# Deviation of top eigenvalue for some tridiagonal matrices under various
moment assumptions

Yi HAN  Department of Pure Mathematics and Mathematical Statistics, University
of Cambridge<EMAIL_ADDRESS>
###### Abstract.

Symmetric tridiagonal matrices appear ubiquitously in mathematical physics,
serving as the matrix representation of discrete random Schrödinger operators.
In this work we investigate the top eigenvalue of these matrices in the large
deviation regime, assuming the random potentials are on the diagonal with a
certain decaying factor $N^{-\alpha}$, and the probability law $\mu$ of the
potentials satisfy specific decay assumptions. We investigate two different
models, one of which has random matrix behavior at the spectral edge but the
other does not. Both the light-tailed regime, i.e. when $\mu$ has all moments,
and the heavy-tailed regime are covered. Precise right tail estimates and a
crude left tail estimate are derived. In particular we show that when the tail
$\mu$ has a certain decay rate, then the top eigenvalue is distributed as the
Frechet law composed with some deterministic functions. The proof relies on
computing one point perturbations of fixed tridiagonal matrices.

Supported by EPSRC grant EP/W524141/1.

##  1\. Introduction

The theory of random Schrödinger operators gained much success in describing
various phenomena in mathematical physics. A 1-d random Schrödinger operator
in its finite volume version can be described as the following operator
defined on $\\{0,1,\cdots,N\\}$

$H_{N}\varphi(k)=\varphi(k-1)+\varphi(k+1)+\mathfrak{a}(k)\varphi(k),\quad\varphi(0)=\varphi(N)=0,$
(1.1)

where $\mathfrak{a}(k),k=1,\cdots,N$, are certain random potentials. It
corresponds to the matrix representation

$H_{N}=H_{N}^{\infty}+\Lambda_{N}^{0},$ (1.2)

where

$H_{N}^{\infty}=\begin{pmatrix}0&1&0&\cdots&0\\\ 1&0&1&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\ 0&\cdots&1&0&1\\\
0&\cdots&0&1&0\end{pmatrix},$ (1.3)

and

$\Lambda_{N}^{0}=\operatorname{diag}\left(\mathfrak{a}(1),\cdots,\mathfrak{a}(N)\right).$
(1.4)

Under some slight assumptions on the potentials $\mathfrak{a}(\cdot)$, [20]
proves that the bulk eigenvalues of $H_{N}$ converge to a Poisson process
after an appropriate re-scaling. Further, assuming that $\mathfrak{a}$ has a
doubly exponential type upper tail distribution, it is known [6] that
$\lambda_{1}(N)$, the top eigenvalue of $H_{N}$, converges to a Gumbel
distribution of max-order class.

To investigate the transition from localized phase to de-localized phase, a
variant of $H_{N}$ was introduced in [15] (for a modified model) and [16]
where the potentials are vanishing with $N$, i.e.

$H_{N}^{\alpha}=H_{N}^{\infty}+\Lambda_{N}^{\alpha},\quad\Lambda_{N}^{\alpha}=\operatorname{diag}(N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)).$
(1.5)

The transition from localized to de-localized phase has been verified the bulk
statistics at the critical value $\alpha=\frac{1}{2}$, see for example [21].

In this work we investigate the asymptotics of the top eigenvalue of
$H_{N}^{\alpha}$ with decaying diagonal potentials (1.5). When the potentials
$\mathfrak{a}$ are not too heavy-tailed, it is easy to check that the top
eigenvalue of $H_{N}^{\alpha}$ converges to 2 almost surely, and when
$\mathfrak{a}$ has heavier tails, the top eigenvalue can be arbitrarily large
with positive probability. We identify the large deviation profile of the top
eigenvalue in the former light-tailed case, then identify the threshold of
criticality, and finally derive the law of the top eigenvector in the critical
case. When $\mathfrak{a}(\cdot)$ has even heavier tails, we verify that the
top eigenvalue has a Poisson distribution.

This work is partially inspired by the recent progress in large deviation of
Wigner matrices. When the matrix has sub-Gaussian entries with a bound on its
Laplace transform, the large deviation profile of the largest eigenvalue was
derived via spherical integral in [11], where it was the collective behavior
of random potentials, instead of the behavior of one outlier potential, that
accounts the large deviation profile. When the random variables have a tail
heavier than Gaussian, the large deviation of spectral statistics was derived
in [7] (and see [3] for deviation of the largest eigenvalue in this regime)
where the deviation is accounted for by few outlier potentials taking very
large values. For heavy-tailed wigner matrices, the distribution of the top
eigenvalue was also well studied by many authors. When matrix entries have
finite fourth moment, the top eigenvalue sticks to the edge of semicircle law
[4], and the rescaled top eigenvalue has a Tracy-Widom fluctuation at the
edge[19]. When the matrix entries do not have a finite fourth moment, it is
verified in [24] and [2] that the largest eigenvalue has a Poisson
distribution. Finally, when the density of the matrix entries have a tail
decay $x^{-4}dx$, the distribution of the top eigenvalue is explicitly
determined in [8]. For the random Schrödinger operator $H_{N}^{\alpha}$, we
will show that the large deviation profile, and the asymptotic distribution of
the top eigenvalue is always governed by outlier potentials of $\mathfrak{a}$
taking unusually large values.

We also study a model of close relevance: consider the matrix

$G_{N}^{\infty}=\begin{pmatrix}0&\sqrt{(N-1)/N}&0&&\\\
\sqrt{(N-1)/N}&0&\sqrt{(N-2)/N}\\\ &\ddots&\ddots&&\\\ &&0&\sqrt{2/N}&0\\\
&&\sqrt{2/N}&0&\sqrt{1/N}\\\ &&0&\sqrt{1/N}&0\end{pmatrix}.$ (1.6)

This model is closely related to the matrix model of beta ensembles,
introduced by Domitriu and Edelman in [10]. Indeed, $G_{N}^{\infty}$ can be
thought of the zero temperature model in this beta-parameterized matrix family
(3.2). We are also interested in random operators of the following form

$G_{N}^{\infty}+\Lambda_{N}^{\alpha},\quad\Lambda_{N}^{\alpha}=\operatorname{diag}(N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)),$
(1.7)

but in order to foster better analytic treatment, we will instead work with an
orthogonal invariant version

$G_{N}^{\alpha}\overset{\text{def}}{=}G_{N}^{\infty}+U_{N}\Lambda_{N}^{\alpha}U_{N}^{t},$
(1.8)

where $U_{N}$ is a Haar distributed orthogonal matrix of size $N$, independent
of the random variables $\mathfrak{a}(\cdot)$, and $U_{N}^{t}$ denotes the
transpose of $U_{N}$. We will verify that the large deviation profiles of the
top eigenvalue of $H_{N}^{\infty}$ and $G_{N}^{\infty}$ are very similar,
except that the precise expression of large deviation rate function is
different.

###  1.1. Statement of main results

The main results of this paper are outlined as follows:

######  Theorem 1.1.

Consider the random operator $H_{N}^{\alpha}$ defined in (1.5), and denote by
$\lambda_{1}(N)$ the largest eigenvalue of $H_{N}^{\alpha}$. Assume the random
potentials $\mathfrak{a}(1),\cdots,\mathfrak{a}(N)$ are i.i.d. distributed as
a real-valued symmetric random variable $\mathfrak{a}$. Then

  1. (1)

(Weibull distribution) Assume that for some $\beta>0$, some $C>0$,
$0<C_{1}<C_{2}<\infty$ we have for any $t>1$,

$C_{1}\exp(-Ct^{\beta})\leq\mathbb{P}(\mathfrak{a}>t)\leq
C_{2}\exp(-Ct^{\beta}),$ (1.9) $\mathbb{P}(\mathfrak{a}<-t)\leq
C_{2}\exp(-Ct^{\beta}),$ (1.10)

then for any $\lambda>2$, we have the upper tail estimate

$\lim_{N\to\infty}\frac{\log(\mathbb{P}(\lambda_{1}(N)>\lambda))}{N^{\alpha\beta}}=-C\left(\sqrt{\lambda^{2}-4}\right)^{\beta},$
(1.11)

and for any $\lambda<2$, we have the lower tail estimate: there exists some
constants $C_{2},C_{3}>0$, and when $\beta<2$ there exists some sufficiently
small $c>0$ such that

$\mathbb{P}(\lambda_{1}(N)<\lambda)\leq\begin{cases}C_{2}e^{-C_{3}N^{\alpha\beta+1}},\quad\beta\in[2,\infty),\\\
C_{2}e^{-C_{3}N^{\alpha\beta+c}},\quad\beta\in(0,2).\end{cases}$ (1.12)

The constant $C_{3}>0$ depends on $\lambda$, and can be written as an explicit
function of $\lambda$. See Remark 2.2 for a discussion.

  2. (2)

(Sub-critical) Assume that for some $\beta>\frac{1}{\alpha}$, $C>0,C_{1}>0$ we
have for all $t>1$,

$\mathbb{P}(\mathfrak{a}>t)=Ct^{-\beta},$ (1.13)
$\mathbb{P}(\mathfrak{a}<-t)\leq C_{1}t^{-\beta},$ (1.14)

then for any $\lambda>2$, we have the upper tail estimate:

$\lim_{N\to\infty}N^{\alpha\beta-1}\mathbb{P}(\lambda_{1}(N)>\lambda)=\left(\sqrt{\lambda^{2}-4}\right)^{-\beta},$
(1.15)

and for any $\lambda<2$ we have the lower tail estimate: when $\beta>2$, for
any $\lambda<2$ and any $\epsilon>0$ we can find $C_{2}=C_{2}(\epsilon)>0$
depending on $\epsilon$ and $\lambda$ such that

$\mathbb{P}(\lambda_{1}(N)<\lambda)\leq(C_{2}N)^{-(\alpha(\beta-\epsilon)+1)}$
(1.16)

and when $\beta\in(0,2]$, for any $\lambda<2$ we can find sufficiently small
$c>0$ and some $C_{2}=C_{2}(\beta,c,\lambda)$ such that

$\mathbb{P}(\lambda_{1}(N)<\lambda)\leq(C_{2}N)^{-c}.$ (1.17)

  3. (3)

(Critical regime) Assume there exists some $C>0,C_{1}>0$ such that for any
$t>1$,

$\mathbb{P}(\mathfrak{a}>t)=Ct^{-\frac{1}{\alpha}},$ (1.18)
$\mathbb{P}(\mathfrak{a}<-t)\leq C_{1}t^{-\frac{1}{\alpha}},$ (1.19)

then almost surely

$\lambda_{1}(N)\overset{law}{\to}\sqrt{\xi^{2}+4},\quad N\to\infty,$ (1.20)

where $\xi$ is a non-negative-valued random variable that satisfies, for any
$\lambda>0$,

$\mathbb{P}(\xi>\lambda)=1-e^{-C\lambda^{-\frac{1}{\alpha}}}.$ (1.21)

  4. (4)

(Randomness dominating) Assume that for some $0<\beta<\frac{1}{\alpha}$,
$C>0,C_{1}>0$ we have for all $t>1$,

$\mathbb{P}(\mathfrak{a}>t)=Ct^{-\beta},$ (1.22)
$\mathbb{P}(\mathfrak{a}<-t)\leq C_{1}t^{-\beta},$ (1.23)

then

$N^{\alpha-\frac{1}{\beta}}\lambda_{1}(N)\overset{law}{\to}\xi,\quad
N\to\infty,$ (1.24)

where $\xi$ is a non-negative valued random variable satisfying for any
$\lambda>0$,

$\mathbb{P}(\xi>\lambda)=1-e^{-C\lambda^{-\beta}}.$ (1.25)

######  Remark 1.2.

The upper tail estimates in cases (1) and (2) are sharp, yet the lower tail
estimates derived in cases (1) and (2) are crude and may not be optimal. A
related example is the lower tail of KPZ equation with narrow wedge initial
condition [25], where the exact lower tail estimate was derived via computing
the exponential moment of the Airy-2 process, which admits an interpretation
via the stochastic Airy operator. The stochastic Airy operator is closely
related to the theme of this paper, yet it is defined via Brownian motion and
thus a lot of analytical techniques (like the Girsanov transform) can be used.
In our paper, we consider different moment assumptions on the random
variables, so that the analytical techniques in the context of stochastic Airy
operator are not available to us for deriving a sharper lower tail estimate.

There has also been some recent investigations on the matrix model (1.5),
where a scaling limit of top eigenvalue (when $\alpha=\frac{3}{2}$) was
derived in [13], and the bulk localizing properties were investigated in [12]
when the random variables have heavy tails.

We can also derive the limiting distribution of the second, third, fourth,
etc. largest eigenvalues of $H_{N}^{\alpha}$, denoted by $\lambda_{i}(N),i\geq
2$. All the derivations depend on the distribution of the largest value of
$\mathfrak{a}(1),\cdots,\mathfrak{a}(N)$ and the finite rank perturbation
formula, so that explicit distributions of the second, third, fourth, etc.
eigenvalues can be written out explicitly.

In cases (1) and (2), we can check that the deviation probabilities of
$\lambda_{i}(N),i\geq 2$ are negligible compared to $e^{-N^{\alpha\beta}}$ and
$N^{1-\alpha\beta}$ respectively, and is thus negligible compared to deviation
of $\lambda_{1}(N)$. In cases (3) and (4) however, deviations of
$\lambda_{i}(N),i\geq 2$ have the same magnitude as that of $\lambda_{1}(N)$,
and we have the following point process characterization:

######  Theorem 1.3.

In case (3) of Theorem 1.1, for any $\epsilon>0$, the largest eigenvalues of
$H_{N}^{\alpha}$ in $(2+\epsilon,\infty)$ converges to a random point process
which is the image under the map $x\mapsto\sqrt{x^{2}+4}$ of the Poisson
process on $[0,\infty)$ with intensity $\frac{C}{\alpha
x^{\frac{1}{\alpha}+1}},\quad x>0.$

In case (4) of Theorem 1.5, the point process

$N^{\alpha-\frac{1}{\beta}}\lambda_{1}(N)\geq
N^{\alpha-\frac{1}{\beta}}\lambda_{2}(N)\geq\cdots,$

where $\lambda_{1}(N)\geq\lambda_{2}(N)\geq\cdots$ denote the eigenvalue of
$H_{N}^{\alpha}$ in decreasing order, converges to the Poisson point process
on $[0,\infty)$ with intensity $\frac{C\beta}{x^{\beta+1}},\quad x>0.$

######  Remark 1.4.

The Poisson distribution claimed in (4) has been verified for several other
random matrix models with heavy tail distribution, see [24]. Meanwhile, in all
these claims, the assumption that the tail estimate for $\mathfrak{a}$ holds
for all $t>1$ can be easily changed to holding for all $t$ larger than a fixed
number.

In the following we summarize the results on top eigenvalues concerning the
random operator (1.8). We first introduce a convenient notation: for any
$\lambda\geq 2$, denote by $f(\lambda)$ the solution to

$f(\lambda)+\frac{1}{f(\lambda)}=\lambda$

that satisfies $f(\lambda)\geq 1$.

######  Theorem 1.5.

Consider the random operator $G_{N}^{\alpha}$ defined in (1.8), where $U_{N}$
is a Haar distributed unitary matrix independent of the random variables
$\mathfrak{a}(\cdot)$. We denote by $\lambda_{1}(N)$ the largest eigenvalue of
$G_{N}^{\alpha}$. Assume the random potentials
$\mathfrak{a}(1),\cdots,\mathfrak{a}(N)$ are i.i.d. distributed as a real-
valued symmetric random variable $\mathfrak{a}$. Then

  1. (1)

(Weibull distribution) Assume that the random variables $\mathfrak{a}(\cdot)$
satisfy Assumptions (1.9), (1.10) Then for any $\lambda>2$, we have the upper
tail estimate

$\lim_{N\to\infty}\frac{\log(\mathbb{P}(\lambda_{1}(N)>\lambda))}{N^{\alpha\beta}}=-C\left(f(\lambda)\right)^{\beta},$
(1.26)

and for any $\lambda<2$ we have the lower tail estimate as in (1.12).

  2. (2)

(Sub-critical) Assume that the random variables satisfy (1.13) and (1.14).
Then for any $\lambda>2$, we have the upper tail estimate:

$\lim_{N\to\infty}N^{\alpha\beta-1}\mathbb{P}(\lambda_{1}(N)>\lambda)=\left(f(\lambda)\right)^{-\beta},$
(1.27)

and for any $\lambda<2$ we have the lower tail estimates in exactly the same
form as in (1.16), (1.17) (though the numerical constants may be slightly
different).

  3. (3)

(Critical regime) Assume that the random variables satisfy (1.18), (1.19) Then

$\lambda_{1}(N)\overset{law}{=}\begin{cases}2,&\xi\leq 1,\\\
\xi+\frac{1}{\xi},&\xi>2,\end{cases}$ (1.28)

where $\xi$ is a non-negative-valued random variable that satisfies, for any
$\lambda>0$,

$\mathbb{P}(\xi>\lambda)=1-e^{-C\lambda^{-\frac{1}{\alpha}}}.$ (1.29)

More generally, the largest eigenvalues of $G_{N}^{\alpha}$ in $(2,\infty)$
converges to a random point process which is the image under the map $x\mapsto
x+\frac{1}{x}$ of the Poisson process on $[0,\infty)$ with intensity
$\frac{C}{\alpha x^{\frac{1}{\alpha}+1}},\quad x>0.$

  4. (4)

(Randomness dominating) Assume that the random variables satisfy (1.22),
(1.23). Then the top eigenvalue $\lambda_{1}(N)$ satisfies (1.24). Meanwhile,
the point process

$N^{\alpha-\frac{1}{\beta}}\lambda_{1}(N)\geq
N^{\alpha-\frac{1}{\beta}}\lambda_{2}(N)\geq\cdots$

converges to the Poisson point process on $[0,\infty)$ with intensity
$\frac{C\beta}{x^{\beta+1}},\quad x>0,$ where
$\lambda_{1}(N)\geq\lambda_{2}(N),\cdots\geq$ denote the eigenvalues of
$G_{N}^{\alpha}$ in decreasing order.

######  Remark 1.6.

The matrix model $G_{N}^{\infty}$ (1.6) is closely related to the Gaussian
beta ensemble. The small deviations of top eigenvalue of Gaussian beta
ensembles were derived in [18]. Note that the precise large deviation rate
function was not derived in [18]. In this paper we derive the precise large
deviation rate function, but for a modified model with orthogonal invariance.
From a different perspective, there has been a number of recent works
concerning deviations of the top eigenvalue of Gaussian beta ensembles at the
high temperature limit, including [22] and [23]. These works mainly consider
the double scaling regime of the Gaussian beta ensemble in which $\beta\to 0$
and $N\beta\to\infty$, and is very different from the models we investigate in
this paper where there is only one scaling.

###  1.2. Plan of the paper

In Section 1.2 we prove Theorem 2.2 and 1.3. Then in Section 3 we prove
Theorem 1.5.

##  2\. Proof of Theorem 1.1

To prove Theorem 1.1, we first need the following computation on the top
eigenvalue of diagonally perturbed matrices.

######  Lemma 2.1.

Given real numbers $\lambda_{1},\cdots,\lambda_{N}$ satisfying the following
assumptions: there exists some (small) $c\in(0,1)$ such that

  * •

$M=\max_{i\in[N]}\\{\lambda_{1},\cdots,\lambda_{N}\\}\in(0,\infty)$ for a
unique $i\in[N]$, and there exists some $\epsilon>0$ such that for any other
$j\neq i$, $\lambda_{j}\leq M-\epsilon$.

  * •

For any $i\in[1,N]$ such that $|\lambda_{i}|>N^{-c}$, at any site $j\in[1,N]$
where $|i-j|<N^{2c}$ the value must satisfy $|\lambda_{j}|\leq N^{-c}.$

  * •

There are at most $N^{1.5c}$ indices $i\in[1,N]$ such that
$|\lambda_{i}|>N^{-c}.$

Then

  1. (1)

If $M$ is achieved in the middle of $[1,N]$ in the sense that for any
$i\in[1,\lfloor N^{c}]\rfloor\cup[N-\lfloor N^{c}\rfloor,N]$, we have
$|\lambda_{i}|\leq N^{-c}$, then the largest eigenvalue of
$H_{N}^{\infty}+\operatorname{diag}(\lambda_{1},\cdots,\lambda_{N})$ converges
to

$\sqrt{M^{2}+4}$

as $N$ tends to infinity.

  2. (2)

In any case, the largest eigenvalue of
$H_{N}^{\infty}+\operatorname{diag}(\lambda_{1},\cdots,\lambda_{N})$ is not
larger than $\sqrt{M^{2}+4}(1+o(1))$ as $N$ tends to infinity.

###### Proof.

Denote by $\Lambda_{N}=\operatorname{diag}(\lambda_{1},\cdots,\lambda_{N})$
and

$\Lambda_{N}^{*}=\operatorname{diag}(\lambda_{1}1_{|\lambda_{1}|>N^{-c}},\lambda_{2}1_{|\lambda_{2}|>N^{-c}},\cdots,\lambda_{N}1_{|\lambda_{N}|>N^{-c}}).$

Then as $N\to\infty$ the top eigenvalue of $H_{N}^{\infty}+\Lambda_{N}$
converges to the top eigenvalue of $H_{N}^{\infty}+\Lambda_{N}^{*}$, so it
suffices to compute the top eigenvalue of the latter matrix. Recall that
$\lambda>2$ is an eigenvalue of $H_{N}^{\infty}+\Lambda_{N}^{*}$ and only if

$\det(\lambda\operatorname{I}_{N}-H_{N}^{\infty}-\Lambda_{N}^{*})=0.$

Since $\lambda>2$ and eigenvalues of $H_{N}^{\infty}$ lie in $(-2,2)$, this
implies

$\det(\operatorname{I}_{N}-(\lambda\operatorname{I}_{N}-H_{N}^{\infty})^{-1}\Lambda_{N}^{*})=0.$

At this point we use [14] to compute
$R_{ij}:=((\lambda\operatorname{I}_{N}-H_{N}^{\infty})^{-1})_{ij}$. Denote by
$\lambda^{*}=\operatorname{arccosh}(\frac{1}{2}\lambda)$, then the expression
is

$R_{ij}=\frac{\cosh((N+1-|i-j|)\lambda^{*})-\cosh((N+1-i-j)\lambda^{*})}{2\sinh(\lambda^{*})\sinh((N+1)\lambda^{*})}.$

In the following we first consider case (1). From properties of trigonometric
functions, we can verify that for any $i\in[N^{c},N-N^{c}]$, we have

$|R_{ii}-\frac{1}{2\sinh(\operatorname{arccosh}(\frac{1}{2}\lambda))}|\to
0,\quad N\to\infty.$

Meanwhile, for any such $i$, we have

$|R_{ij}|\leq e^{-N^{2c}}\quad\text{ for any }|j-i|>N^{2c}.$ (2.1)

Then for any $j\in[1,N]$ such that $|\lambda_{j}|<N^{-c}$, the $j$-th column
of $\operatorname{I}_{N}-R\Lambda_{N}^{*}$ must be the unit vector with 1 on
the $j$-th row and $0$ otherwise. We can thus use Gauss elimination to
simplify the matrix $\operatorname{I}_{N}-R\Lambda_{N}^{*}$, in the following
steps: (i) for any $j\in[1,N]$ such that $|\lambda_{j}|>N^{-c}$, we can use
Gauss elimination so that any $(i,j)$-th entry of
$\operatorname{I}_{N}-R\Lambda_{N}^{*}$ with $0<|i-j|<N^{2c}$ is turned to
zero, and (ii) more generally any $(i,j)$-th entry where
$|\lambda_{i}|<N^{-c}$ is turned to zero. This follows from our assumption on
the distribution of the values $\lambda_{i}$. Finally, (iii) any non-diagonal
entry $(i,j)$ of this matrix after our Gaussian elimination has absolute value
bounded by $e^{-N^{2c}}$ thanks to (2.1), and there are at most $N^{1.5c}$
possible choices of $i$ and $j$ by assumption (as each such $i$ corresponds to
a set with $|\lambda_{i}|>N^{-c}$).

We can now easily compute $\det(\operatorname{I}_{N}-R\Lambda_{N}^{*})$. By
the Gaussian elimination procedure, this boils down to computing the
determinant of a diagonal matrix (with bounded entries) perturbed by a sub-
matrix of size $N^{1.5c}$ with each entry bounded by $e^{-N^{2c}}$. By
Stirling’s formula, the contribution of the determinant that involves terms
from this perturbed sub-matrix is at most $(N^{1.5c})!e^{-N^{2c}}=o(1)$ as
$N\to\infty$. Therefore we only need to compute the determinant of the
diagonal part, and the largest solution $\lambda>2$ to
$\det(\operatorname{I}_{N}-R\Lambda_{N}^{*})=0$ converges to the $\lambda>2$
that satisfies

$2\sinh(\operatorname{arccosh}(\frac{1}{2}\lambda))=M,$

as $N\to\infty$. This implies, for $\lambda>2$, that $M=\sqrt{\lambda^{2}-4}$
and completes the proof for case (1).

For case (2), all the reductions in the previous step work as well, and the
only difference is that for $i\in[1,N^{c}]$ or $[N-N^{c},N]$ necessarily we
have, as $N\to\infty$,

$R_{ii}\to\frac{1-e^{-2\min(i,N+1-i)\lambda_{*}}}{2\sinh(\lambda_{*})}\leq\frac{1}{2\sinh(\operatorname{arccosh}(\frac{1}{2}\lambda))},$

and in this case the top eigenvalue is no more than $\sqrt{M^{2}+4}$. ∎

Now we prove Theorem 1.1.

We first verify the upper tail claims in case (1), (2) and (3).

###### Proof of Theorem 1.1, upper tail of case (1).

Assume that case (1) of Theorem 1.1 holds. Then for any $\lambda>0$,

$\displaystyle\mathbb{P}\left(\max\\{N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)\\}\geq\lambda\right)$
$\displaystyle=1-\mathbb{P}(\mathfrak{a}(1)\leq N^{\alpha}\lambda)^{N},$ (2.2)
$\displaystyle=1-(1-\mathbb{P}(\mathfrak{a}(1)>N^{\alpha}\lambda))^{N}.$

Noting that $\mathfrak{a}$ has the Weibull distribution, we conclude that

$\frac{\mathbb{P}\left(\max\\{N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)\\}\geq\lambda\right)}{N\exp(-CN^{\alpha\beta}\lambda^{\beta})}\in[C_{1},C_{2}](1+o(1)).$
(2.3)

Then we argue that the maximum in (2.2) is achieved at only one index $i$.
Indeed, for any $\lambda>0$,

$\mathbb{P}(\text{ there are two indices
}i:N^{-\alpha}\mathfrak{a}(i)\geq\lambda)\leq\frac{N^{2}-N}{2}\exp(-CN^{2\alpha\beta}\lambda^{2\beta})$
(2.4)

which is negligible compared to $\exp(-CN^{\alpha\beta})$.

We further check the other assumptions of Lemma 2.1 hold with very high
probability: for example, the possibility that there are indices $i,j$ such
that $|i-j|<N^{2c}$ such that
$N^{-\alpha}\mathfrak{a}(i)>N^{-c},N^{-\alpha}\mathfrak{a}(j)>N^{-c}$ is at
most

$N\cdot N^{2c}\cdot\exp(-CN^{2(\alpha-c)\beta}),$ (2.5)

which is negligible compared to $\exp(-CN^{\alpha\beta})$ if $c>0$ is chosen
small enough. Meanwhile, the possibility that there are more than $N^{1.5c}$
sites $i$ with $N^{-\alpha}\mathfrak{a}(i)>N^{-c}$ is also
$o(\exp(-CN^{\alpha\beta}))$ as can be seen from taking a union bound.

Now we can apply Lemma 2.1 to complete the proof as follows:

$\displaystyle\mathbb{P}(\lambda_{1}(N)>t)$
$\displaystyle\geq\mathbb{P}(\max_{i}N^{-\alpha}\mathfrak{a}(i)>{\sqrt{(t+o(1))^{2}-4}})$
(2.6)
$\displaystyle\cdot\mathbb{P}(\operatorname{argmax}_{i}N^{-\alpha}\mathfrak{a}(i)\in[N^{c},N-N^{c}])(1+o(1))$
$\displaystyle\geq
N\exp(-CN^{\alpha\beta}(\sqrt{(t+o(1))^{2}-4})^{\beta})(1+o(1))$

where $t-o(1)$ stands for some $N$-dependent constant smaller than $t$ but
converging to $t$ as $N\to\infty.$ We utilized the independence of
$\mathfrak{a}(i)$ and the fact that

$\mathbb{P}(\operatorname{argmax}_{i}N^{-\alpha}\mathfrak{a}(i)\in[N^{c},N-N^{c}])=1+o(1).$

We also have the reverse inequality

$\displaystyle\mathbb{P}(\lambda_{1}(N)>t)$
$\displaystyle\leq\mathbb{P}(\max_{i}N^{-\alpha}\mathfrak{a}(i)>{\sqrt{(t-o(1))^{2}-4}})$
(2.7) $\displaystyle+\mathbb{P}(\text{Assumptions in Lemma \ref{lemma2.1} not
all satisfied with }\lambda_{i}=N^{-\alpha}\mathfrak{a}(i))$
$\displaystyle=N\exp(-CN^{\alpha\beta}(\sqrt{(t-o(1))^{2}-4})^{\beta})(1+o(1)),$

since if none of the potentials $N^{-\alpha}\mathfrak{a}(i)$ are larger than
$\sqrt{t^{2}-4}$, then Lemma 2.1 implies it would be impossible to have
$\lambda_{1}(M)>t$. This completes the proof of case (1), the upper tail.∎

###### Proof of Theorem 1.1, upper tail of case (2).

Now we consider case (2), where the potentials $\mathfrak{a}(i)$ have heavier
tails. We see that

$\displaystyle\mathbb{P}\left(\max\\{N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)\\}\geq\lambda\right)$
$\displaystyle=1-(1-\mathbb{P}(\mathfrak{a}(1)>N^{\alpha}\lambda))^{N}$ (2.8)
$\displaystyle=1-(1-CN^{-\alpha\beta}\lambda^{-\beta})^{N}$
$\displaystyle=CN^{1-\alpha\beta}\lambda^{-\beta}(1+o(1))$

recalling the assumption that $\alpha\beta>1$. Similar to the previous case,
one can check that on the event that (2.8) holds, the assumptions of Lemma 2.1
hold with probability $1+o(1)$. Therefore we have the analogue of estimates
(2.6), (2.7) in this setting of heavier tails:

$\mathbb{P}(\lambda_{1}(N)>t)\geq
CN^{1-\alpha\beta}\sqrt{(t+o(1))^{2}-4}^{-\beta}(1+o(1)),$ (2.9)

and

$\displaystyle\mathbb{P}(\lambda_{1}(N)>t)$ $\displaystyle\leq
CN^{1-\alpha\beta}\sqrt{(t-o(1))^{2}-4}^{-\beta}(1+o(1))$ (2.10)
$\displaystyle+\mathbb{P}(\text{Assumptions in Lemma \ref{lemma2.1} not all
satisfied with }\lambda_{i}=N^{-\alpha}\mathfrak{a}(i))$ $\displaystyle\leq
CN^{1-\alpha\beta}\sqrt{(t-o(1))^{2}-4}^{-\beta}(1+o(1)),$

justifying the upper tail in case (2). ∎

###### Proof of Theorem 1.1, upper tail of case (3).

Now we consider case (3). Under the critical moment condition (1.18),

$\displaystyle\mathbb{P}\left(\max\\{N^{-\alpha}\mathfrak{a}(1),\cdots,N^{-\alpha}\mathfrak{a}(N)\\}\geq\lambda\right)$
$\displaystyle=1-(1-\mathbb{P}(\mathfrak{a}(1)>N^{\alpha}\lambda))^{N}$ (2.11)
$\displaystyle=1-(1-CN^{-1}\lambda^{-\frac{1}{\alpha}})^{N}$ $\displaystyle\to
1-e^{-C\lambda^{-\frac{1}{\alpha}}},\quad N\to\infty.$

Moreover, with the claimed event (2.11) occurs, it is easy to check that the
assumptions in Lemma 2.1 hold with probability $1-0(1)$. Therefore we have the
following two-sided inequalities: for any $t>2$,

$\displaystyle
1-e^{-c(\sqrt{t+o(1))^{2}-4})^{-\frac{1}{\alpha}}}(1+o(1))\leq\mathbb{P}(\lambda_{1}(N)>t)\leq
1-e^{-c(\sqrt{t-o(1))^{2}-4})^{-\frac{1}{\alpha}}}(1+o(1))$ (2.12)

where as before, $t-o(1)$ represents some constants $t(N)$ smaller than $t$
but converging to $t$ in the limit. The upper tail claim in case (3) now
follows. ∎

###### Proof of Theorem 1.1, Poisson distribution in case (4).

The argument is similar to [24].

We first compute that for any $\lambda>0$,

$\displaystyle\mathbb{P}\left(\max\\{N^{-\frac{1}{\beta}}\mathfrak{a}(1),\cdots,N^{-\frac{1}{\beta}}\mathfrak{a}(N)\\}\geq\lambda\right)$
$\displaystyle=1-(1-\mathbb{P}(\mathfrak{a}(1)>N^{\alpha}\lambda))^{N}$ (2.13)
$\displaystyle=1-(1-CN^{-1}\lambda^{-\frac{1}{\beta}})^{N}$ $\displaystyle\to
1-e^{-C\lambda^{-\frac{1}{\beta}}},\quad N\to\infty.$

That is,

$\mathbb{P}(\max_{i}N^{-\alpha}\mathfrak{a}(i)\geq
N^{\frac{1}{\beta}-\alpha}\lambda)\to 1-e^{-C\lambda^{-\frac{1}{\beta}}},$

and one can easily check that the second largest value among
$N^{-\frac{1}{\beta}}\mathfrak{a}(i)$ is strictly smaller than the first
largest, with probability tending to one. Assuming that $i\in[n]$ is the site
where the maximum is achieved in (2.13), then we can choose a unit vector such
that $f_{i}=(0,\cdots,0,1,\cdots,0)$, i.e. $f_{i}$ is 1 on the $i$-th
coordinate and is zero everywhere else. Then one easily sees, assuming
$\max_{i}N^{-\frac{1}{\beta}}\mathfrak{a}(i)=\lambda$, that

$H_{N}^{\alpha}f_{i}=N^{\frac{1}{\beta}-\alpha}\lambda f_{i}(1+o(1)),$

so by perturbation theorem of Hermitian operators,
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$ has en eigenvalue at
$\lambda(1+o(1))$. Meanwhile, observe that for such a choice of $\lambda$, we
have $N^{\frac{1}{\beta}}-\lambda$ is the value of the matrix $H_{N}^{\sigma}$
(which is also sparse), so the largest eigenvalue of
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$ cannot exceed $\lambda(1+o(1))$.
This completes the proof of part (4).

∎

###### Proof of Theorem 1.1, Lower tails in cases (1),(2) and (3)..

The last step we need to verify is that the possibility for $\lambda_{1}(N)$
to be smaller than 2 is much smaller than the possibility specified in each
case (1),(2),(3), justifying our claims on the left tail.

For this purpose, note that if $\lambda_{1}(N)<2-\delta<2$ for some
$\delta>0$, then

$d_{2}(\mu_{H_{N}^{\alpha}},\mu_{H_{N}^{\infty}})>C(\delta)>0$

for some constant $C(\delta)>0$, where $d_{2}$ denotes the 2-Wasserstein
distance on $\mathbb{R}$ defined for two probability measures $\mu,\nu$ via

$d_{2}(\mu,\nu)=\inf_{\pi\in\mathcal{C}(\mu,\nu)}\left(\int_{\mathbb{R}^{2}}|x-y|^{2}d\mu(x,y)\right)^{1/2},$

(with $\mathcal{C}(\mu,\nu)$ the space of probability measures on
$\mathbb{R}^{2}$ with first marginal $\mu$ and second marginal $\nu$), and
$\mu_{H_{N}^{\alpha}}$,$\mu_{H_{N}^{\infty}}$ respectively denote the
empirical measure of eigenvalues of $H_{N}^{\alpha}$ and $H_{N}^{\infty}$.

Using the trivial coupling, we have

$\displaystyle d_{2}(\mu_{H_{N}^{\alpha}},\mu_{H_{N}^{\infty}})$
$\displaystyle\leq(\frac{1}{N}\sum_{i=1}^{N}|\lambda_{i}^{\alpha}-\lambda_{i}^{\infty}|^{2})^{1/2}$
(2.14)
$\displaystyle\leq\frac{1}{\sqrt{N}}(\operatorname{Tr}(H_{N}^{\infty}-H_{N}^{\alpha})^{2})^{\frac{1}{2}}$
$\displaystyle=\frac{1}{N^{\alpha+\frac{1}{2}}}\left(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\right)^{\frac{1}{2}}.$

where the last step follows from Hoffman-Wielandt inequality, see for example
[1], Lemma 2.1.19 and also [9].

We begin with case (1) of Theorem 2.2, where $\mathfrak{a}(\cdot)$ is assumed
to have a Weibull distribution. First assume (1a) that $\beta=2$. Then we
proceed with the following computation: by Markov’s inequality, for any $d>0$,

$\displaystyle\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\geq
N^{2\alpha+1})$ $\displaystyle\leq
e^{-CdN^{2\alpha+1}}\mathbb{E}\left[\exp(d\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2})\right]$
(2.15)
$\displaystyle=e^{-CdN^{2\alpha+1}}\mathbb{E}[\exp\left(d|\mathfrak{a}(i)|^{2}\right)]^{N}$
$\displaystyle\leq C_{2}e^{-C_{3}N^{2\alpha+1}}$

for some constants $C_{2},C_{3}>0$, where the last step we used the tail decay
of the law of $\mathfrak{a}$.

Then assume (1b) that $\beta>2$, then from Jensen’s inequality

$\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\leq\left(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{\beta}\right)^{\frac{2}{\beta}}N^{1-\frac{2}{\beta}}$
(2.16)

we similarly deduce that, for a different $C^{\prime}>0$,

$\displaystyle\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\geq
CN^{2\alpha+1})\leq\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{\beta}\geq
C^{\prime}N^{\frac{\beta}{2}(2\alpha+\frac{2}{\beta})})\leq
C_{2}e^{-C_{3}N^{\alpha\beta+1}},$ (2.17)

where in the last steep we used Markov’s inequality (see (2.15)) and the tail
estimate in the law of $\mathfrak{a}$.

Finally assume (1c) that $\beta<2$. In this case the above reasoning is no
longer effective. We end up with a much weaker estimate of the following form:
for some $c>0$ to be determined,

$\displaystyle\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\geq
CN^{2\alpha+1})$
$\displaystyle\leq\mathbb{P}(\max_{i\in[1,N]}|\mathfrak{a}(i)|\geq
CN^{\alpha+c})$ (2.18)
$\displaystyle+\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{\beta}\geq
CN^{2\alpha+1-(2-\beta)(\alpha+c)})$ $\displaystyle\leq
C_{2}e^{-C_{3}N^{(\alpha+c)\beta}}+C_{2}e^{-C_{3}N^{\alpha\beta+1+(\beta-2)c}},$

and a careful choice of (sufficiently small) $c>0$ completes the proof.

Now we consider case (2) of Theorem 2.2, i.e. the random variables
$\mathfrak{a}(\cdot)$ have heavy tails yet $\alpha\beta>1$. Fix some
sufficiently small $\epsilon>0$, then $\mathfrak{a}$ has finite
$\beta-\epsilon$-th moment. Denote by $\beta^{\prime}=\beta-\epsilon$. In case
(2a), (2b) that $\beta^{\prime}\geq 2$, we get similar to the previous proof
that

$\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\geq
CN^{2\alpha+1})\leq\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{\beta^{\prime}}\geq
C^{\prime}N^{\frac{\beta^{\prime}}{2}(2\alpha+\frac{2}{\beta^{\prime}})})\leq(C_{2}N)^{-(\alpha\beta^{\prime}+1)}.$
(2.19)

In case (2c) that $\beta^{\prime}<2$, we deduce as previous that for
sufficiently small $c>0$,

$\displaystyle\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{2}\geq
CN^{2\alpha+1})$
$\displaystyle\leq\mathbb{P}(\max_{i\in[1,N]}|\mathfrak{a}(i)|\geq
CN^{\alpha+c})$ (2.20)
$\displaystyle+\mathbb{P}(\sum_{i=1}^{N}|\mathfrak{a}(i)|^{\beta^{\prime}}\geq
CN^{2\alpha+1-(2-{\beta^{\prime}})(\alpha+c)})$ $\displaystyle\leq
C_{2}N^{1-\beta^{\prime}(\alpha+c)}+(C_{2}N)^{-(\alpha\beta+1+(\beta-2)c)}.$

This completes the proof.

######  Remark 2.2.

We can also obtain the explicit dependence on $\lambda$ concerning the left
tail rate function. Notice that $\mu_{H_{N}^{\infty}}$ converges to the
arcsine law $\mu_{as}$ on $[-2,2]$, we can deduce that if
$\lambda_{1}(N)\in(-2,2),$ then

$d_{2}^{2}(\mu_{H_{N}^{\alpha}},\mu_{as})\geq\int_{\lambda_{1}(N)}^{2}\frac{(x-\lambda_{1}(N))^{2}dx}{\pi\sqrt{4-x^{2}}}$

and when $\lambda_{1}(N)<-2$ we similarly have

$d_{2}(\mu_{H_{N}^{\alpha}},\mu_{as})\geq-2-\lambda_{1}(N).$

Then we can plug this more precise estimate into (2.14), (2.15) and apply
Markov’s inequality. Overall we get a left tail estimate with explicit
dependence on $\mu$.

The same procedure applies to the matrix model $G_{N}^{\alpha}$, and one only
needs to replace the arcsine law $\mu_{as}$ by the semicircie law $\mu_{sc}$,
which is the limiting spectral measure of $G_{N}^{\infty}.$

∎

Finally we prove Theorem 1.3. The ideas are similar to the proof of [24],
Theorem 1.2. We begin with a lemma:

######  Lemma 2.3.

For any $0<c_{1}<d_{1}<c_{2}<d_{2}<\cdots<c_{k}<d_{k}<\infty$, let
$I_{l}=(c_{l},d_{l})$.

Then in case (3) of Theorem 2.2, the number of $N^{-\alpha}\mathfrak{a}(i)$,
$i\in[N]$ in $(I_{l})_{l=1}^{k}$ forms a Poisson point process on $[0,\infty)$
with intensity $\frac{C}{\alpha x^{\frac{1}{\alpha}+1}}$ on $[0,\infty)$.

In case (4) of Theorem 2.2, the number of
$N^{-\frac{1}{\beta}}\mathfrak{a}(i)$ in $(I_{l})_{l=1}^{k}$ forms a Poisson
point process on $[0,\infty)$ with intensity $\frac{C\beta}{x^{\beta+1}}$ on
$[0,\infty)$.

This lemma is close to Proposition 1 of [24]. Its verification is a
generalization of computations in (2.11) and (2.13) and can be found for
example in [17], Theorem 2.3.1.

###### Proof of Theorem 1.3.

In case (3) of Theorem 1.3, for any $\epsilon>0$ with probability $1-o(1)$
there are only a bounded number of $i\in[N]$ such that
$N^{-\alpha}\mathfrak{a}(i)>\epsilon$, and with probability $1-o(1)$ the
configuration $\lambda_{i}=N^{-\alpha}\mathfrak{a}(i),i\in[N]$ satisfies the
assumptions of Lemma 2.1, with any $i$ such that
$N^{-\alpha}\mathfrak{a}(i)>\epsilon$ lying in $[N^{c},N-N^{c}]$ for a small
$c$. In this case, the computations in Lemma 2.1 provide us with a one-to-one
correspondence between the large $N$-limit of some
$N^{-\alpha}\mathfrak{a}(i)>\epsilon$ and the large $N$-limit of a large
eigenvalue of $H_{N}^{\alpha}$ via the map $x\mapsto\sqrt{x^{2}+4}$. This
completes the proof.

In case (4) of Theorem 1.3, the idea of proof is similar to [24], Theorem 1.2.
We claim that the large $N$-limit of the configuration of
$N^{-\frac{1}{\beta}}\mathfrak{a}(i)$ precisely correspond to the large
$N$-limit of the largest eigenvalues of
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$. This correspondence has been
verified for the top eigenvalue in the proof of Theorem 1.1, and repeating
that proof for the second, third, etc. largest point in the limit (considering
approximate eigenfunctions supported on only one index, and using perturbation
to show existence of an eigenvalue) shows that each these limit points
correspond to the $N\to\infty$ limit of an eigenvalue of $H_{N}^{\alpha}$, so
we have identified $k$ distinct values as the $N\to\infty$ limit of
eigenvalues of $N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$. To show that they
are indeed the largest $k$ limiting eigenvalues of
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$, consider the $k$ indices $i$ such
that $N^{-\frac{1}{\beta}}\mathfrak{a}(i)$ achieves the $k$ largest values for
$i\in[1,N]$ and consider $H_{N}^{\alpha,k}$, which is $H_{N}^{\alpha}$ with
these $k$ rows and columns removed. Then the largest eigenvalue of
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha,k}$ converges to the $k+1$-st largest
point in the limiting configuration of $N^{-\frac{1}{\beta}}\mathfrak{a}(i)$.
Meanwhile, by interlacing, this limit is at least as large as the $k+1$-th
largest eigenvalue of $N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$. Thus we have
set up the correspondence between the top $k$ eigenvalues of
$N^{\alpha-\frac{1}{\beta}}H_{N}^{\alpha}$ and the largest $k$ points in the
Poisson point process, for each $k$. ∎

##  3\. Proof of Theorem 1.5

The proof of Theorem 1.5 is essentially the same as that of Theorem 1.1,
except that we need to prove the following result on unitarily purturbed
matrices.

######  Lemma 3.1.

Recall the deterministic matrix $G_{N}^{\infty}$ defined in (1.6). Let

$\Lambda_{N}=\operatorname{diag}(\lambda_{1},\cdots,\lambda_{N}),$

and let $U_{N}$ be a Haar distributed unitary matrix. Let
$M=\max_{i\in[N]}\lambda_{i}$. Assume that for any $\epsilon>0$ there are only
finitely many $i\in[1,N]$ (the number does not grow with $N$) such that
$|\lambda_{i}|>\epsilon$. Then as $N\to\infty$, the largest eigenvalue of

$G_{N}^{\infty}+U_{N}\Lambda_{N}U_{N}^{t}$

converges almost surely to

$\begin{cases}2\quad M<1,\\\ M+\frac{1}{M},\quad M\geq 1.\end{cases}$ (3.1)

###### Proof.

Consider the matrix $G_{N}^{\beta}$ defined as

$G_{N}^{\beta}=\frac{1}{\beta}\begin{pmatrix}N(0,2)&\chi_{(N-1)\beta}&0&\cdots&\cdots&0\\\
\chi_{(N-1)\beta}&N(0,2)&\chi_{(N-2)\beta}&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\\\
0&\cdots&\chi_{3\beta}&N(0,2)&\chi_{2\beta}&0\\\
0&\cdots&0&\chi_{2\beta}&N(0,2)&\chi_{\beta}\\\
0&\cdots&\cdots&0&\chi_{\beta}&N(0,2)\end{pmatrix},$ (3.2)

where the $N(\cdot)$ are centered normal distributions and the $\chi_{\cdot}$
are chi distributions with the specified parameter. Thanks to subGaussian
tails of these random variables, once we have proved that the top eigenvalue
of $G_{N}^{\beta}+U_{N}\Lambda_{N}U_{N}T$ converges almost surely to some
deterministic limit as $N\to\infty$, the same conclusion holds for the top
eigenvalue of $G_{N}^{\infty}+U_{N}\Lambda_{N}U_{N}T$.

The matrix $G_{N}^{\beta}$ is the matrix representation for Gaussian beta
ensembles obtained in Dumitriu-Edelman [10]. In particular, the empirical
measure of eigenvalues of $G_{N}^{\beta}$ converges to the semicircle law on
$[-2,2]$ as $N\to\infty$ and that the largest and smallest eigenvalues of
$G_{N}^{\beta}$ converge to $-2$ and $2$ respectively.

Thanks to our assumption on the magnitude of $\lambda_{i}$, we may assume
without loss of generality that $\Lambda_{N}$ has finite rank (that is, we set
$\lambda_{i}$ to be 0 if $|\lambda_{i}|<c$, and the resulting top eigenvalue
of the modified matrix differs by at most $c$ to the top eigenvalue of the
original matrix, and we set $c\to 0$). Since $U_{N}\Lambda_{N}U_{N}^{t}$ is
orthogonal invariant, we may use the main result of [5] to conclude that the
top eigenvalue of $G_{N}^{\beta}+U_{N}\Lambda_{N}U_{N}^{t}$ converges to (3.1)
almost surely. The same conclusion holds for
$G_{N}^{\infty}+U_{N}\Lambda_{N}U_{N}^{t}$.

∎

Now we complete the proof of Theorem 1.5.

###### Proof.

The proof of Theorem 1.5 is exactly the same as the proof of Theorem 1.1 and
1.3, and one simply needs to replace Lemma 2.1 by Lemma 3.1.

For the lower tails, we follow exactly the same computation as in Theorem 1.1.
For the upper tails in cases (1), (2) and (3), we only need to check that the
assumptions of Lemma 3.1 are satisfied with probability $1-o(1)$: indeed, we
may work under assumption (1.18) where $\mathfrak{a}$ has the heaviest tails
and deduce that

$\mathbb{P}(\text{ there exists $k$ different
}i\in[N]:N^{-\frac{1}{\alpha}}\mathfrak{a}(i)>\epsilon)\leq
C_{\epsilon}\tbinom{k}{N}(N)^{-k}\leq\frac{C_{\epsilon}}{k!}\to 0,\quad
k\to\infty,$ (3.3)

so that almost surely there are only finitely many $i$ such that
$N^{-\frac{1}{\alpha}}\mathfrak{a}(i)>\epsilon$. Then by Lemma 3.1, the
largest eigenvalue is governed by the largest value of the potentials
$N^{-\frac{1}{\alpha}}\mathfrak{a}(i),i\in[N].$ The distribution of this
maximum is already derived in (2.2), (2.8) and (2.11). This completes the
proof.

∎

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