CAREER: Probability and Mathematical Statistical Mechanics

概率论是对随机过程的数学研究。数理统计力学使用概率方法来理解物理现象，如物质从固态到液态和气态的转变，或流体通过多孔介质的流动。该项目将重点关注数理统计力学中的两个核心问题，即：(a)随机景观中的距离和(b)高维晶格上相变简化模型的大尺度行为。教育活动将包括培训研究生，组织暑期学校，以及开发适合在惩教环境中教育成人的概率和统计学习材料。研究的第一个方向将是利用鲁棒耦合技术研究稳态kardar - paris - zhang模型中的普遍波动，特别关注相互作用扩散和聚合物模型。我们的目标是研究耦合方法在多大程度上适用于KPZ类中的一般模型。第一步将是通过解析方法研究与O'Connell-Yor半离散聚合物的多路径扩展相关的三角形系统，以及它们在极限形状边缘的缩放。第二个目标是构建高维渗流中的结垢极限，包括高维渗流中距离的联合结垢极限。一个目标是在宏观距离上获得点集合到原点距离的联合缩放极限。这与分支随机游走的相应限制相匹配。另一个值得关注的缩放限制是聚类度量。这些研究将通过簇扩展和解耦技术来实现，这些技术类似于现有的二维渗透技术，但它们在高维中依赖于完全不同的机制。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。

Probability theory is the mathematical study of random processes. Mathematical statistical mechanics uses probability methods to understand physical phenomena such as transitions from solid to liquid and gaseous phases of matter, or the flow of fluid through a porous medium. This project will focus on two central problems in mathematical statistical mechanics relating to (a) distances in random landscapes and (b) the large scale behavior of a simplified model of a phase transition on high-dimensional lattices. Educational activities will include training of graduate students, organization of a summer school, and the development of probability and statistics learning materials suitable for teaching adults in correctional environments.The first direction of research will be investigation of universal fluctuations in stationary Kardar-Parisi-Zhang models using robust coupling techniques, focusing especially on interacting diffusions and polymer models. The goal is to investigate the extent to which coupling methods apply to general models expected to lie in the KPZ class. A first step will be the study, via analytic methods, of triangular systems relating to multi-path extensions of the O'Connell-Yor semi-discrete polymer, and their scaling at the edge of the limit shape. A second objective is the construction of scaling limits in high dimensional percolation, including joint scaling limits of distances in high dimensional percolation. One goal is to obtain the joint scaling limit of distances to the origin for collections of points at macroscopic distance. This is expected to match the corresponding limit for branching random walk. Another scaling limit of interest is that of the cluster measure. These investigations will be enabled by cluster extension and decoupling techniques akin to those existing for 2-dimensional percolation, but which nonetheless rest on completely different mechanisms in high dimensions.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.