Problems in Particle and Interface Models

这个项目研究模拟复杂的相互作用行为的随机过程，例如相互作用的粒子系统和界面模型。所研究的粒子系统的基本例子是排除型过程和哈默斯利过程。该项目研究的特殊情况包括瓶颈、交通拥堵、不同阶段之间的边界、个体之间的优先顺序以及其他干扰个体粒子的不规则情况。界面模型描述了一种相相对于另一种相的生长或衰减。这类数学模型的例子有最后通过模型、递增序列模型和行军士兵模型。所涉及的问题是界面的速度、最终形状和粗糙度。这个项目的结果将涉及这些模型的宏观行为以及围绕预期行为的波动和偏差。我们周围世界的一个基本特征是，自然过程发生在几个不同的尺度上。大尺度特征是通过较小尺度上的大量算子的综合作用形成的。这可以在物理过程、生物过程和人造过程中观察到。例如，单个流体颗粒的无组织运动和碰撞构成了河流的流量；个体层面的竞争和合作决定了种群的进化；车辆在高速公路上的行为决定了交通流量。许多不同的复杂系统共享共同的组织和行为基本原则。这样的共享特征可以抽象为数学模型，对这些模型的分析将对许多特定的应用产生影响。这个项目是关于这类模型的数学研究，这个项目的结果是更好地理解复杂系统的某些类模型的行为。

This project studies stochastic processes that model complex, interacting behavior, such as interacting particle systems and interface models. Basic examples of particle systems studied are exclusion-type processes and Hammersley's process. Special situations studied in this project include bottlenecks, traffic jams, boundaries between different phases, priorities between individuals, and other irregularities that disturb the individual particles. An interface model describes the growth or decay of one phase relative to another. Examples of mathematical models in this category are last-passage models, models of increasing sequences, and marching soldiers models. The questions addressed are the speed, the eventual shape, and the roughness of the interface. The results of this project will concern the macroscopic behavior of these models and the fluctuations and deviations around the expected behavior. A basic feature of the world around us is that natural processes occur at several different scales. Large scale features are formed through the combined effect of a vast number of operators on a smaller scale. This can be observed in physical processes, biological processes, and man-made processes. For example, the disorganized motions and collisions of individual fluid particles make up the flow of a river; competition and cooperation at the level of individuals determines the evolution of a population; and the behavior of vehicles on a freeway determines the flow of traffic. Many different complex systems share common basic principles of organization and behavior. Such shared features can be abstracted in mathematical models, and analysis of these models then has implications for many particular applications. This project is about the mathematical study of such models, and the outcome of this project is a better understanding of the behavior of certain classes of models of complex systems.