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Self-avoiding walk and its continuum'limit on fractals and geometric figures defined by conformal mappings

Research Project

Project/Area Number 09640255
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 InstitutionSHINSHU UNIVERSITY

Principal Investigator

HATTORI Kumiko  Shinshu University, Faculty of Science, Assistant Professor, 理学部, 助教授 (80231520)

Co-Investigator(Kenkyū-buntansha) NAKAYAMA Kazuaki  Shinshu University, Faculty of Science, Assistant, 理学部, 助手 (20281040)
KAMIYA Hisao  Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (80020676)
INOUE Kazuyuki  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (70020675)
ABE Koujun  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)
ASADA Akira  Shinshu University, Faculty of Science, Professor, 理学部, 教授 (00020652)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1998: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1997: ¥1,900,000 (Direct Cost: ¥1,900,000)
Keywordsfractals / Self-avoiding walk / stochastic process / continuum limit / sample path / Hausdorff measure / diffusion / ハウスドルフ速度 / ハウスドルフ次元 / ランダムフラクタル
Research Abstract

(1) We dealt with the continuum limits of self-avoiding walks on fractals and studied geometric properties of their sample paths. The trajectory of a sample path is regarded as a multi-type random construction. We first developed a general theorem on the 'exact Hausdorff dimension' for a wide class of multi-type random constructions.
Our general theorem deals with multi-type random constructions with almost sure Hausdorff dimension D (usually, Hausdorff dimensions of random constructions are determined almost surely) and with zero D-dimensional Hausdorff measure. It determines dimension functions which give positive and finite Hausdorff measures, which we call exact Hausdorff dimensions, for a wide class of constructions.
As an application of this theorem, we considered a model of self-avoiding walk called the 'branching model' on the d-dimensional Sierpinski gasket. We showed the existence of the continuum limit and then determined the exact Hausdorff dimensions.
(2) We considered anisot … More ropic diffusions on the 2-dimensional Sierpinski carpet, which is an infinite-ramified fractal, and showed that the isotropy is asymptotically restored as the scale in which we see the diffusions gets larger. This can be shown in terms of restoration of isotropy of anisotropic resistance networks on the pre-Sierpinski carpet. This phenomenon of restoration of isotropy is unique and of interest in the sense that it does not happen in a homogenious space such as the Eucledian spaces, but occurs only in inhomogenious spaces such as fractals.
Using Grant-in-Aid, we bought books on fractals, Hausdorff measures, probability theory, ergodic theory etc, and also computer software to be used for electronic communication and writing papers.
The Grant also enabled us to meet in person researchers in close fields to discuss and collect information on random constructions, Hausdorff and Packing measures of geometic figures constructed using conformal mappings, which helped us much get insight and hints for future developments of our research. Less

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report

Research Products

(21 results)

All Other

All Publications (21 results)

  • [Publications] 服部久美子: "Weak homogenization of anisotropic diffusion on' pre-Sierpioski carpets" Commun.Math.Phys. 188. 1-27 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 浅田明: "Hodge operators on mapping spaces" Group21:Physical Applications and Mathematical Aspects of Geometry. (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 浅田明: "Spectre invariants and Geometry of mapping spaces" Contemporary Mathematics,Geometric Aspects of Partial Differential Equations. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 阿部孝順: "Volumes of Compact Symmetric Spaces" Tokyo Math Jour.20. 87-105 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 井上和行: "A stochastic model for a dam with non-additive in put" Proceedings of SAP'98. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 中山一昭: "Motion of Curves in hyperboloid in the Minkowski space" J.Phys.Soc.Jpn.67. 3031-3037 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Hattori: "Weak homogenization of anisotropic diffusion on pre-Sierpinski carpets" Commun.Math.Phys.188. 1-27 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Asada: "Hodge operators on mapping spaces" Group 21 : Physical Applications and Mathematical Aspects of Geometry. (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Asada: "Spectre invariants and geometry of mapping spaces" Contemporary Mathematics, Geome-tric Aspects of Partial Differen-tial Equations.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Abe: "Volumes of compact symmetric spaces" Tokyo Math.Jour.20. 87-105 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Inoue: "A stochastic model for a dam with non-additive input" Proceedings, of SAP'98.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Nakayama: "Motion of curves in hyperboloid in the Minkowski space" J.Phys.Soc.Jpn.67. 3031-3037 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 中山一昭: "Motion of curves in Hyperbolod in the Minkowski space" Journal phys.Soc.Jpn. 67巻9号. 3031-3037 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 井上和行: "A stochastic model for a dam with non-additive input" Proceedings of SAP'98. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 井上和行: "非加法的な流入を伴うダムの確率過程" 統計数理研究所共同研究リポート. 109. 13-16 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 浅田明: "Spectre invariants and geometry of mapping spaces" Contemporary Mathematics, Geometric Aspects of Partial Differential Eqations. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 阿部 孝順: "Realization of spaces E_6/(U (1) Spin (10),E_7/(U (1) E_6),E_8/(U (1) E_7) and their volumes" Tokyo Math.Jour.20. 73-86 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 阿部 孝順: "Volumes of compact symmetric spaces" Tokyo Math.Jour.20. 87-105 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 阿部 孝順: "Invariant forms on the exceptional symmetric spaces FII and EIII" Proceedings of 1996 Korea-Japan conference on transformation group theory. 3-16 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 浅田 朗: "Hodge operators of mapping spaces ; Localstudy" BSG Proceedings 1 ; Global Analysis,Differential Geometry,Lie Algebras. 11-20 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 浅田 朗: "Hodge operators on mapping spaces" Group 21 : Physical Applications and Mathematical Aspects of Geometry. 925-928 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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