• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Spectral measures of random matrices and universality of random Jacobi matrices

Research Project

Project/Area Number 16K17616
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionTohoku University (2017-2018)
Kyushu University (2016)

Principal Investigator

Trinh Khanh Duy  東北大学, 数理科学連携研究センター, 准教授 (00726127)

Research Collaborator Nakano Fumihiko  
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
KeywordsGaussian beta ensembles / radom Jacobi matrices / spectral measures / semi-circle law / classical beta ensembles / orthogonal polynomials / random Jacobi matrices / Wigner's semi-circle law / Gaussian fluctuation / Wishart beta ensembles / Jacobi beta ensembles / localization / random matrix theory / random Jacobi matrix / beta ensembles
Outline of Final Research Achievements

Gaussian beta ensembles, a natural generalization of Gaussian orthogonal/unitary/symplectic in terms of the joint probability density functions, are now realized as eigenvalues of random symmetric tridiagonal matrices, called Jacobi matrices, with independent entries. In this research, we establish several new spectral properties of Gaussian beta ensembles such as convergence to a limit and Gaussian fluctuations around the limit of the spectral measures and of the empirical distributions. Approaches which are mainly based on the random matrix model are also applicable to a large class of random Jacobi matrices.

Academic Significance and Societal Importance of the Research Achievements

This research establishes several new spectral properties for a random matrix model so called Gaussian beta ensembles. Approaches are also new and are applicable to a large class of random Jacobi matrices.

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (22 results)

All 2019 2018 2017 2016

All Journal Article (9 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 8 results,  Open Access: 3 results,  Acknowledgement Compliant: 2 results) Presentation (13 results) (of which Int'l Joint Research: 5 results,  Invited: 10 results)

  • [Journal Article] On spectral measures of random Jacobi matrices2018

    • Author(s)
      Trinh Khanh Duy
    • Journal Title

      Osaka J. Math.

      Volume: 55 Pages: 595-617

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics2018

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

      Journal of Statistical Physics

      Volume: 173 Issue: 2 Pages: 295-321

    • DOI

      10.1007/s10955-018-2131-9

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime2018

    • Author(s)
      Goel Akshay、Trinh Khanh Duy、Tsunoda Kenkichi
    • Journal Title

      Journal of Statistical Physics

      Volume: 174 Issue: 4 Pages: 865-892

    • DOI

      10.1007/s10955-018-2201-z

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Limit theorems for persistence diagrams2018

    • Author(s)
      Hiraoka Yasuaki、Shirai Tomoyuki、Trinh Khanh Duy
    • Journal Title

      The Annals of Applied Probability

      Volume: 28 Issue: 5 Pages: 2740-2780

    • DOI

      10.1214/17-aap1371

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Distributions of the determinants of Gaussian beta ensembles2017

    • Author(s)
      Khanh Duy Trinh
    • Journal Title

      RIMS Kokyuroku

      Volume: 2023 Pages: 77-85

    • Related Report
      2017 Research-status Report
  • [Journal Article] Global Spectrum Fluctuations for Gaussian Beta Ensembles: A Martingale Approach2017

    • Author(s)
      Khanh Duy Trinh
    • Journal Title

      J. Theor. Probab.

      Volume: 印刷中 Issue: 3 Pages: 1-18

    • DOI

      10.1007/s10959-017-0794-9

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Limit theorems for persistence diagrams2017

    • Author(s)
      Yasuaki Hiraoka, Tomoyuki Shirai and Khanh Duy Trinh
    • Journal Title

      Annals of Applied Probability

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] A remark on the convergence of Betti numbers in the thermodynamic regime2017

    • Author(s)
      Trinh Khanh Duy
    • Journal Title

      Pac. J. Math. Ind.

      Volume: 9 Issue: 1

    • DOI

      10.1186/s40736-017-0029-0

    • NAID

      40021482133

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On spectral measures of random Jacobi matrices2017

    • Author(s)
      Trinh Khanh Duy
    • Journal Title

      Osaka J. Math.

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Some aspects of beta ensembles at high temperature2019

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Spectra of Random Operators and Related Topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Gaussian beta ensembles in global regime2019

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      MSJ Spring Meeting 2019
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Classical beta ensembles at high temperature2018

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Random matrices and their applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Classical beta ensembles in global regime2018

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Seminar on spectral theory, Gakushuin University
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Global spectral properties of Gaussian beta ensembles2018

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      大阪大学確率論セミナ
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On infinite Jacobi matrices related to beta ensembles2018

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      The 16th annual Korea-Japan Workshop on Algebra and Combinatorics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On associated orthogonal polynomials2018

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      Workshop on "Random matrices, determinantal processes and their related topics"
    • Related Report
      2017 Research-status Report
  • [Presentation] Poisson statistics for Gaussian beta ensembles at high temperature2017

    • Author(s)
      Khanh Duy Trinh
    • Organizer
      2017年度確率論シンポジウム
    • Related Report
      2017 Research-status Report
  • [Presentation] Law of Large Numbers for Persistence Diagrams2017

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Topological Data Analysis and Related Topics
    • Place of Presentation
      東北大学、仙台
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On i.i.d. Jacobi matrices and Gaussian beta ensembles at high temperature2016

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      第152回学習院大学スペクトル理論セミナー
    • Place of Presentation
      学習院大学、東京
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Global asymptotic behaviours of Gaussian beta ensembles2016

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      First Kyushu-UNSW Joint Workshop on the Mathematics underpinning Industry and Innovation
    • Place of Presentation
      University of New South Wales, Sydney, Australia
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On local statistics of random Jacobi matrices2016

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Spectra of Random Operators and Related Topics
    • Place of Presentation
      慶應義塾大学、横浜
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Some results on Gaussian beta ensembles at high temperature2016

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      XII Brunel-Bielefeld Workshop on Random Matrix Theory
    • Place of Presentation
      Brunel University London, London, England
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research

URL: 

Published: 2016-04-21   Modified: 2020-03-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi