Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

There is currently a wide interest in the statistics of point processes, with applications ranging from ecology [74], finance [11], signal processing [85], statistical and condensed-matter physics [99], quantum optics [10] all the way to machine learning [67]. To characterize such point processes, a useful and common tool is the so called full counting statistics (FCS). If we denote by $\mathcal{S}$ the full system and by $\Omega \subset \mathcal{S}$ a sub-system of $\mathcal{S}$, the FCS addresses the fluctuations of the total number of points $\mathcal{N}_{\Omega}$ inside the region Ω. The simplest example is probably the Poisson point processes where, in the homogeneous case, the points are just uniformly and independently distributed over $\mathcal{S}$, with a finite density ρ [34]. In this case, $\mathcal{N}_{\Omega}$ is simply a Poisson random variable of parameter $\rho |\Omega|$, where $|\Omega|$ denotes the size of the sub-domain Ω. A remarkable property of the Poisson point processes is that the mean and the variance of $\mathcal{N}_{\Omega}$ do coincide, i.e. $\mathbb{E}[\mathcal{N}_{\Omega}] = \operatorname{Var}(\mathcal{N}_\Omega)$, indicating that the fluctuations are very large here. Hence the mean does not carry so much information about the random variable $\mathcal{N}_{\Omega}$.

In fact, in many relevant systems in mathematics and physics the corresponding point processes are much more rigid, in the sense that $\operatorname{Var}(\mathcal{N}_\Omega)/\mathbb{E}[\mathcal{N}_{\Omega}] \to 0$ as $|\Omega| \to \infty$. Such systems are generically called hyperuniform [88, 99] and they have drawn a lot of attention during the last twenty years [98]. A prominent example of such hyperuniform systems is provided by determinantal point processes (DPP), which include for instance complex random matrix ensembles, non-interacting trapped fermions [35] and many others, see e.g. [56, 59] and references therein. In the context of fermions, FCS turns out to be particularly interesting since, in many cases, the variance $\operatorname{Var}(\mathcal{N}_\Omega)$ is proportional to the entanglement entropy between Ω and its complement $\overline{\Omega}$ in $\mathcal{S}$ [21–23, 65, 97]. This property is all the more interesting since entanglement entropy is very difficult to measure experimentally, whereas FCS is much more accessible for fermionic systems. In particular the recently developed quantum Fermi microscopes [29, 54, 86] allow to take 'pictures' of the positions of the fermions, in particular in two-dimensional systems where most of these experiments are carried out. In fact, it was recently realised that several non-interacting trapped fermionic systems, in one and two dimensions, are related to Gaussian random matrix ensembles with complex entries [36]. In particular, in two-dimensions, it was shown that the positions of fermions in a two-dimensional rotating harmonic trap are in one-to-one correspondence with the eigenvalues of Gaussian random matrices belonging to the complex Ginibre ensemble [73]. This physical situation is akin to the well known lowest Landau level problem of electrons in a plane and in the presence of a perpendicular magnetic field [32, 45, 48]. This connection with the physics of the lowest Landau levels has naturally motivated the study of the FCS in the complex Ginibre ensemble, denoted here as GinUE [1, 6, 7, 14, 25, 27, 28, 39, 73, 90] (for a recent review see [16]), as well as some natural extensions of it, including the higher Landau levels [68, 69, 91, 94], related to the so-called poly-analytic Ginibre ensemble [53]. We also refer to [78, 79] and references therein for earlier work on the counting statistics of Hermitian random matrix ensembles and its applications to one-dimensional systems of trapped fermions. For such models, analytical progress is possible thanks to the fact that the underlying point processes are DPPs, for which very powerful analytical tools are available, already for a finite number of points N, see [56, 59].

From a mathematical perspective it is natural to ask how robust the predictions of the complex Ginibre ensemble are, i.e. the question of their universality. There are at least two directions how this question of universality can be addressed. First, one can replace the Gaussian weight in the space of matrix elements with a more general potential, which leads to normal random matrix ensembles, see e.g. [16, section 5]. A second direction is to consider the 'cousins' of the GinUE, with real (GinOE) or quaternion matrix elements (GinSE) with independent Gaussian distributions, without further symmetry constraints, see [17]. Both ensembles lead to Pfaffian point-processes instead, which are technically much harder to study than DPP.

In this paper, we will consider both of these universality questions. Currently, it is not clear how to directly relate the ground state wave function of a Schrödinger equation with an anharmonic potential to a complex ensemble of random matrices with a non-Gaussian distribution in two dimensions. Nonetheless, there have been several examples in one dimension, see e.g. [33, 72, 93] as well as the review [36], where such a connection could be established. Note also that anharmonic potentials have recently been studied in the context of the quantum Hall effect [82]. Besides, it is not known if the Pfaffian point processes obtained for the GinOE and GinSE enjoy a direct quantum mechanical realisation. However, establishing a mathematically rigorous universality statement within all three ensembles, including normal matrices, gives rise to the expectation that the FCS might also be robust in more general quantum Hamiltonians.

It has been shown previously that the local statistics agrees for all complex eigenvalue correlation functions in all three Ginibre ensembles, away from the real line, both in the bulk and at the edge of the support of the spectrum (see e.g. [4, 13]). Here, the support is determined by the circular law on a disc of radius $\sqrt{N}$. (In the latter case, we sometimes rescale the model so that the radius of the circular law is 1.) This does not imply though, that the FCS statistics also agrees, when considering a domain Ω of macroscopic extent. Due to the (partial) rotational invariance in these three ensembles, and due to the map to a rotating trap in two-dimensions, it is most natural to choose a centred disc D_R of radius R for the domain Ω. Here, one has to distinguish three different regimes (see figure 1 for illustration): the origin regime, when the radius $R = {O}(1)$, ⁵ the bulk regime when $1\ll R\lt\sqrt{N}$, and the edge regime when $R\approx \sqrt{N}$ is in the vicinity of the edge of support. When taking the limit $R \gg 1$ contact can be made between the origin and bulk limit.

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It is useful to recall the universality results that have been obtained so far for the Ginibre ensembles. In [73], the mean, the variance and all higher order cumulants were determined for the GinUE and its extension to rotational invariant potentials $W(|z|)$, the random normal matrix ensembles in the GinUE class. (See also [8] for recent work on the case of the potential W with a hard wall.) In this case, the cumulants are given in terms of poly-logarithms for finite and infinite N. These results were extended to normal symplectic ensembles in the GinSE class for the mean and variance in [1]. It was found that when measuring the variance in units of the mean density given by the Laplacian of the potential $\Delta W(R)$ at radius R, the linear growth of the variance in the bulk is universal, i.e. the GinUE and the normal GinSE coincide. The same was found for the universal scaling function of the variance at the edge of the spectrum. (Here, we use the convention $\Delta = \partial \bar{\partial}$, a quarter of the usual Laplacian.) On the other hand, in the origin regime, the variance depends on the singularities (or zeros) of the potential W and on the repulsion from the real axis present in the GinSE class.

In this work we will address the third class, the GinOE, and complete the FCS for the GinSE. First, we analytically determine the mean and the variance in the GinOE in the origin regime. For the bulk and edge regime we formulate precise conjectures for the GinOE to be universal, and provide numerical evidence for these conjectures. We make contact between the origin and bulk regime by taking the large radius limit analytically. Because, in the GinOE, correlations between real, complex and both kind of eigenvalues have to be considered separately, the computation of the mean and the variance will occupy a substantial part of the paper. In the last part we explicitly determine all cumulants for finite and large-N in the planar symplectic ensembles with rotationally invariant potentials W in the GinSE class, and prove their universality in the bulk and edge scaling limit. Indeed, we show that they coincide with those found for the random normal matrix ensembles.

The remainder of the paper is organised as follows. In the next section 2, we briefly recall some basic properties of the three Ginibre ensembles, including their joint densities and correlation functions of complex eigenvalues (see appendix for details for the GinOE). The following main results are presented there. In section 2.1 the mean number of eigenvalues in a centred disc is given for finite N, outside the limiting support, and in the large N limit at the origin. This includes new results for the GinOE. Section 2.2 is devoted to the limiting variance. In the origin limit new expressions are given for the GinSE and GinUE. For the GinOE the main theorem 2.6 states the limiting variance at the origin for asymptotically large radius. We also discuss the issue of universality for all three ensembles there and formulate conjectures for the behaviour of the variance in the GinOE in the bulk, at the edge and outside the droplet. These conjectures are supported by numerics. Finally section 2.3 summarises our findings for the FCS in planar symplectic ensembles. The proofs for the results from section 2.1 on the mean are given in section 3. The results on the variance from section 2.2 are proved in section 4, and the universal FCS for planar symplectic ensembles is given in section 5. We conclude in section 6 and list a number of open problems.

In order to present our main results we briefly recall the Ginibre ensembles together with the quantities we need. For the GinOE more details are presented in the appendix. In terms of matrices G, the Ginibre ensembles consists of random matrices that have N² independent, centred normal real, complex or quaternion matrix elements, $g_{jk}\in\mathcal{N}_F(0,(N\beta)^{-\frac12})$, without further symmetry constraint. Here the Dyson index $\beta = 1,2,4$ labels the field $F = \mathbb{R},\mathbb{C},\mathbb{H}$, respectively.

The joint probability distribution function (jpdf) of complex eigenvalues $z_1,\dots,z_N$ of the GinUE respectively GinSE, normalised by the partition function $\mathcal{Z}_N^{(2,4)}$, reads

$$\begin{align} \mathcal{P}_N^{\left(2\right)}\left(z_1,\dots,z_N\right)& = \frac{1}{\mathcal{Z}_{N}^{\left(2\right)}} \prod_{j>k = 1}^{N} |z_j-z_k|^2 \prod_{l = 1}^N e^{-N|z_l|^2}, \end{align} \tag{ 2.1 }$$

$$\begin{align} \mathcal{P}_N^{\left(4\right)}\left(z_1,\dots,z_N\right) & = \frac{1}{\mathcal{Z}_{N}^{\left(4\right)}} \prod_{j>k = 1}^{N} |z_j-z_k|^2 |z_j-\overline{z}_k|^2 \prod_{l = 1}^N |z_l-\overline{z}_l|^2 e^{-2N|z_l|^2}. \end{align} \tag{ 2.2 }$$

For β = 4 note that an N × N quaternion valued matrix G has an infinite number of eigenvalues, however they decompose into N equivalence classes of the form $[z_j] = \{q^{-1} z_j q \mid q \in \mathbb{H} \}$ for some $z_j \in \mathbb{C}$ and $j = 1, \dots, N$, see [80] for a discussion. Furthermore it holds that $[z_j] \cap \mathbb{C} = \{z_j, \overline{z}_j\}$ and the 2N complex numbers $\{z_1, \overline{z}_1, \dots, z_N, \overline{z}_N\}$ are precisely the eigenvalues of the complex $2N \times 2N$ representation of G. In (2.2) we give the jpdf for N complex eigenvalues only, as the remaining N are fixed by complex conjugation. In principle, we could restrict the jpdf to the upper-half plane $\mathbb{C}_+: = \{z \mid z\in\mathbb{C}, \mathrm{Im}(z)\gt0\}$, because the jpdf is invariant under complex conjugation, see [64]. However, in much of the literature [2, 61, 80] the jpdf is given on the full complex plane $\mathbb{C}$, the reason being that it is not difficult to find the complex eigenvalue correlation functions in the lower half plane as well.

For the GinOE with N × N real matrices G, we have to distinguish k real and 2l (non-real) complex eigenvalues, coming in conjugated pairs. Because of $N = k+2l$, k is always of the same parity as N, otherwise the probability of finding k such real eigenvalues is zero. The corresponding jpdfs for fixed k and l are not immediately needed here and are presented in the appendix for completeness. They depend on k real and on l complex eigenvalues, with the remaining l given by complex conjugation. In contrast to β = 4 it is not straightforward to write the correlation functions in the entire complex plane, see however [95], due to sign-functions present in the jpdf. Therefore, as most authors, we restrict ourselves to the upper half plane (except for the density, see below). Notice that all three Ginibre ensembles correspond to a two-dimensional Coulomb gas at inverse temperature $(k_\mathrm{B} T)^{-1} = 2$ [41, 43, 77, 89]. In the following, we will use the notation $\beta = 1, 2, 4$ to describe respectively the GinOE, the GinUE and the GinSE, but it does not have the interpretation of an inverse temperature. This is in contrast to the Hermitian random matrices, where the Gaussian Orthogonal/Unitary/Symplectic ensembles correspond to the Coulomb gas with respective β-values of 1, 2, and 4. (See also a recent work [81] for a non-Hermitian β-ensemble.)

The k-point correlation functions at finite-N, denoted by bold face, are defined as

$$\begin{equation} \textbf{R}_{N,k}^{\left(\beta\right)}\left(z_1,\dots,z_k\right): = \frac{N!}{\left(N-k\right)!}\prod_{j = k+1}^N\int_{\mathbb{C}} \mathcal{P}_N^{\left(\beta\right)}\left(z_1,\dots,z_N\right) \,\mathrm{d}A\left(z_j\right), \end{equation} \tag{ 2.3 }$$

for $\beta = 2,4$, where $\mathrm{d}A(z) = d^2z/\pi$ is the area measure. Notice that in part of the literature the flat measure without a factor of π is used instead. For β = 1 we refer to the discussion in the appendix.

As a consequence we obtain the following determinantal, (respectively Pfaffian), point processes:

• GinUE: The N complex eigenvalues in (2.1) form a DPP with kernel K_N [49],
  $$\begin{equation} \textbf{R}_{N,k}^{\left(2\right)}\left(z_1,\dots,z_k\right) = \det \left[ K_N\left(z_j,z_l\right) \right]_{j,l = 1}^k, \end{equation} \tag{ 2.4 }$$
  where
  $$\begin{equation} K_N\left(z,w\right) = e^{-\frac{N}{2}\left(|z|^2+|w|^2\right)+Nz\bar{w}}NQ\left(N,Nz\bar{w}\right) = e^{-\frac{N}{2}\left(|z|^2+|w|^2\right)+Nz\bar{w}}N \sum_{j = 0}^{N-1} \frac{\left( N z \bar{w} \right)^j}{j! } . \end{equation} \tag{ 2.5 }$$
  Here,
  $$\begin{equation} Q\left(n+1,z\right) = \frac{\Gamma\left(n+1,z\right)}{\Gamma\left(n+1\right)} = e^{-z}e_n\left(z\right) \end{equation} \tag{ 2.6 }$$
  is the normalised incomplete Gamma-function, that relates to the incomplete exponential $e_n(z) = \sum_{j = 0}^{n}\frac{z^j}{j!}$.
• GinSE: The N complex eigenvalues in (2.2) form a Pfaffian point process [61, 80]
  $$\begin{equation} \textbf{R}_{N,k}^{\left(4\right)}\left(z_1,\ldots, z_k\right) = N^{k/2}\prod_{j = 1}^{k} \left(\bar{z}_j-z_j\right) \, {\textrm{Pf}} \left[ e^{-N|z_j|^2-N|z_l|^2} \begin{pmatrix} \kappa_N\left(z_j,z_l\right) & \kappa_N\left(z_j,\bar{z}_l\right) \\ \kappa_N\left(\bar{z}_j,z_l\right) & \kappa_N\left(\bar{z}_j,\bar{z}_l\right) \end{pmatrix} \right]_{j,l = 1,\ldots k }, \end{equation} \tag{ 2.7 }$$
  where the $2\times2$ matrix valued kernel above contains the skew-kernel
  $$\begin{equation} \kappa_N\left(z,w\right) : = N\sum_{j = 0}^{N-1}\sum_{l = 0}^{j}N^{k+l+1}\frac{z^{2j+1}w^{2l}-w^{2j+1}z^{2l}}{\left(2k+1\right)!!\,\left(2l\right)!!}. \end{equation} \tag{ 2.8 }$$

For the GinOE we instead give the densities $\textbf{R}_{N,1,\mathbb{C}}^{(1)}(z)$ and $\textbf{R}_{N,1,\mathbb{R}}^{(1)}(x)$ for the non-real complex and real eigenvalues, respectively, as examples. They were first derived by [37], respectively [38].

• GinOE: The 1-point densities of complex ($z = x+iy\in\mathbb{C}$) (respectively, real ($x\in\mathbb{R}$)) eigenvalues are
  $$\begin{align} \textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(z\right) & = \sqrt{2N\pi} N|y| \operatorname{erfc}\left(\sqrt{2N}|y|\right) e^{2Ny^2} Q\left(N-1,N|z|^2\right), \end{align} \tag{ 2.9 }$$
  $$\begin{align} \textbf{R}_{N,1,\mathbb{R}}^{\left(1\right)}\left(x\right) & = \sqrt{\frac{N}{2\pi}} Q\left(N-1,Nx^2\right) + \frac{N^{N/2} }{2^{N/2 } \Gamma\left(\frac{N}{2}\right) } e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right), \end{align} \tag{ 2.10 }$$
  where
  $$\begin{align*} \operatorname{erfc}\left(z\right) = 1- \operatorname{erf}\left(z\right) = \frac{2}{\sqrt{\pi} } \int_z^\infty e^{-t^2}\,\mathrm{d}t \end{align*}$$
  is the complementary error function and
  $$\begin{equation} P\left(n+1,z\right) = 1-Q\left(n+1,z\right) = \frac{\gamma\left(n+1,z\right)}{\Gamma\left(n+1\right)} , \qquad \gamma\left(a,z\right) = \int_0^z t^{a-1} e^{-t}\,\mathrm{d}t \end{equation} \tag{ 2.11 }$$
  is the second regularised incomplete Gamma-function. Notice that in (2.9) we defined the density of complex eigenvalues in the full complex plane $\mathbb{C}$. In general, we will also encounter mixed correlation functions, $\textbf{R}_{N,k,\mathbb{R},l,\mathbb{C}}^{(1)}$, of k real and l complex eigenvalues. For these higher order correlation functions, which are needed for the computation of the variance, it is not so obvious where to put absolute values around the imaginary parts, see however [95]. For the general structure of the Pfaffian point process on the upper half plane $\mathbb{C}_+$, and the definition of general correlation functions among real, complex, or both kind of eigenvalues, we refer to the appendix.

Notice that due to the definition (2.3) for k = 1, together with (2.9) and (2.10), the densities are normalised to N on the full complex plane, rather than to unity,

$$\begin{equation} \int_{\mathbb{C}} \textbf{R}_{N,1}^{\left(\beta = 2,4\right)}\left(z\right) \,\mathrm{d}A\left(z\right) = N = \int_{\mathbb{C}} \textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(z\right) \,\mathrm{d}A\left(z\right)+\int_{\mathbb{R}} \textbf{R}_{N,1,\mathbb{R}}^{\left(1\right)}\left(x\right) \,\mathrm{d}x. \end{equation} \tag{ 2.12 }$$

Furthermore, our normalisation is such that for $N\gg1$ the support of the (complex) densities (also called droplet) converges to the circular law with radius unity.

2.1. Expected number in a centred disc in the Ginibre ensembles

2.1.1. Finite-N Ginibre ensembles.

We denote by $E_N^{(\beta)}(a)$ the expected number of eigenvalues in the centred disc D_a of radius a. For the GinOE, we split this into the contribution of complex eigenvalues $E_{N,\mathbb{C}}^{(1)}(a)$, and write $E_{N,\mathbb{R}}^{(1)}(a)$ for the expected number of real eigenvalues in the interval $(-a,a)$. Note that the droplet has radius a = 1. In other words, we define

$$\begin{equation} E_N^{\left(\beta = 2,4\right)}\left(a\right) : = \int_{D_a} \textbf{R}_{N,1}^{\left(\beta = 2,4\right)}\left(z\right) \,\mathrm{d}A\left(z\right), \qquad E_N^{\left(1\right)}\left(a\right) : = E^{\left(1\right)}_{N, \mathbb{C}}\left(a\right)+E^{\left(1\right)}_{N,\mathbb{R}}\left(a\right), \end{equation} \tag{ 2.13 }$$

where

$$\begin{equation} E^{\left(1\right)}_{N,\mathbb{C}}\left(a\right): = \int_{D_a} \textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(z\right) \,\mathrm{d}A\left(z\right), \qquad E^{\left(1\right)}_{N,\mathbb{R}}\left(a\right) : = \int_{-a}^a \textbf{R}_{N,1,\mathbb{R}}^{\left(1\right)}\left(x\right) \,\mathrm{d}x . \end{equation} \tag{ 2.14 }$$

We first obtain analytic expressions for the expected number of eigenvalues. For this, recall that the (generalised) hypergeometric function ${}_pF_q$ is given by [83, chapter 16]

$$\begin{equation} {}_pF_q\left( a_1,\dots,a_p; b_1,\dots,b_q ; z \right) = \sum_{k = 0}^\infty \frac{\left(a_1\right)_k \dots \left(a_p\right)_k }{\left(b_1\right)_k \dots \left(b_q\right)_k } \frac{z^k}{k!}, \end{equation} \tag{ 2.15 }$$

where $(x)_k = x (x+1) \dots (x+k-1)$ is the Pochhammer symbol.

Proposition 2.1 (Expected number of eigenvalues for finite-N). For each $a \unicode{x2A7E} 0$ and $N\in\mathbb{N}$, we have the following:

• (i)  
  GinUE: We have

  $$\begin{equation} \begin{aligned} E_N^{\left(2\right)}\left(a\right) & = \sum_{k = 0}^{N-1} P\left(k+1,Na^2\right) = Na^2+N\left(1-a^2\right) \,P\left(N,Na^2\right)-\frac{\left(Na^2\right)^N\, e^{-Na^2} }{\left(N-1\right)! }. \end{aligned} \end{equation} \tag{ 2.16 }$$
• (ii)  
  GinSE: We have

  $$\begin{equation} \begin{aligned} \quad E_N^{\left(4\right)}\left(a\right) & = \sum_{k = 0}^{N-1} P\left(2k+2,2Na^2\right) = \frac12 E_{2N}^{\left(2\right)}\left(a\right)- \frac{e^{-2Na^2} }{2} \sum_{k = 0}^{N-1} \frac{\left(2Na^2\right)^{2k+1} }{\left(2k+1\right)!} \\ & = \frac12 E_{2N}^{\left(2\right)}\left(a\right)- \frac{e^{-2Na^2} }{2} \left( \sinh\left(2Na^2\right)- \frac{\left(2Na^2\right)^{2N+1} }{\left(2N+1\right)!} \, {}_1F_2\left( 1; N+1, N+\frac32 ; N^2a^4 \right) \right). \end{aligned} \end{equation} \tag{ 2.17 }$$
• (iii)  
  GinOE: We have

  $$\begin{equation} \begin{aligned} E_{N,\mathbb{C}}^{\left(1\right)}\left(a\right) & = \sum_{k = 0}^{N-2} P\left(k+1,Na^2\right) - 2 \int_0^{\sqrt{N}a } r\, e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) Q\left(N-1,r^2\right)\,\mathrm{d}r \end{aligned} \end{equation} \tag{ 2.18 }$$

  and for even N,

  $$\begin{align} \begin{aligned} E_{N,\mathbb{R}}^{\left(1\right)}\left(a\right) & = \sqrt{\frac{2}{\pi} } \frac{a\sqrt{N} \, \Gamma\left(N-1,Na^2\right)+\gamma\left(N-\tfrac12, Na^2\right) }{\left(N-2\right)! } \\ &\quad +\frac{1 }{\left(N-2\right)!! } \int_{0}^{\sqrt{N}a } e^{-x^2/2 } x^{N-1} \operatorname{erf}\left( \frac{x}{\sqrt{2} }\right)\,\mathrm{d}x \\ &\quad - \frac{1}{2^{N/2} \, \left(N-2\right)!! } \sum_{k = 1}^{N/2-1 } \frac{ \left(N+2k-3\right)!! }{2^{k-1/2} \, \left(2k-1\right)!! } P\left( \frac{N-1}{2}+k, Na^2\right). \end{aligned} \end{align} \tag{ 2.19 }$$

We emphasize that proposition 2.1 (i) and (ii) are given in [1, proposition 3.1] in a slightly different form. For N odd the corresponding expressions in proposition 2.1 (iii) are given in lemma 3.1 in the main text.

For β = 1, the $N \to \infty$ leading order asymptotic behaviours are given by

$$\begin{equation} E_{N,\mathbb{C}}^{\left(1\right)}\left(a\right) \sim N \min\left(a^2,1\right) \ ,\qquad E_{N,\mathbb{R}}^{\left(1\right)}\left(a\right) \sim \sqrt{\frac{2N}{\pi} } \min\left(a,1\right). \end{equation} \tag{ 2.20 }$$

This corresponds to the results in [38]. We also mention that the function

$$\begin{equation} \left( 1-e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) \right) Q\left(N-1,r^2\right)\ ,\quad \mbox{with}\ r = |z|, \end{equation} \tag{ 2.21 }$$

in the integrand of (2.18) corresponds to the radial density of the complex eigenvalues of the GinOE, where the angular integral has been performed.

As an application of proposition 2.1, we obtain the following corollary.

Corollary 2.2 (Expected number of eigenvalues outside the limiting support). As $N\to \infty$, we have

$$\begin{align} N-E_N^{\left(\beta = 1,2\right)}\left(1\right) & = \frac{\sqrt{N} }{\sqrt{2\pi} }+o\left(\sqrt{N}\right), \end{align} \tag{ 2.22 }$$

$$\begin{align} N- E_N^{\left(4\right)}\left(1\right) & = \frac{\sqrt{N} }{2\sqrt{\pi} }+o\left(\sqrt{N}\right), \end{align} \tag{ 2.23 }$$

For β = 1 we also have the more detailed asymptotics

$$\begin{equation} E_{N,\mathbb{C}}^{\left(1\right)}\left(1\right) = N - \frac32 \sqrt{\frac{2}{\pi} } \sqrt{N}+o\left(\sqrt{N}\right), \qquad E_{N,\mathbb{R}}^{\left(1\right)}\left(1\right) = \sqrt{\frac{2}{\pi} } \sqrt{N}+o\left(\sqrt{N}\right). \end{equation} \tag{ 2.24 }$$

Notice the difference by a factor $1/\sqrt{2}$ for β = 4 comparing (2.22) and (2.23). Let us stress that in [75], the behaviour (2.22) was shown in the more general context of the elliptic GinUE. Furthermore, for the GinUE, more precise asymptotic behaviour is available in [26], see also remark 2.15. We mention that the order $O(\sqrt{N})$ is closely related to the order of the semi-large gap probabilities, see [15, 27]. Let us also note that for the GinUE case, the optimal convergence rate of $O(N^{-1/2})$ to the circular law was established in [51]. This result was further generalised to the products of GinUE matrices as well [58].

In figure 2(A), the different contributions of real and complex eigenvalues to the expected number are plotted for β = 1. In figure 2(B), the deviation from the total number of eigenvalues N is compared in all three ensembles.

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2.1.2. Infinite Ginibre ensembles in the origin scaling limit.

The origin scaling limit is defined by rescaling all variables in the k-point correlation functions as $z_j^{\prime} = \sqrt{N}z_j$, $j = 1,\dots,k$, and in particular the radius of the disc for the FCS as $R = \sqrt{N} a$. In other words we define

$$\begin{equation} {R}_{k}^{\left(\beta = 2,4\right)}\left(z_1^{\prime},\dots,z_k^{\prime}\right): = \lim_{N\to\infty}N^{-k}\ \textbf{R}_{N,k}^{\left(\beta = 2,4\right)}\left(z_1 = z_1^{\prime}/\sqrt{N},\dots,z_k = z_k^{\prime}/\sqrt{N}\right). \end{equation} \tag{ 2.25 }$$

For the GinOE we have to distinguish the different dimensions of the eigenvalues:

$$\begin{align} {R}_{k,\mathbb{C}}^{\left(1\right)}\left(z_1^{\prime},\dots,z_k^{\prime}\right)&: = \lim_{N\to\infty}N^{-k}\ \textbf{R}_{N,k,\mathbb{C}}^{\left(1\right)}\left(z_1 = z_1^{\prime}/\sqrt{N},\dots,z_k = z_k^{\prime}/\sqrt{N}\right), \end{align} \tag{ 2.26 }$$

$$\begin{align} {R}_{k,\mathbb{R}}^{\left(1\right)}\left(x_1^{\prime},\dots,x_k^{\prime}\right)&: = \lim_{N\to\infty}N^{-k/2}\ \textbf{R}_{N,k,\mathbb{R}}^{\left(1\right)}\left(x_1 = x_1^{\prime}/\sqrt{N},\dots,x_k = x_k^{\prime}/\sqrt{N}\right). \end{align} \tag{ 2.27 }$$

Likewise, for the mixed correlation functions $\textbf{R}_{N,k,\mathbb{R},l,\mathbb{C}}^{(1)}$ defined in the appendix, we have for the origin scaling limit for k = 1 real and l = 1 complex eigenvalue

$$\begin{equation} {R}_{1,\mathbb{R},1,\mathbb{C}}^{\left(1\right)}\left(x_1^{\prime},z_1^{\prime}\right): = \lim_{N\to\infty}N^{-\frac{1}{2}-1}\ \textbf{R}_{N,1,\mathbb{R},1,\mathbb{C}}^{\left(1\right)}\left(x_1 = x_1^{\prime}/\sqrt{N},z_1 = z_1^{\prime}/\sqrt{N}\right). \end{equation} \tag{ 2.28 }$$

Consequently, we define the limiting mean at the origin as

$$\begin{equation} E^{\left(\beta = 2,4\right)}\left(R\right) : = \int_{D_R} R_1^{\left(\beta = 2,4\right)}\left(z\right) \,\mathrm{d}A\left(z\right), \qquad E^{\left(1\right)}\left(R\right) : = E^{\left(1\right)}_{\mathbb{C}}\left(R\right)+E^{\left(1\right)}_{\mathbb{R}}\left(R\right), \end{equation} \tag{ 2.29 }$$

where

$$\begin{equation} E^{\left(1\right)}_{\mathbb{C}}\left(R\right): = \int_{D_R} R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) \,\mathrm{d}A\left(z\right), \qquad E^{\left(1\right)}_{\mathbb{R}}\left(R\right) : = \int_{-R}^R R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) \,\mathrm{d}x . \end{equation} \tag{ 2.30 }$$

These have the interpretation as the expected number of eigenvalues in a disc of radius R. Furthermore, for the radial densities of complex eigenvalues on $\mathbb{C}$ we write

$$\begin{equation} \widehat{R}_1^{\left(\beta = 2,4\right)}\left(r\right): = \frac{1}{2\pi} \int_0^{2\pi} R_1^{\left(\beta = 2,4\right)} \left( r e^{i\theta} \right)\,d \theta, \qquad \widehat{R}_{1,\mathbb{C}}^{\left(1\right)}\left(r\right): = \frac{1}{2\pi} \int_0^{2\pi} R_{1,\mathbb{C}}^{\left(1\right)} \left( r e^{i\theta} \right)\,d \theta. \end{equation} \tag{ 2.31 }$$

Proposition 2.3 (Limiting mean in the origin scaling limit). We have

$$\begin{align} &E^{\left(2\right)}\left(R\right) = R^2, \end{align} \tag{ 2.32 }$$

$$\begin{align} & E^{\left(4\right)}\left(R\right) = R^2-\frac14+\frac14 e^{-4R^2}, \end{align} \tag{ 2.33 }$$

$$\begin{align} &E^{\left(1\right)}\left(R\right) = R^2 +\frac12- \frac12 e^{2R^2 } \operatorname{erfc}\left(\sqrt{2}R\right). \end{align} \tag{ 2.34 }$$

In particular, we have

$$\begin{equation} E^{\left(1\right)}\left(R\right) = E^{\left(1\right)}_{\mathbb{C}}\left(R\right)+ E^{\left(1\right)}_{\mathbb{R}}\left(R\right), \end{equation} \tag{ 2.35 }$$

where

$$\begin{equation} E_{\mathbb{C}}^{\left(1\right)}\left(R\right) = R^2-\sqrt{\frac{2}{\pi} } R +\frac12- \frac12 e^{2R^2 } \operatorname{erfc}\left(\sqrt{2}R\right), \qquad E_{\mathbb{R}}^{\left(1\right)}\left(R\right) = \sqrt{\frac{2}{\pi} } R . \end{equation} \tag{ 2.36 }$$

Furthermore, the respective radial densities read

$$\begin{equation} \widehat{R}_1^{\left(2\right)}\left(r\right) = 1, \qquad \widehat{R}_1^{\left(4\right)}\left(r\right) = 1-e^{-4r^2}, \qquad \widehat{R}_{1,\mathbb{C}}^{\left(1\right)}\left(r\right) = 1-e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right). \end{equation} \tag{ 2.37 }$$

The expansion of the exact results from proposition 2.3 in the limits R → 0 and $R\to\infty$ is collected in table 1. Recalling the rescaling of the origin limit with $R^2 = Na^2$, the large-R limit makes contact with the bulk regime where $0\lt a\lt1$. Note that all three ensembles show a R²-behaviour in the large R limit, however $E(R) = R^2$ is exact only for the GinUE. Furthermore, close to the origin, we observe an R⁴-behaviour for the GinSE, which arises from the repulsive effect of the real axis. The linear behaviour for the GinOE stems from the real eigenvalues, while the number of complex eigenvalues in (2.36) reveals a cubic behaviour

$$\begin{equation} E^{\left(1\right)}_{\mathbb{C}}\left(R\right) = \sqrt{\frac{2}{\pi} } \frac{4}{3} R^3 + O\left(R^5\right), \qquad R \to 0. \end{equation} \tag{ 2.38 }$$

Figure 3 illustrates the expected number through plots for all three ensembles.

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Table 1. Comparison of the exact results for the expected mean number and its expansion for the small and large R limit for all three Ginibre ensembles.

	GinOE	GinUE	GinSE
$E^{(\beta)}(R)$	$R^2 +\dfrac12- \dfrac12 e^{2R^2 } \operatorname{erfc}(\sqrt{2}R)$	R2	$R^2-\dfrac14+\dfrac14 e^{-4R^2}$
R → 0	$\sqrt{\dfrac{2}{\pi} } R+O(R^3)$	R2	$2R^4+O(R^6)$
$R \to \infty$	$R^2 +\dfrac12+O\Big(\dfrac{1}{R}\Big)$	R2	$R^2-\dfrac14 +O(e^{-4R^2})$

2.2. Number variance of the infinite Ginibre ensembles

In this subsection we will only consider the large-N limit for the number variance.

2.2.1. Infinite Ginibre ensembles in the origin scaling limit.

Similarly to (2.29) and (2.30) we define the limiting variance in the origin limit as

$$\begin{equation} \begin{aligned} \operatorname{Var} \, \mathcal{N}^{\left(\beta = 2,4\right)}\left( D_R \right) &: = \int_{D_R } R_{1}^{\left(\beta = 2,4\right)}\left(z\right)\,\mathrm{d}A\left(z\right) \\ &\quad +\int_{D_R^2 } \left( R_{2}^{\left(\beta = 2,4\right)} \left(z,w\right)-R_{1}^{\left(\beta = 2,4\right)}\left(z\right) R_{1}^{\left(\beta = 2,4\right)}\left(w\right) \right) \,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right). \end{aligned} \end{equation} \tag{ 2.39 }$$

Likewise, for β = 1, we define

$$\begin{equation} \operatorname{Var} \mathcal{N}^{\left(1\right)}\left(D_R\right) : = \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)} \left(D_R\right) + 2\,\operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) + \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right), \end{equation} \tag{ 2.40 }$$

where

$$\begin{align} &\operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) : = \int_{-R}^R R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right)\,\mathrm{d}x +\int_{\left(-R,R\right)^2} \left(R_{2,\mathbb{R}}^{\left(1\right)}\left(x,y\right)-R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) R_{1,\mathbb{R}}^{\left(1\right)}\left(y\right)\right)\, \mathrm{d}x\,\mathrm{d}y, \end{align} \tag{ 2.41 }$$

$$\begin{align} &\operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) : = \int_{D_R } R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right)\,\mathrm{d}A\left(z\right) +\int_{D_R^2 }\left( R_{2,\mathbb{C}}^{\left(1\right)} \left(z,w\right)-R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(w\right) \right)\,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right), \end{align} \tag{ 2.42 }$$

$$\begin{align} &\operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) : = \int_{D_R} \int_{-R}^R \left( R_{1,\mathbb{R},1,\mathbb{C}}^{\left(1\right)}\left(x,z\right)- R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) \right) \,\mathrm{d}x \,\mathrm{d}A\left(z\right). \end{align} \tag{ 2.43 }$$

In the proof of proposition 4.3 we will explain how these expressions are determined through integrals over the upper half plane only, where the limiting correlations functions given in the appendix are used.

Let us first recall some special functions that will be used to express the number variance. The modified Bessel function I_ν of the first kind is given by [83, chapter 10]

$$\begin{equation} I_\nu\left(z\right): = \sum_{k = 0}^{\infty} \frac{\left(z/2\right)^{2k+\nu}}{k! \,\Gamma\left(k+\nu+1\right)}. \end{equation} \tag{ 2.44 }$$

Proposition 2.4 (Number variance of the infinite GinUE and GinSE in the origin limit). We have the following equivalent expressions for the limiting variance at the origin for $\beta = 2,4$.

• (i)  
  GinUE: We have

  $$\begin{align} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) & = \sum_{j = 1}^\infty P\left(j,R^2\right) Q\left(j,R^2\right) \end{align} \tag{ 2.45 }$$

  $$\begin{align} & = R^2 e^{-2 R^2} \left( I_0\left(2 R^2\right) + I_1\left(2 R^2\right) \right) , \end{align} \tag{ 2.46 }$$

  with the following limiting expressions:

  $$\begin{align} &\operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) \sim \dfrac{R}{\sqrt{\pi}}, \qquad R \to \infty, \end{align} \tag{ 2.47 }$$

  $$\begin{align} & \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) \sim R^2, \qquad R \to 0. \end{align} \tag{ 2.48 }$$
• (ii)  
  GinSE: We have

  $$\begin{align} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = \sum_{k = 1}^\infty P\left(2k,2R^2\right) Q\left(2k,2R^2\right) \end{align} \tag{ 2.49 }$$

  $$\begin{align} & = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) - {}_1 F_2 \left( 1/2; 1,3/2; -4 R^4 \right) \right) \end{align} \tag{ 2.50 }$$

  with the limiting expressions:

  $$\begin{align} & \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) \sim \dfrac{R }{\sqrt{2\pi} }, \qquad R \to \infty, \end{align} \tag{ 2.51 }$$

  $$\begin{align} & \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) \sim 2 R^4, \qquad R \to 0. \end{align} \tag{ 2.52 }$$

The first expression (2.45) was derived in [73], the second expression (2.46) in [90, theorem 1.3], where it is written as

$$\begin{equation} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) = \frac{R}{\pi} \int_0^{4R^2} \left(1-\frac{x}{4R^2}\right)^{1/2} x^{-1/2} e^{-x} \,\mathrm{d}x. \end{equation} \tag{ 2.53 }$$

We will present an alternative derivation of this result compared to [90]. The expression (2.46) is equivalent to (2.53) due to [52, 3.364.1 and 3.366.1].

The expression (2.49) for β = 4 was shown in [1] as a large-N limit of the number variance (2.94) of the finite symplectic Ginibre ensemble in the origin regime. The second expression (2.50) is new and can be rewritten as

$$\begin{align} & \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right)\nonumber\\ &\quad = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) - J_0\left(4R^2\right) \frac{\textbf{H}_{-1}\left(4R^2\right)}{2/\pi}-J_{1}\left(4R^2\right) \frac{\textbf{H}_0\left(4R^2\right)}{2/\pi} \right) , \end{align} \tag{ 2.54 }$$

where

$$\begin{equation} J_\nu\left(z\right) = \sum_{k = 0}^\infty \left(-1\right)^k \frac{\left(z/2\right)^{2k+\nu}}{k! \Gamma\left(k+\nu+1\right) } \end{equation} \tag{ 2.55 }$$

is the Bessel function of the first kind and

$$\begin{equation} \textbf{H}_\nu\left(z\right) = \sum_{k = 0}^\infty \left(-1\right)^k \frac{\left(z/2\right)^{2k+\nu+1} }{\Gamma\left(k+\frac32\right) \Gamma\left(\nu+k+\frac32\right) } \end{equation} \tag{ 2.56 }$$

is the Struve function [83, Chapters 10, 11].

Remark 2.5. Let $c(R): = R^2 e^{-2 R^2} ( I_0(2 R^2) + I_1(2 R^2) )$. Then we have

$$\begin{equation} c^{^{\prime}}\left(R\right) = 2 R e^{-2R^2} I_0\left(2R^2\right), \qquad c\left(R\right) = \frac{1}{8} c^{^{\prime\prime}}\left(R\right)+\left( R-\frac{1}{8R} \right) c^{^{\prime}}\left(R\right), \end{equation} \tag{ 2.57 }$$

which follows from $I_0^{\prime}(x) = I_1(x)$ and $xI_1^{\prime}(x) = xI_0(x)-I_1(x)$. We mention that the function c(R) also appears in several different contexts on the studies of the Ginibre ensembles, see e.g. [24, proposition 1] and [18, theorem 2.1].

For the limiting variance and covariance of the GinOE at the origin we give explicit expressions as functions of R in proposition 4.3, containing up to two-fold integrals. Here, let us state just their asymptotic behaviour.

Theorem 2.6 (Asymptotic number variance of the infinite GinOE in the origin limit). As $R \to \infty$, we have

$$\begin{align} &\operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \sim \frac{2\sqrt{2}-2 }{\sqrt{\pi}} R, \end{align} \tag{ 2.58 }$$

$$\begin{align} & \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) \sim \frac{2\sqrt{2} }{\sqrt{\pi} } R, \end{align} \tag{ 2.59 }$$

$$\begin{align} &\operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) \sim -\frac{2\sqrt{2}-2 }{\sqrt{\pi}} R . \end{align} \tag{ 2.60 }$$

In particular, put together we have

$$\begin{equation} \operatorname{Var} \mathcal{N}^{\left(1\right)}\left(D_R\right) \sim \frac{2}{\sqrt{\pi}} R. \end{equation} \tag{ 2.61 }$$

Figure 4 illustrates the variance for the three ensembles.

Remark 2.7. The negative covariance between real and complex eigenvalues can be intuitively understood as follows. Looking at the joint density (A.1), it vanishes both when a complex eigenvalue (pair) becomes real, from the factor in the second line, and when it comes close to a real eigenvalue, from the two factors in the first line. The same negative correlation can also be expected on a larger scale away from the origin, e.g. from the expected number outside the limiting support in corollary 2.2, equation (2.24): Whenever the number of real eigenvalues increases (by an $O(\sqrt{N})$), the number of complex eigenvalues has to decrease by the same amount, in order to keep the total balance in (2.22). This is in agreement with the total variance vanishing in conjecture 2.10.

Remark 2.8. By proposition 2.3 we have that as R → 0,

$$\begin{equation} E^{\left(4\right)}\left(R\right) \sim 2R^4, \qquad E_{\mathbb{R}}^{\left(1\right)}\left(R\right) \sim \sqrt{\frac{2}{\pi} } R. \end{equation} \tag{ 2.62 }$$

On the one hand, it follows from (2.52) and (4.46) that as R → 0,

$$\begin{equation} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) \sim 2 R^4, \qquad \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \sim \sqrt{\frac{2}{\pi} } R. \end{equation} \tag{ 2.63 }$$

Thus, one can observe the behaviours:

$$\begin{equation} \lim_{R \to 0} \frac{\operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right)}{E^{\left(4\right)}\left(R\right)} = \lim_{R \to 0} \frac{\operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right)}{E_{\mathbb{R}}^{\left(1\right)}\left(R\right)} = 1, \end{equation} \tag{ 2.64 }$$

which is a characteristic feature of the Poisson point processes. Indeed, for the GinUE and GinSE, one can show that not only the variance but also all the higher cumulants are asymptotically the same as their mean value in this regime. This follows from the asymptotic results on the FCS (see [73] for the GinUE case and theorem 2.14 for the GinSE case)—see the remark 2.18 below. Furthermore, the above asymptotic behaviour again indicates the possibility of Poissonian statistics also for the GinOE in the origin limit.

Remark 2.9. By the asymptotic behaviours (2.47), (2.51) and (2.61), we have for all three Ginibre ensembles the following universal behaviour in the bulk of the spectrum

$$\begin{equation} \operatorname{Var} \, \mathcal{N}^{\left(\beta\right)}\left( D_R\right) \sim \frac{R}{\sqrt{\pi}} \frac{2}{\mathfrak{c}\left(\beta\right)} \qquad R \to \infty, \end{equation} \tag{ 2.65 }$$

where $\mathfrak{c}(\beta)$ is given by

$$\begin{equation} \mathfrak{c}\left(\beta\right) = \begin{cases} 1 & \beta = 1, \\ 2 & \beta = 2, \\ 2\sqrt{2} & \beta = 4. \end{cases} \end{equation} \tag{ 2.66 }$$

By comparison with remark 2.11 below one can notice that this asymptotic formula is consistent with the bulk scaling limit (2.74), with $a = R/\sqrt{N}$.

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2.2.2. Conjectured bulk and edge behaviour for the GinOE.

In this subsection we consider again the initial definition of the Ginibre ensembles at the beginning of this section 2, that is without rescaling the distances as in the origin limit. Hence, the limiting support of the spectrum is the centred disc D_a with radius a = 1. Let $\mathcal{N}_a$ be the number of eigenvalues in D_a . For the finite GinOE it can be written as

$$\begin{equation} \mathcal{N}_a = \mathcal{N}_a^{\mathbb{C}} + \mathcal{N}_a^{\mathbb{R}}, \end{equation} \tag{ 2.67 }$$

where $\mathcal{N}_a^{\mathbb{C}}$ (resp., $\mathcal{N}_a^{\mathbb{R}}$) is the number of complex (resp. real) eigenvalues.

We mention that the counting statistics of real eigenvalues of the (elliptic) GinOE have been studied in several recent works [18, 20, 40, 42, 44, 92]. Additionally, a very recent work [50] has also investigated the counting statistics of complex eigenvalues. We also refer the reader to [30, 31, 66, 84] and references therein for the study of fluctuations of the linear statistics.

For the GinUE and GinSE the limiting variance $\operatorname{Var} \mathcal{N}_a^{(\beta = 2,4)}$ is known in the bulk ($0\lt a\lt1$) and edge scaling limits. It was shown to be universal for non-Gaussian rotational invariant potentials in [73] and [1] respectively, see remark 2.11 below.

For the GinOE, as in (2.40) we have

$$\begin{equation} \operatorname{Var} \mathcal{N}_a^{\left(1\right)} = \operatorname{Var} \mathcal{N}_a^{\mathbb{C}}+ \operatorname{Var} \mathcal{N}_a^{\mathbb{R}} + 2 \operatorname{Cov}\left( \mathcal{N}_a^{\mathbb{C}}, \mathcal{N}_a^{\mathbb{R}} \right). \end{equation} \tag{ 2.68 }$$

Then we have the following conjecture for the scaling limits of each contribution to the different scaling regimes:

Conjecture 2.10 (GinOE number variance scaling limits). We expect the following as $N \to \infty$.

• Bulk: For $a \in (0,1)$ fixed, we have
  $$\begin{align} \frac{\operatorname{Var} \mathcal{N}_a^{\mathbb{C}} }{\sqrt{N/\pi} } \sim 2\sqrt{2} \, a , \quad \frac{\operatorname{Var} \mathcal{N}_a^{\mathbb{R}} }{\sqrt{N/\pi} } \sim \left(2\sqrt{2}-2\right) a, \quad 2 \dfrac{ \operatorname{Cov}\left( \mathcal{N}_a^{\mathbb{C}}, \mathcal{N}_a^{\mathbb{R}} \right) }{ \sqrt{N/\pi} } \sim -2\left(2\sqrt{2}-2\right) \, a. \end{align} \tag{ 2.69 }$$
  In particular, for the total variance we obtain
  $$\begin{equation} \frac{\operatorname{Var} \mathcal{N}_a^{\left(1\right)} }{\sqrt{N/\pi} } \sim 2\, a . \end{equation} \tag{ 2.70 }$$
• Edge: In the scaling limit $a = 1-\mathcal{S}/\sqrt{2N}$, we have
  $$\begin{equation} \frac{\operatorname{Var} \mathcal{N}_a^{\left(1\right)} }{\sqrt{N/\pi} } \sim 2 f\left(\mathcal{S}\right) , \qquad f\left(\mathcal{S}\right) : = \sqrt{2\pi } \int_{-\infty}^{\mathcal{S}} \frac{\operatorname{erfc}\left(t\right)\operatorname{erfc}\left(-t\right)}{4} \,\mathrm{d}t. \end{equation} \tag{ 2.71 }$$
• Outside: For a > 1 fixed, we have
  $$\begin{equation} \frac{\operatorname{Var} \mathcal{N}_a^{\mathbb{C}}}{\sqrt{N/\pi}} \sim 2\sqrt{2}-2,\qquad \frac{\operatorname{Var} \mathcal{N}_a^{\mathbb{R}}}{\sqrt{N/\pi}} \sim 2 \sqrt{2}-2, \qquad 2\frac{\operatorname{Cov}\left( \mathcal{N}_a^{\mathbb{C}}, \mathcal{N}_a^{\mathbb{R}} \right) }{\sqrt{N/\pi} } \sim -2\left(2\sqrt{2}-2\right), \end{equation} \tag{ 2.72 }$$
  leading to the vanishing of the total variance
  $$\begin{equation} \frac{\operatorname{Var} \mathcal{N}_a^{\left(1\right)} }{\sqrt{N/\pi} } \sim 0. \end{equation} \tag{ 2.73 }$$

Let us mention that the bulk part of conjecture 2.10 is supported by theorem 2.6 and the outside part by [46], these are also summarised in table 2. All parts of our conjecture are supported by numerical simulations, see figure 5.

Remark 2.11 (Bulk and edge universality). Conjecture 2.10 suggests that the number variance for all three Ginibre ensembles is universal in the bulk and at the edge and takes the form

$$\begin{equation} \mathfrak{c}\left(\beta\right) \frac{\operatorname{Var} \mathcal{N}_a^{\left(\beta\right)} }{\sqrt{N/\pi} } \sim \begin{cases} \displaystyle 2 a & a \in \left(0,1\right)\quad \mbox{(bulk)}, \\ 2 f\left(\mathcal{S}\right) & a = 1-\mathcal{S}/\sqrt{2N} \quad \mbox{(edge),} \end{cases} \end{equation} \tag{ 2.74 }$$

where $\mathfrak{c}(\beta)$ is given in (2.66). The variance formula for the GinUE $(\beta = 2)$ and GinSE $(\beta = 4)$ can be found in [1, theorem 1.2] for more general potentials $W(|z|)$, where the variance and scaling variable $\mathcal{S}$ have to be rescaled by $1/\sqrt{\Delta W(a)}$, the inverse mean level spacing, to preserve the universal form. More detailed asymptotic behaviours for these cases can be found in [14, 26].

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Table 2. Leading order asymptotic behaviour of the individual contributions to the number variance in the GinOE in the bulk with fixed $0\lt a\lt1$ and outside the support for a > 1.

	$\dfrac{\operatorname{Var} \mathcal{N}_a^{\mathbb{C}}}{\sqrt{N/\pi}}$	$\dfrac{\operatorname{Var} \mathcal{N}_a^{\mathbb{R}}}{\sqrt{N/\pi}}$	$2\,\dfrac{ \operatorname{Cov}( \mathcal{N}_a^{\mathbb{C}}, \mathcal{N}_a^{\mathbb{R}} ) }{ \sqrt{N/\pi} }$	$\dfrac{\operatorname{Var} \mathcal{N}_a^{(1)}}{\sqrt{N/\pi}}$
$a\in (0,1)$	$2\sqrt{2} \, a$	$(2\sqrt{2}-2)\,a$	$-2(2\sqrt{2}-2)\, a$	$2\, a$
a > 1	$2\sqrt{2}-2$	$2\sqrt{2}-2$	$-2(2\sqrt{2}-2)$	0

2.3. FCS of planar symplectic ensembles

For comparison we present both the FCS for the non-Gaussian extension of the GinUE and GinSE class, the so-called planar ensembles. The determinantal Coulomb gas ensemble (random normal matrix ensemble) is defined by the joint density (2.1), replacing the Gaussian potential $|z|^2$ in the exponent by a rotationally invariant external potential, $W \equiv W_N : \mathbb{C} \to \mathbb{R}$ which possibly depends on N. Its integrable structure (2.4) remains valid, with the kernel given in terms of planar orthogonal polynomials with respect to e^−NW , see [16].

Likewise, the same generalisation is made in the Pfaffian Coulomb gas ensemble based on the joint density (2.2), called planar symplectic ensembles. Also here, the integrable structure (2.7) remains, with the corresponding skew-kernel (2.8) given in terms of planar skew-orthogonal polynomials, see [17].

Following [1], we shall use the following terminology.

Definition 2.12. A rotationally invariant potential $W_N(z) = g_N(|z|)$ will be called suitable, if it satisfies the following conditions:

• $g_N(r) \gg \log r$ as $r \to \infty$;
• $g_N \in C^2(0,1]$;
• $r g^{^{\prime}}_N(r)$ increases on $(0,\infty)$;
• $\lim\limits_{r \to 0} r g_N^{^{\prime}}(r) = 0$, and $\lim_{N \to \infty}g^{^{\prime}}_N(1) = 4$.

Note that condition (3) is equivalent to $\Delta W_N(z) \gt 0$ for z ≠ 0 (here, $\Delta = \partial_z \overline{\partial_z} = \frac14 (\partial_x^2+\partial_y^2)$ for $z = x+iy$ is one-quarter of the usual Laplacian) and condition (4) implies that the limiting droplet is the unit disk.

Let us write

$$\begin{equation} h_j: = \int_{\mathbb{C}} e^{-N W_N\left(z\right)}\,|z|^{2j}\,\mathrm{d}A\left(z\right) = 2 \int_0^\infty e^{-N g_N\left(r\right)}\,r^{2j+1}\,\mathrm{d}r, \end{equation} \tag{ 2.75 }$$

for the non-vanishing jth moments, which are at the same time the squared norms of planar orthogonal polynomials (monomials). We also write

$$\begin{align} \begin{aligned} h_{j,1}\left(a\right)&: = \int_{|z| < a } e^{-N W_N\left(z\right)} |z|^{2j}\,\mathrm{d}A\left(z\right) = 2 \int_0^a e^{-N g_N\left(r\right)} r^{2j+1} \,\mathrm{d}r, \\ h_{j,2}\left(a\right)&: = \int_{|z| > a } e^{-N W_N\left(z\right)} |z|^{2j}\,\mathrm{d}A\left(z\right) = 2 \int_a^\infty e^{-N g_N\left(r\right)} r^{2j+1} \,\mathrm{d}r \end{aligned} \end{align} \tag{ 2.76 }$$

for the truncated moments or squared norms. We also denote

$$\begin{equation} \mathcal{L}_{j}\left(a\right) = \frac{h_{j,1}\left(a\right) }{h_j }, \qquad \mathcal{M}_{j}\left(a\right) = \frac{h_{j,2}\left(a\right) }{h_j }. \end{equation} \tag{ 2.77 }$$

Let us recall that $\mathcal{N}_a : = \# \{z_j: |z_j| \unicode{x2A7D} a \}$ denotes the number of particles in the centred disc D_a of radius $a.$ Here,

$$\begin{equation} \operatorname{Li}_s\left(x\right) = \sum_{k = 1}^{\infty} k^{-s} x^k, \quad |x|<1, \end{equation} \tag{ 2.78 }$$

is the definition of the polylogarithm function, that also converges for $|x| = 1$ and $\mathrm{Re}(s)\gt1$. It is given by analytic continuation otherwise.

Proposition 2.13 (Finite-N expressions of the cumulant generating functions). Let W be a suitable potential, then we have the following for $a \unicode{x2A7E} 0$ and $N \in \mathbb{N}$.

• (i)  
  (Random normal matrix ensembles). We have for the moment generating function

  $$\begin{equation} \mathbb{E}_{N,W}^{\left(\beta = 2\right)}\left[ e^{u \mathcal{N}_a } \right] = \prod_{j = 0}^{N-1} \left(e^{u} \mathcal{L}_{j}\left(a\right)+ \mathcal{M}_{j}\left(a\right) \right) . \end{equation} \tag{ 2.79 }$$

  Furthermore, for any $p \unicode{x2A7E} 2$, the pth cumulant of $\mathcal{N}_a$ has the expression

  $$\begin{equation} \kappa_p^{\left(\beta = 2\right)}\left(a\right) = \left(-1\right)^{p+1} \sum_{j = 0}^{N-1} \operatorname{Li}_{1 - p}\left( 1 - \frac{1}{\mathcal{L}_j\left(a\right)} \right). \end{equation} \tag{ 2.80 }$$

  We also have for the mean

  $$\begin{equation} E_{N,W}^{\left(\beta = 2\right)}\left(a\right): = \mathbb{E}_{N,W}^{\left(\beta = 2\right)}\left[\mathcal{N}_a\right] = \sum_{j = 0}^{N-1} \mathcal{L}_j\left(a\right). \end{equation} \tag{ 2.81 }$$
• (ii)  
  (Planar symplectic ensembles). We have

  $$\begin{equation} \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a } \right] = \prod_{j = 0}^{N-1} \left( e^{u}\mathcal{L}_{2j+1}\left(a\right)+ \mathcal{M}_{2j+1}\left(a\right) \right). \end{equation} \tag{ 2.82 }$$

  Furthermore, for any $p \unicode{x2A7E} 2$, the pth cumulant of $\mathcal{N}_a$ has the expression

  $$\begin{equation} \kappa_p^{\left(\beta = 4\right)}\left(a\right) = \left(-1\right)^{p+1} \sum_{j = 0}^{N-1} \operatorname{Li}_{1 - p}\left( 1 - \frac{1}{\mathcal{L}_{2j+1}\left(a\right)} \right). \end{equation} \tag{ 2.83 }$$

  For the mean, we have

  $$\begin{equation} E_{N,W}^{\left(\beta = 4\right)}\left(a\right): = \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[\mathcal{N}_a\right] = \sum_{j = 0}^{N-1} \mathcal{L}_{2j+1}\left(a\right). \end{equation} \tag{ 2.84 }$$

Note that proposition 2.13 (i) was shown in [73].

Theorem 2.14 (FCS of the planar symplectic ensembles). Let W be a suitable potential. Then as $N \to \infty$, the following holds.

• (i)  
  ${\mathbf{Bulk:}}$ For $a\in (0,1)$ fixed and $p \unicode{x2A7E} 2$, we have

  $$\begin{equation} \lim_{N \to \infty} \sqrt{\frac{2}{N \Delta W\left(a\right)} } \, \kappa_p^{\left(\beta = 4\right)}\left(a\right) = a\,\kappa_p^{\textrm{bulk}}, \end{equation} \tag{ 2.85 }$$

  where

  $$\begin{equation} \begin{aligned} {\kappa_{p}^{\mathrm{bulk}}} : = &\, \left(-1\right)^{p + 1} {\int_{-\infty}^{\infty}} \operatorname{Li}_{1 - p}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) \, \mathrm{d}x \\ = & \begin{cases} \displaystyle - {\int_{-\infty}^{\infty}} \operatorname{Li}_{1 - p}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) \, \mathrm{d}x, & \textrm{if }p\,\,\, \textrm{is even}, \\ 0 , & \textrm{if }p\,\,\, \textrm{is odd}. \end{cases} \end{aligned} \end{equation} \tag{ 2.86 }$$
• (ii)  
  ${\mathbf{Edge:}}$ For $\mathcal{S} \in \mathbb{R}$, let

  $$\begin{equation} a = 1-\frac{\mathcal{S} }{\sqrt{2 \Delta W\left(1\right) N} }. \end{equation} \tag{ 2.87 }$$

  Then we have

  $$\begin{equation} \lim_{N \to \infty} \sqrt{\frac{2}{N \Delta W\left(1\right) } } \, \kappa_p^{\left(\beta = 4\right)}\left(a\right) = \kappa_p^{\textrm{edge}}\left(\mathcal{S}\right), \end{equation} \tag{ 2.88 }$$

  where

  $$\begin{equation} \kappa_p^{\textrm{edge}}\left(\mathcal{S}\right) : = \left(-1\right)^{p + 1} \int_{-\infty}^{\mathcal{S}} \operatorname{Li}_{1 - p}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) \, \mathrm{d}x. \end{equation} \tag{ 2.89 }$$

Remark 2.15 (FCS of the Mittag–Leffler ensembles). The Mittag–Leffler ensembles are two-parameter generalisations of the Ginibre ensembles associated with the potential of the form

$$\begin{equation} W^{\mathrm{ML}}\left(z\right) = \alpha |z|^{2b} - \frac{2c}{N} \log|z|, \qquad \left(\alpha,b>0, \,\, c>-1\right). \end{equation} \tag{ 2.90 }$$

For the complex Mittag–Leffler ensembles, a very precise FCS was obtained in a recent work by Charlier [26, corollary 1.6]. We recall that the FCS is equivalent to the problem of deriving a large-N expansion of the partition functions having jump-type singularities (see [19] and references therein for various literature on the partition functions of random normal matrix and planar symplectic ensembles). As a consequence, by [17, proposition 5.10], the FCS of the complex Mittag–Leffler ensembles induces that of the symplectic Mittag–Leffler ensembles (with different parameters). For instance, the FCS of the symplectic Ginibre ensemble follows from the FCS of the complex Mittag–Leffler ensemble associated with the linear potential $W^{\mathrm{ML}}(z) = 2|z|$.

In figure 6 we illustrate proposition 2.13 and theorem 2.14 for $p = 3,4$ using the potentials

$$\begin{align} W_N^{\left( m \right)}\left( z \right){\textrm{ }}: = - \frac{1}{N}\log G_{0,m}^{m,0}\left( {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} - \\ {{\mkern 1mu} {\mathbf{0}}} \end{array}\left| {\begin{array}{*{20}{l}} {{{\left( {2N} \right)}^m}|z{|^2}} \end{array}} \right.} \end{array}} \right),\qquad m \in ,\end{align} \tag{ 2.91 }$$

$$\begin{align} W^{\mathrm{tr}}\left(z\right) &: = \begin{cases} -2 \tilde{c} \, \log \left( 1-\dfrac{|z|^2}{1+{\tilde{c} } } \right) &\mathrm{if } |z| \unicode{x2A7D} \sqrt{1+\tilde{c}}, \\ \infty &\mathrm{otherwise}, \end{cases} \qquad \tilde{c}>0. \end{align} \tag{ 2.92 }$$

Here, $G^{m,n}_{p,q}$ is the Meijer G-function defined by

$$\begin{align} G_{p,q}^{m,n}\left( {\begin{array}{*{20}{l}} {\begin{array}{*{20}{l}} {{a_1}, \ldots ,{a_p}}\\ {{b_1}, \ldots ,{b_q}} \end{array}\left| z \right.} \end{array}} \right): = \frac{1}{{2\pi i}}\oint_L {{z^s}} \frac{{\prod\limits_{j = 1}^m \Gamma \left( {{b_j} - s} \right)\prod\limits_{j = 1}^n \Gamma \left( {1 - {a_j} + s} \right)}}{{\prod\limits_{j = 1 + m}^q \Gamma \left( {1 - {b_j} + s} \right)\prod\limits_{j = 1 + n}^p \Gamma \left( {{a_j} - s} \right)}}{\mkern 1mu} {\textrm{d}}s, \end{align} \tag{ 2.93 }$$

where the integration contour L in $\mathbb{C}$ depends on $\{a_j\}, \{b_j\}$, see e.g. [83, chapter 16]. The potential (2.91) describes the product of m symplectic Ginibre matrices (m = 1: GinSE), see e.g. [57]. On the one hand, the potential (2.92) stems from the truncated symplectic unitary ensemble, see e.g. [63].

Example 2.16. For p = 2 it follows from $\operatorname{Li}_{-1}(1-1/x) = x(x-1)$ that for the GinSE

$$\begin{equation} \operatorname{Var} \mathcal{N}_a^{\left(\beta = 4\right)} = \sum_{j = 0}^{N-1} \mathcal{L}_{2j+1}\left(a\right) \mathcal{M}_{2j+1}\left(a\right). \end{equation} \tag{ 2.94 }$$

This formula appears in [1, proposition 1.1]. Furthermore we have

$$\begin{equation} - \operatorname{Li}_{-1}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) = \frac{1-\operatorname{erf}\left(x\right)^2}{4}, \qquad \int_{\mathbb{R}} \frac{1-\operatorname{erf}\left(x\right)^2}{4} \,\mathrm{d}x = \frac{1}{\sqrt{2\pi} }. \end{equation} \tag{ 2.95 }$$

Therefore, one can observe that theorem 2.14 recovers [1, theorem 1.2].

Remark 2.17 (Origin limit of the cumulants). In the origin limit where we rescale $R = \sqrt{N} \, a$, we obtain for cumulants of the GinSE

$$\begin{equation} \lim_{N \to \infty} \kappa_p^{\left(\beta = 4\right)}\left(\frac{R}{\sqrt{N}}\right) = \left(-1\right)^{p+1} \sum_{j = 0}^{\infty} \operatorname{Li}_{1 - p}\left( -\frac{\Gamma\left(2j+2, 2R^2\right)}{\gamma\left(2j+2, 2R^2\right)} \right). \end{equation} \tag{ 2.96 }$$

This follows from (2.83), where the convergence of the infinite sum is assured by the Gamma-function asymptotics [83, equation (8.11.4)]. An analogous formula for the GinUE was already shown in the supplementary material of [73]. Note that the origin limit is not universal, e.g. for the potentials in (2.90)–(2.92) different results were obtained in [1, propositions 1.4, 1.5 and 1.7], respectively.

Remark 2.18. It is interesting to study the behaviour of the cumulants for the GinUE and the GinSE, given respectively in equations (2.80) and (2.83) in the limit a → 0, with fixed N. Indeed, in this limit, one has

$$\begin{equation} {\mathcal L}_j\left(a\right) = \frac{e^{-Ng\left(0\right)}}{h_j} \frac{a^{2j+2}}{2j+2}\left(1 + o\left(1\right)\right), \quad a \to 0 \;. \end{equation} \tag{ 2.97 }$$

Note that for $n \in \mathbb{N},$ it follows from the inversion formula [60, equation (1)]

$$\begin{equation} \operatorname{Li}_{-n}\left(x\right) = \left(-1\right)^{n-1} \operatorname{Li}_{-n}\left(1/x\right) = \left(-1\right)^{n-1} \sum_{k = 1}^\infty \frac{k^n}{x^k} , \end{equation} \tag{ 2.98 }$$

that as $x \to 0,$

$$\begin{equation} {\mathrm{Li}}_{1-p}\left(-x\right) = \frac{\left(-1\right)^{p+1}}{x} + O\left(\frac{1}{x^2} \right), \quad p \unicode{x2A7E} 2 \quad {\mathrm{and}} \quad p \in \mathbb{N}. \end{equation} \tag{ 2.99 }$$

Therefore one finds, as a → 0,

$$\begin{equation} \kappa_p^{\left(\beta = 2\right)}\left(a\right) = \frac{e^{-Ng\left(0\right)}}{2h_0} a^2\left(1 + o\left(1\right)\right), \qquad \kappa_p^{\left(\beta = 4\right)}\left(a\right) = \frac{e^{-Ng\left(0\right)}}{4h_1} a^4\left(1 + o\left(1\right)\right) \;. \end{equation} \tag{ 2.100 }$$

On the other hand, from equations (2.81) and (2.84) one finds

$$\begin{align} \begin{aligned} & E_{N,W}^{\left(\beta = 2\right)}\left(a\right) = {\mathcal L}_0\left(a\right)\left(1+o\left(1\right)\right) = \frac{e^{-Ng\left(0\right)}}{2h_0} a^2\left(1+o\left(1\right)\right), \\ & E_{N,W}^{\left(\beta = 4\right)}\left(a\right) = {\mathcal L}_1\left(a\right)\left(1+o\left(1\right)\right) = \frac{e^{-Ng\left(0\right)}}{4h_1} a^4\left(1+o\left(1\right)\right) \;. \end{aligned} \end{align} \tag{ 2.101 }$$

Therefore, by comparing equations (2.100) and (2.101) we see that as a → 0, all the cumulants coincide with the mean value, i.e.

$$\begin{equation} \lim_{a \to 0} \frac{\kappa_p^{\left(\beta = 2\right)}\left(a\right)}{E_{N,W}^{\left(\beta = 2\right)}\left(a\right)} = \lim_{a \to 0} \frac{\kappa_p^{\left(\beta = 4\right)}\left(a\right)}{E_{N,W}^{\left(\beta = 4\right)}\left(a\right)} = 1, \qquad p \unicode{x2A7E} 2 \;, \end{equation} \tag{ 2.102 }$$

which shows, as anticipated in the remark 2.8, that ${\mathcal N}_a$ indeed becomes a Poissonian random variable in this limit a → 0, keeping N fixed. We emphasize that the behaviour of the cumulants found in the limit $N \to \infty$, keeping $a \in (0,1)$ fixed in theorem 2.14, is quite different. Indeed, although the functional dependence on the variable a of all the cumulants is similar, the coefficients $\kappa^{\mathrm{bulk}}_p$ in (2.86) have a non-trivial dependence on p. This shows that in this limit the fluctuations of ${\mathcal N}_a$ are clearly non-Poissonian (see [70] for a similar discussion in the context of the GinUE).

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Our main results are summarized in table 3.

Table 3. Summary of our main results and (some) references to known results.

		GinUE		GinSE
		Ginibre	Potential $W(\lvert z\rvert)$	Ginibre	Potential $W(\lvert z\rvert)$	GinOE
Expectation	Finite N	[1, proposition 3.1] proposition 2.1(i)	[70], [1, proposition 1.1] proposition 2.13(i)	[1, proposition 3.1] proposition 2.1(i)	[1, proposition 1.1] proposition 2.13(ii)	proposition 2.1(iii) lemma 3.1
	Origin	[73] proposition 2.3	Depends on W e.g. [1, section 1.3]	proposition 2.3	Depends on W e.g. [1, section 1.3]	proposition 2.3
	Bulk	[1, proposition 3.1], [26, corollary 1.6 (a)]	via [55]	[1, proposition 3.1]	via [12] [1, corollary 1.3]	[38] equation (2.20)
	Edge	[26, corollary 1.6 (b)], [73], corollary 2.2	via [9]	via remark 2.15 corollary 2.2	Unknown	Unknown corollary 2.2
Variance	Finite N	[70, 73]	[70], [1, proposition 1.1]	[1, equation (3.8)]	[1, proposition 1.1]	remark 4.4
	Origin	[73] proposition 2.4(i)	Depends on W e.g. [1, section 1.3]	proposition 2.4(ii)	Depends on W e.g. [1, section 1.3]	proposition 4.3
	Bulk	[70, 73], [1, theorem 1.2], remark 2.11		[1, theorem 1.2], remark 2.11		conjecture 2.10 (theorem 2.6)
	Edge	[73], [1, theorem 1.2], remark 2.11		[1, theorem 1.2], remark 2.11		conjecture 2.10
Cumulants	Finite N	[70, 73], proposition 2.13(i)		proposition 2.13(ii)		Partly unknown, cf [42]
	Origin	[73]	Depends on W	remark 2.17	Depends on W	Unknown
	Bulk	[70, 73]	[70]	theorem 2.14(i)		Unknown
	Edge	[73]	[73] with [71]	theorem 2.14(ii)		Unknown

3.1. Finite Ginibre ensembles

The lemmas 3.1 and 3.3 below imply proposition 2.1.

Lemma 3.1. For N odd,

$$\begin{equation} \begin{aligned} E_{N,\mathbb{R}}^{\left(1\right)}\left(a\right)& = \frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{\left(2k\right)!! } P\left(k+\frac12, Na^2\right) +P\left(\frac{N}{2}, \frac{Na^2}{2}\right) \\ &\quad - \frac{1}{2^{N/2} \, \left(N-2\right)!! } \sum_{k = 0}^{\left(N-3\right)/2 } \frac{\left(N+2k-2\right)!! }{2^k \, \left(2k\right)!!} P\left( \frac{N}{2}+k, Na^2\right). \end{aligned} \end{equation} \tag{ 3.1 }$$

For N even,

$$\begin{align} E_{N,\mathbb{R}}^{\left(1\right)}\left(a\right)& = \frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{\left(2k\right)!! } P\left(k+\frac12, Na^2\right) +\frac{N^{N/2} }{\left(N-2\right)!! } \int_{0}^a e^{-\frac{N}{2} x^2 } x^{N-1} \operatorname{erf}\left(\sqrt{\frac{N}{2}}x\right)\,\mathrm{d}x\nonumber\\ & \quad -\frac{1}{2^{N/2} \, \left(N-2\right)!! } \sum_{k = 1}^{N/2-1 } \frac{ \left(N+2k-3\right)!! }{2^{k-1/2} \, \left(2k-1\right)!! } P\left( \frac{N-1}{2}+k, Na^2\right). \end{align} \tag{ 3.2 }$$

Note that for any $N \in \mathbb{N}$

$$\begin{equation} \frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{\left(2k\right)!! } P\left(k+\frac12, Na^2\right) = \sqrt{\frac{2}{\pi} } \frac{a\sqrt{N} \, \Gamma\left(N-1,Na^2\right)+\gamma\left(N-\tfrac12, Na^2\right) }{\left(N-2\right)! }. \end{equation} \tag{ 3.3 }$$

Remark 3.2. If $a = \infty$, $E_{N,\mathbb{R}}^{(1)} = E_{N,\mathbb{R}}^{(1)}(\infty)$ corresponds to the expected number of real eigenvalues. For $a = \infty$, we have that for N odd,

$$\begin{align} E_{N,\mathbb{R}}^{\left(1\right)} & = 1+\frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{\left(2k\right)!! }- \frac{1 }{2^{N/2 } \left(N-2\right)!! } \sum_{k = 0}^{\left(N-3\right)/2 } \frac{\left(N+2k-2\right)!! }{2^k \,\left(2 k\right)!!} \end{align} \tag{ 3.4 }$$

whereas for N even, using the series expansion of the error function inside the integral we obtain

$$\begin{align} \begin{aligned} E_{N,\mathbb{R}}^{\left(1\right)} & = \frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{\left(2k\right)!! } + \frac{2}{\sqrt{\pi} } \frac{\Gamma\left( \frac{N+1}{2} \right) }{\Gamma\left( \frac{N}{2} \right) } {}_2 F_1\left( \frac12, \frac{N+1}{2} ; \frac32; -1 \right) \\ &\quad - \frac{1}{2^{N/2} \, \left(N-2\right)!! } \sum_{k = 1}^{N/2-1 } \frac{ \left(N+2k-3\right)!! }{2^{k-1/2} \, \left(2k-1\right)!! }. \end{aligned} \end{align} \tag{ 3.5 }$$

One can check that these correspond to the expressions [38]

$$\begin{equation} E_{N,\mathbb{R}}^{\left(1\right)} = \begin{cases} \displaystyle 1+ \sqrt{2} \sum_{k = 1}^{\left(N-1\right)/2} \frac{\left(4k-3\right)!!}{\left(4k-2\right)!! } &N\,\,\,\,\,\, \textrm{odd}, \\ \displaystyle \sqrt{2} \sum_{k = 0}^{N/2-1} \frac{\left(4k-1\right)!!}{\left(4k\right)!! } &N\,\,\,\,\,\, \textrm{even}. \end{cases} \end{equation} \tag{ 3.6 }$$

Proof of lemma 3.1. Inserting the 1-point function $\textbf{R}_{N,1,\mathbb{R}}^{(1)}(x)$ of the real eigenvalues from (2.10) into the definition of $E_{N,\mathbb{R}}^{(1)}(a)$ from (2.14) gives

$$\begin{equation} E_{N,\mathbb{R}}^{\left(1\right)}\left(a\right) = \int_{-a}^{a} \left(\sqrt{\frac{N}{2\pi}} Q\left(N-1,Nx^2\right) + \frac{N^{N/2} }{2^{N/2 } \Gamma\left(\frac{N}{2}\right) } e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) \right)\,\mathrm{d}x, \end{equation} \tag{ 3.7 }$$

for any $N \in \mathbb{N}.$ For the first term we use

$$\begin{align*} Q\left(N-1,Nx^2\right) = e^{-Nx^2} \sum_{j = 0}^{N-2} \frac{N^k x^{2k}}{k!}, \qquad \Gamma\left(k+\frac12\right) = \sqrt{\pi} \frac{\left(2k-1\right)!!}{2^k}, \end{align*}$$

and obtain

$$\begin{align} \begin{aligned} &\sqrt{\frac{N}{2\pi}}\int_{-a}^a Q\left(N-1,Nx^2\right)\,\mathrm{d}x = \sqrt{\frac{N}{2\pi}} \int_{-a}^{a} e^{-N x^2 } \sum_{k = 0}^{N-2} \frac{N^k x^{2k} }{k!}\,\mathrm{d}x \\ &\quad = \frac{1}{\sqrt{2\pi} } \sum_{k = 0}^{N-2} \frac{\gamma\left(k+\frac12, Na^2\right) }{k!} = \frac{1}{\sqrt{2\pi} } \sum_{k = 0}^{N-2} \frac{\Gamma\left(k+\frac12\right) }{k!} P\left(k+\frac12, Na^2\right) \\ &\quad = \frac{1}{\sqrt{2} } \sum_{k = 0}^{N-2} \frac{\left(2k-1\right)!! }{2^{k} k! } P\left(k+\frac12, Na^2\right) . \end{aligned} \end{align} \tag{ 3.8 }$$

On the other hand, using integration by parts we obtain

$$\begin{align} \begin{aligned} \sqrt{\frac{N}{2\pi}}\int_{-a}^a Q\left(N-1,Nx^2\right)\,\mathrm{d}x & = \frac{1}{\sqrt{2\pi} } \int_{0}^{Na^2} \frac{Q\left(N-1,x\right) }{\sqrt{x} }\,\mathrm{d}x \\ & = \sqrt{\frac{2}{\pi} } \frac{a\sqrt{N} \, \Gamma\left(N-1,Na^2\right)+\gamma\left(N-\tfrac12, Na^2\right) }{\left(N-2\right)! }. \end{aligned} \end{align} \tag{ 3.9 }$$

This implies the identity (3.3).

For the second term we need to distinguish between even and odd N because

$$\begin{align*} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) & = \begin{cases} \displaystyle 1- e^{-\frac{N}{2}x^2 } \sum_{k = 0}^{\left(N-3\right)/2 } \frac{\left(N x^2/2\right)^k }{k!} &\textrm{for odd }N, \\ \displaystyle \operatorname{erf}\left(\sqrt{\frac{N}{2}}|x|\right)-\frac{e^{-\frac{N}{2}x^2 } }{\sqrt{\pi} } \sum_{k = 1}^{N/2-1} \frac{\left(N x^2/2\right)^{k-1/2} }{\left(1/2\right)_k } &\textrm{for even }N, \end{cases} \end{align*}$$

see [83, section 8.4]. Therefore if N is odd,

$$\begin{align*} &\int_{-a}^a e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) \,\mathrm{d}x \\ &\quad = 2\int_{0}^a e^{-\frac{N}{2} x^2 } x^{N-1}\,\mathrm{d}x -2\sum_{k = 0}^{\left(N-3\right)/2 } \frac{\left(N /2\right)^k }{k!} \int_{0}^a e^{-N x^2 } x^{N-1+2k} \,\mathrm{d}x \\ &\quad = 2^{N/2} N^{-N/2} \gamma\left(\frac{N}{2}, \frac{Na^2}{2}\right) - N^{-N/2}\sum_{k = 0}^{\left(N-3\right)/2 } \frac{1 }{2^k \,k!} \gamma\left( \frac{N}{2}+k, Na^2\right) \end{align*}$$

and consequently

$$\begin{align*} & \frac{N^{N/2} }{2^{N/2 } \Gamma\left(\frac{N}{2}\right) } \int_{-a}^a e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) \,\mathrm{d}x \\ &\quad = P\left(\frac{N}{2}, \frac{Na^2}{2}\right) - \sum_{k = 0}^{\left(N-3\right)/2 } \frac{\Gamma\left(\frac{N}{2}+k\right) }{2^{k+N/2} \Gamma\left(\frac{N}{2}\right)\,k!} P\left( \frac{N}{2}+k, Na^2\right) \\ &\quad = P\left(\frac{N}{2}, \frac{Na^2}{2}\right) - \frac{1}{2^{N/2} \, \left(N-2\right)!! } \sum_{k = 0}^{\left(N-3\right)/2 } \frac{\left(N+2k-2\right)!! }{2^{2k} \,k!} P\left( \frac{N}{2}+k, Na^2\right) \end{align*}$$

where we used [83, equation (5.4.2)] to write the Gamma-functions in terms of double factorials. This implies (3.1), since $2^k k! = (2k)!!$.

On the other hand, if N is even,

$$\begin{align*} & \int_{-a}^a e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) \,\mathrm{d}x \\ &\quad = 2\int_{0}^a e^{-\frac{N}{2} x^2 } x^{N-1} \operatorname{erf}\left(\sqrt{\frac{N}{2}}x\right)\,\mathrm{d}x -\frac{2}{\sqrt{\pi} }\sum_{k = 1}^{N/2-1 } \frac{\left(N /2\right)^{k-1/2} }{\left(1/2\right)_k } \int_{0}^a e^{-N x^2 } x^{N-2+2k} \,\mathrm{d}x \\ &\quad = 2\int_{0}^a e^{-\frac{N}{2} x^2 } x^{N-1} \operatorname{erf}\left(\sqrt{\frac{N}{2}}x\right)\,\mathrm{d}x -\frac{ \sqrt{2} }{\sqrt{\pi} } N^{-N/2} \sum_{k = 1}^{N/2-1 } \frac{1 }{\left(2k-1\right)!! } \gamma\left( \frac{N-1}{2}+k, Na^2\right), \end{align*}$$

and including the normalisation we have

$$\begin{align*} &\frac{N^{N/2} }{2^{N/2 } \Gamma\left(\frac{N}{2}\right) } \int_{-a}^a e^{-\frac{N}{2} x^2 } |x|^{N-1} P\left( \frac{N-1}{2}, \frac{Nx^2}{2} \right) \,\mathrm{d}x \\ &\quad = \frac{N^{N/2} }{2^{N/2-1 } \Gamma\left(\frac{N}{2}\right) } \int_{0}^a e^{-\frac{N}{2} x^2 } x^{N-1} \operatorname{erf}\left(\sqrt{\frac{N}{2}}x\right)\,\mathrm{d}x \\ &\qquad - \frac{1 }{2^{\left(N-1\right)/2 } \Gamma\left(\frac{N}{2}\right) } \frac{1}{\sqrt{\pi} } \sum_{k = 1}^{N/2-1 } \frac{ \Gamma\left(\frac{N-1}{2}+k\right) }{\left(2k-1\right)!! } P\left( \frac{N-1}{2}+k, Na^2\right). \end{align*}$$

This yields (3.2), after rewriting the Gamma-functions using [83, equation (5.4.2)] in the second term. □

Lemma 3.3. For $N \in \mathbb{N}$, we have for the radial density at finite-N

$$\begin{equation} \frac{1}{N}\frac{1}{2\pi}\int_{0 }^{2\pi} \textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(\frac{r e^{i\theta}}{\sqrt{N}} \right)\,\mathrm{d}\theta = \left( 1-e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) \right) Q\left(N-1,r^2\right) . \end{equation} \tag{ 3.10 }$$

This leads to

$$\begin{align} \begin{aligned} E_{N,\mathbb{C}}^{\left(1\right)} \left(a\right) & = \sum_{k = 0}^{N-2} P\left(k+1,Na^2\right) - 2 \int_0^{\sqrt{N}a } e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) Q\left(N-1,r^2\right)\,\mathrm{d}r. \end{aligned} \end{align} \tag{ 3.11 }$$

Proof. The 1-point function $\textbf{R}_{N,1,\mathbb{C}}^{(1)}(x)$ of the complex eigenvalues is given by

$$\begin{equation} \frac{1}{N}\textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(\frac{z}{\sqrt{N}}\right) = \sqrt{{2}{\pi} } |y| \operatorname{erfc}\left(\sqrt{2}|y|\right) e^{2y^2} Q\left(N-1,|z|^2\right). \end{equation} \tag{ 3.12 }$$

Using

$$\begin{equation} \operatorname{erfc}\left(\sqrt{2}y\right) = 2 \sqrt{\frac{2}{\pi} } \int_{-\infty}^0 e^{-2\left(t-y\right)^2}\,\mathrm{d}t, \end{equation} \tag{ 3.13 }$$

with $y = r\sin{\theta}$, we obtain for the integral (3.10)

$$\begin{align} \begin{aligned} \frac{1}{N2\pi}\int_{0 }^{2\pi} \textbf{R}_{N,1,\mathbb{C}}^{\left(1\right)}\left(\frac{r e^{i\theta}}{\sqrt{N}} \right)\,\mathrm{d}\theta & = \frac{2}{\pi} Q\left(N-1,r^2\right)\int_{-\infty}^0 \int_{0 }^{2\pi} |y| e^{-2\left(t-|y|\right)^2+2y^2 } \,\mathrm{d}\theta \,\mathrm{d}t \\ & = \frac{4r}{\pi} Q\left(N-1,r^2\right)\int_{-\infty}^0 e^{-2t^2} \int_{0}^{\pi} \left(\sin \theta\right) e^{4 t r\sin \theta } \,\mathrm{d}\theta \,\mathrm{d}t \\ & = \frac{4r}{\pi} Q\left(N-1,r^2\right)\int_{-\infty}^0 e^{-2t^2} \sum_{l = 0}^\infty \frac{\left(4tr\right)^l}{l!}\frac{2^{l+1}\Gamma\left(\frac{l}{2}+1\right)^2}{\Gamma\left(l+2\right)} \,\mathrm{d}t \\ & = \frac{4r}{\pi} Q\left(N-1,r^2\right)\sum_{l = 0}^\infty \frac{\left(8r\right)^l}{l!}\frac{\Gamma\left(\frac{l}{2}+1\right)^2}{\Gamma\left(l+2\right)}\frac{\left(-1\right)^l\Gamma\left(\frac{l+1}{2}\right)}{2^{\frac{l+1}{2}}}, \end{aligned} \end{align} \tag{ 3.14 }$$

using the elementary integrals

$$\begin{equation} \int_0^{\pi} \left(\sin \theta\right)^{l+1} \,\mathrm{d}\theta = \frac{2^{l+1} \Gamma\left(\frac{l}{2}+1\right)^2}{\Gamma\left(l+2\right)}, \qquad \int_{-\infty}^0 e^{-2t^2} t^l \,\mathrm{d}t = \frac{\left(-1\right)^l}{2^{1+\frac{l+1}{2}}} \Gamma\left(\frac{l+1}{2}\right). \end{equation} \tag{ 3.15 }$$

The duplication formula

$$\begin{equation} \Gamma\left(2z\right) = \frac{1}{\sqrt{\pi}} 2^{2z-1} \Gamma\left(z\right)\Gamma\left(z+\tfrac12\right), \end{equation} \tag{ 3.16 }$$

can be used to further simplify the sum, and we obtain

$$\begin{align} \sqrt{\pi}\sum_{l = 0}^\infty \frac{2^{\frac{3}{2}l-\frac{1}{2}}\left(-r\right)^l\Gamma\left(\frac{l}{2}+1\right)}{\Gamma\left(l+2\right)}& = \sqrt{\pi} \sum_{k = 0}^\infty \frac{2^{3k-\frac{1}{2}r^{2k}\Gamma\left(k+1\right)}}{\Gamma\left(2k+2\right)} -\sqrt{\pi} \sum_{k = 0}^\infty \frac{2^{2k+1}r^{2k+1}\Gamma\left(k+ \frac{3}{2}\right)}{\Gamma\left(2k+3\right)}\nonumber\\ & = \sqrt{\pi} \sum_{k = 0}^\infty \frac{2^{2k-\frac{1}{2}}r^{2k}}{\left(2k+1\right)!!} -\pi \sum_{k = 0}^\infty \frac{2^{k-1}r^{2k+1}}{\left(k+1\right)!} , \end{align} \tag{ 3.17 }$$

after splitting the sum into even and odd indices l and applying once more the duplication formula (3.16). Using the expansion of the error function [83, equation (7.6.2)],

$$\begin{equation} \operatorname{erf}\left(z\right)e^{z^2} = \frac{2}{\pi}\sum_{n = 0}^\infty \frac{2^nz^{2n+1}}{\left(2n+1\right)!!} \end{equation} \tag{ 3.18 }$$

and $\operatorname{erfc}(z) = 1-\operatorname{erf}(z)$ we obtain the following identity

$$\begin{align} \frac{\pi}{4r}\left(1-e^{2r^2}\operatorname{erfc}\left(\sqrt{2}r\right) \right) & = \frac{\pi}{4r}\left( 1-\sum_{n = 0}^\infty \frac{\left(2r\right)^n}{n!}+\frac{2}{\sqrt{\pi}}\sum_{n = 0}^\infty\frac{\left(2r\right)^{2n}\sqrt{2}r}{\left(2n+1\right)!!}\right), \end{align} \tag{ 3.19 }$$

which agrees with the sum in (3.17). This yields (3.10).

After having determined the radial density we can compute the mean in (3.11) as

$$\begin{align} \begin{aligned} E_{N,\mathbb{C}}^{\left(1\right)} \left(a\right) & = 2 \int_0^{\sqrt{N}a } r \left( 1-e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) \right) Q\left(N-1,r^2\right)\,\mathrm{d}r \\ & = 2 \sum_{k = 0}^{N-2} \int_0^{\sqrt{N}a } e^{-r^2}\frac{r^{2k+1}}{k!} \,\mathrm{d}r - 2 \int_0^{\sqrt{N}a } e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) Q\left(N-1,r^2\right)\,\mathrm{d}r \\ & = \sum_{k = 0}^{N-2} P\left(k+1,Na^2\right) - 2 \int_0^{\sqrt{N}a } e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) Q\left(N-1,r^2\right)\,\mathrm{d}r. \end{aligned} \end{align} \tag{ 3.20 }$$

 □

Proof of corollary 2.2. For $\beta = 2,4$ these are immediate consequences of proposition 2.1 (i) and (ii). To be more precise, for β = 2, it immediately follows from the Stirling's formula. On the other hand, for β = 4, the behaviour (2.23) follows from β = 2 case, when estimating the second term in (2.17) given by an exponential times a truncated sine-hyperbolic to be of order unity.

It remains to show the case β = 1. For this, using the well-known asymptotic behaviour

$$\begin{align*} \operatorname{erfc}\left(x\right) \sim \frac{e^{-x^2} }{\sqrt{\pi} x }, \qquad x \to \infty, \end{align*}$$

we have

$$\begin{align*} 2 \int_0^{\sqrt{N} } r\, e^{2r^2} \operatorname{erfc}\left(\sqrt{2}r\right) Q\left(N-1,r^2\right)\,\mathrm{d}r & = 2 N \int_0^{1 } t\, e^{2t^2N} \operatorname{erfc}\left(\sqrt{2N}\,t \right) Q\left(N-1,t^2N\right)\,\mathrm{d}t \\ & \sim \sqrt{\frac{2}{\pi} N} \int_0^{1 } Q\left(N-1,t^2N\right)\,\mathrm{d}t \\ & = \sqrt{\frac{2}{\pi} N} \left( \frac{\gamma\left(N-\frac12, N\right) }{ \sqrt{N} \, \Gamma\left(N-1\right) } +Q\left(N-1,N\right) \right) \sim \sqrt{\frac{2}{\pi} N}. \end{align*}$$

In the last line, we have used integration by parts. Using the asymptotics of the incomplete Gamma-functions [83, equation (8.10.11)], each summand inside the bracket gives $1/2$, which yields the final result. Combining proposition 2.1, (2.20), and (2.22) for β = 2, we obtain the desired asymptotic behavior (2.22) for β = 1. □

3.2. Infinite Ginibre ensembles at the origin

Proof of proposition 2.3. Note that for β = 2, it is obvious since $R_1^{(2)}(z) = 1$ [49]. For $\beta = 2,4$ we use

$$\begin{equation} E^{\left(\beta\right)}\left(R\right) = 2 \int_0^R r \widehat{R}_1^{\left(\beta\right)}\left(r\right)\,\mathrm{d}r. \end{equation} \tag{ 3.21 }$$

For $\beta = 4,$ note that [61, 80]

$$\begin{equation} R_1^{\left(4\right)}\left(z\right) = 4y F\left(2y\right), \qquad F\left(z\right) = e^{-z^2} \int_0^z e^{t^2}\,\mathrm{d}t, \end{equation} \tag{ 3.22 }$$

where F is Dawson's integral function. Then the desired identity (2.33) follows from

$$\begin{align*} E^{\left(4\right)}\left(R\right)& = \int_{D_R} 4y F\left(2y\right) \,\mathrm{d}A\left(z\right) = \frac{4}{\pi} \int_{0}^R r^2 \,\mathrm{d}r \int_0^{2\pi} \mathrm{d}\theta \, \sin\left(\theta\right) F\left(2r \sin\left(\theta\right)\right). \end{align*}$$

Using the series expansion of Dawson's function [83, equations (7.5.1) and (7.6.3)]

$$\begin{equation} F\left(z\right) = \sum_{k = 0}^\infty \frac{\left(-1\right)^k2^k}{\left(2k+1\right)!!}z^{2k+1}, \end{equation} \tag{ 3.23 }$$

we obtain for the angular integral

$$\begin{align*} \int_0^{2\pi} \mathrm{d}\theta \, \sum_{k = 0}^\infty\frac{\left(-1\right)^k2^k}{\left(2k+1\right)!!}\left(2r\right)^{k+1}\sin\left(\theta\right)^{2k+2} & = \sum_{k = 0}^\infty\frac{\left(-1\right)^k2^{5k+4}\Gamma\left(k+\frac{3}{2}\right)^2}{\left(2k+1\right)!!\,\Gamma\left(2k+3\right)}r^{2k+1} \\ & = \pi \sum_{k = 0}^\infty \frac{\left(-1\right)^k\left(2r\right)^{2k+1}}{\left(k+1\right)!}, \end{align*}$$

using again (3.15) and (3.16). The sum can be easily done and we are lead to

$$\begin{align} E^{\left(4\right)}\left(R\right) = \frac{4}{\pi} \int_0^R r^2 \frac{\pi}{2r}\left(1-e^{-4r^2}\right) \,\mathrm{d}r = R^2-\frac14+\frac14 e^{-4R^2}. \end{align} \tag{ 3.24 }$$

For β = 1, the radial density in the large-N limit follows from lemma 3.3, with $Q(N-1,r^2)$ replaced by 1. This gives $E_{\mathbb{C}}^{(1)}(R)$ from integration by parts:

$$\begin{align*} E_{\mathbb{C}}^{\left(1\right)} \left(R\right) & = 2 \int_0^{R} r \left( 1-e^{2r^2}\left(1 -\operatorname{erf}\left(\sqrt{2}r\right)\right) \right) \,\mathrm{d}r \\ & = R^2-\frac{1}{2}e^{2R^2}+\frac{1}{2}+\frac{1}{2}e^{2R^2}\operatorname{erf}\left(\sqrt{2}R\right)-\int_0^R \frac{1}{2}e^{2r^2}\frac{2\sqrt{2}}{\sqrt{\pi}}e^{-2r^2}\mathrm{d}r. \end{align*}$$

On the other hand, $E_{\mathbb{R}}^{(1)}(R)$ follows from $R_{1,\mathbb{R}}^{(1)}(x) = 1/\sqrt{2\pi}$ [37]. □

It is also possible to obtain direct proofs for β = 1 based on the limiting form of $R_{1,\mathbb{C}}(z)$ and the integral representation of the complementary error function. We omit these for brevity here.

4.1. Number variance of the GinUE at the origin

The number variance of the GinUE is written as

$$\begin{equation} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) = \int_{D_R} \int_{D_R^c} |K\left(z_1,z_2\right)|^2\,\mathrm{d}A\left(z_1\right)\,\mathrm{d}A\left(z_2\right) , \end{equation} \tag{ 4.1 }$$

see e.g. [1].

Proof of proposition 2.4 (i). Note that

$$\begin{align} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) & = \int_{D_R} \,\mathrm{d}A\left(z_1\right) \int_{D_R^c} \,\mathrm{d}A\left(z_2\right) e^{-|z_1|^2-|z_2|^2+z_1 \bar{z}_2+ \bar{z}_1 z_2 } . \end{align} \tag{ 4.2 }$$

It follows from the orthogonality with respect to the angular integration that

$$\begin{align} \begin{aligned} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) & = \int_{D_R} \,\mathrm{d}A\left(z_1\right) \int_{D_R^c} \,\mathrm{d}A\left(z_2\right) e^{-|z_1|^2-|z_2|^2 } \sum_{j = 0}^\infty \frac{ \left(z_1 \bar{z}_2+ \bar{z}_1 z_2\right)^j }{j! } \\ & = \int_{D_R} \,\mathrm{d}A\left(z_1\right) \int_{D_R^c} \,\mathrm{d}A\left(z_2\right) e^{-|z_1|^2-|z_2|^2 } \sum_{j = 0}^\infty \frac{ |z_1|^{2j} |z_2|^{2j} }{\left(2j\right)! } \binom{2j}{j}. \end{aligned} \end{align} \tag{ 4.3 }$$

This gives

$$\begin{align} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) & = \sum_{j = 0}^\infty \int_{D_R} \frac{|z_1|^{2j} e^{-|z_1|^2} }{j! } \,\mathrm{d}A\left(z_1\right) \int_{D_R^c} \frac{|z_2|^{2j} e^{-|z_2|^2} }{j! } \,\mathrm{d}A\left(z_2\right) = \sum_{j = 1}^\infty P\left(j,R^2\right) Q\left(j,R^2\right), \end{align} \tag{ 4.4 }$$

which is the first expression in (2.45).

On the other hand, note that by (4.3), upon inserting the series (2.44) we have

$$\begin{align} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) & = \int_{D_R} \,\mathrm{d}A\left(z_1\right) \int_{D_R^c} \,\mathrm{d}A\left(z_2\right) e^{-|z_1|^2-|z_2|^2 }I_0\left(2|z_1z_2|\right) \end{align} \tag{ 4.5 }$$

$$\begin{align} & = \int_{0}^R dr_1 f\left(R,r_1\right) , \end{align} \tag{ 4.6 }$$

where we have defined

$$\begin{equation} f\left(R,r_1\right): = 4 \int_R^\infty e^{-r_1^2-r_2^2} r_1 r_2\,I_0\left(2r_1r_2\right)\,dr_2. \end{equation} \tag{ 4.7 }$$

Using Leibniz' rule, we obtain

$$\begin{align} \frac{d}{\mathrm{d}R} \int_0^R f\left(R,r_1\right)\,dr_1 & = f\left(R,R\right) + \int_0^R \frac{d}{\mathrm{d}R}{\,}f\left(R,r_1\right)\,\mathrm{d}r_1 \end{align} \tag{ 4.8 }$$

$$\begin{align} & = 4\int_R^\infty e^{-R^2-r_2^2} Rr_2\,I_0\left(2Rr_2\right)\,dr_2 -4 \int_0^R e^{-r_1^2-R^2} r_1 R I_0\left(2r_1R\right)\,dr_1 \end{align} \tag{ 4.9 }$$

$$\begin{align} & = 4R e^{-R^2} \left( \int_R^\infty-\int_0^R\right) e^{-x^2} x I_0\left(2Rx\right)\,\mathrm{d}x = 4R e^{-R^2} g\left(R\right), \end{align} \tag{ 4.10 }$$

defining

$$\begin{equation} g\left(R\right): = \left( \int_R^\infty-\int_0^R\right) e^{-x^2} x I_0\left(2Rx\right)\,\mathrm{d}x. \end{equation} \tag{ 4.11 }$$

It is determined by the following two known integrals [52, equations (6.631.4), (6.631.8)], respectively

$$\begin{align} \int_0^\infty xe^{-x^2}I_0\left(2xR\right)& = \frac{1}{2}e^{R^2} \end{align} \tag{ 4.12 }$$

$$\begin{align} \int_0^R x e^{-x^2} I_0\left(2xR\right)& = \frac{1}{4}\left( e^{R^2}-e^{-R^2}I_0\left(2R^2\right)\right). \end{align} \tag{ 4.13 }$$

This yields for (4.10)

$$\begin{equation} 4R e^{-R^2} g\left(R\right) = 4R e^{-R^2} \left( \int_0^\infty-2\int_0^R\right) e^{-x^2} x I_0\left(2Rx\right)\,\mathrm{d}x = 2Re^{-2R^2}I_0\left(2R^2\right). \end{equation} \tag{ 4.14 }$$

On the one hand, the integral of (4.8) over $[0,R]$ trivially gives (4.6). On the other hand, using the previous result (4.14), we obtain from the same integral

$$\begin{equation} \operatorname{Var} \, \mathcal{N}^{\left(2\right)}\left( D_R\right) = \int_0^R 2xe^{-2x^2}I_0\left(2x^2\right)\mathrm{d}x = R^2e^{-2R^2}\left(I_0\left(2R^2\right)+I_1\left(2R^2\right)\right), \end{equation} \tag{ 4.15 }$$

thanks to the anti-derivative given in remark 2.5. The asymptotic behaviours (2.47) and (2.48) follow from the well-known asymptotic behaviours of the modified Bessel-function

$$\begin{equation} I_\nu\left(x\right) \sim \begin{cases} \dfrac{e^x }{\sqrt{2\pi x} }, & x \to \infty, \\ \dfrac{x^\nu}{2^\nu \nu! }, & x \to 0. \end{cases} \end{equation} \tag{ 4.16 }$$

This completes the proof. □

Remark 4.1. In order to prepare for the proof in the next subsection, where we will not have all integrals as in (4.13) at hand, we present a different determination of g(R) starting from (4.11). It contains a nice expression closely related to the generating function for $Q(k+1,R^2)$.

Inserting the series (2.44) for the Bessel function into the definition (4.11) we obtain

$$\begin{align*} g\left(R\right)& = \frac12\sum_{k = 0}^\infty \frac{R^{2k}}{k!} \left( Q\left(k+1,R^2\right)-P\left(k+1,R^2\right) \right) = \sum_{k = 0}^\infty \frac{R^{2k}}{k!} Q\left(k+1,R^2\right)- \frac{e^{R^2} }{2}. \end{align*}$$

Furthermore, using the expression

$$\begin{equation} Q\left(k+1,x\right) = e^{-x} \sum_{l = 0}^k \frac{x^l}{l!}, \end{equation} \tag{ 4.17 }$$

we have

$$\begin{align} \sum_{k = 0}^\infty \frac{R^{2k}}{k!} Q\left(k+1,R^2\right) & = e^{-R^2} \sum_{k = 0}^\infty \frac{R^{2k}}{k!} \sum_{l = 0}^k \frac{R^{2l} }{l!} = e^{-R^2 } \sum_{m = 0 }^\infty \frac{1}{m!} \sum_{k = \lceil \frac{m}{2}\rceil }^{m} \binom{m}{k} R^{2m}. \end{align} \tag{ 4.18 }$$

Here, we have introduced the variable $m = k+l$, with $k\unicode{x2A7E} l = m-k\unicode{x2A7E}0$ and $m\unicode{x2A7E} k$, resulting into the summation boundaries indicated, and $\lceil \frac{m}{2}\rceil$ is the ceiling function. The following two identities follow from the Binomial Theorem, by relabelling one of the sums in the first identity, and using it together with Pascal's identity to derive the second:

$$\begin{equation} 2\sum_{k = m}^{2m} \binom{2m}{k} = 2^{2m} + \binom{2m}{m}, \qquad \sum_{k = m+1}^{2m+1} \binom{2m+1}{k} = 2^{2m} . \end{equation} \tag{ 4.19 }$$

Splitting the sum in (4.18) into even and odd summands, we obtain

$$\begin{align*} \sum_{k = 0}^\infty \frac{R^{2k}}{k!} Q\left(k+1,R^2\right)& = \sum_{m = 0 }^\infty \frac{1}{\left(2m\right)!} \left( 2^{2m-1} +\frac{1}{2}\binom{2m}{m}\right) \left(R^2\right)^{2m} + \sum_{m = 0 }^\infty \frac{1}{\left(2m+1\right)!} 2^{2m} \left(R^{2}\right)^{2m+1} \\ & = \frac12 \sum_{m = 0}^\infty \frac{\left(2R^2\right)^m}{m!} +\frac12 \sum_{m = 0}^\infty \frac{\left(R^2\right)^{2m} }{\left(m!\right)^2 } = \frac12 e^{2R^2} + \frac12 I_0\left(2R^2\right). \end{align*}$$

This leads again to (4.14).

4.2. Number variance of the GinSE at the origin

For the GinSE, we have

$$\begin{equation} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) = -\int_{D_R }\,\mathrm{d}A\left(z\right) \int_{D_R^c }\,\mathrm{d}A\left(w\right)\, R_2^{\left(4\right),c}\left(z,w\right), \end{equation} \tag{ 4.20 }$$

where $R_2^{(4),c}(z,w) = R_2^{(4)}(z,w)-R_1^{(4)}(z)R_1^{(4)}(w)$ is the connected 2-point function.

Proof of proposition 2.4 (ii). For completeness we present $R_2^{(4),c}(z,w)$ in terms of the limiting kernel at the origin [61, 80]

$$\begin{equation} \kappa\left(z,w\right) = e^{z^2+w^2}\operatorname{erf} \left(z-w\right). \end{equation} \tag{ 4.21 }$$

Inserted this into the limit of (2.7) defined in (2.25), we obtain

$$\begin{equation} R_1^{\left(4\right)}\left(z\right) = \left(\bar{z}-z\right)e^{-2|z|^2 } \kappa\left(z,\bar{z}\right) \end{equation} \tag{ 4.22 }$$

and

$$\begin{equation} R_2^{\left(4\right)}\left(z,w\right) = \left(\bar{z}-z\right)\left(\bar{w}-w\right) e^{-2|z|^2-2|w|^2 } \left( \kappa\left(z,\bar{z}\right) \kappa\left(w,\bar{w}\right)-|\kappa\left(z,w\right)|^2+|\kappa\left(z,\bar{w}\right)|^2 \right). \end{equation} \tag{ 4.23 }$$

This leads to

$$\begin{align} \begin{aligned} R_2^{\left(4\right),c}\left(z,w\right)& = -\left(\bar{z}-z\right)\left(\bar{w}-w\right) e^{-2|z|^2-2|w|^2 } \left( |\kappa\left(z,w\right)|^2-|\kappa\left(z,\bar{w}\right)|^2 \right) \\ & = -\pi \left(\bar{z}-z\right)\left(\bar{w}-w\right) e^{-2|z|^2-2|w|^2 } \left( \Big| e^{z^2+w^2} \operatorname{erf}\left(z-w\right) \Big|^2- \Big| e^{z^2+\bar{w}^2} \operatorname{erf}\left(z- \bar{w} \right) \Big|^2 \right). \end{aligned} \end{align} \tag{ 4.24 }$$

In the following, we will use a different expression, directly following from the skew-orthogonal polynomial formalism [61, 80], where the angular integrals are more easily performed:

$$\begin{equation} \kappa\left(z,w\right) = \sum_{0 \unicode{x2A7D} l \unicode{x2A7D} k \unicode{x2A7D} \infty } \frac{2^{k+l+1} }{\left(2k+1\right)!!\, \left(2l\right)!! } \left( z^{2k+1} w^{2l} - w^{2k+1 } z^{2l} \right) = : G\left(z,w\right)-G\left(w,z\right), \end{equation} \tag{ 4.25 }$$

see e.g. [2]. Following the proof in [1, section 2.2], using the complex conjugation symmetry and the orthogonality, we have

$$\begin{align*} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = 2\int_{D }\,\mathrm{d}A\left(z\right) \int_{D^c }\,\mathrm{d}A\left(w\right)\,\left(\bar{z}-z\right)\left(\bar{w}-w\right) e^{-2|z|^2-2|w|^2 } |\kappa\left(z,w\right)|^2 \\ & = -4\int_{D }\,\mathrm{d}A\left(z\right) \int_{D^c }\,\mathrm{d}A\left(w\right)\,\left(zw+\bar{z}\bar{w}-z \bar{w}-\bar{z}w\right) e^{-2|z|^2-2|w|^2 } G\left(z,w\right) G\left( \bar{w} , \bar{z} \right). \\ & = 4\int_{D }\,\mathrm{d}A\left(z\right) \int_{D^c }\,\mathrm{d}A\left(w\right)\, \bar{z}w e^{-2|z|^2-2|w|^2 } G\left(z,w\right) G\left( \bar{w} , \bar{z} \right), \end{align*}$$

where only the third term in the second line can be shown to contribute [1]. The angular integrals project onto a single sum,

$$\begin{align} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = \sum_{k = 0}^\infty \frac{2^{4k+4}}{\left(2k+1\right)!^2} \int_{D }\,\mathrm{d}A\left(z\right) \int_{D^c }\,\mathrm{d}A\left(w\right)\, e^{-2|z|^2-2|w|^2 } |z|^{4k+2} |w|^{4k+2} \end{align} \tag{ 4.26 }$$

$$\begin{align} & = \sum_{k = 0}^\infty P\left(2k+2,2R^2\right) Q\left(2k+2,2R^2\right), \end{align} \tag{ 4.27 }$$

which leads to (2.49).

It remains to show (2.50). For this, note that the series expansion for the Bessel function of first kind,

$$\begin{equation} J_0\left(z\right) = \sum_{k = 0}^\infty\frac{\left(-1\right)^k\left(z/2\right)^{2k}}{\left(k!\right)^2}, \end{equation} \tag{ 4.28 }$$

together with that for the modified Bessel function $I_0(z)$ in (2.44), can be used to rewrite the sum in (4.26) as the following difference:

$$\begin{align} \begin{aligned} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = 2\int_{D }\,\mathrm{d}A\left(z\right) \int_{D^c }\,\mathrm{d}A\left(w\right)\, e^{-2|z|^2-2|w|^2 } \left( I_0\left(4|zw|\right)- J_0\left(4|zw|\right) \right) \\ & = 8 \int_0^R \,dr_1 \int_R^\infty \,dr_2\, e^{-2r_1^2-2r_2^2 } r_1r_2\left( I_0\left(4r_1r_2\right)- J_0\left(4r_1r_2\right) \right). \end{aligned} \end{align} \tag{ 4.29 }$$

The first integral follows from proposition 2.4 (i), by substituting $R^2\to2R^2$:

$$\begin{align*} 8 \int_0^R \,dr_1 \int_R^\infty \,dr_2\, e^{-2r_1^2-2r_2^2 } r_1r_2 I_0\left(4r_1r_2\right) = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) \right). \end{align*}$$

Next, we evaluate the second integral

$$\begin{equation} 8 \int_0^R \,dr_1 \int_R^\infty \,dr_2\, e^{-2r_1^2-2r_2^2 } r_1r_2 J_0\left(4r_1r_2\right) = \int_0^R \,dr_1 f\left(R,r_1\right), \end{equation} \tag{ 4.30 }$$

where we defined

$$\begin{equation} f\left(R,r_1\right): = 8 \int_R^\infty e^{-2r_1^2-2r_2^2} r_1 r_2\,J_0\left(4r_1r_2\right)\,dr_2. \end{equation} \tag{ 4.31 }$$

As in the previous subsection we obtain

$$\begin{align*} \frac{d}{\mathrm{d}R} \int_0^R f\left(R,r_1\right)\,dr_1 & = f\left(R,R\right) + \int_0^R \frac{d}{\mathrm{d}R}{\,}f\left(R,r_1\right)\,dr_1 \\ & = 8R \int_R^\infty e^{-2R^2-2r_2^2} r_2\,J_0\left(4Rr_2\right)\,dr_2 -8R \int_0^R e^{-2r_1^2-2R^2} r_1 J_0\left(4Rr_1\right)\,dr_1 \\ & = 8R e^{-2R^2} g\left(R\right) \end{align*}$$

with

$$\begin{equation} g\left(R\right): = \left( \int_R^\infty-\int_0^R\right) e^{-2x^2} x J_0\left(4Rx\right)\,\mathrm{d}x, \end{equation} \tag{ 4.32 }$$

now containing the function J₀ instead of I₀. Although the first integral in (4.12) can be analytically continued, we did not find the second integral with J₀ in the literature. Thus, we follow the same strategy as in remark 4.1, with (4.28) instead:

$$\begin{align} \begin{aligned} g\left(R\right) & = \left( \int_R^\infty-\int_0^R\right) e^{-2x^2} x \sum_{k = 0}^\infty \frac{\left(-1\right)^k \left(2Rx\right)^{2k} }{k!^2 } \,\mathrm{d}x \\ & = \frac{1}{4}\sum_{k = 0}^\infty \frac{\left(-1\right)^k\left(2R^2\right)^k}{k!} \left( Q\left(k+1,2R^2\right)- P\left(k+1,2R^2\right) \right) \\ & = \frac{1}{2}\sum_{k = 0}^\infty \frac{\left(-1\right)^k\left(2R^2\right)^k}{k!} Q\left(k+1,2R^2\right)-\frac14 e^{-2R^2} . \end{aligned} \end{align} \tag{ 4.33 }$$

Similarly to (4.18) we can thus write

$$\begin{align*} \frac{1}{2}\sum_{k = 0}^\infty \frac{\left(-1\right)^k\left(2R^2\right)^k}{k!} Q\left(k+1,2R^2\right) & = \frac{1}{2} e^{-2R^2} + \frac{1}{2} e^{-2R^2} \sum_{m = 1}^\infty \frac{1}{m!} \sum_{k = \lceil \frac{m}{2}\rceil }^{m} \left(-1\right)^k \binom{m}{k} \left(2R^2\right)^{m}. \end{align*}$$

Similarly to the remark 4.1 we can use two combinatorial identities, that follow from the Binomial theorem, this time with alternating coefficients. For that reason the terms 2^m are absent, and we obtain

$$\begin{align} & 2\sum_{k = m }^{2m} \left(-1\right)^k \binom{2m}{k} = \left(-1\right)^m \binom{2m}{m}, \qquad \left(m \unicode{x2A7E} 1\right), \end{align} \tag{ 4.34 }$$

$$\begin{align} &\sum_{k = m+1 }^{2m+1} \left(-1\right)^k \binom{2m+1}{k} = - \left(-1\right)^m \binom{2m}{m}, \qquad \left(m \unicode{x2A7E} 0\right). \end{align} \tag{ 4.35 }$$

Using this, we have

$$\begin{align} \begin{aligned} &\frac{1}{2}\sum_{k = 0}^\infty \frac{\left(-1\right)^k\left(2R^2\right)^k}{k!} Q\left(k+1,2R^2\right) \\ & \quad = \frac{e^{-2R^2}}{2}\left(1+ \sum_{m = 1 }^\infty \frac{1}{\left(2m\right)!} \left(-1\right)^m\frac{1}{2}\binom{2m}{m} \left(2R^2\right)^{2m} + \sum_{m = 0 }^\infty \frac{1}{\left(2m+1\right)!} \left(-1\right)^{m+1}\binom{2m}{m} \left(2R^{2}\right)^{2m+1} \right) \\ & \quad = \frac{e^{-2R^2}}{4} \left(1+ J_0\left(4R^2\right) \right) - e^{-2R^2} \,R^2\,{}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 R^4 \right) . \end{aligned} \end{align} \tag{ 4.36 }$$

The last step follows from (4.28) and from spelling out the hypergeometric function defined in (2.15), in this case as

$$\begin{equation} {}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 x^4 \right) = \sum_{k = 0}^\infty \frac{\left(-1\right)^k \left(2x^2\right)^{2k} }{\left(k!\right)^2\left(2k+1\right) }. \end{equation} \tag{ 4.37 }$$

Inserted into (4.33) we thus obtain

$$\begin{equation} g\left(R\right) = \frac14 e^{-2R^2} J_0\left(4R^2\right) - e^{-2R^2} \,R^2\,{}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 R^4 \right). \end{equation} \tag{ 4.38 }$$

Therefore, we have shown that

$$\begin{align} \begin{aligned} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) \right) \\ &\quad -2 \int_0^R x e^{-4x^2} \left( J_0\left(4x^2\right) -4 x^2\,{}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 x^4 \right) \right)\,\mathrm{d}x. \end{aligned} \end{align} \tag{ 4.39 }$$

It follows from

$$\begin{equation} \frac{\partial}{\partial x} \left(x^2 {}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 x^4 \right)\right) = 2x J_0\left(4x^2\right) \end{equation} \tag{ 4.40 }$$

and $\frac{\partial}{\partial x}(e^{-4x^2}) = -8xe^{-4x^2}$, that the integrand in the integral (4.39) is a total derivative. This leads to the final answer

$$\begin{align*} \operatorname{Var} \mathcal{N}^{\left(4\right)}\left(D_R\right) & = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) \right) -R^2e^{-4R^2}\,{}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -4 R^4 \right). \end{align*}$$

The asymptotic behaviours (2.51) and (2.52) follow from (4.16) and that of the hypergeometric function [83, section 16.11(ii)] and

$$\begin{equation} {}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -x \right) = \frac{1}{2\sqrt{x}}+O\left(x^{-\frac{3}{2}}\right), \qquad x \to \infty, \end{equation} \tag{ 4.41 }$$

and

$$\begin{equation} {}_1 F_2 \left( \frac{1}{2}; 1,\frac{3}{2}; -x \right) = 1 +O\left(x\right) , \qquad x \to 0. \end{equation} \tag{ 4.42 }$$

 □

To derive (2.54), observe from (4.40) that

$$\begin{align} \begin{aligned} R^2 {}_1 F_2 \left( 1/2; 1,3/2; -4 R^4 \right) & = \int_0^R 2t J_0\left(4t^2\right)\,\mathrm{d}t = \frac{1}{4}\int_0^{4R^2} J_0\left(s\right)\,\mathrm{d}s \\ & = R^2 J_0\left(4R^2\right)+\frac{\pi}{2} R^2 \left( J_1\left(4R^2\right) \textbf{H}_{0}\left(4R^2\right)-J_{0}\left(4R^2\right) \textbf{H}_1\left(4R^2\right) \right), \end{aligned} \end{align} \tag{ 4.43 }$$

where we used [52, 6.511.6]. Then the expression (2.54) follows from the recurrence relation [83, equation (11.4.23)]

$$\begin{equation} \textbf{H}_{\nu-1}\left(z\right)+\textbf{H}_{\nu+1}\left(z\right) = \frac{2\nu}{z} \textbf{H}_\nu\left(z\right) + \frac{\left(z/2\right)^\nu}{\sqrt{\pi} \, \Gamma\left(\nu+\frac32\right) }, \end{equation} \tag{ 4.44 }$$

with ν = 0.

Remark 4.2. Combining propositions 2.4 (i) and (ii), we also obtain

$$\begin{align} \sum_{k = 0}^\infty P\left(2k+1,2R^2\right) Q\left(2k+1,2R^2\right) = R^2 e^{-4R^2 } \left( I_0\left(4 R^2\right) + I_1\left(4 R^2\right) + {}_1 F_2 \left( 1/2; 1,3/2; -4 R^4 \right) \right). \end{align} \tag{ 4.45 }$$

The left-hand side of this identity appears in the number variance of the induced GinSE, see [1, proposition 1.4].

4.3. Number variance of the GinOE at the origin

In this subsection, we show theorem 2.6.

Proposition 4.3 (Expression of the number variance of the infinite GinOE). For any radius $R \in [0, \infty)$ we have the following.

• (i)  
  We have for the real eigenvalues

  $$\begin{align} \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) = 2 \sqrt{\frac{2}{\pi} } R - \frac{2}{\sqrt{\pi}} R \operatorname{erf}\left(2R\right) -\frac12 \operatorname{erf}\left( \sqrt{2}R \right) + \frac14 \operatorname{erf}\left( \sqrt{2}R \right)^2 + \frac{1}{\pi} \left( 1 -e^{-4R^2} \right) .\nonumber\\ \end{align} \tag{ 4.46 }$$
• (ii)  
  We have for the mixing between real and complex eigenvalues

  $$\begin{align} \operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) & = \frac{2}{\pi} \int_{0}^R \operatorname{erfc}\left(\sqrt{2}v\right) v \,e^{v^2} \quad \Bigg( e^{-\left(\hat{a} - R\right)^2}-e^{-\left(\hat{a} + R\right)^2}\nonumber\\ & \quad + \sqrt{\pi}\left(\hat{a} - R\right) \operatorname{erf}\left(\hat{a} - R\right) -\sqrt{\pi} \left(\hat{a} + R\right) \operatorname{erf}\left(\hat{a} + R\right) \Bigg)\,\mathrm{d}v, \nonumber \end{align} \tag{ 4.47 }$$

  where $\hat{a} \equiv \hat{a}(R) = \sqrt{R^2-v^2}.$
• (iii)  
  We have for the complex eigenvalues

  $$\begin{equation} \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) = 2 E_{\mathbb{C}}^{\left(1\right)}\left(R\right) + 4 \left( I_-\left(R\right) - I_+\left(R\right) \right) \end{equation} \tag{ 4.48 }$$

  where

  $$\begin{equation} I_\pm\left(R\right): = I_{\pm,1}\left(R\right) + I_{\pm,2}\left(R\right) + I_{\pm,3}\left(R\right) + I_{\pm,4}\left(R\right). \end{equation} \tag{ 4.49 }$$

  Here,

  $$\begin{align*} I_{\pm,1}\left(R\right) & = \frac{1}{2\pi} \int_0^R \int_0^R \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 \pm y_2\right)^2} e^{-\left(a+b\right)^2} \left(1+\left(y_1 \pm y_2\right)^2\right) \,\mathrm{d}y_2 \,\mathrm{d}y_1, \\ I_{\pm,2}\left(R\right) & = -\frac{1}{2\pi} \int_0^R \int_0^R \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 \pm y_2\right)^2} e^{-\left(a-b\right)^2} \left(1+\left(y_1 \pm y_2\right)^2\right) \,\mathrm{d}y_2 \,\mathrm{d}y_1, \\ I_{\pm,3}\left(R\right) & = \frac{1}{4\sqrt{\pi}} \int_0^R \int_0^R \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 \pm y_2\right)^2} \left(a+b\right)\operatorname{erf}\left(a+b\right)\\ &\quad \times \left(1+2\left(y_1 \pm y_2\right)^2\right) \,\mathrm{d}y_2 \,\mathrm{d}y_1, \\ I_{\pm,4}\left(R\right) & = -\frac{1}{4\sqrt{\pi}} \int_0^R \int_0^R \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}(\sqrt{2}y_2) e^{(y_1 \pm y_2)^2} \\ & \quad \times (a-b)\operatorname{erf}(a-b) (1+2(y_1\pm y_2)^2) \,\mathrm{d}y_2 \,\mathrm{d}y_1, \end{align*}$$

  where $a \equiv a(R) = \sqrt{R^2-y_1^2}$ and $b \equiv b(R) = \sqrt{R^2-y_2^2}$.

Remark 4.4 (Number variance of the finite GinOE). The counterpart of proposition 4.3 for the finite GinOE can be computed analogously. This involves leveraging the 1- and 2-point correlation functions of the real and complex eigenvalues, along with the mixed real and complex $(1,1)$-point function available in the [13, 46, 95]. The resulting formula requires straightforward yet lengthy computations. Nonetheless, we do not include it here because it exceeds the scope of this paper.

Proof of proposition 4.3. We first show (i). Note that

$$\begin{equation} \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) = \int_{-R}^R R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right)\,\mathrm{d}x +\int_{\left(-R,R\right)^2} \left(R_{2,\mathbb{R}}^{\left(1\right)}\left(x,y\right)-R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) R_{1,\mathbb{R}}^{\left(1\right)}\left(y\right)\right)\, \mathrm{d}x\,\mathrm{d}y, \end{equation} \tag{ 4.50 }$$

see e.g. [46] or [80, chapter 16]. Since

$$\begin{equation} R_{2,\mathbb{R}}^{\left(1\right)}\left(x,y\right) = S_{\mathbb{R}}\left(x,x\right)S_{\mathbb{R}}\left(y,y\right)-D_{\mathbb{R}}\left(x,y\right)\tilde{I}_{\mathbb{R}}\left(x,y\right)-S_{\mathbb{R}}\left(x,y\right)S_{\mathbb{R}}\left(y,x\right), \end{equation} \tag{ 4.51 }$$

as given in the appendix, we have

$$\begin{equation} \begin{aligned} \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right)& = E_{\mathbb{R}}^{\left(1\right)}\left(R\right)-\int_{\left(-R,R\right)^2}\left( S_{\mathbb{R}}\left(x,y\right)^2 + \tilde{I}_{\mathbb{R}}\left(x,y\right) D_{\mathbb{R}}\left(x,y\right)\right)\, \mathrm{d}x\,\mathrm{d}y. \end{aligned} \end{equation} \tag{ 4.52 }$$

With the limiting matrix elements given in (A.13)–(A.15), note that

$$\begin{align} \begin{aligned} -\int_{\left(-R,R\right)^2} \operatorname{sgn}\left(x-y\right) D_{\mathbb{R}}\left(x,y\right) \,\mathrm{d}x\,\mathrm{d}y & = \int_{\left(-R,R\right)^2} \operatorname{sgn}\left(x-y\right) \frac{\partial}{\partial y} S_{\mathbb{R}}\left(x,y\right) \,\mathrm{d}x\,\mathrm{d}y \\ & = \int_{-R}^R \left[ \int_{-R}^R \frac{\partial}{\partial y} S_{\mathbb{R}}\left(x,y\right) \,\mathrm{d}y -2 \int_{x}^R \frac{\partial}{\partial y} S_{\mathbb{R}}\left(x,y\right) \,\mathrm{d}y \right] \,\mathrm{d}x \\ & = \int_{-R}^R 2S_{\mathbb{R}}\left(x,x\right)-S_{\mathbb{R}}\left(x,R\right)-S_{\mathbb{R}}\left(x,-R\right) \,\mathrm{d}x. \end{aligned} \end{align} \tag{ 4.53 }$$

Furthermore, since

$$\begin{align*} & \int_{-R}^R \int_x^y S_{\mathbb{R}}\left(t,x\right)\,\mathrm{d}t \frac{\partial}{\partial y} S_{\mathbb{R}}\left(x,y\right)\,\mathrm{d}y \\ &\quad = S_{\mathbb{R}}\left(x,y\right) \int_x^y S_{\mathbb{R}}\left(t,x\right)\,\mathrm{d}t \Big|_{-R}^R- \int_{-R}^R S_{\mathbb{R}}\left(x,y\right)^2 \,\mathrm{d}y \\ &\quad = S_{\mathbb{R}}\left(x,R\right) \int_x^R S_{\mathbb{R}}\left(t,x\right)\,\mathrm{d}t- S_{\mathbb{R}}\left(x,-R\right) \int_{x}^{-R} S_{\mathbb{R}}\left(t,x\right)\,\mathrm{d}t - \int_{-R}^R S_{\mathbb{R}}\left(x,y\right)^2 \,\mathrm{d}y, \end{align*}$$

we have

$$\begin{align} \begin{aligned} \operatorname{Var} \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) & = 2 E_{\mathbb{R}}^{\left(1\right)}\left(R\right) -2 \int_{\left(-R,R\right)^2} S\left(x,y\right)^2\,\mathrm{d}x\,\mathrm{d}y -\frac12 \int_{-R}^R \,\mathrm{d}x\, \left( S\left(x,R\right)+S\left(x,-R\right)\right) \\ & \quad + \int_{-R}^R\,\mathrm{d}x \left[ S\left(x,R\right) \int_x^R S\left(t,x\right)\,\mathrm{d}t + S\left(x,-R\right) \int_{-R}^{x} S\left(t,x\right)\,\mathrm{d}t \right]. \end{aligned} \end{align} \tag{ 4.54 }$$

Since

$$\begin{align*} \int_{\left(-R,R\right)^2 } S\left(x,y\right)^2 \,\mathrm{d}x\,\mathrm{d}y & = \frac{1}{2\pi} \int_{\left(-R,R\right)^2 } e^{-\left(x-y\right)^2}\,\mathrm{d}x\,\mathrm{d}y = \frac{1}{2\pi} \left( 2\sqrt{\pi} R \operatorname{erf}\left(2R\right) -1 +e^{-4R^2} \right), \end{align*}$$

we have together with $E_{\mathbb{R}}^{(1)}(R)$ from proposition 2.3

$$\begin{equation} \begin{aligned} 2 E_{\mathbb{R}}^{\left(1\right)}\left(R\right) -2 \int_{\left(-R,R\right)^2} S\left(x,y\right)^2\,\mathrm{d}x\,\mathrm{d}y & = 2 \sqrt{\frac{2}{\pi} } R - \frac{2}{\sqrt{\pi}} R \operatorname{erf}\left(2R\right) + \frac{1}{\pi} \left( 1 -e^{-4R^2} \right). \end{aligned} \end{equation} \tag{ 4.55 }$$

Similarly, by direct computation, we have

$$\begin{equation} -\frac12 \int_{-R}^R \,\mathrm{d}x\, \left( S\left(x,R\right)+S\left(x,-R\right)\right) = -\frac12 \operatorname{erf}\left( \sqrt{2}R \right) \end{equation} \tag{ 4.56 }$$

and

$$\begin{equation} \int_{-R}^R\,\mathrm{d}x \left[ S_{\mathbb{R}}\left(x,R\right) \int_x^R S_{\mathbb{R}}\left(t,x\right)\,\mathrm{d}t + S_{\mathbb{R}}\left(x,-R\right) \int_{-R}^{x} S\left(t,x\right)\,\mathrm{d}t \right] = \frac14 \operatorname{erf}\left( \sqrt{2}R \right)^2. \end{equation} \tag{ 4.57 }$$

By inserting the last three equations into (4.54), we obtain proposition 4.3 (i).

Next, we show (ii). By definition, we have

$$\begin{align} \begin{aligned} & \operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) = 2 \, \operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(H_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) \\ & \quad = 2\int_{H_R} \frac{\mathrm{d}u\,\mathrm{d}v}{\pi} \int_{-R}^R \,\mathrm{d}x \, \left( R_{1,\mathbb{R},1,\mathbb{C}}^{\left(1\right)}\left(x,u+iv\right)- R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(u+iv\right) \right) , \end{aligned} \end{align} \tag{ 4.58 }$$

where $H_R = \mathbb{C}_+ \cap D_R$ is the semi-disc of radius R in the upper-half plane. On the one hand, it follows from (A.16) that

$$\begin{equation} R_{1,\mathbb{R},1,\mathbb{C}}^{\left(1\right)}\left(x,u+iv\right)- R_{1,\mathbb{R}}^{\left(1\right)}\left(x\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(u+iv\right) = -\operatorname{erfc}\left(\sqrt{2}v\right) v\, e^{-\left(x-u\right)^2+v^2 } . \end{equation} \tag{ 4.59 }$$

Note that

$$\begin{align*} & \int_{-\hat{a}}^{\hat{a}} \,\mathrm{d}u \int_{-R}^R \,\mathrm{d}x\, e^{-\left(x-u\right)^2 } \\ & \quad = \sqrt{\pi}\left(\frac{e^{-\left(\hat{a}+ R\right)^2}-e^{-\left( \hat{a}-R\right)^2} }{\sqrt{\pi}}+ \left(\hat{a} + R\right) \operatorname{erf}\left(\hat{a} + R\right) - \left(\hat{a} - R\right) \operatorname{erf}\left(\hat{a} - R\right) \right), \end{align*}$$

where we use $\hat{a} \equiv \hat{a}(R) = \sqrt{R^2-v^2}$ for the bounds of the u integral. Therefore we obtain

$$\begin{align*} \operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(H_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) & = -\frac{1}{\pi} \int_{0}^R \operatorname{erfc}\left(\sqrt{2}v\right) v \,e^{v^2}\Bigg( e^{-\left(\hat{a} + R\right)^2}-e^{-\left(\hat{a} - R\right)^2}\\ & \quad+ \sqrt{\pi}\left(\hat{a} + R\right) \operatorname{erf}\left(\hat{a} + R\right) -\sqrt{\pi} \left(\hat{a} - R\right) \operatorname{erf}\left(\hat{a} - R\right) \Bigg)\,\mathrm{d}v. \end{align*}$$

Finally, we show (iii). Note that

$$\begin{align} \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) & = \int_{D_R } R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right)\,\mathrm{d}A\left(z\right) +\int_{D_R^2} \left( R_{2,\mathbb{C}}^{\left(1\right)} \left(z,w\right)-R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(w\right) \right) \,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right) \end{align} \tag{ 4.60 }$$

$$\begin{align} & = E_{\mathbb{C}}^{\left(1\right)}\left(R\right) +\int_{D_R^2} \left( R_{2,\mathbb{C}}^{\left(1\right)} \left(z,w\right)-R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(w\right) \right) \,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right). \end{align} \tag{ 4.61 }$$

Since

$$\begin{equation} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) = 2 \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(H_R\right), \end{equation} \tag{ 4.62 }$$

we have

$$\begin{equation} E_{\mathbb{C}}^{\left(1\right)}\left(R\right) = 2 \int_{H_R } R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right)\,\mathrm{d}A\left(z\right) , \quad \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) = 4 \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(H_R\right). \end{equation} \tag{ 4.63 }$$

Here,

$$\begin{align} \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(H_R\right) & = \int_{H_R } R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right)\,\mathrm{d}A\left(z\right)+ \int_{H_R^2 } \left( R_{2,\mathbb{C}}^{\left(1\right)} \left(z,w\right)-R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(w\right) \right) \,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right). \end{align} \tag{ 4.64 }$$

By (A.10), we have

$$\begin{align} \begin{aligned} & R_{2,\mathbb{C}} \left(z_1,z_2\right)-R_{1,\mathbb{C}}\left(z_1\right) R_{1,\mathbb{C}}\left(z_2\right) \\ & \quad = \frac{\pi}{2} \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{-\left(x_1-x_2\right)^2} \\ & \qquad \times \left[ - \left( \left(x_1-x_2\right)^2 +\left(y_1+y_2\right)^2 \right) e^{\left(y_1+y_2\right)^2 }+ \left( \left(x_1-x_2\right)^2 +\left(y_1-y_2\right)^2 \right) e^{\left(y_1-y_2\right)^2 } \right], \end{aligned} \end{align} \tag{ 4.65 }$$

where $z_k = x_k+iy_k$ ($k = 1,2$) with $y_k\gt0.$ Using the Gaussian integrals

$$\begin{align} \begin{aligned} & \int_{-a}^{a} \int_{-b}^{b} e^{-\left(x_1-x_2\right)^2} \left( \left(x_1-x_2\right)^2 + c \right) \,\mathrm{d}x_2 \,\mathrm{d}x_1 \\ & \quad = \left( e^{-\left(a+b\right)^2} - e^{-\left(a-b\right)^2} \right) \left(1+c\right) + \frac{\sqrt{\pi}}{2} \left( \left(a+b\right)\operatorname{erf}\left(a+b\right) - \left(a-b\right)\operatorname{erf}\left(a-b\right) \right) \left(1+2c\right), \end{aligned} \end{align} \tag{ 4.66 }$$

we obtain

$$\begin{align} \int_{H_R ^2 } \left( R_{2,\mathbb{C}}^{\left(1\right)} \left(z,w\right)-R_{1,\mathbb{C}}^{\left(1\right)}\left(z\right) R_{1,\mathbb{C}}^{\left(1\right)}\left(w\right) \right) \,\mathrm{d}A\left(z\right)\,\mathrm{d}A\left(w\right) = I_-\left(R\right) - I_+\left(R\right). \end{align} \tag{ 4.67 }$$

Now the proposition follows. □

To prove theorem 2.6 (iii), we first show the following lemma.

Lemma 4.5. As $R \to \infty$, we have

$$\begin{align} I_{-,1}\left(R\right) & = O\left(R^4 e^{-2R^2}\right), & I_{-,2}\left(R\right) & = O\left(1\right), \end{align} \tag{ 4.68 }$$

$$\begin{align} I_{-,3}\left(R\right) & = \frac{\pi \left(2 \sqrt{2} - 1\right) - 4}{4 \pi^{3/2}} R + O\left( \frac{1}{R} \right), & I_{-,4}\left(R\right) & = O\left( \frac{1}{R^2} \right), \end{align} \tag{ 4.69 }$$

and

$$\begin{align} I_{+,1}\left(R\right) & = O\left( \frac{1}{R^2} \right),& \qquad\qquad\qquad\qquad\qquad\qquad I_{+,2}\left(R\right) = -\frac{R}{4 \sqrt{\pi}} + o\left(R\right), \end{align} \tag{ 4.70 }$$

$$\begin{align} I_{+,3}\left(R\right) & = \frac{1}{2} R^2 - \frac{4 + \pi \left(2 \sqrt{2} - 1\right)}{4 \pi^{3/2}} R + o\left(R\right), & \qquad I_{+,4}\left(R\right) = -\frac{R}{4 \sqrt{\pi}} + o\left(R\right). \end{align} \tag{ 4.71 }$$

A comparison with numerics shows that the o(R) error terms for $I_{+,2}(R)$, $I_{+,3}(R)$ and $I_{+,4}(R)$ are in fact of order O(1) for $I_{+,2}(R)$, and $O(\sqrt{R})$ for $I_{+,3}(R)$ and $I_{+,4}(R)$. While we do not need a more precise asymptotic analysis for theorem 2.6, we note that these subleading behaviours can be derived analytically in a similar way as our analysis of $I_{+,3}(R)$.

Proof. To analyse $I_{-,1}(R)$ and $I_{-,2}(R)$, we first note that

$$\begin{align*} \left(y_1 - y_2\right)^2 - \left(a \pm b\right)^2 = 2 \left(y_1^2 + y_2^2\right) - 2 y_1 y_2 - 2 R^2 \left( 1 \pm \sqrt{1 - \frac{y_1^2 + y_2^2}{R^2} + \frac{y_1^2 y_2^2}{R^4}} \right) \end{align*}$$

and that for any $y_1,y_2 \in [0,R],$

$$\begin{align*} 0 \unicode{x2A7D} \sqrt{1 - \frac{y_1^2 + y_2^2}{R^2} + \frac{y_1^2 y_2^2}{R^4}} \unicode{x2A7D} 1 - \frac{y_1^2 + y_2^2}{2 R^2}. \end{align*}$$

Combining these with elementary estimates

$$\begin{align*} \operatorname{erfc}\left(\sqrt{2} y\right) \unicode{x2A7D} e^{-2 y^2}, \qquad 1 + \left(y_1 - y_2\right)^2 \unicode{x2A7D} 1 + R^2, \qquad e^{-2 y_1 y_2} \unicode{x2A7D} 1, \end{align*}$$

we obtain

$$\begin{align} |I_{-,1}\left(R\right)| \unicode{x2A7D} \frac{1}{2 \pi} \int_0^R \int_0^R e^{-2 \left(y_1^2 + y_2^2\right)} \, e^{2 \left(y_1^2 + y_2^2\right) - 2 y_1 y_2 - 2 R^2} \left(1 + R^2\right) \,\mathrm{d}y_1 \,\mathrm{d}y_2 \unicode{x2A7D} \frac{1}{2 \pi} R^4 e^{-2 R^2}. \end{align} \tag{ 4.72 }$$

The integrand of $I_{-,2}(R)$ is bounded by the integrable function $e^{-(y_1+y_2)^2} (1 + (y_1 - y_2)^2)$ and thus

$$\begin{align} \lvert I_{-,2}\left(R\right)\rvert \unicode{x2A7D} -\frac{1}{2\pi} \int_0^\infty \int_0^\infty e^{-\left(y_1+y_2\right)^2} \left(1 + \left(y_1 - y_2\right)^2\right) \,\mathrm{d}y_2 \,\mathrm{d}3y_1 = \frac{1}{3 \pi}. \end{align} \tag{ 4.73 }$$

These give (4.68).

For $I_{-,3}(R)$ and $I_{-,4}(R)$, we observe that for $y_1,y_2 \unicode{x2A7E} 0,$

$$\begin{align*} \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1-y_2\right)^2} \unicode{x2A7D} e^{-\left(y_1^2+y_2^2\right)}. \end{align*}$$

After expanding the other factors in the integrand using a Taylor series, it can be observed that the error of the approximation is polynomial in y₁ and y₂. As a result, the integral over this error will remain bounded as $R \to \infty.$ Note that

$$\begin{align*} a + b = \sqrt{R^2-y_1^2} + \sqrt{R^2-y_2^2} = 2 R + O\left(\frac{1}{R}\right), \qquad \operatorname{erf}\left(a+b\right) = 1 + O\left(e^{-4R^2}\right). \end{align*}$$

Using these, we have

$$\begin{align} I_{-,3}\left(R\right) & = \frac{1}{4\sqrt{\pi}} \int_0^\infty \int_0^\infty \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 - y_2\right)^2} \left( 2 R + O\left(\frac{1}{R}\right) \right)\nonumber\\ &\quad \left(1+2\left(y_1 - y_2\right)^2\right) \,\mathrm{d}y_2 \,\mathrm{d}y_1 = \frac{R}{2\sqrt{\pi}} \frac{\pi \left(2 \sqrt{2} - 1\right) - 4}{2 \pi} + O\left(\frac{1}{R}\right). \end{align} \tag{ 4.74 }$$

Here, the double integral has been evaluated as follows. For the inner integral over y₂, we obtain from an integration by parts

$$\begin{align*} \begin{aligned} & \int_0^\infty \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 - y_2\right)^2}\left(1+2\left(y_1 - y_2\right)^2\right) \,\mathrm{d}y_2\\ & \quad = \left[ \operatorname{erfc}\left(\sqrt{2}y_2\right)\left(y_2-y_1\right)e^{\left(y_2-y_1\right)^2}\right]_0^\infty+ \frac{2\sqrt{2}}{\sqrt{\pi}}\int_0^\infty e^{-2y_2^2}\left(y_2-y_1\right)e^{\left(y_2-y_1\right)^2} \,\mathrm{d}y_2 \\ & \quad = e^{y_1^2}\left(y_1+\frac{2\sqrt{2 }}{\sqrt{\pi}}e^{y_1^2}\left( \frac{1}{2}e^{-y_1^2}-y_1\sqrt{\pi}\operatorname{erfc}\left(y_1\right)\right)\right) \;. \end{aligned} \end{align*}$$

The remaining integral over y₁ contains three parts:

$$\begin{align} \int_0^\infty\operatorname{erfc}\left(\sqrt{2}y_1\right) e^{y_1^2}\left(y_1+\sqrt{\frac{2}{\pi}}-2\sqrt{2}y_1\operatorname{erfc}\left(y_1\right)e^{y_1^2}\right)\,\mathrm{d}y_1. \end{align} \tag{ 4.75 }$$

For the first term we obtain [52, 6.238.1]

$$\begin{equation} \int_0^\infty\operatorname{erfc}\left(\sqrt{2}y_1\right) e^{y_1^2}y_1\,\mathrm{d}y_1 = -\frac{1}{2}+\frac{\sqrt{2}}{2}. \end{equation} \tag{ 4.76 }$$

For the third term we have

$$\begin{align*} & -2\sqrt{2}\int_0^\infty\operatorname{erfc}\left(\sqrt{2}y_1\right)\operatorname{erfc}\left(y_1\right) e^{2y_1^2}y_1\,\mathrm{d}y_1\\ & \quad = -2\sqrt{2}\left[ \operatorname{erfc}\left(\sqrt{2}y_1\right)\operatorname{erfc}\left(y_1\right)\frac{1}{4}e^{2y_1^2}\right]_0^\infty+2\sqrt{2}\int_0^\infty \left( \operatorname{erfc}\left(\sqrt{2}y_1\right)\frac{-2}{\sqrt{\pi}}e^{-y_1^2}-\frac{2\sqrt{2}}{\sqrt{\pi}}e^{2y_1^2}\operatorname{erfc}\left(y_1\right) \right)\\ & \qquad \times \frac{1}{4}e^{2y_1^2}\,\mathrm{d}y_1 = \frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{\sqrt{\pi}}\int_0^\infty \operatorname{erfc}\left(\sqrt{2}y_1\right)e^{y_1^2}\,\mathrm{d}y_1-\frac{2}{\sqrt{\pi}}\int_0^\infty \operatorname{erfc}\left(y_1\right)\,\mathrm{d}y_1. \end{align*}$$

Here, the second integral cancels the second term in (4.75), and the last integral can be evaluated using [52, 6.281.1], to give $1/\sqrt{\pi}$. Putting all together yields the value given in (4.74).

Similarly, it follows for $I_{-,4}(R)$ from

$$\begin{align*} a - b = O\left(\frac{1}{R}\right), \qquad \operatorname{erf}\left(a-b\right) = O\left(\frac{1}{R}\right), \end{align*}$$

that we can estimate $I_{-,4}(R) = O(1/R^2)$ without further computation. We have shown (4.69).

It remains to show (4.70) and (4.71). For $I_{+,1}(R)$, using

$$\begin{align*} \exp\left(\left(y_1+y_2\right)^2 - \left(a+b\right)^2\right) \unicode{x2A7D} \exp\left(3\left(y_1^2+y_2^2\right) - 2R^2\right), \qquad 1+\left(y_1+y_2\right)^2 \unicode{x2A7D} 1+4R^2, \end{align*}$$

one can decouple the integrals

$$\begin{align} \begin{aligned} |I_{+,1}\left(R\right)| &\unicode{x2A7D} \frac{1}{2\pi} \int_0^R \int_0^R \operatorname{erfc}\left(\sqrt{2}y_1\right) \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{3\left(y_1^2+y_2^2\right)-2R^2} \left(1+4R^2\right) \,\mathrm{d}y_2 \,\mathrm{d}y_1 \\ & = \frac{1+4R^2}{2\pi} \left( e^{-R^2} \int_0^R \operatorname{erfc}\left(\sqrt{2} y\right) e^{3y^2} \,\mathrm{d}y \right)^2. \end{aligned} \end{align} \tag{ 4.77 }$$

When R is large, the function $y \mapsto \operatorname{erfc}(\sqrt{2} y) e^{3y^2}$ reaches its maximum value within the interval $[0,R]$ at y = R. A saddle-point approximation for $f(y) = \log[\operatorname{erfc}(\sqrt{2}y)]+3y^2$, with $f^{^{\prime}}(y)\sim 2y$ for large y thus gives

$$\begin{align*} \int_0^R \operatorname{erfc}\left(\sqrt{2} y\right) e^{3y^2} \,\mathrm{d}y &\sim \frac{e^{R^2}}{\sqrt{2\pi}R}\int_0^R e^{\left(y-R\right)2R}\,\mathrm{d}y = \frac{e^{R^2}}{\sqrt{2\pi}R}\frac{\left(1-e^{-2R^2}\right)}{2R}. \end{align*}$$

Inserted into (4.77) this demonstrates that $I_{+,1}(R) = O(1/R^2)$.

For $I_{+,2}(R)$, $I_{+,3}(R)$, and $I_{+,4}(R)$, it is convenient to use light-cone coordinates

$$\begin{align*} \begin{cases} u = y_1 - y_2, \\ v = y_1 + y_2, \end{cases} \qquad \textrm{i.e.} \qquad \begin{cases} y_1 = \frac{1}{2} \left(v + u\right) , \\ y_2 = \frac{1}{2} \left(v - u\right). \end{cases} \end{align*}$$

From these transformations, we have $\mathrm{d}y_2 \,\mathrm{d}y_1 = \frac{R}{2} \mathrm{d}u \,\mathrm{d}s$, where the integration domains are given by

$$\begin{equation} s \in \left[0, 2\right], \qquad u \in \left[R \lvert s - 1\rvert - R, R - R \lvert s - 1\rvert\right]. \end{equation} \tag{ 4.78 }$$

Note that as $R \to \infty$, we have

$$\begin{align*} \operatorname{erfc}\left( \frac{Rs+u}{\sqrt{2}} \right) \operatorname{erfc}\left( \frac{Rs-u}{\sqrt{2}} \right) e^{R^2 s^2} \left(1 + c R^2 s^2\right) = \frac{2 c }{\pi} e^{-u^2} +O\left(\frac{1}{R^2}\right) \end{align*}$$

and

$$\begin{align*} a + b & = \sqrt{R^2-\left( \frac{Rs+u}{2} \right)^2} + \sqrt{R^2-\left( \frac{Rs-u}{2} \right)^2} = \sqrt{4-s^2} \, R+O\left(\frac{1}{R}\right), \\ a - b & = -\frac{us}{\sqrt{4-s^2}}+O\left(\frac{1}{R^2}\right). \end{align*}$$

Using this, the s-integral in $I_{+,2}(R)$ is dominated by contributions from $0 \lt s \lt 2$. Therefore, $|s-1|\lt1$ and thus we can set the bounds of the u-integral in (4.78) to $\pm\infty$ and arrive at the following integral which is solvable:

$$\begin{align} I_{+,2}\left(R\right) = -\frac{R}{2 \pi^2} \int_0^2 \int_{-\infty}^{\infty} e^{-\frac{4}{4-s^2} u^2} \,\mathrm{d}u \,\mathrm{d}s + o\left(R\right) = -\frac{R}{4 \sqrt{\pi}} + o\left(R\right). \end{align} \tag{ 4.79 }$$

For $I_{+,3}(R)$, following the same strategy as for $I_{+,2}(R)$, and together with $\operatorname{erf}(a + b) = 1 + O(e^{-R^2})$, we obtain the leading order asymptotic behaviour

$$\begin{equation} I_{+,3}\left(R\right) \sim \frac{R^2}{2\pi^{3/2}} \int_0^2 \int_{-\infty}^{\infty} \sqrt{4-s^2} \, e^{-u^2} \,\mathrm{d}u \,\mathrm{d}s = \frac{R^2}{2}. \end{equation} \tag{ 4.80 }$$

To derive the subleading terms we write

$$\begin{align*} I_{+,3}\left(R\right) = \frac{R^2}{2} - I_{+,3}^{A}\left(R\right) - I_{+,3}^{B}\left(R\right) + O\left(1\right), \end{align*}$$

where we add and subtract terms, such that the limits of the following two integrals can be obtained more easily:

$$\begin{align*} I_{+,3}^{A}\left(R\right) & = \frac{R^2}{2} - \frac{R}{8\sqrt{\pi}} \int_0^{2R} \int_{\lvert v-R\rvert-R}^{R-\lvert v-R\rvert} \frac{4}{\pi} e^{-u^2} \sqrt{4-\frac{v^2}{R^2}} \,\mathrm{d}u \,\mathrm{d}v, \\ I_{+,3}^{B}\left(R\right) & = \frac{R}{8\sqrt{\pi}} \int_0^{2R} \int_{\lvert v-R\rvert-R}^{R-\lvert v-R\rvert} \left[ \frac{4}{\pi} e^{-u^2} - \operatorname{erfc}\left( \frac{v+u}{\sqrt{2}} \right) \right.\\ & \quad \times \left. \operatorname{erfc}\left( \frac{v-u}{\sqrt{2}} \right) e^{v^2} \left(1+2 v^2\right) \right] \sqrt{4-\frac{v^2}{R^2}} \,\mathrm{d}u \,\mathrm{d}v. \end{align*}$$

In the last integral we have switched back to the unscaled v-coordinate. Note that the v-integral in $I_{+,3}^{A}(R)$ is dominated by contributions from small v, and we obtain from the u-integral

$$\begin{align} \begin{aligned} I_{+,3}^{A}\left(R\right) & = \frac{R^2}{2} - \frac{R}{2\pi} \int_0^{2R} \operatorname{erf}\left(R-|v-R|\right) \sqrt{4-\frac{v^2}{R^2}} \,\mathrm{d}v\\ & = \frac{R}{2\pi} \left(\int_0^{R}+\int_R^{2R}\right) \operatorname{erfc}\left(R-\lvert v-R\rvert\right) \sqrt{4-\frac{v^2}{R^2}} \,\mathrm{d}v \\ & = \frac{R}{2\pi} \int_0^{\infty} \operatorname{erfc}\left(v\right) \left( 2 + O\left(\frac{1}{R^2}\right) \right) \,\mathrm{d}v + o\left(R\right) = \frac{1}{\pi^{3/2}} R + o\left(R\right). \end{aligned} \end{align} \tag{ 4.81 }$$

In the second step we can neglect the second integral on $[R,2R]$, as the error function then leads to an exponential suppression. This yields the last line, giving a contribution to the linear order in R. Similarly, the v-integral in $I_{+,3}^{B}(R)$ is dominated by small v. The limiting integral is easier to compute in the original $(y_1,y_2)$-coordinates where we use $\sqrt{4-(y_1+y_2)^2/R^2} = 2 + O(\frac{1}{R^2})$ to derive

$$\begin{align*} I_{+,3}^{B}\left(R\right) & = \frac{R}{2\sqrt{\pi}} \int_0^\infty \int_0^\infty \left[ \frac{4}{\pi} e^{-\left(y_1-y_2\right)^2}- \operatorname{erfc}\left(\sqrt{2}y_1\right) \right.\\ & \quad \times \left. \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 + y_2\right)^2} \left(1+2\left(y_1 + y_2\right)^2\right) \right] \,\mathrm{d}y_2 \,\mathrm{d}y_1 + o\left(R\right) . \end{align*}$$

We evaluate the double integral analogously to $I_{-,3}(R)$ above. The integral over the first term gives

$$\begin{equation*} \frac{4}{\pi} \int_0^\infty e^{-\left(y_1-y_2\right)^2} \,\mathrm{d}y_2 = \frac{2}{\sqrt{\pi}} \operatorname{erfc}\left(-y_1\right). \end{equation*}$$

Via integration by parts we obtain for the second term

$$\begin{align*} \int_0^\infty \operatorname{erfc}\left(\sqrt{2}y_2\right) e^{\left(y_1 + y_2\right)^2} \left(1+2\left(y_1 + y_2\right)^2\right) \,\mathrm{d}y_2 & = -y_1e^{y_1^2} +\int_0^\infty\frac{2\sqrt{2}}{\sqrt{\pi}}e^{-2y_2^2}\left(y_1+y_2\right)e^{\left(y_1 + y_2\right)^2} \,\mathrm{d}y_2\\ & = -y_1e^{y_1^2} +2\sqrt{2}y_1e^{2y_1^2}\operatorname{erfc}\left(-y_1\right)+\sqrt{\frac{2}{\pi}}e^{y_1^2}. \end{align*}$$

The integral over y₁ now contains four parts

$$\begin{equation} \begin{aligned} I_{+,3}^{B}\left(R\right) & = \frac{R}{2\sqrt{\pi}} \int_0^\infty \left[ \frac{2}{\sqrt{\pi}}\operatorname{erfc}\left(-y_1\right)+ \operatorname{erfc}\left(\sqrt{2}y_1\right) y_1e^{y_1^2} - \operatorname{erfc}\left(\sqrt{2}y_1\right)2\sqrt{2}y_1e^{2y_1^2}\right.\\ & \quad \times \left.\operatorname{erfc}\left(-y_1\right) -\operatorname{erfc}\left(\sqrt{2}y_1\right)\sqrt{\frac{2}{\pi}}e^{y_1^2}\right]\,\mathrm{d}y_1. \end{aligned} \end{equation} \tag{ 4.82 }$$

For the third part of the integral in the first line we have, again after integration by parts

$$\begin{align*} & -\int_0^\infty 2\sqrt{2}y_1e^{2y_1^2}\operatorname{erfc}\left(\sqrt{2}y_1\right)\operatorname{erfc}\left(-y_1\right)\,\mathrm{d}y_1 \\ & \quad = \frac{\sqrt{2}}{2} +\int_0^\infty \frac{\sqrt{2}}{2}e^{2y_1^2}\left( -\frac{2\sqrt{2}}{\sqrt{\pi}}e^{-2y_1^2}\operatorname{erfc}\left(-y_1\right)+\operatorname{erfc}\left(\sqrt{2}y_1\right)\frac{2}{\sqrt{\pi}}e^{-y_1^2} \right)\, \mathrm{d}y_1. \end{align*}$$

The first integral cancels the first term in line one of (4.82) and the second integral cancels the fourth term in line two of (4.82). We are left with the second term in line one of (4.82), which follows from (4.76). We thus arrive at

$$\begin{equation} I_{+,3}^{B}\left(R\right) = \frac{R}{2\sqrt{\pi}} \left( -\frac{1}{2}+\sqrt{2}\right) + o\left(R\right). \end{equation} \tag{ 4.83 }$$

Combining the leading and subleading terms, we obtain the asymptotic formula for $I_{+,3}(R)$ in (4.71).

Finally, for $I_{+,4}(R)$ we use that as $R \to \infty,$

$$\begin{equation*} a - b = -\frac{us}{\sqrt{4-s^2}} +O\left(\frac{1}{R^2}\right), \qquad \operatorname{erf}\left(a - b\right) = \operatorname{erf}\left( -\frac{us}{\sqrt{4-s^2}} \right) +O\left(\frac{1}{R^2}\right). \end{equation*}$$

Since the $I_{+,4}(R)$ integral is dominated by $0 \lt s \lt 2$, as before we set the bounds of the u-integral to $\pm\infty$, do an integration by parts and conclude

$$\begin{align} \begin{aligned} I_{+,4}\left(R\right) & = -\frac{R}{2 \pi^{3/2}} \int_0^2 \frac{s}{\sqrt{4-s^2}} \int_{-\infty}^{\infty} u e^{-u^2} \operatorname{erf}\left( \frac{us}{\sqrt{4-s^2}} \right) \,\mathrm{d}u \,\mathrm{d}s + o\left(R\right) \\ & = -\frac{R}{\pi^2} \int_0^2 \frac{s^2}{4-s^2} \int_{-\infty}^{\infty} e^{-\frac{u^2}{4-s^2}}\,\mathrm{d}u\,\mathrm{d}s + o\left(R\right) \\ & = -\frac{R}{4\pi^{\frac{3}{2}}} \int_0^2 \frac{s^2}{\sqrt{4-s^2}} \,\mathrm{d}s + o\left(R\right) = -\frac{R}{4 \sqrt{\pi}} + o\left(R\right). \end{aligned} \end{align} \tag{ 4.84 }$$

This completes the proof. □

We are now ready to show theorem 2.6.

Proof of theorem 2.6. Equation (2.58) follows from proposition 4.3 (i) by series expansion.

For (4.47), recalling $\hat{a} = \sqrt{R^2-v^2}$, we have as $R \to \infty,$

$$\begin{align*} & \int_{0}^R \operatorname{erfc}\left(\sqrt{2}v\right) v \,e^{v^2} \left( e^{-\left(\hat{a} - R\right)^2}-e^{-\left(\hat{a} + R\right)^2}+ \sqrt{\pi}\left(\hat{a} - R\right) \operatorname{erf}\left(\hat{a} - R\right) -\sqrt{\pi} \left(\hat{a} + R\right) \operatorname{erf}\left(\hat{a} + R\right) \right)\,\mathrm{d}v \\ & \quad \sim -\sqrt{\pi} \int_{0}^R \operatorname{erfc}\left(\sqrt{2}v\right) v \,e^{v^2} \left(\hat{a} + R\right) \operatorname{erf}\left(\hat{a} + R\right) \,\mathrm{d}v \sim -2\sqrt{\pi} R \int_{0}^R \operatorname{erfc}\left(\sqrt{2}v\right) v \,e^{v^2} \,\mathrm{d}v. \end{align*}$$

Then by using proposition 4.3 (ii) and (4.76) we obtain equation (2.60),

$$\begin{align*} \operatorname{Cov} \left( \mathcal{N}_{\mathbb{C}}^{\left(1\right)}\left(D_R\right), \mathcal{N}_{\mathbb{R}}^{\left(1\right)}\left(D_R\right) \right) \sim -2 \frac{\sqrt{2}-1 }{\sqrt{\pi}} R. \end{align*}$$

Finally, we show equation (2.59). It follows from lemma 4.5 that

$$\begin{equation} 4 \left( I_-\left(R\right) - I_+\left(R\right) \right) = -2 R^2 + 4\sqrt{\frac{2}{\pi}} R + o\left(R\right). \end{equation} \tag{ 4.85 }$$

On the one hand, by proposition 2.3,

$$\begin{equation} 2 E_{\mathbb{C}}^{\left(1\right)}\left(R\right) = 2 R^2-2 \sqrt{\frac{2}{\pi} } R +O\left(1\right) , \qquad R \to \infty. \end{equation} \tag{ 4.86 }$$

Combining this with proposition 4.3 (iii), we obtain

$$\begin{equation} \operatorname{Var} \mathcal{N}_{\mathbb{C}}^{\left(1\right)} \left(D_R\right) \sim 2 \sqrt{\frac{2}{\pi} } R. \end{equation} \tag{ 4.87 }$$

Equation (2.61) follows simply from (2.40). □

Recall that $W \equiv W_N$ is a (possibly N-dependent) rotationally invariant potential that satisfies the conditions in definition 2.12 and that $g: [0, \infty) \to \mathbb{R}$ with $W(z) = g(\lvert z\rvert)$.

Proof of proposition 2.13 (ii). We start by writing out the Laplace transform of $\mathcal{N}_a$ as

$$\begin{align*} \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a} \right] & = \int_{\mathbb{C}^N} e^{u \sum_{k = 1}^{N} \chi_a\left(z_k\right)} \,\mathrm{d}\mathcal{P}_{N,W}^{\left(4\right)} \left(\boldsymbol{z}\right) \\ & = \frac{1}{Z_{N,W}^{\left(\beta = 4\right)}} \int_{\mathbb{C}^N} \det\begin{bmatrix} z_l^{k - 1} \\ \overline{z}_l^{k - 1} \end{bmatrix}_{k \unicode{x2A7D} 2 N, l \unicode{x2A7D} N} \, \prod_{j = 1}^N \left(z_j-\overline{z}_j\right) e^{u \chi_a\left(z_j\right) -N W\left(z_j\right)} \,\mathrm{d}A\left(z_j\right), \end{align*}$$

where $\chi_a(z) = 1$ if $|z| \unicode{x2A7D} a$ and $\chi_a(z) = 0$ otherwise. Next, we simplify the N-fold integration via de Bruijn's formula,

$$\begin{equation} \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a } \right] = \frac{N! }{Z_{N,W}^{\left(\beta = 4\right)}} \, {\textrm{Pf}}\left[ \int_{\mathbb{C}} \left(z^{k - 1} \overline{z}^{l - 1} - z^{l - 1} \overline{z}^{k - 1}\right) \left(z-\overline{z}\right) e^{u \chi_a\left(z\right) -N W\left(z\right)} \,\mathrm{d}A\left(z\right) \right]_{k, l = 1}^{2N}, \end{equation} \tag{ 5.1 }$$

see e.g. [3, remark 2.5]. Due to the radial symmetry of W, we can further simplify the integrals in the Pfaffian, see also [80, 87]. Namely, by multiplying out $(z^{k - 1} \overline{z}^{l - 1} - z^{l - 1} \overline{z}^{k - 1}) (z-\overline{z})$ and using that $e^{u \chi_a(z) -N W(z)}$ only depends on the radius $|z|$, we obtain for the entries

$$\begin{align} \begin{aligned} & \int_{\mathbb{C}} \left(z^{k - 1} \overline{z}^{l - 1} - z^{l - 1} \overline{z}^{k - 1}\right) \left(z-\overline{z}\right) e^{u \chi_a\left(z\right) -N W\left(z\right)} \,\mathrm{d}A\left(z\right) \\ & \quad = 2 \int_0^\infty \left(r^k r^{l - 1} \delta_{k, l - 1} - r^{k - 1} r^l \delta_{k - 1, l} - r^l r^{k - 1} \delta_{k - 1, l} + r^{l - 1} r^k \delta_{k, l - 1}\right) e^{u \chi_a\left(r\right) -N g\left(r\right)} r \,\mathrm{d}r \\ & \quad = 4 \left(\delta_{k, l - 1} - \delta_{k, l + 1}\right) \int_0^\infty r^{k + l} e^{u \chi_a\left(r\right) -N g\left(r\right)} \,\mathrm{d}r. \end{aligned} \end{align} \tag{ 5.2 }$$

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

$$\begin{align} \begin{aligned} \int_0^\infty r^{k + l} e^{u \chi_a\left(r\right) -N g\left(r\right)} \,\mathrm{d}r & = \int_0^\infty r^{k + l} e^{-N g\left(r\right)} \,\mathrm{d}r - \left(1 - e^{u}\right) \int_0^a r^{k + l} e^{-N g\left(r\right)} \,\mathrm{d}r \\ & = \frac{1}{2} h_{\frac{k + l - 1}{2}} - \left(1 - e^{u}\right) \frac{1}{2} h_{\frac{k + l - 1}{2},1}\left(a\right). \end{aligned} \end{align} \tag{ 5.3 }$$

Note that the indices are integer-valued for $k = l - 1$ and $k = l + 1$. Combining (5.1)–(5.3), we arrive at

$$\begin{equation} \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a } \right] = \frac{N!}{Z_{N,W}^{\left(\beta = 4\right)}} {\textrm{Pf}}\left[ 2 \left(\delta_{k, l - 1} - \delta_{k, l + 1}\right) \left( h_{\frac{k + l - 1}{2}} - \left(1 - e^{u}\right) h_{\frac{k + l - 1}{2},1}\left(a\right) \right) \right]_{k, l = 1}^{2N}. \end{equation} \tag{ 5.4 }$$

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):

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

On the other hand, it follows from [3, remark 2.5 and corollary 3.3] that

$$\begin{equation} Z_{N,W}^{\left(\beta = 4\right)} = N! \, 2^N \prod_{j = 0}^{N-2} h_{2 j+ 1}. \end{equation} \tag{ 5.5 }$$

This leads to

$$\begin{equation} \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a } \right] = \prod_{j = 0}^{N-1} \left( 1 - \left(1 - e^{u}\right) \frac{h_{2j+1,1}\left(a\right)}{h_{2j+1}} \right). \end{equation} \tag{ 5.6 }$$

Using $h_{j} - h_{j,1}(a) = h_{j,2}(a)$, we obtain the desired identity (2.82).

The expression (2.83) follows along the lines of the supplementary material in [73] using the series expansion

$$\begin{equation} \log \mathbb{E}_{N,W}^{\left(\beta = 4\right)}\left[ e^{u \mathcal{N}_a } \right] = u \, E_{N,W}^{\left(\beta = 4\right)}\left(a\right)+ \sum_{p = 2}^{\infty} \kappa_p^{\left(\beta = 4\right)}\left(a\right) \, \frac{u^p}{p!}. \end{equation} \tag{ 5.7 }$$

Note that the expression (2.84) was obtained in [1, proposition 1.1]. □

Proof of theorem 2.14. In [1, subsection 4.1], it was shown that if g satisfies the conditions in definition 2.12, then in the bulk $a\in (0,1)$ the function $\mathcal{L}_k(a)$ has the large-N asymptotic

$$\begin{equation} \mathcal{L}_k\left(a\right) \sim \frac{1}{2} \operatorname{erfc}\left( \frac{s}{\sqrt{2 \Delta W\left(a\right)} \, a} \right), \qquad k = \frac{1}{2}\left(N a g^{^{\prime}}\left(a\right) - 1\right) + \sqrt{N} s, \end{equation} \tag{ 5.8 }$$

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

$$\begin{equation} \mathcal{L}_{2j+1}\left(a\right) \sim \frac{1}{2} \operatorname{erfc}\left( \frac{2 j - \frac{1}{2} N g^{^{\prime}}\left(a\right) a + \frac{3}{2}}{\sqrt{2 \Delta W\left(a\right) N} \, a} \right). \end{equation} \tag{ 5.9 }$$

Now we insert the asymptotic behavior of $\mathcal{L}_{2j+1}(a)$ into the finite-N cumulant formula (2.83). Note that the argument of the polylogarithm is then of the form

$$\begin{equation} 1 - \frac{1}{\frac{1}{2} \operatorname{erfc}\left(x\right)} = \frac{\operatorname{erfc}\left(x\right) - 2}{\operatorname{erfc}\left(x\right)} = -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)}, \quad \text{with } x = \frac{2 j - \frac{1}{2} N g^{^{\prime}}\left(a\right) a + \frac{3}{2}}{\sqrt{2 \Delta W\left(a\right) N} \, a}. \end{equation} \tag{ 5.10 }$$

Next we replace the sum over j in the range $|2j+1-\frac{1}{2}(N a g^{^{\prime}}(a) - 1)|\unicode{x2A7D}\sqrt{N} s$ that contributes to the sum by a Riemann integral over x, and we obtain

$$\begin{equation} \kappa_p^{\left(\beta = 4\right)}\left(a\right) \sim \left(-1\right)^{p + 1} \frac{\sqrt{2 \Delta W\left(a\right) N} \, a}{2} \int_{-\infty}^{\infty} \operatorname{Li}_{1 - p}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) \, \mathrm{d}x \end{equation} \tag{ 5.11 }$$

as $N \to \infty$. Note that the prefactor comes from the change of integration variable j → x. The parity in (2.86) comes from that of the integrand under $x\to-x$, using the inversion formula (2.98).

Similarly, for the edge case when a is given by (2.87), it follows that

$$\begin{equation} \kappa_p^{\left(\beta = 4\right)}\left(a\right) \sim \left(-1\right)^{p + 1} \frac{\sqrt{2 \Delta W\left(a\right) N} \, a}{2} \int_{-\infty}^{\mathcal{S}} \operatorname{Li}_{1 - p}\left( -\frac{\operatorname{erfc}\left(-x\right)}{\operatorname{erfc}\left(x\right)} \right) \, \mathrm{d}x \end{equation} \tag{ 5.12 }$$

as $N \to \infty$. Since we sum up to $j = N - 1$, the integration will stop at some value $x \lt \infty$ now, which can be derived as in [1, subsection 4.1]. Note that we use $4 + g^{^{\prime\prime}}(1) = 4 \Delta W(1)$ after we expand the $g^{^{\prime}}(a)$-term. □

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 FCS 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 FCS 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 [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.

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.

The numerical data generated for part of the figures are merely an illustration of our entirely theoretical findings. These data are available upon request from the authors.

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 (2.3). 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 × N matrix from the GinOE can have k real eigenvalues $x_1,\dots,x_k\in\mathbb{R}$ and 2l complex (non-real) eigenvalues that come in complex conjugate pairs, $z_1,\dots,z_l,\bar{z}_1,\dots,\bar{z}_l\in\mathbb{C}\setminus \mathbb{R}$. 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 [17, section 2.3]. 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 $\mathbb{C}_+$, with $\mathrm{Im}(z_1),\dots, \mathrm{Im}(z_l)\gt0$. 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]:

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

where the normalisation constant reads

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

Note that in the second line of (A.1) 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 (2.9) and (2.10) in the main part we use a scaling with N. Note also that the first line in (A.1) 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 (2.2). A Coulomb gas interpretation of the jpdf (A.1) at inverse temperature $2 = 1/(k_\mathrm{B} T)$ was given in [43].

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

$$\begin{equation} \mathcal{P}_{N}^{\left(1\right)}\left(w_1,\dots,w_N\right) = \sum_{\substack{k=0\\ N-k{\,\,}\text{even}}}^N \mathbf{P}_{N,k,\left(N-k\right)/2}^{\left(1\right)}\left(x_1,\dots,x_k,z_1,\dots,z_{\left(N-k\right)/2}\right). \end{equation} \tag{ A.3 }$$

There are two equivalent ways of defining correlation functions of real, complex or mixed eigenvalues. The first introduced in [46, 95] is to add extra source terms to the joint densities (A.1) to define generalised partition functions

$$\begin{equation} \mathbf{Z}_{N,k,l}\left[u,v\right]: = \prod_{i = 1}^k \int _{\mathbb{R}}\mathrm{d}x_i u\left(x_i\right) \prod_{j = 1}^l \int _{\mathbb{C}_+}\mathrm{d}A\left(z_j\right) v\left(z_j\right) \mathbf{P}_{N,k,l}^{\left(1\right)}\left(x_1,\dots,x_k,z_1,\dots,z_l\right), \end{equation} \tag{ A.4 }$$

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

$$\begin{equation} \mathcal{Z}_N\left[u,v\right]: = \sum_{\substack{k=0\\ N-k{\,\,}\text{even}}}^N \mathbf{Z}_{N,k,\left(N-k\right)/2}\left[u,v\right] \end{equation} \tag{ A.5 }$$

as follows:

$$\begin{align} \mathbf{R}_{N,k,\mathbb{R}}^{\left(1\right)}\left(x_1,\dots,x_k\right) &: = \frac{1}{\mathcal{Z}_N\left[1,1\right]}\frac{\delta^k}{\delta u\left(x_1\right) \cdots\delta u\left(x_k\right)} \left. \mathcal{Z}_N\left[u,1\right]\right|_{u = 1}, \end{align} \tag{ A.6 }$$

$$\begin{align} \mathbf{R}_{N,l,\mathbb{C}}^{\left(1\right)}\left(z_1,\dots,z_l\right) &: = \frac{1}{\mathcal{Z}_N\left[1,1\right]}\frac{\delta^k}{\delta v\left(z_1\right) \cdots\delta v\left(z_l\right)} \left. \mathcal{Z}_N\left[1,v\right]\right|_{v = 1}, \end{align} \tag{ A.7 }$$

$$\begin{align} \mathbf{R}_{N,k,\mathbb{R},l,\mathbb{C}}^{\left(1\right)}\left(x_1,\dots,x_{k},z_1,\dots,z_l\right) &: = \frac{1}{\mathcal{Z}_N\left[1,1\right]}\frac{\delta^{k+l}}{\delta u\left(x_1\right) \cdots\delta u\left(x_k\right)\delta v\left(z_1\right) \cdots\delta v\left(z_l\right)} \left. \mathcal{Z}_N\left[u,v\right]\right|_{u = 1,v = 1}, \end{align} \tag{ A.8 }$$

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 $l\unicode{x2A7D} L\unicode{x2A7D} (N-K)/2$ and $k\unicode{x2A7D} K = N-2L$ real eigenvalues, for a given N with fixed k and l, was used in [13]

$$\begin{equation} \begin{aligned} \mathbf{R}_{N,k,\mathbb{R},l,\mathbb{C}}^{\left(1\right)}\left(x_1,\dots,x_{k},z_1,\dots,z_l\right) & = \frac{2^l}{\mathbf{Z}_{N,k,l}\left[1,1\right]} \sum_{\substack{k=0\\ N-K{\,\,}\text{even}}}^N \sum_{L = l}^{\left(N-K\right)/2}\frac{1}{\left(K-k\right)!\,\left(L-l\right)!}\\ & \quad \times \prod_{i = 1}^{K-k} \int _{\mathbb{R}}\mathrm{d}x_i\prod_{j = 1}^{L-l} \int _{\mathbb{C}_+}\mathrm{d}A\left(z_j\right) \mathbf{P}_{N,K,L}^{\left(1\right)}\left(x_1,\dots,x_K,z_1,\dots,z_L\right), \end{aligned} \end{equation} \tag{ A.9 }$$

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 $\mathbb{R}^2$ and a similar product on $\mathbb{C}_+^2$, 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 (2.9) and (2.10), in order to compute the mean number of eigenvalues in a (one- respectively two-dimensional) disc of radius R, see proposition 2.1 and lemma 3.1 for N even and odd.

Instead, we will directly present the resulting correlation functions [13, 46, 95, 96] in the large-N limit and in the origin scaling limit defined in (2.26)–(2.28), that will be used in the computation of the limiting number variance:

• GinOE complex k-point function in the origin limit
  $$\begin{equation} R_{k,\mathbb{C}}^{\left(1\right)}\left(z_1,\cdots, z_k\right) = \prod_{j = 1}^{k} \left(2\pi i \operatorname{erfc}\left(\sqrt{2}y_j\right)\right) \, {\textrm{Pf}} \left[ \begin{pmatrix} S_{\mathbb{C}}\left(\bar{z}_j,\bar{z}_l\right) & S_{\mathbb{C}}\left(\bar{z}_j,z_l\right) \\ S_{\mathbb{C}}\left(z_j,\bar{z}_l\right) & S_{\mathbb{C}}\left(z_j,z_l\right) \end{pmatrix} \right]_{j,l = 1,\cdots k }, \end{equation} \tag{ A.10 }$$
  where
  $$\begin{equation} S_{\mathbb{C}}\left(z,w\right) : = \frac{z-w}{2\sqrt{2\pi}} e^{-\left(z-w\right)^2/2}, \end{equation} \tag{ A.11 }$$
  see e.g. [47, equation (6.21)].
• GinOE real k-point function in the origin limit
  $$\begin{equation} R_{k,\mathbb{R}}^{\left(1\right)}\left(x_1,\cdots, x_k\right) = {\textrm{Pf}} \left[ \begin{pmatrix} D_{\mathbb{R}}\left(x_j,x_l\right) & S_{\mathbb{R}}\left(x_j,x_l\right) \\ -S_{\mathbb{R}}\left(x_l,x_j\right) & \tilde{I}_{\mathbb{R}}\left(x_j,x_l\right) \end{pmatrix} \right]_{j,l = 1,\cdots k }, \end{equation} \tag{ A.12 }$$
  where
  $$\begin{equation} S_{\mathbb{R}}\left(x,y\right): = \frac{1}{\sqrt{2\pi} } e^{-\left(x-y\right)^2/2} \end{equation} \tag{ A.13 }$$
  and
  $$\begin{align} D_{\mathbb{R}}\left(x,y\right)&: = -\frac{\partial}{\partial y} S_{\mathbb{R}}\left(x,y\right) = - \frac{x-y}{\sqrt{2\pi}} e^{-\left(x-y\right)^2/2}, \end{align} \tag{ A.14 }$$
  $$\begin{align} \tilde{I}_{\mathbb{R}}\left(x,y\right)&: = \int_{x}^{y} S_{\mathbb{R}}\left(t,y\right)\,\mathrm{d}t +\frac12 \operatorname{sgn}\left(x-y\right) = -\frac12 \operatorname{erf} \left( \frac{x-y}{\sqrt{2}} \right) +\frac12 \operatorname{sgn}\left(x-y\right), \end{align} \tag{ A.15 }$$
  see e.g. [17, equation (2.36)].
• GinOE mixed real and complex (k, l)-point function in the origin limit [13]
  $$\begin{align} R_{k,\mathbb{R},l,\mathbb{C}}^{\left(1\right)}\left(x_1,\dots,x_{k},z_1,\dots,z_{l}\right) = {\textrm{Pf}} \begin{bmatrix} \left[\mathcal{K}_{\mathbb{R}}^{\left(1\right)}\left(x_j,x_l\right)\right]_{j,l = 1}^{k} & \left[\mathcal{K}_{\mathbb{R},\mathbb{C}}^{\left(1\right)}\left(x_j,z_l\right)\right]_{\substack{j = 1,\dots,k\\ l = 1,\dots,l}} \\ \left( - \left[\mathcal{K}_{\mathbb{R},\mathbb{C}}^{\left(1\right)}\left(x_j,z_l\right)\right]_{\substack{j = 1,\dots,k\\ l = 1,\dots,l}} \right)^T & \left[\mathcal{K}_{\mathbb{C}}^{\left(1\right)}\left(z_j,z_l\right)\right]_{j,l = 1}^{l} \end{bmatrix}\nonumber\\ \end{align} \tag{ A.16 }$$
  where
  $$\begin{align} &\mathcal{K}_{\mathbb{C}}^{\left(1\right)}\left(z,w\right): = 2\pi i \left(\operatorname{erfc}\left(\sqrt{2}\operatorname{Im} z\right) \operatorname{erfc}\left(\sqrt{2} \operatorname{Im} w\right) \right)^{1/2} \begin{pmatrix} S_{\mathbb{C}}\left(\bar{z},\bar{w}\right) & S_{\mathbb{C}}\left(\bar{z},w\right) \\ S_{\mathbb{C}}\left(z,\bar{w}\right) & S_{\mathbb{C}}\left(z,w\right) \end{pmatrix}, \end{align} \tag{ A.17 }$$
  $$\begin{align} &\mathcal{K}_{\mathbb{R}}^{\left(1\right)}\left(x,y\right): = \begin{pmatrix} D_{\mathbb{R}}\left(x,y\right) & S_{\mathbb{R}}\left(x,y\right) \\ -S_{\mathbb{R}}\left(x,y\right) & \tilde{I}_{\mathbb{R}}\left(x,y\right) \end{pmatrix}, \end{align} \tag{ A.18 }$$
  $$\begin{align} &\mathcal{K}_{\mathbb{R},\mathbb{C}}^{\left(1\right)}\left(x,z\right) : = \frac{1}{\sqrt{2}} \left(\operatorname{erfc}\left(\sqrt{2} \operatorname{Im} z\right) \right)^{1/2} \begin{pmatrix} \left(z-x\right) e^{-\left(x-z\right)^2/2} & i \left(\bar{z} - x\right) e^{-\left(x-\bar{z}\right)^2/2} \\ - e^{-\left(x-z\right)^2/2} & -i e^{-\left(x-\bar{z}\right)^2/2} \end{pmatrix}. \end{align} \tag{ A.19 }$$

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.