# Spectral density of complex eigenvalues and associated mean eigenvector
self-overlaps at the edge of elliptic Ginibre ensembles

Mark J. Crumpton & Tim R. Würfel  Department of Mathematics, King’s College
London, London WC2R 2LS, UK

###### Abstract

We consider the density of complex eigenvalues, $\rho(z)$, and the associated
mean eigenvector self-overlaps, $\mathcal{O}(z)$, at the spectral edge of
$N\times N$ real and complex elliptic Ginibre matrices, as $N\to\infty$. Two
different regimes of ellipticity are studied: strong non-Hermiticity, keeping
the ellipticity parameter $\tau$ fixed and weak non-Hermiticity with
$\tau\rightarrow 1$ as $N\rightarrow\infty$. At strong non-Hermiticity, we
find that both $\rho(z)$ and $\mathcal{O}(z)$ have the same leading order
behaviour across the elliptic Ginibre ensembles, establishing the expected
universality. In the limit of weak non-Hermiticity, we find different results
for $\rho(z)$ and $\mathcal{O}(z)$ across the two ensembles. This paper is the
final of three papers that we have presented addressing the mean self-overlap
of eigenvectors in these ensembles.

Keywords: non-Hermitian random matrices, real elliptic Ginibre ensemble,
complex elliptic Ginibre ensemble, bi-orthogonal eigenvectors, eigenvector
overlaps, edge statistics, strong non-Hermiticity, weak non-Hermiticity

##  1 Introduction

Statistical properties of eigenvalues and eigenvectors have attracted a wealth
of attention in the field of Random Matrix Theory (RMT) over recent decades,
see reviews such as [1, 2, 3, 4, 5] for a more detailed introduction and more
information. Such spectral studies are well established in the case of
Hermitian matrices, which are equal to their own adjoint. Therein, eigenvalues
are purely real and the eigenvectors form an orthogonal basis, leading to
trivial correlations between eigenvectors, thus focusing attention to
eigenvalues only. Celebrated results for Hermitian random matrices include,
for example, the Tracy-Widom distribution [6] and the Wigner semi-circle law
[7]. However, such results are much harder to obtain in the realm of non-
Hermitian matrices, due to the presence of non-trivial correlations between
eigenvectors, which are not constrained further by rotational symmetry.

Initially, we focus on the eigenvalues, $z_{n}$, of $N\times N$ non-Hermitian
matrices, which are not solely constrained to the real line, but may take
complex values. The most natural object to study is $\rho(z)$, the mean
spectral density at a point $z$ in the complex plane, which is defined as

$\rho\,(z)\equiv\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\delta(z-z_{n})\bigg{\rangle}\
,$ (1.1)

where $\delta(\cdot)$ is the Dirac $\delta-$function and $\langle\cdot\rangle$
stands for the expectation value with respect to the jpdf of a particular
random matrix ensemble. The most popular ensembles within which one can study
this density are the _Ginibre ensembles_ [8], featuring independent,
identically distributed (iid) Gaussian entries. These entries can be real,
complex or even quaternion and the corresponding ensembles are abbreviated as
GinOE, GinUE and GinSE respectively. For such matrices it is well-known that,
in the limit of large $N$, the complex eigenvalues converge to a uniform disc
in the complex plane [8], with less trivial density behaviour at the edge of
this disc [9, 10]. One can extend these ensembles into _elliptic Ginibre
ensembles_ \- abbreviated as eGinOE, eGinUE and eGinSE respectively - through
the introduction of correlations between off-diagonal elements, using a single
parameter $\tau\in[0,1)$ [11, 12]. In such ensembles, as $N$ becomes large,
the complex eigenvalues are now constrained to an elliptic droplet, with
dimensions dependent upon $\tau$.

Focusing on complex eigenvalues of eGinOE and eGinUE matrices, finite-$N$ and
asymptotic results for the mean spectral density and other statistical
properties of eigenvalues are well-known for much of the complex plane, see
e.g. [2, 4, 5, 13, 14, 15, 16, 17]. Typically, large $N$ asymptotic results
are calculated for two regimes of $\tau$, these are the limits of _strong non-
Hermiticity_ , with fixed $\tau$, and _weak non-Hermiticity_ , where $\tau\to
1$ as $N\to\infty$, as introduced in [18, 19, 20]. To the best of our
knowledge, the density of complex eigenvalues at the edge of the elliptic
droplet has not yet been previously studied at strong or weak non-Hermiticity
for the eGinOE. On the other hand, explicit results for the eGinUE exist in
the work of Lee and Riser [15] for strong non-Hermiticity and in the work of
Garciá-Garciá et. al. [16] for weak non-Hermiticity. In the limit of strong
non-Hermiticity, universality is expected among all members of the class of
real and complex non-Hermitian matrices with iid entries, in the bulk and at
the edge of the droplet [21]. See also [22, 23, 24] for more recent works on
universality of eigenvalue correlation functions in the bulk. In the present
paper, among other tasks, we aim to provide a more detailed study of this
feature at the spectral edge of real and complex elliptic Ginibre matrices.

For real Ginibre-type matrices, a macroscopic proportion of the eigenvalues
are constrained to the real line and so one must consider the density of
complex and real eigenvalues separately. The latter having already been
studied at finite-$N$ and in several asymptotic regimes in the GinOE and
eGinOE, see [13, 25, 26] for more. Additionally, the spectral density of the
induced extensions of the real and complex Ginibre ensembles has been studied
at finite-$N$ and in various large $N$ limits by Fischmann et al. [27]. There,
for both ensembles, they observed a bulk spectrum of eigenvalues existing in
an annulus as $N\to\infty$, similar to what was observed in [28, 29]. Annular
spectra are also a feature of the ensemble of sub-unitary matrices studied at
finite-$N$ and asymptotically by Wei and Fyodorov and Bogomolny respectively
[30, 31].

On the other hand, the study of statistical properties of eigenvectors is also
an interesting avenue. In general, the eigenvalues of a random matrix $X$ have
multiplicity one, thus one can safely write that $X=S\Lambda S^{-1}$. Here,
$\Lambda$ is a diagonal matrix containing the eigenvalues of $X$ and the
columns (rows) of $S$ $(S^{-1})$ are the right (left) eigenvectors of $X$,
denoted as $\bm{v}_{R,i}$ and $\bm{v}_{L,i}^{\dagger}$, which satisfy
$\bm{v}_{L,i}^{\dagger}X=z_{i}\bm{v}_{L,i}^{\dagger}$ and
$X\bm{v}_{R,i}=z_{i}\bm{v}_{R,i}$ respectively. Within this setup, the left
and right eigenvectors obey a bi-orthogonality condition, such that
$\bm{v}_{L,i}^{\dagger}\bm{v}_{R,i}=\delta_{ij}$. However, providing that $X$
is not normal (i.e. it does not commute with its own adjoint), a relationship
of this nature does not exist within the sets of left and right eigenvectors.
As such, it is the relations within sets of left and right eigenvectors that
can be used as a measure of non-normality. In order to quantitatively study
this effect, Chalker & Mehlig introduced the now well-known matrix of overlaps
with entries

$\mathcal{O}_{mn}=\left(\bm{v}^{\dagger}_{L,m}\bm{v}_{L,n}\right)\left(\bm{v}^{\dagger}_{R,n}\bm{v}_{R,m}\right)\quad
m,n=1,2,\ldots,N\ ,$ (1.2)

such that the so-called self-overlaps, $\mathcal{O}_{nn}$, are the diagonal
entries of this matrix [32, 33]. Within their works, they studied the large
$N$ asymptotic limits of the connected ensemble averages

$\mathcal{O}(z)\equiv\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\mathcal{O}_{nn}\
\delta(z-z_{n})\bigg{\rangle}\hskip 21.33955pt\text{and}\hskip
21.33955pt\mathcal{O}(z_{1},z_{2})\equiv\bigg{\langle}\frac{1}{N^{2}}\sum_{\begin{subarray}{c}m,n=1\\\
m\neq n\end{subarray}}^{N}\mathcal{O}_{mn}\ \delta(z_{1}-z_{m})\
\delta(z_{2}-z_{n})\bigg{\rangle}\ ,$ (1.3)

finding that, within the spectral bulk of the GinUE, $\mathcal{O}(z)\sim
N(1-|z|^{2})/\pi$ and zero otherwise. This showed that the mean self-overlap
in non-normal matrices is macroscopically larger than the corresponding value
in normal matrices, where $\mathcal{O}_{nn}=1$ for all $n=1,\ldots,N$.

The findings of Chalker & Mehlig prompted further study of the self-overlap in
a few other ensembles of random non-normal matrices. For example, in
subsequent years they extended their own work to consider the eGinUE [33] and
Janik et al. introduced a new method to calculate the mean self-overlap from
the spectral Green’s function [34], independently verifying the results of
Chalker & Mehlig. Recently, this field received a further boost in attention
as the papers by Bourgade & Dubach [35] and Fyodorov [36] simultaneously
extended the work of Chalker & Mehlig to obtain the full distribution of the
mean self-overlap, for both finite-$N$ and in various asymptotic limits.
Additionally, Bourgade & Dubach used techniques from free probability to
derive statistics of the off-diagonal overlap, whereas Fyodorov utilised
incomplete Schur decomposition [37] to derive the distribution of the self-
overlap associated with real eigenvalues in the GinOE. These papers led to a
recent surge in interest in the self-overlap of eigenvectors in Ginibre
ensembles with results that include: the distribution of the self-overlap
associated with real eigenvalues in the eGinOE at finite-$N$ and in different
asymptotic regimes [26, 38], the mean self-overlap associated with complex
eigenvalues in the GinOE, eGinUE and eGinOE [17, 39] and the mean self-overlap
and off-diagonal overlap in the GinSE [40]. Other results in the study of
eigenvectors of random matrices and the self-overlap include [41, 42, 43, 44,
45, 46, 47]. For a broader review of results and applications of the overlap
matrix, one can consult [17, 39] and references therein, which lay the
groundwork for the present paper.

In spite of recent developments, to the best of our knowledge, results have
yet to be derived for the mean self-overlap associated with complex
eigenvalues at the edge of both the eGinUE and eGinOE at strong and weak non-
Hermiticity. Therefore, the main aim of this paper is to provide results in
these limits for both ensembles. Note that edge statistics are an important
area of study as they smoothly interpolate the statistics of eigenvalues
inside and outside of the bulk, whilst also having applications to chaotic
scattering [48, 49, 50]. As part of our presentation of such results, it is
convenient to introduce the mean conditional self-overlap

$\mathbb{E}\left(z\right)\equiv\mathbb{E}\left(\mathcal{O}_{nn}\,|\,z=z_{n}\right)=\frac{\mathcal{O}(z)}{\rho\,(z)}\
,$ (1.4)

as this is the object that we will mostly be concerned with when making
comparisons to numerical data. This object is particularly useful in this work
as it serves to verify results for both the density of complex eigenvalues and
the associated mean self-overlap of eigenvectors.

The remainder of this manuscript is arranged as follows. In Section 2.1 we
provide the necessary mathematical definitions of the eGinOE and eGinUE,
whilst reviewing some relevant existing results and important terminology.
Then, in Section 2.2, we outline our main results for the density of complex
eigenvalues in the eGinOE and the mean self-overlap associated with such
eigenvalues in the eGinUE and eGinOE. This is done at the edge of the elliptic
droplet for strong and weak non-Hermiticity in Section 2.2.1 and Section 2.2.2
respectively. We then proceed to review some open problems in Section 2.3
before, finally, in Section 3 we provide proofs of our main results.

##  2 Statement & Discussion of Main Results

###  2.1 Remarks on Elliptic Ensembles

Complex-valued matrices within the eGinUE have the following joint probability
density function (jpdf) with respect to the flat Lebesgue measure,
$dX=\prod_{i,j=1}^{N}dX_{ij}d\bar{X}_{ij}$,

$\mathcal{P}_{\text{eGinUE}}(X)\,dX=\frac{1}{D_{N,\tau}}\exp{-\frac{1}{1-\tau^{2}}\Tr\left(XX^{\dagger}-\tau\
{\rm Re\,}X^{2}\right)}dX\ ,$ (2.1)

with normalisation constant
$D_{N,\tau}=\pi^{N^{2}}\left(1-\tau^{2}\right)^{\frac{N^{2}}{2}}$ [12, 51]. On
the other hand, real-valued matrices in the eGinOE have the following jpdf
with respect to the flat real Lebesgue measure

$\mathcal{P}_{\text{eGinOE}}(X)\,dX=\frac{1}{C_{N,\tau}}\exp{-\frac{1}{2(1-\tau^{2})}\Tr\left(XX^{T}-\tau
X^{2}\right)}dX\ ,$ (2.2)

with normalisation constant
$C_{N,\tau}=(2\pi)^{N^{2}/2}\left(1+\tau\right)^{\frac{N(N+1)}{4}}\left(1-\tau\right)^{\frac{N(N-1)}{4}}$
[12, 38]. In both the eGinUE and eGinOE the parameter $\tau$ quantifies the
correlation between pairs of off-diagonal elements, with $\tau=0$
corresponding to uncorrelated GinUE or GinOE matrices and $\tau=1$ yielding
Hermitian or real symmetric matrices. The parameter $\tau$ also controls the
shape of the droplet to which the eigenvalues converge in the complex plane.
For large $N$, the eGinOE and eGinUE have been studied in two distinctly
different regimes of $\tau$, namely strong and weak non-Hermiticity,
abbreviated as SNH and WNH respectively. At SNH, $\tau\in[0,1)$ and remains
fixed as $N\to\infty$, whereas at WNH, $\tau$ asymptotically approaches unity
as $N$ becomes large, via $\tau=1-(\pi\alpha)^{2}/2N$, where $\alpha$ remains
finite.

The density of complex eigenvalues at finite-$N$ has previously been found in
both the eGinUE and eGinOE, see [2, 4] for the eGinUE and [25, 52] for the
eGinOE. In the eGinUE, the result is given by

$\rho^{(\text{eGinUE},c)}_{N}(z)=\frac{1}{\pi}\frac{1}{\sqrt{1-\tau^{2}}}\exp\left\\{-\frac{|z|^{2}-\tau\
\text{Re}(z^{2})}{1-\tau^{2}}\right\\}\sum_{k=0}^{N-1}\frac{\tau^{k}}{k!}{\rm
He\,}_{k}\left(\frac{z}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right)\ ,$ (2.3)

and correspondingly, in the eGinOE,

$\rho_{N}^{(\text{eGinOE,c})}(z)=\sqrt{\frac{2}{\pi}}\frac{\imaginary(z)}{1+\tau}\
\exp\left\\{\frac{\imaginary(z)^{2}-\real(z)^{2}}{1+\tau}\right\\}\text{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\
|\imaginary(z)|\right)P_{N-2}\ ,$ (2.4)

such that

$P_{N}=\frac{1}{\bar{z}-z}\sum_{k=0}^{N}\
\frac{\tau^{k+\frac{1}{2}}}{k!}\bigg{[}{\rm
He\,}_{k+1}\left(\frac{\bar{z}}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{z}{\sqrt{\tau}}\right)-{\rm
He\,}_{k+1}\left(\frac{z}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right)\bigg{]}\ .$ (2.5)

Note that in the above equations we have introduced the Hermite polynomials

${\rm
He\,}_{k}(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dy\,\exp{-\frac{1}{2}y^{2}}\,\left(x+iy\right)^{k}=k!\sum_{m=0}^{[k/2]}\frac{(-1)^{m}}{(k-2m)!m!}\frac{x^{k-2m}}{2^{m}}$
(2.6)

and the complementary error function, $\text{erfc}(x)=1-\erf(x)$, where
$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$. One should also note
that the density of real eigenvalues in the eGinOE is known at finite-$N$ and
in various asymptotic regimes, see [25, 54] for results at SNH and [20, 26,
55] for WNH. In this manuscript we will not discuss the density of real
eigenvalues, or associated self-overlaps further, instead focusing on complex
eigenvalues and their eigenvector self-overlaps.

It is well-known that, for large $N$, the complex eigenvalues of both
ensembles fill the interior of an ellipse in the complex plane, with semi-
major and semi-minor axes given by $\sqrt{N}(1+\tau)$ and $\sqrt{N}(1-\tau)$
respectively [12, 51]. In the case of the eGinUE, this ellipse features two
very distinct non-zero regimes of eigenvalue density for both SNH and WNH, a
spectral bulk and an edge region. On the other hand, the eGinOE features both
of these regions as well as an additional region of eigenvalue depletion close
to the real line, due to the presence of a macroscopic number of real
eigenvalues. These regimes are depicted in both ensembles for both strong and
weak non-Hermiticity in Figure 1.

Figure 1: Large $N$ density of complex eigenvalues in the eGinOE (left) and
eGinUE (right) in the strong (top) and weak (bottom) non-Hermiticity regimes,
with relevant scaling regions labelled. Strong non-Hermiticity plots are shown
for $\tau=0.25$ and weak non-Hermiticity plots are shown for $\alpha=1$. In
all plots, the density is depicted on a rainbow scale and the solid black
ellipse has semi-major and semi-minor axes $\sqrt{N}(1+\tau)$ and
$\sqrt{N}(1-\tau)$ respectively. Each heatmap is plotted using the finite-$N$
expression for the density of complex eigenvalues in the associated ensemble
when $N=125$. This diagram was originally presented in [17].

The other main object of interest in this work is the mean self-overlap
between left and right eigenvectors. This object was recently calculated at
finite-$N$ and for $\tau\in[0,1)$ in [17] for both the eGinUE and eGinOE. In
the eGinUE it was found that

$\mathcal{O}^{(\textup{eGinUE},c)}_{N}(z)=\rho^{(\textup{eGinUE},c)}_{N}(z)+(1-\tau^{2})\bigg{[}\,\rho^{(\textup{eGinUE},c)}_{N-1}(z)+(N-2)\,\rho^{(\textup{eGinUE},c)}_{N-2}(z)-R_{N-3}\,\bigg{]}\
,$ (2.7)

where $\rho^{(\textup{eGinUE},c)}_{N}(z)$ is defined in Eq. (2.3) and

$\begin{split}R_{N}&\equiv\frac{1}{\pi}\ \frac{1}{\sqrt{1-\tau^{2}}}\
\exp\left\\{-\frac{|z|^{2}-\tau\,{\rm
Re\,}(z^{2})}{1-\tau^{2}}\right\\}\sum_{k=0}^{N}\ \frac{k\ \tau^{k}}{k!}\,{\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{z}{\sqrt{\tau}}\right)\ .\end{split}$ (2.8)

Correspondingly, in the eGinOE it was found that

$\displaystyle\mathcal{O}^{(\textup{eGinOE},c)}_{N}(z)$
$\displaystyle=\frac{1}{\pi}\ \sqrt{\frac{1-\tau}{1+\tau}}e^{-\frac{{\rm
Re\,}(z)^{2}}{1+\tau}-\frac{{\rm
Im\,}(z)^{2}}{1-\tau}}\bigg{[}1+\sqrt{\frac{\pi(1-\tau^{2})}{2}}e^{\frac{2{\rm
Im\,}(z)^{2}}{1-\tau^{2}}}\,\frac{1}{2|{\rm Im\,}(z)|}\
\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\ |{\rm
Im\,}(z)|\right)\bigg{]}$
$\displaystyle\times\bigg{[}P_{N-2}+(1-\tau^{2})\bigg{(}P_{N-3}+(N-3)P_{N-4}-T_{N-4}\bigg{)}\bigg{]}\
,$ (2.9)

where $P_{N}$ is defined in Eq. (2.5) and

$T_{N}\equiv\frac{1}{\bar{z}-z}\sum_{k=0}^{N}\ \frac{k\
\tau^{k+\frac{1}{2}}}{k!}\bigg{[}{\rm
He\,}_{k+1}\left(\frac{\bar{z}}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{z}{\sqrt{\tau}}\right)-{\rm
He\,}_{k+1}\left(\frac{z}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right)\bigg{]}\ .$ (2.10)

######  Remark 2.1.

This manuscript will not discuss the density of complex eigenvalues or the
mean self-overlap in the bulk or depletion regime, for neither SNH nor WNH,
but will focus solely on the edge region. Results for the bulk region of the
eGinOE and eGinUE and the depletion regime of the eGinOE can be found in [17,
Section 2]. One can also find more information on the correlation structure
and methods of sampling matrices in the eGinUE and eGinOE within [17].
Additionally, for a review of other results available in the eGinUE and eGinOE
one can consult [4] and [5] respectively.

In the work by Lee and Riser [15], the authors consider the density of
eigenvalues at the edge of the eGinUE in the strong non-Hermiticity regime.
Here, they derive the leading order term in $N$ and the first two subleading
correctional terms. In their setting, they find that the leading order term is
independent of the curvature of the ellipse and solely depends on the
perpendicular distance from the ellipse edge. Additionally, the density of
complex eigenvalues at the edge of the eGinUE at weak non-Hermiticity can be
found within the work by Garciá-Garciá et. al. [16], who derive the
correlation kernel of the corresponding eigenvalue point process in this
limit, from which the density follows straightforwardly. These results provide
useful points of comparison and so we state their findings as part of the
following section.

###  2.2 Statement of Main Results

In this section, we outline the main results and definitions that are central
to this paper. We begin by providing results relevant to the strong non-
Hermiticity regime and then move onto the weak non-Hermiticity regime. At the
beginning of each section, we provide a definition of the edge in each
specific regime of $\tau$.

####  2.2.1 Strong non-Hermiticity Regime

In order to state our main results for the eGinUE and eGinOE at strong non-
Hermiticity, we first carefully define the scaling necessary to describe the
edge region of the droplet. A point on the boundary of the elliptic droplet is
defined as

$z_{0}=\sqrt{N}(1+\tau)\cos(\theta)+i\sqrt{N}(1-\tau)\sin(\theta)\ ,$ (2.11)

where in the eGinUE we have $0\leq\theta<2\pi$. However, in order to avoid the
edge of the depletion regime in the eGinOE, we require that
$|\sin(\theta)|\sim O(1)$. The unit normal vector from the point $z_{0}$ can
be presented in the form

$\widehat{\bm{n}}=\frac{1}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}\left(\begin{matrix}(1-\tau)\cos(\theta)\\\
(1+\tau)\sin(\theta)\end{matrix}\right)\ ,$ (2.12)

which we utilise to define another point in the vicinity of the edge, denoted
as $z$. Specifically, we use a small perturbation, $\eta\,\widehat{\bm{n}}$,
from $z_{0}$, such that we can write the real and imaginary parts of $z$ as

$\displaystyle{\rm
Re\,}(z)=\sqrt{N}(1+\tau)\cos(\theta)+\frac{\eta(1-\tau)\cos(\theta)}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}=\sqrt{N}x_{e}+w_{x}$
(2.13) $\displaystyle{\rm
Im\,}(z)=\sqrt{N}(1-\tau)\sin(\theta)+\frac{\eta(1+\tau)\sin(\theta)}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}=\sqrt{N}y_{e}+w_{y}\
.$

In order to derive the density of complex eigenvalues for the edge of the
eGinUE at SNH, one can utilise the above definition of $z$ in the finite-$N$
result for the density, Eq. (2.3), for a fixed $\tau\in[0,1)$ and then take
$N\to\infty$. This yields

$\displaystyle\rho_{\text{SNH,edge}}^{\text{(eGinUE,c)}}(\eta)=\lim_{N\to\infty}\rho_{N}^{\text{(eGinUE,c)}}\Big{(}\big{(}\sqrt{N}x_{e}+w_{x}\big{)}+i\big{(}\sqrt{N}y_{e}+w_{y}\big{)}\Big{)}=\frac{1}{2\pi(1-\tau^{2})}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\,\eta\right)\
,$ (2.14)

which is independent of the angle $\theta$ and only depends on $\eta$. The
above result for the eGinUE coincides with one found by Lee and Riser [15],
and our next statement extends its validity for the case of the eGinOE,
confirming the expected universality of the density of complex eigenvalues at
the edge.

######  Theorem 2.2.

Let $X$ be an $N\times N$ matrix drawn from the eGinOE via Eq. (2.2). Let $z$
be a complex eigenvalue of $X$, scaled in the strong non-Hermiticity limit,
where $\tau\in[0,1)$ is fixed and $z$ obeys Eq. (2.13). In this limit, as $N$
becomes large, the mean density of complex eigenvalues reads

$\displaystyle\rho_{\textup{SNH,edge}}^{\textup{(eGinOE,c)}}(\eta)=\lim_{N\to\infty}\rho_{N}^{\textup{(eGinOE,c)}}\Big{(}\big{(}\sqrt{N}x_{e}+w_{x}\big{)}+i\big{(}\sqrt{N}y_{e}+w_{y}\big{)}\Big{)}=\frac{1}{2\pi(1-\tau^{2})}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\,\eta\right)\
.$ (2.15)

In Figure 2 we plot the normalised density of complex eigenvalues with
$\eta>0$,

$\widetilde{\rho}^{\,\text{(ensemble)}}_{\,\text{SNH,edge}}(\eta)=\frac{\rho_{\text{SNH,
edge}}^{\text{(ensemble)}}(\eta)}{\int_{0}^{\infty}\rho_{\text{SNH,
edge}}^{\text{(ensemble)}}(\eta^{\prime})\,d\eta^{\prime}}\ ,$ (2.16)

for both the eGinUE and eGinOE, with $\tau=0.25$ and $\tau=0.75$. The
universality of this density is evident from the Figure as the results for the
eGinUE and eGinOE are nearly identical.

Figure 2: Normalised density of complex eigenvalues at the edge of the eGinUE
(left) and eGinOE (right) at strong non-Hermiticity as a function of $\eta$,
for $\tau=0.25$ (red) and $\tau=0.75$ (green). In both plots the solid
(dotted) line is plotted using Eq. (2.16) with $\tau=0.25$ (0.75). Each
histogram is generated by retaining eigenvalues with $\eta>0$, from a sample
containing $O(10^{7})$ eigenvalues of eGinUE/eGinOE matrices of size $N=1000$.
In the eGinOE plot, eigenvalues with $|\sin(\theta)|\sim O(1)$ are not
included, so as to avoid the depletion regime.

One can then extend this analysis so as to consider the mean self-overlap of
eigenvectors at the edge, in both the eGinUE and eGinOE. This is achieved by
starting from the finite-$N$ results in Eqs. (2.7) and (2.9) and choosing the
edge to be defined as in Eq. (2.13). Keeping $\tau\in[0,1)$ fixed and taking
$N\to\infty$, one is able to obtain the leading order behaviour of the mean
self-overlap at the edge in the eGinUE and eGinOE.

######  Theorem 2.3.

Let $X$ be an $N\times N$ matrix drawn from the eGinUE via Eq. (2.1). Let $z$
be a complex eigenvalue of $X$, scaled in the strong non-Hermiticity limit,
where $\tau\in[0,1)$ is fixed and $z$ obeys Eq. (2.13). In this limit, as $N$
becomes large, the mean self-overlap of the associated left and right
eigenvectors reads

$\displaystyle\mathcal{O}_{\textup{SNH,edge}}^{\textup{(eGinUE,c)}}(\eta,\theta)=$
$\displaystyle\lim_{N\to\infty}\frac{1}{\sqrt{N}}\mathcal{O}_{N}^{\textup{(eGinUE,c)}}\Big{(}\big{(}\sqrt{N}x_{e}+w_{x}\big{)}+i\big{(}\sqrt{N}y_{e}+w_{y}\big{)}\Big{)}$
(2.17) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{1+\tau^{2}-2\tau\cos(2\theta)}{1-\tau^{2}}}\left[\frac{1}{\sqrt{2\pi}}\exp{\frac{-2\eta^{2}}{1-\tau^{2}}}-\frac{\eta}{\sqrt{{1-\tau^{2}}}}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\eta\right)\right]\
.$

Correspondingly, if one considers the same quantity in the eGinOE, with the
additional constraint in Eq. (2.13) that $|\sin(\theta)|\sim O(1)$, one finds
that:

$\displaystyle\mathcal{O}_{\textup{SNH,edge}}^{\textup{(eGinOE,c)}}(\eta,\theta)=$
$\displaystyle\lim_{N\to\infty}\frac{1}{\sqrt{N}}\mathcal{O}_{N}^{\textup{(eGinOE,c)}}\Big{(}\big{(}\sqrt{N}x_{e}+w_{x}\big{)}+i\big{(}\sqrt{N}y_{e}+w_{y}\big{)}\Big{)}$
(2.18) $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{1+\tau^{2}-2\tau\cos(2\theta)}{1-\tau^{2}}}\left[\frac{1}{\sqrt{2\pi}}\exp{\frac{-2\eta^{2}}{1-\tau^{2}}}-\frac{\eta}{\sqrt{{1-\tau^{2}}}}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\eta\right)\right]\
.$

This directly leads to the following Corollary:

######  Corollary 2.4.

In the strong non-Hermiticity limit, for both the eGinUE and eGinOE, the mean
weighted conditional self-overlap of eigenvectors associated with a complex
eigenvalue $z$, scaled via Eq. (2.13), reads:

$\displaystyle\widetilde{\mathbb{E}}^{\textup{(eGin,c)}}_{\textup{SNH,edge}}(\tilde{\eta})=$
$\displaystyle\frac{1}{2\sqrt{(1-\tau^{2})(1+\tau^{2}-2\tau\cos(2\theta))}}\frac{\mathcal{O}_{\textup{SNH,edge}}^{\textup{(eGin,c)}}(\eta,\theta)}{\rho_{\textup{SNH,edge}}^{\textup{(eGin,c)}}(\eta)}$
(2.19) $\displaystyle=$
$\displaystyle\frac{1}{\textup{erfc}(\sqrt{2}\tilde{\eta})}\left[\frac{1}{\sqrt{2\pi}}\exp{-2\widetilde{\eta}^{\,2}}-\widetilde{\eta}\,\textup{erfc}(\sqrt{2}\widetilde{\eta})\right]\
.$

Here we have used eGin as a collective shorthand to denote both the eGinOE and
eGinUE and $\tilde{\eta}=\eta/\sqrt{1-\tau^{2}}$.

These results establish universality of the leading order behaviour of the
mean self-overlap at the edge for the eGinUE and eGinOE, in the strong non-
Hermiticity regime. Note that, in contrast to the mean density, the limiting
mean self-overlap in both ensembles is dependent on both $\eta$ and $\theta$.
In Figure 3, we compare these findings for the edge in the eGinUE and eGinOE
at strong non-Hermiticity to direct numerical simulation of the mean weighted
conditional self-overlap. These plots demonstrate a good agreement between
theory and simulations when $N=2500$, which is a much larger value of $N$ than
was required for previous results pertaining to the spectral bulk [17]. This
is due to the fact that the leading order asymptotic term in the mean self-
overlap is proportional to $\sqrt{N}$ at the edge, as opposed to $N$ in the
bulk.

Figure 3: Mean weighted conditional self-overlap of eigenvectors associated
with complex eigenvalues at the elliptic edge in the eGinUE (left) and eGinOE
(right) at strong non-Hermiticity, for a range of $\tau$ (coloured markers).
The solid black line in both plots is the universal prediction for the mean
weighted conditional self-overlap from Eq. (2.19). Each coloured marker is a
numerical average of self-overlaps associated with complex eigenvalues within
$\pm 1/\sqrt{N}$ of the specified $\eta$, taken from a set containing
$O(10^{6})$ eigenvalues of eGinUE and eGinOE matrices of size $N=2500$. Note
that, in order to avoid the depletion regime in the eGinOE, we do not consider
eigenvalues with imaginary component smaller than one.

######  Remark 2.5.

Setting $\tau=0$ in Eqs. (2.14), (2.15), (2.17) and (2.18) accurately
reproduces results pertaining to the edge of the classical Ginibre ensembles,
GinUE and GinOE, which are independent of $\theta$, as outlined in [39].
Therefore, one might have suspected similar angular independence to be
observed for the mean self-overlap in the elliptic Ginibre ensembles. However,
upon further consideration, the angular dependence in Eqs. (2.17) and (2.18)
can be reconciled by noting that in [17, 39] the self-overlap is
systematically reduced as eigenvalues move further away from the origin.
Hence, in an elliptic ensemble at fixed $\eta$, it is unsurprising that there
is a non-trivial dependence on $\theta$.

####  2.2.2 Weak non-Hermiticity Regime

In the limit of weak non-Hermiticity, the correlation parameter $\tau$
approaches unity asymptotically as $N\to\infty$, hence the droplet is reduced
to a thin strip, close to the real line. We define this regime such that

$\tau=1-\frac{\kappa}{N}\,,\hskip 28.45274pt\text{and}\hskip
28.45274pt\kappa=\frac{(\pi\alpha)^{2}}{2}\ ,$ (2.20)

where the choice of $\kappa$ is to allow one to make connections with existing
literature results [17, 53]. We choose to parameterise the edge in the same
way as in [16], such that

$z=2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\ .$ (2.21)

Strictly speaking, this should be referred to as the rightmost edge of the
ellipse, however, due to symmetry arguments, these results also hold at the
left-most edge, where ${\rm Re\,}(z)=-2\sqrt{N}-xN^{-1/6}$. For simplicity we
focus on $y>0$ henceforth, note however that results can also be extrapolated
to negative $y$ via simple symmetry arguments. In the above edge scaling at
weak non-Hermiticity, it is known that, in the eGinUE, the limiting density of
complex eigenvalues reads

$\displaystyle\rho_{\textup{WNH,edge}}^{\textup{(eGinUE,c)}}(x,y)=$
$\displaystyle\lim_{N\to\infty}\frac{1}{N^{2/3}}\rho_{N}^{\textup{(eGinUE,c)}}\bigg{(}z=2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\bigg{)}=\frac{1}{\sqrt{\pi\kappa}}e^{-\frac{y^{2}}{\kappa}}\int_{0}^{\infty}dt\,\textup{Ai}(x+t)^{2}\
,$ (2.22)

where we have made use of the well-known Airy function defined as

$\textup{Ai}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\,\exp{i\left(\frac{t^{3}}{3}+zt\right)}=\frac{1}{2\pi
i}\int_{\leavevmode\hbox to3.3pt{\vbox
to6.11pt{\pgfpicture\makeatletter\hbox{\hskip 1.19998pt\lower-1.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}{}\pgfsys@moveto{1.085pt}{-1.05788pt}\pgfsys@lineto{-0.81375pt}{1.6275pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}{}{{
{\pgfsys@beginscope\pgfsys@setlinewidth{0.32pt}\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{}
{}{}{} {}{}{}
\pgfsys@moveto{-1.19998pt}{1.59998pt}\pgfsys@curveto{-1.09998pt}{0.99998pt}{0.0pt}{0.09999pt}{0.29999pt}{0.0pt}\pgfsys@curveto{0.0pt}{-0.09999pt}{-1.09998pt}{-0.99998pt}{-1.19998pt}{-1.59998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{-0.81375pt}{1.6275pt}\pgfsys@lineto{1.57185pt}{3.98926pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.71065}{0.70354}{-0.70354}{0.71065}{1.57185pt}{3.98924pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}dp\,\exp{\frac{p^{3}}{3}-zp}\
,$ (2.23)

such that  is a contour that starts at $\infty$ with an argument of $-\pi/3$
and ends at $\infty$ with an argument of $\pi/3$. The statement below provides
the counterpart of the density in Eq. (2.22), for the case of the eGinOE,
which, to the best of our knowledge, has not yet been studied in the regime
defined above.

######  Theorem 2.6.

Let $X$ be an $N\times N$ matrix drawn from the eGinOE, via Eq. (2.2). Let $z$
be a complex eigenvalue of $X$, scaled in the weak non-Hermiticity limit,
where $\tau=1-\kappa/N$ and $z$ obeys Eq. (2.21). In this limit, as $N$
becomes large, the mean density of complex eigenvalues reads

$\displaystyle\rho_{\textup{WNH,edge}}^{\textup{(eGinOE,c)}}(x,y)=$
$\displaystyle\lim_{N\to\infty}\rho_{N}^{\textup{(eGinOE,c)}}\Big{(}2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\Big{)}$
(2.24) $\displaystyle=$
$\displaystyle\,y\,\textup{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\int_{0}^{\infty}dt\bigg{[}\textup{Ai}^{\prime}(x+t)^{2}-\textup{Ai}(x+t)\,\textup{Ai}^{\prime\prime}(x+t)\bigg{]}\
.$

This is given in terms of $\textup{Ai}^{\prime}(w)$ and
$\textup{Ai}^{\prime\prime}(w)$, which denote the first and second derivatives
of the Airy function, defined in Eq. (2.23), evaluated at the point $w$.

Note that both densities given in Eqs. (2.22) and (2.24) can have their $x$
and $y$ dependencies factorised such that
$\rho_{\text{WNH,edge}}^{\textup{(eGin,c)}}(x,y)=\rho^{\textup{(eGin,c)}}_{\text{WNH},X}(x)\rho^{\textup{(eGin,c)}}_{\text{WNH},Y}(y)$.
This indicates that, at WNH, the leading order behaviour of the real and
imaginary components of edge eigenvalues are independent variables. Therefore,
we introduce the normalised density of the real and imaginary components,
through the use of

$\widetilde{\rho}^{\textup{(eGin,c)}}_{\text{WNH},X}(x)=\frac{\rho^{\textup{(eGin,c)}}_{\text{WNH},X}(x)}{\int_{S_{R}}\rho^{\textup{(eGin,c)}}_{\text{WNH},X}(x^{\prime})dx^{\prime}}\hskip
14.22636pt\text{and}\hskip
14.22636pt\widetilde{\rho}^{\textup{(eGin,c)}}_{\text{WNH},Y}(y)=\frac{\rho^{\textup{(eGin,c)}}_{\text{WNH},Y}(y)}{\int_{-\infty}^{\infty}\rho^{\textup{(eGin,c)}}_{\text{WNH},Y}(y^{\prime})dy^{\prime}}\
,$ (2.25)

where $S_{R}$ is some subset of the real line. In Figure 4 we plot the above
distributions and compare them to results from direct numerical simulation of
large eGinUE and eGinOE matrices, choosing to use $S_{R}=[-5,1]$.

Figure 4: Normalised density of scaled real (left) and imaginary (right)
components of complex eigenvalues at the edge of the elliptic droplet for
eGinOE (green) and eGinUE (red) matrices. This is shown for the weak non-
Hermiticity regime of $\tau$, with fixed $\alpha=0.5$. Each plot contains
normalised histograms of eigenvalues obtained through numerical
diagonalisation of $O(10^{7})$ eGinUE and eGinOE matrices of size $N=1000$.
Solid (dashed) lines are generated from the theoretical density of complex
eigenvalues for the eGinUE (eGinOE) and normalised according to Eq. (2.25).

We now consider the mean self-overlap at the edge for both the eGinUE and
eGinOE at WNH. In contrast to what was found in the case of SNH, the results
here do not turn out to be universal across the two ensembles.

######  Remark 2.7.

At weak non-Hermiticity in the eGinUE, it can be seen that the mean self-
overlap of eigenvectors associated with complex edge eigenvalues,
asymptotically approaches the density, i.e. $\mathcal{O}(z)\approx\rho(z)$ as
$N\to\infty$. As a consequence of this, the mean conditional self-overlap goes
to unity, i.e, $\mathbb{E}\to 1$. For this reason, in the eGinUE, we make the
choice to study the mean _shifted_ conditional self-overlap,
$\mathcal{S}=\mathbb{E}-1$, so as to obtain a non-trivial result to compare to
numerical data.

######  Theorem 2.8.

Let $X$ be an $N\times N$ matrix drawn from the eGinUE via Eq. (2.1). Let $z$
be a complex eigenvalue of $X$, scaled in the weak non-Hermiticity limit,
where $\tau=1-\kappa/N$ and $z$ obeys Eq. (2.21). In this limit, as $N$
becomes large, the mean shifted conditional self-overlap of the associated
left and right eigenvectors reads

$\displaystyle\mathcal{S}^{\textup{(eGinUE,c)}}_{\textup{WNH, edge}}(x)=$
$\displaystyle\lim_{N\to\infty}N^{2/3}\bigg{(}\mathbb{E}^{\textup{(eGinUE,c)}}_{N}\left(z=2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\right)-1\bigg{)}=\frac{2\kappa\int_{0}^{\infty}dt\,t\,\textup{Ai}(x+t)^{2}}{\int_{0}^{\infty}dt\,\textup{Ai}(x+t)^{2}}\
,$ (2.26)

where $\textup{Ai}(\cdot)$ denotes the Airy function, defined in Eq. (2.23).

When considering the mean self-overlap at the edge of the eGinOE at WNH, one
finds contrasting behaviour to the eGinUE in the same limit. Namely, due to
the presence of the depletion regime, the mean self-overlap does not approach
the density as $N$ becomes large. Hence, in this case, it does make sense to
study the conventional mean conditional self-overlap, with the result given by
the next Theorem:

######  Theorem 2.9.

Let $X$ be an $N\times N$ matrix drawn from the eGinOE via Eq. (2.2). Let $z$
be a complex eigenvalue of $X$, scaled in the weak non-Hermiticity limit,
where $\tau=1-\kappa/N$ and $z$ obeys Eq. (2.21). In this limit, as $N$
becomes large, the mean conditional self-overlap of the associated left and
right eigenvectors reads

$\displaystyle\mathbb{E}_{\textup{WNH,edge}}^{\textup{(eGinOE,c)}}(y)=$
$\displaystyle\lim_{N\to\infty}\mathbb{E}^{\textup{(eGinOE,c)}}_{N}\bigg{(}z=2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\bigg{)}$
(2.27) $\displaystyle=$
$\displaystyle\frac{1}{y\,\textup{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)}\bigg{[}\sqrt{\frac{\kappa}{\pi}}\exp{\frac{-y^{2}}{\kappa}}+\frac{\kappa}{2y}\textup{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\bigg{]}\
.$

In Figure 5, we plot the mean shifted conditional self-overlap for the eGinUE
and the mean conditional self-overlap for the eGinOE and compare to direct
numerical simulation of the self-overlap of left and right eigenvectors
associated with complex eigenvalues at the edge. In both plots, one finds a
good agreement with theoretical predictions at large $N$, with minor
discrepancies attributed to finite-$N$ effects. This is particularly relevant
to the results pertaining to the eGinUE (left-hand plot).

######  Remark 2.10.

The results outlined in this section show that, when comparing the eGinUE and
eGinOE, the density of complex eigenvalues and the mean self-overlap are not
universal quantities at the edge, in the limit of weak non-Hermiticity. This
phenomenon is not unexpected and follows from the fact that the eGinOE has a
macroscopic fraction of real eigenvalues and a region of eigenvalue depletion
close to the real line, whereas all the eigenvalues are complex in the case of
the eGinUE.

Figure 5: Conditional self-overlap of eigenvectors associated with complex
eigenvalues at the edge of elliptic Ginibre ensembles at weak non-Hermiticity,
with fixed $\alpha=0.5$ for large finite-$N$. Left: mean shifted conditional
self-overlap in the eGinUE,
$\mathcal{S}^{\text{(eGinUE,c)}}_{\text{WNH,edge}}(x)$, as a function of $x$.
Right: mean conditional self-overlap in the eGinOE,
$\mathbb{E}^{\text{(eGinOE,c)}}_{\text{WNH,edge}}(y)$, as a function of $y$.
The solid black line is plotted according to Eqs. (2.26) and (2.27) in the
left and right hand plots respectively. Each coloured marker is a numerical
average of self-overlaps observed in simulation of eGinUE/eGinOE matrices,
taken from a sample containing $O(10^{7})$ eigenvalues. In simulation, self-
overlaps associated with a fixed $x$ are obtained by retaining all $x$ within
$\pm 0.1$ of the desired $x$, for all $y$. The opposite is true for a fixed
$y$.

###  2.3 Discussion of Open Problems

Despite many recent advances in the study of eigenvalues and eigenvectors in
the Ginibre ensembles, some questions still remain open. For example, in terms
of the eigenvalue density at the edge in the strong non-Hermiticity regime,
the analytic extension of Eqs. (2.14) and (2.15) to the eGinSE remains an open
problem to the best of our knowledge. Although some progress has already been
made in this direction, stemming from works such as [56, 57, 58]. In contrast,
there is no such universality to be extended in the realm of weak non-
Hermiticity, therefore, the edge density of complex eigenvalues is very much
an open question in the WNH regime of the eGinSE.

Similar questions remain as to the universality of the mean self-overlap. One
would conjecture that the results of Theorem 2.3 could be extended to the
eGinSE at SNH, due to the expected universality. On the other hand, at WNH,
the corresponding results in the eGinSE are entirely unknown. Furthermore, in
comparison to the eGinOE and eGinUE, results for the mean self-overlap are
more scarce in the eGinSE, as questions remain over a suitable finite-$N$
expression for the mean self-overlap and various other results at strong and
weak non-Hermiticity.

Additionally, it is natural to wish to build upon results for the mean self-
overlap by considering higher moments of $\mathcal{O}_{nn}$, or even the
entire distribution. For example, the finite-$N$ distribution of self-overlaps
associated with complex eigenvalues remains an open problem in the: GinOE,
GinSE and all elliptic Ginibre ensembles. Finally, one can also consider
statistics of the off-diagonal entries of the overlap matrix, Eq. (1.3), so as
to ascertain if there is a universality that can be linked to pre-existing
results found, for example, in [32, 35, 40].

##  3 Derivation of Main Results

###  3.1 Preliminaries for Analysis

Before we go on to prove our main results, it is convenient for us to briefly
introduce some important identities that will be commonly utilised throughout
our proofs. The first of these is the following integral representations of
the Hermite polynomials,

$\displaystyle{\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right)=\frac{i^{k}}{\sqrt{2\pi\tau}}\exp{\frac{\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\exp{-\frac{t_{1}^{2}}{2\tau}-i\frac{\bar{z}}{\tau}t_{1}}\,t_{1}^{k}\,\tau^{-k/2}$
(3.1)

and the corresponding representation of ${\rm He\,}_{k}(z/\sqrt{\tau})$, where
we choose to exchange $i\to-i$. Upon introducing these integral
representations in all proofs, one must then utilise the incomplete
$\Gamma$-function, defined as

$\Gamma\left(N+1,x\right)=\Gamma\left(N+1\right)\,e^{-x}\,\sum_{k=0}^{N}\frac{x^{k}}{k!}=x^{N+1}\int_{1}^{\infty}du\,u^{N}e^{-ux}\
,$ (3.2)

where $\Gamma(N)=(N-1)!$. Finally, for our proofs at weak non-Hermiticity, it
is useful to employ the following asymptotic behaviour of the incomplete
$\Gamma$-function

$\Gamma(m+1,x)=e^{-x}\frac{x^{m+1}}{x-m}\left[1+O\left(\frac{m}{(x-m)^{2}}\right)...\right]\
,$ (3.3)

valid for $x\gtrsim m$ and $m\gg 1$, cf. [16].

###  3.2 Strong non-Hermiticity proofs

####  3.2.1 Proof of Theorem 2.2 \- Density in the eGinOE

In order to derive the density of complex eigenvalues at the edge in the
eGinOE at strong non-Hermiticity, we start from the corresponding finite-$N$
equation, given in Eqs. (2.4) and (2.5). Initially, the prefactors can be
simplified by noting that, in this regime, ${\rm Im\,}(z)\sim O(\sqrt{N})$.
This can be used alongside the large argument behaviour of the complementary
error function, $\text{erfc}(x)\sim e^{-x^{2}}/(\sqrt{\pi}x)$ for $x\gg 1$, to
show that,

$\sqrt{\frac{2}{\pi}}\frac{{\rm Im\,}(z)}{1+\tau}\ \exp\left\\{\frac{{\rm
Im\,}(z)^{2}-{\rm
Re\,}(z)^{2}}{1+\tau}\right\\}\text{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\
|{\rm
Im\,}(z)|\right)\approx\frac{1}{\pi}\sqrt{\frac{1-\tau}{1+\tau}}\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}}\
.$ (3.4)

At this point, we introduce the integral representations of the Hermite
polynomials given in Eq. (3.1) and then utilise the series representation of
the incomplete $\Gamma$-function in Eq. (3.2). After some manipulation, the
density can be written as

$\displaystyle\rho_{N}^{\textup{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{-1}{4\pi^{2}\tau}\frac{1}{{\rm
Im\,}(z)}\sqrt{\frac{1-\tau}{1+\tau}}\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}$
(3.5)
$\displaystyle\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\,(t_{1}+t_{2})\,\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}+\frac{i}{\tau}\left(zt_{2}-\bar{z}t_{1}\right)}\frac{\Gamma(N+1,t_{1}t_{2})}{\Gamma(N+1)}\
.$

We now make use of the substitution $\sqrt{N}p=t_{1}+t_{2}$ and
$\sqrt{N}q=t_{1}-t_{2}$, such that

$t_{1}^{2}+t_{2}^{2}=\frac{N(p^{2}+q^{2})}{2}\ ,\hskip
28.45274ptt_{1}\,t_{2}=\frac{N(p^{2}-q^{2})}{4}\ ,\hskip
28.45274ptdt_{1}\,dt_{2}=\frac{N}{2}\,dp\,dq\ .$ (3.6)

Next, we utilise the integral representation of the incomplete
$\Gamma$-function from Eq. (3.2) and insert the parameterisation of the edge
given in Eq. (2.13), yielding

$\displaystyle\rho$
${}_{N}^{\textup{(eGinOE,c)}}(z)\approx\frac{-N^{N+3/2}}{8\pi^{2}\tau\Gamma(N)}\frac{1}{{\rm
Im\,}(z)}\sqrt{\frac{1-\tau}{1+\tau}}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{1}^{\infty}du\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{p(p^{2}-q^{2})}{4}e^{-\frac{i\sqrt{N}w_{x}q+\sqrt{N}w_{y}p}{\tau}}$
$\displaystyle\exp{N\left[\frac{(1-u)(p^{2}-q^{2})}{4}-\frac{p^{2}+q^{2}}{4\tau}-\frac{i(1+\tau)\cos(\theta)q}{\tau}-\frac{(1-\tau)\sin(\theta)p}{\tau}+\ln(\frac{u(p^{2}-q^{2})}{4})\right]}\
.$ (3.7)

To obtain leading order asymptotic behaviour of the above equation, one must
locate the maximum of the function in square brackets. We find that the
dominant extremal point is given by
$\left(p^{*},q^{*},u^{*}\right)=\left(-2\sin(\theta),-2i\cos(\theta),1\right)$,
thus, by slightly deforming the contour, we expand around the maximum using

$p=-2\sin(\theta)+\frac{r}{\sqrt{N}}\hskip
28.45274ptq=-2i\cos(\theta)+\frac{s}{\sqrt{N}}\hskip
28.45274ptu=1+\frac{v}{\sqrt{N}}\ .$ (3.8)

The expansion around this maximum within the leading exponential is a rather
cumbersome process and so we relegate the details to Appendix A. Upon
completion of this procedure, we utilise Eq. (A.7) and find that

$\displaystyle\rho_{N}^{\textup{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{N^{N}e^{-N}}{4\pi^{2}\tau\Gamma(N)}\frac{\sin(\theta)}{{\rm
Im\,}(z)}\sqrt{\frac{1-\tau}{1+\tau}}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}+\frac{2\sqrt{N}(w_{y}\sin(\theta)-w_{x}\cos(\theta))}{\tau}-\frac{N(\tau+\cos(2\theta))}{\tau}}$
$\displaystyle\int_{0}^{\infty}dv\,e^{-\frac{v^{2}}{2}}\int_{-\infty}^{\infty}dr\int_{-\infty}^{\infty}ds\exp{-\frac{r^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{w_{y}r}{\tau}+rv\sin(\theta)}$
(3.9)
$\displaystyle\exp{-\frac{s^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{iw_{x}s}{\tau}-isv\cos(\theta)+irs\cos(\theta)\sin(\theta)}\
,$

where, in order to extract the leading order behaviour, we have used that

$\frac{p(p^{2}-q^{2})}{4}\to\frac{1}{4}\left(\frac{r}{\sqrt{N}}-2\sin(\theta)\right)\left[\left(\frac{r}{\sqrt{N}}-2\sin(\theta)\right)^{2}-\left(\frac{s}{\sqrt{N}}-2i\cos(\theta)\right)^{2}\right]=-2\sin(\theta)+O\left(N^{-1/2}\right)\
.$ (3.10)

Therefore, one must calculate the coupled Gaussian integrals over $r$ and $s$
using the following identity

$\int_{-\infty}^{\infty}dr\,\exp{-\alpha r^{2}+\beta
r}\int_{-\infty}^{\infty}ds\,\exp{-as^{2}+bs+crs}=\frac{2\pi}{\sqrt{4\alpha
a-c^{2}}}\exp{\frac{\alpha b^{2}+b\beta c+a\beta^{2}}{4\alpha a-c^{2}}}\ ,$
(3.11)

where in our case, with the help of Mathematica, the prefactor and exponent
can be evaluated as

$\frac{2\pi}{\sqrt{4\alpha
a-c^{2}}}=\frac{4\pi\tau}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}$ (3.12)

and

$\frac{\alpha b^{2}+b\beta c+a\beta^{2}}{4\alpha
a-c^{2}}=\frac{\eta^{2}\Big{(}\tau-\cos(2\theta)\Big{)}-2v\eta\tau\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}+v^{2}\tau^{2}\Big{(}\tau-\cos(2\theta)\Big{)}}{\tau(1+\tau^{2}-2\tau\cos(2\theta))}\
,$ (3.13)

respectively. Thus, after some additional manipulations on the first
exponential, again involving Mathematica, this yields that

$\displaystyle\rho_{N}^{\textup{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{N^{N}e^{-N}}{\pi\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}\Gamma(N)}\frac{\sin(\theta)}{{\rm
Im\,}(z)}\sqrt{\frac{1-\tau}{1+\tau}}e^{-\frac{\eta^{2}(\tau(\tau^{2}+3)-(1+3\tau^{2})\cos(2\theta))}{\tau(1-\tau^{2})(1+\tau^{2}-2\tau\cos(2\theta))}+\frac{\eta^{2}(\tau-\cos(2\theta))}{\tau(1+\tau^{2}-2\tau\cos(2\theta))}}$
(3.14)
$\displaystyle\int_{0}^{\infty}dv\exp{-\frac{v^{2}}{2}+\frac{v^{2}\tau(\tau-\cos(2\theta))}{(1+\tau^{2}-2\tau\cos(2\theta))}-\frac{2v\eta}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}}\
.$

At this point, one can complete the square on the terms inside the exponential
of the final integral, such that

$\displaystyle-\frac{v^{2}}{2}+\frac{v^{2}\tau(\tau-\cos(2\theta))}{(1+\tau^{2}-2\tau\cos(2\theta))}-$
$\displaystyle\frac{2v\eta}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}$ (3.15)
$\displaystyle=-\frac{1-\tau^{2}}{2(1+\tau^{2}-2\tau\cos(2\theta))}\left(v+\frac{2\eta\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}{1-\tau^{2}}\right)^{2}+\frac{2\eta^{2}}{1-\tau^{2}}\
.$

Additionally, the prefactors can be simplified by employing Stirling’s formula
($\Gamma(N)=\sqrt{2\pi}N^{N-1/2}e^{-N}$) and using that, to leading order,
$\imaginary(z)=\sqrt{N}(1-\tau)\sin(\theta)$. This produces

$\displaystyle\rho_{N}^{\textup{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{1}{\pi\sqrt{2\pi(1-\tau^{2})(1+\tau^{2}-2\tau\cos(2\theta))}}\int_{0}^{\infty}dv\,e^{-\frac{1-\tau^{2}}{2(1+\tau^{2}-2\tau\cos(2\theta))}\left(v+\frac{2\eta\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}{1-\tau^{2}}\right)^{2}}$
(3.16) $\displaystyle=$
$\displaystyle\frac{1}{\pi\sqrt{2\pi}(1-\tau^{2})}\int_{\frac{2\eta}{\sqrt{1-\tau^{2}}}}^{\infty}du\,\exp{-\frac{u^{2}}{2}}\
.$

Thus, one can complete the proof by using the definition of the complementary
error function,
$\text{erfc}(x)=\sqrt{\frac{2}{\pi}}\int_{\sqrt{2}x}^{\infty}e^{-\frac{z^{2}}{2}}dz$,
to see that, for large $N$, the density is given by

$\rho_{N}^{\textup{(eGinOE,c)}}(z)\approx\frac{1}{2\pi(1-\tau^{2})}\text{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\eta\right)\
.$ (3.17)

####  3.2.2 Proof of Theorem 2.3 \- Self-Overlap in the eGinUE

Over the course of the next two sections, we provide a proof of the
universality of the leading order behaviour of the mean self-overlap of
eigenvectors at strong non-Hermiticity in the edge regime of the eGinUE and
eGinOE. Focusing first on the eGinUE, we begin by analysing the finite-$N$
equation for the mean self-overlap, given in Eq. (2.7). Since $1-\tau^{2}\sim
O(1)$ and $\rho_{N}^{\textup{(eGinUE,c)}}(z)\sim O(1)$ as $N\to\infty$ at SNH,
cf. [15], we see that the mean self-overlap in this region can be obtained
through

$\lim_{N\to\infty}\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)=(1-\tau^{2})\lim_{N\to\infty}\left[N\rho_{N+1}^{\textup{(eGinUE,c)}}(z)-R_{N}\right]\
.$ (3.18)

Similarly to the previous section, we now introduce the integral
representations of the Hermite polynomials from Eq. (3.1) and evaluate the
summations using the definition of the incomplete $\Gamma$-function in Eq.
(3.2), thus

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)\approx$
$\displaystyle\frac{\sqrt{1-\tau^{2}}}{2\pi^{2}\tau\Gamma(N)}\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}$
(3.19)
$\displaystyle\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}+\frac{i}{\tau}\left(zt_{2}-\bar{z}t_{1}\right)}\Big{(}\Gamma(N+1,t_{1}t_{2})-t_{1}t_{2}\Gamma(N,t_{1}t_{2})\Big{)}\
.$

One may now make use of the integral representation of the incomplete
$\Gamma$-function in Eq. (3.2), such that

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)\approx$
$\displaystyle\frac{\sqrt{1-\tau^{2}}}{2\pi^{2}\tau\Gamma(N)}\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{1}^{\infty}du\,\frac{u-1}{u}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\,\,t_{1}t_{2}$
(3.20)
$\displaystyle(ut_{1}t_{2})^{N}\exp{(1-u)t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}+\frac{i}{\tau}\left(zt_{2}-\bar{z}t_{1}\right)}\
.$

We now make the same change of variables as defined in the vicinity of Eq.
(3.6), with $\sqrt{N}p=t_{1}+t_{2}$ and $\sqrt{N}q=t_{1}-t_{2}$, such that

$\displaystyle\mathcal{O}$
${}_{N}^{\textup{(eGinUE,c)}}(z)\approx\frac{N^{N+2}\sqrt{1-\tau^{2}}}{4\pi^{2}\tau\Gamma(N)}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{1}^{\infty}du\,\frac{u-1}{u}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{p^{2}-q^{2}}{4}e^{-\frac{i\sqrt{N}w_{x}q+\sqrt{N}w_{y}p}{\tau}}$
$\displaystyle\exp{N\left[\frac{(1-u)(p^{2}-q^{2})}{4}-\frac{p^{2}+q^{2}}{4\tau}-\frac{i(1+\tau)\cos(\theta)q}{\tau}-\frac{(1-\tau)\sin(\theta)p}{\tau}+\ln(\frac{u(p^{2}-q^{2})}{4})\right]}\
,$ (3.21)

where $w_{x}$ and $w_{y}$ are defined in Eq. (2.13). As in the previous
section, the extremal point is located at
$\left(p^{*},q^{*},u^{*}\right)=\left(-2\sin(\theta),-2i\cos(\theta),1\right)$,
hence we deform the contour and use the substitutions defined in Eq. (3.8).
Following the steps outlined in Appendix A and utilising that
$(p^{2}-q^{2})/4\to 1+O(N^{-1/2})$, we find that

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)\approx$
$\displaystyle\frac{\sqrt{1-\tau^{2}}}{4\pi^{2}\tau}\frac{N^{N}e^{-N}}{\Gamma(N)}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}+\frac{2\sqrt{N}(w_{y}\sin(\theta)-w_{x}\cos(\theta))}{\tau}-\frac{N(\tau+\cos(2\theta))}{\tau}}$
$\displaystyle\int_{0}^{\infty}dv\,v\,e^{-\frac{v^{2}}{2}}\int_{-\infty}^{\infty}dr\int_{-\infty}^{\infty}ds\exp{-\frac{r^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{w_{y}r}{\tau}+rv\sin(\theta)}$
(3.22)
$\displaystyle\exp{-\frac{s^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{iw_{x}s}{\tau}-isv\cos(\theta)+irs\cos(\theta)\sin(\theta)}\
.$

The coupled Gaussian integrals over $r$ and $s$ can be evaluated using Eqs.
(3.11) to (3.13), Stirling’s formula can be applied and one can simplify the
exponential prefactor using Mathematica. Eventually this produces

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)\approx$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{N}{2\pi}\frac{1-\tau^{2}}{1+\tau^{2}-2\tau\cos(2\theta)}}e^{-\frac{\eta^{2}(\tau(\tau^{2}+3)-(1+3\tau^{2})\cos(2\theta))}{\tau(1-\tau^{2})(1+\tau^{2}-2\tau\cos(2\theta))}+\frac{\eta^{2}(\tau-\cos(2\theta))}{\tau(1+\tau^{2}-2\tau\cos(2\theta))}}$
(3.23)
$\displaystyle\int_{0}^{\infty}dv\,v\,\exp{-\frac{v^{2}}{2}+\frac{v^{2}\tau(\tau-\cos(2\theta))}{(1+\tau^{2}-2\tau\cos(2\theta))}-\frac{2v\eta}{\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}}\
.$

If one now completes the square, exactly as in Eq. (3.15), and employs the
identity

$\int_{0}^{\infty}dx\,x\exp{-\frac{a}{2}(x+b)^{2}}=\frac{1}{a}\exp{-\frac{ab^{2}}{2}}-b\sqrt{\frac{\pi}{2a}}\text{erfc}\left(b\sqrt{\frac{a}{2}}\right)\
,$ (3.24)

it becomes clear that

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinUE,c)}}(z)\approx$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{N}{2\pi}\frac{1-\tau^{2}}{1+\tau^{2}-2\tau\cos(2\theta)}}\int_{0}^{\infty}dv\,v\,e^{-\frac{1-\tau^{2}}{2(1+\tau^{2}-2\tau\cos(2\theta))}\left(v+\frac{2\eta\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}{1-\tau^{2}}\right)^{2}}$
(3.25) $\displaystyle=$
$\displaystyle\frac{\sqrt{N}}{\pi}\sqrt{\frac{1+\tau^{2}-2\tau\cos(2\theta)}{1-\tau^{2}}}\left[\frac{1}{\sqrt{2\pi}}\exp{\frac{-2\eta^{2}}{1-\tau^{2}}}-\frac{\eta}{\sqrt{{1-\tau^{2}}}}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\eta\right)\right]\
.$ (3.26)

Thus, by a simple rescaling and taking the limit of $N\to\infty$ we arrive at
Eq. (2.17), concluding the proof.

####  3.2.3 Proof of Theorem 2.3 \- Self-Overlap in the eGinOE

The proof of Theorem 2.3 in the eGinOE follows steps that are similar to the
ones outlined in the previous two proofs. With that in mind, we proceed to
highlight some key points of the calculation, dealing with large amounts of
the technical details through the use of previous material. We begin by
considering the form of the finite-$N$ result for the mean self-overlap
presented in Eq. (2.9). Immediately, one can note that, in the limit of large
$N$, the term containing the error function becomes negligible and the
dominant term within the square bracket is $NP_{N}-T_{N}$, therefore,

$\lim_{N\to\infty}\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)=\frac{(1-\tau)\sqrt{1-\tau^{2}}}{\pi}\lim_{N\to\infty}\left[\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}}\Big{(}NP_{N}-T_{N}\Big{)}\right]\
.$ (3.27)

Using the definitions of $P_{N}$ and $T_{N}$, Eqs. (2.5) and (2.10)
respectively, the integral representations of Hermite polynomials, Eq. (3.1),
and the series representation of the incomplete $\Gamma$-function, Eq. (3.2),
one finds that

$\displaystyle\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{(1-\tau)\sqrt{1-\tau^{2}}}{4\pi^{2}\tau\,\Gamma(N)}\left(\frac{-1}{{\rm
Im\,}(z)}\right)\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}$
(3.28)
$\displaystyle(t_{1}+t_{2})\,\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}+\frac{i}{\tau}\left(zt_{2}-\bar{z}t_{1}\right)}\Big{(}\Gamma(N+1,t_{1}t_{2})-t_{1}t_{2}\Gamma(N,t_{1}t_{2})\Big{)}\
.$

Introducing integral representations of the incomplete $\Gamma$-functions,
according to Eq. (3.2), one then arrives at

$\displaystyle\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{(1-\tau)\sqrt{1-\tau^{2}}}{4\pi^{2}\tau\,\Gamma(N)}\left(\frac{-1}{{\rm
Im\,}(z)}\right)\exp{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{0}^{\infty}du\,\frac{u-1}{u}$
(3.29)
$\displaystyle\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\,(t_{1}+t_{2})\,t_{1}t_{2}\,(ut_{1}t_{2})^{N}\,\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}+\frac{i}{\tau}\left(zt_{2}-\bar{z}t_{1}\right)}$

and upon employing the usual substitutions of $\sqrt{N}p=t_{1}+t_{2}$ and
$\sqrt{N}q=t_{1}-t_{2}$, we find that

$\displaystyle\mathcal{O}$
${}_{N}^{\text{(eGinOE,c)}}(z)\approx\frac{-N^{N+5/2}(1-\tau)\sqrt{1-\tau^{2}}}{8\pi^{2}\tau\,\Gamma(N)\,{\rm
Im\,}(z)}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{0}^{\infty}du\,\frac{u-1}{u}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{p(p^{2}-q^{2})}{4}$
(3.30) $\displaystyle
e^{-\frac{i\sqrt{N}w_{x}q}{\tau}-\frac{\sqrt{N}w_{y}p}{\tau}}\exp{N\left[\frac{(1-u)(p^{2}-q^{2})}{4}-\frac{p^{2}+q^{2}}{4\tau}-\frac{i(1+\tau)\cos(\theta)q}{\tau}-\frac{(1-\tau)\sin(\theta)p}{\tau}+\ln(\frac{u(p^{2}-q^{2})}{4})\right]}\
,$

with $w_{x}$ and $w_{y}$ defined in Eq. (2.13). As in the previous sections,
this integral is dominated at the maximum of the final exponential located at
$\left(p^{*},q^{*},u^{*}\right)=\left(-2\sin(\theta),-2i\cos(\theta),1\right)$.
For this reason, we expand around the maximum using the substitutions defined
in Eq. (3.8), as detailed in Appendix A, yielding

$\displaystyle\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{N^{N+1/2}(1-\tau)\sqrt{1-\tau^{2}}\sin(\theta)}{8\pi^{2}\tau\,\Gamma(N)\,{\rm
Im\,}(z)}e^{-\frac{|z|^{2}-\tau\real(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}\frac{2\sqrt{N}(w_{y}\sin(\theta)-w_{x}\cos(\theta))}{\tau}-\frac{N(\tau+\cos(2\theta))}{\tau}}$
$\displaystyle\int_{0}^{\infty}dv\,v\,e^{-\frac{v^{2}}{2}}\int_{-\infty}^{\infty}dr\int_{-\infty}^{\infty}ds\exp{-\frac{r^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{w_{y}r}{\tau}+rv\sin(\theta)}$
(3.31)
$\displaystyle\exp{-\frac{s^{2}(1-\tau\cos(2\theta))}{4\tau}-\frac{iw_{x}s}{\tau}-isv\cos(\theta)+irs\cos(\theta)\sin(\theta)}\
.$

To complete this derivation, we evaluate the coupled Gaussian integrals using
exactly the same procedure as in Eqs. (3.11) to (3.13) and we simplify the
prefactors using Stirling’s formula and ${\rm
Im\,}(z)\approx\sqrt{N}(1-\tau)\sin(\theta)$. One can then complete the square
on the resulting exponential using Eq. (3.15), yielding

$\displaystyle\mathcal{O}_{N}^{\textup{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{1}{\pi}\sqrt{\frac{N}{2\pi}\frac{1-\tau^{2}}{1+\tau^{2}-2\tau\cos(2\theta)}}\int_{0}^{\infty}dv\,v\,e^{-\frac{1-\tau^{2}}{2(1+\tau^{2}-2\tau\cos(2\theta))}\left(v+\frac{2\eta\sqrt{1+\tau^{2}-2\tau\cos(2\theta)}}{1-\tau^{2}}\right)^{2}}$
(3.32) $\displaystyle=$
$\displaystyle\frac{\sqrt{N}}{\pi}\sqrt{\frac{1+\tau^{2}-2\tau\cos(2\theta)}{1-\tau^{2}}}\left[\frac{1}{\sqrt{2\pi}}\exp{\frac{-2\eta^{2}}{1-\tau^{2}}}-\frac{\eta}{\sqrt{{1-\tau^{2}}}}\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\eta\right)\right]\
,$ (3.33)

where the final integral has been evaluated using Eq. (3.24). Thus, by
rescaling and taking the limit of $N\to\infty$ we arrive at the desired result
of Eq. (2.18).

###  3.3 Weak non-Hermiticity proofs

####  3.3.1 Proof of Theorem 2.6 \- Density in the eGinOE

We now move on to prove the results pertaining to the weak non-Hermiticity
regime, outlined in Section 2.2.2, and begin by considering the density of
complex eigenvalues at the edge for the eGinOE. In order to derive the result
in Eq. (2.24), one starts from the finite-$N$ equation in Eq. (2.3) and can
immediately say that

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{i}{2\sqrt{2\pi}}\
\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\,\exp{\frac{{\rm
Im\,}(z)^{2}-{\rm Re\,}(z)^{2}}{1+\tau}}$ (3.34)
$\displaystyle\sum_{k=0}^{N}\frac{\tau^{k+\frac{1}{2}}}{k!}\left[{\rm
He\,}_{k+1}\left(\frac{\bar{z}}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{z}{\sqrt{\tau}}\right)-{\rm
He\,}_{k+1}\left(\frac{z}{\sqrt{\tau}}\right){\rm
He\,}_{k}\left(\frac{\bar{z}}{\sqrt{\tau}}\right)\right]\ ,$

where we have utilised the definition of $P_{N}$ and the parameterisation of
the edge at WNH from Eqs. (2.5) and (2.13) respectively. One can now introduce
the integral representations of the Hermite polynomials from Eq. (3.1) and
carry out the summation using the definition of the incomplete
$\Gamma$-function from Eq. (3.2), yielding

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{-1}{2(2\pi)^{3/2}}\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\exp{\frac{{\rm
Im\,}(z)^{2}-{\rm Re\,}(z)^{2}}{1+\tau}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}$
(3.35)
$\displaystyle\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\,(t_{1}+t_{2})\,\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}-\frac{i\bar{z}t_{1}}{\tau}+\frac{izt_{2}}{\tau}}\,\frac{\Gamma(N+1,t_{1}t_{2})}{\Gamma(N+1)}\
.$

In order to make the above equation more amenable to saddle point analysis, we
rescale $(t_{1},t_{2})\to\sqrt{N}(t_{1},t_{2})$ and employ the asymptotic
behaviour of the incomplete $\Gamma$-function, given in Eq. (3.3), producing

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\left(\frac{-N^{N+1/2}}{2(2\pi)^{3/2}\Gamma(N)}\right)\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\exp{\frac{{\rm
Im\,}(z)^{2}-{\rm
Re\,}(z)^{2}}{1+\tau}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}$
(3.36)
$\displaystyle\frac{(t_{1}+t_{2})t_{1}t_{2}}{t_{1}t_{2}-1}\exp{-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)}\exp{-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)}\
.$

Therefore, one must locate and expand around the saddle points of

$g_{1}(t_{1})=\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\tau\sqrt{N}}-\ln(t_{1})\hskip
28.45274pt\text{and}\hskip
28.45274ptg_{2}(t_{2})=\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\tau\sqrt{N}}-\ln(t_{2})\
.$ (3.37)

Analysis of the above functions yields that the saddle point is located at
$(t_{1}^{*},t_{2}^{*})=(-i,i)$ and that the second derivatives both vanish
there, i.e.
$g_{1}^{\prime\prime}(t_{1}^{*})=g_{2}^{\prime\prime}(t_{2}^{*})=0$. Hence, we
must expand around the maximum using

$t_{1}=-i-\frac{p}{N^{1/3}}\hskip 56.9055pt\text{and}\hskip
56.9055ptt_{2}=i+\frac{q}{N^{1/3}}\ .$ (3.38)

With the help of Mathematica, it is possible to carefully Taylor expand the
terms inside the exponential and, after some algebraic steps, we find that

$\displaystyle\frac{{\rm Im\,}(z)^{2}-{\rm Re\,}(z)^{2}}{1+\tau}+$
$\displaystyle\frac{z^{2}+\bar{z}^{2}}{2\tau}-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)$
(3.39)
$\displaystyle\to-N+\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx+\frac{\frac{x^{2}}{2}+(p-q)y+i(p+q)\kappa}{N^{1/3}}+O(N^{-2/3})\
.$

Additionally, it is easy to see that the remaining part of the integrand can
be written as

$\frac{(t_{1}+t_{2})t_{1}t_{2}}{t_{1}t_{2}-1}\to\frac{p-q}{i(p+q)}+O(N^{-1/3})\
.$ (3.40)

Returning to the density, this means that

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\left(\frac{-N^{N-1/6}e^{-N}}{2(2\pi)^{3/2}\Gamma(N)}\right)\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{p-q}{i(p+q)}$
(3.41)
$\displaystyle\exp{\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx}\left(1+\frac{\frac{x^{2}}{2}+(p-q)y+i(p+q)\kappa}{N^{1/3}}\right)$

and one can see that the only term which contributes to this integral is the
term proportional to $(p-q)^{2}$. Therefore, we obtain

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{-1}{2(2\pi)^{2}}y\,\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{(p-q)^{2}}{i(p+q)}\exp{\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx}\
,$ (3.42)

where Stirling’s formula has been utilised to simplify the prefactors. If one
now makes the change of variables $(p,q)\to i(p,q)$ and deforms the contour
from along the imaginary axis to a curved contour, , starting at infinity with
an argument of $-\pi/3$ and ending at infinity with an argument of $\pi/3$, it
can be seen that

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
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{}{}{}{}{}{}{{}}\pgfsys@moveto{-0.81375pt}{1.6275pt}\pgfsys@lineto{1.57185pt}{3.98926pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@transformcm{0.71065}{0.70354}{-0.70354}{0.71065}{1.57185pt}{3.98924pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}dq\,\frac{(p-q)^{2}}{p+q}\exp{\frac{p^{3}}{3}-px+\frac{q^{3}}{3}-qx}\
.$ (3.43)

Utilising $a^{-1}=\int_{0}^{\infty}dt\,e^{-at}$, for $\real(a)>0$, it becomes
clear that

$\displaystyle\rho_{N}^{\text{(eGinOE,c)}}(z)\approx$
$\displaystyle\frac{-1}{2}y\,\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\frac{1}{(2\pi
i)^{2}}\int_{0}^{\infty}dt\int_{\leavevmode\hbox to3.11pt{\vbox
to5.77pt{\pgfpicture\makeatletter\hbox{\hskip 1.01375pt\lower-1.25787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}{}\pgfsys@moveto{1.085pt}{-1.05788pt}\pgfsys@lineto{-0.81375pt}{1.6275pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-0.81375pt}{1.6275pt}\pgfsys@lineto{1.57185pt}{3.98926pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.71065}{0.70354}{-0.70354}{0.71065}{1.57185pt}{3.98924pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}dp\int_{\leavevmode\hbox
to3.11pt{\vbox to5.77pt{\pgfpicture\makeatletter\hbox{\hskip
1.01375pt\lower-1.25787pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}{}\pgfsys@moveto{1.085pt}{-1.05788pt}\pgfsys@lineto{-0.81375pt}{1.6275pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-0.81375pt}{1.6275pt}\pgfsys@lineto{1.57185pt}{3.98926pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.71065}{0.70354}{-0.70354}{0.71065}{1.57185pt}{3.98924pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}dq\,(p-q)^{2}e^{\frac{p^{3}}{3}-p(x+t)+\frac{q^{3}}{3}-q(x+t)}$
(3.44) $\displaystyle=$
$\displaystyle\,y\,\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\int_{0}^{\infty}dt\bigg{[}\text{Ai}^{\prime}(x+t)^{2}-\text{Ai}(x+t)\,\text{Ai}^{\prime\prime}(x+t)\bigg{]}\
,$ (3.45)

where $\text{Ai}^{\prime}(\cdot)$ and $\text{Ai}^{\prime\prime}(\cdot)$ are
the first and second derivatives of the Airy function, defined in Eq. (2.23),
with general $k$-th derivative

$\text{Ai}^{(k)}(z)=\frac{\partial^{k}}{\partial
x^{k}}\text{Ai}(x)\bigg{|}_{x=z}=\frac{(-1)^{k}}{2\pi i}\int_{\leavevmode\hbox
to3.11pt{\vbox to5.77pt{\pgfpicture\makeatletter\hbox{\hskip
1.01375pt\lower-1.25787pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}
{}{}{}\pgfsys@moveto{1.085pt}{-1.05788pt}\pgfsys@lineto{-0.81375pt}{1.6275pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-0.81375pt}{1.6275pt}\pgfsys@lineto{1.57185pt}{3.98926pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.71065}{0.70354}{-0.70354}{0.71065}{1.57185pt}{3.98924pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}dp\,p^{k}\exp{\frac{p^{3}}{3}-pz}\
.$ (3.46)

####  3.3.2 Proof of Theorem 2.8 \- Self-Overlap in the eGinUE

We now move on to the mean self-overlap of eigenvectors at the edge for the
eGinUE at weak non-Hermiticity. Considering the finite-$N$ form of the mean
self-overlap, Eq. (2.7), one can see that the term proportional to
$\rho_{N-1}^{\textup{(eGinUE,c)}}(z)$ cannot possibly contribute to the
leading order asymptotic behaviour, since $1-\tau^{2}\sim O(1/N)$. Following a
very similar method to the previous section, one can also see that, to leading
order, $R_{N}\approx N\rho_{N+1}$. Therefore, at the edge of the eGinUE in the
weak non-Hermiticity regime, $\mathcal{O}\approx\rho$ and consequently
$\mathbb{E}\to 1$ as $N\to\infty$. For this reason, we choose to consider the
leading order deviation from $\mathbb{E}=1$, quantified by

$\delta_{N}=(1-\tau^{2})\Big{(}N\rho_{N+1}^{\textup{(eGinUE,c)}}(z)-R_{N}\Big{)}\
,$ (3.47)

such that for large $N$, $\mathcal{O}_{N}\approx\rho_{N}+\delta_{N}$. Using
the definitions of $\rho_{N}^{\textup{(eGinUE,c)}}(z)$ and $R_{N}$ from Eqs.
(2.3) and (2.8), introducing the integral representations of the Hermite
polynomials from Eq. (3.1) and evaluating the summation using the definition
of the incomplete $\Gamma$-function in Eq. (3.2), one finds that

$\displaystyle\delta_{N}\approx$
$\displaystyle\frac{1}{\sqrt{N}\pi^{2}}\sqrt{\frac{\kappa}{2}}\frac{1}{\Gamma(N)}\exp{-\frac{|z|^{2}-\tau{\rm
Re\,}(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}$
(3.48) $\displaystyle\hskip 28.45274pt\hskip
28.45274pt\exp{t_{1}t_{2}-\frac{t_{1}^{2}+t_{2}^{2}}{2\tau}-\frac{i\bar{z}t_{1}}{\tau}+\frac{izt_{2}}{\tau}}\,\bigg{[}\Gamma(N+1,t_{1}t_{2})-t_{1}t_{2}\,\Gamma(N,t_{1}t_{2})\bigg{]}\
.$

We now rescale $(t_{1},t_{2})\to\sqrt{N}(t_{1},t_{2})$ and then employ the
asymptotic behaviour of the incomplete $\Gamma$-function from Eq. (3.3), to
see that

$\displaystyle\delta_{N}\approx$
$\displaystyle\frac{N^{N+3/2}}{\pi^{2}}\sqrt{\frac{\kappa}{2}}\frac{1}{\Gamma(N)}e^{-\frac{|z|^{2}-\tau{\rm
Re\,}(z^{2})}{1-\tau^{2}}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}\,t_{1}t_{2}\bigg{[}\frac{1}{Nt_{1}t_{2}-N}-\frac{1}{Nt_{1}t_{2}-N+1}\bigg{]}$
(3.49)
$\displaystyle\exp{-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)}\exp{-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)}\
.$

As previously, these integrals are dominated in the region of
$(t_{1},t_{2})=(-i,i)$, with the second derivative of the functions inside the
exponential being zero at the saddle point. Hence, we employ the substitution

$t_{1}=-i-\frac{p}{N^{1/3}}\hskip 56.9055pt\text{and}\hskip
56.9055ptt_{2}=i+\frac{q}{N^{1/3}}\ ,$ (3.50)

and find that

$\delta_{N}\approx\frac{N^{N-\frac{1}{2}}e^{-N}}{\pi^{2}\Gamma(N)}\sqrt{\frac{\kappa}{2}}\exp{\frac{-y^{2}}{\kappa}}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{e^{\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx}}{(p+q)^{2}}\
,$ (3.51)

since

$\displaystyle-\frac{|z|^{2}-\tau{\rm Re\,}(z^{2})}{1-\tau^{2}}+$
$\displaystyle\frac{z^{2}+\bar{z}^{2}}{2\tau}-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)$
(3.52)
$\displaystyle\to-N-\frac{y^{2}}{\kappa}+\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx+O(N^{-1/3})$

and

$\frac{1}{Nt_{1}t_{2}-N}-\frac{1}{Nt_{1}t_{2}-N+1}=-\frac{1}{(p+q)^{2}N^{4/3}}+O(N^{-5/3})\
.$ (3.53)

Note that one can employ finer asymptotics of the incomplete $\Gamma$-function
to verify that higher order terms in the expansion do not contribute to the
leading term of the above equation, see e.g. [59, Eq. (8.11.7)]. One now
rescales $(p,q)\to i(p,q)$, deforms the contours of the $p$ and $q$ integrals
to  and utilises $a^{-2}=\int_{0}^{\infty}dt\,t\,e^{-at}$, for $\real(a)>0$.
This leads to

$\delta_{N}\approx
2\sqrt{\frac{\kappa}{\pi}}\exp{\frac{-y^{2}}{\kappa}}\int_{0}^{\infty}dt\,t\,\text{Ai}(x+t)^{2}$
(3.54)

and so by introducing
$\mathcal{S}_{N}=(\mathcal{O}_{N}-\rho_{N})/\rho_{N}=\mathbb{E}_{N}-1\approx\delta_{N}/\rho_{N}$,
we find that

$\displaystyle\mathcal{S}^{\textup{(eGinUE,c)}}_{\textup{WNH, edge}}(x)=$
$\displaystyle\lim_{N\to\infty}N^{2/3}\bigg{(}\mathbb{E}^{\textup{(eGinUE,c)}}_{N}\left(z=2\sqrt{N}+\frac{x}{N^{1/6}}+i\frac{y}{N^{1/2}}\right)-1\bigg{)}=\frac{2\kappa\int_{0}^{\infty}dt\,t\,\textup{Ai}(x+t)^{2}}{\int_{0}^{\infty}dt\,\textup{Ai}(x+t)^{2}}\
.$ (3.55)

####  3.3.3 Proof of Theorem 2.9 \- Self-Overlap in the eGinOE

The proof of our final theorem relies heavily on the techniques outlined in
Section 3.3.1. For this reason we only outline key differences in this section
and leave large amounts of the technical detail as an exercise for the reader.
Considering the finite-$N$ form of the mean self-overlap, Eq. (2.9), it can be
seen that, at the edge, as $N$ becomes large, $\mathcal{O}_{N}$ does not
approach $\rho_{N}$ in the eGinOE. This is in contrast to what we observed in
the eGinUE. Initially, one utilises Eq. (2.21) in the prefactors to note that

$\Bigg{[}1+\sqrt{\frac{\pi(1-\tau^{2})}{2}}\ \exp{\frac{2{\rm
Im\,}(z)^{2}}{1-\tau^{2}}}\,\frac{1}{2|{\rm Im\,}(z)|}\
\textup{erfc}\left(\sqrt{\frac{2}{1-\tau^{2}}}\ |{\rm
Im\,}(z)|\right)\Bigg{]}\to
1+\frac{\sqrt{\pi\kappa}}{2y}\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\
.$ (3.56)

Then, following a process similar to Section 3.3.1, we find that, to leading
order, $NP_{N}\sim T_{N}$, therefore the leading order behaviour of the mean
self-overlap is given by

$\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)\approx\left[1+\frac{\sqrt{\pi\kappa}}{2y}\exp{\frac{y^{2}}{\kappa}}\text{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\right]Q_{N}\
,$ (3.57)

where we have introduced

$Q_{N}=\frac{1}{\pi}\ \sqrt{\frac{1-\tau}{1+\tau}}\exp{-\frac{{\rm
Re\,}(z)^{2}}{1+\tau}-\frac{{\rm Im\,}(z)^{2}}{1-\tau}}P_{N}\ .$ (3.58)

At this point, we introduce the definition of $P_{N}$ and the integral
representations of the Hermite polynomials, then carry out the summation, in
exactly the same way as in Eqs. (3.34) to (3.36), eventually yielding

$\displaystyle Q_{N}\approx$ $\displaystyle\frac{N^{N+1/2}}{(2\pi
i)^{2}}\sqrt{\frac{\kappa}{2}}\frac{1}{y\Gamma(N)}\exp{-\frac{{\rm
Re\,}(z)^{2}}{1+\tau}-\frac{{\rm
Im\,}(z)^{2}}{1-\tau}+\frac{z^{2}+\bar{z}^{2}}{2\tau}}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{\infty}dt_{2}$
(3.59)
$\displaystyle\frac{(t_{1}+t_{2})t_{1}t_{2}}{t_{1}t_{2}-1}\exp{-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)}\exp{-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)}\
.$

Once again, this double integral is dominated in the vicinity of
$(t_{1},t_{2})=(-i,i)$ and so we employ the substitutions in Eq. (3.38),
insert the result from Eq. (3.40) and utilise the following expansion

$\displaystyle-\frac{{\rm Re\,}(z)^{2}}{1+\tau}-\frac{{\rm
Im\,}(z)^{2}}{1-\tau}$
$\displaystyle+\frac{z^{2}+\bar{z}^{2}}{2\tau}-N\left(\frac{t_{1}^{2}}{2\tau}+\frac{i\bar{z}t_{1}}{\sqrt{N}\tau}-\ln(t_{1})\right)-N\left(\frac{t_{2}^{2}}{2\tau}-\frac{izt_{2}}{\sqrt{N}\tau}-\ln(t_{2})\right)$
(3.60)
$\displaystyle\to-N-\frac{y^{2}}{\kappa}+\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx+\frac{\frac{x^{2}}{2}+(p-q)y+i(p+q)\kappa}{N^{1/3}}+O(N^{-2/3})\
,$

which helps us to write down

$\displaystyle Q_{N}\approx$
$\displaystyle\frac{1}{2y}\sqrt{\frac{\kappa}{\pi}}\frac{N^{1/3}}{(2\pi
i)^{2}}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dq\,\frac{p-q}{i(p+q)}e^{\frac{ip^{3}}{3}+ipx+\frac{iq^{3}}{3}+iqx}\left(1+\frac{\frac{x^{2}}{2}+(p-q)y+i(p+q)\kappa}{N^{1/3}}\right)\
.$ (3.61)

We then realise that the only non-zero contribution comes from the term
proportional to $(p-q)^{2}$. The integral can then be evaluated by changing
$(p,q)\to i(p,q)$, deforming the contours of the $p$ and $q$ integrals to  and
recognising the derivatives of the Airy function, Eq. (3.46). Returning to the
mean self-overlap this yields

$\mathcal{O}_{N}^{\text{(eGinOE,c)}}(z)\approx\bigg{[}\sqrt{\frac{\kappa}{\pi}}\exp{\frac{-y^{2}}{\kappa}}+\frac{\kappa}{2y}\textup{erfc}\left(\frac{y}{\sqrt{\kappa}}\right)\bigg{]}\int_{0}^{\infty}dt\Big{[}\textup{Ai}^{\prime}(x+t)^{2}-\textup{Ai}(x+t)\textup{Ai}^{\prime\prime}(x+t)\Big{]}\
,$ (3.62)

and so we can now prove the result in Theorem 2.9 by dividing the above
expression by Eq. (2.22).

#### Acknowledgements

We would like to thank Yan V. Fyodorov for interesting discussions and for
offering feedback on this manuscript. We would also like to thank Wojciech
Tarnowski for useful discussions. This research has been supported by the
EPSRC Grant EP/V002473/1 “Random Hessians and Jacobians: theory and
applications”.

##  Appendix A Expansion around Maximum of SNH Exponential

To derive our main results at strong non-Hermiticity, one must expand of
function

$\displaystyle H\equiv
N\bigg{[}\frac{(1-u)(p^{2}-q^{2})}{4}-\frac{p^{2}+q^{2}}{4\tau}-\frac{i(1+\tau)\cos(\theta)q}{\tau}-\frac{(1-\tau)\sin(\theta)p}{\tau}+\ln(\frac{u(p^{2}-q^{2})}{4})\bigg{]}\
,$ (A.1)

which arises in the leading exponential of Eqs. (3.7), (3.21) and (3.30), in
the vicinity of

$p=-2\sin(\theta)+\frac{r}{\sqrt{N}}\hskip
28.45274ptq=-2i\cos(\theta)+\frac{s}{\sqrt{N}}\hskip
28.45274ptu=1+\frac{v}{\sqrt{N}}\ ,$ (A.2)

as $N$ becomes large. Immediately, inserting the above substitutions and
performing simple manipulations on the terms outside the logarithm, we find
that

$\displaystyle H\approx$
$\displaystyle\frac{N}{\tau}\Big{(}(1-\tau)\sin^{2}(\theta)-(1+\tau)\cos^{2}(\theta)\Big{)}-\frac{(1-\tau)p^{2}}{4\tau}-\frac{(1+\tau)q^{2}}{4\tau}-N\Bigg{\\{}\frac{1}{4}\left(1+\frac{v}{\sqrt{N}}\right)\bigg{\\{}\left(\frac{r}{\sqrt{N}}-2\sin(\theta)\right)^{2}$
$\displaystyle-\left(\frac{s}{\sqrt{N}}-2i\cos(\theta)\right)^{2}\bigg{]}-\ln(\frac{1}{4}\left(1+\frac{v}{\sqrt{N}}\right)\bigg{[}\left(\frac{r}{\sqrt{N}}-2\sin(\theta)\right)^{2}-\left(\frac{s}{\sqrt{N}}-2i\cos(\theta)\right)^{2}\bigg{]})\Bigg{\\}}\
.$ (A.3)

To evaluate the term in the curly brackets we write that

$\frac{1}{4}\left(1+\frac{v}{\sqrt{N}}\right)\bigg{[}\left(\frac{r}{\sqrt{N}}-2\sin(\theta)\right)^{2}-\left(\frac{s}{\sqrt{N}}-2i\cos(\theta)\right)^{2}\bigg{]}=1+\nu\
,$ (A.4)

with

$\nu=\frac{is\cos(\theta)-r\sin(\theta)+v}{\sqrt{N}}+\frac{{r}^{2}-{s}^{2}+4v\big{(}is\cos(\theta)-r\sin(\theta)\big{)}}{4N}+\frac{v({r}^{2}-{s}^{2})}{N^{3/2}}\
,$ (A.5)

expanded in powers of $N$. Thus, if we Taylor expand the term of the form
$1+\nu-\ln(1+\nu)\approx 1+\frac{\nu^{2}}{2}$ in Eq. (A.3) and retain the
leading order terms in $N$, we find that

$1+\nu-\ln(1+\nu)\approx
1+\frac{1}{N}\left[\frac{{r}^{2}\sin^{2}(\theta)-{s}^{2}\cos^{2}(\theta)+v^{2}}{2}+isv\cos(\theta)-rv\sin(\theta)-irs\cos(\theta)\sin(\theta)\right]\
.$ (A.6)

Therefore, after some simple manipulations, involving trigonometric
identities, one can see that

$\displaystyle H\approx-N-$
$\displaystyle\frac{N\big{(}\tau+\cos(2\theta)\big{)}}{\tau}-\frac{v^{2}}{2}-\frac{{r}^{2}(1-\tau\cos(2\theta))}{4\tau}$
(A.7)
$\displaystyle+rv\sin(\theta)-\frac{{s}^{2}(1-\tau\cos(2\theta))}{4\tau}-isv\cos(\theta)+irs\cos(\theta)\sin(\theta)\
.$

This result is utilised in the proofs of all three of our new results at
strong non-Hermiticity. As an example, it is used in the proof of Theorem 2.2
to arrive at Eq. (3.9) from Eq. (3.7).

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