Noncommutative Toda Chains, Hankel Quasideterminants And Painlev'e II Equation
Vladimir Retakh, Vladimir Rubtsov
我们构造解决方案的无限Toda系统和一个类似的Painlev'e II方程的非交换微分除环的Hankel矩阵的拟行列式。
We construct solutions of an infinite Toda system and an analogue of the Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices.
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Quasideterminant solutions of NC Painlevé II equation with the Toda solution at $ n=1 $ as a seed solution in its Darboux transformation
在本文中，我构造的达布变换的非交换Toda解在$ n=1 $的线性系统的帮助下，其兼容性条件产生零曲率表示的相关的非线性微分方程组。通过将$ n=1 $上的Toda解作为达布变换的种子解，得到了非对易Painlevé II方程的拟行列式解.进一步通过迭代，将种子解的达布变换推广到$ N$ -th形式.最后给出了量子Painlevé II方程的零曲率表示，它明确地包含了普朗克常量$ \hbar $，当$ \hbar \rightarrow 0 $时，系统退化为经典Painlevé II。
In this article non-abelian version of quantum Painlevé II equation is presented with Its quasideterminant solutions has been derived by using the Darboux transformations. This non-abelian quantum Painlevé II equation may be considered as a specific case of its purely noncommutatie analogue presented by V. Retakh and V. Rubtsov . In these computations the quantum Painlevé II symmetric form with commutation relations presented by H. Nagoya are applied to derive Nonabelian quantum Painlevé II equation and a new commutation relation between variable $z$ and the solution $ f(z)$ such as $ z f - f …
This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo-Miwa-Okamoto $σ$-form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced …
Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations
本文通过将tau函数与汉克尔行列式关联构造Painlevé V方程有理解，分析其极点（对应多项式tau函数的根）的渐近行为，证明次数无穷大时根渐近填充解析弧界定区域并给出量子化条件下的近似位置。
Lax pair representation and Darboux transformation of NC Painlevé-II equation
将潘勒韦方程推广到非对易空间是可积系统理论中的一个重要研究方向，探讨Painlevé方程的一些重要性质，Lax表示，达布变换以及它们与可积方程的关系也是一个有趣的问题.本文研究了最近由V. Retakh和V. Rubtsov提出的非对易Painlevé二阶方程的Lax公式、达布变换和拟行列式解。
Generating Function Associated with the Determinant Formula for the Solutions of the Painleve' II Equation
本文考虑Painleve' II方程通解的Hankel行列式公式。我们表明，生成函数的条目的汉克尔行列式有关的渐近解在无穷远的线性问题的Painleve' II方程描述的isomonodromic变形。
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 determina…
This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension of Chen and Pruessner \cite{Chen2005}, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlevé IV. In addition, we consider the large $n$ behavior of the corresponding orthogonal polynomials and prove that the…
Nonisospectral deformations of noncommutative Laurent biorthogonal polynomials and matrix discrete Painlevé-type equations
本文通过非交换洛朗双正交多项式的非等谱变形建立其与矩阵离散潘勒韦方程的联系，并导出矩阵离散潘勒韦型方程。
Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight
我们研究了与权重$$w(x;t)=(1-x^2)^β(t^2-x^2)^αh(x),~x\in(-1,1),$$相关的Hankel行列式，其中$β>-1$，$α+β>-1$，$t>1$，$h(x)$在包含$x\in[-1,1]$的$[-1,1]$和$h(x)>0$的域中解析。基于Deift-Zhou非线性最速下降分析，研究了Hankel行列式$n\to \infty$和$t\to 1$的双标度极限.我们得到的Hankel行列式的渐近近似，评价的Jimbo-Miwa-Okamoto $σ$ -功能的Painlevé III方程。得到了扰动雅可比多项式的首项系数和递推系数的渐近性。
We study the Hankel determinant generated by a singularly perturbed Jacobi weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{x^{2}}},\;\;\;\;\;\;x\in[-1,1],\;\;α>0,\;\;t\geq 0. $$ If $t=0$, it is reduced to the classical symmetric Jacobi weight. For $t>0$, the factor $\mathrm{e}^{-\frac{t}{x^{2}}}$ induces an infinitely strong zero at the origin. This Hankel determinant is related to the Wigner time-delay distribution in chaotic cavities. In the finite $n$ dimensional case, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ by using the ladder operator approach. We show that the Hankel de…
Hankel Determinant structure of the Rational Solutions for Fifth Painlevé Equation
利用Umemura多项式构造了第五类Painlevé方程有理解的Hankel行列式表示.我们的构造给出了$ n\geq 0$的Umemura多项式$σ_{n}$的Hankel行列式公式的显式形式。此外，我们还利用Heun融合函数的对数导数计算了元素的生成函数。
Hankel Determinants for a Gaussian weight with Fisher-Hartwig Singularities and Generalized Painlevé IV Equation
本文研究了带Fisher-Hartwig奇点的高斯权重Hankel行列式，通过推导阶梯算子并引入辅助量构建差分方程与Riccati方程，最终证明其满足广义Painlevé IV方程。
Random Matrix Models, Double-Time Painlevé Equations, and Wireless Relaying
本文对多天线无线通信场景进行了深入研究，在该场景中，中间中继站接收到的微弱信号被放大，然后转发到最终目的地。决定系统性能的关键量是目的地的信噪比（SNR）γ的统计特性。在对编码结构的某些假设下，最近的工作已经通过其矩生成函数来表征SNR分布，所述矩生成函数根据经由变形的拉盖尔权重生成的某个汉克尔行列式。在这里，我们采用两种不同的方法来描述汉克尔行列式。首先，我们利用正交多项式满足的阶梯算子给出了一个“双时间”Painlevé微分方程的精确刻画，在一定的限制下，它减少到Painlevé V。其次，我们采用Dyson的库仑流体方法推导出一个封闭的形式近似的汉克尔行列式。这两个特征用于推导γ的累积量的封闭形式表达式，并计算工程感兴趣的性能量。
Generalized Charlier Recurrence Coefficients: Hankel Formulas, Laguerre--Freud Dynamics, and Painlevé V Connections
本文研究广义Charlier多项式递推系数，通过Hankel行列式与Laguerre--Freud动力学给出其公式，并联系Painlevé V方程，数值验证大次数渐近行为。
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…
We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlevé II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of…
Hankel Determinants for a Deformed Laguerre Weight with Multiple Variables and Generalized Painlevé V Equation
该文研究多变量变形拉盖尔权重的汉克尔行列式，借助梯级算子等方法推导相关方程与系综特征值平衡密度，建立其与广义Painlevé方程的联系。
In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble. This problem originates from the largest or smallest eigenvalue distribution of the degenerate Laguerre unitary ensemble. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight. By applying the ladder operators to our problem, we obtain two auxiliary quantities $R_n(t)$ and $r_n(t)$ and show that they satisfy the coupled Riccati equations, from which we find that $R_n(t)$ satisfies the Painlevé V equation. Furthermore, we prove that $σ_{n}(…
Hankel Determinant for a Perturbed Laguerre Weight with Pole Singularities and Generalized Painlevé III' Equation
本文研究了带有极点奇点的扰动拉盖尔权函数的Hankel行列式，推导了其与广义Painlevé III'方程的关联。
Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
本文研究带跳跃间断的拉盖尔权重生成的 Hankel 行列式，通过阶梯算子与 Riemann-Hilbert 方法，证明其对数导数满足推广的 Painlevé V 方程，并与耦合 Painlevé V 系统及广义 Painlevé III 方程建立联系。
We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,γ)=e^{-z^2+tz}|z-t|^γ(A+Bθ(z-t))$, where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $γ>-1$ and $z\in\textbf{R}$. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility conditions, we obtain a series of difference and differential equations to describe the relations among $α_n$, $β_n$, $R_n(t)$ and $r_n(t)$. Especially, we find that the auxiliary quantities $R_n(t)$ and $r_n(t)$ satisfy the coupled Riccati equations, and $R_n(t)$ satisfies a part…
Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
本文研究首一半经典 Laguerre 多项式，证明非对角递推系数满足 Chazy II，并结合库仑流体与离散系统得到递推系数、次首项系数及 Hankel 行列式的大 n 渐近展开。
Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System
该研究针对带跳跃间断点的高斯权重生成的Hankel行列式，借助阶梯算子与补充条件推导其对数导数满足的二阶偏微分方程，进一步通过Riemann-Hilbert问题建立辅助量与耦合Painlevé IV系统解的关联。
Zeros of large degree Vorob'ev-Yablonski polynomials via a Hankel determinant identity
本文给出了Vorob'ev-Yablonski多项式$\mathcal{Q}_n(x),x\in\mathbb{C}$平方的一种新的Hankel行列式表示.这些多项式是第二个潘勒韦方程$u_{xx}=xu+2u^3+α$的有理解的主要成分。作为新恒等式的一个应用，我们通过渐近分析与上述Hankel行列式相关的一组（伪）正交多项式，研究了$\mathcal{Q}_n(x)$作为$n\rightarrow\infty$的零分布。我们的方法再现了Buckingham和Miller \cite{BM}最近在相同背景下获得的结果，其中使用了PII方程的Jimbo-Miwa Lax表示及其渐近分析。
The discrete Painlevé XXXIV hierarchy arising from the gap probability distributions of Freud random matrix ensembles
该研究针对Freud酉系综对称间隙概率分布问题，通过梯算子方法推导间断Freud权正交多项式递推系数的方程，首次在随机矩阵理论中发现离散Painlevé XXXIV层级，并明确相关系数间的关系。
Asymptotics of determinants of Hankel matrices via non-linear difference equations
E.海涅在19世纪研究了一个系统的正交多项式与重量$\left[x(x-α)(x-β)\right]^{-\frac{1}{2}}$，$x\in[0,α]$，$0<α<β$。C. J. Rees在1945年提出的，与重量$\left[(1-x^2)(1-k^2x^2)\right]^{-\frac{1}{2}}$、$x\in[-1,1]$、$k^2\in(0,1)$相关联。这些也被称为椭圆正交多项式，因为权重的矩可以用椭圆积分表示。这样的正交多项式是非常感兴趣的，因为相应的汉克尔行列式，取决于一个参数$k^2$，其中$0<k^2<1$是$τ$ ~功能的一个特定的Painlevé VI，其特殊情况是相关的枚举问题所产生的弦理论。我们表明，递推系数，表示为$β_n(k^2),\;n=1,2,\dots$ ;和${\rm p}_1(n,k^2)$，系数$x^{n-2}$的一元多项式正交关于广义版本的重量研究里斯，$$ (1-x^2)^α(1-k^2x^2)^β,\;\;x\in[-1,1],\;α>-1,\;β\in \mathbb{R}, $$满足二阶非线性差分方程。基于差分方程的大$n$展开与相关汉克尔行列式的前导项的已知渐近性相结合时，产生了汉克尔行列式的完全渐近展开。还讨论了Painlevé方程以及由Rees发现的线性二阶微分方程的推广。
The Hilbert series of N=1 SO(N_c) and Sp(N_c) SQCD, Painlevé VI and Integrable Systems
本文提出了一种新的方法来计算具有SO（N_c）和Sp（N_c）规范群的4dN =1超对称QCD的Hilbert级数。结果表明，这样的希尔伯特级数可以重铸的Hankel矩阵的行列式。借助于随机矩阵理论的结果，这种Hankel行列式可以被精确地和渐近地估计。从而得到了关于一般色味数的Hilbert级数的几个新结果。我们表明，希尔伯特级数引起家庭的合理解决方案，回文分子，Painlevé VI方程。由于这种潘勒韦方程的存在，存在描述SO（N_c）和Sp（N_c）SQCD的模空间的可积Hamilton系统.对于每个系统，我们明确地陈述了相应的哈密顿量和椭圆曲线族。结果表明，这类椭圆曲线与4维N=2 SU（2）规范理论的Seiberg-Witten曲线具有相同的形式.
Non-Abelian discrete Toda chains and related lattices
该研究推导了具有拟行列式解与多形式Lax对的二维离散Toda格非阿贝尔类似物，探讨了其与多类低维格点非阿贝尔模型的关联，并研究了离散Toda方程到Somos-N序列再到离散Painlevé方程体系的非交换类比。
Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants
本文研究广义Freud权对应的正交多项式与Hankel行列式，揭示离散Painlevé I hierarchy的统一结构，推导相关微分方程与恒等式，得到这些量的完全渐近展开并以十次Freud权为例说明。
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…
We study the joint probability generating function for $k$ occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case $k = 1$. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probab…
Gap Probability Distribution of Gaussian Unitary Ensembles and Painlevé V Equation
本文研究由特定偶权函数生成的 Hankel 行列式与高斯酉系综间隙概率的关系，通过阶梯算子推导出差分方程和常微分方程，并在极限下将其约化为 Painlevé V 方程的 σ-形式。
All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé equation in generic form depends on two parameters $m$ and $n$, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as $n\in\mathbb{Z}$ increases and how the patterns vary with $m\in\mathbb{C}$. This study suggests that it is reasonable to consider the rational solutions in the li…
Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
本文通过引入对数项研究扩展Gross-Witten-Wadia酉矩阵模型，利用Deift-Zhou非线性最速下降法求解Riemann-Hilbert问题，建立了Toeplitz行列式的渐近近似，并与非齐次Painlevé II方程的解相联系。
Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
本文研究受扰高斯权重下的正交多项式与 Hankel 行列式，通过 Painlevé V、差分方程及 Dyson 库仑流体方法建立递推系数的大 $n$ 渐近展开。
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $β_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal…
Painlevé V and time dependent Jacobi polynomials
在本文中，我们研究最简单的变形序列的正交多项式，即取代原来的（或参考）的重量$w_0(x)$定义在一个区间上的$w_0(x)e^{-tx}.$。这是一个众所周知的事实，在这样的变形下的递归系数表示为$α_n$和$β_n$演变在$t$根据户田方程，产生的时间相关的正交多项式，使用Sogo的术语。所得到的“时间相关”雅可比多项式满足线性二阶方程。我们将证明该颂歌的系数与特定的Painlevé V密切相关。此外，我们还证明了与“时间相关”雅可比权重相关的一元正交多项式的$z^{n-1}$系数满足，直到$t,$中的翻译Jimbo-Miwa $σ$ -当递归系数$P_{V};$取决于$t$中的平移和线性分数变换$P_{V}(α^2/2,-β^2/2, 2n+1+α+β,-1/2).$时，组合一对非线性差分方程和一对Toda方程。这将反过来使我们能够证明，与一类Toeplitz加Hankel算子相关的某个Fredholm行列式与Painlevé方程有联系。
Linear systems, Hankel products and the sinh-Gordon equation
本文研究连续时间线性系统的 Hankel 积分算子与 Fredholm 行列式，通过构造状态空间上的代数结构获得 sinh-Gordon 方程的解，并揭示其 tau 函数满足 Painléve III' 方程且对应可由 Coulomb 流体方法求解的随机矩阵模型。
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