Growth and Motion in a Random Medium

该奖项支持对描述不规则环境中复杂相互作用、生长和运动的数学模型进行基础研究。这些数学系统也包含随机的不可预测性。目标是发现控制这些系统的一般数学规律。这些系统在小尺度和大尺度上表现得非常不同。因此，了解小规模进化的不同规则如何导致不同的大规模系统行为是很重要的。这种数学研究可以阐明的现实世界现象包括车辆的运动、通信网络中的数据包、管道中的流体颗粒、流体在多孔介质中传播的润湿过渡、在人群中传播的流行病或流体中聚合物链的波动。从长远来看，理解复杂的相互作用对科学和工程，从而对社会有着深远的影响。数学家、自然科学家、社会科学家和工程师都在密切地、同时地研究这类在提案中描述的数学系统。本项目研究随机介质中生长和运动的数学模型。例子包括第一通道渗透、角落生长模型、随机环境中的随机游走和定向聚合物模型。这项工作的目标是在数学上对这些模型的行为进行严格的描述，并为它们的分析开发健壮的工具。具体目标包括极限形状的规律性，最优路径的性质，如其长度，波动和几何特征，用变分公式和熵描述大规模极限，以及描述复杂随机几何对象的概率分布，如测地树和竞争界面。在这项工作中采用的方法是严格的数学研究，辅以实验计算机模拟。该奖项反映了美国国家科学基金会的法定使命，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 award supports fundamental research on mathematical models that describe complex interactions, growth and motion in an irregular environment. These mathematical systems incorporate also random unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at small scales and large scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale system-wide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. Over the long term understanding complex interactions has profound implications for science and engineering and thereby for society. Mathematical systems of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. This project investigates mathematical models of growth and motion in random media. Examples include first-passage percolation, the corner growth model, random walk in random environment, and directed polymer models. The objectives of this work are mathematically rigorous descriptions of the behavior of these models and the development of robust tools for their analysis. Specific goals include regularity of limit shapes, properties of optimal paths such as their length, fluctuations and geometric features, descriptions of large scale limits in terms of variational formulas and entropy, and descriptions of probability distributions of complicated random geometric objects such as trees of geodesics and competition interfaces. The methods employed in this work are those of rigorous mathematical research, aided by experimental computer simulation.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.