Random Media in and Beyond Two Dimensions

这个项目涉及研究真实的物理现象的数学模型。其中包括所谓的渗透模型，它描述了多孔介质中流体的流动、不断增长的细菌感染或长聚合物链。另一个要研究的模型是Bak-Tang-维森费尔德模型，这是自组织临界性（SOC）的经典模型，其中简单的局部规则生成复杂的大规模模式。SOC被用来解释太阳耀斑和神经元放电等明显不同的现象之间的共性。在这些领域中，许多最成功的解决问题的方法只在某些特殊情况下有效;例如，某些具有特殊对称性质的二维简化模型。该项目的一个主要目标是开发强大的数学技术，以解释这种系统的行为-例如，在广泛的物理有趣的情况下，一滴液体的表面有多光滑。在临界伯努利渗流（其中一个随机删除刚好足够的边缘，使一个无限的图形有限），PI将研究图形组件的缩放限制，并给出图形距离和电阻的尖锐渐近。在第一次通过渗流（其中一个随机扰动图中的边长）中，PI将研究图距离的波动和长测地线的曲折度。在Bak-Tang-维森费尔德模型（一种相互作用的粒子系统）中，PI将研究不同时刻粒子运动之间的相关性。一些相关的问题以前似乎更容易处理的特殊设置，如某些“可解决的”增长模型或在某些方面。该项目将采用可靠的方法--随机几何技术、测量方法的集中度等--这些方法适用于最广泛的可能环境，并提取出感兴趣现象的一般原理。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。

This project involves the study of mathematical models of real physical phenomena. Among these are so-called percolative models, which describe the flow of fluid in a porous medium, a growing bacterial infection, or a long polymer chain. Another model to be studied is the Bak-Tang-Wiesenfeld model, the classic model of self-organized criticality (SOC), where simple, local rules generate complex, large-scale patterns. SOC has been used to explain commonalities between such apparently dissimilar phenomena as solar flares and neuron firing. Many of the most successful approaches to problems in these areas are valid only in certain special cases; for instance, certain two-dimensional percolative models with particular symmetry properties. A main goal of the project is the development of robust mathematical techniques which explain the behavior of such systems -- for instance, how smooth is the surface of a spreading drop of fluid -- in a broad range of physically interesting cases.The specific research problems to be studied include the following. In critical Bernoulli percolation (where one randomly removes just enough edges to make an infinite graph finite), the PI will study scaling limits of graph components and give sharp asymptotics for graph distances and electrical resistances. In first-passage percolation (where one randomly perturbs edge lengths in a graph), the PI will study fluctuations of graph distances and the tortuosity of long geodesics. In the Bak-Tang-Wiesenfeld model (an interacting particle system), the PI will study the correlation between particle movements at different moments of time. Some related problems have previously seemed more tractable in special settings like certain "solvable" growth models or in certain dimensions. This project will employ robust methods – stochastic geometric techniques, concentration of measure methods, and others – which apply to the widest range of possible settings and which distill the general principles underlying phenomena of interest.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.