2019 Fiscal Year Research-status Report
Multi-aspects of beta ensembles and related random matrix models
| Project/Area Number |
19K14547
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| Research Institution | Waseda University |
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Principal Investigator |
Trinh Khanh・Duy 早稲田大学, 理工学術院, 准教授(任期付) (00726127)
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | beta ensembles / random matrix theory / beta Laguerre ensembles / local statistics |
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| Outline of Annual Research Achievements |
This research studies spectral properties of beta ensembles and related random matrix models in case the inverse temperature beta is allowed to vary with the system size. We obtain the following results.
(1) For beta Laguerre ensembles, one of three classical beta ensembles on the real line, we completely describe the global asymptotic behavior of the empirical distribution, that is, the convergence to a limit distribution and Gaussian fluctuations around the limit. Beta Laguerre ensembles are generalizations of the distribution of the eigenvalues of Wishart matrices or Laguerre matrices, two types of random matrices in statistics, in terms of the joint density. They are now realized as eigenvalues of a random tridiagonal matrix model. For the proof, we make use of the random matrix model and extend some ideas used in the case of Gaussian beta ensembles, another classical beta ensembles.
(2) For general beta ensembles on the real line in a high temperature regime, the regime where beta tends to zero at the rate of the reciprocal of the system size, we show that the local statistics around any fixed reference energy converges to a homogeneous Poisson point process. We prove the Poisson statistics by analyzing the joint density with the help of some estimates from the theory of large deviation principle.
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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
We have obtained two results: the global asymptotic behavior for beta Laguerre ensembles and the local asymptotic behavior for general beta ensembles in a high temperature regime. The former completely describes the limiting behavior of the empirical distribution of beta Laguerre ensembles for varying parameter beta. The latter shows the universality of a Poisson statistics in a high temperature regime.
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| Strategy for Future Research Activity |
We continue to study spectral properties of beta ensembles and related random matrix models. In particular, the plan of this year is to deal with a dynamic version of beta ensembles.
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| Causes of Carryover |
One business trip was canceled and will be moved to the next fiscal year.
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