Discrete Potential Theory and Applications

该项目致力于纯数学的基础研究。它是基于被视为抽象数学工具的电网络理论和图上随机行走理论之间的相互作用而得到充分研究的。这一框架内的四个具体主题由该项目解决：--图和群上随机行走的泊松边界；我们遵循由Kesten引发的群论中的活跃传统，其中关于群的结果是通过考虑群上的随机游走并将其行为或与其相关联的边界的结构与群的代数性质联系起来而间接获得的。该项目得益于申请人最近的一个强有力的结果，即提供了泊松边界的标准，以及利用Lovasz等人最近的石墨子理论，将随机有限图而不是随机游动与群相关联的新想法。艾尔--Benjamini&Schramm意义下的离散保角一致化我们试图加强申请人的一个新结果，与上述结果相关，它回答了Benjamini&Schramm的一个问题。这种加强将在泊松边界上提供新的结果。--极值和随机有限图的覆盖时间与覆盖成本之间的关系。图的覆盖时间是数学和计算机科学中的一个重要概念，甚至被物理学家研究过，但它很难计算，甚至很难近似。利用申请人引入的覆盖成本的概念，我们试图简化许多类图的覆盖时间的近似，将其分解为两个步骤：证明它接近覆盖成本，以及计算(可证明更容易处理的)覆盖成本。这两个主题位于不同的数学领域，所有这些都是近年来出现的大量研究活动。它们通过电子网络、随机行走及其相互作用的总主题相互联系，并共享更精细的相互连接。该项目旨在通过分别为每个分专题产生新的结果以及建立或加强它们之间的联系来作出贡献。该项目的结果将引起几个研究团体的兴趣，包括图论、概率论、(离散)势理论和群论。

The project is devoted to basic research in pure mathematics. It is based on the well-studied interplay between the theory of electrical networks -seen as abstract mathematical tools- and the theory of random walks on graphs. Four specific topics within this framework are addressed by the project:-- The Poisson boundary for random walk on graphs and groups;We follow an active tradition in group theory, triggered by Kesten, where results about groups are obtained indirectly by considering a random walk on the group and relating its behaviour, or the structure of a boundary associated to it, to the algebraic properties of the group. The project benefits from a recent strong result of the applicant providing a criterion for the Poisson boundary, as well as a novel idea of associating a random finite graph rather than a random walk with a group, exploiting the recent theory of graphons by Lovasz et. al. -- Discrete conformal uniformization in the sense of Benjamini & Schramm;We seek to strengthen a new result of the applicant, related to the above, that answered a question of Benjamini & Schramm. Such a strengthening will provide new results on the Poisson boundary.-- The relationship between the cover time and the cover cost in extremal and random finite graphs.The cover time of a graph is an important concept in mathematics and computer science, and is even studied by physicists, but it is very hard to compute or even approximate. Using the concept of cover cost that the applicant introduced, we seek to simplify the approximation of the cover time for many classes of graphs by breaking it down into two steps: showing that it is close to the cover cost, and computing the (provably more tractable) cover cost.These topics lie in different areas of mathematics, all of which have seen a lot of research activity in recent years. They are interlinked by the general theme of electrical networks, random walks, and their interplay, and share further finer interconnections. The project aims to contribute by producing new results individually for each sub-topic as well as by establishing or strengthening connections between them. The project's results will be of interest to several research communities, including Graph Theory, Probability, (discrete) Potential Theory and Group Theory.