Interacting Particle Systems

小行星9703830 这个项目涉及到一些与接触过程有关的问题， 排斥过程，选民模型和反应扩散过程。的接触过程 在齐次树上有两个临界值，因此有三个不同的参数区域 感兴趣 从这个项目的角度来看，最重要的是中间的一个， 弱生存。 在这一地区，建议延长Liggett的早期 构造径向对称的平稳分布，更一般地， 研究所有极值平稳分布集合的结构。 一种方法 这是一个由他提出的关于某种增长的规模的猜想。 参数. 另一个问题是确定较大临界值的渐近性， 树的配位数增加。 最后，还有一个问题， 树的有限部分上的接触过程消失所需的时间，以及这如何取决于 参数值。 要考虑的排除过程是一维的， 不对称的，均匀的，除了一个键，它比其他键慢。 的 问题是要确定这种慢键对一组结构的影响。 平稳分布 投票模型和反应扩散过程的问题 处理遍历和非遍历的区域。 物理学和生物学的许多领域都涉及到这样一种情况，即大量的代理人在时间上以这样一种方式进化，即每个代理人的进化受到其他代理人状态的影响。 例如，代理人可以是晶体中的原子，或者是宿主种群中的寄生虫。 在许多这样的情况下，有一些量（比如温度）会影响相互作用发生的方式，在许多情况下，当这个量发生微小变化时，系统的演化就会发生剧烈变化。 被称为相互作用粒子系统的领域是研究这类问题的概率论的分支。 本项目中要研究的模型是这一领域中出现的较重要的模型之一。接触过程与感染传播相关，排斥过程模型粒子运动，选民模型具有社会政治性 解释和反应扩散过程模型系统，其中生长和 两个动作都发生了。 平稳分布描述了可能的长时间 一个系统的状态。 每一类模型的主要问题是确定它们是什么， 以及它们的结构如何取决于指定强度的量。 互动

9703830 Liggett This project involves a number of problems concerning contact processes, exclusion processes, voter models, and reaction diffusion processes. The contact process on a homogeneous tree has two critical values, and hence three distinct parameter regions of interest. The most important from the point of view of this project is the middle one, in which there is weak survival. In this region, it is proposed to extend Liggett's earlier construction of radially symmetric stationary distributions, and more generally, to investigate the structure of the set of all extremal stationary distributions. One approach to this proceeds via a conjecture made by him concerning the size of a certain growth parameter. Another problem is to determine the asymptotics of the larger critical value as the coordination number of the tree increases. Finally, there is the issue of how long it takes for the contact process on a finite part of the tree to die out, and how this depends on the parameter value. The exclusion process to be considered is one dimensional and asymmetric, and homogeneous except for one bond, which is slower than the others. The problem is to determine the effect of this slow bond on the structure of the set of stationary distributions. The problems for voter models and reaction diffusion processes deal with regions of ergodicity and nonergodicity. A significant number of areas of physics and biology involve situations in which a large number of agents evolve in time in such a way that the evolution of each agent is affected by the states of the others. The agents could be atoms in a crystal, or parasites in a host population, for example. In many of these situations, there is some quantity (temperature, say) which affects the way in which the interaction occurs, and in many cases, the evolution of the system is changed drastically when this quantity changes by a small amount. The area known as interacting particle systems is the branch o f probability theory that studies such issues. The models to be studied in this project are among the more important which arise in this area. The contact process is related to spread of infection, exclusion processes model particle motion, voter models have a socio-political interpretation, and reaction diffusion processes model systems in which growth and motion both occur. Stationary distributions are descriptions of the possible long time states of a system. A main issue for each class of models is to determine what these are, and how their structure depends on the quantity which specifies the strength of the interaction.