2016 Fiscal Year Research-status Report
Spectral measures of random matrices and universality of random Jacobi matrices
| Project/Area Number |
16K17616
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| Research Institution | Kyushu University |
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Principal Investigator |
Trinh Khanh・Duy 九州大学, マス・フォア・インダストリ研究所, 助教授 (00726127)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | random Jacobi matrices / Gaussian beta ensembles / Wishart beta ensembles / Jacobi beta ensembles / spectral measures / localization |
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| Outline of Annual Research Achievements |
The purpose of this research is to exploit new spectral properties of Gaussian beta ensembles, in particular, and of random Jacobi matrices, in general. Let me mention two main achievements last year.
(i) I propose a universal approach to study the limiting behavior of the spectral measures of random Jacobi matrices. As an application to Gaussian beta ensembles, the convergence and Gaussian fluctuation around the semicircle distribution are derived. It can be also applicable to Wishart and Jacobi beta ensembles.
(ii) Consider Gaussian beta ensembles in the regime that the parameter beta tends to zero with additional constraint, we show the Gaussian fluctuations for eigenvalues and Poisson statistics for the bulk statistics. This is a joint work with Professor Fumihiko Nakano (Gakushuin University).
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
A general approach to study spectral measures of random Jacobi matrices was proposed and some connection with classical theory of Random Jacobi matrices was established. Two papers were written, one has been accepted to be published.
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| Strategy for Future Research Activity |
The next plan is to study the limiting behavior of Gaussian beta ensembles in a double scaling regime and related random Jacobi matrices. We are interested in both the global (linear statistics) and local (bulk statistics) limiting behaviors.
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| Causes of Carryover |
The plan of buying some books is changed into next years.
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| Expenditure Plan for Carryover Budget |
To buy some books related to probability theory.
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