# Partition functions of determinantal and Pfaffian Coulomb gases with
radially 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>,  Nam-Gyu Kang  School of Mathematics, Korea Institute
for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>and  Seong-Mi Seo  Department of Mathematics, Chungnam
National University, 99 Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of
Korea<EMAIL_ADDRESS>
###### Abstract.

We consider random normal matrix and planar symplectic ensembles, which can be
interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian
structures, respectively. For general radially symmetric potentials, we derive
the asymptotic expansions of the log-partition functions up to and including
the $O(1)$-terms as the number $N$ of particles increases. Notably, our
findings stress that the formulas of the $O(\log N)$\- and $O(1)$-terms in
these expansions depend on the connectivity of the droplet. For random normal
matrix ensembles, our formulas agree with the predictions proposed by Zabrodin
and Wiegmann up to a universal additive constant. For planar symplectic
ensembles, the expansions contain a new kind of ingredient in the
$O(N)$-terms, the logarithmic potential evaluated at the origin in addition to
the entropy of the ensembles.

Sung-Soo Byun and Nam-Gyu Kang were partially supported by Samsung Science and
Technology Foundation (SSTF-BA1401-51) and by the National Research Foundation
of Korea (NRF-2019R1A5A1028324). Sung-Soo Byun was partially supported by a
KIAS Individual Grant (SP083201) via the Center for Mathematical Challenges at
Korea Institute for Advanced Study. Nam-Gyu Kang was partially supported by a
KIAS Individual Grant (MG058103) at Korea Institute for Advanced Study. Seong-
Mi Seo was partially supported by the National Research Foundation of Korea
(2019R1A5A1028324, NRF-2022R1I1A1A01072052).

##  1\. Introduction and main results

The Coulomb gas ensemble in the complex plane is governed by the law

(1.1)
$dP_{N}^{(\beta)}(z_{1},\dots,z_{N}):=\frac{1}{Z_{N}^{(\beta)}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{\beta}\prod_{j=1}^{N}e^{-\frac{\beta
N}{2}Q(z_{j})}\,dA(z_{j}),$

where $N$ is the number of particles, $\beta$ is the inverse temperature and
$dA(z):=d^{2}z/\pi$ is the area measure. Here, $Q:\mathbb{C}\to\mathbb{R}$ is
called the confining/external potential that satisfies suitable potential
theoretic conditions. We refer to [28, 44, 49] and references therein for
recent developments of two-dimensional Coulomb gases. Contrary to (1.1), the
configurational canonical Coulomb gas ensemble in the upper-half plane [39]
(cf. [14, Appendix A]) has an additional complex conjugation symmetry (i.e.
the particles come in complex conjugate pairs) and is governed by the law

(1.2)
$d\widetilde{P}_{N}^{(\beta)}(z_{1},\dots,z_{N}):=\frac{1}{\widetilde{Z}_{N}^{(\beta)}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{\beta}|z_{j}-\bar{z}_{k}|^{\beta}\prod_{j=1}^{N}|z_{j}-\bar{z}_{j}|^{\beta}\,e^{-\beta
NQ(z_{j})}\,dA(z_{j}).$

In (1.1) and (1.2), the normalization constants

(1.3) $\displaystyle Z_{N}^{(\beta)}$
$\displaystyle:=\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{\beta}\prod_{j=1}^{N}e^{-\frac{\beta
N}{2}Q(z_{j})}\,dA(z_{j}),$ (1.4) $\displaystyle\widetilde{Z}_{N}^{(\beta)}$
$\displaystyle:=\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{\beta}|z_{j}-\bar{z}_{k}|^{\beta}\prod_{j=1}^{N}|z_{j}-\bar{z}_{j}|^{\beta}\,e^{-\beta
NQ(z_{j})}\,dA(z_{j})$

that make (1.1) and (1.2) probability measures are called partition functions.
Furthermore, the logarithm of a partition function (divided by $N^{2}$) is
often called free energy.

For the special value $\beta=2$, (1.1) and (1.2) represent joint probability
distributions of the random normal matrix and planar symplectic ensembles,
respectively. In particular, if $Q(z)=|z|^{2}$, these correspond to the
complex and symplectic Ginibre ensembles [31]. An important feature of this
special value $\beta=2$ is that, due to the Vandermonde determinant terms, the
ensembles (1.1) and (1.2) form _determinantal_ and _Pfaffian_ point processes
in the plane [28], respectively. In other words, all their correlation
functions can be expressed in terms of the (pre-)kernel of planar
(skew-)orthogonal polynomials. In the sequel, for $\beta=2$, we omit the
superscript $(\beta)$ in (1.3) and (1.4), and simply write $Z_{N}\equiv
Z_{N}^{(2)}$ and $\widetilde{Z}_{N}\equiv\widetilde{Z}_{N}^{(2)}$.

We mention that the definition of partition functions (1.3) and (1.4) is more
common in the statistical physics community. On the other hand, in the random
matrix theory community, another widely used convention for the (canonical)
partition functions is

(1.5)
$\mathcal{Z}_{N}:=\frac{1}{N!}Z_{N},\qquad\widetilde{\mathcal{Z}}_{N}:=\frac{1}{N!}\widetilde{Z}_{N},$

see e.g. [28, Section 1.4]. The prefactor $1/N!$ in (1.5) allows writing
$\mathcal{Z}_{N}$ and $\widetilde{\mathcal{Z}}_{N}$ in terms of a structured
determinant and Pfaffian, respectively.

In this work, we study the asymptotic expansions of $Z_{N}$ and
$\widetilde{Z}_{N}$ as $N\to\infty$.

###  1.1. Summary of previous results

Before introducing our results, let us summarize some known results on the
asymptotics of $Z_{N}^{(\beta)}$ for general $\beta$ and $Q$. Cf. the
literature on $\widetilde{Z}_{N}^{(\beta)}$ is much more limited.

  * •

(Zabrodin-Wiegmann prediction) In [54], it was predicted that the partition
function $Z_{N}^{(\beta)}$ has an asymptotic expansion of the form

(1.6) $\log Z_{N}^{(\beta)}=C_{0}N^{2}+C_{1}N\log N+C_{2}N+C_{3}\log
N+C_{4}+O(\frac{1}{N}).$

Furthermore, they proposed explicit formulas for the constants $C_{j}\equiv
C_{j}(\beta,Q)$ $(j=0,\dots,4)$ depending on $\beta$ and $Q$, cf. (1.29). (See
also [15, 36] for a similar prediction in a different setup.) Incidentally,
the formulas for $C_{3}$ and $C_{4}$ in [54] have been controversial as
pointed out for instance in [47, 50].

  * •

(Asymptotic of the leading order $O(N^{2})$-term) It was shown in [33, Theorem
2.11] and [16, Theorem 1.1] (among others) that as $N\to\infty$,

(1.7) $\log Z_{N}^{(\beta)}=-\frac{\beta}{2}N^{2}I_{Q}[\mu_{Q}]+o(N^{2}).$

Here $\mu_{Q}$ is Frostman’s equilibrium measure [46], a unique probability
measure that minimizes the weighted logarithmic energy

(1.8)
$I_{Q}[\mu]:=\iint_{\mathbb{C}^{2}}\log|z-w|\,d\mu(z)\,d\mu(w)+\int_{\mathbb{C}}Q\,d\mu.$

  * •

(Asymptotic up to the $O(N)$-term) It was shown by Leblé and Serfaty [42,
Corollary 1.1] that as $N\to\infty$,

(1.9) $\log
Z_{N}^{(\beta)}=-\frac{\beta}{2}N^{2}I_{Q}[\mu_{Q}]+\frac{\beta}{4}N\log
N-\Big{(}C(\beta)+\Big{(}1-\frac{\beta}{4}\Big{)}E_{Q}[\mu_{Q}]\Big{)}N+o(N),$

where $C(\beta)$ is a constant independent of the potential $Q$ and

(1.10) $E_{Q}[\mu_{Q}]:=\int_{\mathbb{C}}\log(\Delta Q)\,d\mu_{Q}$

is the entropy associated with $\mu_{Q}$. Here,
$\Delta:=\partial\bar{\partial}$ is the quarter of the usual Laplacian. We
refer the reader to [10, 50] for the expansion (1.9) with quantitative error
bounds.

Beyond the general cases mentioned above, for $\beta=2$ with a specific (and
fundamental from the random matrix theory viewpoint) potential, there have
been several works on the precise asymptotic expansion of the partition
functions, see e.g. [19, 20] and references therein. This type of potential
usually contains certain singularities. As a result, the asymptotic expansions
of the associated partition functions are more complicated (for instance, some
non-trivial $O(\sqrt{N})$ terms appear as well). Several topics in this
direction will be discussed in a separate remark at the end of the next
subsection.

###  1.2. Main results

We study asymptotic behaviors of $Z_{N}$ and $\widetilde{Z}_{N}$ for the
exactly-solvable case where $Q$ is radially symmetric. Our main findings are
summarized as follows.

  1. (i)

We derive the large-$N$ expansions of $\log Z_{N}$ and $\log\widetilde{Z}_{N}$
up to and including the $O(1)$-terms.

  2. (ii)

In the large-$N$ expansions, the formulas of the $O(\log N)$\- and
$O(1)$-terms depend on whether the limiting spectrum is an _annulus_ or a
_disc_ , see Theorems 1.1 and 1.2, respectively, cf. Figure 1. (This
distinction is crucial in the asymptotic analysis but seems not considered in
[54].)

  3. (iii)

For the partition function $Z_{N}$ of random normal matrix ensembles, our
expansions (1.18) and (1.24) up to the $O(N)$-terms agree with the formula
(1.9) with $\beta=2$. Furthermore, we verify from (1.24) that the asymptotic
formula given in [54, Eqs.(1.2), (C.7)] holds up to an additive constant
(1.35).

  4. (iv)

For the partition function $\widetilde{Z}_{N}$ of planar symplectic ensembles,
the asymptotic formulas (1.19) and (1.25) are new to the best of our
knowledge. Contrary to (1.9), the $O(N)$-terms in these expansions contain not
only the entropy but also the logarithmic potential (1.15).

Figure 1. Eigenvalues of complex Ginibre (left) and complex induced Ginibre
(right) matrices where $N=1000$.

Let us be more precise in introducing our results. It is well known [11, 33]
that under some mild assumptions on $Q$, as $N\to\infty$, the empirical
measures $\frac{1}{N}\sum_{j=1}^{N}\delta_{z_{j}}$ of (1.1) and (1.2) weakly
converge to $\mu_{Q}$, which takes the form

(1.11) $d\mu_{Q}=\Delta Q\cdot\mathbbm{1}_{S}\,dA.$

Here $S\equiv S_{Q}$ is a certain compact subset of $\mathbb{C}$ called the
_droplet_ , see Figure 1.

We consider the case where the external potential $Q$ is radially symmetric,
i.e. $Q(z)=q(|z|)$ for some function $q$ defined in $[0,\infty)$. We assume
the basic growth condition

(1.12) $\liminf_{|z|\to\infty}\frac{Q(z)}{2\log|z|}>1,$

which guarantees that $Z_{N},\widetilde{Z}_{N}<+\infty.$ Furthermore, we
assume that $Q$ is smooth, subharmonic in $\mathbb{C}$, and strictly
subharmonic in a neighborhood of the droplet. In terms of the function $q,$
the latter conditions are equivalent to the requirements that $rq^{\prime}(r)$
is increasing on $(0,\infty)$, and strictly increasing in a neighborhood of
the droplet, cf. (2.3). Under the above assumptions, the droplet is given by

(1.13) $S=\mathbb{A}_{r_{0},r_{1}}:=\\{z\in\mathbb{C}:r_{0}\leq|z|\leq
r_{1}\\},$

where $r_{0}$ is the largest solution to $rq^{\prime}(r)=0$ and $r_{1}$ is the
smallest solution to $rq^{\prime}(r)=2$, see [46, Section IV.6]. (We mention
that the annular droplets often appear in non-Hermitian random matrix theory,
see e.g. [32].) In particular, if $r_{0}=0$, we denote
$\mathbb{D}_{r_{1}}=\mathbb{A}_{0,r_{1}}.$ Henceforth, we keep the assumptions
on $Q$ described above.

For a radially symmetric potential $Q$, by using (1.11) and (1.13), one can
show that the energy $I_{Q}[\mu_{Q}]$ in (1.8) is given by

(1.14) $I_{Q}[\mu_{Q}]=q(r_{1})-\log
r_{1}-\frac{1}{4}\int_{r_{0}}^{r_{1}}rq^{\prime}(r)^{2}\,dr.$

Similarly, in terms of the logarithmic potential

(1.15) $U_{\mu}(z)=\int\log\frac{1}{|z-w|}\,d\mu(w),$

we have

(1.16) $U_{\mu_{Q}}(0)=-\int_{S}\log|w|\,d\mu_{Q}(w)=-\log
r_{1}+\frac{q(r_{1})-q(r_{0})}{2}.$

See [46, Section IV.6] for more details.

For the annular droplet case, we have the following.

######  Theorem 1.1.

(Large-$N$ expansion of the partition functions: annular droplet case) Suppose
that $r_{0}>0$, i.e. the droplet $S$ in (1.13) is an annulus. Let

(1.17)
$F_{Q}[\mathbb{A}_{r_{0},r_{1}}]:=\frac{1}{12}\log\Big{(}\frac{r_{0}^{2}\Delta
Q(r_{0})}{r_{1}^{2}\Delta
Q(r_{1})}\Big{)}-\frac{1}{16}\Big{(}r_{1}\frac{({\partial}_{r}\Delta
Q)(r_{1})}{\Delta Q(r_{1})}-r_{0}\frac{({\partial}_{r}\Delta Q)(r_{0})}{\Delta
Q(r_{0})}\Big{)}+\frac{1}{24}\int_{r_{0}}^{r_{1}}\Big{(}\frac{{\partial}_{r}\Delta
Q(r)}{\Delta Q(r)}\Big{)}^{2}\,r\,dr.$

Then as $N\to\infty,$ the following holds.

  1. (i)

(Random normal matrix ensemble) We have

(1.18) $\begin{split}\log Z_{N}&=-N^{2}I_{Q}[\mu_{Q}]+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N\\\
&\quad+\frac{1}{2}\log
N+\frac{\log(2\pi)}{2}+F_{Q}[\mathbb{A}_{r_{0},r_{1}}]+O(N^{-1}).\end{split}$

  2. (ii)

(Planar symplectic ensemble) We have

(1.19)
$\begin{split}\log\widetilde{Z}_{N}&=-2N^{2}I_{Q}[\mu_{Q}]+\frac{1}{2}N\log
N+\Big{(}\frac{\log(4\pi)}{2}-1-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}N\\\
&\quad+\frac{1}{2}\log
N+\frac{\log(2\pi)}{2}+\frac{1}{2}F_{Q}[\mathbb{A}_{r_{0},r_{1}}]+\frac{1}{8}\log\Big{(}\frac{\Delta
Q(r_{0})}{\Delta Q(r_{1})}\Big{)}+O(N^{-1}).\end{split}$

Using the convention (1.5) together with (2.30), our result can also be
rewritten as

(1.20)
$\begin{split}\log\mathcal{Z}_{N}&=-N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N+F_{Q}[\mathbb{A}_{r_{0},r_{1}}]+O(N^{-1}).\end{split}$

and

(1.21)
$\begin{split}\log\widetilde{\mathcal{Z}}_{N}&=-2N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(4\pi)}{2}-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}N\\\
&\quad+\frac{1}{2}F_{Q}[\mathbb{A}_{r_{0},r_{1}}]+\frac{1}{8}\log\Big{(}\frac{\Delta
Q(r_{0})}{\Delta Q(r_{1})}\Big{)}+O(N^{-1}).\end{split}$

We mention that these formulas (1.20) and (1.21) as well as the formulas
(1.26) and (1.27) below are more convenient to compare with some asymptotic
results in the previous literature [19, 20].

Notice that the term $\log(r_{1}/r_{0})$ is the extremal length of the annulus
(1.13), see e.g. [29, p.142]. It is worth pointing out that a characteristic
difference between the expansions (1.18) and (1.19) is the appearance of the
logarithmic potential $U_{\mu_{Q}}(0)$ in the $O(N)$-term of (1.19). This
additional term can be interpreted as

(1.22) $-\int_{S}\log|w-\bar{w}|\,d\mu_{Q}(w)$

after a proper renormalization. The interpretation (1.22) is natural from the
perspective of the repulsion term $|z_{j}-\bar{z}_{j}|^{\beta}$ in (1.2) and
is closely related to the notion of the next-order energy, see e.g. [43]. (We
thank T. Leblé for pointing out this.)

In Subsection 4.1 we present an example of Theorem 1.1 for the Mittag-Leffler
ensembles from which we expect that the error terms $O(N^{-1})$ are optimal.

For the disc droplet case, we have the following.

######  Theorem 1.2.

(Large-$N$ expansion of the partition functions: disc droplet case) Suppose
that $r_{0}=0$, i.e. the droplet $S$ in (1.13) is a disc. Let

(1.23)
$\begin{split}F_{Q}[\mathbb{D}_{r_{1}}]:=\frac{1}{12}\log\Big{(}\frac{1}{r_{1}^{2}\Delta
Q(r_{1})}\Big{)}-\frac{1}{16}r_{1}\frac{({\partial}_{r}\Delta
Q)(r_{1})}{\Delta
Q(r_{1})}+\frac{1}{24}\int_{0}^{r_{1}}\Big{(}\frac{{\partial}_{r}\Delta
Q(r)}{\Delta Q(r)}\Big{)}^{2}\,r\,dr.\end{split}$

Then as $N\to\infty,$ the following holds.

  1. (i)

(Random normal matrix ensemble) We have

(1.24) $\displaystyle\begin{split}\log
Z_{N}&=-N^{2}I_{Q}[\mu_{Q}]+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N+\frac{5}{12}\log
N\\\
&\quad+\frac{\log(2\pi)}{2}+\zeta^{\prime}(-1)+F_{Q}[\mathbb{D}_{r_{1}}]+O(N^{-\frac{1}{12}}(\log
N)^{3}).\end{split}$

  2. (ii)

(Planar symplectic ensemble) We have

(1.25)
$\displaystyle\begin{split}\log\widetilde{Z}_{N}&=-2N^{2}I_{Q}[\mu_{Q}]+\frac{1}{2}N\log
N+\Big{(}\frac{\log(4\pi)}{2}-1-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}N+\frac{11}{24}\log
N\\\
&\quad+\frac{\log(2\pi)}{2}+\frac{1}{2}\zeta^{\prime}(-1)+\frac{1}{2}F_{Q}[\mathbb{D}_{r_{1}}]+\frac{5}{24}\log
2+\frac{1}{8}\log\Big{(}\frac{\Delta Q(0)}{\Delta
Q(r_{1})}\Big{)}+O(N^{-\frac{1}{12}}(\log N)^{3}).\end{split}$

Here $\zeta$ is the Riemann zeta function.

Again, using the convention (1.5), we have

(1.26)
$\displaystyle\begin{split}\log\mathcal{Z}_{N}&=-N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}\,N-\frac{1}{12}\log
N\\\
&\quad+\zeta^{\prime}(-1)+F_{Q}[\mathbb{D}_{r_{1}}]+O(N^{-\frac{1}{12}}(\log
N)^{3})\end{split}$

and

(1.27)
$\displaystyle\begin{split}\log\widetilde{\mathcal{Z}}_{N}&=-2N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+\Big{(}\frac{\log(4\pi)}{2}-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}N-\frac{1}{24}\log
N\\\
&\quad+\frac{1}{2}\zeta^{\prime}(-1)+\frac{1}{2}F_{Q}[\mathbb{D}_{r_{1}}]+\frac{5}{24}\log
2+\frac{1}{8}\log\Big{(}\frac{\Delta Q(0)}{\Delta
Q(r_{1})}\Big{)}+O(N^{-\frac{1}{12}}(\log N)^{3}).\end{split}$

In Subsection 4.2, we provide an example of Theorem 1.2 for truncated unitary
ensembles. Contrary to Theorem 1.1, the error terms in Theorem 1.2 do not
coincide with the expected optimal orders $O(N^{-1})$. Our error bounds
originate from a decomposition of the analytic expressions of
$Z_{N},\widetilde{Z}_{N}$ (see Subsection 1.3), which depends on sufficiently
large but seemingly arbitrary number $m_{N}>0.$ (Such a decomposition was not
necessary for the proof of Theorem 1.1.) Later, we choose $m_{N}=N^{1/6}$ that
gives rise to the control of the total error bounds presented in Theorem 1.2.
We mention that such error estimates also naturally appeared in similar
computations, see e.g. [13, 19, 20]. Nevertheless, we expect that the
estimates can be improved with more effort.

In terms of the function $\chi:=\frac{1}{2}\log\Delta Q$, one can rewrite
(1.23) as

(1.28)
$F_{Q}[\mathbb{D}_{r_{1}}]=\frac{1}{12}\log\Big{(}\frac{1}{r_{1}^{2}}\Big{)}-\frac{1}{12\pi}\oint_{\partial
S}\kappa\,\chi\,ds-\frac{1}{4}\int_{S}\Delta\chi\,dA+\frac{1}{12}\int_{S}|\nabla\chi|^{2}\,dA,$

where $S=\mathbb{D}_{r_{1}}$ and $\kappa=1/r_{1}$ is the curvature of the
boundary, see [54, p.8960] and (1.34). Here, the third term
$\int_{S}\Delta\chi\,dA$ on the right-hand side of (1.28) is known as a “zero
mode” of the loop operator (cf. [54, Eq.(5.26)]), whereas the fourth term
corresponds to the Dirichlet energy of $\chi$.

We end this subsection by giving some crucial remarks on our theorems.

######  Remark (Comparison with Zabrodin-Wiegmann formula).

We compare our formula (1.24) with the prediction by Zabrodin and Wiegmann.
For $\beta=2$ and a radially symmetric $Q$ associated with a disc droplet of
radius $r_{1}$, the asymptotic formula (1.6) is written in [54,
Eqs.(1.2),(C.7)] as

(1.29) $\begin{split}\log
Z_{N}&=F_{0}N^{2}+F_{1/2}N+F_{1}+c(N)+O(N^{-1}),\qquad\Big{(}c(N):=\log
N!-\frac{1}{2}N\log N+\gamma N\Big{)}\\\ &=F_{0}N^{2}+\frac{1}{2}N\log
N+\Big{(}\gamma-1+F_{1/2}\Big{)}N+\frac{1}{2}\log
N+\frac{\log(2\pi)}{2}+F_{1}+O(N^{-1}),\end{split}$

where $\gamma$ is a “numerical” constant [54, p.8938] (that is not explicitly
presented). (For reader’s convenience, let us mention that in [54], the
authors use a different convention for $\beta$ so that $\beta=2$ in our case
corresponds to $\beta=1$ in [54]. Furthermore, the Planck constant $\hbar$ in
[54] is identified as $1/N$.) The coefficients $F_{0},F_{1/2}$ and $F_{1}$ in
(1.29) are given by

(1.30) $\displaystyle F_{0}$
$\displaystyle:=\pi\int_{0}^{r_{1}^{2}}(W_{\textrm{rad}}(x)-W_{\textrm{rad}}^{\prime}(x)x\log
x)\sigma_{\textrm{rad}}(x)\,dx,$ (1.31) $\displaystyle F_{1/2}$
$\displaystyle:=-\frac{\pi}{2}\int_{0}^{r_{1}^{2}}\sigma_{\textrm{rad}}(x)\log(\pi\sigma_{\textrm{rad}}(x))\,dx,$
(1.32) $\displaystyle F_{1}$
$\displaystyle:=\frac{1}{12}\log\Big{(}\frac{1}{r_{1}^{2}}\Big{)}-\frac{1}{6}\chi_{\textrm{rad}}(r_{1}^{2})-\frac{1}{4}r_{1}^{2}\chi_{\textrm{rad}}^{\prime}(r_{1}^{2})+\frac{1}{3}\int_{0}^{r_{1}^{2}}x(\chi_{\textrm{rad}}^{\prime}(x))^{2}\,dx,$

where

(1.33)
$W_{\textrm{rad}}(r):=-Q(\sqrt{r}),\qquad\sigma_{\textrm{rad}}(r):=\frac{1}{\pi}\Delta
Q(\sqrt{r}),\qquad\chi_{\textrm{rad}}(r):=\frac{1}{2}\log\Delta Q(\sqrt{r}).$

Using (1.33), it is straightforward to check that the formulas (1.30), (1.31)
and (1.32) can be identified as

(1.34) $F_{0}=-I_{Q}[\mu_{Q}],\qquad F_{1/2}=-\frac{1}{2}E_{Q}[\mu_{Q}],\qquad
F_{1}=F_{Q}[\mathbb{D}_{r_{1}}],$

where $I_{Q}$, $E_{Q}$ and $F_{Q}[\mathbb{D}_{r_{1}}]$ are given by (1.14),
(1.10) and (1.23). (Cf. the identification of $F_{0}$ follows from the
computation (2.23) below.) Then by letting $\gamma=\log(2\pi)/2$, one can
deduce from (1.34) that the asymptotic formula (1.29) agrees with our result
(1.24) up to the additive terms

(1.35) $-\frac{1}{12}\log N+\zeta^{\prime}(-1).$

We remark that the asymptotic expansion of the partition function of the
complex Ginibre ensemble was presented in [15, 52], where one can also observe
the term $\zeta^{\prime}(-1)$ in (1.35). Indeed, the term $\zeta^{\prime}(-1)$
and its generalizations have appeared in similar situations in the Hermitian
matrix theory, see [23, Remark 1.3, Proposition 1.4], [18, Theorem 1.1], [21,
Theorem 1.2]. Interestingly, the coefficients of the $\zeta^{\prime}(-1)$ term
depend on the connectivity of the droplet (i.e. the number of disjoint
intervals in this case) and the number of hard edges.

######  Remark (Non-triviality of the limit $r_{0}\to 0$).

The formulas (1.24) and (1.25) cannot be recovered by simply taking the limit
$r_{0}\to 0$ of (1.18) and (1.19). Namely, it is obvious that as $r_{0}\to 0$,
the terms

(1.36) $-\frac{1}{12}\log\Big{(}\frac{1}{r_{0}^{2}\Delta
Q(r_{0})}\Big{)}+\frac{1}{16}r_{0}\frac{({\partial}_{r}\Delta
Q)(r_{0})}{\Delta Q(r_{0})}$

do not correspond to the terms (1.35). (One may however notice that for the
standard microscale $r_{0}=O(1/\sqrt{N})$, at least the $-\frac{1}{12}\log N$
term in (1.35) follows.)

From the viewpoint of the proof, the origin of (1.35) and (1.36) is
essentially similar in the sense that these terms arise from the asymptotic
behaviors of the summand in (1.38) of _lower degrees_. Nevertheless, it is
essential (but seems not discussed in [54]) that these asymptotic behaviors
depend on whether the droplet is contractible or not, i.e. for the radially
symmetric potentials, disc or annulus. We remark that the contractible case
requires considerably more analysis than the other case, see the following
subsection for more discussion.

######  Remark (Invariance of the $O(1)$-terms under the dilation).

For $a>0$, let $Q_{a}(z):=Q(z/a)$. Then the droplet associated with $Q_{a}$ is
given by $\\{z\in\mathbb{C}:ar_{0}\leq|z|\leq ar_{1}\\}$, where $r_{0}$ and
$r_{1}$ are given in (1.13). Then it follows from (1.17) and (1.23) that

(1.37)
$F_{Q}[\mathbb{A}_{r_{0},r_{1}}]=F_{Q_{a}}[\mathbb{A}_{ar_{0},ar_{1}}],\qquad
F_{Q}[\mathbb{D}_{r_{1}}]=F_{Q_{a}}[\mathbb{D}_{ar_{1}}].$

This in turn means that the $O(1)$-terms in the expansions in Theorems 1.1 and
1.2 are invariant under the dilation $\\{z_{j}\\}\mapsto\\{a\cdot z_{j}\\}$.
The property (1.37) can be expected from the analytic expression (1.38) below.

######  Remark (Weight function with singularities and classical problems in
random matrix theory).

In Theorems 1.1 and 1.2, we focus on the weight function $e^{-NQ}$ without any
kind of singularities. In contrast, if a specific singularity is allowed for
the weight function, the problems of deriving asymptotic expansions of the
associated partition function are (when combined with Theorems 1.1 and 1.2)
equivalent to several classical problems in random matrix theory.

To be more concrete, we list various problems in this direction. If the weight
function has a _hard-edge_ inside the droplet, the associated partition
function provides the large gap (hole) probability, see [1, 2, 7, 12, 19, 27,
30, 35] and references therein. The weight function with a _jump-type
singularity_ gives rise to the moment generating function of the disc counting
function. It has been extensively studied in recent years [4, 8, 13, 17, 20,
22]. (We also refer to [26, 40, 41, 51] for physical motivations of these
problems from the counting statistics of rotating free fermions.) Finally, a
_root-type singularity_ arises in the study of the log-characteristic
polynomials [13, 24, 53].

We stress that the literature mentioned above is limited mainly to a
particular model, such as the Ginibre ensemble, when deriving precise
asymptotic results or to the leading order asymptotic when considering general
potentials. We expect that Theorems 1.1 and 1.2 provide the building blocks
for obtaining precise asymptotic results on the problems mentioned above with
general radially symmetric potentials.

######  Remark (Planar point processes with a general external potential $Q$).

For a general potential $Q$ beyond a radially symmetric one, the asymptotic
behaviors of planar orthogonal polynomials (of sufficiently large degrees)
with respect to $e^{-NQ}\,dA$ were recently obtained in [34]. We expect that
this will be helpful to extend Theorem 1.1 (i) to a general potential $Q$
associated with a “non-contractible” droplet. On the other hand, for the
extension of Theorem 1.2 (i), it is required to derive asymptotics of
orthogonal polynomials of lower degrees as well.

Such a generalization of Theorems 1.1 and 1.2 (ii) for planar symplectic
ensembles seems at present far from being solved. More precisely, in order to
obtain an analytic expression of $\widetilde{Z}_{N}$, it is required to
construct the associated skew-orthogonal polynomial. However, for a non-
radially symmetric potential, this construction has been known only in a few
special cases [3, 37] (cf. see [6] for a possible generality).

###  1.3. Outline of the proof

In this subsection, we outline the proofs of our main results. Using the
determinantal (resp., Pfaffian) structure and de Bruijn’s type formulas, one
can express $Z_{N}$ (resp., $\widetilde{Z}_{N}$) in terms of the
(skew-)orthogonal norms. Consequently, since $Q$ is radially symmetric, we
find

(1.38) $\log Z_{N}=\log N!+\sum_{j=0}^{N-1}\log
h_{j},\qquad\log\widetilde{Z}_{N}=\log
N!+\sum_{j=0}^{N-1}\log(2\widetilde{h}_{2j+1}),$

where

(1.39)
$h_{j}:=\int_{\mathbb{C}}|z|^{2j}\,e^{-NQ(z)}\,dA(z),\qquad\widetilde{h}_{j}:=\int_{\mathbb{C}}|z|^{2j}\,e^{-2NQ(z)}\,dA(z).$

These formulas can be found for instance in [20, Lemma 1.7] and [6, Remark
2.5]. In particular, for planar symplectic ensembles, we have used the
explicit construction of skew-orthogonal polynomials associated with radially
symmetric potentials, see [6, Corollary 3.3].

In order to obtain the large-$N$ expansions of partition functions up to the
$O(1)$-terms, we need to derive asymptotic behaviors of $h_{j}$ and
$\widetilde{h}_{j}$ up to the first subleading terms, for which we apply
Laplace’s method. For this purpose, let $r_{\tau}$ be a unique number
$r_{\tau}$ such that $r_{\tau}q^{\prime}(r_{\tau})=2\tau$ for $0\leq\tau\leq
1$. Such a function $\tau\mapsto r_{\tau}$ plays an important role in
Laplace’s method, and we defer more explanations to Subsection 2.1. In the
asymptotic expansions of $h_{j}$ and $\widetilde{h}_{j}$, one should
distinguish the following two cases depending on a small constant
$\varepsilon>0$.

  * •

Case 1: $r_{j/N}\gg N^{-\varepsilon}$. For the annular droplet case where
$r_{0}>0$, this case covers all $j=0,1,\dots,N-1$ (Lemma 2.1). On the other
hand, for the disc droplet case where $r_{0}=0$, this case covers only
$j=m_{N},m_{N}+1,\dots,N-1$ for $m_{N}=N^{\epsilon}$ with some $\epsilon>0$
(Lemma 3.2).

  * •

Case 2: $r_{j/N}\ll N^{-\varepsilon}$. This covers the remaining disc droplet
case with $j=0,1,\dots,m_{N}-1$ (Lemma 3.1). Notably, the asymptotic expansion
involves gamma functions in this case.

Furthermore, we apply the Euler-Maclaurin formula (see e.g. [45, Section
2.10]) to precisely analyze the summations in (1.38)

(1.40)
$\begin{split}\sum_{j=m}^{n}f(j)=\int_{m}^{n}f(x)\,dx+\frac{f(m)+f(n)}{2}+\sum_{k=1}^{l}\frac{B_{2k}}{(2k)!}\Big{(}f^{(2k-1)}(n)-f^{(2k-1)}(m)\Big{)}+R_{l},\end{split}$

where $B_{k}$ is the Bernoulli number defined by

(1.41) $\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}$

and $R_{l}$ is an error term with the following estimate

$|R_{l}|\leq\frac{2\zeta(l)}{(2\pi)^{l}}\int_{m}^{n}|f^{(l)}(x)|\,dx.$

Here $\zeta$ is the Riemann zeta function. In particular, for the disc droplet
case, in the summation of lower degrees $j=0,1,\dots,m_{N}-1$, we consider the
Barnes $G$-function [45, Section 5.17]

$G(z+1)=(2\pi)^{z/2}e^{-(z+z^{2}(1+\gamma))/2}\prod_{n=0}^{\infty}\big{(}1+\frac{z}{n}\big{)}^{n}e^{-z+z^{2}/(2n)}$

and use its asymptotic expansion [45, Eq.(5.17.5)]:

(1.42) $\displaystyle\begin{split}\log G(z+1)=\frac{z^{2}\log
z}{2}-\frac{3}{4}z^{2}+\frac{\log(2\pi)z}{2}-\frac{\log
z}{12}+\zeta^{\prime}(-1)+O(\frac{1}{z^{2}}),\qquad(z\to\infty).\end{split}$

This asymptotic expansion leads to the appearance of the Riemann zeta function
in Theorem 1.2.

### Plan of the paper

The rest of this paper is organized as follows. In Section 2, we prove Theorem
1.1. Subsection 2.1 is devoted to deriving asymptotic behaviors of $h_{j}$ and
$\widetilde{h}_{j}$ using Laplace’s method. Then we show Theorem 1.1 (i) on
random normal matrices in Subsection 2.2 and Theorem 1.1 (ii) on planar
symplectic ensembles in Subsection 2.3. Section 3 is structured in parallel
with a goal to show Theorem 1.2 albeit it requires considerably more
computations compared to those in Section 2. In Section 4, we present examples
of Theorems 1.1 and 1.2 for the Mittag-Leffler and truncated unitary ensembles
whose partition functions can be explicitly expressed in terms of well-known
special functions.

##  2\. Proof of Theorem 1.1

In this section, we prove Theorem 1.1. Throughout this section, we assume that
$r_{0}>0.$

###  2.1. Asymptotics of the orthogonal norm

We first introduce an auxiliary function $V_{\tau}$ in $(0,\infty)$

(2.1) $V_{\tau}(r):=q(r)-2\tau\log r.$

With the following choices of $\tau=\tau(j),\widetilde{\tau}(j)$

(2.2) $\tau(j):=\frac{j}{N},\qquad\widetilde{\tau}(j):=\frac{j}{2N},$

the integrands in $h_{j},\widetilde{h}_{j}$ (1.39) can be expressed in terms
of $V_{\tau}:$

$r^{2j}e^{-Nq(r)}=e^{-NV_{\tau(j)}(r)},\qquad
r^{2j}e^{-2Nq(r)}=e^{-NV_{\widetilde{\tau}(j)}(r)}.$

For a radially symmetric potential $Q$, we represent $\Delta Q$ in terms of
$q$ as

(2.3) $4\Delta
Q(z)|_{z=r}=\frac{1}{r}(rq^{\prime}(r))^{\prime}=\frac{q^{\prime}(r)}{r}+q^{\prime\prime}(r).$

Differentiating (2.1), we have

(2.4)
$\displaystyle\begin{split}V_{\tau}^{\prime}(r)=q^{\prime}(r)-\frac{2\tau}{r},\qquad
V_{\tau}^{\prime\prime}(r)&=4\Delta
Q(r)-\frac{1}{r}V_{\tau}^{\prime}(r),\qquad
V_{\tau}^{(3)}(r)=4{\partial}_{r}\Delta Q(r)-\frac{4}{r}\Delta
Q(r)+\frac{2}{r^{2}}V_{\tau}^{\prime}(r),\\\
V_{\tau}^{(4)}(r)&=4{\partial}_{r}^{2}\Delta Q(r)+\frac{12}{r^{2}}\Delta
Q(r)-\frac{4}{r}{\partial}_{r}\Delta
Q(r)-\frac{6}{r^{3}}V_{\tau}^{\prime}(r).\end{split}$

We now set the stage to apply Laplace’s method. Since $rq^{\prime}(r)$ is
strictly increasing inside the droplet, for $0\leq\tau\leq 1$, there exists a
unique number $r(\tau)$ such that

(2.5) $V_{\tau}^{\prime}(r(\tau))=0,\qquad
V_{\tau}^{\prime\prime}(r(\tau))>0.$

Moreover, by (2.3) and the relation (2.5), it follows that

(2.6)
$\frac{dr(\tau)}{d\tau}=\frac{2}{(rq^{\prime}(r))^{\prime}}\Big{|}_{r=r(\tau)}=\frac{1}{2r(\tau)\Delta
Q(r(\tau))}>0.$

Thus $r(\tau)$ is an increasing function of $\tau$. On the other hand,
$r(1)=r_{1},r(0)=r_{0}$, where $r_{0},r_{1}$ are given in (1.13). Therefore,
we denote $r_{\tau}=r(\tau)$ making the notation consistent with (1.13). We
also mention here that $r_{\tau}$ corresponds to the outer radius of the so-
called “$\tau$-droplet” [34]. By (2.4) and (2.5), $r_{\tau}$ satisfies

(2.7) $r_{\tau}q^{\prime}(r_{\tau})=2\tau.$

In particular, $q^{\prime}(r_{0})=0$ and $r_{1}q^{\prime}(r_{1})=2.$

Let

(2.8) $\mathfrak{B}_{1}(r):=-\frac{1}{32}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{(\Delta Q(r))^{2}}-\frac{19}{96r}\frac{{\partial}_{r}\Delta
Q(r)}{(\Delta Q(r))^{2}}+\frac{5}{96}\frac{({\partial}_{r}\Delta
Q(r))^{2}}{\Delta Q(r)^{3}}+\frac{1}{12r^{2}}\frac{1}{\Delta Q(r)}.$

Here, the subscript $1$ is added to $\mathfrak{B}$ to emphasize that this
function appears as the first subleading term of the asymptotic expansion of
orthonormal polynomials, see Lemma 2.1 below. Indeed, function
$\mathfrak{B}_{1}$ is closely related to function $\mathfrak{B}_{\tau,1}$ in
[34, Theorem 1.3].

######  Lemma 2.1.

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

  * •

For each $j$ with $0\leq j\leq N-1$,

(2.9)
$h_{j}=N^{-\frac{1}{2}}e^{-NV_{\tau(j)}(r_{\tau(j)})}\Big{(}\frac{{2\pi}r_{\tau(j)}^{2}}{\Delta
Q(r_{\tau(j)})}\Big{)}^{\frac{1}{2}}\Big{(}1+\frac{1}{N}\mathfrak{B}_{1}(r_{\tau(j)})+O(N^{-2})\Big{)}.$

  * •

For each $j$ with $0\leq j\leq 2N-1$,

(2.10)
$\widetilde{h}_{j}=(2N)^{-\frac{1}{2}}e^{-2NV_{\widetilde{\tau}(j)}(r_{\widetilde{\tau}(j)})}\Big{(}\frac{{2\pi}r_{\widetilde{\tau}(j)}^{2}}{\Delta
Q(r_{\widetilde{\tau}(j)})}\Big{)}^{\frac{1}{2}}\Big{(}1+\frac{1}{2N}\mathfrak{B}_{1}(r_{\widetilde{\tau}(j)})+O(N^{-2})\Big{)}.$

###### Proof.

It suffices to show the first assertion as the second one follows by replacing
$N$ with $2N.$

Write $\delta_{N}:=\log N/\sqrt{N}$. As seen in (2.1), (2.5) and (2.6), the
function $V_{\tau}$ has a global minimum at $r=r_{\tau}$ and
$r_{\tau(j+\frac{1}{2})}-r_{\tau(j)}=O(N^{-1})$. We have

$V_{\tau(j+\frac{1}{2})}(r)\geq
V_{\tau(j+\frac{1}{2})}(r_{\tau(j)}+\delta_{N})=V_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})+\frac{1}{2}V_{\tau(j+\frac{1}{2})}^{\prime\prime}(r_{\tau(j+\frac{1}{2})})(r_{\tau(j)}+\delta_{N}-r_{\tau(j+\frac{1}{2})})^{2}+O(\delta_{N}^{3})$

for all $r$ with $r>r_{\tau(j)}+\delta_{N}$ since $rq^{\prime}(r)$ is
increasing in $(0,\infty)$. A similar estimate holds for
$r<r_{\tau(j)}-\delta_{N}$. Thus we deduce from the estimate

(2.11)
$V_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})-V_{\tau(j)}(r_{\tau(j)})=O(N^{-1})$

that there exists a positive number $c>0$ such that for all
$|r-r_{\tau(j)}|>\delta_{N}$

(2.12) $\displaystyle V_{\tau(j+\frac{1}{2})}(r)-V_{\tau(j)}(r_{\tau(j)})\geq
c\,\delta_{N}^{2}.$

We split $h_{j}$ in (1.39) into two integrals

$\displaystyle
h_{j}=\int_{0}^{\infty}e^{-NV_{\tau(j)}(r)}2r\,dr=\int_{|r-r_{\tau(j)}|<\delta_{N}}e^{-NV_{\tau(j)}(r)}2r\,dr+\int_{|r-r_{\tau(j)}|>\delta_{N}}e^{-NV_{\tau(j)}(r)}2r\,dr.$

Using (1.12), we choose sufficiently large $M>0$ such that

(2.13) $\int_{0}^{\infty}\Big{(}r^{2}e^{-q(r)}\Big{)}^{M}\,r\,dr<+\infty.$

We then use (2.12) and (2.13) to find an error estimate for the second
integral

(2.14)
$\displaystyle\begin{split}&\quad\int_{|r-r_{\tau(j)}|>\delta_{N}}e^{-NV_{\tau(j)}(r)}2r\,dr=e^{-NV_{\tau(j)}(r_{\tau(j)})}\int_{|r-r_{\tau(j)}|>\delta_{N}}e^{-N\big{(}V_{\tau(j)}(r)-V_{\tau(j)}(r_{\tau(j)})\big{)}}\,2r\,dr\\\
&\leq
e^{-NV_{\tau(j)}(r_{\tau(j)})}e^{-c(N-M)\delta_{N}^{2}}\int_{|r-r_{\tau(j)}|>\delta_{N}}e^{-M\big{(}V_{\tau(j)}(r)-V_{\tau(j)}(r_{\tau(j)})\big{)}}\,2r\,dr=e^{-NV_{\tau(j)}(r_{\tau(j)})}\epsilon_{N},\end{split}$

where $\epsilon_{N}=O(e^{-c(\log N)^{2}})$ for some $c>0$ and the
$O$-constants are bounded uniformly for all $\tau$ with $0\leq\tau\leq 1$. We
deduce from the asymptotic expansion of $V_{\tau}(r)$ near the critical point
$r_{\tau}$ and (LABEL:ex_error) that

$\displaystyle
h_{j}=e^{-NV_{\tau(j)}(r_{\tau(j)})}\Big{(}\int_{-\delta_{N}}^{\delta_{N}}e^{-2N\Delta
Q(r_{\tau(j)})r^{2}}e^{-N(\frac{1}{3!}V_{\tau(j)}^{(3)}(r_{\tau(j)})r^{3}+\frac{1}{4!}V_{\tau(j)}^{(4)}(r_{\tau(j)})r^{4}+O(r^{5}))}2(r_{\tau(j)}+r)\,dr+\epsilon_{N}\Big{)}.$

A change of variables gives that

(2.15) $\begin{split}&e^{NV_{\tau(j)}(r_{\tau(j)})}h_{j}\\\
=&\frac{2\,r_{\tau(j)}}{\sqrt{N}}\int_{-\sqrt{N}\delta_{N}}^{\sqrt{N}\delta_{N}}e^{-2\Delta
Q(r_{\tau(j)})r^{2}-\frac{1}{\sqrt{N}}\frac{1}{3!}V_{\tau(j)}^{(3)}(r_{\tau(j)})r^{3}-\frac{1}{N}\frac{1}{4!}V_{\tau(j)}^{(4)}(r_{\tau(j)})r^{4}+O(N^{-\frac{3}{2}}r^{5})}\big{(}1+\frac{r}{r_{\tau(j)}\sqrt{N}}\big{)}\,dr+\epsilon_{N}.\end{split}$

Using the Taylor series expansion of the function

$\displaystyle
e^{-\frac{1}{\sqrt{N}}\frac{1}{3!}V_{\tau(j)}^{(3)}(r_{\tau(j)})r^{3}-\frac{1}{N}\frac{1}{4!}V_{\tau(j)}^{(4)}(r_{\tau(j)})r^{4}},$

we have the asymptotic expansion

(2.16)
$\begin{split}&\frac{\sqrt{N}}{2r_{\tau(j)}}e^{NV_{\tau(j)}(r_{\tau(j)})}h_{j}\\\
=&\int_{-\infty}^{\infty}e^{-2\Delta
Q(r_{\tau(j)})r^{2}}\Big{[}1-\frac{1}{N}\Big{(}\frac{1}{6r_{\tau(j)}}V_{\tau(j)}^{(3)}(r_{\tau(j)})+\frac{1}{24}V_{\tau(j)}^{(4)}(r_{\tau(j)})\Big{)}r^{4}+\frac{1}{N}\frac{1}{72}(V_{\tau(j)}^{(3)}(r_{\tau(j)}))^{2}r^{6}\Big{]}\,dr+O(N^{-2})\end{split}$

since the odd terms vanish in the integral, leaving only the even terms.

Combining (2.4) with the elementary Gaussian integrals

(2.17)
$\int_{\mathbb{R}}e^{-2ar^{2}}\,dr=\sqrt{\frac{\pi}{2}}\frac{1}{a^{1/2}},\qquad\int_{\mathbb{R}}e^{-2ar^{2}}r^{4}\,dr=\sqrt{\frac{\pi}{2}}\frac{3}{16}\frac{1}{a^{5/2}},\qquad\int_{\mathbb{R}}e^{-2ar^{2}}r^{6}\,dr=\sqrt{\frac{\pi}{2}}\frac{15}{64}\frac{1}{a^{7/2}},$

we obtain the desired asymptotic behavior after some straightforward
computations. ∎

The following elementary integration will be helpful later.

######  Lemma 2.2.

We have

(2.18)
$\begin{split}\int_{S}\mathfrak{B}_{1}\,d\mu_{Q}=F_{Q}[\mathbb{A}_{r_{0},r_{1}}]-\frac{1}{4}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta
Q(r_{0})}\Big{)}+\frac{1}{3}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)},\end{split}$

where $F_{Q}[\mathbb{A}_{r_{0},r_{1}}]$ is given in (1.17).

###### Proof.

By (2.8) and (1.11), we have

$\begin{split}\int_{S}\mathfrak{B}_{1}\,d\mu_{Q}&=\frac{1}{6}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)}-\frac{19}{48}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta
Q(r_{0})}\Big{)}-\frac{1}{16}\int_{r_{0}}^{r_{1}}\Big{[}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{\Delta Q(r)}-\frac{5}{3}\Big{(}\frac{{\partial}_{r}\Delta Q(r)}{\Delta
Q(r)}\Big{)}^{2}\Big{]}\,r\,dr.\end{split}$

Then the lemma follows using integration by parts

(2.19)
$\begin{split}&\quad\int_{r_{0}}^{r_{1}}\Big{[}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{\Delta Q(r)}-\frac{5}{3}\Big{(}\frac{{\partial}_{r}\Delta Q(r)}{\Delta
Q(r)}\Big{)}^{2}\Big{]}\,r\,dr\\\ &=r_{1}\,\frac{({\partial}_{r}\Delta
Q)(r_{1})}{\Delta Q(r_{1})}-r_{0}\,\frac{({\partial}_{r}\Delta
Q)(r_{0})}{\Delta Q(r_{0})}+\log\Big{(}\frac{\Delta Q(r_{0})}{\Delta
Q(r_{1})}\Big{)}-\frac{2}{3}\int_{r_{0}}^{r_{1}}\Big{(}\frac{{\partial}_{r}\Delta
Q(r)}{\Delta Q(r)}\Big{)}^{2}\,r\,dr.\end{split}$

∎

###  2.2. Random normal matrix ensemble

In this subsection, we prove Theorem 1.1 (i). By Lemma 2.1, we have

(2.20) $\log h_{j}=-NV_{\tau(j)}(r_{\tau(j)})+\frac{1}{2}\Big{(}\log(2\pi
r_{\tau(j)}^{2})-\log N-\log\Delta
Q(r_{\tau(j)})\Big{)}+\frac{1}{N}\mathfrak{B}_{1}(r_{\tau(j)})+O(N^{-2}).$

In the following lemmas, we analyze the asymptotic behavior of the partial sum
of each term in (2.20).

######  Lemma 2.3.

As $N\to\infty$, we have

(2.21)
$\sum_{j=0}^{N-1}V_{\tau(j)}(r_{\tau(j)})=NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)+\frac{1}{6N}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}+O(N^{-3}).$

###### Proof.

The sequence $\tau$ in (2.2) can be extended to the function on $[0,N]$:
$\tau(t)=t/N.$ Using the Euler-Maclaurin formula (1.40) and (2.2), we have

(2.22)
$\displaystyle\begin{split}\sum_{j=0}^{N-1}V_{\tau(j)}(r_{\tau(j)})&=\int_{0}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt-\frac{1}{2}\big{(}V_{\tau(N)}(r_{\tau(N)})-V_{\tau(0)}(r_{\tau(0)})\big{)}\\\
&\quad+\frac{1}{12}\Big{[}{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=N}-{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=0}\Big{]}+O(N^{-3}).\end{split}$

Here, we also used the second Bernoulli number $B_{2}=1/6$, which can be
easily seen from the definition (1.41). For the first term on the right-hand
side of (2.22), the change of variables $s=r_{\tau(t)}$, the definition (2.1)
of $V_{\tau}$, the definition (2.2) of $\tau$, the formula (2.6) of
$dr/d\tau$, and the equation (2.7) $r_{\tau}q^{\prime}(r_{\tau})=2\tau$ for
$r_{\tau}$ give that

$\frac{1}{N}\int_{0}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt=2\int_{r_{0}}^{r_{1}}\big{(}q(s)-sq^{\prime}(s)\log
s\big{)}s\Delta Q(s)\,ds.$

We use the polar coordinate system to represent the first term on the right-
hand side of the above equation as

$2\int_{r_{0}}^{r_{1}}sq(s)\Delta Q(s)\,ds=\int_{S}Q\cdot\Delta Q\,dA.$

By (2.3), the method of integration by parts, and the relation
$r_{1}q^{\prime}(r_{1})=2,q^{\prime}(r_{0})=0$ (see (2.7)), the second term is
simplified to

$-2\int_{r_{0}}^{r_{1}}s^{2}q^{\prime}(s)\log s\Delta
Q(s)\,ds=-\frac{1}{4}\int_{r_{0}}^{r_{1}}\log
s\cdot((sq^{\prime}(s))^{2})^{\prime}\,ds\\\ =-\log
r_{1}+\frac{1}{4}\int_{r_{0}}^{r_{1}}s(q^{\prime}(s))^{2}\,ds.$

Applying the method of integration by parts again to the last integral,

$\frac{1}{4}\int_{r_{0}}^{r_{1}}s(q^{\prime}(s))^{2}\,ds=\frac{1}{2}q(r_{1})-\frac{1}{4}\int_{r_{0}}^{r_{1}}q(s)(sq^{\prime}(s))^{\prime}\,ds.$

Using the formula (1.14) of $I_{Q}[\mu_{Q}]$, the representation (2.3) of
$\Delta Q$ in terms of $q$, and the equation (2.7)
$r_{\tau}q^{\prime}(r_{\tau})=2\tau$ for $r_{\tau}$, we have

(2.23)
$\displaystyle\begin{split}\frac{1}{N}\int_{0}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt&=\int_{S}Q\cdot\Delta
Q\,dA-\log
r_{1}+\frac{1}{2}q(r_{1})-\frac{1}{4}\int_{r_{0}}^{r_{1}}q(s)(sq^{\prime}(s))^{\prime}\,ds\\\
&=\frac{1}{2}\int_{S}Q\cdot\Delta Q\,dA-\log
r_{1}+\frac{1}{2}q(r_{1})=I_{Q}[\mu_{Q}].\end{split}$

For the next term on the right-hand side of (2.22), we observe that

(2.24)
$V_{\tau(N)}(r_{\tau(N)})-V_{\tau(0)}(r_{\tau(0)})=V_{1}(r_{1})-V_{0}(r_{0})=q(r_{1})-q(r_{0})-2\log
r_{1}=2U_{\mu_{Q}}(0),$

where we have used (1.16). To analyze the remaining term, we use the Leibniz
rule and obtain

$\displaystyle\begin{split}{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=N}&=\frac{1}{N}\big{[}{\partial}_{\tau}(q(r_{\tau})-2\log
r_{\tau})\big{|}_{\tau=1}-2\log r_{1}\big{]},\\\
{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=0}&=\frac{1}{N}\big{[}{\partial}_{\tau}q(r_{\tau})\big{|}_{\tau=0}-2\log
r_{0}\big{]}.\end{split}$

It follows from $r_{1}q^{\prime}(r_{1})=2$, $r_{0}q^{\prime}(r_{0})=0$ and the
formula (2.6) of $dr/d\tau$ that

(2.25)
$\displaystyle\begin{split}&\quad{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=N}-{\partial}_{t}V_{\tau(t)}(r_{\tau(t)})\big{|}_{t=0}\\\
&=\frac{1}{N}\Big{[}\frac{q^{\prime}(r_{1})}{2r_{1}\Delta Q(r_{1})}-2\log
r_{1}-\frac{1}{r_{1}^{2}\Delta Q(r_{1})}-\frac{q^{\prime}(r_{0})}{2r_{0}\Delta
Q(r_{0})}+2\log
r_{0}\Big{]}=\frac{2}{N}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}.\end{split}$

Combining (2.22), (2.23), (2.24), and (LABEL:sum_Vtau_EM_3), the proof is
complete. ∎

######  Lemma 2.4.

As $N\to\infty$, we have

(2.26) $\sum_{j=0}^{N-1}\log\Delta
Q(r_{\tau(j)})=NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(r_{0})}\Big{)}+O(N^{-1}),$

and

(2.27) $\sum_{j=0}^{N-1}\log
r_{\tau(j)}=-NU_{\mu_{Q}}(0)-\frac{1}{2}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)}+O(N^{-1}).$

###### Proof.

As in Lemma 2.3, we apply the Euler-Maclaurin formula (1.40) and obtain

(2.28) $\displaystyle\begin{split}\sum_{j=0}^{N-1}\log\Delta
Q(r_{\tau(j)})&=\int_{0}^{N}\log\Delta
Q(r_{\tau(t)})\,dt-\frac{1}{2}\log\Big{(}\frac{\Delta Q(r_{1})}{\Delta
Q(r_{0})}\Big{)}\\\ &\quad+\frac{1}{12}\Big{[}{\partial}_{t}\log\Delta
Q(r_{\tau(t)})\big{|}_{t=N}-{\partial}_{t}\log\Delta
Q(r_{\tau(t)})\big{|}_{t=0}\Big{]}+O(N^{-3}).\end{split}$

It follows from a change of variables and (2.6) that the first term on the
right-hand side of (2.28) gives the entropy term

(2.29) $\int_{0}^{N}\log\Delta
Q(r_{\tau(t)})\,dt=N\int_{r_{0}}^{r_{1}}\log\Delta Q(s)2s\Delta
Q(s)\,ds=N\int_{S}\log\Delta Q\,d\mu_{Q}.$

By the chain rule and the formula (2.6) of $dr/d\tau$ again, we also observe
that

$\displaystyle{\partial}_{t}\log\Delta
Q(r_{\tau(t)})\big{|}_{t=N}-{\partial}_{t}\log\Delta
Q(r_{\tau(t)})\big{|}_{t=0}=\frac{1}{2N}\Big{(}\frac{({\partial}_{r}\Delta
Q)(r_{1})}{r_{1}(\Delta Q(r_{1}))^{2}}-\frac{({\partial}_{r}\Delta
Q)(r_{0})}{r_{0}(\Delta Q(r_{0}))^{2}}\Big{)}=O(N^{-1}).$

Combining all of the above, we obtain (2.26). The equation (2.27) follows
similarly. ∎

We are now ready to prove the first assertion of Theorem 1.1.

###### Proof of Theorem 1.1 (i).

Combining Lemmas 2.3 and 2.4 with (2.20), we obtain

$\displaystyle\sum_{j=0}^{N-1}\log h_{j}$
$\displaystyle=-N^{2}I_{Q}[\mu_{Q}]-\frac{N}{2}\log
N+N\Big{(}\frac{\log(2\pi)}{2}-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}$
$\displaystyle\quad-\frac{1}{3}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)}+\frac{1}{4}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta
Q(r_{0})}\Big{)}+\int_{S}\mathfrak{B}_{1}\,d\mu_{Q}+O(N^{-1}),$

where $\mathfrak{B}_{1}$ is given by (2.8). Then the desired asymptotic
expansion (1.18) follows from

(2.30) $\log N!=N\log N-N+\frac{1}{2}\log
N+\frac{1}{2}\log(2\pi)+O(\frac{1}{N}),\qquad(N\to\infty)$

(see e.g. [45, Eq.(5.11.1)]) and Lemma 2.2. ∎

###  2.3. Planar symplectic ensemble

In this subsection, we prove Theorem 1.1 (ii).

Recall that by Lemma 2.1,

(2.31)
$\log\widetilde{h}_{j}=-2NV_{\widetilde{\tau}(j)}(r_{\widetilde{\tau}(j)})+\frac{1}{2}\Big{(}\log(\pi
r_{\widetilde{\tau}(j)}^{2})-\log N-\log\Delta
Q(r_{\widetilde{\tau}(j)})\Big{)}+\frac{1}{2N}\mathfrak{B}_{1}(r_{\widetilde{\tau}(j)})+O(N^{-2}).$

In the following lemmas, we derive asymptotic expansions for the partial sum
of each term on the right-hand side of (2.31). We obtain the following as a
counterpart of Lemma 2.3.

######  Lemma 2.5.

As $N\to\infty$, we have

(2.32)
$\sum_{j=0}^{N-1}V_{\widetilde{\tau}(2j+1)}(r_{\widetilde{\tau}(2j+1)})=NI_{Q}[\mu_{Q}]-\frac{1}{12N}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}+O(N^{-3}).$

###### Proof.

Applying Lemma 2.3 by replacing $N$ with $2N$, we have

(2.33)
$\sum_{j=0}^{2N-1}V_{\widetilde{\tau}(j)}(r_{\widetilde{\tau}(j)})=2NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)+\frac{1}{12N}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}+O(N^{-3}).$

It follows from
$V_{\widetilde{\tau}(2j)}(r_{\widetilde{\tau}(2j)})=V_{\tau(j)}(r_{\tau(j)})$
and Lemma 2.3 that

(2.34)
$\sum_{j=0}^{N-1}V_{\widetilde{\tau}(2j)}(r_{\widetilde{\tau}(2j)})=NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)+\frac{1}{6N}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}+O(N^{-3}).$

Then (2.32) follows from (2.33) and (2.34). ∎

######  Lemma 2.6.

As $N\to\infty$, we have

$\sum_{j=0}^{N-1}\log\Delta
Q(r_{\widetilde{\tau}(2j+1)})=NE_{Q}[\mu_{Q}]+O(N^{-1}),$

and

$\sum_{j=0}^{N-1}\log r_{\widetilde{\tau}(2j+1)}=-NU_{\mu_{Q}}(0)+O(N^{-1}).$

###### Proof.

By Lemma 2.4, we have

(2.35) $\displaystyle\sum_{j=0}^{2N-1}\log\Delta Q(r_{\widetilde{\tau}(j)})$
$\displaystyle=2NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(r_{0})}\Big{)}+O(N^{-1}),$ (2.36)
$\displaystyle\sum_{j=0}^{2N-1}\log r_{\widetilde{\tau}(j)}$
$\displaystyle=-2NU_{\mu_{Q}}(0)-\frac{1}{2}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)}+O(N^{-1}).$

Along the lines of Lemma 2.4, one can also show that

(2.37) $\displaystyle\sum_{j=0}^{N-1}\log\Delta Q(r_{\widetilde{\tau}(2j)})$
$\displaystyle=NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(r_{0})}\Big{)}+O(N^{-1}),$ (2.38)
$\displaystyle\sum_{j=0}^{N-1}\log r_{\widetilde{\tau}(2j)}$
$\displaystyle=-NU_{\mu_{Q}}(0)-\frac{1}{2}\log\Big{(}\frac{r_{1}}{r_{0}}\Big{)}+O(N^{-1}).$

This completes the proof. ∎

We now prove the second assertion of Theorem 1.1.

###### Proof of Theorem 1.1 (ii).

By Lemmas 2.5 and 2.6, we have

$\displaystyle\sum_{j=0}^{N-1}\log(2\widetilde{h}_{2j+1})$
$\displaystyle=-2N^{2}I_{Q}[\mu_{Q}]-\frac{N}{2}\log N+\frac{N}{2}\log
2+\frac{N}{2}\log(2\pi)$ $\displaystyle\quad-
NU_{\mu_{Q}}(0)-\frac{N}{2}E_{Q}[\mu_{Q}]+\frac{1}{2}\int_{S}\mathfrak{B}_{1}\,d\mu_{Q}+\frac{1}{6}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}+O(N^{-1}).$

Combining (1.38), (2.30), and Lemma 2.2 completes the proof. ∎

##  3\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. Throughout this section, we let
$r_{0}=0.$

###  3.1. Asymptotics of the orthogonal norm

Let $\delta_{N}=N^{-1/2}\log N$ and $m_{N}=N^{\epsilon}$ for $0<\epsilon<1/5$.
As explained in Subsection 1.3, for the disc droplet case, the asymptotic
behaviors of $h_{j}$ and $\widetilde{h}_{j}$ depend on whether the degree $j$
is sufficiently small or not, see Lemmas 3.1 and 3.2, respectively.

######  Lemma 3.1.

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

  * •

For $j=0,1,\dots,m_{N}-1$, we have

$\displaystyle\log
h_{j}=-Nq(0)-(j+1)\log\Big{(}\frac{N}{2}q^{\prime\prime}(0)\Big{)}+\log
j!+O\big{(}N^{-\frac{1}{2}}(j+1)^{\frac{3}{2}}(\log N)^{3}\big{)}.$

  * •

For $j=0,1,\dots,2m_{N}-1$, we have

$\displaystyle\log\widetilde{h}_{j}=-2Nq(0)-(j+1)\log(Nq^{\prime\prime}(0))+\log
j!+O\big{(}N^{-\frac{1}{2}}(j+1)^{\frac{3}{2}}(\log N)^{3}\big{)}.$

###### Proof.

The second assertion is an immediate consequence of the first one. Recall that
$\tau(j)=j/N$. Let

$r^{*}_{\tau}:=r_{\tau}\cdot\log N,$

where $r_{\tau}$ is given in (2.5) or (2.7),
$r_{\tau}q^{\prime}(r_{\tau})=2\tau$. We consider the decomposition

$\displaystyle h_{j}$
$\displaystyle=\int_{0}^{\infty}2r^{2j+1}e^{-Nq(r)}\,dr=\int_{0}^{r^{*}_{{\tau(j+\frac{1}{2})}}}2r^{2j+1}e^{-Nq(r)}\,dr+\int_{r^{*}_{\tau(j+\frac{1}{2})}}^{\infty}2r^{2j+1}e^{-Nq(r)}\,dr.$

Due to strict subharmonicity of $Q$ in a neighborhood of the droplet,
$r_{\tau}$ defined in (2.5) satisfies $r_{\tau}=O({\tau}^{\frac{1}{2}})$ as
$\tau\to 0$. Since the function $V_{\tau}$ in (2.1) has a global minimum at
$r=r_{\tau}$ and increases in $(r_{\tau},\infty)$, for all
$r>r^{*}_{\tau(j+\frac{1}{2})}$, we have

(3.1) $\displaystyle\begin{split}V_{\tau(j+\frac{1}{2})}(r)&\geq
V_{\tau(j+\frac{1}{2})}(r^{*}_{\tau(j+\frac{1}{2})})=q(r^{*}_{\tau(j+\frac{1}{2})})-\frac{2j+1}{N}\log(r^{*}_{\tau(j+\frac{1}{2})})\\\
&=q(0)+\frac{1}{2}q^{\prime\prime}(0)(r^{*}_{\tau(j+\frac{1}{2})})^{2}-\frac{2j+1}{N}\log(r^{*}_{\tau(j+\frac{1}{2})})+O(r^{*}_{\tau(j+\frac{1}{2})})^{3}.\end{split}$

Here, we also have used $q^{\prime}(0)=0$, which follows from (2.3) and the
fact that $\Delta Q(z)\in(0,\infty)$ near the origin. Using (3.1), it follows
that

$\displaystyle\quad
e^{Nq(0)}\int_{r^{*}_{\tau(j+\frac{1}{2})}}^{\infty}2r^{2j+1}e^{-Nq(r)}\,dr=\int_{r^{*}_{\tau(j+\frac{1}{2})}}^{\infty}2\,e^{-N(V_{\tau(j+\frac{1}{2})}(r)-q(0))}\,dr$
$\displaystyle\leq
e^{-c_{1}(N-M)(r^{*}_{\tau(j+\frac{1}{2})})^{2}}(r^{*}_{\tau(j+\frac{1}{2})})^{(2j+1)\frac{N-M}{N}}\int_{r^{*}_{\tau(j+\frac{1}{2})}}^{\infty}2\,e^{-M(V_{\tau(j+\frac{1}{2})}(r)-q(0))}\,dr=:\epsilon_{N}(j)$

for some $c_{1}>0$. Here, $M$ is given by (2.13). Since
$r_{\tau}=O({\tau}^{\frac{1}{2}})$, there exists $c_{2}>0$ such that

$\displaystyle\epsilon_{N}(j)=O\Big{(}(r^{*}_{\tau(j+\frac{1}{2})})^{(2j+1)\frac{N-M}{N}}\cdot
e^{-c_{2}(j+\frac{1}{2})(\log
N)^{2}}\Big{)}=O\Big{(}\big{(}\tfrac{2j+1}{2N}(\log
N)^{2}\big{)}^{(2j+1)\frac{N-M}{2N}}e^{-c_{2}(j+\frac{1}{2})(\log
N)^{2}}\Big{)}.$

Therefore we obtain

$\displaystyle h_{j}$
$\displaystyle=\int_{0}^{\infty}2r^{2j+1}e^{-Nq(r)}\,dr=\int_{0}^{r^{*}_{\tau(j+\frac{1}{2})}}2r^{2j+1}e^{-Nq(r)}\,dr+e^{-Nq(0)}\epsilon_{N}(j)$
$\displaystyle=\int_{0}^{r^{*}_{\tau(j+\frac{1}{2})}}2r^{2j+1}e^{-N(q(0)+\frac{1}{2}q^{\prime\prime}(0)r^{2})+O(N(r^{*}_{\tau(j+\frac{1}{2})})^{3})}\,dr+e^{-Nq(0)}\epsilon_{N}(j)$
$\displaystyle=e^{-Nq(0)}\Big{[}N^{-(j+1)}\int_{0}^{\infty}2r^{2j+1}e^{-\frac{1}{2}q^{\prime\prime}(0)r^{2}}\,dr\,\big{(}1+O(N(r^{*}_{\tau(j+\frac{1}{2})})^{3})\big{)}+\epsilon_{N}(j)\Big{]}$
$\displaystyle=e^{-Nq(0)}\Big{[}N^{-(j+1)}\Gamma(j+1)\Big{(}\frac{1}{2}q^{\prime\prime}(0)\Big{)}^{-(j+1)}\,\Big{(}1+O\Big{(}N^{-\frac{1}{2}}\big{(}j+\tfrac{1}{2}\big{)}^{\frac{3}{2}}(\log
N)^{3}\Big{)}\Big{)}+\epsilon_{N}(j)\Big{]},$

where $\epsilon_{N}(j)$ is negligible. This completes the proof. ∎

Recall that the sequences $\tau(j)$ and $\widetilde{\tau}(j)$ are given by
(2.2): $\tau(j):=j/N,\widetilde{\tau}(j):=j/(2N)$ and the function
$\mathfrak{B}_{1}$ is given by (2.8):

$\begin{split}\mathfrak{B}_{1}(r):=-\frac{1}{32}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{(\Delta Q(r))^{2}}-\frac{19}{96r}\frac{{\partial}_{r}\Delta
Q(r)}{(\Delta Q(r))^{2}}+\frac{5}{96}\frac{({\partial}_{r}\Delta
Q(r))^{2}}{\Delta Q(r)^{3}}+\frac{1}{12r^{2}}\frac{1}{\Delta
Q(r)}.\end{split}$

Recall also that $V_{\tau}$ is given by (2.1) $V_{\tau}(r):=q(r)-2\tau\log r$
and $r_{\tau}$ is given by (2.5) or (2.7)
$r_{\tau}q^{\prime}(r_{\tau})=2\tau$. As a counterpart of Lemma 2.1, we show
the following lemma.

######  Lemma 3.2.

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

  * •

For $j=m_{N},m_{N}+1,\dots,N-1$, we have

$\log h_{j}=-NV_{\tau(j)}(r_{\tau(j)})+\frac{1}{2}\log\Big{(}\frac{2\pi
r_{\tau(j)}^{2}}{N\Delta
Q(r_{\tau(j)})}\Big{)}+\frac{1}{N}\mathfrak{B}_{1}(r_{\tau(j)})+O(j^{-\frac{3}{2}}(\log
N)^{\alpha}).$

  * •

For $j=2m_{N},2m_{N}+1,\dots,2N-1$, we have

$\log\widetilde{h}_{j}=-2NV_{\widetilde{\tau}(j)}(r_{\widetilde{\tau}(j)})+\frac{1}{2}\log\Big{(}\frac{\pi
r_{\widetilde{\tau}(j)}^{2}}{N\Delta
Q(r_{\widetilde{\tau}(j)})}\Big{)}+\frac{1}{2N}\mathfrak{B}_{1}(r_{\widetilde{\tau}(j)})+O(j^{-\frac{3}{2}}(\log
N)^{\alpha}).$

Here $\alpha>0$ is a small constant.

###### Proof.

This lemma can be shown in a similar way to Lemma 2.1. Recall that $r_{\tau}$
satisfies $r_{\tau}=O(\tau^{\frac{1}{2}})$ as $\tau\to 0$. Note that for
$d\geq 1$

(3.2)
$|V_{\tau}^{(d)}(r_{\tau})|=|q^{(d)}(r_{\tau})+(-1)^{d}\,2\tau\,(d-1)!\,r_{\tau}^{-d}|\leq
C_{1}(1+\tau^{-\frac{d}{2}+1}),$

where $C_{1}>0$ is a constant that can be taken uniformly for all $\tau$. We
now split the integral for $h_{j}$ by

(3.3) $\displaystyle h_{j}$
$\displaystyle=\int_{0}^{\infty}2r^{2j+1}e^{-Nq(r)}\,dr=\int_{0}^{\infty}2re^{-NV_{\tau(j)}(r)}\,dr$
(3.4)
$\displaystyle=\int_{r_{\tau(j)}-\delta_{N}}^{r_{\tau(j)}+\delta_{N}}2re^{-NV_{\tau(j)}(r)}\,dr+\int_{|r-r_{\tau(j)}|>\delta_{N}}2re^{-NV_{\tau(j)}(r)}\,dr$

and first compute the integral over the outer region. For $m_{N}\leq j<N$, we
have

$r_{\tau(j+\frac{1}{2})}-r_{\tau(j)}=O((jN)^{-\frac{1}{2}}).$

Since $V_{\tau}$ is increasing in $(r_{\tau},\infty)$, the Taylor series
expansion for $V_{\tau}$ gives

(3.5) $\displaystyle\begin{split}V_{\tau(j+\frac{1}{2})}(r)&\geq
V_{\tau(j+\frac{1}{2})}(r_{\tau(j)}+\delta_{N})\\\
&=V_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})+\frac{1}{2}V^{\prime\prime}_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})\big{(}r_{\tau(j)}+\delta_{N}-r_{\tau(j+\frac{1}{2})}\big{)}^{2}+O(\tau(j)^{-\frac{1}{2}}\delta_{N}^{3})\end{split}$

for all $r>r_{\tau(j)}+\delta_{N}$. Here,
$O(\tau(j)^{-\frac{1}{2}}\delta_{N}^{3})=O(N^{-1}j^{-\frac{1}{2}}(\log
N)^{3})$. Similarly, since $V_{\tau}$ is decreasing in $(0,r_{\tau})$, we have

(3.6) $\displaystyle\begin{split}V_{\tau(j+\frac{1}{2})}(r)&\geq
V_{\tau(j+\frac{1}{2})}(r_{\tau(j)}-\delta_{N})\\\
&=V_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})+\frac{1}{2}V^{\prime\prime}_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})\big{(}r_{\tau(j)}-\delta_{N}-r_{\tau(j+\frac{1}{2})}\big{)}^{2}+O(\tau(j)^{-\frac{1}{2}}\delta_{N}^{3})\end{split}$

for all $r<r_{\tau(j)}-\delta_{N}$. Using the Taylor series for $V_{\tau}$
again, we have

(3.7)
$\displaystyle\begin{split}V_{\tau(j+\frac{1}{2})}(r_{\tau(j+\frac{1}{2})})-V_{\tau(j)}(r_{\tau(j)})&=V_{\tau(j)}(r_{\tau(j+\frac{1}{2})})-V_{\tau(j)}(r_{\tau(j)})-\frac{1}{N}\log
r_{\tau(j+\frac{1}{2})}\\\ &=O((jN)^{-1})-\frac{1}{N}\log
r_{\tau(j+\frac{1}{2})}.\end{split}$

Thus, it follows from (3.5), (3.6), and (3.7) that

(3.8) $\displaystyle\begin{split}&\quad
e^{NV_{\tau(j)}(r_{\tau(j)})}\int_{|r-r_{\tau(j)}|>\delta_{N}}2r^{2j+1}e^{-Nq(r)}\,dr=\int_{|r-r_{\tau(j)}|>\delta_{N}}2e^{-N\big{(}V_{\tau(j+\frac{1}{2})}(r)-V_{\tau(j)}(r_{\tau(j)})\big{)}}\,dr\\\
&\leq e^{-c_{1}(\log N)^{2}}e^{\frac{N-M}{N}\log
r_{\tau(j+\frac{1}{2})}}\int_{|r-r_{\tau(j)}|>\delta_{N}}2e^{-M\big{(}V_{\tau(j+\frac{1}{2})}(r)-V_{\tau(j)}(r_{\tau(j)})\big{)}}\,dr=O(e^{-c_{2}(\log
N)^{2}})\end{split}$

for some $c_{1},c_{2}>0$. Here $M$ is given by (2.13). For the integral near
the critical point $r_{\tau(j)}$, we use the Taylor series expansion to obtain

$\displaystyle\quad\int_{r_{\tau(j)}-\delta_{N}}^{r_{\tau(j)}+\delta_{N}}2re^{-NV_{\tau(j)}(r)}\,dr$
$\displaystyle=e^{-NV_{\tau(j)}(r_{\tau(j)})}\int_{-\delta_{N}}^{\delta_{N}}2(r_{\tau(j)}+t)\,e^{-N(\frac{1}{2}V_{\tau(j)}^{\prime\prime}(r_{\tau(j)})t^{2}+\frac{1}{6}V_{\tau(j)}^{(3)}(r_{\tau(j)})t^{3}+\frac{1}{24}V_{\tau(j)}^{(4)}(r_{\tau(j)})t^{4}+O(\tau(j)^{-\frac{3}{2}}|t|^{5}))}dt$
$\displaystyle=e^{-NV_{\tau(j)}(r_{\tau(j)})}\frac{1}{\sqrt{N}}\int_{-\sqrt{N}\delta_{N}}^{\sqrt{N}\delta_{N}}2e^{-2\Delta
Q(r_{\tau(j)})t^{2}}\,\Big{(}r_{\tau(j)}+\frac{t}{\sqrt{N}}\Big{)}$
$\displaystyle\hskip
100.00015pt\Big{(}1-\frac{V_{\tau(j)}^{(3)}(r_{\tau(j)})}{6\sqrt{N}}t^{3}-\frac{V_{\tau(j)}^{(4)}(r_{\tau(j)})}{24N}t^{4}+\frac{(V_{\tau(j)}^{(3)}(r_{\tau(j)}))^{2}}{72N}t^{6}+\epsilon^{\prime}_{N,1}\Big{)}\,dt$

where $\epsilon^{\prime}_{N,1}=O(j^{-\frac{3}{2}}(\log N)^{\alpha})$ for some
$\alpha>0$. Combining the all of the above, we obtain

$\displaystyle
h_{j}=e^{-NV_{\tau(j)}(r_{\tau(j)})}\Big{[}\frac{1}{\sqrt{N}}\Big{(}\frac{2\pi
r_{\tau(j)}^{2}}{\Delta
Q(r_{\tau(j)})}\Big{)}^{\frac{1}{2}}\Big{(}1+\frac{1}{N}\mathfrak{B}_{1}(r_{\tau(j)})+\epsilon_{N,1}\Big{)}+\epsilon_{N,2}\Big{]}$

where $\epsilon_{N,1}=O(j^{-\frac{3}{2}}(\log N)^{\alpha})$ and
$\epsilon_{N,2}=O(e^{-c_{2}(\log N)^{2}})$. ∎

###  3.2. Random normal matrix ensemble

In this subsection, we show Theorem 1.2 (i). According to the asymptotic
expansions of $h_{j}$ given in Lemmas 3.1 and 3.2, we analyze the summation in
(1.38) through the decomposition

(3.9) $\sum_{j=0}^{N-1}\log h_{j}=\sum_{j=0}^{m_{N}-1}\log
h_{j}+\sum_{j=m_{N}}^{N-1}\log h_{j}.$

The asymptotic behaviors of each summation on the right-hand side of (3.9) are
given in Lemma 3.3 and 3.7, respectively.

######  Lemma 3.3.

As $N\to\infty$, we have

$\displaystyle\sum_{j=0}^{m_{N}-1}\log h_{j}$
$\displaystyle=-m_{N}Nq(0)-\frac{m_{N}(m_{N}+1)}{2}\Big{(}\log
N+\log\big{(}\frac{1}{2}q^{\prime\prime}(0)\big{)}\Big{)}$
$\displaystyle\quad+\frac{1}{2}m_{N}^{2}\log
m_{N}-\frac{3}{4}m_{N}^{2}+\frac{\log(2\pi)}{2}m_{N}-\frac{1}{12}\log
m_{N}+\zeta^{\prime}(-1)+O(m_{N}^{-2}+N^{-\frac{1}{2}(1-5\epsilon)}(\log
N)^{3}).$

###### Proof.

By Lemma 3.1, we have

$\displaystyle\sum_{j=0}^{m_{N}-1}\log
h_{j}=-m_{N}Nq(0)-\frac{m_{N}(m_{N}+1)}{2}\Big{(}\log
N+\log\big{(}\frac{1}{2}q^{\prime\prime}(0)\big{)}\Big{)}+\log
G(m_{N}+1)+O(N^{-\frac{1}{2}(1-5\epsilon)}(\log N)^{3}),$

where $G$ is the Barnes $G$-function that can be defined recursively by

(3.10) $G(z+1)=\Gamma(z)G(z),\qquad G(1)=1.$

Now lemma follows from (1.42). ∎

######  Lemma 3.4.

As $N\to\infty$, we have

$\displaystyle\sum_{j=m_{N}}^{N-1}V_{\tau(j)}(r_{\tau(j)})$
$\displaystyle=NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)-\frac{1}{6N}\log r_{1}$
$\displaystyle\quad-
m_{N}q(0)-\frac{3}{4}\frac{m_{N}^{2}}{N}+\frac{1}{2N}\Big{(}m_{N}^{2}-m_{N}+\frac{1}{6}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+\frac{m_{N}}{2N}+O(N^{-\frac{1}{2}(3-5\epsilon)}).$

###### Proof.

By applying the Euler-Maclaurin formula (1.40), we have

(3.11)
$\displaystyle\begin{split}\sum_{j=m_{N}}^{N-1}V_{\tau(j)}(r_{\tau(j)})&=\int_{m_{N}}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt-\frac{1}{2}\Big{(}V_{\tau(N)}(r_{\tau(N)})-V_{\tau(m_{N})}(r_{\tau(m_{N})})\Big{)}\\\
&\quad+\frac{1}{12}\Big{(}{\partial}_{t}(V_{\tau(t)}(r_{\tau(t)}))\big{|}_{t=N}-{\partial}_{t}(V_{\tau(t)}(r_{\tau(t)}))\big{|}_{t=m_{N}}\Big{)}+O(N^{-1-2\epsilon}).\end{split}$

Here we have used
${\partial}_{t}^{3}(V_{\tau(t)}(r_{\tau(t)}))\big{|}_{t=m_{N}}=O(N^{-3}(\tau(m_{N}))^{-2})=O(m_{N}^{-2}N^{-1})$
and $B_{2}=1/6$. By the change of variables $s=r_{\tau(t)}$ and the formula
(1.14) of $I_{Q}[\mu_{Q}]$, we obtain

(3.12)
$\displaystyle\quad\int_{m_{N}}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt=2N\int_{r_{\tau(m_{N})}}^{r_{1}}(q(s)-sq^{\prime}(s)\log
s)s\Delta Q(s)\,ds$ (3.13) $\displaystyle=N\Big{(}\frac{1}{2}\int_{S\setminus
S_{\tau(m_{N})}}\\!\\!\\!Q\cdot\Delta Q\,dA-\log r_{1}+(\tau(m_{N}))^{2}\log
r_{\tau(m_{N})}+\frac{1}{2}(q(r_{1})-\tau(m_{N})\cdot
q(r_{\tau(m_{N})}))\Big{)}$ (3.14)
$\displaystyle=NI_{Q}[\mu_{Q}]-\frac{N}{2}\int_{S_{\tau(m_{N})}}Q\cdot\Delta
QdA+\frac{m_{N}^{2}}{N}\log
r_{\tau(m_{N})}-\frac{m_{N}}{2}q(r_{\tau(m_{N})}).$

Observe here that

(3.15)
$r_{\tau}=\Big{(}\frac{2\tau}{q^{\prime\prime}(0)}\Big{)}^{\frac{1}{2}}+O(\tau)=\Big{(}\frac{\tau}{\Delta
Q(0)}\Big{)}^{\frac{1}{2}}+O(\tau)\qquad\textup{as }\tau\to 0.$

Thus we have

(3.16) $\displaystyle\log r_{\tau(m_{N})}$
$\displaystyle=\frac{1}{2}\log\Big{(}\frac{\tau(m_{N})}{\Delta
Q(0)}\Big{)}+O(\tau(m_{N})^{\frac{1}{2}})=\frac{1}{2}\log\Big{(}\frac{\tau(m_{N})}{\Delta
Q(0)}\Big{)}+O(N^{-\frac{1}{2}(1-\epsilon)}),$ (3.17) $\displaystyle
q(r_{\tau(m_{N})})$
$\displaystyle=q(0)+\frac{1}{2}q^{\prime\prime}(0)(r_{\tau(m_{N})})^{2}+O((\tau(m_{N}))^{\frac{3}{2}})=q(0)+\tau(m_{N})+O(N^{-\frac{3}{2}(1-\epsilon)}),$

and

$\displaystyle\frac{1}{2}\int_{S_{\tau(m_{N})}}Q\cdot\Delta Q\,dA$
$\displaystyle=\frac{1}{4}\int_{0}^{r_{\tau(m_{N})}}q(s)\cdot(sq^{\prime}(s))^{\prime}\,ds=\frac{1}{2}\tau(m_{N})\cdot
q(r_{\tau(m_{N})})-\frac{1}{4}\int_{0}^{r_{\tau(m_{N})}}s(q^{\prime}(s))^{2}\,ds$
$\displaystyle=\frac{1}{2}\tau(m_{N})\cdot
q(r_{\tau(m_{N})})-\frac{1}{4}(q^{\prime\prime}(0))^{2}\int_{0}^{r_{\tau(m_{N})}}s^{3}\,ds+O(r_{\tau(m_{N})}^{5})$
$\displaystyle=\frac{1}{2}\tau(m_{N})\cdot
q(r_{\tau(m_{N})})-\frac{1}{4}\tau(m_{N})^{2}+O(N^{-\frac{5}{2}(1-\epsilon)}).$

Combining all of the above asymptotic expansions, we obtain

(3.18)
$\displaystyle\begin{split}\int_{m_{N}}^{N}V_{\tau(t)}(r_{\tau(t)})\,dt&=NI_{Q}[\mu_{Q}]-m_{N}q(r_{\tau(m_{N})})+\frac{1}{4}\frac{m_{N}^{2}}{N}+\frac{m_{N}^{2}}{N}\log
r_{\tau(m_{N})}+O(N^{-\frac{1}{2}(3-5\epsilon)})\\\
&=NI_{Q}[\mu_{Q}]-m_{N}q(0)-\frac{3}{4}\,\frac{m_{N}^{2}}{N}+\frac{m_{N}^{2}}{2N}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\frac{1}{2}(3-5\epsilon)}).\end{split}$

Furthermore, it follows from the formula (1.16) of $U_{\mu_{Q}}(0)$, (3.16)
and (3.17) that

(3.19) $\displaystyle\begin{split}&\quad
V_{\tau(N)}(r_{\tau(N)})-V_{\tau(m_{N})}(r_{\tau(m_{N})})=q(r_{1})-2\log
r_{1}-q(r_{\tau(m_{N})})+2\tau(m_{N})\cdot\log r_{\tau(m_{N})}\\\
&=q(r_{1})-2\log
r_{1}-q(0)-\frac{m_{N}}{N}+\frac{m_{N}}{N}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\frac{3}{2}(1-\epsilon)})\\\
&=2U_{\mu_{Q}}(0)-\frac{m_{N}}{N}+\frac{m_{N}}{N}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\frac{3}{2}(1-\epsilon)}).\end{split}$

Similarly, we have

(3.20)
$\displaystyle\begin{split}{\partial}_{t}(V_{\tau(t)}(r_{\tau(t)}))\big{|}_{t=N}-{\partial}_{t}(V_{\tau(t)}(r_{\tau(t)}))\big{|}_{t=m_{N}}&=\frac{2}{N}(\log{r_{\tau(m_{N})}}-\log{r_{1}})\\\
&=\frac{1}{N}\log\Big{(}\frac{m_{N}}{N\Delta Q(0)}\Big{)}-\frac{2}{N}\log
r_{1}+O(N^{-\frac{1}{2}(3-\epsilon)}).\end{split}$

Now lemma follows from (3.11), (3.18), (3.19) and (3.20). ∎

######  Lemma 3.5.

As $N\to\infty$, we have

(3.21) $\sum_{j=m_{N}}^{N-1}\log\Delta
Q(r_{\tau(j)})=NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(0)}\Big{)}-m_{N}\log\Delta
Q(0)+O(N^{-\frac{1}{2}(1-\epsilon)}),$

and

(3.22) $\sum_{j=m_{N}}^{N-1}\log
r_{\tau(j)}=-NU_{\mu_{Q}}(0)+\frac{m_{N}}{2}-\frac{1}{2}\log
r_{1}-\Big{(}\frac{m_{N}}{2}-\frac{1}{4}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\epsilon}).$

###### Proof.

Using the Euler-maclaurin formula (1.40),

$\displaystyle\sum_{j=m_{N}}^{N-1}\log\Delta Q(r_{\tau(j)})=$ $\displaystyle
N\int_{S\setminus S_{\tau(m_{N})}}\log\Delta
Q\,d\mu_{Q}-\frac{1}{2}\log\Big{(}\frac{\Delta Q(r_{1})}{\Delta
Q(r_{\tau(m_{N})})}\Big{)}$
$\displaystyle+\frac{1}{12}\big{(}{\partial}_{t}\log\Delta
Q(r_{\tau(t)}))\big{|}_{t=N}-{\partial}_{t}\log\Delta
Q(r_{\tau(t)}))\big{|}_{t=m_{N}}\big{)}+o(N^{-1}\tau(m_{N})^{-\frac{1}{2}}).$

We verify from (3.15) that

$\displaystyle\int_{S_{\tau(m_{N})}}\log\Delta Q\,d\mu_{Q}$
$\displaystyle=\Delta Q(0)\cdot\log\Delta Q(0)\cdot
r_{\tau(m_{N})}^{2}+O(\tau(m_{N})^{3})=\frac{m_{N}}{N}\log\Delta
Q(0)+O(N^{-3(1-\epsilon)})$

and

$\displaystyle{\partial}_{t}\log\Delta
Q(r_{\tau(t)})\big{|}_{t=m_{N}}=\frac{1}{2N}\frac{{\partial}_{r}\Delta
Q(r_{\tau(m_{N})})}{r_{\tau(m_{N})}(\Delta
Q(r_{\tau(m_{N})}))^{2}}=O(N^{-1}\tau(m_{N})^{-\frac{1}{2}})=O(N^{-\frac{1}{2}(1+\epsilon)}).$

Observe that $\log\Delta Q(r_{\tau(m_{N})})=\log\Delta
Q(0)+O(r_{\tau(m_{N})})$. Combining above equations, we obtain the first
assertion. Similarly, by using the Euler-Maclaurin formula (1.40), (3.16), and
(3.17), we have

$\displaystyle\sum_{j=m_{N}}^{N}\log r_{\tau(j)}$
$\displaystyle=N\int_{S\setminus
S_{\tau(m_{N})}}\log|z|\,d\mu_{Q}-\frac{1}{2}\log\Big{(}\frac{r_{1}}{r_{\tau(m_{N})}}\Big{)}+O(m_{N}^{-1})$
$\displaystyle=\frac{N}{2}\Big{(}2\log r_{1}-2\tau(m_{N})\log
r_{\tau(m_{N})}-q(r_{1})+q(r_{\tau(m_{N})})\Big{)}-\frac{1}{2}\log\Big{(}\frac{r_{1}}{r_{\tau(m_{N})}}\Big{)}$
$\displaystyle=N\Big{(}\log
r_{1}-\frac{q(r_{1})-q(0)}{2}+\frac{1}{2}\tau(m_{N})\Big{)}-\frac{1}{2}\log
r_{1}-\Big{(}\frac{m_{N}}{2}-\frac{1}{4}\Big{)}\log\Big{(}\frac{\tau(m_{N})}{\Delta
Q(0)}\Big{)}+O(N^{-\epsilon}),$

which completes the proof. ∎

######  Lemma 3.6.

As $N\to\infty$, we have

$\displaystyle\frac{1}{N}\sum_{j=m_{N}}^{N-1}\mathfrak{B}_{1}(r_{\tau(j)})$
$\displaystyle=F_{Q}[\mathbb{D}_{r_{1}}]+\frac{1}{3}\log
r_{1}-\frac{1}{12}\log\Big{(}\frac{m_{N}}{N}\Big{)}-\frac{1}{4}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(0)}\Big{)}$ $\displaystyle\quad+\frac{1}{6}\log\Delta
Q(0)+O(N^{-\epsilon}+N^{-\frac{1}{2}(1-\epsilon)}),$

where $F_{Q}[\mathbb{D}_{r_{1}}]$ is given in (1.23).

###### Proof.

Observe that

$\displaystyle\sum_{j=m_{N}}^{N-1}\mathfrak{B}_{1}(r_{\tau(j)})$
$\displaystyle=N\int_{S\setminus
S_{\tau(m_{N})}}\mathfrak{B}_{1}\,d\mu_{Q}+O(\tau(m_{N})^{-1}).$

By (3.15) and (3.16),

$\displaystyle\quad\int_{S\setminus
S_{\tau(m_{N})}}\\!\\!\\!\mathfrak{B}_{1}\,d\mu_{Q}=\frac{1}{6}\log\Big{(}\frac{r_{1}}{r_{\tau(m_{N})}}\Big{)}-\frac{19}{48}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta
Q(r_{\tau(m_{N})})}\Big{)}-\frac{1}{16}\int_{r_{m_{N}}}^{r_{1}}\Big{[}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{\Delta Q(r)}-\frac{5}{3}\Big{(}\frac{{\partial}_{r}\Delta Q(r)}{\Delta
Q(r)}\Big{)}^{2}\Big{]}r\,dr$ $\displaystyle=\frac{1}{6}\log
r_{1}-\frac{1}{12}\log\Big{(}\frac{\tau(m_{N})}{\Delta
Q(0)}\Big{)}-\frac{19}{48}\log\Big{(}\frac{\Delta Q(r_{1})}{\Delta
Q(0)}\Big{)}-\frac{1}{16}\int_{0}^{r_{1}}\Big{[}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{\Delta Q(r)}-\frac{5}{3}\Big{(}\frac{{\partial}_{r}\Delta Q(r)}{\Delta
Q(r)}\Big{)}^{2}\Big{]}r\,dr+O(N^{-\frac{1}{2}(1-\epsilon)}).$

Thus we have

$\displaystyle\frac{1}{N}\sum_{j=m_{N}}^{N-1}\mathfrak{B}_{1}(r_{\tau(j)})$
$\displaystyle=\frac{1}{6}\log
r_{1}-\frac{1}{12}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}-\frac{19}{48}\log\Big{(}\frac{\Delta Q(r_{1})}{\Delta
Q(0)}\Big{)}$
$\displaystyle\quad-\frac{1}{16}\int_{0}^{r_{1}}\Big{[}\frac{{\partial}_{r}^{2}\Delta
Q(r)}{\Delta Q(r)}-\frac{5}{3}\Big{(}\frac{{\partial}_{r}\Delta Q(r)}{\Delta
Q(r)}\Big{)}^{2}\Big{]}r\,dr+O(N^{-\epsilon}+N^{-\frac{1}{2}(1-\epsilon)}).$

Now lemma follows from (LABEL:Delta_Q_partial_integral). ∎

######  Lemma 3.7.

As $N\to\infty$, we have

$\displaystyle\sum_{j=m_{N}}^{N-1}\log h_{j}$
$\displaystyle=-N^{2}I_{Q}[\mu_{Q}]+\frac{N-m_{N}}{2}\log\Big{(}\frac{2\pi}{N}\Big{)}+Nm_{N}q(0)+\frac{3}{4}m_{N}^{2}-\frac{1}{2}NE_{Q}[\mu_{Q}]$
$\displaystyle\quad-\frac{1}{2}m_{N}(m_{N}+1)\log\Big{(}\frac{1}{\Delta
Q(0)}\Big{)}+F_{Q}[\mathbb{D}_{r_{1}}]-\frac{1}{2}\Big{(}m_{N}^{2}-\frac{1}{6}\Big{)}\log\Big{(}\frac{m_{N}}{N}\Big{)}+O(N^{-\frac{1}{12}}(\log
N)^{3}).$

###### Proof.

By Lemma 3.2, we have

$\displaystyle\sum_{j=m_{N}}^{N-1}\log h_{j}=$
$\displaystyle\sum_{j=m_{N}}^{N-1}\Big{(}-NV_{\tau(j)}(r_{\tau(j)})+\log
r_{\tau(j)}-\frac{1}{2}\log\Delta
Q(r_{\tau(j)})+\frac{1}{N}\mathfrak{B}_{1}(r_{\tau(j)})\Big{)}$
$\displaystyle+\frac{(N-m_{N})}{2}\log\Big{(}\frac{2\pi}{N}\Big{)}+O(m_{N}^{-\frac{1}{2}}(\log
N)^{\alpha})+O(N^{-\frac{1}{2}(1-5\epsilon)}).$

Now Lemmas 3.2, 3.4, 3.5 and 3.6 complete the proof. Here, for the error term,
we take $\epsilon=1/6$ so that $\epsilon/2=(1-5\epsilon)/2=1/12$. ∎

We are now ready to prove the first assertion of Theorem 1.1.

###### Proof of Theorem 1.2 (i).

By combining Lemmas 3.3, 3.7 and (3.9), we obtain

(3.23) $\sum_{j=0}^{N-1}\log h_{j}=-N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N-\frac{N}{2}\Big{(}E_{Q}[\mu_{Q}]-\log(2\pi)\Big{)}-\frac{\log
N}{12}+\zeta^{\prime}(-1)+F_{Q}[\mathbb{D}_{r_{1}}]+O(N^{-\frac{1}{12}}(\log
N)^{3}).$

Note here that all the terms involving $m_{N}$ in Lemmas 3.3 and 3.7 vanish.
Then the desired asymptotic expansion (1.24) follows from (1.38) and (2.30).
This completes the proof. ∎

###  3.3. Planar symplectic ensemble

In this subsection, we prove the second assertion of Theorem 1.2.

As a counterpart of Lemma 3.3, we have the following.

######  Lemma 3.8.

As $N\to\infty$, we have

$\displaystyle\sum_{j=0}^{m_{N}-1}\log\widetilde{h}_{2j+1}$
$\displaystyle=-2m_{N}Nq(0)+m_{N}^{2}\log m_{N}+\Big{(}\log
2-\frac{3}{2}-\log(Nq^{\prime\prime}(0))\Big{)}m_{N}^{2}$
$\displaystyle\quad+\frac{1}{2}m_{N}\log m_{N}+\Big{(}\log
2-\frac{1}{2}+\frac{1}{2}\log\pi-\log(Nq^{\prime\prime}(0))\Big{)}m_{N}$
$\displaystyle\quad-\frac{1}{24}\log m_{N}+\frac{5}{24}\log
2+\frac{1}{2}\zeta^{\prime}(-1)+O(m_{N}^{-1}+N^{-\frac{1}{2}(1-5\epsilon)}(\log
N)^{3}).$

###### Proof.

By Lemma 3.1 and (4.9), we have

$\displaystyle\log\widetilde{h}_{2j+1}=-2Nq(0)-(2j+2)\log(Nq^{\prime\prime}(0))+\log\Big{(}\frac{2^{2j+1}}{\sqrt{\pi}}\Gamma(j+1)\Gamma(j+\tfrac{3}{2})\Big{)}+O(N^{-\frac{1}{2}(1-3\epsilon)}).$

Thus we have

$\displaystyle\sum_{j=0}^{m_{N}-1}\log\widetilde{h}_{2j+1}$
$\displaystyle=-2m_{N}Nq(0)-m_{N}(m_{N}+1)\log(Nq^{\prime\prime}(0))+m_{N}^{2}\log
2-\frac{m_{N}}{2}\log\pi$
$\displaystyle\quad+\log\Big{(}G(m_{N}+1)\frac{G(m_{N}+\frac{3}{2})}{G(\frac{3}{2})}\Big{)}+O(N^{-\frac{1}{2}(1-5\epsilon)}).$

Now lemma follows from the asymptotic expansion (1.42) of the Barnes $G$
function and

(3.24)
$G(\tfrac{1}{2})=2^{\frac{1}{24}}\,\exp\Big{(}\frac{3}{2}\zeta^{\prime}(-1)\Big{)}\pi^{-\frac{1}{4}},\qquad
G(\tfrac{3}{2})=G(\tfrac{1}{2})\Gamma(\tfrac{1}{2})=G(\tfrac{1}{2})\sqrt{\pi}.$

∎

######  Lemma 3.9.

As $N\to\infty,$ we have

$\displaystyle\sum_{j=m_{N}}^{N-1}V_{\widetilde{\tau}(2j+1)}(r_{\widetilde{\tau}(2j+1)})$
$\displaystyle=NI_{Q}[\mu_{Q}]+\frac{1}{12N}\log r_{1}$ $\displaystyle\quad-
m_{N}q(0)-\frac{3}{4}\frac{m_{N}^{2}}{N}+\frac{1}{4N}\Big{(}2m_{N}^{2}-\frac{1}{6}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\frac{1}{2}(3-5\epsilon)}).$

###### Proof.

By using Lemma 3.4 with $N\to 2N$, we have

$\displaystyle\sum_{j=2m_{N}}^{2N-1}V_{\widetilde{\tau}(j)}(r_{\widetilde{\tau}(j)})$
$\displaystyle=2NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)-\frac{1}{12N}\log r_{1}$
$\displaystyle\quad-2m_{N}q(0)-\frac{3}{2}\frac{m_{N}^{2}}{N}+\frac{1}{4N}\Big{(}4m_{N}^{2}-2m_{N}+\frac{1}{6}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+\frac{m_{N}}{2N}+O(N^{-\frac{1}{2}(3-5\epsilon)}).$

On the other hand, by the Euler-Maclaurin formula (1.40), we have

(3.25)
$\displaystyle\begin{split}\sum_{j=m_{N}}^{N-1}V_{\widetilde{\tau}(2j)}(r_{\widetilde{\tau}(2j)})&=\frac{1}{2}\int_{2m_{N}}^{2N}V_{\widetilde{\tau}(t)}(r_{\widetilde{\tau}(t)})\,dt-\frac{1}{2}\big{(}V_{\widetilde{\tau}(2N)}(r_{\widetilde{\tau}(2N)})-V_{\widetilde{\tau}(2m_{N})}(r_{\widetilde{\tau}(2m_{N})})\big{)}\\\
&\quad+\frac{1}{12}\Big{(}{\partial}_{t}(V_{\widetilde{\tau}(2t)}(r_{\widetilde{\tau}(2t)}))\big{|}_{t=N}-{\partial}_{t}(V_{\widetilde{\tau}(2t)}(r_{\widetilde{\tau}(2t)}))\big{|}_{t=m_{N}}\Big{)}+O(N^{-1-2\epsilon}).\end{split}$

Following the proof of Lemma 3.4, we have

(3.26)
$\displaystyle\begin{split}\sum_{j=m_{N}}^{N-1}V_{\widetilde{\tau}(2j)}(r_{\widetilde{\tau}(2j)})&=NI_{Q}[\mu_{Q}]-U_{\mu_{Q}}(0)-\frac{1}{6N}\log
r_{1}\\\ &\quad-
m_{N}q(0)-\frac{3}{4}\frac{m_{N}^{2}}{N}+\frac{1}{2N}(m_{N}^{2}-m_{N}+\frac{1}{6})\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+\frac{m_{N}}{2N}+O(N^{-\frac{1}{2}(3-5\epsilon)}),\end{split}$

which completes the proof. ∎

######  Lemma 3.10.

As $N\to\infty$, we have

(3.27) $\sum_{j=m_{N}}^{N-1}\log\Delta
Q(r_{\widetilde{\tau}(2j+1)})=NE_{Q}[\mu_{Q}]-m_{N}\log\Delta
Q(0)+O(N^{-\frac{1}{2}(1-\epsilon)}),$

and

(3.28) $\sum_{j=m_{N}}^{N-1}\log
r_{\widetilde{\tau}(2j+1)}=-NU_{\mu_{Q}}(0)+\frac{m_{N}}{2}-\frac{m_{N}}{2}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\epsilon}).$

###### Proof.

By Lemma 3.5 with $N\to 2N$, we have

$\displaystyle\sum_{j=2m_{N}}^{2N-1}\log\Delta Q(r_{\widetilde{\tau}(j)})$
$\displaystyle=2NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(0)}\Big{)}-2m_{N}\log\Delta
Q(0)+O(N^{-\frac{1}{2}(1-\epsilon)}),$
$\displaystyle\sum_{j=2m_{N}}^{2N-1}\log r_{\widetilde{\tau}(j)}$
$\displaystyle=-2NU_{\mu_{Q}}(0)+m_{N}-\frac{1}{2}\log
r_{1}-\Big{(}m_{N}-\frac{1}{4}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\epsilon}).$

Following the proof of Lemma 3.5, we also have

$\displaystyle\sum_{j=m_{N}}^{N-1}\log\Delta Q(r_{\widetilde{\tau}(2j)})$
$\displaystyle=NE_{Q}[\mu_{Q}]-\frac{1}{2}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(0)}\Big{)}-2m_{N}\log\Delta
Q(0)+O(N^{-\frac{1}{2}(1-\epsilon)}),$ $\displaystyle\sum_{j=m_{N}}^{N-1}\log
r_{\widetilde{\tau}(2j)}$
$\displaystyle=-NU_{\mu_{Q}}(0)+m_{N}-\frac{1}{2}\log
r_{1}-\Big{(}m_{N}-\frac{1}{4}\Big{)}\log\Big{(}\frac{m_{N}}{N\Delta
Q(0)}\Big{)}+O(N^{-\epsilon}).$

This completes the proof. ∎

######  Lemma 3.11.

As $N\to\infty$, we have

$\displaystyle\frac{1}{2N}\sum_{j=m_{N}}^{N-1}\mathfrak{B}_{1}(r_{\widetilde{\tau}(2j+1)})=\frac{1}{2}F_{Q}[\mathbb{D}_{r_{1}}]+\frac{1}{6}\log
r_{1}-\frac{1}{24}\log\Big{(}\frac{m_{N}}{N}\Big{)}-\frac{1}{8}\log\Big{(}\frac{\Delta
Q(r_{1})}{\Delta Q(0)}\Big{)}+\frac{1}{12}\log\Delta Q(0)+O(m_{N}^{-1}).$

###### Proof.

This lemma follows along the same lines of Lemma 3.6. ∎

######  Lemma 3.12.

As $N\to\infty$, we have

$\displaystyle\sum_{j=m_{N}}^{N-1}\log\widetilde{h}_{2j+1}$
$\displaystyle=-2N^{2}I_{Q}[\mu_{Q}]-\frac{N}{2}\log
N+N\Big{(}\frac{\log\pi}{2}-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}$
$\displaystyle\quad+\frac{3}{2}m_{N}^{2}-\Big{(}m_{N}^{2}-\frac{1}{24}\Big{)}\log\Big{(}\frac{m_{N}}{N}\Big{)}+m_{N}\Big{(}2Nq(0)+\frac{1}{2}+\frac{1}{2}\log\Big{(}\frac{m_{N}}{\pi}\Big{)}\Big{)}$
$\displaystyle\quad-\Big{(}m_{N}^{2}+\frac{1}{8}\Big{)}\log\Big{(}\frac{1}{\Delta
Q(0)}\Big{)}+\frac{1}{2}F_{Q}[\mathbb{D}_{r_{1}}]+\frac{1}{8}\log\Big{(}\frac{1}{\Delta
Q(r_{1})}\Big{)}+O(m_{N}^{-\frac{1}{2}}(\log N)^{\alpha}).$

###### Proof.

Note that by Lemma 3.2, we have

$\displaystyle\sum_{j=m_{N}}^{N-1}\log\widetilde{h}_{2j+1}=$
$\displaystyle\sum_{j=m_{N}}^{N-1}\Big{(}-2NV_{\widetilde{\tau}(2j+1)}(r_{\widetilde{\tau}(2j+1)})+\log
r_{\widetilde{\tau}(2j+1)}-\frac{1}{2}\log\Delta
Q(r_{\widetilde{\tau}(2j+1)})+\frac{1}{2N}\mathfrak{B}_{1}(r_{\widetilde{\tau}(2j+1)})\Big{)}$
$\displaystyle+\frac{(N-m_{N})}{2}\log\Big{(}\frac{\pi}{N}\Big{)}+O(m_{N}^{-\frac{1}{2}}(\log
N)^{\alpha}).$

The lemma now follows from Lemmas 3.9, 3.10 and 3.11. ∎

We now finish the proof of Theorem 1.2.

###### Proof of Theorem 1.2 (ii).

Combining Lemmas 3.8 and 3.12, after long but straightforward simplifications,
we obtain

$\displaystyle\quad\sum_{j=0}^{N-1}\log(2\widetilde{h}_{2j+1})=N\log
2+\sum_{j=0}^{m_{N}-1}\log\widetilde{h}_{2j+1}+\sum_{j=m_{N}}^{N-1}\log\widetilde{h}_{2j+1}$
$\displaystyle=-2N^{2}I_{Q}[\mu_{Q}]-\frac{1}{2}N\log
N+N\Big{(}\frac{\log(4\pi)}{2}-U_{\mu_{Q}}(0)-\frac{1}{2}E_{Q}[\mu_{Q}]\Big{)}-\frac{1}{24}\log
N$ $\displaystyle\quad+\frac{5}{24}\log
2+\frac{1}{2}\zeta^{\prime}(-1)+\frac{1}{2}F_{Q}[\mathbb{D}_{r_{1}}]+\frac{1}{8}\log\Big{(}\frac{\Delta
Q(0)}{\Delta Q(r_{1})}\Big{)}+O(m_{N}^{-1}+N^{-\frac{1}{2}(1-5\epsilon)}(\log
N)^{3}).$

Again, it is noteworthy that all the terms in Lemmas 3.8 and 3.12 involving
$m_{N}$ cancel each other. Now the asymptotic behavior (1.25) follows from
(1.38) and the asymptotic expansion (2.30) of $\log N!$ with $\epsilon=1/6$. ∎

##  4\. Examples: Mittag-Leffler ensemble and truncated unitary ensemble

This section presents examples of our Theorems 1.1 and 1.2 for some well-known
planar point processes.

###  4.1. Mittag-Leffler ensemble

Let us consider the potential

(4.1) $Q(z)=|z|^{2\lambda}-2c\log|z|,\quad c>0.$

The models (1.1) and (1.2) associated with the potential (4.1) are known as
the Mittag-Leffler ensemble [9]. We refer to [13, 19, 22] and [5] for recent
studies on complex and symplectic Mittag-Leffler ensembles, respectively.
Using (1.13), we have

(4.2) $r_{0}=\Big{(}\frac{c}{\lambda}\Big{)}^{\frac{1}{2\lambda}},\qquad
r_{1}=\Big{(}\frac{1+c}{\lambda}\Big{)}^{\frac{1}{2\lambda}},\qquad\Delta
Q(z)=\lambda^{2}|z|^{2\lambda-2}.$

In particular, by (4.2), the Mittag-Leffler ensemble (4.1) falls into the
class considered in Theorem 1.1.

By direct computations using (1.10), (1.14) and (1.16), we have

(4.3) $\displaystyle I_{Q}[\mu_{Q}]$
$\displaystyle=\frac{1}{2\lambda}\log\Big{(}\frac{c^{c^{2}}}{(1+c)^{(1+c)^{2}}}\Big{)}+\frac{1+2c}{2\lambda}\Big{(}\log\lambda+\frac{3}{2}\Big{)},$
(4.4) $\displaystyle E_{Q}[\mu_{Q}]$
$\displaystyle=\frac{1+\lambda}{\lambda}\log\lambda+\frac{1-\lambda}{\lambda}\Big{(}1+\log\Big{(}\frac{c^{c}}{(1+c)^{1+c}}\Big{)}\Big{)}.$

It also follows from (1.17) and

(4.5) $r\frac{(\partial_{r}\Delta)Q(r)}{\Delta Q(r)}=2\lambda-2,\qquad
r^{2}\Delta Q(r)=\lambda^{2}r^{2\lambda}$

that

(4.6)
$F_{Q}[\mathbb{A}_{r_{0},r_{1}}]=\Big{(}\frac{\lambda}{6}-\frac{(\lambda-1)^{2}}{6}\Big{)}\log\Big{(}\frac{r_{0}}{r_{1}}\Big{)}=-\frac{\lambda^{2}-3\lambda+1}{12\lambda}\log\Big{(}\frac{c}{1+c}\Big{)}.$

On the other hand, by using (1.38),

(4.7)
$h_{j}=2\int_{0}^{\infty}r^{2j+1+2cN}e^{-Nr^{2\lambda}}\,dr=\frac{1}{\lambda}N^{-\frac{j+1+cN}{\lambda}}\Gamma\Big{(}\frac{j+Nc+1}{\lambda}\Big{)},$

and the analogous formula for $\widetilde{h}_{j}$, we have

(4.8)
$Z_{N}=\frac{N!}{\lambda^{N}}N^{-\frac{(2c+1)N^{2}+N}{2\lambda}}\prod_{j=0}^{N-1}\Gamma\Big{(}\frac{j+Nc+1}{\lambda}\Big{)},\qquad\widetilde{Z}_{N}=\frac{N!}{(\lambda/2)^{N}}\,(2N)^{-\frac{(2c+1)N^{2}+N}{\lambda}}\prod_{j=0}^{N-1}\Gamma\Big{(}\frac{j+Nc+1}{\lambda/2}\Big{)}.$

Furthermore, using the multiplication theorem of gamma function ([45, Eq.
(5.5.6)])

(4.9)
$\Gamma(nz)=(2\pi)^{\frac{1-n}{2}}n^{nz-\frac{1}{2}}\prod_{k=0}^{n-1}\Gamma(z+\tfrac{k}{n})$

and the characteristic property (3.10) of the Barnes $G$-function, we have
that for $\frac{1}{\lambda}\in\mathbb{N}$,

(4.10)
$Z_{N}=N!\,N^{-\frac{(2c+1)N^{2}+N}{2\lambda}}(2\pi)^{(\frac{1}{2}-\frac{1}{2\lambda})N}\Big{(}\frac{1}{\lambda}\Big{)}^{(\frac{1}{2\lambda}+\frac{c}{\lambda})N^{2}+(\frac{1}{\lambda}+\frac{1}{2}-\frac{1}{2\lambda})N}\prod_{k=0}^{\frac{1}{\lambda}-1}\frac{G(N+Nc+1+\lambda
k)}{G(Nc+1+\lambda k)}.$

Similarly, for $\frac{2}{\lambda}\in\mathbb{N}$, we have

(4.11)
$\widetilde{Z}_{N}=N!\,(2N)^{-\frac{(2c+1)N^{2}+N}{\lambda}}(2\pi)^{(\frac{1}{2}-\frac{1}{\lambda})N}\Big{(}\frac{2}{\lambda}\Big{)}^{(\frac{1}{\lambda}+\frac{2c}{\lambda})N^{2}+(\frac{2}{\lambda}+\frac{1}{2}-\frac{1}{\lambda})N}\prod_{k=0}^{\frac{2}{\lambda}-1}\frac{G(N+Nc+1+\frac{\lambda}{2}k)}{G(Nc+1+\frac{\lambda}{2}k)}.$

Then by using (1.42), one can directly check that the partition functions
(4.10) and (4.11) satisfy the expansions (1.18) and (1.19) with (4.3), (4.4)
and (4.6).

###  4.2. Truncated unitary ensemble

We now consider the potential

(4.12)
$Q(z)=\begin{cases}-\alpha\,\log\Big{(}1-\dfrac{|z|^{2}}{R^{2}(1+\alpha)}\Big{)}&\textup{if
}|z|\leq R\sqrt{1+\alpha},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\infty&\textup{otherwise},\end{cases}\qquad\alpha,R>0.$

The models associated with (4.12) correspond to the truncated unitary
ensembles at strong non-unitarity [38, 55]. These models provide a one-
parameter generalization of the Ginibre ensembles that can be recovered in the
extremal case, i.e. $\lim_{\alpha\to\infty}Q(z)=|z|^{2}/R^{2}$. (See [25, 48]
and references therein for recent works on these models.) In this case, we
have

(4.13) $r_{0}=0,\qquad r_{1}=R,\qquad\Delta
Q(z)=\frac{R^{2}\,\alpha(1+\alpha)}{(R^{2}(1+\alpha)-|z|^{2})^{2}}.$

From (4.13), we see that the truncated unitary ensembles are contained in the
class covered in Theorem 1.2. (The hard edge condition imposed in (4.12)
outside the droplet does not harm the proof of Theorem 1.2.)

It is easy to verify from (1.10), (1.14) and (1.16) that

(4.14) $\displaystyle I_{Q}[\mu_{Q}]$
$\displaystyle=-\frac{\alpha}{2}-\frac{\alpha(2+\alpha)}{2}\log\Big{(}\frac{\alpha}{1+\alpha}\Big{)}-\log
R,$ (4.15) $\displaystyle E_{Q}[\mu_{Q}]$
$\displaystyle=-2-(1+2\alpha)\log\Big{(}\frac{\alpha}{1+\alpha}\Big{)}-2\log
R.$

Since

(4.16) $r\frac{(\partial_{r}\Delta)Q(r)}{\Delta
Q(r)}\Big{|}_{r=R}=\frac{4r^{2}}{R^{2}(1+\alpha)-r^{2}}\Big{|}_{r=R}=\frac{4}{\alpha},\qquad
R^{2}\Delta Q(R)=\frac{1+\alpha}{\alpha},$

we deduce from (1.23) that

(4.17)
$F_{Q}[\mathbb{D}_{R}]=\frac{1}{12}\log\Big{(}\frac{\alpha}{1+\alpha}\Big{)}-\frac{1}{4\alpha}+\frac{1}{3}\Big{(}\frac{1}{\alpha}+\log\Big{(}\frac{\alpha}{1+\alpha}\Big{)}\Big{)}=\frac{1}{12}\Big{(}\frac{1}{\alpha}+5\log\Big{(}\frac{\alpha}{1+\alpha}\Big{)}\Big{)}.$

Notice here that $F_{Q}[\mathbb{D}_{R}]$ is independent of $R$, which is
consistent with the invariance of the $O(1)$-terms of (1.24) and (1.25) under
the dilation, see the remark below Theorem 1.2.

Using the Euler’s beta integral

(4.18)
$\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\,dt,\qquad(a,b>0),$

the orthogonal norm $h_{j}$ is computed as

(4.19)
$h_{j}=\int_{0}^{R\sqrt{1+\alpha}}r^{2j+1}\Big{(}1-\frac{r^{2}}{R^{2}(1+\alpha)}\Big{)}^{\alpha
N}\,dr=R^{2j+2}(1+\alpha)^{j+1}\frac{\Gamma(\alpha
N+1)\Gamma(j+1)}{\Gamma(\alpha N+j+2)}.$

Then by (1.38) and (3.10), we have

(4.20)
$Z_{N}=N!\,R^{N(N+1)}(1+\alpha)^{\frac{N^{2}}{2}+\frac{N}{2}}\,\Gamma(\alpha
N+1)^{N}\,\frac{G(N+1)G(\alpha N+2)}{G(\alpha N+N+2)}.$

Similarly, by (1.38) and the duplication formula of the gamma function (i.e.
(4.9) with $n=2$),

(4.21) $\begin{split}\widetilde{Z}_{N}&=N!\,R^{2N(N+1)}2^{-2\alpha
N^{2}}\,(1+\alpha)^{N^{2}+N}\,\Gamma(2\alpha N+1)^{N}\\\ &\quad\times
G(N+1)\frac{G(N+\frac{3}{2})}{G(\frac{3}{2})}\frac{G(\alpha N+2)}{G(\alpha
N+N+2)}\frac{G(\alpha N+\frac{3}{2})}{G(\alpha N+N+\frac{3}{2})}.\end{split}$

Then by using (1.42), it is again straightforward to check that the partition
functions (4.20) and (4.21) satisfy the expansions (1.24) and (1.25) with
(4.14), (4.15) and (4.17). In the extremal case where $\alpha\to\infty$, the
expansion of the partition function $Z_{N}$ of the complex Ginibre ensemble
appears in [52].

### Acknowledgements

We thank Christophe Charlier and Thomas Leblé for helpful discussions.

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