Reconstruction of the theory of pont processes and Gibbs random fields and its application to equlibrium processes, fermion processes etc.

• Project/Area Number: 22540188
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: Takahashi Yoichiro 東京大学, 生産技術研究所, 名誉教授 (20033889)
• Project Period (FY): 2010-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 点過程(確率点場) / Gibbs測度 / 一般化Fock空間 / フェルミオン過程 / fermion(行列式)過程 / 平衡過程 / 大偏差原理 / カオスの伝播 / 点過程 / ギブス測度 / 一般化 / マルコフ過程の急減少(カットオフ)現象

Outline of Final Research Achievements

A new general method to construct a Fock space representation associated with a given point process or random point field is introduced. It gives a simple description of important random fields such as equilibrium processes, Gibbs point fields, fermion(determinantal) processes and boson(permanental) processes.
In addition, the rapid mixing or cut-off phenomenon in Markov chains such as Ehrenfest model is shown to be a special case of the propagation of chaos (in the theory of Boltzmann equation), and which leads to a short proof without tedious computation.

Research Products

• [Journal Article] カオスの伝播とカットオフ現象 エーレンフェスト模型の場合 (2013)
  • Author(s): 髙橋陽一郎
  • Journal Title: 数理解析研究所講究録 Volume: 1855巻 Pages: 236-244
  • Open Access
• [Journal Article] Unitary matrices and random permutations : conjecture and degenerated Laplacian
  • Author(s): Y.Takahashi
  • Journal Title: "Number-theoretic Functions and their Probablistic Aspects--2010 Kyoto", RIMS Kokyuroku Bessatsu 2012 Volume: (未定)(掲載確定)
  • Peer Reviewed
• [Presentation] カオスの伝播とカット・オフ現象 (2012)
  • Author(s): 高橋陽一郎
  • Organizer: 研究集会「無限粒子系、確率場の諸問題VIII」
  • Place of Presentation: 奈良市
  • Invited
• [Presentation] カオスの伝播とカット・オフ：エーレンフェスト模型の場合 (2012)
  • Author(s): 高橋陽一郎
  • Organizer: 数理解析研究所研究集会: 確率論シンポジウム
  • Place of Presentation: 京都
  • Invited
• [Presentation] Does a unitary matrix determines a random permutation? (2011)
  • Author(s): Y.Takahashi
  • Organizer: Control and Noise in Dynamical Systems (Part 2)
  • Place of Presentation: Sapporo, Japan(招待講演)
  • Year and Date: 2011-08-25
• [Presentation] Unitary matrices and probability (2010)
  • Author(s): Yoichiro Takahashi
  • Organizer: The international conference Functions in Number Theory and Their Probabilistic Aspects
  • Place of Presentation: Kyoto University, Kyoto, Japan
  • Year and Date: 2010-12-13
• [Book] 確率論入門II (2015)
  • Author(s): 池田信行、小倉幸雄、髙橋陽一郎、眞鍋昭治郎
  • Total Pages: 381
  • Publisher: 培風館