2007 Fiscal Year Final Research Report Summary
Self-avoiding processes and self-repelling processes on fractals
| Project/Area Number |
16540101
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
|
|---|
| Research Institution | Tokyo Metropolitan University (2007)
Shinshu University (2004-2006) |
|---|
Principal Investigator |
HATTORI Kumiko Tokyo Metropolitan University, Graduate School of Science and Technology, Professor (80231520)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KAMIYA Hisao Shinshu University, Department of Science, associate professor (80020676)
|
|---|
| Project Period (FY) |
2004 – 2007
|
| Keywords | fractal / self-avoiding walk / self-repelling walk / self-attracting walk / mean-square displacement / recurrence / renormalization group / expected return time |
|---|
| Research Abstract |
We constructed a family of self-repelling walks on the pre-Sierpinski gasket and on the 1-dimensional Euclidean space, respectively, which continuously interpolates between the simple random walk and a self-avoiding walk It is a one-parameter family with parameter u, and u=0 corresponds to a self avoiding walk, u=1 to the simple random walk and 0<u<1 to self-repelling walks The asymptotic behaviors of the walks have been obtained in terms of displacement exponents and a law of iterated logarithms. The result can further be extended to self-attracting walks, with u>1. Our method is based on renormalization group and we found that we can construct more general stochastic chains, using this method. The asympotitic behaviors are obtained in a parallel manner.
We studied also the recurrence of the stochastic chains constructed by renormalization group method and obtained a sufficient condition for recurrence. In particular, we proved the above mentioned family of self-repelling and self-attracting walks are recurrent if u>0. We also proved that there is a positive constant c>1 such that the expected return time to the origin is infinite for 0<u<c. This implies that there is a unique, sigma-finite, ergodic invariant measure on the infinite-length path space on the Sierpinski gasket and the 1-dimensional Euclidean space.
|
-
-
-
-
-
-
-
-
[Book] フラクタル幾何学2006
Author(s)
ファルコナー著、服部 久美子・村井 浄信 共訳
Total Pages
428
Publisher
共立出版
Description
「研究成果報告書概要(和文)」より
