Study of critical phenomena by lace expansion and renormalization group

花边扩展和重正化群的临界现象研究

• 批准号: 13640112
• 负责人: HARA Takashi
• 金额: $ 2.11万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640112/
• 关键词: percolation; self-avoiding walk; critical phenomena; renormalization group; Ising model; hierarchical model; two-point function; critical dimension; ランダムクラスターモデル; 臨界点; 特性関数

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.

我们已经研究了随机几何模型的临界现象（自避免行走，渗流，格子动物...）以数学上严格的方式，使用重整化群和蕾丝展开技术。我们的主要结果如下：1.等级自旋模型的严格重整化群分析。我们严格地进行了四维层次伊辛模型的重整化群变换，并证明了它的连续性限制高斯（即“平凡”）。我们还找到了一个与重整化群变换等价的偏微分方程。2.自避免行走、渗流和格点动物临界两点函数的严格渐近估计。我们证明了只要系统维数足够大，它们的渐近行为与简单随机游动的渐近行为相同。3.我们对四维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“对数修正”的存在性。4.我们目前正在分析一类随机集团模型的临界行为，该模型是在渗流和格点动物的基础上建立的。我们的目标是确定其临界尺寸。

项目成果

T.Hattori, T.Tsuda: "Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski Gasket"J.Statist.Phys.. 109. 39-66 (2002)

T.Hattori、T.Tsuda：“d 维 Sierpinski 垫片上自回避路径的重正化群分析”J.Statist.Phys.. 109. 39-66 (2002)

Takashi Hara, Tetsuya Hattori, Hiroshi Watanabe: "Thrviality of hierarchical Ising model in four dimensions"Commun.Math.Phys.. 220. 13-40 (2001)

Takashi Hara、Tetsuya Hattori、Hiroshi Watanabe：“四个维度的分层伊辛模型的有效性”Commun.Math.Phys.. 220. 13-40 (2001)

Takashi Hara, Remco van der Hofstad, Gordon Slade: "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models"Ann.Prob.. 31. 349-408 (2003)

Takashi Hara、Remco van der Hofstad、Gordon Slade：“临界两点函数和展开高维渗滤及相关模型的花边展开”Ann.Prob.. 31. 349-408 (2003)

T.Hattori, T.Tsuda: "Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpinski Gasket."J.Statist.Phys.. 109. 39-66 (2002)

T.Hattori、T.Tsuda：“d 维 Sierpinski 垫片上自回避路径的重正化群分析。”J.Statist.Phys.. 109. 39-66 (2002)

Takashi Hara, Tetsuya Hattori, Hiroshi Watanabe: "Triviality of hierarchical Ising model in four dimensions"Commun. Math. Phys.. 220. 13-40 (2001)

Takashi Hara、Tetsuya Hattori、Hiroshi Watanabe：“四个维度中分层伊辛模型的琐碎性”Commun。