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

本研究的短期目的是从重整化群数学可能性的一个未达到的宏伟目标的角度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标是为重整化群作为随机模型的数学分析找到线索。本课题的主要结果如下：1.四维层次Ising模型的平凡性证明了四维层次Ising模型重整化群的临界轨线的存在性。轨迹收敛到高斯不动点。远离高斯不动点的全局轨迹分析通过严格的计算机辅助证明完成。结果表明，在4维时空中，Ising模型唯一可用的连续极限量子场论是无相互作用的自由场。2. Sierpinski垫片上的自排斥过程在一维空间和Sierpinski垫片上，我们发现了一个单参数的连续非平凡自排斥过程族，它连续地内插自排斥过程，避免过程和布朗运动。通过在相应的重整化群中引入一个内插参数，发现了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，它严格地揭示了d维垫片上自回避路径的渐近行为.所有的结果都符合研究项目的目的，因为它们是关于轨迹分析与尺寸垫片之间的严格关系的结果。重正化群与随机模型的渐近行为，以及重正化群轨迹的全局分析。

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.