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2018 Fiscal Year Annual Research Report

Spectral measures of random matrices and universality of random Jacobi matrices

Research Project

Project/Area Number 16K17616
Research InstitutionTohoku University

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)

Project Period (FY) 2016-04-01 – 2019-03-31
Keywordsclassical beta ensembles / orthogonal polynomials
Outline of Annual Research Achievements

Three classical beta ensembles on the real line, beta-Hermite, beta-Laguerre and beta-Jacobi ensembles, are realized as eigenvalues of three different types of random symmetric tridiagonal matrices. Note that symmetric tridiagonal matrices, also called Jacobi matrices, arise naturally in the theory of orthogonal polynomials on the real line. Indeed, from a Jacobi matrix, one forms a sequence of polynomials and under some conditions, there is a unique probability measure under which the polynomials are orthogonal. The unique probability measure is also referred to as the spectral measure of a Jacobi matrix.

When the parameter beta is fixed, the empirical distribution of eigenvalues of the three ensembles converges to the semi-circle distribution, Marchenko-Pastur distributions and Kesten-McKay distributions, respectively. These results can be intuitively seen via the random matrix models and spectral measures.

In this research, we found a further relation between classical beta ensembles and Hermite, Laguerre and Jacobi orthogonal polynomials. The relation reads as, in the regime where beta is proportional to the inverse of the system size, the empirical distribution converges to the probability measure of a corresponding associated Hermite, Laguerre and Jacobi orthogonal polynomials.

  • Research Products

    (8 results)

All 2019 2018

All Journal Article (4 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 4 results,  Open Access: 2 results) Presentation (4 results) (of which Int'l Joint Research: 2 results,  Invited: 4 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

    • 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 Pages: 295~321

    • DOI

      https://doi.org/10.1007/s10955-018-2131-9

    • 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 Pages: 865~892

    • DOI

      https://doi.org/10.1007/s10955-018-2201-z

    • 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 Pages: 2740~2780

    • DOI

      http://dx.doi.org/10.1214/17-AAP1371

    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] Some aspects of beta ensembles at high temperature2019

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

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

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

    • Author(s)
      Trinh Khanh Duy
    • Organizer
      Seminar on spectral theory, Gakushuin University
    • Invited

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Published: 2019-12-27  

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