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Multi-aspects of beta ensembles and related random matrix models

Research Project

Project/Area Number 19K14547
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 12010:Basic analysis-related
Research InstitutionWaseda 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
Project Status Completed (Fiscal Year 2022)
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)
Keywordsbeta 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
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.

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).

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.

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (17 results)

All 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (1 results) Journal Article (7 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 7 results,  Open Access: 2 results) Presentation (9 results) (of which Int'l Joint Research: 3 results,  Invited: 3 results)

  • [Int'l Joint Research] Vietnam National University, Hanoi(ベトナム)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Limit theorems for moment processes of beta Dyson’s Brownian motions and beta Laguerre processes2023

    • Author(s)
      Nakano Fumihiko、Trinh Hoang Dung、Trinh Khanh Duy
    • Journal Title

      Random Matrices: Theory and Applications

      Volume: - Issue: 03

    • DOI

      10.1142/s2010326323500053

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals2022

    • Author(s)
      Can Van Hao、Trinh Khanh Duy
    • Journal Title

      Electronic Journal of Probability

      Volume: 27 Issue: none Pages: 1-40

    • DOI

      10.1214/22-ejp759

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Beta Jacobi Ensembles and Associated Jacobi Polynomials2021

    • Author(s)
      Trinh Hoang Dung、Trinh Khanh Duy
    • Journal Title

      Journal of Statistical Physics

      Volume: 185 Issue: 1

    • DOI

      10.1007/s10955-021-02832-z

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Beta Laguerre ensembles in global regime2021

    • Author(s)
      Trinh Hoang Dung、Trinh Khanh Duy
    • Journal Title

      Osaka J. Math.

      Volume: 58(2) Pages: 435-450

    • NAID

      120007046105

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Beta Laguerre processes in a high temperature regime2021

    • Author(s)
      Trinh Hoang Dung、Trinh Khanh Duy
    • Journal Title

      Stochastic Processes and their Applications

      Volume: 136 Pages: 192-205

    • DOI

      10.1016/j.spa.2021.03.002

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Poisson Statistics for Beta Ensembles on the Real Line at High Temperature2020

    • Author(s)
      Fumihiko Nakano、Khanh Duy Trinh
    • Journal Title

      Journal of Statistical Physics

      Volume: 179 Issue: 2 Pages: 632-649

    • DOI

      10.1007/s10955-020-02542-y

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Beta Laguerre ensembles in global regime2020

    • Author(s)
      Hoang Dung Trinh, Khanh Duy Trinh
    • Journal Title

      Osaka J. Math.

      Volume: 未定

    • NAID

      120007046105

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Gaussian beta ensembles and orthogonal polynomials2022

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Rigorous Statistical Mechanics and Related Topics
    • Related Report
      2022 Annual Research Report
    • Invited
  • [Presentation] Mean-field limit for stochastic processes related to classical beta ensembles2022

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Interplay of partial differential equations and stochastic processes
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Orthogonal polynomials and classical beta ensembles2022

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Annual Conference on Probability and Related Topics
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gaussian beta ensembles and associated Hermite polynomials2021

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      UniMelb-Bielefeld RMT Seminar
    • Related Report
      2021 Research-status Report
  • [Presentation] Global spectrum fluctuations for Gaussian beta ensembles2021

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Danwakai, Tohoku University
    • Related Report
      2021 Research-status Report
  • [Presentation] On the moment method for beta Wishart processes2020

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Spectra of Random Operators and Related Topics
    • Related Report
      2019 Research-status Report
  • [Presentation] On beta Laguerre ensembles at varying temperature2019

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Japanese-German Open Conference on Stochastic Analysis 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Local statistics for beta ensembles at high temperature2019

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      The 18th Symposium Stochastic Analysis on Large Scale Interacting Systems
    • Related Report
      2019 Research-status Report
  • [Presentation] On Wishart processes2019

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      One-day Symposium: Hydrodynamic limit and related topics
    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2024-01-30  

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