9703815 Mountford The subject of interacting particle systems is of particular interest when a given process has multiple stationary distributions. In this case it is of interest to know which initial configurations converge to which stationary distributions. For the contact process the complete convergence theorem provides the perfect answer to the above question. This solution may be carried over to a large class of supercritical, attractive, reversible nearest particle systems. Such a complete convergence theorem cannot hold for the critical reversible nearest particle systems as finite systems die out almost surely. Mountford is interested in examining the above question in this case. The convergence question for the voter model is well understood and a good description of the stationary distributions is given, but it remains difficult to address large deviation and percolation issues. Mountford wishes to investigate these and similar questions. Mountford will also continue his work on the structure of level sets for the Brownian sheet, in collaboration with Professor Robert Dalang. Professor Mountford is continuing his work on interacting particle systems. Interacting particle systems are processes where a large (often infinite) population have associated variables that evolve in random fashion; a particular individual's variable will typically evolve according to its value and those of its close neighbors. For instance the population may be trees in an orchard, the variable values may be sick or healthy and an individuals value may change from sick to healthy or vice-versa depending on the health and sickness values of the immediately surrounding trees. These processes provide a rich family of models for the spread of disease, the propagation of knowledge, the establishment of consensus. The subject of interacting particle systems for infinite populations becomes particularly interesting when the system admits distinct equilibria. The n one important issue is to say which (if any) equilibrium a system will converge to, when starting from a given initial configuration. Another important area of research is to describe the distinct equilibria. If the population is large but finite, then although usually there is only one equilibrium, multiple equilibria in the corresponding infinite population translate into the existence of initial configurations for the finite system that take a very long time to reach equilibrium.
