# Determinantal Coulomb gas ensembles with a class of discrete rotational
symmetric potentials

Sung-Soo Byun  Center for Mathematical Challenges, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>and  Meng Yang  Department of Mathematical Sciences,
University of Copenhagen, Copenhagen, Universitetsparken 5, 2100 København Ø,
Denmark<EMAIL_ADDRESS>
###### Abstract.

We consider determinantal Coulomb gas ensembles with a class of discrete
rotational symmetric potentials whose droplets consist of several disconnected
components. Under the insertion of a point charge at the origin, we derive the
asymptotic behaviour of the correlation kernels both in the macro- and
microscopic scales. In the macroscopic scale, this particularly shows that
there are strong correlations among the particles on the boundary of the
droplets. In the microscopic scale, this establishes the edge universality.
For the proofs, we use the nonlinear steepest descent method on the matrix
Riemann-Hilbert problem to derive the asymptotic behaviours of the associated
planar orthogonal polynomials and their norms up to the first subleading
terms.

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. Meng
Yang was supported by VILLUM FOUNDEN research grant no. 29369 and the NNF
grant NNF20OC0064428.

##  1\. Introduction and main results

We consider a configuration $\\{z_{j}\\}_{1}^{N}$ of $N$ points in
$\mathbb{C}$ with joint probability distribution

(1.1)
$d\mathbf{P}_{N}=\frac{1}{Z_{N}}\prod_{j>k}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}e^{-NQ(z_{j})}\,dA(z_{j}),\qquad
dA(z):=\frac{d^{2}z}{\pi},$

where $Z_{N}$ is the normalisation constant and $Q:\mathbb{C}\to\mathbb{R}$ is
a suitable function called external potential. The ensemble (1.1) corresponds
to the eigenvalue system of the random normal matrix model, which can be
interpreted as the two-dimensional Coulomb gas ensemble at a specific inverse
temperature $\beta=2$. For a recent account of the theory and various topics
on the Coulomb gas ensemble, we refer the reader to [41] and references
therein.

By definition, the $k$-point correlation function $R_{N,k}$ of the system
(1.1) is given by

(1.2)
$R_{N,k}(z_{1},\cdots,z_{k}):=\frac{N!}{(N-k)!}\int_{\mathbb{C}^{n-k}}\mathbf{P}_{N}\prod_{j=k+1}^{N}dA(z_{j}).$

The normalised $1$-point function $\frac{1}{N}R_{N,1}$ corresponds to the
macroscopic density of the model. It is well known that as $N\to\infty$, the
empirical measure of $\\{z_{j}\\}_{1}^{N}$ converges to Frostman’s equilibrium
measure, see e.g. [25, 5]. In particular, the system $\\{z_{j}\\}_{1}^{N}$
tends to occupy certain compact set $S$ called the droplet.

The $k$-point function $R_{N,k}$ can be effectively analysed in terms of the
correlation kernel. To be more concrete, let $p_{k}\equiv p_{k,N}$ be the
$k$:th orthonormal polynomial with respect to the weighted Lebesgue measure
$e^{-NQ}\,dA$:

(1.3)
$\int_{\mathbb{C}}p_{j}(z)\overline{p_{k}(z)}e^{-NQ(z)}\,dA(z)=\delta_{jk},$

where $\delta_{jk}$ is the Kronecker delta. We write

(1.4)
$K_{N}(z,w)=e^{-\frac{N}{2}(Q(z)+Q(w))}\sum_{j=0}^{N-1}p_{j}(z)\overline{p_{j}(w)}$

for the weighted reproducing kernel of analytic polynomials (of degree less
than $N-1$) in $L^{2}(e^{-NQ}\,dA)$. Then the $k$-point function $R_{N,k}$ in
(1.2) is expressed as

(1.5)
$R_{N,k}(z_{1},\cdots,z_{k})=\det\Big{[}K_{N}(z_{j},z_{l})\Big{]}_{j,l=1}^{k}.$

We mention that the correlation kernel can be defined up to a sequence of
cocycles, i.e.

(1.6)
$\det\Big{[}K_{N}(z_{j},z_{l})\Big{]}_{j,l=1}^{k}=\det\Big{[}g_{N}(z_{j})\,\overline{g_{N}(z_{l})}\cdot
K_{N}(z_{j},z_{l})\Big{]}_{j,l=1}^{k},$

where $g_{N}$ is a continuous unimodular function.

Due to the property (1.5), the system (1.1) is also called the determinantal
Coulomb gas ensemble. Moreover, this naturally calls for the investigation of
various asymptotic behaviours of $K_{N}$ as $N\to\infty$. Here, one has to
distinguish two cases, the macroscopic scale and the microscopic scale.

The asymptotic behaviour in the microscopic scale is closely related to the
universality principle in random matrix theory. To describe the local
statistics of the model at a given base point $p\in S$, one needs to
investigate the asymptotic behaviour of the function

(1.7) $(z,w)\mapsto K_{N}\Big{(}p+\frac{e^{i\theta}\,z}{\sqrt{N\Delta
Q(p)}}\,,p+\frac{e^{i\theta}\,w}{\sqrt{N\Delta Q(p)}}\Big{)}.$

Here if $p\in\partial S$, the angle $\theta\in[0,2\pi)$ is chosen so that
$e^{i\theta}$ is outer normal to $\partial S$ at $p$, and otherwise
$\theta=0.$ We remark that the specific choice of the rescaling factor
$\sqrt{N\Delta Q(p)}$ in (1.7) (which is often called the “unfolding”) comes
from the fact that $\frac{1}{N}R_{N,1}(p)\sim\Delta Q(p)$.

For the bulk case when $p\in\operatorname{Int}S$, it was shown in [8] that for
a general external potential $Q$,

(1.8) $K_{N}\Big{(}p+\frac{z}{\sqrt{N\Delta Q(p)}}\,,p+\frac{w}{\sqrt{N\Delta
Q(p)}}\Big{)}\to G(z,w):=e^{z\bar{w}-\frac{|z|^{2}}{2}-\frac{|w|^{2}}{2}}.$

Here $\operatorname{Int}S$ stands for the interior of $S$, the largest open
set of $S$, and the universal scaling limit $G$ in (1.8) is called the Ginibre
kernel [32]. For the edge case when $p\in\partial S$, it was shown in a fairly
recent work [35] that for a general external potential $Q$,

(1.9) $K_{N}\Big{(}p+\frac{e^{i\theta}\,z}{\sqrt{N\Delta
Q(p)}}\,,p+\frac{e^{i\theta}\,w}{\sqrt{N\Delta Q(p)}}\Big{)}\to
G(z,w)\,\frac{1}{2}\operatorname{erfc}\Big{(}\frac{z+\bar{w}}{\sqrt{2}}\Big{)}.$

The class of potentials $Q$ covered in [35] is quite general but dependent on
the topology of the associated droplet.

Turning to the macroscopic scale, recently, Ameur and Cronvall [7] made
significant results on the asymptotic behaviour of $K_{N}(z,w)$. For the
Ginibre ensemble with $Q(z)=|z|^{2}$, they obtained a precise asymptotic
result. Namely, it was obtained in [7, Theorem 1.1] that

(1.10)
$K_{N}(z,w)=\sqrt{\frac{N}{2\pi}}\frac{1}{z\bar{w}-1}(z\bar{w})^{N}e^{N-\frac{N}{2}(|z|^{2}+|w|^{2})}\cdot\Big{(}1+O(\frac{1}{N})\Big{)},$

where $z\not=w$ and $z\bar{w}$ is outside the Szegő curve

(1.11) $\mathcal{S}_{1}:=\\{z\in\mathbb{C}:|z|\leq 1,|z\,e^{1-z}|=1\\}.$

Here, we intentionally add the subscript $1$ since (1.11) can be realised as a
special case of $\SS_{a}$ in (1.13) below with $a=1$. We stress that [7,
Theorem 1.1] indeed provides a closed form of large-$N$ expansions of $K_{N}.$
Let us also mention that (1.10) can also be interpreted as an asymptotic
result of the incomplete gamma function with complex argument, see [7, Section
1.4] and (A.6). (Cf. this was crucially used in a recent work [20].)

Beyond the Ginibre ensemble, Ameur and Cronvall considered general external
potential $Q$ and derived the uniform asymptotic behaviour of $K_{N}(z,w)$ for
$z,w$ outside the droplet, see [7, Theorem 1.3]. (We also refer to [3, 31, 42]
for similar results on the elliptic Ginibre ensemble.) In particular, they
showed that there are strong correlations among the particles on the boundary
of the droplet. One of the main ingredients in their proof is the asymptotic
behaviour of planar orthogonal polynomials (1.3) due to Hedenmalm and Wennman
[35].

The above-mentioned results were mainly obtained for the case where the
external potential $Q$ is fixed, i.e. independent of $N$. Nevertheless, the
case when $Q$ depends on $N$ is also interesting in particular in the context
of the insertion of point charges [11] also known as the induced ensembles
[30] or spectral singularities [36]. (Another important example that
$N$-dependence of the potential being crucial is the almost-Hermitian regime,
see e.g. [6].)

Furthermore, in [35] (and also in the follow-up paper [34]), the asymptotic
behaviours of planar orthogonal polynomials were constructed in terms of a
conformal map from the outside the droplet onto the outside the unit disc.
Accordingly, the asymptotic result in [35] was obtained for the potential $Q$
whose associated droplet is simply connected as a domain on the Riemann
sphere. As a consequence, the edge universality (1.9) in [35] as well as the
Szegő type asymptotic behaviour in [7] were obtained under the assumption that
the associated droplet does not have several disconnected components.

In this work, we aim to provide concrete examples of asymptotic results for
the ensembles with a class of $N$-dependent potentials associated with
disconnected droplets, see Figure 1.

Figure 1. Illustration of the lemniscate archipelago and zooming process

###  1.1. Main results

We now precisely introduce our models. It is more convenient to begin with a
special case when removing the discrete rotational symmetry. In this case, the
model corresponds to the induced Ginibre ensemble [30] with the potential

(1.12) $Q_{c}(z)\equiv Q_{N,c}(z):=|z|^{2}-\frac{2c}{N}\log|z-a|,$

where $c>-1$ and $a\geq 0$. From the statistical physics point of view, we
insert a point charge $c$ at a given point $a$. When $c$ is an integer, the
ensemble (1.1) with the potential (1.12) can also be realised as the Ginibre
ensemble conditioned to have eigenvalue $a$ with multiplicity $c$.

The orthogonal polynomials associated with (1.12) reveal a discontinuity at
$c=0$. Namely, if $c=0$, since the orthogonal polynomials are simply given by
monomials, all the zeros are located at the origin. On the other hand, in
[38], it was shown that for any $c\not=0$ and $a>1$, the zeros of orthogonal
polynomials tend to occupy the _limiting skeleton_ (also known as _mother
body_ , cf. [33])

(1.13)
$\mathcal{S}_{a}:=\Big{\\{}z\in\mathbb{C}:\log|z|-a\operatorname{Re}z=\log\Big{(}\frac{1}{a}\Big{)}-1\,,\,\operatorname{Re}z\leq\frac{1}{a}\Big{\\}}.$

Note that $\SS_{a}$ crosses the point $1/a$. The limiting skeleton $\SS_{a}$
plays an important role in the asymptotic behaviours of the orthogonal
polynomials. See Figure 2 for the shape of $\SS_{a}$.

(a) $a=4$

(b) $a=2$

(c) $a=4/3$

Figure 2.  The plots display $\SS_{a}$. The red dots show the origin and
$1/a$. The green dashed lines indicate the branch cuts in Theorem 1.1.

In our first result, we obtain the following asymptotic behaviour of $K_{N}$
in the macroscopic scaling.

######  Theorem 1.1.

(Macroscopic asymptotic of the induced Ginibre ensemble) Let $Q$ be the
induced Ginibre potential (1.12) with $a>1$ and $c>-1$ $(c\not=0)$. Suppose
that $z$ and $w$ are outside $\SS_{a}$, and $|z-w|>\delta$ for some
$\delta>0$. Then we have

(1.14)
$K_{N}(z,w)=\sqrt{\frac{N}{2\pi}}\frac{1}{z\bar{w}-1}\,\Big{(}\frac{z}{1-az}\frac{\bar{w}}{1-a\bar{w}}\Big{)}^{c}(z\bar{w})^{N}|(z-a)(w-a)|^{c}e^{N-\frac{N}{2}(|z|^{2}+|w|^{2})}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

Here the branch cuts for the variables $z$ and $\bar{w}$ are the line segment
$[0,1/a]$.

Note that if we formally put $c=0$, the formula (1.14) corresponds (1.10). We
mention that the condition $z$ and $w$ being outside the limiting skeleton was
also considered in [3] for the elliptic Ginibre ensemble. (In this case, the
limiting skeleton is a line segment connecting two foci of the ellipse.)

In the spirit of the edge universality (1.9), we obtain the following.

######  Theorem 1.2.

(Boundary scaling limits of the induced Ginibre ensemble) Let $Q$ be the
induced Ginibre potential (1.12) with $a>1$ and $c>-1$. Let $p$ be a point on
the unit circle. Then as $N\to\infty$, we have

(1.15)
$\frac{1}{N}K_{N}\Big{(}p+\frac{p\,z}{\sqrt{N}}\,,\,p+\frac{p\,w}{\sqrt{N}}\Big{)}\to
G(z,w)\,\frac{1}{2}\operatorname{erfc}\Big{(}\frac{z+\bar{w}}{\sqrt{2}}\Big{)},$

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

We now discuss the ensemble with discrete rotational symmetry. For $a\geq 0$
and $d\in\mathbb{N}$, let

(1.16) $V(z)=\frac{1}{d}|z^{d}-a|^{2},\qquad V_{c}(z)\equiv
V_{N,c}:=V(z)-\frac{2c}{N}\log|z|.$

We refer to [10, 21, 27] and references therein for recent studies on such
models. Note that the induced Ginibre potential (1.12) corresponds to (1.16)
with $d=1$ up to a translation. It is well known that the droplet $S_{V}$
associated with the potential $V$ is given by

(1.17) $S_{V}:=\\{z\in\mathbb{C}:|z^{d}-a|\leq 1\\},$

see e.g. [14, Lemma 1]. The density with respect to $dA$ is given by

(1.18) $\Delta V(z)=d\,|z|^{2d-2}.$

Due to the explicit formula (1.17), one can easily notice that if $a<1$,
$S_{V}$ is connected. On the other hand, if $a>1$, $S_{V}$ consists of
$d$-connected components that we call the _lemniscate archipelago_ following
[7], see Figure 1.

We denote by $q_{j,N}^{c}$ the orthonormal polynomials associated with the
weighted measure $e^{-NV_{c}}\,dA$:

(1.19)
$\int_{\mathbb{C}}q_{j,N}^{c}(z)\overline{q_{k,N}^{c}(z)}|z|^{2c}e^{-NV(z)}\,dA(z)=\delta_{jk}.$

For $a>1$, it was shown in [13, 38] that as $j\to\infty$, the (non-trivial)
zeros of $q_{j,N}^{c}$ tend to accumulate on the curve

(1.20)
$\mathcal{S}_{a}^{d}:=\Big{\\{}z\in\mathbb{C}:\log|z^{d}-a|+a\operatorname{Re}z^{d}=\log\Big{(}\frac{1}{a}\Big{)}-1+a^{2}\,,\,\operatorname{Re}z^{d}\geq
a-\frac{1}{a}\Big{\\}}.$

Notice that (1.20) and (1.13) are related by the mapping $z\mapsto a-z^{d}$.
See Figure 3 for the shape of $\SS_{a}^{d}$.

(a) $d=2$

(b) $d=3$

(c) $d=4$

Figure 3.  The plots display $\partial S_{V}$ (black) and
$\mathcal{S}_{a}^{d}$ (blue), where $a=1.1$. The red dots indicate
$(a-\frac{1}{a})^{1/d}\omega^{k}$ and $a^{1/d}\omega^{k}$. The green dashed
lines are the branch cuts in (1.22).

Let us consider the associated correlation kernel

(1.21)
$K_{N}^{c}(z,w):=|zw|^{c}e^{-\frac{N}{2}(V(z)+V(w))}\sum_{j=0}^{N-1}q_{j,N}^{c}(z)\overline{q_{j,N}^{c}(w)}.$

The kernel (1.21) corresponds to the reproducing kernel (1.4) associated with
the potential $Q=V_{c}.$ We derive the asymptotic behaviours of $K_{N}^{c}$ in
the macroscopic scale.

######  Theorem 1.3.

(Macroscopic asymptotic of the lemniscate archipelago) Let $d>1$, $a>1$ and
$c>-1$ be fixed. Suppose that $z$ and $w$ are outside $\SS_{a}^{d}$, and
$|z-w|>\delta$ for some $\delta>0$. If $c=0,1,\dots,d-1$, we further assume
that $(z^{d}-a)(\bar{w}^{d}-a)$ is outside $\SS_{1}$. Then as $N\to\infty,$ we
have

(1.22)
$\displaystyle\begin{split}K_{dN}^{c}(z,w)&=d\sqrt{\frac{N}{2\pi}}\frac{((z^{d}-a)(\bar{w}^{d}-a))^{N}}{(z^{d}-a)(\bar{w}^{d}-a)-1}\,|z\bar{w}|^{c}e^{N-\frac{dN}{2}(V(z)+V(w))}\\\
&\quad\times\Big{(}\frac{z^{d}-a}{az^{d}+1-a^{2}}\frac{\bar{w}^{d}-a}{a\bar{w}^{d}+1-a^{2}}\Big{)}^{\frac{c}{d}}((z^{d}-a)(\bar{w}^{d}-a))^{\frac{1}{d}-1}\\\
&\quad\times\frac{(az^{d}+1-a^{2})(a\bar{w}^{d}+1-a^{2})-z^{d}(z^{d}-a)\bar{w}^{d}(\bar{w}^{d}-a)}{(az^{d}+1-a^{2})^{\frac{1}{d}}(a\bar{w}^{d}+1-a^{2})^{\frac{1}{d}}-z(z^{d}-a)^{\frac{1}{d}}\bar{w}(\bar{w}^{d}-1)^{\frac{1}{d}}}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

Here the branch cuts for the variables $z$ and $\bar{w}$ are given by the
combination of $d$ line segments connecting $(a-\frac{1}{a})^{1/d}\omega^{k}$
and $a^{1/d}\omega^{k}$, where $\omega=e^{2\pi i/d}$ and $k=0,1,\dots,d-1.$

Note that by (1.16) and (1.17), we have

(1.23) $|z^{d}-a|=1,\qquad V(z)=\frac{1}{d},\qquad(z\in\partial S_{V}).$

Then as an immediate consequence of Theorem 1.3, we obtain that for
$z,w\in\partial S_{V}$,

(1.24)
$\displaystyle\begin{split}|K_{dN}^{c}(z,w)|&=d\sqrt{\frac{N}{2\pi}}\Big{|}\frac{(z\bar{w})^{c}}{(z^{d}-a)(\bar{w}^{d}-a)-1}\Big{(}(az^{d}+1-a^{2})(a\bar{w}^{d}+1-a^{2})\Big{)}^{-\frac{c}{d}}\Big{|}\\\
&\quad\times\Big{|}\frac{(az^{d}+1-a^{2})(a\bar{w}^{d}+1-a^{2})-z^{d}(z^{d}-a)\bar{w}^{d}(\bar{w}^{d}-a)}{(az^{d}+1-a^{2})^{\frac{1}{d}}(a\bar{w}^{d}+1-a^{2})^{\frac{1}{d}}-z(z^{d}-a)^{\frac{1}{d}}\bar{w}(\bar{w}^{d}-1)^{\frac{1}{d}}}\Big{|}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

Thus one can notice that $K_{dN}^{c}(z,w)=O(\sqrt{N})$ for $z,w\in\partial
S_{V}$, which indicates that there are strong correlations among the particles
on the boundary of the droplets.

To provide a physical realisation of Theorem 1.3, let us consider the Berezin
kernel

(1.25)
$B_{N}(z,w):=\frac{R_{N,1}(z)R_{N,1}(w)-R_{N,2}(z,w)}{R_{N,1}(z)}=\frac{|K_{N}^{c}(z,w)|^{2}}{R_{N,1}(z)}.$

For a given point $z$, the function $w\mapsto B_{N}(z,w)$ corresponds to the
probability density of the ensemble conditioned to have a particle at $z.$ See
Figure 4 for the graphs of $B_{N}$.

(a) $d=2$

(b) $d=3$

(c) $d=4$

Figure 4.  The plots display the approximation of the graphs $w\mapsto
B_{N}(z,w)$ in (1.25) (for $w$ away from $z$), where
$z=(a-1)^{\frac{1}{d}}\in\partial S_{V}$ and $a=1.1$. Here $c=0$ and $N=600$.
For the approximation, we use (1.22) and the fact that $R_{N,1}(z)\sim N\Delta
V(z)/2$.

We remark that the asymptotic behaviour (1.22) may involve special functions
(in the subleading terms) with certain periodicity such as Jacobi theta
functions as observed in [26] for a rotationally symmetric ensemble. We refer
to [28, 18] for a discussion on similar situations on Hermitian matrix model.

In our final result, we derive the boundary scaling limits.

######  Theorem 1.4.

(Boundary scaling limits of the lemniscate archipelago) Let $p\in\partial
S_{V}$ and choose $\theta$ so that $e^{i\theta}$ is outer normal to $\partial
S_{V}$ at $p$. Then as $N\to\infty$, we have

(1.26) $\frac{1}{dN\Delta
V(p)}K_{dN}^{c}\Big{(}p+\frac{e^{i\theta}\,z}{\sqrt{dN\Delta
V(p)}}\,,\,p+\frac{e^{i\theta}\,w}{\sqrt{dN\Delta V(p)}}\Big{)}\to
G(z,w)\,\frac{1}{2}\operatorname{erfc}\Big{(}\frac{z+\bar{w}}{\sqrt{2}}\Big{)},$

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

We emphasise that Theorem 1.4 provides an example of edge universality for the
ensembles with disconnected droplets that are not covered in [35].

###  1.2. Outline of the proofs

The overall strategy of the proofs is as follows.

  * •

We use the multi-fold transform of the correlation kernels (Lemma 2.1) that
relates those of $Q_{c}$ in (1.12) and of $V_{c}$ in (1.16). Due to this
property, one can easily derive Theorems 1.3 and 1.4 from Theorems 1.1 and
1.2, respectively.

  * •

We apply the generalised Christoffel-Darboux formula (Proposition 2.2) that
allows expressing the correlation kernel only in terms of three monic
orthogonal polynomials (of degree $N-1$, $N$, and $N+1$) and their norms.

  * •

Using the steepest descent method to the Riemann-Hilbert problem developed in
[12, 38], we derive the asymptotic behaviours of orthogonal polynomials
(Proposition 2.3) and norms (Lemma 2.4) up to the first subleading terms.
Combined with the Christoffel-Darboux formula, these lead to Theorems 1.1 and
1.2.

The overall strategy described above was introduced in [24] to obtain the
microscopic limit of the correlation kernel at multi-criticality $a=1$. We use
this strategy when $a>1$ together with new asymptotic behaviours of orthogonal
polynomials and their norms (Proposition 2.3 and Lemma 2.4). These are
probably of interest by themselves in the spirit of several works [13, 40, 15,
19] on Riemann-Hilbert analysis for planar orthogonal polynomials.

The rest of this paper is organised as follows. In Section 2, we present the
overall strategy of the proofs in more detail and show our main results.
However it requires Proposition 2.3 and Lemma 2.4 that are only shown in the
following section. For the proofs, in Section 3, we use the nonlinear steepest
descent method to the Riemann-Hilbert problem associated with the orthogonal
polynomials. In Appendix A, we present the proofs of Proposition 2.3 and Lemma
2.4 for the exactly solvable case $c=1$ using well-known properties of some
special functions.

##  2\. Proofs of main results

In this section, we present the overall strategy of the proofs and show the
main results. In Subsections 2.1 and 2.2 we introduce the multi-fold transform
(Lemma 2.1) and the generalised Christoffel-Darboux formula. In Subsection
2.3, we present asymptotic behaviours of orthogonal polynomials (Proposition
2.3) and the norms (Lemma 2.4). In Subsection 2.4, we prove Theorems 1.1 and
1.2. In the last subsection, we show Theorems 1.3 and 1.4.

###  2.1. Multi-fold transform

We write $p_{j,N}^{c}$ for the orthonormal polynomials satisfying

(2.1)
$\int_{\mathbb{C}}p_{j,N}^{c}(z)\overline{p_{k,N}^{c}(z)}|z|^{2c}e^{-N|z-a|^{2}}\,dA(z)=\delta_{jk}.$

Then the orthogonal polynomials $q_{j,N}^{c}$ in (1.19) is related to
$p_{j,N}^{c}$ as

(2.2) $q_{dj+l,dN}^{c}(z)=\sqrt{d}\,z^{l}p_{j,N}^{\frac{c+l+1}{d}-1}(z^{d}),$

see e.g. [15, Section 3] and [24, Section 2]. We now define the correlation
kernel

(2.3)
$\widehat{K}_{N}^{c}(z,w)=(z\bar{w})^{c}e^{-\frac{N}{2}(|z-a|^{2}+|w-a|)^{2}}\sum_{j=0}^{N-1}p_{j,N}^{c}(z)\overline{p_{j,N}^{c}(w)}.$

Notice that we use $(z\bar{w})^{c}$ instead of $|zw|^{c}$.

By (2.2), we have the following multi-fold transform relation, see [24,
Section 2] for more detail. (Cf. this idea appeared also in [29, Proposition
2.1], see [1] for the chiral setup.) Recall that $K_{N}^{c}$ is given by
(1.21).

######  Lemma 2.1.

We have

(2.4)
$K_{dN}^{c}(z,w)=d(z\bar{w})^{d-1}\Big{(}\frac{|zw|}{z\bar{w}}\Big{)}^{c}\sum_{l=0}^{d-1}\widehat{K}_{N}^{\frac{c+l+1}{d}-1}(z^{d},w^{d}).$

Note that in the left-hand side of (2.4) we use $dN$ instead of $N$. This is
indeed the key observation for such a transform. Due to Lemma 2.1, it suffices
to derive the asymptotics of $\widehat{K}_{N}^{c}$.

###  2.2. Christoffel-Darboux formula

One can compute asymptotics of $\bar{\partial}_{w}\widehat{K}_{N}^{c}(z,w)$ by
virtue of the Christoffel-Darboux formula in [24, Theorem 3.2].

For this, we set some notations. Let $P_{j}\equiv P_{j}^{c}$ be the _monic_
orthogonal polynomial satisfying

(2.5)
$\displaystyle\int_{\mathbb{C}}P_{j}(z)\overline{P_{k}(z)}|z-a|^{2c}e^{-N|z|^{2}}\,dA(z)=h_{j}\,\delta_{jk},$

where $h_{j}$ is the (squared) orthogonal norm. Note that we have the
following relation

(2.6) $p_{j,N}^{c}(z)=\frac{1}{\sqrt{h_{j}}}P_{j}(a-z).$

We denote

(2.7)
$W(z)=(z-a)^{c},\qquad\psi_{j}(z):=W(z)P_{j}(z),\qquad\phi_{j}(z):=W(z)\frac{P_{j}(z)}{h_{j}}.$

Let us define

(2.8)
$\widetilde{K}_{N}^{c}(z,w):=((z-a)(\bar{w}-a))^{c}e^{-Nz\bar{w}}\sum_{j=0}^{N-1}\frac{P_{j}(z)\overline{P_{j}(w)}}{h_{j}}=e^{-Nz\bar{w}}\sum_{j=0}^{N-1}\overline{\phi_{j}(w)}\psi_{j}(z).$

The kernel $K_{N}$ in (1.4) with $Q$ given by (1.12) is written in terms of
$\widetilde{K}_{N}^{c}$ as

(2.9)
$K_{N}(z,w)=\Big{(}\frac{|(z-a)(\bar{w}-a)|}{(z-a)(\bar{w}-a)}\Big{)}^{c}e^{-\frac{N}{2}(|z|^{2}+|w|^{2}-2z\bar{w})}\widetilde{K}_{N}^{c}(z,w).$

Note also that

(2.10)
$\widetilde{K}_{N}^{c}(a-z,a-w)=(z\bar{w})^{c}e^{-N(a-z)(a-\bar{w})}\sum_{j=0}^{N-1}p_{j,N}^{c}(z)\overline{p_{j,N}^{c}(w)}$

Thus it is related to $\widehat{K}_{N}^{c}$ in (2.3) as

(2.11)
$\widehat{K}_{N}^{c}(z,w)=e^{-\frac{N}{2}(|z-a|^{2}+|w-a|^{2}-2(z-a)(\bar{w}-a))}\widetilde{K}_{N}^{c}(a-z,a-w).$

The following version of the Christoffel-Darboux formula was obtained in [24,
Theorem 3.2].

######  Proposition 2.2.

(Christoffel-Darboux formula) Suppose that $a\not=0$ and that

(2.12) $\langle z\psi_{j}|\phi_{0}\rangle\neq 0,\qquad\phi_{j}(a)\neq
0,\qquad\text{for all }j.$

Then we have the following form of the Christoffel-Darboux identity:

(2.13)
$\displaystyle\begin{split}{\bar{\partial}}_{w}\widetilde{K}_{N}^{c}(z,w)&=e^{-Nz\bar{w}}\frac{1}{\tfrac{N+c}{N}h_{N-1}-h_{N}}{\bar{\partial}}_{w}\overline{\psi_{N}(w)}\Big{(}\psi_{N}(z)-z\psi_{N-1}(z)\Big{)}\\\
&\quad-e^{-Nz\bar{w}}\frac{P_{N+1}(a)}{P_{N}(a)}\frac{N\,h_{N}/h_{N-1}}{\tfrac{N+c+1}{N}h_{N}-h_{N+1}}\overline{\psi_{N-1}(w)}\Big{(}\psi_{N+1}(z)-z\psi_{N}(z)\Big{)}.\end{split}$

This formula plays a key role in performing asymptotic analysis for Theorems
1.1 and 1.2. We also refer to [37, 2, 16, 22, 20, 23] for various Christoffel-
Darboux type identities involving certain differential operators.

###  2.3. Fine asymptotic behaviours of orthogonal polynomials and norms

Recall that the monic polynomial $P_{j}$ satisfies the orthogonality condition
(2.5). The weighted orthogonal polynomial $\psi_{j}$ is given by (2.7). We
obtain the strong asymptotic behaviour of $\psi_{j}$ up to the first
subleading terms.

######  Proposition 2.3.

Let $a>1$ and $c>-1$. Then for $z\in\mathbb{C}$ outside $\mathcal{S}_{a}$ in
(1.13), we have

(2.14) $\displaystyle\psi_{N-1}(z)$
$\displaystyle=z^{N+c-1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\cdot\Big{[}1-\frac{c}{1-az}\Big{(}\frac{1+c}{2}\frac{1}{1-az}+\frac{c}{1-a^{2}}-1\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]},$
(2.15) $\displaystyle\psi_{N}(z)$
$\displaystyle=z^{N+c}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\cdot\Big{[}1-\frac{c}{1-az}\Big{(}\frac{1+c}{2}\frac{1}{1-az}+\frac{c}{1-a^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]},$
(2.16) $\displaystyle\psi_{N+1}(z)$
$\displaystyle=z^{N+c+1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\cdot\Big{[}1-\frac{c}{1-az}\Big{(}\frac{1+c}{2}\frac{1}{1-az}+\frac{c}{1-a^{2}}+1\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.$

In particular, we have

(2.17) $\displaystyle\psi_{N}(z)-z\psi_{N-1}(z)$
$\displaystyle=z^{N+c}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\cdot\frac{c}{az-1}\frac{1}{N}\cdot\Big{(}1+O(\frac{1}{N})\Big{)},$
(2.18) $\displaystyle\psi_{N+1}(z)-z\psi_{N}(z)$
$\displaystyle=z^{N+1+c}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\cdot\frac{c}{az-1}\frac{1}{N}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

We emphasise that the leading terms in Proposition 2.3 were obtained in [38,
Theorem 2]. Note that the terms (2.17) and (2.18) appear in the Christoffel-
Darboux formula (2.13). For these terms, we should extend [38, Theorem 2] up
to the first subleading $O(1/N)$ terms.

Notice that if $c=0$, then $\psi_{j}(z)=z^{j}.$ Thus in this case Proposition
2.3 trivially holds.

To apply the Christoffel-Darboux formula (2.13), one should also derive the
asymptotic behaviours of the orthogonal norms $h_{j}$ in (2.5).

######  Lemma 2.4.

Let $a>1$ and $c>-1$. Then we have

(2.19) $\displaystyle h_{N-1}$
$\displaystyle=e^{-N}\sqrt{\frac{2\pi}{N}}a^{2c}\cdot\Big{[}1+\Big{(}\frac{c}{a^{2}-1}+\frac{c(c-1)}{2}+\frac{1}{12}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]},$
(2.20) $\displaystyle h_{N}$
$\displaystyle=e^{-N}\sqrt{\frac{2\pi}{N}}a^{2c}\cdot\Big{[}1+\Big{(}\frac{c}{a^{2}-1}+\frac{c(c-1)}{2}+\frac{1}{12}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]},$
(2.21) $\displaystyle h_{N+1}$
$\displaystyle=e^{-N}\sqrt{\frac{2\pi}{N}}a^{2c}\cdot\Big{[}1+\Big{(}\frac{c}{a^{2}-1}+\frac{c(c-1)}{2}+\frac{13}{12}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.$

In particular, for $c\not=0$, we have

(2.22) $\displaystyle\frac{1}{\tfrac{N+c}{N}h_{N-1}-h_{N}}$
$\displaystyle=\frac{1}{c}\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,e^{N}\cdot\Big{(}1+O(\frac{1}{N})\Big{)},$
(2.23) $\displaystyle\frac{N\,h_{N}/h_{N-1}}{\tfrac{N+c+1}{N}h_{N}-h_{N+1}}$
$\displaystyle=\frac{1}{c}\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{5}{2}}\,e^{N}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

Note that (2.22) and (2.23) appear in the Christoffel-Darboux formula (2.13).
We remark that for $c=0$, we have

$h_{j}=\int_{\mathbb{C}}|z|^{2j}e^{-N|z|^{2}}\,dA(z)=2\int_{0}^{\infty}r^{2j+1}e^{-Nr^{2}}\,dr=\frac{j!}{N^{j+1}}.$

Thus by Stirling’s formula, one can directly check that Lemma 2.4 holds for
$c=0.$

###  2.4. Proofs of Theorems 1.1 and 1.2

Combining the Christoffel-Darboux formula (Proposition 2.2) with Proposition
2.3 and Lemma 2.4, we show Theorem 1.1.

###### Proof of Theorem 1.1.

By the transform (2.9), it suffices to show that for $z$ and $\bar{w}$ outside
$\SS_{a}$ in (1.13),

(2.24)
$\widetilde{K}_{N}^{c}(z,w)=\sqrt{\frac{N}{2\pi}}\frac{1}{z\bar{w}-1}\,\Big{(}\frac{z-a}{1-az}\frac{\bar{w}-a}{1-a\bar{w}}\Big{)}^{c}\cdot(z\bar{w})^{N+c}e^{N-Nz\bar{w}}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

Using Proposition 2.3, we have

(2.25)
${\bar{\partial}}_{w}\overline{\psi_{N}(w)}=N\bar{w}^{N+c-1}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)},\qquad\overline{\psi_{N-1}(w)}=\bar{w}^{N+c-1}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

By [38, Theorem 2], we also have

(2.26) $\frac{P_{N+1}(a)}{P_{N}(a)}=a\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

Therefore by (2.17) and (2.18), we obtain

(2.27)
${\bar{\partial}}_{w}\overline{\psi_{N}(w)}\Big{(}\psi_{N}(z)-z\psi_{N-1}(z)\Big{)}=\frac{c}{az-1}z^{N+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}$

and

(2.28)
$\begin{split}\overline{\psi_{N-1}(w)}\Big{(}\psi_{N+1}(z)-z\psi_{N}(z)\Big{)}=\frac{c}{az-1}\frac{1}{N}z^{N+1+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

Then by Lemma 2.4, we have

(2.29)
$\begin{split}&\quad\frac{1}{\tfrac{N+c}{N}h_{N-1}-h_{N}}{\bar{\partial}}_{w}\overline{\psi_{N}(w)}\Big{(}\psi_{N}(z)-z\psi_{N-1}(z)\Big{)}\\\
&=\frac{1}{c}\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,e^{N}\frac{c}{az-1}z^{N+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}\\\
&=\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,e^{N}\frac{1}{az-1}z^{N+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{1-az}\frac{\bar{w}-a}{1-a\bar{w}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}\end{split}$

and

(2.30)
$\begin{split}&\quad\frac{P_{N+1}(a)}{P_{N}(a)}\frac{N\,h_{N}/h_{N-1}}{\tfrac{N+c+1}{N}h_{N}-h_{N+1}}\overline{\psi_{N-1}(w)}\Big{(}\psi_{N+1}(z)-z\psi_{N}(z)\Big{)}\\\
&=a\,\frac{1}{c}\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{5}{2}}\,e^{N}\frac{c}{az-1}\frac{1}{N}z^{N+1+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{z-\frac{1}{a}}\Big{)}^{c}\Big{(}\frac{\bar{w}-a}{\bar{w}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}\\\
&=\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,e^{N}\frac{az}{az-1}z^{N+c}\bar{w}^{N+c-1}\Big{(}\frac{z-a}{1-az}\frac{\bar{w}-a}{1-a\bar{w}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

Now it follows from the Christoffel-Darboux formula (Proposition 2.2) that

(2.31)
${\bar{\partial}}_{w}\widetilde{K}_{N}^{c}(z,w)=-\frac{N^{\frac{3}{2}}}{\sqrt{2\pi}}\,\Big{(}\frac{z-a}{1-az}\frac{\bar{w}-a}{1-a\bar{w}}\Big{)}^{c}\cdot
z^{N+c}\bar{w}^{N+c-1}e^{N-Nz\bar{w}}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.$

Integrating this equation, we obtain

(2.32)
$\widetilde{K}_{N}^{c}(z,w)=\sqrt{\frac{N}{2\pi}}\frac{1}{z\bar{w}-1}\,\Big{(}\frac{z-a}{1-az}\frac{\bar{w}-a}{1-a\bar{w}}\Big{)}^{c}\cdot(z\bar{w})^{N+c}e^{N-Nz\bar{w}}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}+f_{N}(z)$

for some function $f_{N}$ depending only on $z$. Due to the symmetry
$\widetilde{K}_{N}^{c}(z,w)=\widetilde{K}_{N}^{c}(w,z)$, it follows that
$f_{N}$ is a constant function. Furthermore, by combining the exterior
estimate

$\widetilde{K}_{N}^{c}(z,z)\to 0,\qquad\mbox{as }z\to\infty$

that holds in general (see [9, Section 4.1.1]) and the elementary inequality

$\det\begin{bmatrix}\widetilde{K}_{N}^{c}(z,z)&\widetilde{K}_{N}^{c}(z,w)\\\
\widetilde{K}_{N}^{c}(w,z)&\widetilde{K}_{N}^{c}(w,w)\end{bmatrix}\geq
0,\qquad\mbox{i.e.}\qquad\widetilde{K}_{N}^{c}(z,z)\widetilde{K}_{N}^{c}(w,w)\geq|\widetilde{K}_{N}^{c}(z,w)|^{2},$

one can observe that $\widetilde{K}_{N}^{c}(z,w)\to 0$ as $z\to\infty$. Thus
we conclude (2.24). ∎

###### Proof of Theorem 1.2.

By (2.9), it suffices to show that

(2.33)
$\frac{1}{N}\widetilde{K}_{N}^{c}\Big{(}p+\frac{p\,z}{\sqrt{N}}\,,\,p+\frac{p\,w}{\sqrt{N}}\Big{)}\to\frac{1}{2}\operatorname{erfc}\Big{(}\frac{z+\bar{w}}{\sqrt{2}}\Big{)}.$

To lighten notations, let us write

(2.34) $z_{p}:=p\,\Big{(}1+\frac{z}{\sqrt{N}}\Big{)},\qquad
w_{p}:=p\,\Big{(}1+\frac{w}{\sqrt{N}}\Big{)}.$

First note that

(2.35) $e^{-Nz_{p}\bar{w}_{p}}=e^{-N-\sqrt{N}(z+\bar{w})-z\bar{w}}.$

We also have

(2.36)
$z_{p}^{N+c}=p^{N+c}\,e^{\sqrt{N}z-\frac{z^{2}}{2}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)},\qquad
z_{p}^{N+1+c}=p^{N+1+c}\,e^{\sqrt{N}z-\frac{z^{2}}{2}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Combining these asymptotics with Proposition 2.3, we have

(2.37) $\displaystyle\psi_{N}(z_{p})-z_{p}\psi_{N-1}(z_{p})$
$\displaystyle=p^{N+c}\,e^{\sqrt{N}z-\frac{z^{2}}{2}}\Big{(}\frac{p-a}{p-\frac{1}{a}}\Big{)}^{c}\cdot\frac{c}{ap-1}\frac{1}{N}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)},$
(2.38) $\displaystyle\psi_{N+1}(z_{p})-z_{p}\psi_{N}(z_{p})$
$\displaystyle=p^{N+1+c}\,e^{\sqrt{N}z-\frac{z^{2}}{2}}\Big{(}\frac{p-a}{p-\frac{1}{a}}\Big{)}^{c}\cdot\frac{c}{ap-1}\frac{1}{N}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}$

and

(2.39) $\displaystyle\overline{\psi^{\prime}_{N}(w_{p})}$
$\displaystyle=N\,\bar{p}^{N+c-1}e^{\sqrt{N}\bar{w}-\frac{\bar{w}^{2}}{2}}\Big{(}\frac{\bar{p}-a}{\bar{p}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)},$
(2.40) $\displaystyle\overline{\psi_{N-1}(w_{p})}$
$\displaystyle=\bar{p}^{N+c-1}e^{\sqrt{N}\bar{w}-\frac{\bar{w}^{2}}{2}}\Big{(}\frac{\bar{p}-a}{\bar{p}-\frac{1}{a}}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Then by Lemma 2.4, we obtain

(2.41) $\begin{split}&\quad
e^{-Nz_{p}\bar{w}_{p}}\frac{1}{\tfrac{N+c}{N}h_{N-1}-h_{N}}\overline{\psi^{\prime}_{N}(w_{p})}\Big{(}\psi_{N}(z_{p})-z_{p}\psi_{N-1}(z_{p})\Big{)}\\\
&=e^{-\frac{(z+\bar{w})^{2}}{2}}\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,\Big{(}\frac{\bar{p}-a}{\bar{p}-\frac{1}{a}}\Big{)}^{c}\,\Big{(}\frac{p-a}{p-\frac{1}{a}}\Big{)}^{c}\cdot\frac{p}{ap-1}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}\\\
&=e^{-\frac{(z+\bar{w})^{2}}{2}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,\cdot\frac{p}{ap-1}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.\end{split}$

Here we have used that

(2.42) $\Big{|}\frac{p-a}{ap-1}\Big{|}=\Big{|}\frac{p-a}{a-\bar{p}}\Big{|}=1,$

which follows from $|p|=1$.

Similarly, we have

(2.43) $\begin{split}&\quad
e^{-Nz_{p}\bar{w}_{p}}\frac{P_{N+1}(a)}{P_{N}(a)}\frac{N\,h_{N}/h_{N-1}}{\tfrac{N+c+1}{N}h_{N}-h_{N+1}}\overline{\psi_{N-1}(w_{p})}\Big{(}\psi_{N+1}(z_{p})-z_{p}\psi_{N}(z_{p})\Big{)}\\\
&=e^{-\frac{(z+\bar{w})^{2}}{2}}\,\frac{1}{a^{2c}}\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,\Big{(}\frac{\bar{p}-a}{\bar{p}-\frac{1}{a}}\Big{)}^{c}\,\Big{(}\frac{p-a}{p-\frac{1}{a}}\Big{)}^{c}\cdot\frac{ap^{2}}{ap-1}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}\\\
&=e^{-\frac{(z+\bar{w})^{2}}{2}}\,\frac{1}{\sqrt{2\pi}}\,N^{\frac{3}{2}}\,\cdot\frac{ap^{2}}{ap-1}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.\end{split}$

Therefore by Proposition 2.2, we obtain that

(2.44)
${\bar{\partial}}_{w}\Big{[}\frac{1}{N}\widetilde{K}_{N}^{c}(z_{p},w_{p})\Big{]}=-\frac{1}{\sqrt{2\pi}}e^{-\frac{(z+\bar{w}^{2})}{2}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

This gives that

(2.45)
$\frac{1}{N}\widetilde{K}_{N}^{c}(z_{p},w_{p})=\frac{1}{2}\operatorname{erfc}\Big{(}\frac{z+\bar{w}}{\sqrt{2}}\Big{)}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)},$

which leads to the desired convergence (2.33). Here the integration constant
is determined similarly above by the fact that
$\frac{1}{N}\widetilde{K}_{N}^{c}(z_{p},w_{p})\to 0$ as $z\to+\infty$. ∎

###  2.5. Proofs of Theorems 1.3 and 1.4

We derive Theorem 1.3 from Theorem 1.1.

###### Proof of Theorem 1.3.

By Theorem 1.1 and (2.11), we have

(2.46)
$\begin{split}\widehat{K}_{N}^{c}(z,w)&=\sqrt{\frac{N}{2\pi}}\frac{1}{(a-z)(a-\bar{w})-1}\,\Big{(}\frac{z}{a^{2}-1-az}\frac{\bar{w}}{a^{2}-1-a\bar{w}}\Big{)}^{c}\\\
&\quad\times\Big{(}(z-a)(\bar{w}-a)\Big{)}^{N+c}e^{N-\frac{N}{2}(|z-a|^{2}+|w-a|^{2})}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

By (2.4), we obtain

(2.47)
$\begin{split}K_{dN}^{c}(z,w)&=d\sqrt{\frac{N}{2\pi}}\frac{((z^{d}-a)(\bar{w}^{d}-a))^{N}}{(z^{d}-a)(\bar{w}^{d}-a)-1}\,|zw|^{c}e^{N-\frac{dN}{2}(V(z)+V(w))}\\\
&\quad\times\sum_{l=0}^{d-1}\Big{(}\frac{z^{d}-a}{az^{d}+1-a^{2}}\frac{\bar{w}^{d}-a}{a\bar{w}^{d}+1-a^{2}}\Big{)}^{\frac{c+l+1}{d}-1}(z\bar{w})^{l}\cdot\Big{(}1+O(\frac{1}{N})\Big{)}.\end{split}$

Now (1.22) follows from straightforward computations.

∎

Finally, we derive Theorem 1.4 from Theorem 1.2.

###### Proof of Theorem 1.4.

By (2.4) and (1.18), we have

(2.48) $\begin{split}&\quad\frac{1}{dN\Delta
V(p)}K_{dN}^{c}\Big{(}p+\frac{e^{i\theta}\,z}{\sqrt{dN\Delta
V(p)}}\,,\,p+\frac{e^{i\theta}\,w}{\sqrt{dN\Delta V(p)}}\Big{)}\\\
&=\frac{1}{d^{2}Np^{2d-2}}K_{dN}^{c}\Big{(}p+\frac{e^{i\theta}\,z}{d\,p^{d-1}\sqrt{N}}\,,\,p+\frac{e^{i\theta}\,w}{d\,p^{d-1}\sqrt{N}}\Big{)}\\\
&=\frac{1}{dN}\sum_{l=0}^{d-1}\widehat{K}_{N}^{\frac{c+l+1}{d}-1}\Big{(}p^{d}+\frac{e^{i\theta}\,z}{\sqrt{N}}\,,\,p^{d}+\frac{e^{i\theta}\,w}{\sqrt{N}}\Big{)}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.\end{split}$

Therefore Theorem 1.4 follows from Theorem 1.2. ∎

##  3\. Riemann-Hilbert analysis and fine asymptotic behaviours

In this section, we derive fine asymptotic behaviours of the orthogonal
polynomials (Proposition 2.3) and the orthogonal norms (Lemma 2.4). Subsection
3.1 is devoted to the recalling the matrix-valued Riemann-Hilbert problem
developed in [12] and the transforms introduced in [38]. Based on the Riemann-
Hilbert analysis in Subsections 3.2 and 3.3, we prove Proposition 2.3 and
Lemma 2.4.

###  3.1. Outline of the Riemann-Hilbert analysis

Let us briefly recall the Riemann-Hilbert analysis in [38, 12] (see also [39,
40] for its generalisation) that was developed to derive the asymptotic
behaviours of the orthogonal polynomials $P_{n}.$ We also refer the reader to
[15, 13] for similar studies in different settings. This will be used in the
following subsection to derive fine asymptotic behaviours of $P_{n}$.

Let $\Gamma$ be a simple closed curve that encloses the line segment
$[0,a]\in\mathbb{C}$ with counterclockwise orientation. Let the analytic
function $w_{n,N}$ on $\mathbb{C}\setminus[0,a]$ be defined by

(3.1) $w_{n,N}(z):=\Big{(}\frac{z-a}{z}\Big{)}^{c}\frac{e^{-Naz}}{z^{n}},$

where we choose the principal branch.

Define the matrix function $Y(z)$ by

(3.2) $Y(z):=\begin{bmatrix}P_{n}(z)&\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\frac{P_{n}(s)w_{n,N}(s)}{s-z}\,ds\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ Q_{n-1}(z)&\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\frac{Q_{n-1}(s)w_{n,N}(s)}{s-z}\,ds\end{bmatrix},$

where $Q_{n-1}$ is a unique polynomial of degree $n-1$ satisfying

$\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\frac{Q_{n-1}(s)w_{n,N}(s)}{s-z}\,ds=\frac{1}{z^{n}}\cdot\Big{(}1+O(\frac{1}{z})\Big{)}.$

Then it was shown in [12, Section 3] that $Y(z)$ is a unique solution to the
Riemann-Hilbert problem

(3.3) $\begin{cases}Y(z)\mbox{ is holomorphic in
$\mathbb{C}\setminus\Gamma$},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
Y_{+}(z)=Y_{-}(z)\begin{bmatrix}1&w_{n,N}(z)\\\
0&1\end{bmatrix},\quad&z\in\Gamma,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
Y(z)=\Big{(}I+O(\frac{1}{N})\Big{)}\begin{bmatrix}z^{n}&0\\\
0&z^{-n}\end{bmatrix},\quad&z\to\infty.\end{cases}$

Here $Y_{\pm}(z)$ are the boundary values on the sides of the corresponding
contour. Since $P_{n}(z)=[Y(z)]_{11}$, we aim to analyse the solution to the
Riemann-Hilbert problem (3.3). For this purpose, we shall introduce several
transforms of (3.3).

First, let us define $g$ by

(3.4) $g(z)=\begin{cases}\log z,&z\in\overline{{\rm
Ext}{\mathcal{S}}_{a}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
az+\log\beta-a\beta,&z\in{\rm Int}{\mathcal{S}}_{a}.\end{cases}$

Here and in the sequel, we write $\beta=1/a$. The function $g$ is a building
block to define

(3.5) $\phi(z)=az+\log z-2g(z)+l,\qquad l=\log\beta-a\beta,$

which satisfies $\operatorname{Re}\phi(z)=0$ for $z\in{\mathcal{S}}_{a}$.

Following the nonlinear steepest descent method that applied to the above
Riemann-Hilbert problem for $Y$, we define

(3.6)
$Z(z):=e^{\frac{-Nl}{2}\sigma_{3}}Y(z)e^{-Ng(z)\sigma_{3}}e^{\frac{Nl}{2}\sigma_{3}}\begin{bmatrix}1&0\\\
\star\big{(}\frac{z}{z-a}\big{)}^{c}e^{N\phi(z)}&1\end{bmatrix},$

where

$\star=\begin{cases}1,&\mbox{when $z\in U\cap{\rm Ext}\Gamma$},\\\
-1,&\mbox{when $z\in U\cap{\rm Int}\Gamma$},\\\ 0,&\mbox{when $z\notin
U$}.\end{cases}$

Here $U$ is a neighbourhood of $\SS_{a}$. Then by (3.3), the matrix function
$Z$ satisfies the following Riemann-Hilbert problem

(3.7) $\begin{cases}Z_{+}(z)=Z_{-}(z)\begin{bmatrix}1&0\\\
\big{(}\frac{z}{z-a}\big{)}^{c}e^{N\phi(z)}&1\end{bmatrix},\quad&z\in\partial
U,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
Z_{+}(z)=Z_{-}(z)\begin{bmatrix}0&\big{(}\frac{z-a}{z}\big{)}^{c}\\\
-\big{(}\frac{z}{z-a}\big{)}^{c}&0\end{bmatrix},\quad&z\in\Gamma\cap U,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
Z_{+}(z)=Z_{-}(z)\begin{bmatrix}1&\big{(}\frac{z-a}{z}\big{)}^{c}e^{-N\phi(z)}\\\
0&1\end{bmatrix},\quad&z\in\Gamma\setminus U,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ Z(z)=I+O(\frac{1}{N}),\quad&z\to\infty,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ Z(z)\mbox{ is holomorphic},&\mbox{otherwise}.\\\ \end{cases}$

Next, we define the global parametrix

(3.8)
$\Phi(z)=\begin{cases}\begin{bmatrix}\big{(}\frac{z}{z-\beta}\big{)}^{c}&0\\\
0&\big{(}\frac{z-\beta}{z}\big{)}^{c}\end{bmatrix},\quad&z\in{\rm
Ext}\Gamma,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\begin{bmatrix}0&\big{(}\frac{z-a}{z-\beta}\big{)}^{c}\\\
-\big{(}\frac{z-\beta}{z-a}\big{)}^{c}&0\end{bmatrix},\quad&z\in{\rm
Int}\Gamma.\end{cases}$

Then $\Phi$ satisfies the following Riemann-Hilbert problem

(3.9)
$\begin{cases}\Phi_{+}(z)=\Phi_{-}(z)\begin{bmatrix}0&\big{(}\frac{z-a}{z}\big{)}^{c}\\\
-\big{(}\frac{z}{z-a}\big{)}^{c}&0\end{bmatrix},\quad&z\in{\mathcal{S}}_{a},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\Phi(z)=I+O(\frac{1}{N}),\quad&z\to\infty,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \Phi(z)\quad\mbox{is holomorphic},&\mbox{otherwise}.\\\ \end{cases}$

Note that we let $\Gamma$ match $\SS_{a}$ for $z\in U$ and away from a small
neighborhood of $\beta$.

Near the point $\beta$, the jump matrices of $\Phi$ do not converge to those
of $Z$. Therefore one needs the local parametrix around $\beta$ that satisfies
the exact jump conditions of $Z$. Moreover, we shall construct a rational
matrix function $R$ such that the improved global parametrix, $R\Phi$, matches
the local parametrix better. This construction is called “partial Schlesinger
transform” [17], and it was used in [12] to obtain the strong asymptotics of
$P_{n}$. Here we use it to derive fine asymptotic behaviours of the orthogonal
polynomials (Proposition 2.3) and the orthogonal norms (Lemma 2.4).

Let $D_{\beta}$ be a disk neighborhood of $\beta$ with a fixed radius such
that the map $\zeta:D_{\beta}\to\mathbb{C}$ given by

(3.10) $\zeta:=\sqrt{2N(a(z-\beta)-\log
z+\log\beta)}=a\sqrt{N}(z-\beta)(1+O(z-\beta))$

is univalent.

$\Gamma$$\Gamma$$U$$U$ Figure 5. The jump contours of $P(z)$ in $D_{\beta}$.
$\Gamma$ are the blue curves, $U$ are the shaded region bounded by the green
curves.

We now define $P:D_{\beta}\to\mathbb{C}^{2\times 2}$ that satisfies the
following Riemann-Hilbert problem

(3.11)
$\begin{cases}P_{+}(z)=P_{-}(z)\begin{bmatrix}1&e^{-\frac{\zeta(z)^{2}}{2}}\\\
0&1\end{bmatrix},\quad&z\in\Gamma\setminus U,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ P_{+}(z)=P_{-}(z)\begin{bmatrix}1&0\\\
e^{\frac{\zeta(z)^{2}}{2}}&1\end{bmatrix},\quad&z\in\partial U\cap{\rm
Ext}\Gamma,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
P_{+}(z)=P_{-}(z)\begin{bmatrix}1&0\\\
e^{-\frac{\zeta(z)^{2}}{2}}&1\end{bmatrix},\quad&z\in\partial U\cap{\rm
Int}\Gamma,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
P_{+}(z)=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}P_{-}(z)\begin{bmatrix}0&1\\\
-1&0\end{bmatrix},\quad&z\in\Gamma\cap U,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ P_{+}(z)=e^{-c\pi i\sigma_{3}}P_{-}(z)e^{c\pi
i\sigma_{3}},\quad&z\in\mathbb{R},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
P(z)\quad\mbox{is holomorphic},&\mbox{otherwise}.\\\ \end{cases}$

and the boundary condition, $P(z)\sim I$ on $\partial D_{\beta}$. Using the
Riemann-Hilbert problem (3.11) for $P$, one can notice that the matrix
function

(3.12)
$\Phi(z)\Big{(}\frac{z-a}{z}\Big{)}^{\frac{c}{2}\sigma_{3}}P(z)\Big{(}\frac{z-a}{z}\Big{)}^{-\frac{c}{2}\sigma_{3}}$

satisfies the jump conditions of $Z$ in (3.7).

Finally, let us define $W$ by

(3.13) $W(z):=\zeta(z)^{-c\sigma_{3}}SP(z)T(\zeta(z))^{-1}S^{-1},$

where $T$ is a diagonal matrix function

(3.14)
$T(\zeta)=\begin{cases}\exp\big{(}\frac{\zeta^{2}}{4}\sigma_{3}\big{)},\quad&|{\arg}\zeta|<3\pi/4,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\exp\big{(}-\frac{\zeta^{2}}{4}\sigma_{3}\big{)},\quad&\mbox{otherwise},\end{cases}$

and $S$ is a piecewise constant matrix

(3.15) $S=\begin{cases}I,&{\rm Im}\zeta<0\cap|{\arg}\zeta|<3\pi/4,\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ e^{c\pi i\sigma_{3}}\quad&{\rm Im}\zeta>0\cap|{\rm
arg}\zeta|<3\pi/4,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \begin{bmatrix}0&1\\\
-1&0\end{bmatrix},\quad&{\rm Im}\zeta<0\cap|{\rm arg}\zeta|\geq 3\pi/4,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ e^{c\pi i\sigma_{3}}\begin{bmatrix}0&1\\\
-1&0\end{bmatrix},\quad&{\rm Im}\zeta>0\cap|{\rm arg}\zeta|\geq
3\pi/4.\end{cases}$

Then $W$ satisfies the following jump conditions,

(3.16) $\begin{cases}W_{+}(z)=W_{-}(z)\begin{bmatrix}1&1-e^{2c\pi i}\\\
0&1\end{bmatrix},\quad&\zeta(z)\in\mathbb{R}^{+},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ W_{+}(z)=W_{-}(z)\begin{bmatrix}1&0\\\ e^{-2c\pi
i}&1\end{bmatrix},\quad&\zeta(z)\in i\mathbb{R}^{+},\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ W_{+}(z)=W_{-}(z)\begin{bmatrix}e^{2c\pi i}&e^{2c\pi i}-1\\\
0&e^{-2c\pi i}\end{bmatrix},\quad&\zeta(z)\in\mathbb{R}^{-},\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ W_{+}(z)=W_{-}(z)\begin{bmatrix}1&0\\\
-1&1\end{bmatrix},\quad&\zeta(z)\in i\mathbb{R}^{-}.\end{cases}$

###  3.2. Asymptotic behaviours of orthogonal polynomials

In this subsection, we prove Proposition 2.3.

###### Proof of Proposition 2.3.

Recall that the parabolic cylinder function $D_{-c}$ is given by

(3.17)
$D_{-c}(\zeta):=\frac{e^{\frac{\zeta^{2}}{4}}}{i\sqrt{2\pi}}\int_{\varepsilon-i\infty}^{\varepsilon+i\infty}e^{-\zeta
s+\frac{s^{2}}{2}}s^{-c}\,ds,\qquad\varepsilon>0,$

see e.g. [43, Chapter 12]. Using this, we define
$\mathcal{W}:\mathbb{C}\setminus(\mathbb{R}\cup
i\mathbb{R})\to\mathbb{C}^{2\times 2}$ by

(3.18)
$\mathcal{W}(\zeta):=\begin{cases}\begin{bmatrix}D_{-c}(\zeta)&\frac{i\sqrt{2\pi}e^{\frac{c\pi
i}{2}}}{\Gamma(c)}D_{-1+c}(i\zeta)\\\ -\frac{\Gamma(c+1)}{\sqrt{2\pi}e^{c\pi
i}}D_{-1-c}(\zeta)&e^{-\frac{c\pi
i}{2}}D_{c}(i\zeta)\end{bmatrix},&-\frac{\pi}{2}<\arg(\zeta)<0,\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\
\begin{bmatrix}D_{-c}(\zeta)&-\frac{i\sqrt{2\pi}e^{\frac{3c\pi
i}{2}}}{\Gamma(c)}D_{-1+c}(-i\zeta)\\\ -\frac{\Gamma(c+1)}{\sqrt{2\pi}e^{c\pi
i}}D_{-1-c}(\zeta)&e^{\frac{c\pi
i}{2}}D_{c}(-i\zeta)\end{bmatrix},&0<\arg(\zeta)<\frac{\pi}{2},\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ \begin{bmatrix}e^{-c\pi
i}D_{-c}(-\zeta)&-\frac{i\sqrt{2\pi}e^{\frac{3c\pi
i}{2}}}{\Gamma(c)}D_{-1+c}(-i\zeta)\\\ \frac{\Gamma(c+1)}{\sqrt{2\pi}e^{2c\pi
i}}D_{-1-c}(-\zeta)&e^{\frac{c\pi
i}{2}}D_{c}(-i\zeta)\end{bmatrix},&\frac{\pi}{2}<\arg(\zeta)<\pi,\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ \begin{bmatrix}e^{c\pi
i}D_{-c}(-\zeta)&\frac{i\sqrt{2\pi}e^{\frac{c\pi
i}{2}}}{\Gamma(c)}D_{-1+c}(i\zeta)\\\
\frac{\Gamma(c+1)}{\sqrt{2\pi}}D_{-1-c}(-\zeta)&e^{-\frac{c\pi
i}{2}}D_{c}(i\zeta)\end{bmatrix},&\pi<\arg(\zeta)<\frac{3\pi}{2}.\end{cases}$

This function is used to define

(3.19) $W(\zeta)=H(z){\mathcal{W}}(\zeta(z)),$

where $H(z)$ is a unimodular holomorphic matrix function on $D_{\beta}$ that
will be determined later.

By [38, Lemma 7], the function ${\mathcal{W}}(\zeta(z))$ satisfies the jump
conditions of $W$ in (3.16), and the asymptotic behaviour

(3.20)
$F(\zeta(z)):={\mathcal{W}}(\zeta)\zeta^{c\sigma_{3}}e^{\frac{\zeta^{2}}{4}\sigma_{3}}=I+\frac{C_{1}}{\zeta}+\frac{C_{2}}{\zeta^{2}}+O(\frac{1}{\zeta^{3}}),\qquad(|\zeta|\to\infty),$

where

(3.21) $C_{1}=\begin{bmatrix}0&\frac{\sqrt{2\pi}e^{c\pi i}}{\Gamma(c)}\\\
\frac{-\Gamma(c+1)}{\sqrt{2\pi}e^{c\pi i}}&0\end{bmatrix},\qquad
C_{2}=\begin{bmatrix}-\frac{c(c+1)}{2}&0\\\ 0&\frac{c(c-1)}{2}\end{bmatrix}.$

Moreover, by [38, Lemma 9], for any positive integer $L$, there exists a
positive integer $k$ such that $F(\zeta)$ can be decomposed into

(3.22) $F(\zeta)F_{1}(\zeta)^{-1}\dots
F_{k}(\zeta)^{-1}=I+O(\frac{1}{\zeta^{L}}).$

In particular, $F_{1}$ and $F_{2}$ are given by

(3.23)
$F_{1}(\zeta)=I+\frac{1}{\zeta}\begin{bmatrix}0&\frac{\sqrt{2\pi}e^{c\pi
i}}{\Gamma(c)}\\\ 0&0\end{bmatrix},\qquad
F_{2}(\zeta)=I+\begin{bmatrix}-\frac{c(c+1)}{2\zeta^{2}}&\frac{\sqrt{2\pi}e^{c\pi
i}c^{2}(c+1)^{2}}{4\Gamma(c+1)\zeta^{3}}\\\
-\frac{\Gamma(c+1)}{\sqrt{2\pi}e^{c\pi
i}\zeta}&\frac{c(c+1)}{2\zeta^{2}}\end{bmatrix}.$

Given $\\{F_{j}\\}_{j=1}^{k}$, the sequences $\\{H_{j}\\}$ and $\\{R_{j}\\}$
can be obtained inductively. Assume that $H_{j-1}$ is unimodular holomorphic
and nonvanishing at $\beta$. When $j=1$, we choose $H_{0}(z)=I$. We define

(3.24)
$\widetilde{F}_{j}(z):=\Big{(}\frac{z-a}{z}\Big{)}^{\frac{c}{2}\sigma_{3}}\Big{(}\frac{z\zeta(z)}{z-\beta}\Big{)}^{c\sigma_{3}}H_{j-1}(z)F_{j}(\zeta(z))H_{j-1}(z)^{-1}\Big{(}\frac{z-a}{z}\Big{)}^{-\frac{c}{2}\sigma_{3}}\Big{(}\frac{z\zeta(z)}{z-\beta}\Big{)}^{-c\sigma_{3}}.$

Given $\widetilde{F}_{j}$ as above, by [38, Lemma 10], the unique rational
matrix function $R_{j}$ can be constructed explicitly such that its only
singularity is at $\beta$, $R_{j}(\infty)=I$, and
$R_{j}(z)\widetilde{F}_{j}(z)^{-1}$ is holomorphic at $\beta$.

We define $R_{1}$, a unimodular meromorphic matrix function with a simple pole
at $\beta$, by

(3.25)
$R_{1}(z)=I+\frac{\sqrt{2\pi}(a^{2}-1)^{c}}{N^{1/2-c}a\Gamma(c)(z-\beta)}\begin{bmatrix}0&1\\\
0&0\end{bmatrix}.$

Using $R_{1}$ and $F_{1}$ in (3.23), set

(3.26)
$H_{1}(z):=\Big{(}\frac{z-a}{z}\Big{)}^{-\frac{c}{2}\sigma_{3}}\Big{(}\frac{z\zeta(z)}{z-\beta}\Big{)}^{-c\sigma_{3}}R_{1}(z)\Big{(}\frac{z-a}{z}\Big{)}^{\frac{c}{2}\sigma_{3}}\Big{(}\frac{z\zeta(z)}{z-\beta}\Big{)}^{c\sigma_{3}}F_{1}(\zeta(z))^{-1}.$

Then $H_{1}$ is unimodular and holomorphic at $\beta$.

Next, let us write

(3.27)
$R_{2}(z)=I+\begin{bmatrix}\frac{c_{11}}{z-\beta}+\frac{c_{12}}{(z-\beta)^{2}}&\frac{c_{21}}{z-\beta}+\frac{c_{22}}{(z-\beta)^{2}}+\frac{c_{23}}{(z-\beta)^{3}}\\\
\frac{c_{31}}{z-\beta}&\frac{c_{41}}{z-\beta}+\frac{c_{42}}{(z-\beta)^{2}}\end{bmatrix},$

where $c_{jk}$’s are some constants. Using $H_{1}(z)$ in (3.26),
$\widetilde{F}_{j}$ in (3.24) with $j=2$ and the condition that
$R_{2}(z)\widetilde{F}_{2}(z)^{-1}$ is holomorphic at $\beta$, we have

(3.28)
$R_{2}(z)=N^{\frac{c}{2}\sigma_{3}}\Big{(}I+\begin{bmatrix}\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})&O(\frac{1}{N})\\\
O(\frac{1}{\sqrt{N}})&O(\frac{1}{\sqrt{N}})\end{bmatrix}\Big{)}N^{-\frac{c}{2}\sigma_{3}}.$

Moreover, by (3.22), we have $F_{k}(\zeta)=I+O(\zeta^{-3})$ for $k\geq 3$.
Then by [38, Corollary 1], when $z\in\partial D_{\beta}$ we have

$R_{k}(z)\dots R_{3}(z)=N^{\frac{c}{2}\sigma_{3}}(I+O(N^{-3/2})).$

Combining the above equation with $R_{1}$ in (3.25) and $R_{2}$ in (3.27), for
$z\in D_{\beta}$, we have

(3.29) $\displaystyle R(z)$ $\displaystyle=R_{k}(z)\dots R_{1}(z)$
$\displaystyle=N^{\frac{c}{2}\sigma_{3}}\Big{(}I+\begin{bmatrix}\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})&\frac{\sqrt{2\pi}(a^{2}-1)^{c}}{\sqrt{N}a\Gamma(c)}\frac{1}{z-\beta}+O(\frac{1}{N})\\\
O(\frac{1}{\sqrt{N}})&O(\frac{1}{\sqrt{N}})\end{bmatrix}\Big{)}N^{-\frac{c}{2}\sigma_{3}}.$

Note in particular that

(3.30)
$[R(z)]_{11}=1+\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}}).$

We define $Z^{\infty}(z)$ by

(3.31) $Z^{\infty}(z):=\begin{cases}R(z)\Phi(z),&z\notin D_{\beta},\\\
\Phi(z)\Big{(}\frac{z-a}{z}\Big{)}^{\frac{c}{2}\sigma_{3}}P(z)\Big{(}\frac{z-a}{z}\Big{)}^{-\frac{c}{2}\sigma_{3}},&z\in
D_{\beta}.\end{cases}$

By the proof of [38, Theorem 2], we have

(3.32)
$Z(z)=\Big{(}I+O\big{(}\frac{1}{N^{\infty}}\big{)}\Big{)}Z^{\infty}(z),$

where the error bound $O(\frac{1}{N^{\infty}})$ means $O(\frac{1}{N^{k}})$ for
arbitrary integer $k$. Note that the error bound is uniform over any compact
subset of the corresponding region.

Using (3.6), for $z$ outside $\mathcal{S}_{a}$, we have

(3.33)
$\displaystyle\begin{split}Y(z)&=e^{\frac{Nl}{2}\sigma_{3}}Z(z)e^{-\frac{Nl}{2}\sigma_{3}}e^{Ng(z)\sigma_{3}}=e^{\frac{Nl}{2}\sigma_{3}}\Big{(}I+O(\frac{1}{N^{\infty}})\Big{)}\,R(z)\Phi(z)e^{-\frac{Nl}{2}\sigma_{3}}z^{N\sigma_{3}},\end{split}$

where the second equality follows from (3.32) and (3.31). Here $l$ is given by
(3.5). Then by (3.30), we obtain

(3.34) $\displaystyle P_{N}(z)$
$\displaystyle=[Y(z)]_{11}=z^{N}\Big{(}\frac{z}{z-\beta}\Big{)}^{c}[R(z)]_{11}\cdot\Big{(}1+O(\frac{1}{N^{\infty}})\Big{)}$
$\displaystyle=z^{N}\Big{(}\frac{z}{z-\beta}\Big{)}^{c}\Big{(}1+\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})\Big{)}\cdot\Big{(}1+O(\frac{1}{N^{\infty}})\Big{)},$

which leads to (2.15). For (2.14) and (2.16), we shall use the relation

(3.35)
$P_{n,N}(z;a)=\Big{(}\frac{n}{N}\Big{)}^{\frac{n}{2}}P_{n,n}\Big{(}\sqrt{\frac{N}{n}}z,\sqrt{\frac{N}{n}}a\Big{)}.$

Using (3.35), we have

$\displaystyle\quad
P_{N-1}(z)=\Big{(}\frac{N-1}{N}\Big{)}^{\frac{N-1}{2}}P_{N-1,N-1}\Big{(}\sqrt{\frac{N}{N-1}}z,\sqrt{\frac{N}{N-1}}a\Big{)}$
$\displaystyle=z^{N-1}\Big{(}\frac{\sqrt{\frac{N}{N-1}}z}{\sqrt{\frac{N}{N-1}}z-\sqrt{\frac{N-1}{N}}\beta}\Big{)}^{c}\Big{(}1+O(\frac{1}{N^{\infty}})\Big{)}$
$\displaystyle\quad\times\Big{(}1+\frac{c^{2}\sqrt{\frac{N-1}{N}}\beta/(\sqrt{\frac{N}{N-1}}z-\sqrt{\frac{N-1}{N}}\beta)}{N\sqrt{\frac{N}{N-1}}a(\sqrt{\frac{N-1}{N}}\beta-\sqrt{\frac{N}{N-1}}a)}-\frac{c(c+1)\sqrt{\frac{N-1}{N}}\beta}{2N\sqrt{\frac{N}{N-1}}a}\frac{1}{(\sqrt{\frac{N}{N-1}}z-\sqrt{\frac{N-1}{N}}\beta)^{2}}+O(\frac{1}{N^{2}})\Big{)}.$

This gives

(3.36) $\displaystyle P_{N-1}(z)$
$\displaystyle=z^{N-1}\Big{(}\frac{z}{z-\beta}\Big{)}^{c}\Big{(}1-\frac{c}{Na(z-\beta)}+O(\frac{1}{N^{2}})\Big{)}$
$\displaystyle\quad\times\Big{(}1+\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})\Big{)}\big{(}1+O(\frac{1}{N^{\infty}})\big{)}$
$\displaystyle=z^{N-1}\Big{(}\frac{z}{z-\beta}\Big{)}^{c}\Big{(}1+\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}-\frac{c}{Na(z-\beta)}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})\Big{)},$

which leads to (2.14). Similarly, we obtain

(3.37) $\displaystyle P_{N+1}(z)$
$\displaystyle=\big{(}\frac{N+1}{N}\big{)}^{\frac{N+1}{2}}P_{N+1,N+1}\big{(}\sqrt{\frac{N}{N+1}}z,\sqrt{\frac{N}{N+1}}a\big{)}$
$\displaystyle=z^{N+1}\Big{(}\frac{z}{z-\beta}\Big{)}^{c}\Big{(}1+\frac{c^{2}\beta}{Na(\beta-a)}\frac{1}{z-\beta}+\frac{c}{Na(z-\beta)}-\frac{c(c+1)\beta}{2Na}\frac{1}{(z-\beta)^{2}}+O(\frac{1}{N^{2}})\Big{)},$

which gives (2.16). This completes the proof. ∎

###  3.3. Asymptotic behaviours of orthogonal norms

In this subsection, we prove Lemma 2.4.

###### Proof of Lemma 2.4.

By [12, Proposition 7.1], we have

(3.38)
$h_{n}=-\frac{1}{\pi}\frac{\Gamma(c+n+1)}{2iN^{c+n+1}}\frac{\widetilde{h}_{n}}{P_{n+1,N}(0)},\qquad\widetilde{h}_{n}\equiv\widetilde{h}_{n,N}(a):=\int_{\Gamma}P_{n,N}(z)^{2}w_{n,N}(z)\,dz.$

Here $w_{n,N}$ is given by (3.1). Using (3.35), we also have

(3.39)
$\widetilde{h}_{n,N}(a)=\Big{(}\frac{n}{N}\Big{)}^{\frac{n+1}{2}}\widetilde{h}_{n,n}\Big{(}\sqrt{\tfrac{N}{n}}a\Big{)}.$

By [38, Theorem 2], for $z\in\operatorname{Int}\mathcal{S}_{a}\setminus U$, we
have

(3.40)
$P_{N}(z)=-\frac{\beta^{N}\sqrt{2\pi}(a^{2}-1)^{c}e^{Na(z-\beta)}}{N^{\frac{1}{2}-c}a\Gamma(c)(z-\beta)}\Big{(}\frac{z-\beta}{z-a}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Recall here that $\beta=1/a$. Combining (3.35) and (3.40), we have

(3.41) $\displaystyle P_{N+1}(z)$
$\displaystyle=\Big{(}\frac{N+1}{N}\Big{)}^{\frac{N+1}{2}}P_{N+1,N+1}\Big{(}\sqrt{\tfrac{N}{N+1}}z,\sqrt{\tfrac{N}{N+1}}a\Big{)}$
$\displaystyle=-\frac{(\frac{N+1}{N}\beta)^{N+1}\sqrt{2\pi}(\frac{N}{N+1}a^{2}-1)^{c}e^{Na(z-\frac{N+1}{N}\beta)}}{(N+1)^{\frac{1}{2}-c}\frac{N}{N+1}a\Gamma(c)(z-\frac{N+1}{N}\beta)}\Big{(}\frac{z-\frac{N+1}{N}\beta}{z-a}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}$

and

(3.42) $\displaystyle P_{N+2}(z)$
$\displaystyle=\Big{(}\frac{N+2}{N}\Big{)}^{\frac{N+2}{2}}P_{N+2,N+2}\Big{(}\sqrt{\tfrac{N}{N+2}}z,\sqrt{\tfrac{N}{N+2}}a\Big{)}$
$\displaystyle=-\frac{(\frac{N+2}{N}\beta)^{N+2}\sqrt{2\pi}(\frac{N}{N+2}a^{2}-1)^{c}e^{Na(z-\frac{N+2}{N}\beta)}}{(N+2)^{\frac{1}{2}-c}\frac{N}{N+2}a\Gamma(c)(z-\frac{N+2}{N}\beta)}\Big{(}\frac{z-\frac{N+2}{N}\beta}{z-a}\Big{)}^{c}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Using (3.33), we have

(3.43) $\displaystyle\widetilde{h}_{N}$ $\displaystyle=-2\pi
i\lim_{z\to\infty}z^{N+1}[Y(z)]_{12}=-2\pi
i\lim_{z\to\infty}z^{N+1}[R(z)]_{12}\Big{(}\frac{z-\beta}{z}\Big{)}^{c}\frac{\beta^{N}}{z^{N}e^{Na\beta}}\cdot\Big{(}1+O(\frac{1}{N^{\infty}})\Big{)}$
$\displaystyle=-2\pi
i\lim_{z\to\infty}z^{N+1}\Big{(}\frac{\sqrt{2\pi}(a^{2}-1)^{c}}{N^{\frac{1}{2}-c}a\Gamma(c)}\frac{1}{z-\beta}+O(\frac{1}{N})\Big{)}\Big{(}\frac{z-\beta}{z}\Big{)}^{c}\frac{\beta^{N}}{z^{N}e^{Na\beta}}\cdot\Big{(}1+O(\frac{1}{N^{\infty}})\Big{)}$
$\displaystyle=-2\pi
i\frac{\beta^{N}}{e^{Na\beta}}\cdot\Big{(}\frac{\sqrt{2\pi}(a^{2}-1)^{c}}{N^{\frac{1}{2}-c}a\Gamma(c)}+O(\frac{1}{N})\Big{)}.$

Combining (3.39) and (3.43), we have

(3.44) $\displaystyle\widetilde{h}_{N-1}$
$\displaystyle=\Big{(}\frac{N-1}{N}\Big{)}^{\frac{N}{2}}\widetilde{h}_{N-1,N-1}\Big{(}\sqrt{\tfrac{N}{N-1}}a\Big{)}$
$\displaystyle=-2\pi
i\frac{\beta^{N-1}\big{(}\frac{N-1}{N}\big{)}^{N-\frac{1}{2}}}{e^{(N-1)a\beta}}\cdot\Big{(}\frac{\sqrt{2\pi}(\frac{N}{N-1}a^{2}-1)^{c}}{(N-1)^{\frac{1}{2}-c}\sqrt{\frac{N}{N-1}}a\Gamma(c)}+O(\frac{1}{N})\Big{)}$

and

(3.45) $\displaystyle\widetilde{h}_{N+1}$
$\displaystyle=\Big{(}\frac{N+1}{N}\Big{)}^{\frac{N+2}{2}}\widetilde{h}_{N+1,N+1}\Big{(}\sqrt{\tfrac{N}{N+1}}a\Big{)}$
$\displaystyle=-2\pi
i\frac{\beta^{N+1}\big{(}\frac{N+1}{N}\big{)}^{N+1-\frac{1}{2}}}{e^{(N+1)a\beta}}\cdot\Big{(}\frac{\sqrt{2\pi}(\frac{N}{N+1}a^{2}-1)^{c}}{(N+1)^{\frac{1}{2}-c}\sqrt{\frac{N}{N+1}}a\Gamma(c)}+O(\frac{1}{N})\Big{)}.$

Substituting (3.43) and (3.41) with $z=0$ into (3.38), we obtain

(3.46)
$h_{N}=\frac{\Gamma(N+c+1)}{N^{N+c+1}}\Big{(}\frac{a^{2}-1}{1-\frac{N+1}{Na^{2}}}\Big{)}^{c}\Big{(}\frac{N+1}{N}\Big{)}^{\frac{1}{2}-c}\frac{e}{(\frac{N+1}{N})^{N+1}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Similarly, it follows from (3.44), (3.40) and (3.45), (3.42) that

(3.47) $\displaystyle h_{N-1}$
$\displaystyle=\frac{\Gamma(N+c)}{N^{N+c}}\Big{(}\frac{\frac{N}{N-1}a^{2}-1}{1-\frac{1}{a^{2}}}\Big{)}^{c}\Big{(}\frac{N}{N-1}\Big{)}^{\frac{1}{2}-c}\frac{e}{(\frac{N}{N-1})^{N}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)},$
(3.48) $\displaystyle h_{N+1}$
$\displaystyle=\frac{\Gamma(N+c+2)}{N^{N+c+2}}\Big{(}\frac{\frac{N}{N+1}a^{2}-1}{1-\frac{N+2}{Na^{2}}}\Big{)}^{c}\Big{(}\frac{N+2}{N+1}\Big{)}^{\frac{1}{2}-c}\frac{e}{(\frac{N+2}{N+1})^{N+2}}\cdot\Big{(}1+O(\frac{1}{\sqrt{N}})\Big{)}.$

Now (2.19), (2.20) and (2.21) follow from straightforward computations using
Stirling’s formula. ∎

##  Appendix A Asymptotic analysis for the exactly solvable case $c=1$

As a concrete example, we study the case $c=1$ in this appendix. For this
special case, Proposition 2.3 and Lemma 2.4 can be achieved using asymptotic
behaviours of some well-known special functions instead of using the Riemann-
Hilbert analysis. Thus for the readers who are not familiar with Riemann-
Hilbert analysis, we provide direct proofs for this exactly solvable case.

We also remark that indeed, the value $c=1$ also reveals a phase transition in
a sense that as the degree of the orthogonal polynomials increases, their
zeros approach $\mathcal{S}_{a}$ in (1.13) from $\operatorname{Ext}\SS_{a}$
for $c>1$, and from $\operatorname{Int}\SS_{a}$ for $c<1$, see [38, p.308].

For $c=1$, we have

(A.1)
$P_{k}(z)=\frac{1}{z-a}\Big{(}z^{k+1}-e^{aN(z-a)}\frac{Q(k+1,Na\,z)}{Q(k+1,Na^{2})}a^{k+1}\Big{)}$

and

(A.2) $h_{k}=\frac{(k+1)!}{N^{k+2}}\frac{Q(k+2,Na^{2})}{Q(k+1,Na^{2})},$

see [24, Subsection 3.2] and [4, Section 3]. Here

(A.3)
$Q(a,z):=\frac{\Gamma(a,z)}{\Gamma(a)}=\frac{1}{\Gamma(a)}\int_{z}^{\infty}t^{a-1}e^{-t}\,dt$

is the regularised incomplete gamma function.

Using this explicit representation, we show Proposition 2.3 and Lemma 2.4 for
$c=1$.

###### Proof of Proposition 2.3 for $c=1$.

By (A.1), we have

(A.4)
$\psi_{k}(z)=z^{k+1}-e^{aN(z-a)}\frac{Q(k+1,Na\,z)}{Q(k+1,Na^{2})}a^{k+1},$

which gives

(A.5)
$\begin{split}\psi_{N}(z)-z\psi_{N-1}(z)=e^{aN(z-a)}a^{N}\Big{(}z\,\frac{Q(N,Na\,z)}{Q(N,Na^{2})}-a\,\frac{Q(N+1,Na\,z)}{Q(N+1,Na^{2})}\Big{)}.\end{split}$

We first recall the asymptotic behaviours of $Q$. It follows from [7, Theorem
1.1] that

(A.6)
$\begin{split}Q(N,Nz)&=e^{-Nz}\frac{N^{N}}{N!}\frac{z^{N}}{z-1}\Big{[}1-\frac{z}{(1-z)^{2}}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\\\
&=\frac{1}{\sqrt{2\pi
N}}\,e^{N-Nz}\frac{z^{N}}{z-1}\Big{[}1-\Big{(}\frac{1}{12}+\frac{z}{(1-z)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\end{split}$

for $z$ outside $\mathcal{S}_{1}$. Note that if $z$ is outside
$\mathcal{S}_{a}$, then $az$ is outside $\mathcal{S}_{1}$. Then we have

(A.7) $\displaystyle Q(N,Naz)$ $\displaystyle=\frac{1}{\sqrt{2\pi
N}}\,e^{N-Naz}\frac{(az)^{N}}{az-1}\Big{[}1-\Big{(}\frac{1}{12}+\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.$

This gives that

(A.8)
$\frac{Q(N,Naz)}{Q(N,Na^{2})}=e^{-aN(z-a)}\Big{(}\frac{z}{a}\Big{)}^{N}\frac{a^{2}-1}{az-1}\Big{[}1+\Big{(}\frac{a^{2}}{(1-a^{2})^{2}}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.$

Similarly, we have

(A.9)
$\begin{split}Q(N+1,Naz)=\frac{e^{N-Naz}}{\sqrt{2\pi(N+1)}}\,\frac{(az)^{N+1}}{az-1}\Big{[}1+\Big{(}\frac{5}{12}-\frac{1}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.\end{split}$

This gives

(A.10)
$\begin{split}\frac{Q(N+1,Naz)}{Q(N+1,Na^{2})}&=e^{-aN(z-a)}\Big{(}\frac{z}{a}\Big{)}^{N+1}\frac{a^{2}-1}{az-1}\Big{[}1+\Big{(}\frac{1}{(1-a^{2})^{2}}-\frac{1}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.\end{split}$

We also have

(A.11)
$\begin{split}Q(N+2,Naz)=\frac{e^{N-Naz}}{\sqrt{2\pi(N+2)}}\,\frac{(az)^{N+2}}{az-1}\Big{[}1-\Big{(}\frac{2}{1-az}+\frac{1}{12}+\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]},\end{split}$

which leads to

(A.12)
$\frac{Q(N+2,Naz)}{Q(N+2,Na^{2})}=e^{aN(a-z)}\Big{(}\frac{z}{a}\Big{)}^{N+2}\frac{a^{2}-1}{az-1}\Big{[}1+\Big{(}\frac{2}{1-a^{2}}+\frac{a^{2}}{(1-a^{2})^{2}}-\frac{2}{1-az}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.$

Therefore by (A.4), we have

(A.13)
$\begin{split}\psi_{N-1}(z)&=z^{N}-z^{N}\frac{1-a^{2}}{1-az}\Big{[}1+\Big{(}\frac{a^{2}}{(1-a^{2})^{2}}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\\\
&=z^{N}\Big{[}\frac{z-a}{z-\frac{1}{a}}-\frac{1-a^{2}}{1-az}\Big{(}\frac{a^{2}}{(1-a^{2})^{2}}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.\end{split}$

Similarly, we have

(A.14)
$\begin{split}\psi_{N}(z)&=z^{N+1}-z^{N+1}\frac{1-a^{2}}{1-az}\Big{[}1+\Big{(}\frac{1}{(1-a^{2})^{2}}-\frac{1}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\\\
&=z^{N+1}\Big{[}\frac{z-a}{z-\frac{1}{a}}-\frac{1-a^{2}}{1-az}\Big{(}\frac{1}{(1-a^{2})^{2}}-\frac{1}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\end{split}$

and

(A.15)
$\begin{split}\psi_{N+1}(z)&=z^{N+2}-z^{N+2}\frac{1-a^{2}}{1-az}\Big{[}1+\Big{(}\frac{2}{1-a^{2}}+\frac{a^{2}}{(1-a^{2})^{2}}-\frac{2}{1-az}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}\\\
&=z^{N+2}\Big{[}\frac{z-a}{z-\frac{1}{a}}-\frac{1-a^{2}}{1-az}\Big{(}\frac{2}{1-a^{2}}+\frac{a^{2}}{(1-a^{2})^{2}}-\frac{2}{1-az}-\frac{az}{(1-az)^{2}}\Big{)}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{]}.\end{split}$

Now the proof is complete. ∎

###### Proof of Lemma 2.4 for $c=1$.

By (A.7), (A.9) and (A.11), we have

(A.16) $\displaystyle\frac{Q(N+1,Na^{2})}{Q(N,Na^{2})}$
$\displaystyle=a^{2}\Big{(}1+\frac{1}{a^{2}-1}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)},$
(A.17) $\displaystyle\frac{Q(N+2,Na^{2})}{Q(N+1,Na^{2})}$
$\displaystyle=a^{2}\Big{(}1+\frac{2-a^{2}}{a^{2}-1}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)}.$

Similarly, we have

(A.18)
$\frac{Q(N+3,Na^{2})}{Q(N+2,Na^{2})}=a^{2}\Big{(}1+\frac{3-2a^{2}}{a^{2}-1}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)}.$

By (A.2) and Stirling’s formula, we obtain

(A.19) $\displaystyle h_{N-1}$
$\displaystyle=\frac{N!}{N^{N-1}}\frac{Q(N+1,Na^{2})}{Q(N,Na^{2})}=e^{-N}\sqrt{\frac{2\pi}{N}}\,\frac{Q(N+1,Na^{2})}{Q(N,Na^{2})}\cdot\Big{(}1+\frac{1}{12}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)},$
(A.20) $\displaystyle h_{N}$
$\displaystyle=\frac{(N+1)!}{N^{N+2}}\frac{Q(N+2,Na^{2})}{Q(N+1,Na^{2})}=e^{-N}\sqrt{\frac{2\pi}{N}}\,\frac{Q(N+2,Na^{2})}{Q(N+1,Na^{2})}\cdot\Big{(}1+\frac{13}{12}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)},$
(A.21) $\displaystyle h_{N+1}$
$\displaystyle=\frac{(N+2)!}{N^{N+3}}\frac{Q(N+3,Na^{2})}{Q(N+2,Na^{2})}=e^{-N}\sqrt{\frac{2\pi}{N}}\,\frac{Q(N+3,Na^{2})}{Q(N+2,Na^{2})}\cdot\Big{(}1+\frac{37}{12}\frac{1}{N}+O(\frac{1}{N^{2}})\Big{)}.$

This completes the proof. ∎

### Acknowledgements

This work was highly motivated by the recent work [7] of Yacin Ameur and
Joakim Cronvall, and we thank them for stimulating conversations. It is also
our pleasure to thank Christophe Charlier and Seung-Yeop Lee for helpful
discussions.

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