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Probabilistic Analysis of Gibbs measures associated with the integrable system

Research Project

Project/Area Number 14540110
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 InstitutionKwansei Gakuin University

Principal Investigator

CHIYONOBU Taizo  Kwansei Gakuin Univ., School of Science and Technology, Ass.Professor, 理工学部, 助教授 (50197638)

Co-Investigator(Kenkyū-buntansha) YABUTA Kozo  Kwansei Gakuin Univ., School of Science and Technology, Professor, 理工学部, 教授 (30004435)
YAMANE Hideshi  Kwansei Gakuin Univ., School of Science and Technology, Ass.Professor, 理工学部, 助教授 (80286145)
ICHIHARA Kanji  Kansai Uni., Faculty of Engineering, Professor, 工学部, 教授 (00112293)
TAMURA Yozo  Keio Univ., Science and Technology, Ass.Professor, 理工学部, 助教授 (50171905)
OSADA Hirofumi  Kyushu Univ., Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20177207)
Project Period (FY) 2002 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)
Keywordsintegrable system / Gibbs measure / limit theorems / network / ポリヤの壺 / 長距離可積分系 / 大偏差原理 / ラプラス法
Research Abstract

We have obtained results on two topics. First, we study the asymptotic behavior of the partition function of a finete Gibbs measure with repulsive interaction potentials. The function is a modification of the one studied by G.Gallavotti & C.Marchioro. While their approach depends on the integrability of the model, we apply the Laplace method to give an logarithmic asymptotic of the partition function. This is of some interest for the theory of the large deviation principle in the sense that the partition function has the different order of asymptotics from the one given by Varadhan's asymptotic formula. Our result is weaker than theirs in the sense that it gives only the logarithmic asymptotics. But we believe it is still worth noting down here, because we view the integal from a totally different perspective ; namely, we treat as a Laplace asymptotics and thus our method here is applicable to not only the potentials which constitutes the integrable Hamiltonian system, but to more gene … More ral singular potentials. Theorem is of interest from the point of view of the theory of large deviations because it gives the different order 1/n^2 from the ordinary order 1/n, which is the one when Varadhan's asymptotic formula can be applied.
Secondly, we study on networks. Many real networks in the world have the power-law degree distribution One of the most famous model that generates SF networks is ‘BA model' proposed by Barab'asi and Albert. BA model has two properties. One is the growth property and the other is the it preferential attachment property such that a new added vertex is connected to an old vertex randomly chosen with the probability proportional to its degree. However, all of real-world networks are not generated by these properties. This network model based on thresholding of the summed vertex weights, which belongs to the subclass of a model such that the connection by edges are determined by interactions of vertices that are endowed with intrinstic weights. We focused on the mechanism by which the weight distribution is generated. We proved the weight distribution asymptotically tends to the geometric distribution which generates on the preferential attachment property, and we show by the numerical experiments that a graph generated by the threshold model when the weight distribution is geometric distribution is a SF network. Less

Report

(5 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (16 results)

All 2006 2005 2004 2002

All Journal Article (16 results)

  • [Journal Article] On Laplace asymptotic formula for Gibbs measures with repulsive interactions.2006

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Kwansei Gakuin University Natural Sciences Review 10

      Pages: 11-25

    • NAID

      110004910802

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Operator semi-self-similar processes and their space-scaling matrices,2006

    • Author(s)
      T.Saigo
    • Journal Title

      Ststist. Probab.Letters 76

      Pages: 675-681

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Laplace asymptotic formula for Gibbs measures with repulsive interactions2006

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Kwansei Gakuin Univ. Natural Sciences Review 10

      Pages: 11-25

    • NAID

      110004910802

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Operator semi-self-similar processes and their space-scaling matrices2006

    • Author(s)
      T.Saigo
    • Journal Title

      Ststist.Probab.Letters 76

      Pages: 675-681

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A probabilistic analysis of the modified Barabasi-Albert model for scale-free networks2005

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Kwansei Gakuin University natural sciences review 9

      Pages: 1-16

    • NAID

      110004595530

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Homogenization of disconnected selfsimilar fractal sets in R2005

    • Author(s)
      H.Takahashi
    • Journal Title

      Tokyo J. Math 28

      Pages: 83-101

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A probabilistic analysis of the modified Barbasi-Albert model for scale-free networks2005

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Kwansei Gakuin Univ. Natural Sciences Review 9

      Pages: 1-16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Homogenization on disconnected selfsimilar fractal sets in R2005

    • Author(s)
      H.Takahashi
    • Journal Title

      Tokyo J.Math. 28

      Pages: 83-101

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A probabilistic analysis of the modified Barabasi-Albert model for scale-free networks2005

    • Author(s)
      T.Chiyonobu, D.Harada, H.Miwa
    • Journal Title

      Kwansei Gakuin University natural sciences review, 9

      Pages: 1-16

    • NAID

      110004595530

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Fourier-Ehrenpreis integral formula for harmonic function2004

    • Author(s)
      H.Yamane
    • Journal Title

      J. Math. Soc. Japan 56・3

      Pages: 9-18

    • NAID

      10013358903

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Laplace method approach for Gibbs measures with strong mcan-field potentials2004

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Proceedings of the 5~<th> Tunisia_Japan Symposium on Culture, Science and Technology

      Pages: 15-18

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fourier-Ehrenpreis integral formula for harmonic functions2004

    • Author(s)
      H.Yamane
    • Journal Title

      J.Math.Soc.Japan 56, no.3

      Pages: 9-18

    • NAID

      10013358903

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Laplace method approach for Gibbs measures with storng mean-field potentials2004

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Proceedings of the 5^<th> Tunisia-Japan Symposium on Culture, Science and Technology

      Pages: 15-18

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Laplace method approach for Gibbs measures with storng mean-field potentials2004

    • Author(s)
      T.Chiyonobu
    • Journal Title

      Proceedings of the 5^<th> Tunisia_Japan Symposium on Culture, Science and Technology

      Pages: 15-18

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Birth and death processes in randomly fluctuation environments2002

    • Author(s)
      K.Ichihara
    • Journal Title

      Nagoya Math J 166

      Pages: 29-50

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Birth and death processes in randomly fluctuation environments2002

    • Author(s)
      Ichihara
    • Journal Title

      Nagoya Math J. 166

      Pages: 29-50

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary

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Published: 2002-04-01   Modified: 2016-04-21  

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