Studies in Interacting Particle Systems and Statistical Mechanics

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0071766Schonmann Schonmann plans to 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 research are the ergodic behavior and the relaxation patterns of several interacting particle systems, including Ising models, contact processes and other lattice spin systems. Percolation theory is also a major object of concern in this proposal, because of its strong relations to these processes and its simplicity, which makes it a paradigm in the study of probabilistic models with many components. A substantial part of the proposal refers to the study of the above mentioned processes on arbitrary graphs -- an important current trend in the area. Several of the proposed problems have their motivation in phenomena observed in nature as, for example, phase transitions, critical behavior, metastable behavior of systems in the vicinity of phase transition regions and domain formation and growth. On a more informal and broad level one can see the objects of interest in this research in the following fashion. A system contains a large number of similar components and each component interacts in a simple way with a few other ones, which are near to it in some sense. 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 (e.g., the rate of infection between neighboring individuals in our 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.