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Harmonic analysis on graphs and discrete groups, and scaling limit for probability models

Research Project

Project/Area Number 13640175
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOkayama University

Principal Investigator

HORA Akihito  Okayama University, Faculty of Environmental Science and Technology, Associate Professor, 環境理工学部, 助教授 (10212200)

Co-Investigator(Kenkyū-buntansha) MURAI Joshin  Okayama University, Graduate School of Humanities and Social Sciences, Assistant, 大学院・文化科学研究科, 助手 (00294447)
SASAKI Toru  Okayama University, Faculty of Environmental Science and Technology, Lecturer, 環境理工学部, 講師 (20260664)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsspectrum of graph / scaling limit / central limit theorem / distance-regular graph / representation of symmetric group / quantum probability / harmonic analysis / method of quantum decomposition / ヤング図形 / 代数的確率論 / 対称群 / 相互作用フォック空間 / 確率モデル
Research Abstract

The aim of this project is to read out statistical properties of huge systems characterized by a certain symmetry from the viewpoint of asymptotic spectral analysis and scaling limits by using the methods of harmonic analysis and representation theory. We obtained concrete results as follows.
1. We computed scaling limits for the spectral distributions of adjacency operators on graphs in the framework of quantum central limit theorem. We introduced Gibbs states as well as vacuum states on distance-regular graphs and investigated the limit picture in low temperature and high degrees especially for Johnson graphs. The result is described in terms of the interacting Fock space associated with Meixner polynomials. Interesting distributions are derived in the limit by using combinatorial structure of creators and annihilators.
2. We established a general theory for spectral analysis of graphs by the method of quantum decomposition. We revealed a connection of asymptotic characteristic values of regular graphs with the parameters of interacting Fock spaces. The limit distributions are systematically described by using methods of orthogonal polynomials and Green functions beyond computation of individual spectral limits. The item here is closely related to a joint work with Nobuaki Obata at Tohoku University.
3. We obtained an extension (a quantization) of Kerov's central limit theorem for irreducible characters and the Plancherel measure as an asymptotic aspect of representations of the symmetric groups. Since the result goes out of the framework of interacting Fock spaces, we introduced a modification of the usual Young graph as well as creators and annihilators on it.

Report

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

    (33 results)

All Other

All Publications (33 results)

  • [Publications] A.Hora: "Asymptotic spectral analysis on the Johnson graphs in infinit degree and zero temperature limit"Interdisciplinary Information Sciences. 10. 1-10 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Higuchi, J.Murai, J.Wang: "The Dobrushin-Hryniv. theory for the two-dimensional lattice Widom-Rowlinson model"Advanced Studies in Pure Mathematics. 39. 233-281 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Scaling limit for Gibbs states of Johnson graphs and resulting Meixner classes"Infinite Dimensional Analysis, Quantum Probability, and Related Topics. 6. 139-143 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora, N.Obata: "Quantum decomposition and quantum central limit theorem"Quantum Probability and White Noise Analysis. 17. 284-305 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group"Springer Lecture Notes in Mathematics. 1815. 77-88 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 洞 彰人: "ヤング図形の極限形状とゆらぎにまつわる漸近的組合せ論"数理解析研究所講究録. 1310. 85-104 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : method of quantum de composition"Journal of Mathematical Physics. 44. 71-88 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Noncommutative aspect of central limit theorem for the irreducible characters of the symmetric groups"Quantum Probability and White Noise Analysis. 16. 318-328 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 洞 彰人, 尾畑 伸明: "量子分解法による隣接作用素のスペクトル解析I"数理解析研究所講究録. 1291. 11-44 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] 村井 浄信: "相分離クラスタの確率過程"電子情報通信学会誌. 85. 639-643 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Gibbs state, quadratic embedding, and central limit theorem on large graphs"Quantum Information III. 67-74 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "The symmetric groups and algebraic central limit theorems"数理解析研究所講究録. 1227. 145-153 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Asymptotic spectral analysis on the Johnson graphs in infinite degree and zero temperature limit"Interdisplinary Information Sciences. 10. 1-10 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Higuchi, J.Murai, J.Wang: "The Dobrushin-Hryniv theory for the two-dimensional lattice Widom-Rowlinson model"Advanced Studies in Pure Mathematics. 39. 233-281 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Scaling limit for Gibbs states of Johnson graphs and resulting Meixner classes"Infinite Dimensional Analysis, Quantum Probability and Related Tonics. 6. 139-143 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora, N.Obata: "Quantum decomposition and quantum central limit Theorem"Quantum Probability and White Noise Analysis. 17. 284-305 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group"Springer Lecture Notes in Mathematics. 1815. 77-88 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Asymptotic combinatorics related to the limit shape and its fluctuation of Young diagrams (in Japanese)"RIMS Kokyuroku. 1319. 85-104 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs method of quantum decomposition"Journal of Mathematical Physics. 44. 71-88 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Noncommutative aspect of central limit theorem for the irreducible characters of the symmetric groups"Quantum Probability and White Noise Analysis. 16. 318-328 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora, N.Obata: "Spectral analysis for adjacency operators by the method of quantum decomposition I (in Japanese)"RIMS Kokyuroku. 1291. 11-44 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] J.Murai: "Stochastic processes for clusters of phase separation (in Japanese)"Denshi Joho Tsushin Gakkai. 85. 639-643 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Gibbs state, quadratic embedding, and central limit theorem on large graphs"Quantum Informational. III. 67-74 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "The symmetric groups and algebraic central limit theorems"RIMS Kokyuroku. 1227. 145-153 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Hora: "Scaling limit for Gibbs states of Johnson graphs and resulting Maixner classes"Infinite Dimensional Analysis, Quantum Probability, and Related Topics. 6,No.1. 139-143 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Hora, N.Obata: "Quantum decomposition and quantum central limit theorem"Quantum Probability and White Noise Analysis. 17. 284-305 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Hora: "A noncommutative version of Kerov's Gaussian limit for the Plancherel measure of the symmetric group"Springer Lecture Notes in Mathematics. 1815. 77-88 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 洞 彰人: "ヤング図形の極限形状とゆらぎにまつわる漸近的組合せ論"数理解析研究所講究録. 1310. 85-104 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"Journal of Mathematical Physics. 44・1. 71-88 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Hora: "Noncommutative aspect of central limit theorem for the irreducible characters of the symmetric groups"Quantum Probability and White Noise Analysis. 16. 318-328 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 洞 彰人, 尾畑 伸明: "量子分解法による隣接作用素のスペクトル解析I"数理解析研究所講究録. 1291. 11-44 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Hora: "Gibbs state, quadratic embedding, and central limit theorem on large graphs"Quantum Information III. 67-74 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Hora: "The symmetric groups and algebraic central limit theorems"数理解析研究所講究録. 1227. 145-153 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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