arXiv subjects

Shulin Lyu

Publications and source records attributed to Shulin Lyu.

14 recordsLinked to original sources

arXiv2026

Recurrence Coefficients of the Orthogonal Polynomials for Oscillatory Jacobi-type Weight Functions

We study two classes of oscillatory Jacobi-type weight functions $x^c(1-x^2)^{\lambda-1/2}\exp(i\zeta x)$, $x\in[-1,1], \lambda>-1/2, c\in\{0,1\}$. Other restrictions are imposed on $\lambda$ and $\zeta$ to guarantee the existence of the associated orthogonal polynomials. By using the ladder operators established in the recent literature for monic orthogonal polynomials associated with Jacobi-type weight functions and three compatibility conditions, we derive two coupled difference equations satisfied by the three-term recurrence coefficients. Compared with the existing results, these equations are structurally simpler and of lower order. Once the initial values are determined, the recurrence coefficients can be computed at any stage through the difference equations. The obtained expressions enable us to conjecture the symbolic forms for the recurrence coefficients.

math-ph↗

arXiv2026

Hankel Determinant for a Perturbed Laguerre Weight with Pole Singularities and Generalized Painlev\'e III' Equation

We study the Hankel determinant for the weight $x^{\alpha}{\rm exp}(-x-t_1/x-t_2/x^2), x\in[0,+\infty)$, with $\alpha>-1,~t_1\in\mathbb{R}\setminus\{0\}, ~t_2>0.$ Compared with the weight $x^{\alpha}{\rm e}^{-x-t_1/x}$ studied in prior work (where $\alpha,t_1>0$), the range of $\alpha$ in our work is extended and the parameter $t_2$ introduces a ``stronger" zero at the origin. This leads to more varied behavior of the Hankel determinant, and the interplay between $t_1$ and $t_2$ introduces uncertainty and complexity into the analysis. By using a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions, we show that the recurrence coefficients are expressed in terms of four auxiliary quantities which satisfy a system of difference equations that can be iterated. We also establish two coupled second order partial differential equations (PDEs) satisfied by two of the auxiliary quantities, which are reduced to a Painlev\'{e} III$^\prime$ equation when $t_2\rightarrow0^+$. Moreover, the logarithmic derivative of the Hankel determinant is shown to satisfy a second order six degree PDE which is reduced to the $\sigma$-form of the Painlev\'{e} III$^{\prime}$ equation when $t_2\rightarrow0^+$. Under suitable double scaling, we obtain the limiting forms of the above PDEs and deduce the equilibrium density of the eigenvalues for the unitary ensemble. We extend our analysis to the Hankel determinant for $x^{\alpha}\exp(-x-\sum_{k=1}^m t_k/x^k)$ with $m=3$. For general $m$, we outline a derivation that leads, at least in principle, to the PDE satisfied by the logarithmic derivative of the Hankel determinant.

math-ph↗

arXiv2025

Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials

In the literature concerning the Laguerre-type weight function $x^\lambda w_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function $x^{\alpha}(1-x)^{\beta}w_0(x), x\in[0,1]$, with $w_0(x)$ continuously differentiable, the parameters $\lambda,\alpha,\beta$ are usually constrained to be strictly positive to ensure the validity of the results. Recently, in [C. Min and P. Fang, Physica D 473 (2025), 134560 (9pp)], the ladder operators for the monic Laguerre-type orthogonal polynomials with $\lambda>-1$ were derived by exploiting the orthogonality properties. The quantities $A_n$ and $B_n$, which appear as coefficients in the ladder operators, exhibit different expressions compared with the previous ones for $\lambda>0$. In this paper, we construct an alternative deduction by making use of the Riemann-Hilbert problem satisfied by the orthogonal polynomials. Moreover, we employ both derivation strategies mentioned above to produce the ladder operators for the monic standard and shifted Jacobi-type orthogonal polynomials with $\alpha,\beta>-1$. When $\lambda,\alpha,\beta$ are restricted to positive values, our expressions of $A_n$ and $B_n$ are consistent with those in prior work. We present examples to validate our findings and generalize the existing conclusions, established by using the three compatibility conditions of the ladder operators and differentiating the orthogonality relations for the monic orthogonal polynomials, from $\lambda,\alpha,\beta>0$ to $\lambda,\alpha,\beta>-1$.

math.CA↗

arXiv2024

Hankel Determinants for a Deformed Laguerre Weight with Multiple Variables and Generalized Painlev\'{e} V Equation

We study the Hankel determinant generated by the moments of the deformed Laguerre weight function $x^{\alpha}{\rm{e}}^{-x}\prod\limits_{k=1}^{N}(x+t_k)^{\lambda_k}$, where $x\in \left[0,+\infty \right)$, $\alpha,t_k >0, \lambda_k\in\mathbb{R}$ for $k=1,\cdots ,N$. By using the ladder operators for the associated monic orthogonal polynomials and three compatibility conditions, we express the recurrence coefficients in terms of the $2N$ auxiliary quantities which are introduced in the ladder operators and shown to satisfy a system of difference equations. Combining these results with the differential identities obtained from the differentiation of the orthogonality relations, we deduce the Riccati equations for the auxiliary quantities. From them we establish a system of second order PDEs which are reduced to a Painlev\'{e} V equation for $N=1$. Moreover, we derive the second order PDE satisfied by the logarithmic derivative of the Hankel determinant, from which the limiting PDE is obtained under a double scaling. When $N=1$, these two PDEs are reduced to the $\sigma$-form of a Painlev\'{e} V and III equation respectively. When the dimension of the Hermitian matrices from the corresponding deformed Laguerre unitary ensemble is large and $\lambda_k\ge0$, based on Dyson's Coulomb fluid theory, we deduce the equilibrium density for the eigenvalues.

math-ph↗

arXiv2024

Gap Probability Distribution of Gaussian Unitary Ensembles and Painlev\'{e} V Equation

We consider the Hankel determinant generated by the moments of the even weight function ${\rm e}^{-x^2}(A+B\theta(x^2-a^2)), x\in(-\infty,+\infty), a>0, A\ge0, A+B\ge0$. It is intimately related to the gap probability of the Gaussian unitary ensembles on $(-a,a)$ or $(-\infty,-a)\cup(a,+\infty)$. We derive the ladder operators for the monic polynomials orthogonal with respect to this weight function and three supplementary conditions. By using them and differentiating the orthogonality relation, we get difference and Riccati equations for the two auxiliary quantities introduced in the ladder operators. From these equations, we obtain a second-order difference equation and a second-order second-degree ordinary differential equation satisfied by the coefficient of the three-term recurrence relation for the monic orthogonal polynomials. Moreover, we establish the second-order fourth-degree ordinary differential equation satisfied by the logarithmic derivative of the Hankel determinant. As $n\to\infty$ and $a\to0^{+}$ such that $\tau=2\sqrt{2n}a$ is fixed, this equation is reduced to the $\sigma$-form of a Painlev\'{e} V equation in the variable $\tau$.

math-ph↗

arXiv2023

Hankel Determinants for a Gaussian weight with Fisher-Hartwig Singularities and Generalized Painlev\'{e} IV Equation

We study the Hankel determinant generated by a Gaussian weight with Fisher-Hartwig singularities of root type at $t_j$, $j=1,\cdots ,N$. It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing $2N$ auxiliary quantities $\{R_{n,j}, r_{n,j}, j=1,\cdots,N\}$, we build a series of difference equations. Furthermore, we prove that $\{R_{n,j}, r_{n,j}\}$ satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by $\{R_{n,j}, j=1,\cdots,N\}$, which reduces to a Painlev\'{e} IV equation for $N=1$. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized $\sigma$-form of a Painlev\'{e} IV equation.

math-ph↗

arXiv2023

Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} IV System

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $\sigma$-form of a Painlev\'{e} IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlev\'{e} IV system.

math-ph↗

arXiv2022

Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} V System

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at $t_k, k=1,\cdots,m$. By employing the ladder operator approach to establish Riccati equations, we show that $\sigma_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $\sigma$-from of Painlev\'{e} V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express $\sigma_n$ in terms of solutions of a coupled Painlev\'{e} V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and Lax pair. In addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes to $\infty$, the scaled $\sigma_n$ is shown to satisfy a generalized Painlev\'{e} III equation.

nlin.SI↗

arXiv2020

The Smallest Eigenvalue Distribution of the Jacobi Unitary Ensembles

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^α(1-x)^β,~x\in[0,1],~α,β>0$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval $[t,1]$ is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel-kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval $(-a,a),a>0,$ is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight $(1-x^2)^β, x\in[-1,1]$.

math-ph↗

arXiv2020

Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. {\bf 42} (2019), 301--321] where a second order PDE was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in [X. Wu and S. Xu, arXiv: 2002.11240v2] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as $n\rightarrow\infty$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.

math-ph↗

arXiv2020

Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble

We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^γ\mathrm{e}^{-4nx},\;x\in[0,\infty),\;γ>-1$, lie in the interval $[0,α]$. By using previous results for finite $n$ obtained by the ladder operator approach of orthogonal polynomials, we derive the large $n$ asymptotics of the largest eigenvalue distribution function with $α$ ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when $γ=0$.

math-ph↗

arXiv2018

Painlevé III$'$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$ w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0. $$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlevé III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.

math-ph↗

arXiv2018

Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $α=β$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $α=-1/2,$ and $α=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^β$ and $x^{-1/2}(1-x)^β$ The $σ$ function, namely, the derivative of the log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE, satisfies the Jimbo-Miwa-Okamoto $σ$ form of $P_{V}$ and $P_{VI}$, although in the shift Jacobi case, with the weight $x^α(1-x)^β,$ the $β$ parameter does not show up in the equation. We also obtain the asymptotic expansions for the smallest eigenvalue distributions of the Laguerre unitary and Jacobi unitary ensembles after appropriate double scalings, and obtained the constants in the asymptotic expansion of the gap probablities, expressed in term of the Barnes $G-$ function valuated at special point.

math-ph↗

arXiv2015

The largest eigenvalue distribution of the Laguerre unitary ensemble

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are between 0 and t, i.e., the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely αn(t) and \b{eta}n(t), as well as three auxiliary quantities, denoted by rn(t), Rn(t) and sigma n(t). We establish the second order differential equations for both beta n(t) and rn(t). By investigating the soft edge scaling limit when alpha = O(n) as n ! 1 or alpha is finite, we derive a PII , the sigma-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy equation for rho(t) = d/dt(capsigma(t) with capsigma(t) satisfying the sigma-form of PIV or PV . 1 I

math-ph↗