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.

我们讨论了鉴于几何形状，尤其是大偏差特性，在晶体晶格上进行了长时间的行为，并将其与有限图的第一个同源性组中的有理凸多面体联系起来，这是显着的组合特征。晶体晶格具有图形距离的度量结构。通过更改距离的规模，我们获得了一个参数的度量空间家族。该序列的Gromov-Hausdorff极限称为晶格无穷大的渐近锥。随着量表的周期性，由于晶格晶格的周期性，渐近锥存在，我们根据组合数据明确确定了其单位球。我们还发表了一篇有关Sugaku Expository Crystal Lattice的离散几何分析的调查文章，Amer.math.soc.math.soc。在那里，我们讨论了在晶体晶格上随机行走的光谱特性和几何形状，例如大数量定律，中心极限定理，磁性schroedinger operators of非交通性几何形状的磁性schroedinger oberators的光谱。

项目成果

Springer

• DOI: 10.1021/es00003a702
• 发表时间: 2006-02
• 期刊: Z Psy.Praxis
• 作者: Nathalie Jacobs

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

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

Random walks and Geometry of a crystall lattice

晶格的随机游走和几何形状

• 发表时间: 2005
• 作者: Kotani;Motoko;M.Kotani;M.Kotani
• 通讯作者: M.Kotani

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