Study of critical phenomena by lace expansion and renormalization group

我们研究了随机几何模型的临界现象(自回避行走、渗流、格子动物……)。)以严格的数学方式，使用重整化群和花边扩展技术。我们的主要结果如下：1.分层自旋模型的重整化群分析。严格地进行了四维层次伊辛模型的重整化群变换，证明了它的连续性限制了它的高斯性(即“平凡”)。得到了与重整化群变换等价的偏微分方程解。2.自回避行走、渗流和晶格动物临界两点函数的渐近估计。我们证明了，只要系统的维度足够大，它们的渐近行为与简单随机游动的渐近行为是相同的。3.我们对四维的分层弱自排斥游动进行了重整化群分析。我们的结果证明了所谓的“对数修正”的敏感性的存在。4.我们目前正在分析一类随机簇模型的临界行为，该模型内插渗流和晶格动物。我们的目标是确定它的关键维度。

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.