Restoration of symmetry in stochastic models on fractals

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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.