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宏大未达目标出发，分析高维垫片上自避路径的渐近性，这意味着本研究的更高目的是为重整化群作为随机模型的数学分析寻找线索。项目期间的主要成果如下：1.项目成果四维分层Ising模型的琐碎性。证明了四维分层Ising模型重整化群的临界轨迹的存在性。轨迹收敛于高斯不动点。通过严密的计算机辅助证明，完成了远离高斯不动点的全局轨迹分析。结果提出了一个未经证实的猜想，即在4个时空维度中，从Ising模型中得到的唯一连续统极限量子场论是不相互作用的自由场。谢尔宾斯基垫片的自排斥过程。在一维空间和Sierpinski垫片上，我们发现了连续的非平凡自排斥过程的单参数族，它连续地插入了自回避过程和布朗运动。发现是通过在相应的重整化群中引入插值参数来完成的。d维垫片上自避路径的渐近行为。我们完成了一个重整化群公式，该公式严格地蕴涵了d维垫片上自避路径的渐近行为。这些结果都符合本课题的研究目的，因为它们是关于重整化群的轨迹分析与随机模型的渐近行为之间的严格关系的结果，并且研究的重点是重整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.