The operator $Q_{\delta}AQ_{\delta}$ is bounded from below by $-\|A\|$ which
is estimated in (4.21). We thus conclude that for all $N$ large enough:

$Q_{\delta}H^{\prime}+Q_{\delta}AQ_{\delta}\geq
E_{\delta/2}-\|A\|\geq-N\left(\beta_{c}-\delta/2+2\Gamma\sqrt{\alpha(1-\alpha)}+o_{\Gamma,\alpha}(1)\right)\geq-N\left(\beta_{c}-\delta/4\right).$

Consequently, the Schur complement matrix

$S_{\delta}(E):=\left(Q_{\delta}H^{\prime}+Q_{\delta}AQ_{\delta}-E\right)^{-1}$

is well defined on $Q_{\delta}\ell^{2}(\mathcal{Q}_{N})$ and bounded,
$\|S_{\delta}(E)\|\leq(E_{\delta/4}-E)^{-1}$ for any $E<E_{\delta/4}$.

The spectrum of $H$ below $E_{\delta/4}=-N\left(\beta_{c}-\delta/4\right)$ is
thus characterized using Schur’s method, which yields:

  1. 1.

$E<E_{\delta/4}$ is an eigenvalue of $H$ if and only if
$E\in\operatorname{spec}\left(P_{\delta}H^{\prime}-P_{\delta}AS_{\delta}(E)AP_{\delta}\right)$.

  2. 2.

The $\ell^{2}$-normalized eigenvector $\psi$ corresponding to $E$ and $H$
satisfies:

$\displaystyle(P_{\delta}H^{\prime}-EP_{\delta})\psi$
$\displaystyle=P_{\delta}AS_{\delta}(E)AP_{\delta}\psi$ $\displaystyle
Q_{\delta}\psi$ $\displaystyle=-S_{\delta}(E)AP_{\delta}\psi.$ (4.24)

We now proceed with the completion of the proof of the assertion on the
spectrum and eigenvectors separately.

Spectrum: The spectrum of $H$ below $E_{\delta/8}$ is determined through the
above Schur complement method. Since for all $E\leq E_{\delta/8}$ at at some
$C=C(\Gamma,\delta)<\infty$ and $c=c(\alpha)>0$

$\|P_{\delta}AS_{\delta}(E)AP_{\delta}\|\leq\|S_{\delta}(E)\|\
\|AP_{\delta}\|^{2}\leq C\ N^{3}\,e^{-2cN},$ (4.25)

the eigenvalues below $E_{\delta/8}$ thus coincide with the eigenvalues of
$P_{\delta}H^{\prime}$ below this energy up to an error, which is
exponentially small in $N$ [13, Corollary 3.2.6]. Since the eigenvalues of
$P_{\delta}H^{\prime}$ are given by (4.3), the assertion in Theorem 1.5
follows.

Eigenvectors: We now concentrate our attention on energies below
$E_{s}=-N(\beta_{c}-s)$ with $s\in(0,\delta/8]$ small enough such that
$2\beta_{c}s<c$ with the decay rate $c>0$ from (4.25). This ensures that
$e^{-\alpha cN}<e^{-2\beta_{c}sN}\eqqcolon r(s)$ for all sufficiently large
$N$. According to the spectral averaging Lemma 4.4, since $s\leq\delta/8$ and
the condition (4.7) is monotone in $\delta$ the event

$\left\\{\forall\,\boldsymbol{\sigma},\boldsymbol{\sigma}^{\prime}\in\mathcal{L}_{\beta_{c}-s},\boldsymbol{\sigma}\neq\boldsymbol{\sigma}^{\prime}:\quad\operatorname{dist}\left(\operatorname{spec}_{E_{s}}H_{\alpha
N}(\boldsymbol{\sigma}),\operatorname{spec}_{E_{s}}H_{\alpha
N}(\boldsymbol{\sigma}^{\prime})\right)>r(s)\right\\}$ (4.26)

has probability of at least $1-4e^{-s^{2}N}-e^{-cN}$ for some $c>0$. We may
therefore assume its occurrence.

Perturbation theory based on the above Schur complement analysis and (4.25)
(combined with the characterization of eigenvalues established in Theorem 1.5)
then guarantees that the eigenvector $\psi$ of $H$ corresponding to the
eigenvalue
$E=U(\boldsymbol{\sigma})+\Gamma^{2}N/U(\boldsymbol{\sigma})+\mathcal{O}(N^{-1/4})$,
which is uniquely characterized by
$\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-s}$, is norm-close to the
ground-state eigenvector $\psi_{\boldsymbol{\sigma}}$ of $H_{\alpha
N}(\boldsymbol{\sigma})$, i.e.,

$\displaystyle\|\psi-\psi_{\boldsymbol{\sigma}}\|$
$\displaystyle\leq\|P_{\delta}\psi-\psi_{\boldsymbol{\sigma}}\|+\|Q_{\delta}\psi\|$
$\displaystyle\leq\frac{\|P_{\delta}AS_{\delta}(E)AP_{\delta}\|}{r(s)}+\|S_{\delta}(E)\|\
\|AP_{\delta}\|\leq C\ e^{-cN}.$ (4.27)

Here the inequalities combine (2)–(4.26). The rest of the claim on the
$\ell^{2}$-estimates of the eigenvectors then follows from the respective
properties of $\psi_{\boldsymbol{\sigma}}$ established in Lemma 4.3. The event
$\Omega_{N,\Gamma,\delta}^{\textup{loc}}$ is then defined by specifying a
value for $\alpha_{0}=\alpha_{0}(\Gamma,\delta)$ and intersecting
$\Omega_{N}(\Gamma,\delta,\alpha_{0})$ with (4.26).

∎

###  4.5 Proof of Theorem 1.7

All assertions concerning the $\ell^{2}$-properties of the ground-state can
easily be collected from the proof of Theorem 1.5.

###### Proof of Theorem 1.7 – $\ell^{2}$-properties.

According to Theorem 1.5 for all events aside from one of exponentially small
probability, there is some $\boldsymbol{\sigma}_{0}\in\mathcal{Q}_{N}$ such
that the ground state eigenvector is approximated by
$\|\psi-\delta_{\boldsymbol{\sigma}_{0}}\|_{\ell^{2}}^{2}=\mathcal{O}_{\Gamma}\left(\frac{1}{N}\right)$.
The estimate $\mathcal{O}_{\Gamma}\left(\frac{1}{N}\right)$ does not depend on
$\delta$ anymore as we may fix $\delta$, if we only consider the ground state.
This will be always assumed in the following. Moreover, the ground-state
energy is
$E=U(\boldsymbol{\sigma}_{0})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma}_{0})}+\mathcal{O}_{\Gamma}\left(N^{-1/4}\right)$,
where $U(\boldsymbol{\sigma}_{0})$ is one of the REM’s extremal energies for
which we may assume that

$\left|U(\boldsymbol{\sigma}_{0})+\beta_{c}N\right|\leq\mathcal{O}(\sqrt{N}),\quad\text{and
hence }\quad\left|E+\beta_{c}N\right|\leq\mathcal{O}(\sqrt{N})$ (4.28)

at the expense of excluding another event of exponentially small probability
stemming from deviations to the known extremal statistics of the REM, cf.
(1.7).

It thus remains to establish the assertion on the first order perturbation
$\xi\in\ell^{2}(\mathcal{Q}_{N})$. That $\langle\xi|H\xi\rangle$ agrees with
the ground state energy up to order $o_{\Gamma}(1)$ is a result of a simple
calculation and a comparison with the above formula for $E$. It remains to
prove $\|\psi-\xi\|^{2}=\mathcal{O}_{\Gamma}(N^{-2})$. To this end, we revisit
the proof of Theorem 1.5. From the validity of the global
$(\beta_{c}/2,\delta,\alpha)$-deep hole scenario specified there and in view
of (4.9), it suffices to show

$\left|\psi(\boldsymbol{\sigma_{0}})-\sqrt{1-\frac{\Gamma^{2}}{\beta_{c}^{2}N}}\right|^{2}=\mathcal{O}_{\Gamma}(N^{-2})\quad\text{and}\quad\sum_{\boldsymbol{\sigma}\in
S_{1}(\boldsymbol{\sigma}_{0})}\left|\psi(\boldsymbol{\sigma})-\frac{\Gamma}{\beta_{c}N}\right|^{2}=\mathcal{O}_{\Gamma}(N^{-2}).$
(4.29)

For a proof of these assertions we use the eigenvalue equation (4.16) on
$S_{1}(\boldsymbol{\sigma}_{0})$ together with
$\psi({\boldsymbol{\sigma}_{0}})=1+\mathcal{O}_{\Gamma}(N^{-1})$ . If we pick
$\boldsymbol{\sigma}\in S_{1}(\boldsymbol{\sigma}_{0})$, this yields

$\begin{split}\psi({\boldsymbol{\sigma}})-\frac{\Gamma}{\beta_{c}N}&=-\frac{\Gamma}{E-U(\boldsymbol{\sigma})}\left(1+\mathcal{O}_{\Gamma}(N^{-1})\right)-\frac{\Gamma}{\beta_{c}N}\\\
&=\frac{\Gamma
U(\boldsymbol{\sigma})}{\beta_{c}N(E-U(\boldsymbol{\sigma}))}+\mathcal{O}_{\Gamma}(N^{-3/2}).\end{split}$

Here the last step also relied on the estimate
$|U(\boldsymbol{\sigma})|\leq\varepsilon N$ valid in the
$(\varepsilon,\delta,\alpha)$-deep hole scenario, as well as (4.28). With a
suitable constant $C=C(\Gamma)<\infty$, we then have

$\sum_{\boldsymbol{\sigma}\in
S_{1}(\boldsymbol{\sigma}_{0})}\left|\psi(\boldsymbol{\sigma})-\frac{\Gamma}{\beta_{c}N}\right|^{2}\leq\frac{C}{N^{4}}\sum_{\boldsymbol{\sigma}\in
S_{1}(\boldsymbol{\sigma}_{0})}U(\boldsymbol{\sigma})^{2}+\mathcal{O}_{\Gamma}(N^{-2})$

An exponential Chebychev-Markov estimate leads to
${\mathbb{P}}(N^{-2}\sum_{\boldsymbol{\sigma}\in
S_{1}(\boldsymbol{\sigma}_{0})}(U(\boldsymbol{\sigma})^{2}-N))\geq 1)\leq
e^{-cN}$ for some $c>0$. Thus, except for an event of exponentially small
probability the second claim in (4.29) holds. Since $\psi$ is
$\ell^{2}$-normalized, this leads to

$\psi(\boldsymbol{\sigma_{0}})^{2}=1-\sum_{\boldsymbol{\sigma}\in
S_{1}(\boldsymbol{\sigma}_{0})}\psi(\boldsymbol{\sigma})^{2}+\mathcal{O}_{\Gamma}(N^{-2})=1-\frac{\Gamma^{2}}{\beta_{c}^{2}N}+\mathcal{O}_{\Gamma}(N^{-2}),$

which readily implies the first claim in (4.29). ∎

For a proof of the $\ell^{1}$-estimate on the ground state eigenfunction, we
need to sharpen estimates on the large-deviation geometry of the REM. To this
end we define for $\varepsilon,\delta>0$ the following tripartition of the
Hamming cube:

$\begin{split}A_{1}(\varepsilon)&\coloneqq\\{\boldsymbol{\sigma}\in\mathcal{Q}_{N}\,|\,|U(\boldsymbol{\sigma})|\leq\varepsilon
N\\}\\\
A_{2}(\varepsilon,\delta)&\coloneqq\\{\boldsymbol{\sigma}\in\mathcal{Q}_{N}\,|\,\varepsilon
N<|U(\boldsymbol{\sigma})|\leq(\beta_{c}-\delta)N\\}\\\
A_{3}(\delta)&\coloneqq\\{\boldsymbol{\sigma}\in\mathcal{Q}_{N}\,|\,|U(\boldsymbol{\sigma})|>(\beta_{c}-\delta)N\\}.\
\end{split}$

A modification of ideas used in the proof of Lemma 4.2 and [50, Lemma 2]
yields:

######  Lemma 4.6.

For any $\varepsilon>0$ there exist $K=K(\varepsilon)\in{\mathbb{N}}$ and a
family of events $\Omega_{\varepsilon,N}$ such that for $N$ large enough

  1. (i)

For any $\boldsymbol{\sigma}\in A_{2}(\varepsilon,\delta)\cup A_{3}(\delta)$:
$\displaystyle|B_{4}(\boldsymbol{\sigma})\cap(A_{2}(\varepsilon,\delta)\cup
A_{3}(\delta))|\leq K$ on $\Omega_{\varepsilon,N}$.

  2. (ii)

${\mathbb{P}}(\Omega_{\varepsilon,N})\geq 1-2^{-N}$.

###### Proof.

Let $\Omega_{\varepsilon,N,K}$ be the event, where the assertion (i) holds
true with constant $K$. It remains to show that the complement satisfies
${\mathbb{P}}(\Omega_{\varepsilon,N,K}^{c})\leq 2^{-N}$ for an appropriate
choice for $K$ and $N$ large enough. To this end we estimate

$\begin{split}&\quad\,{\mathbb{P}}(\Omega_{\varepsilon,N,K}^{c})={\mathbb{P}}(\exists\
\boldsymbol{\sigma}\in A_{2}(\varepsilon,\delta)\cup
A_{3}(\delta)\;\text{s.t.}\;|B_{4}(\boldsymbol{\sigma})\cap(A_{2}(\varepsilon,\delta)\cup
A_{3}(\delta))|\geq K)\\\
&\leq\sum_{\boldsymbol{\sigma}_{0}\in\mathcal{Q}_{N}}{\mathbb{P}}(|U(\boldsymbol{\sigma}_{0})|\geq\varepsilon
N)\,{\mathbb{P}}(\exists\,K-1\text{ different
}\boldsymbol{\sigma}_{1},\ldots\boldsymbol{\sigma}_{K-1}\in
B_{4}(\boldsymbol{\sigma}_{0})\backslash\\{\boldsymbol{\sigma}_{0}\\}\,\text{s.t.}\\\
&\mkern 400.0mu\,|U(\boldsymbol{\sigma}_{j})|\geq\varepsilon N\text{ for
}j=1,\ldots,K-1)\\\
&\leq\binom{N^{4}}{K-1}\,2^{N}\,{\mathbb{P}}(|U(\boldsymbol{\sigma}_{0})|\geq\varepsilon
N)^{K}\leq N^{4K}2^{N}e^{-KN\varepsilon^{2}/2},\end{split}.$

Here the second line is a consequence of the union bound and the third line
follows from the independence and a simple counting argument. Choosing $K>4\ln
2/\varepsilon^{2}$, we see that
${\mathbb{P}}(\Omega_{\varepsilon,N,K}^{c})<2^{-N}$ for $N$ large enough.

∎

As a final preparation, we also need the following elementary observation on
the size of large deviation sites.

######  Lemma 4.7.

For any $\delta\in(0,\beta_{c})$ and all $N$:

${\mathbb{P}}\left(|A_{3}(\delta)|\geq 2e^{\beta_{c}\delta N}\right)\leq
e^{-N\delta^{2}/2}.$

###### Proof.

The cardinality $|A_{3}(\delta)|$ is a sum of $2^{N}$ independent Bernoulli
variables with success probability
$p={\mathbb{P}}(|U(\boldsymbol{\sigma})|>(\beta_{c}-\delta)N)\leq
2e^{-\frac{1}{2}(\beta_{c}-\delta)^{2}N}$ such that
$\mathbb{E}[|A_{3}((\delta))|]=2^{N}p$. The claim thus follows from a standard
Markov estimate. ∎

We are finally ready to finish the proof of our main result in the
localization regime.

###### Proof of Theorem 1.7 – $\ell^{p}$-properties.

We first observe that the claims on the $\ell^{p}$-norms immediately follow
from the claimed $\ell^{1}$-norm asymptotics (1.25). To see this, recall that
$\psi(\boldsymbol{\sigma}_{0})=1+o_{\Gamma}(1)$ for some
$\boldsymbol{\sigma}_{0}\in\mathcal{Q}_{N}$, and that
$\psi(\boldsymbol{\sigma})\leq c\,N^{-1}$ for all
$\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}$. Hence for any $1<p<\infty$:

$1+o_{\Gamma}(1)\leq\|\psi\|_{\ell^{p}}^{p}\leq
1+\frac{c^{p-1}}{N^{p-1}}\sum_{\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}}\psi(\boldsymbol{\sigma})\leq
1+\frac{c^{p-1}}{N^{p-1}}\|\psi\|_{\ell^{1}}=1+o_{\Gamma,p}(1).$

It therefore remains to establish the $\ell^{1}$ claim (1.25).

Recalling that the ground state wavefunction $\psi$ is positive, we can write
$\|\psi\|_{\ell^{1}}=\sum_{\boldsymbol{\sigma}}\psi(\boldsymbol{\sigma})$. The
eigenvalue equation for $\psi$ leads to

$\begin{split}E\sum_{\boldsymbol{\sigma}}\psi(\boldsymbol{\sigma})&=+\Gamma\sum_{\boldsymbol{\sigma}}(T\psi)(\boldsymbol{\sigma})+\sum_{\boldsymbol{\sigma}}(U\psi)(\boldsymbol{\sigma})=-\Gamma
N\sum_{\boldsymbol{\sigma}}\psi(\boldsymbol{\sigma})-\sum_{\boldsymbol{\sigma}}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma})\\\
&=-\Gamma
N\sum_{\boldsymbol{\sigma}}\psi(\boldsymbol{\sigma})+U(\boldsymbol{\sigma}_{0})\psi(\boldsymbol{\sigma}_{0})+\sum_{\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma}).\end{split}$
(4.30)

The second equality follows from the fact that each $\boldsymbol{\sigma}$ has
$N$ neighbors. The main idea is now to show that the remainder term
$\sum_{\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma})$
can be controlled by the other two terms on the right side. Here, we use the
tripartition $A_{1}(\varepsilon),A_{2}(\varepsilon,\delta),A_{3}(\delta)$ of
the configuration space and bound the contribution of each $A_{i}$ separately.

In the following, we fix $\delta,\alpha>0$ small enough, such that the REM
satisfies a global $(\beta_{c}/2,\delta,\alpha)$-deep hole scenario with a
probability which is exponentially close to one. Moreover, we pick
$\varepsilon>0$ arbitrary and fix $K=K(\varepsilon)\in\mathbb{N}$ the
assertions of Lemma 4.6 hold on a joint event on which the global
$(\beta_{c}/2,\delta_{0},\alpha_{0})$-deep hole scenario applies as well. This
event still has a probability of at least $1-e^{-c(\delta,\alpha)N}$ with some
$c(\delta,\alpha)>0$, which is independent of $\varepsilon$.

Contribution of $A_{1}(\varepsilon)$: In this case we use the trivial
estimate, $\left|\sum_{\boldsymbol{\sigma}\in
A_{1}}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma})\right|\leq\varepsilon
N\|\psi\|_{\ell^{1}}$.

Contribution of $A_{3}(\delta)$: We only consider $\delta\leq\delta_{0}$, such
that configurations $\boldsymbol{\sigma}\in
A_{3}(\delta)\setminus\\{\boldsymbol{\sigma}_{0}\\}$ lie outside the ball
$B_{\alpha_{0}N}(\boldsymbol{\sigma}_{0})$. In particular, there is some $c>0$
such that for all $N$ large enough and all $\boldsymbol{\sigma}\in
A_{3}(\delta)\setminus\\{\boldsymbol{\sigma}_{0}\\}$ the ground state is
uniformly bounded, $|\psi(\boldsymbol{\sigma})|\leq e^{-cN}$. We now pick
$\delta\coloneqq\min\\{\delta_{0},c/(4\beta_{c})\\}$ and shrink the considered
event such that $|A_{3}(\delta)|\leq 2\ e^{Nc/4}$. According to Lemma 4.7 this
event still has a probability greater than $1-e^{-c(\delta,\alpha)N}$ with
some $c(\delta,\alpha)>0$. On this event, we conclude for all $N$ large enough

$\sum_{\boldsymbol{\sigma}\in
A_{3}(\delta)\setminus\\{\boldsymbol{\sigma}_{0}\\}}|U(\boldsymbol{\sigma})|\psi(\boldsymbol{\sigma})\leq
e^{-Nc/2}.$

Contribution of $A_{2}(\varepsilon,\delta)$: We first consider the
configurations in $A_{2}(\varepsilon,\delta)$ close to the center
$\boldsymbol{\sigma}_{0}$, which we estimate for $N$ large by

$\begin{split}\sum_{\boldsymbol{\sigma}\in A_{2}(\varepsilon,\delta)\cap
B_{4}(\boldsymbol{\sigma}_{0})}|U(\boldsymbol{\sigma})|\psi(\boldsymbol{\sigma})&\leq|A_{2}(\varepsilon,\delta)\cap
B_{4}(\boldsymbol{\sigma}_{0})|\max_{\boldsymbol{\sigma}\in
B_{4}(\boldsymbol{\sigma}_{0})\backslash\\{\boldsymbol{\sigma}_{0}\\}}|U(\boldsymbol{\sigma})|\psi(\boldsymbol{\sigma})\leq
C\,K\end{split}$

with some $C=C(\Gamma)$. We use $|U(\boldsymbol{\sigma})|\leq\beta_{c}N/2$ due
to the validity of the global $(\beta_{c}/2,\delta,\alpha)$-deep hole scenario
as well as the pointwise bound $\psi(\boldsymbol{\sigma})\leq CN^{-1}$ for all
$\boldsymbol{\sigma}\in B_{4}(\boldsymbol{\sigma}_{0})$ with
$\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}$.

It remains to consider $\boldsymbol{\sigma}\in
A_{2}(\varepsilon,\delta)\setminus B_{4}(\boldsymbol{\sigma}_{0})$. The
eigenvalue equation reads

$|E-U(\boldsymbol{\sigma})|\psi(\boldsymbol{\sigma})=\Gamma\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime}).$

Since $E\leq(\beta_{c}-\delta/2)N$ for $N$ large enough, we obtain for
$\boldsymbol{\sigma}\in A_{2}(\varepsilon,\delta)$ the bound

$\psi(\boldsymbol{\sigma})\leq\frac{2\Gamma}{\delta
N}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime}).$

The essence of the following argument is that the value of any
$\psi(\boldsymbol{\sigma})$ is comparable to the mean on the corresponding
$S_{1}(\boldsymbol{\sigma})$ sphere and, thus, $\psi$ cannot take especially
large values on $A_{2}(\varepsilon,\delta)$. To make this intuition precise,
we separate the $A_{3}(\delta)$ configurations, which we possibly encounter in
the spherical mean and repeat the procedure for the remaining
$\boldsymbol{\sigma}^{\prime}\in S_{1}(\boldsymbol{\sigma})$. This leads to

$\begin{split}\psi(\boldsymbol{\sigma})&\leq\frac{2\Gamma}{\delta
N}\sum_{\boldsymbol{\sigma}^{\prime}\in S_{1}(\boldsymbol{\sigma})\cap
A_{3}(\delta)}\psi(\boldsymbol{\sigma}^{\prime})+\frac{4\Gamma^{2}}{\delta^{2}N^{2}}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})\setminus
A_{3}(\delta)}\sum_{\boldsymbol{\sigma}^{\prime\prime}\in
S_{1}(\boldsymbol{\sigma}^{\prime})}\psi(\boldsymbol{\sigma}^{\prime\prime})\\\
&\leq\frac{2\Gamma}{\delta N}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})\cap
A_{3}(\delta)}\psi(\boldsymbol{\sigma}^{\prime})+\frac{4\Gamma^{2}}{\delta^{2}N}\psi(\boldsymbol{\sigma})+\frac{8\Gamma^{2}}{\delta^{2}N^{2}}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{2}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime}),\end{split}$

which for $N$ large enough implies

$\psi(\boldsymbol{\sigma})\leq\frac{4\Gamma}{\delta
N}\sum_{\boldsymbol{\sigma}^{\prime}\in S_{1}(\boldsymbol{\sigma})\cap
A_{3}(\delta)}\psi(\boldsymbol{\sigma}^{\prime})+\frac{16\Gamma^{2}}{\delta^{2}N^{2}}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{2}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime}).$

We now further shrink the considered event to ensure that $\|U\|_{\infty}\leq
2\beta_{c}N$ holds true. This happens for all but an event of exponentially
probability, cf. (1.6). Thus, for $N$ large enough

$\begin{split}\left|\sum_{\boldsymbol{\sigma}\in A_{2}\setminus
B_{4}(\boldsymbol{\sigma}_{0})}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma})\right|&\leq
2\beta_{c}N\sum_{\boldsymbol{\sigma}\in A_{2}\setminus
B_{4}(\boldsymbol{\sigma}_{0})}\psi(\boldsymbol{\sigma})\\\
&\leq\sum_{\boldsymbol{\sigma}\in A_{2}\setminus
B_{4}(\boldsymbol{\sigma}_{0})}\Big{(}\frac{8\beta_{c}\Gamma}{\delta}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})\cap
A_{3}(\delta)}\psi(\boldsymbol{\sigma}^{\prime})+\frac{16\beta_{c}\Gamma^{2}}{\delta^{2}N}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{2}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime})\Big{)}\\\
&\leq\frac{8\beta_{c}K\Gamma}{\delta}\sum_{\boldsymbol{\sigma}\in
A_{3}(\delta)\setminus\\{\boldsymbol{\sigma}_{0}\\}}\psi(\boldsymbol{\sigma})+\frac{16\beta_{c}\Gamma^{2}K}{\delta^{2}N}\|\psi\|_{\ell^{1}}\\\
&\leq
e^{-Nc/4}+\frac{16\beta_{c}\Gamma^{2}K}{\delta^{2}N}\|\psi\|_{\ell^{1}}.\end{split}$

In the third line we used the observation that each configuration
$\boldsymbol{\sigma}\in\mathcal{Q}_{N}$ appears in the summation at most $K$
times due to Lemma 4.6. The last step is a consequence of our exponential
bound on $\psi(\boldsymbol{\sigma})$ on the $A_{3}(\delta)$-configurations.

Combining the partial results on each $A_{J}$, we arrive with some
$C=C(\Gamma)$ at the bound

$|\sum_{\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}}U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma})|\leq(2\varepsilon+\mathcal{O}_{K,\Gamma}(N^{-1}))N\|\psi\|_{\ell^{1}}+4CK$

which is valid on with probability of at least $1-e^{-cN}$ with some $c>0$
which in independent of $\varepsilon$. Since $\varepsilon>0$ was arbitrary,
the claimed convergence now follows from (4.30). ∎

###  4.6 Proof of Corollary 1.6

Corollary 1.6 is a direct consequence of Theorem 1.5.

###### Proof of Corollary 1.6.

We recall [39] that for random measures $(\mu_{n})_{n\in{\mathbb{N}}},\mu$ on
${\mathbb{R}}$ the almost sure convergence
$\lim_{n\to\infty}\int\,f\,d\mu_{n}=\int\,f\,d\mu$ for any compactly supported
continuous function $f\in C_{c}({\mathbb{R}})$ does imply the almost sure weak
convergence $\mu_{n}\to\mu$, since $C_{c}({\mathbb{R}})$ has a countable dense
subset.

Let $\delta>0$ be small enough and $\Omega_{N,\Gamma,\delta}^{\textup{loc}}$
the sequence of events from Theorem 1.5. Moreover, for $M\in{\mathbb{N}}$ we
set
$\Omega_{\Gamma,M}^{\infty}\coloneqq\bigcap_{N=M+1}\Omega_{N,\Gamma,\delta}^{\textup{loc}}$.
We now pick $f\in C_{c}({\mathbb{R}})$ with
${\operatorname{supp}\,}(f)\subset(a,b)$ for some $a,b\in{\mathbb{R}}$, and
let $E\in\operatorname{spec}H$ be an eigenvalue such that
$s_{N}(E,\Gamma)^{-1}\in(a,b)$, i.e.,

$-\beta_{c}N+\frac{\ln(N\ln
2)+\ln(4\pi)}{2\beta_{c}}-\frac{\Gamma^{2}}{\beta_{c}}-\frac{b}{\beta_{c}}<E<-\beta_{c}N+\frac{\ln(N\ln
2)+\ln(4\pi)}{2\beta_{c}}-\frac{\Gamma^{2}}{\beta_{c}}-\frac{a}{\beta_{c}}.$

By Theorem 1.5 on $\Omega_{\Gamma,M}^{\infty}$ such an $E$ corresponds for
$N\geq M$ to a unique potential value $U(\boldsymbol{\sigma})$ via

$E=U(\boldsymbol{\sigma})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma})}+o_{\Gamma}(1)=U(\boldsymbol{\sigma})-\frac{\Gamma^{2}}{\beta_{c}}+o_{\Gamma}(1)$

with a $o_{\Gamma}(1)$ bound, which is independent of the large deviation site
$\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}$. In particular, we see
that $s_{N}^{-1}(E,\Gamma)=s_{N}^{-1}(U(\boldsymbol{\sigma}))+o_{\Gamma}(1)$
and, hence, we have on $\Omega_{\Gamma,M}^{\infty}$

$\int\,f\,d\left(\sum_{E\in\operatorname{spec}H}\delta_{s_{N}^{-1}(E,\Gamma)}\right)=\sum_{E\in\operatorname{spec}H}f(s_{N}^{-1}(E,\Gamma))=\sum_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}f(s_{N}^{-1}(U(\boldsymbol{\sigma})+o_{\Gamma}(1)).$

Since $f$ is continuous and ${\operatorname{supp}\,}(f)\subset(a,b)$, the last
expression coincides in the limit $N\to\infty$ with the known almost sure
converge (1.20) of the REM process. Thus, we have established the almost sure
convergence on $\Omega_{\Gamma,M}^{\infty}$ for any $M\in{\mathbb{N}}$ and,
thus, also on $\cup_{M\in{\mathbb{N}}}\Omega_{\Gamma,M}^{\infty}$ . Since the
probability of the complements of $\Omega_{N,\Gamma,\delta}^{\textup{loc}}$
are summable in $N$, the Borel-Cantelli lemma yields
${\mathbb{P}}(\cup_{M\in{\mathbb{N}}}\Omega_{\Gamma,M}^{\infty})=1$ which
concludes the proof. ∎

##  5 Free energy asympotics

The main idea for our proof of Theorem 1.10 is simple. We exploit that the
partition function is determined by the eigenvalues close to the thermal
averages

$\langle
U\rangle^{\text{cl}}_{\beta}\coloneqq\frac{{\operatorname{Tr}\,}Ue^{-\beta
U}}{{\operatorname{Tr}\,}e^{-\beta U}}\quad\text{or}\quad\langle
T\rangle^{\text{pm}}_{\beta}\coloneqq\frac{{\operatorname{Tr}\,}Te^{-\beta
T}}{{\operatorname{Tr}\,}e^{-\beta T}},$ (5.1)

depending on the phase, cf. Proposition 1.1. To determine their behavior we
consider the local region around
$\boldsymbol{\sigma}\in\mathcal{L}_{\varepsilon}$, where $\varepsilon>0$ has
to be allowed to be arbitrarily small. In this case, we cannot guarantee
anymore that all balls $B_{R}(\boldsymbol{\sigma})$ are disjoint. However, we
will show that this is still true for isolated large deviations
$\boldsymbol{\sigma}\in\mathcal{L}_{\varepsilon}$, which are in the majority.
Then, we establish the order-one corrections of Theorem 1.5 for those isolated
large deviations. Based on these results, we prove Theorem 1.10 via a suitable
approximation argument using auxiliary operators on cut domains of the
configuration space.

###  5.1 Basic large deviations

We start by recording some standard facts in the statistical mechanics of the
pure REM and pure paramagnet.

######  Proposition 5.1.

  1. 1.

For any $\beta\geq 0$ we have $\langle
T\rangle^{\text{pm}}_{\beta}=-N\tanh\beta$. Moreover, for any $\beta\geq
0,\delta>0$ there exists some $c=c(\beta,\delta)>0$ such that

$\frac{{\operatorname{Tr}\,}\mathbbm{1}_{[-N(\tanh\beta+\delta),-N(\tanh\beta-\delta)]}(T)e^{-\beta
T}}{{\operatorname{Tr}\,}e^{-\beta T}}\geq 1-e^{-cN}.$ (5.2)

  2. 2.

For $\beta<\beta_{c}$ we have almost surely $\langle
U\rangle^{\text{cl}}_{\beta}=-(\beta+o(1))N$. Moreover, for $\beta<\beta_{c}$
and $\delta>0$ there exists some $c=c(\beta,\delta)>0$ such that

$\frac{{\operatorname{Tr}\,}\mathbbm{1}_{[-N(\beta+\delta),-N(\beta-\delta)]}(U)e^{-\beta
U}}{{\operatorname{Tr}\,}e^{-\beta U}}\geq 1-e^{-cN}$ (5.3)

except for an exponentially small event.

  3. 3.

For $\beta>\beta_{c}$ we have almost surely $\langle
U\rangle^{\text{cl}}_{\beta}=-(\beta_{c}+o(1))N$. Moreover, for
$\beta>\beta_{c}$ and $\delta>0$ there exists some $c=c(\delta)>0$ such that

$\frac{{\operatorname{Tr}\,}\mathbbm{1}_{(-\infty,-N(\beta_{c}-\delta)]}(U)e^{-\beta
U}}{{\operatorname{Tr}\,}e^{-\beta U}}\geq 1-e^{-cN}.$ (5.4)

The proof for the expressions of the thermal averages $\langle
T\rangle^{\text{pm}}_{\beta},\langle U\rangle^{\text{cl}}_{\beta}$ is by
differentiating the explicit formulas for the pressure with respect to $\beta$
(see (1.3)). The results on the concentration of the Gibbs measure are then of
Cramér type and follow from the usual convexity estimates of the (explicit)
free energy.

###  5.2 Spectral analysis on clusters

In the proofs of Theorem 1.5 and 1.7 we derived the order-one correction of
the energy levels $U(\boldsymbol{\sigma})$ for large deviations
$\boldsymbol{\sigma}\in\mathcal{L}_{\varepsilon}$ with
$\varepsilon\approx\beta_{c}$ from a local analysis on non-overlapping balls
$B_{R}(\boldsymbol{\sigma})$ of some radius $R$. For the proof of Theorem 1.10
however, we need good control on all eigenvalues with energy below
$-\varepsilon N$ with $\varepsilon>0$ arbitrary and, thus, the large deviation
set $\mathcal{L}_{\varepsilon}$ has to be considered for any $\varepsilon>0$.
The balls $B_{R}(\boldsymbol{\sigma})$ then have a nonempty intersection and
the aim of this subsection is to deal with this modified situation.

Let us introduce some definitions and notation.

######  Definition 5.2.

Let $\varepsilon>0$ and $k\in\mathbb{N}_{0}$. We denote
$\boldsymbol{\sigma}\overset{k}{\sim}\boldsymbol{\sigma}^{\prime}\iff
d(\boldsymbol{\sigma},\boldsymbol{\sigma}^{\prime})\leq 2k+2$. We call a set
$G\subset\mathcal{L}_{\varepsilon}$ _$(k,\varepsilon)$ -connected_ (with
respect to $\overset{k}{\sim}$) if for any
$\boldsymbol{\sigma},\boldsymbol{\sigma}^{\prime}\in G$ there exists a
sequence
$\boldsymbol{\sigma}=\boldsymbol{\sigma}^{(0)},\boldsymbol{\sigma}^{(1)},\cdots,\boldsymbol{\sigma}^{(m)}=\boldsymbol{\sigma}^{\prime}$
such that $\boldsymbol{\sigma}^{(i)}\in G$ and
$\boldsymbol{\sigma}^{(i)}\overset{k}{\sim}\boldsymbol{\sigma}^{(i+1)}$ for
all $0\leq i\leq m-1$. If $G\subset\mathcal{L}_{\varepsilon}$ is
$(k,\varepsilon)$-connected and for any $(k,\varepsilon)$-connected
$G^{\prime}$ with $G\subset G^{\prime}\subset\mathcal{L}_{\varepsilon}$ it
follows $G=G^{\prime}$, we call $G$ a _$(k,\varepsilon)$ -component_. We
denote the family of $(k,\varepsilon)$-components of
$\mathcal{L}_{\varepsilon}$ by $\mathcal{G}_{k,\varepsilon}$.

We call $\boldsymbol{\sigma}\in\mathcal{Q}_{N}$ _$(k,\varepsilon)$ -isolated_
if $G=\\{\boldsymbol{\sigma}\\}\in\mathcal{G}_{k,\varepsilon}$ and
$I_{k,\varepsilon}$ denotes the collection of $(k,\varepsilon)$-isolated
configurations.

We remark that the case $k=0$ coincides with the notion of ’gap-connected’
used in [50, 49, 51].

The large deviations $\mathcal{L}_{\varepsilon}$ is naturally decomposed in
its components, i.e.,
$\mathcal{L}_{\varepsilon}=\cup_{G\in\mathcal{G}_{k,\varepsilon}}G$. Moreover,
we define for each $(k,\varepsilon)$-component $G$ the corresponding cluster

$C_{k}(G)\coloneqq\bigcup_{\boldsymbol{\sigma}\in
G}B_{k}(\boldsymbol{\sigma}).$

By construction $d(C_{k}(G),C_{k}(G^{\prime}))\geq 2$ for different
$k$-components $G\neq G^{\prime}$.

We start with a combinatorial lemma which shows that the size of
$(k,\varepsilon)$-components remains bounded and that most
$(k,\varepsilon)$-components are isolated.

######  Lemma 5.3.

Let $\varepsilon>0$ and $k\in{\mathbb{N}}_{0}$ be fixed, but arbitrary.

  1. 1.

There exists an $M=M(k,\varepsilon)\in{\mathbb{N}}$ such that

$\Omega_{N,M}(\varepsilon,k)\coloneqq\left\\{\max_{G\in\mathcal{G}_{k,\varepsilon}}|G|\leq
M\right\\}.$ (5.5)

occurs with probability ${\mathbb{P}}(\Omega_{N,M}(k,\varepsilon))\geq
1-e^{-cN}$ for some $c>0$.

  2. 2.

Let $\varepsilon<a<b$ and $b<\beta_{c}$. Then for all events, but one of
exponentially small probability:

$\frac{|\mathcal{L}_{a,b}\cap I_{k,\varepsilon}^{c}|}{|\mathcal{L}_{a,b}|}\leq
e^{-\varepsilon^{2}N/4},$ (5.6)

where $\mathcal{L}_{a,b}\coloneqq\mathcal{L}_{a}\cap\mathcal{L}_{b}^{c}$ and
$(\cdot)^{c}$ indicates the complement of that set.

  3. 3.

Let $a=\beta_{c}-\delta$ and suppose that
$\delta(2\beta_{c}-\delta)<\varepsilon^{2}$. Then, besides of an exponentially
small event

$\mathcal{L}_{a}\cap I_{k,\varepsilon}^{c}=\emptyset.$ (5.7)

###### Proof.

For a proof of the first assertion, we estimate for any $M\in\mathbb{N}$:

$\begin{split}{\mathbb{P}}(\max_{G\in\mathcal{G}_{k,\varepsilon}}|G|\geq
M)&\leq{\mathbb{P}}(\exists\,\boldsymbol{\sigma}\in\mathcal{Q}_{N}\text{ s.t.
}|B_{(2k+2)M}(\boldsymbol{\sigma})\cap\mathcal{L}_{\varepsilon}|\geq M)\\\
&\leq
2^{N}\binom{|B_{(2k+2)M}|}{M}e^{-\frac{1}{2}\varepsilon^{2}MN}.\end{split}$

The first inequality follows from the definition of $k$-connectedness: any
$k$-connected set with size $M$ is contained in some ball
$B_{(2k+2)M}(\boldsymbol{\sigma})$ and if
$\max_{G\in\mathcal{G}_{k,\varepsilon}}|G|\geq M$ there exists a $k$-connected
set with size $M$ (e.g. as a subset of the component with maximal size). The
second line follows from the union bound, the independence of the random
variables $U$ and the standard Gaussian tail estimate. As the binomial
coefficient is a polynomial in $N$ the claim follows for $M>2\ln
2/\varepsilon^{2}$.

For a proof of the second assertion, we rewrite
$|\mathcal{L}_{a,b}|=\sum_{\boldsymbol{\sigma}}Z_{\boldsymbol{\sigma}}$, where
$Z_{\boldsymbol{\sigma}}$ are iid Bernoulli variables with success probability

$p_{N}\coloneqq{\mathbb{P}}(\boldsymbol{\sigma}\in\mathcal{L}_{a,b})\geq\frac{\sqrt{N}(b-a)}{\sqrt{2\pi}}\
e^{-Nb^{2}/2}$ (5.8)

by the standard Gaussian tail estimates. Since $b<\beta_{c}$, the average size
$\mathbb{E}[|\mathcal{L}_{a,b}|]=2^{N}p_{N}$ is exponentially large and, by a
Markov estimate, the same applies to all events aside from one of super-
exponentially small probability, i.e., ${\mathbb{P}}(|\mathcal{L}_{a,b}|\leq
2^{N-1}p_{N})\leq e^{-e^{CN}}$ for some $C>0$. Similarly, we note that the
conditional probability
$\mathbb{P}_{\boldsymbol{\sigma}}:=\mathbb{P}\left(\cdot|\\{\boldsymbol{\sigma}\\}^{c}\right)$
of the configuration to not be $(\varepsilon,k)$-isolated equals the
probability to find on $B^{o}_{2k+2}\coloneqq
B_{2k+2}(\boldsymbol{\sigma})\backslash\\{\boldsymbol{\sigma}\\}$ another
large deviation in $\mathcal{L}_{\varepsilon}$ and hence

$\mathbb{P}_{\boldsymbol{\sigma}}(\exists\ \boldsymbol{\sigma}^{\prime}\in
I_{k,\varepsilon}^{c}\cap
B^{o}_{2k+2})\leq|B_{2k+2}|e^{-N\varepsilon^{2}/2}\leq
N^{2k+2}e^{-N\varepsilon^{2}/2}$

by the union bound and the Gaussian tail estimate again. This allows us to
estimate

$\mathbb{E}\left[|\mathcal{L}_{a,b}\cap
I_{k,\varepsilon}^{c}|\right]=\sum_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}\mathbb{E}\left[\mathbbm{1}[\boldsymbol{\sigma}\in\mathcal{L}_{a,b}]\,\mathbb{P}_{\boldsymbol{\sigma}}(\boldsymbol{\sigma}\in
I_{k,\varepsilon}^{c})\right]\leq 2^{N}p_{N}\,N^{2k+2}e^{-N\varepsilon^{2}/2}$
(5.9)

with $p_{N}$ from (5.8). Excluding the event on which $|\mathcal{L}_{a,b}|\leq
2^{N-1}p_{N}$, we thus arrive at

$\displaystyle{\mathbb{P}}\left(|\mathcal{L}_{a,b}\cap
I_{k,\varepsilon}^{c}|\geq e^{-\varepsilon^{2}N/4}|\mathcal{L}_{a,b}|\right)$
$\displaystyle\leq{\mathbb{P}}\left(|\mathcal{L}_{a,b}\cap
I_{k,\varepsilon}^{c}|\geq
e^{-\varepsilon^{2}N/4}2^{N-1}p_{N}\right)+e^{-e^{CN}}$
$\displaystyle\leq\frac{e^{\varepsilon^{2}N/4}}{2^{N-1}p_{N}}\mathbb{E}\left[|\mathcal{L}_{a,b}\cap
I_{k,\varepsilon}^{c}|\right]+e^{-e^{CN}}$

by a Chebychev-Markov estimate. Inserting the bound (5.9) completes the proof
of the second claim.

For the last assertion, we note that by Lemma 4.2 the condition on $\delta$
implies that for $\alpha>0$ small enough a global
$(\varepsilon,\delta,\alpha)$-deep hole scenario occurs with probability
exponentially close to one. ∎

The next lemma establishes the spectral properties of the restriction
$H_{C_{k}(G)}$ of the QREM Hamiltonian to the Hilbert space
$\ell^{2}(C_{k}(G))$ of a cluster corresponding to
$G\in\mathcal{G}_{k,\varepsilon}$. For its formulation, we define for
$\delta>0$ the spectral projections

$P_{\delta}(G)\coloneqq\mathbbm{1}_{(-\infty,-\delta N)}(H_{C_{k}(G)}),\quad
Q_{\delta}(G)\coloneqq\mathbbm{1}-P_{\delta}(G).$

Recall the events $\Omega_{N}^{u}$ defined in (4.3) and
$\Omega_{N,M}(k,\varepsilon)$ defined in (5.5).

######  Lemma 5.4.

Let $\varepsilon>0$ and $k\geq 2$. On the event
$\Omega_{N}^{u}\cap\Omega_{N,M}(\varepsilon,k)$ the following assertions are
valid for all $N$ large enough:

  1. 1.

$\displaystyle\max_{G\in\mathcal{G}_{k,\varepsilon}}\|H_{C_{k}(G)}-U_{C_{k}(G)}\|=\mathcal{O}_{\Gamma,k,M}(\sqrt{N})$.

  2. 2.

If $\psi$ is an $\ell^{2}$-normalized eigenfunction of $H_{C_{k}(G)}$ with
$\langle\psi,\,H_{C_{k}(G)}\psi\rangle\leq-\frac{3}{2}\varepsilon N$, we have

$|\psi(\boldsymbol{\sigma})|=\mathcal{O}_{\Gamma,k,M,\varepsilon}(N^{-\operatorname{dist}(G,\boldsymbol{\sigma})}),$
(5.10)

and

$\||\mathbbm{1}_{C_{k}(G)\backslash
G}\psi\|^{2}=\mathcal{O}_{\Gamma,k,M,\varepsilon}(N^{-1}),\qquad\|\mathbbm{1}_{\partial
C_{k}(G)}\psi\|^{2}=\mathcal{O}_{\Gamma,k,M,\varepsilon}(N^{-k}),$ (5.11)

where $\mathbbm{1}_{\partial C_{k}(G)}$ is the natural projection onto the
boundary of $C_{k}(G)$. In particular, all estimates are independent of $\psi$
and $G$.

  3. 3.

$\displaystyle\sup_{G\in\mathcal{G}_{k,\varepsilon}}\sup_{\boldsymbol{\sigma}\in\mathcal{L}_{2\varepsilon}\cap
G}\langle\delta_{\boldsymbol{\sigma}}|Q_{3\varepsilon/2}(G)\delta_{\boldsymbol{\sigma}}\rangle=\mathcal{O}_{\Gamma,k,M,\varepsilon}(N^{-1})$.

  4. 4.

If $G=\\{\boldsymbol{\sigma_{0}}\\}$ is $(k,\varepsilon)$-isolated and
$U(\boldsymbol{\sigma}_{0})\leq-2\varepsilon N$, then the ground state energy
of $H_{C_{k}(G)}$ is given by

$E_{\boldsymbol{\sigma_{0}}}\coloneqq\inf\operatorname{spec}H_{C_{k}(G)}=U(\boldsymbol{\sigma}_{0})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma}_{0})}+\mathcal{O}_{\Gamma,k,\varepsilon}(N^{-1/4}).$
(5.12)

###### Proof.

  1. 1.

We write $H_{C_{k}(G)}=U_{C_{k}(G)}+\Gamma T_{C_{k}(G)}$ and recall that
$C_{k}(G)$ is a union of at most $M$ Hamming balls
$B_{k}(\boldsymbol{\sigma})$ with $\boldsymbol{\sigma}\in G$. Thus, by the
triangle inequality and Lemma 2.1 we obtain $\|T_{C_{k}(G)}\|\leq
Mc_{k}\sqrt{N}$, and hence the claim.

  2. 2.

We introduce the modified spheres $S_{r}(G)$ for $0\leq r\leq k$,

$S_{r}(G)\coloneqq C_{r}(G)\setminus C_{r-1}(G)=\\{\boldsymbol{\sigma}\in
C_{k}(G)\,|\,\text{dist}(\boldsymbol{\sigma},G)=r\\}$

and for the eigenvector $\psi$ the maximal values on the spheres,
$s_{r}\coloneqq\max_{\boldsymbol{\sigma}\in
S_{r}(G)}|\psi(\boldsymbol{\sigma})|$. We use the convention $S_{0}(G)=G$ and
note that $S_{k}(G)=\partial C_{k}(G)$. Moreover, we observe that for any
$\boldsymbol{\sigma}\in S_{r}(G)$ and $1\leq r\leq k$:

$|S_{1}(\boldsymbol{\sigma})\cap S_{r}(G)|\leq M,\quad
r\leq|S_{1}(\boldsymbol{\sigma})\cap S_{r-1}(G)|\leq rM,\quad
N-(r+1)M\leq|S_{1}(\boldsymbol{\sigma})\cap S_{r+1}(G)|\leq N-r.$

We now use the eigenvalue equation

$-E\psi(\boldsymbol{\sigma})=\Gamma\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})}\psi(\boldsymbol{\sigma}^{\prime})-U(\boldsymbol{\sigma})\psi(\boldsymbol{\sigma}),$

to derive the claimed decay estimate. Inserting the above geometric bounds
into the eigenvalue equation, we obtain for all $1\leq r\leq k$ with the
convention $s_{k+1}=0$:

$-Es_{r}\leq\Gamma rMs_{r-1}+\Gamma(N-r)s_{r+1}+(\varepsilon N+\Gamma
M)s_{r}.$ (5.13)

We now claim that for all $1\leq r\leq k$ and $N$ large enough

$s_{r}\leq\frac{2M\Gamma k}{|E|-\varepsilon N-\Gamma M}s_{r-1}.$

This is immediate from (5.13) in case $r=k$ (even without the factor $2$). In
case $1\leq r<k$, the bound is proven recursively. If the inequality holds for
$r+1$, then (5.13) implies

$|E|s_{r}\leq\Gamma rMs_{r-1}+\left(\frac{2N\Gamma^{2}k}{|E|-\varepsilon
N-\Gamma M}+\varepsilon N+\Gamma M\right)s_{r},$

and hence the claimed inequality for all $N$ large enough. Since $s_{0}\leq
1$, this establishes (5.10) by iteration.

The first claim in (5.11) follows from (5.10) using $|G|\leq M$. Indeed, for
some $C=C(\Gamma,k,M,\varepsilon)$

$\sum_{\boldsymbol{\sigma}\neq\boldsymbol{\sigma}_{0}}|\psi(\boldsymbol{\sigma})|^{2}\leq
C\sum_{r=1}^{K}N^{-2r}\left|\\{\boldsymbol{\sigma}\,|\,\operatorname{dist}(\boldsymbol{\sigma},G)=r\\}\right|\leq
CM\,\sum_{r=1}^{K}N^{-r}=\frac{CM}{N-1}.$

Since $|\partial C_{k}(G)|\leq MN^{k}$ the second inequality in (5.11) follows
similarly.

  3. 3.

We will repeatedly make use of a coupling principle which follows from 1.,
namely the fact that the eigenvalues of $H_{C_{k}(G)}$ and $U_{C_{k}(G)}$
agree up to a uniform error of order $\mathcal{O}_{\Gamma,k,M}(N^{1/2})$.
Since $|\mathcal{L}_{\varepsilon}\cap G|\leq|G|\leq M$, this implies that
$\dim P_{3/2\varepsilon}(G)\leq M$ for any component
$G\in\mathcal{G}_{k,\varepsilon}$ if $N$ is chosen large enough. By the
pigeon-hole principle, for any component $G$ we find some $3\varepsilon/2\leq
a=a(G)\leq 2\varepsilon$ such that

$P_{a-\varepsilon/(2M)}(G)-P_{a+\varepsilon/(2M)}(G)=\mathbbm{1}_{(-(a+\varepsilon/(2M))N,-(a-\varepsilon/(2M))N}(H_{C_{k}(G)})=0.$
(5.14)

Since $Q_{3\varepsilon/2}(G)\leq Q_{a}(G)$, it is enough to prove the
assertion with $Q_{3\varepsilon/2}(G)$ replaced by $Q_{a}(G)$.

To this end, we fix a component $G\in\mathcal{G}_{k,\varepsilon}$ and observe
that the coupling principle and (5.14) yield

$|\mathcal{L}_{a}\cap G|=\dim P_{a}(G)\eqqcolon m_{a}$ (5.15)

with a natural number $m_{a}\leq M$. We denote by
$\psi_{1},\ldots\psi_{m_{a}}$ the normalized low energy eigenfunctions of
$H_{C_{k}(G)}$ corresponding to $P_{a}(G)$, which form an orthonormal basis
for this subspace. The first inequality in (5.11) bounds the contribution of
each eigenfunction to $C_{k}(G)\setminus G$. Moreover, the eigenvalue equation
readily implies for $\boldsymbol{\sigma}\in G\setminus\mathcal{L}_{a}$:

$|\psi_{j}(\boldsymbol{\sigma}))|\leq\frac{\Gamma}{|E_{j}-U(\boldsymbol{\sigma})|}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})}|\psi_{j}(\boldsymbol{\sigma}^{\prime}))|\leq\frac{2M\Gamma}{\varepsilon
N}\sum_{\boldsymbol{\sigma}^{\prime}\in
S_{1}(\boldsymbol{\sigma})}|\psi_{j}(\boldsymbol{\sigma}^{\prime}))|\leq\frac{2M\Gamma}{\varepsilon\sqrt{N}},$

with $E_{j}=\langle\psi_{j},\,H_{C_{k}(G)}\psi_{j}\rangle\leq-aN$ the
eigenvalue corresponding to $\psi_{j}$. The second inequality follows from
(5.14) and the last step is a consequence of the Cauchy-Schwarz inequality and
$\|\psi_{j}\|=1$. As $|G\setminus\mathcal{L}_{a}|\leq M$, we also conclude
that

$\sum_{\boldsymbol{\sigma}\in
C_{k}(G)\setminus\mathcal{L}_{a}}|\psi_{j}(\boldsymbol{\sigma})|^{2}\leq\frac{C}{N}$

with a uniform $C=C(\Gamma,k,M,\varepsilon)<\infty$. We thus learn that
$\sup_{j}\|\mathbbm{1}_{\mathcal{L}_{a}\cap G}\ \psi_{j}-\psi_{j}\|^{2}\leq
C/N$. Lemma 5.5 below shows that (with $P=\mathbbm{1}_{\mathcal{L}_{a}\cap G}$
and $F=P_{a}(G))$

$\sup_{\boldsymbol{\sigma}\in\mathcal{L}_{a}\cap
G}\langle\delta_{\boldsymbol{\sigma}}|Q_{a}(G)\delta_{\boldsymbol{\sigma}}\rangle\leq\|Q_{a}(G)\mathbbm{1}_{\mathcal{L}_{a}\cap
G}\|\leq\|P_{a}(G)-\mathbbm{1}_{\mathcal{L}_{a}\cap G}\|\leq
4(\sqrt{m_{a}}+m_{a})^{2}\frac{C}{N}\leq 16M^{2}\frac{C}{N}.$

Since $a\leq 2\varepsilon$, this proves the claim.

  4. 4.

By the Rayleigh-Ritz variational principle, we have
$E_{\boldsymbol{\sigma}_{0}}\leq-2\varepsilon N$, and hence the results of 2.
apply to the corresponding ground state wavefunction
$\psi\in\ell^{2}(C_{k}(G))$. By (5.11) this ensures
$\psi(\boldsymbol{\sigma}_{0})=1+O_{\Gamma,k,\varepsilon}(N^{-1/2})$.
Following the steps in analysis (4.2)–(4.2) of the eigenfunction equation, in
which we use (5.10) and the assumed bound on $u$, we thus conclude that (4.3)
remains valid. This concludes the proof of (5.12).

∎

In the proof of Lemma 5.4 we used the following result on finite-rank
projections:

######  Lemma 5.5.

Suppose $\mathcal{H}$ is a finite-dimensional Hilbert space, $P$ an orthogonal
projection of rank $m$ and $f_{1},f_{2},\ldots f_{m}$ a sequence of $m$
orthonormal vectors in $\mathcal{H}$, which span the projection $F$. If for
some $c<\infty$

$\max_{j=1,\ldots,m}\|Pf_{j}-f_{j}\|\leq c,$ (5.16)

then $\|P-F\|\leq(m+2\sqrt{m})c$.

###### Proof.

We employ the triangle inequality $\|P-F\|\leq\|PF-F\|+\|PF-PFP\|+\|P-PFP\|$
and bound the three terms on the right-hand side individually. For the first
term we invoke that $PF-F$ vanishes on the orthogonal complement
${\operatorname{Im}\,}F^{\perp}$ and, thus, a Frobenius norm estimate yields

$\|PF-F\|\leq\sqrt{\sum_{j=1}^{m}\|(P-F)f_{j}\|^{2}}=\sqrt{\sum_{j=1}^{m}\|Pf_{j}-f_{j}\|^{2}}\leq\sqrt{m}c.$

Our bound on the second term, relies on the norm estimate for the first term,
$\|PF-PFP\|=\|P(F-FP)\|\leq\|F-FP\|=\|PF-F\|\leq\sqrt{m}c$, where we used that
$\|P\|=1$ for the first bound and applied the elementary identity
$\|A\|=\|A^{\ast}\|$. For the last term, we employ the operator inequality
$0\leq PFP\leq P$ and the fact that the operator norm is bounded by the trace
norm $\|\cdot\|_{1}$:

$\displaystyle\|P-PFP\|$
$\displaystyle\leq\|P-PFP\|_{1}={\operatorname{Tr}\,}P-{\operatorname{Tr}\,}PFP=\sum_{j=1}^{m}\langle\psi_{j},(1-P)\psi_{j}\rangle\leq
mc.$

This completes the proof. ∎

###  5.3 Proof of Theorem 1.10

Before we dive into the details of the proof, we fix some notation, which
makes our strategy transparent. For $k\in{\mathbb{N}}$ and $\varepsilon>0$, we
will use the restricted Hamiltonian corresponding to the collection of all
clusters $C_{k}(G)$,

$H^{(c)}\coloneqq\bigoplus_{G\in\mathcal{G}_{k,\varepsilon}}H_{C_{k}(G)}$

acting on the complete Hilbert space $\ell^{2}(\mathcal{Q}_{N})$. We further
denote by

$P_{\varepsilon}^{(c)}\coloneqq\mathbbm{1}_{(-\infty,-3N\varepsilon/2)}(H^{(c)})=\bigoplus_{G\in\mathcal{G}_{k,\varepsilon}}P_{3\varepsilon/2}(G),\quad
Q_{\varepsilon}^{(c)}\coloneqq\mathbbm{1}-P_{\varepsilon}^{(c)}$

the spectral projections of $H^{(c)}$. The factor $3/2$ is motivated by the
third assertion of Lemma 5.4. The subspace corresponding to $P_{\delta}^{(c)}$
represents the ”localized” part of the QREM and $Q_{\delta}^{(c)}$ corresponds
to the ”delocalized” part. Corresponding to this block decomposition, we set
the diagonal parts of $H$ as well as their partition functions:

$\displaystyle H^{(1)}\coloneqq
P_{\varepsilon}^{(c)}HP_{\varepsilon}^{(c)},\quad H^{(2)}\coloneqq
Q_{\varepsilon}^{(c)}HQ_{\varepsilon}^{(c)},$ (5.17) $\displaystyle
Z_{N}^{(j)}(\beta,\Gamma)\coloneqq 2^{-N}{\operatorname{Tr}\,}_{j}e^{-\beta
H^{(j)}},\quad j=1,2.$ (5.18)

Here the traces ${\operatorname{Tr}\,}_{j}(\cdot)$ run over the natural
subspaces $P_{\varepsilon}^{(c)}\ell^{2}(\mathcal{Q}_{N})$ in case $j=1$, or
$Q_{\varepsilon}^{(c)}\ell^{2}(\mathcal{Q}_{N})$ in case $j=2$, on which
$H^{(j)}$ acts non-trivially.

The key observation is now that $P_{\varepsilon}^{(c)}$ commutes with the
restriction of $H$ to the clusters $H^{(c)}$. If we denote by $A$ the
adjacency matrix between the inner and outer boundaries of the clusters
$C_{k}(G)$, we see

$P_{\varepsilon}^{(c)}HQ_{\varepsilon}^{(c)}=\Gamma
P_{\varepsilon}^{(c)}AQ_{\varepsilon}^{(c)}.$ (5.19)

We recall that $d(C_{k}(G),C_{k}(G^{\prime}))\geq 2$ for two different
components $G\neq G^{\prime}\in\mathcal{G}_{k,\varepsilon}$, which implies
that the adjacency matrix $A$ is a direct sum of operators $A_{C_{k}(G)}$
corresponding to each cluster $C_{k}(G)$. This in turn yields

$\|A\|\leq Mc_{k}\sqrt{N}$ (5.20)

by Lemma 2.1 on the event on which the assertions in Lemma 5.3 apply. We
further observe that $A$ only acts nontrivially on the boundaries $\partial
C_{k}(G)$. Exploiting the decay estimate (5.11) from Lemma 5.4, we arrive at

$\|P_{\varepsilon}^{(c)}HQ_{\varepsilon}^{(c)}\|=\mathcal{O}_{\Gamma,k,M,\varepsilon}(N^{-(k-1)/2}).$

We conclude that for any $k\geq 2$:

$Z_{N}(\beta,\Gamma)=e^{o_{\Gamma,k,M,\varepsilon}(1)}(Z_{N}^{(1)}+Z_{N}^{(2)}).$
(5.21)

The proof of Theorem 1.10 now reduces to an analysis of $Z_{N}^{(1)}$ and
$Z_{N}^{(2)}$.

###### Proof of Theorem 1.10.

Since our claims in case $\beta=0$ and $\Gamma=0$ are trivial, we fix
$\beta,\Gamma>0$ away from the phase transition, and pick

$0<\varepsilon<\frac{1}{8}\min\\{\beta,\beta_{c},\Gamma\tanh\beta\Gamma,\min\\{1,\beta^{-1}\\}\ln\cosh\beta\Gamma\\}.$

In the following, we will only work on the event
$\Omega_{N,\beta_{c}}^{\textup{REM}}\cap\Omega_{N}^{u}\cap\Omega_{N,M}(\varepsilon,k)$,
where the conditions of Lemma 5.3 are valid at $k\geq 2$ and some $M$.
According to Lemma 5.4 and (4.4) as well as (1.7), this event can be chosen to
have a probability of at least $1-e^{-cN}$.

We now proceed in four steps. We first analyze the localized part
$Z_{N}^{(1)}$. As a second and third step we derive an upper and lower bound
for $Z_{N}^{(2)}$. The last part then collects these estimates.

Step 1 – Analysis of $Z_{N}^{(1)}$: Let us first remark that
$H^{(1)}=P_{\varepsilon}^{(c)}HP_{\varepsilon}^{(c)}=P_{\varepsilon}^{(c)}H^{(c)}P_{\varepsilon}^{(c)}$
and hence $H^{(1)}P_{\varepsilon}^{(c)}=H^{(c)}P_{\varepsilon}^{(c)}$. It thus
remains to consider the low energy spectrum of $H^{(c)}$. We abbreviate

$u\coloneqq u(\beta)\coloneqq\lim_{N\to\infty}\langle
U\rangle_{\beta}^{\text{cl}}/N=\begin{cases}-\beta,&\beta\leq\beta_{c},\\\
-\beta_{c},&\beta>\beta_{c},\end{cases}$

by Proposition 5.1. Since $8\varepsilon<-u$, the dominant energy levels of $U$
are not effected by the projection $P_{\varepsilon}^{(c)}$. We now split
$Z_{N}^{(1)}$ into the contribution arising from energy levels within
$J_{\delta}:=[(u-\delta)N,(u+\delta)N]$ with
$0<\delta<\min\\{-u/2-3\varepsilon/4,\varepsilon^{2}/(16\beta)\\}$ arbitrarily
small, and a remainder:

$\begin{split}Z_{N}^{(1)}(\beta,\Gamma)=2^{-N}\left({\operatorname{Tr}\,}e^{-\beta
H^{(1)}}\mathbbm{1}_{J_{\delta}}(H^{(1)})+{\operatorname{Tr}\,}e^{-\beta
H^{(1)}}\mathbbm{1}_{(-\infty,-3\varepsilon N/2)\setminus
J_{\delta}}(H^{(1)})\right).\end{split}$

The second term is estimated using Lemma 5.4 and subsequently Lemma 5.1, which
yields some $c=c(\beta,\delta)>0$ such that for all sufficiently large $N$:

${\operatorname{Tr}\,}e^{-\beta H^{(1)}}\mathbbm{1}_{{\mathbb{R}}\setminus
J_{\delta}}(H^{(1)})\leq\ e^{\beta\mathcal{O}_{\Gamma,k,M}(\sqrt{N})}\
{\operatorname{Tr}\,}e^{-\beta U}\mathbbm{1}_{{\mathbb{R}}\setminus
J_{\delta/2}}(U)\leq e^{-cN}\,2^{N}Z_{N}(\beta,0).$ (5.22)

The remaining term is decomposed further into the contribution of isolated and
non-isolated clusters:

${\operatorname{Tr}\,}e^{-\beta
H^{(1)}}\mathbbm{1}_{J_{\delta}}(H^{(1)})=\sum_{G\in\mathcal{I}_{k,\varepsilon}}{\operatorname{Tr}\,}e^{-\beta
H_{C_{k}(G)}}\mathbbm{1}_{J_{\delta}}(H_{C_{k}(G)})+\sum_{G\in\mathcal{G}_{k,\varepsilon}\setminus
I_{k,\varepsilon}^{c}}{\operatorname{Tr}\,}e^{-\beta
H_{C_{k}(G)}}\mathbbm{1}_{J_{\delta}}(H_{C_{k}(G)}).$ (5.23)

Since
$\sup_{G\in\mathcal{G}_{k,\varepsilon}}\|H_{C_{k}(G)}-U_{C_{k}(G)}\|\leq\mathcal{O}_{\Gamma,k,M}(\sqrt{N})$
by Lemma 5.4, we bound the second term for all sufficiently large $N$ as
follows:

$\sum_{G\in\mathcal{G}_{k,\varepsilon}\setminus
I_{k,\varepsilon}^{c}}{\operatorname{Tr}\,}e^{-\beta
H_{C_{k}(G)}}\mathbbm{1}_{J_{\delta}}(H_{C_{k}(G)})\leq e^{-\beta
N(u-\delta)}\sum_{G\in\mathcal{G}_{k,\varepsilon}\setminus
I_{k,\varepsilon}^{c}}{\operatorname{Tr}\,}\mathbbm{1}_{J_{2\delta}}(U_{C_{k}(G)})\\\
\leq e^{-N\varepsilon^{2}/4}e^{3\beta
N\delta}\sum_{G\in\mathcal{G}_{k,\varepsilon}}{\operatorname{Tr}\,}e^{-\beta
U_{C_{k}(G)}}\mathbbm{1}_{J_{2\delta}}(U_{C_{k}(G)})\leq
e^{-cN}\,2^{N}Z_{N}(\beta,0).$ (5.24)

At the expense of throwing out another event of exponentially small
probability, we consult Lemma 5.3 and assume in case $u>-\beta_{c}$ the
validity of (5.6) with $a=-u-2\delta$ and $b=-u+2\delta$ and in case
$u=-\beta_{c}$ the validity of (5.7) with $a=-u-2\delta$. This guarantees that
non-isolated clusters are exponentially rare. The last inequality is a
consequence of the choice of $\delta$ and of the fact our definition of the
partition function $Z_{N}$ includes a normilisation by $2^{-N}$.

Then, the first term on the right side of (5.23) can be expressed using the
energy correction formula (5.12) for isolated large deviations. At the expense
of excluding or including small subintervals at the boundary of $J_{\delta}$,
which are negligible in comparison to the main term by Lemma 5.1, this first
term is of the form

$\sum_{\boldsymbol{\sigma}\in
I_{k,\varepsilon}\cap\mathcal{L}_{u-\delta,u+\delta}}e^{-\beta(U(\boldsymbol{\sigma})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma})}+o(1))}=S-R,$

where, similarly to (5.24), the remainder is again bounded using (5.6):

$R\coloneqq\sum_{\boldsymbol{\sigma}\in
I_{k,\varepsilon}^{c}\cap\mathcal{L}_{u-\delta,u+\delta}}e^{-\beta(U(\boldsymbol{\sigma})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma})}+o(1))}\leq
e^{-cN}\,2^{N}Z_{N}(\beta,0).$

The main term is

$S\coloneqq\sum_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}e^{-\beta
U(\boldsymbol{\sigma})+\frac{\Gamma^{2}N}{U(\boldsymbol{\sigma})}+o(1))}\mathbbm{1}[U(\boldsymbol{\sigma})\in
J_{\delta}].$

By definition of $J_{\delta}$ and since the REM’s partition function
concentrates around $u$ by Lemma 5.1, $S$ equals $2^{N}Z(\beta,0)$ plus an
error which is bounded by $e^{-cN}2^{N}Z(\beta,0)$.

In summary, in this first step we have shown that for any $\delta>0$ small and
$N$ large enough:

$e^{-\frac{\beta\Gamma^{2}}{u-\delta}+o(1)}Z_{N}(\beta,0)\leq
Z_{N}^{(1)}(\beta,\Gamma)\leq
e^{-\frac{\beta\Gamma^{2}}{u+\delta}+o(1)}Z_{N}(\beta,0).$ (5.25)

Step 2 – Upper bound on $Z_{N}^{(2)}$: We write $U_{\varepsilon}^{<}\coloneqq
U\mathbbm{1}_{U\geq-2\varepsilon N}$ and $U_{\varepsilon}^{>}\coloneqq
U\mathbbm{1}_{U>-2\varepsilon N}$ as well as $U_{\varepsilon}\coloneqq
U\mathbbm{1}_{|U|\leq 2\varepsilon N}$, and estimate using the Jensen-Peierls
inequality [12]:

$\begin{split}2^{N}Z_{N}^{(2)}(\beta,\Gamma)&={\operatorname{Tr}\,}_{2}e^{-\beta
Q_{\varepsilon}^{(c)}[\Gamma
T+U_{\varepsilon}^{<}+Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}]Q_{\varepsilon}^{(c)}}\\\
&\leq{\operatorname{Tr}\,}Q_{\varepsilon}^{(c)}\ e^{-\beta[\Gamma
T+U_{\varepsilon}^{<}+Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}]}\leq{\operatorname{Tr}\,}e^{-\beta[\Gamma
T+U_{\varepsilon}+Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}]}.\end{split}$

The last inequality follows from a trivial extension of the trace and the
monotonicity of eigenvalues in the potential, $U_{\varepsilon}^{<}\geq
U_{\varepsilon}$. From Lemma 5.4 we learn that
$\max_{\boldsymbol{\sigma}\in\mathcal{L}_{2\varepsilon}}\|Q_{\varepsilon}^{(c)}\delta_{\boldsymbol{\sigma}}\|^{2}\leq
CN^{-1}$. Moreover, if $\boldsymbol{\sigma}\in C_{k}(G)$ for some component
$G$, the projection $Q_{\varepsilon}^{(c)}\delta_{\boldsymbol{\sigma}}$ has
only support on $C_{k}(G)$. As any cluster $C_{k}(G)$ has at most $M$
configurations $\boldsymbol{\sigma}\in\mathcal{L}_{2\varepsilon}$, these
observations result in the norm estimate

$\|Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}\|\leq
CM\|U\|_{\infty}N^{-1}.$

Since on the event considered we also have $\|U\|_{\infty}\leq 2\beta_{c}$ and
the operator $Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}$
only acts non trivially on the clusters $C_{k}(G)$, we thus conclude that for
some $D\in(0,\infty)$:

$Q_{\varepsilon}^{(c)}U_{\varepsilon}^{>}Q_{\varepsilon}^{(c)}\geq
V\coloneqq-D\mathbbm{1}_{C}=-D\sum_{G\in\mathcal{G}_{k,\varepsilon}}\mathbbm{1}_{C_{k}(G)},\quad\mathcal{C}\coloneqq\bigcup_{G\in\mathcal{G}_{k,\varepsilon}}C_{k}(G)$

To summarize, we have thus shown that $Z_{N}^{(2)}(\beta,\Gamma)\leq
2^{-N}{\operatorname{Tr}\,}e^{-\beta[\Gamma T+U_{\varepsilon}+V]}$.

From here, there are at least two possible ways to continue the proof. One
could show that the potential $U_{\varepsilon}+V$ meets the requirements of
Theorem 3.4. Then, one needs to control V, which is a little bit technical.
Instead, we will employ a convexity argument. To this end, we introduce for
$\lambda\in{\mathbb{R}}$ the family of pressures and corresponding
Hamiltonians on $\ell^{2}(\mathcal{Q}_{N})$:

$\Phi_{N}(\beta,\Gamma,\lambda)\coloneqq\ln
2^{-N}{\operatorname{Tr}\,}e^{-\beta H(\lambda)},\qquad
H(\lambda)\coloneqq\Gamma T+U_{\varepsilon}+\lambda V$ (5.26)

The pressure $\Phi_{N}(\beta,\Gamma,\lambda)$ is convex [63] in $\lambda$, and
$\lambda=1$ is the case of interest.

Let us first discuss the case $\lambda=0$ in which case Theorem 3.4 is
applicable with $W=U_{\varepsilon}$. Since $\|U_{\varepsilon}\|_{\infty}\leq
2\varepsilon N$ and
$\mathbb{E}\left[U_{\varepsilon}(\boldsymbol{\sigma})^{2}\right]\leq
N(1-e^{-2\varepsilon^{2}N})\leq N$, Theorem 3.4 guarantees that all
eigenvalues of $\Gamma T+U_{\varepsilon}$ below $E<-4\varepsilon N$, counted
with multiplicity, are shifted with respect to the eigenvalues $E$ of $\Gamma
T$ to $E+\frac{N}{E}+o(1)$. Since $\langle\Gamma
T\rangle^{\text{pm}}_{\beta}=-N\Gamma\tanh\beta\Gamma\leq-8\varepsilon N$,
Lemma 5.1 allows to spectrally focus the partition function onto an interval
around $\langle\Gamma T\rangle^{\text{pm}}_{\beta}$ of arbitrarily small size
$0<\delta<\Gamma\tanh\beta\Gamma-4\varepsilon$. A similar argument as in Step
1, then yields for all sufficiently large $N$:

$\Phi_{N}(\beta,\Gamma,0)\leq
N\ln\cosh\beta\Gamma+\frac{\beta}{\Gamma\tanh\beta\Gamma-\delta}+o(1).$

Next, we consider general parameters $\lambda$. Recall that
$\mathbbm{1}_{\mathcal{C}}$ stands for the orthogonal projection onto the
subspace of the union of all clusters, and that
$\mathbbm{1}_{\mathcal{C}^{c}}$ is the orthogonal complement. In terms of the
operator $A$ introduced in (5.19), the norm estimate (5.20) yields:
$\|H(\lambda)-\mathbbm{1}_{\mathcal{C}}H(\lambda)\mathbbm{1}_{\mathcal{C}}-\mathbbm{1}_{\mathcal{C}^{c}}H(\lambda)\mathbbm{1}_{\mathcal{C}^{c}}\|\leq\|A\|\leq
C\sqrt{N}$ and hence

${\operatorname{Tr}\,}e^{-\beta H(\lambda)}\leq
e^{C\sqrt{N}}{\operatorname{Tr}\,}e^{-\beta(\mathbbm{1}_{\mathcal{C}}H(\lambda)\mathbbm{1}_{\mathcal{C}}+\mathbbm{1}_{\mathcal{C}^{c}}H(\lambda)\mathbbm{1}_{\mathcal{C}^{c}})}=e^{C\sqrt{N}}\left[{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}}e^{-\beta(\mathbbm{1}_{\mathcal{C}}H(\lambda)\mathbbm{1}_{\mathcal{C}})}+{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}^{c}}e^{-\beta(\mathbbm{1}_{\mathcal{C}^{c}}H(\lambda)\mathbbm{1}_{\mathcal{C}^{c}})}\right]$

at some $C<\infty$, which is independent of $N$ and $\lambda$. Each of the
traces in the right side is now estimated separately:

$2^{-N}{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}}e^{-\beta(\mathbbm{1}_{\mathcal{C}}H(\lambda)\mathbbm{1}_{\mathcal{C}})}\leq
e^{\beta\|\mathbbm{1}_{\mathcal{C}}H(\lambda)\mathbbm{1}_{\mathcal{C}}\|}\leq\exp\left(\beta\left(C\Gamma\sqrt{N}+2\varepsilon
N+\lambda D\right)\right)$

where we used the triangle inequality for the operator norm as well as (5.20)
again. Since $H(\lambda)$ and $H(0)$ agree on
$\mathbbm{1}_{\mathcal{C}^{c}}\,\ell^{2}(\mathcal{Q}_{N})$, we also have

$2^{-N}{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}^{c}}e^{-\beta\mathbbm{1}_{\mathcal{C}^{c}}H(\lambda)\mathbbm{1}_{\mathcal{C}^{c}}}=2^{-N}{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}^{c}}e^{-\beta\mathbbm{1}_{\mathcal{C}^{c}}H(0)\mathbbm{1}_{\mathcal{C}^{c}}}\leq
2^{-N}{\operatorname{Tr}\,}\mathbbm{1}_{\mathcal{C}^{c}}e^{-\beta H(0)}\leq
e^{\Phi_{N}(\beta,\Gamma,0)}.$

The first inequality relied on the Jensen-Peierls estimate, which allows to
pull down the projections [12].
Since $\Phi_{N}(\beta,\Gamma,0)>4\beta\varepsilon N$, the correction to the
pressure at $\lambda_{0}\coloneqq\frac{\varepsilon N}{D}$, is still of order
$\mathcal{O}(\sqrt{N})$:

$\Phi_{N}(\beta,\Gamma,\lambda_{0})\leq\Phi_{N}(\beta,\Gamma,0)+C\sqrt{N}.$

We are now in the situation to exploit convexity:

$\Phi_{N}(\beta,\Gamma,1)\leq(1-\lambda_{0}^{-1})\
\Phi_{N}(\beta,\Gamma,0)+\lambda_{0}^{-1}\Phi_{N}(\beta,\Gamma,\lambda_{0})\leq
N\ln\cosh\beta\Gamma+\frac{\beta}{\Gamma\tanh\beta\Gamma-\delta}+o(1).$

Step 3 – Paramagnetic lower bound: To show that the upper bound of Step 2 is
also an asymptotic lower bound, it is more convenient to work with the full
partition function $Z_{N}$, which by (5.21) is a lower bound on $Z_{N}^{(2)}$
up to a multiplicative error of $e^{o(1)}$.

For an estimate on $Z_{N}$, we split the potential
$U=U_{\varepsilon}+V_{\varepsilon}$, where $V_{\varepsilon}\coloneqq
U\mathbbm{1}[|U|>2\varepsilon]$. The pressure $\Phi_{N}(\beta,\Gamma,0)$ of
$H(0)=\Gamma T+U_{\varepsilon}$, which is defined in (5.26), was already
analyzed in Step 2. Here, we now consider the following family of Hamiltonians
$H(\lambda)=\Gamma T+U_{\varepsilon}+\lambda V_{\varepsilon}$, which differs
from the one in Step 2. By a slight abuse of notation, we nevertheless denote
the corresponding pressure again by
$\Phi_{N}(\beta,\Gamma,\lambda)\coloneqq\ln{\operatorname{Tr}\,}2^{-N}{\operatorname{Tr}\,}e^{-\beta
H(\lambda)}$. The convexity of the pressure in $\lambda$ is again the basis
for our argument.

Since on the event considered, we may assume $\|U\|\leq 2\beta_{c}N$, the
potential $W=U_{\varepsilon}+\lambda V_{\varepsilon}$ meets the requirements
of Theorem 3.4 with $\|W\|_{\infty}\leq
N\max\\{2\varepsilon,2\lambda\beta_{c}\\}$ and
$\mathbb{E}\left[W(\boldsymbol{\sigma})^{2}\right]\leq
N(1-e^{-2\varepsilon^{2}N})\leq N$. Thus if $\lambda<\varepsilon/\beta_{c}$
the eigenvalues of $H(\lambda)$ below $E<-4\varepsilon N$, counted with
multiplicity, are shifted with respect to the eigenvalues $E$ of $\Gamma T$ to
$E+\frac{N}{E}+o(1)$ and we have
$\Phi_{N}(\beta,\Gamma,\lambda)=\Phi_{N}(\beta,\Gamma,0)+o(1)$. Fixing
$\lambda_{0}\coloneqq\frac{\varepsilon}{2\beta_{c}}<1$, convexity implies:

$\Phi_{N}(\beta,\Gamma,1)\geq\Phi_{N}(\beta,\Gamma,0)+\frac{1}{\lambda_{0}}(\Phi_{N}(\beta,\Gamma,\lambda_{0})-\Phi_{N}(\beta,\Gamma,0))=N\ln\cosh(\beta\Gamma)+\frac{\beta}{\Gamma\tanh\beta\Gamma+\delta}+o(1).$

The last step, which holds for all $\delta>0$ sufficiently small, is the
desired lower bound, again relied on an explicit estimate based on the
concentration of the partition function of $\Gamma T$ around energies near
$-\Gamma\tanh\beta\Gamma$, cf. Proposition 5.1.

Step 4 – Completing the proof. Away from the first-order phase transition at
$\Gamma=\Gamma_{c}(\beta)$ described in Proposition 1.1 and on the event on
which Step 1-3 are valid, the partition function (5.21) is either dominated by
the REM-term $Z_{N}^{(1)}$ in case $\Gamma<\Gamma_{c}(\beta)$, or by the
paramagnetic term $Z_{N}^{(2)}$ in case $\Gamma>\Gamma_{c}(\beta)$.

More precisely, in case $\Gamma<\Gamma_{c}(\beta)$ and since the probability
of the event, which is excluded in Step 1–3, is exponentially small in $N$ and
hence summable, we conclude for any $\varepsilon>0$:

$\sum_{N\geq
1}{\mathbb{P}}\left(\left[\Phi_{N}(\beta,\Gamma)-\Phi_{N}(\beta,0)+\frac{\beta\Gamma^{2}}{u(\beta)}\right|>\varepsilon\right)<\infty.$
(5.27)

The claimed almost-sure convergence then follows by a Borel-Cantelli argument.
The analogous argument establishes the claim in case
$\Gamma>\Gamma_{c}(\beta)$. ∎

##  Appendix A Proof of Proposition 1.8 and 1.9

In this section, we sketch the modifications necessary for covering the
critical case. For any $\varepsilon>0$ we introduce the symmetrized large-
deviation set

$\mathcal{S}_{\varepsilon}\coloneqq\\{\boldsymbol{\sigma}\in\mathcal{Q}_{N}\,|\,|U(\boldsymbol{\sigma})|\geq\varepsilon
N\\}.$

For parameters $\varepsilon,\alpha,\delta>0$, we use the notion of a
symmetrized global $(\varepsilon,\alpha,\delta)$-deep-hole scenario if
$\mathcal{L}_{\beta_{c}-\delta}$ is replaced by
$\mathcal{S}_{\beta_{c}-\delta}$ in Definition 4.1. As is evident from the
proof of Lemma 4.2, due to the symmetry of the Gaussian distribution, if
$\varepsilon,\alpha,\delta>0$ satisfy (4.1) then the event

$\Omega_{N}^{s}(\varepsilon,\alpha,\delta):=\left\\{\text{$U$ satisfies a
symmetrized global $(\varepsilon,\alpha,\delta)$-deep-hole scenario}\right\\}$

still has a probability bounded by $1-e^{-c(\varepsilon,\alpha,\delta)N}$ for
all sufficiently large $N$. We will again pick $\varepsilon=\beta_{c}/2$ and
the values of $\alpha,\delta$ will be given (implicitly) as we go along the
proof. Similarly as in the proof of Theorem 1.5, under this scenario, we work
with the auxiliary Hamiltonian

$H^{\prime}=\left(\bigoplus_{\boldsymbol{\sigma}\in\mathcal{S}_{\beta_{c}-\delta}}H_{\alpha
N}(\boldsymbol{\sigma})\right)\bigoplus H_{r},$ (A.1)

which is well-defined, since the balls $B_{\alpha N}(\boldsymbol{\sigma})$ are
disjoint for two different configurations in $\mathcal{S}_{\beta_{c}-\delta}$.
That we work with $\mathcal{S}_{\beta_{c}-\delta}$ instead of
$\mathcal{L}_{\beta_{c}-\delta}$ is due to purely technical reasons: we want
to ensure that the restriction of $U$ to the complement of the balls
$H_{\alpha N}(\boldsymbol{\sigma})$ is still a symmetric random variable so
that we can apply Theorem 3.4.

The first main step in the proof of Proposition 1.8 and 1.9 is to show that
for $\Gamma\simeq\beta_{c}$ there are $c,r>0$ (which may still depend on
$\alpha,\delta$) such that

$\|(H-H^{\prime})\mathbbm{1}_{(-\infty,(\Gamma-r)N}(H^{\prime})\|\leq
e^{-cN}.$

However, the paramagnetic part causes a challenge, since the less energy
levels we cut out (i.e. choosing a smaller parameter $\delta>0$), the low
energy states of $H_{r}$ become less delocalized. Indeed our bound from
Proposition 3.5 becomes trivial for $\Gamma=\beta_{c}$ and $r\leq\delta$. To
overcome this problem, we use the improved delocalization estimate from
Theorem 3.8, which deals with $H_{\delta}=\Gamma
T+U\mathbbm{1}_{|U|\leq(\beta_{c}-\delta)N}$ on $\ell^{2}(\mathcal{Q}_{N})$.
We first show that for $\delta>0$ small enough the low energy states of
$H_{r}$ and $H_{\delta}$ agree up to an experientially small error. Here and
in the following we always assume that the parameter $\delta>0$ coincides in
the definition of $H_{r}$ and $H_{\delta}$. Then, we show that the low
eigenvalues of $H$ and $H^{\prime}$ again coincide besides an exponential
error. From there, we establish Proposition 1.8 and 1.9.

Step 1: Comparison of the low energy spaces of $H_{r}$ and $H_{\delta}$:  We
abbreviate

$B_{\cup}\coloneqq\bigcup_{\boldsymbol{\sigma}\in\mathcal{S}_{\beta_{c}-\delta}}B_{\alpha
N}(\boldsymbol{\sigma}),\qquad
B_{\cup}^{+}\coloneqq\bigcup_{\boldsymbol{\sigma}\in\mathcal{S}_{\beta_{c}-\delta}}B_{\alpha
N+1}(\boldsymbol{\sigma}).$

and recall that $H_{r}$ is the restriction of $H$ onto the complement of
$B_{\cup}$. In the following, $\hat{H}_{r}$ stands for its canonical extension
to $\ell^{2}(\mathcal{Q}_{N})$. Then, the spectra of $H_{r}$ and $\hat{H}_{r}$
agree outside zero. Theorem 3.8 guarantees the existence of some
$\eta=\eta(\delta)>0$ and some $c>0$, which is independent of $\delta$, such
that for all $N$ large enough:

  1. 1.

the spectral projection
$P_{\delta}(\eta)\coloneqq\mathbbm{1}_{(-\infty,-(\Gamma-\eta)N)}(H_{\delta})$
is finite-dimensional, and the eigenvalue below $-(\Gamma-\eta)N$ are
$(2n-N)\Gamma+\frac{N}{(2n-N)\Gamma}+\mathcal{O}_{\Gamma}(N^{-1/4})$ with
$2n\leq\eta N/\Gamma$.

  2. 2.

$\max_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}\langle\delta_{\boldsymbol{\sigma}}\,|\,P_{\delta}(\eta)\,\delta_{\boldsymbol{\sigma}}\rangle\leq
e^{-2cN}$.

Since $\hat{H}_{r}$ and $H_{\delta}$ only differ on $B_{\cup}^{+}$, the
triangle inequality thus yields

$\displaystyle\|(H_{\delta}-\hat{H}_{r})P_{\delta}(\eta)\|$
$\displaystyle\leq\sum_{\boldsymbol{\sigma}\in
B_{\cup}^{+}}\|(H_{\delta}-\hat{H}_{r})\delta_{\boldsymbol{\sigma}}\|\
\|P_{\delta}(\eta)\delta_{\boldsymbol{\sigma}}\|\leq(\beta_{c}+\Gamma)N\
|B_{\cup}^{+}|\max_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}\sqrt{\langle\delta_{\boldsymbol{\sigma}}\,|\,P_{\delta}(\eta)\,\delta_{\boldsymbol{\sigma}}\rangle}$
$\displaystyle\leq|\mathcal{S}_{\beta_{c}-\delta}|\
e^{N\gamma(\alpha)+o(N)}e^{-cN}.$

Consulting Lemma 4.7, since the set $A_{3}(\delta)$ agrees with
$\mathcal{S}_{\beta_{c}-\delta}$, the cardinality is bounded
$|\mathcal{S}_{\beta_{c}-\delta}|\leq 2e^{\beta_{c}\delta N}$ except for an
exponentially small event. We now use that $c$ is independent of $\delta$. We
thus find some (possibly) smaller parameters $\alpha,\delta>0$ such that
$\beta_{c}\delta+\gamma(\alpha)<c/2$. It then follows that for some
$\eta=\eta(\delta)>0$ and all $N$ large enough:

$\|(H_{\delta}-\hat{H}_{r})P_{\delta}(\eta)\|\leq e^{-cN}.$

In particular, the ’off-diagonal’ blocks
$P_{\delta}(\eta)\hat{H}_{r}Q_{\delta}(\eta)$ and
$Q_{\delta}(\eta)\hat{H}_{r}P_{\delta}(\eta)$ with
$Q_{\delta}(\eta)=1-P_{\delta}(\eta)$ are bounded in norm by $e^{-cN}$. The
spectrum of $\hat{H}_{r}$ hence agrees up to the error $e^{-cN}$ with those of
the block matrices $P_{\delta}(\eta){H}_{\delta}$ and
$Q_{\delta}(\eta)\hat{H}_{r}Q_{\delta}(\eta)$. In order to see that the low-
energy spectrum is entirely determined by the first block, we need to derive a
lower bound on ground state energy of the second block. We will prove below
that for sufficiently large $N$:

$\inf\operatorname{spec}Q_{\delta}(\eta)\hat{H}_{r}Q_{\delta}(\eta)\geq-(\Gamma-\eta/2)N.$
(A.2)

Perturbation theory based on the Feshbach-Schur method and the fact that the
spectrum of $H_{\delta}$ is discrete with eigenvalue clusters which are
separated from each other by gaps of at least $1$, then yields that
$\|\mathbbm{1}_{(-\infty,-(\Gamma-\eta_{0})N)}(\hat{H}_{r})-P_{\delta}(\eta_{0})\|\leq
Ce^{-cN}$ for some $\eta_{0}=\eta_{0}(\delta)>0$ and all sufficiently large
$N$, and hence

$\max_{\boldsymbol{\sigma}\in(B_{\cup})^{c}}\langle\delta_{\boldsymbol{\sigma}}\,|\,\mathbbm{1}_{(-\infty,-(\Gamma-\eta_{0})N)}(H_{r}),\delta_{\boldsymbol{\sigma}}\rangle\leq\max_{\boldsymbol{\sigma}\in\mathcal{Q}_{N}}\langle\delta_{\boldsymbol{\sigma}}\,|\,P_{\delta}(\eta_{0})\,\delta_{\boldsymbol{\sigma}}\rangle+Ce^{-cN}\leq(1+C)\,e^{-cN}.$

It thus remains to proof (A.2). We will employ a counting argument based on
the min-max principle. It rests on the following observations. For any $E<0$
the eigenvalues of $\hat{H}_{r}$ and $H_{r}$ below $E$ agree. The same applies
to $\hat{H}_{r}$ and the energy form
$q_{r}^{\infty}:\ell^{2}(\mathcal{Q}_{N})\to{\mathbb{R}}\cup\infty$:

$q_{r}^{\infty}(\psi)\coloneqq\begin{cases}\langle\psi,\,H_{r}\psi\rangle,&\text{
if }\psi\in\ell^{2}((B_{\cup})^{c}),\\\ +\infty&\text{ otherwise
}.\end{cases}$

Denoting by $\hat{H}_{r}^{\infty}$ the corresponding operator, which results
in driving the potential values on $B_{\cup})$ to infinity (see e.g. [62]), we
thus have $H_{\delta}\leq\hat{H}_{r}^{\infty}$ and hence
${\operatorname{Tr}\,}\mathbbm{1}_{(-\infty,E)}(\hat{H}_{r})={\operatorname{Tr}\,}\mathbbm{1}_{(-\infty,E)}(\hat{H}_{r}^{\infty})\leq{\operatorname{Tr}\,}\mathbbm{1}_{(-\infty,E)}(H_{\delta})$.
However, since we have shown above that $\hat{H}_{r}$ is $e^{-cN}$-norm close
to $P_{\delta}(\eta){H}_{\delta}+Q_{\delta}(\eta)\hat{H}_{r}Q_{\delta}(\eta)$
and all eigenvalues of $P_{\delta}(\eta){H}_{\delta}$ below $-(\Gamma-\eta)N$
are already accounted for at this energy, we conclude the validity of (A.2)
(even with $\eta/2$ replaced by $\eta-e^{-cN}$).

Step 2: Comparison of the low energy spectra of $H$ and $H^{\prime}$:  We want
to show that the low energies of $H$ and $H^{\prime}$ coincide up to an
exponentially small error. We abbreviate
$P_{\delta}^{\prime}(\eta)\coloneqq\mathbbm{1}_{(-\infty,-(\Gamma-\eta)N)}(H^{\prime})$
and $Q_{\delta}^{\prime}(\eta)=\mathbbm{1}-P_{\delta}^{\prime}(r)$. As in the
first step, by the Feshbach-Schur method it suffices to show

$\|(H^{\prime}-H)P_{\delta}^{\prime}(\eta)\|\leq e^{-cN},\quad
Q_{\delta}^{\prime}(\eta)HQ_{\delta}^{\prime}(\eta)\geq-(\Gamma-\eta/2)NQ_{\delta}^{\prime}(\eta)$

for some $\eta,c>0$. To this end, let $\eta<\eta_{0}$ with $\eta_{0}$ from the
first step. Due to the direct-sum structure of $H^{\prime}$ we have

$\begin{split}\|(H^{\prime}-H)P_{\delta}^{\prime}(\eta)\|&\leq\|(H^{\prime}-H)\mathbbm{1}_{(B_{\cup})^{c}}\mathbbm{1}_{(-\infty,-(\Gamma-\eta)N)}(H_{r})\|\\\
&+\left\|(H^{\prime}-H)\mathbbm{1}_{B_{\cup}}\mathbbm{1}_{(-\infty,-(\Gamma-\eta)N)}\left(\left(\bigoplus_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}H_{\alpha
N}(\boldsymbol{\sigma})\right)\right)\right\|\end{split}$

Note that we have replaced $\mathcal{S}_{\beta_{c}-\delta}$ by
$\mathcal{L}_{\beta_{c}-\delta}$ in the above direct sum, as only those balls
contain low energy states. That the first summand is exponentially small for
$t$ small enough follows from the same argument we used to compare the low
energy projections of $H_{r}$ and $H_{\delta}$ in the first step. The second
summand is exponentially small, too. We simply recall the exponential decay
estimate of the localized wavefunctions from Theorem 1.5. Moreover, we have

$Q_{\delta}^{\prime}(\eta)HQ_{\delta}^{\prime}(\eta)\geq-(\Gamma-t)NQ_{\delta}^{\prime}(\eta)+Q_{\delta}^{\prime}(\eta)(H-H^{\prime})Q_{\delta}^{\prime}(\eta)\geq-(\Gamma-\eta+2\sqrt{\alpha(1-\alpha)}+o(N))Q_{\delta}^{\prime}(\eta),$

where we used that $H-H^{\prime}$ is the direct sum of the hopping operators
on the boundaries $\partial B_{\alpha N}(\boldsymbol{\sigma})$ and then used
the norm estimate from Lemma 2.1. If we choose $\alpha$ small enough, we
arrive at
$Q_{\delta}^{\prime}(\eta)HQ_{\delta}^{\prime}(\eta)\geq-(\Gamma-t/2)NQ_{\delta}^{\prime}(\eta)$.
We conclude that the energy levels $E<-(\Gamma-\eta/4)N$ of $H$ and
$H^{\prime}$ agree up to an at most exponentially small shift $e^{-cN}$ for
$t$ small enough.

Step 3: Proof of Proposition 1.8: To conclude the proof of Proposition 1.8, it
remains to evaluate for $\Gamma=\beta_{c}$ the probability
${\mathbb{P}}(E_{\boldsymbol{\sigma}_{0}}<\inf\operatorname{spec}H_{r})$,
where $E_{\boldsymbol{\sigma}_{0}}$ denotes the minimal ball energy. As we are
only interested in the order of this probability, we simplify the calculations
and write $x\simeq y$ if $x/y=\mathcal{O}(1)$. Then,

$\begin{split}{\mathbb{P}}(E_{\boldsymbol{\sigma}_{0}}<\inf\operatorname{spec}H_{r})&\simeq{\mathbb{P}}(\min
U<-\beta_{c}N)\simeq{\mathbb{P}}(\min U<s_{N}(\ln(N)/2))\\\
&=1-(1-2^{-N}e^{-\ln(N)/2})^{2^{N}}\simeq
1-e^{-1/\sqrt{N}}\simeq\frac{1}{\sqrt{N}}.\end{split}$

For the first line we recall that $E_{\boldsymbol{\sigma}_{0}}=\min
U+\mathcal{O}(1)$ and $\inf\operatorname{spec}H_{r}=-\Gamma N+\mathcal{O}(1)$.
Moreover, we recall the definition of $s_{N}$ from (1.8) and its connection to
the extreme value process of $U$ via (1.7). The remaining expressions can
derived by elementary calculus. This is enough to conclude the assertions on
the ground state energy.

It remains to show that aside from an exponentially small event, the ground
state wave function is localized if
$E_{\boldsymbol{\sigma}_{0}}<\inf\operatorname{spec}H_{r}$ and otherwise
delocalized. To this end, we observe case $\inf\operatorname{spec}H_{r}$ and
$E_{\boldsymbol{\sigma}_{0}}$ are independent random variables. Thus, the
spectral averaging method from Lemma 4.4 shows that for any $c>0$ there exists
some $b(c)>0$ such that

$|{\mathbb{P}}(\min_{\boldsymbol{\sigma}}E_{\boldsymbol{\sigma}}-\inf\operatorname{spec}H_{r})|\leq
e^{-cN})\leq e^{-b(c)N}$

for $N$ large enough. The Feshbach-Schur formula then yields that except for
an exponentially small event the ground state function $\psi$ of $H$ satisfies
$\|\psi-\psi_{\boldsymbol{\sigma}_{0}}\|\leq e^{-cN}$ if
$E_{\boldsymbol{\sigma}_{0}}<\inf\operatorname{spec}H_{r}$, and otherwise
$\|\psi-\psi_{r}\|\leq e^{-cN}$ with the ground state wave functions
$\psi_{\boldsymbol{\sigma}_{0}}$ of $H_{\alpha N}(\boldsymbol{\sigma}_{0})$
and $\psi_{r}$ of $H_{r}$. From the previous steps, it follows that
$\psi_{\boldsymbol{\sigma}_{0}}$ satisfies the assertions of Theorem 1.5 and
that $\psi_{r}$ is exponentially delocalized and satisfies the first assertion
of Theorem 1.5.

Step 4: Proof of Proposition 1.9. Since

$\min_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}E_{\boldsymbol{\sigma}}(\Gamma=0)<\inf\operatorname{spec}H_{r}(\Gamma=0)\text{
and
}\min_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}E_{\boldsymbol{\sigma}}(\Gamma=2\beta_{c})>\inf\operatorname{spec}H_{r}(\Gamma=2\beta_{c}),$

by continuity of the eigenvalues of the involved matrices there exists some
$\Gamma_{N}$ such that

$\min_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}E_{\boldsymbol{\sigma}}(\Gamma_{N})=\inf\operatorname{spec}H_{r}(\Gamma_{N}).$

For this $\Gamma_{N}$ the functions
$\min_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}E_{\boldsymbol{\sigma}}(\Gamma_{N})$
and $\inf\operatorname{spec}H_{r}(\Gamma_{N})$ in particular agree up to order
one, i.e., (1.27) holds. Since $H$ and $H^{\prime}$ agree at these low
energies up to an exponentially small shift, we conclude that
$\Delta_{N}(\Gamma_{N})$ is exponentially small. The minimum is attained at a
possibly different $\Gamma_{N}^{\star}$. However, since still
$\min_{\boldsymbol{\sigma}\in\mathcal{L}_{\beta_{c}-\delta}}E_{\boldsymbol{\sigma}}(\Gamma_{N}^{\star})-\inf\operatorname{spec}H_{r}(\Gamma_{N}^{\star})=o(1)$,
we have $\Gamma_{N}^{\star}-\Gamma_{N}=o(N^{-1})$ as claimed. ∎

Acknowledgments This work was supported by the DFG under EXC-2111 – 390814868.

## References

  * [1] J. Adame, S. Warzel. Exponential vanishing of the ground-state gap of the QREM via adiabatic quantum computing, J. Math. Phys. 56: 113301 (2015).
  * [2] A. Adhikari, C. Brennecke. Free energy of the Quantum Sherrington-Kirkpatrick spin-glass model with transverse field. J. Math. Phys. 61: 083302 (2020).
  * [3] M. Aizenman, J. L. Lebowitz, D. Ruelle. Some Rigorous Results on the Sherrington-Kirkpatrick Spin Glass Model. Commun. Math. Phys. 112: 3-20 (1987).
  * [4] M. Aizenman, M. Shamis, S. Warzel. Resonances and partial delocalization on the complete graph, Ann. Henri Poincare 16: 1969-2003 (2015).
  * [5] M. Aizenman, S. Warzel. Random Operators: Disorder Effects on Quantum Spectra and Dynamics, AMS 2015.
  * [6] L. Avena, O. Gün, M. Hesse. The parabolic Anderson model on the hypercube. Stoch. Proc. Appl. 130, 3369-3393 (2020).
  * [7] E. Baake, M. Baake, and H. Wagner. Ising quantum chain is equivalent to a model of biological evolution. Phys. Rev. Lett., 78: 559–562 (1997). Erratum: Phys. Rev. Lett. 79: 1782 (1997)
  * [8] E. Baake, H. Wagner. Mutation–selection models solved exactly with methods of statistical mechanics. Genet. Res. 78: 93–117 (2001).
  * [9] C. L. Baldwin, C. R. Laumann, A. Pal, A. Scardicchio. The many-body localized phase of the quantum random energy model, Phys. Rev. B 93: 024202 (2016).
  * [10] G. Ben Arous, A. Bovier, V. Gayrard. Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Comm. Math. Phys., 235: 379–425 (2003).
  * [11] G. Ben Arous, A. Bovier, V. Gayrard. Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys., 236: 1–54 (2003).
  * [12] F. A. Berezin. Convex operator functions. Math. USSR Sbornik 17, 269–277 (1972). Mat. Sbornik 88, 268–276 (1972).
  * [13] R. Bhatia. Matrix Analysis. Springer, 1997.
  * [14] G. Biroli, D. Facoetti, M. Schiró, M. Tarzia, P. Vivo. Out-of-equilibrium phase diagram of the quantum random energy model. Phys. Rev. B 103: 014204 (2021).
  * [15] M. Biskup, W. König. Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails. Commun. Math. Phys. 341, 179–218 (2016).
  * [16] B. Bollobás, J. Leed, S. Letzter. Eigenvalues of subgraphs of the cube, Eur. J. Comb. 70: 125–148 (2018).
  * [17] A. Bovier. Statistical Mechanics of Disordered Systems. A Mathematical Perspective. Cambridge University Press, 2012.
  * [18] A. Bovier, I. Kurkova, M. Löwe. Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab., 30: 605–651 (2002).
  * [19] A. Bovier, I. Kurkova. Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist., 40:439–480 (2004).
  * [20] A. Bovier, I. Kurkova. Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Statist., 40:481–495 (2004).
  * [21] A.J. Bray, M.A. Moore. Replica theory of quantum spin glasses. J. Phys. C, Solid State Phys. 13: L655 (1980).
  * [22] A. Burin. Localization and chaos in a quantum spin glass model in random longitudinal fields: Mapping to the localization problem in a Bethe lattice with a correlated disorder. Ann. Phys. 529: 1600292 (2017).
  * [23] D. Capocaccia, M. Cassandro, P. Picco. On the existence of thermodynamics for the generalized random energy model. J. Stat. Phys. 46:493–505 (1987).
  * [24] J. C̃erný, T. Wassmer. Aging of the metropolis dynamics on the random energy model, Probab. Theory Related Fields 167: 1–51 (2015).
  * [25] N. Crawford. Thermodynamics and universality for mean field quantum spin glasses, Commun. Math. Phys. 274: 821–839 (2007).
  * [26] D.M. Cvetković, M. Doob. H. Sachs. Spectra of graphs: Theory and applications, 3rd ed., Johann Ambrosius Barth, Heidelberg, 1995.
  * [27] B. Derrida. Random energy model: limit of a family of disordered models, Phys. Rev. Lett. 45: 79-82 (1980).
  * [28] B. Derrida. Random energy model: an exactly solvable model of disordered systems, Phys. Rev. B 24: 2613-2326 (1981).
  * [29] B. Derrida. A generalization of the random energy model that includes correlations between the energies, J. Phys. Lett. 46: 401–407 (1985).
  * [30] B. Derrida, E. Gardner. Solution of the generalized random energy model, J. Phys. C 19: 2253–2274 (1986).
  * [31] L. Faoro, M. V. Feigelman and L. Ioffe. Non-ergodic extended phase of the quantum random energy model, Annals of Physics 409: 167916 (2019).
  * [32] Y.V. Fedorov, E.F. Shender. Quantum spin glasses in the Ising model with a transverse field. JETP Lett. 43: 681 (1986).
  * [33] J. Friedman, J. P. Tillich. Generalized Alon-Boppana theorems and error-correcting codes. Siam J. Descrete Math. 19: 700–718 (2005).
  * [34] A. Galves, S. Martinez, P. Picco. Fluctuations in Derrida’s random energy and generalized random energy models. J Stat Phys 54: 515–529 (1989).
  * [35] V. Gayrard. Aging in Metropolis dynamics of the REM: a proof. Probab. Theory Relat. Fields 174, 501-551 (2019).
  * [36] V. Gayrard, L. Hartung. Dynamic phase diagram of the REM. In: V. Gayrard, L.P. Arguin, N. Kistler, I. Kourkova (eds) Statistical Mechanics of Classical and Disordered Systems. StaMeClaDys 2018. Springer Proceedings in Mathematics & Statistics, vol 293. Springer, Cham 2019.
  * [37] Y. Y. Goldschmidt. Solvable model of the quantum spin glass in a transverse field. Phys. Rev. B 41: 4858 (1990).
  * [38] J. Hermisson, O. Redner, H. Wagner. E. Baake. Mutation-selection balance: ancestry, load, and maximum principle. Theor. Popul. Biol. 62: 9–46 (2002).
  * [39] O. Kallenberg. Foundations of modern probability, Springer-Verlag, New York 2002.
  * [40] J. F. C. Kingman. A simple model for the balance between selection and mutation. J. Appl. Probab. 15: 1–12 (1978).
  * [41] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, M. Amini. A random matrix model with localization and ergodic transitions. New J. Phys. 17:122002, 2015.
  * [42] M.R. Leadbetter, G. Lindgren, H. Rootzén. Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer-Verlag, New York, 1983.
  * [43] D. A. Levin, Y. Peres, E. L. Wilmer, J. Propp, D. B. Wilson. Markov Chains and Mixing Times. 2nd ed. AMS 2017.
  * [44] T. Jörg, F. Krzakala, J. Kurchan, A.C. Maggs. Simple glass models and their quantum annealing. Phys. Rev Lett. 101: 147204 (2008).
  * [45] T. Jörg, F. Krzakala, G. Semerjian, F. Zamponi. First-order transitions and the performance of quantum algorithms in random optimization problems. Phys. Rev. Lett. 104: 207206 (2010).
  * [46] C. R. Laumann, A. Pal, A. Scardicchio. Many-body mobility edge in a mean-field quantum spin glass, Phys. Rev. Lett. 113: 200405 (2014).
  * [47] H. Leschke, C. Manai, R. Ruder, S. Warzel. Existence of replica-symmetry breaking in quantum glasses, Phys. Rev. Lett. 127: 207204 (2021).
  * [48] H. Leschke, S. Rothlauf, R. Ruder, W. Spitzer. The free energy of a quantum Sherrington-Kirkpatrick spin-glass model for weak disorder, J. Stat. Phys. 182: 55 (2021).
  * [49] C. Manai and S. Warzel. Phase diagram of the quantum random energy model, J. Stat. Phys. 180: 654–664 (2020).
  * [50] C. Manai and S. Warzel. The quantum random energy model as a limit of p-spin interactions, Rev. Math. Phys. 33: 2060013 (2020).
  * [51] C. Manai, S. Warzel. Generalized random energy models in a transversal magnetic field: free energy and phase diagrams, Probab. Math. Phys. 3: 215–245 (2022).
  * [52] C. Manai, S. Warzel. The de Almeida-Thouless Line in Hierarchical Quantum Spin Glasses, J. Stat. Phys. 186:14 (2022).
  * [53] M. Mézard, G. Parisi, M. A. Virasoro. Spin Glass Theory and Beyond. World Scientific, Singapore (1987).
  * [54] E. Olivieri, P. Picco. On the existence of thermodynamics for the random energy model. Comun. Math. Phys. 96: 125-144 (1984)
  * [55] R. I. Oliveira. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. Preprint http://arXiv.org/abs/0911.0600, Feb. 2010.
  * [56] D. Panchenko. The Sherrington-Kirkpatrick model. Springer, 2013.
  * [57] G. Parisi. The order parameter for spin glasses: a function on the interval 0-1, J. Phys. A: Math. Gen., 13: 1101–1112 (1980).
  * [58] P. Ray, B. K. Chakrabarti, A. Chakrabarti. Sherrington–Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations, Phys. Rev. B 39, 11828–11832 (1989).
  * [59] D. Ruelle. A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108: 225–239 (1987).
  * [60] I. G. Szendro, M. F. Schenk, J. Franke, J. Krug, J. A. G. M. de Visser. Quantitative analyses of empirical fitness landscapes. J. Stat. Mech. Theory Exp. 2013:P01005 (2013).
  * [61] G. De Tomasi, I. M. Khaymovich, F. Pollmann, S. Warzel. Rare thermal bubbles at the many-body localization transition from the Fock space point of view, Phys. Rev. B 104: 024202 (2021).
  * [62] B. Simon, A Comprehensive Course in Analysis, Part 4. Operator Theory, American Mathematical Soc., 2015
  * [63] B. Simon. The statistical mechanics of lattice gases, Princeton Univ. Press 1993.
  * [64] V. N. Smelyanskiy, K. Kechedzhi, S. Boixo, S. V. Isakov, H. Neven, B. Altshuler. Nonergodic delocalized states for efficient population transfer within a narrow band of the energy landscape. Phys. Rev. X 10: 011017 (2020).
  * [65] S. Suzuki, J. Inoue, B. K. Chakrabarti. Quantum Ising phases and transitions in transverse Ising models, 2nd ed., Springer 2013.
  * [66] M. Talagrand. Concentration of measure and isoperimetric inequalities in product spaces, Publ. Math. IHES 81:73–205 (1995).
  * [67] M. Talagrand. Mean field models for spin glasses (Vol I+II), Springer 2011.
  * [68] T. Tao. Topics in random matrix theory, AMS 2012.
  * [69] J. A. Tropp. An introduction to matrix concentration inequalities, Foundations and Trends in Machine Learning 8:1–230 (2015).
  * [70] P. von Soosten, S. Warzel. Non-ergodic delocalization in the Rosenzweig-Porter model, Lett. Math. Phys. 109: 905-922 (2019).
  * [71] T. Yamamoto, H. Ishii. A perturbation expansion for the Sherrington–Kirkpatrick model with a transverse field, J. Phys. C 20, 6053–6060 (1987).

Chokri Manai and Simone Warzel
MCQST & Zentrum Mathematik
Technische Universität München