\begin{split}
&\quad \int_{\C} (z^{k - 1} \overline{z}^{l - 1} - z^{l - 1} \overline{z}^{k - 1}) (z-\overline{z}) e^{u \chi_a(z) -N W(z)} \,dA(z)
\\
&= 2 \int_0^\infty (r^k r^{l - 1} \delta_{k, l - 1} - r^{k - 1} r^l \delta_{k - 1, l} - r^l r^{k - 1} \delta_{k - 1, l} + r^{l - 1} r^k \delta_{k, l - 1}) e^{u \chi_a(r) -N g(r)} r \,dr
\\
&= 4 (\delta_{k, l - 1} - \delta_{k, l + 1}) \int_0^\infty r^{k + l} e^{u \chi_a(r) -N g(r)} \,dr. \label{entries in Pf}
\end{split}
\end{align}

Since $e^{u \chi_a(r)} = 1 - (1 - e^{u}) \chi_a(r)$, we obtain for the integral

\begin{align}
\begin{split} \label{int in Pf}
\int_0^\infty r^{k + l} e^{u \chi_a(r) -N g(r)} \,dr
& = \int_0^\infty r^{k + l} e^{-N g(r)} \,dr - (1 - e^{u}) \int_0^a r^{k + l} e^{-N g(r)} \,dr \\
& = \frac{1}{2} h_{\frac{k + l - 1}{2}} - (1 - e^{u}) \frac{1}{2} h_{\frac{k + l - 1}{2},1}(a).
\end{split}
\end{align}

Note that the indices are integer-valued for $k = l - 1$ and $k = l + 1$.
Combining (<ref>), (<ref>) and (<ref>), we arrive at

\begin{equation}
\mathbb{E}_{N,W}^{(\beta=4)}\Big[ e^{ u \NN_a } \Big]
= \frac{N!}{Z_{N,W}^{(\beta=4)}}  \Pf\Big[ 2 (\delta_{k, l - 1} - \delta_{k, l + 1}) \big( h_{\frac{k + l - 1}{2}} - (1 - e^{u}) h_{\frac{k + l - 1}{2},1}(a) \big) \Big]_{k, l =1}^{2N}.
\end{equation}

The Kronecker deltas inside the Pfaffian lead to an antisymmetric matrix whose only non-zero entries appear above and below the main diagonal.
The Pfaffian of such a matrix is the product of every second entry above the main diagonal (i.e. $l = k - 1$ and then only take the even $k$):

\begin{align*}
&\quad \Pf\Big[ 2 (\delta_{k, l - 1} - \delta_{k, l + 1}) \big( h_{\frac{k + l - 1}{2}} - (1 - e^{u}) h_{\frac{k + l - 1}{2},1}(a) \big) \Big]_{k, l =1}^{2N}
\\
&= \prod_{ \substack{k=2 \\ k:\textup{ even}} }^{2 N} 2 \big( h_{k-1} - (1 - e^{u}) h_{k-1,1}(a) \big) = 2^N \prod_{j = 0}^{N-1} \big( h_{2j+1} - (1 - e^{u}) h_{2j+1,1}(a) \big).
\end{align*}

On the other hand, it follows from <cit.> that

\begin{equation}
Z_{N,W}^{(\beta=4)} = N! \, 2^N \prod_{j = 0}^{N-2} h_{2 j+ 1}.
\end{equation}

This leads to

\begin{equation}
\mathbb{E}_{N,W}^{(\beta=4)}\Big[ e^{ u \NN_a } \Big]= \prod_{j = 0}^{N-1} \Big( 1 - (1 - e^{u}) \frac{h_{2j+1,1}(a)}{h_{2j+1}} \Big).
\end{equation}

Using $h_{j} - h_{j,1}(a) = h_{j,2}(a)$, we obtain the desired identity (<ref>).

The expression (<ref>) follows along the lines of the supplementary material in [73] using the series expansion

\begin{equation}
\log \mathbb{E}_{N,W}^{(\beta=4)}\Big[ e^{ u \NN_a } \Big] =u \, E_{N,W}^{(\beta=4)}(a)+ \sum_{p = 2}^{\infty} \kappa_p^{(\beta=4)}(a) \,  \frac{u^p}{p!}.
\end{equation}

Note that the expression (<ref>) was obtained in <cit.>.

In <cit.>, it was shown that if $g$ satisfies the conditions in Definition <ref>, then in the bulk $a\in (0,1)$ the function $\mathcal{L}_k(a)$ has the large-$N$ asymptotic

\begin{equation}
\mathcal{L}_k(a) \sim \frac{1}{2} \erfc\Big( \frac{s}{\sqrt{2 \Delta W(a)} \, a} \Big), \qquad k = \frac{1}{2}(N a g'(a) - 1) + \sqrt{N} s,
\end{equation}

and is exponentially suppressed for $|k-\frac{1}{2}(N a g'(a) - 1)|>M\sqrt{N}$ for large $M$.
Hence for $k = 2j+1$ we get

\begin{equation}
\mathcal{L}_{2j+1}(a) \sim \frac{1}{2} \erfc\Big( \frac{2 j - \frac{1}{2} N g'(a) a + \frac{3}{2}}{\sqrt{2 \Delta W(a) N} \, a} \Big).
\end{equation}

Now we insert the asymptotic behavior of $\mathcal{L}_{2j+1}(a)$ into the finite-$N$ cumulant formula (<ref>).
Note that the argument of the polylogarithm is then of the form

\begin{equation}
1 - \frac{1}{\frac{1}{2} \erfc(x)} = \frac{\erfc(x) - 2}{\erfc(x)} = -\frac{\erfc(-x)}{\erfc(x)}, \quad
\text{ with } x = \frac{2 j - \frac{1}{2} N g'(a) a + \frac{3}{2}}{\sqrt{2 \Delta W(a) N} \, a}.
\end{equation}

Next we replace the sum over $j$ in the range
$|2j+1-\frac{1}{2}(N a g'(a) - 1)|\leq\sqrt{N} s$ that contributes to the sum
by a Riemann integral over $x$, and we obtain

\begin{equation}
\kappa_p^{(\beta=4)}(a) \sim (-1)^{p + 1} \frac{\sqrt{2 \Delta W(a) N} \, a}{2} \int_{-\infty}^{\infty} \Li_{1 - p}\Big( -\frac{\erfc(-x)}{\erfc(x)} \Big) \, dx
\end{equation}

as $N \to \infty$.
Note that the prefactor comes from the change of integration variable $j \to x$.
The parity in (<ref>) comes from that of the integrand under $x\to-x$, using the inversion formula (<ref>).

Similarly, for the edge case when $a$ is given by (<ref>), it follows that

\begin{equation}
\kappa_p^{(\beta=4)}(a) \sim (-1)^{p + 1} \frac{\sqrt{2 \Delta W(a) N} \, a}{2} \int_{-\infty}^{\SS} \Li_{1 - p}\Big( -\frac{\erfc(-x)}{\erfc(x)} \Big) \, dx
\end{equation}

as $N \to \infty$.
Since we sum up to $j = N - 1$, the integration will stop at some value $x < \infty$ now, which can be derived as in <cit.>.
Note that we use $4 + g''(1) = 4 \Delta W(1)$ after we expand the $g'(a)$-term.

§ CONCLUSIONS AND OPEN QUESTIONS

In this paper we have analysed the counting statistics of the number of eigenvalues in a centred disc of radius $R$ in the real and symplectic Ginibre ensembles. For the symplectic ensemble we determined the full counting statistics by computing the generating function for all cumulants at finite matrix dimension $N$, for a large class of rotationally invariant non-Gaussian potentials. In the large-$N$ limit we could prove that the symplectic ensemble is in the same universality class as its complex counterpart. This holds both in the bulk regime, when the radius is smaller but of the same order as the edge of the limiting support of the eigenvalues, and in the edge regime when choosing the radius in the vicinity of the edge.

For the real Ginibre ensemble, the counting statistics is much more involved, as the correlations of its real and complex eigenvalues have to be considered separately, including their covariance. Therefore, we restricted ourselves to the Gaussian potential and computed the mean number of points at finite and large-$N$, as well as the variance in the large-$N$ limit, in both cases in the vicinity of the origin. In the large radius limit, leading into the bulk regime, we found a matching for both quantities with the universality class in the two other ensembles. In a small radius expansion, however, all three ensembles differ, reflecting their different repulsion of eigenvalues from the real axis.
For the bulk and edge regime, based on numerical simulations we conjectured the same universality found for the symplectic ensemble to hold for the real ensemble. It remains an open problem to prove these conjectures, and to extend our computations to the full counting statistics for all cumulants within the real ensemble. This would in principle give access to the full distribution of the number of eigenvalues in a disk of radius $R$. The large deviation regime of this distribution was studied using Coulomb gas techniques in Ref. [5]. However, one expects that this large deviation regime does not carry any information about the cumulants, which as in the GinUE case [70], are probably determined by some “intermediate deviation regime” of the distribution. Because of the intricate Pfaffian structure of the GinOE with two kinds of eigenvalues (complex and real) this remains a challenging task.

Our analysis was driven by the question of universality of the counting statistics, within the three different ensembles and for non-Gaussian deformations of the weight function. While all three Ginibre ensembles enjoy an interpretation as a static Coulomb gas in two dimensions, only for the complex Ginibre ensemble we know of a map to a quantum Hamiltonian for fermions in a rotating harmonic trap, or equivalently in the presence of a magnetic field. It remains an open question if such a map exists also for the real and symplectic Ginibre ensembles studied here, which both lead to Pfaffian point processes.

§.§ Acknowledgements

The work of Gernot Akemann was partly funded by the Deutsche Forschungsgemeinschaft (DFG) grant SFB 1283/2 2021 – 317210226.
Sung-Soo Byun was partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-51) and by the POSCO TJ Park Foundation (POSCO Science Fellowship).
This work was initiated during the Fourth ZiF Summer School in August 2022, and we wish to express our gratitude for the hospitality of the Centre for Interdisciplinary Research ZiF.

§ INTEGRABLE STRUCTURE OF THE GINOE: JPDF AND ORIGIN LIMIT

In this appendix we recall some aspects of the integrable structure of the GinOE in terms of Pfaffian determinants.
First, we give the joint density of real and complex eigenvalues of the GinOE. This allows us to define the correlation functions of real, complex or both types of eigenvalues as in (<ref>). Second, we provide the origin scaling of all three types of correlation functions as they will be needed in the computation of the number variance of the GinOE in this limit.
For more details we refer to
[17] as well as to the original literature.

A real $N\times N$ matrix from the GinOE can have $k$ real eigenvalues $x_1,\dots,x_k\in\R$ and $2l$ complex (non-real) eigenvalues that come in complex conjugate pairs, $z_1,\dots,z_l,\bar{z}_1,\dots,\bar{z}_l\in\C\setminus \R$. Because of the sum $N=k+2l$,  $k$ has to be of the same parity as $N$, i.e. both are either even or odd, from here on. Consequently, $N-k$ is always even.
For the probability $p_{N,k}$ that exactly $k$ out of $N$ eigenvalues are real, closed form expressions were given for instance in [46, 62], see also <cit.>.
Furthermore, since the eigenvalues in the lower half  plane are completely fixed by complex conjugation of those in the upper half plane, and in particular are not independent, we can restrict ourselves to $l$ complex eigenvalues in the upper half plane $\C_+$, with $\Im(z_1),\dots,\Im(z_l)>0$.
The partial joint density for such eigenvalues with given $N$ with fixed $k$ real and $l$ complex
eigenvalues was derived independently by Lehmann and Sommers [76] and Edelman [37]:

\begin{equation} \label{jpdf-kl-1}
\begin{split}
\mathbf{P}_{N,k,l}^{(1)}(x_1,\dots,x_k,z_1,\dots,z_l)
&= C_{N,k,l}\prod_{i>j}^k|x_i-x_j| \prod_{i=1}^k\prod_{j=1}^l(x_i-z_j)(x_i-\bar{z}_j)\prod_{i>j}^l|z_i-z_j|^2|z_i-\bar{z}_j|^2 \\
&\quad \times \prod_{j=1}^li^{-1}(z_j-\bar{z}_j)\prod_{j=1}^k e^{-x_j^2/2}\prod_{j=1}^l\erfc\left(\frac{z_j-\bar{z}_j}{i\sqrt{2}}\right)e^{-\frac12 (z_j^2+\bar{z}_j^2)},
\end{split}
\end{equation}

where the normalisation constant reads

\begin{equation}
\label{CNkDef}
C_{N,k,l}:= \frac{2^{l-N(N+1)/4}}{k!\,l!\prod_{j=1}^{N}\Gamma(j/2)}.
\end{equation}

Note that in the second line of (<ref>) we do not scale the exponent with $N$ (to be consistent with the references mentioned above), hence the limiting support is the unit disk with radius $\sqrt{N}$.
However for (<ref>) and (<ref>) in the main part we use a scaling with $N$.
Note also that the first line in (<ref>) together with the first factor in the second line is proportional to $|\Delta_{k+2l}(x_1,\dots,x_k,z_1,\dots,z_l,\bar{z}_1,\dots,\bar{z}_l)|$. That part of the joint density that only depends on the $z_1,\dots,z_l$ is reminiscent of the jpdf of the GinSE (<ref>). A Coulomb gas interpretation of the jpdf (<ref>) at inverse temperature $2=1/(k_B T)$ was given in [43].

The full joint density of $N$ eigenvalues $w_1,\dots,w_N\in\C_+\cup\R$, where it is not specified if they are real or complex,  is then given by summing over all sectors,

\begin{equation}
\mathcal{P}_{N}^{(1)}(w_1,\dots,w_N)=
\sum_{\substack{k=0\\N-k \text{ even}}}^N
\mathbf{P}_{N,k,(N-k)/2}^{(1)}(x_1,\dots,x_k,z_1,\dots,z_{(N-k)/2}).
\label{jpdf1}
\end{equation}

There are two equivalent ways of defining correlation functions of real, complex or mixed eigenvalues. The first introduced in [95] and [46] is to add extra source terms to the joint densities (<ref>) to define generalised partition functions

\begin{equation}
\mathbf{Z}_{N,k,l}[u,v]:= \prod_{i=1}^k \int _{\R}dx_i u(x_i) \prod_{j=1}^l \int _{\C_+}dA(z_j) v(z_j)  \mathbf{P}_{N,k,l}^{(1)}(x_1,\dots,x_k,z_1,\dots,z_l),
\end{equation}

and then to take functional derivatives of the full generalised partition function

\begin{equation}
\mathcal{Z}_N[u,v]:=
\sum_{\substack{k=0\\N-k \text{ even}}}^N
\mathbf{Z}_{N,k,(N-k)/2}[u,v]
\end{equation}

as follows:

\begin{align}
\mathbf{R}_{N,k,\R}^{(1)}(x_1,\dots,x_k)
&:= \frac{1}{\mathcal{Z}_N[1,1]}\frac{\delta^k}{\delta u(x_1) \cdots\delta u(x_k)} \left. \mathcal{Z}_N[u,1]\right|_{u=1}, \label{RkR1def}\\
\mathbf{R}_{N,l,\C}^{(1)}(z_1,\dots,z_l)
&:= \frac{1}{\mathcal{Z}_N[1,1]}\frac{\delta^k}{\delta v(z_1) \cdots\delta v(z_l)} \left. \mathcal{Z}_N[1,v]\right|_{v=1}, \label{RlC1def}\\
\mathbf{R}_{N,k,\R,l,\C}^{(1)}(x_1,\dots,x_{k},z_1,\dots,z_l)
&:= \frac{1}{\mathcal{Z}_N[1,1]}\frac{\delta^{k+l}}{\delta u(x_1) \cdots\delta u(x_k)\delta v(z_1) \cdots\delta v(z_l)} \left. \mathcal{Z}_N[u,v]\right|_{u=1,v=1}, \label{RkRlC1Def}
\end{align}

for the real-real, complex-complex, and real-complex correlation functions, respectively. Obviously, the last line contains the first, respectively second when setting $k=0$ or $l=0$, respectively.
Alternatively, the following sum over $L$ complex eigenvalues with $l\leq L\leq (N-K)/2$ and $k\leq K=N-2L$ real eigenvalues, for a given $N$ with fixed $k$  and $l$, was used in [13]

\begin{equation}
\begin{split}
\mathbf{R}_{N,k,\R,l,\C}^{(1)}(x_1,\dots,x_{k},z_1,\dots,z_l)
\sum_{\substack{K=k\\N-K \text{ even}}}^N
\sum_{L=l}^{(N-K)/2}\frac{1}{(K-k)!\,(L-l)!}\\
&\times \prod_{i=1}^{K-k} \int _{\R}dx_i\prod_{j=1}^{L-l} \int _{\C_+}dA(z_j)  \mathbf{P}_{N,K,L}^{(1)}(x_1,\dots,x_K,z_1,\dots,z_L),
\end{split}
\end{equation}

where we only give the most general correlation function.

All three types of correlation functions can be expressed as a Pfaffian determinant of a matrix valued skew-kernel. The latter is given in terms of polynomials that are skew-orthogonal with respect to a combination of an antisymmetric product on $\R^2$ and a similar product on $\C_+^2$, and we refer to [46] for details.
We will not reproduce them here as for finite-$N$ we only need the real and complex 1-point functions given in (<ref>) and (<ref>), in order to compute the mean number of eigenvalues in a (one- respectively two-dimensional) disc of radius $R$, see
Proposition <ref> and Lemma <ref> for $N$ even and odd.

Instead, we will directly present the resulting correlation functions  [46, 95, 13, 96] in the large-$N$ limit and in the origin scaling limit defined in (<ref>), (<ref>) and (<ref>), that will be used in the computation of the limiting number variance:

* GinOE complex $k$-point function in the origin limit

\begin{equation} \label{R k GinOE c}
R_{k,\C}^{(1)}(z_1,\cdots, z_k) =\prod_{j=1}^{k} (2\pi i \erfc(\sqrt{2}y_j)) \, \Pf \Big[
\begin{pmatrix}
S_\C(\bar{z}_j,\bar{z}_l)   &  S_\C(\bar{z}_j,z_l)
\\
S_\C(z_j,\bar{z}_l) & S_\C(z_j,z_l)
\end{pmatrix}  \Big]_{ j,l=1,\cdots k },
\end{equation}

\begin{equation} \label{S bulk}
S_\C(z,w) := \frac{z-w}{2\sqrt{2\pi}} e^{-(z-w)^2/2},
\end{equation}

see e.g. <cit.>.

* GinOE  real $k$-point function  in the origin limit

\begin{equation} \label{R k GinOE r}
R_{k,\R}^{(1)}(x_1,\cdots, x_k) =\Pf \Big[
\begin{pmatrix}
D_\R(x_j,x_l)   &  S_\R(x_j,x_l)
\\
-S_\R(x_l,x_j) & \tilde{I}_\R(x_j,x_l)
\end{pmatrix}  \Big]_{ j,l=1,\cdots k },
\end{equation}

\begin{equation}\label{SR def}
S_\R(x,y):= \frac{1}{ \sqrt{2\pi} } e^{-(x-y)^2/2}
\end{equation}

\begin{align}\label{DR def}
D_\R(x,y)&:= -\frac{\pa}{\pa y} S_\R(x,y) = - \frac{x-y}{\sqrt{2\pi}} e^{-(x-y)^2/2},
\\
\begin{split}
\label{IR def}
\tilde{I}_\R(x,y)&:= \int_{x}^{y} S_\R(t,y)\,dt +\frac12 \operatorname{sgn}(x-y)
= -\frac12 \erf \Big( \frac{x-y}{\sqrt{2}} \Big) +\frac12 \operatorname{sgn}(x-y), %= \frac12 \operatorname{sgn}(x-y) \erfc\Big( \frac{|x-y|}{\sqrt{2}} \Big),
\end{split}
\end{align}

see e.g. <cit.>.

* GinOE mixed real and complex $(k,l)$-point function in the origin limit [13]

\begin{equation}
\label{R k1k2 rc}
= \Pf \begin{bmatrix}
[\mathcal{K}_\R^{(1)}(x_j,x_l)]_{j,l=1}^{k} &
[\mathcal{K}_{\R,\C}^{(1)}(x_j,z_l)]_{\substack{j=1,\dots,k\\ l=1,\dots,l}} \\
\Big ( - [\mathcal{K}_{\R,\C}^{(1)}(x_j,z_l)]_{\substack{j=1,\dots,k\\ l=1,\dots,l}} \Big )^T &
\end{bmatrix}
\end{equation}

\begin{align}
&\mathcal{K}_{\C}^{(1)}(z,w):= 2\pi i (\erfc(\sqrt{2}\im z) \erfc(\sqrt{2} \im w) )^{1/2}  \begin{pmatrix}
S_\C(\bar{z},\bar{w})   &  S_\C(\bar{z},w)
\\
S_\C(z,\bar{w}) & S_\C(z,w)
\end{pmatrix},
\\
&\mathcal{K}_\R^{(1)}(x,y):= \begin{pmatrix}
D_\R(x,y)   &  S_\R(x,y)
\\
-S_\R(x,y) & \tilde{I}_\R(x,y)
\end{pmatrix},
\\
&\mathcal{K}_{\R,\C}^{(1)}(x,z) := \frac{1}{\sqrt{2}} (\erfc(\sqrt{2} \im z) )^{1/2}
\begin{pmatrix} (z-x) e^{-(x-z)^2/2} & i (\bar{z} - x) e^{-(x-\bar{z})^2/2} \\
- e^{-(x-z)^2/2} &
-i e^{-(x-\bar{z})^2/2} \end{pmatrix}.
\end{align}

Notice that in the large-$N$ limit in the bulk of the spectrum the GinOE becomes a DPP that agrees with the GinUE [13]. The same holds true for the GinSE [4]. However, as the integration domain for the computation of the number variance in the bulk limit includes the real axis, this does not automatically imply that the variances of all three ensembles agree as well.

[1]
G. Akemann, S.-S. Byun, and M. Ebke.
Universality of the number variance in rotational invariant
two-dimensional Coulomb gases.
J. Stat. Phys., 190(1):34, 2023.

[2]
G. Akemann, S.-S. Byun, and N.-G. Kang.
Scaling limits of planar symplectic ensembles.
SIGMA Symmetry Integrability Geom. Methods Appl., 18(7):40,

[3]
G. Akemann, M. Ebke, and I. Parra.
Skew-orthogonal polynomials in the complex plane and their
Bergman-like kernels.
Comm. Math. Phys., 389(1):621, 2022.

[4]
G. Akemann, M. Kieburg, A. Mielke, and T. Prosen.
Universal signature from integrability to chaos in dissipative open
quantum systems.
Phys. Rev. Lett., 123(25):254101, 2019.

[5]
R. Allez, J. Touboul, and G. Wainrib.
Index distribution of the Ginibre ensemble.
J. Phys. A, 47(4):042001, 2014.

[6]
Y. Ameur, C. Charlier, J. Cronvall, and J. Lenells.
Disk counting statistics near hard edges of random normal matrices:
the multi-component regime.
Preprint arXiv:2210.13962, 2022.

[7]
Y. Ameur, C. Charlier, J. Cronvall, and J. Lenells.
Exponential moments for disk counting statistics at the hard edge of
random normal matrices.
J. Spectr. Theory (to appear), arXiv:2207.11092, 2022.

[8]
Y. Ameur, C. Charlier, and P. Moreillon.
Eigenvalues of truncated unitary matrices: disk counting statistics.
Preprint arXiv:2305.08976, 2023.

[9]
Y. Ameur, N.-G. Kang, and N. Makarov.
Rescaling Ward identities in the random normal matrix model.
Constr. Approx., 50(1):63, 2019.

[10]
R. Bardenet, A. Feller, J. Bouttier, P. Degiovanni, A. Hardy, A. Rançon,
B. Roussel, G. Schehr, and C. I. Westbrook.
From point processes to quantum optics and back.
Preprint arXiv:2210.05522, 2022.

[11]
L. Bauwens and N. Hautsch.
Modelling financial high frequency data using point processes.
In Handbook of financial time series, pages 953–979. Springer,

[12]
F. Benaych-Georges and F. Chapon.
Random right eigenvalues of Gaussian quaternionic matrices.
Random Matrices Theory Appl., 1(2):1150009, 18, 2012.

[13]
A. Borodin and C. D. Sinclair.
The Ginibre ensemble of real random matrices and its scaling
Comm. Math. Phys., 291(1):177, 2009.

[14]
S.-S. Byun and C. Charlier.
On the characteristic polynomial of the eigenvalue moduli of random
normal matrices.
Preprint arXiv:2205.04298, 2022.

[15]
S.-S. Byun and C. Charlier.
On the almost-circular symplectic induced Ginibre ensemble.
Stud. Appl. Math., 150(1):184, 2023.

[16]
S.-S. Byun and P. J. Forrester.
Progress on the study of the Ginibre ensembles I: GinUE.
Preprint arXiv:2211.16223, 2022.

[17]
S.-S. Byun and P. J. Forrester.
Progress on the study of the Ginibre ensembles II: GinOE and
Preprint arXiv:2301.05022, 2023.

[18]
S.-S. Byun, N.-G. Kang, J. O. Lee, and J. Lee.
Real eigenvalues of elliptic random matrices.
Int. Math. Res. Not., 2023(3):2243, 2023.

[19]
S.-S. Byun, N.-G. Kang, and S.-M. Seo.
Partition functions of determinantal and Pfaffian Coulomb gases
with radially symmetric potentials.
Comm. Math. Phys., 401(2):1627, 2023.

[20]
S.-S. Byun, L. Molag, and N. Simm.
Large deviations and fluctuations of real eigenvalues of elliptic
random matrices.
Preprint arXiv:2305.02753, 2023.

[21]
P. Calabrese, P. Le Doussal, and S. N. Majumdar.
Random matrices and entanglement entropy of trapped Fermi gases.
Phys. Rev. A, 91(1):012303, 2015.

[22]
P. Calabrese, M. Mintchev, and E. Vicari.
Entanglement entropy of one-dimensional gases.
Phys. Rev. Lett., 107(2):020601, 2011.

[23]
P. Calabrese, M. Mintchev, and E. Vicari.
Exact relations between particle fluctuations and entanglement in
Fermi gases.
Europhys. Lett., 98(2):20003, 2012.

[24]
T. Can.
Random Lindblad dynamics.
J. Phys. A, 52(48):485302, 2019.

[25]
L. Charles and B. Estienne.
Entanglement entropy and Berezin–Toeplitz operators.
Comm. Math. Phys., 376(1):521, 2020.

[26]
C. Charlier.
Asymptotics of determinants with a rotation-invariant weight and
discontinuities along circles.
Adv. Math., 408:36, 2022.

[27]
C. Charlier.
Large gap asymptotics on annuli in the random normal matrix model.
Math. Ann. (Online), 2023.

[28]
C. Charlier and J. Lenells.
Exponential moments for disk counting statistics of random normal
matrices in the critical regime.
Nonlinearity, 36(3):1593, 2023.

[29]
L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr,
T. Lompe, and M. W. Zwierlein.
Quantum-gas microscope for fermionic atoms.
Phys. Rev. Lett., 114(19):193001, 2015.

[30]
G. Cipolloni, L. Erdős, and D. Schröder.
Fluctuation around the circular law for random matrices with real
Electron. J. Probab., 26:1, 2021.

[31]
G. Cipolloni, L. Erdős, and D. Schröder.
Central limit theorem for linear eigenvalue statistics of
non-Hermitian random matrices.
Comm. Pure Appl. Math., 76(5):946, 2023.

[32]
N. R. Cooper.
Quantum Hall states of ultracold atomic gases.
Many-Body Physics with Ultracold Gases: Lecture Notes of the Les
Houches Summer School: Volume 94, July 2010, 94:189, 2012.

[33]
F. D. Cunden, F. Mezzadri, and N. O'Connell.
Free fermions and the classical compact groups.
J. Stat. Phys., 171(5):768, 2018.

[34]
D. J. Daley and D. Vere-Jones.
An introduction to the theory of point processes. Vol. I.
Elementary theory and methods.
Probability and its Applications. Springer, New York, second edition,

[35]
D. S. Dean, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Noninteracting fermions at finite temperature in a d-dimensional
trap: Universal correlations.
Phys. Rev. A, 94(6):063622, 2016.

[36]
D. S. Dean, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Noninteracting fermions in a trap and random matrix theory.
J. Phys. A, 52(14):144006, 2019.

[37]
A. Edelman.
The probability that a random real Gaussian matrix has k real
eigenvalues, related distributions, and the circular law.
J. Multivar. Anal., 60(2):203, 1997.

[38]
A. Edelman, E. Kostlan, and M. Shub.
How many eigenvalues of a random matrix are real?
J. Amer. Math. Soc., 7(1):247, 1994.

[39]
M. Fenzl and G. Lambert.
Precise deviations for disk counting statistics of invariant
determinantal processes.
Int. Math. Res. Not., 2022(10):7420, 2022.

[40]
W. FitzGerald and N. Simm.
Fluctuations and correlations for products of real asymmetric random
Ann. Inst. Henri Poincare (B) Probab. Stat (to appear)
arXiv:2109.00322, 2021.

[41]
P. J. Forrester.
Log-gases and Random Matrices (LMS-34).
Princeton University Press, Princeton, 2010.

[42]
P. J. Forrester.
Diffusion processes and the asymptotic bulk gap probability for the
real Ginibre ensemble.
J. Phys. A, 48(32):324001, 2015.

[43]
P. J. Forrester.
Analogies between random matrix ensembles and the one-component
plasma in two-dimensions.
Nucl. Phys. B, 904:253–281, 2016.

[44]
P. J. Forrester.
Local central limit theorem for real eigenvalue fluctuations of
elliptic GinOE matrices.
Preprint arXiv:2305.09124, 2023.

[45]
P. J. Forrester and G. Honner.
Exact statistical properties of the zeros of complex random
J. Phys. A, 32(16):2961, 1999.

[46]
P. J. Forrester and T. Nagao.
Eigenvalue statistics of the real Ginibre ensemble.
Phys. Rev. Lett., 99(5):050603, 2007.

[47]
P. J. Forrester and T. Nagao.
Skew orthogonal polynomials and the partly symmetric real Ginibre
J. Phys. A, 41(37):375003, 19, 2008.

[48]
A. Garcia-Garcia, S. Nishigaki, and J. Verbaarschot.
Critical statistics for non-Hermitian matrices.
Phys. Rev. E, 66(1):016132, 2002.

[49]
J. Ginibre.
Statistical ensembles of complex, quaternion, and real matrices.
J. Math. Phys., 6(3):440–449, 1965.

[50]
A. Goel, P. Lopatto, and X. Xie.
Central limit theorem for the complex eigenvalues of Gaussian
random matrices.
Preprint arXiv:2306.10243, 2023.

[51]
F. Götze and J. Jalowy.
Rate of convergence to the Circular Law via smoothing
inequalities for log-potentials.
Random Matrices Theory Appl., 10(3):2150026, 25, 2021.

[52]
I. S. Gradshteyn and I. M. Ryzhik.
Table of integrals, series, and products.
Elsevier/Academic Press, Amsterdam, 2014.

[53]
A. Haimi and H. Hedenmalm.
The polyanalytic Ginibre ensembles.
J. Stat. Phys., 153(1):10, 2013.

[54]
E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and
S. Kuhr.
Single-atom imaging of fermions in a quantum-gas microscope.
Nature Physics, 11(9):738, 2015.

[55]
H. Hedenmalm and N. Makarov.
Coulomb gas ensembles and Laplacian growth.
Proc. Lond. Math. Soc. (3), 106(4):859, 2013.

[56]
J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág.
Determinantal Processes and Independence.
Probab. Surv., 3:206, 2006.

[57]
J. R. Ipsen.
Products of independent quaternion Ginibre matrices and their
correlation functions.
J. Phys. A, 46(26):265201, 16, 2013.

[58]
J. Jalowy.
Rate of convergence for products of independent non-Hermitian
random matrices.
Electron. J. Probab., 26:24, 2021.

[59]
K. Johansson.
Random matrices and determinantal processes.
Les Houches Summer School Session LXXXIII, Mathematical
Stastical Mechanics. Elsevier/Academic Press, Amsterdam, 2006.

[60]
A. Jonquière.
Note sur la série $\sum _ {n= 1}^{\infty}\frac {x^ n}{n^ s} $.
Bull. Soc. Math. France, 17:142, 1889.

[61]
E. Kanzieper.
Eigenvalue correlations in non-Hermitean symplectic random
J. Phys. A, 35(31):6631, 2002.

[62]
E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe, and O. Zaboronski.
What is the probability that a large random matrix has no real
Ann. Appl. Probab., 26(5):2733, 2016.

[63]
B. A. Khoruzhenko and S. Lysychkin.
Truncations of random symplectic unitary matrices.
Preprint arXiv:2111.02381, 2021.

[64]
B. A. Khoruzhenko and H.-J. Sommers.
Non-hermitian random matrix ensembles.
Chapter 18 in The Oxford Handbook of Random Matrix Theory. In:
Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford University Press,
Oxford, arXiv:0911.5645, 2011.

[65]
I. Klich and L. Levitov.
Quantum noise as an entanglement meter.
Phys. Rev. Lett., 102(10):100502, 2009.

[66]
P. Kopel.
Linear statistics of non-Hermitian matrices matching the real or
complex Ginibre ensemble to four moments.
Preprint arXiv:1510.02987, 2015.

[67]
A. Kulesza, B. Taskar, et al.
Determinantal point processes for machine learning.
Foundations and Trends® in Machine Learning,
5(2–3):123, 2012.

[68]
M. Kulkarni, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Density profile of noninteracting fermions in a rotating
two-dimensional trap at finite temperature.
Phys. Rev. A, 107(2):023302, 2023.

[69]
M. Kulkarni, S. N. Majumdar, and G. Schehr.
Multilayered density profile for noninteracting fermions in a
rotating two-dimensional trap.
Phys. Rev. A, 103(3):033321, 2021.

[70]
B. Lacroix-A-Chez-Toine, J. A. M. Garzón, C. S. H. Calva, I. P. Castillo,
A. Kundu, S. N. Majumdar, and G. Schehr.
Intermediate deviation regime for the full eigenvalue statistics in
the complex Ginibre ensemble.
Phys. Rev. E, 100(1):012137, 2019.

[71]
B. Lacroix-A-Chez-Toine, A. Grabsch, S. N. Majumdar, and G. Schehr.
Extremes of 2d coulomb gas: universal intermediate deviation regime.
J. Stat. Mech., 2018(1):013203, 2018.

[72]
B. Lacroix-A-Chez-Toine, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Statistics of fermions in a d-dimensional box near a hard wall.
Europhysics Letters, 120(1):10006, 2017.

[73]
B. Lacroix-A-Chez-Toine, S. N. Majumdar, and G. Schehr.
Rotating trapped fermions in two dimensions and the complex Ginibre
ensemble: Exact results for the entanglement entropy and number variance.
Phys. Rev. A, 99(2):021602, 2019.

[74]
R. Law, J. Illian, D. F. Burslem, G. Gratzer, C. Gunatilleke, and
I. Gunatilleke.
Ecological information from spatial patterns of plants: insights from
point process theory.
Journal of Ecology, 97(4):616–628, 2009.

[75]
S.-Y. Lee and R. Riser.
Fine asymptotic behavior for eigenvalues of random normal matrices:
Ellipse case.
J. Math. Phys., 57(2):023302, 29, 2016.

[76]
N. Lehmann and H.-J. Sommers.
Eigenvalue statistics of random real matrices.
Phys. Rev. Lett., 67(8):941, 1991.

[77]
M. Lewin.
Coulomb and Riesz gases: the known and the unknown.
J. Math. Phys., 63(6):77, 2022.

[78]
R. Marino, S. N. Majumdar, G. Schehr, and P. Vivo.
Phase transitions and edge scaling of number variance in Gaussian
random matrices.
Phys. Rev. Lett., 112(25):254101, 2014.

[79]
R. Marino, S. N. Majumdar, G. Schehr, and P. Vivo.
Number statistics for $\beta$-ensembles of random matrices:
applications to trapped fermions at zero temperature.
Phys. Rev. E, 94(3):032115, 2016.

[80]
M. L. Mehta.
Random Matrices, volume 142 of Pure and Applied
Elsevier/Academic Press, Amsterdam, third edition, 2004.

[81]
F. Mezzadri and H. Taylor.
A matrix model of a non-Hermitian $\beta$-ensemble.
Preprint arXiv:2305.13184, 2023.

[82]
B. Oblak, B. Lapierre, P. Moosavi, J.-M. Stéphan, and B. Estienne.
Anisotropic quantum hall droplets.
Preprint arXiv:2301.01726, 2023.

[83]
F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Editors).
NIST Handbook of Mathematical Functions.
Cambridge University Press, Cambridge, 2010.

[84]
S. O’Rourke and D. Renfrew.
Central limit theorem for linear eigenvalue statistics of elliptic
random matrices.
J. Theoret. Probab., 29:1121, 2016.

[85]
A. R. Paiva, I. Park, and J. C. Principe.
A reproducing kernel hilbert space framework for spike train signal
Neural computation, 21(2):424, 2009.

[86]
M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan,
K. Wooley-Brown, S. Blatt, and M. Greiner.
Site-resolved imaging of fermionic li 6 in an optical lattice.
Phys. Rev. Lett., 114(21):213002, 2015.

[87]
B. Rider.
Order statistics and Ginibre's ensembles.
J. Stat. Phys., 114:1139, 2004.

[88]
A. Scardicchio, C. E. Zachary, and S. Torquato.
Statistical properties of determinantal point processes in
high-dimensional Euclidean spaces.
Phys. Rev. E, 79(4):041108, 2009.

[89]
S. Serfaty.
Microscopic description of Log and Coulomb gases.
In Random matrices, volume 26 of IAS/Park City Math.
Ser., pages 341–387. Amer. Math. Soc., Providence, RI, 2019.

[90]
T. Shirai.
Large deviations for the fermion point process associated with the
exponential kernel.
J. Stat. Phys., 123(3):615, 2006.

[91]
T. Shirai.
Ginibre-type point processes and their asymptotic behavior.
J. Math. Soc. Japan, 67(2):763, 2015.

[92]
N. Simm.
Central limit theorems for the real eigenvalues of large Gaussian
random matrices.
Random Matrices Theory Appl., 6(1):1750002, 18, 2017.

[93]
N. R. Smith, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Full counting statistics for interacting trapped fermions.
SciPost Physics, 11(6):110, 2021.

[94]
N. R. Smith, P. Le Doussal, S. N. Majumdar, and G. Schehr.
Counting statistics for noninteracting fermions in a rotating trap.
Phys. Rev. A, 105(4):043315, 2022.

[95]
H.-J. Sommers.
Symplectic structure of the real Ginibre ensemble.
J. Phys. A, 40(29):F671, 2007.

[96]
H.-J. Sommers and W. Wieczorek.
General eigenvalue correlations for the real Ginibre ensemble.
J. Phys. A, 41(40):405003, 24, 2008.

[97]
H. F. Song, C. Flindt, S. Rachel, I. Klich, and K. Le Hur.
Entanglement entropy from charge statistics: Exact relations for
noninteracting many-body systems.
Phys. Rev. B, 83(16):161408, 2011.

[98]
S. Torquato.
Hyperuniform states of matter.
Phys. Rep., 745:1–95, 2018.

[99]
S. Torquato and F. H. Stillinger.
Local density fluctuations, hyperuniformity, and order metrics.
Phys. Rev. E, 68(4):041113, 2003.