Towards deeper understanding of renormalization group and lace expansion

我的研究成果可以分为三个方面：(1)自避行走、渗流、格树和动物的关键两点函数分析。我将蕾丝展开技术发挥到极致，并以数学上严格的方式，在高维中推导出这些模型的两点函数的渐近形式。并对两点函数的子导行为进行了详细的估计。(2) p-q模型临界行为分析。p -q模型有两个参数p和q，当q=1-p时与渗流重合，当q=1时与格动物重合。这个模型很有趣，因为渗透的临界维数是6(6)，而晶格动物的临界维数是8 (8)；p-q模型展示了哪个关键维度？我们首先为这个模型建立了蕾丝展开和van den Berg-Kesten不等式。然后我们证明了该模型在高维中表现出与晶格动物相同的临界行为。我们的分析还表明，p-q模型的关键维度是正确的(8)，而不是6(6)。(3)四维层次自回避行走分析。我们在四个维度上研究了分层自回避行走模型的临界行为。利用层次结构，我们得到了一个可严格分析的重整化群递归。结果，我们证明了该模型具有所谓的对数修正。

My research results can be classified into three:(1) Analysis of critical two-point functions in self-avoiding walk, percolation, and lattice trees and animals.I used the lace expansion technique to the extreme, and derived the asymptotic form of two-point functions of these models in a mathematically rigorous manner, in high dimensions. I also derived a detailed estimates on sub-leading behavior of two-point functions.(2) Analysis of the critical behavior of p-q model.P-q model has two parameters, p and q, and coincides with percolation when q=1-p, and coincides with lattice animals when q=1. This model is interesting, because percolation has the critical dimension six (6) and lattice animal has eight (8); which of the critical dimension does the p-q model exhibit? We first established the lace expansion and van den Berg-Kesten inequalities for this model. We then proved that the model exhibits the same critical behavior as that of lattice animals in high dimensions. Our analysis also suggests that the critical dimension of the p-q model is right (8), not six(6).(3) Analysis of hierarchical self-avoiding walk in four dimensions.We investigated the critical behavior of hierarchical self-avoiding walk model in four dimensions. By making use of hierarchical structure, we could derive a renormalization group recursion which can be rigorously analyzed. As a result, we proved the model exhibits the so-called logarithmic corrections.