Growth and Motion in a Random Medium

This project will conduct 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, which appear quite different at small scales and at large scales. It is important to understand how different kinds of small-scale evolution lead to different large-scale systemwide behaviors. Real-world phenomena modeled by these mathematical systems include the motion of vehicles in traffic, data 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. Understanding complex interactions has profound implications for science and engineering and thereby for society. This project also produces educational materials on the mathematics of random phenomena and provides training for young researchers. This project investigates mathematical models of growth and motion in random media, such as first-passage percolation, the corner growth model, and directed polymer models. The outcomes of this project are mathematically rigorous descriptions of the behavior of mathematical models of growth and robust tools for their analysis. A central question pursued in this project is to understand which mathematical growth models have product-form invariant distributions. Other goals of this project include descriptions of the environment around an optimizing path in terms of functions that can be computed from the environment, the joint probability distributions of geodesic trees and competition interfaces, and the regularity properties of large scale limit shapes. A long-term goal is the extension of the results for two-dimensional growth models to higher dimensions where the behavior of these models is much more complicated. 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.