| Project/Area Number |
13640175
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Okayama University |
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Principal Investigator |
HORA Akihito Okayama University, Faculty of Environmental Science and Technology, Associate Professor, 環境理工学部, 助教授 (10212200)
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| 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)
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| Project Period (FY) |
2001 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| 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)
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| Keywords | spectrum 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 |
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| 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.
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