{"title":"扰动LUE和汉克尔矩阵的奇异线性统计量","authors":"Dan Wang, Mengkun Zhu, Yang Chen","doi":"10.1063/5.0143858","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A singular linear statistic for a perturbed LUE and the Hankel matrices\",\"authors\":\"Dan Wang, Mengkun Zhu, Yang Chen\",\"doi\":\"10.1063/5.0143858\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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.\",\"PeriodicalId\":50141,\"journal\":{\"name\":\"Journal of Mathematical Physics Analysis Geometry\",\"volume\":\"23 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Physics Analysis Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0143858\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Physics Analysis Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1063/5.0143858","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.