Towards deeper understanding of renormalization group and lace expansion

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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.