N^{1/3+o(1)}$ by Proposition 3.2 (i) and Proposition 3.1 (ii), recalling the
condition (3.2). We hence conclude (ii) using Lemma E.2.

For the proof of (iii), we first note that
$H^{(X)}\mathbf{v}_{r}(x)=H\mathbf{v}_{r}(x)$ since $X\cap B_{r}(x)=\emptyset$
and $\operatorname{supp}\mathbf{v}\subset B_{r-1}(x)$. To estimate
$(H-\mathbb{E}H-\Lambda(\alpha_{x}))\mathbf{v}_{r}(x)$, we now apply (ADK19, ,
Proposition 5.1). From the assumption $\log d\ll r\leqslant\frac{\log N}{6\log
d}$, $\lvert S_{1}(x)\rvert\leqslant 10\log N$ by Proposition 3.1 (i) and the
lower bound in (1.5), we conclude that the condition on $r$ in (ADK19, ,
Proposition 5.1) is satisfied141414See (ADK19, , eq. (5.1)) for the definition
of $r_{x}$.. The condition $\log d\ll r\leqslant\frac{\log N}{6\log d}$,
$2+\kappa\leqslant\alpha^{*}\leqslant\alpha_{x}$ for $x\in\mathcal{V}$ and
Proposition 3.1 (i) imply that the event from (ADK19, , eq. (5.4)) occurs with
high probability. Therefore, using the lower bound on $d$ from (1.5) and Lemma
B.2, we obtain from (ADK19, , Proposition 5.1) that

$\lVert(H-\mathbb{E}H-\Lambda(\alpha_{x}))\mathbf{v}_{r}(x)\rVert\lesssim(\log\log
N)^{-1}\bigg{(}\log\bigg{(}\frac{10\log N}{d}\bigg{)}\bigg{)}^{1/2}=o(1)$

with high probability. Using that $\operatorname{supp}\mathbf{v}_{r}(x)\subset
B_{r-1}(x)$ and $\mathbb{E}H_{xy}=\frac{\sqrt{d}}{N}\mathbbm{1}_{x\neq y}$, we
deduce that
$\lVert(\mathbb{E}H)\mathbf{v}_{r}(x)\rVert\leqslant\sqrt{\frac{d}{N}\lvert
B_{r}(x)\rvert}=o(1)$, where in the last step we used the upper bound on $r$
and Proposition 3.1 (ii). We hence conclude that
$\lVert(H^{(X)}-\Lambda(\alpha_{x}))\mathbf{v}_{r}(x)\rVert=o(1)$ by recalling
$H^{(X)}\mathbf{v}_{r}(x)=H\mathbf{v}_{r}(x)$.

We now turn to the proof of (iv). For the given $r\in\mathbb{N}$, we choose
$\mathbf{q}$ as in (ADK21, , Proposition 6.1). Then (ADK21, , Proposition 6.1)
yields $\lVert\mathbf{q}-\mathbf{e}\rVert\lesssim d^{-1/2}$ and
$\lVert(H-\sqrt{d})\mathbf{q}\rVert\lesssim d^{-1/2}$. In order to estimate
$\lVert\mathbf{w}_{1}-\mathbf{q}\rVert$, we use Lemma E.2 with $M=H$,
$\widehat{\lambda}=\sqrt{d}$ and $\mathbf{v}=\mathbf{q}$ whose assumptions we
check next. From (5.24) and eigenvalue interlacing, we conclude that
$\lambda_{2}(H)\ll\sqrt{d}$. Therefore, there is $\Delta\asymp\sqrt{d}$ such
that $H$ has a unique eigenvalue in $[\sqrt{d}-\Delta,\sqrt{d}+\Delta]$. Since
$\lVert(H-\sqrt{d})\mathbf{q}\rVert\lesssim d^{-1/2}$ as explained above, the
conditions of Lemma E.2 with $\Delta\asymp\sqrt{d}$ and $\varepsilon\asymp
d^{-1/2}$ are satisfied and we obtain
$\lVert\mathbf{w}_{1}-\mathbf{q}\rVert\lesssim d^{-1}$.

Finally, from (ADK21, , Proposition 6.1), we know the inclusion
$\operatorname{supp}\mathbf{q}\subset\big{(}\bigcup_{x\in\mathcal{U}}B_{r+1}(x)\big{)}^{c}$,
where

$\mathcal{U}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{cases}\big{\\{}x\in[N]\mathrel{\vbox{\hbox{.}\hbox{.}}}\alpha_{x}\geqslant
2+\xi^{1/4}\big{\\}}&\text{ if }d>(\log N)^{3/4}\\\
\big{\\{}x\in[N]\mathrel{\vbox{\hbox{.}\hbox{.}}}\alpha_{x}\geqslant\mathfrak{a}/5\big{\\}}&\text{
if }d\leqslant(\log N)^{3/4}\end{cases}$ (D.2)

with $\xi$ from (5.17) and $\mathfrak{a}$ the solution of
$h(\mathfrak{a}-1)=\frac{\log N}{d}$ (compare with (E.1) below).

We now show that $\mathcal{V}\subset\mathcal{U}$. If $d>(\log N)^{3/4}$, the
inclusion $\mathcal{V}\subset\mathcal{U}$ is obvious as $\xi=o(1)$ by (5.21)
and $\alpha_{x}\geqslant 2+\kappa$ by (2.1) and (1.6). For $d\leqslant(\log
N)^{3/4}$, we note that a short analysis of the definition of $\mathfrak{a}$
reveals that $\mathfrak{a}=\frac{t}{\log t}(1+o(1))$ with
$t\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{\log N}{d}$ if
$t\geqslant C$ for some sufficiently large constant $C$. By possibly
increasing $C$, we conclude from Lemma B.1 that
$\alpha^{*}(\mu)\geqslant\mathfrak{a}/5$ if $\mu\in[0,1/5)$ and $t\geqslant
C$. This implies $\mathcal{V}\setminus\mathcal{W}$ if $d\leqslant(\log
N)^{3/4}$ and completes the proof of Proposition 3.4 (iv). ∎

######  Remark D.2.

We note that in Proposition 3.4 (iv), actually
$\operatorname{supp}\mathbf{q}\subset\big{(}\bigcup_{x\in\mathcal{U}}B_{r+1}(x)\big{)}^{c}$
holds as shown in the above proof.

###### Proof of Lemma D.1.

We first recall $\xi$ and $\xi_{u}$ from (5.17). Throughout the proof, we
choose
$\tau\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1+\xi^{1/2}$.
Note that $\xi=o(1)$, $\tau=1+o(1)$ and $\xi_{\tau-1}=o(1)$ by (5.21).

The proof relies on an analogous argument in (ADK20, , proof of Proposition
3.12), to which we refer in the following. We begin by introducing some
notation. Following (ADK20, , Section 3.1), we define
$\mathcal{V}_{\alpha}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in[N]\mathrel{\vbox{\hbox{.}\hbox{.}}}\alpha_{x}\geqslant\alpha\\}$
for $\alpha>0$. We set
$\mathcal{U}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathcal{V}_{2+\xi^{1/4}}\setminus\mathcal{V}_{\alpha^{*}(\nu)}$;
we note that the set $\mathcal{U}\cup\mathcal{V}_{\alpha^{*}(\nu)}$ was
denoted by $\mathcal{V}$ in ADK20 , see (ADK20, , eqs. (1.9) and (3.3)).
Observe that $\mathcal{V}_{\alpha^{*}(\nu)}\subset\mathcal{V}_{2+\xi^{1/4}}$
by the definition of $\alpha^{*}$ in (1.6). From (5.23) and $\lVert
M^{(\mathcal{V}_{\alpha^{*}(\nu)})}\rVert\leqslant\lVert M\rVert$ for any
matrix $M$, we conclude that
$(H-\mathbb{E}H)^{(\mathcal{V}_{\alpha^{*}(\nu)})}=(\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}+o(1)$,
where $o(1)$ is in the sense of operator norm. Therefore, in order to prove
(D.1), it suffices to prove that, for any constant $\delta>0$, the matrix
$(\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}$ does not have an
eigenvalue larger than $\Lambda(\alpha^{*}(\nu))+\delta$.

Suppose, by contradiction, that $\lambda$ is an eigenvalue of
$(\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}$ with associated
normalized eigenvector $\mathbf{w}$ such that
$\lvert\lambda\rvert>\Lambda(\alpha^{*}(\nu))+\delta$. Let $x\in\mathcal{U}$.
Since
$(\operatorname{supp}\mathbf{v}^{\tau}_{\sigma}(x))\cap\mathcal{V}_{\alpha^{*}(\nu)}=\emptyset$
by the definition of $\mathbf{v}_{\sigma}^{\tau}(x)$ in (5.18), (ADK20, ,
Proposition 3.1 (i)) and $\alpha^{*}(\nu)\geqslant 2+\kappa\geqslant\tau$, we
deduce from the definition of $\widehat{H}^{\tau}$ in (5.20) that
$\mathbf{v}_{\sigma}^{\tau}(x)$ is an eigenvector of
$(\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}$ with eigenvalue
$\sigma\Lambda(\alpha_{x})$. As
$\Lambda(\alpha_{x})\leqslant\Lambda(\alpha^{*}(\nu))$ for any
$x\in\mathcal{U}$, we conclude that
$\mathbf{w}\perp\mathbf{v}^{\tau}_{\sigma}(x)$ for all $x\in\mathcal{U}$ and
$\sigma=\pm$. Since
$\operatorname{supp}\mathbf{w}\subset(\mathcal{V}_{\alpha^{*}(\nu)})^{c}$, we
deduce from (5.23), (ADK20, , Proposition 3.13)151515We stress that the
definition of $H$ in ADK20 differs from that in the current paper; see (ADK20,
, Definition 3.6). and (1.5) that

$\lvert\lambda\rvert=\lvert\langle\mathbf{w}\mspace{2.0mu},(\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}\mathbf{w}\rangle\rvert=\lvert\langle\mathbf{w}\mspace{2.0mu},\widehat{H}^{\tau}\mathbf{w}\rangle\rvert\leqslant
1+\tau+(1+o(1))\sum_{x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}}\alpha_{x}\langle\mathbf{1}_{x}\mspace{2.0mu},\mathbf{w}\rangle^{2}+o(1)\,.$
(D.3)

Note that $\tau=1+o(1)$ and $\Lambda(\alpha^{*}(\nu))\geqslant 2$ imply
$2\tau\leqslant\Lambda(\alpha^{*}(\nu))+o(1)$.

We will now apply Lemma 5.15 for each
$x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}$. Note that
$\lvert\lambda\rvert\geqslant\Lambda(\alpha^{*}(\nu))+\delta>2+o(1)=2\tau+C\xi$.
In particular, there is a constant $c\equiv c_{\delta}\in(0,1)$ such that
$\frac{2\tau+C\xi}{\lvert\lambda\rvert}\leqslant 1-c$. Owing to the
disjointness of the balls
$(B_{2r_{\star}}^{\tau}(x))_{x\in\mathcal{V}_{\tau}}$ (see (ADK20, ,
Proposition 3.1 (i))) and the locality of $\widehat{H}^{\tau}$, we have
$((\widehat{H}^{\tau})^{(\mathcal{V}_{\alpha^{*}(\nu)})}\mathbf{w})|_{B_{2r_{\star}}^{\tau}(x)}=(\widehat{H}^{\tau}\mathbf{w})|_{B_{2r_{\star}}^{\tau}(x)}$
for all $x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}$. Hence,
for these $x$, (5.36) holds. If
$x\in\mathcal{V}_{2+\xi^{1/4}}\setminus\mathcal{V}_{\alpha^{*}(\nu)}$ then
$\mathbf{w}\perp\mathbf{v}_{\pm}^{\tau}(x)$ as demonstrated above. Thus, all
conditions of Lemma 5.15 are satisfied for each
$x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}$. Therefore,
Lemma 5.15 and the disjointness of the balls
$(B_{2r_{\star}}^{\tau}(x))_{x\in\mathcal{V}_{\tau}}$ (compare with the
similar argument in (5.38)) yield

$\sum_{x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}}\langle\mathbf{1}_{x}\mspace{2.0mu},\mathbf{w}\rangle^{2}\leqslant\bigg{(}1-\frac{2\tau+C\xi}{\lvert\lambda\rvert}\bigg{)}^{-4}\bigg{(}\frac{2\tau+C\xi}{\lvert\lambda\rvert}\bigg{)}^{2r_{\star}}\leqslant
c^{-4}(1-c)^{2r_{\star}}\ll\frac{d}{\log N}\,,$

where we used $\frac{2\tau+C\xi}{\lvert\lambda\rvert}\leqslant 1-c$ in the
second step and $r_{\star}\asymp\sqrt{\log N}$ as well as (1.5) in the last
step. Hence, from Proposition 3.1 (i), we conclude
$\sum_{x\in\mathcal{V}_{\tau}\setminus\mathcal{V}_{\alpha^{*}(\nu)}}\alpha_{x}\langle\mathbf{1}_{x}\mspace{2.0mu},\mathbf{w}\rangle^{2}=o(1)$.
Using this in (D.3) implies
$\lvert\lambda\rvert\leqslant\Lambda(\alpha^{*}(\nu))+o(1)$, in contradiction
with the assumption on $\lambda$. We therefore conclude (D.1). ∎

##  Appendix E Tools

In this appendix we summarize several well known results used throughout this
paper.

### E.1. Bennett’s inequality

Define the function $h\colon[0,\infty)\to[0,\infty)$ by

$h(a)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(1+a)\log(1+a)-a\,.$
(E.1)

We use the notation $\mathcal{P}_{\mu}$ to denote a Poisson random variable
with parameter $\mu\geqslant 0$, and $\mathcal{B}_{n,p}$ to denote a binomial
random variable with parameters $n\in\mathbb{N}^{*}$ and $p\in[0,1]$. The
following estimate is proved in (BLM13, , Section 2.7).

######  Lemma E.1 (Bennett).

For $0\leqslant\mu\leqslant n$ and $a>0$ we have

$\mathbb{P}(\mathcal{B}_{n,\mu/n}-\mu\geqslant a\mu)\leqslant\mathrm{e}^{-\mu
h(a)}\,,\qquad\mathbb{P}(\mathcal{B}_{n,\mu/n}-\mu\leqslant-a\mu)\leqslant\mathrm{e}^{-\mu
a^{2}/2}\leqslant\mathrm{e}^{-\mu h(a)}\,,$

and $\frac{a^{2}}{2(1+a/3)}\leqslant h(a)\leqslant\frac{a^{2}}{2}$. By taking
$n\to\infty$, the same estimates hold with $\mathcal{B}_{n,\mu/n}$ replaced
with $\mathcal{P}_{\mu}$.

### E.2. Perturbation theory

The following lemma contains simple perturbation estimates for approximate
eigenvalues and eigenvectors. Its proof can be found in (ADK21, , Lemma 4.10).

######  Lemma E.2.

Let $M$ be a real symmetric matrix. Let $\varepsilon$, $\Delta>0$ satisfy
$5\varepsilon\leqslant\Delta$. Suppose that $M$ has a unique eigenvalue,
$\lambda$, in $[\widehat{\lambda}-\Delta,\widehat{\lambda}+\Delta]$ for some
$\widehat{\lambda}\in\mathbb{R}$. Let $\mathbf{w}$ be a corresponding
normalized eigenvector of $M$. If there exists a normalized vector
$\mathbf{v}$ such that
$\lVert(M-\widehat{\lambda})\mathbf{v}\rVert\leqslant\varepsilon$ then, for
some $\sigma\in\\{\pm\\}$,

$\lambda-\widehat{\lambda}=\langle\mathbf{v}\mspace{2.0mu},(M-\widehat{\lambda})\mathbf{v}\rangle+O\bigg{(}\frac{\varepsilon^{2}}{\Delta}\bigg{)},\qquad\qquad\lVert\mathbf{w}-\sigma\mathbf{v}\rVert=O\bigg{(}\frac{\varepsilon}{\Delta}\bigg{)}\,.$

### E.3. Resolvent of adjacency matrix of $\mathbb{N}^{*}$

Define the infinite tridiagonal matrix

$M\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}0&1&0&\\\
1&0&1&\ddots\\\ 0&1&0&\ddots\\\ &\ddots&\ddots&\ddots\end{pmatrix}\,,$ (E.2)

which we regard as a bounded operator on $\ell^{2}(\mathbb{N}^{*})$. Note that
$M$ is the adjacency matrix of $\mathbb{N}^{*}$ with the graph structure
induced by considering adjacent numbers as neighbours.

######  Lemma E.3.

Let $t>2$. Then $1-t^{-1}M$ is invertible with convergent Neumann series and
explicit inverse

$(1-t^{-1}M)^{-1}_{1j}=t\bigg{(}\frac{2}{t+\sqrt{t^{2}-4}}\bigg{)}^{j}$

for any $j\in\mathbb{N}^{*}$.

###### Proof.

By the Schur test, $\lVert M\rVert\leqslant 2$. Therefore, $t>2$ implies the
invertibility of $1-t^{-1}M$ in $\ell^{2}(\mathbb{N}^{*})$ and the convergence
of the Neumann series representation of its inverse. In particular,
$a=(a_{j})_{j\in\mathbb{N}^{*}}\in\ell^{2}(\mathbb{N}^{*})$ with
$a_{j}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(1-t^{-1}M)^{-1}_{1j}$
for any $j\in\mathbb{N}^{*}$. With
$e_{1}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\delta_{1j})_{j\in\mathbb{N}^{*}}$,
the definition of the resolvent yields $(1-t^{-1}M)a=e_{1}$ and, thus,

$a_{1}-t^{-1}a_{2}=1,\qquad\qquad\begin{pmatrix}a_{j+1}\\\
a_{j}\end{pmatrix}=T(t)\begin{pmatrix}a_{j}\\\ a_{j-1}\end{pmatrix},\qquad
T(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}t&-1\\\
1&0\end{pmatrix}$ (E.3)

for any $j\in\mathbb{N}$ satisfying $j\geqslant 2$. The transfer matrix $T(t)$
has the two eigenvalues, $\gamma$ and $\gamma^{-1}$, where

$\gamma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{2}{t+\sqrt{t^{2}-4}}\,.$

As $t>2$, we have $\gamma<1$ and, hence, $a\in\ell^{2}(\mathbb{N}^{*})$
implies $a_{j}=\gamma a_{j-1}=\gamma^{j-1}a_{1}$ for any $j\geqslant 2$.
Together with the first relation in (E.3), we obtain
$a_{1}=\frac{2t}{t+\sqrt{t^{2}-4}}$, which completes the proof. ∎

##  Appendix F The expected optimal range for $\mu$

In this appendix we give a heuristic argument that yields the expected optimal
size of the localized phase, by providing the optimal upper bound on the
exponent $\mu$. We also explain how the conclusion of this argument relates to
that of tarzia2022 .

The optimal value for $\mu$ can already be seen from the simple case where
$\mathcal{V}=\mathcal{W}=\\{x,y\\}$ consists of two vertices, $x$ and $y$,
which are are in resonance (i.e. with normalized degrees close to each other).
In that case, as in Proposition 5.3, we obtain with high probability

$(H-\lambda(x))\mathbf{u}(x)=\varepsilon_{y}(x)\mathbf{1}_{y}\,,\qquad\lvert\varepsilon_{y}(x)\rvert\leqslant
N^{-1/2+o(1)}\,,$

where the estimate on $\varepsilon_{y}(x)$ follows as in the proof of
Proposition 5.3.

We conclude that the tunnelling amplitude between $x$ and $y$ is with high
probability bounded by $N^{-1/2+o(1)}$. By Mott’s criterion, we therefore
expect localization at vertex $x$ whenever the typical eigenvalue spacing
$N^{-\rho_{b}(\lambda(x))+o(1)}$ is much larger than $N^{-1/2+o(1)}$, i.e.
$\rho_{b}(\lambda(x))<\frac{1}{2}$. This condition holds precisely for the
$N^{\mu}$ largest eigenvalues of the semilocalized phase for any
$\mu<\frac{1}{2}$ (see Appendix C). This conclusion coincides with that of
tarzia2022 , obtained by a different argument.

The preceding argument has two important shortcomings. First, it only
considers a single pair of resonant vertices, instead of the $N^{\mu+o(1)}$
vertices associated with the localized phase. Second, it assumes that the
eigenvalue spacing around $\lambda(x)$ is typical, i.e. $N^{-\mu}$, which is
true for most eigenvalues in the localized phase but is false for all
eigenvalues simultaneously.

To address these shortcomings, we define a random matrix model that
heuristically captures the main features of the localized phase. Define the
$\mathcal{W}\times\mathcal{W}$ matrix $M=(M_{xy})_{x,y\in\mathcal{W}}$ through

$M_{xy}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\langle\mathbf{u}(x)\mspace{2.0mu},H\mathbf{u}(y)\rangle\,.$

Recall from Proposition 2.3 that $(\mathbf{u}(x))_{x\in\mathcal{W}}$ are
approximate eigenvectors of $H$, and that all eigenvalues of $H$ in the
localized phase arise by perturbation from the approximate eigenvalues
$(\lambda(x))_{x\in\mathcal{W}}$. Hence, localization is tantamount to the
matrix $M$ being close to diagonal.

We make the simplifying assumption that the vectors
$(\mathbf{u}(x)\mathrel{\vbox{\hbox{.}\hbox{.}}}x\in\mathcal{W})$ are
orthogonal, i.e. we neglect their overlaps for the purpose of this heuristic
argument. As above, the proof of Proposition 5.3 implies that

$M_{xy}\approx\delta_{xy}\,\lambda(x)+O(N^{-1/2+o(1)})$

with high probability for all $x,y\in\mathcal{W}$, since (under the
orthogonality assumption)

$\displaystyle M_{xy}$
$\displaystyle\approx\delta_{xy}\,\lambda(x)+(1-\delta_{xy})\,\langle\mathbf{u}(x)\mspace{2.0mu},(H-\lambda(y))\mathbf{u}(y)\rangle$
$\displaystyle=\delta_{xy}\,\lambda(x)+(1-\delta_{xy})\,\varepsilon_{x}(y)\langle\mathbf{u}(x)\mspace{2.0mu},\mathbf{1}_{x}\rangle$
$\displaystyle\approx\delta_{xy}\,\lambda(x)+(1-\delta_{xy})\,\varepsilon_{x}(y)\,,$

where we used that
$\langle\mathbf{u}(x)\mspace{2.0mu},\mathbf{1}_{z}\rangle=0$ for any
$z\in\mathcal{W}\setminus\\{x\\}$.

Hence, the random variables $M_{xy}$ are typically of order $N^{-1/2+o(1)}$.
Assuming that they are mean zero, independent up to the symmetry constraint
$M_{xy}=M_{yx}$, and have light enough tails, we therefore arrive at the
following toy model for $M$:

$M\approx
M(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=D+\sqrt{t}\,W\,,\qquad
t=N^{\mu-1+o(1)}\,,$ (F.1)

where
$D=\operatorname{diag}(\lambda(x)\mathrel{\vbox{\hbox{.}\hbox{.}}}x\in\mathcal{W})$
and $W$ is an independent $\mathcal{W}\times\mathcal{W}$ Wigner matrix,
normalized so that its entries have variance $1/\lvert\mathcal{W}\rvert$ (and
hence $W$ has typically a norm of order $1$).

The matrix $M(t)$ from (F.1) is a deformed Wigner matrix, the kind of which
has been extensively studied in the random matrix theory literature. Assuming
that $W$ is a GOE matrix (i.e. with a Gaussian law), the law of $M(t)$ is
governed by Dyson Brownian motion at time $t$ starting from the diagonal
matrix $D$. We refer for instance to VW1 ; benaych2021eigenvectors ;
benigni2020eigenvectors ; LS1 ; von2019non . Although not stated explicitly in
these works, the analysis of Dyson Brownian motion, as for instance in
benigni2020eigenvectors , implies that, with high probability, for any
$x\in\mathcal{W}$, the eigenvector associated with the eigenvalue
$\lambda(t,x)$ of the $\mathcal{W}\times\mathcal{W}$ matrix $M(t)$ remains
localized for

$t\ll\lvert\mathcal{W}\rvert\,\Delta(x)^{2}\,,$ (F.2)

where
$\Delta(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\min_{y\neq
x}\lvert\lambda(x)-\lambda(y)\rvert$ is the eigenvalue spacing of $D$ around
$\lambda(x)$. Indeed, the eigenvalues $\lambda(t,x)$ of $M(t)$ satisfy

$\mathrm{d}\lambda(t,x)=\sqrt{\frac{2}{\lvert\mathcal{W}\rvert}}\,\mathrm{d}B(t,x)+\frac{1}{\lvert\mathcal{W}\rvert}\sum_{y\neq
x}\frac{\mathrm{d}t}{\lambda(t,x)-\lambda(t,y)}\,,$

where $(B(t,x)\mathrel{\vbox{\hbox{.}\hbox{.}}}t\geqslant 0,x\in\mathcal{W})$
is a family of independent standard Brownian motions. Heuristically, we
therefore see immediately that if (F.2) holds then
$\lvert\lambda(t,x)-\lambda(0,x)\rvert\ll\Delta(x)$, i.e. the shift in
$\lambda(x)$ is negligible compared to $\Delta(x)$. A similar calculation
using Dyson Brownian motion for the eigenvectors (see e.g.
benigni2020eigenvectors ) shows that they remain localized under the condition
(F.2). Another heuristic way of arriving at the condition (F.2) is to notice
that under it all finite-order corrections in perturbation theory to
$\lambda(t,x)$ are $o(\Delta(x))$, and similarly all finite-order corrections
in perturbation theory to the eigenvectors are $o(1)$.

We remark that the condition (F.2) is strictly weaker than Mott’s criterion,
which reads $t\ll\Delta(x)^{2}$. Hence, for the model (F.1), Mott’s criterion
is sufficient but not necessary for localization. Heuristically, this
discrepancy stems from the special relationship between the mean-field matrix
$W$ and the diagonal eigenvector basis of $D$, in which all entries of $W$ are
of order $N^{-\mu/2}$, while the norm of $W$ is of order $1$.

The typical eigenvalue spacing of $D$ is $N^{-\mu+o(1)}$, i.e.
$\Delta(x)=N^{-\mu+o(1)}$ for most $x\in\mathcal{W}$. In contrast, we expect
the minimal eigenvalue spacing $\min_{x\in\mathcal{W}}\Delta(x)$ to be
$N^{-2\mu+o(1)}$, as follows from basic extreme value theory (cf. the birthday
paradox) and the assumption that the random variables
$(\lambda(x)\mathrel{\vbox{\hbox{.}\hbox{.}}}x\in\mathcal{W})$ are
independent. Using $t=N^{\mu-1+o(1)}$ and
$\lvert\mathcal{W}\rvert=N^{\mu+o(1)}$, from the condition (F.2) we therefore
expect the following behaviour for the top $N^{\mu}$ eigenvectors.

  1. (i)

For $\mu<\frac{1}{2}$ most eigenvectors are localized.

  2. (ii)

For $\mu<\frac{1}{4}$ all eigenvectors are localized.

For $\frac{1}{4}\leqslant\mu<\frac{1}{2}$, while most eigenvectors are
localized, we expect some eigenvectors to hybridize with others associated
with resonant vertices. These hybridized eigenvectors have a divergent
localization length (1.14). In fact, a more refined analysis of Dyson Brownian
motion in (F.1), as performed for instance in benigni2020eigenvectors , yields
a precise picture of the hybridized structure of such eigenvectors. We shall
not go into further details in this appendix.

In particular, the optimal bound for $\mu$ in Theorem 1.1 is expected to be
$\frac{1}{4}$. In our proof we obtain the bound $\frac{1}{24}$ because both
our upper bound on the tunnelling amplitude (Proposition 2.3) is worse than
$N^{-1/2+o(1)}$ and because our lower bound on the eigenvalue spacing
(Proposition 2.4) is worse than $N^{-2\mu+o(1)}$. Both of these bounds,
however, can be somewhat improved by a suitable refinement of our proof.

## Acknowledgements

JA acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No.
895698 as well as funding from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1,
project-id 390685813. RD acknowledges funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 884584). AK acknowledges funding from the
European Research Council (ERC) and the Swiss State Secretariat for Education,
Research and Innovation (SERI) through the consolidator grant ProbQuant. JA
and AK acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 715539), funding from the Swiss National Science Foundation
through the NCCR SwissMAP grant, and support from the US National Science
Foundation under Grant No. DMS-1928930 during their participation in the
program _Universality and Integrability in Random Matrix Theory and
Interacting Particle Systems_ hosted by the Mathematical Sciences Research
Institute in Berkeley, California during the Fall semester of 2021.

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Johannes Alt, University of Bonn, Institute for Applied Mathematics,
<EMAIL_ADDRESS>
Raphaël Ducatez, Université Lyon 1, Institut Camille Jordan,
<EMAIL_ADDRESS>
Antti Knowles, University of Geneva, Section of Mathematics,
<EMAIL_ADDRESS>