Mathematical Sciences: Studies in Interacting Particle Systems and Brownian Motion

[400644] Liggett Liggett、Mountford和Schonmann教授将从事与概率论两个领域有关的几个项目：相互作用粒子系统和布朗运动。其中第一个是对经历相变的大型物理系统演化的概率模型的研究，而第二个是一个更经典的模型，在本世纪的大部分时间里，它在概率论中发挥了核心作用。在第一个领域中要考虑的问题是：在分离单相和多相参数区域的临界值上获得良好的界限，以及确定几种模型（包括随机Ising模型和最近粒子系统）收敛到平衡的速率。在第二部分，我们将研究布朗表的能级集的详细性质。概率论中一个相对较新的和非常活跃的领域被称为相互作用粒子系统。它致力于分析物理系统演化的概率模型，这些系统可以处于两个或多个阶段-例如金属（磁化或不磁化）或水（冰，水，蒸汽）。这一领域的一些调查结果也可应用于其他领域，如队列和疾病传播的研究。Liggett、Mountford和Schonmann教授将继续研究这一领域的一些基本问题。其中之一是确定各种相发生的物理参数（温度，压力等）区域，以及相描述发生变化的点附近系统的行为。另一个是研究系统接近平衡的速度。

9400644 Liggett Professors Liggett, Mountford and Schonmann will work on several projects related to two areas of Probability Theory: Interacting Particle Systems and Brownian Motion. The first of these is the study of probabilistic models for the evolution of large physical systems which undergo phase transitions, while the second is a more classical model which has played a central role in Probability for much of this century. Among the problems to be considered in the first area are: obtaining good bounds on the critical values which separate the single phase from the multiphase parameter region, and determining rates of convergence to equilibrium in several models, including the stochastic Ising model and nearest particle systems. In the second area, detailed properties of the level sets of the Brownian sheet will be investigated. A relatively new and very active area of Probability Theory is known as Interacting Particle Systems. It is devoted to the analysis of probabilistic models for the evolution of physical systems which can be in two or more phases - pieces of metal (magnetized or not) or water (ice, water, steam) for example. Some results of investigations in this area also have applications in other areas, such as the study of queues and transmission of disease. Professors Liggett, Mountford and Schonmann will continue their work on some of the basic problems in this field. One of them is the determination of the regions of physical parameters (temperature, pressure, etc.) in which various phases occur, and the behavior of the systems near points at which the phase description changes. Another is a study of the speed with which systems approach equilibrium.