Interacting Particle Systems

这项研究解决了被称为相互作用粒子系统的几个问题，主要涉及排斥过程和自旋系统。第一个问题涉及对称排除过程和负相关不等之间的关系。这将类似于正关联和自旋系统之间已知的重要联系。第二种尝试理解非对称排斥过程动态中的局部变化可以产生全球影响的方式。第三个问题是确定在遍历的一维自旋系统中加入大量的对称排斥相互作用是否(以及何时)会使组合过程变得非遍历。最后，研究了随机树或非齐次树上的可逆随机感染模型。相互作用的粒子系统是概率论的一个分支，它研究存在大量个人(或粒子、汽车、分子、细胞、细菌、计算机网络中的消息等)的情况下的随机模型。以不同的方式相互作用。这种类型的模型出现在许多不同的科学领域，如上面括号中的列表所示。排除过程已被用于对粒子运动、计算机网络、交通流量以及与DNA-RNA相关的问题进行建模。自旋系统已被用来对磁性、传染病传播和经济系统等进行建模。这一研究领域的主要问题是了解局部进化规则如何影响系统的长期全局行为。该项目将从多个方面促进这一理解。

This research addresses several problems in the area known as interacting particle systems, and deals primarily with the exclusion process and spin systems. The first problem involves relationships between symmetric exclusion processes and negative correlation inequalities. This would be analogous to the important known connections between positive correlations and spin systems. The second attempts to understand the way in which local changes in the dynamics of the asymmetric exclusion process can have global effects. The third problem is to determine whether (and when) adding a large amount of a symmetric exclusion interaction to an ergodic one dimensional spin system can render the combined process nonergodic. The final one is to study reversible stochastic infection models on random or inhomogeneous trees. Interacting particle systems is a branch of probability theory that studies random models for situations in which there is a large number of individuals (or particles, cars, molecules, cells, bacteria, messages in a computer network,...) that interact in various ways. Models of this type arise in many different areas of science, as indicated in the parenthetical list above. The exclusion process has been used to model particle motion, computer networks, traffic flow, and issues related to DNA-RNA. Spin systems have been used to model magnetism, spread of infection, and economic systems among others. The main issue in this area of research is to understand how the local evolution rules affect the long time global behavior of the system. This project will contribute to this understanding in a number of ways.