Stochastic Systems with Complex Interactions and Random Environments

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This project studies mathematical models of random paths in random environments and percolation models that describe random growth. 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. There are two main directions of research. (i) The study of the combinatorics, probability distributions and large deviations of 1+1 dimensional exactly solvable models in the Kardar-Parisi-Zhang universality class. Outcomes of this research include rigorous verification of fluctuation exponents conjectured in the physics literature, new connections between combinatorics and models from probability and statistical physics, and better understanding of the integrable structures underlying exact solvability. (ii) The study of more general models of random paths and growing clusters with probabilistic tools. This work will lead to variational descriptions of limiting free energies and limit shapes, and the identification of structures in these models that enable us to study their universal fluctuation properties. The grand goal is to establish some universal properties for models beyond the narrow set of explicitly solvable cases. This project investigates mathematical models that describe complex interactions, growth, 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 laws that govern such systems. These systems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale systemwide behavior. Real-world phenomena that such mathematical studies can illuminate include 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. Over the long term understanding 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.