2022 Fiscal Year Annual Research 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 Jacobi ensembles / Gaussian fluctuations / beta Jacobi processes |
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| Outline of Annual Research Achievements |
Continue to study beta Jacobi ensembles in a high temperature regime, we establish central limit theorems for polynomial test functions of the empirical distributions.
In a high temperature regime, in a previous work, we have shown that the empirical distribution of the eigenvalues converges weakly to a limiting probability measure which is related to a new model (Model III) of associated Jacobi polynomials. We now obtain a result on Gaussian fluctuations around the limit, or central limit theorems involving orthogonal polynomials. The idea is to use a dynamical approach by studying beta Jacobi processes, the dynamical version of beta Jacobi ensembles. We refine a moment method at the process level which has been successfully used in the study of Gaussian beta ensembles and beta Laguerre ensembles.
The Jacobi case is technically more difficult than the Gaussian case and the Laguerre case. Detailed arguments involve: result on the freezing regime in which the system size is fixed while the parameter beta tends to infinity, duals of Jacobi polynomials, joint convergence of stochastic processes and their initial data. Two important new ideas here are: (i) dealing with stationary beta Jacobi processes whose stationary distribution is nothing but the corresponding beta Jacobi ensembles; and (ii) showing that in the limit, the initial conditions are independent of the rest of the process.
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