Gaussian unitary ensemble with boundary spectrum singularity and $\sigma$-form of the Painlev\'{e} II equation

TL;DR

This paper analyzes the asymptotic behavior of the Gaussian unitary ensemble with boundary spectrum singularities, revealing connections to Painlevé equations and deriving limits for eigenvalue distributions.

Contribution

It introduces a novel analysis of perturbed Gaussian ensembles with boundary singularities, linking asymptotics to Painlevé equations and deriving new double scaling limits.

Findings

Asymptotics of Hankel determinants expressed via Painlevé XXXIV and Painlevé II equations.

Double scaling limits of eigenvalue distributions and correlation kernels obtained.

Asymptotic properties of Painlevé functions studied in the context of random matrix theory.

Abstract

We consider the Gaussian unitary ensemble perturbed by a Fisher-Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlev\'{e} XXXIV equation and the $\sigma$-form of the Painlev\'{e} II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlev\'{e} XXXIV functions and the $\sigma$-form of the Painlev\'{e} II equation are also studied.

Equations

$${\rho }_n({\lambda }_1,...,{\lambda }_n)=\frac{1}{Z_{n,\alpha ,\omega }}\prod _{i=1}^{n}w({\lambda }_i)\prod _{1\le i<j\le n}^n|{\lambda }_i-{\lambda }_j{|}^2,$$

$$w(x;\alpha,\mu,\omega)=e^{-x^{2}}|x-\mu|^{2\alpha}\left\{\begin{array}[]{cc}1&x\leq\mu\\ \omega&x>\mu,\end{array}\right.$$

$${\rho }_n({\lambda }_1,...,{\lambda }_n)=\det [K_n({\lambda }_i,{\lambda }_j){]}_{1\le i,j\le n},$$

$$K_n(x,y)=\sqrt{w(x)w(y)}\sum _{k=0}^{n-1}{P}_k(x)P_k(y),$$

$$z{\pi }_n(z)={\pi }_{n+1}(z)+a_n{\pi }_n(z)+b_n^2{\pi }_{n-1}(z).$$

$$H_n(\mu ;\alpha ,\omega )$$

$$Z_{n,\alpha ,\omega }=n!H_n(\mu ;\alpha ,\omega ).$$

$$\mathrm{Pro}({\lambda }_{max}\le \mu |\lambda =\mu \text{mbox}isaneigenvaluewithmultiplicity\alpha )=\frac{H_n(\mu ;\alpha ,0)}{H_n(\mu ;\alpha ,1)}.$$

$$\mathrm{Pro}\left({\lambda }_{max}^{\mathrm{R}}\le \mu |\lambda =\mu \text{mbox}isaneigenvaluewithmultiplicity\alpha \right)=\frac{H_n(\mu ;\alpha ,\omega )}{H_n(\mu ;\alpha ,1)}.$$

$$\underset{n\to \infty }{\lim }\mathrm{Pro}\left(\sqrt{2}{n}^{1/6}\left({\lambda }_{max}-\sqrt{2n}\right)\le s\right)=\exp \left(-{\int }_s^{\infty }(x-s)q^2(x)dx\right),$$

$$q_{xx}=xq+2q^3,$$

$$q(x)\sim \mathrm{Ai}(x),x\to +\infty .$$

$$\mathrm{Pro}\left({\lambda }_{max}\le \sqrt{2n}+\frac{s}{\sqrt{2}{n}^{1/6}}\right)=\frac{H_n(\mu ;0,0)}{H_n(\mu ;0,1)}.$$

$$\mathrm{Pro}\left({\lambda }_{max}\le \sqrt{2n}+\frac{s}{\sqrt{2}{n}^{1/6}}\right)=\frac{H_n(\mu ;0,1-k^2)}{H_n(\mu ;0,1)}.$$

$$\mathrm{Pro}\left({\lambda }_{max}\le \sqrt{2n}+\frac{s}{\sqrt{2}{n}^{1/6}}\right)=\exp \left(-{\int }_s^{\infty }(x-s)q^2(x)dx\right)(1+o(1)),$$

$$q(x)\sim k\mathrm{Ai}(x),x\to +\infty .$$

$$({\sigma }^{\prime \prime }{)}^2+4({\sigma }^{\prime }{)}^3-4s({\sigma }^{\prime }{)}^2+4{\sigma }^{\prime }\sigma -(2\alpha {)}^2=0,$$

$$\sigma (s)\sim \frac{s^2}{4}+\frac{16{\alpha }^2-1}{8}{s}^{-1},s\to -\infty$$

$$\sigma (s)\sim -2\alpha s^{1/2}-\frac{{\alpha }^2}{s},s\to +\infty$$

$${\sigma }^{\prime }=-q^2,$$

$$\begin{array}{rl}\sigma (s)=& -2\alpha s^{1/2}\left[\sum _{k=0}^m{a}_k{s}^{-\frac{3}{2}k}+O\left(s^{-\frac{3(m+1)}{2}}\right)\right]\\ & +\left(e^{2\pi i\alpha }-\omega \right)\frac{\Gamma (2\alpha +1)}{2^{3+6\alpha }\pi }{s}^{-(3\alpha +1)}{e}^{-\frac{4}{3}{s}^{3/2}}\left[1+O\left(s^{-\frac{1}{4}}\right)\right],\end{array}$$

$$\sigma (s)=\frac{s^2}{4}+\frac{16{\alpha }^2-1}{8}{s}^{-1}+O(s^{-5/2})\text{mbox}ass\to -\infty$$

$$\sigma (s)=$$

$$+O(s^{3|\mathrm{Re}\beta |-\frac{5}{2}})\text{mbox}ass\to -\infty$$

$$u_{ss}=\frac{u_s^2}{2u}+4u^2+2su-\frac{(2\alpha {)}^2}{2u},$$

$$\begin{array}{rl}u(s)=& \frac{\alpha }{\sqrt{s}}\left[\sum _{k=0}^m{c}_k{s}^{-\frac{3}{2}k}+O\left(s^{-\frac{3(m+1)}{2}}\right)\right]\\ & +\left(e^{2\pi i\alpha }-\omega \right)\frac{\Gamma (2\alpha +1)}{2^{2+6\alpha }\pi }{s}^{-(3\alpha +\frac{1}{2})}{e}^{-\frac{4}{3}{s}^{3/2}}\left[1+O\left(s^{-\frac{1}{4}}\right)\right],\end{array}$$

$$u(s)=-\frac{s}{2}+\frac{16{\alpha }^2-1}{8}{s}^{-2}+O(s^{-7/2})\text{mbox}ass\to -\infty$$

$$u(s)=\frac{1}{\sqrt{-s}}\left[i\beta +\frac{1}{2}\frac{\Gamma (1+\alpha -\beta )}{\Gamma (\alpha +\beta )}{e}^{i\theta (s;\alpha ,\beta )}+\frac{1}{2}\frac{\Gamma (1+\alpha +\beta )}{\Gamma (\alpha -\beta )}{e}^{-i\theta (s;\alpha ,\beta )}\right]+O(s^{3|\mathrm{Re}\beta |-2})$$

$$\frac{d}{d{\mu }_n}\ln H_n({\mu }_n;\alpha ,\omega )=\sqrt{2n}\left(2\alpha +\frac{\sigma (s)}{n^{1/3}}+\frac{\alpha (u(s)+s)}{n^{2/3}}+O\left(\frac{1}{n}\right)\right),$$

$$\begin{array}{rl}& \underset{n\to +\infty }{\lim }\mathrm{Pro}\left({\lambda }_{max}^{\mathrm{R}}\le {\mu }_n|\lambda ={\mu }_n\text{mbox}isaneigenvaluewithmultiplicity\alpha \right)\\ & =\underset{n\to +\infty }{\lim }\frac{H_n({\mu }_n;\alpha ,\omega )}{H_n({\mu }_n;\alpha ,1)}\\ & =\exp \left({\int }_s^{+\infty }\sigma (t;\alpha ,1)-\sigma (t;\alpha ,\omega )dt\right),\end{array}$$

$$a_n=-\frac{u(s)}{2^{1/2}}{n}^{-1/6}+O(n^{-1/2})$$

Full text

Gaussian unitary ensemble with boundary spectrum singularity and $\sigma$-form of the Painlevé II equation

Xiao-Bo Wua, Shuai-Xia Xub111Corresponding author (Shuai-Xia Xu). E-mail address: xushx3@mail.sysu.edu.cn * and Yu-Qiu Zhaoc*

( *a**School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China

bInstitut Franco-Chinois de l’Energie Nucleaire, Sun Yat-sen University, GuangZhou 510275, China

cDepartment of Mathematics, Sun Yat-sen University, GuangZhou 510275, China*)

**Abstract ** We consider the Gaussian unitary ensemble perturbed by a Fisher-Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the $\sigma$-form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the $\sigma$-form of the Painlevé II equation are also studied.

2010 Mathematics subject classification: 33E17; 34M55; 41A60

Keywords: Painlevé XXXIV equation, perturbed Gaussian unitary ensemble, Hankel determinant, orthogonal polynomials, Riemann-Hilbert approach

1 Introduction and statement of results

We consider the perturbed Gaussian Unitary Ensemble (pGUE) defined by the following joint probability density function of the eigenvalues

[TABLE]

with $w(x)$ being the Gaussian weight perturbed by a Fisher-Hartwig singularity, that is,

[TABLE]

where $\alpha>-\frac{1}{2}$, the constant $\omega\in\mathbb{C}\setminus(-\infty,0)$, and the partition function $Z_{n,\alpha,\omega}$ is a normalization constant. It is readily seen that there is a singularity at $x=\mu$, simultaneously of root type and jump type. The case where $\mu=\mu_{n}$ approaches the soft edge $\sqrt{2n}$ is of particular interest to us.

It is well known that (1.1) admits a determinantal form

[TABLE]

where

[TABLE]

and $P_{k}(x)$ are orthonormal polynomials with respect to the weight (1.2). For $\alpha>-\frac{1}{2}$ and $\omega\geq 0$, the system of orthogonal polynomials are well defined. We will prove later that the orthogonal polynomials $P_{n}$ are also well defined for $\alpha>-\frac{1}{2}$, $\omega\in\mathbb{C}\setminus(-\infty,0)$ and $n$ large enough. Let $P_{n}(x)=\gamma_{n}\pi_{n}(x)$, $\gamma_{n}$ being the leading coefficient of the orthonormal polynomial, we have the three-term recurrence relation

[TABLE]

Let $H_{n}(\mu;\alpha,\omega)$ be the Hankel determinant with respect to (1.2), that is,

[TABLE]

then the normalization constant $Z_{n,\alpha,\omega}$ in (1.1) is related to the Hankel determinant via

[TABLE]

The pGUE arises naturally in the statistics of eigenvalues of Gaussian unitary ensemble of random matrices. For $\alpha\in\mathbb{N}$ and $\omega=1$, the pGUE can be interpreted as the probability density function of the classical Gaussian unitary ensemble under the condition that $\mu$ is an eigenvalue with multiplicity $\alpha$ [11, 14]. Moreover, the distribution of the largest eigenvalue of this conditioning Gaussian unitary ensemble can be expressed as the ratio of two Hankel determinants with different parameters defined in (1.6)

[TABLE]

Generally, if we remove each eigenvalue of the conditioning Gaussian unitary ensemble with probability $\omega\in[0,1]$, then the distribution of the remaining largest eigenvalue is described as

[TABLE]

The study of thinning and conditioning random processes can be found in [1, 2, 3].

For the Gaussian unitary ensemble, there is the celebrated Tracy-Widom formula for the distribution of the largest eigenvalue [21]

[TABLE]

where $q(x)$ is the Hastings-Mcleod solution to the Painlevé II equation

[TABLE]

determined by the boundary condition

[TABLE]

It is noted that, if we take $\alpha=0$ and $\mu=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$ in (1.1), then

[TABLE]

The Tracy-Widom distribution holds for general random matrix ensembles [6], and is thus termed a universality property.

In [1], Bogatskiy, Claeys and Its studied the thinned Gaussian unitary ensemble. By removing each eigenvalue of the Gaussian unitary ensemble with probability $1-k^{2}$, they arrive at a thinning process. In our notations, the distribution of the largest particle of the thinning process can by expressed as

[TABLE]

The large $n$ asymptotic approximation of the distribution was proved in [1],

[TABLE]

where $q(x)$ is the Ablowitz-Segur solution to the Painlevé II equation determined by the boundary condition

[TABLE]

The Tracy-Widom type formula (1.11) was obtained in [1] by studying the asmptotics of the orthogonal polynomials with respect to (1.2) via the Riemann-Hilbert approach, specifying $\alpha=0$. This system of orthogonal polynomials has also been considered by Xu and Zhao [24], in which the asymptotics of recurrence coefficients were obtained, also by applying the Riemann-Hilbert approach.

In [16], Its, Kuijlaars and Östensson studied pGUE (1.1) in the case $\omega=0$. When the algebra singularity $\mu$ is close to the soft edge, or, more precisely, when $\mu=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$ with mild $s$, the asymptotics of the correlation kernel (1.4) was found and characterised by the solution to a Lax pair, related to a certain Painlevé XXXIV transcendent. The authors of [16] showed that the Painlevé XXXIV kernel is valid for quite general ensembles with root-typed singularity $|z-\mu|^{2\alpha}$, $\mu$ being close to the soft edge.

Earlier in [12], Forrester and Witte considered pGUE (1.1) with the parameter $2\alpha\in\mathbb{N}$ and $\omega=0,\;1$. By applying the Okamoto $\tau$-function theory, they found that the asymptotics of the quantities $H_{n}(\mu;\alpha,\omega)$ in (1.6), with $\mu=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$, can be expressed in terms of solutions of the following Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation:

[TABLE]

with the boundary condition

[TABLE]

for $\omega=0$, and

[TABLE]

for $\omega=1$. In [12], the authors also noted that for $\alpha=0$,

[TABLE]

and they further obtained (1.11)-(1.12). In [13] and [25, 26], asymptotic formulas have been derived of the Hankel determinants associated with the Jacobi weight perturbed by a Fisher-Hartwig singularity close to the hard edge $x=1$, involving the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé III equation. The Painlevé equations play an important role in the asymptotic study of the Hankel determinants; see [5, 14, 22, 23].

In this paper, we consider pGUE (1.1) with Fisher-Hartwig singularity of both root type and jump type simultaneously at $\lambda=\mu_{n}$, which approaches the soft edge $\sqrt{2n}$ at a certain speed such that $\sqrt{2}n^{1/6}(\mu_{n}-\sqrt{2n}\,)\to s$ with finite $s$, as $n\to\infty$. We derive the asymptotics of the Hankel determinant, the recurrence coefficients and the correlation kernel in the double scaling limit. The asymptotic results are expressed in terms of a family of solutions to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation and the Painlevé XXXIV equation. The asymptotic properties of this family of solutions are also investigated.

1.1 Statement of main results

1.1.1 Asymptotics of solutions to the Painlevé XXXIV equation and the $\sigma$-form of the Painlevé II equation

Our first result is on the asymptotics of a family of solutions to the Painlevé XXXIV equation and the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation. The asymptotics at positive infinity are derived by using known results on the second Painlevé transcendents in [9]. These solutions are used to describe the asymptotics of the Hankel determinants and the recurrence coefficients.

THEOREM 1**.**

Let $\alpha>-\frac{1}{2}$ and $\omega\in\mathbb{C}\setminus(-\infty,0)$, then for $s\in\mathbb{R}$, there is an analytic solution to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation (1.13), uniquely determined by the boundary condition as $s\to\infty$ and $\mathop{\rm arg}\nolimits s\in(-\frac{\pi}{3},\frac{\pi}{3}]$, namely

[TABLE]

where $m$ is a positive integer, $a_{0}=1$, and $a_{1}=\frac{\alpha}{2}$. Furthermore, the asymptotic behavior of the solution at negative infinity is

[TABLE]

for $\omega=0$, and

[TABLE]

for $\omega=e^{-2\pi i\beta}$ with $|\mathop{\rm Re}\nolimits\beta|<1/2$, where $\theta(s;\alpha,\beta)=\frac{4}{3}|s|^{3/2}-\alpha\pi-6i\beta\ln 2-3i\beta\ln|s|$.

THEOREM 2**.**

Let $\alpha>-\frac{1}{2}$ and $\omega\in\mathbb{C}\setminus(-\infty,0)$, then for $s\in\mathbb{R}$, there is an analytic solution to the Painlevé XXXIV equation

[TABLE]

uniquely determined by the boundary condition as $s\to\infty$ and $\mathop{\rm arg}\nolimits s\in(-\frac{\pi}{3},\frac{\pi}{3}]$, namely

[TABLE]

where $m$ is a positive integer, $c_{0}=1$ and $c_{1}=-\alpha$. Furthermore, the asymptotic behavior of the solution at negative infinity is

[TABLE]

for $\omega=0$, and

[TABLE]

as $s\to-\infty$ for $\omega=e^{-2\pi i\beta}$ with $|\mathop{\rm Re}\nolimits\beta|<1/2$, where $\theta(s;\alpha,\beta)=\frac{4}{3}|s|^{3/2}-\alpha\pi-6i\beta\ln 2-3i\beta\ln|s|$.

1.1.2 Asymptotics of the Hankel determinant and recurrence coefficients

Our second result is on the asymptotics of the Hankel determinant, characterized by the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation.

THEOREM 3**.**

Let $\alpha>-\frac{1}{2}$, $\omega\in\mathbb{C}\setminus(-\infty,0)$ and $\mu_{n}=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$, we have the following asymptotic formula of the Hankel determinant

[TABLE]

where $\sigma(s)$ and $u(s)$ are the analytic solutions to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation (1.13) and the Painlevé XXXIV equation, determined respectively by the boundary condition (1.17) and (1.21).

The above theorem can be applied directly to obtain the double scaling limit of the distribution of the largest eigenvalue in the conditioning and thinning procedure in GUE; cf. (1.7). The limit distribution is connected to the perturbed GUE (1.1), and can also be expressed in terms of the Hankel determinant in (1.6).

COROLLARY 1**.**

Let $\alpha>-\frac{1}{2}$, $\omega\in\mathbb{C}\setminus(-\infty,0)$ and $\mu_{n}=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$ with finite $s$, we have the following double scaling limit for the distribution of the largest eigenvalue remaining in the thinning procedure of the conditioning GUE defined in (1.7):

[TABLE]

where $\sigma(s;\alpha,\omega)$ are a family of analytic solutions to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation (1.13), determined by the boundary condition (1.17), and depending on the parameter $\alpha$ and $\omega$.

For $\alpha=0$, the Jimbo-Miwa-Okamoto $\sigma$-form is related to the Painlevé II equation by $\sigma^{\prime}(s)=-q(s)^{2}$; see (1.16), we can accordingly derive from (1.25) and (1.17) the Tracy-Widom distribution of the largest eigenvalue for GUE (1.8)-(1.10) and for the thinned GUE (1.11)-(1.12), which have previously been obtained in [21] and [1], respectively.

We also have the following asymptotic approximations of the recurrence coefficients in (1.5), in terms of the Painlevé XXXIV transcendent.

THEOREM 4**.**

Let $\alpha>-\frac{1}{2}$, $\omega\in\mathbb{C}\setminus(-\infty,0)$ and $\mu_{n}=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$, we have

[TABLE]

and

[TABLE]

as $n\to\infty$, where $u(s)$ is the Painlevé XXXIV transcendent (1.20), analytic for $s\in\mathbb{R}$ and subject to the boundary condition (1.21).

1.1.3 Asymptotics of the correlation kernel

Let \left(\begin{array}[]{c}\psi_{1}(x;s)\\ \psi_{2}(x;s)\\ \end{array}\right) be the unique solution to the linear differential equation

[TABLE]

subject to the boundary conditions

[TABLE]

and

[TABLE]

The $\Psi$-kernel is defined as

[TABLE]

Then, our result on the correlation kernel can be stated as follows.

THEOREM 5**.**

Let $\alpha>-\frac{1}{2}$, $\omega\in\mathbb{C}\setminus(-\infty,0)$ and $\mu_{n}=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$, we have the limit of the correlation kernel

[TABLE]

for $v_{1},v_{2}\in\mathbb{R}\setminus\{0\}$.

The rest of the paper is organized as follows. In Section 2, we introduce the Riemann-Hilbert problem (RH problem, for short) for the Painlevé XXXIV equation. The existence of a unique solution to the RH problem is proved and thus the related Painlevé XXXIV tanscendents and $\sigma$-form of the Painlevé II equation are pole free on the real axis. The asymptotics of the Painlevé XXXIV functions are also discussed in this section. In Section 3, we state the RH problem for the orthogonal polynomials associated with the weight (1.1). We show that the Hankel determinant of finite size satisfies the $\sigma$-form of the Painlevé IV equation. A differential identity for the Hankel determinant is also derived. The large-$n$ asymptotic analysis of the RH problem for the orthogonal polynomials is then carried out by applying the Deift-Zhou method. In Section 4, using the asymptotic results of the RH problem for the orthogonal polynomials obtained in the previous section, we obtain the asymptotics of the Hankel determinant and the recurrence coefficients, expressed in terms of the $\sigma$-form of the Painlevé II equation and the Painlevé XXXIV tanscendents. An asymptotic formula of the correlation kernel are also proved in this section. These furnish proofs of Theorems 3, 4 and 5. The proofs of the rest theorems, namely Theorems 1 and 2, are left to the Appendix. In the Appendix, the asymptotics of the Painlevé XXXIV tanscendents and $\sigma$-form of the Painlevé II equation as $s\to+\infty$ are derived by analyzing the RH problem related to the Painlevé XXXIV equation.

2 Riemann-Hilbert problem for the Painlevé XXXIV equation

The matrix-valued RH problem for the Painlevé XXXIV equation is as follows; see [9, 16, 17].

(a)   $\Psi(\zeta;s)$ ($\Psi(\zeta)$, for short) is analytic in $\mathbb{C}\backslash\bigcup_{j=1}^{4}\Sigma_{j}$; see Fig. 1 for the jump contours;

(b)   $\Psi(\zeta)$ satisfies the jump condition

[TABLE]

(c)   As $\zeta\rightarrow\infty$

[TABLE]

where $M_{0}=\frac{I+i\sigma_{1}}{\sqrt{2}}$, $\vartheta=\vartheta(\zeta,s)=\frac{2}{3}\zeta^{3/2}+s\zeta^{1/2}$, and $\sigma_{j}$ are the Pauli matrices, namely,

[TABLE]

(d)   As $\zeta\rightarrow 0$, $\zeta\in\Omega_{j}$ for $j=1,2,3,4$,

[TABLE]

or

[TABLE]

where \sigma_{+}=\left(\begin{array}[]{cc}0&1\\ 0&0\\ \end{array}\right), $\Psi_{0}(\zeta)$ is analytic at $\zeta=0$, and $E_{j}$ are certain constant matrices; see [16, (3.21)].

PROPOSITION 1**.**

For $\alpha>-\frac{1}{2}$ and $\omega\in\mathbb{C}\setminus(-\infty,0)$, there exits a unique solution to the above Riemann-Hilbert problem for $\Psi(\zeta;s)$.

Proof.

The existence of the solution $\Psi(\zeta)$ is obtained by proving a vanishing lemma; see [17] and [24] for a proof of the vanishing lemma. ∎

Now we observe that the entries of $\Psi(\zeta)$ are closely related with equations (1.13) and (1.20) as follows:

PROPOSITION 2**.**

Let

[TABLE]

then $\sigma(s)$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation (1.13). Also,

[TABLE]

satisfies the Painlevé XXXIV equation (1.20). Moreover, for $\alpha>-\frac{1}{2}$ and $\omega\in\mathbb{C}\setminus(-\infty,0)$, the scalar functions $\sigma(s)$ and $u(s)$ are analytic for $s\in\mathbb{R}$.

Let \widehat{\Psi}(\zeta;s)=\left(\begin{array}[]{cc}1&0\\ i(\sigma(s)-\frac{s^{2}}{4})&1\\ \end{array}\right)\Psi(\zeta;s), then $\widehat{\Psi}(\zeta;s)$ satisfies the following Lax pair system of differential equations

[TABLE]

see [16, Lemma 3.2].

It is well known that the Painlevé XXXIV equation (1.20) is related to the Painlevé II equation

[TABLE]

by the formulas

[TABLE]

see [9, (5.0.2)(5.0.55)]. Moreover, the solution $u(s)$ associated with the model RH problem, as stated in Proposition 2, is related to a family of special solutions to the second Painlevé equation (2.18) corresponding to the Stokes multipliers $\{s_{1}=-e^{-2\pi i\alpha},s_{2}=\omega,s_{3}=-e^{2\pi i\alpha}\}$; cf. [9, 11.3], see also [16, 24] for the relation between the model RH problem for $\Psi$ and the RH problem for the second Painlevé equation. Thus we may extract the asymptotics of $u(s)$, knowing that of $q(s)$.

It is also known that the solutions $q(s)$ corresponding to the Stokes multipliers $\{s_{1}=-e^{-2\pi i\alpha},s_{2}=\omega,s_{3}=-e^{2\pi i\alpha}\}$ is uniquely determined by the following asymptotic behavior

[TABLE]

as $\mathop{\rm arg}\nolimits(-s)\in(-\frac{\pi}{3},\frac{\pi}{3}]$ and $s\to\infty$, [9, (11.5.56)]. The first few coefficients are

[TABLE]

and

[TABLE]

see [9, (11.5.58) and (11.5.68)]. A combination of the asymptotics of $q(s)$ and the relation (2.19) will then gives the asymptotic approximations of $u(s)$ and $\sigma(s)$ as $s\to\infty$ and $\mathop{\rm arg}\nolimits s\in(-\frac{\pi}{3},\frac{\pi}{3}]$, as stated in (1.17) and (1.21).

We also mention that, for $\alpha=0$ and $\omega=1$, the solution $q(x)$ determined by (2.20) or the Stokes multipliers $\{s_{1}=-1,s_{2}=1,s_{3}=-1\}$ is the classical solution

[TABLE]

cf. [9, (11.4.15)]. Thus by (2.19) the Painlevé XXXIV function corresponding to the parameter $\alpha=0$ and $\omega=1$ is trivial, that is, $u(x)=0$; see also [24, Lemma 3].

The asymptotics of $q(s)$ as $s\to+\infty$ can be found in, e.g. [9, Chapter 10]. However, the leading asymptotics of $q(s)$ is canceled out when we substitute it into (2.19). Therefore, the leading term alone is not enough to derive the asymptotics of $u(s)$; see also the discussion in [16, 17]. We will derive the asymptotics of $u(s)$ and $\sigma(s)$ as $s\to-\infty$ by using the RH problem for $\Psi$. Details will be given in the Appendix.

3 Nonlinear steepest descend analysis of orthogonal polynomials

In this section, we start by presenting the RH problem $Y(z;\mu,n)$ for the orthogonal polynomials associated with the weight function (1.2). Then we show that the logarithmic derivative of the Hankel determinants can be expressed in terms of the solutions to this RH problem. For $n$ fixed, by relating the solution $Y(z;\mu,n)$ to the Lax pair for the Painlevé IV equation, we show that the logarithmic derivative of the Hankel determinant satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé IV equation. Then we perform a nonlinear steepest descent analysis of the RH problem for $Y$ via the Deift-Zhou method [8, 6, 7] .

3.1 Riemann-Hilbert problem for orthogonal polynomials

The RH problem for the orthogonal polynomials with respect to (1.2) is as follows.

(Y1)   $Y(z)$ is analytic in $\mathbb{C}\backslash\mathbb{R}$;

(Y2)   $Y(z)$ satisfies the jump condition

[TABLE]

where $w(x)$ is defined in (1.2);

(Y3)   The behavior of $Y(z)$ at infinity is

[TABLE]

(Y4)   The behavior of $Y(z)$ at $\mu$ is

[TABLE]

For $\alpha>-\frac{1}{2}$ and $\omega\geq 0$, the system of orthogonal polynomials with respect to (1.2) are well defined. It follows from the Sochocki-Plemelj formula and Liouville’s theorem that the unique solution of the RH problem for $Y$ is given by

[TABLE]

where $\pi_{n}(z)$ is the monic orthogonal polynomial with respect to the weight $w(x)$ defined in (1.2); see [10]. For general $\omega\in\mathbb{C}\setminus(-\infty,0)$, we will prove later that for $n$ large enough, the RH problem for $Y$ can be solved and thus the polynomials orthogonal with respect to (1.2) exist for large $n$.

Now we introduce

[TABLE]

then $\Phi(z,\mu)$ satisfies constant jumps on the real axis and has a regular singularity at $z=0$ and an irregular singularity at infinity, with

[TABLE]

Thus, it is related to the Garnier-Jimbo-Miwa Lax pair $\Phi_{\mathrm{PIV}}$ for the Painlevé IV equation with the parameters $\theta_{\infty}=-(\alpha+n)$ and $\theta_{0}=\alpha$ by

[TABLE]

see [9, Theorem 5.2,Proposition 5.4] and [18, Eq.(C.30)-Eq.(C.34)]. From this relation, we get the $\sigma$-form of the Painlevé IV equation [18, Eq.(C.30)-Eq.(C.37)].

PROPOSITION 3**.**

Let

[TABLE]

then $\sigma_{\mathrm{IV}}$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé IV equation

[TABLE]

At the end of this section, we state and prove a differential identity for the Hankel determinant associated with the weight $w(x)$ in (1.2). The identity serves as the start point of our asymptotic study of the Hankel determinant. We also show that the logarithmic derivative of the Hankel determinant satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé IV equation, as was first obtained in [12].

LEMMA 1**.**

Let $H_{n}$ be the Hankel determinant associated with $w(x;\mu)$ in (1.1), then

[TABLE]

and $\sigma_{\mathrm{IV}}(\mu)=\frac{d}{d\mu}\ln H_{n}$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé IV equation

[TABLE]

Proof.

By (1.6) and taking logarithmic derivative with respect to $\mu$, we have

[TABLE]

where

[TABLE]

Taking derivative with respect to $\mu$ on both sides of (3.13) and using the orthogonality and the definition of the weight (1.2), we get

[TABLE]

It follows from (3.12), (3.14) and the Christoffel-Darboux formula that

[TABLE]

Write $\pi_{n}(x)=x^{n}+p_{n}x^{n-1}+\cdots$, we have the following decomposition into the orthogonal system $\{\pi_{k}\}_{k=0}^{n}$,

[TABLE]

By using the orthogonality, we have $\frac{d}{d\mu}\ln H_{n}=2p_{n}$. Then (3.10) follows from (3.4), and the fulfilment of (3.11) follows from Proposition 3. ∎

3.2 The transformations $Y\rightarrow T\to S$

From this section on, we take $\mu:=\mu_{n}=\sqrt{2n}+\frac{s}{\sqrt{2}n^{1/6}}$ in the weight function (1.2). We introduce the transformation

[TABLE]

where the $g$-function

[TABLE]

and $l=\-1-2\ln 2$, with the logarithm taking the branch $\mathop{\rm arg}\nolimits(z-x)\in(-\pi,\pi)$. We also introduce the $\phi$-function

[TABLE]

where the branches are chosen such that $\sqrt{z^{2}-1}$ is analytic in $\mathbb{C}\setminus[-1,1]$, taking positive values on $(1,+\infty)$, $z+\sqrt{z^{2}-1}$ maps $\mathbb{C}\setminus[-1,1]$ onto the outside of the unit disk, with $z+\sqrt{z^{2}-1}\sim 2z$ as $z\to\infty$, and the logarithm taking the principal branch. The $\phi$-function and $g$-function are related by the phase condition

[TABLE]

So defined, $T$ is normalized at infinity in the sense that $T(z)=I+O(1/z)$, and satisfies the jump condition

[TABLE]

where $\lambda=\frac{\mu_{n}}{\sqrt{2n}}=1+\frac{s}{2n^{2/3}}$, $\theta(x)=\omega$ for $x>0$ and $\theta(x)=1$ for $x<0$.

To remove the oscillation of the diagonal entries of the jump matrix on $(-1,1)$, we deform the interval $[-1,\lambda]$ into a lens-shaped region as illustrated in Fig. 2, and introduce the following transformation

[TABLE]

It is easy to see that $S$ satisfies the jump conditions

[TABLE]

and

[TABLE]

We note that for $n$ large, the jumps for $S$ is close to the identity matrix, except the one on $[-1,\lambda]$. Here both cases, namely $\lambda<1$ and $\lambda>1$, are possible.

3.3 Global Parametrix

The global parametrix solves the following approximating RH problem, with only the jump along $(-1,\lambda)$ remains:

(N1)   $N(z)$ is analytic in $\mathbb{C}\backslash[-1,\lambda]$;

(N2)

[TABLE]

(N3)

[TABLE]

A solution of the RH problem can be constructed explicitly as

[TABLE]

where

[TABLE]

where $\varsigma(z)$ is analytic in $\mathbb{C}\setminus[-1,\lambda]$, and $\varsigma(z)\sim 1$ as $z\to\infty$.

The Szegő function $d(z)$ in (3.27) is analytic in $\mathbb{C}\setminus[-1,\lambda]$ and satisfies

[TABLE]

see [20]. Indeed, the Szegő function can be written explicitly as

[TABLE]

where $\mathop{\rm arg}\nolimits(z+1)\in(-\pi,\pi)$, $\mathop{\rm arg}\nolimits(z-\lambda)\in(-\pi,\pi)$ and the logarithm takes the principal branch with the branch cut along $(-\infty,0)$.

3.4 Local parametrix at $z=-1$

The local parametrix at $z=-1$ satisfies the following RH problem.

(a)   $P^{(-1)}(z)$ is analytic in $U(-1,r)\backslash\Sigma_{S}$; see Fig. 2 for the contours;

(b)   $P^{(-1)}_{+}(z)=P^{(-1)}_{-}(z)J_{S}(z),~{}~{}z\in\Sigma_{S}\cap U(-1,r)$, where $J_{S}(z)$ are the jump matrices of $S(z)$ in (3.23);

(c)   $P^{(-1)}(z)N^{-1}(z)=I+O(1/n)$ for $z\in\partial U(-1,r)$.

The local parametrix can be constructed as follows:

[TABLE]

where $E_{-1}(z)$ is a certain matrix-valued function analytic in $U(-1,r)$, $\Phi_{A}$ is the Airy parametrix defined as

[TABLE]

with the regions indicated in Fig. 3 and M=\sqrt{2\pi}e^{\frac{1}{6}\pi i}\left(\begin{array}[]{cc}1&0\\ 0&-i\\ \end{array}\right); cf. [7, (7.9)] and [4, Ch. 7.6]. The Airy parametrix $\Phi_{A}$ satisfies the asymptotic condition at infinity

[TABLE]

To match $P^{-1}(z)$ with the global parametrix $N(z)$, we choose

[TABLE]

3.5 Local parametrix at $z=\lambda$

We seek a local parametrix $P^{(1)}(z)$ near $\lambda=\frac{\mu_{n}}{\sqrt{2n}}=1+\frac{s}{2n^{2/3}}$. The parametrix solves the following RH problem:

(a)   $P^{(1)}(z)$ is analytic in $U(1,r)\backslash\Sigma_{S}$; see Fig. 2 for the contours;

(b)   On $\Sigma_{S}\cap U(1,r)$, $P^{(1)}(z)$ satisfies the same jump condition as $S(z)$,

[TABLE]

(c)   $P^{(1)}(z)$ satisfies the following matching condition on $\partial U(1,r)$:

[TABLE]

(d)   The behavior at $z=\lambda$ is the same as that of $S(z)$.

It is observed that $P^{(1)}(z)e^{-n\phi(z)\sigma_{3}}(z-\lambda)^{\alpha\sigma_{3}}$ solves a RH problem with the same constant jumps as the model RH problem (2.13). We take the conformal mapping

[TABLE]

from a neighborhood of $z=1$ to that of the origin. Then for $r$ small enough, the local parametrix ${P}^{(1)}(z)$ can be constructed as follows

[TABLE]

where

[TABLE]

To verify the matching condition (3.34), in view of (2.14), we can write the exponent of (3.36) as

[TABLE]

for $|z-1|=r$ and $n\to\infty$. Now substituting it in (2.14) and (3.36), we have the matching condition (3.34).

3.6 The final transformation: $S\rightarrow R$

The final transformation is defined as

[TABLE]

From the matching condition (3.34), we have

[TABLE]

Thus

[TABLE]

where the error bound is uniform for $z$ in whole complex plane.

4 Proof of the main theorems

The differential identity (3.10) provides an expression of the logarithmic derivative of the Hankel determinant, in terms of the large-$z$ behavior of $Y(z)$; see (3.2). From (3.4), one can also write the recurrence coefficients in (1.5) as

[TABLE]

where $Y_{1}$ and $Y_{2}$ are the coefficients in the large-$z$ expansion of $Y(z)$ in (3.2). In this section, by using the asymptotics of $Y$ obtained in Section 3, we proceed to derive the large-$n$ asymptotic behavior of the Hankel determinant, the recurrence coefficients and the correlation kernel.

Tracing back the sequence of transformations $Y\to T\to S\to R$, given in (3.17), (3.22) and (3.39), we have

[TABLE]

As $z\rightarrow\infty$,

[TABLE]

where the matrix-valued coefficients $G_{1}=0$ and $G_{2}=\left\{-\frac{n}{2}\int_{-1}^{1}x^{2}\varphi(x)dx\right\}\sigma_{3}$. Denoting

[TABLE]

and

[TABLE]

and substituting (4.4) and (4.5) in (4.2), we have

[TABLE]

and

[TABLE]

To evaluate $N_{1}$ and $N_{2}$, we expand the Szegő function (3.29) at infinity,

[TABLE]

We also expand $N_{0}(z)$ in (3.28) at infinity,

[TABLE]

where

[TABLE]

see [24]. Substituting (4)-(4.10) in (3.27) gives (4.4), with

[TABLE]

and

[TABLE]

A combination of (2.14), (3.36) and (3.37) yields the following more precise description of the jump for $R(z)$ on $\partial U(1,r)$:

[TABLE]

where $F_{n}(z)$ is defined in (3.5). In deriving (4), use has been made of the fact that $n^{2/3}f(\lambda)=s+O(n^{-2/3})$. From the approximation of $F_{n}$ in (3.5), we further have

[TABLE]

where

[TABLE]

Alternatively, $N_{0}(z)$ in (3.28) can be written as

[TABLE]

Accordingly, for $|z-1|=r$ we have

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

By (3.29), we have

[TABLE]

and hence

[TABLE]

Thus

[TABLE]

where $R^{(1)}$ and $R^{(2)}$ satisfy the scalar RH problems

[TABLE]

From (4.15) and (4.19), we know that $r_{1}(z,t)$, $r_{3}(z,t)$ and $r_{4}(z,t)$ have a simple pole at $z=\lambda$ and $r_{2}(z,t)$ is analytic at $z=\lambda$. By Cauchy’s theorem, we get

[TABLE]

and

[TABLE]

where $k_{j}(s)=\mathrm{Res}_{z=\lambda}r_{j}(z,\lambda)$ for $j=1,3,4$, $k_{2}(s)=\lim_{z\rightarrow\lambda}r_{2}(z,\lambda)$ and $k_{5}(s)$ arises from the residue of the term $\eta$ in $\Delta_{2}(z)$ and $R^{(1)}_{-}(z)\Delta_{1}(z)$. After some careful calculations, we get from (4.19) and (4.15)

[TABLE]

Expanding $R^{(1)}(z)$ and $R^{(2)}(z)$ into Laurent series at infinity

[TABLE]

then, from (4.24) and (4.25) we have

[TABLE]

where use has been made of the estimate $2k_{4}(s)-k_{1}(s)k_{2}(s)=O(n^{-2/3})$, and the identities

[TABLE]

cf. (2.16) for $\sigma_{+}$.

4.1 Proof of Theorem 3: asymptotics of the Hankel determinant

Substituting (4.6) into the differential identity (3.10), we have

[TABLE]

Replacing $N_{1}$ and $R_{1}$ with (4.11) and (4) yields

[TABLE]

where $n^{2/3}(\lambda-1)=\frac{s}{2}+O(n^{-2/3})$, and $\sigma(s)=\hat{\sigma}(s)+\frac{s^{2}}{4}$ is the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation; see Proposition 2. Using the relation $u(s)=-\sigma^{\prime}(s)$, we obtain (1.24), as stated in Theorem 3.

4.2 Proof of Theorem 4: asymptotics of the recurrence coefficients

It follows from (4.1) and (4.6) that

[TABLE]

From (4.9), (4) and (4.30), we get

[TABLE]

By the definition of $k_{j}(s)$ in (4.26), we obtain the asymptotic approximation of $b_{n}$,

[TABLE]

where $u(s)=-\sigma^{\prime}(s)$ satisfies the Painlevé XXXIV equation.

In a similar way, we compute the asymptotics of the recurrence coefficient $a_{n}$. First, a combination of (4.1), (4.6) and (4.7) gives

[TABLE]

Noting that $(G_{2})_{12}=0$, $R_{1}=R_{2}=O(n^{-1/3})$, $(N_{1})_{11}=-(N_{1})_{22}$, and $(N_{1})_{12}=\frac{i}{4}(1+\lambda)d^{2}(\infty)=O(1)$, the formula (4.33) is then turned into

[TABLE]

From (4.11) and (4), we have

[TABLE]

By (4.11) and (4), we get

[TABLE]

and

[TABLE]

Thus, (4.2) is further simplified to

[TABLE]

Substituting in the relation $u(s)=-\sigma^{\prime}(s)$, we obtain the asymptotic approximation (1.26) of the recurrence coefficient $a_{n}$, as stated in Theorem 4.

4.3 Proof of Theorem 5: asymptotics of the correlation kernel

Now Christoffel-Darboux formula applies, and the correlation kernel (1.4) can be written as

[TABLE]

From (3.4), the kernel can be expressed in terms of $Y(z)$, such that

[TABLE]

where $\lambda=\mu/\sqrt{2n}$. Appealing to the transformation (3.17) and the phase condition (3.20), we arrive at

[TABLE]

for $x,y\in\mathbb{R}$. Tracing back the transformation (3.22), (3.39) and (3.36), we have

[TABLE]

for $x\in(\lambda-\delta,\lambda)$, where $\delta\in(0,r)$ is a constant, and

[TABLE]

for $x\in(\lambda,\lambda+\delta)$. Since $R(z)$ and $E(z)$ are analytic in $(\lambda-\delta,\lambda+\delta)$, one has

[TABLE]

From (3.35) we see that

[TABLE]

Thus, inserting the expressions (4.39)-(4.40) and the estimates (4.41)-(4.42) into (4.38), we arrive at

[TABLE]

where

[TABLE]

and

[TABLE]

Acknowledgements

The work of Shuai-Xia Xu was supported in part by the National Natural Science Foundation of China under grant numbers 11571376 and 11201493, GuangDong Natural Science Foundation under grant numbers 2014A030313176. Yu-Qiu Zhao was supported in part by the National Natural Science Foundation of China under grant number 11571375.

Appendix. Asymptotics of the $\sigma$-form of Painlevé II and the Painlevé XXXIV function

In the Appendix, we provide the asymptotics of Jimbo-Miwa-Okamoto $\sigma$-form of the Painlevé II equation and the Painlevé XXXIV function as $s\rightarrow-\infty$. The proof is similar to that of [17].

For negative $s$, take a re-scaling of variable

[TABLE]

Then $A(z)$ is analytic in $\mathbb{C}\backslash{\cup_{j=1}^{4}\Sigma_{j}}$ and shares the same jumps and the same behavior at [math] with $\Psi(\zeta)$, $j=1,2,3,4$. Here $\Sigma_{j}$ are the same rays as illustrated in Fig. 1. According to the large-$z$ behavior of $\Psi(z)$ in (2.14), one has

[TABLE]

where $\tau=(-s)^{3/2}$. Using (2.14) and (2.17), the $\sigma$-form can be expressed in terms of $A(z)$,

[TABLE]

A.4 Case 1: $\omega=0$

In the case, there is no jump for $\Psi(\zeta)$ along $\zeta\in\Sigma_{1}$; cf. (2.13) and Fig. 1. Denote

[TABLE]

then

[TABLE]

and $B(z)=A(z)e^{-\tau h(z)\sigma_{3}}$ solves the following RH problem.

(B1) $B(z)$ is analytic in $\mathbb{C}\backslash\Sigma_{B}$, $\Sigma_{B}=\Sigma_{2}\cup\Sigma_{3}\cup\Sigma_{4}$; cf. Fig. 1 for the contours;

(B2) $B(z)$ satisfies the jump condition

[TABLE]

(B3) $B(z)=\left(I+O(z^{-1})\right)z^{-\sigma_{3}/4}\frac{1}{\sqrt{2}}(I+i\sigma_{1})$ as $z\rightarrow\infty$;

(B4) $B(z)$ has the same behavior at [math] as $A(z)$ does.

The jump matrices are close to the identity matrix except the one on $(-\infty,0]$. Thus we arrive at the approximating RH problem.

(a)   $B^{(\infty)}(z)$ is analytic in $\mathbb{C}\backslash(-\infty,0]$;

(b)   $B^{(\infty)}(z)$ satisfies the jump condition

[TABLE]

(c)   $B^{(\infty)}(z)=\left(I+O\left(1/z\right)\right)z^{-\sigma_{3}/4}\frac{1}{\sqrt{2}}(I+i\sigma_{1})$ as $z\rightarrow\infty$.

The solution is straightforward. Explicitly we can write down

[TABLE]

It is observed that $B^{(\infty)}(z)$ fails to approximate $B(z)$ at the origin since they demonstrate different singularities there. Hence a parametrix has to be brought in. The local parametrix at the origin is constructed in terms of the Bessel functions as

[TABLE]

where

[TABLE]

see [19]. To match the local parametrix $P^{(0)}(z)$ with the outside parametrix $B^{(\infty)}$, we choose

[TABLE]

which is analytic for $|z|<\frac{3}{2}$. Then from (A.5) and (A.6) we have

[TABLE]

where $U_{0}$ is a disk centered at the origin with radius less than $3/2$, and $\tau=(-s)^{3/2}\to+\infty$ as $s\to-\infty$.

In the final transformation, we define

[TABLE]

Then $C(z)$ is analytic in $\mathbb{C}\backslash\Sigma_{C}$ and $J_{C}(z)=I+O(\tau^{-1})$. Here $\Sigma_{C}$ is the union of $\partial U_{0}$ and the parts of $\Sigma_{2}$ and $\Sigma_{4}$ outside of $U_{0}$. Thus $C(z)=I+O(1/\tau)$, uniformly in the complex plane. Take the expansion as $z\to\infty$

[TABLE]

Noting that (A.7) actually holds in each small annulus around the origin, we have

[TABLE]

Substituting it into (A.3), we obtain the asymptotic approximation of $\sigma(s)$

[TABLE]

A.5 Case 2: $\omega=e^{-2\beta\pi i}$ with $|\mathop{\rm Re}\nolimits\beta|<1/2$

In [17], Its, Kuijlaars and Östenssonthe obtained the leading asymptotics of the family of Painlevé XXXIV functions corresponding to $\omega=e^{-2\beta\pi i}$, $|\mathop{\rm Re}\nolimits\beta|<1/2$; see [17, A.41]. The $\sigma$-form and the Painlevé XXXIV function are related by $u(x)=-\sigma^{\prime}(x)$. Unfortunately, to find the $O(1/s)$ term in the asymptotics of the $\sigma$-form by using the above relation, we need the $O(1/s^{2})$ term asymptotic approximation of $u(s)$ which is not available in [17, A.41]. In this section, we compute the asymptotics of the $\sigma$-form directly using its relation with $\Psi$ established in Proposition 2.

Instead of $h(z)$ in (A.4), we introduce in this case the function

[TABLE]

It is readily seen that

[TABLE]

This time, we define

[TABLE]

Then, $B(z)$ satisfies the RH problem.

(B1) $B(z)$ is analytic in $\mathbb{C}\backslash\Sigma_{B}$, where $\Sigma_{B}=\cup^{4}_{j=2}\Sigma_{j}\cup(0,1)\cup(1,+\infty)$; cf. Fig. 1 and Fig. 4;

(B2) $B(z)$ satisfies the jump condition

[TABLE]

(B3) $B(z)=\left(I+O(z^{-1})\right)z^{-\sigma_{3}/4}\frac{1}{\sqrt{2}}(I+i\sigma_{1})$ as $z\rightarrow\infty$;

(B4) $B(z)$ has the same behavior at [math] as $A(z)$ does.

We take the transformation

[TABLE]

where the regions are illustrated in Fig. 4.

Then $C(z)$ satisfies the following RH problem.

(C1) $C(z)$ is analytic in $\mathbb{C}\backslash\Sigma_{C}$, with $\Sigma_{C}$ consisting of all contours illustrated in Fig. 4;

(C2) $C(z)$ satisfies the jump conditions

[TABLE]

and

[TABLE]

(C3) $C(z)=\left(I+O(z^{-1})\right)z^{-\sigma_{3}/4}\frac{1}{\sqrt{2}}(I+i\sigma_{1})$ as $z\rightarrow\infty$;

(C4) $C(z)$ has the same behavior at [math] as $A(z)$ does.

The jumps (A.14) of $C(z)$ are close to the identity matrix for large $\tau$, while those in (A.13) are not. Therefore, we have an approximating RH problem.

(a)   $C^{(\infty)}(z)$ is analytic in $C\backslash(-\infty,1]$;

(b)   $C^{(\infty)}(z)$ satisfies jump conditions

[TABLE]

(c)   $C^{(\infty)}(z)=\left(I+C^{(\infty)}_{1}/z+O(z^{-2})\right)z^{-\sigma_{3}/4}\frac{1}{\sqrt{2}}(I+i\sigma_{1})$ as $z\rightarrow\infty$.

The solution of the above RH problem is given by

[TABLE]

where the square root and the logarithm take the principal branches with their branch cuts along $(-\infty,1)$ and $(-\infty,0)$, respectively.

We note that $C^{(\infty)}(z)$ fails to approximate $C(z)$ near $z=0$ and $z=1$. Thus we need to construct parametrices at these two points. First, we seek a local parametrix defined in $U(1,r)$ and satisfied the following RH problem.

(a)   $C^{(1)}(z)$ is analytic in $U(1,r)\backslash\Sigma_{C}$; see Fig. 4 for $\Sigma_{C}$;

(b)  $C^{(1)}(z)$ satisfies the jump condition $P^{(1)}_{+}(z)=P^{(1)}_{-}(z)J_{C}(z)$ for $z\in\Sigma_{C}\cap U(1,r)$, where $J_{C}(z)$ are the jump matrices for $C(z)$, given in (A.13) and (A.14);

(c)  The matching condition $C^{(1)}(z)=(I+O(1/\tau))C^{(\infty)}(z)$ holds on $\partial U(1,r)$.

The local parametrix can be constructed in terms of the Airy function

[TABLE]

where $\Phi_{A}(\zeta)$ and $h(z)$ are defined in (3.31) and (A.10), respectively. To match $C^{(1)}(z)$ with the outside parametrix $C^{(\infty)}(z)$, we choose the analytic factor

[TABLE]

Next, we turn to $z=0$, we seek a local parametrix at the origin of the form

[TABLE]

where $\tau=(-s)^{3/2}$,

[TABLE]

and $E(z)$ is analytic in a neighborhood $U(0,r)$. The function $\Phi$ in (A.16) is to be constructed by using arguments from [5, Sec. 4]. To begin with, let

[TABLE]

Then $\Phi^{(\mathrm{CHF})}(\zeta)$ satisfies the following jump condition on contours illustrated in Fig. 5,

[TABLE]

Such a RH problem with constant jumps has been treated in [15] and [5], with the solution being constructed explicitly for $\zeta\in I$ (cf. Fig. 5) as

[TABLE]

where the constant matrix C_{1}=2^{\beta\sigma_{3}}e^{\beta\pi i\sigma_{3}/2}\left(\begin{array}[]{cc}e^{-i\pi(\alpha+2\beta)}&0\\ 0&e^{i\pi(2\alpha+\beta)}\end{array}\right) and $\psi(a,b,z)$ is the confluent hypergeometric function; see [5, Proposition 4.1]. The solution in the other sectors are determined by using the jump relation in (A.18). Then the function $\Phi^{(\mathrm{CHF})}(\zeta)$ possesses the following asymptotic behavior as $\zeta\to\infty$ (see [5, (4.42)]):

[TABLE]

where $\mathop{\rm arg}\nolimits\zeta\in(-\pi/{2},~{}{3}\pi/{2})$.

Now we take the function $E(z)$ in (A.16) as

[TABLE]

with $\mathop{\rm arg}\nolimits f(z)\in(0,2\pi)$. Then it is readily verified that $E(z)$ is analytic in $U(0,r)$ for sufficiently small $r$, and the following matching condition is fulfilled on $\partial U(0,r)$:

[TABLE]

In the final transformation, we put

[TABLE]

In this case, on the remaining contour consisting of $\partial U(0,r)$, $\partial U(1,r)$, and the parts of $\Sigma_{C}\setminus(-\infty,1]$ (see Fig. 4) outside of these two circles, the jump $J_{D}(z)=D_{-}(z)^{-1}D_{+}(z)=I+O\left(\tau^{2|\mathop{\rm Re}\nolimits\beta|-1}\right)$. Thus $D(z)=I+O\left(\tau^{2|\mathop{\rm Re}\nolimits\beta|-1}\right)$ uniformly for $z$ in the complex plane. Expanding $D(z)$ as

[TABLE]

we get from (A.5) that

[TABLE]

with $\theta(s;\alpha,\beta)=\frac{4}{3}|s|^{3/2}-\alpha\pi-6i\beta\ln 2-3i\beta\ln|s|$ for $\omega=e^{-2\pi i\beta}$ with $|\mathop{\rm Re}\nolimits\beta|<1/2$. Tracing back the transformations $A(z)\to B(z)\to C(z)\to D(z)$, we have

[TABLE]

Consequently, in view of (A.2), one has

[TABLE]

Thus, a combination of (2.14), (A.1), (A.24) and (A.26) gives the asymptotic approximation of $\sigma(s)$ in (1) as $s\to-\infty$. The asymptotics of $u(s)$ in (1.23) as $s\to-\infty$ follows from the relation $u(s)=-\sigma^{\prime}(s)$.