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.

我们分别在前谢尔宾斯基垫片和一维欧几里得空间上构建了一系列自排斥游走，它在简单随机游走和自回避游走之间连续插值，它是一个参数为 u 的单参数族，u=0 对应于自回避游走，u=1 对应于简单随机游走，0<u<1 对应于自排斥游走 游走的渐近行为已通过以下方式获得位移指数和迭代对数定律。结果可以进一步扩展到自吸引游走，且 u>1。我们的方法基于重整化群，我们发现使用这种方法可以构造更一般的随机链。以并行的方式得到了渐近行为。我们还研究了重正化群方法构造的随机链的递推性，得到了递推的充分条件。特别是，我们证明了如果 u>0，上述自排斥和自吸引游走系列是经常出现的。我们还证明了存在一个正常数 c>1，使得当 0<u<c 时，返回原点的预期时间是无限的。这意味着在谢尔宾斯基垫片上的无限长路径空间和一维欧几里德空间上存在唯一的西格玛有限遍历不变测度。

项目成果

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
• 作者: K. Hattori; T. Hattori
• 通讯作者: T. Hattori

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

前谢尔宾斯基垫片上自排斥游走和自吸引游走的位移指数

• 发表时间: 2005
• 作者: K.Hattori; T.Hattori
• 通讯作者: T.Hattori

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

在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
• 作者: M.Denker; K.Hatttori
• 通讯作者: K.Hatttori

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

在前谢尔宾斯基垫片和 Z 上反复出现自排斥和自吸引行走

• 发表时间: 2008
• 作者: M. Denker; K. Hattori
• 通讯作者: K. Hattori

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
• 作者: M. Denker;K. Hattori
• 通讯作者: K. Hattori