Probability, as a field, tries to address the question of how large complex random systems behave. An important class of probabilistic models deal with growth in random media. These can be used to model how a cancer grows in a particular organ, how a car moves through traffic on a highway, how neurons move through the brain, or how disease spreads through a population. The purpose of this project is to understand important models for growth in random media, both in terms of developing statistical distributions associated with the models, and in terms of understanding in what sort of systems these models are relevant. The project will leverage tools that the PI has been developing from a number of areas of mathematics to solve problems which were previously inaccessible.
Stochastic partial differential equations, random walks in random media, interacting particle systems, six vertex model, and Gibbs states are active areas of study within probability, equilibrium / non-equilibrium statistics physics, combinatorics, analysis and representation theory. This project touches on problems in and draws upon tools from each of these areas. In particular, this project will (1) Develop a new Markov duality based method to prove convergence of microscopic models (including the six vertex model, dynamic ASEP and ASEP with inhomogeneous jump rates) to the KPZ equation, (2) Prove tail and large deviation bounds on the KPZ equation and related processes, and use these for applications like the slow bond problem, (3) Study scaling behavior for random walks in random environments and develop relationships between the FKPP and KPZ equations, as well as study the uniqueness of Gibbsian line ensembles. Through marrying methods from integrable probability and stochastic analysis, the PI will solve problems in both areas which were previously inaccessible from either approach alone.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.