Topics in Percolationand Particle Models

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9803267 Newman This research is in the general area of Probability Theory with special emphasis on a number of stochastic models with interesting spatial structure. One specific topic is first- passage percolation based on d-dimensional Poisson point processes, with particular emphasis on results related to the existence and nonexistence of semi-infinite and doubly infinite geodesics. A second major part of the work concerns a class of voter models in random environments which exhibit the intriguing phenomenon of "chaotic time dependence." This is when the model (which is a Markov process in time) has multiple invariant distributions and the distribution at time t (starting from a randomly selected initial state) does not converge as t tends to infinity. A smaller, but significant, part of the research involves continuum scaling limits of minimal spanning trees in the plane. In the general area of probability theory, an increasingly important role has been played in recent years by systems in which random effects are observed in the spatial structure rather than in (or in addition to) the behavior as a function of time (as in models of equity prices). Some of these models, such as first-passage percolation, have arisen separately in multiple contexts, such as fluid flow in porous media (which is relevant for example to modeling of pollutant dispersion in aquifers), polymer structure, and other parts of materials science. There are also connections to combinatorial optimization (as in the minimal spanning tree problem, which is a variant of the classical traveling salesman problem of designing optimal routings) and thus to various parts of theoretical computer science. The research under this grant concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. One particular issue of study is the connection between randomness in the spatial environment and chaotic depe ndence in time.