Restoration of symmetry in stochastic models on fractals

目前研究项目的直接目标是继续我们的研究，以恢复分形上的异质性，with emphasis on tracing the global structure of the orbits of 'renormalization group' dynamical systems.The ultimate purpose is to find an entially 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 the the the epoc数学物理学，特别是量子场理论中的数学物理学。主要研究结果表明，当前项目的期限如下：1我们扩展了我们对Sierpinski气体到abc气体和尺度irregular abb气体的线性单尺寸差异的分析。以下是一个例子，用于任何对称性的全球恢复都不可能发生或不准确的自我相似性。我们对Sierpinski地毯上的异向扩散进行了类似的分析，这是一个极其精简的分形的典型例子。2我们衍生出了Chiral U（1）异常，是量子场理论的唯一数学现象，来自第一原则。这是第一个数学上合理的证明，作为晶格量子场理论的连续极限，与威尔逊术语3使用Taubelian类型理论的Y。Kasahara，我们发现了一个关于独立随机变量的特定加权总和的极限理论，并将其应用于zeta函数的值分布的非对称评估。4我们发现了一个新的椭圆解决方案，用于杨氏-巴克斯特方程的新面孔与2N-2真实参数。We also found the intertwining relation between the face model and the Z-D2N-symmetric vertex model of Belavin。

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.