby it naturally produces
isotropic quantities (see Lemma 5.3 below). Hence, the argument for
$\Psi_{k}^{\rm av}$ cannot be self-contained and must rely on $\Psi_{k}^{\rm
iso}$, which in fact will not involve the averaged local laws at all.

We fix some further notation. We have an initial Wigner matrix
$H^{(0)}:=H^{(\bm{v})}$ and iteratively define

$H^{(\gamma)}:=H^{(\gamma-1)}-\frac{1}{\sqrt{N}}\Delta_{V}^{(\gamma)}+\frac{1}{\sqrt{N}}\Delta_{W}^{(\gamma)},$
(5.7)

a sequence of Wigner matrices for $\gamma\in[\gamma(N)]$, where we
denoted181818Observe that in this normalization, the non-zero entries of
$\Delta_{V}^{(\gamma)}$ and $\Delta_{W}^{(\gamma)}$ are of order one random
variables.

$\displaystyle\Delta_{V}^{(\gamma)}:=\sqrt{N}\frac{E^{(ij)}(H^{(\bm{v})})_{ij}+E^{(ji)}(H^{(\bm{v})})_{ji}}{1+\delta_{ij}}\quad\text{and}\quad\Delta_{W}^{(\gamma)}:=\sqrt{N}\frac{E^{(ij)}(H^{(\bm{w})})_{ij}+E^{(ji)}(H^{(\bm{w})})_{ji}}{1+\delta_{ij}}\,.$
(5.8)

Here, $\phi\big{(}(i,j)\big{)}=\gamma$ and $E^{(ij)}$ denotes the matrix whose
matrix elements are zero everywhere except at position $(i,j)$, i.e.
$(E^{(ij)})_{k\ell}=\delta_{ik}\delta_{j\ell}$. The denominator
$1+\delta_{ij}$ is introduced to account for the factor of two in the
numerator occurring for diagonal indices. Note that $H^{(\gamma)}$ and
$H^{(\gamma-1)}$ differ only in the $(i,j)$ and $(j,i)$ matrix elements, and
they can be written as

$\displaystyle
H^{(\gamma-1)}=\widecheck{H}^{(\gamma)}+\frac{1}{\sqrt{N}}\Delta_{V}^{(\gamma)}\quad\text{and}\quad
H^{(\gamma)}=\widecheck{H}^{(\gamma)}+\frac{1}{\sqrt{N}}\Delta_{W}^{(\gamma)}$
(5.9)

with a matrix $\widecheck{H}^{(\gamma)}$ whose matrix element is zero at the
$(i,j)$ and $(j,i)$ positions. Similarly, we denote the corresponding
resolvents at spectral parameter $z_{j}\in{\mathbb{C}}\setminus{\mathbb{R}}$
by

$G_{j}^{(\gamma)}:=(H^{(\gamma)}-z_{j})^{-1}\,,\quad
G_{j}^{(\gamma-1)}:=(H^{(\gamma-1)}-z_{j})^{-1}\,,\quad\text{and}\quad\widecheck{G}_{j}^{(\gamma)}:=(\widecheck{H}^{(\gamma)}-z_{j})^{-1}\,.$
(5.10)

Observe that, at each step $\gamma$ in the replacement procedure, the
deterministic approximation to a resolvent chain involving $G^{(\gamma)}$ is
the same. This is because only the first two moments of the matrix elements of
$H^{(\gamma)}$ determine this approximation, symbolically denoted by $M$, via
the _Matrix Dyson Equation (MDE)_ , see, e.g., [32]. For a chain in the
checked resolvents $\widecheck{G}$, the approximating $M$ is _in principle_
differing from the non-checked ones, simply because the self-energy operator
$\widecheck{\mathcal{S}}^{(\gamma)}[R]={\mathbb{E}}[\widecheck{H}^{(\gamma)}R\widecheck{H}^{(\gamma)}]$
associated with $\widecheck{H}^{(\gamma)}$ is no longer exactly the averaged
trace $\langle\cdot\rangle$. However, since this discrepancy introduces an
error of size $1/N$ in the MDE, which is a stable equation, this will not be
visible in the local laws (5.4). Therefore, we shall henceforth ignore this
minor point and shall just define the normalized differences

$\Psi_{k}^{{\rm av},(\gamma)}\,,\quad\widecheck{\Psi}_{k}^{{\rm
av},(\gamma)}\,,\quad\Psi_{k}^{{\rm
iso},(\gamma)}(\bm{x},\bm{y})\,,\quad\text{and}\quad\widecheck{\Psi}_{k}^{{\rm
iso},(\gamma)}(\bm{x},\bm{y})\,,$

exactly as in (5.2)–(5.3), but with $G_{j}$ replaced by $G_{j}^{(\gamma)}$ and
$\widecheck{G}_{j}^{(\gamma)}$, respectively. We emphasize again that the
deterministic counterparts in all of the normalized differences are the
_same_.

We can now turn to the actual proof.

###  5.2. Part (a): Proof of the isotropic law

In this first part, we exclusively work with isotropic quantities and we shall
hence drop the superscript iso in the entire Section 5.2. As already mentioned
above, we shall prove the claim by a _double induction_ on $k$ and the moment
$p$ taken of $\Psi_{k}$, i.e. ${\mathbb{E}}|\Psi_{k}|^{p}$.

Thereby, the primary induction parameter is $k$ and our goal is to show that,
if for some $k\in{\mathbb{N}}$ we have

$\max_{\gamma\leq\gamma(N)}\Psi^{(\gamma)}_{k^{\prime}}+\max_{\gamma\leq\gamma(N)}\widecheck{\Psi}^{(\gamma)}_{k^{\prime}}\prec
1\,,\qquad\forall\,k^{\prime}\in\\{0,...,k-1\\}\,,$ (5.11)

then also

$\max_{\gamma\leq\gamma(N)}\Psi^{(\gamma)}_{k}+\max_{\gamma\leq\gamma(N)}\widecheck{\Psi}^{(\gamma)}_{k}\prec
1\,.$ (5.12)

Within the proof of (5.12), for a fixed $k$, we will then crucially use that
the $\prec$-bound is equivalent to controlling arbitrarily high moments
${\mathbb{E}}|\Psi_{k}|^{p}$ up to an $N^{\xi}$-error for an arbitrarily small
$\xi>0$. Therefore, we use another secondary induction on the moment $p$. More
precisely, in order to establish (5.12) from (5.11), our goal is to show that,
for any fixed $k\in{\mathbb{N}}$, if for some $p\in{\mathbb{N}}$ we have that

$\max_{\gamma\leq\gamma(N)}\big{\|}\Psi^{(\gamma)}_{k}\big{\|}_{p-1}+\max_{\gamma\leq\gamma(N)}\big{\|}\widecheck{\Psi}^{(\gamma)}_{k}\big{\|}_{p-1}\lesssim
N^{\xi}$

for any $\xi>0$, then also

$\max_{\gamma\leq\gamma(N)}\big{\|}\Psi^{(\gamma)}_{k}\big{\|}_{p}+\max_{\gamma\leq\gamma(N)}\big{\|}\widecheck{\Psi}^{(\gamma)}_{k}\big{\|}_{p}\lesssim
N^{\xi}\,$ (5.13)

holds for any $\xi>0$, where implicit constants depend on $k,p$ and $\xi$.
Here for a random variable $X$ we used the definition
$\|X\big{\|}_{p}:=[{\mathbb{E}}|X|^{p}]^{1/p}$.

To summarize, as the _induction hypothesis_ , given some arbitrary fixed
$p,k\in{\mathbb{N}}$, we will assume that

$\max_{\gamma\leq\gamma(N)}\Psi^{(\gamma)}_{k^{\prime}}+\max_{\gamma\leq\gamma(N)}\widecheck{\Psi}^{(\gamma)}_{k^{\prime}}\prec
1\quad\text{and}\quad\max_{\gamma\leq\gamma(N)}\big{\|}\Psi^{(\gamma)}_{k}\big{\|}_{p-1}+\max_{\gamma\leq\gamma(N)}\big{\|}\widecheck{\Psi}^{(\gamma)}_{k}\big{\|}_{p-1}\leq
C_{k,p,\xi}N^{\xi}$ (5.14)

hold uniformly for all ${k^{\prime}\in\\{0,...,k-1\\}}$ and $\xi>0$ with an
appropriate $N$-independent constant. Then we will conclude (5.13).

The overall _base case_ ($k=1$, $p=1$) is easy to verify: it solely consists
of the usual isotropic law (the first estimate in (5.14) for $k^{\prime}=0$)
and the trivial bound ${\mathbb{E}}|\Psi_{1}|^{0}=1$ (the second estimate in
(5.14) for $k=1$ and $p=1$).

We start with two arbitrary but fixed bounded deterministic vectors
$\|\bm{x}\|,\|\bm{y}\|\lesssim 1$ and introduce the set

$I_{\bm{x}\bm{y}}:=\\{\bm{x},\bm{y}\\}\cup\\{{\bm{e}}_{a}:a\in[N]\\}\subset{\mathbb{C}}^{N}$
(5.15)

of vectors, which will naturally arise along the argument (see (5.32) below),
where ${\bm{e}}_{a}$ denotes the standard basis vector in the coordinate
direction $a$. Note that the cardinality of $I_{\bm{x}\bm{y}}$ is $N+2$. After
defining191919Here, $p$ is a superscript, not a power.

$\Omega_{k}^{p}(\gamma):=\max_{\bm{u},\bm{v}\in
I_{\bm{x}\bm{y}}}\|{\Psi}_{k}^{(\gamma)}(\bm{u},\bm{v})\|_{p}^{p}\,$ (5.16)

(we omitted the dependence on $\bm{x},\bm{y}$ in the notation, as they are
considered fixed along the whole argument), the principal goal of the
induction step is to prove the following proposition.

######  Proposition 5.2 (Gronwall estimate).

Fix $p,k\in{\mathbb{N}}$ and assume (5.14) holds. Then, for any $\xi>0$, there
exist some constants $C_{1},C_{2}>0$ (depending on $p$, $k$, and $\xi$, but
independent of $N$, $\bm{x}$, and $\bm{y}$) such that

$\Omega_{k}^{p}(\gamma_{0})\leq
C_{1}\frac{1}{N^{2}}\sum_{\gamma<\gamma_{0}}\Omega_{k}^{p}(\gamma)+C_{2}N^{\xi}$
(5.17)

for every $\gamma_{0}\in[\gamma(N)]$.

Note that (5.17) is a discrete Gronwall inequality for
$\Omega_{k}^{p}(\gamma)$. Hence, having Proposition 5.2 at hand (note that, in
particular, $\Omega_{k}^{p}(0)\leq C_{2}N^{\xi}$), we obtain

$\max_{\gamma\leq\gamma(N)}\Omega_{k}^{p}(\gamma)\leq
C_{2}\mathrm{e}^{C_{1}}N^{\xi}\leq C_{3}(k,p,\xi)N^{\xi}\,,$ (5.18)

uniformly in $\bm{x}$ and $\bm{y}$ and all choices of spectral parameters and
traceless deterministic matrices, which then implies the $\Psi$-part of
(5.13). In the next subsections we present auxiliary results necessary for the
proof of Proposition 5.2 which will then be concluded in Section 5.2.5. The
$\widecheck{\Psi}$-part of (5.13) and thus the induction step will finally be
completed in Section 5.2.6.

In order to simplify notation, we shall henceforth drop the subscripts for all
resolvents and deterministic matrices, i.e. write $G_{j}=G$ and $A_{j}=A$
instead.

####  5.2.1. Preliminaries

The fundamental building block of our proof is the following elementary lemma
on resolvent expansion. Note that we need to express
$G^{(\gamma-1)},G^{(\gamma)}$ in terms of the ”unperturbed” resolvent
$\widecheck{G}^{(\gamma)}$ of $\widecheck{H}^{(\gamma)}$ that has zero
elements in the $\gamma$-th position, and conversely, we need to express
$\widecheck{G}^{(\gamma)}$ in terms of both ”perturbed” resolvents using
$\Delta_{V}^{(\gamma)}$ and $\Delta_{W}^{(\gamma)}$ from (5.8) as
perturbations, see (5.10). We work with finite resolvent expansions up to some
order $m$, independent of $N$, to be determined later. The last term therefore
always contains the original resolvent as well and it will have to be
estimated deterministically by its norm but if $m$ is large enough this will
be affordable.

######  Lemma 5.3 (Resolvent expansions).

For every fixed $m\in{\mathbb{N}}$, it holds that

$\widecheck{G}^{(\gamma)}=\sum_{\ell=0}^{m}N^{-\ell/2}\big{(}G^{(\gamma)}\Delta_{W}^{(\gamma)}\big{)}^{\ell}G^{(\gamma)}+N^{-(m+1)/2}\big{(}G^{(\gamma)}\Delta_{W}^{(\gamma)}\big{)}^{m+1}\widecheck{G}^{(\gamma)}$
(5.19a) and
${G}^{(\gamma)}=\sum_{\ell=0}^{m}(-1)^{\ell}N^{-\ell/2}\big{(}\widecheck{G}^{(\gamma)}\Delta_{W}^{(\gamma)}\big{)}^{\ell}\widecheck{G}^{(\gamma)}+(-1)^{(m+1)}N^{-(m+1)/2}\big{(}\widecheck{G}^{(\gamma)}\Delta_{W}^{(\gamma)}\big{)}^{m+1}{G}^{(\gamma)}\,.$
(5.19b)

These relations also hold verbatim when replacing $G^{(\gamma)}\to
G^{(\gamma-1)}$ and $\Delta_{W}^{(\gamma)}\to\Delta_{V}^{(\gamma)}$. ∎

We now expand each $G^{(\gamma)}$ in

$\big{|}\Psi_{k}^{(\gamma)}(\bm{x},\bm{y})\big{|}^{p}=\left(\frac{N\eta}{\rho}\right)^{p/2}N^{-pk/2}\big{|}\big{(}(G^{(\gamma)}A)^{k}G^{(\gamma)}-M_{[1,k+1]}\big{)}_{\bm{x}\bm{y}}\big{|}^{p}$
(5.20)

and each $G^{(\gamma-1)}$ in

$\big{|}\Psi_{k}^{(\gamma-1)}(\bm{x},\bm{y})\big{|}^{p}=\left(\frac{N\eta}{\rho}\right)^{p/2}N^{-pk/2}\big{|}\big{(}(G^{(\gamma-1)}A)^{k}G^{(\gamma-1)}-M_{[1,k+1]}\big{)}_{\bm{x}\bm{y}}\big{|}^{p}$
(5.21)

according to (5.19b) (for some $m\geq 4$ to be determined below, depending on
$p$ and $k$; see (5.49)) and sort the resulting terms by their power
$r=0,1,2,...$ of $N^{-1/2}$. Then we take the expectation with respect to
$w_{ij}$ and $v_{ij}$, respectively (recall that
$\phi\big{(}(i,j)\big{)}=\gamma$), and use the moment matching condition
(3.12). As a result, we find that the terms with a prefactor $N^{-r/2}$ for
$r=0,1,2,3$ are algebraically _exactly the same_ for both (5.20) and (5.21).
The conclusion of this argument is formalized in the following lemma.

######  Lemma 5.4.

For any fixed $(i,j)\in[N]^{2}$ with $i\leq j$ and $\gamma=\phi(i,j)$ we have
that

$\displaystyle{\mathbb{E}}_{w_{ij}}\big{|}\Psi_{k}^{(\gamma)}(\bm{x},\bm{y})\big{|}^{p}=\sum_{r=0}^{3}N^{-r/2}\alpha_{k,r}^{(\gamma)}(\bm{x},\bm{y})\big{|}\widecheck{\Psi}_{k}^{(\gamma)}(\bm{x},\bm{y})\big{|}^{p-r}+\text{\rm
higher order terms}$ (5.22)
$\displaystyle{\mathbb{E}}_{v_{ij}}\big{|}\Psi_{k}^{(\gamma-1)}(\bm{x},\bm{y})\big{|}^{p}=\sum_{r=0}^{3}N^{-r/2}\alpha_{k,r}^{(\gamma)}(\bm{x},\bm{y})\big{|}\widecheck{\Psi}_{k}^{(\gamma)}(\bm{x},\bm{y})\big{|}^{p-r}+\text{\rm
higher order terms}$ (5.23)

for some _identical_ coefficients $\alpha_{k,r}^{(\gamma)}(\bm{x},\bm{y})$
independent of $v_{ij}$ and $w_{ij}$ whose precise values are (mostly)
irrelevant. Here ”higher order terms” denote terms with prefactor $N^{-r/2}$
with $r\geq 4$.

In the following Sections 5.2.2–5.2.4, preparing the conclusion of the proof
of Proposition 5.2 in Section 5.2.5, we will discuss the higher order terms in
(5.22) and (5.23). These have to be estimated individually by size when we
will consider the difference of (5.22) and (5.23). Recall that, we will
eventually compare $\Psi_{k}^{(0)}(\bm{x},\bm{y})$ and
$\Psi_{k}^{(\gamma(N))}(\bm{x},\bm{y})$ in $\gamma(N)=O(N^{2})$ many steps,
which is why the higher order terms must all be bounded by $1/N^{2}$, roughly
said. More precisely, we will use the following telescopic summation: For
every $\gamma_{0}\in[\gamma(N)]$, it holds that

$\left|\|\Psi_{k}^{(\gamma_{0})}(\bm{x},\bm{y})\|_{p}^{p}-\|\Psi_{k}^{(0)}(\bm{x},\bm{y})\|_{p}^{p}\right|\leq\sum_{1\leq\gamma\leq\gamma_{0}}\left|\|\Psi_{k}^{(\gamma)}(\bm{x},\bm{y})\|_{p}^{p}-\|\Psi_{k}^{(\gamma-1)}(\bm{x},\bm{y})\|_{p}^{p}\right|\,.$
(5.24)

In the next Section 5.2.2, we will explain the term with $r=4$ in Lemma 5.4,
i.e. with $N^{-2}$-prefactor, in detail. All other higher order terms with
$r\geq 5$ but still involving only the resolvent $\widecheck{G}^{(\gamma)}$
are completely analogous, in fact easier (see Section 5.2.3 later for some
detail). Afterwards, in Section 5.2.4, we will discuss, how the maximal order
$m$ of the resolvent expansion (5.19b) has to be chosen in order to
accommodate the remainder term involving a non-checked resolvent
$G^{(\gamma)}$ (resp. $G^{(\gamma-1)}$).

Throughout the following argument we shall focus on the higher order terms in
(5.22), the treatment of (5.23) is exactly the same. Whenever it does not lead
to confusion, we shall henceforth drop the superscript $\gamma$.

####  5.2.2. Fourth order terms in Lemma 5.4

The goal of the current Section 5.2.2 is to show that the terms of order $r=4$
arising in the telescopic summation (5.24) can be bounded by the rhs. of
(5.17).

In the following, we denote (cf. (5.8))

$\Delta=\Delta^{(\gamma)}=\frac{E^{(ij)}+E^{(ji)}}{1+\delta_{ij}}$ (5.25)

and find, similarly to (5.8), after taking the full expectation, the $r=4$
(i.e. $1/N^{2}$) prefactor of the higher order terms in (5.22) to be bounded
by (a constant times)

${\mathbb{E}}\sum_{d=1}^{4\wedge
p}\big{|}\widecheck{\Psi}_{k}(\bm{x},\bm{y})\big{|}^{p-d}\left(\frac{N\eta}{\rho}\right)^{d/2}N^{-dk/2}\sum_{4\Delta\,\leadsto\,d}\big{|}\underbrace{\big{(}...\Delta...\Delta...\big{)}_{\bm{x}\bm{y}}\ldots\big{(}...\Delta...\big{)}_{\bm{x}\bm{y}}}_{\text{four}\,\Delta\;\text{in
a total}\;d\,\text{chains}}\big{|}\,.$ (5.26)

Here $d$ counts the number of formerly ”intact” resolvent chains
$\big{(}(\widecheck{G}A)^{k}\widecheck{G}\big{)}_{\bm{x}\bm{y}}$, which have
been ‘destroyed’ by at least one replacement
$\widecheck{G}\to\widecheck{G}\Delta\widecheck{G}$ due to the expansion
(5.19b). The symbol

$\sum_{4\Delta\,\leadsto\,d}$ (5.27)

indicates that we sum over all possibilities to destroy exactly $d$ chains by
four $\Delta$’s. Note that a chain may be ”destroyed” by more than one
$\Delta$, therefore $d$ may be less than four. After using the explicit form
of $\Delta$, altogether we arrive at a finite sum of $4+d$ chains.

######  Example 5.5.

For example, for $d=1$ we have that

$\begin{split}&\sum_{4\Delta\,\leadsto\,1}\big{|}\big{(}...\Delta...\Delta...\Delta...\Delta...\big{)}_{\bm{x}\bm{y}}\big{|}\\\
&=\sum_{\begin{subarray}{c}k_{1},...,k_{5}\geq 0:\\\
\sum_{l}k_{l}=k\end{subarray}}\big{|}\big{(}(\widecheck{G}A)^{k_{1}}\widecheck{G}\Delta(\widecheck{G}A)^{k_{2}}\widecheck{G}\Delta(\widecheck{G}A)^{k_{3}}\widecheck{G}\Delta(\widecheck{G}A)^{k_{4}}\widecheck{G}\Delta(\widecheck{G}A)^{k_{5}}\widecheck{G}\big{)}_{\bm{x}\bm{y}}\big{|}\\\
&=\sum_{\begin{subarray}{c}k_{1},...,k_{5}\geq 0:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[\big{|}\big{(}(\widecheck{G}A)^{k_{1}}\widecheck{G}\big{)}_{\bm{x}\bm{e}_{i}}\big{(}(\widecheck{G}A)^{k_{2}}\widecheck{G}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}(\widecheck{G}A)^{k_{3}}\widecheck{G}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}(\widecheck{G}A)^{k_{4}}\widecheck{G}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}(\widecheck{G}A)^{k_{5}}\widecheck{G}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}+...\right]\end{split}$
(5.28)

with the neglected summands being analogous, only having different
distributions of $\bm{e}_{i}$ and $\bm{e}_{j}$ occurring, which can be
produced by the structure of $\Delta$.

For general $d$, in each of the $4+d$ resolvent chains in the rhs. of (5.26),
we now add and subtract the corresponding deterministic $M$-term,
$(\widecheck{G}A)^{k}\widecheck{G}=((\widecheck{G}A)^{k}\widecheck{G}-M_{k+1})+M_{k+1}$
(see also (5.33) below), schematically written as $G=(G-M)+M$. In the sequel,
we will distinguish the following two complementary cases:

  * Case (i):

At least $d$ of the $d+4$ resolvent chains are replaced by their fluctuating
part, $G-M$.

  * Case (ii):

At least five of the $d+4$ resolvent chains are replaced by their
deterministic counterpart, $M$.

Case (i): In case (i), we first separate those possibilities from (5.27),
where the destruction of the $d$ chains
$\big{(}(\widecheck{G}A)^{k}\widecheck{G}\big{)}_{\bm{x}\bm{y}}$ in fact
_preserves_ $d$ resolvent chains each with $k$ traceless matrices $A$, but
with deterministic vectors, which are not $\bm{x}$ and $\bm{y}$. This happens
when all four $\Delta$’s are placed at the ends of the chains. For example, if
$d=1$, we separate these possibilities as

$\begin{split}\widecheck{G}_{\bm{x}\bm{e}_{i}}\widecheck{G}_{\bm{e}_{j}\bm{e}_{j}}\widecheck{G}_{\bm{e}_{i}\bm{e}_{i}}\widecheck{G}_{\bm{e}_{j}\bm{e}_{j}}\big{(}(\widecheck{G}A)^{k}\widecheck{G}\big{)}_{\bm{e}_{i}\bm{y}}&+...\quad\text{or}\\\
\widecheck{G}_{\bm{x}\bm{e}_{i}}\widecheck{G}_{\bm{e}_{j}\bm{e}_{j}}\big{(}(\widecheck{G}A)^{k}\widecheck{G}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\widecheck{G}_{\bm{e}_{j}\bm{e}_{j}}\widecheck{G}_{\bm{e}_{i}\bm{y}}&+...\,.\end{split}$
(5.29)

In the following, we shall focus on the first exemplary term in (5.29). Its
fluctuating part

$\big{(}(\widecheck{G}A)^{k}\widecheck{G}-M_{[1,k+1]}\big{)}_{\bm{e}_{i}\bm{y}}$
(5.30)

can then be paired with the leftover $\big{(}N\eta/\rho\big{)}^{1/2}N^{-k/2}$
in (5.26) and thereby produces a further full
$\big{|}\widecheck{\Psi}_{k}^{(\gamma)}(\bm{e}_{j},\bm{y})\big{|}$; the
remaining terms coming from a single resolvent in (5.29) are simply estimated
by one,

$|\widecheck{G}_{\bm{u}\bm{v}}|\prec 1\,,\qquad\bm{u},\bm{v}\in
I_{\bm{x}\bm{y}}\quad\mbox{cf.~{}\eqref{eq:Ixy}}\,,$ (5.31)

by the usual isotropic law (5.5). All these terms stemming from (5.26) and
constituting a full $\big{|}\widecheck{\Psi}_{k}^{(\gamma)}\big{|}$ (or
$\big{|}\widecheck{\Psi}_{k}^{(\gamma)}\big{|}^{d}$ for general $d\in[4\wedge
p]$) can then be estimated by

$\widecheck{\Omega}_{k}^{p}(\gamma):=\max_{\bm{u},\bm{v}\in
I_{\bm{x}\bm{y}}}\|\widecheck{\Psi}_{k}^{(\gamma)}(\bm{u},\bm{v})\|_{p}^{p}\,.$
(5.32)

Now, after having separated the possibilities from (5.27), where the
destruction preserves $d$ resolvent chains with $k$ deterministic matrices in
between, we are left with those which solely create _strictly shorter_ chains
by the procedure $4\Delta\leadsto d$. These terms can entirely be treated by
our _induction hypothesis_ (5.14): The power of
$\widecheck{\Psi}_{k}^{(\gamma)}$ has been reduced by (at least) one (cf. the
second estimate in (5.14)) and
$\widecheck{\Psi}_{k^{\prime}}^{(\gamma)}+\Psi_{k^{\prime}}^{(\gamma)}\prec 1$
uniformly in $\gamma$ for _strictly_ shorter chains, $k^{\prime}<k$, has
already been shown (first estimate in (5.14)).

######  Example 5.6.

Writing

$M_{j-i+1}\equiv M_{[i,j]}\quad\text{for}\quad 1\leq i<j\leq k+1\,,$ (5.33)

with a slight abuse of notation, we estimate the $d=1$ term in (5.26) (after
having split off the cases when one of the $k_{l}$’s equals $k$ and all others
are zero in (5.29)) as

$\begin{split}{\mathbb{E}}&\big{|}\widecheck{\Psi}_{k}(\bm{x},\bm{y})\big{|}^{p-1}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\times\\\
&\ \times\sum_{\begin{subarray}{c}0\leq k_{l}\leq k-1:\\\
\sum_{l}k_{l}=k\end{subarray}}\Big{[}\big{|}\big{(}(\widecheck{G}A)^{k_{1}}\widecheck{G}-M_{k_{1}+1}\big{)}_{\bm{x}\bm{e}_{i}}\big{(}M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}\\\
&\quad+\big{|}\big{(}(\widecheck{G}A)^{k_{1}}\widecheck{G}-M_{k_{1}+1}\big{)}_{\bm{x}\bm{e}_{i}}\big{(}(\widecheck{G}A)^{k_{2}}\widecheck{G}-M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}+...\Big{]}\\\
\lesssim\,&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\sum_{\begin{subarray}{c}0\leq
k_{l}\leq k-1:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[\left(\frac{\rho}{N\eta}\right)^{1/2}N^{\sum_{l}k_{l}/2}+\left(\frac{\rho}{N\eta}\right)N^{\sum_{l}k_{l}/2}+...\right]\lesssim
N^{\xi}\,,\end{split}$ (5.34)

where analogous summands (i.e. having further $G-M$ factors instead of $M$, or
other arrangements of standard basis vectors $\bm{e}_{i},\bm{e}_{j}$ stemming
from (5.25)) are again indicated by dots. In the first estimate, we used that
$\big{|}(M_{j+1})_{\bm{u}\bm{v}}\big{|}\lesssim N^{j/2}$ for all
$\bm{u},\bm{v}\in I_{x,y}$ from Lemma 2.3 (b) together with the induction
hypothesis (5.14).

In the general case, $d\geq 1$, the argument works analogously to the above
example: The minimal number of $d$ fluctuating terms carrying an
$(\rho/N\eta)^{1/2}$-factor cancel the leftover $(N\eta/\rho)^{d/2}$-factor in
(5.26). The remaining $N^{k_{l}/2}$-factors can then be handled by a simple
power counting.

Overall, we find that, all the terms in (5.26) summarized in Case (i), can be
bounded by

$C_{1}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}$ (5.35)

for some positive constants $C_{1},C_{2}>0$, which shall henceforth be used
generically, i.e. their value might change from line to line (but remain
uniformly bounded in $\gamma$).

Case (ii): For the second case, we recall that all the purely deterministic
terms are _independent_ of $\gamma$, i.e., as emphasized above, at each
replacement step the deterministic approximation to a resolvent chain is the
same. However, it is _not_ sufficient to just estimate every $M$-term blindly
via $\big{|}(M_{j+1})_{\bm{u}\bm{v}}\big{|}\lesssim N^{j/2}$, as done in
(5.34). Instead, we need to _gain from the summation_ in (5.24) over all
replacement positions. This is the main new element of our proof compared with
previous GFT arguments.

######  Example 5.7.

We again look at our $d=1$ example. Using the notation (5.33), we find the
_trivial estimate_

$\begin{split}&{\mathbb{E}}\big{|}\widecheck{\Psi}_{k}(\bm{x},\bm{y})\big{|}^{p-1}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\sum_{\begin{subarray}{c}0\leq
k_{l}\leq k:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[\big{|}\big{(}M_{k_{1}+1}\big{)}_{\bm{x}\bm{e}_{i}}\big{(}M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}+...\right]\\\
\lesssim\,&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\sum_{\begin{subarray}{c}0\leq
k_{l}\leq k:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[N^{\sum_{l}k_{l}/2}+...\right]\lesssim
N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}\,,\end{split}$ (5.36)

where we again used the induction hypothesis (5.14) and
$\big{|}(M_{j+1})_{\bm{u}\bm{v}}\big{|}\lesssim N^{j/2}$. This bound is off by
a factor $(N\eta/\rho)^{1/2}$, which we will now improve on.

Indeed, the point in _gaining from the summation_ is that, although at each
individual step $\gamma$, the deterministic terms in (5.36) might be large,
_on average_ over $\gamma$ their contribution is bounded. More precisely,
fixing one constellation of $k_{l}$’s in (5.36) and using
${\mathbb{E}}\big{|}\widecheck{\Psi}_{k}\big{|}^{p-1}\lesssim N^{\xi}$ , we
find the average of the first line in (5.36) over all $i,j\in[N]$ to be
bounded by (a constant times)

$\begin{split}&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\frac{1}{N^{2}}\sum_{i,j}\left[\big{|}\big{(}M_{k_{1}+1}\big{)}_{\bm{x}\bm{e}_{i}}\big{(}M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}+...\right]\\\
\lesssim\,&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}\frac{1}{N^{2}}\sum_{i,j}\left[\frac{\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{|}}{N^{k_{1}/2}}+...\right]\\\
\lesssim\,&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}\frac{1}{N}\sqrt{N}\left[\frac{\sqrt{\big{(}|M_{k_{1}+1}|^{2}\big{)}_{\bm{x}\bm{x}}}}{N^{k_{1}/2}}+...\right]\lesssim\,N^{\xi}\left(\frac{\eta}{\rho}\right)^{1/2}\lesssim
N^{\xi}\,.\end{split}$ (5.37)

To go from the first to the second line, we used
$\big{|}(M_{j+1})_{\bm{u}\bm{v}}\big{|}\lesssim N^{j/2}$ for all but the first
$M$ factor. Next, we used a Schwarz inequality for the $i$-summation, which
involves the off-diagonal term $(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}$:

$\sum_{i}\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{|}\leq\sqrt{N}\left(\sum_{i}\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{|}^{2}\right)^{1/2}\leq\sqrt{N}\sqrt{\big{(}|M_{k_{1}+1}|^{2}\big{)}_{\bm{x}\bm{x}}}\,.$
(5.38)

In the penultimate estimate, we used that

$\sqrt{\big{(}|M_{j+1}|^{2}\big{)}_{\bm{u}\bm{u}}}\lesssim N^{j/2}\,,$ (5.39)

as follows from the fact that $N^{j/2}$ is in fact the operator norm bound for
$M_{j+1}$, and the final estimate in (5.37) simply used the general fact
$\eta/\rho\lesssim 1$.

We point out that we even could have gained another $1/\sqrt{N}$-factor from
the $i$-summation by not estimating
$\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}$ trivially by $N^{k_{5}/2}$ but
using

$\begin{split}\sum_{i}\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{(}M_{k_{5}+1}\big{)}_{\bm{e}_{i}\bm{y}}\big{|}&\leq\left(\sum_{i}\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{|}^{2}\right)^{1/2}\left(\sum_{i}\big{|}(M_{k_{5}+1})_{\bm{e}_{i}\bm{y}}\big{|}^{2}\right)^{1/2}\\\
&\leq\sqrt{\big{(}|M_{k_{1}+1}|^{2}\big{)}_{\bm{x}\bm{x}}}\sqrt{\big{(}|M_{k_{5}+1}|^{2}\big{)}_{\bm{y}\bm{y}}}\,.\end{split}$
(5.40)

instead of (5.38). However, we do not need this additional factor $1/\sqrt{N}$
here. Finally, note that the $j$-summation in (5.37) would have been useless,
since the $j$-terms are diagonal. The summation gain is effective only for
off-diagonal terms as in (5.38).

The above example indicates the following general mechanism: After estimating
all the $G-M$-type terms with the aid of the induction hypothesis (5.14), and
estimating the $M$-factors just trivially by their size, we are left with an
excess $(N\eta/\rho)^{u/2}$-factor, for some $u\in[4]$. In order to remove
this leftover factor, we need at least $u$ _(collectively) summable bounded
$M$-terms_ like

$\frac{\big{|}(M_{k_{1}+1})_{\bm{x}\bm{e}_{i}}\big{|}}{N^{k_{1}/2}}$ (5.41)

in (5.37) (see also (5.39)). In fact, each of these collectively summable
factors will gain one $1/\sqrt{N}$ compared to the trivial estimate, like the
one in (5.36). Here, the notion ”collective” refers to particular index
structures, which allow an effective summation. Denoting terms like (5.41)
symbolically by $M_{\bm{x}\bm{e}_{i}}$ for brevity, by _(collectively)
summable bounded $M$-terms_ we mean the following possible index structures

$\begin{split}u=1\,:&\qquad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}|\quad\text{or}\quad\sum_{i,j}|M_{\bm{e}_{j}\bm{y}}|\quad\text{or}\quad...\\\
u=2\,:&\qquad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{j}\bm{y}}|\quad\text{or}\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{i}\bm{y}}|\quad\text{or}\quad...\\\
u=3\,:&\qquad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{i}\bm{y}}||M_{\bm{e}_{j}\bm{y}}|\quad\text{or}\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{j}\bm{y}}|^{2}\quad\text{or}\quad...\\\
u=4\,:&\qquad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{x}\bm{e}_{j}}||M_{\bm{e}_{i}\bm{y}}||M_{\bm{e}_{j}\bm{y}}|\quad\text{or}\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}|^{2}|M_{\bm{e}_{j}\bm{y}}|^{2}\quad\text{or}\quad...\end{split}$
(5.42)

where dots are always indicating other similar terms, obtained from trivial
exchanges $\bm{x}\leftrightarrow\bm{y}$ or $i\leftrightarrow j$.

In principle, every summation over $i$ and $j$ potentially gains a full
$1/N$-factor each – provided that there are enough $M$’s with suitable indices
as in (5.42). The existence of $u$ _collectively summable bounded $M$-terms_
then ensures that of this potential $1/N^{2}$-improvement at least a
$1/N^{u/2}$-gain is effective. More precisely, as an example, for the first
column of terms in (5.42) we have that

$\begin{split}u=1\,:&\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}|\leq
N^{3/2}\left(\sum_{i}|M_{\bm{x}\bm{e}_{i}}|^{2}\right)^{1/2}\lesssim
N^{2-1/2}\\\
u=2\,:&\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{j}\bm{y}}|\leq
N\left(\sum_{i}|M_{\bm{x}\bm{e}_{i}}|^{2}\right)^{1/2}\left(\sum_{j}|M_{\bm{e}_{j}\bm{y}}|^{2}\right)^{1/2}\lesssim
N^{2-2/2}\\\
u=3\,:&\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{e}_{i}\bm{y}}||M_{\bm{e}_{j}\bm{y}}|\\\
&\qquad\quad\leq
N^{1/2}\left(\sum_{i}|M_{\bm{x}\bm{e}_{i}}|^{2}\right)^{1/2}\left(\sum_{i}|M_{\bm{e}_{i}\bm{y}}|^{2}\right)^{1/2}\left(\sum_{j}|M_{\bm{e}_{j}\bm{y}}|^{2}\right)^{1/2}\lesssim
N^{2-3/2}\\\
u=4\,:&\quad\sum_{i,j}|M_{\bm{x}\bm{e}_{i}}||M_{\bm{x}\bm{e}_{j}}||M_{\bm{e}_{i}\bm{y}}||M_{\bm{e}_{j}\bm{y}}|\\\
&\qquad\quad\leq\left(\sum_{i}|M_{\bm{x}\bm{e}_{i}}|^{2}\right)^{1/2}\left(\sum_{i}|M_{\bm{e}_{i}\bm{y}}|^{2}\right)^{1/2}\left(\sum_{j}|M_{\bm{e}_{j}\bm{y}}|^{2}\right)^{1/2}\left(\sum_{j}|M_{\bm{x}\bm{e}_{j}}|^{2}\right)^{1/2}\lesssim
N^{2-4/2}\end{split}$ (5.43)

by application of Schwarz inequalities like in (5.38)–(5.40) and using that
$\|M\|\lesssim 1$. We point out that the $\eta/\rho\leq 1$ factor within each
excess $(N\eta/\rho)^{1/2}$ would not be able to compensate for excess
$N$-factors; but the _gains from the summation_ are obtained solely on the
level of $N$’s.

It follows from a simple counting argument (or simply by considering all cases
directly), that for any $u\in[4]$, we find an appropriately summable index
structure within the at least five purely deterministic terms, as in
(5.42)–(5.43). Hence, we deduce that

$\eqref{eq:fourthorder}\leq
C_{1}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}\bigg{(}1+\sum_{u=1}^{4}\left(\frac{N\eta}{\rho}\right)^{u/2}\left|\big{[}u\,\text{sum.
bdd.}\,M\text{-terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}\right|\bigg{)}\,,$
(5.44)

where

$\big{[}u\,\text{sum. bdd.}\,M\text{-terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}$
(5.45)

stands symbolically for a product of _$u$ collectively summable bounded
deterministic terms_, like (5.41), for which we have just shown the following.

######  Lemma 5.8.

It holds that

$\sum_{\gamma\in[\gamma(N)]}\left|\big{[}u\,\text{\rm sum. bdd.}\,M\text{\rm-
terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}\right|\lesssim N^{2-u/2}\,.$ (5.46)

Combining (5.44) with (5.46), this concludes the argument for the fourth order
terms in (5.22).

####  5.2.3. Further higher order terms in Lemma 5.4

Just as in the previous Section 5.2.2, the goal of the current Section 5.2.3
is to show that the terms of order $r\geq 5$ arising in the telescopic
summation (5.24) can be bounded by the rhs. of (5.17).

For these other higher order terms in (5.22) with $r\geq 5$ and involving
_only_ $\widecheck{G}$ (and not ${G}$), the two cases distinguished above for
$r=4$ generalize to the following.

  * Case (i’):

At least $d$ of the $d+r$ resolvent chains are replaced by their fluctuating
part, $G-M$.

  * Case (ii’):

At least $r+1$ of the $d+r$ resolvent chains are replaced by their
deterministic counterpart, $M$.

For Case (i’), we separate a $1/N^{2}$-prefactor and find that the remaining
part can be estimated by

$C_{1}N^{-(r-4)/2}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}N^{-(r-4)/2}\,,$
(5.47)

completely analogously to (5.35). In fact, we gain an additional
$N^{-(r-4)/2}\ll 1$ factor in both terms. This reflects the idea that more
$G-M$ terms are better because their presumed bounds carry a factor
$(\rho/N\eta)^{1/2}$ (encoded in the prefactor $(N\eta/\rho)^{1/2}$ in the
definition of $\Psi_{k}^{\rm iso}$ in (5.3)).

For Case (ii’), we include the additional $N^{-(r-4)/2}$ (after having
separated a $1/N^{2}$-prefactor) into our counting of the leftover
$(N\eta/\rho)^{u/2}$-factor (recall the discussion below (5.41)). In this way,
we find that the maximal number of such leftover factors is $r-(r-4)=4$.
Hence, for every $u\in[4]$, we find an appropriately summable index structure,
completely analogously to (5.42), and deduce that (leaving out the separated
$1/N^{2}$-prefactor)

$\begin{split}r^{\rm th}\,&\text{order term in}\,\eqref{eq:gammaexp}\\\ &\leq
C_{1}N^{-(r-4)/2}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}\bigg{(}N^{-(r-4)/2}+\sum_{u=1}^{4}\left(\frac{N\eta}{\rho}\right)^{u/2}\left|\big{[}u\,\text{sum.
bdd.}\,M\text{-terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}\right|\bigg{)}\,,\end{split}$
(5.48)

which can be directly incorporated into (5.44) after adjusting the constants.
Note that while the contributions form Case (i’) improve by larger $r$, the
terms from Case (ii’) that carry many $M$-factors, do not.

Combining (5.48) with (5.46), this concludes the argument for the higher order
terms in (5.22).

####  5.2.4. Truncation of the resolvent expansion

It remains to discuss the truncation terms, which involve _not_ only
$\widecheck{G}$, but also ${G}$, i.e. the order $m\in{\mathbb{N}}$ for the
truncation of the resolvent expansion (5.19b). Also here, our goal is to show
that the contribution of these terms arising in the telescopic summation
(5.24) can be bounded by the rhs. of (5.17). After expanding each resolvent in
(5.20) via (5.19b), for every fixed $q\geq 1$, we collect those terms which
contain the final summand in (5.19b) (the _truncation term_), and hence ${G}$
exactly $q$ times. For these terms with $q\geq 1$ fixed, we then proceed as
follows: Estimate those chains within the truncation term in which ${G}$
appears trivially by norm, $\|G\|\leq 1/\eta$ (note that there are at most
$k+1$ resolvents in such chains and we can afford estimating all of them by
$1/\eta$ not just the last one $G$) and use
$\|A\|\leq\sqrt{N}\langle|A|^{2}\rangle^{1/2}$ (recall that we assumed
$\langle|A|^{2}\rangle^{1/2}=1$ around (5.2)–(5.3)), and treat the other
factors by our induction hypothesis (5.14) (resulting in an $N^{\xi}$ factor).

In this way, we conclude the estimate

$\big{[}q\,\text{truncation terms}\big{]}\lesssim
N^{\xi}\frac{(N\eta/\rho)^{p/2}}{\big{(}N^{\frac{m+1}{2}}\big{)}^{q}}\left(\frac{N^{k/2}}{\eta^{k+1}}\right)^{q}=\frac{N^{\xi}}{N^{2q}}\frac{1}{N^{p(q-1)/2}}\left(\frac{\eta}{\rho}\right)^{p/2}\frac{1}{(N\eta)^{(k+1)q}}\lesssim\frac{N^{\xi}}{N^{2}}$
(5.49)

when choosing $m=p+3k+5$, where in the last step we used that
$\eta/\rho\lesssim 1$ and $N\eta\gg 1$. We remark that $(N\eta/\rho)^{p/2}$ in
(5.49) comes from the prefactor of $\Psi_{k}$,
$(N^{\frac{m+1}{2}}\big{)}^{-q}$ from the cumulant order of the truncation
terms and $\big{(}N^{k/2}/\eta^{k+1}\big{)}^{q}$ from the trivial bounds.

####  5.2.5. Proof of Proposition 5.2

As mentioned above (5.25), the treatment of the higher order terms in (5.23)
is identical to our discussion above. Therefore, summarizing Sections
5.2.2–5.2.4, we have proven the following.

######  Lemma 5.9.

Fix $p,k\in{\mathbb{N}}$ and assume that the induction hypothesis (5.14)
holds. Then, for every $\gamma\in[\gamma(N)]$, we have that

$\left|\|\Psi_{k}^{(\gamma)}(\bm{x},\bm{y})\|_{p}^{p}-\|\Psi_{k}^{(\gamma-1)}(\bm{x},\bm{y})\|_{p}^{p}\right|\leq\frac{C_{1}}{N^{2}}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}\frac{N^{\xi}}{N^{2}}\bigg{(}1+\sum_{u=1}^{4}\left(\frac{N\eta}{\rho}\right)^{u/2}\left|\big{[}u\,\text{\rm
sum. bdd.}\,M\text{\rm-
terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}\right|\bigg{)}\,,$

where $\big{[}u\,\text{\rm sum. bdd.}\,M\text{\rm-
terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}$ is understood as explained below
(5.45).

Next, employing the telescopic summation from (5.24) we find that

$\|\Psi_{k}^{(\gamma_{0})}(\bm{x},\bm{y})\|_{p}^{p}\leq
C_{1}\frac{1}{N^{2}}\sum_{\gamma<\gamma_{0}}\widecheck{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}+\frac{N^{\xi}}{N^{2}}\sum_{\gamma<\gamma_{0}}\bigg{(}\sum_{u=1}^{4}\left(\frac{N\eta}{\rho}\right)^{u/2}\left|\big{[}u\,\text{\rm
sum. bdd.}\,M\text{\rm-
terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}\right|\bigg{)}$ (5.50)

after having absorbed $\|\Psi_{k}^{(0)}(\bm{x},\bm{y})\|_{p}^{p}$ into
$C_{2}N^{\xi}$ by our initial assumption that we have multi-resolvent local
laws (5.4) for the Wigner matrix $H^{(\bf v)}=H^{(0)}$. We are left with
discussing the first and last term on the rhs. of (5.50).

For the first term, we rely on the following lemma, which says that, in
particular, we can replace each $\widecheck{\Omega}_{k}^{p}(\gamma)$ in (5.50)
by $\Omega_{k}^{p}(\gamma)$, absorbing the additional error into $C_{2}$.

######  Lemma 5.10.

Fix $p,k\in{\mathbb{N}}$. Then, for every fixed $\gamma\in[\gamma(N)]$, the
expressions (recall (5.32))

$\Omega_{k}^{p}(\gamma)\,,\quad\Omega_{k}^{p}(\gamma-1)\,,\quad\text{and}\quad\widecheck{\Omega}_{k}^{p}(\gamma)$

are comparable up to an additive error of order $N^{\xi}$ for arbitrarily
small $\xi>0$.

###### Proof.

We give a sketch of the simple argument based on Lemma 5.9 in combination with
Lemma 5.4: Similarly to the proof of Lemma 5.9, we first expand $G^{(\gamma)}$
(resp. $G^{(\gamma-1)}$) in $\|\Psi_{k}^{(\gamma)}(\bm{x},\bm{y})\|_{p}^{p}$
(resp. $\|\Psi_{k}^{(\gamma-1)}(\bm{x},\bm{y})\|_{p}^{p}$) by means of (5.19b)
and realize that $\alpha_{k,0}^{(\gamma)}(\bm{x},\bm{y})=1$ in (5.22)–(5.23).
The various terms arising in the expansion (now for all $r\geq 1$ and not only
for $r\geq 4$) are dealt with as explained in Sections 5.2.2–5.2.4.

However, there is a major simplification, since we do not need to gain from
the summation as in Case (ii) in Section 5.2.2: The maximal excess power $u$
of the leftover $(N\eta/\rho)^{1/2}$-factor is bounded by the order $r$ of the
expansions in (5.22)–(5.23) (simply because at order $r$, there are at most
$d=r$ destroyed resolvent chains), such that the characteristic
$1/N^{r/2}$-factor at order $r$ balances this excess. Finally, we take a
maximum over all $\bm{u},\bm{v}\in I_{\bm{x},\bm{y}}$ for all
$\|\widecheck{\Psi}_{k}^{(\gamma)}(\bm{u},\bm{v})\|_{p}^{p}$ arising through
the expansion (see (5.32)).

This finishes the sketch of the proof of Lemma 5.10. ∎

For the last term in (5.50), we extend the summation
$\sum_{\gamma<\gamma_{0}}$ to all indices $i,j\in[N]$; it is an upper bound as
we only sum positive terms. Then, for every fixed $u\in[4]$, we need to gain
from this summation of $\big{[}u\,\text{\rm sum. bdd.}\,M\text{\rm-
terms}\big{]}^{(\gamma)}_{\bm{x},\bm{y}}$ over all $\gamma\in[\gamma(N)]$
precisely $N^{-u/2}$ compared to the naive $N^{2}$-size of the summation. This
was achieved in Lemma 5.8 by the index structure (5.42) of the factors and
application of several Schwarz inequalities (5.43).

Hence, combining (5.50) with Lemma 5.10 and Lemma 5.8, we find that

$\|\Psi_{k}^{(\gamma_{0})}(\bm{x},\bm{y})\|_{p}^{p}\leq
C_{1}\frac{1}{N^{2}}\sum_{\gamma<\gamma_{0}}{\Omega}_{k}^{p}(\gamma)+C_{2}N^{\xi}\,.$

Since the rhs. is independent of the elements in $I_{\bm{x}\bm{y}}$ (recall
(5.32)), we can as well maximize over those on the lhs. and arrive at
Proposition 5.2. ∎

####  5.2.6. Conclusion of the induction step

Having Proposition 5.2 and hence (5.18) at hand, we can immediately deduce

$\max_{\gamma\leq\gamma(N)}\widecheck{\Omega}_{k}^{p}(\gamma)\lesssim N^{\xi}$

from Lemma 5.10 above. This proves the $\widecheck{\Psi}$-part of (5.13) and
thus finishes the induction step.

Therefore, using uniformity of this bound, we conclude the proof of the
isotropic multi-resolvent local laws (3.14).

###  5.3. Part (b): Proof of the averaged law.

The general idea of the proof of the averaged law is exactly the same as in
the previous section: We replace all matrix elements one-by-one in
$\gamma(N)\sim N^{2}$ steps and sum up the changes over all positions
$\gamma\in[\gamma(N)]$ (cf. (5.24)). However, there are a several (minor)
differences in the averaged case compared to Section 5.2, which we will
explain in the following.

Since both, averaged and isotropic normalized differences, (5.2) and (5.3),
appear, we shall henceforth reintroduce the superscripts av and iso. Moreover,
contrary to the isotropic proof, in this part it is sufficient to consider an
arbitrary fixed $k\in{\mathbb{N}}$ and perform a _single induction_ on the
moment $p$ taken of $\Psi_{k}^{\rm av}$, i.e. ${\mathbb{E}}|\Psi_{k}^{\rm
av}|^{p}=\|\Psi_{k}^{\rm av}\|_{p}^{p}$. We point out that the induction on
$k$ used in the previous section is not needed, because the proof of the
isotropic laws has already been concluded (see (5.53) later). Hence, as the
_induction hypothesis_ , we will assume that

$\max_{\gamma\leq\gamma(N)}\|\Psi_{k}^{\rm
av,(\gamma)}\|_{p-1}+\max_{\gamma\leq\gamma(N)}\|\widecheck{\Psi}_{k}^{\rm
av,(\gamma)}\|_{p-1}\lesssim N^{\xi}$ (5.51)

holds uniformly in traceless matrices for all $\xi>0$, and our goal is to
prove the same relation with $p$ replacing $p-1$. The base case is thus simply
the trivial bound ($p=1$) given by ${\mathbb{E}}|\Psi_{k}^{\rm av}|^{0}=1$. To
ease notation, just as in Section 5.2, we will drop the subscripts for all
resolvents and deterministic matrices, i.e. write $G_{j}=G$ and $A_{j}=A$
instead. Moreover, whenever it does not lead to confusion, we will drop all
further sub- and superscripts.

Completely analogously to Section 5.2, we use resolvent expansions from Lemma
5.3 to prove the exact agreement of the orders $r\in\\{0,1,2,3\\}$ as in Lemma
5.4. For the higher order terms (again focusing on the most critical fourth
order ones, see Section 5.2.2), we argue completely analogously to (5.26), but
now we have an additional effect: Whenever an intact averaged chain gets
destroyed by a replacement $G\to G\Delta G$ from a derivative, we obtain (a
sum of) isotropic chains with a $1/N$ prefactor from the normalization of the
trace, i.e.

$\langle(GA)^{k}\rangle\longrightarrow\langle
G\Delta(GA)^{k}\rangle=\frac{1}{N}\big{(}(GA)^{k}G\big{)}_{\bm{e}_{i}\bm{e}_{j}}+\frac{1}{N}\big{(}(GA)^{k}G\big{)}_{\bm{e}_{j}\bm{e}_{i}}\,.$
(5.52)

In this way, the analogue of (5.26) reads

${\mathbb{E}}\sum_{d=1}^{4\wedge p}\big{|}\widecheck{\Psi}_{k}^{\rm
av}(\bm{x},\bm{y})\big{|}^{p-d}\left(\frac{N\eta}{\rho}\right)^{d/2}N^{-d(k/2-1)}\frac{1}{N^{d}}\sum_{(4-d)\Delta\,\leadsto\,d}\big{|}\underbrace{\big{(}...\Delta...\Delta...\big{)}_{\bm{e}_{i}\bm{e}_{j}}\cdot...\cdot\big{(}...\Delta...\big{)}_{\bm{e}_{j}\bm{e}_{i}}}_{(4-d)\;\Delta\;\text{in
a total}\;d\,\text{iso chains}}\big{|}\,,$ (5.53)

where the isotropic chains referred to in (5.53), are precisely those obtained
in (5.52). In particular, one $\Delta$ has already been ”used” for each
destroyed averaged chain, hence only $(4-d)$ $\Delta$’s are placed in the
isotropic chains (recall (5.27)). Observe that, after writing
$N^{-d(k/2-1)}/N^{d}=N^{-dk/2}$, beside from the unit vectors in the isotropic
chains, the structure of (5.53) is exactly the same for (5.26).

Next, in each of the resulting four resolvent chains in the rhs. of (5.53), as
before we add and subtract the corresponding $M$-term, again schematically
written as $G=(G-M)+M$. Exactly as in the previous section, we have to
distinguish two cases.

  * Case (i):

At least $d$ of the $4$ resolvent chains are replaced by their fluctuating
part, $G-M$.

  * Case (ii):

At least $5-d$ of the $4$ resolvent chains are replaced by their deterministic
counterpart, $M$.

Case (i): First, we note that, since there are only strictly lower moments of
$\Psi_{k}^{\rm av}$ appearing in (5.53) after the resolvent expansion, we can
directly employ the _induction hypothesis_ (5.51), i.e. there is no
possibility of preserving the destroyed chains unlike in (5.29). Therefore, by
additionally applying the already established isotropic laws from the previous
section in combination with $\big{|}(M_{j+1})_{\bm{u}\bm{v}}\big{|}\lesssim
N^{j/2}$ (recall also (5.33)), we find that

$\text{Case (i) terms of \eqref{eq:fourthorderAV}}\lesssim
N^{\xi}\sum_{d=1}^{4\wedge
p}\left(\frac{N\eta}{\rho}\right)^{d/2}N^{-dk/2}\left[N^{dk/2}\left(\frac{\rho}{N\eta}\right)^{d/2}+...\right]\lesssim
N^{\xi}\,,$ (5.54)

indicating terms with more than $d$ factors of $G-M$ by dots. This concludes
the discussion of Case (i).
Case (ii): For the second case, we again recall that all purely deterministic
terms are independent of the replacement step $\gamma$. Moreover, completely
analogously to Case (ii) in Section 5.2.2, it is not sufficient to just
estimate every isotropic $M$-term blindly – instead we again need to _gain
from the summation_ over all replacement positions. We again illustrate this
by an example.

######  Example 5.11.

We first consider $d=1$ and use the notation (5.33). Then, by means of the
induction hypothesis (5.51), we have the trivial estimate

$\begin{split}&{\mathbb{E}}\big{|}\widecheck{\Psi}^{\rm
av}_{k}(\bm{x},\bm{y})\big{|}^{p-1}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\sum_{\begin{subarray}{c}0\leq
k_{l}\leq k:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[\big{|}\big{(}M_{k_{1}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{|}+...\right]\\\
\lesssim\,&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\sum_{\begin{subarray}{c}0\leq
k_{l}\leq k:\\\
\sum_{l}k_{l}=k\end{subarray}}\left[N^{\sum_{l}k_{l}/2}+...\right]\lesssim
N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}\,,\end{split}$ (5.55)

analogously to (5.36). Again, this bound is off by a factor
$(N\eta/\rho)^{1/2}$, which can be improved on by averaging over all
replacement position.

Compared to the isotropic case, we can no longer gain from summing over off-
diagonal terms of the form $M_{\bm{x}\bm{e}_{i}}$. Instead, now we sum over
squares of terms of the form $M_{\bm{e}_{i}\bm{e}_{i}}$ and estimate it by

$\sum_{i}\big{|}M_{\bm{e}_{i}\bm{e}_{i}}\big{|}^{2}\leq\sum_{i,j}\big{|}M_{\bm{e}_{i}\bm{e}_{j}}\big{|}^{2}\leq\sum_{i}\big{(}|M|^{2}\big{)}_{\bm{e}_{i}\bm{e}_{i}}=N\langle|M|^{2}\rangle\,,$
(5.56)

similarly to (5.38)–(5.40). Note that (5.56) is better than the trivial bound,
which would give $N\|M\|^{2}$. The key for exploiting this improvement is the
following lemma, the proof of which is given in Appendix A.

######  Lemma 5.12.

Using the assumptions and notations from Lemma 2.3 and the normalization
$\langle|A_{i}|^{2}\rangle=1$, we have that

$\big{\langle}\big{|}\mathcal{M}(z_{1},A_{1},...,A_{k},z_{k+1};\mathfrak{I}_{k+1})\big{|}^{2}\big{\rangle}\lesssim
N^{k}\left(\prod_{i\in\mathfrak{I}_{k+1}}\rho_{i}\right)^{2}\left[\left(\frac{\max_{i\in[k+1]}\big{(}\rho_{i}+\mathbf{1}(i\notin\mathfrak{I}_{k+1})\big{)}}{N\ell}\right)^{2}\vee\frac{1}{N}\right]\,.$
(5.57)

Applying (5.57) for $k=k_{l}$ and $\mathfrak{I}_{k_{l}+1}=\emptyset$ (recall
(2.10), (2.15), and (5.33)), we see the bound

$\langle|M_{k_{l}+1}|^{2}\rangle\lesssim
N^{k_{l}}\left[\left(\frac{\rho}{N\eta}\right)^{2}\vee\frac{1}{N}\right]\,.$
(5.58)

We remark that this estimate is better by the factor
$\big{[}\big{(}N\eta/\rho\big{)}^{-2}\vee N^{-1}\big{]}\ll 1$ compared to the
naive norm bound
$|(M_{k_{l}+1})_{\bm{u}\bm{v}}|^{2}\leq\|M_{k_{l}+1}\|^{2}\lesssim N^{k_{l}}$
from Lemma 2.3 (b) employed in (5.55). Hence, fixing one constellation of
$k_{l}$’s in (5.55), we find the average of the first line in (5.55) over all
$i,j\in[N]$ to be bounded by

$\begin{split}&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\frac{1}{N^{2}}\sum_{i,j}\left[\big{|}\big{(}M_{k_{1}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{2}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{(}M_{k_{3}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{(}M_{k_{4}+1}\big{)}_{\bm{e}_{j}\bm{e}_{j}}\big{|}+...\right]\\\
\lesssim&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\frac{1}{N^{2}}\left[\prod_{l\in[4]}\left(\sum_{i}\big{|}\big{(}M_{k_{l}+1}\big{)}_{\bm{e}_{i}\bm{e}_{i}}\big{|}^{2}\right)^{1/2}+...\right]\\\
\lesssim&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{1/2}N^{-k/2}\frac{1}{N^{2}}\left[\left(\prod_{l\in[4]}N^{k_{l}+1}\left[\left(\frac{\rho}{N\eta}\right)^{2}\vee\frac{1}{N}\right]\right)^{1/2}+...\right]\\\
\lesssim&N^{\xi}\left[\left(\frac{\rho}{N\eta}\right)^{7/2}\vee\left(\frac{\eta}{\rho}\right)^{1/2}\frac{1}{N^{3/2}}\right]\lesssim
N^{\xi}\,.\end{split}$ (5.59)

To go from the first to the second line, we employed a trivial Schwarz
inequality. To go to the penultimate line, we used (5.56) with
$M=M_{k_{l}+1}$. For the final estimate, we employed
$(\prod_{l\in[4]}N^{k_{l}+1})^{1/2}=N^{k/2+2}$.

Next, we consider one example for $d=4$, where all four resolvent chains are
replaced by their deterministic counterpart. In this case, the analog of
(5.59) reads

$\begin{split}&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{2}N^{-2k}\frac{1}{N^{2}}\sum_{i,j}\left[\big{|}\big{(}M_{k+1}\big{)}_{\bm{e}_{i}\bm{e}_{j}}\big{(}M_{k+1}\big{)}_{\bm{e}_{j}\bm{e}_{i}}\big{(}M_{k+1}\big{)}_{\bm{e}_{i}\bm{e}_{j}}\big{(}M_{k+1}\big{)}_{\bm{e}_{j}\bm{e}_{i}}\big{|}+...\right]\\\
\lesssim&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{2}N^{-k}\frac{1}{N^{2}}\left[\sum_{i,j}\big{|}\big{(}M_{k+1}\big{)}_{\bm{e}_{i}\bm{e}_{j}}\big{|}^{2}+...\right]\\\
\lesssim&N^{\xi}\left(\frac{N\eta}{\rho}\right)^{2}N^{-k}\frac{1}{N^{2}}\left[N^{k+1}\left[\left(\frac{\rho}{N\eta}\right)^{2}\vee\frac{1}{N}\right]+...\right]\lesssim
N^{\xi}\left[\frac{1}{N}\vee\left(\frac{\eta}{\rho}\right)^{2}\right]\lesssim
N^{\xi}\,.\end{split}$

To go from the first to the second line, we estimated two factors of $M_{k+1}$
by their norm, $\big{|}(M_{k+1})_{\bm{u}\bm{v}}\big{|}\lesssim N^{k/2}$. Next,
to go to the third line, we employed (5.56) and Lemma 5.12. The final estimate
used $\eta/\rho\lesssim 1$.

The above examples showcase the general mechanism for the terms in Case (ii):
After estimating all the $(G-M)$-type terms with the aid of the induction
hypothesis (5.51), we are left with an excess $(N\eta/\rho)^{u/2}$-factor, for
some $u\in[4]$. Analogously to (5.41)–(5.42), this leftover factor is then
controlled by _gaining from the summation_ like in (5.57). We skip the simple
counting argument ensuring this gain.

The treatment of the further higher order terms and the truncation of the
resolvent expansion is completely analogous to Sections 5.2.3 and 5.2.4,
respectively. Therefore, by telescopic summation like in (5.24), we find that

$\max_{\gamma\leq\gamma(N)}\|\Psi_{k}^{\rm
av,(\gamma)}\|_{p}^{p}+\max_{\gamma\leq\gamma(N)}\|\widecheck{\Psi}_{k}^{\rm
av,(\gamma)}\|_{p}^{p}\lesssim\|\Psi_{k}^{\rm
av,(0)}\|_{p}^{p}+N^{\xi}\lesssim N^{\xi}$

where in the last step we absorbed $\|\Psi_{k}^{\rm av,(0)}\|_{p}^{p}$ into
$N^{\xi}$ by our initial assumption that we have multi-resolvent local laws
(5.4) for the matrix $H^{(\bm{v})}=H^{(0)}$. The checked version is obtained
completely analogously to Lemma 5.10.

This completes the proof of the induction step. We have thus finished the
argument for the averaged case and hence the proof of Proposition 3.4. ∎

###  5.4. The case $\mathfrak{I}_{k}\neq\emptyset$ (resp.
$\mathfrak{I}_{k+1}\neq\emptyset$)

In this section, we explain how to adjust the above argument for proving
Proposition 3.4 in the case that at least one of the resolvents in the chains
of interests

$\langle\mathcal{G}_{1}A_{1}...\mathcal{G}_{k}A_{k}\rangle\qquad\text{and}\qquad\big{(}\mathcal{G}_{1}A_{1}...\mathcal{G}_{k}A_{k}\mathcal{G}_{k+1}\big{)}_{\bm{x}\bm{y}}$

is an imaginary part, i.e. $\mathcal{G}_{i}=\mathrm{Im}\,G_{i}$ for at least
one index $i\in[k]$ (resp. $i\in[k+1]$). Recall the local laws for the average
and isotropic chain from (3.13) and (3.14), respectively. Compared to the case
of no imaginary parts, handled in the previous Sections 5.2–5.3, there are now
two changes: First, the bound contains the product
$\prod_{i\in\mathfrak{I}}\rho_{i}$ (instead of one). Second, the smallness
factor $(N\eta/\rho)^{-1/2}$ from before is now replaced by $(N\ell)^{-1/2}$

For adjusting the first change, the simple but key insight is, that when
applying the resolvent expansion from Lemma 5.3 to both $G$ and $G^{*}$ in
$\mathrm{Im}\,G=\frac{1}{2\mathrm{i}}(G-G^{*})$, we can always ”restore”
exactly one $\mathrm{Im}\,G$ on the rhs. More precisely, taking (5.19b) for
concreteness and using $\Delta=\Delta^{*}$, we have that

$\begin{split}\mathrm{Im}\,G=\frac{1}{2\mathrm{i}}\big{[}G-G^{*}\big{]}&=\frac{1}{2\mathrm{i}}\bigg{[}\left(\widecheck{G}-N^{-1/2}\widecheck{G}\Delta\widecheck{G}+N^{-1}\widecheck{G}\Delta\widecheck{G}\Delta\widecheck{G}+...\right)\\\
&\qquad\qquad-\left(\widecheck{G}^{*}-N^{-1/2}\widecheck{G}^{*}\Delta\widecheck{G}^{*}+N^{-1}\widecheck{G}^{*}\Delta\widecheck{G}^{*}\Delta\widecheck{G}^{*}+...\right)\bigg{]}\\\
&=\mathrm{Im}\,\widecheck{G}-N^{-1/2}\big{(}\mathrm{Im}\,\widecheck{G}\Delta\widecheck{G}+\widecheck{G}^{*}\Delta\mathrm{Im}\,\widecheck{G}\big{)}\\\\[2.84526pt]
&\qquad\qquad+N^{-1}\big{(}\mathrm{Im}\,\widecheck{G}\Delta\widecheck{G}\Delta\widecheck{G}+\widecheck{G}^{*}\Delta\mathrm{Im}\,\widecheck{G}\Delta\widecheck{G}+\widecheck{G}^{*}\Delta\widecheck{G}^{*}\Delta\mathrm{Im}\,\widecheck{G}\big{)}+...\end{split}$

In this way, the imaginary parts in the original chain are ”preserved” by the
resolvent expansion. Recall that
$|\mathrm{Im}\,\widecheck{G}_{\bm{u}\bm{v}}(z)|\prec\rho(z)$ (as a consequence
of (5.5) for $N|\mathrm{Im}\,z|\rho(z)\gg 1$; recall $N\hat{\ell}\gg 1$),
which improves (5.31). In particular, using Lemma 2.3, we find that the factor
$\big{(}\prod_{i\in\mathfrak{I}}\rho_{i}\big{)}^{-d}$ stemming from the
correct normalisation of the analog of $\Psi_{k}^{\rm av/iso}$ in (5.2)–(5.3)
and thus appearing in the expression analogous to (5.26) is naturally
compensated by a product of $\rho$’s stemming from the destroyed chains.

For adjusting to the second change, it suffices to replace every $\eta/\rho$
appearing in Sections 5.2–5.3 by $\ell$ and realize that the complement of the
interesting regime, i.e. the regime $\ell\geq 1$ is already proven in
Proposition 3.1.

##  Appendix A Additional technical results

In this section we prove several additional technical results which are used
in the main sections.

###  A.1. Bounds on the deterministic approximations

###### Proofs of Lemma 2.3 and the claim in Remark 2.6 (ii).

We will first proof the following stronger bound in Lemma A.1, from which we
immediately deduce Lemma 2.3 and the claim in Remark 2.6 (ii). The proof of
the following lemma is given at the end of the current section.

######  Lemma A.1.

Fix $k\geq 1$. Consider spectral parameters
$z_{1},...,z_{k}\in{\mathbb{C}}\setminus{\mathbb{R}}$ and traceless matrices
$A_{1},...,A_{k}\in{\mathbb{C}}^{N\times N}$, and define for every $j\in[k]$

${\eta}_{j}:=|\mathrm{Im}\,z_{j}|,\qquad\rho_{j}:=\frac{1}{\pi}|\mathrm{Im}\,m_{\rm
sc}(z_{j})|,\qquad\ell:=\min_{j}\big{[}\eta_{j}(\rho_{j}+\mathbf{1}(j\notin\mathfrak{I}_{k}))\big{]}\,.$

Then, for every $1\leq s\leq\lfloor k/2\rfloor$, it holds that

$\left|\sum_{\begin{subarray}{c}\pi\in\mathrm{NC}([k]):\\\
|\pi|=k+1-s\end{subarray}}\langle\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k-1})A_{k}\rangle\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k})}[S]\right|\lesssim\left(\prod_{j\in\mathfrak{I}_{k}}\rho_{j}\right)\frac{1}{\ell^{s-1}}\prod_{\begin{subarray}{c}S\in
K(\pi)\\\ |S|\geq 2\end{subarray}}\prod_{j\in
S}\left\langle|A_{j}|^{|S|}\right\rangle^{\frac{1}{|S|}}\,.$ (A.1)

with $m_{\circ}^{(\mathfrak{I})}[S]$ being defined above (2.16). For
$s>\lfloor k/2\rfloor$ the lhs. of (A.1) equals zero.

For the proof of Lemma 2.3 (a) and the claim in Remark 2.6 (ii) concerning
(2.20) we use that $\langle|A|^{p}\rangle^{1/p}\leq
N^{\frac{p-2}{2p}}\langle|A|^{2}\rangle^{1/2}$ for any $p\geq 2$, and hence

$\text{rhs. of}\,\eqref{eq:Mboundstrong}\lesssim
N^{k/2-1}\left(\prod_{j\in\mathfrak{I}_{k}}\rho_{j}\right)\left(\prod_{j=1}^{k}\langle|A_{j}|^{2}\rangle^{1/2}\right)\frac{1}{(N\ell)^{s-1}}\,.$

This shows that, in particular, all terms with $s>1$ in (A.1) are explicitly
smaller than the error term in (2.20), where we used that $N\ell\gg 1$. The
$s=1$ term exactly constitutes the deterministic approximation in (2.23), i.e.
the sum in (A.1) contains exactly one term

$\sum_{\begin{subarray}{c}\pi\in\mathrm{NC}([k]):\\\
|\pi|=k\end{subarray}}\langle\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k-1})A_{k}\rangle\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k})}[S]=\bigg{(}\prod_{j\in\mathfrak{I}_{k}}\mathrm{Im}\,m_{j}\bigg{)}\bigg{(}\prod_{j\notin\mathfrak{I}_{k}}m_{j}\bigg{)}\langle
A_{1}...A_{k}\rangle\,.$

Here we used that $|\pi|=k$ implies that the Kreweras complement consists of
the full set, $K(\pi)=[k]$.

Finally, for the proof of Lemma 2.3 (b) and the claim in Remark 2.6 (ii)
concerning (2.21) (i.e. the corresponding isotropic bounds) we argue
completely analogously to Section 4.2. ∎

It remains to prove Lemma A.1.

###### Proof of Lemma A.1.

Fix an arbitrary non-crossing partition $\pi\in\mathrm{NC}([k])$ consisting of
$|\pi|=k+1-s$ blocks.

First, note that, in order to get a non-vanishing partial trace

$\langle\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k-1})A_{k}\rangle=\prod_{S\in
K(\pi)}\left\langle\prod_{j\in S}A_{j}\right\rangle$

the minimal size of a block $S\in K(\pi)$ is two (using that the $A_{i}$’s are
traceless). Therefore, by application of Hölder’s inequality,

$\begin{split}\big{|}\langle\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k-1})A_{k}\rangle\big{|}\leq\prod_{\begin{subarray}{c}S\in
K(\pi)\\\ |S|\geq 2\end{subarray}}\prod_{j\in
S}\left\langle|A_{j}|^{|S|}\right\rangle^{\frac{1}{|S|}}\,.\end{split}$ (A.2)

In order to estimate $\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k})}[S]$, we
recall the Möbius inversion formula [16, Lemma 2.16]

$m_{\circ}^{(\mathfrak{I}_{k})}[S]=m^{(\mathfrak{I}_{k})}[S]+\sum_{\begin{subarray}{c}\pi\in\mathrm{NC}(S)\\\
|\pi|\geq 2\end{subarray}}(-1)^{|\pi|-1}\left(\prod_{T\in
K(\pi)}C_{|T|-1}\right)\prod_{U\in\pi}m^{(\mathfrak{I}_{k})}[U]$ (A.3)

where $C_{n}$ is the $n^{\rm th}$ Catalan number. Hence, it suffices to bound
the iterated divided differences $m^{(\mathfrak{I}_{k})}[S]$ for a subset
$S\subset[k]$ as

$\left|m^{(\mathfrak{I}_{k})}[S]\right|\lesssim\frac{\prod_{i\in\mathfrak{I}_{k}\cap
S}\rho_{i}}{\ell^{|S|-1}}$ (A.4)

which is a direct consequence of the integral representation (2.16). Indeed,
combining (A.3) with (A.4) and using that the sum in (A.3) is restricted to
partitions of $S$ with at least two blocks, we obtain

$\left|\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k})}[S]\,\right|\lesssim\left(\prod_{i\in\mathfrak{I}_{k}}\rho_{i}\right)\frac{1}{\ell^{s-1}}$
(A.5)

where we additionally used that the original non-crossing partition
$\pi\in\mathrm{NC}([k])$ consists of exactly $k+1-s$ blocks. Combining (A.5)
with (A.2), we conclude the proof of (A.1).

For $s>\lfloor k/2\rfloor$, we note that the Kreweras complement $K(\pi)$
necessary contains singletons, and hence the lhs. of (A.1) vanishes since
$\langle A_{i}\rangle=0$. ∎

We conclude this section by giving the proof of Lemma 5.12.

###### Proof of Lemma 5.12.

The principal idea of the proof is very similar to the previous ones given in
this section, hence we provide only a brief argument.

Recalling (2.15)–(2.16), we have that

$\big{\langle}\big{|}\mathcal{M}(z_{1},A_{1},\dots,A_{k},z_{k+1};\mathfrak{I}_{k+1})\big{|}^{2}\big{\rangle}\lesssim\sum_{\pi\in\mathrm{NC}([k+1])}\big{\langle}\big{|}\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k})\big{|}^{2}\big{\rangle}\left|\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k+1})}[S]\right|^{2}\,.$
(A.6)

Next, analogously to Lemma A.1 above, we decompose the summation over all
partitions $\pi$ into groups, where $|\pi|=k+2-s$ with $1\leq
s\leq\lceil(k+1)/2\rceil$ is fixed (note that $\lfloor\cdot\rfloor$ got
replaced by $\lceil\cdot\rceil$ due to the presence of a non-traceless
identity matrix). Moreover, for fixed $s$ we distinguish two cases in (A.6)
(recall (2.9)): For Case (i), we assume that the unique block
$\mathfrak{B}(k+1)\in K(\pi)$ containing $k+1$ contains no other elements,
i.e. $\mathfrak{B}(k+1)\setminus\\{k+1\\}=\emptyset$. For Case (ii), we assume
that $\mathfrak{B}(k+1)\setminus\\{k+1\\}\neq\emptyset$.
Case (i). First, we note that necessarily $s\geq 2$ in this case. Then, we
have that

$\big{\langle}\big{|}\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k})\big{|}^{2}\big{\rangle}\leq\left(\prod_{\begin{subarray}{c}S\in
K(\pi)\setminus\mathfrak{B}(k+1)\\\ |S|\geq 2\end{subarray}}\prod_{j\in
S}\left\langle|A_{j}|^{|S|}\right\rangle^{\frac{1}{|S|}}\right)^{2}\leq\left(\frac{N^{k/2}}{N^{s-1}}\right)^{2}\,,$

analogously to (A.2). Since in Case (i), $z_{1}$ and $z_{k+1}$ are always
together in one block $S\in\pi$ with $|\pi|=k+2-s$, we obtain

$\left|\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k+1})}[S]\,\right|^{2}\lesssim\left[\frac{\left(\prod_{i\in\mathfrak{I}_{k+1}}\rho_{i}\right)\wedge\max_{i\in[k+1]}\rho_{i}}{\ell^{s-1}}\right]^{2}$

analogously to (A.5) by means of (A.3) and the integral representation (2.16).
The additional $\wedge\max_{i\in[k+1]}\rho_{i}$, which is effective only for
$\mathfrak{I}_{k+1}=\emptyset$, comes from the estimate

$\int_{\mathbb{R}}\frac{\rho(x)}{|x-z_{1}|\,|x-z_{k+1}|}\mathrm{d}x\lesssim\frac{\rho_{1}\vee\rho_{k+1}}{\ell}\,,$

easily obtained by a Schwarz inequality.
Case (ii). In this case, the above estimates of the two factors in (A.6)
modify to

$\big{\langle}\big{|}\mathrm{pTr}_{K(\pi)}(A_{1},\ldots,A_{k})\big{|}^{2}\big{\rangle}\leq\left(\prod_{\begin{subarray}{c}S\in
K(\pi)\\\ |S|\geq 2\end{subarray}}\ \left(\prod_{j\in
S_{1}}\left\langle|A_{j}|^{2(|S_{1}|-1)}\right\rangle^{\frac{1}{2(|S_{1}|-1)}}\right)\left(\prod_{i=2}^{s}\prod_{j\in
S_{i}}\left\langle|A_{j}|^{|S_{i}|}\right\rangle^{\frac{1}{|S_{i}|}}\right)\right)^{2}\,,$

assuming that $S_{1}=\mathfrak{B}(k+1)$, and

$\left|\prod_{S\in\pi}m_{\circ}^{(\mathfrak{I}_{k+1})}[S]\,\right|^{2}\lesssim\left[\frac{\prod_{i\in\mathfrak{I}_{k+1}}\rho_{i}}{\ell^{s-1}}\right]^{2}\,.$

Putting the two cases together and using $\langle|A|^{p}\rangle^{1/p}\leq
N^{\frac{p-2}{2p}}\langle|A|^{2}\rangle^{1/2}$ for any $p\geq 2$ together with
$N\ell>1$ and the normalization $\langle|A_{j}|^{2}\rangle=1$, we find that

$\big{\langle}\big{|}\mathcal{M}(z_{1},A_{1},\dots,A_{k},z_{k+1};\mathfrak{I}_{k+1})\big{|}^{2}\big{\rangle}\lesssim
N^{k}\,\left(\prod_{i\in\mathfrak{I}_{k+1}}\rho_{i}\right)^{2}\left[\left(\frac{\max_{i\in[k+1]}\big{(}\rho_{i}+\mathbf{1}(i\notin\mathfrak{I}_{k+1})\big{)}}{N\ell}\right)^{2}+\frac{1}{N}\right]\,.$

∎

###  A.2. Complex moment matching

In order to conduct the third step of our proof, the Green function comparison
(GFT) of Proposition 3.4, we need to guarantee the moment matching condition
(3.12) of the single entry distributions. For real random variables (or
complex ones with independent real and imaginary parts), the argument ensuring
this (and even an approximately matching fourth moment) is standard (see,
e.g., [30, Lemma 16.2]) and based on an explicit construction of a
distribution supported on three points in ${\mathbb{R}}$. However, for general
complex random variables, this construction is not sufficient; we now present
its complex variant.

Let $Z$ be a complex random variable and denote its moments by

$m_{i,j}=m_{i,j}(Z):={\mathbb{E}}\big{[}\overline{Z}^{i}Z^{j}\big{]}\qquad\text{for}\qquad
i,j\in{\mathbb{N}}_{0}\,,$ (A.7)

and call $i+j$ the _order_ of $m_{i,j}$. Clearly $m_{0,0}=1$ and
$m_{i,j}=\overline{m}_{j,i}$, so we can focus on $m_{i,j}$ with $i\leq j$.

######  Lemma A.2.

Let $m_{0,2},m_{0,3},m_{1,2}\in{\mathbb{C}}$ with $|m_{0,2}|\leq 1$. Then
there exists a complex random variable $Z$ supported on at most eleven points
$z_{1},...,z_{11}\in{\mathbb{C}}$, such that its moments (A.7) are given by

$m_{0,1}(Z)=0\,,\quad m_{1,1}(Z)=1\,,\quad m_{0,2}(Z)=m_{0,2}\,,\quad
m_{0,3}(Z)=m_{0,3}\,,\quad\text{and}\quad m_{1,2}(Z)=m_{1,2}\,.$ (A.8)

######  Remark A.3.

A generalized version of this problem (constructing an atomic measure with
arbitrary number of prescribed moments), known as the _truncated complex
$K$-moment problem_, has been solved by Curto and Fialkow in [24]. To keep our
result self-contained, we give a simple independent proof for the special case
of three moments that we need here.

Having Lemma A.2 at hand, one can easily see that there exists a random
variable that has the prescribed first three moments and it has an independent
Gaussian component of given variance $\gamma>0$. More precisely, given
$m_{0,1}=0$, $m_{1,1}=1$, $m_{0,2}$, $m_{0,3}$, and $m_{1,2}$ with
$|m_{0,2}|\leq 1$ as the set of moments of $\chi_{\mathrm{od}}$, we look for a
representation of $Z$ in the form

$Z:=(1-\gamma)^{1/2}Z^{\prime}+\gamma^{1/2}\xi_{G}\quad\text{with}\quad\gamma\in(0,1)\quad\text{fixed}$

with some random variable $Z^{\prime}$ to be constructed, where $\xi_{G}$ is a
centered complex Gaussian random variable having second moments
$m_{0,2}(\xi_{G})=m_{0,2}$ and $m_{1,1}(\xi_{G})=1$. The moments of
$Z^{\prime}$ thus satisfy the relations

$m_{i,j}=(1-\gamma)^{(i+j)/2}m_{i,j}(Z^{\prime})+\gamma^{(i+j)/2}m_{i,j}(\xi_{G})\quad\text{with}\quad
1\leq i+j\leq 3.$ (A.9)

In particular, $|m_{0,2}(Z^{\prime})|=|m_{0,2}|\leq 1$, so the moment sequence
$m_{i,j}(Z^{\prime})$ from (A.9) satisfy the only nontrivial condition of
Lemma A.2. Therefore, by Lemma A.2, we can construct the random variable
$Z^{\prime}$. Finally, we remark that all random variables involved have
arbitrarily high moments (cf. Assumption 2.1). This moment matching argument
shows how to choose the distribution of the initial condition $W_{0}$ of the
Ornstein-Uhlenbeck flow (3.3) so that after time $T=\gamma$ it will match with
the distribution of the original matrix $W$ up to three moments.

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

We only outline the construction of the points
$z_{1},...,z_{11}\in{\mathbb{C}}$, the precise computations are a simple
exercise in calculus and linear algebra and hence omitted.

We set $z_{11}=0$ to be the origin. The remaining ten points are then placed
on five lines through the origin, carrying two points each, i.e. we put

$z_{j}=r_{j}\mathrm{e}^{\mathrm{i}\varphi_{j}}\quad\text{and}\quad
z_{11-j}=\hat{z}_{j}:=-\hat{r}_{j}\mathrm{e}^{\mathrm{i}\varphi_{j}}\quad\text{with}\quad
r_{j},\hat{r}_{j}\geq 0\,,\varphi_{j}\in[0,2\pi)\quad\text{for}\quad
j\in[5]\,.$

For simplicity, we can even prescribe four of the five angular variables in
such a way that the corresponding points lie on the real and imaginary axis
and the two diagonals, i.e. set $\varphi_{j}:=j\pi/4$ for $j\in[4]$.

We then take the law of $Z$ to be of the form

$\sum_{j\in[5]}\big{(}p_{j}\delta_{z_{j}}+\hat{p}_{j}\delta_{\hat{z}_{j}}\big{)}+\bigg{(}1-\sum_{j\in[5]}(p_{j}+\hat{p}_{j})\bigg{)}\delta_{0}$

for weights $p_{j},\hat{p}_{j}\geq 0$ satisfying
$\sum_{j\in[5]}(p_{j}+\hat{p}_{j})\leq 1$. As mentioned above, it is a simple
exercise to show that the remaining parameters
$r_{j},\hat{r}_{j},p_{j},\hat{p}_{j}\geq 0$ for $j\in[5]$ and
$\varphi_{5}\in[0,2\pi)$ can be chosen in such a way to accommodate (A.8).
More precisely, taking $A_{j}:=p_{j}r_{j}=\hat{p}_{j}\hat{r}_{j}\geq 0$ for
$j\in[5]$ (this ensures $m_{0,1}(Z)=0$), $r_{5}=\hat{r}_{5}$, and using our
choices of $\varphi_{j}=j\pi/4$ for $j\in[4]$, the two complex conditions
$m_{0,3}(Z)=m_{0,3}$ and $m_{1,2}(Z)=m_{1,2}$ turn into four real linear
equations for the variables $C_{j}:=B_{j}(r_{j}-\hat{r}_{j})\in{\mathbb{R}}$
for $j\in[4]$ with $B_{j}:=A_{j}(r_{j}+\hat{r}_{j})\geq 0$. The determinant of
this linear systems can easily seen to be non-vanishing and it thus determines
the difference variables $r_{j}-\hat{r}_{j}\in{\mathbb{R}}$ for $j\in[4]$.
Finally, the independent variables $\varphi_{5}\in[0,2\pi)$ and
$B_{j}:=A_{j}(r_{j}+\hat{r}_{j})\geq 0$ for $j\in[5]$ can easily be chosen to
satisfy $m_{1,1}(Z)=1$ and $m_{0,2}(Z)=m_{0,2}$. ∎

###  A.3. Additional proofs for Section 4

###### Proofs of Lemmas 4.1 and 4.8.

The claim of Lemma 4.1 follows by multi-linearity from Lemma 4.8.

For the proof of Lemma 4.8, we will use a _tensorization argument_ (or _meta
argument_) similar to [23] and [20, Proof of Lemma D.1]. Throughout this proof
the size $N$ of $W$ is fixed. For $d\in{\mathbb{N}}$ consider the
$(Nd)\times(Nd)$ Wigner matrix ${\bm{W}}^{(d)}$, i.e. the entries of
${\bm{W}}^{(d)}$ have variance $1/(Nd)$. Let ${\bm{W}}_{t}^{(d)}$ be the
Ornstein-Uhlenbeck flow as in (3.3) with initial condition
${\bm{W}}_{0}^{(d)}={\bm{W}}^{(d)}$, and define its resolvent
${\bm{G}}_{i,t}^{(d)}:=({\bm{W}}_{t}^{(d)}-z_{i,t})^{-1}$, then the
deterministic approximation of the resolvent is still given by $m_{1}$, the
Stieltjes transform of the semicircular law.

We now explain that also the deterministic approximation of products of
resolvents and deterministic matrices is unchanged. For $1\leq i\leq k$,
define ${\bm{A}}_{i}^{(d)}:=A_{i}\otimes I_{d}$, with $I_{d}$ denoting the
$d$–dimensional identity, then for ${\bm{M}}_{[1,k],t}^{(d)}$ defined as in
(2.10) with ${\bm{M}}_{i,t}^{(d)}$ and ${\bm{A}}_{i}^{(d)}$ we have

${\bm{M}}_{[1,k],t}^{(d)}:={\bm{M}}^{(d)}(z_{1,t},{\bm{A}}_{1}^{(d)},\dots,{\bm{A}}_{k-1}^{(d)},z_{k,t})=M(z_{1,t},A_{1},\dots,A_{k-1},z_{k,t})\otimes
I_{d}.$ (A.10)

Fix $0<s<t$, then integrating (4.5) for the bold faced resolvent and
deterministic matrices, in time from $s$ to $t$ and taking the expectation we
obtain

$\begin{split}&\langle{\bm{M}}_{[1,k],t}^{(d)}\bm{A}_{k}\rangle-\langle{\bm{M}}_{[1,k],s}^{(d)}\bm{A}_{k}\rangle\\\
=&-{\mathbb{E}}\langle(\bm{G}_{[1,k],t}-{\bm{M}}_{[1,k],t}^{(d)})\bm{A}_{k}\rangle+{\mathbb{E}}\langle(\bm{G}_{[1,k],s}-{\bm{M}}_{[1,k],s}^{(d)})\bm{A}_{k}\rangle+\frac{k}{2}\int_{s}^{t}{\mathbb{E}}\langle\bm{G}_{[1,k],r}\bm{A}_{k}\rangle\,\mathrm{d}r\\\
&+\sum_{i,j=1\atop
i<j}^{k}\int_{s}^{t}{\mathbb{E}}\langle\bm{G}_{[i,j],r}\rangle\langle\bm{G}_{[j,i],r}\rangle\,\mathrm{d}r+\sum_{i=1}^{k}\int_{s}^{t}{\mathbb{E}}\langle\bm{G}_{i,r}-m_{i,r}\rangle\langle\bm{G}^{(i)}_{[1,k],r}\bm{A}_{k}\rangle\,\mathrm{d}r+\frac{\sigma}{Nd}\sum_{i,j=1\atop
i\leq
j}^{k}\int_{s}^{t}{\mathbb{E}}\langle\bm{G}_{[i,j],r}\bm{G}_{[j,i],r}^{\mathfrak{t}}\rangle\,\mathrm{d}r\,.\end{split}$
(A.11)

Using the global law in Proposition 3.1 and (A.10), and taking the limit
$d\to\infty$, this implies that for $|\mathrm{Im}\,z_{i}|\gtrsim 1$ we have

$\langle M_{[1,k],t}A_{k}\rangle-\langle
M_{[1,k],s}A_{k}\rangle=\frac{k}{2}\int_{s}^{t}\langle
M_{[1,k],r}A_{k}\rangle\,\mathrm{d}r+\sum_{i,j=1,\atop
i<j}^{k-1}\int_{s}^{t}\langle M_{[i,j],r}\rangle\langle
M_{[j,i],r}\rangle\,\mathrm{d}r.$ (A.12)

Finally, dividing (A.12) by $t-s$ and taking the limit $s\to t$, we conclude
the proof of Lemma 4.8. ∎

###### Proof of Lemma 4.4.

The proof of this lemma is very similar to [18, Lemma 3.3]. Hence we give the
argument only for the case where $k$ is even, if $k$ is odd the proof is
completely analogous. Moreover, for notational simplicity we henceforth drop
the time dependence and the precise indices of $G$’s and $A$’s, i.e. write
$\mathrm{Im}\,G\equiv\mathrm{Im}\,G_{i}$, $A\equiv A_{j}$,
$\rho\equiv\rho_{i}$ and so on. Then, by application of the general bound

$|\langle B_{1}B_{2}B_{3}B_{4}\rangle|\leq
N\prod_{i=1}^{4}\langle|B_{i}|^{2}\rangle^{1/2}\quad\text{for all}\quad
B_{i}\in{\mathbb{C}}^{N\times N}$

applied to
$B_{i}=\sqrt{|\mathrm{Im}\,G|}A(\mathrm{Im}\,GA)^{k/2-1}\sqrt{|\mathrm{Im}\,G|}$
and with the aid of (2.17), we find that

$\begin{split}\Phi_{2k}&=\frac{\sqrt{N\hat{\ell}}}{N^{k-1}\,\rho^{2k}\,\langle|A|^{2}\rangle^{k}}\big{|}\big{\langle}(\mathrm{Im}\,GA)^{2k}-\widehat{{M}}_{[\hat{1},\widehat{2k}]}A\big{\rangle}\big{|}\\\
&\lesssim\sqrt{N\hat{\ell}}+\frac{\sqrt{N\hat{\ell}}}{N^{k-1}\,\rho^{2k}\,\langle|A|^{2}\rangle^{k}}N\big{|}\big{\langle}(\mathrm{Im}\,GA)^{k}\big{\rangle}\big{|}^{2}\\\
&\prec\sqrt{N\hat{\ell}}+\frac{\sqrt{N\hat{\ell}}}{N^{k-1}\,\rho^{2k}\,\langle|A|^{2}\rangle^{k}}N\left[N^{k/2-1}\rho^{k}\langle|A|^{2}\rangle^{k/2}\left(1+\frac{\phi_{k}}{\sqrt{N\hat{\ell}}}\right)\right]^{2}\\\
&\lesssim\sqrt{N\hat{\ell}}+\frac{\phi_{k}^{2}}{\sqrt{N\hat{\ell}}}\,.\end{split}$

We remark that, in order to bound
$\big{\langle}(\mathrm{Im}\,GA)^{k}\big{\rangle}$ in terms of $\phi_{k}$, we
added and subtracted the corresponding $M$-term and used the assumption that
$\Phi_{k}\prec\phi_{k}$. ∎

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