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. 对于一类具有几乎确定的豪斯多夫维数 D（通常随机构造的豪斯多夫维数几乎确定）且具有零 D 维豪斯多夫测度的多类型随机构造，我们得到了精确豪斯多夫维数的一般定理。该定理确定了给出正且有限豪斯多夫测度的维数函数。我们将定理应用于d维Sierpinski垫片上自回避过程的样本路径轨迹，可以将其视为随机分形。 2．我们通过对前谢尔宾斯基垫片上的自排斥行程进行缩放限制，在谢尔宾斯基垫片上构建了一个单参数自排斥过程族。我们证明了我们的模型在谢尔宾斯基垫片上连续插值布朗运动和自回避过程。也就是说，我们证明了在定律收敛意义上，过程在参数上是连续的，并且决定过程速度的指数在参数上也是连续的。我们还建立了自我排斥过程的迭代对数定律。我们的方法也可以应用于一维欧几里得空间，并且我们获得了一类新的一维自排斥过程。