# Mean left-right eigenvector self-overlap in the real Ginibre ensemble

T. R. Würfel  Department of Mathematics, King’s College London, London WC2R
2LS, UK  M. J. Crumpton  Department of Mathematics, King’s College London,
London WC2R 2LS, UK  Y. V. Fyodorov  Department of Mathematics, King’s College
London, London WC2R 2LS, UK

###### Abstract

We study analytically the Chalker-Mehlig mean diagonal overlap
$\mathcal{O}(z)$ between left and right eigenvectors associated with a complex
eigenvalue $z$ of $N\times N$ matrices in the real Ginibre ensemble (GinOE).
We first derive a general finite $N$ expression for the mean overlap and then
investigate several scaling regimes in the limit $N\rightarrow\infty$. While
in the generic spectral bulk and edge of the GinOE the limiting expressions
for $\mathcal{O}(z)$ are found to coincide with the known results for the
complex Ginibre ensemble (GinUE), in the region of eigenvalue depletion close
to the real axis the asymptotic for the GinOE is considerably different. We
also study numerically the distribution of diagonal overlaps and conjecture
that it is the same in the bulk and at the edge of both the GinOE and GinUE,
but essentially different in the depletion region of the GinOE.

Keywords: non-Hermitian random matrices, real Ginibre ensemble, bi-orthogonal
eigenvectors, eigenvector overlaps, eigenvalue depletion, bulk and edge
statistics

##  1 Introduction

Random matrices of finite size $N\times N$ are often categorized in terms of
their global symmetries, influencing statistical properties of their
eigenvalues and eigenvectors. The two major categories to distinguish are
matrices which are self-adjoint (Hermitian) from their non-self-adjoint (non-
Hermitian) counterparts. The former are defined as satisfying the condition
$H=H^{\dagger}\equiv\bar{H}^{T}$, with † standing for Hermitian conjugation, T
denoting the transpose of the matrix and the bar standing for the complex
conjugation of its entries. All Hermitian matrices are by definition normal,
with vanishing commutator $[H,H^{\dagger}]=0$, which ensures that $H$ is
diagonalizable by a unitary transformation. The Hermiticity of $H$ ensures all
ensuing eigenvalues $\lambda_{n}$ to be necessarily real.

On the other hand, _non-Hermitian_ matrices, which are denoted by $X$, may or
may not be normal, in the latter case $XX^{\dagger}\neq X^{\dagger}X$. Such
matrices generically have the majority of the eigenvalues $z_{n}$
($n=1,\ldots,N$) located in the full complex plane $\mathbb{C}$, although some
of them may still remain on the real line $\mathbb{R}$. The non-normality of
such matrices is well known to lead to important numerical issues, when the
matrix size $N$ grows. For example, keeping a fixed precision of calculations
might not be sufficient, as some eigenvalues can be ‘ill-conditioned’, i.e.
they are much more sensitive to a perturbation of the matrix entries, in
contrast to eigenvalues of normal matrices, which are relatively robust.

For a random matrix $X$, we can safely assume that all its $N$ eigenvalues
have multiplicity one, ensuring that $X$ is still diagonalizable by a
transformation to its eigenbasis: $X=S\Lambda S^{-1}$, with $\Lambda={\rm
diag\,}(z_{1},\ldots,z_{N})$. However for non-normal matrices the matrix $S$
ceases to be unitary: $S^{\dagger}\neq S^{-1}$. The associated eigenvalue
problem then splits and one needs to distinguish between _left_ (row)
eigenvectors, satisfying
$\mathbf{x}_{L_{n}}^{\dagger}X=z_{n}\mathbf{x}_{L_{n}}^{\dagger}$ and _right_
(column) eigenvectors, which instead satisfy
$X\mathbf{x}_{R_{n}}=z_{n}\mathbf{x}_{R_{n}}$. The right eigenvectors are the
columns of the matrix $S$, whereas the left eigenvectors are the rows of
$S^{-1}$, and generically $\mathbf{x}_{R_{n}}\neq\mathbf{x}_{L_{n}}$. In this
representation, the sets of left and right eigenvectors automatically satisfy
the bi-orthonormality condition
$\mathbf{x}_{L_{n}}^{\dagger}\mathbf{x}_{R_{m}}=\delta_{nm}$, for
$n,m=1,\ldots,N$, but non-unitarity of $S$ implies that
$\mathbf{x}_{L_{n}}^{\dagger}\mathbf{x}_{L_{m}}\neq\delta_{nm}$ and similarly
$\mathbf{x}_{R_{n}}^{\dagger}\mathbf{x}_{R_{m}}\neq\delta_{nm}$. The first
object that allows qualitative insights into the non-orthogonality of the
eigenvectors, is the overlap matrix, with entries
$\mathcal{O}_{nm}=\left(\mathbf{x}_{L_{n}}^{\dagger}\mathbf{x}_{L_{m}}\right)\left(\mathbf{x}_{R_{m}}^{\dagger}\mathbf{x}_{R_{n}}\right)$.
In physical literature the diagonal entries, $\mathcal{O}_{nn}$, are generally
referred to as diagonal overlaps, or self-overlaps. In numerical analysis
literature their square roots are sometimes referred to as the eigenvalue
condition numbers, as they characterize the sensitivity of the eigenvalue
$z_{n}$ to a perturbation of the entries of $X$, see e.g. [1, 2]. To this end,
consider a matrix $X^{\prime}=X+\varepsilon P$, the second term reflecting an
error made with respect to the entries of $X$, which are stored with finite
machine precision. $P$ here can be chosen to have its $2$-norm fixed to
$||P||_{2}=1$, such that the strength of such a perturbation is measured via
the real parameter $\varepsilon$. Then, the change in eigenvalue $z_{n}$ of
$X$ incurred by a small $\varepsilon$ can be characterized via

$\frac{dz_{n}(\varepsilon)}{d\varepsilon}\bigg{|}_{\varepsilon=0}:=\dot{z}_{n}(0)=\mathbf{x}_{L_{n}}^{\dagger}P\mathbf{x}_{R_{n}}\
,$ (1.1)

which can be derived using the bi-orthonormality of the corresponding left-
and right-eigenvectors. By using the Cauchy-Schwartz inequality, the magnitude
of the shift can be seen to satisfy the following bound:

$|\dot{z}_{n}(0)|=|\mathbf{x}_{L_{n}}^{\dagger}P\mathbf{x}_{R_{n}}|\leq|\mathbf{x}_{L_{n}}|\
||P||_{2}\
|\mathbf{x}_{R_{n}}|=\sqrt{(\mathbf{x}_{L_{n}}^{\dagger}\mathbf{x}_{L_{n}})(\mathbf{x}_{R_{n}}^{\dagger}\mathbf{x}_{R_{n}})}=\mathcal{O}_{nn}^{1/2}\
,$ (1.2)

where $\mathcal{O}_{nn}$ are the diagonal entries of the overlap matrix. We
indeed see that the magnitude of the perturbation is essentially controlled by
$\mathcal{O}_{nn}^{1/2}$, which assumes the minimal value only if $X$ is
normal and all $O_{nn}=1$. In the realm of non-normal matrices one can find
numerous examples where $\mathcal{O}_{nn}\gg 1$, indicating that the
associated eigenvalues are extremely sensitive to perturbations, see e.g. [1,
2]. In fact, random matrices can be used to provide a regularization of
eigenvalue condition numbers of highly non-normal, non-random matrices, see
for example [3, 4, 5] for more information.

For $X$ taken to be a random matrix from a specified probability measure, the
statistics of the entries of the overlap matrix $\mathcal{O}_{mn}$ become an
important object of study. The simplest nontrivial choice is to assume that
all entries $X_{ij}$ are mean-zero, independent, identically distributed
(i.i.d.) Gaussian numbers, which can be real, complex, or quaternion. This
defines the three classical Ginibre ensembles [6]. Understanding of statistics
of the overlap matrix in this setting has been influenced heavily by the
seminal papers of Chalker and Mehlig [7, 8] who treated the _complex Ginibre_
ensemble. We denote the latter ensemble as GinUE in the rest of the paper, to
distinguish it from its real (GinOE) and quaternion (GinSE) counterparts, the
nomenclature relating to the classical Dyson Hermitian ensembles - GUE, GOE
and GSE. In particular, Chalker and Mehlig addressed the statistics of the
overlap matrix $\mathcal{O}_{mn}$ via considering the following single-point
and two-point correlation functions

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

where the angular brackets stand for the expectation value with respect to the
probability measure associated with the ensemble in question, GinUE for the
particular case studied by Chalker and Mehlig. Here $\delta(z-z_{n})$ is the
Dirac delta mass at the eigenvalue $z_{n}$, so that the empirical density of
eigenvalues in the complex plane $z$ reads
$\rho^{(\text{emp})}_{N}(z)=\frac{1}{N}\sum_{n=1}^{N}\delta(z-z_{n})$. It is
evident that $\mathcal{O}(z)$ describes the conditional expectations of
$\mathcal{O}_{nn}$ and we can define the mean conditional self-overlap as

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

where $\rho\,(z)$ is the mean spectral density, defined via

$\rho\,(z)\equiv\langle\ \rho_{N}^{(\text{emp})}(z)\
\rangle=\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\delta(z-z_{n})\bigg{\rangle}\
.$ (1.5)

Recall that, as $N\to\infty$, the asymptotic mean eigenvalue density is
nonvanishing and uniform only inside the unit circle in the complex plane:
$\rho(z)\approx 1/\pi$ for $|z|^{2}<1$ and zero otherwise [9]. Chalker and
Mehlig were able to extract the leading asymptotic behaviour of
$\mathcal{O}(z)$ and $\mathcal{O}(z_{1},z_{2})$ as $N$ tends to $\infty$.
Choosing the variance of the entries of $X$ to be $1/N$, they found that
$\mathcal{O}(z)\approx N\left(1-|z|^{2}\right)/\pi$ inside the unit disk
$|z|^{2}<1$ and zero otherwise. Consequently, one should expect
$\mathcal{O}_{nn}\sim N$ for eigenvalues inside the disk, which is thus
parametrically larger than the value for normal matrices, where
$\mathcal{O}_{nn}=1$. Note that only the rescaled correlator
$\widetilde{\mathcal{O}}(z)=\mathcal{O}(z)/N$ is well-defined and finite in
the limit $N\to\infty$.

The Chalker-Mehlig (CM) correlators and other aspects pertaining to
eigenvector non-orthogonality in random matrix ensembles have attracted
growing interest in the theoretical physics community over the past decades,
see e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The main
interest being in extending CM correlators from the GinUE to more general
classes of random matrix models. Apart from the important issues of eigenvalue
stability, the eigenvector non-orthogonality is known to play an important
role in describing transient behaviour in complex systems with classical
dynamics and related questions [24, 25, 26, 27, 28, 29, 30], as well as in
producing intriguing features in their quantum counterparts [31, 32, 33].
Another strong motivation comes from the field of quantum chaotic wave
scattering, where non-Hermitian random matrix ensembles, different from
Ginibre, play a prominent role, see e.g. [34, 35, 36, 37] for some background
information. The overlap matrix $\mathcal{O}_{mn}$ appears naturally in many
scattering observables, for example in the derivation of decay laws [38], in
the ‘Petermann factors’ describing excess noise in open laser resonators [11,
39], in issues of increased sensitivity of resonance widths to small
perturbations [15, 16] and in the shape of reflected power profiles [23]. In
that context both $\mathcal{O}(z)$ and $\mathcal{O}(z_{1},z_{2})$ have been
studied theoretically [11, 12, 13, 15, 23] and experimentally [16, 40, 41].

In a not unrelated development, mathematically rigorous studies of CM
correlators became a field of considerable activity more recently, see e.g.
[42, 43, 44]. The usage of free probability techniques [10] to compute the
self-overlap, Eq. (1.3), allowed extensions to invariant ensembles with non-
Gaussian weights [20, 21]. Since $\mathcal{O}(z)$ is known in the GinUE at
finite matrix size [7, 8] it also became possible to compute it for products
of small Ginibre matrices [19]. It was shown that one- and two-point functions
of eigenvector overlaps, conditioned on an arbitrary number of eigenvalues in
the GinUE, lead to determinantal structures [45]. Deep insights into Dysonian
dynamics related to eigenvalues have been made, from different angles, in [17,
46, 47, 48]. Lower [4, 49] and upper [3, 5, 50] bounds on diagonal eigenvector
overlaps have been provided for a fixed non-Hermitian matrix perturbed by
random matrices with i.i.d. entries, even beyond Gaussian case. Some other
properties of eigenvectors of non-normal random matrices have also been
studied rigorously, see [51, 52, 53, 54].

Finally, let us mention that in addition to analysing the correlators in Eq.
(1.3), Chalker and Mehlig put forward a conjecture for the far tail behaviour
of the distribution of the random variable $\mathcal{O}_{nn}$ for the GinUE as
$N\rightarrow\infty$. Based on both numerical evidence and simple eigenvalue
repulsion arguments, illustrated by a solvable $2\times 2$ GinUE matrix, they
predicted that for large overlaps the probability density of
$\mathcal{O}_{nn}$ must exhibit a power law tail proportional to
$1/\mathcal{O}_{nn}^{3}$. Such tail would make all the positive integer
moments beyond $\mathcal{O}(z)$ divergent. This conjecture has been rigorously
proved by two different approaches in Bourgade and Dubach [47] and Fyodorov
[55], in fact recovering the full form of the probability density beyond the
tail region. While the work of Bourgade and Dubach proceeded on studying the
off-diagonal correlator $\mathcal{O}(z_{1},z_{2})$ for the GinUE, the paper by
Fyodorov revealed that for real eigenvalues of the GinOE the diagonal overlaps
$\mathcal{O}_{nn}$ are distributed with an even heavier probability density
tail, decaying as $1/\mathcal{O}_{nn}^{2}$. This implies that for real Ginibre
matrices the mean self-overlap $\mathcal{O}(z)$ is divergent on the real line.
Understanding both the distribution of $\mathcal{O}_{nn}$ and the CM mean
$\mathcal{O}(z)$ for complex eigenvalues of the GinOE remained however an
outstanding problem.

In the present paper we make the first step towards addressing the above
issues, and present the results for the mean diagonal CM correlator
$\mathcal{O}(z)$ for the GinOE in the complex plane, first at finite $N$ and
then in various scaling regimes as $N\gg 1$. We will also systematically
compare our findings with both results for complex eigenvalues in the GinUE
and for real eigenvalues in the GinOE.

The rest of the paper is organized as follows. We present our main findings in
Section 2, in particular, the mean self-overlap at finite matrix size $N$ is
given in Theorem 2.5. Asymptotic results for the mean self-overlap, as
$N\to\infty$, are given in the bulk, Corollary 2.6, at the spectral edge,
Corollary 2.7, and in an eigenvalue depleted region of the droplet in
Corollary 2.8. We then compare with similar results for the GinUE, and use
numerical simulations, both to corroborate analytical results and provide
insights into yet analytically unavailable distributions of $O_{nn}$ for the
GinOE. Finally, the corresponding proofs of our findings are presented in
Section 3 for finite $N$ \- Theorem 2.5 \- and in Section 4 for the asymptotic
results - Corollary 2.6, 2.7 and 2.8.

##  2 Statement and Discussion of Main Results

In order to state our main results, we start by introducing the necessary
notation and statements about random real Ginibre matrices in Section 2.1. We
then present our main results for the mean self-overlap in Section 2.2 for
both finite matrix size $N$ and in several large $N$ regimes of the complex
plane. We follow up with a discussion about connections to previously known
results, comparisons to numerical simulations and open problems.

###  2.1 Remarks on real and complex Ginibre ensembles

######  Definition 2.1.

Let $G=\left(G_{ij}\right)_{i,j=1}^{N}$ be an $N\times N$ matrix containing
i.i.d. real Gaussian entries with mean zero and unit variance, such that the
off-diagonal entries $G_{ij}$ and $G_{ji}$ are uncorrelated. The joint
probability density function (JPDF) of matrices $G$ is defined with respect to
the flat Lebesgue measure, $dG=\prod_{i,j=1}^{N}dG_{ij}$, via

$P_{\text{GinOE}}(G)\ dG=\frac{1}{C_{N}}\
\exp\left[-\frac{1}{2}\Tr\left(GG^{T}\right)\right]\ dG,\quad
C_{N}=(2\pi)^{N^{2}/2}\ \text{.}$ (2.1)

The corresponding ensemble is called the _real Ginibre_ ensemble, and is
denoted as GinOE. Its counterpart for matrices with complex i.i.d. entries is
the _complex Ginibre_ ensemble (GinUE). For recent reviews of available
results on the GinUE and GinOE see [56, 57]. The remarks below provide a
discussion of main facts on both ensembles which will be of direct relevance
for the present paper.

######  Remark 2.2.

Generically, the spectrum of GinOE matrices consists of real eigenvalues
$\lambda\in\mathbb{R}$ and complex eigenvalues $z=x+iy,\,y\neq 0$. Complex
eigenvalues of real matrices, $G$, always appear in conjugated pairs, i.e.
$\bar{z}=x-iy$ is an eigenvalue of $G$ iff $z$ is an eigenvalue of $G$.
Correspondingly, the mean spectral density, Eq. (1.5), necessarily has the
form $\rho\,(z)=\rho^{(c)}(z)+\delta(y)\rho^{(r)}(x)$, where $\rho^{(c/r)}(z)$
describes the mean density of complex/real eigenvalues, respectively. On the
other hand, with probability one, the eigenvalues of GinUE matrices are
complex without conjugate counterparts, due to the entries of GinUE matrices
being complex.

In Figure 1, we show examples of the spectra of real Ginibre matrices for
three different values of $N$. As is well-known, the majority of complex
eigenvalues lie inside a circle of radius $\sqrt{N}$, with a subleading
fraction (of the order of $1/\sqrt{N}$ for $N\gg 1)$ of purely real
eigenvalues. As $N$ tends to $\infty$ the support of the spectrum approaches a
uniform disc.

Figure 1: Spectra of real Ginibre matrices for three different values of $N$,
$N=20$ (left), $N=100$ (centre) and $N=1000$ (right). Each plot contains 1,000
samples of eigenvalues and has a solid black line depicting a circle of radius
$\sqrt{N}$. Complex eigenvalues are shown in red and real eigenvalues are
shown in blue.

######  Remark 2.3.

Results for the mean densities (both real and complex) are well known in the
GinOE and GinUE at finite $N$ [6, 58, 59, 60]. The mean density of complex
eigenvalues is of particular interest in this work and is given for finite $N$
in the GinOE by

$\begin{split}\rho_{N}^{(\text{GinOE,c})}(z)&=\sqrt{\frac{2}{\pi}}\ |y|\
e^{2y^{2}}\ \text{erfc}\left(\sqrt{2}\ |y|\right)\
\frac{\Gamma\left(N-1,|z|^{2}\right)}{\Gamma\left(N-1\right)}\ ,\end{split}$
(2.2)

see e.g. [57, Eq. (2.46)] and [59, Theorem 6.2]. In the above, $\Gamma(N)$
denotes the standard Euler $\Gamma$-function and $\Gamma(N,|z|^{2})$ denotes
the incomplete (upper) $\Gamma$-function defined as

$\Gamma\left(N,a\right)=\Gamma\left(N\right)\ e^{-a}\
\sum_{k=0}^{N-1}\frac{a^{k}}{k!}=\int_{a}^{\infty}du\ u^{N-1}\ e^{-u}\ .$
(2.3)

We have also made use of the complementary error function
${\text{erfc}(x)=1-\erf(x)}$, where
$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$.

In the GinUE, several equivalent representations for the mean density can be
found in e.g. [6], [44, Eq. (20)], [56, Proposition 2.2] and [60, Eq.
(18.2.11)], which we prefer to write as

$\rho^{(\text{GinUE},c)}_{N}(z)=\frac{1}{\pi}\
\frac{\Gamma\left(N,|z|^{2}\right)}{\Gamma\left(N\right)}=\frac{1}{\pi}e^{-|z|^{2}}\sum_{n=0}^{N-1}\frac{|z|^{2n}}{n!}\
.$ (2.4)

The mean density of _real_ eigenvalues in the GinOE is also known at finite
$N$, see e.g. [58, 61], but is not needed for our purposes.

######  Remark 2.4.

It is worth discussing the large $N$ asymptotic behaviour of the mean
eigenvalue density in more detail. A comparison between the densities of
complex eigenvalues in the GinOE and GinUE at large $N$ is shown in Figure 2
using heatmaps. The heatmap associated with the GinUE features two distinct
scaling regimes: a spectral bulk inside the disk and edge along the disk
circumference. Similar regimes are also seen in the plot for the GinOE.

Figure 2: Large $N$ density of complex eigenvalues in the GinOE and GinUE with
relevant scaling regions labelled. Left: GinOE. Right: GinUE. In both diagrams
the solid black circle within the heatmap has radius $\sqrt{N}$ and the
density is depicted on a rainbow scale. This diagram explains schematically
what is meant by the bulk, edge and depletion regimes of large real and
complex Ginibre matrices. Each heatmap is plotted using the analytic
expression for the density of complex eigenvalues in the associated ensembles
when $N=250$.

Firstly, in both ensembles it is easy to show that the limiting mean density
of complex eigenvalues, after rescaling $z=\sqrt{N}w$, converges to the
uniform density:

$\rho_{\text{bulk}}^{(\textup{GinUE},c)}(w)=\rho_{\text{bulk}}^{(\textup{GinOE},c)}(w)=\frac{1}{\pi}\
,$ (2.5)

in the generic bulk of the unit disk, where $|w|<1$. Similarly, at the edge of
the disk, given by $|z|=\sqrt{N}+\eta$, the density for both ensembles can be
shown to be given by

$\rho_{\text{edge}}^{(\textup{GinUE},c)}(\eta)=\rho_{\text{edge}}^{(\textup{GinOE},c)}(\eta)=\frac{1}{2\pi}\text{erfc}\left(\sqrt{2}\eta\right)\
.$ (2.6)

In addition, the heatmap in the GinOE case shows yet another scaling regime
close to the real axis, essentially for heights of the fraction $\sim
1/\sqrt{N}$ of the disk radius. In that region the density of eigenvalues is
reduced when compared to the spectral bulk value. The origin of such a
depletion is clearly seen analytically from the presence of the factor $|y|$
in Eq. (2.2). Henceforth the scaling associated with this region of the
complex plane will be referred to as the _depletion_ regime. Inside the bulk
depletion regime, i.e. $z=\sqrt{N}x+iy$, with $|x|<1$ and $y\sim O(1)$, the
mean GinOE density in the limit $N\to\infty$ converges to

$\rho_{\text{depletion}}^{(\textup{GinOE},c)}(y)=\sqrt{\frac{2}{\pi}}\ |y|\
e^{2y^{2}}\text{erfc}\left(\sqrt{2}\ |y|\right)\ ,$ (2.7)

see e.g. [57, Eq. (2.42)], cf. Eq. (2.2).

After the digressions on the mean densities, let us come back to our main
objects of interest, the left and right eigenvectors. Given the complex
eigenvalue $z_{n}$ we denote the associated left-eigenvector by
$\mathbf{x}_{L_{n}}^{\dagger}$ and the right-eigenvector by
$\mathbf{x}_{R_{n}}$. The overlap matrix
$\mathcal{O}_{nm}=\left(\mathbf{x}_{L_{n}}^{\dagger}\mathbf{x}_{L_{m}}\right)\left(\mathbf{x}_{R_{m}}^{\dagger}\mathbf{x}_{R_{n}}\right)$
of left- and right-eigenvectors is used to define the mean self-overlap
$\mathcal{O}(z)$ as in Eq. (1.3) and its conditional companion $\mathbb{E}(z)$
as in Eq. (1.4). Note that $\mathbb{E}\left(z\right)$ is particularly useful
when comparing theoretical predictions with numerical data.

Results for the mean self-overlap have been established in the GinUE for the
entire complex plane, see [7, 8, 44, 47, 55] for several equivalent forms. For
our purposes we present it as

$\mathcal{O}^{(\text{GinUE,c})}_{N}(z)=\frac{1}{\pi}\left[\frac{\Gamma(N,|z|^{2})}{(N-1)!}(N-|z|^{2})+\frac{|z|^{2N}}{(N-1)!}e^{-|z|^{2}}\right]\
.$ (2.8)

For convenience of the reader we give a derivation of Eq. (2.8) in the
Appendix A, using the results in [55] as a starting point. One can further
easily find the bulk asymptotics of this expression, which gives back the
original Chalker-Mehlig result, as well as the corresponding edge asymptotics
for the scaling $|z|=\left(\sqrt{N}+\eta\right)$ given by

$\lim_{N\to\infty}\frac{1}{\sqrt{N}}\mathcal{O}_{\textup{edge}}^{\textup{(GinUE,c)}}(z)=\frac{1}{\pi}\left(\frac{1}{\sqrt{2\pi}}\
e^{-2\eta^{2}}-\eta\ \textup{erfc}\left(\sqrt{2}\ \eta\right)\right)\ .$ (2.9)

In the GinOE, so far, results were limited to real eigenvalues only, i.e. when
$z=x$, see [55, 62]. In the next subsection, we provide results for the mean
self-overlap $\mathcal{O}(z)$ of eigenvectors associated with complex
eigenvalues in the GinOE at finite $N$ and in several large $N$ scaling
regions, dictated by the corresponding scalings of the mean eigenvalue density
discussed above.

###  2.2 Statement of Main Results for GinOE and comparison to GinUE

Relegating the proofs and technical details to Section 3, we present our main
findings below. The following theorem gives the mean self-overlap
$\mathcal{O}(z)$ of eigenvectors associated with a complex eigenvalue $z$ for
the real Ginibre ensemble at finite matrix size $N$.

######  Theorem 2.5.

Let $G$ be an $N\times N$ random matrix drawn from the GinOE, distributed
according to Eq. (2.1) in Definition 2.1. The mean self-overlap, Eq. (1.3),
associated with a complex eigenvalue $z$ at finite matrix size $N$ is given by

$\begin{split}\mathcal{O}^{(\textup{GinOE},c)}_{N}(z)&=\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\mathcal{O}_{nn}\
\delta(z-z_{n})\bigg{\rangle}_{\textup{GinOE},N}=\frac{1}{\pi}\
\left(1+\sqrt{\frac{\pi}{2}}\ \exp\left[2y^{2}\right]\ \frac{1}{2|y|}\
\textup{erfc}\left(\sqrt{2}\ |y|\right)\right)\\\ &\times\bigg{[}\
\frac{\Gamma\left(N-1,|z|^{2}\right)}{(N-2)!}\bigg{(}N-1-|z|^{2}\bigg{)}+\frac{|z|^{2(N-1)}}{(N-2)!}\
e^{-|z|^{2}}\bigg{]}\ .\end{split}$ (2.10)

Comparing Eq. (2.10) with its GinUE counterpart Eq. (2.8), we see that they
almost share the term inside the square brackets (with the change $N\to N-1$).
The result in the GinUE is fully rotationally symmetric as there is no
$y-$dependent factor present. In the GinOE however, the self-overlap is
dependent on the distance to the real axis and so is not rotationally
symmetric.

In Figure 3, we compare the mean conditional self-overlap, Eq. (1.4), for
eigenvectors associated with purely imaginary eigenvalues, $z=iy$, in the
GinOE and GinUE at finite $N$, as predicted by our theory and as seen in
numerical simulations. As is evident, in the region close to the real line,
the mean conditional self-overlap is much larger in the GinOE than the GinUE,
implying that the GinOE has a higher degree of non-normality.

Figure 3: Mean conditional self-overlap $\mathbb{E}_{N}(z)$ associated with
purely imaginary eigenvalues in the GinOE and GinUE, for three different
values of $N$. Theoretical predictions are red dashed lines (GinOE) and blue
solid lines (GinUE). Triangular (circular) markers represent numerical results
for the GinOE (GinUE). The mean self-overlap is measured by averaging self-
overlaps of the closest $O(10^{3})$ eigenvalues with respect to the chosen
value of $y$. Numerically averaged values are taken from a data set containing
$O(10^{8})$ samples of the self-overlap.

Next, we provide results for the large $N$ asymptotic behaviour of the mean
self-overlap in the bulk, at the spectral edge and in the depletion regime, as
defined in the previous section.

######  Corollary 2.6.

For a complex eigenvalue $z=\sqrt{N}w$, where $w=x+iy$ with $|w|<1$ while
keeping $|y|\gg N^{-1/2}$ as $N\to\infty$, the limiting scaled mean self-
overlap in the bulk is given by

$\mathcal{O}_{\textup{bulk}}^{\textup{(GinOE,c)}}(w)\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\
\mathcal{O}_{N}^{\textup{(GinOE,c)}}\left(\sqrt{N}w\right)=\frac{1}{\pi}\left(1-|w|^{2}\right)\
\Theta\left[1-|w|^{2}\right]\ ,$ (2.11)

where $\Theta\left[a\right]$ is the Heaviside function, which is equal to one,
when $a>0$ and zero otherwise.

This is nothing else but the original Chalker-Mehlig asymptotic for the mean
self-overlap in the spectral bulk of the GinUE. This relation has been tested
in both the GinOE and GinUE using numerical simulation, with the results shown
in Figure 4. In the case of the GinOE, one can see that deep inside the bulk
region the formula Eq. (2.11) is accurate for $N\geq O(10^{2})$, indicating
that the theory is accurate as $N\to\infty$.

Figure 4: Numerical simulation of the mean conditional self-overlap,
$\mathbb{E}_{\text{bulk}}(w)$, within the bulk of the GinOE and GinUE for
large $N$ compared to the Chalker and Mehlig result (black lines). Left:
$\mathbb{E}_{\text{bulk}}(w)$ in the bulk of GinOE as a function of $|w|$ for
a range of different values of $N$ (coloured markers). We obtain results by
considering the self-overlaps for eigenvalues within $\pm 1/\sqrt{N}$ of each
value of $|z|$. Right: $\mathbb{E}_{\text{bulk}}(w)$ of eigenvectors of size
$N=250$ associated with purely imaginary eigenvalues in the GinOE (red
triangles) and GinUE (blue circles). Numerical results for each value of $N$
are taken from a sample with $O(10^{6})$ GinOE and GinUE matrices.

One observes a crucial difference between the two ensembles when considering
eigenvalues close to the real axis. This can be seen when measuring the mean
self-overlap of eigenvectors associated with purely imaginary eigenvalues in
the bulk for all $\text{Im}(z)\in[-\sqrt{N},\sqrt{N}]$. In the case of the
GinUE, the Chalker-Mehlig result holds for all $\text{Im}(z)$. In the GinOE
however, the depletion of eigenvalues close to the real line leads to
considerable deviations from the Chalker-Mehlig formula. This effect will be
accounted for by treating this region more carefully in Corollary 2.8.
However, before doing so, we consider the edge of the droplet and find the
following Corollary, in agreement with its GinUE counterpart, given in Eq.
(2.9).

######  Corollary 2.7.

For a complex eigenvalue $z=\left(\sqrt{N}+\eta\right)e^{i\theta}$ satisfying
$|\sin\theta|\sim\mathcal{O}(1)$ and $\eta>0$ the limiting scaled mean self-
overlap at the edge reads

$\mathcal{O}_{\textup{edge}}^{\textup{(GinOE,c)}}(\eta)\equiv\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}}\
\mathcal{O}_{N}^{\textup{(GinOE,c)}}\left(\left(\sqrt{N}+\eta\right)e^{i\theta}\right)=\frac{1}{\pi}\left(\frac{1}{\sqrt{2\pi}}\
e^{-2\eta^{2}}-\eta\ \textup{erfc}\left(\sqrt{2}\ \eta\right)\right)\ .$
(2.12)

We thus see that, away from the real axis, both the bulk asymptotic and the
edge asymptotic of the mean diagonal overlap is shared between the GinOE and
GinUE. Note that when approaching the boundary of the droplet, the mean self-
overlap turns out to be parametrically weaker, which is reflected in rescaling
with $1/\sqrt{N}$ instead of $1/N$ to obtain a non-trivial limit as
$N\to\infty$.

The result for the mean self-overlap at the spectral edge of the GinOE has
been considered in Figure 5. One can see from this figure that as $N$
increases, the agreement between the theoretical and numerically observed mean
self-overlaps becomes better.

Figure 5: Left: Limiting density of complex eigenvalues at the edge of the
GinOE droplet. Right: Mean conditional self-overlap,
$\mathbb{E}_{\text{edge}}(\eta)$, as a function of eigenvalue moduli,
$|z|=\sqrt{N}+\eta$, for a range of different values of $N$. The limiting
theoretical prediction of $\mathbb{E}_{\text{edge}}(\eta)$ for the GinOE
(solid black line) is compared to numerical simulations for different values
of $N$ (coloured markers). For each value of $\eta$, numerical averages are
obtained by considering all eigenvalues with a modulus between $\pm
1/\sqrt{N}$ of $(\sqrt{N}+\eta)$. Samples of the self-overlap are taken from a
data set generated from $O(10^{6})$ GinOE matrices of each used value of $N$.

Finally, we present the asymptotic results for the depletion region of the
droplet in the GinOE.

######  Corollary 2.8.

For a complex eigenvalue $z=x+i\xi$, such that $\xi\sim\mathcal{O}(1)$ the
limiting scaled mean self-overlap in the depleted region close to the origin,
i.e. $x\sim\mathcal{O}(1)$, reads

$\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,origin}}(\xi)\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\
\mathcal{O}_{N}^{\textup{(GinOE,c)}}\left(x+i\xi\right)=\frac{1}{\pi}\left(1+\sqrt{\frac{\pi}{2}}\
\frac{1}{2|\xi|}\ e^{2\xi^{2}}\ \textup{erfc}\left(\sqrt{2}\
|\xi|\right)\right)\ .$ (2.13)

Rescaling instead $x=\sqrt{N}\delta$, the limiting scaled mean self-overlap in
the depleted region becomes

$\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,strip}}(\delta,\xi)\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\
\mathcal{O}_{N}^{\textup{(GinOE,c)}}\left(\sqrt{N}\delta+i\xi\right)=\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,origin}}(\xi)\left(1-\delta^{2}\right)\Theta\left[1-\delta^{2}\right]\
.$ (2.14)

In order to demonstrate numerically the appropriate scale on which the
depletion regime should be studied, the mean density of purely imaginary
eigenvalues in the GinOE is plotted in Figure 6. This illustrates a region of
reduced eigenvalue density in the GinOE when $O(10^{-1})<y<O(10^{1})$ before
reaching an approximately constant value inside the bulk. Figure 6 also shows
the mean conditional self-overlap, Eq. (1.4), close to the origin and in a
small strip close to the real line.

Figure 6: Large $N$ density of complex eigenvalues and mean conditional self-
overlap $\mathbb{E}_{\text{dep}}(z)$ in the depletion regime of the GinOE.
Left: Comparison of the limiting density of purely imaginary eigenvalues in
the GinOE (red) and GinUE (blue). Centre: Theoretical prediction of
$\mathbb{E}_{\text{dep}}^{\text{(origin)}}(\xi)$ close to the origin (red
line), compared to numerical simulations (red triangles). Right: Mean
conditional self-overlap
$\mathbb{E}_{\text{dep}}^{\text{(strip)}}(\delta,\xi)$, with
$z=\sqrt{N}\delta+i\xi$, for three different fixed values of $\xi$ in a strip
close to the real line. In the centre and right hand plots the solid black
line represents the Chalker and Mehlig result. Theoretical predictions
(coloured lines) are compared to numerical simulations (coloured markers) of
self-overlaps generated from $O(10^{7})$ GinOE matrices of size $N=500$.

When considering the mean self-overlap in this region, we start by considering
eigenvalues close to the origin, i.e $z=x+i\xi$ with $x$ and $\xi\sim O(1)$.
Here, one can see that Eq. (2.13) is independent of $x$ and depends purely on
the imaginary component $\xi$. On the other hand, when considering the mean
self-overlap along a rectangular strip close to the real line, i.e. taking
$x\to\sqrt{N}\delta$, as in Eq. (2.14), there is now an explicit dependence on
$\delta$. Essentially, one can interpret this expression as a scaled version
of the Chalker-Mehlig result, that describes the increased mean self-overlap
in the region of eigenvalue depletion close to the real line. Note that as
$\xi$ becomes comparable to $\sqrt{N}$, i.e. the eigenvalue is inside the
bulk, $\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,origin}}(z)\to
1/\pi$ and we, unsurprisingly, recover the Chalker-Mehlig result.

###  2.3 Numerical Simulations of the distribution of the diagonal overlap and
discussion of open questions

We have already seen that, in the limit of large $N$, the mean self-overlap of
eigenvectors in the spectral bulk and at the edge is the same for both the
GinOE and GinUE. However, despite these similarities, there is a discernible
difference in behaviour of the mean self-overlap in these two ensembles due to
the existence of the depletion regime in the GinOE. It is natural to expect
that a similar picture should hold not only for the first moment, but the
whole distribution of the diagonal overlaps. As we do not yet have the
analytic expression for such a distribution, $\mathcal{P}(t,z)$ where
$t=\mathcal{O}_{nn}-1$, for the GinOE in the complex plane, we proceed with
briefly reviewing the results for complex eigenvalues of the GinUE and real
eigenvalues of the GinOE, following the work [55]. The equation for the
limiting JPDF of the eigenvector self-overlap in the bulk of the GinUE reads:

$\displaystyle\mathcal{P}^{\text{(GinUE,c)}}_{\text{bulk}}\left(s,w\right)=\frac{(1-|w|^{2})^{2}}{\pi
s^{3}}e^{-\frac{1-|w|^{2}}{s}}\Theta[1-|w|^{2}]\ ,$ (2.15)

with $s=t/N$ and $z=\sqrt{N}w$, whereas the distribution at the spectral edge
is given by

$\displaystyle\mathcal{P}^{\text{(GinUE,c)}}_{\text{edge}}\left(\sigma,\eta\right)=$
$\displaystyle\
\frac{1}{2\pi\sigma^{5}}e^{-\frac{\Delta^{2}}{2\sigma^{2}}}\bigg{\\{}\frac{e^{-2\delta^{2}}}{\pi}\left(2\sigma^{2}-\Delta\right)-\frac{1}{\sqrt{2\pi}}\left(4\delta\sigma^{2}-\Delta(2\delta+\sigma)\right)\text{erfc}\left(\sqrt{2}\delta\right)$
(2.16)
$\displaystyle+\frac{e^{2\delta^{2}}}{2}\left(\Delta^{2}-\sigma^{2}\right)\text{erfc}^{2}\left(\sqrt{2}\delta\right)\bigg{\\}}\
,$

where $\sigma=t/\sqrt{N}$ and $\Delta=1-2\sigma\eta$. In [55] one also finds
an expression for the density of the self-overlap of GinOE eigenvectors
associated with purely real eigenvalues in the bulk, $z=\sqrt{N}x$, which is
given by

$\displaystyle\mathcal{P}^{\text{(GinOE,r)}}_{\text{bulk}}\left(s,x\right)=\frac{(1-x^{2})}{2\sqrt{2\pi}}\frac{e^{-\frac{1-x^{2}}{2s}}}{s^{2}}\Theta[1-x^{2}]\
.$ (2.17)

To compare with numerical simulations, the above distributions must also be
normalised with respect to the mean spectral density. We denote the normalised
distribution as $\widetilde{\mathcal{P}}(t,z)=\mathcal{P}(t,z)/\rho(z)$.

In Figure 7, we compare distributions of the self-overlap of eigenvectors in
the GinOE and GinUE. This is done in three different ways. Firstly, we
consider the theoretical distributions in the complex bulk of the GinUE and
real bulk of the GinOE (at $x=0$), in comparison to a numerically observed
distribution in the depletion regime of the GinOE. We also compare the
distribution of eigenvector self-overlaps in the complex bulk of the GinOE to
the theoretical limiting distribution of self-overlaps in the bulk of the
GinOE. Finally, we make a comparison between the distribution of the self-
overlap at the edge of the GinUE and a numerically measured distribution at
the edge of the GinOE.

Figure 7: Numerical simulation of the distribution of the eigenvector self-
overlaps in the GinOE compared to theoretical distributions for the GinOE and
GinUE. In each plot the solid black line is the limiting distribution of the
self-overlap in a region of the GinUE and the red shaded area shows a
normalised histogram of self-overlaps observed in numerical simulation of
$O(10^{7})$ GinOE matrices of size $N=500$. Left: Depletion regime of the
GinOE compared to the bulk of the GinUE and bulk real eigenvalues in the GinOE
(blue line). Centre: Bulk of the GinOE compared to the bulk of the GinUE.
Right: Edge region of the GinOE compared to the edge of the GinUE.

The data indeed seems to confirm that the GinUE distribution accurately
describes the corresponding quantities for the GinOE away from the real axis.
In the depletion regime around the real axis the GinOE is not described by the
limiting distribution associated with either complex eigenvalues in the bulk
of the GinUE, Eq. (2.15), or eigenvalues in the real bulk of the GinOE, Eq.
(2.17). It seems that the tails of the distribution of self-overlaps in the
depletion regime is described better by Eq. (2.17) than Eq. (2.15), indicating
that the corresponding distribution may have a $1/s^{2}$ tail (at least as an
intermediate asymptotics) as opposed to $1/s^{3}$. Next, considering the edge
region it is apparent that there is a small discrepancy between theory and
numerical simulations, however we attribute this to finite $N$ effects and
expect this deviation to approach zero as $N\to\infty$.

Rigorously proving equivalence of the distribution of the GinOE self-overlap
for asymptotics within the bulk and at the edge for the GinOE remains an
outstanding challenge. Characterizing the distribution of eigenvector diagonal
overlaps in the depletion regime remains a completely open question. Another
interesting extension of this work would be to study the off-diagonal averaged
overlap, see Eq. (1.3), which is not yet known for the GinOE, neither for
complex nor for real eigenvalues.

In a separate paper, the results of this work will be extended to the mean
self-overlap in the complex plane of the real elliptic Ginibre ensemble,
interpolating between GinOE and GOE. In particular, this should allow us to
study the depletion region in the new non-trivial scaling regime of weak non-
Hermiticity [66, 67, 68].

##  3 Derivation of Main Results

Our strategy for the proof of the main results outlined in the previous
section amounts to using the (incomplete) Schur decomposition with respect to
pairs of complex eigenvalues of a real random matrix $G$. In this way we
reduce the problem of evaluating the mean self-overlap in the complex plane to
calculating the expectation value of a certain determinant, which is
eventually implemented using Grassmann integration. In the process, we adapt
techniques used in similar circumstances in [55] and [62], which will help to
prove the asymptotic results in Section 4.

###  3.1 Eigenvectors of real matrices via incomplete Schur decomposition

Let $z=x+iy$ (with $y\neq 0$) be a complex eigenvalue of the $N\times N$ real,
non-Hermitian matrix $G$. The associated left and right eigenvectors are
denoted by $\mathbf{x}_{L}^{\dagger}$ and $\mathbf{x}_{R}$ respectively.
Following the influential paper by Edelman [59], we employ the _incomplete
Schur decomposition_ of $G$ with respect to $z$, which is

$G=Q\widetilde{G}Q^{T}\quad\text{with}\quad\widetilde{G}=\left(\begin{matrix}\begin{matrix}x&b\\\
-c&x\\\ \end{matrix}&W\\\ 0&G_{2}\\\ \end{matrix}\right)\text{,}\
bc>0\text{,}\ b\geq c\text{,}\ y=\sqrt{bc}\text{,}$ (3.1)

where real symmetric $Q$ is a so-called Householder reflection matrix such
that $Q^{2}=\leavevmode\hbox{\small 1\kern-3.8pt\normalsize 1}_{N}$. The block
matrix $W$ is real and has size $2\times(N-2)$, i.e. $W=\left(\mathbf{w}_{1}\
\text{,}\ \mathbf{w}_{2}\right)^{T}$, where $\mathbf{w}_{1}$, $\mathbf{w}_{2}$
are vectors with $N-2$ real i.i.d. entries following standard normal
distributions. The matrix $G_{2}$ in this decomposition is essentially an
$(N-2)\times(N-2)$ dimensional real Ginibre matrix. The Jacobian of the
transformation is presented in [59] and the integration measure changes
according to

$\begin{split}dG&=2(b-c)\ \det\left[(x\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}\right]\ dx\ db\ dc\ dW\ dG_{2}\ dS\
,\end{split}$ (3.2)

where $dS$ denotes the volume element of the Stiefel manifold originating from
the Householder reflection matrix $Q$. After simple manipulations we find the
probability measure, defined in terms of the new variables, reads:

$\begin{split}P_{\text{GinOE}}(G)dG=&\
C_{N}^{\prime}\det\left[(x\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}\right]e^{-\frac{1}{2}\Tr G_{2}G_{2}^{T}}\\\ &\
(b-c)e^{-\frac{1}{2}\left(2x^{2}+b^{2}+c^{2}+\Tr WW^{T}\right)}dx\ db\ dc\ dW\
dG_{2}\ ,\end{split}$ (3.3)

where the constant $C_{N}^{\prime}$ now reads

$\begin{split}C_{N}^{\prime}=2\
\frac{(2\pi)^{-\frac{1}{2}(N-1)^{2}}}{\sqrt{2\pi}\Gamma(N-1)}\ .\end{split}$
(3.4)

The next step is to determine the left and right eigenvectors in terms of the
Schur decomposition variables, which is done adapting the method of [55],
where the case of real GinOE eigenvalues was considered. For $z=x+iy$, the
eigenvalue problems read

$G\mathbf{x}_{R}=z\mathbf{x}_{R}\quad\text{and}\quad\mathbf{x}_{L}^{\dagger}G=z\mathbf{x}_{L}^{\dagger}\
.$ (3.5)

Applying the incomplete Schur decomposition from Eq. (3.1), we introduce
$\widetilde{\mathbf{x}}_{L}^{\dagger}\equiv\mathbf{x}_{L}^{\dagger}Q$ and
$\widetilde{\mathbf{x}}_{R}\equiv Q\mathbf{x}_{R}$ and see that the eigenvalue
problems for $\widetilde{G}$ can be rewritten as
$\widetilde{\mathbf{x}}_{L}^{\dagger}\widetilde{G}=z\widetilde{\mathbf{x}}_{L}^{\dagger}$
and $\widetilde{G}\widetilde{\mathbf{x}}_{R}=z\widetilde{\mathbf{x}}_{R}$.

The left-right diagonal overlap, corresponding to the eigenvalue $z$ is
obviously invariant under the incomplete Schur decomposition, i.e.

$\mathcal{O}_{z}=\left(\mathbf{x}_{L}^{\dagger}\mathbf{x}_{L}\right)\left(\mathbf{x}_{R}^{\dagger}\mathbf{x}_{R}\right)=\left(\widetilde{\mathbf{x}}_{L}^{\dagger}\widetilde{\mathbf{x}}_{L}\right)\left(\widetilde{\mathbf{x}}_{R}^{\dagger}\widetilde{\mathbf{x}}_{R}\right)\
,$ (3.6)

hence we can continue the calculation of the mean self-overlap using
$\widetilde{\mathbf{x}}_{L}^{\dagger}$, $\widetilde{\mathbf{x}}_{R}$ and
$\widetilde{G}$ instead of $\mathbf{x}_{L}^{\dagger}$, $\mathbf{x}_{R}$ and
$G$.

The incomplete Schur decomposition gives us the forms of
$\widetilde{\mathbf{x}}_{R}$ and $\widetilde{\mathbf{x}}_{L}^{\dagger}$. By
construction,
$\widetilde{\mathbf{x}}_{L}^{\dagger}\widetilde{\mathbf{x}}_{R}=1$ and it is
easy to check that the eigenvectors must have the following structure:

$\widetilde{\mathbf{x}}_{R}=\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\\
i\sqrt{\frac{c}{b}}\\\ \mathbf{0}_{N-2}\\\
\end{matrix}\right)\quad\text{and}\quad\widetilde{\mathbf{x}}_{L}^{\dagger}=\frac{1}{\sqrt{2}}\left(\begin{matrix}1\\\
i\sqrt{\frac{b}{c}}\\\ \sqrt{2}\ \mathbf{b}_{N-2}\\\
\end{matrix}\right)^{\dagger}\ ,$ (3.7)

with some $\mathbf{b}_{N-2}$ yet to be determined. Substituting the above into
the eigenvalue equation,
$\widetilde{\mathbf{x}}_{L}^{\dagger}\widetilde{G}=z\widetilde{\mathbf{x}}_{L}^{\dagger}$,
we find that

$z\mathbf{b}_{N-2}^{\dagger}\overset{!}{=}\frac{1}{\sqrt{2}}\left(\mathbf{w}_{1}^{T}-i\sqrt{\frac{b}{c}}\mathbf{w}_{2}^{T}\right)+\mathbf{b}_{N-2}^{\dagger}G_{2}\quad\Leftrightarrow\quad\mathbf{b}_{N-2}^{\dagger}=\frac{1}{\sqrt{2}}\left(\mathbf{w}_{1}^{T}-i\sqrt{\frac{b}{c}}\mathbf{w}_{2}^{T}\right)\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}-G_{2}\right)^{-1}\ .$ (3.8)

Using the above relation, the self-overlap $\mathcal{O}_{z}$ associated with a
complex eigenvalue $z$ in the GinOE becomes

$\begin{split}\mathcal{O}_{z}&=\left(\widetilde{\mathbf{x}}_{L}^{\dagger}\widetilde{\mathbf{x}}_{L}\right)\left(\widetilde{\mathbf{x}}_{R}^{\dagger}\widetilde{\mathbf{x}}_{R}\right)=\frac{1}{4}\left(2+\frac{c^{2}+b^{2}}{bc}\right)+\frac{1}{2}\left(1+\frac{c}{b}\right)\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\
.\end{split}$ (3.9)

To perform the ensemble average we follow [59] and change variables from $b$
and $c$ to $y=\sqrt{bc}$ and $\delta=b-c>0$, implying

$\begin{split}db\ dc&=\frac{2y}{\sqrt{\delta^{2}+4y^{2}}}\ dy\ d\delta\
\text{,}\quad
b^{2}+c^{2}=\delta^{2}+2y^{2}\quad\text{and}\quad\frac{b}{c}=\exp\left[\ 2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\ \text{,}\end{split}$
(3.10)

so that the relevant JPDF, Eq. (3.3), now takes the form

$\begin{split}P_{\text{GinOE}}(G)dG=&\
C_{N}^{\prime}\det\left[(x\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}\right]\exp\left[-\frac{1}{2}\Tr\left(G_{2}G_{2}^{T}+WW^{T}\right)\right]\\\
&\ \times\exp\left[-x^{2}\right]\ \exp\left[-y^{2}\right]\
\exp\left[-\frac{1}{2}\delta^{2}\right]\
\frac{2y\delta}{\sqrt{\delta^{2}+4y^{2}}}\ dx\ dy\ d\delta\ dW\ dG_{2}\
.\end{split}$ (3.11)

Defining the matrix

$\begin{split}B&\equiv\left(z\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2}\right)^{\dagger}\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}-G_{2}\right)\ ,\end{split}$ (3.12)

we first express $\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)$ as

$\begin{split}\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)=&\frac{1}{2}\mathbf{w}_{1}^{T}\
B^{-1}\ \mathbf{w}_{1}+\frac{1}{2}\exp\left[\ 2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\mathbf{w}_{2}^{T}\ B^{-1}\
\mathbf{w}_{2}\\\ &+\frac{1}{2}\ i\ \exp\left[\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\bigg{[}\mathbf{w}_{1}^{T}\
B^{-1}\ \mathbf{w}_{2}-\mathbf{w}_{2}^{T}\ B^{-1}\ \mathbf{w}_{1}\bigg{]}\
,\end{split}$ (3.13)

which when substituted into Eq. (3.9) implies

$\begin{split}\mathcal{O}_{z}=\widetilde{c}_{1}+\widetilde{c}_{2}\
\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\ ,\end{split}$ (3.14)

where the constants read

$\begin{split}\widetilde{c}_{1}&=\frac{1}{4}\left(2+\frac{\delta^{2}+2y^{2}}{y^{2}}\right)\quad\text{and}\quad\widetilde{c}_{2}=\frac{1}{2}\left(1+\exp\left[-2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\right)\
\text{.}\end{split}$ (3.15)

With these formulas in hand, we now proceed to proving our main theorem in the
next section.

###  3.2 Proof of Theorem 2.5

Using the equivalence of different indices $n$ under the ensemble averaging,
the mean self-overlap for the GinOE can be written as follows:

$\begin{split}\mathcal{O}_{N}^{(\textup{GinOE},c)}(z)&=\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\mathcal{O}_{nn}\
\delta(z-z_{n})\bigg{\rangle}_{\text{GinOE},N}=\bigg{\langle}\mathcal{O}_{\widetilde{z}}\
\delta(z-\widetilde{z})\bigg{\rangle}_{\textup{GinOE},N}\ \text{,}\end{split}$
(3.16)

where the two-dimensional $\delta$-function of complex argument, $z=x+iy$,
should be interpreted as the product of two one-dimensional
$\delta$-functions, containing its real and imaginary parts respectively, i.e.
$\delta(z-\widetilde{z})=\delta(x-\widetilde{x})\delta(y-\widetilde{y})$. This
allows us to use the incomplete Schur decomposition as presented in the
previous section, in particular using results from Eq. (3.11) to Eq. (3.15).
The average splits into integrations over $G_{2}$, $W$, $x$, $y$ and $\delta$
according to Eq. (3.11). The two $\delta$-functions make the integrations over
$x$ and $y$ trivial, meaning that the next non-trivial task is to perform the
integration with respect to the matrix $W$. We have $\Tr
WW^{T}=\mathbf{w}_{1}^{T}\mathbf{w}_{1}+\mathbf{w}_{2}^{T}\mathbf{w}_{2}$ and
$dW=d\mathbf{w}_{1}d\mathbf{w}_{2}$, which allows us to perform the Gaussian
averages with respect to vectors $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ instead
of $W$. We define the average with respect to $\mathbf{w}$ of an observable
$\mathcal{A}(\mathbf{w})$ as

$\bigg{\langle}\mathcal{A}(\mathbf{w})\bigg{\rangle}_{\mathbf{w}}\equiv\frac{1}{(2\pi)^{\frac{N-2}{2}}}\int
d\mathbf{w}\exp\left[-\frac{1}{2}\mathbf{w}^{T}\mathbf{w}\right]\mathcal{A}(\mathbf{w})\
\text{,}$ (3.17)

normalized in such a way that $\langle\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}\rangle_{\mathbf{w}}=1$. The mean self-overlap for
the GinOE then can be written as

$\begin{split}&\bigg{\langle}\mathcal{O}_{\widetilde{z}}\
\delta(z-\widetilde{z})\ \bigg{\rangle}_{\textup{GinOE},N}=C_{N}^{\prime}\
(2\pi)^{N-2}\ \exp\left[-\left(x^{2}+y^{2}\right)\right]\int d\delta\
\frac{2y\delta}{\sqrt{\delta^{2}+4y^{2}}}\
\exp\left[-\frac{1}{2}\delta^{2}\right]\\\ &\times\int dG_{2}\
\det\left[(x\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}\right]\exp\left[-\frac{1}{2}\Tr G_{2}G_{2}^{T}\right]\
\bigg{\langle}\bigg{\langle}\widetilde{c}_{1}+\widetilde{c}_{2}\
\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}\
.\end{split}$ (3.18)

The computation of the double average over $\mathbf{w}_{1}$, $\mathbf{w}_{2}$
is performed in the next step, using the following Lemma.

######  Lemma 3.1.

Let $\mathbf{w}_{1}$, $\mathbf{w}_{2}$ be two vectors, each of length $N-2$,
with independent real variables as entries and $X$ be an $N-2$ dimensional
matrix. Then

$\begin{split}&\bigg{\langle}\mathbf{w}^{T}\ X\
\mathbf{w}\bigg{\rangle}_{\mathbf{w}}=\Tr
X\quad\textup{and}\quad\bigg{\langle}\bigg{\langle}\mathbf{w}_{1}^{T}\ X\
\mathbf{w}_{2}\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}=\bigg{\langle}\bigg{\langle}\mathbf{w}_{2}^{T}\
X\
\mathbf{w}_{1}\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}=0\
\text{,}\end{split}$ (3.19)

with the average defined in Eq. (3.17) and
$\mathbf{w}\in\\{\mathbf{w}_{1},\mathbf{w_{2}}\\}$.

The verification of this Lemma is straightforward and can be done by
exploiting the Gaussian nature of the integrals involved. This allows us to
show the validity of the next Lemma.

######  Lemma 3.2.

The average over $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ of the object
$\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)$ reads

$\begin{split}\bigg{\langle}\bigg{\langle}\widetilde{c}_{1}+\widetilde{c}_{2}\
\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}&=\frac{1}{2}\left(2+\frac{\delta^{2}}{2y^{2}}\right)\bigg{[}1+\Tr\left[B^{-1}\right]\bigg{]}\
.\end{split}$ (3.20)

###### Proof.

The verification amounts to using the expression of
$\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)$, in Eq. (3.13), in
conjunction with Lemma 3.1, such that:

$\begin{split}\bigg{\langle}\
\bigg{\langle}\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}&=\frac{1}{2}\bigg{\langle}\mathbf{w}_{1}^{T}\
B^{-1}\ \mathbf{w}_{1}\bigg{\rangle}_{\mathbf{w}_{1}}+\frac{1}{2}\exp\left[\
2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\bigg{\langle}\mathbf{w}_{2}^{T}\
B^{-1}\ \mathbf{w}_{2}\bigg{\rangle}_{\mathbf{w}_{2}}\\\
&=\frac{1}{2}\left(1+\exp\left[\ 2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\right)\Tr B^{-1}\
.\end{split}$ (3.21)

Then exploiting the fact that
$\cosh^{2}\left[\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]=1+\frac{\delta^{2}}{4y^{2}}$,
alongside the definition of $\tilde{c}_{2}$ in Eq. (3.15) this yields:

$\begin{split}\frac{\widetilde{c}_{2}}{2}\left(1+\exp\left[\ 2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\right)=\frac{1}{2}\bigg{[}1+\cosh\left[2\
\text{arcsinh}\left(\frac{\delta}{2y}\right)\right]\bigg{]}=1+\frac{\delta^{2}}{4y^{2}}.\end{split}$
(3.22)

Finally, applying the definition of $\tilde{c}_{1}$ from Eq. (3.15), leads to

$\begin{split}\bigg{\langle}\bigg{\langle}\widetilde{c}_{1}+\widetilde{c}_{2}\
\left(\mathbf{b}_{N-2}^{\dagger}\mathbf{b}_{N-2}\right)\bigg{\rangle}_{\mathbf{w}_{2}}\bigg{\rangle}_{\mathbf{w}_{1}}&=\widetilde{c}_{1}+\frac{1}{2}\Tr\left[B^{-1}\right]\left(2+\frac{\delta^{2}}{2y^{2}}\right)=\frac{1}{2}\left(2+\frac{\delta^{2}}{2y^{2}}\right)\bigg{[}1+\Tr\left[B^{-1}\right]\bigg{]}\
,\end{split}$ (3.23)

which concludes the proof of this Lemma. ∎

With Lemma 3.2, we are able to express the mean self-overlap at finite $N$ in
such a way, that the remaining average over $G_{2}$ becomes tractable.
Starting from Eq. (3.18) and applying Lemma 3.2 we obtain

$\begin{split}\bigg{\langle}\mathcal{O}_{\widetilde{z}}\
\delta(z-\widetilde{z})\ \bigg{\rangle}_{\textup{GinOE},N}&=C_{N}^{\prime}\
(2\pi)^{N-2}\ \exp\left[-\left(x^{2}+y^{2}\right)\right]\int d\delta\
\frac{2y\delta}{\sqrt{\delta^{2}+4y^{2}}}\
\exp\left[-\frac{1}{2}\delta^{2}\right]\frac{1}{2}\left(2+\frac{\delta^{2}}{2y^{2}}\right)\\\
&\times\int dG_{2}\ \det\left[(x\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}\right]\exp\left[-\frac{1}{2}\Tr
G_{2}G_{2}^{T}\right]\bigg{[}1+\Tr\left[B^{-1}\right]\bigg{]}\ .\end{split}$
(3.24)

The integrand, involving the matrix $G_{2}$, can be written in terms of the
matrix $B$ from Eq. (3.12), noticing that

$\begin{split}\det B&=\det\left[\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{N-2}-G_{2}\right)^{\dagger}\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}_{N-2}-G_{2}\right)\right]=\det\left[(x\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}-G_{2})^{2}+y^{2}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}\right]\\\
&=\det\left[\begin{matrix}0&i(z\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2})\\\ i(\bar{z}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N-2}-G_{2}^{T})&0\\\ \end{matrix}\right]\ ,\end{split}$ (3.25)

as well as

$\frac{\partial}{\partial\mu}\det\left(\mu\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N-2}+B\right)\bigg{|}_{\mu=0}=\det B\ \Tr B^{-1}\
.$ (3.26)

In fact, it is more convenient to introduce a block-matrix $M$ via

$M\equiv\left(\begin{matrix}\sqrt{\mu}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}&i\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}-G\right)\\\
i\left(\bar{z}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}-G^{T}\right)&\sqrt{\mu}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}\\\ \end{matrix}\right)\ ,$ (3.27)

where each block is of size $N\times N$ and $\mu$ is a real parameter. It is
easy to see that for $\mu=0$ we then have

$\det M=\det B\quad\text{and}\quad\frac{\partial}{\partial\mu}\det M\
\bigg{|}_{\mu=0}=\det B\ \Tr B^{-1}\ .$ (3.28)

Now we define the averaging over $G_{2}$ as

$\bigg{\langle}\mathcal{A}(G_{2})\bigg{\rangle}_{G_{2}}\equiv C_{N-2}^{-1}\int
dG_{2}\ \exp\left[-\frac{1}{2}\Tr\left(G_{2}G_{2}^{T}\right)\right]\
\mathcal{A}(G_{2})\ ,$ (3.29)

where the constant $C_{N-2}$ is given in Eq. (2.1) to ensure the correct
normalization: $\langle\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}\rangle_{G_{2}}=1$. The expression in Eq. (3.24) therefore becomes

$\begin{split}\bigg{\langle}\mathcal{O}_{\widetilde{z}}\
\delta(z-\widetilde{z})\ \bigg{\rangle}_{\textup{GinOE},N}&=C_{N}^{\prime}\
(2\pi)^{N-2}\ C_{N-2}\ \exp\left[-\left(x^{2}+y^{2}\right)\right]\int d\delta\
\frac{y\delta}{\sqrt{\delta^{2}+4y^{2}}}\
\exp\left[-\frac{1}{2}\delta^{2}\right]\left(2+\frac{\delta^{2}}{2y^{2}}\right)\\\
&\times\bigg{[}\ \bigg{\langle}\det M\bigg{\rangle}_{G_{2}}\
\bigg{|}_{\mu=0}+\frac{\partial}{\partial\mu}\bigg{\langle}\det
M\bigg{\rangle}_{G_{2}}\ \bigg{|}_{\mu=0}\ \bigg{]}\ .\end{split}$ (3.30)

The analysis of the average of the determinant of the matrix $M$ with respect
to the GinOE of size $N-2$ is done using the standard representation of the
determinant via Berezin integration over anti-commuting Grassmann variables
[63]:

$\int D(\bm{\Phi},\bm{\chi})\exp\left[-\bm{\Phi}^{T}M\bm{\chi}\right]=\det M\
\text{,}$ (3.31)

where
$D(\bm{\Phi},\bm{\chi})=d\bm{\phi}_{1}d\bm{\chi}_{1}d\bm{\phi}_{2}d\bm{\chi}_{2}$
and $\bm{\Phi}^{T}=\left(\bm{\phi}_{1},\bm{\phi}_{2}\right)^{T}$,
$\bm{\chi}^{T}=\left(\bm{\chi}_{1},\bm{\chi}_{2}\right)^{T}$. The outcome is
provided by the following Proposition.

######  Proposition 3.3.

Let $M$ be the matrix defined in Eq. (3.27), $z=x+iy$ is a complex number and
$G$ is an $N\times N$ real Ginibre matrix. The average over the determinant of
$M$, with respect to the real Ginibre matrix $G$, can be expressed in the
following form:

$\displaystyle\bigg{\langle}\det
M\bigg{\rangle}_{G}=\frac{1}{\pi}\int_{0}^{\infty}dr\leavevmode\nobreak\ r\
e^{-r^{2}}\int_{0}^{2\pi}d\theta\bigg{[}\mu+|z|^{2}+2\sqrt{\mu}r\cos(\theta)+r^{2}\bigg{]}^{N}$
(3.32) $\displaystyle=\int_{0}^{\infty}dR\
e^{-R}\bigg{[}\left(R+|z|^{2}\right)^{2}+\mu^{2}+2\mu(|z|^{2}-R)\bigg{]}^{\frac{N}{2}}\,P_{N}\left(\frac{\mu+R+|z|^{2}}{\sqrt{\left(R+|z|^{2}\right)^{2}+\mu^{2}+2\mu(|z|^{2}-R)}}\right)\
,$ (3.33)

where $P_{N}(t)$ is a Legendre polynomial defined via the identity [64,
8.913.3]

$P_{N}(t)=\frac{1}{\pi}\int_{0}^{\pi}d\theta\
\left(t+\sqrt{t^{2}-1}\cos{\theta}\right)^{N}\ .$ (3.34)

###### Proof.

Using the integration over vectors with $N$ anticommuting components each,
$\bm{\phi}_{1}$, $\bm{\phi}_{2}$, $\bm{\chi}_{1}$, $\bm{\chi}_{2}$, with
entries $\chi_{1,i}$ $(i=1,2,...N)$ and so forth, we start with writing Eq.
(3.31) in the explicit form for our particular choice:

$\bigg{\langle}\det M\bigg{\rangle}_{G}=(-1)^{N}\expectationvalue{\int
D\bm{\Phi}D\bm{\chi}\exp{-\left(\begin{matrix}\bm{\phi}_{1}^{T}&\bm{\phi}_{2}^{T}\end{matrix}\right)\left(\begin{matrix}\sqrt{\mu}\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}&i\left(z\leavevmode\hbox{\small
1\kern-3.8pt\normalsize 1}_{N}-G\right)\\\
i\left(\bar{z}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}-G^{T}\right)&\sqrt{\mu}\leavevmode\hbox{\small 1\kern-3.8pt\normalsize
1}_{N}\end{matrix}\right)\left(\begin{matrix}\bm{\chi}_{1}\\\
\bm{\chi}_{2}\end{matrix}\right)}}_{G}\ ,$ (3.35)

which upon using
$\bm{\phi}_{1}^{T}G\bm{\chi}_{2}=-\Tr[G\bm{\chi}_{2}\otimes\phi_{1}^{T}]$ can
be written as

$\displaystyle\bigg{\langle}\det M\bigg{\rangle}_{G}=(-1)^{N}\int$
$\displaystyle
D\bm{\Phi}D\bm{\chi}e^{-\sqrt{\mu}\bm{\phi}_{1}^{T}\bm{\chi}_{1}-\sqrt{\mu}\bm{\phi}_{2}^{T}\bm{\chi}_{2}-iz\bm{\phi}_{1}^{T}\bm{\chi}_{2}-i\bar{z}\bm{\phi}_{2}^{T}\bm{\chi}_{1}}\expectationvalue{e^{-i\Tr[G\bm{\chi}_{2}\otimes\bm{\phi}_{1}^{T}]-i\Tr[G^{T}\bm{\chi}_{1}\otimes\bm{\phi}_{2}^{T}]}}_{G}\
.$ (3.36)

The expectation value in the above integrand can be evaluated using the
following identity

$\expectationvalue{e^{-\Tr[G\bm{A}]-\Tr[G^{T}\bm{B}]}}_{G}=\exp{\frac{1}{2}\Tr[\bm{A}^{T}\bm{A}]+\frac{1}{2}\Tr[\bm{B}^{T}\bm{B}]+\Tr[\bm{A}\bm{B}]}\
,$ (3.37)

which can be found in [55, Eq. (3.15)]. In our case we have that
$\bm{A}=i\bm{\chi}_{2}\otimes\bm{\phi}_{1}^{T}$ and
$\bm{B}=i\bm{\chi}_{1}\otimes\bm{\phi}_{2}^{T}$, so that

$\Tr[\bm{A}^{T}\bm{A}]=\Tr[\bm{B}^{T}\bm{B}]=0\quad\text{and}\quad\Tr[\bm{A}\bm{B}]=\left(\bm{\phi}_{1}^{T}\bm{\chi}_{1}\right)\left(\bm{\phi}_{2}^{T}\bm{\chi}_{2}\right)\
,$ (3.38)

yielding

$\bigg{\langle}\det M\bigg{\rangle}_{G}=(-1)^{N}\int
D\bm{\Phi}D\bm{\chi}\exp{-\sqrt{\mu}\bm{\phi}_{1}^{T}\bm{\chi}_{1}-\sqrt{\mu}\bm{\phi}_{2}^{T}\bm{\chi}_{2}-iz\bm{\phi}_{1}^{T}\bm{\chi}_{2}-i\bar{z}\bm{\phi}_{2}^{T}\bm{\chi}_{1}+\left(\bm{\phi}_{1}^{T}\bm{\chi}_{1}\right)\left(\bm{\phi}_{2}^{T}\bm{\chi}_{2}\right)}\
.$ (3.39)

The exponential of the term quartic in anticommuting variables can be re-
expressed using a Hubbard-Stratonovich transformation of the form

$e^{ab}=\frac{1}{2\pi}\int d\bar{q}dqe^{-|q|^{2}-(aq+b\bar{q})}\ ,$ (3.40)

so that after changing the order of integration and performing the Gaussian
integrals over anticommuting vectors we arrive at

$\displaystyle\bigg{\langle}\det M\bigg{\rangle}_{G}$
$\displaystyle=\frac{(-1)^{N}}{2\pi}\int d\bar{q}dqe^{-|q|^{2}}\int
D\bm{\Phi}D\bm{\chi}e^{-\sqrt{\mu}\bm{\phi}_{1}^{T}\bm{\chi}_{1}-\sqrt{\mu}\bm{\phi}_{2}^{T}\bm{\chi}_{2}-iz\bm{\phi}_{1}^{T}\bm{\chi}_{2}-i\bar{z}\bm{\phi}_{2}^{T}\bm{\chi}_{1}-q\bm{\phi}_{1}^{T}\bm{\chi}_{1}-\bar{q}\bm{\phi}_{2}^{T}\bm{\chi}_{2}}$
(3.41) $\displaystyle=\ \frac{1}{2\pi}\int
d\bar{q}dqe^{-|q|^{2}}\left[\left(\sqrt{\mu}+q\right)\left(\sqrt{\mu}+\bar{q}\right)+|z|^{2}\right]^{N}\
.$ (3.42)

Employing the change of variables to polar coordinates, $q=re^{i\theta}$ and
$\bar{q}=re^{-i\theta}$, yields Eq. (3.32), then using $r^{2}=R$ and the
definition of a Legendre polynomial, Eq. (3.34), yields the second form, Eq.
(3.33). ∎

######  Remark 3.4.

Proposition 3.3 allows one to compute the two terms in Eq. (3.30) which
require averages with respect to $G_{2}$. Note that for $\mu=0$ the
determinant of the matrix $M$ can be written as a product of two
characteristic polynomials of Ginibre matrices. The average of a product of
two characteristic polynomials is proportional to the associated kernel at
equal arguments for complex eigenvalues in the GinOE, and is well-known, see
e.g. [60, Eq. (18.5.40)]. However, we need a slightly more general average
involving the derivative over $\mu$. We state the results we need in the next
Corollary.

######  Corollary 3.5.

With the average taken as in Eq. (3.29) using the $N-2$ sized GinOE matrix
$G_{2}$, we have

$\bigg{\langle}\det M\bigg{\rangle}_{G_{2}}\ \bigg{|}_{\mu=0}=e^{|z|^{2}}\
\Gamma\left(N-1,|z|^{2}\right)$ (3.43)

and

$\frac{\partial}{\partial\mu}\bigg{\langle}\det M\bigg{\rangle}_{G_{2}}\
\bigg{|}_{\mu=0}=\big{(}N-2-|z|^{2}\big{)}\ e^{|z|^{2}}\
\Gamma(N-1,|z|^{2})+|z|^{2(N-1)}\ .$ (3.44)

where we have used the incomplete Gamma-function defined in Eq. (2.3).

###### Proof.

The proof of Eq. (3.43) is immediate from Eq. (3.33) after changing $N\to
N-2$, using $P_{N-2}(1)=1$ [65, Table 18.6.1] and the identity [65, 8.6.5]

$\int_{0}^{\infty}\,e^{-R}(R+|z|^{2})^{N-2}\,dR=e^{|z|^{2}}\Gamma(N-1,|z|^{2})\
.$ (3.45)

In order to prove Eq. (3.44), we replace $N\to N-2$ in Eq. (3.33), then
differentiate over $\mu$ using

$\frac{\partial}{\partial\mu}\left(\frac{\mu+R+|z|^{2}}{\sqrt{\left(R+|z|^{2}\right)^{2}+\mu^{2}+2\mu(|z|^{2}-R)}}\right)\bigg{|}_{\mu=0}=\frac{2R}{(R+|z|^{2})^{2}}$
(3.46)

and also $P_{N-2}^{\prime}(1)=(N-2)(N-1)/2$, which follows per induction from
[64, 8.939.6]. Collecting all terms gives

$\begin{split}&\frac{\partial}{\partial\mu}\bigg{\langle}\det
M\bigg{\rangle}_{G_{2}}\ \bigg{|}_{\mu=0}=(N-2)\int_{0}^{\infty}\,dR\ e^{-R}\
(R+|z|^{2})^{N-4}\ \bigg{[}|z|^{2}+R(N-2)\bigg{]}\\\
&=(N-2)\left[(N-2)\int_{0}^{\infty}\,dR\
e^{-R}\left(R+|z|^{2}\right)^{N-3}-(N-3)\ |z|^{2}\ \int_{0}^{\infty}\,dR\
e^{-R}\left(R+|z|^{2}\right)^{N-4}\right]\\\ &=(N-2)\
e^{|z|^{2}}\bigg{[}(N-2)\ \Gamma(N-2,|z|^{2})-(N-3)\ |z|^{2}\
\Gamma(N-3,|z|^{2})\bigg{]}\ ,\end{split}$ (3.47)

where we have again utilised Eq. (3.45). By now substituting the following
identity for incomplete $\Gamma$-functions, see e.g. [64, 8.356.2],

$m\ \Gamma\left(m,x\right)=\Gamma\left(m+1,x\right)-e^{-x}x^{m}\ ,$ (3.48)

in Eq. (3.47), both for $m=N-2$ and $m=N-3$, one finds that

$\frac{\partial}{\partial\mu}\bigg{\langle}\det M\bigg{\rangle}_{G_{2}}\
\bigg{|}_{\mu=0}=\big{(}N-2-|z|^{2}\big{)}\ e^{|z|^{2}}\
\Gamma(N-1,|z|^{2})+|z|^{2(N-1)}\ ,$ (3.49)

which concludes our proof of Eq. (3.44). ∎

We can now finish the proof of Theorem 2.5. Going back to Eq. (3.30) and
utilising the results from Corollary 3.5 for the averages over $G_{2}$, we get
the following expression:

$\begin{split}\bigg{\langle}\mathcal{O}_{\widetilde{z}}\
\delta(z-\widetilde{z})\bigg{\rangle}_{\textup{GinOE},N}=&\ C_{N}^{\prime}\
(2\pi)^{N-2}\ C_{N-2}\ \ \frac{1}{2y}\int d\delta\ \delta\
\sqrt{\delta^{2}+4y^{2}}\ \exp\left[-\frac{1}{2}\delta^{2}\right]\\\
&\times\bigg{[}\Gamma(N-1,|z|^{2})\
\big{(}N-1-|z|^{2}\big{)}+|z|^{2(N-1)}e^{-|z|^{2}}\bigg{]}\ .\end{split}$
(3.50)

Finally, calculating the remaining integral over $\delta$, using [64,
3.382.4], Eq. (3.48) and [65, 8.4.6], as

$\frac{1}{2y}\ \int_{0}^{\infty}d\delta\ \delta\ \sqrt{\delta^{2}+4y^{2}}\
e^{-\frac{1}{2}\delta^{2}}=1+\sqrt{\frac{\pi}{2}}\ e^{2y^{2}}\ \frac{1}{2|y|}\
\text{erfc}\left(\sqrt{2}\ |y|\right)$ (3.51)

and collecting all multiplicative constants together yields

$\begin{split}\mathcal{O}^{(\textup{GinOE},c)}_{N}(z)&=\bigg{\langle}\frac{1}{N}\sum_{n=1}^{N}\mathcal{O}_{nn}\
\delta(z-z_{n})\bigg{\rangle}_{\textup{GinOE},N}=\frac{1}{\pi}\
\left(1+\sqrt{\frac{\pi}{2}}\ \exp\left[2y^{2}\right]\ \frac{1}{2|y|}\
\textup{erfc}\left(\sqrt{2}\ |y|\right)\right)\\\ &\times\bigg{[}\
\frac{\Gamma\left(N-1,|z|^{2}\right)}{(N-2)!}\bigg{[}N-1-|z|^{2}\bigg{]}+\frac{|z|^{2(N-1)}}{(N-2)!}\
e^{-|z|^{2}}\bigg{]}\ ,\end{split}$ (3.52)

thus proving the Theorem 2.5.

##  4 Asymptotic Analysis for large matrix size

In this Section, we prove the large $N\gg 1$ asymptotic results in various
scaling regimes, which are stated in Corollaries 2.6, 2.7 and 2.8. The
starting point is always the finite $N$ result in Theorem 2.5, from there we
then perform our asymptotic analysis via Laplace’s method.

######  Remark 4.1.

The asymptotic analysis at $N\rightarrow\infty$ becomes simpler, after
defining the function

$\Theta_{N}^{(M)}(x)\equiv\frac{\Gamma(N-M+1,Nx)}{\Gamma(N-M+1)}\ ,$ (4.1)

which appears in the expression at finite $N$ for the mean self-overlap in the
GinOE. We observe that, for real $x$ and fixed $M$, this function is bounded
for all $N$ by $1$, i.e. $\Theta_{N}^{(M)}(x)\leq 1$. Furthermore, in the
limit where $N\rightarrow\infty$ for a fixed, real $x$ and fixed non-negative
integer $M$, the following holds:

$\lim_{N\rightarrow\infty}\Theta_{N}^{(M)}(x)=\Theta[1-x]\ ,$ (4.2)

where $\Theta[x]$ is the Heaviside step function. We will also need the
following asymptotic formula, see e.g. [55, Eq. (2.9)]:

$\lim_{N\rightarrow\infty}\frac{\Gamma\left(N-1,N+2\delta
N^{1/2}\right)}{\Gamma\left(N-1\right)}=\frac{1}{\sqrt{2\pi}}\int_{2\delta}^{\infty}dv\
\exp\left[-\frac{v^{2}}{2}\right]=\frac{1}{2}\text{erfc}\left(\sqrt{2}\delta\right)\
.$ (4.3)

For a fixed, finite $\delta$ we have from the definition of the incomplete
$\Gamma$-function that:

$\lim_{N\rightarrow\infty}\frac{\Gamma\left(N-1,\delta\right)}{\Gamma\left(N-1\right)}=\
e^{-\delta}\ \sum_{k=0}^{\infty}\frac{\delta^{k}}{k!}=1\ .$ (4.4)

We will also use the following large $N$ asymptotic behaviour of the
complementary error function $\text{erfc}(x)$, see e.g. [65, 7.12.1]

$\text{erfc}\left(\sqrt{2}y\right)=\frac{e^{-2y^{2}}}{y\sqrt{2\pi}}\sum_{m=0}^{\infty}(-1)^{m}\frac{(2m-1)!!}{(4y^{2})^{m}}\approx\frac{e^{-2y^{2}}}{y\sqrt{2\pi}}\
,$ (4.5)

in particular, the above implies that as $y\rightarrow\sqrt{N}y$ and
$N\rightarrow\infty$, the following term vanishes

$\begin{split}e^{2y^{2}}\ \frac{1}{2y}\
\text{erfc}\left(\sqrt{2}y\right)&\rightarrow e^{2Ny^{2}}\
\frac{1}{2\sqrt{N}y}\
\frac{e^{-2Ny^{2}}}{\sqrt{N}y\sqrt{2\pi}}=\frac{1}{2Ny^{2}\sqrt{2\pi}}\overset{N\gg
1}{\longrightarrow}0\ .\end{split}$ (4.6)

We now start with the asymptotic analysis in the bulk of the GinOE and give
the proof of Corollary 2.6.

###### Proof.

Starting from Eq. (2.10), we introduce the scaling $z\rightarrow\sqrt{N}w$,
with $w=x+iy$ such that $|w|<1$ and keeping $|y|>N^{1/2}$ to avoid the
depletion regime. This transforms the Eq. (2.10) to

$\begin{split}\frac{1}{N}\mathcal{O}_{N}^{(\textup{GinOE},c)}(\sqrt{N}w)&=\frac{1}{\pi\
N}\ \left(1+\sqrt{\frac{\pi}{2}}\ \exp\left[2\ N\ y^{2}\right]\ \frac{1}{2\ N\
|y|}\ \textup{erfc}\left(\sqrt{2}\ N\ |y|\right)\right)\\\ &\times N\
\bigg{[}\
\frac{\Gamma\left(N-1,N|w|^{2}\right)}{\Gamma\left(N-1\right)}\bigg{[}1-\frac{1}{N}-|w|^{2}\bigg{]}+\frac{1}{N}\frac{N^{N-1}\
|w|^{2(N-1)}}{(N-2)!}\ e^{-N\ |w|^{2}}\bigg{]}\ .\end{split}$ (4.7)

The last term in the above expression vanishes, because $|w|<1$ and
$N^{N-2}/(N-2)!\rightarrow 0$ as $N\rightarrow\infty$ via Stirling’s formula.
From Remark 4.1 we deduce that the term containing the error function vanishes
and the ratio of $\Gamma$-functions can be identified with
$\Theta^{(2)}_{N}(|z|^{2})$, which leaves us with an expression of the form

$\begin{split}\frac{1}{N}\mathcal{O}_{N}^{(\textup{GinOE},c)}(\sqrt{N}w)&\rightarrow\frac{1}{\pi}\
\Theta^{(2)}_{N}(|w|^{2})\bigg{[}1-\frac{1}{N}-|w|^{2}\bigg{]}\ .\end{split}$
(4.8)

Taking the limit straightforwardly results in

$\mathcal{O}_{\textup{bulk}}^{\textup{(GinOE,c)}}(w)\equiv\lim_{N\rightarrow\infty}\frac{1}{N}\
\mathcal{O}_{N}^{\textup{(GinOE,c)}}\left(\sqrt{N}w\right)=\frac{1}{\pi}\left(1-|w|^{2}\right)\
\Theta\left[1-|w|^{2}\right]\ ,$ (4.9)

in full accordance with the result of Chalker & Mehlig in the bulk of the
GinUE [7]. ∎

We now proceed with calculating the large $N$ asymptotic behaviour of the mean
self-overlap at the edge, still keeping away from the depletion regime of the
GinOE, providing a proof of Corollary 2.7.

###### Proof.

For the edge of the spectrum we consider the scaling

$z=x+iy=\left(\sqrt{N}+\eta\right)e^{i\theta}=\left(\sqrt{N}+\eta\right)\cos\left(\theta\right)+i\left(\sqrt{N}+\eta\right)\sin\left(\theta\right)\
,$ (4.10)

which implies that the imaginary part $y$ is to be replaced by

$y=\left(\sqrt{N}+\eta\right)\sin\left(\theta\right)\overset{N\gg
1}{\approx}\sqrt{N}\ \sin\left(\theta\right)\ ,$ (4.11)

with the condition that $|\sin\left(\theta\right)|\sim\mathcal{O}(1)$ to avoid
the depletion regime. Together with Remark 4.1, this implies immediately that
the term containing the error function will again vanish in this scaling
limit. Also, as $N$ becomes large, the absolute value of $z$ becomes

$|z|^{2}=\left(\sqrt{N}+\eta\right)^{2}=N+2\eta\
\sqrt{N}+\eta^{2}\overset{N\gg 1}{\approx}N+2\eta\ \sqrt{N}\ .$ (4.12)

Recalling the finite $N$ result in Eq. (2.10), the limiting mean self-overlap
at the edge becomes

$\displaystyle\frac{1}{\sqrt{N}}\ \mathcal{O}_{N}^{(\textup{GinOE},c)}(z)$
$\displaystyle=\frac{1}{\pi}\ \bigg{[}\ \frac{\Gamma\left(N-1,N+2\eta\
\sqrt{N}\right)}{\Gamma\left(N-1\right)}\bigg{(}-\frac{1}{\sqrt{N}}-2\eta\bigg{)}$
(4.13) $\displaystyle+\frac{1}{\sqrt{N}}\ \frac{1}{(N-2)!}\ \left(N+2\eta\
\sqrt{N}\right)^{N-1}\ e^{-N-2\eta\ \sqrt{N}}\bigg{]}\ .$

Now, by invoking Eq. (4.3) in Remark 4.1 we see that the first term tends to
$-\eta\ \text{erfc}\left(\sqrt{2}\eta\right)$. In the second term, after using
the Stirling approximation $(N-2)!\sim\sqrt{2\pi}\ e^{-N}\ N^{N-3/2}$, we take
the limit, straightforwardly arriving at

$\lim_{N\to\infty}\frac{1}{\sqrt{N}}\frac{(N+2\eta\sqrt{N})^{N-1}}{(N-2)!}e^{-N-2\eta\sqrt{N}}=\lim_{N\to\infty}\frac{1}{\sqrt{2\pi}}\left(1+\frac{2\eta}{\sqrt{N}}\right)^{N-1}e^{-2\eta\sqrt{N}}=\frac{1}{\sqrt{2\pi}}e^{-2\eta^{2}}\
.$ (4.14)

Combining the two contributions thus gives the required result:

$\mathcal{O}_{\textup{edge}}^{\textup{(GinOE,c)}}(\eta)=\frac{1}{\pi}\left(\frac{1}{\sqrt{2\pi}}\
e^{-2\eta^{2}}-\eta\ \textup{erfc}\left(\sqrt{2}\ \eta\right)\right)\ .$
(4.15)

∎

Finally we consider the depletion region covered by Corollary 2.8 and proceed
to give its proof.

###### Proof.

We consider the limiting behaviour of the mean self-overlap of eigenvectors
associated with eigenvalues of the form $z=x+i\xi$, where
$\xi\sim\mathcal{O}(1)$ as $N\rightarrow\infty$. If $x\sim\mathcal{O}(1)$,
i.e. we stay close to the origin as $N\rightarrow\infty$, then we have that
$|z|^{2}\sim\mathcal{O}(1)$. Recalling Eq. (2.10) and inserting into it that
$z=x+i\xi$, we find that

$\begin{split}\mathcal{O}_{N}^{(\textup{GinOE},c)}(z)&=\frac{1}{\pi}\
\left(1+\sqrt{\frac{\pi}{2}}\ \exp\left[2\xi^{2}\right]\ \frac{1}{2|\xi|}\
\textup{erfc}\left(\sqrt{2}\ |\xi|\right)\right)\\\ &\times\bigg{[}\
\frac{\Gamma\left(N-1,x^{2}+\xi^{2}\right)}{\Gamma\left(N-1\right)}\bigg{[}N-1-x^{2}-\xi^{2}\bigg{]}+\frac{\left(x^{2}+\xi^{2}\right)^{N-1}}{(N-2)!}\
e^{-x^{2}-\xi^{2}}\bigg{]}\ .\end{split}$ (4.16)

Multiplying with overall factor $1/N$, noticing that the final term vanishes
as $N\to\infty$ and recalling Remark 4.1, one arrives at

$\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,origin}}(\xi)=\frac{1}{\pi}\left(1+\sqrt{\frac{\pi}{2}}\
\frac{1}{2|\xi|}\ e^{2\xi^{2}}\ \textup{erfc}\left(\sqrt{2}\
|\xi|\right)\right)\ ,$ (4.17)

thus reproducing Eq. (2.13).

If we scale the real part as $x=\sqrt{N}\delta$ instead, the limiting self-
overlap acquires an additional contribution from the $\Gamma$-function terms
in the second line in Eq. (4.16). This can be seen starting from the finite
$N$ expression

$\begin{split}\mathcal{O}_{N}^{(\textup{GinOE},c)}(z)&=\frac{1}{\pi}\
\left(1+\sqrt{\frac{\pi}{2}}\ \exp\left[2\xi^{2}\right]\ \frac{1}{2|\xi|}\
\textup{erfc}\left(\sqrt{2}\ |\xi|\right)\right)\\\ &\times\bigg{[}\
\frac{\Gamma\left(N-1,N\delta^{2}+\xi^{2}\right)}{\Gamma\left(N-1\right)}\bigg{[}N\left(1-\delta^{2}\right)-1-\xi^{2}\bigg{]}+\frac{N^{N-1}\left(\delta^{2}+\xi^{2}\right)^{N-1}}{(N-2)!}\
e^{-N\ \delta^{2}-\xi^{2}}\bigg{]}\ ,\end{split}$ (4.18)

then utilising Remark 4.1, which leads to an additional
$\Theta\left[1-\delta^{2}\right]$. Employing an overall scaling by $1/N$ to
ensure a non-trivial limit completes the proof and we get Eq. (2.14) for the
strip in the depleted region, i.e.

$\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,strip}}(\delta,\xi)=\mathcal{O}^{\textup{(GinOE,c)}}_{\textup{depletion,origin}}(\xi)\left(1-\delta^{2}\right)\Theta\left[1-\delta^{2}\right]\
.$ (4.19)

∎

#### Acknowledgements

We would like to thank Gernot Akemann, Mario Kieburg and 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 Consistency Check: $\mathcal{O}(z)$ in the GinUE

In the limit of $N\rightarrow\infty$ it is natural to expect the statistics of
complex eigenvalues and eigenvectors of the real and complex Ginibre ensembles
to match in the bulk and at the edge of the disk. For the GinUE, the finite
$N$ joint probability density function of an eigenvalue $z=x+iy$, with $y\neq
0$, and the associated self-overlap of the eigenvectors, $\mathcal{O}$ has
been derived in a rather complicated form in [55], which reads:

$\displaystyle\mathcal{P}^{\text{(GinUE,c)}}_{N}(\mathcal{O},z)=\frac{1}{\pi\Gamma(N)\Gamma(N-1)}\frac{e^{\frac{|z|^{2}}{\mathcal{O}}}}{\mathcal{O}^{3}}\left(\frac{\mathcal{O}-1}{\mathcal{O}}\right)^{N-2}\left[D_{1}^{(N)}+|z|^{2}\frac{D_{2}^{(N)}}{\mathcal{O}}+|z|^{4}\frac{d_{1}^{(N)}}{\mathcal{O}^{2}}\right]\
,$ (A.1)

where

$\displaystyle D_{1}^{(N)}=$ $\displaystyle\leavevmode\nobreak\
|z|^{4}(N-1)(N-2)d_{1}^{(N-1)}+\left[(N-1)N-2|z|^{2}(N+|z|^{2})\right]d_{1}^{(N)}$
(A.2) $\displaystyle\leavevmode\nobreak\
-|z|^{2}(N-2)(N-|z|^{2})d_{2}^{(N-1)}+|z|^{2}d_{2}^{(N)}$ $\displaystyle
D_{2}^{(N)}=$ $\displaystyle\leavevmode\nobreak\
2Nd_{1}^{(N)}-|z|^{2}(N-2)d_{2}^{(N-1)}$ (A.3) $\displaystyle d_{1}^{(N)}=$
$\displaystyle\leavevmode\nobreak\
\Gamma\left(N-1,|z|^{2}\right)\Gamma\left(N+1,|z|^{2}\right)-\Gamma\left(N,|z|^{2}\right)\Gamma\left(N,|z|^{2}\right)$
(A.4) $\displaystyle d_{2}^{(N)}=$ $\displaystyle\leavevmode\nobreak\
\Gamma\left(N-1,|z|^{2}\right)\Gamma\left(N+2,|z|^{2}\right)-\Gamma\left(N,|z|^{2}\right)\Gamma\left(N+1,|z|^{2}\right)\
.$ (A.5)

For completeness, we check below that its first moment produces the known mean
self-overlap at finite $N$.

By definition, the mean self-overlap is the first moment of the above JPDF and
therefore can be obtained by integration. Hence, we have

$\mathcal{O}^{\text{(GinUE,c)}}_{N}(z)=\int_{1}^{\infty}d\mathcal{O}\
\mathcal{O}\int_{\mathbb{C}}d^{2}z\ \delta(z-\tilde{z})\
\mathcal{P}^{\text{(GinUE,c)}}_{N}(\mathcal{O},z)\ ,$ (A.6)

which when written explicitly has the following form:

$\displaystyle\mathcal{O}^{\text{(GinUE,c)}}_{N}(z)=$
$\displaystyle\frac{1}{\pi\Gamma(N)\Gamma(N-1)}\int_{1}^{\infty}d\mathcal{O}\
\frac{e^{\frac{|z|^{2}}{\mathcal{O}}}}{\mathcal{O}^{2}}\left(\frac{\mathcal{O}-1}{\mathcal{O}}\right)^{N-2}\left[D_{1}^{(N)}+|z|^{2}\frac{D_{2}^{(N)}}{\mathcal{O}}+|z|^{4}\frac{d_{1}^{(N)}}{\mathcal{O}^{2}}\right]$
(A.7) $\displaystyle=$
$\displaystyle\frac{1}{\pi\Gamma(N)\Gamma(N-1)}\left[D_{1}^{(N)}I_{1}+|z|^{2}D_{2}^{(N)}I_{2}+|z|^{4}d_{1}^{(N)}I_{3}\right]\
,$ (A.8)

where the integrals $I_{1}$, $I_{2}$ and $I_{3}$ are defined below and can be
readily evaluated in terms of the incomplete $\Gamma$-function:

$\displaystyle I_{1}=$ $\displaystyle\int_{1}^{\infty}d\mathcal{O}\
e^{\frac{|z|^{2}}{\mathcal{O}}}\frac{\left(\mathcal{O}-1\right)^{N-2}}{\mathcal{O}^{N}}=\frac{e^{|z|^{2}}}{|z|^{2(N-1)}}\left(\Gamma(N-1)-\Gamma(N-1,|z|^{2})\right)$
(A.9) $\displaystyle I_{2}=$ $\displaystyle\int_{1}^{\infty}d\mathcal{O}\
e^{\frac{|z|^{2}}{\mathcal{O}}}\frac{\left(\mathcal{O}-1\right)^{N-2}}{\mathcal{O}^{N+1}}=\frac{\left(|z|^{2N}-e^{|z|^{2}}(N-|z|^{2}-1)\left[\Gamma(N)-\Gamma(N,|z|^{2})\right]\right)}{(N-1)|z|^{2N}}$
(A.10) $\displaystyle I_{3}=$ $\displaystyle\int_{1}^{\infty}d\mathcal{O}\
e^{\frac{|z|^{2}}{\mathcal{O}}}\frac{\left(\mathcal{O}-1\right)^{N-2}}{\mathcal{O}^{N+2}}=\frac{\Gamma(N-1)}{|z|^{2(N+1)}\Gamma(N+2)}\bigg{(}(N+2-N^{2})|z|^{2(N+1)}+(N+1)|z|^{2(N+2)}$
(A.11)
$\displaystyle+e^{|z|^{2}}\left(|z|^{4}-2(N-1)|z|^{2}+N(N-1)\right)\left[\Gamma(N+2)-(N+1)\Gamma(N+1,|z|^{2})\right]\bigg{)}\
.$

To combine all the above results we utilised Mathematica software to
manipulate the resulting expressions, finally finding that

$\mathcal{O}^{(\textup{GinUE},c)}_{N}(z)=\frac{1}{\pi}\left[\frac{\Gamma(N,|z|^{2})}{(N-1)!}(N-|z|^{2})+\frac{|z|^{2N}}{(N-1)!}e^{-|z|^{2}}\right]\
,$ (A.12)

which can be shown to be equivalent to the forms appearing in the literature,
see e.g. [44]. The corresponding edge and bulk scaling limits can be
straightforwardly recovered using essentially the same steps as employed for
the GinOE case.

## References

  * [1] Trefethen, L.N.; Embree, M. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, USA, 2005.
  * [2] Trefethen, L.N.; Trefethen, A.E.; Reddy, S.-C.; Driscoll, T.A. Hydrodynamic stability without eigenvalues. Science 1993 261, 578–-584.
  * [3] Banks, J; Kulkarni, A; Mukherjee, S.; Srivastava, N. Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture. Commun. Pure Applied Math. 2021 LXXIV, 2114–-2131
  * [4] Cipolloni, C.; Erdös, L.; Henheik, J.; Schröder, D. Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices. arXiv 2023, arXiv:2301.03549
  * [5] Jain, V.; Sah, A.; Sawhney, M. On the real Davies’ conjecture. Ann. Probab. 2021 49 3011–3031.
  * [6] Ginibre, J. Statistical ensembles of complex, quaternion and real matrices. J. Math. Phys. 1965 6 , 440–449.
  * [7] Chalker, J.-T.; Mehlig, B. Eigenvector Statistics in Non-Hermitian Random Matrix Ensembles. Phys. Rev. Lett. 1998 81, 3367–3370.
  * [8] Mehlig, B; Chalker, J.-T. Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles. J. Math. Phys. 2000 41, 3233–3256
  * [9] Girko, V.L. Circular law. Theor. Probab. and Applic. 1985 29, 694–706.
  * [10] Janik, R.A; Nörenberg, W.; Nowak, M.A.; Papp G.; Zahed, I. Correlations of eigenvectors for nonHermitian random matrix models. Phys. Rev. E 1999 60, 2699 – 2705
  * [11] Schomerus, H; Frahm, K; Patra, M; Beenakker, C.W.J. Quantum limit of the laser line width in chaotic cavities and statistics of residues of scattering matrix poles. Physica A 2000 A278, 469 –496
  * [12] Mehlig, B.; Santer, M. Universal eigenvector statistics in a quantum scattering ensemble. Phys. Rev. E 2001 63, 020105(R).
  * [13] Fyodorov Y.V.; Mehlig, B. Statistics of resonances and nonorthogonal eigenfunctions in a model for single-channel chaotic scattering. Phys. Rev. E  2002 66, 045202.
  * [14] Goetschy, A,; Skipetrov, S.E. Non-Hermitian Euclidean random matrix theory. Phys. Rev. E  2011 84, 011150.
  * [15] Fyodorov, Y.V; Savin, D.V. Statistics of resonance width shifts as a signature of eigenfunction non-orthogonality. Phys. Rev. Lett.  2012 108, 184101.
  * [16] Gros J.-B.; Kuhl U. ; Legrand O.; Mortessagne F.; Richalot E.; Savin, D.V. Experimental width shift distribution: a test of nonorthogonality for local and global perturbations. Phys. Rev. Lett.  2014 113, 224101.
  * [17] Burda, Z; Grela, J.; Nowak, M.A.; Tarnowski, W.; Warchol, P. Dysonian dynamics of the Ginibre ensemble. Phys. Rev. Lett.  2014 113, 104102
  * [18] Burda, Z; Grela, J.; Nowak, M.A.; Tarnowski, W.; Warchol, P. Unveiling the significance of eigenvectors in diffusing non-Hermitian matrices by identifying the underlying Burgers dynamics. Nucl. Phys. B  2015 897, 421–-447.
  * [19] Burda, Z.; Spisak, B.J.; Vivo, P. Eigenvector statistics of the product of Ginibre matrices. Phys. Rev. E  2017 95(2), 022134.
  * [20] Belinschi, S.; Nowak, M.A; Speicher, R; Tarnowski, W. Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem. J. Phys. A: Math. Theor. 2017 50, 105204.
  * [21] Nowak, M.A. ; Tarnowski, W. Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach. JHEP 2018 2018, 152.
  * [22] Akemann, G.; Förster, Y.-P.; Kieburg, M. Universal eigenvector correlations in quaternionic Ginibre ensembles. J. Phys. A: Math.Theor. 2020 53, 145201, 26 pp.
  * [23] Fyodorov Y.V.; Osman, M. Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities. J. Phys. A: Math. and Theor. 2022 55 , 224013.
  * [24] Gangulia, S.; Huhc, D.; Sompolinsky, H. Memory traces in dynamical systems. Proc. Natl. Acad. Sci. USA 2008 105 18970–-18975.
  * [25] Grela, J. What drives transient behavior in complex systems? Phys. Rev. E  2017 96, 022316.
  * [26] Erdös, L; Krüger, T; Renfrew, D. Randomly coupled differential equations with correlations. arXiv 2019, arXiv:1908.05178
  * [27] Marti, D.; Brunel N; and Ostojic, S. Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks. Phys. Rev. E 2018 97, 062314.
  * [28] Tarnowski, W.; Neri, I.; Vivo, P. Universal transient behavior in large dynamical systems on networks. Phys. Rev. Res. 2020 2, 023333.
  * [29] Gudowska-Nowak, E.; Ochab, J; Chialvo, D.; Nowak M.A.; Tarnowski, W. From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior. Neural Comput. 2020 32, 395–423.
  * [30] Fyodorov, Y.V.; Gudowska-Nowak, E., Nowak, M.A.; Tarnowski, W. Fluctuation-dissipation relation for non-Hermitian Langevin dynamics. manuscript in preparation.
  * [31] Cipolloni, G.; Kudler-Flam, J. Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble. Phys. Rev. Lett.  2023 130, 010401.
  * [32] Cipolloni G.; Kudler-Flam, J. Non-Hermitian Hamiltonians Violate the Eigenstate Thermalization Hypothesis. arXiv 2023 arXiv:2303.03448
  * [33] Ghosh, S.; Kulkarni M.; Roy, S. Eigenvector Correlations Across the Localisation Transition in non-Hermitian Power-Law Banded Random Matrices. arXiv 2023, arXiv:2304.09892
  * [34] Fyodorov, Y.V.; Sommers, H.-J. Statistics of resonance poles, phaseshifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. 1997 38 1918–-1981.
  * [35] Fyodorov, Y.V.; Sommers, H.-J. Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A: Math. Theor. 2003 36, 3303–3347.
  * [36] Rotter, I. A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 2009 42, 153001.
  * [37] Fyodorov, Y.V; Savin, D.V. Resonance scattering in chaotic systems. chapter 34. In: G. Akemann, J. Baik, P. Di Francesco (eds.), The Oxford Handbook of Random Matrix Theory p. 703. Oxford University Press, Oxford, 2011.
  * [38] Savin, D.V; Sokolov, V.V. Quantum versus classical decay laws in open chaotic systems. Phys. Rev. E 1997 56, R4911–-R4913.
  * [39] Patra, M.; Schomerus H.; Beenakker, C.W.J. Quantum-limited linewidth of a chaotic laser cavity. Phys. Rev. A 2000 61, 023810.
  * [40] Davy, M.; Genack, A.Z. Selectively exciting quasi-normal modes in open disordered systems. Nature Comm. 2018 9, 4714.
  * [41] Davy, M; Genack, A.Z. Probing nonorthogonality of eigenfunctions and its impact on transport through open systems. Phys. Rev. Research 2019 1, 033026.
  * [42] Dubach, G. On eigenvector statistics in the spherical and truncated unitary ensembles. Electron. J. Probab. 2021 26, 1–29.
  * [43] Dubach, G. Explicit formulas concerning eigenvectors of weakly non-unitary matrices. Electron. Commun. Probab. 2023 28 1–11.
  * [44] Walters, M.; Starr, S. A note on mixed matrix moments for the complex Ginibre ensemble. J. Math. Phys. 2015 56, 013301.
  * [45] Akemann, G.; Tribe, R.; Tsareas, A.; Zaboronski, O. On the determinantal structure of conditional overlaps for the complex Ginibre ensemble. Random Matrix Th. Appl. 2020, 9, 2050015.
  * [46] Grela, J.; Warchol, P. Full Dysonian dynamics of the complex Ginibre ensemble. J. Phys. A: Math. Theor. 2018 51, 425203.
  * [47] Bourgade, P.; Dubach, G. The distribution of overlaps between eigenvectors of Ginibre matrices, Probab. Theory Relat. Fields 2020 177 397–-464.
  * [48] Yabuoku, S. Eigenvalue processes of Elliptic Ginibre Ensemble and their overlaps. Int. J. of Math. Ind.  2020 12, 2050003.
  * [49] Cipolloni, G; Erdös L.;Schröder, D. On the condition number of the shifted real Ginibre ensemble. SIAM J. Matr. Anal. Applic. 2022 43, 1469–1487.
  * [50] Erdös, L; Ji, H.C. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. arXiv 2023, arXiv:2301.04981
  * [51] Benaych-Georges, F.; Zeitouni, O. Eigenvectors of non normal random matrices. Electron. Commun. Probab. 2018 23, 1-12.
  * [52] Crawford, N.; Rosenthal, R. Eigenvector correlations in the complex Ginibre ensemble. Ann. Appl. Probab. 2022 32, 2706-2754.
  * [53] Esaki, L.; Katori, M,; Yabuoku, S. Eigenvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion. arXiv 2023, arXiv:2306.00300
  * [54] Movassagh, R. Eigenvalue Attraction. J. Stat. Phys. 2016 162, 615–-643.
  * [55] Fyodorov, Y.V. On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry. Commun. Math. Phys. 2018 363, 579–603.
  * [56] Byun, S.S; Forrester, P.J. Progess on the study of the Ginibre ensembles I: GinUE, arXiv 2022, arXiv:2211.16223
  * [57] Byun, S.S; Forrester, P.J. Progess on the study of the Ginibre ensembles II: GinOE and GinSE, arXiv 2023, arXiv:2301.05022
  * [58] Edelman, A.; Kostlan E.; Shub, M. How many eigenvalues of a random matrix are real?, J. Amer. Math. Soc. 1994 7, 247–267.
  * [59] Edelman, A. The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law, J. Multivar. Anal. 1997 60, 203–-232.
  * [60] Khoruzhenko, B.A.; Sommers, H.-J. Non-Hermitian ensembles. Chapter 18 in The Oxford Handbook of Random Matrix Theory. In: G. Akemann, J. Baik, P. Di Francesco, (eds.) Oxford University Press, 2011.
  * [61] Forrester, P.J.; Nagao, T. Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 2007 99, 050603.
  * [62] Fyodorov, Y.V.; Tarnowski, W. Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble, Ann. Henri Poincaré 2021 22, 309–330.
  * [63] Berezin, F.A. The Method of Second Quantization, Academic Press, New York, 1966.
  * [64] Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integers, Series and Products 7th ed. Academic Press, New York (2000).
  * [65] Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark C.W. (eds.), NIST Handbook of Mathematical Functions. Cambridge University Press 2010, Cambridge.
  * [66] Fyodorov, Y.V.; Khoruzhenko, B.A.; Sommers, H.-J. Almost-Hermitian random matrices: eigenvalue density in the complex plane. Phys. Lett. A. 1997 226, 46-52.
  * [67] Fyodorov, Y.V.; Khoruzhenko, B.A.; Sommers, H.-J. Almost Hermitian random matrices: crossover from Wigner-Dyson to Ginibre eigenvalue statistics. Phys. Rev. Lett. 1997 79, 557–560.
  * [68] Fyodorov, Y.V.; Khoruzhenko, B.A.; Sommers, H.-J. Universality in the random matrix spectra in the regime of weak non-Hermiticity. Ann. Inst. H. Poincaré Phys. Théor. 1998m 68, 449––489.