Spectral Analysis of infinite grapgs with discrete group actions

具有离散群作用的无限图形的谱分析

• 批准号: 16340013
• 负责人: KOTANI Motoko
• 金额: $ 7.88万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340013/
• 关键词: 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

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.

从几何的角度讨论了晶格上周期随机游动的长时间行为，特别是大偏差性质，并将其与有限图的第一同调群中的有理凸多面体联系起来，作为显著的组合特征，.晶格具有具有图形距离的度规结构。通过改变距离的尺度，我们得到了一个单参数度量空间族。序列的Gromov-Hausdorff极限称为晶格无穷远处的渐近锥体。当尺度趋于零时，由于晶格的周期性，渐近圆锥的存在，我们用组合数据明确地确定了它的单位球。我们还发表了Sugaku说明书，amer.Math.Soc的一篇关于晶格离散几何分析的综述文章。在这里，我们讨论了晶格上随机游动的谱性质和几何性质，如大数定律、中心极限定理、大偏差和磁薛定谔算子与非对易几何的谱。

项目成果

Volume collapsed three-manifolds with a lower curvature bound

• DOI: 10.1007/s00208-005-0667-x
• 发表时间: 2003-04
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: T. Shioya;Takao Yamaguchi

A note on N. Konno's paper on quantum walk : "Continuous-time quantum walks on trees in quantum probability theory

N. Konno 关于量子行走的论文的注释：“量子概率论中树上的连续时间量子行走

• 发表时间: 2006
• 期刊: Infinite Anal. Quantum. Probab. Relat.Top. 9
• 作者: N.Obata

Ourdiane, Habib Quantum Levy Laplacian and associated heat equation

Ourdiane、Habib Quantum Levy Laplacian 和相关热方程

• 发表时间: 2007
• 期刊: J. Funct. Anal 249, no. 1
• 作者: Ji;Un Cig; obata;Nobuaki
• 通讯作者: Nobuaki

Quantum probability and spectral analysis of graphs. With a foreword by Luigi Accardi

量子概率和图谱分析。

• 发表时间: 2007
• 期刊: Theoretical and Mathematical Physics
• 影响因子: 1
• 作者: Hora;Akihito; Obata;nobuaki
• 通讯作者: nobuaki

Discrete analysis of crystal lattices

晶格的离散分析

• 发表时间: 2006
• 期刊: Sugaku Expositions 19
• 作者: Kotani;motoko
• 通讯作者: motoko