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Interacting Infinite Particle Systems and Random Matrices

Research Project

Project/Area Number15540106
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionCHIBA UNIVERSITY

Principal Investigator

TANEMURA Hideki  Chiba University, Faculty of Science, Professor, 理学部, 教授 (40217162)

Co-Investigator(Kenkyū-buntansha) NAKAGAMI Jyunichi  Chiba University, Faculty of Science, Professor, 理学部, 教授 (30092076)
NAGISA Masaru  Chiba University, Faculty of Science, Professor, 理学部, 教授 (50189172)
KONNO Norio  Yokohama National University, Faculty of Engineering, Professor, 工学研究院, 教授 (80205575)
KATORI Makoto  Chuo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60202016)
Project Period (FY) 2003 – 2006
Project Status Completed(Fiscal Year 2006)
Budget Amount *help
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 2006 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2005 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2004 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 2003 : ¥900,000 (Direct Cost : ¥900,000)
Keywordsrandom matrix / noncolliding process / Brownian motion / Bessel process / generalized meander / determinantal process / Dyson process / Airy process / determinantal point process / stochastic epidemic model / duality / coexistence / Domany-Kinzel model / vicious walkers / determinantal random point process / Brownian meander / non-colliding process
Research Abstract

It is known that the distribution of particle positions in Dyson's model of Brownian motions coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. We considered a system of noncolliding Brownian motions, in which the noncolliding condition is imposed in a finite time interval (0,T]. This is a temporally inhomogeneous diffusion process and in the limit T to infinity, it converges to Dyson's model. We constructed such a Hermitian matrix-valued process that its eigenvalues are identically distributed with the particle positions of our system of noncolliding Brownian motions.
As an extension of the theory of Dyson's models for the standard Gaussian random-matrix ensembles, we made a systematic study of hermitian matrix-valued processes. In addition to the noncolliding Brownian motions, we introduced noncolliding systems of generalized meanders and showed that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems.
Then we proved that these non-colliding diffusions are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we showed that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions.

Report

(5results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report

Research Products

(31results)

All 2007 2006 2005 2004 2003 Other

All Journal Article Book Publications

  • [Journal Article] Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals.2007

    • Author(s)
      Makoto Katori
    • Journal Title

      Probability Theory and Related Fields 138

      Pages : 113-156

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals.2007

    • Author(s)
      M.Katori, H.Tanemura
    • Journal Title

      Probability Theory and Related Fields 138

      Pages : 113-156

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Random Matrices and Non-colliding processes.2006

    • Author(s)
      Makoto Katori
    • Journal Title

      Invitation to Mathematical Physics(Yuusei-sha) 28

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multi-state epidemic processes on complex networks2006

    • Author(s)
      N.Konno, N.Masuda
    • Journal Title

      Journal of Theoretical Biology 243

      Pages : 64-75

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Nonintersecting paths, noncolliding diffusion processes and Representation theory2005

    • Author(s)
      M.Katori, H.Tanemura
    • Journal Title

      数理解析研究所講究録 1438

      Pages : 12-25

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Non-colliding systems of Brownian particles as Pfaffian process2005

    • Author(s)
      M.Katori
    • Journal Title

      数理解析研究所講究録 1422

      Pages : 12-25

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Journal of Mathematical Physics 45

      Pages : 3058-3085

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Dualities for the Domany-Kinzel model2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Journal of Theoretical Probability 17

      Pages : 131-144

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Coexistence results for a spatial stochastic epidemic model2004

    • Author(s)
      Norio Konno
    • Journal Title

      Markov Processes and Related Fields 10

      Pages : 367-376

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Infinite systems of non-colliding Brownian particles2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Advanced studies in Pure Mathematics 39

      Pages : 283-306

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems.2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Journal of Mathematical Physics 45

      Pages : 3058-3085

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dualities for the Domany-Kinzel model.2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Journal of Theoretical Probability 17

      Pages : 131-144

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Coexistence results for a spatial stochastic epidemic model.2004

    • Author(s)
      Norio Konno
    • Journal Title

      Markov Processes and Related Fields 10

      Pages : 367-376

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Infinite systems of non-colliding Brownian particles.2004

    • Author(s)
      Makoto Katori
    • Journal Title

      Advanced studies in Pure Mathematics 39

      Pages : 283-306

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Noncolliding Brownian motions and Harish-Chandra formula2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Elect. Comm. Probab. 8

      Pages : 112-121

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Functional central limit theorems for vicious walkers2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Stochastics and Stochastics Reports 8

      Pages : 369-390

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dynamical correlations among vicious random walkers2003

    • Author(s)
      Taro Nagao
    • Journal Title

      Physics Letters A307/1

      Pages : 29-35

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Vicious walks with a wall, noncolling meanders, and chiral and Bogoliubov-de Gennes random matrix2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Physical Review E68

      Pages : 021112

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Localization transition of q-friendly walkers2003

    • Author(s)
      Hideki Tanemura
    • Journal Title

      Probability Theory and Related Fields 125

      Pages : 593-608

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Noncolliding Brownian motions and Harish-Chandra formula.2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Electronic Communications in Probability 8

      Pages : 112-121

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Functional central limit theorems for vicious walkers.2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Stochastics and Stochastics Reports 8

      Pages : 369-390

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dynamical correlations among vicious random walkers.2003

    • Author(s)
      Taro Nagao
    • Journal Title

      Physics Letters A307/1

      Pages : 29-35

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Vicious walks with a wall, noncolling meanders, and chiral and Bogoliubov-de Gennes random matrix.2003

    • Author(s)
      Makoto Katori
    • Journal Title

      Physical Review E68

      Pages : 021112

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Localization transition of q-friendly walkers.2003

    • Author(s)
      Hideki Tanemura
    • Journal Title

      Probability Theory and Related Fields 125

      Pages : 593-608

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Book] "ランダム行列と非衝突過程",『数理物理への誘い6』2006

    • Author(s)
      香取眞理, 種村秀紀
    • Total Pages
      28
    • Publisher
      遊星社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Publications] Makoto Katori: "Infinite systems of non-colliding Brownian particles"Adv.Stud.Pure Math.. 39. 283-306 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Makoto Katori: "Functional central limit theorems for vicious walkers"Stoch.Stoch.Rep.. 75. 369-390 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Makoto Katori: "Noncolliding Brownian motions and Harish-Chandra formula"Elect.Comm.Probab.. 8. 112-121 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Makoto Katori: "Vicious walks with a wall, noncolling meanders, and chiral and Bogoliubov-de Gennes random matrix"Phys.Rev.. E68. 021112 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Hideki Tanemura: "Localization transition of d-friendly walkers"Probab.Th.Rel.Fields. 125. 593-608 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Taro Nagao: "Dynamical correlations among vicious random walkers"Phys.Lett.. A307/1. 29-35 (2003)

    • Related Report
      2003 Annual Research Report

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Published : 2003-04-01   Modified : 2016-04-21  

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