# Spherical induced ensembles with symplectic symmetry

Sung-Soo Byun  Center for Mathematical Challenges, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>and  PETER J. FORRESTER  School of Mathematical and
Statistics, The University of Melbourne, Victoria 3010, Australia
<EMAIL_ADDRESS>
###### Abstract.

We consider the complex eigenvalues of the induced spherical Ginibre ensemble
with symplectic symmetry and establish the local universality of these point
processes along the real axis. We derive scaling limits of all correlation
functions at regular points both in the strong and weak non-unitary regimes as
well as at the origin having spectral singularity. A key ingredient of our
proof is a derivation of a differential equation satisfied by the correlation
kernels of the associated Pfaffian point processes, thereby allowing us to
perform asymptotic analysis.

###### Key words and phrases:

Planar symplectic ensemble, Spherical induced Ginibre ensemble, Universality,
Skew-orthogonal polynomials, Asymptotic analysis.

###### 2020 Mathematics Subject Classification:

Primary 60B20; Secondary 33C45

Sung-Soo Byun was supported by Samsung Science and Technology Foundation
(SSTF-BA1401-51), by the National Research Foundation of Korea
(NRF-2019R1A5A1028324) and by a KIAS Individual Grant (SP083201) via the
Center for Mathematical Challenges at Korea Institute for Advanced Study.
Funding support to Peter Forrester for this research was through the
Australian Research Council Discovery Project grant DP210102887.

##  1\. Introduction and Results

The footprints of the universality in non-Hermitian random matrix theory began
with the work [35] of Ginibre. There, three classes of Gaussian random
matrices with complex, real, and quaternion elements were introduced, and they
are now called the Ginibre ensembles. Although the eigenvalues of the matrices
in each symmetry class all follow the universal circular law at the
macroscopic level, their statistical properties are quite different from many
perspectives. For instance, in the complex symmetry class, the real axis is
not special due to the rotational invariance. On the other hand, in the real
and quaternion cases, there exist microscopic attraction and repulsion
respectively along the real axis.

The difference among these three symmetry classes can also be found in their
integrable structures. More precisely, the eigenvalues of the complex matrices
form determinantal point processes, whereas those of the real and quaternion
matrices form Pfaffian point processes. Furthermore, while the correlation
kernels of the complex matrices can be written in terms of the planar
orthogonal polynomials, their counterparts for the real and quaternion
matrices are described in terms of the (less understood) planar skew-
orthogonal polynomials.

Due to the more complicated integrable structures of Pfaffian point processes,
it is not surprising that the local universality classes (i.e. scaling limits
of all eigenvalue correlation functions) were first investigated in the
complex symmetry class. Indeed, the bulk scaling limit of the complex Ginibre
ensemble was already introduced in the work [35] of Ginibre. On the other
hand, the edge scaling limit of the complex Ginibre ensemble was discovered in
[30]. For the real symmetry class, the bulk and edge scaling limits of the
Ginibre ensemble were investigated in [53, 32, 11]. Finally, for the
quaternion case, the bulk scaling limit was first introduced in the second
edition of Mehta’s book [50] and later rediscovered by Kanzieper [39]. In
contrast, the edge scaling limit in this symmetry class was discovered only
recently in [3]. (See also [47] for an alternative derivation for the
$1$-point function.)

From the above philosophy, it is not surprising again that the universality
principle was first established in the complex symmetry class. Among plenty of
works in this direction, the bulk universality of random normal matrix
ensembles was obtained in [9]. More recently, the edge universality of these
models was obtained in [37], where the authors developed a general asymptotic
theory of the planar orthogonal polynomials. However, the literature on the
universality in the other symmetry classes are more limited.

Nevertheless, there have been several recent works on the scaling limits of
_planar symplectic ensembles_ , which are contained in the symmetry class of
the quaternion Ginibre ensemble. For instance, the universal scaling limits of
the symplectic elliptic Ginibre ensemble at the origin were obtained in [5]
and were extended in [15] along the whole real axis. Furthermore, non-standard
universality classes under the presence of certain singularities have been
discovered as well. To name a few, the scaling limits at the _singular origin_
were studied in [3] for the Mittagg-Leffler ensemble (a generalisation of the
symplectic induced Ginibre ensemble), in [38] for the product ensembles and in
[2, 5] for the Laguerre ensembles. The boundary scaling limits under the _hard
edge_ type conditions were investigated in [40] for the truncated symplectic
unitary ensembles and in [16] for the Ginbire ensemble with boundary
confinements. Beyond the above-mentioned cases, the scaling limits of the
models interpolating one- and two-dimensional ensembles have also been
studied. In this direction, the scaling limits of the symplectic elliptic
Ginibre ensemble in the _almost-Hermitian_ (or _weakly non-Hermitian_) regime
were derived in [39, 16, 7]. Very recently, the scaling limits of the
symplectic induced Ginibre ensemble in the _almost-circular_ (or _weakly non-
unitary_) regime were obtained in [13]. While the almost-Hermitian [34, 33, 4]
and almost-circular [8, 17] ensembles have the same bulk scaling limits in the
complex symmetry class, those are different in the symplectic symmetry class
in the vicinity of the real line due to the lack of the translation
invariance; see [13] for further details.

In this work, we study the symplectic induced spherical ensembles with the
goal to derive their scaling limits in various regimes and to establish the
universality of these point processes. The symplectic induced spherical
ensemble $\mathbf{G}$ of size $N$ is defined by the matrix probability
distribution function proportional to

(1.1)
$\frac{\det(\mathbf{G}\mathbf{G}^{\dagger})^{2L}}{\det(\mathbf{1}_{N}+\mathbf{G}\mathbf{G}^{\dagger})^{2(n+N+L)}}.$

Here $n$ and $L$ are parameters, with $n\geq N$ and $L\geq 0$ also possibly
dependent on $N$. In particular, if $n=N$, $L=0$, the model (1.1) is known as
the spherical ensemble with symplectic symmetry. The name “spherical”
originates from the fact that their eigenvalues tend to be uniformly
distributed on the unit sphere under the (inverse) stereographic projection;
see e.g. [31, 42]. And as discussed in the ensuing text, the term symplectic
symmetry relates to an invariance of the underlying Gaussian matrices. To
realise the matrix probability distribution (1.1), following [23] and [49,
Appendix B] first introduce a particular $(N+L)\times N$ random matrix
$\mathbf{Y}$ with each entry itself a $2\times 2$ matrix representation of a
quaternion. The matrix is said to have quaternion entries for short; see e.g.
[26, Section 1.3.2]. The specification of $\mathbf{Y}$ is that
$\mathbf{Y}=\mathbf{X}\mathbf{A}^{-1/2}$, where $\mathbf{X}$ is an
$(N+L)\times N$ standard Gaussian matrix with quaternion entries (also
referred to as a rectangular quaternion Ginibre matrix), while $\mathbf{A}$ is
an $N\times N$ Wishart matrix with quaternion entries. More explicity,
$\mathbf{A}=\mathbf{Y}^{\dagger}\mathbf{Y}$ where $\mathbf{Y}$ is an $n\times
N$ rectangular quaternion Ginibre matrix; see e.g. [26, Def. 3.1.2]. In terms
of such $\mathbf{Y}$, and a Haar distributed unitary random matrix with
quaternion entries (i.e. a symplectic unitary matrix $\mathbf{U}$; see e.g.
[21]), define $\mathbf{G}=\mathbf{U}(\mathbf{Y}^{\dagger}\mathbf{Y})^{1/2}$.
It is the random matrix $\mathbf{G}$ which has matrix distribution (1.1); the
corresponding eigenvalues, which must come in complex conjugate pairs, are
used in producing the plots of Figure 1.

(a) $L=N$, $n=2N$

(b) $L=\frac{N^{2}}{\rho^{2}}-N$, $n=\frac{N^{2}}{\rho^{2}}$, $\rho=\sqrt{10}$

(c) $L=1$, $n=N$

(d) $L=N$, $n=2N$

(e) $L=\frac{N^{2}}{\rho^{2}}-N$, $n=\frac{N^{2}}{\rho^{2}}$, $\rho=\sqrt{10}$

(f) $L=1$, $n=N$

Figure 1.  The plots (A)–(C) display the eigenvalues of G for $N=100$ and
$200$ realisations with different values of $n$ and $L$. In all figures
(A)–(C), the local repulsion along the real axis is visible. The plots (D)–(F)
show a sample of the eigenvalues of G for $N=1000$ projected onto the unit
sphere.

Several fundamental properties of the symplectic induced spherical ensemble
were discovered by Mays and Ponsaing [49]. (We also refer to an earlier work
[48] on the induced spherical ensemble with orthogonal symmetry.) In
particular, it was shown in [49, Section 3] that the joint probability
distribution $\mathbf{P}_{N}$ of its independent eigenvalues
$\bm{\zeta}=\\{\zeta_{j}\\}_{j=1}^{N}$ is given by

(1.2) $d\mathbf{P}_{N}(\bm{\zeta})=\frac{1}{N!Z_{N}}\prod_{1\leq j<k\leq
N}\lvert\zeta_{j}-\zeta_{k}\rvert^{2}\lvert\zeta_{j}-\overline{\zeta}_{k}\rvert^{2}\prod_{j=1}^{N}\lvert\zeta_{j}-\overline{\zeta}_{j}\rvert^{2}e^{-2NQ(\zeta_{j})}\,dA(\zeta_{j}),$

where $dA(\zeta):=d^{2}\zeta/\pi$, and $Z_{N}$ is the normalisation constant.
While the independent eigenvalues should have each $\zeta_{j}$ in the upper
half complex plane, relaxing this condition leaves (1.2) unaltered and
simplifies the presentation. Here the potential $Q$ is given by

(1.3) $Q(\zeta):=\frac{n+L+1}{N}\log(1+|\zeta|^{2})-\frac{2L}{N}\log|\zeta|.$

We remark that the distribution (1.2) can be interpreted as a two-dimensional
Coulomb gas ensemble [28, 41] with additional complex conjugation symmetry;
see also Appendix A.

We first briefly recall the macroscopic property of the ensemble (1.2).
Combining the convergence of the empirical measure [10] and the basic facts
from the logarithmic potential theory [52, Section IV.6], one can see that as
$N\to\infty$, the eigenvalues $\bm{\zeta}$ tend to be distributed on the
droplet

(1.4) $S=\\{\zeta\in\mathbb{C}:r_{1}\leq|\zeta|\leq r_{2}\\},\qquad
r_{1}=\sqrt{\frac{L}{n}},\qquad r_{2}=\sqrt{\frac{N+L}{n-N}}$

with the density

(1.5) $\frac{n+L}{N}\frac{1}{(1+|\zeta|^{2})^{2}}.$

This property was also shown in [49, Section 6] using a different method. We
also refer to [20] and references therein for recent works on the equilibrium
measure problems on the sphere under the insertion of point charges.

For detailed statistical information about the ensemble (1.2), we study its
$k$-point correlation function

(1.6)
$\mathbf{R}_{N,k}(\zeta_{1},\dots,\zeta_{k}):=\frac{N!}{(N-k)!}\int_{\mathbb{C}^{N-k}}\mathbf{P}_{N}(\bm{\zeta})\prod_{j=k+1}^{N}dA(\zeta_{j}).$

The following proposition provides useful formulas to analyse the correlation
functions (1.6).

######  Proposition 1.1 (Analysis at finite-$N$).

For any $N,L\geq 0$, $n\geq N$, and $k\in\mathbb{N}$, the following holds.

  * (a)

Eigenvalue correlation functions at finite-$N$.
We have

(1.7)
$\displaystyle\mathbf{R}_{N,k}(\zeta_{1},\dots,\zeta_{k})={\textup{Pf}}\Big{[}\omega(\zeta_{j})\omega(\zeta_{l})\begin{pmatrix}\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\zeta_{l})&\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\bar{\zeta}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\zeta_{l})&\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\bar{\zeta}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{\zeta}_{j}-\zeta_{j}),$

where

(1.8)
$\displaystyle\omega(\zeta)=\frac{|1+\zeta^{2}|^{n+L-\frac{1}{2}}}{(1+|\zeta|^{2})^{n+L+1}}.$

Here, the skew-kernel $\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta)$ is given
by

(1.9) $\displaystyle\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta)$
$\displaystyle=\frac{1}{((1+\zeta^{2})(1+\eta^{2}))^{n+L-\frac{1}{2}}}\Big{(}\widehat{\bm{G}}_{N}(\zeta,\eta)-\widehat{\bm{G}}_{N}(\eta,\zeta)\Big{)},$

where

(1.10)
$\widehat{\bm{G}}_{N}(\zeta,\eta):=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}.$

  * (b)

Generalised Christoffel-Darboux formula.
The skew-kernel $\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta)$ satisfies

(1.11)
$\partial_{\zeta}\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta)=\frac{1}{(1+\zeta^{2})^{n+L+\frac{1}{2}}}\Big{(}\textup{I}_{N}(\zeta,\eta)-\textup{II}_{N}(\zeta,\eta)-\textup{III}_{N}(\zeta,\eta)\Big{)},$

where

$\displaystyle\textup{I}_{N}(\zeta,\eta):=\frac{(1+\zeta\eta)^{2n+2L-1}}{(1+\eta^{2})^{n+L-\frac{1}{2}}}(2n+2L+1)(n+L)\sum_{k=0}^{2N-1}\binom{2n+2L-1}{k+2L}\mathfrak{p}^{k+2L}(1-\mathfrak{p})^{2n-k-1},$
$\displaystyle\textup{II}_{N}(\zeta,\eta):=\frac{\zeta^{2N+2L}}{2^{2L+2n}}\frac{\pi\,\Gamma(2n+2L+2)}{\Gamma(N+L+\frac{1}{2})\Gamma(n-N)\Gamma(n+L+\frac{1}{2})}\sum_{k=0}^{N-1}\binom{n+L-\frac{1}{2}}{k+L}\mathfrak{q}^{k+L}(1-\mathfrak{q})^{n-k-\frac{1}{2}},$
$\displaystyle\textup{III}_{N}(\zeta,\eta):=\frac{\zeta^{2L-1}}{2^{2L+2n}}\frac{\pi\,\Gamma(2n+2L+2)}{\Gamma(n+\frac{1}{2})\Gamma(L)\Gamma(n+L+\frac{1}{2})}\sum_{k=0}^{N-1}\binom{n+L-\frac{1}{2}}{k+L+\frac{1}{2}}\mathfrak{q}^{k+L+\frac{1}{2}}(1-\mathfrak{q})^{n-k-1}.$

Here

(1.12)
$\mathfrak{p}:=\frac{\zeta\eta}{1+\zeta\eta},\qquad\mathfrak{q}:=\frac{\eta^{2}}{1+\eta^{2}}.$

######  Remark 1.2.

We stress that Proposition 1.1 (a) is a direct consequence of the general
theory of planar symplectic ensembles [39] and the explicit formula of skew-
orthogonal polynomials associated with the potential (1.3) that can be found
in [27, Proposition 4]. (Cf. See [38, p.7] and [5, Corollary 3.2] for a
construction of skew-orthogonal polynomials associated with general radially
symmetric potentials.)

Nevertheless, the crux of Proposition 1.1 is the transforms (1.8) and (1.9) in
the expression (1.7), which lead to a simple differential equation (1.11)
stated in Proposition 1.1 (b). To be more concrete, let us mention that in
general, one strategy for performing an asymptotic analysis on a double
summation appearing in the skew-orthogonal polynomial kernel is to derive a
“proper” differential equation satisfied by the kernel; see [39, 2, 3, 15,
13]. (Such a differential equation for the two-dimensional ensemble is broadly
called the generalised Christoffel-Darboux formula [3, 15, 46]). However, if
we do not take well-chosen transforms, the resulting differential equation may
be difficult to analyse, cf. see [49, Section 6.2] for a similar discussion on
the spherical induced ensemble.

We also mention that the inhomogeneous term $\textup{I}_{N}(\zeta,\eta)$ in
(1.11) corresponds to the kernel of the complex counterpart [24]. Such a
relation has been observed not only for the two-dimensional ensembles [2, 3,
15, 13] but also for their one-dimensional counterparts [1, 57].

######  Remark 1.3.

The terms on the right-hand side of (1.11) are indeed expressed in a way that
one can easily derive their asymptotic behaviours. More precisely, the
summations in these terms can be written in terms of the incomplete beta
functions (see (3.48), (3.49) and (3.50)) whose asymptotic behaviours are well
understood. This fact will play an important role in the proof of Theorem 1.4
below.

Let us now introduce our main results on various scaling limits of the induced
spherical ensembles. From the microscopic point of view, we first mention that
the origin is special since there exists an insertion of a point charge (i.e.
$\frac{4L}{N}\log|\zeta|$ term in (1.3), or equivalently the charge $Nq_{1}$
at the north pole in the sphere picture as given by (A.5)), which is also
known as the _spectral singularity_ ; see e.g. [43]. Furthermore, the local
statistics of the ensemble depends on whether the width of the droplet $S$ in
(1.4) is of order $O(1)$ or $O(1/N)$. The former is called _strong non-
unitarity_ and the latter is called _weak non-unitarity_ (or _almost-circular
regime_). In summary, we should distinguish the following three different
regimes.

  * (a)

At regular points in the limit of strong non-unitarity. (Cf. Figure 1 (A).)
This means the case where the width of the droplet $S$ in (1.4) is of order
$O(1)$, and the zooming point $p\in\mathbb{R}$ we look at the local statistics
is away from the origin. To investigate this regime, we set the parameters as

(1.13) $L=aN,\qquad n=(b+1)N,\qquad\mbox{with fixed}\quad a,b\geq 0,$

which in the Coulomb gas picture of the Appendix A corresponds to the external
charges at the poles being proportional to $N$. Note that with this choice of
the parameters, the inner and outer radii in (1.4) satisfy

(1.14) $r_{1}=\sqrt{\frac{a}{b+1}}+O(\frac{1}{N}),\qquad
r_{2}=\sqrt{\frac{a+1}{b}}+O(\frac{1}{N})\qquad\mbox{as }N\to\infty.$

  * (b)

At regular points in the limit of weak non-unitarity. (Cf. Figure 1 (B).)
This means the case where the droplet $S$ is close to the unit circle and its
width is of order $O(1/N)$. For this regime, we set

(1.15) $L=\frac{N^{2}}{\rho^{2}}-N,\qquad
n=\frac{N^{2}}{\rho^{2}},\qquad\mbox{with fixed}\quad\rho>0.$

This choice of parameters implies that we impose strong charges (proportional
to $N^{2}$) both at the origin and the infinity which makes the droplet close
to the unit circle. Indeed, one can see that

(1.16) $r_{1}=1-\frac{\rho^{2}}{2N}+O(\frac{1}{N^{2}}),\qquad
r_{2}=1+\frac{\rho^{2}}{2N}+O(\frac{1}{N^{2}}),\qquad\mbox{as }N\to\infty.$

  * (c)

At the singular origin. (Cf. Figure 1 (C).)
This covers the case where the droplet contains the origin, i.e. $r_{1}=o(1).$
For this, we set

(1.17) $L>0\quad\mbox{fixed },\qquad n=(b+1)N.$

Here the charge at the north pole in the Coulomb gas picture of the Appendix A
is $O(1)$. Then we have

(1.18) $r_{1}=O(\frac{1}{N}),\qquad
r_{2}=\frac{1}{\sqrt{b}}+O(\frac{1}{N}),\qquad\mbox{as }N\to\infty.$

It is convenient to introduce and recall some notations to describe the
scaling limits. Let us define

(1.19) $f_{z}(u):=\frac{1}{2}\operatorname{erfc}(\sqrt{2}(z-u)).$

Recall that the two-parametric Mittag-Leffler function $E_{a,b}(z)$ is given
by

(1.20) $E_{a,b}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(ak+b)}.$

We also write $W(f,g):=fg^{\prime}-gf^{\prime}$ for the Wronskian. For a given
$p\in\mathbb{R}$, we set

(1.21) $\delta:=\frac{n+L}{N}\frac{1}{(1+p^{2})^{2}},$

which corresponds to the density (1.5) of the ensemble at the point $p$. Now
we are ready to state our main results. Without loss of generality, it
suffices to consider the case $p\geq 0.$

######  Theorem 1.4 (Scaling limits of the eigenvalue correlations).

For a fixed $p\geq 0$, let

(1.22)
$R_{N,k}(z_{1},\dots,z_{k}):=\mathbf{R}_{N,k}\Big{(}p+\frac{z_{1}}{\sqrt{N\delta}},\dots,p+\frac{z_{k}}{\sqrt{N\delta}}\Big{)},$

where $\mathbf{R}_{N,k}$ and $\delta$ are given by (1.6) and (1.21). Then the
following holds.

  * (a)

At regular points in the limit of strong non-unitarity.
Let $L=aN$, $n=(b+1)N$ with fixed $a\geq 0$, $b>0$. Let $p>0$ be fixed. Then
as $N\to\infty$,

(1.23)
$R_{N,k}(z_{1},\dots,z_{k})={\textup{Pf}}\Big{[}e^{-|z_{j}|^{2}-|z_{l}|^{2}}\begin{pmatrix}\kappa_{(\textup{s})}(z_{j},z_{l})&\kappa_{(\textup{s})}(z_{j},\bar{z}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\kappa_{(\textup{s})}(\bar{z}_{j},z_{l})&\kappa_{(\textup{s})}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})+o(1),$

uniformly for $z_{1},\ldots,z_{k}$ in compact subsets of $\mathbb{C}$, where

(1.24)
$\kappa_{(\textup{s})}(z,w):=\sqrt{\pi}e^{z^{2}+w^{2}}\int_{E}W(f_{w},f_{z})(u)\,du,\quad
E=\begin{cases}(-\infty,\infty)&\textup{if }r_{1}<p<r_{2},\\\
(-\infty,0)&\textup{if }p=r_{1}\textup{ or }p=r_{2}.\end{cases}$

Here $f_{z}$ is given by (1.19).

  * (b)

At regular points in the limit of weak non-unitarity.
Let $L=aN$, $n=(b+1)N$ with fixed $a\geq 0$, $b>0$. Let $p=1$. Then as
$N\to\infty$,

(1.25)
$R_{N,k}(z_{1},\dots,z_{k})={\textup{Pf}}\Big{[}e^{-|z_{j}|^{2}-|z_{l}|^{2}}\begin{pmatrix}\kappa_{(\textup{w})}(z_{j},z_{l})&\kappa_{(\textup{w})}(z_{j},\bar{z}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\kappa_{(\textup{w})}(\bar{z}_{j},z_{l})&\kappa_{(\textup{w})}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})+o(1),$

uniformly for $z_{1},\ldots,z_{k}$ in compact subsets of $\mathbb{C}$, where

(1.26)
$\kappa_{(\textup{w})}(z,w):=\sqrt{\pi}e^{z^{2}+w^{2}}\Big{(}\int_{-a}^{a}W(f_{w},f_{z})(u)\,du+f_{w}(a)f_{z}(-a)-f_{z}(a)f_{w}(-a)\Big{)},\quad
a=\frac{\rho}{2\sqrt{2}}.$

  * (c)

At the singular origin.
Let $L\geq 0$ be fixed and $n=(b+1)N$ with fixed $b>0$. Let $p=0$. Then as
$N\to\infty$,

(1.27)
$R_{N,k}(z_{1},\dots,z_{k})={\textup{Pf}}\Big{[}e^{-|z_{j}|^{2}-|z_{l}|^{2}}\begin{pmatrix}\kappa_{(\textup{o})}(z_{j},z_{l})&\kappa_{(\textup{o})}(z_{j},\bar{z}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\kappa_{(\textup{o})}(\bar{z}_{j},z_{l})&\kappa_{(\textup{o})}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})+o(1),$

uniformly for $z_{1},\ldots,z_{k}$ in compact subsets of $\mathbb{C}$, where

(1.28)
$\kappa_{(\textup{o})}(z,w)=2(2zw)^{2L}\int_{0}^{1}s^{2L}\Big{(}ze^{(1-s^{2})z^{2}}-we^{(1-s^{2})w^{2}}\Big{)}E_{2,1+2L}((2szw)^{2})\,ds.$

Here, $E_{a,b}$ is the two-parametric Mittag-Leffler function (1.20).

The limiting kernel of the form (1.24) was introduced in [3, Theorem 2.1] as a
scaling limit of the planar symplectic Ginibre ensemble. Here
$E=(-\infty,\infty)$ corresponds to the bulk case, whereas $E=(-\infty,0)$
corresponds to the edge case. (We also refer to [3, Remark 2.4] for more
discussions on the role of the integral domain $E$.) Therefore Theorem 1.4 (a)
shows that in the limit of strong non-unitarity, the spherical induced
symplectic Ginibre ensemble is contained in the universality class of the
planar Ginibre ensemble. Note also that for the bulk case when
$E=(-\infty,\infty)$, the integral in (1.24) can be further simplified, which
gives rise to the expression

(1.29)
$\kappa_{(\textup{s})}(z,w)=\sqrt{\pi}\,e^{z^{2}+w^{2}}\operatorname{erf}(z-w),\qquad\mbox{if}\quad
E=(-\infty,\infty).$

This form of the kernel appeared in [50, 39].

The limiting kernel (1.26) was introduced very recently in [13, Theorem 1.1
(b)] as a scaling limit of the planar induced symplectic ensemble in the
almost-circular regime. Thus Theorem 1.4 (b) also establishes the universality
in this regime. An interesting feature of the kernel (1.26) is that it
interpolates the bulk scaling limits of the symplectic Ginibre ensemble which
form Pfaffian point processes ($\rho\to\infty$) and those of the chiral
Gaussian unitary ensemble ($\rho\to 0$) which form determinantal point
processes. We refer to [13, Remark 1.4 and Proposition 1.5] for more details
about this interpolating property.

Finally, the limiting kernel of the form (1.28) appeared in [3, Theorem 2.4
and Example 2.6] (with $c=2L$, $\lambda=1$) as a scaling limit of the planar
induced symplectic Ginibre ensemble at the origin having spectral singularity.
Therefore Theorem 1.4 (c) again shows the universality and also asserts that
under the insertion of a point charge, it is the strength of the charge (i.e.
$4L$ in (1.3)) that determines the universality class. We also mention that if
$L=0$, one can see from $E_{2,1}(z^{2})=\cosh(z)$ that the kernel (1.28)
agrees with the kernel (1.29). Furthermore, it follows from the relation

(1.30) $\displaystyle
2E_{2,1+c}(z^{2})=E_{1,1+c}(z)+E_{1,1+c}(-z)=e^{z}z^{-c}P(c,z)+e^{-z}(-z)^{-c}P(c,-z),$

where

(1.31) $P(c,z):=\frac{1}{\Gamma(c)}\int_{0}^{z}t^{c-1}e^{-t}\,dt,\qquad(c>0)$

is the (regularised) incomplete gamma function, that we have an alternative
representation

(1.32)
$\kappa_{(\textup{o})}(z,w)=\int_{0}^{1}\Big{(}ze^{(1-s^{2})z^{2}}-we^{(1-s^{2})w^{2}}\Big{)}\Big{(}e^{2szw}P(2L,2szw)+(-1)^{-2L}e^{-2szw}P(2L,-2szw)\Big{)}\,ds.$

In Theorem 1.4, we have focused on the scaling limits along the real axis,
i.e. $p\in\mathbb{R}.$ In general, it can be expected that away from the real
axis (i.e. $p\in\mathbb{C}\setminus\mathbb{R}$), the scaling limits of the
ensemble (1.2) become determinantal with the correlation kernel of its complex
counterpart; see [6] for a heuristic discussion for this. (Such a statement
was shown in [13] for the planar induced symplectic ensemble.) For the
spherical induced symplectic Ginibre ensemble, the scaling limits away from
the real axis was studied in [49, Section 6], where the authors derived the
universal $1$-point functions.

Further points for investigation are also suggested. One is the study of the
so called hole probability, i.e. the probability that a prescribed region is
free of eigenvalues. In the case of the complex Ginibre ensemble, this was
first investigated long ago in [36], and in a generalised form has been the
subject of a number of recent works [45, 44, 22, 19, 18, 14]. Another is the
study of fluctuation formulas associated with linear statistics; see the
recent review [29, Section 3.5] and references therein, and Appendix B for
results relating to (1.2) in the case of radial symmetry.

The rest of this paper is organised as follows. Section 2 begins with the
finite $N$ result of Proposition 1.1 and then identifies a rescaling of the
correlation functions valid to leading order in $N$. Next, in Proposition 2.2,
the large $N$ form of the differential equation of Proposition 1.1 in the
various regimes of interest for Theorem 1.4 is given. The proof of this result
is deferred until Section 3. The final new result of Section 2, Lemma 2.3, is
to present the solutions of the limiting differential equations. Section 2
concludes by showing how these various results can be assembled to prove
Theorem 1.4. The main content of Section 3 is the proofs of Propositions 1.1
and 2.2, stated but not proved in earlier sections.

### Acknowledgements

The authors thank Markus Ebke for the help with numerical simulations.

##  2\. Proof of Theorem 1.4

This section culminates in the proof of Theorem 1.4. For reader’s convenience,
we first present a summary of the strategy.

  * (i)

In Lemma 2.1, we first obtain the structure of the correlation function which
contains the rescaled skew-kernel $\widetilde{\kappa}_{N}$ in (2.1) and
Gaussian terms. This follows from the explicit formula given in Proposition
1.1 (a).

  * (ii)

In Proposition 2.2, we derive the asymptotic behaviour of
$\partial_{z}\widetilde{\kappa}_{N}(z,w)$. For this, we use the Christoffel-
Darboux formula in Proposition 1.1 (b) and compute the asymptotic expansions
of its inhomogeneous terms.

  * (iii)

In Lemma 2.3, we solve the differential equation appearing in Proposition 2.2,
which gives rise to the explicit formulas of the limiting correlation kernels
in Theorem 1.4.

As already mentioned in the final paragraph of the above section, the proofs
of Propositions 1.1 and 2.2 are given separately in Section 3.

We begin with deriving the basic structure of the correlation functions using
Proposition 1.1 (a).

######  Lemma 2.1 (Structure of the correlation functions).

Let

(2.1)
$\widetilde{\kappa}_{N}(z,w):=\frac{1}{(1+p^{2})^{3}\,(N\delta)^{3/2}}\widetilde{{\bm{\varkappa}}}_{N}\Big{(}p+\frac{z}{\sqrt{N\delta}},p+\frac{w}{\sqrt{N}}\Big{)},$

where $\widetilde{{\bm{\varkappa}}}_{N}$ is given by (1.9). Recall that
$R_{N,k}$ is given by (1.22). Then under the same assumptions on Theorem 1.4
and in each case (a),(b),(c), we have

$\displaystyle R_{N,k}(z_{1},\dots,z_{k})$
$\displaystyle={\textup{Pf}}\Big{[}e^{-|z_{j}|^{2}-|z_{l}|^{2}}\begin{pmatrix}e^{z_{j}^{2}+z_{l}^{2}}\widetilde{\kappa}_{N}(z_{j},z_{l})&e^{z_{j}^{2}+\bar{z}_{l}^{2}}\widetilde{\kappa}_{N}(z_{j},\bar{z}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
e^{\bar{z}_{j}^{2}+z_{l}^{2}}\widetilde{\kappa}_{N}(\bar{z}_{j},z_{l})&e^{\bar{z}_{j}^{2}+\bar{z}_{l}^{2}}\widetilde{\kappa}_{N}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})+O(\frac{1}{\sqrt{N}}),$

uniformly for $z_{1},\ldots,z_{k}$ in compact subsets of $\mathbb{C}$, as
$N\to\infty$.

###### Proof.

Recall that the weight function $\omega$ is given by (1.8). Then it follows
from direct computations that in each case (a),(b),(c) of Theorem 1.4,

(2.2)
$\omega\Big{(}p+\frac{z}{\sqrt{N\delta}}\Big{)}=(1+p^{2})^{-\frac{3}{2}}e^{-|z|^{2}+\frac{1}{2}(z^{2}+\bar{z}^{2})}+O(\frac{1}{\sqrt{N}}),\qquad\mbox{as
}N\to\infty.$

For given $p\geq 0$, we write

(2.3) $\zeta_{j}=p+\frac{z_{j}}{\sqrt{N\delta}},$

where $\delta$ is given by (1.21). Then by combining Proposition 1.1 with
(1.22), (2.2) and (2.3), we obtain

$\displaystyle\quad
R_{N,k}(z_{1},\dots,z_{k})=\frac{1}{(N\delta)^{k}}\mathbf{R}_{N,k}(\zeta_{1},\dots,\zeta_{k})$
$\displaystyle={\textup{Pf}}\Big{[}\frac{e^{-|z_{j}|^{2}-|z_{l}|^{2}}}{(1+p^{2})^{3}(N\delta)^{3/2}}\begin{pmatrix}e^{z_{j}^{2}+z_{l}^{2}}\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\zeta_{l})&e^{z_{j}^{2}+\bar{z}_{l}^{2}}\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\bar{\zeta}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
e^{\bar{z}_{j}^{2}+z_{l}^{2}}\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\zeta_{l})&e^{\bar{z}_{j}^{2}+\bar{z}_{l}^{2}}\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\bar{\zeta}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})+O(\frac{1}{\sqrt{N}}).$

Now lemma follows from (2.1). ∎

The next step is to derive the asymptotic behaviour of the derivative of
$\widetilde{\kappa}_{N}$ in (2.1). This step crucially relies on Proposition
1.1 (b).

######  Proposition 2.2 (Large-$N$ expansions of the differential equations).

As $N\to\infty$, the following hold.

  * (a)

Under the setup of Theorem 1.4 (a),

(2.4) $\partial_{z}\widetilde{\kappa}_{N}(z,w)=F_{\textup{(s)}}(z,w)+o(1),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$, where

(2.5) $F_{\textup{(s)}}(z,w):=\begin{cases}\displaystyle
2\,e^{-(z-w)^{2}}&\textup{if }r_{1}<p<r_{2},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \displaystyle
e^{-(z-w)^{2}}\operatorname{erfc}(z+w)-\frac{e^{-2z^{2}}}{\sqrt{2}}\operatorname{erfc}(\sqrt{2}w)&\textup{if
}p=r_{1},r_{2}.\end{cases}$

  * (b)

Under the setup of Theorem 1.4 (b),

(2.6) $\partial_{z}\widetilde{\kappa}_{N}(z,w)=F_{\textup{(w)}}(z,w)+o(1),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$, where

(2.7)
$\displaystyle\begin{split}F_{\textup{(w)}}(z,w)&:=e^{-(z-w)^{2}}\Big{(}\operatorname{erfc}(z+w-\tfrac{\rho}{\sqrt{2}})-\operatorname{erfc}(z+w+\tfrac{\rho}{\sqrt{2}})\Big{)}\\\
&\quad-\frac{1}{\sqrt{2}}\Big{(}e^{-(\sqrt{2}z-\frac{\rho}{2})^{2}}+e^{-(\sqrt{2}z+\frac{\rho}{2})^{2}}\Big{)}\Big{(}\operatorname{erfc}(\sqrt{2}w-\tfrac{\rho}{2})-\operatorname{erfc}(\sqrt{2}w+\tfrac{\rho}{2})\Big{)}.\end{split}$

  * (c)

Under the setup of Theorem 1.4 (c),

(2.8) $\partial_{z}\widetilde{\kappa}_{N}(z,w)=F_{\textup{(o)}}(z,w)+o(1),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$, where

(2.9) $\displaystyle
F_{\textup{(o)}}(z,w):=2\,e^{-(z-w)^{2}}P(2L,2zw)-\frac{2\sqrt{\pi}}{\Gamma(L)}z^{2L-1}e^{-z^{2}}P(L+\tfrac{1}{2},w^{2}).$

Here $P$ is the regularised incomplete gamma function (1.31).

The proof of this proposition will be given in the next section.

Finally, we solve the differential equations appearing in Proposition 2.2. The
following lemma is an immediate consequence of several results established in
[3, 13].

######  Lemma 2.3.

Let

(2.10) $\displaystyle\mathcal{K}_{\textup{(s)}}(z,w)$
$\displaystyle:=e^{-z^{2}-w^{2}}\kappa_{\textup{(s)}}(z,w),$ (2.11)
$\displaystyle\mathcal{K}_{\textup{(w)}}(z,w)$
$\displaystyle:=e^{-z^{2}-w^{2}}\kappa_{\textup{(w)}}(z,w),$ (2.12)
$\displaystyle\mathcal{K}_{\textup{(o)}}(z,w)$
$\displaystyle:=e^{-z^{2}-w^{2}}\kappa_{\textup{(o)}}(z,w),$

where $\kappa_{\textup{(s)}}$, $\kappa_{\textup{(w)}}$ and
$\kappa_{\textup{(o)}}$ are given by (1.24), (1.26) and (1.28). Then the
following hold.

  * (a)

For a given $w\in\mathbb{C}$, the function
$z\mapsto\mathcal{K}_{\textup{(s)}}(z,w)$ is a unique solution to

(2.13)
$\partial_{z}\mathcal{K}_{\textup{(s)}}(z,w)=F_{\textup{(s)}}(z,w),\qquad\mathcal{K}_{\textup{(s)}}(z,w)|_{z=w}=0,$

where $F_{\textup{(s)}}$ is given by (2.5).

  * (b)

For a given $w\in\mathbb{C}$, the function
$z\mapsto\mathcal{K}_{\textup{(w)}}(z,w)$ is a unique solution to

(2.14)
$\partial_{z}\mathcal{K}_{\textup{(w)}}(z,w)=F_{\textup{(w)}}(z,w),\qquad\mathcal{K}_{\textup{(w)}}(z,w)|_{z=w}=0,$

where $F_{\textup{(w)}}$ is given by (2.7).

  * (c)

For a given $w\in\mathbb{C}$, the function
$z\mapsto\mathcal{K}_{\textup{(o)}}(z,w)$ is a unique solution to

(2.15)
$\partial_{z}\mathcal{K}_{\textup{(o)}}(z,w)=F_{\textup{(o)}}(z,w),\qquad\mathcal{K}_{\textup{(o)}}(z,w)|_{z=w}=0,$

where $F_{\textup{(o)}}$ is given by (2.9).

###### Proof.

By (1.29), the first assertion (a) for the bulk case when $E=(-\infty,\infty)$
is trivial. For the edge case when $E=(-\infty,0)$, this was shown in [3,
p.21].

The second assertion (b) was shown in [13, Subsection 3.2].

Finally, the third assertion (c) follows from [3, Section 4]. More precisely,
the equation in the statement (c) is a special case of [3, Eq.(4.4)] with
$\lambda=1,c=2L$ up to a trivial transformation. Here, we also use the
relation (1.30). Then this differential equation was solved in [3, Subsection
4.2]. ∎

Let us now combine the results introduced above and finish the proof of
Theorem 1.4.

###### Proof of Theorem 1.4.

Combining Proposition 2.2, Lemma 2.3 and [15, Lemma 3.10], we obtain that

(2.16)
$\widetilde{\kappa}_{N}(z,w)=e^{-z^{2}-w^{2}}\begin{cases}\kappa_{\textup{s}}(z,w)+o(1)&\textup{for
the case (a)},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\kappa_{\textup{w}}(z,w)+o(1)&\textup{for the case (b)},\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ \kappa_{\textup{o}}(z,w)+o(1)&\textup{for the case
(c)}.\end{cases}$

Here we also have used (2.10), (2.11) and (2.12). The theorem now follows from
Lemma 2.1 and (2.16). ∎

##  3\. Proof of Propositions 1.1 and 2.2

In this section we present the proofs of Propositions 1.1 and 2.2. Both these
results have been used in the proof of Theorem 1.4.

###  3.1. Skew-orthogonal polynomial kernels

Using the general theory on planar symplectic ensembles and skew-orthogonal
polynomials, we first show the following lemma.

Recall that the potential $Q$ and the correlation function $\mathbf{R}_{N,k}$
are given by (1.3) and (1.6).

######  Lemma 3.1.

We have

(3.1)
$\mathbf{R}_{N,k}(\zeta_{1},\dots,\zeta_{k})={\textup{Pf}}\Big{[}e^{-N(Q(\zeta_{j})+Q(\zeta_{l}))}\begin{pmatrix}{\bm{\varkappa}}_{N}(\zeta_{j},\zeta_{l})&{\bm{\varkappa}}_{N}(\zeta_{j},\bar{\zeta}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
{\bm{\varkappa}}_{N}(\bar{\zeta}_{j},\zeta_{l})&{\bm{\varkappa}}_{N}(\bar{\zeta}_{j},\bar{\zeta}_{l})\end{pmatrix}\Big{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{\zeta}_{j}-\zeta_{j}),$

where

(3.2)
$\begin{split}{\bm{\varkappa}}_{N}(\zeta,\eta):=\bm{G}_{N}(\zeta,\eta)-\bm{G}_{N}(\eta,\zeta),\end{split}$

and

(3.3)
$\begin{split}\bm{G}_{N}(\zeta,\eta)&:=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+1}\eta^{2l}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}.\end{split}$

We stress that Lemma 3.1 is an immediate consequence of [27, Proposition 4].
Nevertheless, as this lemma is crucially used in the present work, we briefly
recall the proof.

###### Proof of Lemma 3.1.

First, let us consider the ensemble (1.2) with a general potential $Q$. Define
the skew-symmetric form

$\langle
f,g\rangle_{s}:=\int_{\mathbb{C}}\Big{(}f(\zeta)g(\bar{\zeta})-g(\zeta)f(\bar{\zeta})\Big{)}(\zeta-\bar{\zeta})e^{-2NQ(\zeta)}\,dA(\zeta).$

Then the skew-orthogonal polynomial $q_{m}$ of degree $m$ is defined by the
condition: for all $k,l\in\mathbb{N}$

(3.4) $\langle q_{2k},q_{2l}\rangle_{s}=\langle
q_{2k+1},q_{2l+1}\rangle_{s}=0,\qquad\langle
q_{2k},q_{2l+1}\rangle_{s}=-\langle
q_{2l+1},q_{2k}\rangle_{s}=r_{k}\,\delta_{k,l}.$

Here, $\delta_{k,l}$ is the Kronecker delta. Then it is well known [39] that
the correlation function (1.6) is of the form (3.1) with the canonical skew-
kernel

(3.5)
${\bm{\varkappa}}_{N}(\zeta,\eta):=\sum_{k=0}^{N-1}\frac{q_{2k+1}(\zeta)q_{2k}(\eta)-q_{2k}(\zeta)q_{2k+1}(\eta)}{r_{k}},\qquad\bm{G}_{N}(\zeta,\eta):=\sum_{k=0}^{N-1}\frac{q_{2k+1}(\zeta)q_{2k}(\eta)}{r_{k}}.$

Thus it suffices to compute the skew-orthogonal polynomials.

Let us now consider a general radially symmetric potential
$Q(\zeta)=Q(|\zeta|)$. We write

(3.6) $h_{k}:=\int_{\mathbb{C}}|\zeta|^{2k}e^{-2NQ(\zeta)}\,dA(\zeta).$

for the (squared) orthogonal norm. Then it follows from [5, Corollary 3.2]
that

(3.7) $q_{2k+1}(\zeta)=\zeta^{2k+1},\qquad
q_{2k}(\zeta)=\zeta^{2k}+\sum_{l=0}^{k-1}\zeta^{2l}\prod_{j=0}^{k-l-1}\frac{h_{2l+2j+2}}{h_{2l+2j+1}},\qquad
r_{k}=2h_{2k+1}.$

We now turn to the potential (1.3). For this

(3.8)
$h_{k}=2\int_{0}^{\infty}\frac{r^{2k+4L+1}}{(1+r^{2})^{2(n+L+1)}}\,dr=\frac{\Gamma(k+2L+1)\Gamma(2n-k+1)}{\Gamma(2n+2L+2)},$

and so by using (3.7), we obtain

(3.9)
$q_{2k}(\zeta)=\frac{\Gamma(k+L+1)}{\Gamma(k-n+\frac{1}{2})}\sum_{l=0}^{k}(-1)^{k-l}\frac{\Gamma(l-n+\frac{1}{2})}{\Gamma(l+L+1)}\zeta^{2l},\qquad
r_{k}=\frac{2\Gamma(2k+2L+2)\Gamma(2n-2k)}{\Gamma(2n+2L+2)}.$

(Note that these formulas were also derived [27, Proposition 4].) Combining
(3.5), (3.9) and the basic functional relations

(3.10) $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi
z)},\qquad\Gamma(2z)=\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma(z+\tfrac{1}{2})$

of the gamma function, we obtain

(3.11)
$\begin{split}&\quad\bm{G}_{N}(\zeta,\eta)=\sum_{k=0}^{N-1}\frac{\Gamma(2n+2L+2)}{2\Gamma(2k+2L+2)\Gamma(2n-2k)}\frac{\Gamma(k+L+1)}{\Gamma(k-n+\frac{1}{2})}\sum_{l=0}^{k}(-1)^{k-l}\frac{\Gamma(l-n+\frac{1}{2})}{\Gamma(l+L+1)}\zeta^{2k+1}\eta^{2l}\\\
&=\Gamma(2n+2L+2)\sum_{k=0}^{N-1}\frac{\Gamma(n-k+\frac{1}{2})}{2\Gamma(2k+2L+2)\Gamma(2n-2k)}\frac{\Gamma(k+L+1)}{\pi(-1)^{k-n}}\sum_{l=0}^{k}(-1)^{k-l}\frac{\Gamma(l-n+\frac{1}{2})}{\Gamma(l+L+1)}\zeta^{2k+1}\eta^{2l}\\\
&=\Gamma(2n+2L+2)\sum_{k=0}^{N-1}\frac{1}{2^{2L+2n+1}\Gamma(k+L+\frac{3}{2})\Gamma(n-k)}\sum_{l=0}^{k}(-1)^{n-l}\frac{\Gamma(l-n+\frac{1}{2})}{\Gamma(l+L+1)}\zeta^{2k+1}\eta^{2l}.\end{split}$

This completes the proof. ∎

###  3.2. Proof of Propositions 1.1

Let

(3.12)
$\widehat{{\bm{\varkappa}}}_{N}(\zeta,\eta):=\widehat{\bm{G}}_{N}(\zeta,\eta)-\widehat{\bm{G}}_{N}(\eta,\zeta)=\Big{(}(1+\zeta^{2})(1+\eta^{2})\Big{)}^{n+L-\frac{1}{2}}\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta),$

where $\widehat{\bm{G}}_{N}$ and $\widetilde{{\bm{\varkappa}}}_{N}$ are given
by (1.10) and (1.9).

The key step to prove Proposition 1.1 (b) is the following lemma.

######  Lemma 3.2.

We have

$\displaystyle(1+\zeta^{2})\partial_{\zeta}\widehat{{\bm{\varkappa}}}_{N}(\zeta,\eta)$
$\displaystyle=2\zeta\Big{(}n+L-\frac{1}{2}\Big{)}\widehat{{\bm{\varkappa}}}_{N}(\zeta,\eta)+\frac{\Gamma(2n+2L+2)}{2}\sum_{k=0}^{2N-1}\frac{\zeta^{k+2L}\eta^{k+2L}}{\Gamma(k+2L+1)\Gamma(2n-k)}$
$\displaystyle\quad-\frac{\pi\,\Gamma(2n+2L+2)}{2^{2L+2n}\Gamma(N+L+\frac{1}{2})\Gamma(n-N)}\zeta^{2N+2L}\sum_{k=0}^{N-1}\frac{\eta^{2k+2L}}{\Gamma(n-k+\frac{1}{2})\Gamma(k+L+1)}$
$\displaystyle\quad-\frac{\pi\,\Gamma(2n+2L+2)}{2^{2L+2n}\Gamma(n+\frac{1}{2})\Gamma(L)}\zeta^{2L-1}\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}}{\Gamma(n-k)\Gamma(k+L+\frac{3}{2})}.$

###### Proof.

Let us first compute $\partial_{\zeta}\widehat{\bm{G}}_{N}(\zeta,\eta)$. Note
that

$\displaystyle\quad\partial_{\zeta}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\zeta\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L-1}\eta^{2l+2L}}{\Gamma(k+L+\frac{1}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\frac{\zeta^{2L}\eta^{2L}}{\Gamma(L+\frac{1}{2})\Gamma(n)\Gamma(n+\frac{1}{2})\Gamma(L+1)}+2\zeta\sum_{k=0}^{N-2}\sum_{l=0}^{k+1}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k-1)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}.$

Here, we have

$\displaystyle\quad
2\zeta\sum_{k=0}^{N-2}\sum_{l=0}^{k+1}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k-1)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\zeta\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k-1)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad-2\zeta\sum_{l=0}^{N-1}\frac{\zeta^{2N+2L-1}\eta^{2l+2L}}{\Gamma(N+L+\frac{1}{2})\Gamma(n-N)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad+2\sum_{k=1}^{N-1}\frac{\zeta^{2k+2L}\eta^{2k+2L}}{\Gamma(k+L+\frac{1}{2})\Gamma(n-k)\Gamma(n-k+\frac{1}{2})\Gamma(k+L+1)}.$

Therefore we obtain

$\displaystyle\quad\partial_{\zeta}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\zeta\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k-1)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad-2\sum_{l=0}^{N-1}\frac{\zeta^{2N+2L}\eta^{2l+2L}}{\Gamma(N+L+\frac{1}{2})\Gamma(n-N)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad+2\sum_{k=0}^{N-1}\frac{\zeta^{2k+2L}\eta^{2k+2L}}{\Gamma(k+L+\frac{1}{2})\Gamma(n-k)\Gamma(n-k+\frac{1}{2})\Gamma(k+L+1)}.$

Note here that by (1.10), we have

$\displaystyle\quad\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k-1)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{(n-k-1)\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{((n+L-\frac{1}{2})-(k+L+\frac{1}{2}))\zeta^{2k+2L+1}\eta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\Big{(}n+L-\frac{1}{2}\Big{)}\widehat{\bm{G}}_{N}(\zeta,\eta)-\frac{1}{2}\zeta\partial_{\zeta}\widehat{\bm{G}}_{N}(\zeta,\eta).$

Combining all of the above equations, we conclude

(3.13)
$\displaystyle\begin{split}\partial_{\zeta}\widehat{\bm{G}}_{N}(\zeta,\eta)&=2\zeta\Big{(}n+L-\frac{1}{2}\Big{)}\widehat{\bm{G}}_{N}(\zeta,\eta)-\zeta^{2}\partial_{\zeta}\widehat{\bm{G}}_{N}(\zeta,\eta)\\\
&\quad-\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n}}\sum_{l=0}^{N-1}\frac{\zeta^{2N+2L}\eta^{2l+2L}}{\Gamma(N+L+\frac{1}{2})\Gamma(n-N)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}\\\
&\quad+\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n}}\sum_{k=0}^{N-1}\frac{\zeta^{2k+2L}\eta^{2k+2L}}{\Gamma(k+L+\frac{1}{2})\Gamma(n-k)\Gamma(n-k+\frac{1}{2})\Gamma(k+L+1)}.\end{split}$

Next, we compute $\partial_{\zeta}\widehat{\bm{G}}_{N}(\eta,\zeta)$. By
similar computations as above, we have

$\displaystyle\quad\partial_{\zeta}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2L-1}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n+\frac{1}{2})\Gamma(L)}$
$\displaystyle\quad+2\zeta\sum_{k=0}^{N-1}\sum_{l=0}^{k-1}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l-\frac{1}{2})\Gamma(l+L+1)}.$

Here, the last term is rearranged as

$\displaystyle\quad\sum_{k=0}^{N-1}\sum_{l=0}^{k-1}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l-\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l-\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad-\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2k+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-k-\frac{1}{2})\Gamma(k+L+1)}.$

This gives rise to

$\displaystyle\quad\partial_{\zeta}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=2\zeta\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l-\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle\quad+2\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2L-1}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n+\frac{1}{2})\Gamma(L)}$
$\displaystyle\quad-2\zeta\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2k+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-k-\frac{1}{2})\Gamma(k+L+1)}.$

By using (1.10), we also have

$\displaystyle\quad\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l-\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{(n-l-\frac{1}{2})\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n+1}}\sum_{k=0}^{N-1}\sum_{l=0}^{k}\frac{((n+L-\frac{1}{2})-(l+L))\eta^{2k+2L+1}\zeta^{2l+2L}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-l+\frac{1}{2})\Gamma(l+L+1)}$
$\displaystyle=\Big{(}n+L-\frac{1}{2}\Big{)}\widehat{\bm{G}}_{N}(\eta,\zeta)-\frac{1}{2}\zeta\partial_{\zeta}\widehat{\bm{G}}_{N}(\eta,\zeta).$

Therefore we have shown that

(3.14)
$\displaystyle\begin{split}\partial_{\zeta}\widehat{\bm{G}}_{N}(\eta,\zeta)&=2\zeta\Big{(}n+L-\frac{1}{2}\Big{)}\widehat{\bm{G}}_{N}(\eta,\zeta)-\zeta^{2}\partial_{\zeta}\widehat{\bm{G}}_{N}(\eta,\zeta)\\\
&\quad+\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n}}\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2L-1}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n+\frac{1}{2})\Gamma(L)}\\\
&\quad-\pi\frac{\Gamma(2n+2L+2)}{2^{2L+2n}}\sum_{k=0}^{N-1}\frac{\eta^{2k+2L+1}\zeta^{2k+2L+1}}{\Gamma(k+L+\frac{3}{2})\Gamma(n-k)\Gamma(n-k-\frac{1}{2})\Gamma(k+L+1)}.\end{split}$

Now lemma follows from (3.13), (3.14) and (3.12). ∎

We now finish the proof of Proposition 1.1.

###### Proof of Proposition 1.1.

We first show the first part. By combining Lemma 3.1 with (1.3), (1.9) and
(3.2), we obtain

$\displaystyle\quad\mathbf{R}_{N,k}(\zeta_{1},\dots,\zeta_{k})={\textup{Pf}}\bigg{[}\frac{1}{((1+|\zeta_{j}|^{2})(1+|\zeta_{l}|^{2}))^{n+L+1}}$
$\displaystyle\begin{pmatrix}((1+\zeta_{j}^{2})(1+\zeta_{l}^{2}))^{n+L-\frac{1}{2}}\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\zeta_{l})&((1+\zeta_{j}^{2})(1+\bar{\zeta}_{l}^{2}))^{n+L-\frac{1}{2}}\widetilde{{\bm{\varkappa}}}_{N}(\zeta_{j},\bar{\zeta}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
((1+\bar{\zeta}_{j}^{2})(1+\zeta_{l}^{2}))^{n+L-\frac{1}{2}}\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\zeta_{l})&((1+\bar{\zeta}_{j}^{2})(1+\bar{\zeta}_{l}^{2}))^{n+L-\frac{1}{2}}\widetilde{{\bm{\varkappa}}}_{N}(\bar{\zeta}_{j},\bar{\zeta}_{l})\end{pmatrix}\bigg{]}_{j,l=1}^{k}\prod_{j=1}^{k}(\bar{\zeta}_{j}-\zeta_{j}).$

Then (1.7) follows from the basic properties of Pfaffians. For instance, we
have

(3.15)
$\mathbf{R}_{N,1}(\zeta)=\frac{|1+\zeta^{2}|^{2n+2L-1}}{(1+|\zeta|^{2})^{2n+2L+2}}\,\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\bar{\zeta})\,(\bar{\zeta}-\zeta)$

and

(3.16)
$\displaystyle\begin{split}\mathbf{R}_{N,2}(\zeta,\eta)&=\frac{|1+\zeta^{2}|^{2n+2L-1}|1+\eta^{2}|^{2n+2L-1}}{(1+|\zeta|^{2})^{2n+2L+2}(1+|\eta|^{2})^{2n+2L+2}}\\\
&\quad\times\Big{(}\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\bar{\zeta})\widetilde{{\bm{\varkappa}}}_{N}(\eta,\bar{\eta})-|\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\eta)|^{2}+|\widetilde{{\bm{\varkappa}}}_{N}(\zeta,\bar{\eta})|^{2}\Big{)}\,(\bar{\zeta}-\zeta)\,(\bar{\eta}-\eta).\end{split}$

This establishes Proposition 1.1 (a).

For the second assertion, recall that $\mathfrak{p}$ and $\mathfrak{q}$ are
given in (1.12) and that

(3.17) $\binom{r}{k}=\frac{\Gamma(r+1)}{\Gamma(k+1)\Gamma(r-k+1)}.$

Then after some straightforward computations using Lemma 3.2, (3.17) and the
transform (3.12), the desired formula (1.11) follows. ∎

###  3.3. Proof of Proposition 2.2

It remains to show Proposition 2.2 to validate the proof of Theorem 1.4. We
begin with the following lemma which is the rescaled version of Proposition
1.1 (b).

######  Lemma 3.3.

We have

(3.18)
$\partial_{z}\widetilde{\kappa}_{N}(z,w)=\textup{I}_{N}^{(1)}(z,w)\textup{I}_{N}^{(2)}(z,w)-\textup{II}_{N}^{(1)}(z,w)\textup{II}_{N}^{(2)}(z,w)-\textup{III}_{N}^{(1)}(z,w)\textup{III}_{N}^{(2)}(z,w),$

where

(3.19)
$\displaystyle\textup{I}_{N}^{(1)}(z,w):=\frac{1}{(1+p^{2})^{3}}\frac{1}{(N\delta)^{2}}\frac{1}{1+\zeta^{2}}\frac{(1+\zeta\eta)^{2n+2L-1}}{((1+\zeta^{2})(1+\eta^{2}))^{n+L-\frac{1}{2}}}\frac{\Gamma(2n+2L+2)}{2\Gamma(2n+2L)},$
(3.20)
$\displaystyle\textup{II}_{N}^{(1)}(z,w):=\frac{1}{(1+p^{2})^{3}}\frac{1}{(N\delta)^{2}}\,\frac{\zeta^{2N+2L}}{(1+\zeta^{2})^{n+L+\frac{1}{2}}}\frac{\pi\,\Gamma(2n+2L+2)/2^{2L+2n}}{\Gamma(N+L+\frac{1}{2})\Gamma(n-N)\Gamma(n+L+\frac{1}{2})},$
(3.21)
$\displaystyle\textup{III}_{N}^{(1)}(z,w):=\frac{1}{(1+p^{2})^{3}}\frac{1}{(N\delta)^{2}}\frac{\zeta^{2L-1}}{(1+\zeta^{2})^{n+L+\frac{1}{2}}}\frac{\pi\,\Gamma(2n+2L+2)/2^{2L+2n}}{\Gamma(n+\frac{1}{2})\Gamma(L)\Gamma(n+L+\frac{1}{2})},$

and

(3.22)
$\displaystyle\textup{I}_{N}^{(2)}(z,w):=\sum_{k=0}^{2N-1}\binom{2n+2L-1}{k+2L}\Big{(}\frac{\zeta\eta}{1+\zeta\eta}\Big{)}^{k+2L}\Big{(}\frac{1}{1+\zeta\eta}\Big{)}^{2n-k-1},$
(3.23)
$\displaystyle\textup{II}_{N}^{(2)}(z,w):=\sum_{k=0}^{N-1}\binom{n+L-\frac{1}{2}}{k+L}\Big{(}\frac{\eta^{2}}{1+\eta^{2}}\Big{)}^{k+L}\Big{(}\frac{1}{1+\eta^{2}}\Big{)}^{n-k-\frac{1}{2}},$
(3.24)
$\displaystyle\textup{III}_{N}^{(2)}(z,w):=\sum_{k=0}^{N-1}\binom{n+L-\frac{1}{2}}{k+L+\frac{1}{2}}\Big{(}\frac{\eta^{2}}{1+\eta^{2}}\Big{)}^{k+L+\frac{1}{2}}\Big{(}\frac{1}{1+\eta^{2}}\Big{)}^{n-k-1}.$

Here,

(3.25)
$\zeta=p+\frac{z}{\sqrt{N\delta}},\qquad\eta=p+\frac{w}{\sqrt{N\delta}}.$

###### Proof.

This is an immediate consequence of (2.1) and (1.11). ∎

We need to analyse the right-hand side of (3.18). For this, we need the
following lemma.

######  Lemma 3.4.

Recall that
$\textup{I}_{N}^{(1)},\textup{II}_{N}^{(1)},\textup{III}_{N}^{(1)}$ are given
by (3.19), (3.20), (3.21) and $\zeta,\eta$ are given by (3.25). As
$N\to\infty$, the following hold.

  * (a)

Under the setup of Theorem 1.4 (a), we have

(3.26)
$\displaystyle\textup{I}_{N}^{(1)}(z,w)=2\,e^{-(z-w)^{2}}+O(\frac{1}{\sqrt{N}}),$
(3.27)
$\displaystyle\textup{II}_{N}^{(1)}(z,w)=\begin{cases}\sqrt{2}\,e^{-2z^{2}}&\textup{if
}p=r_{2},\\\ O(e^{-N\epsilon})&\textup{otherwise},\end{cases}$ (3.28)
$\displaystyle\textup{III}_{N}^{(1)}(z,w)=\begin{cases}\sqrt{2}\,e^{-2z^{2}}&\textup{if
}p=r_{1},\\\ O(e^{-N\epsilon})&\textup{otherwise},\end{cases}$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$. Here, $\epsilon>0$ is
a small constant.

  * (b)

Under the setup of Theorem 1.4 (b), we have

(3.29)
$\displaystyle\textup{I}_{N}^{(1)}(z,w)=2\,e^{-(z-w)^{2}}+O(\frac{1}{N}),$
(3.30)
$\displaystyle\textup{II}_{N}^{(1)}(z,w)=\sqrt{2}\,e^{-(\sqrt{2}z-\frac{\rho}{2})^{2}}+O(\frac{1}{N}),$
(3.31)
$\displaystyle\textup{III}_{N}^{(1)}(z,w)=\sqrt{2}\,e^{-(\sqrt{2}z+\frac{\rho}{2})^{2}}+O(\frac{1}{N}),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$.

  * (c)

Under the setup of Theorem 1.4 (c), we have

(3.32)
$\displaystyle\textup{I}_{N}^{(1)}(z,w)=2\,e^{-(z-w)^{2}}+O(\frac{1}{N}),$
(3.33) $\displaystyle\textup{II}_{N}^{(1)}(z,w)=O(e^{-N\epsilon}),$ (3.34)
$\displaystyle\textup{III}_{N}^{(1)}(z,w)=\frac{2\sqrt{\pi}}{\Gamma(L)}z^{2L-1}e^{-z^{2}}+O(\frac{1}{N}),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$. Here, $\epsilon>0$ is
a small constant.

###### Proof.

This follows from long but straightforward computations repeatably using
Stirling’s formula. The most non-trivial part is the computations for
$\textup{II}_{N}^{(1)}$ and $\textup{III}_{N}^{(1)}$ under the setup of
Theorem 1.4 (a). In this case, we have

$\displaystyle\textup{II}_{N}^{(1)}(z,w)$
$\displaystyle=\Big{(}\frac{(1+a+b)^{1+a+b}}{(1+a)^{1+a}b^{b}}\frac{p^{2+2a}}{(1+p^{2})^{1+a+b}}\Big{)}^{N}\exp\Big{(}\frac{2(1+a-bp^{2})}{p\sqrt{1+a+b}}\,z\sqrt{N}\Big{)}$
$\displaystyle\quad\times\Big{[}\exp\Big{(}-\frac{1+a+(3+3a+b)p^{2}-bp^{4}}{(1+a+b)p^{2}}z^{2}\Big{)}\frac{(1+p^{2})^{1/2}}{(1+a+b)^{1/2}}\sqrt{2b}+O(\frac{1}{\sqrt{N}})\Big{]}$

and

$\displaystyle\textup{III}_{N}^{(1)}(z,w)$
$\displaystyle=\Big{(}\frac{(1+a+b)^{a+b+1}}{a^{a}(1+b)^{1+b}}\frac{p^{2a}}{(1+p^{2})^{1+a+b}}\Big{)}^{N}\exp\Big{(}\frac{2(a-(1+b)p^{2})}{p\sqrt{1+a+b}}\,z\sqrt{N}\Big{)}$
$\displaystyle\qquad\times\Big{[}\exp\Big{(}-\frac{a+(1+3a+b)p^{2}-(1+b)p^{4}}{(1+a+b)p^{2}}z^{2}\Big{)}\frac{(1+p^{2})^{1/2}}{(1+a+b)^{1/2}}\frac{\sqrt{2a}}{p}+O(\frac{1}{\sqrt{N}})\Big{]}$

as $N\to\infty$. Then the desired asymptotic expansion follows from the fact
that

$r_{1}=\sqrt{\frac{a}{b+1}}+O(\frac{1}{N}),\qquad
r_{2}=\sqrt{\frac{a+1}{b}}+O(\frac{1}{N})\qquad\mbox{as }N\to\infty.$

∎

######  Lemma 3.5.

Recall that
$\textup{I}_{N}^{(2)},\textup{II}_{N}^{(2)},\textup{III}_{N}^{(2)}$ are given
by (3.22), (3.23), (3.24) and $\zeta,\eta$ are given by (3.25). As
$N\to\infty$, the following holds.

  * (a)

Under the setup of Theorem 1.4 (a), we have

(3.35) $\displaystyle\textup{I}_{N}^{(2)}(z,w)=\begin{cases}1+o(1)&\textup{if
}r_{1}<p<r_{2}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\frac{1}{2}\operatorname{erfc}(z+w)+o(1)&\textup{if
}p=r_{1},r_{2},\end{cases}$ (3.36)
$\displaystyle\textup{II}_{N}^{(2)}(z,w)=\begin{cases}1+o(1)&\textup{if
}r_{1}<p<r_{2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\frac{1}{2}\operatorname{erfc}(\sqrt{2}w)+o(1)&\textup{if
}p=r_{1},r_{2},\end{cases}$ (3.37)
$\displaystyle\textup{III}_{N}^{(2)}(z,w)=\begin{cases}1+o(1)&\textup{if
}r_{1}<p<r_{2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\frac{1}{2}\operatorname{erfc}(\sqrt{2}w)+o(1)&\textup{if
}p=r_{1},r_{2},\end{cases}$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$.

  * (b)

Under the setup of Theorem 1.4 (b), we have

(3.38)
$\displaystyle\textup{I}_{N}^{(2)}(z,w)=\frac{1}{2}\Big{(}\operatorname{erfc}(z+w-\tfrac{\rho}{\sqrt{2}})-\operatorname{erfc}(z+w+\tfrac{\rho}{\sqrt{2}})\Big{)}+o(1),$
(3.39)
$\displaystyle\textup{II}_{N}^{(2)}(z,w)=\frac{1}{2}\Big{(}\operatorname{erfc}(\sqrt{2}w-\tfrac{\rho}{2})-\operatorname{erfc}(\sqrt{2}w+\tfrac{\rho}{2})\Big{)}+o(1),$
(3.40)
$\displaystyle\textup{III}_{N}^{(2)}(z,w)=\frac{1}{2}\Big{(}\operatorname{erfc}(\sqrt{2}w-\tfrac{\rho}{2})-\operatorname{erfc}(\sqrt{2}w+\tfrac{\rho}{2})\Big{)}+o(1),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$.

  * (c)

Under the setup of Theorem 1.4 (c), we have

(3.41) $\displaystyle\textup{I}_{N}^{(2)}(z,w)=P(2L,2zw)+o(1),$ (3.42)
$\displaystyle\textup{II}_{N}^{(2)}(z,w)=P(L,w^{2})+o(1),$ (3.43)
$\displaystyle\textup{III}_{N}^{(2)}(z,w)=P(L+\tfrac{1}{2},w^{2})+o(1),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$. Here $P$ is the
regularised incomplete gamma function.

###### Proof.

Recall that $\mathfrak{p}$ and $\mathfrak{q}$ are given by (1.12). We first
present a probabilistic proof of the lemma, which requires in particular that
$z,w\in\mathbb{R}$. This is instructive as it clearly shows the appearance of
the $\operatorname{erfc}$ and the incomplete gamma functions in the context of
the normal and Poisson approximations of the binomial distributions. We then
extend the validity to complex $z,w$ by Vitali’s theorem. This latter strategy
has been applied in related settings in e.g. [12, proof of Theorem 3.3].

Let $X\sim B(2n+2L-1,\mathfrak{p})$ be the binomial distribution assuming that
$2n+2L-1$,$2L$ are integers and $z,w\in\mathbb{R}$. Then
$\textup{I}_{N}^{(2)}$ can be rewritten as

$\displaystyle\quad\textup{I}_{N}^{(2)}(z,w)={\mathbb{P}}(2L\leq X\leq
2N+2L-1)$
$\displaystyle={\mathbb{P}}\Big{(}\frac{2L-(2n+2L-1)\mathfrak{p}}{\sqrt{(2n+2L-1)\mathfrak{p}(1-\mathfrak{p})}}\leq\frac{X-(2n+2L-1)\mathfrak{p}}{\sqrt{(2n+2L-1)\mathfrak{p}(1-\mathfrak{p})}}\leq\frac{2N+2L-1-(2n+2L-1)\mathfrak{p}}{\sqrt{(2n+2L-1)\mathfrak{p}(1-\mathfrak{p})}}\Big{)}.$

Let us consider the setup of Theorem 1.4 (a). Then as $N\to\infty$,

$\displaystyle\frac{2N+2L-1-(2n+2L-1)\mathfrak{p}}{\sqrt{(2n+2L-1)\mathfrak{p}(1-\mathfrak{p})}}=\frac{\sqrt{2}(1+a-bp^{2})}{p\sqrt{1+a+b}}\sqrt{N}-\frac{(1+p^{2})(1+a+bp^{2})}{\sqrt{2}(1+a+b)p^{2}}(z+w)+O(\frac{1}{\sqrt{N}}),$
$\displaystyle\frac{2L-(2n+2L-1)\mathfrak{p}}{\sqrt{(2n+2L-1)\mathfrak{p}(1-\mathfrak{p})}}=\frac{\sqrt{2}(a-(1+b)p^{2})}{p\sqrt{1+a+b}}\sqrt{N}-\frac{(1+p^{2})(a+(1+b)p^{2})}{\sqrt{2}(1+a+b)p^{2}}(z+w)+O(\frac{1}{\sqrt{N}}),$

uniformly for $z,w$ in compact subsets of $\mathbb{C}$. Recall that
$r_{1}\sim\sqrt{\frac{a}{b+1}}$ and $r_{2}\sim\sqrt{\frac{a+1}{b}}$. Then by
the Gaussian approximation of the binomial distribution, as $N\to\infty$, we
obtain

(3.44) ${\mathbb{P}}(2L\leq X\leq
2N+2L-1)\sim\begin{cases}{\mathbb{P}}(-\infty\leq Z\leq\infty)&\textup{if
}r_{1}<p<r_{2},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ {\mathbb{P}}(-\infty\leq
Z\leq-\sqrt{2}(z+w))&\textup{if }p=r_{1},r_{2},\end{cases}$

where $Z$ is the standard normal distribution. This gives rise to the desired
asymptotic behaviour (3.35). All other asymptotic formulas (3.36), (3.37),
(3.38), (3.39), (3.40) involving the $\operatorname{erfc}$ function follow
along the same lines.

Under the setup of Theorem 1.4 (c), we have

(3.45)
$\mathfrak{p}=\frac{zw}{1+b}\frac{1}{N}+O(\frac{1}{N^{2}}),\qquad\mbox{as
}N\to\infty.$

Thus the binomial distribution $X$ is approximated by the Poisson distribution
with intensity $\lambda=(2n+2L-1)\mathfrak{p}\sim 2zw$. Since the regularised
incomplete gamma function is the cumulative distribution function of the
Poisson distribution, we have

(3.46) ${\mathbb{P}}(2L\leq X\leq 2N+2L-1)\sim P(2L,2zw),\qquad\mbox{as
}N\to\infty,$

which leads to (3.41). The other asymptotics (3.42), (3.43) follow in a
similar way.

We now turn to the case with general parameters. In general, the functions
$\textup{I}_{N}^{(2)},\textup{II}_{N}^{(2)},\textup{III}_{N}^{(2)}$ can be
written in terms of the (regularised) incomplete beta function

(3.47)
$I_{x}(a,b):=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$

as

(3.48)
$\displaystyle\textup{I}_{N}^{(2)}(z,w)=I_{\mathfrak{p}}(2L,2n)-I_{\mathfrak{p}}(2N+2L,2n-2N),$
(3.49)
$\displaystyle\textup{II}_{N}^{(2)}(z,w)=I_{\mathfrak{q}}(L,n+\tfrac{1}{2})-I_{\mathfrak{q}}(N+L,n-N+\tfrac{1}{2}),$
(3.50)
$\displaystyle\textup{III}_{N}^{(2)}(z,w)=I_{\mathfrak{q}}(L+\tfrac{1}{2},n)-I_{\mathfrak{q}}(N+L+\tfrac{1}{2},n-N).$

For integer valued cases, the expressions (3.48), (3.49), (3.50) easily follow
from

(3.51) $I_{x}(m,n-m+1)=\sum_{j=m}^{n}\binom{n}{j}x^{j}(1-x)^{n-j},$

see e.g. [51, Eq.(8.17.5)]. In general, these follow from the definition of
the hypergeometric function in series form

(3.52)
${}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum_{s=0}^{\infty}\frac{\Gamma(a+s)\Gamma(b+s)}{\Gamma(c+s)\,s!}z^{s},$

and the relation

(3.53)
$I_{x}(a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\frac{x^{a}(1-x)^{b-1}}{a}{}_{2}F_{1}\Big{(}1,1-b\,;a+1\,;\frac{x}{x-1}\Big{)}.$

The significance of (3.48)–(3.50) is that $I_{x}(a,b)$ as defined by (3.47) is
an analytic function of $x$ in the cut complex-$x$ plane
$\mathbb{C}\backslash(-\infty,0)$ provided $|x|<1$. Appealing to Vitali’s
theorem on uniform convergence inside of a domain $\mathcal{C}$ for sequences
of analytic functions on $\mathcal{C}$ (see [56]) then allows the result
proved for $z,w$ real to be extended to compact sets of the complex plane.
Here a relevant point is the scaling of these variables by $\sqrt{N}$ as
required by (3.25), ensuring that for $z,w$ in compact subsets of
$\mathbb{C}$, the limiting sequence remains in the domain of analyticity. ∎

######  Remark 3.6.

For $z,w$ real, the uniform asymptotic expansions of the incomplete beta
function (3.47) that can be found in [51, Section 8.18] and [55, Section
11.3.3]. A method to extend these to the complex plane using a direct argument
can be found in [54, Section 5].

We now finish the proof of Proposition 2.2.

###### Proof of Proposition 2.2.

This immediately follows by substituting the asymptotic expansions in Lemmas
3.4 and 3.5 into the identity (3.18). ∎

## Appendix A

Consider an eigenvalue probability density function for $2N$ eigenvalues in
the complex plane, coordinates $\\{\zeta_{j}\\}_{j=1}^{N}$, specified by

(A.1) $\frac{1}{Z_{2N}}\prod_{1\leq j<k\leq
2N}|\zeta_{k}-\zeta_{j}|^{2}\prod_{l=1}^{2N}e^{-NQ(|\zeta_{l}|)}.$

Here the radial potential $Q(|\zeta_{l}|)$ is to be taken as general, subject
only to the normalisation $Z_{2N}$ being well defined. Suppose next that in
this functional form only $\\{\zeta_{j}\\}_{j=1}^{N}$ are independent, with
$\zeta_{j+N}=\bar{\zeta}_{j}$ ($j=1,\dots,N$). Then (A.1) reduces to a
probability density function for $N$ eigenvalues specified by

(A.2) $\frac{1}{\tilde{Z}_{N}}\prod_{1\leq j<k\leq
N}|\zeta_{k}-\zeta_{j}|^{2}\prod_{l=1}^{N}e^{-2NQ(|\zeta_{l}|)}.$

We see that (1.1) relates through this prescription to (A.1) with
$Q(|\zeta_{l}|)$ given by (1.3). In this Appendix, following [24], we revise
the interpretation of (A.1) in terms of the Boltzmann factor for a certain
one-component two-dimensional Coulomb system, features of which are then
inherited by (A.2).

The first point to note is the mapping of (A.1) from the complex plane to the
unit diameter Riemann sphere specified by

(A.3) $z=e^{i\phi}\tan(\theta/2),\qquad 0\leq\phi\leq
2\pi,\>0\leq\theta\leq\pi,$

which geometrically corresponds to a stereographic projection from the south
pole. Introducing the Cayley-Klein parameters

$u=\cos(\theta/2)e^{i\phi/2},\qquad v=-i\sin(\theta/2)e^{-i\phi/2},$

and with $dS$ denoting an element of the surface area of the sphere, a
straightforward calculation shows

(A.4) $\prod_{l=1}^{m}\Big{(}{|z_{l}|^{2}\over
1+|z_{l}|^{2}}\Big{)}^{q_{1}m}{1\over(1+|z_{l}|^{2})^{q_{2}m+m+1}}\prod_{1\leq
j<k\leq m}|z_{k}-z_{j}|^{2}\,dz_{1}\cdots dz_{m}\\\
=\prod_{l=1}^{m}|v_{l}|^{2q_{1}{m}}|u_{l}|^{2q_{2}{m}}\prod_{1\leq j<k\leq
m}|u_{k}v_{j}-u_{j}v_{k}|^{2}\,dS_{1}\cdots dS_{m},$

where $m:=2N$. The relevance of (A.4) is that with $Q(|\zeta_{l}|)$ given by
(1.3), the left hand side of (A.4) results with

(A.5) $q_{1}={2L\over m},\qquad q_{2}={(2n-m)\over m}.$

The parameters $q_{1},q_{2}$ have an electrostatic interpretation on the RHS
of (A.4). This comes about by first recalling the fact that two points
$(\theta,\phi)$ and $(\theta^{\prime},\phi^{\prime})$ on a sphere of unit
diameter, the solution of the charge neutral Poisson equation at
$(\theta,\phi)$ due to a unit charge at $(\theta^{\prime},\phi^{\prime})$, is
(see e.g. [26, Eq. (15.108)]) given in terms of the corresponding Cayley-Klein
parameters by

(A.6)
$\Phi((\theta,\phi),(\theta^{\prime},\phi^{\prime}))=-\log|u^{\prime}v-uv^{\prime}|.$

Let there be $m$ unit charges with coordinates $(\theta,\phi)$ interacting
pairwise on the sphere via the potential (A.6). This gives an energy

(A.7) $U_{0}=-\sum_{1\leq j<k\leq m}\log|u_{j}v_{k}-u_{k}v_{j}|.$

Suppose that at the north pole there is a fixed charge $mq_{1}$, and at the
south pole there is a fixed charge $mq_{2}$. The interaction with the mobile
unit charges gives an energy

(A.8)
$U_{1}=-mq_{1}\sum_{j=1}^{m}\log|v_{j}|-mq_{2}\sum_{j=1}^{m}\log|u_{j}|.$

We see that forming the Boltzmann factor $e^{-\beta(U_{0}+U_{1})}$ gives, for
$\beta=2$, precisely the right hand side of (A.4).

A spherical cap about the north pole with azimuthal angle $\theta$ has surface
area $\pi\sin^{2}(\theta/2)$. With a charge $mq_{1}$ at the north pole, the
value of $\theta$, $\theta_{q_{1}}$ say, which corresponds to a uniform
neutralising background charge $-mq_{1}$ in the spherical cap is such that

$\sin^{2}(\theta_{q_{1}}/2)={q_{1}\over q_{1}+q_{2}+1}.$

Here the left hand side is the proportion of the total surface area of the
sphere which is in the spherical cap. On the right hand side the ratio is
obtained by dividing the charge at the north pole by the total charge. Mapped
to the complex plane using (A.3), this gives a radius $r_{q_{1}}$ such that
$r_{q_{1}}^{2}=q_{1}/(1+q_{2})$ — note that this corresponds to $r_{1}$ in
(1.4). An analogous calculation for the spherical cap about the charge
$mq_{2}$ at the south pole corresponding to a charge neutral region region
leads to $r_{q_{2}}^{2}=(1+q_{1})/q_{2}$, which corresponds to $r_{2}$ in
(1.4).

## Appendix B

Here consideration is given to fluctuation formulas for linear statistics
relating to (1.2). A linear statistic is the random function
$B:=\sum_{j=1}^{N}b(\zeta_{j})$. The mean $\mu_{N,B}$ can be expressed in
terms of the eigenvalue density ($1$-point correlation)
$\mathbf{R}_{N,1}(\zeta)$ according to

$\mu_{N,B}=\int_{\mathbb{C}}b(\zeta)\mathbf{R}_{N,1}(\zeta)\,dA(\zeta).$

The variance can be expressed in terms of the one and two point correlations
according to

(B.1)
$\sigma_{N,B}^{2}=\int_{\mathbb{C}}dA(\zeta_{1})\,b(\zeta_{1})\int_{\mathbb{C}}dA(\zeta_{2})\,b(\zeta_{2})\\\
\times\Big{(}\mathbf{R}_{N,2}(\zeta_{1},\zeta_{2})-\mathbf{R}_{N,1}(\zeta_{1})\mathbf{R}_{N,1}(\zeta_{2})+\mathbf{R}_{N,1}(\zeta_{1})\delta(\zeta_{1}-\zeta_{2})\Big{)}.$

The most appropriate scaling regime to analyse a linear statistic is the
macroscopic limit. The density then has the large $N$ form given by (1.5),
supported on the region $S$ specified by (1.4), and so in this setting

(B.2)
$\mu_{N,B}\sim(n+L)\int_{S}{b(\zeta)\over(1+|\zeta|^{2})^{2}}\,dA(\zeta).$

In particular, this shows the mean is extensive, being proportional to $N$. In
contrast, in the macroscopic limit the variance is expected to be independent
of $N$, under the assumption that $b$ is smooth. The full distribution is
expected to be a Gaussian. Heuristic reasoning from the Coulomb gas viewpoint
underlying these predictions can be found in e.g. [26, Section 14.4].

The limit formulas for the correlations functions of Theorem 1.3 relate to
local rather than global scaling. Upon global scaling the functional form
relating to the correlations in (B.1) is not expected to be well defined as a
function, but rather to take the form of a distribution; see [26, Section
15.4]. In fact in the particular case that $b$ is smooth and a function of the
distance from the origin only, it is possible to compute the limiting form of
the variance indirectly, by considering the large $N$ form of the
characteristic function.

######  Proposition B.1.

Consider the radially symmetric linear statistic
$B=\sum_{j=1}^{N}b(|\zeta_{j}|)$ in relation to the induced symplectic induced
spherical ensemble as specified by (1.2) and (1.3). Let $\hat{P}_{N,B}(k)$
denote the corresponding characteristic function. We have that for large $N$

(B.3)
$\hat{P}_{N,B}(k)=e^{ik\tilde{\mu}_{N,B}-k^{2}\tilde{\sigma}_{B}^{2}/2+o(1)},$

where $\tilde{\mu}_{N,B}$ denotes the RHS of (B.2) and with $S$ defined as in
(B.2)

(B.4) $\tilde{\sigma}_{B}^{2}={1\over 8}\int_{S}||\nabla
b(|\zeta|)||^{2}\,dA(\zeta).$

###### Proof.

(Sketch) For the most part we follow the method given in [25] for the
analogous setting in the case of the complex Ginibre ensemble, although (B.6)
is a crucial ingredient made possible by a recent finding in the literature.

By definition

$\hat{P}_{N,B}(k)=\Big{\langle}\prod_{l=1}^{N}e^{ikb(|\zeta_{j}|)}\Big{\rangle},$

where the average is with respect to (1.2). Define

(B.5) $u_{l}:=\int_{0}^{\infty}r^{4l+3}e^{-2NQ(r)}\,dr,\qquad
u_{l}(b):=\int_{0}^{\infty}r^{4l+3}e^{-2NQ(r)}e^{ikb(r)}\,dr.$

According to the theory of radially symmetric skew orthogonal polynomials in
the complex plane [5, Corollary 3.3] we have

(B.6) $\hat{P}_{N,B}(k)=\prod_{l=0}^{N-1}u_{l}(b)/u_{l}.$

To the integrals in (B.4) we now apply Laplace’s method of asymptotic
analysis. For large $n,L,l$ this method begin by writing in each integrand

$r^{4l+3}e^{-2NQ(r)}=e^{-2(n+L+1)\log(1+r^{2})+(4(L+l)+3)\log r}=:e^{f(r)}$

then expands the integrand about the value of $r$ which maximises the
exponent, $r_{l}$ say. An elementary calculation shows

$r_{l}=\sqrt{L+l\over n-l},\qquad
f^{\prime\prime}(r_{l})=-{8(n-l)^{2}\over(n+L)}.$

Expanding the integrands in (B.5) about this point to second order in the
exponent shows

$\hat{P}_{N,B}(k)\sim\prod_{l=1}^{N}e^{ikb(r_{l})}e^{-k^{2}(b^{\prime}(r_{l}))^{2}/(2|f^{\prime\prime}(r_{l})|)}\sim
e^{ik\tilde{\mu}_{N,B}}e^{-k^{2}\tilde{\sigma}_{B}^{2}/2},$

where $\tilde{\mu}_{N,B}$ is given by the RHS of (B.2) and

$\tilde{\sigma}_{B}^{2}={1\over
4}\int_{r_{1}}^{r_{2}}r(b^{\prime}(r))^{2}\,dr.$

The RHS of this latter expression is equivalent to (B.4). ∎

The large $N$ functional form (B.3) for the characteristic function of $B$
implies that the centred linear statistic $B-\tilde{\mu}_{N,B}$ is a mean zero
Gaussian with variance given by (B.4). The structure of the latter is familiar
from the study of the fluctuations associated with a linear statistic for the
complex Ginibre ensemble; see the recent review [29, Section 3.5].

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