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Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
Chao Min1; Yang Chen2
2019-01-25
Source PublicationarXiv
ISSN2331-8422
Abstract

In this paper, we study the large N behavior of the moment-generating function (MGF) of the linear statistics of N × N Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite N Fredholm determinant expression of the MGF of the linear statistics [21], we find the large N asymptotics of the MGF associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then we obtain the mean and variance of the suitably scaled linear statistics. We show that there is an equivalence between the large N behavior of the MGF of the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to the statistical equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In the end, we use the Coulomb fluid method to obtain the mean and variance of another type of linear statistics in GUE, which reproduces the result of Basor and Widom [7].

KeywordRandom Matrix Ensembles Linear Statistics Moment-generating Function Airy Kernel Mean And Variance Asymptotics
Language英语
Fulltext Access
Document TypeJournal article
CollectionFaculty of Science and Technology
DEPARTMENT OF MATHEMATICS
Corresponding AuthorYang Chen
Affiliation1.School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/T 7714
Chao Min,Yang Chen. Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel[J]. arXiv,2019.
APA Chao Min,&Yang Chen.(2019).Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel.arXiv.
MLA Chao Min,et al."Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel".arXiv (2019).
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