Problems in Particle and Interface Models

这个项目研究随机过程，模拟复杂的，相互作用的行为，如相互作用的粒子系统和界面模型。粒子系统研究的基本例子是排斥型过程和Hammersley过程。在这个项目中研究的特殊情况包括瓶颈，交通堵塞，不同阶段之间的边界，个体之间的优先级，以及其他干扰单个粒子的不规则性。界面模型描述了一个相相对于另一个相的生长或衰减。这类数学模型的例子有最后一次通过模型、递增序列模型和行军士兵模型。解决的问题是速度、最终形状和界面粗糙度。这个项目的结果将关注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。大规模的特征是通过大量操作者在较小规模上的组合效应形成的。这可以在物理过程、生物过程和人造过程中观察到。例如，单个流体粒子的无序运动和碰撞构成了河流的流动;个体水平上的竞争和合作决定了种群的进化;高速公路上车辆的行为决定了交通流量。许多不同的复杂系统都有共同的组织和行为的基本原则。这些共同的特点可以抽象的数学模型，这些模型的分析，然后有许多特定的应用程序的影响。这个项目是关于这类模型的数学研究，这个项目的成果是更好地理解复杂系统的某些类别的模型的行为。

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.