Studies in Interacting Particle Systems and Statistical Mechanics

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9703814 Schonmann The Principal Investigator will work on various projects in the areas of interacting particle systems, statistical mechanics, percolation, and other related subjects in Probability Theory. The main objects of interest in this proposal are the ergodic behavior and the relaxation patterns of several interacting particle systems, including Ising models and other lattice spin systems, contact processes on arbitrary graphs, growth models and cellular automata. Several of the proposed problems have their motivation in phenomena observed in nature as, for example, metastable states of systems in the vicinity of phase transition regions. The Principal Investigator will study problems in interacting particle systems. These are models for systems which contain a large number of similar components and in which each component interacts in a simple way with a few other ones. For instance one can think of the atoms in a crystal interacting when they are close enough, or individuals in a large population which may suffer or not from some infection with contamination occurring between neighbors. The local interaction may involve a substantial amount of randomness, but since the system as a whole is large, one expects and indeed observes a much more predictable behavior at the global level. A great deal of interest in such systems stems from the fact that the qualitative behavior of the whole system may depend in non-trivial ways on parameters which are built into the local interactions, but which only affect those in a smooth way (for example, the rate of infection between neighboring individuals in the second example above). The general issues of interest are then the understanding of which sort of equilibria can be reached by such types of stochastic dynamics, how an equilibrium state is reached, and how the parameters which affect the local rules may affect the answers to these questions.