Critical and Subcritical Growth Models

该奖项支持的项目涉及研究随机增长的数学模型。一些实例包括细菌或肿瘤扩散，以及流体流过多孔介质。研究问题集中在最佳增长路径的几何方面，以及增长的总体规模和速度。这项工作与数学和物理学的其他领域有关，比如无序磁体的结构，以及计算机科学中的满意度问题。本课题以概率论和数学物理为题，以首次通过逾渗（FPP）和伯努利逾渗（Bernoulli percolation）等扩散型增长模型为中心，开展本科生、研究生和博士后研究人员的工作，并为研究生提供研究训练的机会。这些模型是在20世纪50年代引入的，但尽管研究人员数十年的努力，它们的许多基本性质仍然难以捉摸。提出的项目包括确定伯努利渗流中盒交叉路径的分形性质和标度极限，随机噪声对临界FPP中的时间渐近性的影响，以及亚临界（通常）FPP中生长集的几何和拓扑结构。预计在这些研究中获得的结果将影响流行病模型、无序自旋系统和聚合物模型的工作。该奖项反映了NSF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。

The projects supported by this award involve studying mathematical models for random growth. Some examples include bacterial or tumor spread, and fluid flow through porous media. The research questions center on geometric aspects of optimal growth paths, as well as the overall size and speed of growth. The proposed work has connections to other areas of mathematics and physics, like the structure of disordered magnets, and satisfaction problems from computer science. The projects call for work by undergraduate and graduate students, as well as postdoctoral researchers, and provide research training opportunities for graduate students.This project contains questions in probability theory and mathematical physics, and centers on percolation-type growth models including first-passage percolation (FPP) and Bernoulli percolation. These are models that were introduced in the 1950's, but despite decades of effort by researchers, many of their fundamental properties remain elusive. The proposed projects include determination of fractal properties and scaling limits of box-crossing paths in Bernoulli percolation, the effect of random noise on passage-time asymptotics in critical FPP, and the geometry and topological structure of the growing set in sub-critical (usual) FPP. It is expected that results obtained in these studies will affect work on epidemic models, disordered spin systems, and polymer models.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.