arXiv subjects

Yu-Qiu Zhao

Publications and source records attributed to Yu-Qiu Zhao.

At least 19 recordsLinked to original sources

arXiv2025

Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlev\'e IV transition

In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the $(x,t)$-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlev\'e IV equation.

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arXiv2025

Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model

We consider the asymptotics of the partition function of the extended Gross-Witten-Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a $\tau$-function sequence of the Painlev\'e III' equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta-function and the Barnes $G$-function. A third-order phase transition in the leading terms of the asymptotic expansions is also observed.

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arXiv2025

Asymptotics of Fredholm determinant solutions of the noncommutative Painlev\'e II equation

In this paper, we study the asymptotic behavior of a family of pole-free solutions to the noncommutative Painlev\'e II equation. These particular solutions can be expressed in terms of the Fredholm determinant of the matrix version of the classical Airy operator, which are analogous to the Hastings-McLeod solution and the Ablowitz-Segur solution of the classical Painlev\'e II equation. Using the Riemann-Hilbert approach, we derive the asymptotics of the Fredholm determinant and the associated particular solutions $\beta(\vec{s})$ to the noncommutative Painlev\'e II equation in the regime $\vec{s}=\left(s+\frac{\tau}{\sqrt{-s}},s-\frac{\tau}{\sqrt{-s}}\right)$ with $\tau\ge 0$ and $s\to-\infty$. The solutions depend on a two by two Hermitian matrix with eigenvalues in the interval $(-1,1)$. The asymptotics are expressed in terms of one parameter family of special solutions of the classical Painlev\'e V equation. Furthermore, we derive the asymptotics, including the connection formulas, for this one parameter family of solutions of the Painlev\'e V equation both as $ix\to -\infty$ and $x\to 0$.

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arXiv2024

On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with $n$ discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the $n$ discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes $G$-function.

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arXiv2024

Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation

In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the $(i,j)$-entry being the modified Bessel functions of order $i-j-ν$, $ν\in\mathbb{C}$. When the degree $n$ is finite, we show that the Toeplitz determinant is described by the isomonodromy $τ$-function of the Painlevé III equation. As a double scaling limit, %In the double scaling limit as the degree $n\to\infty$, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlevé II equation with parameter $ν+\frac{1}{2}$. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point $z=-1$, where the $ψ$-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

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arXiv2023

Asymptotics of the deformed higher order Airy-kernel determinants and applications

We study the one-parameter family of Fredholm determinants $\det(I-ρ^2\mathcal{K}_{n,x})$, $ρ\in\mathbb{R}$, where $\mathcal{K}_{n,x}$ stands for the integral operator acting on $L^2(x,+\infty)$ with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy-Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the $n$-th member of the Painlevé II hierarchy. Using the Riemann-Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as $x\to -\infty$ for $0<|ρ|<1$ and $|ρ|>1$, respectively. In the case of $0<|ρ|<1$, we are able to calculate the constant term in the asymptotic expansion of the determinants, while for $|ρ|>1$, the relevant asymptotics exhibit singular behaviors. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

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arXiv2023

Clarkson-McLeod solutions of the fourth Painlevé equation and the parabolic cylinder-kernel determinant

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like $κD_{α-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $κ$ is some real constant and $D_{α-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as $x\to-\infty$. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the $σ$-form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the $σ$-form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel.

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arXiv2022

Recent Advances in Asymptotic Analysis

This is a survey article on an old topic in classical analysis. We present some new developments in asymptotics in the last fifty years. We start with the classical method of Darboux and its generalizations, including an uniformity treatment which has a direct application to the Heisenberg polynomials. We then present the development of an asymptotic theory for difference equations, which is a major advancement since the work of Birkhoff and Trjitzinsky in 1933. A new method was introduced into this field in the nineteen nineties, which is now known as the nonlinear steepest descent method or the Riemann-Hilbert approach. The advantage of this method is that it can be applied to orthogonal polynomials which do not satisfy any differential or difference equations neither do they have any integral representations. As an example, we mention the case of orthogonal polynomials with respect to the Freud weight. Finally, we show how the Wiener-Hopf technique can be used to derive asymptotic expansions for the solutions of an integral equation on a half line.

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arXiv2022

Singular asymptotics for the Clarkson-McLeod solutions of the fourth Painlevé equation

We consider the Clarkson-McLeod solutions of the fourth Painlevé equation. This family of solutions behave like $κD_{α-\frac{1}{2}}^2(\sqrt{2}x)$ as $x\rightarrow +\infty$, where $κ$ is an arbitrary real constant and $D_{α-\frac{1}{2}}(x)$ is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we obtain the singular asymptotics of the solutions as $x\to-\infty$ when $κ\left( κ-κ^*\right )>0$ for some real constant $κ^*$. The connection formulas are also explicitly evaluated. This proves and extends Clarkson and McLeod's conjecture that when the parameter $κ>κ^*>0$, the Clarkson-McLeod solutions have infinitely many simple poles on the negative real axis.

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arXiv2021

Isomonodromy sets of accessory parameters for Heun class equations

In this paper, we consider the monodromy and, in particularly, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. While the isomonodromy sets of accessory parameters for the Heun class equations are described by the Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly-confluent Heun equation and the reduced biconfluent Heun equation.

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arXiv2020

Asymptotics of the Charlier polynomials via difference equation methods

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained respectively in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.

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arXiv2020

Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system

We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four. This leads to a Painlevé-type representation of the confluent hypergeometric-kernel determinant. Moreover, the large gap asymptotics, including the constant terms, are derived by evaluating the total integral of the Hamiltonian. In particular, we reproduce the large gap asymptotics of the confluent hypergeometric-kernel determinant obtained by Deift, Krasovsky and Vasilevska, and the sine-kernel determinant as a special case, including the constant term conjectured earlier by Dyson.

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arXiv2017

Real solutions of the first Painlevé equation with large initial data

We consider three special cases of the initial value problem of the first Painlevé equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.

math.CA↗

arXiv2017

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

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 $σ$-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 $σ$-form of the Painlevé II equation are also studied.

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arXiv2016

Proof of a conjecture of Granath on optimal bounds of the Landau constants

We study the asymptotic expansion for the Landau constants $G_n$, \begin{equation*} πG_{n}\sim \ln(16N)+γ+\sum^{\infty}_{k=1}\frac{α_k}{N^k} ~~\mbox{as} ~ n\rightarrow\infty, \end{equation*} where $N=n+1$, and $γ$ is Euler's constant. We show that the signs of the coefficients $α_{k}$ demonstrate a periodic behavior such that $(-1)^{\frac {l(l+1)} 2} α_{l+1}< 0$ for all $l$. We further prove a conjecture of Granath which states that $(-1)^{\frac {l(l+1)} 2} \varepsilon_l(N)<0$ for $l=0,1,2,\cdots$ and $n=0,1,2,\cdots$, $\varepsilon_l(N)$ being the error due to truncation at the $l$-th order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the form \begin{equation*} \ln(16N)+γ+\sum_{k=1}^{p}\frac{α_{k}}{N^{k}}<πG_{n}<\ln(16N)+γ+\sum_{k=1}^{q}\frac{α_{k}}{N^{k}} \end{equation*} for all $n=0,1,2\cdots$, all $p=4s+1,\; 4s+2$ and $q=4m,\; 4m+3$, with $s=0,1,2,\cdots$ and $m=0, 1, 2,\cdots$.

math.CA↗

arXiv2016

Special functions, integral equations and Riemann-Hilbert problem

We consider a pair of special functions, $u_β$ and $v_β$, defined respectively as the solutions to the integral equations \begin{equation*} u(x)=1+\int^\infty_0 \frac {K(t) u(t) dt}{t+x} ~~\mbox{and}~~v(x)=1-\int^\infty_0 \frac{ K(t) v(t) dt}{t+x},~~x\in [0, \infty), \end{equation*} where $K(t)= \frac {1} π\exp \left (- t^β\sin\frac {πβ} 2\right )\sin \left ( t^β\cos\frac{πβ} 2 \right )$ for $β\in (0, 1)$. In this note, we establish the existence and uniqueness of $u_β$ and $v_β$ which are bounded and continuous in $[0, +\infty)$. Also, we show that a solution to a model Riemann-Hilbert problem in Kriecherbauer and McLaughlin [Int. Math. Res. Not., 1999] can be constructed explicitly in terms of these functions. A preliminary asymptotic study is carried out on the Stokes phenomena of these functions by making use of their connection formulas. Several open questions are also proposed for a thorough investigation of the analytic and asymptotic properties of the functions $u_β$ and $v_β$, and a related new special function $G_β$.

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arXiv2016

Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.

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arXiv2015

Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight

We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2)^β(t^2-x^2)^αh(x),~x\in(-1,1),$$ where $β>-1$, $α+β>-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for $x\in[-1,1]$. In this paper, based on the Deift-Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as $n\to \infty$ and $t\to 1$. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto $σ$-function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.

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