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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Keith Crank
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University of California Los Angeles
Los Angeles
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