# Free energy expansions of a conditional GinUE and
large deviations of the smallest eigenvalue of the LUE

Sung-Soo Byun  Department of Mathematical Sciences and Research Institute of
Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
<EMAIL_ADDRESS>,  Seong-Mi Seo  Department of Mathematics, Chungnam
National University, 99 Daehak-ro, Yuseong-gu, Daejeon 34134, Republic of
Korea<EMAIL_ADDRESS>and  Meng Yang  Department of Mathematics, School of
Sciences, Great Bay University, Dongguan, 523000, China<EMAIL_ADDRESS>
###### Abstract.

We consider a planar Coulomb gas ensemble of size $N$ with the inverse
temperature $\beta=2$ and external potential $Q(z)=|z|^{2}-2c\log|z-a|$, where
$c>0$ and $a\in\mathbb{C}$. Equivalently, this model can be realised as $N$
eigenvalues of the complex Ginibre matrix of size $(c+1)N\times(c+1)N$
conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$.
Depending on the values of $c$ and $a$, the droplet reveals a phase
transition: it is doubly connected in the post-critical regime and simply
connected in the pre-critical regime. In both regimes, we derive precise
large-$N$ expansions of the free energy up to the $O(1)$ term, providing a
non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture
made for general planar Coulomb gas ensembles. As a consequence, our results
provide asymptotic behaviours of moments of the characteristic polynomial of
the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore,
by combining with a duality formula, we obtain precise large deviation
probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our
proof is based on a refined Riemann-Hilbert analysis for planar orthogonal
polynomials using the partial Schlesinger transform.

##  1\. Introduction

###  1.1. Models and numerology

In this work, we obtain precise asymptotic behaviours up to the $O(1)$ term in
the context of the following interrelated topics.

  * •

Zabrodin-Wiegmann prediction on the partition functions of planar Coulomb gas
ensembles: a case study for a conditional complex Ginibre ensemble breaking
the rotational symmetry.

  * •

Asymptotic behaviours of moments of the characteristic polynomial of the
complex Ginibre ensemble.

  * •

Large deviation probabilities of the smallest eigenvalue of the Laguerre
unitary ensemble.

Due to a certain duality relation (Proposition 1.1) these topics are indeed
equivalent, and readers may find a particular viewpoint most interesting based
on their individual interests. In the aforementioned topics, a specific phase
transition occurs, yielding distinct geometric/probabilistic implications in
each context, see Subsection 1.3 for details, cf. Figures 1 and 2.

Let us be more precise in introducing our models and formulations.

####  1.1.1. GinUE and its characteristic polynomials

We begin with the complex Ginibre ensemble (GinUE) $\textbf{{G}}_{N}$, an
$N\times N$ matrix whose entries are given by independent centered complex
Gaussian random variables with variance $1/N$, see [29, 30] for recent
reviews. It is well known that the eigenvalues $\\{z_{j}\\}_{j=1}^{N}$ of
$\textbf{{G}}_{N}$ follow the joint probability distribution

(1.1) $\frac{1}{Z_{N}^{\rm
Gin}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}e^{-N|z_{j}|^{2}}\,dA(z_{j}),\qquad
Z_{N}^{\rm Gin}=N!\frac{G(N+1)}{N^{N(N+1)/2}},$

where $dA(z)=d^{2}z/\pi$ is the area measure and $G$ is the Barnes
$G$-function [91, Section 5.17]. Here, $Z_{N}^{\rm Gin}$ is the normalisation
constant, known as the partition function that makes (1.1) a probability
measure. As $N\to\infty$, the eigenvalues $\\{z_{j}\\}_{j=1}^{N}$ tend to be
uniformly distributed on the unit disk, known as the circular law, see e.g.
[67] for a recent progress. Note here that we have a simple weight function
$e^{-N|z|^{2}}$ in (1.1), which enables explicit computations of the GinUE
statistics. For instance, the evaluation of $Z_{N}^{\rm Gin}$ follows from
Andréief’s formula together with the norm of the associated orthogonal
polynomial (which is in this case monomial).

For the GinUE model, we shall investigate the moment of its characteristic
polynomial

(1.2)
$\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-z)\Big{|}^{2\gamma},\qquad\gamma\geq
0.$

For a fixed value of $\gamma\geq 0$, the asymptotic behaviours of (1.2) were
recently obtained by Webb and Wong [108] for the bulk regime $|z|<1$, and by
Deaño and Simm [50] for the edge regime $|z|=1+O(1/\sqrt{N})$. These
asymptotic behaviours can be applied to construct a Gaussian multiplicative
chaos measure [95], see also [77]. We mention that the moment of the
characteristic polynomial in Hermitian random matrix theory has been
extensively studied, see e.g. [107, 21, 78, 35, 74, 40, 39]. Along the similar
spirit of [108, 50], we study the asymptotic behaviours of (1.2), but instead
of a fixed $\gamma$, we consider the exponentially varying regime
$\gamma=O(N)$. Namely, we examine the regime where $\gamma$ is scaled as
$\gamma=cN$ for a fixed parameter $c>0$. In this case, a phase transition
occurs at the critical value $|z|=\sqrt{c+1}-\sqrt{c}$, and we will
investigate all the regimes that arise.

####  1.1.2. Partition function of a determinantal Coulomb gas

As a second formulation, we consider a conditional point process. For this
purpose, we fix a parameter $c>0$ and consider the GinUE of size
$(c+1)N\times(c+1)N$ conditioned to have deterministic eigenvalue $a\geq 0$
with multiplicity $cN$. Then the remaining $N$ random eigenvalues
$\\{z_{j}\\}_{j=1}^{N}$ follow the distribution

(1.3)
$\frac{1}{Z_{N}(a,c)}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}|z_{j}-a|^{2cN}e^{-N|z_{j}|^{2}}\,dA(z_{j}).$

This model is called the induced GinUE [52, 14] and its partition function
$Z_{N}(a,c)$ is also called a massive partition function in the quantum
chromodynamics related literature [6]. Note that if $c>0$, the ensemble is
rotationally symmetric only for the case $a=0$, and in this case, again
explicit computations lead to

(1.4)
$Z_{N}(0,c)=N!\frac{G(N+cN+1)}{G(cN+1)}N^{-(c+\frac{1}{2})N^{2}-\frac{1}{2}N}.$

From the statistical physics viewpoint, the model (1.3) can be realised as a
planar Coulomb gas with inverse temperature $\beta=2$ (also known as the
random normal matrix model [12, 41]) and the external potential

(1.5) $Q(z)=|z|^{2}-2c\log|z-a|.$

We mention that potentials of this type are sometimes called Hele-Shaw
potentials, see [15] for a recent work. As $N\to\infty$, the ensemble (1.3)
tends to be uniformly distributed on a certain droplet $S\equiv S_{Q}$, see
Figure 1. The precise shape of the droplet is characterised in [16], and it
reveals a topological phase transition, see Subsection 1.3. In the theory of
Coulomb gases, the large-$N$ expansion of the free energy $\log Z_{N}(a,c)$ is
a fundamental topic [113, 81], as the coefficients of this expansion provide
essential potential theoretic/geometric properties of the model, see
Subsection 1.2 for more details.

(a) Post-critical; $a=1/4$

(b) Critical; $a=1/2$

(c) Pre-critical; $a=1$

Figure 1. Illustration of the droplet, where $c=9/16$. The black dot indicates
the point $a.$

####  1.1.3. Laguerre unitary ensemble and its smallest eigenvalue

The third formulation is in the context of a classical Hermitian random matrix
model [54]. We consider an $N\times N$ Wishart matrix, also known as the
Laguerre unitary ensemble (LUE)
$\textbf{W}_{N}=\textbf{R}_{N}\textbf{R}_{N}^{*},$ where $\textbf{R}_{N}$ of
size $N\times(\alpha+1)N$ is a rectangular complex Ginibre matrix. Here
$\alpha\geq 0$ is the rectangular parameter. Then the joint probability
distribution of eigenvalues $\\{\lambda_{j}\\}_{j=1}^{N}$ of $\textbf{W}_{N}$
is proportional to

(1.6)
$\prod_{j>k=1}^{N}|\lambda_{j}-\lambda_{k}|^{2}\prod_{j=1}^{N}\lambda_{j}^{\alpha
N}e^{-N\sum_{j=1}^{N}\lambda_{j}},\qquad(\lambda_{N}>\dots>\lambda_{1}>0).$

It is well known that the as $N\to\infty$, the empirical measure of the LUE
follows the Marchenko-Pastur distribution

(1.7)
$\frac{1}{2\pi}\frac{\sqrt{(\lambda_{+}-x)(x-\lambda_{-})}}{x}\cdot\mathbbm{1}_{[\lambda_{-},\lambda_{+}]}(x),\qquad\lambda_{\pm}:=(\sqrt{\alpha+1}\pm
1)^{2}.$

In this context, we focus on the statistics of the smallest eigenvalue
$\lambda_{1}$. Such a statistic of the LUE, or a more general sample
covariance matrix where the Gaussian entries are replaced by an i.i.d. random
variable, finds several applications in random matrix theory, see e.g. [111,
18, 62] and references therein.

(a) Pulled; $t=\lambda_{-}-1$

(b) Critical; $t=\lambda_{-}$

(c) Pushed; $t=\lambda_{-}+1$

Figure 2. Illustration of the Marchenko-Pastur law (1.7) and constrained
spectral density (2.45), where $\alpha=5$. Here, the vertical (full red) line
indicates the hard wall $x=t$ of the LUE.

####  1.1.4. Duality relation

We now discuss the equivalence of the above three formulations. First, notice
that by definition, the characteristic polynomial and partition functions are
related as

(1.8)
$\displaystyle\mathbb{E}\Big{|}\det(G_{N}-z)\Big{|}^{2cN}=\frac{1}{Z_{N}^{\rm
Gin}}\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}|z-z_{j}|^{2cN}e^{-N\,|z_{j}|^{2}}\,dA(z_{j})=\frac{Z_{N}(|z|,c)}{Z_{N}^{\rm
Gin}}.$

The following equivalence was introduced in [57, 90]. This is a restatement of
[50, Proposition 3.1] after several transformations, and for the reader’s
convenience, we provide the details in Section 6.

######  Proposition 1.1 (Duality relation).

Let $\lambda_{1}$ be the smallest eigenvalue of the LUE in (1.6). For a fixed
$c>0$, we put $\alpha=1/c$. Then for any $x\in\mathbb{R}$, we have

(1.9)
$\displaystyle\begin{split}\mathbb{P}\Big{[}\lambda_{1}>\frac{x^{2}}{c}\Big{]}&=e^{-cN^{2}x^{2}}\frac{Z_{N}(x,c)}{Z_{N}(0,c)}\bigg{|}_{N\to
N/c}.\end{split}$

Due to this duality relation and the well-known asymptotic expansion (3.16) of
the Barnes $G$-function, it becomes evident that the asymptotic behaviours of
the three aforementioned formulations are equivalent. We mention that the
duality relation originates from the supersymmetry method, cf. [90, 61, 59].
Remarkably, this relation expresses the integral (1.2) over an $N\times N$
non-Hermitian random matrix in terms of an integral over a
$\gamma\times\gamma$ Hermitian random matrix. In our present case, where
$\gamma=cN$, we further make the change of variables $cN\mapsto N$, resulting
in the formula (1.9) with the parameter $\alpha=1/c$ of the LUE. It is
noteworthy that such a duality relation finds application in various contexts
of random matrix theory, see e.g. [86, 99, 100].

###  1.2. Free energy expansion; Zabrodin-Wiegmann prediction

Among the above three formulations, let us discuss the free energy expansion
from a viewpoint of a more general Coulomb gas theory. In general, the
partition function of the random normal matrix model is given by

(1.10) $\displaystyle
Z_{N}^{V}:=\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}e^{-N\,V(z_{j})}\,dA(z_{j}),$

where $V:\mathbb{C}\to\mathbb{R}$ is a given external potential.

To describe the asymptotic behaviour of $Z_{N}^{V}$, we recall some potential
theoretic notions [96]. Given a compactly supported probability measure $\mu$
on $\mathbb{C}$, the weighted logarithmic energy $I_{V}[\mu]$ associated with
the potential $V$ is given by

(1.11)
$I_{V}[\mu]:=\int_{\mathbb{C}^{2}}\log\frac{1}{|z-w|}\,d\mu(z)\,d\mu(w)+\int_{\mathbb{C}}V\,d\mu.$

For a general $V$, there exists a unique minimizer $\sigma_{V}$ called the
equilibrium measure. Furthermore, due to Frostman’s theorem, it is of the form

(1.12) $d\sigma_{V}(z)=\Delta
V(z)\,\mathbbm{1}_{S_{V}}(z)\,dA,\qquad\Delta=\partial\bar{\partial},$

where the compact support $S_{V}$ is called the droplet. It is conjectured
that if the droplet is _connected_ , the partition function $Z_{N}^{V}$ has
the asymptotic expansion of the form

(1.13) $\begin{split}\log Z_{N}^{V}&=-I_{V}[\sigma_{V}]N^{2}+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1-\frac{1}{2}\int_{\mathbb{C}}\log(\Delta
V)\,d\sigma_{V}\Big{)}N\\\ &\quad+\frac{6-\chi}{12}\log
N+\frac{\log(2\pi)}{2}+\chi\,\zeta^{\prime}(-1)+\mathcal{F}_{V}+o(1).\end{split}$

Here $\chi$ is the Euler characteristic of the droplet and $\zeta$ is the
Riemann zeta function. We refer the reader to [31, Section 1.1] and [29,
Section 5.4] for the development of the expansion (1.13).

Among the variety of literature on the expansion (1.13), we mention that Leblé
and Serfaty [81] proved the expansion (1.13) up to the order $O(N)$ in the
context of a more general $\beta$ ensemble, see also [101] for quantitative
error bounds. The topology-dependence of the $O(\log N)$ term was introduced
in the work of Jancovici et al. [68, 103] through exactly solvable examples
such as (induced) Ginibre and spherical ensembles. The $O(1)$ term reflects a
conformal geometric property of the droplet. More precisely, in [113],
Zabrodin and Wiegmann made use of the Ward’s identities in conformal field
theory (see e.g. [70, Appendix 6]) and proposed a remarkable prediction,
suggesting that the term $\mathcal{F}_{V}$ can be expressed in terms of the
zeta-regularized determinant of the exterior droplet as

(1.14)
$\mathcal{F}_{V}=-\frac{1}{2}\log\textup{det}_{\zeta}(\Delta_{\mathbb{C}\setminus
S_{V}}),$

see Remark 2.3 for more details.

The expansion of the form (1.13) was obtained in a recent work [31] for a
radially symmetric potential $V$ with $\Delta V>0$ in $\mathbb{C}$. This
strictly sub-harmonic assumption is crucial in [31], as it leads to a droplet
that is either a disc or an annulus. The asymptotic behaviour of the partition
function $Z_{N}^{V}$ has been further investigated in [9] to include the case
where the ensemble exhibits a spectral gap (i.e. $V$ can be such that $\Delta
V<0$). In particular, it was shown in [9] that if the droplet has multiple
components, a non-trivial oscillatory term (a new “displacement term”) emerges
in the $O(1)$ expansion, indicating that the expansion of the form (1.13) does
not hold. Furthermore, in [9], the case with the harmonic measure perturbation
of the order $O(1/N)$ in the potential and Fisher-Hartwig singularities [73]
has been investigated, which are closely related to the fluctuation theorem
[8].

######  Remark 1.2 (Potential with a hard edge).

In the above discussions, we have focused on the case that $V$ is supported on
the whole complex plane, leading to a droplet with a soft edge. In contrast,
if $V$ is confined to a subset of the droplet, the expansion of the free
energy takes on a notably different form. This hard edge regime finds
applications in different contexts including the counting statistics [28, 36,
10, 11, 2] and hole probabilities [53, 5, 1, 60, 37, 38], cf. Remark 2.4.
Moreover, when $V$ is supported on a Jordan curve, the asymptotic behaviour of
the free energy has also been investigated, see e.g. [109, 43, 69] and
references therein.

In this work, we obtain a precise expansion in the form of (1.13) for the
potential given by (1.5). To our knowledge, this provides the first non-
radially symmetric example (other than exactly solvable models such as the
elliptic GinUE) that confirms this conjecture. Indeed, the elliptic GinUE [30,
Section 2.3], indexed by the non-Hermiticity parameter $\tau\in[0,1)$, is the
only known non-radially symmetric example for which the free energy expansion
in the form of (1.13) is known. However, in this case, the free energy is
simply the same as that of the GinUE up to an additive constant of
$\frac{N}{2}\log(1-\tau^{2})$.

Due to the lack of the rotational symmetry, it requires a different approach
compared to [31, 9], and we implement the partial Schlesinger transform [24]
to refine the Riemann-Hilbert analysis in [16, 85, 75]. Furthermore, as
explained in the previous subsection, it follows from Proposition 1.1 that
this result implies the asymptotic behaviours of moments of the characteristic
polynomials of the GinUE, as well as the large-deviation probabilities of the
smallest eigenvalue of the LUE.

###  1.3. Phase transition

As previously mentioned, the model we consider undergoes an interesting phase
transition. We now precisely describe such a transition. Let $S\equiv S_{Q}$
be the droplet in (1.12) associated with the potential $Q$ given in (1.5). The
droplet reveals a topological phase transition at the critical value

(1.15) $c_{\rm cri}:=\frac{(1-a^{2})^{2}}{4a^{2}},\qquad a_{\rm
cri}:=\sqrt{c+1}-\sqrt{c}.$

In the post- and pre-critical regimes, the parameters $a$ and $c$ are assumed
to be fixed (i.e. independent of $N$). The droplet is then given as follows
[16, Section 2].

  * •

Post-critical regime: $c<c_{\rm cri}$, i.e. $a<a_{\rm cri}.$ In this case, the
droplet is given by

(1.16) $S=\overline{\mathbb{D}(0,\sqrt{1+c})}\setminus\mathbb{D}(a,\sqrt{c}),$

where $\mathbb{D}(z,r)$ is a disk with center $z$ and radius $r>0.$

  * •

Pre-critcal regime111Compared to [16], we have replaced the notations:
$\alpha\to q$ and $\rho\to R$.: $c>c_{\rm cri}$, i.e. $a>a_{\rm cri}.$ In this
case, the droplet is a simply connected domain whose boundary is given by the
image of the unit circle under the conformal map

(1.17) $f(z)=R\,z-\frac{\kappa}{z-q}-\frac{\kappa}{q},\qquad
R=\frac{1+a^{2}q^{2}}{2aq},\qquad\kappa=\frac{(1-q^{2})(1-a^{2}q^{2})}{2aq}.$

Note that $R>0$ is the conformal radius of the droplet. Here, $q\equiv q(a)$
satisfies $f(1/q)=a$ and it is given by a unique solution of the algebraic
equation

(1.18) $q^{6}-\Big{(}\frac{a^{2}+4c+2}{2a^{2}}\Big{)}q^{4}+\frac{1}{2a^{4}}=0$

such that $0<q<1$ and $\kappa>0.$

See Figure 1 for the shape of the droplet. From the above description of the
droplet, one can see that the Euler characteristic $\chi$ of the droplet
$S_{Q}$ is given by

(1.19) $\chi=\begin{cases}0&\textup{for the post-critical case,}\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ 1&\textup{for the pre-critical case}.\end{cases}$

This will play an important role in the free energy expansion.

Let us mention that there are some more examples of the droplets revealing a
topological phase transition, see e.g. [27, 3, 17, 44]. A notable feature of
the phase transition is the emergence of a singular boundary point. In our
present case, it is a merging (double) point that falls into the class of
Sakai’s regularity theory [97, 82]. Such singular boundaries are of particular
interest from the viewpoint of the non-standard universality classes, and have
been studied for several different models, see e.g. [23, 26, 89, 13, 32] and
references therein.

Beyond the post- and pre-critical regimes, it is also natural to study the
behaviour of the ensemble (1.3) in the critical regime, see [75] for a recent
work on the local statistics. Let us first define the critical regime.

######  Definition 1.3 (Critical scaling regime).

The critical case corresponds to the scaling regime

$1-a(a+2\sqrt{c})=O(N^{-2/3}).$

We consider a parameter $s\in\mathbb{R}$ describing the critical regime, see
[16, Eq.(1.36)]. Then the parameter $a$ satisfies

(1.20) $\displaystyle a=a_{\rm
cri}-\frac{(\sqrt{c+1}-\sqrt{c})^{1/3}}{2c^{1/6}(c+1)^{1/6}}\frac{s}{N^{2/3}}+O\Big{(}\frac{1}{N^{4/3}}\Big{)}.$

As $s\to+\infty$, the limit coincides with the post-critical regime, while the
opposite limit as $s\to-\infty$ coincides with the pre-critical regime.

Turning to the LUE statistics, note that in the setup in Proposition 1.1 with
$x=a\geq 0$, it follows from (1.7) that

(1.21)
$\frac{x^{2}}{c}<\lambda_{-}=\Big{(}\sqrt{\frac{c+1}{c}}-1\Big{)}^{2},\quad\textup{if
and only if}\quad a<\sqrt{c+1}-\sqrt{c}=a_{\rm cri}.$

Hence, the topological phase transition of the droplet can be naturally
interpreted from the viewpoint of lower and upper large deviation
probabilities. This transition is also called the push-to-pulled transition,
see Figure 2. Furthermore, while the order $O(N^{-2/3})$ in the critical
scaling regime might be less intuitive in the two-dimensional model (1.3), it
becomes clear if we interpret this in the context of the LUE given the
standard square root decay of the Marchenko-Pastur law (1.7) at the soft edge.
In this context, a universal pulled-to-pushed transition of the third order
has been observed, see e.g. [87, 45]. See Remark 2.6 for more details.

##  2\. Main results

In this section, we present our main results. Recall that the partition
function $Z_{N}(a,c)$ is given by

(2.1) $\displaystyle Z_{N}(a,c)\equiv
Z_{N}^{Q}:=\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}e^{-NQ(z_{j})}\,dA(z_{j}),$

where $Q$ is given by (1.5). As discussed in Subsection 1.2, the logarithmic
energy (1.11) should appear in the leading term of the free energy expansion.
We first evaluate the energy associated with the potential $Q$ explicitly.

######  Proposition 2.1 (Evaluation of the energy).

Let $Q$ be given by (1.5). Then we have the following.

  * •

For the post-critical case, we have

(2.2) $I_{Q}[\sigma_{Q}]=\mathcal{I}^{\rm
post}(a,c):=\frac{3}{4}+\frac{3c}{2}+\frac{c^{2}}{2}\log
c-\frac{(c+1)^{2}}{2}\log(c+1)-ca^{2}.$

  * •

For the pre-critical case, we have

(2.3) $\displaystyle\begin{split}I_{Q}[\sigma_{Q}]=\mathcal{I}^{\rm
pre}(a,c)&:=\frac{3}{8}+\frac{a^{2}}{8}+\frac{3}{8a^{2}q^{4}}-\frac{5}{8q^{2}}+\Big{(}\frac{3}{4}+\frac{a^{2}}{8}\Big{)}a^{2}q^{2}-\frac{3a^{4}q^{4}}{8}\\\
&\quad+\log(2aq)+2c\log(2aq^{2})+\log\frac{(1+a^{2}q^{2}-2a^{2}q^{4})^{c^{2}}}{(1+a^{2}q^{2})^{(c+1)^{2}}},\end{split}$

where $q=q(a)$ is given by (1.18).

Furthermore, for a fixed $c>0$, suppose that $a>a_{\rm cri}$. Then we have

(2.4) $\displaystyle\mathcal{I}^{\rm post}(a,c)\leq\mathcal{I}^{\rm
pre}(a,c),$

where the equality holds when $a=a_{\rm cri}$.

See Figure 3 for the graphs of the energy. We mention that the inequality
(2.4) plays an important role in Theorem 2.9 below. The evaluation of energy
is closely related to the equilibrium measure problem. Using potential theory
and conformal mapping methods, Proposition 2.1 will be established in Section
4.

Figure 3. The plot shows the graph of the energy $a\mapsto I_{Q}[\sigma_{Q}]$,
where $c=9/16$. Here, the dotted vertical line represents $a=a_{\rm cri}=1/2$.
The graph (full line) for $a<a_{\rm cri}$ follows (2.2), while for $a>a_{\rm
cri}$ it follows (2.3). The dotted line for $a>a_{\rm cri}$ is the
continuation of the graph (2.2).

Recall that the Bernoulli number $B_{k}$ is given in terms of the generating
function as

(2.5) $\frac{t}{e^{t}-1}=\sum_{k=0}^{\infty}B_{k}\frac{t^{k}}{k!}.$

We are now ready to state our main result.

######  Theorem 2.2 (Free energy expansion for the post- and pre-critical
cases).

Let $Q$ be given by (1.5). Then as $N\to\infty,$ we have

(2.6) $\begin{split}\log Z_{N}(a,c)&=-I_{Q}[\sigma_{Q}]N^{2}+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1\Big{)}N\\\ &\quad+\frac{6-\chi}{12}\log
N+\frac{\log(2\pi)}{2}+\chi\,\zeta^{\prime}(-1)+\mathcal{F}(a,c)+\mathcal{E}_{N},\end{split}$

where $I_{Q}[\mu_{Q}]$ is the energy given in Proposition 2.1, $\chi$ is the
Euler characteristic of the droplet $S_{Q}$ given in (1.19) and $\zeta$ is the
Riemann zeta function. Here $\mathcal{F}(a,c)$ and the error term
$\mathcal{E}_{N}$ are given as follows.

  * •

For the post-critical case, we have

(2.7) $\mathcal{F}(a,c)=\mathcal{F}^{\rm
post}(a,c):=\frac{1}{12}\log\Big{(}\frac{c}{1+c}\Big{)}$

and

(2.8)
$\mathcal{E}_{N}=\sum_{k=1}^{M}\bigg{(}\frac{B_{2k}}{2k(2k-1)}\frac{1}{N^{2k-1}}+\frac{B_{2k+2}}{4k(k+1)}\Big{(}\frac{1}{(c+1)^{2k}}-\frac{1}{c^{2k}}\Big{)}\frac{1}{N^{2k}}\bigg{)}+O(\frac{1}{N^{2M+1}})$

for any $M>0$, where $B_{k}$ is the Bernoulli number.

  * •

For the pre-critical case, we have

(2.9) $\mathcal{F}(a,c)=\mathcal{F}^{\rm
pre}(a,c):=\frac{1}{24}\log\Big{(}\frac{(1+a^{2}q^{2}-2a^{2}q^{4})^{4}}{(1+a^{2}q^{2})^{4}(1-q^{2})^{3}(1-a^{4}q^{6})}\Big{)}$

and $\mathcal{E}_{N}=O(N^{-1}).$

As explained above, up to the $O(N)$ term, Theorem 2.2 is a special case of
[81, Corollary 1.1]. It is obvious that the entropy in the $O(N)$ term of
(1.13) vanishes since $\Delta Q=1$. The constant term $\mathcal{F}(a,c)$
coincides with the prediction (1.13), as we discuss now.

######  Remark 2.3 (Regularized determinant of Laplacian).

Let $0<\lambda_{1,D}\leq\lambda_{2,D}\leq\dots$ be the eigenvalues of the
Dirichlet Laplacian $\Delta$ on a domain $D\subset\mathbb{C}$. Then the
spectral zeta function is defined by

(2.10) $\zeta_{\Delta}(s;D):=\sum\lambda_{j,D}^{-s}.$

This is a building block to define the zeta-regularized determinant of
$\Delta$:

(2.11)
$\textup{det}_{\zeta}(\Delta_{D}):=\exp(-\zeta_{\Delta}^{\prime}(0;D)),$

see e.g. [63] for more details. The spectral determinant can also be expressed
in terms of several different domain functionals in conformal geometry, such
as Brownian loop measure [79, 80], Loewner energy [105, 106], and the Grunsky
operator [102, 69]. It is also used to describe large deviation principles
(see e.g. [105, 92]), aligning with the same spirit as our result, especially
from the viewpoint of the LUE statistics.

If the derivative of the potential $\partial V$ is rational,
$\textup{det}_{\zeta}(\Delta_{\mathbb{C}\setminus S_{V}})$ can be made more
explicit as discussed in [113, Section 6.1]. As a consequence, for the post-
critical case, it can be observed that

(2.12) $\log\textup{det}_{\zeta}(\Delta_{\mathbb{C}\setminus
S})=-\frac{1}{6}\log\Big{(}\frac{c}{1+c}\Big{)}$

since the boundary of the droplet is a union of two circles. For the pre-
critical case222The conformal map $z(w)$ was introduced below Eq.(5.22) of
[113], which in our case is $f(z)$. , let

(2.13) $z_{\pm}=q\pm i\sqrt{\frac{\kappa}{R}}$

be the critical points of the conformal map $f$ given in (1.17). Then we have

(2.14)
$\displaystyle\begin{split}\log\textup{det}_{\zeta}(\Delta_{\mathbb{C}\setminus
S})&=\frac{1}{12}\log\Big{(}\frac{R^{4}\,f^{\prime}(1/z_{+})f^{\prime}(1/z_{-})}{f^{\prime}(1/q)^{2}}\Big{)}=-\frac{1}{12}\log\Big{(}\frac{(1+a^{2}q^{2}-2a^{2}q^{4})^{4}}{(1-q^{2})^{3}(1-a^{4}q^{6})(1+a^{2}q^{2})^{4}}\Big{)}.\end{split}$

Therefore, one can observe that
$\mathcal{F}(a,c)=-\frac{1}{2}\log\textup{det}_{\zeta}(\Delta_{\mathbb{C}\setminus
S})$, confirming the prediction of Zabrodin and Wiegmann.

We mention that in our present case, since we consider the quasi-harmonic
potential $\Delta Q\equiv 1$, the equilibrium measure (1.12) has a flat
metric. On the other hand, if $\Delta Q$ is not a constant, the determinant of
Laplacian with respect to a non-trivial conformal metric or its conformal
transformation law can be obtained via the Polyakov-Alvarez conformal anomaly
formula [94, 7].

######  Remark 2.4 (Absence of the $O(\sqrt{N})$ term for $\beta=2$).

As a side remark, it is worth mentioning the belief for a general $\beta$
ensemble that there exists an $O(\sqrt{N})$ term, with a coefficient
proportional to $\log(\beta/2)$ called the surface tension. This conjecture
was made in an unpublished note of Lutsyshin and first appeared in [34].
However, since the coefficient is expected to be proportional to
$\log(\beta/2)$, the absence of the $O(\sqrt{N})$ term in the determinantal
case $\beta=2$ is, in this context, a statement of prediction. The absence of
the $O(\sqrt{N})$ term has been verified for the rotational symmetry case [31,
9]. However, one might question whether this absence truly results from
$\beta=2$ or from rotational symmetry. Nonetheless, as per Theorem 2.2, one
can observe that even without the rotational symmetry, there is no
$O(\sqrt{N})$ term for $\beta=2$. (It is worth noting however that the
$O(\sqrt{N})$ term does arise for $\beta=2$ when considering the hard wall
constraints of the potential [10, 28, 36, 37]).

We now discuss the critical regime. It can be expected from the duality
relation (Proposition 1.1) that the expansion of the free energy in the
critical regime is closely related to the Tracy-Widom distribution:

(2.15) $F_{\rm
TW}(t):=\exp\Big{(}-\int_{t}^{\infty}(x-t)\bm{q}(x)^{2}\,dx\Big{)},$

where $\bm{q}$ is the Hastings–McLeod solution to Painlevé II equation

(2.16)
$\bm{q}^{\prime\prime}(s)=s\bm{q}(s)+2\bm{q}(s)^{3},\qquad\bm{q}(s)\sim\operatorname{Ai}(s)\quad\mbox{as
}s\to\infty.$

Then we have the following.

######  Proposition 2.5 (Free energy expansion for the critical regime).

For a given fixed parameter $c>0$, let $a$ be scaled as (1.20). Then as
$N\to\infty,$ we have

(2.17) $\displaystyle\begin{split}\log
Z_{N}(a,c)&=-\Big{(}\frac{3}{4}+\frac{3c}{2}+\frac{c^{2}}{2}\log
c-\frac{(c+1)^{2}}{2}\log(c+1)-ca^{2}\Big{)}N^{2}\\\ &\quad+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1\Big{)}N+\frac{1}{2}\log N\\\
&\quad+\frac{\log(2\pi)}{2}+\frac{1}{12}\log\Big{(}\frac{c}{1+c}\Big{)}+\log
F_{\rm TW}(c^{-2/3}s)+O(N^{-2/3}),\end{split}$

where $F_{\rm TW}$ is the Tracy-Widom distribution.

This will be shown in Section 6. Note that the expansion (2.17) is not of the
form (1.13). Namely, by the scaling (1.20), we have

(2.18) $ca^{2}N^{2}=ca_{\rm
cri}^{2}N^{2}+\widetilde{C}_{1}(s)N^{4/3}+\widetilde{C}_{2}(s)N^{2/3}+\widetilde{C}_{3}(s)+O(N^{-2/3})$

for some constants $\widetilde{C}_{k}$ ($k=1,2,3$).

The Painlevé II critical asymptotic behaviours of the associated planar
orthogonal have been discovered in [16, 75]. However, the asymptotic
behaviours presented in [16, 75] are not enough to derive Proposition 2.5,
particularly to derive the $O(1)$ term. On the other hand, Proposition 2.5 can
be readily derived utilizing the duality relation (Proposition 1.1), the
Marchenko-Pastur law (1.7), and the well-established edge universality of the
random Hermitian matrix ensemble [48]. This aligns with the probability
theoretic intuition of the free energy expansion: the law of large numbers
(determining the position of the left edge of the Marchenko-Pastur law) gives
rise to the leading order of the free energy, while fluctuations (governed by
the Tracy-Widom distribution) contribute to the constant term.

######  Remark 2.6 (Free energy expansion under the topology transitions).

Recall the well-known asymptotic behaviours of the Tracy-Widom distribution:
as $x\to+\infty,$

(2.19) $\displaystyle F_{\rm TW}(-x)$
$\displaystyle=\frac{2^{1/24}e^{\zeta^{\prime}(-1)}}{x^{1/8}}e^{-x^{3}/12}\Big{(}1+\frac{3}{2^{6}x^{3}}+O(x^{-6})\Big{)},$
(2.20) $\displaystyle 1-F_{\rm TW}(x)$ $\displaystyle=\frac{1}{32\pi
x^{3/2}}e^{-4x^{3/2}/3}\Big{(}1+O(x^{-3/2})\Big{)}.$

Using this, we have

(2.21) $\lim_{s\to+\infty}\log F_{\rm TW}(c^{-2/3}s)=0.$

Thus in this limit, (2.17) matches with Theorem 2.2 for the post-critical
regime. On the other hand, in the opposite limit, we have

(2.22) $\log F_{\rm TW}(c^{-2/3}s)=\frac{1}{24}\log
2+\zeta^{\prime}(-1)-\frac{1}{8}\log|s|-\frac{|s|^{3}}{12}+O(|s|^{-3}),\qquad
s\to-\infty.$

Note that by (1.20), at least formally, the proper scaling for the pre-
critical regime should be $s=O(N^{2/3})$ in the critical scaling. With this
scaling, one can notice the additional term

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

appearing in Theorem 2.2 for the pre-critical regime when $\chi=1.$

One can observe such a transition in the opposite direction, from the pre-
critical regime. Namely, in the scaling regime (1.20), by (1.18), we have

$\displaystyle
q=1+\frac{s}{4c^{1/6}(c+1)^{1/6}(\sqrt{c+1}-\sqrt{c})^{2/3}}N^{-2/3}+O(N^{-1}).$

This gives that $\mathcal{F}^{\rm pre}$ in (2.9) has the asymptotic expansion

(2.24) $\mathcal{F}^{\rm pre}(a,c)=\frac{1}{12}\log N+O(1).$

Hence, one can again observe the additional $\frac{1}{12}\log N$ term.

We now turn back to the other formulations of the problem. First of all, as a
consequence of Theorem 2.2, we have the asymptotic behaviours of the moments
of characteristic polynomial. For this, we shall use the notations in [16,
Eqs.(2.17),(4.6)]. Let

(2.25)
$F(z)=\frac{1}{2R}\Big{[}z+|\beta|+\sqrt{(z-\beta)(z-\overline{\beta})}\Big{]}$

be the inverse of $f$ in (1.17), where

(2.26) $\beta=f(z_{+})=Rq-\frac{\kappa}{q}+2i\sqrt{\kappa R}$

is a critical value of $f$.

######  Theorem 2.7 (Moments of characteristic polynomial of the GinUE).

Let $\textbf{{G}}_{N}$ be the complex Ginibre matrix of size $N$ and let $c>0$
be fixed. Then as $N\to\infty,$

(2.27)
$\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-z)\Big{|}^{2cN}=N^{\frac{1-\chi}{12}}\,e^{(\chi-1)\zeta^{\prime}(-1)}\,\mathcal{G}(|z|)\,\exp\Big{(}\mathcal{H}(|z|)N^{2}+\widetilde{\mathcal{E}}_{N}\Big{)}$

where $\mathcal{H}(z)$, $\mathcal{G}(z)$ and $\widetilde{\mathcal{E}}_{N}$ are
given as follows.

  * •

If $|z|<\sqrt{c+1}-\sqrt{c}$, we have $\chi=0$,

(2.28) $\mathcal{H}(z)=\mathcal{H}^{\rm
post}(z):=c\,z^{2}-\frac{3c}{2}+\frac{(c+1)^{2}}{2}\log(c+1)-\frac{c^{2}}{2}\log
c$

and

(2.29) $\mathcal{G}(z)=\mathcal{G}^{\rm
post}(z):=\Big{(}\frac{c}{1+c}\Big{)}^{\frac{1}{12}}.$

Here for any $M>0$,

(2.30)
$\widetilde{\mathcal{E}}_{N}=\sum_{k=1}^{M}\frac{B_{2k+2}}{4k(k+1)}\Big{(}\frac{1}{(c+1)^{2k}}-\frac{1}{c^{2k}}-1\Big{)}\frac{1}{N^{2k}}+O(\frac{1}{N^{2M+2}}).$

  * •

If $|z|>\sqrt{c+1}-\sqrt{c}$, we have $\chi=1$,

(2.31) $\displaystyle\begin{split}\mathcal{H}(z)=\mathcal{H}^{\rm
pre}(z)&:=\frac{3}{8}-\frac{z^{2}}{8}-\frac{3F(z)^{4}}{8z^{2}}+\frac{5F(z)^{2}}{8}-\Big{(}\frac{3}{4}+\frac{z^{2}}{8}\Big{)}\frac{z^{2}}{F(z)^{2}}+\frac{3}{8}\frac{z^{4}}{F(z)^{4}}\\\
&\quad+\log\Big{(}\frac{F(z)^{2c^{2}-1}(F(z)^{2}+z^{2})^{(c+1)^{2}}}{(2z)^{2c+1}(F(z)^{4}+z^{2}F(z)^{2}-2z^{2})^{c^{2}}}\Big{)}\end{split}$

and

(2.32) $\mathcal{G}(z)=\mathcal{G}^{\rm
pre}(z):=\Big{(}\frac{F(z)^{4}(F(z)^{4}+z^{2}F(z)^{2}-2z^{2})^{4}}{(F(z)^{2}+z^{2})^{4}(F(z)-1)^{3}(F(z)^{6}-z^{4})}\Big{)}^{\frac{1}{24}},$

where $F$ is given by (2.25). Here $\widetilde{\mathcal{E}}_{N}=O(1/N).$

Note that Theorem 2.7 immediately follows from Proposition 2.1, Theorem 2.2
and Lemma 3.1. Here we have used also the fact that $q=1/F(a)$.

######  Remark 2.8 (Comparison with the small insertion).

Let us stress again that the case $c=O(1/N)$ was studied in [108] for the bulk
case $|z|<1$ and also in [50] for the edge case $|z|=1+O(1/\sqrt{N})$. From
the viewpoint of the induced model (1.3), the regime $c=O(1/N)$ represents the
case in which the point charge insertion is finite. In this situation, the
conditional process does not lead to macroscopic changes, and consequently,
the droplet remains the unit disk of the circular law. It particular, it was
shown in [108] that for the bulk case $|z|<1$,

(2.33)
$\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-z)\Big{|}^{2\gamma}=N^{\frac{\gamma^{2}}{2}}\exp\Big{(}\gamma
N(|z|^{2}-1)\Big{)}\frac{(2\pi)^{\gamma/2}}{G(\gamma+1)}(1+o(1)).$

Note that the leading order in the exponent is $O(N)$, contrasting with the
order of $O(N^{2})$ in (2.27). This difference arises from the fact that the
$O(N^{2})$ term originates from the energy of the equilibrium measure, which
is a macroscopic quantity. Nonetheless, one can observe, at least up to a
multiplicative constant, that the asymptotic formula (2.33) coincides with our
formula (2.27) by simply setting $c=\gamma/N$:

(2.34) $\exp\Big{(}\mathcal{H}^{\rm
post}(|z|)N^{2}\Big{)}\Big{|}_{c=\gamma/N}=N^{\frac{\gamma^{2}}{2}}\exp\Big{(}\gamma
N(|z|^{2}-1)\Big{)}\cdot O(1).$

However, the multiplicative constant term does not match. Nonetheless, if we
make use of the Barnes $G$-function in the asymptotic formula, by (3.16), one
can see that the asymptotic behaviour

(2.35) $\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-z)\Big{|}^{2\gamma}=N^{-\gamma
N}\exp\Big{(}\gamma
N\,|z|^{2}\Big{)}\frac{G(\gamma+N+1)}{G(\gamma+1)G(N+1)}(1+o(1))$

matches both (2.33) and (2.27) for $\gamma$ fixed and $\gamma=cN$,
respectively.

We now discuss the large deviation of the LUE smallest eigenvalue. In general,
the large deviation principles of the extremal eigenvalues of random matrices
have been extensively studied in the literature. For instance, the statistics
of the maximal eigenvalues of the Gaussian ensembles have been studied in [20,
46, 47]. Also, for the Laguerre ensembles, which are closely related to our
present case, there has been extensive work on large deviation probabilities
for both the smallest and largest eigenvalues [4, 72, 93, 104, 111, 112, 110,
88], see Remark 2.10. For such integrable random matrices, the Coulomb gas
approach has been mainly implemented, and essentially, the large deviation
speed and rate can be derived by computing the energy associated with the
potential under the presence of a hard wall. An advantage of this approach is
that it can be applied to a general $\beta$ ensemble [55, 58, 51]. However, it
is limited in deriving the leading-order asymptotic behaviours. In addition,
within the context of large deviation probabilities of the extremal
eigenvalues, there is a universal pulled-to-pushed transition of the third
order, see e.g. [87, 45]. We also point out that such large deviation
probabilities are closely related to the maximal height of $N$ non-
intersecting Brownian excursions [98, 56, 71].

######  Theorem 2.9 (Large deviation probabilities of the smallest eigenvalues
of the LUE).

Let $\lambda_{1}$ be the smallest eigenvalue of the LUE in (1.6). Then As
$N\to\infty$, we have the following.

  * •

(Pulled-regime) If $t<\lambda_{-}$, we have

(2.36) $\log\mathbb{P}\Big{[}\lambda_{1}>t\Big{]}=O(N^{-\infty}).$

Here, $O(N^{-\infty})$ means $O(N^{-m})$ for any positive integer $m.$

  * •

(Pushed-regime) If $t>\lambda_{-}$, we have

(2.37)
$\log\mathbb{P}\Big{[}\lambda_{1}>t\Big{]}=-\Phi(t;\alpha)N^{2}-\frac{1}{12}\log(\alpha
N)+\zeta^{\prime}(-1)+\Psi(t;\alpha)+O(\frac{1}{N}),$

where

(2.38) $\displaystyle\Phi(t;\alpha)$
$\displaystyle=\alpha^{2}\Big{(}\mathcal{I}^{\rm
pre}(\sqrt{t/\alpha},1/\alpha)-\mathcal{I}^{\rm
post}(\sqrt{t/\alpha},1/\alpha)\Big{)},$ (2.39) $\displaystyle\Psi(t;\alpha)$
$\displaystyle=\mathcal{F}^{\rm
pre}(\sqrt{t/\alpha},1/\alpha)-\mathcal{F}^{\rm
post}(\sqrt{t/\alpha},1/\alpha).$

Here, $\mathcal{I}^{\rm pre}$, $\mathcal{I}^{\rm post}$, $\mathcal{F}^{\rm
pre}$ and $\mathcal{F}^{\rm post}$ are given by (2.2), (2.3) (2.7) and (2.9).

Theorem 2.9 follows from Proposition 1.1 and Theorem 2.2 (with $N\mapsto
N/c=\alpha N$). Note that the positivity of the rate function
$\Phi(t;\alpha)>0$ follows from the inequality (2.4). See Figure 4 for the
graph of $\Phi$.

Figure 4. The full red line represents the graph of the rate function
$t\mapsto\Phi(t;\alpha)$, where $\alpha=16/9$ and $t\geq\lambda_{-}=4/9$. The
blue dots indicate the values of the function $t\mapsto S(t)-S(\lambda_{-})$.
Here, one can also observe the asymptotic behaviours (2.40) and (2.41) in the
left and right figures, respectively.

By the explicit formulas (2.2) and (2.3), one can observe that

(2.40)
$\Phi(t;\alpha)\sim\frac{\sqrt{\alpha+1}}{12\lambda_{-}^{2}}\,(t-\lambda_{-})^{3},\qquad
t\to\lambda_{-},$

which agrees with the tail probability of the Tracy-Widom distribution. This
is known as a third-order phase transition appearing in a more general
context, and we refer the reader to [87] for a review. Let us also mention
that in the opposite limit, we have

(2.41) $\Phi(t;\alpha)\sim t-\alpha\log t,\qquad t\to\infty.$

Here, the linear growth $t$ comes from the post-critical energy (2.2), while
the logarithmic corrections $\alpha\log t$ comes from the pre-critical energy
(2.3), cf. (4.32).

######  Remark 2.10 (Comparison with the Katzav-Castillo formula from a
Coulomb gas approach).

In [72], Katzav and Castillo used a Coulomb gas method and derived the leading
order asymptotic behaviour

(2.42)
$\log\mathbb{P}\Big{[}\lambda_{1}>t\Big{]}=-\Big{(}S(t)-S(\lambda_{-})\Big{)}N^{2}+o(N^{2}),\qquad(t>\lambda_{-})$

where $S$ is given by

(2.43)
$\displaystyle\begin{split}S(t)&=\frac{U(t)+t}{2}-\frac{(U(t)-t)^{2}}{32}+\frac{\alpha}{4}(\sqrt{U(t)}-\sqrt{t})^{2}\\\
&\quad-\log\Big{(}\frac{U(t)-t}{4}\Big{)}+\frac{\alpha^{2}}{4}\log\Big{(}t\,U(t)\Big{)}-\alpha(\alpha+2)\log\Big{(}\frac{\sqrt{U(t)}+\sqrt{t}}{2}\Big{)}.\end{split}$

Here,

(2.44)
$U(t)=\frac{4}{3}\Big{(}t+2(\alpha+2)\Big{)}\cos^{2}\Big{(}\frac{\theta+2\pi}{3}\Big{)},\qquad\theta=\arctan\Big{(}\sqrt{\frac{(t+2(\alpha+2))^{3}-27\alpha^{2}t}{27\alpha^{2}t}}\Big{)}.$

Indeed, this value $U(t)$ is the right edge of the _constrained_ spectral
density (see [104])

(2.45)
$\frac{\sqrt{U(t)-x}}{2\pi\sqrt{x-t}}\frac{x-\alpha\sqrt{t/U(t)}}{x}\mathbbm{1}_{[t,U(t)]}(x),$

the density of the LUE (1.6) conditioned on $\lambda_{1}>t$. By comparing
(2.37) and (2.42), we have

(2.46) $\Phi(t;\alpha)=S(t)-S(\lambda_{-}),$

see Figure 4. This identity should follow from the explicit solution of the
cubic equation (1.18) (in the variable $q^{2}$). We also refer to [76, Section
6.2] for the interpretation of the formula (2.42) from the viewpoint of the
recursion scheme. By (2.46), one can observe that our result extends the
result (2.42), incorporating not only polynomial but also constant corrections
of the large deviation probability $\mathbb{P}[\lambda_{1}>t]$. Let us
emphasise that while the Coulomb gas approach yields explicit formula for the
leading-order asymptotic behaviour, applicable not only to the LUE but also to
general Laguerre-$\beta$ ensembles, this approach is difficult to implement
for deriving precise asymptotic behaviours.

In the pulled-regime, the precise order for $O(N^{-\infty})$ in (2.36) is
expected to be exponentially small: for a certain $\widetilde{c}(t)$,

(2.47) $\mathbb{P}\Big{[}\lambda_{1}>t\Big{]}=e^{-\widetilde{c}(t)N}(1+o(1)).$

See [72, Eq.(16)] for the Coulomb gas prediction on the constant
$\widetilde{c}(t)$. We also refer to [55, Eq.(1.4)] for the constant
$\widetilde{c}(t)$ derived from the tail behaviour of the LUE density.
However, capturing such an exponentially decaying behaviour seems challenging
based on the Riemann-Hilbert analysis we implement in this work.

### Organisation of the paper

The rest of this paper is organised as follows. In Section 3, we introduce the
overall strategy and complete the proof of Theorem 2.2. However, two main
ingredients, Proposition 2.1 for the energy evaluations, and Theorem 3.11 for
the fine asymptotic behaviours of the orthogonal polynomials, will be
established in the later sections. Section 4 is devoted to proving Proposition
2.1 based on the logarithmic potential theory and conformal mapping method. In
Section 5, we prove Theorem 3.11 using Riemann-Hilbert analysis and a partial
Schlesinger transform. Section 6 is a separate part, where we provide the
derivation of the duality formula of the form (1.9) and also the proof of
Proposition 2.5 on the free energy expansion in the critical regime.

##  3\. Proof of Theorem 2.2

This section reaches its culmination with the proof of Theorem 2.2. For
reader’s convenience, we begin by providing a summary of the overall strategy.

  * •

As a first step, we express the partition function as an integral of its
derivative with respect to a deformation parameter. Here, the integration
constants are given by the reference partition functions, whose asymptotic
behaviours can be computed precisely (Lemma 3.1). With proper choices of
reference partition functions, this implies that it suffices to derive the
asymptotic expansion of the derivative of free energies (Proposition 3.5), for
which we also need the derivatives of the coefficients (Lemma 3.3).

  * •

Using the $\tau$-function, the derivative of the partition functions can be
expressed in terms of the solution to the Riemann-Hilbert problem for the
associated planar orthogonal polynomial (Lemma 3.8). This allows us to express
the derivative of the partition function in terms of the coefficient of the
solution to the Riemann-Hilbert problem in the asymptotic expansion
(Proposition 3.9).

  * •

As a consequence of the previous step, the free energy expansion can be
derived using the asymptotic expansion of the orthogonal polynomial near
infinity (Proposition 3.10). Then, by refining the Riemann-Hilbert analysis in
[16], we obtain the fine asymptotic behaviour of the orthogonal polynomial
outside the motherbody (Theorem 3.11). Consequently, by computing the residue
near infinity, we complete the proof of Theorem 2.2.

Let us now be more precise in introducing our strategy. Let $p_{j}$ be the
monic polynomial satisfying

(3.1)
$\int_{\mathbb{C}}p_{j}(z)\overline{p_{k}(z)}e^{-NQ(z)}\,dA(z)=h_{j}\,\delta_{jk},$

where $Q$ is given by (1.5) and $\delta_{jk}$ is the Kronecker delta. Then we
have

(3.2) $Z_{N}(a,c)=N!\prod_{j=0}^{N-1}h_{j},$

see e.g. [29, Chapter 5].

We now begin by explaining the above strategy in more detail. Each step in the
above item is given in Subsections 3.1, 3.2, and 3.3, respectively.

###  3.1. Deformations of partition functions

An important idea we employ in the asymptotic analysis of the partition
functions is to consider the partition function $a\mapsto Z_{N}(a,c)$ as a
function of the deformation parameter $a\geq 0$. Next, we require reference
partition functions for which we can compute explicit formulas, along with
their precise asymptotic expansions. The chosen reference partition functions
are in the extremal, rotationally symmetric case. Additionally, it is crucial
to select a reference partition function whose droplet is topologically
equivalent to the droplet of the regime we intend to compute. This choice is
essential to simplify the proof by avoiding the critical regime during this
deformation. Consequently, we set $a=0$ and $a=\infty$ for the post- and pre-
critical cases, respectively. The concrete statement of the above discussion
is provided in the following lemma. We mention that the idea of implementing
deformations of partition functions (or structured determinants) has been
utilized in Hermitian random matrix theory [74, 49, 42] as well as in the
Coulomb gas theory on a Jordan domain [102, 69].

Figure 5. The plot shows deformations of the droplet. The leftmost $(a=0)$ and
rightmost $(a=\infty)$ are the rotationally symmetric cases, for which we use
the associated partition functions as reference. The thick lines indicate the
integral domains in (3.3) and (3.4), respectively.

######  Lemma 3.1 (Deformations and reference partition functions).

We have

(3.3) $\displaystyle\log Z_{N}(a,c)$ $\displaystyle=\log
Z_{N}(0,c)+\int_{0}^{a}\frac{d}{dt}\log Z_{N}(t,c)\,dt$ (3.4)
$\displaystyle=\log Z_{N}^{\rm Gin}+(2c\log
a)N^{2}-\int_{a}^{\infty}\Big{(}\frac{d}{dt}\log
Z_{N}(t,c)-\frac{2cN^{2}}{t}\Big{)}\,dt.$

The reference partition functions are evaluated as

(3.5)
$Z_{N}(0,c)=N!\frac{G(N+cN+1)}{G(cN+1)}N^{-(c+\frac{1}{2})N^{2}-\frac{1}{2}N},\qquad
Z_{N}^{\rm Gin}=N!\,G(N+1)N^{-\frac{1}{2}N^{2}-\frac{1}{2}N},$

and they satisfy the asymptotic behaviours as $N\to\infty$:

(3.6) $\displaystyle\begin{split}\log
Z_{N}(0,c)&=-\Big{(}\frac{3}{4}+\frac{3c}{2}+\frac{c^{2}}{2}\log
c-\frac{(c+1)^{2}}{2}\log(c+1)\Big{)}\,N^{2}+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1\Big{)}N\\\ &\quad+\frac{1}{2}\log
N+\frac{\log(2\pi)}{2}+\frac{1}{12}\log\Big{(}\frac{c}{1+c}\Big{)}\\\
&\quad+\sum_{k=1}^{\infty}\bigg{(}\frac{B_{2k}}{2k(2k-1)}\frac{1}{N^{2k-1}}+\frac{B_{2k+2}}{4k(k+1)}\Big{(}\frac{1}{(c+1)^{2k}}-\frac{1}{c^{2k}}\Big{)}\frac{1}{N^{2k}}\bigg{)},\end{split}$
(3.7) $\displaystyle\begin{split}\log Z_{N}^{\rm
Gin}&=-\frac{3}{4}\,N^{2}+\frac{1}{2}N\log
N+\Big{(}\frac{\log(2\pi)}{2}-1\Big{)}N\\\ &\quad+\frac{5}{12}\log
N+\frac{\log(2\pi)}{2}+\zeta^{\prime}(-1)+\sum_{k=1}^{\infty}\bigg{(}\frac{B_{2k}}{2k(2k-1)}\frac{1}{N^{2k-1}}+\frac{B_{2k+2}}{4k(k+1)}\frac{1}{N^{2k}}\bigg{)},\end{split}$

where $B_{k}$ is the Bernoulli number.

As previously mentioned, we shall use (3.3) for the post-critical regime, and
(3.4) for the pre-critical regime, see Figure 5 for an illustration.

######  Remark 3.2 (Total integration formula).

As an immediate consequence of (3.3), (3.4) and (3.5), it follows that

(3.8) $\displaystyle\begin{split}\int_{0}^{\infty}\Big{(}\partial_{t}\log
Z_{N}(t,c)-\frac{2cN^{2}}{t}\mathbbm{1}_{t>a}\Big{)}\,dt&=(2c\log
a)N^{2}+\log\Big{(}\frac{Z_{N}^{\rm Gin}}{Z_{N}(0,c)}\Big{)}\\\
&=cN^{2}\log(a^{2}N)+\log\Big{(}\frac{G(N+1)G(cN+1)}{G(N+cN+1)}\Big{)}.\end{split}$

###### Proof of Lemma 3.1.

The formula (3.3) is obvious. To see (3.4), let us write

(3.9)
$\displaystyle\widehat{Z}_{N}(a,c):=\int_{\mathbb{C}^{N}}\prod_{j>k=1}^{N}|z_{j}-z_{k}|^{2}\prod_{j=1}^{N}e^{-N\widehat{Q}(z_{j})}\,dA(z_{j}),\qquad\widehat{Q}(z)=|z|^{2}-2c\log\Big{|}\frac{z-a}{a}\Big{|}.$

Note that

(3.10)
$\lim_{a\to\infty}\widehat{Q}(z)=|z|^{2},\qquad\widehat{Z}_{N}(\infty,c)=Z_{N}^{\rm
Gin}.$

Since $e^{-NQ(z)}=e^{-N\widehat{Q}(z)}\,a^{2cN},$ it follows that

(3.11) $\log Z_{N}(a,c)=(2c\log a)N^{2}+\log\widehat{Z}_{N}(a,c).$

Then we have

(3.12) $\begin{split}\log Z_{N}(a,c)&=(2c\log
a)N^{2}+\log\widehat{Z}_{N}(a,c)\\\ &=(2c\log
a)N^{2}+\log\widehat{Z}_{N}(\infty,c)-\int_{a}^{\infty}\partial_{t}\log\widehat{Z}_{N}(t,c)\,dt\\\
&=(2c\log
a)N^{2}+\log\widehat{Z}_{N}(\infty,c)-\int_{a}^{\infty}\Big{(}\partial_{t}\log
Z_{N}(t,c)-\frac{2cN^{2}}{t}\Big{)}\,dt,\end{split}$

which leads to (3.4).

If $a=0$, the potential is radially symmetric, and consequently the associated
orthogonal polynomial is monomial, i.e. $p_{n}(z)=z^{n}.$ The orthogonal norm
is then computed as

(3.13)
$h_{j}=\int_{\mathbb{C}}|z|^{2j}e^{-NQ(z)}\,dA(z)=2\int_{0}^{\infty}r^{2j+2cN+1}e^{-Nr^{2}}\,dr=\frac{\Gamma(j+cN+1)}{N^{j+cN+1}}.$

By using (3.2), we have

(3.14)
$Z_{N}(0,c)=N!\prod_{k=0}^{N-1}\frac{\Gamma(k+cN+1)}{N^{k+cN+1}}=N!\frac{G(N+cN+1)}{G(cN+1)}N^{-(c+\frac{1}{2})N^{2}-\frac{1}{2}N}.$

This also gives (3.5) since $Z_{N}^{\rm Gin}=Z_{N}(0,0)$. Now (3.6) and (3.7)
follow from the asymptotic behaviours of the gamma function

(3.15) $\log N!=\log N+\log\Gamma(N)=\Big{(}N+\frac{1}{2}\Big{)}\log
N-N+\frac{1}{2}\log(2\pi)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)N^{2k-1}}$

as $N\to\infty$, and of the Barnes $G$-function

(3.16) $\displaystyle\begin{split}\log G(z+1)&=\frac{z^{2}\log
z}{2}-\frac{3}{4}z^{2}+\frac{\log(2\pi)z}{2}-\frac{\log
z}{12}+\zeta^{\prime}(-1)+\sum_{k=1}^{\infty}\frac{B_{2k+2}}{4k(k+1)}\frac{1}{z^{2k}},\end{split}$

as $z\to\infty$, see e.g. [91, Eqs.(5.11.1), (5.17.5)]. ∎

By Lemma 3.1, we need to derive the asymptotic behaviours of $\frac{d}{dt}\log
Z_{N}(t,c)$. For this purpose, we need to compute the derivatives of the
coefficients in the expansion (2.6). In the following, we denote by
$\partial_{a}$ and $\bar{\partial}_{a}$ the complex derivatives with respect
to $a$ and $\bar{a}$, respectively. To distinguish the notations further, we
also use $\mathfrak{d}_{a}=\partial_{a}+\bar{\partial}_{a}$ to represent the
operator of differentiation with respect to the real variable $a$. For
instance, $\mathfrak{d}_{a}a^{2}=2a$, whereas $\bar{\partial}_{a}|a|^{2}=a$.

######  Lemma 3.3 (Derivatives of the coefficients in the free energy
expansion).

For the pre-critical case, we have

(3.17) $\displaystyle\begin{split}\mathfrak{d}_{a}\,\mathcal{I}^{\rm
pre}(a,c)=\frac{1}{2a}+a-\frac{1}{aq^{2}}-\frac{a^{3}q^{4}}{2}=-\frac{(1-a^{2}q^{2})(2-q^{2}-a^{2}q^{4})}{2aq^{2}}\end{split}$

and

(3.18) $\displaystyle\begin{split}\mathfrak{d}_{a}\mathcal{F}^{\rm
pre}(a,c)&=-\frac{q^{2}(1-a^{4}q^{4})^{2}}{8a(1-q^{2})(1-a^{4}q^{6})^{2}}.\end{split}$

###### Proof.

Recall that $q\equiv q(a)$ is a function of $a.$ By (1.18), we have

(3.19) $c=\frac{a^{2}q^{2}}{2}+\frac{1}{4a^{2}q^{4}}-\frac{a^{2}+2}{4}.$

By differentiating (1.18) with respect to $a$ and using (3.19), we have

(3.20)
$\frac{q^{\prime}}{q}=\frac{1}{2a}\frac{1+a^{4}q^{4}-2a^{4}q^{6}}{a^{4}q^{6}-1}.$

Using (3.20), one can express derivatives of (2.3) and (2.9) in terms of $q$
and $a$. Then the lemma follows from straightforward computations. ∎

######  Remark 3.4.

Note that by (3.17) and (3.19) , we have

(3.21) $\displaystyle\mathfrak{d}_{a}\,\Big{(}\mathcal{I}^{\rm
pre}(a,c)-\mathcal{I}^{\rm post}(a,c)\Big{)}$
$\displaystyle=\frac{1}{2a}+a-\frac{1}{aq^{2}}-\frac{a^{3}q^{4}}{2}+2ac=\frac{(1-q^{2})^{2}(1-a^{4}q^{4})}{2aq^{4}}>0.$

This implies the inequality (2.4).

By combining Proposition 2.1, Lemma 3.1 and Lemma 3.3, Theorem 2.2 can be
reduced to the following proposition.

######  Proposition 3.5 (Asymptotic expansion of the derivative of free
energies).

As $N\to\infty$, we have the following.

  * (i)

For the post-critical case, we have

(3.22) $\mathfrak{d}_{a}\log Z_{N}(a,c)=2caN^{2}+O(N^{-\infty}).$

  * (ii)

For the pre-critical case, we have

(3.23) $\mathfrak{d}_{a}\log
Z_{N}(a,c)=\frac{(1-a^{2}q^{2})(2-q^{2}-a^{2}q^{4})}{2aq^{2}}\,N^{2}-\frac{q^{2}(1-a^{4}q^{4})^{2}}{8a(1-q^{2})(1-a^{4}q^{6})^{2}}+O(N^{-1}).$

In the following subsection, we further reduce Proposition 3.5 to certain
asymptotic behaviours of the planar orthogonal polynomial (3.1).

###  3.2. Riemann-Hilbert problem and $\tau$-function

We shall implement the $\tau$-function [25], which arises in the context of
the Riemann-Hilbert problem, cf. [16, 83, 85]. The first important idea in
[16] for the asymptotic analysis of planar orthogonal polynomial $p_{n}$ is to
demonstrate an equivalence between the planar orthogonality (3.1) and a
certain orthogonality on a contour. Extensions of such an equivalence can be
found in [22, 84]. As a consequence, one can construct a standard Riemann-
Hilbert problem for $p_{n}$, and we recall it here.

Let us mention that on some occasions, we write both $n$ and $N$. This
distinction has usually been made in Riemann-Hilbert analysis to distinguish
between the degree of the orthogonal polynomial and the number of particles.
However, in our context, we use $n=N$ throughout the paper.

######  Definition 3.6 (Riemann-Hilbert problem $Y$).

Let $\Gamma$ be a simple closed curve enclosing $0$ and $a$. Define the weight
function

(3.24) $\omega_{n,N}(z)=\frac{(z-a)^{Nc}e^{-Naz}}{z^{Nc+n}}.$

We then consider the following Riemann-Hilbert problem for $Y\equiv Y_{n}$:

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

Here $Y_{\pm}(x)$ is the boundary values of $Y$ on $\Gamma$.

Let $q_{n}(z):=q_{n,N}(z)$ be a unique polynomial of degree $n-1$ such that

$\frac{1}{2\pi
i}\int_{\Gamma}\frac{q_{n}(w)\omega_{n,N}(w)}{w-z}dw=\frac{1}{z^{n}}\Big{(}1+{O}\Big{(}\frac{1}{z}\Big{)}\Big{)}.$

Then the matrix function

(3.26) $Y(z)\equiv Y_{n}(z):=\begin{pmatrix}\displaystyle
p_{n}(z)&\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\frac{p_{n}(w)\omega_{n,N}(w)}{w-z}\,dw\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ q_{n}(z)&\displaystyle\frac{1}{2\pi
i}\int_{\Gamma}\frac{q_{n}(w)\omega_{n,N}(w)}{w-z}\,dw\end{pmatrix}$

is a unique solution to the Riemann-Hilbert problem (3.25).

We now define

(3.27) $\widetilde{Y}(z):=Y(z)T(z),\qquad T(z)\equiv
T_{n}(z)=\begin{pmatrix}\Big{(}\dfrac{z-a}{z}\Big{)}^{Nc}{e^{-Naz}}&0\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ 0&z^{n}\end{pmatrix}.$

Then $\widetilde{Y}$ satisfies the following Riemann-Hilbert problem:

(3.28) $\begin{cases}\widetilde{Y}(z)\mbox{\quad is holomorphic},&\quad
z\in\mathbb{C}\setminus\Gamma,\\\
\widetilde{Y}_{+}(z)=\widetilde{Y}_{-}(z)\begin{pmatrix}1&1\\\
0&1\end{pmatrix},&\quad z\in\Gamma,\\\
\widetilde{Y}(z)=\big{(}I+{O}(z^{-1})\big{)}\begin{pmatrix}z^{n}&0\\\
0&z^{-n}\end{pmatrix}\begin{pmatrix}\Big{(}\dfrac{z-a}{z}\Big{)}^{Nc}\dfrac{z^{n}}{e^{Naz}}&0\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ 0&1\end{pmatrix},&\quad
z\to\infty.\end{cases}$

Then the matrices $Y_{N}$ and $T_{N}$ can be used to express the derivative of
the partition function.

######  Definition 3.7 ($\tau$-function).

Following [25, Definition 3.2], the $\tau$-function is defined by

(3.29)
$\partial_{a}\log\tau:=\underset{z=\infty}{\textup{Res}}\,\bigg{[}\operatorname{Tr}\Big{(}Y(z)^{-1}\partial_{z}Y(z)\partial_{a}T(z)T(z)^{-1}\Big{)}\bigg{]}.$

Here $\tau\equiv\tau_{n}$ depends on the degree $n$ of the orthogonal
polynomial $p_{n}$.

######  Lemma 3.8 (Partition function in terms of the Riemann-Hilbert
problem).

We have

(3.30) $\partial_{a}\log Z_{N}(a,c)=\partial_{a}\log\tau_{N}.$

###### Proof.

It follows from [25, Theorem 3.4] that

(3.31)
$\partial_{a}\log\tau_{N}=\underset{z=\infty}{\textup{Res}}\,\bigg{[}\operatorname{Tr}\Big{(}Y_{N}(z)^{-1}\partial_{z}Y_{N}(z)\partial_{a}T_{N}(z)\,T_{N}(z)^{-1}\Big{)}\bigg{]}=\partial_{a}\log\Big{(}\det[\nu_{j+k}]_{j,k=0}^{N-1}\Big{)},$

where

(3.32) $\nu_{k}=\int_{\Gamma}z^{k}\omega_{n,N}(z)\,dz.$

Let us define

(3.33) $\mu_{jk}:=\int_{\mathbb{C}}z^{j}\bar{z}^{k}e^{-NQ(z)}\,dA(z).$

Then by the general theory on determinantal point process, we have

(3.34) $Z_{N}(a,c)=N!\,\det[\mu_{jk}]_{j,k=1}^{N}.$

Note that by [16, (E.1)], we have

(3.35)
$\frac{\det[\mu_{jk}]_{j,k=1}^{N}}{\det[\nu_{j+k}]_{j,k=0}^{N-1}}=\frac{(-1)^{N(N-1)/2}}{\pi^{N}}\prod_{k=0}^{N-1}\frac{\Gamma(cN+k+1)}{2i\,N^{cN+k+1}}.$

Since the right-hand side of this equation does not depend on $a$, it follows
that

(3.36)
$\partial_{a}\det[\mu_{jk}]_{j,k=1}^{N}=\partial_{a}\det[\nu_{j+k}]_{j,k=0}^{N-1}.$

Combining all of the above, the lemma follows. ∎

######  Proposition 3.9 (Partition functions and fine asymptotics of the
orthogonal polynomial).

Suppose that $Y_{N}(z)$ satisfies the strong asymptotic behaviour of the form

(3.37)
$Y_{N}(z)=\bigg{(}I+\frac{1}{z}\begin{pmatrix}\mathfrak{A}_{11}&\mathfrak{A}_{12}\\\
\mathfrak{A}_{21}&-\mathfrak{A}_{11}\end{pmatrix}+O\Big{(}\frac{1}{z^{2}}\Big{)}\bigg{)}\begin{pmatrix}z^{N}&0\\\
0&z^{-N}\end{pmatrix},\qquad z\to\infty.$

for certain coefficients $\mathfrak{A}_{jk}$’s. Then we have

(3.38) $\mathfrak{d}_{a}Z_{N}(a,c)=2N\,\mathfrak{A}_{11}.$

###### Proof.

By (3.27), we have

$\partial_{a}T_{N}(z)T_{N}(z)^{-1}=-N\begin{pmatrix}z+\dfrac{c}{z-a}&0\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ 0&0\end{pmatrix},$

which gives

$\displaystyle\operatorname{Tr}\Big{(}Y_{N}(z)^{-1}\partial_{z}Y_{N}(z)\partial_{a}T_{N}(z)\,T_{N}(z)^{-1}\Big{)}$
$\displaystyle=-N\Big{(}z+\frac{c}{z-a}\Big{)}\Big{[}Y_{N}(z)^{-1}\partial_{z}Y_{N}(z)\Big{]}_{11}.$

Then it follows from the asymptotic behaviour in (3.37) that

$\underset{z=\infty}{\textup{Res}}\,\bigg{[}\operatorname{Tr}\Big{(}Y_{N}(z)^{-1}\partial_{z}Y_{N}(z)\partial_{a}T_{N}(z)\,T_{N}(z)^{-1}\Big{)}\bigg{]}=\underset{z=\infty}{\textup{Res}}\,\Big{[}-N^{2}+\frac{N\mathfrak{A}_{11}}{z}+O\big{(}\frac{1}{z^{2}}\big{)}\Big{]}=N\mathfrak{A}_{11}.$

Combining this with (3.30), the proof of the lemma is complete. ∎

Then by (3.26), Proposition 3.5 can be further reduced to the following.

######  Proposition 3.10 (Asymptotic behaviour of the coefficients).

The orthogonal polynomial $p_{N}$ satisfies the asymptotic behaviour of the
form

(3.39) $p_{N}(z)=z^{N}+\mathfrak{A}_{11}z^{N-1}+O(z^{N-2}),\qquad z\to\infty.$

Furthermore, as $N\to\infty$, the coefficient $\mathfrak{A}_{11}$ satisfies
the following asymptotic behaviour.

  * •

For the post-critical case,

(3.40) $\mathfrak{A}_{11}=caN+O(\frac{1}{N^{\infty}}).$

  * •

For the pre-critical case,

(3.41)
$\mathfrak{A}_{11}=\frac{(1-a^{2}q^{2})(2-q^{2}-a^{2}q^{4})}{4aq^{2}}\,N-\frac{q^{2}(1-a^{4}q^{4})^{2}}{16a(1-q^{2})(1-a^{4}q^{6})^{2}}\frac{1}{N}+O(\frac{1}{N^{2}}).$

In Theorem 3.11 below, we shall prove a stronger statement on the fine
asymptotic behaviours of $p_{N}$.

###  3.3. Fine asymptotic behaviour and proof of Theorem 2.2

We show the fine asymptotic behaviours of the orthogonal polynomial.

######  Theorem 3.11 (Fine asymptotic behaviour of the orthogonal polynomial).

Let $\mathcal{B}$ the curve defined in Definition 4.2 and $g$ be the function
defined in Definition 4.3. Then as $N\to\infty$, we have the following.

  * •

(Post-critical regime) We have

(3.42)
$p_{N}(z)=z^{N}\Big{(}\frac{z}{z-a}\Big{)}^{cN}(1+O(N^{-\infty})),\qquad
z\in\operatorname{ext}(\mathcal{B}).$

  * •

(Pre-critical regime) We have

(3.43) $p_{N}(z)=\Big{(}\sqrt{RF^{\prime}(z)}(1+R_{11}(z))-\frac{\sqrt{\kappa
F^{\prime}(z)}}{F(z)-q}R_{12}(z)\Big{)}e^{Ng(z)}\Big{(}1+O(N^{-2})\Big{)},\qquad
z\in\operatorname{ext}(\mathcal{B})$

where $F(z)$ is given by (2.25), and $R_{11}$ and $R_{12}$ are defined in
Definition 5.8.

Here the error terms are uniform over $z$ in compact sets of
$\mathbb{C}\setminus\mathcal{B}$.

We stress that Theorem 3.11 extends [16, Theorems 1.3 and 1.4] providing more
precise error terms. Let us also mention that the asymptotic behaviour of
_orthonormal_ polynomials outside the droplet has also been studied by
Hedenmalm and Wennman in a more general context [66, 65, 64]. In these works,
an algorithm computing the subleading correction terms was also introduced,
but their explicit evaluations for a given potential require a separate
analysis. In particular, Theorem 3.11 is not a direct consequence of the
general theory developed by Hedenmalm and Wennman, as we need to compute the
_monic_ orthogonal polynomial and also derive the asymptotic behaviour inside
the droplet.

We are now ready to show Theorem 2.2.

###### Proof of Theorem 2.2.

As explained in the previous subsection, it suffices to show Proposition 3.10.

For the post-critical case, it follows from (3.42) that

$p_{N}(z)=\Big{(}z^{N}+Nac\,z^{N-1}+O(z^{N-2})\Big{)}\Big{(}1+O(N^{-\infty})\Big{)},\qquad
z\to\infty.$

This gives rise to (3.40).

Next, let us show the pre-critical case. By Lemma 4.4 below, we have

(3.44)
$e^{Ng(z)}=z^{N}\bigg{(}1-\Big{(}\frac{1}{4a}+\frac{a}{2}-\frac{1}{2aq^{2}}-\frac{a^{3}q^{4}}{4}\Big{)}\frac{N}{z}+O(\frac{1}{z^{2}})\bigg{)}.$

On the other hand, by using [16, Eq.(1.28)],

(3.45)
$F(z)=\frac{z}{R}+\frac{\kappa}{Rq}+\frac{\kappa}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)},\qquad
z\to\infty.$

Therefore we have

(3.46) $RF^{\prime}(z)=1+O(\frac{1}{z^{2}}),\qquad z\to\infty.$

Furthermore, by definition, we have

(3.47) $R_{11}(z)=O(\frac{1}{z}),\qquad R_{12}(z)=O(\frac{1}{z}),\qquad
z\to\infty.$

Then the desired identity (3.41) follows from

(3.48)
$\underset{z=\infty}{\textup{Res}}\,\Big{[}R_{11}(z)\Big{]}=-\frac{q^{2}(1-a^{4}q^{4})^{2}}{16a(1-q^{2})(1-a^{4}q^{6})^{2}}\frac{1}{N}.$

It remains to show (3.48). For this, note that as $z\to\infty,$

$\displaystyle R_{1}(z)R_{2}(z)$
$\displaystyle=\bigg{(}I+\frac{1}{z-\overline{\beta}}R_{2}(z)h_{11}R_{2}(z)^{-1}+\frac{1}{(z-\overline{\beta})^{2}}R_{2}(z)h_{12}R_{2}(z)^{-1}\bigg{)}\bigg{(}I+\frac{h_{21}}{z-\beta}+\frac{h_{22}}{(z-\beta)^{2}}\bigg{)}.$

Here $R_{1},R_{2}$, $h_{11}$, $h_{12}$ and $h_{21},h_{22}$ are given in
Definition 5.8. Then

(3.49)
$\displaystyle\begin{split}\underset{z=\infty}{\textup{Res}}\,\Big{[}R_{11}(z)\Big{]}&=\frac{1}{N}\frac{1+i}{128\sqrt{2}}\bigg{(}\frac{3}{(R\kappa)^{1/4}}\Big{(}\frac{1}{\gamma_{11}^{3/2}}-\frac{i}{\gamma_{21}^{3/2}}\Big{)}+10(R\kappa)^{1/4}\Big{(}\frac{\gamma_{22}}{\gamma_{21}^{5/2}}-\frac{i\gamma_{12}}{\gamma_{11}^{5/2}}\Big{)}\bigg{)}\\\
&=\frac{1+i}{128\sqrt{2}}\frac{1}{(R\kappa)^{1/4}}\bigg{(}\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}-i\,\overline{\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}}\,\bigg{)},\end{split}$

where $\gamma_{11}=\overline{\gamma}_{21}$ and
$\gamma_{12}=\overline{\gamma}_{22}$ are given by (5.11) and (5.12). We claim
that

(3.50)
$\displaystyle\begin{split}&\quad\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}-i\,\overline{\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}}\\\
&=-4\sqrt{2}(1-i)(R\kappa)^{1/4}\frac{q^{2}(1-a^{4}q^{4})^{2}}{a(1-q^{2})(1-a^{4}q^{6})^{2}},\end{split}$

which leads to (3.48). By using (5.11) and (5.12), we have

$\displaystyle\quad\frac{1}{(R\kappa)^{1/4}}\frac{1}{\gamma_{11}^{3/2}}\bigg{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\bigg{)}=\frac{2}{(R\kappa)^{1/4}}\frac{(\overline{\beta}-a)\overline{\beta}}{a(\overline{\beta}-R/q)\sqrt{\overline{\beta}-\beta}}\bigg{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\bigg{)}$
$\displaystyle=\frac{2}{(R\kappa)^{1/4}}\frac{(\overline{\beta}-a)\overline{\beta}}{a(\overline{\beta}-R/q)\sqrt{\overline{\beta}-\beta}}\bigg{(}3-8(R\kappa)^{1/2}i\Big{(}\frac{1}{\overline{\beta}-R/q}+\frac{1}{2(\overline{\beta}-\beta)}-\frac{1}{\overline{\beta}-a}-\frac{1}{\overline{\beta}}\Big{)}\bigg{)}.$

Then

$\displaystyle\quad\frac{1}{(R\kappa)^{1/4}}\bigg{(}\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}-i\,\overline{\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}}\,\bigg{)}$
$\displaystyle=\frac{e^{\frac{\pi i}{4}}}{a(\kappa
R)^{1/2}}\frac{(\overline{\beta}-a)\overline{\beta}}{\overline{\beta}-R/q}\bigg{(}3-8(R\kappa)^{1/2}i\Big{(}\frac{1}{\overline{\beta}-R/q}+\frac{1}{2(\overline{\beta}-\beta)}-\frac{1}{\overline{\beta}-a}-\frac{1}{\overline{\beta}}\Big{)}\bigg{)}$
$\displaystyle\quad-\frac{e^{\frac{\pi i}{4}}}{a(\kappa
R)^{1/2}}\frac{(\beta-a)\beta}{\beta-R/q}\bigg{(}3+8(R\kappa)^{1/2}i\Big{(}\frac{1}{\beta-R/q}+\frac{1}{2(\beta-\overline{\beta})}-\frac{1}{\beta-a}-\frac{1}{\beta}\Big{)}\bigg{)}.$

Note that

$\displaystyle\quad\frac{(\overline{\beta}-a)\overline{\beta}}{\overline{\beta}-R/q}-\frac{(\beta-a)\beta}{\beta-R/q}=\frac{(\overline{\beta}-a)\overline{\beta}(\beta-R/q)-(\beta-a)\beta(\overline{\beta}-R/q)}{(\overline{\beta}-R/q)(\beta-R/q)}$
$\displaystyle=-q(\beta-\overline{\beta})\frac{q|\beta|^{2}+(a-\beta-\overline{\beta})R}{q^{2}|\beta|^{2}-q(\beta+\overline{\beta})R+R^{2}}=-4(\kappa
R)^{1/2}i\,q\,\frac{q|\beta|^{2}+(a-\beta-\overline{\beta})R}{q^{2}|\beta|^{2}-q(\beta+\overline{\beta})R+R^{2}}.$

On the other hand, we have

$\displaystyle\quad\frac{(\beta-a)\beta}{\beta-R/q}\Big{(}\frac{1}{\beta-R/q}+\frac{1}{2(\beta-\overline{\beta})}-\frac{1}{\beta-a}-\frac{1}{\beta}\Big{)}+\frac{(\overline{\beta}-a)\overline{\beta}}{\overline{\beta}-R/q}\Big{(}\frac{1}{\overline{\beta}-R/q}+\frac{1}{2(\overline{\beta}-\beta)}-\frac{1}{\overline{\beta}-a}-\frac{1}{\overline{\beta}}\Big{)}$
$\displaystyle=-\frac{\beta^{2}+(a-2\beta)R/q}{(\beta-R/q)^{2}}-\frac{\overline{\beta}^{2}+(a-2\overline{\beta})R/q}{(\overline{\beta}-R/q)^{2}}+\frac{q}{2}\frac{q|\beta|^{2}+(a-\beta-\overline{\beta})R}{q^{2}|\beta|^{2}-q(\beta+\overline{\beta})R+R^{2}}.$

Therefore we obtain

$\displaystyle\quad\frac{1}{(R\kappa)^{1/4}}\bigg{(}\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}-i\,\overline{\frac{1}{\gamma_{11}^{3/2}}\Big{(}3-10(R\kappa)^{1/2}i\frac{\gamma_{12}}{\gamma_{11}}\Big{)}}\,\bigg{)}$
$\displaystyle=\frac{4\sqrt{2}(1-i)}{a}\bigg{(}2q\,\frac{q|\beta|^{2}+(a-\beta-\overline{\beta})R}{q^{2}|\beta|^{2}-q(\beta+\overline{\beta})R+R^{2}}-\frac{\beta^{2}+(a-2\beta)R/q}{(\beta-R/q)^{2}}-\frac{\overline{\beta}^{2}+(a-2\overline{\beta})R/q}{(\overline{\beta}-R/q)^{2}}\bigg{)}.$

Then the desired identity (LABEL:desired_iden_for_main) follows from (2.26)
and (1.17). ∎

For our main theorem, it remains to prove Proposition 2.1 and Theorem 3.11.
Before that, we discuss the critical case in a separate subsection.

##  4\. Potential theoretic preliminaries

In this section, we prove Proposition 2.1. Before the proof, let us first
consider the radially symmetric case.

######  Remark 4.1 (Energy for the radially symmetric case).

We consider a general radially symmetric potential

(4.1) $W(z)=w(|z|),\qquad w:\mathbb{R}_{+}\to\mathbb{R}.$

Suppose that $\Delta W(z)>0$ in $\mathbb{C}$. Then it follows from a general
theory [96] that the associated droplet $S_{W}$ is given by

(4.2) $S_{W}=\\{z\in\mathbb{C}:r_{0}\leq|z|\leq r_{1}\\},$

where $(r_{0},r_{1})$ is a unique pair of constants satisfying

(4.3) $r_{0}w^{\prime}(r_{0})=0,\qquad r_{1}w^{\prime}(r_{1})=2.$

Furthermore, the energy $I_{V}[\sigma_{Q}]$ is given by

(4.4) $I_{W}[\sigma_{W}]=w(r_{1})-\log
r_{1}-\frac{1}{4}\int_{r_{0}}^{r_{1}}rw^{\prime}(r)^{2}\,dr.$

Using this, we have that for the potential $Q$ in (1.5) with $a=0$,

(4.5)
$I_{Q}[\sigma_{Q}]\Big{|}_{a=0}=\frac{3}{4}+\frac{3c}{2}+\frac{c^{2}}{2}\log
c-\frac{(c+1)^{2}}{2}\log(c+1).$

This coincides with (2.2).

###  4.1. $g$-function

In Riemann-Hilbert analysis, the first step is to normalise the problem so
that it is close to the identity near infinity. For this, we need a proper
function called the $g$-function, which indeed gives the logarithmic energy of
limiting zeros of the orthogonal polynomial.

We first recall the motherbody defined in [16, Section 1.1].

######  Definition 4.2 (Motherbody; limiting skeleton).

The motherbody $\mathcal{B}$ is defined as follows.

  * •

For the post-critical case, let

(4.6)
$\displaystyle\beta=\frac{a^{2}+1-\sqrt{(1-a^{2})^{2}-4a^{2}c}}{2a},\qquad
b=\frac{a^{2}+1+\sqrt{(1-a^{2})^{2}-4a^{2}c}}{2a}.$

Then the simple closed curve $\mathcal{B}$ is defined by the following three
conditions.

    * –

$\mathcal{B}$ is a simple closed curve such that $\beta\in\mathcal{B}$;

    * –

$\mathcal{B}$ contains $0$ and $a$ in its interior;

    * –

The following inequality is satisfied on the curve $\mathcal{B}$

(4.7) $a^{2}\frac{(z-\beta)^{2}(z-b)^{2}}{z^{2}(z-a)^{2}}\,dz<0,$

where $dz$ is the standard differential.

  * •

For the pre-critical case, let $\beta$ be given by (2.26). Let $b=R/q$, where
$R$ and $q$ are given by (1.17) and (1.18).333There is a typo in [16,
Eq.(1.12)], where $b:=\alpha/\rho$ should be replaced with $b:=\rho/\alpha$.
Then $\mathcal{B}$ is defined by the following three conditions.

    * –

$\mathcal{B}$ has the endpoints at $\beta$ and $\overline{\beta}$;

    * –

$\mathcal{B}$ intersects the negative real axis;

    * –

The following inequality is satisfied on the curve $\mathcal{B}$

(4.8) $\frac{(z-b)^{2}(z-\beta)(z-\overline{\beta})}{(z-a)^{2}z^{2}}\,dz<0.$

Here, we use a slight abuse of notation by employing both $\beta$ and $b$ to
denote variables in both post- and pre-critical regimes.

We now recall the $g$-function defined in [16, Definition 1.2].

######  Definition 4.3 ($g$-function).

The $g$-function is defined as follows.

  * •

For the post-critical case,

(4.9) $g(z)=\begin{cases}\displaystyle\log
z+c\log\Big{(}\frac{z}{z-a}\Big{)},&z\in\operatorname{ext}(\mathcal{B}),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
az+\operatorname{Re}\big{(}(1+c)\log\beta-c\log(\beta-a)-\beta
a\big{)},&z\in\mbox{int}(\mathcal{B}).\end{cases}$

We also define

(4.10) $\phi(z)=\int_{\beta}^{z}y(s)\,ds,\qquad y(z)=\pm
a\frac{(z-b)(z-\beta)}{z(z-a)}.$

  * •

For the pre-critical case, let

(4.11) $V(z)=az-c\log(z-a)+(c+1)\log z.$

Write

(4.12)
$y(z)=\frac{a(z-R/q)\sqrt{(z-\beta)(z-\overline{\beta})}}{(z-a)z},\qquad\phi(z)=\int_{\beta}^{z}y(s)\,ds,\qquad(z\in\mathbb{C}\setminus\mathcal{B}).$

Then the $g$-function is defined by

(4.13)
$g(z)=\frac{1}{2}\Big{(}V(z)-\phi(z)+\ell\Big{)},\qquad(z\in\mathbb{C}\setminus\mathcal{B}),$

where the constant $\ell\in\mathbb{R}$ is defined such that

(4.14) $\lim_{z\to\infty}(g(z)-\log z)=0.$

The $g$-function plays an important role in the asymptotic behaviour of
orthogonal polynomials, see Theorem 3.11. This indeed comes from the fact that
if we write $\zeta_{j}$ for the zeros of $p_{n}$ and $\nu$ for their limiting
distribution, we have formally

$\displaystyle
p_{N}(z)=\prod_{j=1}^{N}(z-\zeta_{j})=\exp\Big{(}\sum_{j=1}^{N}\log(z-\zeta_{j})\Big{)}\sim\exp\Big{(}N\int\log(z-\zeta)\,d\nu(\zeta)\Big{)}.$

Thus, the $g$-function is naturally defined by
$\int\log(z-\zeta)\,d\nu(\zeta)$ in a more general context, and Definition 4.3
provides its evaluations in our present context.

For our purposes explained in Section 3, let us compute the asymptotic
behaviour of $e^{Ng(z)}$ as $z\to\infty$. In the end, this will provide the
leading order asymptotic behaviour of the derivative of the partition
function.

######  Lemma 4.4 (Asymptotic behaviour of the $g$-function).

As $z\to\infty$, we have the following.

  * •

For the post-critical case,

(4.15)
$e^{Ng(z)}=z^{N}\Big{(}1+ac\,\frac{N}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)}\Big{)},\qquad
z\to\infty.$

  * •

For the pre-critical case,

(4.16)
$e^{Ng(z)}=z^{N}\bigg{(}1-\Big{(}\frac{1}{4a}+\frac{a}{2}-\frac{1}{2aq^{2}}-\frac{a^{3}q^{4}}{4}\Big{)}\frac{N}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)}\bigg{)},\qquad
z\to\infty.$

######  Remark 4.5.

Notice that the coefficients of the $1/z$ term in Lemma 4.4 coincide with the
leading terms in Proposition 3.10.

###### Proof of Lemma 4.4.

This follows from the definition. For instance, for the pre-critical case, we
have

(4.17)
$2g^{\prime}(z)=a-\frac{c}{z-a}+\frac{c+1}{z}-\frac{a(z-R/q)\sqrt{(z-f(z_{+}))(z-f(z_{-}))}}{(z-a)z}.$

Then it follows that

(4.18)
$g^{\prime}(z)=\frac{1}{z}+\Big{(}\frac{1+2a^{2}}{4a}-\frac{1}{2aq^{2}}-\frac{a^{3}q^{4}}{4}\Big{)}\frac{1}{z^{2}}+O\Big{(}\frac{1}{z^{3}}\Big{)}.$

This gives

(4.19) $g(z)=\log
z-\Big{(}\frac{1}{4a}+\frac{a}{2}-\frac{1}{2aq^{2}}-\frac{a^{3}q^{4}}{4}\Big{)}\frac{1}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)},$

which completes the proof. ∎

To compute the energy (2.3), we need to evaluate the $g$-function at the point
$a.$

######  Lemma 4.6.

For the pre-critical case, we have

(4.20)
$\displaystyle\begin{split}\operatorname{Re}g(a)&=\frac{a^{2}q^{2}}{2}-\frac{1}{2}+\log\Big{(}\frac{1+a^{2}q^{2}}{2aq^{2}}\Big{)}+c\log\Big{(}\frac{1+a^{2}q^{2}}{1+a^{2}q^{2}-2a^{2}q^{4}}\Big{)}.\end{split}$

This lemma is shown in the next subsection.

######  Remark 4.7.

We encounter surprising cancellations when differentiating (4.20) with respect
to $a$. Namely,

(4.21) $\mathfrak{d}_{a}\operatorname{Re}g(a)=aq^{2},$

where we have used (3.20). We currently lack a good intuition for such a
significant simplification.

###  4.2. Robin’s constant and energy

Recall that the equilibrium measure $\mu_{Q}$ satisfies the variational
equality (also known as the Euler–Lagrange equation):

(4.22)
$\displaystyle\int\log\frac{1}{|z-w|}\,d\mu_{Q}(z)+\frac{Q(w)}{2}=C,\qquad\text{if
}w\in S_{Q}$

where $C\equiv C(a)$ is called the modified Robin’s constant, see [96, p.27].
Then we have

(4.23)
$\int_{\mathbb{C}^{2}}\log\frac{1}{|z-w|}\,d\mu_{Q}(z)\,d\mu_{Q}(w)=C(a)-\frac{1}{2}\int
Q(z)\,d\mu_{Q}(z)$

and

(4.24) $\displaystyle\begin{split}I_{Q}[\mu_{Q}]=C(a)+\frac{1}{2}\int
Q(z)\,d\mu_{Q}(z).\end{split}$

By Lemma 4.6 and (4.24), it suffices to show the following lemma to complete
the proof of Proposition 2.1. Note that the inequality (2.4) was shown in
Remark 3.4.

######  Lemma 4.8 (Robin’s constant and energy).

We have the following.

  * •

For the post-critical case, we have

(4.25) $C(a)=\frac{c+1}{2}-\frac{c+1}{2}\log(c+1)$

and

(4.26) $\frac{1}{2}\int Q\,d\mu_{Q}=c+\frac{1}{4}+\frac{c^{2}}{2}\log
c-\frac{c(1+c)}{2}\log(1+c)-ca^{2}.$

In particular, we have (2.2).

  * •

For the pre-critical case, we have

(4.27) $\begin{split}C(a)&=c\log
q-(c+1)\log\Big{(}\frac{1+a^{2}q^{2}}{2aq}\Big{)}+\frac{3}{4}+\frac{a^{2}}{4}+\frac{c}{2}-\Big{(}\frac{a^{2}}{4}+c+\frac{3}{4}\Big{)}a^{2}q^{2}+\frac{a^{4}q^{4}}{2}\end{split}$

and

(4.28) $\begin{split}\frac{1}{2}\int
Q\,d\mu_{Q}&=\frac{3}{8}+\frac{a^{2}}{4}+\frac{c}{2}-\Big{(}\frac{a^{2}}{4}+c+\frac{1}{2}\Big{)}a^{2}q^{2}+\frac{3}{8}a^{4}q^{4}-c\operatorname{Re}g(a).\end{split}$

In particular, we have

(4.29) $\begin{split}I_{Q}[\mu_{Q}]&=c\Big{(}\log
q-\operatorname{Re}g(a)\Big{)}-(c+1)\log\Big{(}\frac{1+a^{2}q^{2}}{2aq}\Big{)}+\frac{5}{8}+\frac{a^{2}}{4}+\frac{1}{4a^{2}q^{4}}-\frac{1}{2q^{2}}+\frac{a^{2}q^{2}}{4}-\frac{a^{4}q^{4}}{8}.\end{split}$

######  Remark 4.9 (Point charge insertion at infinity).

It is easy to check from (1.18) that

(4.30)
$q=\frac{1}{a}\Big{(}1-\frac{c}{a^{2}}+O(\frac{1}{a^{4}})\Big{)},\qquad(a\to\infty).$

This in turn gives that

(4.31)
$R=1+O(\frac{1}{a^{4}}),\qquad\kappa=\frac{c}{a^{2}}+O(\frac{1}{a^{4}}),\qquad
f(z)=z+O(\frac{1}{a}),\qquad(a\to\infty),$

where $R,\kappa$ and $f$ are given by (1.17). Therefore the droplet tends to
the unit disc when $a\to\infty$, cf. Figure 5. Let us also mention that by
(4.30), we have

(4.32) $\lim_{a\to\infty}\Big{(}I_{Q}[\mu_{Q}]+2c\log
a\Big{)}=\frac{3}{4},\qquad\lim_{a\to\infty}\mathcal{F}(a,c)=0.$

From the viewpoint of the free energy expansions (1.13), these asymptotic
behaviours are consistent with (3.7) and (3.11). Furthermore, by (4.14) and
(4.30), we also have

(4.33) $\lim_{a\to\infty}\Big{(}C(a)+c\log
a\Big{)}=\frac{1}{2},\qquad\lim_{a\to\infty}\Big{(}\frac{1}{2}\int
Q(z)\,d\mu_{Q}(z)+c\log a\Big{)}=\frac{1}{4}.$

Note also that by combining Lemma 3.3 and (4.30), one can check that as
$a\to\infty,$

(4.34)
$\partial_{a}I_{Q}[\mu_{Q}]=-\frac{2c}{a}+O(a^{-3}),\qquad\partial_{a}\mathcal{F}(a,c)=-\frac{2c^{2}}{a^{7}}+O(a^{-9}).$

We now prove Lemma 4.8.

###### Proof of Lemma 4.8.

The modified Robin constant was computed in [16, Lemma 7.2]. Let us mention
that the Robin constant $\ell_{\rm 2D}$ in [16] corresponds to $\ell_{\rm
2D}=-2\,C(a).$

It remains to compute $\int Q\,d\mu_{Q}.$ We first show the post-critical case
when $c<c_{\rm cri}$. By [27, Lemma 2.4], we have that for $R>0$ and
$p\in\mathbb{C}$,

(4.35) $\int_{\mathbb{D}(p,R)}\log|z-w|\,dA(z)=\begin{cases}\displaystyle
R^{2}\log|w-p|&\text{if }w\notin\mathbb{D}(p,R),\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \displaystyle R^{2}\log
R-\frac{R^{2}}{2}+\frac{|w-p|^{2}}{2}&\text{otherwise}.\end{cases}$

Using this, we have

$\displaystyle-2c\int_{S}\log|w-a|\,dA(w)$
$\displaystyle=-2c\Big{(}\int_{\mathbb{D}(0,\sqrt{1+c})}\log|w-a|\,dA(w)-\int_{\mathbb{D}(a,\sqrt{c})}\log|w-a|\,dA(w)\Big{)}$
$\displaystyle=c^{2}(\log
c-1)-2c\int_{\mathbb{D}(0,\sqrt{1+c})}\log|w-a|\,dA(w)$
$\displaystyle=c+c^{2}\log c-c(1+c)\log(1+c)-ca^{2}.$

On the other hand, note that

$\displaystyle\begin{split}\int_{S}|w|^{2}\,dA(w)&=\int_{\mathbb{D}(0,\sqrt{1+c})}|w|^{2}\,dA(w)-\int_{\mathbb{D}(a,\sqrt{c})}|w|^{2}\,dA(w)=\frac{1}{2}(c+1)^{2}-\int_{\mathbb{D}(a,\sqrt{c})}|w|^{2}\,dA(w).\end{split}$

Here by Green’s formula, and using the map $z\mapsto w=\sqrt{c}z+a$,

$\displaystyle\begin{split}\int_{\mathbb{D}(a,\sqrt{c})}|w|^{2}\,dA(w)&=\frac{1}{2\pi
i}\int_{|w-a|=\sqrt{c}}\frac{w\bar{w}^{2}}{2}\,dw=\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}\frac{\sqrt{c}}{2}(\sqrt{c}z+a)(\sqrt{c}\bar{z}+a)^{2}\,dz\\\
&=\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}\frac{\sqrt{c}}{2}(\sqrt{c}z+a)(\sqrt{c}/z+a)^{2}\,dz=\frac{1}{2}c^{2}+ca^{2}.\end{split}$

This gives

(4.36) $\int_{S}|w|^{2}\,dA(w)=c+\frac{1}{2}-ca^{2}.$

We have shown the post-critical case.

Next, we consider the pre-critical case when $c>c_{\rm cri}$. By Green’s
formula,

$\displaystyle\int_{S}|z|^{2}\,dA(z)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\partial S}\frac{z\bar{z}^{2}}{2}\,dz=\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}\frac{f(w)f(1/w)^{2}}{2}\,f^{\prime}(w)\,dw,$

where $f$ is given by (1.17). Note that $0<q<1$ and

(4.37) $f(z)=\frac{R\,z(z-z_{0})}{z-q},\qquad
f(1/z)=\frac{R(1-z_{0}z)}{z(1-qz)},\qquad z_{0}=q+\frac{\kappa}{qR}.$

Then by the residue calculus,

(4.38)
$\displaystyle\begin{split}\int_{S}|z|^{2}\,dA(z)&=\underset{w=q}{\textup{Res}}\Big{[}\frac{f(w)f(1/w)^{2}}{2}\,f^{\prime}(w)\Big{]}+\underset{w=0}{\textup{Res}}\Big{[}\frac{f(w)f(1/w)^{2}}{2}\,f^{\prime}(w)\Big{]}\\\
&=\frac{1}{4}\Big{(}1+\frac{1}{a^{2}q^{4}}-\frac{2}{q^{2}}-a^{4}q^{4}+a^{2}(1+2q^{2})\Big{)}.\end{split}$

Here we have used

(4.39) $\frac{1}{q^{4}}=-2a^{4}q^{2}+(a^{2}+4c+2)a^{2}.$

Note that by [16, Lemma 3.7], we have

(4.40) $\int_{S}\log|z-w|\,dA(w)=\operatorname{Re}g(z),\qquad z\not\in S.$

This in particular gives

(4.41) $\int_{S}\log|z-a|\,dA(z)=\operatorname{Re}g(a),$

which completes the proof. ∎

We now prove Lemma 4.6, which completes the proof of Proposition 2.1.

###### Proof of Lemma 4.6.

By [16, Eq.(7.9)], it follows that

(4.42)
$\displaystyle\begin{split}\int_{S}\log|z-w|\,dA(w)=\operatorname{Re}\int_{z_{0}}^{z}\mathcal{S}(\zeta)\,d\zeta-c\log|z-a|+\frac{|z_{0}|^{2}}{2}-C(a),\end{split}$

where $\mathcal{S}$ is the Schwarz function associated with the droplet, see
[16, Definition 2.1]. Choose $z_{0}=f(1).$ Then since

(4.43) $\mathcal{S}(f(z))=f(1/z),$

we have

(4.44) $\displaystyle\int_{z_{0}}^{z}\mathcal{S}(\zeta)\,d\zeta$
$\displaystyle=\int_{1}^{u}f(1/w)f^{\prime}(w)\,dw,\qquad f(u)=z.$

Note that

$\displaystyle f(1/z)f^{\prime}(z)$
$\displaystyle=\frac{(1+a^{2}q^{2})(1-a^{2}q^{2}+2a^{2}q^{4})}{4a^{2}q^{4}}\frac{1}{z}+\frac{1-q^{2}-a^{2}q^{2}+a^{2}q^{4}}{2q}\frac{1}{(z-q)^{2}}+\frac{c}{z-1/q}-\frac{c}{z-q}$
$\displaystyle=\frac{c+1}{z}+\frac{1-q^{2}-a^{2}q^{2}+a^{2}q^{4}}{2q}\frac{1}{(z-q)^{2}}+\frac{c}{z-1/q}-\frac{c}{z-q}.$

On the other hand,

$\displaystyle\quad\log|f(1)-a|-\log|z-a|=\log|f(1)-f(1/q)|-\log|f(u)-f(1/q)|$
$\displaystyle=\int_{1}^{u}\frac{1}{z-q}-\frac{1}{z-1/q}-\frac{1+a^{2}q^{2}}{(1+a^{2}q^{2})z-2a^{2}q^{3}}\,dz.$

Combining the above, we have

$\displaystyle\quad\operatorname{Re}\int_{z_{0}}^{z}\mathcal{S}(\zeta)\,d\zeta-c\log|z-a|+c\log|f(1)-a|$
$\displaystyle=\int_{1}^{u}\frac{c+1}{z}+\frac{1-q^{2}-a^{2}q^{2}+a^{2}q^{4}}{2q}\frac{1}{(z-q)^{2}}-\frac{c}{z-2a^{2}q^{3}/(1+a^{2}q^{2})}\,dz$

Note that

$\displaystyle\int_{1}^{1/q}\frac{c+1}{z}\,dz=-(c+1)\log
q,\qquad\int_{1}^{1/q}\frac{1-q^{2}-a^{2}q^{2}+a^{2}q^{4}}{2q}\frac{1}{(z-q)^{2}}\,dz=\frac{1}{2q}-\frac{a^{2}q}{2},$

and

$\displaystyle\quad-\int_{1}^{1/q}\frac{c}{z-2a^{2}q^{3}/(1+a^{2}q^{2})}\,dz-c\log|f(1)-a|$
$\displaystyle=-c\log\Big{(}\frac{1+a^{2}q^{2}-2a^{2}q^{4}}{q(1+a^{2}q^{2})}\Big{)}+c\log\Big{(}\frac{1+a^{2}q^{2}-2a^{2}q^{3}}{1+a^{2}q^{2}}\Big{)}-c\log\Big{(}\frac{1+a^{2}q^{2}-2a^{2}q^{3}}{2aq^{2}}\Big{)}$
$\displaystyle=-c\log\Big{(}\frac{1+a^{2}q^{2}-2a^{2}q^{4}}{q(1+a^{2}q^{2})}\Big{)}+c\log\Big{(}\frac{2aq^{2}}{1+a^{2}q^{2}}\Big{)}=c\log\Big{(}\frac{2aq^{3}}{1+a^{2}q^{2}-2a^{2}q^{4}}\Big{)}.$

Therefore it follows that

$\displaystyle\quad\lim_{z\to
a}\Big{(}\operatorname{Re}\int_{z_{0}}^{z}S(\zeta)\,d\zeta-c\log|z-a|\Big{)}$
$\displaystyle=\frac{1}{2q}-\frac{a^{2}q}{2}-\frac{(1+a^{2}q^{2})(1-a^{2}q^{2}+2a^{2}q^{4})}{4a^{2}q^{4}}\log
q+c\log\Big{(}\frac{2aq^{3}}{1+a^{2}q^{2}-2a^{2}q^{4}}\Big{)}.$

Then we have shown that

$\displaystyle\operatorname{Re}g(a)$
$\displaystyle=\frac{a^{2}}{8}+\frac{1}{8a^{2}q^{4}}-\frac{1}{4q^{2}}+\frac{a^{2}q^{2}}{2}-C(a)-(c+1)\log
q+c\log\Big{(}\frac{2aq^{3}}{1+a^{2}q^{2}-2a^{2}q^{4}}\Big{)}.$

Now the lemma follows from (4.27). ∎

##  5\. Riemann-Hilbert analysis and Fine asymptotic behaviours

In this section, we show Theorem 3.11.

Recall that $\Gamma$ be the contour in the Riemann-Hilbert problem (3.25). We
deform the contour $\Gamma$ so that it matches with the contour
${\mathcal{B}}$ in Definition 4.2. We choose the contour $\Gamma_{+}$ and
$\Gamma_{-}$ following the steepest descent paths from $\beta$ such that
$\operatorname{Re}\phi(z)<0$ on those contours. Here $\phi$ is given in
Definition 4.3. More precisely, we choose the one inside
$\mbox{int}({\mathcal{B}})$ to be $\Gamma_{+}$, while the one inside
$\operatorname{ext}({\mathcal{B}})$ to be $\Gamma_{-}$. The domains
$\Omega_{\pm}$ are defined by the open sets enclosed by ${\mathcal{B}}$ and
$\Gamma_{\pm}$ respectively.

Let us define the matrix function $A(z)$ by

(5.1)
$A(z):=\begin{cases}e^{-\frac{N\ell}{2}\sigma_{3}}Y(z)e^{-N(g(z)-\ell/2)\sigma_{3}},\quad&z\in\mathbb{C}\setminus(\Omega_{+}\cup\Omega_{-}),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
e^{-\frac{N\ell}{2}\sigma_{3}}Y(z)\begin{pmatrix}1&0\\\
-1/\omega_{n,N}(z)&1\end{pmatrix}e^{-N(g(z)-\ell/2)\sigma_{3}},\quad&z\in\Omega_{+},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
e^{-\frac{N\ell}{2}\sigma_{3}}Y(z)\begin{pmatrix}1&0\\\
1/\omega_{n,N}(z)&1\end{pmatrix}e^{-N(g(z)-\ell/2)\sigma_{3}},\quad&z\in\Omega_{-},\end{cases}$

where $Y$ and $\omega_{n,N}$ are given in Definition 3.6, while $\ell$ and
$g(z)$ are given in Definition 4.3. Here,

(5.2) $\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$

is the third Pauli matrix. Then by (3.25), one can check that $A$ satisfies
the following Riemann-Hilbert problem:

(5.3) $\begin{cases}A_{+}(z)=A_{-}(z)\begin{pmatrix}1&0\\\
e^{N\phi(z)}&1\end{pmatrix},&\quad z\in\Gamma_{\pm},\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ A_{+}(z)=A_{-}(z)\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},&\quad
z\in{\mathcal{B}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
A_{+}(z)=A_{-}(z)\begin{pmatrix}1&e^{-N\phi(z)}\\\ 0&1\end{pmatrix},&\quad
z\in\Gamma\setminus{\mathcal{B}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
A(z)=I+{O}(z^{-1}),&\quad z\to\infty,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
A(z)\mbox{\quad is holomorphic},&\quad\mbox{otherwise}.\end{cases}$

Note that $\operatorname{Re}(\phi(z))$ is negative on $\Gamma_{\pm}$, while it
is positive on $\Gamma\setminus{\mathcal{B}}$. Therefore, when $z$ is away
from ${\mathcal{B}}$, the jump matrices are close to identity exponentially.

Next, we define the global parametrix, cf. [16, Eqs.(4.14), (5.1)].

######  Definition 5.1 (Global parametrix).

The global parametrix $\Phi$ is defined as follows.

  * •

For the post-critical case,

(5.4) $\Phi(z):=\begin{cases}I,&z\in\operatorname{ext}(\mathcal{B}),\\\
\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},&z\in\mbox{int}(\mathcal{B}).\end{cases}$

  * •

For the pre-critical case,

(5.5) $\Phi(z):=\sqrt{RF^{\prime}(z)}\begin{pmatrix}1&\dfrac{\sqrt{\kappa
R}}{RF(z)-Rq}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ -\dfrac{\sqrt{\kappa
R}}{RF(z)-Rq}&1\end{pmatrix}$

where $F$ is given by (2.25).

By construction, $\Phi(z)$ satisfies the following Riemann-Hilbert problem:

(5.6) $\begin{cases}\Phi(z)\mbox{\quad is holomorphic},&\quad
z\in\mathbb{C}\setminus{\mathcal{B}},\\\
\Phi_{+}(z)=\Phi_{-}(z)\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},&\quad
z\in{\mathcal{B}},\\\ \Phi(z)=I+{O}(z^{-1}),&\quad z\to\infty.\end{cases}$

For the remaining Riemann-Hilbert analysis, we should treat post- and pre-
critical cases separately. We shall first discuss the pre-critical case, as it
is the more difficult one.

###  5.1. Pre-critical case

We denote $D_{w}$ as a small disk centered at $w\in\mathbb{C}$ with a finite
radius. Let us first define the local coordinates that will be used
frequently.

######  Definition 5.2 (Local coordinates).

Recall that $\phi$ is given by (4.12). The local coordinate $\zeta$ is defined
by

(5.7) $\frac{4}{3}\zeta(z)^{3/2}:=N(\phi(z)-\phi(\overline{\beta})),\quad z\in
D_{\overline{\beta}},$

cf. [16, Eq.(4.15)]. Here $\zeta$ maps $\Gamma_{b\overline{\beta}}$ into
$\mathbb{R}_{+}$, maps ${\mathcal{B}}$ into $\mathbb{R}_{-}$, maps
$\Gamma_{+}$ into ray $\gamma_{\overline{\beta}}^{+}:=[0,e^{2\pi i/3\infty})$,
and maps $\Gamma_{-}$ into ray $\gamma_{\overline{\beta}}^{-}:=[0,e^{-2\pi
i/3\infty})$. Similarly, we define the local coordinate $\xi$ by

(5.8) $\frac{4}{3}\xi(z)^{3/2}:=N\phi(z),\quad z\in D_{\beta},$

such that $\xi$ maps $\Gamma_{b\beta}$ into $\mathbb{R}_{+}$, maps
${\mathcal{B}}$ into $\mathbb{R}_{-}$, maps $\Gamma_{+}$ into ray
$\gamma_{\beta}^{+}:=[0,e^{-2\pi i/3\infty})$, and maps $\Gamma_{-}$ into ray
$\gamma_{\beta}^{-}:=[0,e^{2\pi i/3\infty})$.

######  Lemma 5.3 (Asymptotic of the local coordinates).

We have

(5.9) $\displaystyle\frac{\zeta(z)}{N^{2/3}}$
$\displaystyle=\gamma_{11}(z-\overline{\beta})+\frac{\gamma_{12}}{2}(z-\overline{\beta})^{2}+O(z-\overline{\beta})^{3},\qquad
z\to\overline{\beta},$ (5.10) $\displaystyle\frac{\xi(z)}{N^{2/3}}$
$\displaystyle=\overline{\gamma_{11}}(z-\beta)+\frac{\overline{\gamma_{12}}}{2}(z-\beta)^{2}+O(z-\beta)^{3},\qquad
z\to\beta,$

where

(5.11) $\displaystyle\gamma_{11}$
$\displaystyle=\frac{1}{2^{2/3}}\,\Big{(}\frac{a(\overline{\beta}-R/q)\sqrt{\overline{\beta}-\beta}}{(\overline{\beta}-a)\overline{\beta}}\Big{)}^{2/3},$
(5.12) $\displaystyle\gamma_{12}$
$\displaystyle=\frac{4}{5}\Big{(}\frac{1}{\overline{\beta}-R/q}+\frac{1}{2(\overline{\beta}-\beta)}-\frac{1}{\overline{\beta}-a}-\frac{1}{\overline{\beta}}\Big{)}\gamma_{11}.$

###### Proof.

By definition, we have

$\displaystyle\zeta^{\prime}(z)$
$\displaystyle=\frac{2}{3}\Big{(}\frac{3N}{4}\Big{)}^{2/3}\,\Big{(}\int_{\overline{\beta}}^{z}y(s)\,ds\Big{)}^{-1/3}\,y(z),\qquad\frac{\zeta^{\prime\prime}(z)}{\zeta^{\prime}(z)}=-\frac{1}{3}\Big{(}\int_{\overline{\beta}}^{z}y(s)\,ds\Big{)}^{-1}\,y(z)+\frac{y^{\prime}(z)}{y(z)}.$

Let us write

$\displaystyle y(z)=h(z)\sqrt{z-\overline{\beta}},\qquad
h(z):=\frac{a(z-R/q)\sqrt{z-\beta}}{(z-a)z}.$

Then we have

$\displaystyle y(z)$
$\displaystyle=h(\overline{\beta})(z-\overline{\beta})^{1/2}\Big{(}1+\frac{h^{\prime}(\overline{\beta})}{h(\overline{\beta})}(z-\overline{\beta})+O((z-\overline{\beta})^{2})\Big{)},$
$\displaystyle y^{\prime}(z)$
$\displaystyle=\frac{1}{2}h(\overline{\beta})(z-\overline{\beta})^{-1/2}\Big{(}1+3\frac{h^{\prime}(\overline{\beta})}{h(\overline{\beta})}(z-\overline{\beta})+O((z-\overline{\beta})^{2})\Big{)}.$

Using these, direct computations give

$\displaystyle\Big{(}\int_{\overline{\beta}}^{z}y(s)\,ds\Big{)}^{-1}$
$\displaystyle=\frac{3}{2}\frac{1}{h(\overline{\beta})}(z-\overline{\beta})^{-3/2}\Big{(}1-\frac{3}{5}\frac{h^{\prime}(\overline{\beta})}{h(\overline{\beta})}(z-\overline{\beta})+O((z-\overline{\beta})^{2})\Big{)},$
$\displaystyle\frac{y^{\prime}(z)}{y(z)}$
$\displaystyle=\frac{1}{2}(z-\overline{\beta})^{-1}+\frac{h^{\prime}(\overline{\beta})}{h(\overline{\beta})}+O(z-\overline{\beta}),$

which leads to

$-\frac{1}{3}\Big{(}\int_{\overline{\beta}}^{z}y(s)\,ds\Big{)}^{-1}\,y(z)=-\frac{1}{2}(z-\overline{\beta})^{-1}-\frac{1}{5}\frac{h^{\prime}(\overline{\beta})}{h(\overline{\beta})}+O(z-\overline{\beta}).$

Then the lemma follows from

$\frac{h^{\prime}(z)}{h(z)}=\frac{1}{z-R/q}+\frac{1}{2(z-\beta)}-\frac{1}{z-a}-\frac{1}{z}.$

∎

We now define the local parametrices.

######  Definition 5.4 (Riemann-Hilbert problems for local parametrices).

We consider the Riemann-Hilbert problems.

  * •

Inside $D_{\overline{\beta}}$, we define the following Riemann-Hilbert
problem:

(5.13) $\begin{cases}[P_{\overline{\beta}}(\zeta)]_{+}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}[P_{\overline{\beta}}(\zeta)]_{-}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\quad&\zeta\in\mathbb{R}_{-},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
[P_{\overline{\beta}}(\zeta)]_{+}=[P_{\overline{\beta}}(\zeta)]_{-}\begin{pmatrix}1&0\\\
e^{\frac{4}{3}\zeta^{3/2}}&1\end{pmatrix},\quad&z\in\gamma_{\overline{\beta}}^{\pm},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
[P_{\overline{\beta}}(\zeta)]_{+}=[P_{\overline{\beta}}(\zeta)]_{-}\begin{pmatrix}1&e^{-\frac{4}{3}\zeta^{3/2}}\\\
0&1\end{pmatrix},\quad&\zeta\in\mathbb{R}_{+},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
P_{\overline{\beta}}(\zeta)=I+O(\zeta^{-3/2}),\quad&\zeta\to\infty,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
P_{\overline{\beta}}(\zeta)=O(\zeta^{-1/4}),\quad&\zeta\to 0.\end{cases}$

Then $\Phi(z)P_{\overline{\beta}}(z)$ satisfies the same jump conditions of
$A(z)$.

  * •

Inside $D_{\beta}$, we define the following Riemann-Hilbert problem:

(5.14) $\begin{cases}[\widehat{P}_{\beta}(\xi)]_{+}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}[\widehat{P}_{\beta}(\xi)]_{-}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\quad&\xi\in\mathbb{R}_{-},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\
[\widehat{P}_{\beta}(\xi)]_{+}=[\widehat{P}_{\beta}(\xi)]_{-}\begin{pmatrix}1&0\\\
-e^{\frac{4}{3}\xi^{3/2}}&1\end{pmatrix},\quad&\xi\in\gamma_{\beta}^{\pm},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
[\widehat{P}_{\beta}(\xi)]_{+}=[\widehat{P}_{\beta}(\xi)]_{-}\begin{pmatrix}1&-e^{-\frac{4}{3}\xi^{3/2}}\\\
0&1\end{pmatrix},\quad&\xi\in\mathbb{R}_{+},\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ \widehat{P}_{\beta}(\xi)=I+O(\xi^{-3/2}),\quad&\xi\to\infty,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\widehat{P}_{\beta}(\xi)=O(\xi^{-1/4}),\quad&\xi\to 0.\end{cases}$

Then $\Phi(z)\widehat{P}_{\beta}(z)$ satisfies the same jump conditions of
$A(z)$.

The jump contours and the corresponding orientations can be seen from Figure
6.

$0$$\gamma_{\overline{\beta}}^{+}$$\gamma_{\overline{\beta}}^{-}$

$0$$\gamma_{\beta}^{-}$$\gamma_{\beta}^{+}$

Figure 6. The jump contours of $P_{\overline{\beta}}(\zeta)$ in
$D_{\overline{\beta}}$ and those of $\widehat{P}_{\beta}(\xi)$ in $D_{\beta}$.

The solution to the above Riemann-Hilbert problems can be constructed using
the Airy parametrices, which we now recall.

######  Definition 5.5 (Airy parametrices).

Let

(5.15)
$y_{j}(\zeta)=\omega^{j}\operatorname{Ai}(\omega^{j}\zeta),\qquad\omega=e^{2\pi
i/3},\quad(j=0,1,2).$

Then we define444Here, there is a minor typo in [16, Eq.(4.23)], where
$5\pi/3$ should be replaced with $4\pi/3$.

(5.16)
${\mathcal{A}}(\zeta):=\sqrt{2\pi}e^{-\frac{i\pi}{4}}\begin{cases}\begin{pmatrix}y_{0}(\zeta)&-y_{2}(\zeta)\\\
y_{0}^{\prime}(\zeta)&-y_{2}^{\prime}(\zeta)\end{pmatrix}e^{\frac{2}{3}\zeta^{3/2}\sigma_{3}},\quad&\arg\zeta\in(0,2\pi/3),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}-y_{1}(\zeta)&-y_{2}(\zeta)\\\
-y_{1}^{\prime}(\zeta)&-y_{2}^{\prime}(\zeta)\end{pmatrix}e^{\frac{2}{3}\zeta^{3/2}\sigma_{3}},\quad&\arg\zeta\in(2\pi/3,\pi),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}-y_{2}(\zeta)&y_{1}(\zeta)\\\
-y_{2}^{\prime}(\zeta)&y_{1}^{\prime}(\zeta)\end{pmatrix}e^{\frac{2}{3}\zeta^{3/2}\sigma_{3}},\quad&\arg\zeta\in(\pi,4\pi/3),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}y_{0}(\zeta)&y_{1}(\zeta)\\\
y_{0}^{\prime}(\zeta)&y_{1}^{\prime}(\zeta)\end{pmatrix}e^{\frac{2}{3}\zeta^{3/2}\sigma_{3}},\quad&\arg\zeta\in(4\pi/3,2\pi)\end{cases}$

and

(5.17)
$\widehat{{\mathcal{A}}}(\xi):=\sqrt{2\pi}e^{-\frac{i\pi}{4}}\begin{cases}\begin{pmatrix}y_{0}(\xi)&y_{2}(\xi)\\\
y_{0}^{\prime}(\xi)&y_{2}^{\prime}(\xi)\end{pmatrix}e^{\frac{2}{3}\xi^{3/2}\sigma_{3}},\quad&\arg\xi\in(0,2\pi/3),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}-y_{1}(\xi)&y_{2}(\xi)\\\
-y_{1}^{\prime}(\xi)&y_{2}^{\prime}(\xi)\end{pmatrix}e^{\frac{2}{3}\xi^{3/2}\sigma_{3}},\quad&\arg\xi\in(2\pi/3,\pi),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}-y_{2}(\xi)&-y_{1}(\xi)\\\
-y_{2}^{\prime}(\xi)&-y_{1}^{\prime}(\xi)\end{pmatrix}e^{\frac{2}{3}\xi^{3/2}\sigma_{3}},\quad&\arg\xi\in(\pi,4\pi/3),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}y_{0}(\xi)&-y_{1}(\xi)\\\
y_{0}^{\prime}(\xi)&-y_{1}^{\prime}(\xi)\end{pmatrix}e^{\frac{2}{3}\xi^{3/2}\sigma_{3}},\quad&\arg\xi\in(4\pi/3,2\pi).\end{cases}$

Using the standard Airy Riemann-Hilbert problem, one can show that the
solution to the above Riemann-Hilbert problems in Definition 5.4 is given by

(5.18) $\displaystyle P_{\overline{\beta}}(\zeta)$
$\displaystyle=e^{\frac{i\pi}{4}\sigma_{3}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\zeta^{\frac{\sigma_{3}}{4}}{\mathcal{A}}(\zeta),$ (5.19)
$\displaystyle\widehat{P}_{\beta}(\xi)$
$\displaystyle=\begin{pmatrix}e^{\frac{i\pi}{4}}&0\\\
0&-e^{-\frac{i\pi}{4}}\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\xi^{\frac{\sigma_{3}}{4}}\widehat{{\mathcal{A}}}(\xi).$

See also [16, Eq.(4.22)].

######  Lemma 5.6 (Asymptotics of local parametrices).

The solutions $P_{\overline{\beta}}$ and $\widehat{P}_{\beta}$ in (5.18) and
(5.19) satisfy the following asymptotic behaviours.

  * •

As $\zeta\to\infty$, we have

(5.20)
$P_{\overline{\beta}}(\zeta)=I+\frac{\Pi_{11}}{\zeta^{3/2}}+\frac{\Pi_{12}}{\zeta^{3}}+O\Big{(}\frac{1}{\zeta^{9/2}}\Big{)},$

where

(5.21) $\Pi_{11}=\frac{1}{8}\begin{pmatrix}1/6&i\\\
i&-1/6\end{pmatrix},\qquad\Pi_{12}=\frac{35}{384}\begin{pmatrix}-1/12&i\\\
-i&-1/12\end{pmatrix}.$

  * •

As $\xi\to\infty$, we have

(5.22)
$\widehat{P}_{\beta}(\xi)=I+\frac{\Pi_{21}}{\xi^{3/2}}+\frac{\Pi_{22}}{\xi^{3}}+O\Big{(}\frac{1}{\xi^{9/2}}\Big{)},$

where

(5.23) $\Pi_{21}=\frac{1}{8}\begin{pmatrix}1/6&-i\\\
-i&-1/6\end{pmatrix},\qquad\Pi_{22}=\frac{35}{384}\begin{pmatrix}-1/12&-i\\\
i&-1/12\end{pmatrix}.$

###### Proof.

Recall the asymptotic expansion of the Airy function: for $|\arg z|<\pi$,

(5.24)
$\operatorname{Ai}(z)\sim\frac{e^{-\frac{2}{3}z^{3/2}}}{2\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{k}}{(\frac{2}{3}z^{3/2})^{k}},\qquad\operatorname{Ai}^{\prime}(z)\sim-\frac{z^{1/4}e^{-\frac{2}{3}z^{3/2}}}{2\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{k}}{(\frac{2}{3}z^{3/2})^{k}},$

where $u_{0}=v_{0}=1$ and for $k=1,2,\dots,$

(5.25) $u_{k}=\frac{(2k+1)(2k+3)\dots(6k-1)}{(216)^{k}\,k!},\qquad
v_{k}=\frac{6k+1}{1-6k}u_{k},$

see [91, Eqs.(9.7.5), (9.7.6)]. Using these, the lemma follows from direct
computations. ∎

Following the definition of the global parametrix in (5.5) we define

(5.26) $\displaystyle\widehat{\textup{{{H}}}}_{\beta}(z)$
$\displaystyle:=\Phi(z)\widehat{S}(z),\qquad\widehat{S}(z):=\begin{pmatrix}e^{\frac{i\pi}{4}}&0\\\
0&-e^{-\frac{i\pi}{4}}\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\xi(z)^{\frac{\sigma_{3}}{4}},$ (5.27)
$\displaystyle\textbf{{H}}_{\overline{\beta}}(z)$
$\displaystyle:=\Phi(z)S(z),\qquad
S(z):=e^{\frac{i\pi\sigma_{3}}{4}}\frac{1}{\sqrt{2}}\begin{pmatrix}1&-1\\\
1&1\end{pmatrix}\zeta(z)^{\frac{\sigma_{3}}{4}}.$

We show that they are holomorphic and find the $N$-dependence.

######  Lemma 5.7 (Asymptotics of the global parametrices).

The global parametrices in Definition 5.1 satisfy the following asymptotic
behaviours.

  * •

As $z\to\overline{\beta}$, we have

(5.28)
$\displaystyle\begin{split}\textup{{H}}_{\overline{\beta}}(z)&=\frac{(\overline{\beta}-\beta)^{1/4}}{2}\begin{pmatrix}\displaystyle(1+i)\gamma_{11}^{1/4}N^{1/6}&\displaystyle\frac{-i}{\sqrt{2}(R\kappa\gamma_{11})^{1/4}N^{1/6}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle(1-i)\gamma_{11}^{1/4}N^{1/6}&\displaystyle\frac{1}{\sqrt{2}(R\kappa\gamma_{11})^{1/4}N^{1/6}}\end{pmatrix}\\\
&\quad-\frac{\gamma_{11}-2i\sqrt{R\kappa}\gamma_{12}}{16(i\sqrt{R\kappa}\gamma_{11})^{3/4}}\begin{pmatrix}\displaystyle
iN^{1/6}&\displaystyle\frac{1}{2(i\sqrt{R\kappa}\gamma_{11})^{1/2}N^{1/6}}\\\
\displaystyle
N^{1/6}&\displaystyle\frac{i}{2(i\sqrt{R\kappa}\gamma_{11})^{1/2}N^{1/6}}\end{pmatrix}(z-\overline{\beta})+O\big{(}(z-\overline{\beta})^{2}\big{)}.\end{split}$

  * •

As $z\to\beta,$ we have

(5.29)
$\displaystyle\begin{split}\widehat{\textup{{H}}}_{\beta}(z)&=\frac{(\beta-\overline{\beta})^{1/4}}{2}\begin{pmatrix}\displaystyle(1+i)\overline{\gamma}_{11}^{1/4}N^{1/6}&\displaystyle\frac{-1}{\sqrt{2}(R\kappa\overline{\gamma}_{11})^{1/4}N^{1/6}}\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle-(1-i)\overline{\gamma}_{11}^{1/4}N^{1/6}&\displaystyle\frac{i}{\sqrt{2}(R\kappa\overline{\gamma}_{11})^{1/4}N^{1/6}}\end{pmatrix}\\\
&\quad-\frac{(i)^{1/4}(\overline{\gamma}_{12}+2i\sqrt{R\kappa}\overline{\gamma}_{11})}{16(R\kappa)^{3/8}\overline{\gamma}_{12}^{3/4}}\begin{pmatrix}\displaystyle
N^{1/6}&\displaystyle\frac{i}{2(R\kappa)^{1/4}\overline{\gamma}_{12}^{1/2}}\\\
\displaystyle
iN^{1/6}&\displaystyle\frac{1}{2(R\kappa)^{1/4}\overline{\gamma}_{12}^{1/2}}\end{pmatrix}(z-\beta)+O\big{(}(z-\beta)^{2}\big{)}.\end{split}$

###### Proof.

This follows from a long but straightforward computations using Lemma 5.3,
(2.26) and (2.25). ∎

We shall apply the partial Schlesinger transform to improve the local
parametrix. For this purpose, we shall implement the following steps:

  * •

Construct rational matrix functions $R_{1}(z)$ with the only pole at
$\overline{\beta}$ and $R_{2}(z)$ with the only pole at $\beta$;

  * •

Construct holomorphic matrix functions $H_{1}(z)$ and $H_{2}(z)$ such that the
modified global parametrix $R_{1}(z)R_{2}(z)\Phi(z)$ matches the local
parametrix $R_{2}(z)H_{1}(z)\Phi(z)P_{\overline{\beta}}(\zeta(z))$ along
$\partial D_{\overline{\beta}}$ and matches the local parametrix
$R_{1}(z)H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z))$ along $\partial
D_{\beta}$.

We now define the rational functions. At first glance, the definition of these
rational functions may appear quite complex, making it challenging to grasp
the underlying motivation. However, this definition is crafted with a specific
purpose: to ensure that in the proof of the theorem below, the error matrix
satisfies a small norm Riemann-Hilbert problem. Notably, this formulation
guarantees the elimination of all terms at the order of $O(N^{-1})$ within
this norm, resulting in an error term of the order $O(N^{-2})$. This becomes
clear in the proof of Theorem 3.11 below.

######  Definition 5.8 (Rational functions).

Let

(5.30) $\displaystyle H_{1}(z)$
$\displaystyle:=I-\frac{1}{48}\textbf{{H}}_{\overline{\beta}}(z)\begin{pmatrix}0&5/\zeta(z)^{2}\\\
-7/\zeta(z)&0\end{pmatrix}\textbf{{H}}_{\overline{\beta}}(z)^{-1}+\frac{h_{11}}{z-\overline{\beta}}+\frac{h_{12}}{(z-\overline{\beta})^{2}},$
(5.31) $\displaystyle H_{2}(z)$
$\displaystyle:=I-\frac{1}{48}\widehat{\textbf{{H}}}_{\beta}(z)\begin{pmatrix}0&5/\xi(z)^{2}\\\
-7/\xi(z)&0\end{pmatrix}\widehat{\textbf{{H}}}_{\beta}(z)^{-1}+\frac{h_{21}}{z-\beta}+\frac{h_{22}}{(z-\beta)^{2}},$

where

(5.32)
$\displaystyle\begin{split}h_{11}&=\frac{1+i}{128\sqrt{2}(R\kappa)^{1/4}\gamma_{11}^{5/2}N}\begin{pmatrix}\displaystyle
3\gamma_{11}-10i\sqrt{R\kappa}\gamma_{12}&\displaystyle\frac{19\gamma_{11}+30i\sqrt{R\kappa}\gamma_{12}}{3}\\\
\displaystyle\frac{19\gamma_{11}+30i\sqrt{R\kappa}\gamma_{12}}{3}&\displaystyle-3\gamma_{11}+10i\sqrt{R\kappa}\gamma_{12}\end{pmatrix},\\\
h_{12}&=\frac{5(R\kappa)^{1/4}}{48\sqrt{2}\gamma_{11}^{3/2}N}\begin{pmatrix}-1+i&1+i\\\
1+i&1-i\end{pmatrix},\qquad h_{21}=\overline{h_{11}},\qquad
h_{22}=\overline{h_{12}}.\end{split}$

These coefficient $h_{jk}$’s are defined in a way that the matrix functions
$H_{1}$ and $H_{2}$ are holomorphic at $\overline{\beta}$ and at $\beta$
respectively.

We further define

(5.33) $R_{2}(z):=I+\frac{h_{21}}{z-\beta}+\frac{h_{22}}{(z-\beta)^{2}}$

and

(5.34)
$R_{1}(z):=I+\frac{1}{z-\overline{\beta}}R_{2}(z)h_{11}R_{2}(z)^{-1}+\frac{1}{(z-\overline{\beta})^{2}}R_{2}(z)h_{12}R_{2}(z)^{-1}.$

Here $R_{1}(z)$ is a rational matrix function with the only pole at
$\overline{\beta}$, and $R_{2}(z)$ is a rational matrix function with the only
pole at $\beta$. Let us also write

(5.35) $R_{1}(z)R_{2}(z):=I+\begin{pmatrix}R_{11}(z)&R_{12}(z)\\\
R_{21}(z)&R_{22}(z)\end{pmatrix}.$

######  Definition 5.9 (Strong asymptotics for the pre-critical case).

We define the strong asymptotics of $A(z)$ by

(5.36) $A^{\infty}(z)=\begin{cases}R_{1}(z)R_{2}(z)\Phi(z),&z\notin
D_{\overline{\beta}}\cup D_{\beta},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
R_{2}(z)H_{1}(z)\Phi(z)P_{\overline{\beta}}(\zeta(z)),&z\in
D_{\overline{\beta}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
R_{1}(z)H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z)),&z\in
D_{\beta}.\end{cases}$

We note that the strong asymptotic in [16] is defined in a similar manner but
without the rational functions. Then, the error analysis in [16] can only be
applied to derive the leading-order asymptotic behaviour of the orthogonal
polynomials. In other words, it is crucial to construct the rational functions
in Definition 5.8 to define (5.36), which yields find asymptotic behaviours.

###### Proof of Theorem 3.11 for the pre-critical case.

We aim to show that

(5.37) $A(z)=\Big{(}I+O(N^{-2})\Big{)}A^{\infty}(z),\qquad
z\in\operatorname{ext}(\mathcal{B}).$

For this, let us define the error function

(5.38) ${\mathcal{E}}(z)=A^{\infty}(z)A^{-1}(z)$

and verify that $\mathcal{E}$ satisfies a small-norm Riemann-Hilbert problem
with error $O(N^{-2})$.

Note that when $z$ is away from $\partial D_{\overline{\beta}}$ and $\partial
D_{\beta}$, the jump of the error matrix is exponentially small in $N$, see
[16, Eq.(4.27)]. On the other hand, by (5.36) and (5.3), we have

(5.39)
${\mathcal{E}}_{+}(z){\mathcal{E}}_{-}(z)^{-1}=\begin{cases}R_{1}(z)\Big{(}H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z))\Phi(z)^{-1}R_{2}(z)^{-1}\Big{)}R_{1}(z)^{-1},&z\in\partial
D_{\beta},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\Big{(}R_{2}(z)H_{1}(z)\Phi(z)P_{\overline{\beta}}(\zeta(z))\Phi(z)^{-1}R_{2}(z)^{-1}\Big{)}R_{1}(z)^{-1},&z\in\partial
D_{\overline{\beta}}.\end{cases}$

We first discuss the case $z\to\partial D_{\beta}$. Note that by (5.26) and
(5.22), we have

(5.40) $\displaystyle H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z))\Phi(z)^{-1}$
$\displaystyle=H_{2}(z)\widehat{\textbf{H}}_{\beta}(z)\Big{(}S(z)^{-1}\widehat{P}_{\beta}(\xi(z))S(z)\Big{)}\widehat{\textbf{H}}_{\beta}(z)^{-1}$
$\displaystyle=H_{2}(z)\widehat{\textbf{H}}_{\beta}(z)\bigg{(}I+\frac{1}{48}\begin{pmatrix}0&5/\xi(z)^{2}\\\
-7/\xi(z)&0\end{pmatrix}+O\Big{(}\frac{1}{\xi(z)^{3}}\Big{)}\bigg{)}\widehat{\textbf{H}}_{\beta}(z)^{-1}$
$\displaystyle=H_{2}(z)\bigg{(}I+\frac{1}{48}\widehat{\textbf{H}}_{\beta}(z)\begin{pmatrix}0&5/\xi(z)^{2}\\\
-7/\xi(z)&0\end{pmatrix}\widehat{\textbf{H}}_{\beta}(z)^{-1}+O\Big{(}\frac{1}{\xi(z)^{3}}\Big{)}\bigg{)}.$

Then it follows from the definition (5.31) of $H_{2}$ that

(5.41) $\displaystyle
H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z))\Phi(z)^{-1}=I+\frac{h_{21}}{z-\beta}+\frac{h_{22}}{(z-\beta)^{2}}+O\Big{(}\frac{1}{\xi(z)^{3}},\frac{1}{N^{2}}\Big{)}.$

Furthermore, by the definition (5.33) of $R_{2}$, we have that as
$z\to\partial D_{\beta}$,

(5.42) $\displaystyle
R_{1}(z)\Big{(}H_{2}(z)\Phi(z)\widehat{P}_{\beta}(\xi(z))\Phi(z)^{-1}R_{2}(z)^{-1}\Big{)}R_{1}(z)^{-1}$
$\displaystyle=R_{1}(z)\Big{(}I+O(N^{-2})\Big{)}R_{1}(z)^{-1}=I+O(N^{-2}).$

Here, we have used the fact that $R_{1}(z)$ is analytic in $D_{\beta}$.

Next, we discuss the case $z\to\partial D_{\overline{\beta}}$. Note that by
(5.26),

(5.43) $\displaystyle\begin{split}&\quad
R_{2}(z)H_{1}(z)\Big{(}\Phi(z)P_{\overline{\beta}}(\zeta(z))\Phi(z)^{-1}\Big{)}R_{2}(z)^{-1}\\\
&=R_{2}(z)H_{1}(z)\textbf{H}_{\overline{\beta}}(z)\Big{(}\widehat{S}(z)^{-1}P_{\overline{\beta}}(\zeta(z))\widehat{S}(z)\Big{)}\textbf{H}_{\overline{\beta}}(z)^{-1}R_{2}(z)^{-1}\\\
&=R_{2}(z)H_{1}(z)\bigg{(}I+\frac{1}{48}\textbf{H}_{\overline{\beta}}(z)\begin{pmatrix}0&5/\zeta(z)^{2}\\\
-7/\zeta(z)&0\end{pmatrix}\textbf{H}_{\overline{\beta}}(z)^{-1}+O\Big{(}\frac{1}{\zeta(z)^{3}}\Big{)}\bigg{)}R_{2}(z)^{-1}.\end{split}$

Furthermore, by (5.30), we have

(5.44) $\displaystyle\quad
R_{2}(z)H_{1}(z)\Big{(}\Phi(z)P_{\overline{\beta}}(\zeta(z))\Phi(z)^{-1}\Big{)}R_{2}(z)^{-1}$
$\displaystyle=R_{2}(z)\bigg{(}I+\frac{h_{11}}{z-\overline{\beta}}+\frac{h_{12}}{(z-\overline{\beta})^{2}}+O\Big{(}\frac{1}{\zeta(z)^{3}}\Big{)}\bigg{)}R_{2}(z)^{-1}$
$\displaystyle=\bigg{(}I+\frac{1}{z-\overline{\beta}}R_{2}(z)h_{11}R_{2}(z)^{-1}+\frac{1}{(z-\overline{\beta})^{2}}R_{2}(z)h_{12}R_{2}(z)^{-1}+O\Big{(}\frac{1}{\zeta(z)^{3}}\Big{)}\bigg{)}.$

Then by the definition (5.34) of $R_{1}$, one can see that as $z\to
D_{\overline{\beta}}$,

(5.45)
$\displaystyle\begin{split}\Big{(}R_{2}(z)H_{1}(z)\Phi(z)P_{\overline{\beta}}(\zeta(z))\Phi(z)^{-1}R_{2}(z)^{-1}\Big{)}R_{1}(z)^{-1}=I+O(N^{-2}).\end{split}$

Combining (5.42) and (5.45), we conclude (5.37). As a consequence of (5.37),
for $z\in\mathbb{C}\setminus(\Omega_{+}\cup\Omega_{-})$, the explicit
transform (5.1) together with the definition (5.36) gives rise to

(5.46) $\displaystyle Y(z)$
$\displaystyle=e^{N\ell/2\sigma_{3}}A(z)e^{tN(g(z)-\ell/2)\sigma_{3}}=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-2})\Big{)}A^{\infty}(z)e^{N(g(z)-\ell/2)\sigma_{3}}$
$\displaystyle=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-2})\Big{)}R_{1}(z)R_{2}(z)\Phi(z)e^{N(g(z)-\ell/2)\sigma_{3}}.$

Then by (5.5) and (5.35), this in turn implies that

(5.47)
$\displaystyle\begin{split}Y(z)&=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-2})\Big{)}\begin{pmatrix}1+R_{11}(z)&R_{12}(z)\\\
R_{21}(z)&1+R_{22}(z)\end{pmatrix}\begin{pmatrix}\sqrt{RF^{\prime}(z)}&\dfrac{\sqrt{\kappa
F^{\prime}(z)}}{F(z)-q}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
-\dfrac{\sqrt{\kappa
F^{\prime}(z)}}{F(z)-q}&\sqrt{RF^{\prime}(z)}\end{pmatrix}e^{N(g(z)-\ell/2)\sigma_{3}}.\end{split}$

In particular, since $p_{n}(z)=[Y(z)]_{11}$, we obtain the desired asymptotic
behaviour (3.43). ∎

###  5.2. Post-critical case

In this subsection, we now discuss the post-critical case, which is much
simpler compared to the pre-critical case.

Recall that for the post-critical case, the global parametrix $\Phi$ is given
by (5.4). Similar to the pre-critical case, we also apply the partial
Schlesinger transform to improve the local parametrix. Namely, we derive a
rational matrix functions $R(z)$ with the only pole at $\beta$, and a
holomorphic matrix function $H(z)$ such that the modified global parametrices

(5.48) $\Phi(z)R(z),\qquad\Phi(z)\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}R(z)\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$

match with the local parametrix $\Phi(z)P(\zeta(z))$ along $\partial
D_{\beta}$. The construction of such $H(z)$ and $R(z)$ with a simple pole at
$\beta$ was described in [16, Section 5].

For the post-critical case, we define the local coordinate $\zeta$ inside
$D_{\beta}$,

(5.49)
$\zeta(z)^{2}=2N\begin{cases}\phi(z),&z\in\operatorname{ext}(\mathcal{B}),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
-\phi(z),&z\in\mbox{int}(\mathcal{B}).\end{cases}$

Here $\phi$ is defined in Definition 4.3. Here the sign is $+$ for
$z\in\operatorname{ext}(\mathcal{B})$ and $-$ for $z\in\rm{int}(\mathcal{B})$.
Note that $\zeta$ maps ${\mathcal{B}}$ into rays $\gamma_{+}\cup\gamma_{-}$,
where $\gamma_{+}:=[0,e^{\pi i/4\infty})$ and $\gamma_{-}:=[0,e^{-\pi
i/4\infty})$, maps $\Gamma_{+}$ into imaginary axis $i\mathbb{R}$.

It can be observed from the definition that as $z\to\beta$, we have the
expansion

(5.50)
$\frac{\zeta(z)}{\sqrt{N}}=\frac{1}{\gamma_{1}}(z-\beta)(1+O(z-\beta)),\qquad\gamma_{1}=\sqrt{\frac{\beta(\beta-a)}{a(b-\beta)}},$

where $\beta$ and $b$ are given in (4.6). Inside $D_{\beta}$, we define the
following Riemann-Hilbert problem

(5.51) $\begin{cases}[P(\zeta)]_{+}=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}[P(\zeta)]_{-}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\quad&\zeta\in\gamma_{+}\cup\gamma_{-},\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ [P(\zeta)]_{+}=[P(\zeta)]_{-}\begin{pmatrix}1&0\\\
e^{\zeta^{2}}&1\end{pmatrix},\quad&\zeta\in i\mathbb{R},\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
P(\zeta)=I+O(\zeta^{-1}),\quad&\zeta\to\infty.\end{cases}$

Then $\Phi(z)P(\zeta(z))$ satisfies the same jump conditions of $A(z)$.
Furthermore, the solution to this Riemann-Hilbert problem is given by

(5.52)
$P(\zeta(z))=\begin{cases}H(z)F(\zeta(z)),&z\in\operatorname{ext}(\mathcal{B}),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \begin{pmatrix}0&-1\\\
1&0\end{pmatrix}H(z)F(\zeta(z))\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},&z\in\mbox{int}(\mathcal{B}),\end{cases}$

where

(5.53) $F(\zeta(z)):=\begin{pmatrix}1&\displaystyle-\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}\frac{e^{s^{2}/2}}{s-\zeta(z)}ds\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ 0&1\end{pmatrix}$

and

(5.54) $H(z):=R(z)\bigg{(}I-\frac{1}{\sqrt{2\pi}\zeta(z)}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\bigg{)},\qquad R(z):=I+\frac{\gamma_{1}}{\sqrt{2\pi
N}}\frac{1}{z-\beta}\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$

$0$$i\mathbb{R}_{+}$$i\mathbb{R}_{-}$$\gamma_{+}$$\gamma_{-}$ Figure 7. The
jump contours of $P(\zeta)$ in $D_{\beta}$.

######  Definition 5.10 (Strong asymptotics for the post-critical case).

We define the strong asymptotics of $A(z)$ by

(5.55)
$A^{\infty}(z)=\begin{cases}\Phi(z)R(z),&z\in\operatorname{ext}(\mathcal{B})\setminus
D_{\beta},\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ \Phi(z)\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}R(z)\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},&z\in\mbox{int}(\mathcal{B})\setminus D_{\beta},\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ \Phi(z)P(\zeta(z)),&z\in D_{\beta}.\end{cases}$

We are now ready to complete the proof of Theorem 3.11.

###### Proof of Theorem 3.11 for the post-critical case.

As before, we define the error function

(5.56) ${\mathcal{E}}(z)=A^{\infty}(z)A^{-1}(z).$

Note that as $z\to D_{\beta}$,

(5.57) $F(\zeta(z))=I+\frac{1}{\sqrt{2\pi}\zeta(z)}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}+O\Big{(}\frac{1}{\zeta(z)^{3}}\Big{)}.$

Therefore, when $z$ is on the boundary of $\partial
D_{\beta}\cap\operatorname{ext}(\mathcal{B})$, we have

$\displaystyle\quad{\mathcal{E}}_{+}(z){\mathcal{E}}_{-}(z)^{-1}=\Phi(z)P(\zeta(z))R(z)^{-1}\Phi(z)^{-1}=H(z)F(\zeta(z))R(z)^{-1}$
$\displaystyle=R(z)\bigg{(}I-\frac{1}{\sqrt{2\pi}\zeta}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\bigg{)}\bigg{(}I+\frac{1}{\sqrt{2\pi}\zeta}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}+O\Big{(}\frac{1}{\zeta^{3}}\Big{)}\bigg{)}R(z)^{-1}=I+O(N^{-3/2}).$

Similarly, one can check that the same error bound holds for $z\in\partial
D_{\beta}\cap\mbox{int}(\mathcal{B})$. One can also check that the error
bounds are exponentially small for $z$ in other regions. Using the small norm
theorem, we obtain

$A(z)=\Big{(}I+O(N^{-3/2})\Big{)}A^{\infty}(z).$

Then for $z\in\operatorname{ext}(\mathcal{B})\setminus D_{\beta}$, it follows
that

$\displaystyle\begin{split}Y(z)&=e^{N\ell/2\sigma_{3}}A(z)e^{N(g(z)-\ell/2)\sigma_{3}}=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-3/2})\Big{)}A^{\infty}(z)e^{N(g(z)-\ell/2)\sigma_{3}}\\\
&=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-3/2})\Big{)}\Phi(z)R(z)e^{N(g(z)-\ell/2)\sigma_{3}}.\end{split}$

This gives

(5.58)
$Y(z)=e^{N\ell/2\sigma_{3}}\Big{(}I+O(N^{-3/2})\Big{)}\begin{pmatrix}1&\dfrac{1}{\sqrt{2\pi
N}}\dfrac{\gamma_{1}}{z-\beta}\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
0&1\end{pmatrix}\begin{pmatrix}\dfrac{z^{Nc+N}}{(z-a)^{Nc}}&0\vskip 3.0pt plus
1.0pt minus 1.0pt\\\
0&\dfrac{(z-a)^{Nc}}{z^{Nc+N}}\end{pmatrix}e^{-N\ell/2\sigma_{3}},$

which in particular yields

$p_{N}(z)=[Y(z)]_{11}=z^{N}\Big{(}\frac{z}{z-a}\Big{)}^{cN}\Big{(}1+O(N^{-3/2})\Big{)}.$

Note that the above partial Schlesinger transform only updates the
$(1,2)$-entry in $Y(z)$, but not the $(1,1)$-entry. Thus, by continuing to use
the partial Schlesinger transform, we can improve the error bound so that it
becomes $O(N^{-m})$ for any $m>0$. Therefore, we conclude (3.42). ∎

##  6\. Duality and critical case

In this section, we establish the free energy expansion for the critical case.
Let us first discuss the duality formula (1.9).

###### Proof of Proposition 1.1.

We consider the LUE of size $(cN)\times(cN)$ with joint probability
distribution proportional to

(6.1)
$\prod_{j>k=1}^{cN}|\widetilde{\lambda}_{j}-\widetilde{\lambda}_{k}|^{2}\prod_{j=1}^{cN}\widetilde{\lambda}_{j}^{N}e^{-N\widetilde{\lambda}_{j}},\qquad(\widetilde{\lambda}_{cN}>\dots>\widetilde{\lambda}_{1}>0).$

Then by [50, Proposition 3.1] with $k=cN$,

(6.2)
$\mathbb{P}\Big{[}\widetilde{\lambda}_{1}>x^{2}\Big{]}=\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-x)\Big{|}^{2cN}N^{cN^{2}}e^{-cN^{2}x^{2}}\frac{G(cN+1)G(N+1)}{G(cN+N+1)}.$

Furthermore by (3.5) and (1.8), it can be rewritten as

(6.3)
$\displaystyle\begin{split}\mathbb{P}\Big{[}\widetilde{\lambda}_{1}>x^{2}\Big{]}&=\mathbb{E}\Big{|}\det(\textbf{{G}}_{N}-x)\Big{|}^{2cN}\,e^{-cN^{2}x^{2}}\,\frac{Z_{N}^{\rm
Gin}}{Z_{N}(0,c)}=e^{-cN^{2}x^{2}}\,\frac{Z_{N}(x,c)}{Z_{N}(0,c)}.\end{split}$

We make a change of variable
$\widetilde{\lambda}_{j}=c\,\widehat{\lambda}_{j}$. Then
$\widehat{\lambda}_{j}$ follows the distribution proportional to

(6.4)
$\prod_{j>k=1}^{cN}|\widehat{\lambda}_{j}-\widehat{\lambda}_{k}|^{2}\prod_{j=1}^{cN}\widehat{\lambda}_{j}^{N}e^{-cN\widehat{\lambda}_{j}},\qquad(\widehat{\lambda}_{cN}>\dots>\widehat{\lambda}_{1}>0).$

If we relabelling $cN\mapsto N$, this follows the LUE (1.6) with $\alpha=1/c$.
Then the duality formula (1.9) follows from

(6.5)
$\mathbb{P}\Big{[}\lambda_{1}>\frac{x^{2}}{c}\Big{]}=\mathbb{P}\Big{[}\widehat{\lambda}_{1}>\frac{x^{2}}{c}\Big{]}\bigg{|}_{N\to
N/c},\qquad\mathbb{P}\Big{[}\widehat{\lambda}_{1}>\frac{x^{2}}{c}\Big{]}=\mathbb{P}\Big{[}\widetilde{\lambda}_{1}>x^{2}\Big{]}=e^{-cN^{2}x^{2}}\,\frac{Z_{N}(x,c)}{Z_{N}(0,c)}.$

This completes the proof. ∎

We now prove Proposition 2.5.

###### Proof of Proposition 2.5.

Note that by (1.20),

$\displaystyle\frac{a^{2}}{c}$
$\displaystyle=\lambda_{-}-\frac{(\sqrt{c+1}-\sqrt{c})^{4/3}}{c^{7/6}(c+1)^{1/6}}\,s\,N^{-2/3}+O(N^{-4/3}).$

Notice here that the Marchenko-Pastur law (1.7) satisfies the behaviour

(6.6)
$\frac{1}{2\pi}\frac{\sqrt{(\lambda_{+}-x)(x-\lambda_{-})}}{x}\sim\frac{\delta}{\pi}\sqrt{x-\lambda_{-}},\qquad
x\to\lambda_{-},$

where

(6.7)
$\delta=\frac{1}{2}\frac{\sqrt{\lambda_{+}-\lambda_{-}}}{\lambda_{-}}=\frac{(\alpha+1)^{1/4}}{(\sqrt{\alpha+1}-1)^{2}}=\frac{c^{3/4}(c+1)^{1/4}}{(\sqrt{c+1}-\sqrt{c})^{2}}.$

Then by the edge universality of the Hermitian unitary ensembles [48], it
follows that

(6.8) $\mathbb{P}\Big{[}(\lambda_{1}-\lambda_{-})(\delta N)^{2/3}>-y\Big{]}\to
F_{\rm TW}(y).$

(In our present case, this also follows from the classical Plancherel-Rotach
asymptotic formula for the Laguerre polynomials.) To be more precise, we have

$\displaystyle\mathbb{P}\Big{[}\lambda_{1}>\frac{a^{2}}{c}\Big{]}$
$\displaystyle=\mathbb{P}\Big{[}\lambda_{1}>\lambda_{-}-\frac{(\sqrt{c+1}-\sqrt{c})^{4/3}}{c^{7/6}(c+1)^{1/6}}\,s\,N^{-2/3}+O(N^{-4/3})\Big{]}$
$\displaystyle=\mathbb{P}\Big{[}(\lambda_{1}-\lambda_{-})(\delta
N)^{2/3}>-\delta^{2/3}\frac{(\sqrt{c+1}-\sqrt{c})^{4/3}}{c^{7/6}(c+1)^{1/6}}s\Big{]}+O(N^{-2/3})$
$\displaystyle=\mathbb{P}\Big{[}(\lambda_{1}-\lambda_{-})(\delta
N)^{2/3}>-\frac{s}{c^{2/3}}\Big{]}+O(N^{-2/3}).$

Therefore we obtain

(6.9) $\mathbb{P}\Big{[}\lambda_{1}>\frac{a^{2}}{c}\Big{]}=F_{\rm
TW}(c^{-2/3}s)+O(N^{-2/3}).$

Then the proposition follows from Proposition 1.1 and (3.6). ∎

While the proof of Proposition 2.5 follows easily from the duality relation as
well as the well-established Hermitian random matrix theory, our overall
strategy presented in Subsections 3.1, 3.2 and 3.3 also works for the critical
case. For this case, again by using the partial Schlesinger transform, one can
obtain the fine asymptotic behaviours of the orthogonal polynomial, which can
be written in terms of the solution to the Painlevé II equation

(6.10) $\mathfrak{q}^{\prime\prime}(s)=s\mathfrak{q}(s)+2\mathfrak{q}(s)^{3}$

with the asymptotic behaviour

(6.11) $\mathfrak{q}(s)=\begin{cases}{\rm
Ai}(s)+{O}\Big{(}\dfrac{e^{-\frac{4}{3}s^{3/2}}}{s^{1/4}}\Big{)},&s\to+\infty,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\sqrt{-\frac{1}{2}s}\Big{(}1+{O}\big{(}s^{-3}\big{)}\Big{)},&s\to-\infty.\end{cases}$

An additional difficulty arises from the fact that, regardless of the
deformation (3.3) or (3.4) we employ, in the critical case, we also need to
address either the post-critical or pre-critical regimes. Nonetheless, our
approach leads to the free energy expansion expressed in terms of the Painlevé
II solution, which should also coincide with the Tracy-Widom distribution.
This would lead to an alternative representation of the Tracy-Widom
distribution, see also [19] for a recently found another representation.

### Acknowledgements

Sung-Soo Byun was partially supported by the POSCO TJ Park Foundation (POSCO
Science Fellowship) and by the New Faculty Startup Fund at Seoul National
University. Seong-Mi Seo was partially supported by the National Research
Foundation of Korea (NRF-2022R1I1A1A01072052). Meng Yang was partially
supported by the Research Grant RCXMA23007 and the Start-up funding YJKY230037
at Great Bay University.

The authors are greatly indebted to Seung-Yeop Lee for several valuable
suggestions on the partial Schlesinger transform during the Riemann-Hilbert
analysis. We would like to express our gratitude to Nick Simm for drawing our
attention to the duality formula, and to Christian Webb for helpful
discussions regarding Remark 2.8. In addition, we extend our thanks to Gernot
Akemann, Yacin Ameur, Christophe Charlier, Peter J. Forrester, Kohei Noda, and
Aron Wennman for their interest and suggestions improving the presentation.

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