Weak Convergence Results for Particle Systems

小行星9703815 相互作用粒子系统的主题是特别感兴趣的，当给定的 过程具有多个平稳分布。 在这种情况下，有兴趣知道 初始配置收敛到哪个平稳分布。 对于接触过程 完全收敛定理为上述问题提供了完美的答案。 这种解决方案可以被带到一个大类的超临界，有吸引力的，可逆的， 最近的粒子系统 这样一个完全收敛定理不能为 临界可逆最近粒子系统作为有限系统几乎肯定会消亡。 芒福德有兴趣在本案中研究上述问题。收敛 问题的选民模型是很好地理解和一个很好的描述平稳 分布，但仍然难以解决大偏差和渗流 问题.芒福德希望研究这些问题和类似的问题。芒福德还将 继续他的工作的结构水平集的布朗单，在合作 罗伯特·达朗教授 芒福德教授正在继续研究相互作用的粒子系统。 相互作用的粒子系统是一个大的（通常是无限的）人口具有 以随机方式演变的相关变量;特定个体的变量将 通常根据其价值及其近邻的价值而演变。 比如 种群可以是果园中的树木，变量值可以是生病的或健康的， 个人价值可能会从生病到健康，反之亦然，这取决于健康状况。 和周围树木的病态值。 这些过程提供了丰富的 疾病传播、知识传播、 建立共识。无限大的相互作用粒子系统 当系统允许不同的均衡时，种群变得特别有趣。 一个重要的问题是说一个系统将收敛到哪个（如果有的话）均衡， 当从给定的初始配置开始时。另一个重要的研究领域是 描述不同的均衡。 如果人口众多但有限，那么尽管通常 在相应的无限总体中只有一个平衡点，多重平衡点 转化为有限系统的初始配置的存在， 需要很长时间才能达到平衡。

9703815 Mountford The subject of interacting particle systems is of particular interest when a given process has multiple stationary distributions. In this case it is of interest to know which initial configurations converge to which stationary distributions. For the contact process the complete convergence theorem provides the perfect answer to the above question. This solution may be carried over to a large class of supercritical, attractive, reversible nearest particle systems. Such a complete convergence theorem cannot hold for the critical reversible nearest particle systems as finite systems die out almost surely. Mountford is interested in examining the above question in this case. The convergence question for the voter model is well understood and a good description of the stationary distributions is given, but it remains difficult to address large deviation and percolation issues. Mountford wishes to investigate these and similar questions. Mountford will also continue his work on the structure of level sets for the Brownian sheet, in collaboration with Professor Robert Dalang. Professor Mountford is continuing his work on interacting particle systems. Interacting particle systems are processes where a large (often infinite) population have associated variables that evolve in random fashion; a particular individual's variable will typically evolve according to its value and those of its close neighbors. For instance the population may be trees in an orchard, the variable values may be sick or healthy and an individuals value may change from sick to healthy or vice-versa depending on the health and sickness values of the immediately surrounding trees. These processes provide a rich family of models for the spread of disease, the propagation of knowledge, the establishment of consensus. The subject of interacting particle systems for infinite populations becomes particularly interesting when the system admits distinct equilibria. The n one important issue is to say which (if any) equilibrium a system will converge to, when starting from a given initial configuration. Another important area of research is to describe the distinct equilibria. If the population is large but finite, then although usually there is only one equilibrium, multiple equilibria in the corresponding infinite population translate into the existence of initial configurations for the finite system that take a very long time to reach equilibrium.