Multi-aspects of beta ensembles and related random matrix models

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14547
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Waseda University
• Principal Investigator: Trinh Khanh Duy 早稲田大学, 理工学術院, 准教授(任期付) (00726127)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: beta ensembles / high temperature regime / orthogonal polynomials / Gaussian fluctuations

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).

Free Research Field

probability theory

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.