This project studies three categories of stochastic processes: interacting particle systems, interface models, and random motion in a random medium. The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior. Examples of particular objects of study are fluctuations and large deviations of the current in the exclusion process and the zero range process, behavior of the exclusion process in a random environment, height fluctuations in the random average process, and quenched central limit theorems for random walk in a random environment. An important phenomenon to clarify is the transition between different universality classes as the parameters of the zero range process are varied.
This project investigates mathematical models that describe complex interactions and motion of particles in an irregular environment. These mathematical systems incorporate randomness to model irregularity and unpredictability. The goal is to unearth general mathematical laws that govern such systems irrespective of less important details. A key point is that these systems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale interactions and motions lead to different large-scale systemwide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles on a freeway, packets making their way through a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, or epidemics advancing through an orchard of trees. Over the longer term understanding complex interactions has wide implications for science and engineering and thereby for society. Models of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists,social scientists, and engineers.
