Study of critical phenomena by lace expansion and renormalization group

我们使用重整化群和蕾丝展开技术，以数学严谨的方式研究了随机几何模型的关键现象（自我避免行走、渗透、格子动物……）。我们的主要结果如下：层次自旋模型的严格重整化群分析。我们对四维的分层Ising模型进行了严格的重整化群变换，并证明了它的连续极限是高斯的(即。“琐碎”)。我们还得到了一个等价于重整化群变换的偏微分方程。自避行走、渗透和格点动物临界两点函数的严格渐近估计。我们已经证明，只要系统维数足够大，它们的渐近行为与简单随机漫步的渐近行为相同。我们在四维上对分层弱自排斥行走进行了重整化群分析。我们的结果证明了所谓的“对数修正”对磁化率的存在。我们目前正在分析一类随机聚类模型的临界行为，该模型插入了渗透和晶格动物。我们的目标是确定它的临界维度。

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.