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

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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.