Geometry of Physical Models Governed by Diffusion

在过去的几十年里，概率论在解释标度极限和大规模现象方面取得了巨大的成功。然而，没有明显比例限制的系统在物理和生命科学中都很丰富，因此有实际需要更好地了解这种系统。这个研究项目的目的是通过研究增加了额外对称性的模型来促进对这类系统的理解，这些模型分享了所研究的经典无序系统的局部行为。研究生和博士后助理将通过参与研究接受培训。本项目研究的主题分为三个子领域：(1)定常扩散限制聚集(DLA)和Hastings-Levitov模型；(2)楔形DLA；(3)随机交织中的化学距离。这些模型对统计力学和数学物理很有意义。它们的共同特征是由扩散或调和测量产生，并试图抓住物理现象的本质，而不是该领域中更经典的模型，如DLA或伯努利渗流。所采用的方法借鉴了不同数学学科的思想和工具，包括复分析、调和分析、微分方程和遍历理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Over the past decades probability theory has had tremendous success in elucidating scaling limits and large-scale phenomena. However, systems that have no apparent scaling limit are abundant in both the physical and life sciences, and there is practical need to better understand such systems. This research project aims to advance understanding of such systems by studying models augmented with additional symmetries that share the local behavior of the classical disordered systems under study. Graduate students and postdoctoral associates will receive training through involvement in the research. The topics under study in this project divide into three subareas: (1) stationary diffusion limited aggregation (DLA) and Hastings-Levitov models, (2) DLA in a wedge, and (3) chemical distance in random interlacements. These models are of interest for statistical mechanics and mathematical physics. They share the features of being generated by diffusion or the harmonic measure, and of attempting to grasp the nature of a physical phenomenon that is not amenable more classical models in the field, such as DLA or Bernoulli percolation. The methods to be employed borrow ideas and tools from various mathematical disciplines, including complex analysis, harmonic analysis, differential equations, and ergodic theory.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.