Growth and Motion in a Random Medium

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This award supports fundamental research on mathematical models that describe complex interactions, growth and motion in an irregular environment. These mathematical systems incorporate also random unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at small scales and large scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale system-wide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. Over the long term understanding complex interactions has profound implications for science and engineering and thereby for society. Mathematical systems of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. This project investigates mathematical models of growth and motion in random media. Examples include first-passage percolation, the corner growth model, random walk in random environment, and directed polymer models. The objectives of this work are mathematically rigorous descriptions of the behavior of these models and the development of robust tools for their analysis. Specific goals include regularity of limit shapes, properties of optimal paths such as their length, fluctuations and geometric features, descriptions of large scale limits in terms of variational formulas and entropy, and descriptions of probability distributions of complicated random geometric objects such as trees of geodesics and competition interfaces. The methods employed in this work are those of rigorous mathematical research, aided by experimental computer simulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.