Studies in Interacting Random Systems

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

0402231 Seppalainen This project studies stochastic processes that model complex, interacting behavior, such as interacting particle systems and interface models. The goal is to obtain mathematical results that describe behavior over large space and time scales in terms of limit theorems, and roughness of interfaces or fluctuations in particle flow by order-of-magnitude exponents, limit distributions, and large deviation rate functions. The emphasis is on finding new phenomena, and on capturing the overall picture of connections between different classes of microscopic rules and the associated large-scale behavior. One particular problem of interest in the context of asymmetric systems is the fluctuations of the current across a characteristic of the partial differential equation that describes the macroscopic evolution of the system. The objective of this research is precise mathematical understanding of certain kinds of macroscopic phenomena that result from interactions among a large number of microscopic components. The project studies mathematical models that include randomness in the behavior of the individual components. These models are abstractions of basic principles of organization and behavior common to many types of complex systems. The goal is to describe the typical large scale behavior of the model, and also to quantify the chances of deviations from the typical behavior. Complex interactions occur all around us in natural and man-made processes. Examples include cars on a freeway, customers in a queue, packets making their way through a communication network, or fluid particles in a tube. Consequently understanding the mathematics of complex interactions has potentially wide implications for science and engineering. Models of the types described in the proposal and others related to them are concurrently studied by mathematicians, natural scientists, social scientists, and engineers.