| Project/Area Number |
16340013
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tohoku University |
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Principal Investigator |
KOTANI Motoko Tohoku University, Tohoku University, Graduate School ofScineces, Professor (50230024)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOYA Takashi Tohoku University, Graduate School ofScineces, Professor (90235507)
IZEKI Hiroyasu Tohoku University, Graduate School ofScineces, Associate Professor (90244409)
OBATA Nobuaki Tohoku University, Graduate School ofInformation Scineces, Professor (10169360)
SUNADA Toshikazu Meiji University, Fucalty ofScinece and Technology, Professor (20022741)
NAYATANI Shin Nagoya University, Graduate School ofMathematics, Professor (70222180)
藤家 雪朗 東北大学, 大学院・理学研究科, 講師 (00238536)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥12,310,000 (Direct Cost: ¥11,500,000、Indirect Cost: ¥810,000)
Fiscal Year 2007: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2006: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2005: ¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2004: ¥3,600,000 (Direct Cost: ¥3,600,000)
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| Keywords | random walk / large deviation / gramov-hausdorff limit / asymptotic cone / crystal lattice / 磁場付き推移作用素 / 非可換空間 / ランダムウォーク / 被覆グラフ / 大偏差原理 / 距離空間 / 磁場付きシュレジンガー作用素 / ヒルベルト バイ モジュール |
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| Research Abstract |
We discuss ed a long time behavior of periodic random walks on a crystal lattice in view of geometry, a large deviation property in particular, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as remarkable combinatorial features,. A crystal lattice has a metric structure with the graph distance. By changing scale of the distance, we obtain a one-parameter family of metric spaces. The Gromov-Hausdorff limit of the sequence is called the asymptotic cone at the infinity of the crystal lattice. As the scale go to zero., because of the periodicity of the crystal lattice, the asymptotic cone exists and we determinded its unit ball explicitely in terms of combinatorial data.
We also published a survey article on discrete geometric analysis of crystal lattice from Sugaku Expository, Amer.Math.Soc. In there, we discussed spectral properties and geometry of random walks on a crystal lattice, such as the law of large number, the central limit theorem, large deviation and spectrum of magnetic Schroedinger operators from non-commutative geometry.
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