扰动LUE和汉克尔矩阵的奇异线性统计量

IF 0.5 4区 数学 Q3 MATHEMATICS Journal of Mathematical Physics Analysis Geometry Pub Date : 2023-08-01 DOI:10.1063/5.0143858
Dan Wang, Mengkun Zhu, Yang Chen
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引用次数: 0

摘要

本文研究了由两个参数的奇异拉盖尔权产生的汉克尔行列式。利用适用于与权相关的单正交多项式的阶梯算子,我们证明了其中一个辅助量是painlevev方程的解,并推导了Hankel行列式的两个对数偏导数的离散σ-形式。利用汉克尔行列式的维数和两个参数的标度极限,将双参数奇异拉盖尔权值的一元正交多项式所满足的二阶微分方程近似为双合流Heun方程。此外,我们利用库仑流体方法和瑞利商,建立了大汉克尔矩阵的最小特征值与两个参数权相关的渐近性质。
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A singular linear statistic for a perturbed LUE and the Hankel matrices
In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated with the weight, we show that one of the auxiliary quantities is a solution to the Painlevé III′ equation and derive the discrete σ-forms of two logarithmic partial derivatives of the Hankel determinant. We approximate the second-order differential equation satisfied by the monic orthogonal polynomials with respect to the singular Laguerre weight with two parameters to the double confluent Heun equation, leveraging the scaling limit for two parameters and the dimension of the Hankel determinant. In addition, we establish the asymptotic behavior of the smallest eigenvalue of large Hankel matrices associated with the weight with two parameters, using the Coulomb fluid method and the Rayleigh quotient.
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来源期刊
CiteScore
0.70
自引率
20.00%
发文量
18
审稿时长
>12 weeks
期刊介绍: Journal of Mathematical Physics, Analysis, Geometry (JMPAG) publishes original papers and reviews on the main subjects: mathematical problems of modern physics; complex analysis and its applications; asymptotic problems of differential equations; spectral theory including inverse problems and their applications; geometry in large and differential geometry; functional analysis, theory of representations, and operator algebras including ergodic theory. The Journal aims at a broad readership of actively involved in scientific research and/or teaching at all levels scientists.
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