Self-avoiding processes and self-repelling processes on fractals

分形上的自回避过程和自排斥过程

• 批准号: 16540101
• 负责人: HATTORI Kumiko
• 金额: $ 1.91万
• 依托单位: Tokyo Metropolitan University (2007)Shinshu University (2004-2006)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 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-16540101/
• 关键词: fractal; self-avoiding walk; self-repelling walk; self-attracting walk; mean-square displacement; recurrence; renormalization group; expected return time; 自己反発ウォーク; 自己吸引ウォーク; 不変測度; シェルピンスキー・ガスケット; 平均2乗距離の指数

We constructed a family of self-repelling walks on the pre-Sierpinski gasket and on the 1-dimensional Euclidean space, respectively, which continuously interpolates between the simple random walk and a self-avoiding walk It is a one-parameter family with parameter u, and u=0 corresponds to a self avoiding walk, u=1 to the simple random walk and 0<u<1 to self-repelling walks The asymptotic behaviors of the walks have been obtained in terms of displacement exponents and a law of iterated logarithms. The result can further be extended to self-attracting walks, with u>1. Our method is based on renormalization group and we found that we can construct more general stochastic chains, using this method. The asympotitic behaviors are obtained in a parallel manner.We studied also the recurrence of the stochastic chains constructed by renormalization group method and obtained a sufficient condition for recurrence. In particular, we proved the above mentioned family of self-repelling and self-attracting walks are recurrent if u>0. We also proved that there is a positive constant c>1 such that the expected return time to the origin is infinite for 0<u<c. This implies that there is a unique, sigma-finite, ergodic invariant measure on the infinite-length path space on the Sierpinski gasket and the 1-dimensional Euclidean space.

分别在pre-Sierpinski gasket和1维欧氏空间上构造了一类连续插值于简单随机游动和自回避游动之间的自排斥游动。这是一个单参数族，参数为u，u=0对应于自回避游动。u=1的简单随机游动和0<u<1的自斥游动.本文用位移指数和迭代律得到了游动的渐近性态.这个结果可以进一步推广到自吸引行走，其中u>1。我们的方法是基于重整化群，我们发现，我们可以构造更一般的随机链，使用这种方法。研究了用重正化群方法构造的随机链的常返性，得到了常返性的一个充分条件。特别地，我们证明了当u>0时，上述自排斥和自吸引游动族是常返的。我们还证明了存在一个正的常数c>1，使得对于0<u<c，期望返回原点的时间是无穷大的。这意味着在Sierpinski垫片上的无限长路径空间和1维欧氏空间上存在唯一的、σ有限的、遍历不变测度。

项目成果

Recurrence of self-repelling and self-attracting walks on the pre-Sierpinski gasket

前谢尔宾斯基垫片上自排斥和自吸引游走的重复出现

• 发表时间: 2008
• 期刊: Stochastics and Dynamics 8
• 作者: M. Denker;K. Hattori;M. Denker and K. Hattori
• 通讯作者: M. Denker and K. Hattori

The hydrodynamic limit of the stochastic ranking processes

随机排序过程的流体动力学极限

• 发表时间: 2008

Stochastic ranking processの流体力学極限

随机排序过程的流体力学极限

• 发表时间: 2008
• 作者: K.Hattori;T.Hattori;服部 久美子
• 通讯作者: 服部 久美子

フラクタル幾何学

分形几何

• 发表时间: 2007
• 作者: K.Falconer;服部久美子;村井浄信
• 通讯作者: 村井浄信

Displacement exponents of self-repelling walks and self-attracting walks on the Sierpinski gasket

Sierpinski 垫片上自排斥游走和自吸引游走的位移指数

• 发表时间: 2005
• 期刊: Journal of Mathematical Sciences University of Tokyo 12
• 作者: K. Hattori;T. Hattori
• 通讯作者: T. Hattori