1998 Fiscal Year Final Research Report Summary
Selfavoiding walk and its continuum'limit on fractals and geometric figures defined by conformal mappings
Project/Area Number 
09640255

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  SHINSHU UNIVERSITY 
Principal Investigator 
HATTORI Kumiko Shinshu University, Faculty of Science, Assistant Professor, 理学部, 助教授 (80231520)

CoInvestigator(Kenkyūbuntansha) 
NAKAYAMA Kazuaki Shinshu University, Faculty of Science, Assistant, 理学部, 助手 (20281040)
KAMIYA Hisao Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (80020676)
INOUE Kazuyuki Shinshu University, Faculty of Science, Professor, 理学部, 教授 (70020675)
ABE Koujun Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)
ASADA Akira Shinshu University, Faculty of Science, Professor, 理学部, 教授 (00020652)

Project Period (FY) 
1997 – 1998

Keywords  fractals / Selfavoiding walk / stochastic process / continuum limit / sample path / Hausdorff measure / diffusion 
Research Abstract 
(1) We dealt with the continuum limits of selfavoiding walks on fractals and studied geometric properties of their sample paths. The trajectory of a sample path is regarded as a multitype random construction. We first developed a general theorem on the 'exact Hausdorff dimension' for a wide class of multitype random constructions. Our general theorem deals with multitype random constructions with almost sure Hausdorff dimension D (usually, Hausdorff dimensions of random constructions are determined almost surely) and with zero Ddimensional 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 selfavoiding walk called the 'branching model' on the ddimensional Sierpinski gasket. We showed the existence of the continuum limit and then determined the exact Hausdorff dimensions. (2) We considered anisot
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ropic diffusions on the 2dimensional Sierpinski carpet, which is an infiniteramified 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 preSierpinski 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 GrantinAid, 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

Research Products
(12 results)