# Concentration of Equilibria and Relative Instability in Disordered Non-
Relaxational Dynamics

Pax Kivimae Courant Institute of Mathematical Sciences, New York University.
Email<EMAIL_ADDRESS>
###### Abstract

We consider a system of random autonomous ODEs introduced by Cugliandolo et
al. [22], which serves as a non-relaxational analog of the gradient flow for
the spherical $p$-spin model. The asymptotics for the expected number of
equilibria in this model was recently computed by Fyodorov [32] in the high-
dimensional limit, followed a similar computation for the expected number of
stable equilibria by Garcia [38].

We show that for $p>9$ the number of equilibria, as well as the number of
stable equilibria, concentrate around their respective averages, generalizing
recent results of Subag and Zeitouni [61, 64] in the relaxational case. In
particular, we confirm that this model undergoes a transition from relative to
absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko
[11].

##  1 Introduction

Understanding the typical behavior of dynamic equilibria (stationary points)
in large disordered complex systems is a far-reaching problem motivated by
numerous applications in fields such as machine learning, ecology, and
economics, among others. The time evolution of such systems may often be
described by a set of coupled first-order nonlinear autonomous ordinary
differential equations (ODEs)

$\frac{dx_{t}}{dt}=\textbf{F}(x_{t}),$ (1.1)

where F is a random vector field on some smooth high-dimensional manifold
where $x_{t}$ takes its values.

Much of the interest in this question began with the celebrated paper of May
[50], who modeled the behavior of large ecological systems around an
equilibrium point as a random linear system, which May then analyzed by
employing tools developed in random matrix theory. They observed that as one
varies the model parameters, the unique equilibrium transitions from being
typically stable to typically unstable at a fixed value, a result which
becomes deterministic in the high-dimensional limit. This result, known as the
May-Wigner transition, has since been verified rigorously in a wide variety of
models and formed the basis of the local study of equilibria in random systems
(see [4] for a modern review).

To study global questions about the dynamics however, one must instead work
with non-linear systems, which may have many competing equilibria of varying
stability. The simplest statistic available in this case is the number of
equilibria, as well as the number of equilibria which are additionally stable.
The computation of these expected counts has seen extensive development in
recent years through application of the Kac-Rice formula (see [31] and the
results therein).

Foundational to recent work in non-relaxational case has been works of Ben
Arous, Fyodorov and Khoruzhenko [35, 11], who considered a non-linear analogue
of May’s model and found the model undergoes two distinct transitions. More
specifically, the expected number of equilibria was computed in [35], who
found that for suitably strong noise, this count would become exponentially
large in the system dimension, whereas for weak noise the system would
typically have only a single equilibrium point. This transition is expected to
indicate the onset of complex behavior in the dynamics.

The behavior within the complex phase was further studied in [11] by
considering the expected number of stable equilibria. They found a further
transition occurs for this quantity, which is specific to the non-relaxational
case. In particular, they found that if the relative strength of the non-
relaxational terms was brought above a certain threshold, the system would
transition from having an exponentially large expected number of stable
equilibria, to simply having none at all. They designated these as the regimes
of ”relative” and ”absolute” instability, respectively. Moreover, by computing
the ratio of these expected counts in the relative instability phase, they
predicted that while the number of stable equilibria typically would grow
exponentially large, the fraction of equilibria which are stable would be
exponentially small.

Similar results have been obtained in other models [8, 46, 68, 8], and in
particular, the recent works of Fyodorov and Garcia [32, 38] found such
analogous transitions in the non-relaxational spherical $p$-spin model. This
model was introduced by Cugliandolo et al. [22] as a generalization of the
spherical pure $p$-spin model, which has served as one of the paradigmatic
models in complex systems. In [32], asymptotics for the expected number of
equilibria were given, which were found to grow exponentially, independent of
the relative strength of the non-relaxational term. The expected number of
stable equilibria however, were found in [38] to not only depend on this
strength, but would transition from growing exponentially large to
exponentially small below a certain threshold, indicating a transition from
relative to absolute instability should occur at this point.

The purpose of this paper is to confirm that such a transition occurs in this
model. To do this, we must understand the typical (quenched) behavior of these
counts, which in general may deviate drastically from what one would predict
from their expected (annealed) value (see [19] and the discussion in [11]).
While powerful heuristics for the typical behavior exist in the physics
literature [55, 23], rigorously computing them, and understanding when they
coincide with their annealed counterparts, has only seen progress quite
recently. In particular, in the relaxational case, major progress was made by
Subag [61], who showed that the number of local minima for the spherical
$p$-spin model concentrates around its average. This result was then extended
by Subag and Zeitouni [64] to the number of critical points. These results
have proven important in establishing properties of the dynamics [39] as well
as properties of corresponding potential landscape [62, 63]. These works have
since been extended to other models in the relaxational case [12, 44, 9]. The
main result of this paper is an extension of these methods, and in particular
the required results in random matrix theory, to the simplest model in the
non-relaxational case.

We now recall the model, introduced in [22] and based on the earlier model
introduction in the $p=2$ case introduced by [21]. Fix a choice of $p\geq 3$
and $\tau\in(-(p-1)^{-1},1)$. Let $\textbf{f}(x)=(f_{k}(x))_{k=1}^{N}$ be the
smooth centered Gaussian vector-valued function on $\mathbb{R}^{N}$ whose
components satisfy

$\mathbb{E}[f_{k}(x)f_{l}(y)]=\delta_{kl}p\left(x,y\right)_{N}^{p-1}+\tau
p(p-1)\left(\frac{y_{l}x_{k}}{N}\right)\left(x,y\right)_{N}^{p-2},\;\;\;1\leq
k,l\leq N,$ (1.2)

where $(x,y)_{N}=N^{-1}\sum_{i=1}^{N}x_{i}y_{i}$ denotes the Euclidean inner
product. We then define F for $x\in S_{N}$ by

$\textbf{F}(x):=-\lambda(x)x+\textbf{f}(x),$ (1.3)

where $S_{N}=\\{x\in\mathbb{R}^{N}:\|x\|^{2}=N\\}$ denotes the $(N-1)$-sphere
and $\lambda(x)=(x,\textbf{f}(x))_{N}$ is a Lagrange multiplier chosen so that

$\textbf{F}(x)\in T_{x}S_{N}=\\{v\in\mathbb{R}^{N}:(v,x)_{N}=0\\}.$ (1.4)

One may alternatively define F by specifying the terms in the following
decomposition of f

$f_{i}(x)=c_{s}\nabla_{i}^{euc}H(x)+\frac{c_{a}}{N}\sum_{j=1}^{N}x_{j}A_{ij}(x),$
(1.5)

where the matrix $A_{ij}(x)$ is anti-symmetric (i.e. $A_{ij}=-A_{ji}$),
$\nabla_{i}^{euc}$ denotes the $i$-th standard Euclidean partial derivative,
and $c_{s},c_{a}$ are normalization constants. When $p=2$, this coincides with
the decomposition of a matrix in symmetric and anti-symmetric parts. In
addition, it is similar in form to the gradient-solenodial (Hodge)
decomposition used to define the models in [11, 35]. In our case, we take $H$
to be a spherical $p$-spin model, and for each of $1\leq i<j\leq N$, takes
$A_{ij}$ to be an independent $(p-2)$-spin model. Then taking
$c_{s}=\sqrt{p^{-1}(1+(p-1)\tau)}$ and $c_{a}=\sqrt{(p-1)(1-\tau)}$, the
components of (1.5) satisfy (1.2), as checked in Lemma A.3 of Appendix A.

Let us denote the number of equilibria of the vector field F as
$\mathcal{N}_{N,tot}$, and similarly for denote the number of stable
equilibria as $\mathcal{N}_{N,st}$. Asymptotics for
$\mathbb{E}[\mathcal{N}_{N,st}]$ were obtained by Fyodorov [32].

######  Theorem 1.1.

[32] We have that

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,tot}])=\frac{1}{2}\log(p-1).$
(1.6)

Note that if one lets $\tau=1$, this coincides with the results of Auffinger,
Ben Arous, and Černý [6] for the spherical $p$-spin model. Building upon this,
Garcia [38] computed asymptotics for $\mathbb{E}]\mathcal{N}_{N,st}]$.

######  Theorem 1.2.

[38] We have that

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,st}])=\frac{1}{2}\log(p-1)-\frac{(p-2)(1+\tau)}{2(1+(p-1)\tau)}=:\theta_{p,\tau}.$
(1.7)

With some mild algebraic manipulation, this shows that for each $p\geq 3$,
there is a critical value

$\tau_{p}:=\frac{(p-2)-\log(p-1)}{(p-1)\log(p-1)-(p-2)}\geq 0,$ (1.8)

such that for $\tau<\tau_{p}$ it is exponentially rare that the system has
even a single stationary equilibrium, while for $\tau>\tau_{p}$ the expected
number of stationary equilibria is exponentially large, but growing at a rate
slower than the expected total number. Note this is in exact analogy with the
transition from relative to absolute instability found in [11]. Our main
result will be the following concentration result, which confirms that this
transition occurs.

######  Theorem 1.3.

Assume $p>9$ and $\tau\in(0,1)$. Then in probability as $N\to\infty$

$N^{-1}\log(\mathcal{N}_{N,tot})\Rightarrow\frac{1}{2}\log(p-1).$ (1.9)

If in addition $\tau\in(\tau_{c},1)$, then in probability as $N\to\infty$

$N^{-1}\log(\mathcal{N}_{N,st})\Rightarrow\theta_{p,\tau},$ (1.10)

so that in particular we have also

$N^{-1}\log(\mathcal{N}_{N,st}/\mathcal{N}_{N,tot})\Rightarrow-\frac{(p-2)(1+\tau)}{2(1+(p-1)\tau)}.$
(1.11)

We prove this by applying the second moment method. Before doing this, it will
be useful to introduce the counts for equilibrium points with a given Lagrange
multiplier. A key motivation for this is that the value taken by the Lagrange
multiplier at an equilibrium point will essentially determine the
equilibrium’s stability. Explicitly, let us define the threshold

$E_{\infty;p,\tau}=\sqrt{p(p-1)(1+\tau)}.$ (1.12)

Then a fixed equilibrium point $x\in S_{N}$ satisfying
$\lambda(x)<-E_{\infty;p,\tau}$ is typically stable, and similarly a
equilibrium point satisfying $\lambda(x)>-E_{\infty;p,\tau}$ is typically
unstable. In particular, we will find it useful to denote for
$u\in\mathbb{R}\cup\\{\infty\\}$, the total number of equilibria $x$, such
that $\lambda(x)<u$ by $\mathcal{N}_{N}(u)$, and similarly for
$\mathcal{N}_{N,st}(u)$. We now recall the following more general form of
Theorem 1.1, also derived in [32].

######  Theorem 1.4.

[32] For $u\in\mathbb{R}$ we have that

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}(u)])=\sup_{v<u}\Sigma^{p,\tau}(v),$
(1.13)

where $\Sigma^{p,\tau}$ is a fixed function recalled in (1.27) below.

An important threshold for the function $\Sigma^{p,\tau}$ will be the unique
value $E_{0;p,\tau}>0$, for which $\Sigma^{p,\tau}(v)<0$ for
$v<-E_{0;p,\tau}$. Note by Markov’s inequality, it is exponentially rare for
there to exist a single equilibrium such that the associated Lagrange multiple
is strictly less than $-E_{0;p,\tau}$. We also observe that
$\Sigma^{p,\tau}(E_{\infty;p,\tau})=\theta_{p,\tau}$, so that $\tau_{p}$ may
equivalently be defined as unique $\tau_{p}$, such that
$E_{\infty;p,\tau}>E_{0;p,\tau}$ for $\tau<\tau_{p}$ and
$E_{\infty;p,\tau}<E_{0;p,\tau}$ for $\tau>\tau_{p}$.

With these thresholds defined, our main result will be the following second
moment estimates.

######  Theorem 1.5.

Assume $\tau\in(0,1)$ and $p>9$. Then for $u\in(-E_{0;p,\tau},\infty]$ with
$u\neq-E_{\infty;p,\tau}$ we have that

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}(u)^{2}]}{\mathbb{E}[\mathcal{N}_{N}(u)]^{2}}=1.$
(1.14)

If in addition $\tau\in(\tau_{c},1)$ and $u<-E_{\infty;p,\tau}$

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,st}(u)^{2}]}{\mathbb{E}[\mathcal{N}_{N,st}(u)]^{2}}=1.$
(1.15)

By a standard argument, (1.14) implies that in probability as $N\to\infty$

$\log(\mathcal{N}_{N}(u))-\log(\mathbb{E}[\mathcal{N}_{N}(u)])\Rightarrow 0.$
(1.16)

In particular, we observe that (1.14) and Theorem 1.1 immediately imply (1.9).
Similarly, by applying $\Sigma^{p,\tau}(-E_{\infty;p,\tau})=\theta_{p,\tau}$,
we may obtain (1.10) from (1.15), Theorem 1.2, and Markov’s inequality.

We will also show the following result on the exponential scale, though with
less restrictions on $(p,\tau,u)$, obtained in the course of proving Theorem
1.5.

######  Theorem 1.6.

For any $u\in\mathbb{R}$ we have that

$\lim_{N\to\infty}N^{-1}\log\left(\frac{\mathbb{E}[\mathcal{N}_{N}(u)^{2}]}{\mathbb{E}[\mathcal{N}_{N}(u)]^{2}}\right)=0.$
(1.17)

We note that Theorem 1.5 actually establishes a much stronger concentration
result than is used by Theorem 1.3. On the other hand, Theorem 1.6 is
insufficient to establish any concentration result.

###  1.1 Methods

We will now discuss the methods we use to show our main result, with a more
complete outline being the focus of Section 2. The starting point will be the
Kac-Rice formula, which will give us an explicit representation of
$\mathbb{E}[\mathcal{N}_{N}(u)^{2}]$ (see Lemma 2.1). The primary difficulty
in analyzing this expression is a term involving the product of absolute
determinants of a pair of conditional Jacobians.

The corresponding expression for the first moment in [32] expresses
$\mathbb{E}[\mathcal{N}_{N}(u)]$ as a fixed multiple of the expected absolute
determinant of a matrix sampled from the elliptic ensemble (recalled below in
Definition 1.7), up to a random multiple of the identity. Fixing this
multiple, the problem of computing the first moment is reduced to producing
asymptotics for the expected absolute characteristic polynomial of the
elliptic ensemble.

To compute the second moment however, we must deal with both the effect the
conditioning has on the Jacobian, as well as the effect of the correlations
between the Jacobians themselves. The first effectively augments the law of
the Jacobians by a low-rank perturbation, so we are reduced to the problem of
computing the expected characteristic polynomial for perturbations of a pair
of correlated matrices both sampled from the elliptic ensemble.

On the exponential scale, we do this by approximating the characteristic
polynomial by the exponential of a more well-behaved linear statistic. Not
only does this make the characteristic polynomial more stable under
perturbation, but using suitable concentration estimates, we may show that
this statistic may be replaced by its expectation, which is insensitive to
correlations. In the symmetric case this method was explored quite fully in
the work of Ben Arous, Bourgade and McKenna [10], who showed how one may
employ this method under a fairly mild set of conditions. We follow this
method, which entails establishing and modifying a variety of results from
symmetric matrix theory to the elliptic ensemble, where the eigenvalues are
less stable under perturbation. This will be the focus of Section 7.

To obtain estimates on the determinant term to vanishing order, as needed for
the proof of Theorem 1.5, will be a more difficult task. The key problem is
that in this case the correlations between the Jacobians are no longer
negligible, as indeed the fluctuations of the determinant are typically on the
order of a polynomial factor. Thus the primary difficulty is to control these
fluctuations, for which our main tool will be a moment bound for the
determinant term which is tight up to a constant.

The primary technical input will be the computation of the higher moments for
the absolute characteristic polynomial of the elliptic ensemble up to constant
order, given in Section 8. Such bounds have a long history, and are known
classically in the Symmetric and Hermitian case [16, 17], as well as for
certain classical groups [20, 49]. Much less is known in the non-symmetric
case though, and instead we must rely on methods more dependent on the
invariance of the ensemble. A more in-depth discussion of the background for
this problem, as well as our methods, are given in Section 8.

We believe one could relax the assumptions of Theorem 1.3 in sight of Theorem
1.6. The restriction on $p>9$ is needed only in the proof of Lemma 2.11 and is
similar in nature to the restriction to $p\geq 32$ in [64], though removing it
is beyond the scope of our methods. The primary reason our proof holds for
this lower value of $p$ is that the characteristic polynomial when $\tau<1$
has weaker fluctuates then in the symmetric case, as well as the related
increase in the typical spacing of eigenvalues from $N^{-1}$ in the symmetric
case to $N^{-1/2}$.

While we believe our methods could be generalized to the case $\tau=0$,
significant changes would be needed to include $\tau<0$. The first problem is
that while the largest real eigenvalue of the elliptic ensemble is typically
$1+\tau$, the largest singular value is typically $1+|\tau|$. This renders a
large portion of Section 7 useless, as the largest singular value no longer
accurately detects when a shift by the identity is positive definite. The
second problem is that the representation of the correlations functions given
in [29], and used in the proof of Proposition 8.1, only holds for $\tau\geq
0$.

###  1.2 Structure

We begin the paper with Section 2, which outlines the general argument and
reduces the proofs of our main results to a series of more technical results
proven in the later sections. The remainder of the paper is then roughly
structured in two parts, the first composed of Section 3-6 and the second of
Sections 7-8.

The first part focuses on results specific to the computation of
$\mathbb{E}[\mathcal{N}_{N}(u)^{2}]$. In Section 3 we will provide our
application of the Kac-Rice formula to the second moment, and give results
which describe the structure of the conditional Jacobian at two points. In
Section 4, we compute the leading order asymptotics for this formula, which is
given by Theorem 2.3. These sections are primarily the standard structure of
the Kac-Rice formalism, as in [32, 38, 6, 61, 51]. In particular, we tried to
structure these sections similarly to Sections 4 and 5 [61], so as to assist
readers familiar with this work. We then refine our results to the constant
order in Section 5, where we prove the tighter asymptotics for the correlated
determinants mentioned above. Then in Section 6 we prove some analytic results
about the limiting complexity functions.

The second part is composed of our results in random matrix theory and is
essentially self-contained. Section 7 contains the results we need to treat
the characteristic polynomial on the exponential scale, as well as outside of
the bulk. In Section 8, we prove our main results for moments of the
characteristic polynomial in the bulk.

Finally, we provide two Appendices. Appendix A contains the majority of the
raw technical computations involving the covariance structure, as well as the
verification of a few miscellaneous results involving them. In Appendix B we
give the exact form of the Kac-Rice formula we use, which is a modification of
the results in Chapter 6 of [7].

###  1.3 Notation

We begin by introducing our counts more formally. We define for a Borel subset
$B\subseteq\mathbb{R}$,

$\mathcal{N}_{N}(B)=\\#\\{\sigma\in
S_{N}:\textbf{F}(\sigma)=0,\lambda(\sigma)\in B\\},$ (1.18)
$\mathcal{N}_{N,st}(B)=\\#\\{\sigma\in
S_{N}:\textbf{F}(\sigma)=0,\lambda(\sigma)\in B,\nabla\textbf{F}(\sigma)\geq
0\\},$ (1.19)

where $\nabla\textbf{F}(\sigma)$ denotes the Jacobian at $\sigma$ taken with
respect to the standard Riemannian metric on $S_{N}$. We observe that
$\mathcal{N}_{N}(u)=\mathcal{N}_{N}((-\infty,u))$ and
$\mathcal{N}_{N,st}(u)=\mathcal{N}_{N,st}((-\infty,u))$, and will henceforth
abandon this notation for uniformity. Moreover, for Borel subsets
$B\subseteq\mathbb{R}$ and $I\subseteq[-1,1]$, we define

$\mathcal{N}_{N,2}(I,B)=\\#\\{(\sigma,\tau)\in
S_{N}^{2}:\textbf{F}(\sigma)=\textbf{F}(\tau)=0,\lambda(\sigma),\lambda(\tau)\in
B,(\sigma,\tau)_{N}\in I\\}.$ (1.20)

We note that $\mathcal{N}_{N,2}([-1,1],B)=\mathcal{N}_{N}(B)^{2}$. We will say
a subset of $\mathbb{R}$ is nice if it is a finite union of open internals,
and similarly we will say a subset of $[-1,1]$ is nice if it is the
intersection of a nice subset of $\mathbb{R}$ with $[-1,1]$.

We now define our complexity functions. For $|\tau|<1$ we define the ellipse
$\mathcal{E}_{\tau}\subseteq\mathbb{R}^{2}$

$\mathcal{E}_{\tau}=\left\\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+\tau)^{2}}+\frac{y^{2}}{(1-\tau)^{2}}\leq
1\right\\},$ (1.21)

and denote the logarithmic potential of the uniform measure on this ellipse as

$\phi_{\tau}(u)=\frac{1}{(1-\tau^{2})\pi}\int_{\mathcal{E}_{\tau}}\log(|x+iy-u|)dxdy.$
(1.22)

For $\tau=1$, we define

$\phi_{1}(u)=\lim_{\tau\to
1}\phi_{\tau}(u)=\frac{1}{2\pi}\int_{-2}^{2}\sqrt{4-x^{2}}\log(|x-u|)dx,$
(1.23)

the standard semi-circle potential. It is well known that (see (4.1) of [18])

$\phi_{\tau}(u)=\frac{u^{2}}{2(1+\tau)}-\frac{1}{2}-I_{\tau}(u)I(|u|\geq
1+\tau),$ (1.24)
$I_{\tau}(u)=\begin{cases}\frac{1}{2(1+\tau)}u^{2}-\frac{|u|(|u|-\sqrt{u^{2}-4\tau})}{4\tau}-\log(\frac{|u|+\sqrt{u^{2}-4\tau}}{2});\;\;\tau\neq
0\\\ -\log u+\frac{1}{2}u^{2}-\frac{1}{2};\;\;\tau=0\end{cases}.$ (1.25)

It will be convenient to define a rescaling of this as well

$\phi_{\tau,p}(u)=\phi_{\tau}\left(\frac{u}{\sqrt{p(p-1)}}\right).$ (1.26)

With this notation the annealed complexity function is given as

$\Sigma^{p,\tau}(u)=\frac{1}{2}+\frac{1}{2}\log(p-1)-\frac{u^{2}}{2p\alpha}+\phi_{\tau,p}\left(u\right).$
(1.27)

For the two-point complexity function, we first define an auxiliary function

$h(r):=\frac{1}{2}\log\left(\frac{1-r^{2}}{1-r^{2p-2}}\right),$ (1.28)

and define

$\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})=1+\log(p-1)+h(r)-\frac{1}{2}(u_{1},u_{2})\Sigma_{U}(r)^{-1}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}+\phi_{\tau,p}\left(u_{1}\right)+\phi_{\tau,p}\left(u_{2}\right),$
(1.29)

where $\Sigma_{U}(r)=\Sigma_{U,p,\tau}(r)$ is a certain 2-by-2 matrix-valued
function defined in (A.62) below.

We will now introduce some notation involving matrices. For an $n$-by-$n$
matrix $A$ we will denote the (algebraic) eigenvalues of $A$ as
$\lambda_{1}(A),\dots,\lambda_{n}(A)$, ordered so that
$\mathrm{Re}(\lambda_{i-1}(A))\leq\mathrm{Re}(\lambda_{i}(A))$. Similarly we
will denote its singular values as $0\leq s_{n}(A)\leq\dots\leq s_{1}(A)$. We
will denote the empirical (spectral) measure for the eigenvalues of $A$ by

$\mu_{A}=\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}(A)}.$ (1.30)

In addition, for any choice of $z\in\mathbb{R}$ and $n$-by-$n$ matrix $A$ we
define the empirical measure for the singular values of $A$ by

$\nu_{A,z}=\frac{1}{n}\sum_{i=1}^{n}\delta_{s_{i}(A-zI)},$ (1.31)

and $\nu_{A}=\nu_{A,0}$. We will use the notation $\|A\|_{F}$ to denote the
Frobenius norm on matrices (i.e. $\|A\|_{F}=\text{\rm Tr}(A^{T}A)$).

We now define the key classical ensemble occurring in this model, namely the
(Gaussian) real elliptic ensemble.

######  Definition 1.7.

For $-1\leq\tau\leq 1$ we will say a $N$-by-$N$ random matrix $A_{N}$ belongs
to the real Elliptic Ensemble $GEE(N,\tau)$ if, when considered as a vector in
$\mathbb{R}^{N^{2}}$, $A_{N}$ is a centered Gaussian random vector whose
covariance is given by

$\mathbb{E}[A_{ij}A_{kl}]=\frac{1}{N}(\delta_{ik}\delta_{jl}+\tau\delta_{il}\delta_{jl}).$
(1.32)

The limiting law for the empirical measure (when $|\tau|<1$) for the elliptic
ensemble was first identified by Girko [41, 42] as the uniform measure on the
ellipse $\mathcal{E}_{\tau}$ defined above. More specifically, it was shown in
[59] (see as well [53]) that for $A_{N}$ sampled from $GEE(N,\tau)$, we have
as $N\to\infty$

$\mu_{A_{N}}\Rightarrow\frac{1}{(1-\tau^{2})\pi}I_{\mathcal{E}_{\tau}}$ (1.33)

in probability, where $I_{\mathcal{E}_{\tau}}$ denotes the indicator function
of the ellipse defined in (1.21). When $\tau=0$ this recovers the circular
law, and when $\tau=1$ we recover the semi-circular law.

##  2 Outline of The Proofs for Theorems 1.5 and 1.6

In this section, we outline the general structure of the proof of our main
results. We begin by giving the exact expressions for the first two moments of
$\mathcal{N}_{N}(B)$ and then introduce Theorem 2.3, which concerns the
computation of $\mathbb{E}[\mathcal{N}_{N}(B)^{2}]$ up to the exponential
scale. We then provide some analytic results about the resulting complexity
functions, from which we derive Theorem 1.6.

After this we will proceed to the proof of Theorem 1.5. Our first step will be
to reduce this proof to Proposition 2.6, which concerns the contribution of
points which are roughly orthogonal. This makes the computation at this scale
possible, as the components appearing in the Kac-Rice formula become roughly
independent. In particular, we are able to prove Theorem 1.5 through a series
of estimates for the absolute determinant term in the Kac-Rice formula.

Our exact formula for $\mathbb{E}[\mathcal{N}_{N,2}(B,I)]$ which will follow
from a standard application of the Kac-Rice formula. This expresses
$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]$ as an integral over
$(\sigma,\sigma^{\prime})\in S_{N}\times S_{N}$, though as the law of F is
isotropic, we will reduce this to an integral depending only on
$r=(\sigma,\sigma^{\prime})_{N}$ by the co-area formula.

######  Lemma 2.1.

Let $(M_{N}^{k}(r))_{k=1,2}$ and $(U^{k}_{N}(r))_{k=1,2}$ denote the random
variables given in Lemma 3.3. These are, respectively, a pair of
$(N-1)$-by-$(N-1)$ random matrices and a pair of random scalars with marginal
law given by $(U^{1}_{N}(r),U^{2}_{N}(r))\stackrel{{\scriptstyle
d}}{{=}}\mathcal{N}(0,N^{-1}\Sigma_{U}(r))$ where $\Sigma_{U}(r)$ is given by
(A.62). Then for nice $B\subseteq\mathbb{R}$ and nice $I\subseteq(-1,1)$

$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]=$ (2.1)
$C_{N}\int_{I}\left(\frac{1-r^{2}}{1-r^{2p-2}}\right)^{\frac{N-2}{2}}\mathcal{G}(r)\mathbb{E}\left[\prod_{k=1,2}|\det(M^{k}_{N}(r))|I(U^{k}_{N}(r)\in
B)\right]dr.$ (2.2)

where here
$\mathcal{G}(r):=(1-r^{2})^{-1/2}(1-(r^{p}-\tau(p-1)r^{p-2}(1-r^{2}))^{2})^{-1/2}$
and

$C_{N}=\mathrm{vol}(S^{N-1})\mathrm{vol}(S^{N-2})\left((N-1)(2\pi
p)^{-1}(p-1)\right)^{(N-1)},$ (2.3)

where $\mathrm{vol}(S^{\ell-1})$ denotes the surface area of the
$(\ell-1)$-dimensional unit sphere

$\mathrm{vol}(S^{\ell-1})=\frac{2\pi^{\ell/2}}{\Gamma(\ell/2)}.$ (2.4)

This will be proven in Section 3, based on the form of the Kac-Rice formula
derived in Appendix B. From this expression, we see that the primary obstacle
for obtaining asymptotics for $\mathbb{E}[\mathcal{N}_{N,2}(B,I)]$ is
understanding the term

$\mathbb{E}\left[\prod_{k=1,2}|\det(\bar{M}_{N}^{k}(r,u_{1},u_{2}))|\right].$
(2.5)

To express how this is done, it will be convenient to recall the corresponding
exact expression for the annealed complexity, which occurs as Theorem 3.1 of
[32].

######  Lemma 2.2.

Let $A_{N-1}$ be sampled from $GEE(N-1,\tau)$ and let $X_{N}$ be an
independent centered random variable with variance $p\alpha/N$. Then for nice
$B\subseteq\mathbb{R}$ we have that

$\mathbb{E}[\mathcal{N}_{N}(B)]=\bar{C}_{N}\mathbb{E}\left[|\det\left(A_{N-1}-\hat{X}_{N}I\right)|I(X_{N}\in
B)\right],$ (2.6)

where here

$\hat{X}_{N}=\left(\frac{N}{(N-1)p(p-1)}\right)^{1/2}X_{N},$ (2.7)
$\bar{C}_{N}=\mathrm{vol}(S^{N-1})\left((N-1)(2\pi)^{-1}(p-1)\right)^{(N-1)/2}.$
(2.8)

Conditioning on the value of $X$, we see that the computation of
$\mathbb{E}[\mathcal{N}_{N}(B)]$ is similarly reduced to the computation of
the expected value of the absolute characteristic polynomial of a matrix
sampled from $GEE(N-1,\tau)$. To relate this to the term relevant to the
second moment, we will condition $M^{k}_{N}(r)$ on the event
$(U^{k}_{N}(r)=u_{1},U^{k}_{N}(r)=u_{2})$, to obtain a pair of matrices
$(\bar{M}^{k}_{N}(r,u_{1},u_{2}))_{k=1,2}$, which we describe in Lemma 3.4
below. We will show in Section 5 that there are rank-$2$ matrices
$E_{N}^{k}(r,u_{1},u_{2})$ such that we may write

$(\bar{M}_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}\stackrel{{\scriptstyle
d}}{{=}}(A_{N-1}^{k}(r)-\hat{u}_{k,N}I+E_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}$
(2.9)

where $(A_{N-1}^{1}(r),A_{N-1}^{2}(r))$ are a pair of correlated
$GEE(N-1,\tau)$ matrices and

$\hat{u}_{k,N}:=\left(\sqrt{\frac{N}{(N-1)p(p-1)}}\right)u_{k}.$ (2.10)

Thus to compare the first and second moments of $\mathcal{N}_{N}(B)$, we want
to show that the contribution to the determinant coming from the perturbation
term $E_{N}^{k}(r,u_{1},u_{2})$, as well as the effect of the correlations
between the matrices $(A_{N-1}^{1}(r),A_{N-1}^{2}(r))$, are both negligible.

On the exponential scale, which is sufficient to obtain the leading order term
of $\mathbb{E}[\mathcal{N}_{N,2}(B,I)]$, this will follow by replacing the
determinant with a suitably regularized analog. The following computation will
be our main result on this scale, whose proof will be the focus of Section 4.

######  Theorem 2.3.

For nice $B\subseteq\mathbb{R}$, and nice $I\subseteq(-1,1)$, we have that

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(B,I)])=\sup_{u_{1},u_{2}\in
B,r\in I}\Sigma_{2}^{p,\tau}(r,u_{1},u_{2}).$ (2.11)

We note that with Theorems 1.4 and 2.3, the proof of Theorem 1.6 is reduced to
a comparison of the resulting complexity functions. To begin we need the
following result which allows us to restrict our attention to points where
$u_{1}=u_{2}$.

######  Lemma 2.4.

For nice $B\subseteq\mathbb{R}$ that intersects
$(-E_{0;p,\tau},E_{0;p,\tau})$, and any nice subset of $I\subseteq[-1,1]$

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(B,I)])\leq\sup_{r\in
I\cap(-1,1),u\in B}\Sigma_{2}^{p,\tau}(r,u,u).$ (2.12)

In particular,

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}(B)^{2}])\leq\sup_{r\in(-1,1),u\in
B}\Sigma_{2}^{p,\tau}(r,u,u).$ (2.13)

This result will be proven in Section 6. We will also need the following
purely analytic result, also proven in Section 6.

######  Lemma 2.5.

Let $u\geq-E_{0;p,\tau}$, and fix any nice $I\subseteq(-1,1)$. Then
$\\{0\\}\in\bar{I}$ if and only if

$\sup_{v\leq u,r\in I}\Sigma_{2}^{p,\tau}(r,v,v)=2\sup_{v\leq
u}\Sigma^{p,\tau}(v),$ (2.14)

with equality replaced by $<$ otherwise.

Theorem 1.6 follows immediately from Theorems 1.4 and 2.3 and Lemmas 2.4 and
2.5.

We now turn to the proof of Theorem 1.5, which concerns the refinement of our
results to the scale $1+o(1)$. In particular, we must return our attention to
understanding (2.5) now up to a term of vanishing order. Even considering this
up to a $O(1)$-term requires significantly more care, as we are now in the
regime in which the correlations of $(A^{1}_{N-1}(r),A^{2}_{N-1}(r))$ may not
only contribute nontrivially but may even diverge in $N$. To illustrate this,
note that if $u=\hat{u}_{1,N}=\hat{u}_{2,N}$ then as $r\to 1$, this amounts to
understanding the ratio

$\frac{\mathbb{E}[|\det(A_{N}-uI)|^{2}]}{\mathbb{E}[|\det(A_{N}-uI)|]^{2}}.$
(2.15)

In the case where $\tau=1$ and $|u|<2$ it follows from the results of [17]
that this quantity grows linearly in $N$.

On the other hand, Lemmas 2.4 and 2.5 show that for any $\rho>0$ and $u\geq-
E_{0;p,\tau}$, we have that

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}((-\rho,\rho)^{c},(-\infty,u))])<\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]^{2}).$
(2.16)

We also show below (see Remark 6.2) that $\Sigma^{p,\tau}$ is strictly
increasing on $(-\infty,0]$ and strictly decreasing on $[0,\infty)$, so for
$\epsilon>0$ and $0\geq u\geq-E_{0;p,\tau}$

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}((-\infty,u-\epsilon))^{2}])<\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}((u-\epsilon,u))^{2}]),$
(2.17)

and for $u>0$

$\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}((-\infty,u)-(-\epsilon,\epsilon))^{2}])<\lim_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N}((-\epsilon,\epsilon))^{2}]).$
(2.18)

In particular, we see that the dominant contribution to
$\mathbb{E}[\mathcal{N}_{N}((-\infty,u)^{2})]$ comes from pairs of equilibria
which are roughly orthogonal and which have roughly the same Lagrangian
multiplier. It is for these points that we compute the contribution up to a
vanishing term.

######  Proposition 2.6.

Let us assume that $\tau\in(0,1)$, and $p>9$ and let us fix $u\in\mathbb{R}$
such that $u\neq\pm E_{\infty;p,\tau}$. Then for any choice of $\rho_{N}>0$
and $a_{N}<b_{N}$ such that $\rho_{N}\to 0$ and $a_{N},b_{N}\to u$

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,2}((-\rho_{N},\rho_{N}),(a_{N},b_{N}))]}{\mathbb{E}[\mathcal{N}_{N}((a_{N},b_{N}))]^{2}}=1.$
(2.19)

Finally we will need the following result, shown in Section 5, which shows
that the dominant contribution to $\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]$
for $u<-E_{\infty;p,\tau}$ comes from stable points.

######  Lemma 2.7.

Assume $\tau\in(0,1)$. Then for any $u<-E_{\infty;p,\tau}$, we have that

$\limsup_{N\to\infty}N^{-1}\log\left(\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))-\mathcal{N}_{N,st}((-\infty,u))]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]}\right)<0.$
(2.20)

Before discussing the proof of Proposition 2.6, we first derive Theorem 1.5 as
a consequence.

###### Proof of Theorem 1.5.

We first prove (1.14). By Jensen’s inequality it suffices to show that

$\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]^{2}}\leq
1.$ (2.21)

We will show this in that case $-E_{0;p,\tau}<u\leq 0$, with the other case of
$u>0$ being similar. As the left-hand side of (2.16) is increasing in $\rho$,
while the right-hand side if decreasing, we may choose $\rho_{N}\to 0$ such
that

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,2}((-\rho_{N},\rho_{N})^{c},(-\infty,u))]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}=0.$
(2.22)

Similarly, by (2.17) we may choose $\epsilon_{N}\to 0$ such that

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u-\epsilon_{N}))^{2}]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}=0.$
(2.23)

Combining these statements sees that the left-hand side of (2.21) is bounded
by

$\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,2}((-\rho_{N},\rho_{N}),(u-\epsilon_{N},u))^{2}]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}\leq\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,2}((-\rho_{N},\rho_{N}),(u-\epsilon_{N},u))^{2}]}{\mathbb{E}[\mathcal{N}_{N}((u-\epsilon_{N},u))^{2}]}.$
(2.24)

Thus Proposition 2.6 completes the proof of (2.21).

To prove (1.15) we note it again suffices to show

$\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))^{2}]}{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))]^{2}}\leq
1.$ (2.25)

By Lemma 2.7 when $u<-E_{\infty;p,\tau}$ we have that

$\lim_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]}{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))]}=1.$
(2.26)

Thus we have

$\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))^{2}]}{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))]^{2}}\leq\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}{\mathbb{E}[\mathcal{N}_{N,st}((-\infty,u))]^{2}}=\limsup_{N\to\infty}\frac{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))^{2}]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u))]^{2}},$
(2.27)

and so (2.25) follows from (2.21). ∎

The focus of the remainder of the section will be the proof of Proposition
2.6. We denote the ratio of the absolute determinants as

$\Delta_{N}(r,u_{1},u_{2}):=\frac{\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N}(r,u_{1},u_{2}))|]}{\prod_{k=1,2}\mathbb{E}\left[|\det(A_{N-1}-\hat{u}_{N,k}I)|\right]},$
(2.28)

where here $A_{N-1}$ is sampled from $GEE(N-1,\tau)$. By expanding the
expressions in Lemmas 2.1 and 2.2 around $r=0$, we obtain the following
result, proven in Section 5.

######  Lemma 2.8.

For any choice of $u\in\mathbb{R}$, $\rho_{N}>0$ and $a_{N}<b_{N}$ such that
$\rho_{N}\to 0$ and $a_{N},b_{N}\to u$, we have that

$\frac{\mathbb{E}[\mathcal{N}_{N,2}((-\rho_{N},\rho_{N}),(a_{N},b_{N}))]}{\mathbb{E}[\mathcal{N}_{N}((a_{N},b_{N}))]^{2}}\leq$
(2.29)
$(1+o(1))\sup_{a_{N}<u_{1},u_{2}<b_{N}}\left(\sqrt{\frac{N}{2\pi}}\int_{(-\rho_{N},\rho_{N})}e^{-N(\frac{r^{2}}{2}+o(r^{3}))}\Delta_{N}(r,u_{1},u_{2})dr\right).$
(2.30)

This reduces us to producing estimates for the size of
$\Delta_{N}(r,u_{1},u_{2})$. The first of these will give us a bound on (2.5)
up to a polynomial order in $N$.

######  Lemma 2.9.

Fix $\tau\in(0,1)$, even $\ell$, $u_{1},u_{2}\in\mathbb{R}$, $r\in(-1,1)$ and
$k=1,2$. Then there is $C$ such that if $|u_{1}|,|u_{2}|<E_{\infty;p,\tau}$

$\frac{\mathbb{E}[|\det(\bar{M}_{N}^{k}(r,u_{1},u_{2}))|^{\ell}]}{\mathbb{E}[|\det(A_{N-1}-\hat{u}_{k,N}I)|]^{\ell}}\leq
CN^{\ell(\ell-1)/4},$ (2.31)

and if $|u_{1}|,|u_{2}|>E_{\infty;p,\tau}$

$\frac{\mathbb{E}[|\det(\bar{M}_{N}^{k}(r,u_{1},u_{2}))|^{\ell}]}{\mathbb{E}[|\det(A_{N-1}-\hat{u}_{k,N}I)|]^{\ell}}\leq
C.$ (2.32)

Moreover, these bounds are uniform over compact subsets of
$(r,u_{1},u_{2})\in(-1,1)\times(\mathbb{R}-\\{\pm E_{\infty;p,\tau}\\})^{2}$.

The proof of Lemma 2.9, given in Section 5, is primarily reliant on the
following bounds for the absolute characteristic polynomial of the elliptic
ensemble in the bulk.

######  Theorem 2.10.

Fix $\tau\in(0,1)$, $\ell\in\mathbb{N}$, and $x\in\mathbb{R}$. Then there is
$C$ such that if $|x|<1+\tau$

$\frac{\mathbb{E}[|\det(A_{N}-xI)|^{\ell}]}{\mathbb{E}[|\det(A_{N}-xI)|]^{\ell}}\leq
CN^{\ell(\ell-1)/4},$ (2.33)

and such that if $|x|>1+\tau$

$\frac{\mathbb{E}[|\det(A_{N}-xI)|^{\ell}]}{\mathbb{E}[|\det(A_{N}-xI)|]^{\ell}}\leq
C.$ (2.34)

Moreover, these bounds are uniform in compact subsets of
$\mathbb{R}-\\{\pm(1+\tau)\\}$.

In the bulk, this result follows from Proposition 8.1 below, which is the
focus of Section 8. Outside of the bulk, this follows from Proposition 7.11,
which is a main result of Section 7.

By the Cauchy-Schwarz inequality, we see under the conditions of Lemma 2.9
there is $C$ such that

$\Delta_{N}(r,u_{1},u_{2})\leq\frac{(\mathbb{E}[|\det(\bar{M}_{N}^{1}(r,u_{1},u_{2}))|^{2}]\mathbb{E}[|\det(\bar{M}_{N}^{2}(r,u_{1},u_{2}))|^{2}])^{1/2}}{\mathbb{E}[|\det(A_{N-1}-\hat{u}_{1,N}I)|]\mathbb{E}[|\det(A_{N-1}-\hat{u}_{2,N}I)|]}\leq
C\sqrt{N}.$ (2.35)

In particular, if we denote
$J_{N}:=(-\rho_{N},\rho_{N})-(-C\sqrt{\log(N)/N},C\sqrt{\log(N)/N})$ and fix
$\epsilon>0$ we have that

$\int_{J_{N}}e^{-N(\frac{r^{2}}{2}-\epsilon|r|^{3})}\Delta_{N}(r,u_{1},u_{2})dr\leq
C\sqrt{N}\int_{J_{N}}e^{-N(\frac{r^{2}}{2}-\epsilon|r|^{3})}dr\leq$ (2.36)
$2|\rho_{N}|CN^{1/2-C/2}e^{CN(\frac{\log(N)}{N})^{3/2}},$ (2.37)

which goes to zero if $C>1$. So we are reduced to considering the case where
$|r_{N}|<C\sqrt{\log(N)/N}$, which is given by the following result.

######  Lemma 2.11.

Assume $p>9$ and $\tau\in(0,1)$, and fix choices of $C>0$ and
$u\in\mathbb{R}-\\{\pm E_{\infty;p,\tau}\\}$. Then for any $a_{N}<b_{N}$ with
$a_{N},b_{N}\to u$ we have that

$\limsup_{N\to\infty}\sup_{a_{N}<u_{1},u_{2}<b_{N}}\sup_{|r|<C\sqrt{\log(N)/N}}\Delta_{N}(r,u_{1},u_{2})\leq
1.$ (2.38)

This lemma will also be proven in Section 5. This result, combined with Lemma
2.8 and (2.37) completes the proof of Proposition 2.6.

##  3 Proof of Lemma 2.1

In this section, we will give a proof of Lemma 2.1. This will follow readily
from a direct application of the Kac-Rice formula, as well as a variety of
computations about the covariance structure of F and its derivatives, which
will be given in Appendix A. The exact form of the Kac-Rice formula we will
apply is given by Lemma B.1 below, which is proven in Appendix B. This is a
modification of the Kac-Rice formula for vector fields given in Chapter 6 of
[7] to the case of a general Riemannian manifold. It is important to note
that, even restricted to the relaxational case, this requires fewer non-
degeneracy conditions than the corresponding formula of [2], which simplifies
the number of conditions we need to verify.

To begin, we perform a normalization on F to simplify the factors which appear
in our analysis. For each $\ell$, let $S^{\ell}:=S^{\ell}(1).$ We define for
$x\in S^{N-1}$

$\textbf{g}(x):=\textbf{f}(\sqrt{N}x),\;\;\textbf{G}(x):=\textbf{F}(\sqrt{N}x),\;\;\zeta(x):=\lambda(\sqrt{N}x).$
(3.1)

We observe that g, G and $\zeta$ are, respectively, a vector-valued function,
vector field, and function on $S^{N-1}$. The zero sets of G and F are related
up to a rescaling, so it will suffice to prove a similar result for G. Indeed
we have that

$\mathcal{N}_{N}(B)=\\#\\{\sigma\in S^{N-1}:\nabla\textbf{G}(x)=0,\zeta(x)\in
B\\},$ (3.2)

and similarly for $\mathcal{N}_{N,2}(B,I)$, so it will suffice to work with
$(\zeta,\textbf{g},\textbf{G})$ instead of $(\lambda,\textbf{f},\textbf{F})$.

Given a (piecewise) smooth orthogonal frame field $E$ on $S^{N-1}$, we define

$\textbf{G}_{E}(x):=((\textbf{G}(x),E_{i}(x)))_{i=1}^{N-1},\;\;\;\nabla_{E}\textbf{G}(x):=(E_{i}(\textbf{G}(x),E_{j}(x)))_{i,j=1}^{N-1},$
(3.3)

where the inner-product here is the standard Riemannian metric on $S^{N-1}$
inherited from the standard embedding into $\mathbb{R}^{N}$. Lastly, we denote
for $r\in[-1,1]$

$\textbf{n}(r)=(0,\dots 0,\sqrt{1-r^{2}},r)\in S^{N-1},$ (3.4)

and the north pole as $\textbf{n}=\textbf{n}(1)$.

We now give the result of a direct application of the Kac-Rice formula, after
which we will simplify the remaining terms to obtain Lemma 2.1.

######  Lemma 3.1.

Let $E$ be an arbitrary (piecewise) smooth orthonormal frame field on
$S^{N-1}$. For any nice $B\subseteq\mathbb{R}$ and nice $I\subseteq(-1,1)$,

$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]=\tilde{C}_{N}\int_{I}(1-r^{2})^{\frac{N-3}{2}}\varphi_{\textbf{G}_{E}(\textbf{n}),\textbf{G}_{E}(\textbf{n}(r))}(0,0)$
(3.5)
$\times\mathbb{E}\big{[}|\det\left(\frac{\nabla_{E}\textbf{G}(\textbf{n})}{\sqrt{(N-1)p(p-1)}}\right)||\det\left(\frac{\nabla_{E}\textbf{G}(\textbf{n}(r))}{\sqrt{(N-1)p(p-1)}}\right)|\times$
(3.6) $I(\zeta(\textbf{n}),\zeta(\textbf{n}(r))\in
B)\big{|}\textbf{G}_{E}(\textbf{n})=\textbf{G}_{E}(\textbf{n}(r))=0\big{]}dr,$
(3.7)

where here $\varphi_{\textbf{G}_{E}(\sigma),\textbf{G}_{E}(\sigma^{\prime})}$
is the density of the random vector
$(\textbf{G}_{E}(\sigma),\textbf{G}_{E}(\sigma^{\prime}))$ at zero and

$\tilde{C}_{N}=\mathrm{vol}(S^{N-1})\mathrm{vol}(S^{N-2})((N-1)p(p-1))^{N-1}.$
(3.8)

Before proving this, we note that the law of the function
$(\zeta,\textbf{g}):S^{N-1}\to\mathbb{R}^{N+1}$ in isotropic in the following
sense: for $R\in O(N)$ we have the distributional equality

$(\zeta,\textbf{g})\stackrel{{\scriptstyle d}}{{=}}(\zeta\circ
R,R^{-1}\textbf{g}\circ R)$ (3.9)

where $\textbf{g}\circ R(x)=\textbf{g}(R(x))$ denotes precomposition by the
rotation $R$. In particular, the same holds for G.

###### Proof of Lemma 3.1.

We define $S^{2}_{N}(I)=\\{(x,y)\in S^{N-1}\times S^{N-1}:(x,y)\in I\\}$,
considered as an open Riemannian submanifold of $S^{N-1}\times S^{N-1}$. In
the notation of Lemma B.1, $\mathcal{N}_{N,2}(B,I)=N_{h,k}(B)$ with vector
field on $S^{2}_{N}(I)$ given by $h(x,y)=(\textbf{G}(x),\textbf{G}(y))$, where
the decomposition is with respect to the identification
$T_{(x,y)}S^{2}_{N}(I)=T_{x}S^{N-1}\oplus T_{y}S^{N-1}$, and
$k(x,y)=(\zeta(x),\zeta(y))$.

One may produce an orthonormal frame field on $\bar{E}$ on $S^{2}_{N}(I)$, by
acting on each coordinate by $E$, which satisfies

$\nabla_{\bar{E}}h(x,y)=\begin{bmatrix}\nabla_{E}\textbf{G}(x)&0\\\
0&\nabla_{E}\textbf{G}(y)\end{bmatrix}$ (3.10)

where the block structure is with respect to the above decomposition of
$T_{(x,y)}S^{2}_{N}(I)$.

We note that if $r\neq\pm 1$, then by Lemma 3.2 below
$(\textbf{G}(\textbf{n}),\textbf{G}(\textbf{n}(r)))$ is non-degenerate.
Employing the above invariance, we see that the vector
$(\textbf{G}(x),\textbf{G}(y))$ is non-degenerate as long as $x\neq\pm y$. In
particular, we may apply Lemma B.1 to obtain that for any orthonormal frame
field on $S^{N-1}$

$\displaystyle\mathbb{E}[\mathcal{N}_{N,2}(B,I)]=\int_{S^{2}_{N}(I)}\varphi_{\textbf{G}_{E}(x),\textbf{G}_{E}(y)}(0,0)\mathbb{E}\big{[}|\det(\nabla_{E}\textbf{G}(x))||\det(\nabla_{E}\textbf{G}(y))|\times$
(3.11) $\displaystyle I(\zeta(x),\zeta(y)\in
B)\big{|}\textbf{G}_{E}(x)=\textbf{G}_{E}(y)=0\big{]}\omega_{S^{N-1}}(dx)\omega_{S^{N-1}}(dy),$
(3.12)

where $\omega_{S^{N-1}}$ is the standard surface measure on $S^{N-1}$. By the
above invariance, the integrand of (3.12) is invariant under rotation of
$S^{N-1}$. By the co-area formulae (see as well pg. 10 of [61]) for any
isotropic integrable function on $S^{N-1}\times S^{N-1}$

$\int_{S^{2}_{N}(I)}f(x,y)\omega_{S^{N-1}}(dx)\omega_{S^{N-1}}(dy)=\mathrm{vol}(S^{N-1})\mathrm{vol}(S^{N-2})\int_{I}(1-r^{2})^{\frac{N-3}{2}}f(\textbf{n},\textbf{n}(r))dr.$
(3.13)

Applying this identity to (3.12) completes the proof. ∎

We now need the following results which describe the various terms appearing
in (3.7), beginning with the following expression for the conditional density.

######  Lemma 3.2.

For any $r\in(-1,1)$, there is an orthonormal frame field
$E=(E_{i})_{i=1}^{N-1}$ such that the following holds: The Gaussian vector
$(\textbf{G}_{E}(\textbf{n}),\textbf{G}_{E}(\textbf{n}(r)))$ is non-
degenerate, with density at the origin given by

$\varphi_{\textbf{G}_{E}(\textbf{n}),\textbf{G}_{E}(\textbf{n}(r))}(0,0)=(2\pi
p)^{-(N-1)}\left(1-r^{2p-2}\right)^{-\frac{N-2}{2}}(1-(r^{p}-\tau(p-1)r^{p-2}(1-r^{2}))^{2})^{-1/2}.$
(3.14)

Moreover, conditional on the event
$(\textbf{G}_{E}(\textbf{n})=\textbf{G}_{E}(\textbf{n}(r))=0)$,
$\sqrt{N}(\zeta(\textbf{n}),\zeta(\textbf{n}(r)))$ is a non-degenerate
centered Gaussian vector, with $N$-independent covariance matrix
$\Sigma_{U}(r)$, whose inverse is given by (A.62) below.

The proof of this result is given in Appendix A. We will now describe the law
of the conditional Jacobian, which we will break into two statements, the
first giving the result after one conditions on the event that the values of G
vanish, and the second after conditioning additionally on the values of
$\zeta$. This is done as the first is more convenient for the proof of Theorem
2.3, while in the proof of Proposition 2.6 we will find it convenient to
employ the latter.

######  Lemma 3.3.

For $r\in(-1,1),$ and with the same choice of $(E_{i})_{i=1}^{N-1}$ as in
Lemma 3.2, the following holds: Conditional on the event

$(\textbf{G}_{E}(\textbf{n})=0,\textbf{G}_{E}(\textbf{n}(r))=0)$ (3.15)

let us denote the law of

$\left(\frac{\nabla_{E}\textbf{G}(\textbf{n})}{\sqrt{(N-1)p(p-1)}},\frac{\nabla_{E}\textbf{G}(\textbf{n}(r))}{\sqrt{(N-1)p(p-1)}},\zeta(\textbf{n}),\zeta(\textbf{n}(r))\right)$
(3.16)

as

$(M^{1}_{N}(r),M^{2}_{N}(r),U^{1}_{N}(r),U^{2}_{N}(r)).$ (3.17)

For $k=1,2$, let us define

$\hat{U}^{k}_{N}(r):=\left(\sqrt{\frac{N}{p(p-1)(N-1)}}\right)U^{k}_{N}(r),$
(3.18)

and write $M^{k}_{N}(r)$ in terms of square $(N-1)$-by-$1$ blocks as

$M^{k}_{N}(r)=\begin{bmatrix}H^{k}_{N}(r)&W^{k}_{N}(r)\\\
V^{k}_{N}(r)^{t}&S^{k}_{N}(r)\\\ \end{bmatrix}-\hat{U}^{k}_{N}(r)I.$ (3.19)

Then
$(H^{k}_{N}(r),W^{k}_{N}(r),V^{k}_{N}(r),S^{k}_{N}(r),U^{k}_{N}(r))_{k=1,2}$
is a centered Gaussian random vector whose covariance satisfies the following
properties:

  * •

The variables $(S^{k}_{N}(r),U^{k}_{N}(r))_{k=1,2}$,
$(V^{k}_{N}(r),W^{k}_{N}(r))_{k=1,2}$, and $(H^{k}_{N}(r))_{k=1,2}$ are
independent.

  * •

We have the distributional equality

$\sqrt{\frac{N-1}{N-2}}\begin{bmatrix}H^{1}_{N}(r)\\\
H^{2}_{N}(r)\end{bmatrix}\stackrel{{\scriptstyle
d}}{{=}}\begin{bmatrix}\sqrt{1-|r|^{p-2}}\hat{G}^{1}_{N}+\mathrm{Sign}(r)^{p}\sqrt{|r|^{p-2}}G_{N}\\\
\sqrt{1-|r|^{p-2}}\hat{G}^{2}_{N}+\sqrt{|r|^{p-2}}G_{N}\end{bmatrix},$ (3.20)

where $\hat{G}^{1}_{N},\hat{G}^{2}_{N},G_{N}$ are i.i.d matrices sampled from
$GEE(\tau,N-2)$.

  * •

The vectors $(V^{k}_{N}(r)_{i},W^{k}_{N}(r)_{i})_{k=1,2}$ for $1\leq i<N-1$
are i.i.d and for fixed $i$

$\sqrt{N-1}(V^{1}_{N}(r)_{i},W^{1}_{N}(r)_{i},V^{2}_{N}(r)_{i},W^{2}_{N}(r)_{i})$
(3.21)

has an $N$-independent covariance function given by the matrix $\Sigma_{Z}(r)$
defined in (A.68).

  * •

$(\sqrt{N-1}S^{k}_{N}(r),\sqrt{N}U^{k}_{N}(r))_{k=1,2}$ has an $N$-independent
covariance function.

######  Lemma 3.4.

Fix $r\in(-1,1)$, and $u_{1},u_{2}\in\mathbb{R}$. With notation as in Lemma
3.4, let us denote

$(\bar{M}^{1}_{N}(r,u_{1},u_{2}),\bar{M}^{2}_{N}(r,u_{1},u_{2}))=\left(M^{1}_{N}(r),M^{2}_{N}(r)\right)\big{|}(U^{1}_{N}(r)=u_{1},U^{2}_{N}(r)=u_{2}).$
(3.22)

For $k=1,2$, let us write $\bar{M}^{k}_{N}(r,u_{1},u_{2})$ in terms of square
$(N-1)$-by-$1$ blocks as

$\bar{M}^{k}_{N}(r,u_{1},u_{2})=\begin{bmatrix}\bar{G}^{k}_{N}(r)&\bar{W}^{k}_{N}(r)\\\
\bar{V}^{k}_{N}(r)^{t}&\bar{S}^{k}_{N}(r,u_{1},u_{2})\\\
\end{bmatrix}-\hat{u}_{N,k}I.$ (3.23)

Then
$(\bar{G}^{k}_{N}(r),\bar{W}^{k}_{N}(r),\bar{V}^{k}_{N}(r),\bar{S}^{k}_{N}(r,u_{1},u_{2}))_{k=1,2}$
is the Gaussian vector determined by the following properties:

  * •

$(\bar{G}^{k}_{N}(r),\bar{W}^{k}_{N}(r),\bar{V}^{k}_{N}(r))_{k=1,2}$ is
independent of $(\bar{S}^{k}_{N}(r,u_{1},u_{2}))_{k=1,2}$ and coincides in law
with the random variable $(H^{k}_{N}(r),W^{k}_{N}(r),V^{k}_{N}(r))_{k=1,2}$
given in Lemma 3.3.

  * •

For $1\leq k,l\leq 2$

$(N-1)\mathbb{E}[\bar{S}^{k}_{N}(r,u_{1},u_{2})]=m^{k}_{1}(r)\hat{u}_{N,1}+m^{k}_{2}(r)\hat{u}_{N,2},$
(3.24)
$(N-1)\mathbb{E}[\bar{S}^{k}_{N}(r,u_{1},u_{2})\bar{S}^{l}_{N}(r,u_{1},u_{2})]=[\Sigma_{S}(r)]_{k,l},$
(3.25)

where $m^{k}_{l}$ and $\Sigma_{S}(r)$ are given by (A.79) and (A.78), and
finally $[\Sigma_{S}(r)]_{k,k}>0$ for $k=1,2$.

The proof of these lemmas will also be given in Appendix A. The proof of Lemma
2.1 follows immediately from Lemmas 3.1, 3.2, and 3.3.

We remark that while the omission of the notation $(u_{1},u_{2})$ in various
factors in Lemma 3.4 is an abuse of notation, we have chosen this omission for
terms that are unaffected by this conditioning, which will significantly
simplify notation in latter sections.

##  4 Proof of Theorem 2.3

The primary purpose of this section is to give the proof of Theorem 2.3, which
is our computation of $\mathbb{E}[\mathcal{N}_{N,2}(I,B)]$ up to a
subexponential term. This will follow analysis of the terms occurring in Lemma
2.1, with the primary obstacle being the terms involving the absolute
determinants.

As outlined above, our method will be to rely on general concentration
results, but this introduces an important technical distinction between the
relaxational and non-relaxational cases, as we are not able to directly use
the empirical spectral measure. The reason for this is that the eigenvalues of
a general matrix are significantly less stable under perturbation than in the
symmetric case (see, for example, the discussion in [14]).

Instead we must rely on singular values, and in particular the following
relationship which holds for any $n$-by-$n$ matrix $A$

$|\det(A-zI)|=\prod_{i=1}^{n}s_{i}(A-zI)=\exp(n\int_{0}^{\infty}\log(x)\nu_{A,z}(dx)).$
(4.1)

The measures $\nu_{A,z}$ satisfy few relations for distinct $z$, unlike in the
case of eigenvalues where the corresponding measures are just translations of
the empirical spectral measure. This complicates the analysis of $\nu_{A,z}$,
and in particular their uniformity, which is explored further in Section 7.

After making this replacement though, treating the perturbation term
$E^{k}_{N}(r,u_{1},u_{2})$ will be possible through the following
deterministic matrix inequalities. To state them, let $\epsilon>0$ and define
the functions

$h_{\epsilon}(x)=\max(x,\epsilon),\;\;\;\log_{\epsilon}(x)=\log(h_{\epsilon}(x)).$
(4.2)

######  Lemma 4.1.

Let $M$ be an $n$-by-$n$ matrix which may be written in terms of square
$(n-1)$-by-$1$ blocks as

$M=\begin{bmatrix}H&W\\\ V^{t}&S\end{bmatrix}.$ (4.3)

Then for any $\epsilon>0$

$|\det(M)|\leq\epsilon^{-1}\sqrt{\|W\|^{2}+S^{2}}\left(\|V\|+\epsilon\right)\prod_{i=1}^{n-1}h_{\epsilon}(s_{i}(H)),$
(4.4) $|\det(M)|\geq|\det(H)||S|(1+\|V\|/s_{n}(M))^{-1}.$ (4.5)

The proof of this result is delayed till the end of this section. We now note
that

$\prod_{i=1}^{n}h_{\epsilon}(s_{i}(A-zI))=\exp\left(\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A,z}(dx)\right).$
(4.6)

We will show in Section 7 that
$\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx)$ satisfies a sub-
exponential concentration estimate (see Lemma 7.5). Applying this, as well as
some tightness results for the elliptic law, we will prove in Section 7 the
following result.

######  Lemma 4.2.

For $|\tau|<1$, and any choice of $\ell,K>0$ we have that

$\lim_{\epsilon\to
0}\limsup_{N\to\infty}\sup_{|z|<K}\left|(N\ell)^{-1}\log(\mathbb{E}[\exp(\ell
N\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))])-\phi_{\tau}(z)\right|=0.$
(4.7)

In addition, for $0<\epsilon<1$ we have that

$\limsup_{N\to\infty}\sup_{|z|\geq
1}\left((N\ell)^{-1}\log(\mathbb{E}[\exp(\ell
N\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))])-(2+\log(|z|))\right)\leq
0.$ (4.8)

With these preliminaries stated, we now prepare for the proof of the upper
bound of Theorem 2.3. From Lemma 4.1 and (4.1) we see that for any
$\epsilon>0$ we have that

$|\det(M^{k}_{N}(r))|\leq\epsilon^{-1}\sqrt{\|W^{k}_{N}(r)\|^{2}+S^{k}_{N}(r)^{2}}\left(\|V^{k}_{N}(r)\|+\epsilon\right)\times$
(4.9)
$\exp\left((N-2)\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{H^{k}_{N}(r),\hat{U}_{k,N}(r)}(dx)\right).$
(4.10)

We recall that $\sqrt{\frac{N-2}{N-1}}H^{k}_{N}(r)$ is distributed as
$GEE(\tau,N-2)$. Thus Lemma 4.2 implies that for any $1>\delta>0$ and $\ell>0$

$\limsup_{\epsilon\to
0}\limsup_{N\to\infty}\sup_{u\in\mathbb{R}}\left(\frac{1}{N\ell}\log(\mathbb{E}[\exp(\ell(N-2)\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{H^{k}_{N}(r),u}(dx))])-\phi_{\tau}^{\delta}(u)\right)\leq
0,$ (4.11)

where here

$\phi_{\tau}^{\delta}(u)=\begin{cases}\phi_{\tau}(u);\;\;\;|u|\leq\delta^{-1}\\\
2+\log(|u|);\;\;\;|u|>\delta^{-1}\end{cases}.$ (4.12)

The following result allows us to control the remaining terms of (4.10),
particularly as $r\to 1$.

######  Lemma 4.3.

For any $m\in\mathbb{N}$ there is $C$ such that for $r\in(-1,1)$

$\mathbb{E}\left[\|W^{k}_{N}(r)\|^{2m}\right]^{1/2m}\leq
C\sqrt{(1-r^{2})},\;\;\;\mathbb{E}\left[|S^{k}_{N}(r)|^{2m}\right]^{1/2m}\leq
C\sqrt{(1-r^{2})},$ (4.13)
$\mathbb{E}\left[\|V^{k}_{N}(r)\|^{2m}\right]^{1/2m}\leq C.$ (4.14)

###### Proof of Lemma 4.3.

Let $(X_{i})_{i=1}$ denote a family of i.i.d standard Gaussian random
variables. Then by Lemma 3.3 we have that

$\mathbb{E}\left[\|W^{k}_{N}(r)\|^{2m}\right]^{1/2m}=\sigma_{2}(r)^{1/2}\mathbb{E}\left[\left(\frac{1}{N-1}\sum_{i=1}^{N-2}X_{i}^{2}\right)^{m}\right]^{1/2m},$
(4.15)

where $\sigma_{2}(r)$ is the covariance function of any entry of
$\sqrt{N-1}W^{k}_{N}(r)$. An expression for this is derived in Appendix A,
namely in equation (A.86). We observe that

$\mathbb{E}[(\sum_{i=1}^{N-2}X_{i}^{2})^{m}]^{1/2m}=\mathbb{E}[(\chi_{N-2}^{2})^{m}]^{1/2m},$
(4.16)

where $\chi_{N-2}^{2}$ denotes a $\chi^{2}$-random variable of parameter
$(N-2)$. Recalling that for $k>3$ (see [57])

$\mathbb{E}[(\chi_{k}^{2})^{m}]=(k-1)(k+1)\cdots(k-3+2m)\leq(k+2m)^{m}\leq(2m)^{m}k^{m},$
(4.17)

we see that for $N>5$

$\mathbb{E}[\|W^{k}_{N}(r)\|^{2m}]^{1/2m}\leq\sigma_{1}(r)^{1/2}(2m)^{1/2}.$
(4.18)

Thus to show the desired claim for $W^{k}_{N}(r)$ it suffices to show there is
$C$ such that $\sigma_{1}(r)\leq C\sqrt{1-r^{2}}$, which is among the
statements of Lemma A.4. A similar argument demonstrates the claim for
$S^{k}_{N}(r)$ and $V^{k}_{N}(r)$. ∎

With these preliminaries completed we are ready to give the proof of the upper
bound of Theorem 2.3.

###### Proof of the upper bound of Theorem 2.3.

Our goal is to show that

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(B,I)])\leq\sup_{r\in
I,u_{1},u_{2}\in B}\Sigma_{2}^{p,\tau}(r,u_{1},u_{2}).$ (4.19)

Let us fix $m\geq 2$ and $\epsilon>0$ and denote $\ell=\ell(m)=m/(m-1)$. By
(4.10) and Hölder’s inequality we have that

$\mathbb{E}\left[\prod_{k=1,2}|\det(M^{k}_{N}(r))|I(U^{k}_{N}(r)\in
B)\right]\leq\mathcal{F}_{N,\epsilon,m}(r)\mathcal{E}_{N,\epsilon,m}(r),$
(4.20)

where here

$\mathcal{E}_{N,\epsilon,m}(r):=\mathbb{E}\left[\prod_{k=1,2}\exp\left(\ell(N-2)\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{H^{k}_{N}(r),\hat{U}_{k,N}(r)}(dx)\right)I(U^{k}_{N}(r)\in
B)\right]^{1/\ell},$ (4.21)
$\mathcal{F}_{N,\epsilon,m}(r):=\epsilon^{-1}\prod_{k=1,2}\mathbb{E}[(\|W^{k}_{N}(r)\|^{2}+S^{k}_{N}(r)^{2})^{m}(\|V^{k}_{N}(r)\|+\epsilon)^{2m}]^{1/2m}.$
(4.22)

We observe as an immediate corollary of Lemma 4.3 and the Cauchy-Schwarz
inequality there is $C:=C(\epsilon,m)$ such that for $r\in(-1,1)$

$\mathcal{F}_{N,\epsilon,m}(r)\leq C\sqrt{1-r^{2}}.$ (4.23)

Noting as well that $\mathcal{G}(r)\leq C(1-r^{2})^{-1}$ we see by Lemma 2.1

$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]\leq
CC_{N}\int_{I}\left(\frac{1-r^{2}}{1-r^{2p-2}}\right)^{(N-2)/2}\mathcal{E}_{N,m,\epsilon}(r)\frac{dr}{\sqrt{1-r^{2}}}.$
(4.24)

Moreover by Hölder’s inequality, and possibly increasing $C$, this is less
than

$CC_{N}\left(\int_{I}\left(\frac{1-r^{2}}{1-r^{2p-2}}\right)^{\ell(N-2)/2}\mathcal{E}_{N,m,\epsilon}(r)^{\ell}\frac{dr}{\sqrt{1-r^{2}}}\right)^{1/\ell}$
(4.25)

Conditioning on $(U^{1}_{N}(r),U^{2}_{N}(r))$, we see by (4.11) that for any
fixed $1>\delta>0$, sufficiently small $\epsilon$ and large $N$, we have that

$\mathcal{E}_{N,m,\epsilon}(r)\leq
e^{N\delta}\mathbb{E}\left[\prod_{k=1,2}\exp\left(\ell(N-2)\phi_{\tau}^{\delta}(\hat{U}_{k,N})\right)I(U^{k}_{N}(r)\in
B)\right]^{1/\ell}.$ (4.26)

With this we define the integral

$I_{N,\ell,\epsilon,\delta}:=\int_{I}\exp(\ell
Nh(r))\mathbb{E}[\prod_{k=1,2}\exp\left(\ell
N\phi_{\tau}^{\delta}(\hat{U}^{k}_{N+2}(r))\right)I(U^{k}_{N+2}(r)\in
B)]\frac{dr}{\sqrt{1-r^{2}}},$ (4.27)

so that

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(B,I)])\leq\delta+\limsup_{N\to\infty}N^{-1}\log(C_{N})+\limsup_{N\to\infty}(N\ell)^{-1}\log(I_{N-2,\ell,\epsilon,\delta}).$
(4.28)

We note that by Stirling’s approximation

$\lim_{N\to\infty}N^{-1}\log(C_{N})=\log(p-1)+1.$ (4.29)

Thus we need only obtain asymptotics for $I_{N,\ell,\epsilon,\delta}$, which
will follow by Varadhan’s Lemma (see Theorem 4.3.1 and Lemma 4.3.6 of [24]).
Let us denote by $(X_{1},X_{2})$ a pair of standard Gaussian random variables
and $R$ an independent random variable on $[-1,1]$ with density
$\pi^{-1}(1-r^{2})^{-1/2}$. We define

$X_{N,i}=(N+2)^{-1/2}X_{i},\;\;\hat{X}_{N,i}=((N+1)p(p-1))^{-1/2}X_{i}.$
(4.30)

and observe the distributional equality

$(U^{1}_{N+2}(r),U^{2}_{N+2}(r))\stackrel{{\scriptstyle
d}}{{=}}\left(X_{1,N},X_{2,N}\right)\Sigma_{U}(r)^{1/2},$ (4.31)

and where $\Sigma_{U}(r)$ is defined in (A.62). A similar result holds for
$\hat{U}^{k}_{N+2}$. If we further denote

$W:=\\{(u_{1},u_{2},r)\in\mathbb{R}^{2}\times
I:(u_{1},u_{2})\Sigma_{U}(r)^{1/2}\in B^{2}\\},$ (4.32)

then we see that

$I_{N,\ell,\epsilon,\delta}=\mathbb{E}\big{[}\exp\left(\ell
N[h(R)+\sum_{k=1,2}\phi_{\tau}^{\delta}\left(\left[(\hat{X}_{1,N},\hat{X}_{2,N})\Sigma_{U}(R)^{1/2}\right]_{k}\right)]\right)\times$
(4.33) $I\left((X_{1,N},X_{2,N},R)\in W\right)\big{]}.$ (4.34)

We now observe that the random variable
$(\hat{X}_{1,N},\hat{X}_{2,N},X_{1,N},X_{2,N},R)$ satisfies a LDP on
$\mathbb{R}^{4}\times[-1,1]$ with good rate function

$J(v_{1},v_{2},u_{1},u_{2},r)=\begin{cases}\frac{u_{1}^{2}}{2}+\frac{u_{2}^{2}}{2};\;\;\;v_{1}=\sqrt{p(p-1)}u_{1},\;v_{2}=\sqrt{p(p-1)}u_{2}\\\
\infty;\;\;\;\;\text{otherwise}\end{cases}.$ (4.35)

To apply Varadhan’s Lemma, we will need to first verify a moment condition.
For this we note that there is $C>0$ such that for $x\in\mathbb{R}$

$\phi^{\delta}_{\tau}(x)\leq C(1+\log(1+|x|)).$ (4.36)

As the entries of $\Sigma_{U}(r)$ are uniformly bounded in $r\in[-1,1]$, we
see that there is $C$ such that for any
$(u_{1},u_{2},r)\in\mathbb{R}^{2}\times[-1,1]$

$\sum_{k=1,2}\phi_{p,\tau}^{\delta}([(u_{1},u_{2})\Sigma_{U}(r)^{1/2}]_{k})\leq
C(1+\log(1+\|(u_{1},u_{2})\|)).$ (4.37)

Observing as well that $h(r)$ is bounded on $[-1,1]$, we see that there is $C$
such for any $\ell^{\prime}>0$

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\exp(\ell^{\prime}N[h(R)+\sum_{k=1,2}\phi_{\tau}^{\delta}\left(\left[(\hat{X}_{1,N},\hat{X}_{2,N})\Sigma_{U}(R)^{1/2}\right]_{k}\right)]\leq$
(4.38)
$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\exp(\ell^{\prime}CN(1+2\log(1+\|(X_{1},X_{2})\|/\sqrt{N})))])<\infty.$
(4.39)

With this moment condition verified, we may now apply Varadhan’s Lemma to
conclude that

$\limsup_{N\to\infty}N^{-1}\log(I_{N,\ell,\epsilon,\delta})\leq$ (4.40)
$\sup_{(u_{1},u_{2},r)\in W}\ell
h(r)-\frac{u_{1}^{2}}{2}-\frac{u_{2}^{2}}{2}+\ell\sum_{k=1,2}\phi_{p,\tau}^{\delta}([(u_{1},u_{2})\Sigma_{U}(R)^{1/2}]_{k})=$
(4.41) $\sup_{(u_{1},u_{2})\in B_{\epsilon}(B),r\in I}\ell
h(r)-\frac{1}{2}(u_{1},u_{2})\Sigma_{U}(r)^{-1}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}+\ell\phi_{p,\tau}^{\delta}(u_{1})+\ell\phi_{p,\tau}^{\delta}(u_{2}).$
(4.42)

Thus taking $m\to\infty$ (so $\ell\to 1$) then $\epsilon\to 0$, and finally
$\delta\to 0$ we see that equations (4.28), (4.29) and (4.42) complete the
proof of (4.19). ∎

With the proof of the upper bound completed, we now proceed to the proof of
the lower bound. Our lower bound on the determinant, as with our upper bound,
will come from Lemma 4.1. Unlike in the case of the upper bound though in this
case, our lower bound is dependent on the lowest eigenvalue of the matrices
$M^{k}_{N}(r)$. This will make it convenient to begin working with the
matrices $\bar{M}^{k}_{N}(r,u_{1},u_{2})$, defined in Lemma 3.4. The primary
tool will be the following result.

######  Lemma 4.4.

For $u_{1},u_{2}\in\mathbb{R}$, $K>0$, and $r\in(-1,1)$ we have that

$\liminf_{N\to\infty}\inf_{\delta,\delta_{1},\delta_{2}\in[-K,K]}N^{-1}\log(\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N}(r+\delta/N,u_{1}+\delta_{1}/N,u_{2}+\delta_{2}/N))])\geq$
(4.43) $\phi_{\tau,p}(u_{1})+\phi_{\tau,p}(u_{2}).$ (4.44)

###### Proof of Lemma 4.4.

Let us take $(r^{N},u_{1}^{N},u_{2}^{N})$ which minimizes the infimum on the
left-hand side of (4.44) and omit this choice from the notation for the
remainder of the proof. It suffices to show that

$\liminf_{N\to\infty}N^{-1}\log(\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N})|])\geq\phi_{\tau,p}(u_{1})+\phi_{\tau,p}(u_{2}).$
(4.45)

By Lemma 4.1 we see that a.s.

$|\det(\bar{M}^{k}_{N})|\geq|\det(\bar{H}^{k}_{N})|(1+\|\bar{W}^{k}_{N}\|/s_{N-1}(\bar{M}^{k}_{N}))^{-1}|\bar{S}^{k}_{N}|.$
(4.46)

For $\epsilon>0$, we define the event

$\mathcal{E}_{\text{small}}:=\\{s_{N-1}(\bar{M}^{k}_{N})>\epsilon
N^{-1},k=1,2\\}.$ (4.47)

By Lemma 7.8, so we may choose $\epsilon>0$ sufficiently small that

$\liminf_{N\to\infty}\mathbb{P}(\mathcal{E}_{\text{small}})\geq 1/2.$ (4.48)

We further denote the event

$\mathcal{E}_{\text{edge}}:=\\{|\bar{S}^{k}_{N}|\geq
N^{-2},\|\bar{V}^{k}_{N}\|\leq N^{2}\\}.$ (4.49)

We recall that $\sqrt{(N-1)}\bar{S}^{k}_{N}$ is a non-degenerate Gaussian of
fixed variance. By this, Lemma 4.3 and Markov’s inequality, we see that
$\mathbb{P}(\mathcal{E}_{\text{edge}})\to 1$ as $N\to\infty$. Note that
conditionally on $\mathcal{E}_{\text{small}}\cap\mathcal{E}_{\text{edge}}$ and
for large enough $N$

$(1+\|\bar{V}^{k}_{N}\|/s_{N-1}(\bar{M}^{k}_{N}))^{-1}|\bar{S}^{k}_{N}|\geq(1+\epsilon
N^{3})^{-1}N^{-2}\geq N^{-6}.$ (4.50)

Now we must estimate $\prod_{k=1,2}|\det(\bar{H}^{k}_{N})|$. By Lemma 3.4, we
see that

$\bar{H}^{k}_{N}\stackrel{{\scriptstyle d}}{{=}}A_{N-2}-\hat{u}_{N,k}I,$
(4.51)

where $A_{N-2}$ is sampled from $GEE(N-2,\tau)$. Thus by Lemma 4.2 and
Markov’s inequality, for any $\delta>0$ (to be chosen later), if denote

$\mathcal{E}_{\text{log}}:=\bigg{\\{}\int_{0}^{\infty}\log_{\delta}(x)\nu_{\bar{H}^{k}_{N}}(dx)\geq\phi_{\tau,p}(u_{k})-\delta,\;\;k=1,2\bigg{\\}},$
(4.52)

then we have that $\mathbb{P}(\mathcal{E}_{\text{log}})\to 1$ as $N\to\infty$.
Conditional on $\mathcal{E}_{\text{log}}$ we have that

$(N-2)^{-1}\log(\prod_{k=1,2}|\det(\bar{H}^{k}_{N})|)=(N-2)^{-1}\log(\prod_{k=1,2}\prod_{i=1}^{N-2}s_{i}(\bar{H}^{k}_{N}))\geq$
(4.53)
$(N-2)^{-1}\sum_{k=1,2}\sum_{i=1}^{N-2}\log\left(\frac{s_{i}(\bar{H}^{k}_{N})}{h_{\delta}(\bar{H}^{k}_{N})}\right)+\phi_{\tau,p}(u_{1})+\phi_{\tau,p}(u_{2})-2\delta.$
(4.54)

To bound the final sum, we note that by Lemma 7.12 there is $c>0$ such that if
we denote

$\mathcal{E}_{\text{med}}:=\\{s_{N-i}(\bar{H}^{k}_{N})\geq ciN^{-1}\text{ for
all }N^{1/2}\leq i\leq N-1\text{ and }k=1,2\\},$ (4.55)

we have $\mathbb{P}(\mathcal{E}_{\text{med}})\to 1$. Moreover on the event
$\mathcal{E}_{\text{med}}$, for sufficiently large $N$ (and assuming
$\epsilon/c<1$)

$\sum_{i=1}^{N-2}\log(s_{i}(\bar{H}^{k}_{N}))\geq[N^{1/2}]\log(\epsilon
N^{-1})+\sum_{i=[N^{1/2}]+1}^{[\epsilon N]}\log(ciN^{-1})+\sum_{i=[\epsilon
N]+1}^{N-1}\log_{\epsilon/c}(s_{N-i}(\bar{H}^{k}_{N}))\geq$ (4.56)
$[N^{1/2}]\log(\epsilon N^{-1})+\sum_{i=1}^{[\epsilon
N]}\log(ciN^{-1})-[\epsilon
N]+\sum_{i=1}^{N-1}\log_{\epsilon/c}(s_{i}(\bar{H}^{k}_{N})).$ (4.57)

We note

$\sum_{i=1}^{[\epsilon N]}\log(ciN^{-1})\geq
N\int_{0}^{\epsilon}\log(cx)dx=N\epsilon\log(c)+N\epsilon(\log(\epsilon)-1).$
(4.58)

Thus if we take $\delta=\epsilon/c$, then on
$\mathcal{E}_{\text{log}}\cap\mathcal{E}_{\text{med}}$

$\sum_{i=1}^{N-2}\log\left(\frac{s_{i}(\bar{H}^{k}_{N})}{h_{\delta}(\bar{H}^{k}_{N})}\right)\geq[N^{1/2}]\log(\epsilon
N^{-1})+N(\epsilon\log(c)+\epsilon(\log(\epsilon)-1))-[\epsilon
N]\log(\epsilon/c)$ (4.59)

Let us denote the right-hand side as $K_{N}(\epsilon)$. Noting that $c$ is
independent of $\epsilon$, we see that

$\lim_{\epsilon\to 0}\lim_{N\to\infty}N^{-1}K_{N}(\epsilon)=0.$ (4.60)

Combining these we see that if we denote
$\mathcal{E}:=\mathcal{E}_{\text{med}}\cap\mathcal{E}_{\text{small}}\cap\mathcal{E}_{\text{med}}\cap\mathcal{E}_{\text{log}}$
then

$N^{-1}\log(\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N})|I_{\mathcal{E}}])\geq
N^{-1}\log(N^{-6}\mathbb{E}[\prod_{k=1,2}|\det(\bar{H}^{k}_{N})|I_{\mathcal{E}}])\geq$
(4.61)
$\phi_{\tau,p}(u_{1})+\phi_{\tau,p}(u_{2})-2\epsilon/c+2N^{-1}K_{N}(\epsilon)+N^{-1}\log(\mathbb{P}(\mathcal{E}))-6N^{-1}\log(N).$
(4.62)

Finally, noting that $\liminf_{N\to\infty}\mathbb{P}(\mathcal{E})\geq 1/2$,
and sending $N\to\infty$ and then $\epsilon\to 0$ completes the proof. ∎

We are now ready to provide the proof of the lower bound of Theorem 2.3.

###### Proof of the lower bound of Theorem 2.3.

It suffices to fix $r^{*}\in I$, and $u_{1}^{*},u_{2}^{*}\in B$, and show that

$\Sigma_{2}^{p,\tau}(r^{*},u_{1}^{*},u_{2}^{*})\leq\liminf_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(I,B)]).$
(4.63)

We will omit this choice from the notation as before. We denote

$R_{N}:=(r^{*},r^{*}+N^{-1})\times(u_{1}^{*},u_{1}^{*}+N^{-1})\times(u_{2}^{*},u_{2}^{*}+N^{-1}),$
(4.64)

and define

$\bar{\varphi}_{N}(r,u_{1},u_{2}):=\frac{1}{\sqrt{(2\pi)^{2}\det(\Sigma_{U}(r))}}\exp\left(-\frac{N}{2}(u_{1},u_{2})\Sigma_{U}(r)^{-1}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}\right)$ (4.65)

which is the density of the Gaussian vector $(U^{1}_{N}(r),U^{2}_{N}(r))$ at
$(u_{1},u_{2})$. Then choosing $N$ sufficiently large that $R_{N}\subseteq
I\times B^{2}$, we see that

$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]\geq$ (4.66)
$C_{N}\int_{R_{N}}e^{(N-2)h(r)}\mathcal{G}(r)\bar{\varphi}_{N}(r,u_{1},u_{2})\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N}(r))|]drdu_{1}du_{2}\geq$
(4.67) $C_{N}N^{-3}\inf_{(r,u_{1},u_{2})\in
R_{N}}e^{(N-2)h(r)}\mathcal{G}(r)\bar{\varphi}_{N}(r,u_{1},u_{2})\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}^{k}_{N}(r))|].$
(4.68)

We see that

$\lim_{N\to\infty}N^{-1}\log\left(\inf_{(r,u_{1},u_{2})\in
R_{N}}e^{(N-2)h(r)}\mathcal{G}(r)\bar{\varphi}_{N}(r,u_{1},u_{2})\right)=h(r^{*})-\frac{1}{2}(u_{1}^{*},u_{2}^{*})\Sigma_{U}(r^{*})^{-1}\begin{bmatrix}u_{1}^{*}\\\
u_{2}^{*}\end{bmatrix}.$ (4.69)

Employing this, Lemma 4.4 and the asymptotics for the constant term given in
(4.29), we obtain (4.68). ∎

This completes the proof of Theorem 2.3. We complete this section with proof
of Lemma 4.1.

###### Proof of Lemma 4.1.

Let us denote

$Y=\begin{bmatrix}H&0\\\ V^{t}&0\end{bmatrix}.$ (4.70)

The difference $M-Y$ has a single nonzero singular value given by
$\sqrt{\|W\|^{2}+S^{2}}$. We note as well that $s_{n}(Y)=0$. Thus by applying
Corollary 7.2 we see that

$|\det(M)|\leq\sqrt{\|W\|^{2}+S^{2}}\prod_{i=1}^{n-1}s_{i}(Y).$ (4.71)

We observe that

$\prod_{i=1}^{n-1}s_{i}(Y)^{2}=\lim_{\delta\to
0}\frac{1}{\delta}\det(Y^{t}Y+\delta I)=\lim_{\delta\to
0}\det(H^{t}H+VV^{t}+\delta I)=\det(H^{t}H+VV^{t}).$ (4.72)

By applying Corollary 7.2 again we see that

$|\det(H^{t}H+VV^{t})|\leq(s_{n-1}(H)+\|V\|)\prod_{i=1}^{n-2}s_{i}(H)\leq\epsilon^{-1}(\|V\|+\epsilon)\prod_{i=1}^{n-1}h_{\epsilon}(s_{i}(H)).$
(4.73)

Where in the last inequality we have used that for $x,y\in(0,\infty)$

$x+y\leq h_{\epsilon}(x)\epsilon^{-1}(\epsilon+y).$ (4.74)

Together these statements establish (4.4). To establish (4.5) we define

$Z=\begin{bmatrix}H&W\\\ 0&S\end{bmatrix}.$ (4.75)

Applying Corollary 7.2 again we see that

$|\det(H)S|=|\det(Z)|\leq(s_{n}(M)+\|V\|)\prod_{i=1}^{n-1}s_{i}(M)=|\det(M)|(1+\|V\|/s_{n}(M)).$
(4.76)

∎

##  5 Proof of Lemmas 2.8, 2.9, 2.7 and 2.11

In this section we will give the results needed to prove Proposition 2.6,
which involve the computation of the second moment at the constant order
scale. We will begin by giving the proof of Lemmas 2.9 and 2.11, and then
proceed to the proofs of Lemmas 2.7 and 2.8.

To begin we need the following basic concentration result, whose proof we
defer to the end of the section.

######  Lemma 5.1.

Fix a Gaussian pair $(X,Y)\sim\mathcal{N}(0,\Sigma)$. For each $N$, let
$N^{-1/2}(X_{N},Y_{N})\in\mathbb{R}^{N}\times\mathbb{R}^{N}$ be the random
vector whose entries are i.i.d copies of $(X,Y)$. Let
$f:\mathbb{R}^{3}\to\mathbb{R}$ be any continuous function of polynomial
growth. Then we have that

$\lim_{N\to\infty}\mathbb{E}[f(\|X_{N}\|,\|Y_{N}\|,(X_{N},Y_{N})^{2})]=f(\Sigma_{11},\Sigma_{22},\Sigma_{12}^{2}).$
(5.1)

We now proceed to the proof of Lemma 2.9. As in the proof of Lemma 8 in [64]
in the relaxational case, we first separate the perturbation term by taking a
cofactor expansion, and then simplify these terms by using rotation
invariance. In the symmetric case, rotation invariance is quite strong, and
this gives a direct recurrence in terms of the characteristic polynomials of
smaller size. In our case however, we instead rely on direct comparison of the
terms in this expansion for both $\bar{M}^{k}_{N}(r,u_{1},u_{2})$ and
$A_{N-1}-\hat{u}_{N,k}I$.

###### Proof of Lemma 2.9.

We will fix and omit the choice of $(k,r,u_{1},u_{2})$ from the notation for
clarity. Additionally, we denote $v=\hat{u}_{N,k}$. With this notation, the
decomposition of $\bar{M}_{N}$ in Lemma 3.4 in terms of square $(N-1)$-by-$1$
blocks is given as

$\bar{M}_{N}=\begin{bmatrix}\bar{H}_{N}&\bar{W}_{N}\\\
\bar{V}_{N}^{t}&\bar{S}_{N}\end{bmatrix}-vI.$ (5.2)

To compare this to $A_{N-1}$ sampled from $GEE(N-1,\tau)$, we will also write
this in terms of square $(N-1)$-by-$1$ blocks as

$A_{N-1}=\begin{bmatrix}\hat{H}_{N}&\hat{W}_{N}\\\
\hat{V}_{N}^{t}&\hat{S}_{N}\end{bmatrix}.$ (5.3)

By Lemma 3.4 we see that $\bar{H}_{N}\stackrel{{\scriptstyle
d}}{{=}}\hat{H}_{N}$, though the laws for the other elements are in general
different. To compare these we will expand both in similar terms starting with
$\bar{M}_{N}$. We first note that a.s.

$\det(\bar{M}_{N}-vI)=\det(\bar{H}_{N}-vI)\left(\bar{S}_{N}-v-\bar{V}_{N}^{t}(\bar{H}_{N}-vI)^{-1}\bar{W}_{N}\right).$
(5.4)

For convenience, let us denote $\bar{G}_{N}=(\bar{H}_{N}-vI)^{-1}$. Then using
the inequality $(x+y)^{\ell}\leq 2^{\ell}(x^{\ell}+y^{\ell})$, and noting that
$\bar{S}_{N}$ is independent of $(\bar{H}_{N},\bar{V}_{N},\bar{W}_{N})$, we
see that

$2^{-\ell}\mathbb{E}[|\det(\bar{M}_{N}-vI)|^{\ell}]\leq\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}]\mathbb{E}[|\bar{S}_{N}-v|^{\ell}]+\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|\bar{V}_{N}^{t}\bar{G}_{N}\bar{W}_{N}|^{\ell}].$
(5.5)

The first term is bounded by $C\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}]$, so
we focus on the second. Let us denote

$\bar{R}_{N}=(\bar{V}_{N},\bar{W}_{N}),\;\;\;\bar{Q}_{N}=\sqrt{\|\bar{V}_{N}\|^{2}\|\bar{W}_{N}\|^{2}-(\bar{V}_{N},\bar{W}_{N})^{2}}.$
(5.6)

As the law of $\bar{H}_{N}$ is invariant under orthogonal conjugation, and
$(\bar{V}_{N},\bar{W}_{N})$ is independent of $\bar{H}_{N}$, we see we may
choose $U\in O(N)$, independent of $\bar{H}_{n}$, such that

$U\bar{V}_{N}=(\|\bar{V}_{N}\|,0,\dots
0),\;\;\;\;U\bar{W}_{N}=\|\bar{V}_{N}\|^{-1}(\bar{R}_{N},\bar{Q}_{N},0,\dots
0).$ (5.7)

Applying $U$ and conjugation invariance of $\bar{H}_{N}$ (and thus
$\bar{G}_{N}$) we see that

$\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|\bar{V}_{N}^{t}\bar{G}_{N}\bar{W}_{N}|^{\ell}]=\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|(U\bar{V}_{N})^{t}\bar{G}_{N}U\bar{W}_{N}|^{\ell}]=$
(5.8)
$\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|[\bar{G}_{N}]_{11}\bar{R}_{N}+[\bar{G}_{N}]_{12}\bar{Q}_{N}|^{\ell}|]$
(5.9)

We will now use that $\ell$ is even, so that we may expand this using the
binomial theorem as

$\sum_{k=0}^{\ell}\binom{\ell}{k}\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|[\bar{G}]_{11}^{\ell-k}[\bar{G}]_{12}^{k}]\mathbb{E}[\bar{R}_{N}^{\ell-k}\bar{Q}_{N}^{k}].$
(5.10)

As $(\bar{V},\bar{W})\stackrel{{\scriptstyle d}}{{=}}(-\bar{V},\bar{W})$ we
see that the terms when $k$ is odd must vanish, so we may write (5.10) as

$\sum_{k=0}^{\ell/2}\binom{\ell}{2k}\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|[\bar{G}_{N}]_{11}^{\ell-2k}[\bar{G}_{N}]_{12}^{2k}]\bar{\Delta}_{N}^{k},\;\;\;\;\;\bar{\Delta}_{N}^{k}:=\mathbb{E}[\bar{R}_{N}^{\ell-2k}\bar{Q}_{N}^{2k}]$
(5.11)

Now the argument may be applied equally well to (5.3) to show that

$2^{-\ell}\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}|\hat{V}_{N}^{t}\hat{G}_{N}\hat{W}_{N}|^{\ell}]\leq\mathbb{E}[|\det(A_{N-1}-vI)|^{\ell}]+\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}]\mathbb{E}[(\hat{S}_{N}-u)^{\ell}]$
(5.12)
$\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}|\hat{V}_{N}^{t}\hat{G}_{N}\hat{W}_{N}|^{\ell}]=\sum_{k=0}^{\ell/2}\binom{\ell}{2k}\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}|[\hat{G}_{N}]_{11}^{\ell-2k}[\hat{G}_{N}]_{12}^{2k}]\hat{\Delta}_{N}^{k},$
(5.13)

where $\hat{G}_{N}$ and $\hat{\Delta}_{N}$ similarly. Now as
$\bar{H}_{N}\stackrel{{\scriptstyle d}}{{=}}\hat{H}_{N}$ we see that

$\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}|[\hat{G}_{N}]_{11}^{\ell-2k}[\hat{G}_{N}]_{12}^{2k}]=\mathbb{E}[|\det(\bar{H}_{N}-vI)|^{\ell}|[\bar{G}_{N}]_{11}^{\ell-2k}[\bar{G}_{N}]_{12}^{2k}].$
(5.14)

Moreover, as $\ell$ is even, we have that
$\bar{\Delta}_{N}^{k},\hat{\Delta}_{N}^{k}\geq 0$. Thus comparing the
determinants has essentially been reduced to comparing $\bar{\Delta}_{N}^{k}$
and $\hat{\Delta}_{N}^{k}$. By applying Lemma 5.1 we see that

$\lim_{N\to\infty}\hat{\Delta}_{N}^{k}=\tau^{\ell-2k}(1-\tau^{2})^{k},\;\;\;\lim_{N\to\infty}\bar{\Delta}_{N}^{k}\leq\|\Sigma_{Z}(r)\|_{\textbf{g}}^{2\ell}.$
(5.15)

We note that $\Sigma_{Z}(r)$ is bounded in $r$, and so as $\tau\in(0,1)$ we
may find $C>0$ such that

$\max_{1\leq k\leq\ell}\bar{\Delta}_{N}^{k}/\hat{\Delta}_{N}^{k}\leq C.$
(5.16)

Enlarging $C$ we may also assume that

$\mathbb{E}[|\bar{S}_{N}-v|^{\ell}]+\mathbb{E}[|\hat{S}_{N}-v|^{\ell}]+1\leq
C.$ (5.17)

Combing both of the expressions with the expression for $\det(\bar{M}_{N})$
and $\det(A_{N-1}-vI)$ we see that

$4^{-\ell}\mathbb{E}[|\det(\bar{M}_{N})|]\leq
C^{2}\left(\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}]+\mathbb{E}[|\det(A_{N-1}-vI)|^{\ell}]\right).$
(5.18)

The submatrix $\hat{H}_{N}$ is proportional to a matrix sampled from
$GEE(N-2,\tau)$, so that

$\mathbb{E}[|\det(\hat{H}_{N}-vI)|^{\ell}]=\left(\frac{N-2}{N-1}\right)^{\ell(N-2)/2}\mathbb{E}[|\det(A_{N-2}-\sqrt{(N-1)/(N-2)}vI)|^{\ell}].$
(5.19)

By applying Proposition 8.1 in the case where $|v|<E_{\infty;p,\tau}$ and
Proposition 7.11 in the case where $|v|>E_{\infty;p,\tau}$, we see that there
is $C$ such that

$\mathbb{E}[|\det(A_{N-2}-\sqrt{(N-1)/(N-2)}vI)|^{\ell}]\leq
C\mathbb{E}[|\det(A_{N-1}-vI)|^{\ell}].$ (5.20)

Applying this and Theorem 2.10 to (5.18) completes the proof. ∎

Before proceeding to the proof of Lemma 2.11, we will need the following
precise form of the decomposition given in (2.9).

######  Lemma 5.2.

Let us fix three i.i.d matrices sampled from $GEE(N-1,\tau)$ and denote them

$(A_{N-1}^{1},A_{N-1}^{2},A_{N-1}).$ (5.21)

Define for $r\in(-1,1)$ and $k=1,2$

$A_{N-1}^{k}(r):=\sqrt{1-|r|^{p-2}}A_{N-1}^{k}+\mathrm{Sign}(r)^{pk}\sqrt{|r|^{p-2}}A_{N-1}.$
(5.22)

Then one may form random matrices $(E_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}$,
correlated to $(A_{N-1},A_{N-1}^{1},A_{N-1}^{2})$, such that a.s.
$\mathrm{rank}(E_{N}^{k}(r,u_{1},u_{2}))\leq 2$ and which satisfy

$(\bar{M}_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}\stackrel{{\scriptstyle
d}}{{=}}(A_{N-1}^{k}(r)+E_{N}^{k}(r,u_{1},u_{2})-\hat{u}_{N,k}I)_{k=1,2}.$
(5.23)

Additionally there are $c,C>0$ such that

$\mathbb{P}\left(\|E_{N}^{k}(r,u_{1},u_{2})\|_{F}\geq
C(r^{p-3}+N^{-1/2}r^{p-4})^{1/2}\right)\leq Ce^{-cN^{1/2}},$ (5.24)

and moreover $c,C$ can be chosen so that this holds over any compact subset of
$(r,u_{1},u_{2})\in(-1,1)\times\mathbb{R}^{2}$.

The construction of $E_{N}^{k}(r,u_{1},u_{2})$ is given at the end of this
section, as is the proof of this lemma. It is primarily reliant on the
technical computations from Appendix A.

We now give the proof of Lemma 2.11. As this provides asymptotics of the
second moment up to vanishing order, we must quite carefully control the
interplay between the fluctuations of the determinant and the strength of the
correlations between the two matrices. The general strategy will be to
introduce a series of rare events, and show that away from these events the
determinant may be bounded by a perturbative argument using Corollary 7.2. We
then show that the contributions coming from each of these events are
negligible by estimating their probabilities, and then applying moment
estimates. The primary tool will be Lemma 2.9 as before.

###### Proof of Lemma 2.11.

As in the proof of Lemma 2.9, it suffices to show this for a fixed sequence
$(r_{N},u_{N}^{1},u_{N}^{2})$, such that $u_{N}^{1},u_{N}^{2}\to u$ and such
that $|r_{N}|<C\sqrt{\log(N)/N}$, which we will omit from the notation.
Through the proof we will take constants $C,c>0$, with $C$ assumed to be
large, and $c$ taken to be sufficiently small, and both of which are allowed
to become larger (or smaller) line by line, but always independently of $N$.

With the notation of Lemma 5.2, we first define the event

$\mathcal{E}_{\text{large}}=\\{s_{1}(A_{N-1})>C\\}\cup\bigcup_{k=1,2}\\{\|E_{N}^{k}\|_{F}>C|r_{N}|^{(p-3)/2}\\}\cup\\{s_{1}(A_{N-1}^{k})>C\\}.$
(5.25)

We also define for $k=1,2$ the events

$\mathcal{E}_{\text{small},k}=\\{s_{N-1}(A_{N-1}^{k}-\hat{u}_{N,k}I)<c|r_{N}|^{(p-3)/2-c}\\}.$
(5.26)

We noting that

$s_{N-1}(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)\geq
s_{N-1}(A_{N-1}^{k}-\hat{u}_{N,k}I)$ (5.27)
$-\left(1-(1-|r_{N}|^{p-2})^{1/2}\right)s_{1}(A_{N-1}^{k})-|r_{N}|^{(p-2)/2}s_{1}(A_{N-1}),$
(5.28)

we see that on
$\mathcal{E}_{\text{large}}^{c}\cap\mathcal{E}_{\text{small},k}^{c}$, and for
large enough $N$, that

$s_{N-1}(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)>c|r_{N}|^{(p-3)/2-c}.$ (5.29)

By Corollary 7.2 we have that for $k=1,2$

$|\det(\bar{M}_{N}^{k})|\leq|\det(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)|\left(1+\frac{\|E_{N}^{k}\|_{F}}{s_{N-1}(A_{N-1}^{k}(r)-\hat{u}_{N,k}I)}\right)^{2},$
(5.30)

Thus conditional on the event
$\mathcal{E}_{\text{large}}^{c}\cap\mathcal{E}_{\text{small},1}^{c}\cap\mathcal{E}_{\text{small},2}^{c}$
we have that

$|\det(\bar{M}_{N}^{k})|\leq|\det(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)|\left(1+C|r_{N}|^{c}\right)^{2}=|\det(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)|(1+o(1)).$
(5.31)

To further simplify the term on the right, we will introduce our final event.
Let us denote $K=[2/c]^{-1}$ and $\epsilon=K^{-1}$ so that $\epsilon<c$. We
define

$\mathcal{E}_{\text{medium}}=\bigcup_{k=1,2}\bigcup_{m=0}^{K-1}\\{s_{N-[N^{\epsilon
m}]+1}(A_{N-1}^{k}-\hat{u}_{N,k}I)>cN^{\epsilon m-1}\\}.$ (5.32)

We observe that for $1\leq m\leq K$, the number of $i$ such that
$N-[N^{\epsilon(m-1)}]<i\leq N-[N^{\epsilon m}]$ is less than $N^{\epsilon
m}$. Then applying Corollary 7.2 again we see that

$\frac{|\det(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)|}{|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|}\leq\prod_{i=1}^{N-1}\left(1+\frac{s_{i}(A_{N-1}^{k}(r_{N})-A_{N-1}^{k})}{s_{N-i}(A_{N-1}^{k}-\hat{u}_{N,k}I)}\right)$
(5.33)
$\prod_{i=1}^{N-1}\left(1+\frac{s_{1}(A_{N-1}^{k}(r_{N})-A_{N-1}^{k})}{s_{N-i}(A_{N-1}^{k}-\hat{u}_{N,k}I)}\right)\leq\prod_{m=0}^{K-1}\left(1+\frac{s_{1}(A_{N-1}^{k}(r_{N})-A_{N-1}^{k})}{s_{N-[N^{\epsilon
m}]+1}(A_{N-1}^{k}-\hat{u}_{N,k}I)}\right)^{N^{\epsilon(m+1)}}.$ (5.34)

We note that on $\mathcal{E}_{\text{large}}^{c}$ we have that

$s_{1}(A_{N-1}^{k}(r_{N})-A_{N-1}^{k})\leq\left(1-(1-|r_{N}|^{p-2}_{N})^{1/2}\right)s_{1}(A_{N-1}^{k})+|r_{N}|^{(p-2)/2}s_{1}(A_{N-1})\leq
C|r_{N}|^{(p-2)/2}.$ (5.35)

Thus conditional on the event
$\mathcal{E}_{\text{large}}^{c}\cap\mathcal{E}_{\text{medium}}^{c}\cap\mathcal{E}_{\text{small},k}^{c}$
we have that

$\frac{|\det(A_{N-1}^{k}(r_{N})-\hat{u}_{N,k}I)|}{|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|}\leq
C\left(1+C|r_{N}|^{1/2+c}\right)^{N^{\epsilon}}\times\prod_{m=1}^{K-1}(1+C|r_{N}|^{(p-2)/2}N^{1-\epsilon
m})^{N^{\epsilon(m+1)}}.$ (5.36)

As $|r_{N}|<C\sqrt{\log(N)/N}$, we see that the first term is $1+o(1)$.
Similarly, as $(p-2)/4>1+\epsilon$, we see that
$|r_{N}|^{(p-2)/2}N^{1+\epsilon}\to 0$, so that each term in the product on
the right is $1+o(1)$. Combining these claims we see that for $k=1,2$ and on
$\mathcal{E}_{\text{large}}^{c}\cap\mathcal{E}_{\text{medium}}^{c}\cap\mathcal{E}_{\text{small},1}^{c}\cap\mathcal{E}_{\text{small},2}^{c}$

$|\det(\bar{M}_{N}^{k})|\leq|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|(1+o(1)).$
(5.37)

In particular, we see that to prove both of our desired claims it is
sufficient to bound the contribution to the determinant coming from each of
these events.

To begin we will recalll some bounds on the probabilies for these events
coming from Section 7. By Lemma 7.6 we see that
$\mathbb{P}(s_{1}(A_{N-1})>C)\leq Ce^{-Nc}$, and similarly for $A_{N-1}^{k}$.
Combining this with the final claim of Lemma 5.2 we conclude that
$\mathbb{P}(\mathcal{E}_{\text{large}})\leq Ce^{-cN^{1/2}}$. By applying Lemma
7.12 we see as well that $\mathbb{P}(\mathcal{E}_{\text{medium}})\leq
Ce^{-cN^{\epsilon}}$. Finally by Corollary 7.9 for $\ell=1,2$ we have that

$\mathbb{P}(\mathcal{E}_{\text{small},\ell})\leq CNr^{(p-3)/2-c}_{N}.$ (5.38)

Now to first let us bound the contribution from an arbitrary event
$\mathcal{A}$. We see from the Cauchy-Schwarz inequality that

$\mathbb{E}\left[\prod_{k=1,2}|\det(\bar{M}_{N}^{k})|I_{\mathcal{A}}\right]\leq\mathbb{P}(\mathcal{A})^{1/2}\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}_{N}^{k})|^{2}]^{1/2}\leq\mathbb{P}(\mathcal{A})^{1/2}\prod_{k=1,2}\mathbb{E}\left[|\det(\bar{M}_{N}^{k})|^{4}\right]^{1/4}.$
(5.39)

Employing Lemma 2.9 we have that

$\prod_{k=1,2}\mathbb{E}[|\det(\bar{M}_{N}^{k})|^{4}]^{1/4}\leq
CN^{3/2}\prod_{k=1,2}\mathbb{E}[|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|].$ (5.40)

In particular, if $N^{3/2}\mathbb{P}(\mathcal{A})^{1/2}\to 0$ we have that

$\limsup_{N\to\infty}\left(\frac{\mathbb{E}[\prod_{k=1,2}|\det(\bar{M}_{N}^{k})|I_{\mathcal{A}}]}{\prod_{k=1,2}\mathbb{E}[|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|]}\right)=0.$
(5.41)

This holds for both $\mathcal{A}=\mathcal{E}_{\text{large}}$ and
$\mathcal{A}=\mathcal{E}_{\text{medium}}$ in either case, so all that remains
is to treat the case of $\mathcal{E}_{\text{small},\ell}$ for $\ell=1,2$. For
this note that for an event $\mathcal{A}$, which is only dependant on
$A_{N-1}^{1}$, we have that

$\mathbb{E}\left[\prod_{k=1,2}|\det(\bar{M}_{N}^{k})|I_{\mathcal{A}}\right]\leq\mathbb{E}[|\det(\bar{M}_{N}^{1})|^{2}]^{1/2}\mathbb{E}[|\det(\bar{M}_{N}^{2})|^{2}I_{\mathcal{A}}]^{1/2}=$
(5.42)
$\left(\prod_{k=1,2}\mathbb{E}[|\det(\bar{M}_{N}^{k})|^{2}]^{1/2}\right)\mathbb{P}(\mathcal{A})^{1/2}\leq
CN^{1/4}\mathbb{P}(\mathcal{A})^{1/2}\prod_{k=1,2}\mathbb{E}[|\det(A_{N-1}^{k}-\hat{u}_{N,k}I)|]$
(5.43)

where in the second to last equality we have employed the independence of
$A_{N-1}^{1}$ from $\bar{M}_{N}^{2}$. A similar inequality holds for events
only involving $A_{N-1}^{2}$. In particular, when the event only depends on a
single $A_{N-1}^{k}$, it suffices to show that
$N^{1/2}\mathbb{P}(\mathcal{A})\to 0$. In view of (5.38) this holds for
$\mathcal{E}_{\text{small},k}$ as $p>9$. ∎

With this done, we will now proceed to the proof of Lemma 2.8, which gives the
expansion of the expressions for the first and second moments around $r=0$.

###### Proof of Lemma 2.8.

By employing the expression for $\mathbb{E}[\mathcal{N}_{N,2}(B,I)]$ given in
Lemma 2.1 and the expression for $\mathbb{E}[\mathcal{N}_{N}(B)]$ given in
Lemma 2.2 we see that

$\frac{\mathbb{E}[\mathcal{N}_{N,2}(B,I)]}{\mathbb{E}[\mathcal{N}_{N}(B)]^{2}}=\bar{C}_{N}^{-2}C_{N}\int_{I}\left(\frac{1-r^{2}}{1-r^{2p-2}}\right)^{\frac{N-2}{2}}\mathcal{G}(r)R_{N}(r)dr,$
(5.44)
$R_{N}(r):=\frac{\mathbb{E}\big{[}\prod_{k=1,2}|\det(M_{N}^{k}(r))|I(U_{N}^{k}(r)\in
B)]}{\mathbb{E}[|\det(A_{N-1}-\hat{X}_{N}I)|I(X_{N}\in B)]^{2}}.$ (5.45)

We see that

$\bar{C}_{N}^{-2}C_{N}=\mathrm{vol}(S^{N-2})/\mathrm{vol}(S^{N-1}).$ (5.46)

By Stirling’s approximation we see that

$\mathrm{vol}(S^{N-2})/\mathrm{vol}(S^{N-1})=(1+O(N^{-1}))\sqrt{N/2\pi}.$
(5.47)

We also note that as $p\geq 3$, $\mathcal{G}(r)=1+O(r^{2})$. Thus (5.45) can
be rewritten as

$(1+O(N^{-1}))\sqrt{\frac{N}{2\pi}}\int_{(-\rho_{N},\rho_{N})}e^{-N(\frac{r^{2}}{2}+O(r^{3}))+O(r^{2})}R_{N}(r)dr.$
(5.48)

To compute $R_{N}(r)$ we recall that $\sqrt{N}(U_{N}^{1}(r),U_{N}^{2}(r))$ is
centered Gaussian vector with covariance matrix $\Sigma_{U}(r)$, whose inverse
is computed in (A.62). We note that $k_{2}(r)=O(r^{p})$,
$k_{1}(r)-b(r)=O(r^{2p-2})$, and $b(r)=1+O(r^{2p-2})$ so that

$\Sigma_{U}(r)^{-1}=\frac{1}{p\alpha b(r)}\begin{bmatrix}k_{1}(r)&k_{2}(r)\\\
k_{2}(r)&k_{1}(r)\end{bmatrix}=(p\alpha)^{-1}I+O(r^{p}).$ (5.49)

In particular the density of the event
$(U_{N}^{1}(r)=u_{1},U_{N}^{2}(r)=u_{2})$ is given by

$\frac{1}{\sqrt{(2\pi)^{2}\det(\Sigma_{U}(r))}}\exp\left(-\frac{N}{2}(u_{1},u_{2})\Sigma_{U}(r)^{-1}\begin{bmatrix}u_{1}\\\
u_{2}\end{bmatrix}\right)=$ (5.50)
$\frac{1+O(r^{p})}{\sqrt{(2\pi)^{2}(p\alpha)^{2}}}\exp\left(-\frac{N}{2p\alpha}(u_{1}^{2}+u_{2}^{2})(1+O(r^{p}))\right).$
(5.51)

So we see conditioning on the energy value in both the numerator and the
denominator in $R_{N}(r)$ that

$R_{N}(r)=e^{-NO(r^{3})}\frac{\int_{B\times
B}\mathbb{E}\big{[}\prod_{k=1,2}|\det(\bar{M}_{N}^{k}(r,u_{1},u_{2}))|]e^{-\frac{N}{2p\alpha}u_{k}^{2}}du_{1}du_{2}}{\int_{B\times
B}\prod_{k=1,2}\mathbb{E}[|\det(A_{N-1}-\hat{u}_{N,k}I)|]e^{-\frac{N}{2p\alpha}u_{k}^{2}}du_{1}du_{2}}.$
(5.52)

Inserting this result into (5.48) and taking the maximum value over
$(u_{1},u_{2})$ completes the proof. ∎

Now give a proof of Lemma 2.7, which is relatively straightforward given
Theorem 2.10.

###### Proof of Lemma 2.7.

For the duration of this proof we will adopt the notation

$\mathcal{N}_{N,unst}(B)=\mathcal{N}_{N}(B)-\mathcal{N}_{N,st}(B)$ (5.53)

for the total number of unstable points. As $\Sigma^{p,\tau}$ is strictly
increasing on $(-\infty,0)$ (see Remark 6.2) we see by Theorem 1.4 that

$\limsup_{N\to\infty}N^{-1}\log\left(\frac{\mathbb{E}[\mathcal{N}_{N}((u-1,u))]}{\mathbb{E}[\mathcal{N}_{N}((-\infty,u-1))]}\right)<0.$
(5.54)

In particular it suffices to show that

$\limsup_{N\to\infty}N^{-1}\log\left(\frac{\mathbb{E}[\mathcal{N}_{N,unst}((u-1,u))]}{\mathbb{E}[\mathcal{N}_{N,st}((u-1,u))]}\right)<0.$
(5.55)

By conditioning on $X_{N}$ in Lemma 2.2 we have that

$\mathbb{E}[\mathcal{N}_{N}((u-1,u))]=\bar{C}_{N}\sqrt{\frac{N}{p\alpha}}\int_{u-1}^{u}\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|]e^{-N\frac{v^{2}}{2p\alpha}}dv.$
(5.56)

A similar application of the Kac-Rice formula to the number of unstable points
yields

$\mathbb{E}[\mathcal{N}_{N,unst}((u-1,u))=\bar{C}_{N}\sqrt{\frac{N}{p\alpha}}\int_{u-1}^{u}\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|I(A_{N-1}\leq\hat{v}_{N}I)]e^{-N\frac{v^{2}}{2p\alpha}}dv.$
(5.57)

Taking the ratio these integrands we see that the left hand side of (5.55) is
bounded by

$\sup_{u-1<v<u}\frac{\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|I(A_{N-1}\leq\hat{v}_{N}I)]}{\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|]}\leq$
(5.58)
$\mathbb{P}(A_{N-1}\leq\hat{u}_{N}I)^{1/2}\sup_{u-1<v<u}\left(\frac{\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|^{2}]^{1/2}}{\mathbb{E}[|\det(A_{N-1}-\hat{v}_{N}I)|]}\right).$
(5.59)

By Lemma 7.6 we see that there are $C,c>0$ such that

$\mathbb{P}(A_{N-1}\leq\hat{u}_{N}I)\leq Ce^{-Nc}.$ (5.60)

Combining this with Theorem 2.10 completes the proof. ∎

We now give a proof of Lemma 5.1, which will be followed by basic
concentration estimates.

###### Proof of Lemma 5.1.

We note that

$\mathbb{E}[(X_{N},Y_{N})^{2}]=N^{-1}(N-1)\Sigma_{12}^{2}+N^{-1}(\Sigma_{12}^{2}+2\Sigma_{11}\Sigma_{22})=\Sigma_{12}^{2}+O(N^{-1}),$
(5.61)

and similarly $\mathbb{E}[(X_{N},Y_{N})^{4}]=\Sigma_{12}^{4}+O(N^{-1})$, so
that $\mathrm{Var}((X_{N},Y_{N})^{2})=O(N^{-1})$. Similarly we have that
$\mathrm{Var}(\|X_{N}\|^{2}),\mathrm{Var}(\|Y_{N}\|^{2})=O(N^{-1})$, so if we
define the event

$\mathcal{A}_{N}=\left\\{\min(|(X_{N},Y_{N})^{2}-\Sigma_{12}^{2}|,|\|X_{N}\|^{2}-\Sigma_{11}^{2}|,|\|Y_{N}\|^{2}-\Sigma_{22}^{2}|)>N^{-1/4}\right\\}$
(5.62)

Chebysev’s inequality shows that $\mathbb{P}(\mathcal{A}_{N})\to 0$ as
$N\to\infty$. We observe that

$\lim_{N\to\infty}\mathbb{E}[f(\|X_{N}\|^{2},\|Y_{N}\|^{2},(X_{N},Y_{N})^{2})I_{\mathcal{A}^{c}_{N}}]=f(\Sigma_{11}^{2},\Sigma_{22}^{2},\Sigma_{12}^{2}),$
(5.63)

and that

$\mathbb{E}[f(\|X_{N}\|^{2},\|Y_{N}\|^{2},(X_{N},Y_{N})^{2})I_{\mathcal{A}_{N}}]\leq\mathbb{P}(\mathcal{A}_{N})^{1/2}\mathbb{E}[f(\|X_{N}\|^{2},\|Y_{N}\|^{2},(X_{N},Y_{N})^{2})^{2}]^{1/2},$
(5.64)

so it suffices to show the final term is bounded in $N$. For this, note that
$|(X_{N},Y_{N})|\leq\|X_{N}\|\|Y_{N}\|$, so as $f^{2}$ is of polynomial
growth, it suffices to show that for each $\ell\in\mathbb{N}$

$\limsup_{N\to\infty}\left(\mathbb{E}[\|X_{N}\|^{2\ell}]+\mathbb{E}[\|Y_{N}\|^{2\ell}]\right)<\infty.$
(5.65)

Observing that $\|X_{N}\|^{2}\stackrel{{\scriptstyle d}}{{=}}\chi_{N}^{2}$,
where $\chi_{N}^{2}$ is a $\chi^{2}$-random variable, this follows from the
estimate above (4.17). ∎

We complete this section by providing the construction of
$E^{k}_{N}(r,u_{1},u_{2})$ and a proof of Lemma 5.2. To begin, we fix three
i.i.d matrices sampled from $GEE(N-1,\tau)$ and denote them
$A_{N-1}^{1},A_{N-1}^{2}$ and $A_{N-1}$. For $1\leq i\leq N-2$ and $k=1,2$ we
define

$\begin{bmatrix}Z_{N-1,i}^{k}(r)\\\
Z_{i,N-1}^{k}(r)\end{bmatrix}:=\left(\mathrm{Sign}(\Sigma^{1,2}_{Z}(r))^{k}|\Sigma^{1,2}_{Z}(r)|^{1/2}\begin{bmatrix}1&\tau\\\
\tau&1\end{bmatrix}^{-1/2}-\mathrm{Sign}(r)^{pk}\sqrt{|r|^{p-2}}\right)\begin{bmatrix}[A_{N-1}]_{(N-1)i}\\\
[A_{N-1}]_{i(N-1)}\end{bmatrix}+$ (5.66)
$\left((\Sigma^{1,1}_{Z}(r)-\Sigma^{1,2}_{Z}(r))^{1/2}\begin{bmatrix}1&\tau\\\
\tau&1\end{bmatrix}^{-1/2}-\sqrt{1-|r|^{p-2}}\right)\begin{bmatrix}[A_{N-1}^{k}]_{(N-1)i}\\\
[A_{N-1}^{k}]_{i(N-1)}\end{bmatrix}.$ (5.67)
$Z_{N-1,N-1}^{k}(r):=\left(\mathrm{Sign}([\Sigma_{S}(r)]_{12})^{k}[|\Sigma_{S}(r)|]_{12}^{1/2}-\mathrm{Sign}(r)^{pk}\sqrt{|r|^{p-2}}\right)[A_{N-1}]_{(N-1)(N-1)}+$
(5.68)
$\left(\left([\Sigma_{S}(r)]_{11}-[\Sigma_{S}(r)]_{12}\right)^{1/2}-\sqrt{1-|r|^{p-2}}\right)[A_{N-1}^{k}]_{(N-1)(N-1)}.$
(5.69)

We finally define an $(N-1)$-by-$(N-1)$ matrix $E_{N}^{k}(r,u_{1},u_{2})$ by
defining for $1\leq i\leq N-2$

$[E_{N}^{k}(r,u_{1},u_{2})]_{i(N-1)}=Z^{k}_{i,N-1}(r),\;\;\;[E_{N}^{k}(r,u_{1},u_{2})]_{(N-1)i}=Z^{k}_{N-1,i}(r),$
(5.70)
$[E_{N}^{k}(r,u_{1},u_{2})]_{(N-1)(N-1)}=m^{k}_{1}(r)\hat{u}_{N,1}+m^{k}_{2}(r)\hat{u}_{N,2}+Z^{k}_{N-1,N-1}(r),$
(5.71)

and letting all other entries be zero.

We first verify the claim (5.23). As both sides are Gaussian, it will suffices
to verify that the Gaussian vector
$(A_{N-1}^{k}(r)+E_{N}^{k}(r,u_{1},u_{2})-\hat{u}_{N,k}I)_{k=1,2}$ has the
mean and covariance structure specified by Lemma 3.4. The nontrivial claims
are those involving the covariance matrices of
$(V_{N}^{k}(r),W_{N}^{k}(r))_{k=1,2}$ and
$(S_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}$. For this, we note that for $1\leq i\leq
N-2$

$\begin{bmatrix}1&\tau\\\
\tau&1\end{bmatrix}^{-1/2}\begin{bmatrix}[A_{N-1}]_{(N-1)i}\\\
[A_{N-1}]_{i(N-1)}\end{bmatrix}\stackrel{{\scriptstyle
d}}{{=}}\mathcal{N}(0,I),$ (5.72)

and similarly for entries involving $A_{N-1}^{k}$. From this we see that the
covariance matrix of
$([A_{N-1}^{k}(r)+E_{N}^{k}(r,u_{1},u_{2})]_{i(N-1)},[A_{N-1}^{k}(r)+E_{N}^{k}(r,u_{1},u_{2})]_{(N-1)i})_{k=1,2}$
for $1\leq i\leq N-2$ is given by

$\begin{bmatrix}\Sigma^{1,1}_{Z}(r)&\Sigma^{1,2}_{Z}(r)\\\
\Sigma^{1,2}_{Z}(r)&\Sigma^{1,1}_{Z}(r)\\\ \end{bmatrix}.$ (5.73)

Thus recalling that $\Sigma_{Z}^{1,1}(r)=\Sigma_{Z}^{2,2}(r)$ and
$\Sigma_{Z}^{1,2}(r)=\Sigma_{Z}^{2,1}(r)$ verifies the first claim. The claim
for $(S_{N}^{k}(r,u_{1},u_{2}))_{k=1,2}$ is similar, which establishes (5.23).

To complete the proof of Lemma 5.2, we need to prove the statements about
$E_{N}^{k}(r,u_{1},u_{2})$. The claim involving the rank is obvious, so we
focus on the latter. The primary tool is the following result, proven in
Appendix A, which quantifies the rate at which $E_{N}^{k}(r,u_{1},u_{2})$
vanishes as $r\to 0$.

######  Lemma 5.3.

For $\epsilon>0$ there is $C>0$ such that for $|r|<1-\epsilon$, $k,l=1,2$, and
$1\leq i\leq N-2$

$\mathbb{E}[(Z_{N-1,i}^{k}(r))^{2}]\leq
CN^{-1}r^{p-3},\;\;\mathbb{E}[(Z_{i,N-1}^{k}(r))^{2}]\leq CN^{-1}r^{p-3}$
(5.74) $\mathbb{E}[(Z_{N-1,N-1}^{k}(r))^{2}]\leq
CN^{-1}r^{p-4},\;\;|m^{k}_{l}(r)|\leq Cr^{p-3}.$ (5.75)

###### Proof of Lemma 5.2.

As noted, we only need to show (5.24). We note that

$\|E^{k}_{N}(r,u_{1},u_{2})\|^{2}_{F}=\sum_{i=1}^{N-1}\left(Z_{i,N-1}^{k}(r)^{2}+Z_{N-1,i}^{k}(r)^{2}\right)+(m_{1}^{k}(r)\hat{u}_{N,1}+m_{2}^{k}(r)\hat{u}_{N,2}+Z^{k}_{N-1,N-1}(r))^{2}.$
(5.76)

By Lemma 5.3 and the independence of these elements, we see that for large
enough $C$, $\|E^{k}_{N}(r)\|^{2}_{F}$ is stochastically dominated by

$CN^{-1}\chi_{N-1}^{2}r^{p-3}+C(\hat{u}_{N,1}+\hat{u}_{N,2})^{2}r^{p-2}+CN^{-1}r^{p-4}X^{2}$
(5.77)

where $\chi^{2}_{N}$ is a $\chi^{2}$-random variable of parameter $N$ and $X$
is a standard Gaussian random variable. By appealing to Cramér’s Theorem, we
see that for $C>1$ there is $c>0$ such that
$\mathbb{P}(N^{-1}\chi_{N-1}^{2}\geq C)\leq Ce^{-cN}$. More simply note that
there are $c,C$ such that $\mathbb{P}(X^{2}\geq N^{1/2})\leq Ce^{-cN^{1/2}}$.
Combining these statements yields the final claim.

∎

##  6 Proofs of Lemmas 2.4 and 2.5

The purpose of this section is to give proofs of the main analytic lemmas
through which we understand the complexity functions. This will primarily
involve understanding the points which achieve their supremums, and as such
will primarily be shown through explicit but non-trivial calculus.

The results are analogs of results of [61, 64] for the case $\tau=1$. Our main
method to establish them for general $\tau$ will be to reduce to the cases of
$\tau=1$ and $\tau=-(p-1)^{-1}$, with the latter often being surprisingly
simple. We will begin with the following prerequisite for the proof of Lemma
2.4

######  Lemma 6.1.

Define $\Sigma_{2}^{p,\tau}(r,u):=\Sigma_{2}^{p,\tau}(r,u,u)$. Then if $B$ is
a connected open subset of either $(-\infty,-E_{\infty;p,\tau})$,
$(-E_{\infty;p,\tau},E_{\infty;p,\tau})$, or $(E_{\infty;p,\tau},\infty)$, we
have for any $r\in(-1,1)$ that

$\sup_{u_{1},u_{2}\in B}\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})\leq\sup_{u\in
B}\Sigma_{2}^{p,\tau}(r,u).$ (6.1)

###### Proof.

This will follow from studying the behavior of the function
$\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})$ in terms of the variable
$\nu=(u_{1}-u_{2})$. Let us first treat the case of
$B\subset(-\infty,-E_{\infty;p,\tau})$. Note that $\phi_{\tau}$ is concave on
$(-\infty,-1-\tau)$, as this set is outside of $\mathcal{E}_{\tau}$. Noting as
well that $\Sigma_{U}(r)$ is positive-definite by construction, we see that
$\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})$ is concave in $(u_{1},u_{2})$ in the
region $(-\infty,-E_{\infty;p,\tau})^{2}$, and in particular, it is concave in
the variable $\nu=u_{1}-u_{2}$, restricted to this region. Now as the function
$\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})$ is symmetric in $(u_{1},u_{2})$, we see
that $\frac{d}{d\nu}\Sigma_{2}^{p,\tau}(r,u,u)=0$, so that for
$u_{1},u_{2}\in(-\infty,-E_{\infty;p,\tau})$, we have that

$\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})\leq\Sigma_{2}^{p,\tau}\left(r,\frac{u_{1}+u_{2}}{2},\frac{u_{1}+u_{2}}{2}\right).$
(6.2)

This demonstrates the claim in the case that
$B\subset(-\infty,-E_{\infty;p,\tau})$, and an identical argument demonstrates
the claim for $B\subset(E_{\infty;p,\tau},\infty)$.

In the case where $B\subset(-E_{\infty;p,\tau},E_{\infty;p,\tau})$, we see
that it is sufficient to show for
$u_{1},u_{2}\in(-E_{\infty;p,\tau},E_{\infty;p,\tau})$ that

$\frac{d^{2}}{d\nu^{2}}\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})<0.$ (6.3)

We note that

$\frac{d^{2}}{d\nu^{2}}\Sigma_{2}^{p,\tau}(r,u_{1},u_{2})=-[\Sigma_{U}(r)^{-1}]_{11}-[\Sigma_{U}(r)^{-1}]_{22}+2[\Sigma_{U}(r)^{-1}]_{12}+$
(6.4)
$\frac{1}{p(p-1)}\phi_{\tau}^{\prime\prime}\left(\frac{u_{1}}{\sqrt{p(p-1)}}\right)+\frac{1}{p(p-1)}\phi_{\tau}^{\prime\prime}\left(\frac{u_{2}}{\sqrt{p(p-1)}}\right)=$
(6.5) $\frac{2}{p\alpha b(r)}(k_{1}(r)-k_{2}(r))+\frac{2}{p(p-1)\tau}$ (6.6)

where in the last step we have used (A.62). Thus by the positivity of $b$,
(6.3) is equivalent to the claim

$j_{\tau,p}(r):=\alpha b(r)-(1+\tau)(p-1)(k_{1}(r)-k_{2}(r))<0.$ (6.7)

We first show that for $|\tau|<1$ and $|r|<1$ we have that

$1-r^{2p-2}+\tau(p-1)r^{p-2}(1-r^{2})>0.$ (6.8)

For this, we observe that

$1-r^{2p-2}-(p-1)|r|^{p-2}(1-r^{2})=(1-r^{2})\bigg{(}\sum_{i=0}^{p-2}r^{2i}-(p-1)|r|^{p-2}\bigg{)}>0,$
(6.9)

as we have that

$\frac{1}{p-1}\sum_{i=0}^{p-2}r^{2i}>|r|^{p-2},$ (6.10)

by the AM-GM inequality applied to the sequence $\\{r^{2i}\\}_{i=0}^{p-2}$.
This establishes (6.8).

Now we observe that

$k_{1}(r)-k_{2}(r)=b(r)+r^{p}(1+r^{p-2})(1-r^{2p-2}+\tau(p-1)r^{p-2}(1-r^{2})),$
(6.11)
$b(r)=(1-r^{2p-2}-\tau(p-1)r^{p-2}(1-r^{2}))(1-r^{2p-2}+\tau(p-1)r^{p-2}(1-r^{2})).$
(6.12)

We have just shown that the common factor of these two terms is positive, so
it suffices to show that

$\bar{j}_{\tau,p}(r):=(1-r^{2p-2}+\tau(p-1)r^{p-2}(1-r^{2}))^{-1}j_{\tau,p}(r)=$
(6.13)
$-(p-2)(1-r^{2p-2}-\tau(p-1)r^{p-2}(1-r^{2}))-(1+\tau)(p-1)r^{p}(1+r^{p-2})<0.$
(6.14)

Now we observe that $\bar{j}_{\tau,p}(r)$ is linear in $\tau$, so it suffices
to verify this claim for $\tau\in\\{-(p-1)^{-1},1\\}$. For $\tau=-(p-1)^{-1}$
we have that

$\bar{j}_{-(p-1)^{-1},p}(r)=-(p-2)(1+r^{p-2})<0.$ (6.15)

The case of $\tau=1$ is treated by [64]. Specifically,
$-\bar{j}_{1,p}(r)=h(r)$, where $h$ is introduced in (2.10) of [64], which
they then show is positive. ∎

######  Remark 6.2.

We note that when $r=0$ we have that
$\Sigma_{2}^{p,\tau}(0,u_{1},u_{2})=\Sigma^{p,\tau}(u_{1})+\Sigma^{p,\tau}(u_{2})$.
In particular, the concavity in $u_{1}-u_{2}$ established in the above proof
shows that the function $\Sigma^{p,\tau}(u)$ is strictly concave for
$u\in(E_{\infty;p,\tau},\infty)$ and
$u\in(-E_{\infty;p,\tau},E_{\infty;p,\tau})$. As the function is even, this
implies it is strictly decreasing on $[0,E_{\infty;p,\tau})$. As in addition
$\phi_{\tau}^{\prime}(1+\tau)=1$, we see that

$(\Sigma^{p,\tau})^{\prime}(E_{\infty;p,\tau})=-\frac{(1+\tau)\sqrt{p(p-1)}}{p\alpha}+\frac{\phi_{\tau}^{\prime}(1+\tau)}{\sqrt{p(p-1)}}=-\frac{(p-2)}{\alpha\sqrt{p(p-1)}}<0,$
(6.16)

so that we see that in fact $\Sigma^{p,\tau}$ is strictly decreasing on
$(0,\infty)$. In addition, as the function is even, it must be strictly
increasing on $(-\infty,0)$.

###### Proof of Lemma 2.4.

Theorem 2.3 applied to $I\cap(-1,1)\subseteq I$ implies the lower bound, so
that it is sufficient to show that

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(I,B)])\leq\sup_{u\in
B,r\in I\cap(-1,1)}\Sigma_{2}^{p,\tau}(r,u).$ (6.17)

For this, we observe that as $\mathcal{N}_{N}(\\{\pm E_{\infty;p,\tau}\\})=0$
a.s, it suffices to treat the case where we can write
$B=\coprod_{i=1}^{m}B_{i}$ as a disjoint union of intervals which satisfy the
conditions of Lemma 6.1. We note as well that
$\mathbb{E}[\mathcal{N}_{N,2}(\\{\pm 1\\},B)]\leq
2\mathbb{E}[\mathcal{N}_{N}(B)]$. Using this and repeatedly applying the
inequality $(x+y)^{2}\leq 2(x^{2}+y^{2})$ we see that

$\mathbb{E}[\mathcal{N}_{N,2}(B,I)]\leq
2^{m}\sum_{i=1}^{m}\left(2\mathbb{E}[\mathcal{N}_{N}(B_{i})]+\mathbb{E}[\mathcal{N}_{N,2}((-1,1)\cap
I,B_{i})]\right).$ (6.18)

Applying Theorem 2.3 and Lemma 6.1 we see that

$\limsup_{N\to\infty}N^{-1}\log(\mathbb{E}[\mathcal{N}_{N,2}(I,B)])\leq\max_{i=1,\dots,m}(\sup_{u\in
B_{i}}\sup_{r\in(-1,1)\cap
I}\max(\Sigma_{2}^{p,\tau}(r,u),\Sigma^{p,\tau}(u)))=$ (6.19) $\sup_{u\in
B}\sup_{r\in(-1,1)\cap I}\max(\Sigma_{2}^{p,\tau}(r,u),\Sigma^{p,\tau}(u)).$
(6.20)

Now we note that $\Sigma^{p,\tau}_{2}(0,u)=2\Sigma^{p,\tau}(u)$, which is
positive for somewhere on $B$ by assumption. In particular,

$\sup_{u\in
B}\sup_{r\in(-1,1)}\max(\Sigma_{2}^{p,\tau}(r,u),\Sigma^{p,\tau}(u))=\sup_{u\in
B}\sup_{r\in(-1,1)}\Sigma_{2}^{p,\tau}(r,u),$ (6.21)

which completes the proof of (6.17). ∎

To prove Lemma 2.5, we need the following identification for the maximizers of
$\Sigma_{2}^{p,\tau}(r,u)$.

######  Lemma 6.3.

Let us define

$u_{th;p,\tau}:=\sqrt{(1+\tau)\frac{\log(p-1)(p-1)p\alpha}{(p-2)}}.$ (6.22)

Then we may extend $\Sigma_{2}^{p,\tau}(r,u)$, for fixed $u$, to a continuous
function of $r\in(-1,1]$, which we denote by $\bar{\Sigma}_{2}^{p,\tau}(r,u)$,
and such that:

  * •

For $u=|u_{th;p,\tau}|$ and $r\in(-1,1)-\\{0\\}$ we have that

$\Sigma_{2}^{p,\tau}(r,u)<\Sigma_{2}^{p,\tau}(0,u)=2\Sigma_{2}^{p,\tau}(0,u).$
(6.23)

  * •

For $|u|<u_{th;p,\tau}$ and $\epsilon>0$, we have that
$\sup_{|r|\geq\epsilon}\Sigma_{2}^{p,\tau}(r,u)<\Sigma_{2}^{p,\tau}(0,u)$.

  * •

For $|u|>u_{th;p,\tau}$ and $\epsilon>0$, we have $\sup_{|r|\leq
1-\epsilon}\Sigma_{2}^{p,\tau}(r,u)<\bar{\Sigma}_{2}^{p,\tau}(1,u)$.

###### Proof of Lemma 6.3.

We begin by simplifying the expressions in
$\Sigma_{2}^{p,\tau}-2\Sigma^{p,\tau}$. Employing (A.62) we may write

$\frac{1}{2}(1,1)\Sigma_{U}(r)^{-1}\begin{bmatrix}1\\\
1\end{bmatrix}=\frac{k_{1}(r)+k_{2}(r)}{p\alpha b(r)}.$ (6.24)

Using the expression for $k_{1},k_{2}$ and $b$ given after (A.62) we see that

$k_{1}(r)+k_{2}(r)-b(r)=(r^{2p-2}-r^{p})(1-r^{2p-2}-(p-1)\tau
r^{p-2}(1-r^{2})).$ (6.25)

By this and (6.12) we see we may cancel a common factor in the expression
(6.24), so that

$\frac{k_{1}(r)+k_{2}(r)}{p\alpha
b(r)}=-g^{p,\tau}(r)+(p\alpha)^{-1},\;\;g^{p,\tau}(r):=\frac{r^{p}(1-r^{p-2})}{1-r^{2p-2}+(p-1)\tau
r^{p-2}(1-r^{2})}.$ (6.26)

If we define then

$\;\;Q^{p,\tau}(r,u):=h(r)+\frac{u^{2}}{p\alpha}g^{p,\tau}(r),$ (6.27)

we see that

$\Sigma_{2}^{p,\tau}(r,u)=Q^{p,\tau}(r,u)+1+\log(p-1)-\frac{u^{2}}{p\alpha}+2\phi_{\tau,p}\left(u\right).$
(6.28)

We see it suffices to prove the corresponding statements for
$Q^{p,\tau}(r,u)$, as the other terms are constant in $r$. To prove the
existence of the desired continuous extension, we note that

$\lim_{r\to
1}Q^{p,\tau}(r,u)=-\frac{1}{2}\log(p-1)+\frac{u^{2}}{p\alpha}\left(\frac{p-2}{2(p-1)(1+\tau)}\right)=-\frac{1}{2}\log(p-1)(1-\frac{u^{2}}{u_{th;p,\tau}^{2}}).$
(6.29)

For notational ease, we will denote the extension of $Q^{p,\tau}$ to
$r\in(-1,1]$ again by $Q^{p,\tau}$.

We next verify that it suffices to show the claims when $r\in[0,1)$. When $p$
is even, this follows from the fact that $Q^{p,\tau}$ is even in $r$. When $p$
is odd, it will suffice to show that for $r\in(0,1)$, $g^{p,\tau}(r)>0$ and
for $r\in(-1,0)$, $g^{p,\tau}(r)<0$. On the other hand by (6.10) the
denominator is positive, so this follows by inspection of the numerator. In
either case, we may now restrict to $r\in[0,1)$.

Next we will show that $g^{p,\tau}$ is increasing in $r\in(0,1)$. We first
note that

$\frac{d}{dr}g^{p,\tau}(r)=\frac{r^{p-1}\bar{h}^{p,\tau}(r)}{(1-r^{2p-2}+(p-1)\tau
r^{p-2}(1-r^{2}))^{2}},$ (6.30)

where here

$\bar{h}^{p,\tau}(r):=(p-(2p-2)r^{p-2})(1-r^{2p-2}+(p-1)\tau
r^{p-2}(1-r^{2}))+$ (6.31) $(r-r^{p-1})((2p-2)r^{2p-3}-(p-1)\tau
r^{p-3}((p-2)-pr^{2})).$ (6.32)

To show that $g^{p,\tau}$ is increasing it suffices to show that
$\bar{h}^{p,\tau}(r)>0$ for $r\in(0,1)$. As $\bar{h}^{p,\tau}(r)$ is linear in
$\tau$, it suffices to verify this in the cases that $\tau=1$ and
$\tau=-(p-1)^{-1}$. Routine calculation gives that

$\bar{h}^{p,1}(r)=p(1-r^{2p-2}-(p-1)r^{2p-4}(1-r^{2}))\geq
p(1-r^{2p-2}-(p-1)|r|^{p-2}(1-r^{2}))>0,$ (6.33)

and similarly that

$\bar{h}^{p,-(p-1)^{-1}}(r)=p(1-r^{p-2})^{2}>0$ (6.34)

This completes the verification that $g^{p,\tau}$ is increasing.

We note from (6.29) that
$Q^{p,\tau}(1,u_{th;p,\tau})=0=Q^{p,\tau}(0,u_{th;p,\tau})$. We will now
reduce to the case of $u=u_{th;p,\tau}$. Indeed, given this claim we note that
for $|u|<u_{th;p,\tau}$, and $r\in(0,1)$, we have that

$Q^{p,\tau}(r,u)<Q^{p,\tau}(r,u_{th;p,\tau})\leq
Q^{p,\tau}(0,u_{th;p,\tau})=Q^{p,\tau}(0,u),$ (6.35)

where in the first inequality we have used that $g^{p,\tau}(r)>0$. Similarly
if $|u|>u_{th,\tau}$ then for $r\in(0,1)$

$Q^{p,\tau}(1,u)=Q^{p,\tau}(1,u_{th;p,\tau})+\frac{(u^{2}-u_{th;p,\tau}^{2})}{p\alpha}g^{p,\tau}(1)\geq
Q^{p,\tau}(r,u_{th;p,\tau})+\frac{(u^{2}-u_{th;p,\tau}^{2})}{p\alpha}g^{p,\tau}(1)=$
(6.36)
$Q^{p,\tau}(r,u)+\frac{(u^{2}-u_{th;p,\tau}^{2})}{p\alpha}(g^{p,\tau}(1)-g^{p,\tau}(r))\geq
Q^{p,\tau}(r,u),$ (6.37)

where in the last step we have used that $g^{p,\tau}(r)$ is increasing in $r$.
This completes the reduction to the case where $u=u_{th;p,\tau}$.

Let us denote $\bar{Q}^{p,\tau}(r):=Q^{p,\tau}(r,u_{th;p,\tau})$, so that we
need only show that for $r\in(0,1)$, we have that $\bar{Q}^{p,\tau}(r)<0$. Our
final reduction will be to the case where $\tau=1$. To do this, we may compute

$\frac{d}{d\tau}\bar{Q}^{p,\tau}(r)=\frac{d}{d\tau}\bigg{(}\frac{u_{th;p,\tau}^{2}}{p\alpha}g^{p,\tau}(r)\bigg{)}=\frac{(p-1)\log(p-1)}{(p-2)}\frac{d}{d\tau}\bigg{(}(1+\tau)g^{p,\tau}(r)\bigg{)}=$
(6.38)
$\frac{(p-1)\log(p-1)}{(p-2)}\bigg{(}\frac{r^{p}(1-r^{p-2})(1-r^{2p-2}-\tau(p-1)r^{p-2}(1-r^{2}))}{(1-r^{2p-2}+(p-1)\tau
r^{p-2}(1-r^{2}))^{2}}\bigg{)}.$ (6.39)

In particular, we see from this and (6.9) we see that
$\frac{d}{d\tau}\bar{Q}^{p,\tau}(r)>0$, so it suffices to verify the case
where $\tau=1$. This coincides with the case addressed in [61]. In particular,
our $\bar{Q}^{p,1}(r)$ coincides with their $Q_{p}^{u_{th}(p)}$, which they
define in their Lemma 7 and Eqn. (6.12). In particular, the verification that
$\bar{Q}^{p,1}(r)<0$ for $r\in(0,1)$ now follows from their verification that
$Q^{u_{th}(p)}_{p}(r)<0$, given in their proof of Lemma 7 (more specifically,
following their Eqn. 6.15). ∎

We are now able to give the proof of Lemma 2.5

###### Proof of Lemma 2.5.

Applying Theorem 2.3 and Lemmas 6.1 and 6.3, we see that

$\lim_{N\to\infty}N^{-1}\log\left(\mathbb{E}[\mathcal{N}_{N,2}(B\cap(-u_{th;p,\tau},u_{th;p,\tau}),I)]\right)\leq\sup_{u\in
B\cap(-u_{th;p,\tau},u_{th;p,\tau}),r\in
I\cap(-1,1)}\Sigma_{2}^{p,\tau}(r,u,u)$ (6.40)
$\lim_{N\to\infty}N^{-1}\log\left(\mathbb{E}[\mathcal{N}_{N,2}(B\cap(-u_{th;p,\tau},u_{th;p,\tau})^{c},I)]\right)\leq\sup_{u\in
B\cap(-u_{th;p,\tau},u_{th;p,\tau})^{c}}\bar{\Sigma}_{2}^{p,\tau}(1,u)).$
(6.41)

We note that by Lemma 6.3 that if $\\{0\\}\in\bar{I}$ then

$\sup_{u\in B,r\in I\cap(-1,1),|u|\leq
u_{th;p,\tau}}2\Sigma_{2}^{p,\tau}(r,u,u)=\sup_{u\in B,|u|\leq
u_{th;p,\tau}}2\Sigma^{p,\tau}(u)$ (6.42)

with strict inequality otherwise. The right hand side of (6.40) is positive by
the assumption that $B\cap(-E_{0;p,\tau},E_{0;p,\tau})\neq\varnothing$, we see
that it suffices to show that for $|u|>u_{th;p,\tau}$

$\bar{\Sigma}^{p,\tau}_{2}(1,u)<0.$ (6.43)

To do this we first need a preliminary fact about $\phi_{\tau}$. We note that
the function $\phi_{\tau}$ is concave for $u\in(1+\tau,\infty)$. We note as
well that $\phi_{\tau}(1+\tau)=\tau/2$ and $\phi_{\tau}^{\prime}(1+\tau)=1$,
so by concavity we have that for $u>1+\tau$

$\phi_{\tau}(u)\leq\frac{\tau}{2}+(u-(1+\tau))=\frac{u^{2}}{2(1+\tau)}-\frac{1}{2}-\frac{1}{2(1+\tau)}(u-(1+\tau))^{2}\leq\frac{u^{2}}{2(1+\tau)}-\frac{1}{2}.$
(6.44)

As this bound is an equality for $|u|\leq 1+\tau$, and the function
$\phi_{\tau}$ is even, we see that for any $u\in\mathbb{R}$

$\phi_{\tau}(u)\leq\frac{u^{2}}{2(1+\tau)}-\frac{1}{2}.$ (6.45)

Employing (6.29) we see that we may write

$\bar{\Sigma}_{2}^{p,\tau}(1,u)=\frac{\log(p-1)}{2}\left(\frac{u^{2}}{u_{th;p,\tau}^{2}}-1\right)+1+\log(p-1)-\frac{u^{2}}{p\alpha}+2\phi_{\tau,p}\left(u\right).$
(6.46)

By (6.45) we have that

$2\phi_{\tau,p}\left(u\right)\leq-1+\frac{u^{2}}{p(p-1)(1+\tau)}=-1+\frac{\log(p-1)\alpha}{(p-2)}\frac{u^{2}}{u_{th;p,\tau}^{2}},$
(6.47)

so we see that

$\bar{\Sigma}_{2}^{p,\tau}(1,u)\leq\frac{1}{2}\log(p-1)+\frac{u^{2}}{u_{th;p,\tau}^{2}}\bigg{(}\frac{\log(p-1)}{2}-\frac{\log(p-1)(p-1)(1+\tau)}{(p-2)}+$
(6.48)
$\frac{\log(p-1)\alpha}{(p-2)}\bigg{)}=\frac{\log(p-1)}{2}\left(1-\frac{u^{2}}{u_{th;p,\tau}^{2}}\right),$
(6.49)

which verifies (6.43). ∎

##  7 Random Matrix Theory and Elliptic Ensembles

In this section we will prove the majority of results we need for the elliptic
ensembles, other than Proposition 8.1. Our first goal will be to give a proof
of Lemma 4.2, which gives the behavior of the characteristic polynomial on the
exponential scale. Our other main result will be Proposition 7.11, which gives
our bounds on the characteristic polynomial outside of the bulk. In the course
of proving these, we will also need two overcrowding results to control the
behavior of the smallest singular value. These are given by Lemmas 7.8 and
7.10. The majority of the results in this section will follow from applying
standard Gaussian concentration inequalities to the forms of the elliptic law
given in [53, 54].

To begin we recall some general facts from linear algebra. The first of these
is the Hoffman-Wielandt inequality (see Lemma 2.1.19 of [5]). This states that
for $n$-by-$n$ symmetric matrices $A$ and $B$ we have that

$\sum_{i=1}^{n}|\lambda_{i}(A)-\lambda_{i}(B)|^{2}\leq\text{\rm
Tr}(A-B)^{2}=\|A-B\|_{\textbf{g}}^{2}.$ (7.1)

We also recall that for an $n$-by-$n$ matrix $M$, the eigenvalues of the
matrix

$\bar{M}:=\begin{bmatrix}0&M\\\ M^{t}&0\end{bmatrix}$ (7.2)

are given by $-s_{1}(M),\dots,-s_{n}(M),s_{n}(M),\dots,s_{1}(M)$. Combining
these two statements we see that for general $n$-by-$n$ matrices $A$ and $B$
we have that

$\sum_{i=1}^{n}|s_{i}(A)-s_{i}(B)|^{2}\leq\|A-B\|_{\textbf{g}}^{2}.$ (7.3)

From this statement, we conclude the following result, which may be compared
to Lemma 2.3.1 of [5].

######  Lemma 7.1.

[5] Let $f:[0,\infty)\to\mathbb{R}$ be a Lipschitz function of norm less than
$C$. Then for any $z\in\mathbb{R}$ the function on $N$-by-$N$ matrices given
by

$A\mapsto\int_{0}^{\infty}f(x)\nu_{A,z}(dx)$ (7.4)

has Lipschitz norm bounded by $C/\sqrt{N}$ (with respect to the Frobenius
norm)

###### Proof.

If $A$ and $B$ are two $N$-by-$N$ matrices then

$|\int_{0}^{\infty}f(x)\nu_{A,z}(dx)-\int_{0}^{\infty}f(x)\nu_{B,z}(dx)|\leq\frac{C}{N}\sum_{i=1}^{N}|s_{i}(A-zI)-s_{i}(B-zI)|\leq$
(7.5)
$\frac{C}{\sqrt{N}}\left(\sum_{i=1}^{N}|s_{i}(A-zI)-s_{i}(B-zI)|^{2}\right)^{1/2}\leq\frac{C}{\sqrt{N}}\|A-B\|_{\textbf{g}}^{2}.$
(7.6)

∎

We will also need a bound on the determinant of the sum of two matrices. In
the case of symmetric matrices this follows from the main result of [27], and
for non-symmetric case it follows by applying the symmetric case to (7.2).

######  Corollary 7.2.

[27] Let $A$ and $B$ be $n$-by-$n$ matrices. Then

$|\det(A+B)|\leq\prod_{i=1}^{n}(s_{i}(A)+s_{n-i}(B)).$ (7.7)

In particular, if $\mathrm{rank}(B)\leq d$ then

$|\det(A+B)|\leq|\det(A)|\left(1+\frac{s_{1}(B)}{s_{n}(A)}\right)^{d}.$ (7.8)

Next we will need to recall some results for the empirical measures
$\nu_{A_{N},z}$. In the course of proving a general form for the elliptic law,
it was shown in [53] (more specifically, in their proof of Theorem 5.2) that
for fixed $z$, the sequence $\nu_{A_{N},z}$ converges a.s. to some
deterministic limit $\nu_{z}$. This measure is studied more extensively in
Appendix C of [54]. We will collect some of the results that we will need as
the following lemma.

######  Lemma 7.3.

[53, 54] For any $|\tau|<1$, and fixed $z\in\mathbb{R}$ we have that

$\mathbb{E}[\nu_{A_{N},z}]\Rightarrow\nu_{z}$ (7.9)

where $\nu_{z}$ is a certain ($\tau$-dependent) measure, characterized by
Lemma C.6 of [54]. Moreover the convergence in (7.9) is uniform for $z$ in any
compact subset of $\mathbb{R}$, and the measure $\nu_{z}$ is compactly-
supported with a continuous density. Finally, this measure is uniformly
bounded in the following sense: for any $K>0$ we may find $C>0$ such that if
we denote the density of $\nu_{z}$ as $\bar{\nu}_{z}$ then

$\sup_{|z|\leq K,x\in\mathbb{R}}|\bar{\nu}_{z}(x)|\leq C.$ (7.10)

###### Proof.

Except for the uniformity claims, all of these results directly follow from
Lemma C.7 of [54] and the discussion immediately prior. Uniform boundedness of
the density follows from observing that all bounds used in their proof of
Lemma C.7 hold uniformly in compact subsets of $z$. Finally, to establish the
uniformity claim for the convergence in (7.9) we observe that for
$x,y\in\mathbb{R}$

$\frac{1}{N}\sum_{i=1}^{N}\left|s_{i}(A_{N}-xI)-s_{i}(A_{N}-yI)\right|\leq$
(7.11)
$\frac{1}{\sqrt{N}}\left(\sum_{i=1}^{N}(s_{i}(A_{N}-xI)-s_{i}(A_{N}-yI))^{2}\right)^{1/2}\leq|x-y|\frac{\|I\|_{F}}{\sqrt{N}}|x-y|,$
(7.12)

where in the last step we have used (7.3). Taking expectations, this statement
easily gives the desired uniformity. ∎

We now note that the limiting analog of the identity

$|\det(A-zI)|=\sqrt{|\det((A-zI)^{*}(A-zI))|},$ (7.13)

yields the following continuum relation (see as well Lemma 2.3 of [53])

$\int_{0}^{\infty}\log(x)\nu_{z}(dx)=\phi_{\tau}(z).$ (7.14)

We will need the following result which will follow from this and the uniform
boundedness of $\nu_{z}$.

######  Lemma 7.4.

For any $|\tau|<1$ and $K>0$ we have that

$\lim_{\epsilon\to 0}\sup_{|z|\leq
K}\left|\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{z}(dx)-\phi_{\tau}(z)\right|=0.$
(7.15)

###### Proof.

We observe that by (7.14) we have that

$|\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{z}(dx)-\phi_{\tau}(z)|=|\int_{0}^{\epsilon}\log(x/\epsilon)\nu_{z}(dx)|.$
(7.16)

By (7.10) we see that there is $C>0$ such that for $|z|<K$

$|\int_{0}^{\epsilon}\log(x/\epsilon)\nu_{z}(dx)|\leq
C\epsilon\int_{0}^{1}\log(|x|)dx.$ (7.17)

Together these establish (7.15). ∎

We will now give a concentration estimate for
$\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx)$.

######  Lemma 7.5.

For any $\epsilon>0$ and $\ell>0$ we have that

$\mathbb{E}[\exp(\ell\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))]\leq\exp\left(\ell
N\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)+8\ell^{2}\epsilon^{-2}\right).$
(7.18)

###### Proof.

We observe that all entries of $A_{N}$ are independent except for the pairs
$([A_{N}]_{ij},[A_{N}]_{ji})$. For each $i,j$, the pair
$([A_{N}]_{ij},[A_{N}]_{ji})$ satisfies the log-Sobolev inequality with
constant $4/N$, by the Bakry-Emery criterion [47]. Thus $A_{N}$, considered as
a random vector in $\mathbb{R}^{N^{2}}$, also satisfies the log-Sobolev
inequality with constant $4/N$. We also note that $\log_{\epsilon}$ has
Lipschitz constant $1/(2\epsilon)$, so by Lemma 7.1 the function

$A\mapsto\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A,z}(dx)$ (7.19)

has Lipschitz constant bounded by $1/(2\epsilon\sqrt{N})$. Thus (7.18) now
follows by Herbst’s argument (see section 2.3 of [47]). ∎

To make use of this, we will need some tail estimates for the largest singular
value. By Corollary 2.3 of [54] $s_{1}(A_{N})$ converges to $1+|\tau|$ a.s..
From this, we are able to obtain the following result by the Borell-TIS
inequality (see Theorem 2.1.1 of [2]).

######  Lemma 7.6.

For $|\tau|<1$ and any $\epsilon>0$ we have that

$\lim_{N\to\infty}\mathbb{E}[s_{1}(A_{N})]=1+|\tau|,$ (7.20)
$\lim_{N\to\infty}\mathbb{E}[I(s_{1}(A_{N})>1+|\tau|+\epsilon)s_{1}(A_{N})]e^{N\epsilon^{2}/32}=0.$
(7.21)

In particular, we have that

$\lim_{N\to\infty}\mathbb{P}(s_{1}(A_{N})>1+|\tau|+\epsilon)e^{N\epsilon^{2}/32}=0.$
(7.22)

###### Proof.

To begin we note that

$s_{1}(A_{N})=\sup_{x,y\in S^{N-1}}|x^{t}A_{N}y|.$ (7.23)

The function $x^{t}A_{N}y$ is a centered Gaussian process on $S^{N-1}\times
S^{N-1}$ with covariance

$\mathbb{E}[(x_{1}^{t}A_{N}y_{1})(x_{2}^{t}A_{N}y_{2})]=N^{-1}\left[(x_{1},x_{2})(y_{1},y_{2})+\tau(x_{1},y_{1})(x_{2},y_{2})\right].$
(7.24)

We observe that on $S^{N-1}\times S^{N-1}$ this is bounded by $2/N$. Thus by
applying the Borell-TIS inequality we see that for any $u>0$

$\mathbb{P}(|s_{1}(A_{N})-\mathbb{E}[s_{1}(A_{N})]|\geq u)\leq
2\exp(-Nu^{2}/4).$ (7.25)

In particular by the Borel-Cantelli lemma, for any $\epsilon>0$ we have a.s.

$\limsup_{N\to\infty}|s_{1}(A_{N})-\mathbb{E}[s_{1}(A_{N})]|<\epsilon.$ (7.26)

As we also know that $s_{1}(A_{N})$ converges to $(1+\tau)$ a.s. we obtain the
first claim.

Now fix $\epsilon>0$ and take $N$ sufficiently large that
$\mathbb{E}[s_{1}(A_{N})]\leq 1+|\tau|+\epsilon/2$. Then

$\mathbb{E}[I(s_{1}(A_{N})>1+|\tau|+\epsilon)s_{1}(A_{N})]=\int_{1+|\tau|+\epsilon}^{\infty}\mathbb{P}(s_{1}(A_{N})>x)dx\leq$
(7.27)
$\int_{\epsilon/2}^{\infty}\mathbb{P}(|s_{1}(A_{N})-\mathbb{E}[s_{1}(A_{N})]|>x)dx\leq
2\int_{\epsilon/2}^{\infty}\exp(-Nx^{2}/4)dx,$ (7.28)

from which the second bound immediately follows. Lastly the final claim
follows by Markov’s inequality. ∎

With these two results, we are able to obtain the following uniform result.

######  Lemma 7.7.

For any choices of $|\tau|<1$, $\ell>0$, $\epsilon>0$ and $K>0$ we have that

$\lim_{N\to\infty}\sup_{|z|<K}\left|(N\ell)^{-1}\log(\mathbb{E}[\exp(\ell\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))])-\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{z}(dx)\right|=0.$
(7.29)

Moreover for $0<\epsilon<1$ we have that

$\lim_{N\to\infty}\sup_{|z|\geq
1}\left((N\ell)^{-1}\log(\mathbb{E}[\exp(\ell\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))])-(2+\log(|z|))\right)\leq
0.$ (7.30)

###### Proof.

By Jensen’s inequality we have that

$\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)\leq(N\ell)^{-1}\log(\mathbb{E}[\exp(\ell\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))]).$
(7.31)

Combining this with Lemma 7.5 we see that

$\lim_{N\to\infty}\sup_{|z|<K}\left|(N\ell)^{-1}\log(\mathbb{E}[\exp(\ell\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{A_{N},z}(dx))])-\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)\right|=0.$
(7.32)

Thus to show (7.29) it suffices to show that

$\limsup_{N\to\infty}\sup_{|z|<K}\left|\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)-\int_{0}^{\infty}\log_{\epsilon}(x)\nu_{z}(dx)\right|=0.$
(7.33)

For this we observe that for $C>\epsilon$

$\left|\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)-\int_{0}^{C}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)\right|\leq$
(7.34)
$\mathbb{E}[I(s_{1}(A_{N}-zI)>C)\log(s_{1}(A_{N}-zI))]\leq\mathbb{E}[I(s_{1}(A_{N})>C-|z|)(s_{1}(A_{N})+|z|)],$
(7.35)

where in the last line we have used that $s_{1}(A_{N}-zI)\leq
s_{1}(A_{N})+|z|$ and that $\log(x)\leq x$. Taking $C$ sufficiently large we
see by Lemma 7.6 that

$\limsup_{N\to\infty}\sup_{|z|<K}\left|\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)-\int_{0}^{C}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)\right|=0.$
(7.36)

This as well as the uniform convergence in Lemma 7.3 completes the proof of
(7.33), and thus of the first claim.

To show (7.30) we note that for $|z|>\max(1,\epsilon)$,

$\int_{0}^{\infty}\log_{\epsilon}(x)\mathbb{E}[\nu_{A_{N},z}](dx)\leq\mathbb{E}[\log_{\epsilon}(s_{1}(A_{N}-zI))]\leq\mathbb{E}[\log_{\epsilon}(s_{1}(A)+|z|)]=$
(7.37)
$\mathbb{E}[\log(s_{1}(A_{N})+|z|)]\leq\mathbb{E}[s_{1}(A_{N})]+\log(|z|).$
(7.38)

where in the last inequality we have used that for $x,y\in(0,\infty)$

$\log(x+y)=\log(x)+\log(1+y/x)\leq y/x+\log(x).$ (7.39)

Using 7.6 we may conclude (7.30) as before. ∎

The proof of Lemma 4.2 now immediately follows from Lemmas 7.3 and 7.7.

Our next step will be to provide our overcrowding results. The first is
essentially Theorem 3.1 of [53], which is based on the ideas of Mark Rudelson
and Roman Vershynin [56, 67]. As we are in the case where the entries are
Gaussian though we are able to get a significantly stronger result by a small
modification of the argument.

######  Lemma 7.8.

Let $X_{N}$ be an $N$-by-$N$ random matrix such that each of the pairs
$([X_{N}]_{ij},[X_{N}]_{ji})$, for $1\leq i,j\leq N$, are independent centered
Gaussian random variables. Let us further assume that there are constants
$M,m>0$ such that $MN^{-1}\geq\mathbb{E}[X_{ij}^{2}]\geq mN^{-1}$ for all
$1\leq i,j\leq N$. Finally let us assume that $F_{N}$ is a sequence of
deterministic $N$-by-$N$ matrices such that $s_{1}(F_{N})\leq M$. Then for any
$\gamma>0$, there is $C>0$ such that

$\mathbb{P}\left(s_{N}(X_{N}+F_{N})\leq N^{-\gamma-1}\right)\leq
CN^{-\gamma}.$ (7.40)

###### Proof.

To begin, we show for sufficiently large $K$, there are $c,C>0$ such that for
any $\epsilon>0$

$\mathbb{P}(s_{N}(X_{N}+F_{N})\leq\epsilon N^{-1/2},s_{1}(X_{N}+F_{N})\leq
K)\leq C(e^{-cN}+\epsilon N^{1/2}).$ (7.41)

Before proceeding we note that the $n^{-1}$ which appears in their Lemma 3.8
appears to be a typo for $n^{-1/2}$, as it appears in Lemma 3.5 of [56] which
they reference. With this noted, following the proof of Theorem 3.1 in [53],
we see that if denote the $i$-th column vector of $X_{N}+F_{N}$ as $V_{i}$,
there are $C,c>0$ such that

$\mathbb{P}(s_{N}(X_{N}+F_{N})\leq\epsilon N^{-1/2},s_{1}(X_{N}+F_{N})\leq
K)\leq C(e^{-cN}+$ (7.42) $N\max_{1\leq i\leq
N}\mathbb{P}(d(V_{i},\mathrm{span}(V_{1},\cdots,V_{i-1},V_{i+1},\cdots,V_{N})\leq
2\epsilon)).$ (7.43)

To bound the final term on the right, we take $i=1$ for notational simplicity.
We notate the components of $X_{N}+F_{N}$ in terms of $1$-by-$(N-1)$ blocks as

$X_{N}+F_{N}=\begin{bmatrix}a_{11}&V^{T}\\\ U&B\end{bmatrix}.$ (7.44)

They then show that the probability on the right may be bounded by

$\mathbb{P}\left(\frac{|(B^{-T}V,U)-a_{11}|}{\sqrt{1+\|B^{-T}V\|^{2}}}\leq
2\epsilon\right).$ (7.45)

On the other hand, we observe that $a_{11}$ is a Gaussian random variable of
variance above $c$, which is independent of $(B,V,U)$. In particular,
conditioning on $(B,V,U)$ and noting that $\sqrt{1+\|B^{-T}V\|^{2}}\geq 1$ we
have that

$\mathbb{P}\left(\frac{|(B^{-T}V,U)-a_{11}|}{\sqrt{1+\|B^{-T}V\|^{2}}}\leq\epsilon\right)\leq\sup_{v\in\mathbb{R}}\mathbb{P}(|v-Z|\leq
N^{1/2}\epsilon),$ (7.46)

where $Z$ is a Gaussian random variable of variance $m$. As $Z$ has a bounded
density on $\mathbb{R}$ we obtain (7.41).

To complete the proof, it suffices show there are $C,c>0$ such that

$\mathbb{P}(s_{1}(X_{N})\geq C)\leq Ce^{-cN}.$ (7.47)

This follows easily from a standard $\epsilon$-net argument. In particular,
from Lemma 2.3.4 and Exercise 2.3.2 of [66] there is universal $K>0$ such that

$\mathbb{P}(s_{1}(X_{N})\geq C)\leq K^{N}\sup_{v,w\in
S^{N-1}}\mathbb{P}(|v^{t}X_{N}w|>C/2).$ (7.48)

As $v^{t}X_{N}w$ is a centered Gaussian random variable with variance bounded
by $2MN^{-1}$, we see that (7.47) follows (7.48) and the Gaussian tail
estimate $\mathbb{P}(|\mathcal{N}(0,1)|>u)\leq 2e^{-u^{2}/2}$. ∎

We record, for the ease of reference in latter sections, the following
corollary of this result.

######  Corollary 7.9.

For any $|\tau|<1$, $u\in\mathbb{R}$, and $\gamma>0$ as well as any choice of
$u_{N}\to u$, then there is $C>0$ such that

$\mathbb{P}\left(s_{N}(A_{N,\tau}-u_{N}I)<N^{-\gamma-1}\right)\leq
CN^{-\gamma}.$ (7.49)

To prove our result for the characteristic polynomial outside of the bulk,
given by Proposition 7.11 below, we need an additional overcrowding result,
which is not tight, but which applies uniformly until zero. We will obtain
this by modifying the methods of [26] in the symmetric case.

######  Lemma 7.10.

Let $X$ be an $N$-by-$N$ random matrix such that for each $1\leq i\leq N$ the
diagonal entry $[X_{N}]_{ii}$ is independent of all other entries, and possess
a density function which is bounded by some $C$. Then for any $\eta>0$

$\mathbb{P}(s_{N}(X)\leq\eta)\leq\mathbb{E}[\\#\\{i:s_{i}(X)\leq\eta\\}]\leq
2\pi CN\eta.$ (7.50)

###### Proof.

We will consider the matrix $\tilde{X}$ defined in (7.2) above. We denote for
$1\leq i\leq N$, $H_{i}=\text{span}(e_{i},e_{N+i})$, taken with the obvious
basis. We note that

$\mathbb{E}[\\#\\{i:s_{i}(X)\leq\eta\\}]\leq\mathbb{E}[\sum_{i=1}^{N}\frac{2\eta^{2}}{\eta^{2}+s_{i}(x)^{2}}]=\eta\mathbb{E}[\mathrm{Im}(\text{\rm
Tr}((\tilde{X}-i\eta I)^{-1}))]\leq$ (7.51)
$\eta\sum_{k=1}^{N}\mathbb{E}[\mathrm{Im}(\text{\rm Tr}([(\tilde{X}-i\eta
I)^{-1}]_{H_{k}}))]\leq\eta N\sup_{1\leq k\leq
N}\mathbb{E}[\mathrm{Im}(\text{\rm Tr}([(\tilde{X}-i\eta I)^{-1}]_{H_{k}}))],$
(7.52)

where in the last expression $[(\tilde{X}-i\eta I)^{-1}]_{H_{k}}$ denotes the
$2$-by-$2$ matrix given by restricting $(\tilde{X}-i\eta I)^{-1}$ to $H_{k}$.
To proceed let us fix $k$. Note by the Schur complement formula
$(\tilde{X}-i\eta I)^{-1}$ is given in terms of $N$-by-$N$ blocks as

$(\tilde{X}-i\eta
I)^{-1}=\begin{bmatrix}-i(I\eta+\eta^{-1}XX^{t})^{-1}&(I+\eta^{-2}XX^{t})^{-1}X\\\
X^{t}(I+\eta^{-2}X^{t}X)^{-1}&-i(I\eta+\eta^{-1}X^{t}X)^{-1}\end{bmatrix}.$
(7.53)

From this we see if write

$[(\tilde{X}-i\eta I)^{-1}]_{H_{k}}=\begin{bmatrix}-iU&B\\\
B&-iV\end{bmatrix}$ (7.54)

then $U,V\geq 0$ and $B\in\mathbb{R}$. Moreover, denote block decomposition of
$\tilde{X}$ with respect to $H_{k}$ and $H_{k}^{\perp}$ (with the basis given
by the remaining coordinate vectors) as

$\tilde{X}=\begin{bmatrix}W&Y\\\
Y^{t}&Z\end{bmatrix},\;\;\;W=\begin{bmatrix}0&X_{ii}\\\ X_{ii}&0\\\
\end{bmatrix}.$ (7.55)

By our assumptions $W$ is independent of $(Y,Z)$. Again using the Schur
complement formula

$\mathbb{E}[\mathrm{Im}(\text{\rm Tr}([(\tilde{X}-i\eta
I)^{-1}]_{H_{k}}))]=\mathbb{E}[\mathrm{Im}(\text{\rm Tr}((W-i\eta I-Y(Z-i\eta
I)^{-1}Y^{t})^{-1}))].$ (7.56)

In particular if we denote $W=B-X_{ii}$, then $X_{ii}$ is independent of
$(U,V,W)$. Now we may write

$\mathbb{E}[\mathrm{Im}(\text{\rm Tr}([(\tilde{X}-i\eta
I)^{-1}]_{H_{k}}))]=\mathbb{E}\left[\frac{U+V}{UV+B^{2}}\right]=\mathbb{E}\left[\frac{U+V}{UV+(W+X_{ii})^{2}}\right].$
(7.57)

By conditioning on their values of $(U,V,W)$ we have the upper bound

$\mathbb{E}\left[\frac{U+V}{UV+(W+X_{ii})^{2}}\right]\leq\sup_{u,v>0,w\in\mathbb{R}}\mathbb{E}\left[\frac{u+v}{uv+(w+X_{ii})^{2}}\right]\leq$
(7.58)
$\sup_{u,v>0,w\in\mathbb{R}}C\int_{-\infty}^{\infty}\frac{u+v}{uv+(x+w)^{2}}dx=C\sup_{v\geq
1}\int_{-\infty}^{\infty}\frac{1+v}{v^{2}+x^{2}}dx$ (7.59)

where in the second to last step we have used that $X_{ii}$ has a bounded
density and in the last step we have used translation and scale invariance of
the integral. Finally noting that

$\sup_{v\geq
1}\int_{-\infty}^{\infty}\frac{1+v}{v^{2}+x^{2}}dx\leq\int_{-\infty}^{\infty}\frac{2}{1+x^{2}}dx=2\pi,$
(7.60)

completes the proof. ∎

With all of these results at hand, we may obtain the following bound on the