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Self-avoiding process on high-dimensional gaskets and uniqueness of fixed point of renormalization group

高维垫片的自回避过程及重正化群不动点的唯一性

基本信息

批准号：
12640116
负责人：
HATTORI Tetsuya
金额：
$ 1.92万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640116/
关键词：
renormalization group gasket self-repelling walk self-avoiding walk law of iterated logarithm displacement exponent Sierpinski gasket self-repelling walk self-avoiding walk law of iterated logarithm displacement exponent renormalizat

项目摘要

The short time purpose of this research was to analyze the asymptotics of self-avoiding paths on the higher dimensional gaskets, from the viewpoint of a grand unreached destination of mathematical possibilities of renormalization group, which implies that the higher aim of this research is to find clues for the renormalization group as a mathematical analysis of stochastic models. Main results in the project term are the following.1.Triviality of 4-dimensional hierarchical Ising models.We proved the existence of a critical trajectory of renormalization group for 4-dimensional hierarchical Ising model. The trajectory converges to the Gaussian fixed point. A global trajectory analysis far from the Gaussian fixed point is done by rigorous computer assisted proofs. The result suggests the unproved conjecture that in 4 space-time dimensions, the only continuum limit quantum field theory available from the Ising model is the non-interacting free field.2.Self-repelling process on the Sierpinski gasket.On 1-dimensional space and on the Sierpinski gasket, we found a one parameter family of continuous non-trivial self-repelling processes which continuously interpolates the self-avoiding process and the Brownian motion. Discovery is done by introducing an interpolating parameter in the corresponding renormalization group.3.Asymptotic behavior of self-avoiding paths on d-dimensional gaskets.We completed a renormalization group formulation which rigorously implies asymptotic behaviors of self-avoiding path on d-dimensional gaskets.All the results fits in the purpose of the research project in that they are results on the rigorous relations between the trajectory analysis of the renormalization group and the asymptotic behaviors of stochastic models, and also in that the studies focus on the global analysis of renormalization group trajectories.

这项研究的短期目的是从重整化群的数学可能性的一个宏大未达目的地的观点出发，分析高维垫片上自回避路径的渐近性，这意味着本研究的更高目的是为重整化群作为随机模型的数学分析寻找线索。本项目的主要结果如下：1.四维分层伊辛模型的平凡性。我们证明了四维分层伊辛模型重整化群的临界轨道的存在性。轨迹收敛到高斯不动点。通过严格的计算机辅助证明，进行了远离高斯不动点的全局轨迹分析。2.在一维空间和Sierpinski垫片上，我们发现了一族连续的非平凡自排斥过程，它连续地内插着自回避过程和布朗运动。通过在相应的重整化群中引入一个内插参数来发现。3.d维垫片上自回避路径的渐近行为。我们完成了一个重整化群公式，它严格地隐含了d维垫片上自回避路径的渐近行为。所有这些结果都符合研究项目的目的，因为它们是重整化群的轨迹分析与随机模型的渐近行为之间的严格联系的结果，也在于研究集中在重整化群轨迹的全局分析上。