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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640255/
• 关键词: fractals; Self-avoiding walk; stochastic process; continuum limit; sample path; Hausdorff measure; diffusion; ハウスドルフ速度; ハウスドルフ次元; ランダムフラクタル; fractals; Self-avoiding walk; stochastic process; continuum limit; sample path; Hausdorff measure; diffusion; ハウスドルフ速度; ハウスドルフ次元; ランダムフラクタル

(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）我们处理在其样品路径的分形和研究二的几何特性上进行自我避免步行的持续限制。样品路径的轨迹被认为是多类随机结构。我们首先为广泛的多类随机结构开发了有关“确切的Hausdorff维度”的一般定理。我们的一般定理处理具有几乎肯定的Hausdorff dimension D（通常，几乎可以确定随机构建体的Hausdorff尺寸）的多类随机构建体）并且具有零D二维Hausdorff测量。它确定了为广泛的构造而言，它具有积极和有限的Hausdorff测量值，我们称之为确切的Hausdorff尺寸。由于该定理的应用，我们考虑了一种自我避免自我避免行走模型的模型，称为“分支模型”在d-digensional sierpinski垫圈上。我们展示了持续极限的存在，然后确定了确切的Hausdorff尺寸。（2）我们考虑了Anisot…在二维Sierpinski地毯上更具弹性差异，这是一个无限型的分形，并且表明同型在较大的差异中是不对键的，这是不对键的。这可以通过恢复到塞皮斯基前地毯上各向异性抗性网络的各向异性恢复。这种各向同性恢复的现象是独一无二的，在某种意义上不发生在诸如欧克里德式空间之类的同质空间中，但仅发生在诸如分形之类的非构态空间中在近距离领域的研究人员中，讨论和收集有关随机构造，豪斯多夫和使用保形映射构建的地貌数字的包装度量的信息，这有助于我们为未来的研究发展提供了深刻的见识和暗示。较少的