2018 Fiscal Year Final Research Report
Spectral measures of random matrices and universality of random Jacobi matrices
| Project/Area Number |
16K17616
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Basic analysis
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| Research Institution | Tohoku University (2017-2018)
Kyushu University (2016) |
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Principal Investigator |
Trinh Khanh Duy 東北大学, 数理科学連携研究センター, 准教授 (00726127)
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| Research Collaborator |
Nakano Fumihiko
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | Gaussian beta ensembles / radom Jacobi matrices / spectral measures / semi-circle law |
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| Outline of Final Research Achievements |
Gaussian beta ensembles, a natural generalization of Gaussian orthogonal/unitary/symplectic in terms of the joint probability density functions, are now realized as eigenvalues of random symmetric tridiagonal matrices, called Jacobi matrices, with independent entries. In this research, we establish several new spectral properties of Gaussian beta ensembles such as convergence to a limit and Gaussian fluctuations around the limit of the spectral measures and of the empirical distributions. Approaches which are mainly based on the random matrix model are also applicable to a large class of random Jacobi matrices.
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| Free Research Field |
Probability theory, Random matrix theory
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| Academic Significance and Societal Importance of the Research Achievements |
This research establishes several new spectral properties for a random matrix model so called Gaussian beta ensembles. Approaches are also new and are applicable to a large class of random Jacobi matrices.
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