example example∎ remark remark∎

# Special Function solutions of Painlevé-IV and the Quartic Model

Ahmad Barhoumi<EMAIL_ADDRESS>Department of Mathematics, Royal Institute of
Technology (KTH), Lindstedtsvägen 25, 10044, Stockholm, Sweden

###### Abstract.

The fourth Painlevé equation, known as Painlevé-IV, is a family of
differential equations depending on two complex parameters and is given by

$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}u}{\operatorname{d\\!}{x^{2}}}}=\dfrac{1}{2u}\left(\mathinner{\dfrac{\operatorname{d\\!}{}u}{\operatorname{d\\!}{x}}}\right)^{2}+\dfrac{3}{2}u^{3}+4xu^{2}+2(x^{2}-\alpha)u+\dfrac{\beta}{u},\quad\alpha,\beta\in{\mathbb{C}}.$

While generic solutions of Painlevé equations are known to be transcendental,
five of the six equations admit solutions written in terms of classical
special functions. In this work, we analyze a particular family of such
solutions of Painlevé-IV given in terms of Parabolic Cylinder functions. We do
so by establishing explicit formulas connecting these solutions and recurrence
coefficients of polynomials orthogonal with respect to the measure
${\mathrm{e}}^{-V(z;t)}{\mathrm{d}}z$ were $V(z;t)=z^{4}/4+tz^{2}/2$. This
particular family of polynomials appears in the study of the Quartic Random
Matrix Model.

###### Key words and phrases:

Painlevé-IV equation, Riemann-Hilbert Analysis, Non-Hermitian Orthogonal
Polynomials.

###### 2020 Mathematics Subject Classification:

Primary: 34M55, 33C47; Secondary: 15B52, 30E15, 34E05,34M50.

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

###### Contents

  1. 1 Introduction
  2. 2 Painlevé-IV and its Bäcklund Transformations
  3. 3 Connecting the Quartic Model, Orthogonal Polynomials, and Painlevé-IV
  4. 4 Preliminaries to the Asymptotic Analysis of $P_{n},\gamma^{2}_{n}$
  5. 5 Riemann-Hilbert Analysis
  6. 6 Conclusion and Discussion
  7. A Connection with Semiclassical Laguerre Polynomials

##  1\. Introduction

This work centers around the interactions of three mathematical objects that
have been the subject of intense study over the last few decades:

  * •

solutions of Painlevé-IV,

  * •

the Quartic One-Matrix Model, and

  * •

Semi-classical orthogonal polynomials.

Each has given rise to a beautiful theory and a large body of work; we do not
dare to attempt an exhaustive exposition of their literature. Instead, we
settle for describing the bare minimum necessary for the reader to appreciate
the connections between them.

###  1.1. Painlevé-IV

By now, the Painlevé equations are fundamental equations in integrable systems
and mathematical physics. While generic solutions of Painlevé equations are
highly transcendental, all but Painlevé-I possess special solutions written in
terms of elementary and/or classical special functions (Airy functions, Bessel
functions, parabolic cylinder functions, etc.). We will focus our attention on
the fourth Painlevé equation, given by

($\textnormal{P}_{IV}$)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}u}{\operatorname{d\\!}{x^{2}}}}=\dfrac{1}{2u}\left(\mathinner{\dfrac{\operatorname{d\\!}{}u}{\operatorname{d\\!}{x}}}\right)^{2}+\dfrac{3}{2}u^{3}+4xu^{2}+2(x^{2}-\alpha)u+\dfrac{\beta}{u},\quad\alpha,\beta\in{\mathbb{C}}.$

Painlevé-IV possesses both rational solutions and solutions written in terms
parabolic cylinder functions, see e.g. [BCH, GLS, MR0896355, MR0217352, O].
Rational solutions of ($\textnormal{P}_{IV}$) appear in various applications
(see [BCH, BM] and references therein), and their large-degree asymptotics
were recently obtained in [BM]. There, a different set of parameters than the
ones in ($\textnormal{P}_{IV}$) is used; they are given by

(1.1) $\alpha=2\Theta_{\infty}-1,\quad\beta=-8\Theta_{0}^{2}.$

These parameters have natural interpretations in the recasting of
($\textnormal{P}_{IV}$) as the condition for isomonodromic deformation for an
associated $2\times 2$ system of differential equations. While we will not use
this fact, we will still adopt these parameters and denote solutions of
($\textnormal{P}_{IV}$), $u(x)$ with parameters given by (1.1) by the triple
$(u(x),\Theta_{0},\Theta_{\infty})$. A different reformulation of
($\textnormal{P}_{IV}$) which we will make heavy use of is the following
symmetric form of Painlevé-IV; see, e.g., [N]. We will follow the notation of
[FW].

######  Theorem 1.1 ([Adler]).

Let $\\{\alpha_{j}\\}_{j=0}^{2}\in{\mathbb{C}}$ be arbitrary constants
satisfying

(1.2) $\alpha_{0}+\alpha_{1}+\alpha_{2}=1.$

Consider the system

($\textnormal{P}^{\textnormal{sym}}_{IV}$)
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}f_{0}}{\operatorname{d\\!}{x}}}$
$\displaystyle=f_{0}(f_{1}-f_{2})+2\alpha_{0},$
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}f_{1}}{\operatorname{d\\!}{x}}}$
$\displaystyle=f_{1}(f_{2}-f_{0})+2\alpha_{1},$
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}f_{2}}{\operatorname{d\\!}{x}}}$
$\displaystyle=f_{2}(f_{0}-f_{1})+2\alpha_{2},$

along with the constraint

(1.3) $f_{0}(x)+f_{1}(x)+f_{2}(x)=2x,$

are equivalent to ($\textnormal{P}_{IV}$). That is, for $j=0,1,2$,
$u(x)=-f_{j}(x)$ solves ($\textnormal{P}_{IV}$) with parameters111Here, the
indices are understood cyclically, i.e. $\alpha_{3}\equiv\alpha_{0}$.

(1.4) $\alpha=\alpha_{j-1}-\alpha_{j+1},\quad\beta=-2\alpha_{j}^{2}.$

Furthermore, given a solution of ($\textnormal{P}_{IV}$),
$(u(x),\Theta_{0},\Theta_{\infty})$, there exists a triple
$(f_{0}(x),f_{1}(x),f_{2}(x))$ and parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})$ satisfying
($\textnormal{P}^{\textnormal{sym}}_{IV}$).

The symmetric form of ($\textnormal{P}^{\textnormal{sym}}_{IV}$) allows one to
deduce various Bäcklund transformation; transformations that map a solution
$(f_{0},f_{1},f_{2})$ to another triple which solves
($\textnormal{P}^{\textnormal{sym}}_{IV}$) with possibly modified parameters.
Their connection with the already known Bäcklund transformations of
($\textnormal{P}_{IV}$) is discussed in Section 2.2. A key role will be played
by three commuting transformations $T_{1},T_{2},T_{3}$, whose explicit
description is given in Section 2.1, which act on the parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})$ is summarized in Table 1.

| $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$
---|---|---|---
$T_{1}$ | $\alpha_{0}-1$ | $\alpha_{1}+1$ | $\alpha_{2}$
$T_{2}$ | $\alpha_{0}$ | $\alpha_{1}-1$ | $\alpha_{2}+1$
$T_{3}$ | $\alpha_{0}+1$ | $\alpha_{1}$ | $\alpha_{2}-1$
Table 1. Action of $T_{i}$ on parameters $\alpha_{j}$.

Moving forward, we will use the notation
$K_{n,m,k}(x)\mathrel{\mathop{\mathchar
58\relax}}=T_{3}^{k}T_{2}^{m}T_{1}^{n}(K_{0}(x))$ where $K$ is replaced by
various relevant functions. Two such functions are the _Jimbo-Miwa-Okamoto
$\tau$\- and $\sigma$-functions_, which will play a key role in our results,
are defined in Section 2.3.

It is well-known (see Section 2.5) that ($\textnormal{P}_{IV}$) (and therefore
($\textnormal{P}^{\textnormal{sym}}_{IV}$)) possesses solutions which can be
written in terms of parabolic cylinder functions and complementary error
function whenever either of following conditions hold:

$\Theta_{0}\pm\Theta_{\infty}=0\quad\text{and}\quad\Theta_{0}\pm\Theta_{\infty}=1.$

Starting with one such solution as a “seed” function and iterating Bäcklund
transformations produces a families of _special function solutions_ of
Painlevé-IV. In this work, we will focus our attention on a particular family
of parabolic cylinder solutions that appear in the Quartic One-Matrix Model
which we now introduce.

###  1.2. Orthogonal Polynomials

Let $\\{P_{n}(z;t,N)\\}_{n=0}^{\infty}$ be the sequence of monic polynomials
orthogonal satisfying

(1.5)
$\int_{{\mathbb{R}}}z^{k}P_{n}(z;t,N){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z=0\quad\text{for}\quad
k=0,1,...,n-1,$

where

(1.6) $V(z;t)\mathrel{\mathop{\mathchar
58\relax}}=\dfrac{z^{4}}{4}+\dfrac{tz^{2}}{2}.$

Computing the polynomial $P_{n}$ amounts to solving an under-determined,
homogeneous system of equations, and so it always exists, but may have degree
smaller than $n$. Henceforth, we denote by $P_{n}$ the _minimal degree monic
polynomial satisfying ( 1.5)_, which is a unique object. When
$t\in{\mathbb{R}}$ (we will always take $N\geq 0$) the measure of
orthogonality in (1.5) is positive which implies $\text{deg}P_{n}=n$ for all
$n\in{\mathbb{N}}$. When $t\in{\mathbb{C}}\setminus{\mathbb{R}}$, we can only
assert that $\text{deg}P_{n}\leq n$. A key observation which we will make
repeated use of is that $\text{deg}P_{n}(z;t,N)=n$ if and only if
$H_{n-1}(t,N)\neq 0$, where

(1.7)
$H_{n-1}(t,N)=\det\left[\int_{\mathbb{R}}s^{i+j}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s\right]_{i,j=0}^{n-1}.$

Orthogonal polynomials are known to satisfy a three-term recurrence relation;
since the weight of orthogonality is even the relation takes the form222Since
$H_{n}(t,N)$ is entire for each $n\in{\mathbb{N}}_{0}$, the roots of
$\prod_{j=0}^{n}H_{j}(t,N)$ forms a discrete, countable subset of
${\mathbb{C}}$. Outside this set (1.8) holds and is then meromorphically
continued to all $t\in{\mathbb{C}}$.

(1.8) $zP_{n}(z;t,N)=P_{n+1}(z;t,N)+\gamma_{n}^{2}(t,N)P_{n-1}(z;t,N).$

where

(1.9) $\gamma_{n}^{2}(t,N)=\dfrac{h_{n}(t,N)}{h_{n-1}(t,N)}$

and $h_{n}$ are the normalizing constants given by

(1.10) $h_{n}(t,N)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\mathbb{R}}P_{n}^{2}(z;t,N){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z=\dfrac{H_{n}(t,N)}{H_{n-1}(t,N)}.$

Since $H_{n}$ are entire functions of $t$, $\gamma_{n}^{2}(t,N)$ and
$h_{n}(t,N)$ are meromorphic functions in $t$. To state our next main result,
we need the following lemma, whose proof is given in Section 3.1.

######  Lemma 1.2.

Let $H_{n}(t,N)$ be as in (1.7). Set $H_{0}^{(e,o)}(t,N)\equiv 1$ and

(1.11) $\displaystyle H^{(e)}_{n}(t,N)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\det\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{t^{i+j}}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{1}{2}}\left({N^{1/2}2^{-1/2}t}\right)\right)\right]_{i,j=0}^{n-1},$
(1.12) $\displaystyle H^{(o)}_{n}(t,N)$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\det\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{t^{i+j}}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{3}{2}}\left({N^{1/2}2^{-1/2}t}\right)\right)\right]_{i,j=0}^{n-1},$

where $D_{\nu}(\diamond)$ is the parabolic cylinder functions (cf.
[DLMF]*Section 12). Then, we have the following identities

(1.13) $\displaystyle H_{2n-1}(t,N)$
$\displaystyle=\dfrac{\pi^{n}2^{2n(n-1)}}{N^{n(2n-1)}}H^{(e)}_{n}(t,N)H^{(o)}_{n}(t,N),$
(1.14) $\displaystyle H_{2n}(t,N)$
$\displaystyle=\dfrac{\pi^{n+\frac{1}{2}}2^{2n^{2}+\frac{1}{4}}}{N^{2n^{2}+n+\frac{1}{4}}}H^{(e)}_{n+1}(t,N)H^{(o)}_{n}(t,N).$

Our first result is that quantities related to the polynomials $P_{n}$ can be
written, as functions of $t$, in terms of solutions to
($\textnormal{P}^{\textnormal{sym}}_{IV}$) and their $\tau$ and $\sigma$
functions in the following manner.

######  Theorem 1.3.

Let $\tau_{n,m,k},\sigma_{n,m,k},$ and
$(f_{0})_{n,m,k},(f_{1})_{n,m,k},(f_{2})_{n,m,k}$ be the Bäcklund iterates of
$\tau_{0,0,0},\sigma_{0,0,0}$ and the solution of
($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,\frac{1}{2},\frac{1}{2})$ given by

$(f_{0}(x),f_{1}(x),f_{2}(x))_{0,0,0}=\left(0,\frac{\varphi^{\prime}(x)}{\varphi(x)},2x-\frac{\varphi^{\prime}(x)}{\varphi(x)}\right),\quad\varphi(x)={\mathrm{e}}^{x^{2}/2}D_{-1/2}(\sqrt{2}x).$

Then,

(1.15) $\displaystyle\tau_{n-1,0,0}(x)$
$\displaystyle={\mathrm{e}}^{-(n-\frac{1}{2})x^{2}}H_{n}^{(e)}(2N^{-1/2}x,N)={\mathrm{e}}^{-(n-\frac{1}{2})x^{2}}H_{n}^{(e)}(2x,1),\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.16) $\displaystyle\tau_{n-1,-1,0}(x)$
$\displaystyle={\mathrm{e}}^{-(n+\frac{1}{2})x^{2}}H_{n}^{(o)}(2N^{-1/2}x,N)={\mathrm{e}}^{-(n+\frac{1}{2})x^{2}}H_{n}^{(o)}(2x,1),\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.17) $\displaystyle(f_{0})_{n,0,0}(x)$
$\displaystyle=-N^{1/2}\gamma_{2n}^{2}(2N^{-1/2}x,N)=-\gamma_{2n}^{2}(2x,1),\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.18) $\displaystyle(f_{1})_{n,0,0}(x)$
$\displaystyle=-N^{1/2}\gamma_{2n+1}^{2}(2N^{-1/2}x,N)=-\gamma_{2n+1}^{2}(2x,1),\vskip
6.0pt plus 2.0pt minus 2.0pt$ (1.19) $\displaystyle(f_{2})_{n,0,0}(x)$
$\displaystyle=2x+N^{1/2}(\gamma_{2n}^{2}(2N^{-1/2}x,N)+\gamma_{2n+1}^{2}(2N^{-1/2}x,N)\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=2x+\gamma_{2n}^{2}(2x,1)+\gamma_{2n+1}^{2}(2x,1).$

Furthermore, writing333Since $V(z;t)$ is an even function, we have
$P_{n}(z;t,N)$ are even(odd) functions whenever $\text{deg}P_{n}$ is even(odd)

$P_{n}(z;t,N)\mathrel{\mathop{\mathchar
58\relax}}=z^{n}+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\mathfrak{p}_{n,n-2k}(t,N)z^{n-2k},$

we have

(1.20) $\displaystyle\sigma_{n-1,0,0}(x)$
$\displaystyle=N^{1/2}\mathfrak{p}_{2n,2n-2}(2N^{-1/2}x,N)-(2n-1)x\vskip 6.0pt
plus 2.0pt minus 2.0pt$
$\displaystyle=\mathfrak{p}_{2n,2n-2}(2x,1)-(2n-1)x,\vskip 6.0pt plus 2.0pt
minus 2.0pt$ (1.21) $\displaystyle\sigma_{n-1,-1,0}(x)$
$\displaystyle=N^{1/2}\mathfrak{p}_{2n+1,2n-1}(2N^{-1/2}x,N)-(2n+1)x\vskip
6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=\mathfrak{p}_{2n+1,2n-1}(2x,1)-(2n+1)x.$

(a) $n=0$

(b) $n=1$

Figure 1.  Poles of $(f_{1})_{n,0,0}(x)$.

We prove Theorem 1.3 in Section 3.3. Figure 1 depicts the poles of
$(f_{1})_{n,0,0}$, and while the figure depicts poles for small $n$, it
appears that large swaths of the plane are free of poles. With Theorem 1.3 in
hand, we can explain the lack of poles and study the large $n$ behavior of
special function solutions of ($\textnormal{P}^{\textnormal{sym}}_{IV}$) by
studying the coefficients $\gamma^{2}_{n}(t,N)$ and the zeros of $H_{n}(t,N)$.
The particular choice of parameters in Theorem 1.3 is motivated by the Quartic
One-Matrix Model, which we now introduce.

###  1.3. The Quartic Model

Let $\mathbf{M}$ be an $n\times n$ Hermitian matrix drawn with respect to the
distribution

(1.22)
$\frac{1}{\tilde{\mathcal{Z}}_{nN}}{\mathrm{e}}^{-N\mathrm{Tr}\left(\mathcal{V}(\mathbf{M})\right)}{\mathrm{d}}\mathbf{M},$

where $\mathcal{V}(x)\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}\to{\mathbb{R}}$ is a real-valued analytic function
satisfying a particular growth condition as $x\to\infty$, and
$\mathcal{V}(\mathbf{M})$ is understood using the spectral theorem. The set of
such matrices is known in the literature as a _unitarily invariant ensemble_ ,
see e.g. [BL]*Chapter 1. Unitarily invariant ensembles can be viewed as
generalizations of the classical Gaussian Unitary Ensemble (GUE) and their
connection with orthogonal polynomials on the real line is classical [Mehta].

The particular case where $\mathcal{V}(x)=\frac{1}{2}x^{2}+\frac{u}{4}x^{4}$
is often referred to as the _Quartic One-Matrix Model_ 444The adjective “One-
Matrix” is there to distinguish this model from the quartic two-matrix model
which has also been the subject of many studies, but will not play a role here
and so we will drop this adjective in the sequel.. The eigenvalues of a matrix
drawn from this ensemble have a joint probability density

${\mathrm{d}}\mu_{nN}(z;u)=\dfrac{1}{\mathcal{Z}_{n}(u,N)}\prod^{n}_{1\leq
k<j\leq
n}(z_{k}-z_{j})^{2}\prod_{j=1}^{n}{\mathrm{e}}^{-N\left(\frac{1}{2}z_{j}^{2}+\frac{u}{4}z^{4}_{j}\right)}{\mathrm{d}}z_{1}\cdots{\mathrm{d}}z_{n}$

and $\mathcal{Z}_{n}(u,N)$ is the eigenvalue partition function given by

(1.23)
$\mathcal{Z}_{n}(u,N)=\int_{{\mathbb{R}}}\cdots\int_{\mathbb{R}}\prod^{n}_{1\leq
k<j\leq
n}(z_{k}-z_{j})^{2}\prod_{j=1}^{n}{\mathrm{e}}^{-N\left(\frac{1}{2}z_{j}^{2}+\frac{u}{4}z^{4}_{j}\right)}{\mathrm{d}}z_{1}\cdots{\mathrm{d}}z_{n}.$

The two partition functions in (1.22) and (1.23) are related by a classical
formula

$\dfrac{\mathcal{Z}_{n}(u,N)}{\tilde{\mathcal{Z}}_{n}(u,N)}=\dfrac{1}{\pi^{n(n-1)/2}}\prod_{j=1}^{n}j!.$

The _free energy_ of this ensemble is defined as

$\mathcal{F}_{n}(u,N)\mathrel{\mathop{\mathchar
58\relax}}=\dfrac{1}{n^{2}}\log\dfrac{\tilde{\mathcal{Z}}_{n}(u,N)}{\tilde{\mathcal{Z}}_{n}(0,N)}=\dfrac{1}{n^{2}}\log\dfrac{{\mathcal{Z}}_{n}(u,N)}{{\mathcal{Z}}_{n}(0,N)}.$

The quantity $\tilde{\mathcal{Z}}_{n}(0,N)$ is the partition function of the
GUE and is given by (see, e.g., [BL]*Page 2)

${\mathcal{Z}}_{n}(0,N)=\left(\dfrac{n}{N}\right)^{n^{2}}\dfrac{(2\pi)^{n/2}}{(2n)^{n^{2}/2}}\prod_{j=1}^{n}j!.$

The Quartic Model was first investigated in [BIPZ, BIZ], where it was observed
that when $n=N$, the free energy (1.25) admits an asymptotic expansion in
inverse powers of $N^{-2}$ and that the coefficient of $N^{-2g}$, as functions
of $u$, serve as generating functions for the number of four-valent graphs on
a Riemann surface of genus $g$. This expansion came to be known as the
_topological expansion_ , and its existence was proven in [MR1953782] for
matrix models with even polynomial potential.

While the model does not make sense when
$u\in{\mathbb{C}}\setminus{\mathbb{R}}_{\geq 0}$, one can arrive at a quantity
that can be readily analytically continued to the complex plane by starting
with $u>0$ and letting

$z_{j}=t^{1/2}\zeta_{j}\quad\text{and}\quad u=t^{-2},\ t>0.$

Then, we arrive at

${\mathcal{Z}}_{n}(u(t),N)=t^{n^{2}/2}\int_{{\mathbb{R}}}\cdots\int_{\mathbb{R}}\prod^{n}_{1\leq
k<j\leq
n}(\zeta_{k}-\zeta_{j})^{2}\prod_{j=1}^{n}{\mathrm{e}}^{-N\left(\frac{t}{2}\zeta_{j}^{2}+\frac{1}{4}\zeta^{4}_{j}\right)}{\mathrm{d}}\zeta_{1}\cdots{\mathrm{d}}\zeta_{n}.$

This expression is convergent for all $t\in{\mathbb{C}}$ and defines an entire
function. We will denote

(1.24) $Z_{n}(t,N)\mathrel{\mathop{\mathchar
58\relax}}=\int_{{\mathbb{R}}}\cdots\int_{\mathbb{R}}\prod^{n}_{1\leq k<j\leq
n}(\zeta_{k}-\zeta_{j})^{2}\prod_{j=1}^{n}{\mathrm{e}}^{-N\left(\frac{t}{2}\zeta_{j}^{2}+\frac{1}{4}\zeta^{4}_{j}\right)}{\mathrm{d}}\zeta_{1}\cdots{\mathrm{d}}\zeta_{n},$

and observe the simple relation
$\mathcal{Z}_{n}(u(t),N)=t^{n^{2}/2}Z_{n}(t,N)=t^{n^{2}/2}Z_{n}(t,N)$ where
$t=u^{-1/2}$. Defining a corresponding free energy by

(1.25) $F_{n}(t,N)\mathrel{\mathop{\mathchar
58\relax}}=\dfrac{1}{n^{2}}\log\dfrac{Z_{n}(t,N)}{{\mathcal{Z}}_{n}(0,N)}$

yields the identity

(1.26) $\mathcal{F}_{nN}(u)=\dfrac{\ln t}{2}+F_{n}(t,N),\quad t=u^{-1/2}.$

Using the classical Heine formula (see, e.g., [MR2191786]*Corollary 2.1.3) we
may write

(1.27) $Z_{n}(t,N)=N!H_{n-1}(t,N).$

Using this and Lemma 1.2, we arrive at the following corollary of Theorem 1.3.

######  Corollary 1.4.

Let $Z_{n,N}(t)$ be as in (1.24) and $\tau_{n,m,k},\sigma_{n,m,k}$ be Bäcklund
iterates of $\tau_{0,0,0},\sigma_{0,0,0}$ corresponding to the initial
solution of ($\textnormal{P}^{\textnormal{sym}}_{IV}$)

$(f_{0},f_{1},f_{2})=\left(0,\frac{\varphi^{\prime}(x)}{\varphi(x)},2x-\frac{\varphi^{\prime}(x)}{\varphi(x)}\right),\quad\varphi(x)={\mathrm{e}}^{x^{2}/2}D_{-1/2}(\sqrt{2}x),$

and the triple of parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,\frac{1}{2},\frac{1}{2})$, then

(1.28)
$Z_{k}(t,N)=\dfrac{N!\pi^{n}2^{2n(n-1)}}{N^{n(2n-1)}}{\mathrm{e}}^{\frac{nN}{2}t^{2}}\begin{cases}(\tau_{n-1,0,0}\tau_{n-1,-1,0})(N^{1/2}2^{-1}t),&k=2n,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{\pi^{\frac{1}{2}}2^{2n+\frac{1}{4}}}{N^{2n+\frac{1}{4}}}{\mathrm{e}}^{\frac{Nt^{2}}{4}}(\tau_{n,0,0}\tau_{n-1,-1,0})(N^{1/2}2^{-1}t),&k=2n+1.\end{cases}$

Furthermore, we have

(1.29)
$\displaystyle(2n)^{2}\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}F_{2n}(t,N)=nNt+\dfrac{N^{1/2}}{2}\left(\sigma_{n-1,0,0}(N^{1/2}2^{-1}t)+\sigma_{n-1,-1,0}(N^{1/2}2^{-1}t)\right),$

and

(1.30)
$(2n+1)^{2}\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}F_{2n+1}(t,N)=\left(n-\frac{1}{2}\right)Nt+\dfrac{N^{1/2}}{2}\left(\sigma_{n,0,0}(N^{1/2}2^{-1/2}t)+\sigma_{n-1,-1,0}(N^{1/2}2^{-1/2}t)\right),$

where $\sigma_{n,0,0}(x),\sigma_{n,-1,0}(x)$ satisfy (2.25) with parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=\left(-n,\frac{1}{2}+n,\frac{1}{2}\right)$,
$\left(-n,\frac{3}{2}+n,-\frac{1}{2}\right)$, respectively.

Seeing (1.27), perhaps it is not surprising that the asymptotic analysis done
in [BI, BGM] relied on the connection between the partition function and
orthogonal polynomials. We now briefly describe some of our results in this
direction.

###  1.4. Asymptotic analysis

Figure 2. Phase diagram for the geometry of $\mu_{t}$, courtesy of Roozbeh
Gharakhloo.

It is well understood that the large $n$ behavior of
$P_{n}(z;t,N),\gamma_{n}^{2}(t,N)$ (and consequently the partition function
$Z_{n}(t,N)$ and the free energy $F_{n}(t,N)$) heavily depends on the choice
of the complex parameter $t$, and particularly on the geometry of the
attracting set for the zeros of $P_{n}(z;t,N)$. The corresponding phase
diagram, shown in Figure 2, was obtained computationally in [BT], and a
rigorous description followed in [BGM]. The regions labeled one-, two-, and
three-cuts correspond to parameters $t$ where the measure $\mu_{t}$ defined by
$n^{-1}\sum_{z\mathrel{\mathop{\mathchar 58\relax}}\
P_{n}(z)=0}\delta_{z}\stackrel{{\scriptstyle*}}{{\to}}\mu_{t}$ is supported on
one, two, or three analytic arcs, respectively. Henceforth we will label these
regions $\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}$, respectively, and
delay a more precise definition to Section 4. The striking resemblance between
Figures 1, 2 is no accident! Heuristically, given a measure of the form
${\mathrm{e}}^{-NV(z)}{\mathrm{d}}z$, one can prove that non-Hermitian
orthogonal polynomials satisfy $\text{deg}P_{n}=n$ for all $n$ large enough in
the one-cut regime. This implies a lack of zeros of $H_{n-1}(t,N)$ (and hence
poles of $\gamma_{n}^{2}(t,N)$). In the setting of the Quartic Model,
asymptotics of $P_{n}$ have been obtained when $t\geq-2$ in [BI] and extended
to $t\in\mathcal{O}_{1}$ in [BGM]. It is slightly more surprising that the
region $\mathcal{O}_{2}$ is also free of zeros of $H_{n-1}(t,N)$, but this is
indeed the case and is basically due to the symmetry of (1.6). We will prove
this as well as asymptotics of $P_{n}(z;t,N),\gamma_{n}^{2}(t,N)$ in Section 5
(see Theorems 4.13, 4.17, respectively). A by-product of this analysis is that
$(f_{0},f_{1},f_{2})_{n,0,0}(x)$ is asymptotically free of poles in
$2^{-1/2}n^{1/2}\mathcal{O}_{2}$, as can be seen from the following corollary.

######  Corollary 1.5.

Let $(f_{0})_{n,m,k},(f_{1})_{n,m,k},(f_{2})_{n,m,k}$ be the Bäcklund iterates
of the solution of ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,\frac{1}{2},\frac{1}{2})$ given by

$(f_{0}(x),f_{1}(x),f_{2}(x))_{0,0,0}=\left(0,\frac{\varphi^{\prime}(x)}{\varphi(x)},2x-\frac{\varphi^{\prime}(x)}{\varphi(x)}\right),\quad\varphi(x)={\mathrm{e}}^{x^{2}/2}D_{-1/2}(\sqrt{2}x).$

Then, as $n\to\infty$,

(1.31) $\displaystyle 2^{1/2}n^{-1/2}(f_{0})_{n,0,0}(2^{-1/2}n^{1/2}x)$
$\displaystyle=x+\sqrt{x^{2}-4}+\mathcal{O}\left(\frac{1}{n}\right),$
$\displaystyle 2^{1/2}n^{-1/2}(f_{1})_{n,0,0}(2^{-1/2}n^{1/2}x)$
$\displaystyle=\dfrac{4}{x+\sqrt{x^{2}-4}}+\mathcal{O}\left(\frac{1}{n}\right),$
$\displaystyle 2^{1/2}n^{-1/2}(f_{2})_{n,0,0}(2^{-1/2}n^{1/2}x)$
$\displaystyle=\mathcal{O}\left(\frac{1}{n}\right),$

and

(1.32) $\displaystyle 2^{1/2}n^{-1/2}\sigma_{n-1,0,0}(2^{-1/2}n^{1/2}x)$
$\displaystyle=\dfrac{x-\sqrt{x^{2}-4}}{2}+\mathcal{O}\left(\frac{1}{n}\right),$
$\displaystyle 2^{1/2}n^{-1/2}\sigma_{n-1,-1,0}(2^{-1/2}n^{1/2}x)$
$\displaystyle=\dfrac{-x+\sqrt{x^{2}-4}}{2}+\mathcal{O}\left(\frac{1}{n}\right),$

locally uniformly for $x\in\mathcal{O}_{2}$. In all cases, $\sqrt{x^{4}-4}$ is
the branch analytic in ${\mathbb{C}}\setminus[-2,2]$ and satisfies
$\sqrt{x^{2}-4}\sim-x$ as $x\to\infty$.

######  Remark 1.6.

An analogous model, known as the Cubic Model was also investigated in [BIPZ]
and is connected with the enumeration of 3-valent graphs on Riemann surfaces.
This corresponds to the specialization
$\mathcal{V}(x)=\frac{1}{2}x^{2}+\frac{u}{3}x^{3}$ in (1.22), and results in
the replacement of the potential $V(z)$ by the polynomial
$\tilde{V}(z)=z^{3}/3+tz^{2}/2$. The astute reader might immediately object
since integrals in (1.5) and the total mass of (1.22) are not convergent. To
make sense of the model, one needs to deform the contour of integration to the
complex plane, at which point the model loses its interpretation as an
ensemble of random Hermitian matrices. However, the enumeration property still
holds as was shown in [BD]. Later, the connection between the partition
function and solutions of Painlevé-II was established in [BBDY] and further
expanded on in [BBDY2]. In many ways, this manuscript can be viewed as the
analog of [BBDY] for the Quartic Model.

We point out that the phase diagram contains three points that are of
particular interest in the mathematical physics literature because of the
presence of “critical phenomenon.” At the level of polynomials $P_{n}$, this
manifests itself in non-generic asymptotic formulas. Asymptotic formulas in
the double-scaling regime $t\to t_{cr}^{II}=-2$ were obtained in [MR1715324]
and involve particular solutions of Painlevé-II. An analogous result, but
involving solutions to Painlevé-I, when $t\to
t_{cr}^{I}=\pm{\mathrm{i}}\sqrt{12}$ were obtained in [DK].

In the finite $n$ setting, it was already known to Shohat [Shohat] that the
recurrence coefficients of polynomials orthogonal with respect to measures of
the form in (1.5) are solutions to integrable equations. Fokas, Its, and
Kitaev realized that the recurrence coefficients $\gamma_{n}^{2}(t,N)$ satisfy
both continuous and discrete Painlevé equations [FIK, FIK2] (see also Magnus’
work[Magnus, Magnus2]). The discrete equation in question is the discrete
version of Painlevé-I and is given by

(1.33)
$\gamma_{n}^{2}(t,N)\left(t+\gamma_{n-1}^{2}(t,N)+\gamma_{n}^{2}(t,N)+\gamma_{n+1}^{2}(t,N)\right)=\dfrac{n}{N}.$

This equation is known in the random matrix theory literature as the _string
equation_ , and is central to the analysis in [MR1715324], for example. While
we will not use it here, we do mention some of its applications to our
analysis in Section 6. A related family of orthogonal polynomials, known as
the semi-classical Laguerre polynomials, also appears in the literature. These
are polynomials $L_{n}(x;t,\ell)$ satisfying

(1.34)
$\int_{0}^{\infty}x^{k}L_{n}(x;t,\ell)x^{\ell}{\mathrm{e}}^{-\left(x^{2}+tx\right)}{\mathrm{d}}x=0\quad\text{for}\quad
k=0,1,...,n-1.$

These polynomials were the subject of study in [BV, FVZ, CJ], where the
connection between their recurrence coefficients and special function
solutions of ($\textnormal{P}_{IV}$) was established. It is not hard to
imagine that polynomials $L_{n}$ and $P_{n}$ are related, see Proposition A.2
in the Appendix.

###  1.5. Overview of the rest of the paper

In Section 2, we recall various properties of the symmetric and Hamiltonian
formulations of ($\textnormal{P}_{IV}$), their various Bäcklund
transformations, and define $\tau_{n,m,k}(x),\sigma_{n,m,k}(x)$. In Section 3,
we derive some auxiliary identities and prove Lemma 1.2 and Theorem 1.3. In
Section 4 we introduce various notions that will be essential to the
asymptotic analysis of $P_{n},\gamma_{n}^{2}$ and state asymptotic formulas
for $P_{n}$ in Theorem 4.13 and for $h_{n},\gamma_{n}^{2}$ in Theorem 4.17 and
deduce Corollary 1.5. Section 5 is dedicated to the proofs of Theorems 4.13,
4.17. We highlight various open questions and directions to extend this work
in Section 6. Finally, we record some observations regarding $P_{n}$ in the
Appendix A.

###  1.6. Acknowledgements

It is the author’s pleasure to thank Pavel Bleher and Maxim Yattselev for
introducing me to the Cubic and Quartic Models, Alfredo Deaño and Peter Miller
for introducing me to special function solutions of Painlevé equations and the
theory surrounding them, Pavel Bleher, Roozbeh Gharakhloo, and Ken McLaughlin
for patiently explaining their work [BGM] to me, and Maurice Duits, Nathan
Hayford, and Roger Van Peski for their helpful comments while writing this
manuscript. A special thank you to Andrei Prokhorov, who was my office mate at
University of Michigan and is infinitely willing to discuss mathematics.

##  2\. Painlevé-IV and its Bäcklund Transformations

We start this section by closely examining Theorem 1.1. While the forward
direction of (1.1) can be readily checked, it is less clear that given $u(x)$
which solves ($\textnormal{P}_{IV}$), one can construct a triple
$(f_{0}(x),f_{1}(x),f_{2}(x))$ which solves
($\textnormal{P}^{\textnormal{sym}}_{IV}$). This can be seen from the
Hamiltonian formulation of ($\textnormal{P}_{IV}$). Indeed, letting

(2.1)
$H_{IV}(x;p,q;\alpha_{0},\alpha_{1},\alpha_{2})=(2p-q-2x)pq-2\alpha_{1}p-\alpha_{2}q,$

we arrive at the system

(2.2)
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}q}{\operatorname{d\\!}{x}}}=\mathinner{\dfrac{\partial{}H_{IV}}{\partial{p}}}$
$\displaystyle=q(4p-q-2x)-2\alpha_{1},$ (2.3)
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}p}{\operatorname{d\\!}{x}}}=-\mathinner{\dfrac{\partial{}H_{IV}}{\partial{q}}}$
$\displaystyle=p(2q-2p+2x)+\alpha_{2}.$

One can readily check that eliminating $p(x)$ from (2.2), (2.3) reveals that
$q(x)$ solves ($\textnormal{P}_{IV}$) with parameters (1.4). On the other
hand, given a solution of ($\textnormal{P}_{IV}$), $u(x)$, the Hamiltonian
system $\\{H,p,q\\}$ is determined by $q(x)=u(x)$ and

(2.4)
$p(x)=\dfrac{1}{4q(x)}\left(2\alpha_{1}+2xq(x)+q^{2}(x)+q^{\prime}(x)\right).$

Letting

(2.5) $f_{1}(x)=-q(x),\quad f_{2}(x)=2p(x),\quad f_{0}(x)=2x+q(x)-2p(x),$

shows that (2.2), (2.3) are equivalent to the last two equations in
($\textnormal{P}^{\textnormal{sym}}_{IV}$). Since (2.5) clearly satisfy the
constraint (1.3), this implies that the functions (2.5) satisfy the first
equation in ($\textnormal{P}^{\textnormal{sym}}_{IV}$).

######  Remark 2.1.

The reader might wonder how, given a solution
$(u(x),\Theta_{0},\Theta_{\infty})$ one might determine the parameters
$\alpha_{0},\alpha_{1},\alpha_{2}$. From (1.1) and (1.4), it is clear that one
must choose $\Theta_{0}=\pm 2\alpha_{j}$ for some $j=0,1,2$. Once this choice
is made, the remaining parameters are uniquely determined by the conditions

$\displaystyle\alpha_{j-1}+\alpha_{j+1}$ $\displaystyle=1-\alpha_{j},$
$\displaystyle\alpha_{j-1}-\alpha_{j+1}$ $\displaystyle=2\Theta_{\infty}-1.$

Changing our choice $\alpha_{j}\mapsto-\alpha_{j}$ amounts to applying the
Bäcklund transformation $s_{j}^{-1}$ which is introduced in the next section.

###  2.1. Bäcklund transformations for the system
($\textnormal{P}^{\textnormal{sym}}_{IV}$)

Much of the discussion below is contained in [NY], see also the already
mentioned [FW]. We begin by recording an obvious symmetry

(2.6) $\pi(f_{j}(x))=f_{j-1}(x),\quad\pi(\alpha_{j})=\alpha_{j-1}$

leaves the system ($\textnormal{P}^{\textnormal{sym}}_{IV}$) and condition
(1.3) invariant.

######  Remark 2.2.

Due to this symmetry of ($\textnormal{P}^{\textnormal{sym}}_{IV}$), a given
solution of ($\textnormal{P}_{IV}$) can be identified with the negative of any
one function in the triple $(f_{0}(x),f_{1}(x),f_{2}(x))$. Moving forward, we
will always write $u(x)=-f_{1}(x)$ to avoid ambiguity.

More nontrivial transformations exist, and they are summarized by the
proposition below.

######  Proposition 2.3.

Let $\\{\alpha_{j}\\}_{j=0}^{2}$ and $\\{f_{j}(x)\\}_{j=0}^{2}$ be as above.
Define the Cartan matrix $A$ and orientation matrix $U$ by

(2.7) $\mathbf{A}=[a_{ij}]_{i,j=0}^{2}=\left[\begin{array}[]{rrr}2&-1&-1\\\
-1&2&-1\\\
-1&-1&2\end{array}\right],\quad\mathbf{U}=[u_{ij}]_{i,j=0}^{2}=\left[\begin{array}[]{rrr}0&1&-1\\\
-1&0&1\\\ 1&-1&0\end{array}\right],$

and consider the functions given by transformations

(2.8) $g_{ij}(x)\mathrel{\mathop{\mathchar
58\relax}}=s_{i}(f_{j}(x))=f_{j}(x)+u_{ij}\dfrac{2\alpha_{i}}{f_{i}(x)},\quad
i,j\in\\{0,1,2\\}.$

Then the triple $(g_{i0},g_{i1},g_{i2})$ satisfy (1.3) and solve
($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\beta_{i0},\beta_{i1},\beta_{i2})$ given by555We slightly abuse notation by
using $s_{i}$ to denote the transformation of both the functions $f_{j}$ and
parameters $\alpha_{j}$.

(2.9) $\beta_{ij}=s_{i}(\alpha_{j})\mathrel{\mathop{\mathchar
58\relax}}=\alpha_{j}-a_{ij}\alpha_{i}.$

######  Remark 2.4.

One can observe that the lines $\alpha_{j}=k\in{\mathbb{Z}}$ define a
triangular lattice on the plane (1.2) which is associated with an
$A_{2}^{(1)}$ root system whose roots, with a slight abuse of notation, are
given by

(2.10)
$\boldsymbol{\alpha}_{0}=\begin{bmatrix}0&-1&1\end{bmatrix}^{T},\quad\boldsymbol{\alpha}_{1}=\begin{bmatrix}1&0&-1\end{bmatrix}^{T},\quad\boldsymbol{\alpha}_{2}=\begin{bmatrix}-1&1&0\end{bmatrix}^{T}.$

The reflection across $\boldsymbol{\alpha}_{j}$ is then given by

(2.11)
$s_{i}(\boldsymbol{\alpha}_{j})=\boldsymbol{\alpha}_{j}-a_{ij}\boldsymbol{\alpha}_{i},$

where $a_{ij}$ are the elements of $\mathbf{A}$ in (2.7). This connection is
the source of the names of the matrices $\mathbf{A},\mathbf{U}$, but will not
be used in the sequel.

It can be checked that $s_{j}$, when acting on triples
$(f_{0}(x),f_{1}(x),f_{2}(x))$, are involutions. Indeed,

(2.12) $\displaystyle s_{0}^{2}(f_{0}(x),f_{1}(x),f_{2}(x))$
$\displaystyle=s_{0}\left(f_{0}(x),f_{1}(x)+\dfrac{2\alpha_{0}}{f_{0}(x)},f_{2}(x)-\dfrac{2\alpha_{0}}{f_{0}(x)}\right)$
(2.13)
$\displaystyle=\left(f_{0}(x),f_{1}(x)+\dfrac{2\alpha_{0}}{f_{0}(x)}+\dfrac{2s_{0}(\alpha_{0})}{f_{0}(x)},f_{2}(x)-\dfrac{2\alpha_{0}}{f_{0}(x)}-\dfrac{2s_{0}(\alpha_{0})}{f_{0}(x)}\right)$
(2.14) $\displaystyle=(f_{0}(x),f_{1}(x),f_{2}(x)).$

In fact, slightly more involved (but direct) computations reveal that
$s_{j},\sigma$ satisfy666This makes the group
$\widetilde{W}\mathrel{\mathop{\mathchar 58\relax}}=\langle
s_{0},s_{1},s_{2},\pi\rangle$ the _extended affine Weyl group of type_
$A_{2}^{(1)}$.

(2.15) $s_{j}^{2}=1,\quad(s_{j}s_{j+1})^{3}=1,\quad\pi^{3}=1,\quad\pi
s_{j}=s_{j+1}\pi.$

Moving forward, we will think of $s_{i},\sigma$ as acting on a triples of
functions where we denote by

$s_{j}(f_{0}(x),f_{1}(x),f_{2}(x))\mathrel{\mathop{\mathchar
58\relax}}=\left(s_{j}(f_{0}(x)),s_{j}(f_{1}(x)),s_{j}(f_{2}(x))\right)$

a triple which solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with
parameters

$s_{j}(\alpha_{0},\alpha_{1},\alpha_{2})=(s_{j}(\alpha_{0}),s_{j}(\alpha_{1}),s_{j}(\alpha_{2})).$

We now construct three transformations:

(2.16) $T_{1}\mathrel{\mathop{\mathchar 58\relax}}=\pi s_{2}s_{1},\quad
T_{2}\mathrel{\mathop{\mathchar 58\relax}}=s_{1}\pi s_{2},\quad
T_{3}\mathrel{\mathop{\mathchar 58\relax}}=s_{2}s_{1}\pi.$

The action of $T_{i}$ on parameters $\alpha_{j}$ is summarized in the Table 1.
One might guess from this table that (with indices interpreted cyclically)

(2.17) $T_{j}T_{j+1}T_{j+2}=1\quad\text{for}\quad j=1,2,3,$

and this is indeed the case as can be verified using (2.15). We verify one
such identity and leave the rest to the reader:

$T_{1}T_{2}T_{3}=\pi s_{2}s_{1}s_{1}\pi s_{2}s_{2}s_{1}\pi=\pi s_{2}\pi
s_{1}\pi=s_{0}\pi^{3}s_{0}=1.$

Similarly, one can verify that $T_{i}T_{j}=T_{j}T_{i}$ for any
$i,j\in\\{1,2,3\\}$ and the identities

(2.18) $T_{j+1}=\pi T_{j}\pi^{-1}.$

Since $s_{j},\pi$ are invertible, it follows that $T_{j}$ are invertible and

(2.19) $T_{1}^{-1}=s_{1}s_{2}\pi^{-1},\quad
T_{2}^{-1}=s_{2}\pi^{-1}s_{1},\quad T_{3}^{-1}=\pi^{-1}s_{1}s_{2}.$

We now investigate the effect of $T_{i}$ on a given solution of
($\textnormal{P}_{IV}$); since the transformations $T_{j}$ are related by
(2.18), we focus our attention on $T_{1}$, which will generate the family of
solutions we are interested in. Given a triple $(f_{0}(x),f_{1}(x),f_{2}(x))$
which solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$), then

(2.20) $\displaystyle T_{1}$ $\displaystyle(f_{0}(x),f_{1}(x),f_{2}(x))=\pi
s_{2}\left(g_{10}(x),g_{11}(x),g_{12}(x)\right)$
$\displaystyle=\pi\left(g_{10}(x)+u_{20}\dfrac{2s_{1}(\alpha_{2})}{g_{12}(x)},g_{11}(x)+u_{21}\dfrac{2s_{1}(\alpha_{2})}{g_{12}(x)},g_{12}(x)+u_{22}\dfrac{2s_{1}(\alpha_{2})}{g_{12}(x)}\right)$
$\displaystyle=\pi\left(g_{10}(x)+\dfrac{2(\alpha_{1}+\alpha_{2})}{g_{12}(x)},g_{11}(x)-\dfrac{2(\alpha_{1}+\alpha_{2})}{g_{12}(x)},g_{12}(x)\right)$
$\displaystyle=\left(g_{12}(x),g_{10}(x)+\dfrac{2(\alpha_{1}+\alpha_{2})}{g_{12}(x)},g_{11}(x)-\dfrac{2(\alpha_{1}+\alpha_{2})}{g_{12}(x)}\right)$
$\displaystyle=\left(f_{2}(x)+\dfrac{2\alpha_{1}}{f_{1}(x)},f_{0}(x)-\dfrac{2\alpha_{1}}{f_{1}(x)}+\dfrac{2(\alpha_{1}+\alpha_{2})f_{1}(x)}{(f_{1}f_{2})(x)+2\alpha_{1}},f_{1}(x)-\dfrac{2(\alpha_{1}+\alpha_{2})f_{1}(x)}{(f_{1}f_{2})(x)+2\alpha_{1}}\right).$

solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0}-1,\alpha_{1}+1,\alpha_{2})$. In a similar fashion, one can check
that

(2.21) $T_{1}^{-1}(f_{0}(x),f_{1}(x),f_{2}(x))\\\
=\left(f_{1}(x)+\dfrac{2\alpha_{0}}{f_{0}(x)}-\dfrac{2(\alpha_{0}+\alpha_{2})f_{0}(x)}{(f_{0}f_{2})(x)-2\alpha_{0}},f_{2}(x)-\dfrac{2\alpha_{0}}{f_{0}(x)},f_{0}(x)+\dfrac{2(\alpha_{0}+\alpha_{2})f_{0}(x)}{(f_{0}f_{2})(x)-2\alpha_{0}}\right)$

solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0}+1,\alpha_{1}-1,\alpha_{2})$. Henceforth, we will use the notation
$T_{1}(f_{j}(x)),T_{1}^{-1}(f_{j}(x))$ to denote the $j$th entry of the hand
side of (2.20), (2.21), respectively.

######  Remark 2.5.

It can occur (see Example 1 below) that either $f_{1}(x)=0$ or
$(f_{1}f_{2})(x)+2\alpha_{1}=0$ holds identically, which causes an
indeterminacy in (2.20). We highlight that the requirement that
$T_{1}(f_{0},f_{1},f_{2})$ be determinate is really a condition on $f_{1}$.
Indeed, given that $(f_{0},f_{1},f_{2})$ satisfies
($\textnormal{P}^{\textnormal{sym}}_{IV}$), (1.3), $f_{2}$ is given in terms
of $f_{1}$ as (this is exactly (2.4))

$f_{2}(x)=\dfrac{1}{2f_{1}(x)}\left(-2\alpha_{1}+2xf_{1}(x)-f_{1}^{2}(x)+f_{1}^{\prime}(x)\right).$

Hence, for $T_{1}(f_{0},f_{1},f_{2})$ to be determinate we must require
$f_{1}(x)\not\equiv 0$ and

$f_{1}^{\prime}(x)-f_{1}^{2}(x)+2xf_{1}(x)+2\alpha_{1}\not\equiv 0.$

We will later show (see Proposition 2.12 below) that the particular family of
solutions we study in this paper is determinate.

###  2.2. Bäcklund transformations for ($\textnormal{P}_{IV}$)

On the level of the parameters $(\Theta_{0},\Theta_{\infty})$, if
$u(x)=-f_{1}(x)$ solves ($\textnormal{P}_{IV}$) with parameters
$\alpha=\alpha_{0}-\alpha_{2}=2\Theta_{\infty}-1$ and
$\beta=-2\alpha_{1}^{2}=-8\Theta_{0}^{2}$, then letting
$\alpha_{1}=2\Theta_{0}$ and $\alpha_{0},\alpha_{2}$ be the corresponding
parameters (see Remark 2.1) then $u_{\searrow}(x)\mathrel{\mathop{\mathchar
58\relax}}=-T_{1}(f_{1}(x))$ solves ($\textnormal{P}_{IV}$) with parameters

${\Theta}_{\infty,\searrow}=\Theta_{\infty}-\frac{1}{2},\quad{\Theta}_{0,\searrow}=\Theta_{0}+\frac{1}{2}.$

It follows from (2.20) that

(2.22)
$u_{\searrow}(x)=\dfrac{(2\alpha_{1})^{2}-8(\alpha_{1}+\alpha_{2})u^{2}(x)-4x^{2}u^{2}(x)-4xu^{3}(x)-u^{4}(x)-4\alpha_{1}u^{\prime}(x)+u^{\prime}(x)^{2}}{2u(x)(-2\alpha_{1}+2xu(x)+u^{2}(x)+u^{\prime}(x))},$

which, upon recalling the definitions of $\alpha_{i}$’s and the constraint
(1.2), agrees with [BM]*Equation 3.50. If, on the other hand, we set
$\alpha_{1}=-2\Theta_{0}$ and $\alpha_{0},\alpha_{2}$ be the corresponding
parameters, then $u_{\swarrow}(x)\mathrel{\mathop{\mathchar
58\relax}}=-T_{1}(f_{1}(x))$ solves ($\textnormal{P}_{IV}$) with parameters

$\Theta_{\infty,\swarrow}=\Theta_{\infty}-\frac{1}{2},\quad\Theta_{0,\swarrow}=\Theta_{0}-\frac{1}{2}.$

The function $u_{\swarrow}$ is again given by the right hand side of (2.22)
which, upon recalling the definitions of $\alpha_{i}$’s and the constraint
(1.2), agrees with [BM]*Equation 3.49. We can extract two more Bäcklund
transformations for ($\textnormal{P}_{IV}$) from the expression (2.21);
letting $\alpha_{1}=2\Theta_{0}$, $\alpha_{0},\alpha_{2}$ be the corresponding
parameters, and $u_{\nwarrow}(x)\mathrel{\mathop{\mathchar
58\relax}}=-T_{1}^{-1}(f_{1})$, then it follows from (2.21) that

(2.23)
$u_{\nwarrow}(x)=\dfrac{(2\alpha_{1})^{2}+8\alpha_{0}u^{2}(x)-4x^{2}u^{2}(x)-4xu^{3}(x)-u^{4}(x)+4\alpha_{1}u^{\prime}(x)+u^{\prime}(x)^{2}}{2u(x)(-2\alpha_{1}+2xu(x)+u^{2}(x)-u^{\prime}(x))},$

which agrees with [BM]*Equation 3.51. In a similar fashion to the previous
paragraph, if instead we set $\alpha_{1}=-2\Theta_{0}$,
$\alpha_{0},\alpha_{2}$ be the corresponding parameters, and
$u_{\nearrow}(x)\mathrel{\mathop{\mathchar 58\relax}}=-T_{1}^{-1}(f_{1})$,
then we find that $u_{\nearrow}(x)$ is given by the right hand side of (2.23).
Plugging in the the definitions of $\alpha_{i}$’s and the constraint (1.2), we
arrive at [BM]*Equation 3.48.

######  Remark 2.6.

We pause here to highlight the result of the above calculation. A given
solution of ($\textnormal{P}_{IV}$) $(u(x),\Theta_{0},\Theta_{\infty})$ can be
embedded into two distinct pairs of solutions of
($\textnormal{P}^{\textnormal{sym}}_{IV}$) by setting $u(x)=-f_{1}(x)$ and
making the choice $\alpha_{1}=\pm 2\Theta_{0}$. This then fixes the triple
$(f_{0},f_{1},f_{2})$ uniquely (see Remark 2.1). Of course these are not the
only triples where the function $u(x)$ appears as $-f_{1}(x)$ since one could
act on this triple by $s_{1}$. Furthermore, one could write triples for which
$u(x)=-f_{2}(x)$ or $u(x)=-f_{0}(x)$ by acting on $(f_{0},f_{1},f_{2})$ by
$\pi^{\pm 1}$. As stated in Remark 2.2, given a solution
$(u(x),\Theta_{0},\Theta_{\infty})$, we will prefer to set $u(x)=-f_{1}(x)$.

###  2.3. The $\sigma$-form of Painlevé-IV

Recalling (2.1), (2.5), we define
$\sigma(x;\alpha_{0},\alpha_{1},\alpha_{2})\equiv\sigma(x)$ by

(2.24)
$\sigma(x)=H(x;p(x),q(x);\alpha_{0},\alpha_{1},\alpha_{2})=\frac{1}{2}(f_{0}f_{1}f_{2})(x)+\alpha_{2}f_{1}(x)-\alpha_{1}f_{2}(x).$

It follows (see [FW]*Proposition 4) that $\sigma(x)$ satisfies the Jimbo-Miwa-
Okamoto $\sigma$-form of $\textnormal{P}_{IV}$:

(2.25)
$\left(\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}\sigma}{\operatorname{d\\!}{x^{2}}}}\right)^{2}-4\left(x\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}-\sigma(x)\right)+4\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}\left(\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}+2\alpha_{1}\right)\left(\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}-2\alpha_{2}\right)=0.$

######  Remark 2.7.

The function $\sigma(x)$ can be directly related to a solution of
($\textnormal{P}_{IV}$). In one direction, given
$(u(x),\Theta_{0},\Theta_{\infty})$ which solves ($\textnormal{P}_{IV}$),
using (2.4) and (2.5) immediately yields an expression for $\sigma(x)$ in
terms of $u(x)$. On the other hand, letting $p,q$ be as in (2.5), it follows
from (2.2), (2.3) and the last two equations in
($\textnormal{P}^{\textnormal{sym}}_{IV}$) that

(2.26)
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}$
$\displaystyle=f_{1}f_{2},$ (2.27)
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}\sigma}{\operatorname{d\\!}{x^{2}}}}$
$\displaystyle=f_{1}f_{2}(f_{2}-f_{1})+2\alpha_{2}f_{1}+2\alpha_{1}f_{2}.$

Using (2.26), (2.24) in (2.27) yields

(2.28)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}\sigma}{\operatorname{d\\!}{x^{2}}}}=2\alpha_{2}f_{1}(x)-(\sigma
f_{1})(x)+\dfrac{1}{f_{1}(x)}\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}\left(\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}+2\alpha_{1}\right).$

To calculate the last term, we note that using (2.24) and (1.3) in (2.24)
implies

(2.29)
$\dfrac{1}{f_{1}(x)}\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}\left(\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}+2\alpha_{1}\right)=(2x-f_{1}(x))\mathinner{\dfrac{\operatorname{d\\!}{}\sigma}{\operatorname{d\\!}{x}}}+2\alpha_{2}f_{1}(x)-2\sigma(x).$

Putting this together yields a solution of ($\textnormal{P}_{IV}$) given by

$u(x)=-f_{1}(x)=\dfrac{\sigma^{\prime\prime}(x)-2x\sigma^{\prime}(x)+2\sigma(x)}{2\sigma^{\prime}(x)-4\alpha_{2}}.$

In the sequel, given a Bäcklund transformation $B\in\langle
s_{0},s_{1},s_{2},\pi\rangle$, we define (sticking with the abuse of notation
in Footnote 5)

(2.30) $B(\sigma(x))\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\left(B(f_{0})B(f_{1})B(f_{2})\right)(x)+B(\alpha_{2})B(f_{1})(x)-B(\alpha_{1})B(f_{2})(x),$

A direct computation using (2.24), (1.2), and (1.3) shows that

(2.31) $\displaystyle s_{0}(\sigma(x))$
$\displaystyle=\sigma(x)+\dfrac{2\alpha_{0}}{f_{0}(x)},$ $\displaystyle
s_{1}(\sigma(x))$ $\displaystyle=\sigma(x)+2\alpha_{1}x,$ $\displaystyle
s_{2}(\sigma(x))$ $\displaystyle=\sigma(x)-2\alpha_{2}x,$
$\displaystyle\pi(\sigma(x))$ $\displaystyle=\sigma(x)-f_{1}(x)+2\alpha_{1}x,$
$\displaystyle\pi^{-1}(\sigma(x))=\pi^{2}(\sigma(x))$
$\displaystyle=\sigma(x)+f_{2}(x)-2\alpha_{2}x.$

Using these we find that

(2.32) $\displaystyle T_{1}(\sigma(x))$
$\displaystyle=\sigma(x)-s_{2}s_{1}(f_{1}(x))=\sigma(x)-T_{1}(f_{2}(x)),$
(2.33) $\displaystyle T_{1}^{-1}(\sigma(x))$
$\displaystyle=\sigma(x)+f_{2}(x).$

While they can be computed directly using (2.31), relation (2.32) allows us to
deduce analogous identities for $T_{2},T_{3}$ by using (2.31) and (2.18) and
we find777One arrives at (2.34) by directly using (2.31); if instead one
starts from (2.32) it is necessary to use the identity
$T_{2}(f_{2})+T_{2}(f_{0})=f_{2}+s_{2}(f_{1})$.

(2.34) $\displaystyle T_{2}(\sigma(x))-\sigma(x)=2x-s_{2}(f_{1}(x)),$ (2.35)
$\displaystyle T_{3}(\sigma(x))-\sigma(x)=-f_{1}(x).$

The final object which will be important for our investigation is the so-
called $\tau$-function, defined up to a constant of integration to be fixed
later (see Proposition 2.8) by

(2.36)
$\sigma(x)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\tau(x).$

The Bäcklund iterates of $\tau$, $B(\tau(x))$, $B\in\langle
s_{0},s_{1},s_{2},\pi\rangle$, are defined analogously to (2.30) by

(2.37)
$B(\sigma(x))=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log
B(\tau(x)).$

Plugging this into (2.32) we see that

(2.38)
$-T_{1}(f_{2}(z))=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log\left(\dfrac{T_{1}(\tau(x))}{\tau(x)}\right).$

A remarkable fact first observed by Okamoto, cf. [O]*Proposition 3.5, is that
the difference $T_{1}(f_{2}(x))-f_{2}(x)$ can be written as a derivative. More
precisely, the third entry in the final expression in (2.20) and
($\textnormal{P}^{\textnormal{sym}}_{IV}$) imply

(2.39) $\displaystyle T_{1}(f_{2}(x))-f_{2}(x)$
$\displaystyle=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\left((f_{1}f_{2})(x)+2\alpha_{1}\right),$

and by applying identity (2.39) to the triple $T_{1}(f_{0},f_{1},f_{2})$ we
get

(2.40)
$f_{2}(x)-T_{1}^{-1}(f_{2}(x))=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\left((f_{0}f_{2})(x)-2\alpha_{0}\right).$

We record one more identity of this type here; by recalling that (2.18) we
arrive at

(2.41)
$f_{1}(x)-T_{3}^{-1}(f_{1}(x))=-\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\left((f_{1}f_{2})(x)-2\alpha_{2}\right).$

These identities will be crucial ingredients in deriving the Toda equations.

###  2.4. Bäcklund iterates and the Toda equation(s)

We will be interested in families of solutions of ($\textnormal{P}_{IV}$)
generated by iterating $T_{1},T_{2},T_{3}$ and their inverses. To this end,
recall our notation $g_{n,m,k}(x)\mathrel{\mathop{\mathchar
58\relax}}=T_{3}^{k}T_{2}^{m}T_{1}^{n}(g_{0}(x))$ for all relevant functions
and $n,m,k\in{\mathbb{Z}}$ ($T_{j}$’s commute so the order of application is
immaterial). When iterating a single $T_{j}$, we will slightly abuse notation
(hopefully without causing confusion) by dropping the fixed indices.

######  Proposition 2.8 ([O], see also [FW]*Proposition 5).

Fix $m,k\in{\mathbb{Z}}$. For all $n\in{\mathbb{Z}}$, let
$\tilde{\tau}_{n,m,k}(x)\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{e}}^{x^{2}(\alpha_{1}+n)}\tau_{n,m,k}(x)$. Then, functions
$\\{\tilde{\tau}_{n}\\}_{n\in{\mathbb{Z}}}$ satisfy the Toda equation:

(2.42)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log\tilde{\tau}_{n,m,k}(x)=\dfrac{\tilde{\tau}_{n+1,m,k}(x)\tilde{\tau}_{n-1,m,k}(x)}{\tilde{\tau}^{2}_{n,m,k}(x)}.$

###### Proof.

We start by writing $\tilde{\tau}_{n}(x)=\tilde{\tau}_{n,m,k}(x)$ noting that

(2.43)
$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{\tilde{\tau}_{n+1}(x)\tilde{\tau}_{n-1}(x)}{\tilde{\tau}^{2}_{n}(x)}=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{{\tau}_{n+1}(x){\tau}_{n-1}(x)}{{\tau}^{2}_{n}(x)}=\sigma_{n+1}(x)+\sigma_{n-1}(x)-2\sigma_{n}(x)\\\
=(\sigma_{n+1}(x)-\sigma_{n}(x))-(\sigma_{n}(x)-\sigma_{n-1}(x)).$

Now, writing $\sigma_{n+1}=T_{1}(\sigma_{n})$ and
$\sigma_{n-1}=T_{1}^{-1}(\sigma_{n})$ and recalling (2.32), (2.33), and (2.39)
yields

(2.44)
$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{\tilde{\tau}_{n+1}(x)\tilde{\tau}_{n-1}(x)}{\tilde{\tau}^{2}_{n}(x)}=f_{2,n}(x)-T_{1}(f_{2,n}(x))=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log((f_{1,n}f_{2,n})+2(\alpha_{1}+n)).$

Finally, it follows from (2.26) that

(2.45)
$(f_{1,n}f_{2,n})+2(\alpha_{1}+n)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}(\sigma_{n}+2x(\alpha_{1}+n))=\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log\left(\tau_{n}{\mathrm{e}}^{x^{2}(\alpha_{1}+n)}\right).$

Plugging (2.45) into (2.44), integrating and exponentiating yields

(2.46)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log\tilde{\tau}_{n}(x)=K_{n,m,k}\dfrac{\tilde{\tau}_{n+1}(x)\tilde{\tau}_{n-1}(x)}{\tilde{\tau}^{2}_{n}(x)},\quad
K_{n,m,k}\in{\mathbb{C}}\setminus\\{0\\}.$

Now, we recall that $\tau_{n}$ is defined by (2.36) up to a constant of
integration, and so we choose this constant so that $K_{n,m,k}=1$ for all
$n\geq 0$. ∎

An even simpler statement holds for tau functions resulting from iterating
$T_{2}$.

######  Proposition 2.9.

Fix $n,k\in{\mathbb{Z}}$. For all $n\in{\mathbb{Z}}$, the functions
$\\{{\tau}_{n,m,k}\\}_{m\in{\mathbb{Z}}}$, there exist constants
$K_{n,m,k}\in{\mathbb{C}}$ such that

(2.47)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log{\tau}_{n,m,k}(x)=K_{n,m,k}\dfrac{{\tau}_{n,m+1,k}(x){\tau}_{n,m-1,k}(x)}{{\tau}^{2}_{n,m,k}(x)}.$

###### Proof.

Beginning in the same was as in the proof of Proposition 2.8, we write

(2.48)
$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{{\tau}_{m+1}(x){\tau}_{m-1}(x)}{{\tau}^{2}_{m}(x)}=(\sigma_{m+1}(x)-\sigma_{m}(x))-(\sigma_{m}(x)-\sigma_{m-1}(x)).$

Writing $\sigma_{m+1}(x)=T_{2}(\sigma_{m}(x)),\
\sigma_{m-1}(x)=T_{2}^{-1}(\sigma_{m}(x))$, we now need identities analogous
to (2.32), (2.33). Applying (2.34) to $\sigma_{m},T_{2}^{-1}(\sigma_{m})$ we
find

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{{\tau}_{m+1}(x){\tau}_{m-1}(x)}{{\tau}^{2}_{m}(x)}=s_{2}\left(T_{2}^{-1}(f_{1,m})\right)-s_{2}(f_{1,m})=T_{3}^{-1}(s_{2}(f_{1,m}))-s_{2}(f_{1,m}),$

where the last equality follows by definition (2.16) of $T_{2},T_{3}$.
Applying identity (2.41) with the triple $s_{2}(f_{0,m},f_{1,m},f_{2,m})$ and
recalling (2.26), (2.36), we have

(2.49)
$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\dfrac{{\tau}_{m+1}(x){\tau}_{m-1}(x)}{{\tau}^{2}_{m}(x)}=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log(f_{1,m}f_{2,m})=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\mathinner{\dfrac{\operatorname{d\\!}{}\sigma_{m}}{\operatorname{d\\!}{x}}}=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log\tau_{m}(x).$

Integrating and exponentiating yields the result. ∎

Finally, for the sake of completeness, we record the Toda equation for
iterates of $T_{3}$ and leave the proof (which essentially relies on (2.35)
and (2.41)) as an exercise.

######  Proposition 2.10.

Fix $n,m\in{\mathbb{Z}}$ and let
$\hat{\tau}_{n,m,k}(x)\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{e}}^{x^{2}(k-\alpha_{2})}{\tau}_{n,m,k}$. For all
$k\in{\mathbb{Z}}$, the functions
$\\{{\hat{\tau}}_{n,m,k}\\}_{k\in{\mathbb{Z}}}$, there exist constants
$C_{n,m,k}\in{\mathbb{C}}$ such that

(2.50)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{2}}}{\operatorname{d\\!}{x^{2}}}}\log{\hat{\tau}}_{n,m,k}(x)=C_{n,m,k}\dfrac{{\hat{\tau}}_{n,m+1,k}(x){\hat{\tau}}_{n,m-1,k}(x)}{{\hat{\tau}}^{2}_{n,m,k}(x)}.$

######  Remark 2.11.

We pause here to make a few observations that will be useful in the sequel. It
is known [GLS]*Theorem 4.1 that solutions of ($\textnormal{P}_{IV}$) can be
continued to meromorphic function in ${\mathbb{C}}$. In particular,
$\sigma(x)$ is a meromorphic function in ${\mathbb{C}}$. Furthermore, a
dominant balance argument reveals that if $x_{0}$ is a pole of $\sigma(x)$,
then

(2.51)
$\sigma(x)=\dfrac{1}{x-x_{0}}+\sigma_{0}-\dfrac{2}{3}(\alpha_{2}-\alpha_{1})(x-x_{0})+\mathcal{O}((x-x_{0})^{2})\quad\text{as}\quad
x\to x_{0},\ \sigma_{0}\in{\mathbb{C}}.$

In view of (2.36), (2.51) implies that roots of $\tau(x)$ are simple.
Furthermore, since $\tilde{\tau}_{n,m,k}$ and $\tau_{n,m,k}$ share the same
roots, roots of $\tilde{\tau}_{n,m,k}$ and $\hat{\tau}_{n,m,k}$ are simple.
Writing (2.42) in the form

$\tilde{\tau}_{n,m,k}^{\prime\prime}\tilde{\tau}_{n,m,k}-(\tilde{\tau}_{n,m,k}^{\prime})^{2}=\tilde{\tau}_{n+1,m,k}\tilde{\tau}_{n-1,m,k},$

one can see that $\tilde{\tau}_{n,m,k}(x),\tilde{\tau}_{n+1,m,k}(x)$ (and
hence ${\tau}_{n,m,k}(x),{\tau}_{n+1,m,k}(x)$) do not share any common roots.
Similar considerations imply the analogous fact about
$\tau_{n,m,k},\tau_{n,m+1,k}$ and $\tau_{n,m,k},\tau_{n,m,k+1}$.

###  2.5. Solutions in terms of special functions

It was already known (see, e.g., [O]*Section 5) that ($\textnormal{P}_{IV}$)
possesses solutions which can be written in terms of classical special
function. One can see this by requiring that ($\textnormal{P}_{IV}$) be
compatible with a Riccati equation. Indeed, suppose we require that $u(x),$
besides satisfying ($\textnormal{P}_{IV}$) also satisfies

(2.52)
$\mathinner{\dfrac{\operatorname{d\\!}{}u}{\operatorname{d\\!}{x}}}=k_{2}(x)u^{2}(x)+k_{1}(x)u(x)+k_{0}(x).$

Then, differentiating this equation and using ($\textnormal{P}_{IV}$) and
comparing coefficients of the indicated terms implies

$\displaystyle u^{3}$ $\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle k_{2}^{2}(x)$ $\displaystyle=1,$ $\displaystyle u^{2}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$ $\displaystyle
2k_{2}^{\prime}(x)+4k_{1}(x)k_{2}(x)-8x$ $\displaystyle=0,$ $\displaystyle u$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$ $\displaystyle
2k_{1}^{\prime}(x)+k_{1}^{2}(x)+2k_{0}(x)k_{2}(x)-4x^{2}+4\alpha$
$\displaystyle=0,$ const. $\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle k_{0}^{\prime}(x)$ $\displaystyle=0,$ $\displaystyle u^{-1}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}$ $\displaystyle
k_{0}^{2}(x)+2\beta$ $\displaystyle=0.$

From these equations we deduce that for $\varepsilon,\upsilon$ satisfying
$\varepsilon^{2}=1=\upsilon^{2}$,

$k_{0}(x)=4\upsilon\Theta_{0},\qquad k_{1}(x)=2\varepsilon x,\qquad
k_{2}(x)\equiv\varepsilon.$

The equation arising from the coefficient of $u$ can now be interpreted as
conditions on the values of $\Theta_{0},\Theta_{\infty}$ which read

(2.53) $2\Theta_{\infty}+2\varepsilon\upsilon\Theta_{0}=1-\varepsilon,$

which yields four well-known conditions, namely

(2.54) $\displaystyle\varepsilon=1$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}\quad\Theta_{\infty}\pm\Theta_{0}=0,$ (2.55)
$\displaystyle\varepsilon=-1$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}\quad\Theta_{\infty}\pm\Theta_{0}=1.$

Under these conditions (2.52) becomes

$\mathinner{\dfrac{\operatorname{d\\!}{}u}{\operatorname{d\\!}{x}}}=\varepsilon
u^{2}(x)+2\varepsilon xu(x)+4\upsilon\Theta_{0}.$

This can be linearized by letting $u(x)=\mathrel{\mathop{\mathchar
58\relax}}-\varepsilon\varphi^{\prime}(x)/\varphi(x)$ and yields

(2.56)
$\varphi^{\prime\prime}-2x\varepsilon\varphi^{\prime}+4\upsilon\varepsilon\Theta_{0}\varphi=0.$

As can be seen from the change of variables
$\varphi=y{\mathrm{e}}^{\varepsilon x^{2}/2}$ and $z=\sqrt{2}x$, equation
(2.56) can be solved in terms of parabolic cylinder functions (cf.
[DLMF]*Section 12) and one finds

(2.57) $\displaystyle\varepsilon=1$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}$ $\displaystyle\varphi(x)$
$\displaystyle={\mathrm{e}}^{x^{2}/2}\left(C_{1}D_{2\upsilon\Theta_{0}}(\sqrt{2}x)+C_{2}D_{-1-2\upsilon\Theta_{0}}({\mathrm{i}}\sqrt{2}x)\right),$
(2.58) $\displaystyle\varepsilon=-1$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}$ $\displaystyle\varphi(x)$
$\displaystyle={\mathrm{e}}^{-x^{2}/2}\left(C_{1}D_{2\upsilon\Theta_{0}}({\mathrm{i}}\sqrt{2}x)+C_{2}D_{-1-2\upsilon\Theta_{0}}(\sqrt{2}x)\right),$

where $D_{\nu}(x)$ denotes the classical parabolic cylinder function
[DLMF]*Eq. 12.2.5. Observe that since $u(x)$ is a logarithmic derivative of
$\varphi$, formulas (2.57), (2.58) generate a one-parameter family of
solutions of ($\textnormal{P}_{IV}$) where the parameter can be chosen to be,
say, $\lambda\mathrel{\mathop{\mathchar
58\relax}}=C_{2}/C_{1}\in\overline{{\mathbb{C}}}$.

By using $u(x)$ constructed in this manner as a “seed” solution, we can now
generate families of solutions of ($\textnormal{P}_{IV}$) by iterating
Bäcklund transformations. However, one must exercise care in doing so since
not all transformations described in Section 2.2 are determinate (cf. Remark
2.5).

######  Example 1.

Let $\varepsilon=-1$ and $u(x)=\varphi^{\prime}(x)/\varphi$ where $\varphi$ is
as in (2.58) and $\Theta_{0},\Theta_{\infty}$ satisfying (2.55). If we take
$u(x)=-f_{1}(x)$ and $\alpha_{1}=2\nu\Theta_{0}$ (see Remark (2.1)), then
direct calculations yield the triple

(2.59)
$(f_{0}(x),f_{1}(x),f_{2}(x))=\left(2x+\frac{\varphi^{\prime}(x)}{\varphi(x)}-4\nu\Theta_{0}\dfrac{\varphi(x)}{\varphi^{\prime}(x)},-\frac{\varphi^{\prime}(x)}{\varphi(x)},4\nu\Theta_{0}\dfrac{\varphi(x)}{\varphi^{\prime}(x)}\right),$

corresponding to parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=\left(1,2\nu\Theta_{0},-2\nu\Theta_{0}\right)$.
Observe that applying $T_{1}$ to the triple (2.59) yields an indeterminacy
since $(f_{1}f_{2})(x)+2\alpha_{1}\equiv 0.$ On the other hand, letting
$\alpha_{1}=-2\nu\Theta_{0}$ does not cause this issue since the corresponding
triple is

$(f_{0}(x),f_{1}(x),f_{2}(x))=\left(2x+\frac{\varphi^{\prime}(x)}{\varphi(x)},-\frac{\varphi^{\prime}(x)}{\varphi(x)},0\right),$

which solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=(1+2\nu\Theta_{0},-2\nu\Theta_{0},0)$888That
$u(x)=-f_{2}(x)=0$ solves ($\textnormal{P}_{IV}$) with parameters
$\alpha=4\nu\Theta_{0}+1$, $\beta=0$ has to be interpreted by first
multiplying both sides of the equation by $u(x)$..

Sufficient conditions for the determinacy of the Bäcklund transformations on
the level of the solutions $u(x)$ were obtained in [BM]; since the triples
$(f_{0},f_{1},f_{2})$ are determined by $f_{1}$ once a choice of $\alpha_{1}$
is made, one can deduce determinacy of iterating $T_{1},T_{1}^{-1}$ and we
summarize this result here.

######  Proposition 2.12 ([BM]*Proposition 11).

Let $u(x)=-\varepsilon\varphi^{\prime}(x)/\varphi(x)=-f_{1}(x)$ with $\varphi$
as in (2.57) when $\varepsilon=1$ (resp. (2.58) when $\varepsilon=-1$) and
$\Theta_{0},\Theta_{\infty}$ satisfying (2.54) (resp. (2.55)).

  1. (a)

If $\varepsilon=\upsilon=1$ and $\alpha_{1}=2\Theta_{0}$, then

     * •

$T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$-2\Theta_{0}\not\in{\mathbb{N}}\cup\\{0\\}=\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{N}}_{0}$,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

  2. (b)

If $\varepsilon=\upsilon=1$ and $\alpha_{1}=-2\Theta_{0}$, then

     * •

$T_{1}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$,

     * •

$T_{1}^{-1}(f_{0},f_{1},f_{2})$ is indeterminate.

  3. (c)

If $\varepsilon=-\upsilon=1$ and $\alpha_{1}=2\Theta_{0}$, then

     * •

$T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

     * •

$T_{1}^{-1}(f_{0},f_{1},f_{2})$ is indeterminate.

  4. (d)

If $\varepsilon=-\upsilon=1$ and $\alpha_{1}=-2\Theta_{0}$, then

     * •

$T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$-2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

  5. (e)

If $\varepsilon=-\upsilon=-1$ and $\alpha_{1}=2\Theta_{0}$, then

     * •

$T_{1}(f_{0},f_{1},f_{2})$ is indeterminate,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

  6. (f)

If $\varepsilon=-\upsilon=-1$ and $\alpha_{1}=-2\Theta_{0}$, then

     * •

$T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$-2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

  7. (g)

If $\varepsilon=\upsilon=-1$ and $\alpha_{1}=2\Theta_{0}$, then

     * •

$T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$-2\Theta_{0}\not\in{\mathbb{N}}_{0}$,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

  8. (h)

If $\varepsilon=\upsilon=-1$ and $\alpha_{1}=-2\Theta_{0}$, then

     * •

$T_{1}(f_{0},f_{1},f_{2})$ is indeterminate,

     * •

$T_{1}^{-n}(f_{0},f_{1},f_{2})$ is determinate for all $n\geq 0$ provided
$-2\Theta_{0}\not\in{\mathbb{N}}_{0}$.

###### Proof.

We describe how the first bullet point of (a) follows from [BM]*Proposition 11
and leave the rest for the reader to verify. When $\varepsilon=\upsilon=1,$
the constraint (2.54) reads $\Theta_{\infty}+\Theta_{0}=0$. Furthermore, the
choice of $\alpha_{1}$ implies $-T_{1}(f_{1}(x))=u_{\searrow}(x)$ (see Section
2.2). We now observe that if $u_{\searrow}$ is determinate, then
$T_{1}(f_{0},f_{1},f_{2})$ is determinate. Indeed, if $u_{\searrow}$ is
determinate, then the denominator of (2.22) is not identically vanishing.
Comparing Remark 2.5 with the denominator of (2.22) we see that the conditions
in Remark 2.5 for determinacy of $T_{1}(f_{0},f_{1},f_{2})$ are satisfied.
Since, by assumption,

$(\Theta_{\infty}-\Theta_{0}-1)\Theta_{0}=-(2\Theta_{0}+1)\Theta_{0}\neq 0,$

we have $u_{\searrow}$ is determinate by [BM]*Proposition 11, and so is
$T_{1}(f_{0},f_{1},f_{2})$.

Finally, application of $T_{1}$ with $\alpha_{1}=2\Theta_{0}$ changes
$\Theta_{0}\mapsto\Theta_{0}+\frac{1}{2}$ and
$\Theta_{\infty}\mapsto\Theta_{\infty}-\frac{1}{2}$, the assumptions on
$\Theta_{0}$ imply
$(\Theta_{\infty}-\Theta_{0}-1-n)(\Theta_{0}+\frac{n}{2})=-(2\Theta_{0}+n+1)(\Theta_{0}+\frac{n}{2})\neq
0$ for all $n\geq 0$ and the claim follows by induction. ∎

With Proposition 2.12 in hand, we can now demonstrate how to generate families
of well-defined special function solutions.

######  Example 2.

Let $\varepsilon=1$ and $\alpha_{1}=-2\upsilon\Theta_{0}$ and
$u(x)=-\varphi^{\prime}(x)/\varphi(x)$. The corresponding triple

(2.60)
$(f_{0},f_{1},f_{2})=\left(0,\frac{\varphi^{\prime}(x)}{\varphi(x)},2x-\frac{\varphi^{\prime}(x)}{\varphi(x)}\right)$

solves ($\textnormal{P}^{\textnormal{sym}}_{IV}$) with parameters
$(\alpha_{0},\alpha_{1},\alpha_{2})=(0,-2\upsilon\Theta_{0},1+2\upsilon\Theta_{0})$.
Plugging this into (2.24) yields

$\sigma_{0,0,0}(x)=\dfrac{\varphi^{\prime}(x)}{\varphi(x)}-2\alpha_{1}x=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log\left({\mathrm{e}}^{-\alpha_{1}x^{2}}\varphi(x)\right).$

In view of (2.36) we have $\tau_{0,0,0}(x)\mathrel{\mathop{\mathchar
58\relax}}=\tau(x)={\mathrm{e}}^{-\alpha_{1}x^{2}}\varphi(x)$ or
$\tilde{\tau}_{0,0,0}(x)=\varphi(x)$ (see Proposition 2.8). It follows from
Proposition 2.12 that $T_{1}^{n}(f_{0},f_{1},f_{2})$ is determinate as long as
$2\Theta_{0}\not\in{\mathbb{N}}_{0}$. Setting $\tau_{-1,0,0}(x)\equiv 1$999It
can be directly verified that this is consistent with applying $T_{1}^{-1}$ to
the triple (2.60), the Desnanot-Jacobi identity implies that the functions

(2.61)
$\tilde{\tau}_{n,0,0}(x;\lambda)=\det\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}\tilde{\tau}_{0,0,0}}{\operatorname{d\\!}{x^{i+j}}}}(x;\lambda)\right]_{i,j=0}^{n}$

satisfy (2.42). In the second part of the paper, we will encounter the
particular case $\upsilon=-1,\Theta_{0}=\Theta_{\infty}=\frac{1}{4}$,
$\alpha_{2}=\frac{1}{2}$, and $\lambda=\infty$, which corresponds to

$\varphi(x)={\mathrm{e}}^{x^{2}/2}D_{-1/2}(\sqrt{2}x),\quad\tilde{\tau}_{0,0,0}(x)=\varphi(x),$

and

(2.62)
$\tilde{\tau}_{n,0,0}(x;\infty)=\det\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{x^{i+j}}}}\left({\mathrm{e}}^{x^{2}/2}D_{-1/2}(\sqrt{2}x)\right)\right]_{i,j=0}^{n}.$

######  Remark 2.13.

We make the following observation which will be used in Section 3. By applying
(2.34) to $T_{2}^{-1}(f_{0},f_{1},f_{2})$ we have

(2.63)
$T_{2}^{-1}(\sigma(x))=\sigma(x)+f_{2}(x)+\dfrac{2\alpha_{1}}{f_{1}(x)}-2x.$

In this particular case, formula (2.63) reads

(2.64)
$T_{2}^{-1}(\sigma(x))=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\log{\mathrm{e}}^{-x^{2}/2}+\dfrac{\varphi(x)}{\varphi^{\prime}(x)}.$

Using [DLMF]*Eqs. 12.8.1-2, we find

(2.65)
$\dfrac{\varphi(x)}{\varphi^{\prime}(x)}=-\dfrac{\sqrt{2}D_{-1/2}(\sqrt{2}x)}{D_{-3/2}(\sqrt{2}x)}=-x+\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\left(D_{-3/2}(\sqrt{2}x)\right)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\left({\mathrm{e}}^{-x^{2}}D_{-3/2}(\sqrt{2}x)\right).$

Combining (2.65) with (2.64) yields

$T_{2}^{-1}(\sigma(x))=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{x}}}\left({\mathrm{e}}^{-x^{2}}D_{-3/2}(\sqrt{2}x)\right)$

or, in other words, (recall Table 1)

$\tilde{\tau}_{0,-1,0}(x;\infty)={\mathrm{e}}^{x^{2}/2}D_{-3/2}(\sqrt{2}x).$

Since $T_{1}T_{2}=T_{2}T_{1}$, we have the remarkable identity

(2.66)
$\tilde{\tau}_{n,-1,0}(x;\infty)=T_{2}^{-1}T_{1}^{n}(\tilde{\tau}_{0,0,0}(x;\infty))=T_{1}^{n}(\tilde{\tau}_{0,-1,0}(x;\infty))=\det\left[\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{x^{i+j}}}}\left({\mathrm{e}}^{x^{2}/2}D_{-3/2}(\sqrt{2}x)\right)\right]_{i,j=0}^{n}.$

To summarize, the above computation reveals that (2.62) is the tau function
associated with the triple $T^{n}_{1}(f_{0},f_{1},f_{2})$ while (2.66) is the
tau function associated with the triple
$T_{2}^{-1}T^{n}_{1}(f_{0},f_{1},f_{2})$. As a final observation, we note that
by (2.17), we have $\tilde{\tau}_{n+j,m+j,k+j}(x)=\tilde{\tau}_{n,m,k}(x)$,
which in this instance yields

$\tilde{\tau}_{n,-1,0}(x;\infty)=\tilde{\tau}_{n+1,0,1}(x;\infty).$

In the remainder of the paper we will be exclusively studying tau functions
corresponding to the choice $\lambda=\infty$ (or $C_{2}=0$ in (2.57), (2.58)),
and so we drop the parameter $\lambda$ and set
$\tau_{n,m,k}(x)\mathrel{\mathop{\mathchar 58\relax}}=\tau_{n,m,k}(x;\infty)$.

##  3\. Connecting the Quartic Model, Orthogonal Polynomials, and Painlevé-IV

In this section, we prove Lemma 1.2 and Theorem 1.3. To do so, we must first
collect some intermediate identities. We start by observing that using
[DLMF]*Eqs. 12.5.1, the evenness of $V$, and the rescaling
$s^{2}=2^{1/2}N^{-1/2}x$ implies

(3.1)
$\int_{\mathbb{R}}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s=2\int_{0}^{\infty}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s=\dfrac{2^{1/4}{\pi}^{1/2}}{N^{1/4}}{\mathrm{e}}^{Nt^{2}/8}D_{-\frac{1}{2}}\left(\dfrac{N^{1/2}t}{2^{1/2}}\right).$

Furthermore, for all $k\in{\mathbb{N}}$ the following identity holds:

(3.2)
$\mathinner{\dfrac{\operatorname{d\\!}{{}^{k}}}{\operatorname{d\\!}{t^{k}}}}\int_{\mathbb{R}}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s=(-1)^{k}\dfrac{N^{k}}{2^{k}}\int_{\mathbb{R}}s^{2k}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s$

Combining (3.1), (3.3), we arrive at

(3.3)
$\int_{{\mathbb{R}}}s^{2k}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s=(-1)^{k}\dfrac{2^{k+\frac{1}{4}}\pi^{1/2}}{N^{k+\frac{1}{4}}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{k}}}{\operatorname{d\\!}{t^{k}}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{1}{2}}\left(\dfrac{N^{1/2}t}{2^{1/2}}\right)\right),$

Although this expression is reminiscent of formula (2.61), Hankel determinants
$H_{n-1}$ have many zero entries. To remedy this, we appeal to the following
beautiful identity.

######  Proposition 3.1.

Let $\\{c_{j}\\}_{j=0}^{\infty}$ be a sequence of variables and set
$c_{k/2}=0$ for all $k\in{\mathbb{N}}\setminus 2{\mathbb{N}}$. Then

(3.4) $\displaystyle\det[c_{\frac{i+j}{2}}]_{i,j=0}^{2n-1}$
$\displaystyle=\det[c_{i+j}]_{i,j=0}^{n-1}\cdot\det[c_{i+j+1}]_{i,j=0}^{n-1},$
(3.5) $\displaystyle\det[c_{\frac{i+j}{2}}]_{i,j=0}^{2n}$
$\displaystyle=\det[c_{i+j}]_{i,j=0}^{n}\cdot\det[c_{i+j+1}]_{i,j=0}^{n-1}.$

######  Remark 3.2.

Note that due to the evenness of $V$, matrices (1.7) are of the form of the
right hand side in (3.4), (3.4). It is instructive to write out (3.4), (3.5);
for example, (3.4) reads

$\det\begin{bmatrix}c_{0}&0&c_{1}&0&c_{2}&\cdots&0&c_{n-1}&0\\\
0&c_{1}&0&c_{2}&0&\cdots&c_{n-1}&0&c_{n}\\\
c_{1}&0&c_{2}&0&c_{3}&\cdots&0&c_{n}&0\\\
&\vdots&\vdots&\vdots&&\ddots&\vdots&\vdots&\vdots\\\
c_{n-1}&0&c_{n}&0&c_{n+1}&\cdots&0&c_{2n-2}&0\\\
0&c_{n}&0&c_{n+1}&0&\cdots&c_{2n-2}&0&c_{2n-1}\end{bmatrix}\\\
=\det\begin{bmatrix}c_{0}&c_{1}&c_{2}&\cdots&c_{n-1}\\\
c_{1}&c_{2}&c_{3}&\cdots&c_{n}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
c_{n-1}&c_{n}&c_{n+1}&\cdots&c_{2n-2}\end{bmatrix}\cdot\det\begin{bmatrix}c_{1}&c_{2}&c_{3}&\cdots&c_{n}\\\
c_{2}&c_{3}&c_{4}&\cdots&c_{n+1}\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
c_{n}&c_{n+1}&c_{n+2}&\cdots&c_{2n-1}\\\ \end{bmatrix}$

###### Proof of Proposition 3.1.

We begin by proving (3.4). Denote $H\mathrel{\mathop{\mathchar
58\relax}}=[c_{\frac{i+j}{2}}]_{i,j=0}^{2n-1}$ and its entries by
$H_{i,j}=c_{\frac{i+j}{2}}$. Let $P_{2n}\subseteq S_{2n}$ be the subgroup
defined by the property that $j$ and $\sigma(j)$ have the same parity for all
$i=0,1,...,2n-1$; such permutations make up a subgroup of the group of parity-
alternating permutations in [Tanimoto]. Note that such permutations factor
into cycles which act on even numbers and cycles which act on odd numbers.
That is, for all $\sigma\in P_{2n}$, there exist $\sigma_{e},\sigma_{o}\in
S_{n}$ such that101010One should interpret $\sigma_{e}\in S_{n}$ as a
permutation acting on the alphabet $0,2,...,2n-2$, while $\sigma_{o}\in S_{n}$
acts on the alphabet $1,3,...,2n-1$. $\sigma=\sigma_{e}\sigma_{o}$. Of course
any such product also yields an element of $P_{2n}$. With this in mind, the
definition of the determinant reads111111Here we interpret elements of
$S_{2n}$ as acting on the alphabet $0,1,...,2n-1$

$\det H=\sum_{\sigma\in
S_{2n}}\text{sgn}(\sigma)\prod_{i=0}^{2n-1}H_{i,\sigma(i)}\\\
=\left(\sum_{\sigma_{e}\in
S_{n}}\text{sgn}(\sigma_{e})\prod_{i=0}^{n-1}H_{2i,\sigma_{e}(2i)}\right)\left(\sum_{\sigma_{o}\in
S_{n}}\text{sgn}(\sigma_{o})\prod_{i=0}^{n-1}H_{2i+1,\sigma_{o}(2i+1)}\right).$

The terms in the last expression can now readily be interpreted as the
determinants in the right hand side of (3.4). The proof of (3.5) follows using
the same argument and noting that the set $\\{0,1,...,2n\\}$ has one more even
numbers than odd numbers. ∎

###  3.1. Proof of Lemma 1.2

To apply Proposition 3.1 to determinants $H_{n}(t,N)$, we set

$c_{i+j}=\int_{\mathbb{R}}s^{2(i+j)}{\mathrm{e}}^{-NV(s;t)}{\mathrm{d}}s=(-1)^{i+j}\dfrac{2^{i+j+\frac{1}{4}}\pi^{1/2}}{N^{i+j+\frac{1}{4}}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{t^{i+j}}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{1}{2}}\left(\dfrac{N^{1/2}t}{2^{1/2}}\right)\right)$

and note that by [DLMF]*Eq. 12.8.9 we have the identity

(3.6)
$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{1}{2}}\left({N^{1/2}2^{-1/2}t}\right)\right)=-{N^{1/2}}{2^{-3/2}}{\mathrm{e}}^{Nt^{2}/8}D_{-\frac{3}{2}}\left({N^{1/2}2^{-1/2}t}\right)$

In particular, (3.3) gives

(3.7)
$c_{i+j+1}=-\frac{2}{N}\mathinner{\dfrac{\operatorname{d\\!}{}c_{i+j}}{\operatorname{d\\!}{t}}}=\dfrac{1}{2^{1/2}N^{1/2}}(-1)^{i+j}\dfrac{2^{i+j+\frac{1}{4}}\pi^{1/2}}{N^{i+j+\frac{1}{4}}}\mathinner{\dfrac{\operatorname{d\\!}{{}^{i+j}}}{\operatorname{d\\!}{t^{i+j}}}}\left({\mathrm{e}}^{Nt^{2}/8}D_{-\frac{3}{2}}\left(\dfrac{N^{1/2}t}{2^{1/2}}\right)\right)$

Pulling out common factors from each row/column and recalling (1.11), (1.12),
we find

(3.8) $\displaystyle\det[c_{i+j}]_{i,j=0}^{n-1}$
$\displaystyle=\dfrac{\pi^{n/2}2^{n/4}}{N^{n/4}}\left(\prod_{k=0}^{n-1}(-1)^{k}\dfrac{2^{k}}{N^{k}}\right)^{2}H_{n}^{(e)}(t,N)=\dfrac{\pi^{n/2}2^{n(n-3/4)}}{N^{n(n-3/4)}}H_{n}^{(e)}(t,N),$
(3.9) $\displaystyle\det[c_{i+j+1}]_{i,j=0}^{n-1}$
$\displaystyle=N^{-n/2}2^{-n/2}\dfrac{\pi^{n/2}2^{n(n-3/4)}}{N^{n(n-3/4)}}H_{n}^{(o)}(t,N)=\dfrac{\pi^{n/2}2^{n(n-5/4)}}{N^{n(n-1/4)}}H_{n}^{(o)}(t,N).$

Finally, we plug (3.8), (3.9) into (3.4), (3.5) to arrive at the result.

###  3.2. More on polynomials $P_{n}(z;t,N)$

When the measure ${\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z$ is a Toda deformation
(see, e.g., [Ismail]*Section 2.8), the quantities
$h_{k}(t,N),\gamma_{k}^{2}(t,N),H_{k}(t,N)$ satisfy various differential
identities in the parameter $t$. While the weight of orthogonality in (1.5) is
not a Toda deformation, we may still derive useful identities which we
summarize here.

######  Lemma 3.3.

Fix $n\in{\mathbb{N}}$ and let $t$ be in the open set such that
$\text{deg}P_{k}=k$ for $k\in{n+1,n,n-1}$ (see Footnote 2) and write

$P_{n}(z;t,N)\mathrel{\mathop{\mathchar
58\relax}}=z^{n}+\sum_{k=1}^{\lfloor\frac{n}{2}\rfloor}\mathfrak{p}_{n,n-2k}(t,N)z^{n-2k}$

be the polynomial satisfying (1.5) and
$h_{k}(t,N),\gamma_{k}^{2}(t,N),H_{k}(t,N)$ be as in (1.10), (1.9), and (1.7),
respectively. The following identities hold:

  1. (a)

$\gamma_{n}^{2}(t,N)=\mathfrak{p}_{n,n-2}(t,N)-\mathfrak{p}_{n+1,n-1}(t,N)$,

  2. (b)

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
h_{n}(t,N)=-\dfrac{N}{2}\left(\gamma_{n}^{2}(t,N)+\gamma_{n+1}^{2}(t,N)\right)$,

  3. (c)

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{n}(t,N)=\dfrac{N}{2}\left(\mathfrak{p}_{n+2,n}(t,N)+\mathfrak{p}_{n+1,n-1}(t,N)\right)$,

  4. (d)

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(e)}_{n}(t,N)=\dfrac{N}{2}\mathfrak{p}_{2n,2n-2}(t,N)$,

  5. (e)

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(o)}_{n}(t,N)=\dfrac{N}{2}\mathfrak{p}_{2n+1,2n-1}(t,N)$,

  6. (f)

$\mathinner{\dfrac{\operatorname{d\\!}{}\mathfrak{p}_{n+1,n-1}}{\operatorname{d\\!}{t}}}(t,N)=\dfrac{N}{2}(\gamma_{n}^{2}\gamma_{n+1}^{2})(t,N)$.

###### Proof.

We begin by noting that since all the quantities below can be meromorphically
continued to all $t\in{\mathbb{C}}$, the identities below hold for all
$t\in{\mathbb{C}}$ where the involved quantities are analytic. Since $N$ is
fixed in what follows, we drop the dependence on it.

  1. (a)

Follows from comparing the coefficient of $z^{n-1}$ in (1.8).

  2. (b)

Using definition (1.10), (1.5), and the fact that $P_{n}$ is monic we have

(3.10)
$\mathinner{\dfrac{\operatorname{d\\!}{}h_{n}}{\operatorname{d\\!}{t}}}=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\left(\int_{\mathbb{R}}P_{n}^{2}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z\right)=-\frac{N}{2}\int_{{\mathbb{R}}}z^{2}P_{n}^{2}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z\\\
=-\frac{N}{2}\int_{{\mathbb{R}}}\left(z^{n+2}+\mathfrak{p}_{n,n-2}(t)z^{n}+\cdots\right)P_{n}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z=-\dfrac{N}{2}(\mathfrak{p}_{n,n-2}(t)-\mathfrak{p}_{n+2,n}(t))h_{n}(t).$

The result now follows by using (a).

  3. (c)

Using the second equality in (1.10), we have

$\log H_{n}(t)=\sum_{k=0}^{n}\log h_{k}(t).$

Using (a), (b) we find

$\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{n}(t)=-\frac{N}{2}\sum_{k=0}^{n}(\gamma_{k}^{2}(t)+\gamma_{k+1}^{2}(t))=\frac{N}{2}\left(\mathfrak{p}_{n+2,n}(t)+\mathfrak{p}_{n+1,n-1}(t)\right).$

  4. (d, e)

We prove these two identities by induction. Note that by (c) and (1.13),
(1.14),

$\displaystyle\frac{N}{2}\mathfrak{p}_{2,0}(t)$
$\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{0}(t)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{1}^{(e)}(t),$
$\displaystyle\frac{N}{2}\left(\mathfrak{p}_{2,0}(t)+\mathfrak{p}_{3,1}(t)\right)$
$\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{1}(t)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{1}^{(e)}(t)+\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{1}^{(o)}(t),$ $\displaystyle\implies$
$\displaystyle\frac{N}{2}\mathfrak{p}_{3,1}(t)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{1}^{(o)}(t).$

This establishes the base case. Assuming (d, e) hold true and using (c) we
have

$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{2n}(t,N)$
$\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(e)}_{n+1}(t,N)+\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(o)}_{n}(t,N)=\dfrac{N}{2}\left(\mathfrak{p}_{2n+2,2n}(t)+\mathfrak{p}_{2n+1,2n-1}(t)\right)$
$\displaystyle\implies$
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(e)}_{n+1}(t,N)=\dfrac{N}{2}\mathfrak{p}_{2n+2,2n}(t)$

and

$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{2n+1}(t,N)$
$\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(e)}_{n+1}(t,N)+\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(o)}_{n+1}(t,N)=\dfrac{N}{2}\left(\mathfrak{p}_{2n+2,2n}(t)+\mathfrak{p}_{2n+3,2n+1}(t)\right)$
$\displaystyle\implies$
$\displaystyle\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H^{(o)}_{n+1}(t,N)=\dfrac{N}{2}\mathfrak{p}_{2n+3,2n+1}(t).$

  5. (f)

Using (1.5) and calculations similar to the proof of (b),

$\displaystyle 0$
$\displaystyle=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\left(\int_{\mathbb{R}}P_{n+1}(z;t)P_{n-1}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z\right)$
$\displaystyle=-\dfrac{N}{2}\int_{{\mathbb{R}}}z^{2}P_{n+1}(z;t)P_{n-1}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z+\int_{\mathbb{R}}\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}P_{n+1}(z;t)P_{n-1}(z;t){\mathrm{e}}^{-NV(z;t)}{\mathrm{d}}z$
$\displaystyle=-\dfrac{N}{2}h_{n+1}(t)+\mathinner{\dfrac{\operatorname{d\\!}{}\mathfrak{p}_{n+1,n-1}}{\operatorname{d\\!}{t}}}h_{n-1}(t).$

The result now follows by recalling (1.9).

∎

###  3.3. Proof of Theorem 1.3

We begin by proving the fist set of equalities in (1.15)-(1.21). Recalling
Example 2.62 and Remark 2.13, we see that

(3.11) $\displaystyle H_{n}^{(e)}(t,N)$
$\displaystyle=\tilde{\tau}_{n-1,0,0}({N^{1/2}}{2^{-1}}t),$ (3.12)
$\displaystyle H_{n}^{(o)}(t,N)$
$\displaystyle=\tilde{\tau}_{n-1,-1,0}({N^{1/2}}{2^{-1}}t).$

Equations (1.15), (1.16) follow directly from the definition of $\tilde{\tau}$
(see Proposition 2.8) and (3.11), (3.12), respectively. To prove (1.20), we
compute the logarithmic derivative of $H_{n}^{(e)}$ in two different ways. On
the one hand, (3.11) and (2.36) imply

(3.13) $\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log
H_{n}^{(e)}(t,N)=\mathinner{\dfrac{\operatorname{d\\!}{}}{\operatorname{d\\!}{t}}}\log\left({\mathrm{e}}^{(\frac{1}{2}+n)\frac{N}{2}t^{2}}\tau_{n,0,0}(N^{1/2}2^{-1/2}t)\right)\\\
=N\left(n+\frac{1}{2}\right)t+N^{1/2}2^{-1/2}\sigma_{n,0,0}(N^{1/2}2^{-1/2}t).$

Combining (3.13) with Lemma 3.3(d) and re-scaling the independent variable
yields (1.20). Formula (1.21) is analogously derived using Lemma 3.3(e).

Recall that by (2.33), we have

(3.14) $(f_{2})_{n,0,0}(x)=\sigma_{n-1,0,0}(x)-\sigma_{n,0,0}(x)$

Using (1.20) and Lemma 3.3(a) immediately yields (1.19). Similarly, to arrive
at (1.18) we recall that by (2.35), we have

(3.15)
$(f_{1})_{n,0,0}(x)=\sigma_{n,0,0}(x)-\sigma_{n,0,1}(x)=\sigma_{n,0,0}(x)-\sigma_{n-1,-1,0}(x)$

where the last equality follows from (2.17). Using (1.20), (1.21), and Lemma
3.3(a) yields (1.18). Condition (1.3) immediately implies formula (1.17).

We now prove the second set of equalities in (1.15)-(1.21). From (1.11),
(1.12), we immediately have

$H_{n}^{(e,o)}(2N^{-1/2}x,N)=H_{n}^{(e,o)}(2x,1).$

Similarly, it follows from (A.4) that121212Again here, we check this identity
for $t\in{\mathbb{C}}$ satisfying the assumptions in the statement of Theorem
1.3, and then conclude it for $t\in{\mathbb{C}}$ be meromorphic continuation.

$N^{1/2}\mathfrak{p}_{n,n-2}(2N^{-1/2}x,N)=\mathfrak{p}_{n,n-2}(2x,1),$

and by Lemma 3.3(a), we have

(3.16) $N^{1/2}\gamma_{n}^{2}(2N^{-1/2}x,N)=\gamma_{n}^{2}(2x,1).$

######  Remark 3.4.

With Theorem 1.3 in hand, one could compute $(f_{0},f_{1},f_{2})_{n,-1,0}$
explicitly. An interesting observation in this direction is that
$(f_{2})_{n,-1,0}$ can be computed in two different ways; one analogous to the
calculation of $(f_{2})_{n,-1,0}$, and another by directly applying
$T_{2}^{-1}$:

$T_{2}^{-1}(f_{0},f_{1},f_{2})=\left(f_{1}+\dfrac{2(\alpha_{0}+\alpha_{1})}{f_{0}-\frac{2\alpha_{1}}{f_{1}}},f_{2}+\frac{2\alpha_{1}}{f_{1}},f_{0}-\frac{2\alpha_{1}}{f_{1}}\right).$

This, along with (3.16) yields the identity

(3.17)
$-N^{1/2}\gamma_{2n}^{2}(2x,N)+\dfrac{2n+1}{N^{1/2}\gamma_{2n+1}^{2}(2x,N)}=2N^{1/2}x+N^{1/2}\gamma_{2n+1}^{2}(2x,N)+N^{1/2}\gamma_{2n+2}^{2}(2x,N).$

We now observe that (3.17) is equivalent to the string equation (1.33) with
odd index.

Theorem 1.3 establishes an explicit connection between the recurrence
coefficients and normalizing constants of $P_{n}$ and solutions of
($\textnormal{P}^{\textnormal{sym}}_{IV}$). This motivates the asymptotic
analysis of the polynomials $P_{n}$ and their recurrence coefficients, which
occupies the remainder of the paper.

##  4\. Preliminaries to the Asymptotic Analysis of $P_{n},\gamma^{2}_{n}$

###  4.1. Potential theory and S-curves

It is well understood that the zeros of polynomials satisfying (1.5)
asymptotically distribute as a certain _weighted equilibrium measure_ on an
_S-contour_ corresponding to the weight function (1.6). Let us start with some
definitions.

######  Definition 4.1.

Let $V$ be an entire function. The logarithmic energy in the external field
$\Re V$ of a measure $\nu$ in the complex plane is defined as

$E_{V}(\nu)=\iint\log\frac{1}{|s-t|}\mathrm{d}\nu(s)\mathrm{d}\nu(t)+\int\Re
V(s)\mathrm{d}\nu(s).$

The equilibrium energy of a contour $\Gamma$ in the external field $\Re V$ is
equal to

(4.1) $\mathcal{E}_{V}(\Gamma)=\inf_{\nu\in\mathcal{M}(\Gamma)}E_{V}(\nu),$

where $\mathcal{M}(\Gamma)$ denotes the space of Borel probability measures on
$\Gamma$.

When $\Re V(s)-\log|s|\to+\infty$ as $\Gamma\ni s\to\infty$, there exists a
unique minimizing measure for (4.1), which is called the weighted equilibrium
measure of $\Gamma$ in the external field $\Re V$, say $\mu_{\Gamma}$, see
[MR1485778]*Theorem I.1.3. The support of $\mu_{\Gamma}$, say $J_{\Gamma}$, is
a compact subset of $\Gamma$. The equilibrium measure $\mu=\mu_{\Gamma}$ is
characterized by the Euler–Lagrange variational conditions:

(4.2) $U^{\mu}(z)+\dfrac{1}{2}\Re V(z)\;\left\\{\begin{aligned} &=\ell,\qquad
z\in J_{\Gamma},\\\ &\geq\ell,\qquad z\in\Gamma\setminus
J_{\Gamma},\end{aligned}\right.$

where $\ell=\ell_{\Gamma}$ is a constant, the Lagrange multiplier, and
$U^{\mu}(z)\mathrel{\mathop{\mathchar
58\relax}}=-\int\log|z-s|{\mathrm{d}}\mu(s)$ is the logarithmic potential of
$\mu$, see [MR1485778]*Theorem I.3.3.

Observe that due to the analyticity of the integrand in (1.5) and (1.24), the
contour of integration can be deformed without changing the value of
$Z_{N}(t)$. Henceforth, we suppose that partition function $Z_{N}(t)$ and
orthogonal polynomials $\\{P_{n}(z;t,N)\\}_{n=0}^{\infty}$ are defined with
$\Gamma\in\mathcal{T}$, where $\mathcal{T}$ is the following class of
contours.

######  Definition 4.2.

We shall denote by $\mathcal{T}$ the collection of all piece-wise smooth
contours that extend to infinity in both directions, each admitting a
parametrization $z(s)$, $s\in{\mathbb{R}}$, for which there exists
$\varepsilon\in(0,\pi/8)$ and $s_{0}>0$ such that

(4.3) $\left\\{\begin{array}[]{ll}|\text{arg}(z(s))|\leq\varepsilon,&s\geq
s_{0},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
|\text{arg}(z(s))-\pi|\leq\varepsilon,&s\leq-s_{0},\end{array}\right.$

where $\text{arg}(z(s))\in[0,2\pi)$.

Despite the above flexibility, it is well understood in the theory of non-
Hermitian orthogonal polynomials, starting with the works of Stahl [stahl-
domains, stahl-structure, stahl-complexortho] and Gonchar and Rakhmanov
[MR922628], that one should use the contour whose equilibrium measure has
support _symmetric_ (with the _S-property_) in the external field $\Re V$. We
make this idea precise in the following definition.

######  Definition 4.3.

The support $J_{\Gamma}$ has the S-property in the external field $\Re V$, if
it consists of a finite number of open analytic arcs and their endpoints, and
on each arc it holds that

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

where $\frac{\partial}{\partial n_{+}}$ and $\frac{\partial}{\partial n_{-}}$
are the normal derivatives from the $(+)$\- and $(-)$-side of $\Gamma$. We
shall say that a curve $\Gamma\in\mathcal{T}$ is an S-curve in the field $\Re
V$, if $J_{\Gamma}$ has the S-property in this field.

It is also understood that geometrically $J_{\Gamma}$ is comprised of
_critical trajectories_ of a certain quadratic differential. Recall that if
$Q$ 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 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 $-Q(z)\mathrm{d}z^{2}$.

In our setting, Kuijlaars and Silva [MR3306308]*Theorems 2.3 and 2.4 already
identified some important features of the function $Q$, which we summarize
below for $V(z;t)$ as in (1.6).

######  Theorem 4.4 ([MR3306308]).

Let $V(z;t)$ be given by (1.6). Then,

  1. (a)

there exists a contour $\Gamma_{t}\in\mathcal{T}$ such that

(4.5)
$\mathcal{E}_{V}(\Gamma_{t})=\sup_{\Gamma\in\mathcal{T}}\mathcal{E}_{V}(\Gamma).$

  2. (b)

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 $\Re\ V(z;t)$ and the measure $\mu_{t}$ is
uniquely determined by (4.4). Therefore, $\mu_{t}$ and its support
$J_{t}\mathrel{\mathop{\mathchar 58\relax}}=J_{\Gamma_{t}}$ are the same for
every $\Gamma_{t}$ satisfying (4.5).

  3. (c)

The function

(4.6)
$Q(z;t)=\left(\frac{V^{\prime}(z;t)}{2}-\int\frac{\mathrm{d}\mu_{t}(x)}{z-x}\right)^{2},\quad
z\in{\mathbb{C}}\setminus J_{t},$

is a polynomial of degree 6.

  4. (d)

The support $J_{t}$ consists of some short critical trajectories of the
quadratic differential $-Q(z;t)\mathrm{d}z^{2}$ and the equation

(4.7)
$\mathrm{d}\mu_{t}(z)=\frac{1}{\pi{\mathrm{i}}}\,Q_{+}^{1/2}(z;t)\mathrm{d}z,\quad
z\in J_{t},$

holds on each such critical trajectory, where
$Q^{1/2}(z;t)=\frac{1}{2}z^{3}+\mathcal{O}(z)$ as $z\to\infty$ (in what
follows, $Q^{1/2}(z;t)$ will always stand for such a branch).

For a description of the structure of the critical graphs of a general
quadratic differential, e.g. [MR0096806, MR743423]. Since $\text{deg}\,Q=6$,
$J_{t}$ consists of one, two, or three arcs, corresponding (respectively) to
the cases where $Q(z;t)$ has two, four, or six simple zeros, respectively.
This depends on the choice of $t$, and the phase diagram is shown in Figure 2.
It is important to point out that the author of [MR1083917] arrived at the
same diagram heuristically. Away from $J_{t}$, one has freedom in choosing
$\Gamma_{t}$. In particular, let

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

where $e\in J_{t}$ is any and the second equality follows from (4.6) (since
the constant $\ell_{\Gamma_{t}}$ in (4.2) is the same for both connected
components of $J_{t}$ and the integrand is purely imaginary on $J_{t}$, the
choice of $e$ is indeed not important). Clearly, $\mathcal{U}(z;t)$ is a
subharmonic function (harmonic away from $J_{t}$) which is equal to zero
$J_{t}$ by (4.2). The trajectories of $-Q(z;t){\mathrm{d}}z^{2}$ emanating out
of the endpoints of $J_{t}$ belong to the set $\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)=0\\}$ and it follows from the variational condition
(4.2) that $\Gamma_{t}\setminus J_{t}\subseteq\\{z\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}(z;t)<0\\}$. However, within the region
$\\{z\mathrel{\mathop{\mathchar 58\relax}}\mathcal{U}(z;t)<0\\}$ the set
$\Gamma_{t}\setminus J_{t}$ can be varied freely.

The geometry of the critical graphs of $-Q(z;t){\mathrm{d}}z^{2}$ were studied
in great detail in [BGM] in all the above cases. The next subsection
summarizes those results.

###  4.2. Critical graph of $-Q(z){\mathrm{d}}z^{2}$

To state the next theorem, we will need the following notation

######  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
$-Q(z;t){\mathrm{d}}z^{2}$ connecting $z_{1}$ and $z_{2}$, oriented from
$z_{1}$ to $z_{2}$, and $\Gamma(z,e^{\mathrm{i}\theta}\infty)$ (resp.
$\Gamma(e^{\mathrm{i}\theta}\infty,z)$) stands for the orthogonal trajectory
ending at $z$, approaching infinity at the angle $\theta$, and oriented away
from $z$ (resp. oriented towards $z$).131313This notation is unambiguous as
the corresponding trajectories are unique for polynomial differentials as
follows from Teichmüller’s lemma.

######  Definition 4.5.

The open sets $\mathcal{O}_{j},j=1,2,3$ are defined by the property that

$t\in\mathcal{O}_{j}\implies Q(z;t)\text{ has $2j$ simple zeros}.$

These sets are shown in Figure 2, where $\mathcal{O}_{j}$ is labelled the
“$j$-cut region.”

The following summarizes results pertaining to the one-cut regime in
[BGM]*Section 2.4.1.

(a) $t=1+{\mathrm{i}}\in\mathcal{O}_{1}$.

(b) $t=1+3.8{\mathrm{i}}\in\mathcal{O}_{1}$.

(c) $t=1+3.92{\mathrm{i}}\in\mathcal{O}_{1}$.

(d) $t=1+4{\mathrm{i}}\in\mathcal{O}_{1}$.

(e) $t=4{\mathrm{i}}\in\mathcal{O}_{1}$.

(f) $t=-1+4{\mathrm{i}}\in\mathcal{O}_{1}$.

(g) $t_{\mbox{cr}}\in\gamma_{3}$.

(h) $t=-1.35+4{\mathrm{i}}\notin\mathcal{O}_{1}$.

Figure 3.  This sequence of figures shows allowable regions in light blue
through which the contour of integration (for the orthogonal polynomials) must
pass, for a varying collection of values of $t$. Regions in light blue are the
$t$-stable lands where $\mathcal{U}(z;t)<0$ and the regions in white are the
$t$-unstable lands where $\mathcal{U}(z;t)>0$. Figures courtesy of Roozbeh
Gharakhloo.

######  Theorem 4.6 ([BGM]).

Let $\mu_{t}$ and $Q(z;t)$ be as in Theorem 4.4. If
$t\in\overline{\mathcal{O}_{1}}$, then

(4.9) $Q(z;t)=\dfrac{1}{4}(z^{2}-z^{2}_{0}(t))^{2}(z^{2}-b_{1}^{2}(t)),$

where

(4.10)
$b_{1}(t)=\sqrt{\dfrac{2}{3}\left(-t+\sqrt{12+t^{2}}\right)}\quad\text{and}\quad
z_{0}(t)=\sqrt{\dfrac{1}{3}\left(-2t-\sqrt{12+t^{2}}\right)},$

with branches chosen so that for $t>-2$,

$z_{0}(t)={\mathrm{i}}y_{0},\quad y_{0}>0\quad\text{and}\quad a_{1}(t)>0.$

Furthermore,

  1. (a)

if $t\in\mathcal{O}_{1}$, then $J_{t}=\Gamma[-b_{1},b_{1}]$ and we take
$\Gamma_{t}=\Gamma({\mathrm{e}}^{{\mathrm{i}}\pi}\infty,-b_{1})\cup
J_{t}\cup\Gamma(b_{1},{\mathrm{e}}^{0\pi{\mathrm{i}}}\infty),$ see Figure
3(A-D),

  2. (b)

if $t\in\gamma_{j},\ j\in\\{3,4\\}$, then $J_{t}=\Gamma[-a_{1},a_{1}]$ and
$\Gamma_{t}=\Gamma({\mathrm{e}}^{\pi{\mathrm{i}}}\infty,(-1)^{j+1}z_{0})\cup\Gamma[(-1)^{j+1}z_{0},-b_{1}]\cup
J_{t}\cup\Gamma[-b_{1},(-1)^{j}z_{0}]\cup\Gamma((-1)^{j}z_{0},{\mathrm{e}}^{0\pi{\mathrm{i}}}\infty)$,
see Figure 3(G),

  3. (c)

if $t\in\gamma_{j},\ j\in\\{1,2\\}$, then the support of $\mu_{t}$ “splits”
and

$J_{t}=\Gamma[-b_{1},(-1)^{j+1}z_{0}]\cup\Gamma[(-1)^{j+1}z_{0},(-1)^{j}z_{0}]\cup\Gamma[(-1)^{j}z_{0},b_{1}],$

and we choose
$\Gamma_{t}=\Gamma({\mathrm{e}}^{\pi{\mathrm{i}}}\infty,-b_{1})\cup
J_{t}\cup\Gamma(b_{1},{\mathrm{e}}^{0\pi{\mathrm{i}}}\infty)$, see Figure 4

  4. (d)

if $t=\pm{\mathrm{i}}\sqrt{12}$, then $\pm z_{0}=\mp a_{1}$ and
$J_{t}=[-b_{1},b_{1}]$ and $\Gamma_{t}$ can be chosen as in (a),

  5. (e)

if $t=-2,$ then $z_{0}=0$ and the support of $\mu_{t}$ “splits into
$J_{t}=\Gamma[-b_{1},0]\cup\Gamma[0,b_{1}]$, see [BGM]*Figure 6. Here, we
choose $\Gamma_{t}$ as in (c).

(a) $t=-1\in\mathcal{O}_{1}$.

(b) $t=-1+{\mathrm{i}}in\mathcal{O}_{1}$.

(c) $t=-1+1.6{\mathrm{i}}\in\mathcal{O}_{1}$.

(d) $t=-1+1.78{\mathrm{i}}\notin\mathcal{O}_{1}$.

Figure 4.  Same as Figure 3, except wihtout the shading. Figures courtesy of
Roozbeh Gharakhloo.

To state an analogous theorem for $t\in\mathcal{O}_{2}$, we will need the
following bit of notation. Let $I_{t}$ be an analytic arc connecting
$-a_{2}(t),a_{2}(t)$ in such a way that it avoids the origin, lies entirely in
the region $\\{\mathcal{U}(z;t)<0\\}$, and for a fixed neighborhood $U_{\pm
a_{2}}$ of $\pm a_{2}(t)$, $I_{t}\cap U_{\pm a_{2}}$ coincides with orthogonal
trajectories emanating from $\pm a_{2}$.The following theorem summarizes the
relevant results from [BGM]*Section 2.4.2.

######  Theorem 4.7 ([BGM]).

Let $\mu_{t}$ and $Q(z;t)$ be as in Theorem 4.4. If
$t\in\overline{\mathcal{O}_{2}}$, then

(4.11) $Q(z;t)=\dfrac{1}{4}z^{2}(z^{2}-a_{2}^{2}(t))(z^{2}-b_{2}^{2}(t)),$

where

(4.12) $a_{2}(t)=\sqrt{-2-t},\quad b_{2}(t)=\sqrt{2-t},$

and $\sqrt{\diamond}$ is the principle branch. Furthermore, we have that the
support of $\mu_{t}$ is given by

(4.13)
$J_{t}=\Gamma[-b_{2},-a_{2}]\cup\Gamma[a_{2},b_{2}]\mathrel{\mathop{\mathchar
58\relax}}=J_{t,1}\cup J_{t,2},$

where $J_{t,1}=\Gamma[-b_{2},-a_{2}]$ and $J_{t,2}=\Gamma[a_{2},b_{2}]$.
Furthermore, we have

  1. (a)

if $t\in\mathcal{O}_{2}$, then we take
$\Gamma_{t}=\Gamma({\mathrm{e}}^{\pi{\mathrm{i}}}\infty,-b_{2})\cup
J_{t,1}\cup I_{t}\cup
J_{t,2}\cup\Gamma(b_{2},{\mathrm{e}}^{0\pi{\mathrm{i}}}\infty)$, see Figure
5(a),

  2. (b)

if $t\in\gamma_{5}\cup\gamma_{6}$, then we take
$\Gamma_{t}=\Gamma({\mathrm{e}}^{\pi{\mathrm{i}}}\infty,-b_{2})\cup
J_{t,1}\cup I_{t}\cup
J_{2,t}\cup\Gamma(b_{2},{\mathrm{e}}^{0\pi{\mathrm{i}}}\infty)$, see Figure
5(b).

(a) $t=-3+{\mathrm{i}}\in\mathcal{O}_{2}$.

(b) $t=-3+1.5{\mathrm{i}}\in\mathcal{O}_{2}$.

(c) $t=-3+1.6{\mathrm{i}}\not\in\mathcal{O}_{2}$.

Figure 5.  Same as in Figure 3. Figures courtesy of Roozbeh Gharakhloo.

Finally, when $t\in\mathcal{O}_{3}$, $Q(z;t)$ has 6 distinct roots. These are
determined by a combination of algebraic “moment conditions” as well as two
integral equations. Since $t\in\mathcal{O}_{3}$ is outside of the scope of
this work, we simply refer to [BGM, BBGMT]. Since we will be primarily
interested in $t\in\mathcal{O}_{2}$, we record in the following remark a more
explicit description of it.

######  Remark 4.8.

It was shown in [BGM]*Section 4.2 that the arcs $\gamma_{5},\gamma_{6}$ shown
in Figure 2 are exactly the two unbounded trajectories in the left half plane
of the quadratic differential

$\frac{1}{4}(z^{2}+4)({\mathrm{d}}z)^{2}.$

These trajectories are asymptotic to lines at angles $\pm 3\pi/4$. This is
equivalent to (see [BGM]*Section 4.2 for details) the zero-set of the real
part of

$\Phi(z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\int_{2}^{z}\sqrt{z^{2}-4}\ {\mathrm{d}}z.$

A moment’s thought with this expression reveals that
$\mathcal{O}_{2}\subseteq\\{z\ \mathrel{\mathop{\mathchar 58\relax}}\
3\pi/4<\text{arg}(z)<5\pi/4\\}$.

###  4.3. Construction of the $g$-function

The first step in Section 5.1 will be to normalize the initial Riemann-Hilbert
problem RHP-${\mathbf{Y}_{n}}$ at $z=\infty$, for which we require a function
$g(z)$, colloquially known as the _$g$ -function_:

(4.14) $g(z;t)\mathrel{\mathop{\mathchar
58\relax}}=\int_{J_{t}}\log(z-x){\mathrm{d}}\mu_{t}(x),$

where, for a fixed $x\in J$, we take the branch cut of $\log(z-s)$ along
$\Gamma_{t}$. It follows from the definition of $g$ that

(4.15) $U^{\mu_{t}}(z)=-\Re(g(z;t))$

and we can now rewrite the Euler-Lagrange conditions:

(4.16) $\displaystyle-\Re g(z;t)+\dfrac{1}{2}\Re
V(z;t)=\left\\{\begin{array}[]{ll}=\ell_{t},&z\in J_{t},\\\
\geq\ell_{t},&z\in\Gamma_{t}\setminus J_{t}.\end{array}\right.$

Furthermore, we note that

(4.17) $g^{\prime}(z;t)=\int_{J_{t}}\dfrac{{\mathrm{d}}\mu_{t}(x)}{z-x}.$

The $g$ function was constructed in [BGM], and we summarize its relevant
features here. In particular, we focus our attention on
$t\in\mathcal{O}_{2}\cup\mathcal{O}_{3}$, since $t\in\mathcal{O}_{1}$ was
already analyzed in [BGM].

####  4.3.1. Two-cut case.

Combining (4.6),(4.11), and (4.17), we conclude that there exists a constant
$\ell^{(2)}_{*}(t)$ such that 141414Comparing (4.18) with (4.16), we note that
$\Re(\ell_{*}^{(2)}(t))=-2\ell_{t}$

(4.18) $g(z;t)=\dfrac{V(z;t)+\ell_{*}^{(2)}(t)+\eta_{2}(z;t)}{2},\quad
z\in{\mathbb{C}}\setminus\Gamma_{t}(-\infty,b_{2}],$

where

(4.19) $\eta_{2}(z;t)\mathrel{\mathop{\mathchar
58\relax}}=-2\int_{b_{2}}^{z}Q^{1/2}(x;t){\mathrm{d}}x,$

where the path of integration is taken to be in
${\mathbb{C}}\setminus\Gamma_{t}(-\infty,b_{2}]$. It will be convenient to
introduce the following notation: for $e\in\\{\pm a_{2},\pm b_{2}\\}$, let

(4.20) $\eta_{2,e}(z)\mathrel{\mathop{\mathchar
58\relax}}=-2\int_{e}^{z}Q^{1/2}(x){\mathrm{d}}x,\quad$

where the path of integration is taken in

  * •

${\mathbb{C}}\setminus\Gamma_{t}(-\infty,b_{2}]$ when $e=b_{2}$,

  * •

${\mathbb{C}}\setminus\Gamma_{t}[-b_{2},\infty)$ when $e=-b_{2}$,

  * •

${\mathbb{C}}\setminus\Gamma_{t}(-\infty,-a_{2}]\cup\Gamma[a_{2},\infty)$ when
$e\in\\{\pm a_{2}\\}$.

Then, it follows from definition (4.19) that $\eta_{2}(z)=\eta_{2,b_{2}}(z)$.
Note that since $\mu_{t}$ in (4.7) is a probability measure, and using the
evenness of $Q^{1/2}_{+}$, we have

$\dfrac{1}{\pi{\mathrm{i}}}\int_{J_{t,1}}Q^{1/2}_{+}(x){\mathrm{d}}x=\dfrac{1}{\pi{\mathrm{i}}}\int_{J_{t,2}}Q^{1/2}_{+}(x){\mathrm{d}}x=\dfrac{1}{2}.$

Using this, we can deduce that

(4.21)
$\eta_{2,b_{2}}(z;t)=\left\\{\begin{array}[]{l}\eta_{2,a_{2}}(z;t)\pm\pi{\mathrm{i}},\\\
\eta_{2,-a_{2}}(z;t)\pm\pi{\mathrm{i}},\\\ \eta_{2,-b_{2}}(z;t)\pm
2\pi{\mathrm{i}}\end{array}\right.\quad z\in{\mathbb{C}}\setminus\Gamma,$

where the plus/minus signs correspond to $z$ on the left/right of
$\Gamma_{t}$, respectively.

It turns out, $\eta_{2}$ (and consequently, all $\eta_{2,e}$) can be written
in terms of elementary functions. Specifically, note that one can rewrite

(4.22) $Q^{1/2}(z;t)=\dfrac{1}{2}zR^{1/2}(z;t)$

where $R^{1/2}(z;t)=\sqrt{(z^{2}-a_{2}^{2})(z^{2}-b_{2}^{2})}$ is the branch
of the square root analytic in ${\mathbb{C}}\setminus J_{t}$ and satisfies
$R^{1/2}(z)=z^{2}+O(z)$ as $z\to\infty$, then one can check that

(4.23)
$\eta_{2}(z;t)=-\dfrac{1}{4}(z^{2}+t)R^{1/2}(z)+\log\left(\dfrac{2z^{2}+t+R^{1/2}(z)}{2}\right),\quad
z\in{\mathbb{C}}\setminus\Gamma_{t}(-\infty,b_{2}].$

Then, using (4.18), the $g$ functions can be written explicitly

(4.24)
$g(z;t)=\dfrac{1}{2}\left(\dfrac{tz^{2}}{2}+\dfrac{z^{4}}{4}+\ell_{*}^{(2)}(t)\right)-\dfrac{1}{8}(z^{2}+t)R^{1/2}(z)+\dfrac{1}{2}\log\left(\dfrac{2z^{2}+t+R^{1/2}(z)}{2}\right).$

Combining this expression with the requirement that

(4.25) $g(z;t)=\log z+O(z^{-1})$

yields

(4.26) $\ell_{*}^{(2)}(t)=\dfrac{t^{2}}{4}-\dfrac{1}{2}.$

Furthermore, $g(z;t)$ has the following jumps

(4.27)
$(g_{+}-g_{-})(z;t)=\left\\{\begin{array}[]{ll}0,&z\in\Gamma[b_{2},\infty),\\\
\eta_{2,+}(z;t),&z\in\Gamma[a_{2},b_{2}],\\\
\pi{\mathrm{i}},&z\in\Gamma[-a_{2},a_{2}],\\\
\eta_{2,+}(z;t),&z\in\Gamma[-b_{2},-a_{2}],\\\
2\pi{\mathrm{i}},&z\in\Gamma(-\infty,-b_{2}],\end{array}\right.$

and

(4.28)
$\left(g_{+}+g_{-}-V-\ell_{*}^{(2)}\right)(z;t)=\left\\{\begin{array}[]{ll}\eta_{2}(z;t),&z\in\Gamma[b_{2},\infty),\\\
0,&z\in\Gamma[a_{2},b_{2}],\\\
\eta_{2,\pm}(z;t)\mp\pi{\mathrm{i}},&z\in\Gamma[-a_{2},a_{2}],\\\
0,&z\in\Gamma[-b_{2},-a_{2}],\\\ \eta_{2,\pm}(z;t)\mp
2\pi{\mathrm{i}},&z\in\Gamma(-\infty,-b_{2}],\end{array}\right.$

where the second-to-last line in both formulas uses the fact that

$\int_{-a_{2}}^{a_{2}}x\sqrt{(x^{2}-a_{2}^{2})(x^{2}-b_{2}^{2})}{\mathrm{d}}x=0.$

The arcs $\Gamma[a_{2},b_{2}]\cup\Gamma[-b_{2},-a_{2}]$ are solutions of
$\Re[\eta_{2}(z;t)]=0$; in fact, it was shown in [BGM]*Section 3.2 that for
all $t\in\mathcal{O}_{2}$, there exists a (non-unique) choice of $\Gamma_{t}$
so that

(4.29) $\Re[\eta_{2}(z;t)]<0,\quad z\in\Gamma_{t}\setminus J_{t}.$

Furthermore, it follows from (4.8), (4.15) , and the subharmonicity of
$\mathcal{U}$ that $\Re[\eta_{2}(z;t)]$ is subharmonic of any point $z_{0}\in
J_{t}$. We then deduce from the maximum principle for subharmonic functions
that every $z\in J_{t,1}\cup J_{t,2}$ is on the boundary of the set
$\Re[\eta_{2}(z;t)]>0$. See Figure 5.

###  4.4. Szegő function

To handle the general double-scaling limit where
$\\{N_{n}\\}_{n=1}^{\infty}\subseteq{\mathbb{N}}$ with $|N_{n}-n|<C$ for a
fixed $C\geq 0$151515As opposed to only the case $N=n$, which does not require
the construction in this section., we will need the _Szegő function_ of
${\mathrm{e}}^{V(z;t)}$. The following proposition defines and summarizes its
properties.

######  Proposition 4.9.

Let $R^{1/2}(z;t)$ be is as in (4.22). Then, the Szegő function

(4.30) $\mathcal{D}(z;t)\mathrel{\mathop{\mathchar
58\relax}}=\exp\left(\dfrac{1}{2}V(z;t)-\dfrac{1}{8}(z^{2}+t)R^{1/2}(z;t)\right).$

is analytic in $\overline{{\mathbb{C}}}\setminus J_{t}$ and has continuous
traces on $J_{t}\setminus\\{\pm a_{2},\pm b_{2}\\}$ which satisfy

(4.31) $\mathcal{D}_{+}(x;t)\mathcal{D}_{-}(x;t)={\mathrm{e}}^{V(x;t)},\quad
x\in J_{t}.$

Furthermore, $\mathcal{D}(z;t)$ is bounded as $z\to e\in\\{\pm a_{2},\pm
b_{2}\\}$, and

(4.32)
$\mathcal{D}(z;t)={\mathrm{e}}^{-\ell_{*}^{(2)}(t)/2}\left(1+\dfrac{1}{4z^{4}}+\mathcal{O}\left(\dfrac{1}{z^{6}}\right)\right)\quad\text{as}\quad
z\to\infty,$

where $\ell_{*}^{(2)}(t)$ is as in (4.26).

######  Remark 4.10.

Henceforth, we let $D(z;t)\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{e}}^{\ell_{*}^{(2)}(t)/2}\mathcal{D}(z;t)$ be the
_normalized Szegő function._

###### Proof.

From (4.22) it follows that $R^{1/2}(z;t)$ is analytic in
${\mathbb{C}}\setminus J_{t}$ and for $z\in J_{t}\setminus\\{\pm a_{2},\pm
b_{2}\\}$ we have $R^{1/2}(z;t)_{+}=-R^{1/2}(z;t)_{-}$. This implies (4.31).
At $z=\infty$, we note that by (4.6) and the choice if the square root, we
have

(4.33)
$Q^{1/2}(z,t)=\dfrac{V^{\prime}(z,t)}{2}-\dfrac{1}{z}+\mathcal{O}\left(\dfrac{1}{z^{3}}\right)\quad\text{as}\quad
z\to\infty.$

By definition of $R^{1/2}$ and (4.33) we find

(4.34)
$(z^{2}+t)R^{1/2}(z;t)=4V(z;t)+4\ell^{(2)}_{*}(t)+\mathcal{O}\left(\frac{1}{z^{2}}\right)\quad\text{as}\quad
z\to\infty.$

Hence, $\mathcal{D}(z;t)$ has a convergent Laurent expansion centered at
$z=\infty$ of the form (4.32). ∎

###  4.5. Riemann surface

We will need an arsenal of functions that naturally live on an associated
Riemann surface, which we introduce here. Let
$R(z;t)=(z^{2}-a_{2}^{2}(t))(z^{2}-b_{2}^{2}(t))$ and

$\mathfrak{S}\mathrel{\mathop{\mathchar 58\relax}}=\left\\{(z,w)\
\mathrel{\mathop{\mathchar 58\relax}}\ w^{2}=R(z)\right\\}.$

Denote by $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{S}\to\overline{\mathbb{C}}$ the natural projection
$\pi({\boldsymbol{z}})=z$ and by $\diamond^{*}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{S}\to\mathfrak{S}$ the natural involution
${\boldsymbol{z}}=(z,w)\implies{\boldsymbol{z}}^{*}=(z,w)^{*}=(z,-w)$. We use
notation ${\boldsymbol{z}},{\boldsymbol{x}},\boldsymbol{a}$ for points on
$\mathfrak{S}$ with natural projections $z,x,a$.

The function $w({\boldsymbol{z}})$, defined by $w^{2}({\boldsymbol{z}})=R(z)$,
is meromorphic on $\mathfrak{S}$ with simple zeros at the ramification points
$\boldsymbol{E}_{t}=\\{\pm\boldsymbol{a}_{2},\pm\boldsymbol{b}_{2}\\}$, double
poles at the points on top of infinity, and is otherwise non-vanishing and
finite. Put

$\boldsymbol{\Delta}\mathrel{\mathop{\mathchar
58\relax}}=\pi^{-1}(J_{t})\quad\text{and}\quad\mathfrak{S}=\mathfrak{S}^{(0)}\cup\boldsymbol{\Delta}\cup\mathfrak{S}^{(1)},$

where the domains $\mathfrak{S}^{(k)}$ project onto
$\overline{\mathbb{C}}\setminus J_{t}$ with labels chosen so that
$w({\boldsymbol{z}})=(-1)^{k}z^{2}+O(z)$ as ${\boldsymbol{z}}$ approaches the
point on top of infinity within $\mathfrak{S}^{(k)}$. For
$z\in\overline{\mathbb{C}}\setminus J_{t}$ we let $z^{(k)}$ stand for the
unique ${\boldsymbol{z}}\in\mathfrak{S}^{(k)}$ with $\pi({\boldsymbol{z}})=z$.
We define a homology basis on $\mathfrak{S}$ in the following way: we let (see
discussion before Theorem 4.7)

$\boldsymbol{\alpha}\mathrel{\mathop{\mathchar
58\relax}}=\pi^{-1}(I_{t})\quad\text{and}\quad\boldsymbol{\beta}\mathrel{\mathop{\mathchar
58\relax}}=\pi^{-1}(J_{t,1}),$

where $\boldsymbol{\alpha}$ is oriented towards $-\boldsymbol{a}_{2}$ within
$\mathfrak{S}^{(0)}$ and $\boldsymbol{\beta}$ is oriented so that
$\boldsymbol{\alpha},\boldsymbol{\beta}$ form the right pair at
$-\boldsymbol{a}_{2}$, see Figure 6. We will often use the notation
$\mathfrak{S}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{S}\setminus\\{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}\\}$.

•$-\boldsymbol{a}_{2}$•$-\boldsymbol{a}_{2}$•$-\boldsymbol{a}_{2}$•$-\boldsymbol{a}_{2}$•$-\boldsymbol{b}_{2}$•$-\boldsymbol{b}_{2}$•$\boldsymbol{b}_{2}$•$\boldsymbol{a}_{2}$•$\boldsymbol{a}_{2}$$<$$<$$\boldsymbol{\alpha}$$\boldsymbol{\alpha}$$<$$<$$\boldsymbol{\alpha}$$\boldsymbol{\alpha}$$<<$$<<$$\boldsymbol{\beta}$$\boldsymbol{\beta}$$<<$$<<$$\boldsymbol{\beta}$$\boldsymbol{\beta}$$\mathfrak{S}^{(0)}$$\mathfrak{S}^{(1)}$
Figure 6. Schematic representation of the surface $\mathfrak{S}$ (shaded
region represents $\mathfrak{S}^{(1)}$), which topologically is a torus, and
the homology basis $\boldsymbol{\alpha},\boldsymbol{\beta}$ .

Surface $\mathfrak{S}$ is genus 1, and so the vector space of holomorphic
differentials is one dimensional and we choose the following basis

(4.35)
$\mathcal{H}(z)=\left(\int_{\boldsymbol{\alpha}}\dfrac{{\mathrm{d}}x}{w({\boldsymbol{x}})}\right)^{-1}\dfrac{{\mathrm{d}}z}{w({\boldsymbol{z}})},$

which is normalized to have period 1 on $\boldsymbol{\alpha}$; it follows by
classical theory (see, e.g. [MR583745]) that when $t\in\mathcal{O}_{2}$,

$\mathrm{B}(t)\mathrel{\mathop{\mathchar
58\relax}}=\int_{\boldsymbol{\beta}}\mathcal{H}\implies\Im\mathrm{B}(t)>0.$

In what follows, we suppress the dependence of ${\mathrm{B}}(t)$ on $t$ and
write ${\mathrm{B}}(t)={\mathrm{B}}$.

####  4.5.1. Abel’s map

Let ${\mathcal{A}}\mathrel{\mathop{\mathchar
58\relax}}\mathfrak{S}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\to{\mathbb{C}}$
be the abelian integral

(4.36) $\mathcal{A}({\boldsymbol{z}})\mathrel{\mathop{\mathchar
58\relax}}=\int_{{\boldsymbol{b}}_{2}}^{\boldsymbol{z}}\mathcal{H},\quad{\boldsymbol{z}}\in\mathfrak{S}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}},$

where the path of integration is taken in
$\mathfrak{S}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$. ${\mathcal{A}}$
is holomorphic in $\mathfrak{S}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$,
and it follows from the definition of ${\mathrm{B}}$ and the normalization of
$\mathcal{H}$ that

(4.37)
$({\mathcal{A}}_{+}-{\mathcal{A}}_{-})({\boldsymbol{x}})=\left\\{\begin{array}[]{rl}-{\mathrm{B}},&{\boldsymbol{x}}\in{\boldsymbol{\alpha}}\setminus\\{-{\boldsymbol{a}}_{2}\\},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
1,&{\boldsymbol{x}}\in{\boldsymbol{\beta}}\setminus\\{-{\boldsymbol{a}}_{2}\\}.\end{array}\right.$

Let
${\mathrm{Jac}}(\mathfrak{S})={\mathbb{C}}/\\{{\mathbb{Z}}+{\mathrm{B}}{\mathbb{Z}}\\}$
denote the Jacobi variety of $\mathfrak{S}$, whose elements are the following
equivalence classes, given $x\in{\mathbb{C}}$,

$[x]=\\{x+k+j{\mathrm{B}}\in{\mathbb{C}}\ \mathrel{\mathop{\mathchar
58\relax}}x\in\ k,j\in{\mathbb{Z}}\\}.$

It follows from (4.37) that the _Abel’s map_
${\boldsymbol{z}}\mapsto[{\mathcal{A}}({\boldsymbol{z}})]$ is well-defined. In
fact, since $\mathfrak{S}$ has genus 1, it is an isomorphism
[MR583745]*Section III.6. We will use this to formulate a Jacobi inversion
problem in Section 4.6.

###  4.6. Jacobi inversion problem

In view of the Abel map being an isomorphism, let ${\boldsymbol{z}}_{[x]}$
denote the unique solution to the equation

$\left[{\mathcal{A}}\left({\boldsymbol{z}}_{[x]}\right)\right]=[x].$

The problem of determining ${\boldsymbol{z}}_{[x]}$ for a given
$x\in{\mathbb{C}}$ is known as the Jacobi inversion problem. We will be
interested in two problems, namely

(4.38)
$\left[{\mathcal{A}}\left({\boldsymbol{z}}_{k,[x_{*}]}\right)\right]=[x_{*}]\quad\text{
where }\quad x_{*}={\mathcal{A}}\left(0^{(k)}\right)+\dfrac{1}{2},\quad
k\in\\{0,1\\}.$

Due to various symmetries of $\mathfrak{S}$, it turns out we can solve this
problem exactly. This is the content of the following proposition.

######  Proposition 4.11.

Let $x_{*}$ be as above, and $k\in\\{0,1\\}$ then
${\boldsymbol{z}}_{k,[x_{*}]}=0^{(1-k)}$.

###### Proof.

Let $\tilde{{\boldsymbol{\alpha}}}$ be an involution-symmetric cycle
homologous to ${\boldsymbol{\alpha}}$ containing $0^{(0)},0^{(1)}$, and note
the following two symmetries:

(4.39)
$\int_{\pm\boldsymbol{a}_{2}}^{{\boldsymbol{z}}}\mathcal{H}=-\int_{\pm\boldsymbol{a}_{2}}^{{\boldsymbol{z}}^{*}}\mathcal{H},\quad\text{and}\quad\int_{\pm\boldsymbol{a}_{2}}^{0^{(k)}}\mathcal{H}=\int_{0^{(k)}}^{\mp\boldsymbol{a}_{2}}\mathcal{H}.$

Indeed, the first equality follows from
$w({\boldsymbol{z}}^{*})=-w({\boldsymbol{z}}^{*})$ while the other follows
from $w(\pi({\boldsymbol{z}}))=w(-\pi({\boldsymbol{z}}))$. Using the first
symmetry and normalization (4.35), for a path contained in
$\mathfrak{S}^{(1)}$ we have that

$\int_{-\boldsymbol{a}_{2}}^{\boldsymbol{a}_{2}}\mathcal{H}=\dfrac{1}{2}.$

Using this and the above symmetries, we have

$\dfrac{1}{2}=\int_{-\boldsymbol{a}_{2}}^{\boldsymbol{a}_{2}}\mathcal{H}=\int_{-\boldsymbol{a}_{2}}^{0^{(1)}}\mathcal{H}+\int_{0^{(1)}}^{\boldsymbol{a}_{2}}\mathcal{H}=-\int_{-\boldsymbol{a}_{2}}^{0^{(0)}}\mathcal{H}+\int_{0^{(1)}}^{\boldsymbol{a}_{2}}\mathcal{H}\\\
=\int_{0^{(0)}}^{-\boldsymbol{a}_{2}}\mathcal{H}+\int_{0^{(1)}}^{\boldsymbol{a}_{2}}\mathcal{H}=\int^{0^{(0)}}_{\boldsymbol{a}_{2}}\mathcal{H}+\int_{0^{(1)}}^{\boldsymbol{a}_{2}}\mathcal{H}.$

So,
$\left[{\mathcal{A}}\left(0^{(0)}\right)-{\mathcal{A}}\left(0^{(1)}\right)\right]=\left[1/{2}\right]$.
Noting that $[1/2]=[-1/2]$ yields the desired result. ∎

In fact, the proof of the above proposition yields a slightly better identity,
namely that

${\mathcal{A}}(0^{(0)})={\mathcal{A}}(0^{(1)})+\dfrac{1}{2},$

which we will use in the following subsection.

###  4.7. Theta functions

Let $\theta(x;{\mathrm{B}})$ be the Riemann theta function associated with
${\mathrm{B}}$, i.e.,

(4.40)
$\theta(x;{\mathrm{B}})=\sum_{k\in{\mathbb{Z}}}\exp\left(\pi{\mathrm{i}}\mathsf{{\mathrm{B}}}k^{2}+2\pi{\mathrm{i}}xk\right),\quad
x\in{\mathbb{C}}.$

The function $\theta(x)$ is holomorphic in ${\mathbb{C}}$ and enjoys the
following periodicity properties:

(4.41)
$\theta(x+j+{\mathrm{B}}m)=\exp\left\\{-\pi{\mathrm{i}}{\mathrm{B}}m^{2}-2\pi{\mathrm{i}}xm\right\\}\theta(x),\qquad
j,m\in{\mathbb{Z}}.$

It is also known that $\theta(x)$ vanishes only at the points of the lattice
$\left[\frac{\mathsf{B}+1}{2}\right]$. Let

(4.42)
$\Theta({\boldsymbol{z}})=\frac{\theta\left({\mathcal{A}}({\boldsymbol{z}})-{{\mathcal{A}}}\left(0^{(1)}\right)-\frac{\mathsf{B}+1}{2}\right)}{\theta\left({\mathcal{A}}({\boldsymbol{z}})-{{\mathcal{A}}}\left(0^{(0)}\right)-\frac{\mathsf{B}+1}{2}\right)}.$

The functions $\Theta({\boldsymbol{z}})$ are meromorphic on
$\mathfrak{S}\setminus\boldsymbol{\alpha}$ with exactly one simple pole at
${\boldsymbol{z}}=0^{(0)}$, and exactly one simple zero at
${\boldsymbol{z}}=0^{(1)}$ ($\Theta$ can be analytically continued to
multiplicatively multi-valued functions on the whole surface $\mathfrak{S}$;
thus, we can talk about simplicity of a pole or zero whether it belongs to the
cycles of a homology basis or not). Moreover, according to (4.37), (4.40), and
(4.42), they possess continuous traces on
$\boldsymbol{\alpha},\boldsymbol{\beta}$ away from $-\boldsymbol{a}_{2}$ that
satisfy

(4.43)
$\Theta_{+}({\boldsymbol{x}})={\mathrm{e}}^{\pi{\mathrm{i}}}\Theta_{-}({\boldsymbol{x}}),\quad{\boldsymbol{x}}\in{\boldsymbol{\alpha}}\setminus\\{-\boldsymbol{a}_{2}\\}.$

###  4.8. More auxiliary functions

Let

(4.44) $\gamma(z)\mathrel{\mathop{\mathchar
58\relax}}=\left(\dfrac{z-b_{2}}{z+b_{2}}\dfrac{z+a_{2}}{z-a_{2}}\right)^{1/4},\quad
z\in\overline{{\mathbb{C}}}\setminus J_{t},$

be the branch analytic outside of $J_{t}$ and satisfying $\gamma(\infty)=1$.
Furthermore, let

(4.45) $A(z)\mathrel{\mathop{\mathchar
58\relax}}=\dfrac{\gamma(z)+\gamma^{-1}(z)}{2},\quad
B(z)\mathrel{\mathop{\mathchar
58\relax}}=\dfrac{\gamma(z)-\gamma^{-1}(z)}{2{\mathrm{i}}},\quad
z\in\overline{{\mathbb{C}}}\setminus J_{t}.$

Both $A(z),B(z)$ are analytic in $\overline{{\mathbb{C}}}\setminus J_{t}$ and
satisfy

(4.46) $A(\infty)=1,\quad\text{and}\quad B(\infty)=0.$

Furthermore, they satisfy

(4.47) $A_{+}(x)=B_{-}(x)\quad\text{and}\quad A_{-}(x)=-B_{+}(x),\quad x\in
J_{t}\setminus\\{\pm a_{2},\pm b_{2}\\}.$

Observe that with the choice of branch of $\gamma$, we have that $B(0)=0$.
Indeed, the equation $(AB)(z)=0$ is equivalent to $\gamma^{4}(z)=1$, which has
two solutions in $\overline{{\mathbb{C}}}$, namely $z=\infty$ and $z=0$. To
see that $z=0$ is a root of $B(z)$ instead of $A(z)$, we’ll show that
$\gamma(0)=1$. To this end, let $w=\gamma^{4}(z)$ and observe that
$\gamma^{4}(J_{t,1})$ is a contour connecting $w=0$ to $w=\infty$. Observe
that

$\gamma^{4}(z)=\gamma^{4}(x)\implies z=x\quad\text{or}\quad
z=-\dfrac{a_{2}b_{2}}{x}.$

Let $L$ denote a contour connecting $z=\infty$ to $z=0$ while avoiding
$J_{t,1}$ and its image under the map $z\mapsto-a_{2}b_{2}/z$. Then,
$\gamma^{4}(L)$ is a closed contour containing $w=1$ and not winding around
the origin. Hence, the continuation of the principle branch of
$\diamond^{1/4}$ along $\gamma^{4}(L)$ is single-valued, and produces
$\gamma(L)$. Finally, since $\gamma(\infty)=1$, we have $\gamma(0)=1$ as
desired.

###  4.9. Statement of asymptotic results

To formulate the following theorem requires the following definition (cf.
[MR3607591]*Definition 4.2).

######  Definition 4.12.

An estimate $f_{n}(z,t)=\mathcal{O}(n^{-\alpha})$ holds $(z,t)$-_locally
uniformly_ for $z\in V$ as $n\to\infty$ if for each compact
$T\subseteq\mathcal{O}_{2}$ and any collection of compact sets
$\\{K_{t}\\}_{t\in K}$ such that $\cup_{t\in T}K_{t}\subseteq V$, there is a
constant $C(V;T)<\infty$ such that

$|f_{n}(z;t)|\leq C(V;T)n^{-\alpha},\quad z\in K_{t},\quad t\in T,$

for all $N$ large enough.

We are now ready to state our main results regarding the asymptotic behavior
of the orthogonal polynomials and their recurrence coefficients and
normalizing constants.

######  Theorem 4.13.

Let $t\in\mathcal{O}_{2}$ and $\\{N_{n}\\}_{n\in{\mathbb{N}}}$ be a sequence
of integers such that there exists $C\in{\mathbb{N}}$ with $|N_{n}-n|<C$ for
all $n\in{\mathbb{N}}$. Then,

(4.48)
$P_{n}(z)={\mathrm{e}}^{ng(z)}\left(\left(D^{(N_{n}-n)}(z)A(z)\cdot\left\\{\begin{array}[]{ll}1,&n\in
2{\mathbb{N}},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{\Theta^{(0)}(\infty)}{\Theta^{(0)}(z)},&n\in{\mathbb{N}}\setminus
2{\mathbb{N}}\end{array}\right\\}\right.+\mathcal{O}\left(\dfrac{1}{n}\right)\right),\quad\text{as}\quad
n\to\infty,$

$(z,t)$-locally uniformly in ${\mathbb{C}}\setminus J_{t}$.

######  Remark 4.14.

Note that the oddness of the function $P_{2n+1}(z;t,N)$ is reflected in the
fact that the leading term in (4.48) vanishes at $z=0$ since $\Theta^{(0)}(z)$
possesses a simple pole there.

Formulas similar to (4.48) can easily be deduced in the interior of $J_{t}$,
but since we do not use these we do not state them here. One immediate
consequence of Theorem 4.13 and (4.25) is that for all $t\in\mathcal{O}_{2}$
and $n$ large enough, $\text{deg}P_{n}=n$. In view of the discussion in
Section 3.2 we have the following corollary.

######  Corollary 4.15.

Let $K\subseteq\mathcal{O}_{2}$ be a compact set and
$\\{N_{n}\\}_{n\in{\mathbb{N}}}$ be a sequence such that for all $n$. Then,
there exists $n_{*}\in{\mathbb{N}}$ large enough so that for all $n>n_{*}$,
$H_{n}(t,N_{n})$ is nonvanishing for $t\in K$, and
$h_{n}(t,N_{n}),\gamma_{n}^{2}(t,N_{n})$ are nonvanishing and bounded for
$t\in K$.

######  Remark 4.16.

The same boundedness and nonvanishing result is already known for
$K\subseteq\mathcal{O}_{1}$; this follows directly from the asymptotic
analysis in [BGM]. In fact, motivated by Figure 1, one might suspect that the
results can be extended to _closed_ subsets of
$\mathcal{O}_{1},\mathcal{O}_{2}$. An analogous story takes place in the Cubic
Model, where boundedness on compact subsets of the one-cut region was shown in
[MR3607591] and upgraded to closed subsets in [BBDY].

The asymptotic analysis in Section 5 also yields asymptotic formulas for
$\gamma_{n}^{2},h_{n}$, which are stated in the following theorem.

######  Theorem 4.17.

Let $t\in\mathcal{O}_{2}$, $\\{N_{n}\\}_{n\in{\mathbb{N}}}$ be as in Theorem
4.13, and $a_{2}(t),b_{2}(t)$ be as in (4.12). Then,

(4.49)
$\gamma^{2}_{n}(t,N_{n})=\left\\{\begin{array}[]{ll}\dfrac{1}{4}(a_{2}(t)-b_{2}(t))^{2},&n\in
2{\mathbb{N}},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{4}{(a_{2}(t)-b_{2}(t))^{2}},&n\in{\mathbb{N}}\setminus
2{\mathbb{N}}.\end{array}\right\\}+\mathcal{O}\left(\dfrac{1}{n}\right)\quad\text{as}\quad
n\to\infty.$

Furthermore, with $\ell^{(2)}_{*}$ given by (4.26), we have

(4.50)
$h_{n}(t,N_{n})=2\pi{\mathrm{e}}^{N_{n}\ell_{*}^{(2)}}\left\\{\begin{array}[]{ll}\dfrac{1}{2}(a_{2}(t)-b_{2}(t)),&n\in
2{\mathbb{N}},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{2}{a_{2}(t)-b_{2}(t)},&n\in{\mathbb{N}}\setminus
2{\mathbb{N}}.\end{array}\right\\}+\mathcal{O}\left(\dfrac{1}{n}\right)\quad\text{as}\quad
n\to\infty.$

Finally, for all $n\in{\mathbb{N}}$ large enough, we have

(4.51) $\displaystyle\mathfrak{p}_{2n,2n-2}(t,N_{2n})$
$\displaystyle={nt}+\dfrac{(a_{2}-b_{2})^{2}}{8}+\mathcal{O}\left(\frac{1}{n}\right)\quad\text{as}\quad
n\to\infty,$ (4.52) $\displaystyle\mathfrak{p}_{2n-1,2n-3}(t,N_{2n})$
$\displaystyle={nt}+\dfrac{3a_{2}^{2}+2a_{2}b_{2}+3b_{2}^{2}}{8}+\mathcal{O}\left(\frac{1}{n}\right)\quad\text{as}\quad
n\to\infty.$

Corollary 1.5 is an immediate consequence of Theorems 1.3 and 4.17. We end
this section with a few remarks:

######  Remark 4.18.

We may deduce (5.33) from (4.51), (4.52), Lemma 3.3, and the identity

$\dfrac{(a_{2}+b_{2})^{2}}{4}=\dfrac{4}{(a_{2}-b_{2})^{2}}.$

The latter follows directly from definitions (4.12).

######  Remark 4.19.

A particular instance of (4.49) has already appeared in the literature; when
$t\in(-\infty,-2)$, (4.49) reduces to

(4.53)
$\gamma^{2}_{n}(t,N_{n})=\left\\{\begin{array}[]{ll}\dfrac{-t-\sqrt{t^{2}-4}}{2},&n\in
2{\mathbb{N}},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\dfrac{-t+\sqrt{t^{2}-4}}{2},&n\in{\mathbb{N}}\setminus
2{\mathbb{N}},\end{array}\right\\}+\mathcal{O}\left(\dfrac{1}{n}\right)\quad\text{as}\quad
n\to\infty,$

which agrees with [MR1715324]*Theorem 1.1 for all $n\in{\mathbb{N}}$ large
enough.

######  Remark 4.20.

We take a moment to point out that the leading orders in (1.31), (1.32) are
consistent with (3.14), (3.15), and (1.3).

##  5\. Riemann-Hilbert Analysis

###  5.1. Initial Riemann-Hilbert problem

Henceforth, we omit the dependence on $t$ whenever it does not cause
ambiguity. In what follows, we will denote matrices with bold, capitalized
symbols (e.g. $\mathbf{Y}$), with the exception of the Pauli matrices:

$\mathbb{I}\mathrel{\mathop{\mathchar 58\relax}}=\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\quad\sigma_{1}\mathrel{\mathop{\mathchar
58\relax}}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},\quad\sigma_{2}\mathrel{\mathop{\mathchar
58\relax}}=\begin{bmatrix}0&-{\mathrm{i}}\\\
{\mathrm{i}}&0\end{bmatrix}\quad\text{and}\quad\sigma_{3}\mathrel{\mathop{\mathchar
58\relax}}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix},$

and will use the notation

$f^{\sigma_{3}}\mathrel{\mathop{\mathchar 58\relax}}=\begin{bmatrix}f&0\\\
0&f^{-1}\end{bmatrix}.$

We are seeking solutions of the following sequence of Riemann-Hilbert problems
for $2\times 2$ matrix functions (RHP-${\mathbf{Y}_{n}}$):

  1. (a)

$\mathbf{Y}_{n}(z)$ is analytic in ${\mathbb{C}}\setminus\Gamma$ and
$\lim_{{\mathbb{C}}\setminus\Gamma\ni
z\to\infty}\mathbf{Y}_{n}(z)z^{-n\sigma_{3}}=\mathbf{I}$;

  2. (b)

$\mathbf{Y}_{n}(z)$ has continuous traces on $\Gamma$ that satisfy

$\mathbf{Y}_{n,+}(x)=\mathbf{Y}_{n,-}(x)\begin{bmatrix}1&{\mathrm{e}}^{-N_{n}V(x)}\\\
0&1\end{bmatrix},$

where, as before, $V(z)$ is given by (1.6) and the sequence $\\{N_{n}\\}$ is
such that $|n-N_{n}|\leq N_{*}$ for some $N_{*}>0$.

The connection of RHP-${\mathbf{Y}_{n}}$ to orthogonal polynomials was first
demonstrated by Fokas, Its, and Kitaev in [FIK2] and lies in the following. If
the solution of RHP-${\mathbf{Y}_{n}}$ exists, then it is necessarily of the
form

(5.1)
$\mathbf{Y}_{n}(z)=\begin{bmatrix}P_{n}(z)&\big{(}\mathcal{C}P_{n}{\mathrm{e}}^{-N_{n}V}\big{)}(z)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
-\frac{2\pi{\mathrm{i}}}{h_{n-1}}P_{n-1}(z)&-\frac{2\pi{\mathrm{i}}}{h_{n-1}}\big{(}\mathcal{C}P_{n-1}{\mathrm{e}}^{-N_{n}V}\big{)}(z)\end{bmatrix},$

where $P_{n}(z)=P_{n}(z;t,N_{n})$ are the polynomial satisfying orthogonality
relations (1.5), $h_{n}=h_{n}(t,N_{n})$ are the constants defined in (1.10),
and $\mathcal{C}f(z)$ is the Cauchy transform of a function $f$ given on
${\mathbb{R}}$, i.e.,

$(\mathcal{C}f)(z)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2\pi{\mathrm{i}}}\int_{\mathbb{R}}\frac{f(x)}{x-z}{\mathrm{d}}x.$

Below, we show the solvability of RHP-${\mathbf{Y}_{n}}$ for all
$n\in{\mathbb{N}}(t,\varepsilon)$ large enough following the framework of the
steepest descent analysis introduced by Deift and Zhou [MR1207209]. The latter
lies in a series of transformations which reduce RHP-${\mathbf{Y}_{n}}$ to a
problem with jumps asymptotically close to identity.

###  5.2. Normalized Riemann-Hilbert problem and lenses

Suppose that $\mathbf{Y}_{n}(z)$ is a solution of RHP-${\mathbf{Y}_{n}}$. Put

(5.2) $\mathbf{T}_{n}(z)\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{e}}^{-n\ell^{(2)}_{*}\sigma_{3}/2}\mathbf{Y}_{n}(z){\mathrm{e}}^{-n(g(z)-\ell^{(2)}_{*}/2)\sigma_{3}},$

where the function $g(z)$ is defined by (4.14) and $\ell^{(2)}_{*}$ was
introduced in (4.18). Then

$\mathbf{T}_{n,+}(x)=\mathbf{T}_{n,-}(x)\begin{bmatrix}{\mathrm{e}}^{-n(g_{+}(x)-g_{-}(x))}&{\mathrm{e}}^{n(g_{+}(x)+g_{-}(x)-V(x)-\ell^{(2)}_{*})+(n-N_{n})V(x)}\\\
0&{\mathrm{e}}^{-n(g_{-}(x)-g_{+}(x))}\end{bmatrix},$

$x\in\Gamma$, and therefore we deduce from (4.25), (4.27), and (4.28) that
$\mathbf{T}_{n}(z)$ solves RHP-${\mathbf{T}_{n}}$:

  * (a)

$\mathbf{T}_{n}(z)$ is analytic in ${\mathbb{C}}\setminus\Gamma$ and
$\lim_{{\mathbb{C}}\setminus\Gamma\ni
z\to\infty}\mathbf{T}_{n}(z)=\mathbb{I}$;

  * (b)

$\mathbf{T}_{n}(z)$ has continuous traces on $\Gamma\setminus\\{\pm a_{2},\pm
b_{2}\\}$ that satisfy

$\mathbf{T}_{n,+}(x)=\mathbf{T}_{n,-}(x)\left\\{\begin{array}[]{ll}\begin{bmatrix}1&{\mathrm{e}}^{n(\eta_{2,+}(x)+(n-N_{n})V(x)}\\\
0&1\end{bmatrix},&x\in\Gamma(e^{{\mathrm{i}}\pi}\infty,-b_{2}),\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
\begin{bmatrix}1&{\mathrm{e}}^{n\eta_{2}(x)+(n-N_{n})V(x)}\\\
0&1\end{bmatrix},&x\in\Gamma(b_{2},{{\mathrm{e}}^{0\pi{\mathrm{i}}}}\infty),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\begin{bmatrix}{\mathrm{e}}^{n\pi{\mathrm{i}}}&{\mathrm{e}}^{n(\eta_{2,+}(x)-\pi{\mathrm{i}})+(n-N_{n})V(x)}\\\
0&{\mathrm{e}}^{-n\pi{\mathrm{i}}}\end{bmatrix},&x\in\Gamma(-a_{2},a_{2}),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\begin{bmatrix}{\mathrm{e}}^{-n\eta_{2,+}(x)}&{\mathrm{e}}^{(n-N_{n})V(x)}\\\
0&{\mathrm{e}}^{n\eta_{2,+}(x)}\end{bmatrix},&x\in\Gamma(-b_{2},-a_{2})\cup\Gamma(a_{2},b_{2}),\end{array}\right.$

Clearly, if RHP-${\mathbf{T}_{n}}$ is solvable and $\mathbf{T}_{n}(z)$ is the
solution, then by inverting (5.2) one obtains a matrix $\mathbf{Y}_{n}(z)$
that solves RHP-${\mathbf{Y}_{n}}$.

To proceed, we observe the factorization of the jump matrices on $J_{t}$

$\begin{bmatrix}{\mathrm{e}}^{-n\eta_{2,+}(x)}&{\mathrm{e}}^{(n-N_{n})V(x)}\\\
0&{\mathrm{e}}^{n\eta_{2,+}(x)}\end{bmatrix}\\\ =\begin{bmatrix}1&0\\\
{\mathrm{e}}^{n\eta_{2,+}(x)-(n-N_{n})V(x)}&1\end{bmatrix}\begin{bmatrix}0&{\mathrm{e}}^{(n-N_{n})V(x)}\\\
-{\mathrm{e}}^{-(n-N_{n})V(x)}&0\end{bmatrix}\begin{bmatrix}1&0\\\
{\mathrm{e}}^{-n\eta_{2,+}(x)-(n-N_{n})V(x)}&1\end{bmatrix}.$

This motivates the transformation: Denote by $J_{\pm}$ smooth contours
connecting $\pm a_{2}$ and $\pm b_{2}$ and otherwise remaining in the region
$\Re[\eta_{2}(z)]>0$, as shown in Figure 7.

$J_{+}$$J_{-}$$J_{+}$$J_{-}$

Figure 7. Schematic representation of $\Gamma\cup J^{\pm}$ superimposed on
Figure 5(A)

Denote the regions bounded by $J_{t}$ and $J_{\pm}$ by $L_{\pm}$ and define

(5.3) $\mathbf{S}_{n}(z)\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{T}_{n}(z)\left\\{\begin{array}[]{ll}\begin{bmatrix}1&0\\\
\mp e^{-n\eta_{2}(z)-(n-N_{n})V(z)}&1\end{bmatrix}&z\in L_{\pm},\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ \mathbb{I},&\text{otherwise}.\end{array}\right.$

Then, $\mathbf{S}$ solves RHP-${\mathbf{S}_{n}}$:

  1. (a)

$\mathbf{S}_{n}(z)$ is analytic in ${\mathbb{C}}\setminus(\Gamma\cup J_{\pm})$
and $\lim_{{\mathbb{C}}\setminus\Gamma\ni
z\to\infty}\mathbf{S}_{n}(z)=\mathbb{I}$,

  2. (b)

$\mathbf{S}_{n}(z)$ has continuous traces on $\Gamma\cup
J_{\pm}\setminus\\{\pm a_{2},\pm b_{2}\\}$ which satisfy

$\mathbf{S}_{n,+}(x)=\mathbf{S}_{n,-}(x)\left\\{\begin{array}[]{ll}\begin{bmatrix}1&{\mathrm{e}}^{n(\eta_{2,+}(x)+(n-N_{n})V(x)}\\\
0&1\end{bmatrix},&x\in\Gamma(e^{{\mathrm{i}}\pi}\infty,-b_{2}),\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\
\begin{bmatrix}1&{\mathrm{e}}^{n\eta_{2}(x)+(n-N_{n})V(x)}\\\
0&1\end{bmatrix},&x\in\Gamma(b_{2},{{\mathrm{e}}^{0\pi{\mathrm{i}}}}\infty),\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\begin{bmatrix}{\mathrm{e}}^{n\pi{\mathrm{i}}}&{\mathrm{e}}^{n(\eta_{2,+}(x)-\pi{\mathrm{i}})+(n-N_{n})V(x)}\\\