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

Self-avoiding process on high-dimensional gaskets and uniqueness of fixed point of renormalization group

Research Project

Project/Area Number 12640116
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 InstitutionNagoya University

Principal Investigator

HATTORI Tetsuya  Nagoya University, Grad School of Maths, Assoc. Prof., 大学院・多元数理科学研究科, 助教授 (10180902)

Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Keywordsrenormalization group / gasket / self-repelling walk / self-avoiding walk / law of iterated logarithm / displacement exponent / Sierpinski gasket / self-repelling walk / self-avoiding walk / law of iterated logarithm / displacement exponent / renormalization group / gelf-avoiding walk / mean sguare displacement / hierarchical model / porous medium equation / mean square displacement / triviality
Research Abstract

The short time purpose of this research was to analyze the asymptotics of self-avoiding paths on the higher dimensional gaskets, from the viewpoint of a grand unreached destination of mathematical possibilities of renormalization group, which implies that the higher aim of this research is to find clues for the renormalization group as a mathematical analysis of stochastic models. Main results in the project term are the following.
1.Triviality of 4-dimensional hierarchical Ising models.
We proved the existence of a critical trajectory of renormalization group for 4-dimensional hierarchical Ising model. The trajectory converges to the Gaussian fixed point. A global trajectory analysis far from the Gaussian fixed point is done by rigorous computer assisted proofs. The result suggests the unproved conjecture that in 4 space-time dimensions, the only continuum limit quantum field theory available from the Ising model is the non-interacting free field.
2.Self-repelling process on the Sierpinski gasket.
On 1-dimensional space and on the Sierpinski gasket, we found a one parameter family of continuous non-trivial self-repelling processes which continuously interpolates the self-avoiding process and the Brownian motion. Discovery is done by introducing an interpolating parameter in the corresponding renormalization group.
3.Asymptotic behavior of self-avoiding paths on d-dimensional gaskets.
We completed a renormalization group formulation which rigorously implies asymptotic behaviors of self-avoiding path on d-dimensional gaskets.
All the results fits in the purpose of the research project in that they are results on the rigorous relations between the trajectory analysis of the renormalization group and the asymptotic behaviors of stochastic models, and also in that the studies focus on the global analysis of renormalization group trajectories.

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] T.Hattori: "Renormalization group approach to a generalization of the law of iterated logarithms for one-dimensional (non-Markovian) stochastic chains"京都大学数理解析研究所講究録. (発表予定). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 服部 哲弥: "d次元ガスケット上のself-avoiding pathのくりこみ群解析"物制研究. (発表予定). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori: "Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski gaskets"Journal of Statistical Physics. 109. 39-66 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] B.Hambly: "Self-repelling Walk on the Sierpinski Gasket"Probability Theory and Related Fields. 124. 1-25 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hara: "Triviality of hierarchical Ising model in four dimensions"Communications in Mathematical Physics. 220. 13-40 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 服部 哲弥: "ランダムクォークとくりこみ群-一つの数理物理学入門-"共立出版(発表予定). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori, K Hattori: "Renormalization group approach to a generalization of the law of iterated logarithms for one-dimensional (non-Markovian) stochastic chains"Kokyuroku (Kyoto Univ.). (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori: "Renormalization group analysis of self-avoiding paths on d-dimensional gaskets"Bussei Kenkyu. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori, T.Tsuda: "Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski gaskets"Journal of Statistical Physics. 109. 39-66 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] B.Hambly, K.Hattori, T.Hattori: "Self-repelling walk on the Sierpinski gasket"Probability Theory and Related Fields. 124. 1-25 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hara, T.Hattori, H.Watanabe: "Triviality of hierarchical Ising model in four dimensions"Communications in Mathematical Physics. 220. 13-40 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori: "Random walk and renormalization group"Kyoritsu publ. (to appear). (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Hattori: "Renormalization group approach to a generalication of…"京都大学数理解析研究所講究録. (発売予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 服部 哲弥: "d次元ガスケット上のself-avoiding pathのくりこみ群解析"物性研究. (発売予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 服部 哲弥: "ランダムウォークとくりこみ群-一つの数理物理学入門-"共立出版(発売予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] B.Hambly: "Self-repelling walk on the Sierpinski gasket"Probability theory and related fields. 124. 1-25 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Hattori: "Renormalization group analysis of the self-avoiding paths・・・"Journal of statistical physics. 109. 39-66 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] B.Hambly, K.Hattori, T.Hattori: "Self-repelling walk on the Sierpinski gasket"Probability theory and related fields. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hara, T.Hattori, H.Watanabe: "Triviality of hierarchical Lsing model in four dimensions"Communications in mathematical Physics. 220. 13-40 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Hara: "Triviality of hierarchical Ising model in four dimensions"Communications in Mathematical Phycics. (出版予定).

    • Related Report
      2000 Annual Research Report

URL: 

Published: 2000-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi