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Restoration of symmetry in stochastic models on fractals

分形随机模型中对称性的恢复

基本信息

批准号：
09640298
负责人：
HATTORI Tetsuya
金额：
$ 2.24万
依托单位：
Nagoya University (1999)Rikkyo University (1997-1998)
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 1999
项目状态：
已结题

项目摘要

A direct aim of the present research project was continue our study on the restoration of isotropy of stochastic models on fractals, with emphasis on tracing the global structure of the orbits of 'renormalization group' dynamical systems.The ultimate purpose is to find an entirely new and general method of analysis of asymptotic properties of stochastic models which contain the essence of the renormalization group philosophy which was an epoc in the mathematical physics, especially in the quantum field theories.Main research results during the term of the present project is as follows :1 We extended our renormalization group analysis of asymptotically one-dimensional diffusions on Sierpinski gasket to abc-gaskets and scale-irregular abb-gaskets. These are examples for which either global restoration of symmetry does not occur or lacks exact self similarity. We applied similar analysis to an anisotropic diffusion on Sierpinski carpet, which is a typical example of infinitely ramified fractals.2 We derived chiral U(1) anomaly, a mathematical phenomena unique to quantum field theories, from first principles. This is the first mathematically rigorous proof of chiral anomaly as a continuum limit of lattice quantum field theories with Wilson terms.3 Using a Taubelian type theorem of Y. Kasahara, we found a limit theorem on a certain weighted sum of independent stochastic variables, and applied it to an asymptotic evaluation of value-distribution of the zeta function.4 We found new elliptic solutions to the Yang-Baxter equation for a new face with 2N-2 real parameters. We also found the intertwining relation between the face model and the ZィイD2NィエD2-symmetric vertex model of Belavin.

本研究项目的一个直接目的是继续研究分形随机模型的各向同性恢复，重点是追踪“重整化群”动力系统轨道的全局结构，最终目的是找到一种全新的、通用的分析随机模型渐近性质的方法，这种方法包含了作为1950年代一个时代的重整化群哲学的精髓。本课题的主要研究成果如下：1将Sierpinski垫片上渐近一维扩散的重整化群分析推广到abc垫片和尺度不规则的abb垫片上。这些都是不发生全局对称性恢复或缺乏精确自相似性的例子。我们将类似的分析应用于Sierpinski地毯上的各向异性扩散，Sierpinski地毯是无限分歧分形的典型例子。2我们从第一性原理导出了手征U（1）反常，这是量子场论特有的数学现象。这是手征反常作为Wilson项格点量子场论连续极限的第一个严格的数学证明。Kasahara等人，我们发现了一个关于独立随机变量加权和的极限定理，并将其应用于zeta函数值分布的渐近估计。4我们发现了具有2N-2个真实的参数的新面的Yang-Baxter方程的新椭圆解。我们还发现了Belavin的Z D2 N D2对称顶点模型与人脸模型之间的交织关系。