Stochastic Systems with Complex Interactions and Random Environments

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This project is fundamental research on mathematical models that describe complex interactions, growth, and motion in an irregular environment. These systems show very different features at small scales and at large scales, and it is important to understand how different rules for small-scale evolution lead to different large-scale system-wide behavior. Mathematical studies can illuminate a wide range of processes, including 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. The goal of this project is to discover general mathematical laws that govern such systems, which are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. Over the long term, understanding these and other complex interactions has profound implications for science and engineering and thereby for society. The project involves the training of Ph.D. students through involvement in the research. This project studies random paths in random environments and random growth models. The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior. The emphasis is on finding universal principles that apply to classes of models that share fundamental characteristics. The main direction of the current proposal is to find and exploit new mathematical structures in these models. These structures are gradient-like functions called cocycles and Markov processes. These objects are selected as extrema of variational formulas that describe the limiting free energies and limit shapes of these models. This project attempts to characterize features of the limiting objects through the variational formulas. The cocycles that solve the variational formulas define invariant versions of the models and can be used to study fluctuations. The overarching goal is to establish universal properties for models beyond the narrow set of explicitly solvable cases.