Limits and Deviations for Interacting Random Systems

这个项目发展了对某些宏观现象的精确的数学理解，这些现象是由大量微观成分相互作用产生的。微观组件的行为具有随机性，因此模型本质上是随机的。该项目涵盖了当前感兴趣的几个模型：最长递增子序列、移动界面、哈默斯利过程、排除过程、冻结无序模型和随机集群模型。研究解决了两个基本问题：(1)是否存在不依赖于随机微观运动的特殊性的可预测的宏观行为？所寻求的答案可以是感兴趣的量的数值，可以是描述界面形状的函数，也可以是描述宏观演化的微分方程。(2)微观演化明显偏离宏观描述的可能性有多大？这个问题的答案是大偏差定理和中心极限定理，可能涉及到寻找系统寻求最大化或最小化的熵函数。在许多情况下，系统的宏观行为取决于微观组件的相互作用。根据研究中的模型，微观成分可以有多种类型：高速公路上的车辆，排队的顾客，或者管道或土壤等多孔介质中的流体颗粒。这个项目的结果是更好地理解了这样一个复杂系统的整体行为。例如，在交通模型中，调查者试图描述两种不同的情况，低密度阶段和高密度阶段，在车辆移动中具有定性不同的模式。在排队模型中，人们寻求对通过网络的客户流的描述，以了解系统是否会变得危险地堵塞和不稳定。这些模型构成了现代通信技术理论基础的一部分。对于运动界面的模型，有趣的属性是界面的速度、最终形状和粗糙度。数学家、物理学家和工程师对这些模型及其近亲进行了深入而同时的研究。在许多情况下，模拟并没有产生关于这种行为的确凿图片。因此，非常需要基础的数学工作来帮助更多的应用型科学家理解这种现象。

9801085SeppalainenThis project develops precise mathematical understanding of certain macroscopic phenomena that result from the interaction of a large number of microscopic components. There is randomness in the behavior of the microscopic components, so the models are stochastic in nature. The project covers several models of current interest: longest increasing subsequences, moving interfaces, Hammersley's process, the exclusion process, models with frozen disorder, and the random-cluster model. The investigation addresses two basic questions: (1) Is there predictable macroscopic behavior that does not depend on the particularities of the random microscopic motions? The answer sought can be the numerical value of an interesting quantity, a function that describes the shape of an interface, or a differential equation that describes the macroscopic evolution. (2) What are the chances that the microscopic evolution deviates noticeably from the macroscopic description? Answers to this question are large deviation theorems and central limit theorems, and may involve finding an entropy function that the system seeks to maximize or minimize.In many situations macroscopic behavior of a system depends on interactions of microscopic components. The microscopic components can be of many types, depending on the model under study: vehicles on a freeway, customers in a sequence of queues, or fluid particles in a pipe or in a porous medium such as soil. The outcome of this project is a better understanding of the overall behavior of such a complex system. For example, in the traffic model the investigator seeks to describe two distinct situations, a low density phase and a high density phase, with qualitatively different patterns in the movement of vehicles. In a queueing model one seeks a description of the flow of customers through the network, to understand whether the system can become dangerously clogged and unstable. Such models form part of the theoretical underpinnings of modern communication technology. For a model of a moving interface the interesting properties are the speed, the eventual shape, and the roughness of the interface. These models and their relatives are intensely and concurrently studied by mathematicians, physical scientists, and engineers. In many cases simulations have not yielded conclusive pictures of the behavior. Consequently there is great need for basic mathematical work to aid the more applied scientists' understanding of such phenomena.