# On Airy solutions of PII and the complex cubic ensemble of random matrices,
II

Ahmad Barhoumi  Department of Mathematics, University of Michigan, East Hall,
530 Church Street, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>Department of
Mathematics, Royal Institute of Technology (KTH), Stockholm, Sweden ,  Pavel
Bleher  Department of Mathematical Sciences, Indiana University Indianapolis,
402 North Blackford Street, Indianapolis, IN 46202, USA<EMAIL_ADDRESS>,
Alfredo Deaño  Department of Mathematics, Universidad Carlos III de Madrid,
Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
<EMAIL_ADDRESS>and  Maxim Yattselev  Department of Mathematical
Sciences, Indiana University Indianapolis, 402 North Blackford Street,
Indianapolis, IN 46202, USA<EMAIL_ADDRESS>
###### Abstract.

We describe the pole-free regions of the one-parameter family of special
solutions of PII, the second Painlevé equation, constructed from the Airy
functions. This is achieved by exploiting the connection between these
solutions and the recurrence coefficients of orthogonal polynomials that
appear in the analysis of the ensemble of random matrices corresponding to the
cubic potential.

###### Key words and phrases:

Airy functions, Painlevé equations, random matrix models

###### 2020 Mathematics Subject Classification:

15B52, 33C10, 33C47

The first author was partially supported by the European Research Council
(ERC), Grant Agreement No. 101002013

The third author acknowledges financial support from Universidad Carlos III de
Madrid (I Convocatoria para la Recualificación del Profesorado Universitario),
from Grant PID2021-123969NB-I00, funded by MCIN/AEI/ 10.13039/501100011033,
and from grant PID2021-122154NB-I00 from Spanish Agencia Estatal de
Investigación.

The research of the last author was supported in part by a grant from the
Simons Foundation, CGM-706591.

##  1\. Introduction

This paper is a study of the so-called special function solutions of PII, the
second Painlevé equation, and is a continuation of our previous work [1].
While generic solutions of Painlevé equations are highly transcendental, it is
known that all but the first Painlevé equation admit special solutions which
can be expressed in terms of elementary or classical special functions. The
second Painlevé equation,

(1) $q^{\prime\prime}(z)=2q^{3}(z)+zq(z)+\alpha,$

possesses both rational solutions and solutions written in terms of the Airy
functions for specific values of the parameter $\alpha\in\mathbb{C}$. In this
work, we are interested in the latter, which are known to be meromorphic
functions of $z\in\mathbb{C}$, with only simple poles. Our goal is to identify
the pole-free regions for these solutions, by connecting these Airy solutions
to the recurrence coefficients for orthogonal polynomials that arise in the
complex cubic random matrix model, where $z$, properly rescaled, appears as a
parameter.

It is heuristically well understood in the field of non-Hermitian orthogonal
polynomials, but requires rigorous technical asymptotic analysis that we shall
carry out in subsequent publications, that the pole-free regions of the
recurrence coefficients as functions of the parameter correspond to the
regions of the parameter plane where the attracting set of zeros of the
orthogonal polynomials (as their degree tends to $\infty$) consists of a
single Jordan arc. This is usually known in the literature as the _one–cut
case_. The main goal of this work is to describe the phase diagram
geometrically, that is, the partition of the parameter $z$ space into
different regions where the geometry of the zero-attracting set is
qualitatively different, thus identifying the pole-free regions for the
recurrence coefficients and for the respective Airy solutions of PII.

This paper is divided into three sections. In Section 2, we give a succinct
but complete construction of the Airy solutions and various related functions.
Section 3 is devoted to a brief description of the cubic random matrix model
and the related orthogonal polynomials. There, we also state the connection
between orthogonal polynomials and the Airy solutions established earlier in
[1]. Finally, in Section 4, we prove the main result of this work describing
the phase diagram for the complex cubic random matrix model. This geometric
analysis is strongly connected with potential theory in the complex plane, and
it is an essential step (the construction of the so-called $g$ function) in
the method of nonlinear steepest descent for asymptotics of orthogonal
polynomials. This full asymptotic study will eventually allow us to prove
rigorously the structure of the pole free regions for Airy solutions of PII.

##  2\. Airy Solutions of PII

It was shown by Gambier [18] that PII has a one-parameter family of solutions
that are expressible in terms of the Airy functions and their derivatives
whenever $\alpha+1/2=n\in\mathbb{Z}$. These are meromorphic functions in the
complex plane whose poles exhibit a well-structured behavior, see Figure 1.

Figure 1. Zeros (blue) and poles (red) of $q_{3}(z;0)$ (left panel) and
$q_{3}(z;\infty)$ (right panel), see (5)–(6) for the meaning of the second
parameter.

The goal of this work is to provide a qualitative description of the pole-free
regions for these solutions. Our approach proceeds through the connection with
the orthogonal polynomials. Arguably one of the first successful
implementations of a recasting of poles of Painelvé solutions as zeros of a
related Hankel determinants was carried out in [4]. To understand this
connection, we need a determinantal representation of these solutions, see
[29] as well as [14, 17], whose derivation is explained in this section.

###  2.1. Airy Solutions

The family of the Airy solutions of PII is highly structured and can be
generated from a single seed solution. To obtain it, compatibility with a
Riccati equation is required, i.e., it is asked that $q(z)$ also satisfies

(2) $q^{\prime}(z)=k_{1}(z)q^{2}(z)+k_{2}(z)q(z)+k_{3}(z)$

for some functions $k_{1}(z),k_{2}(z),k_{3}(z)$ to be determined.
Differentiating both sides of (2) and using (1) reduces the Riccati equation
to an algebraic equation

$(2-2k_{1}^{2}(z))q^{3}(z)-(k_{1}^{\prime}(z)+3k_{1}(z)k_{2}(z))q^{2}(z)-\\\
(k_{2}^{\prime}(z)+k_{2}^{2}(z)+2k_{1}(z)k_{3}(z)-z)q(z)+\alpha-
k_{2}(z)k_{3}(z)-k_{3}^{\prime}(z)=0,$

which must hold identically. Setting the coefficients next to the powers of
$q(z)$ to zero gives that $k_{1}(z)=\epsilon$, $k_{2}(z)=0$, $k_{3}(z)=\alpha
z$ while $\alpha=\epsilon/2$, where $\epsilon^{2}=1$. This turns (2) into

(3) $q_{\epsilon}^{\prime}(z)=\epsilon\big{(}q_{\epsilon}^{2}(z)+z/2\big{)}.$

Choosing $\epsilon=1$ and making a substitution
$q_{1}(z)=-(\log\tau_{1}(z))^{\prime}$, linearizes (3) and yields

$\tau_{1}^{\prime\prime}(z)+\frac{1}{2}z\tau_{1}(z)=0,$

which is a rescaled Airy equation that is solved by linear combinations

(4) $\tau_{1}(z)=C_{1}\mathrm{Ai}(-2^{-1/3}z)+C_{2}\mathrm{Bi}(-2^{-1/3}z),$

where $C_{1},C_{2}\in\mathbb{C}$ are arbitrary constants. Thus, the
corresponding solution of (3) for $\epsilon=1$, i.e. $\alpha=1/2$, is given by

(5)
$q_{1}(z;\lambda)=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}\log\left(C_{1}\,\mathrm{Ai}\left(-2^{-1/3}z\right)+C_{2}\,\mathrm{Bi}\left(-2^{-1/3}z\right)\right),$

where $\lambda=C_{2}/C_{1}\in\overline{\mathbb{C}}$ (note that the presence of
the logarithmic derivative in the definition of $q_{1}(z;\lambda)$ makes it so
that this is truly a one-parameter family parametrized by $\lambda$). The
reader might protest that imposing the Riccati condition is unmotivated, but
this does have an interpretation, see Remark 2.1 further below.

The solutions $q_{n}(z;\lambda)$ of (1) corresponding to $\alpha=n-1/2$,
$n\geq 2$, are now constructed recursively via the Bäcklund transformation

(6)
$q_{n+1}(z;\lambda)=-q_{n}(z;\lambda)-\frac{2n}{2q_{n}^{2}(z;\lambda)+2q_{n}^{\prime}(z;\lambda)+z},$

see for instance [16, 32.7.25]. The solutions for non-positive values of $n$
are then obtained via the symmetry $q_{n}(z;\lambda)=-q_{-n+1}(z;\lambda)$.

###  2.2. Hamiltonian Structure

Each of the first six Painlevé equations can be written as a Hamiltonian
system

$\mathinner{\dfrac{\operatorname{d\\!}{}q}{\operatorname{d\\!}{z}}}=\frac{\partial
H}{\partial
p}\quad\text{and}\quad\mathinner{\dfrac{\operatorname{d\\!}{}p}{\operatorname{d\\!}{z}}}=-\frac{\partial
H}{\partial q},$

see [23, 27, 28]. In the case of PII the Hamiltonian $H_{\mathrm{II}}$ is
equal to

(7)
$H_{\mathrm{II}}(q,p,z;\alpha)=\frac{1}{2}p^{2}-\left(q^{2}+\frac{z}{2}\right)p-\left(\alpha+\frac{1}{2}\right)q.$

Hence, the Hamiltonian system becomes

(8) $q^{\prime}(z)=p(z)-q^{2}(z)-z/2\quad\text{and}\quad
p^{\prime}(z)=2p(z)q(z)+\alpha+1/2.$

Eliminating $p(z)$ from (8) gives (1), while eliminating $q(z)$ from (8)
yields PXXXIV:

(9)
$p^{\prime\prime}(z)=\frac{(p^{\prime}(z))^{2}-(\alpha+1/2)^{2}}{2p(z)}+2p^{2}(z)-zp(z).$

Moreover, if $q_{\alpha+1/2}(z)$ and $p_{\alpha+1/2}(z)$ solve (1) and (9),
respectively, that is, as a pair solve (8), and

(10) $\sigma_{\alpha+1/2}(z)\mathrel{\mathop{\mathchar
58\relax}}=H_{\mathrm{II}}\big{(}q_{\alpha+1/2}(z),p_{\alpha+1/2}(z),z;\alpha\big{)},$

then this function solves SII, the Jimbo-Miwa-Okamoto $\sigma$-form of PII:

(11)
$(\sigma^{\prime\prime})^{2}+4(\sigma^{\prime})^{3}+2\sigma^{\prime}(z\sigma^{\prime}-\sigma)=(\alpha/2+1/4)^{2}.$

Conversely, if $\sigma_{\alpha+1/2}(z)$ solves (11), then the functions

(12)
$q_{\alpha+1/2}(z)=\frac{2\sigma_{\alpha+1/2}^{\prime\prime}(z)+\alpha+1/2}{4\sigma_{\alpha+1/2}^{\prime}(z)}\quad\text{and}\quad
p_{\alpha+1/2}(z)=-2\sigma_{\alpha+1/2}^{\prime}(z),$

solve (1) and (9), respectively, see [23, 27, 28, 29].

######  Remark 2.1.

Imposing the Riccati equation (3) when $\epsilon=-1$, i.e., $\alpha=-1/2$, is
equivalent to seeking a “stationary” solution to the Hamiltonian system where
$p(z)\equiv 0$. To arrive at a similar interpretation for the choice
$\epsilon=1$ requires the following modification of the Hamiltonian, cf.111In
the cited Remark, there is a sign error in the definition of
$\hat{H}_{\mathrm{II}}$ which we correct here. [29, Remark 1.2]. Consider the
change of variables

$\hat{q}(z)\mathrel{\mathop{\mathchar
58\relax}}=q(z),\quad\hat{p}(z)\mathrel{\mathop{\mathchar
58\relax}}=p(z)-2q^{2}(z)-z,$

along with the Hamiltonian

$\hat{H}_{\mathrm{II}}(\hat{p},\hat{q},z;\alpha)\mathrel{\mathop{\mathchar
58\relax}}=H_{\mathrm{II}}(q,p,z;\alpha)+q=\frac{1}{2}\hat{p}^{2}+\left(\hat{q}^{2}+\frac{z}{2}\right)\hat{p}-\left(\alpha-\frac{1}{2}\right)\hat{q}.$

Then, the Hamiltonian system reads

$\hat{q}^{\prime}(z)=\hat{p}(z)+\hat{q}^{2}(z)+z/2\quad\text{and}\quad\hat{p}^{\prime}(z)=-2\hat{p}(z)\hat{q}(z)+\alpha-1/2.$

Eliminating $\hat{p}(z)$ again gives (1), and now we can see that imposing the
Riccati equation (3) with $\epsilon=1$ is equivalent to requiring
$\hat{p}(z)\equiv 0$.

###  2.3. Tau Functions

Let now $q_{n}(z;\lambda)$ be the solutions of (1) given by (5)–(6), while
$p_{n}(z;\lambda)$ and $\sigma_{n}(z;\lambda)$ be the corresponding solutions
of (9) and (11) obtained via (8) and (10), respectively. We may now _define_
$\tau_{n}(z;\lambda)$, up to a multiplicative constant, via the formula

(13)
$\sigma_{n}(z;\lambda)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}\log\tau_{n}(z;\lambda),$

where we take $\tau_{0}\equiv 1$ (as $p_{0}\equiv 1$ due to Remark 2.1,
$\sigma_{0}\equiv 0$ by (7) and (10)) and $\tau_{1}(z;\lambda)$ to be
$\tau_{1}(z)$ from (4) for some choice $C_{1},C_{2}$ such that
$C_{2}/C_{1}=\lambda$ (all such choices lead to the same function
$\sigma_{1}(z;\lambda)$). Then

(14)
$q_{n}(z;\lambda)\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}\log\frac{\tau_{n-1}(z;\lambda)}{\tau_{n}(z;\lambda)}.$

To arrive at expression (14), we need the inverse of the Bäcklund
transformation (6), see e.g. [14, Theorem 2],

(15)
$q_{n-1}(z)=-q_{n}(z)+\dfrac{2(n-1)}{2q_{n}^{\prime}(z)-2q_{n}^{2}(z)-z},$

where we drop the dependence on $\lambda$ for brevity. Combining (6), (15),
and the first equation in (8), one discovers the identity

(16) $p_{n-1}(z)=-p_{n}(z)+2q_{n}^{2}(z)+z.$

Using (16), (15) yields the remarkable relation (cf. [29, Eq. (1.13)3])

(17)
$H_{\mathrm{II}}(q_{n-1}(z),p_{n-1}(z),z;n-3/2)=H_{\mathrm{II}}(q_{n}(z),p_{n}(z),z;n-3/2),$

which, in view of (10) and (13) implies (14).

To derive the determinantal representation for the functions $\tau_{n}(z)$,
observe that they satisfy the Toda equation

(18)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{z^{2}}}}\log\tau_{n}(z)=K_{n}\dfrac{\tau_{n-1}(z)\tau_{n+1}(z)}{\tau_{n}^{2}(z)},\quad
K_{n}\in\mathbb{C}.$

Indeed, on the one hand, (14) implies that

(19)
$q_{n+1}(z)-q_{n}(z)=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}\log\dfrac{\tau_{n+1}(z)\tau_{n-1}(z)}{\tau_{n}^{2}(z)}.$

On the other hand, the Bäcklund transformation (6) together with the first
equation in (8) as well as (12) give that

(20)
$q_{n+1}(z)-q_{n}(z)=-\dfrac{n}{p_{n}(z)}+2q_{n}(z)=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}\log\sigma_{n}^{\prime}(z).$

Combining (19)–(20) with (13) clearly implies (18). Recall now that the tau
functions are defined by (13) only up to a multiplicative constant. Given the
seed functions $\tau_{0}$ and $\tau_{1}$ as described after (13), we normalize
the rest of them so that (18) is satisfied with $K_{n}=1$. This yields the
representation

(21)
$\tau_{n}(z;\lambda)=\mathrm{det}\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{j+k}}}{\operatorname{d\\!}{z^{j+k}}}}\left(C_{1}\,\mathrm{Ai}\left(-2^{-1/3}z\right)+C_{2}\,\mathrm{Bi}\left(-2^{-1/3}z\right)\right)\right]_{j,k=0}^{n-1},$

where $C_{1},C_{2}$ are the same as in (4), because the right-hand sides above
satisfy (18) with $K_{n}=1$ for all $n\geq 1$ according to Dodgson
condensation identity222Also known as Desnanot-Jacobi identity or Sylvester
determinant identity..

The functions $\tau_{n}(z;\lambda)$, which are entire, have an interpretation
as the isomonodromic tau function, see [24], that, in the context of Painlevé
equations, were studied in great detail in [23]. There, the authors made the
following observation.

######  Remark 2.2.

The functions $\tau_{n}(z;\lambda)$ have only simple zeros. Indeed, if $z_{0}$
is a zero of $\tau_{n}(z;\lambda)$ of order $p$, then, in view of (13),
$\sigma_{n}(z;\lambda)$ is a meromorphic function in $\mathbb{C}$ and has a
simple pole at $z=z_{0}$ with residue $p$. A local analysis of (11) as $z\to
z_{0}$ reveals that

$\sigma_{n}(z;\lambda)=\frac{1}{z-z_{0}}+s_{0}-\dfrac{z_{0}}{6}(z-z_{0})-\frac{1}{8}(z-z_{0})^{2}+\cdots,\quad
s_{0}\in\mathbb{C},$

which implies that $p=1$ as claimed.

######  Remark 2.3.

The functions $\tau_{n}(z;\lambda)$ and $\tau_{n+1}(z;\lambda)$ have no zeros
in common for any $n\geq 0$. Indeed, this follows from Remark 2.2 and the fact
that Toda equation (18) can be rewritten in the form

(22)
$\tau_{n}(z;\lambda)\tau_{n}^{\prime\prime}(z;\lambda)-(\tau_{n}^{\prime}(z;\lambda))^{2}=\tau_{n+1}(z;\lambda)\tau_{n-1}(z;\lambda).$

##  3\. Cubic Random Matrix Model

Cubic random matrix model studies statistics of $N\times N$ Hermitian matrices
$M$ drawn from the ensemble given by the probability distribution

$\tilde{Z}_{N}^{-1}(u)\mathrm{e}^{-N\mathrm{Tr}(M^{2}/2-uM^{3})}\mathrm{d}M,$

where $u$ is a parameter and $\tilde{Z}_{N}(u)$ is the normalizing constant
(partition function). This model has been investigated in physical literature
by Brézin, Itzykson, Parisi, and Zuber [12] and Bessis, Itzykson, and Zuber
[7]. An interesting feature of the model is that its free energy possesses an
asymptotic expansion in powers of $N^{-2}$ whose coefficients are functions of
$u$ which themselves admit a Taylor series expansion near $u=0$. It was
observed that the Taylor series of the coefficient of $N^{-2g}$ is the
generating function for the number of three-valent graphs on a Riemann surface
of genus $g$, and so this asymptotic expansion of the free energy came to be
known as the _topological expansion_. The existence of a topological expansion
was shown in [15] for matrix models with even polynomial potential. In the
case of the cubic potential this expansion was obtained in [8] by the middle
two authors of the present work, where they computed the number of three-
valent graphs explicitly for $g=0,1$ and asymptotically for $g>1$.

###  3.1. Connection to Airy Solutions

After a change of variables, see [8], the cubic ensemble turns into the
unitary ensemble of random matrices whose partition function is formally
defined by the matrix integral over the space of $N\times N$ Hermitian
matrices:

$\int_{\mathcal{H}_{N}}\mathrm{e}^{-N\mathrm{Tr}\left(-\frac{1}{3}M^{3}+tM\right)}\mathrm{d}M,$

where $t\in\mathbb{C}$ is a complex parameter. The formal partition function
of the eigenvalues of these matrices can be written as

$\int_{-\infty}^{\infty}\ldots\int_{-\infty}^{\infty}\prod_{1\leq j<k\leq
N}(s_{j}-s_{k})^{2}\,\prod_{j=1}^{N}\mathrm{e}^{-NV(s_{j};t)}\mathrm{d}s_{1}\ldots\mathrm{d}s_{N},$

with the polynomial potential $V(s;t)$ given by

(23) $V(s;t)\mathrel{\mathop{\mathchar 58\relax}}=-\frac{1}{3}s^{3}+st.$

Figure 2. Contours $L_{0}$, $L_{1}$ and $L_{2}$ used in the construction of
$\Gamma$.

This expression is formal because the integrals are divergent and need
regularization. To achieve it, let

(24) $\Gamma=\Gamma(\lambda)\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{0}L_{0}+\alpha_{1}L_{1}+\alpha_{2}L_{2},$

where $L_{k}\mathrel{\mathop{\mathchar
58\relax}}=\\{z\mathrel{\mathop{\mathchar
58\relax}}~{}z=re^{(-1+2k/3)\pi\mathrm{i}},~{}r\in(0,\infty)\\}$ are oriented
towards the origin, see Figure 2, and
$\alpha_{0},\alpha_{1},\alpha_{2}\in\mathbb{C}$ are a one-parameter family of
complex numbers given in terms of the parameter $\lambda$ (see (5)) by

(25)
$\alpha_{0}=\frac{\lambda}{\pi},\quad\alpha_{1}=-\frac{\lambda}{2\pi}+\frac{1}{2\pi\mathrm{i}},\quad\text{and}\quad\alpha_{2}=-\frac{\lambda}{2\pi}-\frac{1}{2\pi\mathrm{i}},$

when $|\lambda|<\infty$ and $\alpha_{0}=1/\pi$,
$\alpha_{1}=\alpha_{2}=-1/2\pi$ when $\lambda=\infty$. Let

$Z_{N}(t;\lambda)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\Gamma}\ldots\int_{\Gamma}\prod_{1\leq j<k\leq
N}(s_{j}-s_{k})^{2}\,\prod_{j=1}^{N}\mathrm{e}^{-NV(s_{j};t)}\mathrm{d}s_{1}\ldots\mathrm{d}s_{N},$

which is well defined (the integrals are convergent) for all values
$t\in\mathbb{C}$. As briefly discussed above, it was shown in [8] that the
topological expansion of the free energy

$F_{N}(t;\lambda)=\frac{1}{N^{2}}\log Z_{N}(t;\lambda)$

is connected to the enumeration of regular graphs of degree 3 on Riemann
surfaces. The partition function $Z_{N}(t;\lambda)$ can also be expressed as
$N!D_{N-1}(t;N,\lambda)$, where

$D_{n}(t;N,\lambda)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{det}\left[\int_{\Gamma}s^{i+j}\mathrm{e}^{-NV(s;t)}\mathrm{d}s\right]_{i,j=0}^{n}.$

The entries of the matrix defining $D_{n}(-t;N,\lambda)$ can be written as

$\displaystyle\int_{\Gamma}s^{k}\mathrm{e}^{-NV(s;-t)}\mathrm{d}s$
$\displaystyle=\frac{1}{N^{k+1/3}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{k}}}{\operatorname{d\\!}{t^{k}}}}\left(\int_{\Gamma}\mathrm{e}^{-V\left(s;-N^{2/3}t\right)}\mathrm{d}s\right)$
$\displaystyle=\frac{1}{N^{k+1/3}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{k}}}{\operatorname{d\\!}{t^{k}}}}\left(\alpha_{0}\pi\,\mathrm{Bi}\left(-N^{2/3}t\right)+(\alpha_{1}-\alpha_{2})\pi\mathrm{i}\,\mathrm{Ai}\left(-N^{2/3}t\right)\right),$

where we used the classical integral representations, see [16, 9.5.4, 9.5.5],

$\mathrm{Ai}(z)=\frac{1}{2\pi\mathrm{i}}\int_{L_{1}-L_{2}}\mathrm{e}^{-V(s;z)}\mathrm{d}s\quad\text{and}\quad\mathrm{Bi}(z)=\frac{1}{2\pi}\left(\int_{L_{0}-L_{1}}+\int_{L_{0}-L_{2}}\right)\mathrm{e}^{-V(s;z)}\mathrm{d}s,$

as well as the condition $\alpha_{0}+\alpha_{1}+\alpha_{2}=0$. Setting
$z=(\sqrt{2}N)^{2/3}t$, we get

(26) $\int_{\Gamma}s^{k}\mathrm{e}^{-NV(s;-t)}\mathrm{d}s\\\
=\frac{2^{k/3}}{N^{(k+1)/3}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{k}}}{\operatorname{d\\!}{z^{k}}}}\left(\alpha_{0}\pi\,\mathrm{Bi}\left(-2^{-1/3}z\right)+(\alpha_{1}-\alpha_{2})\pi\mathrm{i}\,\mathrm{Ai}\left(-2^{-1/3}z\right)\right).$

Equation (26), representation (21) of $\tau_{n}(z;\lambda)$, and the Hankel
structure of the determinant $D_{n}(-t;N,\lambda)$ now yield that

(27)
$D_{n}(-t;N,\lambda)=\frac{2^{n(n+1)/3}}{N^{(n+1)(n+2)/3}}\tau_{n+1}(z;\lambda),\quad
z=(\sqrt{2}N)^{2/3}t.$

An immediate consequence of (27) is that these Hankel determinants satisfy a
version of the Toda equation (22), namely

$D_{n}^{\prime\prime}(t;N,\lambda)D_{n}(t;N,\lambda)-(D_{n}^{\prime}(t;N,\lambda))^{2}=N^{2}D_{n+1}(t;N,\lambda)D_{n-1}(t;N,\lambda).$

Furthermore, if we take $n=N$, then (13) can be rewritten in terms of the free
energy $F_{N}(t;\lambda)$ as

$\sigma_{N}(z;\lambda)={N^{2}}\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{z}}}F_{N}\left(-(\sqrt{2}N)^{-2/3}z;\lambda\right),$

which, of course, means that $p_{N}(z;\lambda)$ and $q_{N}(z;\lambda)$ can be
expressed via $F_{N}(t;\lambda)$ using (12). The above representations can be
further rewritten using the orthogonal polynomials associated to the cubic
ensemble, which we do next.

###  3.2. Non-Hermitian Orthogonal Polynomials

The idea of using orthogonal polynomials in computations relating to random
matrices is classical at this point, see e.g. [26] and references therein. Let
$P_{n}(s;t,N,\lambda)$ be a non-identically zero polynomial of degree at most
$n$ such that

(28)
$\int_{\Gamma}s^{k}P_{n}(s;t,N,\lambda)\mathrm{e}^{-NV(s;t)}\mathrm{d}s=0,\quad
k\in\\{0,\ldots,n-1\\},$

where $\Gamma=\Gamma(\lambda)$ is as in (24)–(25) and we often shall drop the
explicit dependence on $\lambda$ when this dependence is not essential to us.
This family of polynomials for specific values of $\lambda$ was studied by
Huybrechs, Kuijlaars, and Lejon in [20, 21] in the context of complex valued
Gaussian quadrature rules, and by the authors of the present work in [2, 9,
10] in the context of the cubic model.

Due to the non-Hermitian character of the relations in (28), it might happen
that polynomial satisfying (28) is non-unique. In this case we denote by
$P_{n}(s;t,N,\lambda)$ the monic non-identically zero polynomial of the
smallest degree; such a polynomial is always unique. It is a linear algebra
exercise to verify that $\deg P_{n}(s;t,N,\lambda)=n$ if and only if
$D_{n-1}(t;N)\neq 0$. Hence, equality (27) and Remark 2.3 immediately yield
the following observation.

######  Remark 3.1.

For fixed $t\in\mathbb{C}$, $N>0$, $\lambda\in\overline{\mathbb{C}}$, and all
$n\in\mathbb{N}$ it holds that

$\deg P_{n}(\cdot;t,N,\lambda)=n\quad\text{or}\quad\deg
P_{n+1}(\cdot;t,N,\lambda)=n+1.$

Since $P_{n+1}(s;t,N,\lambda)=P_{n}(s;t,N,\lambda)$ when $\deg
P_{n+1}(\cdot;t,N,\lambda)<n+1$, this means that $\deg
P_{n+1}(\cdot;t,N,\lambda)\in\\{n,n+1\\}$ for each $n$.

The determinantal representation of orthogonal polynomials (see, e.g., [38])
yields that

$h_{n}(t;N)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\Gamma}P_{n}^{2}(s;t,N)\mathrm{e}^{-NV(s;t)}\mathrm{d}s=\frac{D_{n}(t;N)}{D_{n-1}(t;N)},$

where $D_{-1}(t;N)\equiv 1$. Since $D_{n}(t;N)$ is an entire function of $t$,
each $h_{n}(t;N)$ is meromorphic in $\mathbb{C}$. Hence, given $n$, the set of
the values $t$ for which there exists $k\in\\{0,\ldots,n\\}$ such that
$h_{k}(t;N)=0$ is countable with no limit points in the finite plane. Outside
of this set a standard argument using (28) shows that

(29)
$sP_{n}(s;t,N)=P_{n+1}(s;t,N)+\beta_{n}(t;N)P_{n}(s;t,N)+\gamma_{n}^{2}(t;N)P_{n-1}(s;t,N),$

and by analytic continuation (29) extends to those values of $t$ for which
$n+1$-st and $n$-th polynomials have the prescribed degrees (that is,
$D_{n-1}(t;N)D_{n}(t;N)\neq 0$), where

(30)
$\gamma_{n}^{2}(t;N)=\frac{h_{n}(t;N)}{h_{n-1}(t;N)}=\frac{D_{n}(t;N)D_{n-2}(t;N)}{D_{n-1}(t;N)^{2}}.$

Denote by $p_{n,n-1}(t;N)$ the coefficient of $P_{n}(s;t,N)$ multiplying
$s^{n-1}$ and set $p_{0,-1}(t;N)\mathrel{\mathop{\mathchar 58\relax}}=0$. It
follows from (29) that

$\beta_{n}(t;N)=p_{n,n-1}(t;N)-p_{n+1,n}(t;N).$

Using the determinantal form for orthogonal polynomials again and the form of
the weight function in our case, we can deduce that

(31) $p_{n,n-1}(t;N)=\frac{1}{N}\frac{\mathrm{d}}{\mathrm{d}t}\log
D_{n-1}(t;N),$

and consequently

(32) $\beta_{n}(t;N)=\frac{1}{N}\left(\frac{\mathrm{d}}{\mathrm{d}t}\log
D_{n-1}(t;N)-\frac{\mathrm{d}}{\mathrm{d}t}\log D_{n}(t;N)\right).$

The following theorem has appeared in [1].

######  Theorem 3.2.

Fix $N\geq 1$. Given $\lambda\in\overline{\mathbb{C}}$, it holds for each
$n\in\mathbb{N}$ that

$\begin{cases}q_{n}(z;\lambda)&\displaystyle=-(N/2)^{1/3}\,\beta_{n-1}\left(-(\sqrt{2}N)^{-2/3}z;N,\lambda\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
p_{n}(z;\lambda)&\displaystyle=-2(N/2)^{2/3}\,\gamma_{n}^{2}\left(-(\sqrt{2}N)^{-2/3}z;N,\lambda\right),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\sigma_{n}(z;\lambda)&\displaystyle=-(N/2)^{1/3}\,p_{n,n-1}\left(-(\sqrt{2}N)^{-2/3}z;N,\lambda\right),\end{cases}$

where $q_{n}(z;\lambda)$ is a solution of (1) given by (5) and (6),
$p_{n}(z;\lambda)$ is the corresponding solution of (9), see (8), while
$\sigma_{n}(z;\lambda)$ is the corresponding solution of (11) determined by
(10).

As we have stated before, our goal is to identify the pole-free regions and
study scaling limits of the functions $q_{n},p_{n},\sigma_{n}$. Our approach
goes through the connection to orthogonal polynomials stated in Theorem 3.2.
In particular, combining this theorem with (30), (31), (32), and Remarks 2.2,
2.3 shows that the poles of the Painlevé function $q_{n}$ in the $z$-plane
correspond to zeros of $D_{n-1}$ and $D_{n}$ in the $t$-plane, i.e., to the
values of $t$ for which $\deg P_{n}=n-1$ or $\deg P_{n+1}=n$, while the poles
of $p_{n}$ and $\sigma_{n}$ correspond to zeros of $D_{n-1}$.

The study of the asymptotic behavior of the polynomials $P_{n}$ as
$n\to\infty$ and for generic values of the parameter $\lambda$ (i.e. for
generic combination of contours, see (24)) is a very technical topic that we
leave to subsequent publications. However, the expected outcome of such a
study is well understood: if the attracting set of zeros of $P_{n}$ consists
of a single Jordan arc, all the zeros of $P_{n}$ remain bounded and so are the
quantities $\beta_{n-1},\gamma_{n},p_{n,n-1}$; otherwise polynomials $P_{n}$
will have a zero exhibiting a _spurious_ behavior leading to the polar
singularities of $\beta_{n-1},\gamma_{n},p_{n,n-1}$.

In fact the above scheme has already been successfully carried out by us in
[1] where we interpreted the asymptotic results of [2, 10] as scaling limits
of $q_{n}(z;\lambda)$ when $\lambda\in\\{0,-\mathrm{i},\mathrm{i}\\}$. These
cases are essentially the same as they correspond to taking the seed function
in (21) to be

$\mathrm{Ai}\left(-2^{-1/3}z\right),\quad\mathrm{Ai}\left(-2^{-1/3}z\mathrm{e}^{2\pi\mathrm{i}/3}\right),\quad\text{and}\quad\mathrm{Ai}\left(-2^{-1/3}z\mathrm{e}^{-2\pi\mathrm{i}/3}\right)$

since $\mathrm{Ai}(z)\pm\mathrm{i}\mathrm{Bi}(z)=2\mathrm{e}^{\pm
2\pi\mathrm{i}/3}\mathrm{Ai}(z\mathrm{e}^{\mp 2\pi\mathrm{i}/3})$, see [16,
Equation (9.2.1)]. They are also qualitatively different from the remaining
ones as can be easily seen from Figure 1. Notice that these cases correspond
to $\Gamma$ in (24) for which $\alpha_{0}\alpha_{1}\alpha_{2}=0$ by (25).
Hence, in the remaining part of the paper we shall always assume that
$\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$.

##  4\. Phase Diagram

In this section we take the first step in understanding the pole-free regions
of Airy solutions of PII corresponding to
$\lambda\notin\\{0,-\mathrm{i},\mathrm{i}\\}$, or, equivalently, the behavior
of the orthogonal polynomials corresponding to $\Gamma$ for which
$\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$. This step consists in describing the
attracting set for the zeros of the orthogonal polynomials, or equivalently
the phase diagram in the parameter space. For the rest of this section, we
assume that $\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$.

Observe that due to the analyticity of the integrand in (28), the chain of
integration $\Gamma$ can be locally varied without changing the orthogonal
polynomials. Hence, there arises a question of the identification of the
contour attracting the zeros of the orthogonal polynomials. The answer to this
question follows from Stahl-Gonchar-Rakhmanov theory of symmetric contours
[34, 35, 36, 19] developed in a compact setting whose implications to
unbounded contours with polynomials external fields were worked out by
Kuijlaars and Silva [25].

###  4.1. Symmetric Contours

Denote by $\mathcal{T}$ the collection of admissible contours defined as
follows: each $T\in\mathcal{T}$ is a finite connected union of $C^{1}$-smooth
Jordan arcs for which there exist $R_{T}>0$ and $\epsilon_{T}\in(0,\pi/6)$
such that $T\setminus\\{|z|\leq R_{T}\\}$ consists of three unbounded Jordan
arcs, say $T_{0},T_{1},T_{2}$, each connecting a point on $\\{|z|=R_{T}\\}$ to
the point at infinity and satisfying $T_{k}\subset S_{k,\epsilon_{T}}$, where

$S_{k,\epsilon}\mathrel{\mathop{\mathchar
58\relax}}=\left\\{z\mathrel{\mathop{\mathchar 58\relax}}\
\left|\mathrm{arg}(z)+\pi-{2k\pi}/{3}\right|<\epsilon\right\\},\quad
k\in\\{0,1,2\\}.$

Notice that $L_{0}\cup L_{1}\cup L_{2}$, see (24), belongs to $\mathcal{T}$.
Let $\mathcal{M}(T)$ be the space of Borel probability measures on
$T\in\mathcal{T}$. The equilibrium energy of $T$ in the external field
$\mathrm{Re}\,V(\cdot;t)$, see (23), is equal to

$\mathcal{E}_{V}(T)=\inf_{\nu\in\mathcal{M}(T)}\left(\iint\log\frac{1}{|s-t|}\mathrm{d}\nu(s)\mathrm{d}\nu(t)+\int\mathrm{Re}\,V(s;t)\mathrm{d}\nu(s)\right).$

Figure 3. Schematic example of an admissible contour $T\in\mathcal{T}$ .

The infimum above is achieved by a unique minimizer, which is called the
weighted equilibrium measure of $T$ in the external field
$\mathrm{Re}\,V(\cdot;t)$, see [33, Theorem I.1.3]. We shall denote this
minimizer $\mu_{T}$ (of course, it is $t$-dependent). The support of
$\mu_{T}$, say $J_{T}$, is a compact subset of $T$. The equilibrium measure
$\mu=\mu_{T}$ is characterized by the Euler–Lagrange variational conditions:

(33) $2U^{\mu}(z)+\mathrm{Re}\ V(z;t)\;\begin{cases}=\ell_{T},&z\in J_{T},\\\
\geq\ell_{T},&z\in T\setminus J_{T},\end{cases}$

for some constant $\ell_{T}$, the Lagrange multiplier, where

$U^{\mu}(z)\mathrel{\mathop{\mathchar
58\relax}}=\int\log\frac{1}{|z-s|}\mathrm{d}\mu(s)$

is the logarithmic potential of $\mu$, see [33, Theorem I.3.3]. These notions
are well defined for any $T\in\mathcal{T}$. However, it is by now well
understood that one should use the contour whose equilibrium measure has
support _symmetric_ (with the _S-property_) in the external field
$\mathrm{Re}\,V(\cdot;t)$. That is, it must hold that the support $J_{T}$
consists of a finite number of open analytic arcs and their endpoints, and on
each arc it must hold that

(34) $\frac{\partial}{\partial
n_{+}}\,\big{(}2U^{\mu_{T}}+\mathrm{Re}\,V\big{)}=\frac{\partial}{\partial
n_{-}}\,\big{(}2U^{\mu_{T}}+\mathrm{Re}\,V\big{)},$

where $\partial/\partial n_{+}$ and $\partial/\partial n_{-}$ are the normal
derivatives from the $(+)$\- and $(-)$-side of $T$. It is said that a curve
$T\in\mathcal{T}$ is an _S-curve_ in the field $\mathrm{Re}\,V(\cdot;t)$, if
$J_{T}$ has the S-property in this field. It is also understood that
geometrically $J_{T}$ is comprised of _critical trajectories_ of a certain
quadratic differential; this was already known to Stahl (see, e.g. [35,
Theorem 1]) and remains a useful recasting of the S-property in the context of
Painlevé transcendents, see e.g. the recent work [13]. Recall that if $Q(z)$
is a meromorphic function, a _trajectory_ (resp. _orthogonal trajectory_) of a
quadratic differential $-Q(z)\mathrm{d}z^{2}$ is a maximal regular arc on
which

$-Q(z(s))\big{(}z^{\prime}(s)\big{)}^{2}>0\quad\big{(}\text{resp.}\quad-Q(z(s))\big{(}z^{\prime}(s)\big{)}^{2}<0\big{)}$

for any local uniformizing parameter. A trajectory is called _critical_ if it
is incident with a _finite critical point_ (a zero or a simple pole of
$-Q(z)\mathrm{d}z^{2}$) and it is called _short_ if it is non-recurrent and
incident only with finite critical points. We designate the expression
_critical (orthogonal) graph of $-Q(z)\mathrm{d}z^{2}$_ for the totality of
the critical (orthogonal) trajectories $-Q(z)\mathrm{d}z^{2}$. The following
theorem is a specialization to $V(z;t)$ of the results of Kuijlaars and Silva
[25, Theorems 2.3 and 2.4].

######  Theorem 4.1.

Let $V(z;t)$ be as in (23). For each $t\in\mathbb{C}$ there exists a contour
$\Gamma_{t}\in\mathcal{T}$ such that
$\mathcal{E}_{V}(\Gamma_{t})=\sup_{T\in\mathcal{T}}\mathcal{E}_{V}(T)$.
Moreover,

  * •

the support $J_{t}$ of the equilibrium measure
$\mu_{t}\mathrel{\mathop{\mathchar 58\relax}}=\mu_{\Gamma_{t}}$ has the
S-property in the external field $\mathrm{Re}\,V(\cdot;t)$;

  * •

the function

(35)
$Q(z;t)=\left(\frac{V^{\prime}(z;t)}{2}-\int\frac{\mathrm{d}\mu_{t}(s)}{z-s}\right)^{2},\quad
z\in\mathbb{C}\setminus J_{t},$

is a polynomial of degree 4;

  * •

the support $J_{t}$ consists of some (possibly not all) short critical
trajectories of the quadratic differential

$\varpi_{t}(z)\mathrel{\mathop{\mathchar 58\relax}}=-Q(z;t)\mathrm{d}z^{2};$

  * •

the equilibrium measure $\mu_{t}$ is absolutely continuous with respect to
Lebesgue measure and

(36)
$\mathrm{d}\mu_{t}(z)=-\frac{1}{\pi\mathrm{i}}\,Q_{+}^{1/2}(z;t)\mathrm{d}z,\quad
z\in J_{t},$

where $Q^{1/2}(z;t)=\frac{1}{2}z^{2}+\mathcal{O}(z)$ as $z\to\infty$ is the
branch holomorphic in $\mathbb{C}\setminus J_{t}$.

######  Remark 4.2.

Recall that every $T\in\mathcal{T}$ is a finite connected union of
$C^{1}$-smooth Jordan arcs. Observe also that
$\mathcal{E}_{V}(T^{\prime})\geq\mathcal{E}_{V}(T)$ if $T^{\prime}\subseteq T$
by the very definition of minimal energy. Hence, every extremal $\Gamma_{t}$
must consist of three semi-unbounded non-intersecting open Jordan arcs
stretching out to infinity within the sectors
$S_{0,\epsilon_{\Gamma_{t}}},S_{1,\epsilon_{\Gamma_{t}}},S_{2,\epsilon_{\Gamma_{t}}}$
that have a common endpoint (loops, intervals of overlap, and chains of Jordan
arcs with a finite terminal point can be removed without decreasing maximality
of minimal energy). Thus, $\Gamma_{t}=\Gamma_{t}^{02}\cup\Gamma_{t}^{12}$,
where $\Gamma_{t}^{ij}$ is a Jordan arc stretching to infinity within the
sectors $S_{i,\epsilon_{\Gamma_{t}}}$ and $S_{j,\epsilon_{\Gamma_{t}}}$.
Therefore,

$\Gamma=\alpha_{0}L_{0}+\alpha_{1}L_{1}+\alpha_{2}L_{2}=\alpha_{0}(L_{0}-L_{2})+\alpha_{1}(L_{1}-L_{2})$

is homologous to $\alpha_{0}\Gamma_{t}^{02}+\alpha_{1}\Gamma_{t}^{12}$, recall
(24)–(25). Hence, we can use the latter chain in (28) to define orthogonal
polynomials $P_{n}(s;t,N,\lambda)$.

######  Remark 4.3.

S-curve $\Gamma_{t}$ cannot be fixed uniquely since one has freedom in
choosing $\Gamma_{t}$ away from $J_{t}$. Indeed, let

(37) $\mathcal{U}(z;t)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Re}\left(2\int_{e}^{z}Q^{1/2}(s;t)\mathrm{d}s\right)=\ell_{\Gamma_{t}}-\mathrm{Re}(V(z;t))-2U^{\mu_{t}}(z),$

where $e\in J_{t}$ is arbitrary and the second equality follows from (35)
(since the constant $\ell_{\Gamma_{t}}$ in (33) is the same for all connected
components of $J_{t}$ and the integrand is purely imaginary on $J_{t}$, the
choice of $e$ is indeed not important). It follows from the variational
condition (33) that $J_{t}$ is a part of the set
$\\{z\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}(z;t)=0\\}$ while
$\Gamma_{t}\setminus J_{t}$ must belong to $\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)\leq 0\\}$ and can be varied freely within this
region. As $\mathcal{U}(z;t)$ is harmonic in $\mathbb{C}\setminus J_{t}$,
$\Gamma_{t}\setminus J_{t}$ belongs to the closure of
$\\{z\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}(z;t)<0\\}$, which means
that such variations are indeed possible (on the other hand, the arcs in
$J_{t}$ border $\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)>0\\}$ on both sides).

######  Remark 4.4.

The set $J_{t}$ and the measure $\mu_{t}$ are unique for each parameter $t$.
Indeed, let $\Gamma_{t}^{*}$ and $\Gamma_{t}^{**}$ be two S-curves
corresponding to the same $t$ with equilibrium measures $\mu_{t}^{*}$ and
$\mu_{t}^{**}$, respectively. Let $\epsilon>0$ be such that both
$\Gamma_{t}^{*}$ and $\Gamma_{t}^{**}$ extend to infinity within
$S_{k,\epsilon}$, $k\in\\{0,1,2\\}$. Choose $R>0$ large enough so that
$J_{t}^{*},J_{t}^{**}\subset\\{|z|<R\\}$ while

$S_{k,\epsilon}\cap\\{|z|\geq R\\}\subset
U=\big{\\{}z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}^{*}(z;t)<0\big{\\}}\cap\big{\\{}z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}^{**}(z;t)<0\big{\\}}.$

According to Remark 4.3, we can modify $\Gamma_{t}^{*}$ and $\Gamma_{t}^{**}$
within $U$ so that they intersect $\\{|z|=R\\}$ at the same three points. Set
$\Gamma_{t,R}^{*}\mathrel{\mathop{\mathchar
58\relax}}=\Gamma_{t}^{*}\cap\\{|z|\leq R\\}$ and define $\Gamma_{t,R}^{**}$
similarly. Since (33) characterizes weighted equilibrium measures,
$\mu_{t}^{*}$ and $\mu_{t}^{**}$ remain being such measures for
$\Gamma_{t,R}^{*}$ and $\Gamma_{t,R}^{**}$, respectively. As (34) also remains
valid for these measures, $\Gamma_{t,R}^{*}$ and $\Gamma_{t,R}^{**}$ are
compact S-curves. Due to Remark 4.2 and our local modification of
$\Gamma_{t}^{*}$ and $\Gamma_{t}^{**}$, $\Gamma_{t,R}^{*}$ and
$\Gamma_{t,R}^{**}$ are homologous to each other and therefore non-Hermitian
orthogonal polynomials on them must be the same. Then it follows from [19,
Lemma 1] that the normalized zero counting measures of these polynomials must
simultaneously converge weak∗ to $\mu_{t}^{*}$ and $\mu_{t}^{**}$, which
establishes the desired claim.

######  Remark 4.5.

It is enough to describe $\mu_{t}$ for $t\in O$ for any $O$ such that
$\mathbb{C}=O\cup\eta O\cup\bar{\eta}O$, where
$\eta=\mathrm{e}^{2\pi\mathrm{i}/3}$. Indeed, it holds that $\eta
T\in\mathcal{T}$ when $T\in\mathcal{T}$. Let $\mu$ be a measure on $T$ and
$\mu^{\eta}$ be the push forward measure of $\mu$ on $\eta T$, that is,
$\mu^{\eta}(A)=\mu(\bar{\eta}A)$ for any Borel subset $A$ of $\eta T$. Since

$U^{\mu^{\eta}}(z)=\int\log\frac{1}{|z-\eta x|}d\mu(x)=U^{\mu}(\bar{\eta}z),$

and $V(z;\bar{\eta}t)=V(\bar{\eta}z;t)$, it follows from (33) that the
weighted equilibrium measure of $\eta T$ in the external field
$\mathrm{Re}\,V(\cdot;\bar{\eta}t)$ is equal to $\mu_{T}^{\eta}$, the push
forward of the weighted equilibrium measure of $T$ in the external field
$\mathrm{Re}\,V(\cdot;t)$. This means that

$\sup_{T\in\mathcal{T}}\mathcal{E}_{V(\cdot;t)}(T)=\sup_{T\in\mathcal{T}}\mathcal{E}_{V(\cdot;\bar{\eta}t)}(T)$

and that if $\Gamma_{t}$ is a symmetric contour from Theorem 4.1 then $\eta
J_{t}=J_{\bar{\eta}t}$. In particular, $Q(z;\bar{\eta}t)=\eta
Q(\bar{\eta}z;t)$.

###  4.2. Phase Diagram

Both global and local structures of the critical graph of quadratic
differentials have been well-studied; see e.g. [22, 30, 37]. Since
$\deg\,Q(z;t)=4$, $J_{t}$ consists of one arc, two arcs, or three arcs with a
common endpoint; the first case occurs when $Q(z;t)$ has two simple and one
double zero, and the other two occur when $Q(z;t)$ has four simple zeros.
Below, we examine when these distinct possibilities happen.

By developing the right hand side of (35) into a Laurent series and recalling
that $Q(z;t)$ is a polynomial, we may write

(38) $Q(z;t)=\dfrac{1}{4}(z^{2}-t)^{2}+z+K(t),$

for some $K(t)\in\mathbb{C}$. To find parameters $t$ for which $\mu_{t}$ is
supported on a single arc (often referred to as a “cut”), we start with the
ansatz

(39) $Q(z;t)=\frac{1}{4}(z-a(t))(z-b(t))(z-c(t))^{2}.$

One could identify values of $K(t)$ for which (38) has multiple roots by
computing the discriminant, but the following approach appears more
beneficial. Comparing coefficients of (38), (39) yields the system (we
suppress the $t$-dependence for brevity)

(40) $\begin{cases}a+b+2c&=0,\\\ ab+c^{2}+2(a+b)c&=-2t,\\\
2abc+(a+b)c^{2}&=-4.\end{cases}$

Letting $x\mathrel{\mathop{\mathchar 58\relax}}=-c=(a+b)/2$ and eliminating
$ab$ from the second and third equation yields

(41) $x^{3}-tx-1=0.$

Hence, the matter of solving the system (40) is now translated into that of
understanding the solutions of (41). Precisely, we seek to identifying values
of $t\in\mathbb{C}$ for which the critical graph of $\varpi_{t}(z)$, as
defined by (39) and (40), admits a contour $\Gamma_{t}\in\mathcal{T}$ such
that $\Gamma_{t}\subset\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)\leq 0\\}$ and $\mathcal{U}(z;t)=0$ holds on a
subset of $\Gamma_{t}$ of finite arclength. It was argued in [10, Section 5]
that this condition can be imposed by requiring absence of a short trajectory
connecting $b$ to $c$, or

$\mathrm{Re}\left(\int_{b}^{c}Q^{1/2}(s;t)\mathrm{d}s\right)\neq 0.$

Thus, transitions in the critical graph correspond to choices of $x$ for which

$\mathrm{Re}\left(\dfrac{2}{3}\int_{-1}^{2x^{3}}\left(1+\frac{1}{s}\right)^{3/2}\mathrm{d}s\right)=0,$

i.e., one needed to study the critical graph of the auxiliary quadratic
differential

$-\left(1+\frac{1}{s}\right)^{3}\mathrm{d}s^{2},$

denoted by $\mathcal{C}$.

$0$$-1$

(a)

$\footnotesize-\sqrt[3]{1/2}$$\Omega_{(0)}$

(b)

$3\cdot 2^{-2/3}\mathrm{e}^{-2\pi\mathrm{i}/3}$$3\cdot
2^{-2/3}\mathrm{e}^{2\pi\mathrm{i}/3}$$O_{(0)}$

(c)

Figure 4. Schematic representation of (A) the critical graph $\mathcal{C}$;
(B) the set $\Delta$ (solid lines) and the domain $\Omega_{(0)}$ (shaded
region); (C) domain $O_{(0)}$ (shaded region).

It was shown in [10] that $\mathcal{C}$ consists of 5 critical trajectories
emanating from $-1$ at the angles $2\pi k/5$, $k\in\\{0,1,2,3,4\\}$, one of
them being $(-1,0)$, two forming a loop crossing the real line approximately
at $0.635$, and two approaching infinity along the imaginary axis without
changing the half-plane (upper or lower), see Figure 4(A). Define

$\Delta\mathrel{\mathop{\mathchar
58\relax}}=\big{\\{}x\mathrel{\mathop{\mathchar
58\relax}}~{}2x^{3}\in\mathcal{C}\big{\\}}$

and let $\Omega_{(0)}$ to be the subset of the right-half plane bounded by
three smooth subarcs of $\Delta$ as on Figure 4(B). Set
$\Omega_{(\pm\mathrm{i})}\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{e}^{\mp
2\pi\mathrm{i}/3}\Omega_{(0)}$. The function $t(x)$ defined by (41) is
holomorphic in each $\Omega_{(\tau)}$,
$\tau\in\\{-\mathrm{i},\mathrm{i},0\\}$, with non-vanishing derivative there.
Put $O_{(\tau)}\mathrel{\mathop{\mathchar 58\relax}}=t(\Omega_{(\tau)})$, see
Figure 4(C), then the inverse map $x_{\tau}\mathrel{\mathop{\mathchar
58\relax}}O_{(\tau)}\to\Omega_{(\tau)}$ exists and is holomorphic,
$\tau\in\\{-\mathrm{i},\mathrm{i},0\\}$, and
$O_{(\pm\mathrm{i})}=\mathrm{e}^{\pm 2\pi\mathrm{i}/3}O_{(0)}$.

$O_{0,-\mathrm{i}}$$O_{0,0}$$O_{0,\mathrm{i}}$$O_{1,-}$$O_{1,-\mathrm{i}}$$O_{1,0}$$O_{1,\mathrm{i}}$

Figure 5. Single cut domains $O_{0,0}$, $O_{0,\mathrm{i}}$, and
$O_{0,-\mathrm{i}}$; double cut domains $O_{1,0}$, $O_{1,\mathrm{i}}$,
$O_{1,-\mathrm{i}}$; and three cuts with a common endpoint domain $O_{1,-}$.
Boundaries of these domains correspond to single cut cases.

Denote by $O_{0}$ the union of the unbounded components of $O_{(0)}\cap
O_{(\mathrm{i})}\cap O_{(-\mathrm{i})}$ and by $O_{1,-}$ the bounded component
of this intersection. Further, let $O_{1,\tau}$ be the complement of the
closure of $O_{(\tau)}$ and $O_{0,\tau}$ be the connected component of $O_{0}$
that lies directly clockwise from $O_{1,\tau}$,
$\tau\in\\{0,\mathrm{i},-\mathrm{i}\\}$, see Figure 5 (numbers $0$ and $1$
stand for the genus of the Riemann surface of $Q^{1/2}(z;t)$). We further set
$O_{1}\mathrel{\mathop{\mathchar 58\relax}}=O_{1,0}\cup O_{1,\mathrm{i}}\cup
O_{1,-\mathrm{i}}\cup O_{1,-}$. Then the following theorem takes place.

######  Theorem 4.6.

For each $t\in\mathbb{C}$, let $\Gamma_{t}\in\mathcal{T}$ be as in Theorem
4.1. Recall that $J_{t}$, the support of the equilibrium measure $\mu_{t}$, is
uniquely determined. It holds that

  * •

for $t\in O_{0}\cup\\{t_{\mathsf{cr}},\eta
t_{\mathsf{cr}},\bar{\eta}t_{\mathsf{cr}}\\}$ the support $J_{t}$ consists of
a single analytic Jordan arc, where $\eta=\mathrm{e}^{2\pi\mathrm{i}/3}$ and
$t_{\mathsf{cr}}=3\cdot 2^{-2/3}$;

  * •

for $t\in O_{1,-}$ the support $J_{t}$ consists of three analytic Jordan arcs
with a common endpoint;

  * •

for $t\in O_{1}\setminus O_{1,-}$ the support $J_{t}$ consists of two analytic
Jordan arcs with no endpoints in common;

  * •

in the remaining cases the support $J_{t}$ consists of two analytic Jordan
arcs with a common endpoint that form a corner at it.

The claims of Theorem 4.6 follows from Theorems 4.10–4.12 further below, in
which we discuss the detailed structure of these contours $\Gamma_{t}$.

Figure 6. Poles (red) and zeros (blue) of $q_{3}(-(3\sqrt{2})^{2/3}z;\infty)$
and the curves $\partial O_{1}$ (black).

######  Remark 4.7.

According to our heuristics, Airy solutions $q_{n}(z;\lambda)$ must be pole-
free on compact subsets of $-(\sqrt{2}n)^{2/3}O_{0}$ for all $n$ large, see
Theorem 3.2 where we take $n=N$ and Figure 6. We emphasize that while the
locations of the poles of $q_{n}(-(\sqrt{2}n)^{-2/3}z;\lambda)$ depend on
$\lambda\notin\\{\pm\mathrm{i},0\\}$, the region which they fill in the large
$n$ limit, $O_{1}$, does not!

###  4.3. One-cut Cases

Below, we describe the structure of the critical trajectories of the quadratic
differential $\varpi_{t}(z)$ as well as our preferred choice of $\Gamma_{t}$,
where we adopt the following convention.

(a)

(b)

(c)

Figure 7. Schematic representation of the critical (solid) and critical
orthogonal (dashed) graphs of $\varpi_{t}(z)$ when (A) $t\in
O_{0,-\mathrm{i}}$, $\mathrm{Im}(t)>0$; (B) $t\in O_{0,-\mathrm{i}}$,
$\mathrm{Im}(t)=0$; (C) $t\in O_{0,-\mathrm{i}}$, $\mathrm{Im}(t)<0$. The bold
curves represent the preferred S-curve $\Gamma_{t}$. Shaded region is the set
$\\{\mathcal{U}(z;t)<0\\}$.

######  Notation.

$\Gamma(z_{1},z_{2})$ (resp. $\Gamma[z_{1},z_{2}]$) stands for the trajectory
or orthogonal trajectory (resp. the closure of) of the differential
$\varpi_{t}(z)$ connecting $z_{1}$ and $z_{2}$, oriented from $z_{1}$ to
$z_{2}$, and $\Gamma\big{(}z,\mathrm{e}^{\mathrm{i}\theta}\infty\big{)}$
(resp. $\Gamma\big{(}\mathrm{e}^{\mathrm{i}\theta}\infty,z\big{)}$) stands for
the orthogonal trajectory incident with $z$, approaching infinity at the angle
$\theta$, and oriented away from $z$ (resp. oriented towards $z$).333This
notation is unambiguous as the corresponding trajectories are unique for
polynomial differentials as follows from Teichmüller’s lemma.

(a)

(b)

(c)

Figure 8. Schematic representation of the critical (solid) and critical
orthogonal (dashed) graphs of $\varpi_{t}(z)$ when (a) $t\in\partial
O_{0,-\mathrm{i}}$, $\mathrm{Im}(t)>0$; (b) $t=3\cdot 2^{-2/3}$; (c)
$t\in\partial O_{0,-\mathrm{i}}$, $\mathrm{Im}(t)<0$. The bold curves
represent the preferred S-curve $\Gamma_{t}$. Shaded region is the set
$\\{\mathcal{U}(z;t)<0\\}$.

######  Remark 4.8.

Theorem 4.10 below gives an “appropriate” choice for a chain of integration in
(28). Each provided chain can be easily modified within the region
$\\{z\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}(z;t)<0\\}$ so that its
point set is connected and corresponds to an S-curve guaranteed by Theorem
4.1. That is, our choice modifies the S-curve only within
$\\{z\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}(z;t)<0\\}$, which is
possible, see Remark 4.3, and allows us to choose the chain of integration
that consists solely of trajectories and orthogonal trajectories of
$\varpi_{t}(z)$.

Figure 9. Schematic representation of the critical (solid) and critical
orthogonal (dashed) graphs of $\varpi_{t}(z)$ when $t\in(\partial
O_{1,-\mathrm{i}}\cap\partial O_{1,-})\setminus\\{t_{\mathsf{cr}},\eta
t_{\mathsf{cr}}\\}$. The bold curves represent the preferred S-curve
$\Gamma_{t}$. Shaded region is the set $\\{\mathcal{U}(z;t)<0\\}$.

######  Remark 4.9.

Recall further that to describe S-curves for $t\notin O_{1}$, it is enough to
describe them for $t\in\overline{O}_{0,-\mathrm{i}}$ and $t\in(\partial
O_{1,-\mathrm{i}}\cap\partial O_{1,-})\setminus\\{t_{\mathsf{cr}},\eta
t_{\mathsf{cr}}\\}$, see Figure 5, due to Remark 4.5.

######  Theorem 4.10.

Let $\mu_{t}$ and $Q(z;t)$ be as in Theorem 4.1,
$J_{t}=\mathrm{supp}(\mu_{t})$. Assume that
$\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$. When
$t\in\overline{O}_{0,-\mathrm{i}}$, the polynomial $Q(z;t)$ is of the form
(39) with $a(t)$, $b(t)$, and $c(t)$ given by

(42) $\begin{cases}a(t)&\mathrel{\mathop{\mathchar
58\relax}}=x_{-\mathrm{i}}(t)-\mathrm{i}\sqrt{2}/\sqrt{x_{-\mathrm{i}}}(t),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ b(t)&\mathrel{\mathop{\mathchar
58\relax}}=x_{-\mathrm{i}}(t)+\mathrm{i}\sqrt{2}/\sqrt{x_{-\mathrm{i}}}(t),\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ c(t)&\mathrel{\mathop{\mathchar
58\relax}}=-x_{-\mathrm{i}}(t),\end{cases}$

where $\sqrt{x_{-\mathrm{i}}}(t)$ is the branch holomorphic in
$O_{(-\mathrm{i})}$ satisfying
$\sqrt{x_{-\mathrm{i}}}(0)=\mathrm{e}^{\pi\mathrm{i}/3}$. In these cases
$J_{t}=\Gamma[a,b]$ and the chain of integration $\Gamma$ in (28) can be
chosen as

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,a\big{)}\cup
J_{t}\cup\Gamma\big{(}b,\mathrm{e}^{-\pi\mathrm{i}/3}\infty\big{)}\right)-\alpha_{2}\left(\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,c\big{]}\cup\Gamma\big{(}c,\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}\right)$

when $\mathrm{Im}(t)>0$, see Figure 7(A) and Figure 8(A); as

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,a\big{)}\cup
J_{t}\cup\Gamma(b,c]\right)+\alpha_{1}\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,c\big{)}-\alpha_{2}\Gamma\big{(}c,\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}$

when $t$ is real, see Figure 7(B) and Figure 8(B);

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,a\big{)}\cup
J_{t}\cup\Gamma\big{(}b,\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}\right)+\alpha_{1}\left(\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,c\big{]}\cup\Gamma\big{(}c,\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}\right)$

when $\mathrm{Im}(t)<0$, see Figure 7(C) and Figure 8(C).

If $t\in(\partial O_{1,-\mathrm{i}}\cap
O_{1,-})\setminus\\{t_{\mathsf{cr}},\eta t_{\mathsf{cr}}\\}$, the support
$J_{t}=\Gamma[a,c]\cup\Gamma[c,b]$ is still a single Jordan arc and the chain
of integration $\Gamma$ in (28) can be chosen as

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,a\big{]}\cup\Gamma(a,c)\right)+\alpha_{1}\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,c\big{]}-\alpha_{2}\left(\Gamma(c,b)\cup\Gamma\big{[}b,\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}\right),$

see Figure 9.

These claims are part of [10, Theorems 3.2 and 3.4], where a particular case
$\lambda=-\mathrm{i}$ ($\alpha_{0}=-\alpha_{2}=1/\pi\mathrm{i}$,
$\alpha_{1}=0$) was analyzed. However, as one can clearly see, the obtained
chains are homologous to $\Gamma$ from (24)–(25). In fact, Theorem 4.10
remains valid as stated for $\lambda=-\mathrm{i}$ as well, but Remark 4.5 is
not applicable to this case. So, S-curves for $\lambda=-\mathrm{i}$ and $t\in
O_{(-\mathrm{i})}\setminus O_{1,-\mathrm{i}}$ are not obtained by rotation of
the ones described above, see [10, Figure 4(a-c)].

###  4.4. Two-cut Cases

While it is possible to analytically continue $a(t),b(t)$, and $c(t)$ in (42)
past the boundary of $O_{0,-\mathrm{i}}$, the critical graph of
$\varpi_{t}(z)$ as defined by (39) no longer admits a curve
$\Gamma_{t}\in\mathcal{T}$ satisfying
$\Gamma_{t}\subset\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)\leq 0\\}$, see [10, Theorems 3.2 and 3.4, and
Figure 4].

Figure 10. Schematic representation of the critical (solid) and critical
orthogonal (dashed) graphs of $\varpi_{t}(z)$ when $t\in O_{1,-\mathrm{i}}$.
The bold curves represent the preferred S-curve $\Gamma_{t}$. Shaded region is
the set $\\{\mathcal{U}(z;t)<0\\}$.

Hence, we are forced to abandon ansatz (39) and have to look for $Q(z;t)$ with
four simple zeros. Conditions (40) are now complemented by the _Boutroux
conditions_ :

(43) $\mathrm{Re}\left(\int_{z_{i}(t)}^{z_{j}(t)}Q^{1/2}(s;t)\
\mathrm{d}s\right)=0,\quad i,j\in\\{0,1,2,3\\},$

for any continuous determination of $Q^{1/2}(z;t)$ on the path of integration,
where $z_{i}(t)$ are the zeros of $Q(z;t)$. Relations (43) follow from (36),
which tells us that the integrals are purely imaginary on the branch cuts of
$Q^{1/2}(z;t)$, and (33), (37), which tell us that the integral between the
cuts (if the cuts are not connected to each other) also must be purely
imaginary so that the real constant $\ell_{\Gamma_{t}}$ is the same on all
(both) cuts. Notice that both remarks made before Theorem 4.10 remain relevant
to the theorem below.

######  Theorem 4.11.

Let $\mu_{t}$ and $Q(z;t)$ be as in Theorem 4.1,
$J_{t}=\mathrm{supp}(\mu_{t})$. Assume that
$\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$. When $t\in O_{1,-\mathrm{i}}$, the
polynomial $Q(z;t)$ is of the form

$Q(z;t)=\frac{1}{4}(z-a_{1}(t))(z-b_{1}(t))(z-a_{2}(t))(z-b_{2}(t))$

with $a_{1}(t)$, $b_{1}(t)$, $a_{2}(t)$, and $b_{2}(t)$ all distinct. The real
and imaginary parts of $a_{i}(t),b_{i}(t)$ are real analytic functions of
$\mathrm{Re}(t)$ and $\mathrm{Im}(t)$, however, at no point of
$O_{1,-\mathrm{i}}$ any of the functions $a_{i}(t),b_{i}(t)$ is complex
analytic. The chain of integration $\Gamma$ in (28) can be chosen as

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,a_{1}\big{)}\cup\Gamma[a_{1},b_{1}]\cup\Gamma\big{(}b_{1},\mathrm{e}^{-\pi\mathrm{i}/3}\infty\big{)}\right)-\\\
\alpha_{2}\left(\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,a_{2}\big{)}\cup\Gamma[a_{2},b_{2}]\cup\Gamma\big{(}b_{2},\mathrm{e}^{\pi\mathrm{i}/3}\infty\big{)}\right),$

see Figure 10 (this fixes our labeling of the zeros of $Q(z;t)$). Moreover, it
holds that

(44) $a_{1}(t),b_{1}(t)\to c(t^{*}),\quad b_{1}(t),a_{2}(t)\to
c(t^{*}),\quad\text{and}\quad a_{2}(t),b_{2}(t)\to c(t^{*})$

as $t\to t^{*}\in\partial O_{1,-\mathrm{i}}$ with $t^{*}\in\partial O_{1,-}$,
$t^{*}\in\partial O_{1,-\mathrm{i}}$, and $t^{*}\in\partial O_{0,0}$,
respectively.

This is [2, Theorem 3.2], where the case $\lambda=-\mathrm{i}$ was analyzed.
As before, one can readily see that the obtained chains are homologous to
$\Gamma$ from (24)–(25).

###  4.5. Trefoil Cases

It only remains to describe what happens when $t\in O_{1,-}$. In this case the
geometries of the critical and critical orthogonal graphs of $\varpi_{t}(z)$
are distinct from all the previous cases.

Figure 11. Schematic representation of the critical (solid) and critical
orthogonal (dashed) graphs of $\varpi_{t}(z)$ when $t\in O_{1,-}$. The bold
curves represent the preferred S-curve $\Gamma_{t}$. Shaded region is the set
$\\{\mathcal{U}(z;t)<0\\}$.

Our approach essentially consists in observing that $L_{0}\cup L_{1}\cup
L_{2}$ is the desired S-contour when $t=0\in O_{1,-}$ and arguing that the
geometric structure of the critical and critical orthogonal graphs cannot
change when $t$ is varied within $O_{1,-}$.

######  Theorem 4.12.

Let $\mu_{t}$ and $Q(z;t)$ be as in Theorem 4.1,
$J_{t}=\mathrm{supp}(\mu_{t})$. Assume that
$\alpha_{0}\alpha_{1}\alpha_{2}\neq 0$. When $t\in O_{1,-}$, the polynomial
$Q(z;t)$ is of the form

$Q(z;t)=\frac{1}{4}(z-z_{0}(t))(z-z_{1}(t))(z-z_{2}(t))(z-z_{3}(t))$

with $z_{i}(t)$, $i\in\\{0,1,2,3\\}$, all distinct. The real and imaginary
parts of $z_{i}(t)$ are real analytic functions of $\mathrm{Re}(t)$ and
$\mathrm{Im}(t)$. The chain of integration $\Gamma$ in (28) can be chosen as

$\alpha_{0}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}}\infty,z_{0}\big{)}\cup\Gamma[z_{0},z_{3}]\right)+\alpha_{1}\left(\Gamma\big{(}\mathrm{e}^{-\pi\mathrm{i}/3}\infty,z_{1}\big{)}\cup\Gamma[z_{1},z_{3})\right)+\\\
\alpha_{2}\left(\Gamma\big{(}\mathrm{e}^{\pi\mathrm{i}/3}\infty,z_{2}\big{)}\cup\Gamma[z_{2},z_{3})\right),$

see Figure 11 (this fixes our labeling of the zeros of $Q(z;t)$). Moreover, it
holds that

$z_{0}(t)\to a(t^{*}),\quad z_{2}(t)\to b(t^{*}),\quad\text{and}\quad
z_{1}(t),z_{3}(t)\to c(t^{*})$

as $t\to t^{*}\in\partial O_{1,-\mathrm{i}}\cap\partial O_{1,-}$.

The rest of this section is dedicated to the proof of Theorem 4.12. Let us
emphasize again that in the studied cases all the zeros of $Q(z;t)$ must be
distinct as coincidence of any two of them leads to the ansatz (39), which
necessarily implies that $a(t),b(t),c(t)$ are as in (42), where
$x_{-\mathrm{i}}(t)$ can be replaced by $x_{0}(t)$ or $x_{\mathrm{i}}(t)$, but
no combination of the critical and critical orthogonal trajectories of the
corresponding differentials form a desirable curve $\Gamma_{t}$ (all the
possible configurations were already examined before, see [10, Theorems 3.2
and 3.4, and Figure 4]).

####  4.5.1. Step 1

Since the Jacobian of the elementary symmetric functions of the zeros of
$Q(z;t)$ is its discriminant and all the zeros are distinct when $t\in
O_{1,-}$, it is enough to examine dependence on $t$ of the coefficients of
$Q(z;t)$. In view of (38), we only need to study the free coefficient $K(t)$.

Fix $t^{*}\in O_{1,-}$ and let $K^{*}=K(t^{*})$ be the corresponding constant
guaranteed by Theorem 4.1 and Remark 4.4. Consider

$Q(z;t,K)\mathrel{\mathop{\mathchar 58\relax}}=\dfrac{1}{4}(z^{2}-t)^{2}+z+K$

for $t,K$ in some neighborhoods of $t^{*}$ and $K^{*}$, respectively. We take
these neighborhoods small enough so that there are disjoint neighborhoods of
$z_{i}(t^{*})$, $i\in\\{0,1,2,3\\}$, the zeros of $Q(z;t^{*})$, each
containing one zero of $Q(z;t,K)$. We label $z_{i}(t,K)$ the zero of
$Q(z;t,K)$ that approaches $z_{i}(t^{*})$ as $(t,K)\to(t^{*},K^{*})$ (these
zeros are smooth functions of $t$ and $K$ as mentioned before). Define

$\mathfrak{S}_{t,K}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{(z,w)\
\mathrel{\mathop{\mathchar 58\relax}}\
w^{2}=Q(z;t,K)\right\\}\subset\mathbb{C}^{2}.$

Since the zeros of $Q(z;t,K)$ are simple, this is an elliptic, i.e., genus 1,
Riemann surface. Select a homology basis on $\mathfrak{S}_{t,K}$, say
$(\bm{\alpha}_{t,K},\bm{\beta}_{t,K})$. We can do it in such a way that the
natural projection $(z,w)\mapsto z$ of the cycle $\bm{\alpha}_{t,K}$ is a
Jordan arc connecting $z_{0}(t,K)$ to $z_{1}(t,K)$ while the natural
projection of the cycle $\bm{\beta}_{t,K}$ is a Jordan arc connecting
$z_{0}(t,K)$ to $z_{2}(t,K)$. Moreover, we can arrange so that these Jordan
arcs coincide for all considered $(t,K)$ outside of small neighborhoods of
$z_{0}(t^{*})$, $z_{1}(t^{*})$, and $z_{2}(t^{*})$. Set

$B_{\bm{\gamma}}(t,K)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\mathrm{Re}\left(\oint_{\bm{\gamma}_{t,K}}w\mathrm{d}z\right),\quad\bm{\gamma}\in\\{\bm{\alpha},\bm{\beta}\\},$

which we consider to be functions of $\mathrm{Re}(t)$, $\mathrm{Im}(t)$,
$u\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Re}(K)$, and
$v\mathrel{\mathop{\mathchar 58\relax}}=\mathrm{Im}(K)$. We further make sure
that the homology bases are selected so that $B_{\bm{\gamma}}(t,K)$ are
continuous functions of their parameters around $(t^{*},K^{*})$. Boutroux
conditions (43) yield that

(45) $B_{\bm{\alpha}}(t,K)=0\quad\text{and}\quad B_{\bm{\beta}}(t,K)=0.$

The Cauchy-Riemann equations allow us to compute the Jacobian

$\displaystyle\mathrm{det}\dfrac{\partial(B_{\bm{\alpha}},B_{\bm{\beta}})}{\partial(u,v)}$
$\displaystyle=\frac{1}{4}\left(-\mathrm{Re}\left(\oint_{\bm{\alpha}}\dfrac{\mathrm{d}z}{w}\right)\mathrm{Im}\left(\oint_{\bm{\beta}}\dfrac{\mathrm{d}z}{w}\right)+\mathrm{Re}\left(\oint_{\bm{\beta}}\dfrac{\mathrm{d}z}{w}\right)\mathrm{Im}\left(\oint_{\bm{\alpha}}\dfrac{\mathrm{d}z}{w}\right)\right)$
$\displaystyle=\frac{1}{4}\mathrm{Im}\left(\oint_{\bm{\alpha}}\dfrac{\mathrm{d}z}{w}\overline{\oint_{\bm{\beta}}\dfrac{\mathrm{d}z}{w}}\right).$

The differential $\mathrm{d}z/w$ is the unique (up to multiplication by a
constant) holomorphic differential on $\mathfrak{S}_{t,K}$ and by Riemann’s
bilinear relation, see [11, Lemma 35.2], we have

$\mathrm{det}\dfrac{\partial(B_{\bm{\alpha}},B_{\bm{\beta}})}{\partial(u,v)}<0.$

Thus, it follows from the Implicit Function Theorem that in some neighborhood
of $t^{*}$ there exists a function $K^{*}(t)$ which is real analytic in
$\mathrm{Re}(t)$ and $\mathrm{Im}(t)$ and such that the equations (45) hold
with $(t,K^{*}(t))$ in this neighborhood of $t^{*}$. We still need to argue
that $K(t)=K^{*}(t)$, To this end, we shall examine the critical graph of
$\varpi_{t}^{*}(z)\mathrel{\mathop{\mathchar
58\relax}}=Q(z;t,K^{*}(t))\mathrm{d}z^{2}$. Before we do this, let us recall
relevant facts of the theory of quadratic differentials.

####  4.5.2. Quadratic Differentials

Given a quadratic differential $\varpi_{t}^{*}(z)$ as above, it holds that

  * •

two distinct (orthogonal) trajectories meet only at critical points, see [37,
Theorem 5.5], which, in this case, are the zeros of $Q(z;t,K^{*}(t))$ and the
point at infinity;

  * •

no finite union of (orthogonal) trajectories can form a closed Jordan curve
because $Q(z;t,K^{*}(t))$ is a polynomial, see [31, Lemma 8.3];

  * •

trajectories of $\varpi_{t}^{*}(z)$ cannot be recurrent (dense in two-
dimensional regions), see [22, Theorem 3.6];

  * •

since infinity is a pole of $\varpi_{t}^{*}(z)$ order 8, the critical
trajectories can approach infinity only in six distinguished directions,
namely, asymptotically to the lines $\\{\mathrm{arg}(z)=-5\pi/6+k\pi/3\\}$,
$k\in\\{0,1,2,3,4,5\\}$;

  * •

there exists a neighborhood of infinity such that any trajectory entering it
necessarily tends to infinity, see [37, Theorem 7.4].

Denote by $\mathcal{G}_{t}^{*}$ the critical graph of $\varpi_{t}^{*}(z)$,
that is, the totality of all the critical trajectories of $\varpi_{t}^{*}(z)$.
In the case of polynomial quadratic differentials, more can be said about the
topological nature of $\mathbb{C}\setminus\mathcal{G}_{t}^{*}$. The complement
of $\mathcal{G}_{t}^{*}$ can be written as a disjoint union of either half-
plane or strip domains, see [22, Theorem 3.5]. Recall that a _half-plane (or
end) domain_ is swept by trajectories unbounded in both directions that
approach infinity along consecutive critical directions. Its boundary is
connected and consists of a union of two unbounded critical trajectories and a
finite number (possibly zero) of short trajectories of $\varpi_{t}^{*}(z)$.
The map $z\mapsto\int^{z}\sqrt{Q(s;t,K^{*}(t))}\mathrm{d}s$ maps end domains
conformally onto half planes $\\{z\in\mathbb{C}\ |\ \mathrm{Re}(z)>c\\}$ for
some $c\in\mathbb{R}$ that depends on the domain, and extends continuously to
the boundary. Similarly, a _strip domain_ is swept by trajectories unbounded
in both directions, but its boundary consists of two disjoint trajectories of
$\varpi_{t}^{*}(z)$, each of which is comprised of two unbounded critical
trajectories and a finite number (possibly zero) of short trajectories. The
map $z\mapsto\int^{z}\sqrt{Q(s;t,K^{*}(t))}\mathrm{d}s$ maps strip domains
conformally onto vertical strips $\\{w\in\mathbb{C}\ |\
c_{1}<\mathrm{Re}(w)<c_{2}\\}$ for some $c_{1},c_{2}\in\mathbb{R}$ depending
on the domain, and extends continuously to their boundaries. The number
$c_{2}-c_{1}$ is known as the _width of a strip domain_ and can be calculated
in terms of $\varpi_{t}^{*}(z)$ as

(46) $\left|\mathrm{Re}\int_{p}^{q}\sqrt{Q(s;t,K^{*}(t))}\mathrm{d}s\right|$

for any two points $p,q$ belonging to different components of the boundary of
the domain.

####  4.5.3. Step 2

Equations (45) imply existence of three short critical trajectories of
$\varpi_{t}^{*}(z)$. Indeed, let $J$ be a Jordan arc that connects a pair of
zeros of $Q(z;t,K^{*}(t))$ and $\bm{\gamma}$ be a cycle on
$\mathfrak{S}_{t,K^{*}(t)}$ whose natural projection is $J$. Then it holds
that

$\mathrm{Re}\left(\int_{J}Q^{1/2}(s;t,K^{*}(t))\mathrm{d}s\right)=\pm\frac{1}{2}\mathrm{Re}\left(\oint_{\bm{\gamma}}w\mathrm{d}s\right),$

where $Q^{1/2}(s;t,K^{*}(t))$ is a continuous branch on $J$ and the sign $+$
or $-$ depends on the choice of this branch as well as the chosen orientations
for $J$ and $\bm{\gamma}$. It is well known, see [11, Theorem 34.3], that

$\oint_{\bm{\gamma}}w\mathrm{d}s=n_{1}\oint_{\bm{\alpha}_{t,K^{*}(t)}}w\mathrm{d}s+n_{2}\oint_{\bm{\beta}_{t,K^{*}(t)}}w\mathrm{d}s+2\pi\mathrm{i}\sum
m_{k}\,\mathrm{res}_{z_{k}}\,(w\mathrm{d}z),$

where $n_{1},n_{2}$ and $m_{k}$ are integers and the sum is taken over all the
residues of the differential $w\mathrm{d}z$. Since the only poles of
$w\mathrm{d}z$ are located at points on top of infinity with residues $\pm 1$,
the last two relations together with (45) imply that

(47) $\mathrm{Re}\left(\int_{J}Q^{1/2}(s;t,K^{*}(t))\mathrm{d}s\right)=0$

for any Jordan arc $J$ connecting any pair of zeros of $Q(s;t,K^{*}(t))$ (that
is, this polynomial has property (43)). Recall now that each zero
$z_{i}^{*}(t)=z_{i}(t,K^{*}(t))$ of $Q(z;t,K^{*}(t))$ is incident with three
critical trajectories of $\varpi_{t}^{*}(z)$. If all three critical
trajectories out of $z_{i}^{*}(t)$ approach infinity, then $z_{i}^{*}(t)$ must
belong to a boundary of at least one strip domain. Let $z_{j}^{*}(t)$ be a
different zero belonging to the other component of the boundary of this strip
domain. Then it follows from (46) and (47) that the width of this strip domain
is $0$, which is impossible. Thus, each zero of $Q(z;t,K^{*}(t))$ must be
coincident with at least one short trajectory. Therefore, either there is a
zero connected by short trajectories to the remaining three zeros or there are
at least two short trajectories connecting two pairs of zeros. In the latter
case, label these zeros by $z_{0}^{*}(t),z_{1}^{*}(t)$ and
$z_{2}^{*}(t),z_{3}^{*}(t)$. If the other two trajectories out of both
$z_{0}^{*}(t)$ and $z_{1}^{*}(t)$ approach infinity, one of these zeros again
must belong to the boundary of a strip domain with either $z_{2}^{*}(t)$ or
$z_{3}^{*}(t)$ belonging to the other component of the boundary. As before,
(46) and (47) yield that the width of this strip domain is $0$, which, again,
is impossible. Thus, in this case there also exists a third short critical
trajectory.

Since short critical trajectories cannot form a loop, they form a connected
set, say $\mathcal{G}_{t}^{\mathrm{sh}}$, that contains all four zeros of
$Q(z;t,K^{*}(t))$ (this set is either a trefoil or a single Jordan arc).
Define

(48) $\mathcal{U}^{*}(z;t)\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Re}\left(2\int_{e}^{z}Q^{1/2}(s;t,K^{*}(t))\mathrm{d}s\right),$

where $e\in\mathcal{G}_{t}^{\mathrm{sh}}$ and $Q^{1/2}(s;t,K^{*}(t))$ is the
branch holomorphic in $\mathbb{C}\setminus\mathcal{G}_{t}^{\mathrm{sh}}$ that
behaves like $\frac{1}{2}z^{2}+O(z)$ as $z\to\infty$. $\mathcal{U}^{*}(z;t)$
is a well defined harmonic function in
$\mathbb{C}\setminus\mathcal{G}_{t}^{\mathrm{sh}}$ by (47) (which is
independent of the choice of $e$). Moreover, this function is either harmonic
across a given short critical trajectory in $\mathcal{G}_{t}^{\mathrm{sh}}$
(this happens to the “middle section” when three short critical trajectories
form a single Jordan arc) or can be harmonically continued across it by
$-\mathcal{U}^{*}(z;t)$. Since the zeros of $Q^{1/2}(s;t,K^{*}(t))$ converge
to the zeros of $Q^{1/2}(s;t^{*})$ as $t\to t^{*}$, we have that
$\mathcal{U}^{*}(z;t)$ converges to $\mathcal{U}(z;t^{*})$ point-wise. Using
the above harmonic continuation to define a single harmonic function on a
double ramified cover of this disk via concatenation of $\mathcal{U}^{*}(z;t)$
and $-\mathcal{U}^{*}(z;t)$ and applying the maximum modulus principle and
[32, Theorem 1.3.10], we deduce that the functions $\mathcal{U}^{*}(z;t)$
converge to $\mathcal{U}(z;t^{*})$ as $t\to t^{*}$ uniformly on compact
subsets of $\mathbb{C}$. Furthermore, we have that

(49) $\mathcal{U}^{*}(z;t)=-\mathrm{Re}(V(z;t))+2\log|z/e|+\\\
\mathrm{Re}\left(2\int_{e}^{z}\left(Q^{1/2}(s;t,K^{*}(t))-\frac{s^{2}-t}{2}-\frac{1}{s}\right)\mathrm{d}s\right),$

where the integrand behaves as $O(z^{-2})$ at infinity and therefore the last
summand above is harmonic at infinity. Thus, using (49) we have that the
functions $\mathcal{U}^{*}(z;t)$ converge to $\mathcal{U}(z;t^{*})$ as $t\to
t^{*}$ uniformly on the whole sphere $\overline{\mathbb{C}}$. In particular,
since the critical graph of $\varpi_{t}^{*}(z)$ is the zero level set of
$\mathcal{U}^{*}(z;t)$, this function can be either positive and negative on
different sides of a trajectory (i.e., it is harmonic across) or be positive
on both sides (i.e., it is subharmonic across) since this is true for
$\mathcal{U}(z;t^{*})$ by (33) and (37). The latter situation happens on the
part of $\mathcal{G}_{t}^{\mathrm{sh}}$ that serves as the branch cut for
$Q^{1/2}(s;t,K^{*}(t))$, which we denote by $J^{*}_{t}$.

####  4.5.4. Step 3

Now, we would like to identify possible structures of the critical graph of
$\varpi_{t}^{*}(z)$. To this end, we shall use what is known as _clock
diagrams_.

(a)

(b)

(c)

(d)

Figure 12. Schematic representation of $\mathcal{G}_{t}^{*}$ (A) or (C) and
its associated clock diagram (B) or (D). Shaded region is the set
$\\{\mathcal{U}(z;t)<0\\}$.

Introduced in [5, Definition 5.4], (topological) clock diagrams are a
schematic way of representing the critical graph of $\varpi_{t}^{*}(z)$ that
allows for a more combinatorial treatment similar in spirit to [3] though more
concrete since we deal with a fixed polynomial potential. In our setting, a
clock diagram consists of the outer hexagon whose vertices correspond to the
distinguished directions at infinity; inner vertices that correspond to the
finite critical points, i.e., zeros of $Q(z;t,K^{*}(t))$, that are connected
to each other and the vertices of the hexagon by edges corresponding to the
critical trajectories of $\varpi_{t}^{*}(z)$; and finally the shaded regions
that correspond to the strip or end domains where $\mathcal{U}^{*}(z,t)<0$. It
follows from (37) that every other edge of the hexagon must border a shaded
region starting with the vertical edge on the left, which corresponds to the
sector between the distinguished direction $5\pi/6$ and $7\pi/6$, see Figure
12.

(a)

(b)

(c)

(d)

Figure 13. The four possible configurations of $\mathcal{G}_{t}^{\mathrm{sh}}$
up to rotation.

Since the critical graph of $\varpi_{t}^{*}(z)$ must contain exactly three
short trajectories not allowing for loops, we immediately see that there are
exactly 4 possible configurations of short trajectories (up to rotation),
shown on Figure 13. Each of these configurations can be attached to the
boundary of the hexagon in 6 different ways since once the direction in which
one trajectory goes to infinity is fixed, the others are uniquely determined
by Teichmüller’s Lemma, see [37, Theorem 14.1], see Figures 14 and 15.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14. The six possible ways to connect $\mathcal{G}_{t}^{\mathrm{sh}}$
from Figure 13(B) to the hexagon.

However, we now observe the importance of the shading. As explained after
(49), $J_{t}^{*}$ must border unshaded regions on both sides whereas any other
trajectory must be the boundary of both a shaded and an unshaded region.
Hence, of the 6 possible ways to connect the finite critical points to the
hexagon, only 3 allow for a valid coloring when
$\mathcal{G}_{t}^{\mathrm{sh}}$ is as on Figures 13(A,B) (and their
rotations), see Figure 14, while

(a)

(b)

(c)

(d)

(e)

(f)

Figure 15. The six possible ways to connect $\mathcal{G}_{t}^{\mathrm{sh}}$
from Figure 13(D) to the hexagon.

there are no valid colorings when $\mathcal{G}_{t}^{\mathrm{sh}}$ is as on
Figures 13(C,D), see Figure 15. In the sequel, we will need to talk about
clock diagrams and critical graphs which are “structurally the same.” To do so
in our setting, we attach to a critical graph $\mathcal{G}$ with critical
points $z_{1},z_{2},z_{3},z_{4}$ the adjacency data

$\left\\{\\{z_{1},A_{1}\\},\\{z_{2},A_{2}\\},\\{z_{3},A_{3}\\},\\{z_{4},A_{4}\\}\right\\},$

where $A_{i}\subset\mathbb{N}$ is a finite set and $k\in A_{i}$ if and only if
$z_{i}$ is on the boundary of the end domain containing the line
$\\{r\mathrm{e}^{\pi k\mathrm{i}/3}\ \mathrel{\mathop{\mathchar 58\relax}}\
r>R\\}$ for $R$ large enough.

######  Definition 4.13.

We say that two critical graphs are structurally equivalent if there exists a
labeling of the critical points such that both graphs have the same adjacency
data. Two clock diagrams are structurally equivalent if their corresponding
critical graphs are structurally equivalent.

With this definition in mind, observe that the clock diagrams on Figures
14(A,D,F) structurally describe the same critical graphs as diagrams on
Figures 16(A,B,C), respectively. Hence, if we were to draw the admissible
diagrams corresponding to a rotation of $\mathcal{G}_{t}^{\mathrm{sh}}$ as on
Figure 13(B), they would again structurally correspond to the critical graphs
depicted on Figures 16(A,B,C). As all admissible diagrams corresponding to
$\mathcal{G}_{t}^{\mathrm{sh}}$ as on Figure 13(A) are structurally the same,
see Figure 16(D), Figure 16 depicts all possible admissible clock diagrams for
the critical graph of $\varpi_{t}^{*}(z)$.

(a)

(b)

(c)

(d)

Figure 16. The only four admissible clock diagrams.

####  4.5.5. Step 4

Since $\mathcal{U}^{*}(z;t)$ is subharmonic in $\mathbb{C}$ and harmonic in
$\mathbb{C}\setminus J_{t}^{*}$, its Laplacian is a positive measure, say
$2\mu_{t}^{*}$, supported on $J_{t}^{*}$. It follows from (49), the Riesz
Decomposition Theorem [32, Theorem 3.7.9] and Liouville’s Theorem for
subharmonic functions [32, Corollary 2.3.4] to identify the harmonic part that

$\mathcal{U}^{*}(z;t)=-\mathrm{Re}(V(z;t))+\ell_{t}^{*}-2U^{\mu_{t}^{*}}(z)$

for some real constant $\ell_{t}^{*}$. The measure $\mu_{t}^{*}$ has mass $1$
as this is the only case when $2U^{\mu_{t}^{*}}(z)$ behaves like $-2\log|z|$
as $z\to\infty$, see (49). Moreover, as we have mentioned before, the very
definition of $\mathcal{U}^{*}(z;t)$ in (48) implies that
$-\mathcal{U}^{*}(z;t)$ is the harmonic continuation of $\mathcal{U}^{*}(z;t)$
across any subarc of $J_{t}^{*}$. This, however, is equivalent to saying that
$\mathcal{U}^{*}(z;t)$ satisfies (34). Hence, if we add to the corresponding
$\mathcal{G}_{t}^{\mathrm{sh}}$, of which $J_{t}^{*}$ is a part, three
unbounded Jordan arcs that lie within $\\{\mathcal{U}^{*}(z;t)<0\\}$ (shaded
regions on Figure 16) to create an element of $\mathcal{T}$, we will obtain a
symmetric contour. Recall now that the uniqueness in Remark 4.4 was not
derived based on the maximization of the minimal energy, which is the defining
property of $J_{t}$ in Theorem 4.1, but based on being an S-curve. Hence,
$J_{t}^{*}$ is $J_{t}$ and $\mu_{t}^{*}$ and $\mu_{t}$. Moreover, this
finishes the proof of real analytic dependence of the zeros of $Q(z;t)$ on
$\mathrm{Re}(t)$ and $\mathrm{Im}(t)$.

Since the function $\mathcal{U}(z;t)$ converge uniformly to
$\mathcal{U}(z;t^{*})$ as $t\to t^{*}$ and the critical graph of
$\varpi_{t}(z)$ can have only one of the four structural forms depicted on
Figure 16, the critical graphs of $\varpi_{t}(z)$ are structurally the same as
the critical graph of $\varpi_{t^{*}}(z)$ for all $t$ in some small
neighborhood of $t^{*}$. As any two points in $O_{1,-}$ can be joined by a
path that can be covered by finitely many such neighborhoods, the structure of
the critical graph of $\varpi_{t}(z)$ must be the same for every $t\in
O_{1,-}$. It readily follows from Remark 4.5 that $K(0)=0$ and therefore

$\varpi_{0}(z)=\frac{z(z^{3}+4)}{4}\mathrm{d}z^{2},$

whose critical graph can be readily determined and is depicted on Figure 11.
Since this structure of the critical graph uniquely determines the structure
of the critical orthogonal graph, this finishes the proof of the theorem
except for the last claim.

####  4.5.6. Step 5

It follows from [6, Theorem 5.11] that if $t$ remains in a bounded set, so do
the zeros of $Q(z;t)$. Fix $t^{*}\in\partial O_{1,-}$ and let $\\{t_{m}\\}$ be
a sequence such that $t_{m}\to t^{*}$ as $m\to\infty$. Restricting to a
subsequence if necessary, we see that there exist $z_{i}^{*}$ such that
$z_{i}(t_{m})\to z_{i}^{*}$ as $m\to\infty$, $i\in\\{0,1,2,3\\}$, where
$z_{i}(t)$ are the zeros of $Q(z;t)$. Clearly, polynomials $Q(z;t_{m})$
converge uniformly on compact subsets of $\mathbb{C}$ to $Q^{*}(z)$, the
polynomial with zeros $z_{i}^{*}$ and leading coefficient $1/4$. With the help
of uniform convergence we can conclude that (47) must be satisfied with
$Q(z;t,K^{*}(t))$ replaced by $Q^{*}(z)$. At this point, we can repeat the
steps that follow (47) (we will need to add clock diagrams corresponding to
possible confluence of the zeros of $Q^{*}(z)$) to conclude that an S-curve
can be constructed out of the critical and critical orthogonal trajectories of
$Q^{*}(z)\mathrm{d}z^{2}$, and then appeal to Remark 4.4 to conclude that
$Q^{*}(z)=Q(z;t^{*})$.

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