Spanning Trees, Matroids and Group-Invariant-Processes

9802663 Lyons这位研究者最近与Benjamini、Peres和Schramm在图的概率过程领域进行了合作，这些图的分布在图的自同构下是不变的。其中一个领域涉及均匀生成树，由于该领域与其他概率领域(如随机游走和狄利克莱特函数)的相互联系，已经出现了大量的新问题。这位调查员提议解决几个这样的问题。另一个领域涉及渗流，既有经典的伯努利渗流，也有更一般的不变渗流。与这位研究者的工作直接相关的最早结果是由物理学家Kirchhoff于1847年发现的，他指出电网络问题精确地与称为生成树的特殊子网络的某些概率性质有关。近年来，这些主题之间的联系蓬勃发展，还发现了与其他基本概率过程的联系，这些过程被称为随机行走和渗流。随机游走模拟了许多类型的随机变化，例如股票市场的价格。渗流是渗透介质中扩散的概率模型，例如地下水渗入土壤。拟议的工作涉及这些领域和其他领域之间的各种联系。研究人员的目标是加深对基础研究的基本理解。

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.