This project studies several classes of stochastic processes that possess complicated interactions and in some cases also inhomogeneous or random environments: interacting particle systems, polymer 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. In polymer models the goal is to settle long-standing open problems on scaling exponents that describe the order of magnitude of the fluctuations of the molecule chain and the free energy. For a class of asymmetric zero range processes this project proposes to prove scaling properties of space-time correlations and current fluctuations that confirm KPZ-type behavior. For a single particle moving in a random environment this project studies large deviations, especially the question of when the rate functions for quenched and averaged large deviations coincide. For a large collection of particles moving in a dynamically evolving environment the goal is to prove distributional limit theorems for the current.
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 discover general mathematical principles that govern such systems. 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, epidemics advancing among individuals in a population, or the fluctuations of a polymer chain in a fluid. Over the longer term understanding these complex interactions has profound 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.
This project investigates mathematical models of microscopic complex interactions and the large-scale phenomena that result from the accumulation of a great number of microscopic events. These mathematical systems are stochastic, that is, they incorporate randomness to model irregularity and unpredictability. Examples of such systems include the motion of a particle in an irregular environment or the growth of a crystal structure by random additions of microscopic pieces at the edges. The goal of the project is to discover general mathematical principles that govern such systems. This project has two main outcomes. The first is the discovery of a tractable mathematical model called the log-gamma polymer in a class of models from statistical physics called directed paths in a random medium. These models describe the movement of a particle in a random potential. Such models have been extremely challenging to study. But through the analysis of the log-gamma polymer important conjectures on the behavior of such models have been rigorously verified, and hence have been turned into mathematical facts from mere conjectures. The second important outcome is the discovery of two formulas for the limiting free energies of mathematical models of random paths in a random potential. These formulas will be a foundation for further study of these models. The broader impacts of this project include the development of educational materials such as books and lecture notes on the mathematics of complex systems, available to the public, and the training of junior scientists who have gone on to employment in academia and industry. Books on large deviation theory, interacting particle systems, and growth models, are available on the PI’s website.
