CAREER: Probability and Mathematical Statistical Mechanics

概率论是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，例如物质从固态到液态和气态的转变，或者流体通过多孔介质的流动。这个项目将集中在数学统计力学的两个中心问题，涉及（a）随机景观的距离和（B）高维晶格上相变的简化模型的大尺度行为。教育活动将包括培养研究生，组织暑期学校，以及开发适合在教养环境中教授成年人的概率和统计学习材料。研究的第一个方向将是使用鲁棒耦合技术调查固定的Kardar-Parisi-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。我们的目标是调查耦合方法在何种程度上适用于一般模型，预计在于KPZ类。第一步将是研究，通过分析方法，三角系统有关的多路径扩展的O 'Connell-Yor半离散聚合物，和他们的缩放在边缘的极限形状。第二个目标是在高维渗流的标度限制，包括联合标度限制的距离在高维渗流的建设。一个目标是获得宏观距离处的点集合到原点的距离的联合缩放限制。预计这将与分支随机游走的相应限制相匹配。另一个令人感兴趣的缩放限制是聚类度量。这些研究将通过类似于现有的二维渗流的集群扩展和解耦技术来实现，但它们仍然依赖于完全不同的高维机制。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.