Random Media in and Beyond Two Dimensions

这个项目涉及对真实物理现象的数学模型的研究。其中有所谓的渗流模型，它描述了流体在多孔介质中的流动，不断增长的细菌感染，或长的聚合物链。另一个需要研究的模型是Bak-Tang-Wiesenfeld模型，这是自组织临界性(SOC)的经典模型，在该模型中，简单的局部规则产生复杂的大规模模式。SOC已经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同点。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渗流模型。该项目的一个主要目标是发展强大的数学技术，在广泛的物理有趣的情况下解释这种系统的行为--例如，扩散的液滴的表面有多光滑。需要研究的具体研究问题包括以下几个。在临界Bernoulli渗流中(其中随机删除刚好足以使无限图有限的边)，PI将研究图分量的比例极限，并给出图距离和电阻的精确渐近性。在首次通过渗流(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。在Bak-Tang-Wiesenfeld模型(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。以前，一些相关问题在特殊环境下似乎更容易解决，比如某些“可解决”的增长模式或某些维度。该项目将采用稳健的方法--随机几何技术、集中测量方法和其他方法--这些方法适用于最广泛的可能环境，并提取利益现象背后的一般原理。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。

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.