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.

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步行是根据位移指数和迭代对数定律获得的。结果可以进一步扩展到自> 1的自我吸引步行。我们的方法基于重新归化组，我们发现我们可以使用此方法构建更通用的随机链。我们还研究了通过重新归一化组构建的随机链的复发，并获得了足够的复发条件。特别是，如果u> 0，我们证明了上述自我重复和自我吸引步行的家族是经常出现的。我们还证明存在一个正常数c> 1，因此对于0 <u <c，对原点的预期返回时间是无限的。这意味着在Sierpinski垫圈和一维欧几里得空间上的无限长度路径空间上有一个独特的，sigma-finite的，恒星的不变度。

项目成果

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

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

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

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

前谢尔宾斯基垫片和 Z 上反复出现自排斥和自吸引游动

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