Self-avoiding walk on the high-dimensional Sierpinski gaskets and random trees

在高维谢尔宾斯基垫片和随机树上自回避行走

基本信息

批准号：
11640110
负责人：
HATTORI Kumiko
金额：
$ 1.92万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640110/
关键词：
fractal self-avoiding walk self-repelling walk Sierpinski gasket scaling limit law of iterated logarithm random fractal Hausdorff measure ランダム・ウォーク Sierpinski Gasket 平均二乗距離確率過程 self-repelling

项目摘要

1. We obtained a general theorem on exact Hausdorff dimensions for a class of multi-type random constructions with almost sure Hausdorff dimension D ( usually, Hausdorff dimensions of random constructions are determined almost surely ) and with zero D-dimensional Hausdorff measure. This theorem determines dimension functions which give positive and finite Hausdorff measures. We applied our theorem to the trajectories of sample paths of self-avoiding processes on the d-dimensional Sierpinski gasket, which can be considered as random fractals.2. We constructed a one-parameter family of self-repelling processes on the Sierpinski gasket, by taking scaling limits of self-repelling walks on the pre-Sierpinski gaskets. We proved that our model interpolates continuously the Brownian motion and a self-avoiding process on the Sierpinski gasket. Namely, we proved that the process is continuous in the parameter in the sense of convergence in law, and that the exponent that determines the speed of the process is also continuous in the parameter. We also established a law of the iterated logarithm of our self-repelling processes. Our approach can be applied also to one-dimensional Euclidean space and we obtained a new class of one-dimensional self-repelling processes.

1。我们在一类具有几乎确定的Hausdorff尺寸D的多类随机构建体（通常，几乎可以肯定地确定了Hausdorff尺寸）的多类随机构建体的精确hausdorff尺寸上，并且具有零d-Digensional hausdorff测量。该定理确定了尺寸函数，从而提供了积极和有限的Hausdorff测度。我们将定理应用于d维sierpinski垫圈上自避免过程的样本路径的轨迹，可以将其视为随机分形2。我们在塞尔平垫圈上建立了一个自我撤销过程的一家参与者家族，通过在塞皮斯基赛前垫片上进行自我依靠步行的限制。我们证明了我们的模型在Sierpinski垫圈上连续插入布朗运动和自我避免的过程。也就是说，我们证明了法律收敛意义上该过程在参数中是连续的，并且确定过程速度的指数在参数中也是连续的。我们还建立了一个自我依据过程的迭代对数法律。我们的方法也可以应用于一维的欧几里得空间，我们获得了一级新的一维自我修复过程。