Growth and Motion in a Random Medium

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This project is fundamental research on mathematical models that describe complex interactions, growth, and motion in an irregular environment with stochastic 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 communication networks, fluid particles in a tube, fluid spreading in a porous medium, epidemics advancing in a population, and the fluctuations of a polymer chain in a fluid. Laboratory experiments have demonstrated that these mathematical models capture essential features of physical reality. 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 provides research training opportunities for graduate students. 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, directed polymer models, and stochastic partial differential equations. 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, ergodic properties, 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.