Spanning Trees, Matroids and Group-Invariant-Processes

生成树、拟阵和群不变过程

• 批准号: 9802663
• 负责人: Russell Lyons
• 金额: $ 7.4万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802663&HistoricalAwards=false
• 关键词: Spanning; Trees; Matroids; Group; Invariant

9802663LyonsThe investigator has collaborated recently with Benjamini, Peres and Schramm in the area of probabilistic processes on graphs whose distributions are invariant under automorphisms of the graph. One of the areas concerns uniform spanning trees, where an enormous number of new questions have opened up because of the interconnections of the field with other areas of probability such as random walks and Dirichlet functions. The investigator proposes to work on several such questions. Another area concerns percolation, both classical Bernoulli percolation and more general invariant percolation. A large number of questions in this area are also proposed.The earliest result directly related to this investigator's work was discovered by the physicist Kirchhoff in 1847, who showed that electrical network problems are precisely related to certain probabilistic properties of special subnetworks called spanning trees. The connections between these topics have flourished in recent years and have also found links to other fundamental probabilistic processes known as random walks and percolation. Random walks model many sorts of random change, such as prices on the stock market. Percolation is a probabilistic model of diffusion through a porous medium, such as groundwater seeping through soil. The proposed work involves various connections among these and other fields. The investigator aims to further the fundamental understanding that is part of basic research.

9802663LyonsThe调查员最近与Benjamini，Peres和Schramm在概率过程领域合作，这些过程在图的自同构下分布不变。 其中一个领域涉及均匀生成树，由于该领域与随机游走和狄利克雷函数等其他概率领域的相互联系，因此出现了大量新问题。 调查员提议就若干此类问题开展工作。 另一个领域涉及渗透，经典的伯努利渗透和更一般的不变渗透。 最早的结果直接与这个调查员的工作是由物理学家基尔霍夫在1847年发现的，他表明，电网络的问题，正是与某些概率性质的特殊子网络称为生成树。 近年来，这些主题之间的联系蓬勃发展，并且还发现了与其他基本概率过程（称为随机游走和渗流）的联系。 随机游走模拟了许多种随机变化，比如股票市场的价格。 渗流是通过多孔介质扩散的概率模型，例如地下水渗透通过土壤。 拟议的工作涉及这些领域和其他领域之间的各种联系。 研究者的目标是进一步了解基础研究的一部分。