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Towards deeper understanding of renormalization group and lace expansion

Research Project

Project/Area Number 16540102
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 InstitutionKyushu University

Principal Investigator

HARA Takashi  Kyushu University, Faculty of Mathematics, Professor (20228620)

Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,540,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Keywordslace expansion / renormalization group / critical phenomena / self-avoiding walk / percolation / p-q model / two-point function / hierarchical model / 階層モデル / イジングモデル
Research Abstract

My research results can be classified into three:
(1) Analysis of critical two-point functions in self-avoiding walk, percolation, and lattice trees and animals.
I used the lace expansion technique to the extreme, and derived the asymptotic form of two-point functions of these models in a mathematically rigorous manner, in high dimensions. I also derived a detailed estimates on sub-leading behavior of two-point functions.
(2) Analysis of the critical behavior of p-q model.
P-q model has two parameters, p and q, and coincides with percolation when q=1-p, and coincides with lattice animals when q=1. This model is interesting, because percolation has the critical dimension six (6) and lattice animal has eight (8); which of the critical dimension does the p-q model exhibit? We first established the lace expansion and van den Berg-Kesten inequalities for this model. We then proved that the model exhibits the same critical behavior as that of lattice animals in high dimensions. Our analysis also suggests that the critical dimension of the p-q model is right (8), not six(6).
(3) Analysis of hierarchical self-avoiding walk in four dimensions.
We investigated the critical behavior of hierarchical self-avoiding walk model in four dimensions. By making use of hierarchical structure, we could derive a renormalization group recursion which can be rigorously analyzed. As a result, we proved the model exhibits the so-called logarithmic corrections.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (5 results)

All 2008 2007 2005

All Journal Article (3 results) (of which Peer Reviewed: 1 results) Presentation (2 results)

  • [Journal Article] Decay of correlations for nearest-neighbor self-avoiding walk,percolation,lattice trees and animals2008

    • Author(s)
      Takashi Hara
    • Journal Title

      Annals of Probability 36

      Pages: 530-593

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Decay of correlations for nearest-neighbor self-avoiding walk, percolation, lattice trees and animals2008

    • Author(s)
      T., Hara
    • Journal Title

      Annals of Probability 36

      Pages: 530-593

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Decay of correlations in nearest-neighbour self-avoiding walk, percolation, and lattice trees and animals2007

    • Author(s)
      Takashi HARA
    • Journal Title

      Annals of Probability (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Presentation] Decay of correlations for nearestneighbor self-avoiding walk,percolation,lattice trees and animals2005

    • Author(s)
      Takashi Hara
    • Organizer
      Some Recent Topics on Stochastic Models
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2005-09-29
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Decay of correlations for nearest-neighbor self-avoiding walk, percolation, lattice trees and animals2005

    • Author(s)
      T., Hara
    • Organizer
      Some Recent Topics on Stochastic Models
    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto University
    • Year and Date
      2005-09-29
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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