Study of critical phenomena by lace expansion and renormalization group

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We have studied critical phenomena of stochastic geometric models (self-avoiding walk, percolation, lattice animals... ) in a mathematically rigorous manner, using renormalization group and lace expansion techniques. Our main results are as follows.1.Rigorous renormalization group analysis of hierarchical spin models. We have rigorously performed the renormalization group transformation for the hierarchical Ising model in four dimensions, and proved that its continuume limit it gaussian (i.e. "trivial"). Also we found a partial differential equation which is equivalent to the renormalization group transformation.2.Rigorous asymptotic estimates of the critical two-point functions for self-avoiding walk, percolation and lattice animals. We have shown that their asymptotic behavior is the same as that of the simple random walk, as long as the system dimension is sufficiently large.3.We have performed a renormalization group analysis of hierarchical weakly self-repelling walks in four dimensions. Our result proves the existence of the so called "logarithmic corrections" for the susceptibility.4.We are currently analyzing critical behavior of a kind of random cluster model, which interpolates percolation and lattice animals. Our goal is to determine its critical dimension.