RUI: Boundary and entropy of random walks on groups

随机漫步的应用出现在不同的科学分支和我们的日常生活中。在博弈论中，随机漫步可以模拟一副牌的洗牌，以找到最优的洗牌迭代。在金融中，股票价格变化的不可预测性可以用随机游走来建模。该项目将应用严格的数学工具来研究随机漫步的长期行为。特别是，以下两个问题将被研究：首先，几何或代数特征如何捕捉随机行走的长期行为。第二，不同的量如何描述随机漫步的长期行为。这项研究将吸引本科生和研究生。这种与学生的接触使他们接触到当代数学领域，而不是常规课程的一部分。这项工作将扩展和完善随机游走不变量（如边界和渐近熵）的现有结果。这些是由群上的随机游走和它们的代数或几何结构之间的相互作用产生的。该项目将提高对泊松边界的理解，泊松边界为给定的概率度量提供了群上有界调和函数的表示。目前的工作将利用最近的发展，如关键时间理论，来确定双曲类群上随机行走的泊松边界。另一个目标是研究泊松边界的商与给定群的子群之间的联系。该研究领域与几何群论、遍历理论和信息论有着密切的联系。该项目由DMS概率计划和促进竞争研究的既定计划（EPSCoR）联合资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Applications of random walks emerge in different branches of science and our daily life. In game theory, random walks can model shuffling a deck of cards to find optimal shuffle iterations. In finance, the unpredictability of price changes of stocks can be modeled by random walks. The project will apply rigorous mathematical tools for studying the long-term behavior of random walks. In particular, the following two problems will be investigated: first, how geometric or algebraic features can capture the long-term behavior of random walks. Second, how different quantities can describe the long-term behavior of a random walk. The research will engage undergraduate and graduate students. This engagement with students exposes them to contemporary areas of mathematics not part of the regular curriculum. The work will expand and refine existing results on random walk invariants such as boundaries and asymptotic entropies. These arise from the interaction between random walks on groups and their algebraic or geometric structures. The project will improve understanding of the Poisson boundary, which provides a representation of bounded harmonic functions on a group for a given probability measure. The current work will leverage recent developments, such as the theory of pivotal times, to identify the Poisson boundary of random walks on hyperbolic-like groups. Another goal is to investigate connections between quotients of the Poisson boundaries with subgroups of a given group. The area of research has strong connections to geometric group theory, ergodic theory, and information theory. This project is jointly funded by the DMS Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.