Self-avoiding walk and its continuum'limit on fractals and geometric figures defined by conformal mappings

共形映射定义的分形和几何图形的自回避行走及其连续体极限

基本信息

批准号：
09640255
负责人：
HATTORI Kumiko
金额：
$ 1.98万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

(1) We dealt with the continuum limits of self-avoiding walks on fractals and studied geometric properties of their sample paths. The trajectory of a sample path is regarded as a multi-type random construction. We first developed a general theorem on the 'exact Hausdorff dimension' for a wide class of multi-type random constructions.Our general theorem deals with 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. It determines dimension functions which give positive and finite Hausdorff measures, which we call exact Hausdorff dimensions, for a wide class of constructions.As an application of this theorem, we considered a model of self-avoiding walk called the 'branching model' on the d-dimensional Sierpinski gasket. We showed the existence of the continuum limit and then determined the exact Hausdorff dimensions.(2) We considered anisot … More ropic diffusions on the 2-dimensional Sierpinski carpet, which is an infinite-ramified fractal, and showed that the isotropy is asymptotically restored as the scale in which we see the diffusions gets larger. This can be shown in terms of restoration of isotropy of anisotropic resistance networks on the pre-Sierpinski carpet. This phenomenon of restoration of isotropy is unique and of interest in the sense that it does not happen in a homogenious space such as the Eucledian spaces, but occurs only in inhomogenious spaces such as fractals.Using Grant-in-Aid, we bought books on fractals, Hausdorff measures, probability theory, ergodic theory etc, and also computer software to be used for electronic communication and writing papers.The Grant also enabled us to meet in person researchers in close fields to discuss and collect information on random constructions, Hausdorff and Packing measures of geometic figures constructed using conformal mappings, which helped us much get insight and hints for future developments of our research. Less

(1) 我们处理了分形上自回避游走的连续体极限，并研究了其样本路径的几何性质。我们首先提出了关于“的”的一般定理。精确的豪斯多夫维数'对于广泛的多类型随机构造。我们的一般定理处理具有几乎确定的豪斯多夫维数 D 的多类型随机构造（通常，随机构造的豪斯多夫维数几乎确定当然），并且具有零 D 维豪斯多夫测度。它确定了给出正且有限的豪斯多夫测度的维数函数，我们将其称为精确豪斯多夫维数，对于广泛的构造。作为该定理的应用，我们考虑了自我模型。 -避免游走称为d维Sierpinski垫片上的“分支模型”我们证明了连续极限的存在，然后确定了精确的Hausdorff维数。（2）我们考虑了。二维谢尔宾斯基地毯上的各向异性扩散，这是一个无限分支的分形，并且表明，随着我们看到扩散的尺度变大，各向同性逐渐恢复，这可以通过恢复来证明。前谢尔宾斯基地毯上各向异性电阻网络的各向同性这种各向同性恢复的现象是独特的并且令人感兴趣，因为它不会发生在同质中。我们利用补助金购买了分形、豪斯多夫测度、概率论、遍历理论等方面的书籍，以及用于电子通信的计算机软件这笔资助还使我们能够与密切领域的研究人员会面，讨论和收集有关随机构造、豪斯多夫和使用共形映射构造的几何图形的包装测量的信息，这对我们的研究的未来发展有很大帮助。