2007 Fiscal Year Final Research Report Summary
Spectral Analysis of infinite grapgs with discrete group actions
| 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)
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| Project Period (FY) |
2004 – 2007
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| Keywords | random walk / large deviation / gramov-hausdorff limit / asymptotic cone / crystal lattice |
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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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[Book] 文庫で読む科学2007
Author(s)
小谷元子
Total Pages
55-66
Publisher
岩波書店
Description
「研究成果報告書概要(和文)」より
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