Spherical Sherrington-Kirkpatrick model for deformed Wigner matrix with fast
decaying edges

Ji Oon Lee

Korea Advanced Institute of Science and Technology
<EMAIL_ADDRESS>
Yiting Li

Korea Advanced Institute of Science and Technology
<EMAIL_ADDRESS>
###### Abstract

We consider the $2$-spin spherical Sherrington–Kirkpatrick model whose
disorder is given by a deformed Wigner matrix of the form $W+\lambda V$, where
$W$ is a Wigner matrix and $V$ is a random diagonal matrix with i.i.d.
entries. Assuming that the density function of the entries of $V$ decays
faster than a certain rate near the edges of its spectrum, we prove the sharp
phase transition of the limiting free energy and its fluctuation. In the high
temperature regime, the fluctuation of $F_{N}$ converges in distribution to a
Gaussian distribution, whereas it converges to a Weibull distribution in the
low temperature regime. We also prove several results for deformed Wigner
matrices, including a local law for the resolvent entries, a central limit
theorem of the linear spectral statistics, and a theorem on the rigidity of
eigenvalues.

###### Contents

  1. 1 Introduction
    1. 1.1 Main contribution
    2. 1.2 Related works
      1. 1.2.1 Spherical Sherrington–Kirkpatrick model
      2. 1.2.2 Deformed Wigner matrix
    3. 1.3 Relation to a signal detection problem
    4. 1.4 Organization of the paper
  2. 2 Model and main results
    1. 2.1 Definition of the model
    2. 2.2 Main results
    3. 2.3 Outline of the proof
    4. 2.4 Remarks
  3. 3 Preliminaries
    1. 3.1 Fluctuation of the largest eigenvalue
    2. 3.2 Local law for the trace of the resolvent
    3. 3.3 Integral representation of the partition function of the SSK model
    4. 3.4 Helffer-Sjöstrand formula
    5. 3.5 Cumulant expansion
    6. 3.6 Marcinkiewicz-Zygmund inequality
    7. 3.7 Some results for symmetric matrices
  4. 4 Local law for resolvent entries: proof of Theorem 3
  5. 5 Central limit theorem for linear statistics: proof of Theorem 4
    1. 5.1 Some auxiliary lemmas
    2. 5.2 Proof of Theorem 4
    3. 5.3 Proof of Lemma 14
    4. 5.4 Proof of Lemma 15
  6. 6 Rigidity of eigenvalues: proof of Theorem 5
  7. 7 SSK model in low temperature: proof of Theorem 1
  8. 8 SSK model in high temperature: proof of Theorem 2
  9. A Analysis on the curve of steepest-descent: proof of Lemma 30
  10. B Proofs of auxiliary lemmas

##  1 Introduction

The Sherrington–Kirkpatrick (SK) model and its variants have been intensively
studied in statistical physics and probability theory to understand the
behavior of spin glass. Its spherical variant, known as the spherical
Sherrington–Kirkpatrick (SSK) model, is defined through the mean-field
Hamiltonian of the form

$\displaystyle-\langle J\sigma,\sigma\rangle,$ (1)

where the disorder $J$ is an $N\times N$ matrix and the spin
$\sigma=(\sigma_{1},\ldots,\sigma_{N})\in
S_{N-1}=\\{(\sigma_{1},\ldots,\sigma_{N})\in{\mathbb{R}}^{N}:\sum\sigma_{i}^{2}=N\\}$.
The SSK model is widely used in various fields of study including high-
dimensional statistics and learning theory.

One of the key features of the SSK model (and the SK model) is the sharp phase
transition of the free energy, defined as

$\displaystyle
F_{N}=F_{N}(\beta)=\frac{1}{N}\log\Bigg{[}\int_{S_{N-1}}\exp\Big{(}\beta\langle\sigma,J\sigma\rangle\Big{)}d\omega_{N}(\sigma)\Bigg{]},$
(2)

where $\beta$ is the inverse temperature and $\omega_{N}$ is the normalized
uniform measure on $S_{N-1}$. When the disorder $J$ is a real Wigner matrix,
it was proved by Crisanti and Sommers [8], and Talagrand [28] that as
$N\to\infty$ the free energy $F_{N}$ converges to

$\displaystyle F_{N}\to F_{W}(\beta):=\begin{cases}\beta^{2}\quad&\text{if
}0<\beta\leq 1/2\\\ 2\beta-\frac{1}{2}\log(2\beta)-\frac{3}{4}\quad&\text{if
}\beta>1/2\end{cases}.$ (3)

The fluctuation of the free energy is also markedly different in the high
temperature case ($\beta<1/2$) and the low temperature case ($\beta>1/2$).
Baik and the first author [1] studied the fluctuation $F_{N}-F_{W}(\beta)$ and
proved that

$\displaystyle\begin{cases}N(F_{N}-F_{W}(\beta))\rightarrow\text{ a normal
distribution}\quad&\text{if }0<\beta<1/2\\\
2^{2/3}(\beta-\frac{1}{2})N^{2/3}(F_{N}-F_{W}(\beta))\rightarrow\text{ the
Tracy--Widom distribution }\quad&\text{if }\beta>1/2\end{cases}$ (4)

where the convergence is in distribution.

Heuristically, the fluctuation of the free energy in the high temperature
regime is affected by all eigenvalues of $J$ through its linear spectral
statistics (LSS), defined by

$\sum_{i=1}^{N}f(\lambda_{i}),$

where $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N}$ are the eigenvalues
of $J$. On the other hand, in the low temperature regime, the fluctuation of
$F_{N}$ is dominated by that of the largest $\lambda_{1}$. Since the
fluctuations of the LSS and the largest eigenvalue are given by a Gaussian and
the Tracy–Widom, respectively, one obtains the phase transition as in (4).
Similar argument also holds for other disorders such as the sample covariance
matrix and the orthogonal invariant ensemble [1].

One natural question about the free energy of the SSK model is whether the
heuristic argument above is universal, i.e., the picture of the all
eigenvalues versus the largest eigenvalue is valid even when the disorder is
not one of the classical random matrix models (Wigner matrix, sample
covariance matrix, and invariant ensemble). To test the universality, we
consider the case where the disorder is of the form

$\displaystyle J=W+\lambda V,$ (5)

where $W$ is a Wigner matrix and $V$ is a random diagonal matrix. Such a
matrix is called a deformed Wigner matrix, and with certain choices of the
parameters, it is known that several key assumptions in [1] are not satisfied,
most notably the square-root decay at the edge of the spectrum and the
Tracy–Widom limit of the largest eigenvalue.

###  1.1 Main contribution

Under the assumption that the decay of the spectrum of $J=W+\lambda V$ is
convex, we prove that there exists a critical inverse temperature $\beta_{c}$
such that

  * •

if $\beta<\beta_{c}$, the fluctuation of $F_{N}$ converges in distribution to
a Gaussian distribution, and

  * •

if $\beta>\beta_{c}$, the fluctuation of $F_{N}$ converges in distribution to
a Weibull distribution

with precise formulas for both limiting distributions, where the limiting
Weibull distribution is originated from the corresponding (Weibull)
distribution of the largest eigenvalue of $J$. This in particular suggests
that the dichotomy between the fluctuation given by the LSS and that by the
largest eigenvalue holds not only for the classical random matrix models but
for more general models. We also prove the limiting free energy $F(\beta)$ for
both regimes.

It should be noted that the order of the fluctuation in the low temperature
regime is $N^{-1/(b+1)}$ for some $b>1$ but that in the high temperature is
$N^{-1/2}$, and hence the fluctuation is larger in the low temperature regime
than in the high temperature regime. This was also true for the SSK model with
the Wigner disorder in (4), though the exact orders of the fluctuations
($N^{-2/3}$ in the low temperature regime and $N^{-1}$ in the high temperature
regime) do not coincide with those for our model.

The main technical difficulty in the proof of the main result is the lack of
several results for $J$, which are crucial in the analysis of the free energy
in [1]. In this paper, we prove the following for $J$:

  * •

a local law for resolvent entries,

  * •

a central limit theorem of linear statistics,

  * •

the rigidity of eigenvalues.

These results are not only important for the understanding of the free energy
but also significant per se in random matrix theory.

###  1.2 Related works

####  1.2.1 Spherical Sherrington–Kirkpatrick model

The SK model was introduced by Sherrington–Kirkpatrick [26] as a mean-field
version of the Edwards–Anderson model [9], which is an Ising-type model of
spin glass. The limiting free energy was first predicted by Parisi [25], which
is now known as the Parisi formula, and later proved by Guerra [11] and
Talagrand [27].

The spherical Sherrington–Kirkpatrick (SSK) model was introduced by
Kosterlitz, Thouless, and Jones [13], where the limiting free energy was
explicitly computed without a rigorous proof. A formula analogous to the
Parisi formula was obtained by Crisanti and Sommers [8] and later proved by
Talagrand [28]. For more recent results on the free energy and its fluctuation
for the SSK model, we refer to [2, 3, 4, 15, 16, 23].

####  1.2.2 Deformed Wigner matrix

Deformed Wigner matrix of the form (5) was first introduced by Pastur [24],
where it was proved that the empirical distribution of (5) converges to a
deterministic probability distribution $\mu_{fc}$ as $N\to\infty$. The
$\mu_{fc}$ is known as the free convolution of $\mu$ and the semicircle
distribution, and assuming that the empirical distribution of $V$ is bounded
and exhibits concave decay at the edge of its spectrum, it is known that
$\mu_{fc}$ exhibits square-root decay at the corresponding edge. In this case,
several key results for the Wigner matrix, including the local law for the
resolvent [18, 21], the delocalization of the eigenvectors and the rigidity of
the eigenvalues [18], the bulk universality [21], the edge universality [21,
20], and the normality of the LSS [12], hold with natural modification.

On the other hand, much less is known for the case where $\mu_{fc}$ does not
exhibit the square-root decay at the edge. It was proved by the first author
and Schnelli [18, 17] that $\mu_{fc}$ decays at the same rates as the
empirical distribution of $V$ if it is convex and $\lambda$ in (5) is larger
than a certain critical value $\lambda_{+}$. (See Lemma 1 for more detail.) In
this case, it is also known that the eigenvectors associated to the extreme
eigenvalues are partially localized [17].

###  1.3 Relation to a signal detection problem

The SSK model is closely related to the problem of detecting the presence of
the rank-one signal in a noisy data matrix. Suppose that the data matrix $M$
is of the form

$M=\sqrt{\lambda}xx^{T}+H,$

where the signal $x\in{\mathbb{R}}^{N}$ and the noise $H$ is an $N\times N$
real symmetric random matrix. When the signal-to-noise (SNR) $\lambda$ is not
large, in order to detect the signal, it is common to analyze the largest
eigenvalue and its associated eigenvalue, which is the principal component
analysis (PCA). In the simplest case where $H$ is a Wigner matrix and
$\|x\|=1$, the following transition for the largest eigenvalue $\lambda_{1}$
of $M$ is known; if $\lambda>1$, $\lambda_{1}$ is strictly larger than $2$ and
separates from the bulk of the spectrum, whereas if $\lambda<1$, $\lambda_{1}$
converges to $2$, the edge of the spectrum, and cannot be distinguished from
the null model ($\lambda=0$).

If the SNR $\lambda$ is below the threshold and the noise is Gaussian, it is
known that no tests can reliably detect the presence of the signal. For this
case, it is natural to consider the hypothesis testing between the null
hypothesis $\lambda=0$ and the alternative $\lambda=\omega$ for some positive
constant $\omega$, which is also known as the weak detection. By the
Neyman–Pearson lemma, the likelihood ratio (LR) test is optimal in the sense
that it minimizes the sum of the Type-I error and the Type II-error. For the
$(i,j)$-entry of the data matrix with $i\neq j$, the ratio of the densities
under the null and the alternative is

$\frac{\exp\big{(}N(M_{ij}-\sqrt{\lambda}x_{i}x_{j})^{2}\big{)}}{\exp\big{(}NM_{ij}^{2}\big{)}}.$

Assuming that the signal is chosen uniformly from the unit sphere $S^{N}$ and
the noise is GOE, the likelihood ratio is given by

$\begin{split}\frac{d{\mathbb{P}}_{1}}{d{\mathbb{P}}_{0}}&:=\int_{S^{N}}\prod_{i<j}\frac{\exp\big{(}N(M_{ij}-\sqrt{\lambda}x_{i}x_{j})^{2}\big{)}}{\exp\big{(}NM_{ij}^{2}\big{)}}\prod_{k}\frac{\exp\big{(}N(M_{ij}-\sqrt{\lambda}x_{i}x_{j})^{2}/2\big{)}}{\exp\big{(}NM_{ij}^{2}/2\big{)}}d\omega^{N}(\sigma)\\\
&=\int_{S^{N}}\prod_{i\neq
j}\exp\left(-N\sqrt{\lambda}M_{ij}x_{i}x_{j}+\frac{N}{2}\lambda
x_{i}^{2}x_{j}^{2}\right)d\omega^{N}(\sigma),\end{split}$ (6)

where $d\omega^{N}$ is the uniform measure on $S^{N}$. Note that the logarithm
of the LR in (6) coincides with the free energy of the SSK model after
shifting and rescaling. In the LR test, if the test statistic
$\frac{d{\mathbb{P}}_{1}}{d{\mathbb{P}}_{0}}<1$ the null hypothesis is
accepted, while it is rejected if
$\frac{d{\mathbb{P}}_{1}}{d{\mathbb{P}}_{0}}>1$. Since the fluctuation of the
LR is equal to the fluctuation of the free energy of the SSK model, it is
possible to prove the optimal error for the weak detection.

If the rank-$1$ signal $xx^{T}$ is perturbed by $U=(U_{ij})$, the ratio of the
densities is changed to

$\begin{split}&\frac{\exp\big{(}N(M_{ij}-\sqrt{\lambda}U_{ij}-\sqrt{\lambda}x_{i}x_{j})^{2}\big{)}}{\exp\big{(}NM_{ij}^{2}\big{)}}\\\
&=\exp\left(-2\sqrt{\lambda}N(M_{ij}-\sqrt{\lambda}U_{ij})x_{i}x_{j}-2\sqrt{\lambda}NM_{ij}U_{ij}+\lambda
Nx_{i}^{2}x_{j}^{2}\right).\end{split}$

Thus, for given $U$ the LR in (6) becomes

$\prod_{i\neq
j}\exp\left(-\sqrt{\lambda}NM_{ij}U_{ij}\right)\int_{S^{N}}\prod_{i\neq
j}\exp\left(-N\sqrt{\lambda}(M_{ij}-\sqrt{\lambda}U_{ij})x_{i}x_{j}+\frac{N}{2}\lambda
x_{i}^{2}x_{j}^{2}\right)d\omega^{N}(\sigma)$ (7)

for which it is required to consider the free energy of the SSK model with
deformed Gaussian interaction. Note that while $U$ is not assumed to be
diagonal, we may diagonalize $U$ in the integrand in (7) for the analysis
since GOE is orthogonally invariant.

###  1.4 Organization of the paper

The rest of the paper is organized as follows: In Section 2, we precisely
define the model and introduce our main results. In Section 3, we list several
important results needed in the proof of main results. In Sections 4, 5, and
6, we prove our results on deformed Wigner matrices - local law for the
resolvent entries, CLT for the linear spectral statistics, and the rigidity of
the eigenvalues, respectively. In Sections 7 and 8, we prove the main theorems
for the low temperature case and the high temperature case, respectively. Some
technical details on the results for the steepest descent curve and the proofs
of some auxiliary lemmas are collected in Appendices.

##  2 Model and main results

###  2.1 Definition of the model

Recall that the disorder $J=W+\lambda V$. Here, $W$ is an $N\times N$ real
Wigner matrix for which we use the following definition:

######  Definition 1.

An $N\times N$ matrix $W=(W_{ij})_{N\times N}$ is a Wigner matrix if

  * •

$\\{W_{ij}|i\leq j\\}$ are independent real-valued random variables.

  * •

$W_{ij}=W_{ji}$.

  * •

${\mathbb{E}}[W_{ij}]=0$, ${\mathbb{E}}[W_{ij}^{2}]=\frac{1+\delta_{ij}}{N}$.

  * •

There exist $\theta>1$ and $\theta^{\prime}>0$ such that

$\displaystyle{\mathbb{P}}(\sqrt{N}|W_{ij}|>x)\leq\theta^{\prime}\exp(-x^{1/\theta})\quad\forall
x\geq 0,N\geq 1\text{ and }i,j\in\\{1,\ldots,N\\}$ (8)

We remark that the subexponential decay condition guarantees the the existence
of all (normalized) moments and an overwhelming-probability bound as follows:

  1. 1.

for any $p\in\\{1,2,\ldots\\}$,

$\displaystyle\sup\limits_{i,j,N}{\mathbb{E}}[|\sqrt{N}W_{ij}|^{p}]<\infty;$
(9)

  2. 2.

if $\epsilon^{\prime}>0$ and $D^{\prime}>0$, then for large enough $N$ we have

$\displaystyle{\mathbb{P}}\Big{(}|W_{ij}|\leq
N^{\epsilon^{\prime}-\frac{1}{2}},\forall
i,j\in\\{1,\ldots,N\\}\Big{)}>1-N^{-D^{\prime}}.$ (10)

We assume that $V$ is a random diagonal matrix whose entries are i.i.d. with
centered Jacobi distribution $\mu$ for which we use the following definitions:

######  Definition 2.

A probability measure $\mu$ is a Jacobi measure on $[-1,1]$ if its density
function is given by

$\frac{d\mu}{dx}=\frac{d(x)}{Z}(1+x)^{a}(1-x)^{b}\mathds{1}_{[-1,1]}(x)$

where

  * •

$a>-1$ and $b>-1$

  * •

$d(x)\in C^{1}([-1,1])$ and $d(x)>0$ on $[-1,1]$.

  * •

$Z$ is the normalization constant: $Z=\int_{-1}^{1}d(x)(1+x)^{a}(1-x)^{b}dx$

  * •

$\mu$ is centered: $\int_{-1}^{1}xd\mu(x)=0$

We also assume that $V$ is independent of $W$. For a given constant
$\lambda>0$, if we denote by $\lambda\mu$ the law of $\lambda v$ where $v$ is
a random variable with law $\mu$, then the empirical measure of $W+\lambda V$
converges to $\mu_{fc}$ as $N\to\infty$, which is given by

$\mu_{fc}:=\mu_{sc}\boxplus(\lambda\mu)$

where $\boxplus$ denotes the additive free convolution and $\mu_{sc}$ denotes
the semicircle distribution. It is known that $\mu_{fc}$ has a density
function, which we will call $\rho_{fc}$; see Remark 2.5 in [17] for the
detail. In the following lemma, we collect the results on $\mu_{fc}$,

######  Lemma 1.

Set

$\lambda_{\pm}=\Big{(}\int_{-1}^{1}\frac{d\mu(x)}{(1\mp
x)^{2}}\Big{)}^{1/2},\quad\tau_{\pm}=\int_{-1}^{1}\frac{d\mu(x)}{1\mp x}.$

There exists $L_{-}<0<L_{+}$ such that $\text{supp}(\mu_{fc})=[L_{-},L_{+}]$.
If $b>1$ and $\lambda>\lambda_{+}$, then

  1. 1.

$L_{+}=\lambda+\frac{\tau_{+}}{\lambda}$,

  2. 2.

$L_{+}+\int\frac{\rho_{fc}(x)dx}{x-L_{+}}=\lambda$, and

  3. 3.

there exists $C_{0}\geq 1$ such that

$\displaystyle\frac{x^{b}}{C_{0}}\leq\rho_{fc}(L_{+}-x)\leq C_{0}x^{b}\text{
for }x\in[0,L_{+}]$ (11)

If $a>1$ and $\lambda>\lambda_{-}$, the statements above hold for $L_{-}$,
with $L_{+}$ and $\tau_{+}$ replaced by $L_{-}$ and $\tau_{-}$ respectively.
In particular,

$\displaystyle\frac{x^{a}}{C_{0}}\leq\rho_{fc}(L_{-}+x)\leq C_{0}x^{a}\text{
for }x\in[0,|L_{-}|]$ (12)

For the proof of Lemma 1, see Lemma 2.3 and Remark 2.6 [17]

###  2.2 Main results

Recall that the free energy of the SSK model at inverse temperature $\beta>0$
is defined by

$F_{N}=F_{N}(\beta)=\frac{1}{N}\log\Bigg{[}\int_{S_{N-1}}\exp\Big{(}\beta\langle\sigma,(W+\lambda
V)\sigma\rangle\Big{)}d\omega_{N}(\sigma)\Bigg{]}$

where $S_{N-1}=\\{(x_{1},\ldots,x_{N}):\sum_{i=1}^{N}x_{i}^{2}=N\\}$ and
$\omega_{N}$ is the (normalized) uniform measure on $S_{N-1}$. We will prove
that the constant

$\displaystyle\beta_{c}=\frac{1}{2}\int\frac{\rho_{fc}(t)}{L_{+}-t}dt.$ (13)

is the critical inverse temperature of the SSK model, i.e., we study the
fluctuation of $F_{N}$ in two cases: $0<\beta<\beta_{c}$ (high temperature
regime) and $\beta>\beta_{c}$ (low temperature regime). We remark that
$\beta_{c}$ is well defined when $b>1$ and $\lambda>\lambda_{+}$ (see Equation
(11)).

Our first main result is the following theorem for the free energy in the low
temperature regime:

######  Theorem 1 (Main theorem: low temperature).

Suppose $\beta>\beta_{c}$, $\lambda>\max(\lambda_{-},\lambda_{+})$, $b>11$ and
$1<a<\frac{b^{2}-6b-7}{4}$. Then the fluctuation of $F_{N}$ converges in
distribution to a Weibull distribution. More precisely,

$\lim\limits_{N\to\infty}{\mathbb{P}}\Big{(}N^{\frac{1}{b+1}}\Big{[}\frac{F_{N}+\frac{1}{2}\log(2e\beta)+\frac{1}{2}\int\log(L_{+}-t)d\mu_{fc}(t)-\beta
L_{+}}{\beta-\beta_{c}}\Big{]}\leq
s\Big{)}=\exp\big{(}-\frac{C_{\mu}(-s)^{b+1}}{b+1}\big{)}\quad\forall s\leq 0$

where
$C_{\mu}=\Big{(}\frac{\lambda}{\lambda^{2}-\lambda_{+}^{2}}\Big{)}^{b+1}\cdot
d(1)\cdot 2^{a}\cdot Z^{-1}$.

Our second main result is for the high temperature regime.

######  Theorem 2 (Main theorem: high temperature).

Suppose $0<\beta<\beta_{c}$, $\lambda>\max(\lambda_{-},\lambda_{+})$, $a>1$
and $b>37/3$. Suppose $\hat{\gamma}$ is the unique point on $(L_{+},+\infty)$
such that $\int\frac{1}{\hat{\gamma}-t}d\mu_{fc}(t)=2\beta$. Then

$2\sqrt{N}\Big{(}F_{N}+\frac{1}{2}\log(2\beta
e)-\beta\hat{\gamma}+\frac{1}{2}\int\log(\hat{\gamma}-t)d\mu_{fc}(t)\Big{)}$

converges in distribution to a centered Gaussian distribution whose variance
is

$\frac{1}{4\pi^{2}}\Big{(}\oint_{\mathcal{C}}(1+m_{fc}^{\prime}(\xi))m_{fc}(\xi)\log(\hat{\gamma}-\xi)d\xi\Big{)}^{2}-\frac{1}{4\pi^{2}}\int_{-1}^{1}\Big{(}\oint_{\mathcal{C}}\dfrac{(1+m_{fc}^{\prime}(\xi))\log(\hat{\gamma}-\xi)}{(\lambda
t-\xi-m_{fc}(\xi))}d\xi\Big{)}^{2}d\mu(t)$

where $m_{fc}(\cdot)$ is the Stieltjes transform of $\mu_{fc}$ and
$\mathcal{C}$ is a counterclockwise path which encloses $[L_{-},L_{+}]$ but
does not enclose $\hat{\gamma}$. Here we take the analytic branch of
$\log(\cdot)$ on ${\mathbb{C}}\backslash(-\infty,0]$ such that
${\text{Im}}\log(\cdot)\in(-\pi,\pi)$.

From Theorems 1 and 2, we immediately obtain the following corollary on the
limiting free energy:

######  Corollary 1.

Suppose $\lambda>\max(\lambda_{-},\lambda_{+})$, $b>37/3$ and
$1<a<\frac{b^{2}-6b-7}{4}$. As $N\to\infty$ we have

$\displaystyle F_{N}\to
F(\beta)=\begin{cases}-\frac{1}{2}\log(2e\beta)-\frac{1}{2}\int\log(L_{+}-t)d\mu_{fc}(t)+\beta
L_{+}\quad&\text{if }\beta>\beta_{c}\\\
-\frac{1}{2}\log(2e\beta)-\frac{1}{2}\int\log(\hat{\gamma}-t)d\mu_{fc}(t)+\beta\hat{\gamma}\quad&\text{if
}0<\beta<\beta_{c}\end{cases}$

in distribution.

From the definitions of $\beta_{c}$ and $\hat{\gamma}$, we see that
$\lim\limits_{\beta\to\beta_{c}-}\hat{\gamma}=L_{+}$ and that
$\lim\limits_{\beta\to\beta_{c}-}F(\beta)=\lim\limits_{\beta\to\beta_{c}+}F(\beta)$.

###  2.3 Outline of the proof

In this paper, we study the fluctuation of $F_{N}$ by following the idea
introduced in [1]. In the low temperature case (i.e., $\beta>\beta_{c}$), we
will show that the leading term of $F_{N}$ is a linear function of
$\lambda_{1}$. Since the fluctuation of $\lambda_{1}$ has size
$O(N^{-\frac{1}{1+b}})$ and converges to a Weibull distribution, so does the
fluctuation of $F_{N}$, as in Theorem 1. In the high temperature case (i.e.,
$0<\beta<\beta_{c}$), the leading term of $F_{N}$ is a linear function of the
quantity

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}f(\lambda_{i}).$ (14)

for some $N$-independent deterministic function $f$. Thus, by the central
limit theorem (see Theorem 4), the fluctuation of $F_{N}$ has size
$O(N^{-1/2})$ and converges to a Gaussian distribution, as in Theorem 2.

For the actual proof, in addition to the known results, we need the local law
for resolvent entries, the central limit theorem for linear statistics, and
the rigidity of eigenvalues. While we prove these results in the current
paper, some of them are not strong enough to directly follow the analysis in
[1]. To overcome the difficulty, we introduce several changes in the detail of
the proof. Most notably, (1) for the low temperature case, instead of proving
a lemma analogous to Lemma 6.4 of [1] that is required to control the integral
of an exponential function along the curve of the steepest descent in Lemma
31, we prove a refined result for the curve in Lemma 30, and (2) for the high
temperature case, instead of controlling the difference
$|\gamma-\hat{\gamma}|$ by applying the rigidity of eigenvalues, we use the
local law to control it as in Lemma 33.

In what follows, we list our new results on the deformed Wigner matrices:

######  Definition 3.

  * •

For any $M>0$ and $\delta>0$, define

$D_{\delta}(M)=\big{\\{}x+{\text{i}}y\big{|}|x|\leq M,N^{-\delta}<|y|\leq
3\big{\\}}$

  * •

For any $z\in{\mathbb{C}}\backslash{\mathbb{R}}$, define

$m_{N}(z)=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\lambda_{i}-z},\quad
G(z)=\frac{1}{W+\lambda V-z}$

where $\lambda_{1}\geq\cdots\geq\lambda_{N}$ are eigenvalues of $W+\lambda V$.

######  Theorem 3 (local law for resolvent entries).

Suppose $M>0$ and $0<\delta\leq\frac{1}{4}$. For any $\epsilon^{\prime}>0$ and
$D^{\prime}>0$, we have for large enough $N$ that

${\mathbb{P}}\Big{(}\max_{i,j}|G_{ij}-\delta_{ij}\cdot\frac{1}{\lambda
v_{i}-z-m_{N}(z)}|\leq
N^{\epsilon^{\prime}-\frac{1}{2}}|{\text{Im}}z|^{-3},\forall z\in
D_{\delta}(M)\Big{)}>1-N^{-D^{\prime}}$

We remark that the local law for the trace of the resolvent was proved by the
first author and Schnelli. See [17] and also Section 3.2 of the current paper.

######  Theorem 4 (CLT for linear statistics).

Let $f(x)$ be a function which is analytic on a neighborhood of
$[L_{-},L_{+}]$. Suppose $a>1$, $b>37/3$ and
$\lambda>\max(\lambda_{+},\lambda_{-})$. Then

$\frac{1}{\sqrt{N}}\Big{(}\sum_{i}f(\lambda_{i})-N\int
f(t)\rho_{fc}(t)dt\Big{)}$

converges in distribution to a centered Gaussian distribution whose variance
is

$\frac{1}{4\pi^{2}}\Big{(}\oint_{\mathcal{C}}f(\xi)(1+m_{fc}^{\prime}(\xi))m_{fc}(\xi)d\xi\Big{)}^{2}-\frac{1}{4\pi^{2}}\int_{-1}^{1}\Big{(}\oint_{\mathcal{C}}\dfrac{f(\xi)(1+m_{fc}^{\prime}(\xi))}{\lambda
t-\xi-m_{fc}(\xi)}d\xi\Big{)}^{2}d\mu(t)$

where $\mathcal{C}$ is a counterclockwise path enclosing $[L_{-},L_{+}]$ such
that $f$ is analytic on a neighborhood of the region bounded by $\mathcal{C}$.

######  Definition 4.

Define the deterministic number $\gamma_{x}=\gamma_{x}(N)$ and
$\hat{\gamma}_{y}=\hat{\gamma}_{y}(N)$ by

$\mu_{fc}([\gamma_{x},+\infty])=\frac{x-\frac{1}{2}}{N}\quad\forall x\in[1,N]$
$\mu_{fc}([\hat{\gamma}_{y},+\infty])=\frac{y}{N}\quad\forall y\in(0,N)$

with the convention that $\hat{\gamma}_{N}=L_{-}$ and
$\hat{\gamma}_{0}=L_{+}$. Here $x$ and $y$ are not necessarily integers.

######  Theorem 5 (Rigidity of eigenvalues).

Suppose $a>1$, $b>3$ and $\lambda>\max(\lambda_{-},\lambda_{+})$. Suppose
$\epsilon\in(\frac{1}{b+1},\frac{1}{4})$. There exists an event
$E_{N}(\epsilon)$ such that

$\displaystyle{\mathbb{P}}(E_{N}(\epsilon))\geq 1-\kappa_{0}(\log
N)^{1+2b}N^{-\epsilon}$ (15)

when $N$ is large enough. Moreover, if $E_{N}(\epsilon)$ holds, then:

  1. 1.

for any $\zeta\in(0,\frac{\frac{1}{4}-\epsilon}{b+1})$ we have

$\displaystyle|\lambda_{i}-\gamma_{i}|\leq N^{-\frac{1}{4}+\epsilon+\zeta
b}\quad\text{when }N\text{ is large enough and
}i\in\mathbb{Z}\cap[\kappa^{\prime}N^{1-\zeta(b+1)},\frac{N}{2}]$ (16)

  2. 2.

for any $\zeta^{\prime}\in(0,\frac{\frac{1}{4}-\epsilon}{a+1})$ we have

$\displaystyle|\lambda_{i}-\gamma_{i}|\leq
N^{-\frac{1}{4}+\epsilon+\zeta^{\prime}a}\quad\text{when }N\text{ is large
enough and
}i\in\mathbb{Z}\cap[\frac{N}{2},N-\kappa^{\prime}N^{1-\zeta^{\prime}(a+1)}].$
(17)

Here $\kappa_{0}>0$ and $\kappa^{\prime}>0$ are constants independent of
$\zeta$ and $\zeta^{\prime}$.

###  2.4 Remarks

As discussed in Introduction, we expect the existence of the dichotomy between
the fluctuation given by the LSS in the high temperature regime and the
fluctuation dominated by the largest eigenvalue in the low temperature regime,
regardless of the choice of various parameters in the deformed Wigner matrix.
The main technical issue is the non-optimality of the local law; if the local
law can be improved, the rigidity result will also be improved and it will be
possible to relax the condition on $a$ and $b$. It is even expected that the
fluctuation of $F_{N}$ would converge to a Gaussian distribution when
$\lambda<\lambda_{+}$, since the fluctuation of $\lambda_{1}$ converges to a
Gaussian distribution in this case. However, we do not attempt to prove the
claim in the current paper.

##  3 Preliminaries

######  Definition 5.

Suppose $\omega$ is a measure on ${\mathbb{R}}$. Define its Stieltjes
transform by

$\displaystyle\int\frac{d\omega(t)}{t-z},\quad\forall
z\in{\mathbb{C}}\backslash\text{supp}(\omega).$ (18)

###  3.1 Fluctuation of the largest eigenvalue

Recall that $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}$ are
eigenvalues of $W+\lambda V$. The following theorem can be found in [17].

######  Theorem 6.

If $b>1$ and $\lambda>\lambda_{+}$, then

$\lim_{N\to\infty}{\mathbb{P}}\Big{(}N^{\frac{1}{1+b}}(L_{+}-\lambda_{1})\leq
s\Big{)}=1-\exp\Big{(}-\frac{C_{\mu}s^{1+b}}{1+b}\Big{)},\quad\forall s\geq 0$

where
$C_{\mu}=\Big{(}\frac{\lambda}{\lambda^{2}-\lambda_{+}^{2}}\Big{)}^{b+1}\cdot
d(1)\cdot 2^{a}\cdot Z^{-1}$ as defined in Theorem 1.

If $a>1$ and $\lambda>\lambda_{-}$, then

$\lim_{N\to\infty}{\mathbb{P}}\Big{(}N^{\frac{1}{1+a}}(\lambda_{N}-L_{-})\leq
s\Big{)}=1-\exp\Big{(}-\frac{C_{\mu}^{\prime}s^{1+a}}{1+a}\Big{)},\quad\forall
s\geq 0$

where
$C_{\mu}^{\prime}=\Big{(}\frac{\lambda}{\lambda^{2}-\lambda_{-}^{2}}\Big{)}^{a+1}\cdot
d(-1)\cdot 2^{b}\cdot Z^{-1}$.

######  Remark 1.

For the second conclusion of Theorem 6, see the sentence above section 2.4.1
in [17]. It can also be proved by replacing $W+\lambda V$ by $-W+\lambda(-V)$.

The next lemma is a direct corollary of (3.22) in [17].

######  Lemma 2.

For any constant $r>0$ we have that

$\lim_{N\to\infty}{\mathbb{P}}\Big{(}\max\limits_{1\leq k\leq
N}|\lambda_{k}|\leq 2+\lambda+r\Big{)}=1.$

Therefore,

$\displaystyle[L_{-},L_{+}]\subset[-2-\lambda,2+\lambda].$ (19)

###  3.2 Local law for the trace of the resolvent

In this subsection we introduce the local law for the trace of the resolvent
obtained in [17].

######  Definition 6.

Suppose $\mu_{N}$ is the empirical measure of $W+\lambda V$:
$\mu_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}$. Let

  * •

$m_{fc}(z)$ be the Stieltjes transform of $\mu_{fc}$:
$m_{fc}(z)=\int\frac{\rho_{fc}(t)}{t-z}dt$ (as mentioned in Theorem 2);

  * •

$\hat{m}_{fc}(z)$ be the Stieltjes transform of
$(\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda v_{i}})\boxplus\mu_{sc}$.

######  Definition 7.

For $z\in{\mathbb{C}}\backslash{\mathbb{R}}$, let

$g_{i}(z)=\frac{1}{\lambda
v_{i}-z-m_{fc}(z)},\quad\hat{g}_{i}(z)=\frac{1}{\lambda
v_{i}-z-\hat{m}_{fc}(z)}.$

######  Lemma 3.

For $z\in{\mathbb{C}}\backslash{\mathbb{R}}$,

$m_{N}(z)=\frac{1}{N}{\text{Tr}}G(z),\quad\hat{m}_{fc}(z)=\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i}(z),\quad
m_{fc}(z)={\mathbb{E}}[g_{i}(z)]=\int\frac{1}{\lambda t-z-m_{fc}(z)}d\mu(t)$
$\displaystyle(1+\hat{m}_{fc}^{\prime}(z))\Big{(}1-\frac{1}{N}\sum\hat{g}_{i}^{2}(z)\Big{)}=1$
(20) $\displaystyle(1+m_{fc}^{\prime}(z))\Big{(}1-\int\frac{d\mu(t)}{(\lambda
t-z-m_{fc}(z))^{2}}\Big{)}=1.$ (21)
$\displaystyle|g_{i}(z)|\leq\frac{1}{|{\text{Im}}z|}\quad\text{and}\quad|\hat{g}_{i}(z)|\leq\frac{1}{|{\text{Im}}z|}$
(22)

###### Proof.

The first conclusion is trivial. The second and third conclusions are direct
corollaries of (2.3) of [18]. The fourth conclusion can be proved by taking
derivatives on both sides of the second conclusion:

$\displaystyle\hat{m}_{fc}^{\prime}(z)=\frac{1}{N}\sum_{i=1}^{N}\Big{(}\frac{1}{\lambda
v_{i}-z-\hat{m}_{fc}(z)}\Big{)}^{\prime}=\frac{1}{N}\sum\frac{1+\hat{m}_{fc}^{\prime}(z)}{(\lambda
v_{i}-z-\hat{m}_{fc}(z))^{2}}.$ (23)

The fifth conclusion can be proved by similarly taking derivatives on both
sides of the third conclusion. The last conclusion is because both
${\text{Im}}m_{fc}(z)$ and ${\text{Im}}\hat{m}_{fc}(z)$ have the same sign as
${\text{Im}}z$. ∎

######  Definition 8.

Suppose $\epsilon\in(0,\frac{11b-9}{2b+2})$. Let $\tilde{v}_{i}$ be the $i$-th
largest one of $\\{v_{1},\ldots,v_{N}\\}$. We define the regions
$\mathcal{D}_{\epsilon}$, $\mathcal{D}_{\epsilon}^{\prime}$ and the events
$\tilde{\Omega}(\epsilon)$, $\Omega_{*}(\epsilon)$ and
$\Omega_{0}(\epsilon,c_{1},c_{2})$ by the following.

  * •

$\mathcal{D}_{\epsilon}=\\{x+{\text{i}}y|-3-\lambda\leq x\leq
3+\lambda,N^{-\frac{1}{2}-\epsilon}\leq y\leq N^{-\frac{1}{1+b}+\epsilon}\\}$

  * •

$\mathcal{D}_{\epsilon}^{\prime}=\\{z\in\mathcal{D}_{\epsilon}||\lambda\tilde{v}_{i}-z-m_{fc}(z)|>\frac{1}{2}N^{-\frac{1}{1+b}-\epsilon},\forall
i\in[20,N]\\}$

  * •

$\tilde{\Omega}(\epsilon)=\\{|m_{N}(z)-\hat{m}_{fc}(z)|\leq
N^{2\epsilon-\frac{1}{2}}\text{ for all
}z\in\mathcal{D}_{\epsilon}^{\prime}\\}$

  * •

$\Omega_{*}(\epsilon)=\\{{\text{Im}}m_{N}(z)\leq
N^{2\epsilon-\frac{1}{2}},\forall z\in\mathcal{D}_{\epsilon}^{\prime}\\}.$

  * •

$\Omega_{0}(\epsilon,c_{1},c_{2})$ is the event on which the following
conditions are satisfied for any $k\in\\{1,\ldots,19\\}$.

    * –

If $j\in\\{1,\ldots,N\\}\backslash\\{k\\}$ then
$N^{-\epsilon-\frac{1}{1+b}}<|\tilde{v}_{j}-\tilde{v}_{k}|<(\log
N)N^{-\frac{1}{1+b}}$. Moreover

$N^{-\epsilon-\frac{1}{1+b}}<|1-\tilde{v}_{1}|<(\log N)N^{-\frac{1}{1+b}}.$

    * –

If $z\in\mathcal{D}_{\epsilon}$ and
$|{\text{Re}}(z+m_{fc}(z)-\lambda\tilde{v}_{k})|=\min\limits_{1\leq i\leq
N}|{\text{Re}}(z+m_{fc}(z)-\lambda\tilde{v}_{i})|$ then

$\frac{1}{N}\sum_{i\in\\{1,\ldots,N\\}\backslash\\{k\\}}\frac{1}{|\lambda\tilde{v}_{i}-z-m_{fc}(z)|^{2}}<c_{1}.$

    * –

If $z\in\mathcal{D}_{\epsilon}$ then
$\Big{|}\frac{1}{N}\sum\limits_{i=1}^{N}\frac{1}{\lambda\tilde{v}_{i}-z-m_{fc}(z)}-\int\frac{d\mu(t)}{\lambda
t-z-m_{fc}(z)}\Big{|}\leq c_{2}N^{\frac{3\epsilon}{2}-\frac{1}{2}}$.

Here $c_{1}\in(0,1)$ and $c_{2}>0$ are constants.

######  Remark 2.

  * •

Notice that $\mathcal{D}_{\epsilon}^{\prime}$ is random but is independent of
$W$.

  * •

We defined $\Omega_{0}(\epsilon,c_{1},c_{2})$ in the same way as Definition
3.5 in [17]. The condition $\epsilon\in(0,\frac{11b-9}{2b+2})$ comes from
(3.20) of [17]. Definition 3.5 in [17] involves a constant $n_{0}$ and we let
$n_{0}=20$ in the current paper.

  * •

[17] requires the entries of the diagonal matrix to be ordered along the
diagonal, so in order to used results in [17], we use $\tilde{v}_{i}$ instead
of $v_{i}$ in the definitions of $\mathcal{D}_{\epsilon}^{\prime}$ and
$\Omega_{0}(\epsilon,c_{1},c_{2})$.

######  Proposition 1.

Suppose $b>1$, $\lambda>\lambda_{+}$ and $\epsilon\in(0,\frac{11b-9}{2b+2})$.
There exist constants $c_{1}\in(0,1)$, $c_{2}>0$, $\nu_{0}>0$, $\nu_{1}>0$ and
$N_{0}>0$ such that:

  1. 1.

${\mathbb{P}}(\Omega_{0}(\epsilon,c_{1},c_{2}))\geq 1-\nu_{0}(\log
N)^{1+2b}N^{-\epsilon}$ for all $N$;

  2. 2.

${\mathbb{P}}(\Omega_{0}(\epsilon,c_{1},c_{2})\backslash\tilde{\Omega}(\epsilon))\leq\exp(-\nu_{1}(\log
N)^{10\log\log N})$ if $N$ is large enough;

  3. 3.

$\Omega_{0}(\epsilon,c_{1},c_{2})$ is measurable with respect to the sigma
algebra generated by the entries of $V$.

###### Proof.

The first two conclusions of Proposition 1 are proved in [17]. See (3.30) and
Proposition 5.1 there. The last conclusion is from the definition of
$\Omega_{0}(\epsilon,c_{1},c_{2})$. ∎

######  Definition 9.

Let $\Omega_{V}(\epsilon)$ be the $\Omega_{0}(\epsilon,c_{1},c_{2})$ with
$c_{1}=c_{1}(\epsilon)$ and $c_{2}=c_{2}(\epsilon)$ properly chosen such that
the conclusions of Proposition 1 hold.

The next lemmas are Lemma 5.5 and Lemma 3.7 in [17].

######  Lemma 4.

Suppose $b>1$, $\lambda>\lambda_{+}$ and $\epsilon\in(0,\frac{11b-9}{2b+2})$.
There exists a constant $\nu_{2}>0$ such that if $N$ is large enough then

${\mathbb{P}}(\Omega_{V}(\epsilon)\backslash\Omega_{*}(\epsilon))\leq\exp(-\nu_{2}(\log
N)^{10\log\log N}).$

######  Lemma 5.

Suppose $b>1$, $\lambda>\lambda_{+}$ and $\epsilon\in(0,\frac{11b-9}{2b+2})$.
If $\Omega_{V}(\epsilon)$ holds and $z\in\mathcal{D}_{\epsilon}^{\prime}$,
then

$|\hat{m}_{fc}(z)-m_{fc}(z)|\leq N^{2\epsilon-\frac{1}{2}}.$

###  3.3 Integral representation of the partition function of the SSK model

The following lemma comes from Lemma 1.3 and (5.25) of [1].

######  Lemma 6.

Suppose $M$ is an $N\times N$ real symmetric matrix with eigenvalues
$\lambda_{1}(M)\geq\cdots\geq\lambda_{N}(M)$. Suppose $\beta>0$. Then

$\int_{S_{N-1}}e^{\beta\langle\sigma,M\sigma\rangle}d\omega_{N}(\sigma)=C_{N}\int_{a_{0}-{\text{i}}\infty}^{a_{0}+{\text{i}}\infty}e^{\frac{N}{2}R_{M}(z)}dz$

where

  * •

$a_{0}$ is an arbitrary constant satisfying $a_{0}>\lambda_{1}(M)$;

  * •

the integration contour is the vertical line from $a_{0}-{\text{i}}\infty$ to
$a_{0}+{\text{i}}\infty$;

  * •

$R_{M}(z)=2\beta z-\frac{1}{N}\sum_{i}\log(z-\lambda_{i}(M))$ where we take
the analytic branch of the $\log$ function such that
${\text{Im}}\log(z-\lambda_{i}(M))\in(-\pi,\pi)$ for all $z$ on the
integration contour;

  * •

$C_{N}=\frac{\Gamma(N/2)}{2\pi{\text{i}}(N\beta)^{\frac{N}{2}-1}}$ where
$\Gamma(z)$ denotes the Gamma function. Moreover,

$C_{N}=\frac{\sqrt{N}\beta}{{\text{i}}\sqrt{\pi}(2\beta
e)^{N/2}}(1+O(N^{-1})).$

###  3.4 Helffer-Sjöstrand formula

The next lemma can be found in Section 11.2 of [10].

######  Lemma 7.

Suppose $\chi:{\mathbb{R}}\to[0,1]$ is $C^{\infty}$ such that $\chi(x)=1$ when
$x\in[-1,1]$ and $\chi(x)=0$ when $x\not\in[-2,2]$. If $f\in
C_{c}^{2}({\mathbb{R}})$, then

$f(t)=\frac{1}{2\pi}\int_{{\mathbb{R}}^{2}}\frac{{\text{i}}yf^{\prime\prime}(x)\chi(y)+{\text{i}}\chi^{\prime}(y)(f(x)+{\text{i}}yf^{\prime}(x))}{t-x-{\text{i}}y}dxdy,\quad\forall
t\in{\mathbb{R}}.$

###  3.5 Cumulant expansion

The next lemma is Lemma 3.2 of [19].

######  Lemma 8.

Suppose $l\in\\{1,2,\ldots\\}$ and $F\in C^{l+1}({\mathbb{R}},{\mathbb{C}})$.
Let $Y$ be a real-valued centered random variable with finite moments up to
the order $l+2$. Let $\mathcal{G}$ be a sigma algebra independent of $Y$. Then

$\displaystyle{\mathbb{E}}[YF(Y)|\mathcal{G}]=\sum_{r=1}^{l}\frac{\kappa^{(r+1)}(Y)}{r!}{\mathbb{E}}[F^{(r)}(Y)|\mathcal{G}]+\mathcal{E}(Y)$

where $\kappa^{(r+1)}(Y)$ denotes the $(r+1)$-cumulant of $Y$. The error term
$\mathcal{E}(Y)$ satisfies:

$\displaystyle|\mathcal{E}(Y)|\leq C_{l}{\mathbb{E}}[|Y|^{l+2}]\sup_{|t|\leq
Q}|F^{(l+1)}(t)|+C_{l}{\mathbb{E}}[|Y|^{l+2}\mathds{1}_{|Y|>Q}]\sup_{t\in{\mathbb{R}}}|F^{(l+1)}(t)|$
(24)

where $Q>0$ is an arbitrary cutoff and $C_{l}$ satisfies
$C_{l}\leq\frac{(p_{0}\cdot l)^{l}}{l!}$ for some absolute constant $p_{0}>0$.

###  3.6 Marcinkiewicz-Zygmund inequality

The next lemma is copied from Lemma D.1, Lemma D.2 and Lemma D.3 of [5]. It is
a version of the Marcinkiewicz-Zygmund inequality.

######  Lemma 9.

Let $X_{1},\ldots,X_{N},Y_{1},\ldots,Y_{N}$ be independent centered random
variables such that for each $p\in\\{1,2,\ldots\\}$ there exists a constant
$\mu_{p}>0$ satisfying

${\mathbb{E}}[|X_{i}|^{p}]^{1/p}\leq\mu_{p},\quad{\mathbb{E}}[|Y_{i}|^{p}]^{1/p}\leq\mu_{p}\quad(1\leq
i\leq N).$

Then for deterministic families $(a_{ij})$ and $(b_{i})$ we have

${\mathbb{E}}[|\sum_{i}b_{i}X_{i}|^{p}]^{1/p}\leq
C_{p}\cdot\mu_{p}\cdot(\sum_{i}|b_{i}|^{2})^{1/2},\quad\forall p\geq 1$
${\mathbb{E}}[|\sum_{ij}a_{ij}X_{i}Y_{j}|^{p}]^{1/p}\leq
C_{p}\cdot\mu_{p}^{2}\cdot(\sum_{ij}|a_{ij}|^{2})^{1/2},\quad\forall p\geq 1$
${\mathbb{E}}[|\sum_{i\neq j}a_{ij}X_{i}X_{j}|^{p}]^{1/p}\leq
C_{p}\cdot\mu_{p}^{2}\cdot(\sum_{i\neq j}|a_{ij}|^{2})^{1/2},\quad\forall
p\geq 1$

where $C_{p}$ is a constant depending only on $p$.

###  3.7 Some results for symmetric matrices

Suppose $\mathtt{M}$ is an $N\times N$ real symmetric matrix. Suppose $T$ is a
subset of $\\{1,\ldots,N\\}$.

######  Definition 10.

  * •

We use $\mathtt{M}^{(T)}$ to denote the $(N-|T|)\times(N-|T|)$ matrix:

$(\mathtt{M}_{ij})_{i,j\in\\{1,\ldots,N\\}\backslash T}$

  * •

For $z\in{\mathbb{C}}\backslash{\mathbb{R}}$ let

$\mathtt{R}(z)=(\mathtt{M}-z)^{-1},\quad\mathtt{R}^{(T)}(z)=(\mathtt{M}^{(T)}-z)^{-1}$

  * •

We also set

$\sum_{i}^{(T)}=\sum_{i:i\not\in T},\quad\sum_{i,j}^{(T)}=\sum_{i:i\not\in
T}\sum_{j:j\not\in T}$

######  Remark 3.

  1. 1.

When $T=\\{i\\}$, we use $(i)$ instead of $(\\{i\\})$ in the above
definitions. Similarly, we write $(ij)$ instead of $(\\{i,j\\})$. We use
$(Ti)$ to denote $(T\cup\\{i\\})$.

  2. 2.

In $\mathtt{M}^{(T)}$ and $\mathtt{R}^{(T)}$ we use the original values of
matrix indices. For example, the indices for the rows and columns of
$\mathtt{M}^{(2)}$ are $1,3,4,\ldots,N$.

  3. 3.

It is easy to see that $\mathtt{R}(z)$ is a symmetric matrix.

######  Lemma 10.

Suppose ${\text{Im}}z_{1}\neq 0$ and ${\text{Im}}z_{2}\neq 0$. Then

  * •

$\frac{d}{dz}\mathtt{R}(z_{1})=\mathtt{R}^{2}(z_{1})$

  * •

$\sum_{j}|\mathtt{R}_{ij}(z_{1})|^{2}=\dfrac{{\text{Im}}\mathtt{R}_{ii}(z_{1})}{{\text{Im}}z_{1}},\quad\forall
i$

  * •

$\big{|}\big{(}\mathtt{R}^{k_{1}}(z_{1})(\mathtt{R}^{\prime})^{k_{2}}(z_{2})\big{)}_{ij}\big{|}\leq\frac{1}{|{\text{Im}}z_{1}|^{k_{1}}|{\text{Im}}z_{2}|^{2k_{2}}},\quad$
for any $i,j\in\\{1,\ldots,N\\}$, $k_{1},k_{2}\in\\{0,1,2,\ldots\\}$.

###### Proof.

The first can be proved by directly taking the derivative. The second
conclusion is the Ward identity, see (3.6) of [5]. For the last conclusion,
suppose

$\mathtt{M}=O\text{diag}(\lambda_{1}(\mathtt{M}),\ldots,\lambda_{N}(\mathtt{M}))O^{T}$

where $\lambda_{1}(\mathtt{M}),\ldots,\lambda_{N}(\mathtt{M})$ are eigenvalues
of $\mathtt{M}$ and $O$ is an orthogonal matrix. Then

$\mathtt{R}(z_{1})=O\text{diag}((\lambda_{1}(\mathtt{M})-z_{1})^{-1},\ldots,(\lambda_{N}(\mathtt{M})-z_{1})^{-1})O^{T}\quad\text{and}\\\
\mathtt{R}^{\prime}(z_{2})=O\text{diag}((\lambda_{1}(\mathtt{M})-z_{2})^{-2},\ldots,(\lambda_{N}(\mathtt{M})-z_{2})^{-2})O^{T}\quad\text{(by
the first conclusion of this lemma)}.$

So

$\big{|}\big{(}\mathtt{R}^{k_{1}}(z_{1})(\mathtt{R}^{\prime})^{k_{2}}(z_{2})\big{)}_{ij}\big{|}=\Big{|}\sum_{r=1}^{N}\frac{1}{(\lambda_{r}(\mathtt{M})-z_{1})^{k_{1}}(\lambda_{r}(\mathtt{M})-z_{2})^{2k_{2}}}O_{ir}O_{jr}\Big{|}\\\
\leq\frac{1}{|{\text{Im}}z_{1}|^{k_{1}}|{\text{Im}}z_{2}|^{2k_{2}}}\sum_{r=1}^{N}|O_{ir}O_{jr}|\leq\frac{1}{|{\text{Im}}z_{1}|^{k_{1}}|{\text{Im}}z_{2}|^{2k_{2}}}\sum_{r=1}^{N}\frac{O_{ir}^{2}+O_{jr}^{2}}{2}=\frac{1}{|{\text{Im}}z_{1}|^{k_{1}}|{\text{Im}}z_{2}|^{2k_{2}}}.$

∎

######  Lemma 11 (Resolvent identities).

  1. 1.

If $i,j,k\not\in T$ and $i,j\neq k$, then

$\displaystyle\mathtt{R}^{(T)}_{ij}=\mathtt{R}_{ij}^{(Tk)}+\dfrac{\mathtt{R}^{(T)}_{ik}\mathtt{R}^{(T)}_{jk}}{\mathtt{R}^{(T)}_{kk}}.$
(25)

  2. 2.

if $i\neq j$ then

$\displaystyle\mathtt{R}_{ij}=-\mathtt{R}_{ii}\mathtt{R}_{jj}^{(i)}\Big{(}\mathtt{M}_{ij}-\sum_{k,l}^{(ij)}\mathtt{M}_{ik}\mathtt{R}_{kl}^{(ij)}\mathtt{M}_{lj}\Big{)}$
(26)

  3. 3.

$\displaystyle\mathtt{R}_{ii}^{-1}=\mathtt{M}_{ii}-z-\sum_{k,l}^{(i)}\mathtt{M}_{ik}\mathtt{R}_{kl}^{(i)}M_{li}$
(27)

  4. 4.

$\displaystyle\frac{\partial\mathtt{R}_{kl}}{\partial\mathtt{M}_{ij}}=\frac{-1}{1+\delta_{ij}}(\mathtt{R}_{ki}\mathtt{R}_{lj}+\mathtt{R}_{kj}\mathtt{R}_{li})$
(28)

###### Proof.

The first conclusion can be found in (3.4) of [5]. The second conclusion is
(5.9) of [5]. The third conclusion is (5.1) of [5]. The fourth conclusion can
be proved by definition. ∎

######  Lemma 12.

If $\mathtt{M}$ is an $N\times N$ real symmetric random matrix such that
$\\{\mathtt{M}_{ij}|i\leq j\\}$ are independent and
${\mathbb{E}}[\mathtt{M}_{ij}^{2}]=\frac{1}{N}$ for $i\neq j$, then

$\frac{1}{\mathtt{R}_{ii}(z)}=-z-\frac{1}{N}{\text{Tr}}\mathtt{R}(z)+\mathtt{M}_{ii}+\frac{1}{N}\sum_{k}\frac{(\mathtt{R}_{ki}(z))^{2}}{\mathtt{R}_{ii}(z)}-\sum_{k,l}^{(i)}\mathtt{M}_{ik}\mathtt{R}_{kl}^{(i)}\mathtt{M}_{li}+\frac{1}{N}\sum_{k}^{(i)}\mathtt{R}_{kk}^{(i)}.$

###### Proof.

This lemma is from Lemma 5.2 of [5] and the fact that
$\sum_{k,l}^{(i)}{\mathbb{E}}\Big{[}\mathtt{M}_{ik}\mathtt{R}_{kl}^{(i)}\mathtt{M}_{li}\Big{|}\mathtt{M}^{(i)}\Big{]}=\frac{1}{N}\sum_{k}^{(i)}\mathtt{R}_{kk}^{(i)}$
(since $\mathtt{M}_{ik}\mathtt{M}_{li}$ is independent of the sigma algebra
generated by the entries of $\mathtt{M}^{(i)}$). ∎

##  4 Local law for resolvent entries: proof of Theorem 3

In this section we follow the idea introduced in [5] to prove Theorem 3.
Recall that $G(z)$ is defined in Definition 3.

######  Lemma 13.

Suppose $i\neq j$. For any $\epsilon^{\prime}>0$, $D^{\prime}>0$, there exists
$N_{0}=N_{0}(\epsilon^{\prime},D^{\prime})>0$ such that if $N>N_{0}$ and
$z\in{\mathbb{C}}\backslash{\mathbb{R}}$ then

${\mathbb{P}}\Big{(}\Big{|}\sum_{k,l}^{(ij)}W_{ik}G_{kl}^{(ij)}W_{lj}\Big{|}\leq
N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k,l}^{(ij)}|G_{kl}^{(ij)}|^{2}}\Big{)}>1-N^{-D^{\prime}}$
${\mathbb{P}}\Big{(}\Big{|}\sum_{k,l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}\Big{|}\leq
N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k\neq
l}^{(i)}|G_{kl}^{(i)}|^{2}}+N^{\epsilon^{\prime}-1}\sum_{k}^{(i)}|G_{kk}^{(i)}|\Big{)}>1-N^{-D^{\prime}}$
${\mathbb{P}}\Big{(}\Big{|}\sum_{k,l}^{(ij)}W_{jk}G_{kl}^{(ij)}W_{lj}\Big{|}\leq
N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k\neq
l}^{(ij)}|G_{kl}^{(ij)}|^{2}}+N^{\epsilon^{\prime}-1}\sum_{k}^{(ij)}|G_{kk}^{(ij)}|\Big{)}>1-N^{-D^{\prime}}$
${\mathbb{P}}\Big{(}\Big{|}\sum_{k}^{(i)}(W_{ik}^{2}-\frac{1}{N})G_{kk}^{(i)}\Big{|}\leq
N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k}^{(i)}|G_{kk}^{(i)}|^{2}}\Big{)}>1-N^{-D^{\prime}}$

where each of $G_{kl}^{(ij)}$ and $G_{kl}^{(i)}$ takes value at $z$.

###### Proof.

  1. 1.

For the first conclusion, suppose $\mathcal{G}_{1}$ is the sigma algebra
generated by entries of $(W+\lambda V)^{(ij)}$. Then $W_{ik}$ and $W_{lj}$ are
independent of $\mathcal{G}_{1}$. Let

$B_{kl}=\frac{G_{kl}^{(ij)}}{\sqrt{\sum_{k,l}^{(ij)}|G_{kl}^{(ij)}|^{2}}}.$

For any natural number $p$ and any sample point $\omega$ in the probability
space, we have

${\mathbb{E}}\Big{[}\Big{|}\sum_{k,l}^{(ij)}W_{ik}B_{kl}W_{lj}\Big{|}^{2p}\Big{|}\mathcal{G}_{1}\Big{]}(\omega)={\mathbb{E}}\Big{[}\Big{(}\sum_{k,l}^{(ij)}W_{ik}B_{kl}W_{lj}\Big{)}^{p}\Big{(}\sum_{k,l}^{(ij)}W_{ik}\bar{B}_{kl}W_{lj}\Big{)}^{p}\Big{|}\mathcal{G}_{1}\Big{]}(\omega)\\\
={\mathbb{E}}\Big{[}\Big{(}\sum_{k,l}^{(ij)}W_{ik}B_{kl}(\omega)W_{lj}\Big{)}^{p}\Big{(}\sum_{k,l}^{(ij)}W_{ik}\bar{B}_{kl}(\omega)W_{lj}\Big{)}^{p}\Big{]}={\mathbb{E}}\Big{[}\Big{|}\sum_{k,l}^{(ij)}W_{ik}B_{kl}(\omega)W_{lj}\Big{|}^{2p}\Big{]}\\\
\leq
C\Big{(}\sum_{k,l}^{(ij)}\Big{|}\frac{B_{kl}(\omega)}{N}\Big{|}^{2}\Big{)}^{p}=CN^{-2p}$

where $C>0$ depends only on $p$. We used (9) and the second conclusion of
Lemma 9 in the inequality. So,

${\mathbb{P}}\Big{(}\Big{|}\sum_{k,l}^{(ij)}W_{ik}G_{kl}^{(ij)}W_{lj}\Big{|}>N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k,l}^{(ij)}|G_{kl}^{(ij)}|^{2}}\Big{)}={\mathbb{P}}\Big{(}\Big{|}N\sum_{k,l}^{(ij)}W_{ik}B_{kl}W_{lj}\Big{|}>N^{\epsilon^{\prime}}\Big{)}\\\
\leq
N^{-2p\epsilon^{\prime}}{\mathbb{E}}\Big{[}\Big{|}N\sum_{k,l}^{(ij)}W_{ik}B_{kl}W_{lj}\Big{|}^{2p}\Big{]}\leq
CN^{-2p\epsilon^{\prime}}.$

Choosing $p$ large enough such that $2p\epsilon^{\prime}>D^{\prime}$ we
complete the proof of the first conclusion.

  2. 2.

For the second conclusion, we use the same argument as above except that the
$\mathcal{G}_{1}$ is replaced by the sigma algebra generated by entries of
$(W+\lambda V)^{(i)}$, the summation is replaced by $\sum_{k\neq l}^{(i)}$ and
$B_{kl}$ is replaced by

$\frac{G_{kl}^{(i)}}{\sqrt{\sum_{k\neq l}^{(i)}|G_{kl}^{(i)}|^{2}}}.$

Then using the third conclusion of Lemma 9 we have for
$N>N_{0}=N_{0}(\epsilon^{\prime},D^{\prime})$:

$\displaystyle{\mathbb{P}}\Big{(}\Big{|}\sum_{k\neq
l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}\Big{|}>N^{\epsilon^{\prime}}\sqrt{\frac{1}{N^{2}}\sum_{k\neq
l}^{(i)}|G_{kl}^{(i)}|^{2}}\Big{)}\leq CN^{-2p\epsilon^{\prime}}\leq
N^{-D^{\prime}}.$ (29)

This together with (10) and the fact that
$\sum_{k,l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}=\sum_{k\neq
l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}+\sum_{k}^{(i)}W_{ik}^{2}G_{kk}^{(i)}$
complete the proof of the second conclusion.

  3. 3.

The third conclusion can be proved in the same way as the second conclusion.

  4. 4.

For the last conclusion, let $X_{k}=NW_{ik}^{2}-1$ and
$Q_{k}=G_{kk}^{(i)}/\sqrt{\sum_{k}^{(i)}|G_{kk}^{(i)}|^{2}}$. Then, similarly
as above, with $\mathcal{G}_{2}$ be the sigma algebra generated by entries of
$(W+\lambda V)^{(i)}$, for any natural number $p$ and any sample point
$\omega$,

${\mathbb{E}}[|\sum_{k}^{(i)}X_{k}Q_{k}|^{2p}|\mathcal{G}_{2}](\omega)={\mathbb{E}}[|\sum_{k}^{(i)}X_{k}Q_{k}(\omega)|^{2p}]\leq
C$

where $C$ depends only on $p$. We used (9) and the first conclusion of Lemma 9
in the last inequality. So for any $\epsilon^{\prime}>0$, $D^{\prime}>0$, if
$N>N_{0}=N_{0}(\epsilon^{\prime},D^{\prime})$, then

$\displaystyle{\mathbb{P}}\Big{(}|\sum_{k}^{(i)}X_{k}G_{kk}^{(i)}|>N^{\epsilon^{\prime}}\sqrt{\sum_{k}^{(i)}|G_{kk}^{(i)}|^{2}}\Big{)}\leq
N^{-2p\epsilon^{\prime}}{\mathbb{E}}[|\sum_{k}^{(i)}X_{k}Q_{k}|^{2p}]\leq
CN^{-2p\epsilon^{\prime}}$

Choosing $p$ large enough such that $2p\epsilon^{\prime}>D^{\prime}$, we
complete the proof.

∎

######  Corollary 2.

For any $M>0$, $\epsilon^{\prime}>0$, $D^{\prime}>0$, if $|{\text{Re}}z|\leq
M$ and $0<|{\text{Im}}z|\leq 3$ then

${\mathbb{P}}\Big{(}\Big{|}\frac{1}{G_{ii}}\Big{|}\leq\frac{3N^{\epsilon^{\prime}}}{|{\text{Im}}z|},\forall
i\in[1,N]\Big{)}>1-N^{-D^{\prime}}$
${\mathbb{P}}\Big{(}\Big{|}\frac{1}{G_{jj}^{(i)}}\Big{|}\leq\frac{3N^{\epsilon^{\prime}}}{|{\text{Im}}z|},\forall
i\neq j\Big{)}>1-N^{-D^{\prime}}$

for large enough $N$.

###### Proof.

By Lemma 11,

$\displaystyle\frac{1}{G_{ii}}=\lambda
v_{i}+W_{ii}-z-\sum_{k,l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li},\quad\frac{1}{G_{jj}^{(i)}}=\lambda
v_{j}+W_{jj}-z-\sum_{k,l}^{(ij)}W_{jk}G_{kl}^{(ij)}W_{lj}.$

This together with (10), Lemma 13 and the facts that

$|\lambda
v_{i}|\leq\lambda,\quad|G_{kl}^{(i)}|\leq\frac{1}{|{\text{Im}}z|},\quad|G_{kl}^{(ij)}|\leq\frac{1}{|{\text{Im}}z|},\quad|z|\leq
M+3\quad(\text{provided }|{\text{Re}}z|\leq M,|{\text{Im}}z|\in(0,3])$

yield the conclusions. ∎

###### Proof of Theorem 3.

Suppose $i\neq j$ and $z\in{\mathbb{C}}\backslash{\mathbb{R}}$. By Lemma 10
and (25)

$|\sum_{k}^{(ij)}G_{kk}^{(ij)}|=|\sum_{k}^{(ij)}\Big{(}G_{kk}-\frac{G_{ki}G_{ki}}{G_{ii}}-\frac{G_{kj}^{(i)}G_{kj}^{(i)}}{G_{jj}^{(i)}}\Big{)}|\\\
\leq\frac{N}{|{\text{Im}}z|}+\frac{|(G^{2})_{ii}-(G_{ii})^{2}-(G_{ij})^{2}|}{|G_{ii}|}+\frac{|(G^{(i)})^{2}_{jj}-(G_{jj}^{(i)})^{2}|}{|G_{jj}^{(i)}|}\leq\frac{N}{|{\text{Im}}z|}+\frac{3}{|{\text{Im}}z|^{2}|G_{ii}|}+\frac{2}{|{\text{Im}}z|^{2}|G_{jj}^{(i)}|}$
(30)

and

$\displaystyle\Big{(}\frac{1}{N^{2}}\sum_{k,l}^{(ij)}|G_{kl}^{(ij)}|^{2}\Big{)}^{1/2}=\Big{(}\sum_{k}^{(ij)}\frac{{\text{Im}}G_{kk}^{(ij)}}{N^{2}{\text{Im}}z}\Big{)}^{1/2}\leq\Big{(}\frac{1}{N|{\text{Im}}z|^{2}}+\frac{3}{N^{2}|{\text{Im}}z|^{3}|G_{ii}|}+\frac{2}{N^{2}|{\text{Im}}z|^{3}|G_{jj}^{(i)}|}\Big{)}^{1/2}$

By Corollary 2, for any $\epsilon^{\prime}>0$, $D^{\prime}>0$, if $N$ is large
enough and $i\neq j$, then

$\displaystyle{\mathbb{P}}\Big{(}\Big{(}\frac{1}{N^{2}}\sum_{k,l}^{(ij)}|G_{kl}^{(ij)}|^{2}\Big{)}^{1/2}\leq\sqrt{\frac{1}{N|{\text{Im}}z|^{2}}+\frac{15N^{\epsilon^{\prime}}}{N^{2}|{\text{Im}}z|^{4}}}\Big{)}>1-N^{-D^{\prime}}\quad\text{provided
}|{\text{Re}}z|\leq M,|{\text{Im}}z|\in(0,3]$ (31)

Now using (26), (10), the first conclusion of Lemma 13 and (31), we have that
for any $\epsilon^{\prime}>0$ and $D^{\prime}>0$, if $N$ is large enough then

${\mathbb{P}}\Big{(}\max_{i\neq
j}|G_{ij}|\leq\frac{N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}}+\frac{N^{\epsilon^{\prime}}}{N|{\text{Im}}z|^{4}}\Big{)}>1-N^{-D^{\prime}}\quad\text{provided
}|{\text{Re}}z|\leq M\text{ and }|{\text{Im}}z|\in(0,3].$

If $\delta\leq\frac{1}{4}$, then
$\frac{N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}}\geq\frac{N^{\epsilon^{\prime}}}{N|{\text{Im}}z|^{4}}$
for all $z\in D_{\delta}(M)$. This together with a classic “lattice” argument
proved that for any $\epsilon^{\prime}>0$ and $D^{\prime}>0$, if $N$ is large
enough then

$\displaystyle{\mathbb{P}}\Big{(}\max_{i\neq
j}|G_{ij}|\leq\frac{N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}},\forall
z\in D_{\delta}(M)\Big{)}>1-N^{-D^{\prime}}.$ (32)

By Lemma 12,

$|G_{ii}-\frac{1}{\lambda v_{i}-z-m_{N}(z)}|=\frac{|G_{ii}|}{|\lambda
v_{i}-z-m_{N}(z)|}|\frac{1}{G_{ii}}-(\lambda v_{i}-z-m_{N}(z))|\\\
=\frac{|G_{ii}|}{|\lambda
v_{i}-z-m_{N}(z)|}\Big{|}W_{ii}+\frac{1}{N}\sum_{k}\frac{(G_{ki}(z))^{2}}{G_{ii}(z)}-\sum_{k,l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}+\frac{1}{N}\sum_{k}^{(i)}G_{kk}^{(i)}\Big{|}\\\
\leq|{\text{Im}}z|^{-2}\Big{(}|W_{ii}|+\frac{|(G^{2})_{ii}|}{N|G_{ii}|}+|\sum_{k\neq
l}^{(i)}W_{ik}G_{kl}^{(i)}W_{li}|+|\sum_{k}^{(i)}(W_{ik}^{2}-\frac{1}{N})G_{kk}^{(i)}|\Big{)}$
(33)

where we used Lemma 10 and the fact that ${\text{Im}}m_{N}$ has the same sign
as ${\text{Im}}z$ in the last step.

By Lemma 10 and (25), we have

$\displaystyle\sqrt{\frac{1}{N^{2}}\sum_{k}^{(i)}|G_{kk}^{(i)}|^{2}}\leq\frac{1}{|{\text{Im}}z|\sqrt{N}}$
(34)

and

$\displaystyle\sqrt{\frac{1}{N^{2}}\sum_{k\neq
l}^{(i)}|G_{kl}^{(i)}|^{2}}\leq\sqrt{\frac{1}{N^{2}}\sum_{k,l}^{(i)}|G_{kl}^{(i)}|^{2}}=\Big{(}\sum_{k}^{(i)}\frac{{\text{Im}}G_{kk}^{(i)}}{N^{2}{\text{Im}}z}\Big{)}^{1/2}\leq\frac{1}{|{\text{Im}}z|\sqrt{N}}$
(35)

If $\epsilon^{\prime}>0$, $D^{\prime}>0$, $|{\text{Re}}z|\leq M$ and
$|{\text{Im}}z|\in(0,3]$, then by (33), (10), Corollary 2, (29) and the last
conclusion of Lemma 13 we have for large enough $N$:

${\mathbb{P}}\Big{(}|G_{ii}-\frac{1}{\lambda
v_{i}-z-m_{N}(z)}|\leq\frac{N^{\epsilon^{\prime}}}{|{\text{Im}}z|^{2}}\Big{[}\frac{1}{\sqrt{N}}+\frac{3N^{\epsilon^{\prime}}}{N|{\text{Im}}z|^{3}}+\sqrt{\frac{1}{N^{2}}\sum_{k\neq
l}^{(i)}|G_{kl}^{(i)}|^{2}}+\sqrt{\frac{1}{N^{2}}\sum_{k}^{(i)}|G_{kk}^{(i)}|^{2}}\Big{]}\Big{)}>1-N^{-D^{\prime}}$

thus by (34) and (35),

${\mathbb{P}}\Big{(}|G_{ii}-\frac{1}{\lambda
v_{i}-z-m_{N}(z)}|\leq\frac{3N^{2\epsilon^{\prime}}}{N|{\text{Im}}z|^{5}}+\frac{5N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}}\Big{)}>1-N^{-D^{\prime}}$

If $\delta\leq\frac{1}{4}$, then
$\frac{N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}}\geq\frac{N^{\epsilon^{\prime}}}{N|{\text{Im}}z|^{5}}$
for all $z\in D_{\delta}(M)$. This together with a classic “lattice” argument
yields that for any $\epsilon^{\prime}>0$ and $D^{\prime}>0$, if $N$ is large
enough then

${\mathbb{P}}\Big{(}|G_{ii}-\frac{1}{\lambda
v_{i}-z-m_{N}(z)}|\leq\frac{N^{\epsilon^{\prime}}}{\sqrt{N}|{\text{Im}}z|^{3}},\text{
for any }z\in D_{\delta}(M)\text{ and }1\leq i\leq
N\Big{)}>1-N^{-D^{\prime}}.$

This together with (32) completes the proof. ∎

##  5 Central limit theorem for linear statistics: proof of Theorem 4

Suppose $f$ is a fixed function satisfying the condition in Theorem 4. In this
section we prove Theorem 4, i.e., the fact that the linear statistics

$\displaystyle\frac{1}{\sqrt{N}}\Big{(}\sum_{i}f(\lambda_{i})-N\int
f(t)\rho_{fc}(t)dt\Big{)}$ (36)

converges in distribution to a Gaussian variable. We use the method introduced
in [12], but we prove Lemma 14 in a different way. The method we prove Lemma
14 is similar as that in [22].

Throughout this section, we assume that the conditions of Theorem 4 are
satisfied.

######  Definition 11.

  * •

Suppose $d\in(0,\frac{1}{2})$ is a constant which is small enough such that
$f$ is analytic on a neighborhood of the rectangular region whose vertices are
$L_{+}+d\pm d{\text{i}}$ and $L_{-}-d\pm d{\text{i}}$.

  * •

Use $\Gamma$ to denote the boundary of the above rectangular region with
counterclockwise orientation.

  * •

Let

$\varpi\in\Big{(}\frac{2}{3(b+1)},\min\Big{(}\frac{1}{14},\frac{1}{10}-\frac{2}{5(b+1)},\frac{1}{8}-\frac{1}{b+1},\frac{1}{9}-\frac{2}{3(b+1)}\Big{)}\Big{)},$
$\varsigma\in[\frac{1}{b+1},\frac{1}{8}-\varpi)\quad\quad\text{and}\quad\varsigma^{\prime}\in\Big{(}0,\min\Big{(}\frac{1}{3}-4\varpi,\frac{1}{2}-7\varpi,\frac{1}{2}-5\varpi-2\varsigma\Big{)}\Big{)}.$

  * •

Let $\Gamma_{+}=\\{z\in\Gamma||{\text{Im}}z|\geq N^{-\varpi}\\}$. The
orientation of $\Gamma_{+}$ is induced from $\Gamma$.

######  Remark 4.

Since $b>37/3$, it is easy to check that the constants $\varpi$, $\varsigma$,
$\varsigma^{\prime}$ exist.

######  Definition 12.

  * •

Let $\sigma(V)$ be the sigma algebra generated by $V$:

$\sigma(V)=\sigma(v_{1},\ldots,v_{N}).$

  * •

Use ${\mathbb{E}}_{N}[\cdot]$ to denote the conditional expectation
${\mathbb{E}}[\cdot|\sigma(V)]$.

######  Lemma 14.

As $N\to\infty$,

$\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\Big{[}{\text{Tr}}G(\xi)-{\mathbb{E}}_{N}[{\text{Tr}}G(\xi)]\Big{]}d\xi\to
0\quad\text{in distribution}.$

We prove Lemma 14 in Section 5.3.

######  Lemma 15.

As $N\to\infty$,

$\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\Big{[}N\hat{m}_{fc}(\xi)-{\mathbb{E}}_{N}[{\text{Tr}}G(\xi)]\Big{]}d\xi\to
0\quad\text{in distribution}.$

We prove Lemma 15 in Section 5.4.

###  5.1 Some auxiliary lemmas

Recall that $g_{i}$ and $\hat{g}_{i}$ are defined in Definition 7.

######  Lemma 16.

There is a constant $C_{d}>0$ depending on $d$ such that

$\min_{i}|\lambda v_{i}-z-m_{fc}(z)|\geq C_{d}\quad\forall z\in\Gamma$

Moreover, $g_{i}(z)$ is analytic on ${\mathbb{C}}\backslash[L_{-},L_{+}]$ for
all $i\in\\{1,\ldots,N\\}$.

###### Proof.

See Appendix B. ∎

######  Definition 13.

Define $M_{d}=\max(|L_{+}+d+2|,|L_{-}-d-2|)$ and

$\displaystyle
B_{N}=\tilde{\Omega}(\varsigma)\cap\\{\max_{i,j}|G_{ij}(z)-\frac{\delta_{ij}}{\lambda
v_{i}-z-m_{N}(z)}|\leq
N^{\varsigma^{\prime}-\frac{1}{2}}|{\text{Im}}z|^{-3},\forall z\in
D_{\varpi}(M_{d})\\}$ (37)

where $\tilde{\Omega}(\varsigma)$ is defined in Definition 9 and
$D_{\varpi}(M_{d})$ is defined by Definition 3. The parameters $\varpi$,
$\varsigma$ and $\varsigma^{\prime}$ are defined in Definition 11.

######  Lemma 17.

  1. 1.

there exists $N_{0}>0$ such that if $N>N_{0}$ then
$\Gamma_{+}\subset\mathcal{D}_{\varsigma}^{\prime}$

  2. 2.

for any $D^{\prime}>0$ we have that if $N$ is large enough then

$\displaystyle{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})<N^{-D^{\prime}}.$ (38)

  3. 3.

if $N$ is large enough and $B_{N}\cap\Omega_{V}(\varsigma)$ holds, then the
following holds for each $\xi\in\Gamma_{+}$:

$|G_{ii}(\xi)-\hat{g}_{i}(\xi)|\leq
N^{\varsigma^{\prime}-\frac{1}{2}}\cdot|{\text{Im}}\xi|^{-3}+N^{2\varsigma-\frac{1}{2}}\cdot|{\text{Im}}\xi|^{-2},\quad|G_{ii}(\xi)|\geq
W^{\prime}|{\text{Im}}\xi|$
$|\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi)|\leq
N^{2\varsigma-\frac{1}{2}}+\frac{3}{W^{\prime}N|{\text{Im}}\xi|^{3}}$
$|G_{ii}(\xi)-g_{i}(\xi)|\leq
N^{\varsigma^{\prime}-\frac{1}{2}}\cdot|{\text{Im}}\xi|^{-3}+2N^{2\varsigma-\frac{1}{2}}\cdot|{\text{Im}}\xi|^{-2}$

where $W^{\prime}$ is a constant in $(0,1)$.

###### Proof.

See Appendix B. ∎

###  5.2 Proof of Theorem 4

###### Proof of Theorem 4.

Let $\varpi^{\prime}$ be a positive constant in
$[\frac{1}{b+1}-\varpi,\varpi/2)$ and

$R_{N}=\Big{\\{}\lambda_{i}\in[L_{-}-\frac{d}{10},L_{+}+\frac{d}{10}],\forall
i\Big{\\}}\cap\Omega_{V}(\varpi^{\prime})\cap\tilde{\Omega}(\varpi^{\prime}).$

See Section 3.2 for the notations. By Definition 8, Definition 9, (19), Lemma
16 and the fact that $d<1/2$, we know that if $N$ is large enough then

$\displaystyle\Gamma\cap\\{z|N^{-\frac{1}{2}-\varpi^{\prime}}\leq{\text{Im}}z\leq
N^{-\varpi}\\}\subset\mathcal{D}_{\varpi^{\prime}}^{\prime}.$ (39)

By Theorem 6 and Proposition 1,

$\displaystyle{\mathbb{P}}(R_{N})\to 1\quad\text{as }N\to\infty.$ (40)

On $R_{N}$ we have

$f(\lambda_{i})=\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{f(\xi)}{\xi-\lambda_{i}}d\xi\quad\text{and}\quad
f(t)=\frac{1}{2\pi{\text{i}}}\oint_{\Gamma}\frac{f(\xi)}{\xi-t}d\xi\quad\forall
t\in[L_{-},L_{+}]$

and then

$\frac{1}{\sqrt{N}}[\sum f(\lambda_{i})-N\int
f(t)\rho_{fc}(t)dt]=\frac{1}{\sqrt{N}}\frac{1}{2\pi{\text{i}}}\Big{[}\oint_{\Gamma}f(\xi)\sum\frac{1}{\xi-\lambda_{i}}d\xi-N\int\oint_{\Gamma}\frac{f(\xi)}{\xi-t}d\xi\cdot\rho_{fc}(t)dt\Big{]}\\\
=\frac{1}{\sqrt{N}2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(Nm_{fc}(\xi)-{\text{Tr}}G(\xi))d\xi=\\\
\frac{1}{\sqrt{N}2\pi{\text{i}}}\int_{\Gamma\backslash\Gamma_{+}}f(\xi)(Nm_{fc}(\xi)-{\text{Tr}}G(\xi))d\xi+\frac{1}{\sqrt{N}2\pi{\text{i}}}\int_{\Gamma_{+}}f(\xi)({\mathbb{E}}_{N}[{\text{Tr}}G(\xi)]-{\text{Tr}}G(\xi))d\xi\\\
+\frac{1}{\sqrt{N}2\pi{\text{i}}}\int_{\Gamma_{+}}f(\xi)(N\hat{m}_{fc}(\xi)-{\mathbb{E}}_{N}[{\text{Tr}}G(\xi)])d\xi+\frac{1}{\sqrt{N}2\pi{\text{i}}}\int_{\Gamma_{+}}f(\xi)(Nm_{fc}(\xi)-N\hat{m}_{fc}(\xi))d\xi\\\
:=P_{0}+P_{1}+P_{2}+P_{3}.$ (41)

######  Lemma 18.

If $\xi\in{\mathbb{C}}\backslash{\mathbb{R}}$ then

$\sqrt{N}(\hat{m}_{fc}-m_{fc})=\\\
(1+m_{fc}^{\prime})\Big{(}\frac{1}{\sqrt{N}}\sum\Big{(}g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)]\Big{)}+\frac{(\hat{m}_{fc}-m_{fc})^{2}}{\sqrt{N}}\sum\hat{g}_{i}g_{i}^{2}+\frac{1}{\sqrt{N}}(\hat{m}_{fc}-m_{fc})\sum(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])\Big{)}.$

###### Proof.

See Appendix B. ∎

Using Lemma 18 for $P_{3}$ (i.e., the last term in (41)), we have that if
$R_{N}$ holds then

$\displaystyle\frac{1}{\sqrt{N}}[\sum f(\lambda_{i})-N\int
f(t)\rho_{fc}(t)dt]=P_{0}+P_{1}+P_{2}+P_{31}+P_{32}+P_{33}$ (42)

where $P_{0}$, $P_{1}$, $P_{2}$ are defined in (41) and

$P_{31}:=\frac{-1}{2\pi{\text{i}}\sqrt{N}}\sum_{i}\int_{\Gamma_{+}}f(\xi)(1+m_{fc}^{\prime})(g_{i}-{\mathbb{E}}g_{i})d\xi,$
$P_{32}:=\frac{-1}{2\pi{\text{i}}\sqrt{N}}\int_{\Gamma_{+}}f(\xi)(1+m_{fc}^{\prime})(\hat{m}_{fc}-m_{fc})\sum_{i}(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])d\xi,$
$P_{33}:=\frac{-1}{2\pi{\text{i}}\sqrt{N}}\int_{\Gamma_{+}}f(\xi)(1+m_{fc}^{\prime})(\hat{m}_{fc}-m_{fc})^{2}\sum_{i}\hat{g}_{i}g_{i}^{2}d\xi.$

$\bullet$ Asymptotic behavior of $P_{0}$. When $R_{N}$ holds, we have:

$|P_{0}|\leq\frac{1}{\sqrt{N}2\pi}\int_{\Gamma\cap\\{|{\text{Im}}z|\leq
N^{-(1+\varpi^{\prime})/2}\\}}|f(\xi)||Nm_{fc}(\xi)-{\text{Tr}}G(\xi)|d\xi\\\
+\frac{1}{\sqrt{N}2\pi}\int_{\Gamma\cap\\{N^{-(1+\varpi^{\prime})/2}\leq|{\text{Im}}z|\leq
N^{-\varpi}\\}}|f(\xi)||Nm_{fc}(\xi)-{\text{Tr}}G(\xi)|d\xi\\\
\leq\frac{1}{\sqrt{N}2\pi}4N^{-\frac{1}{2}-\frac{\varpi^{\prime}}{2}}\sup_{z\in\Gamma}|f(z)|\cdot
4N/d\quad\text{(since $|m_{fc}(\xi)|\leq 2/d$ and $|G_{ii}(z)|\leq 2/d$)}\\\
+\frac{1}{\sqrt{N}2\pi}4N^{-\varpi}\sup_{z\in\Gamma}|f(z)|\cdot N\cdot
2N^{2\varpi^{\prime}-\frac{1}{2}}\quad\text{(by \eqref{eqn44}, Lemma
\ref{lemma:hatm_fc_close_tom_fc} and Definition
\ref{definition:D_epsilon_andD'_epsilon})}\\\ =(\frac{8}{\pi
d}+\frac{4}{\pi})\sup_{z\in\Gamma}|f(z)|(N^{-\varpi^{\prime}/2}+N^{2\varpi^{\prime}-\varpi})=o(1)\quad\text{(since
$\varpi^{\prime}<\varpi/2$)}$ (43)

$\bullet$ Asymptotic behavior of $P_{33}$. Let

$\varpi^{\prime\prime}\in(\frac{1}{1+b},\frac{11b-9}{2b+2})\quad\text{such
that}\quad 4\varpi^{\prime\prime}+\varpi<\frac{1}{2}.$

By the condition on $\varpi$, such $\varpi^{\prime\prime}$ exists. When $N$ is
large enough, we have that
$\Gamma_{+}\subset\mathcal{D}_{\varpi^{\prime\prime}}^{\prime}$ and that if
$\Omega_{V}(\varpi^{\prime\prime})$ holds then by Lemma 5,

$\displaystyle|\hat{m}_{fc}(\xi)-m_{fc}(\xi)|\leq
N^{-\frac{1}{2}+2\varpi^{\prime\prime}}\quad\forall\xi\in\Gamma_{+}$ (44)
$|P_{33}\mathds{1}_{\Omega_{V}(\varpi^{\prime\prime})}|\leq\frac{1}{2\pi\sqrt{N}}|\Gamma|\sup_{z\in\Gamma}|f(z)|(1+\frac{1}{d^{2}})N^{-1+4\varpi^{\prime\prime}}N\cdot
N^{\varpi}\frac{1}{C_{d}^{2}}$

where $C_{d}$ is defined in Lemma 16. Here we used the fact that
$|m_{fc}^{\prime}(\xi)|$ is bounded by $\frac{1}{d^{2}}$ for $\xi\in\Gamma$.
The last inequality together with the facts that
$4\varpi^{\prime\prime}+\varpi<\frac{1}{2}$ and that
${\mathbb{P}}(\Omega_{V}(\varpi^{\prime\prime}))\to 1$ (by Proposition 1)
yield:

$\displaystyle P_{33}\to 0\quad\text{in distribution}.$ (45)

$\bullet$ Asymptotic behavior of $P_{32}$. Let

$W_{N}=\\{\frac{1}{N^{\frac{1}{2}+\varpi^{\prime\prime}}}|\sum_{i=1}^{N}(g_{i}^{2}(\xi)-{\mathbb{E}}[g_{i}^{2}(\xi)])|\leq
1,\quad\forall\xi\in\Gamma\\}.$

######  Lemma 19.

Suppose $a_{1}>0$, $a_{2}>0$ are constants. Then

${\mathbb{P}}\Big{(}\Big{|}\frac{1}{N^{\frac{1}{2}+a_{1}}}\sum_{i=1}^{N}(g_{i}^{2}(\xi)-{\mathbb{E}}[g_{i}^{2}(\xi)])\Big{|}\leq
a_{2},\quad\forall\xi\in\Gamma\Big{)}\to 1\quad\text{as }N\to\infty$
${\mathbb{P}}\Big{(}\Big{|}\frac{1}{N^{\frac{1}{2}+a_{1}}}\sum_{i=1}^{N}(g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)])\Big{|}\leq
a_{2},\quad\forall\xi\in\Gamma\Big{)}\to 1\quad\text{as }N\to\infty$

###### Proof.

See Appendix B. ∎

By Lemma 19 and Proposition 1,
${\mathbb{P}}(\Omega_{V}(\varpi^{\prime\prime})\cap W_{N})\to 1$ as
$N\to\infty$. This together with the fact that

$|\mathds{1}_{\Omega_{V}(\varpi^{\prime\prime})\cap
W_{N}}P_{32}|\leq\frac{1}{2\pi}\int_{\Gamma_{+}}\sup_{z\in\Gamma}|f(z)|(1+\frac{1}{d^{2}})|\hat{m}_{fc}-m_{fc}|N^{\varpi^{\prime\prime}}\frac{1}{N^{\frac{1}{2}+\varpi^{\prime\prime}}}\big{|}\sum_{i}(g_{i}^{2}-{\mathbb{E}}[g_{i}^{2}])\big{|}d\xi\\\
\leq\frac{1}{2\pi}|\Gamma|\sup_{z\in\Gamma}|f(z)|(1+\frac{1}{d^{2}})N^{3\varpi^{\prime\prime}-\frac{1}{2}}=o(1)\quad\text{(by
\eqref{eq90} and the condition on $\varpi^{\prime\prime}$)}$

yield:

$\displaystyle P_{32}\to 0\quad\text{in distribution}.$ (46)

$\bullet$ Asymptotic behavior of $P_{31}$. Let

$U_{N}=\\{\frac{1}{N^{\frac{1}{2}+\frac{\varpi}{2}}}|\sum_{i=1}^{N}(g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)])|\leq
1,\quad\forall\xi\in\Gamma\\}.$

By Lemma 19, ${\mathbb{P}}(U_{N})\to 1$ as $N\to\infty$. This together with
the fact that

$\mathds{1}_{U_{N}}\Big{|}\frac{-1}{2\pi{\text{i}}\sqrt{N}}\sum_{i}\int_{\Gamma\backslash\Gamma_{+}}f(\xi)(1+m_{fc}^{\prime})(g_{i}-{\mathbb{E}}g_{i})d\xi\Big{|}\\\
\leq\mathds{1}_{U_{N}}\cdot\frac{1}{2\pi}\int_{\Gamma\backslash\Gamma_{+}}\sup_{z\in\Gamma}|f(z)|(1+\frac{1}{d^{2}})N^{\varpi/2}\cdot\Big{(}\frac{1}{N^{\frac{1}{2}+\frac{\varpi}{2}}}|\sum_{i=1}^{N}(g_{i}(\xi)-{\mathbb{E}}[g_{i}(\xi)])|\Big{)}d\xi\\\
\leq\frac{1}{2\pi}\sup_{z\in\Gamma}|f(z)|(1+\frac{1}{d^{2}})\cdot
4N^{-\varpi/2}=o(1)\quad\text{(since
$|\Gamma\backslash\Gamma_{+}|=4N^{-\varpi}$)}$

yield:

$\frac{-1}{2\pi{\text{i}}\sqrt{N}}\sum_{i}\int_{\Gamma\backslash\Gamma_{+}}f(\xi)(1+m_{fc}^{\prime})(g_{i}-{\mathbb{E}}g_{i})d\xi\to
0\quad\text{in distribution}.$

So $P_{31}$ has the same limit in distribution as

$\displaystyle\frac{-1}{2\pi{\text{i}}\sqrt{N}}\sum_{i}\oint_{\Gamma}f(\xi)(1+m_{fc}^{\prime})(g_{i}-{\mathbb{E}}g_{i})d\xi.$
(47)

By central limit theorem, (47) converges in distribution to a centered
Gaussian distribution whose variance is

$\text{Var}\Big{(}\frac{-1}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(1+m_{fc}^{\prime}(\xi))g_{i}(\xi)d\xi\Big{)}=\text{Var}\Big{(}\frac{-1}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)\frac{1+m_{fc}^{\prime}(\xi)}{\lambda
v_{1}-\xi-m_{fc}(\xi)}d\xi\Big{)}\\\
={\mathbb{E}}\Big{[}\Big{(}\frac{-1}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)\frac{1+m_{fc}^{\prime}(\xi)}{\lambda
v_{1}-\xi-
m_{fc}(\xi)}d\xi\Big{)}^{2}\Big{]}-\Big{(}{\mathbb{E}}\Big{[}\frac{-1}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)\frac{1+m_{fc}^{\prime}(\xi)}{\lambda
v_{1}-\xi-m_{fc}(\xi)}d\xi\Big{]}\Big{)}^{2}\\\
=\frac{1}{4\pi^{2}}\Big{(}\oint_{\Gamma}f(\xi)(1+m_{fc}^{\prime}(\xi))m_{fc}(\xi)d\xi\Big{)}^{2}-\frac{1}{4\pi^{2}}\oint_{\Gamma}\oint_{\Gamma}\int_{-1}^{1}\dfrac{f(\xi_{1})f(\xi_{2})(1+m_{fc}^{\prime}(\xi_{1}))(1+m_{fc}^{\prime}(\xi_{2}))d\mu(t)}{(\lambda
t-\xi_{1}-m_{fc}(\xi_{1}))(\lambda
t-\xi_{2}-m_{fc}(\xi_{2}))}d\xi_{1}d\xi_{2}\\\
=\frac{1}{4\pi^{2}}\Big{(}\oint_{\Gamma}f(\xi)(1+m_{fc}^{\prime}(\xi))m_{fc}(\xi)d\xi\Big{)}^{2}-\frac{1}{4\pi^{2}}\int_{-1}^{1}\Big{(}\oint_{\Gamma}\dfrac{f(\xi)(1+m_{fc}^{\prime}(\xi))}{\lambda
t-\xi-m_{fc}(\xi)}d\xi\Big{)}^{2}d\mu(t)$ (48)

$\bullet$ Conclusion. The asymptotic behaviors of $P_{0}$, $P_{31}$, $P_{32}$,
$P_{33}$ together with (40), (42), Lemma 14 and Lemma 15 complete the proof of
Theorem 4. We remark that the variance of the imaginary part of
$\frac{-1}{2\pi{\text{i}}}\oint_{\Gamma}f(\xi)(1+m_{fc}^{\prime}(\xi))g_{i}(\xi)d\xi$,
i.e., the left hand side of (48), must be 0. This is because the above
argument show that (47) has the same limit in distribution as the real-valued
random variable (36). ∎

###  5.3 Proof of Lemma 14

###### Proof of Lemma 14.

According to Proposition 1, it suffices to prove that

$\displaystyle
X_{N}:=\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\Big{[}{\text{Tr}}G(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi)\Big{]}\cdot\mathds{1}_{\Omega_{V}(\varsigma)}d\xi$

converges in distribution to zero. Fix $t\in{\mathbb{R}}$. We only need to
show that

$\displaystyle{\mathbb{E}}[\exp({\text{i}}tX_{N})]\to 1\quad\text{as
}N\to\infty.$ (49)

Notice that

$\frac{d}{dt}{\mathbb{E}}[\exp({\text{i}}tX_{N})]=\frac{d}{dt}{\mathbb{E}}[{\mathbb{E}}_{N}[\exp({\text{i}}tX_{N})]]\\\
=\frac{{\text{i}}}{\sqrt{N}}{\mathbb{E}}\Big{[}{\mathbb{E}}_{N}\Big{[}\exp({\text{i}}tX_{N})\int_{\Gamma_{+}}f(\xi)\Big{[}{\text{Tr}}G(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi)\Big{]}\cdot\mathds{1}_{\Omega_{V}(\varsigma)}d\xi\Big{]}\Big{]}\\\
=\frac{{\text{i}}}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\cdot{\mathbb{E}}\Big{[}\mathds{1}_{\Omega_{V}(\varsigma)}{\mathbb{E}}_{N}\Big{[}\exp({\text{i}}tX_{N})[{\text{Tr}}G(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi)]\Big{]}\Big{]}d\xi$
(50)

######  Lemma 20.

Suppose the conditions of Lemma 14 are satisfied. Then we have

$\displaystyle\frac{\partial e^{{\text{i}}tX_{N}}}{\partial
W_{ij}}=e^{{\text{i}}tX_{N}}\frac{-2{\text{i}}t}{\sqrt{N}(1+\delta_{ij})}\int_{\Gamma_{+}}f(\xi)G_{ij}^{\prime}(\xi)d\xi\cdot\mathds{1}_{\Omega_{V}(\varsigma)}$
(51)

Moreover, there exist constants $r_{0}>0$ and $N_{0}>0$ such that if $N>N_{0}$
then

$\displaystyle|\frac{\partial e^{{\text{i}}tX_{N}}}{\partial W_{ij}}|\leq
r_{0}\cdot
N^{\varpi-\frac{1}{2}},\quad|\frac{\partial^{2}(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}^{2}}|\leq r_{0}\cdot
N^{2\varpi-\frac{1}{2}},\quad|\frac{\partial^{3}(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}^{3}}|\leq r_{0}\cdot N^{3\varpi-\frac{1}{2}}.$ (52)

###### Proof.

By Lemma 10,

$\frac{\partial e^{{\text{i}}tX_{N}}}{\partial
W_{ij}}=e^{{\text{i}}tX_{N}}\frac{{\text{i}}t}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\frac{-2}{1+\delta_{ij}}\sum_{k}G_{ik}(\xi)G_{kj}(\xi)d\xi\cdot\mathds{1}_{\Omega_{V}(\varsigma)}\\\
=e^{{\text{i}}tX_{N}}\frac{-2{\text{i}}t}{\sqrt{N}(1+\delta_{ij})}\int_{\Gamma_{+}}f(\xi)G_{ij}^{\prime}(\xi)d\xi\cdot\mathds{1}_{\Omega_{V}(\varsigma)}$

Noticing $|G^{\prime}_{ij}(\xi)|\leq|{\text{Im}}\xi|^{-2}$ we complete the
proof of the first inequality in(52). The other two inequalities in (52) can
be proved similarly by directly taking more derivatives of
$e^{{\text{i}}tX_{N}}$ with respect to $W_{ij}$. ∎

For any $\xi\in{\mathbb{C}}\backslash{\mathbb{R}}$, by the definition

$G(\xi)=\frac{1}{\lambda V+W-\xi}$

we have $(\xi-\lambda v_{i})G_{ii}=-1+(WG)_{ii}=-1+\sum_{j}W_{ij}G_{ij}$. Then
by (28),

$(\xi-\lambda
v_{i}){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G_{ii}(\xi)-{\mathbb{E}}_{N}G_{ii}(\xi))]=\sum_{j}\Big{(}{\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}W_{ij}G_{ij}(\xi)]-{\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}]{\mathbb{E}}_{N}[W_{ij}G_{ij}(\xi)]\Big{)}.$

To use cumulant expansion to study $(\xi-\lambda
v_{i}){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G_{ii}(\xi)-{\mathbb{E}}_{N}G_{ii}(\xi))]$,
we write:

$(\xi-\lambda
v_{i}){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G_{ii}(\xi)-{\mathbb{E}}_{N}G_{ii}(\xi))]\\\
=\sum_{j}\frac{1+\delta_{ij}}{N}\Big{(}{\mathbb{E}}_{N}[\frac{\partial
e^{{\text{i}}tX_{N}}}{\partial
W_{ij}}G_{ij}(\xi)+e^{{\text{i}}tX_{N}}\frac{-1}{1+\delta_{ij}}(G_{ii}(\xi)G_{jj}(\xi)+(G_{ij}(\xi))^{2})]\\\
-{\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}]{\mathbb{E}}_{N}[\frac{-1}{1+\delta_{ij}}(G_{ii}(\xi)G_{jj}(\xi)+(G_{ij}(\xi))^{2})]\Big{)}\\\
+\sum_{j}\dfrac{{\mathbb{E}}[|\sqrt{N}W_{ij}|^{3}]}{2N^{3/2}}\Big{(}{\mathbb{E}}_{N}[G_{ij}(\xi)\frac{\partial^{2}(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}^{2}}]+2{\mathbb{E}}_{N}[\frac{\partial(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}}\frac{\partial G_{ij}(\xi)}{\partial
W_{ij}}]+{\mathbb{E}}_{N}[(e^{{\text{i}}tX_{N}}-{\mathbb{E}}_{N}e^{{\text{i}}tX_{N}})\frac{\partial^{2}G_{ij}(\xi)}{\partial
W_{ij}^{2}}]\Big{)}\\\
+\mathcal{E}_{1}^{(i)}(\xi),\quad\forall\xi\in{\mathbb{C}}\backslash{\mathbb{R}}.$
(53)

According to Lemma 8, there are constants $r_{1}>0$ and $N_{0}>0$ such that if
$N>N_{0}$ then

$\displaystyle|\mathcal{E}_{1}^{(i)}(\xi)|\leq
r_{1}\cdot\frac{1}{N}\cdot\big{(}N^{3\varpi-\frac{1}{2}}\frac{1}{|{\text{Im}}\xi|}+\frac{1}{|{\text{Im}}\xi|^{4}}\big{)}\leq\frac{r_{1}}{N|{\text{Im}}\xi|^{4}}\quad\text{for
any }\xi\in\Gamma_{+}$ (54)

Here we used (9), Lemma 20 and the condition $\varpi<\frac{1}{14}$ to control
$\mathcal{E}_{1}^{(i)}(\xi)$. For convenience we let
$\mathcal{E}_{2}^{(i)}(\xi)$ be the second summation on the right hand side of
(53):

$\mathcal{E}_{2}^{(i)}(\xi):=\sum_{j}\dfrac{{\mathbb{E}}[|\sqrt{N}W_{ij}|^{3}]}{2N^{3/2}}\Big{(}{\mathbb{E}}_{N}[G_{ij}(\xi)\frac{\partial^{2}(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}^{2}}]+2{\mathbb{E}}_{N}[\frac{\partial(e^{{\text{i}}tX_{N}})}{\partial
W_{ij}}\frac{\partial G_{ij}(\xi)}{\partial
W_{ij}}]+{\mathbb{E}}_{N}[(e^{{\text{i}}tX_{N}}-{\mathbb{E}}_{N}e^{{\text{i}}tX_{N}})\frac{\partial^{2}G_{ij}(\xi)}{\partial
W_{ij}^{2}}]\Big{)}.$

Moreover we set:

$\mathcal{E}_{3}(\xi):=\frac{1}{N}\sum_{i}g_{i}(\xi){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G_{ii}^{\prime}(\xi)-{\mathbb{E}}_{N}G_{ii}^{\prime}(\xi))]\\\
+\frac{2{\text{i}}t}{N^{3/2}}\int_{\Gamma_{+}}f(\xi^{\prime})\sum_{i}g_{i}(\xi){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G(\xi)G^{\prime}(\xi^{\prime}))_{ii}]d\xi^{\prime}\mathds{1}_{\Omega_{V}(\varsigma)}-\sum_{i}g_{i}(\xi)\Big{(}\mathcal{E}_{1}^{(i)}(\xi)+\mathcal{E}_{2}^{(i)}(\xi)\Big{)}.$

######  Lemma 21.

For any $\xi\in{\mathbb{C}}\backslash{\mathbb{R}}$ we have:

$(1-\frac{1}{N}\sum_{i}g_{i}^{2}(\xi)){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}({\text{Tr}}G(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi))]\\\
=-\sum_{i}g_{i}(\xi){\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}(G_{ii}(\xi)-g_{i}(\xi))(m_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G(\xi))]\\\
-{\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}]\sum_{i}g_{i}(\xi){\mathbb{E}}_{N}[(G_{ii}(\xi)-g_{i}(\xi))(\frac{1}{N}{\text{Tr}}G(\xi)-m_{fc}(\xi))]+\mathcal{E}_{3}(\xi)$
(55)

###### Proof.

See Appendix B. ∎

Using (28) and Lemma 10, for any $z\in{\mathbb{C}}\backslash{\mathbb{R}}$

$|\frac{\partial^{2}G_{ij}}{\partial
W_{ij}^{2}}|=\frac{|6G_{ij}G_{ii}G_{jj}+2(G_{ij})^{3}|}{(1+\delta_{ij})^{2}}\leq\begin{cases}\frac{1}{|{\text{Im}}z|^{3}}\quad&\text{if
}i=j\\\ \frac{6}{|{\text{Im}}z|^{2}}\max_{i\neq j}|G_{ij}|+2\max_{i\neq
j}|G_{ij}|^{3}\quad&\text{if }i\neq j\end{cases}\\\
\leq\begin{cases}\frac{1}{|{\text{Im}}z|^{3}}\quad&\text{if }i=j\\\
\frac{8}{|{\text{Im}}z|^{2}}\max_{i\neq j}|G_{ij}|\quad&\text{if }i\neq
j\end{cases}$ (56)

By (9), (28), (56), Lemma 10 and Lemma 20, if $\xi\in\Gamma_{+}$ and $N>N_{0}$
then

$\displaystyle|\mathcal{E}_{2}^{(i)}(\xi)|\leq
r_{2}\Big{(}\frac{N^{2\varpi}}{N|{\text{Im}}\xi|}+\frac{1}{\sqrt{N}|{\text{Im}}\xi|^{2}}{\mathbb{E}}_{N}[\max_{j\neq
i}|G_{ij}(\xi)|]\Big{)}$ (57)

where $r_{2}>0$ and $N_{0}>0$ are constants.

By Lemma 10,

$\displaystyle|(G(\xi)G^{\prime}(\xi^{\prime}))_{ii}|\leq|{\text{Im}}\xi|^{-1}|{\text{Im}}\xi^{\prime}|^{-2}\quad\forall\xi,\xi^{\prime}\in{\mathbb{C}}\backslash{\mathbb{R}}.$
(58)

Plugging (58) into the definition of $\mathcal{E}_{3}(\xi)$, by (54), (57) and
the fact that $|g_{i}(\xi)|\leq|{\text{Im}}\xi|^{-1}$, we have

$\displaystyle|\mathcal{E}_{3}(\xi)|\leq
r_{3}\big{(}|{\text{Im}}\xi|^{-5}+N^{2\varpi}\cdot|{\text{Im}}\xi|^{-2}+\frac{\sqrt{N}}{|{\text{Im}}\xi|^{3}}{\mathbb{E}}_{N}[\max_{j\neq
i}|G_{ij}(\xi)|]\big{)},\quad\text{if }\xi\in\Gamma_{+}\text{ and }N>N_{0}$
(59)

where $r_{3}>0$ and $N_{0}>0$ are constants.

######  Lemma 22.

Suppose $\Omega_{V}(\varsigma)$ holds. If $N$ is large enough, then:

$\Big{|}1-\frac{1}{N}\sum_{i}g_{i}^{2}(\xi)\Big{|}\geq\frac{1}{3}|{\text{Im}}\xi|^{2}\quad\forall\xi\in\Gamma_{+}.$

###### Proof.

See Appendix B. ∎

By (55), (59), Lemma 17 and Lemma 22, if $N$ is large enough and
$\xi\in\Gamma_{+}$ then

$|{\mathbb{E}}_{N}[e^{{\text{i}}tX_{N}}({\text{Tr}}G(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi))]\mathds{1}_{\Omega_{V}(\varsigma)}|\\\
\leq
6N^{1+3\varpi}\Big{(}{\mathbb{E}}_{N}[|G_{ii}-g_{i}||m_{fc}-\frac{1}{N}{\text{Tr}}G|\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]+{\mathbb{E}}_{N}[\max_{i\neq
j}|G_{ij}(\xi)|\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{)}+p_{N}(\xi)\\\ \leq
30N^{1+5\varpi}{\mathbb{E}}_{N}[\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]+p_{N}(\xi)$ (60)

where
$p_{N}(\xi)=\frac{8N^{2\varsigma+\varsigma^{\prime}}}{|{\text{Im}}\xi|^{6}}+\frac{16N^{4\varsigma}}{|{\text{Im}}\xi|^{5}}+\frac{2r_{3}}{|{\text{Im}}\xi|^{7}}+\frac{2r_{3}N^{2\varpi}}{|{\text{Im}}\xi|^{4}}+\frac{2r_{3}N^{\varsigma^{\prime}}}{|{\text{Im}}\xi|^{8}}$.

According to (60), (50) and (38), there exist constants $r_{4}>0$ and
$N_{0}>0$ such that if $N>N_{0}$ then

$\Big{|}\frac{d}{dt}{\mathbb{E}}[\exp({\text{i}}tX_{N})]\Big{|}\leq\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}|f(\xi)|\Big{(}30N^{1+5\varpi}{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})+p_{N}(\xi)\Big{)}d\xi\\\ \leq
r_{4}(N^{5\varpi+2\varsigma+\varsigma^{\prime}-\frac{1}{2}}+N^{4\varpi+4\varsigma-\frac{1}{2}}+N^{7\varpi+\varsigma^{\prime}-\frac{1}{2}})$

and therefore

$\displaystyle\Big{|}{\mathbb{E}}[\exp({\text{i}}tX_{N})]-1\Big{|}\leq t\cdot
r_{4}(N^{5\varpi+2\varsigma+\varsigma^{\prime}-\frac{1}{2}}+N^{4\varpi+4\varsigma-\frac{1}{2}}+N^{7\varpi+\varsigma^{\prime}-\frac{1}{2}}).$
(61)

By the conditions on $\varpi$, $\varsigma$ and $\varsigma^{\prime}$ in
Definition 11, the exponent for each term on the right hand side of (61) must
be negative. So (49) is true and we complete the proof of Lemma 14. ∎

###  5.4 Proof of Lemma 15

###### Proof of Lemma 15.

According to Proposition 1, it suffices to prove that

$\displaystyle\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)\Big{[}N\hat{m}_{fc}(\xi)-{\mathbb{E}}_{N}{\text{Tr}}G(\xi)\Big{]}\cdot\mathds{1}_{\Omega_{V}(\varsigma)}d\xi\to
0\quad\text{in distribution}.$ (62)

Let

$Q_{i}=-\lambda v_{i}-W_{ii}+\sum_{p,q}^{(i)}W_{ip}G_{pq}^{(i)}W_{qi}.$

######  Lemma 23.

For any $\xi\in{\mathbb{C}}\backslash{\mathbb{R}}$,
${\mathbb{E}}_{N}[{\text{Tr}}G(\xi)-N\hat{m}_{fc}(\xi)]$ equals:

$(1+\hat{m}_{fc}^{\prime}(\xi))\Big{(}-\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i}^{2}(\xi){\mathbb{E}}_{N}[G_{ii}(\xi)+\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}]+{\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}]\Big{)}\\\
+(1+\hat{m}_{fc}^{\prime}(\xi))\sum_{i=1}^{N}\hat{g}_{i}^{3}(\xi)\Big{(}\frac{2}{N}+\frac{2}{N^{2}}{\mathbb{E}}_{N}[({\text{Tr}}G^{(i)}(\xi))^{\prime}]+\sum_{p}^{(i)}{\mathbb{E}}_{N}[(G_{pp}^{(i)}(\xi))^{2}]({\mathbb{E}}[W_{ip}^{4}]-\frac{3}{N^{2}})+{\mathbb{E}}_{N}[(\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi))^{2}]\Big{)}$
(63)

###### Proof.

See Appendix B. ∎

There exists a constant $u_{1}\in(0,1)$ such that if
$\xi\in\Gamma\backslash{\mathbb{R}}$, then

$|1/\hat{g}_{i}|=|\lambda
v_{i}-\xi-\hat{m}_{fc}|\leq\lambda+|\xi|+\frac{1}{|{\text{Im}}\xi|}\leq\frac{1}{u_{1}|{\text{Im}}\xi|}$

and thus by (22)

$\displaystyle
u_{1}\cdot|{\text{Im}}\xi|\leq|\hat{g}_{i}(\xi)|\leq\frac{1}{|{\text{Im}}\xi|}.$
(64)

By the definition of Stieltjes transform,
$|\hat{m}_{fc}^{\prime}|\leq\frac{1}{|{\text{Im}}\xi|^{2}}$. This together
with (9), Lemma 10 and (64) yield:

$\displaystyle|(1+\hat{m}_{fc}^{\prime})\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i}^{2}(\xi){\mathbb{E}}_{N}[G_{ii}(\xi)]|\leq
2|{\text{Im}}\xi|^{-5},\quad\forall\xi\in\Gamma\backslash{\mathbb{R}}$ (65)
$|(1+\hat{m}_{fc}^{\prime})\sum_{i=1}^{N}\hat{g}_{i}^{3}(\xi)\Big{(}\frac{2}{N}+\frac{2}{N^{2}}{\mathbb{E}}_{N}[({\text{Tr}}G^{(i)})^{\prime}]+\sum_{p}^{(i)}{\mathbb{E}}_{N}[(G_{pp}^{(i)})^{2}]({\mathbb{E}}[W_{ip}^{4}]-\frac{3}{N^{2}})\Big{)}|\\\
\leq
2|{\text{Im}}\xi|^{-5}\Big{(}2+2|{\text{Im}}\xi|^{-2}+|{\text{Im}}\xi|^{-2}(\max_{a,b}{\mathbb{E}}[(\sqrt{N}W_{ab})^{4}]+3)\Big{)}\leq
u_{2}|{\text{Im}}\xi|^{-7},\quad\forall\xi\in\Gamma\backslash{\mathbb{R}}$
(66)

where $u_{2}>0$ is a constant. According to Lemma 17, there are constants
$u_{3}>0$ and $N_{0}>0$ such that if $N>N_{0}$ and $\xi\in\Gamma_{+}$, then

$|(1+\hat{m}_{fc}^{\prime})\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i}^{2}(\xi){\mathbb{E}}_{N}[\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\cap
B_{N}}]|\leq\frac{u_{3}}{|{\text{Im}}\xi|^{5}}{\mathbb{E}}_{N}[|\sum_{p}^{(i)}(G_{ip}(\xi))^{2}|\mathds{1}_{\Omega_{V}(\varsigma)\cap
B_{N}}]\\\
=\frac{u_{3}}{|{\text{Im}}\xi|^{5}}{\mathbb{E}}_{N}[|(G^{2}(\xi))_{ii}-(G_{ii}(\xi))^{2}|\mathds{1}_{\Omega_{V}(\varsigma)\cap
B_{N}}]\leq u_{3}|{\text{Im}}\xi|^{-7}\quad\text{(by Lemma
\ref{lemma:well_known_facts_for_resolvent})}$ (67)
$\displaystyle|(1+\hat{m}_{fc}^{\prime}){\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}\mathds{1}_{\Omega_{V}(\varsigma)\cap
B_{N}}]|\leq
u_{3}(\frac{N^{3\varsigma^{\prime}}}{\sqrt{N}|{\text{Im}}\xi|^{13}}+\frac{N^{6\varsigma}}{\sqrt{N}|{\text{Im}}\xi|^{10}})$
(68)
$\displaystyle|(1+\hat{m}_{fc}^{\prime})\sum_{i=1}^{N}\hat{g}_{i}^{3}(\xi){\mathbb{E}}_{N}[(\hat{m}_{fc}-\frac{1}{N}{\text{Tr}}G^{(i)})^{2}\mathds{1}_{\Omega_{V}(\varsigma)\cap
B_{N}}]|\leq u_{3}N^{4\varsigma}|{\text{Im}}\xi|^{-5}.$ (69)

By Lemma 23 and the conditions in Definition 11, the terms on the left hand
side of (65), (66), (67), (68), (69) all make $o(1)$ contribution to the
quantity in (62). So to prove (62) it suffices to show that

$\displaystyle\frac{1}{N^{3/2}}\int_{\Gamma_{+}}f(\xi)(1+\hat{m}_{fc}^{\prime})\sum_{i=1}^{N}\hat{g}_{i}^{2}(\xi){\mathbb{E}}_{N}[\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]d\xi\to 0\quad\text{in distribution}$ (70)
$\displaystyle\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)(1+\hat{m}_{fc}^{\prime}){\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]d\xi\to 0\quad\text{in distribution}$ (71)
$\displaystyle\frac{1}{\sqrt{N}}\int_{\Gamma_{+}}f(\xi)(1+\hat{m}_{fc}^{\prime})\sum_{i=1}^{N}\hat{g}_{i}^{3}(\xi){\mathbb{E}}_{N}[(\hat{m}_{fc}-\frac{1}{N}{\text{Tr}}G^{(i)})^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]d\xi\to 0\quad\text{in distribution}$ (72)

  * •

Notice that $|\hat{m}_{fc}|$ and $|\frac{1}{N}{\text{Tr}}G^{(i)}|$ are bounded
by $\frac{1}{|{\text{Im}}\xi|}$. By (38), if $N$ is large enough and
$\xi\in\Gamma_{+}$ then

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[(\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}>N^{-5}\Big{)}\leq
N^{5}{\mathbb{E}}\Big{[}\Big{|}{\mathbb{E}}_{N}[(\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\Big{]}\\\ \leq
N^{5}{\mathbb{E}}\Big{[}\Big{|}(\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}\Big{|}\Big{]}\leq
4N^{5+2\varpi}{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash B_{N})<N^{-100}$

which together with a classic “lattice” argument yields

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[(\hat{m}_{fc}(\xi)-\frac{1}{N}{\text{Tr}}G^{(i)}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\leq 2N^{-5},\quad\forall\xi\in\Gamma_{+}\Big{)}\geq 1-N^{-20}.$

This and the facts that $|\hat{m}_{fc}^{\prime}(\xi)|\leq N^{2\varpi}$ and
$|\hat{g}_{i}(\xi)|\leq N^{\varpi}$ on $\Gamma_{+}$ complete the proof of
(72).

  * •

According to (38) and (27), if $N$ is large enough and $\xi\in\Gamma_{+}$,
then

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}>N^{-5}\Big{)}\leq
N^{5}{\mathbb{E}}\Big{[}\Big{|}{\mathbb{E}}_{N}[\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\Big{]}\\\ \leq
N^{5}{\mathbb{E}}\Big{[}\Big{|}\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\Big{|}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}\Big{]}\leq
N^{6+2\varpi}{\mathbb{E}}\Big{[}\Big{|}\frac{1}{G_{ii}(\xi)}\Big{|}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}\Big{]}\\\ \leq
N^{6+2\varpi}\sqrt{{\mathbb{E}}\Big{[}\frac{1}{|G_{ii}(\xi)|^{2}}\Big{]}}\sqrt{{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})}\\\ =N^{6+2\varpi}\sqrt{{\mathbb{E}}\Big{[}|\lambda
v_{i}+W_{ii}-\xi+\sum_{k,l}^{(i)}W_{ik}G_{kl}^{(i)}(\xi)W_{li}|^{2}\Big{]}}\sqrt{{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})}\leq N^{8+3\varpi}\sqrt{{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})}\leq N^{-100}$

which together with (27) and a classic “lattice” argument yields

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[\frac{1}{G_{ii}(\xi)}\sum_{p}^{(i)}(G_{ip}(\xi))^{2}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\leq 2N^{-5},\quad\forall\xi\in\Gamma_{+}\Big{)}\geq 1-N^{-20}.$

This and the facts that $|\hat{m}_{fc}^{\prime}(\xi)|\leq N^{2\varpi}$ and
$|\hat{g}_{i}(\xi)|\leq N^{\varpi}$ on $\Gamma_{+}$ complete the proof of
(70).

  * •

Similarly, according to (38) and (27), if $N$ is large enough and
$\xi\in\Gamma_{+}$, then

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}>N^{-5}\Big{)}\leq
N^{5}{\mathbb{E}}\Big{[}\Big{|}{\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\Big{]}\\\ \leq
8N^{5+3\varpi}\sum_{i}\sqrt{{\mathbb{E}}[\frac{1}{|G_{ii}(\xi)|^{4}}]{\mathbb{P}}(\Omega_{V}(\varsigma)\backslash
B_{N})}<N^{-100}$

which together with (27) and a classic “lattice” argument yields

${\mathbb{P}}\Big{(}\Big{|}{\mathbb{E}}_{N}[\sum_{i=1}^{N}\frac{(G_{ii}(\xi)-\hat{g}_{i}(\xi))^{3}}{(G_{ii}(\xi))^{2}}\mathds{1}_{\Omega_{V}(\varsigma)\backslash
B_{N}}]\Big{|}\leq 2N^{-5},\quad\forall\xi\in\Gamma_{+}\Big{)}\geq 1-N^{-20}.$

This and the facts that $|\hat{m}_{fc}^{\prime}(\xi)|\leq N^{2\varpi}$ and
$|\hat{g}_{i}(\xi)|\leq N^{\varpi}$ on $\Gamma_{+}$ complete the proof of
(71).

∎

##  6 Rigidity of eigenvalues: proof of Theorem 5

In this section we prove the rigidity of eigenvalues in the bulk of the
spectrum. For an eigenvalue $\lambda_{i}$ in the bulk, we show that it is very
close to the deterministic number $\gamma_{i}$ with high probability. Roughly
speaking, the distance between $\lambda_{i}$ and $\gamma_{i}$ is no more than
$N^{-\frac{1}{4}+\frac{1}{1+b}}$ with high probability.

By the definition of $\gamma_{i}$ and $\hat{\gamma}_{i}$ (see Definition 4) we
have that

$\displaystyle\int_{\hat{\gamma}_{i+1}}^{\hat{\gamma}_{i}}\frac{1}{L_{+}-t}d\mu_{fc}(t)\leq\frac{1}{N}\frac{1}{L_{+}-\gamma_{i}}\leq\int_{\hat{\gamma}_{i-1}}^{\hat{\gamma}_{i-2}}\frac{1}{L_{+}-t}d\mu_{fc}(t),\quad\forall
i\in[2,N-1].$ (73)

######  Lemma 24.

There exists a constant $C_{*}\geq 1$ such that

  * •

if $b>1$ and $\lambda>\lambda_{+}$ then
$C_{*}^{-1}\Big{(}\frac{i}{N}\Big{)}^{\frac{1}{1+b}}\leq|L_{+}-\gamma_{i}|\leq
C_{*}\Big{(}\frac{i}{N}\Big{)}^{\frac{1}{1+b}}$;

  * •

if $a>1$ and $\lambda>\lambda_{-}$ then
$C_{*}^{-1}\Big{(}\frac{N-i}{N}\Big{)}^{\frac{1}{1+a}}\leq|L_{-}-\gamma_{i}|\leq
C_{*}\Big{(}\frac{N-i}{N}\Big{)}^{\frac{1}{1+a}}$.

###### Proof.

Suppose $b>1$, $\lambda>\lambda_{+}$. According to Lemma 1, there exists
$C_{*}^{\prime}>1$ such that

$\frac{(L_{+}-x)^{b}}{C_{*}^{\prime}}\leq\rho_{fc}(x)\leq
C_{*}^{\prime}(L_{+}-x)^{b}$

for $x\in[\gamma_{0.99N},L_{+}]$. Therefore if $i\leq 0.99N$, then

$\frac{(L_{+}-\gamma_{i})^{b+1}}{C_{*}^{\prime}(b+1)}=\int_{\gamma_{i}}^{L_{+}}\frac{(L_{+}-x)^{b}}{C_{*}^{\prime}}dx\leq\int_{\gamma_{i}}^{L_{+}}\rho_{fc}(x)dx=\frac{i-\frac{1}{2}}{N}\leq\int_{\gamma_{i}}^{L_{+}}C_{*}^{\prime}(L_{+}-x)^{b}dx=\frac{C_{*}^{\prime}(L_{+}-\gamma_{i})^{b+1}}{(b+1)}.$

If $i>0.99N$, then both $i/N$ and $|L_{+}-\gamma_{i}|$ are of order 1, so the
inequality also holds. This proves the first conclusion. The second conclusion
can be proved in the same way. ∎

######  Definition 14.

For $\epsilon>0$, set

$A_{N}(\epsilon):=\Omega_{V}(\epsilon)\cap\tilde{\Omega}(\epsilon)\cap\Omega_{*}(\epsilon)$

where $\Omega_{V}(\epsilon)$, $\tilde{\Omega}(\epsilon)$ and
$\Omega_{*}(\epsilon)$ are defined in Definition 8 and Definition 9.

######  Lemma 25.

Suppose $b>1$ and $\epsilon\in(\frac{1}{1+b},\frac{11b-9}{2b+2})$. Suppose
$A_{N}(\epsilon)$ holds and $z_{0}$ is in

$\displaystyle\Big{\\{}x+{\text{i}}y\Big{|}|x|\leq
3+\lambda,y\in[\frac{1}{2}N^{-\frac{1}{1+b}-\epsilon},N^{-\frac{1}{1+b}+\epsilon}]\Big{\\}}.$
(74)

If $N$ is large enough, then we have $z_{0}\in\mathcal{D}_{\epsilon}^{\prime}$
and

$\displaystyle|m_{N}(z_{0})-m_{fc}(z_{0})|\leq 2N^{2\epsilon-\frac{1}{2}}.$
(75)

######  Remark 5.

In the condition $\epsilon\in(\frac{1}{1+b},\frac{11b-9}{2b+2})$, the interval
is not empty since $b>1$.

###### Proof.

By the condition $b>1$ it’s easy to see that (74) is contained in
$\mathcal{D}_{\epsilon}$. We notice that ${\text{Im}}z_{0}$ and
${\text{Im}}m_{fc}(z_{0})$ have the same sign, so $|\lambda
v_{i}-z_{0}-m_{fc}(z_{0})|>|{\text{Im}}z_{0}|\geq\frac{1}{2}N^{-\frac{1}{1+b}-\epsilon}$.
Thus $z_{0}\in\mathcal{D}_{\epsilon}^{\prime}$. Finally (75) is from Lemma 5
and the definition of $\tilde{\Omega}(\epsilon)$. ∎

Suppose $\epsilon>0$ and

  * •

$I$ is an interval contained in $(-2.99-\lambda,2.99+\lambda)$ and it may
depend on $N$,

  * •

$\eta_{0}=N^{-\frac{1}{4}+\epsilon}$ and
$\eta_{1}=N^{-\frac{1}{1+b}-\epsilon}$;

  * •

$\chi:{\mathbb{R}}\to[0,1]$ is a $C^{\infty}$ function supported on $[-2,2]$
such that $\chi(x)=1$ when $x\in[-1,1]$;

  * •

$f:{\mathbb{R}}\to[0,1]$ is a smooth $N$-depending function such that

$\displaystyle f(x)=\begin{cases}1\quad&\text{if }x\in I\\\ 0\quad&\text{if
}\text{dist}(x,I)\geq\eta_{0}\end{cases}$ (76)

and $\|f^{\prime}\|_{\infty}\leq C_{f}\cdot\eta_{0}^{-1}$,
$\|f^{\prime\prime}\|_{\infty}\leq C_{f}\cdot\eta_{0}^{-2}$ for some absolute
constant $C_{f}>0$.

######  Remark 6.

It is easy to see that $f$ satisfies the following properties.

  1. 1.

If $\epsilon<\frac{1}{4}$ then
$\text{supp}f\subset(-2.995-\lambda,2.995+\lambda)$ when
$N>N_{0}=N_{0}(\epsilon)$.

  2. 2.

$|\text{supp}f^{\prime}|\leq 2\eta_{0}$, $|\text{supp}f^{\prime\prime}|\leq
2\eta_{0}$.

Recall that $\mu_{N}$ is defined in Definition 6.

######  Lemma 26.

Suppose $b>3$ and $\epsilon\in(\frac{1}{1+b},\frac{1}{4})$. Suppose
$A_{N}(\epsilon)$ hold. Then

$|\int f(t)d\mu_{N}(t)-\int
f(t)d\mu_{fc}(t)|\leq\alpha_{1}(\eta_{0}^{2}+\eta_{1}^{2}\eta_{0})$

for large enough $N$. Here $\alpha_{1}>0$ is a constant depending only on
$C_{f}$.

######  Remark 7.

The condition $\epsilon\in(\frac{1}{1+b},\frac{1}{4})$ is stronger than the
condition $\epsilon\in(\frac{1}{1+b},\frac{11b-9}{2b+2})$ in Lemma 25 because
the condition $b>3$ ensures that $\frac{1}{4}<\frac{11b-9}{2b+2}$.

###### Proof.

According to the Helffer-Sjöstrand formula (see Lemma 7),

$\displaystyle\int f(t)d\mu_{N}(t)-\int f(t)d\mu_{fc}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\Big{(}\int_{{\mathbb{R}}^{2}}\frac{{\text{i}}yf^{\prime\prime}(x)\chi(y)+{\text{i}}(f(x)+{\text{i}}yf^{\prime}(x))\chi^{\prime}(y)}{t-x-{\text{i}}y}dxdy\Big{)}(d\mu_{N}(t)-d\mu_{fc}(t))$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{{\mathbb{R}}^{2}}\Big{[}{\text{i}}yf^{\prime\prime}(x)\chi(y)+{\text{i}}(f(x)+{\text{i}}yf^{\prime}(x))\chi^{\prime}(y)\Big{]}(m_{N}(x+{\text{i}}y)-m_{fc}(x+{\text{i}}y))dxdy$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}{\text{Re}}\Bigg{(}\int_{{\mathbb{R}}^{2}}\Big{[}{\text{i}}yf^{\prime\prime}(x)\chi(y)+{\text{i}}(f(x)+{\text{i}}yf^{\prime}(x))\chi^{\prime}(y)\Big{]}(m_{N}(x+{\text{i}}y)-m_{fc}(x+{\text{i}}y))dxdy\Bigg{)}$
$\displaystyle=$ $\displaystyle K_{1}-K_{2}-K_{3}$

where

$K_{1}={\text{Re}}\Big{(}\frac{{\text{i}}}{2\pi}\int_{{\mathbb{R}}^{2}}\chi^{\prime}(y)(f(x)+{\text{i}}yf^{\prime}(x))(m_{N}(x+{\text{i}}y)-m_{fc}(x+{\text{i}}y))dxdy\Big{)}$
$K_{2}=\frac{1}{2\pi}\int_{{\mathbb{R}}}\int_{|y|\geq\eta_{1}}yf^{\prime\prime}(x)\chi(y){\text{Im}}(m_{N}(x+{\text{i}}y)-m_{fc}(x+{\text{i}}y))dydx$
$K_{3}=\frac{1}{2\pi}\int_{{\mathbb{R}}}\int_{|y|<\eta_{1}}yf^{\prime\prime}(x){\text{Im}}(m_{N}(x+{\text{i}}y)-m_{fc}(x+{\text{i}}y))dydx$

Recall $\epsilon>\frac{1}{1+b}$ and that $A_{N}(\epsilon)$ holds. For
simplicity let $\tilde{m}(z)=m_{N}(z)-m_{fc}(z)$.

  1. 1.

We first estimate $K_{1}$. By (75), Remark 6 and the definition of $\chi$, if
$N$ is large enough then:

$\displaystyle|K_{1}|\leq 2N^{2\epsilon-\frac{1}{2}}\cdot\frac{1}{2\pi}\cdot
2\|\chi^{\prime}\|_{\infty}\int|f(x)|+2|f^{\prime}(x)|dx\leq
CN^{2\epsilon-\frac{1}{2}}=C\eta_{0}^{2}$

where $C>0$ is a constant depending only on $C_{f}$.

  2. 2.

Then we estimate $K_{2}$. Suppose $x+{\text{i}}y$ is in the support of
$yf^{\prime\prime}(x)\chi(y)\mathds{1}_{|y|\geq\eta_{1}}(x+{\text{i}}y)$. Let
$L$ be the counterclockwise circle centered at $x+{\text{i}}y$ with radius
$\min(\frac{y}{2},0.005)$. If $N>N_{0}=N_{0}(\epsilon)$, then $L$ is contained
in (74) and we have by Cauchy’s Theorem and (75) that

$|\partial_{z}\tilde{m}(x+{\text{i}}y)|=\Big{|}\frac{1}{2\pi{\text{i}}}\oint_{L}\frac{\tilde{m}(r)}{(r-x-{\text{i}}y)^{2}}dr\Big{|}\leq
4N^{2\epsilon-\frac{1}{2}}\max(\frac{1}{y},100)$

and therefore

$|K_{2}|\leq\frac{1}{2\pi}\Big{|}\int_{|y|\in[\eta_{1},2]}{\text{Im}}\Big{[}-\int_{{\mathbb{R}}}f^{\prime}(x)\partial_{z}\tilde{m}(x+{\text{i}}y)dx\Big{]}y\chi(y)dy\Big{|}\quad\text{(integral
by parts for $x$)}\\\ \leq\frac{1}{2\pi}\cdot 4N^{2\epsilon-\frac{1}{2}}\cdot
2C_{f}\cdot
2\int_{\eta_{1}}^{2}y\chi(y)\max(\frac{1}{y},100)dy\leq\frac{3200C_{f}}{\pi}N^{2\epsilon-\frac{1}{2}}=\frac{3200C_{f}\cdot\eta_{0}^{2}}{\pi}$

  3. 3.

Finally we estimate $K_{3}$. Notice that both
$y{\text{Im}}m_{N}(x+{\text{i}}y)$ and $y{\text{Im}}m_{fc}(x+{\text{i}}y)$ are
nonnegative and increase when $y\in[0,\eta_{1}]$. So,

$\displaystyle|K_{3}|=$
$\displaystyle\frac{1}{\pi}\Big{|}\int_{{\mathbb{R}}}f^{\prime\prime}(x)\int_{0}^{\eta_{1}}(y{\text{Im}}m_{N}(x+{\text{i}}y)-y{\text{Im}}m_{fc}(x+{\text{i}}y))dydx\Big{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\int_{{\mathbb{R}}}|f^{\prime\prime}(x)|\int_{0}^{\eta_{1}}(y{\text{Im}}m_{N}(x+{\text{i}}y)+y{\text{Im}}m_{fc}(x+{\text{i}}y))dydx$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\int_{{\mathbb{R}}}|f^{\prime\prime}(x)|\int_{0}^{\eta_{1}}(\eta_{1}{\text{Im}}m_{N}(x+{\text{i}}\eta_{1})+\eta_{1}{\text{Im}}m_{fc}(x+{\text{i}}\eta_{1}))dydx$
$\displaystyle=$
$\displaystyle\frac{\eta_{1}^{2}}{\pi}\int_{{\mathbb{R}}}|f^{\prime\prime}(x)|({\text{Im}}m_{N}(x+{\text{i}}\eta_{1})+{\text{Im}}m_{fc}(x+{\text{i}}\eta_{1}))dx$

If $x\in\text{supp}f$, then $x+{\text{i}}\eta_{1}$ is in (74) and thus also in
$\mathcal{D}_{\epsilon}^{\prime}$, provided $N$ is large enough. So we know by
(75) and the definition of $\Omega_{*}(\epsilon)$ that if $N$ is large enough
then

${\text{Im}}m_{N}(x+{\text{i}}\eta_{1})+{\text{Im}}m_{fc}(x+{\text{i}}\eta_{1})\leq
4N^{2\epsilon-\frac{1}{2}}$

and thus

$|K_{3}|\leq\frac{4\eta_{1}^{2}}{\pi}N^{2\epsilon-\frac{1}{2}}\int_{{\mathbb{R}}}|f^{\prime\prime}(x)|dx\leq\frac{8C_{f}\cdot\eta_{1}^{2}}{\pi\eta_{0}}N^{2\epsilon-\frac{1}{2}}=\frac{8C_{f}\cdot\eta_{1}^{2}\cdot\eta_{0}}{\pi}$

The estimates for $K_{1}$, $K_{2}$ and $K_{3}$ together complete the proof of
the lemma. ∎

######  Lemma 27.

Suppose $b>3$ and $\epsilon\in(\frac{1}{1+b},\frac{1}{4})$. Suppose
$A_{N}(\epsilon)$ holds. If $N$ is large enough, then for any (possibly
$N$-depending) interval $J\subset{\mathbb{R}}$,

$|\mu_{N}(J)-\mu_{fc}(J)|\leq 4\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})$

where $\alpha_{2}>0$ is a constant depending only on $C_{f}$.

###### Proof.

First, suppose $J\subset(-2.99-\lambda,2.99+\lambda)$. Define
$h:{\mathbb{R}}\to[0,1]$ to be a smooth function such that

$\displaystyle h(x)=\begin{cases}1\quad&\text{if }x\in J\\\ 0\quad&\text{if
}\text{dist}(x,J)\geq\eta_{0}\end{cases}$ (77)

From Lemma 26, if $N$ is large enough then

$\displaystyle\mu_{N}(J)\leq\int h(x)d\mu_{N}(x)\leq\int
h(x)d\mu_{fc}(x)+\alpha_{1}(\eta_{0}^{2}+\eta_{1}^{2}\eta_{0})\leq\mu_{fc}(J)+2\eta_{0}\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}^{2}+\eta_{1}^{2}\eta_{0})$
(78)

where $\alpha_{1}>0$ is defined in Lemma 26. On the other hand, let

$J^{\prime}=\\{x\in J|\text{the distance between $x$ and the edges of $J$ is
no less than }\eta_{0}\\}$

and define $\tilde{h}$ in the same way as $h$, except that $J$ in(77) is
replaced by $J^{\prime}$. So by Lemma 26, if $N$ is large enough then

$\displaystyle\mu_{N}(J)\geq\int\tilde{h}(x)d\mu_{N}(x)\geq\int\tilde{h}(x)d\mu_{fc}(x)-\alpha_{1}(\eta_{0}^{2}+\eta_{1}^{2}\eta_{0})\geq\mu_{fc}(J)-2\eta_{0}\|\rho_{fc}(x)\|_{\infty}-\alpha_{1}(\eta_{0}^{2}+\eta_{1}^{2}\eta_{0}).$
(79)

We remark that if $|J|<2\eta_{0}$, then $J^{\prime}=\emptyset$, but (79) is
trivial in this case. By (79) and (78),

$\displaystyle|\mu_{N}(J)-\mu_{fc}(J)|\leq\eta_{0}(2\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}+\eta_{1}^{2}))$
(80)

so we complete the proof in the case that $J\in(-2.99-\lambda,2.99+\lambda)$.

Then, suppose $J$ is not necessarily contained in
$(-2.99-\lambda,2.99+\lambda)$. By Lemma 2,

$\mu_{fc}([-2-\lambda,2+\lambda])=1.$

So by (80) we have

$\mu_{N}((-2.99-\lambda,2.99+\lambda)^{c})=1-\mu_{N}((-2.99-\lambda,2.99+\lambda))\\\
\leq
1-\Big{(}\mu_{fc}\Big{(}(-2.99-\lambda,2.99+\lambda)\Big{)}-\eta_{0}(2\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}+\eta_{1}^{2}))\Big{)}=\eta_{0}(2\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}+\eta_{1}^{2}))$

Let $J_{1}=J\cap(-2.99-\lambda,2.99+\lambda)$ and $J_{2}=J\backslash J_{1}$.
So by (80) and the above inequality,

$|\mu_{N}(J)-\mu_{fc}(J)|\leq|\mu_{N}(J_{1})-\mu_{fc}(J_{1})|+|\mu_{N}(J_{2})-\mu_{fc}(J_{2})|\\\
\leq\eta_{0}(2\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}+\eta_{1}^{2}))+\mu_{N}(J_{2})\leq
2\eta_{0}(2\|\rho_{fc}(x)\|_{\infty}+\alpha_{1}(\eta_{0}+\eta_{1}^{2})).$

∎

###### Proof of Theorem 5.

By Proposition 1 and Lemma 4, there exist constants $N_{0}>0$ and $\nu_{0}>0$
such that if $N>N_{0}$ then

${\mathbb{P}}(A_{N}(\epsilon))\geq 1-2\nu_{0}(\log N)^{1+2b}N^{-\epsilon}.$

Suppose $A_{N}(\epsilon)$ holds. Now it suffices to prove (16) and (17).

Suppose $L_{0}$ is the unique point in $[L_{-},L_{+}]$ such that
$\mu_{fc}([L_{0},L_{+}])=2/3$. If $N$ is large enough, then by Lemma 27,

$\mu_{N}([L_{0},L_{+}])\geq\frac{1}{2}$

thus

$\displaystyle\lambda_{i}\geq L_{0},\quad\forall i\in[1,\frac{N}{2}].$ (81)

Define $g(x)$ by

$g(x)=\mu_{fc}([x,+\infty)).$

According to Lemma 27, if $N$ is large enough, then

$|g(\lambda_{i})-g(\gamma_{i})|\leq|g(\lambda_{i})-\frac{i}{N}|+|\frac{i}{N}-g(\gamma_{i})|\leq|\mu_{fc}([\lambda_{i},+\infty))-\mu_{N}([\lambda_{i},+\infty))|+\frac{1}{2N}\\\
\leq 5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2}),\quad\forall i\in[1,N]$
(82)

where $\alpha_{2}>0$ is defined in Lemma 27.

By (11) there is a constant $C>1$ such that

$\displaystyle\frac{(L_{+}-x)^{b}}{C}\leq\rho_{fc}(x)\leq
C(L_{+}-x)^{b},\quad\forall x\in[L_{0},L_{+}]$ (83)

and therefore

$\frac{i}{2N}\leq\frac{i-\frac{1}{2}}{N}=\int_{\gamma_{i}}^{L_{+}}\rho_{fc}(x)dx\leq
C\int_{\gamma_{i}}^{L_{+}}(L_{+}-x)^{b}dx=\frac{C}{b+1}|L_{+}-\gamma_{i}|^{b+1}\quad\forall
1\leq i\leq\frac{N}{2}.$

Then we have

$\displaystyle|\gamma_{i}-L_{+}|\geq\big{(}\frac{i}{2N}\frac{b+1}{C}\big{)}^{\frac{1}{1+b}}\geq
N^{-\zeta},\quad\forall i\in[\frac{2C}{1+b}N^{1-\zeta(b+1)},\frac{N}{2}].$
(84)

We control $|\gamma_{i}-\lambda_{i}|$ in two cases.

  1. Case 1.

Suppose $i\in[\frac{2C}{1+b}N^{1-\zeta(b+1)},\frac{N}{2}]$ and
$\lambda_{i}\leq\gamma_{i}$. By (81), (82), (83) and (84), when $N$ is large
enough, there exists $s\in(\lambda_{i},\gamma_{i})$ such that

$\displaystyle|\gamma_{i}-\lambda_{i}|=\frac{|g(\gamma_{i})-g(\lambda_{i})|}{|g^{\prime}(s)|}\leq\frac{5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})}{\rho_{fc}(s)}\leq\frac{5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})}{(L_{+}-s)^{b}/C}\leq
U_{1}N^{-\frac{1}{4}+\epsilon+\zeta b}$ (85)

where $U_{1}>0$ is a constant.

  2. Case 2.

Suppose $i\in[\frac{2C}{1+b}N^{1-\zeta(b+1)},\frac{N}{2}]$ and
$\lambda_{i}>\gamma_{i}$. By (82), (83), (84) and the definition of $\zeta$,
if $N$ is large enough then

$g(\lambda_{i})\geq
g(\gamma_{i})-5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})\geq\int_{\gamma_{i}}^{L_{+}}C^{-1}(L_{+}-x)^{b}dx-5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})\\\
=\frac{C^{-1}(L_{+}-\gamma_{i})^{b+1}}{b+1}-5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})\geq\frac{C^{-1}}{2(b+1)}N^{-\zeta(b+1)}$
(86)

which implies $\lambda_{i}<L_{+}$ (otherwise $g(\lambda_{i})=0$) and thus

$\displaystyle
g(\lambda_{i})\leq\int_{\lambda_{i}}^{L_{+}}C(L_{+}-x)^{b}dx\leq\frac{C}{1+b}(L_{+}-\lambda_{i})^{b+1}.$
(87)

By (86) and (87), if $N$ is large enough then

$L_{+}-\lambda_{i}\geq(2C^{2})^{\frac{-1}{1+b}}N^{-\zeta},$

so

$\displaystyle|\gamma_{i}-\lambda_{i}|=\frac{|g(\gamma_{i})-g(\lambda_{i})|}{|g^{\prime}(t)|}\leq\frac{5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})}{\rho_{fc}(t)}\leq\frac{5\eta_{0}(\|\rho_{fc}(x)\|_{\infty}+\alpha_{2})}{(L_{+}-\lambda_{i})^{b}/C}\leq
U_{2}N^{-\frac{1}{4}+\epsilon+\zeta b}.$ (88)

Here $t\in(\gamma_{i},\lambda_{i})$ and $U_{2}>0$ is a constant.

(85), (88) and the fact that $\zeta$ can be arbitrarily small complete the
proof of (16). (17) can be proved in the same way. ∎

##  7 SSK model in low temperature: proof of Theorem 1

In this section we follow the idea introduced in [1] to prove Theorem 1.
Because of the results in Lemma 30, we know that if a particle is moving along
the curve of steepest-descent defined in Definition 17, then its
$y$-coordinate is monotone, therefore we do not need a lemma like Lemma 6.4 in
[1].

Throughout this section we suppose the conditions of Theorem 1 hold.

######  Definition 15.

Suppose $\epsilon_{0}>0$ is a constant. Let $s_{0}=s_{0}(\epsilon_{0})>0$ be a
constant such that

$\displaystyle{\mathbb{P}}\big{(}|\lambda_{1}-L_{+}|<s_{0}N^{-\frac{1}{1+b}}\text{
and }|\lambda_{N}-L_{-}|<s_{0}N^{-\frac{1}{1+a}}\big{)}>1-\epsilon_{0}.$ (89)

for large enough $N$. Set

$\Omega_{N}^{0}(\epsilon_{0})=\big{\\{}|\lambda_{1}-L_{+}|<s_{0}N^{-\frac{1}{1+b}}\text{
and }|\lambda_{N}-L_{-}|<s_{0}N^{-\frac{1}{1+a}}\big{\\}}.$

######  Remark 8.

By Theorem 6 the constant $s_{0}$ exists.

######  Definition 16.

Let $R(z)$ be an analytic function defined on
${\mathbb{C}}\backslash(-\infty,\lambda_{1}]$ by

$R(z)=2\beta z-\frac{1}{N}\sum_{i=1}^{N}\log(z-\lambda_{i}).$

Here we take the analytic branch of the $\log$ function such that
${\text{Im}}\log(z-\lambda_{i})\in(-\pi,\pi)$ for all
$z\in{\mathbb{C}}\backslash(-\infty,\lambda_{1}]$. Let $\gamma$ denote the
unique number in $(\lambda_{1},+\infty)$ such that $R^{\prime}(\gamma)=0$.
Equivalently, $\gamma$ is the unique number on $(\lambda_{1},+\infty)$
satisfying $2\beta=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\gamma-\lambda_{i}}$.

######  Lemma 28.

Suppose

  * •

$\epsilon$ is a constant in
$\Big{(}\frac{1}{1+b},\min\Big{(}\frac{1}{4}-\frac{2}{b+1},\frac{1}{4}-\frac{1}{b+1}-\frac{a}{(b+1)^{2}}\Big{)}\Big{)}$;

  * •

$\tau$ is a constant in
$\Big{(}\frac{1}{(b+1)^{2}},\min\Big{(}\frac{\frac{1}{4}-\epsilon}{a+1},\frac{\frac{1}{4}-\epsilon-\frac{1}{b+1}}{a},\frac{\frac{1}{4}-\epsilon-\frac{b+2}{(b+1)^{2}}}{b}\Big{)}\Big{)}$;

  * •

$\tau_{1}<\tau_{0}$ are two constants both in $(1-\tau(b+1),1-\frac{1}{b+1})$.

Suppose $E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})$ holds. There exists
a constant $N_{0}>0$ such that if $N>N_{0}$, then

$\lambda_{1}+\frac{1}{3\beta N}<\gamma<\lambda_{1}+N^{-1+\tau_{0}}.$

######  Remark 9.

According to the conditions $b>11$ and $1<a<\frac{b^{2}-6b-7}{4}$, it is easy
to check that the constants $\epsilon$, $\tau$, $\tau_{0}$ and $\tau_{1}$
exist. The event $E_{N}(\epsilon)$ is defined in Theorem 5.

###### Proof.

Notice that
$R^{\prime}(x)=2\beta-\frac{1}{N}\sum_{i=1}^{N}\frac{1}{x-\lambda_{i}}$ is
increasing on $(\lambda_{1},+\infty)$. Since

$R^{\prime}(\lambda_{1}+\frac{1}{3\beta
N})<2\beta-\frac{1}{N}\frac{1}{(\lambda_{1}+\frac{1}{3\beta
N})-\lambda_{1}}<0=R^{\prime}(\gamma),$

we have that $\lambda_{1}+\frac{1}{3\beta N}<\gamma$.

Suppose that $E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})$ holds. Since
$\frac{\frac{1}{4}-\epsilon-\frac{b+2}{(1+b)^{2}}}{b}<\frac{\frac{1}{4}-\epsilon}{b+1}$,
the $\tau$ satisfies the conditions for $\zeta$ and $\zeta^{\prime}$ in
Theorem 5. According to Theorem 5, when $N$ is large enough, we have

$\displaystyle\begin{cases}|\lambda_{i}-\gamma_{i}|\leq
N^{-\frac{1}{4}+\epsilon+\tau b}\quad&\text{if
}i\in[\kappa^{\prime}N^{1-\tau(b+1)},\frac{N}{2}]\\\
|\lambda_{i}-\gamma_{i}|\leq N^{-\frac{1}{4}+\epsilon+\tau a}\quad&\text{if
}i\in[\frac{N}{2},N-\kappa^{\prime}N^{1-\tau(a+1)}]\end{cases}$ (90)

where $\kappa^{\prime}>0$ is defined in Theorem 5.

To prove $\gamma<\lambda_{1}+N^{-1+\tau_{0}}$ we need

$|\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}-\int\frac{d\mu_{fc}(t)}{L_{+}-t}|\leq
I+II+III$

where

$I=\Big{|}\frac{1}{N}\sum\limits_{i=N^{\tau_{1}}}^{N-\kappa^{\prime}N^{1-\tau(a+1)}}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}-\frac{1}{N}\sum\limits_{i=N^{\tau_{1}}}^{N-\kappa^{\prime}N^{1-\tau(a+1)}}\frac{1}{L_{+}-\gamma_{i}}\Big{|}$
$II=\Big{|}\frac{1}{N}\sum\limits_{i=N^{\tau_{1}}}^{N-\kappa^{\prime}N^{1-\tau(a+1)}}\frac{1}{L_{+}-\gamma_{i}}-\int\frac{d\mu_{fc}(t)}{L_{+}-t}\Big{|}$
$III=\Big{|}\frac{1}{N}\sum_{i<N^{\tau_{1}}}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}+\frac{1}{N}\sum_{i>N-\kappa^{\prime}N^{1-\tau(a+1)}}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}\Big{|}$

Estimation of I. If $N$ is large enough, then for any
$i\in[N^{\tau_{1}},N/2]\subset[\kappa^{\prime}N^{1-\tau(b+1)},\frac{N}{2}]$,

$\Big{|}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}-\frac{1}{L_{+}-\gamma_{i}}\Big{|}|L_{+}-\gamma_{i}|\leq\dfrac{|\lambda_{1}-L_{+}|+|\gamma_{i}-\lambda_{i}|+N^{-1+\tau_{0}}}{|L_{+}-\gamma_{i}|-|L_{+}-\lambda_{1}|-|\lambda_{i}-\gamma_{i}|-N^{-1+\tau_{0}}}\\\
\leq\dfrac{s_{0}N^{-\frac{1}{1+b}}+N^{-\frac{1}{4}+\epsilon+\tau
b}+N^{-1+\tau_{0}}}{C_{*}^{-1}(i/N)^{\frac{1}{1+b}}-s_{0}N^{-\frac{1}{1+b}}-N^{-\frac{1}{4}+\epsilon+\tau
b}-N^{-1+\tau_{0}}}=\dfrac{s_{0}+N^{\frac{1}{1+b}-\frac{1}{4}+\epsilon+\tau
b}+N^{\frac{1}{1+b}-1+\tau_{0}}}{C_{*}^{-1}\cdot
i^{\frac{1}{1+b}}-s_{0}-N^{\frac{1}{1+b}-\frac{1}{4}+\epsilon+\tau
b}-N^{\frac{1}{1+b}-1+\tau_{0}}}\\\
\leq\dfrac{s_{0}+N^{\frac{1}{1+b}-\frac{1}{4}+\epsilon+\tau
b}+N^{\frac{1}{1+b}-1+\tau_{0}}}{C_{*}^{-1}\cdot
N^{\frac{\tau_{1}}{1+b}}-s_{0}-N^{\frac{1}{1+b}-\frac{1}{4}+\epsilon+\tau
b}-N^{\frac{1}{1+b}-1+\tau_{0}}}\leq 2s_{0}C_{*}N^{-\frac{\tau_{1}}{1+b}}.$
(91)

Here we used Lemma 24, (90) and the definition of
$\Omega_{N}^{0}(\epsilon_{0})$ in the second inequality and we used the
conditions on $\tau$ and $\tau_{0}$ in the last inequality. (In particular,
the condition $\tau<\frac{\frac{1}{4}-\epsilon-\frac{b+2}{(b+1)^{2}}}{b}$
implies $\frac{1}{1+b}-\frac{1}{4}+\epsilon+\tau b<0$.) The constant $C_{*}>0$
is defined in Lemma 24.

By a similar argument we can prove (91) for
$i\in[N/2,N-\kappa^{\prime}N^{1-\tau(a+1)}]$. So we have that if $N$ is large
enough, then

$I\leq
2s_{0}C_{*}N^{-\frac{\tau_{1}}{1+b}}\cdot\frac{1}{N}\sum\limits_{i=N^{\tau_{1}}}^{N-\kappa^{\prime}N^{1-\tau(a+1)}}\frac{1}{L_{+}-\gamma_{i}}\\\
\leq
2s_{0}C_{*}N^{-\frac{\tau_{1}}{1+b}}\cdot\sum\limits_{i=N^{\tau_{1}}}^{N-\kappa^{\prime}N^{1-\tau(a+1)}}\int_{\hat{\gamma}_{i-1}}^{\hat{\gamma}_{i-2}}\frac{1}{L_{+}-t}d\mu_{fc}(t)\leq
4\beta_{c}\cdot s_{0}C_{*}N^{-\frac{\tau_{1}}{1+b}}$

where we used (73) in the second inequality and used the definition of
$\beta_{c}$ in the last inequality.

Estimation of II.  If $N$ is large enough, then by(73), (11) and Lemma 24,

$\displaystyle II\leq$
$\displaystyle\int_{L_{-}}^{\hat{\gamma}_{N-\kappa^{\prime}N^{1-\tau(a+1)}-2}}\frac{d\mu_{fc}(t)}{L_{+}-t}+\int_{\hat{\gamma}_{N^{\tau_{1}}}}^{L_{+}}\frac{d\mu_{fc}(t)}{L_{+}-t}\leq
C|\hat{\gamma}_{N-\kappa^{\prime}N^{1-\tau(a+1)}-2}-L_{-}|+\int_{\hat{\gamma}_{N^{\tau_{1}}}}^{L_{+}}C_{0}(L_{+}-t)^{b-1}dt$
$\displaystyle=$ $\displaystyle
C|\hat{\gamma}_{N-\kappa^{\prime}N^{1-\tau(a+1)}-2}-L_{-}|+\frac{C_{0}}{b}|L_{+}-\hat{\gamma}_{N^{\tau_{1}}}|^{b}\leq
C|\gamma_{N-\kappa^{\prime}N^{1-\tau(a+1)}-2}-L_{-}|+\frac{C_{0}}{b}|L_{+}-\gamma_{N^{\tau_{1}}+1}|^{b}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\frac{\kappa^{\prime}N^{1-\tau(a+1)}+2}{N}\Big{)}^{\frac{1}{1+a}}+C\Big{(}\frac{N^{\tau_{1}}+1}{N}\Big{)}^{\frac{b}{1+b}}$

where $C_{0}$ is defined in (11) and $C>0$ is a constant.

Estimation of III.  Since $\lambda_{1}\geq\cdots\geq\lambda_{N}$, we have for
large enough $N$:

$0<\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}\leq
N^{1-\tau_{0}}\quad(1\leq i\leq N/2)$
$0<\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}\leq\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{\lfloor
N/2\rfloor}}\\\ \leq\frac{1}{|L_{+}-\gamma_{\lfloor
N/2\rfloor}|-|\lambda_{\lfloor N/2\rfloor}-\gamma_{\lfloor
N/2\rfloor}|-|L_{+}-\lambda_{1}|-N^{-1+\tau_{0}}}\leq\frac{2}{|L_{+}-\gamma_{\lfloor
N/2\rfloor}|}\quad(N/2\leq i\leq N)$

so

$III\leq
N^{\tau_{1}-\tau_{0}}+\kappa^{\prime}N^{-\tau(a+1)}\cdot\frac{2}{|L_{+}-\gamma_{\lfloor
N/2\rfloor}|}.$

Since
$R^{\prime}(\lambda_{1}+N^{-1+\tau_{0}})=2\beta-\frac{1}{N}\sum_{i=1}^{N}\frac{1}{\lambda_{1}+N^{-1+\tau_{0}}-\lambda_{i}}$,
we know from the estimations of $I$, $II$ and $III$ that if $N$ is large
enough, then

$R^{\prime}(\lambda_{1}+N^{-1+\tau_{0}})\geq
2\beta-\int\frac{d\mu_{fc}(t)}{L_{+}-t}-I-II-III\\\ \geq
2\beta-2\beta_{c}-4\beta_{c}s_{0}C_{*}N^{-\frac{\tau_{1}}{1+b}}-\Big{(}C\Big{(}\frac{\kappa^{\prime}N^{1-\tau(a+1)}+2}{N}\Big{)}^{\frac{1}{1+a}}+C\Big{(}\frac{N^{\tau_{1}}+1}{N}\Big{)}^{\frac{b}{1+b}}\Big{)}\\\
-\Big{(}N^{\tau_{1}-\tau_{0}}+\kappa^{\prime}N^{-\tau(a+1)}\cdot\frac{2}{|L_{+}-\gamma_{\lfloor
N/2\rfloor}|}\Big{)}>0=R^{\prime}(\gamma)$ (92)

therefore $\gamma<\lambda_{1}+N^{-1+\tau_{0}}$ because $R^{\prime}$ is
increasing on $(\lambda_{1},+\infty)$. ∎

######  Lemma 29.

Suppose the assumptions in Lemma 28 hold. Suppose $\tau_{2}$, $\tau_{3}$ and
$\tau^{\prime}$ are constants satisfying:

  * •

$\max(1-\tau(1+b),1-\tau(1+a))<\tau_{2}<1$

  * •

$\max\Big{(}2+(1+b)(-\frac{1}{4}+\epsilon+\tau
b),1-\tau(b+1),1-\frac{a+1}{a}(\frac{1}{4}-\epsilon-\frac{1}{b+1})\Big{)}<\tau_{3}<\frac{b}{b+1}$

  * •

$\frac{1-\tau_{3}}{a+1}<\tau^{\prime}<\min(\frac{\frac{1}{4}-\epsilon-\frac{1}{b+1}}{a},\frac{\frac{1}{4}-\epsilon}{1+a})$

Then we have the following conclusions.

  * •

If $N$ is large enough, then

$\displaystyle\Big{|}\frac{1}{N}\sum_{i=1}^{N}\log(\gamma-\lambda_{i})-\int\log(L_{+}-t)d\mu_{fc}(t)-2\beta_{c}(\gamma-
L_{+})\Big{|}\leq W_{1}\Phi_{N}$

where $W_{1}>0$ is a constant and

$\Phi_{N}=N^{-2+2\tau_{0}-2\frac{\tau_{3}-1}{1+b}}+N^{\frac{-2\tau_{3}}{b+1}}+N^{-\frac{1}{4}+\epsilon+\tau
b+\frac{1-\tau_{3}}{b+1}}+N^{-\frac{1}{4}+\epsilon+\tau^{\prime}a}+N^{\frac{b(\tau_{3}-1)}{b+1}}\Big{(}N^{-\frac{1}{1+b}}+N^{-1+\tau_{0}}\Big{)}+N^{\tau_{3}-1}\log
N.$

  * •

If $N$ is large enough, then

$N^{l-l\tau_{0}-1}\leq\frac{R^{(l)}(\gamma)(-1)^{l}}{(l-1)!}\leq
W_{2}^{l}N^{-1+\tau_{2}+l},\quad l=2,3,\ldots$

where $W_{2}>0$ is a constant.

######  Remark 10.

  * •

From the definition of $\epsilon$ and $\tau$ (see Lemma 28), we see that the
$\tau_{2}$ and $\tau_{3}$ satisfying the conditions exist. Since
$\tau<\frac{\frac{1}{4}-\epsilon-\frac{b+2}{(b+1)^{2}}}{b}$, we have from the
definition of $\tau_{3}$:

$\displaystyle\tau_{3}>1-\tau(b+1)>\frac{3}{4}+\epsilon$ (93)

(93) and the definition of $\tau_{3}$ yields
$\frac{1-\tau_{3}}{a+1}<\min(\frac{\frac{1}{4}-\epsilon-\frac{1}{b+1}}{a},\frac{\frac{1}{4}-\epsilon}{1+a})$,
thus $\tau^{\prime}$ is well defined.

  * •

By (93) and the definitions of $\tau_{0}$ and $\tau_{3}$, we have

$\displaystyle\tau_{3}>1-\tau(1+b)>\frac{3}{4}>(\tau_{0}-1)(b+1)+\frac{3}{2}$
(94)

so

$\displaystyle\frac{1}{b+1}-2+2\tau_{0}-2\frac{\tau_{3}-1}{1+b}<0.$ (95)

By $\tau_{0}<\frac{b}{b+1}$ and the definition of $\tau_{3}$ we have
$\tau_{3}<\frac{b}{b+1}<2-\frac{b+1}{b}\tau_{0}$, so

$\displaystyle\frac{1}{b+1}+\frac{b(\tau_{3}-1)}{b+1}-1+\tau_{0}<0$ (96)

By (95), (96) and the definition of $\tau_{3}$ we see that

$\displaystyle\lim_{N\to\infty}N^{\frac{1}{1+b}}\cdot\Phi_{N}=0.$ (97)

###### Proof.

By Lemma 24, Theorem 5, Lemma 28 and the definition of
$\Omega_{N}^{0}(\epsilon_{0})$, if $N$ is large enough and
$i\in[1+N^{\tau_{3}},N-N^{\tau_{3}}]$, then

$\displaystyle\Big{|}\frac{\gamma-
L_{+}}{L_{+}-\gamma_{i}}\Big{|}\leq\Big{|}\frac{\gamma-\lambda_{1}}{L_{+}-\gamma_{i}}\Big{|}+\Big{|}\frac{\lambda_{1}-L_{+}}{L_{+}-\gamma_{i}}\Big{|}\leq\frac{N^{-1+\tau_{0}}+s_{0}N^{-\frac{1}{1+b}}}{C_{*}^{-1}N^{\frac{\tau_{3}-1}{b+1}}}\stackrel{{\scriptstyle
N\to\infty}}{{\longrightarrow}}0\quad\text{ (by \eqref{eq114})}$ (98)
$\displaystyle\Big{|}\frac{\gamma_{i}-\lambda_{i}}{L_{+}-\gamma_{i}}\Big{|}\leq\frac{N^{-\frac{1}{4}+\epsilon+\tau
b}}{C_{*}^{-1}N^{\frac{\tau_{3}-1}{b+1}}}\mathds{1}_{i\leq
N/2}+\frac{N^{-\frac{1}{4}+\epsilon+\tau^{\prime}a}}{|L_{+}-\gamma_{N/2}|}\mathds{1}_{i\geq
N/2}\stackrel{{\scriptstyle N\to\infty}}{{\longrightarrow}}0\quad\text{ (by
definitions of $\tau_{3}$ and $\tau^{\prime}$)}$ (99)

and thus

$\displaystyle\log(\gamma-\lambda_{i})-\log(L_{+}-\gamma_{i})=\log(1+\frac{\gamma-
L_{+}}{L_{+}-\gamma_{i}}+\frac{\gamma_{i}-\lambda_{i}}{L_{+}-\gamma_{i}})=\frac{\gamma-
L_{+}}{L_{+}-\gamma_{i}}+B_{1}+B_{2}$ (100)

where $|B_{1}|\leq
2\Big{|}\frac{\gamma_{i}-\lambda_{i}}{L_{+}-\gamma_{i}}\Big{|}$ and
$|B_{2}|\leq\Big{|}\frac{\gamma-L_{+}}{L_{+}-\gamma_{i}}\Big{|}^{2}$.

By (73), Lemma 1 and Lemma 24, if $N$ is large enough, then

$\Big{|}\frac{1}{N}\sum\limits_{i=N^{\tau_{3}}+1}^{N-N^{\tau_{3}}}\frac{1}{L_{+}-\gamma_{i}}-2\beta_{c}\Big{|}\leq\Big{|}\int_{\hat{\gamma}_{N-N^{\tau_{3}}+1}}^{\hat{\gamma}_{N^{\tau_{3}}+2}}\frac{d\mu_{fc}(t)}{L_{+}-t}-2\beta_{c}\Big{|}=\int_{L_{-}}^{\hat{\gamma}_{N-N^{\tau_{3}}+1}}\frac{d\mu_{fc}(t)}{L_{+}-t}+\int_{\hat{\gamma}_{N^{\tau_{3}}+2}}^{L_{+}}\frac{d\mu_{fc}(t)}{L_{+}-t}\\\
\leq
W_{3}\int_{L_{-}}^{\hat{\gamma}_{N-N^{\tau_{3}}+1}}(t-L_{-})^{a}dt+C_{0}\int_{\hat{\gamma}_{N^{\tau_{3}}+2}}^{L_{+}}(L_{+}-t)^{b-1}dt\leq
W_{3}|\gamma_{N-N^{\tau_{3}}+1}-L_{-}|^{a+1}+\frac{C_{0}}{b}|L_{+}-\hat{\gamma}_{N^{\tau_{3}}+2}|^{b}\\\
\leq
W_{3}\cdot\frac{N^{\tau_{3}}}{N}+\frac{C_{0}}{b}|L_{+}-\gamma_{N^{\tau_{3}}+2}|^{b}\leq
W_{3}\frac{N^{\tau_{3}}}{N}+\frac{C_{0}}{b}\cdot
C_{*}^{b}\Big{(}\frac{N^{\tau_{3}}+3}{N}\Big{)}^{\frac{b}{1+b}}\\\ \leq
W_{3}N^{\frac{b(\tau_{3}-1)}{b+1}}$ (101)

where $W_{3}>0$ is a constant, $C_{0}>0$ is defined in Lemma 1 and $C_{*}$ is
defined in Lemma 24.

According to Lemma 1, Lemma 24, the definitions of $\gamma_{i}$ and
$\hat{\gamma}_{i}$ and the fact that

$\int_{\hat{\gamma}_{i-1}}^{\hat{\gamma}_{i-2}}\log(L_{+}-t)d\mu_{fc}(t)\leq\frac{1}{N}\log(L_{+}-\gamma_{i})\leq\int_{\hat{\gamma}_{i+1}}^{\hat{\gamma}_{i}}\log(L_{+}-t)d\mu_{fc}(t)\quad(2\leq
i\leq N-1)$

we know that if $N$ is large enough, then

$\Big{|}\frac{1}{N}\sum\limits_{i=N^{\tau_{3}}+1}^{N-N^{\tau_{3}}}\log(L_{+}-\gamma_{i})-\int\log(L_{+}-t)d\mu_{fc}(t)\Big{|}\\\
\leq\int_{L_{-}}^{\hat{\gamma}_{N-N^{\tau_{3}}-2}}|\log(L_{+}-t)|d\mu_{fc}(t)+\int_{\hat{\gamma}_{N^{\tau_{3}}+2}}^{L_{+}}|\log(L_{+}-t)|d\mu_{fc}(t)\\\
\leq
W_{4}\int_{L_{-}}^{\gamma_{N-N^{\tau_{3}}-2}}(t-L_{-})^{a}dt+C_{0}\int_{\gamma_{N^{\tau_{3}}+3}}^{L_{+}}(L_{+}-t)^{b}|\log(L_{+}-t)|dt\\\
\leq
W_{4}N^{\tau_{3}-1}+C_{0}\frac{|L_{+}-\gamma_{N^{\tau_{3}}+3}|^{b+1}}{b+1}\Big{|}\log\big{(}L_{+}-\gamma_{N^{\tau_{3}}+3}\big{)}-\frac{1}{b+1}\Big{|}\\\
\leq W_{4}N^{\tau_{3}-1}\log N$ (102)

where $C_{0}>0$ is defined in Lemma 1 and $W_{4}>0$ is a constant.

By Lemma 28 and the definition of $\Omega_{N}^{0}(\epsilon_{0})$, if $N$ is
large enough, then

$\displaystyle\Big{|}\frac{1}{N}\sum_{i\leq
1+N^{\tau_{3}}}\log(\gamma-\lambda_{i})+\frac{1}{N}\sum_{i\geq
N-N^{\tau_{3}}}\log(\gamma-\lambda_{i})\Big{|}\leq 4N^{\tau_{3}-1}\log N.$
(103)

According to (98), (99), (100), (101), (102), (103), if $N$ is large enough,
then

$\Big{|}\frac{1}{N}\sum_{i=1}^{N}\log(\gamma-\lambda_{i})-\int\log(L_{+}-t)d\mu_{fc}(t)-2\beta_{c}(\gamma-
L_{+})\Big{|}\\\
\leq\Big{|}\frac{1}{N}\sum\limits_{i=1+N^{\tau_{3}}}^{N-N^{\tau_{3}}}\log(\gamma-\lambda_{i})-\frac{1}{N}\sum\limits_{i=1+N^{\tau_{3}}}^{N-N^{\tau_{3}}}\log(L_{+}-\gamma_{i})-\frac{\gamma-
L_{+}}{N}\sum\limits_{i=1+N^{\tau_{3}}}^{N-N^{\tau_{3}}}\frac{1}{L_{+}-\gamma_{i}}\Big{|}\\\
+\Big{|}\frac{\gamma-
L_{+}}{N}\sum\limits_{i=1+N^{\tau_{3}}}^{N-N^{\tau_{3}}}\frac{1}{L_{+}-\gamma_{i}}-2\beta_{c}(\gamma-
L_{+})\Big{|}+\Big{|}\frac{1}{N}\sum\limits_{i=1+N^{\tau_{3}}}^{N-N^{\tau_{3}}}\log(L_{+}-\gamma_{i})-\int\log(L_{+}-t)d\mu_{fc}(t)\Big{|}\\\
+\Big{|}\frac{1}{N}\sum_{i\leq
1+N^{\tau_{3}}}\log(\gamma-\lambda_{i})+\frac{1}{N}\sum_{i\geq
N-N^{\tau_{3}}}\log(\gamma-\lambda_{i})\Big{|}\\\
\leq\Big{(}\frac{N^{-1+\tau_{0}}+s_{0}N^{-\frac{1}{1+b}}}{C_{*}^{-1}N^{\frac{\tau_{3}-1}{b+1}}}\Big{)}^{2}+2\frac{N^{-\frac{1}{4}+\epsilon+\tau
b}}{C_{*}^{-1}N^{\frac{\tau_{3}-1}{b+1}}}+2\frac{N^{-\frac{1}{4}+\epsilon+\tau^{\prime}a}}{|L_{+}-\gamma_{N/2}|}+W_{3}N^{\frac{b(\tau_{3}-1)}{b+1}}|\gamma-
L_{+}|+W_{4}N^{\tau_{3}-1}\log N+4N^{\tau_{3}-1}\log N.$

The above inequality together with

$|\gamma-
L_{+}|\leq|\gamma-\lambda_{1}|+|\lambda_{1}-L_{+}|<N^{-1+\tau_{0}}+s_{0}N^{-\frac{1}{b+1}}$

and the fact that $|L_{+}-\gamma_{N/2}|$ is bounded below by a constant
completes the proof of the first conclusion of the lemma.

For the second conclusion, notice that

$R^{(l)}(z)=\frac{1}{N}\sum_{i=1}^{N}\frac{(-1)^{l}(l-1)!}{(z-\lambda_{i})^{l}},\quad
l=2,3,\ldots,$

so for large enough $N$ we have by Lemma 28 that

$\displaystyle\frac{R^{(l)}(\gamma)(-1)^{l}}{(l-1)!}\geq\frac{1}{N}\frac{1}{(\gamma-\lambda_{1})^{l}}\geq
N^{l-l\tau_{0}-1},\quad l=2,3,\ldots$ (104)

From the conditions on $\tau_{2}$ and $\tau$ we see

$\displaystyle 1>\tau_{2}>1-\tau(1+b)>1+(b+1)(-\frac{1}{4}+\epsilon+\tau b).$
(105)

By Lemma 28, if $N$ is large enough, then

$\displaystyle\frac{1}{N}\Big{(}\sum_{i=1}^{N^{\tau_{2}}}+\sum_{i=N-N^{\tau_{2}}}^{N}\Big{)}\frac{1}{(\gamma-\lambda_{i})^{l}}\leq
3N^{-1+\tau_{2}}\cdot(3\beta N)^{l}=3(3\beta)^{l}N^{-1+\tau_{2}+l},\quad
l=2,3,\ldots$ (106)

By (105) and Lemma 24 if $N$ is large enough and
$i\in[N^{\tau_{2}},N-N^{\tau_{2}}]$, then $|L_{+}-\gamma_{i}|\geq
C_{*}^{-1}N^{\frac{\tau_{2}-1}{1+b}}$,

$|L_{+}-\gamma|\leq|L_{+}-\lambda_{1}|+|\gamma-\lambda_{1}|\leq
s_{0}N^{-\frac{1}{b+1}}+N^{-1+\tau_{0}}\leq(1+s_{0})N^{-\frac{1}{b+1}}\quad\text{(since
$\tau_{0}<\frac{b}{b+1}$)}$
$\displaystyle\text{and}\quad|\lambda_{i}-\gamma_{i}|\leq\begin{cases}N^{-\frac{1}{4}+\epsilon+\tau
b}\quad\text{if }N^{\tau_{2}}\leq i\leq N/2\\\ N^{-\frac{1}{4}+\epsilon+\tau
a}\quad\text{if }N/2\leq i\leq N-N^{\tau_{2}}\end{cases}$ (107)

The above estimations and the fact that $|L_{+}-\gamma_{j}|$ is of order 1 for
$j\geq N/2$ imply that if $N$ is large enough then

$\displaystyle|\gamma-\lambda_{i}|\geq|L_{+}-\gamma_{i}|-|\gamma_{i}-\lambda_{i}|-|L_{+}-\gamma|\geq\frac{1}{2}|L_{+}-\gamma_{i}|\geq\frac{1}{2C_{*}}\Big{(}\frac{i}{N}\Big{)}^{\frac{1}{1+b}},\quad\text{if
}N^{\tau_{2}}\leq i\leq N-N^{\tau_{2}}$

and thus

$\frac{1}{N}\sum_{N^{\tau_{2}}<i<N-N^{\tau_{2}}}\frac{1}{(\gamma-\lambda_{i})^{l}}\leq\frac{1}{N}\sum_{N^{\tau_{2}}<i<N-N^{\tau_{2}}}\frac{(2C_{*})^{l}N^{\frac{l}{1+b}}}{i^{l/(b+1)}}\leq(2C_{*})^{l}N^{\frac{l}{1+b}-1}\int_{\frac{1}{2}N^{\tau_{2}}}^{N}x^{-\frac{l}{1+b}}dx\\\
\leq\begin{cases}(2C_{*})^{l}\log N\quad&\text{if $b$ is an integer and
}l=b+1\\\ \frac{b+1}{b+1-l}(2C_{*})^{l}\quad&\text{if }l<b+1\\\
\frac{b+1}{l-(b+1)}(4C_{*})^{l}N^{(1-\tau_{2})(\frac{l}{b+1}-1)}\quad&\text{if
}l>b+1\end{cases}$ (108)

where $C_{*}$ is defined in Lemma 24. In (108), the coefficient
$\pm\frac{b+1}{b+1-l}$ for the case $l\neq b+1$ is bounded by:

$\displaystyle|\frac{b+1}{b+1-l}|\leq\begin{cases}b+1\quad&\text{if
}b\in\mathbb{Z}\\\ \frac{b+1}{\text{dist}(b,\mathbb{Z})}\quad&\text{if
}b\not\in\mathbb{Z}\end{cases}$

and this bound is independent of $l$. So by the fact
$-1+\tau_{2}+l>(1-\tau_{2})(\frac{l}{b+1}-1)$ and (104), (106) and (108) we
complete the proof. ∎

######  Definition 17.

  * •

Set
$S=\\{x+{\text{i}}y\in{\mathbb{C}}\backslash(-\infty,\lambda_{1}]|{\text{Im}}R(x+{\text{i}}y)=0\\}$.

  * •

Set $S^{+}=\\{x+{\text{i}}y\in S|y>0\\}$, $S^{-}=\\{x+{\text{i}}y\in
S|y<0\\}$.

  * •

For $y$ satisfying $0<|y|<\frac{\pi}{2\beta}$, let $h(y)$ be the unique real
number such that $h(y)+{\text{i}}y\in S$. Set $h(0)=\gamma$.

######  Lemma 30.

  * •

$h(y)$ is well defined. In other words, for any $y$ satisfying
$0<|y|<\frac{\pi}{2\beta}$, there is a unique real number $x$ such that
$x+{\text{i}}y\in S$.

  * •

$S=\\{h(y)+{\text{i}}y|-\frac{\pi}{2\beta}<y<\frac{\pi}{2\beta}\\}$ and
$h(y)\in C^{1}((-\frac{\pi}{2\beta},\frac{\pi}{2\beta}))$.

  * •

$h(y)\leq\gamma$ and the identity holds only when $y=0$.

  * •

If $\frac{1}{4}<c_{0}<\frac{\pi}{2}$, then $h(y)$ is strictly decreasing on
$[\frac{c_{0}}{\beta},\frac{\pi}{2\beta})$.

###### Proof.

See Appendix A. ∎

######  Remark 11.

Since $h(y)$ is $C^{1}$, we can define the integral of continuous functions
along $S$.

######  Lemma 31.

Suppose the conditions in Lemma 29 are satisfied. Set

$\displaystyle
K=-{\text{i}}e^{-\frac{N}{2}R(\gamma)}\int_{\gamma-{\text{i}}\infty}^{\gamma+{\text{i}}\infty}e^{\frac{N}{2}R(z)}dz.$
(109)

There exist constants $N_{0}>0$ and $W_{0}>0$ such that if $N>N_{0}$ then

$N^{-10}\leq K\leq W_{0}.$

###### Proof.

From Lemma 6 we know $K>0$. By the same argument as (6.31) of [1],

$\displaystyle K=$
$\displaystyle-{\text{i}}\int_{\gamma-{\text{i}}\infty}^{\gamma+{\text{i}}\infty}\exp\Big{(}\frac{N}{2}\Big{[}R(z)-R(\gamma)\Big{]}\Big{)}dz=\int_{-\infty}^{\infty}\exp\Big{(}\frac{N}{2}\Big{[}R(\gamma+{\text{i}}t)-R(\gamma)\Big{]}\Big{)}dt$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\exp\Big{(}{\text{i}}\beta
Nt-\frac{1}{2}\sum_{i=1}^{N}\log\Big{(}1+\frac{{\text{i}}t}{\gamma-\lambda_{i}}\Big{)}\Big{)}dt$
$\displaystyle\leq$
$\displaystyle\int_{-\infty}^{\infty}\exp\Big{(}-\frac{1}{4}\sum_{i=1}^{N}\log\Big{(}1+\frac{t^{2}}{(\gamma-\lambda_{i})^{2}}\Big{)}\Big{)}dt$

where we take the absolute value of the integrand to obtain the last
inequality. Since

$0<\gamma-\lambda_{i}\leq\gamma-\lambda_{N}\leq|\gamma-\lambda_{1}|+|\lambda_{1}-L_{+}|+|L_{+}-L_{-}|+|L_{-}-\lambda_{N}|\leq
1+2s_{0}+L_{+}-L_{-},$

we have for $N>4$:

$K\leq\int_{-\infty}^{\infty}\exp\Big{(}-\frac{N}{4}\log\Big{(}1+\frac{t^{2}}{(1+2s_{0}+L_{+}-L_{-})^{2}}\Big{)}\Big{)}dt\leq\int_{{\mathbb{R}}}\Big{(}1+\frac{t^{2}}{(1+2s_{0}+L_{+}-L_{-})^{2}}\Big{)}^{-1}dt<\infty$

and therefore the right hand side of the conclusion is proved.

To prove the left hand side of the conclusion we need Lemma 30. We first claim
that

$\displaystyle\int_{\gamma-{\text{i}}\infty}^{\gamma+{\text{i}}\infty}e^{\frac{N}{2}R(z)}dz=\int_{S}e^{\frac{N}{2}R(z)}dz$
(110)

where the direction on $S$ is from $-\infty-\frac{\pi}{2\beta}{\text{i}}$ to
$-\infty+\frac{\pi}{2\beta}{\text{i}}$. In fact, suppose $r>0$ such that
$|z-\lambda_{i}|>r/2$ for all $|z|=r$, then for
$C_{r}:=\\{z\in{\mathbb{C}}||z|=r,{\text{Re}}z\leq\gamma\\}$ we have

${\text{Re}}R(z)\leq 2\beta\gamma-\log(r/2),\quad\forall z\in C_{r}$

and thus

$\Big{|}\int_{C_{r}}\big{|}e^{\frac{N}{2}R(z)}\big{|}dz\Big{|}\leq
2\pi(\sqrt{2}e^{\beta\gamma})^{N}r^{1-\frac{N}{2}}\to 0\quad\text{as
}r\to\infty.$

Moreover, by the last conclusion of Lemma 30 we have
$\int_{0}^{\pi/(2\beta)}|e^{\frac{N}{2}R(h(y)+{\text{i}}y)}|\sqrt{1+(h^{\prime}(y))^{2}}dy<\infty$.
So (110) is true.

Notice that $R(z)=R(\bar{z})$ for $z\in S$, so by (110)

$K=-{\text{i}}\int_{S}e^{\frac{N}{2}(R(z)-R(\gamma))}dz=-{\text{i}}\Big{(}\int_{S^{+}}e^{\frac{N}{2}(R(z)-R(\gamma))}(dx+{\text{i}}dy)+\int_{S^{-}}e^{\frac{N}{2}(R(z)-R(\gamma))}(dx+{\text{i}}dy)\Big{)}\\\
=2\int_{S^{+}}\exp\Big{(}\frac{N}{2}(R(z)-R(\gamma))\Big{)}dy$ (111)

Now we define

$Q_{N}=\big{\\{}z\in{\mathbb{C}}\big{|}|z-\gamma|<N^{-9}\big{\\}}.$

By Lemma 28, $R(z)$ is analytic on a neighborhood of $\bar{Q}_{N}$, so by
$R^{\prime}(\gamma)=0$ we have for large enough $N$:

$\displaystyle
R(z)-R(\gamma)=\sum_{j=2}^{\infty}\frac{R^{(j)}(\gamma)}{j!}(z-\gamma)^{j},\quad\text{if
}z\in Q_{N}.$ (112)

The next lemma shows that $S$ does not leave $\gamma$ too fast when the $y$
coordinate is small enough.

######  Lemma 32.

When $N$ is large enough, we have

$\displaystyle\\{z\in S^{+}|{\text{Im}}z\in(0,N^{-10})\\}\subset Q_{N}.$ (113)

###### Proof.

See Appendix B. ∎

By Lemma 29, (112) and the definition of $Q_{N}$, if $N$ is large enough,
then:

$\displaystyle|R(z)-R(\gamma)|\leq\sum_{j=2}^{\infty}\frac{|R^{(j)}(\gamma)|}{j!}|z-\gamma|^{j}\leq
2W_{2}^{2}N^{-16},\quad\forall z\in Q_{N}$ (114)

which together with (111) and (113) imply:

$\displaystyle
K=2\int_{S^{+}}\exp\Big{(}\frac{N}{2}(R(z)-R(\gamma))\Big{)}dy\geq
2\int_{0}^{N^{-10}}\exp\Big{(}\frac{N}{2}\cdot(-2W_{2}^{2}N^{-16})\Big{)}dy>N^{-10}.$

So the proof is complete. ∎

###### Proof of Theorem 1.

Recall the definition of $K$ in (109). According to Lemma 6,

$\int_{S_{N-1}}e^{\beta\langle\sigma,(W+\lambda
V)\sigma\rangle}d\omega_{N}(\sigma)=\frac{\sqrt{N}\beta}{\sqrt{\pi}(2\beta
e)^{N/2}}\cdot e^{\frac{N}{2}R(\gamma)}\cdot K\cdot(1+O(N^{-1})).$

Now we choose the constants
$s_{0},\epsilon,\tau,\tau_{0},,\tau_{1},\tau_{2},\tau_{3},\tau^{\prime}$ in
the same way as in (89), Lemma 28 and Lemma 29. Suppose
$E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})$ holds. By Lemma 31, if $N$
is large enough, then

$\displaystyle
F_{N}=\frac{1}{N}\log\Big{(}\int_{S_{N-1}}e^{\beta\langle\sigma,(W+\lambda
V)\sigma\rangle}d\omega_{N}(\sigma)\Big{)}=\frac{R(\gamma)}{2}-\frac{1}{2}\log(2\beta
e)+e_{N}$

where

$\displaystyle|e_{N}|\leq C\log N/N$ (115)

for some constant $C>0$. By Lemma 28 and Lemma 29, if $N$ is large enough and
$E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})$ holds, then

$\displaystyle\Big{|}F_{N}+\frac{1}{2}\log(2e\beta)-\beta\lambda_{1}+\frac{1}{2}\int\log(L_{+}-t)d\mu_{fc}(t)+\beta_{c}(\lambda_{1}-L_{+})\Big{|}\leq|e_{N}|+(\beta+\beta_{c})|\gamma-\lambda_{1}|+W_{1}\Phi_{N}$
$\displaystyle\leq$
$\displaystyle|e_{N}|+(\beta+\beta_{c})N^{-1+\tau_{0}}+W_{1}\Phi_{N}$

and thus

$\displaystyle\Big{|}I_{N}-N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\Big{|}\leq\frac{N^{\frac{1}{b+1}}}{|\beta-\beta_{c}|}\Big{[}|e_{N}|+(\beta+\beta_{c})N^{-1+\tau_{0}}+W_{1}\Phi_{N}\Big{]}$
(116)

where

$I_{N}=N^{\frac{1}{b+1}}\Big{[}\frac{F_{N}+\frac{1}{2}\log(2e\beta)+\frac{1}{2}\int\log(L_{+}-t)d\mu_{fc}(t)-\beta
L_{+}}{\beta-\beta_{c}}\Big{]}.$

Fix $s<0$. Choose $\epsilon_{0}\in(0,|s|/10)$. There exists $\delta_{s}>0$
such that if $|s^{\prime}-s|\leq\delta_{s}$ then

$\displaystyle|\exp(-\frac{C_{\mu}(-s^{\prime})^{b+1}}{b+1})-\exp(-\frac{C_{\mu}(-s)^{b+1}}{b+1})|<\epsilon_{0}.$
(117)

Write $\mathcal{E}_{N}=N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})-I_{N}$. Notice
that

${\mathbb{P}}(I_{N}\leq s)={\mathbb{P}}(\\{I_{N}\leq
s\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0}))^{c})+{\mathbb{P}}(\\{I_{N}\leq
s\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})))\\\
={\mathbb{P}}(\\{I_{N}\leq
s\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0}))^{c})+{\mathbb{P}}(\\{N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s+\mathcal{E}_{N}\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0})))\\\
={\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s+\mathcal{E}_{N})-{\mathbb{P}}(\\{N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s+\mathcal{E}_{N}\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0}))^{c})\\\
+{\mathbb{P}}(\\{I_{N}\leq
s\\}\cap(E_{N}(\epsilon)\cap\Omega_{N}^{0}(\epsilon_{0}))^{c}).$ (118)

If $N$ is large enough, then by (97), (115), (116) and the definition of
$\tau_{0}$, we have $\mathcal{E}_{N}\in(-\delta_{s},\delta_{s})$ and then

${\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s+\mathcal{E}_{N})\in\Big{(}{\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s-\delta_{s}),{\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s+\delta_{s})\Big{)}$ (119)

By (117), (118), (119), (15) and (89) we have:

$|{\mathbb{P}}(I_{N}\leq
s)-{\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq s)|\leq
5\epsilon_{0}\quad\text{when $N$ is large enough}.$

Since $\epsilon_{0}$ can be arbitrarily small, we have by Theorem 6,

$\lim_{N\to\infty}{\mathbb{P}}(I_{N}\leq
s)=\lim_{N\to\infty}{\mathbb{P}}(N^{\frac{1}{b+1}}(\lambda_{1}-L_{+})\leq
s)=1-{\mathbb{P}}(N^{\frac{1}{b+1}}(L_{+}-\lambda_{1})\leq-s)=\exp\big{(}-\frac{C_{\mu}(-s)^{b+1}}{b+1}\big{)}.$

It is easy to check that the above identity is also true when $s=0$. ∎

##  8 SSK model in high temperature: proof of Theorem 2

In this section we use the method introduced in [1] and Theorem 4 to prove
Theorem 2, but we follow a different way to control $|\gamma-\hat{\gamma}|$.
In [1], the tool used to control $|\gamma-\hat{\gamma}|$ is the rigidity of
eigenvalues, but we will use the local law to control $|\gamma-\hat{\gamma}|$
because the rigidity we have here is not strong enough to provide a proper
estimation of $|\gamma-\hat{\gamma}|$.

Throughout this section we assume that the conditions of Theorem 2 are
satisfied.

######  Definition 18.

  * •

Set $\hat{R}(z)=2\beta z-\int\log(z-t)d\mu_{fc}(t)$ analytically defined on
${\mathbb{C}}\backslash(-\infty,L_{+}]$ such that