Interacting Particle Systems

本研究解决了相互作用粒子系统领域的几个问题，主要涉及不相容过程和自旋系统。第一个问题涉及对称排斥过程与负相关不等式之间的关系。这类似于已知的正相关和自旋系统之间的重要联系。第二部分试图理解不对称排斥过程的局部动态变化如何产生全球影响。第三个问题是确定在遍历一维自旋系统中加入大量对称不相容相互作用是否（以及何时）可以使组合过程非遍历。最后是研究随机或非齐次树上的可逆随机感染模型。相互作用粒子系统是概率论的一个分支，研究以各种方式相互作用的大量个体（或粒子、汽车、分子、细胞、细菌、计算机网络中的信息……）的随机模型。这种类型的模型出现在许多不同的科学领域，如上面的括号列表所示。排除过程已被用于模拟粒子运动、计算机网络、交通流和与DNA-RNA相关的问题。自旋系统被用来模拟磁力、感染传播和经济系统等。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。这个项目将在许多方面促进这种理解。

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.