| Project/Area Number |
19K14547
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12010:Basic analysis-related
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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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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | beta ensembles / high temperature regime / orthogonal polynomials / Gaussian fluctuations / beta Jacobi ensembles / beta Jacobi processes / classical beta ensembles / moment method / beta Laguerre processes / random matrix theory / beta Laguerre ensembles / local statistics / random Jacobi matrices |
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| Outline of Research at the Start |
Beta ensembles are objects in random matrix theory, statistical mechanics, potential theory and spectral theory. Among them, three classical beta ensembles on the real line are now realized as eigenvalues of certain random tridiagonal matrices. The parameter beta regarded as the inverse temperature is usually assumed to be fixed. Problems with beta varying have been investigated for some specific beta ensembles recently, leading to some crossover results. This research aims to establish new spectral properties and to provide universal approaches to deal with even the case of beta varying.
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| Outline of Final Research Achievements |
We study beta ensembles on the real line with focusing on the three classical beta ensembles (Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles). In a high temperature regime, we show a universality result at the bulk, that is, around any fixed reference energy, the local statistics converges in distribution to a homogeneous Poisson point process. For the three classical beta ensembles, we completely describe the global behavior, that is, two fundamental results on the convergence to a limit of the empirical distribution (law of large numbers) and Gaussian fluctuations around the limit (central limit theorem). We flexibly use tools from probability theory, spectral theory, theory of orthogonal polynomials and stochastic analysis. The limiting measure in a high temperature regime is related to associated Hermite polynomials (Gaussian case), associated Laguerre polynomials (Laguerre case) and associated Jacobi polynomials (Jacobi case).
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| Academic Significance and Societal Importance of the Research Achievements |
We have developed new approaches to study beta ensembles, especially the three classical beta ensembles. By those approaches, we can completely describe the global and the local asymptotic behavior of beta ensembles in a high temperature regime.
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