Discrete geometric analysis of the scaling limit and its applications

尺度极限的离散几何分析及其应用

• 批准号: 24840002
• 负责人: TANAKA Ryokichi
• 金额: $ 0.92万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Research Activity Start-up
• 财政年份: 2012
• 资助国家: 日本
• 起止时间: 2012-08-31 至 2014-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-24840002/
• 关键词: 大域解析学; 離散幾何解析学; ランダムウォーク; 離散群; 等周定数; グラフ; スケール極限

We studied the random walks on groups and the geometry of random graphs having their origins in statistical physics. Those are related to the central theme of this project. In particular, we focused on the important example in the three-dimensional solvable Lie groups and its discrete subgroups, and showed that the harmonic measure associated with the random walk drastically changes its measure-theoretic property according to the step distribution. We also studied the related problems arising from this subject.Furthermore, we estimated the isoperimetric constant which has an important role in the geometry of the graphs for locally free product of random graphs.

我们研究了群上的随机游动和起源于统计物理的随机图的几何。这些都与这个项目的中心主题有关。特别地，我们聚焦于三维可解李群及其离散子群中的重要例子，并证明了与随机游动相关的调和测度根据步长分布极大地改变了它的测度论性质。此外，我们还对随机图的局部自由积估计了在图的几何中起重要作用的等周常数。

项目成果

Large deviation for a random walk on a group of polynomial growth

一组多项式增长的随机游走的大偏差

• 发表时间: 2012
• 作者: E. Ohtani;S. Kamada;T. Sakai;H. Terasaki;Y. Shibazaki;T. Sakamaki;S. Takahashi;T. Sakairi;H. Fukui;A. Q. Baron;Y. Oba;田中亮吉;Y. Oba;Ryokichi Tanaka;大場康弘;Ryokichi Tanaka;Y. Oba;Ryokichi Tanaka;Ryokichi Tanaka;N. Watanabe;Y. Oba;Ryokichi Tanaka;E.-S. Moon;Ryokichi Tanaka;Makoto Nakashima;Motoo Tange;Y. Oba;Ryokichi Tanaka;Motoo Tange;中島誠;大場康弘;Ryokichi Tanaka
• 通讯作者: Ryokichi Tanaka

Harmonic measures for a solvable group

可解群的调和测度

• 发表时间: 2013
• 作者: 尾坂和哉;大場康弘;香内晃;渡部直樹;田中亮吉
• 通讯作者: 田中亮吉

The triviality of the boundary for nilpotent groups

幂零群边界的平凡性

• 发表时间: 2013
• 作者: 尾坂和哉;大場康弘;香内晃;渡部直樹;Ryokichi Tanaka
• 通讯作者: Ryokichi Tanaka

Random walks on solvable Lie groups and their harmonic measures

可解李群的随机游走及其调和测度

• 发表时间: 2014
• 作者: 加藤潤三;前村奈央佳;吉田沙蘭;Ryokichi Tanaka
• 通讯作者: Ryokichi Tanaka

A global law of large number on nilpotent groups

幂零群的全局大数定律

• 发表时间: 2012
• 作者: E. Ohtani;S. Kamada;T. Sakai;H. Terasaki;Y. Shibazaki;T. Sakamaki;S. Takahashi;T. Sakairi;H. Fukui;A. Q. Baron;Y. Oba;田中亮吉;Y. Oba;Ryokichi Tanaka;大場康弘;Ryokichi Tanaka;Y. Oba;Ryokichi Tanaka;Ryokichi Tanaka
• 通讯作者: Ryokichi Tanaka