Interacting Particle Systems

9703830利格特这个项目涉及接触过程、排斥过程、选民模型和反应扩散过程的一些问题。齐次树上的接触过程有两个临界值，因此有三个不同的感兴趣的参数区域。从这个项目的角度来看，最重要的是中间项目，在这个项目中，生存能力很弱。在这一领域，我们建议推广Liggett关于径向对称平稳分布的早期构造，并更一般地研究所有极值平稳分布集的结构。对于这一点，一种方法是通过他关于某个增长参数大小的猜测来进行的。另一个问题是随着树的协调数的增加，确定较大的临界值的渐近性。最后，还有树的有限部分上的接触过程需要多长时间才会消失的问题，以及这如何取决于参数值。要考虑的排斥过程是一维的，不对称的，并且是均匀的，除了一个键，它比其他键慢。问题是要确定这种慢键对平稳分布集合的结构的影响。投票者模型和反应扩散过程的问题涉及遍历性和非遍历性区域。物理学和生物学的许多领域都涉及这样一种情况，即大量的智能体在时间上进化，每个智能体的进化受到其他智能体状态的影响。例如，这些病原体可能是晶体中的原子，也可能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，当这个量发生很小的变化时，系统的演化就会发生巨大的变化。被称为相互作用的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出现的较重要的模型之一。接触过程与感染的传播有关，排斥过程模拟粒子运动，选民模型具有社会政治解释，反应扩散过程模拟增长和运动都发生的系统。平稳分布是对系统可能的长时间状态的描述。每一类模型的一个主要问题是确定这些模型是什么，以及它们的结构如何取决于指定相互作用强度的量。

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.