Mathematical Study of Critical Phenomena for Statistical Models in Probability

概率统计模型关键现象的数学研究

• 批准号: 11640104
• 负责人: HARA Takashi
• 金额: $ 1.66万
• 依托单位: Nagoya University (2000)Tokyo Institute of Technology (1999)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640104/
• 关键词: percolation; critical phenomena; critical point; continuum limit; scaling limit; infinite cluster; integrated super-Brownian excursion; incipient infinite cluster; ISE; lace expansion

The main purpose of the research was to investigate critical behavior of certain stochastic geometric models that appear in probability theory and statistical mechanics. In particular, we investigated continuum (scaling) limits of incipient infinite clusters of the critical percolation model in high dimensions.We found that a reasonable continuum limit could be obtained for a cluster of size N, if we scale the space proportional to the fourth root of N.Moreover, we calculated the first and second moments of the limiting distributions, and identified them as the first and second moments of the integrated super-Brownian excursion (ISE). This strongly suggests that the continuum limit is in fact ISE.The method of proof uses the lace expansion, which has been used successfully in other contexts. To investigate the particular problem of this research, we calculated generating functions of cluster distributions, and applied Tauberian analysis.

研究的主要目的是探讨出现在概率论和统计力学中的某些随机几何模型的临界行为。特别地，我们研究了高维临界渗流模型初始无限团簇的连续（标度）极限.我们发现，对于一个大小为N的团簇，如果我们按N的四次方根比例标度空间，就可以得到一个合理的连续极限.此外，我们还计算了极限分布的一阶和二阶矩，并将它们识别为积分超布朗偏移（伊势）的一阶矩和二阶矩。这有力地表明，连续极限实际上是伊势。证明的方法使用花边扩展，这已成功地用于在其他情况下。为了探讨本研究的特殊问题，我们计算了集群分布的母函数，并应用Tauberian分析。

项目成果

Z.Li,Tokuzo Shiga and L.Yao: "A reversibility problem for the Fleming-Viot processes"Elect.Comm. in probab. 4. 71-82 (1999)

Z.Li、Tokuzo Shiga 和 L.Yao：“弗莱明-维奥过程的可逆性问题”Elect.Comm。

Takashi Hara and Gordon Slade: "The scaling limit of the incipient cluster in high-dimensional percolation.II.Integrated super-Brownian excursion"J. Math. Phys.,41巻3号 (2000). (印刷中).

Takashi Hara 和 Gordon Slade：“高维渗流中初始星团的标度极限。II. 积分超布朗偏移”J. Math.，第 41 卷，第 3 期（出版中）。 。

Takashi Hara and Gordon Slade: "The scaling limit of the incipient infinite cluster in high dimensional percolation. I.Critical exponents"J.Statist.Phys.. 99. 1075-1168 (2000)

Takashi Hara 和 Gordon Slade：“高维渗透中初始无限星团的标度极限。I.Critical exponents”J.Statist.Phys.. 99. 1075-1168 (2000)

Takashi Hara and Gordon Slade: "The scaling limit of the incipient cluster in high-dimensional percolation.I.Critical exponents"J. Statist. Phy. 印刷中.

Takashi Hara 和 Gordon Slade：“高维渗流中初期星团的缩放极限。I. 临界指数”J. Phy。

Z.Li,Tokuzo Shiga and L.Yao: "A reversibility problem for the Fleming-Viot processes"Elect.Comm.in probab. 4. 71-82 (1999)

Z.Li、Tokuzo Shiga 和 L.Yao：“弗莱明-维奥过程的可逆性问题”Elect.Comm.in 概率。