# Universal scaling limits of the symplectic elliptic Ginibre ensemble

Sung-Soo Byun  Department of Mathematical Sciences, Seoul National University,
Seoul, 151-747, Republic of Korea<EMAIL_ADDRESS>and  Markus Ebke
Faculty of Physics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld,
Germany<EMAIL_ADDRESS>
###### Abstract.

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are
known to form a Pfaffian point process whose correlation kernel can be
expressed in terms of the skew-orthogonal Hermite polynomials. We derive the
scaling limits and the convergence rates of the correlation functions at the
real bulk/edge of the spectrum, which in particular establishes the local
universality at strong non-Hermiticity. Furthermore, we obtain the subleading
corrections of the edge correlation kernels, which depend on the non-
Hermiticity parameter contrary to the universal leading term. Our proofs are
based on the asymptotic behaviour of the complex elliptic Ginibre ensemble due
to Lee and Riser as well as on a version of the Christoffel-Darboux identity,
a differential equation satisfied by the skew-orthogonal polynomial kernel.

###### 2020 Mathematics Subject Classification:

Primary 60B20; Secondary 33C45

The authors are grateful to the DFG-NRF International Research Training Group
IRTG 2235 supporting the Bielefeld-Seoul graduate exchange programme.
Furthermore, Sung-Soo Byun was partially supported by Samsung Science and
Technology Foundation (SSTF-BA1401-51).

##  1\. Introduction and main results

What has been understood about non-Hermitian random matrices with symplectic
symmetry? To name a few, we should start with the celebrated work of Ginibre
[21]. There, he introduced Gaussian random matrices with quaternion entries
(now known as the symplectic Ginibre ensemble) and derived the joint
probability distribution function ${\mathbf{P}}_{N}$ for their eigenvalues
$\bm{\zeta}=(\zeta_{1},\cdots,\zeta_{N})\in\mathbb{C}^{N}$. It is given by

(1.1)
$\displaystyle\begin{split}d{\mathbf{P}}_{N}(\bm{\zeta})&=\frac{1}{Z_{N}}\prod_{j>k=1}^{N}\lvert\zeta_{j}-\zeta_{k}\rvert^{2}\lvert\zeta_{j}-\overline{\zeta}_{k}\rvert^{2}\prod_{j=1}^{N}\lvert\zeta_{j}-\overline{\zeta}_{j}\rvert^{2}e^{-NQ(\zeta_{j})}\,dA(\zeta_{j}),\end{split}$

where $Z_{N}$ is the partition function and
$dA(\zeta):=\tfrac{1}{\pi}d^{2}\zeta$. Here $Q(\zeta)=|\zeta|^{2}$, but one
may consider an arbitrary potential $Q:\mathbb{C}\to\mathbb{R}$ as long as it
satisfies suitable potential theoretic assumptions. From the statistical
physics point of view, this eigenvalue distribution (1.1) can be interpreted
as a two-dimensional Coulomb gas with an additional complex conjugation
symmetry, see e.g. [17, Section 15.9] and [18].

Regarding the macroscopic (global) properties of the symplectic Ginibre
ensemble, it is well known that as $N\to\infty$, the eigenvalues tend to be
uniformly distributed on the centered disc with radius $\sqrt{2}$, known as
the circular law. Such a statement was extended to the ensemble (1.1) with
general $Q$ in [11], where it was shown that as $N\to\infty$, the system tends
to minimise the weighted logarithmic energy functional [33], the continuum
limit of its discrete Hamiltonian. In particular, in the large system, the
eigenvalues condense in a compact set $S\subset\mathbb{C}$ called the droplet.
Moreover, due to a well-known fact from the logarithmic potential theory [33],
it follows that the density inside the droplet with respect to the area
measure $dA$ is given by $\Delta Q/2$, where we use the convention
$\Delta=\partial{\bar{\partial}}$.

Turning to the microscopic (local) properties, contrary to the complex Ginibre
ensemble whose bulk scaling limit was already obtained in Ginibre’s original
work [21], the equivalent level of analysis for the symplectic Ginibre
ensemble was obtained long after. To our knowledge, it first appeared in the
second edition of Mehta’s book [27], where the bulk scaling limit was derived
at the origin of the spectrum. More precisely, it is known that the particle
system (1.1) forms a Pfaffian point process whose correlation functions can be
expressed in terms of a certain $2\times 2$ matrix-valued kernel (see (1.5)
below), and the large-$N$ limit of the kernel was computed in [27]. (See also
[22] for the scaling limits of the products of Ginibre matrices at the
singular point, the origin.)

Beyond the radially symmetric potentials, the elliptic generalisation of the
symplectic Ginibre ensemble was studied by Kanzieper [23]. Such an extension
is called the _elliptic Ginibre ensemble_ and is associated with the potential

(1.2)
$Q(\zeta):=\tfrac{1}{1-\tau^{2}}(|\zeta|^{2}-\tau\operatorname{Re}\zeta^{2}),\qquad\tau\in[0,1).$

Moreover, the associated droplet $S$ is given by the elliptic disc

(1.3)
$S:=\\{x+iy\in\mathbb{C}\,|\,(\tfrac{x}{\sqrt{2}(1+\tau)})^{2}+(\tfrac{y}{\sqrt{2}(1-\tau)})^{2}\leq
1\\},$

which is also known as the elliptic law, see the top part of Figure 1. In
[23], Kanzieper introduced the formalism of skew-orthogonal polynomials to
construct the correlation kernel and showed that the elliptic Ginibre ensemble
can be exactly solved using the Hermite polynomials. Furthermore, based on
this formalism, he obtained the scaling limits at the origin in the maximally
non-Hermitian regime where $\tau=0$ as well as in the almost-Hermitian regime
where $1-\tau=O(N^{-1})$. The analogous result for any fixed $\tau\in[0,1)$
was obtained in [5]. We also refer to [2] for the chiral counterpart of such a
result, which can be solved using the generalised Laguerre polynomials.

Figure 1. Eigenvalues of the elliptic Ginibre ensemble

While the above-mentioned works mainly focused on various scaling limits at
the origin, recently, there have been several studies on the scaling limits of
the symplectic ensembles away from the origin as well. One remarkable result
in this direction was due to Akemann, Kieburg, Mielke, and Prosen [6], where
the authors showed that away from the real axis, the eigenvalue statistics of
the symplectic and complex Ginibre ensembles are equivalent in the large-$N$
limit, see also [31] for a similar statement on the fluctuations of the
maximal modulus. Moreover very recently, the edge scaling limit of the
symplectic Ginibre ensemble at the right/left endpoint of the spectrum was
obtained independently in [4] and [24, 26]. These works can be thought of as
the maximally non-Hermitian analogue of the previous work [7], which
investigated the edge scaling limits of the elliptic Ginibre ensemble in the
almost-Hermitian regime where $1-\tau=O(N^{-\frac{1}{3}})$. However at strong
non-Hermiticity with general (fixed) $\tau\in[0,1)$, the scaling limits along
the real axis as well as their convergence rates have not been discovered yet
and we aim to contribute to these problems.

Let us now be more precise in introducing our main results. In the sequel, we
shall focus on the ensemble (1.1) with the elliptic potential (1.2). It is
well known (see e.g. [23]) that the $k$-point correlation function

(1.4)
$\mathbf{R}_{N,k}(\zeta_{1},\cdots,\zeta_{k}):=\frac{N!}{(N-k)!}\int_{\mathbb{C}^{N-k}}{\mathbf{P}}_{N}(\zeta_{1},\dots,\zeta_{N})\prod_{j=k}^{N}\,dA(\zeta_{j})$

has the structure

(1.5)
$\mathbf{R}_{N,k}(\zeta_{1},\cdots,\zeta_{k})=\prod_{j=1}^{k}(\overline{\zeta}_{j}-\zeta_{j}){\textup{Pf}}\Big{[}e^{-NQ(\zeta_{j})/2-NQ(\zeta_{l})/2}\begin{pmatrix}{\bm{\varkappa}}_{N}(\zeta_{j},\zeta_{l})&{\bm{\varkappa}}_{N}(\zeta_{j},\bar{\zeta}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
{\bm{\varkappa}}_{N}(\bar{\zeta}_{j},\zeta_{l})&{\bm{\varkappa}}_{N}(\bar{\zeta}_{j},\bar{\zeta}_{l})\end{pmatrix}\Big{]}_{j,l=1,\cdots,k}.$

Here the two-variable function ${\bm{\varkappa}}_{N}$ is called the _(skew)
pre-kernel_.

To describe the local statistics at a given point $p\in\mathbb{C}$, it is
convenient to define the rescaled point process $\bm{z}=(z_{1},\dots,z_{N})$
as

(1.6)
$z_{j}:=e^{-i\theta}\sqrt{N\delta}\cdot(\zeta_{j}-p),\qquad\delta:=\dfrac{\Delta
Q(p)}{2}=\frac{1}{2(1-\tau^{2})},$

where $\theta\in\mathbb{R}$ is the angle of the outward normal direction at
the boundary if $p\in\partial S$ (and otherwise $\theta=0$). See Figure 1 for
an illustration of the rescaled process. The correlation function $R_{N,k}$ of
the process $\bm{z}$ is given by

(1.7)
$\displaystyle\begin{split}R_{N,k}(z_{1},\cdots,z_{k})&:=\frac{1}{(N\delta)^{k}}\mathbf{R}_{N,k}(\zeta_{1},\cdots,\zeta_{k}).\end{split}$

In particular, if $p\in\mathbb{R}$, it can be written as

(1.8)
$\displaystyle\begin{split}R_{N,k}(z_{1},\cdots,z_{k})&=\prod_{j=1}^{k}(\bar{z}_{j}-z_{j}){\textup{Pf}}\Big{[}e^{-\frac{N}{2}(Q(p+\frac{z_{j}}{\sqrt{N\delta}})+Q(p+\frac{z_{l}}{\sqrt{N\delta}}))}\begin{pmatrix}\kappa_{N}(z_{j},z_{l})&\kappa_{N}(z_{j},\bar{z}_{l})\\\
\kappa_{N}(\bar{z}_{j},z_{l})&\kappa_{N}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1,\cdots
k},\end{split}$

where the rescaled pre-kernel $\kappa_{N}$ reads

(1.9)
$\displaystyle\begin{split}\kappa_{N}(z,w):=\frac{1}{(N\delta)^{\frac{3}{2}}}{\bm{\varkappa}}_{N}(\zeta,\eta).\end{split}$

In particular, the one-point function $R_{N}\equiv R_{N,1}$ for the rescaled
point process has the form

(1.10)
$R_{N}(z)=(\bar{z}-z)e^{-NQ(p+\tfrac{z}{\sqrt{N\delta}})}\,\kappa_{N}(z,\bar{z}).$

Recall that the $k$:th Hermite polynomial $H_{k}$ is given by

(1.11)
$H_{k}(z):=(-1)^{k}e^{z^{2}}\frac{d^{k}}{dz^{k}}e^{-z^{2}}=k!\sum_{m=0}^{\lfloor
k/2\rfloor}\frac{(-1)^{m}}{m!(k-2m)!}(2z)^{k-2m}.$

Then it follows from [23] that the pre-kernel $\kappa_{N}$ has the canonical
representation

(1.12)
$\displaystyle\begin{split}\kappa_{N}(z,w)&=\sqrt{2}(1+\tau)\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w\Big{)}\\\
&\quad-\sqrt{2}(1+\tau)\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w\Big{)}\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}.\end{split}$

See Subsection 2.1 for further details.

In our first result Proposition 1.1, we obtain a differential equation
satisfied by the pre-kernel $\kappa_{N},$ which can be recognised as a version
of the Christoffel-Darboux formula, cf. [25, 4, 12, 14].

######  Proposition 1.1.

For each $N$, the (canonical) pre-kernel $\kappa_{N}$ satisfies the
differential equation

(1.13)
$\displaystyle\begin{split}\partial_{z}\kappa_{N}(z,w)&=\Big{(}\sqrt{\tfrac{2(1-\tau)N}{1+\tau}}\,p+2(1-\tau)z\Big{)}\kappa_{N}(z,w)\\\
&\quad+2\sqrt{1-\tau^{2}}\,\sum_{k=0}^{2N-1}\frac{(\tau/2)^{k}}{k!}H_{k}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}H_{k}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w\Big{)}\\\
&\quad-2\sqrt{1-\tau^{2}}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}\Big{(}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w\Big{)}.\end{split}$

######  Remark.

It follows from

$(\tfrac{\tau}{2})^{k/2}H_{k}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}\sim(\sqrt{N}p+\sqrt{2}z)^{k},\qquad(\tau\to
0)$

that for $\tau=0$, the equation (1.13) reduces to

(1.14)
$\displaystyle\begin{split}\partial_{z}\kappa_{N}(z,w)&=(\sqrt{2N}p+2z)\kappa_{N}(z,w)\\\
&\quad+2\sum_{k=0}^{2N-1}\frac{(\sqrt{N}p+\sqrt{2}z)^{k}(\sqrt{N}p+\sqrt{2}w)^{k}}{k!}-2\frac{(\sqrt{N}p+\sqrt{2}z)^{2N}}{(2N-1)!!}\sum_{l=0}^{N-1}\frac{(\sqrt{N}p+\sqrt{2}w)^{2l}}{(2l)!!}.\end{split}$

This differential equation was utilized in [4]. See also [1] for a related
statement in the other extremal case when $\tau=1$.

It is worth pointing out that the inhomogenous term in the second line of
(1.13) corresponds to the (holomorphic) kernel of the complex elliptic Ginibre
ensemble with dimension $2N$. Such a relation has been observed in other
models as well, which include the Laguerre [2, 30, 3] and the Mittag-Leffler
ensembles [4, 10, 15].

Let us remark that the counterpart of such a differential equation for the
kernel of the complex elliptic Ginibre ensemble was obtained in [25,
Proposition 2.3]. (See also [14, Subsection 3.1] for an alternative derivation
in a more general framework.) As it allows to perform a suitable asymptotic
analysis, this formula has turned out to be very useful in various situations,
see e.g. [8, 13, 32]. In a similar spirit, we emphasise that Proposition 1.1
can also be applied to other cases including the almost-Hermitian regime.

In our main result Theorem 1.2 below, we establish the large-$N$ asymptotic of
the kernel. This is based on the asymptotic analysis of the differential
equation (1.13), see Corollary 2.1 and Proposition 3.1. Recall that by (1.3),
we have $S\cap\mathbb{R}=[-\sqrt{2}(1+\tau),\sqrt{2}(1+\tau)].$

######  Theorem 1.2.

Let $\tau\in[0,1)$ and $p\in\mathbb{R}$. Then there exists a pre-kernel
$\widetilde{\kappa}_{N}$ such that

(1.15)
$R_{N,k}(z_{1},\cdots,z_{k})=\prod_{j=1}^{k}(\bar{z}_{j}-z_{j})\,{\textup{Pf}}\Big{[}e^{-|z|^{2}-|w|^{2}}\begin{pmatrix}\widetilde{\kappa}_{N}(z_{j},z_{l})&\widetilde{\kappa}_{N}(z_{j},\bar{z}_{l})\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\widetilde{\kappa}_{N}(\bar{z}_{j},z_{l})&\widetilde{\kappa}_{N}(\bar{z}_{j},\bar{z}_{l})\end{pmatrix}\Big{]}_{j,l=1,\cdots
k}$

and that satisfies the following asymptotic behaviours as $N\to\infty$.

  * •

(The real bulk) For $|p|<\sqrt{2}(1+\tau)$, there exists an ${\varepsilon}>0$
such that

(1.16)
$\widetilde{\kappa}_{N}(z,w)=\kappa^{\mathbb{R}}_{\textup{bulk}}(z,w)+\mathcal{O}(e^{-N{\varepsilon}}),$

where

(1.17)
$\kappa^{\mathbb{R}}_{\textup{bulk}}(z,w):=\sqrt{\pi}e^{z^{2}+w^{2}}\operatorname{erf}(z-w).$

  * •

(The real edge) For $p=\pm\sqrt{2}(1+\tau)$ and an arbitrary ${\varepsilon}>0$
it holds

(1.18)
$\widetilde{\kappa}_{N}(z,w)=\kappa^{\mathbb{R}}_{\textup{edge}}(z,w)+\tfrac{1}{\sqrt{N}}\kappa^{\mathbb{R},1/2}_{\textup{edge}}(z,w)+\mathcal{O}(N^{-1+{\varepsilon}}),$

where

(1.19)
$\kappa^{\mathbb{R}}_{\textup{edge}}(z,w):=e^{2zw}\int_{-\infty}^{0}e^{-s^{2}}\sinh(2s(w-z))\operatorname{erfc}(z+w-s)\,ds$

and

(1.20)
$\displaystyle\begin{split}\kappa^{\mathbb{R},1/2}_{\textup{edge}}(z,w)&:=\tfrac{1}{12\sqrt{2}}(\tfrac{1+\tau}{1-\tau})^{3/2}\,e^{z^{2}+w^{2}}\\\
&\quad\times\Big{[}\Big{(}(2z^{2}+\tfrac{1-2\tau}{1+\tau})e^{-2z^{2}}\operatorname{erfc}(\sqrt{2}w)+2\sqrt{\tfrac{2}{\pi}}w\,e^{-2(z^{2}+w^{2})}\Big{)}-\Big{(}z\leftrightarrow
w\Big{)}\Big{]}.\end{split}$

  * •

(The real axis outside the droplet) For $|p|>\sqrt{2}(1+\tau)$, there exists
an ${\varepsilon}>0$ such that

(1.21) $\widetilde{\kappa}_{N}(z,w)=\mathcal{O}(e^{-N{\varepsilon}}).$

In all three cases the error terms are uniform for $z,w$ in compact subsets of
$\mathbb{C}$.

Since the leading terms of the pre-kernels are independent of $\tau$, Theorem
1.2 affirms the local bulk/edge universality along the real axis. For the
complex elliptic Ginibre ensemble, the analogous result on the fine asymptotic
behaviours of the kernels were obtained by Lee and Riser [25]. We also refer
to [19] for similar results on Gaussian symplectic ensemble when $\tau=1$.

(a) $R(z)$ for $p$ at the edge

(b) $R_{N}(z)$ when $\operatorname{Im}z=2$

(c) $\sqrt{N}(R_{N}(z)-R(z))$ when $\operatorname{Im}z=2$

(d) $R_{N}(z)$ when $\operatorname{Re}z=-1$

(e) $\sqrt{N}(R_{N}(z)-R(z))$ when $\operatorname{Re}z=-1$

Figure 2.  The plot (A) shows the graph of the limiting microscopic density
$R$ in (1.26). The plot (B) is the graph of $R_{N}$ and its comparison with
$R$. The plot (C) shows the same graph for $\sqrt{N}(R_{N}-R)$ and $R^{(1/2)}$
in (1.27). In both cases $\tau=1/3$ and $\operatorname{Im}z=2$. Additionally
the numeric limits obtained by the following method are displayed: for given
$z\in\mathbb{C}$ we compute $R_{N}(z)$ for $N=2000,3000,4000,5000$ and
estimate the constants $a,b,c$ in the series representation
$R_{N}(z)=a+b/\sqrt{N}+c/N+\cdots$ via a least squares fit. Then the values of
$a$ are drawn as circles. The same method works for the first correction
$\sqrt{N}(R_{N}(z)-R(z))$. The plots (D),(E) are analogous figures along the
cross-section $\operatorname{Re}z=1$.

######  Remark.

As noted in [4], the limiting pre-kernels in Theorem 1.2 can be written in the
following unified way

(1.22)
$\kappa_{a}^{\mathbb{R}}(z,w):=\frac{e^{z^{2}+w^{2}}}{\sqrt{2}}\int_{-\infty}^{a}e^{-2(z-u)^{2}}\operatorname{erfc}(\sqrt{2}(w-u))-(z\leftrightarrow
w)\,du,\qquad
a=\begin{cases}\infty&\textup{if}\quad|p|<\sqrt{2}(1+\tau),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ 0&\textup{if}\quad|p|=\sqrt{2}(1+\tau),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ -\infty&\textup{if}\quad|p|>\sqrt{2}(1+\tau).\end{cases}$

For an arbitrary $a\in\mathbb{R}$, the pre-kernel $\kappa_{a}^{\mathbb{R}}$
can be obtained by rescaling the process at the $N$-dependent point

(1.23) $p=\pm\sqrt{2}(1+\tau)\mp a\sqrt{\tfrac{2(1-\tau^{2})}{N}}.$

(See also [4, 9] for further motivations on zooming the point process at a
moving location.)

As an immediate consequence of Theorem 1.2, we obtain that the density $R_{N}$
for $p$ in the bulk along the real line is given by

(1.24) $R_{N}(x+iy)=4yF(2y)+\mathcal{O}(e^{-N{\varepsilon}}),\qquad
F(z):=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\,dt.$

Here $F$ is Dawson’s integral function. Similarly, for $p$ at the real edge we
have

(1.25)
$R_{N}(z)=R(z)+\tfrac{1}{\sqrt{N}}R^{(1/2)}(z)+\mathcal{O}(N^{-1+{\varepsilon}}),$

where ${\varepsilon}>0$ is arbitrary,

(1.26)
$R(x+iy)=-2y\int_{-\infty}^{0}e^{-t^{2}}\sin(4sy)\operatorname{erfc}(2x-s)\,ds$

and

(1.27)
$R^{(1/2)}(x+iy)=\tfrac{1}{3\sqrt{2}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}}y\,e^{-4y^{2}}\operatorname{Im}\Big{[}(2z^{2}+\tfrac{1-2\tau}{1+\tau})\operatorname{erfc}(\sqrt{2}\bar{z})\,e^{-2z^{2}}+2\sqrt{\tfrac{2}{\pi}}\,\bar{z}\,e^{-4\operatorname{Re}z^{2}}\Big{]}.$

See Figure 2 for illustrations of the edge density for finite $N$ and in the
limit. We remark that for each $x$, as $y\to\infty,$

$R(x+iy)\sim\tfrac{1}{2}\operatorname{erfc}(2x),\qquad
R^{(1/2)}(x+iy)\sim-\tfrac{1}{\sqrt{\pi}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}}e^{-4x^{2}}y^{2}.$

### Organisation of the paper

The rest of this paper is organised as follows. Section 2 is devoted to
recalling the skew-orthogonal polynomial formalism and to proving Proposition
1.1. In Section 3, we perform the asymptotic analysis of the inhomogenous
terms in (1.13) (Proposition 3.1) and complete the proof of Theorem 1.2. In
Appendix A, we show some preliminary estimates used in Section 3.

##  2\. Generalised Christoffel-Darboux identity

In this section, we derive a version of the Christoffel-Darboux formula. For
this we will use an explicit expression of the pre-kernels in terms of the
Hermite polynomials.

###  2.1. Skew-orthogonal polynomials

Following [23] we express the pre-kernel, and by extension all correlation
functions, in terms of skew-orthogonal polynomials. Define the skew-symmetric
form

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

Let $q_{m}$ be a family of monic polynomials of degree $m$ that satisfy the
following skew-orthogonality conditions with positive skew-norms $r_{k}>0$:
for all $k,l\in\mathbb{N}$

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

Here $\delta_{k,l}$ is the Kronecker delta. Such polynomials exist if
$\int\lvert\zeta\rvert^{m}e^{-NQ(\zeta)}\,dA(\zeta)<\infty$ for all
$m\in\mathbb{N}$, see e.g. [5]. Then the pre-kernel ${\bm{\varkappa}}_{N}$ is
written as

(2.2)
${\bm{\varkappa}}_{N}(\zeta,\eta)=\sum_{k=0}^{N-1}\frac{q_{2k+1}(\zeta)q_{2k}(\eta)-q_{2k}(\zeta)q_{2k+1}(\eta)}{r_{k}}.$

For the elliptic potential $Q$ in (1.2), the associated skew-orthogonal
polynomials $q_{k}$ and their skew-norms $r_{k}$ are given by

$q_{2k}(\zeta)=\Big{(}\frac{2}{N}\Big{)}^{k}k!\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}\zeta\Big{)},\qquad
q_{2k+1}(\zeta)=\Big{(}\frac{\tau}{2N}\Big{)}^{k+\frac{1}{2}}H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}\zeta\Big{)},$

and

$r_{k}=2(1-\tau)^{3/2}(1+\tau)^{1/2}\frac{(2k+1)!}{N^{2k+2}},$

see [23]. Thus the pre-kernel ${\bm{\varkappa}}_{N}$ of the ensemble
$\bm{\zeta}$ has the expression

$\begin{split}{\bm{\varkappa}}_{N}(\zeta,\eta)&=\Big{(}\frac{N}{2(1-\tau)}\Big{)}^{\frac{3}{2}}\sqrt{\frac{\tau}{1+\tau}}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k}}{(2k+1)!!}\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}\\\
&\quad\times\Big{[}H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}\zeta\Big{)}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}\eta\Big{)}-H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}\eta\Big{)}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}\zeta\Big{)}\Big{]}.\end{split}$

Therefore by (1.9), the rescaled pre-kernel $\kappa_{N}$ at the point
$p\in\mathbb{R}$ is given by

(2.3)
$\displaystyle\kappa_{N}(z,w)=\sqrt{2}(1+\tau)\Big{(}F(z,w)-F(w,z)\Big{)},$

where

(2.4)
$F(z,w)=\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z\Big{)}\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w\Big{)}.$

###  2.2. Derivation of a differential equation

With the expression (2.4), we show the Christoffel-Darboux identity
(Proposition 1.1) for the kernel.

###### Proof of Proposition 1.1.

Throughout the proof, let us write

(2.5)
$\zeta=\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}z,\qquad\eta=\sqrt{\tfrac{N}{2\tau}}p+\sqrt{\tfrac{1-\tau^{2}}{\tau}}w$

to lighten notations. It is well known that the Hermite polynomial $H_{j}$
satisfies the following differentiation rule and the three term recurrence
relation:

(2.6) $H_{j}^{\prime}(z)=2jH_{j-1}(z),\qquad
H_{j+1}(z)=2zH_{j}(z)-H_{j}^{\prime}(z).$

Using these formulas, we have

$\displaystyle\quad\partial_{z}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\zeta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle=2\sqrt{\frac{1-\tau^{2}}{\tau}}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k-1)!!}H_{2k}(\zeta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle=\sqrt{\tau(1-\tau^{2})}\sum_{k=1}^{N-1}\frac{(\tau/2)^{k-\frac{1}{2}}}{(2k-1)!!}H_{2k}(\zeta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)+\sqrt{2(1-\tau^{2})}$
$\displaystyle=\sqrt{\tau(1-\tau^{2})}\sum_{k=1}^{N-1}\frac{(\tau/2)^{k-\frac{1}{2}}}{(2k-1)!!}(2\zeta
H_{2k-1}(\zeta)-H_{2k-1}^{\prime}(\zeta))\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)+\sqrt{2(1-\tau^{2})}.$

Thus we obtain

$\displaystyle\quad\partial_{z}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\zeta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle=2\sqrt{\tau(1-\tau^{2})}\,\zeta\sum_{k=0}^{N-2}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\zeta)\sum_{l=0}^{k+1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle\quad-\tau\partial_{z}\sum_{k=0}^{N-2}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\zeta)\sum_{l=0}^{k+1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)+\sqrt{2(1-\tau^{2})}.$

By rearranging the terms,

$\displaystyle\quad\sum_{k=0}^{N-2}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\zeta)\sum_{l=0}^{k+1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle=F(z,w)-\frac{(\tau/2)^{N-\frac{1}{2}}}{(2N-1)!!}H_{2N-1}(\zeta)\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)+\sum_{k=0}^{N-2}\frac{(\tau/2)^{2k+\frac{3}{2}}}{(2k+2)!}H_{2k+1}(\zeta)H_{2k+2}(\eta).$

Therefore we obtain that

$\displaystyle\partial_{z}F(z,w)$
$\displaystyle=2\sqrt{\tau(1-\tau^{2})}\,\zeta F(z,w)-\tau\partial_{z}F(z,w)$
$\displaystyle\quad-\sqrt{\tau(1-\tau^{2})}\frac{(\tau/2)^{N-\frac{1}{2}}}{(2N-1)!!}\Big{(}2\zeta
H_{2N-1}(\zeta)-H_{2N-1}^{\prime}(\zeta)\Big{)}\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)$
$\displaystyle\quad+\sqrt{\tau(1-\tau^{2})}\sum_{k=0}^{N-2}\frac{(\tau/2)^{2k+\frac{3}{2}}}{(2k+2)!}\Big{(}2\zeta
H_{2k+1}(\zeta)-H^{\prime}_{2k+1}(\zeta)\Big{)}H_{2k+2}(\eta)+\sqrt{2(1-\tau^{2})}.$

We now use the three term recurrence relation again and obtain

$\displaystyle\quad\partial_{z}F(z,w)=2\sqrt{\tau(1-\tau^{2})}\,\zeta
F(z,w)-\tau\partial_{z}F(z,w)$
$\displaystyle-\sqrt{2(1-\tau^{2})}\Big{[}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}(\zeta)\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)-\sqrt{2(1-\tau^{2})}\sum_{k=0}^{N-1}\frac{(\tau/2)^{2k}}{(2k)!}H_{2k}(\zeta)H_{2k}(\eta)\Big{]}.$

This leads to

(2.7)
$\displaystyle\begin{split}&\quad\partial_{z}F(z,w)=\Big{(}\sqrt{\frac{2(1-\tau)N}{1+\tau}}p+2(1-\tau)z\Big{)}F(z,w)\\\
&-\frac{\sqrt{2(1-\tau^{2})}}{1+\tau}\Big{[}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}(\zeta)\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta)-\frac{\sqrt{2(1-\tau^{2})}}{1+\tau}\sum_{k=0}^{N-1}\frac{(\tau/2)^{2k}}{(2k)!}H_{2k}(\zeta)H_{2k}(\eta)\Big{]}.\end{split}$

Similarly, we obtain

$\displaystyle\quad\partial_{z}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\eta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\zeta)$
$\displaystyle=2\sqrt{\frac{1-\tau^{2}}{\tau}}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\eta)\sum_{l=1}^{k}\frac{(\tau/2)^{l}}{(2l-2)!!}H_{2l-1}(\zeta)$
$\displaystyle=2\sqrt{\frac{1-\tau^{2}}{\tau}}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\eta)\sum_{l=1}^{k}\frac{(\tau/2)^{l}}{(2l-2)!!}\Big{(}2\zeta
H_{2l-2}(\zeta)-H_{2l-2}^{\prime}(\zeta)\Big{)}$
$\displaystyle=\sqrt{(1-\tau^{2})\tau}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\eta)\sum_{l=0}^{k-1}\frac{(\tau/2)^{l}}{(2l)!!}\Big{(}2\zeta
H_{2l}(\zeta)-H_{2l}^{\prime}(\zeta)\Big{)}.$

Thus we have

$\displaystyle\partial_{z}F(w,z)$
$\displaystyle=\sqrt{(1-\tau^{2})\tau}\sum_{k=0}^{N-1}\frac{(\tau/2)^{k+\frac{1}{2}}}{(2k+1)!!}H_{2k+1}(\eta)\sum_{l=0}^{k}\frac{(\tau/2)^{l}}{(2l)!!}\Big{(}2\zeta
H_{2l}(\zeta)-H_{2l}^{\prime}(\zeta)\Big{)}$
$\displaystyle\quad-\sqrt{2(1-\tau^{2})}\sum_{k=0}^{N-1}\frac{(\tau/2)^{2k+1}}{(2k+1)!}H_{2k+1}(\zeta)H_{2k+1}(\eta).$

Therefore we obtain

$\displaystyle\partial_{z}F(w,z)$
$\displaystyle=\Big{(}\sqrt{2(1-\tau^{2})N}p+2(1-\tau^{2})z\Big{)}F(w,z)-\tau\partial_{z}F(w,z)$
$\displaystyle\quad-\sqrt{2(1-\tau^{2})}\sum_{k=0}^{N-1}\frac{(\tau/2)^{2k+1}}{(2k+1)!}H_{2k+1}(\zeta)H_{2k+1}(\eta),$

which leads to

(2.8)
$\displaystyle\begin{split}\partial_{z}F(w,z)&=\Big{(}\sqrt{\frac{2(1-\tau)N}{1+\tau}}p+2(1-\tau)z\Big{)}F(w,z)-\frac{\sqrt{2(1-\tau^{2})}}{1+\tau}\sum_{k=0}^{N-1}\frac{(\tau/2)^{2k+1}}{(2k+1)!}H_{2k+1}(\zeta)H_{2k+1}(\eta).\end{split}$

Combining all of the above, we conclude

$\displaystyle\partial_{z}\Big{(}F(z,w)-F(w,z)\Big{)}$
$\displaystyle=\Big{(}\sqrt{\frac{2(1-\tau)N}{1+\tau}}p+2(1-\tau)z\Big{)}\Big{(}F(z,w)-F(w,z)\Big{)}$
$\displaystyle\quad+\frac{\sqrt{2(1-\tau^{2})}}{1+\tau}\sum_{k=0}^{2N-1}\frac{(\tau/2)^{k}}{k!}H_{k}(\zeta)H_{k}(\eta)$
$\displaystyle\quad-\frac{\sqrt{2(1-\tau^{2})}}{1+\tau}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}(\zeta)\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(\eta).$

Now Proposition 1.1 follows from the relation (2.3).

∎

To analyse the large-$N$ limit of the pre-kernel, let us introduce

(2.9) $\widehat{\kappa}_{N}(z,w):=\omega_{N}(z,w)\kappa_{N}(z,w),$

where

(2.10)
$\omega_{N}(z,w)=\exp\Big{[}\tau\,\Big{(}p\sqrt{\tfrac{N}{2(1-\tau^{2})}}+z\Big{)}^{2}+\tau\,\Big{(}p\sqrt{\tfrac{N}{2(1-\tau^{2})}}+w\Big{)}^{2}-\Big{(}p\sqrt{\tfrac{N}{1-\tau^{2}}}+\sqrt{2}z\Big{)}\Big{(}p\sqrt{\tfrac{N}{1-\tau^{2}}}+\sqrt{2}w\Big{)}\Big{]}.$

Let us write

(2.11)
$\displaystyle\begin{split}E^{1}_{N}(\xi,\omega)&:=2\sqrt{1-\tau^{2}}\exp\Big{[}N\Big{(}\tfrac{\tau}{2}(\xi^{2}+\omega^{2})-\xi\omega\Big{)}\Big{]}\\\
&\quad\times\sum_{k=0}^{2N-1}\frac{(\tau/2)^{k}}{k!}H_{k}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}H_{k}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)},\end{split}$

and

(2.12)
$\displaystyle\begin{split}E^{2}_{N}(\xi,\omega)&:=2\sqrt{1-\tau^{2}}\exp\Big{[}-N\tfrac{1-\tau}{2}(\xi^{2}+\omega^{2})\Big{]}\\\
&\quad\times\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}.\end{split}$

Then as an immediate consequence of Proposition 1.1, we have the following
corollary.

######  Corollary 2.1.

We have

(2.13)
$\partial_{z}\widehat{\kappa}_{N}(z,w)=2(z-w)\widehat{\kappa}_{N}(z,w)+r_{N}(z,w),$

where

(2.14)
$r_{N}(z,w)=E_{N}^{1}\Big{(}\tfrac{p}{\sqrt{1-\tau^{2}}}+\sqrt{\tfrac{2}{N}}z,\tfrac{p}{\sqrt{1-\tau^{2}}}+\sqrt{\tfrac{2}{N}}w\Big{)}-e^{(z-w)^{2}}E_{N}^{2}\Big{(}\tfrac{p}{\sqrt{1-\tau^{2}}}+\sqrt{\tfrac{2}{N}}z,\tfrac{p}{\sqrt{1-\tau^{2}}}+\sqrt{\tfrac{2}{N}}w\Big{)}.$

The asymptotic expansion of $r_{N}$ will be addressed in the next section.

##  3\. Asymptotic analysis for the correlation kernels

In this section, we analyse the large-$N$ asymptotics of the differential
equation (2.13) and prove Theorem 1.2.

###  3.1. Asymptotics of the inhomogeneous terms

As indicated by Lemma 3.10 below, the key ingredient to prove Theorem 1.2 is
the large-$N$ behaviour of the function $r_{N}$.

######  Proposition 3.1.

The following asymptotics hold as $N\to\infty$.

  * •

(The real bulk) For $|p|<\sqrt{2}(1+\tau)$, there exists an ${\varepsilon}>0$
such that

$r_{N}(z,w)=2+\mathcal{O}(e^{-N{\varepsilon}}).$

  * •

(The real edge) For $p=\pm\sqrt{2}(1+\tau)$ and an arbitrary
${\varepsilon}>0$, it holds

(3.1)
$r_{N}(z,w)=r(z,w)+\tfrac{1}{\sqrt{N}}r^{1/2}(z,w)+O(N^{-1+{\varepsilon}}),$

where

(3.2)
$r(z,w):=\operatorname{erfc}(z+w)-\tfrac{1}{\sqrt{2}}e^{(z-w)^{2}-2z^{2}}\operatorname{erfc}(\sqrt{2}w)$

and

(3.3)
$\displaystyle\begin{split}r^{1/2}(z,w)&:=\tfrac{1}{\sqrt{2}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}}e^{(z-w)^{2}-2z^{2}}\\\
&\quad\times\Big{[}\tfrac{1}{\sqrt{2\pi}}(\tfrac{4}{3}z^{2}-\tfrac{4}{3}zw+\tfrac{2}{3}w^{2}-\tfrac{\tau}{1+\tau})e^{-2w^{2}}-(\tfrac{2}{3}z^{3}-\tfrac{\tau}{1+\tau}z)\operatorname{erfc}(\sqrt{2}w)\Big{]}.\end{split}$

  * •

(The real axis outside the droplet) For $|p|>\sqrt{2}(1+\tau)$, there exists
an ${\varepsilon}>0$ such that

(3.4) $r_{N}(z)=\mathcal{O}(e^{-N{\varepsilon}}).$

In all three cases the error terms are uniform for $z,w$ in compact subsets of
$\mathbb{C}$.

Combining Corollary 2.1 and Proposition 3.1, our main result Theorem 1.2 can
be easily derived by solving the associated differential equations, see
Subsection 3.5. The proof of each statement in Proposition 3.1 is given in the
following subsections.

######  Remark.

For the special case when $p=0$, the limiting kernel was obtained in [5,
Theorem 4.1] using an integral representation of the Hermite polynomials. From
the viewpoint of Proposition 3.1, it also follows from the convergence
$\lim_{N\to\infty}r_{N}=2$. We remark that this convergence can be
alternatively obtained using the classical Mehler-Hermite formula (see e.g.
[29, Eq.(18.18.28)]).

For the reader’s convenience, let us outline the strategy for the proof of
Proposition 3.1.

  * •

We rewrite $E_{N}^{1},E_{N}^{2}$ as

(3.5)
$E_{N}^{1}(\xi,\omega)=E_{N}^{1}(\xi_{0},\omega)+\int_{\xi_{0}}^{\xi}\partial_{\xi}E_{N}^{1}(t,\omega)\,dt,\qquad
E_{N}^{2}(\xi,\omega)=E_{N}^{2}(\xi,\omega_{0})+\int_{\omega_{0}}^{\omega}\partial_{\omega}E_{N}^{2}(\xi,t)\,dt.$

  * •

Using a version of the Christoffel-Darboux identity, we express
$\partial_{\xi}E_{N}^{1}$ and $\partial_{\omega}E_{N}^{2}$ only in terms of a
few orthogonal polynomials (Lemma 3.2).

  * •

Using the strong asymptotics of the Hermite polynomials (Lemma 3.3), we
estimate each term in (3.5) for suitable choices of $\xi_{0}$ and
$\omega_{0}$.

We end this subsection by introducing the main ingredients for the proof of
Proposition 3.1.

######  Lemma 3.2.

For each $N$, we have

(3.6)
$\displaystyle\begin{split}\partial_{\xi}E_{N}^{1}(\xi,\omega)&=\sqrt{2\tau}\Big{(}\frac{\tau}{2}\Big{)}^{2N-1}\frac{\sqrt{N}}{(2N-1)!}\exp\Big{[}N\Big{(}\tfrac{\tau}{2}(\xi^{2}+\omega^{2})-\xi\omega\Big{)}\Big{]}\\\
&\quad\times\Big{[}\tau
H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}H_{2N-1}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}-H_{2N-1}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}\Big{]}\end{split}$

and

(3.7)
$\displaystyle\begin{split}\partial_{\omega}E_{N}^{2}(\xi,\omega)&=-\sqrt{2\tau}(1-\tau)\Big{(}\frac{\tau}{2}\Big{)}^{2N-1}\frac{\sqrt{N}}{(2N-1)!}\exp\Big{(}-N\tfrac{1-\tau}{2}(\xi^{2}+\omega^{2})\Big{)}\\\
&\quad\times
H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}H_{2N-1}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}.\end{split}$

###### Proof.

This lemma immediately follows from the Christoffel-Darboux identity for the
kernel of complex elliptic Ginibre ensembles found in [25, Proposition 2.3].
(See [14, Section 3] for the derivation of such identities in a more general
setup.)

To be more precise, let

(3.8)
$S_{N}(\zeta,\eta):=\sum_{k=0}^{N-1}\frac{(\tau/2)^{k}}{k!}H_{k}(\zeta)H_{k}(\eta).$

Then by (2.11) and (2.12), one can express $E_{N}^{1}$ and $E_{N}^{2}$ in
terms of $S_{N}$. In particular, for $E_{N}^{2}$, we use the realisation

$\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}=\sum_{k=0}^{2N-1}\frac{(i\sqrt{\tau}/2)^{k}}{k!}H_{k}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega\Big{)}H_{k}(0),$

which follows from $H_{2l}(0)=(-1)^{l}(2l)!/l!$ and $H_{2l+1}(0)=0$. Now the
Christoffel-Darboux identity

(3.9)
$\begin{split}\partial_{\zeta}S_{N}(\zeta,\eta)&=\frac{2\tau}{1-\tau^{2}}(\eta-\tau\zeta)S_{N}(\zeta,\eta)+\frac{2}{1-\tau^{2}}\Big{(}\frac{\tau}{2}\Big{)}^{N}\frac{\tau
H_{N}(\zeta)H_{N-1}(\eta)-H_{N-1}(\zeta)H_{N}(\eta)}{(N-1)!}\end{split}$

completes the proof. ∎

The strong asymptotics of the Hermite polynomials can be found for instance in
[25, 34]. (See also [28, 16] for the strong asymptotics of the classical
orthogonal polynomials.) Here we follow the conventions in [25]. To describe
such asymptotic behaviours, for $T>0$, let

(3.10) $l:=T\Big{(}\log\frac{T}{1-\tau^{2}}-1\Big{)}$

and

(3.11)
$g(z):=\log\Big{(}z+\sqrt{z^{2}-F_{0}^{2}}\Big{)}+\frac{z}{z+\sqrt{z^{2}-F_{0}^{2}}}-\log
2-\frac{1}{2},\qquad F_{0}:=2\sqrt{T\tfrac{\tau}{1-\tau^{2}}}.$

We also write

(3.12)
$\psi(z):=\frac{\sqrt{\tau}}{F_{0}}\,\Big{(}z+\sqrt{z^{2}-F_{0}^{2}}\Big{)}.$

######  Lemma 3.3.

(Strong asymptotics of Hermite polynomials) For $T>0$ and $R\in{\mathbb{Z}}$,
we have

(3.13)
$\Big{|}H_{TN+R}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}z\Big{)}\Big{|}=\Big{(}\frac{\tau}{2}\Big{)}^{-\frac{TN+R}{2}}\,\sqrt{\frac{(TN+R)!}{N}}\,\Big{|}e^{N(Tg(z)-l/2)}\Big{|}\,\mathcal{O}(N^{\frac{5}{12}}),$

where the $\mathcal{O}(N^{\frac{5}{12}})$-term is uniform for
$z\in\mathbb{C}$, and

(3.14)
$\displaystyle\begin{split}H_{TN+R}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}z\Big{)}&=\Big{(}\frac{N}{2\pi(1-\tau^{2})}\Big{)}^{\frac{1}{4}}\Big{(}\frac{\tau}{2}\Big{)}^{-\frac{TN+R}{2}}\,\sqrt{\frac{(TN+R)!}{N}}\\\
&\quad\times\psi(z)^{R}\sqrt{\psi^{\prime}(z)}e^{N(Tg(z)-l/2)}\big{(}1+\mathcal{O}(1/N)\big{)},\end{split}$

where the $\mathcal{O}(1/N)$-term is uniform for any compact subset of
$\mathbb{C}\setminus[-F_{0},F_{0}]$.

In the sequel, it is convenient to write

(3.15)
$K_{T}:=\\{z\in\mathbb{C}\,|\,\tfrac{1-\tau}{1+\tau}(\operatorname{Re}z)^{2}+\tfrac{1+\tau}{1-\tau}(\operatorname{Im}z)^{2}\leq
T\\},\qquad\varkappa_{T}:=\tfrac{1}{\sqrt{T}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}},$

and

(3.16) $\Omega(z):=\lvert
z\rvert^{2}-\tau\operatorname{Re}z^{2}-2T\operatorname{Re}g(z)+l.$

In the rest of this section, we will mainly consider the case $T=2$.

###  3.2. Inside the bulk

In this section we prove the first part of Proposition 3.1. By (2.14), it
immediately follows from the next lemma.

######  Lemma 3.4.

Let $x_{0}\in\mathbb{R}$ with $\lvert
x_{0}\rvert<\sqrt{2\frac{1+\tau}{1-\tau}}$. Then there exist a neighbourhood
$U\subset\mathbb{C}$ of $x_{0}$ and an ${\varepsilon}>0$ such that the
following estimates hold uniformly for $\xi,\omega\in U$:

$E_{N}^{1}(\xi,\omega)=2+\mathcal{O}(e^{-N{\varepsilon}}),\qquad
E_{N}^{2}(\xi,\omega)=\mathcal{O}(e^{-N{\varepsilon}}).$

###### Proof.

By Lemma 3.2 and the asymptotic (3.13), there exists a constant $C>0$ such
that

$\displaystyle\lvert\partial_{\xi}E_{N}^{1}(\xi,\omega)\rvert$
$\displaystyle\leq
CN^{5/6}\exp\Big{(}-N\operatorname{Re}\Big{[}\xi\omega-\tfrac{\tau}{2}(\xi^{2}+\omega^{2})-2g(\xi)-2g(\omega)+l\Big{]}\Big{)},$
$\displaystyle\lvert\partial_{\omega}E_{N}^{2}(\xi,\omega)\rvert$
$\displaystyle\leq
CN^{5/6}\exp\Big{(}-N\operatorname{Re}\Big{[}\xi\omega-\tfrac{\tau}{2}(\xi^{2}+\omega^{2})+\tfrac{(\xi-\omega)^{2}}{2}-2g(\xi)-2g(\omega)+l\Big{]}\Big{)}$

for a sufficiently large $N$ and $\xi,\omega\in\mathbb{C}$.

By [25, Lemma D.1], we have that $\Omega(z)\geq 0$ for all $z\in\mathbb{C}$
and that $\Omega(z)=0$ only for $z\in\partial K_{T}$. Notice that
$x_{0}\in\mathring{K}_{2}$ (i.e. (3.15) with $T=2$). Hence there exists a
compact set $V\subset\mathring{K}_{2}$ that contains an open neighbourhood of
$x_{0}$ and $\Omega(z)>2{\varepsilon}$ for all $z\in V$ with some
${\varepsilon}>0$. Consequently we obtain that for $\xi,\omega\in V$ with
$\lvert\xi-\bar{\omega}\rvert<\sqrt{2{\varepsilon}}$,

$\displaystyle\operatorname{Re}\Big{[}\xi\omega-\tfrac{\tau}{2}(\xi^{2}+\omega^{2})-2g(\xi)-2g(\omega)+l\Big{]}=-\frac{1}{2}\lvert\xi-\bar{\omega}\rvert^{2}+\frac{1}{2}\Omega(\xi)+\frac{1}{2}\Omega(\omega)>{\varepsilon}.$

Let us take $U^{1}\subset V\cap\\{z\mid\lvert
z-x_{0}\rvert<\sqrt{{\varepsilon}/2}\\}.$ Then we have

(3.17)
$\partial_{\xi}E_{N}^{1}(t,\omega)=\mathcal{O}(N^{5/6}e^{-N{\varepsilon}}),\qquad(t,\omega\in
U^{1}).$

To analyse $E_{N}^{2}$, notice that for $\xi,\omega\in V$ with
$(\operatorname{Im}\xi)^{2},(\operatorname{Im}\omega)^{2}<{\varepsilon}/2$,

$\displaystyle\operatorname{Re}\Big{[}\xi\omega-\tfrac{\tau}{2}(\xi^{2}+\omega^{2})+\tfrac{(\xi-\omega)^{2}}{2}-2g(\xi)-2g(\omega)+l\Big{]}=-(\operatorname{Im}\xi)^{2}-(\operatorname{Im}\omega)^{2}+\frac{1}{2}\Omega(\xi)+\frac{1}{2}\Omega(\omega)>{\varepsilon}.$

Let $U^{2}\subset
V\cap\\{z\mid\lvert\operatorname{Im}z\rvert<\sqrt{{\varepsilon}/2}\\}$. Then
we have

(3.18)
$\partial_{\omega}E_{N}^{2}(\xi,t)=\mathcal{O}(N^{5/6}e^{-N{\varepsilon}}),\qquad(\xi,t\in
U^{2}).$

Next, we analyse $E_{N}^{1}(\xi_{0},\omega)$ and $E_{N}^{2}(\xi,\omega_{0})$
for suitable choices of $\xi_{0}$ and $\omega_{0}$. For $E_{N}^{1}$, let us
choose $\xi_{0}=\bar{\omega}$. Then $E_{N}^{1}(\bar{\omega},\omega)$
corresponds to the density of the complex elliptic Ginibre ensemble with $2N$
eigenvalues. (See the next subsection for further details.) In particular, by
[25, Lemma D.4], we have

(3.19) $E_{N}^{1}(\bar{\omega},\omega)=2+\mathcal{O}(e^{-N{\varepsilon}}).$

We now claim that

(3.20)
$E_{N}^{2}(\xi,0)=\mathcal{O}(e^{-N{\varepsilon}/2}N^{\frac{1}{6}}),\qquad(\xi\in
U^{2}).$

By (3.13), we have

$\displaystyle\quad\Big{|}\exp\Big{(}-N\frac{1-\tau}{2}\xi^{2}\Big{)}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}\Big{|}$
$\displaystyle=\Big{(}\frac{(2N)!!}{(2N-1)!!\,N}\Big{)}^{\frac{1}{2}}\exp\Big{(}-N\operatorname{Re}\Big{[}\tfrac{1-\tau}{2}\xi^{2}-2g(\xi)+\tfrac{l}{2}\Big{]}\Big{)}\mathcal{O}(N^{\frac{5}{12}}).$

We also have that for $\xi\in V$ with
$(\operatorname{Im}\xi)^{2}<{\varepsilon}$,

$\operatorname{Re}\Big{[}\tfrac{1-\tau}{2}\xi^{2}-2g(\xi)+\tfrac{l}{2}\Big{]}=\frac{1}{2}\Omega(\xi)-\frac{1}{2}(\operatorname{Im}\xi)^{2}>\frac{{\varepsilon}}{2}.$

Notice also that Stirling’s formula yields

$\Big{(}\frac{(2N)!!}{(2N-1)!!\,N}\Big{)}^{\frac{1}{2}}=\mathcal{O}(N^{-\frac{1}{4}}).$

Finally, due to a well-known Taylor series, we have

(3.21)
$\displaystyle\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}(0)=\sum_{l=0}^{N-1}\Big{(}-\frac{\tau}{4}\Big{)}^{l}\,\frac{(2l)!}{(l!)^{2}}=\frac{1}{\sqrt{1+\tau}}+\mathcal{O}\Big{(}\tfrac{\tau^{N}}{\sqrt{N}}\Big{)}$

Combining all of the above with (2.12), we obtain (3.20).

We now choose $U\subset U^{1}\cap U^{2}=V\cap\\{z\mid\lvert
z-x_{0}\rvert<\sqrt{{\varepsilon}/2}\\}$. Then using (3.17), (3.19) for
$E_{N}^{1}$ and (3.18), (3.20) for $E_{N}^{2}$, we obtain

$\displaystyle E_{N}^{1}(\xi,\omega)$
$\displaystyle=E_{N}^{1}(\bar{\omega},\omega)+\int_{\bar{\omega}}^{\xi}\partial_{\xi}E_{N}^{1}(t,\omega)\,dt=2+\mathcal{O}(e^{-N{\varepsilon}})+\mathcal{O}(N^{\frac{5}{6}}e^{-N{\varepsilon}}),$
$\displaystyle E_{N}^{2}(\xi,\omega)$
$\displaystyle=E_{N}^{2}(\xi,0)+\int_{0}^{\omega}\partial_{\omega}E_{N}^{2}(\xi,t)\,dt=\mathcal{O}(N^{\frac{1}{6}}e^{-\frac{N{\varepsilon}}{2}})+\mathcal{O}(N^{\frac{5}{6}}e^{-N{\varepsilon}}),$

where the error terms are uniform for $\xi,\omega\in U$. This completes the
proof. ∎

###  3.3. At the edge

In this subsection, we prove the second assertion of Proposition 3.1.

Let us define

(3.22)
${\mathcal{K}}_{n}(z,\overline{w}):=N\sqrt{1-\tau^{2}}e^{-N(z\overline{w}-\tfrac{\tau}{2}(z^{2}+\overline{w}^{2}))}\sum_{j=0}^{n-1}\frac{(\tau/2)^{j}}{j!}H_{j}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}z\Big{)}\overline{H_{j}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}w\Big{)}}.$

Notice that the function ${\mathcal{K}}_{2N}$ is same as $E_{N}^{1}$ in (2.11)
up to a prefactor.

It is well known that ${\mathcal{K}}_{n}$ corresponds to the pre-kernel of the
complex elliptic Ginibre ensemble with $n$ points, which is known to form a
determinantal point process, see e.g. [20]. As $n,N\to\infty$ while keeping
$n/N=T$ fixed, the elliptic Ginibre ensemble condensates to the droplet
$K_{T}$ in (3.15). Moreover, the asymptotic behaviour of (3.22) on the
diagonal was extensively studied in [25]. Along the same lines as the proof of
[25, Theorem 1.1], one can obtain the following proposition. (See also [32,
Corollary 8.12] for the leading term of the proposition.)

######  Proposition 3.5.

(Cf. [25, Theorem 1.1]) Given $T>0$ and $z_{0}\in\partial K_{T}$, let
$\mathbf{n}=\mathbf{n}(z_{0})\in\mathbb{C}$ be the outer unit normal vector of
$K_{T}$ at $z_{0}$. Then as $n,N\to\infty$ while keeping $n/N=T$ fixed and for
an arbitrary ${\varepsilon}>0$, we have

(3.23)
$\frac{1}{N}{\mathcal{K}}_{n}\Big{(}z_{0}+\tfrac{\xi\mathbf{n}}{\sqrt{N}},\overline{z_{0}+\tfrac{\omega\mathbf{n}}{\sqrt{N}}}\Big{)}=\frac{1}{2}\operatorname{erfc}\Big{(}\frac{\xi+\overline{\omega}}{\sqrt{2}}\Big{)}+\frac{1}{\sqrt{N}}\frac{\varkappa(z_{0})}{\sqrt{2\pi}}\frac{\xi^{2}-\xi\overline{\omega}+\overline{\omega}^{2}-1}{3}e^{-\frac{1}{2}(\xi+\overline{\omega})^{2}}+\mathcal{O}(N^{-1+{\varepsilon}}),$

where $\varkappa(z_{0})$ is the curvature of $\partial K_{T}$ at $z_{0}$. Here
the error bound is uniform for bounded $\xi,\omega$ and over $z_{0}\in\partial
K_{T}$.

We shall need the following elementary estimates. We provide the proof in
Appendix A.1. Recall that $l$ is given by (3.10), $g(z)$ by (3.11), $\psi(z)$
by (3.12) and $\varkappa_{T}$ by (3.15).

######  Lemma 3.6.

Let $T>0$ and $x_{0}=\sqrt{T\frac{1+\tau}{1-\tau}}$. As $N\to\infty$, we have
the following:

  * •

for $0\leq\alpha<1/2$, we have

(3.24)
$\operatorname{Re}\Big{[}\tfrac{1-\tau}{2}s^{2}-Tg(s)+\tfrac{l}{2}\Big{]}\geq
N^{-1+2\alpha},\qquad s\in[0,x_{0}-\sqrt{\tfrac{2}{N}}N^{\alpha}];$

  * •

for $0\leq\nu<1/6$, we have

(3.25)
$\exp\Big{(}-N\Big{[}\tfrac{1-\tau}{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}^{2}-Tg\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}+\tfrac{l}{2}\Big{]}\Big{)}=e^{-2z^{2}}\Big{(}1+\frac{2\sqrt{2}}{3}\varkappa_{T}\frac{z^{3}}{\sqrt{N}}+\mathcal{O}(N^{-1+6\nu})\Big{)}$

uniformly for $z=\mathcal{O}(N^{\nu})$;

  * •

for $0\leq\nu<1/6$, we have

(3.26)
$\displaystyle\begin{split}\sqrt{\psi^{\prime}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}}&=\Big{(}\frac{1+\tau}{T(1-\tau)}\Big{)}^{\frac{1}{4}}\Big{(}1-\frac{\sqrt{2}\tau}{1+\tau}\varkappa_{T}\frac{z}{\sqrt{N}}+\mathcal{O}(N^{-1+2\nu})\Big{)},\\\
\sqrt{\psi^{\prime}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}}\frac{\sqrt{\psi^{\prime}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}s\Big{)}}}{\psi\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}s\Big{)}}&=\Big{(}\frac{1+\tau}{T(1-\tau)}\Big{)}^{\frac{1}{2}}\Big{(}1-\frac{\sqrt{2}}{1+\tau}\varkappa_{T}\frac{s+\tau
z}{\sqrt{N}}+\mathcal{O}(N^{-1+2\nu})\Big{)}\end{split}$

uniformly for $z,s=\mathcal{O}(N^{\nu})$.

We now prove the following lemma, which gives the second assertion of
Proposition 3.1.

######  Lemma 3.7.

Let $x_{0}=\pm\sqrt{2\frac{1+\tau}{1-\tau}}$. Then as $N\to\infty$ and for
arbitrary ${\varepsilon}>0$, we have

(3.27)
$E_{N}^{1}\Big{(}x_{0}\pm\sqrt{\tfrac{2}{N}}z,x_{0}\pm\sqrt{\tfrac{2}{N}}w\Big{)}=\operatorname{erfc}(z+w)+\tfrac{\sqrt{2}\varkappa_{2}}{\sqrt{N}}e^{-(z+w)^{2}}\tfrac{2z^{2}-2zw+2w^{2}-1}{3\sqrt{\pi}}+\mathcal{O}(N^{-1+{\varepsilon}})$

and

(3.28)
$\displaystyle\begin{split}&E_{N}^{2}\Big{(}x_{0}\pm\sqrt{\tfrac{2}{N}}z,x_{0}\pm\sqrt{\tfrac{2}{N}}w\Big{)}=\tfrac{1}{\sqrt{2}}e^{-2z^{2}}\operatorname{erfc}(\sqrt{2}w)\\\
&+\tfrac{\varkappa_{2}}{\sqrt{N}}e^{-2z^{2}}\Big{[}(\tfrac{2}{3}z^{3}-\tfrac{\tau}{1+\tau}z)\operatorname{erfc}(\sqrt{2}w)-\tfrac{1}{\sqrt{2\pi}}(\tfrac{1}{1+\tau}-\tfrac{1+2w^{2}}{3})e^{-2w^{2}}\Big{]}+\mathcal{O}(N^{-1+{\varepsilon}}).\end{split}$

The error terms are uniform for bounded $z,w$.

###### Proof.

We shall consider the case $x_{0}=+\sqrt{2\frac{1+\tau}{1-\tau}}$ as the other
case works analogously.

The expansion (3.27) for $E_{N}^{1}$ immediately follows from Proposition 3.5
using the relation

$E_{N}^{1}(\xi,\omega)=\tfrac{2}{N}{\mathcal{K}}_{2N}(\xi,\omega).$

Thus it suffices to show the expansion (3.28) for $E_{N}^{2}$. To analyse
$E_{N}^{2}$, choose $0<\alpha<\nu<1/6$ and let us write

(3.29) $\displaystyle\begin{split}&\quad
E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,x_{0}+\sqrt{\tfrac{2}{N}}w\Big{)}=E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,0\Big{)}\\\
&+\int_{0}^{x_{0}-\sqrt{\frac{2}{N}}N^{\alpha}}\partial_{w}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,s\Big{)}\,ds+\sqrt{\frac{2}{N}}\int_{-N^{\alpha}}^{w}\partial_{w}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,x_{0}+\sqrt{\tfrac{2}{N}}s\Big{)}\,ds.\end{split}$

We first show that for $z=\mathcal{O}(N^{\nu})$ uniformly,

(3.30)
$E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,0\Big{)}=\sqrt{2}\,e^{-2z^{2}}\Big{[}1+\sqrt{\tfrac{2}{N}}\varkappa_{2}(\tfrac{2}{3}z^{3}-\tfrac{\tau}{1+\tau}z)+\mathcal{O}(N^{-1+6\nu})\Big{]}.$

Note that by (3.14) and Stirling’s formula, we have

$\begin{split}\frac{(\tau/2)^{N}}{(2N-1)!!}H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\xi\Big{)}&=\frac{1}{(2\pi(1-\tau^{2})N)^{1/4}}\frac{\sqrt{(2N)!}}{(2N-1)!!}\sqrt{\psi^{\prime}(\xi)}e^{N(2g(\xi)-l/2)}\Big{(}1+\mathcal{O}(1/N)\Big{)}\\\
&=\frac{1}{(2(1-\tau^{2}))^{1/4}}\sqrt{\psi^{\prime}(\xi)}e^{N(2g(\xi)-l/2)}\Big{(}1+\mathcal{O}(1/N)\Big{)},\end{split}$

where the $\mathcal{O}(1/N)$-term is uniform for $\xi$ in a compact subset of
$\mathbb{C}\setminus[-F_{0},F_{0}]$. Applying the estimates (3.25) and (3.26),
the desired asymptotic behaviour (3.30) follows from the expression (2.12) and
the summation formula (3.21).

Next, we show that for $z=\mathcal{O}(N^{\nu})$ uniformly,

(3.31)
$\int_{0}^{x_{0}-\sqrt{\frac{2}{N}}N^{\alpha}}\partial_{\omega}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,s\Big{)}\,ds=e^{-2z^{2}}\mathcal{O}(N^{\frac{2}{3}}e^{-N^{2\alpha}}).$

For this, we derive the asymptotic behaviour of
$\partial_{\omega}E_{N}^{2}(\xi,s)$ using Lemma 3.2. Note that by (3.13) and
(3.14), we have

(3.32) $\displaystyle\lvert\partial_{\omega}E_{N}^{2}(\xi,s)\rvert$
$\displaystyle=2(1-\tau)(\tfrac{2}{\pi(1-\tau^{2})})^{\frac{1}{4}}\lvert\psi^{\prime}(\xi)\rvert$
(3.33)
$\displaystyle\quad\times\exp\Big{(}N\operatorname{Re}\Big{[}-\tfrac{1-\tau}{2}\xi^{2}+2g(\xi)-\tfrac{l}{2}-\tfrac{1-\tau}{2}s^{2}+2g(s)-\tfrac{l}{2}\Big{]}\Big{)}\,\mathcal{O}(N^{\frac{2}{3}}).$

Moreover, it follows from (3.24), (3.25) and (3.26) that

(3.34)
$\Big{|}\partial_{w}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,s\Big{)}\Big{|}=e^{-2\operatorname{Re}z^{2}}\mathcal{O}(N^{\frac{2}{3}}e^{-N^{2\alpha}})$

uniformly for $0\leq s\leq x_{0}-\sqrt{\frac{2}{N}}N^{\alpha}$ and
$z=\mathcal{O}(N^{\nu})$. Now by using the triangle inequality and integrating
this asymptotic, we obtain (3.31).

Finally, we show that for $z,w=\mathcal{O}(N^{\nu})$ uniformly,

(3.35)
$\displaystyle\begin{split}&\quad\sqrt{\tfrac{2}{N}}\int_{-N^{\alpha}}^{w}\partial_{\omega}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,x_{0}+\sqrt{\tfrac{2}{N}}s\Big{)}\,ds=-\tfrac{1}{\sqrt{2}}e^{-2z^{2}}(1+\operatorname{erf}(\sqrt{2}w))\\\
&\quad-\tfrac{\varkappa_{2}}{\sqrt{N}}e^{-2z^{2}}\Big{[}(\tfrac{2}{3}z^{3}-\tfrac{\tau}{1+\tau}z)(1+\operatorname{erf}(\sqrt{2}w))+\tfrac{1}{\sqrt{2\pi}}(\tfrac{1}{1+\tau}-\tfrac{1+2w^{2}}{3})e^{-2w^{2}}\Big{]}\\\
&\quad+e^{-2z^{2}+2(\operatorname{Im}w)^{2}}\mathcal{O}(N^{-1+7\nu}).\end{split}$

Similarly as above, we use Lemma 3.2 and the asymptotics (3.14), (3.25), which
leads to

$\partial_{w}E_{N}^{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z,x_{0}+\sqrt{\tfrac{2}{N}}s\Big{)}=-\sqrt{\tfrac{2N}{\pi}}e^{-2(z^{2}+s^{2})}\Big{[}1+\sqrt{\tfrac{2}{N}}\,\varkappa_{2}\big{(}\tfrac{2}{3}(z^{3}+s^{3})-\tfrac{s+\tau
z}{1+\tau}\big{)}+\mathcal{O}(N^{-1+6\nu})\Big{]}.$

Here $\varkappa_{2}$ is given by (3.15) with $T=2$. By integration,
(LABEL:EN2_asymp_edge_3) follows from elementary asymptotic behaviour of the
error function (see e.g. [29, Eq.(7.12.1)]).

Combining (LABEL:EN2_edge_decom) with the asymptotics (3.30), (3.31) and
(LABEL:EN2_asymp_edge_3) and using that $z,w$ are bounded (but $\nu>0$), we
obtain the desired expansion (3.28) for $E_{N}^{2}$. This completes the proof.
∎

###  3.4. Outside the bulk

This subsection is devoted to proving the last statement of Proposition 3.1.
Again, this is an immediate consequence of the following lemma.

######  Lemma 3.8.

Let $x_{0}\in\mathbb{R}$ with $\lvert
x_{0}\rvert>\sqrt{2\frac{1+\tau}{1-\tau}}$. Then there exists a neighbourhood
$U\subset\mathbb{C}$ of $x_{0}$ and an ${\varepsilon}>0$ such that the
following estimates hold uniformly for $\xi,\omega\in U$:

$E_{N}^{1}(\xi,\omega)=\mathcal{O}(e^{-N{\varepsilon}}),\qquad
E_{N}^{2}(\xi,\omega)=\mathcal{O}(e^{-N{\varepsilon}}).$

In the proof of Lemma 3.8 we shall use an estimate for Hermite polynomials.

######  Lemma 3.9.

For all $\tau\in(0,1)$, $0\leq l\leq N-1$ and
$\lvert\omega_{0}\rvert\geq\sqrt{2\frac{1+\tau}{1-\tau}}$, we have

(3.36)
$\Big{|}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega_{0}\Big{)}\Big{|}\leq\Big{|}\frac{(\tau/2)^{l+1}}{(2l+2)!!}H_{2l+2}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}\omega_{0}\Big{)}\Big{|}.$

The proof of this lemma is given in Appendix A.2.

###### Proof of Lemma 3.8.

The proof is merely same as that of Lemma 3.4 except the fact that in the
representation (3.5), we choose $\xi_{0}$, $\omega_{0}$ outside of the
droplet. In particular, the estimates of the terms involving the derivatives
of $E_{N}^{1}$ or $E_{N}^{2}$ are same as those in Lemma 3.4. Thus all we need
to analyse is the initial values.

Let us choose a compact subset $V$ of $K^{\mathsf{c}}_{2}$. For $E_{N}^{1}$,
it again follows from [25, Lemma D.4] that for an ${\varepsilon}>0,$

$E_{N}^{1}(\bar{\omega},\omega)=\mathcal{O}(e^{-N{\varepsilon}}),\qquad(\omega\in
V).$

It remains to analyse $E_{N}^{2}$ in (2.12). For the $\xi$-dependent part of
$E_{N}^{2}$, the estimates above work, but the $\omega$-dependent part of
$E_{N}^{2}$ requires special care. Let us choose $\omega_{0}=x_{0}$. Then by
combining Lemma 3.9 and the asymptotics (3.13), we have

(3.37)
$\displaystyle\begin{split}\Big{|}e^{-\tfrac{1}{2}N(1-\tau)x_{0}^{2}}\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}x_{0}\Big{)}\Big{|}&\leq
e^{-\tfrac{1}{2}N(1-\tau)x_{0}^{2}}N\frac{(\tau/2)^{N}}{(2N)!!}\Big{|}H_{2N}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}x_{0}\Big{)}\Big{|}\\\
&=e^{-N\operatorname{Re}[\frac{1}{2}(1-\tau)x_{0}^{2}-2g(x_{0})+\frac{l}{2}]}\sqrt{\tfrac{(2N-1)!!}{(2N)!!\,N}}\,\mathcal{O}(N^{1+\frac{5}{12}}).\end{split}$

Since $x_{0}$ is real, we have

$\mathrm{Re}\Big{[}\tfrac{1}{2}(1-\tau)x_{0}^{2}-2g(x_{0})+\tfrac{l}{2}\Big{]}=\frac{1}{2}\Omega(x_{0})>{\varepsilon}.$

Then by Stirling’s formula, we obtain

$\exp\Big{(}-\tfrac{1}{2}N(1-\tau)x_{0}^{2}\Big{)}\sum_{l=0}^{N-1}\frac{(\tau/2)^{l}}{(2l)!!}H_{2l}\Big{(}\sqrt{N\tfrac{1-\tau^{2}}{2\tau}}x_{0}\Big{)}=\mathcal{O}(N^{\frac{2}{3}}e^{-N{\varepsilon}}),$

which leads to

$E_{N}^{2}(\xi,x_{0})=2\sqrt{1-\tau^{2}}\,\mathcal{O}(N^{\frac{1}{6}}e^{-\frac{1}{2}N{\varepsilon}})\mathcal{O}(N^{\frac{2}{3}}e^{-N{\varepsilon}})=\mathcal{O}(N^{\frac{5}{6}}e^{-\frac{3}{2}N{\varepsilon}}).$

This completes the proof. ∎

###  3.5. Proof of the main theorem

In this subsection, we complete the proof of Theorem 1.2. Due to Corollary 2.1
and Proposition 3.1, it remains to solve the resulting differential equations.
For this, we first need the following lemma, which readily follows from the
basic theory of ordinary differential equations. Indeed, only a
straightforward estimate of the well-known explicit solution to linear
inhomogeneous differential equation is needed for the proof (that we skip).

######  Lemma 3.10.

Let $a,r_{N}:\mathbb{C}\to\mathbb{C}$ be analytic functions,
$f_{0}\in\mathbb{C}$ and for all $N\in\mathbb{N}$ let
$f_{N}:\mathbb{C}\to\mathbb{C}$ be the solution to

$f_{N}^{\prime}=a\,f_{N}+r_{N},\qquad f_{N}(z_{0})=f_{0}.$

If $r_{N}\to r$ uniformly on compact subsets of $\mathbb{C}$, then the limit
$f(z):=\lim_{N\to\infty}f_{N}(z)$ exists for all $z\in\mathbb{C}$, is analytic
and satisfies

$f^{\prime}=a\,f+r,\qquad f(z_{0})=f_{0}.$

Moreover suppose that as $N\to\infty$, the inhomogeneous term $r_{N}$ has the
asymptotic expansion

$r_{N}=r+r^{(\gamma)}/N^{\gamma}+\mathcal{O}(C_{N}),$

where the $\mathcal{O}(C_{N})$-term is uniform on compact subsets of
$\mathbb{C}$. Then

$f_{N}=f+f^{(\gamma)}/N^{\gamma}+\mathcal{O}(C_{N})$

also holds uniformly on compact subsets of $\mathbb{C}$ and the subleading
term $f^{(\gamma)}$ satisfies

$f^{(\gamma)\prime}=a\,f^{(\gamma)}+r^{(\gamma)},\qquad
f^{(\gamma)}(z_{0})=0.$

Using Lemma 3.10, we can now derive the large-$N$ asymptotics of
$\widehat{\kappa}_{N}(z,w)$. For the proof of Theorem 1.2, we recall a notion
of the equivalent kernels. (See also [4].)

######  Remark.

(Equivalent kernels) Consider a $2k\times 2k$ skew-symmetric matrix
$A=(a_{j,k})_{j,k}$ and let $g_{j}\in\mathbb{C}$ for $1\leq j\leq 2k$. The new
matrix $B=(g_{j}g_{k}a_{j,k})_{j,k}$ is again skew-symmetric and we have
${\textup{Pf}}(B)=\prod_{j=1}^{2k}g_{j}\,{\textup{Pf}}(A)$. Furthermore, if
$g_{2l}=1/g_{2l-1}$ for $1\leq l\leq k$, then
${\textup{Pf}}(B)={\textup{Pf}}(A)$. This implies that different kernels can
give rise to the same correlation functions. In particular, we call two pre-
kernels ${\bm{\varkappa}}$ and $\widetilde{{\bm{\varkappa}}}$ _equivalent_ if
there exists a unimodular function $g:\mathbb{C}\to\mathbb{C}$ with
$g(\overline{\zeta})=1/g(\zeta)$ such that
$\widetilde{{\bm{\varkappa}}}(\zeta,\eta)=g(\zeta)g(\eta){\bm{\varkappa}}(\zeta,\eta)$.
Also, we call the combination $c(\zeta,\eta):=g(\zeta)g(\eta)$ a _cocycle_.

###### Proof of Theorem 1.2.

Recall that by definition (2.9), we have
$\kappa_{N}=\widehat{\kappa}_{N}/\omega_{N}$. Note that

$e^{-\frac{N}{2}(Q(p+\frac{z}{\sqrt{N\delta}})+Q(p+\frac{w}{\sqrt{N\delta}}))}\frac{1}{\omega_{N}(z,w)}=e^{-\lvert
z\rvert^{2}-\lvert w\rvert^{2}+2zw}\,\frac{1}{c_{N}(z,w)},$

where the cocycle $c_{N}(z,w)$ is given by

(3.38)
$c_{N}(z,w)=\exp\Big{(}-i\sqrt{2N\tfrac{1-\tau}{1+\tau}}\,p\,\operatorname{Im}z-i\sqrt{2N\tfrac{1-\tau}{1+\tau}}\,p\,\operatorname{Im}w+i\tau\operatorname{Im}z^{2}+i\tau\operatorname{Im}w^{2}\Big{)}.$

Therefore as $N\to\infty$, the equivalent kernels have the uniform limit

(3.39)
$\lim_{N\to\infty}c_{N}(z,w)e^{-\frac{N}{2}(Q(p+\frac{z}{\sqrt{N\delta}})+Q(p+\frac{w}{\sqrt{N\delta}}))}\kappa_{N}(z,w)=e^{-\lvert
z\rvert^{2}-\lvert w\rvert^{2}+2zw}\,\widehat{\kappa}(z,w).$

By Proposition 3.1, one can see that for the leading order, the differential
equation (2.13) coincides with the one in [23] (for the real bulk case) and in
[4] (for the real edge case). Moreover, in both cases, it was shown that the
kernels (1.17) and (1.19) are the associated solution in each case. (Here the
uniqueness follows from the anti-symmetry of the pre-kernel.) On the other
hand, for the case outside the droplet, the differential equation is trivial.
Therefore only the derivation of $\kappa_{\textup{edge}}^{\mathbb{R},1/2}$ in
(1.20) remains.

By Corollary 2.1, Proposition 3.1, Lemma 3.10 and (3.39), the function

$\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(z,w):=e^{-2zw}\,\kappa_{\textup{edge}}^{\mathbb{R},1/2}(z,w)$

satisfies the differential equation

(3.40)
$\partial_{z}\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(z,w)=2(z-w)\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(z,w)+r^{1/2}(z,w).$

This differential equations can be easily solved by virtue of an integrating
factor. Furthermore, the anti-symmetry
$\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(z,w)=-\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(w,z)$
determines the solution as

$\begin{split}\widehat{\kappa}_{\textup{edge}}^{\mathbb{R},1/2}(z,w)&=\tfrac{1}{\sqrt{2}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}}e^{(z-w)^{2}}\\\
&\quad\times\int_{w}^{z}e^{-2t^{2}}\Big{[}\tfrac{1}{\sqrt{2\pi}}(\tfrac{4}{3}t^{2}-\tfrac{4}{3}tw+\tfrac{2}{3}w^{2}-\tfrac{\tau}{1+\tau})e^{-2w^{2}}-(\tfrac{2}{3}t^{3}-\tfrac{\tau}{1+\tau}t)\operatorname{erfc}(\sqrt{2}w)\Big{]}\,dt\\\
&=\tfrac{1}{12\sqrt{2}}(\tfrac{1+\tau}{1-\tau})^{\frac{3}{2}}e^{(z-w)^{2}}\Big{[}\Big{(}(2z^{2}+\tfrac{1-2\tau}{1+\tau})e^{-2z^{2}}\operatorname{erfc}(\sqrt{2}w)+2\sqrt{\tfrac{2}{\pi}}\,we^{-2(z^{2}+w^{2})}\Big{)}-\Big{(}z\leftrightarrow
w\Big{)}\Big{]},\end{split}$

which completes the proof. ∎

##  Appendix A Some preliminary estimates

In this appendix, we show some preliminary estimates used in Section 3.

###  A.1. Proof of Lemma 3.6

The proof of Lemma 3.6 merely follows from the computations given in [25].

First, we prove (3.24). By definitions (3.11), (3.16), we have that for
$s\in\mathbb{R}$,

$\operatorname{Re}\Big{[}\tfrac{1-\tau}{2}s^{2}-Tg(s)+\tfrac{l}{2}\Big{]}=\frac{1}{2}\Omega(s).$

Then [25, Lemma 3.3] implies the existence of a threshold $\delta>0$ such that
the approximation

$\Omega(s)=2\lvert s-x_{0}\rvert^{2}+\mathcal{O}(\lvert s-x_{0}\rvert^{3})$

holds uniformly for $s\in[x_{0}-\delta,x_{0}+\delta]$. Additionally, recall
that $\Omega(s)\geq 0$ for all $s$ and $\Omega(s)=0$ only for $s$ on the
boundary of the droplet, i.e. $s=x_{0}$. Therefore we have an
${\varepsilon}>0$ with $\frac{1}{2}\Omega(s)\geq{\varepsilon}$ for all
$s\in[0,x_{0}-\delta]$. For $s\in[x_{0}-\delta,x_{0}-\sqrt{2/N}N^{\alpha}]$ we
can use the quadratic approximation from above to derive

(A.1) $\frac{1}{2}\Omega(s)\geq
2N^{-1+2\alpha}+\mathcal{O}(N^{-3/2+3\alpha})\geq N^{-1+2\alpha}.$

Here the last inequality holds for large enough $N$ if $\alpha<1/2$.

Next, we show the second assertion (3.25). By [25, Lemma 3.2], we have the
expansion

$Tg\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}=\frac{1-\tau}{2}x_{0}^{2}+\frac{l}{2}+(1-\tau)x_{0}\sqrt{\frac{2}{N}}z-\frac{1+\tau}{N}z^{2}+\frac{2\sqrt{2}\varkappa_{T}}{3}\frac{z^{3}}{N^{3/2}}+\mathcal{O}\Big{(}\frac{z^{4}}{N^{2}}\Big{)}.$

Thus we obtain

$\exp\Big{(}-N\Big{[}-\tfrac{1-\tau}{2}\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}^{2}+Tg\Big{(}x_{0}+\sqrt{\tfrac{2}{N}}z\Big{)}-\frac{l}{2}\Big{]}\Big{)}=e^{-2z^{2}}\exp\Big{[}\tfrac{2\sqrt{2}}{3}\varkappa_{T}\tfrac{z^{3}}{\sqrt{N}}+\mathcal{O}\Big{(}\tfrac{z^{4}}{N}\Big{)}\Big{]},$

which leads to (3.25).

For the last assertion (3.26), we use Taylor expansion around $x_{0}$
analogous to [25, Lemma 3.5]. Note that for $x_{0}\in\mathbb{R}$, we have

(A.2)
$\psi(x_{0})=1,\qquad\psi^{\prime}(x_{0})=\sqrt{\frac{1+\tau}{T(1-\tau)}},\qquad\psi^{\prime\prime}(x_{0})=-\frac{2\tau(1+\tau)}{T(1-\tau)^{2}}.$

By direct computations, this gives rise to (3.26) using the convention
$\varkappa_{T}=\frac{1}{\sqrt{T}}(\frac{1+\tau}{1-\tau})^{3/2}$.

###  A.2. Proof of Lemma 3.9

It is well known that the zeros of the Hermite polynomial of degree $n$ lie in
the interval $(-\sqrt{2n+1},\sqrt{2n+1})$, see e.g. [29, Section 18.16]. Thus
we have excluded all zeros. Additionally since the Hermite polynomials are
positive as $\omega_{0}\to\pm\infty$, we can omit the absolute values.

We simplify the inequality (3.36) by multiplying both sides with
$(2l)!!\,(2/\tau)^{l}$ and applying the three term recurrence relation (2.6)
to $H_{2l+2}$. Let us rewrite the argument of the polynomials as

$\sqrt{N(1-\tau^{2})/(2\tau)}\,\omega_{0}=(1+\tau)y/\sqrt{\tau}.$

Then Lemma 3.9 follows from the fact that for $\tau\in(0,1)$, $0\leq l\leq
N-1$ and $y\geq\sqrt{N}$,

$\Big{(}1+\frac{2l+1}{2l+2}\tau\Big{)}H_{2l}\Big{(}\tfrac{1+\tau}{\sqrt{\tau}}y\Big{)}\leq\frac{\tau(1+\tau)}{2(l+1)}y\,H_{2l+1}\Big{(}\tfrac{1+\tau}{\sqrt{\tau}}y\Big{)}.$

Let us show the slightly stricter inequality

(A.3)
$H_{2l}\Big{(}\tfrac{1+\tau}{\sqrt{\tau}}y\Big{)}\leq\frac{\tau}{2(l+1)}y\,H_{2l+1}\Big{(}\tfrac{1+\tau}{\sqrt{\tau}}y\Big{)}$

by the induction argument. For $l=0$, the inequality (A.3) is obvious since
$H_{0}(x)=1$ and $H_{1}(x)=2x$.

From now on, we shall write $a_{l}=\tau y/(2l+2)$ and
$x=(1+\tau)y/\sqrt{\tau}$ to lighten the notations. Due to the three term
recurrence relation (2.6), it suffices to show that

(A.4) $H_{2l}(x)\leq
a_{l}H_{2l+1}(x)=a_{l}\Big{[}2xH_{2l}(x)-4lH_{2l-1}(x)\Big{]}.$

Again, by the three term recurrence relation, this is equivalent to

(A.5) $H_{2l-2}(x)\leq\frac{2a_{l}(x^{2}-l)-x}{(2l-1)(2a_{l}x-1)}H_{2l-1}(x).$

By the induction hypothesis, we have $H_{2l-2}(x)\leq a_{l-1}H_{2l-1}(x)$.
Therefore to complete the induction step it is enough to show that

(A.6)
$a_{l-1}\leq\frac{2a_{l}(x^{2}-l)-x}{(2l-1)(2a_{l}x-1)},\qquad\text{i.e.}\qquad\frac{\sqrt{\tau}}{2l}\leq\frac{(1+\tau)^{2}y^{2}-\tau
l-(1+\tau)(l+1)}{\sqrt{\tau}(2l-1)\big{(}(1+\tau)y^{2}-(l+1)\big{)}}.$

This inequality (A.6) follows from the straightforward computations using
elementary algebra and we leave it to the interested readers.

### Acknowledgements

The authors are greatly indebted to Gernot Akemann for suggesting the problem,
careful reading, and much appreciated help with improving this manuscript. It
is also our pleasure to thank Seung-Yeop Lee for helpful discussions. The
present work was completed when the authors visited the Department of
Mathematical Sciences at Korea Advanced Institute of Science and Technology,
and we wish to express our gratitude to Ji Oon Lee for the invitation and
hospitality.

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