e,OCP},T}(z,z^{\prime})\,d^{2}z^{\prime}=-\rho_{(1),\infty}^{{\rm e,OCP}}(z).$

This is the edge counterpart of the bulk perfect screening sum rule (4.20).

######  Remark 4.7.

It turns out that in the GinUE case, the determinantal structure together with
(4.23) can be used to show that $\rho(z):=\rho_{(1),\infty}^{{\rm
e,GinUE}}(z)$ (this is (2.28)) satisfies a non-linear equation of infinite
order,

(4.24) $\rho(z)=\sum_{n=0}^{\infty}{|\partial_{z}^{(n)}\rho(z)|^{2}\over n!};$

see [38, § 3.6], where (4.23) is referred to as a mass-one equation. The non-
linear equation is equivalent to the special function function identity

(4.25) ${1\over 4}\Big{(}1+{\rm erf}(\sqrt{2}x)\Big{)}\Big{(}1-{\rm
erf}(\sqrt{2}x)\Big{)}={e^{-4x^{2}}\over\pi}\sum_{n=1}^{\infty}{(H_{n-1}(\sqrt{2}x))^{2}\over
2^{n}n!}\quad(x\in\mathbb{R});$

see [38, § 4.5] for the proof of (4.25). Together with the loop equation, the
identity (4.24) is used in [38] to study the universality of normal matrix
models; see §5.4.

To deduce (3.19) from (3.18) requires that

(4.26) $\lim_{R\to\infty}R^{2}S_{\infty}^{\rm
GinUE}(\mathbf{k}/R)={|\mathbf{k}|^{2}\over 4\pi}.$

In the general $\beta>0$ case of the OCP we denote the bulk scaled structure
factor by $S_{\infty}^{\rm OCP}(\mathbf{k})$. A perfect screening argument
extending the viewpoint which implies (4.20) (for this see e.g. [149,
§15.4.1], [147, §3.2]) predicts

(4.27) $\lim_{R\to\infty}R^{2}S_{\infty}^{\rm
OCP}(\mathbf{k}/R)={|\mathbf{k}|^{2}\over 2\beta\pi}.$

This result, in the slightly different guise of a mesoscopic scaling limit, is
implied by the recent work [247, Th. 1 mesoscopic case, formula for the
variance]. In the cases of the OCP applied to the anomalous quantum Hall
effect ($\beta=2M$ with $M$ odd), the sum rule (4.27) combined with the
Feynman-Bijl formula quantifies the collective mode excitation energy from the
ground state [184]. From the definition of the structure factor, this is
equivalent to the moment formula

(4.28) $\int_{\mathbf{R}^{2}}|\mathbf{r}|^{2}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{2\over\pi\beta},$

known as the Stillinger-Lovett sum rule [317]; see [271] for a derivation
which makes use of (4.22) for $p=1$ and $k=1,2$. Higher order moment formulas
can also be derived,

(4.29) $\int_{\mathbf{R}^{2}}|\mathbf{r}|^{4}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{16\over\pi\beta^{2}}\Big{(}1-{\beta\over
4}\Big{)},\quad\int_{\mathbf{R}^{2}}|\mathbf{r}|^{6}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{0})\,d\mathbf{r}=-{18\over\pi\beta^{3}}\Big{(}\beta-6\Big{)}\Big{(}\beta-{8\over
3}\Big{)};$

in distinction to (4.27) the precise statement of these results depends on the
underlying bulk particle density, which in keeping with GinUE has been assumed
to be ${1\over\pi}$. A linear response argument can be used to derive the
first of these; see e.g. [149, §14.1.1]. It involves the thermal pressure for
the OCP, which by a simple scaling argument takes the value $\rho^{\rm
b}(1-\beta/4)$ (here $\rho^{\rm b}$ denotes the bulk density; for a discussion
of the relation between the thermal and mechanic pressure in the OCP, see
[103]) which explains the appearance of this factor. Mayer diagrammatic
expansion methods were used to first derive the sixth moment condition in
(4.29) [217]. Later a response argument involving variations to the spatial
geometry was used to give an alternative derivation [86]. In keeping with the
relationship between (4.27) and (4.28), the moment formulas (4.29) can be
related to the small $k$ expansion of $S_{\infty}^{\rm GinUE}(\mathbf{k})$,
which must therefore read

(4.30) ${2\pi\beta\over|\mathbf{k}|^{2}}S_{\infty}^{\rm
OCP}(\mathbf{k})=1+\Big{(}{\beta\over 4}-1\Big{)}{|\mathbf{k}|^{2}\over
2\beta}+\Big{(}{\beta\over 4}-{3\over 2}\Big{)}\Big{(}{\beta\over 4}-{2\over
3}\Big{)}\Big{(}{|\mathbf{k}|^{2}\over 2\beta}\Big{)}^{2}+{\rm
O}(|\mathbf{k}|^{6}).$

Evidence is given in [217] that the polynomial structure in $\beta/4$ of the
coefficients in this expansion breaks down at ${\rm O}(|\mathbf{k}|^{6})$.
This is in contrast to the power series expansion of the bulk scaled (density
$1/\pi$ for definiteness) structure factor for the log-gas on a one-
dimensional domain, $S_{\infty}^{(\beta)}(k)$ say. Thus expanding $\pi\beta
S_{\infty}^{(\beta)}(k)/k$ as a power series in $(k/\beta)$, the $j$-th
coefficient is a monic polynomial of degree $j$ in $\beta/2$ [162, 155].

The use of linear response to quantify the change of charge density upon the
introduction of a charge $q$ into the OCP, computing from this the change of
charge by integrating, and requiring by screening that this must equal $-q$
can be used [210] to deduce the Carnie-Chan sum rule [88, 89]

(4.31)
$-\beta\int_{\mathbb{R}^{2}}d\mathbf{r}\bigg{(}\int_{\mathbb{R}^{2}}d\mathbf{r}^{\prime}\,\log|\mathbf{r}^{\prime}|\Big{(}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r},\mathbf{r}^{\prime})+{1\over\pi}\delta(\mathbf{r}-\mathbf{r}^{\prime})\Big{)}\bigg{)}=1.$

Here the integral cannot be interchanged, as according to (4.20) integrating
over $\mathbf{r}$ first would give zero. The validity of (4.31) can be checked
directly for $\beta=2$ in the bulk (i.e. for GinUE in the bulk).

######  Proposition 4.8.

The Carnie-Chan sum rule (4.31) is valid for GinUE in the bulk.

###### Proof.

Denote the integral over $\mathbf{r}^{\prime}$ in (4.31) by $g(\mathbf{r})$.
Assuming only rotation and translation invariance of $\rho_{(2),\infty}^{{\rm
OCP},T}$ implies

$g(\mathbf{r})=\log|\mathbf{r}|\Big{(}\int_{|\mathbf{r}^{\prime}|<|\mathbf{r}|}\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r}^{\prime},\mathbf{0})\,d\mathbf{r}^{\prime}+{1\over\pi}\Big{)}+\int_{|\mathbf{r}^{\prime}|>|\mathbf{r}|}\log|\mathbf{r}^{\prime}|\rho_{(2),\infty}^{{\rm
OCP},T}(\mathbf{r}^{\prime},\mathbf{0})\,d\mathbf{r}^{\prime}.$

Specialising now to $\beta=2$ by substituting (2.24), the use of polar
coordinates and integration by parts allows the integrals to be simplified,
with the result

$g(\mathbf{r})=-{1\over\pi}\int_{|\mathbf{r}|}^{\infty}{e^{-(r^{\prime})^{2}}\over
r^{\prime}}\,dr^{\prime}$

The integration over $\mathbf{r}\in\mathbb{R}^{2}$ of this can be computed by
further use of polar coordinates and integration by parts to confirm (4.31). ∎

Formal use of the convolution theorem in (4.31) shows that it reduces to
(4.27) [285]. In the case of the edge geometry, this approach can be used
[213] to derive from (4.31) in the edge case the edge dipole moment sum rule

(4.32)
$-2\pi\beta\int_{-\infty}^{\infty}dy\,\bigg{(}\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm OCP},T}((0,y),(x^{\prime},y^{\prime}))\bigg{)}=1,$

first derived in [64]. In keeping with the analogous property of (4.31), it is
not possible to interchange the integrations over $y$ and $y^{\prime}$ in this
expression (if it was the LHS would vanish due to the sign change of
$y^{\prime}-y$). In the case $\beta=2$ this is readily verified using the
exact result (2.25).

######  Proposition 4.9.

The edge dipole moment sum rule (4.32) is valid for edge scaled GinUE.

###### Proof.

Substituting (2.25), integration by parts gives

$\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm
GinUE},T}((0,y),(x^{\prime},y^{\prime}))=-{1\over\sqrt{8\pi^{3}}}e^{-2y^{2}}$

(this formula can be found in [208, Eq. (2.41)]). Now integrating over $y$
verifies (4.32) for $\beta=2$. ∎

The exact result (2.26) exhibits a slow decay of
$\rho_{(2),\infty}^{T}((x_{1},y_{1}),(x_{2},y_{2}))$ parallel to the edge.
This was first predicted by Jancovici [209, 211], who on the basis of linear
response argument relating to the screening an oscillatory external charge
density applied at the edge obtained for general $\beta>0$,

(4.33)
$\rho_{(2),\infty}^{T}((x_{1},y_{1}),(x_{2},y_{2}))\mathop{\sim}\limits_{|x_{1}-x_{2}|\to\infty}{f(y_{1},y_{2})\over(x_{1}-x_{2})^{2}},\qquad\int_{-\infty}^{\infty}dy_{1}\int_{-\infty}^{\infty}dy_{2}\,f(y_{1},y_{2})=-{1\over
2\beta\pi^{2}}.$

A derivation of this using the Carnie-Chan sum rule for the OCP confined to a
strip geometry is given in [210]. In keeping with this, in [215] the amplitude
$f(y,y^{\prime})$ in (4.33) is related to the dipole moment of the screening
cloud at the edge by deriving that

(4.34)
$\int_{-\infty}^{\infty}dy^{\prime}\,(y^{\prime}-y)\int_{-\infty}^{\infty}dx^{\prime}\,\rho_{(2),\infty}^{{\rm
e},{\rm
OCP},T}((0,y),(x^{\prime},y^{\prime}))=\pi\int_{-\infty}^{\infty}dy^{\prime}\,f(y,y^{\prime}).$

Then the result for the integral over $y$ on the RHS follows from the dipole
moment sum rule (4.32).

We consider now a global scaling, so that the droplet support is the unit
disk. For large $N$, define the charge-charge surface correlation by

(4.35) $\langle\sigma(\theta)\sigma(\theta^{\prime})\rangle_{N}^{T}:=\int
d\mathbf{n}\int d\mathbf{n}^{\prime}\,\rho_{(2),N}^{{\rm g},{\rm
OCP},T}(\mathbf{r},\mathbf{r}^{\prime}).$

Here $\rho_{(2),N}^{{\rm g},{\rm OCP},T}$ refers to $\rho_{(2),N}^{{\rm
OCP},T}$ computed using global scaling (as specified below (4.18)), then
considering the large $N$ form with edge coordinates (these can be taken as
$(r,\theta)$ with $r\approx 1$. The integration is over the normal direction
$\mathbf{n}$ (here the radial direction); the more general notation has been
used as this quantity can be defined in other geometries e.g. for the elliptic
GinUE [161]. The reasoning leading to (4.33) has been extended [211] to lead
to the prediction

(4.36)
$\langle\sigma(\theta)\sigma(\theta^{\prime})\rangle_{\infty}^{T}=-{1\over
8\pi^{2}\beta\sin^{2}((\theta-\theta^{\prime})/2)}.$

This has been checked at $\beta=2$ using the exact result for GinUE in [104].

######  Remark 4.10.

For the bulk scaled OCP with $\beta$ an even integer, there is a constraint on
the small $r=|\mathbf{r}|$ form of $\rho_{(2),\infty}^{{\rm
b}}(\mathbf{r},\mathbf{0})$. Thus [301]

$\rho_{(2),\infty}^{{\rm b}}(\mathbf{r},\mathbf{0})=r^{\beta}e^{-\beta
r^{2}/4}f(r^{4}),$

for $f(z)$ analytic. Note from (2.24) that for GinUE,
$f(z)={2\over\pi^{2}}{\sinh(\sqrt{z}/2)\over\sqrt{z}}$. Recently consequences
have been found in [300].

##  5\. Normal matrix models

###  5.1. Eigenvalue PDF

A generalisation of the joint element PDF (2.34) is the functional form
proportional to

(5.1) $\exp\Big{(}-{N\over t_{0}}{\rm Tr}(JJ^{\dagger}-2{\rm
Re}\,\sum_{p=2}^{M}t_{p}{\rm Tr}\,J^{p})\Big{)},\quad t_{0}>0;$

note the scaling so that there is a factor of $N$ in the exponent. Use of the
Schur decomposition (2.2) gives a separation of the eigenvalues and the
strictly upper triangular elements in the exponent analogous to (2.4). This
allows the latter to be integrated over, showing that the corresponding
eigenvalue PDF is proportional to

(5.2) $\exp\bigg{(}-{N\over t_{0}}\sum_{j=1}^{N}\Big{(}|z_{j}|^{2}-2{\rm
Re}\sum_{p=2}^{M}t_{p}z_{j}^{p}\Big{)}\bigg{)}.$

However for $t_{M}\neq 0$ this is not normalisable for $M>2$.

A simple remedy is to impose a cutoff by restricting the eigenvalues to a disk
centred about the origin of radius $R$, and restricting to small $t_{0}$ and
values of $t_{2},\dots,t_{M}$ so that the support $\Omega$ (assumed simply
connected) of the density is contained inside this disk [138]. The mean field
argument leading to (2.30) tells us that in relation (5.2), the normalised
density has the constant value $1/\pi t_{0}$ in $\Omega$, where the latter is
such that

(5.3) $W(z)={1\over\pi}\int_{\Omega}\log|z-w|\,d^{2}w,\qquad W(z)={1\over
2}\Big{(}|z|^{2}-2{\rm Re}\sum_{p=2}^{M}t_{p}z^{p}\Big{)},$

for $z$ contained inside of $\Omega$. From this equation the coupling
constants $\\{t_{p}\\}_{p=2}^{M}$ can be related to moments associated with
$\Omega$ [4, 328].

######  Proposition 5.1.

In the above setting, for $p\geq 2$ we have

(5.4) $t_{p}={1\over 2\pi
ip}\int_{\partial\Omega}\bar{z}z^{-p}\,dz=-{1\over\pi
p}\int_{\mathbb{C}\backslash\Omega}z^{-p}\,d^{2}z.$

###### Proof.

By applying $\partial_{z}$ to both sides of (5.3), then introducing $W(w)$ in
the integrand via the identity $2\partial_{w}\partial_{\bar{w}}W(w)=1$ shows

$\partial_{z}W(z)={1\over\pi}\int_{\Omega}{\partial_{w}\partial_{\bar{w}}W(w)\over
z-w}\,d^{2}w.$

Now using the Cauchy-Pompeiu formula (3.22) with $C_{1}$ replaced by $\Omega$,
we see from this that

$\int_{\partial\Omega}{\partial_{w}W(w)\over w-z}\,dw=0.$

Simple manipulation then shows

${1\over 2\pi i}\int_{\partial\Omega}{\bar{w}\over
w-z}\,dw=\sum_{p=2}^{M}pt_{p}z^{p-1}.$

The first equality of (5.4) now follows by power series expanding the LHS with
respect to $z$. The second equality can be deduced from the first by applying
the version of (3.22) valid for $z$ outside $C_{1}$ (chosen as
$\mathbb{C}\backslash\Omega$), for which the LHS is to be replaced by $0$. ∎

The simplest case of (5.2) beyond that corresponding to (2.34) is to take
$M=3$. After scaling, the choice of $t_{3}$ can be fixed. Choosing
$t_{3}={1\over 3}$, results from [234] give that the support of $\Omega$ is
contained in a disk for all $0<t_{0}\leq 1/8$, with the critical value
$t_{0}=1/8$ involving three cusp singularities. Then, the boundary
$\partial\Omega$ is a 3-cusped hypercycloid. For general $M\geq 2$ and
$\Omega$ contained in a disk and simply connected, it is shown in [138] that
$\partial\Omega$ can be parametrised by a Laurent polynomial of the form
$\alpha_{1}w+\alpha_{0}+\cdots+\alpha_{M-1}w^{-M+1}$ with $|w|=1$. Dropping
the assumption that $\Omega$ be simply connected requires the theory of
quadrature domains; see [249] and references therein.

More general than the joint eigenvalue PDF (5.1) is the functional form

(5.5) $\exp\Big{(}-\sum_{j,k=1}^{\infty}c_{jk}{\rm
Tr}\,(J^{j}(J^{\dagger})^{k})\Big{)}=:\exp\Big{(}-{\rm
Tr}\,W(J,J^{\dagger})\Big{)}.$

However unlike the former, substituting the Schur decomposition (2.2) does not
in general lead to a separation of the eigenvalues from the strictly upper
triangular variables. To overcome this, attention can be restricted to the
subset of $N\times N$ complex matrices having the further structure
$[J,J^{\dagger}]=0$, which specifies the matrices as being normal. For normal
matrices, the eigenvectors can be chosen to form an orthonormal set, and so
$J=UDU^{\dagger}$, for $U$ unitary and $D$ the diagonal matrix of eigenvalues.
Moreover, the change of variables from $J$ to $\\{U,D\\}$ gives a
decomposition of measure into separate eigenvalue and eigenvector factors,
with the Jacobian given by (2.5) [101]. Hence the eigenvalue PDF corresponding
to (5.5) in this setting is proportional to

(5.6) $\exp\Big{(}-\sum_{j=1}^{N}W(z_{j},\bar{z}_{j})\Big{)}\prod_{1\leq
j<k\leq N}|z_{k}-z_{j}|^{2}.$

Note that for the joint element PDF (5.1), the same eigenvalue PDF (5.2)
results for $J$ specified on the full space of $N\times N$ complex matrices,
as it does on the restriction to normal matrices as implied by (5.6).

###  5.2. Equilibrium measure

Analogous to (5.1), to obtain a compact eigenvalue support for large $N$, one
considers the case that the weight $e^{-W}$ is exponentially varying in a
sense that $W$ is of order $N$, say $W(z,\bar{z})=NQ(z)$ for a fixed potential
$Q:\mathbb{C}\to\mathbb{R}$. Thus the eigenvalue PDF (5.6) is written as

(5.7) $e^{-H(z_{1},\dots,z_{N})},\quad H(z_{1},\dots,z_{N}):=\sum_{1\leq
j<l\leq N}\log\frac{1}{|z_{j}-z_{l}|^{2}}+N\sum_{j=1}^{N}Q(z_{j}).$

As in Remark 2.5, the macroscopic behaviour of the system (5.7) can be
described using the two-dimensional Coulomb gas interpretation with the help
of logarithmic potential theory. Namely, it is well known in the literature
[216, 199, 92, 93, 29] that for a general $Q$ under suitable potential
theoretic assumptions, the empirical measure $\frac{1}{N}\sum\delta_{z_{j}}$
weakly converges to a unique probability measure $\mu_{Q}$ that minimises

(5.8)
$I_{Q}[\mu]:=\int_{\mathbb{C}^{2}}\log\frac{1}{|z-w|}\,d\mu(z)\,d\mu(w)+\int_{\mathbb{C}}Q\,d\mu;$

cf. (1.10). One may notice that (5.8) can be interpreted as a continuum limit
of the Hamiltonian $H_{N}$ in (5.7) after normalisation, generalising (2.29).
The probability measure $\mu_{Q}$ is called the equilibrium measure and its
support $S_{Q}:=\textup{supp}(\mu_{Q})$ is called the droplet, as previously
remarked. Furthermore, if $Q$ is $C^{2}$-smooth in a neighbourhood of $S_{Q}$,
by Frostman’s theorem [298], $\mu_{Q}$ is absolutely continuous with respect
to the Lebesgue measure and takes the form

(5.9) $d\mu_{Q}(z)=\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\chi_{z\in
S_{Q}}\,d^{2}z.$

In particular, for a rotationally symmetric potential $q(r)=Q(|z|=r)$, the
droplet is of the form $S_{Q}=\\{r_{1}\leq|z|\leq r_{2}\\}$, where
$(r_{1},r_{2})$ are the unique pair of constants satisfying

(5.10) $r_{1}q^{\prime}(r_{1})=0,\quad r_{2}q^{\prime}(r_{2})=2,$

see [298, § IV.6]. Here, we have assumed that $Q(z)$ is strictly subharmonic
in $\mathbb{C}$, which is equivalent to the requirement that $r\mapsto
rq^{\prime}(r)$ is increasing in $(0,\infty).$ Let us mention that all the
explicit macroscopic densities with rotation invariance in the above
subsections of §2 can be obtained as special cases of the formulas (5.9) and
(5.10). For instance, the RHS of (2.45) can be realised as the RHS of (5.9)
with $Q(z)=|z|^{2}-2\alpha\log|z|$. Beyond the case when $Q$ is radially
symmetric, the determination of the droplet, also known as the two-dimensional
equilibrium problem (e.g. the ellipse in § 2.3), is far from being obvious
even for some explicit potentials with simple form; see [45, 47, 232, 14, 113]
and references therein for recent works in this direction.

###  5.3. Partition functions

Continuing the discussion in §4.1, we consider the global scaled, non charge
neutral, partition function

(5.11)
$Z_{N}(\beta;Q)=\frac{1}{N!}\int_{\mathbb{C}}d^{2}z_{1}\cdots\int_{\mathbb{C}}d^{2}z_{N}\,e^{-\frac{\beta}{2}H(z_{1},\dots,z_{N})},$

where $H(z_{1},\dots,z_{N})$ is the Hamiltonian given in (5.7). An explicit
formula for the large $N$ expansion of $Z_{N}(\beta;Q)$ was predicted in
[329]. Fairly recently, it was shown by Leblé and Serfaty [246] that for
general $\beta>0$ and $Q$, $Z_{N}(\beta;Q)$ admits the large $N$ asymptotic
expansion of the form

(5.12) $\log
Z_{N}(\beta;Q)=-\frac{\beta}{2}N^{2}I_{Q}[\mu_{Q}]+\Big{(}\frac{\beta}{4}-1\Big{)}N\log
N-\bigg{(}C(\beta)+\Big{(}1-\frac{\beta}{4}\Big{)}E_{Q}[\mu_{Q}]\bigg{)}N+o(N);$

see also an earlier work [303] on (5.12) up to the $O(N\log N)$ term. The term
$I_{Q}[\mu_{Q}]$ appearing in the leading order asymptotic is the energy (5.8)
evaluated at the equilibrium measure $\mu_{Q}$. In the case of the OCP, up to
a sign it is the quantity appearing in the exponent of $A_{N,\beta}$ in (4.2)
at order $N^{2}$. To see this, we note that for a radially symmetric potential
$q(r)=Q(|z|=r)$ generally, it is evaluated as

(5.13) $I_{Q}[\mu_{Q}]=q(r_{1})-\log
r_{1}-\frac{1}{4}\int_{r_{0}}^{r_{1}}rq^{\prime}(r)^{2}\,dr,$

where $r_{1}$ and $r_{2}$ are the constants specified in (5.10). For the OCP
$Q(z)=|z|^{2}$, which gives $I_{Q}[\mu_{Q}]=3/4$, this indeed being the
coefficient of $N^{2}$ seen in $A_{N,\beta}$. The appearance of the term
$(\beta/4-1)N\log N$ in (5.12), whereas there is a term ${1\over 4}\beta
N^{2}\log N$ in $A_{N,\beta}$ of (4.2) is due to the simple scaling
$z_{j}\mapsto z_{j}/\sqrt{N}$ required to go from the OCP with global scaled
coordinates (as assumed in (5.12)) to the OCP itself (as assumed in (4.2)).
The terms appearing in the $O(N)$ term of (5.12) are the entropy

(5.14) $E_{Q}[\mu_{Q}]:=\int_{\mathbb{C}}\mu_{Q}(z)\,\log\mu_{Q}(z)\,d^{2}z$

associated with $\mu_{Q}$ and a constant $C(\beta)$ independent of the
potential $Q$. Their sum is the free energy per particle, $\beta f(\beta;Q)$.
The expansion (5.12) with quantitative error bounds is also available in the
literature [49, 308].

For the random normal matrix model when $\beta=2$, the determinantal structure
(2.9) allows an explicit expression

(5.15) $Z_{N}(2;Q)=\prod_{j=0}^{N-1}h_{j},$

where $h_{j}$ is the orthogonal norm in (5.18); cf. Proposition 2.1. In
particular, if $Q$ is rotationally symmetric, since $p_{j}(z)=z^{j}$, we have

(5.16) $h_{j}=2\pi\int_{0}^{\infty}r^{2j+1}e^{-Nq(r)}\,dr;$

cf. (2.87). Based on this knowledge together with a Laplace approximation of
the integrals, the precise asymptotic expansion of the free energy up to the
$O(1)$ term was derived in a recent work [78]. This in particular shows that

(5.17) $\log Z_{N}(2;Q)=-N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi^{2})}{2}-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N-\frac{\chi}{12}\log
N+O(1),$

where $\chi$ is the Euler index; cf. (4.6). Let us recall here that $\chi=0$
for the annulus ($r_{0}>0$) and $\chi=1$ for the disk ($r_{0}=0$) geometry.
See [78, §4] and [145, §4] for some concrete examples of (5.17) associated
with the matrix models discussed in §2.

###  5.4. Correlation functions and universality

Turning to the correlation functions $\rho_{(k),N}$ of (5.7), the
determinantal structure (2.9) remains valid with the correlation kernel

(5.18)
$K_{N}(z,w)=e^{-\frac{N}{2}(Q(z)+Q(w))}\sum_{j=0}^{N-1}\frac{p_{j}(z)\overline{p_{j}(w)}}{h_{j}},$

where $p_{j}$ is the monic orthogonal polynomial of degree $j$ with respect to
the weighted Lebesgue measure $e^{-NQ(z)}\,d^{2}z$ and $h_{j}$ is its squared
orthogonal norm; cf. (3.6).

Let us first discuss the asymptotic behaviours of $\rho_{(k),N}$ in the micro-
scale. Given a base point $p\in S_{Q}$ for which we zoom the point process, we
denote by

(5.19) $\delta:=\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\Big{|}_{z=p}$

the mean eigenvalue density at $p$; cf.(5.9). From universality principles
(see e.g. [231]), one expects that for a general $Q$ and $p$ such that
$\delta\in(0,\infty)$ (i.e. the eigenvalue density does not vanish or diverge
at $p$), the universal scaling limit in Proposition 2.4 arises. For the bulk
case when $p\in\textup{Int}(S_{Q})$, such a universality was established by
Ameur, Hedenmalm and Makarov in [35] where they showed that for a fairly
general potential $Q$ under some mild assumptions,

(5.20)
$\displaystyle\frac{1}{(N\delta)^{k}}\rho_{(k),N}\Big{(}p+\frac{z_{1}}{\sqrt{N\delta}},\dots,p+\frac{z_{k}}{\sqrt{N\delta}}\Big{)}\to\det\Big{[}K_{\infty}^{\rm
b}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$

uniformly on compact subsets of $\mathbb{C}$ as $N\to\infty$. See also [55,
57]. In a sequential work [36] the theory of loop equations (or Ward’s
identities) was also used to show the bulk scaling limit (5.20). It says that
for a given test function $\psi$,

(5.21) $\mathbb{E}_{N}W_{N}^{+}[\psi]=0,\quad
W_{N}^{+}[\psi]:=\frac{1}{2}\sum_{j\not=k}\frac{\psi(z_{j})-\psi(z_{k})}{z_{j}-z_{k}}-N\sum_{j=1}^{N}[\partial
Q\cdot\psi](z_{j})+\sum_{j=1}^{N}\partial\psi(z_{j}).$

The functional $W_{N}^{+}$ is also called the stress energy tensor in the
context of conformal field theory [219, Appendix 6]. The identity (5.21)
easily follows from the integration by parts

(5.22)
$\mathbb{E}_{N}[\partial\psi(z_{j})]=\mathbb{E}_{N}[\partial_{z_{j}}H(z_{1},\dots,z_{N})\cdot\psi(z_{j})].$

The approach using Ward’s identities was further developed in [38] and several
related works to study various scaling limits of the random normal matrix
models (see e.g. [39, 30] for the bulk scaling limit at weak non-Hermiticity
(2.40); [40, 306] for the edge scaling limit with boundary confinements
(2.65); [37, 41] for normal matrices with Mittag-Leffler type singularities as
in §2.4 and §2.7) and we refer to [15, Remark 2.9] for an expository summary
of this strategy. In particular, in [38], the rescaled version of Ward’s
identity was introduced and used to show the edge universality: for
$p\in\partial S_{Q},$

(5.23)
$\displaystyle\frac{1}{(N\delta)^{k}}\rho_{(k),N}\Big{(}p+i\mathbf{n}\frac{z_{1}}{\sqrt{N\delta}},\dots,p+i\mathbf{n}\frac{z_{k}}{\sqrt{N\delta}}\Big{)}\to\det\Big{[}K_{\infty}^{\rm
e}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$

where $\mathbf{n}$ is the outer normal vector at $p$ as in Proposition 2.8.
However, the result in [38] has an additional assumption that the limiting
correlation function is translation invariant along the real axis. This
assumption is intuitively natural but hard to rigorously show in general
(except e.g. for the rotationally symmetric potential $Q$). The edge
universality was later then shown by Hedenmalm and Wennman for a wide class of
potentials $Q$ in [201], where they developed an asymptotic theory for general
planar orthogonal polynomials; see also [198] for an alternative approach to
derive the main result in [201] and [200] for a theory developed for the
orthogonal polynomials associated with non-exponentially varying weight. The
asymptotic results in [201] together with (5.18) then leads to (5.23) by the
Riemann sum approximation.

We now turn our attention to the asymptotic behaviours of $\rho_{(k),N}$ in
the macro-scale. Let us begin with the $1$-point function $\rho_{(1),N}$. For
the bulk case when $z$ is interior of the droplet, the asymptotic behaviour of
$\rho_{(1),N}$ is well known in the literature, see [55, 27] and references
therein. It says that under the suitable assumptions on $Q$, there are real-
analytic functions $B_{j}$ such that

(5.24) $\pi\rho_{(1),N}(z)=N\Delta Q(z)+\frac{1}{2}\Delta\log\Delta
Q(z)+N^{-1}B_{2}(z)+\cdots+N^{-k+1}B_{k}(z)+O(N^{-k}),$

where $\Delta:=\partial_{z}\partial_{\bar{z}}$ is the one quarter of the usual
Laplacian. Notice here that the leading order asymptotic of (5.24) is implied
by the Laplacian growth (5.9). As a concrete example, we consider the induced
spherical ensemble (2.52) with $M=\alpha_{1}N$ and $n=\alpha_{1}N-1$, which
can be realised as (5.7) with

$Q(z)=(\alpha_{1}+\alpha_{2}-1)\log(1+|z|^{2})-2(\alpha_{1}-1)\log|z|.$

On the other hand, it follows from (2.54) that with
$\zeta=|z|^{2}/(1+|z|^{2})$,

(5.25)
$\pi\rho_{(1),N}(z)=\frac{|z|^{2(M-N)}}{(1+|z|^{2})^{2}}(M+n-N)\Big{(}I_{\zeta}(M-N,n)-I_{\zeta}(M,n-N)\Big{)}.$

Then using the well-known asymptotic behaviours of the incomplete beta
function, one can observe that for $r_{1}<|z|<r_{2}$, where $r_{1},r_{2}$ are
given in (2.53), the asymptotic behaviour (5.24) holds with

$\Delta
Q(z)=\frac{\alpha_{1}+\alpha_{2}-1}{(1+|z|^{2})^{2}},\qquad\Delta\log\Delta
Q(z)=-\frac{2}{(1+|z|^{2})^{2}}.$

Next, we consider the off-diagonal asymptotic behaviour of the correlation
kernel. For the Ginibre ensemble, equivalently, for the random normal matrix
model (5.7) with $Q(z)=|z|^{2}$, the associated correlation kernel $K_{N}$ in
(5.18) satisfies

(5.26)
$K_{N}(z,w)=\sqrt{\frac{2N}{\pi}}(z\bar{w})^{N}e^{N-\frac{N}{2}(|z|^{2}+|w|^{2})}S(z,w)\Big{(}1+O(\frac{1}{N})\Big{)},\qquad(z\not=w)$

where $S(z,w)$ is the exterior Szegö kernel

(5.27) $S(z,w):=\frac{1}{2\pi}\frac{1}{z\bar{w}-1}.$

From a viewpoint of Proposition 2.2, the asymptotic behaviour (5.26) can be
realised as a uniform expansion of the incomplete gamma function
$z\mapsto\Gamma(N;Nz)$, which is available in the literature in some
particular domains; see e.g. [323] for $|\arg(z-1)|<3\pi/4.$ In a recent work
[34], generalising the classical results, it was shown that the asymptotic
behaviour (5.26) remains valid as long as $z\bar{w}$ is outside the Szegö
curve $\\{z\in\mathbb{C}:|z|\leq 1,|z\,e^{1-z}|=1\\}.$ Note in particular that
if $|z|=|w|=1,$ then (5.26) reads

(5.28)
$K_{N}(z,w)\overset{c}{\sim}\sqrt{\frac{2N}{\pi}}S(z,w)\Big{(}1+O(\frac{1}{N})\Big{)},$

where $\overset{c}{\sim}$ means that the asymptotic expansion holds up to a
sequence of cocycles (in this case $(z\bar{w})^{N}$), which cancel out when
forming a determinant (2.9). From a statistical physics point of view, the
behaviour (5.28) indicates that there are strong correlations among the
particles on the boundary of the droplet, which also shows the slow decay of
correlations at the boundary. For GinUE this is explicit in (4.36). For
elliptic GinUE, the phenomenon was studied in [161], as a test of the
generalisation to more general shaped droplets, when the RHS of (4.36) is
predicted to be given in terms of a certain Green’s function for an
electrostatics problem outside of the droplet, which acts as a macroscopic
conductor [211], [147, Eq. (3.29]. Furthermore, it was obtained by Ameur and
Cronvall [34] that for a general class of potentials $Q$, the associated
correlation kernel $K_{N}$ satisfies

(5.29)
$K_{N}(z,w)\sim\sqrt{2N}\Big{(}\frac{\partial_{z}\partial_{\bar{z}}Q(z)}{\pi}\Big{)}^{1/4}\Big{(}\frac{\partial_{w}\partial_{\bar{w}}Q(w)}{\pi}\Big{)}^{1/4}S(z,w)(1+o(1)).$

For this, the use of a general theory on the orthogonal polynomial due to
Hedenmalm and Wennman [201] was made. We also refer to [17, 80, 31] for more
recent studies in this direction.

######  Remark 5.2.

Let $\\{p_{j}^{(N,R)}(z)\\}_{j=0,1,\dots}$ be the orthogonal polynomials with
respect to the inner product $\langle
f|g\rangle:=\int_{C_{R}}f(z)g(\bar{z})e^{-2NW(z)/t_{0}}$, where $W(z)$ as in
(5.3) and it is assumed that the limiting eigenvalue support $\Omega$
corresponding to (5.2) is contained in $D_{R}$. Consider the probability
distribution $\mathcal{P}_{k}^{(N,R)}$ specified by the eigenvalue PDF
proportional to

(5.30) $\prod_{l=1}^{k}e^{-2NW(z_{l})/t_{0}}\prod_{1\leq j<l\leq
k}|z_{l}-z_{j}|^{2}.$

As a result of the underlying determinantal structure, one has the formula for
$p_{k}^{(N,R)}(z)$ as an expectation with respect to $\mathcal{P}_{k}^{(N,R)}$
(see e.g. [149, proof of Proposition 5.1.3]; in fact such formulae were known
to Heine [318])

$p_{k}^{(N,R)}(z)=\Big{\langle}\prod_{l=1}^{k}(z-z_{l})\Big{\rangle}_{\mathcal{P}_{k}^{(N,R)}}.$

For $k/N=x$ as $k,N\to\infty$ it has been conjectured [137] that the zeros of
$p_{k}^{(N,R)}(z)$ accumulate on certain arcs $\Sigma$ contained in $\Omega$,
with corresponding measure $\mu_{x}^{*}$. Assuming this, it follows that for
$z\in\mathbb{C}\backslash\Omega$

(5.31) ${1\over\pi
t_{0}}\int_{\Omega}\log|z-w|\,d^{2}w=\int_{\Sigma}\log|z-s|\,d\mu_{x}^{*}.$

This identifies $\Sigma$ as the so-called mother body or potential theoretic
skeleton of $\Omega$. A number of works give further developments along these
lines, especially in relation to the strong asymptotics of the planar
orthogonal polynomials based on Riemann-Hilbert analysis. In the case of (5.6)
with a cubic potential, references include [62, 234, 272, 63], while for the
induced Gaussian type weight $2W(z)/t_{0}=|z|^{2}-2c\log|z-a|$ or more
generally $2W(z)/t_{0}=|z|^{2}-2\sum_{j=1}^{M}c_{j}\log|z-a_{j}|$ in (5.30),
see [45, $c=O(1)$], [46, $c=O(1/N)$ and $|a|\not=1$], [60, $c=O(1/N)$ and
$|a|\approx 1$], [251, $c=O(1/N)$ and $|a|\not=1$], [252, $c_{j}=O(1/N)$]. We
also refer to [325, 118, 241] for applications of such strong asymptotics in
the context of the characteristic polynomials of the Ginibre matrix.

##  6\. Further theory and applications

###  6.1. Fermi gas wave function interpretation

We have seen that the rewrite of (1.7), written in the exponential form
(1.10), allows for the GinUE eigenvalue PDF to be interpreted as the Boltzmann
factor for a particular Coulomb gas. If instead of an exponential form we use
(2.12) to rewrite (1.7) as

(6.1)
$\Big{|}\prod_{j=1}^{N}e^{-|z_{j}|^{2}/2}\det[z_{j}^{k-1}]_{j,k=1,\dots,N}\Big{|}^{2},$

then we are lead to an interpretation as the absolute value squared of a
ground state free Fermi quantum many body wave function. Thus inside the
absolute value of (6.1) is a Slater determinant of single body wave functions
$\\{\phi_{l}(z)\\}_{l=0,\dots,N-1}$ with $\phi_{l}(z)=e^{-|z|^{2}/2}z^{l}$.
What remains then is to identify the corresponding one body Hamiltonian for
quantum particles in the plane which have these single body wave functions for
the lowest energy states.

The appropriate setting for this task is a quantum particle confined to the
$xy$-plane subject to a perpendicular magnetic field, $(0,0,B)$, $B>0$.
Fundamental to this setting is the vector potential $\mathbf{A}$, related to
the magnetic field by $\nabla\times\mathbf{A}=(0,0,B)$. The so-called
symmetric gauge corresponds to the particular choice
$\mathbf{A}=(-By,Bx,0)=:(A_{x},A_{y},0)$, which henceforth will be assumed.
Physical quantities in this setting are $m$ (the particle mass), $e$ (particle
charge), $\hbar$ (Planck’s constant), $c$ (speed of light), which together
with $B$ are combined to give $\omega_{c}:=eB/mc$ (cyclotron frequency) and
$\ell:=\sqrt{\hbar c/eB}$ (magnetic length).

Defining the generalised momenta and corresponding raising and lowering
operators by

$\Pi_{u}=-i\hbar{\partial\over\partial u}+{e\over c}A_{u}\>\>(u=x,y),\qquad
a^{\dagger}={\ell\over\sqrt{2}\hbar}(\Pi_{x}+i\Pi_{y}),\qquad
a=(a^{\dagger})^{\dagger},$

allows the quantum Hamiltonian to be written in the harmonic oscillator like
form $H_{B}=\hbar\omega_{c}(a^{\dagger}a+{1\over 2})$ [108]. Important too are
the quantum centre of orbit operators and associated raising and lowering
operators

$U=u-{\ell^{2}\over\hbar}\Pi_{u}\>\>(U=X,Y;\,u=x,y),\qquad
b^{\dagger}={1\over\sqrt{2}\ell}(X-iY),\qquad b=(b^{\dagger})^{\dagger},$

for which $X^{2}+Y^{2}=2\ell^{2}(b^{\dagger}b+{1\over 2})$. The operators
$\\{a,a^{\dagger}\\}$ commute with $\\{b,b^{\dagger}\\}$, implying that $H$
and $X^{2}+Y^{2}$ permit simultaneous eigenstates. A complete orthogonal set
can be constructed using the raising operators according to

(6.2)
$|n,m\rangle={(a^{\dagger})^{n}(b^{\dagger})^{m}\over\sqrt{n!m!}}|0,0\rangle,$

with eigenvalues of $H$ equal to $(n+{1\over 2})\hbar\omega_{c}$ and
eigenvalue of $X^{2}+Y^{2}$ equal to $(2m+1)\ell^{2}$. The ground state
$|0,0\rangle$ is characterised by $a|0,0\rangle=b|0,0\rangle=0$, which can be
checked to have the unique solution $|0,0\rangle\propto
e^{-(x^{2}+y^{2})/4\ell^{2}}$. From this, application of $(b^{\dagger})^{m}$
gives $|0,m\rangle\propto\bar{z}^{m}e^{-|z|^{2}/4\ell^{2}}$, $z=x+iy$. Forming
a Slater determinant with respect to the first $N$ eigenstates of this type
gives (6.1) with $\ell^{2}=1/2$. Generally states with quantum number $n=0$
and thus belonging to the ground state are said to be in the lowest Landau
level. One remarks that the largest eigenvalue of $X^{2}+Y^{2}$ is then
$N-1/2$, which is in keeping with the squared radius of the leading order
support in the circular law.

The above theory of a quantum particle in the plane subject to a perpendicular
magnetic field can be recast to apply to a rotating quantum particle in the
plane [202, 239]. It is further true that the elliptic GinUE PDF (2.33) admits
an interpretation as the absolute value squared of state in the lowest Landau
level, and furthermore the corresponding orthogonal polynomials (2.35) can be
constructed using a Bogolyubov transformation of $\\{b,b^{\dagger}\\}$ [161].
Also, the PDF on the sphere (2.50) permits an interpretation as the absolute
value squared of the ground state wave function for a free Fermi gas on the
sphere subject to a perpendicular magnetic field [193]. Another point of
interest relates to the $N$-body Fermi ground state corresponding to the
quantum Pauli Hamiltonian in the plane with a perpendicular inhomogeneous
magnetic field $B(x,y)$. The Hamiltonian $H_{B}$ defined above then is to be
multiplied by the $2\times 2$ identity matrix, and the spin coupling term
$-(g\hbar/2m)B(x,y){\rm diag}\,(1/2,-1/2)$ added. With $B(x,y)=-{1\over
2}\nabla^{2}W(x,y)$ for some real valued $W$, and with the assumption
$\Phi:=\int B(x,y)\,dxdy<\infty$, the ground state for this model (which is
spin polarised all spins up) permits an exact solution for $g=2$ [5]. The
ground state of normalisable eigenfunctions has degeneracy
$[\Phi/2\pi\hbar]=:N$, with basis of eigenfunctions
$\\{z^{j}e^{W(x,y)/2\hbar}\\}_{j=0}^{N-1}$. This implies the Fermi many body
ground state (6.1) with $e^{-|z_{j}|^{2}/2}$ replaced by
$e^{W(x_{j},y_{j})/2\hbar}$ [4].

######  Remark 6.1.

1\. Upon stereographic projection of the sphere to the plane, it is possible
to write the quantum Hamiltonian for a charge particle in a constant
perpendicular magnetic field in a form unified with the original planar case
[130]. This involves the Kähler metric and potential, and permits a viewpoint
which carries over to further generalise the space to higher dimensional
complex manifolds in $\mathbb{C}^{m}$. A point of interest is that doing so
gives, for the bulk scaling limit of the corresponding $N$ particle lowest
Landau level state, the natural higher dimensional analogue of the kernel
(2.19) [57, 58].
2\. The squared wavefunction for higher Landau levels (say the $r$-th) has
been shown to give rise to the determinantal point process with bulk scaled
kernel

$K_{\infty}^{r}(w,z)=L_{r}^{0}(|w-z|^{2})e^{w\bar{z}}e^{-(|w|^{2}+|z|^{2})/2};$

see e.g. [312, Prop. 2.5]. Allowing for mixing between Landau levels up to and
including level $r$ leads to squared wave functions giving rise to the same
determinantal point process except for the replacement of Laguerre polynomials
$L_{r}^{0}\mapsto L_{r}^{1}$ in the kernel [192]. Extending [239], the precise
mapping between the rotating fermions in the higher Landau levels and the
polyanalytic Ginibre ensemble was established in [235]. Furthermore, its full
counting statistics and generalisations to finite temperature were obtained in
[315, 236].
3\. In the theory of the fractional quantum Hall effect, constructing an anti-
symmetric state with filling fraction of the lowest Landau level $\nu=1/m$,
for $m$ an odd integer, plays a crucial role. To accomplish this, Laughlin
[243] proposed the ground state wave function proportional to

(6.3) $\prod_{l=1}^{N}e^{-|z_{l}|^{2}/4\ell^{2}}\prod_{1\leq j<k\leq
N}(\bar{z}_{k}-\bar{z}_{j})^{m};$

note that with the assumption that $m$ is odd, this is anti-symmetric as
required for fermions. Moreover it belongs to the lowest Landau level as
follows from the theory in the text below (6.2). The absolute value squared of
(6.3) coincides with the Boltzmann factor (1.10) with $\beta=2m$, and the
scaling $z_{l}\mapsto z_{l}/\sqrt{2m\ell^{2}}$. From potential theoretic/
Coulomb gas reasoning, the bulk density is therefore $1/(2m\pi\ell^{2})$. The
factor of $m$ in the denominator is in precise agreement with the requirement
that the filling fraction be equal to $1/m$.
4\. The ground state $N$-body free spinless Fermi gas in the plane, without a
magnetic field but confined by a radial harmonic potential, is also an example
of a determinantal point process for which exact calculations are possible;
see the recent review [117]. However, its statistical state is distinct from
that of GinUE. Thus with a global scaling so that the support is the unit
disk, the density profile as the $d=2$ Thomas-Fermi functional form
${2\over\pi}(1-|z|^{2})\chi_{|z|<1}$, in contrast to the circular law (2.17).
The bulk scaled two-point correlation function (bulk density $1/4\pi$) is
given in terms of the $J_{1}$ Bessel function

$\rho_{(2),\infty}^{\rm hF}(z_{1},z_{2})=\Big{(}{1\over
4\pi}\Big{)}^{2}\bigg{(}1-\Big{(}{2J_{1}(|z_{1}-z_{2}|)\over|z_{1}-z_{2}|}\Big{)}^{2}\bigg{)},$

in contrast to (2.24). This gives a decay proportional to
$1/|z_{1}-z_{2}|^{3}$ of $\rho_{(2),\infty}^{{\rm hF},T}$. Also, the edge
scaled correlation kernel now involves Airy functions [116], rather than the
error function seen in (2.19). Universality results relating to many body free
Fermi ground states in dimension $d\geq 2$ have recently been obtained [119].
We highlight in particular the macroscopic fluctuation theorem for the linear
statistic $G=\sum_{j}g(\mathbf{r}_{j}/R)$, with $g$ assumed sufficiently
smooth and absolutely integrable, in the $R\to\infty$ limit [119, Th. III.2]

(6.4)
${G-{R^{d}\omega_{d}\over(2\pi)^{d}}\int_{\mathbb{R}^{d}}g(\mathbf{r})\,d^{d}\mathbf{r}\over\sigma_{d}R^{(d-1)/2}}\to{\rm
N}[0,\Sigma(g)],\quad(\Sigma(g))^{2}=\int_{\mathbb{R}^{d}}|\hat{g}(\mathbf{r})|^{2}|\mathbf{r}|\,d^{d}\mathbf{r}.$

Here $\omega_{d}=\pi^{d/2}/\Gamma(1+d/2)$ is the volume of the Euclidean ball
in $\mathbb{R}^{d}$, $\omega_{d}/(2\pi)^{d}$ is the bulk density,
$\sigma_{d}^{2}:=\omega_{d-1}/(2\pi)^{d}$ and the Fourier transform has the
definition
$\hat{g}(\xi)={1\over(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot\mathbf{r}}g(\mathbf{r})\,d^{d}\mathbf{r}$.
Note in particular that in contrast to (3.19), the variance of $G$ now
diverges with the scale $R$.

###  6.2. Quantum chaos applications

The pioneering works of Wigner and Dyson relating to the Hermitian random
matrix ensembles was, as noted in §1, motivated by seeking a model for the
(highly excited) energy levels of a complex quantum system. Later, in the
1980’s, as a fundamental contribution to the then emerging subject of quantum
chaos, Bohigas et al. [65] identified the correct meaning of a complex quantum
system not by the number of particles but rather as one for which the
underlying classical mechanics is chaotic. To test this prediction on say the
numerically generated spectrum of a quantum billiard system, the energy levels
(beyond some threshold to qualify as being highly excited) were first unfolded
so that their local density became unity, and then their numerically
determined statistical properties were compared against random matrix
predictions for the appropriate symmetry class; see e.g. [191]. Most popular
among the statistical properties have been the variance for the number of
eigenvalues in a large interval, and the distribution of the spacing between
successive eigenvalues.

A natural extension of these advances is to inquire about the spectrum of a
dissipative chaotic quantum system, which due to the loss of energy need not
be real. This question was taken up by Grobe, Haake and Sommers [189] for the
specific model of a damped periodic kicked top. The quantum dynamics are
specified by a subunitary density operator. It is the spectrum of this
operator, which after unfolding, and considering only those eigenvalues in the
upper half plane away from the real axis (there is a symmetry which requires
that the eigenvalues come in complex conjugate pairs — see the recent paper
[20] for a discussion of this point in a random matrix context) that were
compared in [189] a statistical sense to GinUE. Following from precedents in
the Hermitian case, in the statistical quantity measured was the distribution
of the radial spacing between closest eigenvalues, to be denoted $P^{\rm
s,GUE}(r)$ with the normalisation $\int_{0}^{\infty}P^{\rm s,GUE}(r)\,dr=1$.
This is the quantity $F_{\infty}(0;D_{r})$ of Remark 3.3.1. Recalling (3.10)
we therefore have

(6.5) $P^{\rm s,GUE}(r)=-{d\over
dr}e^{r^{2}}\prod_{j=1}^{\infty}\Big{(}1-{\gamma(j;r^{2})\over\Gamma(j)}\Big{)}.$

It follows that for small $r$, $P^{\rm s,GUE}(r)\sim 2r^{3}$, while it follows
from (3.11) and the comment in the sentence immediately above Remark 3.3 that
for large $r$, $\log P^{\rm s,GUE}(r)\sim-r^{4}/4$. A numerical plot can be
obtained from the functional form (6.5). For the moments the formula $\langle
r^{p}\rangle=p\int_{0}^{\infty}r^{p-1}e^{r^{2}}E_{\infty}(0;r)\,dr$ holds
true. In particular, for the mean we calculate $\langle
r\rangle=1.142929\dots$.

A variation of the closest neighbour spacing for an eigenvalue at $z$ is the
complex ratio $(z^{\rm c}-z)/(z^{\rm nc}-z)$, where $z^{\rm c}$ is the closest
neighbour to $z$, and $z^{\rm nc}$ is the next closest neighbour [296]. An
approximation, with fast convergence properties to the large $N$ form, has
been given recently in [131].

Very recently a non-Hermitian Hamiltonian realisation of GinUE has been
obtained in the context of a proposed non-Hermitian $q$-body Sachev-Ye-Kitaev
(SYK) model, with $N$ Majorana fermions — $N$ large and tuned ${\rm mod}\,8$ —
and $q>2$ and tuned ${\rm mod}\,4$ [175]. On another front, again very
recently, the emergence of GinUE behaviours in certain model many body quantum
chaotic systems in the space direction has been demonstrated [314]. Of
interest in both these lines of study is the so-called dissipative (connected)
spectral form factor

${\rm K}_{N}^{\rm c}(t,s)={1\over N}{\rm
Cov}\,\Big{(}\sum_{j=1}^{N}e^{i(x_{j}t+y_{j}s)},\sum_{j=1}^{N}e^{-i(x_{j}t+y_{j}s)}\Big{)}.$

Making use of the first formula in (3.2) and the finite $N$ form of (2.23),
this can be evaluated in terms of the hypergeometric function ${}_{1}F_{1}$
[256, Eq. (3)] (corrected in [176, Appendix A]; note too that both those
references use global scaled variables, whereas we do not).

######  Proposition 6.2.

We have

(6.6) ${\rm K}_{N}^{\rm c}(t,s)=1\\\ -{1\over
N}\sum_{m,n=0}^{N-1}{(t^{2}+s^{2})^{|m-n|/2}\over n!m!2^{|m-n|}}\bigg{(}{{\rm
max}(m,n)!\over|m-n|!}\,{}_{1}F_{1}\Big{(}{\rm
max}(m,n)+1,|m-n|+1;-{t^{2}+s^{2}\over 4}\Big{)}\bigg{)}^{2}.$

In particular,

$\lim_{N\to\infty}{\rm K}_{N}^{\rm c}(t,s)=1-e^{-(t^{2}+s^{2})/4};$

cf. (3.17).

Also of interest in the many body quantum chaos application is the GinUE
average of $|{\rm Tr}\,X^{k}|^{2}$ for positive integer $k$ [314].

######  Proposition 6.3.

We have

$\Big{\langle}|{\rm Tr}\,X^{k}|^{2}\Big{\rangle}_{\rm
GinUE}={1\over(k+1)(N-1)!}\Big{(}(k+N)!-{N!(N-1)!\over(N-k-1)!}\Big{)}.$

In particular

$\lim_{N,k\to\infty\atop k/N=x}{1\over kN^{k}}\Big{\langle}|{\rm
Tr}\,X^{k}|^{2}\Big{\rangle}_{\rm GinUE}={2\sinh(x^{2}/2)\over x^{2}}.$

###### Proof.

(Sketch) In [314] the average is reduced to
$\int_{\mathbb{C}}dz_{1}\int_{\mathbb{C}}dz_{2}\,\rho_{(2),N}(z_{1},z_{2})z_{1}^{k}\bar{z}_{1}^{k}$.
Earlier, the evaluation of a more general quantity was given in [165,
Corollary 4]. ∎

We conclude this subsection with a brief account of the use of the GinUE in an
ensemble theory of Lindblad dynamics [83, 123, 297]. This relates to the
evolution of the density matrix $\rho_{t}$ for an $N$-level dissipative
quantum system in the so-called Markovian regime, specified by the master
equation $\dot{\rho}_{t}=\mathcal{L}(\rho_{t})$. Here the operator
$\mathcal{L}$ assumes a special structure identified by Lindblad [258], and by
Gorini, Kossakowski, and Sudarshan [186]. Specifically $\mathcal{L}$ consists
of the sum of two terms, the first corresponding to the familiar unitary von
Neumann evolution, and the second to a dissipative part, being the sum over
operators $D_{L}$ (referred to as simple dissipators), represented as
$N^{2}\times N^{2}$ matrices according to

$D_{L}=2L\otimes_{T}L^{\dagger}-L^{\dagger}L\otimes_{T}\mathbb{I}_{N}-\mathbb{I}_{N}\otimes_{T}L^{\dagger}L,$

for some $N\times N$ matrix $L$. Here $A\otimes_{T}B:=A\otimes B^{T}$, where
$\otimes$ is the usual Kronecker product. In an ensemble theory, there is
interest in $F_{N}(t):={1\over N^{2}}\langle{\rm Tr}\,e^{tD_{L}}\rangle_{L}$
[83].

######  Proposition 6.4.

Let $L$ be chosen from GinUE with global scaling. We have

$\lim_{N\to\infty}F_{N}(t)=e^{-4t}\Big{(}I_{0}(2t)+I_{1}(2t)\Big{)}^{2}.$

###### Proof.

(Sketch) Following Can [83], using a diagrammatic calculus, it is first
demonstrated that

$\lim_{N\to\infty}{1\over N^{2}}\langle
D_{L}^{k}\rangle=\lim_{N\to\infty}{(-1)^{k}\over
N^{2}}\Big{\langle}\Big{(}L^{\dagger}L\otimes_{T}\mathbb{I}_{N}+\mathbb{I}_{N}\otimes_{T}L^{\dagger}L\Big{)}^{k}\Big{\rangle}_{L\in{\rm
GinUE}}.$

The average on the RHS, in terms of the eigenvalues $\\{x_{j}\\}$ of
$L^{\dagger}L$, reads
$\langle\sum_{j,l=1}^{N}(x_{j}+x_{l})^{k}\rangle_{L^{\dagger}L}$ and
consequently

$\lim_{N\to\infty}F_{N}(t)=\lim_{N\to\infty}\Big{\langle}{1\over
N^{2}}\sum_{j,l=1}^{N}e^{-t(x_{j}+x_{l})}\Big{\rangle}_{L^{\dagger}L}=\lim_{N\to\infty}\bigg{(}\Big{\langle}{1\over
N}\sum_{j=1}^{N}e^{-tx_{j}}\Big{\rangle}_{L^{\dagger}L}\bigg{)}^{2}.$

The latter is the mean of a linear statistic in the ensemble
$\\{L^{\dagger}L\\}$ (complex Wishart matrices; see e.g. [149, §3.2]). Using
the Marchenko-Pastur law for the global density of this ensemble (see e.g.
[149, §3.4.1]), the stated result follows. ∎

######  Remark 6.5.

(Classification of non-Hermitian matrices) It was commented in the
Introduction that, in distinction to Dyson’s viewpoint based on symmetry
considerations, Ginibre’s study [179] was no similarly motivated. Nowadays
however, it is recognised that a symmetry viewpoint is fundamental to
topological driven effects in non-Hermitian quantum physics [44]. Starting
with [59, 268] and continuing in [223], a classification scheme based on
symmetries with respect to the involutions of transpose, complex conjugation
and Hermitian conjugation, and in which the (anti-)commutation relation
involves unitary matrices satisfying certain quadratic relations in terms of
these involution, has been given. For example, defining the block unitary
matrix $P={\rm diag}\,(\mathbb{I}_{N},-\mathbb{I}_{N})$, and requiring that
the matrix ensemble $\\{A\\}$ have the (anti-)symmetry $A=-PAP$, gives that
each $A$ has the form

(6.7) $A=\begin{bmatrix}0_{N\times N}&X\\\ Y&0_{N\times N}\end{bmatrix}$

for some square matrices $X,Y$. Denoting the eigenvalues of the matrix product
$XY$ as $\\{-z_{j}^{2}\\}$, one sees that the eigenvalues of $A$ are $\\{\pm
iz_{j}\\}$.

In keeping with the viewpoint of this subsection, a basic question are
signatures of the symmetry in the eigenvalue spectrum. For example, in (6.7),
with $X,Y$ GinUE matrices, are the bulk scaled eigenvalues of $A$
statistically distinct from individual GinUE matrices? We know from the
results quoted in the paragraph above Remark 2.18 that the answer in this case
is no. However the answer to this question is yes, if instead the symmetry is
that $A=A^{T}$, for the independent entries of $A$ standard complex Gaussians.
This was demonstrated in [195] by a numerical study of the nearest neighbour
spacing distribution, and the relevance to Lindblad dynamics discussed.

###  6.3. Singular values

One recalls that for a complex square matrix $X$ the squared singular values
are the eigenvalues of $X^{\dagger}X$. For a general ensemble of non-Hermitian
matrices $\\{X\\}$, motivation to study the singular values comes from various
viewpoints. For example, in Remark 2.18.4, singular values (specifically of
product matrices) appeared in the context of Lyapunov exponents. As other
example, one recalls that plus/minus of the singular values are the
eigenvalues of the $2N\times 2N$ Hermitian matrix

$H=\begin{bmatrix}0_{N\times N}&X\\\ X^{\dagger}&0_{N\times N}\end{bmatrix}.$

The importance of this in relation to the eigenvalues of $X$ is that resolvent
associated with $H$ is fundamental to the study of the circular law for the
spectral density beyond the Gaussian case; see e.g. [67, §4.1]. Another piece
of theory is that the condition number $\kappa_{N}$ associated with $X$ is
equal to the ratio of the smallest to the largest singular value [135]. And
from the identity $|\det X|=|\det X^{\dagger}X|^{1/2}$ the distribution of the
modulus of $\det X$ is determined by the singular values.

For the GinUE, the squared singular values $\\{s_{j}\\}_{j=1}^{N}$ say are
known to have for their joint distribution a PDF proportional to

(6.8) $\prod_{j=1}^{N}e^{-s_{j}}\prod_{1\leq j<k\leq N}(s_{k}-s_{j})^{2},\quad
s_{j}\in\mathbb{R}_{+};$

see e.g. [149, Prop. 3.2.2 with $\beta=2$, $n=m=N$]. After scaling by $N$,
almost surely the largest squared singular value has the limiting value $4$
[286]. However, after the same scaling, a simple change of variables in (6.8)
integrated from $(s,\infty)$ in each variable reveals that the smallest
singular value is an exponential random variable with rate parameter $N^{2}$.
Putting these facts together implies that for large $N$, $\kappa_{N}/N$ is
distributed according to the heavy tailed distribution with PDF ${8\over
x^{3}}e^{-4/x^{2}}\chi_{x>0}$ [135]. Also, for $n\times N$ ($n\geq N$)
rectangular GinUE matrices, it is proved in [102] that
$\langle\log\kappa_{N}\rangle<{N\over|n-N|+1}+2.24$, for any $N\geq 2$.

Let $P_{N}(t)$ denote the PDF for the distribution of $|\det X|^{2}$ for GinUE
matrices. Making use of knowledge of the PDF of squared singular values (6.8)
shows that the Mellin transform of $P_{N}(t)$ is equal to the multiple
integral

(6.9) ${1\over
C_{N}}\int_{0}^{\infty}ds_{1}\cdots\int_{0}^{\infty}ds_{N}\,\prod_{j=1}^{N}s_{j}^{s-1}e^{-s_{j}}\prod_{1\leq
j<k\leq N}(s_{k}-s_{j})^{2}=\prod_{j=0}^{N-1}{\Gamma(s+j)\over\Gamma(1+j)}.$

Here the normalisation $C_{N}$ is such that the expression equals unity for
$s=1$, while the evaluation of the multiple integral follows as a special case
of the Laguerre weight Selberg integral; see e.g. [149, Prop. 4.7.3]. As noted
in [166, Eq. (2.17)] (see also [293, Prop. 2.2]), it follows immediately from
this that

(6.10) $|\det X|^{2}\mathop{=}\limits^{\rm d}\prod_{l=1}^{N}{1\over
2}\chi_{2l}^{2}.$

In words this says that the absolute value squared of the determinant of GinUE
matrices is equal in distribution to the product of $N$ independent chi-
squared distributions, with degrees of freedom $2,4,\dots,2N$, each scaled by
a factor of 2. Starting from (6.10), and defining the global scaled GinUE
matrices $X^{\rm g}$ by $X^{\rm g}={1\over\sqrt{N}}X$, the distribution of
$\log|\det X^{\rm g}|^{2}$ can be shown to have leading order mean $-N$,
variance $\log N$, and after recentring and rescaling satisfy a central limit
theorem [293, Th. 3.5]. For a general linear statistic
$\sum_{j=1}^{N}f(z_{j})$ of global scaled GinUE matrices, the leading order
mean is ${N\over\pi}\int_{|z|<1}f(z)\,d^{2}z$. For $f(z)=\log|z|^{2}$, this
gives the stated value of $-N$. Also, we notice that substituting this choice
of $f(z)$ in the variance formula implied by (3.21) gives
${1\over\pi}\int_{|\mathbf{r}|<1}{1\over x^{2}+y^{2}}\,dxdy$, which is not
integrable at the origin, in keeping with the variance actually diverging as
$\log N$.

There is an alternative viewpoint on the result (6.10) which does not require
knowledge of the joint distribution of the singular values (6.8), nor the
evaluation of the multiple integral (6.9). The idea, used in both [293, 166]
and which goes back to Bartlett [48] in the case of real Gaussian matrices, is
to decompose $X$ in terms of its QR (Gram-Schmidt) decomposition. The matrix
of orthonormal vectors $Q$ constructed from the columns of $X$ will for
$X\in{\rm GinUE}$, be a Haar distributed unitary matrix, which we denote by
$U$. The matrix $R=[r_{jk}]_{j,k=1}^{N}$ is upper triangular with diagonal
elements real and positive. One notes

(6.11) $\det X^{\dagger}X=\prod_{j=1}^{N}r_{jj}^{2},$

and so it suffices to have knowledge on the distribution of
$\\{r_{jj}\\}_{j=1}^{N}$ for $X$.

######  Proposition 6.6.

Let $\\{r_{jj}\\}_{j=1}^{N}$ denote the diagonal elements in the QR
decomposition of a GinUE matrix $X$. We have

(6.12) $r_{jj}^{2}\mathop{=}\limits^{\rm d}{1\over 2}\chi_{2j}^{2}.$

###### Proof.

The QR decomposition $X=UR$ gives the corresponding decomposition of measure
(see e.g. [149, Prop. 3.2.5])

$(dX)=\prod_{j=1}^{N}r_{jj}^{2(N-j)+1}(dR)(U^{\dagger}U),$

where as anticipated $(U^{\dagger}U)$ is recognised as Haar measure on the
space of complex unitary matrices. The element distribution of GinUE matrices
is proportional to $e^{-{\rm Tr}\,X^{\dagger}X}=e^{-\sum_{1\leq j\leq k\leq
N}|r_{jk}|^{2}}$. The various factorisations implies that integrating over $U$
and the off diagonal elements of $R$ only changes the normalisation. We then
read off that each $r_{jj}$ has a distribution with PDF proportional to
$r^{2(N-j)+1}e^{-r^{2}}$, which implies (6.12). ∎

Using (6.12) in (6.11) reclaims (6.10).

######  Remark 6.7.

1\. Since with $\\{z_{j}\\}$ the eigenvalues of $X$, $|\det
X|^{2}=\prod_{j=1}^{N}|z_{j}|^{2}$, the fact that the Mellin transform of the
distribution of this quantity is given by the product of gamma functions in
(6.9) implies

(6.13) $\Big{\langle}\prod_{l=1}^{N}|z_{l}|^{2(s-1)}\Big{\rangle}_{\rm
GinUE}^{\rm g}=N^{N(s-1)}\prod_{j=0}^{N-1}{\Gamma(s+j)\over\Gamma(1+j)}.$

Here the superscript "g" indicates the use of global scaling coordinates
$z_{l}\mapsto\sqrt{N}z_{l}$. We observe that knowledge of the induced GinUE
normalisation $C_{n,N}$ in Proposition 2.11 provides a direct derivation of
(6.13). For large $N$ this ratio of gamma functions can be written in terms of
the Barnes $G$-function according to ${G(N+s)\over G(N+1)G(s)}$; see [149, Eq.
(4.183)]. Known asympotics for ratios of the Barnes $G$-function (see e.g.
[149, Eq. (4.185)] then gives that for large $N$, and with $s=\gamma/2+1$ for
convenience,

(6.14) $\Big{\langle}\prod_{l=1}^{N}|z_{l}|^{\gamma}\Big{\rangle}_{\rm
GinUE}^{\rm g}\sim N^{\gamma^{2}/8}e^{-(\gamma/2)N}{(2\pi)^{\gamma/4}\over
G(1+\gamma/2)}.$

This is the special case $z=0$ of an asymptotic formula for
$\langle\prod_{l=1}^{N}|z-z_{l}|^{\gamma}\rangle_{\rm GinUE}^{\rm g}$ given by
Webb and Wong [325, Th. 1.1].
2\. It is a standard result in random matrix theory (see e.g. [286]) that the
density of singular values in (6.8), after the global scaling $s_{j}\mapsto
s_{j}N$, as the particular Marchenko-Pastur form
$\rho^{|rmMP}_{(1),\infty}(x)={1\over 2\pi}({4-x\over x})^{1/2}\chi_{0<x<4}$.
The $k$-th moment of the density is given in terms of the particular mixed
moment of a global scaled Ginibre matrix $\tilde{G}$, $\langle{\rm
Tr}(\tilde{G}^{\dagger}G)^{k}\rangle$. The calculation of these moments for
large $N$ relates to free probability — see the recent introductory text [288]
for the main ideas — and to combinatorics as is seen from the fact that
$\int_{0}^{4}x^{k}\rho^{\rm MP}_{(1),\infty}(x)\,dx=C_{k}$, where $C_{k}$
denotes the $k$-th Catalan number. Works on mixed moments of Ginibre matrices
include [324, 115, 194, 124].
3\. The squared singular values as specified by the PDF (6.8) form a
determinantal points process, being a special case of the classical Laguerre
unitary ensemble; see e.g. [149, Chapters. 3 and 5]. This is similarly true of
the squared singular values of the various extensions of GinUE considered
above: for example in the case of the spherical model and truncated unitary
matrices, it is the classical Jacobi unitary ensemble which arises, while the
singular values of products of GinUE matrices, or of truncated unitary
matrices, gives rise to a class of determinantal point processes called Pólya
ensembles [233, 227, 226, 167]. A notable exception is the singular values of
elliptic GinUE matrices, which form a Pfaffian point process [218].

###  6.4. Eigenvectors

Associated with the set of eigenvalues $\\{\lambda_{j}\\}$ of a Ginibre matrix
$G$ are two sets of eigenvectors — the left eigenvectors $\\{\bm{\ell}_{j}\\}$
such that $\bm{\ell}_{j}^{T}G=\lambda_{j}\bm{\ell}_{j}^{T}$, and the right
eigenvectors $\\{\mathbf{r}_{j}\\}$ such that
$G\mathbf{r}_{j}=\lambda_{j}\mathbf{r}_{j}$. These are not independent, but
rather (upon suitable normalisation), form a biorthogonal set

(6.15) $\bm{\ell}_{i}^{T}\mathbf{r}_{j}=\delta_{i,j}.$

This property follows from the diagonalisation formula $G=XDX^{-1}$, where $X$
is the matrix of right eigenvectors, $D$ the diagonal matrix of eigenvectors,
and $X^{-1}$ identified as the matrix of left eigenvectors. For nonzero
scalars $\\{c_{i}\\}$ we see that (6.15) is unchanged by the rescalings
$\mathbf{r}_{j}\mapsto c_{j}\mathbf{r}_{j}$ and
$\bm{\ell}_{j}\mapsto(1/c_{j})\bm{\ell}_{j}$.

For $N\times 1$ column vectors $\mathbf{u},\mathbf{v}$, define the inner
product $\langle\mathbf{u},\mathbf{v}\rangle:=\bar{\mathbf{u}}^{T}\mathbf{v}$.
The so called overlap matrix has its elements $\mathcal{O}_{ij}$ expressed in
terms of this inner product according to

(6.16)
$\mathcal{O}_{ij}:=\langle\bm{\ell}_{i},\bm{\ell}_{j}\rangle\langle\mathbf{r}_{i},\mathbf{r}_{j}\rangle.$

Note that this is invariant under the mappings noted in the final sentence of
the above paragraph, and for fixed $i$ and summing over $j$ gives $1$. Also,
it follows from (6.16) that the diagonal entries relate to the lengths

(6.17) $\mathcal{O}_{jj}=||\bm{\ell}_{j}||^{2}||\mathbf{r}_{j}||^{2}.$

The square root of this quantity is known as the eigenvalue condition number;
see the introduction to [71] and [112, §1.1] for further context and
references. Significant too is the fact that the overlaps (6.17) appear in the
specification of a Dyson Brownian motion extension of GinUE [188, 71].

Statistical properties of $\\{O_{ij}\\}$ for GinUE were first considered by
Chalker and Mehlig [95, 96]. By the Schur decomposition (2.2), instead of a
GinUE matrix $G$, we may consider an upper triangular matrix $Z$ with the
eigenvalues $\\{z_{j}\\}$ of $G$ on the diagonal, and off diagonal entries
standard complex Gaussians. For the eigenvalue $\lambda_{1}$, the triangular
structure shows that $\bm{\ell}_{1}=(1,b_{2},\dots,b_{N})^{T}$ and
$\mathbf{r}_{1}=(1,0,\dots,0)^{T}$ where for $p>1$ and $b_{1}=1$,
$b_{p}={1\over z_{1}-z_{p}}\sum_{q=1}^{p-1}b_{q}Z_{pq}$. From this last
relation, it follows that with $\bm{\ell}_{1}^{(n)}=(1,b_{2},\dots,b_{n})^{T}$
for $n<N$ we have

(6.18)
$||\bm{\ell}_{1}^{(n+1)}||^{2}=||\bm{\ell}_{1}^{(n)}||^{2}\Big{(}1+{1\over|z_{1}-z_{n+1}|^{2}}\Big{|}\sum_{q=1}^{n}\tilde{b}_{q}Z_{(n+1)q}\Big{|}^{2}\Big{)},\qquad\tilde{b}_{q}:={b_{q}\over\sqrt{\sum_{q=1}^{n}|b_{q}|^{2}}}.$

This has immediate consequence in relation to $\mathcal{O}_{11}$ as shown by
Bourgade and Dubach [71].

######  Proposition 6.8.

Let the eigenvalues $\\{z_{j}\\}$ be given. We have

(6.19) $\mathcal{O}_{11}\mathop{=}\limits^{\rm
d}\prod_{n=2}^{N}\Big{(}1+{|X_{n}|^{2}\over|z_{1}-z_{n}|^{2}}\Big{)},$

where each $X_{n}$ is an independent complex standard Gaussian. Furthermore,
it follows from this that after averaging over $\\{z_{2},\dots,z_{N}\\}$

(6.20) $\mathcal{O}_{11}\Big{|}_{z_{1}=0}\mathop{=}\limits^{\rm d}{1\over{\rm
B}[2,N-1]},$

where ${\rm B}[\alpha,\beta]$ refers to the beta distribution.

###### Proof.

A product formula for $\mathcal{O}_{11}$ follows from (6.17), the fact that
$||\mathbf{r}_{1}||=1$, and by iterating (6.18). This product formula is
identified with the RHS of (6.19) upon noting that a vector of independent
standard complex Gaussians dotted with any unit vector (here
$(\tilde{b}_{1},\dots,\tilde{b}_{n})$) has distribution equal to a standard
complex Gaussian.

In relation to (6.20) a minor modification of the proof of Proposition 2.20
shows that conditioned on $z_{1}=0$, the ordered squared moduli
$\\{|z_{j}|^{2}\\}_{j=2}^{N}$ are independently distributed as
$\\{\Gamma[j;1]\\}_{j=2}^{N}$. Noting too that with each $X_{j}$ a standard
complex Gaussian, $|X_{j}|^{2}\mathop{=}\limits^{\rm d}\Gamma[1;1]$ it follows

$\mathcal{O}_{11}\Big{|}_{z_{1}=0}\mathop{=}\limits^{\rm
d}\prod_{n=2}^{N}\Big{(}1+{\tilde{X}_{n}\over
Y_{n}}\Big{)},\qquad\tilde{X}_{n}\mathop{=}\limits^{\rm
d}\Gamma[1;1],\>Y_{n}\mathop{=}\limits^{\rm d}\Gamma[n;1].$

Next, we require knowledge of the standard fact that
$Y_{n}/(Y_{n}+\tilde{X}_{n})\mathop{=}\limits^{\rm d}{\rm B}[n,1]$.
Furthermore (see e.g. [149, Exercises 4.3 q.1]), for $x\mathop{=}\limits^{\rm
d}{\rm B}[\alpha+\beta,\gamma]$, $y\mathop{=}\limits^{\rm d}{\rm
B}[\alpha,\beta]$, we have that $xy\mathop{=}\limits^{\rm d}{\rm
B}[\alpha,\beta+\gamma]$, which tells us that with
$b_{n}\mathop{=}\limits^{\rm d}{\rm B}[n,1]$ we have
$\prod_{n=2}^{N}b_{n}\mathop{=}\limits^{\rm d}{\rm B}[2,N-1]$.

∎

Dividing both sides of (6.20) by $N$ we see that the $N\to\infty$ is well
defined since $N{\rm B}[2,N-1]\to\Gamma[2,1]$. After scaling the GinUE matrix
$G\mapsto G/\sqrt{N}$ so that the leading eigenvalue support is the unit disk,
an analogous limit formula has been extended from $z_{1}=0$ to any $z_{1}=w$,
$|w|<1$ in [71]. Thus

(6.21) ${\mathcal{O}_{11}\Big{|}_{z_{1}=w_{1}}\over
N(1-|w_{1}|^{2})}\mathop{\to}\limits^{\rm d}{1\over\Gamma[2,1]}.$

In words, with $\mathcal{O}_{11}$ corresponding to the condition number, one
has that the instability of the spectrum is of order $N$ and is more stable
towards the edge. Another point of interest is that the PDF for
$1/\Gamma[2,1]$ is $\chi_{t>0}e^{-1/t}/t^{3}$, which is heavy tailed, telling
us that only the zeroth and first integer moments are well defined. We remark
that limit theorems of the universal form (6.21) have been proved in the case
of the complex spherical ensemble of §2.5, and for a sub-block of a Haar
distributed unitary matrix [128]; see also [282].

The $1/t^{3}$ tail implied by (6.21) has been exhibited from another viewpoint
in the work of Fyodorov [169]. There the joint PDF for the overlap non-
orthogonality $\mathcal{O}_{jj}-1$, and the eigenvalue position $z_{j}$, was
computed for finite $N$. The global scaled limit of this quantity,
$\mathcal{P}^{\rm g}(t,w)$ say, was evaluated as [169, Eq. (2.24)]

(6.22) $\mathcal{P}^{\rm g}(t,w)={(1-|w|^{2})^{2}\over\pi
t^{3}}e^{-(1-|w|^{2})/t},\quad|w|<1.$

Note that for the first moment in $t$ this gives

(6.23) $\int_{0}^{\infty}t\mathcal{P}^{\rm
g}(t,w)\,dt={1\over\pi}(1-|w|^{2}),$

in keeping with a prediction from [95, 96]. This was first proved in [324].

An explicit formula for the large $N$ form of the average of the overlap
(6.16) in the off diagonal case (say $(i,j)=(1,2)$), with the GinUE matrix
scaled $G\mapsto G/\sqrt{N}$, and conditioned on $z_{1}=w_{1}$, $z_{2}=w_{2}$
with $|w_{1}|,|w_{2}|<1$ is also known [95, 96, 71, 22, 112]

(6.24)
$\left\langle\mathcal{O}_{12}\Big{|}_{z_{1}=w_{1},z_{2}=w_{2}}\right\rangle\mathop{\sim}\limits_{N\to\infty}-{1\over
N}{1-w_{1}\bar{w}_{2}\over|w_{1}-w_{2}|^{4}}\bigg{(}{1-(1+N|w_{1}-w_{2}|^{2})e^{-N|w_{1}-w_{2}|^{2}}\over
1-e^{-N|w_{1}-w_{2}|^{2}}}\bigg{)}.$

This formula is uniformly valid down to the scale $N|w_{1}-w_{2}|={\rm O}(1)$.
The large $N$ form of the average value of the diagonal overlap for products
of $M$ global scaled GinUE matrices has been considered in [74, 51], with the
result

(6.25) $\lim_{N\to\infty}{1\over
N}\left\langle\mathcal{O}_{11}\Big{|}_{z_{1}=w}\right\rangle={1\over\pi}|z|^{-2+2/M}(1-|z|^{2/M})\chi_{|z|<1};$

cf. (2.77). We remark that the average values of $\mathcal{O}_{11}$ and
$\mathcal{O}_{12}$, conditioned on multiple eigenvalues are shown to have a
determinantal form in [22, Th. 1], thus exhibiting an integrable structure for
these eigenvector statistics.

For general complex non-Hermitian matrices with independently distributed
entries of the form $\xi_{jk}+i{\zeta}_{jk}$, where each
$\xi_{jk},\tilde{\zeta}$ is an identically distributed zero mean real random
variable of unit variance (this class is sometimes referred to as complex non-
Hermitian Wigner matrices; see e.g. [18]), a line of research in relation to
the normalised eigenvectors is to quantify the similarity with a complex
vector drawn from the sphere embedded in $\mathbb{C}^{N}$ with uniform
distribution. Recent references on this include [264, 267]. For random vectors
on the sphere, there are bounds on the size of the components which rule out
gaps is the spread of the size of the components, referred to in [295] as no-
gaps localisation. As noted in [267, §1.1], the bi-unitary invariance of GinUE
matrices implies individual eigenvectors are distributed uniformly at random
from the complex sphere, and thus with probability close to one have that the
$j$-th largest modulus of the entries is bounded above and below by a positive
constant times $\sqrt{N-j}/N$, for $j$ in the range from $N/2$ up to $N$ minus
a constant time $\log N$.

### Acknowledgements

This research is part of the program of study supported by the Australian
Research Council Discovery Project grant DP210102887. SB was partially
supported by the National Research Foundation of Korea grant
NRF-2019R1A5A1028324, Samsung Science and Technology Foundation grant SSTF-
BA1401-51, and KIAS Individual via the Center for Mathematical Challenges at
Korea Institute for Advanced Study grant SP083201. The authors gratefully
acknowledge Gernot Akemann, Christophe Charlier, Yan Fyodorov, Grégory Schehr
and Aron Wennman for helpful feedback on the first draft of this work.

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