Interacting Particle Systems

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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.